Numerals Emojis & Symbols

⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⡀⠀⠀⠀⢀⣀⣀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⣀⣀⢀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⣀⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠛⢻⣿⠟⠛⠃⠀⠀⠀⠛⢿⡿⠻⣿⠟⠃⠀⠀⠘⠻⣿⡟⢿⣿⠛⣿⡟⠃⠀⠀⠀⠛⣿⡿⠻⣿⡟⠛⠛⢿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⢹⣧⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡄⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣷⣿⠇
⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⣶⡆⠀⠀⠀⣴⣿⣷⣾⣿⣶⡄⠀⠀⢠⣶⣿⣷⣾⣿⣶⣿⣷⡆⠀⠀⠀⣴⣿⣷⣶⣶⣾⣿⣿⣶⣶⣶

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠈⠹⣿⡏⠉⠉⣿⡿⠉⠀⠀⠀⠈⠉⣿⡏⠉⠉⣻⡟⠉⣿⡏⠁⠀⠀⠀⠉⠹⣿⠉⠉⠉⣿⠏⠙⣿⠉⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⢿⡇⠀⣸⡿⠀⠐⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣸⡟⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⢀⣿⠃⠀⠸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⠀⠀⢀⣿⠀⣸⣇
⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠶⣶⣶⡶⠶⢶⣶⣶⢶⣶⣶⣶⣶⢶⣶⣶⠆⠀⠀⠀⢶⣶⣶⢶⣶⣶⢶⣶⣶⠶⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⣿⠇⠀⣿⠀⢸⡏⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⢻⣧⣠⣿⠃⠀⠀⠀⠀⠀⠀⢻⣇⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣸⡟⠀⠀⣿⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣿⠃⠀⠀⣿⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣆
⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣀⣀⣠⣿⣄⣼⣧⣠⣿⣇⡀⠀⠀⠀⣠⣿⣧⣠⣿⣧⣨⣿⣦⣀⠀⠀⠀⢀⣠⣿⣧⣸⣿⣆⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⡀⣀⡀⣀⣀⢀⠀⠀⠀⣀⣀⢀⡀⣀⢀⡀⣀⡀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⡟⢻⣿⠟⢻⣿⠟⠃⠀⠀⠛⢿⣿⠛⢿⣿⠛⢿⡿⠻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣾⠏⠀⢸⣿⠀⠀⠀⠀⠀⠈⢿⣦⣿⠃⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻⣧⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠛⣿⡄⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣾⣿⣷⣾⣿⣶⡄⠀⠀⣶⣾⣿⣶⣿⣿⣶⣾⣷⣶⣿⣷⡄

⠀⠀⠀⠈⣿⡏⠀⠀⠀⢹⡟⠀⢹⡏⠀⠀⠀⠘⣿⠁⢹⣿⠁⢹⡟⠀⠀⠀⠈⣿⠁⠈⣿⡁⢸⡟⠁
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢹⣧⣿⠃
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢿⡏
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠈⠉⠁⠈⠉⠀⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⢹⣏⠀⣿⠏⠀⠀⠀⠙⣿⠁⣹⡟⠀⢹⡏⠀⠀⠀⠘⣿⡁⢹⡟⠁⢹⡟⠀⢹⡏
⠀⠀⠀⠀⠀⠀⣿⣴⡟⠀⠀⠀⠀⠀⢻⣧⣿⠁⠀⢸⡇⠀⠀⠀⠀⢹⣧⣿⠃⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠈⣿⡇⢸⡿⠁⢹⣿⠁⢹⣿⠁⢹⡏⠀⠀⠀⠘⣿⠁⠉⢿⡇⣿⠏⠁⠀⠀⠈⢻⣇⣸⡟⠁
⠀⠀⠀⠀⠸⣧⣾⠇⠀⢸⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢿⡟⠀⠀⢸⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣴⡖⣶⡄⠀⠀⠀⠀⣰⡿⢿⣆
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠁⠉⠉⠀⠉⠉⠀⠀⠀⠈⠉⠁⠈⠉⠁⠉⠉⠁⠀⠀⠈⠉⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣨⣿⠋⠈⣿⡋⠀⠀⠀⠈⢿⣇⣹⡟⠁⢹⣿⠁⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⡟⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⣧⠀⠀⣿⠇⠀⠀⠀⠀⣰⡟⢿⡄⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠈⠉⠉⠀⠀⠀⠈⠉⠁⠈⠉⠁⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⢹⡏⠀⠈⣿⠁⠀⠀⠀⠀⠀⢹⡏⠀⢸⣿⠀⠈⣿⠁⠀⠀⠀⠀⠀⢹⡏⠀⠘⣿⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠦⠀⠀⠀⠀⠀⠠⠾⠷⠤⠴⠿⠦⠀⠀⠀⠀⠠⠾⠷⠤⠾⠿⠤⠴⠿⠦⠀⠀⠀⠀⠠⠾⠷⠄⠀⠀⠾⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⠂⠀⠲⣶⠖⠀⠀⠀⠀⠐⢲⡶⠂⠀⢲⣶⠒⠒⣶⠒⠀⠀⠀⠀⠐⢲⣶⠂⠀⢲⣶⠒⠒⣶⠖⠒⢶⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⡏⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣾⠃⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⢠⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡃⠀⠀⣀⣿⣀⣀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⠁⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣇⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢰⡿⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣸⡇⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢘⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠁⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠇⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠷⠀⠀⠰⠾⠷⠶⠾⠿⠶⠶⠛⠖⠀⠀⠀⠀⠰⠾⠷⠖⠾⠿⠆⠀⠶⠻⠶⠀⠀⠀⠀⠀⠾⠻⠆⠀⠶⠻⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⡖⠀⢐⣶⠒⠒⢲⡖⠂⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⠀⠐⣶⠖⠒⢲⣶⠒⠒⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣼⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣸⡟⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠹⣧⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣃⠀⢈⣿⣄⣀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣏⡀⢀⣻⣆⣀⣸⣿⣀⣀⣿⣀
⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡶⠒⣢⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣾⣿⠀⠻⠿⠛⠁⠘⢿⣿⣿⡿⠈⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣥⣤⠀⠀⢀⡀⠀⣾⣷⡦⠀⣀⡀⠀⠀⣴⣿⡇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠉⠉⢠⣾⣿⣿⡆⠀⠁⢠⣾⣿⣿⡆⠀⢀⣤⣤⣀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⣀⠉⠛⠋⠁⠀⢀⣸⣿⣿⣿⠇⠀⢿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠀⠀⢿⣿⡇⠀⣾⡷⠊⠁⠀⠀⠉⠣⣄⣠⣄⠙⠛⠋⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣤⣤⡀⠉⠁⠀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣇⣀⣈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠹⣿⣿⡇⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣀⣉⡀⠀⠙⢟⣀⠤⠲⣾⣿⣿⡟⠉⠁⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⣴⣿⣿⣿⣿⡄⠀⣨⣤⣤⡀⠀⠉⠋⠀⣴⣷⠀⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢿⣿⣿⣿⣿⠇⠘⠿⣿⡿⠇⠀⣿⣷⡄⠈⢉⣀⣀⡈⠉⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣦⠀⠉⠛⠛⢋⠀⠀⢀⣄⡀⢀⡤⠿⢿⠁⠀⣿⣿⣿⣿⣷⣦⠀⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⣴⣿⣧⠀⣾⣿⡿⠁⠀⠀⠀⠱⡀⠈⠙⠛⠛⠋⠁⠀⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣶⣿⣶⣄⠙⠛⠋⠀⠹⢿⠃⠀⠀⠀⠀⠀⢻⣦⠀⠀⣴⣾⣷⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠛⠛⠛⠋⠀⠰⣿⣿⣷⢸⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⠿⠃⠠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⡄⠀⢀⣤⣀⠙⠻⠟⠀⢇⠀⠀⠀⠀⢀⠏⢁⡀⠀⣠⣶⣶⣄⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣿⣿⣷⡄⢠⣶⣮⡢⣀⣀⣠⣾⠀⠿⠇⠀⠻⠿⠿⠟⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠹⢿⣿⠿⠁⢸⣿⣿⡇⠀⣠⣬⣀⠀⠀⣤⣦⣄⠀⣴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢠⣤⣦⡄⠉⠉⠀⢸⣿⡿⠟⠀⠘⣿⣿⠿⢀⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠀⣼⣿⡄⠀⣰⣾⣿⣦⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠛⠷⠤⠿⠛⠛⠋⠁

⡀⠀⢀⡴⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠸⠸⠀⠂⠘⠂⠀⠀⠀⠀⢻⢦⠈⢇
⠁⠀⠉⠀⠇⢠⠏⠀⠀⢀⣀⣀⣀⣠⠤⠀⠀⠀⠀⢸⠞⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡄⠳⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠛⠒⠲⠤⢤⠀⠀⠀⡴⢻⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢤⡀⣧⠈⢣⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣞⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠂⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡀⠀⢳⡀⠀⠀⢀⡀
⡀⠀⠀⠖⢚⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠳⠀⠉⢀⡀⠢⡙
⡙⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣩⡽⠶⠆⠀⠀⠀⠀⠁⠀⠀⠐⠢⢷⡀⠙
⠀⠀⠀⠀⠀⠀⣠⠞⢀⠀⠀⠀⠀⠀⠀⣠⠜⢂⡀⠀⠀⠀⢠⣤⣤⣄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠑
⠀⠀⠀⠀⢠⣞⡥⠖⠋⣡⡀⠀⢀⡠⠞⢁⠴⢋⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⠈⠋⠁⢀⡴⠊⠁⠀⠀⠈⠡⠞⢁⡴⠋⠀⠀⣀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣉⠭⠟⠉
⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⢦⠘⣆⠀⠀⠀⢸⠃⡏⠘⣄⠆⠀⠘⣋⣀⡤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣦⠀⠀⢸⠀⡇⢀⢿⠀⠀⠀⠉
⠀⠀⠀⣀⣠⠤⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠐⠃⠚⠘⠂⠀⠀⠀⠀⠀⢻⢀⡟
⠀⠀⢋⣡⠤⠖⠚⣉⡄⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡤⠖⠚⠉⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣸⠁
⠀⠀⠀⡶⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣧⠇
⠀⠀⢠⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠀⠀⠰⠦⢤⣀⣀
⠀⢀⡞⢠⠇⠀⠀⠀⠀⠀⠀⢲⡀⠑⢼⠀⠀⠀⠁⠉⠙⠒⠦⢤⠀⠀⠀⠀⠀⠀⠀⠶⠖⠒⠒⠛⠛
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⠓⢄
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠈⡧⠸⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⡀⡀
⠀⠀⠀⠀⣿⡴⠋⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⠀⢇⡇⠀⠀⠀⠀⣄⠀⢦⠹

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠸⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢨⡟⠀⠀⠀⠀⠀⠈⣿⡀⠀⢠⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⣾⠃⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣤⡿⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠃⠀⢹⡇⠀⢹⡇⠀⢹⡇⠀⠀⠀⠀⠀⢹⡇⠀⠈⣿⡀⢠⡿⠁⠀⠀⠀⠀⠀⠈⢿⡄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢠⡿⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡇⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠁⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⢻⡆⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢠⡿⠀⠈⢿⡀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣸⠏⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣸⠏⠀⠈⣿⠁⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡟⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠹⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠹⣇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠛⠙⠋⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠚⠋⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⣶⠀⠀⣶⠀⠀⠀⠀⠀⠀⢴⡆⠀⣶⡆⠀⣶⠄⠀⠀⠀⠀⠀⢰⡆⠀⢴⡆⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣷⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⠁
⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡆⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸
⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⡇⠀⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢠⡄⠀⢤⡄⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⣤⠀⠠⣤⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢠⡄⠀⣤⡄⠀⣤⡄⠀⣤⡄
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣼⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⡞⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢸⠁⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡇⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣷⠀⢰⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⡎⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡄⡸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢰⠇⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡾⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠁⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠉⠁


⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⢸⡇⠀⠀⢣⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⢇⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠈⢇⠀⢠⡿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠈⢇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣾⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⠁⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢳⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⢧⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⣼⠃⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⢇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠘⡆⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠚⠃⠀⠘⠛⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠘⠂⠀⠘⠓⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠘⠂⠀⠚⠇⠀⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⣠⠖⠉⢉⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⣉⠉⠓⢧⠄⠀⠀⠀⠀⠀⠀⡠⠂⢈⣉⠉⠲⢄
⠀⠀⠀⠀⢰⠁⠀⠀⢸⠀⠀⠀⠘⡀⠀⠀⠀⠀⢠⡟⠀⠀⠘⢉⡇⠀⠀⢳⠄⠀⠀⠀⠀⡄⠀⠀⠈⢸⡇⠀⠈
⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⣇⠀⠀⢰⠋⠀⠀⠀⣸⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⡆⠀⠀⡐
⠀⠀⠀⠀⠀⠳⣄⠀⠘⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⢻⣦⡀⠘⠶⠂⢀⣴⠃⠀⠀⠀⠀⠀⢈⢄⡀⠘⠚⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⣀⠤⠴⠶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠦⣤⡀
⠀⠀⠀⠀⢀⡞⠁⠀⡄⢤⠀⠈⠳⡄⠀⠀⠀⠀⢠⡞⠁⠀⣴⠒⠀⠈⠳⡄⠀⠀⠀⠀⢀⡼⠋⠀⣠⢦⠀⠀⠙⣆
⠀⠀⠀⠀⢸⠁⠀⠀⣇⣼⠀⠀⠀⡇⠀⠀⠀⠀⢹⠁⠀⠀⠿⢦⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣿⢤⡀⠀⠀⢹
⠀⠀⠀⠀⠘⣆⠀⠀⠀⢹⠀⠀⣠⠇⠀⠀⠀⠀⠹⣇⠀⠀⢤⣸⠃⠀⢠⠏⠀⠀⠀⠀⠈⢧⡀⠀⢿⣸⠇⠀⢀⡟
⠀⠀⠀⠀⠀⠈⠓⠤⣤⣬⠤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⡴⠊⠁⣀⣀⠉⠳⣄⠀⠀⠀⠀⠀⠀⡠⠚⠉⣀⡀⠉⠓⠄⠀⠀⠀⠀⠀⠀⣴⡔⠋⢉⣀⡉⠙⢧
⠀⠀⠀⠀⣼⠁⠀⠀⢀⡇⠀⠀⠘⡆⠀⠀⠀⠀⢰⠃⠀⠀⡇⣿⠀⠀⠈⡄⠀⠀⠀⠀⢸⡏⠀⠀⢸⠀⡇⠀⠀⢷
⠀⠀⠀⠀⢻⡀⠀⠀⣸⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⡀⠀⠀⡏⣿⠀⠀⢀⠇⠀⠀⠀⠀⠨⣇⠀⠀⢈⠙⡇⠀⠀⡼
⠀⠀⠀⠀⠀⢷⣄⡀⠃⠀⣀⡤⡋⠀⠀⠀⠀⠀⠀⠱⣄⡀⠓⠋⢀⡠⠎⠀⠀⠀⠀⠀⠀⢻⣦⣀⠈⠛⠁⣀⣜⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢠⠶⡄⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⢸⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⣀⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣄⣀⣉⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⢀⣤⣠⣿⣯⣸⣯⣟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠁⠈⠀⠀⢈⠀⢁⡀⠀⠀⠀⠀⠈⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣰⢇⡾⠀⠀⠀⠀⠀⢀⠤⠰⠮⣤⣅⠀⠀⠀⠀⠀⠀⡤⠀⢄⡦⣶⠇⡄⠀⠀⠀⠀⠀⠀⡤⠩⢍⡄⠀⠀⠀⠀⠀⠀⡤⠆⠼⠠⡴⠤
⠀⠀⢀⠴⠊⠀⢀⡁⠀⠀⠀⠀⠰⣃⣀⡤⣄⠀⠈⡆⠀⠀⠀⠀⠀⠓⠀⡴⠄⢀⠎⠀⠀⠀⠀⠀⢀⠜⠀⣤⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⡖⠚⠂⠒
⠀⠀⠐⠶⠲⠂⠀⠐⠀⠀⠀⠀⠀⠈⠁⠀⡴⠀⢠⠃⠀⠀⠀⠀⠀⠠⡊⠀⠀⠛⢦⠀⠀⠀⠀⢀⠎⠀⣌⡸⠀⣇⣀⠀⠀⠀⠀⠀⣧⣀⡉⠑⠢⡄
⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠀⣠⠞⠀⣠⣋⣀⠀⠀⠀⠀⠀⠀⠉⠉⡧⠀⠀⡇⠀⠀⠀⣆⣀⣀⣀⡀⠀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⢹
⠀⠀⠀⠀⠀⡅⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⡏⠉⠉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⢰⡋⠉⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣤⢤⣤⢶⡄⠀⠀⠀⢠⠤⠤⠄⠴⡱⢈⢦⡀⠀⠀⠀⠀⢀⣍⣢⣩⡀⠀⠀⠀⠀⠀⢀⡠⢄⣴⡆⣤⠀⠀⠀⠀⠀⠀⢀⡄⣴⣲⣅⡄
⠀⠀⢀⡞⠁⢀⠴⠒⠖⠁⠀⠀⠀⢸⣀⣀⣀⡀⠀⠀⢐⠇⠀⠀⠀⡜⠁⡠⠂⠀⠙⡆⠀⠀⠀⢀⡎⠀⡠⠒⢦⠀⣱⠀⠀⠀⠀⡰⠋⢠⠄⠢⣁⠘⢦
⠀⠀⡼⠀⠀⠉⠉⠙⢢⡀⠀⠀⠀⠀⠀⢀⠜⠁⢀⠖⠉⠀⠀⠀⠀⢣⡀⠉⠊⢁⡴⠁⠀⠀⠀⠸⡇⠀⠣⠤⠞⠐⢹⡄⠀⠀⢰⠃⢰⠁⠀⠀⢈⡆⠸⡆
⠀⠀⠆⠀⢰⠉⠉⡂⠀⣷⠀⠀⠀⠀⢠⠏⠀⣰⠃⠀⠀⠀⠀⠀⡰⠋⢀⠴⠤⣄⠈⢣⠀⠀⠀⠀⠙⠲⠤⣤⠀⠀⡸⠀⠀⠀⢸⠀⠸⣄⠀⠀⣠⠃⢠⡇
⠀⠀⠸⡀⠈⠲⠖⠁⢠⠟⠀⠀⠀⠀⡎⠀⢰⠇⠀⠀⠀⠀⠀⠀⢣⠀⠈⠢⠴⠋⠀⡸⠀⠀⠀⠀⡶⠒⠚⠁⢀⡤⠁⠀⠀⠀⠈⢧⡀⠈⠓⠛⠁⢀⡼⠁
⠀⠀⠀⠉⠒⠦⠴⠒⠉⠀⠀⠀⠀⠘⠓⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠔⠚⠁⠀⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤
⠀⠀⠀⠐⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⣀⣀⣀⣀⣀⡀⠀⢀⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣁⣀⣐⣀⣀⣀⣀⠀⠀⡀⠀⢀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⣀⣈⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⠿⠿⠇⠀⠀⠀⠻⢿⣿⠿⢿⣿⠿⠂⠀⠀⠀⠿⣿⡿⠿⣿⡿⢿⣿⡿⠇⠀⠀⠀⠻⢿⣿⠿⢿⣿⠿⠿⠿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠰⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⡄⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢐⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣧⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣄⣿⠇
⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣄⣀⡀⠀⠀⠀⣀⣸⣿⣀⣼⣿⣀⠀⠀⠀⠀⣀⣿⣧⣀⣿⣇⣨⣿⣄⡀⠀⠀⠀⣀⣸⣿⣀⣀⣀⣽⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠿⣿⣿⣿⣿⠆⠀⠀⠀⠾⣿⣿⣿⠿⢿⣿⣿⡿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣾⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⣿⡏⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⢠⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢸⣿⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣾⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣿⠇⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡁⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠀⠀⠀⣿⣇⢀⣿⡇
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗

⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡏⠀⣿⡇⠈⣿⡅⢸⣿⠁⠀⠀⠀⠀⠀⢹⣿⠀⠹⣿⡅⣸⣿⠃⠀⠀⠀⠀⠀⠀⠹⣿⡀⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⣸⡿⠀⠀⣿⡇⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢀⣿⠇⠀⠀⣿⡇⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⣿⡇⢀⣿⡃⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿⠋⢹⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢻⣧
⠀⠀⠀⠀⠀⠰⣷⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣟⠉⣿⡟⠉⢹⣿⠉⠁⠀⠀⠉⠹⣿⡏⢙⣿⠏⠉⣿⡟⠉⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣿⣾⡏⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠙⣿⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⡟⠹⣷⡀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣶⣿⣿⣶⣾⣿⣶⡄⠀⠀⣶⣾⣿⣷⣾⣿⣷⣶⣿⣷⣶⣿⣷⣦

⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⢸⡷⠀⢾⠆⠀⠀⠀⠀⠐⣿⠂⠰⡷⠀⠰⡷⠀⠀⠀⠀⠀⢾⡇⠀⢾⡆⠀⢐⡿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⢧⠀⢸⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⠀⠀⣯⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⣄⡏
⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⣸⣇⠀⣸⣆⠀⠀⠀⠀⢀⣿⡀⢀⣧⠀⣠⣇⠀⠀⠀⠀⠀⣸⣇⠀⠀⢀⣿⡁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⢀⣄⠀⢀⣄⠀⠀⠀⠀⢀⣄⠀⠀⣠⡀⠀⣠⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⣼⠁⠀⠀⠀⠀⢹⡄⠀⢸⠋⠀⢸⡇⠀⠀⠀⠀⠈⢿⠀⠀⡿⠁⠀⢹⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠇⠀⠀⠀⠀⠀⠀⢧⠀⡏⠀⠀⠈⡇⠀⠀⠀⠀⠀⠘⡆⢰⠃⠀⠀⢸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠀⠀⠀⠀⠀⠀⠀⠸⣾⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⢸⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠿⠂⠐⠿⠂

⠀⠀⠀⠀⠀⢶⠂⠀⢰⡧⠀⢰⡆⠀⢾⠆⠀⣾⠆⠀⠀⠀⠀⠐⣷⠀⠰⢷⠀⠀⢰⡿⠀⠀⠀⠀⢿⡂⠀⢀⡾⠆
⠀⠀⠀⠀⠀⠘⡇⠀⡾⠀⠀⢸⠇⠀⢸⠀⠀⢹⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢧⣠⠏⠀⠀⠀⠀⠀⠀⠻⣄⡞⠁
⠀⠀⠀⠀⠀⠀⢻⣰⠇⠀⠀⢸⡀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⢷⡀⠀⠀⠀⠀⠀⠀⢠⠿⣄
⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⣸⣇⠀⣸⡀⠀⣿⡀⠀⠀⠀⠀⢀⣿⡀⢀⣴⠃⠀⢻⣄⠀⠀⠀⠀⣠⠏⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⢠⣄⠀⠀⢀⣄⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⢠⠟⠁⠈⢿⠁⠀⠀⠀⠈⠻⡄⠀⡼⠋⠀⢸⡏⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢙⡞⠁⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢳⡀⠀⠀⣸⠀⠀⠀⠀⠀⢀⡞⠙⣆⠀⠀⢠⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⢿⠆⠀⠿⠆⠀⠀⠀⠰⠿⠀⠀⠸⠷⠀⠺⠗⠀⠺⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠶⣶⠲⠀⠀⠀⠰⢶⡖⠆⠖⣶⠶⠀⠀⠀⠲⣶⠶⠀⠶⡶⠖⠰⢶⡶⠂⠀⠀⠰⢶⡖⠂⠖⢶⠶⠐⢶⡖⠆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⣾
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣰⡇
⠀⠀⠀⣀⣿⣀⠀⠀⠀⢀⣸⣇⡀⣀⣿⣀⠀⠀⠀⣀⣿⣀⠀⣀⣟⣀⢀⣸⣇⡀⠀⠀⢀⣸⣇⡀⠀⠀⣸⣿⣀
⠀⠀⠀⠁⠀⠉⠀⠀⠀⠈⠀⠈⠁⠁⠀⠈⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠈⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠈⠹⡏⠁⢉⡟⠉⠀⠀⠀⠉⢹⡏⠀⢹⡟⠉⠈⢹⡏⠁⠀⠀⠉⢹⡏⠁⢹⡟⠉⠈⢹⡏⠁⠉⣿⠉
⠀⠀⠀⠀⠀⠀⢿⠀⣼⠃⠀⠀⠀⠀⠀⠈⣷⠀⣾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⣼⠁⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣧⡏⠀⠀⠀⠀⠀⠀⠀⢹⣶⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠤⠿⠥⠀⠀⠀⠀⠀⠀⠀⠤⠿⠤⠀⠀⠠⠼⠧⠄⠀⠀⠀⠀⠤⠿⠥⠀⠀⠠⠼⠧⠄⠤⠿⠤

⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⣀⣀⣀⢠⣀⣀⡀⣀⣀⣠⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⢀⣀⣀⡄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠇⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⡄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⡿⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠒⠀⠀⠐⠚⠓⠂⠒⠛⠒⠐⠚⠛⠒⠀⠀⠀⠀⠀⠒⠛⠒⠐⠚⠛⠂⠐⠛⠓⠂

⠀⠀⠀⠰⠲⣶⠆⠲⣶⠲⠀⠀⠀⠀⠰⠲⣶⠆⠲⣶⠶⠀⠶⣶⠲⠀⠀⠰⠲⣶⠆⠰⣶⠶⠀⠶⣶⠲⠰⢲⡶⠆
⠀⠀⠀⠀⠀⠹⣆⡼⠃⠀⠀⠀⠀⠀⠀⠀⠹⣆⣼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣦⣼⠋⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⢀⣀⣾⡁⣘⣧⣀⠀⠀⠀⠀⢀⣀⣾⡁⣘⣧⣀⠀⣀⣿⣀⠀⠀⠀⢀⣾⡃⢘⣧⣀⠀⣀⣿⢀⢀⣸⡇⡀
⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠈⠈⠉⠀⠁⠉⠉⠈⠉⠉⠁
⣿⣿⣿⣿⣛⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣻⣿⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⣤⡶⠦⠀⠀⢠⠆⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠐⢶⣦⣤⠤⡤⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⠀⠀⣻⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠂⠀⠀⠀⠀⣻⠀⠀⠀⡀⠀⠀⠈⠀⣠⢞⠝⠁⠀⠐⠂⠀⠀⠀⠈⠻⠰⢿⠄⠀⢀⡴⡻⠁
⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⡀⠀⠀⢸⢸⠀⠐⠐⠁⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠙⢦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⣼⠃⠈⠀⣠⢘⠇⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢐⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⡇⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠑⢄⣤⣤⣠⣤⡦⠈⠀⠀⠀⠀⠁⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⡌⡨⠀⠀⠄⠀⠀⠀⠒⢤⡀⠀⠀⠱⣄⠀⠀⠀⠀⣀⡮⠀⠀⠀⠀⠀⢂⡀⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠶⠶⠷⠇⠀⠀⠀⠀⣤⣤⠀⠀⠹⡦⠀⠀⠈⠉⠓⠐⠚⠋
⠀⠀⠀⠀⠤⠰⠂⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢰⠇⠀⠀⡠⠀⠀⠀⠀⠀⠀⣤⣤⣼⡜⠀⠄⠀⢀⣽⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⣿⠁⠀⠀⠀⠲⠶⠼⠇⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠙⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣠⡴⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠁⢭⡉⠉
⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⠀⠀⣀⢀⣿⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈⠀⣀⢀⣀⢤⣄⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠁⠀⠀⢻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣔⡟⠋⠀⠀⠀⠀⠰⠤⣤⠤⠤⣤⣤⣴⣿⠀⠀⠀⠀⠋⠀⠀⠀⠉⠛⠁⠀⡠⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠐⠈⢉⠁⠁⠀⢀⡀⠀⠀⠀⠀⢠⢪⡁⠉⠉⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⡀⠈⠢⠀⠀⠀⠀⠀⢿⡋⠉⠑⠀⠀⣷⠀⠀⠀⠀⠀⠀⢴⡮⠑⠦⠀⠀⣿⠄⠀⠀⠀⠀⠙⠶⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠎⠈⠁⠀⢸⡇⠀⠀⢣⡀⠀⠁⠀⠀⢀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣔⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⡄⠀⢸⡇⠀⠀⠀⠙⠳⠶⠶⠾⠗⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⠀⠉⣄⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠁⠇⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢯⡎⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠉⠀⠀⠀⡠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢠⣤⠲⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠆⠋⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⢶⡶⠦⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⡋⠉⠀⠀⠀⠈⢀⠈⣁⣀⠁⠀⠀⠀⠉⢙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠓⠀⣄⠀⠀⠀⠀⠀⠸⡆⣿⣿⠀⠀⠀⠀⠀⠀⢀⠚⠈⠓⢤⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠙⢦⠘⣆⠀⠀⠀⠀⠀⢧⣿⣿⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣠⠋⠀⠁⠀⠀⠀⠀⠀⠳⣜⠦⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⡴⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⣨⢦
⠀⠀⡼⠡⠀⠐⠶⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠀⠀⣀⡤⣚⠽⠂⠀⠈⢧
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠁⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢮⠉⠀⠀⠈⠗⠋⠁⠀⠀⠀⠀⠘⡆
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣟⠂⠀⠀⠀⠀⠀⠀⠀⣤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠻⠷⣄⡀⠀⢀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡷⠄⠤⠖⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣟⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣭⣭⡏⠠⠶
⠀⡇⠄⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠠⢸
⠀⡧⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠸⡄⠀⢀⡤⣖⡭⣚⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⢠⠇
⠀⠀⢳⠐⠀⢫⠗⠋⢁⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠮⠝⠀⠂⡞
⠀⠀⠀⢳⡚⠀⢀⡴⠋⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡞
⠀⠀⠀⠀⠱⣌⠀⠀⠀⠀⠀⣠⢫⠏⢠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⣡⠎
⠀⠀⠀⠀⠀⠈⠣⣀⠄⠀⠰⣳⠃⠀⡞⠀⠀⠀⠀⢹⠀⣶⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠠⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⣤⠀⠸⠃⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣤⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀⡀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⢀⣨⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠲⠤⠼⠧⠤⠖⠒⠒⠋⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣻⣿⣟⣿

⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⢹⡏⠙⡟⠀⠀⠀⠀⠀⠘⣟⠉⢻⠋⢹⡟⠀⠀⠀⠀⠀⢹⡏⠈⢿⠀⠀⢸⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧⢀⡎
⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣸⣇⣀⣇⠀⠀⠀⠀⠀⢀⣿⣀⣼⣀⣸⣇⠀⠀⠀⠀⠀⣸⣇⠀⠀⠹⡞

⠀⠀⠰⣶⠂⠀⠐⡶⠒⣶⠂⠀⠀⠰⣶⠀⠀⢰⡖⠒⣶⠂⢶⡖⠀⠀⠲⡖⠀⠀⢲⠖⢲⡶⠒⣶⠖⢲⡖
⠀⠀⠀⠸⣇⠀⡼⠀⠀⣿⠀⠀⠀⠀⢹⡆⠀⡜⠀⠀⣿⠀⢸⡇⠀⠀⠀⢻⡄⢀⡎⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢻⡴⠁⠀⠀⣿⠀⠀⠀⠀⠀⢻⡼⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠈⢷⡜⠀⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠠⢤⡄⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⢤⠄⠠⣤⠄⠀⠀⠀⠀⠀⠀⢤⡤⠀⢠⠤⠠⣤⠤⢤⡤
⠀⠀⠀⠈⢳⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠋⠀⠀⣿⠀⢸⡇
⠀⠀⠀⢀⠎⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠹⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⣧⡀⠀⣿⠀⢸⡇
⠀⠀⠐⠛⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠚⠓⠒⠛⠂⠀⠀⠀⠀⠀⠀⠚⠃⠀⠘⠓⠒⠛⠃⠚⠓

⢀⣤⡄⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣠⣤⡤⠀⣤⣄⡀⢠⣤⠀⣠⣤⣤⣤⣤⣤⣤⣤⠀⣀⣤⠤⠤⣤⣀
⠸⣿⣇⣀⣿⣿⣛⠛⢛⣛⣛⣻⣿⡇⠘⠿⣿⣇⡀⠙⠛⠁⢸⣻⠀⠛⠛⠿⠿⢭⣿⡏⠉⢰⣿⠁⠀⠀⠈⣿⡆
⠰⠿⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠟⠁⠀⣿⣿⡇⠠⣶⣶⠀⢰⢶⠀⠀⣠⠤⠤⢄⡀⠀⠀⠸⣿⡀⠀⠀⢠⣿⠇
⠀⡤⠄⢤⣤⣄⣤⣤⢠⣤⣤⢀⡤⣄⠀⣀⣉⣉⣀⣉⣀⠀⢘⣿⠀⠠⡇⠀⠐⡆⢳⠀⣾⠀⢨⡍⠛⠛⠉⡁
⠀⠦⢄⠀⣿⢹⡷⣿⡹⣧⣿⣿⠀⢸⠀⣿⣿⣿⣿⣿⣿⠀⢰⣦⠀⠀⠙⠒⠚⢁⡟⠀⣿⣀⣸⡆⠀⠐⢺⡇
⠀⠤⠞⠸⠿⠘⠷⠿⠓⠶⠿⠙⠦⠞⠀⠛⠛⠛⣿⣿⣿⠀⠸⠟⠀⠀⢀⣠⠤⠊⠀⠀⠈⠉⢹⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⢦⣄⠀⢖⠚⡓⠀⢠⣿⣿⣿⠀⢤⣰⢲⡄⠀⣠⣀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢸⡇
⠀⠀⠒⠀⠀⠻⣧⣀⠀⢀⣠⡿⠃⢺⠱⡵⠀⢸⣿⣿⡇⠀⠀⡉⣋⠁⠀⣿⡍⠑⠦⣄⣀⡀⣀⣽⠆⠀⢸⡇
⠀⠀⢀⠀⠀⢀⣤⠟⠛⠛⢷⣄⠀⠨⡴⠽⠀⣾⣿⣿⡇⠀⣸⣿⣿⡇⠀⠉⠓⠀⠀⠀⠉⠉⢉⠀⠀⠀⢸⡇
⠀⠀⠠⠀⠀⠘⠿⢦⣤⣤⠾⠟⠀⠶⠾⠓⠀⣿⣿⣿⠀⠀⣿⣿⣿⠃⠘⡯⡟⣹⠃⠿⠙⣜⠁⠙⡄⠀⢸⡇
⠀⠐⣶⠖⠀⢀⠤⡀⢀⣀⢀⣤⣀⣀⡀⠀⢸⣿⣿⡏⠀⢀⣿⣿⣿⠀⠈⠃⠋⠛⠃⠘⠃⠈⠒⠊⠀⠀⠉⠉
⠀⠀⣿⠀⠀⣇⠀⡸⢠⢰⣌⣞⢹⣿⡇⠀⣸⣿⣿⡇⠀⢸⣿⣿⡏⠀⠦⣒⠂⠀⢠⣴⠖⠛⠓⢶⣄⠀⠐⠄
⠀⠀⣿⠀⠀⠀⣉⣀⣀⣀⠈⠉⠠⢤⡀⠀⣿⣿⣿⠃⠀⣿⣿⣿⠇⠀⣞⢞⡃⠀⠘⠻⣦⣄⣤⡾⠋
⠀⠀⣿⠀⠀⢸⡏⠁⠀⠉⠙⠢⣄⣸⡇⠀⣠⣤⡀⠀⠀⣿⣿⣿⠀⠀⣞⢢⡇⠀⣴⡟⠉⠁⠉⠙⣷⡄⠀⠐
⠀⠀⣿⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠻⠾⠉⠃⢸⣿⣿⡟⠀⠀⠾⠶⠇⠀⠉⠛⠶⠶⠶⠞⠋
⠀⠀⣿⠀⠀⠀⣼⣤⣤⡀⠀⣠⠔⠊⠉⠀⠀⣼⣷⠀⢸⣿⣿⣧⣤⣤⡄⣰⠒⣄⣴⠶⡆⣴⢶⡆⢰⡆⣠⠒⠂
⠀⠀⠿⠋⠀⠀⢹⠀⢸⡇⢰⠇⣠⠤⢤⡀⠀⢉⡛⠀⢸⣿⣿⣿⣿⣿⡇⣇⠀⣿⣿⢻⣇⣿⢾⣇⣿⠁⠙⠦⡄
⠀⢀⣠⣤⢤⣤⣈⠀⠸⠃⠸⡄⠧⠀⠀⡇⠀⡿⣧⠀⠈⢉⣉⢉⣉⣉⠁⠙⠒⠃⠛⠞⠃⠛⠟⠛⠛⠓⠐⠒⠃
⢠⣿⠇⠀⠀⠈⣿⡇⠀⠀⠀⠙⠒⠒⠊⠀⠀⢙⡋⠀⠘⠿⠟⢀⣿⣿⠗⠀⣴⣶⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⡇
⠘⣿⡆⠀⠀⢀⣿⡇⢀⣸⣿⣟⣻⣶⣶⣦⠀⣿⣯⠀⣠⣤⣄⠉⣿⣷⣦⠸⣿⣿⣤⣤⣤⣤⣤⣿⣿⠀⢸⣿⡇
⣀⣈⣙⣛⣚⣛⣉⣀⣈⣙⣛⣋⣉⣉⣉⣉⣀⣉⣉⣀⣈⣙⣓⣐⣛⣋⣁⣀⣈⣛⣛⣛⣛⣛⣛⣛⣉⣀⣈⣉⣁
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠈⢻⣿⠋⠁⠀⠀⠀⠈⢻⣿⠟⠁⠉⢻⣿⠋⠀⠀⠀⠀⠙⣿⡟⠉⠈⠻⣿⠟⠁⠈⢻⣿⠋⠁⠀⠀⠈⠙⣿⡟⠁⠈⠙⣿⡏⠁⠀⢹⣿⠋⠁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⡀⢀⣾⠇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣧⣼⡟
⠀⠀⠀⢀⣼⣿⣄⡀⠀⠀⠀⢀⣰⣿⣄⡀⣀⣼⣿⣄⠀⠀⠀⠀⣠⣿⣧⣀⢀⣠⣿⣆⡀⢀⣼⣿⣄⡀⠀⠀⢀⣠⣿⣇⡀⠀⠀⠀⣀⣿⣿⣁
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⢠⣤⣤⣤⡀⢠⣤⣤⣤⡄⠀⠀⠀⠀⢠⣤⣤⣤⠀⢠⣤⣤⣤⡀⢠⣤⣤⣤⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⣤⣤⣤⣤⠀⣤⣤⣤⡄⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠈⢿⡇⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣼⡟⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢀⣿⠋⠀⠀⠈⣿⡏⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠘⣿⡄⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣰⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⣼⡏⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠿⠿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠷⠄⠀⠀⠀⠰⠾⠿⠦⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠧⠄⠀⠀⠀⠴⠿⠷⠦⠠⠾⠿⠷⠄



