Angles Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠏⠻⠛⠋⠙⠀⠛⠛⠀⠛⠛⠉⠛⠟⠛⠋⠙

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠒⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠘⠀⠊⠈⠈⠉⠘⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠌
⠀⠀⠀⠠⠤⠠⠠⠤⠠⠠⠄⠤⠤⠄⠄⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⣀⣀⣀⣀⣀⢀⣀⣀⣀⠀⢠⣀⣀⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⡄⣀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡐⠤⠤⠐⡈
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠐⠒⠒⠒⠂⠒⠂⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠈⠉

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⣠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢴⡲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡴⣤⠤⡀
⠀⠀⠀⠀⠠⢈⠘⡀⠸⠄⠈⡀⢠⡀⠀⠀⠀⠀⡴⠛⢄⣢⡀⠈⠐⢤⡀⠀⠀⢠⡶⠟⠂⠉⠉⠙⢳⣧⠤⠄⠀⠀⠀⠀⢀⣠⠾⢧⠀⣄⣘⣤⠛⢶
⠀⠀⠀⢀⡇⢹⢄⡁⢲⢿⣋⢀⡀⠱⡄⠀⠀⣜⣥⡀⠀⠫⣿⠴⠆⢀⡟⠀⠀⢀⠗⠒⡞⠖⠒⠓⢚⣅⡀⣰⠀⠀⠀⡞⣉⣤⡄⠈⡇⠻⢭⣠⡄⠈⣇
⠀⠀⠀⠈⣆⠈⠙⠳⠛⠛⡩⠟⠓⠆⠈⠀⠀⠀⢹⡇⠀⠀⠋⠀⠀⡟⠀⠀⠀⠈⠦⠤⠗⠳⠤⠦⣼⡿⠟⢰⠀⠀⠀⢷⡈⠹⣭⡴⠗⣳⠾⠛⠳⠆⢹⡄
⠀⠀⠀⠀⢸⠀⠀⠈⠃⠈⠀⠀⠀⠀⠰⠁⠀⠀⣸⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠐⡀⢀⣶⡄⠀⠈⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠃⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⡟⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠁⠈⠛⣦⡰⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠳⡆⠀⠀⠠⠋⡃⠀⠀⠀⠀⠀⠘⢷⠠⠴⠟⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠃

⠀⠀⠀⠀⡀
⠀⠀⠀⡞⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠂⠀⠀⠀⠀⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠻⠉⡀⠀⠀⠀⠲⡖⠁⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠺⣘
⠀⠀⠀⠀⠀⠀⡀⠼⢀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠘⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⣀⢁⣤⠃⠀⠀⠀⠀⠀⢰⠀⠀⢠⠀⠴⢄⠀⠀⠀⡀⠀⠀⠠⠀⡠⠂⡆⠀⠀⠀⠀⠱⠀⢀⠀⠀⠀⠀⠈⠑
⠀⡀⠀⠀⡈⠉⠲⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⡇⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠘⠀⠀⠃⠐⡒⢠⠴⠖⠀⠈⠀⠀⠠⠤⣀⣤
⠀⢈⡴⠊⠀⠀⠀⠀⠠⠉⠒⠄⢀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⡅⠀⡇⠀⡆⠀⠈⡄⠘⠂⠈
⠴⠊⠀⠀⢀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⡠⠘⠒⡢⠤⠀⠂⠁⠀⠀⢁⠄⠐⠒⠒⠴⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠

⡴⠉⠓
⡙⠚⠁
⠙⡄
⠀⠘⢆
⠀⠀⠈⢦
⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠙⣄⣤⠠⠤⠤⢄⠀⠀⠣⠼⠸⠽⠹⠸⠽⠾⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠐⢤⡀⠀⢠⡄⢀⣀⢀⣀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠱⡀⠛⠙⠘⠚⠺⠗⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⣀⣄⣀⣀⣈⣷⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠⡞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢂⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣾⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠘⣇⠀⠀⠈⡗⡏⠀⠀⢀⡟⠀⠀⠹⠃⠄⠀⠲⠒⠎⠀⠀⠻⠃⠀⠀⠗⢄⠀⠀⠖⠲⠀⠀⡠⠚
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠈

⠀⠐⠺⢍⣉⡉⠑⠂⠀⠀⠀⣀⠀⠀⢠⡤⣒⠀⠲⠄⠀⠀⠀⠀⣠⠀⠀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸⠋⠀⠉⢻⡶⠖⠛⠁⠀⠀⠸⠀⠀⠱⢲⠉⠉⠑⡖⠁⠀⠀⠀⠀⠀⠑⠲⠀⠀⠀⢠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠠⣄⣀⠀⢀⣉⡤⠤⣴⠊⠉⠀⠀⠐⠠⣀⣀⠤⠄⣁⣠⠔⠛⡖⠀⠀⠦⡀⠀⠀⠀⠀⠀⢀⠈⠑⠆
⠀⠀⠀⠉⠙⢿⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠎⠱⣀⡠⠃⠀⠀⠀⠈⠒⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠸⣿⣦⣠⣴⡶
⠀⠀⠀⠀⠀⠀⢀⣴⣶⢶⣴⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⢿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠻⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠛⠛⠛⣛⠛⠛⣛⢛⡛⠛⢛⠛⠛⠛⡛⠛⠛⡛⠛⠃⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠚⠚⠤⠺⠆⠘⠸⠅⠀⠞⠾⠳⠛⠴⠻⠜⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠫⠴⠖⠂⠀⠀⠸⠧⠟⠟⠭⠾⠄⠿
⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣼⣧⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⠀⠘⢟⣿⡿⠃⠄⢀⠀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⡀⠀⠀⠀⠉⢉⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⡆⠀⣾⣥⣄⣠⣤⣤⣶⣶⣤⣄⣁⣉⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡐⠄⠐⠶⠠⣭⣙⡿⠀⢀⣠⣤⣤⣶⠀⠔⠛⠁
⠀⠀⠀⠀⠀⠸⠇⠀⢻⣿⣿⠉⣴⠉⠻⣟⢿⣿⠃⣿⢿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⠦⣀⣄⡀⠹⠃⢰⣿⠿⠿⠛⠁⠀⣤⣀
⠀⠀⠀⠀⠀⡗⠀⡀⠀⢹⠿⡤⢻⡇⠀⠉⠉⠁⢀⣿⣷⣶⣾⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣄⡀⠈⠁⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⢹⣧⠤⠀⠀⠀⢲⡿⠁⠀⠀⠀⠀⡀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣴⡛⣿⠀⠀⠈⠻⢁
⠀⠀⠀⠀⠀⠘⣿⠶⠂⠈⠀⠀⠀⠀⢀⣀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠻⠟⠻⣿⣧⣀⡀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⢧⣀⣀⡤⠄⠀⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣻⠟
⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⢿⡋⣴⣿⣿⣆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠚⠃⠘⠿⡏⠁⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣴⣶⣄⠀⠀⠘⢀⣠⣤⣤⣤⡀⣿⣼
⠀⠀⠀⠀⠀⠸⠀⠀⣀⣀⣴⣦⡤⣀⢼⣷⣩⣿⣷⣦⡀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⣿⣿⣿⣿⡇⢀⡀⢸⣿⣿⣿⣿⣷⡙⡿
⠀⠀⠀⠀⠀⠀⠀⢶⣷⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡀⠀⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠂⠀⢈⣻⣿⠟⢰⣿⣷⡌⠻⡿⠋⠉⡟⠻⡀
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣧⠸⢿⡿⣿⣟⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⡇⠀⠀⠀⠉⠉⠁⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⢄⡀⠘⠛⠿⠿⠟⠀⢀⣀⣴⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠸⣇⠀⠀⠀⠀⠀⠀⠼⢁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣏⠹⠙⠅⠭⢷⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠰⠶⠳⢦⣯⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣁⣉⣋⣀⣍⣉⣭⣉⣭⣟⣽⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⣩⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⢠⣿⡄⢠⠤⡄⡄⣦⢰⡗⣴⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣦⠄⠀⠀⠀⠚⠐⠓⠈⠓⠂⠓⠛⠈⠛⠉⠛
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠈⢂⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢤
⠀⣿⣉⠙⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠠⡄⠀⢀⡯⣧⡀⣟⣹⣰⣯⡋⢸⡸⣟⡇⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶
⣠⣏⣈⣴⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠂⠉⠁⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀
⣭⣤⣤⣤⣤⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣶⣯
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡆⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⡇⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⢀⣤⣾⡧⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣠⢀⡘⠻⣿⣿⡗⢠⡛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢛⣭⢠⣝⡻⢸⣿⣷⣤⣙⠃⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣻⣵⢰⣬⣛⠘⣫⣴⢸⣿⡿⠛⣥⡆⢰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣷⣿⣿⣿⠸⣿⣿⣷⡦⠙⠈⢵⣾⢸⣮⠇⢸⣿⣿⣷⡄
⠰⣾⢸⣷⣤⣀⠀⠀⠀⠀⣀⣴⣾⡇⣿⣶⣝⡻⣿⢘⡿⢛⣡⣶⣿⠀⠀⠉⠈⢶⡇⢸⡿⠛⠡⢾⡇
⠀⠀⢸⣿⣿⣿⠗⣀⢀⡺⣿⣿⣿⡇⣿⣿⣿⡿⣂⣀⡐⢿⣿⣿⣿⠀⠀⠀⠀⢀⡀⣀
⠀⠀⢸⠟⡋⢠⣛⠿⢸⣿⣷⣽⡻⠇⠿⣫⠁⣌⠻⢿⣿⣷⣬⡛⠿⠀⢀⣠⣶⣿⡇⣿⣷⣦⣄
⠀⠀⠀⠙⠃⠸⣋⣥⢸⣿⣿⣿⣿⣷⣮⣛⠃⢟⣵⣼⣿⣿⠿⣛⣵⠀⠛⢿⣿⣿⡇⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠉⠛⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡝⠛⣫⣵⣿⣿⣿⠀⠀⠀⠈⠛⠇⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⢀⣴⣾⣿⣿⢀⣤⡆
⠀⠀⠀⠀⠀⣀⣴⡆⢸⣿⣿⣿⣿⣿⣿⡿⢛⣡⣶⣿⣿⣿⣿⣿⣿⢰⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⢩⣶⢰⣦⣙⠻⣿⣿⢰⣬⠃
⠀⠀⠀⠴⣾⣿⣿⡇⢸⣿⣿⣿⠿⡋⡁⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⠁⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⢩⣶⢸⣶⠍⢸⣿⣿⣷⡦⠉⠈⠁
⠀⠀⠀⠀⠈⠙⠻⡇⢸⠟⣫⣵⣿⡇⡿⠓⠈⠛⠿⣿⣿⣿⣿⣿⣿⢸⣿⣦⣄⠀⠀⣿⠿⣫⣵⣿⢸⣷⣮⣌⠰⣿⢸⡿⣛⢁⠶⡇⢸⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡇⠀⠀⠀⠀⣠⢈⡙⠿⣿⣿⣿⢸⣿⣿⠿⢋⡀⣔⡻⢿⣿⣿⢸⣿⣿⡿⠟⣀⠀⠚⠿⢸⢟⣆⢸⣿⣿⡿⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇⠀⠀⠀⠘⠻⢸⣿⣷⣤⡙⠻⠘⣋⣥⣾⣿⡇⣿⣿⣶⣍⡻⠸⠟⣩⣴⣿⣿⠀⠀⠀⠀⠛⠇⠸⠛⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣫⣴⢰⣭⡛⢿⣿⡇⣿⣿⡿⢛⣵⣶⣮⣙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⣴⢰⣮⠛⢸⣿⣿⣷⣮⣁⢫⣵⡆⣦⡍⢹⣿⣿⣷⣮⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢵⣿⢸⣿⣿⣿⣿⣿⣷⣮⣅⠩⣾⣿⣿⠿⣫⣵⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢆⣨⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡇⢸⣿⣿⣿⣿⣿⡿⢛⣥⣶⣿⣿⣿⣿⣿⣿⣿⢸⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡇⢸⣿⣿⠿⣫⡅⣬⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⠘⣫⣴⣿⣿⡇⠛⠉⠀⠈⠙⠿⣿⣿⣿⣿⣿⢸⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⢸⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣧⡀⠀⠀⢀⣼⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠀⣈⣿⣿⣿⣏⣁⠀⠉⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⣀⣤⢶⣿⠟⠋⠁⠀⠀⠈⠙⢿⣿⠶⣤⣄⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠋⠁⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⠙⠻⣦⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣴⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⣦⡀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣄⣠⡿⠁⠀⠀⠀⠀⠈⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠀⠀⠀⠀⠀⠈⢿⣄⣠⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠛⠛⠿⢷⣄⡀
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣄⣤⣤⣤⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⣯⡉⠙⢿⡍⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠷⣦⠀⠀⠀⠀⠀⠀⠀⠛⢻⡿⠛⢛⣿⠟⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠈⠻⣦⡀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢀⡴⠟⠁⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⠛⢶⣄⡀⠀⠀⠹⣇⠀⠀⠀⢀⣼⠇⠀⠀⢀⣠⡶⠛⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠈⠛⠻⠶⣦⣭⣿⣶⣾⣿⣥⣴⠶⠟⠛⠁⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⡾⠋⠉⠁⠈⠉⠛⢷⣄⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠶⠶⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠳⢤⣄⣀⣀⣀⣀⣀⣬⠇⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⡼⠉⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢸⡄⠀⠀⠀⠀⠀⠀⣸⠃⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡳⣄⠀⠀⠀⠀⣰⠏⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠓⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠚⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠃⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠹⣧⣠⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡀⣠⠏⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⢳⣄⣼⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡾⠿⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠙⠟⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⢨⡇
⠀⠀⠀⠀⠀⡇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⡇
⠀⠀⠀⠀⠰⡷⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠀⠀⠀⣀⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠀⠀⠈⠉⠙⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢛⣃⣠⡙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡴⠛⣁⣤⢶⣦⡾⢋⠰⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢋⣴⠾⢛⡋⠙⣡⣾⣏⠻⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡾⢁⡾⠋⠀⠐⠋⠀⠙⢷⡈⢷⡀⠀⠀⠀⢸⣿⣿⣿⡿⢡⡿⢡⣾⣿⣧⣾⣿⣦⡙⣷⡘⣿⣿⣿⣿
⠀⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠁⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⢹⡄⠀⠀⠀⠀⠀⠀⢠⡏⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢻⡆⢿⣿⣿⣿⣿⣿⣿⡟⣸⡇⣼⣿⣿⣿
⠀⠀⠀⠀⠻⣄⠹⠦⢀⣴⣂⣀⡴⠟⣠⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⡌⠭⣬⣹⡿⠋⣿⠿⢋⣴⡟⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠃⢴⡏⠉⣉⣉⣤⠾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣾⡃⢴⣶⡶⠾⠛⣩⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠀⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢲⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⢰⠁⠀⡀⣠⣄⠀⠀⢧
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠠⠀⠀⠀⠷⠾⠿⠤⠤⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠭⠀⠀⡇⠀⣄
⠀⠠⡠⣠⡱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⢄⠀⠀⠀⠀⠀⠀⡠⠄⠊⣀⢴⠏
⠀⠀⡔⠯⢤⠀⠀⠀⠀⠀⡲⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠆⠀⠀⠀⡌⠀⣰⠟⠉⠁
⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⠄⠂⠁⠀⠀⠀⠀⠀⠧⠲⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠀⠀⠀⠀⠀⠀⠀⢀⠼⠜⢿⠉⠀⠀⠲⢀⠈⡘⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢲⠂
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠅⠴⠺
⢒⠆⠀⠀⠈⢠⠒⠄⢳⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⠒⡉⠀⠀⠀⠀⠀⢨⠂
⠈⠀⠀⠀⠈⠉⠔⢁⡼⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣖⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⣽⣳
⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠉⠐⠢⠤⠐⠂⠻⡍⠋⠉⠈⠋⠉⠁⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣨⢯
⠀⠀⠀⠀⠀⢠⢊⡆⠀⠀⠀⠀⠒⡆⠀⠀⠀⠀⠀⠀⠈⠁⠰⠶⡦⣤⡀⣀⡀⠄⡅⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⡱⠀⠀⢠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡓⡾⢿⠄⡀⠀⠈⠀⠀⠀⠊⣠⢄⠙⣆⠀⠀⠀⠀⡁⠀⠀⢿⡺⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡰⡽⠙⣓⣒⢫⠦⡈⠢⡀⠀⠀⠀⠑⠋⡠⢁⡟⠀⠀⠀⠀⠁⠀⠀⠈⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⡇⠀⡎⠀⢀⣀⠀⢈⠴⠇⠀⠀⠀⠀⡎⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⢠⡶⣿⣗⣢⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⠀⠇⠀⡇⠘⢊⡡⡄⡄⠀⠀⠀⠀⡌⠨⡀
⠀⠀⠀⠘⠛⢁⣼⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣁⣡⠀⣾⡫⠊⠀⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣗⣷⠈⠀⠀⠀⣀⠄⠂⠱⠀⠀⠈⠉⠉⠉⠀⠈⠙⠟⠛⠂⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⢀⣥⣥⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠤⠤⠄⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠄⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠹⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠃
⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠖⢃⣥⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⣗⡲⡄
⠀⠀⠀⠀⢀⠾⠛⠛⠗⠐⠾⠛⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢠⣮⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢼⣿⣿⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠈⠢⠒⠋⠚⠃⠀⠀⠀⠀⠀⠀⢠⣾⣷⡞⠮⣟⣛⡥⠛⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠠⠴⠾⠛⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠙⠛⠃



⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⢿⣿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⢿⣤⣼⠇
⠀⠀⠀⠀⠀⠀⣠⣿⡟⢿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⡙⠿⠿⠟⣡⣄
⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⣽⡷⠾⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠙⠃



⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠦⠜⠑⢄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⢀⣄⢾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀⡜⠘⠟⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⢻⢟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢩⠍⡀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡟⢻⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢹⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇⢸⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⡇⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠃⠙⠛⠉⠙⠛⠋⠙⠛⠉⠛⠛⠉⠙⠛⠉⠉⠚⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢠⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⢹⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢹⡟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⢸⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢸⡇⠀⠀⠈⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⢸⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⣿⣿⣿⠿⢿⠿⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣦⠹⣆⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣈⠳⣌⢳⡘⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⠷⣝⠳⣝⠳⣍⡳⢦⣄⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡶⢡⡶⣌⡳⢬⡓⢮⣝⣓⠶⠶⢶⣒⣒⠲⣜⣎⣙⠿⢽⣿⠋⣹⡶⢋⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⡿⣰⢤⣝⡲⢭⣓⠦⠭⠭⠭⣝⣲⠽⣝⣳⢂⡿⣡⠞⣡⠞⣩⡴⢛⣡⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢣⡏⡞⡬⣍⣓⠮⣝⣛⣻⠭⣗⣪⡍⡼⣡⢏⡼⣣⠞⣡⠞⣡⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣼⡹⣻⣱⣨⣝⣒⠶⣒⣭⡇⡜⡼⣱⢃⡞⡴⢣⡞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⢻⡇⡇⣧⢧⢣⢳⢨⡭⡷⣼⢸⢱⢇⡏⡞⣸⢡⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⡟⣼⢱⢇⡏⡾⣸⢸⠑⣛⣛⡘⡜⡜⡎⡇⡇⣿⢼⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣱⢏⡼⣡⢏⡞⡼⢱⢃⣛⠭⢖⡲⠭⣝⣸⡼⣸⣹⢹⢸⢸
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣥⠞⣡⢏⡴⢣⠞⣼⣁⡫⢭⣒⡭⠭⢭⣝⡲⢭⣁⢣⢇⡏⣼⢸
⠀⠀⠀⠀⣤⠶⢛⣡⠶⢋⣵⠞⣡⠟⠴⠯⣌⣛⣶⢭⣑⣒⣛⣛⣓⠲⢭⡓⢮⣙⠺⢸⡵⣿⡀
⠀⠀⠀⠀⠀⠐⠛⠥⣔⡻⡤⢨⣽⣿⣌⠵⣿⡾⣽⣿⣒⣒⣒⠶⢬⣝⠲⣍⡳⣬⡻⠏⡼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠙⠳⢬⡙⢮⡙⢦⡙⢦⡇⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣦⠹⣦⡱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⡌⠻⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦


⣿⣿⣿⣿⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣟⣿⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⣆⠉⠉⢹⡏⠉⠉⢰⠖⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡔⠊⠁⠘⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠀⠀⠀⠋⠈⠑⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠈⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢮⡀⠓⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢁⡵⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⠀⠹⣷⣄⠀⠀⠀⠀⠀⠸⢓⣀⠠⠤⢰⡇⣿⡇⠤⠤⣀⡚⠇⠀⠀⠀⠀⠀⣠⣶⠟⠀⠜⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠈⢻⡻⣦⡀⢀⠔⠊⠁⣠⡀⠀⢸⠁⣿⡇⠀⠀⣠⠈⠙⠢⣄⢀⡴⣊⡿⠋⠀⠀⠀⠀⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠀⠀⠀⠀⠙⣎⠻⣷⣄⡀⠀⠉⠁⠀⣼⠀⣿⣿⠀⠀⠉⠀⠀⣠⠞⣩⣾⠏⠀⠀⠀⠀⠀⠀⠉⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠳⡌⠻⣿⣦⣀⠀⠀⡏⠀⣿⣿⠀⠀⣀⡴⠋⣡⣾⡟⠉⢣⡀⠀⠀⠀⠀⠀⠀⢀⡬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⠆⡜⠀⣀⠀⠙⢦⡈⠻⣿⣷⣤⡇⠀⣿⣿⣦⠞⠁⣠⣾⣿⠋⠀⢀⡀⠱⠐⡶⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠧⠄⠀⠀⠀⠀⠀⠀⡘⠀⠘⠛⠀⠀⠈⠳⣄⠈⠻⣿⠁⠀⣿⣿⡇⣠⣾⣿⠟⠁⠀⠀⠙⠃⠀⢣⠀⠀⠀⠀⠀⠀⠠⠴⠆
⠀⡄⠄⢠⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⣤⣤⣤⣤⣶⣶⣾⣶⣾⣿⣷⣄⣿⣿⠟⠛⠛⠛⠶⠶⠦⠤⣤⣤⣄⣸⣀⣀⠀⠀⠀⠀⠀⠀⡆⠀⢠⢀⡄
⠀⢠⡔⣋⠀⠀⠀⢺⠒⠲⠿⠿⠿⢿⣿⣉⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠋⠉⣉⣿⣿⣥⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⡿⠿⠿⠿⠖⠒⡷⠀⡠⢈⡃
⠀⠉⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⠉⠉⠉⠛⠛⠓⣲⣶⣶⢾⣿⣿⠈⠻⣿⠿⠟⢿⠛⠛⠛⠉⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡖⠁⠀⠀⠀⠀⠀⠀⢸⠀⢰⣶⠀⠀⢀⣾⣿⠟⠁⢸⣿⣿⠀⠀⣿⣷⣄⠈⠳⣄⠀⠀⢴⡆⠀⡎⠀⠀⠀⠀⠀⠀⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠞⠇⢡⡀⠀⠀⣰⣿⠟⠁⣠⠖⠋⣿⣿⠀⢸⠏⠻⣿⣷⣄⠙⢦⡀⠀⠀⡜⠰⠷⠀⠀⠀⠀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠀⠀⠀⠀⠀⠑⣤⣾⠟⢁⡴⠚⠁⠀⠀⣿⣿⠀⢸⠀⠀⠀⠙⠿⣷⣄⠳⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠀⠀⠀⠀⠀⠀⣴⠟⣡⣞⠉⠀⠀⣤⡀⠀⢹⣿⠀⡿⠀⠀⣤⠀⠀⠈⣻⢷⣌⢧⡀⠀⠀⠀⠀⠀⢤⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⣠⣾⡷⠋⠁⠀⠑⠢⢄⣉⠀⠀⢸⣿⠀⡇⠀⠀⢉⣠⠴⠊⠁⠀⠉⠷⣵⣄⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⢀⡴⠟⠁⠀⠀⠀⠀⠀⢸⡶⠈⠉⠉⠘⣿⢸⡇⠉⠉⠈⠲⡇⠀⠀⠀⠀⠀⠈⠻⢷⡀⢳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠄⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠄⣰⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⣦⡠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠃⠤⠤⠼⠧⠤⠤⠘⠓⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣡⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⢀⠤⠐⢀⣀⣀⣐⠂⠤⡀
⠀⠀⡠⡪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠑⢤⡈⠒⠊⣡⠴⠒⠒⡠⠀⢍⡙⠲⣌⠑⡄
⠀⠀⢱⡑⢄⣀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⠃⠀⠀⠀⠀⡅⠠⡴⠁⠄⠀⠀⣀⣀⣀⣀⠀⠈⢳⡈⠆
⠀⠀⠈⡀⡅⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠄⡠⢄⠄⠀⠰⠀⣷⠁⠈⠀⠠⠊⠑⠆⠐⠀⢉⠢⠀⢧⢸
⠀⠀⠀⠀⢊⠑⠂⠄⠀⠀⠀⠀⠀⠀⠤⠐⠂⢩⡟⠊⠀⠈⠀⠀⢀⠀⣿⠀⠰⠀⠀⠂⠀⢀⢄⠀⠈⠁⠀⢸⢸
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⠀⠀⠀⠀⠀⠀⠄⢇⢆⠀⢳⠄⢄⠀⠈⠛⠀⠀⠃⡔⠆⡜
⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⠼⠚⠁⠀⠀⠀⠀⠀⠀⠘⢌⠢⡑⠚⠀⠢⢄⠐⣀⢀⠠⡪⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠑⠴⣯⣤⣤⣤⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡑⠢⠤⠤⠤⠤⢔⡪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡴⢚⣩⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⡟⢠⡿⠛⠙⠻⣿⣦⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⣠⣄⠀⠹⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣧⠸⣇⠀⢀⡇⣽⣿⠆⠀⠀⠀⠀⠀⡄⠀⠘⢿⣿⠟⠀⢀⣿⠁
⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⢟⡿⠃⠀⠀⠀⠀⠀⠈⢿⣷⣌⣉⣩⣴⣿⠟⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠛⠄⢻⢰⠟⣀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⣉⣉⠛⢿⣦
⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢶⡤⠄⣠⣤⣆⢌⠜⢋⠁⠀⠀⠀⠀⢀⣿⡇⢠⡿⢛⠻⣿⣮⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣿⣿⣿⣞⢿⣷⡇⠀⠀⠀⠀⠀⢠⣴⠊⡀⠙⠉⣁⣚⠙⠃⠀⠀⠀⠀⠀⣿⣇⠘⠧⠼⠃⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣵⣿⠟⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃⣧⠑⣦⠉⠋⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣤⢄⡀
⠀⠀⠀⠀⢀⣶⣿⣟⣿⡹⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣉⡉⠙⠳⣄⠀⠀⠀⠀⠀⢠⢿⣿⣿⡿⢿⣿⣾⣢
⠀⠀⠀⠀⢸⣿⠷⣻⣿⣿⣿⡴⣿⡷⠀⠀⠀⠀⠀⢰⣿⡇⢠⢞⣉⡙⢷⡀⠸⠀⠀⠀⠀⠀⣿⣾⡷⠁⠀⠈⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣏⣾⣿⡿⣿⣙⣿⡟⠀⠀⠀⠀⠀⠸⣿⣇⠈⠪⠭⠃⣼⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣄⢀⣀⣼⣇⢯⠇
⠀⠀⠀⠀⠀⠙⢿⣷⣿⣼⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⡾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣻⣿⢿⣿⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⢠⢄⣞⠕⠛⠀⠀⠳⡝⣆⠀⠀⠀⠀⠀⢠⡞⢁⣾⠿⣛⠿⣝⢷⣄⠀⠀⠀⠀⠀⣾⡿⣫⣽⣶⣯⡹⣿⣿⡄
⠀⠀⠀⠀⣾⢾⢼⠀⠀⠀⠀⢸⣻⣿⠀⠀⠀⠀⠀⢸⡇⢸⣏⣾⣿⣷⢸⠈⣿⠀⠀⠀⠀⢸⣿⢳⣿⣯⡂⢈⡿⣸⣿⣷
⠀⠀⠀⠀⠸⣞⢞⠦⣀⣀⡤⣫⢣⡏⠀⠀⠀⠀⠀⠘⣷⡄⠙⠻⠛⠋⠊⣼⡟⠀⠀⠀⠀⠈⢿⡞⣿⣷⣽⣯⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠳⢭⣟⣛⣯⠷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣶⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⢮⣛⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣛⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⣿⣿⡟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡈⠁⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠹⣿⠿⠄⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣛⣛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣤⣤⣤⣬⣙⠻⢷⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⣿⣿⣿⣿⡇⠲⣶⣶⣷⣤⠙⠻⠿⠟⠓⠀⠉⠉⠉⠉⠁⠀⠈⠙⠋
⠀⠀⠀⠀⢀⣠⣦⠀⢠⣤⣤⣤⡀⢰⣶⣶⣷⡄⠙⠿⠿⠟⠃⠈⠙⣩⣭⣭⣤⣄
⠀⠀⠀⠺⠿⢿⣿⡆⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣛⣋⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠛⠛⠂⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣻⣛⣟⣛⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠑⣔⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠆⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⢳⡀⠀⡤⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢧
⠀⠀⠲⡀⠊⢙⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠠⠘⡄⠀⠀⡰⢹⡄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢅⠠⠀⠀⠀⡼⠉⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠗⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⣫⠃⠀⠁⠈⠉⠡⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⡋⢀⡤⠖⠁
⠀⠀⠀⠠⠊⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⡮⠚⠉⠀⠀⢄⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⢻⠃⠀⠀⠀⠀⠀⠀⠀⠁⢉⠎⢀⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⣀⢀⠎⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢠⡄⠊⠙⠛⠼⠗⠃⠉⠒⢀⣤
⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢀⣀⡤⠔⠈⠁⢸⠀⠀⠀⠐⠶⣤⣄⣀⣠⡀
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡃⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⠀⢉⣋⣀⡑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠄⠈⠀⠀⠀⠀⠘⠨⢀⠀⠀⠀⠰⠀⢀⡠⠔⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠼⠚⠃⠀⠀⠀⠀⠀⠑⢠⠀⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣨⢩⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡶⠶⣶⣶⣶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢿⣿⡅⠀⢻⣿⡆⠀⢿⣿⡃⠀⣼⣿⡟⠛⢳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣿⣧⣀⠈⠻⣿⣦⡀⠹⢿⣄⠈⢿⣧⠀⢿⣿⠀⢠⣾⣿⠟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣀⠈⠙⠻⠷⢶⣤⠍⣙⡷⠶⢝⣛⣂⣙⣷⣌⢿⡄⢸⣿⠃⠀⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣏⠀⠉⠛⠛⢛⡛⣻⡷⠋⣴⣿⠋⠀⣾⣿⠋⠁⣠⣤⣬⣤⣿⡄⢿⠀⢸⣿⠋⠁⠀⣹⣦⡀
⠀⠀⠀⠀⢀⡾⠻⠿⠿⠿⠟⢻⠟⣰⣿⠀⠀⠿⣿⣦⣀⡙⠿⣦⡀⢿⣿⠁⢀⣤⣭⣘⠇⢸⡇⢀⣴⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣷⣤⣤⣶⢶⢣⣿⠀⠹⢿⣷⡶⠶⡶⢖⣭⠿⠓⣊⣝⣦⣙⣆⢸⣿⠛⠛⠳⣼⡇⣼⡟⠁⠀⣀⠈⢷⡀
⠀⠀⠀⣼⠏⠉⢉⣡⣴⡇⢸⣿⣦⣤⣤⠶⣲⠏⠰⠿⠷⢤⣈⡻⢯⡀⢨⣍⡂⢿⠀⣴⣶⣌⠣⡿⢀⣴⣿⣿⣿⣿⣧
⠀⠀⢠⣿⣷⡿⠟⢋⢸⡇⠀⠙⠛⣉⣴⢣⣿⣦⠶⣲⠟⢁⠲⢿⡓⠛⠜⣿⠛⢾⢰⡟⠉⠙⢷⡃⣾⠟⠁⠀⠀⠉⢿⡄
⠀⠀⢸⡏⢀⣠⡶⢻⠘⣿⣶⣶⡾⢋⣾⠀⢉⡴⠡⠟⣿⡁⠘⣿⡟⠜⣷⡹⢰⣦⢹⢃⣴⣶⡄⢿⠃⣴⣾⣿⣿⣷⣼⡇
⠀⠀⢸⣿⠿⠋⣡⢺⡄⠈⠉⠉⣰⢏⢻⣷⡟⣸⣤⡎⠿⣤⣄⡸⢁⡄⠃⢡⠟⠻⣏⡾⠋⠙⢿⢈⡾⠋⠁⠀⠈⠙⣿⡇
⠀⠀⢸⣇⣠⣾⠏⡌⢿⣶⣶⣾⠏⡼⣄⠉⢰⣇⠻⠃⣤⣻⣍⠡⠽⣷⢏⣪⣶⢠⢏⣤⣶⣆⢸⢋⣤⣶⣿⣿⣷⡄⢸⡇
⠀⠀⠘⣿⠟⠁⣼⢻⡄⠙⠛⠁⣰⡇⡙⣷⣿⠇⣗⢤⣽⣌⠫⢥⣖⣉⠉⢩⣯⠴⠋⠉⢹⡿⣼⠟⠉⠀⠀⠈⠻⣿⣾⠃
⠀⠀⠀⢻⣤⣾⠏⢠⡻⣶⣤⣶⣿⠃⣿⢦⣉⠀⣸⣎⠪⣭⠤⣴⣛⡫⠶⠷⠶⢿⣿⡆⣸⣣⡴⠾⠿⢿⣿⣦⠀⣿⡟
⠀⠀⠀⠈⢿⡏⢠⣿⢱⣌⠙⠛⠁⢀⣿⠀⣯⢛⠿⠟⠀⠘⣷⣦⠉⠛⣿⣷⠀⣰⢟⣐⣩⣥⣤⣤⡀⠀⠹⣿⢠⡿⠁
⠀⠀⠀⠀⠈⢷⣿⠇⢸⡏⠳⣶⣶⣾⡿⠀⢸⣧⠻⣟⠚⣛⣛⣉⣤⠶⠛⣥⣾⣭⣭⣍⣉⠉⠙⢿⣿⡆⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣾⡇⢻⣦⣉⡉⠀⢀⣼⣿⠀⠘⣷⡌⠛⣶⣌⠙⠳⣶⣤⡀⠉⠻⣿⣷⠀⢸⣿⢃⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣅⣸⡇⢸⡝⠛⠿⠟⠋⠀⣠⣿⣿⠀⠈⣿⣷⠀⠈⣿⣿⠀⠀⣿⡟⢀⣾⣧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⣈⣿⠈⢷⡒⢶⠾⠿⠛⠁⣀⣼⣿⠟⠀⣰⡿⠏⣠⡾⢋⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣼⣷⡌⢻⣏⠙⢿⣉⠩⣴⣖⠛⢭⣤⣉⣻⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⠾⠿⠷⠾⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⣿⡿⠿⢿⠿⠿⡿⠿⠿⢿⣿⠀⠀⢸⢫⢞⢜⢴⡵⡣⡣⣫⣮⢞⠅⠀⠀⠸⠛⢀⣴⣿⠟⢀⣠⣿⠟⠀⣠⣿⠟⠀⣠⣿⠟⠀⣠⣿⠟⠁⣠⣾
⠀⠀⠀⠀⣿⣦⣴⣬⣦⣦⣼⣴⣤⣼⣿⠀⠀⢸⢔⠵⡵⡮⡪⢪⢾⢝⠗⡵⡂⠀⠀⢀⣴⣿⠟⠁⠠⠾⠟⠁⠠⠾⠟⠁⠠⠾⠟⠁⠠⠾⠟⠁⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⣠⠀⠀⣶⡴⡖⡶⢦⣦⢶⣴⡶⡖⡶⢶⢲⢶⠀⠘⠁⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠈⣉⣙⡃⡉⡈⠉⠛⢉⡙⠙⡓⠙⠘⠋⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⣠⠀⠘⠙⠚⠋⠟⠟⠛⠟⠻⠺⠘⠟⠛⠇⠏⠀⠘⠁⣠⣶
⠀⠀⠀⠀⠟⣡⠞⣣⡾⢋⡴⢋⡴⢋⣸⠀⠀⢸⣿⣟⡟⢻⢻⣻⠛⡛⠛⣿⡇⠀⠀⢀⣠⣾⠟⠀⣠⣶⠖⠀⣠⣶⠞⠀⣠⣶⠞⠀⣠⣶⠖⠀⣠⣾⠟⠉
⠀⠀⠀⠀⡞⠃⠾⠋⡴⠋⡴⠋⠴⠟⢹⠀⠀⢸⡿⢷⣷⣶⣾⣿⣶⠶⠶⠿⠃⠀⠀⠸⠛⠁⠠⢾⠟⠁⢠⠾⠟⠁⢠⠾⠟⠁⠠⠾⠟⠁⠠⠾⠟⠁⢠⢶

