𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Wednesday, February 02, 2022
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Generated Thu 3 Feb 02:42:14 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/2022/02/02/


╒═══════════════════ 𝐑𝐄𝐂𝐄𝐍𝐓 𝐁𝐔𝐋𝐋𝐄𝐓𝐈𝐍𝐒 ════════════════════════════════════╕
  Previous bulletins in IPFS (past 21 days, in chronological order):

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QmRXpMsysY15jnTdXabNj8RQWyYQoA287zCaWvJJjNWzB2
QmaSCmshgLNtetvtehrJTkVzA4UKVtdrEf2DcHz8b5gBoV
QmZWJhSMSsSbuTS3JfxxEo3tfs2QfTCENeVipKSaQVU5b2
Qmehhm7o1VbxWg8uyeTcYkk6Hr1Cp4Ho21X2wCnesY9asd
Qmf3TSbMJS9koYs7ieZ1w3WBpaDp6x7rx5xnhC6B134fFL
QmbAVHY5Ges1nF9HNsiNn6FsQW8UdTFHURKDG1QgSKpKw7
QmZnZkCBXLzZg5rkewh3peFLzQ5xAZRm6HpjAvmtf39ELR
QmZeHUaFKnCwMohgf9hT7KAvQEFvHbBZ8MUZTtceSkoLZc
QmQLbqvdSXzQ59gWzwwt9RARegF8vZB68qihXTbNcDNoxy
QmXRhRDa28yw47e5ym8hgWTNSCTtmH5t22GCpvFvaGYDwN
QmNrH8cLM4mYaiBXzhdkhAQm8XM54JzRQa3Z1Ng4a41kcV
QmZjLA2jtgeEhSJihNxjJW3q7KaEU6iyCKve9pmkci3m61
QmTWgjCkngZ2r4jN8fZGmd7bpEdJL3pBfU6vKWFNxPmkYf
QmUa6NW563bcVpbJi9K42Aap4iTTXYmJcu5bTsjAcRc9yG
QmPENr81muU2jHsHXxDP2oXCR4itHu2Pb54BpV4z2a6v8g
QmT1WmWT8oyJF7h6FDnkDWXE6tEV7sUdo6MyboEn1kLiFx
Qmd7o1W8PgtXmeAFmAGFoVreMpLA1AVtvVjUMvVr7zc1YQ
Qma88nEdV64rLJsqR8uVd62B8zwrDvuX6Etg42wsovLPUQ
QmWMn5rx4mwf86jmAsH6KUasKtr2WVybRVa9b2sJECtwpz
QmS1wrYoYvTPNPTvZYppVeYPXTR8rHjtMkwCEYGCEaEWUU


╒═══════════════════ 𝐈𝐍𝐃𝐄𝐗 ═══════════════════════════════════════════════════╕

⦿ AMD Apparently Sends Crates of CPUs to Phoronix (But It’s Definitely Not a Bribe!) | Techrights

⦿ IBM Has Not Been Good to Red Hat (the Company Should Never Have Been Sold) | Techrights

⦿ IRC Proceedings: Tuesday, February 01, 2022 | Techrights

⦿ Linux.com Used to Clock Millions of Hits Every Week, But Linux Foundation Can Barely Keep Thousands | Techrights

⦿ Darkening the Counterpart’s Reputation: Microsoft-Friendly (Sometimes Microsoft-Funded) Media Describing Every Security Issue  -  Even Windows Issues  -  as ’Linux’ | Techrights

⦿ One Year Since We Joined Geminispace and the Effort Has Paid Off | Techrights

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
		http://techrights.org/2022/02/02/epyc-trove/#comments
		http://techrights.org/2022/02/02/ibm-ruins-red-hat/#comments
		http://techrights.org/2022/02/02/irc-log-010222/#comments
		http://techrights.org/2022/02/02/lets-talk-linux-com/#comments
		http://techrights.org/2022/02/02/security-perception-manipulation/#comments
		http://techrights.org/2022/02/02/year-of-geminispace/#comments

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
		http://techrights.org/2022/02/02/easyos-3-3/#comments
		http://techrights.org/2022/02/02/gstreamer-1-18-6/#comments
		http://techrights.org/2022/02/02/weston-10-0-0/#comments

                    ䷩ 𝚕𝚒𝚗𝚎 64

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/02/epyc-trove/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/02/epyc-trove/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ AMD_Apparently_Sends_Crates_of_CPUs_to_Phoronix_(But_It’s_Definitely_Not_a
Bribe!)⠀✐


Posted in Hardware at 7:34 pm by Dr. Roy Schestowitz


Posted earlier today:


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇amd-crate⦈_


Either those ads pay Phoronix a lot of money, or electricy costs have gone down
considerably, or…


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇EPYC_shop⦈_


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇AMD_EPYC_Milan_X_EPYC_7773X_64_Core_Server_CPU⦈_


Summary: Over a decade ago AMD earned some notoriety for the way it was
schmoozing bloggers, so we don’t rule out anything…


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⣿⠀⢀⠠⢊⡗⡊⡼⣁⠀⠈⢊⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣍⡛⠛⠋⠀⠀⠀⣰⣿⣿⠏⠀⣤⣿⣿⣿⣿⣿
⣿⣔⡫⠔⣸⠚⣡⠞⠁⠀⠀⠀⠪⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣿⣿⠏⠀⣼⣻⣿⣿⣿⣿⣿
⣿⠛⣵⢎⣵⠞⢀⠀⠀⠀⠚⠉⠀⠁⣻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣀⣴⣿⣶⣌⠛⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣼⣿⣿⡿⣂⣼⣿⣿⣿⣿⣿⣿⣿
⣿⡓⠄⠛⢁⣾⣿⡅⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠛⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣾⡟⠈⠙⢥⣿⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣵⣿⣿⣿⠃⠀⠀⠀⢀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⡙⠻⡿⠿⠃⠀⠀⠀⢀⣾⠏⢀⠀⢀⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣬⣤⣤⣾⣿⣿⣿⣝⢿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣾⣟⣠⣿⡗⡾⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠉⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⢟⣯⡻⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠀⠀⠀⠀⢠⣿⣿⣳⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣾⣿⡿⢿⣿⣿⣿⣿⣻⣿⣿⡇⠔⠉⠂⠙⢿⣝⢿⡻⣿⢿⡿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡻⠃⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣏⣤⣴⣸⣿⣿⠿⣿⣽⡿⢛⣮⣶⣦⣤⣵⣄⡈⢿⣯⡻⣷⣝⡞⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡝⠁⠀⠀⠀⠀⣰⣿⡿⢁⣾⣿⠿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡟⢿⠟⢑⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣮⡛⢿⣿⣝⢝⢿⡿⠏⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣴⣿⣿⣷⣾⣿⣿⡄⢨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣷⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡔⣂⢰⠢⢴⠀⡆⢰⢀⡍⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠁⠈⠈⠀⠈⠀⠉⠁⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣏⠸⠘⡟⠛⠛⣿⠉⡏⠋⠉⡍⣏⠩⠛⠟⡻⠛⢻⢻⡏⢉⠛⡛⢛⠛⠛⠛⢻⠛⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⣿
⣿⣷⣶⣶⣷⣶⣰⣿⣶⣶⣿⣶⣶⣿⣶⣶⣷⣷⣷⣾⣾⣷⣿⣶⣶⣶⣶⣶⣾⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣠⣀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠛⠛⢟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⡿⠿⣶⠶⠶⠶⡶⠶⣖⡶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣿
⣿⣷⣷⣷⣷⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣟⣙⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⡿⣿⢿⢿⡿⠿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣶⣦⣷⣷⣿⣾⣶⣷⣾⣿
⣿⡏⠙⠋⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣇⣀⣁⣉⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
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                    ䷩ 𝚕𝚒𝚗𝚎 188

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/02/ibm-ruins-red-hat/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/02/ibm-ruins-red-hat/

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Gemini_version_available_♊︎


✐ IBM_Has_Not_Been_Good_to_Red_Hat_(the_Company_Should_Never_Have_Been_Sold)⠀✐


Posted in Deception, GNU/Linux, IBM, Red_Hat at 12:09 pm by Dr. Roy Schestowitz


Video_download_link | md5sum 60c2a78d757c755fa338bb6e347ca54c
IBM does not get Red Hat
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/ibm-ruins-redhat.webm


Summary: More than three years after IBM said it would be buying Red Hat the
latter company is a shadow of its former self and many_IBM_layoffs_carry_on
while Red Hat staff — including many executives — leave in droves


A COUPLE of years ago we wrote about what_IBM_had_done_to_Red_Hat_less_than_2
years_after_the_takeover. We occasionally touched the subject again after that.
It has not improved; it seems to have only worsened. The oversensitive but
racist IBM does not tolerate any criticism!


Today, as in so far in 2022, Red Hat’s Web site is occasionally boosting
Microsoft and other “clown” partners, which make IBM seem a bit like SUSE —
mostly a reseller of proprietary software and seller of IBM’s legacy (very
expensive) systems. We keep posting links about that trend in our Daily Links,
though lately we’ve not composed any actual articles about the subject. In a
nutshell, IBM does to Red Hat as a company what it did to Fedora as a community
and to CentOS as a project. IBM is run (ruined) by truly incompetent managers.
They only care about short-term “compensation” of theirs. They don’t know the
people. They don’t care about the product (they don’t even use it).


“These people actually decide to leave their jobs.”This is very sad to me. It
really is. Those who have been reading this site since 2006 very well know that
we were supportive of Red Hat, which many perceived as the archenemy of SUSE/
Novell back when we campaigned to “Boycott Novell” (it was this site’s name!)
and when Jim Whitehurst became CEO we wrote an open letter to him. He later
offered to do an interview with us — a short interview that we published over
10 years ago.


After the takeover Whitehurst became a president at IBM (a “dream job” maybe;
just maybe… if you worked in the IBM of the 1970s!), but that didn’t last long.
This former CEO left IBM and several other CxO-level folks from Red Hat did the
same, including presidents and vice presidents. It’s a lot like IBM of Ginni
Rometty, except now it happens at Red Hat. It’s alarming because it’s difficult
to run a business with a ‘leadership turnover’ quite so profound and quite so
rapid. Those shake-ups aren’t strategic. These people actually decide to leave
their jobs. Is Chris Wright (CTO) next to quit? CEOs rarely leave or resign
unless there’s a major scandal, as that tends to cause a major PR/morale crisis
and the Board members (for shareholders) don’t like that, so they prefer to
spin the whole thing as a peaceful transition of power from one person to
another person. Maybe that’s even in the contract (CEOs never resign in anger
or ‘ragequit’). Look what happened in Microsoft’s GitHub a couple of months
ago. It’s like a_face-saving_cover-up.


“That’s why many people leave (those who can anyway).”Shown above (as video
posted) is the latest high-profile departure (the financial department). Yet
another top-level executive, the CFO of the company (and a lot more, also a
token ‘activist’), just_leaves. It was announced about a day ago (ignore the
face-saving headline and read between the lines), reaffirming what we’ve long
said about IBM ruining what became of Red Hat after more than 20 years. Sooner
or later IBM too will die (while faking its finances, e.g. by offloading losing
units), so the writings on the wall are generally there. That’s why many people
leave (those who can anyway). It is willful.


“IBM has basically discarded red hat, I’m not sure what the point of the
acquisition even was given how things have been (mis-)handled,” an associate of
ours said today.


Wrong assumptions had been made about Red Hat and then they paid too much
mostly for a brand, not recognising how GPL (or copyleft) actually works. It’s
anti-monopoly. A source once told us that attacks on Linus Torvalds and his
undeserved_'suspension' (very shortly after the announcement of a mega-
takeover) were meant to lower the price of buying Red Hat. That was a year
before quote-mining a decade-old RMS on the most problematic issues, not
technical_or_legal_issues. That’s mentioned in passing in the video above. █


                     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇IBM and the Hat⦈
                            IBM and the (tall) Hat