⠀⠀⠀⠲⢶⣶⠖⠀⠐⢶⣶⠖⠂⠲⣶⣶⠶⠀⠲⣶⣶⠖⠐⠶⣶⡶⠆⠀⠀⠀⠐⢶⣶⡶⠂⠐⢶⣶⠂⠐⣶⡶⠖⠀⠀⠀⠀⠲⢶⣶⠀⠐⣶⡶⠒
⠀⠀⠀⠀⠀⢿⣇⠀⢀⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣦⣼⡟
⠀⠀⠀⠀⠀⠘⣿⡄⣾⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣇⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢠⣾⠏⠘⣿⣄⠀⠀⠀⠀⠀⠀⣠⣿⠃⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠚⠛⠛⠛⠀⠛⠛⠛⠛⠀⠛⠛⠛⠃⠀⠀⠀⠘⠛⠛⠛⠁⠘⠛⠛⠃⠀⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠀⠘⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡁⢈⣿⠟⠉⠈⠙⣿⡟⠁⠀⠀⠀⠈⠙⢿⣏⠀⣹⣿⠋⠁⠈⢻⣿⠋⠀⠉⢻⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⢻⣷⡿⠁⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⡿⣷⡀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡁⢈⣿⣦⣀⢀⣠⣿⣧⡀⠀⠀⠀⢀⣠⣾⣏⠀⣹⣿⣄⡀⢀⣼⣿⣄⠀⣠⣾⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠘⠋⠉⡀⠀⢉⢉⠁⠀⠘⠓⠶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⢀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠰⣄⡈⢳⡀⠀⠐⣷⠀⠀⢸⡆⢸⢸⠀⠐⣾⠂⠀⠀⠀⣰⠃⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠈⠳⣄⢳⡀⠀⠀⠀⠀⠀⠷⠸⠸⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⢰⣶⡀⠀⠀⠈⠳⠛⣀⣤⠴⡖⠲⠒⠒⠖⢲⢦⣤⣀⠀⠾⠁⠀⠀⢐⣶⡆⠀⠀⠈⠛
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⢖⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡲⣄⠀⠀⠈⠁⠀⠀⠀⣀⡄⠀⣄
⠀⠀⠀⣠⠀⠈⠓⠲⠦⣤⣀⡀⠀⣠⢞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡳⣄⠀⢀⣠⢖⣫⠵⠚⠁⠘⠂
⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠁⣼⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣧⠉⠖⠋⠁⠀⠀⠀⠀⢠⡄
⠀⠀⣀⠀⠀⠀⢴⣤⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣤⡦⠀⠀⠀⠷
⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⢰⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡆⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⢀⣀⣤⣤⡤⠶⠂⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡇⢐⣒⣒⣒⣒⣒⡆⢸⡇
⠀⢸⡇⠨⠥⠤⠤⠤⠤⠄⢸⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡇⠨⠯⠭⠭⠭⠭⠇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠇⠀⠀⠀⡀⠀⠀⠀⣸⠂
⠀⠀⢶⠀⠀⠀⠺⠛⢀⣤⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠛⠗⠀⠀⠀⠉
⠀⠀⠘⠃⠀⣀⠴⣚⡭⢖⡭⠄⢻⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡟⠀⠉⠳⢦⣀⠀⠀⠀⣠
⠀⠀⠀⠠⡄⠱⣚⡵⠚⢉⢀⡤⠀⠙⢮⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡱⠋⠀⠀⠛⠖⠦⢬⣝⠆⠀⠋
⠀⠀⠀⠀⠙⠀⠁⠀⢀⡴⠋⢀⡀⠀⢀⠙⠮⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠵⠋⠀⠀⢀⡀⡀⠀⠀⢀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⢦⡐⠋⠀⠸⠿⠁⢠⢏⡞⠀⡀⠉⠛⠳⠧⠴⠤⠤⠦⠼⠖⠛⠉⠀⢠⡀⠀⠀⠨⠿⠇⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣰⢋⠞⠀⢸⠃⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⡐⠣⠏⠀⢀⡟⠀⠠⡿⠀⠀⢸⠀⠸⡆⠀⠠⢿⠄⠀⠀⠀⠀⠹⠆⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠘⠁⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢤⡄⠀⢀⣈⣀⠀⠈⣀⣠⡄⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⢀⣠⠴⠖⠛⠋⠉⠉⠙⠛⢶⣤⡀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣠⣴⣏⠡⠐⠑⣀⣀⣁⣀⠀⠀⠀⠙⢋⢦⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣿⣟⣟⣿⣳⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠈⠙⢿⣧⠀⠀⢸⡇⠄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠛⠿⢾⡧⡀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⢀⣿⡑⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡽⠁⢀⡼⠑⠂⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣁⠄⠀⣠⡾⡛⢁⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣃⠚⠁⣠⣾⡿⠓⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡡⠒⠁⣠⣾⡿⠚⢠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣁⠂⠀⣠⣾⡿⠋⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠔⠁⢠⣾⣿⢿⢁⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠌⠀⢰⣿⣿⠻⠛⠛⠛⠛⠛⠒⠛⠛⠛⠛⡛⢻⣿⣿⣷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⠋⣠⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣯⣾⣿⣿⣿⣿⡷⠿⣿⡿⠿⠝⠛⠻⠟⣿⣽⣿⣷⣄⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣿⣿⣿⣾⣶⣾⣶⣷⣶⣶⣶⣶⣷⣶⣿⣿⣿⣿⣻⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⢀⣴⣄⠀⣀⣦⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣴⣄⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⣠⣤⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢹⠁⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢻⠁⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠉⣿⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡆⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣷⢀⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠐⢿⠆⠀⠀⠀⠀⠀⠐⢿⠗⠀⠰⡿⠂⠀⠀⠀⠀⠀⠀⢾⡷⠀⠐⢿⠗⠀⠰⡿⠂⠀⠀⠀⠀⠀⠀⠺⡿⠂⠀⠀⠀⠰⡿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⡀⠀⠀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⢹⡟⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⢸⡟⠁⠀⢹⡟⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⢈⣿⠓⠀⠘⣿⠁⠀⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⠇⠀⠀⠀⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠇⠀⠀⠀⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠄⠀⠀⠀⢠⣿⠄⠀⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣿⠷⠀⠐⣿⠇⠀⠺⣿⠂⠀⢾⡗⠀⠀⠀⠀⠀⠀⠐⢿⡗⠀⠸⢿⡇⠀⠀⢀⣿⠷⠀⠀⠀⠀⠐⢿⣇⠀⠀⠀⣿⠿
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢠⡟⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⣄⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣸⠁⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡏⠀⠀⠀⠀⣾⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠟⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⢀⣿⡀⠀⣠⣷⡀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⢠⣴⡏⠀⠀⠘⣷⣤⠀⠀⠀⠀⠀⣴⡟⠀⠀⠈⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠁⠀⠈⠋⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⠛⠁⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢐⣾⡆⠀⢴⣷⠄⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⣾⡷⠀⠠⣾⠄⠀⢴⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢀⡾⠃⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⣼⠋⠀⠀⠀⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡞⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠁⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢳⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣄⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢻⣄⡀⠀⢀⣇⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠹⣦⡀⠀⠀⣿⡀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠂⠀⠀⠐⠻⠇⠀⠙⠟⠁⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠛⠏⠀⠈⠛⠁⠀⠙⠟⠁

⣶⣶⣶⣶⣶⡾⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢾⣶⢿⠾⣷⠶⣶⡶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣀⡤⠶⡻⢶⢤⢄⣤⡖⢟⡳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠶⠷⣷⣮⣳⡗⡐⡱⠡⢪⣞⢕⢹⡳⠲⢖⢶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣍⣿⠛⠽⣻⣴⣿⢵⡫⠉⠉⠉⢳⣛⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣌⣿⡄⢸⣺⠀⠀⣷⣥⡤⠔⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣎⣙⠷⢴⡿⣏⣰⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠽⣿⣶⣶⢭⡶⢻⣋⣭⣤⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣧⣼⣥⣮⣾⢿⣟⣿⡍⠝⡶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠝⣿⠛⠁⠁⠃⠀⠘⠆⠱⣄⠀⠁⠊⡜⡹⢲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠔⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠊⡰⠉⡹⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡠⠰⠊⠀⠀⠀⢀⣀⣀⣸⣿⠝⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⢊⠜⢑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⡆⡇⠃⠀⠀⠀⡰⡫⢋⢅⣵⢵⠕⡫⣻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠔⡡⢎⡝⣦
⠀⠀⠀⠀⠀⣠⠚⠁⠀⠣⠉⠀⠀⠀⠀⢸⠋⣈⡵⠝⢳⡧⡪⣮⣏⣕⢉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡴⢋⠔⣡⢷⡀
⠀⠀⠀⢠⠞⠁⠆⠢⠘⠀⠀⠀⠀⠀⠀⢿⡼⣿⣇⠀⢸⡪⡮⡟⠀⠙⢷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣡⠞⣡⢊⠝⡄
⠀⠀⡴⠁⢠⠺⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣕⢟⢷⠿⡪⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠞⡡⢋⠜⡼⡄
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣕⡩⡪⣞⣷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠊⡠⢂⠞⡰⣳
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣪⣾⣵⠟⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⢁⢞⣔⣽⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡄⠀⠀⡼⣊⣼⡇⠈⢿⣮⡪⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⡡⡫⢪⢾⡅
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡱⡻⣦⡀⣿⣿⣿⠁⠀⣸⣯⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⡡⢊⠔⡵⡱⣻⠇
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣊⡿⢻⡿⣯⢞⢟⢟⣽⡻⣮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⢊⠔⡡⢪⢎⡔⣩
⠸⡆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣏⢔⢵⠵⠦⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡫⢖⡡⢊⠔⡡⢊⠴⡏
⠀⠱⡀⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠔⢋⠴⣋⠔⡡⢊⠔⣩⡾
⠀⠀⠱⡀⠰⢘⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⢀⠀⠀⠀⢀⠀⡀⢀⠀⡰⢃⠄⣁⡴⢩⠞⡡⢊⡴⣡⢞⣽⢭⠉⢩⠉⠉⣠⣤⡄
⠀⠀⠀⠘⢦⠈⠸⢈⢠⠠⢀⠀⡀⢀⢀⢀⠜⡰⢃⠜⡡⢊⠤⢊⠴⢊⠴⢁⠔⡠⢂⠞⡱⢡⢮⠎⡰⢁⠞⡰⢋⢜⡵⠟⠛⠒⠒⠒⠒⠒⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⢙⣦⣈⠘⠸⠸⠄⠃⠂⠂⠘⠐⠀⠃⠚⠐⠃⠐⠁⢘⣁⠚⠁⠊⠐⠡⢊⡔⡱⢃⠞⡰⢃⣎⣴⠷⠋⠉⠉⠉⠉⠉⠉⠩⠭⠭⠭⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠁⠉⠉⢲⠦⣤⣤⡄⠒⣶⣶⣲⣶⡶⠾⢿⣿⣭⣤⣬⠭⢭⣉⣁⣰⣋⣞⣰⣧⠾⠶⠿⠭⠤⠤⠤⠤⠤⠬⠥⠤⠭⠭⠭⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⣒⣀⣀⣀⡭⠭⠍⠉⠭⠭⠍⠉⠉⠀⠂⠘⠉⠩⠿⠿⠿⣭⣭⣭⣽⠉⠉⠩⠭⠭⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡿
⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⣼⠁
⠀⠹⣟⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⣠⡾⢻⠇
⠀⠀⢘⣦⡀⠉⠙⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉⠁⠀⣀⡤⠞⠉⣰⣯⡦
⠀⠀⠘⢯⡙⠲⢤⣀⠀⠀⠈⠉⠉⠛⠒⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣶⠶⠚⢛⡖⢀⡤⠞⠉⠀⢀⣀⡤⠴⣒⣋⣭⠀⣠⡾⢋⡼⠁
⠀⠀⠀⢶⡿⣦⡀⠉⠉⣛⣒⡒⠶⠦⠤⣤⣤⣀⡀⠉⠓⢦⡀⠀⢠⣶⡿⠟⢋⣵⣿⣿⣾⣿⠿⠟⣿⣼⡵⠋⠀⣠⠴⠚⠉⠀⠀⠀⣀⠴⠋⠀⠁⣠⣞
⠀⠀⠀⠀⠻⣤⡉⠓⠀⠛⠬⣅⣀⡀⠀⠀⠀⠈⠉⠓⢦⡀⠙⢦⡻⣭⡀⢀⡙⣏⠋⠉⠁⠀⠀⠀⣿⡟⠀⣠⠟⠁⠀⠀⠀⠀⢈⣉⡭⠞⢀⡤⠞⣫⠟
⠀⠀⠀⠀⠀⢰⣿⢦⣄⡀⠐⠮⠭⣉⣀⣀⡀⠀⠀⠀⠀⠉⠢⡀⠹⣄⠙⠳⣝⣪⣳⣴⡄⢰⠀⢸⣿⠀⠞⠁⠀⠀⢀⣀⣉⡭⠽⠂⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣭⣁⣀⠀⠘⠓⠒⠒⠒⠲⠆⣠⡶⢦⣌⣀⠙⡆⠀⠈⠙⣿⣿⣿⣄⣇⣠⣿⣿⡄⠀⠤⠔⠋⠀⠀⠀⣀⣺⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣭⣤⣄⣀⣀⡀⠈⠛⠻⠿⠿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⠿⣧⡀⠀⢀⣠⣶⣾⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⢲⣶⣤⠶⠟⠛⠩⢛⣿⣿⢿⡟⢹⣧⠹⣆⠈⠻⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⣠⡶⢶⡶⣻⡿⠟⢡⠟⠀⠈⡟⢧⠙⢦⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠘⠛⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠁⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢿⠿⠏⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⣠⣶⣖⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⢿⣶⣄⠀⠀⠀⠀⢠⡶⠒⠲⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡧⠀⠀⠀⠀⠀⠀⣶⠴⠶⠶⣴⠂
⠀⠀⠀⠀⠛⣯⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⣿⣿⠀⠀⠀⠀⢈⣀⡀⢀⣿⠇⠀⠀⠀⠀⠀⠀⣰⡏⢸⣿⡇⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠈
⠀⠀⠀⣀⠀⣯⡇⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⡼⠟⠃⢠⡀⠀⠀⠀⠑⠚⠛⢷⢤⠠⠀⠀⠀⠀⣰⠟⣨⢸⣿⡇⢰⡆⠀⠀⠀⠈⠉⠉⠋⢻⡿⣄⠢
⠀⠀⠀⠈⠀⣟⡷⢀⠀⠀⠀⠀⣶⣾⣶⡏⠁⠀⠠⢠⠈⠇⠀⠀⣤⠁⠀⠄⢸⣭⢡⠄⠀⠀⠀⣿⣾⡿⢾⣿⡷⠟⠁⠀⠀⠀⣤⣀⠀⠂⠀⡯⡿⠃
⠀⠀⠀⠀⠰⠿⠷⠄⠀⠀⠀⠀⠱⠟⠿⠿⠿⠿⠿⠿⠀⠀⠀⠘⠛⠳⠤⠴⠛⠃⢿⡀⠀⠀⠀⠛⠛⠳⠾⠿⠷⠞⠀⠀⠀⠈⠛⠷⠦⠤⠾⠛⠁


