Senses Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢰⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⢸⠀⠀⡇⠀⠀⡎⠈⡆⠀⠀⠀⠀⠀⠀⠀⢨⠆⠀⠀⠀⠀⢀⣠⣤⡶⠖⠿⣿⣿⡲⢄
⠀⠀⠀⠀⠀⠀⠘⠀⠘⡀⠈⠀⠀⠇⠀⢰⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠾⣽⠿⠁⠈⢱⡀
⠀⠀⡠⠠⠀⠀⠀⠆⠀⢳⠀⡆⠀⢸⢀⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣀⣀⣀⡤⠶⠐
⠀⠀⠱⡀⠐⠄⠀⠠⠀⠘⡄⡇⠀⢀⡘⠁⢠⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁
⠀⠀⠀⠙⣄⠀⠀⣀⠆⠀⠉⠀⠀⠈⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠐
⠀⠀⠀⠀⠘⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⡹⠳⡄
⠀⠀⠀⠀⠀⠈⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠣⡄⠤⠔⠀⠉⠀⠀⠀⠀⠀⣀⠄⠀⠤⠴⠤⣀⠀⠀⠰⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠀⢀⠜⠁⡠⠤⢤⠠⢀⠀⠑⠀⠀⠱⣄⠀⠀⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠊⣠⡾⠷⠶⣿⠶⠾⢿⣦⡀⠀⠀⠈⠳⡄⠀⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠈⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠉⠁⠘⠀⠉⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠤⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⠉⠙⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣴⠟⠋⠁⢠⣻⣿⠇⠙⢆⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠻⣦⣀⠀⠀⠉⠁⠀⣀⡬⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣤⣤⡤⠴⠚⠉⢦⣀⣠⣴⠶⠛⠛⠋⠉⠉⠙⠛⠳⢶⣤⣀⣠⠎⠑⠢⠤⣤⣤⣤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠴⠀⠘⠀⠑⠂⠀⠀⠀⠀⠀⠀⠈⠠⠀⠄⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠂⠀⠀⠐⠀⣇⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣷
⠀⠀⠀⢀⣠⠤⠖⠒⠒⠢⢤⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡄⠀⠀⠀⠀⢈⠄⠀⠀⠀⠉⠰⢀⠈⠀⣀⡨⠂⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠤⢤⣀
⠀⢀⠔⠉⠀⣀⡀⠐⠀⠀⠀⠈⠑⢆⠀⠀⠸⣇⠀⠀⠰⡔⠀⠀⠀⠤⠀⠀⠀⠁⠂⢤⠀⠰⠋⠉⠀⠀⠀⠀⠀⢀⡿⠀⠀⢀⠖⠉⠀⠀⢀⠀⠀⠀⠀⠈⠑⢄
⢠⠏⠀⠀⠈⠁⠀⠘⠓⡄⠀⠀⠀⠀⢳⡤⠔⢿⡀⠀⠀⠠⡀⠠⠀⢀⡠⣤⣤⠀⣀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠗⠤⣴⠃⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⢧
⡎⠀⠀⠀⠀⠀⠀⠁⠀⡸⠀⠀⠀⠀⠀⡇⠀⠈⢷⡀⠀⠀⠉⢰⣿⣷⣾⠿⠭⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈
⡇⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢳⣄⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠘⡇
⢱⡀⠀⠀⠱⣄⠀⠀⠔⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠳⡄⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣄⣀⣀⣀⣀⣤⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⡴⠃
⠀⠀⠈⠳⢤⣀⣀⠈⠒⣂⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⡀⠀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠛⠛⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠤⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠊⠀⠄⢀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁⡴⠿⠶⢾⠶⠾⢷⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⣀⠀⠛⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⡀⠀⠁⠈⠀⠀⠀⡠⠃⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⠉⠒⠒⠂⠚⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⣀⡠⠤⠖⠉

⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⢀⡤⠤⠦⠤⣄⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣄
⠀⠀⠀⠀⡴⠋⢀⡠⠤⣀⠀⠙⣆⠀⠀⠀⡴⠁⡤⠒⠒⡆⠈⠳⡄⠀⠀⠀⡴⠉⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⢰⡇⣰⠁⣤⣿⣆⠙⡄⢸⠀⠀⢸⠁⠀⡇⠀⠀⠷⡢⠀⢹⠀⠀⢸⠁⣴⣒⣑⢒⣑⡲⡄⢹
⠀⠀⠀⠈⣇⠈⠢⢍⣛⡡⠜⠁⡼⠀⠀⠘⡆⠀⡇⠀⠀⢠⠃⢀⡞⠀⠀⠘⣆⠈⠲⠤⠤⠴⠊⢀⡞
⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⡤⠞⠁⠀⠀⠀⠘⠦⣌⣒⣒⣁⡤⠊⠀⠀⠀⠀⠘⠢⣄⣀⣀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⠤⠶⠦⢤⡀⠀⠈⠉⠀⠀⢀⣠⠤⠶⠦⢤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⢖⠼⢖⢄⠉⢦⠀⠀⠀⣰⠋⠀⢠⠀⢀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠟⡄⣠⡼⠀⠘⡇⠀⠀⡇⠀⢀⠎⠀⠘⡄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡙⠤⣳⠃⠀⣸⠁⠀⠀⢧⠀⠸⠤⠄⠤⠜⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣉⣋⣁⣠⠞⠁⠀⠀⠀⠀⠳⢤⣀⣀⣀⡠⠞⠁

⠀⠀⠀⠀⣴⣖⢒⡆⡄⣠⣠⣄⠀⢀⣤⡀⣤⡄⣠⣤⢀⣤⡀⣤⡄⣠⣄⠀⢰⡆⣤⡄⣠⣄⢠⣤⡀
⠀⠀⠀⠀⠻⠉⠘⠇⠻⠇⠻⠿⠀⠘⠿⠛⠿⠏⠻⠸⠘⠿⠛⠿⠗⠻⠟⠀⠸⠛⠷⠇⠳⠞⠸⠻⠇

⠀⠀⠀⠀⠔⢊⣉⣉⡑⠢⡀
⠀⠀⠀⠀⡴⠁⠀⠀⠙⣆⠹⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⡶⢶⣤⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠠⣸⡀⡇⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠾⠄⠀⣿⣟⣛⣽⣇⣬⠟⠂
⠀⠀⠀⠀⠉⠉⠈⠑⡄⠘⡇⡇⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉
⠀⠀⠀⠸⣷⠀⠀⢀⠆⠀⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠈⠐⠢⠤
⠀⠀⠀⢸⣇⠔⠊⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⡀⠀⢀⣠⠴⣦⠀⢀⠌⠀⠰⠒⢆⠀⡀⠀⢸⠀⢠⠒⢢
⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠈⠀⠀⠀⢸⠀⢸⠀⡇⠀⢸⠀⡌⠀⢸
⠀⠀⠀⠀⠀⣈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢂⢸⠀⠘⠀⡇⠀⢸⢀⠃⠀⡌
⠀⠀⠠⠊⠉⠀⠉⠁⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠈⠀⠀⡇⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠒⠒⠛⠛⠓⠛⠛⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⡇⠀⣇⡇⠀⢠⡆⠀⢰
⠀⠀⠀⠂⠤⠤⠤⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⡇⠀⠈⠁⠀⠈⠀⠀⠘⠀⠀⠀⡠⠄⠒
⠀⠀⠀⡀⠤⠤⢄⡠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠎⠀⢀⠔
⠀⠀⠊⠀⢀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⡠⠁
⠀⠀⡠⠦⠐⠠⠜⠠⠄⠢⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣴⠆⠀⠀⠀⠀⠀⠲⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⠀⣀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠂⠄⣀⡀⠠⠊⠀⠀⠀⠀⠀⠉⠑⠄⠀⠔⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠐⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⠐⠀⡀⠈⠸⣧⡗⠉⠀⠆⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢆⠀⠀⠀⠀⠑⠂⠊⠁⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠢⢄
⠀⠀⡠⠒⠉⠁⠀⠈⠉⠐⢤⡉⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠁⣠⠔⠊⡉⠀⠀⠉⠑⠢⡀
⢀⠊⠀⠀⠔⡈⠭⢅⠢⡀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⡄⠀⠀⠈⠰⠒⠀⠀⠃⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠘⠒⠀⠤⠄
⡇⠀⠀⠀⠀⠁⡐⡐⡠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡯⠀⡏⡇⠌⣭⠄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠄
⠠⡀⠀⠀⢀⠈⠀⢀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠃⠀⠃⠸⠁⠲⠄⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠑⢄⠀⠀⠑⠒⠁⠀⠀⣠⠞⠀⠀⠀⠀⣀⡀⣀⡀⣀⠀⠀⣀⡀⢀⡀⢀⣀⠀⠀⠀⠀⠘⢦⡀⠈⠂⠤⠤⠐⠁⢀⠔⠁
⠀⠀⠀⠉⠐⠒⠀⠐⠒⠉⠀⠀⠀⠀⠀⠘⠦⡄⡧⠄⡟⣄⠀⠧⡄⠠⠄⠳⢬⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠓⠂⠃⠘⠂⠒⠊⠐⠂⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⡀⡀⢀⣀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠎⠉⢂⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠒⠇⠒⠸⠒⠂⠲⠲⠀⠀⡄⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠠⠀⠀⠘⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⠆⠀⠈⠀⠀⠞⠁⠀⠀⠃⠈⠉⠉⠉⠁⠸⠀⠀⠀⡠⠂⠀⠀⠀⠢⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢄⡀⢀⠔⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠢⠤⡀⠀⡠⠤⠂⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠀⠀⢀⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣶⣶⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣄
⠀⠀⠀⠀⠀⠀⡞⠛⡇⠀⣿⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡟⠙⡇⠀⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡴⠚⠛⠓⢦⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⣿⠀⣧⡞⠓⡆⠀⠀⠀⠀⠀⠸⠁⠺⣤⡀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⡇⠀⠃⠀⠃⠀⠛⠀⣿⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡼⠁⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⣾⠀⣀⣤⡀⠀⠀⠀⣠⣄⡀⠘⡇
⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣥⠤⠉⠲⠴⠚⠁⢤⣥⠴⠃
⠀⠀⠀⠀⠀⠀⠙⠒⠲⠖⠒⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠛⠉⠉⠉⠛⢶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣚⡉⠀⠀⠀⠈⠉⠁⠀⠀⠀⢉⣓⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡾⠁⠀⣠⣤⠀⠀⠀⢳⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢮⣉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⣉⡴⠋⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣇⠀⠀⢿⣿⡿⠀⠀⢸⠃⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣄⣀⣠⣤⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣽⣄⠀⠀⠀⠀⠀⣠⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡐⠁⠀⣁⠀⠀⠀⢐⢠⢤⡄⡠⠀⠀⠀⠀⡈⠀⠀⠀⠀⣀⠄⠠⠄⠠⣀⠀⠀⠀⠑⠂⢐⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠡⠁⠀⠐⠄⡐⣛⠂⠈⠂⠀⠀⠊⠀⠀⠀⠀⠀⠉⠉⢀⡀⠉⠉⠀⠀⠀⠂⠁⡼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠂⠑⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠶⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠘⢦⠀⠀⠀⠀⡴⠃⠀⠀⣀⣤⠴⠒⠲⢤⣄⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣀⡔⠊⠁⠀⠀⠀⠀⠀⠈⠑⢤⣀⠘⡗⠒⠒⢺⠃⢀⠔⠋⠀⠀⢀⣤⠀⠀⠀⠈⠓⢄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠙⠨⡍⠉⠉⠉⠉⠉⣁⠔⠉⠀⠀⡇⠀⠀⢸⠀⠳⣄⠀⠀⠀⠸⠿⠟⠀⢀⠀⢀⡴⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⡇⠀⠀⠑⠢⠤⣄⣀⣀⡤⠤⠚⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠢⠄⠀⠠⠊⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠳⢄⣀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⢀⣠⠞⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠓⠲⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⢉⠉⠙⠒⢤⡀
⠀⠀⠀⣠⠞⠉⠀⢠⢢⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠔⢉⡁⢈⡉⠢⡀⠙⢦⡀
⠀⠀⡰⠃⠀⠀⠀⢸⢘⡤⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠇⢰⠁⠀⠀⠙⠄⢰⠀⠀⢳
⠀⢰⠃⠀⠀⠀⠀⢸⠸⠇⠾⠉⠖⡄⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢃⠈⠓⠦⡀⠀⠀⡜⠀⠀⠈⡇
⠀⢸⠀⠀⠀⠸⠈⠶⠀⠀⠀⠀⠀⠂⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢠⠃⢀⠴⠋⠀⡰⠁⠀⠀⢀⡇
⠀⠘⣇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠰⡀⠈⠀⠀⡐⠁⠀⠀⠀⡼
⠀⠀⠘⢆⠀⠀⠀⢣⠀⠀⠀⠀⢀⠇⠀⣠⠏⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠸⡀⠀⡰⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠑⢦⣀⡀⠉⠐⠒⠂⣁⡤⠞⠁⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⡀⢀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣉⣈⣁⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠙⢦⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢰⠁⠀⢡⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⣸⠁⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⢹⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡏⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣧⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡠⠃⠀⠀⠀⠀⠈⢢⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⣃⠤⡀⠀⠀⢀⡤⣄⡇⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠉⠉⠂⠐⠋⠉⢁⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠛⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠙⠛⠲⢤⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⠬⢀⠢⡀⠀⠙⢦⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⢁⠂⠀⢑⠱⠀⠀⠈⢧⠀⠀⡞⠀⠀⢀⠔⠂⠒⠒⠙⠢⡀⠀⠈⣧⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⢸⠀⢸⡇⠀⣀⠤⠴⠖⣦⣴⡒⠦⣌⡀⠀⢸⠀⠀⡇⠀⠀⡜⠀⡀⠀⠀⢀⡀⢳⠀⠀⢸⡇
⠀⠀⠀⠈⣇⠀⠀⠀⢠⡀⠠⠜⠀⡸⠀⠀⠀⡾⠀⠈⣇⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⡾⠀⠀⢷⠀⠀⠈⠈⠁⠀⠀⠈⠁⠉⠀⠀⣸⠁
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠐⠈⡠⠞⠀⠀⢀⡼⠃⠀⠀⠘⣆⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡼⠁⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠈⠓⢦⣄⣉⣁⢀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⡀⢀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⢀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠛⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡴⠫⠀⠈⠁⠀⠀⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⣤⣷⣶⣾⣿⣶⡦⠀⠀⠹⡆⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠻⠿⡿⠋⢹⠡⣶⠈⡏⢦⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⠦⢬⡷⠴⠎⠀⠈⠀⢠⡇⠀⠀⣧⠀⠀⢀⠄⠀⠀⡤⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠘⢦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀⠀⢀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⢀⣀⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠶⠆⠀⢀⡀⡀⢀⢀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢶⣶⣄⠡⠰⠀⠛⡈⠀⣰⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⡟⠀⠁⠄⠂⠛⠀⠛⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

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

⠀⠀⠀⢀⠠⠀⠐⠒⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠈⠱⣄
⢀⠎⠀⠀⡠⠔⡒⠛⠒⠢⢤⡀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠂⠀⠀⠢⠀⠀⠀⠀⠘⣷
⢸⠀⢠⠎⠀⠈⠀⣴⡄⠀⠀⠙⠕⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠘⡀⠐⠤⣀⠐⠀⠈⠀⠀⢀⠠⠂⠀⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠱⡀⠀⠀⠉⠂⠒⠐⠈⠈⠀⠀⡼⣱⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⣰⣻
⠀⠀⠈⠲⢄⣀⡀⠀⢀⣀⣠⡴⢿⡿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢄⠀⠈⠒⢄⡀⠀⠀⠀⣀⣀⣤⢾⡿⠋
⠀⠀⠀⠀⠀⠙⠛⠯⠿⠿⠟⠛⠁⠠⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠉⠛⠯⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡎⠀⠀⠀⠢⡀⠀⠀⠀⢠⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠘⢤⡀⠄⠀⠄⠃⠀⠀⢀⣜⡿⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠒⡄⠌⠐⢤⡀⠀⠑⢤⣀⠀⠀⠀⠀⢀⣀⣴⣯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄
