Fan Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠩⠉⠉⠀⠰⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡡⠖⠉⠉⠉⠉⠓⢦⣀⣀⠁⠀⣀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠀⠀⠀⠀⠀⡠⠔⠂⠁⠙⠘⡆⡰⠋⠉⠐⠠⡢⡙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠑⠢⠀⠀⡠⠊⠀⠀⠀⠀⢀⣄⣁⣁⡀⠀⠀⠀⠀⠙⣦⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠔⠈⠁⢀⠀⠀⠀⠉⠢⡀⠀⠀⠈⢇⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣤⣿⣿⣧⣄⣀⡀⡐⠁⠀⠀⢀⣼⠂⠀⠀⠀⠀⠈⢄⠀⠀⢸⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⣿⣿⠿⠟⠛⠛⠻⣯⣷⡄⢀⡠⠈⠡⡄⠀⠀⠀⠀⠀⠈⡄⠀⡎⣡⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠁⠀⠀⠀⠇⢙⠁⡾⠀⡐⠋⠉⠲⡀⡀⠠⠄⠂⢓⡜⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⢀⡀⠃⠹⣤⢣⡀⠀⡠⠁⠀⠀⠀⠀⠏⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣃⠀⠠⠤⡖⠀⠀⠉⡅⠀⣼⡷⠠⠐⠺⠽⡤⠀⠀⢀⠀⠠⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⠀⠀⢃⠀⠀⠀⠠⠀⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⢨⠛⢆⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡄⠀⠀⠀⠐⣅⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢸⣆⢮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢦⡀⠀⠘⢄⠀⠀⠀⠀⠉⠒⠠⠤⠄⠖⠂⠁⠀⠀⠀⡸⠣⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠤⠤⣀⠀⠁⡀⠉⠢⠤⢀⣑⣤⠐⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⢎⠔⠁
⠀⠀⠀⠀⠀⠚⠋⠀⠀⠈⠉⠉⠓⢾⡇⠀⠈⠢⡀⠀⠀⣿⣧⣬⠓⠦⣀⣀⣀⣀⣨⡤⠖⣋⠔⠁
⠀⠀⠀⠀⠀⠳⣄⠀⠀⢀⣀⣤⡴⢞⡻⠀⠀⠀⠈⠢⠤⠻⠿⣿⡀⠀⠀⠀⠀⢀⡠⠖⠊
⠀⠀⠀⠀⠀⠀⠈⡙⠛⠛⠛⠿⠯⠭⠤⠤⣤⣠⣦⡀⠀⠀⠉⠀⠀⠒⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⠥⠤⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⣀⠠⠄⢀⡉⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠖⠚⠉⠉⠙⠋⠉⠉⠛⠲⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⡀⠀⠀⠀⠀⠀⠀⢀⣄⠠⠤⠤⠤⠄⢀⣀⠈⡙⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡰⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⠑⢦⣄⡀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠈⠙⠦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠊⠙⢦⡀
⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⣄⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⠄⠂⠙⣆⠀⠀⠻⣄
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⡆⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢸⠀⠀⠀⠙⣆
⠀⠀⣼⠃⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⢠⠀⠀⠘⣧⠀⠀⠀⠂⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⣧
⠀⣸⠇⠀⠀⠀⠀⠉⠐⠠⢄⡀⣀⣀⣀⠀⠀⠀⠑⢄⠀⠀⠀⠆⠀⠀⠋⠀⢠⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠄⠒⠀⠉⠸⣇
⢀⡟⠀⠀⠀⠀⣀⡴⠶⠚⠋⠉⠉⠩⢍⡉⠙⠓⠢⠤⣡⣄⠠⠜⠃⠈⠉⠓⠣⢄⣀⠄⠊⠀⠀⠀⠀⠀⣀⠠⠤⢰⠟⠁⠀⠀⠀⠀⠀⠀⢻
⢸⠇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠠⠀⠒⠈⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡟⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⢀⡴⠋
⣧⠀⣼⠇⠐⠒⠒⠂⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠜⠋⠉⠁⠁⠀⠐⠒⠒⠀⠀⠤
⣇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢇⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠉⠀⠀⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠬⠱⠦⢤⣄⠒⠒⠒⠒⠂⢤⡀
⢸⡄⠀⢁⢀⡀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠑⣦⣀⡀⠀⠀⢀⣀⠴⠊⠑⢄⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠆⠀⠀⠀⠀⢠
⠈⣷⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣼⠁⠀⠀⠐⢿⠀⠈⢙⠋⠁⠈⠄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢠⡀⠀⠀⠀⣾
⠀⠸⣆⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢰⠁⠀⠀⠰⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠐⠂⣴⠇
⠀⠀⠻⡄⠀⠙⢄⠀⠀⠀⢀⠔⠁⠀⠀⢠⠇⠀⠀⢠⠁⡀⠁⠆⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢠⠟
⠀⠀⠀⠹⣄⠀⠈⠣⣤⠀⠀⠀⠀⠀⣠⠋⠀⠀⢀⠃⠀⡇⠀⠸⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡠⠃⠀⠀⣠⠏
⠀⠀⠀⠀⠘⢦⡠⠊⠀⠉⠐⠒⠒⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠱⣤⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⣢⡖⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣼⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⡠⠞⠁⠈⢢⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠙⠶⣄⡀⠀⠀⠀⡀⣀⣤⠖⠋⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⡌⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡟⠛⠛⠉⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣀⣧⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⡷⠶⠤⠤⣤⣴⣤⣤⠤⠴⠶⢾⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠈⠛⠲⠤⠤⠤⠴⠖⠛⠁⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡄⣀
⠀⠀⠀⠀⡀⠤⠄⠐⠚⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⢀⢤⡀⠀⡠⣄⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠒⠀⠤⠄⣀
⠀⢠⠖⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⡇⠀⢳⢸⠀⢸⡆⡃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡄
⢠⡟⠀⠀⠀⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠄⠁⠡⠄⠀⠘⠤⠌⠀⠣⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⢻
⣸⡃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⠦⢤⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠘
⠘⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾
⠀⠀⠀⠈⠉⠛⠻⠶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⡶⠾⠟⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡞⠋⠉⠏⠉⠻⠟⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡻⠽⠃⠀⠀⠑⣶⠀⠀⠀⠈⢂⠙⢮⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡡⠊⠀⡄⠀⠀⠀⠠⠀⠐⢄⠀⠀⠀⢢⡄⠻⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡾⣁⡀⠀⣃⢀⣀⣀⣠⠀⠀⠀⢣⠀⡠⠛⡀⠀⠙⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠎⠁⢄⡠⠛⠙⠂⠤⠀⠀⠉⠢⡀⠀⠹⠀⠀⢁⠀⢀⣿⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡎⠀⢠⠞⠀⠀⠀⠀⠀⠈⠐⠠⡀⠱⣤⠀⠐⡀⠀⡷⠉⡏⠱⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢸⡆⠀⢰⡼⠧⢤⡁⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⠄⡸⢀⡟⠂⢺⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠉⠀⠀⠀⠙⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠰⠐⡇⢸⡏⠉⢹⠀⠀⠀⠀⠈⠉⠀⠒⠀⠀⠀⢀⣠⡼⠀⠀⠀⠀⠀⠀⠘⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡇⠸⡇⠀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⢖⣉⣀⠀⠙⢢⣤⣤⣤⣄⠐⣋⣹⡤⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢰⠀⣩⠁⠁⠀⠁⠨⣇⣭⠈⡅⠀⢫⠘⣇⠀⠀⣁⠀⠀⠀⠀⠀⠀⢰⠃⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠘⡄⠘⡀⠀⠀⢀⠔⣿⠈⠄⢱⠀⠀⠉⢈⡱⠞⠙⢆⠀⠀⠀⠀⣠⡯⣶⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢹⡀⢇⢀⠔⠀⠀⠘⢧⡐⢄⣠⠤⠚⠉⠀⠀⢀⠊⠙⠂⠶⣟⡃⢀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⣤⣤⣤⣷⠘⣶⣴⡖⣶⠴⠛⣿⡟⠁⠀⠀⠀⠀⡴⠂⢀⠃⠀⢀⠇⠙⡾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⡀⢣⡘⣏⠁⣩⠀⠀⠿⠀⠀⠀⠀⡀⠌⠀⠀⠂⠀⣠⠞⠀⡾⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⡘⢧⠀⠀⠆⠀⠣⣀⣀⠔⠁⠀⡠⠂⢀⡴⠛⣠⢾⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⣆⠳⣄⡁⠀⢀⠼⠛⠂⢦⠾⠓⠛⡩⠁⠔⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡇⠑⢬⣳⢎⡀⠸⠀⡀⠋⡀⢀⣬⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠉⠚⠱⠾⠷⠤⠴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠉⠈⠢⠄⣀⣀⣀⠀⣀⡠⠃⢀⡀⣤⡒⠰⢄⠀⠀⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣏⠙⠳⣤⠉⣶⠚⠃⠀⠀⠀⠀⢀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠠⠈⣿⣒⠟⠈⠁⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠐⠦⢤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⡤⠔⠊⠉⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⡀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠤⢀⣀⣀⡀⠤⠒⠀⠁⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⠤⠤⠤⠤⠤⠴⠚⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠻⠛⠿⡟⠛⡿⠻⣟⠻⢛⣻⠿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣶⣿⣾⣷⣶⣿⣶⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⠶⠞⠛⠟⠙⠻⣧⢴⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣻⡟⠋⠙⠢⣄⠸⠀⠀⠀⠀⢢⠙⢯⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡵⠊⡈⠁⠀⠀⠀⠈⠗⢄⠀⠀⠀⠀⢳⡀⠙⣦⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⠋⠀⠀⡛⠀⠀⠀⠀⢰⠀⠀⠳⣄⠀⠀⠀⣴⠊⠀⢳⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡞⠠⣤⠀⢀⣀⢀⣤⣤⣤⣼⡀⠀⠀⠈⢆⢀⠞⠁⠆⠀⠀⢻⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢠⠎⠑⠦⣀⡴⠋⠉⠃⠤⣀⠀⠀⠙⠲⡄⠀⢀⢯⠀⠀⠸⠀⠀⣘⣷⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡟⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠹⣆⠎⠀⢣⡀⠀⣷⠞⢹⠉⢣⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣆⠀⠀⢡⢀⣷⡀⢰⠀⢀⢿⣴
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢠⡇⢠⡯⠀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⣠⠞⠉⠉⠉⠻⢆⡘⠀⣿
⠀⠀⠀⠀⠀⠀⠻⡿⠟⠃⢸⠃⢸⣄⠀⢼⠁⠐⠒⠂⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⢧⡀⢼
⠀⠀⠀⠀⠀⣠⡔⠓⠊⠉⣻⠀⢸⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⢲⠤⢶⠁⠀⠀⠀⠀⠀⠀⠀⠘⡃⠸
⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⢸⠀⢸⠀⠀⢀⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⢸⠆⠉⠀⠀⠈⠳⣤⣤⣀⣀⣠⣄⢲⡋⣸⡄⠠⠤⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢨⡆⠘⠁⠀⠀⠀⠀⠀⢸⡏⣿⠈⡇⠀⠘⡇⠹⣆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣽
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢃⠀⣧⠀⠀⠀⠀⢀⣼⡏⢻⡀⢱⠀⠀⠙⠦⠙⠦⣤⠞⢧⡀⠀⠀⠀⠀⠀⢀⠿⣰⠇
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠘⡄⢸⡄⠀⢀⠴⠉⠀⢾⡈⠣⡀⢇⠀⢀⣠⠴⠚⠁⠀⢈⠷⣄⣀⣀⣠⠴⠛⠛⣿⢰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢱⡀⢷⣠⠊⠀⠀⢀⡈⣱⣄⣱⣤⠟⠉⠀⠀⠀⠀⢀⠆⠀⡌⠀⡍⢹⢤⡇⢰⢍⡿
⠀⠀⠀⠀⠀⠈⠳⠤⠶⣤⡶⠶⠶⢷⡈⢿⣶⣶⢦⣬⠉⠋⢹⡟⠁⠀⠀⠀⠀⠀⡰⠃⠀⡜⠀⢀⢃⡟⠀⣁⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣈⣷⡈⢇⣀⣀⡼⠀⠐⢯⡄⠀⠀⠀⠀⣠⠞⠀⠀⡰⠀⠀⣸⡞⠁⢰⠟⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠉⠉⠉⠉⠳⡈⢫⡁⠀⠀⠋⠀⠳⣄⠀⢀⠜⠁⠀⢀⡔⠁⢀⣴⠿⢀⣠⣿⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠙⣆⠹⣄⠀⠀⠀⠀⣈⡷⠣⣤⣤⣤⣮⠴⠒⡿⠃⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠳⣌⠻⣆⡠⠚⠁⡀⠀⢀⠼⠁⠀⢀⠜⣠⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⢿⠀⠈⠙⢮⡕⣦⣄⣠⠔⠁⠀⣆⣤⣷⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠋⠛⠛⠓⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⣀⠠⠖⠋⠁⠈⠳⢤⢄⣀⣀⠀⠀⢀⣀⡔⠁⢀⣀⣠⣖⠒⠢⣄⠀⠀⠀⠈⠑⠦⣄
⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣁⣴⠛⠛⢧⣈⠉⢳⡦⠼⠃⠀⠀⠀⠀⠀⠈⢛⡄
⡞⠡⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⣉⠓⣦⡤⣽⢯⡽⠋⠀⠀⠀⠀⢀⣠⠤⠒⠁⠰
⡇⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠬⠡⠟⠐⠋⠀⠀⠀⢀⣀⠴⠚⠉⠀⠀⠀⠀⡜
⢡⡀⠀⠀⠀⠀⠀⠈⠉⠓⠲⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠊⠉⠀⠀⠀⠀⢀⡠⠐⠉
⠀⠉⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⠤⠤⠤⠴⠚⠉⠁⠀⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠉⠑⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠦⢤⣀⣀⣀⣀⡠⠴⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠂⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠔⠊⠉⠉⠉⠁⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡰⠋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠈⠣⡀
⠀⠀⠀⢀⠞⠁⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠙⣄
⠀⠀⢠⠎⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈⢦
⠀⢀⠏⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⢇
⠀⡞⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈
⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⡜⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠖⠉⠀⠈⠉⠢⡀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⡇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⡠⠊
⢡⠀⠀⠀⠀⠀⠀⠉⠒⠂⠠⠤⠀⠒⠂⠉⠁⠀⠀⠀⠀⠈⠳⡤⢀⠠⠄⠊⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠉
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠖⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠑⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠉⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠉⠉⠁⠀⠒⠒⠒⠒⠀⠉⠁⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠠⢤⠤⠤⠤⠤⠤⠤⢤⠖⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⣘⣀⠀⠀⠀⠀⠀⣸⡀⡀⢀⡀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡋⠈⢡⠀⢠⠊⠐⠀⠀⡎⠉⢢⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⡠⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠒⠁⠀⠀⠁⠈⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣆⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⣉⡉⡉⣉⠉⡙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠁⡔⠊⠡⡅⢀⡈⠤⠤⣥⡉⠒⡄⠙⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢠⠊⠈⢘⣀⠀⡝⠀⢰⠀⠐⡁⢀⠻⠌⠈⢢⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡰⠁⠫⢢⣀⠑⡠⡹⡀⠈⠀⠜⠠⡣⢁⢸⡪⠊⠑⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢀⠷⣀⠶⣈⠑⠽⢫⡼⠛⠉⠙⠛⣬⠰⠑⣁⡧⢀⠤⢅⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠨⠠⠁⠀⠈⠈⠢⡏⠀⠀⠀⠀⠀⠈⡧⢠⠏⠀⠀⠀⢨⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢈⠋⠉⠉⣉⠍⡌⣧⠀⠀⠀⠀⠀⢠⣟⡈⢍⣉⠉⠉⢩⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡮⠔⠈⢁⢠⡢⢘⣳⢤⣄⣤⠴⢋⠢⢢⡀⠈⠳⠢⡎⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠐⡬⡲⢕⠁⣐⠕⠹⠄⢰⠀⡧⠐⢕⡀⠑⢕⡣⡔⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠢⣀⠩⠂⠀⠸⠣⡈⠀⠠⠁⠈⢊⣤⡢⠊⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠓⠄⢜⡁⠀⢰⠑⠒⣃⠦⠅⢃⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣤⣉⣁⣉⣡⣤⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠭⠭⠭⠭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠉⠉⠉⠉⠉⠙⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⢏⠁⠀⠀⠠⠂⡆⢰⠀⡎⠢⠀⠀⠀⣉⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⣀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠁⠉⠉⠀⠀⢀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠤⠤⠤⠤⠤⠴⠶⠒⠚⠋⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⣩⣛⣉⣝⣙⣯⣋⣿⣫⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⢶⡶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⢀⣀⣤⣀⣸⡇⠀⠀⠈⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡉⠀⣀⡴⠟⠋⠉⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠈⢻⣾⠋⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢠⡟⠙⢷⣄⠀⠀⠀⢰⠏⣿⠀⢀⣀⣤⡤⢶⣤⣾⡋⠁⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⡿⠀⠀⠀⠙⣷⡀⠀⢿⡀⣿⡶⠛⠉⢀⣴⠾⠋⠈⠛⣦⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⡇⠀⠀⠀⠀⠈⣻⣶⠾⠿⠿⣦⣠⣴⠟⠁⠀⠀⠀⠀⠈⢷⡀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣘⣧⣀⣀⣀⣀⣰⡟⠁⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠻⣏⡙⠉⢛⣿⡇⠀⠀⠀⠀⠀⢨⣟⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢀⣤⣭⣟⣛⣋⣹⣷⣄⠀⠀⠀⣠⣾⡛⠳⣦⠀⠀⠀⠀⠀⢠⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢰⡟⠁⠀⢈⣽⠟⠋⠀⠉⠛⢻⡟⠉⠈⢻⣆⠹⡇⠀⠀⠀⠀⣼⠃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠸⣇⣠⡶⠟⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⢿⣿⡀⠀⣠⡾⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡿⢿⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⡾⠃⠀⠻⣿⡛⠉⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣙⠳⢶⣤⣤⣤⣤⣤⣾⡷⠟⣿⠀⠀⠀⠀⠈⣻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⢸⡏⠀⣿⡇⠀⢸⡇⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣙⣿⠿⠶⠿⠷⠶⢿⣟⣋⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢀⣿⠉⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠖⠒⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣒⡉⠀⠀⠀⠀⠀⠀⣀⡀⠤⠬⡵⠮⣉⣁⡒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢦⠈⠁⠉⠒⢄⡠⠒⠉⠀⢠⣾⣿⣷⣦⣌⢢⡉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠳⡄⠀⡴⠚⡄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⡵⠂⠀⠑⣄