⠀⠀⠀⣠⡴⠶⢦⣤⠀⠀⠀⠀⠀⢠⡦⠶⢤⣦⣶⠂⠀⠀⠀⠀⢀⡤⠀⠦⣤⡀⠀⠀⠀⠀⠀⣠⡤⠖⠦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠲⢶⣄
⠀⠀⣾⡿⣏⣀⣀⠼⠤⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⣄⣾⣧⣤⣀⣿⡇⠀⠀⠀⠀⡼⣿⠀⠀⡂⢻⡷⠀⠀⠀⠀⠤⣼⣿⠀⠄⠈⣿⣧
⠀⢸⣿⣿⣋⡉⠙⣿⣆⠀⠀⠀⠀⡀⣀⢀⣾⢁⡀⠀⠀⠀⠀⢛⣾⣿⠿⣿⣿⣄⠀⠀⠐⠞⢿⣿⡀⠀⢀⣼⣛⡇⠀⠀⠀⢰⣿⡇⢀⣄⣀⣿⣿
⠀⣘⣿⣿⢛⣉⠀⣼⣿⢀⠀⠀⠀⣉⣹⣿⣿⠛⠀⠀⠀⠀⢀⢸⣿⠑⠓⠈⣿⣿⡗⠀⠀⠘⠁⠈⠛⠛⢛⣿⣿⢷⣶⠀⠀⡀⣿⣿⠈⣛⢻⣿⣿⠶⢆
⠀⣿⠛⠻⠦⣡⡴⢟⣁⠀⠀⠀⠀⠈⢸⣯⣿⡘⠁⠀⠀⠀⢰⠚⠻⢦⣎⣥⠾⠟⠛⠁⠀⠀⠀⠿⡦⣤⣾⡿⠷⠊⠀⠀⠀⠀⠈⠻⢦⣭⠼⠋⠀⠃
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠂

⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⣤⡤⢤⣤⠄⠀⠀⠀⢤⣤⠤⣤⡤⢤⣤⠀⠀⠀⠀⣤⡤⢤⣤⠄⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⡇⠀⣿⠀⠘⡇⠀⠀⠀⠀⢸⡇⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⡇⠀⣿⠀⢀⡇⠀⠀⠀⠀⢸⡇⠀⠈⣷⣸⠃
⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⣸⡇⢀⣿⠀⠀⠀⠀⢀⣇⠀⣿⡀⢘⣇⠀⠀⠀⠀⣸⣇⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠐⣶⠂⠀⢶⠖⠀⠀⠀⠐⢶⠆⠀⢶⡖⠰⣶⠂⠀⠀⠀⠲⣶⠀⠐⣶⠂⢲⡖⠐⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⣾⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢠⡏⠀⠀⠀⠀⠀⠈⣧⢠⡟⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⣸⠇⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇⠀⠀⠀⠀⠀⠀⢿⣸⠇⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧⣿⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢠⣿⡀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣸⣇⢀⣿⡀

⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⣿⠁⢹⡏⠈⣿⠁⢹⡏⠀⠀⠀⠀⢻⡏⠉⢿⡁⢸⡟⠁⠀⠀⠀⠈⢻⡇⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡀⢰⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠈⣧⡾⠁⠀⠀⠀⠀⠀⠀⢷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠃⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⣰⠇⢹⡆⠀⠀⠀⠀⠀⢰⡏⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠀⠀⠼⠧⠠⠿⠄⠼⠧⠀⠀⠀⠀⠾⠧⠰⠿⠄⠠⠿⠄⠀⠀⠀⠠⠿⠄⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⣼⠋⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⣯⠀⣼⠋⠀⣿⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢻⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢻⡄⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠚⠛⠒⠛⠃⠀⠀⠀⠀⠀⠀⠛⠓⠀⠚⠛⠒⠛⠓⠚⠛

⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⣠⡶⠒⢢⡀⠀⠀⠀⠀⢠⡶⠒⣦⠀⠀⠀⠀⢠⡆⠀⠀⢰⠀⠀⠀⠀⡴⠖⠲⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠰⡟⠀⠀⠀⣷⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⣾⠁⠀⢠⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⡼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⢰⠇⠀⠀⣼⠀⠀⠀⠀⢰⣇⡠⠤⢤⣀
⠀⠀⠀⠀⠐⠋⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠸⠋⠉⠹⣆⠀⠀⠉⠉⠛⢻⠇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣠⠋⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⢀⣼⠇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠸⣆⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠐⠒⠛⠒⠀⠀⠀⠀⠛⠓⠒⠒⠒⠒⠂⠀⠀⠘⠛⠒⠒⠛⠁⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠁

⠀⠀⠀⠀⣠⠴⠖⠒⠒⢦⠀⠀⣰⠖⠢⡀⠀⣀⣤⠀⠀⠀⠀⣠⠖⠒⢲⡀⠀⠀⢠⡶⠒⠒⠦⣄⠀⠀⠀⠀⢀⡴⠒⠒⣄
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⠉⣹⠃⠀⠀⠀⢰⡏⠀⠀⣸⠃⠀⠐⠟⠀⠀⠀⠀⠸⣆⠀⠀⢠⡟⠀⠀⠀⢸⡆
⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⢷⣀⣴⠏⠀⠀⠘⣗⡀⠀⠀⠀⣀⣟⠀⠀⣿⠃⠀⠀⠀⢸⡇
⠀⠀⣿⠀⢀⣤⠶⠶⢤⣄⠀⠀⠀⠀⠠⠶⣾⠶⠖⠀⠀⠀⣠⠜⠛⢦⡀⠀⠀⠀⠈⠻⠦⠤⠶⠋⣿⠀⠀⡿⠀⠀⠀⠀⢸⡇
⠀⠀⢹⣴⠋⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⠙⣆⠀⢠⡄⠀⠀⠀⠀⢰⡇⠀⠀⣧⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠻⣄⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣠⠏⠀⠸⣇⠀⠀⠀⢀⡾⠀⠀⠀⢹⡄⠀⠀⣸⠏
⠀⠀⠀⠀⠈⠛⠒⠒⠋⠁⠀⠀⠀⠐⠒⠛⠒⠀⠀⠀⠀⠈⠙⠒⠒⠊⠁⠀⠀⠀⠈⠛⠒⠒⠋⠀⠀⠀⠀⠀⠙⠲⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⠦⢤⡀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠳⣄⠀⢀⠜⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡱⡄⠀⠀⠄⠀⠀⢀⠀⢀⡜⠁⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣦⠤⠬⠷⠶⠥⣤⣿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⣀⣀⣀⠀⠀⠈⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠖⠋⠉⠀⠟⡏⠉⠙⠒⢤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠀⠱⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⣿⠉⣟⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣿⢠⡇⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢻⡷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠃⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣸⠇⣿⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢻⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠋⠲⠤⣤⣤⣠⣤⡤⠴⠚⠧⠤⠤⠔⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠘⠀⠀⠙⠁⠀⠘⠀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡄⣤⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠁⠀⣤⡀⠉⠘⢷⡀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡗⠀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠈⢷⡀⠀⠀⠀⠀⢠⠀⠸⣧⣽⠸⡧⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠐⠓⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⠐⠓
⠀⠀⠠⠄⠀⠠⠶⠶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢛⣭⠶⠛⠁⠀⠀⠤
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⢀⣠⣤⡶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠛⠛
⠗⠀⠀⢘⣋⣉⣀⣀⣀⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣉⣉⣉⣉⣙⠀⠀⠐
⣀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⢀

⠘⠂⠀⠀⠀⠀⠀⠀⣀⣤⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⢠⡀⠀⢀⣤⠶⢛⣩⡴⠞⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠵⠞⣋⣥⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣄⣙⠳⢦⡄
⠀⠀⠈⠁⠀⠀⢿⠉⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠉
⠀⠀⠀⠐⠖⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠖
⠀⠀⠀⠀⠀⢠⠄⠈⠁⠀⠀⠀⠀⠀⢀⡾⢡⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢠⡟⣡⡟⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⡟⣰⠏⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡄⠀⠙⠂⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⣤⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣰⣀⣀⣴⣄⣀⣰⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠈⢻⣿⠋⠁⠀⠀⠀⠈⢻⣿⠛⠁⠉⢻⣿⠋⠀⠀⠀⠀⠙⣿⡟⠋⠈⠻⣿⡟⠁⠈⢻⣿⠛⠁⠀⠀⠈⠛⣿⡟⠁⠈⠙⣿⡏⠁⠀⢹⣿⠋⠁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⡀⢀⣾⠏
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣧⣼⡟
⠀⠀⠀⢀⣼⣿⣄⠀⠀⠀⠀⢀⣸⣿⣄⡀⢀⣼⣿⣀⠀⠀⠀⠀⣀⣿⣧⡀⢀⣠⣿⣇⡀⢀⣸⣿⣄⡀⠀⠀⢀⣠⣿⣇⡀⠀⠀⠀⣀⣿⣿⣁
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⢠⣤⣤⣤⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⢠⣤⣤⣤⠀⢀⣄⣤⣤⡀⢠⣄⣤⣤⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⣠⣤⣤⣄⠀⣠⣄⣤⡄⢀⣤⣠⣤⡄
⠀⠀⠀⠀⠈⣿⡇⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣼⡟⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢠⣿⠋⠀⠀⠈⣿⡏⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠘⣿⡄⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⢰⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣾⡏⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠤⠿⠿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠤⠀⠀⠀⠤⠾⠿⠦⠀⠀⠀⠀⠀⠀⠀⠠⠼⠿⠧⠄⠀⠀⠀⠴⠿⠷⠤⠠⠴⠿⠷⠄



⠀⠀⠀⠲⢶⣶⠶⠀⠰⢶⣶⠶⠀⠲⣶⣶⠶⠀⠶⣶⣶⠆⠀⢶⣶⡶⠆⠀⠀⠀⠰⢶⣶⡶⠂⠰⢶⣶⠆⠀⣶⣶⠶⠀⠀⠀⠀⠶⣶⣶⠀⠰⣶⡶⠖
⠀⠀⠀⠀⠈⣿⡆⠀⢀⣿⠇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢻⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣦⣼⡟⠁
⠀⠀⠀⠀⠀⠘⣿⡀⣾⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣾⠏⠙⣿⣄⠀⠀⠀⠀⠀⠀⢠⣿⠋⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠀⠀⠀⠚⠛⠛⠓⠀⠚⠛⠛⠓⠐⠛⠛⠛⠃⠀⠀⠀⠐⠛⠛⠛⠂⠚⠛⠛⠂⠀⠛⠛⠓⠀⠀⠀⠀⠚⠛⠛⠂⠐⠛⠛⠓



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠁⢈⣿⠟⠋⠈⠻⣿⡟⠁⠀⠀⠀⠈⠙⢿⣏⠀⢹⣿⠛⠁⠘⢻⣿⠛⠀⠙⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⠏⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠈⢿⣶⡿⠃⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠁⢘⣿⣆⡀⠀⣠⣿⣇⡀⠀⠀⠀⠀⣀⣾⣟⠀⢹⣷⣄⡀⢀⣼⣿⣄⠀⣀⣼⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠁⠈⠉⠉⠉⠀⠉⠉⠉⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⢶⠶⣶⡶⠶⠶⣶

⠙⢿⡟⠁⠀⠀⠀⠈⢻⡿⠋⠈⠻⣿⠋⠀⠀⠀⠈⢻⣿⠋⠈⠻⣿⠛⠀⠙⣿⡟⠁⠀⠀⠈⢻⡿⠋⠀⠙⣿⡏⠁⠈⣿⡟
⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣷⡀⣰⡿
⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⣷⣿⠁
⠴⠿⠷⠆⠀⠀⠀⠰⠾⠿⠦⠠⠾⠿⠦⠀⠀⠀⠠⠾⠿⠦⠠⠾⠿⠶⠀⠴⠿⠷⠄⠀⠀⠰⠾⠿⠦⠀⠀⠀⠴⠿⠷⠄


⢀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⠀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣀⣀⣀⠀⣀⣀⣀⡀⢀⣀⣀
⠀⠹⣯⠁⠀⢹⡿⠉⠀⠀⠀⠀⠀⠙⣿⡅⠀⢈⣿⠋⠁⠈⢹⣿⠁⠀⠀⠀⠀⠈⠹⣯⠀⠀⢹⡟⠉⠀⠈⣿⡏⠁⠈⢹⡟
⠀⠀⢻⣇⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣼⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢻⣇⢠⣿⠁⠀⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⢿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⠇⠀⠀⠀⠀⣹⡃⠀⠀⢸⣇
⠀⠀⠐⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠓⠂⠀⠀⠐⠛⠛⠓⠀⠀⠀⠀⠀⠀⠐⠛⠛⠒⠀⠀⠀⠚⠛⠛⠂⠐⠛⠛