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣩⣾⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣿⣿⣿⣿⣿⣿⣇⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⣨⠟⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠯⠀⣰⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢐⢀⣾⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢼⣯⣶⣥⣽⣿⣿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠄⣺⣿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣐⠰⠒⣿⣽⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢙⣫⣼⣚⣊⡘⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣂⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠨⣭⣱⣿⡺⣿⠃⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠛⠛⠛⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠁⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣴⣻⣝⣿⡿⢿⡠⢾⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡷⠀⣾⣿⣄⠀⠀⠀⠨⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣥⡐⠒⠻⢿⣿⣿⡿⣶⣮⣿⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⣦⡀⠠⣭⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣼⣻⢿⡳⣿⣿⣿⣿⣿⠃⣹⣿⡟⣻⣿⠋⢿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⠥⠀⣿⣿⣿⡿⡿⡾⡿⡿⠋⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⠙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣿⣿⣿⢿⣿⡿⠀⡟⣿⠀⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣯⠚⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣌⡉⠛⠛⠐⢹⣦⣄⣷⢽⣜⣧⣿⡀⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢻⣿⣇⣿⣿⢻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣝⠘⣿⣿⣿⡗⢸⣿⣿⣿⣿⣿⣿⣴⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣼⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣘⣶⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⣿⣿⣇⠀⢠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣄⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣸⠟⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠯⠀⣰⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⢴⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢼⣯⣶⣤⣿⣿⡿⠿⠟⡿⣿⣿⣷⣶⣤⣀⠀⣺⣿⠿⠿⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⢀⡀⢘⣁⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢈⢰⣻⣳⣿⠂⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣔⣔⠷⠚⠾⠂⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠋⠛⠛⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⠀⠑⠋⠋⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣴⣻⣿⣿⣿⡿⣆⣼⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡇⠀⣾⣿⣄⠀⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣥⡲⠛⠷⢿⣿⣿⣿⣻⣄⣾⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⣦⡀⠰⣨⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣼⣿⡿⠯⣿⣿⣿⣿⣿⠃⣹⣿⡟⣻⣿⠋⢿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⠥⠀⣿⣿⣿⣿⣿⡯⡟⠿⠋⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⠙⣿⣿⣷⣼⣿⣿⠛⢿⣿⣦⣷⡽⠿⢻⡿⠀⡟⣿⠀⢾⠷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣯⠒⠒⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠛⠓⠑⢩⣦⣄⣷⢽⣜⣧⣿⡀⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢻⣿⣇⣿⣿⣻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣝⠘⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣴⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠸⣿⣿⣯⣽⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣼⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣘⣶⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣾⣿⣿⣿⣿⣿⣇⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣦⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⣸⠟⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠝⠯⠀⣰⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⣤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢼⣯⣶⣥⣿⣿⣿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⣺⣿⠿⠿⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⢀⢀⢘⡅⡠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⢹⣿⣿⣷⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⢀⣻⣛⣿⡂⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠀⣄⣴⣗⠻⠾⠁⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠛⠒⠓⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⠉⠁⠋⠁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣴⣿⣿⣿⡿⣿⣿⢾⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡧⠀⣾⣿⣄⠀⠀⠀⠈⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣥⡀⠓⠱⢿⣿⣿⡿⣖⣽⣿⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⣦⡀⠰⣦⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣼⣿⡿⠷⣿⣿⣿⣿⣿⠃⣹⣿⡿⣻⣿⠋⢿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⠤⠀⣿⢿⣿⣿⡿⡿⡿⠿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⠙⣿⣿⣷⣬⣿⣿⠛⢿⣿⣮⣿⣿⡿⢿⡟⠀⡟⣿⠀⢾⠷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⣿⣿⣿⣿⣿⣿⣯⠀⠒⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠛⠛⠃⢹⣦⣄⣿⢿⣜⣧⣿⡀⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢻⣿⣇⣿⣿⣻⣿⣷⢿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣝⠘⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣴⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣼⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣩⣾⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣶⣾⣿⣿⣿⡇⠀⠀⠠⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⣿⣿⣿⣿⣿⣿⣇⠀⢀⡅⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣨⠟⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠯⠀⣸⡃⠸⣿⢿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢐⠠⢼⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢼⣯⣶⣤⣿⣿⣿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⣸⣿⠿⠿⠿⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢈⣘⡬⢿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠜⡲⣻⣩⣫⠐⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠱⣗⣕⣿⡽⣛⠅⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠛⠛⠛⠁⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠙⠋⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣔⣻⣿⣿⣻⣿⢟⣹⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡧⠀⣾⣿⣄⠀⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣥⡖⠚⠛⢿⣿⣿⣿⣮⣿⣿⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⡿⠀⣰⣿⣿⣿⣦⡀⠨⣦⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣼⣻⡿⢧⣽⣿⣿⣿⣿⠃⣹⣿⡟⣻⣿⠋⢿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⠧⠀⣿⣿⣿⣿⣿⡟⡾⠿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⠙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⣿⣿⣟⢿⡟⠀⡟⣿⠀⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣾⣿⣿⣿⣿⣿⣯⠀⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣍⡉⠛⠛⠛⢨⣦⣄⣷⢽⣜⣧⣿⡀⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢻⣿⣇⣿⣿⣻⣿⣿⢿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣝⠘⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣴⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣄⡈⠘⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣼⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣩⣾⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣶⣾⣿⣿⣿⡇⠀⠀⠠⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⣿⣿⣿⣿⣿⣿⣇⠀⢀⡅⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⣨⠟⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠯⠀⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢐⠠⢬⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣿⣤⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢼⣯⣶⣥⣿⣿⡿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⢾⣿⠿⠿⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣘⡬⢿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠜⡲⣻⣩⣓⠃⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣔⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠱⣗⣕⣿⡽⣟⠆⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠛⠛⠛⠁⠀⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠋⠛⠉⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣴⣻⣽⣿⣿⣿⣕⣿⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡧⠀⣾⣿⣄⠀⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣥⡐⠒⠘⢿⣿⣿⡿⣄⣫⣿⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⣦⠀⠨⣽⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣼⣻⢿⣿⣿⣿⣿⣿⣿⠃⣹⣿⡿⣻⣿⠋⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⠧⠀⣿⣿⣿⣿⣿⡟⢾⡿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⠙⣿⣿⣷⣼⣿⣿⠛⠿⣿⣮⣿⣝⡿⣿⡿⠀⡟⣿⠀⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣾⣿⣿⣿⣿⣿⣯⠀⠒⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠛⠛⠈⢸⣦⣄⣷⢽⣜⣧⣿⡀⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢻⣿⣇⣿⣿⢻⣿⣷⣻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣝⠘⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣴⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣄⡈⠘⠿⣿⣶⣤⡀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣼⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢲⣦⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣿⣿⣿⡇⠀⠀⠐⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣇⠀⢀⡀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣔⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢈⠟⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠯⠍⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢴⣯⣶⣤⣽⣿⡿⠿⠟⠿⣿⣿⣷⣶⣤⣀⠀⣸⣿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⢇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣗⢟⣲⠃⢸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣄⡭⠝⠟⠇⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢞⠣⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠛⠓⠛⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⠈⠉⠋⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⡤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢠⠇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣴⣿⣿⣿⣿⣿⣯⣺⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡟⠁⢸⣿⣷⡀⠀⠀⠨⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣯⣄⡂⠒⠘⢿⣿⣿⣿⣬⣉⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⡠⣿⣿⣿⣷⡀⠈⣷⣾⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣼⣻⠿⠳⣿⣿⣿⣿⣿⠃⣿⣿⡿⣻⣿⠋⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡗⠀⣿⣿⣿⣿⣿⡿⡽⠿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡛⠛⠿⢿⠟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠐⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠒⠂⢠⣦⣄⣿⢹⣔⣧⣿⡀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠀⣿⣿⣯⡿⠉⣿⣿⣿⣿⣿⣿⣷⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⣄⡈⠙⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣷⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣧⠀⢠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣸⡿⡏⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠅⣴⣁⡸⣿⣿⠇⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡿⠿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢴⣯⣶⣥⣿⢿⡿⠿⠟⢿⣿⣿⣷⣾⣤⣀⠀⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠀⠸⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡾⢶⣝⠃⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣼⢛⠿⠿⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠚⠛⠛⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠃⠈⠑⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⢀⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢔⣻⣿⣿⣿⣟⣛⣽⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡿⠁⢸⣿⣷⡀⠀⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣅⡒⠂⠺⢿⣿⣿⣿⣠⣿⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⢀⡠⣿⣿⣿⣷⡀⠸⣷⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣳⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣻⢹⣷⣿⣿⣿⣿⣿⠁⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⡿⡟⢟⡿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣼⣿⣿⠛⠿⣿⣯⡩⣿⣛⣿⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠒⠒⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣅⡉⠓⠛⠈⢡⣦⣤⣷⢻⣔⣧⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢹⣿⣇⣿⣿⣻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠉⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣎⠙⣿⣿⣿⣼⣿⣿⣹⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣦⡄⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣾⣿⣿⣿⣿⣿⣇⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣑⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣸⡿⡏⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠅⣼⡁⠸⣿⣿⠇⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡒⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⣤⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢴⣯⣶⣤⣽⣿⡿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⣼⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⡸⡊⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣷⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢘⣚⣼⣘⠣⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣰⢛⠓⠿⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠖⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠚⠋⠛⠀⠀⠀⠀⢻⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠃⠘⠘⠉⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢀⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣔⣿⣿⣿⣿⣿⣄⣸⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡿⠁⢸⣿⣷⡀⠀⠀⠈⡲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣅⡚⠚⠻⢿⣿⣿⣿⣯⣭⣿⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⢀⡠⣿⣿⣿⣷⡀⠸⣗⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣳⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⠽⢷⣿⣿⣿⣿⣿⠁⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡿⢫⡿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣷⡻⡿⠿⢿⠟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠐⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠂⠙⢢⣦⣤⣷⢹⣔⣧⣿⡀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢹⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣼⣿⣿⣽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣦⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣽⣿⣿⣿⣿⣿⣧⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣸⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠅⣰⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡿⢿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⣠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢴⣯⣶⣥⣽⣿⡿⠿⠟⣿⣿⣿⣷⣶⣤⣀⠀⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠁⡸⠫⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣾⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣛⣾⣛⠇⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣸⢓⠚⠿⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠑⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠛⠛⠛⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠃⠈⠉⠉⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣔⣺⣿⣿⣻⣿⣮⣼⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡿⠁⢸⣿⣷⡀⠀⠀⠈⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣯⣥⡲⠲⠻⢿⣿⣿⣿⣔⣼⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⢈⡠⣿⣿⣿⣷⣀⠰⣷⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⣽⠯⣿⣿⣿⣿⣿⠃⣻⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡯⡯⡿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣮⡻⠿⣿⡿⠟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠘⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠛⠉⠉⢠⣦⣤⣷⢻⣔⣧⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢹⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣎⠙⣿⣿⣿⣴⣿⣿⣹⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⣾⣮⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣴⣷⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣰⣿⣿⣿⣿⣿⣿⣧⠀⢠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⣸⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠅⣰⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡿⠿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠜⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⡂⣾⣾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢴⣯⣶⣥⣿⣿⣿⠿⠟⣿⣿⣿⣷⣶⣤⣀⠀⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⢀⢸⣿⣿⣿⣏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⡀⣿⣿⣻⣿⣛⠯⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⣨⣿⣿⣷⣿⣿⠕⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠋⠛⠛⠛⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⢐⣻⣽⣿⣿⣿⣔⣺⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡿⠁⢸⣿⣷⡀⠀⠀⠈⢲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣥⡒⠚⠟⢿⣿⣿⣿⣷⣹⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⢈⡤⣿⣿⣿⣷⣀⠸⣗⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⠽⠏⣿⣿⣿⣿⣿⠁⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡛⢯⡿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣷⣻⣿⢿⣿⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠘⢒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣅⡉⠓⠂⠈⢡⣦⣤⣿⢻⣔⣧⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢹⣿⣇⣿⣿⣻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣎⠙⣿⣿⣿⣼⣿⣿⣽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣰⣦⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣽⣿⣿⣿⣿⣿⣧⠀⢀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣸⠿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠅⣰⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢴⣯⣶⣤⣼⢿⣿⠿⠟⣿⣿⣿⣷⣶⣤⣀⠄⣸⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠁⡸⠫⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣛⣾⣛⠇⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣸⢓⠚⠿⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠔⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠛⠚⠛⠀⠀⠀⠀⢻⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁⠘⠉⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣰⣻⣽⣿⣻⣿⣍⣾⣿⡇⣿⣿⣿⣿⡟⠛⠛⠉⠀⣠⣼⡿⠁⢸⣿⣷⡀⠀⠀⠈⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣥⡚⠒⠻⢿⣿⣿⣿⣐⣽⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⢀⡠⣿⣿⣿⣷⡀⠨⣯⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣳⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⠽⠟⣿⣿⣿⣿⣿⠁⣻⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡯⢏⠿⠍⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⣿⢿⠿⡿⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠒⠒⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠛⠁⠚⢰⣦⣤⣷⢻⣔⣧⣿⡀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢹⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠃⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣎⠙⣿⣿⣿⣼⣿⣿⣹⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣮⣷⣿⣿⣿⡇⠀⠀⢀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⣩⡟⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠯⠅⣼⣃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢐⣠⣾⣾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢴⣯⣶⣵⣿⣿⣿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠄⣼⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⣸⣿⣿⣿⣂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣃⣿⣿⣭⣿⣟⡧⢸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣤⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⣜⣿⣿⣿⣻⣿⡅⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠛⠛⠛⠛⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣲⣿⣽⣿⢯⡿⢝⣻⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣼⡿⠁⢸⣿⣷⡀⠀⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣅⡒⠒⠘⢿⣿⣿⣿⣐⣟⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⢈⣠⣿⣿⣿⣷⡄⠸⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣳⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣻⠽⢿⣿⣿⣿⣿⣿⠃⣹⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡿⢟⠿⠏⠡⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣧⡛⣟⣛⡻⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠒⢒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠛⠓⠊⢡⣦⣤⣷⢻⣔⣧⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣯⡿⠁⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣯⣿⣿⣿⡿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣇⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣤⣶⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⣨⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠅⣰⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⣤⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢴⣯⣾⣤⣿⣿⡿⠿⠟⠿⣿⣿⣷⣶⣤⣀⠀⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣨⡫⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢙⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⢹⢿⣓⠃⢼⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣰⢋⠟⠿⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠔⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠻⠒⠑⠀⠀⠀⠀⢻⣿⣿⣿⣿⠓⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠈⠉⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢀⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢤⣖⣿⣟⣿⣻⣟⢲⣼⣿⡇⣿⣿⣿⣿⡟⠛⠛⠋⠀⣠⣾⡿⠁⢸⣿⣷⡀⠀⠀⠈⠲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣥⡚⠓⠻⢿⣿⣿⡿⣷⣚⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⢀⡠⣿⣿⣿⣷⣄⠰⣾⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣳⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⠽⠿⣿⣿⣿⣿⣿⠁⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡿⢯⡿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣯⡙⢛⡻⢿⠟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⠘⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠉⠁⢠⣦⣤⣷⢻⣜⣧⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢹⣿⣇⣿⣿⣻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣫⡿⠃⣿⣿⣿⣿⣿⣿⣷⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠘⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣆⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⡧⠀⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣾⣿⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⡀⣨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣨⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠫⠍⣸⣃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢴⣯⣶⣥⣿⣿⣿⠿⠟⣿⣿⣿⣷⣶⣤⣀⠀⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠸⡋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠂⡞⡜⣳⣚⠓⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣒⡧⠳⠳⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡢⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠓⠊⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠃⠈⠑⠉⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢠⡆⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣰⣿⣿⣿⣿⣿⢤⣾⣿⡏⣿⣿⣿⣿⣟⠛⠛⠉⠀⣠⣾⡟⠁⢘⣿⣿⣇⠀⠀⢈⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⢿⣿⣿⣄⡐⠊⠂⢿⣿⡏⡘⣿⢉⣱⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⣸⣿⣿⣿⣧⠀⣧⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣻⠫⡍⣯⣿⣿⣿⣿⠃⣽⣿⡿⣿⣿⠋⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡷⠿⠿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣿⣾⣿⡿⡿⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠐⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠐⢡⣦⣄⣷⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⢹⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣤⣾⣿⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⡀⣨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣨⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠅⣸⡃⠸⣿⢿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡴⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢴⣯⣶⣥⣿⣿⡿⠿⠟⢿⣿⣿⣶⣶⣤⣀⠀⣼⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠰⣋⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢙⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠄⠺⡪⣲⣚⠂⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡢⣒⠛⠻⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠃⠃⠐⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁⠙⠉⠀⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢠⡆⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠤⣲⣿⣿⣿⢿⣿⣔⣾⣿⡏⣿⣿⣿⣿⣟⠛⠛⠉⠀⣠⣾⡿⠀⢘⣿⣿⣆⠀⠀⠀⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣄⡐⠒⠚⢿⣿⡿⣘⣟⢭⣽⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣾⣿⣿⣿⠀⢀⣸⣿⣿⣿⣧⠀⣤⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣋⠓⢀⣝⣿⣿⣿⣿⠃⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡷⣯⡿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣶⡿⠿⡿⣿⡟⠀⡏⣿⠁⢸⡷⠀⢘⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠁⢩⣦⣄⣿⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠂⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⢽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣤⣾⣿⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⡀⣨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣨⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠅⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢴⣯⣶⣤⣵⢾⡻⠟⠛⠿⣿⣿⣷⣶⣤⣀⠀⣨⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⢀⢉⢫⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢙⣿⣿⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢲⣻⢟⡝⠃⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡴⣓⠚⠿⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡳⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠛⠊⠓⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠃⠉⠙⠁⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢠⡇⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢤⡒⣺⣽⣿⣿⣿⢆⢿⣿⡏⣿⣿⣿⣿⣟⠛⠛⠉⠀⣠⣾⡿⠀⢘⣿⣿⣆⠀⠀⠀⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⢿⣿⣿⣅⡘⠓⠀⢿⢿⡿⡚⡏⢿⣻⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⢀⣸⣿⣿⣿⣧⠀⣵⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣛⠻⡁⣿⣿⣿⣿⣿⠃⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡭⢾⠿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣧⡹⢿⠿⢿⠟⠀⡏⣿⠁⢸⡷⠀⢘⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠊⠐⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠁⢰⣦⣄⣿⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠊⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠘⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⣶⣵⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⣶⣶⣿⣿⣿⡇⠀⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣶⣾⣿⣿⣿⣿⣿⣇⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣨⡿⡏⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠅⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣰⣻⣻⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢴⣯⣶⣤⣿⣿⡿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⣾⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⢀⢈⢝⡻⡿⠋⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⢹⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣈⢊⡼⣛⢭⣮⠁⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢡⢶⣞⣣⡽⢿⠄⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢠⡇⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢤⣔⣿⣟⣿⣛⣿⢃⣸⣿⡇⣿⣿⣿⣿⡟⠛⠛⠉⠀⣠⣾⡟⠀⢐⣿⣿⣧⠀⠀⠈⢱⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣄⡀⠃⠚⢟⣛⠻⠟⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⢀⣸⣿⣿⣿⣇⠀⣀⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⡳⠍⣬⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⡭⣽⡿⠯⠈⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣯⡿⡭⢿⢿⡟⠀⡏⣿⠁⢼⡿⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠄⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢻⣦⣄⣿⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠘⣿⣿⣿⡿⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⣶⣵⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣶⣶⣿⣿⣿⡇⠀⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣞⣮⣿⣿⣿⣿⣿⣿⣇⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣨⡿⡏⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠅⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢤⣿⣟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢴⣯⣶⣥⣿⣿⡿⠿⠿⢿⣿⣿⣷⣶⣤⣀⠀⢮⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⡀⢨⠴⡿⡿⡃⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠌⠯⣊⣵⣊⢾⠭⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⢩⣽⣎⣺⠯⣿⠆⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⠓⠀⠁⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣔⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⢠⡗⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡤⣒⣿⣽⣟⣟⣿⣿⣽⣿⡇⣿⣿⣿⣿⡟⠛⠛⠉⠀⣠⣾⡿⠀⢐⣿⣿⣦⠀⠀⠀⢵⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣆⡀⠓⠺⢿⣝⣻⢿⣿⣯⣽⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⢀⣸⣿⣿⣿⣧⠀⣐⣾⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣍⢁⣐⣙⣿⣿⣿⣿⠃⣾⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⡿⣿⠿⠏⠈⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡿⠿⡿⢿⡿⠀⡏⣿⠁⢸⡿⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠀⠄⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠃⢠⣦⣄⣿⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⣶⣵⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⣶⣶⣿⣿⣿⡇⠀⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣲⣾⣿⣿⣿⣿⣿⣇⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣨⡿⡏⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠅⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⢐⣷⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢴⣯⣾⣤⣿⣿⣿⠿⠟⢿⣿⣿⣶⣶⣤⣀⠀⣸⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⢨⢎⢿⢿⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⢂⠭⡑⢃⣕⠝⠾⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠡⣫⣚⣿⠿⣻⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡠⡄⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⢠⡗⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡤⣴⣻⣽⣿⣿⣿⡷⣺⣿⡏⣿⣿⣿⣿⡟⠛⠛⠉⠀⣠⣾⡟⠀⢐⣿⣿⣧⠀⠀⢈⠵⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⢿⣿⣿⣄⡐⠃⠒⢿⣛⠿⣻⣿⣷⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⢀⣸⣿⣿⣿⣇⠀⣠⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⠉⢜⣿⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣽⡿⠏⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣿⡿⡿⢿⣿⡟⠀⡏⣿⠁⢼⡿⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠕⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠈⢠⣦⣄⣿⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣯⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠘⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⣶⣵⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣶⣶⣿⣿⣿⡇⠀⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⣶⣥⣿⣿⣿⣿⣿⣿⣇⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣮⣦⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣨⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠅⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣞⢽⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢴⣯⣶⣥⣿⣿⡿⠿⠟⣿⣿⣿⣷⣶⣤⣀⠄⣺⣿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⣨⠼⢿⣿⢵⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢄⣚⣔⡝⢃⣝⠽⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢡⣶⣦⣾⡿⢾⠄⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⠋⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠟⠛⠛⠛⠛⠋⠉⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⢄⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢠⡇⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢤⣰⣻⣽⣿⣿⣾⢏⣿⣿⡏⣿⣿⣿⣿⡟⠛⠛⠉⠀⣠⣾⡟⠀⢐⣿⣿⣧⠀⠀⠈⢱⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⢿⣿⣿⣄⡐⠃⠐⢾⣝⠫⣿⣿⣟⣾⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⢀⣸⣿⣿⣿⣇⠀⣠⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣻⠉⢸⣽⣿⣿⣿⣿⠁⣽⣿⡿⣿⣿⠋⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣽⡿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣿⡿⠯⢿⢿⡿⠀⡏⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠈⠄⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠊⢩⣧⣄⣿⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠊⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠘⠿⣿⣶⣤⡀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣮⣺⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣵⣾⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠨⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠂⣼⡃⢸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠝⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡤⣵⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢴⣯⣶⣥⣿⣿⣿⠿⠿⣿⣿⣿⣷⣶⣤⣀⠔⣾⣿⠿⠿⠿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠀⣠⣿⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠉⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠨⡿⣿⣿⣿⣿⡏⠸⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣽⡿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠚⠛⠓⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠉⠉⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣤⡆⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣴⣿⣿⣿⣿⣿⢫⣽⣿⡗⣿⣿⣿⣿⣛⠛⠉⠁⠀⣠⣾⡿⠀⠀⣿⣿⣿⡄⠀⠀⢓⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣥⡰⠣⠚⢯⣽⣉⣟⠿⡷⣺⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⣀⣿⣿⣿⣿⡀⣠⣾⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡿⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⣁⣴⣾⣿⣿⣿⣿⠁⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⡿⣿⡿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣾⡻⠟⢿⣿⡟⠀⡏⣿⠁⢼⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠋⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠓⢢⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣾⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢐⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⡿⡟⠉⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠂⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣥⣠⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣤⣽⢿⡿⠿⠟⠿⣿⣿⣶⣶⣤⣀⠀⢬⣿⠿⠿⠿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⡠⡄⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠨⠯⠚⣠⠏⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠕⠘⠓⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠂⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡴⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⡆⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣔⣿⡽⡿⣻⣳⠆⢹⣿⡇⣿⣿⣿⣿⡛⠛⠋⠁⠀⣠⣾⠏⠁⠀⣿⣿⣿⡄⠀⠀⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣤⡐⠒⠲⠿⡯⠿⠷⠯⡽⣾⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠀⣀⢀⣿⣿⣿⣿⡀⣀⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣖⣅⡺⣽⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣯⣿⠧⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡿⡼⡿⠿⡿⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⡐⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠂⢠⣦⣄⣷⢹⣼⣧⣿⡀⢸⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣽⣿⣽⠈⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣢⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠀⢤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⡿⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠂⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣥⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢴⣯⣶⣴⣿⢿⡿⠿⠟⠿⣿⣿⣷⣶⣤⣀⠀⢨⣿⠿⠿⠿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⣠⡤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠸⠮⡈⢠⡂⠘⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠐⢉⠵⠂⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⠽⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠒⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣴⣿⣽⣿⣸⣷⡂⣽⣿⡏⣿⣿⣿⣿⡛⠛⠋⠁⠀⣠⣾⠿⠁⠀⣿⣿⣿⡄⠀⠀⢲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣥⡀⠂⠋⠿⡍⣾⢻⣳⣺⣾⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⢀⣿⣿⣿⣿⡠⣄⣺⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣎⢀⠐⣴⣾⣿⣿⣿⠃⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⡿⣿⠿⠧⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣮⡻⣏⡿⢿⠟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⡂⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠈⢲⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⡀⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣢⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⢀⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⡿⡟⠌⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠃⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣅⣠⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣤⣿⣿⡿⠿⠟⠿⣿⣿⣷⣾⣤⣀⠀⢬⣿⠿⠿⠿⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⣠⡤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠉⢹⣿⣿⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠸⠮⡈⢠⡊⠘⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠐⢉⠵⠃⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠣⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠒⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠀⠀⠀⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣴⣿⣿⣿⣹⣿⡀⣿⣿⡇⣿⣿⣿⣿⡛⠛⠉⠁⠀⣠⣾⠟⠁⠀⣿⣿⣿⡄⠀⠀⠲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣤⡀⠂⠚⠾⡏⠾⠿⠼⣿⣺⣿⣿⠇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠀⣀⣀⣿⣿⣿⣿⡠⣄⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣄⢀⠐⣿⣿⣿⣿⣿⠃⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣿⡿⠧⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣮⡻⠟⡿⡿⠟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⣀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠈⢰⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣽⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⢿⣿⣿⣥⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠠⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⡿⡟⠈⣩⣭⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠂⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣇⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢴⣯⣶⣤⣷⣿⡿⠿⠛⢿⣿⣿⣷⣶⣤⣀⠀⣸⣿⠿⠿⠿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⣈⢆⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠉⢹⣿⣿⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠘⠮⣺⣄⠏⠘⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⠘⠽⡓⠄⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⠭⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠂⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠁⠀⠈⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣴⣿⣿⣿⣽⣻⣁⢼⣿⡇⣿⣿⣿⣿⡛⠛⠉⠁⠀⣠⣾⠿⠀⠀⣿⣿⣿⡄⠀⠈⢲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣤⡀⠂⠹⠾⠧⣮⠥⡟⣾⣺⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠀⣀⣀⣿⣿⣿⣿⡀⣄⣹⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣮⢿⡉⣴⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣿⡿⠧⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡚⠋⡿⠿⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⡐⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠁⢠⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⢿⣿⣿⣥⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠠⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣨⡿⡟⠈⣩⣭⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠂⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣧⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢴⣯⣶⣤⣷⣿⡿⠿⠛⢿⣿⣿⣷⣶⣤⣀⠀⣼⣿⠿⠿⠿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⣈⢆⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠉⢹⣿⣿⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠘⠮⣺⣄⠏⠘⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⠘⠽⡓⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠂⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠀⠀⠈⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣔⣿⣿⣿⣿⣻⣂⢺⣿⡏⣿⣿⣿⣿⡛⠛⠉⠁⠀⣠⣾⠟⠀⠀⣿⣿⣿⡄⠀⠀⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣤⡐⠒⠻⠻⡍⢎⢱⡿⣝⣾⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⢀⣿⣿⣿⣿⡀⣄⣾⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡟⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣏⠴⠨⣿⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⡿⣟⡿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡬⠿⣿⠿⡟⠀⡏⣿⠁⢸⡿⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠒⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠉⢡⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⢿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣹⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣾⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⡿⡟⠉⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠂⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢴⣯⣶⣤⣿⣿⡿⠿⠟⠿⣿⣿⣷⣶⣤⣀⠀⢼⣿⠿⠿⠿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⣠⢄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠉⢹⣿⣿⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠘⠮⣚⣠⠏⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⠘⠝⡾⠇⠀⣿⣿⣿⣿⣿⣿⠂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠂⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠀⠀⠀⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢴⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣴⣿⣿⣿⢿⣿⠖⣿⣿⡇⣿⣿⣿⣿⣛⠛⠋⠁⠀⣠⣾⡟⠁⠀⣿⣿⣿⡄⠀⠀⠱⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣥⡀⠂⠚⠿⠯⠾⠻⠝⢟⣾⣿⣿⠇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⡿⠀⣀⢀⣿⣿⣿⣿⡀⣄⣾⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡿⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣀⣴⡀⣝⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣟⡿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣿⣿⣷⣬⣿⣿⠛⠿⣿⣶⡺⡿⢿⠿⡟⠀⡏⣿⠁⢸⠷⠀⢘⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⡀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠁⢡⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠠⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠂⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢴⣯⣮⣤⣿⣿⡿⠿⠛⢿⣿⣿⣷⣶⣤⣀⠀⣨⣿⠿⠿⠿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⣠⢄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠉⢹⣿⣿⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠘⠮⣚⣠⡏⠚⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⠘⠝⡾⠁⠀⣿⣿⣿⣿⣿⣿⠂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠂⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡖⣿⣟⣿⣻⣽⣄⢾⣿⡇⣿⣿⣿⣿⣛⠛⠋⠁⠀⣠⣾⠏⠀⠀⣿⣿⣿⡄⠀⠀⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣤⡀⠊⠺⠿⠽⠻⢟⢿⣿⣾⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⡿⠀⣀⣀⣿⣿⣿⣿⡀⣄⣾⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣔⡂⣠⣶⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⡿⣿⡿⠯⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣿⣻⣿⡿⠿⡟⠀⡏⣿⠁⢸⡿⠀⢘⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⡀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠘⢢⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣯⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⢻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⢽⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣦⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣿⣿⣿⡇⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⠎⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣄⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣯⣦⣥⣽⣿⡿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⣸⣿⠿⠿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣢⢨⡢⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠸⣣⢟⠯⡲⠘⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠘⠋⡙⠋⠂⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠛⠂⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠟⠉⠈⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢴⣶⣻⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣤⡇⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣴⣿⣿⣿⣤⠊⡐⣾⣿⡇⣿⣿⣿⣿⡛⠛⠁⠀⢀⣴⣿⠟⠀⠀⣽⣿⣿⣆⠀⠈⠱⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⢿⣿⣿⣷⡖⠒⠀⠎⢆⠈⡹⠿⣟⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⣀⣻⣿⣿⣿⡀⣄⣺⣿⣿⣿⣿⢷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡟⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⣿⣥⠬⠿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡻⣝⡿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡸⠀⢛⢻⡟⠀⡏⣿⠁⢸⡿⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠈⢡⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⡃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣿⣇⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⠎⡏⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣸⡃⡸⣿⢿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣠⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣯⣶⣦⣽⣿⡿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⢨⣿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣤⣠⣗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠉⢹⣿⣿⣷⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⢸⢷⡿⠦⠣⠘⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠁⠘⠋⡹⠙⠂⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡕⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠛⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠛⠋⠁⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣤⡆⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣴⣻⣽⢿⡤⣡⡗⣼⣿⡏⣿⣿⣿⣿⡛⠛⠁⠀⢀⣴⣿⠟⠀⠀⣽⣿⣿⣆⠀⠈⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡔⠂⠀⠆⡡⣋⡽⠿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⡿⠀⣀⣀⣻⣿⣿⣿⡀⣄⣾⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡿⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣾⢤⠥⢴⣟⣿⣿⣿⠁⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⡟⣿⣿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡰⠁⢙⣫⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠘⠆⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠈⢠⣦⣄⣿⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣇⠙⣿⣿⣿⣴⣿⣿⣽⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣢⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡤⠞⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠅⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣯⣦⣤⣽⢿⡻⠟⠟⠿⣿⣿⣷⣶⣤⣀⠀⡮⣿⠿⠿⠿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣭⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠉⢹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠨⢵⠘⠌⠃⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⡣⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠤⣉⡚⠝⠙⢛⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠅⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠓⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⠉⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣤⡇⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢔⣿⣿⣿⡵⣌⡛⣹⣿⡏⣿⣿⣿⣿⡛⠛⠁⠀⢀⣼⣿⡟⠁⠀⣽⣿⣿⣆⠀⠈⠲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡔⠁⠀⠨⡷⡯⣳⡟⣿⣼⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⣀⣻⣿⣿⣿⡀⣠⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡟⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣢⠯⠹⠲⠿⣿⣿⣿⠁⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⡿⢿⡿⠯⠈⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡓⠊⠿⢿⡿⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠁⢡⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣟⣿⠿⡿⠿⠿⠾⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣢⣠⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡤⠎⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣯⣶⣥⣿⣿⡿⠟⠻⢿⣿⣿⣷⣶⣤⣀⠀⣼⣿⠿⠿⠿⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣼⠄⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠉⢹⣿⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⢨⣵⠜⠎⡃⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡢⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠨⠩⠏⠻⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠆⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠒⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠓⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠟⠋⠈⠁⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣴⡇⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣴⣪⣽⣿⡢⡦⢥⣼⣿⡏⣿⣿⣿⣿⡛⠋⠁⠀⢀⣼⣿⠿⠁⠀⣽⣿⣿⣆⠀⠈⠲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡔⠚⠀⠬⡵⡌⠕⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠂⣀⣀⣻⣿⣿⣿⡀⣤⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡟⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣷⣚⣓⠒⢿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⡿⣟⡿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡀⠿⡻⣿⡟⠀⡏⣿⠁⢸⡷⠀⢘⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⠊⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢀⣦⣄⣿⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣿⣟⣿⠿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⡿⣿⣿⣥⣄⡀⠐⠿⣿⣶⣤⡀⠁⠚⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣦⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣴⣴⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣢⣀⣬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡤⠎⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⡇⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡿⢿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣯⣦⣤⣽⢿⡻⠟⠟⢿⣿⣿⣶⣶⣤⣀⠀⡸⣿⠿⠿⠿⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣼⠄⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠉⢹⣿⣿⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⢨⣵⠜⠞⠃⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⡤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠨⠩⠏⠹⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠒⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠂⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣴⡇⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢔⣻⣿⣿⠀⡶⠦⢻⣿⡏⣿⣿⣿⣿⡛⠛⠁⠀⢀⣼⣿⡿⠁⠀⣿⣿⣿⣆⠀⠀⠱⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡔⠄⠈⠄⡀⡒⠢⡼⣿⣶⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⣀⣻⣿⣿⣿⡀⣀⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡟⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣳⠭⠑⠚⢿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⡿⣿⡿⠏⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡊⠀⢛⠻⡟⠀⡏⣿⠁⢸⡿⠀⢘⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠘⠋⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠁⢀⣦⣄⣿⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⡿⣿⣿⣥⣄⡈⠐⠿⣿⣶⣤⡀⠁⠚⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⢽⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣦⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣴⣴⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣢⣀⣬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡤⠎⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⡇⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡿⢿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣯⣦⣤⣽⢿⡻⠟⠟⢿⣿⣿⣶⣶⣤⣀⠀⡸⣿⠿⠿⠿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣼⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠉⢹⣿⣿⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⢨⣵⠜⠌⠅⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠨⠩⠏⠻⠀⠀⣿⣿⣿⣿⣿⣿⠂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠅⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠒⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠈⠉⠈⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢲⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣴⡇⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣔⣯⣽⣿⠠⣤⠍⣾⣿⡏⣿⣿⣿⣿⣛⠛⠁⠀⢀⣼⣿⡿⠁⠀⣿⣿⣿⣆⠀⠀⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡤⠀⠸⠌⠁⠀⠣⣜⣟⣾⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⣀⣀⣻⣿⣿⣿⡀⣤⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡟⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣣⢍⠸⠉⢽⣿⣿⣿⠃⣽⣿⡿⣿⣿⠃⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣟⡿⠯⠈⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡛⠀⠛⢛⡟⠀⡏⣿⠁⢸⡿⠀⢘⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠊⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢨⣦⣄⣿⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠊⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⡿⣿⣿⣥⣄⡀⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣢⣠⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡤⠎⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣸⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣯⣦⣤⣽⢿⡿⠟⠟⠿⣿⣿⣷⣶⣤⣀⠀⢼⣿⠿⠿⠿⢟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣼⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⢹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⢨⣵⠜⠌⠅⠸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⡂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠨⠩⠏⠻⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠒⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠓⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⢴⣶⣻⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣤⡇⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢔⣻⣽⣿⣶⢭⢄⣸⣿⡟⣿⣿⣿⣿⡛⠛⠁⠀⢀⣼⣿⡟⠁⠀⣽⣿⣿⣆⠀⠈⢲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡜⠀⠈⠨⣰⣥⢔⢿⣿⣾⣿⣿⡇⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⡿⠂⣀⣀⣻⣿⣿⣿⡠⣔⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡟⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣚⢑⡣⣝⢿⣿⣿⣿⠁⣿⣿⡿⣿⣿⠃⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⡿⣿⣿⠯⠈⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣿⡛⠀⠹⣻⡟⠀⡏⣿⠁⢸⡷⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠂⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣹⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠈⢡⣦⣄⣷⢹⣴⣧⣿⡀⢸⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢹⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣿⣟⣿⠿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⡿⣿⣿⣥⣄⡀⠐⠿⣿⣶⣤⡀⠁⠚⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣾⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣷⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⠀⢀⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣖⢛⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⡃⣸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⡇⠅⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠝⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⢖⣶⡖⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣾⣥⣿⣿⡿⠿⠿⣿⣿⣿⣷⣾⣤⣀⠔⣾⣿⠿⠿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⢸⣷⣿⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡉⢹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢼⣲⣿⣿⣿⣿⡿⢽⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢋⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⣿⣿⣿⣿⠂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠛⠛⠋⠙⠂⠀⠀⠀⢻⣿⣿⣿⡿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠟⠋⠉⠉⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣰⣶⡇⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣔⣪⣽⣿⢄⡦⡡⣾⣿⡇⣿⣿⣿⣯⣉⣁⢀⣀⣴⣾⣿⡟⠀⠀⣿⣿⣿⡄⠀⠀⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡆⠇⠀⠠⡹⣝⡙⣿⣧⣽⣾⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⠀⣀⢀⣽⣿⣿⣿⠀⣤⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡟⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣤⠶⠊⢟⠿⣿⣿⣿⠇⣽⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⡿⣿⣿⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡒⠉⠛⣿⣿⠀⡟⣿⠁⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡈⠂⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣧⣄⣷⢻⣼⣧⣿⡀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣖⣾⣷⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣷⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣷⣿⣿⣿⣿⣿⣿⣷⠀⠠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣖⠛⡯⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⡷⠂⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⢀⠀⠁⠍⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⠦⣶⣖⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⣶⣵⣿⣿⡿⠿⠟⠿⣿⣿⣷⣾⣦⣀⠄⣼⣿⠿⠿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡠⠀⢸⣾⣟⣷⣗⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣷⣻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡢⣳⣿⣿⣿⣿⣿⡇⠁⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠑⠙⠛⠛⠛⠓⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠋⠉⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⣰⣶⡇⠀⣠⣶⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢴⣩⣿⣿⣀⠖⠦⣾⣿⡇⣿⣿⣿⣯⣉⣁⡀⣠⣤⣾⣿⠟⠀⠀⣿⣿⣿⡄⠀⠀⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡎⠇⠀⠢⠶⠰⢆⣼⣧⣼⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⡿⠀⣀⣀⣽⣿⣿⣿⠀⣰⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡟⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣂⠒⢓⠾⠻⣿⣿⣿⠇⣿⣿⡿⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⡿⣟⣿⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡣⠉⠛⢻⣿⠀⡿⣿⠁⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡈⠀⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢨⣧⣄⣷⢻⣼⣧⣿⡀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⡷⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣾⣿⣿⣿⡯⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⠀⢀⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⡖⢛⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⣁⢸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠨⡇⠆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⡒⣾⡖⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣷⣤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⣶⣵⣿⣿⡿⠿⠛⢿⣿⣿⣷⣶⣤⣀⠀⣬⣿⠿⠿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡍⣹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⢶⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣫⣲⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⠛⠛⠛⠉⠃⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠋⠉⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡠⢴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⣰⣾⡇⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣔⣮⣽⣿⡀⠻⠆⣼⣿⡇⣿⣿⣿⣯⣉⣁⣀⣠⣴⣾⣿⡿⠀⠀⣿⣿⣿⡄⠀⠀⠵⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡖⠗⠐⠲⠕⡱⠙⣟⣏⣵⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⠀⣀⣄⣽⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣾⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣀⢍⢹⣿⣿⣿⣿⣿⠇⣿⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⡿⣟⣿⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡙⠰⠛⢻⣿⠀⡿⣿⠁⢾⡷⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠂⣀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢨⣧⣄⣷⢻⣼⣧⣿⡀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⠿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣗⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣂⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣷⣶⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠓⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣵⣶⣿⣿⣿⣿⣿⣿⣷⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣖⢛⡿⠉⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⡃⣸⣿⢿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠅⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠁⠍⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢰⣶⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣯⣾⣵⣿⣿⡿⠿⠟⢿⣿⣿⣷⣾⣤⣀⠐⣾⣿⠿⠿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⣸⣷⣿⣿⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡉⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢢⢲⣿⣿⡿⢿⣻⠽⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡣⡳⡿⣿⣿⣫⣿⠆⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠛⠛⠀⠁⠀⠀⠀⢻⣿⣿⣿⡿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠋⠉⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⣠⡀⠴⣶⡿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⣰⣶⡇⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣔⣾⣿⣟⢥⠀⡁⣻⣿⡇⣿⣿⣿⣯⣉⣁⢀⣀⣤⣾⣿⠿⠀⠀⣿⣿⣿⡄⠀⠈⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡞⠇⠐⠲⡓⠪⡜⣿⣗⣾⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⠀⣀⢄⣽⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡿⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣄⢓⠒⠿⢿⣿⣿⣿⠇⣿⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⡿⣟⡿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡒⠪⠛⣿⣿⠀⡟⣿⠁⢾⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠂⣀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠁⢨⣧⣄⣷⢻⣼⣧⣿⡀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⡿⠿⠿⠾⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⡷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣽⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠓⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣿⣿⣿⣿⣿⣿⣷⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣖⠛⣟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⣧⠃⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⡀⠁⠝⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢐⢖⣶⡖⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣾⣤⣿⣿⡿⠿⠿⢿⣿⣿⣷⣾⣤⣀⡔⣾⣿⠿⠿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣷⣿⣿⣧⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡍⣹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣶⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣋⣴⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠑⠙⠓⠛⠋⠃⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠙⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⢶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⣰⣾⡇⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣔⣪⣽⣿⣍⡭⡤⣼⣿⡇⣿⣿⣿⣯⣉⣁⣀⣠⣴⣾⣿⠟⠀⠀⣿⣿⣿⡆⠀⢀⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡖⠟⠐⠤⡗⠧⠕⢺⣏⣲⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⠀⣀⢀⣽⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡿⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣓⢤⢕⠹⠿⣿⣿⣿⠃⣿⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡀⠁⠛⣿⣿⠀⡿⣿⠁⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⣀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢨⣧⣄⣷⢻⣼⣧⣿⡀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣽⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣶⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣷⣶⣾⣿⣿⡧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⣮⣾⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣶⠛⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⡀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣾⡿⢿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠁⠉⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⢐⣷⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣯⣾⣤⣿⣿⡿⠿⠿⢿⣿⣿⣷⣶⣦⣁⡄⣾⣿⠿⠿⠿⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⣸⣗⣿⣟⡧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡉⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢐⣿⣿⣿⢿⣷⡦⠼⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣐⣿⣿⣿⣯⣯⠏⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠚⠛⠛⠁⠀⠀⠀⠀⢻⣿⣿⣿⡿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉⠉⠉⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢔⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣠⣿⡇⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢴⣻⣿⣿⠂⣀⣕⣼⣿⡏⣿⣿⣿⣯⣉⡀⠀⣀⣠⣿⣿⡟⠀⠀⣿⣿⣿⡄⠀⠀⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡎⠛⠐⠪⡗⢶⣪⣿⣿⣼⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⣀⣀⣿⣿⣿⣿⠠⣴⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡟⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣭⠭⢡⣽⣿⣿⣿⣿⠇⣹⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⡿⣿⡿⣟⣿⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡋⠒⡻⢿⣿⠀⡿⣿⠁⢾⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡀⠀⣀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣦⣄⣷⢻⣜⣧⣿⡀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⡿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⡿⣿⣿⣧⣀⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣷⣮⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣿⣿⣿⡷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣶⠛⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣸⡃⡸⣿⢿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⡇⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⢀⡀⠁⠌⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⢔⣽⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣯⣶⣥⣿⣿⡿⠿⠿⣿⣿⣿⣷⣾⣦⣀⠀⣼⣿⠿⠿⠿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⢀⢸⣎⣿⣟⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡉⢹⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠰⣿⣿⣿⡿⣿⡿⠼⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢄⣸⣿⣿⣿⣿⣿⠕⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠛⠛⠂⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⡠⢖⣾⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⣰⣿⡇⠀⣠⣶⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢴⣻⣻⣿⡑⣖⢪⣽⣿⡇⣿⣿⣿⣯⣉⣁⠀⣀⣠⣿⣿⡟⠀⠀⣿⣿⣿⡆⠀⢈⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡎⠟⠀⠘⠃⣞⢱⣿⣿⣾⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⣀⣀⣽⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡟⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣿⣠⠒⠿⠿⣿⣿⣿⠇⣽⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⡿⣟⠿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡚⠚⣛⣻⣿⠀⡿⣿⠁⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡂⠀⣀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢡⣦⣄⣷⢻⣜⣧⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣷⣮⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣷⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣖⠛⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⡿⠿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠌⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⢐⣿⣏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣯⣾⣥⣿⣿⣿⠿⠿⢿⣿⣿⣷⣶⣦⣀⠔⣾⣿⠿⠿⠿⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⢀⢸⣗⣿⣟⣏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢰⣿⣿⠿⣿⢯⡿⢼⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢄⣰⣿⣷⣿⣿⣿⠍⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠛⠛⠂⠀⠀⠀⠀⢻⣿⣿⣿⡿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠋⠉⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢖⣾⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⢀⣰⣾⡇⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢴⣯⣿⣿⠞⣞⠊⢹⣿⡇⣿⣿⣿⣯⣉⣀⣀⣀⣠⣿⣿⠟⠀⠀⣿⣿⣿⡆⠀⠀⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡎⠟⠐⠊⠁⠊⢁⣿⣻⣿⣿⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⣀⣄⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡟⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣬⣷⠛⡿⣿⣿⣿⣿⠇⣽⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⡿⢟⢿⠏⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡘⠝⠙⣿⡿⠀⡿⣿⠁⢾⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢸⣦⣄⣷⢻⣜⣧⣿⡀⢺⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⢿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⡀⠁⠚⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⣿⣿⣾⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡀⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣷⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠋⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠝⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣜⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣯⣄⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣶⣥⣽⣿⡿⠿⠿⣿⣿⣿⣷⣶⣤⣀⠄⣺⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢖⣼⣿⣿⣿⣿⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣉⣹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⢸⣿⣿⣿⣿⣿⡯⠾⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢬⣽⣿⣿⣽⣿⣿⠝⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠛⠿⠛⠛⠛⠁⠀⠀⢿⣿⣿⣿⡿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⠉⠛⠙⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⢠⣾⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⣠⣾⣿⠃⢀⣴⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣶⣙⡗⣿⠀⡀⢂⣻⣿⡏⣿⣿⣿⣶⣤⣤⣀⣄⣾⣿⣿⡏⠀⣿⣿⣿⣆⠀⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⠂⠐⠺⡇⠀⠸⢿⣟⣿⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⣠⣽⣿⣿⣿⡆⠠⣴⣼⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣇⣶⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣀⠉⠨⢍⠻⠿⣿⣿⠇⣻⣿⡟⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⣿⡿⣿⠿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡀⠀⠀⢼⡿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠒⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢈⣦⣄⣷⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣻⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠘⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣼⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣢⣿⣿⣾⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡂⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡅⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠋⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⡂⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢈⡧⠂⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣔⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣯⣠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣾⣥⣿⣿⣿⠿⠿⢿⣿⣿⣷⣾⣦⣀⠔⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣼⣿⣿⣿⣿⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣹⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⢾⣿⣿⣾⣿⣿⣽⠾⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢬⢾⣿⣾⣿⣾⣿⠎⠃⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠛⠻⠛⠛⠛⠋⠀⠀⢻⣿⣿⣿⡿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⢀⣾⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⣠⣾⣾⠃⢀⣴⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢶⣽⣿⣟⠠⡘⢁⣸⣿⡏⣿⣿⣿⣶⣤⣤⣀⣤⣾⣿⣿⡟⠀⣿⣿⣿⣆⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⠂⠐⠖⡀⠉⠀⠿⣻⣿⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⣠⣽⣿⣿⣿⡆⠀⣴⣽⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣎⣶⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣃⠄⠀⠺⡿⢿⣿⣿⠇⣿⣿⡿⣿⣿⠋⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⣿⡿⣿⠿⠏⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡀⠀⠀⢹⡿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢀⣦⣄⣷⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣼⣿⣿⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣽⣿⣿⣇⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠋⡯⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⡣⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢨⣶⣿⣏⡃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣥⣽⣿⡿⠿⠿⢿⣿⣿⣷⣶⣤⣀⠀⣬⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣰⣰⣾⣿⣿⣯⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣍⣹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⡸⣭⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⣺⣽⣿⣿⣿⣿⡮⠂⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠛⠛⠛⠛⠋⠍⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠙⠛⠉⠋⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⣾⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⣶⣿⠇⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣶⢻⢿⡿⢀⠼⠄⣿⣿⡇⣿⣿⣿⣶⣦⣤⣀⣤⣾⣿⣿⡟⠀⣿⣿⣿⣆⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣦⠒⠂⠀⠀⠦⠄⣿⣯⣶⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⢤⣽⣿⣿⣿⡆⠀⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣏⣶⣿⡿⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣃⠊⠂⠂⠻⣿⣿⣿⠇⣿⣿⡿⣿⣟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⣿⣿⣿⠿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡀⠀⠀⢼⣿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢀⣦⣄⣧⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣼⣿⣿⣹⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣷⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⣾⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠉⡯⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⡃⢸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠢⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡧⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢨⣶⣿⣟⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣤⣽⣿⡿⠿⠿⠿⣿⣿⣷⣶⣤⣀⠀⣬⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡰⣠⣺⣿⣿⣟⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⡍⣹⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⣸⣿⣿⠿⣿⣿⢿⢸⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢧⣿⣿⣯⣟⣿⣟⡋⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠟⠛⠛⠋⠉⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠉⠙⠉⠙⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋⠉⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣠⣾⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⣶⣾⠇⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢶⣻⢻⢿⡀⢁⡀⣾⣿⡇⣿⣿⣿⣶⣦⣤⣀⣤⣾⣿⣿⡟⠀⣿⣿⣿⣆⠀⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⠒⠂⠡⠀⠄⢀⡿⣷⣾⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⢤⣽⣿⣿⣿⡆⠀⣶⣻⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣶⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣃⠚⠖⢖⠼⣿⣿⣿⠇⣿⣿⡿⣿⣟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡿⣿⠿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡀⠐⠈⢾⡿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠂⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢀⣦⣄⣧⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣼⣿⣿⢻⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣷⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⣾⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠉⡯⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⡃⢸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠢⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡧⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢨⣶⣿⣟⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣥⣿⣿⡿⠿⠿⢿⣿⣿⣷⣶⣤⣀⠀⣬⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡰⣠⣺⣿⣿⣟⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡍⣹⣿⣿⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⣸⣿⣿⠿⣿⣿⢿⢸⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢧⣿⣿⣯⣟⣿⣟⡋⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠟⠛⠛⠋⠉⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠛⠉⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⡠⡤⢠⣾⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⣠⣾⣿⠃⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⠶⢻⡿⣿⠤⣸⡄⣻⣿⡇⣿⣿⣿⣶⣤⣤⣀⣠⣾⣿⣿⡏⠀⣿⣿⣿⣆⠀⠀⠀⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣷⣶⠒⠀⢀⠀⡂⢨⠿⢧⣽⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⢤⣽⣿⣿⣿⡆⠀⣶⣾⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣯⣤⣤⣄⠉⠙⢿⣿⣆⣶⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣁⠠⢐⠲⠯⣿⣿⣿⠃⣿⣿⡿⣿⣟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⣿⡿⣿⠿⠯⠡⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡀⠀⠈⢽⣿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣦⣄⣧⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣼⣿⣿⣽⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣷⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⣾⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠋⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠢⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡧⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢨⣶⣿⣟⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣤⣽⣿⣿⠿⠿⢿⣿⣿⣷⣶⣤⣀⠀⣸⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡰⣠⣺⣿⣿⣟⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣍⣹⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⣸⣿⣿⠿⣿⣿⢿⢸⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢧⣿⣿⣯⣟⣿⣟⡋⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠟⠛⠛⠋⠉⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠛⠉⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⣠⣾⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⣠⣶⣿⠃⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣶⢻⣿⡿⢔⠂⠀⣽⣿⡇⣿⣿⣿⣶⣦⣤⣀⣠⣾⣿⣿⡏⠀⣿⣿⣿⣦⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣶⠒⠘⠐⠂⢰⢐⠿⢏⣶⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⢠⣹⣿⣿⣿⡆⠀⣵⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣯⣤⣤⣄⠉⠙⢻⣿⣇⣶⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣀⠶⠜⠀⢜⣿⣿⣿⠇⣿⣿⡿⣿⣟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⣿⡿⣿⠿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡀⠀⠀⢼⣿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠂⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢈⣦⣄⣧⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣼⣿⣿⢹⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣷⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣾⣼⣿⣿⣿⡯⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠉⡯⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⡃⣸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠐⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠁⠈⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢨⣦⣿⣏⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⣧⣤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣥⣿⣿⡿⠿⠻⣿⣿⣿⣷⣶⣤⣀⠐⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢐⢜⣾⣿⣿⣯⡧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣍⣹⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠽⣿⣿⣿⡿⣻⡿⠺⣿⣿⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠖⣾⣿⣿⣾⣿⣿⡅⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠛⠛⠛⠛⠟⠋⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠉⠙⠛⠉⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣠⣾⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⣾⣿⠃⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢶⢻⢿⢿⢈⡨⢆⣿⣿⡇⣿⣿⣿⣶⣦⣤⣀⣤⣾⣿⣿⡏⠀⣿⣿⣿⣆⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣶⠒⠂⠊⡀⠤⠠⢿⣻⣴⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⢠⣽⣿⣿⣿⡆⠀⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣄⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣶⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣃⠦⠀⠲⠺⣿⣿⣿⠃⣿⣿⡿⣿⣟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⣿⡿⣿⠿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡒⠒⠤⢽⣿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠂⠂⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢀⣦⣄⣷⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠘⠿⣿⣶⣤⣀⠁⠚⢿⣷⣶⣿⣏⠙⣿⣿⣿⣼⣿⣿⢹⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣽⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣿⣿⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡠⣪⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠐⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠋⡯⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⡃⢸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠀⢦⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡗⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠜⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣰⣿⡏⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣴⣿⣿⣿⠿⠻⣿⣿⣿⣷⣶⣤⣀⠄⣮⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣑⣽⣾⣿⣿⣟⣗⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣍⣹⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠞⡯⣿⣿⣷⣿⣿⡍⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠙⠛⠛⠛⠛⠛⠀⠀⢻⣿⣿⣿⡿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠙⠙⠛⠉⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢠⣾⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⣠⣾⣿⠃⢀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣶⣿⣿⠷⢀⢥⠀⣿⣿⡇⣿⣿⣿⣶⣦⣤⣀⣠⣾⣿⣿⡟⠀⣿⣿⣿⣆⠀⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣶⠒⠀⠬⡀⣤⠐⣟⣫⣼⣿⣿⡇⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⢀⣤⣽⣿⣿⣿⡆⠀⣶⣾⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣎⣶⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣁⠐⠎⡲⢜⣿⣿⣿⠇⣿⣿⡿⣿⣟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⣿⡿⣟⠿⠯⠁⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣶⡐⠒⠀⢺⡿⠀⡿⡿⠀⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢈⣦⣄⣧⢷⣞⣷⣿⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⣿⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠛⠻⣿⣿⣮⣿⣿⣝⠈⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠋⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠘⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣼⣿⣿⢹⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣯⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣷⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠟⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠀⣸⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣄⢄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⡿⢿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠝⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣹⣿⡩⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣯⣶⣴⣿⣿⡿⠿⠟⠿⣿⣿⣷⣾⣤⣀⠔⣨⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⣠⣿⣾⣿⣿⣇⡖⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⠉⢹⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠸⡵⢷⣿⣿⣾⡯⡸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢛⣧⡯⣿⣿⣿⣷⡦⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⠛⠛⠛⠛⠙⠈⠀⠀⢻⣿⣿⣿⡿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠉⠛⠉⠙⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⢀⣴⡇⠀⣠⣴⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣶⣿⢿⣿⣂⡰⡅⢻⣿⡏⣿⣿⣿⣿⣛⡛⠁⠀⣀⣴⣿⣿⠃⠀⣿⣿⣿⣆⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⣿⣷⡆⠒⠀⠩⢻⣻⢏⢐⣘⣶⣻⣿⡇⢻⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣋⣀⣤⣽⣿⣿⣿⡀⣾⣾⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣇⣾⣿⡟⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣹⠭⢥⡤⡿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⡿⣿⡿⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣮⡛⠽⣧⠿⠿⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣧⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣽⠈⣿⣿⣿⡿⠸⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣍⣹⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡾⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⠀⢠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠟⣿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠀⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣄⢄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡿⢿⣿⣿⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠹⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣹⣿⣩⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣷⣤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣯⣶⣥⣽⣿⡿⠿⠿⠿⣿⣿⣷⣾⣤⣀⠀⣼⣿⠿⠿⠿⢛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⣠⣿⣾⣿⣿⣎⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⠉⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠸⡵⢷⣿⣿⣿⡯⠹⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢛⣧⡯⣿⣿⣿⣿⠎⠈⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⠛⠛⠛⠛⠉⠉⠀⠀⢻⣿⣿⣿⡿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠙⠋⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢀⣴⡗⠀⣠⣴⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⣶⣿⣻⣿⠼⡘⠄⣹⣿⡟⣿⣿⣿⣿⣛⡋⠁⠀⣀⣴⣿⣿⠃⠀⣿⣿⣿⣆⠀⠀⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡆⠐⠚⠓⣮⡜⣘⣳⣒⣷⣾⣿⡇⢻⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣋⢀⣤⣽⣿⣿⣿⡠⣾⣟⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣇⣾⣿⡿⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣺⠫⢭⣽⣽⣿⣿⣿⠃⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡟⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣮⡲⠬⣤⡿⠿⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠒⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢁⣦⣄⣿⢹⣴⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣽⣿⣽⠈⣿⣿⣿⡿⠙⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣻⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣥⠾⣿⣿⣿⣿⣿⣿⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣦⣾⣿⣿⣿⣿⣿⣿⣿⠀⢀⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡥⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠀⣸⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡠⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠜⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⣱⣿⡝⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣯⣶⣶⣿⣿⡿⠿⠟⢿⣿⣿⣷⣾⣤⣀⠀⣺⣿⠿⠿⠿⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢣⣶⣿⣿⣿⣵⣯⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⠉⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⣿⣻⢿⢿⣷⢽⢸⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢛⣇⡯⣾⣿⣷⣿⡵⠀⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⠚⠉⠛⠛⠋⠀⠀⠀⢻⣿⣿⣿⡿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠉⠙⠋⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⢐⣺⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢀⣴⡗⠀⣠⣴⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡠⣶⣿⣟⣿⢷⠨⡇⢻⣿⡏⣿⣿⣿⣿⣛⡛⠉⠀⣀⣴⣿⣿⠁⠀⣿⣿⣿⣆⠀⠀⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⣿⣷⡆⠀⠀⠀⢭⠑⢇⣀⣇⣼⣿⣿⡇⢻⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣋⢀⣤⣽⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢿⣿⣇⣾⣿⡟⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣝⠛⠍⢚⢿⣿⣿⣿⠃⣽⣿⡿⣿⣿⠋⢿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡉⠻⡩⠽⠟⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠃⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣧⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠈⣿⣿⣿⣿⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠟⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⣸⡃⣸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡠⣰⣿⡷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⣾⣵⣷⣿⡿⠿⠟⣿⣿⣿⣷⣾⣤⣀⠔⣾⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣾⣿⣿⣯⣏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⢱⣿⣿⣿⣿⣿⡟⢼⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠞⡸⣿⣿⣯⣿⣿⡯⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠛⠛⠛⠛⠛⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢀⣼⡇⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⡶⣿⡷⠤⠬⠀⠅⣼⣿⡏⣿⣿⣿⣿⣛⠛⠁⠀⣀⣴⣿⡟⠀⠀⣿⣿⣿⣆⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣶⡄⠀⠛⠠⠆⢽⠊⠥⢥⣤⣿⣿⡇⢿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⣀⣄⣹⣿⣿⣿⡀⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡿⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣆⠭⠭⣔⠝⢿⣿⣿⠇⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⡇⣺⣿⣿⣿⣿⡿⣟⡽⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡋⠐⠁⠚⠛⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠢⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣦⣄⣿⢹⣴⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠀⣿⣿⣻⣿⠉⢻⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⡀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⡇⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣼⣵⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⢈⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣲⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠟⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⣸⣃⢸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡀⢴⣴⡥⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⣶⣥⣿⣿⡿⠿⠟⢿⣿⣿⣷⣾⣤⣀⠐⣾⣿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢜⣷⣾⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠉⢹⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡈⢫⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠗⡸⣿⣷⣿⣿⣿⡣⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⠛⠛⠛⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢀⣼⡇⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡴⣾⣷⠔⠻⠐⠀⣹⣿⡟⣿⣿⣿⣿⣛⠛⠁⠀⣀⣴⣿⣿⠁⠀⣿⣿⣿⣆⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡄⠀⠀⠘⢅⣖⠞⣐⢤⣠⣾⣿⡇⢿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⣀⣀⣹⣿⣿⣿⡀⣾⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡟⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣦⠄⠐⢪⡭⣿⣿⣿⠃⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣟⡻⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡉⠚⠃⡛⠛⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠁⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣦⣄⣷⢹⣜⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠂⣿⣿⣻⡿⠉⢻⣿⣿⣿⣿⣿⣿⣟⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡃⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⡇⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣼⣵⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⢈⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣲⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠟⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⣸⣃⢸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠜⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣤⣶⣥⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⣶⣥⣿⣿⣿⠿⠟⢿⣿⣿⣷⣾⣤⣀⠐⣺⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢼⣶⣿⣿⣿⣗⡗⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠉⢹⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡈⢫⣿⣿⡿⣿⣿⡿⠾⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠗⡸⣿⣿⣿⣿⣿⡭⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⠛⠛⠛⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣼⡇⠀⣠⣶⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡶⣾⡧⠠⢠⠆⡁⢸⣿⡟⣿⣿⣿⣿⣛⠛⠁⠀⣀⣴⣿⡟⠀⠀⣿⣿⣿⣆⠀⠈⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡄⠀⠀⠨⠑⢆⠈⡬⣋⣤⣾⣿⡇⢿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⣀⣄⣽⣿⣿⣿⡀⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡟⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣦⠀⠺⠏⣘⣿⣿⣿⠃⣾⣿⡿⣿⣿⠋⢿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡇⣽⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡘⠃⠀⡉⠛⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠁⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢀⣦⣄⣷⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠂⣿⣿⣿⡿⠉⢻⣿⣿⣿⣿⣿⣿⣟⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣾⣿⡇⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣼⣵⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⢈⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣲⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠟⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⣸⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣾⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢐⣶⣾⣥⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣷⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⣶⣦⣿⣿⡿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⣾⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢜⣿⣾⣿⣿⣯⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡉⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡈⢫⣿⣿⢿⢿⡿⡿⠿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠟⡸⣿⣿⣿⣽⣿⡭⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠛⠛⠛⠛⠛⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢀⣼⡇⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⡶⣾⡧⠀⠈⠘⠃⢹⣿⡏⣿⣿⣿⣿⣛⠛⠁⠀⣀⣴⣿⡟⠁⠀⣿⣿⣿⣆⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡄⠀⠂⠨⠍⣽⡠⣭⢀⣠⣿⣿⡇⢿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⣀⣤⣽⣿⣿⣿⡀⣾⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣾⣿⡿⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣤⣪⠭⠡⣗⣙⣿⣿⣿⠇⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡇⣽⣿⣿⣿⣿⡿⣟⡿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣽⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⡉⠉⠐⠛⠟⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠃⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠀⠀⢠⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⢻⣿⣷⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠊⣿⣿⣿⡿⠉⢻⣿⣿⣿⣿⣿⣿⣟⣿⡿⢿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣾⣾⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⣿⣿⣿⣿⣿⣇⠀⠐⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣹⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⣸⡃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢐⣲⣾⣷⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢼⣯⣶⣤⣿⣿⣿⠿⠟⢿⣿⣿⣷⣶⣤⣀⠀⣸⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣤⣾⣿⣿⣿⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠈⢹⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡘⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡝⣺⣿⣿⣿⣿⣿⡯⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠙⠛⠛⠛⠛⠛⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⠉⠉⠉⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢰⡇⠀⣀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⡶⢾⣗⣰⡽⠋⣇⣻⣿⡏⣿⣿⣿⣿⣿⡛⠛⠉⠀⣠⣴⡟⠀⢘⣿⣿⣧⠀⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣥⡀⠐⠓⠰⣴⠿⢊⡁⣦⣾⣿⣿⡇⢿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠃⢀⠸⣿⣿⣿⣇⠠⣷⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣷⢒⢓⣢⣿⣿⣿⣿⠇⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣧⣶⣿⣿⣽⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣿⠿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣼⣿⣿⠛⠿⣿⣦⡛⢝⠻⠟⠛⠀⡏⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⠘⠃⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠋⠒⢣⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠂⣿⣿⣻⣿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣘⣾⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣾⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣵⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣹⡿⣿⠉⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠉⣼⡃⡸⣿⢿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⢀⣆⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠝⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣼⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣤⣿⣿⣿⠿⠿⡿⣿⣿⣷⣾⣤⣀⠀⣸⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⢄⣼⣿⣿⣿⣷⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣨⢺⣟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣮⣪⣿⣿⣽⣿⣿⡯⠁⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠑⠙⠛⠛⠛⠛⠋⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢠⡇⠀⣀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡠⣶⣻⢿⢿⠒⣭⣫⣿⣿⡇⣿⣿⣿⣿⣿⡛⠛⠉⠀⣀⣴⡏⠀⢘⣿⣿⣇⠀⠀⠈⣳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣤⡐⠃⠙⠡⣺⠾⢏⣓⣓⣸⣿⣿⡇⢿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠁⢀⠸⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣫⠿⡭⢽⣿⣿⣿⣿⠇⣾⣿⡿⣿⣿⠋⢿⣿⣿⣿⣷⣶⣿⣿⣽⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣼⣿⣿⠛⠿⣿⣮⡑⣽⠿⠟⠛⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⠘⠠⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠙⠛⢣⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠀⣿⣿⣻⣿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠅⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⣶⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣿⣿⣿⣿⡇⠀⠀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⢰⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣨⡿⡟⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠉⣼⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢄⣼⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣥⣿⣿⡿⠿⠿⠿⣿⣿⣷⣾⣤⣀⠀⣾⣿⠿⠿⠿⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢤⣼⣿⣿⣿⣯⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣨⢸⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣮⣹⣿⣿⣿⣿⣿⡯⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠑⠛⠛⠛⠛⠙⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠙⠉⠉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢠⠇⠀⣀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡠⣴⣻⢽⢿⣯⡵⠳⣽⣿⡇⣿⣿⣿⣿⣟⡛⠛⠉⢀⣀⣴⡟⠀⢘⣿⣿⣇⠀⠀⠈⣳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣥⡰⠖⠋⠻⢮⣗⡝⠥⣶⣼⣿⣿⡇⢿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠃⢀⠸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣇⣿⣿⡟⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣺⠯⠎⢙⣿⣿⣿⣿⠇⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣿⣶⣿⣿⣽⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣼⣿⣿⠛⠿⣿⣾⣍⢩⣿⠟⠛⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⡐⠡⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠁⠈⢡⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠀⣿⣿⣻⡿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣾⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣾⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣷⣾⣿⣿⣿⡇⠀⠈⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣦⣶⣿⣿⣿⣿⣿⣿⣧⠀⢨⣅⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡧⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣩⡿⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠉⣼⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣸⣷⡻⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣢⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣧⣿⣿⣿⠿⠟⣿⣿⣿⣷⣶⣤⣀⠔⣼⣿⠿⠿⠿⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠴⣤⣿⣿⣿⡏⣗⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢸⣿⣿⡿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣭⣺⣭⣯⣟⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠘⠛⠛⠛⠙⠙⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⣠⡇⠀⣀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢴⣿⡼⢿⣯⡬⢹⣻⣿⡇⣿⣿⣿⣿⣿⡛⠛⠉⠀⣀⣶⡏⠀⢘⣿⣿⣇⠀⠀⢈⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣤⡐⠃⠈⠲⣛⡭⠯⣙⣄⣸⣿⣿⡇⢻⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⠁⠀⠸⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣇⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⣫⢡⠼⣻⣿⣿⣿⠃⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣼⣿⣿⠛⠿⣿⣾⡝⢿⣿⡟⠛⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⠘⠠⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠒⠀⢠⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠀⣿⣿⣻⣿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣾⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣷⣾⣿⣿⣿⡇⠀⠈⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣦⣶⣿⣿⣿⣿⣿⣿⣧⠀⢨⣅⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡧⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣩⡿⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠉⣼⣃⠸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣸⣷⡻⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣢⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣧⣿⣿⣿⠿⠟⣿⣿⣿⣷⣶⣤⣀⠔⣼⣿⠿⠿⠿⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠴⣤⣿⣿⣿⡏⣗⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢸⣿⣿⡿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣭⣺⣭⣯⣟⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠘⠛⠛⠛⠙⠙⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠙⠉⠉⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣠⠇⠀⣀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡠⣶⣿⣜⣿⠑⣩⢿⣺⣿⡇⣿⣿⣿⣿⣟⡛⠛⠉⠀⣀⣴⡟⠀⢘⣿⣿⣇⠀⠀⢈⣳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣦⡐⠎⠀⠲⣻⠿⡾⢞⢵⣻⣿⣿⡇⢿⣿⣿⣿⣟⣻⣿⣿⣿⣿⡿⠃⢀⠸⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣯⣤⣤⣄⠉⠙⢻⣿⣇⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⡿⠶⣝⣻⣿⣿⣿⠃⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣷⣶⣿⣿⣽⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣦⣅⣦⣿⡟⠛⠀⡟⣿⠁⢹⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⡐⠣⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠉⠁⢢⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠀⣿⣿⣻⡿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⣰⣾⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⣶⣾⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣰⣯⣿⣿⣿⣿⣿⣿⣧⠀⢠⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣹⡿⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠉⣼⡃⡸⣿⢿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⡀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣇⠂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣸⣾⠿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣧⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣧⣿⣿⣿⠿⠿⣿⣿⣿⣷⣶⣤⣀⠔⣼⣿⠿⠿⠿⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠴⣤⣿⣿⣿⡟⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢸⣿⣿⡿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣭⢺⣭⣯⣾⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠈⠛⠛⠛⠚⠙⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠙⠉⠉⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣀⡇⠀⣀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡠⣶⣿⡽⣿⠫⡥⢝⢸⣿⡇⣿⣿⣿⣿⣿⡛⠛⠉⠀⣠⣴⡟⠀⢘⣿⣿⣇⠀⠀⢀⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣦⡐⠞⠛⠹⢺⣓⣽⡿⣟⣻⣿⣿⡇⢿⣿⣿⣿⣟⣻⣿⣿⣿⣿⡿⠁⠀⠸⣿⣿⣿⣇⠠⣷⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢿⣿⣆⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣿⠡⢄⣽⣿⣿⣿⣿⠃⣿⣿⡿⣿⣿⠋⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣿⣿⠯⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡏⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣷⣼⣿⣿⠛⠿⣿⣮⣩⢹⢿⠟⠛⠀⡟⣿⠁⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⠐⠤⡀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠑⠂⢣⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠀⣿⣿⣻⣿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⣶⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣐⣾⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣧⠀⢀⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣨⡿⡿⠈⣩⣽⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⣼⡃⡸⣿⣿⠇⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡀⠂⡂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⣇⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⢿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠍⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣨⣿⡙⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣯⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢴⣯⣶⣧⣿⣿⣿⠿⠿⣿⣿⣿⣷⣾⣤⣀⠀⣺⣿⠿⠿⠿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⢔⣦⣿⣿⣿⡴⣎⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⢹⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢸⣿⡿⢿⣿⢿⡿⠿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡏⣺⣿⣯⣯⣿⣿⡧⠁⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠙⠛⠛⠛⠚⠋⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠼⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢠⠇⠀⣀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢶⣻⢟⣿⠓⠥⣏⣽⣿⡇⣿⣿⣿⣿⣟⡛⠛⠉⠀⣀⣴⡏⠀⢘⣿⣿⣇⠀⠀⠈⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⢀⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣧⡀⠆⠛⠩⣽⡪⣭⣥⢃⣾⣿⣿⡇⢿⣿⣿⣿⣛⣻⣿⣿⣿⣿⡿⠁⢀⠸⣿⣿⣿⣇⠀⣷⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⠀⣀⣀⣤⣤⢻⣭⣤⣤⣄⠉⠙⢻⣿⣆⣿⣿⡿⣷⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣤⣞⠵⠾⢼⢿⣿⣿⣿⠃⣾⣿⡿⣿⣿⠋⢿⣿⣿⣿⣷⣶⣿⣿⣽⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⣿⣿⠏⠉⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣇⣿⣿⡟⠛⢸⣿⡟⣿⣿⡿⣷⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⢙⣿⣿⣷⣬⣿⣿⠛⠿⣿⣮⣭⢡⠿⠟⠛⠀⡟⣿⠁⢹⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⠘⠡⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⡏⢸⣿⣷⣶⢸⣿⣷⣿⣿⣇⣀⢹⣿⣿⣿⣿⡇⣿⣿⣻⣿⡿⣿⣿⡏⠙⣿⣿⣶⣦⣄⡉⠓⠁⠈⢠⣦⣄⣿⢹⣼⣯⣿⡀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀
⠀⣿⣿⡏⣿⣿⢸⣿⣿⣀⣸⣿⣿⣿⣿⡟⠋⢸⣿⣿⢸⣿⣇⣿⣿⣻⣿⣿⢻⣿⡇⠀⣾⣿⣌⠙⠻⢿⣷⡆⢀⡈⠙⠻⣿⣿⣦⣿⣿⣝⠀⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣷⡿⢿⠾⠿⣿⣿⣿⡿⠿⠛⠿⠇⠀⠘⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠐⠿⣿⣶⣤⣀⠁⠘⢿⣷⣶⣿⣏⠙⣿⣿⣿⣴⣿⣿⣿⡁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 1722

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/02/irc-log-010222/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/02/irc-log-010222/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ IRC_Proceedings:_Tuesday,_February_01,_2022⠀✐


Posted in IRC_Logs at 2:28 am by Needs Sunlight


Also available via the Gemini protocol at:


    * gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-010222.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-010222.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-social-010222.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-010222.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


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CID                                              Description     Object type


                                                 IRC log for
 QmTbbJTLsuocaQdNBqMoxuzxUzWQ6WL9TvutFmNCxZH7PY #boycottnovell  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
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                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell
 QmVSDV8PizHf9JEjaGh7Yj4HoX6vhvAsjUotXwR96AAVVT (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
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 Qmaq2FoeXuCaPxPrw8xBhFhcoJk5oicpFdW7PqsD7QwENF social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
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                                                 IRC log for
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 QmQ9E4iSqqTPYEmZHRkJ2qkQYSJQaExfgHb3FtieJnvA5Z social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 (full IRC log
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                                                 IRC log for
 QmUotrz7nTZsghj67nNnNpYuso51Pc2tUcFzSe4CKxN53e #techbytes      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
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                                                 IRC log for
                                                 #techbytes
 Qmc6yEioiXfk4rjB33jxucvxZA732wCQS4EYNHEDfyqY6M (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
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                                                 IRC log for
 QmTTZMiRLUGvg9szBeZPj2DDtPgcAwT7xug8xi7fwDY2aK #techrights     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
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                                                 IRC log for
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                                                 as plain/ASCII
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                    ䷩ 𝚕𝚒𝚗𝚎 1849

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/02/lets-talk-linux-com/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/02/lets-talk-linux-com/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Linux.com_Used_to_Clock_Millions_of_Hits_Every_Week,_But_Linux_Foundation_Can
Barely_Keep_Thousands⠀✐


Posted in GNU/Linux, Kernel, Marketing at 3:47 pm by Dr. Roy Schestowitz


Let’s Talk… about what happened to Linux.com over the past 3 years


                      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Linux.com hit⦈


                   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Go away, Linux users⦈


Summary: After the Linux_Foundation attacked Linux.com by firing all the staff
and all the editors we’re left with a site called Linux.com that makes Linux
look insignificant, often misused for Spamnil’s crude, self-promotional
activities [1, 2]


⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
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⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
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⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
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⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣹⢻⣍⣥⣭⣻⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣄⠀⠈⠉⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠉⠙⠈⠑⠁⠙⡏⠀⠀⣾⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡿⠿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣡⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣟⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠿⠿⠿⠿⠿⠛⢛⣓⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣻⣫⣽⢻⣍⣥⣭⣻⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣄⠀⠈⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣆⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⡿⠟⠏⠁⠀⠁⠹⠿⠀⠀⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣉⣻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⣿⣿⣿⣶⣦⣀⡀⠀⠉⠉⠉⠙⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠷⠿⠛⠛⠛⢛⣋⣉⣭⣭⣤⣦⠤⠤⠀⠀⠀⠀⠀⣠⣄⣈⣉⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣹⢻⣍⣵⣭⣿⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣆⠀⠈⠉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⡿⠟⠋⠟⠛⠯⠿⠹⡿⠑⢸⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣆⣈⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠛⠛⠉⠉⣉⣩⣽⣽⣿⣿⣿⣿⣿⣿⣇⣄⣀⣄⣴⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣹⢻⣍⣵⣭⣿⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣆⠀⠈⠉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣼⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⠿⠟⠛⠿⠍⠋⠻⠻⡿⠋⢸⣿⣿⣿⣿⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡜⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡿⠹⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡉⠙⠛⠛⠛⠛⠛⠛⠛⠟⠙⠛⠻⠿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⢠⣤⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠉⢉⣩⣽⣿⣿⣿⣿⣿⣿⣿⡟⢀⣀⣀⣀⣀⣀⣈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣹⢻⣍⣥⣭⣻⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠟⠝⠀⠉⠀⠊⠉⠯⠁⣿⣿⣿⣿⣿⣿⡎⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡈⢧⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢷⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⡉⠉⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣭⣉⠉⠁⠉⠉⠉⠀⠈⠙⠛⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣷⣶⣶⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠋⠋⠉⣥⣬⣽⣿⣿⣿⣿⣿⣿⣿⠇⠀⠚⠛⠹⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢊⡷⢆⣀⣰⣦⡤⣠⠶⠠⢾⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡃⢀⣾⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣛⣫⣹⢻⣍⣥⣭⣹⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⢻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣸⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠟⠛⠿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠛⠛⠟⠛⠿⠟⠛⠿⠿⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⠻⢿⡿⠈⠩⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡏⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠉⢹⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡆⠈⢷⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣭⣍⣉⣉⣉⠙⠛⠛⠛⠋⠙⠛⠻⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣿⣿⣷⣶⣿⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠚⠛⢹⢿⣿⣿⣿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣯⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣛⣫⣹⢻⣍⣥⣭⣹⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠟⠿⠿⠿⠿⠿⠟⠟⠛⠟⠟⠻⠻⠛⠛⠛⠛⠛⠟⠟⠟⠛⠛⠿⣿⡿⠿⠿⠛⠀⠀⠀⠀⠀⠄⠃⢸⣿⣿⣿⣿⣿⣿⠏⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⢸⢹⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠈⢿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⢶⠍⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣟⡿⠿⠿⠿⣿⠻⠿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣹⣿⣿⣿⣿⣿⣶⣶⣄⢴⣶⣶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢛⣻⠿⠿⠿⢿⡿⠁⠀⠀⠉⠀⠛⠻⣿⣿⣿⠿⡿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣆⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢐⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣽⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡧⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣾⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠇⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⣿⣷⡄⠀⠀⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠋⣯⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢻⣿⡇⣿⣿⣿⣿⣹⣿⢸⣿⣷⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣻⣿⣿⢻⣤⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀⢿⡽⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢒⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠘⡇⠹⡎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣻⣀⣑⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⢙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡍⠻⡙⢟⠛⠛⠉⠛⠛⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢦⠀⠀⠄⠠⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⢤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠂⣙⣈⣶⣤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⣥⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀
⢠⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀
⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡣⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣥⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣥
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⡇⢨⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⢼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣦⣬⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣷⣾⣷⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣿⣿⢿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇⣿⣶⣿⣷⡄⠀⠀⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠀⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡿⣿⣿⣿⡿⣿⢿⣿⣿⢿⢿⡏⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⡿⢿⠿⠿⠟⢹⣿⡇⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣤⣴⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡶⠀⢹⡏⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣶⣷⡾⠃⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷⠿⠿⠿⠏⠀⠀⠀⠀⠀⠁⠈⠛⠻⣟⠿⡟⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡀⠀⠀⢀⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣦⣴⣶⣆⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⣉⠁⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀
⣂⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣄⡀⠀⠀
⣭⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡰⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣯⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⡇⡨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣿⣿⣿⣿⡿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇⣿⣦⣝⣷⡄⠀⠀⠉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢷⣿⡿⢿⣿⡏⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⠟⠿⠿⠿⠟⢹⣿⡇⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣄⣤⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠉⢿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣼⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⡧⠿⠼⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⠏⠀⠀⠀⠀⠀⠋⠙⢿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣇⡟⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠁⠀⠀⠀⣠⡤⠀⠀⠈⠓⠀⠀⠀⠀⠑⢄⠱⡌⢻⣿⣿⣿⣿⣶⣯⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠂⠘⣆⠹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣶⣦⣶⣶⡄⠀⠙⠛⠛⠛⠛⠛⠛⠻⠛⠟⠿⠿⠛⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀
⣕⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⣀⠀⠀
⣭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡥⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡇⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⢿⣳⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣌⢻⣆⠀⠀⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⣿⣿⣾⣿⠀⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢷⣿⣿⠿⣿⡝⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⢿⢿⡿⠿⠟⢹⣿⡇⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠞⠛⠟⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣼⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⡾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠸⣿⣿⣿⣥⡤⠶⠟⠛⠁⠀⠀⣀⣀⠀⠀⠈⠙⠻⡿⠿⡟⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⣦⡀⠀⠀⠀⢀⠘⢄⠻⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣼⣿⣷⠿⠙⠳⣆⠀⠀⠁⡈⢢⡘⣿⣿⣿⣿⣿⣼⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢟⡛⠁⠀⠀⠁⠀⢠⣆⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣯⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣶⣿⣶⣆⠀⠘⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⢀⡥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⡄⠀⠀⠀⠀
⠔⢻⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀
⣭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣭⣶⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⡇⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡇⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣥⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⢿⣗⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⣬⡻⢦⠀⠀⠉⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠀⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢯⣧⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⠿⣿⡿⠀⠟⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⣿⠿⠿⠿⠿⠟⢹⣿⡇⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟⠉⡝⠉⢩⡿⢻⣦⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠟⣿⣿⠃⠀⠀⠀⠀⠚⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣷⠴⠾⠛⠁⠀⠀⠀⢠⣀⠀⠀⠀⠙⠛⠻⣏⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢀⠘⢦⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣆⠀⠀⠀⢀⠐⣄⢻⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⣿⣿⣿⣿⣷⢄⡀⡌⢡⣀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢩⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢟⠛⠛⠋⠁⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣾⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⣷⣾⣷⣿⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⡀⠀⠀⠀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢇⡀⠀⠀
⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⡀⠀
⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣧⣶⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⡇⠨⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡇⣾⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡚⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡟⢴⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣦⡙⢦⠀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠀⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⣿⣿⣿⣿⢿⢾⣿⡿⢿⣿⡿⠀⠟⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠿⡿⠿⢿⣿⣿⡟⠻⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⢿⠿⠿⠿⡿⠿⢿⡿⢻⣿⡇⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠋⠀⡚⠀⣰⠏⢨⣽⣧⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠈⠟⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣼⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉⠁⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠳⣈⠻⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣤⡀⠀⠀⠢⠈⠂⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠈⠢⡐⠦⡙⢂⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢶⣿⣿⣿⣿⣿⣤⣾⡦⣱⣮⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⠀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢚⢻⡋⠙⠛⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣸⣿⣿⣿⣿⣿⣷⣿⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⢀⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⠀⠀⠀⠀
⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣂⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡗⣨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡇⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⢚⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣦⡙⢦⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣾⣿⠀⣧⡅⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⢿⣿⠿⠿⣿⡿⠀⠟⠃⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠟⢉⡉⢉⠽⠛⠛⠻⠿⠙⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⢿⠿⡿⣿⢿⠟⠿⠟⢹⣿⡇⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⡿⠿⠛⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠘⡴⣾⣷⣶⣤⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢶⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⣛⠛⠛⠛⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣦⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣿⣿⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡆⠀⠀⠀⠀
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣷⣥⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡷⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡇⢿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⢴⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣦⡙⢧⡀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠀⣧⡌⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢠⣿⣿⣱⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⡿⠂⠟⠃⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉⠠⠀⠈⠁⠀⠀⠀⠀⠸⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⣿⠿⠿⠿⣿⠟⢿⡿⢹⣿⡟⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⢛⡛⠋⠛⠛⠉⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠀⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣦⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣿⣿⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⡿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠂⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡋⢀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠋⠈⢈⣦⡙⢧⡀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠘⣯⡕⠁⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠿⠿⠿⠟⠁⠀⠀⠀⣽⣿⣿⣱⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠟⠃⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢻⣿⡇⣿⣿⣿⣿⣹⣿⢸⣿⣷⣾⣿⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡛⠛⠙⠛⠋⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣿⣿⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣶⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⢰⣦⡙⣳⡀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠉⣿⡅⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⢋⠉⠉⠉⠀⠀⠀⠀⠊⠉⠻⣵⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⣶⣶⣾⣿⣯⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⡿⠿⣿⢿⢹⣿⡏⢸⣿⡇⣿⣿⣿⣿⣹⣿⢸⣿⣿⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠉⢠⣤⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠈⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡛⠋⠙⠛⠋⠉⠁⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠀⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢦⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠢⢾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⡀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡖⡐⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡡⢤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣽⣭⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⣽⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⢻⠓⡠⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣤⡙⢷⡀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡐⣭⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠉⠀⠐⠁⠀⠀⠀⠀⠁⠉⠑⢳⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠇⠀⠀⠀⠀⢀⣀⣤⣤⡤⠀⣾⣿⣶⣿⣯⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢿⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣿⣶⣿⣶⠂⠀⠀⠀⠀⠀⠀⣼⡿⠿⠛⠀⠀⠀⠀⢀⡀⢹⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⢠⣤⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢛⡛⠋⠙⠛⠋⠉⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢢⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣿⣾⡀⠀⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡗⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⣿⣿⣿⣿⣿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠤⡀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠘⣽⡄⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡟⠋⠩⠁⠀⠉⠀⠀⠀⠀⠈⠀⠀⠱⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠃⠟⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⢀⡀⣠⣤⡤⠐⣾⣿⣷⣿⣧⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⣿⡿⣿⠿⡿⡿⠿⢿⢹⣿⡇⢼⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣶⣷⣶⠂⠀⠀⠀⠀⠀⠀⣾⣿⠿⠟⠁⠀⠀⠀⠀⡀⢸⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠉⢠⣤⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠈⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢛⡛⠛⠙⠛⠋⠉⠁⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣾⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠀⠄⠠⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣿⣾⡀⠀⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣺⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⡿⢿⠟⠛⠂⠉⠀⠙⠳⠀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠋⣯⡅⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⡟⠋⣠⠞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆⠿⠃⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠁⢺⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⡟⢻⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣿⣶⣿⣶⠂⠀⠀⠀⠀⠀⠀⣿⠟⣻⠿⠃⠀⠀⠀⠀⡀⢻⣿⣿⣷⣶⣤⣤⣉⢑⣴⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠐⠗⠉⠀⠀⠀⠀⡠⠒⠋⢠⣈⣹⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢚⡛⠛⠛⠛⠛⠉⠁⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⣆⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣿⣿⣷⣿⣿⣷⣾⡀⠀⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⠀⠀
⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣚⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣥⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢥⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⣽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣽⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⣻⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣾⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡧⡅⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⠟⣻⠟⠛⠋⠛⠂⢩⣷⡀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠹⣯⡍⠁⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⠟⠉⠘⠃⠀⠃⠀⠀⠀⠀⠈⠲⣍⠓⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠃⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⡏⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠐⠠⠀⠉⠲⣬⡉⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢿⣿⡏⣿⣿⣿⣿⣽⣿⢸⣿⣿⣾⣿⣶⠀⠀⠀⠀⠀⠀⢠⣿⠏⣻⡾⠋⠀⠀⠀⠀⡀⢿⣿⣷⣦⣤⣤⣀⡐⢦⣀⣐⠆⢈⣻⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠉⠁⠉⠁⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⠗⠊⠁⠀⠀⠀⡠⠒⠋⢀⣈⣿⣿⣿⢻⣿⣿⣿⣷⣄⣝⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢚⡛⠛⠛⠛⠋⠉⠁⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣼⣆⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢦⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⣿⣿⣿⣿⣿⣷⣾⡀⠀⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⡿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣭⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣸⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠛⢻⠛⠛⠋⠛⠂⢙⣷⡀⠀⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠘⣯⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁⠈⠁⠀⠊⠀⠀⠀⠀⠈⠳⣍⠓⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⢀⣶⣿⢿⣿⡏⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠠⢀⠙⠲⢬⡉⠻⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⣿⡿⣿⠿⠿⢿⡿⣿⢹⣿⡏⢸⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣷⣶⣷⣶⠀⠀⠀⠀⠀⠀⢠⣿⠏⣻⡾⠉⠀⠀⠀⠀⠀⢿⣿⣿⣦⣤⣤⣀⡒⠦⣀⣐⠢⢈⣙⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⠇⠈⠁⠀⠀⠀⡠⠒⠋⢀⣈⣻⣿⣿⢻⣿⣿⣿⣷⣄⣙⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢚⠛⠛⠛⠛⠋⠉⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠀⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⣆⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢲⠀⠀⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⣿⣿⣿⣿⣿⣷⣾⡀⠀⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣾⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⢿⠟⠉⡜⠉⠉⠀⠛⢂⣙⣷⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠘⣽⡅⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠈⢷⡍⠓⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⢠⣾⣿⢿⣫⠃⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠄⡈⠓⢢⡘⠻⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⠿⣿⡿⣿⠿⠿⣿⡿⣿⢹⣿⡇⢸⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣷⣶⣿⣶⠂⠀⠀⠀⠀⠀⢀⡿⠏⡿⠟⠀⠀⠀⢀⣠⠌⢻⣿⣿⣦⣦⣤⣀⠲⣄⡀⠦⠍⠃⢾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⠃⠁⠀⠀⠀⢀⠤⠛⠋⢠⣈⣹⣿⣿⢹⣿⣿⣷⣬⡙⣷⣦⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠈⠉⠈⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢷⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢚⣛⡛⠛⠛⠉⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⣆⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢦⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣿⣿⣿⣿⣾⣷⣾⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣽⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⢿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣾⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⡻⢁⠜⠉⠉⠀⠉⠀⠙⠳⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠉⣯⡅⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⠃⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡅⠿⠃⠀⠀⠀⠀⠀⣠⣾⡿⣻⡟⠃⠀⠀⠀⠀⣴⣦⣀⡀⠀⠀⣀⠐⠂⣀⣀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⣿⣿⣿⣿⡿⣿⡿⢿⢿⣿⡏⢹⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣿⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⡿⠋⠔⠋⠀⠀⠀⢀⣤⠆⢻⣿⣿⣿⣷⣄⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠴⠋⠉⢠⣈⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠈⠀⠀⠈⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢛⡛⠛⠛⠛⠋⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣷⣾⡀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠄⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣥⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠿⢟⠟⠛⠉⠀⠁⠀⠀⠀⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⣠⣾⡿⣻⡟⠃⠀⠀⠀⠀⢠⣶⣴⣶⡦⠀⣤⣄⡀⠉⣢⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⢿⣿⡏⢺⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣿⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⡟⠉⠐⠉⠀⠀⠀⠀⣠⠄⢻⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠉⢠⣬⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡛⠛⠙⠛⠉⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢦⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣷⣾⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣽⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⢀⡘⠳⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⡌⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠱⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠃⠿⠃⠀⠀⠀⠀⠀⢀⣾⡿⣻⡟⠃⠀⠀⠀⠀⢀⣤⣤⣶⡤⠀⣶⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡏⢹⣿⡇⣿⣿⣿⣿⣽⣿⢹⣿⣿⣶⣿⣶⠂⠀⠀⠀⠀⠀⠀⡟⠹⠜⠉⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⢠⣬⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⣻⠛⠙⠛⠉⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣷⣾⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣯⡁⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣼⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⢠⡈⠳⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠑⣯⡅⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠂⠟⠃⠀⠀⠀⠀⠀⢀⣾⡿⣻⡿⠃⠀⠀⠀⠀⢀⣤⣶⣶⡤⠀⣶⣶⣶⣿⣧⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢿⣿⠿⠿⡿⢿⣿⡏⢻⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣿⣶⣷⣶⠂⠀⠀⠀⠀⠀⠀⡿⠫⠴⠋⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣷⣴⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠉⢠⣤⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠀⠉⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡻⠛⠙⠛⠉⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣷⣾⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣦⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢿⠿⠋⠀⠀⠀⢀⡈⠳⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠛⣯⡅⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠟⠉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠛⠑⣵⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠃⠿⠃⠀⠀⠀⠀⠀⢀⣴⣿⢻⡿⠇⠀⠀⠀⠀⢀⣤⣲⣦⡄⠀⣤⣶⣦⣾⣧⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢻⣿⡏⢻⣿⡏⣿⣿⣿⣿⣹⣿⢸⣿⣿⣶⣿⣶⠂⠀⠀⠀⠀⠀⠀⡿⢋⠴⠋⠀⠀⠀⠀⢀⠀⢺⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⢠⣬⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⣛⠛⠛⠛⠋⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⣿⣿⣷⣾⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀
⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣤⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⡿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠛⠃⠉⠀⠀⠀⠀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠈⣯⡄⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠟⠉⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⠂⠟⠃⠀⠀⠀⠀⠀⢀⣶⣿⢿⡿⠏⠀⠀⠀⠀⢀⣀⢀⣄⠀⠀⡀⠀⠀⠙⢥⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⡿⣿⣿⢿⣿⠿⠿⠿⢻⣿⡇⢸⣿⡇⣿⣿⣿⣿⣹⣿⢸⣿⣷⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⡿⢛⡵⠛⠀⠀⠀⠀⢀⠀⢻⣿⣿⣿⣷⣤⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠉⢠⣈⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡛⠋⠙⠛⠋⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣷⣾⠀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣾⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⣻⠿⠿⠏⣿⣦⡉⠳⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⣯⡍⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠋⠁⠴⠃⠰⠁⠀⠀⠀⠀⢦⠀⠑⠳⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠃⠿⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⢄⠈⠂⠀⠀⠀⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡯⢻⣿⡏⣿⣿⣿⣿⣽⣿⢸⣿⣿⣶⣿⣶⠂⠀⠀⠀⠀⠀⢀⡿⢟⡽⠟⠀⠀⠀⠀⣀⠄⢻⣷⣤⣄⡀⢄⠑⠄⠀⢰⣶⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠠⠚⠉⢠⣈⣿⣿⣿⢹⣆⠠⢄⠺⠂⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠁⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡞⣿⣷⣬⣷⣐⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡻⠋⠙⠛⠉⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣷⣾⡀⠈⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡗⣱⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⣻⠿⢿⡇⣿⣦⡉⠱⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠉⣿⡍⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠋⢀⠼⠁⢰⠁⠀⠀⠀⠀⢧⠙⠵⣷⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠂⠟⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠈⠣⡀⠀⠀⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⢿⣿⡏⢹⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣿⣶⣷⣶⠂⠀⠀⠀⠀⠀⠀⡿⢟⡿⠛⠀⠀⠀⠀⣀⠄⢻⣷⣄⡀⠀⢀⠑⢤⡐⠈⢦⣀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠠⠚⠉⢠⣈⣿⣿⣿⢻⣄⠡⣀⠸⡆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡞⣿⣷⣬⣶⣄⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡻⠋⠉⠛⠉⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣷⣾⠀⠘⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⡿⠋⣻⠟⢿⡟⣿⣦⡉⢳⡀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣥⡌⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⠉⠀⠞⠁⠸⠁⠀⠁⠑⠻⣿⠻⢵⣷⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠃⠿⠃⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠘⢧⠀⠉⠻⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡟⢹⣿⡇⣿⣿⣿⣿⣽⣿⢸⣿⣿⣾⣿⣶⠂⠀⠀⠀⠀⠀⠀⡿⣟⡿⠟⠀⠀⠀⠀⣀⠄⢿⣷⣄⡀⠀⠈⠢⡠⡁⠐⢦⣀⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⢀⠠⠚⠉⢠⣈⣿⣿⣿⢲⣌⠂⠈⠠⡄⠈⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠁⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡞⣿⣷⣄⣥⡀⢕⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢛⡛⠋⠙⠛⠋⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣷⣾⡀⠈⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣥⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⡿⠋⣻⠟⠿⡿⣿⣦⡙⢦⠀⠀⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠉⣯⡅⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⠋⠀⠞⠁⠘⠁⠀⠀⠀⠀⢎⠑⠱⠷⡀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⣿⣿⠁⠿⠃⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠁⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣻⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡏⢻⣿⡇⣿⣿⣿⣿⣽⣿⢸⣿⣿⣶⣿⣶⠂⠀⠀⠀⠀⠀⠀⡿⢟⡿⠟⠀⠀⠀⠀⣀⠄⢿⣷⣤⣀⠀⠀⠀⢱⣄⠀⠻⣿⣿⣿⣿⣿⣹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⠀⠠⠚⠉⢠⣘⣿⣿⣿⢳⣄⠀⢦⡈⣳⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡞⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢶⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⢛⡛⠛⠛⠛⠋⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣆⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣷⣾⠀⠘⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣥⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡗⣼⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⣦⡙⢦⠀⠀⠉⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠉⣯⡄⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠟⠉⠀⠂⠐⠁⠀⠀⠀⠀⠀⠁⢀⣰⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⢀⣶⣿⢿⡿⠏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢿⣬⣿⣿⣿⣿⣿⣻⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣿⣿⣿⢿⣿⠿⣿⡿⢹⣿⡏⢹⣿⡏⣿⣿⣾⣿⣽⣿⢸⣿⣿⣾⣶⣶⠀⠀⠀⠀⠀⠀⠀⡿⢛⠹⠟⠁⠀⠀⠀⢀⠀⢿⣿⣶⡆⣤⣴⣴⣷⣦⣠⣬⣿⣿⣿⣿⣿⣹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠉⢠⣘⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣦⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢶⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⢛⡻⠛⠛⠛⠋⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣆⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⣿⣿⣿⣿⣿⣷⣾⠀⠘⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣯⡿⠀⠀⠀⠀⠀⠀⢤⠀⠀⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠟⣿⡌⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠟⠛⠟⠙⠀⠀⠀⠀⠀⡠⣶⢳⣧⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⠃⠿⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣿⣬⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢽⣿⡏⢺⣿⡇⣿⣿⣿⣿⣿⣿⢸⣿⣿⣾⣿⣶⠂⠀⠀⠀⠀⠀⠀⡾⢛⡱⠟⠃⠀⠀⠀⠀⠀⢠⣶⣼⣶⣦⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣛⣛⠛⠙⠛⠉⠉⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣾⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣾⣿⣿⣿⣿⣿⣷⣾⠀⠘⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⢗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣳⠄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡢⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡮⢕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡋⣸⣾⠛⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢢⣼⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡻⠁⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢦⠀⠀⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡻⣿⡧⢲⢁⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠿⠿⠟⠛⠀⠀⠀⠀⠀⣤⡦⢻⣧⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⢃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣬⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⡏⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⣾⢻⣫⠾⠋⠀⠀⠀⠀⠀⢀⣴⣴⣦⣦⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠉⠉⠉⠁⠈⠈⠈⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠙⠙⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣛⡻⠛⠛⠛⠉⠉⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣾⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⣿⣿⣿⣿⣿⣷⣿⠀⠘⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣦⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣽⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⢦⠀⠀⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣯⡌⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⢿⠿⠋⠀⠀⠀⠀⢀⣦⢻⣧⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠁⠿⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣼⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢹⣿⡇⣿⣿⣿⣿⣹⣿⢸⣿⣿⣶⣿⣶⠂⠀⠀⠀⠀⠀⠀⣾⢿⣫⡿⠏⠁⠀⠀⠀⠀⢀⣤⣠⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠟⠈⠁⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠁⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣛⡛⠛⠛⠛⠉⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⣿⣿⣿⣿⣿⣷⣿⠀⠈⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣮⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡩⣷⣷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣸⡷⠠⠀⠀⠀⠀⠉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡍⠁⠂⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⢠⣦⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠿⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠛⠛⠟⠻⠋⠀⠀⠀⠀⣀⣼⣿⣼⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⡏⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢻⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡿⠟⠚⠛⠁⠀⠀⠀⠀⠀⢠⣴⡄⣆⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣾⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠘⣛⡟⠟⠛⠋⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣾⣿⣿⣿⣿⣿⣿⣇⡀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀
⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣧⡁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣹⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣼⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⣌⢙⢆⠀⠀⠉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠘⣿⡍⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢹⣿⣿⣾⣿⣼⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢽⣿⡏⢻⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣿⣶⣷⣶⠂⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣧⠾⠝⠋⠀⠈⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⣿⠟⠁⠀⠀⠂⠘⠛⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠉⠁⠁⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣾⡶⣶⣤⣤⣦⣤⣼⣿⣿⣿⣿⣿⣿⡇⡟⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠱⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⣿⣿⣿⣿⣿⣿⠟⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⣛⠿⠝⠛⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣷⣄⠀⠀⡀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣾⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⢉⠁⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀
⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡷⣔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣽⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣼⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⣌⢛⣆⠀⠀⠉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠉⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢹⣿⣿⣼⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣿⣿⣿⠿⣿⢿⠿⡿⢹⣿⡏⢸⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣿⣶⣶⣶⠂⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣏⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡽⢿⠹⡏⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢛⣻⣽⣿⣿⠟⠁⠀⠀⠀⣥⠛⠛⠉⠁⠀⠀⢀⣀⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⢫⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣶⣿⣿⣧⣆⣶⣤⣀⣠⣀⡀⠙⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⣿⣿⣿⣿⡿⠟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⠀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⡻⠿⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣷⣄⢀⠀⣀⠀⡀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠀⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀
⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣦⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⢼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⣌⠻⣆⠀⠀⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠈⣽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢹⣿⣿⣽⣿⣼⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣿⣿⣿⢿⣿⣿⣿⢿⢻⣿⡏⢹⣿⡇⣿⣿⣾⣿⣽⣿⢸⣿⣷⣶⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡻⡏⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠉⠀⠉⠉⢻⣿⣿⠟⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠾⠛⠹⠛⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢰⣦⣄⣀⣀⣄⢀⡄⠀⠀⠀⠀⠀⡄⠘⠗⠈⠛⠛⠋⠉⠉⠙⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢦⠀⠐⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡈⢄⠀⠄⠸⣄⣠⣶⣄⣘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀
⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣼⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣿⣿⣿⣿⡿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⢸⣷⣌⢻⣆⠀⠀⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠉⣿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠂⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⣿⣿⣿⣧⣽⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡟⢿⣿⡏⣿⣿⣿⣿⣽⣿⢸⣿⣿⣶⣷⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡟⣯⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠚⢻⣿⣿⣿⣿⣿⣿⣿⠃⠉⠈⠁⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡧⡄⠘⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠊⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠹⢿⣯⣿⣿⣿⣿⣿⣿⠏⠁⠉⠻⢿⣛⠛⠿⠿⠿⠿⠟⠿⠻⠿⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢑⡻⠿⣿⠟⠁⠀⠀⠀⢀⣘⣿⡗⣤⣀⠀⢀⣤⡀⠀⣶⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣦⠀⠀⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣸⣿⣷⣦⣤⣶⣤⣦⣤⣄⠑⠋⠀⠟⠹⠿⣿⣿⣷⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣮⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⡿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣿⢸⣷⣌⢻⡆⠀⠀⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡞⣿⡆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠇⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣻⣿⣿⣻⣧⣽⣿⣿⣿⣿⣿⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⡯⢹⣿⡏⣿⣿⣿⣿⣽⣿⢹⣿⣿⣿⣷⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⡇⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⠀⠉⠉⠉⠉⠀⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠛⠋⠉⠉⠉⠛⠛⠻⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠯⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠲⣦⣄⣀⢀⣴⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢦⠀⠀⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢰⣶⣭⣛⠻⡿⠋⠀⠀⠀⠀⢰⠿⣿⣿⣿⣿⣿⣷⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢋⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡱⢀⡁
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠊⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⣍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣉⣲⡶⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⣤⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣨⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⡿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣶⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⢸⣷⡝⢷⡄⠀⠀⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠫⣿⡝⣙⡄⠀⠀⠀⠀⠀⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡼⣿⡄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣷⣿⣧⣿⣿⣿⣿⣿⣽⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⡏⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠉⠉⠉⠉⠉⠉⠀⠁⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡛⣿⣿⣿⣿⣿⣿⣿⡇⠈⠉⠉⠙⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠈⠃⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⣿⣿⣿⣿⣿⡿⡿⠿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀⡀⠀⠀⠀⠀⣠⡀⠀⣤⣤⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠠⢴⣤⣼⣿⣇⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⣤⣟⠻⠿⣿⣿⣿⣿⣿⠏⠀⠈⠀⣽⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣅⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣵⡠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣹⣿⣛⣫⣘⢹⣍⣅⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣿⡖⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⣤⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢗⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣽⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣿⢸⣧⡙⣷⡄⠀⠀⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠹⡎⠀⠀⠀⠀⠀⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣽⣿⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠿⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣷⢿⣧⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⡏⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣥⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡋⣿⣿⣿⣿⣿⣿⣿⡏⠙⠛⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠘⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣘⣛⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣿⣿⠉⢸⣿⢻⣿⢀⣬⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⠀⠀⠀⠀⣠⢰⣶⣶⣮⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠶⣤⣴⣿⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣦⣤⣄⣖⡛⠛⠿⠿⠿⣿⣿⠃⣀⣀⡀⠀⠀⠀⠙⠛⢿⡳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡇⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⡷⡂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣼⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢾⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣼⣧⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⢀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⢸⣧⡙⣷⡄⠀⠀⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢽⡇⠀⠀⠀⠀⠨⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣹⣿⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠿⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢿⣿⣧⢿⣧⣿⣿⣿⣿⣿⡷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡏⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢹⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⣿⣿⣿⣿⣿⣿⣿⡏⠛⠋⠋⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⣘⣛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠙⠙⠛⠻⠿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠄⣶⣶⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠿⠻⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⠈⠀⠉⠛⠿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣄⣤⡀⠉⠛⠛⠛⠛⢛⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⡇⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣹⣿⣛⣫⣘⢹⣍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣯⣷⡿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣾⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣿⣸⣧⡙⣷⡀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣽⡎⠀⠀⠀⠀⠐⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻⣷⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠿⢃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣏⣿⣧⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⡏⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⢰⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠉⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣂⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣿⣿⠉⢸⣿⢹⣿⢄⣬⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⣠⢠⣤⣤⣴⣶⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣶⣶⣦⣤⢀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⡅⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣹⣿⣛⣫⣘⢹⣍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣶⣿⡗⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⢤⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣾⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢸⣧⡙⣷⡀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣽⣿⡍⠍⡃⠀⠀⠀⠀⠀⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣷⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⣿⣿⡏⣿⣦⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⡏⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣳⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠳⠚⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠋⠙⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢉⡔⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣼⣃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣏⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣅⣉⠉⠙⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣿⣿⠉⢸⣿⢹⣿⢀⣬⡻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣀⠀⢀⡀⠈⠙⠻⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠛⠻⠟⠛⢋⣉⡭⠽⠷⠶⢀⠀⢀⣼⡟⠀⣾⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⣠⣶⣶⣾⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣛⣫⣘⢹⣍⣥⣭⣙⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡫⣦⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣦⣽⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣾⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣸⣧⡙⣷⡄⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣷⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⣿⣿⡏⣿⣶⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⡇⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣿⣿⣾⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⡳⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡯⡿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡿⠾⠞⣿⣿⣿⣿⣿⣿⣿⡿⠈⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠙⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢩⡖⠈⢉⣽⣿⣿⣿⣿⣿⣿⣿⣯⣦⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⣆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣼⡃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣀⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣬⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠛⠛⠿⠛⢛⣛⣯⡭⠷⠆⠀⠀⠀⢀⠀⠀⣶⣶⡆⣤⣬⣽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⣶⣶⡀⢠⣶⣶⣾⣿⣿⣶⣾⣿⣧⣴⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣥⣭⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣾⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣇⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣮⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣽⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣿⣼⣧⡙⣷⡀⠀⠀⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣷⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠇⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⡞⣿⣬⣿⣿⣿⣿⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⡿⡿⠿⢹⣿⡿⢺⣿⡇⣿⣿⣿⣿⣽⣿⢸⣿⣿⣷⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⣴⣿⣯⣿⣧⢿⣿⣿⣿⣿⣿⣿⢭⡏⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡏⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⢯⢽⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠁⠁⠉⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠂⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣣⠄⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⡿⢅⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣭⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣽⣿⣿⣿⣿⣿⣦⣤⡀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠟⣿⠿⠿⠛⠻⠭⢓⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠟⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠀⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⣤⣤⠀⠀⣈⣩⣍⣥⣤⣄⣀⣀⠀⠀⠀⣰⣶⢀⣤⣬⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡆⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣛⣫⣘⢹⡍⣅⣍⣙⣻⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣧⣿⣦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣸⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣬⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣽⣿⣿⣿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⣶⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣾⡏⣾⣦⡹⣧⡀⠀⠀⠉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠐⣯⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣧⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠂⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠛⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣿⣿⣿⠿⠿⢿⣿⠿⢹⣿⡇⢸⣿⡇⣿⣿⣾⣿⣹⣿⢸⣿⣷⣾⣷⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣬⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⠇⢸⣿⠇⣿⣿⣿⡟⣿⣿⢸⣿⢘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⢸⣿⢀⣿⠹⣿⡇⣿⣿⣾⣿⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠋⠁⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢀⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣴⡶⣶⡆⢰⣶⣶⠀⣤⣤⣤⡀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠱⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇⠻⣷⣬⡁⢸⣿⣤⢰⣿⣇⣿⠇⣿⣇⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣯⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠇⣶⡎⣿⡇⣾⣿⠉⢸⣿⢹⣿⢀⣌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣽⣿⣿⣿⣿⣟⠘⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠁⠛⠛⠛⠘⠛⠸⠿⠈⠿⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⣿⣿⣿⢿⠿⠶⡀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠀⠄⠠⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢀⣀⡁⠀⠀⠀⢀⣒⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣽⢻⣍⣥⣭⣿⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣧⡀⠀⠈⠉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⢿⣿⣿⣿⢿⡿⠝⠋⠉⠻⠫⣿⡿⠅⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡏⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢳⣿⡿⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⡁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠉⠉⠛⠋⠉⠑⠨⢭⣅⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣻⣫⣽⢻⣍⣥⣭⣿⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣦⣿⣷⡄⠀⠈⠉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠉⠉⠁⠹⢯⠿⠛⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡏⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢯⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣮⣿⠃⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣿⣿⣷⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠉⠀⠙⠋⠋⠉⠙⠂⣤⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣽⢻⣍⣥⣭⣿⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣧⡄⠀⠈⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠏⠉⠍⠹⠹⢿⠿⠍⠥⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⠿⢿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣶⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣮⣿⠗⢸⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣿⣿⣷⠆⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣿⣶⣦⡄⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠉⠂⠙⠙⠉⠉⠉⠒⢨⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠽⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣹⢻⣍⣵⣭⣿⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣧⡄⠀⠈⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠟⠀⠋⠛⠈⠾⠿⠅⠨⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣶⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣿⠗⠼⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣿⣿⣷⠆⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⣿⣶⣦⣄⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠍⠀⠘⠛⠛⠉⠉⠒⣠⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣻⣫⣹⢻⣍⣵⣭⣻⣻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣷⡄⠀⠈⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠩⠏⠘⠑⠸⠫⠿⠛⠀⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⣶⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣿⡗⢐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⡆⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣼⣿⣿⣿⣿⣿⣶⣦⣄⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠍⠃⠘⠛⠛⠋⠉⠑⣂⠀⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 3196

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(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/02/security-perception-manipulation/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/02/security-perception-manipulation/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Darkening_the_Counterpart’s_Reputation:_Microsoft-Friendly_(Sometimes
Microsoft-Funded)_Media_Describing_Every_Security_Issue_—_Even_Windows_Issues_—
as_‘Linux’⠀✐


Posted in Deception, FUD, GNU/Linux, Microsoft, Security, Windows at 6:22 pm by
Dr. Roy Schestowitz


Video_download_link | md5sum 50e30a74c66758c9b3e9173defb765cd
Microsoft Culprit Spun as Expert
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/security-narrative-inverted.webm


Summary: The perception manipulation campaign (it certainly feels like one)
keeps Linux in the headlines in relation to security issues that aren’t even in
Linux; we know who stands to benefit from it and the people who promote these
messages are sometimes ex-Microsofters [1, 2]


THE Microsoft-connected_CrowdStrike_is_once_again_attacking_Linux in an effort
to sell its so-called ‘products’ and disservices, trying to twist a bunch of
generic or Microsoft issua as (somehow) a “Linux” thing. With the media in
their pockets, and parrots as messengers, they can probably get away with it
too. A VFS-related issue (or Samba) gets spun as “Linux”, attacks on VMware are
spun as a “Linux” issue, and systemd is apparently a kernel now, based on how
media reports on it. The video above takes stock of the media’s campaign of
obfuscation and distraction, apparently compensating for Microsoft’s most
terrible year by shifting negative media attention to Linux and then [cref
telling officials to blame Free software]. █



                    ䷩ 𝚕𝚒𝚗𝚎 3245

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/02/year-of-geminispace/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/02/year-of-geminispace/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ One_Year_Since_We_Joined_Geminispace_and_the_Effort_Has_Paid_Off⠀✐


Posted in Site_News at 1:20 pm by Dr. Roy Schestowitz


Video_download_link | md5sum 8ed5666be758c4245573ad1c1cc4815d
Our Gemini Adoption One Year Later
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/gemini-growth-one-year.webm


Summary: It looks like Geminispace will more than double in size in 2022, based
on the trends we saw in the month of January


IN FEBRUARY last year we set up our Gemini capsule and later_shared_some
instructions/guidance_on_setting_up_one's_own_Gemini_server_at_home. It’s a lot
simpler than setting up a Web site!


“There’s an advantage associated with being an “early adopter”.”As we noted
earlier_this_week, Gemini had expanded a lot in January, a lot more than other
alternatives_to_the_Web (like Web0). As the graphs_shown_in_the_video_make
clear, Geminispace added a net of 73 capsules in the past 2.5 weeks alone, at
least based on Lupa. That’s a gain of 5% in less than 20 days! It took almost 3
years — yes, years — to reach 1,600 and now we’re seeing gains of over 100 per
month. Will Geminispace double in size in 2022? Will it triple in size? We can
only hope because we’ve sunk or invested a lot of time/effort in it. There’s an
advantage associated with being an “early adopter”. We’ve probably served about
2-3 million pages over gemini:// this past year.


As Andre_Alves_Garzia_put_it_last_week: “Since the markup language is so
simple, it lowers the barrier of entry for those wanting to produce content.
There is no need to learn web design and have a pretty site (nothing against
those, I like them but they add friction to newcomers). One can learn all about
Gemtext in less than an hour and be happy. [...] The world is not made richer
by having less options for hypertext networks. Gemini is not the Web, and it
doesn’t want to be the Web. Gemini is its own thing that will co-exist with the
Web.” █



                    ䷩ 𝚕𝚒𝚗𝚎 3304

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_2/2/2022:_EasyOS_3.3_and_Kiwi_TCMS_11.1⠀✐


Posted in News_Roundup at 3:04 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Distributions
          o Devices/Embedded
    * Free_Software/Open_Source
    * Leftovers
    * § GNU/Linux⠀➾


          o § Desktop/Laptop⠀➾


                # ⚓ How_I_switched_to_Ubuntu_after_using_Windows_for_20_years_
                  (the_2022_guide)_|_by_Dmitrii_Eliuseev_|_Jan,_2022_|_Medium⠀⇛


                       I was a Windows user for many years, but after
                       buying a new computer I decided to use Ubuntu as
                       the main OS.


          o § Kernel Space⠀➾


                # § Graphics Stack⠀➾


                      # ⚓ First_Blog_Post_for_SoK_2022!⠀⇛


                             As a painter who also likes software
                             development, I first came across Krita when I
                             was looking for digital painting applications
                             on Windows. Later on, I learned about open
                             source and decided to start contributing as a
                             way to build my development skills while
                             working on something that I myself use. I
                             spent the last few months of 2020 building
                             Krita, first in Windows (a nightmare :P),
                             then in Linux, and started learning Qt since
                             I was already familiar with some basic C++
                             from school.


                      # ⚓ AMD_HIP_Linux_GPU_Acceleration_For_Blender_Delayed_To
                        v3.2_Release_–_Phoronix⠀⇛


                             Last year Blender 3.0 added AMD HIP
                             acceleration to its Cycles X render code with
                             OpenCL having been removed. That AMD HIP
                             support for Blender 3.0 was limited to
                             Windows with plans to then enable Linux
                             support for Blender 3.1. Sadly due to AMD
                             driver delays, that HIP Linux support is
                             postponed to Blender 3.2.


                             Blender 3.1 is due to be released in March
                             and sadly will be lacking AMD HIP support on
                             Linux.


                      # ⚓ Intel_Patches_To_Make_It_Easier_To_Run_Their_Discrete
                        Graphics_On_Arm,_Other_Architectures_–_Phoronix⠀⇛


                             A change currently being evaluated for
                             Intel’s “i915″ Linux kernel graphics driver
                             would make it easier for building driver
                             support for their forthcoming discrete
                             graphics products for targeting other non-x86
                             CPU architectures like Arm


                             Sent out today as a “request for comments”
                             were patches that change the Intel Linux
                             kernel graphics driver to allow it to
                             optionally build without support for
                             integrated graphics — leaving the driver just
                             capable of discrete graphics support. While
                             Intel graphics have traditionally been about
                             their integrated graphics on their
                             processors, Intel is moving hard and fast on
                             bringing up their discrete graphics support
                             under Linux with DG2/Alchemist for Intel Arc
                             graphics cards coming together as well as
                             their Xe HPC accelerator.


                      # ⚓ NVIDIA_510.47.03_Linux_Driver_Released_With_Vulkan
                        1.3_Support,_RTX_3050_Compatibility_–_Phoronix⠀⇛


                             While it was just yesterday NVIDIA released
                             the 470.103.01 Linux driver, today they have
                             made public the 510.47.03 Linux driver as
                             their first stable version in the NVIDIA 510
                             Linux driver series.


                             This is the stabilized version from their
                             prior NVIDIA 510 Linux beta that brought
                             Vulkan dynamic rendering, AV1 VDPAU decode
                             support, and a variety of other updates
                             predominantly driven by Vulkan API
                             improvements. There is also improved GBM API
                             support for what was originally introduced in
                             the 495 driver series.


          o § Benchmarks⠀➾


                # ⚓ NVIDIA_Linux_Gaming_Performance_For_Wayland_vs._X.Org_On
                  Ubuntu_22.04⠀⇛


                       
                       With NVIDIA’s newly-introduced 510 Linux driver
                       series paired with the latest XWayland and a modern
                       Wayland compositor like the newest GNOME/Mutter
                       packages, the NVIDIA (X)Wayland experience is in
                       great shape and delivering comparable performance
                       to a traditional X.Org session. The NVIDIA Wayland
                       support with GBM usage has stabilized and appears
                       to be in good shape for the upcoming Ubuntu 22.04
                       LTS release. Here are some benchmarks of the NVIDIA
                       510 driver on the current state of Ubuntu 22.04
                       LTS.


                       In the past we’ve looked at the Radeon Linux gaming
                       performance for X.Org vs. Wayland. Modern Wayland
                       support in the likes of KDE Plasma and GNOME Shell
                       have worked out well with that open-source Radeon
                       driver stack. Now with the latest NVIDIA drivers,
                       their Wayland support is in good shape too.


          o § Instructionals/Technical⠀➾


                # ⚓ How_to_install_Oracle_Solaris_11.4⠀⇛


                       Some time ago, I wrote an article on how to install
                       OmniOS CE on VirtualBox. OmniOS is a distribution
                       of illumos, which is based on (now discontinued)
                       OpenSolaris, which was based on Oracle Solaris.


                       For this article I’m going back to the “source” to
                       show how to install Oracle Solaris 11.4, screen by
                       screen. Last year I’ve switched from VirtualBox
                       (also from Oracle) to gnome-boxes, but the general
                       procedure should be pretty similar to any VM
                       system. Anyway, to use in production, a real server
                       would do better.


                # ⚓ Convert_PDF_To_EPUB_In_Linux_–_OSTechNix⠀⇛


                       In this modern era, everything is digitized. The E-
                       Books have became mainstream. EBooks are available
                       in various formats such as PDF, EPUB, MOBI, AZW3
                       and IBA Etc. Most e-Book readers does support
                       almost all formats. However, some e-Book reader may
                       not support a specific format. For example, ASW3
                       files will only be viewed by Amazon Kindle. In such
                       cases, we need to convert the e-Books from one
                       format to another supported format. In this quick
                       tutorial, we will see how to easily convert EBooks
                       from PDF to EPUB format (and vice versa) in Linux.


                       There are many online and offline applications
                       exists to convert an EBook from one format to
                       another. For the purpose of this tutorial, we will
                       be using Calibre e-book suite, which let us to
                       convert ebooks from both CLI and via GUI.


                # ⚓ How_To_Install_RethinkDB_server_on_Ubuntu_20.04_LTS_Focal
                  fossa⠀⇛


                       RethinkDB is an open-source NoSQL database for
                       production use, it is meant to simplify the
                       creation and scaling of real-time applications.
                       This distributed NoSQL database intended for
                       storing schemaless JSON documents


                       Although MongoDB is already quite popular in the
                       field of NoSQL databases, however, MongoDB lags
                       behind Cassandra, CouchDB, or Riak in terms of
                       scalability. Well, this means RethinDB has to
                       compete with all of them, well it already trying to
                       provide ease of usage like MongoDB with good
                       scalability.


                # ⚓ How_To_Install_Terraform_on_Fedora_35_–_idroot⠀⇛


                       In this tutorial, we will show you how to install
                       Terraform on Fedora 35. For those of you who didn’t
                       know, Terraform is an open-source infrastructure as
                       a code software tool that provides a consistent CLI
                       workflow to manage hundreds of cloud services.
                       Terraform codifies cloud APIs into declarative
                       configuration files. Terraform is built by
                       Hashicorp and released under Mozilla Public
                       License. It supports public, private as well as
                       hybrid cloud, as of now Terraform supports 145
                       providers, which includes popular providers like
                       AWS, Azure cloud, GCP, Oracle cloud, and many
                       others.


                       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 Terraform on a Fedora 35.


                # ⚓ How_to_install_KDE_Plasma_on_Ubuntu_22.04_Jammy_JellyFish⠀⇛


                       There was a time when KDE was considered as a
                       resource binger Desktop UI but not anymore, it is
                       similar to Gnome in terms of performance but with a
                       much beautiful interface and wide range of
                       applications. It comes already comes as a default
                       GUI in many Linux systems such as OpenSUSE, MX
                       Linux, Kubuntu among others. The main focus here is
                       on the many customization options.


                       Although the KDE desktop is beautiful, a bit more
                       confusing at first glance, however as you start
                       using the same, it is very reminiscent of Windows
                       with its taskbar. This also contains the “Activity
                       Manager” with which different desktop versions can
                       be displayed, for example, to access a photo
                       collection directly. The display of windows or the
                       positioning of widgets – almost every element of
                       the user interface can be customized in KDE.
                       However, the variety of options can quickly
                       overwhelm inexperienced users. You should therefore
                       invest some time in your very own KDE experience.