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣠⣴⣾⣎⠀⠀⡙⠀⢀⡔⢹⣿⣿⣿⣟⠹⠂⢀⣼⡆⠀⠈⢧
⠀⠀⠀⠀⠀⡜⠀⠀⣀⣴⣿⣿⣿⣿⣿⡗⣌⠀⢠⠎⠀⣿⣿⣿⣿⡟⣇⢰⣾⠟⢳⠀⠀⠈⣇
⠀⠀⠀⠀⠰⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣷⣷⣤⢤⣿⣿⣿⠏⢀⣸⡟⠁⠀⠘⡆⠀⠀⢸⡄
⠀⠀⠀⠀⢃⡾⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣟⡁⡋⣠⠝⡃⠁⠀⠀⠀⣇⠀⠀⠀⣇
⠀⠀⠀⠸⣼⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢠⣾⣱⠋⠁⠀⠀⠀⢀⣽⠲⡄⠀⢿
⠀⠀⠀⡇⣿⣼⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣋⣽⣩⡆⠀⣀⡤⠖⡎⠹⠉⠙⠥⣿⠶
⠀⠀⢀⡇⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⡦⣿⣯⣽⣿⣿⣛⣄⣀⡜⠀⡇⠀⠀⠀⡿
⠀⠀⠈⡏⠙⢿⡻⣿⣿⣿⣿⣿⢿⣯⣻⣧⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠃⠀⠀⠀⡇
⠀⠀⠀⣇⠀⠀⠙⠻⢿⣿⣿⣾⣯⣿⣿⣿⣟⣥⣿⣿⣃⠜⡷⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡆⢰⣿⣿⡿⢏⣿⣿⠇⠈⠀⠀⢰⠈⢻⣿⣿⣿⣿⣿⠁⠀⠀⢀⡏
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠐⢷⢪⣿⡟⣰⣿⣿⡟⠀⢧⠀⠀⠘⡆⠀⠙⣿⣿⣿⣏⠀⠀⢀⡞
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠈⣾⠏⣴⣿⣿⣿⠃⠀⠈⣇⠀⠀⢱⠀⠀⠈⣻⣿⠋⠀⢠⠎
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢠⠏⠀⢻⣿⣿⡟⠀⠀⠀⠘⠀⠀⠘⡆⢀⠞⠁⢸⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡼⢀⣲⣞⣿⣿⣷⠞⠛⠛⠒⠦⣄⣀⠵⢅⡀⢀⡤⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣇⣼⣿⣿⣿⣿⣻⣀⡀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣽⣿⣿⣷⣿⣇⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣶⣾⣾⡋⢕⣿⣿⣿⣅⡀
⠀⠀⠀⢠⣾⠟⣿⣿⣿⣿⣿⣿⡏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠟⣉⢿⣦⣿⣾⣿⣿⣿⡽⠇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣯⣥⣄⣀⣉⣿⣿⣿⣋⣹⣿⣿⣿⠟⠀⠱⣶⣿⣠⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠛⠛⠳⠶⠬⠭⠭⠭⠭⠭⠭⠾⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠓⠈⠉⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡴⠋⡄⠀⠀⠀⠀⠀⠈⠐⢦⡌⠳⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣴⠋⢀⡞⠀⠀⠀⢀⡴⠁⠀⠀⠀⠱⣄⠈⢷⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⣼⠁⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⢧⠀⢻⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠜⡏⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢻⣎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠀⡇⠀⣼⣀⠀⠀⠀⠀⣠⠔⠈⠁⠀⠀⠈⣧⠀⢿⡞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠇⠀⠁⠈⠙⢦⡤⠊⠁⠀⠀⠀⠀⠀⠀⢸⡆⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠆⠀⠀⠀⣀⣀⠀⠀⠈⢳⡄⠀⢀⠀⠀⠀⠀⠀⠀⣿⠶⢿⡷⠶⠖⠒⠀⠀⠐⠒⠒⢦⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡔⠉⠉⠀⠀⠀⠀⠀⢷⣀⡤⠶⠒⠂⠂⠀⠀⢸⡆⢸⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠘⡀⠀⠀⠀⠀⠀⡀⠐⢻⡇⠀⠀⠀⠀⠙⠉⠉⢹⡇⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡿⢠⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢸⡀⢣⠀⠀⠀⠀⠀⠀⡀⠀⢰⡇⣀⣀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢀⡇⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⣧⠈⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿⢸⡇⢸⣇⣀⣠⠴⠶⠚⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠘⣷⡀⠈⠉⠉⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⠃⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠉⠳⣤⣀⣆⠀⡀⠈⠳⣴⠦⢄⣤⣄⣀⠴⢀⠀⣸⢣⡟⢰⠏⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠉⣯⠁⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⢠⡏⣼⢡⡟⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠄⢽⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂⢀⡾⣰⣿⠏⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢷⡀⠀⠀⠀⠀⠀⠘⢦⠀⢀⠞⣰⠟⢩⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠙⠄⠀⠀⠀⠀⠀⠀⢐⡡⠞⠁⠀⢸⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠉⠀⠒⠀⠐⠒⠒⠈⠁⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠊⠉⠁⠀⢀⣌⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⢀⠒⠢⣄⠀⠀⠠⠹⣧⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣄⠀⠤⠊⠀⢀⠀⠑⠀⠀⠙⠈⠉⠛⠻⠷⠚⠛⠋⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⡀⠀⠀⠈⠓⢦⣀⠐⠁⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⡳⣄⠀⠀⠀⠉⠓⢦⡀⠀⢀⡠⠃⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣬⣙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣍⡙⠻⠶⣤⣀⡀⠀⠀⠀⠀⠐⣚⡭⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⢦⣄⣀⡈⣀⣀⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⢛⣛⣛⠓⠶⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢋⣡⡴⠶⠛⠛⠉⢉⣹⣍⣉⣙⣛⠲⢦⣍⡻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣡⣶⠋⠁⠀⢀⣠⡴⠞⠋⠉⡅⠀⠉⠉⠙⠛⢾⣽⠷⣬⡙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⠞⠁⠀⠳⣄⣴⠟⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠟⢷⡈⠙⢶⣌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣠⡾⠃⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠎⠀⢈⡷⠀⠀⠈⠻⣦⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣼⡋⠀⠀⠀⠀⠀⠀⠸⣇⠳⡄⠀⠀⠀⠀⡇⠀⠀⠀⣀⣞⣤⡴⠞⠁⠀⠀⠀⢀⡬⠻⣦⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣼⠋⠙⠲⣄⡀⠀⠀⠀⠀⠙⢦⣌⢦⣀⣠⣤⣧⣤⣴⣛⡟⠉⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠘⣧⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠏⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⢀⣽⠟⠋⠉⠀⠀⠀⠈⠉⠻⣦⡀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠸⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠟⠁⣠⣀⠀⠀⠀⠀⢀⣀⠀⠈⣿⠉⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢿⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠺⠉⠉⠀⠀⠀⠀⠿⠉⠃⠀⢸⡟⣋⣉⣉⣙⣛⣦⣀⣀⣀⣀⣸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠗⠒⢋⣴⡶⠾⣯⡉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⢻⡍⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⢰⡟⠁⠀⠀⠀⠙⠶⠶⠿⣇⠀⠀⠀⢤⠀⠀⠀⢀⣤⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⠽⢷⣄⠀⠈⠳⠶⠶⠛⠁⣀⡴⠟⣷⠙⠲⢄⡀⠀⠀⠀⠀⣿⠀⢀⡿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡇⢸⡇⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠈⡿⢶⡶⠴⢶⠖⠛⠙⣄⢀⡿⠀⠀⠀⠉⠲⢄⡀⢀⡿⠀⣸⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣄⢻⡄⣀⠴⠊⠁⠀⠀⠀⠀⠀⡰⠃⣼⠀⠀⢸⠀⠀⠀⠈⢾⡇⠀⠀⠀⠀⠀⠀⢉⡾⢇⣼⢋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡹⣶⠻⣇⠀⠀⠀⠀⠀⠀⠀⣰⠃⢰⡏⠀⠀⢸⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣀⡴⠟⣡⢞⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⢷⣌⠳⣦⣀⠀⠀⠀⣰⣃⡴⠟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⡶⠞⠛⢉⣠⠞⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡷⣤⣉⡛⠛⡿⠋⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣹⣤⠶⢋⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣭⣙⡻⠷⢤⣤⣤⣀⣀⣀⣀⣸⣄⣤⣤⣤⠴⠶⢛⣉⣥⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⣤⣤⣭⣭⣭⣭⣤⣤⣤⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⣿⡛⠒⠲⠒⢚⣿⠛⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⡰⢶⣛⣳⠶⡶⠶⡤⢤⠴⣤⣤⠶⣤⡀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠧⡼⢷⣏⣀⣻⣟⣃⣘⣿⣃⣸⣶⣟⡇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣈⣀⣒⣀⣒⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣒⣒⣒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠯⠭⠭⢽⡟
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡶⠛⠉⠉⠈⠉⡏⠉⠛⠷⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⢩⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⠌⢑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠌⠈⠢⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠴⠋⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠊⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡇⠀⠠⠃⢀⠴⠋⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠂⢄⠈⠢⡈⠀⢀⡇⠄⢀⠔⠃⠀⠀⠀⠀⠐⡀⢰
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠺⠂⠀⠀⠀⠐⠐⢓⡒⠒⠾⠀⠀⠀⠀⠀⠀⠓⠐⠒⠒⠂⠐⠛⠒⡅
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠆⠀⠀⠀⠀⠀⠁⠀⢀⡴⠂⠠⡤⠀⠠⣆⠀⠀⠀⠀⠀⠀⡆⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⢀⠴⠛⠄⠀⠠⠇⠀⠀⠈⠢⡀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠠⠀⠀⢀⡴⠁⠀⢰⢒⣒⠚⡇⠀⠀⠀⠀⠈⠢⡀⠰⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡷⣰⠉⠐⠀⠀⠈⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⡖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡉⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠓⢻⢓⣆⣰⡗⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢀⡀⡀⣀⣤⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠱⢠⠂⠀⡅⠀⠀⠡⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠚⠀⠂⠂⠐⠘⠒⠁⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⠀⡀⠀⢀⠀⡐⡂⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠎⠋⠉⠀⠀⠃⠀⠀⢀⠀⠃⠈⠐⢂⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣈⣤⣴⣶⣶⣶⣦⡄⠀⠀⠀⠈⠀⢠⠀⠂⠀⠐⠠⠃⠤⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⡄⠀⠄⠄⠀⠀⠀⠀⠁⢊⠔⡉⠂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠀⠀⡇⠠⠀⠠⠀⠀⠐⠐⣡⣀⣄⣀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣆⣸⣿⣄⢡⡓⣀⣠⣾⣿⣿⣿⣿⣿⣿⣶⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠂⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠛⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡊⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣤⣤⣤⣄⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠤⠌⠀⠀⢘⠻⠿⢿⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣷⡄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⣀⣐⣒⠚⢻⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⣿⢐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠄⠀⠀⠠⢉⠾⣧⠀⠀⠈⠿⣿⣿⣿⣿⠿⠁⠀⢠⣿⡤⠍⠻⣿⣿⣿⣿⡿⠃⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠐⠂⠈⠀⠀⠁⠢⢙⠉⣣⣄⠀⠀⠈⠉⠉⠀⠀⠀⣠⢟⠿⣶⣀⠀⠈⠿⠟⠛⠀⠉⣘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠁⠀⠀⣤⣶⣦⣶⣶⣿⣿⣿⣿⣦⣤⣄⣤⣤⣶⣿⣝⠋⠀⠀⠈⠑⠌⠀⠀⠀⠁⠐⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠟⠂⠀⠀⠀⠀⢄⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠠⡑⢄⡀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⠄⠢⠀⠈⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⣛⠿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣇⢀⠀⠀⠐⡈⢀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠉⠍⣉⢉⣙⠙⡇⣀⣙⣀⣄⠦⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⠍⠀⠁⠭⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⣤⣤⣤⠖⠲⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⢰⡦⢰⣆⢰⣦⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⡤⢤⣀⡀⣀⣀⡤⢤⡤⠴⠖⠛⠉



⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠂⠈⡍⠙⠒⠁⠁⠠⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢔⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠎⠀⠀⡈⠱⡀
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠁⡀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⡜⠀⠀⠰⠁⠀⡘⠂⠀⢀⡀
⠀⠀⠀⠀⠐⠫⠀⠑⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⠁⠀⡀⠀⠀⠀⠀⠀⡠⠀⢳⡀
⠀⠀⢀⣴⠀⠀⠀⠢⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⡀⢠⠃⠀⡀⠀⡠⠁⠀⡠⠊⠀⠀⠈⠉⠀⡄
⠀⠀⠿⣤⡁⠢⡀⠀⠀⠀⠀⠀⠀⠡⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⡌⠀⠌⠀⠔⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠙⠲⣤⡑⢄⡀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⡄⠀⠀⠐⢀⠄⠀⠊⠀⠐⠁⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢬⣢⡀⠀⢌⣦⡀⣀⣄⢴⠂⡶⢢⡤⣀⡊⡀⠁⠀⠀⠀⠀⠀⠀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣹⠟⢧⠘⣌⢇⢃⢸⢠⢁⠎⢱⢮⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢤⡑⢜⣾⣾⣾⡧⣣⢔⣡⡴⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⡀⠀⠀⠈⠻⢿⣿⣿⣿⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠌⣢⠀⠀⢀⣤⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢇⠒⠠⠤⢘⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⢁⣤⢭⣍⣙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠙⠲⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣀⣤⠤⠤⢤⣀⠀⠀⢹⡄⠀⠀⠀⠀⠈⣇⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡞⠉⠀⠀⠀⠀⠈⣻⣦⣴⣇⠀⠀⠀⠀⢀⡏⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⡼⢩⠶⠶⡍⢧⠀⠀⣠⠞⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡆⠀⠀⢀⣤⠶⢧⡸⢦⡤⢇⡼⠖⠛⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢷⡀⢀⡾⠁⠀⣸⠛⠲⠖⠛⠧⣄⣀⣀⣀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠉⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⠀⠀⠙⠲⠤⠴⣚⣩⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡟⠛⠛⢻⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠾⠒⠒⠒⠒⠷⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⡄⠀⡄⠀⡄⢠⡄⠀⣠⢤⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠁⠀⠁⠀⠁⠈⠁⠀⠙⠋⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⣠⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⣄
⠀⠀⣿⠀⢰⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡆⠀⣿
⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠁⠀⢀⡤⠤⠤⠤⠤⣄⣈⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣠⠴⠚⡆⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠊⢀⡴⠋⠀⠀⠀⡇⠀⡞⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⡤⠴⠒⢦⡀⠑⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⢀⡜⠁⢰⠋⠀⠀⠀⠀⠀⡇⢰⠁⠀⠀⠀⠀⣠⠊⡠⠖⠉⠀⠀⠀⠀⠀⠙⢦⠈⢳⡀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⢀⠎⠀⠀⠈⣇⠀⠀⠀⠀⠀⢧⢸⠀⠀⠀⠀⣰⣣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠱⡀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⡞⢀⡴⣄⠀⠘⡄⠀⠀⠀⠀⠘⡜⡆⠀⠀⢠⣷⠃⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⠤⠗⠀⢳⠀⠀⠀⣿
⠀⠀⣿⠀⠀⢸⠁⣸⠀⠈⠳⣄⠘⢆⠀⠀⠀⠀⠙⢿⣀⠤⣚⣓⡢⢄⣠⣾⠭⠖⠒⠒⠒⠢⠤⣄⡀⠀⠀⠈⡇⠀⠀⣿
⠀⠀⣿⠀⠀⡏⠀⡇⠀⠀⠀⠈⠑⢤⣑⠦⣀⠀⠀⡞⡵⠋⠀⠀⠉⠳⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⢻⠀⠀⣿