⣤⣤⣤⠄⠠⣤⣤⣤⠀⣤⣤⣤⠄⢤⣤⣤⡄⢠⣤⣤⡤⠀⠀⠀⢠⣤⣤⡄⢠⣤⣤⡄⠠⣤⣤⠄⠀⠀⠀⢤⣤⣤⠀⢤⣤
⠀⠹⣧⠀⠀⣼⡏⠀⠀⠀⣿⠁⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⡏⠀⠀⠈⢻⣆⣰⡿⠁⠀⠀⠀⠀⠀⠙⣷⣀⣾⠏
⠀⠀⢻⣇⣸⡟⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠈⣿⣿⠁⠀⠀⠀⢀⣿⡄⠀⠀⣼⣇⠀⠀⣸⣿⠀⠀⠀⠀⠀⣸⣇⠀⠀⢀⣾⠋⠙⣷⡀⠀⠀⠀⠀⠀⣰⡿⠉⢻⣆
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠁⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⡆⠀⢶⣶⠆⠰⢶⣶⠶⠀⠀⠀⠀⠶⣶⡆⠀⣶⡶⠆⠰⣶⣶⠶⠀⢶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣷⣴⡟⠁⠀⠀⢸⣿⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡏⠈⣻⣦⡀⢀⣸⣿⣄⠀⠀⠀⠀⣠⣾⡏⠈⣿⣆⡀⢀⣼⣿⣀⢀⣰⣿⣄
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠋⣀⣀⣈⣉⡉⠉⠉⠉⣩⣿⣿⠏⠀⠀⠀⠀⠿⠿⠿⠿⠿⢿⣿⣿⡆⠀⢀⣀⣀⣀⣀
⣿⣿⣿⡆⠀⣀⣴⣾⣿⣿⣿⣶⣦⡈⠁⠀⢠⣾⣿⣿⣿⣿⣿⣷⠀⠀⠻⢿⣿⣆⠀⠀⣀⣴⣶⣶⣶⣦⣄⠸⣿⣿⡇⢰⣿⣿⣿⣿⣿
⣿⣿⣿⠃⢰⣿⣿⡟⠉⠛⠻⣿⣿⣷⡀⢀⣿⣿⡏⠁⠀⠈⢻⣿⣇⠀⠀⢠⣿⣿⠀⢰⣿⣿⠟⠉⢻⣿⣿⡀⢿⣿⠇⠈⠛⠻⠟⠛⠛
⣿⣿⠏⠀⢸⣿⣿⡀⠀⠀⠀⠸⣿⣿⡇⢸⣿⣿⠁⠀⠀⠀⠘⣿⣿⣷⣾⣿⣿⠟⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣿⠋⠀⠀⠸⣿⣿⣷⣤⣶⣶⣾⣿⣿⡿⢸⣿⣿⣶⣿⣿⣿⣦⡈⠙⠿⠛⠛⣁⡀⠀⢹⣿⣿⠀⠀⢰⣿⣿⡇⠀⣤⡀⠀⠀⠀⠀⣴⣿
⠃⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⣿⣿⡇⢸⣿⣿⣿⠿⠿⢿⣿⣷⠀⠀⣠⣾⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⢀⣿⡇⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠸⣿⣿⡇⠀⠀⣠⣿⣿⢠⣾⣿⣿⣿⡇⠀⠘⣿⣿⣦⣤⣾⣿⣿⠃⢸⣿⡇⠀⠀⣼⣿⣿⡿
⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⡇⠀⠀⠈⠛⠛⠿⠿⠛⠁⠀⠸⣿⠇⠀⣸⣿⣿⡿⠁
⠀⠀⢸⣿⣿⡟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⣉⣙⣛⣛⠛⠁⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠻⡿⠛⠁
⣄⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣀⣠⣤⣤
⠿⠀⣿⣿⣿⣇⣀⣤⣄⣀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠉⢻⣿⣿⡇⠀⠀⠘⣿⣿⠃⠀⠀⢸⣿⣿⡿⠉⠀⠀⣿⣿⣿⡀⠀⠀⠻⠿⠿⠿
⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⡀⠀⠀⢸⣿⣿⣷⣄⡀⢀⣿⣿⣿⡇⠀⠀⠀⢀⣠⣤
⣿⡄⠘⠻⠟⠛⠛⠛⠻⢿⣿⣿⣿⣇⢠⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⢠⣾⡿⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣾⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠸⣿⣿⠁⠀⠀⠀⣿⣿⠇⠀⣰⣿⡿⢡⣾⣦⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⡀⠀⠀⠘⠻⠟⠁⢸
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⣿⣿⠀⠀⠀⢰⣿⣿⠀⣼⣿⣿⣦⣾⣿⡇⠀⢠⣾⣿⣿⠟⠋⠀⠻⣿⣿⡄⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠀⠸⣿⣿⣆⠀⠀⣸⣿⣿⠀⠙⠛⠛⠛⢻⣿⡇⠀⣿⣿⡿⠁⠀⠀⠀⢀⣿⣿⣿⠀⠀⢀⣾⣿⣿
⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⣷⠀⢿⣿⣿⣶⣶⣶⣶⣿⣿⣿⠟⠀⢠⣾⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠹⢿⠏⠀⠀⠙⠻⠿⠿⠿⠿⠿⠟⠉⠀⠀⠘⠛⠛⠛⠿
⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣠⣶⣿⣿⣿⣷⣤⠀⠀⣴⣶⣶⣶⣶⣶⠆⠀⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣄⠀⠀⠀⣠⣶⣿⣿⣷⣦
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⠟⠋⠛⢿⣿⣇⠀⠀⠀⢀⣀⣀⠀⠀⠀⣿⡿⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⠄⠀⢰⣿⣿⠋⠉⠉⢿
⠀⠀⢀⣼⣿⣿⣏⡀⠀⠀⢸⣿⣿⡀⠀⢀⣼⣿⣿⠀⢠⣾⣿⣿⣿⣿⡄⠀⣿⣧⣤⣄⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⣾⣿⣯⣤⣤⣤⣀
⡀⠀⠸⣿⡿⠿⣿⣿⡆⠀⠀⠻⣿⣿⣾⣿⣿⡿⠁⠀⠻⠟⠁⠀⣽⣿⡇⠀⠻⠋⠙⣿⣷⠀⠀⠀⣼⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣼⣿⡇⠀⢀⣼⣿⠟⠻⣿⣿⣧⠀⠀⠀⢀⣴⣾⣿⡟⢀⣴⣦⣀⣤⣿⣿⠀⠀⣸⣿⣿⡏⠀⢀⡀⣿⣿⣿⠀⠀⠈⣿
⢣⣤⣤⣄⣠⣼⣿⡿⠃⠀⣾⣿⡏⠀⠀⠈⣿⣿⡇⣠⣾⣿⣿⣟⣉⣀⣈⣻⣿⡿⠿⠛⠁⠀⢰⣿⣿⡿⠀⢀⣿⣿⢻⣿⣿⣦⣤⣾⣿
⠘⢿⣿⣿⣿⡿⠟⠁⠀⠐⣿⣿⣧⣀⣀⣼⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣥⣤⣼⣿⣿⠈⠻⢿⣿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⡋⠉⠀⣿⣿⣿⣿⣿⡇⠀⣿⣿⡷⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣴⣿⣿⠀⠀⢿⣿⣿⡿⠋⠀⠀⠈⠁⣀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⡄⠘⢿⣿⡿⠀⠀⣀⠉⠁⢀⣤⣄⠀⠀⣾⣿⣿⡄⠙⠛⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⣿⣿⠀⠀⠈⢀⡀⠸⠿⠷⠀⠘⣿⣿⠀⠀⣿⣿⣿⣿⠀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⣿⣿⣿⠀⢸⣿⣿⣿⡆⢀⣤⣤⣄⡈⠁⠀⠀⠈⠛⠛⠁⠀⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢈⣛⠋⠁⢀⣀⠉⠛⠛⠁⢸⣿⣿⠿⣿⣦⠀⠶⣿⣷⠄⢰⣶⡄⠀⢈⡉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠀⠺⠿⠇⢠⣿⠟⠋⠁⠀⠀⠀⠈⢳⣄⡀⠉⠀⠈⠉⠁⢰⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢠⣿⣷⣄⠈⠻⢿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠃⠀⠈⠻⠿⠟⠀⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⢀⣴⣦⠀⢀⣴⣆⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣿⣿⠇⢀⣾⣿⣿⠀⢾⣿⡿⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢁⣤⣀⡙⠋⠀⠘⣿⣿⠇⠀⢀⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⠀⢿⣿⣿⣿⣷⡄⠀⠈⠉⠀⠀⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⠀⠈⢿⣿⣿⡿⠃⠀⣴⣿⣦⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣶⣶⡄⠈⠉⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⣾⣷⡝⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⣄⣰⣶⣦⡈⠀⠀⢠⣾⣧⠀⠙⠛⣁⡀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡷⠀⠀⣿⣿⣿⠀⢠⣾⣿⣿⡇⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠈⠙⠁⠀⣿⣿⣿⡿⠁⠈⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⠀⠀⠙⠛⠋⠀⠀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⠀⠀⢨⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⢀⣤⣶⡆⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢻⣿⡿⠃⠀⠀⠘⣿⣿⣿⡿⠉⠲⢤⣄⣀⣀⣠⠞⠙⠛⠁⠀⠈⠻⠟⠁⠀⠸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠉⠀⣠⣶⣦⡄⠈⠉⠉⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⢠⣾⣿⡆⠀⢀⣤⣤⡀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠹⢿⠟⠁⣠⣴⣶⣶⣦⠀⠙⠿⢿⡿⠏⠀⠀⠻⣿⣿⠇⠀⣿⣿⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠀⠀⠀⣀⡀⠀⠹⣿⣿⣿⣿⠇⠀⠀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠘⠻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣰⣿⣿⣿⣷⣄⠈⠉⠋⠁⢠⣶⣿⣿⣿⣆⠀⣾⣿⣿⣿⡿⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣇⠀⠀⠀⢿⣿⣿⣿⣿⣿⢀⣈⣻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡉⡤⢤⠀⠀⠀⠀⠀⡠⢔⣬⢖⡐⡀⠀⢨⢙⡟⠂⡇
⠀⠀⠀⠀⢀⣠⣄⣤⣖⡔⡔⠀⠀⡠⠨⡠⡞⠁⠀⠈⠙⠾⢶⠀⠀⠀⠀⢠⠇⠀⡥⣕⣋⠉⣷⠃⢁⠈⢹
⠀⠀⢀⡴⠋⢀⡤⠤⡑⠗⢣⢀⡬⠣⢽⡁⠘⠚⠉⢹⠀⠀⢸⠀⠀⠀⠀⣼⠀⠘⠧⣀⠀⡹⠁⢠⢺⠀⢸⡄
⠀⠀⡞⠀⣰⠃⠀⠀⠘⣆⠼⠋⠀⢀⠼⡇⠀⣀⠴⠋⠀⣠⢏⠀⢀⡀⠀⠉⠉⢢⠀⠈⢻⠁⠐⠓⠚⠀⠘⠓⣦
⠀⢸⠃⠀⣇⠀⠀⠀⢰⠛⢦⡖⢻⠀⠀⡗⠉⠁⠀⣤⡚⢁⣮⣴⣥⣵⠖⠲⠤⠞⠀⠀⢸⠦⠤⠤⢤⠀⢠⡖⠟
⠀⠘⣇⠀⠘⠦⠤⠖⠃⠀⡼⠁⢸⠀⠀⣧⣄⣀⠀⠀⡸⠉⢈⠥⢍⠉⢳⣄⣀⣀⣠⠴⠃⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠈⠲⣄⣀⣀⣀⣠⠞⠁⠀⢸⠀⠀⢿⠀⠈⠉⠓⢧⡀⠘⠲⠚⢀⡼⠀⠀⠀⠀⡀⢦⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢳⡖⠚⠀⠀⠀⢠⠖⠉⢀⣀⣀⠉⠲⣄⡠⠶⠶⣤⡸⣤⡀⢠⠚⢄
⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠲⠴⢞⠥⢹⣀⠀⠀⠀⣏⠀⠐⣇⣀⣸⠆⡰⠋⠀⣠⠴⠦⠤⣧⢆⡺⠫⠎⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠾⣤⣀⣀⡀⠈⠉⠁⣸⠤⢤⢡⡛⣐⣇⠈⠉⠁⣰⠁⠀⠀⠉⠉⠓⢤⡇⠀⡔⠉⠙⡆⡸⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⠔⠿⠤⣀⡉⠑⠚⡩⠋⠉⠉⣿⠀⢠⠋⠉⢱⠀⠘⣇⠀⠙⠶⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡴⠋⠀⠀⢠⠔⠊⠀⣠⠊⠀⠀⠀⠀⢹⡀⠈⠳⠴⠊⠀⣸⠉⠲⠤⡤⠴⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡞⠁⠀⠀⠀⠘⠶⢤⡀⠈⠱⡄⠀⠀⠀⠀⠙⠲⠤⠤⠴⠚⠁⠀⠀⣀⡤⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠻⠤⢤⡞⠀⠀⠀⠀⡴⠤⣤⠼⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠶⣶⠶⠀⠀⠀⠰⢶⡶⠆⠲⢶⠶⠀⠀⠀⠰⢶⡶⠆⠶⣶⠶⠀⠶⡶⠆⠀⠀⠐⠶⡶⠆⠰⢲⡶⠆⠶⣶⠶
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣇⣿
⠀⠀⠀⣀⣿⣀⠀⠀⠀⢀⣸⣇⡀⢀⣸⣀⠀⠀⠀⢀⣸⣇⡀⣀⣿⣀⠀⣀⣇⡀⠀⠀⠀⣀⣇⡀⠀⠀⢀⣿⣇
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠀⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠈⠹⣏⠁⢉⡿⠉⠀⠀⠀⠈⢹⡏⠁⢉⡟⠉⠈⢹⡟⠁⠀⠀⠈⠙⣿⠁⠉⣿⠉⠁⠉⣿⠉⠈⢹⡏⠁
⠀⠀⠀⠀⠀⠀⢿⡀⣸⠇⠀⠀⠀⠀⠀⠀⢿⠀⣸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⡟⠀⠀⠀⠀⠀⠀⠀⠸⣧⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢤⠿⢧⠄⠀⠀⠀⠀⠀⠀⢤⠿⢥⠀⠀⠠⡼⠧⡤⠀⠀⠀⠀⢠⠽⢧⠄⠀⠀⡤⠿⡤⠠⡼⠧⡄

⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⣀⣀⢀⣀⣀⡀⣀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣧⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢘⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡞⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠒⠀⠀⠐⠚⠓⠂⠒⠚⠓⠂⠒⠛⠒⠀⠀⠀⠀⠀⠐⠚⠓⠂⠒⠛⠓⠐⠚⠓⠒

⠀⠀⠀⠰⠶⣶⠆⠴⣶⠶⠄⠀⠀⠀⠀⠶⣶⠆⠰⣶⠶⠆⠶⣶⠶⠀⠀⠀⠶⣶⡶⠠⢶⡶⠆⠰⢶⡶⠄⠶⣶⠶
⠀⠀⠀⠀⠀⠹⣆⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣠⡟⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⢿⡀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⢀⣾⠁⠘⣧⡀⠀⠀⠀⠀⠀⢀⣼⠃⢈⣷⡀⠀⠀⣿⢀⠀⠀⠀⠀⣰⡏⠈⢻⡄⡀⠀⣸⡇⠀⡀⣿
⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠁⠉⠉⠉⠀⠀⠀⠉⠉⠉⠈⠉⠉⠁⠈⠉⠉⠁⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⢚⠷⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⡀⠀⣠⠴⠦⡶⣥⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠉⠓⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⣧⣛⣇⡞⠁⠀⣀⣀⠀⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⡻⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢹⡉⠲⠋⠁⢸⡆⠀⢠⡇⠀⠀⠀⣀⠊⣰⢊⡙⢆⠀⠀⠀⢀⠔⠁⠀⠐⠣⠤⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡖⠁⢀⡄⠘⠉⣇⠀⠀⢀⠞⠀⢀⡾⠀⣾⠉⠉⠉⠑⠉⠮⠴⠋⢹⠀⠀⠈⠛⠲⠦⢤⣀⠀⠀⠀⠉⡶
⠀⠀⠀⠀⠀⠀⠀⠹⠶⠋⢹⡀⠀⢸⣄⠴⠃⠀⣠⣾⣤⣀⣿⠤⠤⠤⣤⠄⠀⠐⢁⡼⠀⠀⢀⠀⡠⠀⠀⡾⠀⢠⡟⠒⠃
⠀⠀⠀⠀⠀⢀⠄⣀⠴⡆⠈⣇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠃⠀⢀⡖⠉⣠⣅⣶⢇⠷⣸⡀⠘⠧⣤⡼
⠀⠀⠀⠀⠀⣢⣤⢥⣄⡰⣤⢹⡄⠀⢸⡗⠒⠒⠒⠒⠒⠒⠃⢀⡞⠀⠀⣰⠋⢠⠞⠁⢀⡤⠤⣍⠀⠙⢦
⠀⠀⠀⣰⠋⠀⠀⣀⣡⣀⡏⠀⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠇⢀⡏⠀⢰⠃⠀⠀⠀⢳⠀⠘⡇
⠀⠀⣼⠁⠀⠠⠯⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⢸⠃⡀⣿⡐⠆⠀⠀⢸⠀⠀⣿
⠀⠀⡇⠀⠀⢀⣠⣄⠀⠈⢳⠀⢠⠀⢀⣰⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⡼⣇⢤⡋⣌⡦⢤⣠⠞⠀⠀⡟⠀⠀⠀⢀
⠀⠘⡇⠀⠠⡏⠀⠀⣷⢀⡬⠧⡄⢮⡡⢮⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢁⣭⣬⣀⣸⠁⠀⢀⡼⣃⡤⠶⢶⠡⣰⠐⢶
⠀⠀⢷⡀⠀⠓⠦⠖⠃⢸⣇⣀⣈⡁⠈⠉⡴⠁⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⡇⠀⢸⣳⣦⣤⣤⠤⠶⠋⡼⠁⠀⣀⢬⣉⠫⠥⣎
⠀⠀⠀⠳⢤⣀⣀⣀⣤⠖⠃⡠⠊⠁⢀⡞⠀⠀⠀⣰⣤⣭⣤⣵⠀⠀⠀⠀⠧⠤⣄⡀⠈⠑⣆⠀⠀⢸⡃⠀⢸⡁⠀⢈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢺⣀⣀⡀⠀⠉⠳⣤⠊⠁⣉⠭⡉⠉⢳⡀⠀⠀⠀⠀⠀⡷⠀⠀⢸⠂⠀⠈⢧⡀⠀⠙⠒⠋⠠⠔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣿⠀⠀⠧⡤⠟⠀⣸⠃⠀⡸⠉⠙⠉⠀⠀⢀⡼⢠⠶⠤⣤⣩⠷⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠁⠀⠀⣠⣯⠷⠀⠀⠀⠀⠾⢅⠀⠀⠙⠶⠤⠤⠤⠖⠋⠀⢿⣀⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣤⠤⠴⠚⣽⠁⠀⣰⠚⠛⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⠤⠤⠞⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡤⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⢠⡂⢲⡀⣴⣶⣦⡀⣶⣒⠂⢰⠶⢶⡖⠀⠀⣰⠐⢲⡴⠶⣶⡖⢠⠰⣆⠆⡐⣄⠀⣴⠖⠒⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠈⢁⣼⠹⡿⠀⣽⡇⢬⣉⡗⢰⠀⣾⠁⠀⠀⣿⢠⣬⠀⠾⢿⣄⢐⢸⣿⠇⠒⢸⣧⣹⡗⠒⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⣿⣿⢀⡀⢹⡄⢀⣾⠟⠀⣠⣷⣦⡀⢸⣿⠀⠀⠰⣿⠀⢶⠐⢶⡾⠇⢨⢘⠿⡀⠍⣸⣿⠘⢓⣒⣚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⢸⣿⠈⠃⠚⠰⠿⠿⠿⠇⣿⡇⡘⠃⢨⠭⡄⠀⠀⢸⠀⣿⢸⡇⢸⡇⠑⢾⣶⣷⣾⣿⠏⠸⠿⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⠟⢰⣋⣵⠐⠒⠂⠈⠀⣿⡟⢻⡇⢸⡀⡿⠀⠠⢖⡦⡾⡌⢛⢻⡗⠷⣶⢪⣍⢹⠓⠀⠐⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⡀⣰⠞⢻⠀⡆⠀⠀⠰⠂⣿⣇⣼⡇⢸⠛⣧⠀⢀⡘⣧⠙⠁⡀⡨⡁⢰⡟⠈⠛⣈⠙⡆⢸⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡽⠙⣒⡋⢀⣥⣝⡀⣾⢸⡎⣉⣋⣀⢸⡀⣿⠀⠀⢩⠱⣢⢀⡇⢀⡇⣿⠃⣾⠃⠤⠴⠃⣐⣦⢠⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⢸⣉⡁⣿⠋⣹⣷⠈⠼⠃⢉⣿⠃⢈⣛⡁⠀⠀⠸⠴⠥⠨⡌⠉⠧⣖⠂⣿⠻⣦⡆⣷⠀⣿⣸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣸⠇⣿⠿⢻⣿⢸⢿⡂⠈⢹⡇⠸⢤⡄⠀⠀⣺⣼⠃⡜⢱⡾⠀⣿⠃⣿⠀⣿⠤⠟⠉⠁⠘⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠅⠀⠻⠿⠿⠏⠸⠼⠑⠦⠾⠃⠯⠼⠃⠀⠐⠧⠼⠁⠐⠸⠯⠘⠤⠏⠻⠾⠟⠰⠂⠀⠀⠄⠅⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⢀⣀⠀⣀⠀⣀⣀⢀⣀⠀⣀⣀⢀⡀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡤⠘⢧⣜⠸⣿⡌⣑⡆⡏⢹⢺⣿⢨⡕⣺⣭⠃⠀⢰⡋⠀⢹⠤⣿⡇⠈⠀⠐⠛⠸⠟⠛⣿⡆⣭⣍⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠣⢰⡟⢹⣧⠈⣴⠒⠀⣥⠾⢀⠋⠰⠲⠶⣽⠧⠀⠀⡑⣲⠋⠀⢺⢶⡶⠶⢸⣻⣴⡟⠻⠿⠳⠆⠹⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠄⠠⣼⡟⢰⣿⢹⡆⣧⣴⠸⠠⠇⢁⠎⣿⣷⠀⢀⡴⣧⠀⣦⡉⠸⠟⣷⠰⡿⠻⠿⡿⠿⠇⢈⡟⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣦⣄⢸⣿⠀⠻⠞⢡⣶⠀⡄⢦⢀⣿⢀⣭⡍⠀⠘⠧⣿⠸⠷⡷⠰⣦⣿⠸⣿⢰⣿⠅⢀⠘⠋⢸⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⠻⠿⠋⢸⠀⡇⢻⣿⣦⣅⡜⠀⠋⠞⣶⡄⠀⠀⡉⠵⢚⣶⢴⠶⠶⠶⣶⣴⣆⠀⠀⠔⠀⠤⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣶⠑⢹⡴⣿⢈⡐⡇⣴⠙⠻⢿⣿⣆⠺⠴⠟⠀⠀⢀⡅⢘⣿⢅⡆⠈⣱⢞⣁⣿⣟⣩⣭⠙⢬⠋⣈⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡶⡾⢿⡃⣿⢈⣌⡈⢿⡐⠇⠀⣿⠇⠀⡖⠒⣇⠀⠈⠲⠛⠋⢈⠋⢺⡇⢸⢲⣬⠁⢋⣿⠀⢢⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇⠿⣿⣸⡇⠀⢁⣀⠈⣥⡄⠀⣇⣀⡄⠀⠐⠟⣿⠈⢸⠰⣿⣿⡏⣭⣐⢠⣆⢻⡗⡃⠸⢷⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⢠⡞⠙⡆⣿⢻⡇⠀⠅⠄⠄⠰⡇⠀⡇⠀⡇⠀⢠⡾⢋⠀⢸⢼⣏⢹⡧⡾⢠⡏⡛⡋⢸⢾⠰⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠛⠁⠈⠛⠁⠀⠂⠈⠀⠛⠃⠘⠓⠚⠃⠀⠈⠛⠛⠀⠛⠈⠛⠛⠁⠃⠘⠁⠁⠈⠈⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⣀⣠⠤⠒⣦⣀⣀⣀⡤⠖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠈⠻⣯⠥⠀⠀⠀⢿⠉⣽⠉⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⢳⡀⠀⠀⢸⣠⠇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠈⢀⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢴⣶⡷⠞⠛⠛⠉⠉⠉⠉⢉⣩⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣛⣿⣯⡤⣤⡤⠤⢴⡶⠶⠛⠉⠁⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⣭⡽⠋⡽⠀⢀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡸⠓⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠰⠃⢀⡄⡦⡀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠰⡀⠀⠓⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡄⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⡉⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠱⠼⠘⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠘⡏⠁⠹⡏⠀⢹⠏⠀⠀⠀⠀⠈⣿⠁⠈⣿⠁⠀⠀⠀⠀⠹⡏⠀⢹⠏⠈⢹⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⡿⠀⠀⠀⠀⠀⠀⠸⡇⢸⠇⠀⠀⠀⠀⠀⠀⢿⠀⡿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠀⠚⠂

