𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Thursday, January 13, 2022
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Generated Fri 14 Jan 02:40:34 GMT 2022
Created by Dr. Roy Schestowitz (𝚛𝚘𝚢 (at) 𝚜𝚌𝚑𝚎𝚜𝚝𝚘𝚠𝚒𝚝𝚣 (dot) 𝚌𝚘𝚖)
Full hyperlinks for navigation omitted but are fully available in the originals
The corresponding HTML versions are at 𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈
Latest in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕 and older bulletins can be found at 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕-𝒂𝒓𝒄𝒉𝒊𝒗𝒆𝒔
Full IPFS index in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒊𝒑𝒇𝒔 and as plain text in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒊𝒑𝒇𝒔/𝒕𝒙𝒕
Gemini index for the day: gemini://gemini.techrights.org/2022/01/13/
╒═══════════════════ 𝐑𝐄𝐂𝐄𝐍𝐓 𝐁𝐔𝐋𝐋𝐄𝐓𝐈𝐍𝐒 ════════════════════════════════════╕
Previous bulletins in IPFS (past 21 days, in chronological order):
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QmUnmYrNuwB9WxrYeqcbFyf8Bew66hMkdU1YVc9WsKh4JC
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QmarLrYBMcbMFgab1qAkPGRjRr57niwUpd5PEPbMp8225S
QmYYUiUwcwdiuMNs9EXu5Ci3ELNhZcY4b7JsFUZqp7drnc
QmTWhb5PucmgsMLVAJufczuFnxY37RcgjXGMTczMdhndPd
QmVgY18p35Thp6aWh7cRUuoocqLYGYFW2QnLEJhuqn69Kw
QmcN2LseMFZtt5HtSJMT5h9HDZa4ZNy6B7uWe7gK8fPeF9
QmfH2by4K6HCiZyvyGQ1L5syH6ZSfcHH12ph6TDo82aovk
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QmeizGYhw45pUpaYcuwBfE8NWgRhpFbkNtg8fZ7tGbAMXw
QmfKHqGowjRxdQqskGCLaTJvfCGFi19QHkyd8kwa9PMJ21
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QmQ1r9n2zgVqwtQ4bsQLNP2kbfnhn22xRMWVDD5qUuwkKF
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QmUYb6AJaHEaYvQGdw1Z8bffPkohvdveAhmoUhNFHe4PB6
╒═══════════════════ 𝐈𝐍𝐃𝐄𝐗 ═══════════════════════════════════════════════════╕
⦿ Call a Spade a Spade (Microsoft ’Contributions’ to Linux) | Techrights
⦿ Between January 2021 and January 2022 the Number of Active Gemini Capsules Nearly Quadrupled Based on Publicly-Available Catalogue of Capsules | Techrights
⦿ No Excuses for Using GitHub Anymore | Techrights
⦿ IRC Proceedings: Wednesday, January 12, 2022 | Techrights
⦿ 2022 Commences With Microsoft-Themed (and Microsoft-Connected) FUD Against GNU/Linux | Techrights
䷼ Bulletin articles (as HTML) to comment on (requires login):
http://techrights.org/2022/01/13/call-a-spade-a-spade/#comments
http://techrights.org/2022/01/13/gemini-capsules-almost-2000/#comments
http://techrights.org/2022/01/13/github-excuses-running-out/#comments
http://techrights.org/2022/01/13/irc-log-120122/#comments
http://techrights.org/2022/01/13/microsoft-connected-fud-pr/#comments
䷞ Followed by Daily Links (assorted news picks curated and categorised):
http://techrights.org/2022/01/13/kde-plasma-5-24-beta-released/#comments
http://techrights.org/2022/01/13/networkmanager-1-34/#comments
http://techrights.org/2022/01/13/slackware-linux-15-0-rc3/#comments
䷩ 𝚕𝚒𝚗𝚎 61
╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕
(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/13/call-a-spade-a-spade/#comments
Gemini version at gemini://gemini.techrights.org/2022/01/13/call-a-spade-a-spade/
⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.13.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧
Gemini_version_available_♊︎
✐ Call_a_Spade_a_Spade_(Microsoft_‘Contributions’_to_Linux)⠀✐
Posted in GNU/Linux, Kernel, Microsoft at 8:18 am by Dr. Roy Schestowitz
Video_download_link | md5sum 10150a5e18458913bacdcf12178481f6
Linux is Not Windows
Creative Commons Attribution-No Derivative Works 4.0
http://techrights.org/videos/microsoft-linux-wsl.webm
Summary: Call_a_spade_a_spade; Microsoft does not love Linux and doesn’t try to
help Linux, as it’s still all about Windows and proprietary software with
surveillance, back_doors, and worse_things
THIS morning Phoronix published yet another Microsoft article; within 88
minutes there were already 9 responses/comments and by the time I opened the
forum_section (as per the above video) there were even more than 9. People
respond rationally, adding what Michael Larabel tends to leave out. The
comments are typically more insightful than the articles (when it comes to
Microsoft at least) and it's_not_helping_that_Larabel_took_'freebies'_from
Microsoft.
“Almost nobody is permitted to speak out against this sinister agenda.”In any
event, the video above takes stock of what we_wrote_yesterday_about_Mesa.
Microsoft turns Mesa into a mess, adding to it things that benefit nobody
except Microsoft/Windows. This morning in Daily Links we dubbed_it_"More
Microsoft_Aggression_Against_Linux" and it’s even worse than it seems on the
surface. It’s more nefarious than ever before because it’s done from the
inside; the Linux_Foundation takes money from Microsoft, so it won’t put an end
to it. Instead, Zemlin and the Microsoft employees he has put in charge will
put_Linus_Torvalds_in_mental_care.
Almost nobody is permitted to speak out against this sinister agenda. Even
Larabel seems apprehensive and having known him since Phoronix was a relatively
new site I can say with certainty he does not like and does not trust
Microsoft. Why doesn’t he speak more freely/openly on these matters? Honestly,
candidly, without restraint or self-censorship for fear of alienating sponsors
(corporations) and subscribers (some of whom work for those corporations)… █
䷩ 𝚕𝚒𝚗𝚎 122
╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕
(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/13/gemini-capsules-almost-2000/#comments
Gemini version at gemini://gemini.techrights.org/2022/01/13/gemini-capsules-almost-2000/
⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.13.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧
Gemini_version_available_♊︎
✐ Between_January_2021_and_January_2022_the_Number_of_Active_Gemini_Capsules
Nearly_Quadrupled_Based_on_Publicly-Available_Catalogue_of_Capsules⠀✐
Posted in Free/Libre_Software at 6:31 pm by Dr. Roy Schestowitz
Video_download_link | md5sum 5f2ff3ce230d884f5a9a3537421137c2
Gemini Growth From Jan 2021 to Jan 2022
Creative Commons Attribution-No Derivative Works 4.0
http://techrights.org/videos/gemini-almost-quadrupled.webm
Summary: Geminispace has grown to about 2,000 known capsules and 1,600 of them
are active, permanently online, fully accessible; in January last year these
numbers were about 4 times smaller
ABOUT A year ago 441_active_capsules were registered by Stéphane Bortzmeyer’s
Gemini server, which likely inherited a list/index of known capsules from one
of the search engines, to which people can submit their capsules. Lupa has
since then increased its list of known capsules, both active and inactive. The
sum total will reach 2,000 soon and among those nearly 1,600 are active (we’re
13 short at the moment). We hope this growth will motivate more people to join.
The earlier you join, the greater an advantage your capsule will have.
🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Gemini Capsules almost 2,000⦈ Nearly quadrupling in a single
year isn’t easy. The media wasn’t helping, it continues to ignore us, there’s
no marketing at all (my advocacy is 100% grassroots), and hence we rely mostly
on word of mouth. We want to keep the space ‘non-corporate’, just like Free
software, so this is probably the right way to do it. 11 months ago we
explained_how_to_set_up_one's_own_Gemini_capsule. It ought not take more than
an hour, even if you are not an experienced user. Hosting from home is
possible, though one needs a domain and an IP address that is accessible from
the outside world. █
🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇200 MB of RAM? For a single-tab Web page?⦈
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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣱⣏⣾⣮⣿⣟⣹⣿⣵⣿⣿⣿⣎⣇⣻⣵⣿⣏⣾⣎⣷⣍⣾⣋⣇⣏⣾⣜⣟⣿⣿⣉⣷⣽⣿⣙⣿⣩⣾⣛⣷⣹⣾⣱⣇⣿⣿⣟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⡻⢽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⡿⢿⣿⢿⡿⣿⡿⣿⣿⣿⠿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣻⡿⣟⢿⡟⣿⣿⣿⠿⣿⣿⣿⠻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣵⣼⣿⣿⣽⣮⣿⣽⣽⣾⣿⣿⣿⣮⣷⣽⣵⣾⣿⣽⣭⣯⣿⣿⣮⣯⣾⣿⣿⣯⣶⣽⣿⣿⣷⣽⣿⣿⣿⣽⣾⣯⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣯⣏⢽⣿⡟⣿⢻⢻⣯⡏⡏⣽⣽⢻⢻⣯⣿⣯⣿⡏⣯⣽⣹⣿⣿⢸⢹⣏⢽⡇⡯⣏⢿⣿⢩⡏⡏⣯⣽⣽⢨⠛⣟⣽⣿⢻⢹⢸⣿⢻⢏⣯⣽⢩⢘⣽⢻⣯⡏⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡏⢏⡏⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢽⡋⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣯⣽⣏⣝⣏⣽⣿⣻⣿⣿⣿⡏⣯⣿⣿⣿⣿⣿⣏⣇⣿⣿⣾⣿⣁⣽⣧⣿⣸⣿⣿⣿⣼⣿⣝⣿⡏⣿⣹⣏⣽⣿⣹⡿⣹⣿⣽⣿⣹⢸⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣚⣅⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣟⠿⡻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣹⣯⣽⣉⣏⣯⣽⣽⣿⣿⣿⣯⣻⣇⣿⣩⣏⣏⣿⣏⣯⣯⣿⣿⣿⣻⣿⡏⣯⣿⣿⣾⣿⣀⣽⣦⣿⣸⣿⣸⣿⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢭⡋⡍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢹⣿⣿⢻⣿⣿⣿⣟⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⢹⡏⣟⡋⣻⡏⣿⡏⣿⡋⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠁⠹⠿⠿⣿⣿⣿⣿⣿⣿⡻⠏⢶⣲⡆⠦⠋⠻⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠿⠿⠿⠿⢿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢻⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣁⠀⠀⣄⣤⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⠁⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠃⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣠⣶⣶⣶⣿⣿⣷⣀⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠈⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢋⣤⣤⣀⣬⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠉⠛⠛⠛⠛⠿⢶⣤⣤⣠⣤⣶⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⠻⠛⠋⠉⠉⠉⠙⠛⠲⢶⣾⣿⡿⡿⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠁⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢸⠇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣾⠁⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠁⠀⠂⠠⠤⠶⣟⣻⠇⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡯⢵⡃⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠁⠁⠐⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⡳⠇⠀⠀⠲⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡟⣀⣤⣤⣤
⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣏⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⣀⣷⣷⡻⣸⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⡿⠏⢷⣱⡀⠀⠛⠛⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠿⠿⠿⠿⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣦⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢿⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣂⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢸⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣾⣶⣿⣿⣿⣀⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢋⣤⣤⣀⣬⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠈⠛⠛⠛⠛⠿⢶⣤⣤⣄⣤⣶⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠛⠋⠉⠉⠉⠛⠛⠲⠶⣾⣿⡿⡿⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢸⠇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣾⠁⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠁⠀⠂⠠⠔⠴⡾⡿⠇⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⢽⡯⡂⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠈⠀⠂⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠇⠀⠀⠺⣿⣿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⢀⣜⣷⣳⣋⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⣿⣷⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⡿⡿⠝⠀⣢⢀⠢⠚⠛⠁⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣅⠀⠀⣀⣀⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣶⣶⣿⣿⣿⣀⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣟⣴⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣏⣤⣤⣀⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠛⠛⠛⠛⠿⢦⣤⣤⣄⣤⣾⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣼⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠋⠉⠉⠉⠉⠛⠒⠶⣾⣿⣿⡿⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠠⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣮⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⣿⢿⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠾⢿⠿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⢰⢗⣳⢚⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⢿⣷⣶⣴⣴⣾⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠁⠹⠿⠿⢿⣿⣿⣿⣿⡿⡟⠟⠒⣨⢐⠤⠚⠙⠃⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⠀⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀⠀⣠⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠃⣾⣿⣿⠟⣿⣿⣿⠀⢿⡟⠁⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⢿⠿⠏⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣾⣿⣾⣿⣿⣄⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣏⣤⣤⣄⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠛⠛⠛⠛⠿⢶⣤⣤⣤⣤⣾⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣼⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠉⠉⠙⠒⠶⣾⣿⡿⠿⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢠⠁⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣞⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠁⠀⠂⠀⠄⠐⠂⡓⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡹⣍⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠛⠅⠀⠀⠰⡿⠿⠿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢠⣠⢲⣪⣖⣅⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠿⢿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠁⠹⠿⠿⢿⣿⢿⣿⣿⡿⠿⠋⢧⡤⣂⠀⠋⠛⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠿⠿⠿⠿⢿⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⣴⣦⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡾⠿⠿⠿⣿⣿⣿⠿⢿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠃⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣴⣶⣿⣿⣿⣿⣄⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⣤⣤⣄⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠚⠛⠛⠛⠿⢶⣤⣤⣄⣤⣾⣦⣤⡙⠛⠛⠛⠛⢁⣤⡆⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠛⠋⠉⠉⠉⠉⠙⠒⠲⣾⣿⠿⠿⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠁⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠖⠘⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⣬⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠍⠃⠀⠀⠰⡿⠿⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢀⢤⢀⠄⣢⢀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⢿⣿⣶⣦⣦⣼⣿⣿⣿⣟⣟⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠛⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⡿⠿⠛⢿⡸⡵⠐⠋⠛⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣵⣄⡀⣠⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⣿⣿⣿⢿⠿⠿⠏⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠃⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣿⣿⣿⣿⣿⣄⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⡿⠿⢿⢿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣏⣤⣤⣤⣤⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠈⠛⠛⠛⠛⠿⢶⣤⣤⣤⣤⣾⣦⣤⠙⠛⠛⠛⠛⢁⣤⣆⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠉⠀⠀⠒⠲⢾⣿⠿⠿⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠁⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⡾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡪⡝⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⣸⣟⢄⠀⠀⠀⢀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠗⠅⠀⠀⠲⠿⠿⠿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⣀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⢁⣤⣤⣤
⠀⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢠⣰⢽⣾⣦⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⣿⣷⣶⣶⣽⣿⣿⣿⣿⣯⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠠⠙⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⠻⠿⠿⢿⣿⣿⣿⠿⠛⠫⡮⡲⠢⠉⠋⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠾⠿⠿⠿⠿⢿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣧⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢻⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣅⢀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣿⣿⣿⣿⣿⣤⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣏⣤⣤⣤⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠛⠛⠛⠛⠿⢦⣤⣤⣤⣤⣼⣶⡤⠙⠛⠛⠛⠛⢁⣤⣆⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠉⠀⠀⠐⠚⠿⢿⠿⠟⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠠⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣶⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠐⠻⠿⠿⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⣀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⢁⣤⣤⣶
⠀⠈⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢠⣎⢷⣳⡿⡲⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⢿⣷⣶⣤⣤⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣾⡶⠂⢀⠈⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠈⠙⠛⠛⠿⢿⣿⣿⠟⠟⠀⢛⣗⣟⠁⠉⠉⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣷⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⠿⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣧⣀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠃⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⣠⣾⣿⣿⣿⣿⣿⣧⣄⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣷⣿⠿⠿⠿⠻⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣤⣤⣤⣤⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⢁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠚⠛⠛⠛⠿⢦⣤⣤⣤⣤⣤⣤⡀⠙⠛⠛⠛⠛⢁⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠛⢻⡿⠟⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⢶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⡾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⡀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡡⠀⠀⠀⠀⣠⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠛⠛⠛⢿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠃⣠
⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢀⣄⣽⣳⣻⣧⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣆⠀⠀⠀⠛⢿⢿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⠿⠟⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣀⣀⣀⣠⣮⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣿⠿⠿⢛⣻⣿⣿⣿⠿⠿⠟⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠘⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣠⣾⣿⣿⡿⣿⣿⣷⣄⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⠟⠛⠛⠛⢿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣤⣤⣤⣤⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠐⠛⠛⠛⠿⢦⣤⣤⣤⣤⣤⣄⡀⠙⠛⠛⠛⠛⢁⣤⡇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠋⠉⠁⠀⠀⠀⠀⠀⠛⢻⡿⠃⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⡞⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠛⠊⠁⠉⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣯⡢⠀⠀⠀⠀⢠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠉⠉⠉⠉⠛⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⠏
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡋⣀
⠀⠀⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⡀⣰⢐⡐⠌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⣿⣿⣿⣿⣾⣿⣿⡿⣿⢵⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣆⠀⠀⠀⠛⡿⣟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⠟⠁⠁⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠴⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠿⣿⣿⡿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠿⠟⢛⣛⣿⣿⣿⡿⠿⠟⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣀⣶⣿⣿⣿⣿⣿⣷⣦⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣾⡿⠛⠟⠛⢿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣤⣤⣤⣤⣾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠐⠛⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⣁⣴⡇⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠋⠉⠉⠀⠀⠀⠀⠀⠈⢻⡿⠃⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢠⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠙⠭⠌⠻⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠉⠉⠉⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⡀⣀⡀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⢿⣿⣶⣦⣾⣾⣿⣽⣿⣢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣷⢆⠀⠀⠀⠛⢿⣟⡻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠏⠍⠁⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠧⠄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣢⣀⣀⣤⣖⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⢛⣛⣿⣿⣿⡟⠟⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠏⣀⣴⣿⣿⣿⣿⣿⣿⣦⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣾⡿⠛⠛⠛⠛⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣤⣤⣤⣤⣴⣿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠒⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⣁⣤⡇⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠁⠉⠉⠋⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡢⠅⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣀⣤⡤⡄⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣴⣴⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠿⠿⣿⣶⣴⣶⣿⣿⣿⣿⣾⠷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣷⣿⣶⠦⠀⠀⠀⠻⣿⣛⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⠿⠋⠁⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠆⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣆⣀⣀⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢿⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠿⢛⣛⣻⣿⣿⡿⠟⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣀⣴⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣧⣤⣤⡤⣼⣿⣿⣿⣷⠦⠀⠀⠀⠈⠉⠁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠚⠛⠻⢦⣤⣤⣤⣤⣤⣄⡀⠙⠛⠛⠛⠛⢡⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠯⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠂⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣞⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠙⠉⠈⠋⠋⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢜⡆⠀⠀⠀⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣷⣦⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⢀⢠⣤⣤⣆⡠⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠿⣿⣿⣷⣾⣿⣿⣿⡿⣽⡇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⠑⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡵⣷⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡦⣶⣿⣿⣶⣾⣶⠄⠀⠀⠀⠘⠛⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠁⠁⠀⠛⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠾⠿⠿⠿⢿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢻⣿⣿⡿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣽⣿⣿⣿⡆⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢻⡏⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⠛⣛⣻⣿⣿⡿⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣁⣴⣿⣿⡟⠕⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣾⡿⠛⠛⠛⠓⢾⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣧⣤⣤⠄⠴⣿⣿⣿⡷⠦⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠘⠛⠳⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⢡⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠯⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠔⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣥⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢱⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠨⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⠏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠘⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⣲⠖⣦⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⢿⣿⣿⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠈⠉⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣶⣶⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠛⠉⠙⠛⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⢿⣿⣦⣤⣾⣵⣄⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣷⣶⣶⠄⠀⠀⠀⠘⢻⣟⡝⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠈⠈⠈⠛⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠲⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢿⣿⣿⡿⠿⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⣀⣠⣬⣿⣿⣿⣿⡆⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠛⣛⣛⣿⣿⣿⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢸⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣁⣴⣿⣿⡿⢿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⠛⠛⠛⠛⢻⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣧⣤⣤⡤⠴⣿⣿⣿⣷⠦⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢁⣤⣤⣤⣀⠀⠀⠀⠀⠘⠛⠳⢦⣤⣤⣤⣤⣤⣤⡀⠙⠛⠛⠛⠛⢡⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠭⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠖⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣥⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢱⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠐⠂⠀⢸⣿⣿⣿⢰⣿⣿⡿⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠀⠀⢸⣿⣿⡏⣼⣿⣿⣧⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⡏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⡝⡛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡗⠅⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢎⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⣴⣶⣦⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣀⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⣇⡀⠀⣀⣀
⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⣪⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⢿⣿⣶⣤⣷⣾⣴⡀⠠⠄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢄⢠⢤⡀⠀⠀⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠈⠀⠀⠀⠂⢀⠀⠂⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠶⡦⠀⠀⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⣶⣿⣿⣷⣶⣶⠆⠀⠀⠀⠘⢻⣟⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠁⠉⠘⠛⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠳⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣧⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣤⣄⣤⣴⣮⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠿⢋⣉⣛⣿⣿⣿⠙⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢟⣁⣴⣿⣿⣿⣿⣿⡿⠂⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⠛⠛⠛⣿⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⣤⣤⣤⠰⣿⣿⣿⣷⡦⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠉⢁⣤⣤⣤⣀⠀⠀⠀⠀⠘⠛⠻⢦⣤⣤⣤⣤⣤⣤⡀⠙⠛⠛⠛⠛⠁⣴⣧⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠭⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠲⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⣀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣼⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢀⣀⣀⣤⣤⣴⣶⣾⣿⣾⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷
⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠿⣿⣿⣷⣴⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠐⠓⢤⣝⠛⠛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠈⠺⣿⡶⠀⠀⠁⡀⠄⠀⠀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠲⣦⣈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣶⣄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣷⣶⠀⠀⠀⠀⢹⣿⡿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠙⠙⠻⠿⠿⠿⠏⠀⠀⠘⠐⠑⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡷⠷⠤⣤⢀⣠⣤⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠈⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣰⣾⣿⣿⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣲⣤⣤⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡿⠛⣉⣉⣛⣿⣿⡉⠋⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣠⣾⣿⣿⣿⣿⣿⣿⠆⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⠟⠛⠛⠛⢻⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⣶⣶⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠈⠉⠁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠐⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⣴⣇⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠐⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣞⣿⣃⠀⢀⡀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈⠀⠈⠉⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⣿⣿⣶⣴⣶⣤⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⡀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠓⢦⣌⠛⠛⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡶⠦⠀⠈⡀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⢶⣄⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣦⡀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣶⠂⠀⠀⠀⢹⣿⡿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠏⠀⠀⠈⠚⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠶⠤⣤⢀⣠⣤⣤⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠈⠻⢿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣰⣾⣿⣿⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣠⣴⣴⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡿⠛⣉⣉⣛⣿⣿⡉⠉⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣾⣿⣿⣿⣿⣿⣿⠆⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⠟⠛⠛⠛⢻⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⣶⣶⣾⣿⣿⣿⣿⠷⠆⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠘⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⣴⣧⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠐⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣼⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⣠⣴⣿⣣⠀⢀⡀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⣿⣿⣶⣾⣾⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣦⣤⡀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠓⢦⣌⠛⠛⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡶⠦⠀⠈⣀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠐⢶⣄⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣦⡀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⡶⣶⡶⣖⠶⠿⠿⠇⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⢠⣤⣯⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢷⣿⣶⣴⣄⡀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⠷⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠻⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢋⣠⣤⣴⣿⣿⣧⣀⣑⣡⠍⠀⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠈⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠋⠉⠉⢈⣿⣿⣿⣿⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⣿⣿⣺⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣀⣈⣉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠛⠟⠛⠛⠛⠿⢦⣤⣤⣤⣤⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⡇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢠⣾⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣷⣀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁⠀⠈⠉⠈⠁⠈⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠿⢿⣿⣷⣦⣤⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣿⣷⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠈⠛⢦⣭⠛⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠒⠂⠀⢤⡖⠆⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢿⣷⡦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣷⣶⣶⣶⣷⣲⠶⡶⠠⠈⠀⠁⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠉⠉⠿⠿⠿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣵⣶⣶⣴⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⠀⠀⣰⣿⣿⢿⣿⣦⡀⠈⣀⣀⣀⣀⣈⠻⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡄⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣏⣽⣿⣷⣿⣿⣿⣌⣸⣿⡿⠃⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠄⠀⣀⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠘⠛⠿⠛⠛⠛⠳⠤⣤⣀⣀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⣾⡇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣀⠀⠉⠉⠉⠀⣰⣿⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣄⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠙⠻⠻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠶⢦⢠⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠘⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢺⣿⣷⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠈⠛⠢⠌⠛⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡏⠐⠀⠀⠶⠒⢀⠀⠰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠑⣶⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣷⣿⣾⠿⠔⠀⠀⠀⠈⠿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⡟⠉⠉⠈⠉⠻⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡇⠀⠀⠀⠀⠀⠀⡀⣀⢠⣄⣤⣠⣠⣤⣶⣶⣶⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠳⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠗⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣤⣤⣄⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣈⠙⠻⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⣾⡏⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⡀⠉⠉⠉⢀⣴⣿⣇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠿⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣶⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⡿⡿⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠂⠁⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣄⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡛⠻⣿⣷⡄⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠈⠙⠓⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠾⠉⢀⡀⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⠙⢶⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠗⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠐⢿⠷⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠟⠹⠿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠉⠈⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠦⣤⣀⣀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠛⠛⠛⠛⠂⠠⣤⣤⣤⣤⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⡆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠁⠉⠀⠠⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⠉⠊⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿
⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠠⣶⣠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠟⠃⠀⠀⠀⠘⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠈⠁⠀⠂⠀⠀⠀⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠠⣤⡈⢳⣌⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠈⠁⠒⠛⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢟⣟⠣⠄⠀⠄⠤⠀⠀⠀⠰⠐⠔⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠸⠋⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠄⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠈⠛⠓⠒⠐⠂⠠⢤⣤⣤⣤⣤⣤⠄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠋⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣟⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣯⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿
⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣷⣤⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣬⡙⠻⠟⠂⠀⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣦⣀⠀⠙⠃⣀⠀⠀⠀⣠⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠈⠉⠡⣤⡈⠻⣿⣷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣴⣄⠉⠙⠲⠎⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠖⠀⠠⠢⠔⠀⠀⠀⠐⠂⠀⠀⠀⠃⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠯⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⠿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣠⣾⣿⣿⣿⠿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠘⠃⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣸⣿⣿⣦⣿⣿⣿⣿⣿⣿⠀⠉⠉⠁⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠦⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠈⠛⠒⠒⠒⠛⠶⣦⣤⣤⣤⣄⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣴⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡏⠉⠁⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣍⠛⠟⠃⠀⠀⠈⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣤⠀⠉⠃⣀⡀⠀⢀⣠⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠰⢦⣈⠙⢿⣿⠶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣦⡀⠉⠓⠂⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡷⠒⠐⠐⠢⠲⠴⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟⠋⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣰⣿⣿⣿⣿⠿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠈⠁⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣹⣿⣷⣴⣿⣿⣿⣿⣿⡏⠀⠉⠉⠀⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠈⠛⠒⠒⠒⠒⠶⣦⣤⣤⣤⡄⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣴⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿
⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣍⠛⠟⠃⠀⠀⠈⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⣤⠀⠉⠃⣀⡀⠀⠀⣀⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠰⢦⣈⠙⢿⣟⠶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣦⡀⠉⠓⠒⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠒⠀⠒⠲⠲⠶⠖⠠⠀⠛⠃⠉⠉⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠟⢈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡈⠉⣉⣀⣀⣀⣉⠛⠿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠻⢿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠈⠃⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣧⣼⣿⣿⣿⣿⣿⡿⠀⠉⠉⠀⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠆⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠒⠒⠂⠒⠒⠲⢤⣤⣤⣤⣤⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠙⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⡀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣯⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣀⣀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿
⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⡛⠻⠓⠀⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣤⡀⠈⠛⢂⠀⠀⠀⠀⣠⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⢦⣀⠈⠾⣿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠉⠑⠶⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣾⣿⡿⠿⠒⠠⠦⠦⠶⠶⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⠷⠒⠋⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣾⣿⣿⣿⣿⠟⠋⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡈⠉⣉⣁⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠘⠃⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣈⣿⣿⣀⣸⣿⣿⣿⣿⣿⣿⡟⠿⠿⠇⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠘⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣿⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⢀⡏⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⣿⠃⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠁⠉⠻⣿⣿⣿⣿⣷⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⠺⡿⠆⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡀⠈⠛⢦⡀⠀⠀⠀⠀⡀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠁⠀⠀⠀⠀⢀⡀⠀⢶⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠳⣦⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣷⡶⣶⣦⠀⠠⣾⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⣿⣖⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠂⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢀⣼⣿⣿⣿⠟⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡟⠀⣰⣿⣿⢿⣿⣦⡈⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⣿⣿⡄⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⢀⡀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠰⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⡷⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⡀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⣾⡇⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠉⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⣀⣀⣀⠀⠉⠉⠉⠀⠠⠆⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣤⣄⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠙⠳⣤⡙⠃⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠒⠀⠀⣤⠤⠀⠀⣠⡰⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢶⣈⠙⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⣶⣦⠀⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣷⡶⣶⣦⠀⠀⣶⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⣿⣖⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠰⠂⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⣠⣴⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢀⣾⣿⣿⣿⠟⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡟⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⣿⣿⣄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⢀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠻⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⢰⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⡷⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⡀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⣾⠇⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠉⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⡀⠀⠉⠉⠉⠀⠀⠆⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡗⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣦⣄⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠙⠲⣤⡛⠀⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡕⠒⠀⠀⡔⡄⠀⠀⣠⡰⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢶⣈⠙⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⣷⣦⠀⠉⠑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠸⠿⠿⠋⠉⠁⠙⠙⠻⠹⠿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠈⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣽⣾⣷⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠿⠿⠿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠻⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢹⠇⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣷⡀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⠿⠁⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣴⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⣤⣤⣤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠠⠀⠄⠂⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⡝⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⢀⣤⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⢰⣿⣿⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣠⣠⣤⡀⠉⠉⠉⣠⣾⣿⣿⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢛⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿
⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡐⠺⣿⣦⠄⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣄⠀⠉⠲⠎⠁⠀⠀⠀⠀⠈⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠉⠀⠀⠰⠒⠂⠀⠰⣶⣷⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠓⢶⣌⡹⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡷⠀⠀⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠈⠻⢻⡿⡙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⣿⣽⣷⣶⣾⣶⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠈⣀⣀⣀⣀⣀⠙⠿⡿⠿⠿⠛⠻⠿⠷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠈⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣆⣰⣶⣿⣿⣿⣿⣿⣿⡇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠂⠀⠀⣀⣈⠉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣀⠀⠙⠛⠛⠛⠛⠃⠀⠀⣠⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠉⠉⠀⠀⠀⠀⠤⠠⠄⠛⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⣀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⢀⣾⡏⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⣀⠀⠀⠀⠉⠉⠉⠀⣸⣿⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⡠⣶⢶⡾⠟⠛⠁⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣏⠃⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠀⠉⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡁⡀⠀⠀⠻⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿
⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣔⣶⣤⠀⠈⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡐⢦⣈⣙⠿⠿⠁⠀⠀⠀⠊⠉⣿⣿
⠀⠀⠀⠀⠀⠙⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠉⠉⠀⠀⠀⠀⠀⢠⣄⠙⢉⠙
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠙⢶⣭⣷⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡛⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⡛⠖⠂⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⣉⠍⡉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣶⣀⠈⠀⠀⠀⠉⠿⠟⠛⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠘⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣄⣼⣾⡟⠃⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢟⣿⣿⣆⠀⠀⠀⣾⣿⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⣦⣤⡈⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⢦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⣿⣿⠟⣼⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠦⣤⣤⣤⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢯⣽⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⣿⣿⣿⢥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⡏⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣿⣿⣿⣿⠅⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢀⣾⣶⣶⣦⣦⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠃⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⡿⣿⡛⠛⠛⢿⣿⣿⣿⣿⣿⣧⡀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⠹⣿⣿⣿⡎⠀⠘⠉⠙⠂⠀⠊⠉⠋⠋⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠀⠀⠹⠿⣿⣿⣧⣤⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⠀⠢⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠙⠻⠿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠈⠉⠛⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⣄⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠿⣿⣝⣿⡿⠂⠀⠈⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⠈⠉⢁⣀⠀⠀⣀⣠⣄⠈⠙⢿