⠀⢀⠄⠀⠤⠀⠀⡔⢰⠃⠂⠀⠀⠙⣦⠀⠀⠈⠛⠻⠿⠯⠿⠿⠛⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠈⠀⠀⠰⠑⠆⠁⠈⠀⢀⢈⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠂⠀⠐⠐⠊⠉⠀⠠⡀⠀⢿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢁⠎⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡂⠀⠀⠀⠀⠀⠒⠒⢂⠏⠀⢸
⠀⠐⡀⠀⠀⠘⠀⠀⠀⡠⠂⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠂⠄⠀⠀⠀⠄⠂⠁⠀⢀⣿
⠀⠀⠀⢄⠀⠀⠀⠀⠐⠀⠀⠀⢀⣴⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡾
⠀⠀⠀⠀⠉⠲⢤⣤⢤⡴⣶⣾⡯⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡤⣤⣶⣶⣿⠝⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣛⢒⢖⡶⢛⠟⠛⠟⡻⠟⣻⠟⢛⡿⢛⡻⠛⠻⢳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢈⠹⣳⠚⠒⠛⠚⠓⠚⠓⠛⠓⠒⠛⠒⠛⠒⠛⠒⠛⠛⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠪⡢⣑⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⠢⡙⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡱⡜⣾⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⡑⡕⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣷⡜⢼⠃⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⡜⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⣸⢅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⡌⢜⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⣜⢎⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢦⡀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⢷⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠺⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠳⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⡴⡶⢲⠞⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠙⣴⠇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢾⣤⣠⠥⠾⠶⠞⠳⣤⡀⠀⠀⠀⠀⠀⣀⣤⢾⢱⡎⣧⠟⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⣆⠀⠀⠀⠀⠀⠈⠛⢿⣒⠒⠞⢻⠉⠃⢸⣨⠷⠋⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⢦⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠓⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠤⣤⣤⣤⣤⣤⠤⠤⠖⠚⠉

⠀⠀⢘⣆⢠⣾⠆⠛⠋⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⣾⣆⣀⣀⠀⠀⠀⠀⠸⠿⠃
⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠀⠀⠀⢀⡤⡞⠿⠗⠀⠀⢀⡾⠿⠂⡼⠿⠆⠀⠀⣾⠁⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡴⣦⠀⠰⣞⡇⠀⠀⣵⠏⠀⠀⠀⠈⠙⠁⣤⠀⠀⠀⡜⠁⠀⡼⠁⠀⠀⠀⠠⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣹
⠀⠸⣿⠞⠚⠁⠀⠈⠃⠁⢀⡞⣽⠶⢶⡄⠀⢀⣤⣴⠃⠀⠀⠀⠉⠙⠚⠁⣀⠀⠀⠀⠀⠀⠀⡤⢄⣰⣩⠆⠀⠀⢠⣶⣦⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠸⣆⠧⢁⣸⠇⠀⠈⠙⠃⠀⡠⠤⡀⠀⠀⢀⣴⣏⠀⠀⢀⡤⣄⠀⠙⠊⠉⠀⠀⠀⣠⠏⠈⠁⠀⠈⠁
⠀⠀⠀⠯⠵⠫⠖⠃⠀⠀⠀⠈⣙⠛⠁⠀⢀⠀⠀⠀⠀⢻⠉⠀⠀⠀⠻⣴⠋⠀⠀⣸⣼⠃⠀⢀⣶⠀⠀⠀⣀⠀⢀⣴⣄⠀⢠⣖⡦
⡀⠀⠀⣀⠀⣀⡴⠛⢦⡀⠀⠀⠉⠋⠀⢠⠏⠉⢹⠃⠀⠸⠀⠀⢀⣀⠞⠁⠀⣰⢞⣽⣄⡀⠀⢘⣧⠀⠀⣼⠻⠃⡼⠉⠁⠀⠸⡇
⠃⠀⠀⠻⠟⢡⣤⣠⠜⠻⣄⠀⠀⠀⣀⡼⢀⣄⡟⠀⠀⠀⣠⣄⠻⡟⠀⠀⠰⡇⠏⠀⢨⡇⠀⠘⠀⠀⠰⢿⣗⣺⠃⠀⣼⠃⠀⠃⠠
⠃⠀⠀⠀⠀⠈⠛⡁⠀⣠⠞⠁⠀⠘⠿⠃⠈⠛⠁⠀⣴⡆⠻⣅⠀⣹⣆⠀⠀⢙⠲⠶⠛⠁⠀⡤⣤⠀⠀⢠⣄⠀⠀⣸⣧⠀⠀⠀⢰
⠀⣠⣖⠁⠀⠀⠘⠿⠋⢁⡀⢠⢞⡆⠀⠀⠀⠀⠀⠀⠹⣄⢀⡼⢾⡥⠞⠁⠀⠙⠓⠁⠀⡔⢠⡟⠁⠀⠀⢸⡸⠃⠀⠈⠉
⠈⠁⠋⣀⣀⠀⠀⠀⠀⠸⠿⠋⠉⠀⠀⠀⣠⢾⡅⠀⠀⠈⠯⠖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠈⣉⠀⠀⠀⠰⣉⠇⠀⢠⠏⠙⢲⠆
⢀⡀⠈⠓⢮⣇⠀⠀⠀⠀⢀⣠⣴⣦⠀⠀⠀⠉⠀⢠⡤⠤⣤⠀⠀⠀⣀⣤⡞⠉⢳⡀⠀⠸⡯⠭⢽⠀⠀⢀⣤⣴⠋⣀⣠⠏
⣻⠏⠀⠀⠸⡿⠃⠀⠀⠐⠋⠀⠈⠉⢠⣶⡄⠀⠀⢸⠋⠉⣿⠀⠀⢺⡁⠀⣳⡀⢀⣷⡄⠀⣷⣦⢸⡇⠀⠀⠉⠁⠈⠛⠋⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⢠⣶⡆⢸⠉⠀⢀⣤⡿⠰⣿⠇⠀⠀⠀⢳⡐⠿⠃⠈⠛⠁⠀⢀⠀⠀⠿⠗⠀⠀⠀⠀⡞⣇⠀⠀⠉⠋
⠀⠀⡀⠀⠀⠀⠀⠘⡷⢄⠀⢸⡥⠤⣾⠀⠀⠀⠉⠁⠀⠀⣤⣆⠄⠀⠿⠗⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣤⡀⢈⠽⡄
⠀⣷⣿⠄⠀⠀⠀⠀⠛⠃⠀⠈⠛⠉⠉⠀⠀⠀⢀⣀⠀⠘⠱⠋⠀⠀⣤⡄⠀⠺⣝⣦⠀⢠⣶⡇⠀⢠⣶⡄⢸⠋⠁⠈⠊⠁⠀⣄
⠀⠉⠈⠀⠀⠀⢠⡒⡄⠀⠀⠀⢠⠖⣛⣛⡶⠾⡥⠾⠃⠀⠀⢠⣦⠀⡟⠁⠀⠀⠀⣿⡆⠀⠉⠀⠀⢸⣉⣤⣿⠀⠀⠀⠀⠀⣿⠋
⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⢀⣀⢿⠈⠁⢛⣷⠀⠀⠀⡄⠀⠀⢸⡁⠀⢻⠀⠀⠀⠀⠈⠀⠀⣤⣄⠀⠘⠒⠒⠋⠀⠀⠀⠀⠟⠙
⠐⣿⡟⠀⠀⠀⠘⠃⠀⠀⠈⠫⠘⠛⠲⠛⠁⠀⠀⣼⣥⡀⠀⠘⣧⠴⠞⠀⠀⣀⡀⠀⠀⠀⣿⣉⠀⠀⠀⢀⡄⠀⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠒⡆⠀⠀⠀⠀⠀⠀⠀⠸⢧⡞⠀⢠⣦⡀⠀⠀⠀⡴⠛⠛⠀⠀⠀⠸⠜⠀⠀⣰⡟⠁⠀⠀⣾⠀⠀⡇
⢠⣶⡄⠀⠀⠀⢰⠇⣀⣸⠃⠀⠀⢀⣀⠸⣿⠃⠀⠞⠀⠀⣀⡯⠇⠀⠀⠘⠁⠀⠀⠰⣖⢦⠀⠀⠀⠸⠉⠃⠀⠀⠀⢿⣶⠀⣇
⠋⠉⠁⠀⠀⠿⠟⠀⠻⠛⠀⣠⣤⠸⣟⠀⠘⣦⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢀
⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠙⣏⠀⢈⣶⣚⡭⠇⠀⢠⠤⣄⡴⣻⠆⠀⢰⠯⣟⣷⠀⠈⠛⡁⠀⠀⢸⡭⠃⠀⠀⢀⡞⠓⢦⣤⣀⠈
⠀⢀⡤⠖⢿⠉⠈⢧⠀⠀⠀⠀⠈⢿⣭⠞⠉⠀⣀⠀⠀⠉⠁⠉⠁⢀⣀⡏⣠⣤⡏⠀⠀⣿⣿⠀⠀⠸⡇⠀⢠⣄⡾⢀⣄⡾⠀⢉⡟
⣀⣈⣧⣀⣈⣧⣀⣨⣇⣀⣀⣀⣀⣀⣀⣀⣀⣼⣻⣇⣀⣀⣀⣀⣀⣻⣟⣀⣈⣉⣀⣀⣀⣘⣀⣀⣀⣀⣁⣀⣈⣛⣁⣘⣛⣁⣀⣼⣁
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣤⣶⡿⢆⠘⠛⢳⡾⣿⣿⣇⣤⣀⠆
⠀⠀⣿⣧⣤⣼⡄⢀⣿⣇⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡧
⠀⠀⣼⡿⠉⠛⠟⠛⠛⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠀⢠⣤⡶⠿⢂⠈⠀⠀⠰
⠀⠀⠻⠇⠀⠀⠀⠀⠀⠿⠸⠇⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠉⠀⠀⢺⠃⠶⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠖⠶⠟⠈⢿⠇⠀⡄⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢠⡤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠉⡅
⠀⢀⠀⠀⠀⡌⢰⣶⣦⡴⣤⣤⠟⠀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠸⣄⣀⣄⠃
⠀⠹⠀⠀⠔⠀⠈⠛⢹⠃⠈⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠼⠋⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠤⠀⠀⠀⠹⠀⠀⠀⠅⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠇⠀⠀⣼⣇⠀⠰⠀⠖⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢑⠚⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⣠⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠘⠃⢀⠀⠀⠰⣴⣶⣼⣷⣷⣿⡻⣿⣯⠀⠂⠶⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⡀⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠻⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⡠⠖⠛⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠊⢀⡴⠋⠉⠉⠓⢦⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⣒⣒⣂
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠈⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠈⣿⣿⣿⣧⣽⣿⠛⠶⣤⡀
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠢⣸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠠⢀⣹⣷⣭⣭⣾⣿⣤⡶⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⢦⠀⢹⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⢸⠂⢀⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⣀
⠀⠀⠀⠀⢀⣿⠁⢀⣠⠔⠁⠀⠘⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠘⢧⠜⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠏⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣠⠶⠶⠶⠀⠀⡠⠃⠀⢠⠋⠉⡆⠀⢸⠀⠀⢸⠀⠀⡰⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠀⠀⠐⠊⠉⠀⠀⠀⠸⠀⠀⢸⠀⢸⠀⠀⢸⠀⢀⡇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡇⠀⢸⠀⢸⠀⠀⢸⠀⢸⠁⠀⢸
⠀⠀⠀⠀⢀⡤⠴⠦⠤⠤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⡇⠀⠈⡆⢸⠀⠀⢸⠀⡎⠀⠀⡏
⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⡇⣇⠀⠀⡇⢸⠀⠀⢸⢰⠁⠀⢀⡇
⠀⠀⠐⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢡⢸⠀⠀⢣⢸⠀⠀⢸⡸⠀⠀⢸⠁
⠀⠀⠀⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣼⠀⠀⠘⠊⠀⠀⠘⠃⠀⠀⡸⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⣴⠞⠁⠀⣠
⠀⠀⠀⢀⠔⠊⠉⠑⠒⠒⠋⠉⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣠⡾⠁⠀⣠⡾⠋
⠀⠀⠐⠁⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⡸⠉
⠀⠀⠀⣠⣦⡤⠄⣀⠨⢄⡀⠸⢤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣤⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠈⠀⠛⠛⢦⡀⠀⠀⡠⠛⠛⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠶⠶⣦⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠾⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠰⠀⢀⠀⠀⠀⠀⠘⠯⣭⠉⢉⠉⠉⣉⣉⣩⡍⠀⡄⠀⣤⣀⣀⡓⠞⢄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠄⠈⢀⡀⠀⠀⢀⠁⠐⡛⠀⠆⠈⠛⢿⣿⠏⠀⠈⠀⠀⠈⢻⣿⠟⠋⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢦⠀⠀⢈⠋⠀⢀⣡⣴⣿⠁⠀⠀⠀⠀⠀⠉⠀⠁⢀⣄⠀⠀⠈⠁⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠐⡈⠀⠰⠆⠀⡁⠉⠉⠛⠻⡀⠀⠀⡶⠦⠀⠂⠀⠀⠀⠀⠁⡈⢀⠂⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⣄⡒⣴⠅⣡⠀⡂⠀⠀⠀⠐⣅⡈⢂⠀⠀⠀⠉⠂⢤⣾⣾⣦⡄⢂⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠚⡂⢠⢋⣾⣿⡄⠀⠀⠸⠟⠁⠀⠀⠀⠀⠰⠛⠀⠒⣀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⢀⣿⢀⡍⠠⠂⠚⠋⢉⡉⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣸⠇⠛⠳⡀⠒⠀⠀⠀⠀⣾⣧⣤⡶⠶⠶⠶⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢠⡿⢠⠀⡁⣴⣄⠡⠀⠈⠈⠉⠁⢀⡔⠀⠀⣈⢂⣻⠃⡀⠈⠉⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⣼⠇⣀⢱⢀⣿⣿⠶⠄⠀⢂⠠⠀⠈⠓⢀⠲⢶⣶⡥⠀⠤⠈⡐⠠⡆⠀⣉⠻⢶⣄
⠀⠀⠀⠀⢠⡿⠠⠴⣿⠜⠋⠀⠀⠀⠀⠁⠀⣱⡌⠂⣀⠕⢦⡝⢋⠀⠀⢀⡹⡟⠲⣤⠞⣻⣠⠙⢷⡄
⠀⠀⠀⠀⣾⠃⠒⠀⠅⠐⠀⠀⠀⠰⡇⠀⢐⣿⡇⣁⠐⠶⢿⡀⢀⠀⠀⢀⡀⡽⠆⡁⠶⡇⣀⠀⠈⠻⣦
⠀⠀⠀⢰⡟⠀⠀⢸⣦⡱⣤⡀⠀⠀⡀⠄⢀⣨⣥⣴⣬⣥⡘⢳⡌⠀⠀⠌⣹⡥⠞⠛⠦⣽⡌⠆⠀⠀⡹⣧
⠀⠀⠀⣾⠃⡀⢀⣼⣿⣿⡜⣿⠟⢀⣤⡾⠛⠉⠀⠀⠀⠈⠙⢦⡁⠀⠂⡀⠀⠊⠃⡀⠉⠒⠀⡀⡂⢀⣇⢹⣇
⠀⠀⢰⣟⠈⢀⠀⠂⠀⠀⠈⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢄⣀⣀⡀⠄⠃⠀⣀⣀⡀⡤⣼⣎⠆⢿
⠀⠀⠀⢿⡄⠀⢰⣤⠀⡀⠁⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⣿⡿⣱⢦⣠⢖⡜⣿⡇⠛⠢⠍⡀⢸
⠀⠀⠀⠈⠻⢶⣤⣤⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠈⡔⠀⡜⡟⡀⠑⠌⠃⠀⠀⠆⢼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢦⣄⠓⢰⢀⠀⠀⠱⠼⠂⢀⠠
⠀⠀⠀⣀⣤⣤⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⠦⠉⠓⢤⠄⢀⢐⣡⣾⢠
⠀⣠⡾⢋⠁⠀⠠⠀⠄⢈⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠠⠀⠉⠉⣸
⢰⡟⠀⠂⠀⢠⣤⠀⠊⠂⠠⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠊⠂⢠⣤⠀⠀⠆⣰⡆⠀⠁⠀⢠⡿
⣿⠩⠧⡐⢄⠈⠁⠀⣠⣤⡄⡀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⡀⣠⣠⣀⠀⠙⣀⢀⣼⣿⣧⠈⠄⢀⣾⠃
⣿⠀⠀⠀⠈⠀⠀⢹⠟⡉⠀⢅⡀⢻⣆⠀⠀⠀⠀⠀⢀⣠⡾⠃⢀⠅⠊⠉⢿⣤⠄⠛⠈⠉⠁⠉⠁⢠⡾⠃
⢹⣇⣴⠂⠀⠁⠀⠀⠀⠰⡘⣶⡬⠆⠙⠷⠶⡶⠶⠾⠛⠉⠄⠯⣴⡞⡰⠀⠢⢀⠆⠈⠄⠀⠀⢀⣴⠟⠁
⠀⢻⣆⠀⠀⠰⢋⠒⣐⣐⠔⡞⠀⠂⠄⠀⠀⠀⠂⠀⢀⠀⠂⠀⠘⡔⣀⣐⡐⠚⢸⣧⡈⢀⣴⡿⠋
⠀⠀⠙⢷⣄⠠⠀⠐⠨⡛⢿⡏⠀⠨⠄⢦⢜⡃⢀⣛⢥⡄⠠⠀⠀⣿⠟⡋⠐⡂⠞⣩⣶⠟⠁
⠀⠀⠀⠀⠙⠷⣦⣔⡀⠹⠇⠐⠀⡀⠘⠊⢦⡈⠟⣡⠞⠸⠀⡀⠐⠀⢋⣁⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣤⣄⣀⣉⣁⣀⣉⣥⣤⣤⡶⠶⠟⠛⠉⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⣁⣈⣉⡉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣾⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⢸⣿⣿⣿⣿⣿⣦⡝⠻⣿⣿⣿⣿⣿⣿⣿⡿⢟⣡⣾⡇⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠘⣿⣿⣿⣿⣿⠟⣱⣿⣶⣬⣭⣭⣍⠩⣴⣶⣿⣿⣿⠃⠀⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⣈⣭⡍⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⢉⣭⣁⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣠⡀⠀⠀⢿⣿⡿⣩⣭⢩⣭⡉⣭⣍⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠻⣦⠀⠀⠘⣿⣷⣝⣛⣘⣛⣁⣛⣫⣾⣿⠃⠀⠀⠀⢀⠀⢶⣄⢸⡇⢀⣴⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣇⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣄⡀⢹⣇⠀⠙⢿⣿⠟⠁⢠⡿⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⠈⠙⠿⠄⠀⠀⠀⣬⣭⣭⣭⣥⡀⠀⠀⠀⠀⠉⣛⣿⣿⣄⠀⢸⡇⠀⢀⣼⣷⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣼⣿⣿⣿⣿⣿⣷⣤⠀⢀⠘⠛⠛⠉⠉⠙⠻⣾⣷⡾⠛⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⠿⣫⣾⣿⣐⠿⠶⣦⣤⡶⠟⢻⡟⠛⢷⣤⣴⡶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⣛⣛⣡⣾⣿⣿⣿⡏⣴⠾⢻⡿⠀⠀⣸⣧⠀⠀⢹⣟⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⠼⠇⢠⣾⢿⡿⢷⣄⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠆⠙⠁⠸⡇⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠉⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⡻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠋⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⡌⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣯⣠⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⢈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣴⣿⡿⠋
⠀⢀⣾⣿⡿⠿⠟⠿⢿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠸⣿⣿⣄⡀⠀⠀⣀⣼⣿⡟⠀⢀⣴⣿⡿⠿⠿⠿⣿⣿⣿⡟⠁⢙⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠁⢠⣿⣿⠋⠀⠀⣠⣾⣿⠟⢿⣿⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣴⣿⡿⠃⢀⣠⣾⣿⡟⠁⢀⣴⣿⣿⣿⡀⠀⠀⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠟⠛⠉⠀⣴⣿⡿⠛⢿⣿⣶⣿⡿⠋⢸⣿⣷⠀⠀⣿⣿⡏⠉⢉⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⡇
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣠⣿⡿⠋⠀⠀⣸⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⣿⣿⡇
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠁⣀⣤⣾⣿⠟⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⡏⠀⠀⠀⠀⠰⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣿⣿⡇
⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣀⣀⣀⡀⣿⣿⡇⠀⠀⠀⢀⣾⣿⡿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣄⣀⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢺⣿⣯⡁⠀⠀⣉⣿⣿⠇⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣾⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠈⠀⠈⠠⠀⠄⠠⠀⠀⠀⠀⢀⣤⡀⡀⠠⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⢢⡀⠈⠁⡀⠀⠈⠀⠠⡌⡙⢿⡛⢿⡀⢙⣷⠀⢰⣦⡼⢷⣀
⠀⠀⠐⡆⠀⠀⢨⠁⠀⠀⠀⣦⣦⣄⡀⠀⠀⠂⢥⠰⠌⡛⣁⠀⠼⠏⣾⡿⠛⠉⠀⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠰⠫⢻⢻⠛⢦⡣⠀⠀⠀⢡⠁⠀⠀⠀⢆⠋⠀⡀⠔⢐⣶⣤⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⡄⠎⠁⠀⡀⠸⠦⠀⠀⡄⠈⢃⠀⠂⢁⢣⠂⠁⠂⠀⣀
⠀⠀⠀⠂⣈⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠘⠀⡀⠄⠋⠢⢠⠸⡦⠜⠀⠀⠀⠐⠃⠄⠀⠀⠈⣡⠆⢔⠀⠀⠂
⠀⠀⠀⡾⠁⠀⣴⡏⠐⠂⢤⣤⣀⣂⡙⠀⠴⣤⡈⠠⢖⠚⡠⠅⢂⠄⠀⠂⠨⠄⠄⣠⠔⢾⡇⢴⠂⠐
⠀⠀⠀⠀⢶⣤⣿⡇⠀⠀⠈⠛⢻⠋⠀⠄⠀⡀⠈⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠒⢺⠇
⠀⠀⠀⢀⣰⣿⡏⠀⡓⠊⠐⢀⣈⠀⠠⠎⠀⠀⠨⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠐⠀⠄⣁⣴⡿
⠀⠀⠀⠈⡿⠋⠀⠁⠀⠀⠀⢀⠉⢠⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠙⠂⠰⢿⣽⠇
⠀⠀⠀⢷⠔⠀⢀⣤⣶⡎⣤⠜⠁⠀⢀⠄⣀⡂⣀⠀⠀⠒⠀⠀⠀⡀⠀⢀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠨⠄⠀⢸⣏⣌⣁⠙⢶⠀⠀⢸⠀⢇⠀⠀⠀⣠⠧⠀⠀⠐⢃⠀⠘⠠⣁⡀
⠀⠀⠐⠁⠀⠀⡀⠉⣱⠏⠀⠰⠀⠀⠈⡇⠈⠃⠌⠁⣀⠀⠀⠀⠐⣠⢦⡤⠚⠻⢟⡀⠖⡤⠀⠀⡠