⠀⠀⠀⠀⢤⡄⠀⢤⡤⠀⣤⠄⠠⡤⠀⠀⠠⣤⠀⠠⣤⠄⢤⡤⠀⣤⠄⠠⡤⠀⠀⠀⢤⡤⠠⣤⠠⡤
⠀⠀⠀⠀⠘⣧⠀⣾⠀⠀⣿⠀⠀⡇⠀⠀⠀⢻⡀⢰⡇⠀⢸⡇⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇⠀⢹⣼⠃
⠀⠀⠀⠀⠀⢹⣴⡇⠀⠀⣿⠀⠀⡇⠀⠀⠀⠘⣇⡿⠀⠀⢸⡇⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇⠀⢰⣿⡀
⠀⠀⠀⠀⠀⢈⣿⡀⠀⢀⣿⡀⣠⣇⠀⠀⠀⠀⣻⣇⠀⠀⣸⣇⢀⣿⡀⣠⣇⠀⠀⠀⣸⣇⢀⣿⢈⣧⡀

⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⢀⣀⡀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢷⣸⠏⠀⠀⠀⠀⠀⠀⠸⣇⣾⠁⠀⡏⠀⠀⠀⠀⠀⠹⣇⡾⠁⠈⡇⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡾⢹⡆⠀⠀⠀⠀⠀⠀⢠⡏⢷⡀⠀⡇⠀⠀⠀⠀⠀⢰⡏⢷⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠈⠉⠀⠀⠀⠀⠀⠉⠁⠉⠁⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢫⠏⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠂⠀⠈⠉⠁⠀⠀⠀⡞⡸⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠠⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⢢⠀⡀⠀⠀⠀⠀⠀⣴⡇⠀⠀⡆⣥⣀⡀
⠀⠀⠀⠀⠁⠒⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠐⠁⠀⠀⠀⠤⠄⢠⡤⡀⠀⠀⠀⠀⢱⢙⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⢀⣴⣡⠊⠀⢀⡀⠀⠀⠀⠀⠀⠢⠷⠄⠀⠀⠀⡖⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣮⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠉⠉⠈⠈⠉⠁⠀⠁⠀⠉⠉⠀⠀⠀⠀⠉⠐⠐⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠊⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠖⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢒⠆⠀⠀⠀⠀⠀⠀⠒⢀⡤⡀
⠀⠀⠀⠀⠀⢀⡤⣠⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡌⠀⠀⠀⠀⡠⡴⠀⠀⠀⢱⢹
⠀⢀⠄⠀⠀⠀⠀⠀⠉⠙⢢⠢⡀⠀⠀⠀⠀⠀⠀⢀⣎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠜⠀⠀⠀⠀⠀⠢⠧⠄⠀⠀⡨⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠱⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⡖⡶⠀⠀⠀⠀⠱⡈⡄⠀⠀⠀⠀⠀⠀⡜⠐⠁⠀⠀⠀⠀⠀⢀⠔⡶⠀⠀⠀⠈⢢⠱
⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠎⡐⠀⠀⠀⠀⠀⡇⢇⠀⢀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡸⠠⠁⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⢨⠆⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⢜⡴⠁⠀⠀⠀⠀⠀⠉⠐⠓⠁⠀⠀⣰⢁⠆⠀⠀⠀⠀⣰⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠚⠂⠀⠀⠀⢀⠔⡸
⠀⠀⠀⠀⠀⠀⠴⠶⠾⠷⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣠⡤⠾⠵⠁⠀⠠⠤⠤⠤⠯⠾⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⠤⠤⠤⠴⠶⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠑⢬⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⠀⠀⡠⡺⠋⠀⠀⠀⠀⠀⠀⠀⠱⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣱⣄⠀⠀⠀⢀⢇⡇⠀⠀⠀⠀⡘⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠊⡘⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠎⠀⠀⠀⠀⠀⠀⢡⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⠤⠤⡄⠀⠀⠀⠀⢀⡌⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⢀⣠⣣⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠉⠉⠉⠉

⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠈⠉⣿⠉⠁⠀⠈⣟⢹⠉⠀⠀⢹⡏⣿⢹⡏⠀⠀⢹⡏⢻⡉⢹⡏⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠈⠁⠀⠀⠈⠁⠉⠈⠁⠀⠀⠈⠁⠉⠀⠈⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⢸⠀⠀⠀⢸⡇⣿⢸⡇⠀⠀⢸⡇⠘⣇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈
⠀⠀⠰⠶⠿⠶⠄⠀⠴⠷⠾⠶⠀⠀⠾⠷⠿⠾⠷⠀⠀⠾⠷⠶⠿⠷⠶⠄⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠖⠰⠂⠀⠀⠰⠂⠖⠰⠂⠀⠀⠰⠆⠀⠺⠇
⠀⠀⢀⣤⣤⣤⣤⣄⠀⢀⣤⣤⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⢀⡀⠀⣀⢀⡀⠀⠀⠀⣀⠀⢀⡀⣀⢀⡀
⠀⠀⠀⠸⡇⠀⡟⠀⠀⠀⠘⣇⢀⡟⢸⡇⠀⠀⠀⢿⠀⣼⠁⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠉⠀⠀⠀⠈⡀⠀⡁⠀⠁⠀⠀⠀⠁⠀⠨⠀⠁⠈
⠀⠀⠀⠀⢿⣸⠃⠀⠀⠀⠀⢻⣼⠃⢸⡇⠀⠀⠀⠸⣧⡏⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⣄⠂
⠀⠀⠐⠒⠛⠛⠓⠒⠀⠐⠒⠚⠛⠒⠚⠓⠀⠀⠒⠚⠛⠓⠒⠛⠚⠓⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠒⠀⠀⠀⠐⠚⠛⠂⠚⠃⠀⠀⠀⠒⠛⠒⠐⠛⠚⠃
⠀⠀⠠⣤⡤⠤⣤⢤⡤⣤⢤⡤⠀⠀⢤⡤⣤⡤⣤⠤⠀⠠⢤⡤⣤⡤⠀⠀⠀⠀⠀⠀⠀⢤⠄⠠⡤⢤⠄⡤⢤⠄⠀⠀⢠⡤⢤⠤⣤⠀⠀⠀⢤⡤⢤⠄
⠀⠀⠀⠘⣇⢠⡇⢸⡇⣿⢸⡇⠀⠀⢸⡇⠘⣷⠏⠀⠀⠀⠈⢷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⢻⣾⠀⢸⡇⣿⢸⡇⠀⠀⢸⡇⣰⡟⣆⠀⠀⠀⢀⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡜⠀⢀⠀⡀⢀⠀⠀⠀⠀⡀⢀⠋⡀⠀⠀⠀⢀⡈⢂
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠉⠈⠁⠀⠀⠈⠁⠉⠁⠉⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠲⣶⢶⡶⢶⡶⠀⠰⢶⡶⣶⠶⡶⢶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⡆⠰⠄⠀⠀⢠⠆⠦⠀⠆⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⢸⡇⠀⠀⠈⣿⠃⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⢲⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣹⣆⣸⣇⠀⢀⣰⣏⣧⣀⣇⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⣄⢀⡀⠀⠀⣠⡉⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⣠⣾⠛⠱⣦⠀⠀⠀⠀⣴⠾⣷⠀⠀⠀⠀⠐⠾⠛⠿⣦⠀⠀⠀⠀⣴⡶⠤⡀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠂⠀⠀⠀⢀⣀⣼⠇⠀⠀⠀⢠⣾⠛⣿
⠀⠀⢻⣇⠀⢀⣿⠃⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠈⠉⠙⣷⡀⠀⢴⣿⣥⣼⣿⣤
⠀⠀⠈⠻⠷⠿⠋⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠐⠟⠷⠒⠶⠂⠀⠀⠀⣤⣤⣤⡿⠃⠀⠀⠀⠀⠀⣿