⠀⠀⣿⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠚⠿⢾⢸⠀⠀⠀⠀⠀⠀⢹⢹⣶⡤⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⣿
⠀⠀⣿⠀⠀⣧⠀⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⢧⡀⠀⠀⠀⣠⢏⠏⠀⠉⠳⢍⠳⢄⡀⠀⠀⠀⢸⠀⣸⠀⠀⣿
⠀⠀⣿⠀⠀⢸⡀⠀⠀⠈⠙⠒⠢⠤⠤⠤⠴⣒⡿⠋⠓⠭⣗⣒⡯⠕⣿⣄⠀⠀⠀⠀⠱⡄⠙⢦⡀⠀⡏⢀⡇⠀⠀⣿
⠀⠀⣿⠀⠀⠀⢧⠀⢴⠒⠒⠒⠒⠒⠒⠋⠉⠁⠀⠀⠀⢠⢿⠃⠀⠀⠸⡜⡆⠀⠀⠀⠀⠘⡄⠀⠙⠞⠀⡼⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠈⢆⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠏⠀⠀⠀⠀⡇⢳⠀⠀⠀⠀⠀⢹⡀⠀⠀⡰⠁⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠈⢧⡀⠳⣄⠀⠀⠀⠀⠀⣀⠴⠊⡰⠋⠀⠀⠀⠀⢀⠇⢸⠀⠀⠀⠀⠀⣠⠇⢀⡼⠁⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⢄⠈⠳⠤⠖⠚⠉⠀⣠⠞⠁⠀⠀⠀⠀⠀⡼⠀⢸⠀⠀⠀⣠⠞⠁⡠⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠠⣞⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠸⡤⠖⠋⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡉⠙⠒⠒⠒⠒⠚⠁⠀⢀⣀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿
⠀⠀⣿⠀⠸⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠇⠀⣿
⠀⠀⠙⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⣀⣠⣤⣤⣤⣀⡈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠙⢢⡀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣀⣤⣄⣀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣀⣠⠤⠤⠤⣄⡴⠋⢀⣀⣀⠈⠳⡞⠁⠀⠀⢀⣀⣀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠞⠁⠀⠀⠀⠀⢸⠁⠀⡏⠀⠈⡇⠀⢹⠤⠒⠋⠉⠁⠈⢳⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡏⠀⠀⠀⠀⠀⠀⠘⣇⠀⠑⠦⠖⠁⢀⡞⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⣳⠦⠤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠳⣄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠎⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠙⠲⠶⠶⠋⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⡤⣤⣤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⡶⣖⡷⣖⡶⣗⡶⡖⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⠲⠒⠒⠒⠛⠳⠛⠛⠛⠳⠛⠓⠲⠒⠒⠒⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣩⡭⠴⠶⠶⡶⠲⠯⢭⣝⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⣵⡞⠉⠀⣀⡤⠖⠛⡋⠉⠙⠛⠒⢯⣳⢮⣛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠙⣤⡞⠉⠀⠀⠀⢀⠀⠀⠀⠀⢠⠛⢦⠉⠳⣌⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣴⠋⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⢸⠀⠀⠀⢠⠃⢀⡼⠀⠀⠀⠙⣦⡹⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠃⡼⠓⢄⡀⠀⠀⠀⠘⢧⡑⢄⠀⢀⣸⣀⣠⢶⠟⠛⠉⠀⠀⢀⡠⠊⠁⠹⡌⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡏⣸⠃⠀⠀⠈⠓⠤⣀⠀⠀⣹⠿⠛⠉⠉⠉⠉⠛⢦⡀⣀⠤⠚⠁⠀⠀⠀⠀⢹⡄⢷
⠀⠀⠀⠀⠀⠀⢸⠃⡟⠀⠀⠀⠀⠀⠀⠀⢉⡾⢁⣤⣄⠀⠀⠀⣠⣄⠀⢻⣀⣠⣤⣄⡀⠀⠀⠀⠀⣧⠘⡇
⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⢀⣀⣀⠀⠤⠄⣼⠁⠈⠁⠀⠀⠀⠀⠁⠀⠀⢸⡧⠤⠤⠤⠽⢷⡤⠤⠤⡟⠀⡇
⠀⠀⠀⠀⠀⠀⣿⢸⠀⣠⠞⠉⠉⠓⢦⣤⣾⡀⠀⠀⣀⠀⠀⠀⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢻⠀⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⢹⣼⠀⣿⠀⠀⠀⠀⠀⠀⣈⠷⣄⡀⠙⠶⠴⠞⠁⣠⠞⢿⠓⠤⡀⠀⠀⠀⢸⠇⢰⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠸⡿⣇⢻⡀⠀⠀⣀⠔⠊⠀⠀⠈⡝⣳⠲⠲⠚⠋⠱⣀⡾⠀⠀⠈⠑⠦⣀⡾⢀⡾⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡙⣎⣷⡔⠋⠀⠀⠀⠀⠀⡰⢁⡟⠀⢸⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⡼⢳⢞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⢦⡙⢦⡀⠀⠀⠀⣰⣁⡼⠁⠀⠀⠀⠀⠀⠀⠙⢦⡤⠴⠞⣫⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣦⣍⣛⠒⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣢⡴⢚⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢯⣝⣓⠶⠤⠤⣤⣤⣴⠦⠤⠶⠖⣚⣩⡥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⠛⠓⠒⢲⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣀⠀⠀⢀⣧⣀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠻⢦⣭⣭⡤⠏⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⡤⢦⡤⡤⣄⢀⡠⣀⣀⢄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠥⠽⣟⣀⣷⣟⣘⣷⣇⣳⣾⡇⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣬⣤⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠶⠤⠬⠭⠭⠭⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠏
⠀⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣤⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⢀⣤⣄⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠻⠿⠃⠀⢀⣴⣾⣿⣿⣍⠀⠀⠀⠀⠀⠀⠉⣿⣿⣷⣦⡀⠀⠘⠿⠟⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣼⠟⠉⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠻⣷⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠈⢿⣧⠀⡀⠀⠀⢰⣿⠏⠀⠀⠀⠘⢿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠷⣿⡏⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣷⣶⣴⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⢀⣤⣾⣿⡀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⣾⡿⠟⠋⣿⡇⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡄⠀⠀⠀⣠⣴⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢠⣿⠁⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣷⡀⢀⣾⣿⠋⠀⠈⠻⢶⣤⣤⣤⣤⡶⠿⣿⣦⣀⠀⠀⠀⢀⣾⠏⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣷⣿⡿⠁⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⠈⠙⠿⣿⣷⣶⣾⠏⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢿⣧⣀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣶⡄⠀⠈⠻⢷⣦⣄⣀⠀⢸⣿⣧⠀⠀⣀⣠⣴⡾⠟⠁⠀⢠⣶⣦⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠙⠛⠁⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠈⠛⠋⠀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢐⡂⠀⠭⠭⠀⣐⡂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣊⠥⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠊⠀⠀⢀⡀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠎⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠉⠐⠤⡀⠱⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠂⢠⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠘⡀⠐⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠎⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠃⠤⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠱⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢀⡈⢂⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠣⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢅⠜⠸⠐⠒⠄⡀⠀⠀⠃⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠙⢄⠀⠀⠀⠀⡠⠣⡀⠀⠀⠈⠒⠚⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢡⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠁⠈⠁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡜⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢣⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡔⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⢀⢊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⡒⢄⠀⠀⠈⠒⠤⣀⠀⠀⠀⡠⠔⠁⠀⠀⡠⢒⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⣐⠠⠤⢀⣀⡀⣉⣁⡀⠠⠄⣂⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠲⠀⠀⠖⠒⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠦⠀⠀⠴⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⢠⠒⢫⠀⠀⡝⠒⡄⠀⠀⠈⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠈⠑⠐⠚⠓⠂⠊⠁⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠤⠤⠀⠀⣀⣀⠀⠀⠤⠄⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢾⡷⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⢾⡷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⢠⡿⠃⠀⠀⠀⠈⠉⠻⣷⣄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠⣾⣿⠀⠀⠀⢀⣿⠁⠀⠀⢀⣤⡶⠟⠛⠛⠛⢷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣰⡟⠁⣿⡀⠀⠀⢸⣿⠀⠀⣴⡟⠉⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⠁⠀⠘⣷⡄⠀⠀⣿⣤⣾⣿⣤⣤⣤⣶⣶⣦⣤⣀⠈⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠈⠻⢶⣤⣿⠋⠀⠀⠉⣿⣄⠀⠀⠀⠈⠙⠿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣦⣄⡀⠀⠀⠀⠙⣿⡄⠀⠀⣠⣿⠛⠿⣦⡀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⡀⠉⠛⠻⠿⠿⠛⠛⠛⣿⡿⠛⣿⠀⠀⠘⢿⡄⠀⢀⣿⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣀⣼⠟⠀⠀⣿⡇⠀⠀⠈⣿⣀⣼⠏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⢷⣤⣤⣤⣴⠾⠛⠁⠀⠀⢀⣿⠁⠀⠀⠀⣽⡽⠏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⢿⣦⣀⡀⠀⠀⠀⢀⣾⠃⢀⣠⣴⡾⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢾⡷⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⢾⡷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠚⠓⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⡤⠖⠊⠉⠉⠉⠓⢦⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⢳⡤⠴⠦⢤⡀⣠⠃⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠖⠒⠚⠓⢲⠋⢠⠴⠦⡄⠙⡅⣀⣠⠤⠖⢤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡜⠀⠀⠀⠀⠀⢸⡀⠸⣄⣠⠇⢀⡏⠁⠀⠀⠀⠀⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣇⠀⠀⠀⠀⠀⠀⠑⡤⣄⣠⠴⠊⠀⠀⠀⠀⠀⠀⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⣆⠀⠀⠀⠀⠀⢰⠃⠀⢱⡀⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠳⢄⡀⠀⠀⡼⠀⠀⠀⠙⢄⡀⠀⠀⣠⠔⠁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⣀⣀⠀⠀⣀⣀⣠⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⢹⣿⣿⣿⣿⣿⣿⠏⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣏⣙⣻⣿⡇⠀⠀⠰⠌⣅⠀⣶⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⡈⠀⠀⣿⣯⣿⣧⡤
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⠺⣿⡗⠖⠀⠀⠀⢩⠱⠤⠤⠄⠀⠀⠀⠀⠀⣿⣏⠀⠀⠄⠀⠐⠛⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠚⠄⠀⠀⠀⠐⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣦⡀⢄⣄⡜⠂⠀⢀⣶⣶⣶⡄⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⣔⣵⣦⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⣷⣯⣯⣿⣗⠀⠀⣼⡝⣿⡟⣿⡄⠀⠀⢨⣍⣫⡅⠀⠀⠀⠀⠙⠿⡏⠀⠀⠀⠶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣰⣿⣇⣿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠉⠀⠉
⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠠⢄⠀⢀⠤
⠀⠀⠀⠀⣿⡿⢿⣿⡇⠀⠀⠀⡥⢤⣬⡄⠀⠀⣴⠖⠿⠿⠷⢤⠀⢰⣿⢷⣿⣿⣽⠀⢠⡳⢈⣤⣌⢐⡃
⠀⠀⠀⠀⠻⢿⣟⣿⠏⠀⠀⠀⣿⣿⣏⠇⠀⠀⠦⣠⣀⡀⣀⠴⠀⠀⠙⠛⣹⠛⠉⠀⠀⠁⠿⠿⠿⠏⠁
⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣠⣤⣶⣤⡄⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⢻⡇⣿⢿⡇⠀⠀⠋⣿⡿⣿⡟⠃⠀⠀⠘⣿⣷⠆⠀⠀⢰⣟⠛⠙⠘⣳⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠸⡿⣿⢸⠁⠀⠀⠀⢙⣇⣾⠀⠀⠀⠀⠀⢘⡋⠀⠀⠀⠀⠘⠀⠀⠐⠃⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄⠀⠀⢀⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠲⣄⣤⠞⠂⣡⠈⠒⡶⠚⢀⡃⠺⣄⡀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠌⢔⠀⠐⡄⢄⠰⠀⠀⠃⠀⢠⠁⠀⣰⠉⠉⢁⡟⡹
⠀⠀⠀⠀⠀⠀⡶⡤⠖⠣⡀⠠⢈⣢⠀⠀⠢⡑⡇⡀⡀⠀⠷⠈⠔⠁⠀⢀⠊⣠⢚⢤⣀⢤
⠀⠀⠀⠀⠐⣏⠑⠤⡀⢠⠈⠀⠀⠀⢳⠀⠃⠌⢶⠀⠁⢰⡇⡀⠃⠀⡰⠃⠊⠴⠡⠂⣨⠜⢓⡄
⠀⠀⠀⠀⣠⡓⠀⠤⠌⠶⢄⡀⠉⠐⢄⠡⡀⠀⠈⠀⠀⠀⠈⠀⠀⡐⠃⠅⠂⠀⡤⠊⠀⢀⠸⡄
⠀⠀⣜⠚⠥⢬⡓⢄⠀⢀⠀⡙⠠⡑⢀⣡⠴⡖⠒⡃⠀⠛⠂⠲⢌⡀⠠⠂⠔⠃⠀⠀⠐⠐⢊⠝⢢⡄
⠀⠀⠈⠉⠓⠲⠬⣅⣓⠤⢀⠀⠀⠈⠚⣄⠀⣘⣠⢽⠠⠭⣀⠊⢄⡨⠢⡄⠀⠀⠐⣁⡤⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢭⣐⡪⢀⡚⠢⡈⠌⡀⠠⠉⠔⢄⣑⣤⠼⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠷⣤⠛⢉⢳⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢌⣊⠼⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⣉⢤⣤⡀⠀⠀⢠⠋⠈⠑⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣴⣿⣿⣯⢿⡻⠀⠀⣬⡀⠀⠀⢀⠜⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠒⣴⡿⣿⣿⢿⣵⣮⠁⠀⠀⠇⢈⠆⣰⣷⣶⣤⠀⣀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢰⣿⣿⡷⣡⣟⠿⣿⡷⠤⠼⣴⣫⡿⠞⣛⣯⣷⣷⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢀⣸⠿⢿⠇⣿⣿⣿⣿⠁⠀⠀⢸⡷⡿⠇⠿⠿⠿⣿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠙⠛⠯⠛⠛⣉⠽⣦⣤⣴⡿⡓⠤⣕⡻⣿⣿⡿⠉⠉⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢀⠵⡊⢁⣼⣏⣿⢸⣿⠝⣎⠀⠈⠘⠫⢄⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠤⠚⢱⣿⣿⣾⡍⣼⣿⣨⣷⣿⣽⠆⠀⠀⠀⠀⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠀⠀⠀⠻⢿⣿⣹⣿⣿⡟⣿⡿⠟⢂⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⡌⠉⢹⠋⠑⢩⠀⠀⠀⣣⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠤⣀⣸⣀⣠⣾⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣾⣷⣶⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⢿⣿⣿⣿⡇⠬⢍⡛⢷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⡆⡃⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠠⢄⣀⠀⠀⠀⠀⢀⣀⠠⠤⣊⢔⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣤⣬⣥⣤⣄⣐⣪⣥⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣼⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⣤⣤⣍⣙⠛⠿⣿⣿⣿⣿⣧⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣋⣠⣤⣀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⢙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣸⣿⣿⣿⣿⣿⣷⣤⡀⠀⣿⣿⣿⣿⣇⣴⠟⠁⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠁⠀⢀⣦⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⣀⣤⣼⡿⣯⣤⣴⣦⣤⣽⢿⣿⠇⣀⣴⣿⣿⣇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣀⣴⣾⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⠟⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⠟⠋⢠⣿⣷⣟⠛⠿⠿⠛⣻⣾⡿⠛⠉⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠿⠁⠀⢀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⢀⣴⠟⠹⣿⣿⣿⣿⠀⠉⠻⢿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣅⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠈⠉⠛⠛⣩⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⢻⣿⣿⣿⣿⣶⣬⣉⣙⠛⠛⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⡟⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠟⠁




⣿⣿⠿⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⣻⣿⡿⠿⠛⠉⠉⠉⠉⠉⡏⠉⠉⠉⠙⠛⠛⠻⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⣠⣾⣿⠟⠛⠉⠀⣀⣠⣦⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⢀⡞⠉⠛⢿⣿⣦⡀
⠀⠀⢀⡾⠋⠉⠀⠀⣠⡴⠟⠉⠀⠈⢷⡀⠀⠀⠰⡇⠀⠀⠀⢰⠇⠀⠠⠋⠀⣀⡀⠀⠙⢿⣿⣦⡀
⠀⢠⢫⠀⠀⠀⣠⡾⠫⣀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡇⠀⠀⢀⠟⠀⢀⠴⠛⠉⡩⠛⠻⣦⡀⢻⣿⣷⡀
⠀⢠⠏⠀⠀⣼⠏⠀⠀⠈⠳⢤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⡴⠊⠀⠀⠀⠈⢿⡄⠹⣿⣧
⠀⣿⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⣀⣤⣤⣤⣤⣤⡀⠀⠀⠊⠀⠀⠀⠀⠀⣀⡨⣿⡀⢹⣿⣇
⡆⣿⠀⠤⣿⣤⣤⣄⣀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⡤⠶⠚⠋⠁⠀⢹⡇⠀⣿⣿
⣇⣿⠀⠀⣿⡀⠀⠀⠈⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿
⣿⣿⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠒⠒⠛⠛⠋⠉⢹⡟⠀⢸⣿
⢻⣿⣦⠤⠤⠤⠿⠷⠖⠒⠒⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⡀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣿
⠸⣿⣿⡀⠀⠀⣀⣤⠴⠖⠊⠁⠀⠀⠈⠻⣿⣷⣦⣤⣄⣀⣠⣴⠟⠁⠀⠀⠈⠓⠦⣄⣀⣴⠏⠀⠀⣿⢿
⠀⢻⡽⣷⠚⠋⠉⠀⢀⣠⠴⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⢠⡀⠀⠠⢤⣤⠴⠛⠙⠂⢀⣼⣿⡿
⠀⠈⢿⣿⣧⡀⠴⠚⠉⠁⡎⠀⠀⠀⣠⠊⠀⠀⡀⠀⠀⠀⡄⠀⠀⢀⡘⣆⠀⠀⠀⢠⡀⠀⢀⣾⢫⡟⠁
⠀⠀⠀⠻⣿⣷⡀⠀⣀⠄⣷⠀⡠⠚⠁⠀⠀⢠⡇⠀⠀⢠⢳⡄⠀⣼⠁⠈⠣⠀⠀⠀⠙⣠⣾⣷⡟