⠀⠀⠀⠀⠐⢾⣶⣴⣿⠁⠀⠀⠀⣤⣴⣷⣄⡀⠀⠀⠅⠀⢂⣠⠶⢹⣥⣂⢐⡘⠃⠁⠾⠷⢂⣠⡀
⠀⠀⠀⠶⠀⠀⠻⣿⣃⣀⣀⣀⠈⢨⠟⠉⠉⠍⠈⠀⠸⢂⠯⠀⠩⠀⡟⠟⠅⠈⢢⠲⢟⠛⠋⠉⠀⣠⠀⡀
⠀⠀⠐⠀⢰⠀⠀⠈⡿⠏⠛⠉⠀⢀⡀⠀⡘⢆⠛⠠⠠⢹⣥⢤⠴⣎⠁⠈⢃⡔⠁⠁⡀⣘⡂⣴⡞⠛
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⣰⣾⣿⣗⠀⠄⠈⠦⠤⠀⣸⢕⠥⡊⠀⠑⡤⠋⣀⣤⠅⣬⣭⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⡻⠷⠠⣤⣤⣤⣾⡃⠁⢀⠄⠠⡀⡏⠀⠀⠋⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠉⠢⠁⡡⠠⢴⣶⣶⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠏⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢈⣽⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡙⠻⠟⢋⣤⣤⣌⠙⠿⠟⣉⣤⣤⣉⠻⠿⠋⣡⣤⣤⡙⠻⠟⢋⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣿⣿⣏⣉⣩⣿⣿⣏⣉⣩⣿⣿⣏⣉⣹⣿⣿⣀⠀⠰⣶⠾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⣿⣿⣿⡿⢿⣿⣿⣿⠿⠿⣿⣿⣿⡿⡿⣿⣿⣿⢿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣾⣿⣷⣶⣶⣾⣿⣷⣶⣶⣾⣿⣷⣶⣶⣿⣿⣷⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⢻⠇⠀⠀⠀⠀⠐
⠀⠀⢨⠀⠀⠀⠀⠀⢂
⠀⠀⢸⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠐
⠀⠀⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢸⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⠀⠀⢸⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⢀⣤⣀⣼⣿⣦⡰⠾⠗⠀⠀⢀
⠀⠀⢻⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢸⣼⡿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠐⠀⣈⣿⣿⣛⠏⠙⠩⣇⣀⠀⠄⠸⡀
⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⣠⣿⣿⣿⢿⣿⡿⣿⣿⡀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⣄⣰⣿⣿⣧⠀⠀⠁
⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠨⠟⠛⠉⠀⣼⣿⣯⣿⣯⣷⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣻⠻⡟⣽⣻⣤
⠀⠀⢸⠀⠀⠀⢸⠇⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡉⢻⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣇
⠀⠀⠸⡆⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡳⣿⣿⠟⠀⠀⠈⣿⣾⣷⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣙⣿⣿⣿⡆⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣽⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⡿⣿⣿⣽⣿
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣻⣿⡿⠀⠀⢻⣾⣾⣿⡟⠃⠀⠁
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⣸⣿⣼⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣹⣿⣿⣿⠃⠀⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠛⠛⠿⢿⣤⢶⠀⠀⢼⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⢿⡟⠷⠖⠒⠀⠀⠀⠀⠀⢀⣬⢙⠃⠀⢠⡀⠀⠚⠉⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠶⢶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣀⣴⡿⠟⠉⠁⠀⠀⠀⠉⠙⠻⢿⣦⣄⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠿⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠈⠙⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⣰⡟⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠟⠁⠀⠀⢀⣼⠏⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠀⣠⣴⠶⠶⠿⠿⠟⠛⠋⠁⠀⠀⠀⣠⠞⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⢻⣷⣦⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟
⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣠⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣄⠀⠈⠙⢿⣿⣿⣆⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠈⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⠝⠛⠳⠦⣀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠟⠋⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⡘⠀⠀⠀⡄⠀⢠⠀⠀⠀⠈⠀⠈⢢⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢃⠀⠀⠑⡀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡘⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠈⡄⠀⠀⠈⢆⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠐⢤⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢆⠀⡠⠒⠘⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠇⠀⣀⢀⣘⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠰⡎⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣤⡈⠀⠀⠀⣀⠀⠀⢰⣶⠶⣦⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⣯⣹⠇⠀⠀⠀⠁⠀⡿⢹⡿⣾⡧⣀⣴⣦⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣐⣿⣄⠀⠀⣇⠀⠸⠀⠀⡆⠸⡇⠀⣿⣻⣿⣿⡿⠛⠓⠀⠀⠆⠠⠖⠀⠀⠀⡤⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠁⢹⣶⡀⠘⣿⣆⠀⢸⣇⣷⣄⢠⠏⣿⡟⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡄⢤⠄⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠃⠀⠆⢸⠋⠀⠀⠀⠀⠺⣻⠟⢡⢮⠙⠛⢸⡏⠁⢹⣿⢻⣿⡶⠶⠾⠇⡈⣿⡧⠀⢀⡀⠀⠀⢳⣦⢂⡰⢧⡿⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠲⠶⠈⠀⠀⠀⠀⠀⠀⣿⠀⠈⠈⠀⠀⠀⢧⡀⢸⡇⢸⡟⠁⠀⠀⠈⠁⣿⠀⠀⠘⠂⠐⠁⠈⠀⢸⠁⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⣟⣉⣙⢻⡟⣩⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠂⠀⠇⠂⠐⠆⠆⠂⠸
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡠⠀⣉⣁⣤⣽⣿⣧⣄⣉⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣥⣶⠶⠶⣶⣶⣤⡀⢀⡀
⠀⠀⠀⢺⣶⣿⠷⠶⠶⣶⣶⡶⠶⠶⢿⣷⣾⠂⠀⠀⠀⠀⠀⡀⢼⠟⠋⢰⣆⣤⣿⣿⣿⡟⢿⣷⣄
⠀⠀⠀⠻⣿⣿⠀⠀⠀⠀⠋⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⣞⣿⣦⣀⠈⠻⣿⣿⣿⡿⢁⢾⡟
⠀⠀⠀⠀⠈⠻⢆⠀⠀⠀⠀⠀⠀⢀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠦⠭⠥⠤⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠀⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠂⠀⠀⣴⣿⣏⢿⠇⠀⠀⠀⠀⠀⠠⠠⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⢿
⠀⠀⠀⢀⠀⣀⣀⡾⢭⡙⠇⠺⠀⠀⠀⠀⠀⠄⢄⢀⠀⠤⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠈⣻⡙⢂⠂⠀⡄⢀⠀⠀⠀⠀⠀⡄⠉⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣥
⠀⠀⠀⠀⠰⠿⠷⡋⠀⠀⣀⠊⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢀⠨⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⢀⣼⠗⢆
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⠄⠀⠀⠀⠀⠀⢱⣶⣿⣦⢰⣾⣿⣿⡎
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠤⠄⠄⠠⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⣤⣴⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡕
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣟⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠐⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣻⡿⠿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣶⣿⣿⣿⠿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠏⠁⠀⠀⢸⣿⣿⣿⣧⣤⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣶⣿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⣴⣶⡦⡀⠀⠈⠀⠯⠓⠐⠁⠀⣀⠀⢒⡖⢒⣤
⠀⠀⠀⠀⠀⠀⠀⢇⢀⣴⣄⡀⢝⣶⣿⠇⣿⠐⠢⢠⣶⣲⣴⡶⠀⠀⡆⠘⠁⠈⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣟⣷⣿⣻⡟⠀⢰⡀⢠⣿⣿⣿⣯⠀⠀⠘⣇⢀⠀⡀⢸⡦⠀⠀⠀⢤
⠀⠀⠀⠀⠈⠃⠘⠀⠀⠀⠘⠛⢿⣿⢷⡄⣼⣧⢸⣿⢻⣿⣿⡀⠀⠀⢙⣶⣷⢝⣿⠇⡀⢀⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⢠⣂⣈⣿⣾⣿⡏⠉⢹⣿⡞⣿⣿⠁⠀⠀⠘⠉⠉⠙⠋⠀⠀⠊
⠀⠀⠀⠀⠀⠖⠀⠹⣯⠰⠈⠉⠉⣹⡇⢸⡇⠀⠀⢿⣇⣽⣿⡀⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠶⣶⣶⣶⡶⢶⣶⣶⣶
⠀⠀⠀⣾⣽⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠏⠹⡟⢀⡘⢿⣿⣿
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣆⣰⣷⣌⣡⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣴⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣏⣍⣉⣉⣍⣛⣅⣏⣩⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠞⠁⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣶⣤⣄⣀⣠⣤⣾⠿⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⠶⣦⡄⠀⠀⠀⠀⠹⢾⣿⡿⠋
⠀⠀⠀⡠⠒⡶⠛⢽⣿⡿⠟⠋⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄⠀⠀⠻⣿⣿⠏
⠀⠀⡜⠠⠄⠀⠀⡆⢱⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢡⠀⢤⣭⣤⠀⣜⣿⣿⣷⣦⣤⣀⣤⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣼⣩⣿⣷⠀⠀⠀⠀⠀⢠⣎⣿⣷⡀
⠀⠀⢿⣽⣆⠿⠑⠊⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠈⠻⠿⠟
⠀⠀⠋⠐⠛⠋⠁⠛⠀⠃⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⣀⣠⣄⣀⠀⠈⢳⡀⠀⠀⠀⢀⡞⠁⢠⠴⠚⠓⢢⠀⠈⠳⡄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⡟⠀⡰⠋⣁⣰⣦⡌⠑⣄⠀⣷⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⠤⡄⠀⢻⠀⠀⠀⢰⡇⣀⡤⠴⠲⢤⠖⠢⢤⣀⠘⡇
⠀⠀⠀⠀⠀⣇⠀⢣⡀⠻⣿⣿⠇⢀⠞⠀⣿⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠈⣸⠁⠀⣸⠀⠀⠀⢸⡄⠙⢍⡓⠒⠒⠒⢚⣩⠏⢀⡇
⠀⠀⠀⠀⠀⠹⣄⠀⠉⠒⠦⠴⠚⠁⠀⣰⠃⠀⠀⠀⠸⣆⠀⢸⡀⠀⠀⣠⠇⠀⣠⠏⠀⠀⠀⠀⢷⡀⠀⠉⠙⠓⠋⠉⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠈⠳⢤⣉⣉⣉⣁⣤⠞⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⣤⡤⣤⣤⣀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣠⣤⠤⣤⣀⡀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⣠⠤⠤⣄⠉⠳⣄⠀⠀⠀⠀⠀⣠⠞⠉⠀⡀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡎⣴⠋⠉⠋⢳⠀⠘⣧⠀⠀⠀⢰⠏⠀⠀⢠⠇⠀⢹⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠉⡇⢀⣴⡞⠀⠀⣹⠀⠀⠀⢸⠀⠀⢀⠞⠀⠀⠈⢧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢈⠙⠦⠞⡼⠀⠀⢀⡟⠀⠀⠀⠸⡆⠀⠸⣤⢤⢠⠤⡼⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠓⠶⠚⠁⢀⣠⠞⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⢰⣟⣛⢀⡇⣤⡀⣤⣠⣤⡄⠀⢀⣤⣤⢀⣤⣄⢠⣤⣄⢀⣤⣤⢠⣤⣄⢠⣤⡄⠀⢀⣏⢠⣤⣄⣠⣤⣄⢠⣤⣄
⠀⠀⠀⠀⠀⢸⡏⠉⠸⡧⠸⣿⠇⢿⣿⡇⠀⠸⣿⡿⠸⣯⡿⢸⡇⣿⠸⣿⡷⠺⣿⡿⢼⣿⡷⠀⠸⡿⠸⣧⡗⠻⣤⡿⢸⡇⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠉⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠋⠉⣉⣀⣤⣍⣉⠙⠓⠦⣀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡴⠛⣩⠥⠖⠈⠉⠻⣦⡀⠈⠳⣄
⠀⠀⠀⡴⠁⢀⣴⠾⠿⣷⣶⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣰⠋⠀⡰⠃⠀⠰⣄⡀⠀⡈⢿⡀⠀⠈⢧⡀
⠀⠀⡼⠁⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⢿⣿⣦⡄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠇⠀⢊⡵⠖⢲⣤⠈⠁⠀⢱⢸⡇⠀⠀⠀⢧
⠀⢠⡇⠀⠀⢠⣀⣀⣦⣤⣶⣤⣷⣦⣤⣄⡀⠀⠈⠙⠛⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠙⣆⠀⠀⠸⢸⡇⠀⠀⠀⢸⡆
⠀⢸⡇⠀⠲⣶⣿⣿⠿⠛⢹⣿⣉⣽⣿⢿⡟⢶⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠀⣀⣍⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠸⡇⠀⠀⠀⠀⠹⣆⠀⠈⢷⣝⣛⣯⠞⠁⠀⠙⢦⡀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠋⣴⠄⡴⠀⣼⠁⠀⠀⠀⢸⠇
⠀⠀⢳⡀⠀⠀⠀⠀⠈⠳⠦⣤⣄⣉⣀⣤⠴⠆⠀⠘⠋⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⡾
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣠⡤⠶⠒⠒⠒⠒⠶⢤⣄⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠉⠀⠀⡄⠀⠀⠀⢠⡄⠀⠀⠉⠲⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⠦⣀⣀⠀⠀⠀⠀⢀⣀⡤⠖⠋⢠⠞⠁⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠳⣄⠉⠳⠦⣄⣀⡈⠙⠋⠁⣀⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠹⡄⠀⠀⠀⡞⠁⠀⠀⠀⠀⢀⣴⠀⠀⠈⢷⠀⠀⠀⢠⠏⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠈⠉⠓⠦⣄⠘⢦⡀⠀⠳⠄⠶⠶⠶⠀⠀⠶⠶⠶⠠⠋⠀⢀⡴⠋⣠⠴⠚⠉⠁⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡴⠋⠁⠀⠀⠀⠄⠀⠀⠂⠀⠐⠀⠀⠀⠈⠳⣄
⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⠙⠒⠲⠤⠤⠤⠤⠶⠚⠋⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⡾⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠈⠀⠀⡁⠀⠀⠀⠀⢷
⠀⢰⠇⠀⠀⣀⣤⠞⣫⡅⢀⠈⠉⠉⣡⠀⠈⣝⠳⢦⣀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢸⠀⠛⠛⡉⠦⠤⠤⠶⠾⢧⣴⣴⣽⠷⠶⠦⠴⠶⢊⠟⠛⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠸⡆⠀⠀⠈⢦⡀⠀⠀⠀⠄⠆⠐⠠⠀⠀⠀⠀⡴⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠇
⠀⠀⢳⡀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠤⠤⠤⠴⠒⠋⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡞
⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⠀⠀⣀⣀⣠⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣄⣀⣀⠀⠀⣀⣀⣨⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠻⣿⣿⣿⡿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣱⡟⢎⢻⣿⣿⣿⣦⡙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢰⣿⣿⣬⡇⣿⡿⠻⣿⣿⡎⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣝⣛⣩⣾⠟⣴⠿⣌⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢻⣿⣿⣿⣿⣿⣿⠏⣾⣿⣷⡙⣧⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣎⢿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⡟⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣀⡙⣦⡻⢿⣿⣿⣿⣿⣦⣍⣉⣭⣾⠟⣣⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⣩⣶⣿⣿⢿⣿⣿⡶⣭⣛⡛⠿⠿⠟⣛⣫⣥⡘⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏⣼⣿⣿⡿⠃⣾⣿⣿⣷⠘⢿⣿⣿⣧⢻⣿⣿⣿⣿⣦⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡐⢛⣛⣭⣾⡇⣿⣿⣿⡿⢸⣷⣭⣛⡛⢈⣿⣿⣿⣿⣿⡇⠀⠙⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣣⠞⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢼⣿⡏⣵⣶⡆⣠⣾⣿⣿⣿⣷⣦⢰⣶⣮⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣇⢻⣿⣷⣜⣛⢻⣿⡟⣛⣫⣾⣿⡟⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣍⡛⠿⣿⣿⣿⣿⣿⣿⣌⢿⣦⡙⢿⣿⣿⣷⣶⣾⣿⣿⡿⢏⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣶⣤⣭⣿⣿⣿⣷⣝⠿⣶⣭⣛⣛⡛⣛⣛⣭⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠉⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⢷⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⣷⣶⣾⣇⣀⣤⣴⡶⠶⠶⠶⠶⠶⢶⣤⣄⣀
⠀⠀⠀⢠⣾⠟⢉⣠⣴⣶⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⣿⣥⣶⣿⣿⣿⣿⣿⣿⡿⠁⣀⣤⣶⣶⡿⠿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠙⢷⣦
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⣄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠉⣹⡟⠉⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠈⢿⡄
⢀⣀⡏⠂⠀⢰⡟⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠈⣿⡄