⠀⠀⠀⢰⡶⠶⠶⠀⠀⠀⠀⢀⣤⣴⡶⠀⠀⠀⢠⣤⣴⣤⣴⡄⠀⠀⢀⣤⣤⣴⣤⡀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⢀⣿⣤⣤⡀⠀⠀⠀⢰⣿⣋⣀⣀⠀⠀⠀⠀⠁⠀⣼⡿⠁⠀⠀⠺⣯⠀⣨⣿⠇⠀⠀⢰⣏⠀⠈⣿⡄
⠀⠀⠈⠉⠉⠻⣿⡆⠀⠀⢾⣿⠏⠉⠻⣷⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⣠⡿⠛⠻⣧⡀⠀⠀⠈⠿⠶⢾⣿⡇
⠀⠀⢠⣶⣴⣾⠟⠀⠀⠀⠘⢿⣦⣤⣴⠟⠀⠀⠀⣼⡏⠀⠀⠀⠀⠸⣿⣧⣄⣀⣿⠇⠀⠀⢠⣤⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡂⠀⠀⠀⠀⣜⡁⠔⠻⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡦⠀⠀⠀⠀⠀⠀⠀⠱⢶⡏⢣
⠀⠀⠀⠀⣠⠦⠿⠯⣛⡸⢄⠀⠀⠀⠀⠀⢠⣐⢇⣚⡇⠀⣠⠖⠋⠛⠛⠾⠟⣆⠀⠀⠀⣤⠤⠴⣰⣎⠶⡼⢤⠀⠀⠀⠀⢠⠞⠒⠓⠲⡎
⠀⠀⢀⡞⠁⢠⠴⠒⠦⡀⠀⢳⡀⠀⣀⡴⠋⠀⠀⢸⡇⠰⢇⡀⣠⠶⢤⠀⠀⢸⡆⠀⠀⢧⣀⣀⣈⠀⠉⣰⠏⠀⠀⠀⢠⠏⠀⣰⠂⠀⡇
⠀⠀⡾⠀⢰⠇⠀⠀⠀⢸⡄⠀⣧⠀⣇⣀⣤⡄⠐⢹⡇⠀⠀⠉⠁⠀⣸⠀⠀⣸⠁⠀⠀⠀⣠⠔⠁⠀⣴⠃⠀⠀⠀⣰⠃⢀⡜⢸⠀⠀⡇
⠀⠀⡇⠀⢸⡀⠀⠀⠀⢸⠇⠀⣹⠀⠉⠁⢸⡇⠀⢸⡇⠀⠀⠀⢀⡴⠁⠀⣰⠃⠀⠀⠀⠈⣧⣠⣀⠀⠀⠙⢦⠀⣰⠃⠀⠈⠉⠉⠀⠀⠙⢻⡆
⠀⠀⢷⠀⠀⠳⣤⣠⡴⠋⠀⢀⡟⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⡔⠋⠀⠀⠚⠓⠒⢲⡆⠀⠀⠀⠀⢈⡇⠀⠀⢸⡖⠳⠲⠶⠶⠶⢶⠀⠀⡖⠚⠃
⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⡇⠀⢸⡇⠀⠐⣇⣀⣀⣀⣀⣀⣀⣸⡇⢀⡏⠙⠛⠉⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠒⡤⠒⠄⠀⠀⠀⠀⠀⠀⠠⠁⠐⠈⢣⡀⠀⠀⣠⠤⠤⠴⣐⣾⠱⠄⡧⣤⠀⠀⠀⢠⠀⡠⢦⠀⠀⠀⠀⠀⠀⡐⢁⣶⡉⢃
⠀⠀⠀⠀⡏⠑⠙⠣⠭⠏⠉⡇⠀⠀⠀⣠⠞⠛⠛⠓⠊⣹⠀⠀⣿⠀⠀⠀⠀⠀⠙⠋⠅⢸⠀⠀⣠⣧⣼⣶⣴⣄⠀⠀⠀⡴⠋⠙⢒⣙⠒⠋⠳⡀
⠀⠀⠀⠀⡇⠀⢰⠋⠉⠉⠉⠁⠀⠀⡼⠁⠀⣠⣒⡛⠙⠃⠀⠀⠛⠒⠚⢛⡷⠃⠀⢀⡴⠛⠀⡞⠀⢠⠴⢬⡀⠘⣇⠀⢸⠃⠀⣸⠉⠈⢱⠀⢰⢷
⠀⠀⠀⠀⡇⠀⠈⠑⠒⢦⡀⠀⠀⢸⠁⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⣠⠏⠀⢀⡴⠋⠀⠀⠘⣇⠀⠙⠦⠞⠁⢠⠏⠀⠸⡄⠀⠘⠲⠶⠋⠀⠉⢸
⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⠹⡄⠀⣿⠀⠀⡴⠉⠉⢣⠀⠘⡇⠀⠀⣰⠏⠀⢀⡾⠁⠀⠀⠀⣠⠞⠃⠀⠀⠀⠘⠳⢄⠀⠀⠙⠦⢤⣤⣄⠀⠀⠀⡟
⠀⠀⠀⢠⠤⠤⠤⠞⠀⠀⢠⠇⠀⢸⡀⠀⠳⣄⣠⠞⠀⢠⡇⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⢠⡎⠉⠙⡆⠀⠈⡇⠀⣠⣤⠤⠖⠁⠀⢀⡼⠁
⠀⠀⠀⢾⣀⡀⠀⠀⣀⡤⠋⠀⠀⠀⠳⣄⣀⠀⠀⣀⡤⠏⠀⠀⠸⣥⣤⣤⠇⠀⠀⠀⠀⠘⣇⠀⠀⠓⠒⠚⠁⠀⣠⠇⠀⣷⠀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠴⠚⠁⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⠀⠈⠁⠈⠙⠳⠾⠛⢓⡶⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣄⣀⣀⣀⣞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠖⠲⠦⠤⠶⢦⠀⠀⠀⠀⠀⢀⣠⠼⠿⠿⠿⠿⠿⡿⠿⢤⡀⠀⠀⠀⠀⠐⣏⠉⠓⠚⠉⠙⢲⠆⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⣳⣤⣦⣤⣴⠃⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⢈⣦⣤⣤⣴⣴⡏⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠻⠛⠿⠛⠦⣀⡴⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀⠜⠛⠻⠛⠛⠛⠻⣄
⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡴⠒⠛⠓⠲⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⡴⠃⠀⠀⠀⢀⡤⠶⠄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡴⠦⠄⠀⠀⠀⠀⠹⣄
⠀⠀⢰⠇⠀⠀⠀⠀⠸⣄⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⣱⠀⠀⠀⠀⠀⣸
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠙⠞⠁⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠓⠞⠁⠀⠀⠀⠀⣰⠃
⠀⠀⠚⡼⢦⣄⡀⠀⡀⠀⠀⠀⠀⠀⣀⣨⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⠀⠀⠀⠀⠀⠀⠀⣀⡤⢚⢁⢀
⠀⠀⠀⠀⠁⣠⠟⢉⡽⢋⡽⢛⡽⢛⡥⠋⡠⠋⣹⠗⡶⢲⡶⠒⡶⠒⢲⠖⢲⠖⢲⠿⢲⠟⡵⠋⡼⢫⠟⠲⢒⠖⡲⠋⡽⠣⠛⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠀⠈⠀⠈⠀⠀⠁⠞⠴⠋⠀⠚⠀⠐⠃⠐⠋⠐⠋⠐⠃⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⠀⢠⠀⠀⡀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⠎
⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠤⢀⠀⠀⠀⢢⠰⢰⠀⠀⠀⠀⠰⠄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠲⣌⢆⠀⠀⠘⠤⠼⠀⠀⠀⡠⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠐⠯⠇
⠀⠀⠀⠀⠌⢀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⣀⠐⡀
⠀⠀⠀⡈⠀⠀⠐⠲⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠮⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠘⠀⡀⢀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⡀⢸
⠀⠀⢰⠀⠁⠘⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠁⠸
⠀⠀⠈⠀⠀⠀⢀⡠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢁⠀⢀⠑⡭⢊⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠭⠆⠀⠀⠠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠡⠈⠁⠐⠁⠀⢠⢶⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⡠⠁⠀⠀⠀⢈⣏
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠐⠵⢁⠎⠀⠀⠀⡇⡆⠀⠀⠀⠈⠢⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠒⠈⠀⠀⠀⠀⠇⠐⠀⠀⠀⠀⠐⠂⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⣤⣰⠉⠲⡆⣀⡀⠆⡤⠀⠀⢂⠄⡀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⡄⠀⡀⢸⡤⠣⢽⠄⣶⡿⣠⣖⢋⣬⣩⡨⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣟⡛⠛⠛⠛⠛⠛⠻⠤⠒⠚⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠤⢴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉⠻⢻⡟⠉⠉⠉⢹⠟⠉⠁⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠶⠶⢦⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⠚⠛⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣄⣠⣤⣶⣶⣿⢿⣷⡄⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣛⣻⣿⣿⣶⣶⣾⣿⣍⠉⠁⠀⣴⣿⠵⠞⠛⠉⠁⢀⣠⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⡉⠉⠉⣴⣿⡛⠋⣹⣦⣤⣾⡿⢿⣿⣶⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣄⠘⢿⣿⠿⠿⠛⠋⠁⣰⠏⠀⠀⠀⣠⡿⠋⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣬⣿⣦⣀⡀⠀⠰⠏⠀⣀⣠⣾⣯⣴⠶⠛⢛⣛⣩⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠿⣯⣭⣭⣉⣉⣙⣛⡛⠛⣛⣭⣿⣿⠟⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠋⠉⠀⠘⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣀⣬⣷⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠛⠋⠙⣧⠀⠀⠙⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⢻⡄⠀⠀⢸⣆⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠿⠟⢿⡛⠉⠙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⣷⠀⢀⣴⡟⠁
⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣤⡀⠀⣷⣀⣀⣠⣼⣷⠿⠋
⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⣿⠟⠛⠛⣿⡀
⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠹⡇
⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣶⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⡾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⠲⡀⢄⠂⠀⢠⢲⣤⡟⢇
⠀⠀⠀⠀⠀⡀⠀⣀⢀⢄⠀⠀⠀⠠⠀⣠⠴⠒⠺⢶⣜⡍⡇⠀⠀⠀⠀⢠⠏⠑⠛⢔⡦⠧⣄⡜⠲⠓⠤⡎
⠀⠀⣠⠴⠚⠺⠛⣵⣆⡇⢠⢠⡎⢬⡎⠣⣀⡤⢤⡀⠉⠙⡆⠀⠀⠀⠀⡼⠀⢰⠛⠒⠲⢤⡟⠀⣰⡆⠀⡇
⢀⡜⠁⢠⠔⠚⠓⢎⠀⢹⡤⠋⠉⠹⡇⠀⠀⠀⣠⠃⠀⢠⠇⠀⠀⠀⠀⠧⣄⡈⠑⠦⣀⠞⠀⡼⢡⡇⠀⡇
⣼⠀⢠⠇⠀⠀⠀⢸⢶⠉⢀⣀⠠⠂⣇⣀⠤⠚⠁⢀⡴⢋⠤⣀⠶⡀⣀⠀⠀⣱⠀⠀⢹⠀⠀⠉⠉⠀⠀⠉⢹
⡇⠀⠸⡄⠀⠀⢀⡼⠘⢻⠏⢹⠀⠀⣏⠀⠀⠀⠲⠯⣄⠼⡶⢷⣶⢿⡉⠉⠛⠁⠀⠀⡼⠛⠛⠛⢻⠀⠀⡏⠉
⠹⣄⠀⠈⠓⠒⠉⠀⢀⡞⠀⢸⠀⠀⣿⠲⠤⣄⣀⢰⠁⠀⡖⠒⡄⠀⢳⠦⠤⠤⠴⠊⠀⠀⠀⠀⢸⣄⣠⡇
⠀⠈⠳⠤⣤⣤⣤⠴⠋⠀⠀⢸⠀⠀⢸⠀⠀⠀⠈⠙⢆⠀⠙⠋⢁⣠⠋⠀⠀⡔⠠⠄⢷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⣏⠉⠀⠀⠀⢀⠖⠁⢀⡤⠤⡀⠈⠳⣤⠔⠛⠓⠶⠜⣦⡤⢠⣊⢣⣀
⠀⠀⠀⠀⠀⠀⣼⠉⠓⠚⠼⢏⠶⢨⣄⡀⠀⢀⢸⠀⢀⢻⣀⣀⡿⢀⠞⠁⢀⣤⡒⠒⠶⣣⠦⠽⢛⡊⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠧⢤⣤⣀⡀⠀⠉⠑⣸⠦⠤⡠⣜⢩⣻⡀⠀⠀⣀⡞⠀⠀⢀⣀⠉⠑⢤⡇⠀⡜⠉⠉⡇⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⣠⠖⠻⠤⢄⣉⠉⠒⣩⠟⠉⠉⢩⡇⠀⢰⠉⠈⢳⠀⢸⣇⠀⠙⠲⠖⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠋⠀⠀⢀⠴⠊⠁⢀⡔⠁⠀⠀⠀⠀⢧⠀⠘⠢⠔⠃⠀⣸⠉⠲⢤⣤⠤⡦
⠀⠀⠀⠀⠀⣰⠃⠀⢀⡞⠁⠀⠀⠀⠸⠤⢤⡀⠀⠙⣆⠀⠀⠀⠀⠈⠓⠦⣤⣤⠤⠞⠁⠀⠀⢀⡠⠔⠁⠀⣠
⠀⠀⠀⠀⠰⠧⣤⣤⡞⠀⠀⠀⠀⡤⣤⣀⣠⠇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⢀⣀⣀⠀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣯⣾⣿⣿⣆⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠉⠀⢀⡀⠀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⡇⢸⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠻⠇⢸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⢠⣤⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⡇⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠙⠃⠘⠟⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⣿⠀⢸⢸⢰⠀⡆⢧⢸⠀⠀⡆⡜⠀⢸⢀⠃⡇⠀⡇⡸⢸⢸⠀⢸⢠⠇⡇⣺⠀⢸⠘⣤⠃⠀⢣⡜⠀⠘⣤⠃⡇⠀⢧⡎⡆⡇⠀⢸
⠀⠀⡇⠀⣿⠀⢸⢸⢸⠀⡇⢸⡞⠀⠀⢷⡇⠀⠘⣾⠀⡇⠀⢳⡇⢸⢸⠀⠘⣾⠀⡇⢿⠀⢸⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⡇⠀⣸⡄⡇⡇⠀⢸
⠀⠀⠃⠀⠙⠀⠘⠘⠈⠀⠁⠀⠃⠀⠀⠘⠀⠀⠀⠋⠀⠃⠀⠘⠁⠘⠘⠀⠀⠋⠀⠃⠙⠀⠘⠘⠀⠃⠀⠃⠘⠀⠘⠀⠃⠃⠀⠃⠑⠁⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⣀⣤⠀⢀⠔⠚⢩⠂⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠈⠁⠀⡁⠁⠀⠀⠀⢩⠕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣁⣀⡏⣠⣡⣴⡖⣓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠘⡄⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⣤⣬⣿⣛⣛⣉⣭⠽⠿⣕⠤⣀⠀⠀⠀⠀⠀⡠⠰⡄⠀⠀⠀⠰⠁⠀⠀⢀⠛⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠛⠀⠒⠐⠒⠖⢺⠒⠳⣟⠷⠀⠅⣈⠀⠀⠀⠀⡎⠀⠀⢡⠀⠀⠀⠆⢀⠀⠀⠌⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⢐⡇⠘⣀⡀⠀⠈⠈⢣⢱⡀⠀⠱⠀⠀⢰⠀⠠⠀⢸⡀⠀⢀⡀⠐⡂⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡔⢉⣀⡀⠀⠀⠈⠳⡄⠐⠈⢣⡵⠀⠀⠑⢄⠘⡄⢔⠀⠀⡇⠀⠸⠀⠅⠒⠠⠀⠥⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢪⡀⡃⢸⠉⠑⣀⣁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⢐⠀⠀⢰⠀⡠⠀⠭⠰⠆⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠘⠢⡈⠁⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⣯⣄⠢⠀⢸⠰⢁⡒⡨⠋⡒⠂⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡤⡄⠀⠀⡜⠒⢄⠙⡆⠀⠀⢀⡡⢂⠔⠠⠂⣀⠄⢹⣎⣀⣶⠥⠂⣔⡁⠤⠖⠂⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠘⢄⠘⠒⠀⠀⠀⠊⠀⡘⠀⠀⡠⠒⠡⢊⢔⠞⡡⠖⠉⠀⢀⠤⠒⠒⠒⢤⡎⠀⠀⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡂⠀⠀⠀⠈⠑⠒⡄⡐⠒⠀⠂⠁⠀⠀⣀⠄⡂⢑⠔⠋⠀⠀⠀⠀⠘⢔⠢⣀⡠⠾⠁⠂⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠡⠂⢀⠀⠀⠀⠉⠁⠀⠀⠀⡀⠔⡈⠔⣈⠔⠁⠀⠀⠀⠀⢀⠀⠑⠀⣽⠫⠭⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⣰⡥⠞⠚⠛⠛⠉⠁⠀⠋⠓⠒⠂⠈⠁⠀⠀⠀⠀⢠⠒⠀⣀⠴⣙⠲⠈⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⣠⠺⢅⡒⠠⠈⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⢀⡜⠭⣑⠂⠈⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠈⢀⣤⠪⠌⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡀⠀⠀⠀⠀⢀⡀⡀⠐⠁⠔⡡⠞⠁⠔⡀⠀⠠⠔⠺⠉⠂⠁⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⠄⣀⡀⠀⠨⠊⡠⠊⣠⠊⡀⠊⢡⠊⠐⡡⠀⢀⣀⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠉⠐⠒⠠⠾⣡⣊⣀⠔⡡⠊⢔⠐⠊⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠉⠐⠂⠤⠄⣀⡀⠀⠀⠀⠈⠈⠁⠐⠒⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡁⠄⠂⢀⡀⠀⠉⠑⠒⠠⠄⢄⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢁⠠⠐⠀⣀⠤⠒⢉⡠⠔⢀⡠⠄⢀⠉⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠤⠂⢁⠄⠂⣁⠄⢊⡡⠐⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⠐⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⢀⡠⠀⢂⡀⠄⠒⡀⠄⢒⡀⠤⠐⣈⠠⠐⣀⠡⠐⢈⠡⠐⢂⠡⠀⢂⠨⠐⠂⠡⠄⠂⠀⠀⠂⠁⠐⠀⠡⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡀⡂⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠠⣴⠂⠀⠀⠀⠀⠀⠠
⠀⠀⡀⠔⠁⡃⠀⠀⠀⡴⠉⣉⠙⠱⠀⠀⠀⠀⠸⠀⡈⠉⡝⠀⠀⠀⠀⡐⢁⠁⡇⠀⠀⠀⠀⠈⢈⠽⠅⠅
⠀⠀⠦⢲⠀⠁⠀⠀⠀⠀⠉⢀⠃⢰⠁⠀⠀⠀⠀⣊⡀⠘⢄⠀⠀⠀⡜⠠⠨⠀⠇⡀⠀⠀⠀⢤⣈⠐⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠔⠁⠐⠃⢤⠀⠀⠀⢠⠤⠜⠀⢸⠀⠀⠘⠀⠀⢤⠀⡔⠃⠀⠀⠀⡤⠤⠃⠀⡇
⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠚⠀⠀⠀⠓⠂⠐⠒⠁⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠈⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡠⠤⡌⢄⠀⠀⠀⡤⠤⠠⠮⡡⡄⠀⠀⠀⣀⣠⣩⡀⠀⠀⠀⠀⡠⢠⠌⢄⠀⠀⠀⠀⢀⡤⢤⣬⡀
⠀⢀⠎⠀⠔⣀⠂⠀⠀⠀⠦⠤⡤⠀⣠⠇⠀⠀⢘⠀⣆⠄⢹⠀⠀⠀⡎⢠⠂⢲⢀⡇⠀⠀⡰⢁⠔⠒⢄⠑⡀
⠀⢸⠀⠠⠀⠄⠱⠀⠀⠀⢀⠞⢀⠞⠁⠀⠀⠀⠜⢃⠀⡐⠣⡀⠀⠀⠱⢄⣉⠁⠀⡆⠀⠀⡇⢸⠀⠀⢸⠀⡇
⠀⠘⣄⠐⠒⠃⡸⠀⠀⠀⡞⠀⡞⠀⠀⠀⠀⠘⡄⠘⠤⠂⢀⠇⠀⠀⢰⠒⠊⢀⠔⠁⠀⠀⠳⡀⠑⠒⠁⡠⠃
⠀⠀⠀⠉⠂⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠊⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡠⣄⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠀⢠⡀⠸⠀⠀⠀⠀⢀⣸⠀⣀⠔⠋⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⢇⠦⠀⠀⠀⠳⡀⡇⠀⠀⢀⡄⠈⠛⠭⣀⢄⠀⠀⠐⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢆⣰⠎⠍⠀⠙⣿⣟⣁⠈⠀⠀⠠⠓⡼⠦⠑⠎⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢻⣄⠉⢀⣤⣤⡶⣶⡁⠈⠙⠪⢲⡀⡀⡐⣰⣇⣠⣿⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⢷⣤⠃⣤⣄⠎⠀⠫⣄⣠⡀⠀⠑⣷⡼⠟⣫⡴⠞⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠢⢌⡓⠿⠵⣦⣤⣄⣀⠤⠼⠧⠤⠚⠛⠋⣁⣠⠖⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠐⠒⠚⠋⠉⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠂⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠴⠖⠋⠉⠉⠁⢠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠃⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⠤⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⠡⣄⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠒⠒⠀⠀⠐⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⢀⠈⣠⣾⣤⡄⣀⢀⡀⢇⣶⢀⢀⠀⠀⢀⢉⣤⣾⣤⡀⣀⢀⠀⣅⣶⠠⢀⠀⠀⢀⢉⣤⣾⣤⡀⡀⢀⠠⣅⣶⠠⢀
⠀⠀⠀⠀⢶⣾⣿⠀⢸⠋⢈⠈⠸⣿⣿⡕⠂⠁⠀⢾⡾⠋⠉⠙⢿⠋⢉⠀⠸⣿⡕⠀⠁⠀⢾⣾⠋⠉⠉⢻⠏⢉⠀⢸⣿⡕⠀⠁
⠀⠀⠀⠀⢹⣿⣿⠀⠸⠀⢸⠀⠀⣿⣛⣥⠀⠀⠀⣻⣷⡶⠃⢀⣼⠀⢸⡆⠀⣋⡥⠀⠀⠀⣿⣿⣶⠂⠀⠾⠀⢸⠀⠀⣋⡥
⠀⠀⠀⠀⠀⠛⢟⠀⢠⣀⠘⠀⢀⣹⡟⠂⠀⠀⠀⠈⠋⠀⠀⠘⢻⣀⠘⢀⣠⡟⠀⠀⠀⠀⠈⠋⠁⠁⠀⣠⡀⠘⠀⣠⡟
⠀⠀⠀⠀⠐⠋⣬⠙⡻⠿⠻⠧⠉⠉⠀⠀⠀⠀⠀⠘⢃⣤⠘⡻⠿⠻⠧⠉⠉⠀⠀⠀⠀⠀⠘⢣⣬⠘⡻⠿⠻⠧⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸⣿⣧⣿⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠋⠛⠁⠀⠀⠀⠠⠈⠂⠐
⠀⠀⠀⠀⠀⠰⢀⣠⣀⠀⠀⠁⠀⡄⣀⠀⠀⠀⠀⠀⠰⢀⣠⣀⠀⠀⠁⠀⡄⣀⠀⠀⠀⠀⠀⠖⢀⣤⣀⠀⠀⠁⠀⡀⣄
⠀⠀⠀⠀⣨⣴⣿⡿⠿⢷⡲⠸⠂⠹⣿⡎⠌⠂⠀⣸⣴⠛⠻⠛⢳⡳⠼⠂⢽⣿⡊⠄⠂⠀⣸⣴⠿⠟⠿⣷⡳⠾⠀⢹⣿⡪⠑
⠀⠀⠀⠀⢽⣿⡟⠁⠀⢸⠃⢰⡆⠀⠿⠃⠀⠀⠀⢿⣿⠀⠘⠛⠿⠀⣰⠀⢸⡿⠃⠀⠀⠀⢿⣿⠀⠒⠀⢾⠁⢰⡄⠀⡿⠆
⠀⠀⠀⠀⠘⠿⠁⢘⠁⠈⡀⠸⠇⠀⣾⠍⠀⠀⠀⠙⠿⠓⠛⠀⠀⠀⠿⠀⢘⣾⠉⠀⠀⠀⠙⠯⠀⢾⠀⢨⠀⠸⠇⠀⣾⠉
⠀⠀⠀⠀⠠⠦⡰⠫⢤⣼⢿⡆⠴⠞⠃⠀⠀⠀⠀⠠⠦⡐⠶⢶⣿⢶⡄⠶⠞⠃⠀⠀⠀⠀⠠⠆⡐⠤⢴⣿⢷⡄⠴⠞⠃
⠀⠀⠀⠀⠀⠀⠛⠀⠆⠈⠀⠉⠀⠀⡀⠀⡀⢀⡀⢀⣈⠛⠐⠆⠈⠀⠉⠀⣀⢀⣀⡀⣀⡀⠀⢈⣋⡐⠆⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣸⣿⣸⣇⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣽⡷⣟⣿⣇⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠄⠀⠈⠉⠈⠀⠉⠉⠀⠁⠀⠀⠐⠆⠀⠀⠀⠉⠉⠁⠉⠈⠉⠁⠉⠁⠀⠐⠄
⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠂⠀⡀⢀⠀⠀⠀⠀⠀⢤⠀⣠⠀⠀⠀⠂⠀⡀⣀⠀⠀⠀⠀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢐⣠⣾⣿⣿⣦⣦⠰⠄⢳⣿⢔⡐⠄⠀⢰⣠⡾⠿⢿⣦⣦⡴⠀⢳⣿⢔⡒⠄⠀⣰⣠⣾⣿⣿⣦⠦⠴⠀⢳⣿⢔⠒⠄
⠀⠀⠀⠀⣽⣷⣤⡤⠀⢀⠇⠀⡆⠀⣿⠇⠀⠀⠀⣿⣿⠀⢰⠀⣹⠃⢰⡄⠈⣿⠇⠀⠀⠀⣿⡿⠀⣠⠀⠹⠃⢰⡄⠈⣿⠅
⠀⠀⠀⠀⠸⣿⣿⠃⠀⣾⠁⢸⡇⠀⣴⡚⠀⠀⠀⠺⡿⠀⣴⠀⠙⠀⢸⡇⠀⣴⠒⠀⠀⠀⠺⣿⣄⡈⠀⢠⠀⢸⠇⠀⣴⠓
⠀⠀⠀⠀⢀⣄⠡⢠⣤⣿⣷⡄⢤⡴⠏⠀⠀⠀⠀⢀⡄⠀⢩⣀⣼⣦⡄⢤⡴⠏⠀⠀⠀⠀⢀⡄⠠⢈⣄⣼⣦⡄⢤⡼⠃
⠀⠀⠀⠀⠀⠀⠟⠠⡎⠉⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠠⡎⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠠⡎⠉⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢤⡀⠀⠀⠀⡀⠀⠀⠠⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠻⣾⣄⠀⣧⠀⢂⡀⠀⠀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⢸⣿⣷⣽⣶⣿⣾⡀⠰⣄⡀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣀⡴⣶⣾⣿⣿⡿⣿⠛⠙⠻⣶⣾⣿⣶⣶⣶⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠉⣾⢏⡟⣸⠑⠃⠉⣀⠀⠀⢆⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⣿⡾⠱⠇⢀⡀⠀⠀⠀⢀⣬⣄⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠤⢠⠄⠤⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢰⣷⣤⣤⠖⠛⢋⠉⠉⠉⠉⠉
⠀⠀⠀⢀⣴⣦⣶⣿⣾⣷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠓⠈⠀⠀⢠⡍⠉⠀⠀⠀⣼⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡟⠁⠀⠀⢀⣤⣿⣷⣤⣴⣤⣼⠿⠛⠛⠛⠛⠛⢅
⠀⠀⠀⣿⡿⠋⠟⠿⠟⢻⡇⣇⠀⢸⡶⠖⠂⠀⠀⠀⣀⣤⣾⡟⠀⠀⠀⠀⣾⡿⢿⡏⡟⠋⠁⢂⢺⡆⠀⠀⠀⠀⡨⡇
⠀⠀⠀⡀⢁⠀⡀⣀⢰⢸⣇⢸⠀⠈⣧⣶⣤⣄⣐⣶⣿⣿⠏⠀⠀⠀⠀⠀⢿⣧⣼⡇⢳⣦⣀⣼⣾⣶⣶⣿⣿⣿⣿⡇
⠀⠀⢸⣧⣸⣼⣧⣿⣾⣿⣿⠸⡆⠀⢸⣿⣭⣷⣶⣶⣿⡟⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣾⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⣇⠀⠀⣿⣿⠿⠿⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡄⠀⢸⡇⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠠⣤⣄⢸⣇⢸⡇⠹⣿⡇⠀⠀⠀⠀⠀⠤
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢧⡀⠀⠿⠀⠀⢸⣯⣄⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⣼⣧⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⠀⠀⠀⠀⠈⢸⣿⣿⣶⣤⣀⠀⠀⠀⠈⢹⡿⢻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠘⠈⠛⠿⠿⠿⠿⠛⠛⠛⠛⠋⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣤⣄⣀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣴⠤⠶⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠛⠛⠃⠘⠛⠛⠛⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡇⠀⠀⣺⡅⠀⠰⣿⠄⠀⢘⣧⠀⢰⡟⣧⠀⠈⣹⠃⠀⢸⣿⡅⠀⢻⣽⠀⢈⣿⣏⡇
⢀⣶⠶⢤⠀⡠⠶⣄⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠌⠀⠀⡈⠁⠀⠐⠉⢁⠀⠀⠁⢐⠐⠂⢉⡒⢐⠨⠁⡀⢀⠡⢙⡀
⠈⠙⠶⠎⠉⠱⠶⠋⠸⠇⠺⠇⠻⠇⠿⠃⠿⠘⠿⠘⠿⠸⠗⠸⠟⠸⠇⠺⠇⠿⠇⠿⠘⠿⠸⠿⠸⠟⠸⠏⠹⠇⠸⠇⠿⠇