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢱⣄⠙⢿⣿⣷⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣻⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣄⠉⠻⢶⡟⠋⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣯⣯⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠃⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⣩⡭⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣶⡠⠘⠀⠀⠉⠉⢿⠿⠉⠙⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠟⠁⠀⠀⠀⠀⠀⠘⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠋⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣾⣴⠟⠁⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠟⣿⣿⣆⠀⠀⠀⣾⣿⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⣦⣤⠈⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠛⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣴⣿⣿⠟⣨⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠳⠦⣤⣤⣤⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⢩⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠂⠀⢠⣶⣶⣶⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡆⠀⠈⠿⣿⣿⣿⣿⢿⣿⢿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣄⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⠹⣿⣿⣿⡗⠀⠙⠘⠘⠀⠀⠊⠙⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠀⠀⠙⠿⣿⣿⣧⣤⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣶⣶⣶⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠲⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠉⠛⠻⢿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠉⠙⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⣄⠈⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠻⣿⣽⣿⠿⠃⠀⠈⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⢦⠈⠉⢡⣤⠀⠀⣀⣴⣄⠈⠙⢿
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢰⣦⡉⢻⣿⣷⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣻⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣄⠉⠛⠶⠏⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣯⣯⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⣉⣉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠈⣾⡖⠩⠂⠀⠀⠉⢿⠟⠋⠹⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠘⢿⡷⠆⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣴⣴⡞⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠏⣻⣿⣆⠀⠀⠀⣶⣿⣿⣿⡟⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⣶⣤⠀⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠛⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣴⣿⣿⠏⣱⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠢⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⠠⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠈⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠋⣾⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣾⣿⣿⣿⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⣿⣿⣿⢱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⡏⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣿⣿⣿⣿⡅⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠈⠁⠀⠀⠤⢤⣄⣀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡆⠀⠐⠿⣿⣿⣿⣿⠿⢿⢿⢛⠛⠛⢿⣿⣿⣿⣿⣿⣆⡀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⠀⠀⠉⠓⠈⠐⠀⠉⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⠀⠀⠀⠘⠿⣿⣿⣆⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠊⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠒⢦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠈⠛⠛⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠉⠛⠓⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣧⣄⠈⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡈⠛⣾⣯⡿⠟⠁⠀⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⠶⠈⠉⣥⣤⡀⠀⣀⣴⣄⠈⠉⢿
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⢰⣦⡉⢻⣿⣷⣦⠀
⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⣛⣿⣿⣻⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠈⠛⠷⠋⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠃⠀⠀⠀⠐⣽⡯⠭⠀⠀⠉⠙⣿⣿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⢸⣶⣶⡄⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠋⠁⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣷⡀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣴⣾⣿⣿⣀⣀⠀⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣤⣤⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⠿⢦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠀⢀⣀⣀⣠⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⡿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢠⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠥⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡆⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣆⠀⠉⠙⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠈⢻⣿⣿⣿⠋⠁⠻⠻⠺⠒⠀⠩⠛⠛⠟⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⠀⠀⠀⠻⣿⣿⣷⣤⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠛⠛⠛⠟⠳⠻⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠳⣬⣿⣿⡟⠀⠀⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡗⠦⠈⠉⣡⣦⠀⠀⣀⣤⡌⠉⢿⣿
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠈⢰⣆⠙⢿⣿⣶⡤⠙
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣄⠉⠛⠿⠟⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⡿⣛⣿⣿⣻⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠍⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⣋⡛⡋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠻⠟⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢰⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠘⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣴⣄⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢿⣿⣆⠀⠀⠀⣀⣾⣿⣿⣿⣿⠏⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣴⣿⣿⣿⣄⣀⠀⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠿⣿⣿⢿⣿⣿⣶⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣤⣴⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠀⢀⣀⣀⣀⣄⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣽⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣤⣿⣿⣿⠛⣼⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⢋⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⠫⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡟⠁⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣶⣶⣶⣤⣤⣤⣤⣤⣤⡀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡃⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⡿⠏⠙⢿⣿⣿⣿⣿⣿⣆⡀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⡀⠨⣿⣿⣿⣿⣆⠀⠘⠹⠩⠁⠀⠀⠉⠋⠋⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⡀⠙⢿⣿⣿⣷⣦⣠⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣇⠀⠈⠛⠿⠿⠿⠝⠙⠝⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⣀⠘⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠐⠳⣭⣿⡿⠟⠂⠀⠈⠛⢿⣿⣿⣿
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡷⠦⠈⠉⣡⣤⠀⠀⣀⣴⣄⠉⠙⢿
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⢰⣄⠙⢿⣿⣶⡄⠀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣄⠈⠛⠾⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠟⡛⣿⣿⣻⣿⣻⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠯⣿⣽⣿⣽⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⠉⢭⣏⣏⢍⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⢀⡀⠀⠀⠀⠘⣿⡟⠋⠀⠀⠀⠀⠛⡏⠈⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠸⢿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣈⠹⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣦⣶⣿⣦⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⣻⣿⣇⠀⠀⠀⣴⣿⣿⣿⣿⣿⠏⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣿⣿⣦⣄⡀⣿⣿⣿⣿⣿⠇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣤⣤⣀⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠀⠀⣀⣀⣀⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣤⣿⣿⣿⠋⣼⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⡏⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣷⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠅⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠟⡿⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣄⠀⢹⣿⣿⣿⣿⣿⠿⢻⣿⣋⡀⢙⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠓⠅⠀⠀⠀⠐⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣔⢴⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠓⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣷⣦⡀⠘⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠂⣭⣿⠿⠛⠁⠀⠀⠈⠛⢻⣿⣿
⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠉⠉⠀⣦⠀⠀⠀⣴⡄⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⣮⣿⣷⠄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠐⠊⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⣸⣿⣿⢾⠷⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⣠⣶⣶⡄⠀⠛⠛⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠈⣁⣀⣀⣀⣉⠻⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢏⣸⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⢦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢁⣤⣤⣤⣈⠙⠛⠿⠛⠛⠛⠓⠀⢀⣀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⣾⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣤⡄⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢋⠁⢠⣶⣿⣶⣶⣶⣿⣿⣿⣿⣆⡀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢐⠛⠛⠻⠿⠛⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡿⡿⡿⣿⣿⡿⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⡪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣦⣤⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣿⣿⣧⡄⠈⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠛⣦⠿⠛⠃⠀⠀⠀⠈⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠂⠀⠀⣀⠲⣦⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢦⣉⠳⣮⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡖⠂⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣻⣿⣻⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠀⢶⣿⣿⣿⠏⠁⠀⠈⣿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢸⣿⣿⡶⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠉⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠉⠀⠀⠀⠀⠀⣴⡆⠈⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠸⠌⠀⠈⠉⠀⠀⠀⠀⠀⣤⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣦⡀⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠃⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⣸⣿⣿⣄⣀⠀⢾⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⢋⣩⣿⣿⢯⣽⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣈⠙⠛⠿⠛⠛⠛⠓⠠⢄⣀⣀⣠⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢋⣼⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣾⣿⣷⣶⣾⣿⣿⣿⣿⣆⣠⣈⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠉⢿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣶⣮⡀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣻⣿⣿⡤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣄⣹⡿⠗⠉⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠉⠀⠸⠿⢂⡀⠰⣶⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣈⠙⢶⣮⡙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣿⣻⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⣴⣿⣿⡿⠋⠁⠀⠈⢿⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⡷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠉⠀⠀⠀⠀⠀⢰⠆⠁⠀⠀⠀⣀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⣸⣿⣿⣄⣀⠀⢼⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢉⣉⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠠⢀⣀⣀⣠⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢣⣼⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⢃⣼⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⠁⣷⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⠩⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢠⣾⣿⣷⣶⣾⣿⣿⣿⣿⣆⣤⣈⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢝⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠉⢿⣿⣿⣿⣿⣿⣶⡂⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣶⣧⡀⠈⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⡻⢿⣿⡤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣄⣹⠿⠗⠉⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠸⠿⢂⡀⠐⢶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡈⠙⢶⣮⡙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣽⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠙⣿⣾⣤⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⣶⣿⣿⠿⠁⠀⠀⠈⢿⠿⠛⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⢿⣿⡿⠷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣰⠟⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⢸⣿⣟⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣦⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠸⠁⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣃⣸⣿⣿⣀⣀⠀⣾⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⢉⣩⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠤⢀⣀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⣾⣿⣿⡿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣿⣿⣿⣿⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⣤⣾⣿⣷⣶⣾⣿⣿⣿⣿⣦⠀⢀⣉⣁⡉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⡉⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⢿⣿⣿⣿⣿⣿⣾⠊⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣈⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣆⣀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡻⣿⣷⣤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄⡠⠀⠠⠀⡂⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣄⠙⡿⠮⠙⠁⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⡽⢵⡃⠈⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠋⠀⠰⣶⠖⠀⠰⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢶⣌⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣽⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠏⠈⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⢻⣷⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⢴⣿⣿⡿⠏⠁⠀⠀⠸⢿⠟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢸⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣾⠋⠁⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⢈⣇⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠛⠁⠀⠈⠀⠀⠀⠀⠀⠤⣿⣿⡏⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣷⣶⡀⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⡀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠘⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣀⣿⣿⣧⣀⣀⠀⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣉⣩⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⡟⠋⢁⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠤⢀⣀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⡿⣠⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⡟⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⢠⣴⣿⣿⣷⣶⣾⣿⣿⣿⣿⣆⠀⠈⣉⣁⡉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢝⢻⣿⣿⣿⣿⣿⣿⠟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠻⣿⣿⣿⣿⣿⣿⡦⠃⠀⠈⠁⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣽⠉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣅⣀⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠈⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⢀⠀⡀⢄⢄⡉⠛⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡠⡲⣣⢷⣭⣇⣣⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣜⠻⣿⣷⣤⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣯⣽⢟⢛⢟⠲⠐⠀⠹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣀⠙⠿⠮⠋⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠋⠀⢰⣶⠆⠀⢠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠳⣦⣌⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢔⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⠄⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⣠⣶⣦⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⢰⣿⣿⣿⠛⠉⠀⠀⠀⠻⠛⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡴⠟⠁⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠋⠁⠀⠀⠀⠀⠈⠇⠀⠀⠀⢀⣠⣤⣤⣤⣶⣤⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠻⡿⣿⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⡄⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢁⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣃⣰⣿⣿⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⡟⡉⣩⣽⣿⢯⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠀⢀⣀⣀⣠⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⡋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣼⣿⣿⡿⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⣀⡀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⣀⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⠀⠈⢉⣁⡉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣰⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⣿⣿⣿⣿⣿⣿⣷⡜⠀⠈⠉⠉⠉⠀⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣬⠉⠙⠛⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⡉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣾⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⢀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣉⣀⣀⢀⣐⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠙⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣷⣿⣿⣾⡢⠀⠉⠛⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡁⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡚⠺⣿⣦⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠈⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣄⠈⠛⠶⠌⠁⠀⠀⠀⠀⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠉⠀⠀⢲⡶⠀⠀⢴⣶⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⣤⡉⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣶⡄⠉⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⣿⣻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠫⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⢀⣴⣶⣴⣦⣶⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁⣀⡀⠀⠀⢻⣿⣿⠟⠁⠀⠀⠀⠈⠛⠉⠀⠈⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠹⡿⠿⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠤⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⡄⠂⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠛⠿⡿⢟⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡄⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢁⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣿⣿⣧⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣟⣡⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣈⠙⠻⠟⠛⠛⠛⠓⠀⢀⣀⣀⣠⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⢀⣠⡀⠉⠉⠉⢀⣴⣶⡆⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⡁⢀⣴⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣄⣈⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣢⠃⠈⠉⠉⠉⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣼⡉⠋⠛⠛⠻⠟⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣅⣈⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡁⡽⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡛⠻⣿⡦⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣄⠈⠙⠲⠈⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠀⠀⠲⠂⡀⠀⢶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠳⣤⡉⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⡶⠄⠉⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣠⣶⣶⣶⣾⠷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⣀⣀⣀⡀⠺⣿⢿⠟⠁⠀⠀⠀⠀⠉⠁⠀⠀⠠⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠟⠁⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⠕⠀⣀⣀⣠⣴⣶⣶⣶⣾⣾⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠝⠻⠙⠃⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣷⣄⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢞⣃⣼⣿⣿⣄⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣁⣤⣽⡿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⣈⡉⢁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢁⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠂⠀⠀⣀⣀⣀⣤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⠉⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⣀⣤⡀⠉⠉⠉⢀⣴⣶⡆⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢠⣾⣿⣿⣿⣿⣶⣾⣿⣿⡿⢳⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣯⡇⠀⠈⠉⠉⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣍⠙⠙⠛⠿⠿⠃⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣟⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣄⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣠⣤⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿
⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡙⠻⣷⠆⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣷⣄⡀⠙⠒⠀⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠂⠂⣀⠀⢶⣽⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠉⠳⣦⡍⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⠶⠀⠈⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢠⣶⣿⣿⣿⠿⠟⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣀⣤⣤⣤⡀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡿⠿⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⠈⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡇⠀⠾⠋⠀⠀⠀⠀⢀⣠⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠋⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡆⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣼⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⣿⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣏⣠⣿⣿⣷⣄⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣏⣠⣤⣿⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⢁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⢀⣀⣀⢤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⣿⣿⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⡟⠙⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⣀⣤⡀⠉⠉⠉⢀⣴⣾⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠅⠀⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⡟⢰⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢈⠙⣿⣿⣿⣿⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠙⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣭⣉⠉⠙⠛⠛⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡉⠙⠁⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡗⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿
⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⡀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡙⠛⠿⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⣤⡀⠙⠂⠀⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣢⣴⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⢾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠉⠳⣦⡍⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⣾⣿⣿⣿⠿⠙⠘⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⣤⣤⣤⣤⡀⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⣄⠈⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠺⠁⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠙⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢁⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣃⣴⣿⣿⣷⣤⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣧⣤⣤⣿⠛⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⢀⣀⣀⡤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⠟⠈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⢀⣠⣤⣀⠉⠉⠉⢀⣴⣿⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠡⠀⣴⣾⣿⣿⣿⣿⣶⣿⣿⣿⡏⢰⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⢉⣿⣿⣿⣿⣿⡿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠉⠙⢿⣿⣿⣿⣿⣿⣯⡂⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⡉⠉⠉⠛⠛⠁⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡍⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡀⣀⣸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿
⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢤⡙⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡅⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⣤⡀⠙⠃⠀⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠊⠠⣀⠐⢾⣽⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⡄⠉⠱⣦⡍⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⣴⣿⣿⣿⠟⠛⠀⢹⣿⣾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⢀⣴⣶⣦⡄⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⡿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⡀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠾⠃⠀⠀⠀⠀⢠⣴⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣿⣿⡆⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣃⣴⣿⣿⣿⣤⣄⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠿⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣧⣤⣴⣿⠛⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⢁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠛⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣾⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣀⣄⡀⠉⠉⠉⢀⣴⣾⠇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣴⣿⣿⣿⣿⣿⣶⣿⣿⣿⠏⢰⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⣿⣿⣿⠇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣏⠉⠉⠉⢿⣿⣿⣿⣿⣿⣆⠄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡖⣴⣤⡀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⢀⡀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⡿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿
⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢦⣝⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⢀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⢤⠀⠉⠃⠀⠀⠀⠀⢀⡀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣢⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⣄⠀⢾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⡄⠉⠱⠦⠍⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠀⠈⣿⣷⠷⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠠⣶⣶⣶⣄⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢤⡄⠉⠁⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠖⠀⠘⠋⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠟⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⣻⣿⣇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⣿⣿⣿⣦⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⢿⣿⠯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣧⣤⣴⣿⠋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⠉⠁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⣀⣀⢤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣼⣿⣿⡟⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⠟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣾⣿⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⣀⡀⠉⠉⠉⢀⣠⣶⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠊⢀⣴⣿⣿⣿⣿⣿⣶⣾⣿⣿⠏⢰⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⣿⣿⣿⣿⣿⠇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⠀⠉⠿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⣵⡀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⡈⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡂⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣃⣀⣤⣨⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠑⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠟⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠦⣌⠛⠟⠂⠀⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠡⣨⢴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣗⣤⠀⠉⠁⠀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠠⣄⠈⢺⣿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⠉⠑⠶⠌⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠋⠀⠀⣿⣷⡶⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠠⣴⣶⣶⣦⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠛⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠻⠿⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠀⠀⠀⣀⢀⢀⣀⠀⠀⠀⠄⠂⠀⠛⠁⠀⠀⠀⠀⢠⣶⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⣹⣿⣧⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⣿⣿⣿⣷⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠿⠿⢿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣷⣤⣶⣿⠛⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⢁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⣀⣀⡤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⠿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⢠⣾⣿⣿⣷⣶⣶⣶⣾⣿⡟⠁⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⡽⣿⣿⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠉⠙⢿⣿⣿⣿⣿⣿⡎⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡄⠈⠉⠉⠉⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣯⡀⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡂⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⡂⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣤⣠⣠⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠘⠢⣌⠛⠛⠂⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⣴⣵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⠤⠀⠉⢃⠀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠠⣄⠈⢾⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⠉⠑⠶⠌⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⣠⣿⣷⣿⠟⠋⠀⠀⣿⣷⡶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣶⣶⣶⣦⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⠀⢠⣤⠈⠉⠀⠀⠀⠀⡀⣀⠀⣀⣀⠀⠀⠰⠄⠁⠀⠛⠁⠀⠀⠀⠀⢀⣶⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣦⣴⣶⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢹⣿⣧⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⣿⣿⣿⣷⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠟⢿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣷⣤⣶⣿⠋⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⣀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⠏⣃⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣶⣾⣿⣶⣶⣶⣶⣶⣿⡏⠀⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢹⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠈⠉⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⣶⣤⡀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⡀⡀⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣿⣿⣷⣽⣿⣿⠿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠻⠿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠘⢦⣌⠛⠟⠂⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠀⠀⠀⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣗⢤⠄⠉⠁⡀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⣠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⠠⣄⠈⢿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⠉⠑⠶⠌⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⢹⣿⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣶⣶⣷⣶⣄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⡿⠛⠋⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⡿⠋⠀⠀⠀⠀⠠⣦⠌⠉⠁⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢠⢿⡟⠙⡿⠁⠀⠀⠀⠀⢠⣶⣿⣿⢟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠃⠐⠐⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⢹⣿⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣧⣶⣿⣿⣿⣷⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⡿⠟⠛⢿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣤⣶⣿⠟⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⡁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⣀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⡟⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠅⢰⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⣀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⣾⣿⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣰⣿⣿⡀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠟⢹⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠉⢹⣿⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠉⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠉⢿⣿⣿⣿⣿⣿⡆⠂⠀⠉⠉⠉⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣶⣶⡄⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣧⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣄⣮⣔⣴⣰⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠕⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⢄⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠘⠢⣌⠛⠟⠂⠀⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⢀⣷⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⣤⠄⠉⠃⠀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⡄⠀⡀⣤⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠠⣄⠈⢶⣽⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣶⣄⠉⠑⠶⠍⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠋⠀⠀⢸⣿⣶⡾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢿⣿⣿⣷⣦⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⡿⠛⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⡿⠋⠀⠀⠀⠀⠰⣶⡎⠋⠁⠀⠀⣀⣀⣀⣀⣀⣄⡄⠀⠐⣿⡿⠋⡿⠃⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠈⠐⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢹⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣿⣿⣶⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠛⢻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣶⣾⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣇⠁⠀⠘⠂⠛⠋⠋⠉⠙⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠈⢩⣭⡍⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣭⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣯⡿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠙⢽⣍⣰⣴⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣹⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⣒⠒⠶⠤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣿⣿⡿⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡉⠙⢿⣿⣿⣿⣿⣿⣿⡟⠉⠈⠉⠉⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣤⠈⠉⠉⠛⠃⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡉⠋⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⢀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣽⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠛⠢⣍⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣤⠄⠉⠁⡀⠀⠀⠀⢀⡀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢆⣴⣵⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⢡⣄⠈⢾⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣶⣄⠉⠑⠶⠍⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠁⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠁⠀⠈⣿⡿⡾⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⢻⣿⣿⣿⣶⡌⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡄⠀⣰⡿⠋⠉⠁⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⡿⠃⠀⠀⠀⠀⠘⢿⠇⠈⠀⠀⢀⣀⣀⣄⣀⣀⣄⣄⠀⠠⢾⠿⠁⠾⠁⠀⠀⠀⠀⢠⣾⣿⣿⣟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡛⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣸⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⣿⣿⣿⣿⣶⣦⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⢻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣿⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡘⡀⠱⠉⠓⠉⠈⠈⠉⠁⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⡟⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠅⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⢠⣤⣤⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣷⡁⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠻⣿⣿⣿⣿⡿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡟⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣤⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣦⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣤⣤⣀⡀⠀⠀⠀⢀⣠⣾⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⠿⠷⣶⣦⣤⣤⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣺⣿⣿⣿⣿⣿⡿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⠉⠻⣿⣿⣿⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣧⡄⠀⠉⠉⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣈⠉⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡅⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⢀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣺⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠛⢦⣍⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠂⠀⢀⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣤⠀⠉⠁⣀⠀⠀⠀⢠⡀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⢰⣄⠈⢶⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣦⣄⠉⠱⠶⠍⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠄⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⡀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠘⣿⢿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⢻⣿⣿⣿⣶⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⣰⡿⠋⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⢿⠿⠃⠀⠀⠀⠀⠘⣿⣯⠉⠀⠀⢀⣀⣤⣤⣤⣤⣤⣴⠄⠠⠿⠿⠁⠟⠁⠀⠀⠀⠀⢰⣾⣿⣿⣿⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣷⣾⣿⣿⣿⣿⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠙⢻⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣶⣾⣿⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠎⠀⠇⠉⠉⠁⠁⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣼⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣈⠀⠠⣤⣤⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠇⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⡿⣏⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⢑⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣤⣤⣀⣀⠀⠀⠀⣀⣠⣾⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⠿⠶⣶⣶⠴⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣄⣈⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣷⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣀⣀⠉⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡃⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠃⠀⣀⣀⣈⡉⢉⣉⣉⡉⢉⣉⣉⣉⡙⠿⠿⠟⣋⣉⣉⠻⠿⠟⢉⣁⣀⡉⢉⣉⣉⢉⣉⡉⣉⣈⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⡀⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢠⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡌⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡌⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⢿⣿⣿⣷⠀⣿⣿⢿⣿⡆⠀⠀⣀⣤⣶⣿⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣧⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣄⠀⣿⣿⠀⣻⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣾⣿⣿⣿⣿⣿
⠀⠉⠛⠛⠿⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢹⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠐⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠁⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠐⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⣄⠀⠐⠀⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⠰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣷⣤⠀⠉⢁⡀⠀⠀⠀⢠⡀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⢰⣄⠉⢾⣷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣾⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣄⠉⠓⠶⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⡇⣿⣿⣷⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⣂⡀⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣄⣀⣀⣤⣶⣶⣤⣀⣠⣤⣤⣀⣀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠄⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠀⠀⠀⠀⠿⠟⠅⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⢻⣿⣿⡿⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⣀⣼⡿⠃⠁⠀⠈⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡄⠀⠀⠽⠻⠃⠛⠁⠀⠀⠀⠀⢠⣶⣿⣿⡏⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡍⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣷⣾⣿⣿⣿⣿⣿⣷⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠋⣻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡟⢻⠻⣿⣿⣿⣿⣿⡿⣿⠟⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇⢐⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⣿⣿⡟⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣼⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠠⠤⡤⠄⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠄⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⡟⠟⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣤⣤⣀⣀⠀⠀⠀⣀⣠⣾⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠿⠿⠶⠶⠶⠶⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⡿⠏⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠙⣿⣿⣿⣿⣿⣿⣗⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣶⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡄⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⡉⢉⣉⣈⡉⢉⣉⣉⣉⡙⠿⠿⠟⣉⣉⣉⠻⠿⠟⢉⣁⣀⡉⢉⣉⣉⢉⣉⡉⣁⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣉⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢠⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡌⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡌⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣷⠀⣿⣿⢿⣿⡆⠀⠀⣀⣤⣶⣿⣿
⣿⣿⣿⣿⣿⠆⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣄⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠉⠛⠛⠻⠂⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢹⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠐⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠑⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠐⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⢤⣀⠀⠀⠀⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠇⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⠰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣷⠤⠀⠉⣀⡀⠀⠀⠀⣠⡀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠈⢰⣄⠙⢾⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣴⣶⡄⠉⠑⠶⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣂⡀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣶⣶⣄⣀⣀⣤⣄⣀⣀⣀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⣀⣠⣀⠀⢸⣿⣿⣿⠟⠀⠀⠀⠀⠿⠿⠇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣀⢹⣿⣿⡿⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣇⣠⣼⡿⠁⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠟⠁⠀⠀⠀⠀⠈⠻⣿⡄⠀⢀⣤⣴⣶⣶⣶⣶⣶⡶⠆⡴⡿⠟⠃⠛⠁⠀⠀⠀⠀⠰⣿⣿⣿⣿⣵⠍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣀⣿⣿⣧⣀⣀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⡏⢉⣉⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠻⡟⠿⣟⠿⠿⡿⠿⡿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡈⡈⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡟⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠘⠿⢶⠆⠈⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠻⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣡⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⣿⣆⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣶⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⡟⣻⣿⣖⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣶⣶⣶⣤⣄⣀⣀⣤⣴⣾⣿⠹⣿⣿⠾⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠿⢿⠿⠿⠋⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣻⣿⠿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡂⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣧⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠⠠⠄⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡏⠀⣀⣀⣀⡀⢀⣀⣈⡉⢉⣉⣉⣉⡛⠃⠀⠀⣀⣀⣀⠀⠘⠃⢀⣀⣈⡉⢉⣉⣉⢉⣉⡀⣀⣀⠀⢀⣀⣀⡈⢉⣉⣉⠉⢉⣉⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⡇⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡜⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡜⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣤⣾⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢀⣴⣿⣿⣿⣿⣿
⠀⠉⠙⠛⠛⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠑⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⣤⣄⠀⠾⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠃⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣿⣤⡌⠛⣛⣄⠀⠀⠀⢀⡀⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠋⢡⣄⠘⢾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣆⠉⠳⣦⡝⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡀⠀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣠⣴⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠶⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣂⣠⣄⡀⢸⣿⣿⣿⠟⠁⠀⠀⠨⡿⠿⡟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⣀⢹⣿⣿⡿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣀⣼⡿⠃⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠟⠃⠀⠀⠀⠀⠈⠻⣿⣇⡀⣀⣤⣴⣶⣶⣷⣷⣿⣿⠷⣾⣿⠿⠋⠙⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣁⣿⣿⣿⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⡟⢉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⡿⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡝⠟⠛⠛⠋⠁⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⠻⢷⠇⠀⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠉⠙⢿⣿⣿⣿⡟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⠉⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠻⢿⣧⣀⠀⠀⣀⡀⣀⣀⣀⣠⣴⣶⣶⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⠃⠀⠀⠀⠀⣰⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣶⣶⣶⣦⣤⣤⣴⣶⣾⣿⠿⡿⠟⠒⠲⢶⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣽⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡂⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣗⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡟⠀⣀⣀⣀⡀⢀⣀⣀⡉⢉⣉⣀⣉⡉⠀⠀⠀⣀⣀⣀⠀⠐⠁⢀⣀⣀⡉⢉⣉⣁⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢉⣉⣉⠉⢉⣁⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⡇⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡜⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡜⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣠⣶⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣄⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣴⣾⣿⣿⣿⣿
⠀⠉⠛⠛⠛⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠑⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠀⢀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀⣠⡀⠠⣤⣤⣴⡶⠆⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣿⣤⣽⠛⣛⣄⡀⠀⠀⢀⡀⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠙⠉⠀⠀⠻⠟⢡⣄⠐⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣎⠙⣷⣾⡽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠒⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⠀⠀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣴⣶⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⢶⢶⡷⠒⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣢⣶⣶⣤⣀⠐⣿⣿⣿⡿⠋⠀⠀⠀⣼⡿⣿⠝⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⡲⢮⣤⣙⣿⣿⣿⠿⣿⠊⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣴⣿⠏⠀⠀⠀⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣅⣴⣿⣷⣶⣷⣿⣿⣿⡿⣿⡿⡿⠟⠙⢫⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣠⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡍⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢇⣸⣿⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣛⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⡏⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠛⠛⠋⠉⠁⠀⠀⠁⠁⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣚⣿⡿⠆⠈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⡿⣿⡈⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈⠙⠻⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣡⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣄⣀⣀⣠⣄⣄⣠⣤⣶⣶⣶⣶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⠟⠛⠚⠛⣿⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣧⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⡿⠿⠛⡛⡭⡀⣤⣤⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⣿⣏⠀⠁⠈⠻⠿⠿⣿⣿⣿⣷⣶⣤⣼⠿⠿⠿⠿⠿⠿⠈⠈⠀⠀⠊⠾⢾⣿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠁⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠈⠿⠋⢁⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠈⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣠⣴⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣾⣿⣿⣿⣿
