Coypu Emojis & Symbols

⠀⠀⡀⢰⢷⣾⡆⠀⠂⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠶⢶⣿⡟⣇⠛⠁⠀⣾⣶⣾⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⣿⠿⠋⠀⠀⣼⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠰⣶⡄⠀⠉⣿⡟⠀⠁⠁⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⡟⠁⠀⠀⠈⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠃⠭⠀⡂⢒
⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣤⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠑⠀⢄⠠⠄
⠀⠀⠀⠀⠀⠀⣤⣭⣿⣿⣮⣼⣡⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠄⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢫⣽⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠃⠀⠈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠒⠂⣂⠀⠀⠂⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⡤⣂⢈⠀⠀⠐⠄⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠄⠀⠀⢄⢻⣾⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠀⠀⠀⢲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢀⠀⠀⠀⠀⠔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⢹⣿⣿⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⣐⠆⠌⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⡀⠀⠀⠉⠆⠀⠀⠁⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⢀⠀⠐⠒⠀⡀⠀⠢⠄⠂⠁⠀⣠⠀⠄⢐⡀⠀⠀⠀⢀⣅⠀⠀⠀⠈⠀⠀⠀⠔⠁⠀⠀⢈⠖⢂⠀⠀⠀⠀⠀⠘⡚
⠀⡍⠀⠀⢀⣈⣀⣮⠀⠀⢀⡀⠴⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⢐⠆⠀⣀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠲⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣘⣒⠀⠈⠁⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠺⡄⠀⠀⠀⠀⠓⠀⢄⣄⢠⢠⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⣸⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢸⠙⠈⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⡀⠈⠉⠉⠀⠀⠀⠀⠈⢰⢦⡀⠀⠀⠦⠀⠠⢠⠓⠰⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠁⠀⠀⢀⠱⠒⠆⢤⠀⠘⠂⠉⠔⠤⠢⣀⢄⠀⠀⡀⠀⠀⠀⠀⠀⢠⣠⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠈⢦⠔⢒⣆⡠⠔⠒⠉⠀⠀⠀⡠⢀⡠⠀⣀⠄⣀⣠⣊⠉⠄⠉⠈⠩⣉⣉⠑⠲⠮⠽⠚⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠨⠒⠩⡥⡪⠛⢉⣁⠀⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠐⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣬⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⠀⠁⠐⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠐⠤⠤⠀⠐⠂⠠⢄⣀⡀⠑⠀⡐⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠄⠌⠂⠀⠀⢄⠀⠰⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⡀⠀⠀⡅⢿⢴⡀⠀⠀⠀⠄⡰⠀⢀⡀⠀⠀⢆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠒⠍⠌⠀⠀⠈⠉⢉⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠⠀⠀⠀⠀⠀⢀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⢤⣶⣶⣀⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠤⠼⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⠀⡄⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢀⢀⢀⠠⠀⠀⠀⠀⠀⠀⠃⠇⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠠⠞⡁⠠⢄⡀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣘⡄
⠰⠠⠄⠀⠄⠲⡀⠀⠅⠁⠄⠈⠉⠒⠚⣅⢀⡤⢄⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⢘⠃⠠⠀⠀⠀⠀⠀⠀⢣⠁
⠠⠒⠀⠀⡠⠄⠂⠘⠢⠤⠀⠉⠂⢀⠐⠁⠙⠆⠀⣠⠄⠠⠀⠀⣀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⠀⠀⠧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⡮⠭⠄⠀⠀⠐⢲⠀⠀⠀⠀⠀⠀⢀⠠⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⢈⠁⠀⠠⠊⠁⠑⠰⠆⠦⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡆⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢮⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢀⠀⠀⠄⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⠀⠀⢠⠠⡰⡘⠎⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣄⡀⠀⠸⠇⠀⠚⢰⠙⠘⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠈⠑⠲⠀⠀⠀⠀⠀⡀⠀⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡀⠄⠀⠈⠁⠀⠀⢀⠬⠒⠦⠤⠄⠀⠁⠁⠻⠶⠔⢦⣠⢄⠀⢀⡀⠀⠀⠀⠀⠀⠀⣄⣤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠡⣦⠐⢣⣂⡠⠔⠚⠈⠀⠀⠀⡀⠀⣀⠀⢂⡠⢀⣀⣔⣉⣀⠀⠈⠈⢉⣉⣉⣑⣒⡾⠽⠖⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⣍⢔⠝⠋⣉⣁⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠥⠒⠉
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⣷⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣯⣿⣿⣿⣿⣽⣿⣷⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣻⣷⡽⠀⠀⠀⢀⣀⠀⣀⣴⣤⣴⣶⣶⣾⣷⣶⣶⣶⣦⣤⣀⣠⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠲⣀⠀⡀⢈⠋⡓⠀⠀⣀⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣶⡶⡬⡃⢌⡻⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡄⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠨⣿⠟⠃⠀⠀⡁⢌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠛⠀⣁⣧⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⣨⣿⡀⠀⠀⠀⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣾⣿⣿⣿⣡⣤⡾⣿⣿⣿⣿⣿⣿⣽⡟
⠀⠀⠀⠀⠀⠀⠀⡿⠈⠻⡑⠀⠀⣼⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢿⠿⠋⣾⣽⡟⠯⣤⠄
⠀⠀⠀⠀⠀⠀⡄⠃⠀⠀⠙⡜⣼⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⠉⠀⠠⠀⣰⣿⣿⣿⡟⠋⠏⡁
⠀⠀⠀⠀⠀⠀⠣⠸⠁⠀⠀⢻⠩⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣇⣐⣋⣰⠲⣿⣛⢟⢛⡝⠷⠒
⠀⠀⠀⠀⠀⠀⠀⢸⡼⠓⡄⢙⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣿⣿⣿⡱⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢊⣿⣯⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣎⣿⣿⣿⣿⣿⣿⣿⡏⣏⣅⣨⢉⣹⣭⡌⠉⢀⣁⢺⠓⠓⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣿⡏⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⢿⡧⡅⠈⠙⠻⠿⢿⣿⠿⢏⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡁⢰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠗⠃⠀⠀⠀⠘⠀⠹⢿⠚⢿⡏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⠻⣿⠷⣬⣓⠂⠠⠤⣀⣀⡠⠀⠀⠀⠈⠙⠓⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⡽⢿⣿⣿⣿⣧⣤⣿⣶⣷⣽⣿⣒⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠬⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡉⠉⢉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠉⠁⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣤⣶⣷⣷⣶⣶⣦⣤⣴⣄⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠴⠆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⡗⠠⠀⢠⣿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⣲⣴⠦⡀⠀⡀⠢⡀⠄⠀⠀⣀⠀⠁
⠀⠀⠰⣿⣿⣿⣿⣿⠏⣻⣷⣤⣤⣤⣽⣿⠇⠄⠀⢎⡽⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢋⣤⡍⠙⠒⢽⣿⡶⠬⠦⣭⣦⣤⣾⣧⣉⡁
⠀⠀⠀⣿⣿⣿⣿⡏⠀⢻⣿⠉⠀⠫⠽⠃⠀⠀⠀⡿⠊⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠈⠋⠙⠉⢛⠃⠀⠀⠀⠀⠀⠀⠀⣻⠟⠿⠿⠽⠍⡳
⠀⠀⠀⢸⣿⣿⣿⡿⢀⣼⠋⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⣀⣱⣤⣀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣈⡀⡀
⠀⠀⠠⠺⠏⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢑⡤⢿⡷⡄⠀⠀⠀⠀⠀⠊⢿⣀⣞⠐⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠁⣀⣶⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡿⣷⣄⣠⠀⠀⠀⠀⠀⢠⠉⠁⠀⡄
⠀⠀⠀⢠⣶⠿⣿⣿⣿⡿⣿⡟⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣾⣿⣿⣿⣿⣿⣇⣾⣿⣯⡆⠀⡄⠘⠄⠁⠀⠀⠀⠀⠅
⠀⠀⠀⢮⣧⣀⣈⢉⣿⣿⣿⣿⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢸⣿⡿⢿⣿⣿⣿⣿⣿⣭⢀⣆⠀⠀⠈⠀⠈
⠀⠀⠀⠀⣥⡹⡟⣽⣿⣿⣿⠿⣋⠳⣻⣿⣿⣿⣿⣿⣿⢟⢯⡿⠿⢿⡿⣿⡿⠇⢼⣿⣿⣾⣿⣿⣿⣿⣿⣿⠒⠞⡀⠀⠀⠀⠀⠀⠀⠈⠀⠚
⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣄⣥⡀⣼⣿⣿⣿⣿⣿⣟⢰⡟⢆⠀⠂⢠⡘⡇⣈⢸⣿⢻⣿⣿⣿⣿⣿⡿⠙⠳⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⣼⣿⣿⣿⣿⣿⣧⣻⣤⡠⢤⡿⣾⣿⡝⣿⣼⡿⠘⠃⠄⢀⠀⢠⠄⠤⢷⣤⣬⣔⣂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⡿⣿⣿⣿⠷⠚⠛⠛⠻⠉⢉⣩⣤⣶⣾⡶⠚⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠧⣝⣢⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣸⣿⣀⣭⣿⡟⠋⠀⠘⠛⠀⠶⠶⠀⠁⠀⡺⢿⡿⢿⠟⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⠞⣨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣤⡄⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠨⣻⢿⠃⠀⠀⠀⠀⠀⠀⢠⠂⣤⡞⣼⢟⡇⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠏⣻⣿⣿⠁⠀⠠⠄⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⡓⠒⠥⠤⠤⠄⢐⣠⡖⠋⠀⣾⢠⡙⢣⢏⣾⡉⠈⠘⠛⠋⠀⠀⠐⠒⠒⠒⠘⠃⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠈⠉⠁⠉⠙⠲⠒⠟⠿⠋⠀⢰⡇⠈⠏⠈⠘⠩⣭⣽⣧⣭⡍⠙⢻⣿⠿⣿⣿⣿⣿⣿⣶⣷⡂⠂⠀⠀⠀⠀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠁⠀⢸⡍⠭⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⠤⢄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⢀⡈⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠂⠀⠐⠼⠿⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠸⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⢠⣤⣤⣨⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠘⠋⠉⠀⠀⠀⠀⠀⠀⢀⠠⢀⠄⡀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⢄⠈⠩⠠⡄⠀⠂⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢸⠉⠀⠀⠀⠀⠀⠹⡃
⠀⠐⢀⢀⠄⠐⠘⠢⠠⠄⠒⠈⠀⠀⠁⠀⢠⡀⠠⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⡸⠀⠰⠀⠀⠀⠀⠀⢣
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢖⣂⡀⠀⠀⡦⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⡐⠐⠂⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡠⡰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡟⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠈⠉⠁⠀⠀⠀⠈⠶⣄⡀⠀⠃⠀⠂⡜⠙⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠑⡂⠀⠀⠀⢀⠀⣄⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡀⢀⢀⠀⠀⡀⠤⠐⠉⠀⡙⠂⢀⠀⠈⠉⡉⠶⠒⠤⠴⠤⠠⢄⣀⡤⣔⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠚⠂⠁⠀⠀⠀⠀⡪⢔⢂⣐⠦⠐⠚⠁⠉⠀⠀⠀⠈⠉⠉⠉⠉⣁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠂⠉⠁⠁⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠂⠀⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠃⢄⠀⠀⡠⠂⠀⠘⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠂⠈⠉⠉⢅⠀⠀⠉⠒⠶⢄⠀⢂⠊⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⢔⣀⢪⠁⠀⠈⠲⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⡀⠀⠘⢌⣛⡿⠀⠀⠀⠀⠀⠼⠀⠀⠤⡀⠀⠐⠤⠀⠄⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠁⠒⠁⠀⠀⠀⠀⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠂⠀⠀⠀⠀⠀⡠⡈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠁⣾⣶⣶⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⠒⠠⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢄⡀⡀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⣀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠈⠂⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⢐⣀⠂⠀⠄⣀⠀⠀⣀⠔⠋⠀⢄⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⣀⣣
⠀⠰⢠⠄⠀⠄⠲⡀⠀⠈⠀⠄⠀⠉⠁⠲⢏⡀⣠⣤⡀⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠐⡃⠑⡀⠀⠀⠀⠀⠀⠈⡌
⠀⠀⠆⠀⠀⣀⠤⠠⠀⠧⠄⡄⠈⠁⡁⡀⠎⠈⢻⡀⠈⣠⠀⠄⠀⠀⡀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⢀⠃⠀⢰⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠜⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⡍⠀⠠⠀⠀⢡⢍⡁⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠒⠒⠀⠀⡠⠚⠀⠢⣠⡄⡤⠠⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢦⡃⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⢠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡞⠆⠁⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠁⠀⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡦⡀⠀⠀⢀⢆⠀⢠⢇⡳⠑⡄⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠒⠲⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠤⠀⠀⠀⠀⠀⢣⣀⡀⡀⠀⢀⠦⠀⣗⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⡀⢔⣡⡀⠀⠀⣀⠠⠔⠊⠁⠀⢁⠉⠁⢀⠀⠀⠀⠀⢁⠼⠚⠑⠢⠤⠶⠤⠤⠦⣀⣀⡦⣜⡺⠛⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠁⠠⠗⠒⠉⠀⠀⠀⠀⠀⠠⣋⠴⠆⢀⢴⣄⠴⠖⠓⠒⠀⠈⠉⠁⠀⠀⠐⠒⠒⠋⠉⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢚⠕⣁⠤⠄⠒⠒⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⣟⠛⠿⣻⣿⣿⣿⣦⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⣬⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⣿⣷⣼⣶⡿⠿⣿⣿⣿⡙⠛⠩⠛⠹⠙⣿⣿⣿⣿⣿⣿⢚⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⣿⢋⣕⣿⣍⡝⣟⣿⣷⣙⣿⣿⣿⣷⣄⣤⢸⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡀⠄⣶⣿⣿⣶⣿⣿⠛⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣮⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡴⣾⢿⣟⣒⣚⣋⠁
⠀⠀⠀⠈⠀⠒⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠉⠈⠉⠃
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠋⠉⠉⠛⠛⠏⠋⠀⠀⠀⠈⠛⢻⣿⣿⣿⠿⠿⠏⠁⢈⣈⡀⠀⠀⠀⠰⡿⠟⠻⠟⠉⠉⠛⠻⢿⣷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⡄⠀⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢀⠔⢉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠉⠃⠀⠦⠠⣀⠀⣥⠀⠂⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣄⣀⣠⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⠖⢲⢾⣦⣶⣄⣀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⣸⣿⣷⣶⣤⣀⣤⡄
⠀⠀⢀⣤⣴⣨⣿⣤⣶⣤⣤⣧⣦⡎⠛⠝⢿⠻⠇⠀⠀⠈⠁⠀⠀⠰⣾⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣵⠠
⠀⢰⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⡀⠀⠀⠀⠘⠀⢀⣿⣷⣸⣿⣟⣿⣿⣿⣯⣭⣿⣍⣉⣽⡇
⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠁⠀⢀⣀⣤⣤⣤⣄⣀⠠⠴⠿⡟⠟⠛⠛⠛⠛⠋⠀⠤⠼⠛⠛⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠙⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⠀⠀⢽⣯⣽⠆
⠀⠀⣀⣀⣤⣤⣶⣾⣿⣛⣛⣻⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠶⠢⠀⠀⠀⠀⣀⢶⣟⣧⠒⠂
⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠊⠭⠍⠉⠃⠀⠈⠉⠉⠉⠉⠑⠛⠚⠋⠙⠂⠀⢘⣝⠟⠿⠿⣷⣴⣆
⠀⠈⠷⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣠⣤⣖⠒⢠⡤⠀⠘⢛⠀⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⢤⠤⣀⣀⣤⣤⣿⣿⣿⠏⠀⢉⡉⠉⠉⠙⠛⢿⢿⣿⡋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧
⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠿⣄
⠀⠀⠐⠀⠀⠀⣠⣄⣶⡷⣷⡥⣖⣤⣿⡶⣀⣤⡤⠟⠋⠁⠀⠀⠀⠈⠉⠒⠒⠤⠤
⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠉⠛⠙⠓⠚⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣯⣽⣭⣭⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡏⣴⣿⡾⠟⠚⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⠋⠏⠉⠁⡀⣐⣲⠢⣦⣤⣀⢀⣴⣿⡿⣿⣯⣿⣿⣿⣅⣴⡒⣳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠿⢧⠈⠀⠀⠀⠊⠓⠙⠪⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠻⢟⣭⣷⣄⣠⣂⣄⡀⠐⡴⠀⠀⠐⢒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢠⣾⣿⣿⣿⣿⡿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣥⣼⣆⠤⢦⢜⠩⢟⢻⣿⣿⣿⣿⣿⡿⡻⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠰⣆⠀⠀⣼⣿⣿⣿⣿⣿⠷⡇⢛⢿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠆⠀⠉⣸⣿⣿⣿⣿⣿⣿⠛⠃⠁⢠⣿⣿⣞⣓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠘⠃⣿⠀⠀⣿⣿⣿⣿⣿⣿⣷⠆⡈⠻⡿⠿⠏⡉⠉⢣⠘⠿⠓⠄⢄⠀⢸⣿⣿⣿⡟⠊⠫⡀⣀⣰⣾⣯⣉⡛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⢹⡇⢸⣿⣿⣿⣿⣿⣿⡆⢰⠆⠀⠀⠀⠀⠉⣴⣿⣾⢆⢄⣤⣾⡄⠘⣙⣋⣴⣠⣤⡶⠃⠀⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⡿⢻⣿⣿⣿⣿⢿⣿⣾⣂⣾⣄⠰⠀⠀⠀⠻⣿⣿⣼⣾⣿⣿⣷⣆⣀⢉⣹⣿⠟⢻⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⢂⠀⠀⡉⠩⠛⠿⠟⠛⠛⠋⠉⠉⠉⠉⠘⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⣿⣶⣿⣿⠛⣿⣿⣿⢿⡿⣿⢢⠀⣠⣦⡀
⠀⠀⠀⠀⠀⣀⠠⢄⣀⣀⠀⠐⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⠁
⠀⠀⠀⡤⢤⣥⣶⣶⣶⡿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣋⠻⢿⡡⣦⣄
⠀⠀⠀⠀⠐⠒⢂⣀⣤⣶⡾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣛⢛⢷⡿⢾⣟⣛⣿⣦⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣬⣭⢭⡭⢯⣬⠭⣿⠷⡿⠿⡗⠂⠀⠀⠐⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣄⣤⣶⣿⣿⣿⡷⣦⠬⠿⠮⢉⡛⠹⣯⣭⣿⡥⣬⠿⠧⠬⠱⠤⠤⠤⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠠⠬⠭⠍⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣰⣶⠇⠀⠀⣀⡀⡀⢀⢀⣶⠇⢀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠇⠀⠀⠠⠿⠦⣿⠯⠴⠶⠿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣤⣤⣀
⠀⠀⠀⠀⢀⣿⢳⣿⣿⣿⣿⣿⣷⣼⠀⢀⣠⣴⣾⡟⠀⠀⠀⠀⠀⠈⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣴⡟⠃⠟⣟⠿⣻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⢿⡟⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠁⠑⠀⡲⢬⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢞⢃⡀⢸⣷⣿⣿⣿⣿⣿⣿⣿⠀⣠
⠀⠀⠀⠀⠀⡀⢀⡀⠤⠤⠤⢤⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠈⢿⡄⡿⠀⣿⣿⣿⣿⣿⣿⣿⡖⠁
⠀⠀⠀⣀⣀⡉⠉⠁⠘⠛⢶⣡⣿⣿⣿⣿⣿⣿⣏⢻⣿⣿⡿⠿⡇⠀⣻⣿⣿⣧⠀⠀⣹⢗⠀⢻⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠹⠿⠿⠿⠟⠛⠛⠛⠋⢻⣿⣿⣿⣿⣿⠵⠌⠹⣿⣧⠀⠀⠀⣶⣟⣻⣿⣇⠊⠀⠨⡿⠿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣭⣿⣿⣿⣤⣀⣲⣿⣿⣆⣤⣀⣀⣈⣙⣾⠷⣶⢢⣤⣤⣎⡉⠚⠚⠀⠀⠖⠛⠛
⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠿⠛⠛⠩⠽⠿⠯⠍⠉⠉⠭⣯⣙⡿⠟⠻⠟⢻⣿⡽⠛⠋⠤⠀⠉⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠒⠃⠀⠘⠛⠉⠒⠂⠀⠀⠀⠀⠒⠃

⣤⡤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣀⣼⣷⣿⣾⣿⣽⣿⣾⣿⣷⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡏⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⣆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⢝⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣶⣶⣮⡐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄⠀⠀⠀⣸⣿⣿⣿⣟⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⠟⠃
⠀⠀⠀⠀⠀⠀⢀⠒⢄⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣌⣉⠻⣿⣿⣿⡁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠰⡤⠀⠀⠁⢀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣾⣿⡿⠿⠋⠀⡀⠈⠉⠉⠉⠁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠐⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀

