Hairpin Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣒⠒⡒⠤⣄⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣠⠖⠉⣩⠟⢲⡄⠀⠀⢱⠀⠀⠀⠀⠀⠈⠲⢄⣀⣀⠀⠀⠙⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡴⠁⠀⡴⠁⡴⠃⢸⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡞⠀⣠⠎⢠⠎⠀⠀⡸⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡀⠀⠉⠉⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠎⢀⡴⢁⡴⠁⠀⠀⡴⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⡀⠀⠀⠀⠈⠲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡞⢠⠎⣠⠊⠀⠀⣠⠞⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢄⡀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠀⣸⡴⢃⠔⠁⢀⡤⠊⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⡟⣠⣋⠴⠚⣁⠴⠋
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠉⢉⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠧⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡼⠀⢉⣫⡀⢈⡓⠊⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡄⠀⠀⠀⠀⠀⠀⢘⡆⠀⣸⠑⠚⠀⠉⠁⠙⠶⢢⠉⠉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠔⠁⠀⠀⠀⠀⠀⠀⣏⠉⢸⡉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⡆⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⡩⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠼⠰⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠋⢼⠉
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡧⠉⠹⣄⠤⡀⢀⣄⣀⡖⠒⣡⠤⠜⠁
⠀⠀⠀⠀⠀⠸⣅⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠓⡆⢀⡭⣅⠀⡡⠶⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡎⣉⠉⠉⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠀⠄⢌⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⢨⣉⣉⣉⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠘⠀⠀⠰⠴⠤⠤⠀⠀⠀⣉⣉⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢘⣛⣛⡂⠀⠀⠀⠤⠤⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣼⠀⡇⠰⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠈⠠⠐⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠨⠤⠤⠄⠀⠀⠀⠒⢒⠀⠀⠀⠀⡇⠀⡇⢀⠀⠀⠀⡏⠀⢡⠀⡀⠀⠐⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢘⢒⠒⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⢀⡀
⠀⠀⢰⠀⠀⢨⠭⠤⠤⠄⠀⠀⠀⠐⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠠⠬⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠂⢀
⠀⠀⠀⠀⠀⠀⢫⣉⢉⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠌⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠻⣍⠢⡄⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣉⣒⣒⣶⣤⣶⣶⣤⣶⣦⣤⣶⣦⣴⣦⣤⣴⣒⣛⣀⡀⠀⢇⠈⢄⠈⠳⣄⠀⠀⠈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⡌⢳⡷⣌⠳⣮⡲⣄⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠭⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠮⢭⠥⠤⠆⠀⠀⠰⠝⠄⠻⡎⠦⠙⠿⡦⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠆⠐⠀⢶⣦⠰⡄⢶⣶⢢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠢⠭⠛⠛⠋⠝⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠂⠀⠀⠀⢿⡆⠣⠘⠿⡳⠘⠂⠻⠷⢝⠂⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⡀⠀⠀⠀⠘⡖⠀⠀⠰⣧⠀⠀⠀⠀⠀⠰⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠉⠀⠈⠁⠀⠉⠁⠈⠉⠁⠀⠉⠉⠑⠚⠁⠀⠀⠀⢰⡃⠀⠀⠀⡙⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠒⣒⣶⣶⣶⣶⡶⣶⣶⢶⣶⡶⣶⣶⠶⣶⣖⠒⠲⠀⠀⠀⠀⠈⠃⠀⠀⠀⠐⡀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢤⣄
⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢹
⡸⠋⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼
⢸⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿
⠀⣿⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⢹⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡇
⠀⠸⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼
⠀⠀⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢠⡟
⠀⠀⠻⣄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⣾⠁
⠀⠀⠀⠙⣧⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢘⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⣿⠂
⠀⠀⠀⠠⣏⠀⠀⠀⣽⠂⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠇
⠀⠀⠀⠀⠹⣆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⣹⠆
⠀⠀⠀⠀⢠⡟⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏
⠀⠀⠀⠀⠈⠻⡆⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿
⠀⠀⠀⠀⠀⠀⡿⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀⠀⢀⣿⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠛⠃⠀⠈⠛⠁⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠺⠟
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠈⠋
⣿⣿⡿⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠿⣟⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠟⢋⣡⣾⣿⣿⢳⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟⠋⢀⣴⡿⠋⣹⣿⢏⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠋⠀⣠⣶⠟⠁⢀⣾⠟⢁⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠛⠁⢀⣠⣾⠟⠁⢀⣴⡟⠁⣠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠋⠀⢀⣴⣿⠟⠁⢀⣤⡿⠋⠀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟⠁⠀⣠⣶⣿⠟⠁⢀⣴⣿⠋⠀⢠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠁⢀⣤⣾⣿⠟⠁⢀⣴⣿⠟⠁⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⡟⠁⢀⣴⣿⠟⠁⠀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⣿⠋⣠⣶⣿⡿⠋⠀⢀⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⢀⣤⣾⣿⣿⣿⣿⣷⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⡿⠁⠀⣠⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⢿⣿⡿⠋⣀⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⣾⡿⣿⣿⣶⣄⡀⠀⠀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣥⣾⣿⣿⣦⣾⣿⢿⣿⣿⣿⣿⡿⠟⣿⣷⡀⣴⣾⣿⣿⣷⣍⡙⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣿⣿⠀⠉⠻⣿⣿⣿⣿⡿⠿⠛⢙⣧⣤⣿⣿⣌⠙⢿⣿⠿⠟⠛⠚⠛⠿⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣾⣿⣿⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⡿⠋⣾⡏⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠻⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡛⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣽⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⠾⣿⣷⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣸⣿⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠙⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢾⣿⣿⣯⣿⣿⣿⡄⠀⠀⢀⣴⣾⠿⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⣿⣷⡀⢀⣾⣿⠇⣴⡾⠿⠿⠿⢿⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⠉⠉⠻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣩⣿⣷⠟⠉⣡⣾⠏⠀⠀⠀⠀⠻⣮⣝⡻⣿⣿⣶⡤⠦⠶⢶⣶⣯⣛⠻⣷⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣥⣤⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣽⡻⣿⣷⣄⡀⠈⡻⠿⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠛⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣿⣻⣝⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣭⣭⣽⣿⣿⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⡏⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠋⠀⣠⠞⠁⠸⠿⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⢀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⣀⡤⠖⠉⠀⠀⠀⣠⣴⣶⣿⣿⢿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣀⡴⠚⠁⠀⢀⣠⣤⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⡠⠞⠁⠀⣀⣤⣂⣹⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⡴⠋⠀⣀⣴⣯⣥⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡴⠋⠀⣠⠞⣷⣾⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⢀⡤⢾⣷⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⡴⣿⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣠⣾⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⢀⡙⢦⠀⢠⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠹⡌⣇⢸⠀⠀⠀⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠠⡀⠀⣄⢹⣸⣸⡀⢀⠆⣰⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡀⠀⠀⣄⠈⢦⡘⢸⣿⣟⣡⣯⡞⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠙⠲⣤⣈⣙⠀⡿⠛⠉⠙⣿⣿⡤⠤⠴⠧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⢤⣤⠤⣤⠴⠚⣗⠦⢄⡴⠛⣟⡛⠓⠲⢤⡀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣠⠞⣡⠞⣉⣐⢲⡈⢹⡟⣇⠲⠄⠙⠂⠀⠀⠉⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⡼⣡⠞⢁⣾⣿⣿⠃⣰⢻⡆⠹⡄⠐⣆⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣠⢏⠜⢁⣴⣿⣿⡿⢁⡜⠁⡰⢷⡀⠙⠆⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⣼⡵⠃⣠⣾⣿⡿⠋⡰⠋⡠⠊⠀⠀⢳⡄⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣠⣾⠎⣠⣾⣿⡿⠋⣠⢞⡴⠋⠀⠀⠀⡠⠊⠙⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⣴⠟⢁⣴⣿⣿⠟⣡⢞⡵⠋⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⡾⢋⣴⣿⣿⠟⢁⣼⠟⠉⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣰⠏⣠⣾⣿⠟⢁⡴⠋⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡼⢃⢸⡿⠟⣡⠞⠁⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⡸⠁⠀⡼⠁⠈⣒⡴⠚⠁⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⢠⡇⡆⠸⣇⡴⠞⠉⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⢧⠳⣄⣀⡤⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⢶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣵⣖⠚⠉⠛⠻⣿⣟⡻⢿⡷⡄⠀⠀⠀⠀⠀⠀⠀⣿⢵⣧⠤⠞⠙⠛⠻⢿⣟⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠉⠙⠳⣦⣄⢤⡙⢿⣦⠙⢹⡄⠀⠀⠀⠀⠀⠀⡟⠈⠛⠻⠿⣶⣤⡀⢄⠙⢿⣎⠳⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⢁⣀⠀⠀⠀⣼⠆⠙⢿⣿⡌⣿⣧⠀⡇⠀⠀⠀⠀⠀⢸⢀⣀⠀⠀⢠⣿⣊⣙⢾⣷⡈⣿⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⡘⠸⠇⠀⠀⠀⢸⣿⠿⠿⣿⣿⣹⣿⢠⡇⠀⠀⠀⠀⠀⡘⠹⠅⠀⠀⠀⣏⣩⡽⠿⣿⣷⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠘⠁⠞⣹⠟⣿⣿⡿⡼⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⠀⣼⡟⠡⠞⣹⢿⣿⣿⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⢩⡖⠶⢾⣿⣷⣸⣷⣄⠈⣇⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠄⠀⠈⠙⣖⠋⠁⠀⠀⠀⠀⠀⠀⣸⠟⢿⡷⣿⣿⣿⢿⡟⠀⠉⢳⡖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠒⠲⡖⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠘⠁⣛⣉⣵⠟⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⢒⣒⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⢒⣒⣲⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣵⣖⣚⠉⠙⠛⢿⣿⡛⢿⡷⣄⠀⠀⠀⠀⠀⠀⢸⣷⣷⣖⣒⠉⠉⠛⠿⣯⣟⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠉⠉⢻⣶⣄⢦⡙⢿⣧⡙⠹⡆⠀⠀⠀⠀⠀⡸⠁⠀⠈⠉⠻⣷⣤⡠⣌⢻⣷⡌⠞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⡀⠀⠀⣼⣿⣿⣿⣿⡌⣿⣧⠀⡇⠀⠀⠀⠀⠀⣇⣀⣀⠀⠀⢴⣿⣿⣿⣿⣦⢻⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠸⠇⠀⠀⠀⢸⣿⠿⡛⣿⣿⣹⣿⢠⡇⠀⠀⠀⠀⠀⠉⠗⠀⠀⠀⢸⣿⠿⣛⢿⣿⣼⣿⡇⡟
⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠈⠁⢚⣱⣿⣿⣿⣿⡼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠾⠁⠘⣣⣿⣿⣿⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢵⣤⣀⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣠⠆⠀⠈⠉⢟⠉⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢀⡄⠀⠀⠉⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⡏⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⢞⠉⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠋⠁⠀⠀⠨⠛⢿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⢀⡴⡪⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⠟⡋⠁⠂⠁⠀⠀⠀⠀⠀⠂⢑⢦⡀
⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⣄
⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇
⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⣾⣿⣿⣿⡿⠃⠀⠠⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠄⠀⠂⠻⣿⣿⣿⣷
⠀⠀⣿⣿⠟⡁⠊⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠠⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⠀⠀⣟⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠿⠿⠿⠿⠿⠿⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠳⡁⠀⠀⠀⠀⠀⠀⠀⡀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⣶⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⢤⣄⣀⣀⣀⣀⣀⡤⠤⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⣉⣉⣉⣉⡽⠛⠓⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠯⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠤⣤⣀⡀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠈⠉⠳⢦⣀⣴⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣀⡴⠋⠁⠈⠉⠙⠛⠿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡿⣼⣶⣶⣦⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⣄⣀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡞⠋⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠸⡟⢿⣿⣿⣿⠋⠈⠉⠓⠲⠦⣤⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡌⠻⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠿⠟⠛⠦⣄⡀⠀⣀⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣳⣄⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢉⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣶⣤⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣷⣤⣀⣀⣀⣀⣠⡤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⡤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣾⣿⣿⣾⣿⣿⣿⣦⣀⡉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣻⣿⣿⣿⡜⠿⣿⣏⢿⠣⠿⣻⣿⡟⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⡇⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢄⣀⣀⣀⠤⠐⠲⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⠀⣏⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⣯⠣⠤⠀⠴⠃⠀⠈⠢⠄⠤⢚⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⡀⠀⠀⠀⣿⣧⠀⠀⢀⣤⣤⣤⣀⠀⢀⣾⣿⣧⠄⠀⠀⣠⣿⠛⠒⢰⠇
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠹⡄⠀⠈⣿⣿⣷⣿⠿⠿⠛⠿⠻⣿⣿⣿⣿⠁⠀⠀⣰⣿⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠙⢶⠦⣤⣿⣆⠀⢸⣿⣿⣿⣦⣰⣶⣦⣼⣿⣿⣿⣿⣄⠀⣰⡿⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⢹⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠁⠀⠙⠻⠟⠿⠿⠿⠛⠿⠋⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⠽⠟⠿⠿⢿⣿⣿⡿⠿⠿⠻⠯⡛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡧⣀⡀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣦⣀⠀⠀⠀⣀⡀⢼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠉⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠉⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠸⡀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⠛⠛⠛⢧⠀⠀⠀⠀⠀⠀⢀⠇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢣⡀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⡜⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣸⣿⣿⣷⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⣿⣧⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⢾⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⠶⠶⠶⠶⠛⠛⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣤⣶⠾⠿⢿⣿⣛⠓⠲⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣛⡛⠛⠶⠦⣄⡉⢳⡄⢀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣿⢻⣿⡛⠿⢭⣉⠒⢦⡈⢳⡄⢿⡌⡇⢻⡽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⡟⣆⠙⢿⣕⡲⢬⡻⣆⣷⣹⣷⣿⣿⣿⣾⣷⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣳⣜⢮⣙⢦⣜⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⡋⠁⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠉⠙⠛⠛⠋⠭⠒⠁⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠰⣿⣿⣿⣶⣤⣠⡾⣦⣤⣴⣾⣿⣿⡷⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠻⠿⠿⢿⡿⠟⣤⡜⠿⣿⠿⠿⠿⠃⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⣀⠀⠀⠀⣸⣿⣿⡀⠀⠀⢀⡀⠀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣧⠀⠀⠉⠁⠉⠁⠀⢰⡿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢿⡒⠒⠒⠲⠲⠒⠒⣺⢳⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠘⢯⣭⣤⡀⣠⣬⡽⠟⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⣹⣿⣿⣿⣿⣻⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣽⣿⣿⣿⣿⣿⣿⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣾⡿⠋⠉⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣠⡾⣿⠟⠋⠉⠓⠢⠄⣀⣀⣀⣸⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⣰⠋⢀⡆⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⢰⢠⡏⠀⢸⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠦⣄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⣀⣾⢸⠁⢠⣾⣷⣤⣀⡤⠚⠉⠀⠉⠛⢿⠋⠀⠀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⣤⣾⠉⣿⠈⠓⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣴⠟⠁⢸⣤⢿⣇⠠⠘⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣠⠞⠁⣠⠞⠋⢧⡀⠻⣆⠱⣼⡀⠀⠈⠙⠲⠤⣀⣤⣶⣿⣷⣦⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠞⠁⠀⠈⠀⠀⢀⣨⣷⣶⠿⠳⣬⡓⢤⣀⡀⠀⠀⢻⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡠⡏⠀⠀⣀⡤⠶⠛⠋⠉⠀⠀⠉⠀⠀⢉⣁⣈⣉⣙⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢇⠸⠔⠋⠁⠀⠀⣀⣀⣤⠴⠶⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣛⠋⠉⠉⠉⠉⠙⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠈⠙⠻⢶⣤⡀⠀⠀⠀⢸⡟⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣠⡿⠁⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣴⠿⢿⣄⠀⢀⣠⡶⠟⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⡶⠶⠟⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⢀⣻⣶⠟⠉⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠙⣷⡀⣀⣀⣀⣀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⢀⣠⡾⣿⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⢠⡾⠋⠀⢹⣇⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⣿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣿⡆⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣴⣾⣏⣠⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠉⠻⠿⠿⢿⣿⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠻⣿⣿⣶⣤⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠘⠛⠻⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣤⣀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⠀⠀⠀⢀⣠⣤⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣻⣯⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣇⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠟⠿⣿⠟⢻⠿⢿⢿⠻⣿⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣷⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⠶⠶⠶⠶⠶⠶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⡿⠁⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠘⣄⢀⣀⣀⣀⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣀⡤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⡰⣿
⠀⠀⠀⠀⢸⣿⣆⣠⠴⠋⠁⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠚⠁⣿
⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⡏
⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠛⠛⠋⠉⠉⢹⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣣⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠓⢦⣀⠀⢀⣠⣴⣾⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣌⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⣤⣀⣀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠷⠶⠾⠿⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠉⠉⠛⠻⢿⣿⣿⡿⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣉⣻⣿⣷⣬⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⠛⠛⢿⣿⡁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⡀⠈⠻⣿⣄⠀⣘⣿⣿⣦
⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⣤⣿⣿⣿⣿⣿⡿⣿⣷⡀
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⡿⠿⣿⣏⠁⠀⠹⣿⣷⡀
⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⠿⣿⡇⠀⠀⢻⣿⣆⣀⣀⠹⣿⣷
⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⡿⠟⠛⠉⠀⢀⢈⣿⡇⣠⣴⣿⢿⣿⡟⠿⣿⣿⣿
⠀⢸⡏⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⡜⢸⣿⣿⡿⠋⠀⠀⣿⣿⡄⠀⠹⣿
⠀⣾⡇⣀⣤⣶⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣾⣿⡟⠁⠀⠀⠀⢿⣿⣷⠀⠀⢻
⠀⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⣾⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⣸
⠀⢸⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿
⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣿⡿
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢀⣴⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠠⣄⢤⡰⣄⢦⡰⣄⠠⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣐⣰⣦⣀⣒⠦⣄⣀⣀⣀⡰
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢿⣿⠿⠿⠿⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠗⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⡟⠛⠂
⠀⠀⠀⠀⠀⠀⠘⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠛⠛⠛⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡀⡦⠤⣀⡤⠄⠀⢠
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣭⡻⠿⠗
⠀⠀⠀⠀⠀⠀⠹⡿⠋⠁⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠻⣿⣿⠿⠟⠛⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠉⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣷⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⣄⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣶⣾⣿⣶⣶⣼⣿⣷⢶⣶⡎
⠀⠀⠀⠀⠀⠀⢿⣿⠟⠉⠉⠙⠛⠛⠛⠛⠿⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠿⣿⣿⡿⠛⠉⠋⠀⠀⠀⠀⠀⢸⡿⠟⠁⠀⠀⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣄⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⡈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁



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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡾⠛⢁⣠⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⣿⣿⠟⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⠀⢀⡀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠁⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⣴⣶⣤⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠋⡉⠁⠸⢿⣿⣿⣿⣿⣷⣦⣄⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⣠⣾⣿⡏⢠⣶⣄⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⣼⣿⣿⣧⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡏⢠⣿⣿⣿⣿⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⢉⣩⣭⣵⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡰⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⡜⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⣸⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢸⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⢸⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⢻⡇
⠀⠀⢸⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠃
⠀⠀⠈⣶⡇⠀⠀⠀⠀⠈⠻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠰
⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⡄
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡐
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⣠⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⠿⠿⠀⣀⣀⠀⠀⠠⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⢰⠦⡀⣳⠚⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⣈⠉⠢⡀⠀⠀⠀⠀⠀⣀⠌⣹⣷⡼⠃⣠⡼⢐⠓⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣽⣍⢢⣙⠢⠤⠶⠦⢼⣁⠤⠀⣤⡶⢞⠩⢊⡤⡞⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢝⡛⢿⣓⣚⠓⣄⢿⡴⢚⢕⣈⠴⢞⠧⣄⠸⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡩⠓⢬⡋⢕⠿⣦⣉⢉⡉⠉⢆⠈⠢⡁⡱⣦⢈⣉⡵⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⢊⣵⣡⠴⢚⠭⠓⠬⡊⢝⠶⣄⣱⡜⢤⡀⢈⣮⣪⠳⣼⠀⢀⡧⠖⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⣦⣥⣴⠯⠒⠈⠀⠀⠀⠀⠈⠑⠬⡐⠍⠓⠤⣈⡥⠒⠺⠛⢃⡰⢋⠕⣡⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠦⠴⣍⠁⠀⣀⠤⠒⠒⠦⡄⠀⠀⠀⠈⠑⠠⣐⡿⢀⡶⣉⡽⡫⢒⣥⢾⣅⡀⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⣠⠞⠉⠑⠢⠤⠞⡡⠒⠒⠀⠭⣭⣥⠀⣰⠖⠀⡇⠀⠀⠀⡠⠤⠶⠞⢁⣾⢞⠉⡪⠔⠉⢦⡑⠘⣎
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠉⠒⢚⣡⣎⣁⣬⡵⠶⠇⠒⠒⠒⡊⠛⠛⠭⠙⠛⠉⠉⠀⠀⠀⣼⠠⠚⠉⡠⡫⢀⡕⢺⠀⠀⠀⠀⠙⠲⠟
⠀⠀⠀⠀⠀⠀⠀⣤⠒⢚⡛⠋⠭⠉⢙⣀⠨⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢣⠇⡠⠚⡡⡞⡟⡇⢨
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡠⢐⣿⡋⠔⠉⠀⡿⣷⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠚⢲⠿⠋⠀⠀⠀⠀⢓⡏⠎⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⢰⠈⡇⠀⡂⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡇⡰⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡠⠒⠒⠦⣄⣀⡠⠔⢉⠉⠙⢦⠀⠀⠀⠀⡀⠀⡏⠱⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⡠⠊⣀⣈⡑⠒⠊⠕⠋⠉⠀⢐⣲⣶⢬⣔⡏⣀⡜⠀⠀⠀⠀⡇⠀⣇⡔⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⣉⣉⣴⣯⡤⠴⢾⠛⠛⠂⣈⡉⠭⠅⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠉⠩⠁⢀⣒⠠⠭⠔⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡝⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⠀⢰⡞⣻⡿⠛⢿⡟⢳⡄⠀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⡏⣥⠆⣠⣤⡌⢲⡍⣻⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣼⡷⠿⣟⣷⣿⣄⣀⣰⣿⣾⣻⠿⢾⣧⡀⠀⠀⠀⢸⣿⣿⣿⡿⡃⢋⣀⠶⠈⡀⠻⣿⠟⠨⠁⠆⣠⡓⢙⢿⣿⣿⣿
⠀⠀⠀⠘⢾⠧⣴⣟⣿⣤⣽⣿⣭⣼⣿⣻⣤⠼⡷⠃⠀⠀⠀⢸⣿⣿⣿⣧⡁⣙⠛⠤⠀⠻⠂⠀⠺⠏⠀⠄⠛⣃⢨⣾⣿⣿⣿
⠀⠀⠀⠀⠈⠓⢹⠉⡇⣿⢸⠀⡇⣿⢸⠉⡏⠚⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣽⡆⣶⢸⢸⡇⣿⢸⢰⡇⣶⢰⣵⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⡇⣿⢸⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⢸⢸⡇⣿⢸⠸⡇⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⡇⣿⢸⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⢸⢸⡇⣿⢸⢸⡇⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠀⡇⡏⢸⠀⡇⣿⢸⠀⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣿⣾⣸⡇⣿⣸⣸⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠘⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡷⠺⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠲⠓⡔⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠉⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⣤⠄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣷⣤⣤⣤⣾⣷⣶⣶⣦⡀⠀⠀⠀⢠⣾⣿⣿⠁⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠸⣿⣿⣿⣦⡤⠤⣤⣾⣿⣿⣿⠀⠀⠀⠀⠀⢿⠿⣿⣶⣶⣶⣶⣾⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠙⠻⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠒⠂⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢀⣄⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣶⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠁⠀⠀⠀⠉⣻⣦⡄⠀⠀⠀⠀⢠⣴⣶⣿⣿⣧⣀⣀⣀⣼⣿⣿⣶⣦⡀⠀⢀⣴⣶⣾⣿⣧⡀⠀⠀⣠⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⡀⠀⣀⣴⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠛⠋⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠃⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠦⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠳⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⡀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢴⠄⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠈⠛⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣷⣤⣀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠒⠛⠛⠛⠃⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠗⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⣿⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⡄⢶⣦⣄⡀
⠀⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣷⣦⡀
⠀⠀⢀⣴⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣦
⠀⢠⣾⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷
⢀⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⢰⡀
⣸⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⢡⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⣏⢿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⢻⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⡟⣡⣿⣿⣿⡟⠈⢳⡀⠀⠀⠀⠀⠀⡾⠁⢹⡆⠀⠈⢙⣿⣿⣿⣿⣷⣤⡀
⠘⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⢸⣿⣿⣿⠏⣴⣿⣿⣿⡿⠁⠀⠈⢷⡀⠀⠀⠀⣼⠁⠀⠀⣷⠀⠀⣾⠁⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠘⣿⣿⣿⣷⣦⣍⡛⠿⢿⣿⡇⣿⣿⣿⡇⣼⣿⣿⠏⣼⣿⡿⠟⣫⡄⠀⠀⠀⠈⣷⠀⠀⢰⠇⠀⠀⠀⢹⡄⢰⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣶⢠⣬⣭⣍⢁⣍⣭⡉⣨⣭⣴⣶⡟⢹⠃⠀⠀⠀⠀⠸⡇⢀⡟⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⣸⣿⡻⣿⡿⠿⠿⠿⢟⣡⣿⣿⣿⠇⣾⣿⣿⡇⣿⣿⠟⢯⡀⢸⠀⠀⠀⠀⠀⠀⢻⣼⠃⠀⣀⣤⣶⣿⣿⡿⣿⠋
⠀⠀⠸⣿⣿⣌⠻⢿⡛⠿⣿⣿⣿⣿⠏⣼⣿⣿⠿⠟⠈⠁⠀⠈⢳⣼⠀⠀⠀⢀⣀⣤⣼⣿⣾⣿⢿⠟⣿⡇⣸⣷⠏
⠀⠀⠀⠈⠛⢿⣷⣶⣿⣦⣽⣆⣀⣀⣀⣀⣙⣿⣆⣀⣠⣤⣤⣤⣴⣿⣶⣾⠿⢿⡿⣿⡉⣽⠋⣸⢿⣤⡏⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢿⡟⣿⠿⣿⣻⠿⣟⢻⡿⣟⠛⠛⣻⠿⣍⠉⣩⠏⠙⢦⣠⠟⠀⢀⡿⢧⣴⠏⢠⡟⢧⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⣈⡿⢦⣈⡿⣦⡈⠻⣶⡃⠀⠈⣿⢯⡀⠀⣰⠛⠷⣤⡞⠁⣰⠛⢳⣾⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣧⠈⣿⢧⡀⠙⣾⣇⠙⢳⣾⠃⠀⢙⣾⣇⠀⣠⠟⠳⣶⡃⠀⡾⠉⠳⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣻⣇⠀⠙⣾⡃⠙⢳⣞⠉⠛⢦⣞⠁⠈⣻⢯⣀⣼⠃⠙⣾⣅⠀⣸⠏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢧⡼⠃⠙⣶⡏⠈⠳⣦⡟⠙⢳⣼⡃⠀⡽⠷⣄⣸⠃⠈⢳⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠛⣆⣴⠋⠙⢦⣴⠋⠙⢦⣾⠉⢛⣾⡁⠀⣸⠿⢦⡀⣼⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡼⠳⣄⢀⡼⠛⢦⣠⡟⠉⢳⣾⠁⠙⢳⣏⡀⠀⢹⡿⣄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⢙⣿⡁⠀⣰⠟⢦⣀⡞⠉⠳⢦⡟⠈⠙⢦⣾⠀⠈⠳⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⠏⠀⠻⣾⠃⠀⢀⡟⢧⡀⢀⡞⠙⢦⡀⢰⠏⠳⣄⢠⠏⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⢇⠀⢀⡜⠉⠣⣀⠞⠀⠀⠙⣾⡀⠀⠀⠈⡿⣄⠀⠀⢹⠦⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢳⡞⠀⠀⢠⠟⢦⡀⠀⡼⠁⠙⢦⡀⣸⠁⠈⠑⢤⡟⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠏⠹⣄⢠⠏⠀⠀⠳⣼⠃⠀⠀⠀⠙⣯⠀⠀⠀⢈⣷⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠙⣏⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⡟⢦⡀⠀⣸⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⡾⠻⣦⠀⠀⣰⠇⠻⣆⠀⠀⣸⠇⠀⠙⠛⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠
⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠈⢷⠀⠀⠀⡏
⠀⠀⠀⣴⡡⠤⠒⠒⠒⠒⠲⡾⠤⣄⣸⠃⠀⢀⡴⠦⡀
⠀⢀⠈⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡟⠙⠲⣾⡅⠀⠸⡀
⠀⠘⠀⠀⠀⠀⠀⣠⠶⠉⠀⠀⠀⣸⠃⠀⠀⡧⠈⠑⢄⢠
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⢷⠀⠀⠀⢻⡄
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⡆⠀⠀⢘
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⡇⠀⠀⡘
⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡧⠀⡰⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡧⠞⠁⠀⢀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠒⠄⣰⣋⡀⠀⡀⢀⡀⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠓⠲⠤⣤⣀⣀⣀⣀⣀⣀⣠⡤⠖⠊⢁⠞
⠀⠀⠀⠀⠀⠐⠒⠒⠲⠤⠤⠤⣴⣦⠤⠒⠒⠋⠀⠠⡄⠀⠈⠢⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⣀⣀⣀⣀⠀⠐⠄⠀⠀⠀⠙⠒⠠⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠂⠀⣀⡤⠖⠒⠊⠉⠉⠈⢹⡏⠁⠀⠀⠈⠉⠓⠲⡄
⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⣀⡤⢴⣷⠦⢤⣄⡀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡧⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⠄⡀
⠀⠀⠀⢆⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡃⠁
⠀⠀⠀⠀⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣴⣤⣶⢶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠗⠓⠛⠿⠿⠟⠋⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠠⠞⠁⠀⢀⢅⠀⣀⣀⣀⣙⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⣠⣴⡶⠞⠛⠃⠄
⠀⠀⠀⠀⠀⠀⣾⠻⠻⣿⠛⠛⢻⡾⣿⠏⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠒⠛⠫⢹⠷⠀⡀
⠀⠀⠀⠀⠀⠀⠹⣎⢁⢸⡆⠀⠀⠉⠡⣤⡜⠀⠀⠀⠀⠀⠀⠀⢀⣶⢸⠂⠀⠀⡀⢀⢀⡀⣾⡀⠀⣀⡤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣤⣤⣤⡶⢷⣟⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⠆⡾⢹⡇⠿⣿⢰⡿⡼⢰⡟⠊⠘⣿⡁
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠾⢿⣯⣫⡽⢳⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇⠍⠀⠓⠙⣁⣀⠋⢈⠍⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⡋⠀⡀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢷⡀⢰⡿⡈⢠⡶⡄⢰⠆⡄
⠀⠀⢚⢀⣴⣿⣇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣽⢿⡇⢾⣧⠇⣿⠔⠁
⠀⠀⢸⣿⣿⣿⣿⠀⠂⠠⠀⣼⣿⣿⣿⣿⣿⣿⡷⠀⠀⣤⠁⠀⠀⠀⠀⠀⠠⣋⡴⠏⠛⠈⠀⠀⠀⢸⡇
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠲⠷⠿⣿⣿⣿⣿⣿⠇⠀⠸⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⣴⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢀⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⢿⣿⣿⣿⡟
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⡀
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀
⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣤⠄
⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠑⠀⠀⠀⡀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠐⠂

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣤⣄⠀⠀⠀⠠⠤⠤⢄⣀⣀⣀⡀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣚
⠶⠿⢯⣍⡝⠯⣭⠴⠦⡴⠞⠶⠴⠿⠧⠦⢭⣠⣐⣒⣀⠠⠤⠉⢉⣍⠉⠉⡉⠛⠛⠻⠯⣭⡽⠷⠾⠏⠛⢉⡵⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠲⠷⢬⣉⡂⠶⢯⣉⣙⣿⡆⢾⢹⡇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⢦⣄⣙⡶⠿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣭⣒⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢿⡷





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣆⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣙⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢰⣦⠀⠀⠀⠀⠀⠀⢀⢶⠢⡀
⠀⠀⠀⠋⢹⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡜⢻⠀⠀⠀⠀⠀⢠⠂⠈⢧⠑⡄
⠀⠀⡀⠀⠈⢇⠀⢃⠀⠀⠀⠀⠀⠀⢰⡇⢀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡠⠊⠀⢸⠄⠀⠀⠀⢀⡞⠀⠀⠈⠓⢼
⠀⠀⠁⠀⠀⠈⠲⠸⠀⠀⠀⠀⠀⠀⡾⠉⠼⢁⡴⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠗⠊⢀⡬⠀⢸⠀⠀⠀⠀⢸⡇⠀⣾⡇⠀⢸
⠀⢠⠀⠀⢄⣀⠀⠀⡆⠀⠀⠀⠀⢀⡇⠀⠀⣜⠠⡀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠼⡖⠀⣿⠀⠀⠀⠀⢸⡇⠈⣝⢤⠀⠸⡀
⠀⢸⠀⢀⡺⣍⠀⠀⡇⠀⠀⠀⠀⠘⡇⠀⠀⠁⡰⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⣿⠀⠀⠀⠀⢸⡇⠈⢁⠃⠀⠰⡇
⠀⢸⠀⢐⣿⣗⠀⠀⡇⠀⠀⠀⠀⢀⣇⠀⠀⠀⠃⠀⢠⣧⠀⠀⠀⠀⠀⠀⣰⠀⠀⠐⠁⠀⣿⡆⠀⠀⠀⡾⢧⣀⡆⠀⠀⠸⢳
⠀⣸⠀⠀⣍⡁⠀⠀⣇⠀⠀⠀⠀⣿⠻⠛⠖⠲⠶⠚⠉⢻⠀⠀⠀⠀⠀⢸⠉⠒⠒⢒⣒⡋⢱⡇⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⣿
⢀⣸⠀⠀⠙⠃⠀⠀⣻⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⣠⢞⢫⠆⢸⡇⠀⠀⠀⡇⠀⢀⢜⣍⠀⠀⢹
⢸⡏⠻⠶⡶⠿⠴⠒⢻⡇⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⡴⠁⠠⠀⠰⡇⠀⠀⠀⡇⠀⢈⢞⠋⠀⠀⢸
⠘⡇⠀⠀⢤⠄⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⠀⠀⢸⠀⠀⠘⢁⠈⠀⠀⡇⠀⠀⠀⡇⠀⢡⠎⡝⠀⠀⢸
⠀⡇⠀⠠⢷⣶⠀⠀⠀⡇⠀⠀⣿⣿⠄⣀⣀⣀⣀⣀⠀⠈⢸⡇⠀⠀⢠⣄⠀⠀⠀⠆⠀⠀⢀⢿⡆⠀⢀⡇⠀⠀⠐⠀⠀⢀⡸
⠀⣧⡀⠙⠶⡦⠀⠀⢀⣧⠀⠀⣿⠈⠁⠀⠈⠉⢁⡴⠂⠀⢸⡇⠀⠀⣿⣿⣿⡿⢿⠿⢚⣓⣋⣿⡇⠀⢸⠉⠉⠉⠓⠚⠋⠉
⠈⠉⠉⠉⡉⠁⠈⠀⠀⢸⠀⠀⢹⠀⠀⠀⠀⣠⠎⢀⠔⠀⠈⠇⠀⠀⡿⠋⣩⢾⣵⣾⣿⣯⣷⣿⡇⠀⢸⡇⠀⠀⣰⢷⠃
⢀⠀⠀⢐⠪⡲⡆⠀⠀⢸⡀⠀⢸⠀⠀⣠⠞⠁⡠⠊⠀⠀⠀⠀⠀⢰⡿⣫⠔⣻⣿⣾⣿⣯⡽⣿⡇⠀⢸⠇⠀⣰⢃⡴⠆
⢻⠀⠀⠲⣕⣴⢭⠀⠀⢸⡇⠀⣿⠀⠈⠁⠀⠔⠁⢰⠀⠀⠀⠀⠀⢸⣧⠴⢋⡴⣛⣿⣿⡿⢟⣿⡃⠀⢸⠀⠀⢁⠞⠜
⠀⠀⠀⢘⡢⡭⣗⠀⠀⢸⠃⠀⠙⠀⠀⢀⡎⠀⢀⠌⠀⠀⠀⡀⠀⢸⣷⢚⣡⢖⣯⡻⣽⣞⣢⣿⠀⠀⢸⠂⠀⠎⣰⢋⠁
⠀⠀⠀⢸⣿⣦⠲⠀⠀⢨⡀⠀⡀⠀⠀⠈⠀⠀⠎⠀⠀⠀⠀⠗⠀⢸⡿⣫⠴⣿⣷⣿⣾⣫⣑⢿⠀⠀⣼⠀⠀⠐⠁⠈
⠣⡦⣤⣤⣍⣣⣄⣤⠤⠚⠁⠈⠻⢶⣠⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠘⠿⣷⣖⠋⢻⣫⣽⠥⠜⠋⠁⠀⠘⠶⣤⣤⣠⣤⣤⠴⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠐⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣀⣤⣤⣤⡤⠤⠤⠤⢤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣿⠉⠁⠀⠀⠀⢀⣂⣀⣀⣀⣈⣈⡉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠰⣏⣉⡉⢉⣭⣥⣤⡤⠤⠤⠤⠄⠒⠲⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡿⠤⠤⠤⠤⠤⠄⠒⠒⠒⠚⠛⢛⣚⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⣷⠒⠒⠒⠀⣉⣉⣉⣉⣉⣉⣩⣥⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠉⢓⣤⣀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠰⣏⣈⣉⣉⣉⠉⠭⠭⠤⠤⠤⠤⠤⠠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⠞⠀⠀⠉⠲⢤⣀⣀⠎⠀⠀⠀⠀⠀⠀⠀⢀⡿⠤⠤⠤⠄⠒⠒⠶⠒⠒⣒⣀⣂⣉⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠒⠒⠒⠒⡚⡋⢉⠉⠉⠉⠉⣉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠞⠛⠉⠉⠉⢈⡜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣟⠳⣶⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠱⠋⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⣀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠉⣳⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡾⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾
⠀⠀⠀⠀⢀⢾⡀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⢠⢫⡟⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹
⠀⠀⣰⣳⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃
⢀⣼⣿⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢸
⠾⢿⠋⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢰
⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⡆⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡜⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⢤⣀⣀⡀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡧⠞⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠉⠉⠁⠈⠓⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣭⣭⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣤⡶⠒⠒⠲⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⢗⣶⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢩⠎⣠⣶⣿⣿⣿⣿⣿⣿⠿⠒⠤⣀⡀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⣿⣾⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠈⠉⠒⠦⣤⣤⣤⣤⣷⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⡰⠋⠉⠉⠙⠛⠻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⡹⡄
⠀⠀⠀⠀⠀⠀⠀⢰⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠻⢿⣿⣿⣿⣿⡏⠈⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠒⠦⣤⣀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠈⠛⠿⣿⠁⠀⢿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣠⠏⣀⣤⣤⣤⣤⣬⣉⣓⣶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠤⠤⣀⣀⠀⢀⣾⣷⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣇⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣦⣤⣤⣤⣴⣾⡿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣆⣀⢀⣀⣴⣶⣹⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣼⣿⣿⢟⣟⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣦⣤⣤⣶⣾⣿⣿⣷⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢛⣻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠚⠋⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣷⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉

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

⠀⠀⢀⠤⠂⠀⠙⢏⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢄⡀
⠀⠴⠋⠉⠈⠁⠀⠀⠀⠀⢀⡷⡀⠀⠀⠀⢀⣾⣽⣿⣿⣽⣇⣻⠿⣿⣆⠀⠀⠀⠀⣠⡞⠥⠀⠤⠤⢵⡀⡉⠑⣄
⡌⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⢟⣽⣿⣹⣦⣝⣧⠀⠀⡸⠁⠀⠀⠀⠀⣠⠎⢈⠓⠮⡈⢆
⡇⠀⣀⡴⠊⠁⠀⠀⢰⠀⢰⠀⠀⠀⠀⣿⣿⣿⢟⣯⣾⣿⣿⡏⣿⣟⣿⣯⠀⠀⡇⠀⠀⣀⠴⠊⠁⠀⡸⠀⠀⠈⢾
⢣⡴⠁⠀⠀⠀⠀⢠⠏⠀⠀⡆⡰⠀⠀⢹⣟⣵⣿⣿⣿⣿⡿⣹⣿⣿⣿⡟⠀⠀⢧⢀⠞⠁⠀⠀⠀⢠⠇⠀⢠⠀⡼
⠀⠣⡀⠀⠀⠀⣠⠏⠀⠀⣼⠔⠁⠀⠀⠀⠹⣿⣿⣿⣿⡿⣳⣿⣿⣿⠟⠁⠀⠀⠈⢧⡀⠀⠀⠀⢀⠎⠀⠀⣈⡴⠁
⠀⠀⠈⠑⠢⠶⠥⠤⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣔⣁⣀⡠⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣦⣶⣴⣶⣶⣶⣶⣦⣤⣤⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⢿⣿⣿⣿⣿⡇⣿⣿⣿⡇⣿⣿⠿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣼⣎⣿⣿⣿⣧⣿⣿⣿⡇⣿⡇⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣠⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠛⠛⠛⠒⠓⠒⠒⠚⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠙⠋⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠄⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠐⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢰⡆⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣬⣥⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⣿⠁⠈⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠓⠚⢁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠛⠉⠉⠉⠉⠉⢛⡳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⣀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠉⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⢀⣾⣫⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⢠⣿⣧⣯⣼⣸⠽⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠈⠻⣾⢸⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠹⣾⢰⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣷⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢹⡇⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡇⢸⡽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡟⡇⡟⡷⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⡶⢟⣼⡟
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣥⣳⠻⢹⢹⢹⢻⢻⢻⢻⢹⣉⣧⠷⢋⣾⠁
⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⠶⢼⡼⠬⠼⠾⠛⠋⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⣌⠢⡂⢄⡀⠀⠈⣑⠢⠬⢍⣓⡢⠬⣍⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⢶⣬⣭⣲⣶⣬⣥⣶⡶⠿⠟⠋⠁