⠈⠙⠛⠛⠿⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢀⣤⣤⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠀⢤⣤⡀⠀⣀⣀⡀⣤⣤⣤⣴⣶⠄⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠉⣯⡜⢿⣿⣷⣬⣿⠿⣿⣁⠀⠀⠀⠀⠉⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⣿⣷⡄⠙⠿⠋⠁⠸⢿⡿⣧⡀⢶⣿⣷⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⢠⣿⣿⣿⡆⠀⠀⠀⣤⣴⣶⡝⢻⣶⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⣴⣾⣿⣿⣿⣿⡀⠀⠀⠉⠙⢿⣿⠞⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⣤⡀⠀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣶⣶⣤⣄⣤⣤⣤⣀⣀⣀⣀⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣦⣴⣶⣦⣀⣶⣿⣿⣿⡿⠏⠀⠀⠀⢿⠿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣶⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣴⣿⠿⠊⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠟⠛⠉⠁⠀⠀⠹⢿⣿⣿⣿⣦⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⣶⣤⣄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠁⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⡿⡿⡿⣟⡿⡽⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠻⢿⣿⡿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠏⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡁⠀⠀⠀⠈⠉⠛⣶⣌⠁⣶⣦⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠏⢝⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⢶⣤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠿⠋⠉⠛⠛⠋⠹⠶⣶⣶⣶⣶⣦⣤⣀⣀⣀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉⠛⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠻⠿⠿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠏⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠉⠁⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⢀⣀⣤⣶
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣴⣿⣿⣿⣿
⠀⠙⠛⠛⠛⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣤⣄⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⢀⡀⠀⠀⠀⠠⣤⣤⣴⣶⣦⡄⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠈⣭⣶⣿⣿⣦⣹⣿⣿⣿⡋⠁⠀⠀⠉⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⣿⣷⡍⢻⣿⠟⠉⢾⣿⣿⣆⠰⣶⣿⣶⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⢀⣿⣿⣿⡆⠀⠓⢗⣨⣭⣎⠹⣶⣾⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣷⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⣴⣾⣿⣿⣿⣷⡄⠀⠈⠙⠻⣿⣷⣾⠟⠛⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠄⠀⠤⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⠀⡀⣀⠀⠀⢀⣀⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⣀⢀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢠⣤⣴⣦⣤⠀⣶⣿⣿⣿⠇⠀⠀⠀⠸⠿⠛⣦⠰⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣾⠟⠁⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠈⠛⢿⣿⣿⣧⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣶⣦⡀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣸⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣿⠿⡿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠋⠁⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡻⣿⣿⣿⡟⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣿⠛⠃⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⣛⣿⡻⠛⠞⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠉⢷⣶⣶⣦⣶⣶⣶⣶⣶⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠙⠓⠲⠶⢶⣶⣤⣄⣀⣀⣀⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡌⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠟⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠟⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⠀⣀⣠⣶
⠛⠛⠿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠁⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣤⡀⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠰⣾⣷⣤⣛⣿⣿⡟⠛⠁⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠉⠹⣿⠛⠉⣾⣿⣷⡆⢰⣶⣶⣦⠈⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠐⠀⣈⣩⣍⠻⣶⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣿⠟⠛⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢀⣤⣴⣦⣤⡀⣶⣿⣿⣿⠇⠀⠀⠀⠸⠿⠻⣦⠰⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣴⣾⠟⠁⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠈⠛⢿⣿⣿⣷⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣴⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠄⠀⠀⠀⠈⠉⠉⠉⠙⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣸⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⡿⡿⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⠀⠁⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢛⣿⣿⣿⡛⠻⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣿⡛⠁⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠟⢛⣿⡻⠛⠞⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠙⢿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⣶⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠙⠓⠲⠶⢶⣶⣤⣄⣀⣀⣀⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡔⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡯⡙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣟⡩⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠟⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⠀⣀⣠⣶
⠛⠛⠿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠁⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣤⡀⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠰⣾⣷⣄⣛⣿⣿⣿⠛⠁⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠈⠹⣿⠛⠉⣾⣿⣷⡆⢰⣶⣶⣦⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠒⢂⣈⣩⣍⠻⣶⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣽⠟⠛⠉⠁⠀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⠒⢄⠀⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⣤⣦⣦⣤⣀⢠⣾⣿⣿⡿⠖⠀⠀⠀⣶⡿⠿⠠⠂⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣤⣾⠟⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠹⢿⣿⣿⣯⣶⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠋⠛⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣰⣂⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣶⣦⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⣼⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣛⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⡿⠿⠿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⣿⣿⣿⣟⡉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣽⠟⠋⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⣛⣿⡟⠛⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠉⠻⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣏⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣾⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠙⠛⠲⠶⢶⣶⣤⣤⣀⣀⣄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠍⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡧⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣮⠈⠙⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣏⡉⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣀⣴⣾
⠙⠛⠿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣤⣤⣴⣶⣦⡄⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⣿⣿⣦⣽⣿⢿⣿⡋⠀⠠⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠨⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠙⠿⠉⠁⠿⣿⣿⣇⠸⣾⣿⣷⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠉⢀⣤⣼⣦⡹⣷⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡿⠋⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀⢀⣔⣢⣄⣀⠀⣾⣿⣿⣿⠀⠀⠀⠀⣰⣿⠛⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠢⣒⣳⣾⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣇⣠⣼⡟⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠘⢿⣿⣯⣀⣤⣴⣶⣶⣶⣶⣾⣾⣿⣿⣿⣿⡿⠇⠘⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⣠⣢⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣶⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⣴⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⢿⣿⢿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⣿⣿⣯⣿⣏⠙⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡟⠋⠁⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢉⣵⠝⠑⠋⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠛⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣤⣀⣀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⠛⠒⠶⠶⣶⣤⣤⣀⣀⣄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⠉⠉⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⡉⡀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⢀⣀⣠⣶⣿
⠙⠛⠻⢿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⣤⣤⣴⣶⣦⡄⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⢿⣿⣦⣼⡿⢿⣿⣅⠀⠀⢀⡈⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠨⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠙⠛⡉⠀⠻⣿⢿⣇⠰⣿⣿⣷⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠉⢠⣴⣾⣮⡙⣷⣾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣷⠟⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⢀⣀⣠⣀⣀⠀⣾⣿⣿⣿⠂⠀⠀⠀⣰⣿⠟⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢎⢚⣢⣾⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⢠⣼⡿⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠀⠀⠀⠀⠀⠙⢻⣿⣿⣀⣠⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⠏⠘⡃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣄⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣿⣿⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⢿⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠋⠁⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣟⠛⠋⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⣏⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢉⣷⠟⠛⠓⠓⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠻⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣤⣤⣄⣀⣀⣀⡀⢀⣠⣾⣿⣿⣷⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠁⠈⠉⠉⠉⠙⠛⠲⠶⢶⣦⣤⣄⣠⣄⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣗⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡍⠛⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⠉⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⣀⣤⣴⣿⣿
⠉⠛⠛⠿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣤⣤⣴⣶⡦⠀⠈⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⢿⣿⣦⣼⠟⣿⣯⣄⡐⢒⣢⣄⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠨⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠉⠉⠁⠀⠻⢿⢿⣦⡙⢿⣿⣷⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣦⡙⣿⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⠉⠉⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⣀⣠⣷⣶⣤⣀⢰⣾⣿⣿⡿⠂⠀⠀⠀⣶⡿⠿⠔⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣠⣾⡟⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠈⠛⢿⣿⣿⣭⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⠃⢛⡁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⣿⣄⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣯⠛⠋⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣩⣿⠟⠛⠛⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⠻⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣤⣤⣤⣀⣀⣀⣀⣠⣴⣿⣿⣿⣷⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠀⠀⠈⠉⠉⠉⠛⠲⠶⢶⣦⣤⣤⣤⣄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠙⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣅⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣴⣾⣿⣿
⠉⠙⠛⠻⢿⠇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣤⣤⣴⣶⡶⢀⡀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⠻⣿⣷⣾⠟⣿⣷⣴⠜⣛⣥⣤⡈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠉⣤⡀⠻⣿⢻⣦⣙⢿⣿⣷⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣷⣍⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠟⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣦⠄⣀⡆⠀⠰⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⣄⣤⣾⣴⣤⣀⢰⣿⣿⣿⡷⠀⠀⠀⢠⣾⡿⢷⠆⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⡾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣠⣿⡏⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣴⣤⣶⣾⣾⣶⣶⣶⣾⣿⣿⣿⣿⠿⠉⢫⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣛⣿⣿⣆⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⠟⠛⠛⠛⠻⢿⡟⠟⢟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⢿⣿⣿⣿⣯⣍⠙⠋⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡧⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⢿⣟⢜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡾⠒⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠉⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣰⣿⣿⣿⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⣦⣤⣄⣀⣀⣀⣀⣤⣶⣿⣿⣿⣶⣶⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁⠀⠀⠈⠉⠉⠉⠛⠳⠶⠶⠦⠤⢤⣤⡄⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⠙⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣴⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠁⠙⠛⠻⠿⠇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⢦⣤⣴⣶⠷⠀⣀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠈⠻⣿⣿⡾⠛⣿⣷⣤⠈⣈⣥⣤⡈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠹⢶⠀⠙⣿⢻⣶⣍⣻⣿⣷⣶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣷⣍⣿⣿⡿⠛⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢰⡶⠴⣵⡆⠀⢰⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣠⣤⣴⣿⣿⣶⣄⣺⣿⣿⣿⠗⠀⠀⠀⢰⣾⠿⢷⠄⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣴⣿⠇⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣵⣶⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⠛⠻⠻⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣸⣿⣿⣦⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⠻⠿⠛⠛⠛⠿⠻⠿⡿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣿⣿⣿⣿⡟⠉⠉⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠋⠋⠙⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣶⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣶⢶⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠛⠶⠶⠦⠤⠤⣦⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡏⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠘⠻⠿⠿⡿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠀⠉⠛⠛⠿⠃⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢲⣤⣴⡶⠟⠂⣀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠈⠛⢿⣿⡾⠛⣿⣷⣦⡀⣃⣥⣤⡄⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠉⢶⠂⠙⣿⡹⣷⣍⡻⣿⣷⣶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣎⣿⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠉⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⢰⠖⠢⢰⡆⠀⢠⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣁⣤⣴⣴⣿⣿⣶⣤⣻⣿⢿⣿⠏⠀⠀⠀⢸⣿⡟⢳⡃⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠛⠯⠿⠟⣿⣿⣿⣿⢿⠯⠛⠿⡇⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧⣼⡿⠃⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠟⠋⠁⠩⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣄⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣼⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠟⠻⠛⠙⠛⠛⠛⠻⢿⢿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⠿⣿⡏⠉⠉⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣿⣷⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠙⠛⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠋⠛⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⣀⣀⣀⣤⣶⣶⣿⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣴⣶⣶⣶⣦⣤⣤⣤⣤⣶⣿⣿⣿⣿⡶⠶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠁⠀⠀⠀⠀⠉⠉⠉⠙⠛⠲⠶⠶⠤⠤⠶⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣙⡿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠
⠿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠀⠈⠙⠛⠛⠃⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢲⣤⣴⡶⠟⠂⢀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠈⠛⣿⣿⠾⠛⣿⣷⣤⠐⣋⣥⣤⡄⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠉⠶⠀⠉⣻⡹⣷⣍⣻⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣎⣿⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡀⢸⠒⠢⢤⡆⠀⢠⠀⠘⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣠⢴⣶⣾⣿⣿⣿⣦⣹⡿⠿⠿⠃⠀⠀⠀⠸⣿⡟⣵⡄⠈⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠟⠿⠿⠿⢿⣿⣿⣿⠿⠋⠯⡯⠿⢿⣿⣿⡿⡿⠛⠙⠻⢃⢀⡀⠀⠀⣀⣀⣀⣨⣿⣿⣷⣾⡿⠃⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⢻⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⡿⠋⠉⠀⠘⠆⠀⠀⠀⠀⠿⢿⣿⣿⡿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣷⣆⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠏⠋⠉⠙⠛⠛⠛⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟⠿⠿⠅⠈⠁⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣇⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠛⢻⣿⣿⣶⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠛⢿⣿⣿⡿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠛⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠉⠙⢷⣦⡀⠀⢀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣶⣶⣶⣶⣦⣴⣦⣶⣿⣿⣿⣿⠶⠶⠶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠲⠶⠦⠴⠶⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⠿⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠘⠛⠿⠿⠿⢿⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠
⠿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠀⠀⠉⠛⠛⠃⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣯⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⣦⣤⣴⡶⠗⢀⣀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⠻⣿⣷⠿⠛⣿⣷⣤⠘⣩⣤⣤⡈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠱⡆⠀⠙⣛⢿⣶⣉⢿⣿⣷⣦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣷⣍⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⡀⠸⠆⠀⠰⡆⠀⠰⡀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣷⣿⣿⣿⣿⣿⣿⣾⡟⠉⠉⠁⠀⠀⠀⠀⢿⣇⣼⡇⠀⣸⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠐⠿⠻⠷⠹⣿⣿⣿⣟⢛⠈⣉⣤⣤⣠⣄⣄⣤⣤⣤⣴⣿⣿⡿⣿⡟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣄⣀⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠟⠛⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣷⡀⣀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡏⠐⠁⠐⠉⠉⠋⠟⠛⠛⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⡿⡟⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⠛⠛⠃⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⡷⣧⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠙⠿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡾⠋⠁⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣄⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⡿⠛⠛⠷⠶⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠒⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣩⣙⡿⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠙⠿⠿⠿⠿⣿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤
⠿⠿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣶⣿⣿⣿
⠀⠀⠈⠉⠛⠁⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠐⣦⣤⣴⣶⠆⣀⡀⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠙⢿⣿⣶⠿⢻⣿⣶⣴⠿⣛⣧⣤⠘⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠉⠱⡆⠨⠛⣿⢿⣦⡙⢿⣿⣷⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣶⣍⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠂⡀⡆⠘⡇⠀⠲⣦⠀⠐⡆⠀⢲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
[40A⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⡇⠀⠈⠀⠀⠀⠀⠀⠛⣧⣤⣷⠀⣸⣿⠋⠁⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠻⠿⠟⠋⠈⠟⠻⠳⠙⠻⣿⣿⣿⣯⣀⣠⣴⣶⣶⣶⣶⣴⣶⣾⣶⣿⣿⡿⢻⡟⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠁⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢻⣿⣿⣄⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡟⠿⠿⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⣩⡿⠻⣿⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠛⠿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣾⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⠿⠿⠛⠛⠛⠒⠶⢶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣟⠛⠿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠻⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠻⠿⠿⠿⡿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤
⠛⠿⢿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⣦⣤⣴⣶⠗⢀⢀⠘⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠙⠻⣿⣷⠿⢛⣿⣷⣶⡽⣛⣥⣤⡈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠱⠶⠀⠙⣿⢻⣦⡙⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣽⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠐⣶⠰⠄⢠⡀⠳⠀⠀⣶⡄⠀⠄⠀⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
[40A⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⡇⠀⠈⠀⠀⠀⠀⠀⠛⣧⣤⣷⠀⣸⣿⠋⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠋⠈⠿⠻⠗⠝⠻⣿⣿⣿⣏⣀⣠⣴⣶⣶⣶⣶⣴⣶⣶⣶⣿⣿⡿⠻⠏⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣿⣿⣄⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⡿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⡟⠿⠿⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠃⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣭⡿⠻⣿⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠛⢿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣤⡾⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⠿⠿⠛⠛⠛⠒⠶⢶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠛⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣟⠛⠿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤
⠛⠿⢿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⣦⣤⣴⣶⠗⢀⣀⠘⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⣶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠙⠻⣿⣷⠿⢛⣿⣷⣶⠽⣳⣧⣤⡈⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠱⠶⠈⠛⣿⢻⣦⡙⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣽⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣴⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠰⠄⢢⡀⠳⠀⠀⣴⡄⠀⢄⠀⠘⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
[40A⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣾⣿⣿⣷⣾⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣠⣾⠟⠁⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠁⠀⠈⠈⠁⠀⠈⠹⣿⣿⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⣽⣿⣿⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢿⠛⠻⠛⠟⠿⠿⠿⣿⣿⡿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⠛⠛⠛⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⡭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣨⣼⠋⠻⠋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣠⣀⣀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⠉⠙⠒⠲⠶⣶⣦⣤⣀⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣧⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣏⠛⠛⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠛⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣗⢄⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣾
⠉⠛⠿⠿⣿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⡀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣀⣴⣦⣤⡛⢡⣤⣄⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠈⠛⢿⠟⠛⢱⣿⣿⣟⠉⣥⣾⣧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⣯⡹⣷⣮⣻⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣬⡿⠛⠋⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣦⣄⠀⠀⠀⠐⠀⠀⣠⠄⠀⠀⠐⣶⠀⠀⢰⡀⢆⡀⠀⣷⡄⠀⣆⠀⠈⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
[40A⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣴⣿⣿⣿⣷⣶⣿⣿⠿⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣤⣾⠟⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠁⠀⠁⠑⠀⠀⠀⠈⠿⣿⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠟⠁⠉⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣃⣸⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⠟⠝⠋⠛⠋⠛⠻⠿⠿⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⡟⠛⠛⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣿⣿⣿⣿⠧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⣽⠛⠉⠎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠻⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⠉⠉⠉⠉⠙⠒⠶⢶⣶⣦⣤⣀⡀⠀⢀⡀⠹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣏⠉⠛⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣾
⠉⠛⠿⠿⣿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⣀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⣤⣤⣤⣴⣎⣡⡤⣠⠀⠙⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠙⠿⠟⠛⣱⣿⣿⣿⣃⣴⣾⣿⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢈⣬⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡿⠛⠋⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣶⡖⠀⢰⡀⢶⡀⠀⣵⡄⠀⣦⠀⠈⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
[40A⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣴⣿⣿⣿⣿⣷⣾⣿⠏⠀⠙⠁⠀⠀⢀⠀⠀⢀⣀⣀⣼⣿⣿⣶⣿⠟⠁⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠈⠁⡁⡀⠈⠈⢻⣿⣷⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⢟⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣃⣸⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⡙⠛⠝⠋⠙⠉⠙⠛⠛⠿⢿⠿⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡻⣿⡿⠿⣯⡍⠉⠛⠃⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣇⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣼⠋⠙⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠛⠻⣿⣆⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⣠⣤⣄⣀⣀⠀⠀⠀⢀⣠⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠈⠉⠉⠉⠙⠓⠲⠶⢶⣦⣤⣀⡀⣀⣀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡉⠙⠛⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣾
⠀⠉⠻⠿⠿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢁⣤⡄⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣤⣤⣴⠶⣮⣉⣔⠲⠂⢀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠙⠛⠛⠁⠻⣿⣿⣯⡑⢾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢡⣤⣴⣾⡙⢿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡾⠛⠉⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣄⡀⠀⠀⠀⠆⠀⢀⡄⠀⠀⠀⢳⡦⠂⠀⣆⠰⣆⠀⢰⣦⠀⢰⡆⠈⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
[40A⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣣⣴⣿⣿⣿⣾⣿⣾⣿⠏⠀⠙⠁⡀⣀⣀⠀⢀⢀⣀⣄⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠙⠛⠛⠋⠀⠀⠈⢁⣁⣀⡀⠈⠻⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⢛⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⣼⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡻⠐⠏⠁⠍⠍⠉⠙⠛⠿⠿⠿⡿⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢪⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡻⣿⣿⠿⢿⣭⠋⠛⠃⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⡟⡌⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⣿⣿⡿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣼⠻⠋⠹⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠙⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⣠⣤⣤⣄⣀⡀⠀⠀⣀⣠⣾⣿⣿⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠈⠉⠉⠉⠉⠛⠲⠶⢶⣦⣤⣄⣀⣀⣀⠀⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡙⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡉⠙⠛⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣉⣈⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣶
⠀⠉⠛⠿⠿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠐⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠘⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⣴⣤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⣤⣤⡶⣾⣏⣤⣶⡶⢄⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠙⠛⢛⠋⠘⣿⣿⣿⣉⠶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⣾⣶⣍⢻⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠹⣶⠀⠀⣦⠀⢦⡀⢰⣦⡀⠀⡆⠀⠙⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣤⣶⣷⣶⣿⣾⣿⣾⡿⠟⠀⠙⠋⠀⣀⢀⣀⣀⣀⣠⣤⣼⣿⣿⣷⣾⠟⠁⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠋⠀⠀⠀⠈⣁⢈⡈⡹⡻⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⡟⠋⠁⠀⠀⠆⠀⠀⠀⠀⠿⣿⣿⢿⡿⡿⠋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⣼⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢝⠙⡽⠽⠑⠙⠙⠛⠛⠟⠿⠿⠿⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣟⣿⣿⠿⢿⡍⠉⠙⠃⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⡤⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣼⠛⠋⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠙⠛⢿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣠⣾⣿⣿⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠉⠉⠉⠉⠛⠓⠶⠶⣶⣤⣄⣀⣀⣠⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠉⠙⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⠈⠈⠛⠛⠟⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠈⠛⠿⠿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠈⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⣴⣦⡄⠈⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣤⣶⢶⣿⣡⣴⣧⢄⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠈⠛⠛⡋⡀⣿⣿⣿⣏⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣤⣶⣮⡹⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣦⡀⠀⠀⠀⠰⠀⢀⡤⠀⠀⠀⠸⣶⠐⢄⢶⠈⢦⡀⢀⣶⡄⠀⣆⠀⠉⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣶⣶⢾⣿⣾⡿⠿⠀⠈⠋⠀⠀⡀⢀⣀⣀⣀⣤⣴⣿⣿⣿⣾⠿⠃⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠈⢁⠀⣀⣈⣙⢿⣿⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⠛⣿⡿⠋⠁⠀⠀⠠⠀⠀⠀⠀⢿⡿⣿⢿⡿⡿⠋⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣋⣸⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠍⢟⠹⠏⠍⠉⠉⠝⠛⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣙⣿⣿⠿⢿⡯⠉⠛⠁⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢽⣿⣿⣿⣿⠅⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠻⢿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣰⠟⠋⠉⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠛⠻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣧⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠙⠓⠶⠶⣶⣦⣄⣀⡀⠀⣀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣉⠙⠛⠛⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠛⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠙⠻⠿⢿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⢀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⣴⣦⣄⠀⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣀⣶⣶⣿⠛⣻⣤⣄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠈⠻⣿⠟⠛⢳⣿⣿⣷⠛⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣈⣹⣭⡹⣷⣮⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⣮⡿⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⣄⡀⠀⠀⠀⢰⠀⠀⣰⠀⠀⡀⠘⣶⠐⠀⢲⡈⢧⠀⢈⣷⡀⠀⢲⠀⠉⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣶⣶⣲⣶⣾⣿⣾⡿⠿⠀⠈⠛⠀⢀⠀⣀⡀⣀⣀⣤⣤⣿⣿⣿⣾⡿⠃⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠛⠁⠀⠀⠀⠁⠈⢀⣈⡙⣿⣿⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⠛⣿⡿⠛⠋⠀⠀⠤⠀⠀⠀⠀⠿⡿⣿⣿⣿⡿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⣸⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠍⠋⠻⠟⠙⠋⠛⠛⠛⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡛⣿⣿⠿⣯⡭⠉⠛⠃⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢹⣿⣿⣿⣿⡁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣠⠟⠉⠉⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠛⠻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣿⡟⠉⠛⠉⠉⠉⠉⠓⠲⠶⢶⣶⣤⣀⣀⠀⠀⢀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣹⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣉⠙⠛⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠻⠿⣿⣶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠈⠛⠿⢿⣿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⢀⣀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠘⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⣴⣦⡄⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⡀⣶⣶⣿⠿⠛⣤⣀⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⢻⣿⣷⣿⠚⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⢈⣩⣯⠹⣶⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣭⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⣦⣄⠀⠀⠀⠀⢰⠀⠀⣠⠄⠀⡀⠐⣶⠀⠀⢲⡈⢧⠀⠈⣷⡄⠀⠣⠀⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
[40A⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣶⣶⣾⣿⣿⣿⠿⠀⠘⠛⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣼⡿⠃⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋⠛⠛⠛⠛⠛⠿⠿⠿⠟⠁⠀⠀⠀⢁⢀⣀⣀⣉⣻⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⠻⣿⡿⠛⠋⠀⠀⠤⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠋⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠂⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣸⣿⣿⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠝⠛⠹⠛⠙⠋⠛⠟⠟⠻⠿⣿⡿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡻⣿⣿⣿⣿⣯⠛⠛⠃⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠉⢻⣿⣿⣿⣿⢅⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣱⠟⠃⠑⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠛⠿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣶⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⡿⠛⠛⠉⠉⠉⠙⠒⠶⢶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣮⣝⠛⠛⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠙⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠻⠿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣠⣿⣿
⠀⠀⠀⠀⠁⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⣤⣤⡀⠙⢻⣿⣿
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⣶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⣻⣿⣾⣾⠻⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠔⠘⢻⣯⢿⣶⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣄⡄⠀⠀⠀⠀⢰⠀⠀⣰⡆⠐⡀⠐⣶⠐⠀⢰⡈⢷⠀⠙⣷⣄⠀⢣⠀⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
䷩ 𝚕𝚒𝚗𝚎 3414
╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕
(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/13/github-excuses-running-out/#comments
Gemini version at gemini://gemini.techrights.org/2022/01/13/github-excuses-running-out/
⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.13.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧
Gemini_version_available_♊︎
✐ No_Excuses_for_Using_GitHub_Anymore⠀✐
Posted in Free/Libre_Software, Microsoft at 7:43 am by Dr. Roy Schestowitz
Video_download_link | md5sum dc06b1c110d011003e408492d4d8e895
GitHub as a Force of Occupation
Creative Commons Attribution-No Derivative Works 4.0
http://techrights.org/videos/occupied-by-github.webm
Summary: Software developers become living witnesses to more and more reasons
to abandon Microsoft for good
THE above video covers ongoing_discussions about Microsoft’s misuse of the TOS
(a lot like a CoC) to censor developers if not ban them completely. It’s a
subject we_wrote_about_yesterday. It’s an emerging theme, especially since last
year due to a number of high-profile incidents and cautionary tales of_all
sorts.
Towards the end of the video I speak about “GitHub_Takes_Down_‘Widevine_Dump’
Forks_Following_MPA_Complaint”; it increasingly seems like GitHub isn’t much
more than a Microsoft censorship tool, not just for Microsoft’s agenda but also
the copyright cartel’s. Who on Earth would want this sort of thing for code/Git
hosting? Except those ‘bought’ by Microsoft, e.g. OSI and Linux_Foundation?
Racism_is_fine_with_them. They’re in violation of Open Source and Free software
licences by excluding ethnic groups, too. █
䷩ 𝚕𝚒𝚗𝚎 3462
╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕
(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/13/irc-log-120122/#comments
Gemini version at gemini://gemini.techrights.org/2022/01/13/irc-log-120122/
⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.13.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧
Gemini_version_available_♊︎
✐ IRC_Proceedings:_Wednesday,_January_12,_2022⠀✐
Posted in IRC_Logs at 2:33 am by Needs Sunlight
Also available via the Gemini protocol at:
* gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-120122.gmi
* gemini://gemini.techrights.org/irc-gmi/irc-log-120122.gmi
* gemini://gemini.techrights.org/irc-gmi/irc-log-social-120122.gmi
* gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-120122.gmi
Over HTTP:
🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇H 🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇HTML5_logs⦈_
#techrights_log_as_HTML5 #boycottnovell_log_as_HTML5
🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇H 🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇HTML5_logs⦈_
#boycottnovell-social_log_as_HTML5 #techbytes_log_as_HTML5
🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇t 🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇text_logs⦈_
#techrights_log_as_text #boycottnovell_log_as_text
🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇t 🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇text_logs⦈_
#boycottnovell-social_log_as_text #techbytes_log_as_text
Enter_the_IRC_channels_now
=> =============================================================================
§ IPFS Mirrors⠀➾
CID Description Object type
IRC log for
QmRcBYBSWR9XgF49ygEHRakeZokHWs7TWu9Uyh5VP52KVm #boycottnovell 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
(full IRC log
as HTML)
IRC log for
#boycottnovell
QmQJw3ATfwiwhNhyysUjug2nAuHELsxr5VShkXtHUJfrzd (full IRC log 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
as plain/ASCII
text)
IRC log for
#boycottnovell-
Qmd3UaiLCrvs6sF2LXneYVwhJTnRBriGX31iaPgGwJDmKd social 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
(full IRC log
as HTML)
IRC log for
#boycottnovell-
QmdkSyjtr9EayipNLAqux9HWVvn3WMM4QwhbumxUH44oad social 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
(full IRC log
as plain/ASCII
text)
IRC log for
QmQ2ETTxLsCSYQFJGW74YA1eeR2NfaCVuyp7LjK1LxBNri #techbytes 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
(full IRC log
as HTML)
IRC log for
#techbytes
QmVMJKN3FkCau8fsvAL4HYkBHwoJgWkSEtx2yqGDptnmfQ (full IRC log 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
as plain/ASCII
text)
IRC log for
QmWKnQKBqG5PB6MxeRPurkjes4uiCCEx3G4ciayQyAvrya #techrights 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
(full IRC log
as HTML)
IRC log for
#techrights
QmYL4WeE8tbdMvQps4J8vQjcvsNSMqQDk7idHHbk9CcagV (full IRC log 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
as plain/ASCII
text)
🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇IPFS logo⦈
§ Bulletin for Yesterday⠀➾
Local_copy | CID (IPFS): QmUYb6AJaHEaYvQGdw1Z8bffPkohvdveAhmoUhNFHe4PB6
䷩ 𝚕𝚒𝚗𝚎 3589
╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕
(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/13/microsoft-connected-fud-pr/#comments
Gemini version at gemini://gemini.techrights.org/2022/01/13/microsoft-connected-fud-pr/
⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.13.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧
Gemini_version_available_♊︎
✐ 2022_Commences_With_Microsoft-Themed_(and_Microsoft-Connected)_FUD_Against
GNU/Linux⠀✐
Posted in Deception, FUD, GNU/Linux, Microsoft, Security, Servers at 8:50 pm by
Dr. Roy Schestowitz
Video_download_link | md5sum 0759e712fb7708a8b71cbe9b23ac887f
2022 Starts With Microsoft-Connected Linux FUD
Creative Commons Attribution-No Derivative Works 4.0
http://techrights.org/videos/2022-fud-linux.webm
Summary: A psychopathic Microsoft, aided by operatives inside the mainstream
and so-called ‘tech’ media, keeps spreading old and invalid stigma about
“Linux” and Free software; few people still bother responding to these fact-
free FUD campaigns, which boil down to ‘perception_management’_PR/propaganda
THE smear campaigns against Free software are making a big comeback. As Sam
Varghese put_it just before Christmas, “Log4j attacks remain low-key compared
to infosec industry hype” and yesterday_we_wrote_about_Microsoft_operatives
adding_baseless_stereotypes_to_the_mix, exaggerating the severity of Log4j bugs
compared to boundless breaches affecting Microsoft’s products last year.
Well, “it does feel like these attempts at FUD have intensified recently,”
Psydroid said in IRC an hour or so ago, and “not a day seems to go by without
some new FUD thrown our way…”
🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Much work was done years ago to show the many, if not most,
senior execs in companies are psychopaths (if not necessarily psychotic). I
have realised more recently that companies, themselves, are psychopathic. We
need the equivalent of Asimov's laws applied to companies.⦈ Psydroid and I
share these concerns with countless_others.
In the video above I present new examples (from the past couple of days alone),
alluding along the way to Marcus_Hutchins and under-reported NSA_aspects (those
are not outdated or “old news”) while showing that shameless_marketing by the
Microsoft-connected [1, 2] CrowdStrike accompanies this new wave of Go(Lang)/
Linux FUD [1, 2] in Microsoft-friendly sites, which are also happy to frame
Windows/VMware_problems_as_“Linux”, probably for the second time in about a
month. As if “ESXi servers” are the same as “Linux”… that’s like calling
Photoshop “Windows”.
We should moreover note that lots of such FUD (as mentioned in the above video)
could be found back_when_Microsoft_was_being_clobbered_by_GNU/Linux_on_the
server_side and the rise of mobile (Android/iOS) was making Microsoft
increasingly irrelevant on the client side too. We saw lots_of_FUD_last_year,
including that same_pattern_of_Go(lang)_FUD, and it seems to be making a
comeback as soon as this year starts. What’s behind it? Mostly a PR or
perception manipulation campaign, as the facts don’t justify the alert/hype/
panic. It’s all_about_manipulation_of_opinion, as Mark Kent noted some hours
ago. Profits first, facts never. █
🄸🄼🄰🄶🄴🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽⦇Companies_are_all_psychopathic.___We_need_something_like
Asimov's_laws_for_companies_introduced.___In_the_following_order:__A_company:
1)_never_allows_harm_to_humanity;_2)_never_allows_harm_to_a_person;_3)_obeys
orders_by_humans;__4)protects_its_own_existence.⦈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣷⣾⣷⣆⠀⠿⠿⠷⠿⠇⠟⠽⠿⠿⠀⠯⠇⠛⠿⠎⠯⠻⠿⠿⠽⠿⠿⠽⠯⠻⠫⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠯⠿⠱⠀
⢸⣿⣿⣿⣿⣿⣿⠀⢰⣰⢠⣄⡤⡦⡄⢠⣤⣤⣄⣴⡠⠠⣤⡤⣤⢤⡄⡤⣦⢤⣤⣤⣤⠀⣤⢠⣄⣤⣤⣤⡄⣤⣤⢤⡤⡄⢴⡤⡄⢠⣴⢤⡤⣤⣤⡄⢴⡦⣄⣤⠀⡤⣤⣠⣤⢤⢄⡄⠀⡦⡆⢠⡄⣤⣴⠀⠀⠀
⠀⠻⢿⣿⣿⠿⠃⠀⠈⠉⠈⠁⠉⢁⠁⠈⠈⠈⠉⠉⡈⠀⠁⠉⠉⠉⠁⠉⠉⠉⠁⡉⠉⠀⠋⠈⠉⠉⠉⠉⠁⠉⠙⢛⠉⠁⠈⠉⠁⠈⠉⠈⠉⠁⠉⠁⠈⠁⡁⠉⠀⠁⠉⢉⢉⠈⠚⠘⢀⣁⡁⠈⠁⠉⡈⠁⠀⠀
⠀⠀⠀⠉⠁⠀⠀⠀⠸⠏⠿⠽⠿⠹⡤⠸⠿⠿⠿⠷⠿⠽⠏⠰⠿⠽⠽⠿⠯⠿⠇⠇⠏⠆⠯⠽⠽⠏⠿⡯⠿⠿⠏⠿⠺⠿⠿⠰⠿⠏⠿⠇⢸⠷⠿⣳⠻⠅⠿⠮⢿⡿⠿⠿⠽⠹⠿⠄⢇⠿⠁⠿⠗⠽⠯⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡠⢤⣤⣤⣤⣤⣄⡤⣶⢄⡄⢠⢄⣤⣄⣤⠄⣦⡠⣤⣶⣠⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⡚⡀⠘⠉⠉⡚⠈⠁⠉⠈⠁⠉⠈⠑⠉⠀⠀⠀⠀⠀⡀⡀⠀⢀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠹⠿⠏⠞⠿⠇⠸⠿⠿⠿⠇⠿⠿⠿⠯⠇⠸⠏⠿⠽⠏⠿⠇⠸⠿⠿⠭⠿⠿⠿⠯⢟⡎⠸⠽⠹⠿⠿⠅⠯⠽⠽⠏⠿⣿⠷⠿⠏⠿⠿⠷⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣄⣤⡤⡤⣤⣄⣤⡦⣤⣠⣤⣤⠀⢠⡄⣤⣤⠀⡤⣤⡤⣠⡤⢴⢤⡤⣤⢄⣤⢴⢴⣴⣦⠄⠀⠀⣦⣦⣦⡄⢠⢄⣤⣠⣄⢴⠀⡦⡦⣤⣤⢠⣤⡤⣤⢠⡦⣤⣠⣴⢠⣄⣤⢴⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡉⢉⡁⠁⡉⠉⠉⠁⠉⠉⡉⠉⢃⠈⠁⠁⠉⠀⠋⠉⠑⠋⠉⡉⡈⠉⢙⠉⡉⠈⠉⠉⠉⠁⠁⠀⠉⠉⠉⠁⠈⡈⠉⠉⠁⠉⠀⠉⠁⠉⠉⠈⠉⠉⠋⠉⠁⠉⠉⠉⠈⠁⠉⠈⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠇⠼⠳⠿⠇⠿⠹⠯⠟⠞⠹⠿⠸⠼⠟⠾⠿⠿⠸⠿⡯⢿⠽⠇⠿⠷⠿⠀⠯⠯⠇⠪⠽⠽⠏⠿⢾⠗⠿⠸⠇⠿⠷⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⣾⣷⢠⣤⡤⢠⣤⣤⣤⢠⣤⣤⣤⣤⣤⣤⢤⣠⣤⣤⠄⣴⣶⢶⣶⣄⢠⢤⣠⡠⡤⣠⣠⣤⢤⣄⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⢄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠛⠉⠈⠉⠉⠈⠈⠉⠊⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠋⠁⠈⠁⠃⠈⠉⠈⠉⠁⠈⠈⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠷⠶⠷⠖⠶⠶⠶⠷⠸⠦⠶⠾⠿⠰⢶⣶⡶⠶⠶⠶⠷⢶⡆⠶⣶⠶⠷⠶⠶⠸⠿⠖⠆⠶⢰⠲⠲⠶⠲⠶⠾⠖⠴⠶⠖⠢⠳⠶⠶⠖⢰⠾⠂⠲⠕⠶⠶⣶⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣴⣴⡄⣦⣤⣴⣤⣦⣴⣴⣤⣠⣤⣤⣖⣶⣖⣖⣦⣖⢰⡔⣆⣦⣶⡶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢴⢤⣦⠆⣰⣴⣰⢀⢀⣦⣤⡄⢴⣶⡀⣴⡔⣴⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠸⠇⠐⠯⠇⠿⠀⠟⠿⠸⠇⠘⠿⠃⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣀⣼⣄⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⢀⣀⣠⣀⡀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⠀
⠀⢠⣿⣿⣿⣿⣿⣆⠀⠛⠛⠛⠛⠚⠙⠛⠛⠛⠘⠛⠙⠛⠛⠛⠛⠛⠛⠙⠛⠚⠛⠛⠚⠛⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠛⠀
⠀⢸⣿⣿⣿⣿⣿⡿⠀⠿⠿⡾⢿⡾⠿⣿⠸⠿⠰⣿⠿⠿⠏⠾⠾⠿⠽⠷⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠻⢿⣿⡿⠟⠁⠀⡤⢄⣀⣀⣀⢀⣀⣀⢀⡀⣄⣀⣀⠀⣀⢀⣀⡀⢀⡀⣤⠀⣀⣀⣀⢀⣀⣠⡀⣀⣀⡀⣀⣄⣄⣠⣀⡀⠀⠀⢠⣠⣤⣀⠀⣀⣀⣀⣀⣀⡄⢀⣀⣀⣀⣀⣀⣀⣄⣄⢠⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠚⠃⠛⠸⠊⠛⠘⠃⠛⠛⠛⠀⠛⠛⠙⠃⠘⠃⠛⠀⠗⠛⠫⠏⠓⠚⠋⠛⠿⠛⠛⠓⠃⠛⠑⠒⠀⠀⠈⠋⠋⠛⠀⠃⠛⠛⠛⠛⠃⠘⠛⠛⠛⠃⠛⠛⠓⠛⠘⠃⠻⠾⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣶⣶⢠⠿⣴⣾⢰⡖⣶⣲⣶⠚⣶⡄⣷⣶⣶⡶⣶⡀⡟⣶⣶⠂⣖⣶⣶⣶⢲⣖⣶⣶⡖⣾⣶⣶⣆⢸⡖⣾⣲⢲⣲⣶⣷⣶⣖⣶⣶⣺⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢶⠀⣗⣷⢰⣦⢸⢳⣦⣿⣖⣶⣶⢾⡖⣶⣲⠠⣶⣴⣶⣧⣶⡶⡆⠀⣰⣧⢠⣖⣖⣶⢶⡆⣶⣲⣶⣶⢦⢶⠀⣻⣸⡄⡖⣶⣶⣦⢶⣦⡖⢰⣶⣿⣷⡶⣴⣶⣶⢸⣷⣶⣶⢲⢲⣦⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⣄⣀⢀⣁⣀⢀⡁⣀⢨⣄⡀⣀⠀⣬⢫⠀⣈⢀⡈⡀⣁⣀⣀⢀⡀⣤⢀⣁⢈⣀⡀⢠⣉⣈⢀⣀⣛⡀⢠⣀⡈⢀⡀⢀⣁⢀⣀⣁⣀⣀⢁⡀⡈⢡⣄⡄⢁⡀⣄⢈⡀⣀⣀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠿⠃⠏⠿⠾⠏⠿⠸⠟⠿⠸⠷⡿⠸⠀⠾⠸⠀⠏⠿⠿⠿⠹⠿⠇⠸⠿⠿⠻⠟⠿⠿⠗⠸⠹⠿⠿⠹⠿⠇⠸⠿⠟⠸⠿⠸⡿⠻⠿⠹⠗⠿⠻⠗⠇⠸⠗⠇⠸⠞⠿⠻⠿⡿⠿⠇⠀⠀
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䷩ 𝚕𝚒𝚗𝚎 3707
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Gemini_version_available_♊︎
✐ Links_13/1/2022:_Sparky_5.16,_Fwupd_1.7.4,_and_KDE_Plasma_5.24_Beta
Released⠀✐
Posted in News_Roundup at 12:22 pm by Dr. Roy Schestowitz
🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈
§ Contents⠀➾
* GNU/Linux
o Distributions
o Devices/Embedded
* Free_Software/Open_Source
* Leftovers
* § GNU/Linux⠀➾
o § Audiocasts/Shows⠀➾
# ⚓ OBS_Studio_pulls_in_NVIDIA_as_a_new_sponsor_|
GamingOnLinux⠀⇛
OBS Studio continues growing, with NVIDIA joining
as their latest big sponsor to help this free and
open source project continue getting better.