                       Also, the applications from the KDE community are a
                       lot more extensive and, like the KDE desktop, with
                       a wide range of setting options. Great for power
                       users, unfortunately often too overloaded for
                       newcomers.


                # ⚓ DNS_RPZ:_A_DNS_Firewall_to_filter_Sites_and_Users⠀⇛


                       Hello flocks, today we are going to learn some DNS
                       stuffs. We all know about what’s a DNS, how DNS
                       works, its types, etc. So, wrap things up and let’s
                       dive into our today’s topic about DNS Response
                       Policy Zone (RPZ).


                       [...]


                       DNS Response Policy Zone (RPZ) is a DNS zone which
                       enables DNS administrators to customize policy in
                       DNS servers, so that the server returns modified
                       answers to Client’s DNS queries. In other words,
                       RPZ provides a way to alter a DNS response in real
                       time. It can be used to modify potentially unsafe
                       DNS Data to block communication or provide local
                       data to redirect to a “walled garden”. As we can
                       alter query response or block any domain using RPZ
                       it is also known as DNS firewall.


                # ⚓ Grep_Command_Cheat_Sheet_With_Examples_[Free_PDF
                  Download]⠀⇛


                       Grep is a powerful UNIX command that lets you
                       search inside the file contents on a variety of
                       parameters. It’s specially helpful when you are
                       troubleshooting or debugging.


                       The grep command has a huge number of options and
                       use cases. You probably will never need or use all
                       of them. However, you’ll end up using a handful of
                       grep commands most of the time.


                       This article lists the most common grep commands
                       with quick examples. Linux Handbook already has a
                       detailed article on grep, so I won’t go in depth
                       here. I’ll just the common options and their
                       explanation here.


                # ⚓ Increment_and_Decrement_Variables_in_Bash_–_ByteXD⠀⇛


                       A variable is a placeholder that is used to store
                       any numerical or text value so that it can be used
                       during the execution of a computer program. It is
                       normally used in computer programming languages,
                       but you can also use it in Bash scripts.


                       In this tutorial, you will learn about increment
                       and decrement variables and different ways of
                       incrementing and decrementing variables in Bash
                       scripts.


                       In traditional programming languages, there is a
                       proper syntax to define a variable. Also, you have
                       to specify the well-known data type (integer,
                       float, char, etc.) that defines the type of value
                       the variable would contain and allocates memory for
                       it accordingly.


          o § Games⠀➾


                # ⚓ CodeWeavers_advertising_for_a_Wine_developer_and_Linux
                  gaming_tester_|_GamingOnLinux⠀⇛


                       Looking for a job in the Linux space? Now is your
                       chance! CodeWeavers, the company that sponsors
                       development on the Wine compatibility layer and
                       works with Valve on Steam Play Proton are hiring
                       again.


                       Similar to one they offered before is a “General
                       Wine Developer”. This position needs a programmer
                       to hack away on open source to improve Wine’s
                       ability to run games and all sorts of other
                       software. CodeWeavers say there’s multiple
                       positions available here between working on their
                       own CrossOver UI, Proton for Valve and various
                       other CodeWeavers’ stuff. One of the key points
                       though is you cannot have had exposure to Microsoft
                       code or reverse-engineering of Microsoft software,
                       as everything Wine related has to be totally clean-
                       room.


                # ⚓ Cyberpunk_adventure_Technobabylon_gets_a_modern_native
                  Linux_build_|_GamingOnLinux⠀⇛


                       Wadjet Eye Games and Technocrat Games have returned
                       with a fresh upgrade of Technobabylon, a 2015
                       cyberpunk adventure. This follows on from updates /
                       new Linux ports of Unavowed, Gemini Rue and The
                       Blackwell Bundle.


                # ⚓ Steam_announces_changes_to_sales_for_devs,_next_set_of_sale
                  dates_up_|_GamingOnLinux⠀⇛


                       For developers, the time they have to wait between
                       running sales has been decreased. Instead of having
                       to wait six weeks between sales, they now only have
                       to wait four (28 days). Although, this rule does
                       not apply to the four most major site-wide sales
                       being the Lunar New Year, Summer, Autumn, and
                       Winter sales. Valve are also provided developers
                       with a new tool to manage participating in sale
                       events, where developers get a centralized area to
                       see all the events their games qualify to be
                       included in. Sounds pretty useful.


                # ⚓ Total_War:_WARHAMMER_III_gets_a_nice_long_official_Let’s
                  Play_video_|_GamingOnLinux⠀⇛


                       We don’t yet know the exact date of the Linux
                       version, with it coming “as close to launch day as
                       possible” with the main release date being February
                       17.


                # ⚓ If_you_still_don’t_own_Stellaris_it’s_currently_free_with
                  Amazon_Prime_Gaming_|_GamingOnLinux⠀⇛


                       Quick tip – GOG and Amazon Prime Gaming have teamed
                       up to give subscribers an offer they can’t refuse –
                       another free game! Get a copy of Stellaris from
                       Paradox Interactive / Paradox Development Studios.


                       How to get it? Easy. Just head over to Prime Gaming
                       and click the Games header, then you will see that
                       you can redeem Stellaris on GOG. The offer is open
                       until March 1 and the deal is open to any region
                       that is available with GOG, so you should be good
                       to go. Always nice to get another free game and
                       Stellaris is one where you can happily sink 100s of
                       hours into even without any DLC.


                # ⚓ Valheim_getting_Steam_Deck_tweaks,_more_new_content_teased
                  |_GamingOnLinux⠀⇛


                       Valheim, absolutely one of the best games available
                       natively for Linux is going to get some upgrades
                       for the Steam Deck.


                # ⚓ First_big_Jump_in_Steam_Deck_Verified_Games⠀⇛


                       Today marks the first time where the Steam Deck
                       Verified and Playable games are listed at a much
                       faster rate – nothing like what we have seen in the
                       past week. Just in the past 24 hours since our last
                       refresh, 56 titles have been added in total,
                       bringing the current status to…


          o § Desktop Environments/WMs⠀➾


                # § K Desktop Environment/KDE SC/Qt⠀➾


                      # ⚓ KDE’s_Falkon_Browser_Adds_Screen_Capture_and_PDF
                        Reader_with_its_Latest_Update_in_3_Years⠀⇛


                             If you’re a KDE fan, you must have certainly
                             come across or even used Falkon. So, you must
                             be pleasantly surprised to find out that KDE
                             has managed to release a new major upgrade of
                             their web browser.


                             Unlike other mainstream web browsers, Falkon
                             does not receive frequent updates. And, the
                             latest release is an exciting update, after a
                             gap of almost three years!


                             For those unaware, Falkon is a simple open-
                             source web browser built upon the
                             QtWebEngine. It was initially known as
                             QupZilla, later rebranded to Falkon under
                             KDE.


          o § Distributions⠀➾


                # ⚓ FSF-Tailored_Trisquel_10.0_Released_–_Adds_32-bit_Arm,
                  Defaults_To_GNU_Linux-Libre_5.4_–_Phoronix⠀⇛


                       Trisquel 10.0 was released on Tuesday as the latest
                       major release of this operating system that is one
                       of the few GNU/Linux distributions endorsed by the
                       Free Software Foundation. While it has the blessing
                       of the FSF, it’s a bit behind on the software
                       feature front.


                       Trisquel 10.0 is the FSF-endorsed distribution’s
                       rebase to… Ubuntu 20.04 lTS. Two years after the
                       release of Ubuntu 20.04 LTS and two months ahead of
                       Ubuntu 22.04 LTS, Trisquel 10.0 is now onto this
                       Ubuntu long-term support release. Trisquel 10.0 for
                       meeting the free software requirements defaults to
                       using the GNU Linux-Libre 5.4 kernel by default,
                       which is also a bit unfortunate for a 2022+
                       distribution considering all of the improvements
                       since in recent kernels. There is though 5.8 and
                       5.13 based GNU Linux-libre builds available within
                       Trisquel 10.0 as alternatives but surprisingly not
                       any option based on Linux 5.15 LTS.


                       Besides migrating to an Ubuntu 20.04 LTS base,
                       Trisquel 10.0 adds initial support for ARM devices.
                       However, that’s an “armhf” target. Next the project
                       will be working towards 64-bit Arm and POWER
                       architecture support. They did drop 32-bit x86
                       support though with Trisquel 10.


                # ⚓ Trisquel_GNU/Linux_10.0_“Nabia”_Is_Here_with_Support_for
                  ARM_Devices⠀⇛


                       The Trisquel team released Trisquel GNU/Linux 10.0
                       “Nabia” and we’re going to take a look at it today.


                       Very few Linux distros have the privilege of being
                       endorsed by the Free Software Foundation (FSF). One
                       of those distros is Trisquel GNU/Linux.


                       You can be forgiven if you haven’t heard of
                       Trisquel. It is an Ubuntu-based desktop oriented
                       distro which features free and open source software
                       exclusively. This means that there is no closed
                       source proprietary software of any kind included in
                       this distro. That includes the kernel, all the
                       apps, all the libraries, drivers, everything.


                       Keep in mind that if you wanted to go out and get
                       proprietary drivers, Trisquel doesn’t provide any
                       easy tools for you to use to get those proprietary
                       drivers. Certainly, you’re not gonna like spending
                       a ton of money on that high-end Nvidia graphics
                       card and then having to run the nouveau drivers.


                # § EasyOS⠀➾


                      # ⚓ EasyOS_version_3.3_released⠀⇛


                      # ⚓ EasyOS_Dunfell-series_3.3⠀⇛


                             EasyOS was created in 2017, derived from
                             Quirky Linux, which in turn was derived from
                             Puppy Linux in 2013. Easy is built in woofQ,
                             which takes as input binary packages from any
                             distribution, and uses them on top of the
                             unique EasyOS infrastructure.
                             Throughout 2020, the official release for
                             x86_64 PCs was the Buster-series, built with
                             Debian 10.x Buster DEBs.
                             EasyOS has also been built with packages
                             compiled from source, using a fork of
                             OpenEmbedded (OE). Currently, the Dunfell
                             release of OE has been used, to compile two
                             sets of binary packages, for x86_64 and
                             aarch64.
                             The latter have been used to build EasyOS for
                             the Raspberry Pi4, and first official
                             release, 2.6.1, was in January 2021.
                             The page that you are reading now has the
                             release notes for EasyOS Dunfell-series on
                             x86_64 PCs, also debuting in 2021.
                             Ongoing development is now focused on the
                             x86_64 Dunfell-series. The last version in
                             the x86_64 Buster-series is 2.6.2, on June
                             29, 2021, and that is likely to be the end of
                             that series. Releases for the Pi4 Dunfell-
                             series are still planned but very
                             intermittent.
                             The version number is for EasyOS itself,
                             independent of the target hardware; that is,
                             the infrastructure, support-glue, system
                             scripts and system management and
                             configuration applications.
                             The latest version is becoming mature, though
                             Easy is an experimental distribution and some
                             parts are under development and are still
                             considered as beta-quality. However, you will
                             find this distro to be a very pleasant
                             surprise, or so we hope.


                      # ⚓ Should_EasyOS_be_shipped_as_an_uncompressed_image
                        file?⠀⇛


                             The upcoming Easy 3.3 will be a 616MB
                             download, compressed .img.gz file.


                             However, if I was to build it with a 4MB
                             second partition, and not compress the file,
                             the .img file will be 1 + 639 + 4 +1 equals
                             645MB. Those 1MB at start and end are
                             padding.


                             Then there will be no need to uncompress the
                             file, just one click to open it. But, I don’t
                             know if this is really such a big issue,
                             might put it on the back burner. Reconsider
                             sometime later.


                      # ⚓ More_exotic_cross-compile_path_fixing⠀⇛


                      # ⚓ Hack_to_fix_OE_cross-compile_paths⠀⇛


                             This topic is somewhat esoteric, of interest
                             to developers only. The OpenEmbedded cross-
                             compile environment can cause some unexpected
                             side-effects when the executables are run in
                             the target system.


                # § IBM/Red Hat/Fedora⠀➾


                      # ⚓ Cockpit_Project:_Cockpit_262⠀⇛


                      # ⚓ Technical_debt_lessons_from_the_pandemic_|_The
                        Enterprisers_Project⠀⇛


                             If your technology organization is like most,
                             the volume of requests you receive can
                             outpace your ability to keep up with the
                             architecture you need to support innovation
                             and growth. Over time, this situation can
                             cause technical debt, hindering
                             responsiveness and speed to market.


                             Like many organizations, our company faced
                             challenges during the pandemic that left us
                             with the burden of technical debt. Here’s how
                             we responded to come out ahead.


                      # ⚓ Red_Hat_EMEA_Digital_Leaders_Awards_2021:_And_the
                        regional_winners_are…⠀⇛


                             In our previous post we shared the names of
                             the winners of Red Hat EMEA Digital Leaders
                             Awards across three categories: Open
                             Transformation, Cloud Native, and Hybrid
                             Cloud.


                             The awards were launched in July 2021 in
                             collaboration with Intel and IDC with the
                             goal of identifying innovative and
                             transformative digital leaders using Red Hat
                             technologies and services. The awards shine a
                             spotlight on the innovations of our
                             customers, while focusing on the business
                             value and impact of their projects.


                             Philip Carter, group vice president, European
                             chief analyst and WW C-Suite tech research
                             lead at IDC said: “With close to 120
                             applications, we were able to capture and
                             showcase unique use cases of technology and
                             share some of the best practices for making
                             projects a success.


                             We had the pleasure of reviewing many
                             successful practices across industries in
                             EMEA, from large blue chip organisations to
                             local heroes, all of whom provided incredible
                             insights into their projects and operations.”


                      # ⚓ The_State_of_Customer_Experience_at_Red_Hat:
                        Technical_support_and_subscription_management
                        enhancements⠀⇛


                             In the first post of our three-part series,
                             we discussed enhancements made to several Red
                             Hat products and product documentation based
                             on customer feedback in the past year. In
                             this post, we’ll look at some additional
                             areas where we have been working to improve
                             the experience we deliver to customers and
                             partners.


                             Other areas that we have made improvements to
                             based on what we’ve heard from our customers.
                             Let’s take a look at some of the enhancements
                             we made to the areas of support delivery and
                             subscription management.


                      # ⚓ Protect_secrets_in_Git_with_the_clean/smudge_filter_|
                        Red_Hat_Developer⠀⇛


                             When working on public Git repositories, you
                             need to pay close attention so that you don’t
                             accidentally push secret information such as
                             tokens, private server addresses, personal
                             email addresses, and the like. One of the
                             tools that can help you is Git’s clean/smudge
                             filter.


                      # ⚓ Edge_computing_strategy:_5_potential_gaps_to_watch
                        for_|_The_Enterprisers_Project⠀⇛


                             In financial terms, edge has already arrived:
                             IDC predicts that companies will spend $176
                             billion on edge computing worldwide this
                             year, roughly a 15% increase from 2021.


                             Ultimately, that’s just a (big) number. There
                             may be more qualitative signs of edge
                             computing’s maturation in terms of
                             architectural approaches, technical
                             capabilities, enterprise use cases, security
                             tactics, and more.


                             “Even if we see echoes of older architectures
                             in certain edge computing deployments, we
                             also see developing edge trends that are
                             genuinely new, or at least quite different
                             from what existed previously,” Gordon Haff,
                             technology evangelist at Red Hat, wrote
                             recently in his analysis of edge trends to
                             watch in 2022. “And they’re helping IT and
                             business leaders solve problems in industries
                             ranging from telco to automotive, for
                             example, as both sensor data and machine
                             learning (ML) data proliferates.”


                             IT leaders don’t often tackle business
                             problems without a plan, which is why edge
                             strategies – and related categories like IoT
                             and machine learning – figure prominently on
                             their roadmaps. In Red Hat’s 2022 Global Tech
                             Outlook report, for example, 61% of
                             respondents reported plans to run IoT or edge
                             workloads (or both) in the next 12 months.


                # § Debian Family⠀➾


                      # ⚓ How_to_install_Envoy_Proxy_on_Debian_11_–
                        NextGenTips⠀⇛


                             Envoy is an L7 proxy and communication bus
                             designed for large modern service-oriented
                             architecture. The project was born out of the
                             belief that the network should be transparent
                             to applications. When network and
                             applications problems occur, it should be
                             easy to determine the source of the problem.


                             Envoy is an open-source edge and service
                             proxy, designed for cloud-native
                             applications. Let’s dive in and learn how to
                             install Envoy on Debian 11.


                      # ⚓ Ben_Hutchings:_CI_for_the_Debian_kernel_team⠀⇛


                             Starting just after Christmas, I have been
                             working on CI for all the kernel team’s
                             packages on Salsa. The salsa-ci-team has done
                             great work on producing a common pipeline
                             that is usable for most packages with minimal
                             configuration. However, for some packages a
                             lot more work was required.


                      # ⚓ Ben_Hutchings:_Debian_LTS_work,_January_2022⠀⇛


                             In January I was assigned 24 hours of work by
                             Freexian’s Debian LTS initiative. I worked 16
                             hours, and will carry over the remaining time
                             to February.


                             I sent various backported security fixes for
                             Linux to the stable mailing list, and they
                             have been included in subsequent stable
                             releases. I rebased the linux package on the
                             latest 4.9-stable release, but did not yet
                             upload it.


                      # ⚓ Which_version_of_Debian_consumes_less_RAM_?_–
                        LinuxStoney⠀⇛


                             Among the many Linux distributions that we
                             can get our hands on today, Debian is one of
                             the most loved and used for different
                             reasons. Many users install it not only for
                             its functionality, but also for its interface
                             and external appearance.


                             For many, this is an alternative that is not
                             suitable for users with little experience in
                             Linux environments, but it is also true that
                             it has improved a lot in recent years. Little
                             by little it has been adapting to all types
                             of users and devices, hence its growth and
                             acceptance by the majority. Debian is an open
                             source operating system valid for both
                             corporate environments and end users. Of
                             course, depending on the capabilities of our
                             computer, we can customize the behavior of
                             the software .


                             Although in most cases these Linux
                             distributions are prepared to work smoothly
                             on somewhat more limited computers, we must
                             also take some precautions. Generally, the
                             most modern versions of Windows have higher
                             demands than Linux systems, hence precisely
                             one of its attractions. This is something
                             that refers directly to the consumption that
                             the system makes of both the CPU and the RAM
                             memory of the computer.


                             Precisely because of all this that we are
                             telling you and if you have made the decision
                             to download and install Debian on your PC, we
                             are going to focus on the resource
                             consumption section. More specifically, we
                             are going to talk about the version of this
                             Linux that consumes less RAM . It is evident
                             that all this will be useful in the event
                             that you have an old PC or a somewhat limited
                             one in terms of internal specifications.


          o § Devices/Embedded⠀➾


                # ⚓ Linux_Arcade_Cab_Gives_Up_Its_Secrets_Too_Easily_|
                  Hackaday⠀⇛


                       Sometimes reverse engineering embedded systems can
                       be a right old faff, with you needing to resort to
                       all kinds of tricks such as power glitching in
                       order to poke a tiny hole in the armour, giving you
                       an way in. And, sometimes the door is just plain
                       wide open. This detailed exploration of an off-the-
                       shelf retro arcade machine, is definitely in that
                       second camp, for an unknown reason. [Matthew Alt]
                       of VoidStar Security, took a detailed look into how
                       this unit works, which reads as a great
                       introduction to how embedded Linux is constructed
                       on these minimal systems.


                # § Open Hardware/Modding⠀➾


                      # ⚓ Ultimate_Raspberry_Pi_Computer_Build_–_Water_Cooled
                        CM4_with_NVMe_SSD_–_The_DIY_Life⠀⇛


                             If you’ve been following my projects for a
                             while then you’re probably aware that I like
                             to do unnecessary things, like building a
                             water-cooled Raspberry Pi cluster, or this
                             desktop version of a water-cooled Pi. As cool
                             as these may look, they’re not all that
                             practical, so today we’re going to build
                             something that is a little more practical,
                             although is still a bit unnecessary.


                             We’re going to build the ultimate water-
                             cooled Raspberry Pi desktop computer and try
                             to give it all of the bells and whistles,
                             including front-facing USB ports, a power
                             button, an OLED stats display and an NVME
                             SSD.


                             I’m going to be using a Raspberry Pi Compute
                             Module 4, or CM4 for short, for this build so
                             that I can use the CM4 IO Board for a number
                             of features which we’ll get into in a bit.


                # § Mobile Systems/Mobile Applications⠀➾


                      # ⚓ The_Google_One_VPN_for_Android_just_got_a_little
                        safer_–_Android_Authority⠀⇛


                      # ⚓ Google_bans_nasty_Android_app!_If_you’ve_downloaded
                        it,_delete_it_now_|_Express.co.uk⠀⇛


                      # ⚓ Samsung_Galaxy_Tab_A8_review⠀⇛


                      # ⚓ The_best_chemistry_apps_for_Android_–_Android
                        Authority⠀⇛


                      # ⚓ Best_Free_Android_Apps_[February_2022]:_5_Must-Have
                        Apps_|_NDTV_Gadgets_360⠀⇛


                      # ⚓ Sony’s_4K_OLED_Xperia_PRO_5G_Android_smartphone_sees
                        first_discount_at_$500_off_–_9to5Toys⠀⇛


                      # ⚓ Fly_around_your_phone_with_Android’s_hidden_voice
                        command_center_|_Computerworld⠀⇛


                      # ⚓ How_to_minimise_the_risk_of_smartphone_snooping_
                        (Android_privacy_tips)_–_Cybershack⠀⇛


                      # ⚓ How_to_clear_your_Galaxy,_Pixel_or_other_Android
                        phone’s_cookies_and_cache_–_CNET⠀⇛


                      # ⚓ How_to_Leave_a_Group_Text_on_Android_in_2022⠀⇛


                      # ⚓ You_told_us:_You’re_split_about_plans_for_an_Indian-
                        made_Android,_iOS_rival⠀⇛


                             When you think of mobile phone operating
                             systems, Android and iOS probably come to
                             mind. Beyond these two, very few other OSes
                             enter the public discourse. However, the
                             Indian government is considering changing
                             this by creating a rival to these two
                             industry juggernauts.


                             But is this a good idea or another fool’s
                             errand that many companies have fallen foul
                             of? Well, we asked our readers this in a
                             recent poll. Here are the results.


          o § Free, Libre, and Open Source Software⠀➾


                # ⚓ SteveCo:_Video_from_OLF_2021⠀⇛


                       I had two talks at OLF in December. I just noticed
                       that videos are up on YouTube for both of them.


                # ⚓ Kiwi_TCMS:_Kiwi_TCMS_11.1⠀⇛


                       We’re happy to announce Kiwi TCMS version 11.1.


                # ⚓ Open_source_cloud_platform:_meet_OpenStack⠀⇛


                       Are you looking for an open source cloud platform
                       and you don’t know where to start? Are you getting
                       lost in all the independent rankings and cloud
                       platform comparison pages?


                       Try OpenStack and get your open source cloud
                       platform up and running today. OpenStack works at
                       any scale: from a single workstation to thousands
                       of nodes and installs in minutes.


                       Sounds impossible? Give it a try or continue
                       reading to explore where is it coming from.


                # ⚓ Open_Source_Apache_CloudStack_Powers_Government_Services_in
                  India⠀⇛


                       A division of the government of India went with the
                       open source Apache CloudStack instead of a public
                       cloud provider to bring services to millions of
                       citizens.


                # ⚓ Apache_Month_in_Review:_January_2022⠀⇛


                       Welcome to the latest monthly overview of events
                       from the Apache community.


                # ⚓ joinpeertube.org_–_a_free_decentralized_federated_video
                  platform_alternative_to_centralize_YouTube_Dailymotion
                  Vimeo⠀⇛


                # § Productivity Software/LibreOffice/Calligra⠀➾


                      # ⚓ LibreOffice_7.3_Community_is_better_than_ever_at
                        interoperability⠀⇛


                             LibreOffice 7.3 Community, the new major
                             release of the volunteer-supported free
                             office suite for desktop productivity, is
                             available from https://www.libreoffice.org/
                             download. Based on the LibreOffice Technology
                             platform for personal productivity on
                             desktop, mobile and cloud, it provides a
                             large number of improvements targeted at
                             users migrating from Microsoft Office to
                             LibreOffice, or exchanging documents between
                             the two office suites.


                      # ⚓ LibreOffice_7.3_Released,_This_is_What’s_New⠀⇛


                             This release sees a score of improvements
                             aimed at making it easier for users to
                             migrate from Microsoft Office to LibreOffice,
                             as well as those who regularly swap documents
                             between the two office suites.


                             To this end, LibreOffice devs have spent time
                             developing new features to handle change
                             tracking in tables and text within documents.
                             This, The Document Foundation say will have a
                             “positive impact” on interoperability
                             throughout all Microsoft Office formats.


                      # ⚓ LibreOffice_7.3_Released_With_Better_Interoperability
                        For_Microsoft_Office_Files⠀⇛


                             The Document Foundation has released
                             LibreOffice 7.3 as the newest half-year
                             update to this leading open-source, cross-
                             platform office suite.


                             LibreOffice 7.3 is introducing support in
                             Writer for hyperlinks attached to shapes,
                             Writer better tracks and indicates changes to
                             text, various performance improvements, Bash-
                             like auto-completion for Calc spreadsheets, a
                             new bullet mode editing feature, and much
                             more. As usual, LibreOffice 7.3 has many
                             import/export improvements to its Microsoft
                             Office DOC, DOCX, XLSX, and PPTX file format
                             handling. The Document Foundation is
                             particularly promoting LibreOffice 7.3′s
                             improved interoperability with Microsoft
                             Office files as one of the key pillars of
                             this release.


                      # ⚓ Compiling_LibreOffice_with_Meson_even_further⠀⇛


                             After building the basics of LO on Windows
                             and macOS the obvious next step is to build
                             all of it. This is just grunt work and quite
                             boring at that actually. I almost got it done
                             apart from the fact that at the end I got
                             lazy and skipped those bits that require
                             weird dependencies (of which more later).


                             The end result has approximately 7800
                             individual compilation and linking steps. It
                             takes about 30 minutes on a 16 core Ryzen
                             3700X CPU using Windows and Visual Studio. On
                             a 7 year old 4 core Macbook it takes around 3
                             hours.


                             Quite a bit, actually. The most pertinent
                             would be all the configuration files that get
                             installed. There are a lot of them and they
                             need to be exactly right, otherwise the end
                             result fails to start with cryptic error
                             messages, if any. Unlike code compilation
                             there is no easy way to know in advance what
                             should be done or how things should behave.
                             Ideally you’d get help with people who know
                             the innards of the program and all the
                             configgen bits.


                      # ⚓ LibreOffice_7.3_released⠀⇛


                             Version 7.3 of the LibreOffice “Community”
                             edition is out. “In addition to the majority
                             of code commits being focused on
                             interoperability with Microsoft’s proprietary
                             file formats, there is a wealth of new
                             features targeted at users migrating from
                             Office, to simplify the transition”.


                      # ⚓ Install_LibreOffice_7.3_On_Ubuntu_/_Rocky_Linux_&
                        Fedora_|_Tips_On_UNIX⠀⇛


                             This tutorial will be helpful for beginners
                             to install LibreOffice 7.3 on Ubuntu 20.04
                             LTS, Ubuntu 22.04, LinuxMint 20.3, Rocky
                             Linux 8, AlamLinux 8, and Fedora 35.


                             LibreOffice is open-source software and it is
                             based on OpenOffice.org also an alternative
                             for Microsoft office, unlike others
                             LibreOffice supports lots of tools.


                             LibreOffice suite contains the following
                             Writer, Calc, Impress, Draw, and Math


                             The Document Foundation today (02-02-2022)
                             has released the newer version of LibreOffice
                             7.3 and this is the newest half-year update.


                # § FSFE⠀➾


                      # ⚓ Welcome_to_the_Matrix:_the_FSFE_now_runs_its_own
                        server⠀⇛


                             The FSFE is eager to support its community
                             with diverse Free Software communication
                             channels. We are happy to announce that we
                             have recently added Matrix to this list.
                             After successful beta tests, every FSFE
                             supporter and volunteer can now create their
                             own Matrix account. We will also use our
                             instance for the FSFE’s virtual booth during
                             FOSDEM!


                             For over a decade now, the FSFE has been
                             offering an XMPP (also known as Jabber)
                             server as well as a traditional IRC channel
                             for our community to quickly exchange
                             messages. Today we officially open the gates
                             to our new Matrix server, which has been
                             successfully tested for a few months and has
                             a few advantages over XMPP.


                # § FSF⠀➾


                      # § GNU Projects⠀➾


                            # ⚓ GNU_Guix:_Meet_Guix_at_FOSDEM⠀⇛


                                   As usual, GNU Guix will be present at
                                   FOSDEM this week-end, February 5th and
                                   6th. Due to the pandemic, FOSDEM takes
                                   place on-line for the second year, but
                                   we’re confident the wires will be able
                                   carry enthusiasm to the homes of the
                                   thousands of attendees.


                # § Programming/Development⠀➾


                      # ⚓ Qt_Allstack_I_–_Setup_–_KDAB_–_KDAB_on_Qt⠀⇛


                             Writing mobile apps can be a lot of fun,
                             especially with Qt and QML. But if your
                             background is all about C++ and Qt, writing
                             the entire required stack for a mobile app
                             can be challenging. Adding push
                             notifications, storing information, and
                             having to write that with another language
                             and framework might make you give up. But we
                             got you covered!


                      # ⚓ LLVM/Clang_14_Ends_Feature_Development_With_Better
                        C++20_Support,_Armv9_Added_–_Phoronix⠀⇛


                             LLVM release manager Tom Stellard has
                             branched the LLVM 14.0 code and that of its
                             sub-projects like Clang within the mono
                             repository. As such, LLVM 14.0 feature
                             development is now over with the main Git
                             branch working towards what will become LLVM
                             15.0 later this calendar year.


                             LLVM 14.0 will now undergo weeks of bug
                             fixing and testing before being released as
                             stable likely in March. A first release
                             candidate is expected later this week while a
                             second release candidate in early March. If
                             all goes well, LLVM 14.0 stable could be out
                             in mid-March at the earliest.


                      # ⚓ Robert_Foss:_Git_Alias_function_syntax⠀⇛


                             A basic example of the git alias function
                             syntax looks like this.


    * § Leftovers⠀➾


          o ⚓ Will_A_Kettle_Filled_With_Alcohol_Boil_Dry?_|_Hackaday⠀⇛


                 The average home kettle is set up to switch off
                 automatically when water reaches its boiling point. But
                 would a kettle filled with alcohol, which has a
                 significantly lower boiling point, actually turn off?
                 [Steve Mould] set out to find out.


                 The prediction was that a kettle full of 40% strength
                 vodka would boil dry, as the vodka would evaporate before
                 it actually got to a hot enough temperature to cause the
                 kettle’s cutout mechanism to kick in. The experiment was
                 done outside to minimise the dangers from the ethanol
                 vapor. As it turns out, the vapor from the boiling vodka
                 is about 80% ethanol and just 20% water, so eventually
                 the mixture left in the kettle is mostly water and it
                 boils hot enough to trigger the cutout mechanism.


                 However, the experiment doesn’t end there. Trying again
                 with 99% ethanol, when the fluid started boiling, the
                 kettle switched off even more quickly. So what’s going
                 on?


          o § Science⠀➾


                # ⚓ You_have_a_BIG_cognitive_degradation_problem…⠀⇛


                       The book is about how and why everyone is losing
                       their capacity for concentration. As one of many
                       examples, Hari describes a couple a couple he met,
                       in front of a tourist attraction, looking at a
                       picture of that attraction on a tablet screen,
                       instead of enjoying the real thing they had around
                       them.


                # ⚓ The_first_four_words_you_see_are…_|_Stop_at_Zona-M⠀⇛


                       It’s amazing to which lenghts people will go to
                       work for free for the already extremely rich
                       stockholders of social media companies. Yes, the
                       mere act of having and using a social media account
                       today is equivalent to sending money to them. But
                       this is another level.


                # ⚓ Misleading_Packaging_History:_Maybe_Double-Check_the
                  Label⠀⇛


                       A few years ago, I wrote a tweet, which I imagine
                       is shocking to many of you, but this tweet was
                       quite the tweet. It was just five words, attached
                       to an image that highlighted the risks of
                       misleading packaging, and yet, somehow I apparently
                       captured the zeitgeist of clickbait on the internet
                       in 2018. It was, honestly, weird, and it pulled
                       some crazy numbers (while making my phone totally
                       useless for a while). But now that the dust has
                       long settled on that tweet, I got to thinking about
                       the central theme of it: Why do we put up with
                       misleading packaging? And who decides that
                       something is misleading, anyway? Today’s Tedium
                       looks on the other side of the personal pan pizza
                       that is product design.


                       [...]


                       In a lot of ways, the definition of misleading can
                       be in the eye of the beholder, especially if that
                       beholder knows how to contact a lawyer.


                       Let me give you a fairly recent example. Last year,
                       a woman filed a lawsuit, seeking class action
                       status, over branding on a common product that had
                       been sold for decades that she felt was inaccurate?
                       The product? Morningstar Farms veggie burgers. What
                       was her problem? Simply put, she felt that the term
                       “veggie” implied more vegetables than the burgers
                       actually contained. The patties were largely made
                       of wheat and corn syrup.


                       Kellogg’s, the makers of these popular patties
                       (which have a history that goes back more than a
                       century), defended itself in court, and won a quick
                       dismissal of the case just last week, after a judge
                       agreed with the cereal-maker that “veggie” was a
                       common way to refer to vegetarian products.


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # ⚓ Demand_for_Rapid_Risk_Elimination_for_Linux_at_Scale
                        Fueled_TuxCare’s_Meteoric_Growth_in_2021⠀⇛


                             With ever-changing compliance requirements
                             and shrinking timelines to apply security
                             updates, enterprises find themselves in a
                             constant, multipronged race to remain
                             compliant, competitive, and fortified against
                             increasingly sophisticated threat vectors.
                             Live-patching is one way engineering teams
                             can stay ahead of system vulnerabilities
                             without the downtime.


                      # ⚓ Microsoft:_Keep_your_Windows_PC_online_for_this_long,
                        or_it_won’t_update_properly [Ed: You know Microsoft is
                        in for a severe crisis when even Liam Tung (Microsoft
                        booster) admits Windows_is_a_sordid_chaotic_mess]⠀⇛


                      # § Security⠀➾


                            # ⚓ SureMDM_bug_chain_enabled_wholesale_compromise
                              of_managed_devices⠀⇛


                                   Vulnerabilities in SureMDM could have
                                   been chained to compromise every device
                                   running the popular mobile device
                                   management (MDM) platform within a
                                   targeted enterprise, security
                                   researchers have revealed.


                                   The vendor, Indian tech firm 42 Gears,
                                   has patched the bugs, which led to
                                   remote code execution (RCE) via the web
                                   console, along with RCE, command
                                   injection, hardcoded password, local
                                   privilege escalation, and information
                                   disclosure flaws affecting the Linux
                                   agent.


                            # ⚓ Security_updates_for_Wednesday_[LWN.net]⠀⇛


                                   Security updates have been issued by
                                   CentOS (samba), Debian (apache2 and
                                   python-django), Fedora (kernel and
                                   phpMyAdmin), Mageia (kernel and kernel-
                                   linus), openSUSE (samba), Oracle
                                   (nginx:1.20 and samba), Red Hat
                                   (cryptsetup, java-1.8.0-ibm, kernel,
                                   nodejs:14, rpm, and vim), SUSE (kernel,
                                   python-Django, python-Django1, and
                                   samba), and Ubuntu (cron).


                            # ⚓ Samba_bug_may_allow_code_execution_as_root_on
                              Linux_machines,_NAS_devices_(CVE-2021-44142)_–
                              Help_Net_Security⠀⇛


                                   A critical vulnerability (CVE-2021-
                                   44142) in Samba, a widely used open
                                   source implementation of the Server
                                   Message Block (SMB) networking
                                   protocol, could allow attackers to
                                   execute arbitrary code as root on
                                   affected Samba installations.


                                   Several updated versions of Samba have
                                   been released on Monday, fixing CVE-
                                   2021-44142 and two other flaws (1, 2),
                                   but since the software is included in
                                   most Linux and Unix-like operating
                                   systems (including Apple’s macOS and
                                   macOS Server), users of those are
                                   advised to keep an eye out for specific
                                   updates by those developer teams.


                            # ⚓ Tripwire_Patch_Priority_Index_for_January_2022
                              |_The_State_of_Security⠀⇛


                                   Tripwire’s January 2022 Patch Priority
                                   Index (PPI) brings together important
                                   vulnerabilities for Apache, Open Source
                                   Policy Kit, Adobe, and Microsoft.


                            # ⚓ PostgreSQL:_PostgreSQL_JDBC_42.3.2/42.2.25
                              Security_update⠀⇛


                                   A security advisory has been created
                                   for the PostgreSQL JDBC driver. The
                                   driver provides the facility to
                                   instantiate plugin instances based on
                                   class names provided via
                                   authenticationPluginClassName,
                                   sslhostnameverifier, socketFactory,
                                   sslfactory, sslpasswordcallback
                                   connection properties.


                                   However, the driver did not verify if
                                   the class implements the expected
                                   interface before instantiating the
                                   class.


                            # § Privacy/Surveillance⠀➾


                                  # ⚓ Digital_Transparency:_A_Right_to
                                    Information_Report_for_January_2022⠀⇛


                                         One of our key areas of work is
                                         ensuring that public authorities
                                         respect data privacy and engage
                                         in practices that will ensure
                                         that the right to privacy is
                                         protected. We filed 9 RTI
                                         requests and 1 first appeal with
                                         various authorities this month to
                                         ask for information pertaining to
                                         newly introduced projects which
                                         affect the data privacy of Indian
                                         citizens.


                                         [...]


                                         Another focus of our work is to
                                         ensure that freedom of speech and
                                         expression on the internet is
                                         protected and that unnecessary
                                         censorship does not lead to a
                                         chilling effect on people’s
                                         fundamental rights. For this, we
                                         routinely file RTI requests to
                                         demand accountability for
                                         instances that may hamper free
                                         speech on the internet such as
                                         website blocking or internet
                                         shutdowns.


                                         In the last month, we have filed
                                         1 RTI request to demand
                                         accountability for violations of
                                         free speech on the internet with
                                         the National Internet Exchange of
                                         India on the notice to seek
                                         written approval of NIXI CEO on
                                         .in domain registrations.


                                         We also filed one first appeal
                                         with the Ministry of Information
                                         and Broadcasting for their
                                         incomplete reply dated 14/01/2022
                                         to our original reply requesting
                                         information on the list of
                                         publishers to whom communications
                                         were sent by the Ministry of
                                         Information and Broadcasting.



                    ䷩ 𝚕𝚒𝚗𝚎 4876

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_3/2/2022:_System76_Scheduler_and_GStreamer_1.18.6⠀✐


Posted in News_Roundup at 8:53 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Distributions
          o Devices/Embedded
    * Free_Software/Open_Source
    * Leftovers
    * § GNU/Linux⠀➾


          o § Desktop/Laptop⠀➾


                # ⚓ System76_refreshes_Kudu_Linux-based_laptops_with_Ryzen_9
                  5900HX_processor_and_Nvidia_RTX_3060_GPU⠀⇛


                       Linux-based laptop OEM System76 is updating the
                       Kudu lineup with a 2022 model that combines the
                       high-end performance of the AMD Ryzen 9 5900HX APU
                       with the mid-range Nvidia RTX 3060 GPU and the
                       versatility of a Linux distro. Users can choose
                       among System76’s proprietary Pop!_OS versions or
                       the Ubuntu 20.04 LTS.


                       Besides the powerful APU and GPU combo, the latest
                       Kudu6 laptops feature 15.6-inch screens with 1080p
                       resolution and 144 Hz refresh rate, up to 64 GB of
                       DDR4-3200 RAM, and up to 4 TB of NVMe SSD storage
                       via 2x M.2 slots. Connectivity includes 2.5 GbE +
                       Wi-Fi 6 + BT 5, while the system can support up to
                       three external video sources through a USB-C 3.2
                       Gen 2 with DP 1.4, miniDP 1.4 connector, plus an
                       HDMI video out.


                       Additionally, there are ports such as 1x USB-A 3.2
                       Gen 1, 1x USB-A 3.2 Gen and 1x USB 2.0, plus audio
                       jacks. The new model weighs 4.85 lbs (2.2 kg) and
                       comes equipped with a rather small 48.96 Wh
                       battery. Standard features like a 720p webcam and
                       multi-color keyboard are offered, as well.


                       Price-wise, the base model with 8 GB of RAM and 240
                       GB storage starts at US$1,799, while a fully
                       equipped model that includes mechanical keys and
                       carrying bag would cost $US3,906.


                # ⚓ System76-Scheduler_Is_A_New_Pop!_OS_Rust_Effort_To_Improve
                  Desktop_Responsiveness⠀⇛


                       Quietly making its v1.0 debut today is system76-
                       scheduler as a Rust-written daemon aiming to
                       improve Linux desktop responsiveness and catering
                       to their Pop!_OS distribution.


                       System76 Scheduler describes itself as…


                # ⚓ System76_Releases_AMD-Powered_Kudu_Laptop⠀⇛


                       Denver-based System76 welcomes back a familiar face
                       to their laptop lineup with the Kudu.


                       System76 has finally brought back the popular Kudu
                       laptop workstation powerhouse. This time around the
                       laptop is equipped with a 3rd-gen AMD (Zen 3) Ryzen
                       9 5900HX H-class processor with 8 cores and 16
                       threads. The CPU runs at 3.3 GHz but can be boosted
                       up to 4.6 GHz.


                       The Kudu is paired with NVIDIA RTX 3060 graphics
                       and up to 64GB of Dual Channel DDR4 memory.


                       According to Ben Shpurker, Product Manager at
                       System76, “This combination makes it the perfect
                       machine for creating on the go.”


                       The 15″ FHD matte display and a thin bezel, but for
                       creators of all kinds, the Kudu laptop can manage
                       up to three external displays with one HDMI port,
                       one mini DisplayPort, and one DisplayPort over USB-
                       C.


                # ⚓ 11_Funniest_Linux_Meme_Distros,_Software,_and_Commands_–
                  Linux_Stans⠀⇛


                       We’ve already featured serious Linux distros like
                       distros for servers and distros for programming.
                       Now, it’s time to blow off some steam with a few
                       funny Linux distros and software.


                       All of these Linux distros and software are
                       actually usable, and they have a funny spin to
                       them. We won’t include unique distros like Damn
                       Vulnerable Linux, GoboLinux, or certain religious
                       Linux distros. We might create a different list for
                       them. In this article, we’ll only feature funny
                       distros and software that are just a meme.


          o § Audiocasts/Shows⠀➾


                # ⚓ Hackaday_Invades_the_FLOSS_Weekly_Podcast⠀⇛


                       Regular Hackaday readers will know that we’re big
                       supporters of free/libre and open source software
                       (FLOSS) around these parts. There’s an excellent
                       chance you are too, as so many of the incredible
                       projects you send our way make it a habit to share
                       their innermost details, from firmware source code
                       to the OpenSCAD files that generate its 3D printed
                       components. So when our recently minted Editor in
                       Chief [Elliot Williams] was invited to join This
                       Week in Tech’s FLOSS Weekly podcast, he jumped at
                       the chance to represent our little corner of the
                       Internet to the wider world of open source
                       aficionados.


                # ⚓ Hackaday_(FLOSS_Weekly_666:_Hackaday_–_Elliot_Williams,
                  Open_Source_&_Hacking)⠀⇛


                       Hackaday.com is what its new editor-in-chief,
                       Elliot Williams, calls “a library of Alexandra for
                       fugitive hacks”—and that’s just one of the many
                       deep and quotable things he shares with Doc Searls
                       and co-host Jonathan Bennett, who is also a veteran
                       Hackaday writer. Topics range from hackable gizmo
                       and the hacker mindset, to how great open firmware
                       hacks become bait for cool new hackable products
                       coming out of China.