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

⠀⠀⣀⣤⣶⠿⠿⠿⢷⣦⣄⡀
⢀⣼⡿⠋⠀⠀⠀⠀⠀⠈⠛⢿⣆
⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⣿⡆⠀⠀⠚⠛⣿⣶⣄⠀⠀⠀⠀⢻⡇
⠘⢿⡄⠀⣠⠀⣿⠈⢿⡆⠀⠀⠀⢸⣿
⠀⠈⣷⡀⣿⠀⢻⡆⠈⣿⠀⠀⠀⠀⣿
⠀⠀⠘⣧⠘⣧⠘⣷⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠹⣆⠹⣇⢹⣇⠀⣷⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢻⡄⠹⣆⢻⡄⢿⣧⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢿⡀⠹⡄⢿⡌⢿⣷⣄⠀⠙⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠈⣷⠀⠹⡌⢿⡄⠉⠻⢷⣤⡀⠈⠙⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⡄⢿⡄⠀⠀⠈⠛⢦⣄⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢹⡈⢿⡄⠀⠀⠀⠀⠙⢷⡄⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⡈⢿⡄⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠈⢷⡈⢿⡀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠘⣧⠈⢿⡄⠀⠀⠀⢸⡏⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠸⣇⠀⢿⡄⠀⠀⣼⠇⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠹⡆⠈⢿⡄⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⡄⠈⢿⣄⣿⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢷⡀⠘⣿⣿⣆⠀⠀⠀⠛⠷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⣷⠀⠘⣿⣿⢷⣤⡀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠸⣧⠀⢹⣿⡄⠈⠛⢷⣄⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢹⣆⠀⢻⣿⡄⠀⠀⠙⣷⡀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⢻⡄⠈⢿⣿⡀⠀⠀⠘⣧⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢿⡀⠘⣿⣷⠀⠀⢀⣿⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠈⢷⡀⠸⣿⣧⠀⢸⡿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠘⣧⠀⠹⣿⣧⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠘⣧⠀⠹⣿⣿⡇⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠹⣆⠀⠹⣿⡇⠀⠀⠀⠻⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠹⣆⠀⠹⣷⡀⠀⠀⠀⠈⠻⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢹⡆⠀⠹⣷⡄⠀⠀⢀⡀⠀⠈⠉⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢻⡄⠀⠻⣿⣦⡀⠈⠁⠐⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⢻⡄⠀⠹⣿⢿⣄⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢻⡄⠀⠹⣆⠙⢷⣤⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠹⡄⠀⢹⣧⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠹⡄⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⢹⡄⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⣀⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠦⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠢⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠛⠛⠿⠟⣙⣛⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⣤⡇⢀⣡⣍⣤⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣿⡇⠉⡟⡟⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⡇⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⡇⠀⢸⡅
⠀⠀⠀⠀⠀⣴⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⢸⡄⣿⡇⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣸⠇⣿⡇⠀⡇⠀⣸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠀⠀⠁⠐⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⠤⣀⣤⣴⣶⣿⣿⠿⠿⡋⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠀⠉⠠⠀⠂⠈⠀⠀⡄⣀⣀⠠⢀⣀⣠⣤⣶⣾⣿⣷⢮⠻⠛⠉⠁⣀⣠⣤⣶⢾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⢀⣁⠀⣤⠀⠀⣓⣨⣥⣶⣾⣷⣾⡿⠿⠛⠛⠉⠁⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡴⠋⣽⣿⣷⣿⡿⠿⠿⠛⠋⠉⢁⣸⡄⣦⣶⣾⣿⣿⣿⣿⢹⣿⣿⣿⣿⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠃⠀⠻⠿⢿⣿⣷⣀⣤⣴⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡏⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⡀⠀⠀⠙⠿⣿⣿⣿⡿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠠⠤⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣭⣴⣶⣶⣶⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣛⡶⠤⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣭⡶⠶⠶⠶⠶⢶⣶⣤⣍⣙⠳⢦⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣛⠿⠿⠿⣿⣿⣶⣶⣮⣽⡻⢷⣦⡉⠳⣦⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣦⣭⣙⠻⢿⣿⣦⡈⡛⢦⡈⢳⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠲⠄⠀⠀⠉⠛⠿⣿⣿⣦⡹⣿⣷⡹⣆⢉⠂⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠀⠀⠀⠀⠀⢰⣌⠙⢿⣷⣿⣿⣷⢹⣧⠳⡄⡉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠹⣿⣿⣿⡌⣿⣇⢹⡜⡌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣿⡀⢿⣿⣿⠁⣿⣿⠀⣷⣿⡈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠧⢸⣿⡿⢸⣿⣿⠀⣿⣿⡇⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡿⢡⣿⣿⡿⣸⣿⡟⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣱⣿⣿⡿⣱⣿⠟⢡⣿⢏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢋⣾⡿⠃⣰⠟⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣱⣿⢫⡆⢈⣵⢢⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⣼⣿⢣⣿⢀⣾⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⣿⠀⢿⡏⢸⣿⣸⣿⠀⠁⠀⠛⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣮⡛⠘⣿⣿⣿⣇⠐⢄⣙⣚⣵⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢻⣷⡘⣿⣦⣌⠻⣿⣿⣦⣔⡲⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡙⠷⡈⠙⠿⣷⣦⡙⢿⡟⠿⢿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⡛⢀⣜⣿⣷⣄⠈⢓⣮⡙⢿⡌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⣤⣬⡾⢏⣿⣿⣷⠉⠿⣹⢢⡑⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠁⠀⠀⠀⠀⣠⣶⣿⣿⠿⠋⠀⠈⠴⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⣏⠁⠀⠀⠀⠀⠀⠀⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⠞⣀⢁⡤⣀⣤⡄⣠⣀⠄⡶⢣⣄⡄⠀⠀⣠⣤⢀⣄⡤⣠⣤⠞⠀⠀⣠⢋⣤⡤⠀⣤⢠⣞⡅⣠⢃⣤⣄⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉⠙⠂⠙⠘⠊⠛⠛⠘⠃⠙⡿⠃⠀⠈⠃⠛⠙⠙⠃⠙⠙⠂⠀⢸⡻⠉⠊⠓⠛⠋⠉⠙⠉⠓⠙⠊⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣭⣵⣶⣶⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣯⣽⣟⣛⡶⠄⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣋⣥⡷⠶⠶⠶⠶⣶⣶⣬⣍⣛⠳⢦⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣛⠿⠿⠿⣿⣿⣷⣶⣮⣽⡻⢷⣦⡉⠻⣆⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣶⣤⣍⣙⠻⢿⣿⣦⡈⡛⢦⡈⢳⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠲⠄⠀⠀⠉⠛⠿⣿⣷⣦⡹⣿⣷⡹⣆⢉⠂⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⢰⣌⠙⢿⣿⣿⣿⣷⢹⣧⠳⡄⡉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠻⣿⣿⣿⡄⣿⣇⢻⡜⡌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣷⡀⢻⣿⣿⠁⣿⣿⠀⣷⣻⡈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠧⢸⣿⡟⢰⣿⣿⠀⣿⣿⠇⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⢡⣿⣿⡿⣸⣿⡿⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣱⣿⣿⡿⣱⣿⠟⢁⣾⢟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢋⣾⡿⠋⣰⠟⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣱⣿⢫⡄⢈⣵⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣼⣿⢣⣿⢀⣾⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⡏⢸⣿⣸⣿⠀⠁⠀⠛⡢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣮⡻⠘⣿⣿⣿⣇⠀⢄⣚⣚⣡⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢿⣧⠘⣿⣦⣈⠻⣿⣿⣦⣔⡺⣭⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⣦⡙⢷⡈⠛⢿⣷⣤⠙⢿⡿⠿⢿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡛⢀⢜⣿⣷⣄⠈⢒⣮⡙⢷⡌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⠀⠀⣥⣭⣾⢟⣿⣿⣷⠈⠿⣹⢠⡃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⣠⣴⣿⣿⠿⠋⠀⠈⠥⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣟⠉⠀⠀⠀⠀⠀⠀⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠞⣀⢁⡤⢀⣤⡄⣠⢀⠄⡴⢆⣄⡄⠀⠀⣠⣤⢀⣀⡤⢀⣤⠞⠀⠀⣠⠋⣠⣤⠀⣤⢀⣞⡁⣠⢂⣤⡄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠋⠙⠂⠛⠘⠃⠛⠛⠘⠃⠙⡿⠃⠀⠈⠃⠛⠙⠙⠃⠛⠙⠂⠀⢸⡻⠚⠊⠓⠛⠊⠙⠙⠘⠓⠙⠚⠙⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⡷⣿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣿⣷⣶⣶⣶⣾⣷⡾⠷⠾⠶⠾⠿⣾⡶⠿⡶⠿⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣿⡿⠏⠀⠀⢀⣤⣶⣷⣦⡀⠀⠀⠀⠀⣼⢄⣤⡄
⠀⠀⠒⠒⠒⢒⡖⠒⠒⠒⠒⠚⠛⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⠀⣠⠖⢁⡴⠚⠁⠀⠀⢀⡴⢟⣥⡾⢿⣿⡷⠀⠀⠀⣼⣿⣾⠟
⠀⢤⣤⣤⣴⣿⣷⠀⠀⠀⠀⠀⠀⠈⡿⠟⠉⠉⠃⠐⠒⠛⠁⠀⠀⡰⢁⡾⠋⠀⠀⠀⢀⣴⡫⢖⠵⣫⡶⠟⠁⠀⠀⠀⣰⣿⡿⠋⠀⣠⠜
⠀⠈⣿⣿⣿⣿⣿⣿⣷⣆⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡔⡴⠋⠀⠀⠀⠀⣰⡿⢋⣴⣵⠟⠉⠀⠀⠀⠀⠀⣴⣿⠟⠀⢀⣼⠃
⠀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡀⠀⠀⣼⠞⠁⠀⠀⠀⢀⣾⣿⣶⠟⠋⠀⠀⠀⠀⠀⠀⢀⣾⡿⠉⠀⢀⣾⠃
⠀⠉⠀⠀⠉⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣇⣀⠰⠋⠀⠀⠀⠀⠀⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⢀⣾⡏
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠴⠛⠁⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⢹⡆⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⣶⡀
⠀⠀⠀⠀⠀⠤⢀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⢸⡾⡟⣇⠀⠀⢀⣰⣖⣻⣭⣷⣽⣿⣵⣯⣽⣓⣶⣀⠀⠀⣠⣤⣤⣤⡀⠀⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⣀⠀⠀⠀⠀⠀⠀⠈⣷⣷⣿⠀⠰⣏⣶⣶⣿⣿⣿⣿⣿⢿⣿⣿⣷⣶⣎⡷⠸⣿⢿⣿⢻⡇⠀⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⢸⣼⣼⡆⠀⠀⢹⡇⢻⡀⣿⠸⣿⢸⡇⣸⠁⣿⠁⠀⠀⣿⢸⡿⣸⠃⢸⡏⣿⡸⡇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⢸⣧⢸⡇⣿⠀⣿⢸⡇⣿⢀⣿⠀⠀⠀⢸⣾⣧⡟⠀⣾⢹⣿⡇⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⣿⢸⡇⣿⠀⣿⢸⠇⣿⢸⡇⠀⠀⠀⠀⣿⣿⠇⢠⣿⣸⣾⣇⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠟⠂⠀⠀⢻⠸⡇⢸⠀⣿⢸⠀⡿⢸⠇⠀⠀⠀⠀⢹⡿⠀⠸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠘⠀⠃⠘⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣤⣤⣀⣻⣿⣿⣿⣿⣃⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣇⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠋⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡭⠿⠿⠖⠒⠂⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⣀⣤⡶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⠀⠀⢀⣠⣤⣤⣶⠶⠾⠿⠿⠿⠿⠽⣿⣿⣤⣀⡀
⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠛⠉⠙⠃⠉⠛⠉⠙⠋⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⡀⢈⣀⠉⡰⠀⠀⠀⠀⠀⢂⠤⠤⠤⠠⠦⠴⠀⠀⠀⠀⡀⠠⠌⠄⠙⢷⣄
⠀⠀⠘⠖⠖⠒⠲⠒⠢⠀⠀⠀⡀⠁⠀⠀⠀⢀⠀⠀⠀⡤⡉⠀⠀⠀⠀⠀⠀⠆⠀⠀⠸⡅⠀⠀⠁⠀⠫⠌
⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠰⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣀⣀⣀⣠⣤⢀⠂⠀⠀⠀⢀⡀⢉⡅⡉⡱⠀⠀⠀⠀⢀⡆⠀⠀⢸⣿⣧
⠀⠀⣾⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⡏⠀⠀⠀⠈⠐⠀⠀⠀⠀⠻⠋⠉⠉⠉⠉⡆⠀⠀⠀⢸⣤⠄⠤⠦⠂⠏
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣶⣶⣶⣶⣶⡦⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠲⠶⠶⠶⠶⠶⠶⠶⢒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⡏⣼⡇⣾⣷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⡇⣿⣇⣭⣝⣛⡿⠿⢿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⣿⣿⣿⣿⣿⡇⣿⢹⣿⣿⢿⣿⣿⣶⣶⣮⣭⣙⣛⠿⠿⠿⣛⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⡇⣿⢸⣿⡿⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⣷⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⡇⣿⢸⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡿⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⡇⡿⠰⠮⠭⢭⣛⣛⣛⠿⠶⠶⣦⣤⣼⡇⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠛⠛⠛⠛⠛⠁⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠒⠂⠛⠻⠛⠉

⣿⢿⠿⢿⢿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⣻⣭⣭⣛⠷⡀⠀⠀⠀⠀⠀⢀⣤⣭⣝⡻⠿⢗⡀⠀⠀⠀⠀⠀⢠⣜⣻⠿⢿⣛⣤⡀
⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢷⣝⡿⣟⣻⣭⠁⠀⠀⠀⠀⠀⠘⠿⠿⣛⣭⣽⡭⠃⠀⠀⠀⠀⠀⠘⢟⣯⣭⣭⣝⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉


⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣖⣲⠦⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣯⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢇⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠸⣿⣿⣞⢿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣷⣾⣭⣭⣽⣾⡇⠀⠀⠀⠀⠀⢙⡿⢿⣛⣭⣽⣭⠃
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⠿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⢀⡾⣻⣽⣶⣿⣿⡄⠀⠀⠀⠀⠀⢠⡞⣵⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢠⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢸⡷⣿⣿⣿⣿⢧⡿⠀⠀⠀⠀⠀⢸⣇⢿⣿⣿⣿⢿⡃⠀⠀⠀⠀⠀⣿⣞⡿⠿⢿⣛⣭⡅
⠀⠀⠀⠀⠀⠈⠳⣝⣻⣟⣫⡿⠃⠀⠀⠀⠀⠀⠈⢿⣷⣾⣿⣶⡿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣤⣄⣀⣀⣀⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡸⠤⠤⠀⠀⠀⠒⠒⣒⣒⣚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⣹⠉⠉⠉⠥⠤⠄⠀⠀⠀⢚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⡀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣹⠀⠠⠤⠤⠄⠀⠉⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡜⠑⠢⢄⡀⢀⠎⠀⠀⠀⠀⠀⢱⣂⣀⡀⠈⠉⠩⠭⠤⠤⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡜⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⣱⡄⠒⣒⢒⠂⠀⠀⠉⡉⣩⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠞⠋⠁⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⢫⠙⠲⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢈⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡗⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠟⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡌⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠤⠤⢀⣀⣇⣀⣀⣀⣀⣀⡀⠠⠴⠖⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⢉⣥⠞⠋⠁⠉⠉⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⣷⣄⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠖⠒⠒⢀⣄⣀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠙⠛⠷⠶⠶⣶⣿⠷⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣷⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣴⠾⠟⠛⠉⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣶⠟⠋⠉⠀⠀⣿⡇⠀⠀⠀⠉⠙⠻⢷⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡇⢀⣴⠟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⡀⠀⠀⠀⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣆⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⣄⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣯⣷⣦⣄⠀⠈⢿⣆⠀⢀⠀⠀⣀⣀⠤⣀⡠⣀⣠⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣽⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠟⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠹⠏⠙⠛⠛⠛⠟⠛⠛⠛⠛⠉⠙⠏⠙⠫

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠀⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠀⣴⣢⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⠀⠀⠉⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⡇⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡻⣄⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣝⢦⡀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⡄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢹⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠙⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣷⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⣛⣭⢤⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⣀⠴⠚⠁⢀⡰⠃⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⢉⡠⠀⠀⠀⠀⣠⡴⠞⣉⡴⠚⠉⣳⠀⠹⣄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡠⠖⠋⣁⠠⠔⠊⠁⠀⢀⡤⠖⢋⣡⠴⠊⠁⣀⡴⠚⠁⢀⣀⡿⠀⠀⠀⣸⣧⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠊⠁⠠⠚⠉⠀⢀⣀⡤⠶⢛⣡⠴⠞⠋⢁⡤⠖⠋⠁⣠⣴⣾⡿⡋⠑⠢⠤⡶⠿⢻⣟⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⣄⣀⠤⠖⠋⠉⣀⠴⠚⠉⢀⣠⠔⠊⠁⣀⣤⣶⣿⡿⡋⠉⠇⠈⠉⠀⢀⡠⠴⢻⡟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡀⠖⠋⠉⢀⣠⠴⠚⢉⣀⣤⣶⣿⣿⡿⠋⠙⠏⠆⠀⠀⠀⠀⠠⠐⠈⣰⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⠒⠒⠋⠁⣀⣴⣾⣿⣿⣿⡿⢿⣏⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠎⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣶⣶⣾⣿⣿⣿⣿⡿⡟⠻⡀⠀⠹⣕⠂⠑⠀⠈⠂⠤⣀⣀⣠⠊⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠛⠉⠳⡌⢢⡉⠂⠀⢀⡁⠀⠀⠀⡢⣄⣲⠞⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣏⠻⣟⣆⠁⠀⠙⢆⠈⠀⠀⠀⠙⢦⣠⣑⢪⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢦⡙⠮⠧⠀⠀⡀⠀⠀⠀⠀⡀⢄⣩⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⢄⠀⠠⡀⠈⠑⢄⣀⡰⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⢤⣌⣠⡄⠐⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠉⠛⠿⢿⣶⣶⣶⣶⣿⠿⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢧⠀⠀⣀⣤⣤⣀⣈⣉⣉⣁⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠸⣿⡉⠉⠛⠛⠛⠛⠛⣿⠃⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⢷⣤⣀⣀⣀⣠⡾⠏⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢀⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⡿⠀⠀⠠⠤⢀⣀⣐⡒⣒⡺⠿⠿⠿⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⢛⣉⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⡇⣴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡈⢧⠀⠀⣠⣤⠀⠀⠀⠀⢠⠞⠻⡄⠀⠀⠀⢠⢤⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠂⢰⠃⠈⣇⠀⠀⠀⡏⠀⠀⣧⠀⠀⢀⡏⠀⡇⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣾⣦⣤⣿⡀⢀⣸⣅⠀⠀⢸⣄⣀⣸⣧⣴⣧⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠻⠿⠿⠛⠛⠿⣶⡿⠿⠿⠟⠋⠉⠛⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠂⠀⠀⠀⠀⣀⣴⡾⠛⠉⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠁⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⣠⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠋⠀⢀⣴⣿⣿⣿⣄⠀⠀⢀⣠⣾⣿⣿⠀⠀⠈⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⢿⣿⣿⣧⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠿⠛⠛⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣁⣠⣤⣶⣶⠖⠲⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠁⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠊⠉⠀⠀⠀⠱⡄⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠁⢀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣰⣟⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⡀⠀⠀⢀⡼⢦⡀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⠉⠀⠈⢳⡀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠸⠿⠿⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⢀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢳⠀⢸⣿⣿⣦⣀⣀⣴⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⢸⠀⠀⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⡟⢏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠛⠋⠉⠀⠀⠙⢦⣀⡜⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡘⢄⣀⣀⠀⢀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣀⣴⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⠞⠁⠐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡷⠤⠔⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣲⣯⣭⣩⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣠⠖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠛⢦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⣤⣤⣤⣤⣤⣤⡼⣤⠀⠀⢼⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⠉⠁⠈⠉⠉⠉⢹⠀⠀⣿⠉⠉⠉⠈⠉⠉⠉⠉⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⣤⣤⣤⣤⣤⣤⠞⠀⠀⠹⢤⣤⣤⣤⣤⣤⣤⣤⠼⠃