Announced by the official OBS account on Twitter,
it notes that NVIDIA is a new Diamond sponsor. This
means that NVIDIA will be providing at least
$50,000 a year to the project. This doesn’t mean
NVIDIA has any kind of control over OBS and they
join the likes of Logitech, Twitch, Facebook and
more in helping to fund probably the best way to
record and livestream video on Linux.
# ⚓ BSDNow_437:_Audit_that_package⠀⇛
Using FreeBSD’s pkg-audit, 20 year old bug that
went to Mars, FreeBSD on Slimbook, LLDB FreeBSD
kernel core dump support, Steam on OpenBSD, Cool
but obscure X11 tools, and more
o § Kernel Space⠀➾
# ⚓ Intel_Is_Bringing_an_Important_Feature_to_Linux_Kernel
5.17⠀⇛
Intel is working to provide the Linux kernel with
the ability to allow BIOS updates without a reboot.
Certain compute systems require high Service Level
Agreements (SLAs) where fewer system reboot
firmware updates are required for deploying
firmware changes to address bug fixes, security
updates, and to debug and root cause issues. Ever
since BIOS updates became possible, the process
required rebooting the PC.
Intel is now changing that, thanks to a new part of
the ACPI specification called Platform Firmware
Runtime Update and Telemetry (PFRUT). This allows
for firmware updates to a PC’s BIOS or UEFI without
forcing a reboot. The idea is to reduce downtime,
especially for servers that should ideally remain
available 100 percent of the time.
# ⚓ Intel_Arc_DG2_“Alchemist”_Added_For_Mesa_22.0_But_Code
Disabled_For_Now_–_Phoronix⠀⇛
Intel’s open-source Linux graphics driver
developers have now committed the DG2/Alchemist
graphics card PCI IDs and device information data
to Mesa 22.0 for their OpenGL and Vulkan driver
support, but for now until the Linux kernel support
is baked this is disabled.
Landing today in Mesa 22.0, which has now been
extended by three weeks for additional development,
is adding the DG2 (Alchemist) device information
and the twenty PCI IDs. Yes, there are 20 PCI IDs
for DG2 but not necessarily for all different
models planned for going to market but sometimes
extras are reserved for early engineering samples,
possible but currently unplanned future SKUs, and
similar reasons for reserving more possible IDs per
family than what necessarily appear in retail/OEM
channels.
# ⚓ Linux_Kernel_Patches_Posted_For_Bringing_Up_Tesla’s_Full
Self-Driving_SoC_–_Phoronix⠀⇛
Samsung in partnership with Tesla has posted a set
of 23 patches for enabling Tesla’s Full Self-
Driving (FSD) SoC for the mainline Linux kernel.
The 23 patches get Tesla’s Full Self-Driving SoC so
that it can boot off the upstream Linux kernel
compared to the downstream kernel builds currently
used. The initial Tesla FSD SoC is made up of three
clusters of four Cortex-A72 processor cores and
several extra IP blocks.
# ⚓ Intel_adds_twenty_ARC_Alchemist_GPU_PCI_IDs_to_open-source
Linux_Mesa_drivers⠀⇛
As many as twenty Intel ARC Alchemist GPU PCI
Device IDs appear in the upcoming open-source Linux
Mesa graphics driver update
Intel has not divulged a massive amount of details
on their newest ARC Alchemist discrete graphics
cards, especially during last week’s CES 2022
conference in Las Vegas. With this newest update to
the next Linux Mesa driver, it is appearing that
Intel is diligently working to make sure their
newest GPU will be widely available on more than
just Window’s operating system.
o § Applications⠀➾
# ⚓ Fwupd_1.7.4_Released_with_Support_for_ModemManager_Devices
and_New_Hardware⠀⇛
Fwupd 1.7.4 is here exactly one month after fwupd
1.7.3 and adds firmware branch support for
ModemManager devices, adds the ability for firmware
engineers to patch files at known offsets, and
introduces support for displaying why more devices
are not marked as updatable.
This release also introduces support for more
hardware, including the HP USB-C G2 Dock, Nordic
HID devices using MCUBoot, ThinkPad Thunderbolt 4
Dock, Quectel EG25-G LTE modem, many UF2 devices,
as well as more PixArt devices.
o § Instructionals/Technical⠀➾
# ⚓ How_to_install_and_use_ADB_on_Windows,_Mac,_Linux,_Android,
Chromebooks,_or_even_in_a_browser⠀⇛
If you want to do any number of things that require
access to the so-called Android Debug Bridge (ADB)
or the fastboot tools for Android — sideloading
apps, installing custom ROMs, taking screenshots in
apps that forbid it, or accessing certain hidden
features — you’ll need to get ADB up and running on
your platform of choice first. Fortunately, doing
so is possible virtually on any device at this
point — you can even start ADB from another Android
phone, or a web browser. We’ll help you get set up
no matter what platform you’re on in this guide.
# ⚓ How_To_Install_ImageMagick_7.1.0_In_Ubuntu_20.04_LTS_/
Linux_Mint_|_Tips_On_UNIX⠀⇛
ImageMagick is a free and open-source software used
to create, edit, compose or convert digital images.
It can read and write images in a variety of
formats including 200 images.
It is available for multiple operating systems
Linux, Windows, macOS, iOS, Android, and others.
This tutorial will be helpful for beginners to
install ImageMagick 7.1.0 in Ubuntu 20.04 LTS,
Ubuntu 21.10, LinuxMint 20.3, and Fedora 35
# ⚓ Beginner’s_Guide_to_LibreOffice_Writer_for_Microsoft_Word
Users⠀⇛
This is a collection of LibreOffice Writer
tutorials for beginners published over the years by
the Ubuntu Buzz. It covers the most basic exercises
like basic writing, inserting pictures and tables,
and dealing with page as well as paragraph styles.
They are adapted with the hope to be useful and
suitable for those who have background in Microsoft
Word. We wish you would find Ubuntu, LibreOffice
and Free Software community a safe home full of
happiness and success.
# ⚓ Packaging_LLVM_Snapshots⠀⇛
The release manager for LLVM creates source
tarballs with every new release of LLVM. That is
more or less the result of a git archive operation
on a particular directory in the LLVM mono-
repository. In the downstream Fedora operating
system we take those source tarballs and use them
as input to our build system.
[...]
In order to make it easier for us to migrate to the
next official version of LLVM, I tried to keep the
changes to the original blueprints for a package to
a minimum.
# ⚓ How_to_Install_Liquorix_Kernel_on_Linux_Mint_20_–
LinuxCapable⠀⇛
Liqourix Kernel is a free, open-source general-
purpose Linux Kernel alternative to the stock
kernel with Linux Mint 20. It features custom
settings and new features and is built to provide a
responsive and smooth desktop experience,
especially for new hardware. Liquorix Kernel is
popular amongst Linux Gaming, streaming, and ultra-
low latency requirements and often boasts the
latest Linux Kernels, having multiple branches to
choose from the stable, edge, and development.
For users seeking to have their Linux Mint system
kernel up to date and not wanting to manually
install kernels or use the testing/unstable
repositories, installing a third-party kernel that
may be for you.
In the following tutorial, you will learn how to
import the Liquorix Kernel PPA and install the
latest Linux Kernel on Linux Mint 20.
# ⚓ How_to_Set_up_Laravel_Horizon_Queueing_in_Rocky_Linux_8⠀⇛
Laravel Horizon is an open source dashboard app
that keeps track of Laravel Redis queues. The
Horizon dashboard is a single page application
built using Vue.js. The application is designed to
provide real-time insights into queue workloads,
recent jobs, failed jobs, job retries, throughput
and runtime metrics, and process counts. The
Dashboard provides several statistical data on the
execution times, throughput or failure of the
processes involved, sending notifications if errors
occur.
Laravel Horizon has an excellent code-driven setup
and user interface dashboard for your Laravel
powered Redis queues. Horizon permits you to
effortlessly monitor key metrics of your queue
framework like runtime, and work failures.
# ⚓ [Old]_Isolating_Xwayland_in_a_VM⠀⇛
In my last post, Qubes-lite with KVM and Wayland, I
described setting up a Qubes-inspired Linux system
that runs applications in virtual machines. A
Wayland proxy running in each VM connects its
applications to the host Wayland compositor over
virtwl, allowing them to appear on the desktop
alongside normal host applications. In this post, I
extend this to support X11 applications using
Xwayland.
# ⚓ How_To_Install_Chevereto_on_Debian_11_–_idroot⠀⇛
In this tutorial, we will show you how to install
Chevereto on Debian 11. For those of you who didn’t
know, Chevereto is an image hosting script that
allows you to create your own image hosting
websites like the popular Tinypic and Photobucket.
Chevereto is available in the Free and Paid
versions, Paid version comes with all the features
like storage, banners, likes, followers, social
login, etc, while the Free version is always 6
months behind the paid version. Chevereto comes
with all major features like user accounts, albums,
admin dashboard, HTML 5 drag, and many more.
This article assumes you have at least basic
knowledge of Linux, know how to use the shell, and
most importantly, you host your site on your own
VPS. The installation is quite simple and assumes
you are running in the root account, if not you may
need to add ‘sudo‘ to the commands to get root
privileges. I will show you through the step-by-
step installation of Chevereto on a Debian 11
(Bullseye).
o § Games⠀➾
# ⚓ Escape_Simulator_hits_over_$4M_in_gross_sales_|
GamingOnLinux⠀⇛
Escape Simulator has turned into a big success
story for indie games and developer Pine Studio,
with an announcement about how well it’s done.
Releasing on Steam back in October 2021, writing on
Reddit the team noted how it has managed to hit
“$4M in gross sales in less than two months of
being released on Steam”. It just goes to show that
with the right know-how, indie developers can still
manage to cut through the noise of thousands of
games releasing on Steam all the time.
How did they do it? As the Reddit post explains
that part of the reason is that they hired a good
PR team, and worked to create a good trailer.
Plenty of it comes down to clever marketing,
although it sounds like it did take a fair bit of
time to do, like creating special puzzle rooms for
people who covered the game. There was also some
cross-promotion with other developers, and a lot of
wishlists came from having a demo at some Steam
festivals. Overall, it’s an interesting little look
into what goes on behind the scenes to make a game
a success.
# ⚓ SuperTux_released_free_on_Steam,_an_open_source_classic_|
GamingOnLinux⠀⇛
Giving a nice boost to a classic free and open
source game, SuperTux has now been released on
Steam and it’s free to download and play.
“Run and jump through SuperTux, the sidescrolling
2D platformer starring Tux, the Linux mascot.
Squish and knock out enemies, collect powerups, and
solve platforming puzzles throughout the Icy Island
and the Rooted Forest, as Tux tries to save his
beloved Penny from her kidnapper, Nolok!”
# ⚓ Heck_Deck_is_a_brilliant_fusion_of_bullet-hell_and_a_card
game_|_GamingOnLinux⠀⇛
What do you get when you cross a twin-stick bullet-
hell with a card-game? Heck Deck. It’s not a
particularly long game but the idea is excellent.
It ends up more like a deck-building shooter
strategy game, it’s thoroughly odd to properly pin
it to a genre. Note: key provided by the developer.
Time only moves when you move and you directly
control a little sort-of wiggly-thing. Cards are
you abilities and enemies fire cards. The thing is
though, when you run out of cards you need to crash
yourself into the cards the enemies fire to get
more. It hurts you, but you get a new card to use
(except health cards – they don’t hurt of course).
It’s absolutely genius and I love it.
# ⚓ Godot_Engine_–_Dev_snapshot:_Godot_3.5_beta_1⠀⇛
Godot 3.4 was released 2 months ago, and some of
the major planned features for Godot 3.5 have since
been merged and are now ready for wider testing.
So we’re starting the beta testing phase with this
already significant set of changes, and we’ll have
frequent beta builds to polish them for the stable
release. Some more features are still being worked
on and will be included in future beta builds.
All this work is done by contributors on the side
while our main development focus remains on the
upcoming Godot 4.0 alpha (see our release policy
for details on the various Godot versions).
Jump to the Downloads section.
As usual, you can try it live with the online
version of the Godot editor updated for this
release.
o § Desktop Environments/WMs⠀➾
# ⚓ Your_First_Look_at_System76′s_Rust-Based_Desktop_Experience
–_OMG!_Ubuntu!⠀⇛
If you’ve been itching to hear some progress about
System76’s new desktop environment, you’re in luck!
Developer Eduardo Flores went hands on with early
development versions of several COSMIC components
and written about his findings (with copious amount
of screenshots) on his blog.
“System76’s objective is to create something that
is faster, more customizable, and free of the
limitations of the GNOME desktop environment, and
let’s face it, we’re all curious as to how this
desktop will look like,” Eduardo writes.
And hey: he’s not wrong!
Do keep in mind that everything you see in his (and
this) post is at formative stage. Nothing shown is
final, nothing shown is stable, and nothing shown
is immune to change. Expect the final version of
the Rust-based COSMIC desktop to differ (possibly
majorly) from anything you see here.
With that public service announcement out of the
way, let’s dive in!
# § K Desktop Environment/KDE SC/Qt⠀➾
# ⚓ New_Breeze_Theme_Gives_KDE_Neon_Release_Lots_of
Sparkle⠀⇛
Few desktop environments — and Linux is both
blessed and cursed with a plethora of them —
can be inviting enough to fit the computing
needs of all user scenarios. KDE is one of
them. Even better, the October release of KDE
Neon 5.23 makes it a fitting choice over
other distros running KDE.
This release has a double claim to fame. KDE
Neon 5.23 has components not yet absorbed by
other KDE-based distros. It is also the 25th
Anniversary edition of KDE, first released in
1996.
KDE Neon 5.23 is an Ubuntu-based Linux
distribution featuring the latest KDE Plasma
desktop and other KDE community software.
With an edge over other KDE installations,
the Neon project provides a rapidly evolving
software repository with all the latest KDE
software.
# ⚓ KDE_Plasma_5.24_Beta_Released_with_New_Overview
Effect,_Fingerprint_Support,_and_More⠀⇛
KDE Plasma 5.24 comes with numerous changes
that will make many of you happy. For
example, it introduces the long-anticipated
support for fingerprint readers to unlock the
screen, as well as to authenticate in apps
that require administration password, and
also to authenticate with sudo on the
command-line.
Another interesting feature of the upcoming
KDE Plasma 5.24 desktop environment series is
a brand-new Overview effect that lets users
control their virtual workspaces and find
search results from KRunner, all in one
place. The new Overview effect can be toggled
with the Super+W keyboard shortcut and has a
blurred background by default.
# ⚓ Plasma_5.24_Beta_–_KDE_Community⠀⇛
As is traditional, today we are bringing you
the testing version of KDE’s Plasma 5.24.
Plasma 5.24 Beta is aimed at testers,
developers, and bug-hunters.
To help KDE developers iron out bugs and
solve issues, install Plasma 5.24 Beta and
test run the features listed below. Please
report bugs to our bug tracker.
# ⚓ KDE_Plasma_5.24_Beta_Released_With_Better_Wayland
Support_–_Phoronix⠀⇛
KDE has made available the beta of the
upcoming Plasma 5.24 desktop update ahead of
its planned stable release on 8 February.
KDE Plasma 5.24 Beta brings many Wayland
improvements, refining of the Breeze theme, a
variety of system tray and widget
improvements, continued changes to the KDE
System Settings, a new KWin overview effect,
many Discover improvements, and a ton of
fixes.
# § GNOME Desktop/GTK⠀➾
# ⚓ Human_Interface_Guidelines,_libadwaita_1.0_edition_–
Form_and_Function⠀⇛
After a lot of hard work, libadwaita 1.0 was
released on the last day of 2021. If you
haven’t already, check out Alexander’s
announcement, which covers a lot of what’s in
the new release.
When we rewrote the HIG back in May 2021, the
new version expected and recommended
libadwaita. However, libadwaita evolved
between then and 1.0, so changes were needed
to bring the HIG up to date.
Therefore, over the last two or three weeks,
I’ve been working on updating the HIG to
cover libadwaita 1.0. Hopefully this will
mean that developers who are porting to GTK 4
and libadwaita have everything that they need
in terms of design documentation but, if
anything isn’t clear, do reach out using the
usual GNOME design channels.
In the rest of this post, I’ll review what’s
changed in the HIG, compared with the
previous version.
o § Distributions⠀➾
# § Screenshots/Screencasts⠀➾
# ⚓ Linux_Mint_20.3_“Una”_MATE_overview_|_Stable,_robust,
traditional_–_Invidious⠀⇛
In this video, I am going to show an overview
of Linux Mint 20.3 “Una” MATE and some of the
applications pre-installed.
# § PCLinuxOS/Mageia/Mandriva/OpenMandriva Family⠀➾
# ⚓ Rolling_repos_sync_with_Cooker_(2022/01/13)⠀⇛
Hard working OM Cooker devs are at work
copying Cooker repos to Rolling repos. We do
this when Cooker devs believe we are at a
good point for stability and bug fixing.
There have been a huge amont of bugs fixed.
Especially for KDE or other desktop packages
it is best to wait for the entire process to
complete to avoid problems.
Do not upgrade your Rolling system while this
is in progress.You need to wait until copying
cooker repos to rolling repos is finished.
This process takes some time so we all need
to patient.
# § Debian Family⠀➾
# ⚓ Sparky_5.16_–_SparkyLinux⠀⇛
ISO images of Sparky “Nibiru” of the
oldstable line have been updated up to 5.16.
This release is based on Debian oldstable 10
“Buster”.
All packages upgraded from Debian “Buster”
and Sparky “Nibiru” repos as of January 11,
2022.
System reinstallation is not required; if you
have Sparky 5.x installed, simply keep Sparky
up to date.
New live/install media of the oldstable line
can be downloaded from the download/oldstable
page.
# ⚓ Bits_from_Debian:_New_Debian_Developers_and
Maintainers_(November_and_December_2021)⠀⇛
The following contributors got their Debian
Developer accounts in the last two months:
Douglas Andrew Torrance (dtorrance)
Mark Lee Garrett (lee)
The following contributors were added as
Debian Maintainers in the last two months:
Lukas Matthias Märdian
Paulo Roberto Alves de Oliveira
Sergio Almeida Cipriano Junior
Julien Lamy
Kristian Nielsen
Jeremy Paul Arnold Sowden
Jussi Tapio Pakkanen
Marius Gripsgard
Martin Budaj
Peymaneh
Tommi Petteri Höynälänmaa
# § Canonical/Ubuntu Family⠀➾
# ⚓ Ubuntu_21.04_users_need_to_upgrade_their_systems_this
week⠀⇛
It’s January 13 which means there’s just one
week left until Canonical stops giving out
updates for Ubuntu 21.04 ‘Hirsute Hippo’
which launched last April. As it’s an inter-
LTS release, it only receives 9-months of
updates, and then it’s time to move on. The
best plan for people still running Ubuntu
21.04 is to upgrade to Ubuntu 21.10 and then
to upgrade to Ubuntu 22.04 LTS (not available
until April) before July 14.
If you’re not too sure which Ubuntu version
you’re running, open Settings, and then on
the left-hand pane scroll down to About. This
will open a new page in Settings and you need
to look for OS Name, next to which, will be
the version of Ubuntu you’re running. If
you’re on Ubuntu 21.04 or for some reason
still on Ubuntu 20.10, then you need to get
upgrading.
# ⚓ The_State_of_Robotics_–_December_2021_|_Ubuntu⠀⇛
I will be honest, I thought that December was
going to be a slow month for the robotics
news. With all the holidays, I was not
expecting a month with exciting announcements
or events. And when I was ready to put videos
of robots dancing to Christmas carols, with
Christmas hats and Christmas lights, I found
a month packed with great news!
So let’s dive into our monthly robotics blog
and explore what the last month of 2021
brought us. And if you were looking at our
2021 robotics rewind, this is not it.
o § Devices/Embedded⠀➾
# ⚓ LIVA_Q3_Plus_pocket-sized_AMD_Ryzen_Embedded_mini_PC_sells
for_$242_and_up_–_CNX_Software⠀⇛
ECS LIVA Q3 Plus is another one of those pocket-
sized mini PC’s we’ve seen in the past like the ECS
LIVA Q2, Chuwi Larkbox, GMK NucBox, and others.
Most of those mini PCs are based on Intel Apollo
Lake or Gemini Lake mini PC, but LIVA Q3 Plus comes
with a twist as it is powered by a 15W AMD Ryzen
Embedded V1605B or R1505G processor making it,
according to the Taiwanese company, the world’s
smallest 15W AMD mini PC.
# § Mobile Systems/Mobile Applications⠀➾
# ⚓ Unlocked_Galaxy_Note_20_units_get_Android_12_update
in_the_US_–_SamMobile_–_SamMobile⠀⇛
# ⚓ Android_12L_beta_2_is_here_for_select_Pixel_phones⠀⇛
# ⚓ Android_12_update_tracker:_Early_info_we_know_so
far⠀⇛
# ⚓ Google_Pixel_Android_12_bugs,_problems,_&_issues
tracker⠀⇛
# ⚓ Hot_new_leaks_point_to_Moto_G_Stylus_(2022)_variant
with_5G_and_Android_12_–_PhoneArena⠀⇛
# ⚓ It’s_alive!_Info_about_the_Google_Pixel_Fold_surface
in_Android_12L_beta_|_NextPit⠀⇛
# ⚓ Google_Pixel_6_Review:_a_Premium_Android_Phone_on_a
Budget⠀⇛
# ⚓ Qualcomm_Made_Its_Case_About_Dominating_Premium
Android_Phones_In_2022_At_Its_Recent_Snapdragon
Summit⠀⇛
# ⚓ Xiaomi_11i_review:_Making_one_compromise_too_many_|
Android_Central⠀⇛
# ⚓ Google_Pixel_Fold_still_on_the_way_according_to_hints
in_latest_Android_beta_|_TechRadar⠀⇛
# ⚓ How_To_Stop_Pop-Up_Ads_on_Your_Android:_3_Easy
Methods_Explained⠀⇛
o § Free, Libre, and Open Source Software⠀➾
# ⚓ An_open_source_developer’s_guide_to_12-Factor_App
methodology⠀⇛
The 12-Factor App methodology provides guidelines
for building apps in a short time frame and for
making them scalable. It was created by the
developers at Heroku for use with Software-as-a-
Service (SaaS) apps, web apps, and potentially
Communication-Platform-as-a-Service (CPaaS) apps.
For organizing projects effectively and managing
scalable applications, the 12-Factor App
methodology has powerful advantages for open source
development.
The principles of 12-Factor App methodology are
strict rules that act as building blocks for
developing and deploying SaaS applications, and
they are not constrained to any programming
language or database.
# § Web Browsers⠀➾
# § Mozilla⠀➾
# ⚓ Firefox_Is_Currently_Facing_A_Major_Bug_In
Connection⠀⇛
Users of the open source web browser
have started to report a bug on various
online communities where the browser
simply can’t establish a connection to
any website they wish to visit. Even
when they try to close the browser, it
hangs and keeps running in the
background.
No official statement from Mozilla is
released so far. However, the bug is
already reported and developers and
users are discussing the root cause of
the problem.
Right now, the implementation of HTTP3
protocol in the browser seems to be the
root cause of the problem, as users
suggested that turning it off could fix
the issue completely.
# § SaaS/Back End/Databases⠀➾
# ⚓ PostgreSQL:_pgagroal_1.4.0⠀⇛
The pgagroal community is happy to announce
version 1.4.0.
# § Productivity Software/LibreOffice/Calligra⠀➾
# ⚓ ‘IwlIj_jachjaj!_Incoming_LibreOffice_7.3_to_support
Klingon_and_Interslavic⠀⇛
There’s a good chance you’ve heard of
LibreOffice – OK, yes, and Klingon.
Interslavic, maybe not. Here’s why some of
you should care.
LibreOffice is the continuation of the
moribund OpenOffice project, which had to
change its name because Oracle claimed the
trademark on the old name.
If you’re still using OpenOffice, don’t. It’s
basically dead. Download LibreOffice,
uninstall OpenOffice, then install
LibreOffice instead. It’s completely
compatible because it’s the same program,
just a more modern version – smaller, faster,
less buggy, and more secure.
It’s even handy if you have a legit copy of
Microsoft Office. In your correspondent’s
experience, it’s a lot better at recovering
corrupted or damaged MS Office files than MS
Office itself. (It’s also free, resistant to
MS Office viruses, and legal even for
commercial use.)
And as for the languages? The website already
lists 51, and more are coming. That’s a tiny
fraction of the world’s 7,000-plus languages,
and a language goes extinct every two weeks.
For small communities trying to keep minority
languages alive, being able to write in it is
very important.
# § Programming/Development⠀➾
# ⚓ New_KDReports_Release_–_KDAB⠀⇛
We have just released version 2.1.0 of our
KDReports developer tool product.
KDReports generates printable and exportable
reports from code and from XML descriptions.
Developers write the code that integrates
KDReports with the rest of the application.
External designers or marketing and sales
staff can be tasked with creating the report
structure, or its appearance. The created
reports can be shown in a preview dialog that
is part of KD Reports. They can also be saved
to PDF files or sent directly to a printer.
# § Perl/Raku⠀➾
# ⚓ Do-It-Yourself_warnings_categories⠀⇛
One of the reasons I have not “moved
on” from Perl to some other more
“modern” language is that Perl gives me
such great access to its inner
workings. The Do-It-Yourself Lexical
Pragmas post from a couple weeks ago is
an example of this. Another example is
that Perl lets you tie your own code
into its warnings system.
Tying into the warnings machinery
requires a module. That is, the
interface assumes you are reporting
problems relative to another name space
that invoked your code. Your module can
either add diagnostics to existing Perl
warning categories or actually create
new categories. In either case your
diagnostics are sensitive to the
enablement or disablement of the
category, as well as its fatalization.
* § Leftovers⠀➾
o § Hardware⠀➾
# ⚓ Work_The_World_On_A_555_|_Hackaday⠀⇛
Over the years the humble 555 timer has been used
in so many unexpected places, but there’s a project
from [Frank Latos] which we think may be a first.
On a piece of stripboard sit a pair of 555s, and
instead of the usual passives there are a set of LC
circuits. This is no timer, instead it’s a CW
(Morse) transmitter for the 80 metre amateur radio
band.
One 555 is configured as a feedback oscillator
through a toroidal transformer with a tuned circuit
to set the frequency of oscillation. The other
takes an inverted input from the oscillator to
produce complimentary push-pull outputs from both
555s, which are fed to another transformer that in
turn feeds a low-pass filter and thus the antenna.
o § Integrity/Availability⠀➾
# § Proprietary⠀➾
# § Pseudo-Open Source⠀➾
# § Cloudwashing⠀➾
# ⚓ Let_your_ideas_take_flight_in_the_Arduino
Cloud_Games_|_Arduino_Blog⠀⇛
Arduino’s brand new initiative,
the Arduino Cloud Games, is now
live and accepting submissions.
This new program is a way to
build a community showcase of the
most creative, innovative ideas
that show the vast potential and
scope of connected projects.
Let’s take a look at how you can
get involved, and let your ideas
take flight.
# § Security⠀➾
# ⚓ Security_updates_for_Thursday_[LWN.net]⠀⇛
Security updates have been issued by
Debian (epiphany-browser, lxml, and
roundcube), Fedora (gegl04, mingw-
harfbuzz, and mod_auth_mellon),
openSUSE (openexr and python39-pip),
Oracle (firefox and thunderbird), Red
Hat (firefox and thunderbird), SUSE
(apache2, openexr, python36-pip, and
python39-pip), and Ubuntu (apache-
log4j1.2, ghostscript, linux, linux-
gcp, linux-gcp-5.4, linux-hwe-5.4, and
systemd).
# § Fear, Uncertainty, Doubt/Fear-mongering/
Dramatisation⠀➾
# ⚓ Ransomware_is_being_rewritten_in_Go_for
joint_attacks_on_Windows,_Linux_users [Ed:
Microsoft-friendly media wants us to think
that the programming languages themselves
are to blame for criminals who use them — a
typical_FUD_pattern; this latest FUD comes
[1, 2] from a_Microsoft_proxy,_which_is
also_promoting_nuclear_tensions]⠀⇛
Cyber security researchers have
discovered evidence of a years-
old ransomware strain returning
after being rewritten in Golang –
a cross-platform programming
language capable of reaching a
higher number of users across
different operating systems.
The TellYouThePass ransomware was
first discovered in 2019, however
researchers [sic] at Crowdstrike
have now spotted a new strain
being used as a second-stage
attack following a successful
exploit of the Log4Shell
vulnerability revealed in
December 2021.
# § Privacy/Surveillance⠀➾
# ⚓ Using_Foreign_Nationals_to_Bypass_US
Surveillance_Restrictions⠀⇛
Remember when the US and
Australian police surreptitiously
owned and operated the encrypted
cell phone app ANOM? They
arrested 800 people in 2021 based
on that operation.
New documents received by
Motherboard show that over 100 of
those phones were shipped to
users in the US, far more than
previously believed.
# ⚓ Salvadoran_journalists’_phones_hacked
with_spyware,_report_finds_|_Reuters⠀⇛
The cell phones of nearly three
dozen journalists and activists
in El Salvador, several of whom
were investigating alleged state
corruption, have been hacked
since mid-2020 and implanted with
sophisticated spyware typically
available only to governments and
law enforcement, a Canadian
research institute said it has
found.
The alleged hacks, which came
amid an increasingly hostile
environment in El Salvador for
media and rights organizations
under populist President Nayib
Bukele, were discovered late last
year by The Citizen Lab, which
studies spyware at the University
of Toronto’s Munk School of
Global Affairs. Human-rights
group Amnesty International,
which collaborated with Citizen
Lab on the investigation, says it
later confirmed a sample of
Citizen Lab’s findings through
its own technology arm.
Citizen Lab said it found
evidence of incursions on the
phones that occurred between July
2020 and November 2021. It said
it could not identify who was
responsible for deploying the
Israeli-designed spyware. Known
as Pegasus, the software has been
purchased by state actors
worldwide, some of whom have used
the tool to surveil journalists.
# ⚓ NSO_Spyware_Linked_to_Phone_Hacks_of
Journalists,_Activists_in_El_Salvador⠀⇛
Human rights groups say they have
identified 35 journalists and
activists in El Salvador whose
mobile phones were infected with
spyware manufactured by the
Israeli company NSO Group.
In a statement released on
Wednesday, rights groups Access
Now, Amnesty International and
Citizen Lab said that the people
targeted included employees of
media groups El Faro and Gato
Encerrado, in addition to
employees of regional human
rights and pro-democracy
organizations, such as Cristosal
and Fundación Democracia,
Transparencia y Justicia.
A spokesperson for NSO group
declined to comment on the
specific allegations but said
that the company provides its
technology “only to vetted and
legitimate intelligence agencies
as well as to law enforcement
agencies, who use these systems
under warrants by the local
judicial system to fight
criminals, terrorists and
corruption.”
# ⚓ Journalists_in_El_Salvador_Targeted_With
Spyware_Intended_for_Criminals_–_The_New
York_Times⠀⇛
El Salvador’s leading news
outlet, El Faro, said on
Wednesday that the phones of a
majority of its employees had
been hacked with the spyware
Pegasus, which has been used by
governments to monitor human
rights activists, journalists and
dissidents.
The revelation came just months
after the American government
blacklisted the Israeli firm that
produces Pegasus, the NSO Group,
in an attempt to curb the largely
unregulated global market in
spyware.
According to Citizen Lab at the
University of Toronto’s Munk
School and Access Now, two
cybersecurity watchdogs that
analyzed the phones of El Faro’s
employees, the spyware had been
installed on the phones of 22
reporters, editors and other
employees between July 2020 and
November 2021.
During that time, El Faro was
investigating the Salvadoran
government’s clandestine
connections to the country’s
gangs and corruption scandals.
The government has denied any
connection to local gangs.
# ⚓ NSO_Group_promised_to_stop_selling_tools
to_spy_on_journalists._A_new_report_proves
otherwise_–_Rest_of_World⠀⇛
In July, a consortium of
journalists unveiled the Pegasus
Project: An investigation
detailing how governments across
the world deployed Pegasus
spyware against journalists,
activists, and opposition
politicians. In response, the
founder and CEO of the Israeli
developer of the software, the
NSO Group, vowed it would not
work with countries that violated
human rights and targeted
journalists, and claimed that it
had suspended the software in
five countries that had abused
the malware’s usage, although did
not specify which ones.
Now, a newly published report
from a group of prominent digital
rights organizations suggests
that NSO spyware tools are still
being used against journalists in
El Salvador.
The report, produced in
partnership by Access Now, The
Citizen Lab, Fundación Acceso,
Amnesty International, and other
digital rights groups, found that
Pegasus had been installed and
used to infect the devices of 35
Salvadoran journalists and
activists between July 2020 and
November 2021. The findings have
been analyzed and corroborated by
two of the groups behind the
report.