⠀⠀⠀⠀⠙⢿⣿⣮⡁⠀⠘⣯⡀⠀⠀⠀⠀⡸⡇⠀⠀⢸⠈⣧⡼⠋⠀⠀⠀⠁⠀⣀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⡈⠻⢶⣤⣤⣤⣥⣧⣤⣴⣾⡿⠛⠀⠀⠀⢀⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣤⣤⣨⣇⣀⣀⣸⣤⣤⣶⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⡟⠛⠛⢻⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠒⠓⠒⠒⠚⠓⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢀⡤⠤⠤⠤⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠈⠳⠶⣤⡴⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣷⠀⢰⡶⠖⠶⠶⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣶⠀⠀⠛⠿⠿⠿⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⣶⠀⢴⣄⠁⠀⣡⡤⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠀⠀⢉⣛⣛⣉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠘⡄⢸⣏⣀⢀⣹⡇⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣥⣤⣍⣉⣉⣩⣤⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⠉⠉⠉⠉⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣀⣤⡾⠃⠀⠀⠀⠀⠀⢀⠘⢷⡀
⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢷⡀
⠀⠀⠀⢀⣿⠉⠳⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⠶⠾⠛⠛⠛⠁⠀⠘⣷
⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢀⣀⣤⡤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⢰⣀⣀⣴⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠒⠛⠛⠺⢿⣬⢿⢶⣧⣼⣿⣇⠀⠀⢠⣤⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢄⡀⠀⠀⠀⠀⠀⠀⠈⣦⠱⠹⣿⣼⣿⠀⠀⢸⣿⡇⠀⣾⣷⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣧⠀⠀⠀⠀⠀⠀⢰⣿⣾⣷⣿⣿⣿⠀⠀⢸⠉⡇⠀⢸⠛⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣀⡴⠖⠚⠛⢻⡀⠐⠃⠀⠀⠀⠀⠀⣠⣿⡿⠛⠋⠉⠉⠉⠉⠙⠛⠚⠧⣄⣸⢠⡇⠀⣰⡟
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢻⣷⣶⣶⣶⣶⣿⣿⣿⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣤⡾⠋
⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢻⣿⣿⡟⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⠁⠀⠀⢀⣤⡄
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⢠⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠓⠦⣾⡿⠁
⠀⠀⠀⠈⠳⢤⣤⣀⣤⣤⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⣀⣤⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠚⠛⠲⢤⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠙⠻⠿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⣷⣤⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⡾⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⣠⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣁⡀⠀⠐⢧⡀⠀⠀⢀⣀⠀⠀⢸⣿⠉⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⠒⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠠⣇⡀⢼⣦⠀⠀⠙⠲⠾⠿⠦⠷⠖⠚⠉⠀⠀⠀⢻⣿⣶⣿⣿⡶⠖⠛⠉⠉⠙⠲⡄⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠳⠤⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠶⠶⠾⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣄⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣰⣷⣄⡀⣀⣾⣧⡀⠀⢀⣼⣧⡀⠀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣰⠟⡙⠓⠚⠁⢐⠈⠉⠉⠀⠈⠛⠛⠋⠰⠀⠙⠛⠉⡌⢷⣄⣀⣠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⠉⠉⠉⠀⢣⠀⠀⠀⠘⠀⠀⣀⠀⡆⠀⡀⠀⡆⠀⠀⠀⢰⠁⠀⠈⠉⣰⠉⠷⣄⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⠛⠁⠀⢆⠀⠠⣦⢈⡆⠀⠀⠈⡗⠠⣁⠴⡏⠉⠉⠉⡗⢄⠻⠀⣆⠀⠀⠀⢠⠃⠀⠀⠀⢁⠏⠹⢦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡴⠃⠘⢆⠀⠀⠀⡨⢆⠐⠒⣮⢱⠀⠀⠀⢡⢠⢻⠀⡇⠀⠀⠀⠃⣠⠗⢹⠁⠀⠁⢢⡎⣠⠀⠀⢠⠃⠀⠀⠀⢈⠝⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡏⠐⢄⠀⠀⠀⡈⢢⡠⠊⠀⠈⣆⠀⡇⠹⡆⠀⠀⠘⣤⡸⠀⡇⠀⠀⢸⠀⠁⡆⡀⠀⠀⠀⡎⢱⡠⠤⣴⠃⠀⠀⠀⡠⠃⠀⠀⠀⢈⡷
⠀⠀⠀⢴⣤⠤⠞⠁⠀⠀⠱⡄⢈⣓⣠⠳⡀⠀⠀⠘⡄⠱⠆⠸⡀⢶⠀⡇⠁⠀⡇⡴⡆⠸⠘⠖⢡⠁⠀⠀⣜⠔⢩⠆⡴⠀⠑⡄⢀⠞⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⢀⠞⠣⡀⢸⠣⡙⣄⠰⠲⡘⡄⠀⠀⢃⢸⡆⢠⠀⠀⡇⢡⠁⡆⠀⠀⡬⠎⡙⠰⠁⢤⠜⡜⠀⠀⠀⡸⠁⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⡊⠀⠀⠙⢆⠳⠂⠈⢦⠁⢡⠘⡄⠀⠈⡌⠃⠘⠀⠀⡇⠛⠀⠇⠀⢠⢁⠐⢡⠃⠀⢀⠎⠔⠲⢠⠎⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⡀⠀⡴⢤⠱⣄⠀⠀⠳⡘⠃⠱⡀⠀⢱⣀⣀⡇⠀⠃⠀⢰⠀⠀⡆⠘⢠⠃⠀⢠⠃⣌⢔⡕⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢠⡑⡈⠢⡀⠀⠑⡄⠀⠱⣠⠊⡅⠈⢳⡀⢠⠔⢺⠉⢲⠀⢀⠎⠀⣠⠃⠈⣠⠊⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠙⠀⠘⣦⠄⠼⢆⡀⣷⡀⠰⠀⢠⠳⢣⠀⡆⠀⣾⡠⡞⠉⡲⡅⢀⠔⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠇⠑⢄⠈⢻⠃⢣⣀⣧⠼⡦⢼⠤⣇⣠⠉⡜⠀⡜⠁⡸⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠈⠳⣀⡱⡎⢣⠘⡄⠇⢸⢰⠀⡌⡹⢦⡜⢀⠞⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡘⢣⡘⢄⢃⢳⢸⢸⢸⢠⢡⢃⠎⡷⠃⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡑⢌⢪⡊⡌⡴⣄⣼⣃⢣⠎⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡑⣽⣷⣿⣽⣿⡷⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⠿⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡞⢳⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⣉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣾⣭⣖⣾⣛⣏⣉⡒⠒⠢⢍⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠵⣿⠷⠛⢻⡇⠀⡇⠀⠉⣱⢶⣍⠉⠷⣝⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡕⡡⠋⢇⣀⡤⢾⡗⠢⢧⣀⡼⠁⠀⠈⢳⡞⠉⢦⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⢎⠞⠀⡰⠻⡁⠀⢸⣇⠀⠀⡼⠳⣄⣄⡴⠋⠘⡄⠀⢣⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⢏⠎⠀⡞⠀⠀⢳⣤⠾⢿⠦⣼⠁⢀⣼⣏⠀⠀⠀⢹⡀⠈⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⣼⣼⣀⣼⡀⠀⣰⠋⠻⣀⣿⣶⢋⣷⣿⣅⣸⡀⠀⠀⠀⡇⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⡇⠀⡏⠉⠙⡿⠳⢶⠏⠍⣿⣿⢻⡟⠛⠻⡗⠒⠒⠲⣷⢤⣸⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠙⡟⠟⠿⡯⠭⠽⣧⠶⢺⢷⣴⣿⣿⣾⣏⣛⣻⡟⠛⠉⠉⡟⠉⢹⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⡇⡄⠀⢳⠀⠀⢘⣶⡋⢠⠟⣧⣽⣿⣿⣿⣿⣿⣶⠖⠲⡯⠤⣼⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⢣⠀⠈⢧⡰⠋⠈⢩⣿⠚⣿⠁⢻⣿⣿⣿⡿⠋⠀⣸⠁⢀⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⡜⡆⢠⠞⠳⣄⡰⠃⣿⠀⢸⠀⣠⢿⣿⡇⠀⠀⢠⠃⢀⡞⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣼⣧⡀⢀⡼⠉⠒⢿⠒⠚⡏⠀⠈⣿⣷⠀⡰⠃⢀⠞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡟⢿⡷⣏⠀⠀⠀⢸⠀⠀⣇⠀⠀⣹⣿⡞⢁⡴⠋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⠢⣕⡢⢤⣸⣄⣀⣿⡶⢿⣿⣿⡷⢋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣍⣑⠒⢷⢲⣿⣷⣿⣹⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⣿⡏⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣇⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡿⣟⠛⠛⠛⠛⠻⠎⠈⢻⣿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣙⡓⠦⠼⠂⣀⣀⣤⣤⣶⣿⣿⣿⡟⠋⠉⠉⣩⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣏⣿⣟⣿⣶⣶⣾⣿⣿⣾⣿⣿⣯⡿⠁⠀⠀⢸⣅⣀⣀⣠⡤⠴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⡀⡈⠂⢰⠃⢀⠐⠂⡟⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠈⠁⠀⠀⢂⡉⠠⠆⢀⠀⠀⠀⠃⠀⠐⠈⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠁⠖⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⠠⡆⠀⢠⠀⠸⠀⠰⠂⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⠃⠀⠀⠀⠠⢂⡀
⠀⠀⠀⠀⠀⠀⠄⠐⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠄⠎⠀⢠⠄⠀⠖⠀⡀
⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⠀⡖⠀⣠⠀⠀⢸
⠀⠀⠀⠀⢰⠋⡠⠌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠃⢀⠀⢠⠀⠁⡀⠀⠠⢸⡆
⠀⠀⠀⠈⠋⠠⠁⠀⠐⠁⠠⠀⠀⠀⠀⢀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⢀⢤⢏⡠⢀⠔⢁⠐⠈⠑⠚⢀⡠⠂⠃⢤⡀
⠀⠀⠚⡁⠀⠀⠤⠀⠂⠀⠀⠀⠀⡤⠀⡠⠶⠐⠒⠲⢦⡈⠳⡄⠀⠀⣀⠀⢩⠃⢾⡔⣫⠖⣀⠄⠁⡀⠄⠁⣤⠀⠀⠈⠀⠃
⠀⠀⠢⠁⢘⣀⠀⠂⠀⠀⠀⠀⠈⡠⠊⠀⠀⠀⠀⠀⠀⠙⢆⠘⣆⣶⣍⣁⠾⠇⠊⠐⠃⠶⠵⣏⡀⠠⡖⠀⠀⣀⠠⠀⠋⣤
⠀⠀⠂⠀⢀⣀⠁⠋⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⡀⠃⣀⠀⠀⠀⠛⠺
⠀⠘⠢⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⣀⢴⠃
⠀⠘⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⡀⠀⠀⢄⣸
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⢀⣠⠞⢁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠊⡙
⠀⠈⠀⠃⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⢉⡠⠔⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⠀⠀⢾⠇
⠀⠀⠀⠱⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⣦⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣠⠟
⠀⠀⠀⠠⠃⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⢸⠀⠃⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠿⠹⠶⠗⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠠⣁⣛⠂
⠀⠀⠀⠀⠠⠄⠀⠀⠔⠀⠀⠀⠀⠀⠀⣠⣌⠊⡠⠀⠰⢶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠂⢈⡱⠀⡀⠁⡘⠀⠀⠉⠉⣉⡀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⢀⠐⣁⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠐⠁⠄⡀⠂⠀⠃⠀⡔⠁⡀⠃⢸⡎⡈⠀⢾⠆⠀⢸⡇⢃⡀⠀⡠⠀⠀⢘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠈⠀⠰⠀⡐⢅⠈⠺⠃⠁⠀⠼⠃⢀⠈⢡⠀⠀⣠⡖⡾⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠁⠀⠐⠐⡰⠃⠀⡂⠐⣠⢀⡿⠶⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠀⠖⠒⠒⠚⠀⡿⠛⠿⢶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠤⠄⡀⢤⣄⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠐⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⡶⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⢠⠀⡄⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡆⡃⢸⢠⡀⡇⣸⢠⠇⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⡈⡀⠀⠁⢀⢸⠸⢀⢁⠇⠎⡴⢠⢋⡴⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡈⢸⠈⢰⠘⡜⡘⡰⡱⢃⠊⡱⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠲⢚⢷⣷⣵⡱⡱⡡⢊⠔⡁⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢡⣷⣿⣾⣽⣷⣽⠾⣕⠁⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣾⠿⢿⣿⣿⣿⠊⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠂⣄⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⢨⠀⢐⣬⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⡭⠄⢒⡀⠬⣷⣸⣿⣿⣿⣷⣄⡀⠀⠀⢀⣠⣴⡀⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⢒⡩⠄⣒⠥⡊⢷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡡⠖⣊⠔⡪⢖⡵⡹⡻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡵⣟⢅⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⢊⡵⢊⡴⣫⢊⠜⡜⡝⡝⡝⡗⢻⢛⢃⢯⣏⢎⢎⢢⠣⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⢉⠜⣱⢋⠜⡼⠱⢱⢹⢡⢸⠸⢘⡼⡘⡜⡎⢢⠱⡘⢂⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠜⢡⠏⡼⢠⠇⡏⡘⢸⢸⠸⡎⡇⣧⢱⠘⡄⢣⠘⡄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠡⠏⡸⢀⠇⡎⢸⠀⡇⢇⢸⡀⡧⠘⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢈⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡁⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⢠⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢸⣿⣿⣿⣿⠃⠀⠀⢀⣠⣴⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠟⣩⣤⣶⣶⣤⣍⠛⢀⣤⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣾⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣽⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⡿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠁⣤⣙⠛⠿⠿⠛⣋⣴⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⠟⠋⠁⠀⠀⢠⣿⣿⣿⣿⡧⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠃⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⢈⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡁⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⢦⠀⠀⠀⠀⠀⠀⡔⡶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠘⠲⡤⢄⡤⢴⠚⢀⡇⠛⣄⡀⠀⠀⠀⣀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⠙⡦⢄⡠⢤⠖⠛⠀⠀⡇⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠙⠛⠖⠓⢡⠎⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠳⡀⠈⠀⠠⡀⠀⠀⠀⢧⠀⠀⠑⠁⠀⠀⡎⠀⠀⣤⠀⠀⠀⣠⠃⠀⢼⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡴⠛⠀⠀⠱⡄⠀⠈⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠘⠶⣆⣤⣀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣟⠮⡛⠙⠃⠀⢀⡀⠀⠀⠘⢆⠀⠀⠀⠠⣤⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡴⠁⠀⠀⣀⠀⠀⠀⠉⠀⣁⠔⢫⠇
⠀⠀⠀⠀⠀⠀⠀⠨⡇⠈⠳⢄⠀⠘⠁⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⢳⠀⢠⡀⠀⡇⠀⠀⠀⢐⡞⠀⠀⠀⠀⠉⠀⠀⢀⡤⠚⠁⠀⢼⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⣴⠧⡀⠀⠀⠀⠸⠀⠀⠀⢰⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣀⠀⠀⠘⢳⣀⠀⣀
⠀⠀⠀⡠⢤⡤⠾⠁⡀⠀⠀⠀⠀⠶⠈⠳⢄⠀⠀⠀⠀⠱⡄⠀⠀⠀⡇⠀⠀⡜⠀⠀⢀⠎⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠛⠀⠀⠀⠈⣈⡩⠝⢻⡄
⠀⠀⠀⠙⠲⠦⣤⣈⡛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠘⢤⠦⠀⢱⠀⠀⣧⠀⢠⠃⠀⠀⠠⡶⠖⠁⠀⠀⢀⠀⠀⢀⣀⠤⠖⠊⣉⣠⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣤⣀⡀⠀⠀⢠⡄⠈⠒⢄⠀⠈⢦⠀⠸⠀⢸⠀⡰⠃⠀⢀⠴⠊⠀⠀⠀⢀⣀⠷⠓⣋⣩⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠀⠙⠦⡀⠣⡀⡇⠆⡰⠁⡠⠾⠇⢀⡠⠤⢒⣪⣥⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣌⣓⣵⠼⠾⣴⡩⢤⣒⣭⠵⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⡊⢢⠀⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⣁⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢟⣛⣩⣭⣭⣉⡙⠻⣿⡿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⣯⣾⠟⠛⠉⠉⠉⠛⢿⣷⡌⢷⡜⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣵⡿⢃⣾⣿⣇⡀⠀⠀⠀⠀⠀⢆⢹⣿⠈⣷⢸⡇⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⣼⡿⠁⣾⠋⣼⠋⣻⠷⣦⡀⠀⠀⣼⢸⣿⢀⡏⣸⡏⣿⡉⢻⣧
⠀⠀⠀⠀⠀⠀⣰⡿⣿⡟⢸⣿⢃⣸⣧⣼⠧⡼⠥⠀⠀⢻⣤⣴⣿⣾⣇⠈⠰⣏⣽⡉⢷⡀⢿⣇
⠀⠀⠀⠀⠀⠀⣿⢱⣿⠇⣾⣿⠟⣹⣴⠷⠚⠓⠒⠀⠀⣾⡟⠁⠀⠈⠹⣷⠀⢻⣤⡿⢿⣷⢸⣿
⠀⠀⠀⠀⠀⠘⣿⢸⣿⡀⣿⣷⡾⢋⣴⠞⣫⣭⣶⠶⣶⣿⣇⢧⣤⡄⢀⣿⠶⠟⠁⠀⠀⣿⡏⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣏⣿⡇⢹⣿⣱⡟⢡⣿⡟⡩⠖⠛⠉⠛⣿⣷⣾⣶⣿⡃⠀⠀⠀⠀⢀⣿⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡜⣿⣿⠁⣿⣿⠈⠀⠀⠀⠀⣾⠁⠀⠀⠙⢷⣽⣲⣤⣀⣤⣾⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣜⣿⣆⠹⣿⡄⠀⠀⠀⠈⣿⠀⠀⢀⠀⢄⣙⠛⠿⠟⣛⣡⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣝⢿⣦⣀⡀⢀⣿⣟⣛⣎⣙⣦⣭⣙⣛⣛⣩⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣭⣛⠿⠿⣿⣭⣭⡤⠿⠶⣾⣉⣩⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣶⣶⣤⣤⣤⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡶⣸⣯⣭⠭⢿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣟⣾⢥⡿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢟⣛⣛⣻⠿⠿⠿⠿⠛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡰⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠶⠶⠶⠶⠶⠶⠞⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⢡⠀⠀⠀⠀⠀⡀⠖⠀⠉⠉⠉⠀⠐⠢⡴⡀⠈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠑⠦⣀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⠳⣄⡠⠛⢦
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠐⠀⠀⠀⢣⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢀⠔⠈⢷⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠡⠀⠀⠈⢇⠀⠀⠀⠠⠁⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⡇⠀⠀⠙⣆
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢂⠀⠀⢺⡀⠀⢠⠁⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠀⣠⣤⣠⣤⣤⣀⣀⠈⠢⠀⠀⠈⣄⠀⣀⡇⣠⠂⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠂⠈⠀⠹⡄