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

⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⢠⠀⣶⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠹⡧⠞⠈⢣⣸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠙⢶⠋⠉⢻⠀⣸⠖⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣳⣤⠞⠉⠙⡆⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠘⣆⣀⣠⠏⠁⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠶⡏⠈⠀⣗⢦⠒⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⣤⣄⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢶⡉⠢⣑⡌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡴⠋⢀⡴⠛⢽⣉⣡⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠛⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣧⠞⠉⠉⣦⢀⣩⠞⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣴⠚⠛⢷⠞⠉⠙⢦⣠⠴⠿⣭⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⠇⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⢸⡄⠀⠀⣸⣦⣀⣀⣸⠇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⡴⢋⡽⠃⠈⢹⡉⠁⠈⢻⠉
⢠⡶⠛⠋⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣴⡫⠔⢉⣷⣀⣀⣸⢧⣀⣠⠼
⣿⠀⠀⠀⢰⣶⠦⣄⡈⠳⣴⠖⢶⡀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠘⠉⣤⠾⠋⠀⠀⠉
⢻⡀⠀⢾⣦⣈⡧⠀⢙⣶⣥⠀⠀⢳⡞⠁⠀⠈⣧⠀⠀⠀⠀⠀⢸⣃⣀⣀⡤⠾
⠈⠳⣄⠈⠻⢯⣄⣰⠁⠀⠉⠳⣤⠟⠦⡄⢀⣤⠿⣄⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠈⠳⢤⣀⡈⠉⠳⢤⣀⣾⠁⠀⠀⢸⠉⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠙⠳⢦⡶⠋⠙⢄⣀⣠⠜⢤⣤⣠⡴⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣠⣏⠀⠀⠈⣷⢤⣉⠲⢄⡉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⠦⣤⡴⠷⣤⣈⡉⠓⠻⠆⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⣀⠀⠀⠈⢳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠋⠁⢱⠉⠙⡆⠀⢀⣀⣀⣀⣀⣴⣶⣶⣦⡄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⢃⠀⠘⡄⡔⠚⠊⠉⠉⢻⣭⣥⣶⣿⠟⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⢷⠀⠈⡄⠀⣳⣳⣶⣶⡖⠀⠀⡟⠉⠁
⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣷⣶⢿⡿⠷⠴⡿⠻⠏⠉⠡⠚⠓⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣄⠀⠀⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⡇⠀⠀⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣇⣇⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣘⣛⣛⣛⣛⣛⡀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⢀⣻⣻⣿⣟⣻⣃⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⢀⣤⡼⣦⠄
⠀⠀⠀⠀⠀⣿⢡⣴⡗
⠀⠀⠀⢀⣴⣾⡏⢹⣷⣄
⠀⠀⣠⣿⡥⡾⣇⠀⠀⣿⣧
⠀⢠⣿⣾⣎⡳⣟⠻⡟⢻⡾⣇⠀⠀⠀⠀⠀⢀
⠀⠈⢹⡏⠀⢹⢧⣀⣈⡟⣠⣼⡀
⠀⠀⠈⢷⠶⢷⣾⠛⠛⣟⠉⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠹⣿⣟⠀⣀⣼⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻⣷⡆
⠀⠀⠀⠀⠀⠹⠇⠀⠧⠌⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠻⣷⡄⠹⣿⣿⣿⠹⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣶⣿⡇⠀⠈⠻⣿⡄⢿⡟⠀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢈⠐⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣾⣿⣿⣦⡀⠀⠙⠃⠈⠀⠀⠙⣿⣿⣿⣿⣿⡇⠀⡀⢀
⠀⠀⠐⢀⠀⠂⢁⠀⠐⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣏⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⢹⠈
⠀⠀⠀⠐⠀⠁⢠⠀⠂⡀⠀⠀⠀⠀⠀⠀⢹⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⡟⠀⠀⢌⡀
⠀⠀⠈⠠⠀⠄⠠⠀⠁⠠⠐⠈⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣦⠀⣿
⠀⠀⠀⠀⠐⠂⠒⠀⢁⠠⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣾⣥
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⢿⣟⡟⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠸⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣤⣶⣶⣶⣶⠶⠶⠶⣶⣦⠄⠤⠤⠤⠤⠤⠤⠤⠤⠀⠴⠶⠶⠶⠶⢶⣶⣤⣤⣴⣿⣇⣀⣀
⠀⠀⢀⣀⠴⣶⣾⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢟⣉⠻⠿⣿⣵⡶⢦⣀⡀
⢀⡶⠞⣉⡷⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠈⠉⠷⣦⣉⠳⢶
⣿⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠈⠙⣦
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⣿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⣿⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡿
⠀⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠
⠀⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⠀⣴
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⣿⠀⣿
⠀⣿⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿
⠀⢻⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢀⠏⠀⡟
⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠀⣴
⠀⠀⣿⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿
⠀⠀⣿⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡿⠀⣿
⠀⠀⢹⡀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣼⠁⢠⠟
⠀⠀⠈⣧⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⠀⣼
⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣰⡏⠀⣿
⠀⠀⠀⠻⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠇⢀⠟
⠀⠀⠀⠀⣿⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣠⡟⠀⣾
⠀⠀⠀⠀⣿⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⠇⠀⡿
⠀⠀⠀⠀⠈⣷⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢀⣼⡏⠀⣼
⠀⠀⠀⠀⠀⢿⠀⠀⠈⠻⠷⠶⢦⣤⣀⣀⣤⣀⣨⣧⣀⣀⠀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠿⠿⠟⠉⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠛⠒⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡐⠒⠻⠟⠛⠋⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠛⠙⠛⠛⠛⠛⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠃⢀⡸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠁⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢁⣠⣤⠀⠀⠀⠀⢀⡀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢣⠀⠀⠁⠀⠄⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡇⠙⠀⠀⣴⣿⡶⠀⢠⢠⣴⣦⡇⠀⠀⠀⢠⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⣈⣳⠉⢡⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢡⡞⠀⢄⠀⠀⠀⠀⠆⢀⡄⠀⢀⠆⠀⠀⠀⢠⠴⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡀⠈⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠰⣫⠥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⢀⡀⠃⠀⠐⢄⣀⣠⠁⠸⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⣲⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣪⠊⠀⠀⢱⠀⡤⠄⠒⠦⠖⢻⠚⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠈⠂⢄⠀⣀⣀⠀⢀⠄⣰⠀⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⠀⠀⠀⠀⢡⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⣰⠋⢬⠐⢊⠙⣦⠶⣂⠐⠉⠖⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠄⠀⠠⠀⣰⠇⠀⢈⡭⣤⠏⢀⠊⡍⣿⡷⠹⡆⡐⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⡨⠖⠊⠀⢀⣄⡐⢻⢏⡜⣡⠎⠀⣿⣇⠀⠙⢮⡀⠬⠇⠂
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⡀⢀⠄⠈⡀⠔⢦⣀⡠⠔⠊⠁⠀⠀⢀⡔⠃⠀⠀⢽⢾⠊⠱⢂⠀⡟⢻⠀⢀⡒⠋⠃
⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠐⡀⣠⡊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⡺⣼⡀⢸⣿⠃⠃⠘⠰⣿⠏⠀⠀⣿⠞⡆
⠀⠀⠀⠀⠀⢘⣿⡀⠀⠀⢀⠌⠁⠙⢦⡀⠀⠀⢀⣠⡤⠊⠁⠀⠀⠀⠀⠀⠀⠐⠜⡇⠀⠀⠀⠐⡁⠀⠀⠀⠀⢀⣏⠇⠅
⠀⠀⠀⠀⠀⠀⣿⡇⡠⠐⠁⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣰⡷⠄⠀⠀⠀⠒⠀⠀⠀⡠⠢⠈⣧⡅
⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂⠐⠀⠀⠀⠀⠀⠀⠀⣴⣿⢸⡄⢺⠶⢖⠂⢒⡖⠀⢰⣤⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⡙⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠘⠈⠁⠍⣠⣤⠀⠀⠉⠟⠃⠀⣀⠀⠙⣜⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡸⠀⠂⠀⠀⠀⠀⠀⢀⠈⣀⠘⠉⠀⣴⡤⠀⠀⠘⠟⠂⠀⣈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣂⡀⣀⣀⣀⣀⣑⣂⣐⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠆⠘⠟⠀⠀⠀⠈⠁⠀⣴⡤⠀⠀⠈⠟⠋⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣶⡿⡫⡠⡦⣮⣲⣢⣶⣤⣬⣯⣖⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⠙⢡⣾⠃⠀⠀⠀⠀⠈⠠⠪⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠋⠁⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⢵⢟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢑⢟⢗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢾⣯⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⢑⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⢠⣟⣽⣷⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡟⠀⠀⠀⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢐⣽⣿⡿⠈⠿⣦⠀⠀⠀⠀⣿⡄⣀⣠⣴⣿⡷⠶⠟⠛⠛⠉⠉⠉⢽⣕⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢱⣿⣿⢑⢀⣬⢻⣧⣀⣠⣴⣿⡟⠛⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢁⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣵⣿⣟⠉⠀⢸⣧⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣵⣻⣿⠀⠀⠀⣿⡄⠀⠀⠀⠙⢿⣦⡀⠀⠀⣀⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣐⠀⢻⣷⠀⠀⠀⠀⠈⡹⢿⣶⣴⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⢴⣴⣷⢝⣵⢟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣜⣿⣇⣽⣷⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢜⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣠⠖⠋⠁⠀⠀⠀⠈⠉⠙⠛⢿⣿⡿⠟⠋⠉⠀⠀⠀⠈⠑⠢⣄⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⣀⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣮⣧⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠒⠶⠤⣤⣀⣀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⢀⣀⣠⡤⠴⠖⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡄
⠀⠀⠀⡟⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⡇⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃
⠀⠀⠀⣧⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢻⡀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠁
⠀⠀⠀⠀⠈⢷⡀⠀⢹⣤⣀⣀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⣀⣠⣴⡃⠀⢠⠃
⠀⠀⠀⠀⠀⠈⢷⡀⢸⣿⣿⣿⣿⣿⣶⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⣾⣿⣿⣿⣿⡇⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠻⣌⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠟⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠙⠿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⣤⣀⣀⣀⡀⠀⠀⢀⣀⣀⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⣶⣖⢳⡶⢶⢶⠶⢶⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⡤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠞⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠻⢦⣤⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠹⣦⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣴⠋⠈⠻⣦
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⣧⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡼⠃⠀⠀⠀⠘⢧⡀
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⣧⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢠⡟⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢳⡄⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣀⣴⠟⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠻⣆⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣦⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣴⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⢦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⣎⠉⠙⠢⣼⠀⠀⠀⠀⠀⢼⡠⢖⠉⢹⠀⠀⠀⢠⣼⣿⣟⠛⢿⣿⣿⡿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⡇⠰⠿⠿⠶⠀⠀⠀⠀⠀⠰⠶⣭⡷⠸⡄⠀⢀⣿⣯⣤⣾⣿⣿⣿⣿⣷⣴⣿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⣷⡆⣴⣦⣸⣿⢸⣿⢰⣿⡆⣾⡆⣴⡆⡇⠀⠘⢿⣟⣿⣿⣹⣟⣿⣿⣿⣿⣿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠙⣧⡟⢻⡟⢻⢼⢻⢸⠛⣧⠛⡇⡟⡃⡇⠀⠠⠺⠋⠛⠜⠡⢋⡼⣫⠟⢹⠿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⠁⠸⠃⠸⠏⠘⠞⠀⠿⠀⠻⠁⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠋⣠⠟⠲⠄

⠀⠀⣠⣤⡄⠀⠀⣀⠤⠤⠀⠀⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⠛⠁⣠⠔⠋⠠⠀⠠⠀⠀⠀⡀⠠⠈⠑⢤⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣼⠟⠀⠀⡀⣀⠔⠋⣱⠀⠀⠲⠤⡄
⠀⠀⣠⠞⠁⡀⠃⠤⠀⠀⠀⠀⠀⠃⠀⢀⠄⠀⠙⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠔⠈⠀⠀⠐⠀⠄
⠀⡜⠁⠠⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣀⣈⠍⠀⠀⠀⠃⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠢⡁⡀
⡘⣡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠊⠢
⣿⠀⠂⢀⠂⠀⠀⣀⣄⠀⠀⠀⢀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⢄⠑
⠈⠛⠛⠛⡙⠃⠀⠃⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠄⠀⠐
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠻⠆⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢄⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔
⠘⠆⠀⠘⡄⠀⠀⢰⣮⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠄⠀⠘⠀⠀⠈⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠀⠀⠀⠀⠀⢄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢣⠀⠀⠁

⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⢇⠀⠀⠀⢰⣀⡀⠀⠀⡄⡀⠀⠀⠀⠠
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠉⠓⢀⠤⠞⢠⠐⠀⡀⠈⠀⠉⠉⠀⠈⠐
⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠸⡀⠀⠘⢤⡤⠄⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⢀⣾⠂⠀⠀⠀⠀⠘⣇⠠⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠠⣄⣀⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠁⠀⢸⠇⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠄⢢⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⢀⠼⡋⠉⠳⢹⢰⠀⠀⠀⠀⠀⡠⠚⢉⠄⠀⡀⢀⠢⡀
⠀⢰⡉⠉⠁⡔⠀⠀⠀⠀⠀⣀⠀⠀⢠⢯⠏⠀⠀⠀⠀⠯⠀⠀⠀⠀⣸⠁⡴⠁⠔⠚⡁⠀⢧⢱
⠀⢸⡁⠰⣂⡼⠀⠃⠀⠂⠀⡡⠃⠀⠸⡀⠐⠈⠁⠀⠠⠐⠆⠀⠀⠀⡇⢰⠁⠄⠀⠀⠀⠈⢸⠈⡇
⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢄⠀⠀⠀⠀⠁⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢊⠜

⠀⠀⢀⠤⣤⠀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠠⠠⠦
⠀⠀⡟⠚⠋⠤⣴⠖⠊⠀⠀⠀⠀⢀⡠⠤⠖⠋⢁⡔⡡⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⠐
⠀⠀⠈⠂⠉⠋⠁⠀⠤⠠⠀⠀⠐⠁⠀⣀⣤⠴⠋⣰⠃⠄⠀⠆⢠⡀⠀⠀⠀⠀⣀⠀⠐⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠰⠉⠀⢀⣀⠴⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡄⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠁⠀⠡⠄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠰⢀⠀⡀⠀⢀⣰⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡀⢆
⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⢁⡇⠀⠀⠀⠀⠀⠈⠂⠠⠄⠠⠌⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⡇⢨⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠖⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢀⠜⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢠⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⢰⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢧
⠀⠀⢠⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢇
⠀⠀⡜⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠘⡄
⠀⢀⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠃
⠀⢸⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⢷⡀⠀⠀⠀⠘⣧⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡌⢳⣄⠀⠀⠀⢻⡆⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢡⠀⠙⠷⣶⣤⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣀⣤⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢈⣷⣤⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞
⠀⠀⠀⠀⠘⢤⣤⡶⠶⠟⠛⠋⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠞⢋⠞
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠾⠛⠛⠋⠉⠉⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣴⠟⠛⠿⣍⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣾⠃⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⠀⠀⠈⠙⠲⠤⣄⡀⣀⣀⠤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣛⣟⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣲⣽⣾⣿⣏⠁⠀⠀⠀⠀⣉⠉⢙⡛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣪⡵⠛⠙⡛⠣⣿⠛⠀⠀⠀⠸⣿⣶⣿⣷⣿⣷⣶⣍⡻⣷⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⡟⠀⡐⢀⠻⣄⠻⢰⠀⠀⠀⠀⠈⠙⢿⣿⣟⠻⣿⡝⠻⣮⣿⣿⣦⡀
⠀⠀⠀⣠⢟⡽⠛⣩⣿⣿⣿⣿⣿⡁⠀⢃⢿⡀⠈⠳⢄⣧⡀⠀⠀⠀⠀⠀⠙⣿⣧⠘⢿⡄⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠃⡾⢁⣾⠟⠙⠉⣩⣾⣿⣇⢀⠀⠘⢷⣤⣇⢠⠈⠙⠷⣦⡐⣄⢢⡀⠈⢿⣧⠀⠻⡀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⡇⢸⣧⢠⣴⠟⠁⠸⣟⣿⣦⣷⡀⠀⠙⠻⣿⣦⡘⣶⣄⠙⣾⣆⢳⡄⠘⣿⡇⠀⠀⠀⢀⠸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⠈⠻⣏⠻⣿⣶⣄⡀⠀⠉⠻⢮⣿⣷⣌⢿⡎⣿⡀⢸⣿⠀⢧⡀⠸⣆⢹⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⣿⣿⣶⣦⣄⡈⠙⢿⣾⣿⣼⣧⠀⣿⡇⢸⣧⠀⣿⡌⣷⣿⣷⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠶⢦⡄⠀⠀⠀⠠⠶⠛⣋⣉⣁⡀⠀⢉⠙⠿⣦⡀⠹⣿⣿⣿⠀⢸⣧⢸⣿⡄⢹⣧⣿⢸⣿⣷⡡
⠀⠀⠀⠀⠀⠀⠀⠰⠞⢳⣦⡀⠀⠀⠀⠀⠟⠹⣿⡟⡿⠟⠋⠀⠀⠘⣷⡄⢹⣿⣿⠆⢸⣿⣼⣿⡇⣸⢿⣿⢸⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢘⣿⣿⠀⢸⡟⣿⠿⣷⡿⠘⣿⣸⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⢸⣿⡟⠀⣼⠃⡿⠀⣿⠇⠀⠃⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣸⣿⣾⣿⠇⢰⡟⠀⠀⠀⠟⢸⠇⣼⣿⣿⣿⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣷⣿⢿⣿⡟⢀⣿⠃⡀⠀⡄⢀⣿⣰⢟⣿⣿⡏⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣼⠁⣼⠙⢏⣿⣿⠁⣼⡏⢠⠇⢰⡇⣼⣿⢋⣾⣿⣿⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠟⠃⠀⠀⢠⡏⠀⣿⠀⣼⣿⡇⣼⡿⢠⡟⢠⣿⣿⠟⣡⡾⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⢀⣿⡟⣸⣿⢣⣿⣱⣿⡿⢋⣴⠟⢡⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢸⣿⣧⣿⣣⣿⣿⠟⠋⣴⠟⠁⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠰⢸⣿⣿⣿⣿⡿⠟⠁⠀⠈⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢧⣿⣿⠟⠋⠀⠀⠀⠀⡠⢂⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⡄⡜⣿⡿⠀⣠⠆⠀⣠⠞⣡⡾⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⡼⣷⡸⣽⣷⣾⣧⢎⡾⢃⣴⠏⣴⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⣷⣽⣿⣿⡿⠋⣴⠟⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡜⠻⣿⣿⣷⠞⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⢱⣠⢈⣿⣿⣦⡾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣾⣿⣿⢠⣿⣿⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣷⣾⣿⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⢿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟

⠀⠀⠀⠀⠀⠀⢠⠀⡄⢠⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠀⠠⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠉⠔⠢⣄⠀⠀⠀⠀⢰⠊⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡠⠄⡤⢄⠀⠈⠆⠀⠀⠀⠘⢄⡀⠀⠀⠑⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠎⢀⠜⡠⠊⠃⠀⡆⠀⠀⠀⠀⠀⠈⠉⠑⢄⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢠⠃⡠⢂⠔⠀⢀⠇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⡀⠈⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⡜⢠⢃⠜⡠⠃⠀⡠⢃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⣀⠀⠉⠲⠂
⠀⠀⠀⠀⠀⡸⢀⡧⢂⠜⢀⠤⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦
⠀⠀⠀⠀⢠⠃⠸⠵⠗⢊⠥⠊⠁
⠀⠀⠀⠀⢸⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠒⢢⡊⠑⠤⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠄⠀⠀⠀⢀⣈⢅⣰⠒⠊⠈⠉⠓⢲⠁⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡣⠞⠁⠀⠀⠀⠀⠸⣀⠼⠀⠀⠀⠀⠀⠀⠀⢩⡎⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡤⠊⠀⠀⠀⠀⠀⠀⠀⢏⢀⠗⠀⠀⠀⠀⠀⠀⠀⢠⡟⠻⡄
⠀⠀⠀⠀⠀⢀⡠⠔⠊⣩⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠹⣀⠀⠀⠀⠀⠀⠀⡖⠁⡍⠁
⠀⠀⠀⠀⡔⠉⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⡓⠒⠤⠒⢜⡉⡐⠉⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠙⠒⠁⠈⠁