# ⚓ NSO_Group_Spyware_Targeted_Dozens_of
Reporters_in_El_Salvador_|_WIRED⠀⇛
THE ISRAELI SPYWARE developer NSO
Group has long claimed plausible
deniability when it comes to
misuse of its powerful targeted
surveillance tools. Yet despite
its protestations—and increased
scrutiny from tech companies and
regulators alike—the abuses
continue. The latest revelation
comes from El Salvador, where
NSO’s Pegasus malware was found
on 37 devices belonging to 35
journalists and activists as
recently as November of last
year.
Those findings, jointly published
by a consortium of digital rights
organizations, show that despite
NSO Group’s insistence that its
products are used to track
criminals and terrorists,
governments continue to deploy
them against innocent targets—and
that NSO has done little to rein
in its clients.
# ⚓ Report:_22_journalists_at_Salvadoran_news
site_hit_with_Pegasus_hack⠀⇛
At least 22 journalists from the
independent Salvadoran news site
El Faro were targeted with
telephone spyware, investigators
announced Wednesday, in one of
the most extensive attacks yet
discovered using the Pegasus
software that human rights
advocates say has been abused by
governments around the world.
The journalists were among at
least 35 people in El Salvador
whose iPhones were hacked with
Pegasus between July 2020 and
November 2021, according to an
analysis by the Toronto-based
Citizen Lab and other groups.
Also targeted were human rights
activists and reporters for other
news organizations.
Some devices were penetrated a
dozen or more times, the
investigators said. Óscar
Martínez, El Faro’s news editor,
was hacked 42 times, they said.
The digital news site is known
for its hard-hitting
investigations into the
government of President Nayib
Bukele.
o § Censorship/Free Speech⠀➾
# ⚓ Blocking_access_to_Twitter_in_Nigeria_is_a_flagrant
violation_of_fundamental_rights⠀⇛
After seven months of deliberately blocking access
to Twitter, authorities in Nigeria have today
lifted the ban on the social media platform.
According to media reports, the government
indicated that Twitter had pledged to fulfil
certain conditions which had been discussed behind
closed doors. Twitter also confirmed that it had
been unblocked in the country, but did not
acknowledge the conditions, or indicate if it
planned on fulfilling them.
While Access Now welcomes the government’s decision
to end the Twitter ban, many aspects of this
decision remain unclear, and is appealing to both
parties to be transparent and consultative in in
regard to the discussions that occurred, or are
ongoing to take place, and ensure that the
fundamental rights of the people of Nigeria are not
jeopardized in the process.
“Ending the ban on Twitter in Nigeria is the right
thing to do, but it is incredibly unfortunate that
it took the authorities so many months to do so,”
said Felicia Anthonio, Campaigner and #KeepItOn
Lead at Access Now. “The ban was an unnecessary
attack on fundamental rights, while costing the
country’s economy over a billion USD.”
o § Digital Restrictions (DRM)⠀➾
# ⚓ Canon_Temporarily_Abandons_Smart_Ink_Cartridges_|
Hackaday⠀⇛
An unexpected side effect of the global
semiconductor shortage came to light this week —
Japanese printer manufacturer Canon announced they
are temporarily going to provide consumable ink and
toner cartridges without microchips. Furthermore,
they provided instructions for consumers on how to
bypass the printer’s logic, allowing it to function
even when it incorrectly thinks the ink or toner is
low.
䷩ 𝚕𝚒𝚗𝚎 5145
╒═══════════════════ 𝐃𝐀𝐈𝐋𝐘 𝐋𝐈𝐍𝐊𝐒 ═════════════════════════════════════════════╕
⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.13.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧
Gemini_version_available_♊︎
✐ Links_13/1/2022:_NetworkManager_1.34_and_Everett_3.0.0⠀✐
Posted in News_Roundup at 4:51 pm by Dr. Roy Schestowitz
🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈
§ Contents⠀➾
* GNU/Linux
o Distributions
o Devices/Embedded
* Free_Software/Open_Source
* Leftovers
* § GNU/Linux⠀➾
o § Audiocasts/Shows⠀➾
# ⚓ Linux_Action_News_223⠀⇛
Fedora and Red Hat users are getting a web-based
installer, and a new legal situation for Bitcoin
smells like retro SCO FUD.
# ⚓ Road_Trip_Tech_|_Jupiter_Extras_81⠀⇛
Our road trip machine is loaded up from solar to
networking, the tech that made working, living, and
recording from the road possible for 44 days and
over 2,200 miles.
# ⚓ How_to_install_the_Brave_Browser_on_Elementary_OS_6.0_–
Invidious⠀⇛
In this video, we are looking at how to install the
Brave Browser on Elementary OS 6.0.
# ⚓ Adding_Custom_Entries_To_Your_Linux_Bootloader_–
Invidious⠀⇛
While not something you need everyday every so
often it can be advantageoust to add a custom entry
into your Linux bootloader such as grub or systemd
boot for example to drop yourself into a root shell
whenever you need it.
o § Kernel Space⠀➾
# ⚓ Linux_Preparing_To_Finally_Remove_Support_For_The_a.out
Format_–_Phoronix⠀⇛
Back in 2019 the Linux kernel deprecated a.out
support for that file format used several decades
ago before ELF tookover. Now in 2022 it looks like
that a.out code will be removed from the kernel.
Linux relied on the a.out format until v1.2 in the
mid-90′s when ELF became the popular format for
binaries. While the a.out format hasn’t been widely
used on Linux in many years, it took until 2019 for
the support to be deprecated for running a.out
binaries on x86 32-bit. Compilers and other
toolchain components have moved on from the a.out
file format for years.
o § Applications⠀➾
# ⚓ NetworkManager_1.34_Arrives_with_Better_WireGuard_Support,
Many_Improvements⠀⇛
Almost seven months in development, NetworkManager
1.34 is here to further improve support for the
WireGuard VPN tunnel protocol and implementation by
introducing support for WireGuard profiles to
NetworkManager’s text user interface (nmtui), as
well as improving import of WireGuard profiles with
DNS domain and address family disabled via the
command-line interface (nmcli).
# ⚓ Fwupd_1.7.4_Supports_More_Hardware_For_Firmware_Updating_On
Linux_–_Phoronix⠀⇛
Lead Fwupd/LVFS developer Richard Hughes of Red Hat
today released v1.7.4 for this open-souce utility
to allow firmware updating on Linux of system
motherboards and peripherals.
The Fwupd 1.7.4 release adds firmware branch
support for ModemManager devices, support for
firmware engineers to be able to patch files at
known offsets, and a variety of bug fixes.
# ⚓ SciDAVis_is_an_open-source_application_for_scientific_data
analysis_and_data_visualization⠀⇛
SciDAVis is a free interactive application aimed at
data analysis and publication-quality plotting. It
combines a shallow learning curve and an intuitive,
easy-to-use graphical user interface with powerful
features such as scriptability and extensibility.
Alternative to:
SciDAVis is similar in its field of application to
proprietary Windows applications like Origin and
SigmaPlot as well as free applications like
QtiPlot, Labplot, and Gnuplot.
What sets SciDAVis apart from the above is its
emphasis on providing a friendly and open
environment (in the software as well as the
project) for new and experienced users alike.
o § Instructionals/Technical⠀➾
# ⚓ How_To_Install_Eclipse_IDE_on_Fedora_35_–_idroot⠀⇛
In this tutorial, we will show you how to install
Eclipse IDE on Fedora 35. For those of you who
didn’t know, Eclipse is a multi-language software
development environment comprising an integrated
development environment (IDE) and an extensible
plug-in system. The most popular Eclipse product is
Java Integrated Development Environment (IDE), but
there are other pretty cool IDEs, including our C/
C++ IDE, JavaScript/TypeScript IDE, PHP IDE, and
more.
This article assumes you have at least basic
knowledge of Linux, know how to use the shell, and
most importantly, you host your site on your own
VPS. The installation is quite simple and assumes
you are running in the root account, if not you may
need to add ‘sudo‘ to the commands to get root
privileges. I will show you the step-by-step
installation of the Eclipse IDE on a Fedora 35.
# ⚓ How_to_Quickly_Find_Mouse_Pointer_in_Ubuntu_20.04_/_21.10_|
UbuntuHandbook⠀⇛
For Ubuntu 20.04+ and other Linux with GNOME
desktop (e.g., Fedora workstation, Debian and Arch
Linux), there’s an extension lets you quickly
locate mouse pointer.
The extension is called “Jiggle“. It highlights the
mouse pointer position when it moved rapidly by
applying 3 cool animation effects: Cursor Scaling,
Spotlight, and Fireworks.
# ⚓ How_to_Install_Curl_in_Ubuntu_–_buildVirtual⠀⇛
cURL (Client URL Library) is a command-line tool
for transferring data using various network
protocols, typically HTTP/s, though it supports
many more including ftp, scp and smb. Curl has been
around for quite a while – it was first released in
1996 (though back then it was called httpget).
cURL is a command-line tool for getting or sending
data including files using URL syntax. If you are a
Linux user there is a good chance you have come
across it at some point. Curl is often used as a
way to download files from the internet, although
it is capable of much more.
Quite often it can be found on most Linux
distributions, or if not it is straight forward to
install it. This article shows you how to install
curl on Ubuntu systems if it isn’t already present,
and gives some simple examples to show you how to
use curl and confirm that it is working as
expected.
# ⚓ How_to_install_and_Configure_Java_17_on_Fedora_35_–
NextGenTips⠀⇛
In today’s guide, we are going to learn how we can
install Java SE 17 on Fedora 35. Java is widely
used in programs like Cassandra, Graylog, Wine,
etc.
Java delivers thousands of performance, stability,
and security updates that is the reason why java is
widely used and has a larger community base
worldwide.
# ⚓ How_to_Set_Up_NFS_Server_and_Client_on_Rocky/Alma_Linux_8⠀⇛
NFS(Network File System) is a distributed file
system protocol that allows a user on a client
computer to access files over a computer network
much like local storage is accessed. It is a
popular, cross-platform and distributed file system
protocol used to export local file systems over the
network so that clients can share directories and
files with others over a network and interact with
them as though they are mounted locally. This
distributed file system protocol allows a user on a
client computer to access files over a network in
the same way they would access a local storage
file. Because it is an open standard, anyone can
implement the protocol.
Rocky Linux 8 supports NFS version 3(NFSv3) and 4
(NFSv4). The default NFS version is 4.2 which
features support for Access Control Lists (ACLs),
server-side copy, sparse files, space reservation,
labeled NFS, layout enhancements, and much more.
In this guide, we will learn how to install and
configure the NFS Server and NFS client on Rocky
Linux 8.
# ⚓ Using_Ansible_to_install_and_configure_Elasticsearch_on
Ubuntu_20.04_–_Citizix⠀⇛
In this guide, we will learn how to install and
configure Elasticsearch on Ubuntu 20.04 using
Ansible.
Elasticsearch is a distributed search and analytics
engine built on Apache Lucene. It provides a
distributed, multitenant-capable full-text search
engine with an HTTP web interface and schema-free
JSON documents. Elasticsearch has quickly become
the most popular search engine and is commonly used
for log analytics, full-text search, security
intelligence, business analytics, and operational
intelligence use cases.
Ansible is an open-source software provisioning,
configuration management, and application-
deployment tool enabling infrastructure as code. It
runs on many Unix-like systems, and can configure
both Unix-like systems as well as Microsoft
Windows.
# ⚓ How_to_install_Mysql_Server_8_on_FreeBSD_13_–_Citizix⠀⇛
MySQL is an open-source relational database
management system. Its one of the popular
relational management system.
Mysql is commonly installed as part of the popular
LAMP or LEMP (Linux, Apache/Nginx, MySQL/MariaDB,
PHP/Python/Perl) stack. It implements the
relational model and Structured Query Language
(SQL) to manage and query data.
In this guide we are going to install mysql 8 on
FreeBSD 13.
# ⚓ How_to_install_Godot_Game_Engine_on_a_Chromebook⠀⇛
Today we are looking at how to install Godot Game
Engine on a Chromebook. Please follow the video/
audio guide as a tutorial where we explain the
process step by step and use the commands below.
If you have any questions, please contact us via a
YouTube comment and we would be happy to assist
you!
o § Games⠀➾
# ⚓ Humble_Bundle_retires_all_Mac_and_Linux_games_from_the
Humble_Trove_starting_February_1st_–_Neowin⠀⇛
Notification of the change is abrupt, leaving
gamers with only two weeks to download copies of
their games from Humble Bundle into their humble
bindles. Perhaps the biggest issue is that Humble
Bundle will not deliver updates to these games.
Those who yearn for the latest content update in
this month’s Retired Men’s Nude Beach Volleyball
will be out-of-luck. They will forever be locked
behind the Humble Choice subscription plan and only
be offered to Windows users.
# ⚓ Proton_Experimental_fixes_up_Sea_of_Thieves_voice_chat_|
GamingOnLinux⠀⇛
Today a small bug-fix update went out for Proton
Experimental, as Valve continues readying it up
ahead of the launch of their handheld Steam Deck.
What is Proton? It’s a compatibility layer designed
to run Windows games from Steam on Linux. See more
about it in our full guide.
# ⚓ The_Anacrusis_appears_to_run_on_Linux_with_Proton_but_some
possible_caveats_|_GamingOnLinux⠀⇛
The Anacrusis is a brand new release from Stray
Bombay, and it’s another entry in the swarm-shooter
like Left 4 Dead and Back 4 Blood. Running it on
Linux is possible too. Interestingly, Stray Bombay
was co-founded by former Valve designer Chet
Faliszek who worked on the likes of Half-Life,
Portal and Left 4 Dead.
“The Anacrusis is a four-player, cooperative first-
person shooter set aboard a massive starship
stranded at the edge of explored space. Team up
with your friends in an infinitely-replayable fight
against alien hordes to unlock perks, weapons, and
new ways to play that you can share with your
team!”
# ⚓ Godot_3.5_Beta_1_Brings_Async_Shader_Compilation_&_Caching
–_Phoronix⠀⇛
While we are very eager for Godot 4.0 with
everything that this open-source game engine is
going to deliver on, Godot 3.5 beta is out today
and is a rather nice interim step forward.
Exciting with Godot 3.5 is the cross-platform game
engine bringing asynchronous shader compilation.
Godot 3.5 beta introduces async shader compilation
to reduce stuttering with its OpenGL rendering.
This new implementation uses an “ubershader” that
is compiled on start-up and cached for subsequent
runs. For gamers this ubershader system should mean
less stalling (or ideally none) during gameplay.
o § Distributions⠀➾
# § Canonical/Ubuntu Family⠀➾
# ⚓ 5_Signs_the_Ubuntu_Desktop_Has_Gone_Stale⠀⇛
Ubuntu initially garnered attention for being
a version of Linux that was easy to install
and actually use. But that wasn’t all. Ubuntu
was exciting. Canonical and the Ubuntu
community innovated the desktop, thinking
creatively about what the Linux experience
needed or could become.
Two decades later, Ubuntu has more users, but
that fire doesn’t seem to burn as bright.
Here are some reasons the Ubuntu desktop now
seems relatively dim.
o § Devices/Embedded⠀➾
# ⚓ Now_you_can_live_boot_Linux-based_postmarketOS_on_unlocked
Android_phones_–_Liliputing⠀⇛
Ever wanted to try out a Linux-based operating
system like postmarketOS on your phone… but didn’t
want to overwrite the operating system that’s
already installed? Now it’s possible.
As reported by TuxPhones, a recent postmarketOS
update has added support for “network boot,” which
essentially lets you plug an Android phone with an
unlocked bootloader into your computer with a USB
cable and boot the mobile Linux distro on your
phone. Just unplug and reboot to return to Android.
# ⚓ You_can_now_live-boot_postmarketOS_on_Android_phones⠀⇛
To spread awareness of the Linux ecosystem, the
very first “Live CDs” played a crucial role: by
taking away most of the fear of overwriting their
hard drives, they allowed more people to test and
effectively use any spin of the Operating System,
and by keeping the filesystem in RAM, have any
changes magically disappear on reboot. This has
always been possible, RAM constraints aside,
because desktop BIOSes tend to support easy booting
from external devices.
The situation on Android devices is, however, more
complex. Since most consumer ARM devices are not
allowed to boot by any drive other than the
internal flash storage, the fastboot protocol used
on most Android phones solved this by enabling, in
most of its implementations, commands to boot a
custom kernel (just on unlocked devices).
Furthermore, by unofficial means, even iPhones
supported by projects like checkra1n could
theoretically sideload kernel code from the
modified bootloader.
# ⚓ Automating_the_little_things⠀⇛
As the mainline porting workflow becomes more
streamlined, we found ourselves able to automate
and simplify a lot of things that previously
required a lot of manual work. A great example of
this is the mdss-panel-driver-generator which is
able to convert vendor devicetree panel control
sequences into a driver which meets the
requirements for upstreaming into mainline.
More recently, tools like the new msm-firmware-
loader for postmarketOS, and the potential for
scripts which can extract and package firmware
simply given a link to an OTA update, we might now
have a generation of porters with little/no
knowledge on how firmware works in the context of
downstream / mainline. Qualcomm platforms
differentiate between mdt firmware files where the
firmware is split into many files, and mbn files
where it is squashed into one larger one.
# § Open Hardware/Modding⠀➾
# ⚓ The_2021_Arduino_Open_Source_Report_is_out_|_Arduino
Blog⠀⇛
We’re excited to announce the Arduino Open
Source Report for 2021 is now available,
offering many insights into the development
of our open-source ecosystem during the past
year.
In this retrospective report you’ll learn
about the activities Arduino carried out in
the last twelve months, thanks to the hard
work of the employees, contractors and
volunteers on our team and to the passion of
our vibrant community, fueling our mission
every day.
We’re proud of the many achievements we
celebrated in 2021. It was one of the busiest
and most productive years in Arduino’s
history of commitment to open source.
# § Older in Tux Phones⠀➾
# ⚓ postmarketOS_+_mainline_for_the_OnePlus_5/5T!⠀⇛
The OnePlus 5 is a popular high-end phone
from 2017, featuring the Qualcomm Snapdragon
835 SoC and Adreno 540. It has a 1080p
display and up to 8GB of RAM.
Mainline support for the device has been
around for a while, since early 2020 in fact.
Unfortunately, the Snapdragon 835 SoC it is
based on lacks the same interest upstream
that has benefited SDM845 devices so much,
requiring a lot of work to reach a usable
level of functionality.
Despite that, after a lot of time and effort
from Jami Kettunen and several other
Snapdragon 835 developers, we can finally
welcome a postmarketOS port based on a close-
to-mainline kernel! A surprising amount of
the hardware already works, although there
are a few known stability issues, such as the
requirement to run diag-router, a tool meant
for debugging the modem to prevent WiFi from
crashing.
# ⚓ The_world’s_first_RISC-V_phone_might_be_just_around
the_corner⠀⇛
The RISC-V ISA is commonly divided into two
architectures, namely RV32 and RV64, covering
32-bit and 64-bit register sizes
respectively. And although the RV64 standard
is not yet fully RV32-compatible, the two
are, except from some minor divergences,
closely associated products of the same
family.
In fact, it took some years to see the Linux
kernel running on RV64 (and partially RV32).
The first obstacle was predictably the lack
of suitable and powerful enough silicon for
the new software, and the next – which was
thankfully solved quite fast – was porting
the entire kernel to a new architecture.
But now, with Linux support approaching full
stability on a variety of RISC-V chipsets,
distributions like Alpine starting to add
support for RV64 packages, and an increasing
quantity of chips powering Linux single board
computers which retail for prices as low as
$12, another challenge for most consumer
producers occurs: how to turn this into a
competitive ecosystem for Android
smartphones?
# ⚓ PinePhone_modem_gets_easy_OTA_update_support_via
fwupd⠀⇛
As most readers will know, the PinePhone is
essentially considered to be the reference
mobile Linux hacker’s playground. And while
this achievement may be for the most part
thanks to its flexible software, its hardware
structure, and the Quectel EG25-G baseband in
particular, are no exception. The modem
firmware can be flashed through fastboot over
USB like a regular Android phone, and in
recent months, even an open-source firmware
for it was developed, for what could be the
first time ever on this kind of device,
including audio, GPS and proper power
management. In fact, what stopped this
impressive operating system from spreading
further, or being shipped by default on new
Pines, is not really its lack of maturity,
but rather the set of legal problems that are
associated with re-flashing of mobile
broadband modems. (There are still
instructions on how to flash it, of course)
# ⚓ PinePhone_Pro_released:_specs,_pricing_and_very_first
impressions⠀⇛
It was not three years ago that the PinePhone
was announced. With entry-level, but
relatively modern specifications, video
output through Type-C, and full openness to
custom software and mainline Linux, the
device grew to be possibly even more popular
than expected. Name it, and most Linux
enthusiast will either know it, or own one
themselves.
While its humble specifications did not put
it directly on the same league as Purism’s
Librem 5, the community grew large enough to
test most pieces of modern open-source
software to its form factor.
# ⚓ The_best_Linux_phones_you_can_buy_right_now⠀⇛
As many will know, the main difference
between commercial phones and Linux-native
alternatives is in how the latter is more of
a “pocket computer” that can be used without
firmware limitations. Coupled with often very
open, if often modular, hardware, this
category of devices kind of takes on the PDAs
which we missed since the smartphone era.
Having proper desktop-ready apps and advanced
hardware in your pocket brings not only
somewhat better freedom but also improved
possibilities. This, naturally, if you can
accept the still limited usability of most
mobile Linux software, which is far from the
ease of use and stability of commercial
models.
# ⚓ Short_guide_to_Linux_phone_desktops⠀⇛
If choosing a graphical environment for
traditional Linux computers is already
confusing for some users, deciding one for
Linux phones is twice as hard. Most mobile
interfaces still lack the maturity and
usability of their traditional counterparts,
also because integrating all kinds of desktop
applications without keyboard can be tricky,
given the graphical toolkit fragmentation and
the challenges of functionally scaling most
windows to very small sizes.
Here we will briefly showcase the main mobile
Linux projects, most of which we mentioned in
the past, to give an overview of the
reasonable choices for most users.
# ⚓ More_support_for_UNISOC_and_Spreadtrum_chips_lands_in
the_Linux_kernel⠀⇛
UNISOC (and formerly Spreadtrum) chips have
always been a very popular choice for Android
device manufacturers. However, they were not
commonly known for having wide mainline
compatibility. Similarly to Rockchip and
AllWinner, these chips became popular for
their fast performance and relatively low
prices, to the point of being used on devices
ranges from the (just released) JingPad A1 to
mainstream Android Samsung and Teclast
tablets.
# ⚓ Kupfer_is_a_postmarketOS-like_Arch_Linux_spin_for
phones⠀⇛
In the last years, the number of Linux
distributions aimed at smartphones and
tablets has grown considerably. What was
initially a land of projects like UBPorts,
Armbian and postmarketOS has grown to
interest projects like Manjaro, and several
Debian and Arch spins.
However, the newborn Kupfer does not look
like yet another fork for Linux phones. While
at an early stage, the ambitious project’s
infrastructure mimics that of postmarketOS,
and even shares some early testers and users.
The aim is not only to build a ready-to-flash
Arch with some added mobile packages, but
rather to have a complete pmbootstrap-like
suite of tools (the Docker-based
kupferbootstrap) to easily port new devices,
and maintain existing ones. The first devices
to be supported are Snapdragon 845 phones
(OnePlus 6T, Poco F1 among others) and the BQ
Aquaris X5 – and, interestingly, no PinePhone
yet.
# ⚓ A_SHIFT_in_perspective,_is_this_the_next_step_for
Linux_phones?⠀⇛
You have most likely heard of Fairphone, who
iFixit claim to be “the world’s most
repairable phone”, and certainly lives up to
it. But now, there’s a new kid on the block,
and they have a different approach.
SHIFT are a very small phone manufacturer
based in Germany, they produce quite a few
models and even have a fully convergent
device planned, sort of like the Motorola
Lapdock of yonder years. In this article, I
will discuss my experience with the SHIFT6mq,
and why I think you will hear a lot more
about SHIFT in the future.
# ⚓ A_look_at_Popcorn_Computer’s_new_Pocket_P.C.⠀⇛
The Pocket P.C., short for Pocket Popcorn
Computer, is a handheld created by the
independent hardware manufacturer Popcorn
Computer, and originally announced in the
fall of 2019. This PDA comes with an
integrated keyboard, and runs a pure Linux OS
based on the Allwinner A64 platform.
# ⚓ Another_Xiaomi_device_platform_is_getting_Linux
support⠀⇛
By now, most readers will be aware of the
quick progress of Qualcomm phones in the
mobile Linux ecosystem. This became
especially true with the SDM845 platform,
powering the near-stable Linux ports of
devices like the Oneplus 6/6T and Poco F1
which we often mentioned in the past.
However, after some work, a new family of
popular Android devices is entering the
kernel: the Xiaomi Redmi Note 7 from 2019,
was finally shown to boot the 5.15 kernel,
and should be soon entering the mainline
Linux kernel tree in a set of new patches.
The Note 7 (not that one) is based on the
Snapdragon 660 chipset, a 14nm SoC which is
focused on high performance and power
efficiency. Powered by eight Kryo cores
(custom ARM64), it can be clocked as high as
2.2GHz, supports LPDDR4 memories, top-speed
LTE modems and high-speed I/Os while keeping
a close eye to the power consumption.
# ⚓ Doubling_the_PinePhone_storage_perf_with_a_soldering
iron_and_a_steady_hand⠀⇛
There have been some recent waves in the
Pine64 community, as user Federico Izzo
(@_symmetrist_) found a PinePhone mod which
can roughly double the eMMC speed simply by
moving a single resistor. But, anyone can
write a flashy headline, how the heck does
this actually work?
Federico did a write-up on their blog which
explains where they got the idea from and how
to perform the mod for yourself, as well as
how distros can support the mod. Thanks to
this it is now already supported in
postmarketOS!
To summarise, the PinePhone eMMC ships in the
“DDR52″ timing mode by default, which modes
are allowed depends on the eMMC controller
voltage (this voltage is supplied by the VCC-
PC power rail and provides power to the eMMC
controller). DDR52 is the highest timing mode
supported at 3.3v, which is what the
PinePhone uses. To run this any higher, the
supply must be at 1.8v or 1.2v.
# ⚓ Freedom_and_phone_advice⠀⇛
The cold months are here, people are looking
forward to the holidays, and of course there
is a new FSF Ethical Tech Giving Guide. This
is the yearly Free Software Foundation
article giving advice what hardware to avoid
for reasons of freedom, and which ones are
instead recommended.
This year, the guide starts at the top with
advice for smartphones, which is great since
these account for a large chunk of computing
use these days. Let’s take a closer look at
the advice in this screenshot below.
o § Free, Libre, and Open Source Software⠀➾
# § Programming/Development⠀➾
# ⚓ The_Full-Stack_Developer_Vocabulary(The_Full_List!)_–
DEV_Community [Ed: Many buzzwords hype and pure
bullshit like “serverless”]⠀⇛
If you’re new to this whole full-stack
development thing and get confused by some of
the words people use when talking about full-
stack development, then stick around because
here, I’ll be compiling the most common words
and phrases related to full-stack
development. Of course, it will take much
more time to learn the details of each of the
things I’ll be mentioning. Each of the words
and phrases will be summarized to give you an
idea of what these words and phrases mean, so
that you can understand what people are
referencing with them when talking about
certain areas of full-stack development.
# § Python⠀➾
# ⚓ Everett_v3.0.0_released!_|_Will’s_Blog⠀⇛
Everett is a configuration library for
Python apps.
# § Rust⠀➾
# ⚓ Announcing_Rust_1.58.0_|_Rust_Blog⠀⇛
The Rust team is happy to announce a
new version of Rust, 1.58.0. Rust is a
programming language empowering
everyone to build reliable and
efficient software.
If you have a previous version of Rust
installed via rustup, getting Rust
1.58.0 is as easy as…
* § Leftovers⠀➾
o ⚓ Four_Wheel_Steering,_Always_The_Option,_Never_The_Defining
Feature_|_Hackaday⠀⇛
A couple of weeks ago when it emerged that a new Tesla
might have a four-wheel steering capability, our
colleague Dan Maloney mused aloud as to how useful a
four-wheel steering system might be, and indeed whether
or not one might be necessary at all. This is hardly the
first time four-wheel steering has appeared as the Next
Big Thing on the roads. It’s time to take a look at the
subject and ask whether it’s an idea with a future, or
set to go the way of runflat tyres as one of those
evergreen innovations that never quite catches on.
What’s your dream vehicle? If you’re like me, you have
more than one. There in my lottery-winner’s garage,
alongside the trail bikes and the mobile hackerspace, the
dictator-size Mercedes and the Golf Mk1, will be a
vehicle that by coincidence has four-wheel steering. The
JCB Fastrac is a tractor that can travel across almost
any terrain at full speed, and though I have no practical
use for one and will never own one, I have lusted after
one of these machines for over three decades. Their four-
wheel steering system is definitely unusual, but that
makes it the perfect vehicle with which to demonstrate
four-wheel steering.
o ⚓ Exploring_Tesla_Model_S_High_Voltage_Cabling_|_Hackaday⠀⇛
We ignored the warning and jumped right in. The “high”
voltages in the case of an electric vehicle (EV) like the
Model S is approximately 400 volts. Briefly, external
input via the charge connector can be single or three
phase, 120 or 250 VAC, depending on your region and
charging station. This get boosted to a nominal 400 VDC
bus that is distributed around the various vehicle
systems, including the motors and the battery pack.
o ⚓ Ride-on_Tracked_Vehicle_Is_A_Stout_Metal_Build_|_Hackaday⠀⇛
When we think of tracked vehicles, we normally think of
tanks, or perhaps heavy construction machinery. Meanwhile
the average member of the public is left out of the fun.
[Bob] of [Making Stuff] won’t be one of them, however,
having put together a ride-on tracked vehicle for his own
enjoyment.
The machine is welded together from plenty of steel,
making it more than tough enough to soak up the
punishment of off-road duty. The design features four
suspended buggy wheels on either side running inside
rubber tracks, with a cogged drive wheel at the front.
Propulsion is thanks to a 440 cc DuroMax engine good for
a full 18 horsepower and 26 ft-lbs of torque, driving the
tracks through a differential mounted up front.
o § Hardware⠀➾
# ⚓ Impedance_Matching_Revisited_|_Hackaday⠀⇛
If you are an old hand at RF design, you probably
have a good handle on matching impedance. However,
if you are just getting started with RF, [FesZ
Electronic]’s latest video series on lossless
impedance matching is well worth watching.
Matching is important for several reasons. Maximum
power transfer occurs when the source and load
impedance match. Also, at RF, mismatched impedance
can cause reflections which, again, robs you of
useful power. The video covers some math and then
moves on to LTSpice to simulate a test circuit. But
the part you are really waiting for — the practical
circuits — is about 15 minutes in. Since the values
you need are often oddball, [FesZ] makes his own
adjustable inductors and uses a trimmer capacitor
to adjust the actual capacitance value.
# ⚓ Remoticon_2021_//_Hash_Salehi_Outsmarts_His_Smart_Meter_|
Hackaday⠀⇛
Smart meters form mesh networks among themselves
and transmit your usage data all around. Some of
them even allow the power company to turn off your
power remotely, through the mesh. You might want to
know if any of this information is sesnsitive, or
if the power shutdown system has got glaring
security flaws and random people could just turn
your house off. Hash Salehi has set out to get
inside these meters, and luckily enough for the
rest of us, he was kind enough to share his
findings in Remoticon 2021. It’s a journey filled
with wonderful tidbits about GNU Radio, embedded
devices, and running your own power company inside
a Faraday cage.
The smart meter in question is deployed by a power
company known as Oncor in the Dallas, Texas, area.
These particular meters form an extensive mesh
network using a ZigBee module onboard that allows
them to to pass messages amongst themselves that
eventually make their way to a collector or
aggregator to be uploaded to a more central
location. Hash obtained his parts via everyone’s
favorite online auction house and was surprised to
see how many parts were available. Then, with parts
in hand, he began all the usual reverse engineering
tricks: SDR, Faraday cages, flash chip readers, and
recreating the schematic.
o § Integrity/Availability⠀➾
# § Proprietary⠀➾
# ⚓ IDrive®_360_Adds_Endpoint_Cloud_Backup_Support_for
Linux_Machines⠀⇛
IDrive® 360, a secure, enterprise-scale
endpoint cloud backup solution from IDrive,
has added backup support for Linux machines,
enabling IT admins to schedule and automate
backups for all of their organizational Linux
data into a single account through a unified
web console.
# § Pseudo-Open Source⠀➾
# § Openwashing⠀➾
# ⚓ Baumer,_Infineon,_Qualcomm_Innovation
Center,_Percepio_and_Silicon_Labs_Select
Zephyr_RTOS_for_their_Next_Generation_of
Products_and_Solutions⠀⇛
# ⚓ Baumer,_Infineon,_Qualcomm_Innovation
Center,_Percepio_and_Silicon_Labs_Select
Zephyr_RTOS_for_their_Next_Generation_of
Products_and_Solutions⠀⇛
The Zephyr™ Project announces a
major milestone today with Baumer
joining as a Platinum member and
Infineon Technologies, Qualcomm
Innovation Center, Inc., Percepio
and Silicon Labs joining as
Silver members. These new members
have selected Zephyr RTOS as one
of the key technologies to build
their next generation of
connected products and solutions.
