MARIO Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠉⠉⠉⠉⠉⠉⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⠢⢄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠉⠙⠛⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢠⠞⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠉⠁⠸⠀⠘⢁⡛⢻⣏⠁⠀⠀⠀⠈⣠⣿⣿⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⡠⠁
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡜⠁⠀⠀⢾⣷⣾⣿⠀⠀⠀⠀⠰⠛⢋⣴⠉⢉⠡⠈⠙⠁⠀⠀⠀⠀⠘⠛⠤⢄⣠⠾⢀⡀
⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡑⢿⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠆
⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⣣⣄⠀⠀⠀⡀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠈⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠉⠀⠀⠰
⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣸⣯⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡜⠁⠈⠀⠀⢰⠇
⣼⠇⠀⠀⠀⠀⠀⠈⠉⠶⠶⠃⡘⠁⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣃⣀⣇⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⠀⠀⠀⣠⠜⠁
⢣⡄⠲⣄⠀⠀⠀⠀⠀⢀⣴⡏⠉⠉⠙⢄⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⠟⠒⠒⠊⠁
⠈⡟⠢⢀⣉⠉⠉⣉⡙⠋⠉⡇⠀⠀⠀⠈⠣⡌⠙⠁⣤⣄⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⠎
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠦⡀⠈⠛⠛⠛⠻⠿⠿⠿⠟⠁⠈⠙⠋⠁⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠑⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠜⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⡾⠿⠽⣆⡀
⠀⠀⠀⠀⠉⠁⠒⠂⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢀⣤⡀⣀⣠⣀⠔⠂⠉⠀⠀⠀⠀⠀⠈⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠐⠈⢣⡀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠏⠀⠀⠀⠈⠓⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠈⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠱⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⢠⡿⠁⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠀⠒⠈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠉⠀⠴⠒⠶⠶⠆⠘⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠶⠤⢀⣀⣀⣀⣀⣀⠠⠤⠞⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠁⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢲⣆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠉⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡍⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡁⠀⠂⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴⠄⠀⣀⡀⠀⠴⠊⣹⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣀⣀⣀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠋⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠘⠠⠤⠤⠤⠤⠤⠀⠛⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢀⣤⣤⣀⣀⣠⡀⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢁⢔⣲⣄⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣼⣿⣘⣽⣦⣄⠀⠀⠈⠑⢦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣊⢍⠛⠉⠉⣹⣿⡟⠻⣿⢶⡦⢄⡀⠙⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠊⢰⡟⢻⠖⠁⠀⣿⡔⠉⢛⣄⠈⠑⠪⢕⡦⣀⠀⢀⡿⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠙⢦⠈⢧⡇⠈⣦⣄⠀⠁⠀⢸⣿⡄⠀⠀⢠⣿⣮⣿⡏⠀⣤⠄⢳
⠀⠀⠀⠀⠀⣠⡟⠁⠈⠲⣤⣀⣈⡓⣤⡷⠚⠋⠉⠀⠐⣄⠀⣉⡇⣰⣦⣿⣿⣿⣿⠀⠀⠫⢀⣇⣠⣤⠤⠶⠶⢤⣄⡀
⠀⠀⠀⠀⣰⠋⠘⠛⠲⢦⠞⠁⠀⢹⠏⠀⠀⠀⠀⠀⠀⢈⣭⣭⣶⣿⡿⠀⠉⠛⠁⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢀⣙⣦⣀
⠀⠀⠀⠀⢹⡄⢀⡤⠤⡎⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠈⢳⡄
⠀⠀⠀⠀⢺⡀⠀⠀⠀⡇⠀⠀⢸⠿⣷⣤⣀⣀⣠⣴⣿⣿⡿⠛⠏⢹⢦⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣠⡤⠖⠀⠀⢙⠳⣄
⠀⠀⠀⠀⠈⠳⡄⠀⠀⠙⢲⡖⠋⡴⠛⣿⣿⠛⠻⠟⠋⠁⣠⠊⡰⠃⣸⡆⣧⡴⣏⠀⠀⠀⠀⢀⣤⠞⢻⣇⠀⠀⠀⠀⢀⡄⢦⠘⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢧⣈⣲⡶⠋⠠⠚⢀⡴⠃⠈⠳⣄⡀⢀⣦⡈⠙⠷⢤⠞⣹⠉⠀⢸⡀⠀⠀⠀⢿⠘⡇⢸⠇⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠦⡤⢴⡋⠀⠀⠀⠀⢸⠋⠉⢻⠙⠳⠶⠶⠞⠁⡇⠀⠀⢳⠀⠀⠀⢸⠀⣿⠸⣦⣧⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⠀⠀⠘⡆⠀⠀⠀⠄⢰⡇⣠⠄⣌⢧⡄⠠⠆⠀⢹⠀⣾⠁⠀⣤⠀⠀⠀⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣼⡇⡠⠀⢌⢾⡁⠀⠀⢸⡇⡏⠀⣼⡼⡆⠀⠀⠀⣸⣤⣽⣤⣤⣟⡳⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣱⠀⡎⣇⢳⠀⠀⠘⣇⠳⠤⠟⢰⡇⠀⠀⠀⣿⠤⠤⠀⢀⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣖⠚⠻⢯⡉⠉⠙⠳⢄⡀⠘⠶⠶⠶⠋⠀⠀⠀⣸⡷⠤⣄⠀⠀⠈⠓⢤⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠤⠤⠤⣈⠉⠙⠲⢤⡈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⣷⠀⠀⠀⠈⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠂⠀⠐⠂⠬⡁⠢⢄⠙⢦⡀⢲⡀⢀⡇⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⢠⠏⠀⠀⣀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡀⠀⠀⠀⠀⠁⠠⡙⢦⠙⣾⣃⣾⣤⣤⣴⣾⣿⣁⣀⣀⣠⡤⠞⠋⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⢤⣄⣠⠃⠈⠢⠱⣸⡟⠉⠁⠐⠒⠒⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠍⠉⠓⢦⣤⣥⣿⣃⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠒⠒⠀⠒⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣠⣶⡿⢿⣦⡀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠻⡿⠿⠟⠛⠛⠛⠒⠚⠛⢻⠟⠉⠉⠛⠻⣷⡄
⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢀⣿⡇⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀
⠀⠀⠀⠀⢸⣿⡀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢀⡁⠀⠉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⢿⣧⠀⠈⢂⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠸⡀⠁⠀⠀⠀⢀⡊⠀⠁⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠉⠢⡀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣭⣀⣀⣤⡐⠃⠠⠤⣠⠀⠀⠀⠀⠀⠹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢷⣤⡀⠀⠈⠐⠠⡀⠀⠀⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠀⠀⠈⢻⢿⣿⣿⣿⣿⠟⣡⡄⠀⠀⠀⣉⠓⠆⠀⠐⠘⠧⡀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠐⠁⠀⣻⣿⣿⠿⠐⠛⠣⡀⠀⢰⣿⡇⠀⠀⠀⠄⠀⠀⣀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⢄⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠤⠀⡐⠁⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢄⣀⠀⣀⠌⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⣠⣀⠀⠀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠃⠀⠆⠀⡀⠜⠙⢿⣿⣿⣷⣶⣾⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⣀⣠⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣰⣶⣦⡀⠀⠀⠀⠻⣿⣏⠙⠟⠿⠋⡙⠋⠁⠀⠀⠀⠀⡠⠂⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢀⣿⡏⠙⢿⣶⣄⠀⠀⠹⣿⠷⠨⣐⠀⠀⠀⠀⠀⣀⣤⣾⣶⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⣼⡿⠀⠀⠀⠈⠻⣿⡦⢄⣰⠀⠀⠀⠈⠉⠉⠉⠁⠈⠻⣿⣶⣍⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠃⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⡬⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢦⠹⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣿⣦⣠⣄⣀⣀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⠛⠷⠤⠤⢀⡀⠀⠀⠀⠀⢀⠤⠊⠠⠋⠙⠻⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⡈⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣌⡐⠂⠤⠀⠀⠀⠠⢔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣂⣀⣀⣀⣀⣀⣤⣴⣾⠿⠃⠈⠙⠻⠿⢷⣶⣶⣶⠾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡤⠶⠒⠲⢦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠋⠀⠀⠄⡈⠀⢻⡄⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠸⣄⣄⣀⡀⣀⣠⡼⠁⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⠤⠖⠊⠉⠉⠉⠁⠀⠈⠉⠉⠐⠢⢤⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣞⠁⡀⢀⣀⣀⡀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠳⠄⢠⠇
⠀⢀⣀⠀⠀⢀⡠⡤⢤⣀⡀⢀⡔⠋⣿⡏⠉⠋⢉⣹⣧⠀⠀⠀⠀⣴⢿⣛⠻⠉⠛⡛⣳⠯⣄
⣞⡉⠉⠉⠓⠀⠢⠉⠀⣩⠟⠹⡇⠈⣇⢀⠀⠀⡏⢠⣘⡇⠀⠀⢠⣇⡀⠘⡆⠀⡰⠀⣶⠀⢸
⠈⠉⠙⢢⡀⠀⠀⢸⠀⡇⠀⠀⠹⡄⣅⠇⠀⠀⢧⠸⠿⡇⠀⠀⠘⣿⠃⢠⠁⠀⠓⠄⡿⢀⡟
⠀⠀⠀⠀⢳⠀⠀⠘⡄⠣⡀⠀⠀⠈⣯⠀⣿⣦⣄⣱⠛⠉⠀⠈⠙⠻⣟⣁⣤⣶⠀⢺⠖⠋
⠀⠀⠀⣠⠞⠀⢀⣀⣨⠶⠞⠋⠉⠑⢿⡄⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⣽⣿⡿⠋⢀⡞
⠀⠀⠀⠣⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣹⢦⡀⠙⠛⠢⣄⣀⣀⣤⠞⡿⠿⢦⣠⣾⣙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡏⠈⠓⣤⡀⠀⠀⠀⢀⣼⡀⠀⣀⡭⠀⠋⠉⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢗⡀⡀⢸⡇⢉⣛⣒⣋⡥⠞⠉⣹⠋⠀⢀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⡏⠀⢸⣸⢠⡏⠉⠉⠁⠀⠀⠀⡇⠀⢀⠎⠀⢀⡔⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠈⠙⢶⡛⠁⠈⠳⠤⠤⠖⠒⢧⣄⣙⠒⠚⣄⣠⡞⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⡼⢃⢔⣊⠫⠭⠑⠒⢤⡀⢿⠳⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⣟⠙⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠁⠁⠀⠀⠀⠀⠀⠰⣵⣸⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⡄
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢸⡀⡟⠛⠓⠚⠛⠙⠲⣤⣀⠀⠀⠀⠀⢀⣠⠟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣐⣂⣀⣀⢀⣷⠘⡷⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠚⠫⠥⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠔⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⠛⠛⠋⠛⠻⢿⣷⡄⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⢀⡴⣏⠉⡹⡓⢦⣻⣿⡄⢀⣠⣴⣿⠟⠿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⡞⢠⡅⡘⢡⡘⠄⠻⢿⣿⣿⣿⣛⡛⠻⣶⡌⠛⠛⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⢣⠘⢇⣉⣥⣶⡾⠿⠛⠛⠛⠉⠛⠛⠻⢿⣧⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⢀⣤⡷⠟⠛⠉⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣄⣀⣤⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⢀⣠⡶⠟⢋⢻⣿⣿⡿⠿⠛⠁⠀⠉⠉⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⣀⣤⡾⠛⠁⠀⢰⣿⣧⢻⣧⣀⣀⡀⠀⣀⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠻⠀⠀⠀⣠⣿⣿⣤⣤⠶⠞⠛⠉⣿⣿⡆⠀⠀⠈⣿⡿⠛⠛⠛⠛⢿⣷⣦⣠⣾⣿⠏
⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⣴⡿⠋⢻⣆⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⢿⣿⡇⢀⣴⡶⠿⠛⢿⣧⡀⠈⠻⢷⣄⠀⠀⣄⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃⠀⠠⣄⠀⠻⣷⣤⣀⣾⠟⠀⢸⣟⠻⣷⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⢨⣿⣏⠀⠀⠀⠸⢿⡄⠈⠛⠿⠛⠀⠀⠸⣿⣄⡀⠈⠉⠉⠉⠻⣶⣶⣦⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⠀⠀⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣀⣀⣠⡀⠀⢀⣀⣀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣤⣤⣶⣦⡀⠀⠀⠀⠀⠀⢸⡿⠛⣿⢻⣿⣾⣿⠟⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠈⢿⣄⣼⣿⣿⡿⣫⣾⡟
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⡿⠟⠛⠉⠉⠁⠀⢀⣽⡿⢿⣶⣶⣤⣤⣿⠿⢿⣿⣿⣿⡏⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⣰⣿⡿⠿⠿⠿⢶⣤⡀⠀⣀⣴⠾⠟⠋⠀⢸⣿⠀⠀⠙⣷⣤⣤⣿⡧⠙⣿⣿⣶⣾⣿⠟⠉⠉⠉⠻⣿⣷⣤⡀
⠀⠀⣠⣶⣿⠟⠓⠀⠀⠀⠀⠈⠻⣿⣿⠁⠀⠀⠀⠀⣸⡏⢀⣶⠿⠻⢿⣅⠀⠀⢸⣏⠙⣿⣿⠇⠀⠀⠀⠀⠀⢸⡿⣿⣷
⠀⣰⣿⡟⢀⣴⠀⠀⡀⠀⠀⠈⢻⣿⣧⣀⠀⠀⢀⣰⡟⠀⣿⡏⠀⠀⠀⢻⡆⠀⠘⣿⣆⣸⣿⡆⠀⠀⠀⠀⠀⣸⠃⣿⣿
⠀⣿⣿⠀⢾⠟⢀⣾⡇⠀⠀⠀⠀⢿⡟⠛⠿⠟⠛⠉⠀⠀⠹⣷⣄⣀⣤⡿⠁⠀⠀⠈⠻⣿⠟⠀⠀⠀⠀⠀⢠⠏⣸⣿⡏
⠀⣿⣿⡀⠀⠀⠈⠋⠀⠀⣠⡖⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⢀⢟⣾⣿⡟
⠀⠘⣿⣷⣄⠀⠀⠀⠀⢰⡿⣿⡿⠛⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⠀⠀⠀⠀⣠⢏⣾⣿⠏
⠀⠀⠈⠻⢿⣷⣶⣤⣤⣶⡾⠟⠀⠀⠀⣾⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⣿⣷⣦⣴⣾⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⣹⣿⢏⡇⠀⠀⠀⠀⢸⡇⠀⣿⣿⣶⣤⣤⣤⣤⣤⣤⣶⣾⡿⠟⠋⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⢠⣿⡟⢸⠁⠀⠀⠀⠀⠸⠁⠀⣿⣿⠉⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢸⣿⠇⣿⠀⠀⠀⠀⠀⠀⢠⣼⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣇⣿⡀⠀⠀⠀⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣀⠀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢁⡴⠚⠉⠉⠙⠲⣄⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⡞⢀⣀⣀⠀⠀⠀⠸⡆⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⡴⠞⠛⠉⠉⠉⠉⠛⠛⠶⣧⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡞⢡⣾⠿⣿⡟⠛⠛⠛⣲⣶⣿⣶⡝⢷⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣴⠞⣡⠖⠲⡅⠀⠀⢀⡯⠶⢌⠙⣷⣄⢹⡆⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⣿⠀⡇⠀⣴⣿⣀⠀⡞⠀⣠⣼⡇⠀⠙⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣽⡞⢀⡷⠋⠉⠉⠉⠙⠳⡄⣿⣿⡇⠀⢀⣠⠞⣇⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⣿⣷⣾⡁⠀⠀⠀⠀⠀⠀⢹⠞⠋⣀⣴⣜⡇⢰⠏⣉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠋⠓⢣⡾⠁
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠉⠉⢿⣿⣿⢿⣿⣿⣿⡿⣿⠿⠁⠀⠀⠀⠸⠀⢀⡼
⠀⠀⠀⠀⣼⠃⠈⠻⣦⣀⣀⣤⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢿⡭⠭⠭⢽⠁⠀⠀⠀⢀⡴⢤⡴⠶⠛⠁
⠀⠀⠀⠀⢹⣆⠀⠀⠈⢹⠏⢡⣽⡷⠶⠶⠒⠚⢛⡿⠟⣷⣄⠈⠣⣤⠴⠃⠀⣀⣤⢶⣿⣶⡚⠁
⠀⢀⣤⠞⠉⠹⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡴⠋⠀⣼⠃⠈⠛⠶⠶⢶⡶⠛⠉⠀⣼⠁⠈⠙⠳⣤⡀
⢠⣿⡵⠞⣧⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣠⡿⢤⡀⢰⣇⣀⣀⣀⣀⣀⡞⢁⣀⠀⠀⡟⠀⠀⠀⠀⠈⠙⢦
⣼⢋⡴⠚⢹⡄⠀⠀⠀⠀⣦⣀⣉⡿⢩⡏⠀⢇⡇⡿⠁⠀⠀⠉⠉⣿⡴⠋⠉⠳⣄⡇⢤⡀⠀⠀⠀⠀⠈
⠙⢿⣠⡞⠁⠀⢀⣀⣴⠞⠉⠉⠉⠀⢸⣷⡤⢞⡼⠃⠀⠀⠀⠀⠀⣿⢧⠀⠘⡠⣻⢷⣄⣿⠶⠀⠀⠀⣰
⠀⠀⠉⠛⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢦⣍⣭⠴⠋⠀⠻⢻⡆⣀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡟⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣾⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠛⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠐⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢡⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠊⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣥⣀⡀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣀⣀⣄⣤⣬⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠈⠙⢷⣦⣤⣴⣯⣤⢶⠟⠉⠉⠀⠀⠉⠉⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠶⠦⠴⠶⠶⠶⠶⠶⠚⠛⣷⠤⣄⣀⣀⣀⠀⣀⣀⣀⣠⣼⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠖⠛⠛⠙⠛⠛⠷⠦⣤⣉⣉⣉⣉⣉⣉⣠⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⣹⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢁⣾⣿⣶⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢤⣄⠀⠤⣀⠀⢀⡤⠛⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠈⢹⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⠓⠦⣀⠙⠒⢬⠙⠉⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢧⡈⠳⣤⣉⠳⠄⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⡸⢒⣫⣽⣷⠀⠀⠀⠀⠙⣿⣿⡿⠟⠻⣿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠉⢻⣦⣄⣀⠀⣠⢊⣴⣿⡿⠛⢿⣦⡀⠀⢠⣾⣿⢁⡖⢁⣼⡏⠀⠀⢀⣾⣯⣍⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣣⣾⣿⡏⠀⠀⠀⠙⣷⣄⣻⣿⢇⣾⣴⣿⣿⣇⠀⢠⡿⣡⣴⣿⠇⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡄⠀⠀⠀⠈⣿⣿⣷⡾⠋⠁⠀⠈⢻⣿⣿⢡⣿⣿⡿⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⣸⣿⣿⣧⣀⠀⠀⠀⣰⣿⣿⣿⣿⢛⣁⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⢻⡟⢿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⡿⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠸⣷⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⢀⣾⣿⣹⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠙⣷⡄⠸⣿⣿⣿⣿⡿⠟⠉⠁⠀⣠⣴⣿⣿⣿⣾⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⣿⣿⣿⣧⠈⣍⣉⣉⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⡿⠿⣧⢹⣿⣿⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠟⠋⠀⠀⢹⡎⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠹⣦⡹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡄⠀⠀⠀⠀⠀⣽⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠀⠀⠀⣺⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⠀⣠⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠟⠁⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡿⠟⠿⢿⠛⠛⢁⡀⠻⣿⣿⣿⣿⣷⣄⡀⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠟⠛⠛⠉⠁⠐⣤⣾⠿⣟⠿⣟⠥⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⠔⣲⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠋⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⡼⠁⠀⢻⣯⠼⠓⡛⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢠⡄⠀⢠⡄⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⣸⠁⠀⣴⡟⣳⡄⠼⣛⢦⠂⠀⢠⡾⡙⡄⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠁⠀⠑⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣛
⠀⠀⠀⡇⠀⢸⠃⠉⠰⡷⣠⠋⠀⠉⡄⢻⡗⠀⠈⠒⢊⠗⠀⠀⠈⣇⢀⡄⣀⣀⣠⡴⠋⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢸⡁⢀⣀⠀⠀⠟⠛⠛⠲⣤⡾⠳⣸⠋⠁⢑⡴⠃⠀⢀⡠⠤⣀⠀⠀⠀⠀⠁⠀⠀⠸⡀⠀⡎⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠑
⠀⠀⠀⠙⡏⠘⢱⡀⠈⠉⠙⢭⡽⢠⠛⠓⠶⢶⡾⠋⠉⠒⣇⢀⣴⢾⢷⡀⠀⠀⠀⠀⢏⡙⢧⠠⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆
⠀⠀⠀⠀⠈⠒⠺⣍⣖⣶⠂⠒⠺⣙⠄⠀⠀⢸⠁⢀⠶⡛⢻⡏⢦⣸⣆⣧⠤⠀⠀⠀⡀⣳⠀⠀⠀⢹⣀⠄⢄⠀⠀⠀⠀⠀⠀⠀⡔⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠉⢲⡄⢸⠀⠀⠀⠑⠾⢁⠴⠋⠁⠀⠀⠀⠀⢀⠀⠉⠢⡀⠀⠀⣁⠤⠜⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡦⣄⣴⠏⠀⢀⢧⡄⠀⠀⡴⠃⠀⠀⠀⠀⠀⠐⠷⠘⠇⠀⠀⠈⢦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⡄⠐⠀⣰⠛⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢸⠁⠀⠀⠈⣲⣵⢃⠤⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠰⡉⣇⠀⢣⠀⠀⢸⡆⠀⠀⠀⡟⠃⠈⠀⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⢜⠳⡈⢆⠘⡦⣀⡤⠧⡀⠀⠀⢳⠆⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠆
⠀⠀⠀⠀⠈⠓⢌⠂⠀⠁⢸⠀⡠⠧⠖⠲⠊⠉⠢⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⡀⠀⠛⠀⣀⣀⣀⠠⠠⡌⠐⠢⠄⠀⠹⡲⠦⠤⣄⣀⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⠆⠙⢝⢦⡀⠀⢄⠀⠀⢀⠈⠀⠀⠄⡑⠢⠐⢃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢁⡽⠉⠉⠙⠪⢀⢀⣀⣀⠤⠜⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⡀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠾⠿⠛⠛⠿⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⣀⣤⠶⠶⠶⢤⣄⡀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠁⠀⢀⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⣾⠁⠀⣀⣀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣀⣤⠶⠿⠛⠛⠛⠛⠛⠛⠻⠶⢶⣼⣏⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⣴⠟⢋⣴⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣤⣬⣝⠻⣦⡀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠋⢀⣿⠿⠛⠻⣿⡀⠀⠀⠀⢀⣽⣿⠿⣿⣿⣷⠈⠻⣦⠀⠀⠀⢀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣧⣴⡟⢁⡴⠞⠛⣦⠁⠀⠀⠀⢈⡥⠶⠶⣄⠙⢿⣧⡀⠹⣧⠀⠀⠈⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣬⢿⠀⣼⠁⠐⣡⣼⡇⠀⠀⢠⠏⠀⠐⣀⢹⡆⠈⠈⠻⣶⣿⣇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣏⣿⠀⣿⢀⣥⠿⠿⠿⠶⢦⣼⡀⠀⣼⠿⣿⡇⠀⠀⠀⠈⢉⣿⡀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣴⣿⠋⠀⣿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣆⢿⣿⣿⠇⠀⠀⢠⡴⠞⢹⣇⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⣿⣿⣷⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠛⠁⢀⣤⣦⢸⡧⠀⣾⠋⠁⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣿⣿⣿⣿⠘⢷⡾⢡⡞⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠻⣿⡿⣿⣿⣷⣦⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⡟⠿⠋⠀⠀⠀⡿⠋⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠘⠿⢿⣿⣟⣿⣿⣿⣿⣿⡻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠛⠀⣀⣼⠟
