⡍⠈⠁⠉⠁⠉⠈⠁⠉⠈⠁⠉⠈⠁⠉⠈⠁⠉⠈⠁⢙
⡁⠄⠄⠄⠄⠄⠄⠄⠄⣠⣤⣤⡀⠄⠄⠄⠄⠄⠄⠄⠸
⠃⠄⠄⠄⠄⠄⠄⠄⣰⣿⠋⢽⣷⠄⠄⠄⠄⠄⠄⠄⢸
⠇⠄⠄⠄⠄⠄⠄⢠⣿⡇⣬⡌⢿⣧⠄⠄⠄⠄⠄⠄⢰
⡆⠄⠄⠄⠄⠄⢀⣾⡟⢸⣿⣿⡌⣷⡄⠄⠄⠄⠄⠄⢨
⡄⠄⠄⠄⠄⠄⣼⡿⢱⣿⣆⣻⣧⢹⣿⡀⠄⠄⠄⠄⢘
⡁⠄⠄⠄⠄⣰⣿⢃⣜⣩⣋⣝⣉⣆⢻⣷⠄⠄⠄⠄⠸
⠃⠄⠄⠄⢠⣿⠏⣾⡟⠛⠛⠛⠛⣿⡌⣿⣇⠄⠄⠄⢸
⠇⠄⠄⠄⠘⣿⣾⡿⠁⠄⠄⠄⠄⠹⣿⣾⡟⠄⠄⠄⢰
⡆⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⢨
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀:--.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.---=.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.-+===-.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.:=:+:+-.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀:--+=+=--⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.:--------:⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.----------...⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀..:---------------------:..⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀.:-----------------------------.⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀.-------==--------------.-------⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀..----=.:---------------=.----=.⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.:---=-----------------=---:.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.::-=--------=----------=:.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.:-=--====---=-===-==-+-:..⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀-+=::-+++++++++:::-+:.....⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.-+--:::::=++++:::::-++.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.:=+++*#*............##*+*+.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.-+***##.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀**+++*+.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.:+***#+.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.**++**+.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀.-+*****.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀:++*+*#-⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀.+**++*=.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀-*++*+#.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀-+++***.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.*++*#-⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀=++**#:⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀++***.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀+***#=.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.=***=.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀.+**#=⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.****.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀=**#=.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.****.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀.+*#+⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀:****:.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀:**#:⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀-***+-:...⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀=*#+:⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀-=====-:--=+:..⠀⠀⠀⠀⠀
⠀⠀⠀⠀.:+**+=...⠀⠀⠀⠀⠀⠀⠀⠀⠀.:=*+:.:--++*#+*##%%@@@@#⠀⠀
⠀⠀⠀⠀:==+%*++==+%%*=:.:-++*##%%%%%%@@@@@@#++=+*+-⠀⠀
⠀⠀⠀..:.:-===+**#%%%%%%%%@@@#@@@@@@@#+=+=+++=++++=:
⠀⠀⠀:=++*##*#*%%%%%%%%%%%%%%@@@#===-=+=-==-==+==++-
⠀⠀⠀...:-+---=++++**++*+=-:-=:------::===========-.
⠀⠀⠀⠀⠀⠀...⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀.⠀⠀⠀...................

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⣄⣤⣤⣤⣠⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠋⠙⠋⠙⠋⠉⠉⠛⠛⠛⠿⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡉⠿⣿⣷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⡀⢀⣠⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠻⣿⣿⣄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⣠⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣧⡀⢀⠀⢸⣿⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣿⣿⣿⣄⡀⠘⠢⣄⣀⣀⣀⠀⠈⣿⣿⡆⠀⠀⢹⣿⣧⠀⠀