# ⚓ IBM,_Clemson,_Linux_Foundation_join
forces_for_sustainable_crop_information
platform_|_ZDNet [Ed: Linux Foundation as
openwashing_and_greenwashing_services]⠀⇛
IBM has joined forces with the
Linux Foundation and the Clemson
University Cooperative Extension
Service and AgStack Foundation on
a new project seeking to provide
farmers with detailed information
about how to grow crops
sustainably and deal with the
effects of climate change.
IBM developers began working with
Clemson researchers through the
Call For Code program, which IBM
created as a way to support
people building innovative
technology solutions to
persistent global problems.
# ⚓ Call_for_Code_and_AgStack_open-source_Ag
Recommendations_–_IBM_Developer⠀⇛
Many rural farmers in the U.S. do
not have easy access to necessary
crop and pest management data.
Farmers often rely on information
they receive from Research and
Cooperative Extension Services.
Traditionally, getting this
information required calling or
going in person to Cooperative
Extension Service offices to ask
questions. This approach could be
prohibitive for those farmers not
located near a physical office
location. Even farmers in close
proximity to an Extension Service
office often need faster access
to this information to make
decisions while they are on the
farm.
If data sources were more
accessible to farmers, they could
have the crop management
resources they need in a more
timely manner. Clemson
University’s Cooperative
Extension Service is one such
important data source, providing
cooperatives and farmers with
agriculture recommendations to
improve their yields and day-to-
day farming practices.
Modernizing and digitizing this
kind of data to make it
accessible online helps bring
agriculture recommendations to
farmers when and where they need
it, without having to be in a
particular location.
# § Security⠀➾
# ⚓ Apache_Software_Foundation_statement_on_White
House_Open_Source_Security_Summit⠀⇛
The Apache Software Foundation (ASF)
participated today in a meeting hosted
by the White House to discuss security
of open source software, and how to
improve the “supply chain” of open
source software to better facilitate
the rapid adoption of security fixes
when necessary.
The virtual summit included
representation from a number of
companies and U.S. departments and
agencies. Three representatives of the
ASF participated in the virtual summit,
ASF President David Nalley, VP of
Security Mark Cox, and ASF board member
Sam Ruby.
䷩ 𝚕𝚒𝚗𝚎 6329
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Gemini_version_available_♊︎
✐ Links_13/1/2022:_Slackware_Linux_15.0_RC3_and_More_Microsoft_Aggression
Against_Linux⠀✐
Posted in News_Roundup at 6:41 am by Dr. Roy Schestowitz
🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈
§ Contents⠀➾
* GNU/Linux
o Distributions
o Devices/Embedded
* Free_Software/Open_Source
* Leftovers
* § GNU/Linux⠀➾
o ⚓ LWN’s_unreliable_predictions_for_2022_[LWN.net]⠀⇛
It is 2022 already, and that can only mean one thing:
it’s time for your editor to make a (bigger) fool of
himself by posting a set of predictions for what may come
in the new year. One should never pass up an opportunity
for a humbling experience, after all. There can be no
doubt that interesting things will happen this year;
let’s see how many random darts thrown in that direction
can hit close to the mark.
Starting with something that is, hopefully, fairly
obvious: 2022 will see a wider awareness that maintainers
need support for free-software projects to be healthy. It
has been a while since companies working with free
software realized that they needed to support the
developers of that software; that is the path toward
stronger projects and better influence over how those
projects evolve. But even the projects with the most
economic support struggle to support their maintainers,
and the effects can be felt across the entire community.
The ongoing Log4j debacle is just the latest symptom of
this problem.
Supporting maintainers can be a hard sell for a corporate
manager. Developers can focus most of their time directly
on their employers’ needs, but maintainers have to make
the project work for all participants, including their
employers’ competitors. The value of their contribution
is harder to quantify. But the cost of neglected
maintenance is high and growing, and the smarter
companies will start to figure this out.
This support will also take the form of a greater
willingness to pay for supported free-software products
in areas where that has not generally happened. The
recent announcement that support for GnuPG is selling
well is a case in point. This critical project has
languished for years, depending on donations from
individuals; maintainer Werner Koch is now telling donors
that their support is no longer needed.
o § Audiocasts/Shows⠀➾
# ⚓ The_Linux_Link_Tech_Show_Episode_937⠀⇛
it might pop and crackle but joel has the inside
scoop to stoke your fires
# ⚓ The_Linux_Link_Tech_Show_Episode_938⠀⇛
let Joel tell you how to pop a tent over docker
# ⚓ Linux_Saloon,_the_Next_Generation_of_BDLL_–_CubicleNate’s
Techpad⠀⇛
Being a part of BDLL and doing distro challenges
has introduced me to some incredible people and
opened doors for me. This show with Rocco, Zeb,
Dan, Eric and so many more really allowed me to
have a lot of fun with technology in a whole new
dimension.
In late 2020, Rocco stepped away from BDLL. Later,
with encouragement from a BDLL community member,
Michael Vash, I picked up the torch, and with
Rocco’s permission decided to continue to run with
BDLL. I didn’t want this all to just drift off into
the Internet ether. Largely because of the loyalty
I have towards Rocco and the community.
The name of the rebranded show is Linux Saloon. It
will remain at the same time each Saturday, with
the same or similar cast of characters that has
been around the last few months and to avoid
confusion, I have decided to move it to a new
channel and give it a fresh start.
# ⚓ Multi-Monitor_Video_Editing_–_Purism⠀⇛
Next in our video editing series for the Librem 14,
Gardiner Bryant dives into using multiple monitors.
Video editing is resource heavy on any laptop,
which is why we recommend Librem 14. This video
will help those looking to level up their overall
video production.
o § Kernel Space⠀➾
# ⚓ Linux_5.17_Lands_Big_Rewrite_To_FS-Cache_&_CacheFiles
Driver_Code⠀⇛
Being worked on since early 2020 by Red Hat’s David
Howells has been a rewrite to Linux’s FS-Cache and
CacheFiles code focusing on making it smaller and
simpler while also presenting possible memory/
performance advantages. That major rewrite has been
merged now for Linux 5.17.
# ⚓ Microsoft_Reworks_The_“DXGKRNL”_Driver_It_Wants_To_Get_Into
The_Linux_Kernel [Ed: Microsoft is still aggressing_against
Linux]⠀⇛
Back in 2020 Microsoft announced the DXGKRNL driver
as the kernel driver component for supporting GPU
accelerated use-cases within Windows Subsystem for
Linux (WSL2). That original DXGKRNL driver was
quickly shot down by upstream kernel developers and
various issues raised while now for the past year
Microsoft has been reworking this kernel driver and
on Wednesday published the new version.
[...]
Microsoft was also originally criticized with
DXGKRNL since it relied upon closed-source CUDA and
DirectX user-space components for operation. To
that they now are celebrating the open-source user-
space API support offered by Intel with their
OpenCL / OpenVINO / oneAPI support atop this kernel
driver for use with Intel graphics hardware.
# ⚓ x86_Straight_Line_Speculation_CPU_Mitigation_Appears_For
Linux_5.17_–_Phoronix⠀⇛
The Linux 5.17 kernel is introducing support for
the x86 straight-line speculation “SLS” mitigation
with it becoming increasingly clear modern x86_64
CPUs are susceptible to speculatively executing
linearly in memory past an unconditional change in
control flow.
Back in mid-2020 Straight Line Speculation was made
public for Arm CPUs based on research by Google’s
SafeSide initiative. Arm processors were found to
be able to speculatively execute linearly in memory
beyond what should be an unconditional change in
control flow, such as for exception returns, other
exception generating instructions, unconditional
direct/indirect branches, and function returns. If
speculatively executing a “Spectre revelation
gadget” it could in turn make secrets vulnerable to
revelation through timing analysis.
Following that disclosure, open-source software
quickly mitigated for Arm SLS via GCC and LLVM/
Clang compiler hardening around BLR and RETBR
instructions.
# ⚓ Linux_Serial_Console_Driver_Lands_Patch_For_Possible_~25%
Performance_Improvement_–_Phoronix⠀⇛
It’s not an area of Linux hardware performance we
normally look at, but thanks to a Red Hat engineer
discovering very low serial console performance,
there is an improvement queued up for introduction
in Linux 5.17…
Red Hat’s Wander Lairson Costa was noticing the
serial console throughput on an HP Proliant DL380
Gen9 server was coming in well below expectations:
expecting 10KB/s but with the current Linux kernel
only hitting around 2.5KB/s. The Linux 8250 serial
console driver was taking around 410 microseconds
just to dispatch one single byte.
With the Linux 8250/16550 serial port console
driver, Wander has managed to improve the
performance in 2022. The 16550 UARTs have an on-
chip FIFO buffer to which is now being used on
supported systems by the 8250 serial driver’s write
function.
# ⚓ Intel_Core_i5_12400_“Alder_Lake”:_A_Great_~$200_CPU_For
Linux_Users_Review_–_Phoronix⠀⇛
Formally announced at CES, the Core i5 12400 and
other Alder Lake non-K desktop CPUs are beginning
to appear in retail channels. Last week I was able
to buy an Intel Core i5 12400 “Alder Lake” from a
major Internet retailer for $209 USD — and one week
later there remains availability during these
turbulent supply chain times. The i5-12400 has
wound up being a very nice processor for Linux use
that exceeded my initial expectations.
# ⚓ Zero-copy_network_transmission_with_io_uring_[LWN.net]⠀⇛
When the goal is to push bits over the network as
fast as the hardware can go, any overhead hurts.
The cost of copying data to be transmitted from
user space into the kernel can be especially
painful; it adds latency, takes valuable CPU time,
and can be hard on cache performance. So it is
unsurprising that the developers working with
io_uring, which is all about performance, have
turned their attention to zero-copy network
transmission. This patch set from Pavel Begunkov,
now in its second revision, looks to be
significantly faster than the MSG_ZEROCOPY option
supported by current kernels.
As a reminder: io_uring is a relatively new API for
asynchronous I/O (and related operations); it was
first merged less than three years ago. User space
sets up a pair of circular buffers shared with the
kernel; the first buffer is used to submit
operations to the kernel, while the second receives
the results when operations complete. A suitably
busy process that keeps the submission ring full
can perform an indefinite number of operations
without needing to make any system calls, which
clearly improves performance. Io_uring also
implements the concept of “fixed” buffers and
files; these are held open, mapped, and ready for
I/O within the kernel, saving the setup and
teardown overhead that is otherwise incurred by
every operation. It all adds up to a significantly
faster way for I/O-intensive applications to work.
One thing that io_uring still does not have is
zero-copy networking, even though the networking
subsystem supports zero-copy operation via the
MSG_ZEROCOPY socket option. In theory, adding that
support is simply a matter of wiring up the
integration between the two subsystems. In
practice, naturally, there are a few more details
to deal with.
A zero-copy networking implementation must have a
way to inform applications when any given operation
is truly complete; the application cannot reuse a
buffer containing data to be transmitted if the
kernel is still working on it. There is a subtle
point that is relevant here: the completion of a
send() call (for example) does not imply that the
associated buffer is no longer in use. The
operation “completes” when the data has been
accepted into the networking subsystem for
transmission; the higher layers may well be done
with it, but the buffer itself may still be sitting
in a network interface’s transmission queue. A
zero-copy operation is only truly done with its
data buffers when the hardware has done its work —
and, for many protocols, when the remote peer has
acknowledged receipt of the data. That can happen
long after the operation that initiated the
transfer has completed.
So there needs to be a mechanism by which the
kernel can tell applications that a given buffer
can be reused. MSG_ZEROCOPY handles this by
returning notifications via the error queue
associated with the socket — a bit awkward, but it
works. Io_uring, instead, already has a completion-
notification mechanism in place, so the “really
complete” notifications fit in naturally. But there
are still a few complications resulting from the
need to accurately tell an application which
buffers can be reused.
# ⚓ User-managed_concurrency_groups_[LWN.net]⠀⇛
The kernel’s thread model is relatively
straightforward and performs reasonably well, but
that’s not enough for all users. Specifically,
there are use cases out there that benefit from a
lightweight threading model that gives user space
control over scheduling decisions. Back in May
2021, Peter Oskolkov posted a patch set
implementing an abstraction known as user-managed
concurrency groups, or UMCG. Several revisions
later, many observers still lack a clear idea of
what this patch is supposed to do, much less
whether it is a good idea for the kernel. Things
have taken a turn, though, with Peter Zijlstra’s
reimplementation of UMCG.
One developer reimplementing another’s patch set is
likely to raise eyebrows. Zijlstra’s motivation for
doing that work can perhaps be seen in this
message, where he notes that the UMCG code looked
little like the rest of the scheduler code. He also
remarked that it required “reverse engineering” to
figure out how UMCG was meant to be used. By the
time that work was done, perhaps, it was just
easier to recast the code in the form he thought it
should take.
In truth, the documentation for UMCG is no better
than before — a significant problem for a major
proposed addition to the system-call API. But it is
possible to dig through the code (and a “pretty
rough” test application posted by Zijlstra) to get
a sense for what is going on. In short, UMCG calls
for a multi-threaded application to divide itself
into “server” and “worker” threads, where there is
likely to be one server thread for each CPU on the
system. Server threads make scheduling decisions,
while workers run according to those decisions and
get the actual work done. The advantage of a system
like UMCG is that scheduling can happen quickly and
with little overhead from the kernel — assuming the
server threads are properly implemented, of course.
# § Graphics Stack⠀➾
# ⚓ Mesa_22.0_Pushed_Back_By_Three_Weeks_–_Phoronix⠀⇛
While a lot of open-source OpenGL and Vulkan
driver improvements have been landing in
recent days in anticipation of the Mesa 22.0
code branching and feature freeze for
Wednesday, that deadline has now been
extended by three weeks.
Due to problems merging some merge requests
from GitLab as well as FreeDesktop.org
hosting issues, Mesa 22.0 has been pushed
back. Additionally, some Mesa3D developers
have expressed interest in trying to squeeze
in some remaining patches not yet merged.
o § Applications⠀➾
# ⚓ Plots_is_an_open-source,_free_app_to_visualize_visualize
mathematical_formulas⠀⇛
Plots is a graph plotting app for GNOME. Plots
makes it easy to visualize mathematical formulae.
In addition to basic arithmetic operations, it
supports trigonometric, hyperbolic, exponential and
logarithmic functions, as well as arbitrary sums
and products. It can display polar equations, and
both implicit and explicit Cartesian equations.
o § Instructionals/Technical⠀➾
# ⚓ How_to_Check_MySQL_User_Privileges_in_Linux⠀⇛
The first/fresh installation of a MySQL on any
operating system only considers the root user as
the default database user. The first database
transactions/activities are performed by the root
user only.
Therefore, it is not ideal for any user that needs
access to the MySQL database to gain entry via the
root user credentials. Root user access should be
reserved to the database administrator who will
then use the root user credentials to create
database users and grant privileges to execute
different database queries.
# ⚓ Fix_Firefox_96.0_And_95.0.2_Not_Loading_Websites_With_DNS
Over_HTTPS_Enabled_–_Linux_Uprising_Blog⠀⇛
The latest Firefox 96.0 as well as 95.0.2 have an
issue which prevents the browser from establishing
any connections when DNS over HTTPS (DOH) is
enabled. Simply disabling this option once enabled
doesn’t make the issue go away. Read on to see how
to fix this.
With DNS over HTTPS enabled on Firefox 96.0 and
95.0.2, besides not being able to access any
websites, the browser hangs in the background when
closed. The issue affects Linux, Windows, and macOS
Firefox users alike.
# ⚓ How_to_Increase_Request_Timeout_in_NGINX_–_TecAdmin⠀⇛
Sometimes the long running requests failed with the
error message “504: Gateway Timeout” in NGINX web
server. To solve this issue, you need to increase
request timeout in NGINX server configuration. The
default, NGINX request timeout is 60 seconds. Which
can be increased or decreased by updating the
configuration files.
In this quick FAQ, you will learn to change the
request timeout in NGINX web server.
# ⚓ How_to_Install_ModSecurity_3_&_OWASP_Core_Rule_Set_with
Apache_(HTTPD)_on_Fedora_35_–_LinuxCapable⠀⇛
ModSecurity, often referred to as Modsec, is a
free, open-source web application firewall (WAF).
ModSecurity was created as a module for the Apache
HTTP Server. However, since its early days, the WAF
has grown and now covers an array of HyperText
Transfer Protocol request and response filtering
capabilities for various platforms such as
Microsoft IIS, Nginx, and Apache.
How the WAF works, the ModSecurity engine is
deployed in front of the web application, allowing
the engine to scan the incoming and outgoing HTTP
connections. ModSecurity is most commonly used in
conjunction with the OWASP Core Rule Set (CRS), an
open-source set of rules written in ModSecurity’s
SecRules language and is highly regarded among the
security industry.
# ⚓ How_to_Install_Linux_Kernel_5.16_on_Linux_Mint_20_–
LinuxCapable⠀⇛
Linux kernel 5.16 has many new features, support,
and security. The Linux 5.16 kernel release has a
great new feature, FUTEX2, or futex_watv(), which
aims to improve the Linux gaming experience,
growing considerably with better native Linux
porting for Windows games utilizing Wine.
Other improvements have seen write include improved
write congestion management, task scheduler for CPU
clusters sharing L2/L3 cache, amongst many other
additions. More information can be found on the
Linux 5.16 Kernel release changelog.
# ⚓ How_to_install_and_Configure_HAProxy_load_balancer_on
Debian_11⠀⇛
HAProxy is a free and open source software that
provides a high availability load balancer and
proxy server for TCP and HTTP-based applications
that spreads requests across multiple servers. It
distributes the load among the web and application
servers.
Haproxy is popular for load balancing because of
its efficiency, reliability, and low memory and CPU
footprint. Load balancing is a common solution for
distributing web applications horizontally across
multiple hosts while providing the users with a
single point of access to the service.
It is available for install on major Linux
distributions. In this guide we will learn how to
install and configure HAProxy load balancer on
Debian 11.
# ⚓ The_choose_command_in_Linux⠀⇛
Hello, friends. In this post, you will learn how to
use choose command in Linux. We have tested this
tutorial on Debian 11, but it should work on Ubuntu
20.04 and derivatives.
# ⚓ 3_Ways_to_install_and_use_HandBrake_Ubuntu_22.04_|_20.04
LTS_–_Linux_Shout⠀⇛
HandBrake is an open-source video transcoder
distributed under GPL license, here we learn the
steps to install Handbrake on Ubuntu 22.04 Jammy
JellyFish or Ubuntu 20.04 LTS Focal Fossa.
After a gap of some time, finally, the developers
of Handbrake recently announced the latest version
1.5.1 to install for Linux, Mac, and Windows. The
key purpose of this free and open-source software
is to convert common video files and formats; to
play on smartphones, tablets, TV, game console, PC,
or web browser—nearly anything that supports modern
video formats. It offers tools such as FFmpeg,
x264, and x265 to create new MP4 or MKV video
files. To avoid misunderstandings, the Freeware
Handbrake cannot handle copy-protected DVDs or Blu-
Rays. So you cannot make copies of purchased films
with the software. Otherwise, the open-source
software is ideal for converting videos.
One thing that is particularly important with video
tools, broad format support. Handbrake definitely
offers it, there is hardly a video that cannot be
converted with the tool. Apart from the video,
Handbrake offers a wide range of options for sound
format, bit rate, and image sizes. You can also
apply numerous filters or add subtitles. Once set,
you can also run several videos in a series.
# ⚓ Restricting_SSH_agent_keys_[LWN.net]⠀⇛
The OpenSSH suite of tools for secure remote logins
is used widely within our communities; it also
underlies things like remote Git repository access.
A recent experimental feature for the upcoming
OpenSSH 8.9 release will help close a security hole
that can be exploited by attacker-controlled SSH
servers (e.g. sshd) when the user is forwarding
authentication to a local ssh-agent. Instead of
allowing the keys held in the agent to be used for
authenticating to any host where they might work,
SSH agent restriction will allow users to specify
where and how those keys can be used.
# ⚓ Install_Node.js_on_Raspberry_Pi_4⠀⇛
Node.js is an open-source and cross-platform
server-side JavaScript runtime powered by the
Google Chrome V8 JavaScript engine. Node.js is
mainly used to develop network apps, APIs, and
full-stack web apps. Node.js can also be used to
develop desktop apps and mobile apps.
In this article, I am going to show you how to
install the latest LTS (Long Term Support) version
of Node.js on Raspberry Pi 4 running the Raspberry
Pi OS. So, let’s get started.
# ⚓ How_to_change_canvas_size_in_Inkscape⠀⇛
In Inkscape, after creating a design that is ready
for finalization, you will most likely need to
change the size to accommodate whatever you have
created. For instance, you have designed a logo,
and now you want to finalize the editable vector
copies to deliver them to the intended recipients.
Also, when you create a PDF file, Inkscape only
saves the objects that exists within the page
border. So, understanding how to change the canvas
size is essential if you need to change it
according to your preferences.
This write-up will show you how to change the
canvas size in Inkscape by using the “Documents
Properties” menu. The “Documents Properties” menu
also permits you to change the size of the canvas
with numerical input. You can also customize your
canvas size to fit the specific objects added in
your Inkscape document. So, let’s get this guide
started!
# ⚓ How_to_convert_PNG_to_SVG_in_Inkscape⠀⇛
Vector graphics and Raster graphics are the two
primary types of graphics. Portable Graphic Format
(PNG) files are the raster images built from
discrete colored boxes, called pixels. Pixel
graphics are static and have a predetermined size.
In a raster image, the individual pixels become
more visible as you zoom in or try to magnify them.
On the other hand, Vector graphics are based on
mathematical formulas that specify the graphics
features on the X and Y axes. These formulas are
significantly more dynamic than a sequence of
static boxes or pixels. Scalable Vector Graphics
(SVG) format offers many additional advantages
compared to the PNG, such as being fully editable
by utilizing the vector graphics tools and having
the capability to enlarge its points.
Suppose that you have a logo file in PNG format and
you want to convert it to SVG. What will you do?
You will look out for this feature in popular image
editing software or vector graphics editors such as
Inkscape. Inkscape permits users to save and
convert the PNG or JPG image file into SVG file
format. Today, we will thoroughly demonstrate how
you can convert PNG to SVG in Inkscape. So, let’s
start!
# ⚓ Linux_Command:_Namei_Usage⠀⇛
Linux is a diverse platform to play with a lot of
commands in its shell at one time and does other
work as well. These commands are of diverse use and
purpose. There comes a moment while working in
Linux that you have to find out and know more about
some specific file, its owner, its path, and
content between some folders. One of those unique
and great commands is the “namei” command of the
Linux system. The namei command is used so far to
know more about the specific directory i.e., its
path, location, and a lot more things. Therefore,
today we will be discussing the namei command in
our Ubuntu 20.04 Linux shell. So, let’s have a new
start.
You need to open up your system first and then open
up the terminal console as well. Use the
“Ctrl+Alt+T” for doing so. After the opening of the
terminal, we are ready to utilize our command for
specific purposes. You have to know that the namei
command uses many flags in it for those reasons. If
you want to find out more about the namei command,
just write “namei” in the shell and you will see
the commands info.
# ⚓ Linux_Command:_Bridge_Usage⠀⇛
In Linux systems, the brctl command has been called
a bridge command. The “brctl” term stands for
bridge control. It is the ultimate unique command
to let you see all the current Ethernet bridges in
your system. It may also let you add and create new
Ethernet bridges and make changes to many of them
with few keywords in them. Therefore, we will be
utilizing the Ubuntu 20.04 system to discuss the
brctl bridge command. For that, we have to start
the shell console of our system via “Ctrl+Alt+T”.
So, let’s get started.
We are starting our article with the installation
of bridge utilities. For this, we need to use the
“apt-get” package command in the shell. The keyword
“install” has been used after the apt-get package
and the “bridge-utils” package name is utilized
after that with the “-y” flag to force
installation. At the start of installation, it
probably asks for your sudo password and you have
to add it necessarily. After that, the bridge
utilities will be installed on your Ubuntu system.
o § Desktop Environments/WMs⠀➾
# § K Desktop Environment/KDE SC/Qt⠀➾
# ⚓ KDE_Plasma_5.24_–_Top_New_Features_and_Release
Details⠀⇛
We bring you the top new features of upcoming
KDE Plasma 5.24 desktop in this post and give
you a heads-up on what’s in store.
# § GNOME Desktop/GTK⠀➾
# ⚓ GNOME_42_Planned_for_Ubuntu_22.04,_But_Few_GTK4_Apps
–_OMG!_Ubuntu!⠀⇛
GNOME 42 could still feature in Ubuntu 22.04
when it arrives this April — but don’t expect
to see too many GTK4 apps with it.
Ubuntu developers say they ‘aim’ to include
the bulk of GNOME 42 release in Ubuntu 22.04
but are currently tasked with updating the
GNOME Shell stack to the latest GNOME 41
release.
GNOME 42 is itself under active development
ahead of a planned stable release in March.
The first alpha of GNOME 42 expected to drop
this month and will feature a fair number of
apps ported to and/or taking advantage of
GTK4 and libadwaita.
And it’s those that Ubuntu isn’t keen on
including — not yet.
If this all sound a bit conservative it’s
because it is! Ubuntu 22.04 is an LTS and
Ubuntu has to ship a solid, stable software
set it can confidently commit to supporting
over the next five years (and possibly
beyond).
While upcoming GTK4 ports of Settings and
Files are likely to get a ton of stress
testing by GNOME developers — and plucky
enthusiasts — ahead of the GNOME 42 release,
Ubuntu isn’t certain there’s enough time to
test them well enough, not for inclusion in a
long-term support release.
# ⚓ Ubuntu_22.04_LTS_Aiming_For_GNOME_42,_Avoiding_GTK4
Where_Possiblex⠀⇛
Ubuntu developers have laid out their GNOME
versioning plans for this spring’s release of
Ubuntu 22.04 LTS.
While Ubuntu has been behind upstream when it
comes to GNOME 40+ packaging, with Ubuntu
21.10 they are on GNOME 40 and for April’s
release of Ubuntu 22.04 LTS they are planning
to get to GNOME 42. They are currently
shifting to GNOME Shell 41 and then working
on moving to GNOME 42 updates. GNOME 42 will
be officially out in March and the plan is
for that new upstream release to be powering
this next Ubuntu Long Term Support release.
# ⚓ Zrythm_Switches_to_GTK_4_and_libadwaita_Ahead_of
Other_Digital_Audio_Workstations_(DAWs)_–_It’s_FOSS
News⠀⇛
Now that it’s been a while since GTK 4 was
unveiled, several applications have started
to make the move from GTK 3.
The latest of which is Zrythm. While still in
its alpha phase, this change is incredibly
large and impactful, so let’s take a look at
it!
In case you’re curious, Zrythm is a Digital
Audio Workstation, just like LMMS, Ardour,
and other options in our list of best DAWs.
Zrythm allows users to edit audio, and make
music. It has all the essential features
expected from a DAW. And, it seems to be
properly working with the various audio
servers desktop Linux uses (like Pulseaudio,
Pipewire, etc.).
o § Distributions⠀➾
# § New Releases⠀➾
# ⚓ Kali’s_stable_Docker_image_is_now_named_kali-last-
release⠀⇛
Here is a very quick announcement for users
of the Kali Linux Docker Images.
Until recently we used to have a Docker image
named simply kali, and it was built from the
last versioned release of Kali (e.g. 2019.4,
2020.1, etc.) matching our “kali-last-
snapshot” network repositories branch. In a
way, this is our “stable” release, as it will
only get updates quarterly as it is in
synchronisation with our release cycle.
We still provide this Docker image, but now
it has been renamed from kali to kali-last-
release for clarity.
# § Slackware Family⠀➾
# ⚓ Development_Release:_Slackware_Linux_15.0_RC3⠀⇛
“Good hello, and welcome to the third and
final release candidate for Slackware 15.0.
We’re 99% frozen at this point and are mostly
looking for regression or other bug reports
that might be able to be addressed before
this goes stable. Of course, the management
here reserves the right to make exceptions…
that 5.15.15 kernel version has a nice ring
to it. If your requests didn’t make it into
this iteration, perhaps we will revisit them
for the next -current cycle. Some were just a
little too late but will more than likely be
needed next time (I’m looking at Didier’s
grubconfig), while others are just out of
scope for the main tree where I like to abide
by YAGNI as much as possible. Anyway, let’s
get some testing done and we’ll be there
soon. Enjoy!”
# § IBM/Red Hat/Fedora⠀➾
# ⚓ How_to_troubleshoot_DHCP_communication_problems_on
your_network_|_Enable_Sysadmin⠀⇛
Imagine you have a repurposed enterprise
switch with a Dynamic Host Configuration
Protocol (DHCP) service that you need to
troubleshoot. There is little information
available about the switch’s configuration or
previous deployments. The device is reported
to be functional and should lease Internet
Protocol (IP) address configurations to
clients. However, the attached clients are
not receiving IP configurations from the
switch.
# ⚓ IT_talent:_3_hot_IT_roles_in_2022_and_beyond_|_The
Enterprisers_Project⠀⇛
As organizations kick off 2022 IT hiring, the
demand for IT talent is not slowing down.
Digital transformation leaders and IT
security professionals are in particularly
high demand, driven by digital transformation
and the continuation of remote work.
Many companies that put digital
transformation on hold during the pandemic
are now prioritizing these initiatives and
are seeking top IT professionals to lead
them. Hand-in-hand with digital
transformation initiatives is IT security.
Security continues to be a top priority for
organizations as phishing attempts and
hacking threatens their data.
# ⚓ Get_started_with_Node.js_16_on_OpenShift_|_Red_Hat
Developer⠀⇛
In April 2021, Node.js released its latest
major version, Node.js 16. Code-named
Gallium, it became a long-term support (LTS)
release in October.
Red Hat recently released a fully supported
Node 16 container image. Every Red Hat build
of a Node.js release is tested and supported
on Red Hat OpenShift and Red Hat Enterprise
Linux and is based on a Red Hat Universal
Base Image.
Red Hat runtimes are tested and certified
against various popular development
frameworks and technologies while running on
Red Hat OpenShift and RHEL. We are unable to
test every possible framework and version,
but the specific components, modules, and
frameworks supported on Red Hat’s build of
Node.js can be found on the component details
page as per the Node.js module and framework
support policies.
# ⚓ A_developer’s_guide_to_CI/CD_and_GitOps_with_Jenkins
Pipelines_|_Red_Hat_Developer⠀⇛
CI/CD, or continuous integration and
continuous delivery, is an essential part of
the modern software development life cycle.
Coupled with GitOps, CI/CD allows developers
to release high-quality software almost as
soon as they commit code to a repository such
as GitHub.
Automation is a key factor for implementing
effective CI/CD. In this process, developers
and release engineers create scripts that
have all the instructions needed to test the
code in a source code repository before
putting it into a production environment. The
process is efficient but complex.
Fortunately, there are many tools that lessen
the burden.
Jenkins is one of the most popular tools used
for CI/CD. Jenkins has been around for years
and has undergone numerous revisions, adding
features all along the way. One of the most
transformative features added to Jenkins is
the ability to run Jenkins Pipeline jobs
driven by an automation script stored in a
Jenkinsfile. Developers and release engineers
can use Jenkinsfiles to combine the practices
of CI/CD and GitOps into a unified deployment
process. That’s the focus of this article.
We’ll start with a brief refresher of what
Jenkins is and how it applies to both CI/CD
and GitOps. Then, I’ll guide you through how
to use a Jenkinsfile to create deployments
that combine CI/CD and GitOps.
# ⚓ Another_Fedora_integrity-management_proposal_
[LWN.net]⠀⇛
As is usual for feature proposals, Fedora
program manager Ben Cotton posted it to the
Fedora devel mailing list on behalf of the
feature owner: Roberto Sassu. The change
proposal is also on the Fedora wiki. The new
feature would use the Digest Lists Integrity
Module (DIGLIM) feature, which has been
proposed by Sassu as an addition to the
kernel’s Integrity Measurement Architecture
(IMA). Ensuring that file contents and
metadata do not change in unexpected ways is
IMA’s job; DIGLIM is an optimization of sorts
to IMA.
IMA has a number of different functions, but
at its core it maintains “digests” of file
contents and metadata; these digests are
cryptographic hashes that can be used to
reliably detect file changes. IMA can also
use the digests, in combination with the
system’s Trusted Platform Module (TPM), to
calculate a value that proves that the system
is running a known set of software. That
value can be used to ensure the system has
been securely booted or it can be sent
elsewhere to remotely attest to the state of
the system.
Each file being protected by IMA needs its
digest stored with the file, which is
normally done using extended attributes in
the filesystem. IMA can be configured to
check each file before it is accessed to see
if its digest still matches the stored value;
if not, access can be denied. As files are
assessed, their digest can be submitted to
the TPM to extend a Platform Configuration
Register (PCR); the resulting value is a
reflection of the files measured, but it is
also affected by the order of the accesses.