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠘⢷⣄⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠈⢿⣬⣭⣭⣭⢥⣿⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣴⠾⠟⠁
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠙⢛⣿⠟⠛⠛⢷⣄⣀⣀⣤⣤⣤⣴⣶⣶⣿⣶⣄⠀⠈⢿⡀⠀⢀⡴⠃⠀⠀⠀⢀⣠⣶⣿⣥⣼⠏
⠀⠀⠀⠀⢀⣤⡿⢷⡄⠀⠀⠀⠀⠏⠀⣰⠟⠛⠋⠉⠉⠁⠀⠀⣠⡾⠋⠁⣼⠟⢷⣄⡀⠙⠛⠉⠀⢀⣠⣴⠾⠛⢹⡟⠻⢷⣤⣀
⠀⠀⣠⡶⠛⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣼⠏⠀⠀⠉⠛⠿⠿⠿⣿⡟⠋⠁⠀⠀⣾⠃⠀⠀⠈⠙⠷⣦⣀
⢀⣼⢏⣤⠶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢀⣰⣯⣤⡀⠀⢸⡏⠀⣀⡀⠀⠀⠀⠉⣾⠋⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⢸⣿⠋⢁⣤⠾⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠻⠿⣿⠟⣿⠏⠀⡆⢻⠀⣿⠛⠛⠛⠛⠛⠛⠻⣾⢇⣤⠖⠓⢦⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⣿⣇⣴⠋⠀⣠⠾⠂⠀⠀⠀⠀⢀⣸⣷⣦⣶⡾⠋⢠⣿⠀⠰⢃⡿⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣾⠁⠀⠀⡄⢻⢸⡇⠘⢦⣀⠀⠀⠀⠀⠀⠰⢠
⠘⠻⣧⣠⡾⠃⠀⠀⠀⣀⣠⣴⠟⠉⠀⠀⠀⠀⠀⢨⣿⣦⠴⢛⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣟⢻⡀⠀⠜⣡⠏⣾⢻⣆⣠⣿⠷⠆⠀⠀⠠⢡⣾
⠀⠀⠈⠙⠻⠶⠶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⠛⢋⣡⡾⠃⠀⠻⡟⣿⡄⠀⡀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⡇⣿⣧⣼⢾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⣠⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣿⢀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⣿⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠰⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠉⠙⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣯⣤⣀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣆⣍⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁⠉⠉⠙⠻⢶⣤⣀⡀⠀⢠⣿⠃⢀⣠⣴⡶⠛⠛⠉⠉⠉⠿⠛⠛⠷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⢿⣿⠿⠋⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠶⣤⣤⣤⣶⣶⣶⣶⡶⠶⠶⠷⠟⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⢿⣧⣤⣉⠉⠛⠛⠒⠒⠒⠛⠛⠛⠛⠋⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠶⣶⣶⣤⣤⣤⣴⣶⡶⠾⠿⠛⠁

⠀⠀⠈⠀⢠⠀⠀⢠⠀⡀⢀⠀⠀⠀⠀⠀⢠⠠⣄⠀⠤⡄⠀⠀⠀⠀⡀⠐⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠃⠠⠄⠤⠀⠀⣀⠘⠀⠀⠠⠄⠦⠈⠆⠀⠇⠀⠀⠠⠀⠄⣀⠨⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠻⠶⣶⣤⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⣠⣤⣴⣋⡀⠀⠉⠢⡀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁⣀⠉⠉⠛⠲⢤⡃⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⢉⡉⠍⣙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠔⠉⠀⠀⠈⠉⠑⠢⣀⠉⠳⡄⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⠁⠀⠀⠀⠐⡌⢷⡀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⢠⡆⠒⠒⠒⠈⢙⡗⣄⠘⢆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣀⡤⠶⠞⠛⠛⠛⠛⠲⠦⣬⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⡞⠠⡀⠀⠀⠀⠂⠁⠙⣦⡈⡄⠸⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠀⢀⡴⠋⣁⢤⣰⢖⡚⡉⠀⠒⠢⢄⡀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣇⢰⡆⠀⠀⢠⡔⠀⡀⠈⠳⠶⠢⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠘⡰⢋⠀⡞⠉⠀⠉⢀⠈⠁⠁⠲⡖⡂⠙⢆⠈⢿⡄⠀⠀⠀⠀⠀⢸⣥⠟⠉⠉⠙⠲⠛⠁⠈⠀⠀⠀⡔⠀⣿
⠀⠀⠀⠀⠀⣀⣀⠀⠹⣤⠖⠲⡧⠀⢸⠀⠀⣤⠀⠀⡄⠀⠀⠠⣄⠻⡀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡤⠠⢤⡄⠀⠀⠁⣰⠏
⠀⢀⣀⠀⢸⠋⠈⢷⣸⠇⢰⡀⠃⠀⢸⠀⠀⣿⣶⠿⠄⣀⣼⠇⡤⠀⢿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠘⣅⠉⠑⣲⡂⠁⠀⣠⠞⢠⠞⢿⠋⠁
⣼⠋⠉⠳⣾⠀⠀⢸⠙⣆⠈⠃⠓⠛⠡⢄⠀⡜⠁⠀⠀⠀⠙⡿⠃⢠⠎⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡅⠬⠙⠉⠁⢰⣃⣤⡼⠃
⠻⣄⠀⠀⠹⡆⠀⣸⠄⢌⣹⠊⢁⠀⠀⡄⠉⠣⡀⠀⠀⠀⣠⠧⠐⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢶⠖⢀⣴⡿⢻⠟⠛⣦⣄⣀⣀⡀
⠀⠈⢳⣤⠤⢼⡞⠁⠀⠈⣿⡙⢯⣄⠀⠘⠤⡤⠈⠉⠉⢉⠁⠠⢃⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢰⠃⠀⡩⠁⢣⡏⢠⠁⠀⠈⠁⢈⡿
⠀⠀⣸⠧⠄⡀⢣⣀⣠⠇⡼⢻⠛⠒⠛⠦⣄⠐⠊⣛⠖⣣⣦⣚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠓⢄⡟⢒⡶⠁⠀⣸⣁⠚⠀⠘⠠⣾⠋
⠀⠀⠹⣄⠠⣀⣿⠁⠀⠀⢁⡾⠀⠀⠀⠀⠘⡄⠀⠲⡈⢙⡌⢻⡓⢦⡀⠀⠀⠀⠀⢀⡀⠀⠀⣴⠏⣷⠤⢿⠃⠎⠀⢠⣴⠁⡸⣄⣀⣀⡤⢿⡄
⠀⠀⠀⠈⠓⠦⠤⢴⣤⠶⠿⠄⠒⣶⠀⠀⠀⣵⠀⡠⢿⠀⠈⣆⡙⠄⠹⣄⡤⢎⠛⠝⠙⢶⢿⠁⢠⡏⠐⠁⠀⠈⠉⠉⠈⢆⣿⣌⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⡞⢹⣾⡀⢈⠀⠀⠈⣆⢦⡰⠉⢧⡀⠑⠂⠀⠘⠈⠣⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠙⠉⠀⠈⠳⠭⠟⠀⠀⠀⠀⣿⠀⠀⠀⠙⠓⠦⣄⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡟⠀⢰⡞⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠻⠶⠞⠁⠀⠻⣦⠠⣀⡀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⡴⠚⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⡄⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣧⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⠄⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠤⠤⣀⡀⠀⠠⣿⡇⠀⠀⢀⣀⠤⠽⠷⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠒⠢⣄⡀⢸⠃⠀⠀⠀⣀⡤⠼⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠈⠙⠲⢾⣦⠤⠖⠉⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠉⢹⠖⠶⠖⠋⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠈⣿⢀⣀⠀⠀⠀⣀⣀⣸⡦⣀⡀⠀⠀⠀⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠯⣍⣀⣐⣒⣒⣨⡭⠵⠋⠈⠒⠻⠿⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠟⠛⠋⠙⠶⢬⣭⣖⣲⣦⣭⡽⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⡀⠀⠉⠛⢷⣦⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⡴⠋⠉⠀⠉⠑⣄⢙⣷⣴⠾⠛⠋⠉⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⢸⠁⠀⠀⠀⠀⣠⣼⠟⠋⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠸⡄⠀⠀⣠⡾⠛⠁⠀⣠⠴⠂⠀⠀⠀⠀⣾⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠙⣦⡾⠋⠀⠀⣠⠞⠁⠀⡀⠀⠀⢀⣾⠏⠉⠉⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⣿⡆
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣠⣾⣡⣤⣴⣿⣿⣧⣠⡾⠃⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⣠⡿⠋⠀⠀⢠⣿⣀⣤⣴⢶⣦⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡾⢡⣶⣶⣾⣯⡉⠀⠀⠘⠁⣀⠈⢻⡇⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠿⣿⣦⣼⠟⠀⠀⡀⢠⣿⠿⠋⠁⠀⣀⣸⣧⡀
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣰⡟⠀⣿⠟⠁⣀⣄⣀⠀⠀⠀⣜⠉⠙⢆⣿⡀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠘⢿⣏⡀⠀⠀⠈⠻⣥⠀⣀⡴⠞⠋⠉⠉⣿
⠀⠀⠀⣠⡾⠻⠀⠀⠀⠀⢰⡿⠷⠟⠛⠀⣼⠁⠀⠈⢧⡀⠀⢻⣖⣄⣼⡾⠷⠶⢶⣤⡀⠀⠀⠙⣷⡀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠈⠻⣏⠀⣀⣤⡶⠾⠿⣶
⠀⠀⠀⣿⠁⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⢹⣠⢦⡀⠀⣧⠀⣨⡿⠋⠁⠀⠀⠀⠀⠙⢿⡄⠀⠀⢸⣷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⣠⡿
⠀⠀⠀⢿⡄⠀⠀⠀⠀⣾⠋⢷⣄⠀⠀⠀⠈⢿⣿⡷⠀⣼⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛
⠀⠀⠀⠈⣿⣦⡀⣀⣴⣿⡄⠀⠉⠛⣶⡀⠀⠀⠙⠳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢸⡟⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⢸⣿⢉⣿⠋⠁⠈⠹⣦⡀⠀⣿⡇⢠⣿⣦⣄⣀⠀⠀⣤⣀⠀⠀⠀⠀⠀⣠⣾⣏⠀⠀⣸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⢸⡇⢸⡇⠀⠜⠛⠳⣬⠙⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⢀⣼⠏⠈⠻⣷⣄⠀⠀⠀⠀⠠⣤⣶⠾⠛⠉
⠀⠀⠀⢸⡇⠀⢿⡀⠀⠀⠋⠉⢷⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⡿⣣⡾⠁⠀⠀⠀⠈⠙⠷⣦⣤⣤⣤⣾⣿
⠀⠀⠀⢸⣷⠀⠈⠻⣦⣀⠀⠀⠀⠀⣀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣯⣥⣤⡬⠛⢻⣿⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠉⢸⡿
⠀⠀⠀⠀⣿⡀⠀⠀⠈⠙⠛⠓⠻⣿⠉⢷⡄⠀⠀⠀⠈⣷⠶⠶⠶⢶⣾⠃⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢻⣆⠀⠀⠀⠹⣆⠀⠀⣠⠏⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⣿⣀⣀⣴⣀⣿⡄⠀⠀⠀⠈⠙⠋⠁⢀⣴⣏⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠈⠉⠉⢉⣩⣼⣷⡀⠀⠀⠀⠀⠚⠛⣿⡅⠙⢷⣄⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⣀⣀⣠⣾⢿⡍⠀⠘⢿⡟⠛⠛⠚⠛⠛⠉⢻⡀⠀⠙⣷⡀⣠⡾⠁⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠉⠙⢷⣤⣹⣆⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⢿⣿⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⣿⠉⢻⡶⠶⣦⣼⢧⣄⠀⠀⠀⠀⠈⢷⡀⢀⣠⣿⣧⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣴⠟⠀⠀⣿⠀⠸⣧⠀⠘⣷⣤⣶⠶⠟⠛⣧⣾⣁⣬⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⣤⣼⠏⠀⢰⡟⠀⠀⣿⣶⣶⡶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠛⠻⠶⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⠈⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢐⢂⡀⠂⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡈⠑⠈⢂⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠉⠐⠂⠠⢀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠐⠁⢀⣀⣀⣄⠀⠀⠀⣀⣀⣀⠀⠀⠠⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⣀⣴⠿⠻⢧⡀⠀⠀⢠⠾⠛⠿⡟⠦⣀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⡐⠚⠁⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠂⣠⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡁⠀⠀⠀⠀⢺⣿⠇⠀⠀⠾⣿⡇⠀⠀⠀⠀⠀⡀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠔⢠⣄⠀⠸⠁⠀⠀⠀⠀⠘⠂⢀⣠⡄⠘⣠⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣻⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⣀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠻⠿⢷⣦⣤⣤⣴⣿⠿⠟⠀⠀⢀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣄⢀⠀⠈⠉⠁⠉⠉⠁⠀⠀⣠⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠉⠑⡆⣀⠀⠀⢀⣠⠖⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠐⠀⠀⠀⡀⠀⠉⠉⠁⢀⡀⠀⠀⠁⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣠⠊⠁⠂⠁⠉⠉⠉⠉⠀⢠⠀⠀⢢⡀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡴⠃⠤⠴⠂⠀⠀⠀⠀⠀⠀⠁⠢⠤⠁⠹⡄⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣇⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⣀⡉⠉⢁⣉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠙⠉⠉⢁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠉⠉⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡝⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢣⠀⠀⠄⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠁⡠⢆⡀⡧⢨⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⡇⣎⠀⡰⢄⠈⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⣀⠴⠊⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠒⠒⠂⠠⠤⠔⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡼⠛⠳⣄⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠑⠢⠤⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⡀⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠛⠛⠉⠉⠉⠛⠛⠿⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣴⠾⠛⠛⠛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⡿⠁⢠⣾⣧⠀⢀⣴⡄⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣾⡇⢠⣿⠁⢿⣶⠟⣿⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⣧⣤⣤⣿⣧⣿⡃⠀⠀⠁⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⣠⣴⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠶⣴⣿⣄⣀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⢠⣾⣏⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠻⣿⣍⡉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠉⠛⠷⣶⣤⣀⣀⠀⠀⠀⢀⣾⠋⠀⠉⠙⠛⠿⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢷⣿⢃⣾⣿⣆⠀⠀⠀⠀⠀⠉⠙⢻⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠈⣿⡛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣇⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠈⠙⠻⠷⣦⣤⣤⣤⣤⣶⠾⣿⠿⣦⡀⠀⠀⠀⠀⠘⠷⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠛⠉⠛⠛⠻⠿⣧⣄⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⣸⡟⠈⠛⠛⠛⠿⠶⣶⣄⡀⢸⡟
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢸⣿⡁⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⣤⡶⠀⠀⠉⢻⣾⠇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣀⣠⡶⠟⠋⢿⡆⠀⠀⠀⠙⠻⣷⣀⣴⡿⠃⠀⠀⢠⣾⣿⣥⣤⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣤⣶⠾⠛⠉⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⣿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣶⢶⣶⡶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠷⣦⣄⣀⣀⣀⣀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣠⣄⣀⢀⣠⣾⣟⡻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣋⠀⠀⠈⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠙⠛⣛⣿⠿⠛⠛⠿⠶⢶⣶⣶⣶⣶⡶⠶⣿⣿⣿⠋⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠹⣿⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣴⠟⠻⣦⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⣿⠀⠀⠀⠹⣷⡀⠀⣀⣤⣤⣤⣤⣤⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⠁⠀⠀⣿⠂⠀⠀⠀⣴⡿⠋⠉⠉⠙⠻⣦⡀⢹⣧⠀⠀⠀⣸⣿⠟⠋⠉⠀⠀⠀⠉⠉⠛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢸⣿⡇⠀⠀⣰⡟⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢹⣷⠀⢻⣆⢀⣾⣏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢸⡿⣧⣤⣾⠟⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣻⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⣇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣀⣀⣴⠟⠁⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⢠⣿⣧⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣦⣤⣘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⣆⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⠉⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣤⣄⣿⡆⠀⠀⠀⠙⠿⠁⠀⠈⠻⠏⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⣤⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⠛⠛⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠾⠛⠛⠻⠷⢶⣶⣶⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠟⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⣉⣛⠛⠛⠛⠛⠛⢿⡟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠟⢿⣤⡙⠻⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠟⢋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠉⠉⠉⠉⠛⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⢀⣴⠟⢋⡍⠉⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣾⠃⢀⡾⢿⣄⣠⣾⡆⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⢀⣾⠃⠘⠟⠋⣿⠃⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⣠⣤⡶⠾⠟⠛⠛⠛⠛⠛⠛⠿⠿⢧⣤⣤⣀⣀⣿⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠠⣿⣟⠛⠛⠛⠛⠛⠛⠷⠶⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⢀⣽⠏⠛⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠶⣦⣾⢃⣶⣦⠀⠀⠀⠀⠉⠙⠛⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⢿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⣼⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⡌⠙⠻⠷⣦⣤⣀⣀⣤⣤⠶⣿⢷⣄⠀⠀⠀⠀⠘⠷⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠛⠿⢿⣧⡀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠈⠉⠉⠁⠀⢰⡟⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢨⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⢀⣾⠃⠙⠛⠛⠛⠻⢶⣤⡀⣸⡇
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⣿⣄⡀⠀⠀⣠⡿⠁⠀⠀⣀⣴⠾⠂⠀⠈⠻⣿
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣀⣤⡶⠟⠋⠈⣿⠀⠀⠀⠈⠙⢿⣤⡾⠋⠀⠀⢠⣾⣿⠷⠶⠂⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠾⠛⠋⠁⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⣀⣀⣀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣄⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣤⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠙⠛⠶⠶⠶⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠛⠷⠶⢾⣿⡿⠶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⡟⠛⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⣼⠏⣿⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢿⡄⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⡶⠟⢷⡄⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⢹⡇⠀⠀⠙⣧⡀⢀⣀⣤⣤⣤⣤⣘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⡟⠀⠀⢸⡇⠀⠀⠀⣴⠟⠋⠉⠉⠙⢷⣄⠘⣿⡀⠀⠀⣸⡿⠛⠉⠉⠀⠀⠈⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢸⣿⡇⠀⢀⡾⠁⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢻⡆⠘⣷⣀⣾⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢸⡿⢷⡶⠟⠁⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢀⣾⠃⠀⣸⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣴⠟⠁⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢠⡄⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠷⠾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠘⣷⠀⠀⠀⠘⣿⡿⠀⠀⣿⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠷⢿⡇⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢛⣻⣿⣿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠷⢦⣤⣄⣀⣄⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⠿⠶⠶⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠉⠉⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢶⣤⣀⡀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⣉⣛⡛⠛⠛⠛⢻⡟⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠻⣦⣉⡛⠶⢶⣤⣤⣴⣶⣶⡶⠶⠶⠶⠶⠶⠶⠟⢋⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠿⠿⠟⠻⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⠀⠀⣠⠴⠒⢒⡦⣌⠹⣿⣧⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⢀⡞⢁⠓⢤⠎⠱⡈⠳⣼⣿⣧⣴⣾⣿⣿⣿⣤⣈⠻⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢸⠀⢸⠀⣄⣰⢦⠐⣀⣁⣿⣿⣿⣿⣯⣤⣤⣙⣿⣧⠀⠀⠈⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣦⠘⣢⣤⣴⣾⠿⠛⠛⠉⠉⠁⠀⠀⣉⣉⣛⣻⣿⡀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠛⠉⠁⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧⣄⣠⡄⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠈⠉⠁⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⢰⣿⡆⢿⣿⠁⠀⠀⠀⢀⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡟⠀⠀⠀⠀⣠⣾⡟⠁⣀⣀⣤⣴⠾⠛⣿⣷⡄⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣶⣦⡉⠁⢩⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠃⠀⠀⢀⣼⣿⢿⣿⠛⠋⠉⠉⠀⠀⠀⢿⣿⣷⠀⠀⠀⢠⠟⠋⠀⠀⠀⠀⠈⠻⣿⣿⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⢀⣀⣀⣾⣿⠁⠀⢿⣧⡀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⢿⣿⣇⢠⣴⡿⠛⠛⠉⠻⣿⣦⠀⠀⠙⢻⣷⡄⠀⣰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡏⠀⠀⠤⣄⡀⠘⢿⣷⣄⣀⣿⡟⠁⠀⣿⡟⠻⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠼⢿⣆⠀⠙⠻⠿⠟⠁⠀⠀⢿⣷⡀⠀⠀⠉⠉⠋⠙⢿⣶⣶⣤⣤⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⢀⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⠀⠀⡀⠀⠀⠀⠉⠉⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⣀⠁⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣤⣾⣷⢶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⢿⣿⣿⣿⣯⣽⣿⣦⡀⠀⠀⠀⠀⠀⢸⣯⠀⠙⣧⣿⣿⣿⡿⢁⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠿⠿⠿⣿⣶⣄⣀⠀⠀⠀⠻⣶⣾⣿⣿⣿⣋⣴⣿⠏