⠿⠿⠃⠀⠀⣾⠃⠀⠀⢀⣿⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠸⣿⡆⠀⠀⢸⣧⣴⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣿⡇⠀⠀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⣾⠏⠀⣿⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢹⢹⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⠀⢠⠇⠀⠀⢀⢜⣥
⠀⠀⠀⠀⠀⠘⣿⡀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣾⡾⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣇⣴⠋⠀⠀⠀⠘⢌⣡
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠙⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢋⣴⠿⣿⠁⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⣰⡟⠁⠀⣹⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠀⢀⣾⡿⣿⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋⢹⣇⠀⠘⣿⠀⢹⡀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⢠⡄⠀⠀⢿⣄⠀⠈⠁⠸⣇⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠙⠿⢶⣦⣤⣿⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢾⢳⡄⠀⠀⠀⠀⣠⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠋⠀⠀⠀⠀⢰⣿⠿⠟⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣷⣶⣤⣀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣾⣿⣿⣻⡿⢋⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾⣿⡿⠟⡱⠋⠁⢋⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠁⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣉⣉⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣴⣾⣿⡿⠁⠀⠀⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⡿⠁⠀⠀⠀⣀⣠⣄⣠⣵⣶⣶⣶⣾⡿⠿⠟⠋⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣏⠁⠀⠀⢀⣦⡽⠛⠛⣩⣿⣿⣿⡟⠁⠀⠀⠀⠀⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣬⠋⠀⠀⣸⣿⣿⣿⡿⠀⠀⠀⢀⡴⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠃⠀⠀⢰⡇⠀⠀⢰⣿⡿⠛⠋⠀⠀⣀⠤⢘⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢹⠀⠀⠀⢈⠄⠀⠐⠛⠁⡀⠀⢀⣀⣠⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡂⠀⠀⠀⠘⣰⣿⠿⠉⠉⠉⠛⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⡆⠀⠀⠈⣷⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢻⡀⠀⠀⠈⢧⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⣿⣀⡀⠀⠀⠉⠠⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠘⢿⣿⣦⣄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⠀⠀⠙⢿⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠂⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⠤⢤⣤⡤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠦⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⣤⣤⣀
⠀⠀⠀⠀⢀⠴⠋⠁⠀⣀⣀⣀⡀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⣀⣀⣀⡀⠀⠉⠻⣦⡀
⠀⠀⠀⠔⠁⠀⢠⡶⠋⠉⠁⠈⠉⠳⣆⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢠⡶⠛⠉⠉⠉⠙⠳⣦⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢀⠈⢧⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⢀⠘⣷⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⣇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⣇⠀⠸⣇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢹⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢻⡀⠈⣿
⠀⠀⠀⠀⠀⠹⡛⠲⠤⡤⠤⠤⣄⠀⠀⠹⣌⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠲⠦⠤⠤⠤⣄⠀⠀⠹⣾⡇⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢹⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣧⠀⠀⢹⡇⢸⡇
⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⢸⠄⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣠⡀⠀⠀⠀⠀⠀⢸⠆⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠇⠀⠀⠀⠀⣠⠟⠀⠀⣸⠁⣼⠁
⠀⠀⠀⠀⢰⡿⠀⠀⣀⣤⠴⠋⠀⠀⠀⠀⠃⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⣤⠶⠋⠁⠀⠀⠀⠃⢰⡟
⠀⠀⠀⠀⠸⣇⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⣫⣭⣶⣶⣶⣦⣤⣀
⠀⠉⣵⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢟⣵⣿⣿⡿⠿⠿⠿⢿⣿⣿⣷⣄
⠀⠀⠙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣶⣾⣿⣿⣿⠿⢋⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡆
⠀⠀⠀⠈⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣭⠽⠟⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⠛⠛⠿⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣃⣀⠀⠀⠀⠀⠉⠙⠻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠞⠁⠈⠉⠉⠛⠲⠦⣤⡀⠀⠀⠙⢿
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠈⠓⠤⣀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⠀⠀⣸
⠀⠀⠀⠀⠸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣫⣥⠴⠞⠉
⠀⣠⣤⡀⠀⠘⢿⣿⣿⠆⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠿⠻⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠹⣿⣿⣄⠀⣴⣿⣥⣄⠈⠻⣿⣄
⠀⠀⠈⠛⠋⠀⠈⣻⣿⠿⠛⠊⠉⠹⣆
⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⢀⣴⣿⠇⠘⢧⡀
⠀⠀⣴⠟⠁⠀⠀⠀⠀⠐⠿⠟⠋⣀⣄⠈⠻⣄
⢀⣾⠋⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⡿⠋⠀⠀⠈⠳⣄
⣼⡇⠀⠀⠀⠀⠀⠘⠿⢿⠿⠈⢹⣶⣦⣄⠀⠀⠀⠈⠳⣄
⣿⣇⠀⠀⠀⠀⠀⠰⣿⣦⣄⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠁
⢹⣿⣆⠀⠀⠀⠀⠀⠈⠻⠿⠇⣰⣿⣷⠄
⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏
⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠈⠙⠿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠖⠂

⢠⣶⠶⠶⠶⠶⣶⡄⠀⢀⣤⣀⣤⡀⠀⠀⣶⡶⠶⣶⠶⣶⡆⠀⠀⢀⣄⣠⣤⡀⠀⢰⣶⡶⠶⣶⣶⣶⠀⠀⠀⢀⣄⡀⠀⠀⠀⣶⣶⡶⠲⣶⣶
⢸⣧⣤⣦⣴⣤⣼⡇⠀⠸⠋⠀⠙⡇⠀⠀⣯⣀⠠⡤⢀⣨⡇⠀⢾⣿⣿⣿⣿⠀⠀⢸⣿⠄⠀⠘⠋⢻⠀⠀⠀⣿⣿⣧⠀⠀⠀⣿⣿⠃⠀⢿⣿
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠘⣄⣀⣠⠃⠀⠀⣿⣿⣀⣀⣸⣿⡇⠀⠀⡄⢾⣿⡏⠀⠀⢸⣿⣄⣀⣀⣠⣿⠀⠀⢸⣿⣿⣿⡄⠀⠀⣿⣟⣀⢀⣘⣿
⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠚⠛⠛
⠀⠀⣠⣶⣦⣄⠀⠀⢸⣿⡿⢛⣿⣿⣷⠀⠀⢀⣠⣤⣄⠀⠀⢰⡟⠻⠿⢿⠟⢹⡆⠀⠀⠀⣀⡀⠀⠀⠀⣾⠟⠛⠛⠻⢿⡇⠀⢀⣠⣀⣀⡀
⠀⠀⢻⣄⣨⡟⠀⠀⢸⣿⠀⣾⣿⣿⣿⠀⠀⢼⣿⣿⣿⡇⠀⢸⡿⣷⠆⠀⠰⣿⡇⢀⣴⣥⣤⣬⣦⡄⠀⣿⣠⡄⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣄
⠀⠀⢬⣭⣭⠅⠀⠀⢸⣿⣦⣈⣻⣿⣿⠀⠀⠘⠻⠿⠟⠁⠀⢸⣷⣦⢰⣿⡄⣿⡇⠈⠁⠈⠉⠁⠀⠁⠀⣿⣿⣁⣻⣿⣠⡃⠀⠀⠙⠘⠘⠀⠁
⠀⢀⣀⣀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠁⠀⢀⣀⣀⣀⣀⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⣀⣀⣀⣀
⢸⣿⣿⠟⣿⣿⣿⡆⠀⣾⡛⣦⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⡇⠀⠀⢰⡆⢰⡆⠀⠀⢸⣿⣿⡛⢿⣿⣿⠀⠠⣶⣦⠀⠀⠀⠀⠀⡿⠿⠿⠿⠟⠻
⢸⡿⠁⢀⣈⢻⣿⡇⠀⢉⣿⣧⣴⣷⠄⠀⣿⣛⠉⠀⠀⠙⡇⠀⠀⠠⢽⣿⠄⠀⠀⢸⠟⠉⠀⠀⢈⣿⠀⠀⠸⣿⣷⣄⠀⠀⠀⡇⢀⣀⡀⣰⣿
⢸⣷⣾⣾⣯⣶⣽⠇⠀⠘⢿⣿⡟⠉⠀⠀⣿⣿⣿⣷⣾⣿⡇⠀⠀⠀⠸⠇⠀⠀⠀⢸⣧⣤⣤⣤⣼⣿⠀⠀⠀⠿⠿⡿⠧⠀⠀⣷⣿⣿⣷⣿⣿
⠀⠉⠉⠉⢉⡉⠉⠀⢀⣤⣤⣤⣤⣤⣄⠀⠈⠉⡉⠉⠉⠉⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠉⠉⠉⠉⠉⠁⠀⣠⣤⣤⣤⠤⣤⡀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢈⣿⡆⠀⢸⡋⠉⢻⡿⠃⣿⠀⠀⠈⢻⣆⠀⠀⠀⢸⣿⡿⠀⠉⢹⣿⡅⠀⠀⠰⣿⡷⠀⠀⠀⣿⣿⠻⠳⢠⣿⡇⠀⠀⠀⢻⣦
⠀⢸⣿⠿⣿⡟⠁⠀⢸⣷⣤⣄⠀⣠⣿⠀⠀⠀⠸⣿⣿⠆⠀⢸⣿⡀⠀⠀⢀⣿⡄⠀⠀⣼⣿⣿⡀⠀⠀⣿⣧⡀⠀⢠⣿⡇⠀⠀⢀⣿⣿⣧
⠀⠀⠐⠀⠀⠊⠀⠀⠸⣿⣿⣿⣯⣿⡿⠀⠀⠀⠀⠫⠉⠀⠀⠘⢿⣿⣦⣴⣾⡿⠂⠀⠀⠘⠟⠟⠀⠀⠀⢿⣿⣷⣤⣿⣿⠇⠀⠀⠀⠛⠟⠃
⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⣀⣀⣀
⢸⣿⣿⡏⠹⣿⣿⡅⠀⢀⣀⣼⣄⣀⠀⠀⣿⠟⠉⠉⠙⢿⡇⠀⠀⠀⢀⣰⣆⡁⠀⢸⡿⠛⠓⠂⠾⣿⠀⠀⣠⣾⣶⣄⠀⠀⠀⣿⣿⡇⠈⠹⣿
⢸⣿⣯⠃⠀⣼⣿⡇⠀⣿⣿⣿⣿⣿⡇⠀⣏⠀⠈⠀⠀⠀⡇⠀⢠⣶⣿⣿⠟⠁⠀⢸⡇⠀⠀⠀⢀⣿⠀⠀⣿⣿⣟⣿⡆⠀⠀⣿⣿⠏⠀⠀⣿
⠸⣿⣿⣦⣴⣿⣿⠃⠀⠈⠛⠛⠛⠋⠀⠀⢿⣦⣤⣤⣤⣾⠇⠀⠘⠻⠟⠁⠀⠀⠀⠸⣿⣤⣤⣤⣾⡿⠀⠀⠉⠛⠛⠋⠀⠀⠀⢿⣯⣔⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⢀⡀⢀⠀⠀⢀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡄
⠀⢠⣾⣿⣿⣷⡄⠀⢸⡇⠀⠀⠀⠈⣿⠀⢸⣾⡽⣿⢭⢸⠀⢸⣿⣿⠏⠹⣿⣿⡇⠀⠀⢀⡞⣄⠀⠀⠀⣿⡿⠿⠙⠿⢿⡇⠀⠀⠀⣈⣦⣠⠔
⠀⠘⢿⣿⣿⡿⠁⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⣿⢭⣿⠼⢸⠀⢸⣿⠃⠀⠀⠘⣿⠅⠀⢀⢶⡼⡴⢆⠀⠀⣿⣷⠄⣀⢰⣿⡇⠀⠀⠉⢩⠟⢧⡀
⠀⠀⠀⠉⠁⠀⠀⠀⠸⠷⠶⠶⠶⠾⠟⠀⠈⠈⠉⠉⠉⠘⠀⠘⠷⠶⠶⠶⠶⠿⠃⠀⠈⠉⠉⠉⠉⠀⠀⠻⠿⠾⠿⠷⠿⠇⠀⠀⠀⠈
⢠⣶⣶⠖⠒⣶⣶⡀⠀⠀⢠⣤⡄⠀⠀⠀⡔⢲⣶⠒⣶⠖⠆⠀⣤⡄⢠⡄⠀⣤⠀⢰⣶⠶⠒⠶⢶⣦⠀⠀⢀⠤⠀⠄⡀
⢸⡟⠛⣤⣤⠛⢻⡇⠀⣤⣼⠛⢣⣤⡄⠀⡿⢏⣩⠿⣉⠿⠇⠀⣈⡽⢏⠸⠿⠀⠀⢸⠁⠀⠀⠀⠀⢿⠀⠐⢡⢊⣤⠑⠐
⢸⣧⣤⠛⠛⣤⣼⡇⠀⠛⢻⣤⡜⠛⠃⠀⣷⢎⣙⠶⣉⠶⡇⠀⢉⡱⢎⡱⢶⡉⠀⢸⡄⠀⠀⠀⢀⣾⠀⠐⠈⠈⠉⠔⡠
⠈⠛⠛⠒⠒⠛⠛⠁⠀⠀⠈⠉⠁⠀⠀⠀⠓⠘⠛⠐⠛⠂⠃⠀⠙⠁⠘⠂⠀⠋⠀⠘⠛⠓⠒⠒⠛⠛⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡎⠀⢱⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡁⠈⡄⠀⠇⠀⢸⠀⠀⡘⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⣴⡶⠒⠛⣿⣿⡗⠢⡀
⠀⠀⠀⠀⠀⠀⠠⠀⢱⠀⠀⠀⢸⠀⢀⠃⠀⠇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠈⣟⠁⠀⠀⠐⠭⠿⠃⠀⠘⠆
⠀⠀⡖⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⠀⣜⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠳⠦⣤⢤⣤⡤⠤⠴⠚⠊
⠀⠀⠘⡄⠀⢄⠀⠁⠀⢸⡸⠀⢸⡠⠁⢀⠇
⠀⠀⠀⠘⣆⠀⠐⠜⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⡐
⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠉⢢
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠓⠀⠉⠀⠀⠀⠀⠀⠀⣠⠔⠀⠉⠉⠉⠑⠀⠀⠀⢣⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠀⡐⠁⣠⣎⣉⣭⣁⣦⣄⠀⠀⠀⠑⢄⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⣀⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⢴⡞⠉⠉⠉⠋⠉⠉⠻⡷⠄⠀⠀⠀⠱⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠄⠠⠄⠀⢀⠄⠀⠀⠀⠀⠈⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⡀⠀⢀⣀⡤⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣟⣛⣛⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣤⠀⠀⠀⡄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠻⠿⠏⠈⠻⡿⠋⠘⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠁⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⣠⣦⡀⢀⣴⣦⡀⠀⢀⣴⣶⣦⡄⠀⠠⠀⠀⠀⢠⣶⣶⣤⠀⠀⠄⠠⠀⠀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⠈⠉⡏⠉⠈⢹⡏⠁⠀⠸⣿⣿⣿⡟⠀⠀⠀⠁⠀⢿⣿⣿⣿⠃⠀⠀⠈⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠇⠀⠀⠸⠇⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣷⡄⠀⡀⠀⡀⠀⣴⣾⣿⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣿⣿⠇⠀⠀⠈⠀⠀⢻⣿⣿⡿⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣉⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠉⠙⢷⣄⠀⠀⠀⣠⡴⠖⠻⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠹⣧⣠⡞⠁⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡼⢿⣦⠀⠀⠀⣿⢠⠛⣷⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡷⣾⣿⠀⠀⠀⣿⢺⣿⣿⡇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣼⣇⠀⠀⠀⠈⠻⣿⠟⠀⠀⢰⣟⠀⠛⠛⠀⠀⠀⠀⠀⢠⡿⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⣿⠇⢠⠞⠻⣧⡀⠀⠈⠳⢦⣄⣀⣠⣤⠾⠋⠀⠀⠈⠛⠶⠶⠶⠾⠋⠁⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡄⠉⠉⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠾⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⡠⠀⠉⠀⠈⠙⠀⢀⣠⣶⠞⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠶⠶⠶⠶⠶⡿
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⢸⡷⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⢀⣿⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⢸⡟⠀⠀⠉⠙⠻⠶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠉⠙⠛⠷⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⢄⠈⠙⢦⡀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⢰⠃⠀⠀⠀⠡⠀⠡⠡⠀⠀⢳⡀⠀⡜⠁⠀⣠⠤⠤⠤⠦⣄⠀⠈⣇⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢰⡇⠀⣈⠤⢔⣤⣤⡠⣄⡀⠀⢸⠀⢸⠁⠀⢰⠀⡀⠀⠀⡀⢰⠀⠀⢻
⠀⠀⠀⢻⡀⠀⠀⠤⠠⠐⠀⡜⠀⠀⢰⠃⠈⣇⠀⠀⠠⢀⣀⣀⡀⠀⠀⠀⡼⠀⠘⡆⠀⠈⠀⠁⠀⠀⠁⠈⠀⢀⡟
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠠⠊⠀⢀⡰⠋⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡔⡉⠀⠁⠀⠀⠄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣢⣼⣷⣶⠶⢶⣤⠀⠈⣇⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠉⠻⣏⠙⡅⠿⣨⠉⠢⠀⣽⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠉⠓⠚⠛⠁⠀⠀⢠⠏⠀⠀⢧⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠗⠠⢀⢀⡀⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢻⣷⠀⣸⠀⠁⡀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢶⡾⠟⠀⠈⠀⠀⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠈⠠⠠⠊⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⢄⠠⠀⣂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠔⢈⠇⠀⠀⠀⢘⡆⠀⢸⣄⠀⠀⠀⠀⡁⠢⡀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣉⠳⣌⡠⠔⠁⠀⠀⠀⡔⠁⠙⠋⠉⠀⢱⠀⠀⠀⠀⠤⡀⣊⠴⠓⣄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠂⠀⠀⠠⣄⣀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⢑⠎⠀⠀⠀⠀⠀⢘⠉⠉⠁⠉⠉⠉⠉⡂⠀⠀⠀⠀⠈⠡⢔⠩⠈⠀⠀⠠⠀⠀⠰⠂⠠⠀⠀⠂⠀⠀⠀⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢉⡉⠉⠉⠉⠉⡉⡅⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠤⠀⠂⠀⡆⠐⠀⠀⠀⠀⠀⠈⢠⡌⠀⠘⠁⠌⠃⠀⠀⠀⠀⠀⢀⠂⠀⠀⠖⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⢷⠀⠀⠀⠠⠤⠥⠤⠁⠤⠀⠼⠄⠂⠀⠀⠀⡇⠘⠀⢀⡇⠨⠅⢈
⠀⠀⠀⠀⠀⠀⠀⠤⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠣⠄⠄⠀⠒⠚⠒⠒⠒⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠶⠀⠀⠌⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⠔⠉⡄⠀⠀⠀⠀⠠⠈⠂⠈⠀⠀⢰⠅⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⢰⠠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠟⡠⠔⠊⣁⠴⠀⠀⠀⠐⠁⠈⠀⡤⣃⠀⡀⢀⡀⣀⢀⡀⢀⠀⡀⠀⡁⠀⠀⠀⠀⡀⠠⠄⠘⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠱⠖⠂⠉⠂⠀⠀⡐⠀⢀⠀⠠⠄⠁⠉⠀⠀⠀⠀⠁⠈⠈⠉⠈⠁⠈⠁⠉⠉⠁⠁⠀⠆⠰⡀⠀⠐⡀
⠀⠀⠀⠀⠀⢭⣩⣉⣉⡁⠀⠀⠁⠀⠀⠀⠀⡜⢠⠀⠞⠁⠀⢠⢄⡀⠀⠄⠤⠤⠤⠠⠄⠤⠠⠀⠤⠠⢄⠤⠠⢄⠀⠀⠀⠹⠄⠤⠐⡄