⠀⠀⢀⡠⠐⠟⠠⣀⠀⠀⠀⠀⠀⢠⣶⠒⢲⣶⠒⢲⣆⠀⠀⠀⠀⠀⠰⠀⣀⡀⠰⠀⠀⠀⠀⠀⢀⡤⠖⠚⠒⢦⡀
⠀⣴⣾⣶⣶⣶⣶⣾⣶⡄⠀⠀⠀⢿⣏⣀⣸⣿⣀⣈⡿⠇⠀⠀⠀⠀⣶⢲⣾⣷⣶⡆⠀⠀⠀⠀⡞⢠⢠⣄⡄⣤⢹
⠀⣿⡏⠍⢉⡉⠍⡙⣿⡇⠀⠀⠀⢸⡏⢹⡏⠉⣿⠉⣿⠀⠀⠀⠀⣠⣿⣴⣿⣿⣿⣧⡀⠀⠀⠀⢷⡈⠉⠉⠉⠉⡸
⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⣿⣿⠇⠀⢿⣿⣿⡄⠀⠀⠀⢧⣶⣶⣶⣶⣶⣦⠇⠀⠀⠀⠀⠙⠶⠤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠛⠻⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⢦⡀⠀⠀⠀⠀⠀⠀⢀⣎⡏⣻⣳⠀⠀⠀⠀⠀⢠⣸⣭⠆⣯⢿⣠
⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢤⣼⣥⣤⣤⣤⣝⣤⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠶⠶⠆⣶⠰⠶⠶
⠀⠀⠘⢛⡛⠛⣛⠋⠀⠀⠀⠀⠀⠈⣷⣺⣒⣗⣺⣲⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⢸⣿⡇⣿⢰⣿⣷
⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠸⠾⠧⠧⠿⠞⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠼⠀⠀⠀⠀⠘⠿⠇⠿⠘⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣉⠀⠁⠈⠀⠉⠈⡁⠀⠀⠀⠀⢸⠀⣿⢸⢸⡆⣿⢸⠀⠀⠀⠀⠀⡿⣶⣶⣤⣭⣭⡷⠀⠀⠀⠀⠀⡠⢪⢭⡙⣦
⠀⠤⠰⣒⠄⢀⣠⠄⠄⠀⠀⠀⠀⢸⠀⣿⢸⢸⡇⣿⢸⠀⠀⠀⢀⣶⣶⣶⣶⣶⢾⡍⡇⠀⠀⠀⣿⣾⣶⣾⣶⣷⣾
⠀⠒⠀⣊⠖⣯⡒⠐⠂⠀⠀⠀⠀⢸⠀⣿⢸⢸⡇⣿⢸⠀⠀⠀⢠⣿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠽⠀⠀⠉⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠿⠒⠒⠲⠞⠛⠃⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⡀⢀⡀⢀⣤⣤
⠀⣴⡶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠘⡇⣤⣤⣤⠀⠀⠀⠀⢠⣮⣤⣤⣿⣧⣤⣬⣧⠀⠀⠀⢹⠁⢹⡏⠈⠻⠟
⠀⢸⣧⣇⣆⣇⣸⣿⣶⣐⣆⠀⠀⠀⠀⠀⠷⢿⣿⣿⡇⠀⠀⠀⢸⡇⡀⠀⡉⢁⠀⠀⢸⠀⠀⠀⠈⠀⢈⡁⠀⣈⣁
⠀⠿⣿⣿⠿⠿⠿⠿⠿⣿⣿⠧⠀⠀⠀⠀⣞⣻⠜⠒⠀⠀⠀⠀⢸⡃⠏⠘⠞⠚⠃⠃⢸⠀⠀⠀⠰⣤⡞⠁⠈⠙⠋
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠟⢛⣉⡩⠤⠤⠤⠤⠤⠤⢍⣉⡛⠻⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⣡⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⣌⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢁⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣦⡈⠻⣿⣿⣦
⠀⠀⠀⠀⢀⡾⣿⡿⠁⡰⠛⡽⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣸⡟⠀⠈⢿⠻⢦⠈⢿⣟⢷⡀
⠀⠀⠀⠀⡼⠱⢹⣀⠜⠀⠀⠀⠀⠀⢻⣿⠛⠛⣿⠛⠛⣿⡟⠛⢻⡟⠛⢻⣿⠃⠀⠀⠀⠀⠀⠱⣀⡏⠂⠁
⠀⠀⠀⠈⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠘⣿⡶⣶⣿⣷⡶⣿⣷⣶⣿⣷⡶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠁
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣽⣇⣀⣿⣇⣀⣿⣁⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⣟⣉⣿⣏⣹⣿⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⣇⣠⣴⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠉⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣦⣄⣸⡇
⠀⠀⠀⢀⣤⣾⢿⡿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⣷⣾⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡟⣶⣤⡀
⠀⠀⠀⢿⣿⣿⡎⢧⠀⠃⢉⣊⠣⠉⢈⣁⢃⡘⠐⣈⠙⣉⢊⠁⣂⢂⣘⢓⣀⠋⢉⡋⣉⢡⠀⠀⡾⢱⣿⣿⡿
⠀⠀⠀⠈⢿⣿⣿⡌⢣⠀⣿⡿⣥⣼⡟⢉⣟⢹⣿⣿⣾⣷⣿⣿⣿⣿⡯⢿⣽⣿⡟⠉⣿⠿⠀⡼⢡⣿⣿⣿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣄⠡⡙⠛⠀⠀⠙⠋⠙⠋⠉⠉⠁⠉⠋⠉⠙⠙⠛⠉⠈⠁⠙⠋⠙⢛⠊⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠼⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⢿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⡿⠿⠛⠛⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⡀⣠⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⢸⠉⠩⡍⢉⢉⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⣣⡇⣠⠼⠃⠀⠀⠀⠲⠲⡄⠀⠀⠀⠾⠗⠀⠀⠀⠀⠀⠀⠀⢠⡼⣛⣭⣷⣮⣍⡻⣤
⠀⠀⢸⠀⠀⠹⡿⡿⠀⠀⢼⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⣶⡶⠒⠒⢲⠀⠀⠀⠀⠀⢻⣶⢶⡶⢶⡶⣶⡶⣶⡆⠀⠀⠀⣾⠁⣿⠛⡟⡛⢻⡟⢸⡆
⠀⠀⠸⣤⣤⣤⣤⣤⣤⣤⡾⠀⠀⠀⠀⠀⠀⠘⡿⠿⠿⣿⡿⠻⠿⡟⠀⠀⠀⠀⠀⠀⢿⡾⢳⡞⢳⡞⢶⡿⠀⠀⠀⠈⣿⠀⣿⣚⣛⣷⣸⣇⢸⡏
⠀⢀⣾⣿⣿⣿⠛⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠻⡛⠿⠿⢻⠟⠁⠀⠀⠀⠀⠘⢧⣝⠿⢿⠿⢟⣩⠿
⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠳⠶⠶⠿⠷⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠋⠉

⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤
⠀⠀⡏⢰⣶⣶⡆⡶⢦⡖⠘⡇⠀⠀⠀⠀⠈⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠶⠶⠤⣤⣤⣤⣤⠀⠀⠀⠀⣾⣿⣿⣿⣿⣋⢀⣈⣿⡆
⠀⠀⢧⡘⠷⠛⠳⠃⠘⢀⣰⠇⠀⠀⣠⠤⣤⠤⣿⣶⠶⠦⠤⠄⠀⠀⠀⠀⠀⣿⠁⣴⡶⠶⠶⢤⣤⣝⣛⣀⣤⠀⠀⢸⣿⢫⠶⠶⠶⠶⠶⠶⠶⠶⡄
⠀⠀⠀⠉⠉⠁⢻⣶⣭⡍⠁⠀⠀⠀⠓⠶⠋⠐⠊⠻⣆⠒⠀⠒⠐⠂⠀⠀⠀⣿⠇⡟⠁⠀⣴⡋⢿⡄⠈⠻⣿⠀⠀⠈⣿⢸⣷⣶⣶⣶⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠈⠿⢻⣦⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠓⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿⣧⣄⣘⣯⡞⠁⢀⣴⣿⠀⠀⠀⠀⢸⠀⠶⠶⠶⠆⠠⠶⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠉⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣿⠀⠀⠉⢓⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠈⠳⢤⣘⣉⣉⣭⢹⡇⠀⠀⠀⠀⠀⠸⢋⣴⠏⠛⢿⡿⠋⢸⡇⠀⠀⠀⣠⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⢀⡟
⠀⠀⠀⣠⠞⣡⠀⠉⠉⠉⢸⡇⠀⠀⠀⠀⢀⡴⡏⠚⢷⡾⠋⠀⠀⢸⡇⠀⠀⣾⢣⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣤⣤⢀⡾⠋⠀⠸⣧⠤⢤⡀
⠀⠀⢾⡁⠞⢿⣷⣤⢀⡴⠋⠀⠀⠀⠀⠀⢸⣷⣖⡾⠋⠀⠀⠀⠀⢸⡇⠀⠀⠹⣧⣻⣿⣿⣿⣟⡽⢒⠶⣄⠀⠀⠀⢸⣿⣿⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⢦⡀⢟⡵⠋⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣿⣿⣿⢸⡑⢾⠗⣸⠀⠀⠀⢸⣿⣿⢸⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠈⠙⠋⠀⠙⠒⠚⠁⠀⠀⠀⠘⠛⠛⠘⠓⠒⠒⠒⠒⠒⠛
⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⡞⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⢀⠴⠛⠛
⠀⠀⠀⡾⢻⡟⠿⠟⣿⢻⡆⠀⠀⠀⠀⠲⢠⡶⠶⠶⠶⠶⠶⠶⡆⠀⠀⠀⠀⠀⠀⠘⣇⠀⣀⣠⣴⣦⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⡇⠈⢁⠀⠀⠉⢸⡇⠀⠀⠀⠀⠠⢸⡇⠀⠀⠀⠀⠀⠀⣷⡤⠤⡄⠀⠀⠀⢠⡿⡜⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⡟⣛⣛⡛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⡇⠐⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⣿⣶⣶⣿⠀⠀⠀⠈⢧⢛⣽⣛⠿⢟⣻⡄⠀⠀⠀⠀⡇⠉⠉⠁⢠⣦⠀⣶⠀⡇
⠀⠀⠀⡇⠀⠈⠛⠋⠀⢸⡇⠀⠀⠀⠀⠀⠸⢿⠫⢻⠶⠶⠶⠶⢿⡟⢝⡿⠇⠀⠀⠀⠀⣏⠶⣸⡇⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠇⠀⠿⠀⡇
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⣀⣸⡟⠛⢿⣿⣿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣀⣠⣿⠟⠁⠀⠀⢀⣀⣭⣽⣿⣿⣿⣿⠿⠿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠀⠈⠋⠉⠈⠉⠉⠁⠀⢀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡄⡄⡀⠀⢊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣿⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠲⢦⣴⠾⠿⠟⠋⠉⠉⠉⠙⠛⠿⠿⢿⡷⠞⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣠⣦⣶⠒⠀⠐⣦⣦⣦⣠⣾⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⣿⡿⠏⠀⠀⠀⠂⠀⠙⣿⣿⣿⠁⠀⣠⡀⣰⣣⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⢠⣶⣾⣿⡿⠗⢧⠀⠀⠀⢠⠿⢿⣿⣿⣶⣿⣿⣿⣿⣿⡄⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠉⢳⢿⣾⣫⣶⣦⣾⠀⠀⠀⣸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣥⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠋⠀⢿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⡿⡏⣹⣿⣿⡿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠈⠉⠉⣻⢃⣬⣴⣿⡿⠙⡿⠏⠰⠄⢺⣿⣟⢂⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⣭⡀⠀⣤⣾⣷⣀⣨⠋⢀⣹⣿⣏⣾⣿⣯⣷⠄⣼⠩⠉⠨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠐⣿⣷⠀⢿⣿⡏⠸⣿⡖⠿⠟⣿⣿⠝⣿⣿⡟⢳⣿⠛⠒⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⢿⣧⣀⢿⣇⠈⢿⣷⣴⣼⣿⠏⣠⣿⢿⣰⣿⠯⢄⣶⣆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢞⠹⣿⣿⣶⣿⣷⣦⣿⣿⣿⣤⣶⣿⣿⣿⣿⡿⢷⣾⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢞⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠀⠀⠈⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿⣿⣟⠿⡿⠿⠛⠿⠿⡿⢟⣿⣿⣏⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣿⣿⡟⡼⠁⠀⠀⠀⠀⢹⢘⣿⣿⠏⡌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢷⠀⠘⢻⣟⠁⠀⠀⠀⠀⠀⠀⢛⣿⠳⠀⠰⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣁⠛⢇⠀⠀⠀⠀⠀⢠⠛⠋⣁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⣰⣯⣸⠷⠞⠀⠀⠀⠀⠀⠘⠶⢿⣸⣧⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⢉⣉⡉⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⣰⢠⡆⣰⠀⣆⢰⡄⣆⠀⠀⠀⠀⠀⢰⣦⣤⣤⣶⣴⠂
⠀⠀⠀⠀⠀⠸⠻⠀⣾⣷⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠃⡘⠁⢹⣄⠛⠈⢣⠘⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⠔⠂⠈⠁⠀⠀⠀⠀⠀⣯⣭⣽⣯⣭⣭⣿⣿⡇⠀⠀⠀⠀⠀⣇⣀⣀⣁⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⠿⠭⠿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿


⠀⠀⠀⠐⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⣶⣖⣶⣶⣶⢰⠋⠉
⠀⠀⠀⠀⠀⠻⠟⠀⣴⢲⣦⠀⠀⠀⠀⣭⣤⣼⡟⢻⣷⣤⣬⡅⠀⠀⠀⠀⠀⠹⠿⠿⠿⠇⡟⠀⠀⠀⠀⠀⠀⠀⠘⠙⠈⠉⠸⣶
⠀⠀⠀⠀⠀⣤⢠⠀⢿⣠⡿⠁⠀⠀⠀⣟⠛⠻⢯⡽⠟⠛⢻⡁⠀⠀⠀⠀⠀⠀⣛⡛⢛⣏⠁⠀⠀⠀⠀⠀⠀⢠⣠⣀⣀⣀⣀⣀⣠
⠀⠀⠀⢠⣤⣤⢠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠓⠉⠙⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⠿⠏⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣃⣈⣇⣀⣀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡾⣵⡋⣝⣷⣹⡀⠀⠀⠀⠀⣀⡔⣀⣾⠉⣩⠉⠉⢹⡆⠀⠀⠀⠀⢩⢯⣯⣭⣿⣽⣽⡍
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠁⠀⠀⠀⠀⠀⢷⡻⣝⣢⡿⣸⠁⠀⠀⠀⠀⢿⣿⡿⠿⠷⠷⢶⣥⡾⠀⠀⠀⠀⠀⠘⡟⡅⣧⡟⣓⣹⠁
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⡠⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣄
⠀⠀⠀⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⢻⡆⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⡟⠛⢛⠛⡟⠛⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣄⠀⡀
⠀⠀⠀⡇⠤⠨⠠⡇⢽⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⢶⣟⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⣛⡃⣸⣷⣿⡄
⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠿⠟⠿⠛⠿⠿⣿⠆