⠀⠀⠀⠀⢀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⣼⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇
⠀⠀⠀⠀⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇
⠀⠀⠀⠀⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇
⠀⠀⠀⠀⡇⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣷⣃
⠀⠀⢠⠀⠁⣿⣿⣿⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠉⠉⠉⠉⠉⠉⢑⡆
⠀⠀⢸⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣷⣾⣿⠀⢸⡇
⠀⠀⠘⡇⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠙⠛⠁⠀⢸⡇
⠀⠀⠀⣇⠀⢿⣿⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡜⡄
⠀⠀⠀⠀⠀⠀⣰⢋⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣱⡀
⠀⠀⠀⠀⠀⡼⢡⣿⣿⡿⠁⣤⣤⣤⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤⠸⣿⢿⠗⠃
⠀⠀⠀⠀⠀⢁⡿⢿⡿⠁⠀⣿⠷⠿⠀⠀⠀⠀⠀⠸⠿⣿⠿⠇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠈⢧
⠀⠀⠀⠀⢠⡟⠀⠀⠀⣤⠀⢹⣿⣿⣆⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠴⠿⠿⠿⠤⢀⠀⠘⣇
⠀⠀⠀⣠⠏⠀⠀⠀⠀⢿⠀⢸⣿⣿⣿⢸⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡇⠀⣤⣀⣤⠀⢸⠀⠀⠹⡀
⠀⠀⠐⠋⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⡇⡼⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣧⠀⠙⠿⠋⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠸⣿⣿⣇⡇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠲⢦⣄⣀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠈⠉⠛⢧⡀⠀⠀⠀⠀⠀⣶⣴⡿⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣤⠻⣦⠀⠀⢴⣹⣿⠟⠁⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⣽⠇⠀⣿⠀⠈⢷⣤⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢱⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⣼⠇⠀⠀⠀⠙⣿⠃⠀⢀⣀⣀⣤⣤⡤⠶⠶⠿⠛⠛⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠙⢿⣿⣦⠀⠀⢀⣠⡄⠚⠋⠀⠀⠀⢿⡄⠀⠀⠀⠀⡟⠀⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡴⠶⠒⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣤⣀⠀⠹⣿⣿⠾⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣇⣠⣤⣤⣶⠾⠋⠀⠀⢀⡴⠟⠁⠀⣠⡄⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⠿⣶⣼⡏⠀⠀⠀⠀⢠⣦⡄⠀⣤⡶⠟⠛⠁⠀⢀⡄⣿⣍⡼⠋⠁⠀⠀⣠⡾⠋⠀⣀⣤⣾⡿⠃⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠀⠘⢷⣿⣾⡿⣶⣄⡐⡄⣰⣾⠿⣮⣿⡁⠄⣤⣴⢞⣉⣀⣴⣿⣿⣥⣬⣥⣤⣠⣠⣴⡤⠀⠘⡇
⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⢷⣆⢀⣹⡇⠀⠈⠙⠿⠾⠟⠁⠀⠹⢿⣿⡿⠟⠛⠉⢉⠁⣠⣤⠀⠀⠀⠉⠛⠿⣿⣧⡀⠀⣼⠇
⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣷⣝⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⢀⣀⣠⣴⠾⠋⠁⠀⠀⢀⣀⠀⠀⠀⠈⠀⢰⡏
⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⣤⣴⠾⠖⠚⠛⠛⠓⠲⠿⠿⠿⠿⠷⠶⠶⠶⠤⣤⣤⣀⠀⠀⢀⣸⣿⠿⠛⠋⢉⣤⡴⠶⠖⠛⠛⠉⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢸⠃⠀⢀⣤⠾⠛⢥⣤⣤⣄⣀⣀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣴⣿⣥⣀⡀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇
⠀⠀⠀⣿⣀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠻⠿⣶⣤⣀⠀⠀⠀⠄⠀⠀⠀⠀⢿⡍⠉⠙⠻⣧⡀⠈⣿⠻⣦⡀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⣿⡿⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⣦⣰⡀⠀⠀⠀⠈⣷⣀⠀⠀⠈⢻⡄⢹⡆⠈⠛⢧⣤⠶⠞⠋⠁
⠀⠀⣾⠋⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣤⠀⠀⢸⡟⠛⠶⣤⡈⢿⡈⣷⠀⠀⠀⠙⢦⡀
⠀⠈⣿⡀⠀⠀⠀⠀⠀⣠⣐⣮⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢿⡀⠀⠈⠙⢿⣇⢸⡇⠀⠀⠀⢈⣷
⠀⠀⠈⠛⢦⣀⣀⣞⣛⣭⡽⠟⠛⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⣿⣦⡀⠀⠀⠙⢶⡇⠀⠀⣰⠿⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⣦⣦⣤⣀⠠⠀⠀⢠⣤⡀⢀⠀⠀⠀⠙⢾⣯⡻⣦⡀⠀⢈⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⠟⠉⠙⢷⣄⠀⠀⠀⠀⠉⠳⣎⢻⣦⣸⡇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣄⣀⠀⠙⢷⡜⢛⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡎⠁⠀⠀⠀⠀⠀⠀⠉⢑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠓⠒⠂⠤⢄⣀⣤⣤⣶⣾⣦⡀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠗⠢⢄⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠿⢿⣿⣿⣿⠃⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣣⣀⣀⡀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠉⢃⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⢀⠎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣷⣦⡤⠋⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠙⢿⣿⡿⠿⠛⠉⠁⠒⠠⠄⠄⣠⣾⣿⣿⣿⣿⣿⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣟⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⣘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠿⠟⣛⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣤⣤⣄⡠⠄⠚⣁⠠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠍⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⡛⠛⠭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⢶⠶⣗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⢸⠀⢸
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⢸⠀⢸
⠀⠀⠀⠐⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⡇⣿⡇⢸⠀⣿
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣶⣤⣀⠀⠀⣇⣿⡇⠸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣶⠏⠻⠃⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣰⣷⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⡇⠐⠀⠀⠐⠀⣀⢀⡀⠀⣀⣠⣤⣾⣷⣿⠿⠛⠋⣉⣡⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡇⣤⣠⣄⣀⣥⣴⣶⣿⡿⠟⠛⠋⣉⣠⣤⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡼⠁⢿⣿⣿⣏⢉⣁⣤⣴⣾⣼⣿⣿⣿⣿⢟⡿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠀⠀⢶⣿⣿⣿⣿⣿⣿⡿⡹⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠀⠀⠙⣿⠿⠟⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢹⡲⡀⢹⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠈⠉⠲⡀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣇⣑⡴⠃⠀⣠⡧⢒⠻⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⡻⡑⢄⡘⠦⣀⣠⡤⠤⣰⠋⢀⣀⣈⣉⣤⡴⢚⠡⠂⣠⠔⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢔⠓⢿⣖⡶⢦⣲⢤⡈⢳⣁⣤⠶⡋⠅⢊⡠⠶⡉⠈⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢄⡉⠒⠶⢄⡿⡌⠫⠴⠥⠴⠮⡁⠑⢌⠪⠢⣌⡈⠣⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡱⠁⢈⠶⢬⡀⠍⡻⢶⣖⠒⠤⡀⠹⡀⠀⠳⡐⢌⢻⡐⠢⡞⠉⢳⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⣉⠤⣾⡵⠚⣁⠥⠒⠉⠓⢬⡐⠍⡓⢦⣼⡄⠱⢤⣄⡬⠦⣥⠙⣶⠃⠀⣼⠔⠉⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠰⠯⠒⣜⠡⠒⠉⠀⠀⠀⠀⠀⠀⠈⠑⠬⡑⠨⣙⠢⣄⡠⠖⠒⠚⠚⣁⣠⠞⡡⢊⡤⢮⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡦⠤⢴⣉⠁⠀⢀⣠⠔⠒⠒⠦⡄⠀⠀⠀⠀⠈⠓⠤⣉⠺⠁⣠⢦⣉⡽⢋⠔⢁⡴⣯⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠒⢄⣀⡴⢋⡤⠖⠀⠨⢝⣒⣉⡠⠔⣲⠦⠀⣹⠀⠀⠀⠀⣀⠤⠤⠤⠟⢠⣧⠞⡫⢂⡡⠚⠣⡘⠢⠻⣍⠁
⠀⢸⠉⠙⠢⠤⠤⠎⣠⠔⠂⠒⣢⣤⣴⡉⠠⠤⠤⠔⠖⠺⢛⣒⡧⠬⠄⠒⠃⠀⠀⠀⡜⢁⠴⠒⠂⡰⢋⠕⣩⣔⠉⠀⠀⠀⠈⠦⣀⡾
⠀⢈⣲⣤⡤⠤⠴⠾⢓⣚⠛⠉⠍⢀⣒⡠⠥⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠎⢀⡠⡪⢂⣥⡾⠀⠸⡆
⣾⠁⠀⠀⣒⣀⣉⠡⠤⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⣉⣞⡴⠛⡠⠒⠹⢠⢱⠀⡸⠁
⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢀⣔⡨⣽⢋⠔⠉⠀⠀⡄⢸⣿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⢙⡺⠊⠁⠀⠀⠀⠀⡇⢸⠘⢎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⢠⡷⣾⠀⢸⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠂⠼⠃⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢺⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⣠⠤⠤⠤⣄⠀⠀⠀⠀⠀⢸⠀⢸⡗⡄⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠲⢤⡀⢀⡴⢋⡠⠤⠨⢙⡒⠒⢋⡠⠤⣤⡄⠈⡇⠀⠀⠀⠀⣨⠀⢸⠁⠸⠀⢸
⠀⠀⠀⠀⡞⠙⠲⢄⣀⡠⠊⣠⠄⠒⠤⣈⣁⡔⣉⣀⠀⡠⠤⠬⠟⠷⢶⣈⡃⠤⠜⠁⠀⠀⠀⠀⠛⢠⢸⡴⠃⡴⠃
⠀⠀⠀⠀⠳⢄⣠⣀⣠⡤⠾⠷⠖⡚⠛⠉⠭⢁⣂⣠⠤⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⠁⢸
⠀⠀⢠⡾⠉⠀⠀⠂⠐⣂⣈⠭⠤⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⡄⠈⢣⡀
⠀⠀⠈⠛⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⠳⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⢀⢐⡵⢒⡿⡻⡻⡻⢶⢦⣔⡠⡀⠀⠀⠀⠹⡦⢄⣀⣀⡴⢣⠎⠀⠀⠀⢀⢄⣢⡵⡶⢟⢟⢟⢿⡓⢮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⡆⣎⣴⡫⡪⣪⣞⢕⢕⠕⣤⡙⢮⡢⡀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⢀⢔⡵⢋⣴⡪⡪⡢⣳⣕⢕⣝⣦⣱⢰
⠀⠀⠀⠀⠀⠀⠀⡵⣪⣟⢟⢼⡵⡥⡳⣿⣮⢫⢺⢦⣝⣾⢆⠀⠀⠀⠀⠀⠀⠀⠀⡰⣷⡫⡴⣗⡝⣷⣼⢞⣌⢮⡧⡻⣻⣕⢦
⠀⠀⠀⠀⠀⠀⠀⢇⢿⣿⡵⣷⡮⡪⣺⢞⢕⢕⠕⡡⡫⡻⡎⠆⠀⠀⠀⠀⠀⠀⠰⢱⢟⢝⢌⠪⡪⡪⡳⣕⢕⢵⣺⢮⢿⡿⡸
⠀⠀⠀⠀⠀⠀⠀⠘⣜⢯⣾⢟⢝⢕⢵⡷⣵⣷⣫⡪⢊⢔⣿⣼⠀⠀⠀⠀⠀⠀⣧⣿⡢⡑⢕⣝⣾⣮⢾⡵⡪⡫⡿⣷⡽⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡱⣕⣷⡷⡯⡪⡪⣞⢕⢔⡱⠃⡾⢸⠀⠀⠀⠀⠀⠀⡇⢷⠘⢮⡪⡪⣳⢕⢕⢭⢾⣾⡪⢎⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⠺⣿⢗⣝⡴⡷⢋⢠⠜⣡⣿⡀⠀⠀⠀⠀⢀⣿⣌⠣⣄⡙⢾⢦⣣⣻⣿⠗⣩⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠮⣭⣽⣯⣭⡴⢾⣋⣠⠟⢦⡀⢀⡴⠺⣿⣿⣷⢤⣭⣽⣯⣭⠥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣠⠜⠁⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⣠⣴⠋⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⣤⠞⠁⠈⠳⣤⣼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠞⠁⠀⠀⠀⠀⠈⠳⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⢆⣸⡅⠀⠀⠈⠀⠀⠀⠀⢀⠄⢰⠊⠉⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠠⢀⠀⠤⠈⠀⠀⠀⠀⡱⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣌⣀⣰⣾⠀⠀⠀⠚⢆⣀
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠈⠀⠰⠀⠈⠀⡴⠁⠀⠀⠘⢄
⠀⠀⠀⠀⠀⢸⠀⢱⡀⠄⠆⠂⠀⠀⣠⠼⠗⢄⣰⠀⠀⠈⢗⢀
⠀⠀⠀⠀⠀⢸⠀⠀⠑⣄⣀⠄⠀⠀⠀⠇⠀⠀⣿⠀⠀⠀⠈⠆⠠⣀
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⢀
⠀⠀⠀⠀⠀⠙⡶⠤⠤⠤⠄⠀⠀⠀⠈⢸⡆⠘⢎⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠙⢃⣐⣺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢳⠤⠤⠴⠀⠀⠀⠀⠀⠀⠓⠤⣾⡤⠤⠵⠦⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡜⠀⢀⠀⠀⠀⠀⠀⠀⠈⢦⣜⠃⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⢧
⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢓⣶⣶⣶⠖⠒⠚⠓⢤⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⢸⣄⡁
⠀⠀⠀⠀⠀⣸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠸⠈⠳⣌⡂
⠀⠀⠀⠀⠀⣿⣓⡉⠳⠶⢄⠀⠀⠀⠀⡀⠄⠒⠐⠂⠸⠤⠤⠐⠀⠂⠤⣈⠢⣀⠀⣀⠤⠛⠛⠛⠋⠛⠁
⠀⠀⠀⠀⠀⠈⠈⠉⠛⠛⠉⠀⠀⠀⠀⠑⠠⠄⠀⡀⣀⣀⣀⣀⣀⣀⡀⠤⠃⠀⠉