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
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QmfH2by4K6HCiZyvyGQ1L5syH6ZSfcHH12ph6TDo82aovk
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QmeizGYhw45pUpaYcuwBfE8NWgRhpFbkNtg8fZ7tGbAMXw
QmfKHqGowjRxdQqskGCLaTJvfCGFi19QHkyd8kwa9PMJ21
QmP9Us6BzAW1DtJJVXeJt8AyhFyfNcNAfWtLTvSMzp6Ltp
QmRocUr84sVRrmmhApYkrairZ9gM3urZnz5pkw4h6pxA9b
QmQ1r9n2zgVqwtQ4bsQLNP2kbfnhn22xRMWVDD5qUuwkKF
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QmSRxd1L1Tjg5YFR28UqaVozsZZh3xMVhvDKvnfHxuctNe
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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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣱⣏⣾⣮⣿⣟⣹⣿⣵⣿⣿⣿⣎⣇⣻⣵⣿⣏⣾⣎⣷⣍⣾⣋⣇⣏⣾⣜⣟⣿⣿⣉⣷⣽⣿⣙⣿⣩⣾⣛⣷⣹⣾⣱⣇⣿⣿⣟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⡻⢽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⡿⢿⣿⢿⡿⣿⡿⣿⣿⣿⠿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣻⡿⣟⢿⡟⣿⣿⣿⠿⣿⣿⣿⠻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣵⣼⣿⣿⣽⣮⣿⣽⣽⣾⣿⣿⣿⣮⣷⣽⣵⣾⣿⣽⣭⣯⣿⣿⣮⣯⣾⣿⣿⣯⣶⣽⣿⣿⣷⣽⣿⣿⣿⣽⣾⣯⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣯⣏⢽⣿⡟⣿⢻⢻⣯⡏⡏⣽⣽⢻⢻⣯⣿⣯⣿⡏⣯⣽⣹⣿⣿⢸⢹⣏⢽⡇⡯⣏⢿⣿⢩⡏⡏⣯⣽⣽⢨⠛⣟⣽⣿⢻⢹⢸⣿⢻⢏⣯⣽⢩⢘⣽⢻⣯⡏⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡏⢏⡏⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢽⡋⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣯⣽⣏⣝⣏⣽⣿⣻⣿⣿⣿⡏⣯⣿⣿⣿⣿⣿⣏⣇⣿⣿⣾⣿⣁⣽⣧⣿⣸⣿⣿⣿⣼⣿⣝⣿⡏⣿⣹⣏⣽⣿⣹⡿⣹⣿⣽⣿⣹⢸⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣚⣅⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣟⠿⡻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣹⣯⣽⣉⣏⣯⣽⣽⣿⣿⣿⣯⣻⣇⣿⣩⣏⣏⣿⣏⣯⣯⣿⣿⣿⣻⣿⡏⣯⣿⣿⣾⣿⣀⣽⣦⣿⣸⣿⣸⣿⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢭⡋⡍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢹⣿⣿⢻⣿⣿⣿⣟⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⢹⡏⣟⡋⣻⡏⣿⡏⣿⡋⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠁⠹⠿⠿⣿⣿⣿⣿⣿⣿⡻⠏⢶⣲⡆⠦⠋⠻⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠿⠿⠿⠿⢿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢻⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣁⠀⠀⣄⣤⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⠁⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠃⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣠⣶⣶⣶⣿⣿⣷⣀⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠈⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢋⣤⣤⣀⣬⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠉⠛⠛⠛⠛⠿⢶⣤⣤⣠⣤⣶⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⠻⠛⠋⠉⠉⠉⠙⠛⠲⢶⣾⣿⡿⡿⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠁⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢸⠇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣾⠁⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠁⠀⠂⠠⠤⠶⣟⣻⠇⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡯⢵⡃⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠁⠁⠐⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⡳⠇⠀⠀⠲⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡟⣀⣤⣤⣤
⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣏⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⣀⣷⣷⡻⣸⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⡿⠏⢷⣱⡀⠀⠛⠛⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠿⠿⠿⠿⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣦⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢿⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣂⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢸⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣾⣶⣿⣿⣿⣀⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢋⣤⣤⣀⣬⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠈⠛⠛⠛⠛⠿⢶⣤⣤⣄⣤⣶⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠛⠋⠉⠉⠉⠛⠛⠲⠶⣾⣿⡿⡿⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢸⠇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣾⠁⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠁⠀⠂⠠⠔⠴⡾⡿⠇⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⢽⡯⡂⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠈⠀⠂⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠇⠀⠀⠺⣿⣿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⢀⣜⣷⣳⣋⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⣿⣷⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⡿⡿⠝⠀⣢⢀⠢⠚⠛⠁⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣅⠀⠀⣀⣀⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣶⣶⣿⣿⣿⣀⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣟⣴⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣏⣤⣤⣀⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠛⠛⠛⠛⠿⢦⣤⣤⣄⣤⣾⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣼⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠋⠉⠉⠉⠉⠛⠒⠶⣾⣿⣿⡿⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠠⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣮⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⣿⢿⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠾⢿⠿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⢰⢗⣳⢚⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⢿⣷⣶⣴⣴⣾⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠁⠹⠿⠿⢿⣿⣿⣿⣿⡿⡟⠟⠒⣨⢐⠤⠚⠙⠃⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⠀⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀⠀⣠⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠃⣾⣿⣿⠟⣿⣿⣿⠀⢿⡟⠁⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⢿⠿⠏⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣾⣿⣾⣿⣿⣄⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣏⣤⣤⣄⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠛⠛⠛⠛⠿⢶⣤⣤⣤⣤⣾⣦⣤⡙⠛⠛⠛⠛⢁⣤⡄⠀⣼⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠉⠉⠙⠒⠶⣾⣿⡿⠿⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢠⠁⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣞⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠁⠀⠂⠀⠄⠐⠂⡓⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡹⣍⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠛⠅⠀⠀⠰⡿⠿⠿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢠⣠⢲⣪⣖⣅⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠿⢿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠁⠹⠿⠿⢿⣿⢿⣿⣿⡿⠿⠋⢧⡤⣂⠀⠋⠛⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠿⠿⠿⠿⢿⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⣴⣦⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡾⠿⠿⠿⣿⣿⣿⠿⢿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠃⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣴⣶⣿⣿⣿⣿⣄⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⣤⣤⣄⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠚⠛⠛⠛⠿⢶⣤⣤⣄⣤⣾⣦⣤⡙⠛⠛⠛⠛⢁⣤⡆⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠛⠋⠉⠉⠉⠉⠙⠒⠲⣾⣿⠿⠿⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠁⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠖⠘⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⣬⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠍⠃⠀⠀⠰⡿⠿⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢀⣤⣤⣤
⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢀⢤⢀⠄⣢⢀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⢿⣿⣶⣦⣦⣼⣿⣿⣿⣟⣟⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠛⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⡿⠿⠛⢿⡸⡵⠐⠋⠛⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣵⣄⡀⣠⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⣿⣿⣿⢿⠿⠿⠏⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠃⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣿⣿⣿⣿⣿⣄⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⡿⠿⢿⢿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣏⣤⣤⣤⣤⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠈⠛⠛⠛⠛⠿⢶⣤⣤⣤⣤⣾⣦⣤⠙⠛⠛⠛⠛⢁⣤⣆⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠉⠀⠀⠒⠲⢾⣿⠿⠿⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠁⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⡾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡪⡝⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⣸⣟⢄⠀⠀⠀⢀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠗⠅⠀⠀⠲⠿⠿⠿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⣀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⢁⣤⣤⣤
⠀⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢠⣰⢽⣾⣦⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⣿⣷⣶⣶⣽⣿⣿⣿⣿⣯⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠠⠙⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⠻⠿⠿⢿⣿⣿⣿⠿⠛⠫⡮⡲⠢⠉⠋⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠾⠿⠿⠿⠿⢿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣧⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢻⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣅⢀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠋⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⢋⣠⣶⣿⣿⣿⣿⣿⣤⣀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣏⣤⣤⣤⣤⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠈⠉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠛⠛⠛⠛⠿⢦⣤⣤⣤⣤⣼⣶⡤⠙⠛⠛⠛⠛⢁⣤⣆⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠉⠀⠀⠐⠚⠿⢿⠿⠟⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⣾⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠠⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣶⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠐⠻⠿⠿⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⣀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⢁⣤⣤⣶
⠀⠈⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢠⣎⢷⣳⡿⡲⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⢿⣷⣶⣤⣤⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣾⡶⠂⢀⠈⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠈⠙⠛⠛⠿⢿⣿⣿⠟⠟⠀⢛⣗⣟⠁⠉⠉⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣷⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⠿⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣧⣀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠃⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⣠⣾⣿⣿⣿⣿⣿⣧⣄⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣷⣿⠿⠿⠿⠻⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣤⣤⣤⣤⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⢁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠚⠛⠛⠛⠿⢦⣤⣤⣤⣤⣤⣤⡀⠙⠛⠛⠛⠛⢁⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠛⢻⡿⠟⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⢶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⡾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⡀⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡡⠀⠀⠀⠀⣠⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠛⠛⠛⢿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠃⣠
⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢀⣄⣽⣳⣻⣧⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣆⠀⠀⠀⠛⢿⢿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⠿⠟⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣀⣀⣀⣠⣮⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣿⠿⠿⢛⣻⣿⣿⣿⠿⠿⠟⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠘⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣠⣾⣿⣿⡿⣿⣿⣷⣄⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⠟⠛⠛⠛⢿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣤⣤⣤⣤⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠐⠛⠛⠛⠿⢦⣤⣤⣤⣤⣤⣄⡀⠙⠛⠛⠛⠛⢁⣤⡇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠋⠉⠁⠀⠀⠀⠀⠀⠛⢻⡿⠃⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⡞⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠛⠊⠁⠉⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣯⡢⠀⠀⠀⠀⢠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠉⠉⠉⠉⠛⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⠏
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡋⣀
⠀⠀⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⡀⣰⢐⡐⠌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⣿⣿⣿⣿⣾⣿⣿⡿⣿⢵⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣆⠀⠀⠀⠛⡿⣟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⠟⠁⠁⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠴⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠿⣿⣿⡿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠿⠟⢛⣛⣿⣿⣿⡿⠿⠟⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣀⣶⣿⣿⣿⣿⣿⣷⣦⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣾⡿⠛⠟⠛⢿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣤⣤⣤⣤⣾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠐⠛⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⣁⣴⡇⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠋⠉⠉⠀⠀⠀⠀⠀⠈⢻⡿⠃⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢠⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠙⠭⠌⠻⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠉⠉⠉⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⡀⣀⡀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⢿⣿⣶⣦⣾⣾⣿⣽⣿⣢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣷⢆⠀⠀⠀⠛⢿⣟⡻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠏⠍⠁⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠧⠄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⢿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣢⣀⣀⣤⣖⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⢛⣛⣿⣿⣿⡟⠟⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠏⣀⣴⣿⣿⣿⣿⣿⣿⣦⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣾⡿⠛⠛⠛⠛⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣤⣤⣤⣤⣴⣿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠒⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⣁⣤⡇⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠁⠉⠉⠋⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡢⠅⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣀⣤⡤⡄⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣴⣴⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠿⠿⣿⣶⣴⣶⣿⣿⣿⣿⣾⠷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣷⣿⣶⠦⠀⠀⠀⠻⣿⣛⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⠿⠋⠁⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠆⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠼⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣆⣀⣀⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢿⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠿⢛⣛⣻⣿⣿⡿⠟⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣀⣴⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣧⣤⣤⡤⣼⣿⣿⣿⣷⠦⠀⠀⠀⠈⠉⠁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠚⠛⠻⢦⣤⣤⣤⣤⣤⣄⡀⠙⠛⠛⠛⠛⢡⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠯⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠶⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠂⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣞⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠙⠉⠈⠋⠋⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢜⡆⠀⠀⠀⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣷⣦⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⢀⢠⣤⣤⣆⡠⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠿⣿⣿⣷⣾⣿⣿⣿⡿⣽⡇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⠑⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡵⣷⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡦⣶⣿⣿⣶⣾⣶⠄⠀⠀⠀⠘⠛⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠁⠁⠀⠛⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠰⠾⠿⠿⠿⢿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢻⣿⣿⡿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣽⣿⣿⣿⡆⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢻⡏⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⠛⣛⣻⣿⣿⡿⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣁⣴⣿⣿⡟⠕⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣾⡿⠛⠛⠛⠓⢾⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣧⣤⣤⠄⠴⣿⣿⣿⡷⠦⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠘⠛⠳⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⢡⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠯⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠔⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣥⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢱⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠨⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⠏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠘⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⣲⠖⣦⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⢿⣿⣿⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠈⠉⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣶⣶⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠛⠉⠙⠛⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⢿⣿⣦⣤⣾⣵⣄⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣷⣶⣶⠄⠀⠀⠀⠘⢻⣟⡝⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠈⠈⠈⠛⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠲⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⢿⣿⣿⡿⠿⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⣀⣠⣬⣿⣿⣿⣿⡆⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣾⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠛⣛⣛⣿⣿⣿⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢠⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢸⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣁⣴⣿⣿⡿⢿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⠛⠛⠛⠛⢻⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣧⣤⣤⡤⠴⣿⣿⣿⣷⠦⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢁⣤⣤⣤⣀⠀⠀⠀⠀⠘⠛⠳⢦⣤⣤⣤⣤⣤⣤⡀⠙⠛⠛⠛⠛⢡⣴⣇⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠭⠄⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠖⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⡇⣸⣿⣿⡟⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣥⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢱⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠐⠂⠀⢸⣿⣿⣿⢰⣿⣿⡿⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠀⠀⢸⣿⣿⡏⣼⣿⣿⣧⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⡏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⡝⡛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡗⠅⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢎⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⣴⣶⣦⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣀⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⣇⡀⠀⣀⣀
⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⣪⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⢿⣿⣶⣤⣷⣾⣴⡀⠠⠄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢄⢠⢤⡀⠀⠀⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠈⠀⠀⠀⠂⢀⠀⠂⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠶⡦⠀⠀⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⣶⣿⣿⣷⣶⣶⠆⠀⠀⠀⠘⢻⣟⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠁⠉⠘⠛⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠳⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣧⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣤⣄⣤⣴⣮⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠿⢋⣉⣛⣿⣿⣿⠙⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⢸⠁⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢟⣁⣴⣿⣿⣿⣿⣿⡿⠂⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⠛⠛⠛⣿⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⣤⣤⣤⠰⣿⣿⣿⣷⡦⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠉⢁⣤⣤⣤⣀⠀⠀⠀⠀⠘⠛⠻⢦⣤⣤⣤⣤⣤⣤⡀⠙⠛⠛⠛⠛⠁⣴⣧⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠭⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⡇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠲⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⣀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣼⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢀⣀⣀⣤⣤⣴⣶⣾⣿⣾⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷
⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠿⣿⣿⣷⣴⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠐⠓⢤⣝⠛⠛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠈⠺⣿⡶⠀⠀⠁⡀⠄⠀⠀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠲⣦⣈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣶⣄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣷⣶⠀⠀⠀⠀⢹⣿⡿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠙⠙⠻⠿⠿⠿⠏⠀⠀⠘⠐⠑⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡷⠷⠤⣤⢀⣠⣤⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠈⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣰⣾⣿⣿⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣲⣤⣤⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡿⠛⣉⣉⣛⣿⣿⡉⠋⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣠⣾⣿⣿⣿⣿⣿⣿⠆⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⠟⠛⠛⠛⢻⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⣶⣶⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠈⠉⠁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠐⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⣴⣇⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠐⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣾⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣞⣿⣃⠀⢀⡀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈⠀⠈⠉⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⣿⣿⣶⣴⣶⣤⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⡀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠓⢦⣌⠛⠛⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡶⠦⠀⠈⡀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⢶⣄⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣦⡀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣶⠂⠀⠀⠀⢹⣿⡿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠏⠀⠀⠈⠚⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠶⠤⣤⢀⣠⣤⣤⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠈⠻⢿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣰⣾⣿⣿⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡉⢿⣿⣿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣠⣴⣴⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡿⠛⣉⣉⣛⣿⣿⡉⠉⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠸⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣾⣿⣿⣿⣿⣿⣿⠆⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⠟⠛⠛⠛⢻⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⣶⣶⣾⣿⣿⣿⣿⠷⠆⠀⠀⠀⠈⠉⢁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⡀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠘⠛⠛⠻⢦⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⣴⣧⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⢿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠐⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⠇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣼⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⣠⣴⣿⣣⠀⢀⡀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⣿⣿⣶⣾⣾⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣦⣤⡀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠓⢦⣌⠛⠛⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡶⠦⠀⠈⣀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠐⢶⣄⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣦⡀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⡶⣶⡶⣖⠶⠿⠿⠇⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⢠⣤⣯⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢷⣿⣶⣴⣄⡀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⠷⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠻⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢻⡟⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢋⣠⣤⣴⣿⣿⣧⣀⣑⣡⠍⠀⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠈⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠋⠉⠉⢈⣿⣿⣿⣿⣿⣇⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⣿⣿⣺⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣀⣈⣉⣁⣀⡈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠛⠟⠛⠛⠛⠿⢦⣤⣤⣤⣤⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠙⢻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⡇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢠⣾⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣷⣀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁⠀⠈⠉⠈⠁⠈⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠿⢿⣿⣷⣦⣤⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣿⣷⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠈⠛⢦⣭⠛⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠒⠂⠀⢤⡖⠆⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢿⣷⡦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣷⣶⣶⣶⣷⣲⠶⡶⠠⠈⠀⠁⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠉⠉⠿⠿⠿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣵⣶⣶⣴⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⠀⠀⣰⣿⣿⢿⣿⣦⡀⠈⣀⣀⣀⣀⣈⠻⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡄⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⢹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣏⣽⣿⣷⣿⣿⣿⣌⣸⣿⡿⠃⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠄⠀⣀⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠘⠛⠿⠛⠛⠛⠳⠤⣤⣀⣀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⣾⡇⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣀⠀⠉⠉⠉⠀⣰⣿⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣄⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠙⠻⠻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠶⢦⢠⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠘⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢺⣿⣷⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠈⠛⠢⠌⠛⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡏⠐⠀⠀⠶⠒⢀⠀⠰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠑⣶⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣷⣿⣾⠿⠔⠀⠀⠀⠈⠿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⡟⠉⠉⠈⠉⠻⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡇⠀⠀⠀⠀⠀⠀⡀⣀⢠⣄⣤⣠⣠⣤⣶⣶⣶⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠳⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠗⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣤⣤⣄⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣈⠙⠻⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⣾⡏⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⡀⠉⠉⠉⢀⣴⣿⣇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠿⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣶⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⡿⡿⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠂⠁⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣄⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡛⠻⣿⣷⡄⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠈⠙⠓⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠾⠉⢀⡀⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⠙⢶⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠗⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠐⢿⠷⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠟⠹⠿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠉⠈⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠦⣤⣀⣀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠛⠛⠛⠛⠂⠠⣤⣤⣤⣤⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠙⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⢰⡆⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠁⠉⠀⠠⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⠉⠊⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿
⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠠⣶⣠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠟⠃⠀⠀⠀⠘⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠈⠁⠀⠂⠀⠀⠀⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠠⣤⡈⢳⣌⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠈⠁⠒⠛⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢟⣟⠣⠄⠀⠄⠤⠀⠀⠀⠰⠐⠔⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠸⠋⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠄⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠈⠛⠓⠒⠐⠂⠠⢤⣤⣤⣤⣤⣤⠄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠋⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣟⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣯⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿
⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣷⣤⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣬⡙⠻⠟⠂⠀⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣦⣀⠀⠙⠃⣀⠀⠀⠀⣠⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠈⠉⠡⣤⡈⠻⣿⣷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣴⣄⠉⠙⠲⠎⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠖⠀⠠⠢⠔⠀⠀⠀⠐⠂⠀⠀⠀⠃⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠯⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⠿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣠⣾⣿⣿⣿⠿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠘⠃⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣸⣿⣿⣦⣿⣿⣿⣿⣿⣿⠀⠉⠉⠁⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠦⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠈⠛⠒⠒⠒⠛⠶⣦⣤⣤⣤⣄⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣴⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡏⠉⠁⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣍⠛⠟⠃⠀⠀⠈⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣤⠀⠉⠃⣀⡀⠀⢀⣠⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠰⢦⣈⠙⢿⣿⠶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣦⡀⠉⠓⠂⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡷⠒⠐⠐⠢⠲⠴⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟⠋⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣰⣿⣿⣿⣿⠿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠈⠁⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣹⣿⣷⣴⣿⣿⣿⣿⣿⡏⠀⠉⠉⠀⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠈⠛⠒⠒⠒⠒⠶⣦⣤⣤⣤⡄⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣴⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿
⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣍⠛⠟⠃⠀⠀⠈⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⣤⠀⠉⠃⣀⡀⠀⠀⣀⣤⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠰⢦⣈⠙⢿⣟⠶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣦⡀⠉⠓⠒⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠒⠀⠒⠲⠲⠶⠖⠠⠀⠛⠃⠉⠉⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠟⢈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡈⠉⣉⣀⣀⣀⣉⠛⠿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠻⢿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠈⠃⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣧⣼⣿⣿⣿⣿⣿⡿⠀⠉⠉⠀⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠆⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠒⠒⠂⠒⠒⠲⢤⣤⣤⣤⣤⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠙⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⢸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⡀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣯⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣀⣀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿
⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⡛⠻⠓⠀⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣤⡀⠈⠛⢂⠀⠀⠀⠀⣠⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⢦⣀⠈⠾⣿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠉⠑⠶⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣾⣿⡿⠿⠒⠠⠦⠦⠶⠶⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⠷⠒⠋⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣾⣿⣿⣿⣿⠟⠋⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡿⠀⣰⣿⣿⢿⣿⣦⡈⠉⣉⣁⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠧⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠘⠃⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣈⣿⣿⣀⣸⣿⣿⣿⣿⣿⣿⡟⠿⠿⠇⠀⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠘⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣿⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⢀⡏⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⣿⠃⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠁⠉⠻⣿⣿⣿⣿⣷⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⠺⡿⠆⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡀⠈⠛⢦⡀⠀⠀⠀⠀⡀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠁⠀⠀⠀⠀⢀⡀⠀⢶⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠳⣦⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣷⡶⣶⣦⠀⠠⣾⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⣿⣖⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠂⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢀⣼⣿⣿⣿⠟⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡟⠀⣰⣿⣿⢿⣿⣦⡈⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⣿⣿⡄⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⢀⡀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠰⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⡷⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⡀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⣾⡇⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠉⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⣀⣀⣀⠀⠉⠉⠉⠀⠠⠆⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣤⣄⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠙⠳⣤⡙⠃⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠒⠀⠀⣤⠤⠀⠀⣠⡰⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢶⣈⠙⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⣶⣦⠀⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣷⡶⣶⣦⠀⠀⣶⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⣿⣖⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠰⠂⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⣠⣴⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢀⣾⣿⣿⣿⠟⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡟⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⠿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠹⠏⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⣿⣿⣄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⢀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠻⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⢰⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣾⡷⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⡀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⣾⠇⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠉⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣍⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⡀⠀⠉⠉⠉⠀⠀⠆⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡗⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣦⣄⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠙⠲⣤⡛⠀⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡕⠒⠀⠀⡔⡄⠀⠀⣠⡰⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢶⣈⠙⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⣷⣦⠀⠉⠑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠸⠿⠿⠋⠉⠁⠙⠙⠻⠹⠿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠈⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣽⣾⣷⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠿⠿⠿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠻⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⢹⠇⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣷⡀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⠿⠁⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣴⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠈⠉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⣤⣤⣤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠠⠀⠄⠂⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⡝⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⢀⣤⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⢰⣿⣿⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣠⣠⣤⡀⠉⠉⠉⣠⣾⣿⣿⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢛⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿
⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡐⠺⣿⣦⠄⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣄⠀⠉⠲⠎⠁⠀⠀⠀⠀⠈⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠉⠀⠀⠰⠒⠂⠀⠰⣶⣷⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠓⢶⣌⡹⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡷⠀⠀⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠈⠻⢻⡿⡙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⣿⣽⣷⣶⣾⣶⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠈⣀⣀⣀⣀⣀⠙⠿⡿⠿⠿⠛⠻⠿⠷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠈⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣆⣰⣶⣿⣿⣿⣿⣿⣿⡇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠂⠀⠀⣀⣈⠉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣀⠀⠙⠛⠛⠛⠛⠃⠀⠀⣠⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠉⠉⠀⠀⠀⠀⠤⠠⠄⠛⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⣀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⢀⣾⡏⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⣀⠀⠀⠀⠉⠉⠉⠀⣸⣿⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⡠⣶⢶⡾⠟⠛⠁⠀⠀⠈⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣏⠃⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠀⠉⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡁⡀⠀⠀⠻⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿
⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣔⣶⣤⠀⠈⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡐⢦⣈⣙⠿⠿⠁⠀⠀⠀⠊⠉⣿⣿
⠀⠀⠀⠀⠀⠙⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠉⠉⠀⠀⠀⠀⠀⢠⣄⠙⢉⠙
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠙⢶⣭⣷⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡛⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⡛⠖⠂⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⣉⠍⡉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣶⣀⠈⠀⠀⠀⠉⠿⠟⠛⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠘⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣄⣼⣾⡟⠃⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢟⣿⣿⣆⠀⠀⠀⣾⣿⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⣦⣤⡈⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⢦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⣿⣿⠟⣼⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠦⣤⣤⣤⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢯⣽⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⣿⣿⣿⢥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⡏⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣿⣿⣿⣿⠅⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢀⣾⣶⣶⣦⣦⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠃⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⡿⣿⡛⠛⠛⢿⣿⣿⣿⣿⣿⣧⡀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⠹⣿⣿⣿⡎⠀⠘⠉⠙⠂⠀⠊⠉⠋⠋⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠀⠀⠹⠿⣿⣿⣧⣤⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⠀⠢⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠙⠻⠿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠈⠉⠛⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⣄⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠿⣿⣝⣿⡿⠂⠀⠈⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⠈⠉⢁⣀⠀⠀⣀⣠⣄⠈⠙⢿
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢱⣄⠙⢿⣿⣷⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣻⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣄⠉⠻⢶⡟⠋⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣯⣯⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠃⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⣩⡭⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣶⡠⠘⠀⠀⠉⠉⢿⠿⠉⠙⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠟⠁⠀⠀⠀⠀⠀⠘⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠋⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣾⣴⠟⠁⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠟⣿⣿⣆⠀⠀⠀⣾⣿⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⣦⣤⠈⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠛⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣴⣿⣿⠟⣨⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠳⠦⣤⣤⣤⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⢩⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠂⠀⢠⣶⣶⣶⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡆⠀⠈⠿⣿⣿⣿⣿⢿⣿⢿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣄⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⠹⣿⣿⣿⡗⠀⠙⠘⠘⠀⠀⠊⠙⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠀⠀⠙⠿⣿⣿⣧⣤⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣶⣶⣶⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠲⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠉⠛⠻⢿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠉⠙⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⣄⠈⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠻⣿⣽⣿⠿⠃⠀⠈⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⢦⠈⠉⢡⣤⠀⠀⣀⣴⣄⠈⠙⢿
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢰⣦⡉⢻⣿⣷⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣻⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣄⠉⠛⠶⠏⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣯⣯⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⣉⣉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠈⣾⡖⠩⠂⠀⠀⠉⢿⠟⠋⠹⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠘⢿⡷⠆⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣴⣴⡞⠁⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠏⣻⣿⣆⠀⠀⠀⣶⣿⣿⣿⡟⠀⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⣶⣤⠀⣿⣿⣿⣿⣿⡀⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠛⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣴⣿⣿⠏⣱⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠢⣤⣤⣤⣤⣤⣄⠀⠙⠛⠛⠛⠛⠁⠠⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠈⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠋⣾⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣾⣿⣿⣿⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⣿⣿⣿⢱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⡏⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣿⣿⣿⣿⡅⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠈⠁⠀⠀⠤⢤⣄⣀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡆⠀⠐⠿⣿⣿⣿⣿⠿⢿⢿⢛⠛⠛⢿⣿⣿⣿⣿⣿⣆⡀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⠀⠀⠉⠓⠈⠐⠀⠉⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⠀⠀⠀⠘⠿⣿⣿⣆⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠊⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠒⢦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠈⠛⠛⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠉⠛⠓⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣧⣄⠈⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡈⠛⣾⣯⡿⠟⠁⠀⠀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⠶⠈⠉⣥⣤⡀⠀⣀⣴⣄⠈⠉⢿
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⢰⣦⡉⢻⣿⣷⣦⠀
⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⣛⣿⣿⣻⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠈⠛⠷⠋⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠃⠀⠀⠀⠐⣽⡯⠭⠀⠀⠉⠙⣿⣿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⢸⣶⣶⡄⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠋⠁⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣷⡀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣴⣾⣿⣿⣀⣀⠀⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢻⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣤⣤⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⠿⢦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠀⢀⣀⣀⣠⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⡿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⠇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢠⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠥⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣾⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡆⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣆⠀⠉⠙⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠈⢻⣿⣿⣿⠋⠁⠻⠻⠺⠒⠀⠩⠛⠛⠟⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⠀⠀⠀⠻⣿⣿⣷⣤⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠛⠛⠛⠟⠳⠻⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠳⣬⣿⣿⡟⠀⠀⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡗⠦⠈⠉⣡⣦⠀⠀⣀⣤⡌⠉⢿⣿
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠈⢰⣆⠙⢿⣿⣶⡤⠙
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣄⠉⠛⠿⠟⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⡿⣛⣿⣿⣻⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠍⣭⣯⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⣋⡛⡋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠻⠟⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢰⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣀⠘⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣴⣄⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠀⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢿⣿⣆⠀⠀⠀⣀⣾⣿⣿⣿⣿⠏⠀⣠⣾⣿⣿⠟⠀⢰⣿⣿⣿⢀⣿⣿⡿⠀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣴⣿⣿⣿⣄⣀⠀⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠿⣿⣿⢿⣿⣿⣶⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣤⣴⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣤⣤⣀⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠀⢀⣀⣀⣀⣄⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣽⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⠀⢰⣶⣦⣤⣤⣤⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣤⣿⣿⣿⠛⣼⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⢋⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣃⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣰⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⠫⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡟⠁⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣶⣶⣶⣤⣤⣤⣤⣤⣤⡀⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡃⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⡿⠏⠙⢿⣿⣿⣿⣿⣿⣆⡀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⡀⠨⣿⣿⣿⣿⣆⠀⠘⠹⠩⠁⠀⠀⠉⠋⠋⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⡀⠙⢿⣿⣿⣷⣦⣠⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣇⠀⠈⠛⠿⠿⠿⠝⠙⠝⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⣀⠘⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠐⠳⣭⣿⡿⠟⠂⠀⠈⠛⢿⣿⣿⣿
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡷⠦⠈⠉⣡⣤⠀⠀⣀⣴⣄⠉⠙⢿