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠄⠀⠀⠀⠉⠡⡷⠽⠀⡌⠤⣄⢤⣤⣤⢤⢤⣼⣼⢢⡀⠀⠁⢶⣶⣶⣶⣶⣶⡆⢀⣚⡠⡄⣌⡂
⠀⠀⠀⠀⣀⠀⠀⠀⠤⠘⠀⡄⠾⠟⠳⠆⠀⠀⠀⡏⢁⠢⠁⠀⠀⠈⠿⠛⠛⠛⠛⠀⠀⡜⠁⠽⠏⡀
⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⡇⠘⡃⠛⠘⠛⠃⠻⠈⢹⠀⠀⠀⠀⠀⢸⡿⠉⢽⡇⠀⠀⢢⣄⡀⣴⠃
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣸⡟⠛⠛⠒⠚⠀⡄⠤⠀⢀⠀⠀⠀⠀⠀⠀⣀⡈⢀⡀⠈
⠀⠀⣡⠤⠤⠤⢬⡀⠀⡀⠀⠀⠀⠀⠦⠥⠈⠄⠀⠀⠀⠀⠑⠚⠘⠐⠲⠃⠀⠀⢀⠀⠫⠠⡀⠀⢃⣨⠇
⠀⠀⡇⣀⣝⣉⠘⠐⢣⡁⠀⢀⣥⢧⠀⡄⠀⣠⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠐⠉⢀⣄⠑⡮⢷⠻
⠀⠀⡇⣻⣏⣿⢰⠀⠀⠀⠈⡂⠉⠈⠀⠴⠾⠓⣷⠤⠀⠀⠀⢀⣰⡈⠁⣦⣀⠀⠐⠛⠛⠀⠇⠟⠁
⠀⠀⠃⠛⠛⠛⠐⠀⠀⠀⠀⠳⠄⠐⠈⠃⠀⠀⠈⠀⡆⠀⠀⠀⣂⣒⣴⣀⠃⠀⠀⠀⢀⡀⠀⠠⣄
⠀⠀⠃⠀⠐⠁⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⢐⠢⠀⠀⡀⠒⠒⠂⠒⡒⣒⠒⠀⠐⠂⢀⡈⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠤⠀⢈⡔⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⡏⠃⠀⣀⠀⠀⠀⠳
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣰⣀⣦⠀⡤⠀⠈⠈⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⡤⡀⡄⠐⠆
⠀⠀⠀⠀⣀⢀⠀⠀⢸⡁⠀⠀⠀⣀⠟⣠⠚⠆⠀⠀⠀⠀⠈⠀⠀⠀⢤⣄⠲⡢⢀⣰⣦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⢀⣀⣉⣉⡝⠀⠃⠀⠀⠀⠃⠐⠀⢰⠀⠆⠍⡍⠲⡇⣟⠋
⠀⠀⣀⣀⠀⠀⢀⣀⣀⡄⠀⠀⠀⠀⠈⠑⠒⠃⢀⡀⠀⢀⡀⠀⠀⠤⠤⣄⣣⠠⠛⢀⣀⣄⣀
⠀⠀⠀⠀⢠⡆⢠⠀⢸⠖⠶⠦⡀⠀⠀⣠⡔⣪⠥⣦⢤⡐⢈⢢⠀⠀⠀⠀⠀⠦⠖⠇⡁⡀⠄⠘⠦⠤⠂
⠀⠀⢀⣀⣘⠀⠙⠀⢸⡇⣀⢀⠙⡄⠈⠻⢽⡃⠈⠙⠀⢹⠉⡢⠄⡀⡂⠀⠉⡖⠂⠀⠴⢻⠃⠀⠀⠐
⠀⠀⠹⣅⡽⠁⠀⠀⠈⠉⣏⢹⠟⠁⠀⠀⠈⠓⠤⣀⣀⢈⡠⠁⠈⠀⠀⠀⠀⠁⠀⠉⠐⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡦⢤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢲⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢾⣭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠛⠛⠍⠉⠉⠛⠛⠋⠛⣻⣍⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⡤⠔⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⢀⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠯⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡶⠀⠀⠀⠹⣆⡴⠚⠛⠛⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣄⣀⣀⠴⢲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⣠⡴⠾⡅⠀⢹⢁⣴⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⡿⠶⠤⣤⣁⡤⠛
⠀⠀⠀⠀⠀⣀⡠⠖⠉⠀⠀⠀⢎⣐⡷⢃⠀⠀⢸⡷⢤⣤⡴⠃⠀⠀⠀⠀⠀⠀⢀⡿⠁
⡴⠚⠓⣦⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⠉
⢷⣀⣠⣟⡀⣀⣠⣤⣤⣀⣀⣀⣀⡤⠒⠑⣂⡴⠳⣤⣄⣀⣀⣤⠗⠚⠋⠉⠁
⠀⠈⠀⠀⢉⡟⢠⠃⡼⠀⠀⠀⠉⣩⠋⡽⠉⠀⣠⠃⣼
⠀⠀⠀⠀⠈⠉⠛⠒⠁⠀⠀⢀⠜⢁⡜⠀⢀⡴⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠊⠀⢠⣏⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠿⠿⠿⠿⠿⠿⠿⠖⠾⠶⠿⠿⠿⠿⠿⠿⠗⠒⠒⠒⠲⠷⠿⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⡾⠛⠻⣦⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣼⣷⡶⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⣀⣠⡿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⣵⠟⠁⠀⣿⠀⠈⠻⣮⡉⠛⠷⣤⡀⠀⠀⠀⠻⣦⣈⣉⣉⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣼⠏⠀⠀⠀⣿⠀⠀⠀⠘⣷⡀⠀⠈⠻⣦⡀⠀⠀⠀⠉⠉⠉⠉⢹⡏⠉⣿⠉⠁
⠀⠀⠀⠀⠀⠀⣠⡿⠻⠶⣤⣄⣼⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢘⣧⣠⣤⡶⠞⠻⣦⣤⣤⣤⣤⣤⣤⣼⠇⠀⣿
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢩⡿⠙⠛⠛⠛⠛⣿⠛⠛⠛⠛⠋⢻⡏⠀⠀⠀⠀⠹⣧⣀⣀⣀⣀⣀⣀⣀⣠⡿
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢻⡏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⢾⡷⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢈⣷⣀⣤⣤⣤⣤⣿⣤⣤⣤⣤⣄⣼⡇⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠙⣧⣴⠶⠛⠋⢻⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢩⡟⠙⠛⠷⢦⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢻⣄⠀⠀⠀⣿⠀⠀⠀⢠⡿⠁⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⡻⣦⡀⠀⣿⠀⢀⣴⠟⣁⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⠿⠿⠿⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⠀⠀⣶⢰⠀⠀⢸⣱⠁⠀⣼⡌⠁⡠⠁⠀⢀⣖⠀⠀⢀⡀⠀⢔⣨⣽⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡕⢸⢃⠀⠀⠀⢹⣼⡀⠀⣿⡿⠀⣸⡟⠀⣷⠇⠀⠀⣾⠏⢀⡴⣫⠔⠊⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡌⠀⢸⡘⣆⠀⠀⠀⢻⡇⠀⢹⣇⠀⣿⠃⠀⣿⠀⠀⢰⡟⢠⣯⠞⠁⠀⣀⣀⣤⣄⣀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡇⠀⢸⡇⠈⢣⡀⠀⠀⢻⡀⠘⣿⠀⣿⠀⠀⣿⡆⠀⢸⣷⡟⢁⣤⣾⣟⣻⣯⣭⢀⠉⡏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠸⣿⡀⠘⡟⡆⠀⠙⣦⣀⠀⢷⠀⢿⠀⣿⡇⠀⠸⣷⡀⢺⣟⣴⡿⠛⠋⠁⠀⠀⠀⠀⠙⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⣌⠉⣇⠀⠀⠙⢿⣄⣻⡁⠀⠀⠈⠻⣕⡜⡇⢸⡇⢻⣷⠀⠀⢻⣧⣼⡿⢋⣀⠤⣤⣤⣶⣶⢦⡤⡜⠘⠇⠡⡀
⠀⠀⠀⠀⠀⠀⢄⡙⠚⣟⠪⢷⣶⣤⣤⣹⣿⣇⠀⣤⣤⣤⢬⣿⣿⣸⡇⠈⣿⡆⠀⠀⡙⢽⠷⠋⣠⠝⠒⠚⠂⠒⠒⡺⢀⡀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠠⠿⠿⣿⠻⢶⣄⡀⠈⠉⠉⠻⠋⢺⡗⠛⠛⠛⢺⣦⣤⠀⠀⠀⠉⠀⠀⠀⠀⠣⡈⠀⠀⠀⠀⢀⣠⡞⠀⠀⠂⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⣅⣠⣤⣧⣤⣤⣙⠻⣦⡀⠀⠀⠀⢰⣤⣤⣀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢔⠒⠨⠅⠘⠅⠀⠠⢄⡀⠀⠀⠀⠘⠌⠑⢄
⠀⠀⠀⠀⢸⡿⠛⠉⠀⣾⠀⠉⡓⠶⡃⠀⠀⠀⠞⠈⠉⠟⠀⠀⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠈⢲⡕⠀⠀⠀⠀⠀⠀⢂⠀⠀⠆
⠀⠀⡀⠐⠈⠀⠀⢀⠄⢻⠀⠀⣿⠀⠁⠀⠀⢸⢀⣤⡖⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⢿⡷⠁⠈⠀⠰⠀⠀⠀⠀⠀⠡⠀⠀⢂
⠀⠀⠀⠀⢀⣰⢿⠇⢠⠘⠇⢠⡿⠘⠀⠀⠀⠘⠟⠋⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠘⠀⠀⠀⠘⢦⠀⣼⢃⠎⠀⠀⠀⠀⠀⣀⣶⠀⠀⠀⠀⠆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⡼⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡏⠈⠀⠀⠀⠀⠀⠀⠘⡁⠑⢄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄⡄⠀⠜⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠔⠀⠀⠀⢀⣁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠾⠟⠂⠀⠀⠀⠀⢶⣴
⠀⠀⢀⣁⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⢠⠠⠀⡀⠀⠀⠀⢸⣟⣿⣷⣄⡀⠀⢀⣀⡤⣶⣤⡀
⠀⠀⠀⢱⡌⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠈⠻⣿⣋⣿⡿⣟⣿⡿⣟⡵⣾⣿
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⠀⢀⣴⣿⡟⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣼⣟⣉⡵⢿⣾⡃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣼⢳⠀⠀⠀⢹⣶⣿⡿⠛⠀⠀⠂⠙⠉⠀⠀⠀⠒⠳⢬⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠉⠉⠁⠀⠉⠘⣷
⠀⠀⠀⠀⠀⢹⡏⠈⠉⠁⠀⠸⣿⡇⠘⣧⠀⠀⠀⠉⠉⠀⠀⠀⣶⠀⠠⣄⠀⠀⠀⠀⠀⢀⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠉⠓⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠘⣦⠀⠀⠀⠀⠉⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠴⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠀⣀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣶
⠀⠀⢀⡤⠞⠉⠀⠉⠈⠉⠀⠀⢸⡉⠣⠀⠀⣾⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣼⢻⡟⣿⣿⣇⡿⠿⡿⣿⣿⣿⡿⡟⣿⣿
⠀⢠⡿⠖⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⢦⣤⣴⣿⣇⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣶⣭⣛
⠀⠙⣿⠄⠀⢀⡀⢀⣤⣿⣿⣿⡃⠀⠀⣿⣿⡿⠿⣅⡈⢃⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⣿⣯⣿⠿
⠀⠀⢻⣇⢠⣾⣷⣽⣿⣿⣿⡿⢷⠀⠀⣿⠀⠀⠀⠀⠹⢸⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠐⢸⣿⢛⡿⠿⠿⠛⠋⠁⠀⠘⣆⡀⠀⠀⠀⠀⣼⣂⡜⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⣟⣸⠇⠀⠀⠀⠀⠀⠀⠀⠋⢻⠀⠈⢆⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉
⠀⠀⡀⢿⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⣼⠀⠀⠀⠀⢹⡀⢀⣤⡀⢀⣤⡄⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠓⠺⠟⠃⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠘⠛⠃⠀⠀⠀⠈⠘⠁⠀⠀⠀⠉⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠀⡁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢮⡜⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢳
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣎⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣿⣿⣟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠣⣀⣶⣤⣴⠀⣠⣼⠄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦⣿⣿⣿⣿⡇⠀⠀⠀⠠⣤⠀⠀⠀⠀⠉⡇⠀⠉⠑⢆
⠀⠀⠀⠀⠈⠃⢠⡈⡿⠿⠁⠀⠀⠀⠀⠀⠁⠘⠁⠈⠿⣷⡄⠀⠀⠀⠀⠘⣿⣿⣿⣟⣽⣿⠿⢿⠃⠀⠀⠀⡆⠙⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢠⡀⣀⡼⢧⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⠃⠀⠀⢀⡄⠀⠝⠻⠋⠁⠀⠀⢊⠀⠀⠀⠀⠀⣧⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰⡄
⠀⠀⠀⠀⠀⠈⠛⠻⣿⣷⡀⠀⡿⠿⣶⣴⣶⣾⣧⣀⠀⠁⠀⠀⠀⠘⠀⢰⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠓⠶⠂⢴⣿⣷⣤⣀⣠⣴⢿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡁⡎⠀⠀⠀⠙⠋⠁⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠉⠉⠉⠀⠈⡇⢻⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠷⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠌⠀⠀⠀⠀⠀⠀⠄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠼⠒⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⢀⠀⠀⢠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⡛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠙⣿⡄⣠⣤⣤⣄⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣸⣿⡀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⢸⣿⡟⠁⠈⣿⣆⣴⡾⠟⣋⣉⠉⠀⠀⠉⠉⠙⠛⢿⣷⡄⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣿⡇⠀⠀⣿⡟⠉⣠⣾⠟⠛⣿⡟⠛⠛⠷⢦⣄⡀⠹⣷⠀⠘⣿⣾⠟⠻⣷⣀⠀⠀⠀⢀⣤⣶⣶⣤
⠀⠀⢀⣤⡶⠿⠿⣶⣼⡿⠀⣾⣿⡇⠀⢀⣿⠁⣠⣿⠃⠀⠀⡿⠻⢷⣦⣦⣤⡉⠻⣦⣿⡇⠀⣿⡇⠀⢀⠟⠛⢿⣦⣴⡿⠋⠀⠀⢿⡇
⠀⠀⣾⠏⢠⣾⣶⣄⠻⠇⢸⠟⠻⡇⠀⢸⣿⣿⠿⡏⠀⠀⢸⠁⢀⠈⠋⠀⢹⡿⣶⣽⣿⡇⠀⣿⠇⠰⣾⠀⠀⣸⡿⢋⣴⠃⠀⢀⣼⡏
⠀⢸⣿⠀⠀⢿⣮⣻⣇⢀⡏⠀⣶⡁⠀⠈⡀⢀⠀⢀⠀⠀⠆⠀⣾⣇⠀⠀⡾⠀⠀⣿⣿⠃⢰⠇⠀⢠⡟⠀⠠⠋⣰⡿⠁⠀⢠⣾⠏
⠀⠀⢿⣆⠀⠀⠙⠻⢿⣾⣇⠀⢻⠇⠀⢰⣇⢸⡇⣿⡆⠀⡇⠀⢿⣿⠀⢸⡆⠀⢰⣿⡿⠀⣠⠀⢠⡿⠀⠀⢀⣼⡿⠁⠀⣰⡿⠃
⠀⠀⢈⣿⣷⣤⣀⠀⠀⠈⠙⠀⣀⣼⡄⢸⣿⢸⡇⣿⣧⠀⠃⠀⢸⡿⠀⢸⢇⡄⠀⣿⠃⣰⣿⠀⡾⢡⡇⢀⣾⡿⠁⠀⣴⡿⠁
⠀⠀⠈⢻⣿⠀⣿⣷⣦⣀⠀⠀⠈⢻⣧⢸⣿⡀⠃⣿⡿⢷⣾⣧⠈⠃⣦⠈⣼⣷⠀⠟⢠⣿⣿⡀⣡⣿⡇⣸⣿⠃⠀⣼⣟⣀⣀
⠀⠀⠀⠸⣿⠀⢿⣇⢉⡻⣷⣄⠀⠀⣿⣿⣿⣷⡶⠿⠿⠿⢾⣿⣷⣶⣿⣿⠟⣿⣇⢀⣾⣿⡿⠿⠿⠿⠯⠭⠍⠀⢠⣛⣛⣿⣿⣧⣤⣤⣴⣶⡄
⠀⠀⠀⠀⢻⣧⡈⢿⣦⣍⣈⣿⠆⢀⣿⣿⣏⣀⣀⣀⣀⡀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣜⣯⣵⣶⣶⣶⣶⣶⣤⠀⢀⣄⣀⣀⣉⣉⣭⣭⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣦⣌⣉⣉⣉⣴⡿⠋⠙⠛⠉⠻⢿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⣈⡉⠉⠛⠛⠻⠿⠿⠟⠀⠘⠋⣱⡿⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠙⠛⠛⠿⠿⠷⣶⣶⣶⣦⠀⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢿⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣉⣉⣹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣀⡀⠀⣀⣀⡀⠀⣀⣀⣀⠀⣀⣀⣀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⠿⠿⠇⠀⠿⠿⠇⠀⠿⠿⠟⠀⠿⠿⠿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣤⣤⠀⣤⣤⡄⠀⣤⣤⡄⢠⣤⣤⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⠿⠀⠸⠿⠇⠀⠿⠿⠃⠸⠿⠿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢰⣶⡄⢠⣶⡆⠀⣴⣶⠀⣰⣶⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠉⠁⠀⠉⠁⠀⠉⠉⠀⠉⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⣈⠙⣿⣿⣿⣿⡟⢉⣀⡈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠇⢸⣿⣿⣿⠀⢾⣿⣿⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣁⣤⣿⣿⣿⣿⣷⣄⣉⣁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠤⢾⣿⣿⢩⣤⣾⢡⣦⡙⣿⠀⢿⠁⢹⠇⠸⡏⠈⡏⠠⢼⡇⠴⠌⡟⢡⣦⡿⠠⠴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⣛⣻⣶⣷⣌⣛⣻⣌⣛⣡⣇⣰⣀⣇⣸⣀⣇⣰⣄⣇⣘⣛⣇⣰⣄⣧⣈⣛⣧⣘⣛⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⢟⣿⣶⣃⣠⠀⠀⠀⠀⠀⠀⣰⣯⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⣼⢡⡞⣱⢟⣿⢷⣾⣁⡠⠀⣠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡞⣼⢣⣟⣼⢣⡞⣱⢏⡽⣻⢷⣾⣿⣿⡿⣫⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣿⣽⣷⣿⣾⣷⣿⣾⣵⣯⣾⣵⣯⣿⣿⣿⣾⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢁⡯⠞⠛⠛⠛⠛⠛⠟⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣭⣶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣻⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡠⢒⠒⠒⢦⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⣫⣿⣿⣿⣿⣗⡩⠙⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠠⢟⣿⣿⣿⣿⣟⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⡝⣧⠀⠀⠀⠀⠀⣀⠴⣺⣿⠃⠀⠀⢊⢿⣿⣿⣿⣯⣭⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⣰⣯⠖⠛⢿⣮⣷⣄⠀⢀⠀⠀⢀⡿⢸⣧⠰⣤⡶⣛⡵⣿⣿⡏⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⣟⣿⡧
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠠⣿⣿⢿⣿⢇⣿⣿⣿⠀⠈⠀⠀⠈⣳⣤⣿⡆⢟⡾⣫⣾⣴⣿⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⣀⣾⡃⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡾⡷⠋⠀⠀⠀⠀⠀⠈⣿⣿⡟⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⢾⣭⣍⣉⠙⠒⠶⢤⣄⡀⠈⠉⢉⠉⠀⢀⠀⠀⠀⠀⠀⠀⢰⣽⣿⠃⣿⣿⣻⣿⣿⣿⣿⣶⣤⡀⠀⣸⣿⣿⣿⣿⣾⠏
⠀⠀⠀⢀⣹⣿⣿⣿⣷⣦⣄⠈⡛⢦⡑⣤⣠⣀⠀⠀⠀⠈⠀⠀⢀⠞⣿⡟⣸⣿⣿⡿⣵⣿⣿⠿⠛⠉⢀⣰⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠸⢭⣛⣛⣉⣉⣛⣛⣛⣦⣈⣿⣎⣿⣿⣆⡀⠐⣈⣀⢁⣡⣾⡟⣡⣿⣿⣿⣾⡿⠿⠿⠒⠀⠀⣼⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠙⠻⠿⢯⣭⣍⣉⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣴⣿⡿⠿⢻⠍⠄⠀⠀⢀⣠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣾⣯⣭⣭⣵⣶⣿⠿⠋⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⠿⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⢿⣷⣶⣶⣿⡟⠻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣽⣿⡄⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡦⠀⠀⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⣠⠒⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⣇⠀⢄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠳⠤⢵⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠐⢤⣄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠤⢞⣁⣀⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢸⡀⠀⠀⠠⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⡛⠭⢬⣉⠱⠶⣒⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⠒⠒⠒⠷⣇⠀⠀⠀⠤⣤⣤⣤⠤⠖⠉⠫⠽⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠂⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⡀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡄
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠒⠺⢦⠀⠀⠀⠀⠀⠀⠹⡄⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠹⡈⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠙⢄⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠈⠉⠛⠦⡄
⠀⠀⠀⢀⣤⣴⣖⣒⣒⡚⠳⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢀⡼⠁
⠀⢀⢶⡟⠉⠀⠀⠀⠀⠈⠉⠓⠤⣈⠁⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡹⢦⠄⠀⠀⠀⠀⠙⠒⠋⠉⠛⢐⡆
⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠓⠦⠤⢄⡀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⡠⠤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡁⠀⠀⠀⠀⣰⠥⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠂⣀⠔⠀⠀⠀⣠⠎⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡸⠁⠀⠀⠀⠀⠀⡰⢡⠎⠀⠀⠀⠀⡰⢻⠀⢀⠤⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇⠀⠀⠀⠀⠀⢠⡰⠁⠀⠀⠀⠀⠰⠁⠀⡔⢁⡴⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢃⡰⢾⠒⢩⢔⠜⠉⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢀⠜⠀⠆⢰⠁⠆⠀⠀⠀⠀⠀⡠⢊⠔
⠀⠀⠀⠀⠀⠹⡦⡄⠀⠀⢃⢰⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠀⠀⠸⢀⡌⠀⢠⠀⡜⠀⠀⠀⡠⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠘⡌⡆⠀⠈⢻⣀⠀⠀⠀⠀⠀⢄⣣⡀⠀⠀⢀⡇⡏⠠⠀⢠⠋⠀⠀⠀⠊⡰⠃
⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠈⠦⡁⠄⠀⠀⠀⠀⠈⠑⡦⡄⠀⠃⢃⡰⢀⠆⡀⠀⢀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⠀⠀⠈⠢⣱⠀⠀⠀⠀⢀⠖⣼⠎⠀⢨⠓⣼⠀⡇⠔⠉⠇⠀⣠⡴⠂⠀⠀⠀⢀⠜⡜
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⡀⠀⠀⠀⠀⢠⣟⠀⠀⠠⡄⡿⡸⡌⢀⠜⢔⠟⠉⠀⠀⢀⣤⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⠸⠀⢡⠀⠀⠀⢠⡸⡌⢃⠀⣠⢃⡧⢰⡴⢃⡞⠁⠀⠀⠀⡴⡱⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢎⠲⢄⠀⠀⠀⠀⠑⢴⡀⠀⠀⠀⢹⠸⣀⠟⢡⣎⠔⡏⢠⠻⠀⠀⠀⠀⠀⢦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⠠⣤⣀⣀⠈⠣⣄⡀⠀⠸⢀⡧⣦⡋⠤⠊⠀⢸⡘⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⢦⢜⣌⢣⣦⢈⠜⠇⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⣠⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⡎⣇⠎⡘⢀⡠⢤⠶⠚⢛⠟⠲⠖⠾⠭⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠺⠋⡜⠰⡡⠊⢀⠠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⠀⡄⢷⡡⢊⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢴⣴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⡎⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⡔⠁⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠱⡄
⢧⠀⠀⠀⣄⡇⣀⠀⠀⠀⣘⠀⠀⠀⠀⠈⣣⡤⠒⡐⠒⠢⣄⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠈⠳⠤⠴⢯⡀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠵⢀⣀⠁⠀⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡷⠦⣄⡀⠀⠀⠀⠀⠀⣀⡠⢴⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠙⢳⠒⠛⠙⠿⡤⠞⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡜⠀⠀⠀⠀⠒⠚⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢇⠀⠉⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⣡⠤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣄⡉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢰⡁⢰⡋⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠉⢒⠄⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⢳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢈⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⡰⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⣁⠴
⠀⠀⠀⠀⠀⠀⠸⣈⠒⠃⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⠀⢁⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡇
⠀⠀⠀⢰⠋⠉⠉⠒⢤⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⡴⠚⠋⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠸⡀
⠀⠀⠀⠸⡄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⡄
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡀
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣈⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⠓⠲⢬⣉⠒⠦⣄⣀⠀⠀⠀⢀⣠⠜⢠⠃
⡠⠒⠒⠒⠂⠉⢉⣀⣀⡤⠤⠖⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⣞⣁⡠⠤⠤⠤⠤⡀⠉⠒⠦⣄⣈⣁⣉⣁⣠⠴⠃
⠱⢦⣠⣀⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠋⠉⠛⠧⣤⣄⣀⣤⣤⣤⠤⠤⠤⣄⠀⢀⢀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣏⢿⡸⣻⡇⡇⢸⠃⣶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡼⢚⡿⠙⡃⢓⠛⢼⢃⣾⠋⡠
⠀⠀⠀⠀⠀⠰⣾⣿⢆⡞⡅⣰⢡⠏⣴⠆⡘⠃⣼⠖⠋⢀
⠀⠀⠀⠀⣐⣺⣿⠆⣿⣾⡁⣿⢸⣤⠏⣼⠏⣤⠁⢠⣾⠟⢁⡀
⠀⠀⢀⡈⣩⡭⡍⣰⣧⢿⢳⢿⢌⡏⣼⢋⡼⢃⡴⠈⢡⡶⠛⣡⡀
⠀⣰⣿⡇⠲⣿⣽⡻⠸⠞⠋⢿⢸⢱⢣⢞⡴⠋⣠⠾⣀⠰⠾⢛⣷
⠀⠻⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⢋⡎⣰⠟⣠⠞⣡⡾⣿⢇⠸⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⣠⣴⣶⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⢫⣞⡵⢛⡽⢛⡽⣻⣳⢻⠟⣤⢤⣤⡤⡄⣤⡄⣠⠄⣠⣶⠟⠋⣡⣶⣿⢟⣩⡶⠟⣋⣭⣤⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢛⣁⣴⢟⡴⢋⡾⢱⡏⣾⢠⡟⢸⣿⣼⢁⣿⡿⠋⣼⡿⠁⣠⣾⣿⢟⡵⠟⣡⣴⠟⢛⣩⣥⣶⣶⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⢡⡟⣵⠟⡀⣿⢰⡇⣼⠇⣼⣿⠇⣸⣿⠇⣼⠏⢀⣾⡿⢟⡵⢋⣴⡾⠋⣠⣶⠿⢛⣩⣥⣶⣆⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⣴⢏⣼⢏⡞⢰⡇⣼⠁⣿⠀⣿⡿⠀⣿⣿⡾⠁⢠⣿⠏⣠⠋⣠⡿⠋⣠⡾⢋⣥⣾⠿⢟⣛⣭⡥⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣡⣾⢋⡾⢡⣿⢡⣿⠀⣿⡆⣿⡇⠀⣿⣿⠀⢀⣿⡏⠀⢁⣼⠋⢠⠞⣡⡴⠟⠉⣐⣚⣉⣭⣴⣾⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣵⠟⣱⡿⢣⠟⢹⡇⢻⣧⠸⣿⠀⠘⣿⡀⢸⣿⠀⢀⡾⠁⠀⢁⠞⢋⡴⠞⢛⣋⣩⣭⠽⢿⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⡴⠋⡴⢋⣴⡄⢿⡈⣿⠀⢻⡇⠀⢹⡇⢸⡏⠀⡼⠁⠀⠐⢁⡀⠉⠠⠚⣉⠥⠶⠶⣞⣿⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣞⡵⣛⣅⡹⡌⠇⢸⠀⢸⡇⠀⢸⠃⠸⠁⠀⠀⠀⠠⢴⣿⡿⠷⢤⣀⣠⠴⠶⢛⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢯⣭⣅⠹⠄⠀⠀⠈⠀⠠⠤⠤⠐⠒⠈⠀⠀⠀⠀⢴⠶⠶⠖⡟⠑⠖⣛⣭⡵⣿⡝⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣩⠁⠀⠀⠉⢺⣫⣭⣽⣎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⡆⠀⠀⠀⠀⠙⣿⣯⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⢭⡽⠃⠀⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣽⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⠃⠀⠀⠀⠀⠀⠀⠀⢠⠓⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠃⣸⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⠿⠏⣰⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⢠⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠤⠠⠀⠀⠤⠄

⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⡠⠂⢦⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⡀⠀⠀⠀⠀⢀⡴⠲⣄
⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣻⡟⡟⣿⠀⠀⠀⠀⠀⠛⣿⣿⡿⠃⠀⠀⠀⠀⠘⢇⠙⠟⠛⠁⠀⠀⠀⠀⠈⠧⣤⣞
⠀⠰⢿⣿⠗⠀⠀⠀⠀⠀⠘⠷⠧⠧⠏⠀⠀⠀⠀⠀⠀⠛⠉⠓⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⢀⣰⣶⣶⣶⠖⠀⠀⠀⠀⠀⠰⠩⠡⠂⠀⠀⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⠀⢠⣤⡤⢤⣤⡀⠀⠀⠀⠀⠀⠁⠀⠈
⢸⣿⣿⣿⣿⠿⠀⠀⠀⠀⠃⢃⢘⣘⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠿⣿⣌⣸⣿⠟⠀⠀⠀⠀⠀⠀⠛⠁
⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⢀⣄⣤⣤⡄⠀⠀⠀⠀⠀⢰⠛⠛⢿⡆⠀⠀⠀⠀⠀⢳⡰⣶⣿⡀⠀⠀⠀⠀⢰⣯⣴⣿
⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣯⣭⣯⡇⠀⠀⠀⠀⠀⠘⢿⣶⡾⠃⠀⠀⠀⠀⠀⠀⣷⡭⠒⠃⠀⠀⠀⠀⠸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢠⡀⣠⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⣀⣤⣀
⠀⣤⣤⣸⣿⡇⠀⠀⠀⠀⢸⣿⡄⣿⡏⠀⠀⠀⠀⢿⣿⣿⣿⣇⣈⡆⠀⠀⠀⠠⠀⠀⠀⠸⠃⠀⠀⠀⠀⢸⣻⡛⣟
⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠨⣿⠇⣿⡇⠀⠀⠀⠀⠙⠞⠛⠛⠛⠟⠃⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟


⠠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡤⠤⠀⠤⣄⠀⠀⠀⠀⠀⣠⠤⢤⣄⠀⠀⠀⠀⠀⢰⠒⠒⠐
⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣶⣴⣾⡏⠀⠀⠀⠀⠀⠿⣧⣼⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣀⣀⣈⣁⣉⣀⣀⣀⣀⣀⣸⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣙⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠁⢈⣿⡄⠀⠀⠀⢠⣾⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢠⣸⣿⣇⣀⣤⣴⡿⣷⣇⡠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠋⠉⡡⢾⡇⣿⣯⠄⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠋⠀⠀⠀⠀⣻⠁⣿⣿⣏⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠈⠀⠀⠂⠀⠀⠀⠑⣱⣿⡿⠋⠀⠀⠀⣀⣼⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣡⡿⢿⣿⠿⠛⠉⠀⠁⠛⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠁⠀⠀⢠⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠶⣋⠀⠂⠀⠀⠈⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠘⠀⠀⠀⠀⠘⠿⠿⠗⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⡺⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡞⠁⠀⠀⠀⠒⠒⣧⣈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣤⣤⣤⣴⣶⣤⣩⣍⣤⣴⣶⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠉⠀⠊⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⢀⣱⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⣦⡀⢀⠠⠀⠀⣠⠂⠀⡠⠀⡀⢠⢌⣿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⢀⠀⠛⢧⣇⢀⣠⣴⠃⠀⠀⠄⠄⢁⣼⠿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠄⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⢀⣠⣰⣿⡞⠋⠁⠀⢀⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠷⣿⣿⡃⡀⣠⡴⠋⠉⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⢿⣿⣿⣟⠛⠱⠀⠀⠀⠀⢃⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠁⠀⢈⣵⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣼⣾⣿⣵⣷⣌⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠂⠀⢀⣌⠄⢀⣠⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣟⠋⠙⢛⣛⣳⣶⣶⢟⡛⠛⠿⠾⢶⠆⢀⣀⣤⣼⣴⣾⡿⢛⣩⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠿⠷⠿⠶⠷⠶⠶⠾⣿⣛⣛⣛⣫⣭⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

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

⠀⠀⢸⡀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠐⠒⠂⠂⠂⠀⠀⠀⠀⠀⠀⣰⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⠉⡇
⠀⠀⢸⠀⡐⠐⡀⠀⠒⠀⠀⠀⠀⢸⢙⠘⠙⠉⠋⠃⠉⡇⠀⠀⠀⠀⠀⢹⢀⡂⢠⠀⠀⠀⠀⠀⠀⠈⠈⡠⠚⠀⠃⠄
⠀⠀⢸⠁⠙⠛⠁⣠⣤⡄⠀⠀⠀⢸⣠⣤⢀⣤⣀⣤⣄⡇⠀⠀⠀⠀⠀⢹⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⡾
⠀⠀⠈⠉⠉⠉⠁⠉⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡞⠉⠙⣦⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⠉⠇⠉⠉⡇⠀⠀⠀⠀⠀⢀⡔⠉⣥⠉⠳⡄
⠀⠀⠀⠀⣠⡷⢤⡴⠋⠀⠀⠀⠀⢾⠀⠀⡰⠢⠀⠀⢸⠆⠀⠀⠀⠀⢸⣶⡇⢠⣤⡇⠀⠀⠀⠀⠀⢸⠘⢓⠉⢈⠻⢸
⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⠇⠀⠀⠀⠀⠀⠈⠢⣛⠀⢛⡣⠋
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣌⡠⠂
⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠙⡇⠀⠀⠀⠀⠀⠀⠰⠂⡹⣿⡿⠁⠴
⠀⠀⠤⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⣿⣿⣥⡄⠀⠀⠀⠀⠀⠀⠰⠀⢅⠁