⠀⠀⠀⠀⡾⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠠⠈⠉⠓⠲⡶⠀⠁⠀⠀⠀⠉⠐⢤⠀⠀⠀⠀⡀⠤⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢠⡇⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⢸⠁⢠⡏⡀⠀⠀⠤⠄⠀⠀⠐⠒⠒⠒⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⢒⡤⠶⠋⠀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⢸⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⢸⡆⠀⡇⠀⠀⠀⠀⠀⠀⡀⠠⠐⠂⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠍⠐⣤⣤⣤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⡇⠀⢡⢀⠠⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠓⣤⣀⡀⠀⢀⣀⡤⠊⠱⣄⠀⠀⠀⠀⠀⠂⠄⡀⠀⠈⢣⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢹⡂⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠃⠀⠀⡐⡇⠀⠈⡋⠁⠈⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠂⢠⡀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢧⠀⠘⡄⠀⠀⠀⠀⠀⡠⠖⠁⢀⡎⠀⠀⡰⠀⢷⠀⠀⡇⠀⠀⠀⠆⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠐⡾
⠀⠀⠀⠀⠀⠈⢧⠀⠈⢀⠀⠀⡠⠊⠀⠀⢀⡼⠀⠀⠰⠁⡘⠈⡄⠀⠇⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠠⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⢀⡵⠊⣀⠀⢀⡠⠞⠀⠀⠠⠃⠀⠇⠀⠹⡄⠇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⢀⠀⡴⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢀⡠⠛⢄⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⠂⠀⠀⣾⠀⠀⠀⠀⡎⠑⢦⡀⠀⠀⠀⠂⠀⠀⣠⠔⠋⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⢀⠂⠀⠀⠀⡇⠀⠀⠀⠀⡅⠀⠀⠙⠒⢤⠴⠾⡞⠉⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⣀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⣰⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⡷⠶⠤⣤⣤⣧⣤⣤⠤⠶⢾⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠘⢧⣄⠀⠀⠀⠀⢀⣠⡼⠃⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠈⠉⠋⠉⠋⠉⠁⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠐⠂⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠠⠤⠀⣀⡀
⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠹⠀⠋⠹⠀⠋⠱⠀⠊⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣄
⠀⠀⠀⠀⣸⠁⠀⠈⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠰⠈⠄⠰⠈⠄⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠀⠀⠀⠘⡆
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠤⠤⠤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠊⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⠦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠶⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣴⠖⢶⣴⣞⣛⣳⣶⣛⡛⢷⡴⠶⢦⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡶⣛⣹⡿⠷⠖⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⠶⠿⣏⣉⢻⣄⣤⡀
⠀⠀⠀⠀⠀⣠⣤⣼⣏⠤⠛⠋⢋⠀⠀⠀⠀⣠⠴⣄⣀⡤⠤⡤⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠉⠛⡿⣄⣹⣆⣀⡀
⠀⠀⢀⣄⣼⣡⠴⡋⠁⠀⠀⡤⠤⢦⠞⢉⠽⣖⢢⣅⡽⢀⡴⠁⢀⠀⠀⠀⠀⣀⡃⠀⠀⠀⠀⡐⠀⠀⠙⠻⣍⢹⡆
⠀⣀⣟⡠⠋⠀⠀⠈⢂⠀⠀⠈⠲⠬⢧⣬⢤⠾⢎⣸⡉⠁⠀⠀⢸⠀⠀⠀⠀⢿⠉⡶⠒⠢⣤⠥⢄⡀⠀⠀⠈⡻⣟⠙⢳⡄
⣿⣉⠟⢦⡀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⢻⠤⠖⠋⠁⢃⠀⠀⠀⢸⠀⠀⠀⠀⡌⠑⢧⣐⣏⣫⢱⡄⢹⠤⢤⣔⡀⠈⠳⣼⠃
⠈⢿⣦⡀⠙⢦⡀⠀⠀⠀⠈⢢⠀⡀⠀⠈⢆⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⢠⠁⠀⢺⣈⡟⣻⣩⠶⠯⡠⠄⠊⠀⠀⡠⠛⢷⣄
⠀⠀⠈⠻⣦⡀⠙⣦⡤⠤⣤⠞⢹⡇⠀⠀⠈⢶⠒⢤⡴⠷⣲⣖⣛⠓⢦⡠⡼⢤⠀⠀⡌⠉⠉⠁⠀⡔⠁⠀⠀⡠⠊⠀⠀⣠⡿
⠀⠀⠀⠀⠈⠻⣿⣗⠛⣦⣼⡠⠞⠈⢢⠀⠀⠈⢷⠬⢧⣸⠁⣑⡸⢂⣜⣀⡷⠋⠀⡸⠀⠀⠀⢀⠎⠀⠀⡠⠊⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣅⡏⠳⣄⠀⠀⠑⡄⠀⠈⢆⠀⢇⣏⣡⢧⡇⠀⢸⠀⠀⢰⠁⠀⠀⡠⠃⠀⣠⠞⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡈⠳⣄⠀⠈⢢⠀⠈⢆⠀⢹⠀⠀⡇⠀⡌⠰⣖⠧⣀⣄⣜⢁⡠⠞⠁⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡈⠑⢄⠀⠑⡄⠈⢆⣈⡇⠀⡇⢀⠃⢀⠿⢴⣥⠊⣨⠟⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠵⢶⡺⣶⠉⣼⢳⠀⡇⠸⠀⡜⠀⡴⣿⠟⢁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣝⣟⣿⡵⡍⠌⡄⡇⡆⡘⢀⣾⠟⢁⣤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠙⢮⣮⣇⣱⢰⣡⠟⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡈⠻⠿⢿⣡⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢺⣹⠂⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣶⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠎⠀⢿⣷⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢠⣀⣀⣀⣀⣤⣤⠋⠉⠉⠀⠀⠘⡿⠿⠟⠻⠇⠀⡀⠀⠀⠀⠀⠀⠘⠿⢿⣿⡗⢀⣤⣤⡶⠏
⠀⠀⠀⠈⠉⠉⠉⠉⠉⢡⡶⠀⠀⡄⠀⠀⡆⠀⠀⠠⠀⠚⠁⢰⠂⣼⡆⢀⢖⡐⢤⠨⡀⢨⠁⠀⠀⡀⠀⠼⣿⣶⠄
⠀⠀⠀⠀⢀⣾⡖⡶⣾⣴⣥⣾⣿⣷⣥⣥⣩⣼⣿⣼⣠⣦⣟⣸⣿⣭⣆⣼⢻⣷⠏⣄⣷⣽⣿⡆⢴⣰⣧⠀⠈⠉
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡶⠝
⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⣠⣄⣀⣀
⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⠀⠀⠀⠿⠿⣿⣿⣿⠇
⠀⠀⢠⠄⠛⠋⣥⣤⣄⣀⣀⣠⡄⠀⠀⠈⠉⠛⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⢰⣶⣤⣄
⠀⠀⢸⣦⣴⣶⠋⣇⣤⡄⡄⣠⣇⣀⠀⣾⢠⠀⠰⠟⠛⠛⠁⠀⣀⡀⠀⢷⡌⠋⠉⠉⠀⠀⠀⠀⠀⣠
⠀⠀⠸⢻⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⣾⣧⣤⣴⣂⣄⢠⢠⡆⠙⢃⠀⠀⢀⠀⠀⢀⡸⠿⡿⢀⠈⡟⣀
⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣮⣿⣾⣴⣦⣹⣄⣤⣨⣄⣴⣖⣦⣾⣷⣿⣄
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁
⠀⢀⠀⣄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠘⠛⣿⠲⣄⣤⣶⣤⡄⠀⢸⣶⣤⣀⣀⣀
⠀⠀⠛⣿⠀⠈⢿⣿⣿⢧⣤⡀⣉⠍⠙⠛⠛⠀⠀⠀⠀⢠⣦⡀⢤⣤⣄⣀
⠀⠀⠀⠹⣦⣾⡿⣿⣿⣿⣿⣥⢏⣀⣰⠖⠀⠀⡀⣀⡄⡀⠀⠁⠘⠿⠿⠻⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⣼⣷⣄⣷⡎⣠⠆⠀⡆⠀⣄⠿⠷⠀⠰⢶⡄
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣤⣦⣤⣤⣠⣇⣤⡀⣂⢀⠚⠂
⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠒⠒⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⠋⢑⡀⢤⡀⠒⠒⠖⠒⢀⡠⢀⡈⠝⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠄⠊⢻⣿⣆⠀⠈⠉⠁⠀⠀⠂⠉⠈⠑⠀⡀⢙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⠔⢉⠀⡠⠘⣿⣿⣦⠀⠠⠀⠀⡀⠀⠀⢀⠠⡠⠙⢢⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠠⠈⠐⠡⠀⠈⠀⢹⣿⣿⣧⠁⠈⠤⢀⠀⡀⠀⠑⠀⠂⠂⠡⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡑⠀⠌⠀⡐⠀⠀⠔⢑⠌⢻⣿⣿⡖⠒⣠⠅⡪⠀⠀⠊⠀⠠⠀⠑⠱⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠰⠀⠀⠀⢀⠀⢠⠢⠁⡚⣻⣿⣿⣭⣔⠠⠑⡀⠀⣡⠁⠂⢁⢁⠁⢡⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡆⠀⠀⠀⠈⠀⡆⢃⠱⠘⣿⣿⣿⣿⣿⣿⣤⣴⣴⣥⣦⣬⣄⣀⡀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠃⠀⠀⠀⢀⠀⠇⡌⡠⢀⣿⣿⣿⣿⣿⡿⠿⢿⠿⣿⣿⣿⣿⣿⣿⢿⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠀⠰⠀⠀⠐⠈⠄⠘⢔⣴⣿⣿⡿⣿⣿⡙⠅⡡⠃⠀⡙⡈⠆⡀⠌⡀⡘⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡡⠐⠂⠀⠀⠀⠀⣠⣿⣿⣿⠏⠑⣿⣗⠂⢁⠠⡈⠚⢀⠆⠀⠀⠐⡰⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡠⠐⣁⠠⢁⣾⣿⡿⠟⡡⠒⠉⣿⣯⢉⡒⢉⠠⠔⡠⡠⢊⢄⡔⠁⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠣⣾⠿⠛⠩⠀⠀⠉⡐⣿⣟⠒⠈⠡⢀⠀⠈⠐⣠⠌⠡⣢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠄⠀⠢⢄⡂⠌⠁⠀⣿⣗⠂⡀⠡⢀⡠⠔⠚⠡⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠈⠉⠒⠂⣿⣗⠒⠒⠈⠁⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⣿⡿⠶⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠈⠉⣈⠅⣿⣧⠀⠀⠀⠈⣉⣒⡶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⠘⠐⠃⠀⠉⣄⠛⠻⠄⠀⠀⠠⠔⢒⣀⣬⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⢿⣿⣿⠃⣿⣿⡋⣸⣿⠋⣽⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣽⣿⣿⡎⠫⣿⢧⡌⠋⡧⠙⣹⠘⡿⡏⠡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣯⠻⠿⠯⠳⠄⠐⠀⠀⠉⠀⠈⠐⠰⠃⡾⡿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡛⠢⠄⠀⠀⢠⠄⠀⠀⠐⠀⠀⠀⠔⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣽⡦⣭⡶⠄⠀⠀⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠴⢂⣠⣤⣼⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⡽⣿⣿⡿⠿⠟⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡭⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⣰⣿⣿⡎⢿⢿⡳⣎⠛⣻⣿⣿⠇⣀⠀⠀⢀⠀⠃⠀⠀⠀⠄⠀⠀⠀⢀⣀⣲⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣰⣿⣯⣿⣿⡷⠦⠀⠀⣼⣿⣭⣭⣴⡍⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣥⡌⡙⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⢠⣿⣷⣍⣛⣉⣁⡀⠀⢸⣿⣿⣿⣿⠟⢁⣯⡄⣠⠀⠀⡄⠀⠠⢒⣤⠀⢴⡏⠉⣿⠛⢻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣾⣿⣿⣿⣿⠖⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⠃⢹⣶⡀⡇⠲⣷⡀⢿⣇⢸⣧⢸⡿⢠⣿⠃⣠⣿⠟⠁
⠀⢰⣿⣿⣶⣭⣴⡛⠒⠀⠀⠀⢀⠈⠉⢻⣿⣿⣷⣿⣷⣿⠑⢿⣦⡙⣿⣜⣿⣼⣿⣾⣧⣿⢃⣴⠟⠁
⠀⣼⣿⣿⣿⠿⠟⣡⠖⠀⠀⠀⠈⠀⠀⠀⠉⢿⣿⠿⠟⠙⢿⣶⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⣿⣿⣿⣿⣟⣿⣿⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢶⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⣿⣿⣿⣯⣿⣫⣥⣼⣿⡁⣘⡀⠈⡀⠀⢰⡀⣀⠀⢴⣶⣶⢶⢾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣷⣿⠟⣠⣿⣿⡄⢳⣿⣄⠙⣿⣷⡀⣶⣶⠾⣿⣿⣿⣿⣿⠟⠛⠉
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣴⣿⣿⣿⢰⣿⣿⣷⣤⣶⠿⢛⣽⡿⠋
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣤⣾⠟⠉
⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠦⠒⠃⠀⠀⠀⠀⠐⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠠⢀⠀⢀⠄⠀⣠⣴⣴⣦⣦⣄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠒⠓⠧⢮⣠⣾⣿⣿⣿⣿⣿⣿⣷⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣿⣿⣿⣿⡟⠁⠀⣀⣤⣄⡀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⡿⡿⠀⠀⣾⣿⣿⣿⣿⣦⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢧⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⣟⠻⢿⣿⣿⣿⣿⡇⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⢆⠀⠀⠙⠛⠛⠋⠀⢀⡼⣥⡀⠈⠻⣿⠿⠋⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⣴⣤⣤⣴⣾⣷⣦⣤⣀⣠⣤⣶⠛⠀⠀⠉⠂⠀⠀⠀⠀⠠⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠢⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⡻⢿⣿⣿⣿⣿⣿⣿⢿⡅⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠫⠝⡛⠛⡋⠙⠊⣘⡀⣀⠄⠆⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠠⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣈⢁⡈⣉⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠈⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠤⠤⠔⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠒⠛⠉⠉⠁⠀⠀⠉⠀⠈⠉⠉⠉⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠘⡄⠀⠀⠀⠀⠀⢀⡶⠀⠈⠉⠉⠀⠐⠂⣤⡁⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠑⠦⣄⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢿⠀⠁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⠑⢄⡀⢈⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡠⠻⣅⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠘⡀⠀⠀⢹⡆⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠘⡆⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⣼⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠡⠀⠀⠀⣷⠀⠀⠀⡌⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢃⠀⠀⢸⠇⠀⡐⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣀⣤⠴⠞⠛⠛⠛⠛⠒⠒⠦⢤⣑⣄⠀⡈⠦⠀⠚⠂⠶⢀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⣶⠛⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⣾⠀⠀⢠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⣀⢈⠝⠁⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢀⡀⠤⠐⠂⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⡟⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣇⠀⣿⠤⠄⠐⠒⠒⠀⠀⠀⠉⠉⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡩⠝⠋⠉⠀⠒⠒⠒⠀⠠⠤⠄⠀⣀⡀⠀⣿
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⢀⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠳⠀⠷⠶⠴⠶⠖⠒⠒⠲⠶⢄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⠀⢘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡟⠀⠈⢳⡦⣄⣀⣀⣀⣤⡴⠋⠉⢢⠀⠀⠀⠀⠀⠁⠀⠤⣀⠀⠀⠡⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠘⣇⠈⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡞⠀⠀⢀⠺⡀⠀⠈⠉⠀⠀⢠⠀⠀⠀⠁⢤⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢹⡄⠀⢡⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡸⠀⠀⢀⠄⠘⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠁⣾⠃
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠑⡄⠀⢀⠀⠊⠀⠀⠀⡴⠁⠀⠀⠌⠀⠇⢰⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢠⠃⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢈⠶⠄⡀⠀⢀⣠⠞⠁⠀⠀⡜⠀⢸⠀⠀⢳⡀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠐⢄⠀⡰⠁⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡁⠀⠀⠀⠀⠈⠀⠀⠀⠀⡐⠀⠀⡎⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⡠⠛⣄⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡇⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⢂⠀⠀⠀⣀⠴⠋⠁⠀⣈⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠰⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣬⡶⠖⠋⠁⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⡀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⣾⣄⣀⣀⡀⢀⣀⠀⢀⣀⣀⣸⣧⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠈⠛⠶⢄⣠⣀⡠⠴⠖⠋⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡠⣄⠀⡠⣄⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⣀
⠀⠀⠀⠀⠀⣠⠊⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠈⠀⠹⡀⠁⠈⠆⡃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠲⡀
⠀⠀⠀⠀⣸⠇⠀⠀⠈⠳⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠂⠂⠀⠣⠐⠀⠃⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠶⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⢶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡶⠶⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠈⠙⡉⠉⠁⠐⡢⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣉⠴⠒⠋⠙⠛⠒⢶⣀⢀⡔⠁⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠁⠔⠁⠀⠀⠀⡠⠤⠒⠈⡏⢫⠁⡹⠛⠛⠩⣂⢄⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠓⠄⡀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢁⠈⠦⠁⠀⠀⠀⠀⠑⠻⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠅⢀⡔⠁⠀⠀⠀⢀⠤⠒⠚⠙⠛⠃⠒⠠⢀⠀⠀⠀⠙⣧⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⣤⣧⡎⠀⠀⠀⠀⠔⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠳⡀⠀⠀⠸⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⣾⣿⣿⢿⣿⣶⣶⣤⣎⠀⠀⠤⠤⣾⢏⠀⠀⠀⠀⠀⠀⠱⡀⠀⢰⢣⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣿⡿⠛⢩⠀⠀⠈⠉⡿⣿⡗⢀⠖⠂⢠⡽⢄⠀⠀⠀⠀⠀⠀⣃⢀⡾⠤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠏⠀⠀⠘⠀⠀⠀⢠⠃⠨⡇⢸⠀⢰⠉⠀⠀⢱⠤⠐⠒⠂⠉⢹⠜⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠀⠀⠀⠀⠀⣸⡀⠠⠧⡘⠳⠜⠦⣀⣀⠜⠀⠀⠀⠠⠐⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣇⠤⠤⠐⢺⠀⠉⠁⠈⡀⠀⣼⡿⠒⠒⠘⠓⢷⠆⠀⠀⢁⣀⣠⠁⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢇⠀⠀⠀⠘⡄⠀⠀⠀⠱⠀⡿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠋⢢⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⢆⠀⠀⠀⢰⠀⠀⠀⠀⠱⣅⠀⠀⠀⠀⠀⢠⠀⠀⢀⡴⠁⠀⠀⡇⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢣⡀⠀⠀⠱⡀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠤⠆⠂⠁⠀⠀⠀⢠⢏⡜
⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀⠀⠐⢄⠙⠢⢄⣀⠈⠢⣀⠴⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡴⡡⠊
⡠⠾⠋⠉⠉⠉⠓⠲⠦⣝⡆⠀⠈⠢⡀⠀⠀⠉⣿⣏⣐⠷⣄⠀⠀⠀⠀⠀⢀⣀⡤⠖⡩⠊
⢤⡀⠀⠀⠀⠀⠀⠀⣀⡼⢷⡄⠀⠀⠈⠢⣀⢠⣿⣿⣿⡇⠀⠙⠛⠋⠉⠉⢩⡠⠔⠋
⠀⠙⠲⠤⣤⣶⣞⣿⡭⠶⠊⠀⠀⠀⢀⠀⠈⠁⠒⠭⠍⠃⢀⠀⠠⠤⠔⠒⠉
⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠉⠙⢻⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠖⣛⣿⣿⡄
⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣶⠀⠀⠁⠀⠀⠀⠠⠘⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⣄⣚⠈⠤⠄⠈⠙⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣋⣉⡁⠀⠀⠀⠀⠀⠁⠐⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣿⣿⣿⣿⣿⣿⣿⢇⢰⣤⡀⡰⠀⣠⣤⣤⣤⣄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⣿⣿⣿⣿⣿⡿⠞⠛⠛⠻⢿⣾⣿⣿⣿⣿⣿⣷⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠻⣿⣿⣿⣿⠋⢀⣤⣶⣶⣄⠀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢉⣻⡇⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⢽⣷⡀⠈⠻⠿⠿⠟⠁⢠⣿⣏⠻⣿⣿⡿⢁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⣠⣬⣥⣴⣿⣦⣤⣀⣀⣤⣶⣿⠉⠛⠦⡈⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠑⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢨⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠝⡿⢿⣿⣿⣿⡿⣧⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⣶⠂⢂⣛⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⡍⣭⠉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢄⣀⣀⣀⠀⣀⣀⣠⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⢀⣀⠠⠄⢒⣒⣀⣀⣒⡒⠢⠤⣀⡀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣒⡆⣠⠔⣩⠔⠂⣉⠥⠐⠠⡀⠀⢈⣉⠐⠢⢍⠢⣄⢰⢒⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢨⠎⡤⠊⠀⠀⠫⢄⡀⠀⠀⠘⡄⠘⡄⠈⠑⠢⡑⢌⠱⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠊⠔⠂⠐⠒⢄⡀⠙⡄⠀⠀⡇⠀⡄⠀⠀⢀⠇⠀⠑⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⡜⠀⠀⠀⠀⠀⠈⠢⡈⡀⠀⡅⠠⠁⠀⢀⠎⠀⡔⢆⠘⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡌⠀⠓⠊⠉⠉⠑⠂⢄⡔⠋⠁⠉⠙⢧⣀⠔⠁⣀⠔⠀⠀⢆⢡⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⢀⠔⠂⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠓⠊⠉⠀⠀⠀⢀⡜⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡇⠇⠀⠀⢀⡠⠔⢒⠽⡀⠀⠀⠀⠀⢀⣗⠂⠀⠀⠒⠂⠁⠀⢸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⠘⠄⢠⠋⠀⠔⠁⠀⡸⢲⠂⠠⠲⡁⠀⠉⠒⠂⠀⠒⠙⠀⡆⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢣⠈⠁⢀⠎⠀⠀⢠⠁⢸⠀⠀⢃⠈⠢⡀⠀⠀⠀⢀⠆⡜⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢆⠳⡀⠸⡀⠀⠀⢀⠀⢸⠀⠀⠀⠢⣀⡈⠁⠒⠒⢉⠞⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⡑⢌⠲⢍⠒⠠⠬⠆⠀⢣⡀⠀⢀⡠⠜⠀⡠⠖⡡⢪⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠕⠁⠀⠉⠢⢍⡐⠢⠤⠤⠀⠈⠭⠥⠔⣂⡡⠔⠉⠀⠈⠨⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠒⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣀⡤⣤⣶⣴⣢⣴⣦⣴⣦⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣲⣾⡿⣿⢽⣯⣹⣗⣿⣺⣵⣾⣕⣿⢭⡟⣾⡿⣽⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⣝⣯⣼⣷⣿⡿⣿⣹⣏⣿⣿⣿⢹⣸⡟⣿⡟⣿⣷⣿⣩⡟⣻⣷⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⢿⣉⣷⡾⣿⡻⣧⣿⣟⣿⣻⡟⣿⣿⣿⣿⣿⣿⣷⣿⣼⣟⣾⣷⡿⣿⣿⣼⡟⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢛⣯⣾⢿⣯⢿⣞⣷⣻⣿⣿⣿⣿⣿⢿⢿⣿⡿⡿⡿⣿⣿⣿⣾⣿⣿⣿⢿⣼⢟⣿⣿⣽⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣍⣿⣯⡻⣿⣽⣷⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⢿⣿⣿⡿⣻⡾⣩⡾⣻⣾⢩⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡛⣷⣿⣷⣮⣻⣾⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣟⡿⣿⣿⣿⡾⣫⣾⢿⣕⣿⡧⡄
⠀⠀⠀⠀⠀⠀⠀⢐⣽⣫⣻⣿⣝⡻⣾⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣯⡶⢿⣻⣼⢿⣷⡆
⠀⠀⠀⠀⠀⠀⢀⣿⣯⣹⡿⣶⣭⣟⣿⣿⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⡿⣟⣽⣿⡟⢻⣷
⠀⠀⠀⠀⠀⠀⣼⣯⢧⣿⣟⡿⢾⣽⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⡷⢿⣿⣽⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣟⢾⣿⣭⣿⣿⣿⡯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⣿⣿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⢼⣻⣿⣿⣷⣶⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⠀⠀⢀⡀⠤⠐⣒⡀⠀⢀⣀⣒⠂⠤⢀⡀⠀⠀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡠⢚⠥⠖⠊⠁⢀⠀⠀⠀⠀⠀⠉⠉⠒⠬⡓⢄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡰⣫⠞⠁⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠳⣝⢆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠞⡔⠁⠐⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⢠⠀⠀⠀⠈⢢⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡌⡼⠁⠀⠀⠀⠀⠀⠀⠀⢐⡤⠤⠤⣇⡀⠀⡅⠄⠀⠀⡠⠀⠀⢣⢡⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢰⠀⠈⠀⠀⠉⠀⠐⢢⠔⠉⡀⠀⠀⠀⠈⠲⡃⠀⢀⠄⠀⠀⠐⠀⡆⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠈⠀⠀⠀⡬⠀⠈⠀⠀⠀⠀⠐⠀⢡⡤⠁⠀⠀⠀⠀⠀⡇⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢸⠀⠀⠀⡀⠀⠀⠈⢣⠀⢀⠀⠀⠀⠀⡀⠀⡞⠀⠀⢀⡀⠄⠀⠀⡄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠘⡀⠄⠁⠀⠐⠁⠀⢠⠷⣄⠁⠀⠐⠈⣠⠜⠓⠠⠉⠀⠀⠀⣀⢀⠃⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⢡⠀⠀⠂⠀⠀⠀⠩⠀⠀⢹⠛⠛⠙⠅⠀⠀⠀⠀⠀⠀⠀⢀⡞⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⡣⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⢀⢞⡜⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢌⠢⡀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⢀⠔⡡⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⠮⣑⠠⢄⣀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⠵⠊⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠉⠒⠢⠭⠍⠉⠈⠩⠭⠄⠒⠉⠀⠀⠀⠀⠀⠀⠐⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋







⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡪⠭⠙⠒⠂⠀⠐⠚⠋⠭⣒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠚⠁⡇⢰⠀⡤⠒⠑⣃⡒⠬⣀⠀⠉⠪⡢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠢⡀⠀⡇⠘⡎⠀⠀⠀⠀⠀⠐⠀⠑⢄⠁⠈⢮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠁⢀⠔⠒⠲⡘⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢫⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⢡⢣⠀⠀⠀⠑⡼⠚⠂⠐⠢⣄⠀⠀⠀⠀⠀⠀⡅⠁⠈⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⡇⠈⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⣦⣒⣒⣒⣒⣊⣀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡟⠐⠒⠤⡀⠤⣀⠈⠁⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢡⠀⢇⠀⠀⠀⠀⠑⢤⣀⠀⢀⡠⠞⠀⠀⠀⠀⠈⢂⠀⠑⠄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡆⢈⠢⢄⣀⣀⠤⠊⢠⠋⡍⠀⠀⠀⠀⠀⠀⠄⡘⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⢄⠀⠀⢀⡴⠋⣠⠋⠌⠀⠀⠀⠀⠠⠠⢂⡴⠁⠓⡰⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡣⣈⠁⢀⠔⠁⠘⣄⠀⠀⠀⢀⡠⠔⠫⠀⡠⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢝⡣⢄⡀⠠⠂⠉⠉⠙⢋⣠⠤⣒⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠓⠴⠶⡤⠍⠒⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⠂⠉⣁⠛⣀⠀⠀⠀⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⠀⠐⠤⢈⣛⣉⣚⣁⠀⠤⠔⠚⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⣀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣄⠒⣤⠄⠉⠉⠩⠍⠉⠉⠠⡔⠒⣠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⠮⠓⢺⡁⠀⠀⣘⣤⠴⠒⢺⡷⠤⠤⣠⡃⠀⠀⠈⡗⠊⠵⣢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠁⠻⡄⠀⠀⠀⢧⡴⠋⠘⣇⠀⠀⢨⡄⠀⠀⣸⠋⠑⢦⣜⡁⠀⠀⢠⠟⠊⢕⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⠹⣆⠀⠀⠀⠙⣄⠀⢠⠟⡆⠀⠀⣿⡄⠀⢸⣇⣀⣠⣿⣀⠀⣰⠏⠙⠢⣴⠋⠀⠀⠀⣰⠯⡦⡀
⠀⠀⠀⠀⠀⠀⢔⡑⠁⠀⠈⠳⣄⠀⠀⠘⢦⡾⠀⢹⣄⣠⣽⣷⣲⣿⣿⣿⣿⣿⣿⣿⣿⣶⣠⣴⠋⠳⣄⣠⠞⠁⠀⠀⢊⡢
⠀⠀⠀⠀⡠⡢⠈⠙⢦⡀⠀⠀⠈⢦⡀⠀⢈⣷⡖⠛⣿⠁⠘⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡼⢧⠀⠀⢀⡴⠋⠁⢄⢄
⠀⠀⠀⡰⣭⡀⠀⠀⠀⠉⠲⣄⠀⠀⣹⣶⠉⢹⣷⡀⠈⣷⠀⢹⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣧⢸⣧⠞⠋⠀⠀⠀⢀⣭⢆
⠀⠀⡰⡜⠀⠉⠓⢦⣄⠀⠀⠈⢳⣾⡁⠈⠳⣄⢿⣷⡀⢸⡧⢸⣿⣿⣿⣿⣿⣿⣿⢽⠟⠛⣿⠿⣯⣻⣿⣿⠃⠀⠀⣀⡴⠚⠉⠀⠡⢀
⠀⠠⣹⣄⡀⠀⠀⠀⠈⠙⠢⣤⡁⠀⠙⢦⣀⠈⢾⣧⡻⣆⢹⣀⣿⣿⣿⣿⣿⣿⢡⠎⢀⡜⠁⣀⡴⠋⠀⢘⣦⡔⠋⠁⠀⠀⠀⢀⣠⣎⠄
⠀⢧⠁⠀⠉⠙⠲⢤⣠⣤⣼⡁⠉⠓⢦⣀⠉⠳⣄⠙⢿⣹⣾⣷⡿⠿⠿⣿⣿⣷⠃⡴⠋⣠⠞⠁⣀⡴⠚⠉⠈⣄⣀⣠⠤⠖⠊⠉⠀⠈⡜
⠘⣥⣀⣀⠀⠀⣰⠋⢀⣿⣿⢷⠦⢤⣄⡈⠙⠲⣌⡙⣤⠟⠉⠀⠀⣀⡀⠀⠈⠙⢿⣴⢊⣡⠖⠋⢁⣀⡤⠔⠚⠹⡃⠀⠀⠀⠀⠀⣀⣀⣤
⡄⠁⠀⠈⠉⣹⠛⢻⣿⣿⣿⣿⣦⣀⣀⣉⠙⠲⠤⣽⠃⠀⢠⠖⠋⠉⠉⠛⢶⡀⠀⢻⣯⠤⠒⠋⠉⣀⣀⣠⠤⠴⣿⠒⠒⠋⠉⠉⠁⠀⠈
⣇⣀⣀⣀⣠⣇⣀⣼⣿⣼⣿⣿⣽⣄⣀⣉⣉⣁⣒⡟⠀⢠⡏⠀⠀⠀⠀⠀⠀⣿⡄⠘⣿⣶⣞⣙⣛⣛⣻⣶⣶⣦⣬⣀⣀⣀⣀⣀⣀⣀⣀
⡏⠁⠀⠀⢸⠏⠉⢿⣿⢿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣷⠀⠈⣧⠀⠀⠀⠀⠀⢀⣿⠀⢸⣿⡿⢿⣭⣭⣍⣉⣀⠀⠀⡙⠋⠙⢦⡀⠀⠀⠀⠈
⠁⡆⢀⣀⣸⣤⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠈⠳⣦⣤⣤⡶⠟⠁⣴⣿⣿⣷⣶⣤⣥⣈⡙⠋⠙⠻⣿⠶⠤⢤⢷⣀⣀⡀⢀
⠰⠋⠉⠀⢸⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣷⣦⣄⣀⠀⣀⣠⣴⣾⣾⣿⣿⣿⣿⣿⣿⣟⣿⣷⣶⣾⣤⣀⠀⠀⠀⣇⠀⠉⠙
⠀⠞⡀⢀⣸⡦⠖⠟⠛⣿⣿⣿⣿⣿⡿⣿⣿⡿⢋⣴⠟⣵⢻⡟⢻⠿⣿⡿⣫⠙⣏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⠉⠙⠻⠶⢤⣿⡀⢀⠳
⠀⠘⡹⠉⠀⣇⠀⠀⣠⣼⠿⢿⣿⣫⣿⠟⠉⣠⡾⢁⣾⢣⣿⣇⡟⢠⣿⣿⣻⣇⠈⢆⣽⢿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣄⠀⠀⢠⠇⠈⢋⠃
⠀⠀⠰⣡⠀⣸⣶⠞⠉⠀⠀⣠⡿⢿⣅⣠⡾⠋⠀⣾⡵⢿⠙⣾⠃⢸⣇⢻⣿⠛⣄⢨⣿⣾⣿⣿⣿⣿⣿⣿⣟⠀⠀⠉⠱⣶⣏⡀⢜⠆
⠀⠀⠀⠱⡙⡀⠘⢦⣀⣠⠞⠋⠀⢀⣿⠿⣇⣠⣾⠟⢁⡏⢰⣿⠀⢈⣿⢸⣿⣄⣼⣷⣿⣿⣿⣿⣿⣿⡷⠋⠙⠳⣄⣠⠞⠁⢀⢍⠎
⠀⠀⠀⠀⠈⢔⢄⡤⠛⠳⢄⣀⣠⣞⣁⣠⣼⠟⠛⢤⣾⡀⢸⡇⠀⢸⣟⣈⣿⣿⣿⣿⣿⣿⣿⣿⠞⠉⠻⣄⣀⡴⠛⠙⢦⡠⠢⠃
⠀⠀⠀⠀⠀⠀⠱⢥⠀⠀⢀⡼⠋⠉⠉⢠⠋⠀⠀⣸⠁⠈⢹⠏⠐⠻⣿⠿⣿⣟⠛⠻⣿⠉⠈⢹⣦⣤⠶⠛⢯⡀⠀⢀⡼⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⢏⠀⠀⠀⣠⠃⠀⠀⢠⠃⠀⠀⣸⠀⠀⢸⣿⠀⠸⣟⠙⠛⠻⡟⠛⠉⠙⣆⠀⠀⠀⡹⡂⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡠⢴⠃⠀⠀⢀⡏⠀⠀⠀⣇⠀⠀⢸⣿⠀⠀⣼⠀⠀⠀⢹⠀⠀⠀⠘⡧⢔⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⡤⢾⡀⠀⠀⢠⡏⠀⠀⢸⡿⠀⠀⣿⡆⠀⠀⠀⣷⢤⠒⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠁⠼⡟⠒⢂⣚⣛⣛⣛⣻⠯⠌⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠤⠴⠶⠶⠲⠶⠦⠤⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⣠⣄⢀⣤⡀⣠⣄⢀⣤⡀⠉⠉⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠷⠞⠛⠾⠋⠶⠞⠛⠶⠋⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠚⠛⠛⢻⡟⠛⠛⠓⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⢀⣀⣤⣤⢸⡇⣴⣶⣶⣦⣤⣄⡈⠉⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠙⣆⠀⢀⣠⣴⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣦⠀⣰⠋⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⢆⠹⣿⣿⣿⣿⡿⠿⠿⢈⡁⠿⠿⢿⣿⣿⣿⣿⠏⡰⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⣴⡌⢧⠘⣋⣡⠤⠶⠒⡚⢻⡏⢓⠒⠶⠤⣌⣙⠃⡼⢡⡄⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠟⣡⠼⢿⡉⢴⣾⣿⣿⣿⢸⡇⣿⣿⣿⣷⡦⢉⡿⠧⣌⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⢰⡟⠓⠦⣀⠀⠀⠀⠀⠀⣠⠞⢡⣶⣆⠳⡌⢿⣿⣿⣿⢸⡇⣿⣿⣿⡿⢡⠞⣰⣶⡌⠳⣄⠀⠀⠀⠀⠀⣀⠴⠚⢻⡆
⠀⢠⡟⠀⠀⠀⠀⠙⠲⢤⣀⡞⠁⠀⠀⠙⠻⣦⠹⡄⢻⣿⣿⢸⡇⣿⣿⡟⢠⠏⣴⣿⡟⠀⠀⠈⢳⣀⡤⠖⠉⠁⠀⠀⠀⢻⡄
⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⡟⠱⢦⣄⠀⠀⠀⠀⠀⠘⣄⠻⣿⢸⡇⣿⡟⣠⠋⣼⣿⠟⠀⠀⣠⡴⠞⢻⡄⠀⠀⠀⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠘⢀⣩⣤⣤⣌⡀⠃⣼⠟⢁⣠⠔⠋⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸
⣾⠀⠀⠀⠀⠀⠀⠀⢸⠃⣤⣤⣤⣤⣤⣤⣀⡉⠒⢀⣶⣿⣿⣿⣿⣿⣿⣧⡀⠚⠉⠀⠀⠀⠀⢀⣠⣤⠘⡇⢠⣄
⣿⣀⣀⣀⣀⣀⣘⣃⣬⣈⣛⣛⣛⣛⣛⣛⣛⣙⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣀⣀⣀⣀⣈⣛⣛⣛⣁⣵⣈⣉⣃⣀⣀⣀⣀
⣿⠉⠉⣩⣭⣬⣥⡍⢯⢩⣭⣭⣭⣭⣭⣭⣭⣥⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣬⣭⣭⣭⣭⣭⣥⣭⡍⡽⢩⣬⣭⣭⠉⠉⠉
⢻⠀⠀⣿⣿⣿⣿⣇⢸⡄⣿⣿⣿⣿⣿⠿⢛⣡⠄⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⠠⣌⡛⠿⣿⣿⣿⣿⣿⢠⡇⢸⣿⣿⣿
⢸⡆⠀⣿⣿⣿⣿⣿⡈⣧⠹⡿⠟⣋⡤⠖⣉⣤⣾⡟⢠⠈⡙⠛⠛⢋⠉⡄⢻⣷⣤⣉⠲⢤⣙⠻⢿⠏⣼⢀⣿⣿⣿⣿⠀⠀⢠
⠀⣿⡀⢹⣿⣿⣿⣿⠷⠘⣧⠴⢚⣡⣴⣿⣿⣿⠟⣰⠃⠀⣿⢰⡇⢻⣧⠘⣆⠹⣿⣿⣿⣦⣌⡓⠦⣼⠃⠾⣿⣿⣿⡟⠀⢀⣿
⠀⠘⣧⠀⠹⠟⢋⡡⠖⣋⡙⢧⡘⢿⣿⣿⣿⠏⡴⢃⠀⢰⣿⢸⡇⠘⣿⣷⡘⢦⠹⣿⣿⣿⡿⢃⡼⢋⣙⠲⢬⡙⠻⠁⠀⣼⠃
⠀⠀⠹⣧⡴⠊⢡⣴⣾⣿⣿⣦⠙⢤⡙⠿⠋⡼⢁⣿⡄⢸⡟⠈⠃⣤⠘⣿⣷⡌⢧⠙⠿⢋⡠⠋⣴⣿⣿⣷⣦⣌⠓⢦⣼⠏
⠀⠀⠀⠘⣧⡀⠈⠻⣿⣿⣿⣿⣷⣦⡙⢦⣾⡁⠻⠿⠀⢸⣿⢸⡇⣿⡇⠈⠿⠿⢈⣷⡶⢋⣴⣿⣿⣿⣿⣿⣿⠃⢀⡼⠃
⠀⠀⠀⠀⠈⠻⣄⠀⠀⠙⠿⣿⣿⡿⢁⠞⣠⣍⠓⠲⠤⠬⣅⣸⣇⣨⠥⠴⠖⠛⢩⣄⠳⡈⢿⣿⣿⣿⣿⠟⠁⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠉⢡⠏⠐⠛⠉⠀⠀⠀⣶⣦⢸⡇⣴⣶⠀⠀⠀⠀⠈⠀⠱⡌⠻⠛⠋⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣠⠃⠀⠀⠀⠀⠀⠀⢀⣿⣿⢸⡇⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣤⣀⠀⠀⠀⠀⢸⣿⣿⢸⡇⣿⣿⡇⠀⠀⠀⠀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢦⣤⣤⣥⣼⣧⣬⣥⣤⡴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣿⢁⡈⢉⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠐⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⢋⣷⢄⡀
⠀⠀⠀⠀⢀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⢁⣴⠟⠁⠀⠉⠢⡀
⠀⠀⠀⠻⣥⠄⠒⠒⣉⣱⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠉⢀⡴⠋⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠑⢯⣁⡩⠶⣔⠛⢢⣀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣖⠊⣛⣥⡨⠷⢄⠀⣼⣶⣶⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⢉⣼⡿⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠝⠶⣊⠙⣻⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⢁⣴⠟⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠙⠿⣖⣒⡭⢿⠤⠤⢐⣒⡺⡍⠑⡔⢢⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⣠⠶⠋⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⡹⣆⠈⠆⢿⠿⣿⠛⢸⠀⢱⠴⠁⢀⣠⣤⣶⣿⣿⠿⠛⠉⣀⠴⠊⠁⠀⠀⠀⠀⢀⣼⠟⢉
⠀⠀⠀⠀⠀⠀⢈⠀⣀⣀⠤⢄⠀⠀⢡⢻⠀⢸⠘⢃⣉⣶⣼⣤⣾⣶⣿⣿⡿⠟⠛⠉⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣴⡿
⠀⠀⡠⢤⣤⡲⢯⠭⠔⠒⠊⠉⠀⠀⢸⡿⠄⢺⣿⣿⠿⣉⡩⠷⢤⡚⢫⣄⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣴⡿⢛⠇
⠀⠀⠑⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢣⣤⣤⣼⠶⣖⠊⢙⣧⣤⠵⢤⡀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣠⣶⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠁⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠙⢤⡤⠜⠳⣎⣉⡦⣤⠔⠚⠉⠀⢀⣠⣴⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⡼⠶⣉⣉⠹⠤⠘⠓⢤⣴⠾⠛⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⣀⣀⣀⣬⣷⣤⠴⠖⠚⠉⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⠥⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠍⠉⠛⡷⢦⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡪⢟⠋⠀⠈⠑⣶⠀⠀⠀⠈⢦⠙⢮⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡡⠊⠀⠈⠀⠀⠀⠀⡄⠁⢄⠀⠀⠀⢣⡄⠓⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡾⠁⠀⠀⡀⠀⣀⠀⣀⡀⠀⠀⠃⡀⡠⠚⠀⠀⠙⡎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠎⠑⢄⡠⠟⠛⠡⠄⡀⠈⠙⠢⡀⠀⡸⡀⠀⠐⠀⢀⣿⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡎⠀⢠⠎⠀⠀⠀⠀⠀⠀⠑⠢⡀⠙⣴⠀⠐⡀⠀⡷⠉⡇⠙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣸⠄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢜⡄⠀⢁⣬⣧⢤⡃⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣶⠄⢸⠀⡟⠀⢺⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠉⠀⠀⠀⠙⢞⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅⠴⠴⠎⠀⡏⠈⢹⠀⠀⠀⠀⠉⠀⠒⠂⠠⠤⠀⣀⣀⡜⠀⠀⠀⠀⠀⠀⠘⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⠀⡇⠀⢀⣳⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⡧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣇⢈⠀⡀⣙⢢⣤⣤⣤⣀⠒⣋⣹⡤⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⠀⠋⠀⠀⠀⠁⠨⣇⣭⠘⡅⠀⢹⠘⢧⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠⠁⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠸⠀⠀⠀⣀⠔⢿⠘⡀⢣⠀⠀⠙⢈⡱⠚⠙⣆⠀⠀⠀⠀⣠⣏⣾⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠸⡀⢇⠀⠜⠁⠀⠘⢧⡐⣄⣢⡴⠚⠉⠀⠀⢠⠉⠱⠒⠶⣎⡃⠀⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⡤⣤⣤⣤⣧⠘⣷⣴⡶⣦⠴⠛⢻⡟⠁⠀⠀⠀⠀⡐⠁⠠⠁⠀⢄⡏⠙⡛⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⣀⡀⠀⢣⡘⣏⠁⣨⠆⢠⢻⠀⠀⠀⠀⢀⠌⠀⢀⠃⠀⣰⠞⠀⡾⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠉⠳⡘⢧⠀⠀⡄⠀⠳⡀⢀⠔⠁⠀⡠⠂⢀⡴⠛⣤⢾⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⣎⠳⣄⡇⠀⣠⠜⠋⠒⢲⠖⠒⠊⡙⢁⠰⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠅⠑⢬⡳⣎⡀⠀⢀⠔⠉⡄⣀⣬⡤⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠈⠐⠙⠾⠧⠤⠤⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠈⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠈⠣⠄⣀⣀⣀⣀⣀⠠⠂⢀⣀⣰⡒⠢⢄⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⣏⠙⠳⣤⣉⣷⠘⠃⠀⠀⠀⠀⢀⡹⠆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣂⠬⣷⠶⠟⠉⠁⠀⠀⢀⣀⠄⠚⠁⠀⢠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⢀⡠⠤⠚⠉⠀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠤⠄⣀⣀⡠⠤⠒⠈⠀⠀⠀⠀⢀⡠⠔⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠤⠤⣤⣤⠤⠴⠚⠉⠁