                # ⚓ Steam_Deck_Only_Has_200_Verified_Games?!?_–_Invidious⠀⇛


          o § Kernel Space⠀➾


                # ⚓ Parallel_x86_64_CPU_Bringup_Linux_Patches_Revised_For
                  Quicker_Boot_Times_–_Phoronix⠀⇛


                       The Linux kernel patches that alow bringing up
                       x86_64 CPU cores in parallel when booting the
                       system have been revised for faster boot times on
                       today’s systems from high core count desktops to
                       massive servers.


                       This patch series led by longtime kernel developer
                       David Woodhouse is about allowing secondary x86_64
                       CPU cores to be brought up in parallel. While prior
                       versions of the patch series noted a 15x speed-up,
                       his testing of the new “v4″ patches on other
                       hardware is showing a drop from around 500ms down
                       to 100ms.


                # ⚓ Lenovo’s_Platform_Profile_Support_For_AMD_Systems_On_Linux
                  Has_Been_Busted_–_Phoronix⠀⇛


                       ACPI Platform Profile support on Linux has been
                       useful for catering to balancing your power or
                       performance preferences with modern laptops on
                       Linux. It has worked well in general across various
                       devices tested but it turns out to be a dud
                       currently when it comes to AMD Ryzen powered Lenovo
                       systems.


                       ACPI Platform Profile support allows setting the
                       power/balance/power-savings preference (among other
                       possible profiles) on Linux with recent kernel
                       activity that so far has seen major support
                       available for major laptop brands like Lenovo,
                       Dell, and ASUS. The user’s profile preference can
                       be set via a sysfs interface or desktops like KDE
                       Plasma and GNOME have already added convenient user
                       interfaces for setting it nicely from the system
                       settings area.


                # § Graphics Stack⠀➾


                      # ⚓ Download_NVIDIA_510.47.03_Graphics_Driver_For_Linux_|
                        Itsubuntu.com⠀⇛


                             Finally, NVIDIA 510.47.03 proprietary
                             graphics driver is available for Linux-based
                             operating systems. In this driver software,
                             you can find support for Linux 5.17 kernel
                             series and support for the d Vulkan 1.3
                             graphics API.


                      # ⚓ Mike_Blumenkrantz:_Wew⠀⇛


                             Nothing too exciting. Mostly bug fixes. I
                             managed to sneak ARB_sparse_texture_clamp in
                             for zink just before the branchpoint, so all
                             the sparse texturing features supported by
                             Mesa will be supported by zink. But only on
                             NVIDIA since they’re the only driver that
                             fully supports Vulkan sparse texturing.


                             The past couple days I’ve been doing some
                             truly awful things with gl_PointSize to try
                             and make this conformant for all possible
                             cases. It’s a real debacle, and I’ll probably
                             post more in-depth about it so everyone can
                             get a good chuckle.


          o § Intel⠀➾


                # ⚓ Intel’s_“Software_Defined_Silicon”_Linux_Support_Moving
                  Along_–_Phoronix⠀⇛


                       While Intel has not publicly announced their plans
                       around Software Defined Silicon (SDSi), the Linux
                       kernel patches allowing activation of licensed CPU
                       features is continuing to move forward toward
                       mainline integration.


                # ⚓ Intel_Announces_“Project_Circuit_Breaker”_To_Expand_Its_Bug
                  Bounty_Program_–_Phoronix⠀⇛


                       Intel today is announcing Project Circuit Breaker
                       as a broadening of its bug bounty program for
                       “elite hackers” to help discover areas for
                       improvement within Intel’s hardware/software
                       security.


                # ⚓ Intel’s_Vulkan_Linux_Driver_Adds_Experimental_Mesh_Shader
                  Support_For_DG2/Alchemist_–_Phoronix⠀⇛


                       Intel’s open-source “ANV” Vulkan driver for Linux
                       systems now has experimental support for mesh
                       shaders that work with their forthcoming Intel Arc
                       DG2/Alchemist graphics cards.


          o § Applications⠀➾


                # ⚓ XMPP_and_Mail_Clients_–_WindfluechterNet_Blog⠀⇛


                       I really like XMPP, but I’m a little unhappy about
                       the current general situation of XMPP. I think XMPP
                       could do better if there were some benefits of
                       having an XMPP address. For me one of those
                       benefits is to have the option to have just one
                       address I need to communicate to others. If
                       everything is in place and well-configured, a user
                       can be reached by mail, XMPP and SIP (voice/video
                       calls) by just one address.


                       To address this I would like to see XMPP support in
                       mail clients (MUAs). So when you reply to a mail or
                       write a new one, the client will do a lookup in
                       your addressbook if the address has an XMPP field
                       associated with it and (if not) do a DNS lookup for
                       _xmpp-server._tcp.example.com (or the matching
                       domain part of recipients address). If there is an
                       XMPP address listed in mail header, that JID will
                       be used. When the lookup is successful and an xmpp:
                       protocol handler is configured in the system, the
                       MUA offers an option to begin a chat with the
                       recipient and/or displays the presence status of
                       the recipients (depending on available web-presence
                       or presence subscription).


                # ⚓ Standard_Notes_–_A_tidy_little_note-taking_program⠀⇛


                       Note taking. This is one of those things that
                       everyone does, and often, in a rather unique,
                       individual way. Some people leave themselves
                       reminders using physical sticky notes glued to the
                       door of their fridge. Others have reminders in
                       their calendar. Others yet keep information in text
                       files on their desktop. Or you hire a person whose
                       job is to do it for you. Many ways indeed. How
                       about a dedicated tool?


                       [...]


                       ‘Tis a short review, I admit. But then, Standard
                       Notes delivered on its promise. It’s a program that
                       helps you keep a bunch of notes sorted and
                       organized, you can tag your notes, protect them in
                       case they contain sensitive information, and if you
                       go for a full account, you can then also back them
                       up online. Quite handy.


                       Now, does that mean my days of desktop files are
                       over? Well, no. There’s one more thing that notes
                       are supposed to do – constantly remind you of what
                       you should be doing. Hence the desktopness of the
                       whole idea, in me book. But then, you can also have
                       calendar reminders, alerts and alarms, and you
                       could just keep notes of practical things worth
                       remembering, in which case, notes become bookmarks,
                       citations, or diaries. I don’t think I can solve
                       the bigger philosophical problem of how people are
                       meant to work with notes. If you like it tidy,
                       Standard Notes does a pretty decent job. And we’re
                       done.


          o § Instructionals/Technical⠀➾


                # ⚓ Gathering_Information_on_Your_Wi-Fi_NIC_and_Connection_on
                  Linux_—_Virtualization_Review⠀⇛


                       I have been working with a lot of edge devices and
                       virtual desktop infrastructure (VDI) clients lately
                       and one of the issues that I have encountered is
                       that many of these devices have built-in Wi-Fi that
                       I need to monitor and gather information about.


                       Because many edge devices run Linux — which in turn
                       means a wide variety of Linux versions and GUI
                       interfaces that each supports — I have found it
                       easier to use the command line to monitor the Wi-Fi
                       of these devices.


                       In this article, I will show you some of the
                       command-line tools that I use to monitor Wi-Fi NICs
                       on Linux edge devices.


                # ⚓ Install_osTicket_Ticketing_System_on_Ubuntu_22.04/Ubuntu
                  20.04_–_kifarunix.com⠀⇛


                       In this tutorial, you will learn how to install
                       osTicket Ticketing system on Ubuntu 22.04/Ubuntu
                       20.04. osTicket is an opensource ticketing system.


                # ⚓ How_to_install_Crab_Game_by_Dani_on_a_Chromebook_–_Updated
                  Tutorial⠀⇛


                       Today we are looking at how to install Crab Game by
                       Dani 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.


                       This tutorial will only work on Chromebooks with an
                       Intel or AMD CPU (with Linux Apps Support) and not
                       those with an ARM64 architecture CPU.


                # ⚓ How_to_install_Inkscape_on_Zorin_OS_16_–_Invidious⠀⇛


                       In this video, we are looking at how to install
                       Inkscape on Zorin OS 16. Enjoy!


                # ⚓ How_To_Install_Lighttpd_on_Linux_Mint_20_–_idroot⠀⇛


                       In this tutorial, we will show you how to install
                       Lighttpd on Linux Mint 20. For those of you who
                       didn’t know, Lighttpd is a free, open-source, and
                       high-performance web server designed for speed-
                       critical environments. Lighttpd doesn’t require a
                       lot of memory and CPU usage which makes it one of
                       the best servers for any project that needs speed
                       in deploying web pages. Moreover, it also provides
                       full support for HTTPS.


                       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 a Lighttpd on a Linux Mint 20
                       (Ulyana).


                # ⚓ How_to_Backup_&_Restore_MongoDB_on_Ubuntu_–_TechAcute⠀⇛


                       MongoDB is one of the most renowned NoSQL Database
                       Engines. It is widely regarded as a Database that
                       is highly powerful, scalable, and reliable to use.
                       MongoDB leverages JSON-like documents with optional
                       schemas to function efficiently. MongoDB supports a
                       unique Document-based Data Model that helps
                       developers map data and code. MongoDB further
                       empowers application development by providing a
                       unified and robust API for querying.


                       MongoDB lets you optimize, scale, and deploy your
                       applications seamlessly with built-in sharding,
                       replication, performance tools, and indexing to
                       help you run confidently in production. MongoDB
                       ensures that your performance SLAs are met in any
                       conducive environment irrespective of whether it’s
                       450 million users around the world or your first
                       customer.


                # ⚓ How_to_install_Manjaro_21.2.2_–_Invidious⠀⇛


                       In this video, I am going to show how to install
                       Manjaro 21.2.2


                # ⚓ How_to_Install_Magento_2.4_With_OpenLiteSpeed_on_Ubuntu
                  20.04_–_RoseHosting⠀⇛


                       This blog post is about installing Magento 2.4 with
                       OpenLiteSpeed as a web server on Ubuntu 20.04 OS.


                       Magento is an open-source e-commerce platform
                       written in PHP that uses multiple PHP frameworks
                       such as Symfony and Laminas. OpenLiteSpeed is an
                       open-source web server that offers high
                       performance, security and can be used for handling
                       huge traffic for the website. Magento with a
                       combination of OpenLiteSpeed can provide a very
                       powerful and fast website performance.


                       For this setup, we will need about 30 minutes for
                       everything to be configured properly. Let’s get
                       started!


                # ⚓ Linux_Whatis_Command_–_ByteXD⠀⇛


                       Linux whatis command is used to display the brief
                       manual page description of Linux commands. The
                       manual page contains a detailed description of each
                       specified command or option. So, we can say that it
                       helps you to provide a quick short description of a
                       Linux command. It searches for the arguments from
                       its index databases that are provided with the
                       whatis command to view the short manual page
                       description.


                       We will explore in this article what is the purpose
                       of the whatis command. We can use whatis along with
                       different options.


          o § Desktop Environments/WMs⠀➾


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ GStreamer_1.18.6_stable_bug_fix_release⠀⇛


                             The GStreamer team is pleased to announce
                             another bug fix release in the stable 1.18
                             release series of your favourite cross-
                             platform multimedia framework!


                             This release only contains bugfixes and
                             security fixes, and it should be safe to
                             update from 1.18.x.


                      # ⚓ Jean-François_Fortin_Tam:_Year_MMXVI_in_1_½_minute⠀⇛


                             I completed my 2nd term on the board of the
                             GNOME Foundation, and wrote a report
                             summarizing my duties that year.
                             At the end of these two terms, I decided to
                             leave the GNOME Foundation board, as I felt
                             like I had accomplished what I had to
                             accomplish, and because I wanted to get back
                             some balance in my personal life, as well as
                             the ability to focus on my other non-profit
                             work and professional duties.


                      # ⚓ Christian_Hergert:_GSignalGroup_and_GBindingGroup⠀⇛


                             Some of the first abstractions we made when
                             creating GNOME Builder are now available to
                             everyone in GObject!


                             In particular, writing Text Editors requires
                             tracking lots of information and changes from
                             various sources. Sometimes those changes come
                             from 2nd-degree objects via the object you
                             really care about.


                             For example, with a GtkTextView that might
                             mean tracking changes to a GtkTextBuffer,
                             GtkTextTagTable, or many other application-
                             specific accessory objects through the form
                             of signals and properties.


          o § Distributions⠀➾


                # § New Releases⠀➾


                      # ⚓ Tiny_Core_Linux_13_Released:_Needs_Just_46MB_of_RAM,
                        50MB_of_Disk⠀⇛


                             Tiny Core Linux has been updated to v13.0,
                             following its recently established annual
                             cadence. This veteran ultra light Linux
                             distribution is remarkably compact by today’s
                             standards, requiring a mere 22MB download.
                             However, it installs and presents the user
                             with a fully graphical UI, with a modern
                             Linux kernel, and allows you to install and
                             run modern applications (but keep it simple
                             if you install this OS due to your PC being
                             ancient).


                # § Arch Family⠀➾


                      # ⚓ Arch_Linux_–_News:_Debug_packages_and_debuginfod⠀⇛


                             We are very happy to announce that debug
                             packages are now available in Arch Linux.
                             Debug symbols and source listing are provided
                             through our debuginfod instance which can be
                             utilized by debuggers such as gdb and delve.
                             https://debuginfod.archlinux.org/ A couple of
                             sponsored mirrors are providing the debug
                             repositories while we figure out and
                             communicate the new mirror requirements.


                # § IBM/Red Hat/Fedora⠀➾


                      # ⚓ Máirín_Duffy:_Wacom_calibration_troubleshooting_on
                        Fedora⠀⇛


                             My colleague Madeline Peck have the same
                             laptop that we each got late this past fall.
                             It’s the Lenovo Thinkpad X1 Yoga Gen 6, and
                             it is a dream computer, with integrated Wacom
                             screen and stylus


                             Recently though, Madeline noticed the cursor
                             was a bit off from where she placed the
                             stylus on the screen. The issue only seemed
                             to be happening in Krita, but was enough to
                             cause an issue. I suggested trying the GNOME
                             Wacom calibration tool in GNOME Settings,
                             thinking that even though there was a slim
                             chance it’d help (since the issue only
                             affected Krita), at the very least it
                             wouldn’t do any harm and might improve the
                             X,Y calibration of the tablet.


                             It threw the calibration off a good 4 inches.
                             Repeated calibrations using the tool didn’t
                             improve the issue.


                # § Debian Family⠀➾


                      # ⚓ Raspberry_Pi_OS_enters_the_64-bit_era_(but_32-bit_is
                        still_the_default_option)⠀⇛


                             
                             Raspberry Pi has been using 64-bit processors
                             for its tiny, low-cost computers since the
                             organization launched the Raspberry Pi 3 in
                             early 2016. But up until now the official
                             Raspberry Pi OS has been 32-bit only.


                             Now, after months of beta testing, the folks
                             at Raspberry Pi have released the first
                             official build of Raspberry Pi OS that’s
                             available in either 32-bit or 64-bit
                             versions.


                      # ⚓ Raspberry_Pi’s_Raspbian_OS_Finally_Spins_64-bit
                        Version⠀⇛


                      # ⚓ Raspberry_Pi_OS_(64-bit)⠀⇛


                      # ⚓ Raspberry_Pi_OS_(64-bit)_Now_Available⠀⇛


                             Massive good news for the Raspberry lover as
                             the Raspberry Pi OS 64 bit edition is now
                             available for download. Raspberry Pi OS was
                             available only as a 32-bit operating system.


                             Raspberry Pi Zero 2 W, Raspberry Pi Compute
                             Module 4 (CM4), Raspberry Pi Compute Module 3
                             (CM3), Raspberry Pi Compute Module 3+ (CM3+),
                             Raspberry Pi 4, Raspberry Pi 400, Raspberry
                             Pi 3A+, Raspberry Pi 3B+, and Raspberry Pi 3B
                             will support Raspberry Pi OS.


                      # ⚓ Sparky_news_2022/01⠀⇛


                             The 1st monthly Sparky project and donate
                             report of 2022:
                             – Linux kernel updated up to 5.16.4 & 5.17-
                             rc1
                             – Added to repos: Signal Desktop, Mullvad VPN
                             – NsCDE Desktop updated up to 2.0 (twice);
                             debs built for amd64, i386, armhf & arm64
                             – all sparky services, including repos moved
                             to a new, bigger server
                             – mirror de1.repo.sparkylinux.org canceled
                             – Sparky 5.16 Nibiru of the oldstable line
                             released
                             – Sparky Center LXDE updated so can be
                             installed back to LXDE desktop


                # § Canonical/Ubuntu Family⠀➾


                      # ⚓ Linux_Server_Management_in_2022_|_Ubuntu [Ed: server
                        push of Landscape, which is non-free]⠀⇛


                             Linux server management is an integration of
                             cybersecurity and business objectives. Linux
                             server management at scale is a vastly
                             different activity from interacting with a
                             terminal on one machine. The best Linux
                             server management tools universally offer a
                             server management GUI within a web browser.
                             Implementation details matter, especially in
                             a pay-for-compute world. Sysadmin tools that
                             don’t have a lightweight footprint increase
                             overall compute costs. Some of the most
                             popular open source and free Linux server
                             management tools scale poorly, when managing
                             more than one machine. Requiring system
                             administrators to perform the same tasks
                             repetitiously across several machines
                             increases the time it takes to manage the
                             entire estate. Beyond inefficiency, manual
                             administration also introduces risks
                             associated with human error. Server
                             management should be automated as much as
                             possible, through policy-aware tooling that
                             can define when different groups of machines
                             get patched.


                      # ⚓ The_International_Space_Station_Starts_Minting_NFTs⠀⇛


                             The hardware powering Celestium is located in
                             NASA’s part of the ISS. The servers run
                             Ubuntu Linux, with a custom kernel designed
                             to operate on resource-constrained systems.


                             Despite its seemingly playful nature, the
                             project has a very serious goal: to
                             demonstrate data storage, onboard processing,
                             and edge of network computing as a prototype
                             of lunar-based cloud services.


                             The NFTs minted by Celestium won’t be sold
                             for cash. In order to receive the tokens,
                             individuals must contribute images to a
                             collaborative artwork that will be sent into
                             space in March 2022 as a contemporary
                             portrayal of humanity. They can then exchange
                             their ‘fungible tokens’ into NFTs that
                             signify the ownership of a specific asteroid,
                             and its accompanying AI-generated image.


          o § Devices/Embedded⠀➾


                # ⚓ Amlogic_based_home_automation_hubs_start_at_$52⠀⇛


                       JetHome’s $52-and-up “JetHub H1” and $57-and-up,
                       DIN-rail ready “JetHub D1” home automation hubs run
                       Linux on Amlogic S905W and A113X SoCs,
                       respectively, and provide 10/100 LAN, WiFi/BT, USB,
                       and Zigbee. The D1 adds RS485, 1-Wire, relays, and
                       more.


                       St. Petersburg, Russia based JetHome has launched a
                       pair of low-cost, Arm-based automation controllers
                       with Fast Ethernet, WiFi/Bluetooth, and Zigbee
                       connectivity. The more affordable JetHome JetHub H1
                       is focused on basic home automation while the more
                       I/O-rich, DIN-rail form-factor JetHome JetHub D1
                       supports more sophisticated automation networks.


                # ⚓ A_Portable_Projecting_Pi_For_Education_|_Hackaday⠀⇛


                       We cover a lot of cyberdeck projects here at
                       Hackaday, custom portable computers often built
                       around the Raspberry Pi. It’s not often that we
                       cover a computer that perfectly achieves and
                       exceeds what a cyberdeck is trying to do without
                       being a cyberdeck in any way, but that’s what
                       [Subir Bhaduri] has done. In addressing the need
                       for Indian schoolchildren to catch up on two years
                       of COVID-disrupted schooling he’s created the pπ, a
                       Raspberry Pi, projector, and keyboard all-in-one
                       computer in a neat sheet-metal case that looks as
                       though it might be just another set of spanners or
                       similar. At a stroke he’s effortlessly achieved the
                       ultimate cyberdeck, because this machine is no sci-
                       fi prop, instead it has a defined use which it
                       fulfills admirably.


                # ⚓ Did_You_Know_That_The_Raspberry_Pi_4_Has_More_SPI,_I2C,
                  UART_Ports?_|_Hackaday⠀⇛


                       We’ve gotten used to the GPIO-available functions
                       of Raspberry Pi computers remaining largely the
                       same over the years, which is why it might have
                       flown a little bit under the radar: the Raspberry
                       Pi 4 has six SPI controllers, six I2C controllers,
                       and six UARTs – all on its 40-pin header. You can’t
                       make use of all of these at once, but with up to
                       four different connections wired to a single pin
                       you can carve out a pretty powerful combination of
                       peripherals for your next robotics, automation or
                       cat herding project.


                       The datasheet for these peripherals is pleasant to
                       go through, with all the register maps nicely laid
                       out – even if you don’t plan to work with the
                       register mappings yourself, the maintainers of your
                       preferred hardware enablement libraries will have
                       an easier time! And, of course, these peripherals
                       are present on the Compute Module 4, too. It might
                       feel like such a deluge of interfaces is excessive,
                       however, it lets you achieve some pretty cool stuff
                       that wouldn’t be possible otherwise.


                # § Open Hardware/Modding⠀➾


                      # ⚓ This_DIY_power_bank_can_deliver_up_to_60W_and
                        displays_info_in_real-time⠀⇛


                             The advent of USB-C and the power delivery
                             (PD) standard have allowed for a single cable
                             to carry both large amounts of data and power
                             for a wide range of devices. So, while
                             looking for a PD-capable power bank for his
                             laptop but only coming across expensive
                             units, Instructables user Toby Chui decided
                             to make his own — with additional
                             improvements, of course.


                      # ⚓ Golioth_IoT_development_platform_offers_Zephyr_SDK,
                        support_for_nRF9160,_ESP32,_and_over_100_other
                        platforms_–_CNX_Software⠀⇛


                             There are already plenty of IoT development
                             platforms, but here’s another one with
                             Golioth that relies on a Zephyr SDK, “first-
                             tier” support for Nordic Semiconductor
                             nRF9160 (cellular) and Espressif Systems
                             ESP32-C3 (WiFi), as well as a QEMU-based
                             simulator for easy testing.


                             The use of an open-source Zephyr SDK even
                             enables them to support over 100+ hardware
                             components, and the company, also called
                             Golioth, says their platform scales from one
                             device for evaluation to one million devices
                             during deployment, thanks notably to a free
                             Dev Tier account to get started at no cost.


                      # ⚓ Repairing_Librem_14_Part_1:_RAM/WiFi/Storage/Battery
                        –_Purism⠀⇛


                             The Librem 14 is deliberately made easy to
                             repair and upgrade. By removing just a few
                             screws, you can easily replace the battery,
                             wifi module, RAM, or the M.2 ssd. This is
                             really nice if you already own hardware you’d
                             like to put in a barebones Librem 14 order,
                             or if you want to upgrade the hardware in
                             your current Librem 14.


                      # ⚓ Open_Hardware_5V_UPS_Improves_On_Cheap_Powerbank
                        Design_|_Hackaday⠀⇛


                             Often, we need to power a 5V-craving project
                             of ours on the go. So did [Burgduino], and,
                             unhappy with solutions available, designed
                             their own 5V UPS! It takes a cheap powerbank
                             design and augments it with a few parts vital
                             for its UPS purposes.


                             You might be tempted to reach for a powerbank
                             when facing such a problem, but most of them
                             have a fatal flaw, and you can’t easily tell
                             a flawed one apart from a functioning one
                             before you buy it. This flaw is lack of load
                             sharing – ability to continue powering the
                             output when a charger is inserted. Most
                             store-bought powerbanks just shut the output
                             off, which precludes a project running 24/
                             7 without powering it down, and can cause
                             adverse consequences when something like a
                             Raspberry Pi is involved.


                # § Mobile Systems/Mobile Applications⠀➾


                      # ⚓ Android_13:_Release_Date,_Compatible_Devices,
                        Features_and_More⠀⇛


                      # ⚓ One_UI_4.0_based_on_Android_12_comes_to_Galaxy_Z_Flip
                        5G_users_in_the_US_–_NotebookCheck.net_News⠀⇛


                      # ⚓ The_Google_One_VPN_for_Android_just_got_a_little
                        safer_–_Android_Authority⠀⇛


                      # ⚓ 5_Tips_to_improve_Android_battery_life_–_Geeky
                        Gadgets⠀⇛


                      # ⚓ Top_9_Ways_to_Fix_Keyboard_Typing_Lag_on_Android⠀⇛


                      # ⚓ Genius_Android_trick_can_help_you_find_closest_free
                        internet_in_your_city⠀⇛


                      # ⚓ How_to_use_your_phone_as_an_Android_TV_remote
                        control⠀⇛


                      # ⚓ Scrcpy’s_latest_update_brings_keyboard_and_mouse
                        passthrough_without_USB_debugging⠀⇛


                      # ⚓ Lenovo_Tab_P11_Plus_review:_Striking_the_perfect
                        budget_balance_|_Android_Central⠀⇛


          o § Free, Libre, and Open Source Software⠀➾


                # ⚓ curl_dash-dash-json⠀⇛


                       The curl “cockpit” is yet again extended with a new
                       command line option: –json. The 245th command line
                       option.


                       curl is a generic transfer tool for sending and
                       receiving data across networks and it is completely
                       agnostic as to what it transfers or even why.


                       To allow users to craft all sorts of transfers, or
                       requests if you will, it offers a wide range of
                       command line options. This flexibility has made it
                       possible for a large number of users to keep using
                       curl even as network ecosystems and its (HTTP) use
                       have changed over time.


                # § Events⠀➾


                      # ⚓ CFP:_Binary_T00ls_Summit_2022⠀⇛


                             The Binary T00ls Summit 2022 (https://binary-
                             tools.net/summit) is an informal, technical,
                             online event oriented to authors, users and
                             enthusiasts of FLOSS programs that deal with
                             binary data.
                             This includes binary editors, libraries to
                             encode and decode data, parser generators,
                             binary data description languages and
                             frameworks, binary formats and encodings,
                             assemblers, debuggers, reverse engineering
                             suites, and the like.
                             The goal of this event is for developers to
                             get in touch with each other, introduce their
                             tools, have interesting and hopefully
                             productive discussions, and finally what is
                             most important: to have fun.
                             The venue of the event will be an instance of
                             the Big Blue Button web conferencing system,
                             accessible using Firefox, Chromium and other
                             web browsers. We will be having a single
                             track that will span for three days during a
                             weekend.
                             The summit will be held on the first weekend
                             of March 2022, From Friday 4 to Sunday 6.


                # § Web Browsers⠀➾


                      # § Chromium⠀➾


                            # ⚓ Chrome_Releases:_Stable_Channel_Update_for
                              Desktop⠀⇛


                                   The Chrome team is delighted to
                                   announce the promotion of Chrome 98 to
                                   the stable channel for Windows, Mac and
                                   Linux. Chrome 98 is also promoted to
                                   our new extended stable channel for
                                   Windows and Mac. This will roll out
                                   over the coming days/weeks.


                                   Chrome 98.0.4758.80/81/82 for windows
                                   and 98.0.4758.80 for mac and linux
                                   contains a number of fixes and
                                   improvements — a list of changes is
                                   available in the log. Watch out for
                                   upcoming Chrome and Chromium blog posts
                                   about new features and big efforts
                                   delivered in 98.


                            # ⚓ Google_Chrome_98_Released_With_COLRv1_For
                              Smaller_Emoji_Files_–_Phoronix⠀⇛


                                   Chrome 98 is available today as
                                   Google’s second web browser update of
                                   the new year. The Chrome 98 changes are
                                   mostly on the developer-side but with
                                   some user-impacting differences.


                                   Among the changes in Chrome 98 as
                                   Google’s stable browser update for
                                   February 2022 are:


                                   - COLRv1 color gradient vector fonts
                                   support that is motivated by wanting
                                   smaller emoji files for the web while
                                   delivering better quality. COLRv1 fonts
                                   compress well, are vector-based, and
                                   work well with gradients. Google Chrome
                                   and Google Fonts teams consider the
                                   COLRv1 specification as their successor
                                   format to the Google Noto emoji font.
                                   The emoji font size is around 20% the
                                   size as previously while also having
                                   greater rendering fidelity. More
                                   details on COLRv1 via
                                   developer.chrome.com.


                            # ⚓ Google_Releases_Security_Updates_for_Chrome_|
                              CISA⠀⇛


                                   Google has released Chrome versions
                                   98.0.4758.80/81/82 for Windows and
                                   98.0.4758.80 for Mac and Linux. These
                                   versions address vulnerabilities that
                                   an attacker could exploit to take
                                   control of an affected system.


                      # § Mozilla⠀➾


                            # ⚓ Hacks.Mozilla.Org:_Retrospective_and_Technical
                              Details_on_the_recent_Firefox_Outage [Ed: A
                              reminder that under ‘new’_Mozilla Firefox became
                              spyware that phones “mother ship”]⠀⇛


                                   On January 13th 2022, Firefox became
                                   unusable for close to two hours for
                                   users worldwide. This incident
                                   interrupted many people’s workflow.
                                   This post highlights the complex series
                                   of events and circumstances that,
                                   together, triggered a bug deep in the
                                   networking code of Firefox.


                                   [...]


                                   As part of the incident response
                                   process, we quickly discovered that the
                                   client was hanging inside a network
                                   request to one of the Firefox internal
                                   services. However, at this point we
                                   neither had an explanation for why this
                                   would trigger just now, nor what the
                                   scope of the problem was. We continued
                                   to look for the “trigger” — some change
                                   that must have occurred to start the
                                   problem. We found that we had not
                                   shipped updates or configuration
                                   changes that could have caused this
                                   problem. At the same time, we were
                                   keeping in mind that HTTP/3 had been
                                   enabled since Firefox 88 and was
                                   actively used by some popular websites.


                                   [...]


                                   With the load balancer change in place,
                                   and a special code path in a new Rust
                                   service now active, the necessary final
                                   ingredient to trigger the problem for
                                   users was deep in Necko HTTP/3 code.


                                   When handling a request, the code
                                   looked up the field in a case-sensitive
                                   way and failed to find the header as it
                                   had been lower-cased by viaduct.
                                   Without the header, the request was
                                   determined by the Necko code to be
                                   complete, leaving the real request body
                                   unsent. However, this code would only
                                   terminate when there was no additional
                                   content to send. This unexpected state
                                   caused the code to loop indefinitely
                                   rather than returning an error. Because
                                   all network requests go through one
                                   socket thread, this loop blocked any
                                   further network communication and made
                                   Firefox unresponsive, unable to load
                                   web content.


                # § SaaS/Back End/Databases⠀➾


                      # ⚓ MariaDB_announces_plans_to_go_public_for_$672m_via
                        SPAC_•_The_Register⠀⇛


                             MariaDB Corporation Ab, which sells the
                             popular open source database by the same
                             name, said on Tuesday that it intends to
                             become a public company with the help of
                             Angel Pond Holdings Corporation.


                             The deal, announced in a S-1 filing with the
                             US Securities and Exchange Commissio,n [PDF]
                             describes the Cayman Islands-based biz as a
                             special purpose acquisition company (SPAC).
                             It was formed by Shihuang “Simon” Xie, a co-
                             founder of Alibaba Group, and Theodore Wang,
                             a former Goldman Sachs partner, to raise
                             capital from investors in order to acquire
                             another company.


                             SPAC-driven deals have become popular in
                             recent years partly as a defense against
                             market volatility, which can complicate
                             initial public offerings when companies try
                             to go the traditional route to market. They
                             also provide a clearer exit path for
                             investors, allowing acquired firms to go
                             public more quickly and at less cost, and to
                             negotiate their value directly with the SPAC.


                             However, US government regulators have
                             expressed concern about the rising number of
                             SPAC-based deals. SEC Chairman Gary Gensler
                             last May said, SPACs raise a number of policy
                             questions, like whether retail investors are
                             appropriately protected and how SPACs fit
                             into the SEC mission to maintain fair
                             markets.


                # § Productivity Software/LibreOffice/Calligra⠀➾


                      # ⚓ LibreOffice_7.3_Released_with_an_Attention_for_Those
                        Switching_from_MS_Office⠀⇛


                             The Document Foundation has released
                             LibreOffice 7.3, including a large number of
                             fixes intended to compatibility with
                             Microsoft Office.


                             A new version of the open source office suite
                             LibreOffice has been released earlier today.
                             LibreOffice 7.3 is available for all
                             supported platforms. Core improvements
                             include better interoperability with
                             Microsoft Office document formats,
                             performance improvements, and new handling of
                             change tracking in tables.


                             LibreOffice is one of the best-known open
                             source office suites. It is a free Microsoft
                             Office alternative, complete with a
                             spreadsheet program, database tool,
                             presentation maker, and word processor. The
                             project was forked from OpenOffice after
                             Oracle inherited the latter through its
                             acquisition of Sun Microsystems in 2009.


                             Reading and writing to Microsoft Office file
                             formats, like DOCX, XLSX, and PPTX remains a
                             key ask of this open source office suite and
                             something its millions of users rely on
                             daily. Well, with the new 7.3 release, things
                             get even better.


                      # ⚓ LibreOffice_7.3.0_Released!_How_to_Install_in_Ubuntu
                        20.04,_21.10_&_18.04_|_UbuntuHandbook⠀⇛


                             Ubuntu’s default office suite LibreOffice
                             released new major version 7.3.0 today.
                             Here’s what’s new and how to install it in
                             all current Ubuntu editions.


                             LibreOffice 7.3 provided a large number of
                             improvements to Microsoft Office file formats
                             support. It now loads large DOCX and XLSX/
                             XLSM files and some complex documents faster!


                # § Programming/Development⠀➾


                      # ⚓ Design_and_Prototypical_Implementation_of_an_IRC_Chat
                        Server_in_Erlang_OTP_|_Linux_Journal⠀⇛


                             By the time of this article, digital services
                             provide key functionality to businesses and
                             everyday life. Due to the progress of
                             digitization, the reliance on digital
                             services has been growing rapidly. This
                             process is not only shown by the growing
                             number of inter connected devices that
                             communicate with each other but also by the
                             impact of unavailable services during an
                             incident: On the 4th of October 2021, Meta
                             (former Facebook) and all of its
                             organizations (e.g. Instagram, WhatsApp,
                             Facebook Messenger) were unavailable for up
                             to seven hours. The outage resulted in a huge
                             profit loss for the company and connected
                             businesses. [1]


                             This outrage shows that modern solutions must
                             be designed resiliently to enable service
                             provisioning during incidents. There are
                             multiple solutions to develop high-available
                             and reliable services that can be applied to
                             various levels in a system’s architecture and
                             design. Some programming languages are
                             specifically designed to meet these
                             challenges. The functional programming
                             language Erlang provides inherent
                             functionality to develop these resilient
                             services.


                             Therefore, this article presents the
                             exemplary design and development of a
                             communication service based on the Internet
                             Relay Chat (IRC) protocol in Erlang to
                             investigate its availability features.


                      # § Shell/Bash/Zsh/Ksh⠀➾


                            # ⚓ 50_lines_of_Bash_to_bring_a_Wordle_fan_out_of
                              their_shell_•_The_Register⠀⇛


                                   Of course, those 50 lines do not
                                   include all the gubbins required to
                                   render things on a browser nor the
                                   hooks to send a smug little post out to
                                   social media so your friends can see
                                   how clever you are. However, the
                                   requisite colours are present and
                                   correct, as is the frustration factor
                                   as the attempts mount up without
                                   success.


                                   Thankfully, a thoughtful command line
                                   parameter (unlimit) will up the limit
                                   from six tries to quite a bit more.


                                   We took the script for a spin on a tame
                                   Linux box (in this case, running Ubuntu
                                   20.04 LTS) and can confirm that, after
                                   a quick chmod to make things
                                   executable, tapping in ./wordle.sh got
                                   us our fix.


    * § Leftovers⠀➾


          o ⚓ Digital_transformation_in_Eastern_Europe_|_Stop_at_Zona-M⠀⇛


                 Among many other things, the European Union the EU
                 supports digital transformation in partner countries.
                 This week, for example, they signed a Joint Declaration
                 with Armenia, Azerbaijan, Georgia, Moldova and Ukraine.
                 In this area, that currently suffers from high
                 instability, digitization is a priority and an important
                 element of dialogue at several levels, from physical
                 infrastructures to roaming agreements and training. These
                 are the parts that I personally find more interesting in
                 the full Joint Declaration:


                 A: Modernisation and innovation efforts at all levels of
                 education and training will be pursued. This includes
                 structured cooperation between EU and partner countries’
                 universities, VET institutions and youth organisations,
                 as well as the promotion of youth employment,
                 employability, entrepreneurship and skills, including
                 addressing brain drain.


          o ⚓ Building_Forged_Carbon_Fiber_Wings_For_Radio_Control_Cars_|
            Hackaday⠀⇛


                 When it comes to building decent aerodynamic devices, you
                 want to focus on getting your geometry accurate, and
                 making sure your parts are strong enough to deal with the
                 force they’re generating. This build from [Engineering
                 After Hours] delivers on those fronts, consisting of a
                 high-downforce wing for a small RC car.


                 The video points out that, at best, even a decent RC car
                 will have pretty crappy aerodynamic parts from the
                 factory, with a lift-to-drag (L/D)ratio of 2-3:1 at best.
                 This means that, while they may create some small amount
                 of downforce, they’re also creating plenty of drag at the
                 same time.


                 The dual-element wing designed here is much more
                 efficient, hitting an L/D ratio in the vicinity of 17:1 –
                 a huge improvement. Even a casual eye can note that the
                 design looks a lot more like something you’d see on a
                 full-size car, versus some of the whackier designs seen
                 on toys.


                 The wing is built with a forged carbon fiber process
                 using 3D-printed molds, to give the wing plenty of
                 strength. Given that it’s built for an RC car that can do
                 over 100 mph, making sure the wing is stiff enough to
                 perform at speed is key.


          o § Science⠀➾


                # ⚓ Simple_Setup_Answers_Complex_Question_On_The_Physics_Of
                  Solids_|_Hackaday⠀⇛


                       Thought experiments can be extremely powerful;
                       after all, pretty much everything that [Einstein]
                       came up with was based on thought experiments. But
                       when a thought experiment turns into a real
                       experiment, that’s when things can get really
                       interesting, and where unexpected insights crop up.


                       Take [AlphaPhoenix]’s simple question: “Are solid
                       objects really solid?” On the face of it, this
                       seems like a silly and trivial question, but the
                       thought experiment he presents reveals more. He
                       posits that pushing on one end of a solid metal rod
                       a meter or so in length will result in motion at
                       the other end of the rod pretty much instantly. But
                       what if we scale that rod up considerably — say, to
                       one light-second in length. Is a displacement at
                       one end of the rob instantly apparent at the other
                       end? It’s a bit of a mind-boggler.


                # ⚓ Time_And_Accuracy_In_Las_ATMegas_|_Hackaday⠀⇛


                       Do you ever have to ensure that an exact amount of
                       time passes between two tasks in your
                       microcontroller code? Do you know what’s the
                       difference between precision and accuracy? Today,
                       [Jim Mack] tells us about pushing timers and
                       interrupts to their limits when it comes to
                       managing time, while keeping it applicable to an
                       ever-popular ATMega328P target! Every now and then,
                       someone decides to push the frontiers of what’s
                       possible on a given platform, and today’s rules is
                       coding within constraints of an Arduino
                       environment. However, you should check [Jim]’s post
                       out even if you use Arduino as a swearword – purely
                       for all of the theoretical insights laid out,
                       accompanied by hardware-accurate examples!


                       [...]


                       Keeping our projects true to the passage of time
                       can be an issue, and we’ve been at it for ages –
                       calibrating your RC oscillator is a rite of passage
                       for any ATTiny project. If you ever decide to have
                       an interrupt peripheral help you with timing
                       issues, we’ve gone in-depth on that topic in the
                       past, with a three-part series describing the
                       benefits, the drawbacks and the edgecases of
                       interrupts. Going for a more modern target? Our
                       piece on using interrupts with STM32 is a great
                       path for trying out tools of the modern age.


          o § Hardware⠀➾


                # ⚓ Analog_Computer_Made_From_LEGO_Predicts_Tides_|_Hackaday⠀⇛


                       Although the tides in the ocean are caused by the
                       motion of the Sun and the Moon, both of which are
                       easy to observe, accurately predicting the tide
                       more than a few days in advance turns out to be
                       rather difficult. The math behind the tidal
                       movement is so complex that some of the earliest
                       analog computers were built specifically to perform
                       tide calculations. Sir William Thomson (better
                       known as Lord Kelvin) designed one such “tide-
                       predicting machine”, an impressive arrangement of
                       gears and pulleys, back in the late 19th century.


                # ⚓ The_Wanhao_Duplicator_CNC_Heat_Sealer_|_Hackaday⠀⇛


                       [Thane Hunt] needed to find a way to make a variety
                       of different heat-seal patterns on a fluid heat
                       exchanger made from polyolefin film, and didn’t
                       want all the lead time and expense of a traditional
                       sealing press machined from a steel plate. Pattern
                       prototyping meant that the usual approach would not
                       allow sufficient iteration speed and decided to
                       take a CNC approach. Now, who can think of a common
                       tool, capable of positioning in the X-Y plane, with
                       a drivable Z axis and a controlled heat source? Of
                       course, nowadays the answer is the common-or-garden
                       FDM 3D printer. As luck would have it, [Thane] had
                       an older machine to experiment with, so with a
                       little bit of nozzle sanding, and a sheet of rubber
                       on the bed, it was good to go!


                # ⚓ Defective_3D_Printing_For_Great_Strength_|_Hackaday⠀⇛


                       Most of us want our 3D prints to be perfect. But at
                       Cornell University, they’ve been experimenting with
                       deliberately introducing defects into printed
                       titanium. Why? Because using a post-print treatment
                       of heat and pressure they can turn those defects
                       into assets, leading to a stronger and more ductile
                       printed part.


                       The most common ways to print metal use powders
                       melted together and this leads to tiny pores in the
                       material which weakens the final product. Using Ti-
                       6Al-4V the researchers deliberately made a poor
                       print that had more than the usual amount of
                       defects. Then they applied extreme heat and
                       pressure to the resulting piece. The pressure
                       caused the pores to close up, and changed the
                       material’s internal structure to be more like a
                       composite.


                # ⚓ Know_Audio:_A_Mess_Of_Cables_|_Hackaday⠀⇛


                       We’ve now spent several months in this series
                       journeying through the world of audio, and along
                       the way we’ve looked at the various parts of a Hi-
                       Fi system from the speaker backwards to the source.
                       It’s been an enjoyable ride full of technical
                       detail and examining Hi-Fi myths in equal measure,
                       but now it’s time to descend into one of the
                       simplest yet most controversial areas of audio
                       reproduction. Every audio component, whether
                       digital or analogue, must be connected into
                       whatever system it is part of, and this is the job
                       of audio cables, sometimes referred to as
                       interconnects. They are probably the single
                       component most susceptible to tenuous claims about
                       their performance, with audiophiles prepared to
                       spend vast sums on cables claimed to deliver that
                       extra bit of listening performance. Is there
                       something in it, or are they all the same bits of
                       wire with the expensive ones being a scam? Time to
                       take a look.


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # ⚓ Oracle_Linux_8.5_now_available_through_the_Microsoft
                        Store_|_Windows_Central⠀⇛


                             Lovers of Linux have a new option to use on
                             Windows PCs. Oracle Linux 8.5 recently
                             launched through the Microsoft Store,
                             allowing developers, IT admins, and
                             enterprise users to run another Linux
                             distribution on their PCs. Oracle Linux 8.5
                             installs through the Windows Subsystem for
                             Linux, which is available on both Windows 11
                             and Windows 10.


                      # ⚓ Oracle_Linux_appears_somewhere_unexpected:_The
                        Windows_Store⠀⇛


                             On the last day of January, Oracle Linux 8.5,
                             the current version of Big Red’s RHEL-alike,
                             quietly appeared on the Windows Store.