⠀⠀⠀⠀⠀⠲⠤⢤⠠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠄⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⠂⠀⠀⠤⠁⢀⡆⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠘⢤⠀⠡⠀⠀⠘⢂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠈⠀⠀⠀⢀⠔⠁⠂⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠈⠤⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⣀⠀⠀⠸⠁⠀⠀⠀⢰⠀⠀⠀⢼⠀⠀⠀⡀⠀⠀⠈⠣⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡔⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⠀⠀⠐⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⡜⠀⠀⠀⠀⠀⠰⠒⠀⠒⠐⠀⠂⠀⠂⠐⠄⠢⠤⠄⢄⠠⢄⢤⣠⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡀⢀⡜⠀⠀⠀⠀⠀⠀⢠⠤⠠⠤⠤⠠⠠⠤⠄⠀⠀⠄⠤⠠⠄⢤⢠⡤⡘⠀⠀⠀⠀⠀⠀⠀⠐⡤⢀⠉⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢁⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡎⠀⠅⡠⠂⠀⠐⠃⢀⠀⡆⠈⠠⠁⠘⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠂⠀⣠⠤⠴⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠽⢆⡀⣀⣀⡀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣀⣀⠀⠠⠤⠤⠄⠒⡶⠒⠒⠓⠒⠒⠒⠶⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠓⠛⠋⠉⠉⠁⠀⠀⡅⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⣀⡈⠀⠀⠠⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠉⠛⠲⢤⣀⡔⠋⠉⠉⠙⠲⣄⣀⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣼⠀⠀⠀⠀⢀⣀⠹⡄⠀⣠⡀⠀⠀⢸⡇⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢹⣀⣠⠤⠐⠒⠈⠉⠉⠀⠀⠀⠀⠐⠌⢧⠀⠀⠀⠈⠋⠀⢹⣄⠉⠁⠀⠀⣠⠇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠞⠀⠈⠉⢧⡀⡠⠀⣀⣠⠖⠢⣀⣀⣀⡀⠀⠀⢵⣦⣤⣀⣀⠴⠟⠉⠛⠒⠒⠛⠁⠀⠐⠒⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⢕⣿⠀⠷⠀⠤⢥⠀⠠⣝⠂⣒⢲⣀⡀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠠⠂⠀⢀⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐⠚⠲⠤⠄⠐⢤⣠⠤⢀⡨⠤⢵⣄⣀⠀⠀⠀⠀⣀⣀⡠⢤⢖⡎
⠀⠀⠀⠀⢠⠃⠀⢹⠀⢀⡇⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠉⠉⠙⣆⢀⣏⢸⠃
⠀⠀⠀⠀⢸⠀⠀⠎⠀⣎⠀⠀⠀⢘⡧⠴⠾⠏⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣆⣠⠽⣄⡈⢷⠀⠀⢸⡾⠉⠛
⠀⠀⠀⠀⢨⠄⠀⠀⠀⢸⡆⠀⠀⢻⡀⠀⠀⠀⠀⣿⣀⡤⢤⣀⣀⡀⢀⡀⠀⠀⡏⠀⠉⠀⠀⠠⢯⢸⡀⠀⠘⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢾⠀⠀⠀⠸⡀⠀⠀⠀⢀⣀⣨⠇⠀⠀⠀⠈⠉⢉⠛⠁⠓⡄⠀⠀⠀⠀⠸⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠙⢣⣀⣠⠇⠈⠛⠋⠉⠒⠋⠁⠈⠁⠀⢹⣀⡤⠤⠄⠠⡤⠴⠃⡆⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠱⠀⢀⣀⣀⣀⠀⢀⠀⣀⡀⠀⢠⠋⠑⠒⠁⠒⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⡤⠤⢤⣠⡤⢤⣀⣀⣀⣀⣠⣤⣀⡤⠤⢤⣤⣖⠦⠤⠒⠉⠓⠛⠳⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠢⡀⠀⠀⠈⠙⠦⢤⣄⠀⠀⠀⠈⠉⠑⠤⠤⡀⠀⠀⠈⠉⠙⠦⣤⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠉⠙⢦⡀⠀⠀⠀⠈⠓⠲⣄⠀⠀⠀⠀⠀⠘⠒⠲⣤⣄⡀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁⣤⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣾⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠜⠀⢀⣀⠤⠤⠐⠲⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⡴⠋⠀⠀⠀⢀⣀⠠⠤⠴⠛⢟⠓⠒⠒⠓⠈⠁⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠧⢤⣀⢀⠏⠁⠙⠉⠉⠁⠀⠀⠀⠀⠀⢀⡞⢀⣀⡀⠀⠀⣀⣀⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣥⣤⣀⣀⣀⣀⣤⠤⠤⢤⣤⣤⣤⣤⠤⠴⠖⠛⠉⠙⠛⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠤⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢰⠁⣘⠄⣀⠛⠄⡀⠂⠌⠀⢀⣀⣀⣀⠀⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠌⡘⢀⡏⠔⠂⠚⠂⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⡰⠤⠌⠴⠃⣘⠴⡄⠀⡤⣤⠒⣶⣒⢆⣜⣡⣌⡹⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⡊⠀⠀⠀⠀⠀⣯⣱⡇⠀⠁⢉⣉⣉⢉⡁⢤⢄⠀⢤⠐⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⣑⠦⠄⠀⠒⠚⢶⢯⡇⠀⣭⡿⣼⣛⠾⣽⢃⠎⣐⠢⠌⡁⠀⠉
⠀⠀⠀⠀⠀⠈⡐⠠⡎⠁⠀⣀⡤⣴⡖⣶⣸⢿⡇⠀⣿⣽⣳⣯⣿⢣⣏⡲⢠⠋⠴⠅⠀⡦⢀
⠀⠀⠀⠀⠀⠀⢀⠂⠴⠀⠀⠯⠓⣋⣩⣭⢯⣿⡇⠀⣿⣿⣟⡾⣭⣛⡾⣱⠃⡌⣑⠇⠀⡵
⠀⠀⠀⠀⠀⡀⠂⠌⡰⢇⠀⢰⢏⡟⣼⣟⡿⣽⡇⠀⣿⣿⣿⣽⢧⣟⡳⢧⣝⢰⣁⠄⠀⣇⠂
⠀⠀⠀⠀⠀⢀⠀⠠⢁⠺⡀⠀⢿⡽⠗⠛⠛⠋⠁⠀⣿⣟⡿⣯⣟⡶⠋⠁⠀⠀⠀⠀⠀⡏⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠂⡘⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡾⣹⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⢙⡀⠀⡆⠀⠀⠀⠀⠀⢀⡟⡳⢯⡟⠰⢦⡀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣱⠀⠸⡤⢤⣀⡠⢴⠛⠤⢁⠒⡬⢁⠆⡌⠃⠎⡐⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⢇⠂⠬⣑⠂⡌⠑⠎⡜⠠⢌⡐⠠⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢸⡌⠠⢀⡘⠠⢁⠒⠠⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣀⣤⣤⣔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠚⣿⣽⣿⣿⠙⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡈⠀⠀⠈⡀⢳⠀⠀⠀⠀⠀⠃⠀⠈⠛⠿⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠣⠐⠀⠀⠄⠀⠀⠀⠀⠀⡠⢊⠥⢭⡳⡄
⠀⠀⠀⠀⠀⠄⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠛⠾⠞⠻⢿⣷⡆⠀⠀⠀⠀⠀⢾⠄⠀⠔⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⠀⠀⠀⠀⠀⢹⡙⠀⠀⠀⠀⠀⢰⣀⡀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣦⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠄⠰⢂⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⣼⣿⣿⣧⣶⣀⣶⣢⣀⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣳⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣦⣤⣶⣾⣿⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣴⣾⣆⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠈⢀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣿⡖⠐⠀⠀⠀⠀⣀⣴⣦⠝⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⢻⡿⠿⠿⣿⣿⣟⠛⡿⠋⠉⠀⠈⠳⠶⠍⠉⠁⠀⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢸⣷⣀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣿⣿⡗⢶⣾⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢹⣿⣿⣯⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⣿⣧⠀⢈⠁⠀⠀⠀⠈⣾⣄⠰⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⠈⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣷⡿⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣧⣴⣶⣼⣿⣄⣶⣿⣿⣿⠸⣦⠀⠀⠀⠀⠀⣿⣿⠏⢀⡞⢠⢯⣿⣿⣿⣿⣿⣿⣿⢟⡡⠀⠀⠀⠠⠔⢤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢿⣷⣤⣀⣠⣶⣿⣿⣷⣞⣷⣏⡾⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣄⣲⣥⣙⣿⣿⣿⣿⣿⣿⣿⡾⡻⠂⠈⢻⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡋⠀⠀⠊⠁⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠚⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡇⠐⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠄⠀⠀⠊⠹⢿⣿⣿⣿⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣆⠉⠑⢀⡀⠀⠀⠀⠀⠀⠈⠛⢿⡃⠀⠀⣀⣀⣤⠾⢿⣶⣿⠿⠍⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⠉⠀⠰⠶⠆⠐⠒⠦⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠓⠀⠀⠀⠀⠈⠓⠛⣿⣌⢳⣄⠀⠀⠀⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⣠⣤⠴⠟⢻⣿⡃⢤⠀⠀⣀⣀⠀⢠⣨⣿⣿⠳⣝⠳⢦⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⢰⣾⢿⢻⢈⣁⡀⠉⠉⠁⡙⡇⠉⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡁⠀⠛⣿⣧⣽⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠛⢿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣶⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠉⠂⠠⠀⡋⠀⡀⠀⠀⠀⢻⣿⡆⠀⠀⠀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠈⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠶⠁⣀⠀⠀⠀⠡⠀⠀⢀⠀⠀⣼⢻⡤⠃⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢊⠁⠀⠀⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠶⢦⡀⠀⠀⠀⢀⡴⠂⠀⠸⠆⡈⠳⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠙⡇⠘⣿⠇⣀⠀⢾⠃⠀⠀⠀⠄⠀⠀⣄⠀⠀⠁⠠⠀⠀⠀⠀⢀⡄⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⣋⠀⢹⣈⣛⣀⡀⠀⠀⢠⠄⠂⠃⠠⣽⡄⠀⠀⠀⠀⢀⠀⢀⣼⡇⠀⠻⠶⠀⠀⠋⠀⠀⠀⠀⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠊⢿⣇⠈⠙⢻⠋⠁⠀⠀⣼⠄⠀⠀⠀⢈⣿⣦⠀⠀⣻⡇⠀⣾⡟⠀⢤⠀⣤⣧⠀⠈⠃⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣶⠚⠛⠓⢢⡀⢀⠁⠀⠀⢀⡀⠁⡘⣿⣧⡻⠊⢛⣼⣟⠑⣀⠀⢦⡙⢿⣧⠀⠀⠀⠀⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣤⡊⢀⣤⡇⠛⠀⠀⠀⡿⠁⠀⠀⢘⣿⣅⠀⣨⡿⠯⠈⣿⠀⠙⠵⠟⠁⠀⠀⢠⣿⣿⣿⡷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⢿⣿⠛⠁⠀⠀⠀⢸⡇⠀⠀⠠⡿⠏⠙⠛⠉⠀⠀⠀⢿⠀⠀⠀⠀⠀⠐⠠⠙⠻⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⠰⣿⡀⠀⠀⠀⠀⠀⠀⣴⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠉⠃⢰⠀⠀⠀⠀⠀⠻⠀⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠀⠀⠀⠀⠀⠀⡄⢠⣴⠋⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⠀⠀⠀⡆⠈⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⡴⠀⠀⢹⠄⠀⠀⠐⢢⣿⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁

⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣷⣶⣦⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣀⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠟⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣷⡄⠀⠀⠐⠶⠀⠈⢏⠛⢦⡀⢸⣧⣀⠀⠀⠀⢰⡀⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢹⣿⡆⢀⣀⠀⠤⠤⠼⠿⠶⣇⢸⣎⠻⣆⠀⠀⢸⢷⡸⡿⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⠀⡀⠤⠄⠒⠒⠘⡖⠚⠺⡿⢿⣿⠀⡀⢸⠀⢻⢃⠀⠃⢸⢿⣄⢀⡀⢀⡀
⠶⠦⣶⣶⠞⠛⠛⠓⠢⢀⡴⠊⢸⣿⣿⡉⢀⠤⠾⠂⠉⣁⣩⡅⠈⣇⠀⢻⢽⣿⣿⠀⠚⠈⣏⠉⢹⣀⢹⡆⠀⠛⢸⡆
⠀⠀⠀⠈⠉⠓⠢⠤⣠⣋⠀⡠⢺⣿⣿⠛⢧⣴⣶⣾⣥⣈⠿⠁⠒⡧⠠⠉⡐⠄⣺⡐⢨⣾⡷⠀⢺⠂⠘⣥⣐⠛⣻⠴⣆
⠀⠀⠀⠀⢶⠆⠀⢰⢣⣿⣿⣇⠸⣿⣇⢀⣿⡟⢧⡞⠙⢻⡆⠸⠿⠷⠀⠀⠀⢁⣼⠀⠉⡩⢐⣬⣽⢴⣶⣍⣿⠛⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⣤⣇⠀⢻⠿⠟⡼⠙⢿⣧⡙⡿⠥⠈⠲⣼⠃⠐⢾⡿⠀⠀⠘⠛⠋⠐⠍⡀⠿⠿⠋⠀⠉⠁⢠⡄⠠⣦⣤⠖⠁⢴
⠀⠀⠀⠀⠀⢸⣿⣿⠏⣻⠶⢤⣇⣀⢸⠉⠻⠷⠖⠒⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠜⢵⣶⠎⠈⢀
⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⠀⠀⡇⠉⢙⠓⠶⣤⣀⠀⣶⣷⣀⠟⠀⠀⠀⢀⡀⣀⢀⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⡞⢉⡟
⠀⠀⠀⠀⠀⣿⣿⡏⠀⢸⢀⡴⣷⣶⣼⡀⠰⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣧⠉⡄
⠀⠀⠀⠀⢠⣿⣿⣧⠀⠈⢿⣴⣶⡀⠙⣷⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⣿⢿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢀⡀
⠀⠀⠀⠀⢸⣿⣿⠘⡆⠀⠘⡄⠀⢱⡀⠘⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣶⣶⣞⢲⡖⠀⠀⠀⠀⠀⢠⡆⢀⣀⡿⣶⡆⣾⡏
⠀⠀⠀⠀⠈⢉⡉⠙⠛⣖⠒⠻⡦⠤⠷⣤⣌⣢⠀⠸⣤⡀⣀⠀⠀⠀⠀⢸⡇⢿⡿⣿⢩⠿⣧⠀⠀⠀⠀⠀⠈⠁⡘⠻⢿⣤⠁⠉
⠀⠀⠀⠀⠀⠛⠃⢀⣀⡈⢶⣦⠘⣆⠀⢿⣿⡷⠑⣔⠿⠳⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⠌⠲⡄⠠⡤⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠶⠈⠻⣃⡠⠷⡌⢉⡱⣶⣾⡧⡀⠈⣢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠖⠙⣦⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠢⣀⡼⠫⡀⢈⣻⣁⠈⣿⢋⣉⠫⡉⠀⠀⠀⣴⠇⠀⢰⠗⣼⠇⠜⣿⡠⠀⣨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⣧⠖⠉⢢⡠⠚⠻⣿⠿⠓⢷⠾⠛⢄⠀⠑⢄⠀⠀⡄⠀⠀⢢⠜⣄⡠⠊⢣⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣼⠟⢁⣤⡊⠁⠈⠢⡀⢈⡷⣶⣾⣷⢤⡀⠑⠢⣠⣷⢆⡘⡤⢊⠔⠱⡈⢣⣾⡗⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢺⣧⠹⣷⡄⣀⠴⠊⠓⢄⠈⠙⠯⡀⠈⠢⣀⠹⠫⡀⢉⢘⣅⣠⡾⠋⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⢻⣧⡜⣿⣄⠀⠀⠀⠀⣙⡦⡖⣿⣿⣷⡌⠑⢧⣈⢔⣴⡿⠋⠀⠀⠰⣤⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡘⢿⣦⣠⠔⠊⢁⣀⠈⠛⢍⣩⠗⢆⡘⣷⡿⠋⠀⠀⠘⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠚⠉⠀⠀⣤⣈⠙⠋⢠⣾⠛⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠺⠟⢋⣉⣉⠙⠛⢿⣷⣦⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⣿⣿⣦⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀
⠀⢸⣿⣿⣿⡿⠿⠛⠉⠉⠉⠙⠛⠉⠉⠉⠙⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢀⡉⠑⠢⢀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢐⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡷
⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇
⠀⠀⠻⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⠟⠀⠀⣠⣤⣶⣾⣿⣿⣷⣶⣦⣄⡀⠈⠳⣄⣄⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⠿⠟⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣙⣛⡻⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⡿⠉⠀⢀⠞⠛⣭⣿⣿⣧⠈⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣂⣠⠀⠀⢸⣦⣤⣿⣿⣿⣿⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣤⣀⠀⠀⠹⣿⣿⣿⡿⠋⣀⣠⡾
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣷⣶⣤⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⠟⠁⣀⣤⣶⣾⣿⣿⣿⣶⡶⢤⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠳⣄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢦
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⡀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣦⣬⣍⠂⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡆⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠒⢦⡀⠈⢁⡔⢁⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠶⠖⠚⣛⣛⣉⣉⣉⢭⡽⠃
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣇⠐⡩⢦⡘⠒⣡⡈⠢⡀⡰⣡⠔⣒⣒⢠⠃⠀⠀⠀⠀⠀⠰⠯⣥⠶⡖⠉⠙⢉⠁⠀⠈⢸⡄⠀⡇
⠀⠀⠀⠀⣶⡋⢻⡆⠙⠲⠤⢤⠔⡡⣊⡴⣛⣳⣄⠉⢶⠄⠈⠳⡇⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠠⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢎⡴⣫⣾⢟⣵⣿⣦⡀⠀⠀⡤⠜⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⠄⣿⠀⡇⠀⠀⢸⠃⠀⠀⠀⠃⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢔⣹⠿⣋⣴⡷⡄⠀⠘⢆⠀⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣀⣀⣰⣦⠦⠤⠚⠛⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠻⡥⣪⢟⣭⡾⢟⣦⠀⠀⢳⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣉⣉⣉⠥⠤⠤⠒⠒⠚⠋⠻⣧⣤⣤⣤
⠀⠀⠀⡠⠂⠀⠀⠐⠄⠄⠀⠾⠿⠀⠀⠀⠘⢷⠟⡫⣴⡿⡫⣣⡀⠀⠉⠀⢱⣀⡀⠀⠤⠠⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⢀⢀⠠⠇⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠈⠉⡂⠀⠻⣊⢕⡪⣞⣭⣷⣂⠀⠀⠀⢷⢓⠲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿
⡎⠀⠀⠀⠀⠀⠀⠀⠸⣴⡀⠀⠀⠀⠀⢠⠇⠤⠤⢽⣯⡪⢟⣫⣶⣷⠂⠠⣶⢸⠀⠀⠀⢔⣢⠒⣼⡂⣀⠀⠀⢀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠈⠀⠀⢒⢀⡀⠀⠀⠀⠈⠙⢶⣤⡀⠤⡼⢦⠤⣤⣃⠘⣾⣟⠟⣋⢼⡧⠤⠒⠉⠀⢰⠒⣤⠿⣋⡡⠄⠈⢷⣾⣷⣄⠈⠉⠏⠀⠙⢯⡁
⠀⠀⠀⠈⠁⠙⠦⣄⣀⠀⣰⡪⠛⢧⢠⠇⠈⠲⠃⠈⠉⢙⣶⣿⣿⡉⠀⠀⢀⡤⠒⠉⠱⡜⢧⠀⠐⠂⢀⡼⠈⢿⣿⣆⠎⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡹⠒⠒⠒⠢⣤⠀⣾⣿⣿⠋⣡⠴⠊⠁⠀⠀⠀⢀⣵⢈⡧⢴⠊⡿⣦⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠘⠂⠤⠼⠁⠀⠀⠀⠀⢹⠀⢻⡿⠗⠉⠀⠀⠀⣀⠴⠒⠉⠉⠈⢋⠀⠛⠃⠈⠁⠀⠀⡟⣠⡆⠀⠀⠀⠀⠀⡟⠫⠯⠃
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠉⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠰⣟⠄⠃⠀⠀⠀⠀⠀⠃⠀⢀⣤
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠴⠚⠉⠓⠦⡄⡀⠀⠀⢰⣿⢋⠽⢻⣾⣧⡀⠀⣀⠀⠄⠀⠈⠀⠀⠀⠀⢸⣶⣤⡀⠉