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⢰⣄⠙⢿⣿⣶⡄⠀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣄⠈⠛⠾⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠟⡛⣿⣿⣻⣿⣻⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠯⣿⣽⣿⣽⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⠉⢭⣏⣏⢍⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⢀⡀⠀⠀⠀⠘⣿⡟⠋⠀⠀⠀⠀⠛⡏⠈⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠸⢿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠀⣀⣀⣀⣀⣈⠹⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣦⣶⣿⣦⠀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⣻⣿⣇⠀⠀⠀⣴⣿⣿⣿⣿⣿⠏⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣿⣿⣦⣄⡀⣿⣿⣿⣿⣿⠇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣤⣤⣀⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣀⠙⠛⠟⠛⠛⠛⠓⠀⠀⣀⣀⣀⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣤⣿⣿⣿⠋⣼⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⡏⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣷⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠅⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠟⡿⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣄⠀⢹⣿⣿⣿⣿⣿⠿⢻⣿⣋⡀⢙⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠓⠅⠀⠀⠀⠐⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣔⢴⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠓⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣷⣦⡀⠘⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠂⣭⣿⠿⠛⠁⠀⠀⠈⠛⢻⣿⣿
⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠉⠉⠀⣦⠀⠀⠀⣴⡄⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⣮⣿⣷⠄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠐⠊⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⣸⣿⣿⢾⠷⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⣠⣶⣶⡄⠀⠛⠛⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠈⣁⣀⣀⣀⣉⠻⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢏⣸⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⢦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢁⣤⣤⣤⣈⠙⠛⠿⠛⠛⠛⠓⠀⢀⣀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⣾⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣤⡄⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢋⠁⢠⣶⣿⣶⣶⣶⣿⣿⣿⣿⣆⡀⠀⠉⠁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢐⠛⠛⠻⠿⠛⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡿⡿⡿⣿⣿⡿⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⡪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣦⣤⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣿⣿⣧⡄⠈⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠛⣦⠿⠛⠃⠀⠀⠀⠈⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠂⠀⠀⣀⠲⣦⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢦⣉⠳⣮⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡖⠂⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣻⣿⣻⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠀⢶⣿⣿⣿⠏⠁⠀⠈⣿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢸⣿⣿⡶⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠉⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠉⠀⠀⠀⠀⠀⣴⡆⠈⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠸⠌⠀⠈⠉⠀⠀⠀⠀⠀⣤⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣦⡀⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠃⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⣸⣿⣿⣄⣀⠀⢾⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⢋⣩⣿⣿⢯⣽⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣈⠙⠛⠿⠛⠛⠛⠓⠠⢄⣀⣀⣠⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢋⣼⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣾⣿⣷⣶⣾⣿⣿⣿⣿⣆⣠⣈⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠉⢿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣶⣮⡀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣻⣿⣿⡤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣄⣹⡿⠗⠉⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠉⠀⠸⠿⢂⡀⠰⣶⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣈⠙⢶⣮⡙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣿⣻⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⣴⣿⣿⡿⠋⠁⠀⠈⢿⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⡷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠉⠀⠀⠀⠀⠀⢰⠆⠁⠀⠀⠀⣀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⣸⣿⣿⣄⣀⠀⢼⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢉⣉⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠠⢀⣀⣀⣠⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢣⣼⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⢃⣼⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⠁⣷⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⠩⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢠⣾⣿⣷⣶⣾⣿⣿⣿⣿⣆⣤⣈⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢝⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠉⢿⣿⣿⣿⣿⣿⣶⡂⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣶⣧⡀⠈⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⡻⢿⣿⡤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣄⣹⠿⠗⠉⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠸⠿⢂⡀⠐⢶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡈⠙⢶⣮⡙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣽⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠙⣿⣾⣤⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⣶⣿⣿⠿⠁⠀⠀⠈⢿⠿⠛⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⢿⣿⡿⠷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣰⠟⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⢸⣿⣟⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⡀⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣦⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠸⠁⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣃⣸⣿⣿⣀⣀⠀⣾⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⢉⣩⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠤⢀⣀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⣾⣿⣿⡿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣿⣿⣿⣿⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⣤⣾⣿⣷⣶⣾⣿⣿⣿⣿⣦⠀⢀⣉⣁⡉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⡉⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⢿⣿⣿⣿⣿⣿⣾⠊⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣈⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣆⣀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡻⣿⣷⣤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄⡠⠀⠠⠀⡂⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣄⠙⡿⠮⠙⠁⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⡽⢵⡃⠈⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠋⠀⠰⣶⠖⠀⠰⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢶⣌⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣽⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠏⠈⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⢻⣷⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⢴⣿⣿⡿⠏⠁⠀⠀⠸⢿⠟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢸⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣾⠋⠁⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⢈⣇⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠛⠁⠀⠈⠀⠀⠀⠀⠀⠤⣿⣿⡏⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣷⣶⡀⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⡀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠘⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣀⣿⣿⣧⣀⣀⠀⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣉⣩⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⣁⣀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⡟⠋⢁⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠤⢀⣀⣀⣤⣤⡄⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⡿⣠⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⡟⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⢠⣴⣿⣿⣷⣶⣾⣿⣿⣿⣿⣆⠀⠈⣉⣁⡉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢝⢻⣿⣿⣿⣿⣿⣿⠟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠻⣿⣿⣿⣿⣿⣿⡦⠃⠀⠈⠁⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣽⠉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣅⣀⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠈⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿
⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⢀⠀⡀⢄⢄⡉⠛⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡠⡲⣣⢷⣭⣇⣣⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣜⠻⣿⣷⣤⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣯⣽⢟⢛⢟⠲⠐⠀⠹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣀⠙⠿⠮⠋⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠋⠀⢰⣶⠆⠀⢠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠳⣦⣌⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡦⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢔⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⠄⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⣠⣶⣦⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⢰⣿⣿⣿⠛⠉⠀⠀⠀⠻⠛⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡴⠟⠁⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠋⠁⠀⠀⠀⠀⠈⠇⠀⠀⠀⢀⣠⣤⣤⣤⣶⣤⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠻⡿⣿⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣀⣀⣀⣉⠻⢿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⡄⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢁⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣃⣰⣿⣿⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⡟⡉⣩⣽⣿⢯⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢁⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠓⠀⢀⣀⣀⣠⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⡋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣼⣿⣿⡿⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⣀⡀⠉⠉⠉⢀⣴⣶⣦⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⣀⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⠀⠈⢉⣁⡉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣰⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⣿⣿⣿⣿⣿⣿⣷⡜⠀⠈⠉⠉⠉⠀⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣬⠉⠙⠛⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⡉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣾⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⢀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣉⣀⣀⢀⣐⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠙⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣷⣿⣿⣾⡢⠀⠉⠛⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡁⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡚⠺⣿⣦⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠈⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣄⠈⠛⠶⠌⠁⠀⠀⠀⠀⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠉⠀⠀⢲⡶⠀⠀⢴⣶⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⣤⡉⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣶⡄⠉⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⣿⣻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠫⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⢀⣴⣶⣴⣦⣶⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁⣀⡀⠀⠀⢻⣿⣿⠟⠁⠀⠀⠀⠈⠛⠉⠀⠈⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠹⡿⠿⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠤⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⡄⠂⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠛⠿⡿⢟⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡄⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢁⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣿⣿⣧⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣟⣡⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣤⣤⣤⣈⠙⠻⠟⠛⠛⠛⠓⠀⢀⣀⣀⣠⣤⡀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣦⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⢀⣠⡀⠉⠉⠉⢀⣴⣶⡆⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⡁⢀⣴⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣄⣈⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣢⠃⠈⠉⠉⠉⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣼⡉⠋⠛⠛⠻⠟⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣅⣈⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡁⡽⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡛⠻⣿⡦⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣄⠈⠙⠲⠈⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠀⠀⠲⠂⡀⠀⢶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠳⣤⡉⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⡶⠄⠉⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣠⣶⣶⣶⣾⠷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⣀⣀⣀⡀⠺⣿⢿⠟⠁⠀⠀⠀⠀⠉⠁⠀⠀⠠⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠟⠁⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⠕⠀⣀⣀⣠⣴⣶⣶⣶⣾⣾⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠝⠻⠙⠃⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣷⣄⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢞⣃⣼⣿⣿⣄⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣁⣤⣽⡿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⣈⡉⢁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢁⣤⣤⣤⣈⠙⠛⠟⠛⠛⠛⠂⠀⠀⣀⣀⣀⣤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⠉⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⣀⣤⡀⠉⠉⠉⢀⣴⣶⡆⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢠⣾⣿⣿⣿⣿⣶⣾⣿⣿⡿⢳⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣯⡇⠀⠈⠉⠉⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣍⠙⠙⠛⠿⠿⠃⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣟⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣄⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣠⣤⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿
⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡙⠻⣷⠆⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣷⣄⡀⠙⠒⠀⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠂⠂⣀⠀⢶⣽⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠉⠳⣦⡍⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⠶⠀⠈⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢠⣶⣿⣿⣿⠿⠟⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣀⣤⣤⣤⡀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡿⠿⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⠈⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡇⠀⠾⠋⠀⠀⠀⠀⢀⣠⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠋⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡆⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣼⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⣿⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣏⣠⣿⣿⣷⣄⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣏⣠⣤⣿⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⢁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⢀⣀⣀⢤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⣿⣿⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⡟⠙⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⣀⣤⡀⠉⠉⠉⢀⣴⣾⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠅⠀⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⡟⢰⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢈⠙⣿⣿⣿⣿⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠙⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣭⣉⠉⠙⠛⠛⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡉⠙⠁⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡗⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿
⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⡀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡙⠛⠿⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⣤⡀⠙⠂⠀⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣢⣴⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⢾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠉⠳⣦⡍⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⣾⣿⣿⣿⠿⠙⠘⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⣤⣤⣤⣤⡀⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⣄⠈⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠺⠁⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠙⣉⣁⣀⣈⣉⠻⢿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢁⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣃⣴⣿⣿⣷⣤⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣧⣤⣤⣿⠛⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⣁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⢀⣀⣀⡤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⠟⠈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⢀⣠⣤⣀⠉⠉⠉⢀⣴⣿⡇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠡⠀⣴⣾⣿⣿⣿⣿⣶⣿⣿⣿⡏⢰⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⢉⣿⣿⣿⣿⣿⡿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠉⠙⢿⣿⣿⣿⣿⣿⣯⡂⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⡉⠉⠉⠛⠛⠁⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡍⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡀⣀⣸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿
⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢤⡙⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡅⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⣤⡀⠙⠃⠀⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠊⠠⣀⠐⢾⣽⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⡄⠉⠱⣦⡍⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⣴⣿⣿⣿⠟⠛⠀⢹⣿⣾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⢀⣴⣶⣦⡄⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⡿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⡀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠾⠃⠀⠀⠀⠀⢠⣴⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣿⣿⡆⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣃⣴⣿⣿⣿⣤⣄⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠿⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣧⣤⣴⣿⠛⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⢁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠛⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣾⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣀⣄⡀⠉⠉⠉⢀⣴⣾⠇⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣴⣿⣿⣿⣿⣿⣶⣿⣿⣿⠏⢰⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⣿⣿⣿⠇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣏⠉⠉⠉⢿⣿⣿⣿⣿⣿⣆⠄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡖⣴⣤⡀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⢀⡀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⡿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿
⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢦⣝⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⢀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⢤⠀⠉⠃⠀⠀⠀⠀⢀⡀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣢⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⣄⠀⢾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⡄⠉⠱⠦⠍⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠀⠈⣿⣷⠷⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠠⣶⣶⣶⣄⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢤⡄⠉⠁⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠖⠀⠘⠋⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠟⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⣻⣿⣇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⣿⣿⣿⣦⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⢿⣿⠯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣧⣤⣴⣿⠋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⠉⠁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⣀⣀⢤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣼⣿⣿⡟⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⠟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣾⣿⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⣀⡀⠉⠉⠉⢀⣠⣶⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠊⢀⣴⣿⣿⣿⣿⣿⣶⣾⣿⣿⠏⢰⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⣿⣿⣿⣿⣿⠇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⠀⠉⠿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⣵⡀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⡈⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡂⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣃⣀⣤⣨⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠑⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠟⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠦⣌⠛⠟⠂⠀⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠡⣨⢴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣗⣤⠀⠉⠁⠀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠠⣄⠈⢺⣿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⠉⠑⠶⠌⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠋⠀⠀⣿⣷⡶⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠠⣴⣶⣶⣦⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠛⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠻⠿⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠀⠀⠀⣀⢀⢀⣀⠀⠀⠀⠄⠂⠀⠛⠁⠀⠀⠀⠀⢠⣶⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⣹⣿⣧⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⣿⣿⣿⣷⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠿⠿⢿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣷⣤⣶⣿⠛⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⢁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⣀⣀⡤⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣿⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⠿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⢠⣾⣿⣿⣷⣶⣶⣶⣾⣿⡟⠁⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⡽⣿⣿⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠉⠙⢿⣿⣿⣿⣿⣿⡎⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡄⠈⠉⠉⠉⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣯⡀⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡂⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⡂⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣤⣠⣠⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠘⠢⣌⠛⠛⠂⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⣴⣵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⠤⠀⠉⢃⠀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠠⣄⠈⢾⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⠉⠑⠶⠌⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⣠⣿⣷⣿⠟⠋⠀⠀⣿⣷⡶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣶⣶⣶⣦⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⠀⢠⣤⠈⠉⠀⠀⠀⠀⡀⣀⠀⣀⣀⠀⠀⠰⠄⠁⠀⠛⠁⠀⠀⠀⠀⢀⣶⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣦⣴⣶⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣀⣉⠻⢿⣿⡿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⢀⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢹⣿⣧⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⣿⣿⣿⣷⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠟⢿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣷⣤⣶⣿⠋⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡉⠁⠀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⣀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣼⣿⣿⠃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣿⣿⣿⠇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⠏⣃⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⡿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠃⢠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⡟⢰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣀⠀⠀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣶⣾⣿⣶⣶⣶⣶⣶⣿⡏⠀⣶⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢰⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢹⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠈⠉⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⣶⣤⡀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⡀⡀⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣿⣿⣷⣽⣿⣿⠿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠻⠿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠘⢦⣌⠛⠟⠂⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠀⠀⠀⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣗⢤⠄⠉⠁⡀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⣠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⠠⣄⠈⢿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⠉⠑⠶⠌⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⢹⣿⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣶⣶⣷⣶⣄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⡿⠛⠋⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⡿⠋⠀⠀⠀⠀⠠⣦⠌⠉⠁⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢠⢿⡟⠙⡿⠁⠀⠀⠀⠀⢠⣶⣿⣿⢟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠃⠐⠐⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡇⠀⣰⣿⣿⢿⣿⣦⡀⠉⣉⣁⣀⣈⣉⠻⢿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⡇⠰⠿⣿⡇⣠⣿⣿⡇⢰⣿⣿⡿⣿⣿⣧⠀⢀⣠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⢹⣿⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⡿⠃⣾⣿⣿⠀⣿⣿⣿⠁⣾⣿⣿⠟⣿⣿⣿⠀⠀⠀⢀⣶⣶⣦⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣧⣶⣿⣿⣿⣷⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠁⣠⣾⣿⣿⠟⠁⢰⣿⣿⣿⢀⣿⣿⡿⢀⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢰⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⡿⠟⠛⢿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣾⣿⣿⠟⠁⠀⠀⣸⣿⣿⡇⣸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⣿⣸⣿⣿⣿⣿⠃⣼⣿⣿⠟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣤⣶⣿⠟⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⢿⣿⣿⣿⣷⠀⢿⣿⣿⣃⣿⣿⣿⠃⣿⣿⣿⠀⣼⣿⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⣉⡁⡀⠈⠛⠿⠿⠿⠟⠋⠀⢻⣿⣿⣤⣿⣿⡿⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢻⣿⣿⡇⢸⣿⣿⡿⢿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢁⣤⣤⣤⣈⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⣀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⣾⣿⣿⢃⣿⣿⠏⣿⣿⣿⠀⣿⣿⣿⡇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⠟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣿⣿⡟⢿⣿⣷⡀⢰⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠘⠛⠛⠀⠿⢿⡇⢀⣿⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⡇⣾⣿⣿⠃⣼⣿⣿⡿⠿⠿⠀⠀⠀⣿⣿⣶⣶⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⡟⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⢸⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⣇⣀⠀⠀⠀⢠⣿⣿⣿⠉⣿⣿⡇⠀⣸⣿⣿⣿⣿⡆⠀⢰⣶⣤⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⣾⣿⣿⡇⣸⣿⣿⡏⢰⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣧⣼⣿⣿⠇⠠⣿⣿⡿⣿⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣶⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠅⢰⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢿⣿⣿⣧⣿⣿⣿⠇⣼⣿⣿⡇⠀⠁⣀⠀⠀⣿⣿⣿⠟⣿⣿⣏⠀⣼⣿⣿⢹⣿⣿⡇⠰⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢰⣿⣿⡿⢹⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠈⠙⠻⠿⠿⠟⠉⢀⣿⣿⣿⠇⠀⣾⣿⠀⢸⣿⣿⣿⢰⣿⣿⡟⣰⣿⣿⣏⣸⣿⣿⡇⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠃⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣰⣿⣿⡀⢸⣿⣿⡏⣼⣿⣿⣇⣿⣿⣿⠿⣿⣿⣿⡇⣾⣿⣿⣸⣿⣿⣏⣿⣿⣿⠀⣶⣶⣶⣠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠟⢹⣷⣦⣬⣉⣁⠉⠉⠛⠙⠻⠿⠟⠀⣿⣿⣿⢱⣿⣿⡏⣿⣿⣿⢹⣿⣿⡏⠰⢿⣿⡿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⠛⠃⠿⠿⠇⣿⣿⣿⠁⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠉⢹⣿⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠉⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠉⢿⣿⣿⣿⣿⣿⡆⠂⠀⠉⠉⠉⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣶⣶⡄⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣧⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣄⣮⣔⣴⣰⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠕⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⢄⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠘⠢⣌⠛⠟⠂⠀⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⢀⣷⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⣤⠄⠉⠃⠀⠀⠀⠀⢀⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⡄⠀⡀⣤⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠠⣄⠈⢶⣽⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣶⣄⠉⠑⠶⠍⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠋⠀⠀⢸⣿⣶⡾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢿⣿⣿⣷⣦⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⡿⠛⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⡿⠋⠀⠀⠀⠀⠰⣶⡎⠋⠁⠀⠀⣀⣀⣀⣀⣀⣄⡄⠀⠐⣿⡿⠋⡿⠃⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠈⠐⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢹⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣿⣿⣶⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠛⢻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣶⣾⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣇⠁⠀⠘⠂⠛⠋⠋⠉⠙⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠈⢩⣭⡍⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣭⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣯⡿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠙⢽⣍⣰⣴⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣹⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⣒⠒⠶⠤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣿⣿⡿⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡉⠙⢿⣿⣿⣿⣿⣿⣿⡟⠉⠈⠉⠉⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣤⠈⠉⠉⠛⠃⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡉⠋⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⢀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿
⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣽⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠛⠢⣍⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣤⠄⠉⠁⡀⠀⠀⠀⢀⡀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢆⣴⣵⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⢡⣄⠈⢾⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣶⣄⠉⠑⠶⠍⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠁⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠁⠀⠈⣿⡿⡾⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⢻⣿⣿⣿⣶⡌⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡄⠀⣰⡿⠋⠉⠁⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⡿⠃⠀⠀⠀⠀⠘⢿⠇⠈⠀⠀⢀⣀⣀⣄⣀⣀⣄⣄⠀⠠⢾⠿⠁⠾⠁⠀⠀⠀⠀⢠⣾⣿⣿⣟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡛⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣸⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⣿⣿⣿⣿⣶⣦⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⢻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣿⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡘⡀⠱⠉⠓⠉⠈⠈⠉⠁⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⡟⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠅⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⢠⣤⣤⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣷⡁⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣾⣿⣿⡿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠻⣿⣿⣿⣿⡿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡟⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣤⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣦⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣤⣤⣀⡀⠀⠀⠀⢀⣠⣾⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⠿⠷⣶⣦⣤⣤⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣺⣿⣿⣿⣿⣿⡿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⠉⠻⣿⣿⣿⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣧⡄⠀⠉⠉⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣈⠉⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡅⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⢀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣺⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠛⢦⣍⠛⠟⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠂⠀⢀⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣤⠀⠉⠁⣀⠀⠀⠀⢠⡀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⢰⣄⠈⢶⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣦⣄⠉⠱⠶⠍⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠄⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⡀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠘⣿⢿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⢻⣿⣿⣿⣶⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⣰⡿⠋⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⢿⠿⠃⠀⠀⠀⠀⠘⣿⣯⠉⠀⠀⢀⣀⣤⣤⣤⣤⣤⣴⠄⠠⠿⠿⠁⠟⠁⠀⠀⠀⠀⢰⣾⣿⣿⣿⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣷⣾⣿⣿⣿⣿⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠙⢻⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣶⣾⣿⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠎⠀⠇⠉⠉⠁⠁⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣼⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣈⠀⠠⣤⣤⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠇⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⡿⣏⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⢑⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣤⣤⣀⣀⠀⠀⠀⣀⣠⣾⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⠿⠶⣶⣶⠴⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣄⣈⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣷⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣀⣀⠉⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡃⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠃⠀⣀⣀⣈⡉⢉⣉⣉⡉⢉⣉⣉⣉⡙⠿⠿⠟⣋⣉⣉⠻⠿⠟⢉⣁⣀⡉⢉⣉⣉⢉⣉⡉⣉⣈⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⡀⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢠⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡌⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡌⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⢿⣿⣿⣷⠀⣿⣿⢿⣿⡆⠀⠀⣀⣤⣶⣿⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣧⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣄⠀⣿⣿⠀⣻⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣾⣿⣿⣿⣿⣿
⠀⠉⠛⠛⠿⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢹⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠐⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠁⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠐⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⣄⠀⠐⠀⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⠰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣷⣤⠀⠉⢁⡀⠀⠀⠀⢠⡀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⢰⣄⠉⢾⣷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣾⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣄⠉⠓⠶⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⡇⣿⣿⣷⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⣂⡀⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣄⣀⣀⣤⣶⣶⣤⣀⣠⣤⣤⣀⣀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠄⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠀⠀⠀⠀⠿⠟⠅⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⢻⣿⣿⡿⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⣀⣼⡿⠃⠁⠀⠈⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡄⠀⠀⠽⠻⠃⠛⠁⠀⠀⠀⠀⢠⣶⣿⣿⡏⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡍⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣷⣾⣿⣿⣿⣿⣿⣷⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠋⣻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡟⢻⠻⣿⣿⣿⣿⣿⡿⣿⠟⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇⢐⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⣿⣿⡟⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣼⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠠⠤⡤⠄⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠄⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⡟⠟⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣤⣤⣀⣀⠀⠀⠀⣀⣠⣾⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠿⠿⠶⠶⠶⠶⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⡿⠏⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠙⣿⣿⣿⣿⣿⣿⣗⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣶⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡇⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡄⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⡉⢉⣉⣈⡉⢉⣉⣉⣉⡙⠿⠿⠟⣉⣉⣉⠻⠿⠟⢉⣁⣀⡉⢉⣉⣉⢉⣉⡉⣁⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣉⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢠⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡌⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡌⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣷⠀⣿⣿⢿⣿⡆⠀⠀⣀⣤⣶⣿⣿
⣿⣿⣿⣿⣿⠆⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣄⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠉⠛⠛⠻⠂⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢹⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠐⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠑⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠐⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⢤⣀⠀⠀⠀⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠇⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⠰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣷⠤⠀⠉⣀⡀⠀⠀⠀⣠⡀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠈⢰⣄⠙⢾⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣴⣶⡄⠉⠑⠶⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣂⡀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣶⣶⣄⣀⣀⣤⣄⣀⣀⣀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⣀⣠⣀⠀⢸⣿⣿⣿⠟⠀⠀⠀⠀⠿⠿⠇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣀⢹⣿⣿⡿⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣇⣠⣼⡿⠁⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠟⠁⠀⠀⠀⠀⠈⠻⣿⡄⠀⢀⣤⣴⣶⣶⣶⣶⣶⡶⠆⡴⡿⠟⠃⠛⠁⠀⠀⠀⠀⠰⣿⣿⣿⣿⣵⠍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣀⣿⣿⣧⣀⣀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⡏⢉⣉⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠻⡟⠿⣟⠿⠿⡿⠿⡿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡈⡈⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡟⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠘⠿⢶⠆⠈⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠻⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣡⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⣿⣆⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣶⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⡟⣻⣿⣖⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣶⣶⣶⣤⣄⣀⣀⣤⣴⣾⣿⠹⣿⣿⠾⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠿⢿⠿⠿⠋⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣻⣿⠿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡂⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣧⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠⠠⠄⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡏⠀⣀⣀⣀⡀⢀⣀⣈⡉⢉⣉⣉⣉⡛⠃⠀⠀⣀⣀⣀⠀⠘⠃⢀⣀⣈⡉⢉⣉⣉⢉⣉⡀⣀⣀⠀⢀⣀⣀⡈⢉⣉⣉⠉⢉⣉⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⡇⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡜⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡜⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣤⣾⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢀⣴⣿⣿⣿⣿⣿
⠀⠉⠙⠛⠛⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠑⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⣤⣄⠀⠾⠂⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠃⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣿⣤⡌⠛⣛⣄⠀⠀⠀⢀⡀⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠋⢡⣄⠘⢾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣆⠉⠳⣦⡝⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡀⠀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣠⣴⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠶⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣂⣠⣄⡀⢸⣿⣿⣿⠟⠁⠀⠀⠨⡿⠿⡟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⣀⢹⣿⣿⡿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣀⣼⡿⠃⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠟⠃⠀⠀⠀⠀⠈⠻⣿⣇⡀⣀⣤⣴⣶⣶⣷⣷⣿⣿⠷⣾⣿⠿⠋⠙⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣁⣿⣿⣿⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⡟⢉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⡿⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡝⠟⠛⠛⠋⠁⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⠻⢷⠇⠀⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠉⠙⢿⣿⣿⣿⡟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⣿⣿⠉⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠻⢿⣧⣀⠀⠀⣀⡀⣀⣀⣀⣠⣴⣶⣶⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⠃⠀⠀⠀⠀⣰⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣶⣶⣶⣦⣤⣤⣴⣶⣾⣿⠿⡿⠟⠒⠲⢶⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣽⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡂⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣗⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡟⠀⣀⣀⣀⡀⢀⣀⣀⡉⢉⣉⣀⣉⡉⠀⠀⠀⣀⣀⣀⠀⠐⠁⢀⣀⣀⡉⢉⣉⣁⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢉⣉⣉⠉⢉⣁⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⡇⠀⣿⣿⠿⢷⣿⡟⣿⣿⡌⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⡜⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡜⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣠⣶⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣦⣤⣤⣄⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣴⣾⣿⣿⣿⣿
⠀⠉⠛⠛⠛⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠟⠛⠑⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠀⢀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀⣠⡀⠠⣤⣤⣴⡶⠆⠀⠈⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠛⢿⣿⣤⣽⠛⣛⣄⡀⠀⠀⢀⡀⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠀⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠙⠉⠀⠀⠻⠟⢡⣄⠐⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⠀⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣎⠙⣷⣾⡽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣶⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠒⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⠀⠀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣴⣶⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⢶⢶⡷⠒⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣢⣶⣶⣤⣀⠐⣿⣿⣿⡿⠋⠀⠀⠀⣼⡿⣿⠝⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⡲⢮⣤⣙⣿⣿⣿⠿⣿⠊⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣴⣿⠏⠀⠀⠀⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣅⣴⣿⣷⣶⣷⣿⣿⣿⡿⣿⡿⡿⠟⠙⢫⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣠⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡍⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢇⣸⣿⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣛⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⡏⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠛⠛⠋⠉⠁⠀⠀⠁⠁⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣚⣿⡿⠆⠈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⡿⣿⡈⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈⠙⠻⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣡⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣄⣀⣀⣠⣄⣄⣠⣤⣶⣶⣶⣶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⠟⠛⠚⠛⣿⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣧⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⡿⠿⠛⡛⡭⡀⣤⣤⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⣿⣏⠀⠁⠈⠻⠿⠿⣿⣿⣿⣷⣶⣤⣼⠿⠿⠿⠿⠿⠿⠈⠈⠀⠀⠊⠾⢾⣿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠁⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠈⠿⠋⢁⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠈⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣠⣴⣿
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣾⣿⣿⣿⣿
⠈⠙⠛⠛⠿⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢀⣤⣤⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠀⢤⣤⡀⠀⣀⣀⡀⣤⣤⣤⣴⣶⠄⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡖⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠉⣯⡜⢿⣿⣷⣬⣿⠿⣿⣁⠀⠀⠀⠀⠉⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⣿⣷⡄⠙⠿⠋⠁⠸⢿⡿⣧⡀⢶⣿⣷⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⢠⣿⣿⣿⡆⠀⠀⠀⣤⣴⣶⡝⢻⣶⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⣴⣾⣿⣿⣿⣿⡀⠀⠀⠉⠙⢿⣿⠞⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⣤⡀⠀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣶⣶⣤⣄⣤⣤⣤⣀⣀⣀⣀⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣦⣴⣶⣦⣀⣶⣿⣿⣿⡿⠏⠀⠀⠀⢿⠿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣶⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣴⣿⠿⠊⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠟⠛⠉⠁⠀⠀⠹⢿⣿⣿⣿⣦⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⣶⣤⣄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠁⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⡿⡿⡿⣟⡿⡽⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠻⢿⣿⡿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠏⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡁⠀⠀⠀⠈⠉⠛⣶⣌⠁⣶⣦⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠏⢝⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⢶⣤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠿⠋⠉⠛⠛⠋⠹⠶⣶⣶⣶⣶⣦⣤⣀⣀⣀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉⠛⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠻⠿⠿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠏⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠉⠁⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⢀⣀⣤⣶
⣿⣿⣿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣴⣿⣿⣿⣿
⠀⠙⠛⠛⠛⠃⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣤⣄⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⢀⡀⠀⠀⠀⠠⣤⣤⣴⣶⣦⡄⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠈⣭⣶⣿⣿⣦⣹⣿⣿⣿⡋⠁⠀⠀⠉⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⣿⣷⡍⢻⣿⠟⠉⢾⣿⣿⣆⠰⣶⣿⣶⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⢀⣿⣿⣿⡆⠀⠓⢗⣨⣭⣎⠹⣶⣾⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣷⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⣴⣾⣿⣿⣿⣷⡄⠀⠈⠙⠻⣿⣷⣾⠟⠛⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠄⠀⠤⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⠀⡀⣀⠀⠀⢀⣀⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⣀⢀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢠⣤⣴⣦⣤⠀⣶⣿⣿⣿⠇⠀⠀⠀⠸⠿⠛⣦⠰⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣾⠟⠁⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠈⠛⢿⣿⣿⣧⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣶⣦⡀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣸⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣿⠿⡿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠋⠁⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡻⣿⣿⣿⡟⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣿⠛⠃⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⣛⣿⡻⠛⠞⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠉⢷⣶⣶⣦⣶⣶⣶⣶⣶⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠙⠓⠲⠶⢶⣶⣤⣄⣀⣀⣀⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡌⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠟⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠟⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⠀⣀⣠⣶
⠛⠛⠿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠁⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣤⡀⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠰⣾⣷⣤⣛⣿⣿⡟⠛⠁⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠉⠹⣿⠛⠉⣾⣿⣷⡆⢰⣶⣶⣦⠈⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠐⠀⣈⣩⣍⠻⣶⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣿⠟⠛⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢀⣤⣴⣦⣤⡀⣶⣿⣿⣿⠇⠀⠀⠀⠸⠿⠻⣦⠰⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣴⣾⠟⠁⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠈⠛⢿⣿⣿⣷⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣴⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠄⠀⠀⠀⠈⠉⠉⠉⠙⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣸⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⡿⡿⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⠀⠁⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢛⣿⣿⣿⡛⠻⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣿⡛⠁⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠟⢛⣿⡻⠛⠞⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠙⢿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⣶⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠙⠓⠲⠶⢶⣶⣤⣄⣀⣀⣀⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡔⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡯⡙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣟⡩⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠟⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⠀⣀⣠⣶
⠛⠛⠿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠁⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣤⡀⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠰⣾⣷⣄⣛⣿⣿⣿⠛⠁⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠈⠹⣿⠛⠉⣾⣿⣷⡆⢰⣶⣶⣦⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠒⢂⣈⣩⣍⠻⣶⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣽⠟⠛⠉⠁⠀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⠒⢄⠀⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⣤⣦⣦⣤⣀⢠⣾⣿⣿⡿⠖⠀⠀⠀⣶⡿⠿⠠⠂⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣤⣾⠟⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠹⢿⣿⣿⣯⣶⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠋⠛⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣰⣂⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣶⣦⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⣼⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣛⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⡿⠿⠿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⣿⣿⣿⣟⡉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣽⠟⠋⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⣛⣿⡟⠛⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠉⠻⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣏⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣾⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠙⠛⠲⠶⢶⣶⣤⣤⣀⣀⣄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠍⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡧⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣮⠈⠙⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣏⡉⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣀⣴⣾
⠙⠛⠿⣿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣤⣤⣴⣶⣦⡄⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⣿⣿⣦⣽⣿⢿⣿⡋⠀⠠⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠨⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠙⠿⠉⠁⠿⣿⣿⣇⠸⣾⣿⣷⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠉⢀⣤⣼⣦⡹⣷⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡿⠋⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀⢀⣔⣢⣄⣀⠀⣾⣿⣿⣿⠀⠀⠀⠀⣰⣿⠛⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠢⣒⣳⣾⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣇⣠⣼⡟⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠘⢿⣿⣯⣀⣤⣴⣶⣶⣶⣶⣾⣾⣿⣿⣿⣿⡿⠇⠘⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⣠⣢⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣶⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⣴⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⢿⣿⢿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⣿⣿⣯⣿⣏⠙⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡟⠋⠁⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢉⣵⠝⠑⠋⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠛⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣤⣀⣀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⠛⠒⠶⠶⣶⣤⣤⣀⣀⣄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⠉⠉⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⡉⡀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⢀⣀⣠⣶⣿
⠙⠛⠻⢿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⣤⣤⣴⣶⣦⡄⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⢿⣿⣦⣼⡿⢿⣿⣅⠀⠀⢀⡈⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠨⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠙⠛⡉⠀⠻⣿⢿⣇⠰⣿⣿⣷⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠉⢠⣴⣾⣮⡙⣷⣾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣷⠟⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⢀⣀⣠⣀⣀⠀⣾⣿⣿⣿⠂⠀⠀⠀⣰⣿⠟⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢎⢚⣢⣾⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⢠⣼⡿⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠀⠀⠀⠀⠀⠙⢻⣿⣿⣀⣠⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⠏⠘⡃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣄⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣿⣿⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⢿⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠋⠁⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣟⠛⠋⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⣏⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢉⣷⠟⠛⠓⠓⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠉⠻⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣤⣤⣄⣀⣀⣀⡀⢀⣠⣾⣿⣿⣷⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠁⠈⠉⠉⠉⠙⠛⠲⠶⢶⣦⣤⣄⣠⣄⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣗⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡍⠛⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⠉⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⣀⣤⣴⣿⣿
⠉⠛⠛⠿⣿⡇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠘⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⠀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣤⣤⣴⣶⡦⠀⠈⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⢿⣿⣦⣼⠟⣿⣯⣄⡐⢒⣢⣄⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠀⣿⣿⣤⠨⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠉⠉⠁⠀⠻⢿⢿⣦⡙⢿⣿⣷⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡀⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣦⡙⣿⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⠉⠉⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⣀⣠⣷⣶⣤⣀⢰⣾⣿⣿⡿⠂⠀⠀⠀⣶⡿⠿⠔⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣠⣾⡟⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠈⠛⢿⣿⣿⣭⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⠃⢛⡁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⣿⣄⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣯⠛⠋⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣩⣿⠟⠛⠛⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⠻⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣤⣤⣤⣀⣀⣀⣀⣠⣴⣿⣿⣿⣷⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠀⠀⠈⠉⠉⠉⠛⠲⠶⢶⣦⣤⣤⣤⣄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠙⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣅⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣴⣾⣿⣿
⠉⠙⠛⠻⢿⠇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣤⣤⣴⣶⡶⢀⡀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⠻⣿⣷⣾⠟⣿⣷⣴⠜⣛⣥⣤⡈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠉⣤⡀⠻⣿⢻⣦⣙⢿⣿⣷⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣷⣍⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠟⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣦⠄⣀⡆⠀⠰⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⣄⣤⣾⣴⣤⣀⢰⣿⣿⣿⡷⠀⠀⠀⢠⣾⡿⢷⠆⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⡾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣠⣿⡏⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣴⣤⣶⣾⣾⣶⣶⣶⣾⣿⣿⣿⣿⠿⠉⢫⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣛⣿⣿⣆⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⠟⠛⠛⠛⠻⢿⡟⠟⢟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⢿⣿⣿⣿⣯⣍⠙⠋⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡧⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⢿⣟⢜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡾⠒⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠉⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣰⣿⣿⣿⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⣦⣤⣄⣀⣀⣀⣀⣤⣶⣿⣿⣿⣶⣶⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁⠀⠀⠈⠉⠉⠉⠛⠳⠶⠶⠦⠤⢤⣤⡄⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⠙⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣴⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⡿⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⡼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠁⠙⠛⠻⠿⠇⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⠸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡅⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣇⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⡇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⡇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⢦⣤⣴⣶⠷⠀⣀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠈⠻⣿⣿⡾⠛⣿⣷⣤⠈⣈⣥⣤⡈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣾⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠹⢶⠀⠙⣿⢻⣶⣍⣻⣿⣷⣶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣷⣍⣿⣿⡿⠛⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢰⡶⠴⣵⡆⠀⢰⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣠⣤⣴⣿⣿⣶⣄⣺⣿⣿⣿⠗⠀⠀⠀⢰⣾⠿⢷⠄⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣴⣿⠇⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣵⣶⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⠛⠻⠻⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣸⣿⣿⣦⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⠻⠿⠛⠛⠛⠿⠻⠿⡿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣿⣿⣿⣿⡟⠉⠉⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠋⠋⠙⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣶⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣶⢶⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠛⠶⠶⠦⠤⠤⣦⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡏⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠘⠻⠿⠿⡿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠀⠉⠛⠛⠿⠃⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢲⣤⣴⡶⠟⠂⣀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠈⠛⢿⣿⡾⠛⣿⣷⣦⡀⣃⣥⣤⡄⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠉⢶⠂⠙⣿⡹⣷⣍⡻⣿⣷⣶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣎⣿⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠉⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⢰⠖⠢⢰⡆⠀⢠⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣁⣤⣴⣴⣿⣿⣶⣤⣻⣿⢿⣿⠏⠀⠀⠀⢸⣿⡟⢳⡃⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠛⠯⠿⠟⣿⣿⣿⣿⢿⠯⠛⠿⡇⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧⣼⡿⠃⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠟⠋⠁⠩⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣄⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣼⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠟⠻⠛⠙⠛⠛⠛⠻⢿⢿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⠿⣿⡏⠉⠉⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣿⣷⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠙⠛⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠋⠛⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⣀⣀⣀⣤⣶⣶⣿⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣴⣶⣶⣶⣦⣤⣤⣤⣤⣶⣿⣿⣿⣿⡶⠶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠁⠀⠀⠀⠀⠉⠉⠉⠙⠛⠲⠶⠶⠤⠤⠶⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣙⡿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠
⠿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠀⠈⠙⠛⠛⠃⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣇⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⡝⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣧⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣇⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣇⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢲⣤⣴⡶⠟⠂⢀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠈⠛⣿⣿⠾⠛⣿⣷⣤⠐⣋⣥⣤⡄⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠉⠶⠀⠉⣻⡹⣷⣍⣻⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣎⣿⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡀⢸⠒⠢⢤⡆⠀⢠⠀⠘⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣠⢴⣶⣾⣿⣿⣿⣦⣹⡿⠿⠿⠃⠀⠀⠀⠸⣿⡟⣵⡄⠈⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠟⠿⠿⠿⢿⣿⣿⣿⠿⠋⠯⡯⠿⢿⣿⣿⡿⡿⠛⠙⠻⢃⢀⡀⠀⠀⣀⣀⣀⣨⣿⣿⣷⣾⡿⠃⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⢻⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⡿⠋⠉⠀⠘⠆⠀⠀⠀⠀⠿⢿⣿⣿⡿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣷⣆⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠏⠋⠉⠙⠛⠛⠛⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟⠿⠿⠅⠈⠁⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣇⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠛⢻⣿⣿⣶⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠛⢿⣿⣿⡿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠛⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠉⠙⢷⣦⡀⠀⢀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣶⣶⣶⣶⣦⣴⣦⣶⣿⣿⣿⣿⠶⠶⠶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠲⠶⠦⠴⠶⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⠿⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠘⠛⠿⠿⠿⢿⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣁⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠
⠿⣿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⡼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⢀⣠⣶⣿⣿⣿
⠀⠀⠉⠛⠛⠃⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣯⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⣦⣤⣴⡶⠗⢀⣀⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠘⠻⣿⣷⠿⠛⣿⣷⣤⠘⣩⣤⣤⡈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠱⡆⠀⠙⣛⢿⣶⣉⢿⣿⣷⣦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣷⣍⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⡀⠸⠆⠀⠰⡆⠀⠰⡀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣷⣿⣿⣿⣿⣿⣿⣾⡟⠉⠉⠁⠀⠀⠀⠀⢿⣇⣼⡇⠀⣸⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠐⠿⠻⠷⠹⣿⣿⣿⣟⢛⠈⣉⣤⣤⣠⣄⣄⣤⣤⣤⣴⣿⣿⡿⣿⡟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣄⣀⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠟⠛⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣷⡀⣀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡏⠐⠁⠐⠉⠉⠋⠟⠛⠛⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⡿⡟⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⠛⠛⠃⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⡷⣧⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠙⠿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡾⠋⠁⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣄⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⡿⠛⠛⠷⠶⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠒⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣩⣙⡿⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠙⠿⠿⠿⠿⣿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤
⠿⠿⣿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣶⣿⣿⣿
⠀⠀⠈⠉⠛⠁⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠐⣦⣤⣴⣶⠆⣀⡀⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠙⢿⣿⣶⠿⢻⣿⣶⣴⠿⣛⣧⣤⠘⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠉⠱⡆⠨⠛⣿⢿⣦⡙⢿⣿⣷⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣶⣍⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠂⡀⡆⠘⡇⠀⠲⣦⠀⠐⡆⠀⢲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⡇⠀⠈⠀⠀⠀⠀⠀⠛⣧⣤⣷⠀⣸⣿⠋⠁⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠻⠿⠟⠋⠈⠟⠻⠳⠙⠻⣿⣿⣿⣯⣀⣠⣴⣶⣶⣶⣶⣴⣶⣾⣶⣿⣿⡿⢻⡟⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠁⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢻⣿⣿⣄⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡟⠿⠿⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⣩⡿⠻⣿⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠛⠿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣾⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⠿⠿⠛⠛⠛⠒⠶⢶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣟⠛⠿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠻⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠻⠿⠿⠿⡿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤
⠛⠿⢿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⣦⣤⣴⣶⠗⢀⢀⠘⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠙⠻⣿⣷⠿⢛⣿⣷⣶⡽⣛⣥⣤⡈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠱⠶⠀⠙⣿⢻⣦⡙⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣽⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠐⣶⠰⠄⢠⡀⠳⠀⠀⣶⡄⠀⠄⠀⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⡇⠀⠈⠀⠀⠀⠀⠀⠛⣧⣤⣷⠀⣸⣿⠋⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠋⠈⠿⠻⠗⠝⠻⣿⣿⣿⣏⣀⣠⣴⣶⣶⣶⣶⣴⣶⣶⣶⣿⣿⡿⠻⠏⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣿⣿⣄⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⡿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⡟⠿⠿⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠃⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣭⡿⠻⣿⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠛⢿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣤⡾⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⠿⠿⠛⠛⠛⠒⠶⢶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠛⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣟⠛⠿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤
⠛⠿⢿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣾⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⣦⣤⣴⣶⠗⢀⣀⠘⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⣶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠙⠻⣿⣷⠿⢛⣿⣷⣶⠽⣳⣧⣤⡈⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠈⠱⠶⠈⠛⣿⢻⣦⡙⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣽⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣴⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠰⠄⢢⡀⠳⠀⠀⣴⡄⠀⢄⠀⠘⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣾⣿⣿⣷⣾⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣠⣾⠟⠁⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠁⠀⠈⠈⠁⠀⠈⠹⣿⣿⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⣽⣿⣿⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢿⠛⠻⠛⠟⠿⠿⠿⣿⣿⡿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⠛⠛⠛⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⡭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣨⣼⠋⠻⠋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣠⣀⣀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⠉⠙⠒⠲⠶⣶⣦⣤⣀⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣧⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣏⠛⠛⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠛⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣗⢄⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣾
⠉⠛⠿⠿⣿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⡀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣀⣴⣦⣤⡛⢡⣤⣄⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠈⠛⢿⠟⠛⢱⣿⣿⣟⠉⣥⣾⣧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⣯⡹⣷⣮⣻⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣬⡿⠛⠋⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣦⣄⠀⠀⠀⠐⠀⠀⣠⠄⠀⠀⠐⣶⠀⠀⢰⡀⢆⡀⠀⣷⡄⠀⣆⠀⠈⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣴⣿⣿⣿⣷⣶⣿⣿⠿⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣤⣾⠟⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠁⠀⠁⠑⠀⠀⠀⠈⠿⣿⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠟⠁⠉⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣃⣸⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⠟⠝⠋⠛⠋⠛⠻⠿⠿⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⡟⠛⠛⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣿⣿⣿⣿⠧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⣽⠛⠉⠎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠻⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⠉⠉⠉⠉⠙⠒⠶⢶⣶⣦⣤⣀⡀⠀⢀⡀⠹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣏⠉⠛⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣾
⠉⠛⠿⠿⣿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⣀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⣤⣤⣤⣴⣎⣡⡤⣠⠀⠙⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠙⠿⠟⠛⣱⣿⣿⣿⣃⣴⣾⣿⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢈⣬⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡿⠛⠋⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣶⡖⠀⢰⡀⢶⡀⠀⣵⡄⠀⣦⠀⠈⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣴⣿⣿⣿⣿⣷⣾⣿⠏⠀⠙⠁⠀⠀⢀⠀⠀⢀⣀⣀⣼⣿⣿⣶⣿⠟⠁⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠈⠁⡁⡀⠈⠈⢻⣿⣷⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⢟⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣃⣸⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⡙⠛⠝⠋⠙⠉⠙⠛⠛⠿⢿⠿⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡻⣿⡿⠿⣯⡍⠉⠛⠃⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣇⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣼⠋⠙⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠛⠻⣿⣆⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⣠⣤⣄⣀⣀⠀⠀⠀⢀⣠⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠈⠉⠉⠉⠙⠓⠲⠶⢶⣦⣤⣀⡀⣀⣀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡉⠙⠛⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣾
⠀⠉⠻⠿⠿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢁⣤⡄⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣤⣤⣴⠶⣮⣉⣔⠲⠂⢀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠙⠛⠛⠁⠻⣿⣿⣯⡑⢾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢡⣤⣴⣾⡙⢿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡾⠛⠉⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣄⡀⠀⠀⠀⠆⠀⢀⡄⠀⠀⠀⢳⡦⠂⠀⣆⠰⣆⠀⢰⣦⠀⢰⡆⠈⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣣⣴⣿⣿⣿⣾⣿⣾⣿⠏⠀⠙⠁⡀⣀⣀⠀⢀⢀⣀⣄⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠙⠛⠛⠋⠀⠀⠈⢁⣁⣀⡀⠈⠻⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⢛⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⣼⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡻⠐⠏⠁⠍⠍⠉⠙⠛⠿⠿⠿⡿⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢪⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡻⣿⣿⠿⢿⣭⠋⠛⠃⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⡟⡌⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⣿⣿⡿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣼⠻⠋⠹⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠙⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⣠⣤⣤⣄⣀⡀⠀⠀⣀⣠⣾⣿⣿⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠈⠉⠉⠉⠉⠛⠲⠶⢶⣦⣤⣄⣀⣀⣀⠀⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡙⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡉⠙⠛⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣉⣈⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣶
⠀⠉⠛⠿⠿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠐⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠘⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⣴⣤⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⣤⣤⡶⣾⣏⣤⣶⡶⢄⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠙⠛⢛⠋⠘⣿⣿⣿⣉⠶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⣾⣶⣍⢻⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠹⣶⠀⠀⣦⠀⢦⡀⢰⣦⡀⠀⡆⠀⠙⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣤⣶⣷⣶⣿⣾⣿⣾⡿⠟⠀⠙⠋⠀⣀⢀⣀⣀⣀⣠⣤⣼⣿⣿⣷⣾⠟⠁⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠋⠀⠀⠀⠈⣁⢈⡈⡹⡻⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⡟⠋⠁⠀⠀⠆⠀⠀⠀⠀⠿⣿⣿⢿⡿⡿⠋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⣼⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢝⠙⡽⠽⠑⠙⠙⠛⠛⠟⠿⠿⠿⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣟⣿⣿⠿⢿⡍⠉⠙⠃⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⡤⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣼⠛⠋⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠙⠛⢿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣠⣾⣿⣿⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠉⠉⠉⠉⠛⠓⠶⠶⣶⣤⣄⣀⣀⣠⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠉⠙⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⠈⠈⠛⠛⠟⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠈⠛⠿⠿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⣀⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠈⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⣴⣦⡄⠈⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣤⣶⢶⣿⣡⣴⣧⢄⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠈⠛⠛⡋⡀⣿⣿⣿⣏⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣤⣶⣮⡹⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣦⡀⠀⠀⠀⠰⠀⢀⡤⠀⠀⠀⠸⣶⠐⢄⢶⠈⢦⡀⢀⣶⡄⠀⣆⠀⠉⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣶⣶⢾⣿⣾⡿⠿⠀⠈⠋⠀⠀⡀⢀⣀⣀⣀⣤⣴⣿⣿⣿⣾⠿⠃⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠈⢁⠀⣀⣈⣙⢿⣿⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⠛⣿⡿⠋⠁⠀⠀⠠⠀⠀⠀⠀⢿⡿⣿⢿⡿⡿⠋⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣋⣸⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠍⢟⠹⠏⠍⠉⠉⠝⠛⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣙⣿⣿⠿⢿⡯⠉⠛⠁⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢽⣿⣿⣿⣿⠅⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠻⢿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣰⠟⠋⠉⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠛⠻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣧⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠙⠓⠶⠶⣶⣦⣄⣀⡀⠀⣀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣉⠙⠛⠛⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠛⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠙⠻⠿⢿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⢀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⠃⣴⣦⣄⠀⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣀⣶⣶⣿⠛⣻⣤⣄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠈⠻⣿⠟⠛⢳⣿⣿⣷⠛⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣈⣹⣭⡹⣷⣮⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⣮⡿⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⣄⡀⠀⠀⠀⢰⠀⠀⣰⠀⠀⡀⠘⣶⠐⠀⢲⡈⢧⠀⢈⣷⡀⠀⢲⠀⠉⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣶⣶⣲⣶⣾⣿⣾⡿⠿⠀⠈⠛⠀⢀⠀⣀⡀⣀⣀⣤⣤⣿⣿⣿⣾⡿⠃⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠛⠁⠀⠀⠀⠁⠈⢀⣈⡙⣿⣿⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⠛⣿⡿⠛⠋⠀⠀⠤⠀⠀⠀⠀⠿⡿⣿⣿⣿⡿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⣸⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠍⠋⠻⠟⠙⠋⠛⠛⠛⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡛⣿⣿⠿⣯⡭⠉⠛⠃⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢹⣿⣿⣿⣿⡁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣠⠟⠉⠉⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠛⠻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣿⡟⠉⠛⠉⠉⠉⠉⠓⠲⠶⢶⣶⣤⣀⣀⠀⠀⢀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣹⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣉⠙⠛⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠻⠿⣿⣶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣀⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠈⠛⠿⢿⣿⠟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⢀⣀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠘⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⣴⣦⡄⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⡀⣶⣶⣿⠿⠛⣤⣀⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⡶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⢻⣿⣷⣿⠚⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⢈⣩⣯⠹⣶⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣭⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢰⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⣦⣄⠀⠀⠀⠀⢰⠀⠀⣠⠄⠀⡀⠐⣶⠀⠀⢲⡈⢧⠀⠈⣷⡄⠀⠣⠀⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣶⣶⣾⣿⣿⣿⠿⠀⠘⠛⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣼⡿⠃⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋⠛⠛⠛⠛⠛⠿⠿⠿⠟⠁⠀⠀⠀⢁⢀⣀⣀⣉⣻⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⠻⣿⡿⠛⠋⠀⠀⠤⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠋⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠂⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣸⣿⣿⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠝⠛⠹⠛⠙⠋⠛⠟⠟⠻⠿⣿⡿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡻⣿⣿⣿⣿⣯⠛⠛⠃⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠉⢻⣿⣿⣿⣿⢅⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣱⠟⠃⠑⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠛⠿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣶⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⡿⠛⠛⠉⠉⠉⠙⠒⠶⢶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣮⣝⠛⠛⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠙⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢟⣉⣁⣀⡀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⢀⣀⡀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠻⠿⣿⣿⡟⠀⣿⣿⠿⢷⣿⡟⣿⣿⣼⣿⣿⢿⣿⡄⠀⢰⣿⣿⣿⡇⠀⢠⣿⣿⢻⣿⣼⣿⣿⢸⣿⣧⣿⣿⢰⣿⡟⣿⣿⣼⣿⣿⠀⢸⣿⡿⠿⠀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⠀⣿⣿⢿⣿⡆⠀⠀⠀⣀⣠⣿⣿
⠀⠀⠀⠀⠁⠀⠀⣿⣿⣶⣼⣿⡇⣿⣿⡇⣿⣿⣾⣟⠃⠀⢸⣿⣿⣿⣧⠀⠈⠻⣿⣷⣤⢸⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣭⣭⣽⣿⣿⠀⢸⣿⣷⣶⣤⣤⣤⠀⣿⣿⠀⣿⣿⣿⣿⠀⣿⣿⣾⣿⡃⠀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣷⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⠀⠰⣶⣮⢻⣿⣿⣿⣿⢸⣿⡿⣿⣿⢸⣿⡇⣹⣿⣿⣿⣿⣀⣸⣿⣇⣀⠛⠛⠛⢀⣿⣿⠀⣿⣿⣿⣿⡄⣿⣿⣸⣿⡇⠀⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠛⠻⠟⠛⠘⠛⠛⠘⠛⠃⠀⠛⠛⠃⠛⠛⠀⠀⠛⠻⠿⠛⠙⠛⠛⠘⠛⠃⠛⠛⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠀⠛⠛⠘⠛⠃⠛⠛⠛⠛⢃⣤⣤⡀⠙⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⢰⣶⣾⣿⡿⠃⡀⣀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⢰⣶⣶⢰⣶⣶⣶⣶⣶⢰⣶⣶⣶⣦⠀⠀⢰⣶⣶⣶⣤⢰⣶⣶⣦⠀⣴⣾⣷⣦⡄⣶⣶⣶⡆⣴⣾⣷⣦⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⣻⣿⣾⣾⠻⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠁⣿⣿⣤⢸⣿⣿⣿⡿⠀⠀⢸⣿⣇⣿⣿⣾⣿⣿⣿⠀⣿⣿⠛⠛⠃⣿⣿⣤⡀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠔⠘⢻⣯⢿⣶⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⠉⢸⣿⣿⣿⣿⡄⠀⢸⣿⡟⠋⠉⣿⣿⣿⣿⠀⣿⣿⢻⣿⡇⣿⣿⠉⠁⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⢸⣿⣿⣶⣾⣿⣿⣿⣿⠁⠀⢸⣿⣧⠀⢠⣿⣿⢹⣿⡇⢻⣿⡾⣿⡇⣿⣿⣶⡆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣄⡄⠀⠀⠀⠀⢰⠀⠀⣰⡆⠐⡀⠐⣶⠐⠀⢰⡈⢷⠀⠙⣷⣄⠀⢣⠀⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

                    ䷩ 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

╒═══════════════════ 𝐃𝐀𝐈𝐋𝐘 𝐋𝐈𝐍𝐊𝐒 ═════════════════════════════════════════════╕

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.13.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


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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