⠀⠀⠀⠀⡠⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠺⠂⠈⠉⠞⠆⠀⠀⠀⠀⠀⠂⠆⠁⠐⢊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠁⢒⡈⠈⣓⠀⠀⠀⠀⠀⠀⠀⣬⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⢰⢼⡧
⠀⠀⠻⠿⠿⠃⠸⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠊⠁⡇⠀⠁⠀⠀⠀⠀⠀⠰⡿⢿⢿⡷⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠤⠴⠒⠛⠛⠛⠛⠛⠒⠢⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠈⠉⠑⢦⣤⣀⣤⠖⣻⠽⠛⠋⠉⠀⠀⢀⣈⠁⠀⢀⣠⡟⠻⠆⠨⠻⣗⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⡠⠊⠀⠀⠀⠀⠀⡄⢠⡾⠇⠀⠀⠈⠋⠑⠀⠰⣦⠀⠙⣷
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠐⢤⡁⢨⡇⠀⠀⠀⠀⠀⠛⣆⠀⠁⠠⠀⢉⢦
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡝⡞⠃⠀⣠⠞⠁⠀⠀⠸⠀⠓⠀⠀⠴⣱
⠀⠀⠀⠀⠀⢠⢛⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⣀⣤⢸⡷⠀⢀⠔⠂⢀⠀⢴⢿
⠀⠀⠀⠀⠀⠈⢸⡥⠀⠀⠀⠀⠀⠀⠀⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⠉⠁⠀⠀⠀⠠⠔⠂⢀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⣀⣞⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⠀⠀⠀⠀⠀⠀⠀⠉⢀⢶⣼⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢻⣦⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡈⡇⠀⠀⠀⠀⠀⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢡⠸⣆⠀⠀⠀⠀⠀⠀⠘⠈⡀⡏⠙⠶⣤⣀⡀⠀⢀⣀⡀⠀⠀⠀⠂⠀⠂⢹⡿⣿⣷⣐⣄⠠⡀⠀⠀⠉⠄⠼⣄
⠀⠀⠀⠀⠀⠀⠈⡄⢻⡆⠀⠀⠀⠀⠀⠀⣰⠃⢀⢀⡀⠀⠉⡛⠛⠛⠟⣿⣷⣶⣶⡤⠶⠶⠋⠁⢸⠀⢻⠉⠳⣿⡀⠀⠀⠈⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⡄⠀⠀⠀⠀⠀⢾⠆⠀⠈⣧⠰⡷⢻⢀⣀⢀⡀⠀⢉⡰⠀⠤⠀⠀⠀⢸⠀⠈⢧⠀⠻⣧⠀⠀⠈⣆⡜⠳⣝⢄
⠀⠀⠀⠀⠀⠀⢸⠃⢇⠀⢻⡄⠀⠀⠀⠀⢘⣾⣆⠀⠀⠀⠉⠈⣷⣇⡟⡁⢼⣟⣡⣴⠇⠀⠀⠀⣸⠀⠀⠈⠳⣄⠙⢦⡀⣀⡐⣛⠀⠈⠉⠳
⠀⠀⠀⠀⠀⢀⡇⠀⠈⠆⠀⠻⣄⠀⠀⢲⣾⣿⣿⡆⠀⠀⠀⠀⠈⠁⣾⣧⠤⢴⣾⣿⠃⠀⠀⠀⡏⠀⠀⢠⢎⢈⡿⣦⣽⣿⣅⡎
⠀⠀⠀⠀⠀⢼⠇⠀⠀⠘⣆⠀⠀⣵⣆⢸⡷⠿⠿⣷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⣧⠉⠯⠀⠀⠀⡇⠀⠀⢼⠀⠸⠴⠃⠸⠇⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣸⣇⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⡇⠀⠀⠸⣦⡀⠀⠀⣠⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢹⣿⠘⡇⠀⠀⢸⠀⢠⡸⣶⢣⡇⠀⠀⠀⠀⠀⢧⠀⠐⠂⠀⡇⠀⠀⠀⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⢈⡀⢧⠀⠀⢸⠀⢸⠇⠀⢸⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⢾⣿⠘⡆⠀⢸⠀⠀⣿⣧⣸⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠸⣿⣷⢷⠀⢸⠀⠐⢹⣿⣿⠀⠀⠀⠀⠀⢠⣿⡀⠐⠄⢀⠏⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠯⣽⠀⢰⣿⢸⡆⢸⠀⠄⠡⢄⣹⠀⠀⠀⠀⡠⠎⠀⠀⢈⣀⡈⠀⢠⣾⣶⢫⡾⣶⣿⣶⣿⣶⣶⢶⡶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢘⣛⠀⢳⣸⠀⠀⠌⠛⢹⡀⠀⠀⠐⣄⣀⠀⠀⠙⢻⠁⠀⠈⠉⠉⠈⠉⠿⠉⠈⠙⠃⠉⠈⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠻⡏⠀⠀⢁⡤⢀⣷⡀⠀⠀⠀⠉⠓⠪⠵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠤⠵⠞⠚⠁

⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⣀⡄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡄
⢻⣿⢽⢽⢽⣿⠃⠀⠀⠀⠀⠀⢴⣙⡿⢷⣷⣿⠀⠀⠀⠀⠀⠀⣿⡆⣿⢸⡇⠀⠀⠀⠀⠀⣸⡟⠟⠿⡟⣿⡀⠀⠀⠀⠀⣀⣰⣶⣶⣶⣶
⠈⢭⣭⡭⣭⠟⠀⠀⠀⠀⠀⠀⢠⣟⠋⠿⠶⠾⠃⠀⠀⠀⠀⠀⠋⠁⠉⠈⠃⠀⠀⠀⠀⠀⠘⣷⣾⣶⣷⡟⠀⠀⠀⠀⣾⣤⣿⣿⣿⣿⣿⠨⠄
⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠛⠉


⠀⢴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠀⠀⠀⡔⣫⠕⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤⣀
⠀⡶⠳⠶⣶⣼⠃⠀⠀⠀⠀⠀⣿⣿⡏⢦⣹⣿⣧⠀⠀⠀⠈⢦⡘⣃⢜⠞⣷⣆⠀⠀⠀⢠⣶⣾⣷⣶⣮⣙⡳⠀⠀⠀⠀⠶⠶⠶⠶⢶⡾⠇
⣰⣷⣶⣶⣿⣿⣦⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⠛⠓⠀⠀⠀⠀⠀⠈⠀⢿⣧⣾⠟⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠠⠭⠥⠘⠋⣠


⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡜⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠢⠀⠀⠀⠀⠀⠀⢠⣞⣽⡿⢙⣶⠀⠀⠀⠀⠀⠀⢤⣥⣧⣥⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄
⢿⣻⣟⣿⣻⣟⡿⠀⠀⠀⠀⠀⣾⡛⠛⠛⣛⠻⡇⠀⠀⠀⠀⢸⣿⣷⣴⡿⣿⠃⠀⠀⠀⠀⢸⣿⡷⠰⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⢳⠄
⠸⣽⣧⣿⣼⣯⠇⠀⠀⠀⠀⠀⠻⣴⣾⣧⣽⣶⠇⠀⠀⠀⠀⠀⠙⠷⠷⠟⠁⠀⠀⠀⠀⠀⠸⣿⡧⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠫⠔⠋


⠀⠀⡴⠑⠵⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡃
⠀⣿⡿⠿⣿⣿⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⡆⠀⠀⠀⠠⣶⣂⣸⣿⣏⣹⡷⠀⠀⠀⠠⣶⣂⣙⣟⣉⣹⠷⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⡆
⠀⣿⡐⢒⣽⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠶⠿⠟⠋⣈⣿⡇
⠀⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢁⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠘⠃⠈⠁⠙⢦⡀
⠀⠀⠀⠀⠀⢸⣏⢁⣀⣀⣀⣀⣠⣼⠃⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠉⠉⠀⢻⡄⠀⠀⢰⡏⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⢦⡀⠙⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡞⠁⣠⣤⡄⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢿⣤⠟⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣰⠷⣦⡀⠈⢹⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢿⣄⣴⠟⠀⠘⠳⠾⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⢡⡴⣦⠀⠀⣀⠀⠀⢸⡟⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡄⠸⠶⠾⠁⣼⠋⠻⣦⠈⠙⠋⠀⢉⣙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠃⣀⡀⠀⠀⠛⠶⠾⠋⢀⠀⠀⢸⠋⢹⡇⠀⣤⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣾⣉⣿⠀⢀⣤⣄⠀⠀⣿⣻⡆⠘⠛⠛⠁⠀⠷⣾⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⠉⠁⠀⢼⣅⣨⠗⢀⡈⠉⠃⣀⠀⢰⠞⠻⡆⠀⠀⠀⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⠘⢧⡀⠉⠁⣀⣿⠁⠀⠀⠘⣧⠘⠳⠞⠃⠀⢰⠶⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡋⠀⢀⢹⡆⢸⡏⣿⠙⠛⠛⠃⠘⢷⣤⣤⣄⠀⠙⢶⡏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣾⠀⣷⠀⠉⣿⠀⠀⠀⠀⠀⣦⠙⢦⡟⠀⠀⢸⡇⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢹⡆⠀⣿⠀⠀⠀⠀⠀⠸⡇⠀⠻⣦⣤⡈⢳⣄⢿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣶⠟⠀⣿⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠈⣷⠀⢸⠀⠀⠀⠀⠀⠀⢽⠀⢀⡆⢹⡆⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣿⠀⡾⠁⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠈⡇⢸⡇⠀⠀⠀⠀⢀⡟⠀⣧⠀⠈⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠀⠀⣿⠈⣧⠀⠀⠀⠀⢸⡇⢀⡿⠀⠀⣿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⣿⠀⠀⢀⡿⠀⡟⠀⠀⠀⢀⡾⠁⣾⠁⠀⣠⡟⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠀⢰⣿⣥⣤⡿⠀⠀⠐⠿⠤⠤⠾⠂⠺⠧⠤⠤⠿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠛⠳⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢸⠇⣀⣀⡔⠀⡴⠋⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⠀⠀⠠⠘⢋⡉⠉⠉⠛⠧⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣀⠀⠀⠀⠀⢳⣄⠑⠒⣡⢐⠋⠁⠀⠘⠷⡄⡘⠦⠤⠞⢀⡟
⠀⢀⣰⠀⣥⠂⡆⠀⠀⠉⠛⠋⢹⡅⠶⡀⠸⣘⣆⢼⠛⠶⠴⠖⠋
⠀⢣⠘⠆⠁⢸⣀⢀⡀⠀⠀⠀⢸⠈⠉⠀⠀⢈⡀⣸
⠀⠀⢁⠀⠈⠉⡡⠤⠼⠀⠀⠀⢸⡄⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠙⡆⡞⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⢠⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢀⡞⠈⣇⠀⠀⠀⢸⡟⣆⣀⡀
⠀⠀⠀⣈⠀⠓⠒⠒⢒⡒⢒⡞⠀⠀⠸⡄⠀⠀⢸⠁⠸⣦⣉⣉⡛⠓⠒⠒⢒⠒⣶
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⡞⠀⠀⠀⠀⢷⠀⠀⢸⡇⡀⠘⣆⠀⠀⠉⠉⠉⡿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⡆⠀⢸⠃⠂⠀⠘⣆⠀⠀⠀⢀⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠈⢧⡀⢀⠼⠰⢇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⠀⣸⠶⢾⣇⠀⠀⠀⠀⠀⢳⣁⠀⠀⠀⠇⡈
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡠⠖⠈⠁⠉⠈⣿⣶⣿⣿⡇⠀⠀⠀⠀⠀⢻⡝⠓⠒⠈
⠀⠀⠀⠀⠀⠀⣸⠁⢠⠊⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠈⠢⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢀⡇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢿⠛⠒⠒⠒⠚⢻⢻⠉⠀⠀⠱⡄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣄⡀
⡠⠒⡉⠉⠉⠉⠓⠒⠲⠤⠤⢤⡽⠜⣆⠀⠀⠀⢠⠏⢾⣤⣤⣤⣤⡽⠶⠾⠗⠒⠒⠒⠦⣀⠀⠀⣀⠴⢉⠴⠋⠉⠑⠢⢵
⠓⠶⣅⣠⡆⣰⠶⠤⢤⣀⣀⣀⣀⣀⡼⠀⠀⠀⠘⠲⠶⠤⠤⠤⡤⣤⣤⣤⢤⡀⠙⡆⢑⣏⣉⣉⣡⠜⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡿⠿⠛⠁⠀⠈⡻⢷⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀⣀⣠⣀⣤⣶⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠉⠉⠀⢀⣤⣴⡿⢿⣟⣿⣭⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠓⠈⣨⣶⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⢿⣿⡿⣬⢹⢿⣿⡿⠋⢋⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣵⠿⠛⡟⠉⠀⠀⠑⢭⣦⠴⠟⡼⢡⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣠⡿⠟⠛⠛⠛⠛⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠖⠊⣩⠉⠂⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠸⡿⣆⠀⠀⠀⡀⠈⠢⠄⠀⠀⠐⠊⠀⠀⠊⢁⡠⠜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣷⣌⠲⣄⣠⡈⠲⠤⠤⠀⠀⠀⢀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⡿⠃⠀⠀⢠⣴⣾⠿⣛⠛⠊⢉⣿⣿⠛⠳⠦⢬⣍⣙⣀⣠⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠛⠿⣿⣿⣷⣶⣭⡾⠃⠀⠀⢀⣠⣴⣾⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣀⣤⠶⠚⢿⣿⡿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠈⢷⣇⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣤⣄⣈⢹⣿⣷⡀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢀⣀⣀⣠⣤⣤⣬⠻⣾⣶⣿⣷⢀⣀⡀⠀⡀
⠀⠀⢠⠊⠀⢀⢠⠻⣍⣀⠄⠈⠉⠒⠛⢿⣿⣦⡀⠀⠀⠀⢿⣀⣠⣼⣿⡿⠿⢛⣻⣿⠏⠀⢈⣹⣛⣧⡤⠥⠚⠇⠀⣢⠀⣢⡄
⠀⠀⠸⣇⢸⣬⣿⣷⣌⣻⣶⣤⣐⣲⠶⠦⠽⠿⣷⣦⣤⣀⢀⣿⣿⣿⣿⣋⣭⣽⣿⣿⡶⣛⣫⣭⡭⠁⢀⣀⡔⡼⢠⢇⡖⠉
⠀⠀⠀⠙⢌⢮⣝⣷⣽⣿⣿⣷⣌⡋⠙⠲⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠿⢧⣾⠿⢛⣉⣁⣈⣃⣩⣴⢟⣵⡋⠉
⠀⠀⠀⠀⠀⠉⠚⢿⢿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⣿⣿⣿⣶⣿⣵⣿⣿⣿⣿⣟⡪⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠉⠋⠙⢿⣿⣷⣶⣦⣭⣽⣿⣿⣿⣭⡿⠿⠾⠟⠉⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠛⠳⢦⡀⠀⢀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⠠⠀⠀⠀⠀⠀⠄⠀⠙⠿⣋⠉⠖⣀⠉⠙⠻⣷⣀⣀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠈⠀⠰⣿⣠⡀⠀⠀⠉⠁⠀⠉⢹⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠄⠀⠀⠀⠄⠐⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⢰⡆⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⣀⣤⠴⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠗⠲⠤⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢸⢠⡏
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠸⡏⠻⢦⣀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠹⣄⠀⢹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⢦⣄⡀⠘⣿⠾⠃
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠈⣷⣀⡿
⠀⠀⠀⠀⠀⣼⠃⢀⣤⣤⣤⣤⣤⣤⣤⣶⣊⣉⣍⣿⡆⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣏⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣋⣉⣭⣿⣦
⠀⠀⠀⠀⠀⠹⣆⡈⠛⠦⣤⣀⣀⠀⠀⠀⣀⣀⣤⠴⠞⣋⣥⠶⠛⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣉⠉⠉⠉⢉⣉⣡⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠤⢤⠀⠀⠠⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠁⠀⣀⠀⠙⠄⠀⢠⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⡁⣿⣤⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠅⠅⠀⠀⠈⠘⢉⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠⡀⠀⠀⠀⠀⠀⢘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⠀⠐⢤⣶⣶⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠜⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⠠⠄⠀⠠⠀⠀⠀⠈⠰⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠠⠄⠀⠀⠄⠀⢀⠠⠚⢀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠆⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⣬⡆
⠀⠂⠄⠀⠰⠈⠇⠀⡊⠀⠀⠀⠀⢈⣀⣀⣒⡂⠀⠀⠀⠈⠳⠆⠀⠀⠀⢀⣀⠄⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠈⠁⣀⡠⠒⠚⠂⠑⠂⠐⠒⠈⠁⠀⠀⠀⠀⠀⢠⣀⣂⠲⠄⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠐⠀⠀⠀⠀⠀⠀⣣
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣒⡒⠒⠀⠀⠈⣆⣀⠀⠀⠀⠀⠀⠀⠆⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠡⠀⠤⠀⠀⠀⠀⠁⠚⠘⠒⠁⠁⠀⠀⠀⠀⠀⠀⠠⡀⡄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡱⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠸⢤⡤⠀⠀⠀⠀⠰⢂⡀⠀⠀⠀⡄⠀⢀⡄⢆⢧⡃⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠙⢦⣤⣀⠋⠀⠈⠐⠆⠀⠁⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⢅⡀⠀⠀⠀⢠⡀⠤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠠⠀⢄⡀⠀⠀⠀⣀⠠⠔⠊⠁⠃⠙⠂⠀⠀⠀⠉⠉⢉⠶⠓⠤⢀⡦⢄⣀⣄⡀⢠⣄⡓⡟⠚⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠓⠉⠨⠇⠒⠉⠀⠀⠀⠀⢠⣏⠤⠆⢀⢴⡀⠴⠴⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠂⠉⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠨⢚⢋⡁⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣟⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿
⠛⣿⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⣿⡟⠻⣿⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣿
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣤⡄⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣤⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⣿⣧⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⣿⣿⣿⣿⣿⣿⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢼⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢿⡟⠻⣿⠉⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣿⠟⠻⣿⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⡄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠
⣿⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣧⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣼⣿⣿
⣿⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿
⠛⣿⠋⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠟⠻⣿⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣴⡄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣦⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣾⣿⣧⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢰⣿⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢼⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠛⣿⠉⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠻⣿⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⡄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣶⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠
⣿⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣧⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣾⣿⣿
⣿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣿⣟⣻⣿⣻⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠿⠿⠃⠀⠀⠀⢹⣿⡟⠁⠀⠀⢸⣿⠉⠉⣠⠞⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⣾⣷⠀⠀⠀⢰⣿⣠⠞⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠟⠿⠆⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠿⠿⠿⠿⠿⠇


⠀⠀⠀⠀⠀⢀⣻⣿⣿⡀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⢀⣿⡀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢠⣤⣤⣶⣶⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣠⣬⣿⣥⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠈⢹⡏⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠼⠿⠿⠟⠿⠿⠄⠀⠈⣿⠁⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣶⣄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⣤⣾⠤⠄⢼⣦⡄⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠀⠸⠿⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠀⠀⠀⢀⣶⡀⠀⠀⠀⣴⣶⣤⣶⣦⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠐⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⡄⠀⠀⠈⣿⣿⡟⠁⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠔⢿⣿⡟⠆⠀⠀⢠⣿⣼⣷⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⣶⡆⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠛⠉⠛⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆
⠀⠀⠠⢠⣶⣤⣤⣤⣴⣶⡌⠀⠀⠀⠀⠀⠀⠈⢰⣶⣤⣤⣤⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⣤⣶⡆⠁⠀⠀⠀⠀⠀⠀⢡⣶⣦⣤⣤⣤⣶⡄⠄
⠀⠀⠀⢸⣿⣿⡀⢀⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣶⣿⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣿⣿⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣭⣭⣭⣽⡇
⠀⠀⠀⢸⣿⠋⠙⠋⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣾⣿⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣇⣉⣡⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⢾⡇
⠀⠀⠀⢸⣿⠶⠶⠶⢴⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣒⣛⣚⣓⣺⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣛⣛⣛⣿⣿⡇⡀
⠀⠀⠐⠘⠿⠷⠶⠶⠿⠿⢁⠀⠀⠀⠀⠀⠀⢀⠸⠷⠿⠾⠿⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠾⠇⡀⠀⠀⠀⠀⠀⠀⡈⠿⠿⠿⠿⠿⠿⢃
⠀⠀⠀⠀⠉⠈⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉
⠀⠀⠀⡀⠠⠄⠒⠒⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⠒⠀⠄⢀
⠀⠀⠠⢰⣿⣶⡶⢶⣶⣷⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣶⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⣶⣶⣶⣾⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣶⣶⣿⡆⡀
⠀⠀⠀⢸⣿⣿⠓⠚⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣗⣖⣲⣶⣾⡇⡀
⠀⠀⠀⢸⣿⣻⣿⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠏⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣻⣭⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⢘⣿⡯⡭⢭⣭⣽⡇⠄
⠀⠀⠀⢸⣗⣿⣶⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⣿⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣛⣛⣛⣻⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⡡⠀⠀⠀⠀⠀⠀⠠⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃⠄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠐⢸⣿⣷⡶⣶⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣶⣶⡶⢿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣶⣾⣿⡇
⠀⠀⠀⢸⣿⣭⣷⢣⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⡈⠻⠃⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⢿⣿⣿⡇⠄
⠀⠀⠀⢸⣿⣟⣟⣿⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣼⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡿⢿⡇
⠀⠀⢀⢸⣗⣒⣚⣒⣒⣺⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣿⣧⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⢰⣟⣋⣙⣃⣀⣸⡇⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠌⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠄⢉⠉⠉⠉⠉⠉⠡⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⠀⠉⠈⢀⣀⠠⠀⠀⠀⠀⠀⠀⠠⢀⣐⠈⠉⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠁⠁⠁⣂⡀⠄⠀⠀⠀⠀⠀⠀⠄⣀⡀⠁⠉⠀⣀
⠀⠀⠈⢸⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡿⣿⣿⡇⠁
⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠷⢾⣷⣷⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣹⡿⣿⣿⡇⠁
⠀⠀⠀⢸⣿⣿⣟⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡦⣾⣷⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣟⣿⣻⡇⠁
⠀⠀⠀⢸⣿⣧⣿⣷⣾⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣥⣤⣿⣭⣿⡇⡂
⠀⠀⠀⠀⠩⠉⠍⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⠐⠈⠩⠍⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠉⠍⠍⠁⠀⠀⠀⠀⠀⠀⠀⠂⠩⠭⠉⠉⠉⠉⠐

⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⢸⣏⣉⣩⠩⠃⠀⠀⢸⣿⣿⣿⣿⣆⣠⡀⠀⠀⠀⠀⠀⡈⣄⣀⡄⠀⠀⠀⢠⣶⣿⣾⣶⣆⠀⠀⠀⢰⡭⣌⠱⣘⡆
⠀⠀⠀⠀⠀⠈⢻⡗⠓⣶⠂⠀⠀⠘⠻⠛⠛⠛⠛⠻⠋⠀⠀⠀⠀⠀⠁⠋⠉⠃⠀⠀⠀⠘⡿⣿⣾⣿⠗⠀⠀⠀⠈⢷⣤⡢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠚⠓⠂

⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠐⠑⡀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⣴⣿⣿⣶⣷⡄⠀⠀⠀⡢⠕⢚⡉⠀⢠⠀⠀⠀⠸⡈⠇⡨⠰⣦⡀⠀⠀⠠⢖⣋⣵⣾⡆⠀⠀⠀⠰⣶⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠧⠀⠸⠇⠀⠰⠀⠀⠀⠀⠈⠉⢾⣧⣽⠇⠀⠀⠀⣿⣿⣿⡧⣾⠀⠀⠀⠀⢸⡏⠓⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠐⠶⣶⠞⠀⠀⠀⠀⠀⠀⠀⡀⠚⠢⡀⠀⠀⠀⠀⢀⣀⣛⣢⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢛⣿⣌⣹⣿⡃⠀⠀⠀⠛⠚⣟⠚⠣⠀⠀⠀⠀⠀⣿⢿⢿⣿⢿⡇⠀⠀⠀⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⣼⢋⣠⡝⣇
⠀⠀⠀⠀⠀⠸⠿⠿⠼⠼⠁⠀⠀⠀⠀⠀⠧⠀⠐⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠀⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠙⢤⣁⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⠀⠀⠀⠀⠠⣿⣧⠄⠀⠀⠀⠀⠀⠀⢺⣿⠓⠀⠀⠀⠀⠀⠀⢤⣿⡧⠀⠀⠀⠀⠀⠀⠐⢺⣿⠂
⠀⠀⠀⠀⠀⣿⣿⠉⠀⠀⢹⠀⠀⠀⢰⣶⣦⣴⣶⡆⠀⠀⠀⠀⣶⣶⣭⣶⣶⠀⠀⠀⠀⣶⣶⣤⣶⣶⡆⠀⠀⠀⣠⣶⣬⣵⣶⣄
⠀⠀⠀⠀⠀⣿⣿⣶⣤⣴⣿⠀⠀⠀⠸⠿⣿⣿⣿⠇⠀⠀⠀⠀⠿⣿⣿⣿⡿⠀⠀⠀⠀⠿⢿⣿⣿⠿⠁⠀⠀⠀⠸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣀⠀⠀⠀⢰⠛⠿⠿⠛⠆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⢠⣤⡾⠃
⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⠀⠸⠄⠀⠈⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠠⠠⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⢀⣠⠈⠑⣄⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⢀⣠⠖⠀⣉⣉⠘⢦⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⡀⠀⡠
⠀⠀⠠⠀⠢⡄⠀⠁⠀⠠⡀⠀⠀⠈⠂⣀⠔⠉⠇⠀⣮⣗⣿⠇⠀⢠⠓⡤⠂⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⢠⠊
⠀⠀⠀⠃⠀⠘⡀⢀⠀⠀⠠⠀⠀⠀⠠⠉⠂⠀⠀⠀⠀⠉⠋⠀⣠⠿⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⡼⠀⢀⠈⠀⠀⠈
⠀⠀⠀⠘⡀⠀⠑⡄⠆⠀⣦⠢⠀⠀⠀⡂⠀⠇⠀⠐⣤⠄⠀⡘⠁⠀⠀⢀⠃⠀⠀⠀⠀⠈⢰⡇⡇⢀⠃
⠀⠀⠀⠀⠰⠀⠀⡰⢦⢀⢰⢀⠀⠀⠈⠀⠀⠙⢀⠀⡀⠶⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠗⠛⠚⢶⠀⠀⠂
⠀⠀⠀⠀⠀⢃⠸⡀⠆⠃⠻⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⢀⠉⡳⠊⠀⠈
⠀⠀⠀⠀⠀⠈⡀⠙⢆⠀⠀⠀⠁⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠒⠀⠀⠁⠀⠀⢀⡼⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⢰⠊⠁⢨⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠀⠀⣰⣿⣫⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠠⠐⠼⠇⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⣰⡟⢠⣿⣤⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⢇⣀⡈⣻⣿⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢈⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠄⢀⣀⡀⣀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠾⣷⣶⡔⠲⠶⣶⠂
⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⣀⣠⠾⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢼⣶⣿⣯⠀⠀⠀⢸
⠀⠘⠉⠁⠀⠀⠈⠉⠉⠉⠙⠛⠛⠓⠒⠒⠛⠉⠉⠀⠙⠳⣄⣀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄⠘⣿⣿⠀⠿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠶⠖⠛⠛⠻⣿⡿⠿⣟⣿⣷⣴⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠲⠶⠶⠶⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠖⠛⠒⠦⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⢶⣄⠀⠀⠀⣼⣿⠁⠀⠀⠀⢠⡶⢶⣯⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣴⣿⠟⢀⠀⠀⠀⠀⠻⢿⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣼⣦⣐⣧⡀⣤⠀⣄⣓⣻⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣓⠠⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⣉⠙⠛⠛⠛⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠀⠛⠀⣼⣟⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣦⠄⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣴⠆⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣨⡯⢴⣿⡇⠀⠀⣀⣸⣶⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠉⠀⠀⠙⠻⣷⣿⡿⣿⣷⣿⣿⣷⣄⣀⣀⣀⣀⣠⠴⠛⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠛⠛⣿⡿⠛⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠐⢠⣀⠀⠀⠀⠀⠀⡎⠀⠐⠒⢀⠐⠀⠀⠑⠤⠀⠁⠠⠘⠁⠀⠀⠠⠀⠒⠩⠀⠀⠀⡀
⠠⠀⠸⠍⢴⠀⠀⠀⠀⡇⢀⠤⢀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠠⠄⠒⠁⠀⠀⠀⢤⠀⣀⠀⠀⡀⠠
⠂⠊⣈⠓⠚⠀⠀⠀⠀⡇⠈⠈⠀⠁⠀⠀⠀⠀⠐⠀⠀⡠⣠⠜⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠊⠀⠀⠐⠒⢛⢌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⣀⠀⠀⠀⠀⠫⠋⠛⠛⠛⠛⠛⠛⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⢤⣤⡤
⠀⡤⠀⠀⠁⠀⠁⠀⠀⠈⠳⠤⠤⠤⠴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢻⢛⠛⡟⠀⠀⠀⠀⠀⠀⢦⠠⢠
⡸⠁⠀⠂⠰⢰⠂⠀⠦⢰⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢰⣹⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢉⣈⣀
⠆⠀⢰⠀⡀⣿⠀⢸⠀⡀⠁⠀⠀⠀⠀⠠⡤⠂⠀⠀⠀⢀⣤⠀⠀⠀⠀⠹⠀⠀⠀⠴⠀⠀⠐⠀⠀⠀⠀⠀⠲⠂
⠀⢀⠀⠁⠁⠍⠉⣉⢀⢡⠉⡠⠀⠀⠀⠀⠀⠀⢳⠀⣠⡾⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢀⠄⠒⠂⠀⢄
⠀⠈⠀⠒⠀⢀⣀⢀⠘⡇⢰⠀⠀⠀⠀⡄⠀⢀⣬⣾⠏⠀⠀⠀⠀⠀⠈⣁⠀⢲⠀⠀⠀⠬⠀⠀⣰⠃⠀⠀⠠
⠀⠀⠚⠀⠀⠊⠃⠘⠀⠓⠘⠐⠂⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⡀⠀⠻⠟⠉⠓⢤⡀⠀⠀⠀⢻⠀⠀⠐⠦⠄
⠀⠀⠀⠀⠀⣠⠤⢄⣀⠀⠀⠀⠀⠀⠀⣸⡲⠀⡀⠀⡤⠔⠂⢀⡠⠦⠤⣠⣤⠆⡀⠀⠈⠂⠀⠀⠈⠷⣄⣈⣉⣠⠔
⠤⠀⠤⣤⠚⠁⡠⠀⠈⠃⠀⠀⠀⠠⡄⠈⠁⠀⠀⡸⠁⠀⠀⠀⠙⠒⠐⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠉⠋⠉⠉⢧⠀⣀⣀⢔⠄⠀⢠⣄⡀⠀⠀⠀⠀⣰⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠉⠑⠢⢥⢫⢳⡄⠀⠀⠀⢀⠈⠀⠀⠁⠂⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀
⢀⡰⠤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠄⠀⠀⠉⠉⠀⠀⠀⢴⡏⠀⠤⠄⠀⢨⣦⡀⢢⠀⠀⠀⠀⠐⠒⠰
⡁⠀⠀⠀⣉⠲⢀⠀⠀⠀⠈⠐⠐⠒⠒⠐⠄⠀⠀⠀⠀⠈⠀⠐⢠⠄⡄⣄⠀⠀⠅⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠰
⠀⢙⡖⠉⠀⠀⠂⠀⢂⡀⠐⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠰⠤⠤⠬⠤⠤⠤⠤⠤⠀⠁
⠈⠁⡇⠀⢀⠠⠀⠀⠀⠀⠈⢃⠄⠒⠐⢠⡎⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡄⠀⡀
⠀⠁⠂⠘⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠤⠀⠄⡄⣀⡄⣀⠄⠠⠀⠄⡄⠀⠀⠀⠀⠀⠛⠁⠀⣹⠁⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡑⢋⠑⢫⠂⠰⠂⠀⡁⠀⠀⠀⢀⣤⣶⡤⠴⣧⡀⠀⠀⣤⡀⠀⠀⠈⠑⠢⡄⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⡈⠀⢰⡆⠀⢸⡇⠀⠂⠀⠈⠐⢤⠔⠊⠁⠄⠁

⠀⡄⠂⠢⡀⠀⠠⡀⠀⣡⣿⢸⣇⣀⣉⣫⣥⠤⠞⠀⠀⠀⠀⠘⡆⠙⠀⠀⠀⠀⢸⡄⣿⠿⢿⠴⢶⣿⣿⣿⠀⠀⠀⢠⠖⣲⡐⡎⠼⠳⢏⢼⣄⣣
⠀⢣⠀⠀⢿⣿⢧⣿⠀⡄⠀⠉⠹⢶⠀⣴⣳⣷⣤⣦⣤⣤⣄⠀⢷⠸⠓⠀⠀⢀⣈⡇⢻⣄⣸⣤⣼⣿⣿⡿⡤⠂⠱⢿⣿⣾⣠⣧⠤⠶⢾⣟⣀⣋⡙
⣁⣼⡦⠒⠺⣿⡀⢚⣄⠈⢸⠀⠟⣶⡩⢶⡹⢋⣽⣿⣿⣿⣿⣷⣜⠲⠖⠚⠋⠉⠁⢀⣀⣉⠉⠉⠉⣿⠇⠀⠈⠁⢲⢤⡷⣾⣿⠃⠀⠀⠀⢿⡋⠀⢈
⣿⡍⣷⠀⠈⢿⣬⡍⠈⠀⡆⠰⠷⣾⡿⢿⣼⣷⣬⣻⣿⣿⣿⡿⠿⣞⢓⡄⠀⠀⢰⣿⠛⢿⣧⠀⠈⠛⠀⠀⠀⢠⠁⢀⡜⠉⠀⣹⣿⣯⡉⣉⣿⣷⣾
⡟⠉⢩⠋⠒⢼⡟⠀⡀⡰⠀⠘⠦⢤⣈⣡⣝⡓⠯⢵⣟⣉⣛⣚⣁⠩⢷⠃⠀⣀⣸⠿⠦⠾⠛⠒⠒⠉⡆⠀⠀⠢⣸⠁⠑⣄⣰⢿⣄⡀⡽⠾⡿⣿⣿
⣷⠔⢁⠌⢀⡾⢁⠎⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠖⠶⠯⠽⠶⠒⠁⠀⣿⣧⠀⠀⠀⢀⡤⠤⠖⣤⣥⠀⠀⠀⠀⠀⠀⡠⢻⠞⣿⣿⡇⠀⠁⢸⠉
⠹⣦⣀⣠⠟⢡⠎⠀⡠⢰⣊⢍⣹⣿⣶⣿⡞⡻⣷⣆⣵⣤⣤⣤⣤⠀⣀⠀⢸⣻⡄⠀⢰⣧⣿⣿⣿⣿⣷⣷⡄⠀⠀⠀⠐⠧⠴⣏⣻⡭⢷⠀⡠⠛⡄
⠒⠙⠫⠥⠄⣁⣀⡡⠃⡌⢳⠜⣿⣿⣿⣿⣶⣧⣦⣿⣿⣿⣿⣿⣿⣴⣆⡇⢸⢧⢇⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣷⣄⠀⠀⠄⠀⠉⠀⢀⡈⠉⠀⠀⠈
⠀⣴⠐⠒⣐⡀⠀⠀⡄⡷⣾⡷⣿⣿⣿⣿⣿⡞⠉⠉⡹⣿⣿⣿⣏⡸⡋⡇⢠⢸⢸⢸⣿⣭⣿⣯⣿⣿⣿⣿⣿⣿⠃⠀⠀⢠⠈⠑⠀⠀⠀⢀⣁⠐⡄
⠒⠋⠉⠁⠀⡇⠀⠀⡇⡅⢨⣤⣿⡟⠛⡿⢿⡜⠴⠋⠙⠛⠻⣿⣏⡲⠋⡇⠀⢈⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠆⣈⠀⠀⠈⠀⠈⢉⠔⠠⠁
⠀⣀⡀⠀⠀⢸⠀⠀⡇⠈⣑⣚⠛⠛⠛⠷⠾⠷⠶⠶⢶⣶⣮⣿⡏⢁⡸⠁⠀⠯⢹⣇⣻⣿⣿⣿⣿⣾⣿⣿⡿⠿⠋⠀⠀⠄⠓⢈⢁⡀⠈⠠⡀⠈
⠀⠴⠤⠒⠒⠀⠇⠀⠇⠀⠟⠙⠓⠚⠫⠭⠒⠒⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⡠⠄⠀⡤⣄⠀⣀⠀⠀⡠⢄⠀⡀⠀⢀⠀⠀⢀⡠⡀⢀⣀⠀⢀⡀⠀⣠⢄⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⡗⠂⠀⣄⡼⠀⡧⠄⠀⠳⢤⠀⡗⠒⢻⠀⠀⠘⠦⡀⠀⡇⠀⣜⣡⠀⣇⣠⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⣁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠑⠀⠓⠒⠀⠑⠚⠀⠃⠀⠘⠀⠀⠘⠒⠃⠀⠀⠘⠁⠈⠃⠁⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⢀⡀⡀⢀⡀⠀⡀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀
⡏⣿⣿⣛⣿⡦⡟⢂⠈⠢⠴⠉⠙⠛⠁⢤⡀⠀⠀⢀⣤⢤⣶⣶⠒⠛⣆⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣧⡀
⢽⡻⣟⣣⣼⡞⡳⢰⠀⠀⠀⠄⠀⠀⠀⢾⠓⠀⡄⢻⠀⠀⠉⠁⠀⠀⢹⡀⠇⠠⠄⠠⣴⠒⠀⠀⠀⠀⠤⠤⠅⠂⣀⢠⣤⡤⡀⢸⣿⣷⣿⣿⢿⡄⠄
⡂⠈⣣⣔⡪⠴⢷⡏⠀⣀⣠⠤⢄⠀⠀⠸⣦⡀⢁⠸⡇⠀⠀⠀⠀⠀⠘⡇⠉⠁⢰⣖⣋⣛⠙⠛⠲⠖⠒⠲⠶⠦⠎⢸⣼⣃⠇⢺⣿⣿⣿⣿⣆⠹⣆
⠛⠉⠁⢀⡠⠤⠊⡴⣞⠉⠀⠀⠈⣧⠀⠀⢿⡇⠘⠀⣧⠀⠀⠀⠀⢀⡀⢿⠀⠀⣼⡘⠉⠙⠓⠚⠻⠶⢖⣦⣤⣀⣀⣀⠀⠀⠀⠈⠛⠛⢻⣷⣿⡄⠙
⠉⠒⠈⠁⠀⠀⢀⡘⣎⠀⠀⢀⣀⣼⡀⠀⠸⣿⠀⠀⢸⡄⠲⠀⠀⠘⠁⢸⡄⠀⣧⣰⣲⡦⡤⣽⢯⣛⡃⠾⠶⢬⣶⡍⠙⠚⠛⢷⢶⠆⠀⣩⡏⣿⣀
⠀⢸⣇⠀⠀⠀⣺⠇⠙⢶⣾⡟⠛⡉⢹⡄⠀⢿⡇⠀⠈⡇⢐⣆⠀⠀⠀⠀⣇⢸⡿⢷⣾⢽⣯⣯⣿⣟⢛⣭⡽⣻⣞⠃⠤⠤⠤⣼⣿⠂⢀⣉⠉⡉⣁
⠀⣠⠿⣶⣶⡾⠟⠷⣄⠸⣿⠷⡔⢻⠀⢹⡄⠘⣧⡀⠀⢻⣄⣤⣤⠴⠶⠖⠃⣸⣸⠗⠓⠛⢺⣵⣾⡼⠋⢁⢶⡿⢥⣽⣿⠖⠦⣾⡿⡏⢸⣷⣞⠳⢋
⠘⠁⠀⠀⠀⠀⠀⠀⢠⡄⠻⣄⣵⠶⠛⠋⡀⠈⠻⠇⢀⣀⣤⣄⣤⡀⠀⠀⠀⣿⡃⠀⢠⠎⢡⣾⣉⣁⣨⣡⣮⣷⣖⢻⣿⠿⡾⣻⡆⢣⠈⣼⣓⢏⢶
⠰⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠢⣄⠀⢐⠁⠤⣤⢞⣿⣴⣿⣿⣿⣦⣹⡦⡀⣸⡍⣛⣶⣼⣠⣾⠷⣞⣛⣿⡟⢿⣿⣿⢿⣿⣳⣿⡿⠀⢸⠙⢿⣟⣱⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⣽⠃⠀⠀⡀⠘⣼⣸⣻⣾⣿⢷⣽⡿⠦⡇⠀⠈⠉⠈⠛⠛⠛⠻⠾⢶⣦⡴⢾⣿⣟⣿⣿⣷⣿⡇⠀⢸⣧⣦⣷⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⢀⡶⢰⡆⢲⡄⠀⢀⣰⣁⣑⣄⠀⠀⠐⡦⣤⣤⣤⠀⠀⢠⣮⣤⡷⠀⠀⢀⣚⣊⣑⣓⡂⠀⠀⣤⠄⢠⡆
⠀⠀⣟⡟⢻⣛⡇⠀⠘⣿⣿⣽⣾⠁⠀⠀⠘⡟⢛⠛⠀⠀⢸⣿⣿⣿⠀⠀⠀⣾⣾⣷⣿⡇⠀⠀⠁⠠⡄
⠀⠀⠉⠁⠈⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⢀⡄⠀⠀⠀⢠⢠⣄⡄⣤⡄⠀⠀⠾⠥⣤⣤⠄⠀⠀⠀⠴⡗⣄⠀⠰⠖⢀⣄⠒⠆⠀⠀⠀⣠⡄
⠀⠀⣸⣹⣿⣿⡀⠀⢸⢸⣿⡇⣿⡇⠀⣾⣿⣿⣿⡆⠀⠀⣿⣿⣿⡿⣇⠀⠰⠆⠈⠊⣄⡀⠀⢠⣾⣷⣿⣧⡀
⠀⠘⠝⠿⠛⠟⠇⠀⠘⠈⠀⠁⠉⠃⠀⠛⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠘⠇⠛⠁⠀⠈⠉⠉⠋⠙⠁
⠀⢀⣤⠤⠤⠤⡄⠀⠀⣠⠀⢀⢀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⣤⢤⣤⠀⠀⠀⠀⠀⡀
⠀⢸⠀⠻⠗⠀⡇⠀⠸⣯⠁⠚⡞⠀⠀⠀⢧⢀⣽⠀⠀⠀⢸⠢⣤⢼⠀⠀⢰⡿⣿⢿⢿⡇⠀⠀⢀⡄⢀⡀
⠀⠈⠉⣹⣿⡉⠁⠀⠀⠙⢷⣶⡄⠀⠀⢰⣶⣝⣴⣦⠀⠀⢸⣦⣠⣼⠀⠀⠈⠻⠟⠛⠛⠁⠀⠀⠠⡜⠭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⢚⡟⢢⠀⠀⢰⣾⣶⣾⣷⡄⠀⠀⢀⣷⣆⠀⠀⠀⣀⢠⣾⣁⡀⠀⠀⣶⣶⣶⢰⡆⠀⠀⡟⢻⣿⣿
⠀⠀⣷⣀⣃⢴⡇⠀⠈⢿⣿⣿⠟⠀⠀⠀⣾⠿⣿⠄⠀⠀⣿⢸⣿⣿⡇⠀⠀⠻⢿⡿⠘⠃⠀⠀⣯⣭⣭⣿
⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⠤⣤⠤⢀⠀⠀⡤⢠⠄⣶⡆⠀⠀⣤⣤⠆⠤⡀⠀⠀⣤⣄⣀
⠀⢸⣯⣻⣿⣆⣄⠀⠀⠀⢿⣿⡇⠀⠀⠏⠉⠛⠁⢹⠀⠀⠃⢘⠀⣘⡀⠀⠀⣿⣟⠂⢁⠃⠀⠀⡌⣄⡄⡄
⠀⠈⠛⠉⠉⠑⠉⠀⠀⠀⠿⠏⠁⠀⠀⠣⠁⠩⠩⠽⠀⠀⠱⠋⠀⠉⠁⠀⠀⠈⠻⠄⠃⠀⠀⠀⠂⠐⠒
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠊⣁⡀⡙⡄⠀⢠⢋⣀⢈⠑⠀⠀⢀⣰⢿⣻⣷⠀⠀⢀⣴⡿⣹⡦⠀⢀⣠⣈⣁⣂⡀⠀⢀⣔⣈⣁⣀
⠀⠀⠦⣀⣀⡠⠃⠀⠘⢄⣀⣀⠴⠀⠀⣾⣯⣾⠟⠁⠀⠀⣿⣯⡾⠋⠀⠀⢸⣿⣭⣭⣽⡇⠀⢸⣿⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠇⠀⠀⣀⣤⣤⣤⣀⣀⣀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⡻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⢻⣿⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⡿⠁⡤⠀⠀⡍⢻⣿⠀⠀⠀⠀⣿⣧⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡟⠠⢀⣇⠀⠀⠉⠀⠸⡆⠹⣷⠲⠶⠴⠿⣧⣤⣀⡀⠈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⠁⠀⢘⣧⠀⡇⠂⠀⢠⣼⠀⢻⠀⠀⠀⠀⠀⠈⠉⠙⠓⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠁⣄⣿⡄⠹⠀⣀⠾⣾⠁⢸⠇⠀⠀⠀⠅⠀⠀⠄⠀⠀⠀⠀⠉⠛⠻⠶⣤⣄⡔⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠠⠀⠐⠾⠳⣦⡀⠀⠀⠇⠀⠸⠧⠀⠁⠂⠀⢡⣤⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠐⢂⡈⠅⠀⢹⡷⠀⠈⠳⠄⠀⠀⠀⠀⠀⠸⢿⣿⠟⠀⠀⠀⠀⠀⡀⠀⢤⣤⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣿⠈⠍⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠗⡆⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣹⣿⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠈⠀⠀⠀⠠⠀⠀⠰⠂⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⣀⣤⡶⢾⣿⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠁⠀⠀⠐⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⢀⣠⣦⡶⠛⠋⢠⠀⢸⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡀⠀⢐⠀⠄⠀⠁⢀⠀⠀⡀⣀⣴⡴⠶⢲⣿⣿⣿⣿⡇⠠⠀⠺⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣣⠀⠡⠂⠀⠀⣰⣷⣾⣿⣏⠁⠀⠀⡀⢸⣿⣿⣿⣿⣇⠆⠀⠀⠎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣲⠠⠀⠘⢿⣿⣿⠿⢿⣆⠀⢠⠁⠀⢻⡉⠙⠋⠈⠙⠲⢤⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣿⣛⣹⠂⠀⠀⠀⢉⣛⠳⠶⣬⠷⣌⠁⠄⠀⠙⠦⠬⠤⢬⣁⣀⣬⣽⡷⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠰⠯⣍⡤⠚⠷⠀⡟⢦⣨⢗⣶⣦⣷⣬⣷⠀⣤⠀⣤⡓⠲⡞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣠⣤⣤⣴⠶⠾⣾⠟⠛⠛⠛⠋⠉⠉⢹⢡⡼⢻⡇⣸⠟⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠠⠠⠆⠀⠀⠈⠀⢠⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠂⠀⠀⢀⠀⢀⡀
⢀⣶⠀⠀⠀⠸⠂⠂⠀⣄⠀⠀⢸⡇⠀⠀⢀⣶⣩⢣⠀⠀⠀⠐⠂⠀⠀⠀⠁⡂⠀⠫⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⣿⡟⠀⠀⠀⠀⠀⠀⠊⠷⠀⠀⠘⠁⠀⢀⠀⠅⢣⠈⠂⠀⠀⣠⣾⠀⢀⠀⠐
⢯⠁⠀⠀⠀⠀⠀⠂⠀⢀⣇⠀⣀⠀⠀⠈⠀⠀⠘⠄⠀⠀⢠⣾⠟⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣀⣤⠺⠗⠁⠀⣶⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿⠿⠁⠀⠉⠀⠠⠊⠑⠀⢀⠀⠀⠠⠀⠈
⣣⠀⡀⠀⠀⠀⠀⠉⠻⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠓⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⡎⡀⢹⣦⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠠⡢⢂⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⣐⣃⡔⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠐⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠄⠀⠘
⠸⠾⡏⠢⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠋⠀⠄⠀⠀⠀⠉
⠀⠋⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠀⠀⠀⠐⠀⠀⠀⠘⢀⠀⠈⠀⠒⠀⠀⠤⠀⠀⠀⠈
⠀⠄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⠀⠀⠀⠀⠀⠘⠂⠀⠆⠀⢣⡀⠀⠉⠓⢀⠰⡄⠀⠀⠀⠀⠀⡀⣶⡦⢄⣀⠐
⢃⢠⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡇⠀⢀⢤⠀⠀⠀⠀⠀⠳⠀⠀⡟⠳⢄⡀⠀⢰⠀⠊⠁⠀⠁⠀⠀⡍⠁⠀⠀⠵
⠈⠀⠀⢠⡄⠀⢀⣀⡤⠄⠀⠀⠀⠄⠀⠁⠀⢈⠤⣷⡍⠼⠀⠀⠀⠑⢤⠁⠁⠀⠑⢤⡏⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⡀
⠀⠀⠈⢺⠇⠤⠤⣄⠀⠀⠠⣤⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡀⠀⠀⠈⡇⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠶⠀⢄⠀⠀⠀⠈⢒⠀⡔⠋
⠀⡞⠀⠀⠙⠲⢤⣀⣇⣀⣀⠀⠀⠀⠀⢦⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠀⠀⠢⡀⣾⢴⠆
⠰⠃⠀⠀⠀⠀⠀⢸⡁⠀⠈⠣⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣿⠊
⠀⠀⠀⠠⣂⣀⠀⠈⠓⠦⢤⣧⣄⡀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢃
⢰⠏⠉⠃⠿⣿⠀⠀⠀⠀⣟⠀⠈⣷⠀⠀⠀⠘⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣷
⠀⠀⠀⠀⠀⡆⠁⠀⠀⠀⠈⠒⠤⣿⣀⠀⠀⠀⠂⠀⠀⠀⠒⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛
⠚⠐⠀⡐⠔⠁⠀⠀⢱⠠⠀⠀⠀⠻⣌⡇⠀⠀⠐⠶⠿⣶⣶⣄⣀⡀⠀⠀⠉⠁⠁⠁⠐⠲⠠⠄⠀⠐⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡅⠀⠀⠀⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣄⣨⣀⣀⣀⣈⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠋⠁⠀⠀⠀⠀⠀⠙⠻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡆⠀⢰⣿⣆⠀⠀⠀⠀⠀⢸⡇⣡⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠋⠀⠀⠀⠛⠿⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⡄⢀⠀⠀⠀⠀⠠⠊⡤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⢀⣠⣿⣿⣷⣄⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠙⠛⠛⠻⢿⣿⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀⠀⠘⠛⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⠀⠈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⢀⣼⣿⠟⠋⠙⢮⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠏⢉⠛⠻⡅⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠿⠷⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡤⣤⣤⣀⣀⣀⠀⣀⣤⣶⣾⣵⣰⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⢾⣯⡻⣿⢿⣤⡶⠈⠻⣿⣭⣙⠻⠿⣿⣿⣿⣿⣇⠈⠀⠁⠐⢢⠙⠒⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣷⣍⠓⠄⠀⣠⡄⣼⣹⣿⣿⣶⣤⣙⠻⣿⣧⠀⠞⠓⠀⠘⠀⠒⢢⠳⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠀⠀⠀⠉⢜⣵⡿⡿⠿⢿⣿⣿⣿⣬⣛⢶⣄⣰⠋⠙⠢⢤⠤⠖⠛⠳⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣮⡿⣿⣷⣦⠈⢲⠰⢀⠍⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⣶⣦⣭⣁⡀⠀⠀⠉⠛⢿⣿⣷⣞⣿⣿⠀⣦⣤⣀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠙⠻⣿⣿⣿⡆⠲⡄⢹⡇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠀⠀⠀⠻⠿⠟⠛⠀⠘⢿⣿⣿⣿⣿⡢⠘⢿⣿⣷⠀⣁⣬⣔⣊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⣴⠚⠲⣏⠙⡆⠀⠀⡇⠀⣈⠃⡀⠘⠊⢳⡀⠘⣿⡟⢈⡡⠴⠒⠆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⣝⠢⣄⢈⣧⡘⢄⠐⠒⠊⣉⢹⠔⣉⡀⢈⡇⣼⡿⠡⠋⠠⣒⣚⡛⠋
⠀⠀⠀⠀⠀⠀⢀⠤⡘⢁⢤⡤⣙⢮⡣⠀⠉⠲⣅⣒⣠⠞⢀⣈⣁⣨⣼⣿⠟⠁⣀⠴⠋⣉⢄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠥⣙⡲⠬⠁⠂⣀⣴⣾⣿⣿⣿⣿⣿⡁⠀⠤⠤⠤⣄⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡴⠛⠉⠹⢻⣿⣿⣿⣿⡇⠀⠁⢀⡠⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠄⠀⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⢀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⡃⠀⢠⠊⣴⠞⣉⠙⣾⣉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢁⡠⠐⠊⢹⣾⣷⣆⠈⠃⢸⠛⠒⠦⠄⠂⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠒⢾⡓⠂⠀⠀⠀⠛⠽⠿⠿⠔⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠴⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠋⠙⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠩⠖⠲⠤⠤⠤⠰⠚⠛⠉
⠀⠀⠀⣀⠀⣀⣀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⡝
⢀⠔⠉⡄⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣾⠀⠀⠀⣄⣄⠔⢀
⠘⢤⣀⠀⠀⠉⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠹⠤⠀⠀⠘⡄⠉⠈⠍⠐
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠐⠂⠈⡁⡰⠷⠄⠀⠙⠉⠀⡀⡃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠒⠀⠈⠉⠀⠀⠀⠀⠈⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⡅⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⡄⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⡀⠌⡃⠀⠰⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⢱⠈⣇⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⡀⠠⢀⠸⠀⠄⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠂⠈⠡⡼⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⣿⣿⣇⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠠⠈⡁⠘⢄⣤⢼⠇⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠛⠀⠡⠈⢀⠀⠀⠈⢙⣿⣿⣿⣿⣸⠞⠏⡉⠟⠱⣻⢿⣿⣷⠀⠀⠀⠀⠀⠠⣇⡘⠣⣖⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⢀⠀⠀⠀⠑⠀⠴⢿⣾⣿⣿⣷⣿⣇⢠⣵⣤⠀⢩⢽⢿⣿⡇⠀⠀⢀⣠⡦⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠔⠠⢌⣹⣿⣿⣿⣿⢟⠉⢤⣽⣿⣾⣶⣮⣿⣻⡇⠀⠀⣌⡩⣼⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⡟⣄⠀⢀⢨⣻⣿⣿⣿⣿⣿⠃⠠⠀⣸⣷⣾⣻⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄⠀⠴⣿⣿⣿⣿⣿⣿⣇⢠⣠⣴⡾⣿⣿⣿⣿⣟⣁⠄
⠀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣸⣿⣟⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⢻⢿⡿⢉⠁
⠀⠀⠀⠀⠈⠄⠈⠡⡄⢱⡀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣉⢩⣭⣽⣶⠾⠿⠿⣿⣿⣿⣿⡿⣿⣿⡿⠛⠿⢏⠺⡤⠀⠚⠧
⠀⠀⠀⠀⢀⠐⠂⠄⠐⠀⣧⣃⢸⠀⡀⠀⠀⡄⠀⢠⠀⠀⠀⢁⠈⠀⠁⠀⠀⠀⠀⠀⠀⢸⠿⢿⣿⣯⣥⡶⣄⠀⠀⠀⠻⠷⠀⠎⠉
⠀⠀⠀⠀⢸⠥⠆⠴⡁⠱⣾⣿⣈⣔⠀⠀⠀⢤⠰⡀⠀⡄⠀⠀⠀⠄⢀⠀⠀⠀⠀⠂⠳⣦⣄⣰⡄⠀⡀⠈⠮⣷⣦⠘
⠀⠀⠀⠀⠃⡀⣠⣀⣌⢻⣌⣿⣿⣿⣇⣠⡊⢣⣆⣿⣶⡃⡆⢀⢣⠀⡞⠀⠀⠀⠀⠀⠈⠀⠈⠉⠙⠦⢀⣠⡾⠻⠿⠟⠁⠐⠠⣄
⠀⠀⠀⠀⠈⠀⠄⠀⠲⣵⣿⣿⢿⣿⣿⣷⠖⠀⣿⣾⣿⡖⡼⡇⠾⣢⡵⠖⣒⠒⠒⠒⠖⠒⠒⠒⠚⠚⠉⠉⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢄⢯⣿⣿⣿⣿⣿⠿⣖⣷⣿⣿⡿⣥⣴⣿⣿⡟⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠥⠚⠿⠶⠟⠛⠛⠉⠁⠈