⠀⠀⠀⠀⡄⠀⣀⡀⠀⣬⠀⠀⠹⡆⠀⠀⠀⠈⠳⣍⣉⠀⠀⠀⠀⠀⢄⠈⠳⠀⠀⠸⣷⣝⠊⣠⣿⡴⠋⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⣷⠈⣿⣇⠀⠙⠧⣄⠼⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣬⣉⠀⠀⠃⠀⠀⠈⠓⠶⠿⠿⠅⠀⠰⠆⣉⠢⣄⠀⠀⠀⠀⢠⡿⠿⠟⠁
⠀⠀⠀⠀⢸⡆⢹⣜⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⢱⣶⣿⣿⣿⣿⣿⠟⠉⠛⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠳⢄⠀⠀⡜
⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⢀⣿⠿⠿⠻⣿⣿⣄⣀⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠒⠒⠾⢷⣈⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠉⣀⡤⠔⠊⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⡶⠖⠉⠉⠀⠁⠢⣀⠀⠀⠠⡖⠒⠢⣝⢆⠀⠀⢱⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠰⠋⠈⠀⠙⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣀⠤⠤⠀⢠⣤⣈⡑⢦⡔⠋⠀⠀⠀⠀⢣⠀⠀⠀⠠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢣⠀⠀⠀⠀⡀⡇⠀⠘⢿⣿⣿⠿⠋⠀⠀⠹⡄⠀⡰⠁⠀⢸⡿⠋⠁⠀⡍⠓⠢⠤⠤⠴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠑⠤⠤⠤⠜⠁⠀⠀⠈⢟⠁⣀⠤⠤⠤⣄⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠠⡄⠀⠀⠀⢀⠏⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⡠⠤⡄⠀⠀⠈⢎⠃⠒⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠓⠤⠂⠀⣀⣠⣼⠀⠾⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣴⣾⡿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣹⣛⣩⣙⣙⣟⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣟⣛⣿⣻⣿⣿⣶⣶⡆⢀⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⡯⠡⠏⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⢛⣵⣿⣿⣿⡟⠿⡧⠴⠶⠿⠛⠻⠟⠟⠋⠁⠀⠀⠘⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠿⠟⠿⣿⠀⢿⣾⣿⣿⠉⠀⠠⠤⠀⠠⠄⠀⠀⢀⣶⣿⣷⣤⣶⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⣟⣿⣶⡘⣦⣌⠛⢃⣀⢀⣠⣀⣄⣀⣀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢹⣿⡿⠟⠁⣨⣿⠁⠰⠿⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡏⠀⠀⠀⣠⣶⣿⢠⡇⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣰⣾⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣿⡗⣾⣿⢟⣀⣈⣠⣾⣿⣛⣟⡃⠿⣿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠆⠈⢻⣿⡟⠋⠉⠀⠍⠻⠉⠙⠻⢿⣷⣦⣄⡀⠀⠀⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡁⢰⢃⣼⡟⠀⠐⠲⠦⠄⠀⠀⠀⢊⠄⣩⡙⠟⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡷⢸⠿⡃⠀⠘⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⣡⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣷⡖⠛⢀⠀⠀⠀⠄⡀⠀⠀⠀⠄⠀⠀⠀⠒⠀⠀⠀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢛⣿⠀⢸⣿⡦⡀⠀⠀⡐⢦⣄⡀⠀⠀⢀⣠⣄⠀⠁⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠀⢸⣿⡅⣄⠀⠀⠀⠀⠉⠙⠛⠉⣈⣻⡏⠧⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⢸⣿⣗⡙⣺⣶⠶⠦⣀⡀⢀⣈⡿⢻⣿⣀⠀⠋⡄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢿⣿⢿⣿⣿⣿⣿⣿⢇⡠⠟⠋⣿⣿⣿⣾⣿⢏⠠⡈⡇⡔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡟⠾⢿⡧⠿⠟⢽⡿⣿⡿⠛⠙⠙⣷⣾⣿⣿⡷⣾⠀⠫⠐⣢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣦⠤⣤⠀⣌⠉⢮⡉⠁⠀⠈⠙⠛⠛⠻⠇⢿⡄⢆⣀⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⠛⠉⡰⠛⠂⠛⠀⠀⠘⠛⢶⠀⠀⠀⠀⠀⢺⡏⠉⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣇⣀⣄⣀⠀⠀⠀⠀⢠⡀⣷⣶⣤⣶⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠛⠻⠏⠙⡟⠣⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⡄⠀⢀⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⢠⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠾⠟⠉⠀⠀⠙⠿⠶⠾⠟⠉⠀⠈⠙⠿⠶⠾⠟⠁⠀⠈⠙⠿⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠆⠀⠀⠸⠿⠿⠆⠀⠀⠺⠿⠿⠆⠀⠀⠾⠿⠿⠂⠀⠀⠾⠿⠿⠀⠀⠀⠾⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠺⠿⠿⠃⠀⠀⠺⠿⠿⠂⠀⠀⠻⠿⠿⠂⠀⠀⠻⠿⠟



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠉⠁⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢵⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⢹⡇⡀⠄⠤⠐⠒⠒⠒⠲⠦⢄⣀⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠞⠁⠀⠀⣠⠔⠀⡀⠀⠀⠠⢤⡈⠙⢦⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡰⠋⠉⠳⡄⡀⠄⠄⠁⠀⠁⠂⠀⠀⠐⠁⣀⡼⡅
⣄⡀⠀⠀⠀⡠⠒⠁⠀⠀⠙⠢⡀⠀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠛⠀⠐⠸⡃⠠⠀⠉⠀⠘
⢳⠈⠉⠉⠁⠀⠀⠀⠀⣀⣀⡀⢙⠊⠑⠄⢠⡴⠅⠓⠤⠀⣀⣀⡀⠀⠀⢸⣄⠤⢦⡀⡰
⢸⠀⠀⠀⢀⣠⣶⠟⠛⠉⠉⠉⠻⣄⡠⠔⠚⡁⠀⠀⠀⠀⠀⢉⠛⠒⠒⠀⠀⠀⠀⢯
⠘⠦⠐⠂⢻⠋⢀⠔⠁⠀⠀⣀⠔⠉⠀⢀⡴⠣⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡎⠙⠲⣄
⠀⠀⠀⠀⠈⠷⠁⠀⠀⢀⠞⠁⠀⠀⢠⠎⠀⠀⠈⠑⠒⠤⠤⢤⠤⢤⠤⠤⠒⠚⠁⠹⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣀⠔⠈⠻⢿⣦⣤⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣴⠞⠉⠐⠢⡀
⠀⠀⠀⡼⠁⠀⠀⢀⣠⡌⠉⠉⣧⣟⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡴⢿⣹⠛⠋⢁⣀⠀⠀⠀⠈
⠀⠀⠀⠳⠤⠤⠖⠋⠀⢇⠀⢀⡞⠙⢦⣄⡀⠉⠉⠓⠒⠶⠤⠤⡥⠤⠤⠤⠶⠒⠋⠁⢀⡾⢿⠀⠀⢸⠈⠙⠢⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⠀⠉⠒⠦⢤⣀⣀⡀⠀⡀⠀⠀⣀⣀⠤⠤⠚⢹⠀⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠉⢩⠉⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡳⠄⠀⠀⠀⠀⠀⢀⢼⡄⠀⠀⣀⠀⠴⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠄⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⠀⠀⠀⠀⠀⡟⠀⠀⠀⠠⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⢇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠐⢺⠲⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠁⠀⠈⠑⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⢀⣠⣴⣶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡿⠋⠁⢸⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠛⠁⠀⠀⠀⠀⠈⠉⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡿⠋⢈⡇⣸⠇⣿⣿⣿⡿⠛⠿⠀⠀⠀⠀⠀⠠⠶⠛⠉⠉⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⣠⢎⡴⣡⠞⢁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⣶⠦⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠀⢀⠙⠫⣼⡵⣛⣙⠻⠿⠿⠦⠀⠀⠀⠀⠀⠀⠉⢿⣛⠟⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠈⢧⡀⣸⡄⡇⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠄⠀⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢻⢧⣷⡬⠶⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠀⢀⠤⢖⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡧⠀⠀⠀⠈⠁⣀⢾⡄⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣰⠃⠀⠀⠀⡰⠛⠻⣆⠻⣆⠀⠀⠀⠀⠈⢹⠖⠶⠤⣄⣀⣀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢠⠏⠀⠀⠀⠀⡇⠀⠀⠹⣦⠙⢷⣄⠀⠀⢠⣷⣄⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣼⣧⣤⡴⠃⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⠳⣦⣉⡛⠛⠛⣡⡿⠳⣄
⠀⠀⠀⠀⠀⢰⣏⣴⠶⡞⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠻⣏⠎⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠀⠀⠀⠀⠰⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⠀⣀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠑⣴⠁⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡄⡀⠀⠀⠀⣆⣀⣀⡀⠠⠤⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡄⠀⠀⠷⠀⠀⡄⠉⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠢⢄⣀⣀⠔⠉⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨
⠀⠀⢇⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣻⣿⣦⣤⠤⠀⢠⠤⠐⠒⠒⠂⠈⠉⠉
⡔⠀⠀⢵⡀⠐⠆⠀⡠⠂⠑⠀⠀⠀⠀⠀⠈⡇⢸⣿⣿⠁⠀⠀⠀⡄
⠑⠢⣠⡀⠀⠀⠀⠎⠀⢀⠊⠀⠀⠀⠀⠀⠀⢧⠸⡟⠳⣀⡀⠀⠀⢧⠠⠤⠤⠤⢤⣄⣀
⠰⣁⡀⠴⢤⡀⠄⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠙⠒⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⡀⠀⠀⠀⣀⣠⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠓⠋⠉⠉⠑⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠸⠖⠒⠒⠚⠋⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠓⠢⡤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠤⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢠⡊⠉⣀⠴⢄⠀⠶⢄⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠁⠀⢸⠀⢸⠀⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠈⠑⠒⠤⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⡀⣀⣠⠤⠴⢶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣷⣶⣶⣤⣤⡇⠀⣇⣤⣶⣶⣿⣿⣿⣷⣶⣦⣤⣀

⣷⣷⣿⡟⠁⣠⡶⣿⣿⣿⣿⣮⣷⡍⠻⣿⣶⠶⣖⣻⣻⣛⣰⡆⠦⣬⢈⡛⢲⢤⡙⠑⠆
⣿⣿⠏⢀⠞⡙⠂⠘⢹⣿⣿⣿⣿⣿⣿⣮⣻⣿⠁⣇⠉⠈⠹⠙⢋⠗⡶⣍⡑⠦⠛
⣿⡟⠀⡞⠰⠀⠀⠖⢙⠟⠛⢯⡿⣿⣿⣿⣿⣿⠿⠛⠿⠛⠙⠟⡶⢦⣤⣀⠛⠓⠲⢄⠀⠀⠀⠀⠀⢀⣤
⢭⠇⠀⣧⠁⠀⢻⠎⠁⠀⠀⣾⡿⢿⣾⣿⢿⣿⣯⡙⠿⢦⡘⠷⣷⣿⣭⣇⠀⠀⠀⠀⢣⠀⢀⣠⢾⣿⣽
⣧⣾⡀⢲⡀⡐⠀⠀⢰⣿⡆⢀⣴⣾⣿⣕⢌⣿⠟⣮⡢⡈⠟⢶⣽⠿⣿⣿⣿⣤⠀⠀⢠⠐⢿⣿⣿⣿⣿
⣷⣿⣇⠀⠛⠀⣠⡤⡀⠉⣰⠟⣵⣯⣙⣿⢿⣟⠻⣿⣺⡯⡢⣈⠻⣷⣜⠿⣿⣿⣷⣤⠜⠃⣀⢿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠻⠷⠇⣼⡇⣽⣿⣿⣻⣼⡻⣿⣷⣝⣻⡮⡪⡻⣅⣍⠙⢳⣞⠻⣿⣿⡄⠀⠉⣈⣿⣿⣿
⣿⣿⣿⡇⣠⡖⢦⠀⣸⠿⠗⣿⡿⠛⠛⣿⡿⠿⢿⣟⣛⣿⣛⠊⠪⡳⠕⢌⠹⣷⣮⡉⠙⢄⠀⢀⣸⣿⣿
⣿⣿⣿⠀⠻⣿⠿⠃⣿⠃⠘⣿⣁⣴⣾⡟⢂⣴⡌⣿⠿⠿⢿⣯⣦⡁⠀⠈⠳⣌⠿⡿⠀⠀⠀⠀⢼⣿⣿
⠿⣿⣿⠀⠀⡠⢄⠀⢿⡀⠀⢻⣿⣿⣿⣷⣿⣿⣷⢸⠀⠀⢀⣿⣿⣼⡢⡙⣆⠈⠀⠀⠀⠀⠀⠀⡀⣿⣿
⣷⡿⢿⡄⠸⣷⡾⠂⠸⣧⠀⠈⣿⣿⣦⢿⣿⣿⣿⣤⣷⣤⣾⣿⣿⣿⣿⣮⡺⣷⠄⠀⠀⠀⠀⠀⠑⢎⡛
⡟⣷⣾⣇⠀⠀⣠⠤⠀⢻⣇⠊⠈⢿⣿⣷⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢮⠀⠀⠀⠀⠀⠀⣇⣀⠑
⣟⡿⣿⣿⡄⠐⣿⣷⣶⠀⠁⠀⢀⡀⠀⠙⠿⣿⣾⣿⣿⣿⣄⣀⣽⣿⣿⣷⣤⣼⡇⠀⠀⠀⠀⣨⡃⡝⢴
⣿⣿⣿⣿⣿⣄⠈⠉⠁⠀⢀⣾⣯⣀⣡⠀⠀⠀⢀⣩⡿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡪⣻⣮⡓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⡀⠈⠀⠀⠓⢰⡶⠶⠄⢰⠀⠀⠀⠀⠠⡄⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⠀⠐⡛⣯⣙⣿⣿⢁⡆⢀⡀⠀⠈⢀⡈⠀⢀⣀⡀⠘⢀⡄⢰⡇⠠⡾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣴⣿⣶⣤⣿⣿⣿⣿⡿⠸⡗⢸⠘⢧⣿⣿⣾⣷⣿⡏⠀⠀⣾⠃⠀⢠⠲⠐⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⢻⣿⣿⣿⣿⣛⣿⠅⠀⢷⡜⠠⣼⣿⣿⣿⣿⣷⠁⠀⠀⢿⡆⢧⠀⢤⠀⣿⣷⠆⠄⠀⠐⢲⠄
⠀⠀⠀⠀⠀⠘⠓⠀⠻⠀⠀⠀⠀⠹⠿⢿⣿⣿⣳⣀⢀⣺⣧⣀⣿⡿⢿⣿⣿⣟⠀⠀⠀⠘⣿⣶⣷⣌⠷⣸⠿⠠⠉⠀⠀⠈⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠻⣿⣸⣿⣼⡟⠻⠶⣿⣧⣸⣻⣿⣿⠀⠀⠀⠀⠿⠻⠟⠷⣿⡟⠀⡀⢠⡞⠃⠈⠘⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣼⣿⡟⠀⢻⣷⣶⠾⣿⡿⢻⣿⠇⠀⠀⢿⣿⡅⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢩
⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈⣿⡆⠛⠀⠀⠀⠀⣿⡿⠀⣿⠀⠀⠀⡼⢿⣇⣾⣿⣯⠀⠈⠁⠀⠀⠀⠀⠠⠄⢈⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠉⠃⠀⠀⠂⠀⠈⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠈⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⡆
⠀⠀⠀⠀⢠⠒⢢⠀⠀⢸⠀⠀⢧⠀⠀⠀⡤⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠘⠀⠈⡆⠀⢸⠀⠀⢸⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣤⣶⠿⠛⠉⠈⣹⣿⣿⠈⠱⢆
⠀⠀⠀⠀⠀⢃⠀⠸⡀⢸⡀⠀⢸⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠘⠳⠿⠋⠀⠀⠈⠇
⡖⠐⡄⠀⠀⠘⠀⠀⢣⠀⡇⠀⢸⠀⣸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣤⣀⣀⣀⣠⡤⠴⠚⠒⠁
⠱⡀⠘⢄⠀⠀⠆⠀⠸⡀⡇⠀⢸⢠⠋⠀⣸⠁⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠹⣄⠀⠑⢄⠰⠀⠀⠉⠁⠀⠈⠉⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠠⡀
⠀⠀⠙⢦⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠾⠉⠉⢙⠶⣌
⠀⠀⠀⠈⢷⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠎⠀⠈⡆
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⡄⠀⠀⢀⡠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠄⠘⠀⠀⠀⠀⠀⠙⠓⠈⠉⠀⠀⠀⣀⠔⠀⠀⢀⡴⠊⠁⠈⠉⠁⠉⠒⢄⠀⠀⠱⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⢠⠊⢀⣴⡊⠉⣂⡉⢂⣄⠀⠁⠀⠀⠙⢦⡀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢁⣤⠾⠟⠛⠻⠿⠛⠛⠿⣷⣄⡀⠀⠀⠀⠙⢆⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⢹⠒⠀⠀⠀⠀⠀⠀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠁⠀⠀⠀⠀⠀⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠉⠉⠀⠃⠀⠉⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⣀⣠⠤⠚⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀
⠀⠀⠀⢤⣴⣧⠄⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⣾⡅
⠀⠀⠀⠘⠉⢃⣴⣿⣿⣿⣦⣤⡶⠞⠋⠁⠀⠀⢀⠀⠆⠀⠂⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣔⣾⡯⢤⡿⡝⠛
⠀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⣷⠿⣿⢾⡇⠙⠁⠀⢰
⠀⢿⣿⣿⣿⣿⡿⠛⠿⠿⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣀⣀⠔⠈⠈⠐⡌⠧⠉⠁⠀⠀⠀⡠⠃
⠀⠀⠛⠿⠿⠋⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⣣⣤⡀⠸⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠾⣶⡶⢶⠿⣶⢶⡶⠶⢾⡿⢼⡿⠛⢷⡄⠀⠀⣠⡎⠀⠀⠀⠀⠀⢈⠠⠠⠃
⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⣿⢀⡈⡀⣸⢠⣧⠈⠱⣯⣼⡇⢸⢁⡿⠛⠃⠀⠛⠋⠀⢀⡀⠀⠀⠀⠠⠒⠒⢢
⠀⣀⣠⡄⠀⠀⡀⠀⠀⠀⠐⠂⠀⣿⢸⣿⡇⢹⠈⠋⢸⣷⣄⠙⡇⢸⡀⢿⢶⡄⠀⠀⠀⠀⠈⠡⠀⠘⠁⠠⠟⠒⢲
⠀⠘⠿⢁⡄⢦⠙⠀⠀⠀⠀⠀⠀⢿⡼⠛⠷⠾⢷⣴⠟⢷⣤⠾⢷⡾⠷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⣅
⠀⢀⠔⢤⠙⠀⡀⠀⠀⠀⢰⡆⣾⠀⠀⢰⡆⣶⢠⣶⠀⠀⣶⠀⡷⠀⠀⣶⠀⣶⢰⡇⠀⠀⠀⠠⠀⠀⠀⠀⠔⠠⢬
⠀⠈⠙⠀⠀⠀⠀⡠⠀⠀⠸⠇⠻⠀⠀⠸⠇⠿⠘⠟⠀⠀⠻⠐⠇⠀⠀⠛⠀⠛⠘⠃⠀⠀⠀⠀⣠⣄⠀⠠⠖⠒⢦
⢰⠆⠀⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠠⠧⠤⠤
⢀⡚⠀⠖⠄⠀⡄⠀⠀⠀⢀⣠⣄⣠⣾⣶⠀⢠⠀⠀⢀⠀⠰⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣄⣈⣤
⠀⠉⠉⠲⠒⡑⣉⠤⠖⠦⢺⣿⣿⣿⣟⡟⢀⣾⣿⡖⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢡⣙⢻⢒⡀⠀⠀⠙⠿⣿⠏⠀⠀⠈⠹⠀⠀⠀⠀⠛⠀⡿⠀⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣤⣤⣤⣤⣤⣇⠀⠀⣿⣿⣿⠿⠿⠿⣿⣿⣿⢿⣿⠀⠀⢀⣾⣿⣿⣿⣦⠀⠀⠀⣿⠟⠁⠈⠙⠉⠛⠋⠀⠙
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠁⠀⠀⠀⠀⠘⣿⡿⣸⣿⠀⠀⣼⡏⠀⠁⠀⢻⡇⠀⠀⣿⠀⠀⢠⣆
⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⣇⠀⠀⠀⠀⠀⣄⣉⣰⣿⣿⠀⠀⣿⡄⠀⠀⠀⢠⣿⠀⠀⣿⣷⣶⡈⠁⠤⠀⠀⣴⣶
⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⣿⣷⣦⣤⣤⣴⣿⣿⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⠘⠁⠀⠀⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⣶⣤⣤⣤⣤⣦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣶⣦⡀
⣿⡿⠋⠀⠀⠀⠈⠹⠃⣠⣿⠀⠀⣰⣿⣿⣿⣿⣿⣇⠀⠀⣿⠋⠛⠛⠛⠛⠉⠉⠉⢹⣿⠀⠘⣿⣿⣿⣄⣀⣀⣴⢀
⣿⣅⠀⠀⠀⠀⠀⠁⣼⣿⣿⠀⢠⡟⠻⣿⣟⣿⡿⢿⡄⠀⣿⣦⠤⢤⠤⠤⠤⠀⠀⢸⣿⠀⠀⠹⣿⣿⣿⣿⣿⠟⠈
⣿⣿⣴⣦⣾⣿⣾⣿⣿⣿⣿⠀⠘⢷⣾⣿⣿⣿⣷⣾⠃⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠈⠙⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣶⣀⣠⣶⡄⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⣻⣷⣶⣶⣶⣶⡟⠀⠀⣿⣷⢳⣻⣿⣿⣿⠋⠈⢻
⠀⢠⣾⣿⣿⣿⣿⣿⣷⡄⠀⡿⠁⠀⠀⠈⢿⣿⣯⡉⢉⣿⠀⢾⣿⣿⣿⣿⣿⣿⣿⡦⠀⣿⣿⣷⡧⢿⡿⠗⣻⣿⣾