                      # § Pseudo-Open Source⠀➾


                            # § Openwashing⠀➾


                                  # ⚓ CIQ_Provides_Commercial_Enterprise
                                    Services_for_Apptainer⠀⇛


                                         Singularity, the container that
                                         powers HPC, has been moved into
                                         the Linux Foundation and renamed
                                         to Apptainer, and CIQ is here to
                                         help with the transition


                                  # ⚓ Alpha-Omega_Project_Launched_to_Improve
                                    Open_Source_Software_Security⠀⇛


                                  # ⚓ OpenSSF_Announces_The_Alpha-Omega_Project
                                    to_Improve_Software_Supply_Chain_Security
                                    for_10,000_OSS_Projects⠀⇛


                                  # ⚓ Standard_electric_charging⠀⇛


                                         It’s become an LF Energy project,
                                         Shuli Goodman, Linux Foundation
                                         Energy’s Executive Director,
                                         explained because, “At LF Energy,
                                         we firmly believe that only open
                                         source has the speed and
                                         transformative power to change
                                         industries fast. Our mission is
                                         to decarbonize the global economy
                                         by enabling the transformation of
                                         power systems, the transition to
                                         electric mobility, and the
                                         radical efficiency of the built
                                         environment. In this regard,
                                         EVerest is the perfect addition
                                         to our ecosystem.”


                      # § Security⠀➾


                            # ⚓ New_SureMDM_Vulnerabilities_Could_Expose
                              Companies_to_Supply_Chain_Attacks⠀⇛


                                   A number of security vulnerabilities
                                   have been disclosed in 42 Gears’
                                   SureMDM device management solution that
                                   could be weaponized by attackers to
                                   perform a supply chain compromise
                                   against affected organizations.


                                   Cybersecurity firm Immersive Labs, in a
                                   technical write-up detailing the
                                   findings, said that 42Gears released a
                                   series of updates between November 2021
                                   and January 2022 to close out multiple
                                   flaws affecting both the platform’s
                                   Linux agent and the web console.


                            # ⚓ New_Malware_Used_by_SolarWinds_Attackers_Went
                              Undetected_for_Years [Ed: Microsoft-connected
                              CrowdStrike_once_again_attacking_Linux, trying to
                              twist_a_Microsoft_issue_as_(somehow)_a_“Linux”
                              thing]⠀⇛


                                   According to cybersecurity firm
                                   CrowdStrike, which detailed the novel
                                   tactics adopted by the Nobelium hacking
                                   group last week, two sophisticated
                                   malware families were placed on victim
                                   systems — a Linux variant of GoldMax
                                   and a new implant dubbed TrailBlazer —
                                   long before the scale of the attacks
                                   came to light.


                            # § Privacy/Surveillance⠀➾


                                  # ⚓ Wanna_go_to_a_public_school?_Tell_Google!
                                    |_Stop_at_Zona-M⠀⇛


                                         (*) the official explanation that
                                         Pinassi received for the presence
                                         of Google Analytics would be that
                                         “it is exclusively aimed at
                                         logging for reasons of security
                                         of the access operations” and
                                         that (this is the best part IMHO)
                                         “it does not record and use any
                                         user tracking data” In other
                                         words, it is not clear what is it
                                         used for, period.


          o § Environment⠀➾


                # § Energy⠀➾


                      # ⚓ Electric_cars_are_not_middle_class?_That’s_bad,_IF…⠀⇛


                             Carlos Tavares, CEO of the automaker giant
                             Stellantis, is quite skeptical, and worried
                             too, about the real benefits of electric
                             cars. In a recent interview Tavares made two
                             main critiques, both aimed at ecological
                             benefits of electric cars, and above all
                             against what he described as a forced push,
                             by many governments, to buy electric instead
                             of traditional cars with internal combustion
                             engines (ICE).


                      # ⚓ Underwater_Tanks_Turn_Energy_Storage_Upside-Down_|
                        Hackaday⠀⇛


                             Pumped hydro storage is one of the oldest
                             grid storage technologies, and one of the
                             most widely deployed, too. The concept is
                             simple – use excess energy to pump a lot of
                             water up high, then run it back through a
                             turbine when you want to get the energy back
                             later.


                             With the rise in renewable energy deployments
                             around the world, there is much interest in
                             finding ways to store energy from these
                             often-intermittent sources. Traditional
                             pumped hydro can help, but there is only so
                             much suitable land to work with.


                             However, there could be a solution, and it
                             lurks deep under the waves. Yes, we’re
                             talking about underwater pumped hydro
                             storage!


          o § Finance⠀➾


                # ⚓ Kazakhstan_dispels_a_big_myth_behind_Bitcoin⠀⇛


                       It seems that the recent riots in Kazakhstan are
                       due at least parts of them is strictly linked to
                       the “mining” processes of the cryptocurrency called
                       Bitcoin. On one side, it seems that the arrival in
                       Kazakhstan of many Bitcoin mining datacenters
                       forced to leave China has increased the overall
                       energy consumption of the whole country enough to
                       push the government to raise energy prices, which
                       in turn contributed to cause the riots.


                       [...]


                       Regardless of how things will end up in Kazakhstan,
                       and sincerely wishing the best to all Kazakh
                       people… the specific effect of the riots that I
                       summarized above is great. Maybe Bitcoin or its
                       successors will save the world, or maybe they wont.
                       That is another issue.


                       In both cases, the effects on Bitcoin of the riots
                       in a place many “First World” residents could not
                       locate on a map if their life depended on it are
                       another wonderful, perfect reminder that there is
                       no separate “cyberspace”. There is no digital
                       utopia where digitally savvy people live and trade,
                       untouched by the miseries of the one, real world.
                       We are all in this together, with or without
                       cryptocurrencies.


                # ⚓ When_data_favour_the_rich_|_Stop_at_Zona-M⠀⇛


                       Lies, damn lies and socioeconomical indicators.


                       A recent post by Filippo Celata presents a textbook
                       example of how and why data are not neutral and,
                       regardless of their quality, can tell stories very
                       far froom reality if one just accepts them as they
                       are, without further judgment.


                       On December 31st, 2021 the italian government
                       published the decree that assigns the COVID
                       recovery funds specifically allocated for urban
                       regeneration projects. Celata explains clearly how
                       and why those assignments do not really match local
                       needs.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ The_web_of_power_that_has_less_masters_every_year_|_Stop_at
                  Zona-M⠀⇛


                       Google, Amazon, Meta and Microsoft are weaving what
                       I would call a worrying “Fiber-Optic Web of Power”.
                       Here is why you should be worried too.


                       [...]


                       Whatever answer you give to the question above, one
                       thing is certain: the oceans of the world are
                       becoming more and more digital every year, first at
                       the (privatized!) data level and now at the
                       physical level, thanks exactly to the (privatized,
                       again!) cables that run on their floors. This is
                       even less reassuring, when you also consider how
                       fragile, that is needy of really effective and
                       transparent supervision, those cables really are:
                       we are talking about stuff highly vulnerable to
                       anyone that could “deploy underwater submersibles
                       to destroy them”.


          o § Monopolies⠀➾


                # § Copyrights⠀➾


                      # ⚓ Portrait_of_a_Scaphander_–_The_Public_Domain_Review⠀⇛


                             Brad Fox tells a history-story that follows
                             the fuzzy logics we associate with a “shaggy
                             dog”. An aquatic dog? Maybe. The sea bends
                             everything — even light.



                    ䷩ 𝚕𝚒𝚗𝚎 6728

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.02.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_2/2/2022:_Weston_10.0.0_and_LibreOffice_7.3⠀✐


Posted in News_Roundup at 7:56 am by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Distributions
          o Devices/Embedded
    * Free_Software/Open_Source
    * Leftovers
    * § GNU/Linux⠀➾


          o § Desktop/Laptop⠀➾


                # ⚓ System76_Kudu_is_a_Linux_laptop_with_Ryzen_9_5900HX_and
                  NVIDIA_RTX_3060⠀⇛


                       The System76 Kudu is a laptop with a 15.6 inch,
                       1920 x 1080 pixel matte display with a 144 H screen
                       refresh rate, an AMD Ryzen 9 5900HX 8-core, 16-
                       thread processor, and NVIDIA GeForce RTX 3060
                       graphics. And while most laptops with those specs
                       ship with Windows, the Kudu is designed to run
                       Linux.


                       System76 first introduced the laptop earlier this
                       year, and now the Kudu is available for purchase
                       for $1799 and up.


                # ⚓ CutiePi_Tablet_review:_The_open-hardware_Linux_tablet⠀⇛


                       Over the last years, we have seen many Linux-
                       powered tablets, all of which developed with
                       different approaches to “openness”. For instance,
                       the PineTab trades performance and a somewhat
                       conservative design for a very low selling price
                       and full Linux support, which made its demand high
                       to the point of becoming essentially impossible to
                       find on the used market. The JingPad offered a
                       modern, premium feel and near-flagship performance,
                       but it is more of a Linux userspace over an Android
                       kernel, and it did not particularly open to third-
                       party projects.


                       The CutiePi, on the other hand, seems to hit a
                       sweet spot: it is arguably the most open project
                       among those mentioned, being certified open-
                       hardware (code JP000005) to the point that anyone
                       with a 3D printer, good SMD soldering equipment,
                       and a lot of time on their hands could replicate it
                       entirely at home.


          o § Audiocasts/Shows⠀➾


                # ⚓ Lock_Your_Terminal_Sessions_With_Vlock_–_Invidious⠀⇛


                       Vlock is a nifty little program that allows you to
                       lock out your virtual console (tty) session. This
                       is especially useful for Linux machines which have
                       multiple users with access to the console. You can
                       lock your own session(s) while still allowing other
                       users to use the system on other virtual consoles.,
                       or lock the entire virtual console display.


          o § Kernel Space⠀➾


                # § Graphics Stack⠀➾


                      # ⚓ weston_10.0.0⠀⇛


                             Weston 10.0.0 has been released! This new
                             version contains a whole
                             bunch of new features and improvements. Here
                             are some highlights:

                             - Add building blocks for color management:
                             color transformations, gamma
                               correct blending, color profiles. These are
                             only internal changes invisible
                               to users for now, but will allow to enable
                             color management features in a
                               future release.
                             - Add feedback to linux-dmabuf-unstable-v1,
                             enabling zero-copy scanout in more
                               situations.
                             - libseat support has been added. It will
                             supersede all launchers in a future
                               release.
                             - The test suite has been expanded.
                             - All example clients have been converted to
                             xdg-shell.
                             - Weston can now automatically launch a
                             client after startup.
                             - The wl_shell interface, the fbdev backend,
                             and weston-launch are deprecated.

                             Notes for packagers:

                             - libdrm 2.4.95, libwayland 1.18.0 and
                             wayland-protocols 1.24 are now required.
                             - libpipewire 0.3 is required for the
                             PipeWire remoting plugin.
                             - Support for the deprecated wl_shell
                             interface is now disabled by default (it
                               will be removed in a future release, re-
                             enable it with the Meson option
                               -Ddeprecated-wl-shell=true).
                             - Support for the fbdev backend is now
                             deprecated and disabled by default (it
                               will be removed in a future release, re-
                             enable it with the Meson option
                               -Ddeprecated-backend-fbdev=true). Migrating
                             to KMS is recommended.
                             - Support for weston-launch is now deprecated
                             and disabled by default (it will
                               be removed in a future release, re-enable
                             it with the Meson option
                               -Ddeprecated-weston-launch=true). Systems
                             without systemd can use seatd (or
                               seatd-launch) instead.
                             - A KMS driver supporting universal planes is
                             now required for the DRM backend.

                             Thanks to all contributors!

                             Commit history since RC1 below.

                             Kenny Levinsen (4):
                                   meson: Print deprecation warning for
                             weston-launch
                                   meson: Set weston-launch as disabled by
                             default
                                   meson: Rename to deprecated-weston-
                             launch
                                   ci: Enable weston-launch explicitly for
                             test

                             Simon Ser (2):
                                   clients/simple-dmabuf-feedback: fix
                             bound global versions
                                   build: bump to version 10.0.0 for the
                             official release

                             git tag: 10.0.0
                      # ⚓ Wayland’s_Weston_10.0_Compositor_Released_With_DMA-
                        BUF_Feedback,_Libseat_Support⠀⇛


                             Weston 10.0 has been released as the newest
                             feature update to Wayland’s reference
                             compositor that often works as a proving
                             grounds and compositor showcasing shiny new
                             features for Wayland.


                             Arguably most exciting with Weston 10.0 DMA-
                             BUF feedback support is added to allow zero-
                             copy direct scanout in more situations like
                             with hybrid/multi-GPU laptop setups. Other
                             Wayland compositors have also been quick to
                             support the DMA-BUF feedback extension.


          o § Instructionals/Technical⠀➾


                # ⚓ Ansible_playbook_to_upgrade_Ubuntu/Debian_servers_and
                  reboot_if_needed⠀⇛


                       I realized I’ve never posted this playbook to my
                       blog… I needed to grab it for a project I’m working
                       on, so I figured I’d post it here for future
                       reference.


                       Basically, I need a playbook I can run whenever,
                       that will ensure all packages are upgraded, then
                       checks if a reboot is required, and if so, reboots
                       the server. Afterwards, it removes any dependencies
                       no longer required.


                # ⚓ Cascade_Layers:_First_Contact⠀⇛


                       Earlier this week I learned about CSS Cascade
                       Layers and now I’m all hyped up because I really
                       like the concept. I’m eager to find out how we can
                       use them to improve and rethink the architecture of
                       our styles.


                       I will not explain how CSS Cascade Layers work
                       because Bramus and Stephanie have already done that
                       and they did it much better than I ever could. I
                       just want to get my feet wet and share my first
                       impressions. If you’re new to the topic, read their
                       articles first.


                # ⚓ Read_RSS_feeds_on_Mastodon⠀⇛


                       My daily routine when it comes to staying informed
                       of what happens in the world is to read RSS feeds
                       from various sources. To do so, I created an
                       account on Feedly and use their iOS App. But I
                       noticed that I often “forget” to launch the app ;
                       whereas I check Mastodon several times a day. So I
                       decided to configure a Bot that will parse my RSS
                       feeds and publish them on my Masto Home page.


                # ⚓ Using_an_Ansible_playbook_with_an_SSH_bastion_/_jump_host⠀⇛


                       Since I’ve set this up a number of times, but I
                       just realized I’ve never documented it on my blog,
                       I thought I’d finally do that.


                       I have a set of servers that are running on a
                       private network. That network is connected to the
                       Internet through a single reverse proxy / ‘bastion’
                       host.


                       But I still want to be able to manage the servers
                       on the private network behind the bastion from
                       outside.


                # ⚓ How_I_Manage_Content_With_Jekyll⠀⇛


                       The problem as I see it is two fold. Firstly,
                       inserting images using HTML is a pain, and there’s
                       no real way of doing anything except adding an alt
                       attribute using Markdown.


                       What if you want to add a caption to the image? Or
                       a class? Or make it lazy loading? You’re fresh out
                       of luck if you want to use Markdown, so you will
                       have to revert to good old HTML.


                # ⚓ Install_GrandCMS_on_Ubuntu_20.04_–_Make_your_own_blog_for
                  free_with_this_CMS!⠀⇛


                       GrandCMS is an open source CMS that is based on the
                       OpenCart core. The developer says that he has
                       removed all the commercial stuff and focused on
                       providing an attractive, robust and competent CMS.


                       In addition to this, GrandCMS stands out for being
                       very fast and easy to use so in a matter of minutes
                       we can have a new website up and running.


                       So it is a good idea to try it out and keep it in
                       mind for personal and private projects.


                       Let’s get started.


                # ⚓ How_to_share_files_with_Samba_|_Enable_Sysadmin⠀⇛


                       The Samba project provides file sharing and print
                       services for computers on a network. It uses the
                       Server Message Block and Common Internet File
                       System (SMB/CIFS) protocol, so the services created
                       by running Samba are available to Linux, macOS, and
                       Windows clients. It’s an essential service to run
                       in organizations that support multiple operating
                       systems, and it’s even useful on homogenous
                       networks.


                       It’s not difficult to set up, and all you need is
                       at least one server you want to designate as a
                       file-share host (it doesn’t have to be rack mounted
                       and could even be a dedicated workstation). For
                       client access, Samba is either built into the
                       operating system or easily installed from a
                       repository.


                # ⚓ How_to_install_Clonezilla_on_Ubuntu_21.10_–_NextGenTips⠀⇛


                       Clonezilla is a suite of open-source, disk cloning
                       programs used for bare metal backup Fand recovery
                       and also used during system deployment. Clonezilla
                       server edition uses multicast technologies to
                       deploy a single image file to a group of computers
                       in a local area network.


                       In this tutorial guide, we are going to explore how
                       to install and use Clonezilla on Ubuntu 21.10.


                # ⚓ How_To_Install_Google_Chrome_98_In_Ubuntu_/_Rocky_Linux_&
                  Fedora_|_Tips_On_UNIX⠀⇛


                       Google Chrome is one of the common and most widely
                       used web browsers in the world. It is blazing fast
                       and easy to use with security features.


                       Google Chrome’s newer version is 98 and it is the
                       second web browser update of the year. This release
                       contains changes related to the developer side and
                       some user-impacting differences.


                       This tutorial will be helpful for beginners to
                       install Google Chrome 98 on Ubuntu 20.04 LTS, Linux
                       Mint 20.3, Rocky Linux 8, AlmaLinux 8, and Fedora
                       35.


                # ⚓ How_to_Install_LAMP_Apache,_MySQL,_PHP_on_Debian_11⠀⇛


                       How to Install LAMP Apache, MySQL, PHP on Debian
                       11. In this guide you will learn how to install
                       Apache2, MySQL 8.0 and PHP 8.1.


                       You will also install some common PHP extensions
                       and adjust the PHP configurations. Finally you will
                       secure your setup with Let’s Encrypt SSL and
                       configure HTTPS redirection.


                       This setup is tested on Google cloud, so it will
                       work on all cloud hosting services like AWS, Azure
                       or any VPS or any dedicated servers running Debian
                       11.


                # ⚓ How_to_Install_OpenNMS_Horizon_Network_Monitoring_System_on
                  Debian_11⠀⇛


                       OpenNMS is a free, open-source, and one of the most
                       powerful network monitoring and network management
                       platforms used for monitoring remote devices from a
                       central location. It supports many functionalities
                       including, Provisioning, Services Monitoring, Event
                       managing, chart support, and more. OpenNMS uses
                       SNMP and JMX and gathers information from remote
                       systems. It runs on Linux and Windows operating
                       systems and provides a web-based interface to easy
                       monitoring.


                       In this tutorial, I will show you how to install
                       OpenNMS on Debian 11.


                # ⚓ How_to_Setup_a_Kubernetes_Cluster_with_K3S_in_Rocky_Linux
                  8⠀⇛


                       Kubernetes is an open-source container
                       orchestration system for automating software
                       deployment, scaling, and management. Google
                       originally designed Kubernetes, but the Cloud
                       Native Computing Foundation now maintains the
                       project. It groups containers that make up an
                       application into logical units for easy management
                       and discovery.


                       K3S is a certified lightweight kubernetes built for
                       IoT & Edge Computing. K3s is a highly available,
                       certified Kubernetes distribution designed for
                       production workloads in unattended, resource-
                       constrained, remote locations or inside IoT
                       appliances. K3s is packaged as a single <50MB
                       binary that reduces the dependencies and steps
                       needed to install, run and auto-update a production
                       Kubernetes cluster.


                       In this tutorial, we will set up a kubernetes
                       cluster with K3S in Rocky Linux 8.


                # ⚓ How_to_Use_the_nmcli_Command_to_Manage_NetworkManager⠀⇛


                       The nmcli command, available in many Linux
                       variants, is a command-line tool for managing
                       NetworkManager. Using nmcli, a Linux administrator
                       can perform various tasks, such as managing the
                       network connections and displaying the network
                       interface adapter’s status.


          o § Games⠀➾


                # ⚓ Intel_Arc_GPUs_could_raise_the_bar_for_gaming_on_Linux_|
                  Windows_Central [Ed: Interesting to find it in a site such as
                  this]⠀⇛


                       Intel ARC graphics cards look like they’ll have
                       Resizable BAR (ReBAR) support at or near launch.
                       ReBAR technology can boost the frame rate
                       performance of games that support the feature. That
                       list is rather short at the moment, but support for
                       ReBAR would still be nice to see on Intel’s
                       upcoming ARC GPUs.


                       A report by Phoronix explains that Intel’s open-
                       source developers are working on Linux driver
                       support. ReBAR support is on the way, according to
                       the notes for the most recent kernel patches (via
                       PC Gamer).


                # ⚓ Left_4_Dead_2_gets_updated_ready_for_the_Steam_Deck_|
                  GamingOnLinux⠀⇛


                       Looks like playing Left 4 Dead 2 on the upcoming
                       Steam Deck will be a pretty good experience, with
                       Valve pushing out a fresh update today to be ready
                       for it.


                       Part of all titles getting to the status of
                       Verified, gamepad input needs to be rock solid. Not
                       only that, games need to use the correct controller
                       input icons on-screen too. This upgrade for Left 4
                       Dead 2 brings all that. Valve of course want their
                       own games to be Verified as much as possible too
                       for their own hardware!


                # ⚓ Custom_Macintosh_With_A_Real_486_|_Hackaday⠀⇛


                       Older Apple computers can often be something of a
                       collector’s item, with the oldest fetching an
                       enormously high price in auctions. The ones from
                       the late ’80s and early ’90s don’t sell for quite
                       as much yet, but it’s possible that museums and
                       collectors of the future will one day be clamoring
                       for those as well. For that reason, it’s generally
                       frowned upon to hack or modify original hardware.
                       Luckily, this replica of an Apple Macintosh didn’t
                       harm any original hardware yet still manages to run
                       software on bare metal.


                # ⚓ Arcane:_An_Exceptional_TV_Show⠀⇛


                       “Hey have you watched Arcane yet?” – “Me: Not yet,
                       heard about it on Reddit. It’s based on League of
                       Legends right? I’ve never played the game.” – “Just
                       watch it NOW.”


                       If someone had told me that there was going to be
                       something better than Squid Game to watch on TV in
                       2021, I’d have laughed. Now I would have the admit
                       they were right. The best TV show of the year 2021,
                       in my opinion, is most certainly Arcane, developed
                       as a collaboration between Riot Games (characters
                       and story) the French studio Fortiche (for the
                       production itself. By the way, Fortiche is a casual
                       word to mean strong, and they earned it). You may
                       be aware that Riot Games’ reputation is not so hot
                       lately, but do not let that detract you from
                       checking out Arcane.


          o § Desktop Environments/WMs⠀➾


                # § K Desktop Environment/KDE SC/Qt⠀➾


                      # ⚓ What_a_FreeBSD/KDE_user_misses_on_macOS⠀⇛


                             Ed Maste asked what everyone’s favourite
                             changes were to FreeBSD over the last year. I
                             replied that FreeBSD 13 was a great release
                             for the desktop, and that coupled with KDE
                             even made me miss stuff when I’m working on
                             macOS.


                             A few people asked me to expand on what I
                             meant by that. There’s probably nothing
                             surprising here, but here goes: [...]


          o § Distributions⠀➾


                # ⚓ Tiny_Core_Linux_13.0_released_for_older_or_lower-end_x86
                  hardware⠀⇛


                       Tiny Core Linux is a lightweight (~22MB SIO) Linux
                       distribution with an FLTK (Fast Light Toolkit)/FLWM
                       (Fast Light Windows Manager) desktop and based on
                       the Core Project that integrates a recent Linux
                       kernel, vmlinuz, and a root filesystem with low
                       footprint libraries such as busybox. It’s mostly
                       interesting for older or low-end hardware that may
                       be slow and/or unusable with more common Linux
                       distributions like Ubuntu or Debian.


                       I’ve quickly tried the 32-bit ISO in VirtualBox and
                       was offered to boot it with a GUI or command line,
                       but an option for slow devices waiting for USB for
                       a period of time. The desktop environment (shown in
                       the first screenshot above) is very basic with just
                       five icons are the bottom. The only application
                       installed is the Terminal, but you can add more
                       including Firefox through the menu.


                # § New Releases⠀➾


                      # ⚓ Trisquel_GNU/Linux_10.0_“Nabia”_Released_with_GNU
                        Linux-Libre_5.4_Kernel,_ARM_Support⠀⇛


                             Trisquel GNU/Linux continues to keep the
                             spirit of free software alive in 2022 and the
                             new release promises an Ubuntu 20.04 LTS
                             (Focal Fossa) based computer operating system
                             that doesn’t include any proprietary software
                             or firmware and uses the GNU Linux-libre 5.4
                             LTS kernel rather than Ubuntu’s Linux 5.4 LTS
                             kenel.


                             Dubbed “Nabia,” Trisquel GNU/Linux 10.0 is
                             here more than 15 months after Trisquel GNU/
                             Linux 9.0 “Etiona” with new releases of its
                             privacy-oriented software, including Abrowser
                             96.0 web browser and Icedove 91.5.0 email,
                             calendar and news client as rebranded
                             versions of the Mozilla Firefox and
                             Thunderbird apps.


                      # ⚓ Linux_Lite_5.8_Release_Includes_Neofetch,_Updated
                        Theme,_and_New_Wallpapers_–_It’s_FOSS_News⠀⇛


                             One of the best Windows-like distribution and
                             lightweight distros out there, Linux Lite is
                             an impressive choice for modern computers and
                             old systems.


                             And, to improve the experience, finally,
                             Linux Lite 5.8 is here!


                             The latest release includes a host of changes
                             in terms of UI, with performance tweaks and
                             bug fixes.


                # § BSD⠀➾


                      # ⚓ GoT_all_the_things⠀⇛


                             Hola, I was basically using cvs for a long
                             time for OpenBSD and some Github for open
                             source projects, until this thread on ports
                             made me decide to just move all my repos to
                             GoT, which I had the privilege to see an
                             early version long time ago in a hackthon :).


                             So my setup is pretty easy and I will explain
                             the “migration” and how to keep in sync my
                             “new” GoT repos with upstream providers such
                             as Codeberg, Sourcehut and Github.


                      # ⚓ FreeBSD_13_on_Thinkpad_T460s⠀⇛


                             For some reasons, I decided to use FreeBSD on
                             my laptop. Several times have I tried it in
                             the last few years. Several times have I
                             stopped after the first issue that I felt was
                             sign of it not being built for me.


                             This time, I’ll go for at least a whole month
                             of using it. So I can really decide if I keep
                             using it ; or switch back to OpenBSD.


                # § Arch Family⠀➾


                      # ⚓ First_Arch_Linux_ISO_Powered_by_Linux_Kernel_5.16_Is
                        Now_Available_for_Download⠀⇛


                             Arch Linux 2022.02.01 is now available for
                             download and it comes pre-installed with
                             Linux 5.16 as default kernel for new
                             installations and system rescue/recovery
                             tasks. The ISO image includes Linux kernel
                             5.16.4, but it looks like Linux kernel 5.16.5
                             was also released today and you should expect
                             it to land in the stable archives by the end
                             of the week.


                             In addition to the new kernel, the Arch Linux
                             2022.02.01 snapshot incorporates all the
                             updated packages and security patches that
                             have been released through the distribution’s
                             software repositories during the month of
                             January 2022.


                # § IBM/Red Hat/Fedora⠀➾


                      # ⚓ [Old] CentOS_Linux_8_will_reach_End_of_Life_on
                        December_31,_2021._Here’s_what_that_means.⠀⇛


                             Additionally, with this deadline falling
                             during a time when many of our team, as well
                             as many of our users, will be out of the
                             office, we intend to bend the usual EOL
                             process by keeping this content available
                             until January 31st.


                # § Debian Family⠀➾


                      # ⚓ Full_64-bit_Official_Raspberry_Pi_OS_Now_Generally
                        Available⠀⇛


                             
                             Until today, Raspberry Pi OS (formerly
                             Raspbian) was available only as a 32-bit
                             operating system for all Raspberry Pi models,
                             from Raspberry Pi 1 to the latest Raspberry
                             Pi Zero 2 W. This is was the recommend
                             version offered by the Raspberry Pi
                             Foundation, due to compatibility issues and
                             to avoid customer confusion.


                             During the past few months, since April 2021,
                             the Raspberry Pi Foundation announced a beta
                             release of a full 64-bit version of their
                             Debian-based Raspberry Pi OS, which is now
                             finally out of beta testing and ready for
                             mass deployment.


          o § Devices/Embedded⠀➾


                # § Open Hardware/Modding⠀➾


                      # ⚓ Celebrate_Black_history_this_month_with_code!⠀⇛


                      # ⚓ Some_Small_Updates⠀⇛


                             Hi everyone! The X2100 BIOS has been updated
                             and the only change is the GPIO voltage for
                             the Intel_HDA pin. The change from 3.3v to
                             1.8v should mitigate/eliminate EC issues with
                             new motherboards (<100hrs of power up time).
                             Huge thanks to jwise once again for his
                             contributions.


                      # ⚓ Silicon_die_teardown:_a_look_inside_an_early_555
                        timer_chip⠀⇛


                             Eric Schlaepfer (@TubeTimeUS) recently came
                             across the chip above, with a mysterious part
                             number. He tediously sanded through the epoxy
                             package to reveal the die (below) and
                             determined that the chip is a 555 timer.
                             Signetics released the 555 timer in mid-1972
                             4 and the chip below has a January 1973 date
                             code (7304), so it must be one of the first
                             555 timers. Curiously, it is not labeled 555,
                             so perhaps it is a prototype or internal
                             version.3 I took detailed die photos, which I
                             discuss in this blog post.


                      # ⚓ It’s_dire:_Raspberry_Pi_availability_tracker_is
                        launched⠀⇛


                             André has tried including all the official Pi
                             vendors (who contractually will sell Pi
                             models at MSRP and not the insane markups
                             you’ll find on eBay or Amazon), but some
                             block script-based access, so only about half
                             the vendors are currently accounted for.


                             Checking yesterday, two models were
                             available—across all the 29 models tracked.
                             And today that number’s down to one, a lonely
                             CM4 model with 2 GB of RAM, 32 GB eMMC
                             storage, and WiFi/Bluetooth.


                      # ⚓ Soldering_Workbench_Tour⠀⇛


                             Over the past couple of years I have been
                             upgrading the soldering equipment that I used
                             to mostly repair Amigas and build new Amiga
                             parts. Today I’m going to go through what I
                             use and why I decided to purchase each one.
                             The hope is that will help people trying to
                             make similar purchasing decisions in the
                             future.


                             I have a workshop with a 6 meter kitchen
                             counter on it and my soldering section takes
                             up half that counter. I’m going to go from
                             right to left in this post.


                      # ⚓ LTO_Tape_data_storage_for_Linux_nerds⠀⇛


                             One of the common reasons to look towards LTO
                             Tape is that it’s much cheaper than hard
                             drives, where a 12TB SATA drive costs around
                             £18.00 per TB (at the time of writing), a
                             LTO-8 tape that has the same capacity costs
                             around £7.40 per TB (at the time of writing).
                             That’s a significant price difference. So,
                             you may ask what is the catch?


                      # ⚓ Learn_how_to_make_your_old_pedal_generator_work_with
                        Zwift_|_Arduino_Blog⠀⇛


                             Zwift is a neat cycling simulator app that
                             lets you experience a variety of bicycle
                             adventures from the comfort of your living
                             room. You can participate in bike races,
                             cruise your favorite trails, or go on a
                             leisurely ride through the country. Zwift’s
                             key feature is that it reacts to your
                             pedaling and vice-versa. So an uphill section
                             increases the pedaling difficulty and your
                             in-game speed correlates to your pedaling
                             speed. Gene’s Green Machine found a way to
                             integrate an existing bike with Zwift using
                             Arduino.


                             In order for this to work, you will need an
                             indoor bike setup with a pedal generator. The
                             generator creates resistance and its voltage
                             output is proportional to the gear ratio and
                             pedaling speed. By monitoring the voltage,
                             you can tell Zwift how fast your virtual bike
                             should move. Conversely, Zwift can adjust the
                             real world pedaling resistance based on the
                             virtual conditions. Steering isn’t a factor
                             here, but that isn’t a major part of cycling
                             at speed anyway. You also don’t get to
                             experience the virtual weather elements, but
                             that is a good thing as far as most people
                             are concerned.


                # § Mobile Systems/Mobile Applications⠀➾


                      # ⚓ Updating_SailfishOS_(Suomenlinna)⠀⇛


                             Long story short (read this part of the
                             release notes), I decide to remove the
                             packages the updater complains about and a
                             few more.


                      # ⚓ the_smart_modem⠀⇛


                             I think I’ve mentioned occasionally that
                             various devices, mostly cellular modems, just
                             use the Hayes or AT command set. Recently I
                             obtained a GPS tracking device (made by
                             Queclink) that is, interestingly, fully
                             configured via the Hayes command set. It’s an
                             example of a somewhat newer trend of
                             converging the functionality of IoT devices
                             into the modem baseband. But what is this
                             Hayes command set anyway?


                      # ⚓ YouTube_for_Android_rolls_out_video_player_redesign_–
                        9to5Google⠀⇛


                      # ⚓ Samsung’s_Galaxy_S22_Is_Near_—_Don’t_Buy_a_New
                        Android_Phone_yet⠀⇛


                      # ⚓ Repainter_App_Lets_You_Customize_Android_12_Theme
                        Colors⠀⇛


                      # ⚓ Nokia_7.2_receiving_Android_11_Build_update_|
                        Nokiamob⠀⇛


                      # ⚓ Android_Auto_is_looking_more_like_CarPlay_in_new
                        leaked_images_–_The_Verge⠀⇛


                      # ⚓ 7_BestPhoto_Collage_Apps_for_Android_–_Phandroid⠀⇛


                      # ⚓ Android_app_deals_of_the_day:_Monopoly,_Sheltered,
                        Worms_3,_Twilight_Pro_Unlock,_more_–_9to5Toys⠀⇛


                      # ⚓ How_to_sign_a_document_on_your_Android_smartphone⠀⇛


                      # ⚓ The_Google_One_VPN_for_Android_just_got_a_little
                        safer_–_Android_Authority⠀⇛


          o § Free, Libre, and Open Source Software⠀➾


                # ⚓ How_radical_transparency_is_transforming_open_source
                  healthcare_software⠀⇛


                       At Tidepool, where I work as a Community and Clinic
                       Success Manager, the company’s mission is to make
                       diabetes software more accessible, meaningful, and
                       actionable. Operating in the open is how we achieve
                       that. Tidepool’s diabetes management software is an
                       open source platform free for both clinicians and
                       people impacted by diabetes. And, because the
                       company is a nonprofit, it also operates according
                       to the transparency rules that govern 501(c)(3)
                       organizations.


                       This approach means that open source healthcare
                       software can be driven by the people most affected
                       by it—in the case of Tidepool, the diabetes
                       community. The solutions that come from this
                       process put the needs of individual users first,
                       from the design phase to implementation.


                # ⚓ Best_Free_and_Open_Source_Alternatives_to_Apple
                  Compressor⠀⇛


                       
                       Apple, Microsoft, Alphabet (Google’s parent),
                       Amazon and Facebook dominate the tech landscape.
                       Their dominance is so broad they account for more
                       than 20% of the S&P 500.


                       There are many things to admire about Apple’s
                       hardware and software. Apple make great looking
                       (albeit expensive) hardware. Over the years key
                       successes include the iPhone, iPad, iPod, and the
                       MacBook Air. The company designs its own hardware
                       and software. This gives them the power to make an
                       operating system and suite of apps that are tailor-
                       made and optimized for their hardware. Apple also
                       operates the Apple Music and Apple TV media
                       distribution platforms.


                       macOS is Apple’s proprietary operating system for
                       its line of Macintosh computers. Its interface,
                       known as Aqua, is highly polished and built on top
                       of a BSD derivative (Darwin). There’s a whole raft
                       of proprietary applications that are developed by
                       Apple for their operating software. This software
                       is not available for Linux and there’s no prospect
                       of that position changing.


                # ⚓ How_open_source_supports_businesses’_impact_on_climate
                  change_|_Opensource.com⠀⇛


                       In the coming decade, climate changes will compel
                       the creation of new ecological norms and rules for
                       our economy. The financial industry will be at the
                       center of this transition, determining which
                       companies and technologies are granted the
                       resources they need to spearhead this evolution.


                       So far, objectively evaluating the ethics and
                       environmental impact of individual businesses has
                       proved challenging. Many currently used metrics
                       make it easy for companies to promote a message of
                       ecological responsibility while continuing
                       unsustainable practices, a discordance sometimes
                       referred to as greenwashing. A movement toward
                       measuring corporate sustainability based on
                       quantitative values, open source practices, and
                       open science can potentially circumvent this threat
                       to economic transformation.


                # § Events⠀➾


                      # ⚓ FOSDEM_2022⠀⇛


                             The annual FOSDEM conference is kicking off
                             in a matter of days as software developers
                             prepare for thoughtful discussions on the
                             latest open source developments.


                             This two-day virtual event is organized by
                             volunteers to promote the widespread use of
                             free and open source software. With over 50
                             devrooms and nearly 700 talks, this jam-
                             packed weekend is sure to satisfy the most
                             curious of minds.


                             Within this rigorous lineup, you’ll have the
                             chance to get an in-depth look at Mobian: an
                             open-source project aimed at bringing Debian
                             GNU/Linux to mobile devices. Presented by
                             Collabora’s very own Arnaud Ferraris, he’ll
                             take you through the significance of the
                             project’s advancements and contributions to
                             the FLOSS ecosystem.


                # § Productivity Software/LibreOffice/Calligra⠀➾


                      # ⚓ LibreOffice_7.3_Open-Source_Office_Suite_Officially
                        Released,_This_Is_What’s_New⠀⇛


                             
                             As with previous updates in the LibreOffice
                             7.x series, the LibreOffice 7.3 release is
                             here to further improve document
                             interoperability with proprietary formats
                             from the MS Office suite in an attempt to
                             ease the migration from Microsoft Office to
                             LibreOffice.


                             Highlights include handling of change
                             tracking in tables and when text is moved,
                             which have a positive impact on
                             interoperability with Microsoft Office
                             documents, faster opening of large DOCX and
                             XLSX/XLSM files of over 200 pages, improved
                             rendering speed of some complex documents, as
                             well as improvements to import and export
                             filters.


                # § FSF⠀➾


                      # § GNU Projects⠀➾


                            # ⚓ GNU_findutils_4.9.0_released_[Savannah]⠀⇛


                                   This is to announce findutils-4.9.0, a
                                   stable release.
                                   See the NEWS below for more details.
                                   GNU findutils is a set of software
                                   tools for finding files that match
                                   certain criteria and for performing
                                   various operations on them.
                                   Findutils includes the programs "find",
                                   "xargs" and "locate".
                                   More information about findutils is
                                   available at:

                                   https://www.gnu.org/software/findutils/

                                   Please report bugs and problems with
                                   this release via the the
                                   GNU Savannah bug tracker:

                                   https://savannah.gnu.org/bugs/
                                   ?group=findutils

                                   Please send general comments and
                                   feedback about the GNU findutils
                                   package to the mailing list (<mailto:
                                   bug-findutils@gnu.org):

                                   https://lists.gnu.org/mailman/listinfo/
                                   bug-findutils

                                   There have been 35 commits by 6 people
                                   in the 55 weeks since 4.8.0:
                                     Andreas Metzler (1)     Helge
                                   Kreutzmann (1)
                                     Andrew Gaul (1)         James
                                   Youngman (1)
                                     Bernhard Voelker (33)   Renaud
                                   Pacalet (1)
                                   This release was bootstrapped with the
                                   following tools:
                                      Autoconf 2.69
                                      Automake 1.16.5
                                      M4 1.4.18
                                      Gnulib v0.1-5153-g6ef3d78333
                                   Please consider supporting the Free
                                   Software Foundation in its fund
                                   raising appeal; see .
                                   Thanks to everyone who has contributed!

                                   Have a nice day,
                                   Bernhard Voelker [on behalf of the GNU
                                   findutils maintainers]
                            # ⚓ When_is_GIMP_version_3.0_coming?_–
                              LinuxStoney⠀⇛


                                   Of greater or lesser complexity, almost
                                   everyone now has a photo editor
                                   installed on their computer. Although
                                   Adobe Photoshop is the most popular, we
                                   don’t need to go for a solution of that
                                   level either. Serve as a clear example
                                   of all this, GIMP, a very interesting
                                   alternative solution.


                                   It is not for nothing that these two
                                   programs are considered the most
                                   popular and used all over the world
                                   when it comes to editing our images .
                                   However, there is a clear difference
                                   between the two. While the giant
                                   Adobe’s proposal is a paid program,
                                   with GIMP we are faced with a
                                   completely free open source project. It
                                   is true that for some time now, this
                                   program is considered one of
                                   Photoshop’s main competitors. However,
                                   many consider that, in order to
                                   continue on this path, its top managers
                                   will speed up some movements.


                                   With this, what we really mean is that
                                   for some time the usual users of this
                                   software solution have been waiting for
                                   the arrival of GIMP 3.0 . It’s been too
                                   many months now that this new version
                                   has been talked about, which initially
                                   would revolutionize the future of the
                                   editor. To date, some test versions
                                   have been seen, although it seems that
                                   its developers have not made up their
                                   minds about its launch. At the moment,
                                   the previous versions that are
                                   currently on the market continue to be
                                   improved. We refer specifically to
                                   version 2.1 and later.


                # § Programming/Development⠀➾


                      # ⚓ Git_2.34_has_changed_how_you_configure_fast-forward
                        only_pulls_and_rebasing⠀⇛


                             As of Git 2.34, two things have changed.
                             First, the default behavior of ‘git pull’ is
                             now to abort if it can’t fast-forward the
                             upstream into your local branch (ie, ‘git
                             pull –ff-only’); basically, the previous
                             warning has become an error. Second, the
                             configuration setting of ‘pull.ff only’ now
                             takes priority over ‘pull.rebase true’
                             (although not over an explicit –rebase on the
                             command line). If you have both in a
                             repository with things to rebase, you
                             effectively wind up running ‘git pull –ff-
                             only’, which fails because you have
                             additional local changes that Git thinks
                             would have to be merged. The behavior of
                             ‘pull.ff only’ here may be an accidental bug
                             and is certainly not historical behavior, but
                             we have to deal with the Git release we get,
                             not the one we’d like.


                      # ⚓ Regexes_are_Cool_and_Good⠀⇛


                             No, where regex really shines is in
                             interactive use. When you’re trying to
                             substitute in a single file you have open, or
                             grep a folder, things like that. Readability
                             doesn’t matter because you’re writing a one-
                             off throwaway, and fragility is fine because
                             you’re a human-in-the-loop. If anything goes
                             wrong you will see that and tweak the regex.


                      # ⚓ A_thesis:_most_websites_are_implicitly_designed_with
                        a_short_lifetime⠀⇛


                             I don’t think people building websites
                             explicitly think about the expected lifetime
                             and plan for whatever specific lifetime they
                             expect (at least, not usually). But I do
                             think it influences what technologies they
                             consider and choose, and in particular I
                             think people are influenced by a general
                             feeling that no matter what they do, it’s
                             probably not lasting for all that long. Or,
                             to put things another way, they won’t be
                             stuck with what they’ve built for all that
                             long, left to operate and maintain it no
                             matter what. Something will change enough to
                             force changes regardless of what they do.


                      # ⚓ Fortran_newsletter:_February_2022⠀⇛


                             Welcome to the February 2022 edition of the
                             monthly Fortran newsletter. The newsletter
                             comes out at the beginning of every month and
                             details Fortran news from the previous month.


                      # § Perl/Raku⠀➾


                            # ⚓ A_brief_guide_to_perl_character_encoding⠀⇛


                                   I originally wrote this at work, after
                                   my team spent far too many days yelling
                                   at the computer because of Mojibake.
                                   Thanks to my employer for allowing me
                                   to publish it, and the several
                                   colleagues who provided helpful
                                   feedback. Any errors are, naturally,
                                   not their fault.


                      # § Rust⠀➾


                            # ⚓ Arti_0.0.4_is_released:_Refactoring,_rustls,
                              and_more!⠀⇛


                                   Arti is our ongoing project to create a
                                   working embeddable Tor client in Rust.
                                   It’s nowhere near ready to replace the
                                   main Tor implementation in C, but we
                                   believe that it’s the future.


                                   We’re working towards our 0.1.0
                                   milestone in early March, where our
                                   main current priorities are stabilizing
                                   our APIs, and resolving issues that
                                   prevent integration. We’re planning to
                                   do releases every month or so until we
                                   get to that milestone.


                            # ⚓ Trio_of_Rust_Core_Team_members_take_their
                              leave⠀⇛


                                   There is only drama in the open source
                                   community when the day has a “y” in it,
                                   and sure enough a trio of members have
                                   decided to step back from the Rust Core
                                   Team, including a nine-year veteran of
                                   the language.