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣠⣤⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠶⠤⠞⠁⠀⠕⠫⢹⠧⣡⠉⠁⣤⢙⠚⠛⠉⠛⢦⡀
⠀⠀⠀⢀⣠⠤⠤⣎⣀⢠⢦⡠⡆⠃⠀⢠⡄⠀⠉⡆⠀⠀⠀⠀⡘⠀⠀⠀⠀⢤⠟⠋⠛⠲⢄
⠀⠀⣰⠋⠀⠀⢠⠾⣦⣹⣸⡡⠗⠹⠀⠀⠀⠀⠀⡒⠇⠀⡀⣸⢇⡌⣣⢔⠄⡜⠀⠀⠀⠀⠀⡳⣄⡀
⠀⠀⢧⡀⠛⠀⠲⢍⣽⣿⣽⣿⠭⢬⡗⠀⠀⠀⠀⠇⠀⢔⡓⣾⣶⣷⣷⢒⡾⠂⠀⢠⡄⢀⠎⠀⢠⠟
⠀⠀⠀⠹⣄⠀⠀⠥⠺⢻⡏⡧⡹⡐⠻⠀⠘⠃⠀⠀⠀⠄⢩⡿⣻⡾⣟⡿⡑⠄⠀⠀⡔⠁⢀⡴⠋
⠀⠀⠀⠀⠈⢳⡄⠀⠈⠁⠳⡇⠈⠀⠀⡇⠀⡀⠀⠀⢀⡀⠺⠊⠇⠃⣼⠑⠒⠀⡠⠊⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠙⢦⡘⠃⠀⠱⡀⡠⢸⣹⡌⣱⢺⡇⠈⠁⡇⠀⠉⡜⠉⠀⢀⠜⠁⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠱⣜⣒⣽⡷⣷⣽⣅⠃⢸⠀⠀⡜⠀⠙⡰⠃⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠰⠹⡽⢛⢿⡻⣽⡶⡛⡎⠀⡜⠀⢠⠎⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠻⣹⡸⡧⣸⠉⢰⠱⡞⢀⡔⠁⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⠻⡀⢰⢸⠀⡎⡜⡠⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡱⣌⣮⣴⣽⠝⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⣠⠤⡀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠑⠤⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠲⠛⠉⠉⠉⠉⠉⠻⠓⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢀⠀⡠⠚⡇⠒⠈⠑⠈⠉⡷⠒⠒⠤⠄⡈⢹⠲⢤⡀
⠀⠀⠀⠀⠀⣠⠔⠋⠀⢣⠐⠒⠁⠀⠀⠀⢃⡀⢀⡠⠤⣠⣀⡀⠀⠀⠀⡰⠓⡄⠀⠈⠳⣄
⠀⠀⠀⢀⠞⠁⠀⡐⠀⠃⠱⣀⠤⢠⠊⠈⢹⠀⠀⠀⠀⡙⠀⠈⠊⠑⣴⣁⠀⠀⠈⢂⣀⡈⠳⣄
⠀⠀⡴⠋⠒⣔⠈⠃⠀⠠⠒⠛⢄⠀⠀⠀⢀⠆⠠⡠⠀⠧⣀⣀⠀⡔⠀⠘⠄⢄⠀⠀⣀⢼⡋⠈⢣⡀
⠀⡼⠁⡀⠘⠀⠑⠶⡀⠃⠀⢠⠚⢖⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠀⣀⡀⠀⢘⡲⡎⠁⠀⠨⢄⠀⠳⡀
⡼⢁⠤⠃⠀⠐⠊⠀⠈⢲⢎⠉⠀⠀⢣⠄⠀⠸⠀⠀⡜⠐⢠⠄⠀⠀⠀⣸⢖⠉⠀⠓⠆⠀⠀⠸⣀⠤⢷
⠉⠛⠲⠤⣤⣋⠀⠀⣈⠀⠀⠑⡢⡊⠀⠱⡀⠀⡆⢀⠁⢀⠊⠀⢉⢤⠊⠀⠐⠢⣀⣀⠵⢒⣊⣥⠴⠒⠋
⠀⠀⠀⠀⠀⠉⠙⠛⠦⢤⣀⡐⠁⠈⠂⢄⠐⡄⢠⣨⣠⠃⡀⠔⠁⢀⡱⠤⣒⣨⠥⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣤⣱⣿⡾⠿⣿⡮⣄⣲⡬⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢰⡰⢰⡏⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡶⠛⠉⣙⣄⠀⡉⠹⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠉⡰⢎⠀⠈⡄⠨⢰⡄⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠉⠁⠰⠇⡀⠑⡀⢰⠀⠀⠀⠀⡜⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠠⠄⠀⠠⠄⣆⠐⢄⣡⣌⢀⠜⡠⠊⠀⠀⢀⠞⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠠⣤⠶⠶⠦⣵⠃⠀⠀⠙⣎⠂⡀⠶⢖⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢡⠊⢀⣠⠐⠒⢺⡀⠈⠀⢠⠂⢄⡀⠀⣀⠱⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢐⡰⠁⠀⢠⠒⠁⠌⢹⠹⢃⠈⠢⣀⠀⠀⢀⣃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠣⡀⢠⠁⠀⠘⡠⠃⢆⠢⡣⡀⠀⠀⢀⢏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⢹⠐⠀⢋⠀⠀⠈⠄⠈⠚⠓⢊⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢆⠄⣈⢆⡀⠀⢀⠣⢔⡪⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⢆⠀⠀⡧⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠁⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⢠⢤⣍⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠒⠒⠲⠤⢀⣀⣀⣀⣀⠀⠤⠠⠬⠍⠙⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠁⠉

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⠲⢦⡄⠈⠛⢫⡏⠙⠉⢡⡴⠆⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠙⡇⠀⢀⣸⣧⠴⠶⢾⡷⠦⢤⣼⡃⠀⠀⢸⠃⠘⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠑⠈⢣⡀⠀⠀⢱⡶⠋⠁⣿⠀⠀⢸⡇⠀⠀⣿⠉⠓⢶⣎⠀⠀⠀⡼⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠳⣄⠀⠀⠀⢳⡀⢀⡞⢷⠀⠀⢻⡄⠀⢸⣧⡀⣤⣿⣀⢀⣾⠉⠙⢦⣜⠀⠀⠀⣠⠟⢀⡀
⠀⠀⠀⠀⠀⢀⣦⡁⠀⠀⠈⠧⡀⠀⠀⠱⣾⠁⠘⣧⣤⣼⣷⢶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⡞⠉⠳⣄⠾⠁⠀⠀⢀⣵⠄
⠀⠀⠀⠀⢠⠃⠀⠙⢦⣀⠀⠀⠙⢦⠀⢀⣽⡟⠉⢹⡆⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣣⣶⠻⡆⠀⣀⡴⠋⠀⠘⣄
⠀⠀⢀⡀⠷⣄⠀⠀⠀⠈⠳⣄⠀⣠⠿⣅⠈⣿⣆⠀⢿⡀⢹⣿⣿⣿⣿⣿⣿⣿⠟⣿⡿⣿⣿⣿⣿⣄⣿⠞⠁⠀⠀⠀⣠⠼⢄
⠀⠀⠠⠁⠀⠀⠙⠲⢄⡀⠀⢈⠿⢧⡀⠈⠣⣘⡞⣧⠈⣧⠸⣿⣿⣿⣿⣿⣿⡗⡿⠋⢹⠟⠻⢿⡿⠿⠋⠀⢀⡠⠖⠋⠁⠀⠈⠆⡀
⠀⣌⠷⠤⣄⣀⠀⠀⠀⠈⢳⠯⣀⠀⠙⠦⣄⠙⢿⣌⢷⣹⡇⣿⣿⣿⣿⣿⡏⡼⢁⡴⠋⣠⠔⠋⠀⣀⠽⡞⠉⠀⠀⠀⣀⣠⠤⠾⣁
⢠⡘⠀⠀⠀⠈⢉⡷⠶⣶⣧⡀⠀⠙⠲⣤⡈⠳⣄⡙⣮⣷⠿⠛⠛⠛⠛⠿⣿⣡⢋⡠⠞⢁⡠⠶⠋⠁⢀⣰⣤⠶⠚⠋⠁⠀⠀⠀⢃
⡈⠷⠶⠤⠤⣤⣎⣀⣼⣿⣿⣯⠙⠲⠦⣄⣉⠓⢦⣽⠋⠀⢀⣠⡤⢤⣄⡀⠈⢻⣯⡤⠚⣉⣤⠴⠒⠋⠉⠀⢻⣀⣀⣀⣠⠤⠤⠶⠺
⡦⠀⠀⠀⢀⡏⠀⣙⣿⣿⢿⡿⡖⠲⠤⠤⢌⣉⣾⠃⠀⡰⠋⠀⠀⠀⠈⠹⣆⠀⢻⣿⣉⣡⣤⣤⣶⣖⣚⠋⠹⡏
⣷⠶⠤⠶⢾⠷⠦⣿⣿⣿⣿⣷⣿⣦⣴⢂⣦⣤⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣷⡶⠶⠶⠾⠿⠛⠿⢿⣧⢤⡔⠶⠶⠲⠶⠒
⡇⠀⠀⠀⢸⠀⢀⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠹⣆⡀⠀⠀⣀⣴⠟⢠⣿⣿⣉⡓⠒⠶⠶⢤⣴⣤⣸⣇⡀⠙⣦
⢀⡶⠖⠒⣿⠋⠛⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠛⠛⠋⢁⣴⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣈⣿⠉⠉⠉⠛⡷⠒⠖⢶
⠈⢡⠀⠀⢸⣀⣠⡴⢿⣿⣿⣟⣿⣿⣿⣿⣿⡿⢛⣵⢿⡿⣷⣶⣶⣶⣷⡿⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⠿⠶⣦⣄⣀⣿⠀⠀⡌
⠀⠁⡶⠒⠺⡏⠀⠀⠀⣻⣿⣿⡿⢿⣿⡿⠋⣠⠞⢡⡞⣽⡇⡼⠠⣿⣿⣽⡄⢳⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠉⡟⠒⢶⡈
⠀⠀⠔⡀⠀⢳⣠⠼⠟⠉⠀⣙⣷⡟⠁⣀⡾⠋⢠⣟⣼⡟⣿⡇⢰⡏⣿⣿⢳⡀⢹⣧⡻⣿⣮⢿⣿⣿⣾⡏⠈⠛⠧⣄⣼⠀⢀⠢
⠀⠀⠈⠓⡶⠉⢳⡀⠀⢠⡾⠋⠁⢙⣾⣏⠀⣠⣿⠏⡼⠀⣿⠀⢰⣷⢸⣿⠈⣷⣺⣿⣿⣽⣿⣿⣽⡿⠛⢷⣄⠀⣠⠞⠉⢲⠈
⠀⠀⠀⠀⠁⡄⠀⡹⠾⣃⠀⠀⣠⠞⠁⢈⣿⠿⢁⣰⠃⢀⡿⠀⢸⣏⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣄⠀⢀⡽⠿⣅⠀⢠⠊
⠀⠀⠀⠀⠀⠈⢟⠁⠀⠈⢉⡿⠛⠒⠉⡹⠁⠀⢠⡟⠑⠸⡷⠖⢾⣿⢾⣿⡿⠿⢿⡿⠛⠛⢧⣀⣠⡼⢷⡉⠀⠀⡠⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠂⡴⠋⠀⠀⠀⡼⠁⠀⠀⡾⠀⠀⢸⠃⠀⢸⡟⠀⠙⡿⠲⠴⢷⠤⠒⠛⢫⠀⠀⠀⠙⢦⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣠⡞⠀⠀⠀⣸⠁⠀⠀⣿⠀⠀⢸⡏⠀⠀⣧⠀⠀⠈⣇⠀⠀⠀⢳⡄⡠⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⣠⠇⠀⠀⢠⡯⠀⠀⢸⡇⠀⠀⣿⡀⠀⠀⢸⣄⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⣋⠤⣤⠼⠷⣦⣄⣛⠓⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠦⠶⠖⠒⠒⠶⠶⠴⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⡠⢤⣀⢤⣄⡠⣄⣠⢤⡄⠈⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠁⠀⠀⠀⠀⠓⠚⠙⠛⠋⠓⠛⠙⠚⠃⠀⠀⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣩⣿⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠻⡍⠀⠀⣀⣿⣤⣤⣈⣹⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠉⠓⠶⠶⠖⠚⠁⠀⠀⠀⠀⣠⣾⣏⠀⠀⠀⢳⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⣤⡤⠔⠂⠀⠀⠀⢀⣼⠏⠀⠙⢷⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⠞⠁⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⣾⢧⣠⣴⣶⣦⣽⣷⣜⣿⣿⣿⣿⣿⣿⣿⣡⠞⠁⠀⠀⢀⣠⢿⣆
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⡻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣤⡤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣎⣿⣿⣿⣿⣿⣿⣿⣆⣀⣽⡇
⠀⠀⠀⠈⠉⠉⠓⠦⣄⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⣿⣿⣿⣿⣿⣿⣧⡻⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⡏⠀⣽⡇
⠀⠀⠐⠒⠲⠤⣄⠀⠀⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣀⣹⠿⠿⠿⠛⣻⣿⣿⣶⣿⣾⣿⣿⣿⢿⣿⣿⣿⣿⣇⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⠒⠒⠒⠦⣄⡀⠀⠀⠀⣀⠀⠀⠀⠀⢻⣏⠁⠀⠀⠀⣠⠞⣿⣿⣿⣿⣿⣿⣿⣿⠙⢷⣌⠉⠉⠁⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠻⣧⡀⣠⠞⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠙⢷⣄⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠙⠲⠶⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠘⣿⣿⣿⣿⣿⠿⠟⠛⣆⠀⠀⣠⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⣤⣄⣠⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣾⣅⣀⣀⢀⣿⢀⣀⣀⣼⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣾⣷⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠀⠙⠍⠉⠉⠀⢒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢉⠤⠒⠒⠓⠒⠲⠼⣀⠀⢠⠊⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠁⠔⠁⠀⠀⠀⢀⡠⠤⠐⢛⢳⡁⢐⡶⠒⠒⣂⠄⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⢥⡀⠈⠀⠀⢀⡤⠊⠁⠀⠀⠀⢨⠀⠇⡌⠀⠀⠀⠀⠉⠪⣦⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⠃⠀⡰⠋⠀⠀⠀⠀⢀⡠⠼⡖⡖⠖⠤⠄⡀⠀⠀⠀⠈⢻⡌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⢀⡄⡜⠀⠀⠀⠀⢠⠞⠁⠀⠀⢣⠀⠀⠀⠀⠈⠓⡄⠀⠀⠀⢹⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣴⣿⣿⣿⣦⣤⣀⣀⡔⠁⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢸⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣀⣿⣿⡿⢿⠛⠛⠛⠿⣯⣷⣤⠂⣀⡄⠉⢰⡀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡼⢘⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣹⠟⠁⠀⠆⠀⠀⠀⢀⠃⠙⡇⢰⠃⠀⡰⡋⠉⠢⡀⠀⣀⣠⠠⠤⡗⡲⠋⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡜⠀⠀⠀⠄⠀⠀⠀⢸⠀⠀⠇⢸⣄⢠⡅⠀⡀⠀⡏⠁⠀⠀⠀⣠⡟⠁⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠁⠀⠀⠀⣃⣀⠠⠄⠲⠒⠂⢉⣤⣈⣁⢉⣖⣴⠊⠀⠀⠀⠀⠈⢰⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢺⠒⠂⠉⠁⢃⠀⠀⠀⠀⡄⠀⣾⡟⠀⠉⠀⠈⢹⠛⠒⠀⠀⠀⢤⣌⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⣆⠀⠀⠀⠘⡄⠀⠀⠀⠘⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⢳⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣆⠀⠀⠀⠱⡀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⡇⠀⢀⠴⠃⠀⠀⢸⣧⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠣⡀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⢛⠈⠀⠀⠀⠀⠀⡼⣡⠃
⠀⠀⢀⣀⣤⣤⠤⠤⠤⣄⡀⠀⠱⡀⠈⠲⠤⣀⠈⠢⣄⠐⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⡠⢊⠔⠁
⡴⠟⠋⠉⠉⠉⠉⠛⠲⢤⣿⠀⠀⠈⢦⡀⠀⠀⢈⣿⣇⣹⡲⢦⡀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠔⠁
⢠⡄⠀⠀⠀⠀⠀⠀⢀⣠⠿⣦⠀⠀⠀⠀⠢⣀⢸⣿⣿⣿⡇⠀⠈⠙⠋⠉⠉⠉⣁⠤⠊⠁
⠀⠙⠲⠦⣤⣴⣞⣻⣭⠴⠖⠁⠀⠀⠀⢠⠀⠀⠁⠂⠭⠉⢃⣀⣈⠀⠠⠤⠒⠋
⠀⠀⢰⡇⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⣶⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠶⢒⣛⣷
⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⢀⠠⠐⠑⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⣤⣄⡀⠦⠐⣀⠈⠙⠻⠿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣇⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠉⠉⠉⠉⠑⠊⠉⠉⠉⠉⠐⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠁⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⣀⠠⣎⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢱⠄⣀
⠀⠀⠀⠀⡴⠊⠀⠀⠈⠢⡀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢀⠔⠁⠀⠀⠑⢦
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢡
⠀⡠⠚⠢⢄⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠡⠀⢀⡀⢠⡄⢀⡀⠀⠌⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡠⠔⠓⢄
⡜⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠱⡀⡠⠐⢋⠀⠀⠀⠀⠀⠀⡝⠒⢄⢀⠎⠀⠀⠀⠀⠀⣀⠤⠂⠁
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⢀⠎⠢⡀⠀⢆⠀⠀⠀⠀⡰⠀⢀⠔⠱⡀⠀⠀⡠⠔⠈
⠀⠈⠉⠒⠒⠤⠄⣀⡀⠀⠀⠀⠈⠒⠆⡀⠀⠘⢄⠈⠄⠀⠀⠠⠁⡠⠃⠀⢀⠰⠒⠁⠀⠀⠀⢀⣀⠠⠄⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠼⢀⣀⠑⠢⢄⠑⢜⡄⢠⡣⠊⡠⠔⠊⣀⡀⠦⠄⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠙⢲⠿⠿⠖⠋⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⡄⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿⠇⠀⠸⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⢠⡄⠀⠀⢠⠀⢧⣄⠀⠀⠀⢠⠀⠀⣠⢀⡄⠀⠐⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠉⠧⣷⠄⠙⠣⠈⠓⠀⠀⠀⠀⠀⠀⠐⠀⢁⠾⡤⠸⡿⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠩⠷⣣⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡍⢁⠠⣴⠄
⠀⠀⠀⠀⠀⠀⠄⢠⡆⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠄
⠀⠀⠀⠀⠁⡀⠺⠛⠁⠀⠀⢀⠔⣥⡃⠴⣀⣀⣀⢀⡀⢀⣁⡀⠰⠀⠐⢁⣤⠀⠀⠀⠘⠃⠀⠀⠀⢠
⠀⡀⠃⠀⠀⠁⠀⠀⠀⢰⣾⠟⠆⣀⢴⣺⠋⠁⠁⠀⠀⡞⠉⠻⡗⣦⣐⠟⠁⢀
⠀⣭⣟⠦⠁⠀⠀⠀⠀⣍⢋⠄⡔⠓⠈⠀⢀⠀⢠⠀⢰⠁⠀⠀⠈⠉⠛⠧⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠗
⠀⣀⠈⠀⠆⠀⠀⠀⠠⠁⠀⠀⠈⠂⠁⣄⠈⢇⢠⣇⡹⢠⡄⠀⡠⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⢩⡤⣰⢦
⠐⠐⠦⠄⠀⠀⠀⠠⠤⠄⠠⠀⡀⣀⣤⣪⣧⣿⣻⣿⡕⡀⢙⠻⢄⣠⢤⣤⣀⢀⢥⠄⠚⠀⠀⠈⠽⠷⠿
⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠙⠓⠀⠈⠽⠿⣿⡿⣿⣿⡿⢿⡫⢨⠛⡓⠼⣮⡥⠷⠊⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡤⠤⠒⠁⠁⠨⠉⡁⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣲⢾⣆⣶⣦⣤⣤⣤⣀⣠⠤⢤⣠⠴⠓⠒⠂⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠚⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⢋⣉⣉⡉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⣠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣀⣀⣀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣀⣤⣤⣄⣀⠀⢹⡿⠿⠿⢿⡃⢀⣴⣿⣿⣿⣿⣷⡀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣾⣿⣿⣿⣿⣿⣿⠋⢠⣶⣶⡄⠙⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⠟⢀⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣿⣿⣿⣿⣿⡿⠛⢳⣤⣀⣀⣤⡞⠙⠻⠿⠿⠿⠟⠁⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⠻⠿⠟⠋⠀⠀⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⡀⠀⠉⠛⠛⠛⠋⢁⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⡿⠷⠶⡿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⢀⣤⣶⢶⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⢀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⢀⣴⣿⣾⠟⢻⣼⣿⣽⣿⣿⡦⠀⠁⢰⣿⣉⣷⣾⣽⣿⡄⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣾⣿⣿⣿⣑⣷⣶⣿⣿⠟⠉⠀⠀⠀⢿⣉⣿⡄⣾⣿⣽⡇⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠈⢿⣿⣿⣿⣿⣽⡿
⠀⠀⠈⠙⠛⠻⠿⠿⢳⣾⠞⠁⠀⠀⠀⠀⠀⠈⠻⣽⡇⣿⣴⠟⢀⣤⣴⣶⣦⢤⣀⠀⠀⠈⠙⢿⠟⠁⠀⠀⠀⠀⠂⠀⠀⢹⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠁⠀⣼⣿⣿⢿⣯⣿⣿⣮⣷⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠆⠀⢀⣾⡿⠛⠉⠛⣿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠁⠀⣀⣠⣤⣤⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⠀⠸⣿⡇⠘⣿⢀⣭⡋⠀⠀⠀⠙⣿⣿⡻⠟⣿⣿⣿⠏⠀⠀⣂⡆⡗⠀⢀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣷⣄
⠀⠀⠀⠀⡰⣜⢧⢧⣿⡼⣱⠆⠀⠀⠻⠑⣿⡯⠉⠀⠀⠀⠀⠀⠈⠫⢛⢛⡛⡹⠁⠀⠀⠰⣷⣿⡇⠀⠀⠙⢷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣧
⠀⠀⠀⢬⣙⣪⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⠙⠿⡋⡉⠻⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠆⠀⠀⠀⠀⠀⠀⢹⡶⣄⡀⠀⠀⠀⠀⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣄⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢰⣢⣄⣽⣿⣿⣿⣿⣿⢿⡝⡇⡇⡿⣼⣱⢟⣿⣿⣿⣿⣿⣇⣤⣖⠄⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⡻⢮⡳⣽⣽⣷⣧⣳⡣⣋⡵⣛⣽⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠮⣿⣻⣷⡷⣛⣯⠵⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣾⠛⢻⣿⣿⣿⠿⣛⡛⠛⠉⢛⣛⠿⢿⣿⣿⡿⠛⢻⡎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣶⣾⡟⣫⡄⣿⡰⣾⣿⣿⣶⠸⣿⡦⢩⡻⣿⣶⣾⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠋⢂⣭⣅⡛⢷⡌⣿⣿⣿⠁⡿⢡⣷⣅⠈⢿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠃⢡⣿⣿⣿⣿⣷⡌⢘⡿⠿⠸⣡⣿⣿⣿⡆⠎⢿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⢰⡜⡋⣭⡉⠛⢋⣾⣿⣿⣷⡜⣿⣿⠿⡋⢰⣶⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⢸⡿⢃⣭⣶⣾⡎⣿⣿⣿⣿⢇⣄⣒⣘⣃⣘⢻⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡘⠁⣿⣿⣿⡟⡩⢠⣭⣭⢑⢿⣿⣿⣿⣿⡟⠄⣾⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣷⡀⠙⢿⡟⣰⡇⣿⣿⣿⡌⢶⣍⡛⠿⡛⠐⣼⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡿⣿⣦⡲⠄⣿⣇⢻⣿⣿⣿⡎⢻⡿⠘⣣⣾⠿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⣿⣀⣸⣿⣿⣶⣤⣜⣒⣀⣀⣐⣒⣤⣶⣿⣿⣧⣀⣼⠇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⣭⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⡍⠉⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⣿⣿⣿⣿⣿⣿⡟⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠛⠛⠛⠛⠛⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢐⣂⣉⡈⢁⠀⠀⢁⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠡⠊⠁⠀⠀⢈⠉⡦⡤⠅⣀⣤⠄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡀⠈⠀⠀⡠⠔⠈⠀⠀⢀⢸⠰⠁⠀⠈⠐⠱⣌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠁⢠⠎⠀⠀⠀⢀⠠⠴⠒⠒⠀⠄⡀⠀⠀⠈⢧⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠃⠀⠀⢀⠔⠁⠀⠀⣆⠀⠀⠀⠈⠢⡀⠀⠈⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠿⣷⣶⣇⠀⠀⠀⠾⡃⠀⠀⠀⠀⠀⢡⠀⢀⡟⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣻⠟⠉⠘⠀⠀⠀⡏⠻⠃⡞⠁⡰⠛⠢⡀⠀⢀⡀⠀⡇⡞⠐⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠠⠀⠘⣆⢧⡠⣆⠀⠀⠏⠀⠀⠀⠠⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⣀⣀⠠⢤⠀⠒⠈⡉⠀⣼⡦⠤⠸⠯⣯⠄⠀⠀⠀⠰⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡄⠀⠀⠸⠀⠀⠀⠰⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⡄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⡀⠀⠀⢂⠀⠀⠀⠐⢅⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢸⡰⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢆⡀⠀⠣⡀⠀⠀⠀⠈⠀⠒⠒⠚⠀⠀⠀⠀⢀⢞⠄