According to the DIGLIM proposals (for Fedora
and the patch set for the kernel), parallel
execution during the assessment results in
differing values from the TPM; even if the
same code is used, it may result in a
different attestation value. DIGLIM provides
a mechanism to take a digest value of all of
the files installed, instead, and use that
for calculating the attestation value. Only
files that have digests that were not
included in the overall “installation digest”
would be used to further extend the PCR in
the TPM.
It does so by providing a mechanism to enroll
digest values from the installed files into a
kernel “digest list”, which can then be
consulted as files are accessed. If the
digest of a file appears on the list, it can
be considered to be unchanged and its digest
value does not get submitted to the TPM;
otherwise, the file has been modified or was
not included in the digest list at all, so
access could be denied and the file’s digest
added into the attestation value. The latter
would likely mean that the system fails its
attestation.
# § Debian Family⠀➾
# ⚓ Debian_vs._CentOS_|_FOSS_Linux⠀⇛
It is a massive deal for any organization to
finalize a Linux distribution. Even for an
individual, it matters a lot which version of
Linux they end up running on their system.
Debian and CentOS are two different versions
of Linux which have some similarities and
differences. Today we will compare them to
decide which one will work best for you.
In addition to checking out the similarities,
we will also look at the differences in the
builds of both Debian and CentOS, their
management tools, community support,
upgrading, and a few more crucial features
that define an OS. So let us dive right into
it and first look at what these operating
systems are comprised of.
# § Canonical/Ubuntu Family⠀➾
# ⚓ Ubuntu_22.04_LTS_to_leverage_zswap_to_run_on
Raspberry_Pi_4_with_2GB_RAM_–_CNX_Software⠀⇛
Canonical used to recommend Raspberry Pi 4
with at least 4GB RAM to run Ubuntu Desktop,
but Ubuntu 22.04 LTS should run more smoothly
on the Raspberry Pi 4 2GB as the company has
enabled zswap by default to allow the Linux
operating system to run better on systems
with less memory.
Canonical explains that zswap is essentially
a compression tool.
o § Devices/Embedded⠀➾
# ⚓ Bitcoin_Mining_ASICs_Repurposed_To_Keep_NTP_Server_On_Track
|_Hackaday⠀⇛
They say time is money, but if that’s true, money
must also be time. It’s all figurative, of course,
but in the case of this NTP server heater powered
by Bitcoin mining dongles, money actually does
become time.
This is an example of the lengths to which Network
Time Protocol aficionados will go in search of
slightly better performance from their NTP servers.
[Folkert van Heusden], having heard that thermal
stability keeps NTP servers happy, used a picnic
cooler as an environmental chamber for his Pi- and
GPS-based NTP rig. Heat is added to the chamber
thanks to seven Block Erupter ASIC miner dongles,
which are turned on by a Python script when a
microcontroller sends an MQTT message that the
temperature has dropped below the setpoint.
# ⚓ OnLogic_reveals_quartet_of_Alder_Lake_systems_with_up_to_14
LAN_ports⠀⇛
OnLogic unveiled a “Karbon 800 Series” of 4x
embedded PCs based on Intel’s up to 16-core 12th
Gen Alder Lake-S CPUs with up to 64GB RAM and
options including PCIe Gen 4 x16, hot-swap SATA,
4G, Wi-Fi 6E, -40 to 70°C, and up to 14 LAN ports.
OnLogic has announced four Karbon 800 Series
embedded computers that run Windows or Linux 20.04
LTS on Intel’s recently announced 12th Gen Alder
Lake platform. This is really a pre-announcement as
there are relatively few details and the systems
will not ship until Q2.
# § Open Hardware/Modding⠀➾
# ⚓ Raspberry_Pi_CM4_Nano_industrial_mini_PC_supports
wide_temperature_range,_12-18V_DC_input_–_CNX
Software⠀⇛
If you ever wanted a mini PC similar to
Raspberry Pi 4 but working within a wider
temperature range and supply voltage, as well
as a few extra features, the Raspberry Pi CM4
Nano industrial mini PC with a metal
enclosure might be worth looking at.
Based on the EDATEC CM4 Nano carrier board,
the mini PC supports Raspberry Pi CM4 with up
to 8GB RAM, 32GB storage, optional WiFi 5 and
Bluetooth 5.0, and offers one HDMI port, a
flat cable HDMI + Touchscreen connector,
Gigabit Ethernet, three USB 3.0 ports and
more. It works in settings with -25 to +60°C
ambient temperature and offers a 12-18V DC
input.
# ⚓ Raspberry_Pi_system_can_detect_viruses_on_other
devices_without_use_of_software⠀⇛
A team of researchers at the Institute of
Computer Science and Random Systems has built
a non-software-based virus detection system
using a Raspberry Pi, an H-field probe and an
oscilloscope to detect electromagnetic wave
signatures of multiple types of viruses. The
team presented its system and test results at
last month’s ACM Machinery’s Annual Computer
Security Applications Conference and
published a paper describing their system on
ACM’s Research Article page.
The idea behind the new system is that
running software generates electromagnetic
waves. And each piece of software generates
its own unique wave patterns due to the way
the software executes its code. The
researchers took advantage of this knowledge
and began using an H-field probe to capture
wave patterns of known computer viruses
running on various devices and viewed the
results on an oscilloscope. They saw
oscilloscope patterns that were unique to
individual viruses as they were running. The
researchers used that information to program
a Raspberry Pi to identify data from the
other two devices to recognize known virus
wave patterns, using the system as a virus
detector. To determine if a virus is running
on a computer, IoT device or smartphone, a
user places the H-field probe close enough to
the device to read the electromagnetic waves
that are generated. The Raspberry Pi then
reports on whether it found any viruses, and
if so, which ones. Testing found the system
capable of detecting 99.82% of generic
malware, along with a benign virus type.
# ⚓ Arduino_And_An_OLED_Make_This_Space_Invaders_Cabinet
Tiny_|_Hackaday⠀⇛
For as simple as it appears now, Space
Invaders was one machine from the Golden Age
of video games that always seemed to have a
long line waiting for a chance to lose a
couple of quarters. And by way of celebrating
the seminal game’s influence, [Nick Cranch]
has executed what might just be the world’s
smallest Space Invaders replica.
It appears that this started mainly as an
exercise in what’s possible with what’s on
hand, which included a couple of quite small
OLED displays. For the build photos it looks
like there’s an Arduino Nano running the
show; [Nick] relates that the chosen hardware
proved challenging, and that he had to hack
the driver library to make it work. Once he
got a working game, [Nick] didn’t rest on his
laurels. Rather, he went the extra mile and
built a miniature cabinet to house everything
in.
o § Free, Libre, and Open Source Software⠀➾
# § FSF⠀➾
# § GNU Projects⠀➾
# ⚓ GNU_Guix:_Announcing_the_second_online_Guix
Days⠀⇛
The Guix hackers are very happy to
announce the second online Guix Days
Conference on 19 & 20 February 2022.
This conference is open to everyone and
will be held entirely online. Want to
speak? Submit your proposal!
[...]
In addition to the format you would
like to choose, please describe your
session with 10 lines or more (for
lightning talks, at least 1 sentence).
Once you have sent your proposal, you
will be notified in the following days
whether your talk will be part of the
Guix Days. Submit earlier to get more
time to prepare your session!
Even for live presentation, please
prepare a back-up pre-recorded talk, so
we can play it if you cannot attend or
have a technical problem during the
Guix days. The deadline for short
presentations (5 minutes) is February
16.
We welcome all kinds of topics from the
community, especially your own
experience with Guix, your cool
projects that involve Guix in some way,
infrastructure around guix
(translations, continuous integration,
…), and any subject you feel should be
discussed during the conference.
We particularly encourage people who
consider themselves part of a group
underrepresented in Guix and the
broader free software movement to
submit a talk. Do not hesitate to get
in touch with the organizers at guix-
days@gnu.org if unsure or if you would
like guidance on how to prepare your
talk.
# § Programming/Development⠀➾
# ⚓ Arduino_IDE_Creates_Bootable_X86_Floppy_Disks_|
Hackaday⠀⇛
Arguably the biggest advantage of the Arduino
ecosystem is how easy it is to get your code
running. Type a few lines into the IDE, hit
the button, and in a few seconds you’re
seeing an LED blink or some text get echoed
back over the serial port. But what if that
same ease of use didn’t have to be limited to
microcontrollers? What if you could use the
Arduino IDE to create computer software?
That’s exactly what boot2duino, a project
developed by [Jean THOMAS] hopes to
accomplish. As you might have guessed from
the name, the code you write in the Arduino
is turned into a bootable floppy disk image
that you can stick into an old PC. After a
few seconds of beeping and grinding your
“Hello World” should pop up on the monitor,
and you’ve got yourself the world’s biggest
Arduino.
# ⚓ Moving_librsvg’s_documentation_to_gi-docgen_–
Federico’s_Blog⠀⇛
Librsvg’s documentation tooling is pretty
ancient. The man page for rsvg-convert is
written by hand in troff, and the C library’s
reference documentation still uses the
venerable gtk-doc.
As part of the modernization effort, I have
turned the man page into a reStructuredText
document, and the C API documentation into
gi-docgen. This post describes how I did
that.
[...]
Gtk-doc assumed that magic happened somewhere
in developer.gnome.org to generate the
documentation and publish it. Gi-docgen
assumes that your project publishes it with
Gitlab pages.
Indeed, the new documentation is published
there — you can see how it is generated in
.gitlab-ci.yml. Note that there are two jobs:
the reference job generates gi-docgen’s HTML
in a public/Rsvg-2.0 directory, and the pages
job integrates it with the Rust API
documentation and publishes both together.
# § Python⠀➾
# ⚓ How_to_Get_Return_Code_from_Process_in_Python
Subprocess_Execution?⠀⇛
A process is a name for a running
program. Memory, lists of files, and a
program counter that takes account of
the instructions being implemented, and
a call stack that retains the local
variables are all part of each
process’s system state. A process
normally processes statements one after
the other in a single command flow
sequence known as the process’s main
thread. The program only does one thing
at any given moment. Our computer is
always running subprocesses. Every
action we take on our computer entails
activating a subprocess. Even if we are
constructing a basic “hello world”
program in Python. Even if you have
been programming for a while, you might
not know the concept of a subprocess.
The principles of the subprocess will
be covered in this article, as well as
how to use the Python subprocess
standard library.
# § Shell/Bash/Zsh/Ksh⠀➾
# ⚓ Leveling_the_playing_field_for_non-native
speakers⠀⇛
# § Rust⠀➾
# ⚓ Learn_Rust_in_2022_|_Opensource.com⠀⇛
If you’re going to explore Rust this
year, download our free Rust cheat
sheet, so you have a quick reference
for the basics.
# ⚓ This_Week_In_Rust:_This_Week_in_Rust_425⠀⇛
# § Java⠀➾
# ⚓ 8_Best_Free_and_Open_Source_Java_Object-
Relational_Mapping_Software_–_LinuxLinks⠀⇛
Object–relational mapping (ORM) is a
programming technique for converting
data between incompatible type systems
using object-oriented programming
languages. This creates, in effect, a
“virtual object database” that can be
used from within the programming
language.
In essence, ORM is a design pattern for
converting (wrapping) that data stored
within a relational database into an
object that can be used within an
object oriented language. It creates a
layer between the language and the
database, helping programmers work with
data without the OOP paradigm.
Compared to traditional techniques of
exchange between an object-oriented
language and a relational database, ORM
often reduces the amount of code that
needs to be written. It standardizes
interfaces reducing boilerplate and
speeding development time. Advocates of
ORMs claim they increase productivity,
improve application design, reuse code
and maintain the application over time.
On the other hand, ORM suffers the
disadvantage of the abstraction
obscuring what’s happening in the code.
And over-use of ORM software can
produce poorly designed databases.
There are a good range of ORM software
available. Here’s our recommendations
summarised in a legendary ratings
chart.
# ⚓ How_to_install_Oracle_Java_SE_17_on_Ubuntu
20.04_–⠀⇛
In todays guide, we are going to learn
how we can install Java SE 17 on Ubuntu
20.04. Java is widely used in programs
like Cassandra, Graylog, Wine etc.
Java delivers thousands of performance,
stability, and security updates that is
the reason why java is widely used and
has a larger community base worldwide.
* § Leftovers⠀➾
o ⚓ Lucille_Clifton_and_the_Task_of_Remembering⠀⇛
Lucille Clifton’s poetry has seeped into the public
consciousness. These days I can’t open Instagram without
seeing the telltale lines of won’t you celebrate with me
commemorating a birthday or an accomplishment, or some
other of Clifton’s poems fitting neatly into a white
square. The reissue of her 1976 memoir, Generations,
comes at a time of renewed interest in the poet’s life
and works. Her daughter Sidney is currently leading the
process of transforming the family’s home in Baltimore
into a community space to commemorate her parents’
artistry (her father, Fred, was a sculptor and
philosopher). Meanwhile, the novelist and poet Honorée
Fanonne Jeffers is at work on a biography. It is clear
that Clifton’s legacy continues to blossom.
o ⚓ Downstream_From_Del_Rio⠀⇛
Del Rio, which is across the Rio Grande from Ciudad
Acuña, Mexico, hosts a smaller border crossing than those
350 miles downriver in the lower Rio Grande valley and
those 400 miles upriver around El Paso. In early
September, thousands of Haitian and other Latin American
migrants began arriving and crossing the shallows of the
river to set up an improvised camp under a bridge. By
mid-month, the camp had grown to a maximum of some 15,000
people, without adequate water and sanitation. The
migrants were blocked from entering the town to buy food
and supplies, which forced them to cross the river to buy
them in Ciudad Acuña. Conditions in the encampment were
called “deplorable” by the United Nations.
On September 19, Border Patrol officers on horseback
tried to physically block families with children crossing
the river to bring supplies back to the camp, which had
previously been allowed. Videos of the aggressive use of
force against peaceful migrants went viral and provoked
widespread condemnation as an echo of historical racist
aggression against Black people. The Biden administration
disavowed the enforcement operation and initiated an
investigation, which is ongoing as of early January.
o ⚓ CEO_Of_$10_Billion_Open-Source_Company_Elastic_Steps_Down,
President_Leaves⠀⇛
Shay Banon, co-founder of the $10 billion open-source
company Elastic, who is known for being one of the most
outspoken critics of Amazon’s dominant position in the
cloud computing sector, is stepping down as CEO and
chairman of the company.
o ⚓ Cost_of_Attrition⠀⇛
How would we think about retention if we could visualise
the full impact of someone leaving our team?
Beware looking at teams on a spreadsheet. If you have a
hiring rate matching attrition rate it might look like
the team health is maintained. It’s probably not.
Tracking tenure by team and average tenure in team can be
interesting proxy indicators. Teams can be growing but
have dropping tenure.
o ⚓ Illuminating_Origami_Is_Just_Around_The_Corner_|_Hackaday⠀⇛
Pop-up greeting cards are about to get a whole lot more
interesting. Researchers at Seoul National University in
Korea have created glowing 3D objects with a series of
prototypes that fold thin QLED (Quantum Dot LED) sheets
like origami. They used a CO2 laser to etch “fold lines”
in the QLED so the sheets could be formed into 3D shapes.
The bends are actually rounded, but at 5μm they appear to
be sharp corners and the panels continue to illuminate
across the fold lines for at least 500 folds. Some glow
in solid colors, while others use smaller addressable
areas to create animated matrix displays of patterns and
letterforms. See the short video after the break, read
the Physics World article or to see all the prototypes
and dig into details of the full research paper in Nature
(freed from the paywall by SharedIt).
o § Science⠀➾
# ⚓ Melbourne-led_research_team_finds_new_way_to_build_quantum
devices⠀⇛
The new technique – developed by Professor David
Jamieson and co-authors from the University of New
South Wales, Helmholtz-Zentrum Dresden-Rossendorf,
Leibniz Institute of Surface Engineering, and RMIT
– embeds single atoms in silicon wafers, one-by-
one, mirroring methods used to build conventional
devices.
o § Education⠀➾
# ⚓ American_teacher_fired_from_Taiwan_cram_school_for_catching
COVID⠀⇛
One the reasons the school listed for terminating
him was that it had conducted “thorough research
about Covid cases and how it’s still very possible
for patients to relapse even after three weeks.” It
cited the case of a “girl who came back from Japan”
with COVID and claimed that after finishing 21 days
in quarantine, her PCR test came back positive
because she had “relapsed.”
o § Hardware⠀➾
# ⚓ Stout_Peristaltic_Pump_Fabricated_From_Scratch_|_Hackaday⠀⇛
The peristaltic pump is perhaps most well known for
its ability to pump fluids without the pump
mechanism coming into contact with the working
fluid. This is key for food-safe applications and
other situations where a pump could contaminate the
fluid. [Maciej Nowak] has built a great example of
such a pump, crafted out of aluminium from scratch.
The build video covers the machining process in
detail, showing how the aluminium body was
fabricated on the lathe before installing bearings
and a silicone hose. The pump shaft was then
fabricated, along with a set of brass rollers to
press along the tube, creating the pumping action.
The rollers were also lubricated in order to reduce
friction on the tubing. Powering the pump is a
small DC motor, sending drive via a small toothed
belt, giving the finished build quite an industrial
look.
o § Health/Nutrition/Agriculture⠀➾
# ⚓ Prevention_Centers_Save_Lives,_But_Only_Radical_Change_Will
End_Overdose_Crisis⠀⇛
# ⚓ Single-Payer_Bill_Leaps_First_Major_Hurdle_in_California⠀⇛
Single-payer advocates cheered Tuesday after
legislation in California to create a first-in-the-
nation universal healthcare system took a decisive
step forward.
“As the single-payer movement continues to gain
momentum, we signal to corporate interests that
enough is enough.”
# ⚓ Québec_PM_Vows_Health_Tax_for_the_Unvaxxed⠀⇛
As the Omicron-driven surge in coronavirus
infections strains their nation’s healthcare
resources, Canadian progressives are balancing
urgent public health concerns with respect for
civil liberties after the province of Québec said
Tuesday that it would begin levying fines on
residents who refuse Covid-19 vaccinations.
“I think the government has still not exhausted
other alternatives that are more equitable and more
fair.”
# ⚓ Bernie_Sanders_Demands_Refunds_for_Seniors_Hit_by_Medicare
Premium_Hike⠀⇛
# ⚓ How_the_Concerns_of_Teachers_Have_Been_Misrepresented_in
Omicron_Reporting⠀⇛
In-person return plans were disrupted at schools in
Milwaukee (Milwaukee Journal-Sentinel, 1/2/22) and
Atlanta (WXIA, 1/3/22). Chicago canceled school
after the Chicago Teachers Union “approved a labor
action to work remotely due to safety concerns as
Covid-19 and its Omicron variant surge in the city”
(WTTW, 1/4/22).
The Chicago labor dispute has drawn the most
eyeballs, as it is the third-largest US city, but
teacher unionists nationwide are indicating that
the Omicron surge is pushing school systems to
their breaking points. In San Francisco, the system
saw as many as 600 educators out of work, with the
union blasting the district for its severe deficit
in Covid-19 testing kits (Mission Local, 1/6/22).
The left caucus of New York City’s United
Federation of Teachers reported 10,000 student
absences and 2,000 staff absences, criticizing the
mayor for inadequate “baseline testing” (Twitter,
1/6/22), while dozens of New York state and city
lawmakers have demanded a remote option for the
city’s schools (Twitter, 1/6/22).
# ⚓ On_Medicare_Limiting_Coverage_for_Alzheimer’s_Drug⠀⇛
If the administration takes no action, Medicare
recipients will continue to see their biggest
premium increase in history, all because of
Biogen’s greed. That cannot be allowed to happen.
Beyond the incredibly high price, Aduhelm has not
been proven to be effective by the scientific
community. It was rejected for coverage by the
Veterans Health Administration and at least a half
a dozen private health insurance companies in the
United States, while 10 out of the 11 experts on
the Food and Drug Administration’s advisory council
voted against approval of the treatment.
# ⚓ Brownstone_Institute_embraces_its_inner_antivaxxer⠀⇛
The Great Barrington Declaration was published in
October 2020 by three scientists brought together
by the American Institute for Economic Research
(AIER), an antimask, anti-“lockdown,” anti-
(vaccine) mandate “free market” libertarian “think
tank.” AIER is but one of many such astroturf
groups that have been sowing doubt about collective
public health interventions to slow the spread of
COVID-19, but the Great Barrington Declaration was
among the most successful efforts by any of them,
at least when it comes to influencing the policies
of major governments. At the time, I characterized
the Declaration as eugenics (or at least eugenics-
adjacent), given that, in a time before vaccines
against COVID-19, it proposed, in essence, a “let
‘er rip” strategy for the coronavirus, at least to
let it rip through the “healthy” population (in
order to prevent economic damage) while using
“focused protection” to keep those at highest risk
of severe disease and death safe. Never mind that,
as I pointed out, it’s impossible to keep the
vulnerable safe when a deadly virus is spreading
unchecked through the rest of the population, and,
unsurprisingly, public health experts were very
much opposed to this strategy. Such a strategy was
thus nothing more than a big “screw you” to those
at the highest risk from the pandemic. Last year,
AIER begat the “spiritual child” of the Great
Barrington Declaration, a new think tank named the
Brownstone Institute founded by former AIER
Editorial Director Jeffery Tucker, who bragged
about being in the “room where it happened” as the
Great Barrington Declaration was drafted.
# ⚓ Gorsuch_Goes_Maskless_at_Supreme_Court,_Increasing
Colleagues’_COVID_Risk_Levels⠀⇛
# ⚓ Sanders_Reintroduces_‘N95_Mask_for_All’_Legislation⠀⇛
Sen. Bernie Sanders on Wednesday announced the
reintroduction of legislation to get free N95 masks
to all Americans to “prevent death and suffering”
as the coronavirus pandemic continues to rage amid
an explosive surge in cases driven by the highly
contagious Omicron variant.
The Masks for All Act, which has 15 co-sponsors in
the Senate including Democratic Sens. Elizabeth
Warren and Ed Markey of Massachusetts, would ensure
that all individuals—from college students to
medical professionals to the unhoused—would receive
a free package of three N95 respirator masks.
# ⚓ Sanders_Introduces_Bill_to_Send_N95s_to_Everyone_as_Biden
Considers_Mask_Plan⠀⇛
# ⚓ Biden_Under_Fire_for_Resisting_Calls_to_Distribute_N95
Masks_to_All⠀⇛
The Biden administration faced growing backlash
Wednesday for resisting calls from public health
experts and progressives to distribute N95 masks to
all U.S. households to help fight the Omicron wave,
which is pushing new coronavirus infections to
record-shattering highs nationwide.
An unnamed senior administration official sparked
outrage by claiming in an interview with Politico
that because “half the country won’t wear any
mask,” widespread distribution of high-quality face
coverings would be pointless.
# ⚓ Sanders_Demands_Refunds_for_Seniors_Hit_by_Medicare_Premium
Hike⠀⇛
Sen. Bernie Sanders on Tuesday demanded refunds for
seniors who have been hit by the 2022 Medicare
premium hike after federal health officials
recommended limiting the program’s coverage of
Aduhelm, the unproven and expensive Alzheimer’s
drug responsible for a large chunk of the premium
increase.
In a statement, Sanders said CMS officials’
preliminary decision Tuesday to restrict coverage
of Biogen’s Aduhelm to patients taking part in
approved clinical trials was “an important step
forward.” CMS’ final decision on the drug is
expected by April.
o § Integrity/Availability⠀➾
# § Proprietary⠀➾
# § Pseudo-Open Source⠀➾
# § Privatisation/Privateering⠀➾
# § Linux Foundation⠀➾
# ⚓ Please_Join_Us_In_The_January_2022
SPDX_Community_SBOM_DocFest⠀⇛
SPDX was designed for tools
to produce and consume SBOM
documents. A decade of
experience has shown us
that tools may interpret
fields differently – a file
may be a valid syntactic
SPDX SBOM, but different
tools may fill in different
values.
By coming together as a
community to examine the
output of multiple tools
and to compare/contrast the
results, we can refine the
guidance to tool vendors
and improve the robustness
of the ecosystem sharing
SPDX documents.
Historically, these events
were called Bake-offs, but
we’ve evolved them into
“DocFests.”
# § Security⠀➾
# § Fear, Uncertainty, Doubt/Fear-mongering/
Dramatisation⠀➾
# ⚓ TellYouThePass_ransomware_returns_as_a
cross-platform_Golang_threat [Ed: Microsoft
boosters’ site tries_to_associate_Go_with
what_people_could_in_theory_code_using_Go,
as_if_it’s_guilty_of_it]⠀⇛
Golang is a programming language
first adopted by malware authors
in 2019 due to its cross-platform
versatility.
# § Privacy/Surveillance⠀➾
# ⚓ Interview_With_Reinard_Mortlock_–_Livex⠀⇛
Reinard Mortlock: We wanted to
start a software company to
enable startups and SME’s to get
access to world-class software
development without the massive
development cost typically
charged by the software industry.
# ⚓ VICTORY:_Google_Releases_“disable_2g”
Feature_for_New_Android_Smartphones⠀⇛
What is 2G and why is it
vulnerable?2G is the second
generation of mobile
communications, created in 1991.
It’s an old technology from a
time when standards bodies did
not account for certain risk
scenarios such as rogue cell
towers and the need for strong
encryption. As years have gone
by, many vulnerabilities have
been discovered in 2G.
There are two main problems with
2G. First, it uses weak
encryption between the tower and
device that can be cracked in
real time by an attacker to
intercept calls or text messages.
In fact, the attacker can do this
passively without ever
transmitting a single packet. The
second problem with 2G is that
there is no authentication of the
tower to the phone, which means
that anyone can seamlessly
impersonate a real 2G tower and a
phone using the 2G protocol will
never be the wiser.
Cell-site simulators sometimes
work this way. They can exploit
security flaws in 2G in order to
intercept your communications.
Even though many of the security
flaws in 2G have been fixed in
4G, more advanced cell-site
simulators can downgrade your
connection to 2G, making your
phone susceptible to the above
attacks. This makes every user
vulnerable—from journalists and
activists to medical
professionals, government
officials, and even law
enforcement.
# ⚓ Livestreamed_Hearing_Friday:_EFF_Will_Ask
Court_to_Issue_Judgment_Against_SFPD_for
Illegally_Spying_on_Protesters_Marching_in
Support_of_Black_Lives⠀⇛
# ⚓ ABC_to_go_ahead_with_compulsory_iview
logins_despite_privacy_concerns⠀⇛
iview is a service that allows
viewers to see programs that have
already been broadcast, or in
some case which are yet to go to
air. All content provided is paid
for with taxpayer funds as the
ABC is a government-owned body.
Many commercial TV channels have
similar services, but they
require registration as this is
used to monetise the service.
An ABC spokesperson told iTWire
on the phone this morning that
the introduction of logins would
go ahead as had been indicated
last year, when the company
decided to postpone a plan to
introduce the measure on 1 July
2021.
# ⚓ Pegasus_attacks_in_El_Salvador:_spyware
used_to_target_journalists_and_activists_–
Access_Now⠀⇛
NSO Group’s Pegasus spyware has
been used to infect the devices
of 35 Salvadoran journalists and
activists between July 2020 and
November 2021. This breaking
information, initially flagged by
journalists who tested their
devices using Amnesty
International’s Mobile
Verification Toolkit, was
analyzed and corroborated by
Access Now’s Digital Security
Helpline, The Citizen Lab,
Amnesty International, Front Line
Defenders, Fundación Acceso, and
SocialTIC. Read The Citizen Lab’s
technical report.
# ⚓ Polish_Gov’t_Finally_Admits_It_Deployed
NSO_Malware,_Pretends_Targeting_Of
Opposition_Leaders_Isn’t_Abusive⠀⇛
Poland — like far too many
countries — has a Pegasus
problem. The highly intrusive
(and highly effective) phone
malware sold by Israel’s NSO
Group for the ostensible purpose
of tracking down terrorists and
other deadly criminals has been
observed (yet again) being
deployed to track government
critics and political opponents.
# ⚓ Exposed:_civil_society_condemns_use_of
Pegasus_in_El_Salvador_to_spy_on
journalists_and_activists_–_Access_Now⠀⇛
We, the undersigned
organizations, condemn the use of
NSO Group’s Pegasus technology in
El Salvador for the surveillance
of journalists and civil society,
as initially flagged by El Faro
and Gato Encerrado, and confirmed
through a joint investigation by
Access Now, Front Line Defenders,
The Citizen Lab, Amnesty
International, Fundación Acceso,
and SocialTIC. Although, to date,
it has not been established who
the perpetrator of this
surveillance is, NSO Group has
repeatedly claimed it only sells
Pegasus technology to
governments.
These attacks are particularly
alarming, as several of the
infections occurred after the
Pegasus Project revelations
became public in July of 2021,
indicating that those behind the
spyware attacks were aware of,
but ignored, the widespread
denouncement of Pegasus use,
including by international human
rights NGOs and UN experts and
officials.
# ⚓ Project_Torogoz:_Extensive_Hacking_of
Media_&_Civil_Society_in_El_Salvador_with
Pegasus_Spyware_–_The_Citizen_Lab⠀⇛
This report describes the results
of a collaborative investigation
into the abuse of NSO Group’s
Pegasus spyware to target members
of the press and civil society in
El Salvador. The investigation
led to the identification of 35
Pegasus-infected individuals (37
devices) among members of El
Salvador’s media and civil
society.
Our investigation began in
September 2021 when a group of
independent journalists contacted
Access Now’s Digital Security
Helpline after testing their
devices using the Amnesty
International Security Lab’s
Mobile Verification Toolkit (MVT)
tool to detect Pegasus spyware.
The resulting investigation was a
collaboration between the Citizen
Lab and Access Now, with
investigative assistance and case
referrals from Frontline
Defenders, SocialTIC, and
Fundación Acceso. We asked
Amnesty International’s Security
Lab to conduct an independent
review of our analysis for a
sample of cases, and they have
confirmed our findings.
o § Defence/Aggression⠀➾
# ⚓ The_“Credibility”_Factor_and_Biden’s_Foreign_Crises⠀⇛
Barely six months after the chaotic US withdrawal
from Afghanistan, President Joe Biden now confronts
three crises—in Ukraine, Taiwan, and Iran—that
could easily erupt into US military action abroad.
Biden would no doubt prefer to avoid such an
outcome, but he is under enormous pressure from
both Democrats and Republicans in Congress to
demonstrate “resolve” in these disputes, thereby
overcoming the catastrophic loss of “credibility”
supposedly suffered by Washington as a result of
the “debacle” in Afghanistan. How successful Biden
will prove in resisting this pressure will largely
determine whether this country will avoid being
dragged into another military quagmire—and one that
could prove far more deadly than the one in
Afghanistan.
# ⚓ Opinion_|_Social_Cohesion_Is_Vital,_and_We’re_Losing_It⠀⇛
The United States is tumbling toward socio-
political crisis. Here are just a few of the
distress signals recently visible:
# ⚓ What_If_Nuclear_Deterrence_Fails?⠀⇛
“It’s not absolutely foolproof, but it has
protected us all from nuclear war for 75 years.”
There is just one obvious problem with this
statement. In order for deterrence to work, it has
to be absolutely 100 percent foolproof. The
consequence of it being less than that is beyond
catastrophic. It could amount to the end of life on
earth as we know it. That’s one hell of a gamble.
And it’s a gamble that is not morally defensible on
any level. It’s one that should never be taken.
# ⚓ Belarusian_peacekeeping_insignia_adds_to_confusion_as_UN
criticizes_Kazakhstan_over_troops_wearing_blue_helmets⠀⇛
Kazakhstan has come under criticism from the United
Nations after troops deployed to protect strategic
infrastructure amid a crackdown on protests in
Almaty were seen wearing blue helmets reserved for
UN peacekeepers. According to UN spokesperson
Stéphane Dujarric, Kazakhstan’s Permanent Mission
to the United Nations has offered assurances that
the issue has been resolved. However, as reported
by RFE/RL’s Kazakh service, the presence of forces
from Belarus’s peacekeeping company has caused
further confusion, as their insignia closely
resembles the UN emblem.
# ⚓ Putin’s_trigger:_Ten_years_after_they_first_caused_the
Russian_authorities_to_clutch_their_pearls,_Pussy_Riot_has
been_almost_entirely_forced_out_of_the_country⠀⇛
In November 2021, the feminist protest group Pussy
Riot turned 10 years old. For the entirety of the
group’s existence, the Russian authorities (among
others) have been trying their damnedest to shut
them up. After staging a “punk prayer” in Moscow’s
Christ the Savior Cathedral in 2012, three Pussy
Riot activists were sentenced to two years in
prison. After a demonstration at the 2018 World
Cup, other Pussy Riot members, who ran onto the
field in police uniforms, were arrested — and the
group’s unofficial spokesman Pyotr Verzilov was
promptly poisoned. In the last two years, arrests
and prosecutions targeting Pussy Riot activists
have only become more frequent. Just last month,
members Maria Alyokhina and Lyusya Shtein went on
hunger strike while serving two-week stints in
jail. Meduza special correspondent Kristina
Safonova spoke with past and present members of the
group to find out who exactly they were in 2011 —
and who they are now.