                                   It has been a busy few months for the
                                   Rust project. The entire moderation
                                   team quit in November 2021. The
                                   resignation was, according to the post
                                   in GitHub on the matter, “in protest of
                                   the Core Team placing themselves
                                   unaccountable to anyone but
                                   themselves.”


                                   This week has kicked off with three of
                                   Core Team taking their leave. Florian
                                   Gilcher, a Core Team observer and
                                   project director on the Rust foundation
                                   board, is departing to focus on his
                                   company.


                                   Pietro Albini, meanwhile, reckoned he
                                   had too much to do, and is off to
                                   “focus on other parts of the project.”


          o § Standards/Consortia⠀➾


                # ⚓ A_toy_DNS_resolver⠀⇛


                       Hello! I wrote a comic last week called “life of a
                       DNS query” that explains how DNS resolvers work.


                       In this post, I want to explain how DNS resolvers
                       work in a different way – with a short Go program
                       that does the same thing described in the comic.
                       The main function (resolve) is actually just 20
                       lines, including comments.


                       I usually find it easier to understand things work
                       when they come in the form of programs that I can
                       run and modify and poke at, so hopefully this
                       program will be helpful to some of you.


                       The program is here: https://github.com/jvns/tiny-
                       resolver/blob/main/resolve.go


    * § Leftovers⠀➾


          o ⚓ Heeding_James_Joyce’s_“Ulysses”⠀⇛


                 The mythical figures in Homer’s epic – Ulysses is the
                 Latin name for Homer’s hero Odysseus – are reincarnated
                 in the lives of the Irish working-class. Ulysses, the
                 Greekking of Ithaca, whose ruse of the Trojan Horse made
                 him the architect of the victory against Troy, who spent
                 ten years trying to get home after ten years at war and
                 slaughtered the suitors who besieged his wife and ravaged
                 his court during his absence, becomes in Joyce’s hands
                 Leopold Bloom, a 38-year-old ad canvasser for the
                 nationalist newspaper Freeman’s Journal. Leopold, whose
                 father was an observant Hungarian Jew, throughout the
                 novel mourns his infant son Rudy, who died over a decade
                 earlier, a loss that severed his sexual relations with
                 his wife Molly. Ulysses’ son Telemachus, who grew up
                 without his father and who, when he reached manhood, left
                 Ithaca to search for Ulysses, becomes Stephen Dedalus, a
                 fictionalized version of Joyce’s precocious younger self.
                 Penelope, the loyal wife of Ulysses, is reinvented as
                 Molly, the wife of Leopold Bloom, who during the day has
                 a tryst with her lover, Hugh “Blazes” Boylan, and whose
                 approximately 22,000-word monologue, one of the greatest
                 in literature, affirming the sanctity of love and life –
                 along with graphic descriptions of digestion, orgasms,
                 and farts – concludes the book.


                 “Unimpressive as Bloom may seem in so many ways,” writes
                 Joyce’s biographer Richard Ellman, “unworthy to catch
                 marlin or countesses with Hemingway’s characters, or to
                 sop up guilt with Faulkner’s, or to sit on committees
                 with C.P Snow’s, Bloom is a humble vessel elected to bear
                 and transmit unimpeached the best qualities of the mind.
                 Joyce’s discovery, so humanistic that he would have been
                 embarrassed to disclose it out of context, was that the
                 ordinary is the extraordinary.”


          o ⚓ Of_Mice_and_Men⠀⇛


                 This recent controversy on Maus’s suitability (or not)
                 for Tennessee schoolchildren, though, has reinforced an
                 unease I’ve had for quite a while on the uses and misuses
                 of the extermination of European Jewry.


                 It should be said that the fact that the Holocaust is
                 treated as such a matter of importance in the United
                 States—which is, after all, a Christian country—is
                 something that needs to be acknowledged. Considering the
                 brutality meted out to Jews over the centuries, this is
                 no small thing. (And the Holocaust and Nazi oppression
                 were not just directed at Jews, of course.)


          o ⚓ Hunter_S._Thompson: Literary_Gentleman_from_Louisville⠀⇛


                 Thompson always had a nose for notoriety as well as for
                 the news, though he didn’t really find himself until 1970
                 at the age of 33 when Hinckle published in the pages
                 of Scanlon’s, his sensational article, “The Kentucky
                 Derby Is Decadent and Depraved.” Alas, the magazine soon
                 folded. Steadman provided the art for Thompson’s expose
                 of Louisville’s filthy rich. Apparently, he drew the
                 illustrations with lipstick and eyeliner.


                 Peter Richardson tells the riveting story of the Hinckle/
                 Steadman/Thompson triumvirate in Savage Journey
                 (University of California Press; $27.95), a biography, in
                 which he also traces what he calls the “Weird Road to
                 Gonzo.” True, the road could be bumpy with twists and
                 turns, but Thompson’s journey looks and feels a lot more
                 civilized than Richardson makes it seem. At times, Hunter
                 —as friends called him— even appeared to be a gentleman
                 of the old school, especially with his long cigarette
                 holder and shiny bald head and in photos of him sitting
                 next to Senator George McGovern in 1972. Gonzo was a
                 genuine part of his act; much as Beat was a genuine part
                 of Kerouac’s act.


          o ⚓ The_Militant_Passion_of_Emma_Tenayuca⠀⇛


                 Some knew her as la Pasionaria de Texas—the Texas
                 Passionflower. Others called her Red Emma. But most of
                 the people she fought alongside just called her
                 “comrade.” Emma Tenayuca was born in San Antonio, Texas,
                 in 1916 to parents of Spanish and Comanche descent, and
                 spent her childhood learning about Mexican identity, the
                 evils of Jim Crow, and revolution from her grandfather.
                 Emma was also constantly running off to Plaza del Zacate
                 in San Antonio’s Milam Park to listen to anarchists and
                 activists speak about politics and workers’ rights from
                 their soapboxes. Her first experience on the picket line
                 came when she was only 16. In 1933, she joined a group of
                 Mexican women workers from the H.W. Finck Cigar Company,
                 who were out on a wildcat strike over low wages and
                 unsanitary working conditions. The teenager was horrified
                 to witness the violent police response to the strike, and
                 was arrested herself. That early baptism into the labor
                 struggle convinced the young Tejana that she’d found her
                 place—and her purpose.


          o ⚓ ‘The_Thaw’_in_black_and_white_Photographer_Vladimir_Lagrange
            chronicled_the_Soviet_1960s_and_1970s._Meduza_looks_back_at_his
            main_works.⠀⇛


                 Vladimir Lagrange, who died in Moscow on January 22 at
                 the age of 82, was an icon of national photography with
                 work published in both the Soviet and foreign press. From
                 1959 to 1963, he worked in the news photography division
                 of TASS. Until 1989, he was a special photo correspondent
                 for the magazine Sovetsky Soyuz. He also worked for the
                 magazine Rodina and in the Moscow bureau of the French
                 agency Sipa Press. Following a memorial service on
                 January 25, Meduza is publishing some of Lagrange’s
                 classic images, with comments by Natalia Grigorieva-
                 Litvinskaya, the director of the Lumière Gallery, which
                 hosted an exhibit in 2019 titled “Lagrange Street.”


          o ⚓ The_Great_Social_Media_Retreat⠀⇛


                 A few weeks ago I changed my life on a whim. I deleted my
                 Reddit and Twitter accounts. To be fair, I have deleted
                 my Twitter account before, so perhaps that one isn’t a
                 big deal. My Reddit account on the other hand was
                 practically a prized possession. So what caused me to
                 delete it?


                 People. All of them. Myself included.


          o ⚓ Life_in_the_neighborhood⠀⇛


                 I’ve worked from home since 1998. All along I’ve hoped
                 many more people would enjoy the privilege and share in
                 the benefits. Now that it’s finally happening, and seems
                 likely to continue in some form, let’s take a moment to
                 reflect on an underappreciated benefit: neighborhood
                 revitalization.


          o ⚓ Titanium_Edition⠀⇛


                 I still recall how hard it was to find where the right
                 song was, especially once I had had a few drinks.
                 Carrying around a stack of CDs wasn’t fun either. Music
                 discovery was hard and friends, TV channels and radio
                 stations were my main sources of ideas. Pirating music
                 wasn’t exactly easy either, given how slow and expensive
                 internet was back then. I think my first subscription for
                 Internet over LAN was $20/month for something like 300MB/
                 month total traffic. We’d exchange music with friends and
                 I’d spend a lot of time scouring the LAN (usually with
                 the help of DC). Younger people here probably have no
                 idea what I’m talking about. I’m trying to make a point,
                 though – because it was so hard to build a music
                 collection, one had a lot of pride in theirs. Efforts
                 give meaning to stuff.


          o ⚓ Radio_Amateurs_&_Skywatchers_Rejoice,_Sat_Operators_Worry:_Solar
            Storm_Incoming_|_Hackaday⠀⇛


                 How do you look back over your life and divide it up?
                 Maybe by decades, cultural moments, or geopolitical
                 events. For radio amateurs with older callsigns there’s a
                 temptation to do so by solar cycles, as the roughly 11-
                 year period of the Sun’s activity had a huge effect on
                 radio propagation through the charge it creates in the
                 upper atmosphere. We’re now in solar cycle 25, numbered
                 since the 18th century when the science of solar
                 observation began, and as never before we’re surrounded
                 by information from experts such as [Dr. Tamitha Skov],
                 the so-called [Space Weather Woman]. When she says
                 something is on the way we listen, so a recent Tweet
                 predicting a direct hit from a solar storm with a good
                 probability of auroras in lower latitudes is very much
                 worth sharing.


          o § Science⠀➾


                # ⚓ The_Controversy_Surrounding_Hybrid_Cryptography⠀⇛


                       Despite the risks of greenfield cryptographic
                       algorithms, the NSA has begun recommending a
                       strictly-PQ approach to cryptography and have
                       explicitly stated that they will not require hybrid
                       designs.


                # ⚓ The_Perils_and_Promises_of_Virtual_Reality⠀⇛


                       Thus begins David Chalmers enchanting new
                       book Reality + which engages imaginatively with
                       philosophical issues connected to Virtual Reality
                       (VR).


                       In his book, Chalmers argues for:


          o § Education⠀➾


                # ⚓ Where_are_All_the_Substitute_Teachers?⠀⇛


                       Some schools have even called on parents to step in
                       to provide adult supervision in classrooms. In New
                       Mexico, the governor has asked National Guard
                       members to serve as substitute teachers.


                       Normally schools hire substitutes to cover teacher
                       absences. But there are so many teachers out with
                       COVID-19 that the demand is much higher than usual.


          o § Hardware⠀➾


                # ⚓ An_old_photo_of_a_very_large_BBS⠀⇛


                       Nearly 30 years ago, I was given this picture. I
                       wanted to present it in a post all by itself to
                       give it a chance to “sink in” before I went and
                       used it in an analogy. First, I’ll just show the
                       image and explain what you’re looking at here.


                # ⚓ One_machine_can_go_pretty_far_if_you_build_things
                  properly⠀⇛


                       I’ll admit: when I first saw that picture in 1993,
                       I was in awe of the scale. It had never occurred to
                       me what kind of layout you would need to run some
                       of these systems at that level of scale. Some small
                       part of me probably wanted to at least experience
                       that, but I never actually went that far.


                # ⚓ Renaissance-Style_Drone_Would_Make_Da_Vinci_Proud_Four
                  Times_Over_|_Hackaday⠀⇛


                       For as much of a genius as Leonardo da Vinci
                       obviously was, modern eyes looking upon his
                       notebooks from the 1400s tend to see his designs as
                       somewhat quaint. After all, his concept of a
                       vehicle armored with wood would probably only have
                       survived the archers and pikemen of a Renaissance
                       battlefield, and his curious helicopter driven by
                       an Archimedes screw would certainly never fly,
                       right?


                       Don’t tell that to [Austin Prete] and his team from
                       the University of Maryland, who’ve built a da
                       Vinci-style quadcopter that actually flies. Called
                       the “Crimson Spin”, the quad is based on a standard
                       airframe and electronics. Details are sparse — the
                       group just presented the work at a vertical flight
                       conference — but it appears the usual plastic props
                       are replaced with lightweight screws made from wire
                       and some sort of transparent plastic membrane.
                       Opposing pairs of screws have the opposite
                       handedness, which gives the quad yaw control.
                       There’s a video embedded in the link above that
                       shows the quad being tested both indoors and out,
                       and performing surprisingly well. We’d imagine that
                       Crimson Spin might not do so well on a windy day,
                       given the large wind cross-section those screws
                       present, but the fact it got off the ground at all
                       is cool enough. It kind of makes you wonder where
                       we’d be today if da Vinci had access to BLDCs.


                # ⚓ Rainbow_DIP_Switch_Is_The_Coolest_Way_To_Configure_Your
                  Project_|_Hackaday⠀⇛


                       Oftentimes, when programming, we’ll put
                       configuration switches into a config file in order
                       to control the behaviour of our code. However,
                       having to regularly open a text editor to make
                       changes can be a pain. This colorful little DIP
                       switch dongle from [Glen Aikins] makes for a fun
                       alternative solution.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Data_Highlights_‘Egregious’_Pandemic_Profiteering_by_US
                  Food_and_Oil_Giants⠀⇛


                       A new analysis released Tuesday ahead of a
                       congressional hearing on pandemic-era price gouging
                       shows that U.S. corporations in the food and energy
                       sectors—from Tyson to Exxon Mobil—are pushing
                       higher costs onto consumers while raking in ever-
                       increasing revenues and handing executives massive
                       pay packages.


                       Conducted by the advocacy group Food & Water Watch
                       (FWW), the analysis spotlights the fact that
                       skyrocketing food and energy—specifically
                       gasoline—prices have been major contributors to the
                       overall rise of inflation in the U.S.
                       Between December 2019 and December 2021, the
                       nation’s Consumer Price Index (CPI) jumped by 8.5%.


                # ⚓ ‘Five_Alarm_Crisis’:_Teachers_Union_Warns_Pandemic_Burnout
                  Destroying_Schools⠀⇛


                       New polling out Tuesday reveals pervasive burnout
                       among the nation’s educators as pandemic-related
                       staffing shortages and other difficulties drive a
                       potential exodus from the teaching profession.


                       Released by the National Education Association, the
                       nation’s largest union of any profession, the
                       member survey shows, according to NEA president
                       Becky Pringle, that “after persevering through the
                       hardest school years in memory, America’s educators
                       are exhausted and increasingly burned out.”


                # ⚓ The_Limits_of_Understanding_the_Pandemic_Philosophically⠀⇛


                       The Covid-19 pandemic has brought death to the
                       forefront of daily life. Whether in the
                       overwhelming moments of intimate grief or the
                       omnipresent background hum of irrepressible dread,
                       the virus’s deadly ramifications have been
                       impossible to avoid. It is the ways in which we
                       have responded to this crisis, however, that
                       philosopher Byung-Chul Han’s latest work, The
                       Palliative Society, wants to take up as the basis
                       for cultural analysis. For, as Han writes in the
                       opening to his book, “Our relation to pain reveals
                       what kind of society we are.” Written in German in
                       2020 and now translated into English by Daniel
                       Steuer, this short book (or perhaps, more
                       accurately, long essay) attempts to position our
                       social reaction to the pandemic as being of a piece
                       with our contemporary inability to reckon with
                       pain. Han seeks to utilize an extended reading of
                       our relation to pain as a means of opening up a
                       critique of our social, cultural, and political
                       lives. But this work demonstrates both the
                       possibilities and the pitfalls of taking up such a
                       weighty historical moment in media res. Stuck
                       somewhere between genuine insight and banal
                       truisms, radical critique and reactionary
                       entrenchment, The Palliative Society is never quite
                       able to position the pandemic as a historic event,
                       using it merely to confirm an argument that feels
                       already well-rehearsed.


                # ⚓ Theranos_Verdict:_In_the_U.S.,_It_Is_Fine_to_Lie_to
                  Consumers_but_Not_to_Investors⠀⇛


                       The evidence presented during the trial showed that
                       Theranos technology did not work, and Holmes, while
                       fully aware of it, knowingly falsified the results
                       and forged reports. These “doctored reports” showed
                       that major pharmaceutical companies endorsed her
                       products, and even the U.S. military was using
                       Theranos equipment in the field.


                       Holmes got major names in the industry to invest
                       almost a billion dollars in Theranos. The investors
                       included the Walton family, who owns Walmart;
                       Rupert Murdoch, the major media mogul; the family
                       of Betsy DeVos, who was the former secretary of
                       education under the Trump administration; Larry
                       Ellison, the founder of Oracle; and many other
                       people with deep pockets. Meanwhile, Theranos’
                       board of directors also had dazzling names
                       including former U.S. secretaries of state Henry
                       Kissinger and George Shultz and former U.S.
                       secretaries of defense James Mattis and William
                       Perry.


                # ⚓ Opinion_|_Abortion:_What_a_Complex_Web_They_Weave⠀⇛


                       Did you know?


                # ⚓ Native_American_tribes_agree_to_$590M_settlement_with
                  Johnson_&_Johnson,_drug_distributors_over_opioid_epidemic⠀⇛


                       “The Native American population has suffered some
                       of the worst consequences of the opioid epidemic of
                       any population in the United States. Indeed,
                       American Indians have suffered the highest per
                       capita rate of opioid overdoses,” the tribal
                       leadership committee said in a statement filed with
                       the court. “American Indians and Alaska Natives had
                       the highest drug overdose death rates in 2015 and
                       the largest percentage increase in the number of
                       deaths over time from 1999-2015 compared to other
                       racial and ethnic groups.”


                       Johnson & Johnson agreed to pay $150 million over
                       the next two years while not admitting liability or
                       wrongdoing. The company defended its promotion of
                       the medications.


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # ⚓ How_to_Delete_Your_Spotify_Account_–_Plus_5
                        Alternative_Music_Streaming_Services⠀⇛


                             If you’re ready to switch to one of many
                             Spotify alternatives, you can delete your
                             Spotify account. Just don’t forget to export
                             all of your playlists with a service like
                             Soundiiz. If you have Spotify Premium, you’ll
                             need to head to the Spotify website to get
                             the process started.


                      # ⚓ Lock-in_and_Multi-Cloud⠀⇛


                             But that’s today. Let’s think ahead to 2030
                             and zero in on on a hypothetical future
                             version of AWS. First, there was that
                             unfortunate accident involving one of Elon’s
                             satellites and the Blue Origin rocket Jeff
                             was on. And Andy, a multibillionaire via that
                             Amazon-CEO package, bought out the rest of
                             the Kraken owners, then put in a surprise
                             grab and owns the Seahawks too. He’s
                             discovered that being a sports mogul beats
                             the hell out of testifying on live TV to
                             Pramila Jayapal.


                             In this scenario, activist investors and big-
                             time PE players have won a majority on the
                             Amazon board and they don’t want any of their
                             business units missing any chances to turn
                             screws that maximize revenues. Data egress
                             charges ratchet up every corner, as do high-
                             level proprietary services like Kinesis and
                             Lambda.


                             A whole lot of customers have lost control of
                             their cloud-computing spend; they pay
                             whatever their AWS account manager tells them
                             they’re gonna pay.


                      # ⚓ FBI_says_cyber_actors_could_‘disrupt’_Beijing
                        Olympics,_Paralympics [iophk: Windows TCO]⠀⇛


                             The FBI noted it was not aware of any
                             specific cyberattacks that were planning on
                             being carried out but emphasized that
                             athletes and others associated with the Games
                             should remain vigilant, including by
                             recommending that people leave their personal
                             devices at home and use a temporary phone
                             while in China.


                      # ⚓ Cyber-attack_strikes_German_fuel_supplies [iophk:
                        Windows TCO]⠀⇛


                             It has declared “force majeure” for the
                             majority of its inland supply activities in
                             Germany.


                             The declaration of force majeure excuses a
                             company from contractual agreements when an
                             extraordinary event occurs which is beyond
                             its control.


                      # ⚓ What_I_Learned_When_My_Parents_Got_Arrested⠀⇛


                             My parents, who had owned a software-resale
                             business, had nothing to do with the piracy
                             ring itself. Instead, they were charged with
                             30 counts of conspiracy, mail fraud, wire
                             fraud, and money laundering in what
                             authorities described as a scheme to defraud
                             Microsoft of millions of dollars by obtaining
                             discounted academic software under false
                             pretenses.


                             Microsoft initially alleged that it lost as
                             much as $100 million, and the local newspaper
                             ran with that number. Overnight, everyone
                             heard that my parents had been scooped up in
                             a massive anti-​piracy raid. Their faces were
                             plastered on the FBI website for days, and
                             people assumed I was somehow sitting on $100
                             million stashed away in domestic and foreign
                             accounts.


                             Oh, how I wish it were true. I really could
                             have used that money in the weeks and months
                             that followed.


                      # ⚓ [Old] How_to_Set_Up_SSH_2fa_(Two-Factor
                        Authentication)⠀⇛


                             This tutorial guides you through setting up
                             Google Authenticator PAM to enable SSH 2FA
                             for users connecting to a Linux server. We’ll
                             use nano as our editor in examples.


                      # § Security⠀➾


                            # ⚓ Update_now:_Samba_prior_to_4.13.17_hit_with
                              remote_root_code_execution_bug_|_ZDNet⠀⇛


                                   Samba has fixed a vulnerability in all
                                   versions of its software prior to
                                   version 4.13.17 that allowed for a
                                   remote actor to execute code as root,
                                   thanks to an out-of-bounds heap read
                                   write vulnerability.


                            # § Privacy/Surveillance⠀➾


                                  # ⚓ Snowden_Slams_Ongoing_Impunity_for_NSA’s
                                    Domestic_Spying⠀⇛


                                         Exiled U.S. whistleblower and
                                         former National Security Agency
                                         contractor Edward Snowden on
                                         Tuesday called out years of
                                         impunity for the NSA violating
                                         Americans’ civil liberties and
                                         privacy rights.


                                         Snowden’s tweet came in response
                                         to CNN reporting that the NSA
                                         “failed to follow both court-
                                         approved and internal procedures
                                         designed to prevent officials
                                         from using a controversial
                                         foreign surveillance law to
                                         inappropriately monitor
                                         Americans’ communications.”


                                  # ⚓ ID.me_Finally_Admits_It_Runs_Selfies
                                    Against_Preexisting_Databases_As_IRS
                                    Reconsiders_Its_Partnership_With_The
                                    Company⠀⇛


                                         Tech company ID.me has made
                                         amazing inroads with government
                                         customers over the past several
                                         months. Some of this is due to
                                         unvetted claims by the company’s
                                         CEO, Blake Hall, who has asserted
                                         (without evidence) that the
                                         federal government lost $400
                                         billion to fraudulent COVID-
                                         related claims in 2020. He also
                                         claimed (without providing
                                         evidence) that ID.me’s facial
                                         recognition tech was sturdy,
                                         sound, accurate, and backstopped
                                         by human review.


                                  # ⚓ Suicide_Hotline_Collected,_Monetized_The
                                    Data_Of_Desperate_People,_Because_Of_Course
                                    It_Did⠀⇛


                                         Another day, another privacy
                                         scandal that likely ends with
                                         nothing changing.


                                  # ⚓ Chicago_Cops_Love_Them_Some_Facebook
                                    Sharing,_According_To_Internal_Facial
                                    Recognition_Presentation⠀⇛


                                         Somewhere between the calls to
                                         end encryption and calls to do
                                         literally anything about crime
                                         rate spikes at this time of year,
                                         at this time of day, in [insert
                                         part of the country], localized
                                         entirely within [add geofence]
                                         lies the reality of law
                                         enforcement. While many continue
                                         to loudly decry the advent of by-
                                         default encryption, the reality
                                         of the situation is people are
                                         generating more data and content
                                         than ever. And most of it is less
                                         than a warrant away.


                                  # ⚓ Birthday_wishes_inadvertently_give_away
                                    private_information_online⠀⇛


                                         Dilara Kekulluoglu at the
                                         University of Edinburgh, UK, and
                                         her colleagues found more than 18
                                         million Twitter posts that
                                         mentioned “happy birthday” in a
                                         45-day period. Of those, 2.8
                                         million directly mentioned a
                                         user, so they could be used to
                                         ascertain an individual’s
                                         birthday. More than 66,000 of
                                         these tweets also gave away the
                                         age of the user, and therefore
                                         their full date of birth.


                                         Only around 2 per cent of the
                                         Twitter users mentioned in those
                                         posts shared their birth years on
                                         their profiles, so the team warns
                                         that well-wishers are exposing
                                         this information for users who
                                         haven’t proactively shared it
                                         themselves.


                                  # ⚓ The_EARN_IT_Act_is_back_in_Congress⠀⇛


                                         The original EARN IT Act’s
                                         sponsors touted endorsements from
                                         the National Center for Missing
                                         and Exploited Children and other
                                         groups, and it took aim at a
                                         surge of online child sexual
                                         abuse content. But the bill was
                                         strongly condemned by the
                                         American Civil Liberties Union
                                         and many advocates for online
                                         civil liberties, sex workers, and
                                         LGBT rights — who argued that it
                                         would encourage companies to
                                         abandon strong encryption and
                                         privacy protections for users.
                                         Some of these groups, including
                                         Fight For the Future, have issued
                                         statements against the revived
                                         bill as well. The bill is
                                         scheduled to be discussed by the
                                         Senate Judiciary Committee on
                                         Thursday.


                                  # ⚓ EU:_Mass_travel_surveillance:_no_problem,
                                    says_court⠀⇛


                                         The Passenger Name Record (PNR)
                                         Directive was agreed in April
                                         2016 and mandates the
                                         surveillance and profiling of
                                         almost all air passengers
                                         entering, travelling within, or
                                         leaving the EU. Passenger data
                                         has to be transmitted by travel
                                         companies and airlines to
                                         Passenger Information Units
                                         (PIUs) operated by national
                                         police forces.


                                  # ⚓ A_cautious_green_light_for_technology-
                                    driven_mass_surveillance⠀⇛


                                         This cursory analysis sheds light
                                         on some of the AG’s Opinion’s
                                         shortcomings. It thus follows
                                         that the CJEU should deviate from
                                         Pitruzzella’s recommendations.
                                         The PNR Directive, due to the
                                         severity of its effects and its
                                         inherent inefficiency in
                                         fulfilling its stated purpose,
                                         produces disproportionate
                                         interferences with Articles 7 and
                                         8 CFR. It ought to be
                                         invalidated.


                                  # ⚓ IRS_plan_to_scan_your_face_prompts_anger
                                    in_Congress,_confusion_among_taxpayers⠀⇛


                                         Millions of Americans could soon
                                         have to scan their faces to
                                         access their Internal Revenue
                                         Service tax accounts, one of the
                                         government’s biggest expansions
                                         yet of facial recognition
                                         software into people’s everyday
                                         lives.


                                         For now, taxpayers can still file
                                         their returns the old-fashioned
                                         way; the IRS began accepting
                                         returns for 2021 earnings on
                                         Monday, encouraging electronic
                                         filing.


                                         But by this summer, anyone
                                         wanting to access their records —
                                         including details about child tax
                                         credits, payment plans or tax
                                         transcripts — on the IRS website
                                         could be required to record a
                                         video of their face with their
                                         computer or smartphone, and send
                                         it to the private contractor
                                         ID.me to confirm their identity.


                                  # ⚓ Facial_recognition_for_all_online_IRS
                                    accounts:_What’s_up_with_ID.me?⠀⇛


                                         The IRS uses ID.me — a third-
                                         party service with a thorough
                                         registration process that
                                         includes uploading a “video
                                         selfie” through ID.me’s facial
                                         recognition software — to manage
                                         new signups for the agency’s
                                         online tax services.


                                         Soon, everyone who wants to use
                                         IRS services online — including
                                         viewing and making payments
                                         online, updating a mailing
                                         address, and accessing the Child
                                         Tax Credit Update Portal — will
                                         need to register through ID.me.


                                  # ⚓ The_IRS_is_reportedly_looking_for_ID.me
                                    alternatives_amid_privacy_concerns⠀⇛


                                         Last year, the IRS announced its
                                         plans to start requiring people
                                         who file their taxes online to
                                         register with third-party facial
                                         recognition company ID.me.
                                         Through the service, users will
                                         have to submit a video selfie
                                         using a webcam or mobile device
                                         to verify their identity. The IRS
                                         is supposed to roll out the
                                         program this summer.


                                         Although ID.me previously claimed
                                         it only uses one-to-one facial
                                         matching, which involves matching
                                         a user’s face with images of the
                                         same face, ID.me CEO Blake Hall
                                         admitted it uses technology that
                                         matches faces against a larger
                                         database. This only exacerbates
                                         privacy concerns — politicians,
                                         the American Civil Liberties
                                         Union, and digital rights
                                         advocates have already spoken out
                                         about the IRS’ use of the
                                         software.


                                  # ⚓ Website_fined_by_German_court_for_leaking
                                    visitor’s_IP_address_via_Google_Fonts⠀⇛


                                         Earlier this month, a German
                                         court fined an unidentified
                                         website €100 ($110, £84) for
                                         violating EU privacy law by
                                         importing a Google-hosted web
                                         font.


                                         The decision, by Landgericht
                                         München’s third civil chamber in
                                         Munich, found that the website,
                                         by including Google-Fonts-hosted
                                         font on its pages, passed the
                                         unidentified plaintiff’s IP
                                         address to Google without
                                         authorization and without a
                                         legitimate reason for doing so.
                                         And that violates Europe’s
                                         General Data Protection
                                         Regulation (GDPR).


                                  # ⚓ Website_operator_fined_for_using_Google
                                    Fonts_“the_cloudy_way”⠀⇛


                                         The court agreed, demanded that
                                         the website operator start
                                         hosting fonts locally, and
                                         awarded the complainant damages
                                         of €100 (about $110).


                                         The court’s argument doesn’t seem
                                         to be suggesting any and all
                                         other third party “widget
                                         linking” is now considered
                                         illegal in Germany (or, more
                                         particularly, in the region where
                                         this court holds sway), but only
                                         that websites are expected to
                                         host content locally if that’s
                                         easily possible: [...]


                                  # § Confidentiality⠀➾


                                        # ⚓ Why_I_won’t_use_Let’s_Encrypt⠀⇛


                                               Why do not I forcefully
                                               redirect people from HTTP
                                               to HTTPS?


                                               First reason is that I just
                                               do not have geographically
                                               distributed servers under
                                               my control. Most of my
                                               websites are hosted both on
                                               my own home server and on
                                               some VPS in another city. I
                                               can not set up TLS on VPS,
                                               because its hosting company
                                               obviously will have access
                                               to all of its internals,
                                               including TLS private keys.
                                               Currently my websites have
                                               two IP addresses, with two
                                               independent HTTP servers,
                                               but with only single HTTPS
                                               server and single dummy
                                               TCP-proxy located on VPS.
                                               If my main server is down,
                                               then the whole HTTPS is
                                               down too. If I give TLS
                                               private keys to the hosting
                                               company, then what is the
                                               point of using TLS and
                                               lying that it can
                                               authenticate the endpoint
                                               domain?


                                               Second reason is that it is
                                               not my responsibility to
                                               impose user the desired
                                               security protocol usage.
                                               Possibly there is already
                                               IPsec transport session,
                                               transparently securing the
                                               link. Possibly we use some
                                               overlay network like
                                               Yggdrasil, where TLS is
                                               just pointless. My tarballs
                                               and Git tags are always
                                               signed with OpenPGP key.
                                               TLS even won’t give any
                                               metadata privacy there
                                               because of known tarballs/
                                               pages sizes.


                                               Next point is that there
                                               just can not be some
                                               objectively absolutely
                                               valid and proper global
                                               scale PKI certificate.
                                               Because its validity fully
                                               depends on your exact point
                                               of view, specifically on
                                               what trust anchors you use
                                               and validation rules you
                                               apply.


          o § Defence/Aggression⠀➾


                # ⚓ McEnany’s_Texts_Lead_to_Request_for_Ivanka_Trump_to_Speak
                  With_Jan._6_Committee⠀⇛


                # ⚓ More_Than_a_Dozen_HBCUs_Faced_Bomb_Threats_on_First_Day_of
                  Black_History_Month⠀⇛


                # ⚓ NATO_Nonsense_and_Panic⠀⇛


                       Many or most U.S. media overlooked it – that is,
                       buried it. Or emasculated it. In Germany they
                       couldn’t fully ignore it – though unpleasant as a
                       messy cat cadaver near the red carpet at a major
                       film event. This was no star-studded premiere,
                       however, but a dangerous, frightening political and
                       military program, and the disturbing element was
                       not a raggedy, dead alley-cat but the elegantly-
                       uniformed boss of the German Navy. Yet here, too, a
                       demise was involved – that of the vice-admiral’s
                       career!


                       What sin earned him such a fate? When asked about
                       the month-long NATO-Washington campaign against
                       Putin and Russia, based only on vague, dubious
                       assumptions and prophesies by anonymous experts yet
                       rushing headlong toward military catastrophe, this
                       top-level expert had the temerity to puncture the
                       foundation of the whole campaign with one word:
                       “Nonsense!”


                # ⚓ International_Support_Grows_for_Mexico’s_Lawsuit_Against_US
                  Gun-Makers⠀⇛


                       In a big boost to the Mexican government’s historic
                       federal lawsuit against American gun-makers, 13
                       U.S. states, the District of Columbia, two
                       countries, a coalition of attorneys general, and
                       numerous advocacy groups on Monday filed or joined
                       amicus briefs supporting Mexico’s litigation, which
                       seeks to hold weapons manufacturers accountable for
                       the violence they facilitate. 


                       “The defendant gun manufacturers send guns to
                       Mexico, where transnational drug cartels use them
                       to inflict violence on both sides of the border.”


                # ⚓ Opinion_|_How_Much_More_Deadly_Will_the_Next_Variant_of
                  Trumpism_Be?⠀⇛


                       Imagine that you were experiencing all of this (and
                       by this, I mean our lives right now) as if it were
                       a novel, à la Daniel Defoe’s A Journal of the
                       Plague Year.  The famed author of Robinson Crusoe —
                       Defoe claimed it had been written by the fictional
                       Crusoe himself — was five years old in 1665. That
                       was when a year-long visitation of the bubonic
                       plague decimated London. It probably killed more
                       than 100,000 of that city’s residents or 15% of its
                       population. As for Defoe, he published his
                       “journal” in 1722, 57 years later. He wrote it,
                       however, as if he (or his unidentified protagonist)
                       had recorded events as they were happening in the
                       way that all of us, whatever our ages, have been
                       witnessing the ravages of the many variants of
                       Covid-19 in our own all-too-dismantled lives.


                # ⚓ Opinion_|_North_Korea,_Perpetual_Victim_of_the_US_Military-
                  Industrial_Complex⠀⇛


                       It seems hard to believe that in these possible end
                       times in the midst of a global pandemic with an
                       endless succession of catastrophic climate
                       disasters and thousands of nuclear weapons poised
                       and pointed in the U.S. and Russia, ready to
                       destroy life on earth, we are beset by a bought,
                       corrupted mainstream media that assaults us with
                       the “wrongdoings” of Russia and China, and most
                       recently North Korea, with barely a mention in
                       their assaultive reporting of how the U.S. might be
                       the cause in the matter.  Nor do they report on the
                       many remedies that have been rejected by the United
                       States in its drive for global domination. Instead
                       of promoting the critical opportunities we must now
                       seize—all nations and peoples of the world—to work
                       cooperatively to save Mother Earth, the western
                       news reports serve up a steady daily diet of the
                       harm that could be inflicted upon an innocent
                       United States, echoing shades of the dreadful 1950s
                       McCarthy Era in a new Cold War II and maybe World
                       War III. 


                # ⚓ A_Police_Car_Hit_a_Kid_on_Halloween_2019._The_NYPD_Is
                  Quashing_a_Move_to_Punish_the_Officer.⠀⇛


                       It was a little more than two years ago that I
                       really started learning about the reality of police
                       oversight in America.


                       On Halloween night in 2019, my wife and our then-6-
                       year-old daughter were walking home from trick-or-
                       treating in our Brooklyn neighborhood when they saw
                       a New York Police Department car go the wrong way
                       down a street and smack into a teenager, who fell
                       and then ran away.


                # ⚓ Amid_US-Russia_Escalation,_Activists_Emphasize_NATO_Is_“Not
                  About_Security”⠀⇛


                # ⚓ ‘We_have_other_problems’_As_the_world_worries_about_an_all-
                  out_war_between_Russia_and_Ukraine,_Meduza_reports_from_the
                  border_with_the_Donbas⠀⇛


                       Since the fall of 2021, media outlets around the
                       world have been discussing the possibility of
                       another Russian invasion of Ukraine. Moscow has had
                       troops massed near the Ukrainian border for several
                       months now — in response, NATO countries, led by
                       the United States, have deployed additional forces
                       to Eastern Europe and are threatening the Kremlin
                       with sanctions. What do people living near the
                       Russian border with Ukraine think about these
                       escalating tensions? For Meduza, journalist Gleb
                       Golod traveled to towns and villages in Russia’s
                       Rostov region to find out.


                # ⚓ 100+_Anti-War_Groups_Demand_Biden_End_Brinkmanship_With
                  Russia⠀⇛


                       More than 100 advocacy organizations representing
                       millions of people across the U.S. demanded Tuesday
                       that the Biden administration take immediate steps
                       to defuse tensions with Russia as the two nuclear-
                       armed powers remain perilously close to war over
                       Ukraine.


                       “We call upon President Biden to end the U.S. role
                       in escalating the extremely dangerous tensions with
                       Russia,” the progressive groups said in a joint
                       statement spearheaded by CodePink and
                       RootsAction.org. “It is gravely irresponsible for
                       the president to participate in brinkmanship
                       between two nations that possess 90% of the world’s
                       nuclear weapons.”


                # ⚓ The_Exit_From_the_Ukraine_Crisis_That’s_Hiding_in_Plain
                  Sight⠀⇛


                       The crisis over Ukraine grows simultaneously more
                       dangerous and more absurd. Russia has amassed more
                       than 100,000 troops on the Ukrainian border,
                       demanding that NATO not admit Ukraine and stop its
                       expansion east. Russian officials want those
                       demands answered immediately, but President
                       Vladimir Putin also says he won’t make war.


                # ⚓ Russia_Gains_Much_From_Threats_to_Invade_Ukraine,_But_Knows
                  That_Actually_Doing_So_Would_be_a_Disaster⠀⇛


                       Russian troops might be able to capture Kyiv in a
                       week, but this would only be the start of a long
                       war that Russia would find it impossible to win. A
                       more limited Russian offensive in east Ukraine –
                       such as seizing a land corridor between the Russian
                       separatist Donbas and Russian annexed Crimea – is
                       scarcely a more attractive option. It would push
                       the rest of Ukraine further into the embrace of
                       Nato, which would be exactly the opposite of what
                       Russia wants.


                       President Putin may be vindictive and
                       unpredictable, but he has never overplayed his hand
                       as Russian leader in military conflicts from
                       Chechnya in 1999 to Syria in 2015. In all cases,
                       Western hopes and expectations that he was plunging
                       into a quagmire were disappointed.


                # ⚓ More_Than_100_Antiwar_Groups_Call_on_Biden_to_End
                  Brinksmanship_With_Russia⠀⇛


                # ⚓ “It’s_Not_About_Security”:_Belgian_Peace_Activist_Says_NATO
                  Has_Outlived_Its_Purpose⠀⇛


                       To speak about the key role NATO is playing in the
                       Ukraine crisis, we speak with Ludo De Brabander,
                       spokesperson of the peace organization Vrede vzw in
                       Belgium, where NATO is headquartered. De Brabander
                       says NATO has outlived its purpose, and touches on
                       how activists in NATO countries like Belgium are
                       pushing against narratives in the media that war
                       with Russia is necessary.


                # ⚓ Germany_Refuses_to_Send_Arms_to_Ukraine_Despite_Pressure
                  from_U.S._&_NATO⠀⇛


                       Germany’s new coalition government is refusing to
                       send lethal weapons to Ukraine but has offered to
                       send over 5,000 combat helmets to protect Ukrainian
                       soldiers in case of a Russian attack. The move has
                       been ridiculed as the U.S. and other NATO countries
                       continue to send military support to Ukraine. In
                       response, German Chancellor Olaf Scholz has
                       promised his country will stay in tune with
                       European Union and NATO policies toward Russia. To
                       speak more about Germany’s stance toward Ukraine,
                       we’re joined by German peace activist and executive
                       director of the International Peace Bureau Reiner
                       Braun, who calls on the European Union to establish
                       a more “common politics with Russia” to prevent a
                       war in Central Europe. He also says war in the
                       region could result in use of nuclear weapons that
                       would lead to “the end of Europe.”


                # ⚓ Ukrainians_Doubt_a_Russian_Invasion_Is_Imminent_as_U.S.
                  Peace_Groups_Urge_Biden_to_Halt_Escalation⠀⇛


                       The United States and Russia sparred on Monday over
                       the crisis in Ukraine at the United Nations
                       Security Council. Meanwhile, U.S. senators are
                       preparing to unveil a bill that would target
                       Russian President Vladimir Putin, Russian banks and
                       other entities with sanctions. To discuss the
                       Ukraine crisis, we’re joined by the co-founder of
                       CodePink, Medea Benjamin, who says “we need the
                       voice of the American people” to oppose U.S.
                       escalation and also calls on U.S. progressives to
                       vocalize their opposition to fueling a war in
                       Europe. We also speak with Ukrainian sociologist
                       Volodymyr Ishchenko, who says Ukrainian
                       intelligence does not see a Russian invasion as
                       likely or economically wise for Russia.


                # ⚓ How_US_meddling_split_Sudan,_creating_an_oil_republic
                  drowning_in_poverty_and_conflict⠀⇛


                # ⚓ Nigeria_and_Indonesia_cement_defence_ties⠀⇛


                       Following a meeting between the Indonesian
                       Ambassador to Nigeria Usra Harahap and the
                       executive vice-chairman of Nigeria’s National
                       Agency for Science and Engineering Infrastructure
                       (NASENI), Mohammed Sani Haruna, both countries
                       accepted a draft letter of intent on bilateral
                       defence cooperation.


                       The NASENI’s mission is to further the transfer of
                       technologies to Nigeria in various areas including
                       defence and aeronautics. Haruna told The Guardian
                       Nigeria that Indonesia would assist the country in
                       developing “aircraft, both civil and military, the
                       armoured personnel carrier, and other equipment
                       needed by the military”, with approval from the
                       Nigerian Federal government.


                # ⚓ Taiwan_Detects_Chinese_Electronic_Warfare_Aircraft_Near
                  Island⠀⇛


                       The Taiwanese MND stated that the aircraft flew on
                       Monday between Taiwan and Pratas, an island in the
                       northern South China Sea, which the government
                       controls in Taiwan.


                # ⚓ Ambitious_China_gears_up_to_flex_power_in_the_conflict-
                  riven_Horn_of_Africa⠀⇛


                       The Horn of Africa – located in the easternmost
                       corner of the African continent, takes its name
                       from the horn-shaped land formation at the southern
                       end of the Red Sea and on the Gulf of the Aden.
                       Five of the region’s seven countries – the Sudan,
                       Eritrea, Djibouti, Somalia and landlocked Ethiopia,
                       are situated looking into the Indian Ocean south of
                       the Arabian Peninsula. Located on the main shipping
                       route for the transport of oil from the Persian
                       Gulf to Europe and the United States, the Horn of
                       Africa is considered one of the most strategically
                       important regions in the world. Sadly the region is
                       endowed with rivers, lakes, forests and livestock,
                       and has untapped rich deposits of natural resources
                       including gold, petroleum, salt and natural gas
                       etc. Yet its two hundred million people remain one
                       of the poorest on the earth.


                # ⚓ Accused_American_ISIS_leader_‘highly_intelligent’_and
                  ‘marksman’:_Former_friend⠀⇛


                       One former friend, who said she last spoke to
                       Fluke-Ekren more than 10 years ago, painted a
                       picture of a woman who was close with her family
                       but then became increasingly radicalized. Fluke-
                       Ekren was arrested in Syria, where she moved a
                       decade ago and married a “prominent” ISIS leader,
                       according to court documents.