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⡿⠿⠶⣶⡿⠿⠛⣿⣿⢿⣿⣇⣀⡀⠀⣠⣶⣾⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⣼⣿⡿⠿⠛⠿⠿⣶⣤⣀⠀⢀⣤⡶⠿⠛⠉⠀⠀⣿⡇⠀⠀⠘⣿⣤⣤⣾⣿⠇⠙⣿⣿⣿⣾⣿⡿⠋⠁⠀⠈⠉⠻⣿⣷⣦⡀
⠀⠀⠀⣠⣾⣿⠿⠓⠂⠀⠀⠀⠀⠉⢻⣿⣿⡏⠀⠀⠀⠀⠀⢀⣿⠃⢀⣶⡿⠿⢿⣏⠀⠀⠀⢸⣿⠈⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⢀⣾⣿⠏⠁⣠⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⡀⠀⠀⠀⠀⢀⣼⠏⢠⣿⡏⠀⠀⠀⠹⣷⠀⠀⠸⣿⣆⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⣿
⠀⣼⣿⡟⠀⣼⡿⠀⣠⣾⠀⠀⠀⠀⠀⢿⣿⢿⣶⣶⣶⡾⠟⠁⠀⠘⣿⣇⠀⠀⠀⣰⡿⠀⠀⠀⠹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⠏⢰⣿⣿
⠀⣿⣿⡇⠀⠙⠁⠀⣿⠏⠀⠀⢀⣀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⠟⠁⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⢀⡟⣠⣿⣿⠏
⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣠⣾⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⢀⡞⣼⣿⣿⠋
⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠾⢛⣿⡟⠋⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣀⠀⠀⠀⠀⣠⠞⣼⣿⣿⠃
⠀⠀⠀⠙⠻⣿⣿⣶⣶⣤⣶⣶⠿⠋⠀⠀⠀⣰⡟⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⣿⣿⣷⣶⣶⣿⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⣹⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣶⣤⣤⣀⣀⣀⣀⣤⣤⣶⣿⡿⠟⠋⠀⠀⠈⠉⠛⠛⠛⠛⠛⠻⠟⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⡏⠀⠀⠀⠀⠀⠀⡏⠀⢀⣿⣿⠛⠻⠿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣧⣄⣀⣀⣠⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠉⠉⠉⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠄⢠⢛⣄⣠⡛⡄⠠⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⢀⢂⠋⠱⠏⢱⣘⣤⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡦⠽⠤⠤⠤⠤⠭⠴⢄⡀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡔⠃⠀⠀⣀⣠⡤⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀⠀⠀⠑⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡎⠀⢀⣴⣾⠿⠻⢿⣄⠀⠀⠀⠀⣴⠟⠛⠻⣯⠙⢦⣄⠈⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⡴⠾⠉⠈⠀⠀⠀⡀⠌⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠙⠷⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢠⠞⠉⢻⠀⣆⠀⠀⢀⠀⣴⣶⣾⡀⠀⠀⢸⣶⣶⡆⠀⠀⠀⠀⠀⠂⠘⠉⠙⣆
⠀⠀⠀⠀⠀⠀⢺⠀⠀⢺⠀⠉⠀⠀⠀⠀⢹⡿⠛⠉⠀⠀⠉⠻⣿⠃⠀⠀⠀⠀⢈⡀⢰⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠹⡴⠋⢀⣶⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡃⢀⣠⣶⡀⠈⢡⡞⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⡇⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⢰⠇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⠀⠀⠉⢻⣿⣿⣷⣄⡀⠀⢀⣀⣴⣿⣿⣿⠟⠁⠀⠀⣸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠉⠉⠉⠿⠿⠿⠿⠿⠿⠏⠉⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⢶⣤⣀⡀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣀⣤⡶⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⡀⠀⠈⠙⣦⣀⠀⠀⠀⠁⣀⣤⡞⠉⠀⢁⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇⠀⠀⠀⡇⠈⠉⠉⠉⠉⠁⢸⠃⠀⠀⠘⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢠⡠⠴⢤⠀⡗⠒⠲⠶⠶⠒⠒⠻⠃⠠⠠⢀⡇⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⢇⠀⢀⢹⣦⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⢠⡿⠉⠒⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠁⠘⣧⠀⠀⠀⠀⠀⠈⣧
⠀⢀⣰⡏⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢘⣧⡀
⠀⢯⠀⠉⠑⠲⠶⠶⠖⢛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠒⠒⠶⠶⠚⠉⠀⡿
⠀⣾⠛⠲⠶⠶⠶⠶⠖⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⡷⠶⠤⠤⠤⠶⠶⠛⣧
⡸⠁⠀⠂⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠉⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢧⡀⠀⠀⢀⡀⢀⠀⠀⢰⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡄⠀⠀⣀⣀⠀⠀⠀⢀
⠈⢳⡀⠀⠒⠀⢨⡆⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⢠⠇⠀⢰⡅⠀⠇⠀⢀⡿
⠀⠀⠛⠦⠤⠴⠛⠳⠦⠼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠈⠁⠀⠀⠀⠀⢠⡇⠘⢦⡤⠞⠳⢦⣄⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⠤⠞⠋⠁⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠉⠉⠑⠒⠒⠒⠶⠶⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠻⢕⣂⠤⠀⣀⡀⢠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢖⣀⣀⣐⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢀⠔⠉⣿⢟⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⡎⠀⣠⡾⣉⠀⠶⣀⠄⠀⣠⡶⡀⠀⠀⠀⠀⢸⠀⠀⠠⠟⠀⠼⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇
⠀⠀⢸⠀⢰⠋⠋⢽⣄⠞⠈⢂⠰⣷⠁⠐⠀⡲⠄⠀⠘⡄⢀⣀⣀⣀⡴⠃⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢏⢀⣠⡀⠰⣛⡛⣒⣴⠞⣃⠗⠁⢱⡔⠀⢀⠤⠤⢄⠀⠀⠀⠁⠀⣀⣇⠀⡎⢈⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠣⠈⣹⣀⠀⠈⣑⣞⡌⠙⠛⠳⡟⠁⢈⣹⣠⡶⢛⢧⠀⠀⠀⠀⢦⠹⠀⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠜⡃
⠀⠀⠀⠀⠈⠙⠒⠻⡁⠀⠁⠺⢄⠀⢸⠀⠘⠩⣁⡇⢑⠾⠘⠃⠀⠀⠀⠰⢇⠀⠀⠱⠒⢀⠆⠀⠀⠀⠀⠀⢨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⠀⣴⠋⠘⣆⠀⠀⢈⠜⠁⠀⠀⠀⠠⣄⢰⠀⠀⠑⣄⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢇⠀⠈⣨⠣⢤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⣀⣠⣄⠀⠀⢰⠁⠀⠀⢱⣷⢃⠤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⡇⢇⠀⡆⠀⠸⡆⠀⠀⢘⠁⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠘⢌⠢⠣⠸⠂⠖⢁⡄⣀⣸⠧⡀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃
⠀⠀⠀⠀⠀⠱⡄⠀⠈⠀⠮⠁⠀⠀⠀⠈⠂⠀⡀⠈⢦⢄⣀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠢⡄⠲⡲⣉⠉⢀⠀⠁⠀⠀⠀⠈⠢⠄⠝⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣈⣩⠏⠉⠑⠪⠀⠀⠀⠤⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢁⢔⣲⣄⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣼⣿⣘⣽⣦⣄⠀⠀⠈⠑⢦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣊⢍⠛⠉⠉⣹⣿⡟⠻⣿⢶⡦⢄⡀⠙⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠊⢰⡟⢻⠖⠁⠀⣿⡔⠉⢛⣄⠈⠑⠪⢕⡦⣀⠀⢀⡿⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠙⢦⠈⢧⡇⠈⣦⣄⠀⠁⠀⢸⣿⡄⠀⠀⢠⣿⣮⣿⡏⠀⣤⠄⢳
⠀⠀⠀⠀⠀⣠⡟⠁⠈⠲⣤⣀⣈⡓⣤⡷⠚⠋⠉⠀⠐⣄⠀⣉⡇⣰⣦⣿⣿⣿⣿⠀⠀⠫⢀⣇⣠⣤⠤⠶⠶⢤⣄⡀
⠀⠀⠀⠀⣰⠋⠘⠛⠲⢦⠞⠁⠀⢹⠏⠀⠀⠀⠀⠀⠀⢈⣭⣭⣶⣿⡿⠀⠉⠛⠁⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢀⣙⣦⣀
⠀⠀⠀⠀⢹⡄⢀⡤⠤⡎⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠈⢳⡄
⠀⠀⠀⠀⢺⡀⠀⠀⠀⡇⠀⠀⢸⠿⣷⣤⣀⣀⣠⣴⣿⣿⡿⠛⠏⢹⢦⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣠⡤⠖⠀⠀⢙⠳⣄
⠀⠀⠀⠀⠈⠳⡄⠀⠀⠙⢲⡖⠋⡴⠛⣿⣿⠛⠻⠟⠋⠁⣠⠊⡰⠃⣸⡆⣧⡴⣏⠀⠀⠀⠀⢀⣤⠞⢻⣇⠀⠀⠀⠀⢀⡄⢦⠘⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢧⣈⣲⡶⠋⠠⠚⢀⡴⠃⠈⠳⣄⡀⢀⣦⡈⠙⠷⢤⠞⣹⠉⠀⢸⡀⠀⠀⠀⢿⠘⡇⢸⠇⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠦⡤⢴⡋⠀⠀⠀⠀⢸⠋⠉⢻⠙⠳⠶⠶⠞⠁⡇⠀⠀⢳⠀⠀⠀⢸⠀⣿⠸⣦⣧⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⠀⠀⠘⡆⠀⠀⠀⠄⢰⡇⣠⠄⣌⢧⡄⠠⠆⠀⢹⠀⣾⠁⠀⣤⠀⠀⠀⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣼⡇⡠⠀⢌⢾⡁⠀⠀⢸⡇⡏⠀⣼⡼⡆⠀⠀⠀⣸⣤⣽⣤⣤⣟⡳⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣱⠀⡎⣇⢳⠀⠀⠘⣇⠳⠤⠟⢰⡇⠀⠀⠀⣿⠤⠤⠀⢀⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣖⠚⠻⢯⡉⠉⠙⠳⢄⡀⠘⠶⠶⠶⠋⠀⠀⠀⣸⡷⠤⣄⠀⠀⠈⠓⢤⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠤⠤⠤⣈⠉⠙⠲⢤⡈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⣷⠀⠀⠀⠈⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠂⠀⠐⠂⠬⡁⠢⢄⠙⢦⡀⢲⡀⢀⡇⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⢠⠏⠀⠀⣀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡀⠀⠀⠀⠀⠁⠠⡙⢦⠙⣾⣃⣾⣤⣤⣴⣾⣿⣁⣀⣀⣠⡤⠞⠋⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⢤⣄⣠⠃⠈⠢⠱⣸⡟⠉⠁⠐⠒⠒⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠍⠉⠓⢦⣤⣥⣿⣃⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠒⠒⠀⠒⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠐⢤⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠻⠛⠟⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⡀⠘⢦⣈⣷⣼⣇⠀⠀⠀⢀⣠⣤⠶⠖⢛⣛⣛⡛⠶⣤⡀⠀⠀⠀⠀⠀⠀⢸⣦⣶⠐⠒⠰⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠹⣦⡾⠉⠀⠀⢹⣧⡴⠞⠋⠁⠀⢠⠎⣥⠶⢦⣠⡕⢄⠻⣆⠀⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣧⡀⢀⡿⢧⣤⠴⠀⣸⠏⠀⠀⣀⡤⣴⡚⠒⠒⠒⠦⣄⡻⢸⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠙⠋⠁⠀⠀⣀⡴⠋⠀⣠⠾⠧⠞⠛⠿⡶⠶⠦⣄⡀⠙⢯⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠙⠂⠀⠀⠀⠀⠈⢉⡏⢹⡟⣷⠀⢀⠖⢉⢳⡀⠀⠀⠰⠟⠿⣷⢳⡀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⡤⠶⢿⠀⡟⢠⠞⠀⡞⠀⣱⢶⡇⠀⣰⡋⠍⢳⠹⣦⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠘⣧⠗⢣⣾⣄⢹⡼⠛⠉⠙⠳⣯⡟⠄⡸⠀⠀⣸⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣧⠀⠀⠀⠀⠀⡟⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⠈⣯⠔⠁⢠⡞⢹⡦⣞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣇⠀⠀⠀⠀⢷⠀⠀⠙⢿⢿⣷⣤⣀⣀⣀⣴⣷⣶⣾⣧⡇⣼⠆⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠹⣆⠀⠀⠀⠈⢧⡀⠀⢸⣦⣍⣉⠻⠿⢿⣿⣿⠟⠉⢀⡟⠁⠀⡼⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⡦⠀⢀⡴⠋⢹⡄⠘⣏⠀⠉⢻⣿⠞⠀⠀⠀⣀⡾⠷⠶⠛⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣼⠃⣰⠋⠀⠀⠀⡿⢦⣀⡉⠉⢉⣀⣤⠴⠶⣞⠉⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡏⡿⣡⠖⠒⣄⢰⣧⣀⠈⠉⠉⠉⣹⣇⣀⠀⡏⠳⣄⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡃⢀⢄⡟⣼⢀⡼⠛⠛⠚⠋⠉⠉⠉⠉⠓⢦⣘⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣯⣼⠃⣼⠁⠀⡴⠀⢰⠀⢠⠀⠀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣤⣴⡇⠀⣿⠀⢸⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣸⣦⣠⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠉⠉⠉⠉⠉⠁⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡰⠋⠀⠉⠂⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠞⠃⡄⠠⠀⠤⣤⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠘⠁
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣠⠖⠀⠀⢾⣦⣾⡇⠀⠀⠀⠰⠿⢫⡝⠙⠏⠙⠗⠀⠀⠀⠀⠒⠢⣄⣀⡴⢁
⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡱⡏⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⡄⢿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⠈⠱
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡤⢚⡵⣀⠀⠀⠄⠘⠀⠀⠀⠀⠀⢀⠀⠐⠈⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠎⠀⠀⠀⠂
⢰⡇⠀⠀⠀⠀⢠⡀⠀⢀⢧⠜⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁⠀⠀⢀⠞
⡟⠀⠀⠀⠀⠀⠀⠈⠉⠁⡆⠀⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣿⡇⠀⠀⠀⡄⠀⠀⠀⣠⠊
⢇⠘⣄⠀⠀⠀⠀⠀⢀⡴⠓⠚⠳⡘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⡤⠤⠤⠚⠁
⢸⠇⠌⣑⡒⠐⣀⠲⠋⠳⠀⠀⠀⠑⣌⠛⠂⣄⡀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠋⠉⠀⠀⠀⣠⠎
⠀⢤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠳⢄⠈⠛⠛⠛⠿⠿⠿⠛⠉⠙⠋⠁⠀⠀⣀⠠⠘⠁
⠀⠀⠱⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠛⠓⠷⣄
⠀⠀⠀⠀⠀⠈⠁⠐⠒⠠⠤⡀⠀⠀⠀⠂⠀⠀⠀⠉⠐⠒⠂⠴⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡐⢦⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡗⠀⠀⠑⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠈⢲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣾⠁⠈⠷⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣜⠳⠄⣀⣀⣀⣀⡀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢉⣤⡀⠀⠀⠀⠀⠀⢀⡠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠉⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠑⠢⠤⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠐⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⠊⢁⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠐⠠⡄⠀⠀⠀⠀⠄⢚⡅⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⠀⠄⠚⠉
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠟⠁⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡉
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢀⠀⠀⠀⠀⠀⢀⡄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⣀⠴⠁

⠀⠀⡀
⠀⠀⠀⠐⠆⠀⠰⠀⠀⠄⢀⠀⠀⠀⠆⠰⡀⠐⢆⠀⠀⠀⠠⠠⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠈⠐⠂⠃⠀⠀⠂⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⡉⠙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡶⠞⠛⠛⠖⠤⣄⡈⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠛⡛⠿⣦⡀⠀⠀⠀⠀⠀⠀⢸⠋⡠⠐⠀⠂⠒⠂⢄⠉⠓⢄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠠⠂⠁⠀⠁⠨⠻⡄⠀⠀⠀⠀⠀⠸⣆⡀⠀⣠⠀⠒⠲⠆⢙⡦⡀⠣⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢀⡠⠴⠒⠛⠛⠛⠓⠲⠦⣷⠀⠀⠀⠀⠀⠀⠀⠉⠲⡏⠠⠀⡀⠀⠊⠁⠹⣆⢡⠘⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⡇⣠⠞⣁⡤⢤⡞⠏⡁⡀⠀⠂⢄⠈⠻⣆⠀⠀⠀⠀⠀⢀⠼⣇⣼⣀⡀⣠⡆⠀⠀⠈⠚⠓⢤⠏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡰⠡⢈⠋⠀⠀⠐⠄⠀⠀⠘⢨⡄⠑⠀⢹⡆⠀⠀⠀⠀⠘⡞⠁⠀⠀⠉⠉⠀⠀⠀⠀⡴⠁⡸
⠀⠀⠀⠀⡴⠒⣆⢸⠋⡈⠙⠀⢸⠀⢸⣷⣰⡌⠀⣠⠌⡂⢱⣤⡶⠟⠀⠀⠀⠀⠀⠠⠛⠦⢄⡀⠄⠊⠈⡿⢀⣤⡤⠞⠁
⣰⠞⠲⣤⡇⠀⢸⢻⡀⠳⠠⠴⢚⡀⠈⠟⠉⠁⠉⠻⡴⠃⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣖⢛⠲⠖⠊⢠⢋⣈⡇
⢻⡀⠀⠘⣇⠀⢸⠤⣙⡖⠂⠀⠀⡄⠘⢄⠀⠀⠀⢠⡄⢴⠇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡤⢀⣤⡿⡻⠟⣦⣄⢀⣀
⠀⠙⣦⠤⠼⡷⠁⠀⠸⣏⣪⡤⠀⠐⠤⠤⡉⠁⢉⠁⠜⣱⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡞⠀⠨⠁⢧⡁⡄⠀⡘⠉⣩⡇
⠀⠀⡿⠒⠠⣱⡤⠴⢡⠋⡏⠉⠛⠒⣄⠑⠭⢟⣡⢶⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣴⠉⡶⠀⢀⡾⣄⠉⠀⠁⢿⠁
⠀⠀⠹⣤⡢⠾⠀⠀⣀⡼⠁⠀⠀⠀⠸⡀⠈⢢⠠⢆⠱⡝⢦⡀⠀⢀⣠⣤⣄⣀⡼⠃⣷⣚⠏⠈⠤⠤⠿⠰⡜⠦⠤⠖⢹⡆
⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠀⣹⠀⠀⡰⣇⡔⠘⠄⠀⢳⢈⡀⡳⡎⠑⢌⠀⠈⠏⢇⢸⠃⠀⠀⠀⠀⠀⠀⠛⣿⢦⣄⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠢⠞⠁⠸⣑⣰⠂⠀⠀⠲⡟⠀⠘⠲⢤⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢨⡇⠀⡴⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⡼⠁⠀⠀⠀⠀⠈⠳⠴⠃⠀⠻⡆⠠⣀⠀⠀⠀⠀⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡴⠊⠁⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣇⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⡇⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⣁⠀⠀⠀⢰⣿⠀⠀⠀⠀⣘⣧⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠥⣀⠀⠀⡇⠀⠀⠀⢀⣘⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠈⠙⠲⢤⣿⣀⡤⠒⠉⠀⠀⠀⠹⡄⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⠺⡦⠤⠴⠚⠉⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⢻⡄⣀⠀⠀⢀⣀⣀⡧⢀⡀⠀⠀⠀⠀⠀⣀⣽
⠀⠀⠀⠀⠀⠀⠈⠻⠥⣤⣤⣬⠬⠽⠚⠁⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠛⠲⠶⠾⠛⠛⠉⠙⠲⠮⣭⣭⣤⠽⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠞⠙⠛⠛⠛⠋⠑⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠄⢐⣤⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠈⠙⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠂⠀⢎⣴⡳⢞⣧⠸⡀⠐⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠇⢠⡋⡘⠁⠙⠋⠰⣆⣙⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⡼⣧⣤⣤⣤⣤⣬⣥⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣠⡤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⣄⡀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⠀⠒⠀⢀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡴⠁⠀⠀⢀⣠⣶⣶⣾⠛⠉⠉⠉⠉⠉⠉⢉⣿⣿⣷⣶⣦⡄⠀⠀⠙⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸⠀⠀⣠⡴⣿⠿⠛⠛⠿⣷⡀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠿⣦⠙⠷⣦⡀⢰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢠⡶⡞⠁⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢷⣴⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢲⡎⠀⡇⠀⠀⠀⠀⠀⣀⣼⣴⡇⠀⠀⠀⢠⣦⣤⣡⡄⠀⠀⠀⠀⠀⠀⣆⠈⣶⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠂⣠⡇⠀⡇⠀⠀⠀⠃⠀⢿⣿⣿⡷⠄⠐⠀⠼⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⠀⠀⣀⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢸⡇⢀⡁⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠈⠙⣯⡀⠀⠀⠀⠀⠀⠸⣤⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⣷⠞⠁⢠⣾⣦⣄⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⣀⣤⣾⣷⡀⠀⠀⢾⡃⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⢻⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣸⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢾⡀⠀⠀⠙⠿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠉⠀⠀⠀⣿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠛⠿⠿⠿⣿⣷⣶⣶⣶⣾⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠛⠋⣉⠉⠙⢛⡛⠋⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢻⡷⢦⣄⡀⠀⠀⠀⢀⠀⠀⣀⠠⠀⠀⠀⣀⡴⢾⡟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠈⠙⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⡟⠉⠀⠀⢁⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠉⠙⠛⠓⠒⠚⠛⠉⢹⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⡤⣼⡄⠀⠀⣀⠀⡆⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⢸⡔⠉⠉⠱⢀⡇⠀⠈⠉⠉⠉⠉⠀⠀⠘⠀⡌⠉⠉⠰⣧⡀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣰⣟⠃⠀⠀⣠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢣⡀⠀⢠⠘⣷⣄⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠰⠋⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠖⠁⠀⢿⡄⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⢰⠛⠛⠷⢦⣀⣀⣀⣀⣀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣹⣧⣀⣀⣀⣀⣀⣤⠶⠛⠻⡄
⠀⠘⣦⣀⡀⠀⠀⠉⠉⠉⠁⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠀⠉⠉⠁⠀⠀⣀⣴⠃
⠀⣼⠋⠉⠙⠛⠛⠛⠛⠛⠛⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⣀⠠⠄⠀⠀⠀⠀⢸⡟⠛⠛⠒⠛⠛⠛⠛⢉⠹⣦
⣸⠇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠉⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠹⣦⠀⠀⠀⢀⠖⠐⢦⠀⠀⢰⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⡆⠀⠀⡴⠒⠲⡀⠀⠀⢀⣴
⠀⠙⣷⠀⠀⠉⠀⠀⣾⡀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⣼⠁⡞⠁⠀⢀⣷⠀⠀⠋⠀⠀⣼⠋
⠀⠀⠈⠳⠶⠤⠴⠞⠉⠳⠦⠴⠇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠳⣤⡴⠞⠛⠷⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣠⡤⠶⠋⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠳⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣿⡋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠛⠻⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⠻⢾⣉⠠⠄⠀⣀⣀⠀⢤⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡤⠤⠤⠤⢤⣐⣶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠋⠁⠁⠀⠁⠈⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡀⢤⠐⠦⡈⠄⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠐⢰⠀⠈⠀⣠⠘⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⢀⡀⠄⠙⠀⠀⠀⠀⠀⠀⠁⠀⠤⢀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⠀⠊⠁⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠢⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⠂⠀⢀⣰⡾⢿⣧⡀⠀⠀⠀⠀⣴⠿⠿⣿⠲⣄⡀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢸⣠⠞⠉⠉⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠛⣤⠁⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⠆⢸⠀⠀⠀⠀⢠⣰⣴⠀⠀⠀⣠⣆⣤⠀⠀⠀⠀⠀⠀⠐⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⠈⠀⠀⠀⠀⠘⡿⠋⠉⠉⠉⠙⢿⠇⠀⠀⠀⠀⢀⡀⡤⠈⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢣⠖⠁⣴⣄⣀⠸⠀⠀⠀⠀⠀⠀⠀⢃⣀⣠⣴⡄⠀⢣⠏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⣸⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⢸⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠀⠙⢿⣿⣷⣤⣀⣀⣀⣤⣾⣿⣿⠟⠀⠀⠀⣸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠙⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡟⠢⢄⣀⣀⠤⠔⠋⠀⠀⠀⡄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠁⢀⡀⠀⣇⣀⣀⣀⣀⣀⡀⡄⠀⡀⡀⢀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢠⡎⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢳⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣠⠟⠠⡠⠴⠇⠁⠀⠀⠀⠀⠀⠀⠘⠰⠦⢄⠂⠻⡆⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠙⠒⠦⠤⠤⠐⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⠤⠤⠔⠒⠉⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠶⠦⠤⠤⠴⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠤⠄⣠⡤⠤⠖⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠂⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⡜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠑⠀⣠⡄⠀⢇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⡘⢰⠁⠀⣠⡀⠐⠁⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁⠙⠒⠋⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠁⠙⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡠⠔⠊⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠛⠒⠶⠶⠤⠤⣤⣠⢴⣿⠉⠛⠉⠈⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠙⢦⣀⠤⠀⡀⢀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢆⣀⡒⠶⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⡤⢦⡠⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠍⣙⣓⣄⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣰⢶⠉⠙⢛⠻⣌⣦⡀⠘⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠿⢼⡆⠀⢸⡇⢠⠖⢱⠤⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣤⣿⠿⠀⠖⠰⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠛⠻⠫⠟⠀⠀⣠⠴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠲⣬⣥⢖⢯⠏⠙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢳⡀⡀⡔⠉⠙⠦⠟⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣉⡧⠷⢤⣀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⠀⠀⠈⠻⠟⠋⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠐⢢⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⠀⠘⡆⠀⠀⢀⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠙⢻⡄⠶⠋⠁⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠤⠤⢤⠸⣄⡀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠙⠚⠫⠭⠭⠥⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢍⢒⡠⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠴⠾⣍⣳⣸⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠒⢢⣄⣭⣶⢄⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣰⠿⠏⠁⠛⢍⠻⢤⣱⣄⠀⠙⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠿⢤⣷⠀⠀⣾⡇⠀⡤⠋⣦⣀⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⡀⣐⣾⣷⠀⢧⠔⣤⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣴⣾⣿⠿⣟⠁⠀⠀⠀⣁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣌⡀⠑⠩⠁⢀⣠⣴⢯⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠩⠀⠀⢈⣓⡶⢞⠫⣱⠞⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⢣⣀⡀⡴⠁⠀⠙⠦⠛⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣙⣧⣳⢤⣀⠀⠀⠀⢀⡴⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠛⠁⠀⠀⠻⣿⣲⠟⠉⠙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠉⢷⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⡇⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢹⡀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠓⠶⣾⣁⣤⠶⠚⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⣛⣉⡭⠥⠭⣷⠤⠤⠤⠤⠤⠤⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠠⣄⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣀⠀⠳⣄⣘⣧⣼⡇⠀⠀⠀⠀⣀⣤⡴⠶⠖⠒⠓⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡦⠰⠆⠰⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠈⢷⣠⡟⠉⠁⠈⢹⣆⣤⠶⠛⠉⠀⢀⡔⣉⡴⢦⣈⣑⢦⠙⢦⡀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣶⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡄⠀⢸⣿⣄⣀⡤⠀⣸⠏⠁⠀⠀⣀⡤⠼⠶⠾⠦⠬⣁⡹⠄⡇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠛⠛⠛⠁⠉⠁⢀⡴⠋⠀⣠⠶⣯⡵⠿⢿⣦⣤⣤⣀⠀⠙⢶⡁⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠢⠀⠀⠀⠀⠀⠘⢹⡟⢲⡾⡇⠀⠀⡤⠒⠢⡙⠀⠀⢈⡿⢾⣶⡽⡄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣀⣀⣠⡴⣾⢑⡞⢀⡿⠀⡼⠀⢊⣬⡇⠀⢀⡴⡒⢢⡸⣇⢹⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠉⠉⠉⠀⠀⢻⣘⡧⢏⣴⡀⢷⣠⠾⠞⠷⢦⣾⣳⠰⢀⡇⠈⠙⡇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡀⠀⠀⠀⠀⢈⡟⠀⠸⣿⣿⣾⠁⠀⠀⠀⠀⠘⣧⣁⠞⠀⣠⠞⣇⣠⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⢷⠀⠀⠀⠀⠘⡇⠀⠀⢿⣿⣿⣦⡀⠀⠀⠀⢀⣿⣅⣤⣤⣳⠀⣏⡉⢷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣧⠀⠀⠀⠀⢻⡄⠀⠀⣾⣙⠿⠿⢷⣶⣶⣿⣿⣿⠿⠏⣷⠞⠁⠀⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⢧⡀⠀⠀⣠⡿⣦⠀⢸⡟⠛⠲⣤⣭⣭⠏⠉⠁⠀⢀⣿⣀⣠⠞⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣼⠃⢀⡾⠋⠀⠘⣧⡀⠑⠦⠤⠼⠛⠁⢀⣀⣠⡴⠛⠉⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⠧⣴⢋⣀⣀⠀⢠⡏⠙⠲⠦⠤⠴⠖⡟⠉⠉⣿⢦⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⢣⠏⠁⡈⡇⣸⠛⢶⣤⣤⣀⡤⠼⠷⠦⢤⣿⡀⠻⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣤⣐⡵⢣⡏⣰⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⢙⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢯⣤⣿⠀⠀⡞⠀⢸⠀⢰⡇⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⣧⠀⢹⡀⠈⣧⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠳⠶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢷⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣠⡾⢋⠤⠒⠒⠢⢄⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠁⠉⠉⠛⠷⣧⡀⠀⠀⠀⠀⢣⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠘⢷⡄⠀⠀⢀⣤⡶⠛⣿⠀⠀⠀⠀⠀⠢⣄⠈⠛⢦⡀⠀⠀⢸⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣰⡏⠉⢿⡄⠀⠀⠈⢿⣄⣴⠟⠁⠀⠀⢻⡆⠀⢀⣤⣄⣀⣈⢳⡄⠀⠙⢦⡤⠃⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣼⠏⠙⣧⣴⠾⠛⠒⠲⠄⠀⢻⡇⠀⠀⠀⠀⠈⢻⣄⡿⠛⠻⡏⠉⠉⢛⣦⣶⣦⡿⣆⠀⠀⠀⠀⠀⢿⣷⣄
⠀⠀⠀⠀⣠⣿⡄⢠⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣀⣀⣽⣡⠊⢱⠀⠀⢀⡩⠭⣿⣝⠇⠘⢧⠀⠀⠀⠀⢸⣇⠹⣧
⠀⠀⠀⠠⣿⠈⠻⣦⣙⣷⠶⢶⠀⠀⠀⠀⠀⣸⢧⣤⡾⠋⠉⠉⠙⠳⣾⡽⠀⢀⠎⠀⣀⣸⠙⠷⣤⣈⣧⠀⠀⠀⠈⣿⠀⠹⣧
⠀⠀⠀⠀⠙⢦⣄⣀⣸⠇⠀⠈⠀⠀⠀⠀⠀⠁⢸⡟⠀⠀⠀⠀⠀⠀⠈⠛⠀⢸⠀⢸⣷⡟⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⠳⣄⢹⡆
⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⠚⣋⣠⡀⣠⠶⠚⠋⢹⡄⠀⠀⠁⠀⠹⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡋⠁⠀⣀⣤⠾⠋⠀⢿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⢇⡟⠀⠀⣀⣼⠷⣤⣀⠀⠀⢠⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠻⣏⠁⠀⠀⠀⠈⣿⠹⣿⡿⢿⣿⣿⡿⢿⣿⡿⠋⠉⠁⠸⣦⡴⠚⢁⣀⠀⠀⢹⣦⠶⠋⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢷⣄⠀⠀⠀⠘⣧⡀⠀⠻⣟⣛⣚⣛⣹⡇⠀⠀⠀⠀⠀⠀⣰⠟⠂⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠙⢷⣄⠀⠀⢈⣻⣆⠀⠹⣇⡀⣀⡞⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⣀⡴⠋⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢙⣷⡾⠛⢉⡽⢷⣄⡀⠉⠉⣠⣤⠶⣶⡶⢶⣾⣉⠛⠛⣻⡏⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢀⣤⡶⣿⠁⢀⣴⣯⡤⠤⠬⠿⣿⣟⡉⢀⣞⡁⠀⠀⠀⠙⠻⣿⡉⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣁⡀⣸⢡⡿⠄⠀⠀⠀⠠⠦⠍⠈⠙⢿⠉⠻⣶⢤⣤⡀⠀⠈⠻⣦⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣡⡏⠀⠀⠀⠀⠀⠲⠒⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣧⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⠀⠀⠀⣀⡄⠀⣸⣇⡀⠀⠀⠀⣀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠟⠛⠻⠶⠾⠋⠉⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⣀⣀⣀⠙⠻⣦⡀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣠⠊⠀⠀⠀⠉⢦⣘⣿⡶⠟⠋⠉⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⡇⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢷⡀⠀⣠⡾⠋⠀⢀⡤⠞⠁⠀⠀⠀⢠⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣹⡾⠋⠀⢀⣴⠋⠀⣀⣄⡀⠀⣠⡿⠁⠀⠈⠙⠻⢷⣦⣀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣿⠂
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣼⠏⢀⣀⣴⣿⠷⠶⢶⡿⠿⢷⣶⠟⠁⠀⠀⠀⠀⠀⢀⣨⣝⣷⣤⡀⠀⣴⡟⠁⠀⠀⣰⣿⣤⡶⠞⠻⣷⡄
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢠⡾⢡⣾⡿⠿⠿⠄⠀⠀⠈⢀⡤⣄⢹⡆⠀⠀⠀⠀⠀⢰⣟⠁⠈⠙⢻⣿⣾⠏⠀⠀⢤⣰⡿⠋⠁⠀⣀⣤⠾⣷⡄
⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢠⣿⣁⣼⡟⠀⡤⠶⢦⡀⠀⠀⣯⣀⠈⢷⣿⡀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠙⠻⣦⡀⠀⠀⠉⢳⣄⣴⠟⠉⠀⢀⣸⣟
⠀⠀⠀⣾⠋⠁⠀⠀⠀⢀⡿⠉⠉⠉⠁⢸⡅⠀⠀⠙⡆⠀⠹⣯⣶⠿⠛⠛⠛⠻⣶⡄⠀⠀⠙⣷⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠙⢧⣠⡴⠞⠛⠉⢻⣷
⠀⠀⠰⣿⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠘⣷⣏⣦⠀⢹⠀⢼⠟⠁⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⡾⠃
⠀⠀⠀⢻⣆⠀⠀⠀⢸⡏⠈⠳⣤⣀⠀⠀⠘⢿⣟⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⢰⡿⢷⣴⠶⠿⠿⣄⠀⠀⢹⣧⠀⣠⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣿⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⢸⡇⢸⡇⠀⣠⣤⣙⠷⣤⣼⠇⢀⣿⣿⣿⣶⣤⣤⣿⣶⣤⣄⣀⣤⣴⣾⣿⡇⠀⣴⠟⢿⣦⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⠟⠋
⠀⠀⠀⣿⡇⢸⣇⠀⠀⠠⠽⢷⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡾⠋⠀⠀⠙⠷⣦⣀⠀⠀⠈⣹⣿⠉
⠀⠀⠀⢹⡇⠀⠻⣆⠀⠀⠀⠀⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠋⠈⠻⠿⢿⡟⣡⠟⠁⠀⠀⠀⠀⠀⠈⢙⣿⠿⠟⢛⣿
⠀⠀⠀⠸⣷⠀⠀⠈⠛⠶⣤⣤⣤⡾⢦⡀⠀⠀⠉⠙⣿⣉⣉⣙⣛⣹⠋⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢸⡏
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⣸⠇⠀⢻⡄⠀⠀⠀⢹⡄⠀⠀⢙⡟⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⢰⡏⠀⢀⡄⣻⡄⠀⠀⠀⠙⠶⠶⠋⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠈⠛⠟⠛⠛⣫⣿⠀⠀⠀⠀⠀⠤⠶⣾⠋⠻⣦⡀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⣠⣶⡟⠉⠙⢷⡶⠶⠶⠶⠶⠶⠛⣷⠀⠈⢷⣄⠀⣠⡟⠁⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⠛⠛⢻⣄⣹⣆⠀⠀⠻⣄⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⣷⣏⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⢻⡟⢻⣶⢶⣄⣽⣦⣀⠀⠀⠀⠀⠘⣆⠀⢀⣹⣿⡀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⣾⠃⠀⣿⠀⠹⣧⠀⠙⣧⣤⣤⡶⠶⠻⣆⣟⣉⣽⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧⣀⣠⡟⠀⢀⡿⠀⠀⣿⣧⣤⣴⡶⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠛⠳⣦⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣠⣤⣤⣾⡟⠛⣿⣤⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⠙⠛⠉⠉⠉⠀⠀⠉⠉⡹⠁⠀⠀⠈⠹⣷⡀
⠀⠀⠀⠀⢠⣿⠃⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⣾⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⣿⣀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠸⠛⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠸⣷⡀⠉⠆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠹⣷⣄⠈⠂⠀⠀⠀⠐⠀⠀⠀⠀⣀⣹⣀⣀⣀⠐⠃⠀⢀⡀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢾⣿⣿⣿⡃⢠⣿⣿⣿⣿⣿⣿⣿⡿⠿⣦⣀⠀⠈⠑⠠⡀⠀⠈⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⢉⠿⣿⣿⣿⣿⠏⣡⡀⠀⠀⢈⡙⠒⠀⠐⠘⠣⡀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⠟⠐⠛⠣⠀⠀⣿⡿⠀⠀⠀⠂⠀⢐⡄⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠐⠤⢿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠒⡀⠈⡀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠄⠠⠆⠀⠀⡸⣿⣷⣄⣀⣀⣀⣠⣦⣤⣤⣤⣦⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⢘⣁⠠⠊⠀⠙⣿⣿⡿⣿⣿⣿⢿⣿⠿⠛⠁⠀⠀⠀⠄⢶⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⠿⣷⣄⡀⠀⠈⢿⣷⣌⠉⠁⠀⠀⠀⠀⠀⠀⣠⣊⠀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣸⡿⠀⠈⠙⢿⣦⣄⠀⢋⠁⠁⠂⠤⠤⠤⠤⠒⣿⣿⣟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣶⡿⠃⠀⠀⠀⢀⣿⠏⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⢠⡘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠁⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣄⣤⣀⣀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠛⠷⠄⠀⢀⠀⠀⠀⠀⢀⠠⠊⠐⠉⠛⢿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⣇⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣷⣥⣂⡀⠀⠀⠀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣬⣤⣤⣤⣤⣴⣶⡾⠟⠋⠀⠀⠉⠛⠛⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣉⣡⠤⣄⡉⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⠿⢶⣿⣁⠞⠁⠀⠀⠀⠈⢢⡀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠁⠀⠀⠀⠀⡀⠈⠙⢷⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡀⠀⠀⠀⠻⣧⡀⠀⢀⣠⡶⠟⠉⢹⡇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠻⣦⡀⠀⢀⡇⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠀⠘⣷⡄⠀⠀⠀⠹⣷⣴⡿⠋⠀⠀⠀⠘⣿⡀⠀⢀⣤⣤⣀⣀⣈⣳⡄⠀⠈⠻⣦⠞⠀⠀⠀⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⣾⠋⠉⣿⣤⡶⠾⠛⠓⠲⠆⠀⠙⣿⡀⠀⠀⠀⠀⠀⠘⣷⣄⡿⠛⠛⢿⠉⠉⠉⢛⣦⣶⣶⣜⢷⡄⠀⠀⠀⠀⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⣠⣿⡀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠈⣿⢡⠖⠳⡆⠀⠀⠀⣋⣍⣻⣟⢿⠀⠻⣆⠀⠀⠀⠀⠀⣿⡌⢻⣆
⠀⠀⠀⠀⢰⡟⠙⢷⣄⠻⢷⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿⣄⢀⣴⠾⠛⠛⠛⠷⣯⣤⣲⠇⠀⢠⠎⠁⠀⢹⡟⢦⣄⡀⠹⣆⠀⠀⠀⠀⢹⡇⠀⢻⣧
⠀⠀⠀⠀⠘⣷⡀⠀⠙⠛⣿⠋⠀⠻⠀⠀⠀⠀⠀⠐⠋⢹⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⡟⠀⢰⣛⣿⠃⠀⠉⠙⠛⢻⡄⠀⠀⠀⢸⣷⣤⡀⢻⡆
⠀⠀⠀⠀⠀⠈⠙⠳⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣻⣿⠏⠀⠀⠀⣀⣠⡼⣷⡀⠀⠀⠸⠇⠙⢿⡌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣠⣤⠄⠀⠀⠀⠀⣠⡾⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⣀⣨⣭⣤⣶⡆⢰⡟⠉⠉⠀⠸⣇⠀⠀⠀⠀⠀⠘⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣀⣠⣴⠿⠋⠀⠀⢻⣿⣷⣦⣤⣄⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢇⡼⠁⠀⢀⣤⠶⠿⠶⢦⡄⠀⠀⣰⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠻⣯⡀⠀⠀⠀⠀⠀⣿⡘⢿⣿⠿⣿⣿⣿⡿⠿⢿⣿⠟⠉⠉⠁⠘⢷⣤⠖⠋⣀⣤⠄⠀⠀⢻⡶⠞⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠻⣦⡀⠀⠀⠀⠘⢷⡄⠀⠘⢻⡟⠛⠒⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠂⠀⠀⠀⣼⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⣦⡀⠀⢻⡏⠉⠉⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀⡼⠋⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠘⢿⣦⣤⡶⠟⠛⣷⣄⠀⠉⠓⠚⠋⠀⣀⣠⣤⣤⣀⣤⡶⠶⠶⠶⢶⡟⠋⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⢀⣠⣶⠟⠉⠀⣰⠞⠉⠙⠳⢶⣦⣴⡶⠛⠋⣩⠞⠋⠉⠙⠻⢶⣤⣴⠾⠃⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡾⠟⠛⢻⠀⣤⠾⠛⠛⠛⠉⠉⠉⢻⡻⢦⣄⣼⣧⣤⡀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⢂⡞⣰⠟⠛⠀⠀⠀⠀⠘⠛⠂⠀⠀⠈⠣⠀⠘⣿⠛⠷⠶⠄⠀⠀⠘⢿⣄⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣼⡇⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⣠⡄⠀⣸⡏⠀⠀⠀⠀⠀⠀⢀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢶⣶⡶⠾⣿⣤⣴⠟⠛⠻⠿⠷⠶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⡤⠖⠛⠉⠓⢦⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣀⡀⠬⠷⠦⠀⠀⠶⠿⢁⣀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡤⢻⣶⠒⠶⠒⢲⡆⠉⠉⠉⢉⣷⡶⢶⠲⠦⠾⣴⣋
⡔⠛⠛⠒⠆⠡⠘⠈⢛⠗⢺⠀⢸⠁⡀⠀⡞⢉⢙⡀⠀⠀⣾⠁⠙⡆⠀⠰⢠⡕⠈⡇
⠉⠛⠒⣄⠀⠀⢠⠀⡇⠀⠀⢣⣸⡹⠁⠀⢧⠸⠿⡇⠀⠀⢿⠟⢀⠅⠀⠣⠀⡇⣰⠃
⠀⠀⠀⠘⡆⠀⠀⢆⠣⡀⢀⠀⢹⡅⢸⣷⣄⡱⠋⠉⠀⠉⠚⣟⣉⣤⣶⠀⡿⠚⠁
⠀⠀⠀⡔⠁⢀⣠⡬⠗⠚⠉⠉⠙⢧⡈⠛⣿⣇⠀⠀⠀⠀⠀⣸⣿⡟⠋⢠⡇
⠀⠀⠀⠣⠜⠁⠀⠀⠀⠀⠀⠀⠀⢰⢿⢦⡈⠉⠓⠤⠤⠤⠞⡹⠛⠳⡴⠯⣭⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠊⠀⢈⡷⢤⣀⣀⣠⢔⣧⣀⡴⠒⠀⠀⠘⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠖⠠⢸⠃⡴⠤⠴⠒⠉⠀⡸⠁⠀⡐⠀⠀⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠻⣄⣼⠟⠘⣆⣀⣀⣀⣤⠀⠳⣀⣴⠀⠀⡞⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢠⡞⡁⠤⢤⡤⠤⣄⠈⢷⢤⠀⠀⠉⠉⠀⠈⠙⠒⠒⠦⢵⣽⡧⠴⠃
⠀⠀⠀⠀⠀⢀⣿⠈⠊⠉⠀⠀⠀⠀⣱⡼⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⢦⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⡄
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢹
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡄⡟⠙⠛⠉⠉⠓⠦⣄⣀⠀⠀⠀⣠⠾⢁⡾
⠀⠀⠀⠀⠀⠀⠀⢷⠠⠤⠤⠭⠥⠤⠤⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠙⠳⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⢋⣭⢉⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⡾⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣴⡟⠻⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⡶⣌⠙⠢⢌⡒⡴⠊⠀⠀⠸⣇⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠸⡁⠲⣍⡑⢦⠈⠀⢀⠀⠀⢠⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠹⣦⡀⠁⠀⠀⠀⠀⢨⠖⣻⣿⡆⠀⠀⠀⠙⣿⣿⠿⠟⢿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⠔⣡⣾⠟⠛⢿⣆⠀⢀⣾⡿⢡⠞⣠⣽⠀⠀⢀⣿⣯⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣼⣿⣯⠀⠀⠀⠙⣷⣼⣿⢣⣿⠾⠿⣿⣀⢠⡟⣤⣾⣿⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣦⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠈⣿⣿⣵⣿⣿⠃⠀⢀⣾⣿⣿⣿⣿⣿⡛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⢰⣿⣿⣿⣦⣄⣠⣴⣿⣿⣿⣯⣴⣶⣶⠈⠻⣿⣿⣿⣿⡟⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣸⣿⢻⣿⣿⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠙⣷⡀⢻⣿⣿⣿⣿⡿⠛⠋⠀⣀⣴⣶⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣶⣿⣿⣆⠉⣙⣋⠁⢀⣤⣴⣾⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣳⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠛⠁⠘⣧⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠘⢧⡹⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠄⠀⠀⠀⠀⣸⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣦⣠⣴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⡇⠀⠀⠀⢈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣶⣿⣿⣿⣿⣿⠿⢿⡿⠿⠋⠀⠸⣿⣿⣿⣷⣄⡀⣀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠛⠉⠁⠐⣤⣾⢿⡛⣿⠣⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⡼⠀⣇⢈⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡡⣒⣲⣿⠞⡁⠀⠀⢳⢀⣀⣇⣀⠧⠼⢤⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢇⠻⡜⢁⠎⣠⣤⣀⡏⠣⣀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⣹⣡⣿⠊⠉⠐⣿⠶⠒⠺⡇⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠻⡇⠀⠰⣦⠀⠋⠀⠀⠀⢸⠒⠦⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡼⠞⠀⠀⠀⠨⠁⣠⣀⣀⣤⣿⡂⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡟⠒⠊⣢⠻⠿⣿⢿⣿⢟⠋⢠⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢠⠋⠩⣽⠲⠜⠀⠀⠘⡀⠈⡟⢰⣗⠲⢧⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⣀⡠⢤⡤⠤⣤⣈⣨⠴⣣⠖⠉⠺⣶⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠈⠳⣰⠊⠻⣧⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠺⣉⠉⠑⠚⠋⠉⠢⡀⡠⠖⣏⣠⢛⡿⠾⠼⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠚⠲⠬⠝⠁⠀⠀⠀⠐⢯⡀⠀⡏⠵⠋⠀⣀⡀⠈⣆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢉⠗⢀⣀⣀⣠⠤⠖⠠⠴⠃⠀⢱⣄⡔⢉⣉⣉⠑⢿⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣔⡭⠒⠒⢍⢪⡖⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⡏⠀⠀⠀⠀⣿⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠧⢄⣀⠀⡰⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⠝⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡎⠈⠓⠤⠤⢴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⡁⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⣄⣀⡤⠊