# ⚓ Seyed_Mohammad_Marandi_on_the_Iran_Deal_and_the
Assassination_of_Soleimani⠀⇛
# ⚓ To_Avert_‘Global_Nuclear_Holocaust,’_US_Groups_Demand
Abolition_of_ICBMs⠀⇛
More than 60 U.S. organizations issued a joint
statement Wednesday calling for the total
elimination of the country’s land-based nuclear
missiles, warning that the weapons are both an
enormous waste of money and—most crucially—an
existential threat to humankind.
Organized by the advocacy groups RootsAction and
Just Foreign Policy, the statement argues that
intercontinental ballistic missiles (ICBMs) are
“uniquely dangerous, greatly increasing the chances
that a false alarm or miscalculation will result in
nuclear war.”
# ⚓ #Ethiopia_#TigrayGenocide_#ExcuseForRegimeChangeWar⠀⇛
# ⚓ Lethal_Autonomous_Weapons_Systems_and_the_Fight_to_Contain
Them⠀⇛
Science fiction? Not really. It could happen
tomorrow. The technology already exists.
In fact, lethal autonomous weapons systems (LAWS)
have a long history. During the spring of 1972, I
spent a few days occupying the physics building at
Columbia University in New York City. With a
hundred other students, I slept on the floor, ate
donated takeout food, and listened to Alan Ginsberg
when he showed up to honor us with some of his
extemporaneous poetry. I wrote leaflets then,
commandeering a Xerox machine to print them out.
# ⚓ Protest_Song_Of_The_Week:_‘Guardian_Angel_Platoon’⠀⇛
The Guardian Angel Platoon is the moniker of
Canadian veteran, activist, and singer-songwriter
Dennis MacKenzie. He released the self-titled album
in 2021, right before Canada’s Remembrance Day.
The album is a conceptual work that chronologically
charts MacKenzie’s journey as a soldier in
Afghanistan. It deals with sobering topics such as
PTSD, trauma during the war, and mistreatment
afterward. It also discusses overlooked issues in
connection with veterans.
# ⚓ FTA:_Fuck_the_Aggression⠀⇛
Vietnam was already an unpopular war by the time
Fonda went on a political jaunt to NVN. Still, she
gave a radio broadcast from Hanoi in August 1972
that some veterans, to this day, regard as
propaganda bordering on treason, and yet which I
find quite lovely and moving, accentuating a
people’s culture and humanity, de-demonizing them.
Of course, she gave more than one broadcast and one
speech. Here’s an excerpt of things she saw on her
tour of war:
Fonda’s more infamous speech describing the locals
as humans is sharp. Here is her speech presented
before Congress during hearings they held on her
travels to the North Country. (It’s preceded by
Congressional denunciations that provide a glimpse
at the poulter zeit geist.) It is full of
humanistic observations of the North Vietnamese
people.
# ⚓ ‘Not_an_easy_discussion’_Russian_and_NATO_officials_say
they_are_far_from_agreement_after_talks_in_Brussels⠀⇛
Russian and NATO officials convened for talks at
the alliance’s headquarters in Brussels on
Wednesday, January 12. The meeting came two days
after diplomats from Russia and the U.S. held
similar talks in Geneva. Both discussions were
largely inconclusive. Washington and NATO are
attempting to push back against sweeping security
proposals put forward by Russia in December, while
also trying to deter Moscow from launching a full-
fledged attack on Ukraine. Speaking to the press
after Wednesday’s talks, Russian Deputy Foreign
Minister Alexander Grushko and NATO Secretary
General Jens Stoltenberg said the two sides remain
divided on fundamental issues. Meduza summarizes
their comments here.
# ⚓ Swing_State_Trumpers_Forged_Electoral_Letters_in
Harebrained_Scheme_to_Overturn_Biden’s_Win⠀⇛
Pro-Trump groups in at least five states sent the
government forged certificates of ascertainment
declaring Trump the recipient of the state’s 2020
electors. MSNBC’s Rachel Maddow reported on the
falsified documents Tuesday night, noting that the
fake certifications, which were obtained by
watchdog group American Oversight, have “almost the
exact same wording” to the real documents.
# ⚓ Opinion_|_The_Overthrow_of_American_Democracy:_A_Scorecard
for_Trump’s_Next_Coup⠀⇛
“Liberty lies in the hearts of men and women. When
it dies there, no constitution, no law, no court
can do much to help it…The freedom to do as one
likes leads straight to its overthrow.” —Judge
Learned Hand, 1944
# ⚓ Revealed:_The_Billionaires_Funding_the_Coup’s_Brain_Trust⠀⇛
The Claremont Institute, once a little-known think
tank often confused with the liberal-arts college
of the same name, has emerged as a driving force in
the conservative movement’s crusade to use bogus
fraud claims about the 2020 election to rewrite
voting laws and remake the election system in time
for the 2022 midterms and 2024 presidential
election. Most infamously, one of the group’s legal
scholars crafted memos outlining a plan for how
then-Vice President Mike Pence could potentially
overturn the last election.
# ⚓ Drone_drops_explosives_on_civilians’_camp_in_Michoacán—and
films_the_attack⠀⇛
A video that shows explosives being dropped on a
civilian encampment in a forest in Michoacán has
been posted to social media, one of multiple
attacks on civilians in the Tierra Caliente
municipality of Tepalcatepec on Monday.
The footage was filmed by a drone from which the
explosives were believed dropped by members of the
Jalisco New Generation Cartel (CJNG), who were
allegedly operating the unmanned aerial vehicle.
o § Transparency/Investigative Reporting⠀➾
# ⚓ Trump_Can’t_Take_the_Heat,_Hangs_Up_on_NPR_When_Host_Pushes
Back_on_Election_Lies⠀⇛
Inskeep began to ask another question, but he
realized mid-sentence that Trump had already hung
up.
# ⚓ Pressed_on_his_election_lies,_former_President_Trump_cuts
NPR_interview_short⠀⇛
The interview was six years in the making. Trump
and his team have repeatedly declined interviews
with NPR until Tuesday, when he called in from his
home in Florida. It was scheduled for 15 minutes,
but lasted just over nine.
After being pressed about his repeated lies about
the 2020 presidential election, Trump abruptly
ended the interview.
o § Environment⠀➾
# ⚓ Environmental_Justice_Activists_Want_NJ_Gov._to_Vote_No_New
Gas-Fired_Power_Plant_in_Newark⠀⇛
In Newark, New Jersey, residents of the largely
Black and Latinx community of Ironbound are calling
on Governor Phil Murphy to stop plans to build a
$180 million gas-fired power plant that could
worsen the poor local air quality and exacerbate
the climate crisis. As the Passaic Valley Sewerage
Commission holds a vote to begin construction on
Thursday, activists are urging the governor to
enforce the environmental justice law that he
passed last year. “If we don’t set a good precedent
for New Jersey, what does that mean for the country
and other states that are trying to pass similar
laws?” says Maria Lopez-Nuñez, member of the White
House Environmental Justice Advisory Council.
# ⚓ It’s_Time_to_Hold_Law_Firms_Accountable_for_Their_Role_in
Climate_Change⠀⇛
For too long, law firms have been given carte
blanche for their contributions to the climate
crisis: They lobby on behalf of the fossil fuel
industry, file the paperwork necessary for carbon-
emitting projects, and litigate cases against
indigenous and frontline communities. With over
1,500 lawyers in offices around the world, Gibson,
Dunn, & Crutcher LLP is regarded as one of the top
law firms in the United States. Like many of its
peers, Gibson Dunn profits from squashing class
action lawsuits and labor organizing drives,
keeping shareholders from reforming corporate
practices and debtors from getting their day in
court, and shielding US companies from
accountability for their actions overseas and from
regulation at home. Even by the standards of the
legal industry, however, Gibson Dunn’s behavior is
notorious. In 2007, the Montana Supreme Court
rebuked the firm for engaging in “actual malice”
and “legal thuggery,” and a Delaware judge recently
described its pretrial practices as constituting
“fraud.”
# ⚓ Opinion_|_How_Corporate_Greed_Fuels_Killer_Tornadoes⠀⇛
In its ranking of business values, corporate
America proudly provides a special place for
elevated moral behavior. That place is the trash
can.
# ⚓ Why_Words_Matter_in_the_Fight_Against_Climate_Change⠀⇛
# ⚓ ‘Don’t_Look_Up’:_Hollywood’s_Primer_on_Climate_Denial
Illustrates_5_Myths_That_Fuel_Rejection_of_Science⠀⇛
By Gale Sinatra, University of Southern California
and Barbara K. Hofer, Middlebury. This article is
republished from The Conversation under a Creative
Commons license. Read the original article.
Every disaster movie seems to open with a scientist
being ignored. “Don’t Look Up” is no exception – in
fact, people ignoring or flat out denying
scientific evidence is the point.
# ⚓ Climate_change:_thawing_permafrost_a_triple-threat⠀⇛
Another study warns that methane and CO2 escaping
from long-frozen soil could accelerate warming and
overwhelm global efforts to cap the rise in Earth’s
temperature at livable levels.
Exposure of highly combustible organic matter no
longer locked away by ice is also fuelling
unprecedented wildfires, making permafrost a triple
threat, the studies report.
# § Energy⠀➾
# ⚓ Peak_period_[cryptocurrency]_mining_makes_up_1
percent_of_all_electricity_consumption⠀⇛
Nordcoin Mining manager Hermes Brambat told
ERR that their consumption is close to 1.5 MW
and there could be some 10 companies in
Estonia with comparable consumption.
Estonia’s total peak consumption in a colder
winter month is close to 1,500 MW, meaning
cryptocurrency mining makes up a percentage
point of all consumption at peak periods.
# ⚓ Jack_Dorsey_Launches_Bitcoin_Legal_Defense_Fund_to
Protect_Open_Source_Developers_–_Bitcoin_News⠀⇛
The letter explains that interested parties
with questions or concerns can email the fund
team and mentions the email domain
“bitcoindefensefund.org.” The site appears to
be currently under construction, with a
message from domain host Namebright stating
that the site is “coming soon.”Of course, the
Bitcoin Legal Defense Fund became trending
topics on social media After the open letter
was published.
# ⚓ Used_To_Free_Electricity,_Kosovo’s_Bitcoin_Miners_Are
Now_Facing_Difficult_Times_After_Ban⠀⇛
Energy prices have soared across Europe amid
a spike in demand for natural gas as
economies recover from the COVID-19 pandemic
and fresh tensions with Russia, which
supplies one third of Europe’s gas.
# ⚓ Will_The_CW_Be_a_Streaming_Wars_Casualty?⠀⇛
National broadcast TV networks don’t go on
the market very often. So when the news broke
that ViacomCBS and WarnerMedia were shopping
a majority stake in The CW network, and that
local TV giant Nexstar was the lead bidder,
eyebrows were instantly raised. The deal,
assuming it goes through (one source familiar
with the talks says that while they were
advanced, they could fall apart), would
reshape the network TV landscape at a time
when the very idea of what network TV should
be is in question.
# § Overpopulation⠀➾
# ⚓ Natural_population_growth_in_minus_by_5,300_people⠀⇛
Last year, 13,138 people were born and 18,445
people died in Estonia, which means that
there’s negative natural population growth of
5,307 people, the Ministry of Interior
announced.
o § Finance⠀➾
# ⚓ More_Than_8,000_Kroger_Grocery_Workers_Strike_in_Colorado⠀⇛
On the heels of a new report showing significant
financial insecurity, including homelessness, among
workers at Kroger grocery stores, more than 8,000
of the chain’s employees in Colorado went on strike
Wednesday to demand fair wages and better
healthcare benefits.
“The companies were thriving, but our workers
didn’t thrive. Know what our workers got? Covid.
Attacked. Beat up. Spit on. Slapped. Overworked.
And the company? They did great.”
# ⚓ Thousands_of_Workers_at_Kroger-Owned_Grocery_Stores_in
Colorado_Are_on_Strike⠀⇛
# ⚓ Report_Debunks_Manchin’s_Inflation_Argument_Against_Build
Back_Better⠀⇛
To justify obstructing one of his party’s top
legislative priorities, Democratic Sen. Joe Manchin
of West Virginia has repeatedly claimed that the
Build Back Better Act would exacerbate rising
inflation.
“The House-passed Build Back Better Act would make
crucial investments to lower inflation and cut
household costs.”
# ⚓ Manchin_Signals_Unwillingness_Toward_Backing_Biden’s_Calls
for_Filibuster_Reform⠀⇛
# ⚓ Job_Growth_Under_Biden_and_Trump⠀⇛
It’s not unusual for there to be substantial
differences between the surveys, but these are
extraordinary. In the last months, the household
survey has shown an increase in employment of
1,741,000. By comparison, the increase in jobs in
the establishment survey has been just 448,000.
While these divergences are striking, they largely
disappear if we look over a longer period. Over the
last year, the household survey shows employment is
up by 6,092,000. The establishment survey shows a
gain of 6,448,000 jobs.
# ⚓ The_Nonprofit_College_That_Spends_More_on_Marketing_Than
Financial_Aid⠀⇛
Baker College sells itself as a place where
students thrive and lives are transformed: “a haven
for those who dream big.”
From humble beginnings as a small business school
in Flint, Baker rose to become the largest private
college in Michigan, forging a presence in online
learning and in Michigan towns where many students
thought a college degree was beyond their grasp.
For decades, the school’s marketing touted low
costs and employment rates of nearly 100% for job-
seeking graduates — making the dream seem both
affordable and achievable.
o § AstroTurf/Lobbying/Politics⠀➾
# ⚓ Opinion_|_The_Perilous_Condition_of_American_Democracy_at
the_Hands_of_the_Republican_Party⠀⇛
New York Times columnist David Brooks has
breathlessly pronounced to his millions of readers
Jan. 7 a crucial new discovery about the rising
threat to American democracy, which he says, has
been unfortunately distorted by election watchdogs,
Democrats, and major media.
# ⚓ Ron_Johnson_Breaks_His_Term-Limit_Pledge_in_Order_to_Keep
Serving_the_Billionaire_Class⠀⇛
Ron Johnson, the Wisconsin senator who has caused
jaws to drop by promoting Covid vaccine skepticism
while at the same time suggesting that gargling
with mouthwash might help beat the virus, is
running for reelection after pledging to quit at
the end of his current term.
# ⚓ How_the_Left_Alienates_Jews⠀⇛
Eons ago—in 2019—Linda Sarsour, Tamika Mallory, and
Bob Bland stepped down from their leadership
positions on the Women’s March board after a series
of self-inflicted wounds. Aside from the widespread
mismanagement that starved state chapters of
funding and alienated them over trademark wars, the
leadership’s failure to grapple with its own anti-
Semitism (i.e. cozying up to Louis Farrakhan then
offering the weakest possible denunciation of his
racist, homophobic vitriol under the guise of
intersectionality) exposed a gaping ignorance that
many, especially Jewish women, simply could not
abide.1
# ⚓ Five_Starbucks_Locations_Have_Filed_to_Unionize_Over_the
Past_Six_Days⠀⇛
# ⚓ Republicans_in_5_States_Forged_Electoral_College_Documents,
MSNBC_Host_Says⠀⇛
# ⚓ When_Under_Pressure,_Tories_Go_“Anti-Woke”⠀⇛
For one of the richest countries in the world— and
one that is much less populated than US, Brazil,
India, Russia, and Mexico— these numbers make
dismal reading.
The pressure caused by the Omicron variant on
hospitals is not so much on the uptake of ICU beds
(it causes less serious illness than preceding
variants), but because so many staff are sick and
unable to work as a result of the much more
contagious Omicron variant.
# ⚓ Progressives_to_Clinton_and_Other_Corporate_Democrats:
‘Back_Off’_on_Election_Advice⠀⇛
Progressives on Wednesday dismissed arguments from
corporate Democrats that the party should avoid
“going too far left”—despite mounting evidence that
voters are in desperate need of—and demanding—bold,
far-reaching policies and social programs.
An article published Wednesday at The Hill quoted a
recent NBC News interview with former Democratic
presidential candidate Hillary Clinton, who lost
the 2016 presidential election to former President
Donald Trump.
# ⚓ AIPAC_Goes_PAC_and_SuperPAC_to_Cover_its_Tracks_as_it
Targets_Progressives⠀⇛
The American Israel Public Affairs Committee
(AIPAC), the leading organization of the Israel
lobby in the United States, launched two political
action committees (PACs) last month in a move
largely seen as an attempt to retain control amid a
political climate becoming increasingly more
critical of Israel.
# ⚓ Timeline_of_Filibuster_Helps_Explain_Why_So_Many_Say_It_Now
Needs_to_Go⠀⇛
A new timeline documenting the history of the
Senate filibuster shows how the rule, which now
requires a 60-vote supermajority to advance most
legislation, has been used to protect ruling-class
interests for over two centuries and makes the case
that the future of democracy in the U.S. depends on
reforming it.
“The Senate MUST end the filibuster and pass voting
rights legislation. Our democracy depends on it.”
# ⚓ What’s_Not_to_Like_About_Ike⠀⇛
Although the 50s were generally a somnambulant era,
there were seismic forces below and above the
surface that would explode in the decade of the
1960s. The 50s weren’t as staid as they seemed,
with a mass movement for civil rights that
accompanied Brown V. Board of Education and the
Birmingham bus boycott. Closer to home, and unknown
to me, or to the vast majority of others, were the
ongoing nonviolent protests at the nuclear
submarine base and manufacturing facility in New
London, Connecticut. There were people going to
federal prison during that time, including women,
who far outpaced the nascent feminist movement in
radicalism and nonviolence. The peace collective
that protested in New London was located in nearby
Voluntown, Connecticut, which would become the
scene of a violent encounter from an armed right-
wing group. How much do times actually change?
The late David Halberstam’s The Fifties (1993) is
a good place to start for a sweeping view of the
decade of the 1950s and the change it presaged.
# ⚓ Opinion_|_Biden_Is_Calling_for_Urgent_Action_on_Voting
Rights—Will_Congress_Listen?⠀⇛
President Joe Biden’s recent visit to Atlanta,
Martin Luther King, Jr.’s hometown, focused
national attention on a somber fact: the legacy of
the civil rights movement is threatened by recent
and ongoing attacks on voting rights.
# ⚓ ‘Victory’:_Ohio_Supreme_Court_Strikes_Down_GOP_Partisan
Gerrymandering⠀⇛
Democracy defenders on Wednesday cheered a ruling
by the Ohio Supreme Court that invalidated
Republican-drawn state legislative district maps,
which a majority of the justices found were
unconstitutionally gerrymandered against the will
of the state’s voters.
“The General Assembly maps entrenched a GOP
supermajority and flouted clear partisan fairness
requirements in the Ohio constitution.”
# ⚓ Statues_Down!⠀⇛
# ⚓ Although_Democrats_May_Benefit_From_Redistricting,_Midterm
Outlook_Remains_Grim⠀⇛
# ⚓ Lani_Guinier_Taught_Me_Almost_Everything_I_Know_About
Voting_Rights⠀⇛
Harvard Law professor and icon Lani Guinier passed
away on Friday at the age of 71. When I heard the
news, I was reminded of a line from Macbeth: “She
should have died hereafter; there would have been
time for such a word”—not because I’m a budding
authoritarian, but because the line reflects a
sadness that grief over a momentous passing must be
tabled due to an upcoming battle.
# ⚓ Activists_Say_Greed,_Neglect_Are_to_Blame_for_Bronx_Fire
That_Killed_17_People⠀⇛
# ⚓ As_Officials_Blame_Tenants_After_17_Die_in_Bronx_Fire,
Activists_Say_Greed_&_Neglect_Are_to_Blame⠀⇛
A massive fire in an apartment building in the
Bronx, New York, killed 17 people, including eight
children, on Sunday. The city is blaming the fire
on a malfunctioning space heater. Housing advocates
say the real issue is the lack of safe, affordable
public housing, citing lack of heat provided by the
building during subzero winter temperatures and
poor fire safety systems. Tenants and activists
note one of the building’s co-owners is a member of
Mayor Eric Adams’s transition team, and are
demanding an extension to the eviction moratorium
set to expire on January 15. “All of them are
really asking for accountability, not just from the
state and city agencies but first and foremost from
their landlord and the building owners,” says
reporter Claudia Irizarry Aponte, who covers the
Bronx for the nonprofit newsroom The City.
# ⚓ Joe_Biden_Delivers_the_Speech,_and_the_Fire,_on_Voting
Rights_He_Should_Have_Brought_Last_July⠀⇛
“I’m tired of being quiet!” President Joe Biden
told a crowd at Atlanta University Center, which
unites the city’s four historically Black
universities, on Tuesday afternoon, in a speech
that was supposed to represent Democrats’ new push
for federal voting rights legislation. It raised
the question: Who’s been keeping him quiet on
voting rights?
# ⚓ Biden_Backs_Filibuster_Reform_to_Pass_Voting_Rights_Bills
After_Sustained_Grassroots_Pressure⠀⇛
We go to Atlanta, Georgia, where President Biden
and Vice President Harris spoke on Tuesday to
pressure Congress to pass critical voting rights
legislation. Biden endorsed changing the Senate
rules to prevent a minority of senators from
filibustering the bills. We speak to two leaders in
the voting rights movement about the importance of
passing the bills, particularly for people of
color. “Right now 40 senators can stop 100 senators
from having a vote, and that is absolutely unheard
of anywhere else in our democracy,” says Ben
Jealous, who attended Biden’s speech and is
president of People for the American Way and former
president of the NAACP. Biden should prioritize
voting rights and “follow up the speech yesterday
with actions,” says Cliff Albright, co-founder and
executive director of Black Voters Matter, who
boycotted Biden’s address.
# ⚓ Nigeria_lifts_its_ban_on_Twitter_after_7_months⠀⇛
The Nigerian government has lifted its ban on
Twitter in the West African country, seven months
after the country’s more than 200 million people
were shut out of the social media network.
Nigerian President Muhammadu Buhari directed that
Twitter’s operations will resume in the country on
Thursday, according to the director-general of the
country’s National Information Technology
Development Agency. Kashifu Inuwa Abdullahi said
that was only after Twitter agreed to meet some
conditions, including opening an office in Nigeria.
o § Misinformation/Disinformation⠀➾
# ⚓ Fact-checkers_urge_YouTube_to_fight_disinformation⠀⇛
Videos containing false information had gone “under
the radar of YouTube’s policies, especially in non-
English speaking countries”, they said in an open
letter to YouTube chief Susan Wojcicki.
o § Censorship/Free Speech⠀➾
# ⚓ Rachel_Riley_Libel_Ruling_is_the_Latest_Judicial_Attack_on
Political_Speech⠀⇛
Assaults on the right to speak, and to think, are
coming thick and fast. But the significance – and
coordination – of all these developments is being
largely missed.
# ⚓ Content_Moderation_Case_Study:_Facebook_Knew_About
Deceptive_Advertising_Practices_By_A_Group_That_Was_Later
Banned_For_Operating_A_Troll_Farm_(2018-2020)⠀⇛
o § Civil Rights/Policing⠀➾
# ⚓ Florida_GOP_Bill_Proposes_‘Cruel_and_Dangerous’_15-Week
Abortion_Ban⠀⇛
Progressives on Wednesday condemned a bill
introduced in Florida’s GOP-controlled Legislature
that would ban abortions after 15 weeks—with no
exceptions for incest or rape—as the latest salvo
in Republicans’ nationwide attack on reproductive
rights.
“There is nothing ‘reasonable’ with controlling
decisions about my pregnancy.”
# ⚓ New_Illinois_Law_Says_Cops_Need_A_Warrant_To_Grab_Data_From
(Some)_Third_Parties⠀⇛
The state of Illinois continues to provide more
protection than the US Constitution. Its privacy
laws exceed what has been determined to be
“reasonable” violations of privacy by decades of
court precedent. This has allowed it to go after
companies for violating state laws, even when the
collections being prosecuted would likely be legal
under the Supreme Court-created “Third Party
Doctrine.”
# ⚓ Opinion_|_Remembering_Dr._King’s_Message_of_Hope_in_These
Dark_Times⠀⇛
2022 has begun with melancholy, as our country sees
the pandemic reach new heights. Meanwhile our
crises of climate, democracy, and inequality seem
more entrenched than ever.
# ⚓ The_Re-Assassination_of_Martin_Luther_King,_Jr.⠀⇛
A year to the day before he was assassinated Martin
Luther King Jr., a Baptist pastor, publicly defined
the war in Vietnam as a civil rights issue on April
4, 1967, in an address titled Beyond Vietnam: A
Time to Break Silence to a meeting of Clergy and
Laity Concerned about Vietnam at Riverside Church
in New York City. In doing so, King uttered the
following prescient statement.
The war in Vietnam is but a symptom of a far deeper
malady within the American spirit, and if we ignore
this sobering reality we will find ourselves
organizing clergy-and laymen-concerned committees
for the next generation. … In 1957 a sensitive
American official overseas said that it seemed to
him that our nation was on the wrong side of a
world revolution. … I am convinced that if we are
to get on the right side of the world revolution,
we as a nation must undergo a radical revolution of
values. We must rapidly begin the shift from a
“thing-oriented” society to a “person-oriented”
society. When machines and computers, profit
motives and property rights are considered more
important than people, the giant triplets of
racism, materialism, and militarism are incapable
of being conquered.
# ⚓ Child_Porn_Probe_of_Billionaire_Businessman_Denny_Sanford
Continues_at_State_and_Federal_Level,_Court_Records_Show_—
ProPublica⠀⇛
Federal and state authorities are still actively
investigating billionaire T. Denny Sanford for
possession of child pornography, according to new
court records.
In 2020, ProPublica first reported that South
Dakota authorities had started investigating the
state’s richest man and had referred the matter to
the U.S. Department of Justice. But it was not
clear what the DOJ did with the referral or whether
state investigators were still pursuing the case.
# ⚓ Yazidis_Laud_France,_Sweden_for_Launching_Joint_Probe_to
Prosecute_IS_Fighters⠀⇛
The two European countries formed a joint
investigation team last week to look into crimes
against humanity and war crimes committed against
Yazidis by foreign militants linked to IS during
the group’s ruthless rule over parts of Iraq and
Syria.
French and Swedish investigation efforts are being
coordinated by the European Union Agency for
Criminal Justice Cooperation (Eurojust). The group
said the joint team seeks to organize those efforts
and enable information and evidence to be shared
more effectively.
# ⚓ A_year_on,_has_Trump_benefited_from_a_Twitter_ban?⠀⇛
In theory Mr Trump will be allowed back onto
Facebook in a year’s time, on 7 January 2023 to be
exact.
o § Internet Policy/Net Neutrality⠀➾
# ⚓ FCC_Politely_Tells_ISPs_To_Stop_Abusing_Covid_Broadband
Relief_Program_To_Rip_Off_Poor_People⠀⇛
During the COVID crisis the FCC launched the
Emergency Broadband Benefit (EBB program), which
gives lower income Americans a $50 ($75 for those
in tribal lands) discount off of their broadband
bill. Under the program, the government gives money
to ISPs (not exactly ideal given the industry’s
history of fraud), which then dole out discounts to
users if they qualify. But (surprise), many found
that big ISPs erected cumbersome barriers to
actually getting the service, or worse, actively
exploited the sign up process to force struggling
low-income applicants on to more expensive plans
once the initial contract ended. Very on brand.
# ⚓ Internet_Shutdown_Rules:_Gauhati_HC_on_IFF’s_application⠀⇛
IFF filed an intervention application in
proceedings challenging the constitutionality of
the Temporary Suspension of Telecom Services
(Public Emergency or Public Safety) Rules, 2017
issued under the Indian Telegraph Act, 1885. These
rules empower the Union and State Governments to
suspend internet services. But these rules stand on
a tenuous legal footing – they confer unbridled
powers to governments, and they are beyond the
scope, ambit and intent of Sections 5(2) and 7 of
the Telegraph Act. On 23rd December 2021, the
Gauhati High Court has agreed to hear our
application after listening to submissions from our
counsel, Mr Anubhab Atreya.
[...]
In June 2020, IFF filed an intervention application
in the proceedings initiated by Mr Bhuyan, to
assist the Gauhati High Court in determining the
constitutionality of the 2017 Rules. In the
intervention application, we pointed out, firstly,
that the constitutional validity of the 2017 Rules
has not been considered by the courts. In fact, the
Supreme Court in Anuradha Bhasin vs Union of India,
(2020) 3 SCC 637 interpreted the 2017 Rules but
stated that it was not concerned with its
constitutionality since the parties therein had not
canvassed arguments on the same (Paragraph 84).
Secondly, we provided details regarding IFF and
stated that one of the core objectives of IFF was
to advocate and defend freedom of speech and
expression, and access to information in the
digital era. We also listed the cases where IFF had
provided legal assistance to other individuals/
organizations and those where IFF had previously
intervened. We provided these details to the
Gauhati High Court to demonstrate that IFF has
long-standing expertise in freedom of speech, and
has responsibly engaged with authorities on the
issue of internet shutdowns.
o § Monopolies⠀➾
# ⚓ ‘Major_Win’:_Judge_Says_Suit_to_Break_Up_Facebook_Empire
Can_Proceed⠀⇛
A federal judge ruled Tuesday that the Federal
Trade Commission’s revised antitrust lawsuit
against Meta Platforms, the parent company of
Facebook, can move forward—a potentially
significant blow to the social media empire, which
sought to have the case dismissed.
In an amended complaint filed last August, the FTC
provided additional data and stronger details to
back up its allegations that Facebook has
maintained a monopoly on social networking services
for the past decade by “illegally acquiring
innovative competitors and burying successful app
developers.”
# ⚓ FTC’s_Second_Antitrust_Attempt_Against_Facebook_Gets_Past
The_First_Hurdle⠀⇛
As you’ll recall, at the end of 2020, the FTC filed
an antitrust case against Facebook. Last summer,
the district court dismissed the case, noting that
the complaint was “legally insufficient,” and
didn’t really back up its central claims. Based on
that, the FTC went back to the drawing board and
filed an amended complaint last August. As we
noted, the amended complaint was better than the
first one — which was heavy on narrative, but
little on support to back it up. The amended
complaint had more in it, though we still felt that
the market definition was odd, and some of the
complaint seemed to undermine other parts of it.
# ⚓ The_World_Handled_A_‘Wordle’_Ripoff_Just_Fine_Without_Any
IP_Action⠀⇛
In the video game space, it has become commonplace
to see creators freak out over “rip-offs” and
“clones” of their games when the targets of their
ire are actually not rip-offs or clones at all.
This typically comes down to the all to common
confusion over whether you can own or protect ideas
versus specific expression. Typically in these
stories, it turns out someone is complaining that
they’re seeing a similar idea in other games,
whether it’s first person shooters that share
common features, the explosion of battle royale
games, or even just artwork.
# ⚓ Facebook_Objection_Dismissed,_Glo_Fiber_Expanding,_Utopia’s
Timmerman_Advocates_Gigi_Sohn⠀⇛
Facebook’s attempts to convince the court to
dismiss the Federal Trade Commission’s
anticompetition case against it have been rejected
by D.C. District Court Judge James Boasberg, which
advocacy group Public Knowledge said in a Tuesday
press release is “great news.”
Facebook, now called Meta, filed a complaint in
October asking the court to dismiss the case that
alleges the company is a monopoly power that
controls over 60 percent of the “person social
networking services” market. But the court
effectively ruled that there is evidence that can
move the case forward against the company.
# ⚓ Judge_says_the_FTC’s_Meta_monopoly_lawsuit_can_go_forward⠀⇛
The FTC suit against Meta is one of several US
government efforts to curtail the monopoly power of
major tech companies, including a Department of
Justice lawsuit against Google proceeding under
antitrust chief Jonathan Kanter.
# § Copyrights⠀➾
# ⚓ GitHub_Takes_Down_“Widevine_Dump”_Forks_Following_MPA
Complaint⠀⇛
The Motion Picture Association has asked
GitHub to remove a collection of scripts that
allow people to rip content from popular
streaming services such as Netflix, Disney+,
and Amazon Prime. The tools in question
bypass the Widevine copy protection,
violating the DMCA, the group argues.
Hundreds of forks of the “Widevine Dump” code
were also targeted and removed by GitHub.
# ⚓ PrimeWire_Down:_Streaming_Site_Prepares_To_Counter
Domain_Seizures⠀⇛
After being targeted in a lawsuit filed by
Hollywood and Netflix, pirate streaming site
PrimeWire appears to be digging in for the
long haul. In preparation for imminent domain
seizures, the site is now advertising a new
service that will provide up-to-date
information on where the official platform
can be accessed in the future.
# ⚓ Open_Minds_Podcast:_Hessel_van_Oorschot_of_Tribe_of
Noise_&_Free_Music_Archive⠀⇛
In this episode, CC’s Ony Anukem sits down
for a conversation with Hessel van Oorschot,
founder and “Chief of Noise” of the online
music business Tribe of Noise. Tribe of Noise
is a music community that connects artists,
fans, and professionals. Founded in 2008 in
The Netherlands, its main objective is to
create fair and sustainable business
opportunities for talented artists.
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