⠀⠀⠀⢀⣶⠖⠒⣲⣶⡶⠒⣲⣶⡶⠒⣲⣶⡶⠒⢲⣶⡶⠒⢲⣶⡶⠒⢲⣄⠀⠀⣠⢶⣶⣶⣶⣶⣶⣶⢶⢶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⠶⣶
⠀⠀⠀⠘⠃⣠⣾⡿⠁⣠⣾⡿⠋⣠⣾⡿⠋⣠⣼⡿⠋⢀⣴⡿⠋⢀⣴⡿⠋⠀⠀⢺⢵⣿⢿⢗⢽⢿⢿⢵⢿⢿⢷⢿⠿⢿⢯⡿⡿⠯⡿⡿⡿⣮⣿⣾⠁
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠀⠀⣿⢿⣝⠅⠀⢠⣀⣤⣠⡄⣠⠀⣀⣄⣄⣄⣤⣀⡄⣤⡄⠀⣺⣯⣺
⠀⠀⠀⠀⠀⠀⠀⣾⡇⢻⡟⢻⡏⣿⡋⣾⣾⡏⢻⠋⡇⣾⢻⣼⣧⣿⠀⠀⠀⠀⠀⣻⢵⣽⠅⠀⠿⠷⠿⠼⠥⠿⠆⢿⠿⠿⠿⠷⠿⠇⠿⠇⠀⣺⣺⣻
⠀⠀⠀⠀⠀⠀⠠⠟⠿⠸⠇⠸⠇⠿⠤⠿⠹⠇⠸⠀⠗⠻⠼⠸⠟⠿⠀⠀⠀⠀⠀⣺⢹⣝⠄⠀⡖⣴⣤⣦⢶⠐⡆⣶⣆⣤⠖⡶⢠⣶⣴⡄⠀⣸⣫⢼⡁
⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⢀⣀⡀⠀⢀⣀⡀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⣺⢼⣿⡅⠀⠉⠉⠉⠉⠈⠁⠁⠁⠉⠉⠁⠁⠈⠉⠁⠁⠀⣼⣿⣺⡀
⠀⠀⠀⢠⣶⡿⠋⢀⣴⡿⠋⢀⣴⡿⠋⢀⣴⡿⠋⢀⣴⡿⠋⢀⣴⡿⠋⢀⣴⠀⠀⣺⢫⣾⣿⣯⣿⣿⣯⣻⣿⣿⣿⣪⣿⡿⣯⣯⣿⡿⣿⣾⣿⣟⣻⣼⡂
⠀⠀⠀⠈⠛⠐⠒⠛⠛⠒⠒⠛⠛⠀⠒⠛⠛⠒⠒⠛⠛⠐⠒⠛⠛⠐⠒⠛⠋⠀⠀⠘⠛⠛⠛⠛⠛⠛⠚⠛⠛⠚⠚⠚⠛⠚⠚⠛⠛⠚⠛⠛⠚⠓⠛⠚
⠀⠀⠀⠀⣀⣀⣀⢀⣀⡀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⢀⣀⡀⠀⠀⢀⣀⠀⣀⣀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⢟⡵⡯⣫⢞⢝⡵⡫⢫⢞⠝⡵⡫⢪⢟⠽⡽⡫⢫⢟⠽⡽⡫⢎⠷⠀⠀⠞⢁⣼⠟⣡⡾⠋⣠⡿⢋⣴⡇⠀⠀⡏⣠⡾⠋⣠⡿⢃⣴⠟⢁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢴⠰⢠⣦⣤⠶⡆⡆⣴⠀⣦⣶⣴⢠⡄⠀⠀⠀⠀⠀⠀⠸⠛⠁⠾⠋⠠⠾⠋⠰⠟⠁⡆⠀⠀⡟⠋⠀⠞⠋⠰⠟⠁⠰⠛⠁
⠀⠀⠀⠀⣀⢀⣀⡀⣉⣉⢈⡀⡀⣉⣈⢉⡁⡉⣉⢈⢉⡉⡉⣈⢁⢀⡀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠈⠷⠯⠮⠞⠵⠵⠫⠮⠞⠵⠵⠫⠮⠞⠵⠵⠫⠮⠞⠵⠱⠫⠞⠕⠇⠀⠀⠀⢐⣟⣞⡇⢸⣶⣞⡏⡟⠀⡇⠀⠀⡇⠀⣷⣴⡆⢰⣶⣶⢺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣨⣅⣨⣌⣬⣁⣄⠀⡇⠀⠀⡇⠀⡭⣈⣡⣤⢥⡍⣬⡀
⠀⠀⠀⠠⢟⣴⢟⣴⠟⣰⠟⣡⠟⣡⠟⣡⠟⣡⡿⣣⡾⢃⡾⢋⡼⢋⡼⢋⣦⠀⠀⠀⠘⣧⠼⠯⢿⡿⣽⠮⡷⠀⡇⠀⠀⡇⠀⠯⠼⠿⠸⠸⠇⠿⠅
⠀⠀⠀⠀⠋⠁⠈⠁⠈⣡⣈⡅⢈⣁⣌⣡⠈⣍⣠⣉⣀⣉⣀⡉⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠇⠘⠛⠚⠘⠀⠃⠓⠓⠛⠗⠛⠃⠀⠀⠀⠀⡀⠀⠀⠂⣠⡶⠂⣠⡶⢀⣴⠖⢀⣴⡇⠀⠀⡇⣠⡶⢀⣴⠖⢀⣴⠆⣠⡶
⠀⠀⠀⠰⣟⣡⠞⣡⡞⣡⡞⣡⡞⣣⡾⣃⡶⣋⡴⢋⡴⢋⣴⢋⣴⢋⣴⢟⣥⠀⠀⣾⢋⣠⡾⢋⣴⡟⣁⣴⠟⣡⡆⠀⠀⡟⠋⣴⠟⢁⣴⠟⢡⡾⠋⢀
⠀⠀⠀⠀⠈⠁⠀⠁⠀⠁⠀⠉⠀⠉⠀⠉⠀⠉⠀⠈⠀⠈⠀⠈⠁⠈⠁⠈⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⡴⣤⡀⢀⣠⢶⡄⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠉⠁⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠉⠉⠛⠛⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠾⠯⠴⠶⠶⠶⠶⠦⠤⠶⠶⠶⠶⠦⠽⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⣤⣤⣤⣤⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣀⣀⣀⣀⣠⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣤⠀⠀⠀⠀⠀⣀⣴⡟⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣿⣷⡶⢿⡀⠀⠀⠠⣿⢴⣦⡏⠀⠀⠀⣠⡾⢿⣏⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⠀⢸⣦⠾⣾⡇⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠉⣻⣆⡽⠋⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⣠⣤⣤⡤⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⣰⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⢤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣠⡶⠛⠋⠙⠛⢶⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡏⢀⡾⠛⠛⢷⡀⢹⡆⢸⡇
⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠸⣇⠀⠀⣸⠇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠶⣤⡀⠀⠘⣧⡀⠻⣄⡈⠛⠛⢁⣠⠟⢀⣼⠃⠀⢀⣤⠶⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠳⣤⣈⠙⠛⠛⠋⣁⣤⠞⠁⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠄⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠂
⠀⠀⠀⠀⠀⠛⡂⠄⡁⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢀⠄⠐⢃
⠀⠀⠀⠀⠀⠀⠐⡀⠈⠐⠠⡈⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⡁⠄⠊⠀⠠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⠠⢀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡠⠐⠈⠀⠀⠀⠠⠁
⠀⠀⠀⠀⡆⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠑⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠐⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠤⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠘
⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠒⠄⢀⠀⠀⠠⠄⠀⠀⠀⠀⠄⢰⠀⠄⠀⠀⠀⢀⠤⠀⠀⢀⠀⠀⠂⠁⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠑⣄⠀⠈⠀⢸⠀⠈⠄⠀⡐⠁⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡐⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢸⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠈⠀⠀⠀⢀⠠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⠀⢄⠀⠀⠀⠈⡀
⠀⠠⠁⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⠂⢀⠀⠐
⢀⡅⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⡀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢤
⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈
⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠈⠀⢀⠂⠀⠈⠀⠀⢀⠠⠊⠁⠀⠀⠀⠀⠀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠑⣄⡔⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⠁⠀⠀⢁⠰⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡿⠿⠛⠛⣿⣇⠀⢠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠉⠉⠉⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⢿⣿⠀⠀⠹⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣦⡀
⠀⠀⣄⡀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡆⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⢿⣿⣷⣦⣄⣀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠟⠛⠛⠁⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⢸⣿⡄⠉⠛⠿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⢀⣾⡟⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⢻⣿⡀⠀⠀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⣀⣀⣀⣠⣤⣤⣿⣷
⠀⠀⠀⠀⠸⣿⣧⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶
⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢻⣿⡆
⠀⠀⠀⢿⣿⠛⠛⠛⠉⠉⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠀⠀⠈⣿⣇
⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⢀⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣼⡿⠁⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⣀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⢀⣤⣤⣴⣶⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠛⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠸⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⢀⣀⣀⣤⣤⣤⣶⣶⣾⡿⠿⠿⠟⠛⠃⠀⢹⣿⣤⣤⣶⣾⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡙⣌⠫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠩⢋⢅⠢⠒⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡥⢊⢉⠀⠐⠫⡪⢉⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠅⡂⡁⠉⢠⠀⢧⠠⠗⣊⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡒⢝⣺⠃⠀⠀⠀⠀⠱⡀⢊⠥⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⠬⡪⢤⠏⠀⣠⢤⢤⡀⠀⠑⡎⠤⢒⠬⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠣⠊⠄⣪⠃⠀⠀⠀⠴⢂⠙⠀⠀⠈⢆⠓⠀⢙⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡂⢅⡶⠃⠀⠀⠀⠀⡈⢀⢿⠀⠀⠀⠀⢢⡉⣕⠮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢕⡊⡵⠃⠀⠀⠀⠀⠀⡈⠑⣤⠀⠀⠀⠀⠀⠱⡈⢙⡂⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣓⠊⠁⡾⠁⠀⠀⠀⠀⠀⠀⢖⣑⡿⠀⠀⠀⠀⠀⠀⠙⢄⠈⠩⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠣⣐⢤⡿⠁⠀⠀⠀⠀⠀⠀⠀⡻⢅⡄⠀⠀⠀⠀⠀⠀⠀⠈⢣⠨⢐⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠝⠊⡝⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠲⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠝⣒⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠐⣈⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡨⠤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡁⠊⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⣒⢠⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠠⢀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠥⢴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠯⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠓⢃⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⢒⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⡦⢲⠘⠄
⠀⠀⠀⠀⠀⠀⢰⡆⡨⠒⡪⠓⢰⣶⣶⠶⢶⡦⢤⡤⢤⠤⠤⠄⢤⡤⠠⠖⠲⠖⠲⠒⡒⠲⠖⡲⢒⡲⣒⠀⠒⠒⣢⡛⠉⡪⣺⢴
⠀⠀⠀⠀⠀⠀⠈⢧⣀⠊⠀⠚⠋⠁⠀⠈⠁⠈⠑⠈⢀⣀⣀⣈⣁⣉⣀⣃⣐⣉⣀⣉⣀⣀⣊⣊⣁⣉⣀⣋⣐⣉⣂⢐⣉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠧⠋⣩⣼⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠈⢿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢐⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡼⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣪⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠪⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠨⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⡿⠿⠿⠿⠿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠋⠀⠀⠀⠈⠛⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⣸⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⢶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠁⠀⢀⠜⠁⠀⠀⠀⢺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠈⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⢨⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢀⠀⠘⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠠⢧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠁⠀⠀⠀⡀⠀⠠⠀⠂⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠸⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣩⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠦⢘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⡟⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣸⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⡏⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⢸⠇⠀⠀⠀⠤⠤⠄⠑⠤⡀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠸⡇
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠦⣀⠀⠀⠃⠀⣧⡀⣠⠶⠖⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⣩⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⣶⣶⣶⣤⣤⡀⠜⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄⠀⠀⠀⣴⡾⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛

⣿⠿⣿⢿⠿⣿⡟⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠛⠿⣿⣿⣿⠿⠿⢿⡿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢶⡀⢀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣠⣼⣷⣿⣴⠀⠀⠀⠀⠰⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⢂⡄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⠀⣀⣤⣶⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡋⠀⠀⢸⣿⣿⣿⣿⣿⠟⠛⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣷⣤⡈⠻⣿⣿⣿⡿⢄⠀⠀⢺⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠷⢿⣿⣿⠅⢠⢇⣠⣤⡿⢿⡁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢹⣷⣶⣿⣿⣿⣟⢿⡜⣧⣀⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣇⢿⡌⣷⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡾⢿⣿⣿⣿⣿⣿⣾⣷⣸⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣠⣶⣬⣍⣠⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⣮⣭⣏⣰⣿⡟⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣈⣿⣿⣿⣿⣿⣿⡽⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠇⠹⢿⣿⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⡖⠒⠶⠶⠒⠒⠓⠒⢚⠓⠒⠒⠒⠒⠒⠒⠒⠲⠖⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣬⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⡶⠶⠶⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣲⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣧⣥⣤⣥⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣀⣀⡀⢀⠀⠀⡀⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡿⠛⡿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠀⡭⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠄⣽⠀⣀⣠⠤⠖⡚⠍⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢄⣀⡤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠬⠥⠝⠒⡟⠙⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠸⡋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠇⣿⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠆⢸⠙⣇⠀⢐⡽⠊⣡⣇⣠⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⢬⣢⠸⠙⢢⡇⡈⠉⠙⠟⠂⠄⠁⠘⠰⠊⠿⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣋⡥⠶⠶⢬⣝⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⡼⢡⣾⠟⠻⢷⣌⢧⢹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⡇⣿⡇⠀⠀⢸⣿⢸⢇⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣌⠻⢶⡶⠟⣣⠞⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⣓⣒⣋⣥⣾⣿⣿⣿⣟⣭⣥⣬⣽⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠘⢻⣿⣿⡿⢹⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣋⣤⣾⣿⣿⡿⠁⠾⠿⣿⡏⠈⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⣼⣿⣷⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠴⠒⠉⠉⠀⠉⠉⠑⠢⣝⢦⡀
⠀⠀⠀⠀⠀⢠⠞⠡⠂⠀⠀⠙⢦⣀⣀⣀⣀⣀⢀⣀⣀⢀⡀⠀⠀⠀⠀⠀⢠⢎⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡀
⠀⠀⠀⠀⠀⠇⠀⠀⡴⠊⠁⠄⠀⣿⠟⠋⠀⠀⠀⠀⠘⣿⣦⣍⠙⡆⠀⠀⡜⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡇
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠃⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠝⠋⠁⠀⣳⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠆
⠀⠀⠀⢸⡇⠱⡀⠀⠀⠉⠁⢀⡼⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠒⠁⠀⠀⢧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢾⢥⠇
⠀⠀⠀⠀⠛⠒⠋⠑⠓⠒⠚⠉⠀⠙⣄⢄⠢⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⢸⢼⡀⠀⠀⠀⠀⠀⠀⣀⡤⡼⠟⠀⡎⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡷⠒⡄⠀⠀⠀⣾⡋⠀⡠⠒⠼⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠃⠀⠘⡄⠀⠀⣠⡟⡴⠀⢀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⣩⡥⠭⠭⣉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢄⠀⢱⠀⣴⠋⢸⡀⠀⠂⣠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠖⣡⡴⠋⠁⠀⠀⠀⠀⠀⠙⢄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠖⠉⠘⠷⣦⠀⣽⠔⠚⠁
⠀⠀⠀⠀⠀⠀⢰⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⡾⠁
⠀⠀⠀⠀⠀⡰⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠋
⠀⠀⠀⠀⢰⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢸⠃
⠀⠀⠀⠀⡆⠀⠰⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢹⠋⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠖⠚⠒⢶⣄
⠀⠀⠀⢰⠀⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠏⠀⠀⣠⠶⠛⡟⠛⠙⢦⠙⡇⢀⠴⠀⠈⡎⠙⠲⢤⣀
⠀⠀⠀⠘⡄⠀⠀⠎⢰⠀⠀⡠⠒⠉⠉⠒⠉⠀⡰⠃⠀⠀⠀⢧⠀⢸⠀⠀⠀⠀⡆⠑⢄⣀⣀⠴⠓⠓⠒⠲⢯⡙⢦
⠀⠀⠀⠀⣷⣤⠴⢲⣧⠀⢘⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⡞⠋⠀⢸⠀⠀⠀⣰⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⢌⢇
⠀⠀⠀⠀⣿⢿⠀⣿⢾⣤⣤⣧⡀⣀⠤⠊⠀⠀⠀⠀⠀⠈⠋⠛⠛⠛⠧⡀⠘⣧⠤⠾⠋⠉⠉⠉⠉⠙⠷⠶⠶⠶⢚⣿⡟⠃
⠀⠀⠀⠀⢸⡘⠓⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣼⡓⠦⠤⢤⣤⡤⠤⠖⣺⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠶⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⠛⠛⠋⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠉⠀⠀⠀⠠⠀⠀⠜⠡⠀⠀⠄⠀⠀⠀⠉⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠄⠐⠀⢀⣶⠐⠂⠠⡀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⣤⡊⠀⠐⠈⠀⠀⢀⣿⠀⠈⠰⠀⠈⠀⣀⡀⠁⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠠⠂⡈⠋⢀⡀⠀⠀⠀⢸⠏⠀⢀⠃⠀⢀⠸⠛⠁⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠠⠃⠄⡀⠀⠈⢻⡢⢄⡀⣿⡀⡆⣀⠤⡺⠋⠀⠈⠄⢀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠂⠀⠀⠈⠉⠒⠠⣹⡄⣹⠿⡧⣟⣡⠞⠁⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⠀⠶⠶⢶⣶⣯⣭⣭⣤⣤⡇⣶⣶⢹⠿⠿⠷⠶⠶⣦⣤⣤⡄⠀⠄⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠉⠀⠀⡀⠀⠀⠀⠀⠉⠉⢻⡟⣳⢤⣤⡞⠲⡂⠀⠉⠁⠀⠀⠀⡀⠀⠁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠠⠀⠆⠂⠁⠀⣰⠿⠊⠉⢸⣿⠛⠲⢼⣆⠀⠁⠂⠶⢠⠁⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠡⡈⢂⡀⠈⠁⠀⠀⠀⣸⡟⠀⠀⠀⠙⠃⣀⠊⠠⠀⠀⠐⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠸⠿⡁⠢⠀⢀⠀⠀⢻⡇⠀⠀⣀⠠⠀⠹⠟⠀⠀⠂⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠐⠠⢌⠀⠀⢹⠃⠀⢀⠠⠀⠈⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣶⣶⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣤⣤⣤⡤⣤⡶⣉⠖⢿⠛⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠦⠤⣤⣤⣤⡤⠄⣀⣀⡀
⠀⠀⢀⡀⣤⣄⣰⣶⣴⣾⣿⣿⣿⠃⣿⠀⣿⠀⣹⠀⣿⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠶⠶⠦⣤⣀⡀
⠀⡜⣤⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⠃⣿⠃⣿⡆⣿⠀⣿⠀⢘⠀⢈⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦
⣧⡈⢶⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡆⣿⡆⣿⡇⣿⠀⣿⠀⢸⠀⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣼⣿⡤⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⡇⣿⡇⣿⠀⣿⠀⢸⠀⠸⡇⠀⡇⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡤⠚
⢼⣿⢣⠀⠹⡀⢻⡟⢻⡿⠿⠿⣿⣷⣿⣧⣿⣇⣿⣀⣿⣄⣸⣆⣠⡇⠀⣇⠀⢸⡀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠠⣤⣤⣄⣠⣴⢶⣶⢶⡏⠀⣿⠅
⣾⣧⡜⠀⢸⠃⢸⡇⠀⡇⠀⠀⠈⠀⠀⠁⠈⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠋⣿⠙⡟⢻⣿⢹⣿⢻⣿⣿⣿⢸⣿⢸⡇⠀⣿
⣿⣿⣷⠀⢰⠀⢸⡇⠘⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⣥⠀⣇⠀⡇⢸⣷⢸⡿⢸⣿⣿⣿⢾⣿⢸⡇⠀⣿
⣿⣽⡿⠀⢸⡇⢸⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡟⠀⡇⢸⡏⢸⡷⣾⣿⣿⣿⣿⣷⢸⣇⠀⣿
⣿⡿⣿⠀⢸⡇⢸⡇⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠉⠀⡟⠀⡇⠀⡇⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⣿
⣿⣟⣥⠀⢸⡇⢸⡇⠀⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣉⠀⣧⣠⡷⠰⣷⣬⣥⣼⣿⣿⡿⠿⢿⣽⣿⢸⡇⠀⣿
⣿⡏⣿⡆⢸⡄⢸⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢫⣰⣾⣿⣿⡟⢻⡿⢸⣿⠍⢹⡅⠀⠓⠀⠉⠡⠝⠛⠷⡀⢿⡀
⣿⣼⣿⡇⢸⡇⢸⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⡗⢸⣷⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠄⠻⡀
⣿⣿⣿⠇⢸⠇⢸⡇⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣽⣿⣿⣿⣿⣿⡇⢸⡟⢘⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⡶⢸⡆⢸⡆⢀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⢸⡧⢸⣿⠀⢸⡇
⣿⣿⣿⠁⢸⡇⢸⡇⠈⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⣿⣿⣿⣿⡇⢸⡧⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⠀⢸⡇⠸⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⠀⢸⡇⠀⢀
⣿⠏⣿⡆⢸⡇⠐⡇⠀⡇⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⢠⣿⠀⢸⡇
⣿⣆⣿⡅⢸⡇⢸⡇⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢻⣿⣿⣿⣿⣿⣿⡇⢸⡇⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⣯⣄⣿⣇⢸⡇⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡹⣿⣿⣿⣿⣿⡇⢸⡇⢸⣿⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸
⣿⣿⣿⡇⠘⡇⠀⡇⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠷⣮⡻⣿⣿⣿⣧⣸⣇⣸⣿⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⠀⢀⣱⠞⣻
⣿⣏⣿⡇⢸⡇⠀⡇⠀⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⡛⠫⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⡀⡀⠠⣄⣳⣴⣿⣿⠀⣿
⣿⡿⣿⡇⢸⡇⠀⡇⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠻⠃⠀⠀⠉⠐⠟⠉⣭⡭⣭⡽⢯⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⠂⣼
⣿⣷⣿⡇⢺⡇⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⢼⣿⣿⣿⣷⣿⣿⣿⣏⠀⠙
⣿⠇⣿⡇⢸⡇⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⣼⣯⣹⣿⣿⣿⣿⣿⣿⠀⣾
⣿⡦⣿⡷⢸⡇⠀⡇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⢿⡀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⢸⡇⣸⣿⢿⣿⣿⣿⣿⣿⣷⣀⡞
⣿⡇⣿⡇⢈⡇⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣾⡇⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⢸⠇⣼⡧⣼⣿⣿⣿⡿⣿⡿⠄⠰
⣿⡷⣿⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⢸⠇⢸⡧⢘⣿⣿⣿⡟⣿⡇
⣿⡇⣿⡇⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⢘⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⢸⡇⢸⣿⢻⣿⡇⣿⡇⠀⡀
⣿⡇⣿⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣫⣷⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⢾⡇⢸⣿⢸⣿⣇⣿⡇⢰⠁
⣿⠋⣿⡇⠀⡇⠀⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⡇⠀⣿⢻⡆⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⢸⣿⠘⣿⠉⣿⡇⠀⠈
⣿⡤⣿⡇⠀⡇⠀⠃⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢨⡇⢨⡇⠀⣿⠘⣿⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇⢈⣿⠘⣿⣶⣿⡇
⣿⣷⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⢸⡇⠀⣽⠰⣿⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⠀⠈⠟⠀⣩⠀⢻⡏⠉
⣿⠁⣽⡇⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠀⠀⠀⠀⡇⢸⡇⢰⣿⠀⣿⢸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇⢠⣶⠀⣿⠀⣼⡇
⣿⡖⣿⡇⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠃⢸⠀⠀⠀⢸⡇⠀⠃⠸⡏⠀⣷⢸⣿⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠈⠀⢸⡄⢸⡿⠀⣿⠁⣿⡇
⣿⣿⣿⡇⠀⣷⠀⡇⠀⡄⠀⠀⠀⡀⢠⡄⠘⠀⢸⠀⠘⠀⠀⠀⠘⡇⠀⡄⠀⡇⠀⣿⢸⣿⣾⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⢸⡇⠸⡇⢸⣿⠀⣿⠀⠐
⣿⣾⣿⡷⠀⣧⠀⡇⠀⣀⠀⡄⠀⠃⢸⠀⠰⠀⢸⠀⠀⠀⠀⠀⠠⡇⢰⡇⠀⣧⠀⣿⢈⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠸⠃⢰⣇⠈⣿⠀⣿⡇⠰⣀
⣿⣟⣉⡧⠀⣷⠀⣿⠀⣿⠀⡗⠀⡇⢸⠀⢀⠀⢸⠀⢀⠀⠀⡀⢀⡇⢨⡇⢰⣧⢀⣿⢸⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣧⣴⡿⣤⣿⣶⠞⣿
⣿⣿⣿⣷⢺⣿⠀⣯⠀⣿⠀⡇⠀⡇⢸⠀⠸⠀⢸⠀⠸⠀⠀⠃⠘⡇⠘⠇⠸⣿⠀⣏⢸⣿⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠑⣿⢸⡏⠀⠘
⣿⣿⣿⣿⢸⣿⠀⣿⠀⣿⠀⡇⠀⡇⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⡿⢸⣿⢸⣿⢾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⡟⠀⡅⠀⠖
⢿⣿⣿⣿⣼⣿⠀⣿⠀⣿⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇⠀⡇⠀⣿⠀⣷⢸⣿⢸⣿⢾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠾⡟⠀⡇⠀⠀⠀⢠
⠘⣿⣿⣿⢹⣿⠀⡿⠀⣿⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠏⠀⡏⢸⣿⢸⣿⢻⣿⢸⡿⢹⣿⣿⣏⣿⣯⣿⡟⡟⠀⠃⠐⡇⠀⠁⠀⢻
⠀⠈⢿⣿⣸⣿⠀⣧⠀⣿⠀⡷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⡇⠘⣿⢸⣿⢘⣿⢾⣷⢸⣿⣿⣿⣿⠋⣿⠀⡷⠀⡦⢐⡇⠀⠁⡴⠋
⠀⠀⠀⠙⠻⣿⣀⣿⠀⣿⠀⣷⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⡇⠐⣿⢸⣿⢸⣿⢸⣿⢸⣿⣿⡏⣿⠁⣿⠀⡇⠀⡇⢈⣇⣼⠟
⠀⠀⠀⠀⠀⠈⠙⠿⢤⣽⣀⣿⠀⡇⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⡇⠀⣇⠀⡗⠰⣿⢸⣿⢸⣿⢸⡇⢸⡇⢿⡇⣿⠁⣿⠀⣇⣸⡷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠷⢠⣧⣠⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⣿⠀⣇⠀⡿⠸⠟⠸⡇⢸⡇⣸⣇⣼⣅⠿⠖⠛⠙⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣋⣈⣤⣤⣧⣴⣦⣤⣾⣤⣷⣤⣥⣼⣄⣈⣋⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠛⠉⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣈⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠋⠀⠀⠀⣠⣴⣾⣿⣿⡿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠷⣦⣌⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⣿⠟⠋⠁⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣤⣉⠛⢦⣙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⣠⡶⠟⠋⢉⣁⣀⣀⡀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣶⣌⡳⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⣰⣿⣿⣿⠟⠀⠀⠀⡠⢚⣡⣴⡾⠿⠿⠿⠿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠉⠻⣿⣿⣿⣮⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⣼⣿⣿⡿⠃⠀⠀⢀⣪⡴⠋⣁⣀⣤⣤⣀⣀⡀⠀⠀⠉⠻⢿⣿⣿⣶⣄⠀⠀⠈⢻⣿⣿⣷⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⠁⠀⠀⢀⠜⣡⠾⠟⠛⠛⠛⠿⢿⣿⣿⣷⣤⡀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⠈⠮⡙⢶⣤⣤⣤⣤⣴⣾⣿⣿⡿⠛⠁⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⣰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⢻⣿⣿⣷⡄⠀⠀⠈⢝⠲⣄⡉⠙⠛⠛⠋⠉⠁⠀⠀⣀⣀⣛⣛⣻⠛⠋⠀⠀⢀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠱⢬⡙⠻⢷⣶⣶⣶⣶⣿⣿⣿⣿⠛⠋⠉⠀⠀⣀⣰⣿⣿⣿⣛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⠙⠷⣤⣄⣈⡉⠉⠉⠁⠀⠀⢀⣀⣤⣴⣿⣿⣿⠿⢋⡵⡫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠈⠻⣿⣿⣿⣦⣄⡀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⠛⢉⣤⠞⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣄⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⢋⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣤⣀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⢉⣡⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣾⣿⣿⡆⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣿⣿⣿⡇⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⣼⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⡄⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡀⣴⡄⢠⣦⢰⡄⣴⠶⠄⣴⣶⠆⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠾⠳⠗⠿⠼⠏⠿⠏⠷⠼⠇⠿⠭⠅⠿⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃
⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣒⣒⣒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠎⠁⠀⠉⠱⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡀⡠⠚⠉⠙⠳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠮⣵⡆⠡⠶⠶⠷⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⢀⣠⢤⠶⠰⠶⠲⠶⠦⡤⣄⣀⠉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠴⣋⡭⠴⠒⣟⠋⠙⠛⠛⠓⠧⢦⣎⣍⡖⠶⣤⣀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⣴⣻⠴⠋⠅⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣶⢤⡉⠲⢥⡀⡑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣰⡿⠋⠀⠂⠀⠀⠀⠀⠀⠐⠉⣳⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠓⢤⡙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢀⣀⠀⠙⢦⠙⣄⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡰⠛⠳⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⢊⡽⠃⠀⠀⠀⠳⡘⣎⢄⠡
⠀⠀⠀⠀⠀⠀⠀⠏⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⣀⠤⠔⠒⠉⢁⢠⠠⣡⡞⠁⢰⠒⢲⠤⡤⠾⣺⠎⠄⠂
⠀⠀⠀⠀⠀⠀⡘⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢾⣩⣄⡆⢐⠊⡄⡰⢈⡾⠁⠀⠀⠀⢻⠏⠀⠀⠄⢹⣇⠆⡀
⠀⠀⠀⠀⠀⠀⡇⡆⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠃⢎⣷⣹⡜⠀⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠌
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⠒⠞⡛⢦⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡏⡔⡁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠧⠀⠀⡠⠿⡀⠀⠀⠀⢠⠊⠀⢀⠠⢐⠀⢀⡠⡔⠋⠁⠀⠀⠀⠀⠀⠉⠐⠠⠀⠀⠀⠀⢐⢀⡗⡰⡃
⠀⠀⠀⠀⠀⠀⠇⠸⡀⠸⡟⠀⠉⠒⠓⠀⠀⡰⠁⠀⠀⢘⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡘⡱⢠
⠀⠀⠀⠀⠀⠀⠘⡄⠱⡀⢫⠀⠀⠀⠀⢀⠎⢐⣡⡴⠞⠉⠀⠀⠀⠀⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢀⣟⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠣⡀⠙⢤⠀⠰⣷⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢀⠀⠀⠀⠀⠀⠀⠨⢫⣿⠎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠙⣄⠀⠙⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⠛⡽⠀⠀⠀⠀⠀⠀⠀⢠⠔⡵⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡐⠱⢦⣀⠈⠑⠲⣤⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⠀⣀⡸⠖⡁⢌⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣆⡎⡕⡢⣤⣀⠈⠙⠒⠶⠤⠤⢠⢀⣄⣤⡤⠤⠽⠖⠖⠒⡉⠕⢐⡤⠖⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣸⢩⢓⢺⢦⠤⣀⣂⣂⡂⠂⠀⣂⣀⣘⣨⠤⠶⢊⣡⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠺⠶⠤⣤⣴⣸⣉⣉⣁⣁⣄⣌⡬⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠄⡠⡂⠔⡠⢒⡠⠒⡠⢐⡠⢀⠤⢀⠀⡠⢀⡉⢉⠉⣉⠄⣀⠄⡠⢄⡠⢂⠤⡀⠄⡠⢀⠄⡀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠊⠐⠁⠔⠡⠊⠡⠊⠔⠉⠖⠡⠊⠡⠊⠡⠊⠐⠁⠔⠡⠊⠰⠊⠞⠰⠋⠔⠑⠁⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⢀⣤⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣄⡀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠘⢿⣧
⠀⢀⣀⣀⣀⣀⣀⣀⣼⡿⠀⠀⣰⣿⣃⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠘⣿⣇
⠀⠈⠻⣿⡿⠿⠿⠿⠿⠃⠀⠀⠿⠿⠿⠿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⢹⣿⡀
⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠘⣿⡇
⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠈⠛⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠛⠁⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⣿⣿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⣴⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠷⠦


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⢤⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢸⡋⣷⢹⣋⢹⡇⣷⢸⢿⣴⡿⢷⢀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣥⣾⣿⣧⣼⣿⣬⣿⣿⣿⣿⣤⣽⣯⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⣾⣿⣿⣷⣦⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣼⣿⣿⣿⣿⣿⣿⣧⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣾⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⡄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⣿⣿⣿⣿⡟⠀⠀⠀⠀⣻⣿⣿⣿⣿⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢷⡀
⠀⠀⠀⠀⠀⢠⡞⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢳⡄
⠀⠀⠀⠀⣠⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠹⣄
⠀⠀⠀⣰⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠹⣆
⠀⠀⣴⠃⣰⣿⡿⠿⣿⣿⠿⢿⣿⡿⢿⣿⣿⠿⣿⣿⣿⡿⠿⠿⠿⢿⣿⡿⣿⣿⣿⡿⠿⡿⠿⣿⢿⣿⣿⣿⠿⣿⣿⡿⠿⠿⢿⣿⣿⣆⠘⣦
⢀⡼⠃⣰⣿⣿⣿⠀⢻⡏⢀⠘⣿⠁⣸⣿⠃⣤⠘⣿⣿⡇⢰⣶⣦⠀⢹⡇⠈⠻⣿⡇⠀⡇⠀⣿⠀⠙⢿⣿⠀⣿⠏⢠⣴⣶⣤⣿⣿⣿⣆⠘⢧
⣼⠁⣼⣿⣿⣿⣿⣇⠈⠁⣾⡄⠋⢠⣿⠇⠀⠛⠀⠸⣿⡇⢀⣍⠁⢴⣿⡇⢸⣦⡈⠃⠀⡇⠀⣿⠀⣷⣄⠙⠀⣿⡀⠸⣿⣤⠀⢸⣿⣿⣿⣧⠈
⣿⠀⣿⣿⣿⣿⣿⣿⣀⣰⣿⣧⣀⣼⣏⣠⣿⣿⣿⣄⣹⣇⣸⣿⣧⣀⣻⣇⣸⣿⣿⣦⣀⣇⣀⣿⣀⣿⣿⣷⣄⣿⣷⣤⣀⣁⣠⣾⣿⣿⣿⣿
⠹⣆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣰
⠀⠈⠳⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠁⣿⡎⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠳⣦⣀⠀⢠⣿⣿⣿⣇⠀⢀⣠⡶⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣾⣿⡟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣀⢀⡀⣀⠀⢸⠇⠀⢀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡀⠀⠀⣿⠀⠀⣀⣤⡄
⠀⠀⠀⠀⢸⣾⢷⡟⠀⠒⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠖⠂⠀⣸⣒⡂
⠀⠀⠀⠀⠀⠁⠈⠁⠀⢸⡇⠀⠀⠀⠈⠉⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠉⠀⠀⠀⠀⡿⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣴⣿⠿⢻⣿⣿⣿⣿⠻⣿⣷⡀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠼⠋⠁⠀⠘⣿⣿⣿⡏⠀⠀⠙⠳⠄⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣄⢿⢃⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⢐⡠⣶⠀⠠⣶⣤⣶⠀⠀⠀⠀⠀⣦⣤⠀⠀⢠⡶⣶⡀⣶⢆⠀⢠⣶⣦⣶⡶⠀⠀⢠⡄⣤⠄⠀⠀⠀⠀⠀⠰⠀⢀⣀
⠸⠛⠳⣿⢰⠷⡏⠀⢈⣀⣄⡠⢄⣐⣿⣿⠀⢀⣸⣧⡍⠀⠰⡆⠀⢰⢹⡇⢹⣿⠀⢀⠿⠀⡿⠀⠀⠀⠀⠀⠀⣤⣤⣸⡏
⠀⠀⠀⣽⠈⢶⠓⣀⠈⢉⡛⠋⣾⣏⡿⠀⡀⠀⠻⣿⠅⠐⣛⣷⡌⠸⢸⡇⣸⣷⣦⣱⣤⠰⣦⠄⠀⠀⠀⠤⠀⠹⠿⠏⣧
⠰⣆⠀⢾⠖⣿⣿⣿⠀⠸⠿⣾⡆⣀⣽⣧⠀⠀⠸⠟⠀⠀⠈⠰⠀⢦⢠⣄⣛⣿⠇⠉⠿⣄⢨⣵⡦⠀⠀⠀⠀⠀⠠⠖⠿
⠠⢿⢆⡼⣶⣾⣿⡟⢳⡄⠀⠛⢣⣿⣿⣿⡇⠀⣹⢾⠀⠀⠀⠸⣷⣾⡮⠟⢛⣯⢄⠀⠀⠙⡴⠁⠠⠐⠀⠀⠀⠀⠀⣴⣄
⠀⣯⠈⢠⣿⣾⡍⠀⠈⢇⡀⣤⣀⣿⣿⣿⣥⡐⠁⠘⢂⢀⣤⡀⠘⠘⢸⣀⡌⠁⣄⢠⡼⡞⡟⠦⠀⠀⢠⡾⣯⣿⣎⠀⢀
⢸⣷⣦⣼⣿⣿⡇⢶⡆⠸⠟⢹⣿⠿⢿⣧⢍⡴⡄⠁⠀⠈⠛⠁⣠⣶⣸⡶⠃⠠⡩⢀⣀⡖⢡⣦⠀⣈⡉⠀⢀⣧⡾⠀⠎
⢸⡿⢻⣟⠛⠉⠀⢀⠀⠀⣴⡌⠉⠀⠀⠀⣤⣤⣤⠆⣀⣀⠀⠀⠈⠉⠛⠷⠂⠀⠹⠟⣛⠁⠀⠉⢰⠏⠄⢀⡬⠉⣿⣶⣾
⠈⢀⣹⣯⣀⢻⣶⣾⣷⡇⠉⠁⠀⢰⡦⠸⣿⣿⣿⡀⠀⠀⠠⣠⡁⠀⠀⠀⠰⣤⢾⣿⡿⢨⣿⣧⠀⡿⡃⠈⠅⠀⠀⣸⣿
⠀⢸⣿⣿⣿⡄⢛⣿⣿⣆⣴⣄⣿⡏⣭⣙⠋⣻⣀⣥⡀⠀⣰⠀⠉⠀⠀⠀⠰⢾⠀⢹⣧⠈⠁⠀⠀⣄⢹⣀⠀⠀⢠⣍⠈
⠰⡈⣿⣿⣿⣤⣿⣻⠿⢭⡿⠟⠇⠁⠀⡋⠁⠉⣽⣿⠃⢸⣯⠀⠀⢸⣧⣀⢀⡀⢀⣾⣿⡀⢀⠀⢘⣻⣿⠀⠉⠞⠋⠻⣀
⠈⠃⠀⠀⣸⣿⣿⣿⣀⣼⠇⠀⠠⣦⠀⠙⣄⠰⠛⣿⡄⢸⡄⠇⠀⢸⣿⠟⠿⠟⢸⣿⠀⠈⣿⣧⠿⠟⠀⠀⡀⠀⠀⣀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⡶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣤⣶⣶⣾⠱⣶⣶⣤⣦⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⢀⣌⡀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢻⣿⣷⣿⠀⠀⠀⠀⠀⢠⡿⠁⠀⢰⣞⡭⠯⣭⣳⡂⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢀⣀⣛⣛⣉⣉⣻⣤⣾⣿⣿⣿⣄⡀⠀⠀⠀⠾⡇⠀⣀⣾⡁⢀⠀⠀⢈⣷⣀⠀⢸⠷
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠉⣷⣤⣤⣴⣶⠁⠀⠀⠀⠀⢻⡆⠀⠀⠈⠧⠀⠀⠀⠾⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣄⠀⠈⠳⠦⣦⠴⠞⠁⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢣⡟⠛⠛⠉⠋⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣀⠀⠀⠀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⣀⡀
⠀⠀⠀⠀⠀⠉⠙⣿⠁⠀⠀⠁⣿⡆⠀⠈⣿⠏⠉⠁⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠄⠀⣿⣷⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⠀⠀⠀⢴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣦⡙⣦⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣈⠳⣌⠳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠳⣌⠳⣌⠳⣌⡓⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⢲⣬⡓⢮⡛⠦⣝⣓⠶⠾⢽⣗⣲⣲⢿⣯⣙⣿⣿⣿⡟⢃⣿⠝⣳⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⡿⣿⢠⣝⡲⢭⣓⠦⠭⡭⠭⢽⣛⡶⢽⣛⠓⣶⢃⡴⢋⡴⢋⣥⠞⣋⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢹⢣⡏⡇⣍⣓⠮⣝⣓⣒⡯⢽⣲⡭⢡⢏⡞⣱⢏⡴⢋⡴⢋⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⡾⡼⡼⡽⡬⣝⣒⠶⢖⣪⡭⣰⢳⢏⡞⡼⢡⠟⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡇⡇⣇⢳⢳⢱⡹⡨⣭⣵⢰⢣⡇⡏⡾⢸⢣⡏⡾⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⣸⢃⡇⡾⣸⢸⢣⠇⢟⣛⡊⣎⢇⢧⢧⢹⢸⢸⢇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⣴⢣⡞⡼⣱⢣⠏⣚⡫⠵⠶⠭⣝⣘⣞⡞⡞⡼⡾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣡⠞⣡⠞⣱⢏⡼⣱⣃⣚⠯⣗⣺⠭⢭⣝⡲⢮⣙⢸⣱⢳⣇⡏⡇
⠀⠀⠀⠀⣤⠶⠛⣩⡴⢛⣡⠞⣡⠞⢡⠯⢤⣍⣓⠮⢭⣗⣒⣒⣒⠲⢭⣓⠮⣝⠃⡿⣾⠿⡇
⠀⠀⠀⠀⠈⠐⠻⢯⣔⣻⣡⣼⣿⣿⣿⣍⣻⣷⠼⠯⣽⣗⡲⠶⢭⣝⠲⣍⡳⢬⡛⠧⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⢮⡙⢦⡙⢦⡙⢦⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡙⢦⡉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠻⣌⠻⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣀⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠘⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⢀⣼⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣄
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢀⣠⣴⣄⣠⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣄⣠⣦⣄⡀
⠀⠀⠀⠀⠈⠙⠻⠋⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡶⠒⠲⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣾⣷⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣄⣠⢼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠛⠛⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠖⢻⡟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠿⠿⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⣛⣡⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷⠆⠶⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣋⣑⣛⣛⣃⣛⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⡆⢰⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣦⣄⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⣉⠙⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠛⢉⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⡰⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠖⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠳⣄⡀⠀⠀⡀⢹⡀
⠀⠀⠀⠀⡤⠤⢤⣀⠀⢸⠀⠀⢸⠈⠉⠉⠉⣉⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⢹⡁⠀⡇⠀⠀⣀⡠⠤⣄
⠀⠀⠀⠀⠱⣄⠀⠈⢹⠞⠓⢲⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⢳⡤⣷⠒⠉⠁⠀⣠⠋
⠀⠀⠀⠀⢠⠞⠓⠦⠹⣀⢀⡸⠃⠀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⡏⠀⠀⡇⠀⠴⠾⢤
⠀⠀⠀⠀⠈⠳⢄⣀⡀⠈⠹⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢹⠖⠚⠁⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⢠⡋⠉⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠈⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠉⠒⣲⠖⠀⠈⠢⣄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⡴⠋⠀⠐⠲⡒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢶⠖⠀⠈⠙⠲⠤⣄⣀⣀⣀⣀⣀⣀⡠⠴⠚⠉⠀⠘⠲⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠔⡺⠁⢀⣴⡟⢣⡈⠉⠉⢁⡜⢻⣦⡀⠈⡗⠲⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠋⡼⠀⠀⠈⠉⠉⠁⠀⠀⢳⡙⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣳⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉⠉⠙⠧⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢳⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⢠⡖⠚⠋⠉⠉⠉⠉⠙⠓⠲⠤⣄⡀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠦⣌⡉⠓⠲⠦⣄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠑⡆
⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⢁⡦⣀
⠀⠀⠀⠀⠀⣰⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠀⠈⠓⢤⡀
⠀⠀⠀⠀⢰⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣉⡷⠄
⠀⠀⠀⢀⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠶⡄⠀⢹⡀
⠀⠀⠀⢸⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⡇
⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧
⠀⠀⠀⢻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⢸⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡇
⠀⠀⠀⠈⣇⠀⠘⠶⠒⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣸⠁
⠀⠐⢾⣉⠁⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠈⠳⢤⡀⠀⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⠺⢁⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠲⠦⣤⣈⡛⠲⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠈⠙⠲⠦⢤⣄⣀⣀⣀⣀⣠⡤⠼⠃⠀⠀⠀⢈⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⢧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣀⢀⡞⠀⠂⠄⠀⠀⠀⠀⠀⠓⢸⢃⠘⣿⠁⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠏⡜⣀⢣⢣⠰⣿⣟⣞⣷⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢸⡇⠠⢸⡄⠀⠀⠂⠀⠐⣡⢎⠈⠐⢿⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⡘⣡⢫⡌⣎⢦⠹⡜⣞⢦⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠀⠐⢨⠀⠄⠃⠜⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠡⠙⢯⡜⡌⢆⠳⡝⢎⠳⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠑⡈⢞⣜⢦⡑⣜⠦⣑⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⢈⣽⣿⣶⣶⣦⣤⡕⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⢆⠻⢷⣝⣮⡓⢬⡓
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢰⠐⠂⠀⢉⠀⡀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⢄⡙⢿⣟⣲⢬
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⠀⠀⠀⡀⠀⠢⠀⠈⠄⢀⢈⣅⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣦⣍⠢⢍⡑⠻
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⢀⠣⣄⠀⠈⠀⢀⠀⠀⢀⣴⡾⠋⢀⡀⢴⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡎⣿⢻⡟⣶⣬⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⠈⠑⠂⠀⠀⠀⣠⣿⣿⠃⠀⠁⠀⣼⡀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⢸⠃⡇⣽⡈⡇⣏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⣠⣿⣿⣿⡁⣀⠲⣈⠰⣿⠀⠠⣤⣔⠶⣯⠀⠀⠀⠀⡀⠐⣒⣤⢌⡓⠤⣙⠳⣄⡀⠀⠀⠀⢸⠀⡅⣿⣧⡄⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣶⣶⠏⠸⢻⣝⢚⣤⣍⠀⢸⡟⠒⠂⠒⠉⠄⣐⠀⠀⠀⠀⠀⠀⠉⡛⠿⣟⣷⣄⡑⠆⡙⢦⠀⠀⢸⡄⡇⢻⣿⡇⡏⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠍⠀⡀⠡⠈⠠⠉⠉⠀⡀⢾⠇⠀⠠⠤⠀⣀⠉⠀⠀⠀⠀⢀⡐⠂⠤⣓⠤⡙⠿⡻⣦⡙⢆⠳⣄⠈⣇⢣⠘⣿⣧⢱⢳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠘⢿⣥⠀⠀⠀⠀⠀⠀⠐⠮⠕⡂⢽⣆⠀⠤⠈⠁⣂⠨⠭⠥⠬⠭⢔⣒⠩⣐⢄⡑⢎⡲⡙⢿⣫⡤⠱⠘⢆⠸⡌⣆⠹⣷⢇⠳⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠜⢳⡚⠐⠀⠀⠀⠀⢤⣒⣮⣽⣿⠁⣐⣓⣂⡀⠤⠐⣛⣛⣛⣓⡢⠭⣛⠮⡓⢝⠢⡑⡜⢆⠙⡵⢃⠀⠈⢆⠹⡌⢆⠻⣯⣷⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⠹⣢⠔⣀⣴⣞⣿⣟⠓⣽⣿⣧⠶⠶⠶⢶⣞⡛⣯⣭⣭⣿⣻⢷⣮⡳⣌⢦⠓⡱⡘⡆⢀⢱⢫⡆⠠⠸⡀⠙⣌⠳⡜⠷⣮
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠄⠉⠀⠀⣙⡳⠮⣛⣿⣟⠻⠿⠿⠿⠷⠾⣙⣻⠿⢿⣛⢿⣿⣟⢷⡜⣆⢡⢱⢣⠀⠄⡄⡞⣱⠀⡆⢇⠀⠈⠳⢌⠲⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡶⣼⡻⢯⣿⣇⣬⠉⠉⠉⠉⢭⣄⠒⢾⡖⢬⠳⣹⣿⡼⣷⠸⡇⡇⡎⠀⢸⢠⠀⡟⡀⢠⢸⠀⠀⠀⣼⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠠⠤⢬⡛⠦⠝⠛⠻⡛⠶⢛⣛⣛⣛⠶⠮⣹⣿⢿⡇⡆⣿⢸⡃⣿⠀⠇⡇⡇⡄⢸⢸⢰⡇⡇⢸⢸⠀⠀⢸⠃⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡠⠬⠛⡳⠒⠒⠚⠛⠋⣅⠪⢒⠕⣠⣻⣼⢇⢿⢀⡆⡇⡇⠁⢨⢸⠸⡇⡇⢸⢸⠃⠀⢸⢀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡩⢔⠲⢬⣭⣉⡉⠉⠛⠓⠮⢑⣡⢞⣵⡿⣫⣾⢃⡞⡴⠱⠸⠀⡌⡆⣦⢿⠀⡄⡼⠀⠀⢸⠘⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢲⣸⡕⢊⡥⣒⢲⣶⣦⣤⣭⣽⣻⡾⡿⢫⣾⠟⣡⢚⠞⡡⣣⢃⡆⡔⢰⡝⡎⣰⢡⠇⠀⠀⠘⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⣒⡲⠶⣶⣶⣦⣤⣿⣿⣃⡔⣡⣶⢿⣛⣿⣿⣯⣽⣭⣭⡷⠶⢛⣡⢞⡵⡡⢊⠔⣡⢎⠌⣰⡷⡝⣠⢀⡎⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠔⠉⡳⠭⢛⣒⣶⣿⣿⣿⣿⣿⣷⣿⣎⣿⠿⢋⣭⣔⡲⠶⠶⠶⠶⢯⡝⠒⠥⠂⠄⣡⠊⠕⣣⣾⣿⠟⡱⢣⡞⠀⠀⠀⠀⠀⠀⢣⡘
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠁⠀⢀⣡⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⡽⠓⠈⢉⡉⠉⠐⠒⠒⠂⠉⢂⡠⢀⠁⠀⢄⢙⠿⣩⠞⣵⠋⠀⠀⣀⣀⣠⢴⣓⡈⠿
⠀⠀⠀⠀⠀⢀⠀⠈⡄⠀⢀⢮⢾⠟⣿⣿⢟⣻⣭⣭⣭⣿⣿⣿⣿⣧⣿⡐⡴⢉⠄⠒⠤⠤⠤⢤⣬⠍⠐⠂⣀⠠⢀⠼⣡⠈⡥⠛⠀⣠⠔⣭⡔⠂⠲⣝⢶⣭⣙
⠀⠀⠀⠀⠀⠈⠀⡇⢘⠀⡞⣇⢣⡸⢏⡴⡫⠛⠂⢉⠉⠉⠛⣿⢿⣿⣷⣿⡐⡑⢊⢤⡐⠶⠶⡶⢶⢶⣬⡙⠢⢘⠠⢀⡠⠊⢀⡠⢖⣵⣮⢖⡮⣝⡛⣿⣛⡢⠬
⠀⠀⠀⠀⠀⠀⢠⢁⡄⠀⠇⣽⢈⡇⢸⢱⠁⡀⠊⡴⢮⢙⣻⣿⣿⣿⣿⣿⣿⡀⣾⡷⣪⠍⢒⣒⣒⣒⣀⣀⣭⡥⠖⠉⠀⡴⢫⢞⢡⠾⣿⡗⣩⠖⠒⣿⣗⣈⣭
⠀⠀⠀⠀⢀⠴⡻⣿⣿⡧⣼⣧⣾⣧⡾⣸⢀⡇⢸⢸⠰⣾⣟⢿⣿⣿⣿⣿⣿⣿⣿⠐⣡⠚⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢰⣕⣇⢯⣻⢝⣻⣿⡕⣨⣤⣾⣿⣟⣛
⠀⠀⠀⢀⢇⢺⡁⡔⣰⣿⣿⢿⣿⣿⣿⣷⣮⣷⣄⣿⣸⣿⣥⡆⡞⠋⠛⠿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣱⣋⠖⣩⣿⣟⣽⣲⣿⣷⠺
⠀⠀⠀⢸⢸⡇⡇⠀⢻⣿⡟⣿⢣⡿⡟⠙⡁⣰⢟⢿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠙⠿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⢋⠼⣻⣿⣿⣧⣾⡟⣹⣿⣟⣭⣿⣟⣛
⠀⠀⠀⠀⢯⣾⣵⣆⢿⣼⣧⣿⡸⣿⢇⢷⡁⣵⢀⠜⣷⢻⠉⡽⠛⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡤⢺⡝⢾⣿⣿⣮⣵⣮⣿⡝⡿⣿⣿⣿⣟⣽⣿⣯⣾⣷⣶
⠀⠀⠀⠀⠀⠉⠛⢿⢿⣿⣿⣿⣿⣿⣿⣞⣷⣜⢧⣜⣿⣾⣷⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣱⢫⣾⢿⠿⣿⣿⣿⣿⡧⢞⠝⣹⣿⣿⣿⣿⣜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣸⡄⣅⢸⣿⡟⡻⢿⣿⡟⠳⢻⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣻⡜⣧⢣⡞⡱⢫⣽⣿⣿⣶⣾⣽⣿⣿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣼⣥⣹⡙⢁⣾⢯⣿⣽⡟⣼⣄⠂⠸⢸⠁⠁⡿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡴⡻⢿⣿⣿⣿⣼⣏⣠⣷⢫⣿⡿⣿⣿⣿⣟⣫⣾⢟⣿⣿⣿⣿⡏
⠀⠀⠀⡰⣺⣿⣿⢿⣿⣿⣿⣿⣿⣾⣿⣼⣽⣅⣀⢀⣞⠄⣠⣳⡜⡾⢩⠇⠀⠀⠀⠀⠀⢸⡀⣇⢻⡹⢏⡟⡿⢿⣿⣿⣿⣏⣿⡟⡽⣻⣿⣿⣱⣫⣿⡟⣼⣿⣧
⠀⠀⣼⢱⡹⡝⢁⣾⢿⣿⣿⢟⣿⠿⠛⡩⣹⡿⢿⣿⣿⣿⣿⣿⣽⣷⠏⠀⠀⠀⠀⠀⠀⢸⣧⡏⣼⣇⡾⣸⢰⣿⣯⣿⢿⣿⣿⣿⣾⣿⣫⢿⣿⣿⣯⣼⣟⣿⣻
⠀⠀⣷⢸⠇⡇⢸⣼⡏⡟⣿⢸⢌⠀⡄⠰⢹⠀⣞⣷⢳⠁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⣽⣿⢻⣿⣿⣧⡝⠻⠿⣯⠞⠁
⠀⠀⠹⣎⡾⣵⣄⢯⡾⣷⣿⣎⢾⡶⡷⢦⢻⢶⠼⡿⣿⣷⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣧⣻⣿⣿⣯⡟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⢸⢠⡇⣿⢸⣷⣇⣿⣿⢸⡇⡆⢹⢘⠰⠀⡇⣿⠰⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢯⣿⣿⣿⣟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣛⠁
⠀⠀⠀⣷⡇⣿⡿⢸⡇⣿⢿⡏⣸⡇⢉⢸⢸⣶⠀⣷⢸⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡏⣿⢻⢻⡏⡟⣿⣿⢻⡟⣿⣿⠛⢻⡟⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⢿⡿⣿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣺⣇⣿⣲⣇⣿⣸⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⠰⣼⡆⣾⢱⢸⠠⢌⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⡟⢻⣟⠛⣻⣿⠛⢻⣿⣿⠟⠁⠀⠀⠈⠻⣿⣿⣿⡓⣸⡇⣿⡸⣸⣰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢰⣶⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⣿⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠸⡏⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠐⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⣿⠁⢀⣀⣄⣀⡀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣷⡄
⣿⣿⣧⠀⠀⢀⣴⣿⣿⣿⡆⠀⠀⣀⣤⣄⡀⠀⠈⠛⠛⠁⠀⣿⣿⣿⣿⣷⠀⠀⠸⣿⣿⣿⠿⠛⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⡁⠠⣾⣿⣿⣿⠇⠀⢀⣀⣀⠀⠀⠻⣿⣿⠏⢁⣀⡀⠀⠈⠉⠁⣠⣤⣄⡀⠀⠈⠻⠿⠿⠛⠛⠛⠁

⣿⣿⣿⡇⠀⠈⠻⣿⣿⣿⠇⠀⠹⣿⣿⣿⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣷⠀⠀⣼⣿⣿⣿⣿⠄⠀⠀⢀⣠⣤⣄
⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠿⠿⠇⢰⣿⣿⣷⠀⠈⠿⠿⠋⣁⠀⠀⠹⠿⠟⠛⠋⠀⠀⢠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣠⣶⣶⡄⠀⣰⣿⣿⡇⠀⣴⣾⣷⠀⢀⣠⡀⠈⠿⠟⢠⣴⣦⡀⠀⣾⣿⣿⡆⠀⢀⣴⣷⣦⡀⠀⠸⣿⣿⢿⣿⣿⠃
⣦⡀⠀⢰⣿⣿⣿⣧⠀⠻⣿⣿⡷⠀⠻⣿⡿⠀⣿⣿⡇⣤⣤⡄⠘⠿⠟⣁⡀⠙⠛⣉⡀⠀⢺⣿⣿⣿⠇⠀⠀⢀⣀⡀
⣿⣧⣀⠈⠻⢿⣿⠿⠁⠀⠀⠉⠀⣠⣄⠀⣠⣤⠈⣉⡀⠻⠟⢰⣶⡄⠸⡿⠟⠀⢼⣿⣿⠆⠀⣠⣤⡀⠀⢀⣾⣿⣿⣿⡆
⣿⣿⣿⠀⠀⠀⠀⠀⢀⣶⣷⡆⢸⣿⣿⠈⠿⡿⠘⢿⠇⣶⡆⣈⠉⣶⠆⠰⣿⠷⠀⣴⣦⡀⢸⣿⣿⣿⠀⠈⠻⠿⢿⣿⠟
⠛⠋⠁⠀⢠⣶⣷⡄⠸⢿⣿⡿⠀⠈⠉⢠⣤⢠⣶⠠⣶⢨⣄⡛⢴⠀⠾⡂⢴⡶⠈⣛⡛⠁⣠⣤⡄⠀⢀⣤⣶⣦⠀⠀⠀⠀⣀
⠀⢀⡀⠀⢸⣿⣿⣿⡇⠀⢀⡀⢰⣿⣧⠙⠛⢁⡍⣴⠰⠦⠬⣝⢢⠩⠌⣏⠰⠶⢘⠻⠇⠈⠛⠿⠟⠀⠸⢿⣿⣿⡇⠀⢠⣿⣿
⣼⣿⣿⣦⣄⠉⠉⠉⠀⠀⣿⣿⣦⠉⣉⠰⣿⡎⣉⠴⢜⡨⠍⠂⠁⢁⢣⠃⡝⢃⡛⠗⠐⢿⣿⠀⢠⣶⣶⡆⠈⠉⠀⠀⠈⠻⠿
⢿⣿⣿⣿⣿⠂⠀⣴⣶⣄⡉⠉⠁⠀⢿⠿⠀⣶⡍⣡⢋⡖⣦⡀⠀⠀⠭⢚⡰⠎⣃⡸⣿⠆⢀⣀⣈⠻⣿⠏⠀⢠⣶⣶⣶⣦
⠈⠙⠛⠉⠁⠀⠀⢻⣿⣿⠟⠀⢾⣿⡶⠀⣶⡦⠰⠶⢘⡊⣇⢆⠪⢭⣫⢩⡵⡶⠙⠃⣠⣄⠛⣿⡿⠀⢀⣀⡀⠘⢿⣿⣿⡿
⣿⣷⠀⠀⢠⣶⣶⣤⣄⠀⢀⣴⣤⣅⠀⣴⣶⡄⠺⠟⠨⡦⠹⢆⡛⠶⢰⡆⣶⡄⣿⡦⠹⠿⠁⣀⡀⠀⣿⣿⣿⡇⠀⠙⠛⠁
⣿⠟⠀⠀⠘⣿⣿⣿⡿⠀⠀⠻⡿⠟⠀⣈⣩⡀⠺⠿⠀⠺⢇⣈⠡⣾⢠⣤⡈⣤⡄⢠⣦⣄⠸⣿⣿⡆⠈⠿⠿⠃⠀⢀⣀⡀
⠁⠀⠀⣀⣀⠈⠉⠁⠀⠀⣶⣦⣶⣄⠀⠻⠿⠃⠐⣿⡿⠀⠻⠟⣠⣤⠈⠋⠐⠿⠟⠘⠿⠟⠀⠛⠛⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠻⣿⡿⠃⠀⣴⣶⣿⡆⠀⣴⣿⡇⠈⠛⠋⣸⣿⣆⢠⣶⣄⠀⣠⣶⣄⠀⢰⣿⣿⣷⣦⠈⢻⣿⣿⣿
⠀⠀⠈⢻⣿⣿⡿⠋⠀⠀⣠⣄⣤⣤⡀⠈⠛⠛⣀⣀⠉⠉⣡⣾⣦⠀⠉⠋⠁⢾⣿⡿⠃⣹⣿⣿⡗⠀⣿⣿⣿⠏⠀⠈⠻⠿⠋
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⠀⠀⠻⠿⠟⢠⣾⣷⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠈⠛⠁
⠀⢠⣾⣿⣷⣿⣿⣷⠀⠀⠈⠙⠛⠁⣀⠀⠈⠛⠛⠋⣠⣤⣄⠀⠀⣿⣿⣿⠇⢰⣿⣿⣧⠀⠀⣴⣶⣦⡀⠀⠀⣴⣾⣿⣶⣄
⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣶⣿⣿⣷⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠈⣿⣿⣿⣿⡆⠀⢿⣿⣿⣿⣿⣷
⠀⠀⠈⠛⢿⠿⠋⠀⠀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠛⠿⠿⠛⠀⣶⣿⣶⡀⠀⠈⠉⠉⠀⠀⠸⣿⣿⣿⠟⠁⠀⢼⣿⣿⣿⣿⠏
⠀⠀⠀⢀⡀⢀⣤⣤⣄⠀⠀⠀⠉⠙⠋⠀⣠⣤⣄⠀⠀⠀⢰⣾⣿⣿⣿⡇⠀⢀⣠⣄⡀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠚⠛⠛⠓⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠶⠋⢉⠀⠀⢸⠀⠀⠀⠉⠛⢦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⡞⠁⠀⠀⣿⠳⣄⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢠⡟⠀⠀⠀⠀⢸⡄⠈⠻⣦⡀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⢸⣧⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠒⠒⠂⠀⠀⠀⠁⠀⠸⠇⣀⣤⡀⠀⠀⠐⠒⢺⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡄⠀⠀⠀⠀⠀⠀⢠⣶⠛⠋⢸⡇⠀⠀⠀⠀⢸⠇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠈⣇⠀⠀⠀⣠⡟⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠻⣄⠀⠀⠀⠀⠀⢀⠀⠈⠳⡿⠀⢀⡴⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠛⠶⣤⣀⣀⣸⣃⣀⣠⣤⠞⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠉⠉⠉⠉⠉⠁⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠶⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⡀⠀⢀⡀⡀⡀⡀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⢀⠀⢀⢀⡀⠀⡀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠑⠀⠀⠁⠂⠐⠂⠀⠀⠐⠀⠁⠁⠈⠀⠀⠂⠀⠀⠀⠓⠂⠐⠂⠐⠀⠂