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣛⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣥⣾⠟⠛⣻⡿⣷⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠿⠿⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡽⣣⡾⠋⠀⠀⠺⠵⣿⡟⣿⣆⡛⠻⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠖⢰⡟⠁⠀⣠⡴⠾⠿⢛⣻⣿⣿⣿⣷⣦⡙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⠀⠀⣾⡇⣠⣞⣩⣤⠶⠛⢿⣽⢦⡀⠀⣈⣿⠇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⣄⠙⣿⣷⢀⣿⡷⣿⡇⠀⠀⢾⡇⠀⠀⢰⣿⣿⣟⠋⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣹⣽⣆⣿⢿⡟⢁⣤⠀⠛⠀⠀⠈⠀⢠⡾⠋⠉⠉⠻⣷⡘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⣼⠟⠙⠛⠁⣿⠁⠏⠠⠀⠀⠀⢀⠀⠀⠸⣧⣀⠀⠀⣠⣿⠃⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣴⣋⣁⣀⠀⠀⠀⠹⣷⣄⣠⣴⣆⠀⠘⠀⢠⠀⣠⠹⠟⣻⡿⠃⣺⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⡟⠁⣴⣾⣿⣿⣿⠿⠿⠻⠿⣦⣤⣭⣭⠾⠿⣷⢶⡤⣤⣤⣥⣤⡾⠋⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⣠⡿⠋⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⣀⡿⠀⠃⠈⢿⣿⣟⣦⣀⣠⣤⣀⠻⣿⣿⣿
⠀⠀⢰⣿⢦⡾⠋⢈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⣿⡿⠀⠀⠀⠀⠈⡏⢹⠙⣿⠛⠙⢿⡇⢘⣿⣿
⠀⠀⠸⠟⠋⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠘⣿⡀⠀⠀⠀⡀⢀⡿⣯⣷⣿⣿⡙⢦⢡⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⢀⣠⣴⡿⠛⠁⠀⣾⢛⡿⠶⢶⠞⣷⢿⢱⣀⣸⣿⡅⠑⠀⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡾⠿⠛⠉⠀⠀⠀⠘⣷⡟⠀⠀⢺⣿⣿⠈⠑⠿⠛⢸⣷⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣸⡟⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡜⣆⠀⠀⠀⣠⣶⢾⣿⡿⠟⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣯⣽⣿⠿⠿⠛⢙⣿⠿⠷⢶⣦⡀⢀⣀⠳⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠉⠻⣿⣿⣶⣤⣴⣾⡿⠶⠶⢾⢿⣣⣼⣿⣿⡔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣤⣴⣶⠿⠛⠉⠉⠉⠉⠀⠀⠈⠛⠉⠁⠀⠁⣼⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣛⠉⣉⠉⠙⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⣵⣠⣦⠱⡀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣦⡀⠀⠀⠀⣠⣴⠶⠾⠟⠛⠻⠾⠍⣁⠹⢆⠇⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣦⡀⠀⠻⢿⣿⣿⣶⣶⣤⣀⡀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠳⣄⠀⠀⠀⠉⢉⣿⣵⠈⠙⠳⢦⣄⣀⠀⠉⠢⡀⠀⠀⠈⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠂⠀⠀⠘⢷⣴⡾⠛⠛⠛⠿⡀⠀⢰⣿⠀⠉⠛⠛⠒⡿⢦⡀⠀⠈⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣰⠏⠁⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⢠⠖⠁⢈⠷⠄⠤⢄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢹⣦⣤⠤⠘⣷⡀⠀⠀⠀⠀⢀⣀⣤⠴⠊⣷⠀⠈⢦⡴⠋⢀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣿⣁⠀⠀⣀⡼⠻⣖⠒⠒⠊⠀⠀⠀⠀⡴⠏⠀⠀⠀⠀⠀⠇⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣄⠀⠈⠓⢶⣦⣤⣶⣤⢾⡃⠀⠀⠀⢀⣴⣶⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠈⠳⣝⢿⣀⣸⠇⠀⢀⣴⣿⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡶⡟⠛⠲⠶⠖⠚⠛⣧⠀⠈⠙⣳⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⢿⣷⣶⢏⣏⣀⣀⣠⠤⢤⠀⠀⠀⠈⢳⡴⠚⠉⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠀⠀⠀⠀⠹⣿⣾⣿⣿⣯⣝⣦⢈⡇⠀⠄⠀⠘⡷⠀⠀⠀⠀⢄⠲⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⡀⠀⠀⠀⢀⣾⠟⠥⡒⡄⠈⢻⣏⠀⠀⠀⠈⢸⠃⠀⡀⠀⠷⠈⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠡⠀⠀⠀⢸⡇⠀⠀⠈⠞⠆⠀⣿⠀⠀⠀⠀⠘⢦⣤⠿⠄⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢂⡀⠀⣸⡇⠀⠀⠀⠈⠘⠀⢸⠀⠀⢀⣀⣤⠞⢛⣷⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⡽⣿⣄⠁⠈⢉⣁⣠⡿⠿⠿⠿⠭⠖⠚⠉⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣿⢼⣷⡶⠟⠛⠋⠀⠀⠀⠀⠀⣀⡀⣼⠁⡾⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢷⡿⠀⠀⢍⠉⠉⠛⢛⢟⠉⠉⠉⢹⡇⠹⣄⣷⣄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣷⠀⡀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢸⡇⠀⣿⣄⣸⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣄⣐⠀⠀⠀⠀⠣⠤⠁⠀⠀⣸⡇⠀⣿⠋⠁⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠓⠒⠒⠖⠒⠒⠛⠉⠉⠀⠀⣿⣳⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢼⡧⣾⣾⣵⣦⣇⡴⠚⠇⣀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⢠⣛⠟⠁⢈⢠⣟⡋⢸⣏⣥⣤⢐⣿⡽⠃⠰⢀⠔⡆
⠀⠀⠀⣦⠀⠀⣴⣿⣇⣼⣦⣿⣷⠟⣿⠾⢫⡿⠳⠟⠉⠀⣴⣿⠊⠀⣷⠀⢀⢀
⠀⠀⠀⣆⠑⠤⡏⡁⢹⣿⠋⠘⣁⠀⣥⢀⡞⠀⠀⠀⣠⣞⣿⠃⠀⠀⠻⡶⠗⠚⠦⡄⢀⡠⠊⣸
⠀⠀⠀⠸⣦⡼⣿⣷⣷⢃⣤⠀⣿⣸⣿⣿⣿⣄⠐⠻⢛⡟⠻⣷⣄⣀⣶⠃⠀⢄⠀⣼⠉⠀⢴⡇
⠀⠀⠀⠀⠘⣇⢈⠻⣿⣸⣧⣼⢿⣿⣿⣿⣽⡿⠿⠛⠻⠧⡀⠈⠛⢿⠃⠆⠀⡾⠕⠁⠀⠀⠀⣃
⠀⠀⠀⠀⠀⢸⣿⣷⠍⠳⢏⣁⣿⣿⡿⣹⡏⠀⠀⠀⠀⠀⡘⣆⢀⡎⠀⠀⠘⣴⠇⠀⠀⠀⢦⠋⣆⠀⡄
⠀⠀⠀⠀⠀⢸⢻⡿⠃⠀⠈⢻⡿⠶⠊⠛⠃⠀⢀⠀⠀⡠⠛⠛⠉⠀⠀⠀⠀⣧⠖⠂⣀⢠⠆⠀⣸⢿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⢀⡇⢀⣀⣀⠀⠀⣀⣴⠋⢠⣾⡿⠉⠀⣼⡿⠒⠗⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠘⡀⠀⢰⣔⣪⠄⠀⠀⠀⣠⣸⡏⠁⠀⣸⣧⡉⠁⢤⡀⠓⠿⠟⠻⢿⠀⢉⣵⣢⡿⢸⢂⣆⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠉⢿⠱⡟⣿⠶⠶⠰⡿⠈⠉⠁⣰⡞⠛⣿⠿⣧⣴⣿⡿⢿⣿⣿⠶⣶⣿⣿⣿⡗⣣⣿⣼⠋
⠀⠀⠀⠀⠀⢠⠒⢿⠁⠀⠀⣀⡀⠀⠀⢀⣠⡰⠟⠛⠀⠋⢱⣘⠯⠙⣅⠾⢿⣿⣶⣽⣿⡿⣫⣾⣿⡿⣿⢀
⠀⠀⠀⠀⢠⠣⠛⠀⡷⠶⠿⣿⠋⠛⠒⠒⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣨⣿⡿⠙⡏⣹⣿⣿⣇⡷⠒⠁
⠀⠀⠀⢠⠃⠀⠀⠈⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠸⢻⣿⠀⠀⠀⣿⡿⠀⡈⢧
⠀⠀⠀⠈⢂⣲⡶⠀⢀⡴⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⣼⡆⠀⠀⣿⣇⡘⠁⠈⠑⠄⡀
⠀⠀⠀⠀⠀⠈⠃⠓⢤⣄⣠⢾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⣷⣾⣿⣧⠀⠀⣿⣿⣷⡀⢦⣤⢤⡬⠦
⠀⠀⠀⠀⠀⠀⢰⣿⡷⣽⣿⣿⣼⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠾⣿⡄⣰⣿⣿⣿⠖⢋⡀⠒
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢹⣿⣿⣦⠀⠄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠹⣿⡏⣹⡟⢄⠈
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⢸⣿⣿⣿⣶⡄⢀⠀⠀⠀⢀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⡚⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠈⠛⣿⣿⣿⣦⡀⠀⠠⠴⣿⠘⠀⠀⠀⠀⠀⠀⠀⢀⡦⣿⣿⣇⠘⠃⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡒⠀⢁⣠⣼⣿⣿⣿⢻⡟⢿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠠⢿⣷⣿⣿⣿⣷⣦⣤⠄
⠀⠀⠀⠀⠀⠀⢀⡠⠘⢁⡁⠈⠅⠉⠿⢿⣿⡞⣷⠢⠙⠻⢿⣸⣂⠀⠀⠀⠀⠀⢔⣦⣿⣿⣻⡏⠯⠍⠒
⠀⠀⠀⠀⠀⢀⣈⡀⠠⢤⡀⡀⢀⣀⣀⣤⣼⡷⣿⠠⡦⠀⢸⠏⠛⠻⣦⣾⣿⣶⣾⣿⣻⡁⠀⣠⣠⣌⢀
⠀⠀⠀⠀⠀⠈⢈⢣⣴⣾⣿⠻⠿⠞⠛⠉⠙⠂⠁⠀⠀⠀⡠⠊⠉⠀⣤⠬⠩⠩⠆⠾⠛⣇⠀⠏⠻⠏⠩
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣈⠀⠀⢀⠀⣀⢠⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⣾⡏⡀⠀⢿⢻⣇⡀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠉⠙⠛⠦⠞⠋⠉⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⢠⣾⣿⡿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠟⣩⡽⠿⢷⣦⣿⣿⢃⣾⣤⡆⢹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣾⣭⣉⡇⢸⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠰⠶⠞⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⢀⡀⠀⠉⢹⣿⣷⡄⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣉⣴⠾⠛⠛⠛⢿⡁⠀⠀⣾⡏⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡏⠀⠀⠀⠀⠀⠁⠀⠀⡿⠃⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠀⢾⣿⣿⣿⠋⠉⠈⠙⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣤⣤⣶⣿⣿⣿⣿⣿⡿⠀⠀⠛⠛⠁⠀⢀⡴⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⡀⠀⠘⣄⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡙⢿⣿⣿⠃⠀⠀⢀⣴⣿⣿⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠈⣻⣿⣿⣷⣿⣥⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⡟⠁⠈⢻⣿⣿⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣰⡄⠀⢤⡀⠹⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⣀⣸⣷⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠛⠋⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⣿⣿⣿⣿⣿⣿⣿⣧⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢛⠳⣦⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣼⡫⡡⡉⠻⣦
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠞⣦⣯⣺⣷⣾⣿⣽⠛⢶⣄⠀⢠⡴⡟⢹⠏⢹⡶⠒⠺⠧⢼⡅⡐⠀⠘⣧
⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⠧⠓⠋⢉⣠⣤⣾⡿⣦⡀⣿⢴⡞⢆⠑⠈⠠⠞⢻⣿⣷⡲⢤⡈⠢⡀⠀⢹⡆
⠀⠀⠀⣀⣼⠁⠀⡠⠊⢁⣠⣔⠊⠡⣿⣏⣉⡠⢠⣼⠏⠈⠛⢦⡀⠀⢀⡞⠉⠉⠻⠁⣼⠈⢒⢴⡀⠈⠙⣆
⠀⠀⣼⠋⠀⣀⡜⠹⡉⠁⠀⢿⠀⠐⠉⠀⠈⢻⡾⠋⠀⠀⠀⠀⠹⢯⡤⢷⣄⣀⣀⣀⣀⡀⢇⡡⠗⠢⣴⠏
⠀⠀⠹⣶⠉⢬⡙⢄⡸⠃⡖⠢⠤⣄⣀⠀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠻⡌⠲⣶⡤⣀⡴⠃⢀⡪⠁⣠⠇
⠀⠀⠀⢿⣄⠀⠃⡀⠀⠀⠈⣦⣀⣀⣉⣭⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠘⣗⠁⢀⣼⣭⡿⠋⠁
⠀⠀⠀⢀⣩⡿⠿⠿⢶⣤⣀⣱⣤⣾⣿⠟⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠋⡝⣹⠛⠶⣤⣀⣀⣀⣀⡤⡦⣄
⠀⣠⡾⠛⠓⢤⣠⠖⠉⠈⠀⣙⣧⠟⢹⢷⣾⠋⠉⠛⣷⡄⠀⠀⠀⠀⠀⣼⣹⢠⠟⢲⡇⠑⣄⡀⢸⠉⠉⣛⠀⣂⠩
⣼⢋⡄⣀⠀⠐⣯⠄⠠⠞⠰⡁⢈⠇⠈⢧⡇⠀⠀⢠⣻⡏⠀⠀⠀⠀⠀⢸⡁⠈⠓⠊⠈⠒⢻⡟⠷⣦⡀⠀⣤⣤⠴
⢻⣄⠀⠁⢠⣦⠿⢦⣄⠀⠀⠈⠁⠀⠀⣼⠁⠀⢠⣷⡟⠁⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢀⣼⣅⠀⠀⠙⢦⣸⠇
⠀⠉⠛⣶⡞⠁⠀⠈⢸⣇⣀⣀⣠⣴⠞⠛⠷⠶⢷⠟⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣳⠶⣖⡋⠁⠈⣧
⠀⠀⢠⡟⡇⠀⠀⢀⣼⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡜⠁⠀⠀⣰⡟
⠀⠀⠈⠿⣧⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⡾⣦⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠿⢿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣀⣀⣀⠀⠀⠀⠈⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⡿⠿⢿⣿⣷⣄⠀⠀⣼⣿⠋⡴⠋⣡⠀⠀⠙⠳⡄⠀⠀⠀⠀⠙⠻⣿⣶⡄
⠀⠀⠀⣠⣾⣿⠿⠛⠋⠉⣠⡶⠟⠛⠻⢿⣿⣶⣿⡇⡜⢀⣼⣿⣧⣴⣿⠀⢹⡄⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⣴⣿⡟⠁⠀⠀⠀⢸⣿⣶⠶⠶⠿⠿⠿⠻⠿⢷⣧⣼⣟⠹⠟⢹⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⣿⣿⡇⠀⠀⠀⠠⣿⣥⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢮⣍⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⣿⣿⡇⠀⠲⠶⠞⠛⠻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠈⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⡿⢁⡀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠙⢿⣿⣦⣀⣀⣀⠀⠀⠀⢀⣀⣠⣿⢱⣿⣿⠀⠀⠀⠈⠙⢳⣦⣤⣀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⢹⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⣠⣾⠿⠛⠛⠛⠛⠿⣿⡟⠀⠀⠀⠀⣰⣿⡇⠈⠙⠛⠶⠶⣤⣤⣤⣿⣦⠀⠀⠀⠀⠀⡿⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⠘⢷⡀⠀⠀⠀⠁⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⢘⣧⣀⣀⣀⡀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⣾⠋⠁⠀⠀⢀⡾⠋⠁⠀⠈⠙⢷⣄⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠾⠛⠉⢸⣿⠀⠀⠘⣿⡀⣀⣴⠟⠀⢀⣤⠶⠀⠀⠀⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⡶⣤⠤⠶⠾⠛⠁⠀⠀⠀⠀⠀⢀⣸⠏⠀⠀⠀⠈⠛⠋⠁⠀⢰⡟⠃⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⣷⣤⣤⣴⢶⣤⣤⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⡀⠀⢀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠙⢧⡙⢧⡎⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⣾⣭⣭⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⣻⣦⣙⠶⠦⠾⠃⠀⠀⠀⣀⣤⣾⡿⠟⠛⠛⠛⠛⠿⢿⣷⣶⣄⡀
⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⣀⣿⣿⠟⣩⠟⠻⠶⢶⣶⠶⠛⠛⣿⠏⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⣤⣀
⠀⢀⣠⣾⣿⡟⠉⠁⠀⠈⠉⠻⣿⣷⣾⣿⣿⡿⠁⢾⣧⣤⣀⣠⡟⠁⠀⠀⢀⡟⠀⠀⠀⠉⠻⢶⣤⣀⠀⠀⣠⣴⠾⠟⠛⠛⠻⢿⣷⡄
⢠⣿⣿⢿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⢸⡇⠀⠀⠀⢀⣬⠿⠶⢦⣄⠀⠘⣧⠀⠀⠀⠀⠀⠀⢹⣿⣷⡟⠋⠀⠀⠀⠀⠀⠀⠼⢿⣷⣤⡀
⢸⣿⣧⠸⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣼⠃⠀⠀⢠⡟⠀⠀⠀⠀⢹⡄⠀⢻⣄⠀⠀⠀⠀⠀⠀⢿⣿⣠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠙⢿⣿⡄
⠸⣿⣿⡀⠹⡄⠀⠀⠀⠀⠀⠀⠘⣿⣧⡼⠋⠀⠀⠀⠘⣇⠀⠀⠀⠀⢸⡇⠀⠀⠙⠷⣤⣤⣤⣤⡴⢿⡟⠁⠀⠀⠀⠀⢠⣄⠀⢸⣷⡀⠘⣿⣿
⠀⢻⣿⣷⡀⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⠳⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠘⣿⡆⠀⠙⠃⠀⣿⣿
⠀⠀⠹⣿⣷⡄⢳⡀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣆⠀⠐⢶⡀⠀⠀⠈⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠈⣿⣿⡄⢻⡄⠀⠀⠀⠀⠀⢨⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣟⠉⠀⠈⠛⢿⣿⠻⠗⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠘⣿⣿⣄⣹⣦⣤⣤⣤⣾⣿⡿⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⢹⡆⠀⠀⠀⠀⠻⢷⣤⣤⣤⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠛⠛⠛⠛⠁⠀⠀⠉⠻⢿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣸⣿⡇⠀⠀⣇⠀⠀⠀⠀⠀⠈⣇⠹⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⠀⠙⠀⠀⠀⠀⠀⠀⢸⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⣀⣀⣀⣠⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠛⠁

⠀⠀⠀⣶⣶⣶⣶⡶⣀⣀⣀⣀⣶⡶⢶⣾⣿⣧⣀⣀
⠀⠀⢀⣭⣿⡿⣿⣶⠛⠛⢛⣿⣿⠃⠘⣿⣿⡛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⣹⠛⠒⠚⠛⠓⠒⠚⠛⣥⠖⠒⢺⣿⠓⠒⢲⡤⠤⠤⠤⠴⠒⠚⠲⠦⣤⣠⠖⢋⣉⡆
⠀⠀⠀⣇⣀⣀⣀⣀⣀⡀⠀⠀⡿⠀⠀⣾⢿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠰⣏⠁⣀⣀⡀
⠀⠀⠀⠉⣯⣅⠀⠉⠋⠀⢀⣰⣇⣀⣀⣿⡬⠀⠀⢸⣟⡿⠋⠀⠀⠀⠀⠀⣿⠛⠓⢤⣈⣹⠁⢾⣅
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⢉⣽⠷⡀⠉⠉⠀⠀⣠⣍⠉⠀⠀⣀⣼⠀⠀⠀⣿⠀⠀⠀⢈⠃⢀⠀⢧⠤
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠉⠛⠾⠾⠛⠋⠉⠛⠛⠛⠋⠙⠛⠛⠛⠛⠀⠀⠀⠶⠤⠟⠦⣤⠄
⠀⠀⠀⠀⢸⣿⣷⠀⢙⠀⣺⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢰⢮⣍⡉⠀⠈⠀⠈⠋⠉⠀⢀⡀⣀⣀⣀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⡀⠈⠉⠓⠦⣄
⠀⠀⠀⣿⢨⠶⡫⠶⣄⠀⠀⡠⠞⠋⠉⠈⠀⠀⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠈⠑⢆
⠀⠀⠀⢹⡾⣏⣶⣡⡋⢳⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠐⣀⠉⠁⠀⠀⠀⠁⠀⠓⠂⢠⠀⠀⠀⠈⠃
⠀⠀⠀⢀⣿⣿⡿⣶⣷⢼⡇⠀⠀⠀⠀⠀⠀⠐⠂⠐⣷⠏⠘⠉⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠂
⠀⠀⠀⣿⡟⢻⡟⠉⢹⣎⢧⠀⠀⠀⠀⣠⠀⢀⠀⣠⡏⠀⣀⠀⠀⠀⠀⣴⢻⢷⡄⠀⠀⠀⠂⣄⠀⢀⡴
⠀⠀⠀⢻⡀⠺⣿⣀⡼⠛⠀⣱⣆⠁⠈⠀⠀⠀⠀⡛⢁⢸⣿⣷⠀⠀⠀⠉⠙⠛⠀⠀⠀⠀⠀⠉⣾⠘⢦
⠀⠀⠀⠈⠙⠀⠀⠀⣠⠶⢾⡟⣳⣄⣀⣀⣠⣤⡞⢱⣿⣮⡛⠻⣦⣀⠀⠀⠀⠀⠀⢠⣤⣶⡆⢰⡏⠀⢸
⠀⠀⠀⠀⠉⠥⠤⡀⣀⡰⠴⠊⠙⣺⠿⠿⢿⣿⡗⠈⠉⠙⢿⣷⣮⣭⣻⡶⠶⣶⠶⠿⠿⠟⠀⢿⣀⡴⠃
⠀⠀⢀⠀⢀⠀⠰⠑⢈⠵⠶⡶⢸⠃⠀⠀⣀⠿⢠⡀⠀⠀⢲⡄⠀⠀⠉⠛⠿⣷⣶⣶⣶⣾⠇⢸⠃
⠀⠀⠸⠀⢀⠁⠀⠀⠀⠈⠐⢧⣾⠀⠀⠀⠉⢣⠇⠁⠀⠀⡀⠘⠦⣄⠀⢀⠀⠘⠛⠛⠛⢉⠘⠻⡀
⠀⠀⠀⡘⠞⢠⠎⢈⡅⠐⢢⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⣤⣀⠀⠤⢸⣿⡇⠀⠀⠐⡇
⠀⢠⠈⠛⠀⢈⣀⠈⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠀⢀⣾⡟⠀⠀⠀⣞⠁
⠀⠀⠉⠠⡠⠎⡫⠤⢐⡁⢷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⡴⠀⠢⡐⠿⠿⣿⣦⣤⢶⡀⠑⠄
⠀⠀⠀⠀⣵⡿⢷⣰⠿⠿⣿⡗⠲⡄⢻⣶⣶⡶⠶⠶⠾⣿⣿⢡⣈⣁⠀⢀⠁⠀⢀⠁⠀⢙⣾⠳⡀⠸
⠀⠀⠘⣰⡏⠀⢶⡷⠆⠀⠈⡷⠊⢻⣦⡙⠁⠀⠀⠀⠀⠈⡡⠂⠄⠃⠀⡸⠀⢐⠇⠀⣤⠚⠁⠀⠛⢱
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⠗⠚⠀⠀⠀⠀⠐⠈⠀⠀⠀⢀⠀⠀⢀⠼⡃⢀⣀⠀⠀⠀⠀⢺
⠀⢀⠐⠀⢷⣴⠄⠂⠐⠐⢻⣦⡵⠤⠄⠤⠀⠀⠀⠀⠀⠀⢀⣂⣈⣉⠐⢆⡿⢜⡏⡁⢀⣶⡄⠘⢲⠏
⠀⠘⢴⡿⠃⣙⠂⠲⠟⢿⡶⠿⡗⠒⢤⣀⡀⠀⠀⠀⠀⢸⠿⠛⠛⠛⠿⣿⡆⠈⣾⣿⠤⣤⣴⣟⡸⠎⢠
⠀⠀⠀⠀⠘⠆⠀⠀⠀⠺⡝⠂⠉⠊⠀⣠⠃⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠈⣷⢮⠀⡝⠲⠂⠼⠻⠅
⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⡀⠄⡀⢡⡀⠀⠀⠀⣺⡀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠀⣤⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠀⠺⠀⠄⠀⠀⠙⠒⠒⢿⣿⠷⠤⢄⣀⣀⡠⠔⠁⠀⠉⠀⠀⠀⠀⠂
⠀⠀⠀⢰⣆⡆⣶⣖⡆⠠⢤⠀⣰⣆⠀⢤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠄⠀⠤
⠀⠀⠀⠚⠈⠂⠓⠓⠃⠐⠚⠂⠈⠋⠀⠋⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠛⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠖⠶⠶⣏⡔⠉⠀⠈⠑⡄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣄⠀⠀⠀⣀⡴⢾⡇⠀⠀⠀⠠⡈⠙⢦⣀⠀⠀⢸⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠹⣆⣤⠞⠉⠀⠸⡇⠀⢀⣀⡀⠈⢦⡀⠙⢧⣠⠎⠀⠀⠀⠸⣧
⠀⠀⠀⠀⣼⠋⢹⣤⠶⠛⠒⠒⠀⠹⣇⠀⠀⠀⠀⠻⣦⡟⢛⠻⠉⠉⣻⣶⣶⣽⢦⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⣠⠿⣆⠸⣇⠀⠀⠀⠀⠀⠀⢹⡄⠀⣀⣤⣤⣾⣜⣉⡇⠀⣠⠖⠺⣯⡙⠈⢷⡀⠀⠀⢸⡇⠹⣆
⠀⠀⠀⢻⣄⠙⠓⣾⠋⠹⠀⠀⠀⠀⠘⢻⡾⠁⠀⠀⠀⠙⢿⡅⢠⠇⢠⣲⡇⠉⠓⠚⢧⠀⠀⠀⣷⣄⢹⡆
⠀⠀⠀⠀⠉⠻⣞⠋⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠧⡽⠟⡀⢀⣠⡤⠾⡇⠀⠀⠇⠙⣧⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢳⡖⠁⠀⣀⡴⠟⠉⣿⣦⣀⣀⢀⣀⣤⣾⣷⣶⣾⣿⣿⡾⠀⠀⣀⣿⣤⡀⠀⠀⣸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⣍⠀⠀⠀⠸⣏⠿⠿⢿⣿⣿⠿⣿⡿⠋⠉⠘⠧⠴⠊⣡⠄⠀⢹⡶⠞⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠙⢷⣄⠀⠀⠹⣦⡀⠙⣯⡭⠭⢽⡇⠀⠀⠀⠀⠀⠾⠋⠀⠀⣸⠇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠙⢷⣀⣠⡴⢿⣄⠘⠢⠴⠊⠀⣀⣀⣀⣠⡤⣤⣤⣴⠞⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⣠⣴⠛⠁⣴⣋⣉⣳⣶⣶⠞⠋⡽⠛⠉⠙⠳⣦⡶⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠁⣹⢰⡯⠉⠉⠀⠠⣜⠋⠓⢾⠛⢶⣤⣄⡀⠀⠙⣦⡀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣱⡏⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⣴⣇⣠⡿⢤⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠿⠿⠟⠻⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠛⠉⠀⠀⣠⠴⠒⢒⡦⣌⠹⣿⣧⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⢀⡞⢁⠓⢤⠎⠱⡈⠳⣼⣿⣧⣴⣾⣿⣿⣿⣤⣈⠻⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢸⠁⢸⠀⣄⡰⢦⠙⣀⣡⣿⣿⣿⣿⣯⣤⣤⣙⣿⣧⠀⠀⠈⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⣢⣤⣴⣾⠿⠛⠛⠉⠉⠁⠀⠀⣉⣉⣛⣻⣷⡀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠛⠉⠁⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧⣤⣠⡄⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠈⠉⠁⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⣰⣿⣎⢿⣿⠁⠀⠀⠀⢀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡟⠀⠀⠀⠀⣠⣾⡟⠀⣀⣀⣤⣴⠾⠛⣿⣷⡄⠀⠀⠀⢹⣿⣿⣾⣿⣿⣿⣿⣷⣦⡉⠁⢩⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠃⠀⠀⢀⣼⣿⢿⣿⠛⠉⠉⠁⠀⠀⠀⢿⣿⣷⠀⠀⠀⢠⠟⠋⠀⠀⠀⠀⠈⠹⣿⣿⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⢀⣀⣀⣾⣿⠁⠀⢿⣧⡀⠀⠀⠀⠀⠀⠘⢿⡏⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⢿⣿⡇⢀⣴⡿⠛⠛⠉⠛⣿⣦⠀⠀⠙⢻⣷⡀⠀⣰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡏⠀⠀⠤⣄⡀⠘⢿⣷⣄⣀⣿⡟⠁⠀⣿⡟⠿⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠼⢿⣆⠀⠙⠻⠿⠟⠁⠀⠀⢿⣷⡀⠀⠀⠉⠉⠉⠙⢷⣶⣶⣤⣤⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠀⠀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⠀⠀⡀⠀⠀⠀⠉⠉⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⡀⠁⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣤⣾⣷⠶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⢿⣿⣿⣿⣯⣽⣿⣦⡀⠀⠀⠀⠀⠀⢸⣧⠀⠙⣧⣿⣿⣿⡿⢁⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠿⠿⠿⣿⣶⣄⣀⠀⠀⠀⠻⣶⣾⣿⣿⣿⣋⣴⣿⠏
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⡿⠿⠶⣶⡿⠟⠛⣿⣿⢿⣿⣇⣀⡀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⣴⣿⡿⠿⠟⠿⠿⢶⣤⣀⠀⢀⣤⡶⠿⠛⠉⠀⠀⣿⡇⠀⠀⠘⣷⣤⣤⣼⡿⠆⠙⣿⣿⣿⣾⣿⡿⠋⠁⠀⠈⠉⠻⣿⣷⣦⡀
⠀⠀⠀⣠⣾⣿⠿⠓⠂⠀⠀⠀⠀⠉⠻⣿⣿⡏⠀⠀⠀⠀⠀⢀⣿⠃⢀⣶⡿⠿⢿⣏⠀⠀⠀⢰⣿⠈⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⢀⣾⣿⠏⠀⣠⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⡀⠀⠀⠀⠀⢀⣼⠏⢠⣿⡏⠀⠀⠀⠹⣷⠀⠀⠸⣿⣆⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⣿
⠀⣸⣿⡟⠀⣼⡿⠀⣠⣾⠀⠀⠀⠀⠈⢿⣿⢿⣶⣶⣶⡾⠟⠁⠀⠘⣿⣇⠀⠀⠀⣰⡿⠀⠀⠀⠹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⠇⢰⣿⣿
⠀⣿⣿⡇⠀⠙⠁⠰⣿⠏⠀⠀⢀⡀⠀⢸⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⠟⠁⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⢀⡟⣠⣿⣿⠏
⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⣠⣾⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⢀⡞⣼⣿⣿⠋
⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠾⢛⣿⠟⠋⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣄⠀⠀⠀⠀⣠⠟⣼⣿⣿⠁