⠀⠙⢿⣿⣿⣿⣿⡛⣿⠟⠀⣧⠀⠀⠀⠀⠈⠛⢛⣵⣿⣿⠀⠀⠙⢿⣿⣿⣿⡽⠋⠁⠀⣿⣿⣿⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠃⣼⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛
⡿⠻⢿⡿⠿⠟⠿⢿⡿⠟⢿⠀⣠⣤⣄⠀⠀⠀⣠⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣀⣤⣀⡀
⣿⣿⣶⡄⣀⠀⠀⠀⣿⣿⣿⠀⠻⣥⣼⣷⣶⣶⣧⣠⡿⠀⣿⡛⢿⣿⠟⠉⠉⠉⠙⠻⣿⠀⢀⣼⣿⣿⣿⣿⣿⣇⡀
⣿⣿⣿⠀⠟⠂⠀⡀⢻⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⡿⠋⣤⡀⠘⠀⠀⠀⣠⣿⠀⢸⣭⣿⣿⣿⣿⣿⣮⠃
⣿⣿⣿⡀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠻⢿⣿⡿⠟⠀⠀⠀⣿⣷⣾⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠛⠻⢿⣿⣿⠟⠋
⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿
⠀⠀⠀⣶⣦⢰⣿⠀⠀⠀⠀⣿⣟⡉⠛⠋⠉⠉⠋⢠⣽⣿⠀⣤⢤⣤⣴⣶⣦⣤⡤⣤⠀⣿⠟⠛⠁⠀⠀⠀⠀⠙⣿
⠀⠀⠀⢈⣹⣿⣇⣤⡀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠈⣿⣿⠸⣇⣸⣿⣿⣿⣿⣿⣧⣸⠇⣿⠆
⠀⠀⠠⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡀⠓⢰⣿⡆⠚⢠⣿⣿⠀⠀⠹⣯⣿⣿⣿⣽⠃⠀⠀⣿⣤⠀⠐⠂⠀⠐⠂⠀⢾
⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⣿⣿⣿⣶⣤⣤⣤⣶⣿⣿⣿⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⣿⣿⣤⣄⣀⣠⣀⣠⣴⣾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡿⢿⠿⠿⠿⡿⠿⠿⠿⠿⠿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿⠿⠿⠿⠿⡿⣿⠿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣉⡉⣍⠙⠻⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⣿⣻⣼⣽⣿⣿⡿⣿⣽⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠃⡴⠛⠁⠀⠈⠹⢦⠀⣿⣿
⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣽⣷⣿⣿⣿⣯⣹⣿⣧⣿⡿⢿⡿⢿⣿⡾⢿⣹⣿⣿⣿⣧⣼⡇⠀⠀⠀⠀⠀⠈⣧⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣶⣿⢺⢿⣿⣷⣾⣿⣿⣶⣿⡟⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣛⣿⣿⣷⣿⣿⣿⣛⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠶⣦
⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⣀⠈⣹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁⣀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣴⠟⠙⠻⢋⣉⣽⡷⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠷⠶⣾⣯⣉⡙⠟⠛⠻⣦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣧⣀⣤⠾⠟⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⠻⠷⣤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣁⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢷⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠲⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⠳⠄⠀⠉⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠆
⠀⢀⣿⠀⢀⣿⠀⠀⠀⠀⡾⠁⠀⠷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠿⠀⠈⢧⠀⠀⠀⠀⣿⠀⠀⣿⡀
⢰⡟⠁⠀⠀⣟⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⠀⠀⠈⢻
⠈⠙⣧⠀⠀⣿⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⡇⠀⢸⠂⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠀⣼⠋
⣠⡾⠃⠀⢠⡾⠷⢶⣾⡇⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠀⢻⠀⠀⡇⠀⢼⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⢸⣧⡶⠾⢿⡄⠀⠘⢷
⠻⣦⠀⠀⣿⠁⠀⠈⣿⡁⠀⠀⠀⠀⠀⠀⠀⠶⣦⡀⠀⠈⠀⠀⠁⠀⠈⠀⢀⣤⠶⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠈⣿⠀⠀⣴
⠀⣿⠇⠀⢹⣆⠀⠀⠸⡇⢀⣴⠞⠛⣻⠆⠰⣦⣬⣿⠦⠀⠀⠀⠀⠀⠀⠰⣿⣧⣤⠦⠰⣿⠛⠳⣦⡀⣸⡇⠀⠀⣸⡏⠀⠘⣿
⠀⣿⣀⠀⠀⠹⣦⣀⠀⣿⠸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣸⠇⣿⠀⣀⣴⠏⠀⠀⣀⣿
⠀⠈⠙⢻⡆⠀⠈⠉⠛⣿⠀⠈⠛⢋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡙⠛⠉⠀⣿⠛⠋⠁⠀⢠⡟⠋⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄⠀⢰⡏⣻⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡟⢻⡆⠀⢠⡟⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⣌⡛⠋⠀⠐⠋⠈⠛⠚⠋⠻⠶⠶⠟⠙⠛⠞⠁⠙⠂⠀⠙⢛⣡⣴⣿⡁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⠁⠀⠉⠛⣷⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣾⠛⠉⠀⠈⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⢻⡆⠀⠀⣿⠀⠀⢀⣼⣿⡙⠛⠋⢙⣿⣛⠛⣛⣿⠿⠟⠛⠛⠉⣿⠉⠀⠈⣻⡷⠀⠀⠘⣿⠀⠀⣰⡟⠉
⠀⠀⠀⠀⠀⣸⡇⠀⠀⢷⡄⠀⢸⡿⠄⠀⣀⡴⠟⠋⠙⠛⠛⠷⠶⣤⣤⣀⣸⡏⠀⠀⢀⣹⡆⠀⠀⣠⡿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⣿⠀⢘⣷⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⣼⡿⠀⠀⠀⣿⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢻⣄⠀⠙⠛⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⣆⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⡄⠀⠉⠳⣦⣀⡀⠀⠀⠈⠛⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⢠⡴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠛⠛⠻⠛⠛⠉⠁⠀⠉⠛⠛⠛⠶⠶⣦⣤⡴⠟⠁⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⡀⠀⠀⠀⠀⢠⠏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠲⡄⠀⠀⠀⠀⠀⢸⡀⠀⣧⠀⠀⠀⠀⢸⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠹⡄⠀⠀⠀⠀⠈⣇⠀⡟⠀⠀⠀⠀⣿⢀⡞⠀⠀⠀⠀⠀⢀⡴⠚⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢤⡀⠀⠀⠀⠱⣄⠀⢳⡀⠀⠀⠀⠀⠸⣀⡇⠀⠀⠀⠀⢹⡞⠀⠀⠀⠀⢀⡴⠋⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢆⠀⠀⠀⠈⢦⡀⢇⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⣴⣫⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⢧⡀⠀⠀⠀⠙⠾⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣳⡀⠀⠀⠀⢀⡠⠖⠋⠉⠁⠀⠀⠀⠉⠉⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⣠⠖⠉⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠲⠤⢤⣀⡀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⣤⡖⠒⠒⠓⠒⠚⠓⠲⡄
⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠒⠒⠒⠋⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠠⣶⠶⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠀⣀⠴⠋⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣰⠃⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⡶⡀⠀⠀⠀⠀⠀⢧⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡴⠁⠀⠀⠀⢠⠏⢸⠀⠀⠀⠀⢸⠃⠙⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣠⠞⠀⠀⠀⠀⢀⡎⠀⢸⠀⠀⠀⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⠀⠙⠦⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⠀⠀⣼⠀⠀⡞⠀⠀⠀⠀⢸⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡼⠁⠀⠀⠀⠀⠀⢧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣠⠦⣤⣤⣦⠰⡆⡄⢠⢰⣦⢴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠙⠛⣈⠁⣙⠋⡘⠙⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⡟⡾⢧⢹⠹⣿⢸⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣚⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣋⠀⣀⠜⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡡⠚⠉⠲⡄⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡤⠴⠖⠻⡇⠀⠀⢀⡿⢁⣨⣿⡶⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⣘⡦⠤⣞⣰⠋⡰⢦⠈⡀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⢠⠀⠈⠁⠀⠀⠀⠙⢦⣙⣚⡰⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠄⣈⠁⠀⣤⣄⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢦⡼⠁⠳⡀⠀⠀⠀⠀⠀⡐⣒⠂⣠⠿⠼⣔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡰⢒⢢⡹⡀⠀⠀⠀⠀⠀⠀⡤⠟⢲⡖⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⣇⠹⠤⠞⢠⣇⣀⣀⡀⠀⠀⠀⡗⣒⣚⣓⣒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣈⡓⠒⢚⣉⡀⠀⢀⡀⠀⠀⡀⡇⡇⢴⡦⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣋⡀⠈⣓⡋⠀⣉⠓⠉⠀⣀⡼⠃⡇⣇⣒⣒⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠳⠶⠶⠶⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡉⠙⣦⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⠏⠐⠀⢀⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⢱⡄⠀⣴⠃⠀⠂⢀⠌⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣀⣴⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢀⡇⠠⠀⠀⡌⠀⢠⡴⠋⠩⠙⠛⢿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡇⡼⠐⠀⠀⠀⠀⠀⡞⠰⠀⠂⠘⢷⡀⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⡅⠀⠀⠈⣿⠁⠀⠀⠀⡘⠀⢀⡇⢶⠀⠀⠀⠀⠂⠀⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡆⠀⠀⡇⡀⠀⠀⡏⢀⠀⠀⢠⢁⠀⢸⡇⠀⣠⠴⡄⠀⣸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⢰⣧⠟⠀⢀⠛⡄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⣿⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⠀⠈⠀⣹⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⡀⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⢳⡈⠳⠦⣄⢰⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⢧⠀⡇⠀⠀⠀⠀⠀⢀⠀⠠⠴⠀⠀⠂⠂⢻⣄⠀⠀⠁⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠘⡄⡇⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⢹⠛⢷⣀⣀⣲⠡⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡀⠄⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠁⠀⠀⠈⠉⠉⡇⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡃⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡗⠀⠄⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠠⠀⠤⠄⠀⠀⠠⡄⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠉⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⣀⠀⡀⣀⡀⣀⣀⢀⠀⡀⣀⡀⢀⣀⠀⠀⣀⠀⡀⡀⢀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠫⣨⠇⡇⢝⡦⠈⠇⠸⣠⠇⠗⢯⢸⣒⠀⠨⣓⠆⠟⢧⢸⠸⠳⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⡴⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣱⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⢸⠁⠀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢹⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣐⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⠾⠃
⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠐⠦⡠⠔⠲⢤⠤⠒⠦⡤⠔⠲⢤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣛⣛⣭⣍⣛⣫⣭⣝⣛⣭⣭⣛⣋⡅
⠀⠄⠀⠀⠀⠀⠠⠄⠀⠤⠀⠠⠤⠀⠤⠄⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣛⣿⣟⣻⣿⣟⣿⣿⣛⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠀⠒⠀⠐⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣽⣿⣯⣽⣿⣭⣿⣯⣽⣿⣯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡉⣿⣿⣿⣿⡏⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣡⡶⠞⠳⠶⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⣿⣿⣿⣿⡇⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠶⢿⣿⣷⡌⣷⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⣸⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⣿⡆⣿⣿⢃⡿⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢋⣴⠋⣴⠞⣡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠸⣯⣉⠻⣿⣿⠟⢋⣽⡗⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣼⣿⣾⢃⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣴⣶⣬⣩⣴⣶⣴⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⣿⡟⣸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠟⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠻⢿⣶⣤⣤⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡿⠿⠿⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡏⣴⠛⡛⣶⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇⠻⢦⡴⠏⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣿⣿⡿⢛⣛⡛⠿⣿⣶⣶⣿⣿⣿⣿⠟⠙⣿⣇⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠘⣿⡏⣴⣿⣿⣿⣷⣦⣉⡛⠛⠛⣋⣡⣾⣿⡌⠿⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⣀⡛⢡⣿⣿⠿⠛⣛⠛⠿⣿⣛⣛⠛⠿⣿⣿⣿⣄⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣿⣿⣿⡟⣡⣾⣿⣿⣿⣷⣎⠻⣿⣿⣷⣌⢻⣿⣿⡏⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠈⠻⡿⢠⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡄⠿⠋⠀⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣂⡤⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣋⣙⠻⣿⣿⡟⢸⠟⣿⣉⡛⢿⡆⠀⠀⠀⠀⠀⠀⢤⣄⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠉⠀⠀⣀⣤⣶⣶⣿⢃⣿⡇⣾⠩⠌⣷⢸⣿⢁⡏⢸⡋⠌⣻⠈⣿⠸⣷⣶⣦⣄⡀⠀⠈⠳⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⠃⠀⠀⣠⣾⣿⣿⣿⣿⡟⣸⣿⣷⣌⠓⣾⣇⣾⡏⢸⣿⣌⡛⠞⣋⣼⣿⡇⢻⣿⣿⣿⣿⣦⡀⠀⠈⢷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⡿⠡⡿⠿⠿⠿⠿⠿⢿⠿⠡⠄⠿⡿⠿⠿⠿⠿⠿⡿⠘⠿⠿⠿⠿⠿⠿⠄⠀⠈⠁



⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⢀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣴⡟⠛⠛⠛⠛⠛⠛⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⢀⣾⢋⠀⠀⠀⠀⠀⠀⠀⠠⠛⡿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡯⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣿⠿⠿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⣿⠏⠉⠐⠃⠀⠀⠀⠀⠀⠀⠻⠇⢋⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣧⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡃⢿⣷⣤⣤⣤⣄⣀⣤⣀⠀⠀⠀⠈⠑⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⠈⠉⣉⡙⠋⠻⣿⣿⣿⡇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢿⣷⣌⡉⢠⡍⡻⢧⣿⣿⣿⡇⠀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠀⢻⣿⣿⣷⡔⢳⣼⣜⣿⣿⠃⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⢀⠼⠀⠀⡿⠛⠠⠹⣟⣳⣽⡿⠁⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡡⢂⡄⣠⠇⠀⢠⡾⠛⠛⠉⠀⠀⠀⢰⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠃⣴⠋⠀⢰⡏⠀⠀⠀⠀⣀⣤⣴⣿⠿⠃⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢀⡼⠃⠀⠀⠀⠀⠙⣩⣯⠏⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠻⠿⠷⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⣿⡿⠛⠉⠀⠀⠀⠈⠙⠻⣿⡻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠛⠛⠿⣦⣤⣶⠿⠛⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⢀⣴⠟⠋⠀⣼⠏⠀⠀⣠⡶⠶⢶⣦⡀⠀⠘⣷⡀⠉⠻⢶⣄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⢀⣴⠟⠁⠀⠀⢰⣿⠀⠀⣼⠏⠀⠀⠀⠈⣿⡀⠀⢸⡇⠀⠀⠀⠙⢷⣄⠀⢠⣤⣼⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣧⣤⡄