⡶⠶⢶⠶⡶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⢶⣶⣶⣶⢶⡶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⡀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢿⡷⠈⢻⡄
⠀⠀⠀⠀⠀⢀⡾⠃⣸⣀⠀⢀⣷⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠰⠜⠠⠤⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣴⣮⣴⡄⠀⠻⣄
⠀⠀⠀⠀⢠⡾⠁⠊⠉⠙⣥⠋⠉⠑⠀⢻⡄⠀⠀⠀⠀⠀⢀⡾⠃⢠⣦⡀⠠⠺⣔⠁⠄⠻⣆⠀⠀⠀⠀⠀⠀⣼⠋⣠⣘⣿⣿⣿⣯⣠⣄⠹⣦
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠤⠿⠦⠀⠀⠀⠀⠻⣆⠀⠀⠀⢀⡾⠁⠀⠘⠿⠷⠆⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡼⠃⠈⠻⠟⠀⠉⠉⠀⠻⠟⠀⠙⣧
⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠃⣆⡀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣼⠇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣠⣴⣦⣄⢻⡄
⠀⠀⠀⠀⠀⢀⡾⠃⢠⣽⣿⣶⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⣯⣴⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⣇⣼⣇⣼⠇⠻⣆
⠀⠀⠀⠀⢠⡾⠁⠀⣼⡿⠛⢿⣿⡀⠀⢻⡄⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠉⢰⠇⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣼⠃⠀⣴⣬⣛⣛⡵⡆⠀⠹⣦
⠀⠀⠀⢠⡟⠀⠀⠀⠻⡅⠀⠠⠟⠀⠀⠀⠻⣆⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡾⠃⠀⠀⠻⠚⠉⠉⠳⠇⠀⠀⠘⣧
⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⣿⡇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠘⣿⡇⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠠⢄⠜⢄⢀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣾⣷⢀⡠⣾⣧⠹⣆
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠻⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⢀⡾⠁⠂⠀⡀⡘⠃⢥⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⣼⠃⠈⠉⠁⣻⣋⠉⠉⠀⠹⣦
⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡾⠃⠀⠀⠀⠰⠿⠿⠆⠀⠀⠀⠘⣧
⠀⠀⠀⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠄⠀⠀⠀⠘⠲⡶⣶⡄
⠀⠀⠀⠀⠀⣇⠀⢸⣛⡉⠀⢉⣀⣀⣒⣀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡌⠂⠠⣄⡀⠰⠻⣷⠀⠀⢀⠀⣀⣀⢀⡴⠖
⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⠟⢋⠛⢻⡿⢿⣿⠛⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣥⣦⡀⣁⠉⢻⡇⢰⣾⣿⣿⡿⠃⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠙⣿⣄⢼⡆⠀⠙⠚⠃⢰⣿⣧⣤⣶⣶⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⣿⣍⠀⠘⠋⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠘⣾⣭⠅⠀⠀⠐⣾⠃⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣄⢒⡗⠀⠀⠘⠟⡁
⠀⠀⠀⠀⠀⢻⣧⠄⠀⠀⠀⠀⠀⢀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡻⠟⠻⣿⣎⣀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⣇⣀⡠⠤⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡿⠃
⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠐⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡆⣠⣤⣄⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡜⣿⣿⠟⠂⢇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣈⡓⠀⠀⠠⠄⢂⣀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⣀⠀⠀⣰⣶⢿⣥⣤⡀⠉⠂⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣷⡜⢿⡇
⠀⠀⠀⠀⠀⠀⢀⣹⣾⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡆⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠙⢻⣿⡿⢃⣾⣧⠸⣿⡿⠿⢻⡿⡞⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⢿⡷⠶⠒⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠈⠉⠁⣼⣿⣿⣧⠈⠀⠀⠈⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⡇⠈⠁⠙⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⡀⠀⠿⠿⡿⠿⠀⠀⣀⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⢠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡁⣿⠀⠀⠀⠀⠀⠀⡾⢁⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⡿⠙⠉⠥⠼⣿⣿⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡥⠐⠒⠳⢶⡭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠛⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⢀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣤⣤⣤⣤⣤⣴⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡌⢳⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠹⣦⠻⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠶⣟⠢⣌⡑⢒⣴⠶⢋⣠⡴⠾⠛⢋⣁⣤⣤⣴⣾⠷⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡴⣌⠳⣌⠳⣌⡓⠶⢤⡤⠴⣒⣮⢗⠯⡣⣽⢊⡼⢋⡴⢋⣵⠞⣋⣥⡶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣱⢏⢛⢮⣙⠲⢭⣛⣛⣻⢭⣻⡺⡖⣵⢠⢏⡞⡴⢋⡴⣋⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⣫⢞⠓⠮⢭⣒⡲⣶⣒⣭⡮⢺⣥⡇⠏⡞⡼⣱⢏⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣳⠬⣛⡓⣒⡾⢿⢖⡿⣺⡆⢎⢹⢸⠑⡇⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⣽⢿⣏⣭⢤⢶⡒⣪⣄⢇⢎⢧⢧⣣⣧⢻⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢷⢳⢱⠹⣱⣵⣹⢘⡭⣦⣷⣑⣕⣝⢕⣋⢇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼⢸⢃⠇⣿⣿⡘⡯⡫⢖⣚⣛⡺⠤⢧⡵⡣⣪⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣴⢃⡟⡼⡸⢸⢙⣿⣻⡪⢕⣒⣒⠮⢍⣓⠦⣵⣱⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢛⣡⠞⣡⢏⡼⣱⠃⢟⣿⡻⢟⡻⠭⠭⢭⣝⡳⢬⡓⢦⣱⢣⢟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⠶⠛⣋⣥⠶⢋⣵⠞⣥⣟⡠⣝⢮⣻⣕⡯⠝⠒⠛⠛⠓⢦⣝⢶⣝⢦⡙⠏⡼⠁
⠀⠀⠐⠛⠿⠿⠿⢛⡛⠉⢩⣤⣴⡶⠛⠉⡰⠾⠿⠤⣌⣓⠮⠝⠚⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⢦⡹⣦⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢷⡘⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⢷⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀
⣿⡟⡉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣴⣦⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣾⣿⣿⣷⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢁⣾⣿⣿⣿⣿⣷⡈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢠⣿⡟⠛⠛⠛⢻⣿⣿⡄⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣰⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣰⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣆⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣼⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣆
⠀⠀⠀⠀⠀⠀⣼⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣧
⠀⠀⠀⠀⠀⣼⣿⣄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣠⣿⣧
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠘⢿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣾⠛⠻⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣴⣦⠈⢿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⠀⢸⣿⣧⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⢿⣿⠀⠈⢻⣆
⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⠀⢸⣿⣿⣷⡈⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢸⣿⠀⠀⠀⢻⣦
⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⠛⠻⣿⣿⣿⣷⡄⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⣬⣧⡀⠀⠀⠀⢻⣷⡀
⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⡟⣁⣀⣀⣀⣀⣙⣛⣁⣀⣀⣀⣀⣹⣷
⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢿⡷⠀⠰⣿⠇⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠙⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠋⠀⢀⣴⡄
⠀⠀⠀⠀⣠⣀⠀⠉⠁⠀⠀⣴⣦⠀⠀⠿⠿⠿⣿⣿⣷⣶⣤⣀⠀⠉⠁⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠛⠋⠀⢀⣴⣦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣷⣄⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⡧⠰⣿⠟
⠀⠀⣶⡆⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠠⣶⡆⠀⠀⠀⠀⠀⢀⣤⣄⢠⣤⣄
⠀⠀⠈⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠈⠻⠟
⠸⡿⠆⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠾⡿
⢀⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣤⣤⣀⣀⣀
⠿⠿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⢿⣿⡿⢿⣿⣿⣿⣷⣦⣄
⠀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⣫⣿⡿⠋⠀⢸⣿⡇⠀⠈⢻⣿⣯⠛⢿⣿⣦⡀
⠘⠿⠃⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠀⢰⣿⡟⠁⠀⠀⢸⣿⡇⠀⠀⠀⠹⣿⣧⠀⠉⣻⣿⣦⡀
⠀⠀⣤⡄⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⢿⣿⣶⣿⣿⣀⣀⣀⣀⢸⣿⣇⢀⣀⣀⣀⣽⣿⣷⣿⡿⠿⣿⣷⡀
⠀⠀⠛⠃⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⣽⣿⠟⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠛⣿⣿⠁⠀⠀⠘⣿⣷
⠀⠀⠀⠀⣾⡷⠀⠀⠙⠻⣿⣷⣦⣤⣀⣠⣿⡟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠉⠛⠻⠿⢿⣿⣇⣀⣀⣀⣠⣿⣏⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⣶⠀⠀⣀⣼⣿⡿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⢿⣿⡇⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⣿⣿⡀⠀⢀⣀⣀⣸⣿⣇⣀⣀⣀⠀⠀⣼⣿⠃⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⣀⣠⣼⣿⣿⣿⣿⠿⠿⢿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣤⣀⠀⣼⣿⠏
⠀⠀⠀⠀⢀⣤⣄⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠛⢿⣿⡄⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣾⣿⠛⠻⢿⣿⣿⠏
⠀⠀⠀⠀⠈⠻⠿⠀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⡈⢿⣿⣄⠀⠀⢸⣿⡇⠀⠀⢀⣾⣿⠃⣠⣶⣿⠟⠁
⠀⠀⠀⠀⢀⣴⣶⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣿⣿⣷⣄⣸⣿⣇⣠⣴⣿⣿⣷⣿⡿⠛⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠠⠄⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⣄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⢰⠏⠁
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠠⠀⠀⡄⠀⠀⡀⠀⠀⢀⠘⠄⠀⠠⠂⣠⣀⡰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢂⣀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠈⠁⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⡤⠚⡭⠝⣿⡤⠿⠟⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⢈⠀⣀⡄
⠀⠀⠀⠀⠀⢀⡀⢸⡄⠀⣀⣀⡬⠤⠦⠤⠔⠒⡶⠂⠀⠈⣆⠬⣃⢄⠀⠀⠂⠀⠐⠶⠋⠁
⠀⠀⠀⠈⠀⠀⠣⣰⡠⠀⢈⡷⠀⠀⠀⠀⢠⠞⠁⠀⢠⡀⠀⠀⠈⢢⠑⡄⠀⠉⢠⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠉⡴⠚⠁⣀⡤⡄⢀⠜⠁⡠⠀⠀⢸⠀⠀⠉⠀⠀⠇⠁⠀⠀⠈⠁⠿⠆⠀⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡼⠁⠀⠹⠃⡀⠀⠈⠄⠀⠈⡆⠀⢀⣀⢎⠜⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⡀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣿⠸⠉⠂⠅⠀⠀⠄⠐⠄⣷⣤⣤⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣠⠟⠀⠀⠀⡀⡀⢌⠀⠀⠀⡏⠀⠀⠀⠄⣀⠐⠙⠳⢶⣬⣅⡨⡍⠉⢷⢦
⠀⠀⠀⡁⠀⡀⠀⠀⠈⠚⠁⠀⠀⠀⢀⠄⠈⠀⠀⠂⢼⡄⠀⠀⢠⣀⠀⠄⠀⢰⠀⠈⠁⢛⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠰⠀⢠⡂⠀⢀⢔⠝⠁⠀⠀⠉⠙⢦⣄⣀⣀⡀⠀⢠⠏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠇⠀⠑⠁⠀⠀⠰⠶⠀⠀⠀⠈⠁⠉⢹⠞⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢒⣶⠶⠒⢓⣒⡶⠶⠒⢢⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⡾⠋⣁⡴⠞⢉⣠⠴⠛⢉⣡⠴⠖⢛⣉⣥⠤⠶⠛⠓⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣿⡿⠋⢠⡾⠋⣠⡶⠋⣡⡴⠚⣉⡤⠖⢛⣉⡥⠴⠒⢛⣋⣩⣭⠤⠽⠦⣀
⠀⠀⠀⠀⠀⠀⢀⡴⢻⣇⠀⣿⠁⢰⡟⠀⣴⠋⢠⡾⠉⣠⠞⢉⡤⠞⣉⡥⠶⢚⣋⣭⠥⠴⠖⠒⣒⣛⣛⣳⣄
⠀⠀⠀⠀⠀⡴⠻⣦⡀⠹⣾⣿⠀⢺⡁⢸⡇⢀⡟⢀⡾⢁⡴⢋⡴⢚⣡⠴⣚⣩⠤⠖⣚⣋⣩⡭⠭⠥⠤⠤⠬⠷⡄
⠀⠀⠀⢀⡞⠛⢦⣈⠙⢦⣈⠻⣆⠘⢧⠈⢧⠘⣇⢸⡇⡾⢠⠏⡴⣋⠔⣊⠥⢒⡪⠭⠤⠖⣒⣒⣒⣒⣒⣒⣒⡒⠛⢆
⠀⠀⢠⣎⡙⠳⢤⣍⡛⠦⢬⣙⠪⢷⣌⡓⢬⡳⣜⢎⢣⢃⠾⠘⠔⠕⠩⠒⠍⢐⠈⠩⢭⠉⣐⣒⠒⠤⠤⠬⠭⣭⣉⣙⣧
⠀⢀⡾⣤⣉⡓⠲⠤⢍⣙⣒⠂⠭⠕⠂⢈⡓⠪⠄⢁⡁⡀⣀⡀⣁⠀⠀⠐⠤⠥⠀⠀⠀⣐⡒⠠⠭⠭⣉⣓⡒⠶⠤⣬⣉⣆
⠀⡼⠲⠦⢬⣍⣙⣓⡒⠒⠬⠭⠭⠬⠅⢀⡀⠀⠒⠀⢀⠀⠁⡔⠉⠀⠀⠀⡂⠈⠀⣀⠬⣐⡒⠭⣍⡓⠲⢬⣍⡛⠲⢤⣌⡙⡄
⢠⣏⣛⠒⠒⠶⠦⠤⠭⠭⠭⠭⠭⠭⠕⣒⣒⠠⢕⣢⢕⡫⣪⢢⢀⡘⡔⣌⢮⡲⣝⠢⣝⠲⣍⡓⠦⣍⠓⠦⣌⡙⠳⢦⣈⠙⢧
⢸⢭⣍⣉⣉⣙⣛⣓⣒⣒⣚⣋⣉⡭⠭⠴⢒⣊⠥⢖⣩⠖⣡⠏⣼⠀⡇⢸⡄⢳⡈⢷⡈⠳⣄⠙⢦⡈⠻⣦⡈⠛⢦⡀⠙⢷⣼
⢸⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⢒⣚⣉⣥⡴⠖⢋⣥⠞⢁⡼⠁⣼⠃⢸⡇⢸⡇⠈⣷⠀⢹⣆⠈⢻⡄⠈⢿⣄⠈⠻⣦⡀⢙
⢸⣛⣓⣒⣒⣒⣒⣒⣒⣲⣿⡿⠿⠟⠛⢉⣡⡴⠞⠋⣠⡶⠋⣠⡾⠃⢠⡟⠀⣸⡇⠀⣿⡆⠀⣿⡀⠀⣿⡄⠀⢻⣆⠀⠹⣷⣸
⢸⡉⠉⠉⢉⣉⣉⣉⣁⣠⣤⡤⠶⠞⠛⢉⣁⣤⠶⠛⢁⣠⡾⠋⢀⣴⠟⠀⣰⡿⠀⢠⣿⠀⠀⣿⠇⠀⢸⡇⠀⠘⣿⡄⠀⢹⣿
⠈⡟⠛⠛⠋⠉⠉⢉⣉⣀⣠⣤⡴⠶⠛⠋⢁⣀⣤⠶⠛⠁⣠⣴⠟⠁⢀⣼⠟⠀⢠⣿⠃⠀⣸⡿⠀⠀⣾⡇⠀⠀⣿⡇⠀⠘⡏
⠀⢹⠾⠟⠛⠛⠛⠛⠉⠉⣁⣀⣠⣤⠶⠞⠛⠉⢀⣠⣴⠟⠋⠀⣠⣶⠟⠁⢀⣴⡿⠃⠀⣰⣿⠃⠀⢰⣿⠃⠀⢠⣿⠇⠀⢸⠁
⠀⠀⢷⣴⣶⠶⠶⠶⠟⠛⠋⠉⠁⣀⣠⣤⡶⠞⠛⠉⢀⣠⣴⠿⠋⠁⢀⣴⡿⠋⠀⢀⣼⡿⠁⠀⣠⣿⠏⠀⠀⣼⡿⠀⢠⠇
⠀⠀⠈⢧⣀⣀⣠⣤⣤⣶⠶⠿⠛⠋⠉⠀⣀⣠⣴⡾⠟⠋⠁⢀⣤⣾⠟⠋⠀⢀⣴⡿⠋⠀⠀⣴⣿⠋⠀⠀⣼⣿⠃⢠⠏
⠀⠀⠀⠀⠻⡉⠉⠁⠀⢀⣀⣠⣤⣶⠾⠿⠛⠉⠁⢀⣠⣴⡾⠟⠋⠀⠀⣠⣶⡿⠋⠀⠀⣠⣾⠟⠁⠀⢀⣾⡿⠃⡰⠃
⠀⠀⠀⠀⠀⠘⢾⡿⠟⠛⠋⠉⠁⠀⢀⣀⣤⣶⡾⠟⠛⠉⠀⢀⣠⣶⡿⠛⠁⠀⢀⣴⣾⠟⠁⠀⢀⣴⣿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣶⡾⠿⠛⠛⠉⠀⠀⣀⣠⣴⣾⠿⠋⠁⠀⢀⣠⣶⡿⠛⠁⠀⢀⣴⣿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⣀⣀⣤⣴⣶⠿⠟⠋⠉⠀⠀⣀⣤⣾⡿⠛⠉⠀⠀⣠⣴⣿⣟⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢏⣁⠀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⣠⣴⣿⠿⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣛⣵⣶⣶⣶⣒⣒⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣀⠀⠀⠀⠀⢰⡇⢠⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⣍⠛⠶⣄⡀⣾⠀⢸⡇⠀⠈⣧⠀⠀⣀⣤⠴⢺⡿⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢷⣄⠀⠙⢧⠀⢸⡇⠀⠀⣸⠖⠛⠉⣠⠞⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢧⡀⠈⢇⠸⡇⠀⣰⠃⠀⣠⠞⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠧⣄⡀⠀⠙⣦⡘⣦⡇⢠⠇⣠⠞⠁⠀⢀⣰⣃
⠀⠀⢀⡀⠀⡀⠀⣀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠀⠀⠀⠉⠛⠶⢬⣻⣿⣧⣿⣞⣡⠴⠖⠋⠉⠈⠉⠙⠛⠲⠦⢤⣄⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠈⣿⣼⢷⣼⠋⠀⢠⣴⣶⣯⣭⣤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⣶⣾⣿⣿⣿⣿⣖⠲⠶⠒⠶⠶⠶⠶⠶⠲⠶⠶⠾⢿⣿⣶⠦⠀⠀⣿⠤⠤⠁
⠀⠀⠀⠸⠇⠈⠟⠀⠀⠀⠈⠉⠛⠓⠲⠦⢤⣤⣀⣀⣀⣠⡴⠖⠋⣩⠞⡟⣿⢻⠳⣌⠙⠓⠦⣄⡀⣀⣀⣤⡤⠶⠖⠛⠉⠉⠀⠀⠀⠀⠻⠶⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠁⠀⣠⠞⠁⡸⠁⣿⠈⢧⠈⢳⣄⠀⠀⢹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠞⠁⠀⢠⠇⠀⣿⠀⠈⢧⠀⠙⢷⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠞⠁⠀⠀⣀⡞⠀⠀⣿⠀⠀⢘⣧⣄⠀⠙⢧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣁⣤⡴⠖⠛⠉⣧⠀⠀⣿⠀⠀⣸⠁⠈⠛⠶⣄⡙⢦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⢠⡏⠀⠀⠀⠀⠀⠉⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⠉⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠈⢿⠿⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠎⢀⣤⠴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢚⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⡟⢻⡷⠿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣄⣻⢿⣍⣹⣾⣷⠞⠿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠁⠐⠚⢱⣾⣋⣹⠿⣟⠳⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⡆⠀⠀⠀⠈⠛⠻⣿⣴⡟⢿⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢻⣶⠀⢀⣴⡇⠀⠉⠉⠁⣠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⣀⠀⠈⠘⠛⠿⠀⡞⠹⡧⠀⣠⣴⠞⠋⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠿⠿⠷⣦⣦⡀⠀⣿⣴⡷⠟⠁⣡⡔⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡶⠤⣀⢹⣿⣄⢠⣀⠈⠉⠻⣷⣍⡔⣠⣴⣿⠟⠁⣰⣿⠃⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⡿⣧⡀⢸⡄⢿⣷⡄⠹⣿⣷⣄⡈⢻⣵⣿⠿⠋⣀⣼⡿⣡⠎⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡀⠀⠀⠈⠛⠿⠷⠀⠻⣿⣧⣀⠻⢿⣿⣆⠛⠓⠒⣻⣯⣅⢸⡁⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣧⠇⠀⠀⠀⢀⡴⠒⠰⢖⡶⢉⡝⢛⣒⠈⢡⣶⣶⣄⣿⠛⣿⣨⣽⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢷⣼⢿⣿⢀⣘⢛⣛⡛⢚⣿⠋⢰⣾⣿⢏⡆⠘⢿⣍⠙⣿⣀⡿⠋⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡀⣼⣿⠟⢷⣼⣧⡼⠿⣿⣦⠀⠀⣠⡞⠁⣴⡿⠿⠃⣼⢧⣾⡿⠛⢳⣿⣿⣿⠚⠻⣶⣦
⠀⠀⠀⠀⠀⣐⣛⠓⠒⠉⣀⡍⣻⡿⠚⡿⣿⠷⣶⡟⠋⢀⣴⠟⠀⠞⠋⠀⣠⣾⠟⢘⣿⠿⣶⡟⢙⣟⠉⣷⡶⠿⠟
⠀⠀⢀⣴⣿⣥⣬⣿⠆⣸⢻⠛⢿⣷⢿⣇⣼⢷⣾⣿⣠⣿⣫⣤⣤⣦⣶⠿⠛⣡⠟⢙⡻⠰⣿⣷⡾⢻⣶⣾⡇
⠀⠀⠀⠙⠛⠿⠗⠃⠘⠛⠁⠀⠀⠀⠘⠿⠟⠀⠀⠉⠉⠉⠉⠈⠉⠀⠐⠒⠋⠒⠚⠋⠁⠀⠉⠉⠀⠀⠉⠉
⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣟⣻⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⡀
⠀⠀⠀⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡄
⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣲⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⣉⣉⣉⡛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡶⠋⣉⣀⣉⠙⢶⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾⠀⣼⠋⠀⠉⣷⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⡀⢻⣄⣀⣠⡟⠀⡿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠳⣤⣈⣉⣁⣤⠞⢡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠳⢦⣭⣉⣭⡤⠞⠋⠀⠀⠀⠀⠀⡤⠶⠚⠛⠳⠄⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣤⠶⢛⣵⠟⠁⠀⠀⢀⣾⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠖⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⣳⡆⣙⣶⠄⠀⠀⡾⣿⠀⠀⠀⣼⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣠⣾⠿⠚⠉⠀⠀⠀⣼⠡⣯⣤⣄⠀⣿⠘⠷⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠚⠋⠀⠀⠀⠀⠀⠀⠸⠷⢶⡆⣼⠃⠀⢿⠴⢾⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⢿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⣠⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣦⣄
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠙⣦⢹⡧⣄⢼⡦⢠⡄⣤⢠⡤⡄⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⠟⢸⣦⠟⠸⣷⠀⣧⣼⣤⣭⡗⠻⣭⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⠉⣉⣁⣀⣐⡒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⢀⠠⠒⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⢖⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⣹⣆⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢼⡀⠀⠀⠀⠀⠀⠟⠀⠀⣠⠔⠞⢥⠏⠀⠀⠀⠀⠀⠰⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣳⡀⠀⠀⠀⢐⣘⠄⢰⠁⢠⠚⠁⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠂⠉⠀⠀⠀⠸⡆⡇⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠸⡆⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢀⣧⠀⠀⠀⠀⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣻⠀⢀⡀⠈⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡬⠽⠶⣚⡫⠭⠝⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣩⢷⢈⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡪⡸⡄⠀⠀⠀⠀⠀⢸⠄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣈⣉⣉⣷⣦⡇⡀⣇⠀⠀⠀⠀⠀⠀⢻⡿⠿⠦⠾⠶⣀⣀⠉⠀⣸⡇
⠀⠀⡀⠀⠀⠀⢀⠤⠄⠈⢙⣒⣛⣟⣛⣛⣛⣋⣭⣍⣻⣿⣿⠯⣾⠀⠀⠀⢀⠀⢀⡈⠲⠾⣿⡿⠟⠛⠛⠻⠿⠟⡠⢄⡀
⣐⠓⠹⠯⢉⡉⢉⢙⣿⡙⠛⣒⠷⠿⠶⠶⠾⠟⠛⣛⣻⣭⣿⣿⡟⠀⣸⡇⠈⢿⣦⡙⠳⣶⣦⣬⣉⣐⣒⡒⠒⠚⠁⠀⠀⠁⣲
⠀⠀⠀⠐⠚⠓⠒⠀⠉⠻⣷⣤⣀⡿⠿⠿⠿⠿⠿⠿⠟⣋⠕⢁⡠⡾⢋⡟⠀⣸⠽⢿⣆⠘⢯⡉⠛⠛⠛⠛⠻⡇⣀⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⢀⡠⠔⠊⣠⣴⠿⠋⠀⣸⠃⢠⡟⠀⠀⠹⢷⣦⣈⠲⢄⡀⠀⠘⠛⠻⠑⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠤⠂⢁⣠⣴⠿⠚⠁⠀⠀⡰⠃⣰⡿⠁⠀⠀⠀⠀⠉⠛⢷⣦⣌⡑⠢⠤⣤⠎⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠈⠑⠊⢀⣤⣶⠽⠊⠁⠀⠀⠀⢀⡜⠁⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡶⡀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⠀⠀⢀⣉⣲⣄⠀⠀⠀⠀⢠⠞⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣤⣴⡶⠶⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠛⠿⠟⠛⠁⣰⠑⠂⠤⠋⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢀⠀⠀⠀⠉⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣢⣥⣴⡶⣟⠟⠉

⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁
⣷⣄⠀⠀⠙⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁
⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⡟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⡋⠀⠀⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⣾⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⡁⠀⢀⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⠟⠁
⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣶⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣯⡀⠀⣠⣾⣿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⡿⠿⠿⢿⣿⣿⠿⠿⠿⣿⣿⣿⡿⠿
⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣾⣿⣿⠟⢹⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣧⡀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾
⢀⣴⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣾⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿
⣿⣿⡿⠃⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⡀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⣠⣾⣿⣿⠟⠁
⡿⠋⠀⠀⣠⣾⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⣸⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣠⣾⣿⣿⠟⠁⠀⢀
⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⢀⣤⣾
⣠⣾⣿⣿⠟⠁⠀⢀⣴⣿⣦⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣧⡀⠀⣿⣿⣿⠀⠀⢸⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿
⣿⣿⠟⠁⠀⢀⣴⣿⣿⡿⠋⠀⠀⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⣿⣿⣿⠀⠀⢸⣿⣿⠀⢀⣴⣿⣿⡿⠋
⠟⠁⠀⠀⠴⠿⠿⠿⠋⠀⠀⠠⠾⠿⠆⠀⠀⠈⠻⠿⠿⠷⠄⠀⠀⠙⠿⠿⠿⠦⠀⠀⠈⠻⠿⠿⠷⠄⠀⠀⠙⠿⠿⠿⠿⠟⠀⠀⠸⠿⠿⠴⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠔⠺⠷⡄⡼⠗⠒⠄⠀⠀⠀⠐⣾⣯⡟⢿⣧⠆⠀⠀⠀⠀⣀⣀⣀⣨⣿⣀⣀⣀⡀⠀⠀⠀⢰⣿⣥⣴⡆
⠀⠀⠀⠀⠀⠂⣿⡤⠃⠀⠀⠀⠀⠀⢲⣿⠋⠁⠘⢻⣄⡄⠀⠀⠀⠸⣿⣿⠏⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠘⠁⠑⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠀⡼⠟⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢰⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢠⣦⣀⣴⣶⣦⣄⣴⡆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠀⠀⠀⠀⠀⠀⣀⣦⣤⡀
⠀⠀⢠⣤⠾⢤⣿⣤⠿⢤⡄⠀⠀⠀⠀⠀⣿⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠅
⠀⠀⠀⠁⠀⠈⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠛⠁⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠒⠒⠀⠀⠀⠀⠀⠘⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣿⠃⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣣⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⢻⠀⠀⠀⠀⣀⢸⣷⣼⡆
⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢷⠆⠹⣧⠀⠀⠀⠀⠀⠀⠀⣾⢉⡟⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣆⡀
⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣟⣁⣀⣘⣀⣀⣙⣧⡀⠀⠀⠀⠀⠀⠁⠈⡷⠀⠀⠀⠀⠀⠀⣿⣿⢿⢻⣿⡿
⠀⠀⠀⠘⠻⠿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⠛⠁

⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⡰⣂⣀⡒⠄⠀⠀⠀⠀⠀⠰⢰⠀⡴
⠀⠀⢀⣀⠀⢒⣿⠖⠀⢀⡀⠀⠀⠀⠀⣼⠏⢸⡌⢿⡄⠀⠀⠀⠀⠀⠀⣿⠛⡿⢻⡶⠀⠀⠀⠀⠀⠀⠂⡈⠠⠔⣁
⠀⠀⠈⠉⠓⢤⣛⡡⠖⠋⠉⠀⠀⢀⣾⠃⠀⢾⡇⠀⢻⣆⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⣶⡦⠡⢀⡀⠄
⠀⠀⠀⠀⠲⠋⠉⠙⠶⠂⠀⠀⠀⠺⠷⠶⠶⠶⠶⠶⠶⠿⠂⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠰⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠖⠒⢦⣤
⠀⠀⠀⠀⢹⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢻⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠚⠛⣛⡛⠛⠂⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⣀⣀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣾⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢲⣾⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠋⠀⠀⠀⢀⡀⠀⣻⣿⣿⡃⠛⢻⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠄⠀⠀⠀⠀⠐⣾⣿⣧⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠹⠛⠇⠀⠀⠀⣼⣷⣇⠉⠙⠀⠀⠀⢀⠀⢠⣤⣆⡀⣲⣾⣆⠀⣶⣤⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣷⣤⠄⠀⠀⠀⣿⣿⣷⠀⠀⠀⣠⣴⡘⠙⠁⠀⠀⣠⣄⢸⣿⣧⠘⠿⠿⠈⠹⠟⠇⠈⣿⢿⡟⠡⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠄⠀⠀⠀⣈⠋⠉⠀⠀⣀⡙⠛⠁⠀⢀⣶⡄⠻⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣀⠀⠀⠀⣽⣿⣿⠀⠀⠐⠿⠟⠀⠀⣴⡦⠉⠁⠀⣠⡀⣶⡆⠸⡿⠒⣿⡷⢠⣶⣦⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⣠⣭⡀⠀⠀⣿⡷⠀⠀⣶⡆⠁⣀⡰⠗⠙⠁⢀⠀⣀⠀⣀⠀⠀⠀⠉⠁⠸⢿⡿⢀⣰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣀⠀⠀⠀⠛⠟⠃⠀⢠⣶⠀⠀⠾⠆⠀⣴⠉⢁⣠⡔⠗⠛⠁⠛⠘⠛⠰⣿⢆⣤⡀⠀⠀⠀⠸⢿⡿⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠦⠀⠀⢰⣿⡆⠀⢠⣥⠀⠐⡟⠀⣜⠃⡰⠊⣀⠤⠒⠓⠚⠲⠦⣤⠀⡀⠀⠛⢃⣶⣦⠀⠀⠀⠀⣼⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣠⠀⠀⠀⣨⣤⠀⠈⣭⡀⢈⣏⠐⡇⢰⠀⠊⠀⠂⠉⢉⡑⠢⢆⡀⠘⠛⣤⡀⠀⠉⢉⣶⣦⠀⠀⠀⠙⠁⣦⣴⣇⡀
⠀⠀⠀⠀⣹⣶⣿⣄⠀⠀⠐⢻⢿⠏⠀⠀⠙⢫⣄⠀⢩⣦⠀⢷⡀⢇⢨⠀⠀⠀⠀⠀⡀⣌⠲⡈⠱⢆⡀⠉⣶⡦⠀⠈⠛⢩⣤⡄⠀⠀⠐⢿⣿⣿⠁
⠀⠀⠀⠈⠹⡿⠿⠅⠀⠀⠀⠀⢠⣶⣷⠄⠀⠙⢋⣤⠀⠰⢇⡈⠱⣈⠢⠀⠀⠀⠀⠀⠀⠈⡆⠹⡀⠘⣧⡀⠀⣴⣶⠀⠀⠛⠿⠏⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠄⠀⠀⠙⠉⣴⣦⠀⠈⠋⣴⡄⠈⠑⢦⣌⡑⠒⢐⣀⠄⡠⢸⠁⡗⠀⣥⠀⢩⣤⠀⢈⣵⣤⠀⠀⢠⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠈⠻⡟⢛⣀⣄⣀⠀⠈⠙⠋⣴⣦⠀⠀⠘⠛⢴⢄⡄⣈⣩⣡⠴⠊⣰⠏⢠⠏⠀⠶⠀⢨⣥⠀⠀⠙⣋⡀⠀⠀⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣏⢀⡀⠀⠀⠉⠉⠺⡷⢠⣤⢀⣀⢀⡀⣀⣠⡄⠓⠁⣤⠛⠀⣘⠓⠀⣈⣍⠀⠀⠻⣿⠇⠀⠀⢴⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⡧⢠⣄⡀⠀⠀⠈⠁⠘⠋⠘⠃⠉⠀⣠⡰⠶⠉⢀⣈⠛⠀⢀⣙⠛⠀⠀⣤⣶⡄⠀⠀⠘⠋⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠹⢿⠓⢼⣾⡆⣴⣦⠠⣶⡄⠿⠇⠙⠁⠀⣤⣜⠛⠀⠀⠸⠿⠇⠀⠀⢛⠟⠁⠀⠀⠰⣾⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⣶⡇⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠉⠀⠀⡀⠠⣶⣦⠙⠃⠀⠀⢾⣿⠆⠀⠀⣹⣿⣷⠀⠀⠀⠐⠛⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⡿⠛⢤⣿⣿⣇⠀⣤⣤⡄⠀⣄⣤⡀⢠⣶⡄⠸⣿⠇⠉⠁⠀⠀⣴⣶⣌⠀⠀⠀⣀⣠⠛⠉⠀⠀⠠⣤⣼⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠁⠘⢿⢿⡏⠐⢿⠿⠇⠙⠟⠃⠀⠀⠀⠀⣀⣶⣤⠘⠛⠁⠀⠀⠠⢿⣿⡟⠀⠀⠀⠠⠿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⠸⠿⡟⠁⠀⠀⠀⣤⣴⡈⠁⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⢤⣾⣦⡄⠽⢿⠟⠂⠀⠀⠀⠀⠀⠚⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⠚⢿⠟⠃⠀⠀⠀⠀⠀⢀⣠⣷⣶⡇⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡖⣃
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡘⠁⠐⡆
⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⠇⠀⠀⠸⡀
⣿⠂⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⢸⠸⠀⡀⠀⢳
⣿⠂⠀⠀⠀⠙⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⠀⠀⠀⠀⡏⠀⠀⠃⠀⠈⡇
⣿⠀⠀⠡⣿⡤⡀⠀⠀⠐⠦⡀⠀⠀⠀⠠⠚⠁⠀⠀⢡⠄⢰⢀⠰⠀⠀⠀⠀⢡
⡇⠅⠀⠀⠻⣿⣄⠀⠀⡀⠀⠈⡢⡀⣰⠃⠀⠠⣶⣈⣹⡆⣯⠞⠔⠀⠀⠐⠀⢸
⠇⠁⠀⠀⠈⠙⢿⣶⠀⠑⠀⠀⠹⣿⠃⠀⢀⣴⠿⡟⣾⡿⣿⠑⠂⠉⠒⠀⠀⢸
⢀⠀⠀⠀⠀⠀⠈⢿⣦⡈⡇⠀⠒⢾⠀⠠⣬⣵⡶⢟⣽⣷⠿⣗⣀⡀⠀⠀⠀⡟
⠘⡀⠀⠈⠀⠄⢄⠈⠩⣿⣿⣷⣾⢋⣆⠀⠉⢢⠴⢋⡯⠹⡆⢘⢆⡁⠀⠄⡐
⠀⢱⠀⠀⠀⢀⠤⣴⣶⣮⣿⣿⣿⡷⣣⣯⡏⠀⠀⣸⠃⠀⠱⣦⣤⣥⡴⠊
⠀⠀⢣⠀⡀⠀⠤⢴⢿⣿⣿⣿⠿⣿⣭⠞⠀⢀⣾⠃⠀⠀⠀⢀⣾⡍⠊⠁⠀⠀⠀⢤
⠀⠀⠀⠑⢌⣦⣷⢿⣻⣿⡿⠘⢧⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠸⣿⡏⣴⣇⣀⣀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⢸⡆⠀⠀⡴⠃⠀⠀⠀⢀⠀⠀⢻⣾⣿⣯⡥⠄⠀⠀⠀⠠⠌⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠙⣡⣤⠖⠓⠀⣩⡬⠩⢛⣷⣦⣭⣝⡓⠦⣄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⠀⠀⠰⡋⢛⡴⣿⡟⠀⡄⠀⠈⠻⠒⣄⡻⢤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠁⠀⠀⠀⠀⠀⠐⠎⠀⠀⠺⡁⠘⠁⠀⠀⡀⠀⠀⠈⣢⡁⣾⣥⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡂⢤⣀⡀⠈⠈⠙⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⠀⢀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⣤⠈⢷⡄
⠀⠀⠀⠀⠀⠀⣴⠏⢰⠁⣀⡀⣳⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢾⣿⣿⡖⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⣿⣁⡀⠈⢻⡄
⠀⠀⠀⠀⠀⣼⢃⡤⢾⢿⣥⡽⢿⠦⡈⢷⡀⠀⠀⠀⠀⠀⠀⣰⠏⣀⣀⡈⣿⡏⣀⣀⡈⢻⡄⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠛⢛⣿⠁⠀⠀⠻⣆
⠀⠀⠀⢀⣼⠃⠈⠀⠘⢦⣿⡤⠏⠀⠀⠈⢳⡄⠀⠀⠀⠀⣴⠋⠀⢿⣿⣿⠋⢻⣿⣿⠇⠀⢻⣄⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠹⣆
⠀⠀⠀⣾⣁⣀⣀⣐⣐⣚⣉⣙⣒⣒⣀⣀⣀⣻⡄⠀⠀⣼⣃⣀⣀⣀⣙⣃⣀⣀⣙⣋⣀⣀⣀⣹⡆⠀⠀⣴⣏⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣙⣧
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⢠⡞⠀⢀⣼⣆⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⡤⢤⡀⠈⣧⡀
⠀⠀⠀⠀⠀⣠⡟⠀⢠⠃⣿⣿⣆⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⢠⡟⠱⣦⣴⣶⣶⣤⠶⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡏⣹⢗⣺⣏⢹⠈⢷⡀
⠀⠀⠀⠀⣰⠏⠀⠀⣷⣴⣿⣿⣿⠀⠀⠘⣧⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⡿⢿⠿⡿⠀⠀⠈⢷⡀⠀⠀⠀⠀⢠⡾⠁⠀⡟⢻⣿⣻⡟⢻⡀⠈⢻⡄
⠀⠀⠀⣴⠋⠀⠀⠀⠘⠻⠿⠿⠋⠀⠀⠀⠘⢷⡀⠀⠀⣰⠏⠀⠀⠸⠇⠡⣦⡾⠁⠾⠂⠀⠈⢷⡄⠀⠀⢠⡟⠀⠀⠀⠉⠉⢧⠸⠋⠉⠀⠀⠀⠻⣄
⠀⠀⠀⠛⠖⠖⠶⠲⠶⠒⠶⠲⠲⠒⠲⠒⠶⠚⠃⠀⠀⠛⠖⠲⠲⠒⠖⠶⠲⠖⠶⠒⠶⠲⠖⠞⠃⠀⠀⠙⠗⠶⠒⠲⠖⠖⠶⠳⠶⠒⠲⠖⠖⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢀⡀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣿⣿⠀⠹⣆
⠀⠀⠀⠀⠀⢀⡾⠁⣼⣷⣄⣴⣷⡄⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣴⣞⣷⡟⣶⣤⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⣿⡿⠀⠀⠙⣧
⠀⠀⠀⠀⢠⡟⠁⠀⢈⣿⣿⣿⣏⠀⠀⠻⣆⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿⣿⣿⣿⣿⡧⠀⠹⣦⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧
⠀⠀⠀⣠⡟⠀⠀⠀⠻⠿⠋⠻⠿⠃⠀⠀⠹⣦⠀⠀⠀⢠⡾⠁⠀⠈⠛⠳⣾⣷⠛⠛⠀⠀⠘⣧⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠷

⠀⠀⠀⠀⠀⢠⢤
⣶⣶⣶⣶⡶⠣⣦⠱⣶⣶⣶⣶⡶⠶⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢲⣶⣶⣶⣶⡶⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡿⠁⠀⡇⠀⠹⣿⣿⣿⣿⣀⣀⣍⣈⣠⣇⣀⣏⣇⣈⡀⣹⣷⣨⡆⣵⣀⣀⣇⣈⣀⣉⣻⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⢻⣁⣀⣀⡃⣀⣀⣹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣴⣤⣴⣦⣴⣤⣦⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣴⣤⣤
⣿⣿⣏⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⢻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⣻⣿⣿⠈⣿⡛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣉⡤⠗⠚⠉⣽⣿⣿⠀⠸⣏⠉⠓⠲⢌⡙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢏⡤⠚⠁⠀⠀⠀⢰⣿⣿⣿⠀⠀⢻⡄⠀⠀⠀⠉⠲⣌⡹⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣠⠊⠰⣤⣀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠘⣧⠀⠀⢀⣠⣴⠎⠳⡌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⡼⠁⠀⠀⠹⣿⣿⣶⣼⣿⣿⣿⣿⠀⠀⠀⢹⣦⣶⣿⣿⠃⠀⠀⠘⣆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡸⠁⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡃⠀⠀⠀⠀⠘⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⣇⣠⣤⠶⠞⠛⠉⠉⠀⠀⠙⢿⣿⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄⣹⡀⣿
⠀⠀⠀⢶⠀⣦⢠⡆⢀⣀⣤⡴⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⣟⡃
⠀⠀⠀⠘⡾⠹⡾⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠁⠀⣏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠙⡟⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣿⣿⣦⡀⠀⠀⢀⣀⣠⣤⠶⠞⠛⣹⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣳⠀⠀⠀⠀⠈⣹⣿⣿⣏⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⡍⠁⠀⠀⠀⢠⡏⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢧⡀⠀⠀⣰⣿⣿⡿⢿⡄⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⡄⠀⠀⢀⠞⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠱⣄⠰⠟⠋⠁⠀⠈⣷⠀⠀⠀⣿⣿⣿⡿⠀⠀⠈⠙⠿⠄⡰⠋⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣎⠳⣄⡀⠀⠀⠀⢸⣇⠀⠀⣿⣿⣿⠃⠀⠀⠀⢀⡤⠊⣱⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡙⠲⡤⣄⣀⣿⡀⠀⣿⣿⣏⣀⣠⡴⠚⣉⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⣬⣹⣧⠀⣿⣿⣉⣥⣴⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣆⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄⣀⣤⡶⠶⢦⣄⡀⠀⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⠏⣡⣶⢶⣦⡈⢷⡄⠀⠀⠀⠀⠀⠀⢻⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠰⣿⡁⠀⣹⡷⢸⡇⠀⠀⠀⠒⠛⠛⠛⠿⠇
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠹⣦⡙⠿⠿⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⣿⣿⣿⣿⣷⣌⡛⠻⠶⠛⠋⠀⠀⠀⠀⠐⢦⣄⣀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠈⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡇⠀⠈⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠈⢿⣿⣷⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣄⡀⠀⠀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠂

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡟⣉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠾⣿⣿⣿⡿⠷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢻⣿⡿⠋⠀⣼⣿⡟⠁⢀⣴⣿⡿⠛⠛⢛⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣿⠇⠀⠸⣿⡷⠖⣈⣭⡽⠶⠒⢚⣛⣋⣀⢀⣴⣿⡿⠟⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⢿⡿⢋⣴⡿⠉⢠⣾⡿⠋⠀⣴⣿⡿⠛⠉⣉⣉⣉⣋⡀⢀⣤⣶⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢻⣿⠇⣴⡏⠀⣾⣿⡃⠀⢸⣿⣧⡤⠖⠛⠛⢀⣀⣸⣿⣟⠋⠉⠉⠛⠻⠟⠉⠀⠀⠈⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣼⠃⣸⣿⡀⠀⢻⡿⠃⣰⣿⣟⡋⠥⠀⣒⣒⣛⣻⣿⠿⠯⢍⡘⠻⠿⠿⢶⣄⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⡿⢻⡟⣸⡇⠀⢸⣿⣷⣤⠂⢠⣾⣿⠛⢁⣠⣴⠿⠟⠛⠛⠛⠉⠙⠓⠦⣌⡑⢶⣤⣠⣈⡙⠍⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⢰⣿⡇⣸⠁⣿⣷⡀⠀⠙⣿⠏⢀⣴⠟⢡⣶⣿⠟⠋⠉⠀⣠⡤⣬⣝⣲⢦⣄⣤⡙⢷⣍⠻⡿⠿⣦⣺⣿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⢀⣿⣿⡏⣿⠀⠘⣿⣿⡦⢸⣿⠀⣼⠋⢀⣽⡿⠋⣁⣤⠾⠟⡛⠛⠿⢿⢿⡻⢿⡻⣿⣷⡙⢷⡀⢀⡈⢻⡉⠉⠉⠙⢻⣿⡀
⠀⠀⠀⠀⢸⣿⢠⠀⣿⣧⠀⠈⢙⡃⣾⡏⢠⡏⠀⣾⡿⢁⡾⢫⣶⣿⣛⣛⣛⠳⡦⣄⠉⢦⠹⡌⢿⢻⡄⢻⣜⣿⣦⢱⣶⣶⣦⣄⣻⡇
⠀⠀⠀⠀⣾⣿⣿⡄⠹⣿⣿⣶⣾⡇⢿⡇⣾⡇⠀⣿⠇⡞⢠⢯⡾⢉⢠⠄⣉⠑⣄⠘⣧⠀⢁⢸⡈⠆⢻⡄⢿⡇⠘⣏⠛⠛⠻⢿⣿⣿
⠀⠀⠀⠀⣿⣟⠹⣷⡀⠈⠉⠛⢹⡇⢸⡇⢻⣇⠀⣿⠀⡇⠛⢸⣣⣎⠟⠛⣻⡇⢸⡆⢸⣿⢸⡆⣧⢠⠈⣿⠈⡟⣧⢹⣤⣤⣀⠀⢻⣿
⠀⠀⠀⠀⢿⣿⡆⠻⣿⣶⣤⣤⡄⣷⠈⣿⠘⣿⡀⢹⣆⢧⡀⠘⣷⣿⣿⣏⣿⠟⣸⡇⢸⣿⢸⡇⣿⢸⠀⣿⡄⡇⢻⡞⠻⠿⣿⣷⣼⣿
⠀⠀⠀⠀⢸⣿⢿⡄⠈⠛⠻⠿⠻⡘⣷⠘⢇⠈⢿⣆⠙⣮⣷⣬⡪⢝⣛⣫⣥⣾⠟⣡⣿⢏⡿⢸⣿⣿⠀⣿⡇⣇⢸⣥⣀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠈⣿⣌⢿⣶⣤⣀⣀⣠⣿⡜⢷⡈⢧⡈⠻⣷⣌⡛⠮⣟⣛⡛⠛⠉⠥⣚⡵⢋⡾⢁⡞⣼⡇⢀⣿⢰⡿⠸⢿⣿⣷⣄⢸⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠙⠻⠿⣿⠿⠟⠙⢆⠹⣦⣽⢷⣦⣉⠛⠳⠶⠶⠦⠶⠖⢛⣡⡴⢋⡠⠋⣰⣿⠃⣰⠇⢸⣧⡀⠀⠈⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣀⠀⠀⣀⣤⣦⣑⢦⣝⠳⢭⣙⠛⠲⠶⠶⠶⠾⢛⣋⣭⡶⠋⠀⣺⣿⠅⣰⣿⠃⠛⢿⣿⣦⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣝⠿⢿⣿⣿⡿⠟⠉⠛⡯⣟⠶⠤⣤⣉⣒⣲⣶⣿⣿⠯⠅⣀⠤⣾⡟⡋⢈⢽⣷⣦⠀⠀⢻⣿⣇⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⠀⠀⣀⣠⣾⣿⠖⠀⢠⣤⣴⣿⣿⣿⠷⠒⢀⣠⠴⢟⠿⣾⣤⠁⠀⢻⣿⣧⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⢿⡿⠿⠟⠋⠀⢀⣾⣿⠇⠀⢰⣶⡖⠉⠩⣶⣿⠀⠀⢿⣿⡇⠀⢸⣿⣿⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣤⣤⣤⣶⣿⡿⠋⠀⢀⣾⣿⡇⠀⢠⣿⣿⠀⠀⣼⣿⡇⢀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣏⣁⣀⣠⣴⣿⡿⠋⠀⣠⣾⣿⠋⢀⣴⣿⣿⣷⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣶⣶⣿⣿⣿⠷⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡖⠀⠀⣴⣷⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣤⠖⢉⣴⡶⠦⠤⠤
⠀⠀⠀⠀⢀⡄⠀⠈⣿⣿⠃⠀⣼⣿⡟⠁⠀⣀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠸⣣⠞⢡⣾⠏⠁⣰⣾⡟⠛⠒⠶⢤
⠀⠀⠀⢠⣿⣿⡀⠀⣿⣿⠀⢸⣿⠋⢀⣤⣾⣿⠿⠿⠛⠃⠀⠀⠀⠀⠀⢀⡀⢻⡟⢰⢟⣴⠋⣡⣶⡟⠋⢉⣉⣙⡻⠿⢶⣤⡀
⠀⠀⠀⠈⠻⣿⣷⡀⠸⣿⡀⢸⠇⣠⡿⠛⢉⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠸⣷⢸⣇⠈⣾⡇⣠⠟⢋⡥⠶⡾⣭⣛⠻⢶⣄⣈⠻⡄
⠀⠀⣶⣄⠀⠈⠻⣷⣄⠙⢧⢸⡠⢋⡴⠞⠛⠛⠛⠻⠿⣿⣿⡇⠀⠀⢰⣤⣦⠸⣿⠀⣿⢣⣿⠔⡵⣛⣿⣛⢦⡈⢣⣄⠹⡿⣷⡘
⠀⠰⣿⣿⣷⣦⣄⡈⠙⠳⠦⢅⠀⠐⢌⡛⠻⢷⣶⣤⣀⠀⠈⠁⠀⠀⢈⠉⢹⣄⢻⡄⢸⠘⢿⡀⠀⡯⣹⢻⡆⣿⡈⣿⡆⣅⠈⣧
⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⢋⣵⢃⡎⣧⠙⣷⣄⠈⠻⣿⣷⣦⡀⠀⠀⠸⠿⣆⠻⡌⢿⣿⣦⡈⠳⣵⣦⡶⠞⢁⣿⠀⢸⡇⢿⡇⠘
⠀⠀⠰⣶⣶⣶⣶⣶⣶⡿⠛⢁⣾⠁⣿⡇⠘⣿⣦⠀⠘⢿⡿⠁⠀⠀⠀⢶⡾⢷⣽⣄⡈⠈⠻⠿⢶⠭⠤⠖⣿⡿⢁⣾⠀⢸⢇⠇
⠀⠀⠀⠹⠿⠛⠛⠉⠁⣀⣴⣿⠃⢀⣿⡇⠀⢹⣿⣇⠀⠈⠁⠀⠀⠀⠀⠀⢠⣶⣍⡉⠛⠿⠿⠷⠖⠶⢶⣛⡋⠀⣹⡿⢠⡿
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡟⠁⠀⣼⣿⡇⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡯⢛⡓⠒⢶⡶⠶⠟⢋⣠⠴⣫⠄⠜⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⢀⣼⣿⣿⠁⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⣭⠍⣤⡞⢻⠅⠚⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⠛⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⣶⣤⣍⠙⠲⣤⡀⢀⡀
⠀⠀⠀⠀⢀⣴⠟⠉⣀⣤⠤⠴⠦⠤⢤⣀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣄⠈⠻⣷⡄⠘⣿⡄⠹⣆
⠀⠀⠀⣠⡟⢁⣴⠟⠁⣀⣀⣤⣀⣀⠀⠈⠑⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠞⢉⣀⣀⣀⣀⠉⠻⣷⡄⠹⣿⠀⢸⣿⠀⢿⡆⢀
⠀⠀⢰⡟⢠⡟⢁⣴⠟⠉⠀⡀⠀⠈⠑⠢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠉⠙⠛⠻⣦⡈⣿⠀⣿⠀⣼⡏⠀⣾⡇⢸⡆
⠀⠀⣿⠁⣿⢁⣾⢁⣴⠟⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⣴⣶⠿⠟⠛⠶⢬⡳⡸⣰⢃⣴⠟⢀⣼⡟⠁⣸⡇
⠀⠀⣿⢸⣿⢸⡇⢸⡇⣴⣖⣂⡀⢀⣀⡀⣤⣤⡀⣤⣤⢰⣶⡦⠀⠀⠈⣰⡿⠋⢀⣤⡶⠟⣛⡥⣺⡮⡵⢭⡤⠶⠟⠋⣀⣼⡟⢠⠄
⠀⠀⣿⡆⣿⡜⣿⡸⣷⣝⡿⢟⣣⣿⡿⢣⣿⡿⣸⣿⡟⣼⣿⡇⠀⠀⠀⣿⠁⣰⣿⠋⢠⡾⢋⡾⢱⠱⣍⠳⠶⣶⣶⠿⠟⠋⣠⡾
⠀⠀⠘⣿⡜⣷⣜⢿⣮⣛⣿⠿⢟⣋⣵⣿⡟⣱⣿⡿⣱⣿⡟⠀⠀⠀⠀⢿⠀⣿⠇⢠⣿⠃⢸⡇⠸⣧⠙⠷⣦⣤⣤⣤⣴⡾⠟
⠀⠀⠀⠘⢿⣮⡻⣷⣽⣛⡿⠿⠿⢟⣛⣵⣾⡿⢋⣴⣿⠟⠀⠀⠀⠀⠀⠈⠀⢿⡄⢸⣿⠀⢸⣷⡀⠹⣷⣤⣀⠉⠉⠉⣁⡤⠊
⠀⠀⠀⠀⠀⠙⢿⣾⣽⣛⡿⠿⠿⠿⢟⣛⣭⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢻⣧⡀⠹⣷⣄⡈⠙⠛⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠻⠦⣌⡙⠛⠛⠒⠒⠉