⠠⢁⠜⡿⡕⠌⣟⣵⡚⣷⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⠥⡾⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠁⠰⣯⡓
⠀⠘⠦⠅⠱⡀⠈⠀⠞⠓⠀⠄⠀⠀⠀⠀⠀⠖⠚⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀⠄⠀⠀⠀⠀⠀⣡⡄⠀⠀⠀⣸⠀⠐⠶⠋⠠
⠀⠀⠀⠀⠀⠈⠓⠤⣀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⠁⣤⡀⠀⠀⠙
⠀⠀⢀⠤⠄⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⢈⡈⠀⣉⡈⡀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡂⠀⣰⠂⡄⠀⠀⠀⠀⠀⠀⠸⠔⡲⡛⠉⣻⠑⠜⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⣀⠀⠀⠀⠀⠀⠴⢊⠃
⠀⠀⠀⠀⠀⠀⠰⠘⠁⠁⠀⢈⣬⠤⢤⡀⠀⠀⠀⠀⢀⠎⠀⣔⣺⠆⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⢄⡈⠘⠁⢀⡀⠀⠀⢀⠀⣻⡅
⠀⠀⠀⠀⠀⢰⠉⣉⢽⡷⠚⠻⡇⢕⠶⠁⠀⠀⠀⠀⢸⠀⠠⢬⣽⠆⠀⠀⠀⠀⠀⠀⠀⢈⣉⡭⢶⠿⣤⣀⣈⠁⠂⠀⠸⠄⠈
⠀⠀⠀⠀⠀⠘⠊⣡⠋⠞⢰⣤⡑⠂⠡⠀⠀⠀⠀⠀⠾⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠂⣀⣁⠄⢠⠏⣩⠉⡆
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢼⣭⣽⡙⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡀⢸⣽⡹⣯⠈⠂⠊⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢤⣸⣹⡽⠀⢠⠀⢆⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠿⣄⠀⠀⡱⣯⣎⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠠
⣑⠄⠀⠀⠀⠀⢸⠁⠀⠀⠸⠙⠀⠀⣉⠀⠞⠀⠀⠀⠀⠀⣈⣉⠛⠀⠀⠀⠀⠀⠈⠑⠤⣀⠁⠀⢀⠆⢀⠂⠀⠀⠀⠀⢀⠠⠀⠉⣤
⠕⠠⡀⠀⠀⠀⠘⠓⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠰⠁⠰⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠄⠠⠎⠀⠀⠀⠀⠀⣂⠄⣶⡞⠉
⠆⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡜⡙⠑⣰
⣫⡷⠀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⣤⡰⠁⠀⠀⠀⠀⠤⢄⠀⠀⠀⠠⠀⠶⠒⡏⠄⠀⠀⠀⠀⠀⠀⠀⡤⠚⠀⠀⠀⠀⡇⠀⠛
⠉⢂⠀⠊⠁⠀⠀⢀⡀⠀⠀⠀⠠⠀⢀⡿⠉⣫⣤⠄⣀⠀⠲⢀⠀⠀⠀⢀⠁⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠸⡄
⠂⠁⠀⠀⠀⠀⢠⡤⠖⠔⠀⠀⢨⢀⠋⣿⡐⠘⣧⠿⢱⡧⠀⠌⠀⠀⠀⣬⠀⠙⠋⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⢹⣤
⠀⠀⠀⠀⠀⠀⠍⢠⣦⠀⠀⠀⠀⠘⠤⠇⠱⣀⠀⠀⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡤⠈⠠
⠀⠀⠀⠀⠀⠀⡄⠐⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⢀⠠⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⢠⢀⣦⢻⡕⢻⠆⠀⠘⣦⠀⢀⠈⠁⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢂⣤⣠⣤⡆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣷⡎⠈⢋⠓⢋⠀⢀⡴⠋⠁⠀⠀⠰⣾⡁⢸
⣻⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣨⠞⠍⣀⠸⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠆⠀⠙⡶⢷⣒⠒⠊⠁⠀⠀⠀⠀⠠⡐⢸⠀⠺
⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢮⣝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠘⠤⣈⠢⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠎⠁⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡼⠀⠐⢦⡬⠇⢀⠈⡀⠀⠀⠀⠀⠀⠃⣤⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠤⢠⣄⠼⠀⠀⠀⠀⠀⠐⠳⠀⠄⠈⠀⠀⠤⠀⠁⠀⠀⢀⠡⣴⡞⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠀⢈⡈
⠐⠤⠀⣰⠏⠑⣂⣄⠀⠀⠠⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⢱⠀⣼⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠟⢉⠹⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⠾⠛⠁⠈⠙⢛⣛⣛⣻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠥⠶⢶⣦⣀⣴⠖⠻⠶⠛⠉⢉⣽⡿⠁⣀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠈⠁⠀⠀⠀⠀⠀⣰⣿⣿⡾⠟⠛⠋⠉⠉⠙⠛⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⡀⠀⣀⣀⣤⣴⣿⣿⠿⠶⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠿⠿⣿⡭⠾⠛⠙⠿⣷⣤⣀⠀⠀⠀⠈⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⡀⠀⠀⠀⠀⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⣤⣄⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢻⣿⡀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣷⣦⣄⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠀⣿⣇⠙⢿⣷⡄⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢸⣿⡇⣿⣿⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⣿⡇⢿⣿⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣘⣿⡇⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢻⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣿⠀⠀⣿⣿⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⢸⣿⡇⠀⢸⣿⡆⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⣿⣷⠀⠀⢿⣷⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⣸⣿⡆⠀⣸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⠛⠋⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄
⠀⠀⠀⠀⠀⠘⣿⣧⣄⣀⣈⣻⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠉⠛⠛⠻⠿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⠶⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠰⠋⡀⠘⢶⠀⠀⢤⠶⠎⡐⢆⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡠⠖⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠈⠀⠀⡆⠀⠀⠈⢻⡆⠀⠀⠀⠀⢀⠴⠲⢦⡀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⢡⠴⠒⠲⢦⡀⠀⣬⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡇⠀⣴⠶⢄⣆⢸⣠⣶⣄⠀⢀⡇⠀⠀⠀⠀⡏⠀⠀⠀⢹⠀⠀⠠⠊⠁⠉⢢⠀⠀⢐⣴⡏⠀⠀⠀⠀⢳⣤⣾⠀⢠⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠱⣀⣹⣦⠞⠉⠉⢧⣼⠟⢀⡞⠀⢀⣠⠴⠚⠁⣠⣾⣦⠸⠦⡴⠁⠀⠀⠀⢀⡇⠀⠀⠈⢳⡀⠀⣼⡿⢶⠀⠀⣣⡬⠿⣤⠁⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢉⠉⢻⢀⡴⠋⠀⠀⠀⢣⣹⣇⡽⠃⠀⠀⠀⣾⠿⡇⡸⠀⠀⠀⠀⢠⠟⠢⢘⠿⠚⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢀⠤⠊⠀⠀⠙⣄⡀⠀⠀⣠⡏⠁⣼⠏⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠄⠿⠤⢿⡁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘⠳⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣴⠒⠋⠁⠀⠀⠀⠀⠀⠘⡈⢉⢉⡜⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠀⠁⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⡄⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸⡀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠒⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠀⠀⣀⣀⠽⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠋
⢸⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠁⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀
⠸⣤⣤⡤⠶⠚⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠉⠀⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣀⡤⠤⠿⣄⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠠⡾⠴⠒⠋⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠦⣤⣤⡤⠊⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠛⠉⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⣄⠀⣀⣠⠤⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣀⣠⡤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠆⠀⠀⠆

⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⣶⣥⣄⠀⣷⠀⠀⠀⠘⠇⣀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢤⡟⢀⠀⠀⠀⡰⠋⡟
⠀⠀⠸⠏⢿⡋⣠⠟⠀⠀⠀⢘⡟⠟⠀⠀⠀⠀⣠⠞⡁⠀⠀⠀⠀⣄⢀⣶⠈⠀⠀⠘⠓⠚⣟⠂
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⣿⣉⠉⠀⠀⠀⠀⠀⣠⣎⣉⠁⠀⠀⠀⠈⠉⣹⠃⠀⠀⠀⣞⠉⣷⠀⠀⠀⠀⣴⠋⠙⣦⡀
⠀⠀⠀⠉⠨⣿⣿⠀⠀⠀⣸⣿⣯⠌⢹⠄⠀⠀⠀⣰⠃⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠙⠦⢾⣿⡗
⠀⠀⠀⠲⠴⠛⠉⠀⠀⠀⠈⠙⠓⠔⠋⠀⠀⠀⠐⠋⠀⠀⠀⠀⠘⠳⠤⠞⠁⠀⠀⠀⠐⠦⠔⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⠒⡞⠋⠉⠉⠉⠻⡉⠹⣏⠉⠉⢹⠉⠙⢯⡭⠟⡿⠓⠒⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣠⠋⠀⢹⡀⠀⠀⠀⣠⢇⡤⣌⠉⠉⠁⢀⣀⡀⠀⠀⣇⠀⢀⣠⠴⠊⠀⣈⣓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠓⠒⠉⣀⣤⡀⠀⠙⠒⠒⠋⠁⣏⠀⣈⠆⠀⣰⠋⠀⢹⠀⢀⣈⣉⠉⢠⠤⣄⢸⡁⠈⡇⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⣆⠀⢰⠁⠀⠙⢦⠀⡴⠦⡄⠀⣀⣉⣁⣀⠀⠸⣄⣀⡞⢠⡏⠀⠈⢧⠈⠳⠞⠀⠙⠞⠁⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⠸⡀⠀⠀⢘⡆⠳⠚⠁⢞⠀⠀⠀⠀⢹⠄⠀⢀⡀⠈⢇⠀⠀⣸⠀⢰⠉⠉⣹⠄⣰⠋⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⠀⣀⣑⡶⠖⠋⣠⠤⢤⡀⠘⣆⠀⠀⢀⡜⠀⠸⡉⢹⠀⠈⠓⠚⠁⠀⠈⢓⣚⣁⡀⠘⢦⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⢭⣆⣰⣁⣀⣀⣷⣀⣈⣓⣚⣩⣖⣒⣦⡈⠁⢀⡴⠒⠢⡄⠀⢰⠋⠀⠀⠙⡆⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠆⢸⠀⠀⠀⠘⡆⠘⢦⣠⡤⠖⢃⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢺⠀⠀⠀⢀⡇⣀⡤⣄⠀⣴⠉⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠳⡜⢆⣠⠴⠋⡰⠁⠀⣸⠂⠘⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢹⠀⠀⠀⠀⠘⠒⠚⢁⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣉⣓⠦⠴⠒⢁⣠⠴⠚⠋⠓⢦⠀⡇⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡏⠀⠈⠉⠲⡀⣎⠀⠀⠀⠀⠀⢸⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⠀⢹⡀⠀⠀⠀⡇⠈⠙⠒⠶⠤⠖⠋⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠙⢦⠙⠲⠶⠚⣡⠖⠓⡄⢰⡋⢱⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢧⡀⠀⠀⢀⡼⠁⢀⠀⠰⣇⠀⢀⡿⠀⠉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠑⢦⠉⠉⠉⠉⡴⠋⠉⠹⡄⠈⠉⣩⢄⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠀⠀⡸⠀⣠⠤⡀⢣⡀⠀⢀⠇⠀⠀⡇⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⣍⠉⠀⠸⣁⢀⡇⠀⠉⠒⢋⣠⠤⢄⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠹⡄⠀⠈⣉⣀⠀⢀⡔⠋⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣰⠃⡰⠋⠀⠈⡆⢎⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠦⣤⡤⠞⠁⠘⠥⠤⠴⠞⠁⠘⢦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠒⠒⢲⠀⢀⡤⠖⠚⠉⢳⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣏⣀⣞⣀⡠⠴⠚⠁