⠀⠀⠈⠻⣦⡀⠀⠀⠸⣿⠀⠀⢻⣇⠀⠀⠀⢠⣿⠁⠀⢸⡇⠀⠀⢀⣴⡿⠋⠀⠀⠉⣿⠉⠉⠉⠉⣿⡏⠉⠉⠉⠙⠻⣶⠿⠋⠉⠉⠉⢹⣿⠉⠉⠉⢉⣿⠉
⠀⠀⠀⠀⠈⠻⢷⣄⡀⠻⣦⠀⠀⠙⠻⠶⠾⠛⠁⠀⣠⡿⠁⣠⣴⠟⠉⠀⠀⠀⠀⠀⠹⣧⣀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣿⣷⣦⣀⣀⠀⢀⣀⣤⣾⣿⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣴⣶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⠉⠙⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣴⡾⠋⠉⠀⢀⣀⠀⠈⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣿⠀⠀⣿⠁⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⢠⡿⠋⠀⣠⡶⠟⠛⠛⠛⢷⣦⡀⠈⢻⣆⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⣿⡷⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⣿⠃⠀⣼⠏⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢿⡆⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠸⡿⠀⠰⣿⣤⣄⡀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢀⣤⣶⣾⡇⠀⢸⣿⠀⠀⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠀⠀⠀⣼⠇⠀⣸⡇⠀⢸⡇⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣼⠏⠀⣰⡟⠀⠀⢸⡇⠀⢸⡇⠀⠸⡿⠀⠀⠿⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣼⡏⠀⢰⡿⠁⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠀⣰⡟⠀⢠⡿⠁⠀⠀⠀⢸⡇⠀⠘⠿⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠐⠿⠛⠋⠁⠀⣰⡟⠀⢠⡿⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣸⡇⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄
⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣿⠃⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢀⣿⠁⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣠⡾⠃⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⣀⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⣀⠀⠀⠀⢀⣀⣤⡾⠟⠁⠀⠀⠀⠈⢿⣤⡀⠀⠀⢀⣀⣤⣴⣶⣾⣧⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠉⠀⠈⠉⠉⠉⠁

⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠⠀⠀⠀⠀⠀⢀⠀⢤⠄⠂⢀⢀⣄⡀⠀⠂⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⢀⠠⣔⣈⠰⠶⠆⡙⠙⠉⠐⠊⠁⠁⠀⠀⠀⠀⢀⠈⠠⠆⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠀⢉⠀⠐⠉⠀⠔⡫⠅⠈⢀⠠⠆⠀⠈⠉⠀⠀⠉⠀⠀⠓⠠⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⠀⠀⠉⠀⠀⠁⠐⠂⠈⠀⠄⡠⠔⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⢀⡐⠂⠁⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠
⠀⠀⠛⢀⠀⠈⠁⠀⠄⠀⡀⡠⠚⠁⠀⠀⠀⠀⠀⠐⣀⣠⣄⣠⠀⠀⠈⠑⢄⠂⠀⠀⠑⠀⠀⠀⠁⠃
⠀⠀⠀⠈⠀⠀⠀⠊⠀⡀⠊⠀⠀⠀⣠⠔⠀⠀⠐⠈⠀⣠⠉⠉⣛⡾⣄⠄⠀⠑⡀⠀⠀⠠⢀⠀⠀⠀⠀⠘
⠀⠀⠀⠰⡄⠀⡆⠀⡜⠁⠀⠀⠀⠊⠀⣀⠀⢚⠓⠁⠀⢃⢀⠀⣦⠑⢚⠃⡀⠀⠀⠀⠀⠘⠀⠀⠈⠈⠀⠀⠆
⠀⠀⠀⠀⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠠⡅⠀⠂⠀⡄⢀⢀⡀⠀⠈⢢⠀⠡⠀⠀⠀⢠⣳⠄⡃⠀⠀⠈⡀
⠀⠀⢱⠀⡁⠀⠱⢸⠀⠀⠀⠒⠀⠀⠀⠁⢸⠀⠠⠥⠀⣐⠘⠈⠀⠀⠞⠁⠁⠀⠀⠀⠙⠁⡇⠀⠀⠒⠀⠀⠜
⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⡀⠀⢀⠀⠐⠃⠜⢄⠀⠀⠁⠉⠀⠀⢀⠔⠤⠀⡤⠀⡀⠀⠀⠀⠩⢀⠂⠸
⠀⠀⠀⠀⠀⠀⠁⠀⠐⢄⠀⢀⢄⢀⠆⠀⠈⠓⠁⡂⡠⢀⠠⡈⠴⠀⠀⠠⠁⠀⠂⠬⠀⠀⠘
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠈⢧⡅⣄⠠⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⡐⠈⡳⣶⣁⠆⠠⠄⠂⠀⠄⠀⠀⠀⠀⠀⠀⡠⠁⠀⠁⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⣤⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠙⠗⠀⠌⠀⠀⠀⠀⠀⠀⠀⡄⠈⠑⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⢂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠃⠠⠀⠀⢀⡁⡄⠀⠀⠀⡀⠠⠆⢂⡀⠁⠀⠀⠀⠀⠀⠀⢀⠐⠁⠒⠀⠄
⠀⠀⠁⠀⠀⠀⢀⡤⠑⠄⠀⠀⠀⠰⠀⠀⢀⠀⠠⢀⡀⣀⣤⡜⠶⠒⠉⣠⠀⡐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠉⠁⠀⢂⠀⡡⠔⠋⠀⠀⠀⠀⠀⠀⠐⠀⠃⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠘⠁⠀⠻⠄⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠁

⡟⠉⣿⣛⣻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣟⣻⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣆⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣧⡀⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣾⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠿⠿⠀⢹⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⡙⠻⠿⢿⣿⣿⠟⢀⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠟⠛⠋⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢀⣾⡏⠙⣿⡆⠀⠀⠀⣰⡟⠀⠀⠀⢀⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢸⣿⠆⠀⣿⣷⠀⠀⣼⠏⠀⠀⠀⠀⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⣿⡆⢀⣿⠇⢀⣾⠋⣀⣤⣤⡀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠉⠛⠁⣠⡿⠁⣸⣿⠀⢸⣿⡄⠸⣿⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣴⠟⠀⠀⣿⣿⠀⢸⣿⡇⠀⠻⣿⣦⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⢀⣼⠋⠀⠀⠀⠘⢿⣄⣼⠟⠀⠀⠀⠙⢿⣷⣤⣀⠀⠉⠛⠛⠿⠿⢛⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠙⠻⢿⣿⣷⣶⣶⣶⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⠏⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢠⣶⡀
⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡷⠃⣿⣿⣿⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⢸⣼⡇⠀⠀⢰⣄
⣿⣿⣿⣿⣿⣦⣶⣶⣿⣿⠋⣀⣾⣿⣿⠟⢹⣿⣿⡟⢙⣿⡇⠀⠀⠀⠀⠀⠀⡟⠉⢹⡀⠀⣴⣿⡋⠀⠀⢠⣼⠈⠀⢰⣶
⣿⣿⣿⣿⠟⢻⣏⣁⣿⡇⢸⣥⣦⠘⣿⣷⣿⣿⣿⠗⣿⣿⡇⠀⠀⠀⠀⢀⣤⡇⠰⡿⠇⢸⣯⡞⡟⢻⡄⠈⠉⠀⠀⣾⡇
⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡮⢿⣹⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢙⡦⢷⠞⠁⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣘⣹⣿⣿⠉⣹⣏⣀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠻⣿⡾⠀⠀⣾⡷⢰⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⠿⠟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⢂⠄⡠⠊⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠛⢐⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⣴⡁⠈⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⣀⠀⡐⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⡇⠃⠀⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢙⠀⠀⠀⠀⠀⠘⠶⠫⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⡠⠛⢷⣦⡀⠀⠀⠀⡆⠀⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡐⠁⠀⠀⠙⢿⣄⠀⠀⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠹⣧⠀⣤⣄⡀⠀⠀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠤⠙⡘⠙⠻⢿⣷⣄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠠⡀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠐⠀⠐⠠⢀⡀⠀⠀⠀⢀⡠⠊⠀⠉⠉⠉⠁⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣷⣶⣾⣿⣿⡿⠗⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠀⠀⠀⠀⠰⠀⠁⣀⠀⠀⠀⠀⠀⠀⡔⡠⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣦⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢹⣇⠀⠛⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⢾⡇⠈⣷⠀⠀⠀⠀⠙⣷⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⣴⡏⠀⠀⠀⠀⢀⡿⢸⠇⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⠹⡄⠀⠀⠀⢸⡇⣿⠀⢀⣴⣾⣿⣏⣻⣝⣿⠿⣛⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⠀⣿⢱⡇⢰⣟⣿⣿⣿⢧⢻⣽⡾⣿⣚⣿⣯⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢻⡄⢠⣿⣼⢁⣿⣷⣿⣿⣿⡞⢯⢷⣛⢩⣭⣷⣾⡖⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣿⡾⠻⣿⢛⡿⣿⣮⣿⠛⣧⢘⣼⣿⣿⣿⠿⣿⣿⠿⣿⣄⡀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣞⣿⡿⢫⣿⣁⠢⠩⣟⢳⠞⣵⡿⣹⣿⣷⡿⢟⡿⣩⡞⣫⢶⡝⢯⠶⣩⠿⡛⣿⣭⣛⣵⣾⡍⢟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡳⣟⣭⣿⢁⣿⢣⣥⢉⣿⢂⠇⣼⠏⣽⣫⣿⢋⡜⣧⠞⣥⢚⡴⢫⠝⣥⠿⣡⡜⢯⣩⣿⣟⣽⡴⡾⡟⣮⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡼⣱⣟⣾⡧⣜⡏⣗⡮⡙⡙⠾⠽⡟⣼⣳⣿⣋⣏⡸⡏⡶⣹⢎⡸⢇⢿⠙⡤⠹⣧⠚⣇⠴⢻⣿⣷⣻⣿⣱⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣯⣽⣿⣗⣿⣗⣶⣽⡃⢯⣿⣳⣲⢦⣆⣿⣿⣸⠇⣸⣀⢇⡘⢢⡜⢆⣗⠈⡆⠹⡖⢫⣨⣼⡿⣿⣭⣿⣬⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡷⠟⠛⠛⠛⠛⠻⠷⣧⣽⡒⣬⡼⠟⠉⠉⠉⠻⢶⣦⡟⡴⠣⡜⠳⡜⢻⡆⢡⠜⣧⣾⣥⣾⡷⠿⠷⠷⣷⣯⣀
⠀⠀⠀⠀⢀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣤⣑⣮⣉⣮⣽⡾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⢠⡾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷⣦⡀⠀⠀⠀⠀⣠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⣆⣤⣤⣴⡏⠀⠀⢈⡷⠀⠀⠀⢀⣀⣤⣤⢶⡶⠶⠶⢶⣤⣤⣀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠙⣦⣀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀⠙⢷⣄⣀⣏⠀⢀⣿⣥⣤⡴⠞⢁⣤⡶⠟⠋⠉⠠⠀⠈⠀⠀⠀⠀⢀⠈⠙⣻⣶⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠉⠻⡟⠛⠛⠉⢉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣶⡾⠛⢹⣿⣿⡙⢷⣤⣶⠞⠋⡁⠀⠀⠃⠀⢀⠀⢠⠄⠀⠂⠀⠁⠀⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠂⠀⠀⠀⠀⢁⠀⠀⠈⠀⠀⣿⠋⠀⢀⣾⠋⠙⣷⠀⠻⣧⣀⠀⠄⠀⠐⠂⠀⠈⠀⠀⠀⢀⠄⠀⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠉⠈⣿⣤⣀⣼⠏⠁⠀⢻⣦⠶⠟⠃⠀⠀⠀⢀⡀⠀⠰⠀⠘⠁⠀⡀⢠⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠨⠙⠫⠀⠀⠀⠰⠀⠂⠂⠀⢈⠁⠀⠠⠤⠀⠄⠐⠐⠁⠀⠁⣀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢈⡀⠈⡀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠂⠘⠁⠘⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠄⠀⠒⠀⠘⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣡⣀⠀⡀⠀⠁⠀⠈⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠓⠲⠷⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⠛⠛⠉⠁


⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⡷⣶⣶⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⠄⠉⠈⠁⠈⠁⠋⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠄⠀⠀⠀⠀⠀⠀⢀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠸⣿
⠀⠀⠀⠀⢀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⡇⠀⠀⠀⢀⣠⣤⣴⣶⣶⡞⠛⢻⣿⠿⢿⣶⣶⣦⣄⠀⢸⠁⠀⢠⡿
⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⣇⠀⣠⣶⣿⣿⡟⠁⣴⣦⣿⠀⣿⣿⡿⠀⣿⣟⣿⠻⣷⡿⠀⠀⣼⠇⠀⠀⢠
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠹⠄⠀⠀⠀⠹⣿⣾⠋⢡⣿⠛⣷⣄⣨⣴⠏⠀⠈⠉⠓⣾⠟⢻⡜⣧⡹⡀⠀⠀⠋⠀⠀⣠⡿
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣿⡷⢀⣾⡟⠀⠈⠉⠀⠐⢶⣦⣤⣤⡾⠋⠀⠀⢻⣌⠳⣽⣶⣤⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⣦⣴⣾⣿⠟⣡⣟⣼⠃⠀⣀⣤⠶⣾⢋⣹⣿⣷⣶⣤⡀⠀⠈⣿⣿⣾⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠟⠻⣿⡿⠾⠿⢿⣿⣀⠰⣿⣧⣴⠟⠋⠁⠀⠀⠀⠈⢋⣠⠾⣋⣿⣟⣉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⣠⣶⠾⠛⠛⠻⠿⡟⠻⢿⣶⡶⠶⠶⠶⠶⠶⣟⣋⣵⡾⠛⠉⠉⠛⢷⡄
⢴⣤⣤⣀⣀⣀⣤⡾⠋⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠳⠶⣤⣴⡟⠉⠉⠛⠀⠀⠀⠀⠀⠀⢿⡄
⠈⣧⠀⠀⠀⠉⠀⠀⠀⠀⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣿⠀⢀⣀⡬⠛⠛⠓⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢹⣆⠀⠀⠀⠀⠀⣀⣼⠏⠀⢿⡄⠀⠀⠀⢀⣤⡾⠋⠀⠀⣼⠿⠛⠉⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠛⠿⠶⠶⠿⠟⠋⠁⠀⠀⠈⣿⡷⠶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣼⠿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡝⢲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣾⠏⢀⣾⠃
⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⢀⣷⣄⠙⠻⠷⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣴⡿⠛⠛⢋⣁⣤⣾⡏⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⡏⠉⠙⠛⠛⠛⠛⠛⠉⣸⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢠⡿
⠀⠀⣆⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣾⠇
⠀⠀⠸⣧⡀⠀⠀⢻⣧⠀⠀⠀⠀⢈⣷⣤⣤⣀⣀⣀⡀⣀⠀⢀⣀⣀⣸⡀⡀⣀⡀⠀⢀⣀⣴⣿⠀⠀⢀⣾⠃⠀⢀⣴
⠀⠀⠀⠘⢿⣦⡀⠀⠻⣧⡀⠀⠀⢸⡇⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⣿⠀⠀⠚⠁⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠆⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠰⠿⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⠟⠋⠀⠀⠀⠀⠀⠙⢿⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⢀⣴⠟⢿⣶⡶⣶⣶⣶⣶⣾⣿⠿⠷⠦⣄⡀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠛⠻⣍⢻⣿⣿⣿⠟⠛⠋⠁⣹⡆
⠀⠀⠀⠀⢀⣿⡃⠀⠀⠙⠛⠛⠋⠿⠛⠉⠀⠀⠀⠀⠈⢻⣆⠀⠀⢠⣟⠁⠀⠀⠀⠀⠀⠈⠙⠁⣀⣀⣤⣤⠶⢾⣿⠇
⠀⠀⠀⠀⠸⣿⣝⣓⠶⠦⢤⣤⣤⣄⣀⣀⣤⣤⣤⠴⠶⢛⡿⠀⠀⠸⣿⡿⠶⠶⠶⠶⠶⠛⢛⣛⣉⣩⣤⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⢶⣤⣤⣤⣤⣴⡶⠶⠟⠋⠀⠀⠀⠀⠈⠛⠛⠛⠚⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠤⠤⠀⠀⠀⡀⠄⠀⠀⠀⠀⠄⢀⠀⠄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠐⠂⠐⠒⠀⢀⠐⠀⠀⠈⠐⠠⡀⠀⢀⣀⣤⣤⠀⢈⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠂⠐⠠
⠀⠀⠀⠀⠀⠒⡄⠄⡰⡋⠄⠂⠀⠀⡀⠀⠀⢠⡾⣶⣄⡀⠀⠘⡈⠛⠻⣿⣿⣷⣶⡀⠀⠀⡀⠀⠀⠀⠀⢐⠁⠀⠈⠈⠀⠀⡀
⠀⠀⠠⠀⠀⡈⠀⠀⢁⠀⠄⠀⢡⠀⠡⠄⠠⣈⠀⠀⠀⡄⠀⢁⠑⠀⠀⠈⠉⠁⢻⣿⡀⠊⠀⠂⢀⠀⠀⡜⢂⠀⠀⢀⡂⠀⢄
⠀⠀⠀⠀⠈⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⠒⠀⢎⠙⣧⣀⠀⢀⠌⢣⠀⠀⠀⠀⢀⣸⣿⠇⠀⠀⠀⠂⠈⠞⠀⠄⠠⠀⠒⠉⠀⠄⠢
⠀⠀⠀⠀⠈⢃⠁⠀⠀⢡⠀⢠⠀⠐⠀⠈⡈⠀⢠⡈⠻⣿⠀⡀⢀⠋⠀⠡⡈⠛⠉⠉⠀⠈⠀⣘⣁⠜⢀⠀⠀⠀⡇⢄⠐⠀⡀
⠀⠀⠀⠀⠂⢈⢂⠱⣿⣿⣆⠀⠄⠀⠁⡀⠀⠠⠀⠕⣄⠙⡆⠁⡀⠇⢀⠀⠈⠀⠀⡠⠋⠙⣠⢠⡀⠀⠄⢀⠠⢀⣇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⠈⣷⡙⠿⠿⡦⠐⠀⠀⠀⠀⠀⢀⣼⠟⣠⣿⠂⠀⠀⡌⠐⠠⠀⢀⡂⠀⠀⢄⠶⢀⣀⡴⠁⠀⠀⠀⡀⠀⢀⠐⠰
⠀⠀⢀⠀⢀⠀⢤⡜⢹⣄⠲⣤⠠⣄⣀⣠⡥⠶⠟⣡⡔⠙⢿⣤⣤⡌⠀⠑⠠⠀⠐⡆⠀⠀⡘⠐⣼⠁⢺⡈⠀⠀⠴⠁
⠀⠀⠂⠀⣴⠰⣿⣇⠻⢿⣧⣘⢂⣿⣟⣿⠟⠒⠓⠈⡙⢢⡰⠟⢅⣠⢶⡷⢠⠀⣰⠁⢁⢀⠃⠼⠳⠞⠉⣿⡶⡃⠀⡀⢠⡠⠀⠀⡀
⠀⠀⠐⠀⠘⣧⡙⠿⣿⣿⠿⢫⡜⣿⠟⠁⠀⠀⠀⠀⠀⠂⠔⠁⣠⠁⡸⠃⡢⠔⠁⠀⠀⠒⠌⠀⠀⠀⣠⡿⠁⢊⠕⠄⠅⠀⠠⠁
⠀⡄⠀⠐⠄⠹⢿⣶⣶⡶⠾⢿⡷⠀⠀⠀⠀⠀⠠⠊⠄⠀⠀⠀⠌⠉⠀⠊⠐⠀⠀⢀⠀⠈⠀⠀⡠⣾⡟⠁⠀⠈⠐⢂⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣴⡆⣧⡈⠃⠀⠀⠀⠀⠀⠀⠈⠁⢂⠀⠱⠀⠀⠂⠀⠀⢀⠁⠀⠰⢰⣷⡌⡄⠀⡔⡀⠞⠁⠀⠘⣷⡀⡀
⠀⠀⠀⠀⠀⠀⣿⡿⠓⠄⠙⠛⠋⣡⣄⠀⠀⠀⠀⠨⠀⠀⠀⠐⠀⣰⡀⠀⠀⢄⠀⠀⠁⠀⠀⠈⣭⡇⢸⡰⢧⣴⣉⢂⠀⣰⡿⠃⠁
⠀⠀⠀⠀⢀⠀⡛⠁⠀⠖⢢⢻⡿⠛⠋⠁⢄⠈⠂⢀⠒⠄⠀⠀⠦⠐⠈⢤⡀⠀⠀⠀⠀⠁⠀⣸⡿⢁⠇⠘⡦⡙⣇⣠⢸⠉
⠀⠀⠀⠀⠀⡐⢤⣴⡊⠀⢀⡌⠀⠀⢀⡀⠀⠐⢢⡤⠓⠀⠁⡅⡈⠉⠂⢔⠙⠒⠦⡄⠀⠦⢮⠩⠔⢍⠃⠠⣌⣷⢨⠄⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠸⢋⡵⠋⡰⠀⠀⠄⡛⠋⠀⠀⢁⠀⠁⠀⠀⠛⠀⠀⡷⠄⠰⡀⢀⡁⠀⠀⢀⢀⡀⠐⢄⡔⠛⢶⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠆⠈⠣⣀⠀⠀⠀⢀⠁⣶⡞⠀⠀⡀⠀⠀⠀⠠⠄⠂⡱⠁⠀⣘⠼⠀⠙⣄⠔⠃⠂⠁⡜⡀⢀⠀⠂⡄
⠀⠀⠀⠆⠀⡉⢐⠂⡆⠀⠠⢁⣾⣿⣶⡾⣿⠸⣿⠁⠀⠀⠘⠿⢀⠀⠀⠀⢨⡀⠀⠠⢉⠇⠀⠙⠏⣤⣤⡤⢊⣼⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠂⡑⠘⡃⡇⠀⢀⠸⢻⡿⠛⠛⣿⣆⢻⣷⠄⠀⠈⠠⠀⠀⣤⣄⠀⣼⣦⣄⠁⠀⠠⠒⠦⢩⣭⣶⡿⠁⠠⠁⠀⠀⠈
⠀⠀⠈⠈⠀⢀⠜⠡⠈⡀⠀⠂⠹⠵⠾⠃⣾⣿⣦⡀⠨⣁⣶⣶⣶⡿⠟⢁⣼⣿⣿⣿⣷⣦⡈⠀⠀⠊⠈⢉⠀⢠
⠀⠀⠈⠂⠠⠄⠀⠁⠀⠐⠦⣄⢀⠀⠀⠄⠙⠋⣹⣿⡶⠤⠍⠩⠄⠀⠀⠀⠈⡣⠊⠉⠻⢿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠈⠀⠒⠀⠠⠀⠀⠀⠈⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