⠀⠀⠀⠙⠻⣿⣿⣶⣦⣤⣶⣶⠿⠋⠀⠀⠀⣰⡟⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⣿⣿⣷⣶⣶⣿⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⣹⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣦⣤⣤⣀⣀⣀⣀⣤⣤⣶⣿⡿⠟⠋⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⡏⠀⠀⠀⠀⠀⠀⡇⠀⢀⣿⣿⠛⠻⠿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣧⣄⣀⣀⣠⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠨⠙⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣀⣀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⣄⠀⠀⠀⠘⠏⢥⣼⣶⣿⣷⣿⣿⣿⣿⣿⠿⠇⠀⢀⣠⣤⣶⣶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⢿⣯⠁⠁⣴⡿⣿⣟⠻⣦⣶⣿⣟⣿⣏⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣤⠀⠀⣥⣦⡀⠉⣻⣿⠛⢷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠉⠀⠀⠉⠛⠁⠀⠈⠉⣠⡈⠻⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣷⣄⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣦⣬⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣷⣿⣿⣿⣿⣿⡀⢀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⡿⠿⣿⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠠⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⣴⡿⣷⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣌⣥⣈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁
⠀⠀⠀⠀⠀⠀⡀⠙⠁⢀⣀⡄⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣀⣠⠋⠀⢀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠙⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠑⠠⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⣈⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⠀⠉⢈⣉⣉⣩⣿⡷⠀⠀⠀⠊⠉⠉⠀⠀⠉⠉⠙⢛⡻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡾⠿⠛⠛⠛⠛⠛⠿⢿⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠻⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⢵⣷⣼⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣂⣀⣀⣷⣀⣀⣀⣂⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣛⠻⣿⣛⣻⣟⣿⣿⣫⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠒⠲⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠰⢆⠀⡠⠦⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢠⢹⠾⣝⠿⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢦⡌⢀⣈⣀⣹⡄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡠⠤⠘⠂⠀⠀⠉⠉⠁⠀⠈⠓⠠⠤⣀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⣠⠋⠀⠀⢠⣤⣶⣶⡊⠉⠉⠉⠉⠉⢉⣿⣿⣶⣦⣄⡀⠀⠉⡄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠃⢀⡠⢾⡿⠛⠉⠛⠷⡀⠀⠀⠀⢠⠟⠋⠉⠙⠻⡌⠙⢷⡄⢰⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠀⠶⡟⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⢻⠀⡆⠀⠀⠀⠀⢸⣿⣿⣇⡀⠀⠀⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⠀⠃⠀⠀⠀⠀⠈⣿⠛⠉⠁⠀⠈⠉⠻⣿⠃⠀⠀⠀⠀⢈⣀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢻⡴⠊⢀⣶⣄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠸⠃⢀⣠⣶⡄⠀⠉⣼⠃⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠸⡇⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠺⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⡇⠀⠀⠙⢻⣿⣿⣷⣄⡀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠋⠀⠀⠀⡿⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠙⠛⠛⣻⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠘⠉⠙⠲⣦⡀⠀⠀⠀⠀⠊⠁⢀⡴⠚⠋⢹⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡇⠀⠀⠀⣉⠙⠶⠦⠤⠤⠶⠞⢻⠀⠀⠀⠀⡆⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⣤⢀⣀⣀⣀⣀⣀⣀⣨⡄⠀⠀⠀⢡⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⡌⠁⠈⠂⡌⠀⠈⠉⠉⠉⠉⠀⠈⠀⠎⠀⠉⢿⣄⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⢎⣃⣀⣀⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⣎⠿⣆⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⣴⠏⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢹⡆⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⣀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢀⣨⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠙⠛⠛⠛⠋⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠓⠛⠛⠉⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⣾⢉⠛⠒⠶⠶⠶⠶⠚⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠗⠶⠶⠶⠶⠶⠖⠛⣷⡀
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⡀⠀⠀⢰⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠠⠅⠀⢰⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠀⢸⠀⠈⡆⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⡴⠟⣆⣀⣹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠈⠁⠀⠀⠀⠀⣼⠀⣇⣀⣀⠾⢦⣀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⠶⠋⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢤⣄⣀⣀⣀⠀⠀⠀⠀⣠⡷⠤⠤⠴⠒⠚⠉⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠙⠳⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠛⢯⣔⣂⣀⡤⣀⣠⣰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣧⣤⣤⣤⣽⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣷⣦⣤⣤⣄⣀⠈⠁⠀⠀⠉⠛⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣿⡏⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡻⣝⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⢠⣿⡿⣥⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⠓⠶⠦⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠈⠀⠀⠀⢰⣷⠄⠀⠀⠠⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛

⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⣺⠧⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⣀⣀⣠⢤⠒⠉⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⡀⢀⠴⠚⠅⠀⠀⢈⣖⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣰⠀⠀⠠⡀⡴⠀⠐⠺⢉⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢮⠀⠢⣀⡀⠀⠀⠀⠀⠈⢀⣁⠤⠞⢁⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠉⠑⠒⠢
⠀⠀⠀⠀⠀⠀⢀⠤⠶⠒⠒⠚⠋⠒⠒⠒⢢⡇⣠⣀⣧⡠⠤⠤⠤⠴⠟⢳⠋⠛⠉⠀⠀⠘⠓⠢⡄⠉⠓⠪⢂⣠⡮⠁⠀⠀⠀⠀⠀⠘⠒⡄
⠀⠀⠀⠀⠀⠀⣜⡀⠄⠀⠠⠤⠀⢀⣀⣠⠞⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⢲⡞⠀⠀⠀⠀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠻⡤⢂⣤⡄⢀⡼⢋⡭⠟⠀⠀⠘⠀⠀⢀⡠⡤⠒⠚⠛⠛⠳⠷⠤⣤⠀⣦⣹⠋⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⡴⠛⠛⣜⠤⠏⣴⠋⠀⠀⠀⠀⠘⠤⣤⢴⠃⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⢀⠴⠟⠀⡠⠊⠈⠳⣤⠯⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠤⠔⠒⠉
⠀⠀⡰⠃⠀⠀⠰⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣀
⠀⠀⢻⣀⠀⢀⣼⣤⡀⠀⠀⠁⠀⠀⢀⣠⢴⣀⠰⠞⠁⠀⠙⢷
⠀⠀⠀⠉⠛⠋⠁⠀⠈⠳⢦⠤⠶⠚⠉⠀⠀⠈⠳⡄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⣂⡀⣀⢀⢀⠀⣀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡼⠛⠋⠻⣆⠀⠀⠀⠀⠐⠆⠚⠞⠓⢃⣾⠄⢸⠀⣀⢠⣼⠐⢽⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠀⠉⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠚⠳⠖⠺⠛⠻⠻⠟⢻⡟⠛⠟⠛⠿⠛⠿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠛⠛⠛⠉⠀⠀⠀⠀⠠⢸⠀⠀⠐⢀⢈⠸⠀⠸⢸⡇⠀⠂⠁⡇⡇⠰⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣶⣶⣶⣾⣶⣾⣶⣾⡷⣶⣷⣶⣶⣶⣷⢶⣷⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⣀⣄⣄⢀⡀⠀⠀⣲⡀⠀⠀⠀⠲⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠉⠀⠀⢠⣄⠀⠙⠂⢀⠈⠀⠁⣉⠋⣷⡆⠀⠀⣬⠖⠠⣇⠀⠈⡲⠄⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣄⠀⠀⣀⠀⠀⠄⣆⢬⡼⠓⠀⢠⣰⡧⣀⣀⣻⣇⡀⠀⠀⠀⢀⠉⢀⠙⣄⠈⠉⡒⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⠀⣧⡄⡈⣥⢾⣜⡓⢉⣁⡀⠀⡀⠈⠁⠀⠀⠀⠈⢛⣁⠐⠤⠤⢀⡉⠁⠳⢽⡄⠀⡇⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⠀⢰⡈⢳⣶⣿⣦⣿⠿⢋⠁⡠⢈⣨⠍⣁⠐⠦⠀⠀⠀⠀⠀⠈⢥⡆⢀⠀⠨⡇⠢⣀⠀⣿⠢⣅⠀⠀⠹⣿⠀⢀⡄
⠀⠀⠀⢀⠀⠀⠀⠎⠀⣠⣿⡏⢱⢋⣴⠥⠐⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢀⡆⣷⢀⠈⠀⢹⠆⢺⣷⡀⢣⢹⣦⠞⢹⡆
⠀⠀⠀⢸⡄⠀⡄⠀⣸⠿⣸⠷⣿⠁⣀⢂⡂⣔⣬⣰⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢿⠓⠾⢬⣤⡈⢧⡑⣼⢠⣿⣦⣿⠁
⠀⠀⠈⡀⢧⣠⠓⣤⣿⢟⡟⣸⡿⠬⠋⠣⠑⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⠌⢳⣄⠸⡋⠙⢫⣷⣼⠁⢸⣿⠁⢰
⠀⢢⠀⢧⠸⣧⠑⣾⢧⡾⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠙⣿⡗⢾⣿⣊⣋⠄
⠀⠈⣧⢨⠃⠸⣿⣣⡾⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⣄⣿⣧⠸⣿⠈⣯⠆
⠀⠀⠘⣷⣀⠀⢻⣿⡡⣺⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣌⢻⣿⠀⣿⣇⡳⠂
⢦⡀⢰⡛⣿⣦⣾⣯⣠⣿⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⠹⣿⠳⣿⣧⣟⠁
⠈⢷⡌⡾⠰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣱⣿⣧⣿⣿⠃
⠀⠈⢷⡸⡇⢸⣿⣿⠁⣀⡀⠀⣤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⣿⠇
⠀⠀⠘⣿⠷⣸⣿⣿⠘⡿⠛⡸⡯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⠀⠈⠣⣼⣿⣿⣿⣆
⠠⣼⡇⣹⣯⢸⣿⣿⡏⠀⠐⣮⣿⢅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢷⣫⣴⣶⣼⣿⣿⡇⢸
⠀⢨⣟⡵⢻⣿⣿⣿⣇⠀⢐⡫⢷⢝⡂⠀⢄⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⣷⡋⠡⣼⣿⣿⡇⢸
⠀⠀⢻⣫⠌⢻⣿⣿⣿⡆⠀⢩⠽⣝⣃⣨⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣗⡂⢀⣿⣿⣿⡃⣾⠀⢸⠁
⠀⠀⠈⠧⣞⢄⢹⣿⣿⣿⣄⠈⠬⣯⢲⣵⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠬⢽⣿⠒⣾⣿⣿⣿⣶⠃⠀⠅
⠀⠀⠀⠀⠻⢎⢔⣿⣿⣿⣿⣦⠨⣧⣾⣿⢓⣂⣤⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣯⣄⠀⡄⠀⢀⣦⣄⢺⣿⣃⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢏⣿⣿⣿⣿⣿⣷⣾⣿⠻⠛⢻⣟⣿⣿⣤⣤⠀⠀⠀⢀⠀⠀⠐⡩⣻⣽⠇⡇⣇⡼⣻⠏⡽⢷⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠿⣿⣿⣷⣬⣧⣼⣽⣽⣿⣿⣀⠀⣈⣄⣄⣀⣈⡴⠖⠓⣚⡛⠛⣿⣾⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠷⣤⣀⡀⢹⣿⣟⠁⠀⠀⠀⡀⢀⠌⠋⠙⠻⣧⡊⠁⢀⣅⠚⢻⣛⣠⣿⣿⣿⣿⣿⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⠿⢿⣯⣭⣭⠾⢿⣤⣀⢀⠠⠊⢀⣱⡤⢶⣿⠖⡛⠛⠋⠛⠛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣿⣿⣧⣼⣾⣾⠾⠷⣦⠾⡉⠐⢣⡏⠀⠀⠀⣈⡥⠖⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣙⣇⣀⣜⣁⣀⣈⣠⣎⣈⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⠛⠉⠀⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢏⡴⠒⣖⠲⢦⡀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⣴⣾⠿⣶⣶⣶⠶⣷⣦⡀⠀⢀⣀⣀⣀⣿⡇⡞⠀⣸⡿⠁⠀⠙⡄⠀⠀⠀⢹⣿⡄
⠀⠀⢀⣤⣴⣾⡟⠀⠀⠘⡿⠁⠀⠈⣿⣿⣿⠿⠿⠛⠛⠛⠿⢧⣴⣿⣷⣦⠀⢠⠇⠀⠀⠀⠀⢻⣿⡄
⠀⠀⢸⡏⠀⠙⢷⠀⠀⠀⡇⠀⠀⢰⡿⠿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠉⠻⢥⣴⠏⠀⠀⠀⠀⠀⠈⣿⣧
⢀⣴⠿⢷⡄⠀⠈⠃⠀⠀⠀⠀⣀⣿⠀⠀⣼⠟⠛⢻⣿⡿⠳⠶⣤⡀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⢹⣿⡄
⢸⣧⡀⠀⠙⠂⠀⠀⠀⠀⠐⠋⠁⠀⠀⢀⣿⣄⣠⣿⣿⣾⠇⠀⠀⠙⡳⣄⠀⠀⠙⢦⠀⠀⠀⠀⢸⣿⣷⣤⡀
⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠛⠻⢿⡿⠀⠀⠀⣼⡟⠈⠛⠶⣤⣈⣳⠀⠀⠀⠘⠃⠙⢿⣷⡀
⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⡿⠁⠀⠀⢀⡀⠉⢹⣧⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠈⢿⣿⠖⠒⠒⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⡟⠛⠛⠋⢹⡆⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣤⡴⠚⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⢧⣀⣀⣤⡿⠋⠉⠛⢶⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡀⠘⣏⠉⠛⠛⠛⠒⠒⠒⠛⠋⠉⠉⢹⡆⠀⠀⠉⠉⢡⣶⠒⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠈⠳⠶⣤⡶⢦⣀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠹⣧⠸⡍⠛⢿⠿⠟⠋⠀⠀⢀⣤⣶⣄⣈⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠙⢧⡙⠶⠟⠀⠀⠀⠀⣴⠟⠁⠀⠈⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⠀⠈⢿⡄⠀⠀⠀⠀⣼⣿⣶⣶⡶⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣴⣾⠟⠛⢻⠟⢻⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⡞⠁⠀⣴⠋⠀⡾⣿⠉⠛⠻⢿⣶⣤⣀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣤⣶⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⡇⠘⠶⠤⠼⠁⠀⢸⠇⠙⣇⠀⠀⠀⠉⠛⠿⠿⣿⡿⠿⣿⡿⠿⠛⠋⠉⠴⠋⢀⡨⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣸⠀⠀⣠⠖⠋⠳⣆⢻⡀⠀⠘⢷⣄⡀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠐⠛⠃⠀⠐⠚⠁⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⠏⠀⠀⣇⠀⠀⠀⣸⠸⣇⠀⠀⠈⣿⣿⣷⣤⣀⣿⠀⠀⠀⠀⠀⠲⠶⠆⠀⠀⠰⠒⠋⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠘⠳⠤⠶⠋⠀⠈⠳⢤⣤⣿⣿⡏⠛⢿⣿⣄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠟⠻⣷⣄⠀⠀⠀⠰⣶⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣀⣀⠀⠀⠀⣀⣠⣴⡿⠋⠙⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠙⠛⠿⠿⠿⣿⠻⠛⠋⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⠂⠀⢀⣿⠷⠤⠶⠃⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⢀⣾⢃⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⣀⣠⣼⣷⣄⡀⠀⣀⣠⡶⠟⣡⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣶⣶⣶⣿⣿⣿⣭⣭⣥⣶⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⡰⢋⡰⠢⣈⢢⠀⠀⠀⠀⡐⠣⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠔⠄⡀
⠀⠀⠀⠒⠂⠀⠀⠀⠀⣿⠵⠖⠲⠶⣵⡇⠀⢀⠘⠓⢨⡉⠆⠀⠀⠀⠀⠐⣄⡈⠀⠀⠀⠀⠘⢤⠂⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣤⡏⢙⣧⣭⡕⠀⠀⠳⠺⠛⠁⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡄⡈⠋⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢭⡩⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢆⡈⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠓⠖⠂⠑⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠀⠀⠀⠀⠠⢤⠲⡀⠀⠀⠀⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣤⠀⠀⠀⠀⢀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠱⠶⠁⠢⠀⠀⠀⠀⣸⢩⠅⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⢉⡙⠚⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠑⠘⠀⠀⠀⠠⠀⠈⡠⠠⠔⠐⠄⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠇⠀⠀⠁⠀⠀⠰⠊⠠⠀⠉⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠂⠀⠀⠘⡀⠀⠐⠐⠖
⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠅⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠈⠀⠀⠘⠂⠀⠚
⠀⢀⠄⠱⠞⠁⠈⠂⠀⠀⠀⠀⠀⠠⡤
⠀⢰⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡒⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠦⠀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⢰⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡆
⠀⠀⢨⣹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠁⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣠⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣧⣄
⣼⣿⠟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠻⣿
⢻⣿⣦⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣴⣿
⠀⠙⣻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠉
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢻⣿⡇⠀⢰⣿⡟⠉⠙⣿⣷⠀⠀⣿⣿⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡗⠀⠀⢴⣦⠈⠉⢠⣶⡄⠀⢸⣿⡇
⠀⠀⣼⣾⡅⠀⠘⢿⣿⣦⣾⣿⠏⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⠀⠀⠈⠉⣠⣤⠈⠉⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⢀⣀⣸⣿⣿⣀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠘⠻⢿⣿⣿⠿⠟⠀⠀⣿⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣇⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠈⠿⠇⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⠇
⠀⠀⠀⠙⠻⢿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣻⣿⠟⠋
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠆⠘⠃⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⣇⠀⠀⣿⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠁⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣀⣠⣦⡀⠀⢀⠀⠀⠀⠈⠉⠀⡀⠀⠀⠀⠄⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠙⢉⡟⠁⠀⠀⠀⠀⢤⠀⠀⠀⣷⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⢰⣤⣠⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⣿⡄⠀⢰⣿⡇⠀⠀⢠⠀⠀⠀⣀⣂
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⢰⠛⡚⣅⠀⠀⠀⠀⢸⣿⡯⠀⠀⡞⠃⠀⠀⣿⣧⠀⠀⠈⠉
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠻⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠁⢶⣀⠀⠀⠀⠀⠹⡿⡛⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠄⠈⢆⡀⠀⠀⠐⢍⣣⡀⠀⠻⠿⠿⢷⣿⣽⠄⠀⠘⠀⠶⠀⠀⠀⠄⡠⠀⠀⠢⣄⣠⡆⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠉⠁⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠖⠛⠇⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠁⠀⢈⣴⠛⣹⣤⠀⠀⠀⣠⠖⢀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡩⠘⠁
⠀⠀⢠⠀⠀⠀⠀⠀⢼⣦⡄⠀⠀⠀⠀⠈⢹⡿⣿⡇⠀⠀⢀⣒⣀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠀⠀⢰⠶⠶⠆⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠶⠦⠄⠀⠀⠀⣅
⠀⠀⠀⠀⠰⣛⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠛⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣀⣀⣀⠀⠀⠀⠉⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠿⢿⣿⣿⣦⡀⠀⠀⣼⡿⢃⡴⠛⣉⠉⠉⠙⠢⡀⠀⠀⠀⠀⠉⠻⣿⣶⣄
⠀⠀⠀⢀⣴⣿⣿⠿⠛⠛⢁⣴⡶⠿⠿⢿⣿⣶⣼⣿⠃⡞⠀⣴⣿⣆⣴⣿⠀⢹⡀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⢠⣿⣿⠋⠀⠀⠀⢠⣿⣧⣴⣶⠶⠶⠿⠿⠿⣿⣼⣅⣾⡟⢹⠿⢻⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⣿⣿⡇⠀⠀⠀⠀⣼⣟⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣜⣋⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⣿⣿⡇⠀⠤⣤⣤⠿⠿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠋⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢰⣿⡇⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠈⢿⣿⡋⠙⠉⠁⣠⣴⣶⡶⠶⣿⣷⣿⣿⡇⠀⠀⠀⠀⢠⣶⡏⠙⠳⢶⣤⣄⣀⣀⣈⣻⣆⠀⠀⠀⠀⠀⢸⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⣾⡿⠋⠀⠀⠀⠀⠀⠉⠛⢆⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠉⣿⠙⣷⡀⠀⠀⠀⠸⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⣠⡶⠟⠁⠀⠀⢀⣾⠟⠛⠛⠶⣦⡀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠞⢻⣷⠀⠀⠻⣦⠀⠀⠀⣠⠟⠀⠀⢀⡀⠀⠈⠹⣦⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⣀⡀⠀⠀⣀⣠⡴⠛⠛⠛⠉⠉⠀⠀⢠⡿⠀⠀⠀⠻⢶⠴⠞⠁⠀⣠⣾⠉⠁⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣀⣤⡀⠀⠀⢀⡀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣍⠙⣿⡷⢾⠟⡻⠛⠛⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣦⣤⣤⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡈⢷⣌⠻⣆⡀⠀⢠⡟⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣶⣾⣿⣿⣟⡋⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⢻⣿⣦⣽⢷⣬⣉⣛⣋⣀⣀⣀⣤⣴⣾⣿⣿⠉⠁⠀⠀⠉⠉⠙⠻⠿⣿⣦⣄⡀
⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⢿⣿⣷⣄⠀⣀⣠⣾⣿⠟⣡⡾⠋⠉⠛⣻⠟⠉⠉⠁⣼⠁⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣶⣶⣤⡀
⠀⣠⣴⣿⣿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⠃⠐⠻⠶⠶⠤⣾⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⢷⣦⡄⢀⣤⡶⠟⠋⠉⠉⠈⠙⢿⣿⡄
⢰⣿⣿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⣼⠀⠀⠀⠀⣴⠞⠛⠛⠳⣆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣶⡄
⢸⣿⣿⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⡟⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡆⠀⠘⣧⡀⠀⠀⠀⠀⠀⢸⣿⣧⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢿⣿
⠘⣿⣿⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⠛⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣸⠇⠀⠀⠈⠛⠶⠦⢤⠤⠶⢛⡿⠁⠀⠀⠀⠀⠀⣶⡄⠀⢻⣷⡄⠈⣿
⠀⠹⣿⣿⡄⠸⣆⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠸⣿⠆⠀⠉⠁⠀⣿
⠀⠀⠙⢿⣿⣦⠹⣆⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⣷⣀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⢻⣿⣇⠘⣆⠀⠀⠀⠀⠀⢀⣽⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠁⠀⠀⠈⠛⣿⡟⠻⠃⠀⠀⠀⠀⠀⣠⣼⣿⠟
⠀⠀⠀⠀⠈⣿⣿⣦⣈⣷⣦⣤⣤⣶⣿⡿⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠈⣇⠀⠀⠀⠀⠈⠻⣷⣤⣤⣤⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠛⠛⠛⠋⠉⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠸⡄⢿⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣇⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⢀⣀⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠉⠉⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠺⠗⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠊⡱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡔⢢⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠡⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠌⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣽⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣯⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣧⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠠⠁⣀⣠⣂⠀⡤⠃⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⢀⠈⢸⡆⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠃⠄⠀⠁⠀⠁⠏⠀⠀⡤⠂⣠⠾⠅⠀⠀⠊⠭⠋⠀⣈⠉⠛⢤⡀⣧⡀⠀⠘⣿⢃
⠀⠀⠀⠀⠀⠀⠀⢠⢦⡟⠀⢁⠞⠁⠀⣢⠎⠀⣂⣴⡤⠊⠁⠀⠈⠈⠁⠀⠀⠀⣸⣿⣿⡧⠀⠙⢿⡄⠑⢀⣿⡈⣞
⠀⠀⠀⠀⠀⠀⡰⠃⡿⠀⢸⠃⣠⡶⢋⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⡄⠘⠀⠉⣷⣾⣿⣧⡿⡄
⠀⠀⠀⠀⠀⣰⠁⢰⡇⠀⣿⣟⠁⣰⣿⠡⠄⠀⠀⠀⠀⠀⠀⠒⠠⠤⣭⡽⠦⠤⡿⣿⣿⣿⣧⠀⠀⠀⣌⣿⣿⣿⡇⠘⡄⠀⡀
⠀⠀⠀⠀⢠⣿⠀⢸⣧⣰⡿⠈⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⢦⡀⠀⠀⣿⣿⣿⡇⠀⢸⣠⠃
⠀⠀⠀⠀⠘⣿⠀⠀⣿⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣿⣿⣿⡇⢀⡾⠁
⠀⠀⠀⠀⠀⢻⡄⠀⣿⣿⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣼⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⠃⠀⡊⣉⣷⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⣧⠀⢀⠀⠀⠀⠀⠀⢈⣽⡇⠀⣿⠀⠀⣄⡟⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠛⠻⠿⡿⠶⡷⠶⠶⣤⣬⣾⣿⣸⣿⣶⡶⠿⡿⣿⠟⠋⠉⢙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠿⣧⣤⣤⣴⣷⣤⣧⣦⣀⠠⠬⣿⣿⠛⡲⣗⡂⢠⢠⣿⠀⠀⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣨⣿⣾⣿⠁⠀⠀⠴⠿⠿⢭⣭⡭⠀⠼⠿⠷⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠶⠒⠾⠖⠛

⠀⠀⠀⠀⠀⠀⠖⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⢑⣄⡤⢠⡀⠀⠀⠀⠀⠆⠈⢰⣄⠑⠀⠀⢣⠀⠀⠀⠀⠀⠀⠐⠃⢀⠐⠀⠀⠀⠂⡀⠠
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠀⢠⠞⢀⡄⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠
⠀⠀⠀⡤⠀⠀⢀⠈⠀⠀⢈⠀⠀⠀⠠⠁⠀⠀⢀⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠤
⠀⠀⠀⠉⠀⠀⠀⠉⠈⠀⠀⡄⠀⠀⠀⢀⠤⢄⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠞⠈⣀⡀⠄⠀⠀⠀⠀⢀⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠊⠉⠆⠀⠀⠀⠉⡎⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠄⠀⡀⠀⠀⠀⠀⠢⡀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠆⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠈⠠⡀⠊⢲⣀⣀⠴⠧⠤⠃⠀⠀⠀⠀⠀⠀⠈⡆⠠⠔⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠑⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠙⠂
⠀⠀⠀⢀⠤⠐⠒⠒⠀⢄⠀⠀⠀⠀⠀⠀⢀⠀⠰⠖⠒⠠⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠄⢀
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠤⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠘⠄
⠀⠀⠃⠀⠀⢠⠀⢠⠀⠀⠀⠃⠀⠀⠀⠁⢀⠀⠀⡀⢀⠀⢀⠀⠁⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠈⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠰⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡓⠀⢉⠀⠃⣀⢈⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠉⠈⠲⡀⠀⠀⠀⠀⠠⠚⠂⠐⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⢿⠋⡙⡟⠉⡉⠻⠋⠻⠛⠃⠀⠀⠀⠀⠐⠀⠇⠀⠀⠘⠂⢀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠈⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣤⠄⠀⢀⡁⠈⠀⡁⢸⠀⣤⠀⠛⢀⡆⠀⠀⠀⢀⢀⡀⢤⠖⠀⠻⠑⠀⠀⠀⠀⢀⠀⡀⣤⡶⠐⠻
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠃⠀⠚⠛⠋⠘⠛⠛⠙⠛⠛⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠐⠄⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠚⠁
⠀⠀⠀⠀⠐
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠄⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢘⠀⠀⠛⠉⠉⠉⠉⠙⠀⠀⢉⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⣀⠄⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⡃⢘⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢘⠀⠀⣁⠀⠀⠂⠀⣀⠀⠀⢈⠀⠀⢘⠀⠀⠀⠀⠁⡐⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠛
⠀⠀⣀⣠⣈⣡⣴⣆⣀⣀⣤⣀⠀⠀⠘⠄⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⠀⠀⢀⠔⠒⠒⢦⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⢀⡀⠀⠄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠔⠉⠁⢀⡀⢘⠀⠐⢄⠀⠀⠀⠑⠄⠀⠛⠘⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⠁⠈⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⢰⠁⠀⠁
⠀⠀⢐⣇⣀⣀⣀⣀⣀⣀⣘⡇⠀⠀⠣⣀⠀⠀⠀⠀⣀⣀⡠⠞⠀⠀⠀⠀⠘⠈⠀⠀⠉⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠑⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡆⠰⣤⠀⠀⠀⣿⠀⠄⢸⠉⠲⠤⠤⠶⢊⡴⠋⠁⠀⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⠙⠳⢶⡚⠁
⠀⠀⠀⠀⠧⠀⢤⢤⠤⢤⣥⡀⢀⠸⠀⠀⠀⠀⣰⠋⠀⠀⠀⣰⠋⠳⣄⣠⠞⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢤⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠶⢤⡾⠁⠀⠀⠀⢠⠇⣠⣄⠈⠁⣠⣆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠈⠉⠉⠉⠙⠲⠤⣤⣤⣤⠤⠖⠋⠀⢀⡞⠀⠀⠀⠀⠀⣟⣠⠏⠈⠳⠾⠁⠸⠦⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⠦⣤⣠⣤⠤⠶⢤⣤⣤⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⣀⣬⠥⠶⠒⠚⠛⠛⠛⠛⠒⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡴⠚⠉⠁⣀⣤⠤⠴⠶⠶⠶⠶⠶⠤⢤⣄⣀⡀⠉⠙⠶⣄⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠋⢀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠉⠛⠶⣤⣀⠙⢦⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡴⣿⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣷⡄⠀⢼⡉⠛⠾⣇⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡤⠤⡖⠒⠲⣄⠀⠀⠀⢈⡟⢰⡏⣾⡟⢉⠝⢿⣿⡇⠀⠀⠀⡿⠿⢋⠔⠀⠢⡙⢿⠀⠀⠻⣄⠀⠈⠙⢦⣀⣸⣇⣤⣀⠀⡀
⠀⢀⡴⠛⠛⢋⠀⠀⠈⢂⠀⠘⡆⠀⢀⡾⠛⣿⠀⠋⢠⠁⢰⣶⡀⠀⠀⠀⠀⠀⠀⡌⢀⣾⣦⠰⠈⠀⠀⠀⢹⡆⢀⡼⠋⠉⡽⠁⠀⡨⠋⠉⠳⣤⣤⡀
⣰⠛⠢⡀⠀⠀⠑⡀⠀⣴⠛⠋⠉⠛⠻⣄⠀⡟⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠃⠀⠀⠀⠀⣇⣾⣥⠶⠚⠓⠶⢴⡁⠀⠀⡐⠀⠀⠙⢦
⡟⠀⠀⠐⠄⠀⠀⠐⠀⢸⡆⠀⠀⠀⠀⢸⣂⡇⠀⠀⠠⢸⣿⣯⠷⠶⠶⠶⢤⡀⠀⡄⣿⣿⣿⣷⣇⠀⠀⠀⠀⠿⣿⠁⠀⠀⠀⠀⢨⠇⠀⡐⠀⠀⠀⠔⠉
⠻⣄⠀⠀⠈⣀⣠⠴⠓⠁⠙⠶⠖⠁⣠⠏⡿⠀⢠⣦⢘⡾⠋⠀⠀⠀⠀⠀⠀⠙⢦⢐⣹⡿⠟⠁⣿⠀⠀⠀⠀⠀⢹⡄⠀⢦⣀⡴⠛⠦⣴⡀⠀⠀⠊⠀⢀
⠀⢹⡉⠉⠉⠀⠀⠀⠀⠀⠈⢀⣠⠞⠁⢰⣇⠀⣼⠝⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⢘⣿⠀⣀⠀⠀⠀⠀⢹⣦⡀⠀⠐⠄⠀⠀⠈⠑⠒⠶⢲⡏
⠀⠀⠳⣤⡀⠀⠀⠀⠀⠐⠈⢹⡇⠀⠀⠈⣏⠙⢾⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠰⣿⠟⠋⠁⠀⠀⠀⢀⡟⣀⣽⡗⠤⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⣇⠀⠀⢀⣀⣠⠤⠶⠺⢧⣀⡀⠀⠹⡄⠀⠻⣆⠻⢦⣀⡀⠀⢀⣀⣠⠞⠃⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡞⠛⣁⡼⢤⣄⣀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠙⠻⡏⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠓⠻⣄⠀⠈⠛⠶⡾⠿⠟⠛⠛⠳⠶⠶⠚⠋⡗⠀⠀⠀⠀⠀⠀⣠⠞⠛⠉⠁⠀⠀⠀⠉⠙⠓⠲⣶⠶⠃
⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠹⠒⠲⠶⠶⠶⠒⠚⠙⠚⠁⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⣷⢤⡀⠀⠀⠠⣄⣀⡤⠀⠀⠀⢠⠄⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⢦⢄⡀⠀⠀⣰⠀⠈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠙⣦⣀⠀⠀⠀⠀⠀⣀⡴⠿⡟⠁⢰⡇⠀⠀⠀⢠⡦⠤⠶⠒⠛⠉⠁
⠀⠘⢆⠉⠓⠞⢁⣖⣒⣒⠒⠒⠉⣩⠛⠛⠓⠒⠲⡇⠀⢿⠀⠀⡇⠉⢙⣛⣛⣛⣋⣉⣀⣘⡇⠀⢸⡇⠀⣠⠴⠞⠓⠒⠲⢤⣄
⢀⡠⠚⠀⠀⠀⢿⡷⠞⠉⠀⠀⠀⠑⢤⡀⠀⠀⠀⡇⠀⢸⠀⣠⠿⠚⠋⠉⠉⠁⠀⠀⠉⠉⠛⢒⣼⣤⠞⠁⢀⠤⠤⣀⡀⠀⠈⢳⡀
⠯⠀⢰⠞⠓⢄⣘⣳⠶⢾⠀⣴⠗⢶⡆⠽⠂⠀⠀⣧⡤⢾⠟⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢹⠀⡔⠁⠂⠄⡀⠙⢦⠀⠀⢳
⠀⢠⣋⣴⠞⠟⠁⣰⡒⠚⢟⠷⢤⡀⢹⡀⠀⠀⠀⣿⠀⠸⣄⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣴⠇⡼⠁⠒⠤⢀⠀⠑⠺⡆⠀⣸
⢀⡼⠁⢠⠟⠒⠚⠁⣅⠀⠀⡸⠂⠛⢮⡀⠀⠀⠀⢹⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠒⠠⢀⡀⠈⠐⢰⠃⢠⡟⠉⠉⠉⡏⠉⠉⠉⡋
⠟⠒⠒⣡⠤⢄⠀⢀⣀⣉⠉⠀⠀⠀⡞⠙⣆⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡏⠑⠢⢄⣀⠈⠑⠢⣞⡴⠋⠉⠉⠋⠉⠉⠉⠋⠉⠉
⢱⡄⢸⠁⠀⠈⣷⠋⠀⠈⢣⢴⠂⠐⠷⣂⣼⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢸⡃⠂⢄⡀⠈⠉⠒⣼⠋⠀⠀⠐⠒⠓⠒⠖⠒⠓⠒⠖
⠻⢤⣌⣦⣀⣴⣋⣦⣄⣤⣋⣦⣠⣤⠴⠞⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠳⣄⡀⠈⠁⢂⡴⠏⠠⡄⠀⠆⠀⡤⠀⠒⠀⠤⠀⠂
⠀⠀⠀⠀⣨⡏⠀⠙⢦⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡬⠉⠛⠛⠉⠛⠛⠉⠀⠀⠀⣇⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠤⠤⠤⠤⠤⠤⢴⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠐⠒⠚⠛⠛⠓⠂⠐⠂⠀⠉⠁⣾⣿⣿⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠴⠇⠈⠀⣀⣀⠀⠤⠀⠉⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣛⣤⡟⢁⡄⠈⡁⠀⠀⠀⢀⣼⡿⠋⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢸⠀⡐⠀⠀⣀⠀⠀⠘⠶⠊⠀⢿⡿⠀⠛⠁⠀⢁⠀⠠⠔⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢸⠀⢢⠄⠈⠀⠘⢋⠛⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⣭⠈⠉⠀⢄⠀⠆⠀⠀⠀⠀⠈⠉⠉⠛⠛⣿⠏
⠀⠀⠀⠀⢸⡀⠶⠈⠋⠀⠛⠛⠁⠀⠀⠀⠀⢀⠙⣷⡄⠀⢀⠘⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠚⢿⡄⠀⠘⠀⠀⠀⠀⠀⠸⠤⠤⠀⣴⡦⢤⣀⢠⠄⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠑⢠⡀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣤⠂⣀⠛⠇⣼⣿⢏⠖⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠉⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠛⠀⠲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠂⠐⠛⠋⣾⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠗⠀⠀⠀⠀⠈⣳⡌⠒⠛⠒⡒⠩⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢻⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠐⠞⠉⠘⠉⠉⠈⠉⠀⠀⠙⣶⠁⡀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢻⡇⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠀⠰⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⣌⠉⠀⢀⡴⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠘⠻⠟⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡤⠤⡶⢦⣄⣤⣀⣴⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⣬⣀⣄⣀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⢠⡤⠤⢤⡤⣤⣤⣤⢤⣤⡤⢤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣈⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⢉⠋
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠆⢁⡤⠒⠉⣉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣍⠀⠉⠙⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡀⠀⠛⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠱⢀⡀⠀⠃⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠟⠛⠛⢻⣿⠛⠛⠛⠉⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠼⢓⡀⢀⡀⠰⡆⠀⢈⡇⣨⡀⠀⠷⠀⢀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢋⠀⠘⠃⣴⣦⣄⠛⠃⠉⠑⠆⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠗⠚⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⣷⠆⠀⢀⠀⠘⠶⠄⢀⡙⢋⡀⢀⡀⠀⠀⡄⠀⠉⠀⠀⣠⡿⠁⠀⠀⠀⣰⡉⠀⠉⠀⡴⠆⡀⠠⠏⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠠⡏⢦⡠⠈⠙⢹⡛⠀⠀⠈⠁⠘⠋⢹⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣇⡀⠑⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠒⠒⣶⣤⡄⠀⠀⠀⠀⠀⠀⠋⠙⠈⠁⠀⠀⠀⠠⡇
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠰⡶⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠘⠛⠇⣠⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⢠⡇⠀⢀⣀⠳⠶⢄⣀⣤⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠁⠀⠶⠂⠈⠃⠾⠉⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢯⡀⠤⠀⠀⠨⠃⠀⠀⠀⠠⠤⠈⠉⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠰⢤⣾⠟⣠⡴⠚⠟⢫⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣠⡀⠀⠀⣸⢇⠀⠋⠠⠤⠐⢛⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣤⠀⠀⠀⠊⠀⠀⠀⠀⠀⡈⠀⠁⢀⠀⠌⠀⡀⢠⠀⠁⠈⡼⠟⠲⠶⡄⠀⠙⢦⡎⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠐⠛⠃⠚⠀⠀⠀⠀⠀⠀⠘⣁⠀⠘⢿⠖⠁⢠⡄⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠸⠦⠤⠴⠦⠴⠴⠷⠖⠒⠒⠒⠒⠒⠾⠛⠒⠙⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠶⠒⠒⠖⠲⠖⠒⠒⠲⡷
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠒⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡏⠀⢀⡴⠞⠉⠉⠉⠲⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⢄⠀⠈⠀⠀⢈⡀⠤⠄⠚⠛⠋⠀⠀⠀⠀⣀⣀⣠⣤⣈⡉⠉⠉⠉⠙⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣦⡤⠐⠂⠀⣴⠊⠁⡀⢠⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡀⠀⣄⢡⣶⡌⢡⣶⠃⢸⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⡄⠀⢀⣀⡀⠳⣤⠁⡼⢿⣧⠼⠋⠁⠀⣷⠀⠤⠀⠀⠀⣰⠿⠉⠉⠀⠀⠀⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠧⠘⣧⠄⠠⡆⠀⠀⠥⢤⢁⣀⠑⠄⢀⠛⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢸⡀⠄⠀⠀⠀⠀⠘⢀⠀⢀⠀⢹⣏⠑⠒⠚⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣔⡀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡀⠀⠒⡉⣀⢠⣿⠃⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢩⣀⣿⡿⠀⣿⣷⡀⠐⠉⡆⠘⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠲⠀⠄⠈⠋⢹⡷⠞⠉⠛⠇⠈⠀⠀⠀⠀⠀⠀⣤⠀⢀⣤⠉⠸⠿⠟⠄⣠⠶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠁⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⡟⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⡀⠀⠁⣀⢄⡠⣤⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡆⠀⢩⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠈⠒⢾⡏⢀⡝⠂⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠁⠀⠀⠢⡀⣠⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡠⢾⣥⡶⢈⣁⡹⠆⠀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡁⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⠀⠀⢰⠌⠀⠀⠀⠀⠀⠉⡆⠀⠘⣷⣀⣠⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡇⠀⠀⠐⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣀⣶⡀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠏⠈⣀⣉⣼⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡀⠀⠀⣿⠿⢯⡀⢿⣿⠿⣿⠋⠀⠀⠀⠀⠀⠀⣌⣳⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⣡⠋⠁⠉⢣⠀⢀⣏⣀⠀⢹⣼⣷⣴⣿⣤⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⡇⠀⣰⣿⣿⡄⠀⣿⣿⣧⢠⡇⣿⣷⣶⣤⣄⠀⠀⠀⠀⢀⡀⢀⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⣇⠀⢻⣿⣿⠂⠀⢻⣿⠿⠚⠛⠛⠛⠿⣿⣿⣧⠀⢠⠀⣸⣿⡟⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠟⠻⢿⣿⣿⣿⣿⣷⡄⠀⣀⡙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣠⡾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃⠤⣄⡀⠈⢿⣿⣿⣿⠿⠃⠀⣿⣿⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠴⠿⣦⠀⠙⠛⠛⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣦⣄⣀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⢿⣿⠿⣿⡿⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⠀⠸⣿⣿⣿⣿⣃⣤⣴⠾⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠿⠛⠉⠀⠀⠀⢀⡀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠠⢤⠀⠀⠀⠈⠙⠻⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢶⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠈⠓⠂⠀⠀⠀⡀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⣠⣴⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠘⣷
⠀⠀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡀⠀⠰⣆⠀⠀⠈⢧⡀⠀⣿
⠀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣶⣶⠀⠀⢿⠀⠀⠘⣧⡀⠀⢘⡿⠛⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣄⣀⣾⣷⣤⣤⡿⠛⠛⠛⠁
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣄
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⠛⠿⣿⣷⡀
⠀⠀⠀⠈⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⢶⣭⣍⣛⣛⠛⠛⠛⢛⣛⣻⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢏⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡷⣏⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣷⣦⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏



⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣤⣄⠀⣿⠀⠀⡖⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⠀⣿⢠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⡿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠰⢎⡱⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠥⠉⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⣀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⢸⣿⣿⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠁⣿⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⣿⠀⠈⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠤⠚⠁

⠀⢀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠀⢀⡄⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢮⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⢣⡄⠀⠀⠀⣤⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⢀⠀⠈⠁⠠⠂⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀⣀⡴⡼⠁⠀⡠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⢀⠊⢹⣷⠻⡆⠸⣴⠏⠠⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⡄⠰⠏⠀⠀⡀⠀⠀⠀⢀⠀⠀⣿⢷⣿⣿⣱⢿⣾⢷⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠆⠀⠀⠀⠂⠀⠀⠁⠀⠀⢀⠈⠀⠀⡹⣾⣿⡿⠁⣾⢃⠌⣰⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⢀⡀
⠀⢸⣀⡆⠀⡀⠀⠀⠀⢀⠤⡀⠀⠑⠀⠀⠁⢿⣿⡇⢠⠛⡏⣴⠿⠒⠀⣴⠁⠀⠄⢠⠈⠀⠀⠈⠙
⠀⠐⠀⠀⠈⠇⠀⢀⠔⣋⡀⠄⠁⠷⠛⢷⡉⠷⣼⣷⡶⢻⣠⡿⣢⠴⠆⠀⠀⠀⢶⠂⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠖⠈⣩⠖⠈⠀⠀⠀⠀⠀⠀⠚⠳⣦⣴⡬⠛⠛⠿⣵⠄⠀⠀⠛⠓⠚⣁⡴⠆⠰⢤⠤⠄
⠀⠰⡄⢰⠀⡔⠁⠀⠐⣦⡠⡄⠀⠀⠀⠀⠐⠉⣙⡕⠊⠉⠑⣖⠓⣞⡤⡶⠐⣊⠁
⠀⠀⠄⡌⡀⠀⠀⠀⠀⠈⠻⣶⣃⡀⠀⠀⠀⠀⡿⠀⠴⠀⠛⠹⡆⣉⢷⡄⢀⢈⠙⠢⣀⠀⠀⠀⣠⠄⠶⠀⡄⠀⠀⠀⠄
⠀⠰⡎⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠇⠀⠀⢀⡝⠳⡄⠰⠟⠀⢿⣸⠎⠷⠆⢘⣋⡄⢹⠗⠂⠉⠐⠖⠒⠈⠀⢠⠾⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠀⠉⠒⣰⣷⡶⠟⠊⣠⣔⣻⣧⢈⣁⣙⣧⠳⣄⢦⣄⠠⠀⠉⣸⢇⣀⡀
⠀⢰⠆⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⣼⠆⣠⣤⡀⢠⠞⢢⣴⢇⣼⠿⠿⡄⣉⡻⣹⡆⠈⠴⣉⣦⡰⠞⠁⠀⠈⢉⠆
⠀⢨⡂⡇⠘⢇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⡧⠴⠇⠀⠳⢶⣢⣶⣶⣿⣷⠄⣀⠿⢿⠙⢦⣛⡶⡘⠗⣤⠸⣄⠀⠰⠆⠈
⠀⢠⠍⠀⠀⠈⠷⢀⠀⠀⠀⠀⠀⠈⢉⠅⡲⣓⢛⠶⠹⠃⠌⢹⡄⣀⠙⠶⡫⠄⠰⢇⠠⡏⢳⠚⣤⠀⠙⢿⡓⠳⣄
⠀⢠⠀⣆⠐⠀⣄⠀⠙⠙⠢⠄⣠⣖⣁⡴⢉⠋⡽⢛⢄⣼⢳⣷⡾⠙⠥⠀⡿⣤⡚⠁⠀⠿⠀⠃⢻⡉⠩⣈⠣⢤⣁
⠀⠈⠀⠉⠠⡀⠈⠀⠀⢀⠀⠉⠉⣠⠞⣴⠋⣼⠳⠟⣘⡣⠙⠁⢀⡐⠂⠤⠭⢙⣡⡴⠂⠀⠀⠙⢮⣹⠶⠺⢿⠂⣉⠓⡀
⠀⠀⢀⠀⠰⠁⠀⠀⠴⣿⠀⠀⠀⠀⠘⠀⠢⠙⠒⠎⠁⠴⠣⠾⣦⡀⠀⠀⠀⠈⠳⠶⢞⡉⣤⡄⡀⠈⢷⡒⠦⠀⢉⡜
⠀⠀⠈⠂⠀⠀⠀⠀⢠⣄⠙⠆⣰⡆⠀⠂⠀⠀⠈⠀⠀⠀⢰⡞⢻⣿⡧⠀⠀⠀⠀⠀⠈⡡⠟⠐⣻⣿⡀⠘⢦⠁⠈⠓
⠀⠈⠀⢀⠄⢰⡆⠀⠰⣿⠀⠘⠉⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠⡿⢇⣈⣿⣀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠏⠿⣄⠈⠼