⠀⠀⠀⠀⢀⣠⠤⠶⠶⢾⣑⡒⢤⠀⠑⡄⠉⠒⠂⢬⡶⢀⡄⠀⠀⠀⠀⠀⠀⠀⣠⠔⠡⠊
⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⢉⣻⡄⠀⠈⠀⠀⠀⣿⣿⣿⠈⠓⠦⠄⠠⠖⠓⣉⠔
⠀⠀⠀⠀⠀⠓⠦⣤⣤⣶⣾⡯⠵⠋⠀⠀⢀⠀⠉⠉⠭⠛⢂⠀⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉⠉⢻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⢐⢾⡽⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⡀⠀⡀⠌⠙⠻⢿⣿⠇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢇⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠴⠦⣤⡾⠛⠛⢦⣠⠖⠒⢦⣠⡴⠤⣄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢦⡾⠋⠉⠻⣇⠀⠀⣼⠁⠀⠀⠈⣧⠀⠀⣰⠋⠀⠀⠘⡟⠋⠉⠻⡆
⠀⠀⠀⠀⠀⠀⣠⠴⠦⣼⠃⠀⠀⡇⠀⠀⠀⠘⢣⡀⡏⠀⠀⠀⠠⠛⡄⣰⡇⠀⠀⠀⠀⣷⠀⢀⡴⠛⠉⠻⣆
⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠙⢦⡀⣷⠀⠀⠀⠀⠀⠻⡟⠂⠀⡀⠀⠀⢳⡏⠀⠀⣠⠄⠀⣿⡴⠏⠀⠀⠀⢀⡟⠛⠛⣶⡀
⠀⠀⠀⣴⠛⠉⢷⠀⠀⠀⠀⠰⠉⢿⡀⠀⠈⡄⠀⠀⢱⡀⠀⣇⠄⠀⣸⠀⠀⢀⡇⠀⠀⡟⠀⠀⠀⠀⠀⢸⠃⠀⣀⡼⠷⠶⣤
⣠⡶⠶⠷⣤⣀⠈⢇⡀⠀⠀⠀⠀⠀⠻⣄⡀⠸⡀⠀⠈⣇⠀⡏⠀⠀⣸⡄⠀⣼⠀⠐⢺⠃⠐⣤⠀⠀⢀⣏⣴⠋⠁⠀⠀⠀⣽⠇
⣯⠀⠀⠀⠀⠈⠙⠞⢧⡀⠀⠱⡀⢀⠈⢯⣅⡀⢳⠀⢠⢏⠀⢷⠀⠀⣟⢷⠀⡏⣠⣖⠋⠀⣰⠧⠂⢠⡞⠁⠘⢀⠀⠀⠀⣰⠟⠛⢶⡀
⠈⢧⡀⠀⠀⠠⡀⠀⠀⠘⠷⡄⠘⢮⠀⠀⣟⠛⠺⣇⡿⠋⡇⢸⠂⢰⢾⡌⢶⠃⣩⠜⠀⡰⠃⢀⢰⠋⠀⢀⡴⠃⠀⢀⣰⠃⢀⣀⣸⣧⡀
⠀⠀⠳⣎⣁⣀⡈⠑⢄⡀⠀⡟⢳⡌⠳⣄⠘⣆⠀⢸⡃⠈⢻⢸⡀⣸⠀⠁⣾⡾⣷⢀⡼⠃⡰⢻⡞⠀⡴⠿⠆⠀⢀⡼⠛⡟⠉⠁⠀⠀⢹
⠀⠀⠀⠈⠙⣯⣀⠀⡈⠙⠢⣄⠈⠟⠳⣝⢦⣘⣆⡨⠗⠒⠾⡏⣉⡷⠒⠚⠓⣤⠧⢾⣠⠞⠁⠘⡡⠊⠀⣀⣠⠼⠋⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠙⠻⢦⡀⠀⠈⠳⢤⡀⢈⣺⡁⢈⣆⣠⡶⡖⣏⠉⡗⡶⢲⢦⠯⣄⡴⠓⠒⢶⡋⠀⢘⡹⠟⠀⠀⠀⠀⠀⠀⠀⢰⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⡽⠁⣠⡼⢧⡀⢂⠣⣡⢸⠀⠟⣀⠆⡞⢀⠞⢒⣦⣄⠀⣧⡖⠋⠀⣀⡤⠛⠉⠀⢀⣩⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣼⡁⢢⠑⣝⢦⣣⠙⡈⣇⢀⠋⡼⡠⢃⣞⠵⣃⢜⣻⡀⠈⡖⠊⢁⣠⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⣾⣦⡙⢧⠡⢿⡈⣰⠟⡰⢋⡥⣊⡵⢯⣺⡧⠴⠓⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠟⣉⣉⠓⣮⣾⣷⡾⠷⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⣧⣮⠇⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣠⡴⠋
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣒⣮⣭⣭⣍⡉⠉⠉⠑⠲⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⣴⣾⣿⣿⣿⣿⣿⣿⠗⠶⠦⣤⡀⠀⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠙⠦⡀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠔⠁⠈⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢰⠏⠀⠀⠀⠈⠻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠧⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⣶⣶⣿⣿⣿⣶⣶⣶⣾⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣷⣤⣄⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⡸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠦⣤⣤⠴⠚⠁⠀⠉⠻⣿⣿⣿⣿⣿⣿⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⡏⣾⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠙⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠟⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⠻⢿⡋⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢀⡠⠞⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠙⠒⠤⢄⣀⣀⣻⣿⣀⣀⡠⠤⠒⠋⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⠀⠈⠀⢏⡿⠈⠁⠀⣀⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡛⢻⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠊⠉⡀⠀⠉⠑⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠖⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢏⠁⠸⡀⠀⠇⠀⠀⠀⢠⠀⢀⠆⠀⡨⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢣⢀⣈⣆⠀⠱⣀⣨⣤⣬⣤⣼⣀⡘⠀⡰⠁⢠⡊⢈⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⢴⢿⣶⣝⣻⢶⡊⢫⠈⡄⢸⠀⠇⠠⠉⣹⣪⣕⢿⢟⣿⡶⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠠⣻⣿⣧⢩⡟⣕⣿⣕⡀⢆⢡⢸⢐⠀⡓⡿⡫⢋⣼⡳⣟⢟⣽⡿⣆⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⣺⠿⣾⣰⣯⢿⢯⢷⡱⡜⠌⠄⣘⣀⠘⢜⢔⡵⣻⢯⡼⡾⡯⣡⣽⣾⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⢺⣾⣗⢇⣏⣿⣛⡒⠬⢪⣶⣫⣽⣯⣝⣶⡅⡪⠕⠻⢿⢻⣽⣿⣥⣶⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠲⢼⢩⣥⣝⣒⣂⡀⣿⣾⣿⣿⣿⣿⣷⣿⠈⠥⠭⠙⠙⠓⡗⠛⠋⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⢹⣁⡀⠶⠒⢀⡀⢿⡹⣿⣿⣿⣿⢟⡿⡐⠩⠍⣁⠀⢒⠗⠒⠂⠐⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠒⠈⠉⠉⣧⠤⠒⢈⠥⢊⢄⠙⠾⠯⠽⠷⢋⢠⠢⢍⠒⠤⣈⡹⠁⠒⠤⠄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠔⠊⠁⣈⢦⠊⣠⠖⡡⢊⠌⡌⠆⠀⢆⢆⢣⡑⢄⠑⠢⣼⠉⠒⠤⣀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠔⠊⠀⡠⠛⢅⡔⣥⢪⣎⣼⢰⣸⣜⣾⣿⣷⢄⡱⢎⠀⠑⠤⡀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡠⠊⠀⡠⢊⣟⢾⢯⣼⣇⣻⣻⣛⣼⠷⡞⠉⢆⠀⠑⢄⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡔⠁⢀⢷⣿⢏⣾⢾⣾⡏⣾⡶⢿⠞⡄⠀⠣⣀⡤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢮⣀⣉⠺⠿⡆⣿⡇⠿⠟⢇⢀⣸⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢳⣷⣖⣾⣶⡞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣽⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣯⣿⣿⣛⣿⣿⣿⣿⣻⣿⣻⣿⣿⣽⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣰⢲⡄⠀⠀⢰⣟⡇⠀⠀⢠⡶⡆⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢿⣾⣇⠀⠀⣸⣿⣄⠀⠀⣠⣿⣇⠀⢀⣸⣿⡁⠀⠀⢸⣥⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢻⣶⣇⣀⣀⣼⠛⡍⠛⠛⠁⢸⠈⠛⠛⠋⣧⠈⠙⠋⢹⠂⠙⠒⠚⣻⢿⡀⠀⢀⣸⣽⠆
⠀⠀⠀⠀⠀⠀⢯⣾⣧⣀⣀⡴⠋⢏⠉⠉⠀⠀⢧⠀⠀⠀⠸⠀⠀⠀⠀⣿⠀⡤⢤⣸⢠⠠⣀⣄⡇⠈⠙⠛⢋⠿⣇⠀⠀⢀⡶⣆
⠀⠀⠀⣀⣀⣀⣠⠟⠹⡍⠉⣀⣀⠘⡆⠠⠶⢄⠸⡀⠀⢀⡤⣇⠀⠀⠀⡇⠀⢧⠠⡟⢦⠎⣡⣼⡙⡆⠀⢀⡜⠀⠙⠓⠒⢻⣿⠁⠀⠀⣀⣀
⠀⣠⠞⠉⠫⡉⠁⠀⠀⢸⣎⠠⠦⢱⠘⡜⢇⣘⡆⢧⠀⢸⠀⡏⡇⠀⠀⡇⠀⠈⢳⠯⠾⡄⢧⣃⣡⠇⠀⡜⢀⠀⠀⠀⡰⠁⠻⣤⣤⣤⣿⠟
⢾⡁⠀⠀⠀⠙⢄⠀⠀⢸⡈⢆⠒⠊⢀⣱⡤⠿⣧⡸⡠⣌⣢⣷⠼⢄⣀⡇⠀⠀⢸⢀⠤⢻⣎⣀⠀⠀⡼⡅⣜⢳⢀⡜⠀⢠⠀⠀⡠⠊⣿
⠀⠙⣦⡀⠀⠀⠈⠣⡀⠀⠙⠚⢷⠪⠉⡇⢷⠒⡄⡉⢛⠉⠁⢸⠀⣔⠈⡷⡀⠀⡆⠈⠉⢹⠁⣉⠵⡾⠓⠲⠾⢾⠏⢀⠖⠉⢉⠟⠅⠀⠘⢦⣤⣤⣾
⠀⠀⠈⠻⣄⠀⠀⠀⠘⢆⠀⠀⠀⢣⡁⠈⠒⢻⠉⠀⠸⡀⣀⡼⠗⠶⣿⡗⠓⣤⡥⠤⠶⠟⡟⡡⠴⢅⠀⠀⡰⠃⠱⡌⢀⣴⣉⣱⢰⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠳⡄⠀⠀⠱⡀⠀⠈⢆⢰⣂⣿⠁⠀⡇⠧⠋⡇⠈⢸⠀⢚⡼⢰⡁⣵⡻⢸⠀⡔⡥⠤⢤⣸⡏⠘⢌⣡⠞⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⢦⠀⠀⠙⡄⠀⠈⣆⢸⠸⣋⡇⢃⠀⠀⠁⠀⡉⠀⢠⠇⣏⣷⠧⠔⢁⠞⠀⠙⣤⡞⢠⣧⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠑⣄⠀⠘⢆⠀⠹⡄⠑⢟⠁⢸⠀⢀⠂⢀⠇⠀⡜⠀⢡⠃⠀⣠⠋⠀⢀⠔⠁⠉⠉⣽⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢢⡀⠈⢆⠀⠱⡀⠸⡀⠘⡀⢸⠀⢸⠀⢰⠁⢀⠏⠀⡴⠁⠀⡰⠋⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠑⢄⠈⢣⠀⢡⠀⢇⠀⡇⢸⠀⡚⠀⡎⢀⠎⢀⠜⠀⣠⠊⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠣⡀⠳⡀⢣⠸⡀⡇⢸⠀⡇⣸⢀⠎⢠⠊⢀⠞⠁⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⢄⠱⡌⢆⢇⢰⢸⢸⢀⠃⡜⡰⢃⡴⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠳⡘⡜⣞⡸⢸⣌⡘⣜⡜⡡⠊⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⢞⣾⣷⣿⣷⣿⢯⢞⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠹⣿⣿⣿⣿⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⣠⠤⡈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠑⠒⠃⡼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⡶
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣟⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣦⣄⡀⡀⠀⠀⣠⣶⣿⠿⣷⣦⣄⠀⢠⣴⣶⡿⢿⣶⣦
⠀⠀⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⣿⣇⢹⡇⠀⠀⠀⣀⣤⣤⣸⣿⡿⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⠿⠿⠿⠿⠿⢿⣤⣿⣛⠿⠿⠿⠿⠛
⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠁⣿⡏⢹⣿⠈⠁⠀⠀⠀⠹⣿⡿⠟⣿⣿⣸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠏
⠀⠛⠛⠿⣿⡿⠛⠛⠃⠀⠀⠀⠀⠀⠘⡿⠁⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡅⠀⠀⠀⣤⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛⠂

⠀⠀⠀⠰⣀⣀⢰⣂⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⣤⡄⠀⢀⣀⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠫⣿⡯⠋⣠⣤⡝⠆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢘⣶⣿⣿⠛⣿⣿⡛⡄
⠀⠀⠀⠈⢻⣿⣅⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⠇⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⡿⠽⣿⣿⣏⣀⠿⣿⠃
⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠰⣜⠿⠟⢁⣾⣿⣦⣄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢈⣿⣧⣞⣙⠿⣿⣿⠖⠁
⠠⠮⠀⠀⠉⠉⠭⠍⠉⠉⠀⠀⠥⠄⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠴⠿⠟⠂⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗

⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡀
⠀⠀⠀⣍⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣯⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠘⠳⣦⠄⢠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡷⠘⣿⣿⠷⣟⢀⡤⣾⡇⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠈⠑⣿⣿⢿⣿⣿⣿
⠀⣤⣾⣿⣻⣿⣦⡄⠀⠀⠀⠀⣰⣿⣿⣯⡻⠿⠤⠼⢟⣶⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠯⣰⣤⣀
⠀⣿⠋⢸⣿⠈⢻⡇⠀⠀⠀⠀⠙⢿⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠇⠀⠀⠀⠀⠀⠘⠛⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠃⠀⠀⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣷⣦⣤⡀
⢀⣾⣿⣿⣿⣹⣿⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⢿⣶⣤
⠸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣀⣴⣹⣛⡟⢻⣛⣏⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢧⠀⠈⠉⠙
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⢻⣽⣽⢿⣿⣿⠄⠀⠀⠀⠀⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⡸⠠⡄⢆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⣬⡉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠉⢹⡏⠉⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠾⠛⠛⠛⠛⠛⠛⠷⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⣀⣠⣤⡶⢿⠿⠟⠛⠻⠷⢶⣄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⣤⡶⠟⠋⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⣠⡾⠋⢑⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⡰⣀⣤⣤⣤⣤⣤⣀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⣴⠏⠀⣴⡿⠉⠑⢄⠀⠀⠀⠀⠀⠀⣀⣤⠼⡗⠛⢻⡿⠉⠛⠛⠲⠶⢤⣴⡿⠛⠉⠀⠀⠀⠈⠙⠻⣦⣀⠀⠹⣦
⠀⠀⢀⣾⠃⢀⣾⠋⠀⠀⠀⠀⠑⢄⠀⣠⠴⠛⠁⠀⠀⢰⠀⣾⡇⠀⠀⠀⠀⢀⣾⠏⠙⠦⣄⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠼⣷⡀
⠀⠀⣾⠃⠀⣾⠃⠀⠀⠀⠀⠀⠀⣠⠟⢅⠀⠀⠀⠀⠀⠀⠆⣿⡇⠀⠀⠀⢀⣾⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡠⠔⠻⣷⠀⠘⣷⡀
⠀⣼⠇⠀⢰⣿⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠑⢄⠀⠀⠀⠀⠰⢻⡇⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠘⣦⡤⠂⠁⠀⠀⠀⢹⣇⠀⠸⣧
⢠⡿⠠⠄⢸⣿⡀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠑⢄⢀⣤⡶⠿⠛⠛⠷⢾⣿⡀⠀⠀⠀⢀⡠⠔⠊⠈⢧⠀⠀⠀⠀⠀⠀⣿⡀⠀⢻
⣸⡇⠀⠀⠀⠻⣯⡉⠃⠒⢶⠧⢀⣀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡤⠚⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢹⡇⠀⠸
⣿⠀⠀⠀⠀⠀⠙⠿⣦⣄⣿⠀⠀⠀⠈⠁⠒⠂⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠛⠛⠛⠿⠿⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⢼⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣤⣀⡀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣼⡇
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣀⢤⣾⡿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠑⢤⣉⠉⠛⠿⣦⣤⡏⠀⠀⠀⠀⢀⣿⠁⠀⢰
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡠⠔⢊⣴⠿⠋⠀⢈⡿⠷⢶⣤⣤⡶⠾⣿⡇⠀⠀⠀⠀⠑⠢⢄⢈⣿⣧⡀⠀⠀⠀⣼⠏⠀⠀⣼
⠀⢻⡆⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣶⠾⠛⠁⠀⠀⢀⡞⠀⠀⠀⠈⡇⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⢠⡟⠳⢽⣷⣤⣠⣾⠋⠀⠀⢰⡟
⠀⠀⢿⡄⠀⠀⣀⣿⡋⠁⠀⠀⠀⠙⢦⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣆⠀⠀⢀⡴⠋⠀⠀⠀⠈⠙⠫⣀⠀⠀⢠⡿⠁
⠀⠀⠈⢿⡖⠋⠀⠸⣧⠀⠀⠀⠀⠀⠀⠙⠲⣤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡿⠁
⠀⠀⠀⠀⠻⣆⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⡠⠋⠉⠓⠶⠤⣤⣄⣠⣄⣠⣤⠤⠶⠚⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢷⣄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠈⢣⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⡜⠀⠙⠻⠷⣦⣤⣀⣀⠀⠀⠀⡀⣀⣀⣤⡾⠛⠁⠀⠀⠀⠀⢣⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠉⠉⠛⣿⠛⠻⠟⠛⣿⠁⠀⠀⠀⢀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⢶⣿⣤⣤⣤⣤⣿⡶⠶⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⢩⡿⠁⠸⣯⡇⢿⡿⠸⣽⠇⠈⢿⡍⠙⠛⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⣦⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠾⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⢤⣤⡼⡟⠓⣶⠋⣿⠉⠙⡖⠋⡟⠙⣦⠴⢲⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠛⢺⡇⢿⠀⢸⡆⣧⠀⡏⠀⡟⠀⢸⠃⢰⠃⢠⡏⢠⡏⠀⣼⠋⣩⠟⢦
⠀⠀⠀⠀⠀⢠⣿⡛⣧⠘⣇⠀⣧⢸⡄⢸⡇⢿⠀⡇⢀⡇⠀⡾⠀⡟⠀⡾⢀⡞⠀⡼⠃⣰⠋⢀⡼⠛⣻⠷⡄
⠀⠀⠀⠀⠀⢸⡈⢷⠘⣆⠹⡄⢹⡈⡇⠀⡇⢸⠀⡇⢸⠇⢠⡇⢸⠃⣸⢁⡾⠀⣼⠃⡼⠁⣠⠞⢀⡼⠁⢀⡽⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠻⡌⢧⠹⡄⢳⡈⣇⠻⠀⣷⢠⢀⡇⣸⠀⣸⠁⡟⢰⠇⣼⠁⡼⢁⡾⠁⡴⠋⣰⠋⢀⡴⠋⢀⣠⠞⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠹⡌⣧⢻⡈⣧⢹⣰⡆⢻⠸⢸⡇⡄⠀⡟⢈⢁⡟⠰⠃⡼⢁⡞⢀⡞⢁⡞⠁⣴⠏⢀⡴⠋⠁⠀⣠⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣄⢧⢨⡌⣇⠁⢸⠀⢸⡇⠁⢸⠃⠋⡼⠱⠃⡼⢁⠋⣰⠋⣰⠋⣠⠞⢀⡴⠋⠀⣀⡴⠋⠁⠀⣈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠊⣇⠁⢹⡀⢸⡆⢸⡇⠀⣾⠀⢰⠃⢀⡼⠁⢋⡼⣡⠌⣡⠞⢁⡴⠋⢀⡤⠞⠁⠀⣠⡴⠚⠉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠸⣆⠘⡇⠈⡇⢸⡇⢠⡇⢠⡏⢀⡾⠁⣠⠏⠀⣠⠞⢁⡄⠉⣠⠖⠋⢀⣠⠶⠋⠁⢀⣠⡴⠞⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⢻⠀⡇⢸⠀⣸⠁⡾⢀⡞⢁⡴⠃⣠⠞⠁⠀⢀⡴⠋⢁⡤⠖⠋⢀⣠⡴⠞⠋⠁⣀⣠⡤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢳⡘⡇⢷⢸⠀⡏⣸⢃⡞⢠⠞⢀⡾⠁⢀⣤⠞⠉⠰⠚⢀⣠⠴⠚⠉⣁⣠⠴⠒⠋⠉⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣧⢻⢸⣼⢸⢣⢇⡞⡴⢋⡴⠋⣠⠖⠋⠀⢀⣠⠴⢚⣉⠤⠴⠚⢋⣁⣠⠤⠖⠚⠋⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣾⣿⣿⣞⡵⣋⡴⠛⢁⣠⠴⠚⠉⠀⣀⣩⡤⠶⠒⠛⢉⣁⣀⣤⡤⠴⠶⠒⠚⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣯⢞⣡⠴⢚⣉⣤⠤⠖⠚⠋⠉⠐⠒⢐⣛⣉⣉⣭⡤⠤⠤⠶⠖⠒⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣒⣋⣭⠥⠤⠶⠶⠒⠚⠛⣛⣉⢉⣉⣁⣀⣀⣀⣠⣤⣤⣤⠤⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⡛⠿⣿⣿⣿⣟⣲⠶⠶⠶⠶⠦⠤⠤⠤⠤⡤⣤⣬⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠉⠓⢶⣬⣙⠻⠯⣍⣛⠒⠶⠤⣤⣄⣀⡀⠒⠂⠤⠤⢤⣤⣀⣀⣀⣀⡈⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠈⠛⠲⢤⣉⠙⠲⠦⣄⡀⠉⠭⡝⠓⠲⠦⠤⣄⣀⣀⠉⠉⠉⠙⠛⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣼⡯⢻⣶⢤⣌⡙⠒⠦⣤⣈⡉⠉⠛⠒⠶⢤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠯⣝⡲⢭⣗⡶⢤⣈⠉⠛⠲⠦⣤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢭⣓⠺⢽⣓⡦⢤⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠮⣍⣓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