                # ⚓ Turkey_Helps_ISIS_Attack_Rojava⠀⇛


                       It’s no accident that the time bomb exploded on the
                       fourth anniversary of Turkey’s invasion and
                       occupation of Afrin in 2018. Turkey’s support of
                       ISIS and other jihadis is an open secret. A Turkish
                       drone even bombed an SDF vehicle that was racing to
                       Hasaka to help recapture ISIS prisoners. According
                       to an October 2021 report on ISIS sleeper cells,
                       raids resulting in the arrest of ISIS members
                       brought to light documents showing links to Turkish
                       intelligence, suggesting that the two coordinate
                       closely. The SDF believes the Hasaka attack was
                       organized by Turkish forces and their proxies in
                       territories Turkey occupied in 2019, including
                       Serekaniye and Tal Abyad, and the plan was for the
                       ISIS prisoners to flee there and regroup.


                # ⚓ Trump_Steals_a_Strategy_From_Mussolini’s_Playbook⠀⇛


                       That was a dangerous signal in and of itself, as it
                       came from a former president with a track record of
                       abusing his pardon authority for political
                       purposes. It suggested a willingness to employ the
                       power of the presidency to circumvent legitimate
                       and necessary legal accountability for those who
                       launched a deadly attack on the US Capitol, and for
                       himself—as the man who told those at the January 6
                       rally to head to the Capitol and “fight like hell”
                       to overturn the results of the 2020 election.


                # ⚓ Trump_Says_He_Would_Consider_Pardons_for_Jan._6_Defendants
                  if_Elected⠀⇛


                       At least 700 people have been arrested in
                       connection with the Jan. 6 [insurrection],
                       including 11 who have been charged with seditious
                       conspiracy. Some have said they believed they were
                       doing Mr. Trump’s bidding.


          o § Environment⠀➾


                # ⚓ As_China_replants_its_own_forests,_it_is_destroying_the
                  world’s⠀⇛


                       These developments might be interpreted as proof of
                       China’s dedication to protecting trees. Is there
                       substance to such an interpretation, or is the
                       declared appreciation for forests merely a mirage?


                       If history is any guide, reality hews to the
                       latter. That’s because China has a very long record
                       of razing forests to feed its development.
                       Alongside its economic rise in recent decades, it
                       has taken on an outsize and accelerating role in
                       global deforestation.


                # ⚓ First_Latin_American_e-waste_report_published⠀⇛


                       The report was produced by the Sustainable Cycles
                       (SCYCLE) Programme, co-hosted by the UN University
                       (UNU) and the UN Institute for Training and
                       Research (UNITAR). It covers the 13 countries
                       participating in the UNIDO-GEF LAC E-waste project:
                       Argentina, Bolivia, Chile, Costa Rica, Ecuador, El
                       Salvador, Guatemala, Honduras, Nicaragua, Panama,
                       Peru, Uruguay and Venezuela.


                # ⚓ Biden_Praised_for_Reversing_Trump_Effort_to_Unleash_More
                  Toxic_Air_Pollution⠀⇛


                       The Biden administration on Monday won strong
                       praise from public health defenders following the
                       Environmental Protection Agency’s move to restore
                       the legal basis for regulating mercury and other
                       toxic air pollutants from coal- and oil-fired power
                       plants.


                       “Administrator Regan has taken a critical step
                       forward in restoring vital public health
                       protections.”


                # ⚓ After_Passing_‘Point_of_No_Return’_in_2014,_Hot_Oceans_Are
                  Now_‘New_Normal’⠀⇛


                       A study published Tuesday revealed that ocean
                       heatwaves fueled by human-caused climate change
                       passed the “point of no return” in 2014 and have
                       become the “new normal.”


                       “These dramatic changes we’ve recorded in the ocean
                       are yet another piece of evidence that should be a
                       wake-up call to act on climate change.”


                # § Energy⠀➾


                      # ⚓ North_Carolina_Fertilizer_Plant_Fire_Forces_Thousands
                        to_Evacuate⠀⇛


                             Thousands of residents who live within a one-
                             mile radius of a fertilizer plant in Winston-
                             Salem, North Carolina were ordered to
                             immediately evacuate after a fire broke out
                             at the facility Monday night.


                             Although firefighters initially battled the
                             blaze at the Winston Weaver Co. fertilizer
                             plant, located at 4440 N. Cherry St., the
                             crews were pulled back after about an hour
                             and a half due to the explosion risk, Fire
                             Chief Trey Mayo said during a Tuesday morning
                             press conference.


                      # ⚓ Climate_Groups_Urge_Biden_Not_to_Fight_Ruling_Against
                        Oil_and_Gas_Lease_Sale⠀⇛


                      # ⚓ Biden_Urged_Not_to_Fight_Court_Ruling_Against_Massive
                        Oil_and_Gas_Lease_Sale⠀⇛


                             As the fossil fuel industry clamors for an
                             appeal, the Biden administration on Tuesday
                             faced pressure from environmentalists to
                             adhere to a judge’s decision blocking a
                             massive oil and gas lease sale in the Gulf of
                             Mexico, the site of the catastrophic
                             Deepwater Horizon spill.


                             “We urge you to comply with the court’s
                             ruling and not appeal the court’s decision,”
                             more than 70 climate groups wrote in a letter
                             to President Joe Biden and Interior Secretary
                             Deb Haaland. “The [Department of the
                             Interior] should not continue to defend
                             unlawful drilling for oil and gas in public
                             waters in appellate court given the impacts
                             on our climate, clear violations of federal
                             environmental standards, and public
                             commitments made by President Biden to end
                             the practice.”


                      # ⚓ Protests_Erupt_in_Argentina_Over_Plan_for_Offshore
                        Oil_Drilling⠀⇛


                             On January 4, thousands of people took to the
                             streets of Mar del Plata, a coastal city
                             roughly 250 miles south of Buenos Aires,
                             Argentina. They were there to protest the
                             plans by Norwegian oil company Equinor to
                             begin offshore oil exploration later this
                             year. 


                             They held signs that read “the sea is ours!”
                             and “an ocean free of oil,” and they chanted,
                             shouted, and sang. The protests were focused
                             in Mar del Plata, a beach town closest to the
                             offshore blocks, but spread to other cities
                             in the province and around the country. 


                      # ⚓ Lobbyist_Behind_The_Sun’s_Petrol_Car_Ban_Survey
                        Admits_He_Used_‘Loaded’_Question⠀⇛


                             A survey supposedly showing public opposition
                             to the government’s plans to phase out petrol
                             and diesel cars was carried out by a motoring
                             lobbyist who admits he “doesn’t care” that
                             the poll involved a “loaded” question.


                             The Sun reported on Saturday that “fuming
                             voters” oppose the UK’s 2030 ban on new
                             fossil fuel-powered cars, with three in five
                             people saying they would not vote for an MP
                             who supports it. 


                      # ⚓ Oil_Is_Killing_the_Planet_—_And_Driving_Inflation⠀⇛


                             Today’s inflation isn’t just caused by a
                             post-pandemic rebound in fuel prices, but a
                             long-term exhaustion of oil production. We
                             need to end our dependency on fossil fuels
                             without it becoming the pretext for another
                             wave of austerity.


                      # ⚓ The_Case_Against_[Cryptocurrency]⠀⇛


                             I’ve long been extremely skeptical of
                             [cryptocurrency]1, regardless of all the hype
                             surrounding it, and me being surrounded by
                             many people who are very passionate about it.
                             Seems there’s no shortage of believers in
                             [cryptocurrency] and blockchain in the tech
                             circles, and no shortage of people who are
                             drawn to “get rich quick” schemes. I’ve been
                             meaning to write a bit of thoughts on the
                             subject for a while now, but Martin O’Leary
                             saved me some work by writing a fantastic
                             article that perfectly reflects my own
                             sentiment. In a nutshell: [...]


                      # ⚓ Should_I_get_my_paycheck_in_crypto?_Experts_weigh_in
                        as_America_Competes_Act_looms⠀⇛


                             In December 2020, Russell Okung became the
                             first NFL player to be paid part of his
                             salary in Bitcoin. Other pro-athletes have
                             followed suit, with Aaron Rodgers announcing
                             in October 2021 that he would also be paid
                             partially in Bitcoin and Trevor Lawrence
                             placing his signing bonus in a cryptocurrency
                             account in April 2021.


                      # ⚓ Bitcoin_miners_pressed_on_climate_impact,_power
                        consumption⠀⇛


                             Cryptocurrencies have come under increasing
                             fire for the industry’s power consumption,
                             which is now comparable to the entire country
                             of Argentina.


                             The letters raise the stakes in Senator
                             Warren’s campaign to crack down on wasteful
                             Bitcoin operations after she sent a similar
                             request in December to Greenidge Generation
                             Holdings, which powers its upstate New York
                             facility with a natural gas plant.


                      # ⚓ We’re_in_a_Climate_Crisis_—_But_the_Pipelines_Keep
                        Coming⠀⇛


                             The Spire case is exceptional in a number of
                             ways, and yet experts say the project’s
                             problems exemplify everything that is wrong
                             with the natural gas pipeline approval
                             process.


                      # ⚓ Meta_abandons_its_cryptocurrency_venture,_Diem⠀⇛


                             The Facebook-backed digital currency project
                             Diem announced Monday the winding down and
                             $182-million sale of its technology, capping
                             a years-long initiative that drew significant
                             concern from regulators.


                      # ⚓ Facebook_Is_Reportedly_Trying_To_Sell_Off_Its_Botched
                        Cryptocurrency⠀⇛


                             Facebook made a huge splash in 2019 when it
                             announced Libra, a “stable digital
                             cryptocurrency” meant to revolutionize global
                             financial technologies. Even the likes of
                             Uber, Spotify, PayPal, Visa, and Mastercard
                             were on board, signing on as corporate
                             partners with a $10 million stake each.


                             But the ambitious venture imploded after
                             facing fierce backlash from US regulators and
                             even governments around the world.


                # § Overpopulation⠀➾


                      # ⚓ Anti-natalism_(and_other_problems)⠀⇛


                             My main annoyance is, and I’ve (anecdote
                             alert) encountered this time and time again,
                             is people who are like “I don’t need to eat
                             plants or cut down on air travel because I’m
                             not gonna have kids”… and then they do end up
                             having kids.


                             It’s such a science-fiction, implausible,
                             scapegoat of a solution.


          o § Finance⠀➾


                # ⚓ Government_Contracting:_the_Next_Big_Battleground_for_a
                  More_Equal_America?⠀⇛


                # ⚓ Opinion_|_No_Quick_Fixes:_Working_Class_Politics_From_Jim
                  Crow_to_the_Present⠀⇛


                       With a new book out Tuesday reflecting on his years
                       growing up within—and moving through—the racist
                       apartheid order that loomed over the U.S. South in
                       the 1950s and 60s during his upbringing and early
                       adulthood, political scientist Adolph Reed Jr.
                       offers his unique perspective on the Jim Crow era,
                       but please do not call it a memoir.


                # ⚓ A_Wall_Street_Veteran_Speaks_Out_–_Bubbles_&_the_Planet⠀⇛


                       Nowadays, a lot of the air has come out of hot
                       stocks of the covid era, losing 30%-40%-50%, or
                       more, in only a couple of months, e.g. Netflix
                       falling from $700 down to $390/share within 3
                       weeks. That’s a lot of hot air wheezing out
                       quickly. As Joe Granville, market technician 1923-
                       2013, famously said: They (investors) are “bag-
                       holders.”


                       The title of the Bloomberg Grantham article refers
                       to more than overvalued stocks, wherein he claims a
                       Goldilocks period over “the past 25 years is
                       ending, and the world needs to prepare for a future
                       of inflation, slower growth, and labor shortages,”
                       as stated in the Bloomberg Front Row interview.


                # ⚓ More_Thoughts_on_the_Great_Inflation_Debate⠀⇛


                       There are three main components to the Summers-
                       Furman (SF) argument. (Their arguments are not
                       identical, so I’m being a bit unfair to both in
                       trying to mash them together.)  The first is that
                       the Biden administration provided excessive
                       stimulus to the economy with the American Recovery
                       Act (ARA) passed by Congress last February. They
                       argue that demand for goods and services far
                       exceeded the economy’s ability to supply them,
                       leading to a sharp uptick in the rate of inflation.


                       Second, the SF position is that this jump in
                       inflation has unmoored inflationary expectations.
                       While households and businesses had long come to
                       expect low and stable inflation, the surge in
                       inflation we saw in the last year has changed
                       people’s expectations. Just to be clear, this is
                       more Summers’s concern than Furman’s. Also, he
                       views this as a serious risk, but not a necessary
                       outcome from the current situation.


                # ⚓ We_Can’t_Let_Build_Back_Better_Be_Replaced_With_Manchin’s
                  Wish_List⠀⇛


                # ⚓ Poor_People’s_Campaign_Denounces_Dems’_Push_for_Means-
                  Tested_Child_Tax_Credit⠀⇛


                       Leaders of the Poor People’s Campaign on Tuesday
                       blasted Sen. Joe Manchin’s proposal to means-test a
                       child tax credit that expired because the Senate
                       hasn’t passed the House-approved Build Back Better
                       Act—largely due to opposition from the West
                       Virginia Democrat.


                       “Sen. Manchin and others want to do away with it
                       now—not because it didn’t work, but because it
                       did.”


                # ⚓ Inflation_Policies_Must_Deal_With_Impact_of_Rising_Food
                  Prices_on_the_Poor⠀⇛


                # ⚓ Amid_Push_for_Ban,_Lawmakers_Traded_$355_Million_of_Stock
                  in_2021⠀⇛


                       As the coronavirus pandemic continued to roil the
                       U.S. economy and take lives in 2021, congressional
                       lawmakers and their family members went on a stock
                       buying and selling spree, trading an estimated $355
                       million worth of shares in Facebook, Apple, and
                       other prominent corporations.


                       According to a new analysis of financial
                       disclosures by Capitol Trades and reporting from
                       MarketWatch, at least 113 members of Congress have
                       divulged stock transactions completed by themselves
                       or members of their families in 2021.


                # ⚓ ‘Puerto_Rico_Hasn’t_Had_the_Opportunity_to_Develop_Its_Own
                  Economic_Future’⠀⇛


                       Janine Jackson interviewed the Center for Popular
                       Democracy’s Natalia Renta about the Puerto Rican
                       debt deal for the January 28, 2022, episode of
                       CounterSpin. This is a lightly edited transcript.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Nature_on_the_Political_Chopping_Block⠀⇛


                # ⚓ New_5-Minute_Video_Summarizes_Joe_Manchin’s_‘Brazen’
                  Corruption⠀⇛


                       Although right-wing Sen. Joe Manchin’s financial
                       conflicts of interest have been well-documented, a
                       new video released Monday details how the West
                       Virginia Democrat’s “brazen” corruption has
                       derailed his party’s immensely popular economic
                       policies.


                       Not only did Manchin take more than $1 million from
                       corporate-tied PACs last year as he watered down
                       and obstructed the Build Back Better Act (BBB), but
                       he “repeatedly timed his key attacks on [President
                       Joe] Biden’s agenda to occur at events with his
                       largest corporate donors,” according to pro-worker
                       media group More Perfect Union, which summarizes
                       its research in the following five-minute video.


                # ⚓ Pro-government_media_outlets_receive_preference_at
                  government_briefings⠀⇛


                       Regardless of viewership, popularity, or
                       readership, the government gives preference to pro-
                       government journalists and their media outlets
                       during its regular press conferences. Critical
                       media outlets, such as Telex, are left standing at
                       the back of the line. This became clear when we
                       carefully looked through all the government
                       briefings in 2021 and noted the order in which
                       journalists present were addressed by the
                       politicians leading the events. Translation by
                       Dominic Spadacene


                # ⚓ Big_tech_should_reimburse_victims_of_online_scams:_British
                  lawmakers⠀⇛


                       Big tech companies whose online platforms carry
                       advertisements for scams should be made to
                       reimburse victims, British lawmakers said, as part
                       of wider efforts to combat a growing epidemic of
                       online fraud in Britain.


                       While banks have signed up for a voluntary code to
                       reimburse fraud victims who do enough to protect
                       themselves, there is not sufficient regulation
                       governing social media and other websites where
                       victims are often first lured in, Mel Stride,
                       chairman of the cross-party Treasury committee,
                       told Reuters


                # ⚓ Twitter’s_algorithm_favours_the_political_right,_a_recent
                  study_finds⠀⇛


                       In this climate, Twitter commissioned a study to
                       understand whether their algorithm may be biased
                       towards a certain political ideology. While Twitter
                       publicised the findings of the research in 2021,
                       the study has now been published in the peer-
                       reviewed journal PNAS.


                # ⚓ Nord_Stream_2:_Russia-Germany_gas_pipeline_becomes_a
                  geopolitical_lever⠀⇛


                       The crisis surrounding Ukraine has been a harsh
                       reminder to Europeans of just how dependent they
                       are on Russian energy supplies. While the European
                       Union weighs its options for a united and robust
                       response to Russia if Vladimir Putin decides to
                       invade Ukraine, the bloc is feeling a new sense of
                       unease over its dependence on Russian oil and gas.


                # ⚓ Nord_Stream_2:_The_gas_pipeline’s_second_power_struggle⠀⇛


                       Germany is almost totally reliant on natural gas
                       imports, with Russia accounting for more than half
                       of supplies in 2020, according to IHS Markit.


                       Europe’s No.1 economy needs to wean itself off coal
                       and nuclear as part of its energy transition, and
                       wants to use natural gas as a bridge until it can
                       build or import enough renewable energy.


          o § Misinformation/Disinformation⠀➾


                # ⚓ Neil_Young,_Joe_Rogan,_and_a_Swedish_Billionaire⠀⇛


                       I think part of the reason why I feel like I’m not
                       communicating well when I write about Spotify is I
                       focus so much on the negative aspects of the
                       operation.  Another reason is I tend to forget that
                       just because it is by far the most popular music
                       streaming platform on the planet, what this means
                       in practice is that there are hundreds of millions
                       of mostly young people around the world for whom
                       listening to music means opening Spotify, but then
                       there are billions of other people around the
                       world, largely middle-aged or older folks, who have
                       never used a music streaming app of any kind.


                       So I’d just like to put it out there now at the
                       outset that I completely understand why Spotify is
                       so popular, and I use the platform regularly,
                       despite all my complaints about the way the company
                       is run.  I am also a regular user of so many other
                       platforms that are run by other very large
                       corporations with predatory corporate practices,
                       including but not limited to Facebook, Google,
                       Uber, and Airbnb.


                # ⚓ Joe_Rogan_row_puts_cost_of_Spotify_podcasts_under_investor
                  microscope⠀⇛


                       Spotify investors will focus on how much its
                       ‘Netflix for audio’ strategy is costing the
                       streaming service when it reports fourth-quarter
                       results on Wednesday against the backdrop of
                       several angry high-profile artists withdrawing
                       their tracks.


                       Over the past four years, Spotify has spent more
                       than $1 billion on podcasts such as Joe Rogan’s in
                       an attempt to triumph over rival music subscription
                       services from Apple and Amazon.


                # ⚓ The_Joe_Rogan_controversy_is_what_happens_when_you_put
                  podcasts_behind_a_wall⠀⇛


                       We’re moving away from a world in which a podcast
                       player functions as a search engine and toward one
                       in which they act as creators and publishers of
                       that content. This means more backlash and room for
                       questions like: why are you paying Rogan $100
                       million to distribute what many consider to be
                       harmful information? Fair questions!


                       This is the cost of high-profile deals and attempts
                       to expand podcasting’s revenue. Both creators and
                       platforms are implicated in whatever content’s
                       distributed, hosted, and sold, and both need to
                       think clearly about how they’ll handle inevitable
                       controversy.


                # ⚓ Joe_Rogan_Issues_Absolutely_Terrible_Apology_For_Covid
                  Misinfo⠀⇛


                       He posted a video on his Instagram on Sunday
                       wherein he addressed the criticism that his show
                       has spread COVID misinformation. While he was quick
                       to offer a full apology to Spotify, he stopped
                       short of a complete mea culpa to Young and
                       Mitchell, nevermind the public to whom he’s been
                       spreading extremely dubious fear about vaccines.


                # ⚓ Oh_Spotify…⠀⇛


                       If you’ve been living under a rock for the last few
                       days, you wouldn’t have heard about the shit show
                       that is Joe Rogan sharing COVID misinformation on
                       his podcast and Spotify doing nothing about it.


                       Needless to say, this has caused uproar all over
                       the Internet and it seems that people are
                       boycotting the platform, with Neil Young being the
                       most vocal on the matter.


                # ⚓ Spotify’s_Joe_Rogan_Problem_Isn’t_Going_Away⠀⇛


                       Mr. Rogan could double down on Covid-19
                       misinformation, daring Spotify to de-platform him
                       and casting himself as a “victim of the woke mob,”
                       censored for speaking too many uncomfortable
                       truths. He could wriggle out of his Spotify deal
                       and head back to YouTube and to the other platforms
                       that used to carry his show. (He could even go to a
                       right-wing social network like Gettr or Parler, but
                       I’m guessing he’d prefer an audience.)


                # ⚓ Spotify_Isn’t_Really_About_the_Music_Anymore⠀⇛


                       The company’s deal with Rogan was part of a larger
                       strategic shift whose implications listeners may
                       not fully understand. Since launching in 2008,
                       Spotify has transformed the music world by helping
                       make on-demand streaming a reality for millions of
                       listeners and rescuing the industry’s coffers from
                       a years-long decline. But Spotify pays most of its
                       revenues from songs back to labels and artists and
                       has rarely turned a profit. In 2019, the company
                       announced a new focus on “audio,” meaning recorded
                       books, live chats, and the booming medium of
                       podcasts. Spotify began paying out millions in
                       exclusive deals with such creators as Rogan, the
                       Obamas, Bill Simmons’s Ringer network, and Prince
                       Harry and Meghan Markle.


                # ⚓ Spotify_is_more_than_just_a_bystander_to_Joe_Rogan’s
                  misinformation⠀⇛


                       In response to the growing boycott fronted by Young
                       and Mitchell, Spotify has promised to amend its
                       protocols, applying a “content advisory” to podcast
                       episodes that discuss covid-19, which the company
                       says will link to reliable sources about the
                       pandemic. It’s a move straight from the social
                       media company playbook, reminiscent of Facebook and
                       Twitter’s efforts to rein in misinformation that
                       could cause real-life harm to its users. It’s a
                       response that might befit Spotify if an unknown
                       podcaster uploaded an episode exclaiming the
                       benefits of ivermectin, the horse-deworming drug
                       that Rogan has falsely claimed helps treat covid-
                       19.


                       But Rogan isn’t an unknown podcaster uploading to a
                       platform, getting paid fractions of a cent per
                       stream. Rogan is on Spotify’s payroll to the tune
                       of $100 million.


                # ⚓ Insult_atop_injury_to_Syrian_photographer_as_China_diplomat
                  misuses_war_photos⠀⇛


                       Lijian, seeking a correction and apology for the
                       use without permission of his work and the
                       misrepresentation of the images. Zhao’s Jan. 24
                       tweet said: “This is 20 years of war, America’s
                       consequences for children in ‘Afghanistan’.”
                       Suleiman noticed it on Jan. 27.


                       “These large and small shells are the Syrian Assad
                       regime supported by Russia…the legacy of attacks
                       against Syrian civilians and children,” wrote
                       Suleiman, who at 23 has spent half his life with
                       his country embroiled in a brutal civil war.


                       “He did not contact me and did not apologize after
                       deleting the tweet,” Suleiman told RFA’s Mandarin
                       Service in an interview by text, translated from
                       Arabic to English.


                # ⚓ Spotify_CEO_Daniel_Ek_Officially_Responds_to_Neil_Young’s
                  Pullout_—_Here’s_His_Complete_Statement⠀⇛


                # ⚓ Spotify’s_Joe_Rogan_Mess_Exposes_Larger_Rift_Over_Rules,
                  Streaming_Revenues⠀⇛


                       But as Spotify is under the spotlight for its
                       handling of Rogan — who released an episode in
                       December featuring a vaccine skeptic who baselessly
                       claimed people were being “hypnotized” to believe
                       facts about COVID, sparking 270 medical pros to
                       write an open letter in protest — the rift is
                       exposing issues that artists and content creators
                       have long been frustrated about. Even Young, who
                       left “because Spotify is spreading fake information
                       about vaccines,” added elsewhere in his exit letter
                       on his website that Spotify “continues to peddle
                       the lowest quality in music reproduction.”


                # ⚓ Graham_Nash_and_India_Arie_join_Neil_Young’s_Spotify
                  boycott_over_‘problematic’_Rogan⠀⇛


                       Nash declared his support for his Crosby, Stills,
                       Nash & Young bandmate Tuesday after “having heard
                       the COVID disinformation spread by Joe Rogan on
                       Spotify.” Young previously pulled much of his music
                       library from the app after Spotify failed to cut
                       ties with Rogan, a controversial podcast host known
                       for fueling conspiracy theories about COVID-19
                       treatments and vaccines.


                       “I am requesting that my solo recordings be removed
                       from the service,” Nash said in a statement. “There
                       is a difference between being open to varying
                       viewpoints on a matter and knowingly spreading
                       false information which some 270 medical
                       professionals have derided as not only false but
                       dangerous.


                # ⚓ Fact_check:_WikiLeaks_did_not_release_footage_that_proves
                  moon_landing_staged⠀⇛


                       The claim was fact-checked by Reuters and found to
                       be false.


          o § Censorship/Free Speech⠀➾


                # ⚓ As_the_Right_Censors_Public_Libraries,_Families_Are_Forming
                  Banned_Book_Clubs⠀⇛


                # ⚓ Eleanor_Goldfield_and_Nolan_Higdon_Return_to_the_Program_–
                  The_Project_Censored_Show⠀⇛


                       Eleanor Goldfield is a creative radical,
                       journalist, artist, and organizer. She is co-host
                       of the Common Censored podcast with Lee Camp. She
                       also produced a documentary, Hard Road of Hope,
                       about West Virginia communities confronting
                       pollution from fracking. The Craig Murray article
                       she mentions (about Julian Assange) can be found
                       here.


                # ⚓ YouTube_Dusts_Off_Granular_National_Video_Blocking_To
                  Assist_YouTuber_Feuding_With_Toei_Animation⠀⇛


                       Hopefully, you will recall our discussion about one
                       YouTuber, Totally Not Mark, suddenly getting
                       flooded with 150 copyright claims on his YouTube
                       channel all at once from Toei Animation. Mark’s
                       channel is essentially a series of videos that
                       discuss, critique, and review anime. Toei Animation
                       produces anime, including the popular Dragon Ball
                       series. While notable YouTuber PewDiePie weighed in
                       with some heavy criticism over how YouTube protects
                       its community in general from copyright claims, the
                       real problem here was one of location. Matt is in
                       Ireland, while Toei Animation is based out of
                       Japan. Japan has terrible copyright laws when it
                       comes to anything resembling fair use, whereas
                       Ireland is governed by fair dealing laws. In other
                       words, Matt’s use was just fine in Ireland, where
                       he lives, but would not be permitted in Japan.
                       Since YouTube is a global site, takedowns have
                       traditionally been global.


                # ⚓ Cal_State_Poly,_Pomona_Professor_Accused_of_Anti-Semitism_–
                  Censored_Notebook⠀⇛


                       Jaime Scholnick, a Los Angeles-based artist and
                       adjunct professor at California State Polytechnic
                       University, Pomona is under pressure from
                       university administration as a result of unfounded
                       accusations of “anti-Semitism” by a former student.


                # ⚓ How_to_Get_Teenagers_to_Read_Important_Books?_Ban_Them.⠀⇛


                       The real reason I wanted to read The Catcher in the
                       Rye was it had been banned from the library. I knew
                       the librarian kept one copy behind her desk, and I
                       was determined to get it. She reluctantly handed it
                       to me. I read it voraciously.


                # ⚓ Swedish_publishers_demand_China_free_jailed_bookseller_Gui
                  Minhai⠀⇛


                       A group of Swedish publishers on Monday demanded
                       Beijing release Chinese-Swedish bookseller Gui
                       Minhai, who was sentenced to 10 years in prison in
                       2020 on charges of illegally providing intelligence
                       abroad.


                       Gui, one of five Hong Kong-based booksellers known
                       for publishing salacious titles about China’s
                       political leaders, has been at the centre of
                       diplomatic tensions between Stockholm and Beijing
                       for more than six years.


                # ⚓ Controversial_House_Bill_1134_would_limit_classroom
                  curriculum⠀⇛


                       “Contrary to some social media post, HB1134 was NOT
                       adopted into law. It simply passed out of the House
                       after multiple amendments (some comments have
                       reflected the original draft). It now moves over to
                       the Senate for further consideration, amendment, or
                       to die. I appreciate the passion shown around this
                       legislation and welcome the attention and debate
                       it’s drawn. Bullet points for HB1134 as it stands
                       today: [...]


                # ⚓ University_of_Hong_Kong_covers_up_33-year-old_Tiananmen
                  tribute_painted_on_bridge,_weeks_after_removing_statue⠀⇛


                       Residents at Swire Hall painted the slogan on the
                       bridge after the Tiananmen crackdown in 1989. A
                       bloody military intervention ended months of
                       student-led demonstrations on June 4 that year. It
                       is estimated that hundreds, perhaps thousands, died
                       in Beijing when the People’s Liberation Army
                       cleared the protests.


                       Every year since then, HKU students have retouched
                       the slogan leading up to the anniversary. The 20-
                       character inscription reads: “Souls of martyrs
                       shall forever linger despite the brutal massacre;
                       Spark of democracy shall forever glow for the
                       demise of evils.”


                # ⚓ Chinese_censors_give_Fight_Club_movie_a_new_ending_where
                  police_win⠀⇛


                       The first rule of Fight Club in China? Don’t
                       mention the original ending. The second rule of
                       Fight Club in China? Change it so the police win.


          o § Freedom of Information/Freedom of the Press⠀➾


                # ⚓ Pirates_nominate_whistleblower_Julian_Assange_for_the_Nobel
                  Peace_Prize⠀⇛


                       The Pirates in the European Parliament have
                       submitted the nomination of Julian Assange for the
                       Nobel Peace Prize to the Norwegian Nobel Committee.
                       For the Pirates, the case of Assange, who is
                       currently imprisoned in the United Kingdom, is a
                       symbol for the oppression of freedom of speech and
                       the public’s right to information


                # ⚓ Foreign_journalists_in_China_face_lawsuits,_intimidation,
                  state-backed_trolling_attacks,_media_group_says⠀⇛


                       Threats of legal action, online troll campaigns and
                       dwindling numbers after the expulsion of colleagues
                       — foreign journalists in China are facing
                       “unprecedented hurdles” from efforts to discredit
                       independent reporting, a press group said Monday.


                       Beijing appears to be “encouraging a spate of
                       lawsuits”, or the threat of legal action against
                       foreign journalists, often filed long after sources
                       agreed to interviews, the Foreign Correspondents’
                       Club of China (FCCC) said in its annual report.


                # ⚓ Apply_For_The_2022_Marvel_Cooke_Fellowship_For_Abolition
                  Journalism⠀⇛


                       Shadowproof is proud to welcome applications for
                       the 2022 Marvel Cooke Fellowship. 


                       The fellowship’s mission is to fund reporting from
                       writers of color on abolition movements around the
                       world. We are specifically looking for journalists
                       who have a history of engagement and experience
                       with abolition, and who produce journalism with an
                       abolitionist ethic.


                # ⚓ Criminalising_journalists_for_telling_the_truth_is_in
                  nobody’s_interest⠀⇛


                       State manipulation of the media is why journalism
                       has developed beyond the mainstream print and
                       broadcast worlds, which seem to have lost the
                       ability to challenge authority and embarrass
                       governments and big business. Today, though, there
                       is a new breed of journalist operating in the realm
                       of social media; much to the mainstream media’s
                       annoyance, we are seeing the continued growth of a
                       new and very valid kind of journalism, accompanied
                       by a vicious backlash to discredit the new media
                       kids on the block.


          o § Civil Rights/Policing⠀➾


                # ⚓ Opinion_|_The_Absurdity_of_Protecting_Kids_From_the
                  Holocaust_Narrative_‘Maus’⠀⇛


                       Nudity and bad language.  That’s a Tennessee school
                       board’s excuse for banning Art Spiegelman’s graphic
                       novel “Maus.”  Not Holocaust denial.  Not
                       antisemitism.


                # ⚓ ‘Shame_on_Them’:_DOJ_Will_Not_Reopen_Tamir_Rice_Case⠀⇛


                       The mother of Tamir Rice, who was shot to death at
                       age 12 by a Cleveland, Ohio police officer,
                       condemned the U.S. Department of Justice’s decision
                       not to reopen her son’s case.


                       “Shame on them,” Samaria Rice told Buzzfeed News
                       Monday after receiving a letter from the DOJ
                       regarding the Biden administration’s decision. “I
                       think they’re pitiful and pathetic, and at this
                       point no one is going to get justice when it comes
                       to police shootings in America. It’s disgusting I
                       don’t have an indictment for my 12-year-old son.”


                # ⚓ Anti-BDS_Law_in_Texas_Violates_Free_Speech_Rights,_Federal
                  Judge_Rules⠀⇛


                       Progressives welcomed a federal judge’s recent
                       ruling that Texas cannot forbid an engineering firm
                       hired by Houston city officials from boycotting
                       Israel to protest its subjugation of
                       Palestinians—and vowed to keep fighting until the
                       state’s anti-boycott law is thrown out completely.


                       As The Texas Tribune reported Monday:


                # ⚓ Opinion_|_US_Supreme_Court_Nomination_Debates_Are
                  Dangerous⠀⇛


                       Justice Breyer has announced his resignation from
                       the Supreme Court and a Biden/Harris pick is
                       expected to win confirmation. The Democrats’ choice
                       is unlikely to shift the court’s balance, we’re
                       told, but the media’s milquetoast reassurance
                       misses the point that when it comes to balance
                        it’s not just the court, it’s also the public
                       debate that’s shifted dramatically to the right.
                       And nomination fights have a nasty habit of playing
                       a big role in that.


                # ⚓ Progressive_Urge_Biden_to_Nominate_Justice_on_Side_of
                  Workers,_Not_Corporations⠀⇛


                       Following a number of U.S. Supreme Court rulings
                       favoring powerful corporations, progressives are
                       demanding that President Joe Biden nominate a
                       successor to Justice Stephen Breyer who will
                       represent the interests of working people,
                       continuing the trend he began with his nominations
                       to lower federal courts.


                       The one potential nominee the White House has
                       confirmed is on the president’s shortlist—U.S.
                       District Judge J. Michelle Childs—is not reassuring
                       labor advocates that their call will be answered.


                # ⚓ Opinion_|_The_Moral_Panic_of_Critical_Race_Theory_Is_a
                  Right-Wing_Tool⠀⇛


                       The moral panic currently sweeping America about
                       Critical Race Theory (CRT) has been covered ad
                       nauseum by the press and commentators across the
                       political spectrum.  That’s what typically happens
                       with moral panics (more on that in a moment). 


                # ⚓ Mayoral_Candidate_Faces_Backlash_After_Saying_She_Wants_to
                  “Eradicate”_Islam⠀⇛


                # ⚓ The_Hypocrisy_of_American_Islamophobia⠀⇛


                       Fox News anchor Tucker Carlson excused one of the
                       leaders of the extremist Oath Keepers organization
                       implicated in the January 6 insurrection by
                       describing him as “a devout Christian.” It’s safe
                       to surmise that he wouldn’t have offered a similar
                       defense for a Muslim American. Since September 11,
                       and even before that ominous date, they have
                       suffered bitterly from discrimination and hate
                       crimes in this country, while their religion has
                       been demonized. During the first year of the Trump
                       administration, about half of Muslim Americans
                       polled said that they had personally experienced
                       some type of discrimination.


                # ⚓ Massachusetts_Court_Says_Breathaylzers_Are_A-OK_Less_Than
                  Three_Months_After_Declaring_Them_Hot_Garbage⠀⇛


                       Breathalyzers are like drug dogs and field tests:
                       they are considered infallible right up until
                       they’re challenged in court. Once challenged, the
                       evidence seems to indicate all of the above are
                       basically coin tosses the government always claims
                       to win. Good enough for a search or an arrest when
                       only examined by an interested outsider who’s been
                       subjected to warrantless searches and possibly
                       bogus criminal charges. But when the evidentiary
                       standard is a little more rigorous than roadside
                       stops, probable cause assertions seem to start
                       falling apart.


                # ⚓ Tibetan_former_political_prisoner_in_failing_health_still
                  watched_by_police⠀⇛


                       “He is succumbing to weak health due to the
                       beatings he received in prison, and is now in
                       critical condition with both his legs crippled,”
                       RFA’s source said, citing contacts in the Draggo
                       area and speaking on condition of anonymity to
                       protect his sources.


                # ⚓ ‘Low-Skill’_Workers_Don’t_Exist,_But_Low-Wage_Workers_Do⠀⇛


                       On day four of his tenure as mayor of New York
                       City, Eric Adams made a gaffe that drew the ire of
                       many who know the lie of “low-skill” labor.
                       Advocating for the return of workers to their
                       midtown offices, he said, “My low-skill workers, my
                       cooks, my dishwashers, my messengers, my shoe-shine
                       people, those who work at Dunkin’ Donuts — they
                       don’t have the academic skills to sit in a corner
                       office.” His poor choice of words drew backlash
                       from across the internet (myself included). He
                       rephrased his poor choice of words soon after, and
                       Twitter moved on.


                       But the harm perpetuated by the myth of low-skill
                       labor lingers. It’s an old lie, one that has
                       implications much bigger than what Mayor Adams was
                       alluding to. Historically, enforcing the idea that
                       a worker is low-skilled has proved to be an
                       excellent way to justify suppressing their wages.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ Techdirt_Podcast_Episode_309:_Remembering_The_SOPA_Fight,
                  With_Rep._Zoe_Lofgren⠀⇛


                       As many of you know, last week we hosted an online
                       event for the latest Techdirt Greenhouse edition,
                       all about looking back on the lessons learned from
                       the 2012 protests against SOPA and PIPA. Our
                       special guest was Rep. Zoe Lofgren, one of the
                       strongest voices in congress speaking out against
                       the disastrous bills, who provided all kinds of
                       excellent insight into what happened then and
                       what’s happening now. In case you missed it, for
                       this week’s episode of the podcast (yes, we’re
                       finally back with new episodes!) we’ve got the full
                       conversation and Q&A from the event.


                # ⚓ Senate’s_New_EARN_IT_Bill_Will_Make_Child_Exploitation
                  Problem_Worse,_Not_Better,_And_Still_Attacks_Encryption⠀⇛


                       You may recall the terrible and dangerous EARN IT
                       Act from two years ago, which was a push by
                       Senators Richard Blumenthal and Lindsey Graham to
                       chip away more at Section 230 and to blame tech
                       companies for child sexual abuse material (CSAM).
                       When it was initially introduced, many people
                       noticed that it would undermine both encryption and
                       Section 230 in a single bill. While the supporters
                       of the bill insisted that it wouldn’t undermine
                       encryption, the nature of the bill clearly set
                       things up so that you either needed to encrypt
                       everything or to spy on everything. Eventually, the
                       Senators were persuaded to adopt an amendment from
                       Senator Patrick Leahy to more explicitly attempt to
                       exempt encryption from the bill, but it was done in
                       a pretty weak manner. That said, the bill still
                       died.


                # ⚓ Senators’_‘Myths_&_Facts’_About_EARN_IT_Is_Mostly_Myths,
                  Not_Facts⠀⇛


                       I already wrote a long post earlier about the very
                       very real problems with the EARN IT Act — namely
                       that it would make the problem of child sexual
                       abuse material significantly worse by repeating the
                       failed FOSTA playbook, and that it would attack
                       encryption by making it potential “evidence” in a
                       case against a tech company for any CSAM on its
                       site. But with the bill, the sponsors of the bill,
                       Senators Richard Blumenthal and Lindsey Graham,
                       released a “Myth v. Fact” document to try to
                       counter the criticisms of EARN IT. Unfortunately,
                       the document presents an awful lot of “myths” as
                       “facts.” And that’s a real problem.


          o § Monopolies⠀➾


                # ⚓ Google_parent_company_Alphabet_broke_$200_billion_in_annual
                  revenue_for_the_first_time⠀⇛


                       Google parent company Alphabet hit a new record for
                       annual revenue in 2021, showing no ill effects from
                       the lingering coronavirus pandemic or ongoing
                       issues with the global supply chain. For full-year
                       2021, the company saw a 41 percent year-over-year
                       jump in revenue to $257 billion. The company
                       reported revenue of $75.3 billion in the fourth
                       quarter of 2021, up 32 percent from the year
                       earlier.


                # § Patents⠀➾


                      # § Software Patents⠀➾


                            # ⚓ Podcast_Episode:_Saving_Podcasts_from_a_Patent
                              Troll⠀⇛


                                   That’s exactly the situation faced by a
                                   group of podcasters several years ago,
                                   including comedian Marc Maron, who
                                   hosts the WTF podcast. They were being
                                   threatened by a patent troll called
                                   Personal Audio, which claimed that
                                   podcasters were infringing on a patent
                                   that covered “disseminating media
                                   content in a serialized sequence.”


                                   On this episode of How to Fix the
                                   Internet, Marc Maron and his producer
                                   Brendan McDonald join EFF’s Cindy Cohn
                                   and Danny O’Brien to talk about how
                                   they decided not to give in, and how
                                   podcasters came together to support
                                   EFF’s work to defeat the patent.


                                   Click below to listen to the episode
                                   now, or choose your podcast player:


                # § Copyrights⠀➾


                      # ⚓ Anime_YouTuber_Battle_Against_Copyright_Results_In
                        New_YouTube_Rule⠀⇛


                             YouTube decided not to honor Toei’s removal
                             request because it would violate the
                             platform’s fair use policy. Instead, YouTube
                             asked Toei to provide further justification
                             for why Fitzpatrick’s channel should be
                             removed. Rather than provide that evidence,
                             Toei used YouTube’s automated claim system to
                             have the videos blocked.


                      # ⚓ U.S._Seeks_Significant_Prison_Sentence_for_SPARKS
                        Member_to_Deter_Other_Pirates⠀⇛


                             Later this month, a 52-year old British man
                             will be sentenced by a New York federal court
                             for his role in the SPARKS piracy Scene
                             group. Mr. Bridi, who was extradited to the
                             US from Cyprus previously pleaded guilty.
                             While the defense argues that a reduced
                             sentence is warranted, the U.S. Government is
                             asking the court to award a significant 27 to
                             33-month term to deter other pirates.


                      # ⚓ Anti-Piracy_Group_Received_$290K_in_Settlements_from
                        Usenet_Pirates_in_2021⠀⇛


                             Dutch anti-piracy group BREIN has revealed
                             that during 2021, it took down five Usenet
                             indexing platforms and approached 38
                             uploaders of content for settlement. Overall,
                             BREIN collected cash payments of $290K from
                             pirates, with settlements presented as an
                             alternative to protracted and expensive legal
                             battles.


                      # ⚓ Episode_1:_Open_Culture_VOICES_–_Medhavi_Gandhi⠀⇛


                             We are excited to share the first episode of
                             Open Culture VOICES, a vlog series of short
                             interviews with open GLAM (galleries,
                             libraries, archives, and museums) experts
                             from around the world. The Open Culture
                             Program at Creative Commons aims to promote
                             better sharing of cultural heritage in GLAMs
                             collections. With Open Culture VOICES, we’re
                             thrilled to bring you various perspectives
                             from dozens of experts speaking in many
                             different languages on what it’s like to open
                             up heritage content online. In this
                             interview, we hear from Medhavi Gandhi,
                             founder of The Heritage Lab, India’s first
                             digital platform that connects museums and
                             citizens through engaging campaigns, content
                             and resources.


                      # ⚓ That’s_A_Wrap_On_The_Public_Domain_Game_Jam!_Check
                        Out_All_This_Year’s_Great_Entries⠀⇛


                             Last night at midnight, we reached the end of
                             Gaming Like It’s 1926, our fourth annual
                             public domain game jam celebrating the new
                             works that entered the public domain this
                             year. At final count, we got 31 entries
                             representing a huge variety of different
                             kinds of digital and analog games!


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