⠀⠀⠀⠈⢀⣈⠹⠀⡀⠁⣾⠀⠀⠃⠐⠀⠀⠀⠃⠀⠀⠀⢷⣄⣤⠙⣟⠻⡶⣦⣤⣄⣰⡁⠀⢸⣿⣿⠛⡇⢠⠀⠀⠀⡀
⠀⠀⡄⠀⠈⠟⠀⢠⡁⠢⠘⠄⠀⠀⠐⣦⠠⡀⠀⢀⠀⠀⠀⠀⢠⡁⣘⣇⠠⡏⡹⢏⡿⢻⢦⡏⠀⠀⠀⡀
⠀⠀⠁⢀⡈⠀⣀⣾⠑⠊⠀⠀⠀⠀⠈⠈⡠⡏⠀⠘⠀⠀⠀⠀⠀⣉⣹⠟⢻⡷⠇⠈⠱⡟⢬⣿⣶⠟⣿⣀
⠀⠀⠀⠸⠓⠁⠙⢉⠀⠀⠠⢦⠀⢠⡀⠄⠡⣁⠀⠀⠀⠀⠀⠀⠈⠉⠀⣤⣼⢁⢦⡀⠆⢻⡆⣿⠻⣿⣽⣯⣶⡖⠀⠠
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠈⣇⠄⠻⢠⠀⢈⡇⠀⠐⠀⠀⠀⠀⠀⠀⠤⠜⣡⡜⠀⠈⠀⡌⢛⣦⡁⠈⠉⢻⡎⣠⠀⠇
⠀⠀⠀⠀⢳⡤⠤⣄⡈⠀⠀⠈⠀⡀⣦⠈⢀⣊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠁⠻⡄⠀⠀⠸⡇⠙⠻⣦⣡⣤⢹⡗⠀⡀
⠀⠠⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠞⠉⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠙⠄⠀⠀⣷⢀⡀⠟⣀⠀⠀⠀⠀⠁
⠀⢰⣤⠐⠒⣋⠟⣚⣡⡾⠲⣟⣵⠞⣍⣓⣶⣤⣼⡵⠦⠠⠀⢠⠀⠠⠄⠠⠤⠆⠤⠤⠔⠉⣉⣠⣾⠷⢦⠟⠷⣦⡄
⠀⢸⡟⠛⠛⠟⠛⠉⠏⠑⠛⠋⠉⠀⡁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠙⠋⠴⠁
⠀⠸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⣀⡀⢘⡀⢀⠐⠀⡀
⠀⢠⡀⡀⢀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋⠁⠁⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⣒⣖⢢⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠓⢛⠳⠼⢧⡁⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣀⣠⠭⢶⡶⣿⠓⢄⠀⠘⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣏⢭⠀⠐⢒⡎⠑⢚⡧⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠛⠂⠠⣼⣇⠐⡏⡠⢛⠒⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⣠⣶⣶⣿⠛⠀⠙⠁⠏⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⠋⠙⢥⡛⠁⠀⣠⡖⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠋⠙⢲⣤⣴⢾⣫⡽⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣆⢀⢀⡞⠁⠙⢦⠷⠋⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠈⣻⣜⣧⣄⡀⠀⠀⢀⡴⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⠋⠀⠀⠸⢿⣷⠞⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⡀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠙⡆⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢻⡀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⠧⢤⣀⣈⠇⠀⣀⣠⠤⠼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⢹⠖⠙⠉⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠤⠤⠤⢤⣘⣦⣀⣀⠀⠀⢀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠁⠙⠒⠓⠯⠭⠥⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣋⡉⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣌⣵⡴⣎⢳⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⢛⠓⠲⠿⣤⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢀⣉⣱⣦⣴⣿⡷⢄⠀⠀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⡿⠿⠀⠀⢫⢭⡙⠧⣄⣵⣄⠀⠘⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣧⣼⡆⠠⠁⢴⣷⠀⢀⡴⠛⣆⣀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠃⠀⠲⢼⣧⡄⢸⡀⡰⣉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⢀⣀⣴⣶⣿⣿⠟⠁⠈⠉⠀⠟⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡻⠟⠻⣟⣩⠿⠂⠀⠀⢀⡴⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠻⠦⣄⡀⠒⠂⣠⣴⠲⣟⠲⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⠐⠀⠀⠀⡠⠉⡿⠂⡉⢀⠓⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢣⠀⡀⢠⠏⠀⠀⠙⠂⠚⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠙⢃⣘⣦⣀⠀⠀⠀⠀⠀⢀⡔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠶⠿⠃⠈⠀⠘⣿⣶⣦⡴⠖⠛⠶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⢷⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠛⠓⠲⢤⣄⣼⠃⢀⣠⡤⠖⠒⠚⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣻⠋⠙⠋⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⣀⣀⣠⡤⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠤⠖⠒⠒⠚⢿⣉⣗⣒⡒⠒⢒⣚⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⢀⠀⣀⢀⣀⠀⢀⠀⣀⢀⣀⣀⣀⡀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢒⠚⡒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢠⡟⠛⢻⣆⢹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⠉⡩⢝⠆⠀⠀⣫⢭⡉⠙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⡼⠀⣸⡇⠀⢰⣧⠀⢳⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡆⡀⠑⠀⡿⠛⠛⠛⢻⡀⠞⢀⢰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠰⣿⣦⣼⡀⠀⠀⠀⢀⣧⣴⣿⡇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠹⣿⣿⣿⣶⣶⣶⣿⣿⣿⠟⠀⣽⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡀⠀⠈⠛⠛⠉⠛⠛⠁⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⣦⣤⣀⣀⣀⣤⣤⣆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⣿⠃⠀⠀⠀⠈⣿⣿⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠊⠈⢿⣿⣿⣿⣿⣿⣿⠁⠙⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⣤⠀⢧
⠀⢀⣀⣀⠀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣴⠿⠶⠾⠿⠿⠿⠿⠿⠿⡿⠷⠶⠿⢦⡼
⠀⠈⠈⠉⠀⠈⠁⠉⠉⠈⠁⠈⠈

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠴⢆⠄⠀⣀⠀⠀⢀⡴⠂⢀⢠⠲⠉⠄⠀⠀⠀⢤⣤⡀⢀⣀⢀⠀⠀⢀⡀⠀⠀⢀⡀⠠⠔⠒⠄
⠀⠀⠄⠀⠀⠀⢠⠀⠐⡀⠘⠓⠊⠀⠘⠀⠂⠀⣋⠀⢀⢦⡀⠀⡀⠐⠀⠀⣟⠐⣿⡇⠀⠀⠐⠀⠀⣯⠐⡱⠀⠈⠷⠣⠀⠉⠰
⠀⠀⢰⠀⠁⠀⢀⠀⢀⠃⠀⢠⡤⠀⠀⠀⠀⠐⢈⡘⠣⠞⢡⡤⡀⢤⠃⡀⠉⠐⠛⠁⠠⠒⡆⠀⠀⠀⠉⠀⠀⠀⠀⠃
⠀⠀⠡⠔⠀⢤⠀⡀⠊⠀⠀⠠⣴⠶⢂⠀⠀⠀⠀⠨⠀⠻⠋⠐⠀⠈⢐⣉⠤⠔⣒⡀⠀⠀⠀⠀⠀⠸⠭⠀⢀⠐⢀⣀
⠀⠄⡄⠐⠀⠀⠀⠁⠀⢀⠀⠀⠓⠚⠋⠀⠀⠀⠀⠒⠉⢀⡠⠤⠒⠉⢡⠔⠂⠁⠀⢆⠀⠐⠂⠀⠃⠀⠀⠂⠠⠡⠀⠀⠂⠐⡄
⠀⠀⠁⠀⠀⠀⠰⠔⠂⢁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢺⠁⢀⠄⠓⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡄⠀⠇⠘⠠⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠀⠀⠀⢀⡄⠖⡠⠈⡄⠀⠀⢠⠀⠀⠀⠃⡀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⢁⡀⠀⢁⡈⢉⠓⡄⣠⠀⡀⡀⠀⠀⠀⠀⠛⠐⠆⢈⡉⠤⠒⠉⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠈⠁⢀
⠀⠀⣀⠰⠖⠊⠁⠈⢂⢱⠀⠈⠦⣉⠠⢇⡴⠀⠁⠀⢂⠈⢄⣠⠤⠒⠊⠁⠀⠀⠀⠈⠲⡈⢁⡀⠀⠀⠀⡄⢤⣄⠲⡄⢰⣋⣶
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠆⢳⡀⠀⠀⠄⠚⠁⠀⢀⣢⣤⣶⣆⠀⠀⠀⠀⠄⠀⠀⡀⢠⠀⠄⠀⠘⠃⠀⠀⠳⡀⠃⠀⠀⠀⠃
⠀⠀⠘⣇⠰⡀⠀⠀⠀⠘⢀⢳⡀⠀⣀⠷⡀⠀⠘⠟⢋⣉⡄⠀⢀⡇⠀⠀⠀⠀⠀⠄⠠⠀⠇⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⢂⡄
⠀⠠⠃⠸⡆⠀⡄⠄⠀⡬⡈⠈⢧⠀⠈⠺⠁⠀⠀⠀⠈⠀⠀⣹⡉⣱⠖⠀⠀⠑⠀⢀⠁⠀⠀⠀⠀⠘⠀⡀⠀⠀⡀⠂⠄⠤⡀
⠀⠀⠀⠀⢻⡘⠀⣴⠁⠈⠀⠀⢸⡇⠀⠀⡖⢢⡄⠸⠄⠀⠀⠁⡟⠈⠁⠀⠀⠀⠀⠀⠀⠀⢰⢾⣷⣦⡄⢳⠀⠒⢠⡶⡅⣀⠘
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠃⠀⠀⢀⡤⠔⠒⢠⠀⠀⠐⠪⠔⣶⠀⠀⠑⠦⣠⠉⠃⠀⠀⢻⣄⠹⠚
⠀⠀⣐⠶⠲⢤⣄⠀⠀⠀⠀⣤⠀⠒⡀⠀⠀⠀⠀⣠⠞⡿⣗⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⢉⠀⠂⠀⠐
⡀⠀⣉⣦⣘⣂⠌⠣⠀⢂⠈⠀⡸⠀⢡⠀⠀⠜⠀⣋⠜⣰⠋⢉⠈⠙⢦⠀⠀⠀⠀⡿⢀⡀⠀⠀⠀⡀⠄⠀⠀⠀⠈⠂
⡏⠢⡀⠙⠻⠉⠒⡁⠀⠈⠤⠀⠝⠀⠋⠀⠀⠀⠀⠁⠀⢿⡠⠘⡏⠢⣸⡆⠰⣠⠆⣷⠊⠘⠂⠀⠠⠀⠀⡀⡀⠀⠐⣄⠠⠀⠄
⠁⠀⢴⣨⡆⠀⢀⠔⠁⠊⡄⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠓⠤⡧⠀⠈⠀⠀⠀⣀⡏⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠰⠀⠢⡀⢠⠀⡄⠀⢉⠀⠈⠁⠳⡆⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠆⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠈⠢⡀⠀⢰⠈⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⢀⣀⣇⣄⣀⠀⠈⣁⠀⠤⠘⠁⠀⠀⢁⠤⡖⡀
⠀⠴⢲⠖⣄⠀⠀⠀⠀⠀⠁⠈⢀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⢰⣉⠇⠀⠀⠀⠀⢨⡄⠢⠗⢵
⠀⠐⠉⣢⢌⡁⠠⠀⠀⠂⠀⠀⠀⡄⢃⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡏⠀⠉⠀⠀⠀⠀⠀⠢⣤⠔⢦⠺
⠀⠀⠉⠁⠚⠀⠀⠒⠀⢫⠀⠀⠀⠀⠀⠺⡦⠁⠀⠀⠀⠀⠀⠀⠀⢠⠠⡆⠀⠈⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠃
⠄⠀⠀⡀⢦⠀⠀⠀⠀⢀⠰⠖⢤⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠼⠲⠃⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄
⠀⢀⣀⡤⣦⠀⠈⠀⠀⠸⢿⣶⣦⡆⠘⠜⠂⠀⠀⠀⠀⠀⠀⢀⠤⣖⠂⣲⢦⣀⠀⠃⠀⠀⢰⡋⠀⠀⠀⠀⡶⠀⠸⣆
⡄⠀⡋⠶⠘⠃⠀⠀⠐⠀⡄⠈⠁⠀⠀⠐⢠⠐⡀⠀⠀⠀⠀⣯⠊⠁⣸⢧⡞⡵⠃⠀⣄⠀⠈⣧⠀⢸⠑⠘⠠⠤⠁⢹⡄⢠⠤
⠈⠀⠁⠀⠀⠠⠃⠀⠀⠀⠅⣀⠀⢀⠀⠌⠰⠀⠈⠀⠀⠀⠀⣖⠉⣟⢁⡸⢿⡇⠀⠸⣥⠇⠀⠘⣆⠃⡁⠀⠀⠀⠐⡄⢷⠈⠈
⠀⠀⢀⠀⠀⠀⠀⠐⡄⠀⠀⢀⡀⠈⠢⠀⠀⠐⠀⠀⠠⠀⠀⠯⠖⠊⠋⠴⠋⠀⠀⠀⡈⠠⠀⠀⠘⣆⢣⠀⠀⠀⠀⠐⠈⢧
⠀⡶⢻⡆⢰⡆⠺⠎⠹⠀⠀⡀⠒⠀⠐⣀⠀⠈⠐⠀⠁⢀⣀⠤⠒⠢⠘⠡⠓⠀⠀⡼⢣⠄⠒⢄⠀⠘⡄⢆⠀⠀⢀⡠⡀⠞
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠈⠁⣀⠤⠒⠊⢡⡄⣀⠀⠀⠁⠀⠀⠀⠀⠞⠨⡀⠋⠐⠃⠀⠘⠀⠈⠀⠁
⠀⠀⠀⠈⠛⠐⢀⠀⠀⠈⠆⠈⡀⠠⠀⠈⣩⠰⠂⠉⠑⠄⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠀⠒⠈⡇⠀⠀⢠
⠀⢀⠀⠠⠀⠀⠈⠀⠠⠀⠰⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⣉⠠⠃⠀⡦⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⠀⠠⡄⠀⠀⠀⠁
⠀⠰⠀⠀⠀⠀⠄⠁⠀⠀⠀⢀⠀⢀⠀⠈⣣⠀⠀⢀⠠⠀⢀⣈⠤⠖⠊⢁⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠊⠀⢀⠀⠀⠀⠀⠨⠬
⠀⠀⠀⠀⠒⢂⡘⠒⢀⠄⡄⠨⠀⠀⠀⠀⠰⣒⡨⠴⢒⠈⠁⠀⠀⢀⣔⠉⠤⠀⠀⠀⠀⢣⣥⡞⠀⠀⠀⠀⢈⠀⠶⠀⠠⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠙⠐⡡⡀⠸⠏⣩⢖⡑⠀⡠⠀⠀⠀⠠⠴⠀⠀⢠⠁⠀⠃⠀⠀⠀⠆
⠀⡄⠀⠰⣎⣄⠀⠀⣔⠉⠠⠀⢀⡀⠀⠄⢸⣿⠀⣛⠀⠈⠁⠐⠀⠘⢷⠊⠅⢳⠀⢀⠠⡀⠀⢀⡀⡀⠸⡀⠀⠆⠀⠀⠀⠡
⠀⠤⣉⡠⠏⠈⠃⠀⠈⠂⠁⠀⠀⠂⠒⠂⠘⠿⠦⠉⠀⠀⠠⠀⡄⠆⠂⢀⠔⠋⠀⠈⠒⠁⠸⠣⠬⠁⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢋⣬⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣋⣴⣿⣿⣿⣿⣿⢟⣿⣯⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣗⣿⣽⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣇⠐⢿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣒⣸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⣿⣟⠋⡽⠋⠉⢻⣿⡿⠿⣿⡿⠛⣿⣿⣿⣿⣿⡟⣭⢹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣓⣽⣿⣿⣿⣿⣇⠀⠀⣼⡿⠁⠀⢈⡇⢀⣼⡷⠿⢿⣿⣿⣶⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠋⠈⠙⣿⣷⣾⣿⣷⣿⣿⠿⠿⠿⠋⢀⡴⢊⣽⣿⣿⡿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⣠⠴⠚⠙⠻⣿⣿⣿⣿⣿⣷⣶⡶⠋⠉⠉⠁⢿⣿⣿⣆⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣧⣿⣿⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢠⡧⢠⡀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⢀⠀⠀⠀⣇⢠⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠁⠀⠀⠀⣴⠈⠀⠀⣄⢀⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡸⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⠓⠀⠀⠂⣴⠈⠀⠈⠃⠠⡃⠀⠀⢧⡀⠀⠀⠀⠈⠁⠀⠁⢀⡤⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠄⠑⠀⠋⠁⠀⠠⣅⣠⡄⣀⣙⠶⠔⠢⡀⡀⠀⠠⡎⠀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠜⢀⠀⠀⠰⠀⠀⠀⣦⠀⡤⠐⠈⠸⣅⢀⡀⠤⠄⠀⡃⠀⠀⠴⣠⡭⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠓⠀⡣⠀⠀⢰⠘⣠⠀⠀⠀⠀⠈⠀⢹⠐⠂⡘⠀⢀⠀⡸⠁⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢈⠄⠙⠀⠀⠈⠧⣈⢢⡀⠀⠀⠀⠀⠊⠃⢼⠀⢀⣰⠂⠄⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠂⢐⡂⠀⠈⠉⠙⠒⠲⡄⠀⠁⠀⠀⠀⠀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⡄⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⢳⡷⢄⡀⡤⡀⠀⠀⠀⠀⠀⠀⣆⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠁⢩⠔⠉⠀⠀⠌⣦⣀⢀⡀⢀⠴⠋⠀⠀⠀⠀⣠⠔⠀⠀⠋⠀⠅⠈⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠃⠀⠀⠀⣠⣼⣍⣤⠠⠔⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣠⠊⠙⠁⢠⠘⠖⠈⠑⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠂⢺⣿⣤⠆⡈⠉⢀⣀⡀⠀⠃⠀⣄⣰⠖⠂⠀⠀⠂⡀⠀⠀⢀⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠀⣉⠉⢸⣿⣶⠿⠉⠉⠢⠀⠀⠗⠚⠗⠲⣤⠄⣖⡿⢿⣿⣧⠖⠚⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠈⢣⡀⠨⠋⠀⠀⠐⢲⡀⠜⠀⠀⢀⠀⣾⢧⡞⢹⣿⣄⣴⠀⠀⠀⠺⢚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠈⠙⠳⣧⠀⠂⢉⠘⠀⠘⠂⢪⡔⠅⠌⢙⡛⠦⠤⠄⢉⡁⠀⢸⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⡸⢧⣤⡠⢒⡏⠀⠀⠀⠀⠀⡀⠉⢁⠀⠂⠂⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙⠒⠢⢤⡤⠖⠚⠳⢄⠁⡄⠀⠀⠈⠀⠀⠐⠀⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⠠⢄⠉⠉⡇⢸⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠣⡰⠀⠠⢄⠁⠀⢀⠳⠄⢄⠀⠀⠀⠀⣀
⠀⠀⠀⡀⠐⠲⢂⣊⡴⠊⠀⢺⣆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠗⠀⠠⣰⠗⠁⢈⣠⠀⣰⠓⣀⡀⠀⢆⠀⡦
⠀⠀⠐⠣⠀⢐⠒⡏⢸⣀⠂⠁⡈⠂⡤⢳⣄⡴⠐⣥⡤⠤⠚⠁⠀⠀⠀⢧⡐⠉⠁⠠⣞⠀⠘⢫⡅⠦⡀⢠⠁
⠀⠀⠀⠀⢱⠉⠃⡇⢸⡰⠀⠀⡧⠊⠀⢸⡴⢁⡆⠡⠣⠄⠒⡐⠂⢶⠛⠃⢇⠀⠃⠄⢨⠄⡀⠋⡀⠀⠐⠆
⠀⠀⠀⠀⠘⠀⠂⠁⠘⠃⠀⠀⠃⠀⠀⠘⠁⠘⠁⠀⠋⠀⠛⠘⠁⠘⠃⠀⠘⠀⠀⠘⠘⠘⠁⠀⠐⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠛⠉⠉⠉⠉⠉⠉⠛⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⢀⠤⠦⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡠⢊⠁⠀⠀⡀⠀⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡔⠀⡄⢈⠀⠄⠀⡀⠈⠄⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠇⠸⠀⠌⠁⠔⠀⠐⡄⢈⠄⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠲⠀⠀⣀⣀⣀⣈⣁⣠⣼⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⣀⠴⠒⠚⠉⠉⠁⠀⣀⣀⣀⣤⣤⣤⣤⣤⣭⣉⣛⣲⣤⣀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠒⠉⢀⣀⣤⣤⠖⠒⠋⠉⠉⠉⠀⠀⣴⣿⠿⠛⠛⠓⠀⠀⠈⡿⣿⣶⣤⡤⠀⠰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⣋⣤⣶⡿⡿⠿⢿⠿⢿⣦⡀⠀⠀⠀⠠⣾⢋⣁⣀⣀⣀⠀⠀⠀⠀⣿⣿⣿⡏⠀⢀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣸⣿⡇⠀⢰⣃⣀⣀⣈⡁⠀⠀⠀⠀⠊⠉⣿⢿⣿⢿⠄⠀⠀⢸⣿⣿⣿⡇⢠⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣿⡇⠀⠘⠉⢻⡿⣿⣷⡄⠠⠤⠤⣤⠀⠉⠬⠭⠋⠀⢀⡀⠀⠙⢻⡿⣷⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⢿⣇⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⡖⢀⣀⣤⣾⣿⡇⠀⠀⠈⠛⠋⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢻⠀⠀⠈⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⣄⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣬⡿⣷⣶⣶⣿⣦⣀⠀⠀⠀⢀⣠⣾⣿⣿⠿⠟⠁⠀⠀⠀⠀⣰⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣉⠛⢿⣿⣿⣿⣿⣿⣿⠿⠿⢿⠿⠗⢠⠃⠀⠀⠀⣠⣾⣯⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠈⢂⠉⠉⠀⠈⠉⠱⠶⠃⠀⠀⠀⠀⠀⢀⡤⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠚⠀⢼⡓⢲⣤⣄⣀⡀⠀⠀⠀⣀⣠⠴⠞⢋⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⣅⠙⠻⢿⡟⠋⠉⠡⢴⣖⠀⠉⠻⢿⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠠⠀⠁⠀⠀⠀⠀⢀⠀⠈⠲⠋⠻⣧⡀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠞⠀⠀⠘⢿⡆⠀⠀⠻⣦⣤⣴⣶⣶⣶⣿⠗⠚⣿⣶⠛⠲⢄
⠀⠀⠀⠀⠀⠀⢸⠳⡀⠣⡀⠀⠀⠀⠀⢤⣇⣠⠔⠃⠀⠀⠀⠀⢈⣷⣼⠋⠉⠻⣿⡉⠁⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠇⠀⠙⠦⣀⠁⠘⢋⡠⡾⠁⠀⠀⠀⠀⠀⢠⣶⣾⠟⣿⡀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⠁⠀⠀⠀⠀⠀⠴⠻⠟⠃⠀⠈

⡏⢋⠝⡍⠉⠉⠉⠉⠉⠉⠉⠉⢉⡝⠙⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠈⡆⠘⢦⡤⢤⢠⡤⢤⡄⢀⡾⠀⠀⠀⡇
⡇⠀⢹⡲⣌⢻⡜⣼⠀⢸⣢⡟⢁⣼⡃⠀⡇
⡇⠀⠀⢳⠹⠧⠛⢿⠄⣶⡏⠀⠚⠚⠡⠀⡇
⡇⠀⠀⠈⢲⠦⠬⠀⣶⠭⠭⠐⢒⣀⢸⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠿⠿⠿⣶⣦⣄⡀
⡇⠀⠀⠀⠈⣶⣶⡆⡼⠳⢜⠻⣿⣿⡀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣄⠀⠀⠀⣰⡿⢋⡠⠄⠤⢀⠀⠀⠈⠙⠻⣷⣄⡀
⡇⠀⠀⠀⠀⢋⡿⡰⠁⠀⠀⠙⠦⣝⠷⢸⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⢋⣩⣽⣿⣷⣤⣰⣿⠡⣋⢔⡣⠐⠏⠱⠄⠀⠀⠀⠈⠻⣿⣆
⡇⠀⠀⠀⠀⠸⣠⠃⠀⠀⠀⠀⠀⠈⠓⢮⣇⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣿⣥⣴⠶⠾⠿⠿⢧⣿⣒⠈⠖⠃⠠⠀⠂⠀⠀⠀⠀⠀⠈⢻⣷⡀
⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠶⣄⣀⡈⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⣿⣶⣤⣤⣄⣀⠀⠀⠀⠉⠉⠓⠶⢮⣤⠁⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠛⠀⠈⠉⠙⠳⣤⣤⡀⠀⠀⢿⣷⠀⠙⠛⠋⠙⠻⣿⣿⣿⣿⣿⠷⢦⣄⡀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⢹⣷
⡇⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠠⡄⠀⢿⠀⢷⣄⠀⠈⢿⣷⣀⠀⠀⠀⠀⠀⠈⠉⣿⢡⣶⠀⠉⠛⠶⣤⣀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢿⣇
⡇⠀⠀⡟⠳⣄⠀⢠⠟⠀⠀⢀⡤⠠⠄⢀⠀⠀⣄⠄⠂⠋⠚⠲⣄⢻⣿⡉⠁⢀⣴⠾⠟⠛⠷⢾⡿⠀⠀⠀⠀⣼⡟⠛⠶⢦⣤⣤⣤⣽⣄⠀⠀⠀⠘⠿⢷⣦
⡟⠳⢦⡇⠀⠈⠓⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠘⠈⠻⢿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢠⠏⠙⣆⠀⠀⠀⠀⠀⠙
⣧⡴⠠⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣠⠴⠋⠀⠀⣼⠶⠴⠦⣄⡀
⡦⠁⠀⠀⠈⢤⠀⠀⠀⢠⠈⠀⠀⠀⠐⠀⠁⠠⡁⣴⠲⣆⠘⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠞⢻⡄⠀⠻⡄⠀⣠⠞⠁⢀⡀⠀⠀⠻⣴
⡇⠀⠀⠀⠀⠀⠙⣗⠀⠀⠀⠰⠄⢄⠀⠈⠀⠀⠓⠠⢄⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣤⠴⠖⠋⠉⠉⠀⠀⢀⡼⠁⠀⠀⠙⠚⠁⠀⣰⠋⠀⠀⠀⠀⣿
⡇⠀⢀⠀⢀⠀⣠⠗⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠄⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣼⡟
⡇⠀⠈⠀⢀⠁⠽⡀⠀⠀⠀⠀⠠⢀⣀⡀⣻⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠙⢿⠲⢾⡛⠒⠛⣻⠀⠀⠀⠀⠀⠀⣠⡶⠦⣤⣤⣤⣶⡿⠋
⡇⠀⠀⠙⢇⠀⢆⡀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈⠛⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⠳⣄⡙⠦⣠⠟⠀⠀⠀⣀⣤⡾⠿⠿⠿⠿⢿⣯⣄⡀
⡇⠀⠀⠀⠌⢰⠋⠀⠉⠂⢀⠀⠀⠠⡌⢰⣄⣀⡀⢁⡋⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⡀⠀⠀⢈⣿⡿⣫⠟⠓⠶⡶⠖⠚⢻⠏⠻⣄⡀⠀⠀⠀⠀⠀⠙⠻⣷⣤
⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⢠⢉⡞⠀⠀⠉⠉⠀⠀⣠⣴⣿⠟⠁⠀⠀⠈⠹⣿⣶⣿⢿⠋⠐⠓⠦⠤⠞⠁⠀⠀⣿⠀⠀⠀⠉⠓⢦⣀⢀⡤⠖⠛⠋⠉
⡇⠀⠀⠀⠀⠀⠘⡀⠀⡘⠀⠔⠀⠀⢮⣀⠀⠀⠀⠀⠀⠀⣿⡏⢿⠀⠀⠀⠀⠀⠀⢹⣿⠁⢸⠃⠀⠀⡰⠊⠉⠙⢦⠀⢹⡀⠀⠀⠀⠀⠘⣿⡏
⡇⠀⠀⠀⠀⠀⠀⢱⠂⠈⠀⣀⣀⣀⣀⡉⣳⡄⠀⠀⠀⠀⣿⣧⠘⡆⠀⠀⠀⠀⠀⢸⣧⣠⠏⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠙⢤⣀⣀⣀⡠⣾⠋⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠘⡄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⠶⠤⠶⠋⠀⠀⠀⠀⠈⠁⠁⢠⡏⠀⠀⠀⠀⠀⢿
⡏⠉⠁⠂⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠘⡀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣷⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠑⣄⡀⢀⣀⣴⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠉⠀⠈⢻⣍⣁⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠿⠿⠿⠛⠁⠈⠻⢿⣷⣦⣄⣀⡀⠀⠀⢀⣀⣀⣿⡅⠀⠃⠀⠀⠀⠈⠹⠛⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠛⠛⣿⡇⠀⠸⠀⠀⠀⠀⠀⡇⢹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠈⠀⠀⢃⠀⠀⠀⠁⠀⢀⣐⠘⠀⠠⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⢀⣀⣀⢸⡞⢀⣈⣀⣇⣤⣤⣀⣁⣀⠆⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢀⣾⣴
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠒⠉⠈⠉⠉⠁⢀⠉⠉⠉⠁⠀⠀⠀⠉⠉⡱⠞⠋⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⠿⠋⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢀⡄⠀⠀⢸⠀⠀⠀⢨⠀⠀⠀⠀⠀⠘⡄⠠⠀⠈⠆⠐⠀⠀⡇⠀⢸⡇⠀⠀⠀⠠⠄
⡇⠀⠀⠀⠀⠀⠀⠀⣒⡋⠀⡀⠀⢀⣠⣀⡋⠀⠀⠀⠀⡄⣀⠀⣠⠆⠀⠀⢀⠀⢀⣰⠄⡀⠀⣈⡔⠀⢀⡴⠀⠘⡇⠀⠀⣀⡼
⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠉⠁⠀⠀⠈⠈⠉⠉⠉⠁⠀⠈⠁⠀⠉⠉⠉⠉⠁⠀⠈⠈⠁⠀⠀⠁⠉⠈⠉⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠿⠿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢡⣦⣤⣶⠈⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠟⠈⠁⢻⡇⢀⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣦⣬⣄⣺⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⣭⣹⢿⣿⡛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢈⡇⠀⠀⣴⠿⢛⠛⠿⠉⠻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢿⠃⠀⠀⠁⣰⣿⡆⠀⠀⢀⣴⣿⣿⡿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠑⠢⠀⠀⠀⠀⠘⠿⠀⠀⠰⣿⣿⡟⠁⢀⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡆⠀⠛⠋⠀⢰⠋⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⢀⣀⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡟⠛⢿⡿⠟⠛⠛⠉⠀⠀⠀⢀⣴⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠢⡄⠀⠀⠀⠀⠀⣈⠷⢦⣄⣀⣀⣀⣠⣤⠖⠚⢡⣾⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⡟⠉⠳⣤⣶⠟⠁⢠⣿⣿⣿⣿⣿⡿⠋⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠶⢤⡇⠀⢠⣿⣿⣷⡤⣿⣿⣿⣿⣿⡿⣡⣤⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠹⣿⣿⣇⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⠀⣀⣠⣿⣿⡏⠘⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠈⣉⣻⡏⠉⠉⠉⠉⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⢛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⠞⠉⠀⠀⠀⠀⠀⠀⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⡀⠀⠀⠀⢀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠉⠀⣠⡄⢀⣄⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⢠⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⣯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡘⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣁⣀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠘⠆⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⢠⠋⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