⣿⣿⠿⠛⠛⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣏⡟⠉⢙
⠛⠛⠂⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠚⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠷⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⢶⣤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠄⢌⡅⠘⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⡀⠀⠀⠀⠀⠐⢀⣀⡁⠀⠛⠛⠀⠀⠀⠘
⠀⠀⠀⠀⠄⠀⠀⠀⠈⠉⠁⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠁⠀⠀⠀⠀⠀⠈
⠀⠰⠀⠐⢀
⠀⢠⡀⣠⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠘⠇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⠂⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡄⣂⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⣠⡥⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠀⠈⠀⠁⠀⠀⠄⠀⠁⠠⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⠀⠁⢀⠀⠀⣐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢘⠆
⠀⠔⠉⠀⠈⠑⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢈⢟⠄⠀⠀⠀⠀⠀⠴⠤⠀⠀⠄⠀⠀⠐⠓⠄⣠⠄⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠈⠁⠘⠁
⠀⠑⠄⠀⠀⠔⠀⠀⠈⠉⠁⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣄⠀⣦⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠐⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠂⠀⠀⠐⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⣀⣀⡀⠀⠀⠈⠢⠄⠀⣠⠒⠁⢀⠜⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠜⠀⠈⠉⠉⢤⠀⠀⠀⠉⡳⠲⡀⠘⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⢂⣭⠢⠁⠀⠈⠑⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⡐⢆⠀⠀⠸⣘⣷⣿⠀⠀⠀⠀⠐⣠⠃⠀⠤⢄⠀⠀⠣⢀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠐⠓⠂⠈⠀⠀⠀⠀⠉⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀⠦⡀⠀⠀⠀⠀⠀⡠⠺⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠁⠀⠐⢤⣦⣤⣔⣈⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠤⣀⣸⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢘⢪⡏⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⠀⠀⠀⠀⠐⠈⠂⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠉⠴⠤⠉⠈⠒⠠⣀⠠⠔⠋⣡⡀⡐⢤⠄⠁⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠓⠀⠀⠀⠐⡦⢆⠀⠀⠀⠀⠀⠀⠀⡦⣇
⠚⡰⠒⠀⠘⠈⠱⠀⢀⠆⠀⠂⢀⡀⠀⠙⢻⣄⣠⠖⠢⡈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣠⠐⠁⠀⠀⠀⠀⢨⡅⠈⡆⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠓⢁⠄⣠⠔⠒⠒⠚⠓⠰⠄⠀⠘⣀⣄⡊⠀⠘⠆⣀⢠⣆⠐⠀⠀⣀⡄⠀⠊⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⠅⠀⠰⠂⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⣶⠭⠭⠅⠀⠀⠐⣔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢖⡒⠒⠀⢀⠠⠚⠉⠐⢤⡦⢤⠤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⠀⠀⠀⠀⠀⠂⢀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢳⡸⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⢸⠀⠀⢀⣠⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡀⣼⠳⠉⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠀⠈⠁⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⡀⠀⠀⠀⣰⡀⠀⢨⠆⢳⡗⡏⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠷⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠧⠤⢤⡌⠁⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣄⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠁⡀⠄⡊⠁⠀⠀⠀⢀⡠⠤⠊⠉⠒⠳⠆⠐⠉⠀⠘⠊⠒⠛⣦⡴⠤⣀⠀⣄⡀⠀⣀⠀⠀⢀⠀⠦⣾⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢁⢮⠔⠋⢲⣀⠥⠒⠊⠁⠀⠀⠀⢀⠶⢂⡠⠔⢑⡠⠀⣀⣠⣊⡡⠤⠤⠤⠭⠥⠭⠭⣭⣍⠒⠿⠼⠖⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡨⠖⠩⣉⢔⠭⠛⠉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣱⣣⠔⠊⠉

⠀⢀⡀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠠⠠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠈⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⡀⠁⠀⢀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠆⠀⠢⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⠤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄
⠀⣀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⡀⠐⢀⡀⠉⠈⠀⠀⠠⠀⠐⢂⡀
⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠔⠠⠀⠀⠀⡆⠂⠒⠀⠒
⠰⠀⠀⠀⠀⠱⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⠠⠀⠃⠀⠀⠁⢤⠀⠀⣄⠀⠀⠀⠮⠔⠀⠀⢹⠨⠅⠏⠭⢨⠁⠀⢁⠀⠀⠉⠀⠀⠁⠀⠀⠇⠀⠰⠀⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠂
⣀⠀⠀⠠⠄⢀⡀⠀⠀⠀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⡀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⢀⠀⠀⠀⠀⡀
⠁⠘⠁⠐⠀⠀⠀⠀⠀⣀⡀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⢡⡀⠀⠀⠀⠀⠀⠀⢄⠐⠄⠀⠅⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠒⢾⠀⠁⢸⠁⠀⠀⠀⠀⠀⠀⠊⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⡒
⠀⠀⠀⠂⠀⠀⠀⠀⠈⠏⠉⠉⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⣀⡀⠀⠀⢀⢀⣤⠀⡀⠀⠀⠠⣄⠀⠀⠀⢀
⡆⠀⠐⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠠⠀⠀⠀⠋⠉⠉⠉⠉⠉⠃⠀⠐⠁⠉⠐⠉⡈⠀⠀⠀⠀⢄⠀⠠⠂
⠓⠀⠚⠀⠀⢀⡀⠀⠀⠀⠈⡶⠒⢐⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠖⠶⠀⠀⢀⠈⠁⢙⠀⠃⠃⠀⠀⠀⠀⢰
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠁⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠰⠆⠀⠘⠀⠀⠈⠉⠭⠉⠍⠈⠀⠀⠀⠑⠛⠾⠛⠂⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠅⠀⠀⠀⠀⠈⠀⠀⢡⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⢁⡀⠀⠀⢀⡀⠀⠠⠠⠤
⠀⠲⠐⠀⠀⠄⠀⠀⠀⠀⠀⢸⡤⢤⡀⠀⠀⠐⠐⠂⠒⠐⠀⠀⠀⠀⠠⠤⠀⠸⡂⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢠⠗⠰⠋⠀⠀⠀⠀⡇
⠀⠀⠤⠀⠀⠀⠀⠀⢘⠐⣶⠈⡃⠀⠁⠀⠀⠀⠠⠓⠁⠠⠀⠀⠀⠄⠀⠀⠀⠄⠰⠀⠀⠢⠄⠠⠰⠤⠅⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠠
⣀⣀⣀⣀⡀⠀⠀⠀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡿⠀⠀⠀⠷⢠⠄⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⢲⢲⠀⠀⠀⠀⠈⠉⡄⢀⠀⠀⠀⠀⠀⠀⠨⠄⡀
⠀⡆⡆⣰⠸⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡁⠁⠀⠀⠀⠀⠀⠅⠀⠀⠀⠠⠀⠀⢰⠀⡄⠀⡇⠀⠀⠀⠀⠀⠢⡀⠀⠂
⢄⠃⠃⠘⠈⠈⠀⠀⢀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡂⠰⠆⠐⠀⠀⠀⠀⠘⠀⠃⡀⠃⠐⠀⠀⠈⠀⠀⠈⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠻⣯⢰⣿⡎⡇⢸⠇⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠿⡞⠸⠃⠇⠾⢠⡏⣰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⣴⣾⢏⢄⡞⡁⣠⢡⡞⢠⡌⠐⠟⢀⣮⠴⠂
⠀⠀⠀⠀⠀⣤⣾⣋⣣⡏⣼⠁⡇⢸⢀⡿⢡⡷⢀⠛⠁⢀⣴⡷
⠀⠀⠀⠠⠖⣒⡲⠧⣸⢿⣿⢠⢿⢸⡿⢡⡿⢁⡾⢁⡀⠟⢉⣤⡾
⠀⠀⣴⡆⢚⡻⡽⡰⡏⣏⣯⠾⣼⢸⢃⡟⣰⢟⣴⠟⣠⡄⠛⣁⣴⡿⡄
⠀⣼⣿⣧⣤⠽⠟⠓⠁⠉⠀⠀⠋⡜⣸⠐⣡⠟⢡⡾⢋⣥⡦⠙⣁⣪⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣱⢣⡾⢃⡴⣋⣴⢟⡯⣾⣷⡹⣯⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⢒⣡⣴⣿⠟⣛⣭⡵⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣱⡯⠞⢉⡴⣫⡾⣫⢏⡟⣽⢙⣷⢲⣶⡆⡖⢰⣶⣰⠞⣠⣾⠟⠉⣠⣴⣿⠿⣫⣴⠟⢋⣥⣴⡶⠿⠷⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣋⣤⡶⢋⡼⢋⡼⠃⣾⢡⡏⢸⡇⢸⣿⣼⠁⣿⣿⠋⣰⣿⠋⢠⣾⣿⠟⣡⡾⢋⣤⡾⠟⢉⣤⣶⣶⡿⠷⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠟⢠⡟⣱⡿⡡⢸⡇⣼⠁⣿⠁⣾⣿⡇⢸⣿⡇⣰⠟⠁⣰⣿⠟⣡⠞⢁⣴⠟⠉⣠⣾⠿⢛⣩⣴⣶⣾⡆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣴⠏⣼⡟⣰⢃⣾⢀⡿⢰⣿⠀⣿⡿⠀⣾⣿⡿⠃⠀⣼⡿⠁⡴⠋⣴⡟⠁⣠⡾⢋⣡⣾⡿⠟⣛⣭⣵⠆⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣼⠟⣼⠃⣼⡟⣼⣧⢸⣿⡄⣿⡇⠀⢿⣿⡇⠀⣸⣿⠁⠈⢀⡾⠋⣠⠞⢁⣴⠟⠋⠁⣚⣉⣩⣥⣶⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣡⡾⢃⣾⠟⣰⠃⢿⡄⣿⣇⠸⣿⠀⠘⣿⡇⠀⣿⡏⠀⢠⡟⠁⠀⢁⡴⠋⣠⠶⠟⢛⣛⣉⣭⠭⢿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡛⣠⡾⢃⡾⣡⣦⠘⣷⠘⣿⠀⢿⡇⠀⠸⣧⠀⣿⠁⠀⡞⠀⠀⡰⢋⡀⠚⠁⠴⠚⣉⣭⣴⠶⢾⡿⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⣴⢋⡴⢟⡉⢧⢹⡄⢿⠀⢸⡇⠀⢀⡏⠀⠟⠀⠈⠀⠀⠀⣠⢾⣛⣦⣄⠀⢈⣀⣤⡶⠾⢻⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣾⣛⣛⠂⣆⠀⠈⠀⠘⠃⠀⠘⢀⣀⠠⠄⠒⠀⠀⠀⠘⣉⣩⣥⡭⣹⠋⣥⠤⠶⣛⣹⣧⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⢻⣿⣓⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠤⠤⡇⠀⠀⠘⢫⡭⠷⢛⣣⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⡽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣚⣛⡇⠀⠀⠀⠀⠙⣟⡽⠿⡇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⢧⡗⠀⠀⠀⠀⠀⠘⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣭⣛⠟⠁⠀⠀⠀⠀⠀⠀⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠹⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⢠⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⠀⢸⣽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⣷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠟⢁⣼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡛⣽⣻⣉⣝⣛⣛⣋⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⢠⡁⠔⠁⡔⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⢻⠳⠾⠔⠅⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡌⠙⡣⠖⠌⠀⢑⣳
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠰⣧⢀⡤⠂⢠⠄
⠀⠀⠀⠀⠀⢀⠈⢦⣠⢀⠴⢛⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠑⠀⢠⣴⡀⣀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⡄⢈⢳⠆⣐⣥⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠦⣀⣬⣩⡉⣠⢔⣨⠅
⠀⠀⠀⠀⠀⠀⠫⣀⡉⢅⡤⡰⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣀⣀⡈⣭⡌⢈⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠑⣭⡙⣆⢀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢈⢣⣠⡇⠈⡀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠘⢣⣷⠀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⠀⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢩⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠁⣼⠄⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢉⠋⠀⠗⠉⠀⠀⠀⠀⠀⠀⡀⠀⢈⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⠋⠉⠘⠀⠀⠀⠀⠀⣀⠀⠑⢄⠉⡺⠠⢈⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠚⢠⡾⠀⠈⡠⠀⠀⠀⠀⠀⡄⠈⢊⠀⣺⠗⠆⡀
⠀⠀⠀⠀⠀⠀⠁⢤⡀⢰⠆⠠⠮⡄⠀⠀⠀⠀⠄⠈⠀⢠⠝⡀⠀⣊
⠀⠀⠀⠀⠀⠀⢈⡀⠀⣸⠂⠙⠛⠠⠂⠀⠀⢀⠁⠢⢴⡇⠤⠤⢂⠁
⠀⠀⠀⠀⠀⠄⡀⠁⠌⠸⡖⠊⠤⡱⠀⠀⠀⢬⢢⡀⣾⠆⠥⠴⢠
⠀⠀⠀⠀⠀⠀⠈⠐⠨⠘⠿⠆⠀⠀⠀⠈⠣⠀⠪⢎⡇⣀⡤⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⢰⣯⡁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀⣀⣀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠜⢿⠣⡀⠲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠄⡈⠄⠀⡄⠀⠉⠸⣟⣥⠕⠋⠀⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠌⠀⠄⠘⢎⢣⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣾⣶⣴⣼⡇⠀⡘⢀⠀⠉⠐⠋⠀⠀⠀⠋⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠙⢆⠀⠀⠀⠀⠠⠸⣿⣽⢻⣽⣿⣿⣦⠖⠀⠀⠀⠀⠀⠀⠄⢠⠆⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠓⠀⠀⠈⠀⠀⠠⠀⠠⠀⠀⠒⠀⠀⠱⠺⡿⠿⠛⠁⠀⠘⢦⣤⣤⣤⣤⣦⡤⣠⡎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠻⢶⣤⡀⠀⠀⠀⠈⣿⣭⣿⣿⣵⣿⠃
⠀⠀⠀⠀⣷⠀⠀⣤⡂⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣦⣄⠀⠀⠀⠀⠠⠍⠛⠳⠶⣶⠾⠋⠙⠋⠛⠋⠛⠃⠤⠶⣤⣼⣦⣠
⠀⠀⠀⠀⢸⣇⠀⠀⠈⠂⠐⠊⢀⣴⣶⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠂⡽⠀⠀⠀⠉⠛⢿
⠀⠀⠀⠀⠀⠻⣷⣄⣀⠀⠀⢀⣾⣿⣿⡍⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⢀⠀⠀⡀⠘⠲⣤⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣿⣿⣿⣿⣧⠀⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⡿⢥⣿⣿⣿⣿⣿⣿⣦⡤⠊⠀⠀⠉⠁⠀⠹⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⢹⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⡟⣡⣶⠁⠀⠀⠀⠀⠀⠉⠁⠈⠿⡆
⠀⠀⠀⠂⠀⠀⠘⣇⠀⠀⠀⠘⣿⣿⣿⢸⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠋⠁⠉⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣄⠀⠀⢿⣻⠀⠀⣀⣸⣿⣿⣹⣿⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⠿⣋⣡⣤⣶⣦⣤⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⣷⠘⣿⣆⠸⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⢈⣷⡎⣿⡶⡀⠉⠉⠉⠛⠻⠟⠲⠿⠟⠉⠉⠙⠿⣿⠿⠛⠋⠋⠛⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠠⠙⠁
⠀⠀⠀⠀⠀⠘⢿⣷⡸⣷⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠟⢀⣀⣮⣤⢞⠁
⠀⠀⢀⣀⣠⣤⣾⣿⣷⡹⣿⣿⠀⣀⡀⣀⣄⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠙⠛⠉⠙
⠀⠀⠀⠾⠿⠿⠿⢿⣿⣵⣿⣿⣾⣿⣷⡿⠿⠿⢿⡿⣦⠀⣠⣶⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⠴⠖⠂⠛⢿⣷⣻⡿⣿⡿⠤⠶⢄⣼⠗⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢻⣵⡷⣌⠉⠻⣿⣾⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⣻⣿⣿⣿⣿⣿⡟
⠀⠀⣠⣴⠶⠒⢈⣿⠃⢰⢿⡈⢿⣄⠻⡆⠘⢿⣷⣿⣿⣿⣿⣿⣿⣿⢛⡟⢻⡿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⣶⣶⣦⡄
⠀⠀⠁⠀⢀⣴⣾⣿⡀⠸⣷⣏⡀⣧⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣽⣿⣴⣿⣿⣷⣮⢿⣿⡿⡿⠋⢀⣈⣙⣁⣹⣿⡿⠿⠤⣟⣁
⠀⠀⠀⣠⡾⢻⣯⣽⣷⡀⢺⣯⡥⠈⣛⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⣸⣿⣷⠉⣿⣿⣿⣿⡶⣌⣀⠘⠿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⣿⣧
⠀⠀⠿⠿⠟⠻⣿⣻⣿⣿⣦⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣦⠀⠀⠘⣷⣤
⠀⠀⠀⠀⠀⣹⣋⠛⠉⠉⠁⠀⠀⠈⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢫⣄⣼⣿⣿⣿⣇⣠⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⡿⠿⠛⠛⠛⠛⠁⠁⠉⠀⠀⠀⠀⠀⠉⠉⠻⣿⢯⠁
⠀⠀⠀⠀⠀⠠⢤⣤⣀⣀⡀⣀⣤⣴⠾⠟⢛⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠁⠀⢀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣸⣣⣴⣂⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠐⠊⠉⠛⠛⠿⡿⠋⠀⠀⠀⠀⠄⠀⠈⠉⠙⠛⠻⠿⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢂⡼⠂⠀⠀⠀⠀⠀⠀⠆⠀⣰⠆⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁⠀⠀⠀⣄⠀⠀⠊⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠀⢄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠆⡤⠆⠀⠀⠀⡄⠀⢠⡀⢹⣀⠃⠀⠀⢌⠁⢠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢠⣥⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡅⠀⠀⠀⠀⠀⢁⠈⢁⡏⠀⢀⠀⠀⠢⢔⠋⠀⠀⠀⠀⠀⠀⠲⠀⠰⠄⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣘⠇⠀⢲⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠀⠀⠀⡈⠛⠛⢇⡀⠛⠀⠀⠀⡻⢀⡒⠼⠀⠀⢀⠀⠐⠷⠐⣆⡰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣠⠇⠀⠀⢃⠀⠘⢷⢀⣤⠟⠃⠀⠠⣾⢻⠀⡶⠛⠄⠀⢀⣶⣠⡿⠎⢙⠲⠂⠀⣰⡄⢠⣄⣴⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠘⢻⡄⠱⡞⠉⢻⣇⢰⡾⠋⠀⠀⠀⠶⣤⢠⣴⠁⢰⡄⠆⠈⢷⡟⠀⢀⡀⢠⣄⠀⡿⠃⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣤⡀⠀⠘⣿⢾⡇⠀⣀⠀⣼⠷⡅⢀⣼⡷⠟⠀⠞⢀⣾⣁⣴⠿⠄⣸⠄⢈⡶⠋⠹⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⣠⣌⣳⡀⠘⣇⢸⣷⣼⢇⣤⣼⣿⢋⡸⠇⡀⣀⠸⣦⡾⠉⢤⡍⠁⣠⣥⠾⠟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠁⠀⠐⠟⠉⠙⢿⣆⠙⣿⠓⣾⣿⣿⠇⣹⣂⡼⠉⢁⣴⣿⡉⣈⣯⣵⡞⢋⡠⠀⠐⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣙⣧⡼⡀⢻⠟⢹⠋⣌⢋⡤⣤⣿⣿⣿⣾⢿⣏⠀⠘⠛⠋⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⢣⠷⡄⠀⠐⠀⠿⠙⠳⣿⣿⠋⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⣼⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⢀⢀⠀⠸⣿⣧⡤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢸⡟⠀⠀⢸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⣰⠃⠀⠀⠀⠼⣟⡠⠀⠘⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⡇⠀⠀⠀⠀⢲⣏⡀⠀⠀⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣽⠗⠁⠀⠈⠢⣠⣿⠀⠀⠀⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⣿⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠘⣻⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⠧⠀⠀⠀⡀⠈⣿⠃⠀⠀⠀⡼⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⠀⠀⠀⠀⠀⠙⢿⠃⠀⠀⠀⠈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠐⣦⡄⠀⠀⠀⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⢡⣌⢻⣿⣦⡶⠀⠲⣿⣿⣿⡶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠈⠉⠙⠀⠈⠀⠁⢈⣋⣿⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⠁⠀⠀⠀⠀⣶⣾⡶⠆⠠⡾⢿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢿⡀⢀⠀⣀⠀⣀⣉⣀⣀⠀⣀⠀⣈⣉⣅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣧⠉⠀⠉⠀⠸⣿⣿⡿⠛⠉⠛⢿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣀⠀⠀⠀⠀⡈⣿⣷⠀⠀⠀⠠⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⢸⡏⣠⠀⠀⢀⠈⢿⣅⡀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡃⢀⠀⠀⠙⣿⡯⠀⠀⠀⠐⢿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⢻⠀⠀⠀⠀⢉⣿⠃⠀⠀⠛⠛⣿⡟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢷⣄⠀⠀⠾⣿⡛⠆⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢇⠀⠀⠀⣽⣯⡀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢈⣷⠦⡀⠀⣸⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢣⠀⠐⠿⣤⠀⣾⣩⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠐⢶⣍⣃⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣀
⠀⠀⠀⠀⣾⣄⣀⣠⣴⣶⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣗
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⢻⡁⠙⣆
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠙⣿⠏⠉⠉⠙⠋⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣇⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠏⠀⡿⠛⠋⠉⠛⠛⠛⠛⠛⠃⠸⠋⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⠟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠀⠈⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣄⡀
⠀⠀⠀⠋⢿⣷⣶⣦⣶⣶⣦⡀⠀⠀⠀⣠⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠘⣿⡿⠿⠿⣿⣏⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣿⠁⠀⠀⠀⠹⠋⠉⠉⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⢠⣤⡿⠿⢻⣿⠿⠿⠛⡱⠂
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠛⠀⠀⠀⠀⠘⠚⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠁

⠀⠀⠀⢠⣶⣶⣾⣷⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢠⠾⢿⣶⣄⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣄⣤⣿⡿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⢿⡿⣿⣿⠿⣿⠁
⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠙⠉⠀⠟⠆⠀⠼⠃⠿⠀⠀⠀⠀⠀⠀⢸⠟⠻⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢘⡂
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠄⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⢼⣤⠖⠄⠀⠀⠀⠀⢰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⠀⠀⠀⠀⠠⠀⠠⢂⣴⡿
⠀⠀⢀⢰⣘⡶⠀⠀⠀⠀⠀⠀⠀⣀⣀⡁⠀⡀⠀⠀⠀⠀⠂⠠⣾⡋⢀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠐⢀⣴⡿⠋⡀
⠀⠀⠀⠀⣿⠁⠀⠁⠀⠀⠀⢸⡿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠁⠴⠫⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠃⠀⠀⠀⠀⢰⡿⠿⠿⠶⠋⠀⠀⠀⠀⠀⠿⠋⠇⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠄⠁⢖⠀⠀⠀⠀⠀⠀⠀⠰⠄⢙⠀⠤⠀⠀⠀⠀⠀⣴⣻⠛⣿⣧⠀⠀⠀⠀⠀⣰⢋⠈⢉⣷⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⡼⠉⠄⠉⢣
⠀⠀⢀⠀⢨⠀⠀⠀⠀⠀⠀⠐⠆⠀⢀⠀⠐⠆⠀⠀⠀⠀⠹⣶⣞⣾⣿⠄⠀⠀⠀⠀⠹⣧⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡸
⠀⠀⠁⠀⠻⠀⠀⠀⠀⠀⠀⠀⠘⠂⢤⠀⠒⠀⠀⠀⠀⠀⠀⣹⣿⡿⠛⠀⠀⠀⠀⠀⠀⢹⣿⡿⠛⠂⠀⠀⠀⠀⠀⠀⠈⠀⠠⡄⠀⠀⠀⠀⠈⠙⠒⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⣶⠞⠛⠲⣶⠀⠀⠀⠀⠀⠀⣤⣶⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⢾⡇⠀⡀⠀⣹⠇⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠈⣿⡄⠀⠀⢠⣿⠀⠀⠀⠀⡸⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣿⠟⠃⠀⠀⠀⠀⠀⠿⢦⣤⠴⠿⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⠀⠀⠀⠀⠀⠸⠟⠀⠀⠿⠇⠀⠀⠀⠀⠷⠤⠦⠤⠦⠼⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⣰⠊⢉⠙⢦⡀⠀⠀⠀⠀⢸⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⣰⣿⠽⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠖⠉⠃⠲⡄
⠀⠈⡉⠀⢈⠀⢀⡁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⡇⠑⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢾
⠀⠀⠳⢄⣀⣠⠜⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠘⠦⣠⡄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⢠⡞⠛⣦⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡆⠀⠀⠀⠀⠀⣴⠿⣯⢻⣆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⡄
⠀⠘⠳⡴⢿⠖⢦⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣶⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⢮⣉⠟⠲⠚⠀⠀⠀⠀⠀⠀⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⡄⠂⢴⠄⠀⡀⠀⠀⠀⠀⢰⠿⣹⠿⣦⠀⠀⠀⠀⠀⣰⣿⠛⣿⣧⠀⠀⠀⠀⠀⢠⣾⣿⢟⣷⠀⠀⠀⠀⠀⡀⠀⣤⣤⡀⢀⡀⠀⠀⠀⣀⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢸⣷⣧⣾⣿⠄⠀⠀⠀⠀⢹⣿⣶⣿⣿⡆⠀⠀⠀⠀⠸⣯⣙⣻⣿⡆⠀⠀⠀⠀⣷⡄⢻⡿⢁⣾⠇⠀⠀⠀⠿⡛⠿⢿⣯⠛⠃
⠀⠀⠂⠀⠺⠂⡀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠂⠀⠀⠀⠀⠀⢸⣿⣿⠿⠂⠀⠀⠀⠀⢸⣿⠀⠁⢸⡏⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⣀⣀⡀⠀⡀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡆⠘⠛⠛⠛⠛⠛⠃⢰⡆
⠀⠀⠀⢰⢶⣦⣒⣠⡄⣂⣤⣤⣀⣤⣠⣖⣠⣤⢴⢄⣂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠇⣀⣀⣀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠘⠘⠛⠛⠃⠘⠛⠃⠃⠋⠒⠛⠛⠘⠛⠘⠒⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣼⣿⣿⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣀⠀⠀⢀⡀⣀⡀⣀⣀⠀⣀⡀⢀⣀⢀⡀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡀⣀⠀⠀⣰⣿⣿⣿⣆
⠀⠀⠀⠻⠯⠈⠁⠧⠜⠧⠟⠏⠏⠻⠹⠹⠻⠯⠛⠘⠥⠸⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠉⠉⠉⠉⠉⠉⠉⠻⣿⠟⠉
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⢸⠿⠿⢇⡇⣿⡍⢏⢸⠱⣏⡇⢯⢹⢾⣹⢾⢹⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣿⣷⠀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⠟⠛⢛⣛⣋⣙⣛⡛⠛⠻⠿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⢋⣡⣴⠶⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠶⣦⣌⡙⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⢋⣠⣾⡟⢋⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡙⢻⣷⣄⡙⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⢃⣴⣿⣿⣿⣾⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣀⣤⣿⣦⡈⢿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⢳⡿⠂⣠⣿⣿⣿⣿⣿⡿⠻⠻⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⣼⣿⣿⣿⣿⣿⣄⠙⣯⣜⣧
⠀⠀⠀⠀⡘⠂⠅⣧⣴⣿⣿⣿⣿⣿⣿⣷⠀⣶⣶⡆⢰⣶⣶⠀⢰⣶⡆⢀⣶⣶⠆⢰⣿⣿⣿⣿⣿⣿⣿⣦⣤⠈⠐⢆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣉⣁⠀⣉⣉⠀⢈⣉⡁⢀⣉⡉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠿⠿⠀⠿⠿⠀⠹⠿⠃⠸⠿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠠⣶⡄⢠⣶⠀⢠⣦⠀⣴⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠉⢁⣈⣉⣀⣈⣉⡀⠉⣁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣴⣤⠙⢿⣿⣿⠏⢠⣤⡌⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠻⠟⢀⣾⣿⣿⣆⠘⠿⠋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⢰⣾⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣷⣄
⠀⠀⠀⠘⣿⣿⣿⡌⢿⣿⣿⡟⣛⣿⣿⠟⣛⡿⣛⠻⡟⢻⡟⢻⠟⢿⠛⣿⢛⣻⠛⠛⡿⢛⣻⢛⣻⣿⣿⡿⢡⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⡄⠻⣿⡇⠲⣯⣭⣘⠻⣆⠻⢃⡇⣄⣰⣸⣀⣀⡄⣿⠰⢾⢠⡔⣇⠻⢿⡀⢾⣿⠟⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠷⣄⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠚⠛⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠱⠂⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣶⣾⡶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡠⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⢘⢆⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⣀⡄⠀⠀⠀⠀⢘⡆⠀⠀⠀⢠⠙⠀⠀⠀⡀⠀⠀⠀⠐⡝⠀⠀⠈⠠⢠⠉⠀⣀⣴⠀⠀⠀⠀⠀⣒⡀⠀⠀⠀⢠⠋⠀⠀⠀⢄⠀⠀⠀⠐⡙⠀⠀⠀⠀⣬
⠀⠉⠉⠑⣀⠀⠀⠀⠈⣑⡏⠠⡀⠓⡀⠀⠀⢀⠙⣀⠀⠀⠐⢒⣸⡋⠤⡈⢓⠀⠀⠁⠈⠁⡙⢀⡀⠈⢉⣸⡋⢀⡈⢒⠀⠀⠀⠀⠀⡁⠆⠀⠐⢂⣺⠉⢄⠘⢂
⡶⣔⡭⠊⠉⣳⠀⠀⠀⠹⠀⠀⠇⠀⠙⡦⣞⡕⠈⠉⡲⠀⠀⠀⠛⠀⠐⠏⠈⠹⠦⣞⠕⠈⠉⡶⠀⠀⠀⠟⠀⠰⠇⠈⢱⠴⣊⠕⠉⠉⡶⠀⠀⠈⠗⠀⠸⡇⠈
⠀⣣⠄⠀⢤⡷⠶⠂⠙⢀⠀⠴⠤⠀⠀⠘⡡⠈⠀⢵⡷⠖⠂⠙⣀⠀⠴⠤⠐⠀⢘⡤⠈⠀⣼⠶⠖⠀⠛⣀⠀⠶⠤⠀⠀⢘⡤⠁⠀⣼⠴⠖⠈⠓⣀⠀⠦⠄
⠀⠀⢴⣀⣼⡆⠀⠦⠀⣼⠄⠀⢠⣀⣀⠀⠀⣖⣀⣾⡄⠐⠄⠀⡽⠄⠀⢠⣀⡀⠀⠀⣞⣀⣶⡄⠐⠄⢀⡽⠀⠀⢠⣀⡀⠀⠀⣆⣀⣾⡀⠐⠄⢀⡧⠀⠀⢤⣀
⠀⠀⠛⠍⠀⠠⡤⠀⣬⣁⡄⠀⢄⣫⠀⠀⠀⢫⠙⠀⠡⡤⢀⣬⣠⡄⠀⢀⣣⠀⠀⠀⢫⠉⠀⢡⠄⢀⣬⣠⠄⠠⣈⣣⠀⠀⠀⠫⠃⠀⢠⠄⢀⣌⣠⠄⠠⣘⣇
⠠⡀⠓⠀⠀⠀⠀⠑⢉⠸⠁⣀⠄⢹⡃⠀⡈⠓⠀⠀⠀⠀⠀⢉⠺⠁⣠⠀⣙⠃⢠⡈⠓⠀⠀⠀⠈⠐⣍⠮⢁⣠⠈⣹⠁⢠⡈⠃⠀⠀⠀⠈⠀⡁⠎⢀⣠⠈⣹

⠀⠀⠀⢀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤
⠀⠀⣤⣨⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⡇⢤⣼⣿⡟⣠⣀⣄⣀⣀⣀⣬⣙⣿⣇⢀⣠⣄⣀⣐⣒⠈
⠀⠀⡁⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠰⠶⠿⢷⣿⡿⠿⠿⠾⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣿⣾⣿⣿⣧⣤⣄⡀
⠀⠀⠀⢸⣿⣱⣿⣿⣿⣿⣿⣿⣴⡆⡀⠀⣠⣤⣾⡟⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢀⣴⠿⢱⠿⡿⣿⢷⣿⣿⣿⣽⣱⣶⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠛⠛⡌⠠⣄⡙⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠠⣿⣱⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠁⠐⠂⠉⠐⠈⣉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢱⣴⡆⢰⡟⢻⣿⣿⣿⣿⣿⣿⣿⡇⣠⠃
⠀⠀⠀⠴⠤⠴⠌⣩⣭⣉⡝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠙⠇⣸⠁⠟⣿⣿⣿⣿⣿⣿⣿⣟⣀
⠀⢴⣶⣦⣤⣤⣤⣤⣤⣽⣾⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣛⠻⠇⠉⠿⣿⠿⠷⠆⠀⡴⠿⡶⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠘⠛⠛⠛⠛⠋⠉⠉⠉⠀⠸⠿⣿⣿⣿⣿⣷⠳⣆⢺⣿⣿⡄⠀⠀⢸⣷⣯⣴⠶⠶⠄⠀⠶⠶⠞⢾⣿⣿⣿⡿⢿⣿⣿⣿⣇
⠀⠀⢀⠀⠠⠄⠀⠀⠀⢀⣀⣠⣷⣶⣽⣿⣿⣿⣤⣈⣓⣿⣿⣷⣤⣤⣄⣀⣹⣛⣷⠗⢺⠓⣶⣷⣾⣉⠙⠛⠓⠉⠀⠒⠚⠛⠋
⠀⠀⠀⠀⣀⣤⠴⠶⠾⠛⠛⠋⠩⠽⠿⠿⠷⠆⠀⠉⠭⠍⣙⣻⠟⠛⠿⠿⢿⣿⡵⠟⠛⠠⠀⠈⠉⠁
⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⠤⠐⠒⠒⠀⠛⠛⠒⠲⠦⠀⠀⠀⠀⠠⠔⠂

⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⣶⣶⣶⣶⣶⣶⡶⣶⣴⣶⡶⣶⣶⣶⣶⣶⣦⣴⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⣿⣿⣿⣿⣿⡿⡿⡇⢹⣿⣿⣿⣿⢻⡆⣿⢹⠛⢟⠻⠟⢡⡾⢏⣻⣿⣿⠯⣿⠻⣿⠿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡏⠛⠃⠀⡄⢸⣿⢸⣿⣿⠸⡏⢹⡿⠀⠈⣁⢷⠘⢗⡶⡴⢿⣇⡄⣿⣀⣟⣦⠌⡁⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡇⠀⠁⢳⡆⠈⣿⢸⣿⣷⠀⠀⢸⡟⢆⠀⢠⣾⣿⣿⠁⠀⣿⢿⡃⣿⠀⣏⣾⡆⡇⢸⣿⣿⣯⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⡟⡧⠉⠀⠀⠩⣷⠀⠛⢸⣾⣿⠉⠀⠀⠁⠀⠀⠀⠘⠋⠁⠀⠀⠙⠟⠀⣿⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⣷⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⠹⠃⠃⠀⠀⠀⠀⠁⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣠⣤⣼⣤⣤⣿⣽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣹⣟⣿⣿⣿
⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠫⣿⣿⣿⡿⢿⣿⡿⣿⣿⡟⢛⣻⣿⣿⠋⠰⣶⣜⠿⢿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠇⠈⣽⣿⣿⣦⠏⠈⠟⠉⠉⢨⠝⠟⠋⠁⠰⠿⢿⠿⣓⠺⣿⣿⣿⣿⣿⣿⣿
⢸⡿⠀⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⡾⣿⣽⣿⣿⣿⣿⣷⣱⣒⣿⣿⣿⣯⢈⣰⠊⣖⣨⣮⣾⣄⣄⣾⣶⣾⣁⣀⡿⣛⣛⣿⣿⣿⣿⣿⣿
⠸⠗⠷⠇⠁⠐⠺⠇⠒⠂⠀⠀⠀⣠⣷⣾⣿⣿⣿⡿⣿⣧⣿⣿⣿⣍⣭⣿⣿⣿⣿⣿⣷⣿⣷⣿⣾⣿⣿⣿⣿⣿⣏⣙⣭⡟⢐⣹⡏⠀⠀⠀⠀⣁⣀
⢠⣤⣤⡄⠀⣀⣤⣤⡤⢄⣤⣤⣾⣿⣿⣿⠏⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⡿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣶⣶⣶⣾⣿⣿
⢸⡧⠤⠤⠤⠤⣬⣴⣾⣿⡿⠿⠭⠿⠷⠶⠶⠒⠒⠒⠒⠒⠒⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣷⣿⣿⣿⣾⣿⣷⣭⣭⣉⣈⠉⠉⠉⠉⠁⠀⢉
⠀⠀⠀⠀⠀⣾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠃
⣤⣀⠐⠛⠽⠿⢦⣤⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠇
⠛⠒⠀⠀⠀⠀⠀⠙⠷⠦⣈⣹⡯⣤⣶⣞⠓⠄⠀⠀⠄⢤⣄⠀⠀⠒⠛⠶⣄⠀⠀⠀⣴⣾⣿⣿⡟
⠀⢉⠉⡁⠀⠉⠁⠀⣉⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣨⠉⠂⠀⠀⠀⠌⠙⠛⣝⡻⠿⢿⣿⠿⢿⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⡠⠀⠒⠂⠀⠶⠶⠚⢷⣶⡭⠉⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠄⠘⠂⠀⠶⠀⠀⠀⠄⠀⠢⠀⢉⠒⠀⡦⢤⡶⢤⡐⠛⠛⠳⢦⡴⠤
⠀⠀⠀⠤⢤⣬⣁⣀⣠⣄⣈⠁⠈⠩⠷⠶⠶⣶⠤⠤⠐⠀⠀⠀⠀⢠⣴⣤⣀⡀⠀⠢⠤⠚⠋⢁⣀⣀⠀⠀⠀⠠⠶⠀⠀⠐⠄⠀⠀⠀⠐⠖⠀⠤
⡴⠦⢄⣀⡤⠌⠙⠛⠻⠦⠉⠓⠦⠄⠠⢦⣄⡁⠀⠶⢾⣿⣒⣒⡛⠋⠉⠁⠀⢀⣀⠀⠁⠒⠲⠟⠛⠛⠓⣶⣤⡤⠀⠀⠀⠀⠀⠀⠤⠶⠛⠓⠦⠠⣄
⣠⣀⣤⣴⣿⣷⣤⣤⣄⡛⠃⢀⣠⣤⣤⡀⠉⠛⠒⠀⠀⠈⠉⠉⠉⠉⠙⠿⠿⠿⠷⠶⠶⢦⣤⡄⠤⠄⠀⠀⠀⠶⠶⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄
⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠟⠛⠿⠿⣟⠛⢛⣛⠳⠶⠄⠀⠀⠀⠀⠀⠀⣀⣠⣬⣁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⠀⣀⣠⡤⠶⠈⠀⢀⡀
⠀⣀⣤⣤⣤⣄⣀⣉⣁⣀⣀⠀⠀⢀⣀⠀⠈⠉⠀⣀⣀⣀⠠⠠⠖⠒⠀⠈⢉⣴⠤⢶⣦⡄⠀⠀⠀⣀⣀⣀⣠⣴⣶⠿⣟⣉⣭⣡⣤⠀⠀⠀⠉⠀⣠
⠀⣉⣽⣿⣿⡿⠛⠛⠛⠛⠛⠷⣄⣀⣀⠀⠀⠀⠉⠁⠈⠛⠗⠂⠀⠒⠒⠆⠀⠀⠀⠈⠀⠀⢀⣉⣉⣀⣀⡀⠀⠀⣀⣠⣼⣿⠁⠀⢀⣤⢤⣤⠴⠞⠋
⠀⠛⠿⠿⠿⠿⠿⠶⠶⣤⣤⣤⣈⣙⠛⠶⠤⠄⠉⠙⠛⠀⠀⠤⠤⠤⠖⠂⠀⠹⣦⣶⠶⠿⠯⠭⠀⠀⠠⠤⠭⠭⠍⠉⠁⢀⠀⠈⠀⠀⢀⣀⣀
⠀⣐⣒⠀⠦⣤⣄⣀⣀⣠⣼⠿⠛⠛⠛⠒⠒⠒⠒⠚⠛⣛⣳⣶⣤⣤⣴⣶⣶⡶⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⣤⣤⣤⣶⣶⣶⢾
⣿⣿⣿⠿⠶⣄⡀⠉⠉⣉⣁⡀⢀⣠⣤⣄⣉⠉⠀⠀⠈⠉⠉⠉⢉⣥⡶⠶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠄⠀⠀⢘⣛⠛⠋⢉⣉⣥⣌
⣤⣴⣶⣶⣶⣿⣿⡛⠛⠛⠛⠿⠛⠉⠉⠉⠉⢉⠛⠋⠉⠁⣀⣠⠀⠀⠴⢶⠶⠖⠛⠀⠀⢀⣀⡠⠼⠁⠀⠀⠀⠀⣠⠴⠶⠀⠀⠈⠙⠻⠿⠿⠿⠃⠈
⣿⣿⡿⠿⠿⠿⠟⠛⠻⠶⠶⠶⠒⠂⣠⠴⠚⠻⠿⠛⠒⠉⠉⠀⠈⠙⠂⠠⠖⠚⠋⠉⢋⣩⣥⢤⡤⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠂⠀⢀⡄⠀⣠⣄⣠⣴⣾⣿⣿⣿⠗⢀⣀⣀⡀⠀⠀⣀⣀⡴⠋⠀⠠⠴⠂⠴⣾⣶⣦⠀⠀⠀⣹⣿⣿⣶⣦⣄
⠀⠀⢠⣤⣄⣀⣠⠔⠂⠀⣀⡀⠀⠉⠁⠀⠀⣈⡽⠟⠛⠛⠛⢉⣠⡾⠋⠀⢀⣠⠴⠛⠉⢙⣿⠟⠀⠀⠦⣄⠀⠈⠛⠋⠀⠀⠙⠛⠋⠉⠉⢉⣡⣄⣀
⠀⠰⠛⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠁⠠⠖⠂⠀⠈⠋⠉⠀⠀⠈⠉⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠂⠀⠀⠈⠉⠉⠉⠉

⢻⣽⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢇⢰⠺⠿⠿⠻⢧⠀⡀⢹⢿⡿⢿⣿⣿⣿⣿⣿⠧⣸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡿⢏⠂⠉⠀⢹⣸⠀⠀⠀⢀⠀⠀⢂⠘⡄⢇⠀⠀⢹⡏⠛⢁⣸⣿⣿⣿⣿⣿⡿
⣿⣿⣿⠛⣿⢿⡟⠉⠀⠁⢠⣀⣨⣢⣀⠀⠈⠉⠀⠠⣄⠘⠀⠀⠈⠀⠀⡈⠆⠀⠀⠁⠀⠀⣍⣁⡀⠹⣿⣿⢠
⣿⣿⣿⡆⠁⠈⠀⠀⠀⠀⢹⡟⠛⢻⣿⣿⣕⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⣀⡀⠀⠠⠘⡘⢸⣤⠈⠃⣼
⣿⣿⣿⡇⢹⡀⢀⠀⠀⠀⢸⠷⣄⠈⠉⠙⢻⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠀⠀⠀⢻⢾⡋⣦⣸⣿
⣿⣿⣿⠃⠀⢡⠈⠀⠘⢣⡈⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣴⣶⣿⣾⣵⣦⣄⣼⣇⢣⣿⣿
⣿⣿⠃⠀⠀⠈⢁⡀⠀⢘⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠋⠉⠙⠟⠀⣙⣿⡿⠆⢾⡟⠻
⣿⡷⠀⡰⠀⠀⠀⠁⠀⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠛⠹⠋⢀⡟⡇
⣿⠁⣰⠃⡄⢠⠁⢸⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣴⣇
⣿⡴⣫⠎⠀⠈⡀⢸⣇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⣿⢸
⣿⡣⢃⠀⠀⠀⡇⢸⡼⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⣿⡿⢻
⣿⠁⠆⠀⠀⠀⣱⣿⣷⡄⢀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⣿⣿⣿⣵⢡
⣧⡴⠀⠀⠀⠀⠘⢿⠟⠀⠂⠘⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣿⣿⣿⣿⠿
⣿⣄⠀⠀⠀⢠⠀⢸⠀⠠⠄⠓⢌⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⣿⣿⣿⡿⢿⣿⣿⣿⣿⢣⣾
⣿⣿⠆⠀⠀⠈⠀⢸⡄⠀⠀⢠⠀⠳⠘⠂⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⢸⠓⠻⣿⣿⢼⣿⣿⣿⣳⡋⣾
⣿⣿⣧⠀⠀⠀⠀⢸⠀⠀⠀⠘⢂⠀⠀⠀⠈⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠹⡿⣒⡻⢿⠿⢻⣿⣿
⣿⣿⣿⣆⠀⠀⠀⢺⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⡇⢀⡆⣸⣿⣟
⣉⣉⡉⢁⣤⡎⠀⣴⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⣷⣿⣿⢿
⣿⣯⣴⣿⣿⣿⣾⡛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢄⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣤⡄⢠⣾⣷⣶⣦⣦⣤⣾⣧⣤⣤⣼⣦⣴⣶⣾⣿⣿⣟⠙⢿⣿⣿⣿⣿⣿⣿⣿

⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠐⠂⠰⠲⠶⠶⠶⠀⠒⠀⠠⠦⠀⢻⠛⢐⣙⢁⠊⣉
⠀⠂⠀⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠤⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠀⠀⠈⠀⣐⣒⣀⡀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡀⢘⡁⠹⢡⠄⠭⠍⠉⠁⠀⡁⠀⣀⠀⠀⣀⣠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠐⠀⠀⠀⠉⠉⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⠀⠠⡆⠀⠀⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠙
⠀⠀⣠⣬⣀⣴⣒⣺⣟⣚⢳⠂⡀⠒⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣶⠐⠀⠀⠘⠧⠈⠀⠀⠀⢀⠨⠞⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁
⠄⣀⠋⠘⠡⠖⠊⠁⠀⢠⠠⠀⠠⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡽⣇⠘⠳⠀⠀⢰⣯⣾⣿⣽⡇⠀⠀⢋⡤⠖⠈⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⢠⣤⣤⡤
⣤⣷⡀⠀⠀⢛⣿⣃⡀⠉⠀⠘⠛⠒⠀⡴⠀⣾⠐⡆⠀⠀⠀⠀⢀⠀⠐⠂⠀⣤⡍⠀⠀⠘⢻⣿⣿⠏⣷⣿⠄⠀⠀⠘⠃⠀⠘⠃⢠⡘⡋⢻⠀⠀⠸⠸⠿⢿⣷
⣤⡽⠃⠀⠀⡀⢁⡀⠀⣰⠶⠆⠦⠄⠀⣬⢠⡽⢋⡂⠜⠡⢀⠤⠼⠏⠀⠶⡄⠀⢀⠰⠶⠀⣰⢿⣿⢻⣿⠋⡀⠀⡙⠛⠀⠐⠆⠀⠀⠁⠠⠋⠀⠲⣶⣶⣤⡼⠷
⣈⢀⣀⠀⠀⡁⢀⢄⣼⡄⠢⠒⠒⠒⠀⠁⠀⠀⠀⠀⣀⠀⠀⠐⢒⣀⠀⠓⠁⠚⢉⣀⠀⠀⣭⡏⠀⠸⡬⠂⠀⠀⠁⠀⢈⡀⣀⠀⡁⢀⠀⠀⠀⠄⢿⣿⣿⣷⣦
⠀⠀⣏⠀⠘⢣⣼⡖⠐⠛⠑⠐⠂⠀⠈⠀⠀⠀⡄⠀⠉⠀⠤⠀⠀⠀⠀⣴⣾⣦⠈⠁⣠⠀⠉⠀⠀⠀⢀⠀⠦⠀⠀⠀⠀⠰⡁⣰⣁⡀⠀⠀⠐⠛⠶⣿⣿⣿⣉
⠛⠋⠉⠁⢠⠨⢥⡬⠣⠤⡍⠉⠁⠀⠀⣉⠀⠏⠀⠀⢀⠄⠀⠀⠒⣄⠀⠙⠛⠋⠀⠀⠁⠀⠀⠸⠇⠁⠈⠗⠀⠐⠉⠉⠃⠀⠄⠀⠐⠒⠒⠂⠀⠀⠀⠈⠛⠛⠛
⢈⡉⠉⠁⠁⠀⣎⢘⢀⢰⢀⣄⠀⠀⠐⠁⠉⠠⠀⠀⠈⠀⠔⠒⠀⠁⠸⠀⢀⣀⣀⣠⣂⠀⢋⣀⣀⣀⠀⠠⠐⠀⡤⣼⣤⠈⠀⠉⠀⠈⠀⠀⠀⠁⠈⠉⠀⠀⣀
⠀⠀⣀⣰⠀⠀⠼⠿⢧⣏⢩⣽⠀⠀⠀⠄⠀⡨⠀⠀⣴⠀⠀⠁⠑⠦⠀⢠⣤⣍⠁⢨⣤⠾⠁⢤⠀⠀⠴⠀⠤⡄⠀⠤⠀⠀⠒⣒⣶⣦⠶⠠⠡⠆⠀⢀⠀⠉⡉
⣀⣔⡀⠀⠀⠀⠒⠁⣜⣉⠿⠋⠀⠀⠐⠄⠤⢀⠀⠀⠀⠈⠐⣁⠀⠀⠀⠸⠯⠤⠐⠤⠔⠴⠦⠤⠠⠔⠰⠦⠤⠤⠄⠠⠀⠀⠀⠀⢀⡠⠶⠒⠒⢢⡀⠀⠠⢀⣁
⠀⡀⢤⠆⠄⠀⣀⢈⣙⣛⣈⠃⠂⠀⢀⠀⠀⠈⠀⠀⠰⠂⠀⠀⠈⠳⠀⠐⡂⠀⢀⢃⢦⣴⠒⠂⠐⠞⠓⠓⠀⠀⠀⠤⣴⣶⣶⣿⣇⣀⠤⠔⠒⠋⠀⠀⠣⠹⣭
⠛⠁⠛⠐⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠛⠀⣤⠀⠚⠛⠛⠃⠉⠈⠁⢀⠀⠀⠠⠚⠿⠕⣿⣋⣁⡀⠀⢀⣀⠀⠀⠀⠀⠀⠙
⢀⣀⣀⣠⣤⡾⢿⣦⠀⠀⠀⠁⠻⠖⠻⠒⠶⠇⠀⠀⠀⠠⠔⢀⠀⢻⣿⣿⣿⡇⣴⡄⢰⣞⣷⡄⠠⠀⠁⠊⠑⡄⠁⠀⠀⠘⠀⠇⠛⠛⠛⠛⣁⠀⡜⠳
⠀⠀⠀⡒⣾⣿⠽⡇⢀⠀⢀⠀⠀⢀⢂⠐⠂⠒⠀⠀⠀⠀⠬⠀⠀⢻⣿⣿⠗⠟⠛⠁⠨⡉⠋⠀⠰⣀⠀⠀⣀⠌⠀⠀⠐⠒⠃⢁⢋⠀⠀⠂⠈⠀⢀⣠⣤⡀
⠀⠀⠀⣨⡟⢁⣿⣟⣨⠀⠈⢩⡍⠍⠩⣁⡠⠤⠀⠁⢀⡀⠀⠀⡠⠄⠀⠀⠀⠠⠤⠤⡁⠁⠸⠍⠀⠈⠐⠊⠁⠀⠀⠉⠉⠀⠀⠈⠁⠈⠁⠀⡀⠀⣎⠰⢈⡹
⢦⠀⠀⠨⠿⠻⠁⠀⠀⠀⠀⠀⠘⠃⠀⠈⠀⠀⠀⠐⠉⣿⢀⡴⠁⠀⣠⢲⠟⠘⠆⠴⢁⣤⡄⢄⡾⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣘⠀⣀⠀⠀⡈⠁⠒⠁
⢨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣾⣿⣧⣾⡁⠈⠁⠈⠀⠰⠆⡷⠀⠈⠀⠠⣶⠄⠀⠉⡀⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠄⠀⠀⠰⠆⢠⡒
⢿⠁⠂⢠⣀⢀⣄⠀⡄⠀⠀⠀⠀⠀⡽⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠶⡶⠖⠒⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠀⠀⠑
⡾⠀⣀⡀⠀⣄⣉⠀⠀⣀⣀⠀⠀⢀⠀⢩⡉⣡⠀⡴⠀⠀⠀⠊⠓⠒⠲⡀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣍
⣟⠐⢄⡼⠠⢻⡇⠆⣼⠃⠀⠰⣆⠀⠀⣠⣿⣿⣿⣷⣄⠀⢠⠒⠀⠚⣏⡉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠃⠀⣀⡀⠀⣉⣀⠀⠀⠀⠥⠤⠄⢀⠤⠮⣿⣿⣿⡿⠋⠁⠀⠀⠃⠀⠀⠻⠈⠀⠀⠈⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢩⠘⠳⡌⠄⣭⣶⢭⡶⡄⢀⡀⢀⡀⠀⠸⠋⠋⠙⢆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠⡤⠀⠲⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠛⠎⡁⠟⠁⠒⠋⠙⠉⠖⠁⠀⠐⠀⠀⠈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠙⠀⢆⣘⠂⠀⠀⣿⣿⣛⣿⣿⡇⠀⠀⠀⣀⡤⠆⢤⡄⢀⠀⠀⠀⣄
⠠⠄⠀⠠⠤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠘⠨⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠗⠀⠀⠘⠿⢿⣿⠟⠀⠀⢈⠰⡏⠀⠀⠀⢻
⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠰⠶⠖⠚⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢳⣄⣀⣤⠗⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠾⣍⠁⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠉⠓⠂⠀⠘⠀⠀⠀⠀⠀⠃⣠⠇⠀⠚⠛⠓⠂⠀⠀⠀⠀⠀⠀⠁⠠

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣽⣇⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠉⠀⠉
⠀⠀⠠
⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⢸⣷⣤⣤⠀⠀⢠⣾⣿⣿⣿⣶
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠿⠟⠟⠹⠇

⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⢿⡿⢿⡇⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠘⠀⠀⠇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠹⠄⠱

⠀⠀⣀⣤⣤⣤⣄⠀⠀⢠⣶⠄⠀⠀⠀⠀⠠⢶⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠴⠄⢀
⠀⠰⠿⠿⠿⠿⠿⠀⠀⠙⢿⣷⡶⠤⠀⠀⠀⠀⢛⠛⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⣠⡀
⠀⠀⣠⣴⣶⣶⣶⠀⠀⠈⢻⣿⣿⠋⠀⠀⠀⣰⣿⣿⣿⠋
⠀⠀⠉⠁⠃⠈⠉⠀⠀⠀⠈⣿⡏⠀⠀⠀⠰⢷⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁
⣿⣟⣻⠿⢿⠿⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀⣀⡀⠀⢠⣤⣤⣾⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢙⣦⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠗⠒⠤⢴⡽⠃
⠀⠀⠀⠀⢀⠘⢿⣿⣤⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⣿⣿⡿⠿⡿⢿⣿⣿⣿⣟⠿⠛⠙⠂⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠈⠀⠠⣄⣿⡄⠀⠀⠀⠀⣼⣿⣿⣿⠛⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⠀⠀⣀⡠⠃⡀
⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⢀⣀⠤⠞⠱⢼⡿⣧⣈⣁⣠⣤⠰⠤⠄⣠⣠⣤⣸⡄⢣⣤⣦⡉⢒⡻⠅⠀⠈⠋⡡⢪
⠀⠀⠀⢀⡠⠖⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠙⠈⠉⠩⠉⠀⠀⠐⠀⠺⠻⠿⠿⠿⢦⡟⠗⠂⠀⠀⠀⠀⠀⠀⠁⠅⠂
⠀⠀⢰⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠡⠁
⠀⠀⢻⠀⠾⣿⣿⣾⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠘⠦⣌⣀⣚⠙⠑⡠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡄⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠓⠆⠦⢀⡂⢀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠄⢙⡋⡰⡀⢀⢰⣆⣄⡠⠄⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡈⠠⠀⠁

⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠳⣄
⠀⠀⣠⠖⠉⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⢄⠀⠀⠙⢦
⠴⠊⠁⠀⡠⠒⠁⡈⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠐⢄⡀⠁⠀⡀
⠀⠀⠴⠈⠀⠀⠈⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂
⠀⠀⠀⠑⠄⠀⠀⠀⢠⢸⠀⠀⡀⠈⢄⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢀⠀⢀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⢀⠎⢘⡠⠊⠀⢀⢀⠉⠂⠄⡀⠀⢀⠀⠀⠀⢄⠱⣄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡠⠔⠊⢀⠤⢄⠀⠈⠳⣼⠀⢣⠀⠀⠀⢠
⠀⠀⠀⠀⠁⠀⠀⠁⠠⠋⠀⢀⡄⠈⢢⡉⠢⢀⠈⠢⡀⠀⠀⢀⣈⡓⢄⡉⠁⠀⢠⡀⠀⠤⡀⡠⠊⢀⠠⠊⢁⠔⠁⢀⡄⠀⠈⠣⠀⠁⠀⠀⠇
⠀⠀⠀⠀⡄⢠⠋⢢⠄⠠⢆⠀⠪⡢⡀⠑⢄⠀⠁⠐⠉⠀⠀⠀⠀⠀⠉⠉⠁⣀⢤⠀⠑⢄⠈⠗⠂⠁⢀⠔⠁⢀⢖⠕⠀⡠⡆⠐⢤⠊⢡⠀⠇
⠀⠀⠀⢀⠁⠸⠀⠀⢂⠀⢃⠱⡀⠐⢌⠢⡀⠑⢄⠀⡜⠀⡜⠑⠀⠰⡆⠀⡖⠀⠀⡄⠀⠀⠀⠀⠀⡰⠋⢀⠔⡡⠂⢀⠜⠰⠁⢠⠃⠀⠸⡀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠆⠈⢂⠀⠱⡈⠂⠊⠰⠀⠠⠀⡀⠀⡄⠁⠀⡇⢡⠀⠡⠀⠀⠀⠀⠀⠂⠐⢁⠜⠀⡠⠃⠀⠃⠀⠆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠈⡄⠀⠑⢄⠈⡆⠀⡆⠀⠂⠀⡇⠀⡇⠀⠀⠃⠀⠆⠀⠃⠀⠀⠀⠀⠀⢰⠁⡠⠊⠀⠀⠎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠑⠚⠀⠀⠁⠀⠀⠀⠀⠂⠀⠈⠄⠜⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠴⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠃⠀⠀⠀⠀⠀⣀⣤⠶⠒⠒⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⣛⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠠⠼⠍⣁⣠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢒⡲⠿⣷⣶⣶⣶⣶⣶⣾⣿⣷⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣀⠔⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠨⢭⠭⢳⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣔⡀⠈⠙⠋⣉⣿⣿⣿⣷⠒⠒⠒⠂⠒⠒⢤⣲⠄⠚⠁⠀⣴⠋⠀⠀⠀⠀⡠⠠⡠⢄⠈⠭⠩⢥⡙⣆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠈⠉⣿⣿⣿⣿⡿⠅⡀⠤⠁⠀⠀⠀⠈⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⡆⠀⡂⠠⠠⠀⠥⠥⠥⠼⣧
⠀⠀⠀⢀⠴⠶⠷⢤⣤⣀⣀⡀⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠄⠀⠄⢈⠀⡀⠀⣐⣒⣒⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠛⠛⠧⠤⢤⣀⣀⠀⢨⢰⠀⠀⠀⠸⠇⠀⠀⠀⠂⠀⠢⠄⡀⠀⣀⢀⠅⠠⠀⢐⣒⣒⡂⢻⡀
⠀⠀⠀⠀⠀⣀⢤⣤⣀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠠⠲⣺⠆⠀⢂⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠀⠔⠁⠀⠀⠤⠶⠤⠘⣇
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠁⠈⣿⣿⣿⣷⡶⠤⠄⢀⡀⡥⠀⠐⠈⠀⠀⠰⣌⠣⢴⠀⣀⣠⡽⡞⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠹⡶⠤⠀⠀⠲⠛⢥⡖⢦⠀⠀⡀⠀⠀⠸⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠱⣬⠑⠂⠂⠐⠀⠿⠾⠆⠁⠓⠣⠄⠓⠹⠃⠀⠀⠊⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠴⡁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⠀⠤⠤⠤⢀⡀⠀⢸⠀⠀⠀⠀⢢⡀⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⢧⡀⠀⠀⠀⠁⠀⠀⠀⢠⣠⡄⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⢀⠤⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⡔⠁⠀⢀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⣄⠗⠀⠀⠀⢰⢀⡴⡏
⠀⠀⠸⠀⠀⡔⠉⠀⠀⠀⠂⠤⢀⣀⣀⠀⠀⠀⠀⢮⣄⠀⠀⢘⣿⣄⠀⠀⢸⢇⠀⠙⠒⢲⣄
⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠤⠤⣀⠀⠀⣘⣎⠁⠈⠳⡄⠈⠘⠛⢲⠒⠋⠁
⠀⠀⠇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠈⠁⠐⠚⠉
⠀⠀⠘⣄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠒⢄⡀⠈⠐⠢⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⣈⠔⠊
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⠤⣀⣀⣈⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⣃⣉⡡⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠂