Channel Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣤⣶⣶⣶⣶⣶⣶⣤⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠷⠿⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠷⠶⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⡿⠿⠿⣷⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⣶⣶



⣿⡿⣛⣿⡿⣿⣿⡿⢿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣥⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡤⠒⠒⠒⠒⢀⣀⣈⣉⣉⣉⣉⣈⣉⣉⣉⣉⡉⠭⠭⠭⠭⠭⠥⠭⠭⠭⠭⠽⠿⠭⠭⠽⠐⠂⠰⠄⠬⠭⠭⠭⠭⠭⠵⣒⢤⣄
⣾⢸⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠊⢂⠘⡇⣿⣷
⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡇⣿⣾
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣷⢸⣽
⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣼
⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢿
⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣽
⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣾
⡏⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡾
⡇⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡼
⡇⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣰
⡇⠀⡧⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡵
⡇⠀⣇⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣵
⣷⠀⡇⢋⢄⢄⡀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣵
⢸⠀⠳⠄⠀⠈⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠁⣾⢾
⠀⠈⠀⠀⠒⠒⠀⠠⠤⠄⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠉⠉⠉⠉⠉⠉⠀⢸⣿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣛⣛⣛⣛⣛⣛⣊⣁⣈⣑⣒⣒⣒⣒⣒⣒⣒⣊⣉⣉⣉⣉⣛⣛⣚⣛⣿⣯⣷⠄
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣤⣤⣤⠀⠀⣿⣿⣿⣿⣿⠀⠀⣠⣤⣤⣤⡄⠀⣼⣿⣿⣀⣀⢀
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣇⣀⣼⣿⣿⣿⣇⠀⣿⣿⢿⣿⣿⣆⣄⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢉⣀⡈⠉⠉⣁⣀⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠁⣰⣆⠈⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⢰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠙⠛⠃⠀⠀⠘⠛⠉⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⢋⣴⣦⣹⣿⠸⠿⠿⢸⣿⡟⢠⡈⣿⣿⠀⡈⢿⠀⣿⡇⠰⠶⣾⣿⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⠟⣿⡈⠻⠟⢻⣿⢰⣶⣷⢸⡿⢀⣍⣁⠘⣿⠀⣿⣆⠀⣿⡇⠰⠶⠿⣿⠀⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⢻⠉⠉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠘⠉⠛⠉⠉⠀⠀⠀⠘⠛⠛⠉⠀⠀⠙⠿⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⣿⣿⣿⣿⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣈⣿⣿⣿⣀⣀⣸
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣘⣷⣤⣤⣤⣤⣤⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠉⠉⠉⣿⣿⣿⠉⠉⠉
⠀⢀⣠⣶⠿⠋⠉⠉⠉⠉⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠻⣷⡀⠀⠀⠀⠀⠀⣿
⣾⣿⣿⡷⢶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡆⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⣿
⣿⡇⠀⣸⡟⠛⠛⠛⠛⠷⢶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢘⣿
⣿⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⣇⣴⣿⣿⣿⣦⡀⣿⠀⠀⠀⠀⢸⣿
⣿⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⢸⣿
⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⢠⣿⠈⢿⣿⣿⠟⠁⠀⣿⠀⠀⠀⠀⢸⣿
⣿⣿⠀⣿⠀⠀⠀⣀⣤⡾⠟⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁⠀⠀⢸⣿⣶⡦⣴⡆⣴⣶⠀⣿⡀⠀⠀⠀⢸⣿
⢹⡇⠀⣿⡇⠀⠸⠛⠁⢀⣼⠏⠀⠀⠀⠀⠀⣀⣀⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⣾⣦⣿⠄⣾⣧⠀⣿⡇⠀⠀⠀⢸⡏
⣿⡇⠀⣿⡇⠀⠀⢀⣤⣿⣃⣤⣤⣴⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⢸⣿⣼⣤⣷⣤⣿⡅⠀⣿⡇⠀⠀⠀⢸⡇
⢸⡿⠀⢹⡇⠀⠀⠈⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⢸⣿⢈⣁⣩⣬⣭⣅⠀⢻⡇⠀⠀⠀⢸⡇
⢸⣧⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⠐⣲⠶⠶⠶⠒⠂⢸⣿⠀⠀⠀⣾⡇
⠘⣿⡀⢸⣿⣠⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣭⣭⣭⣭⣭⠁⢸⣿⡀⠀⠀⣿⠃
⠀⣿⡇⠈⠛⠉⠉⡝⢯⣉⣛⡛⢛⡛⠛⣛⡛⣛⠛⠛⠛⠛⠿⠿⠿⠿⠶⢶⣶⣾⡿⢀⣓⣛⣷⣶⣶⡖⢸⣿⢿⣦⣼⡿
⠀⣿⣇⣀⣀⣀⣀⠙⠚⠚⠛⠓⠚⠛⠂⠋⠷⠿⠴⠴⣶⢶⣴⣦⣆⣤⡤⡤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠈⠉
⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣮⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿
⠀⠀⠀⣿⣿⣿⡿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣏⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣄⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣶⣒⢶⢄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠀⠀⠀⡞⠛⠛⢲⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄
⠀⠀⢸⣩⣏⣽⣩⣍⡇⠀⠀⠀⠀⠐⢫⠵⣒⠶⣍⠃⠀⠀⠀⠀⣰⣶⣿⣷⣄⠀⢈⠂⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾⠳⢪⢱⢽⣧
⠀⠀⠘⢶⣷⠾⣴⣶⠃⠀⠀⠀⠀⠀⠀⠘⢩⠙⠀⠀⠀⠀⠀⠀⠙⡿⠿⠿⠋⠒⠃⠀⠀⠀⠀⠀⠰⡇⢠⠀⢸⠀⠀⠀⠀⠀⠀⠻⣮⣖⣲⣽⠏
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⣤⣶⠿⡳⣄⠀⠀⠀⠀⠀⢀⣤⣴⣿⢿⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣤⣤⣿⣼⣯⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⡀
⠀⠀⠘⠿⡿⠀⢯⠻⠇⠀⠀⠀⠀⠘⠟⢏⠀⢱⠽⠇⠀⠀⠀⠀⠀⡦⡤⡤⡤⡤⡇⠀⠀⠀⠀⠀⣠⣼⣧⣿⣥⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠄⠀⠀⠀⠀⠀⠀⣸⣶⣆⣀⣁⣶⣾⠀⠀⠀⠀⠀⠈⡿⠩⡿⠀⠀⠀⠀⠀⠀⠀⠿⠛⠿⠿⠿⠇


⠀⠀⢀⣄⠀⠀⡴⠒⡆⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⠸⣿⠁⠘⢄⠴⠇⠀⠀⠀⠀⢠⡖⣶⡶⠒⠲⣆⠀⠀⠀⠀⢠⣼⣇⢀⣟⢳⡏⠀⠀⠀⠀⢰⣿⣷⢸⣿⣿⣧⠀⠀⠀⠀⠀⢋⣍⢉⣉⣉⠹
⠀⠀⠀⠙⢷⣄⣠⣄⠀⠀⠀⠀⠀⢸⡏⠻⣇⠐⢀⣧⠀⠀⠀⠀⢹⣧⣿⡼⣿⣿⠀⠀⠀⠀⠀⠀⢩⡏⠈⠛⠻⠇⠀⠀⠀⠀⠀⠘⠛⠘⠉⠙⢀
⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡎⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⡰⠋⠉⠉⢢⠀⠀⠀⠀⠀⠀⢀⢤⣢⣷⣏⢄⠀⠀⠀⠀⢠⣶⡶⠶⠶⢶⣦⠀⠀⠀⠀⢀⣤⣶⣯⣩⡿⠶⠂
⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢴⢇⢻⣿⣿⠿⡫⠂⠀⠀⠀⢸⡇⠙⠦⠞⠁⣿⠀⠀⠀⠀⢸⡒⠨⠭⠥⢒⡫
⠀⠀⠙⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢣⣣⡩⠗⠁⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠘⠦⠭⣑⣪⠵⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⣬⠰⠀⠈⠉⣹⣿⣿⣿⣿⣿⡷⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠉⣶⡐⠰⡆⣦⠀⢠⡀⣄⣀⠀⢀⡜⠛
⣷⣾⣶⣁⠀⠀⠨⠷⠿⡿⠿⢿⢂⡀⣾⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠿⡤⠀⠛⠸⠃⠨⠹⠎⠉⣻⣿⡁⢠
⠟⡿⠻⠄⠀⠡⠁⠀⠀⠀⠀⠀⠉⠉⠁⢀⣁⠀⢀⡆⢴⣠⣀⠠⡤⠶⢆⠀⠀⠠⠀⠀⡁⠀⠀⠀⠀⠰⡞⠁⠸⡇⠈
⢻⣏⣛⠙⠃⣀⡀⠀⣀⠀⠠⠤⠄⠀⠀⠀⡉⠀⠈⠀⠈⠋⠓⠜⠀⠁⠈⠀⠀⢈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣾⣛⣛⣃⡀⠀⠀⠀⣀⣠⣴⡶⡃⠀⠀⠰⠃⠀⠉⠉⠀⠀⡌⣍⠁⠉⠀⠉⠁⠀⠄⠀⡇⢀⡷⣴⣆⣶⣿⡳⣿⠀⣐
⠉⠉⠉⠉⠡⢳⠄⡉⠉⡏⠁⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠃⠈⠀⠀⠓⠊⠉⠛⠛
⠀⠀⠀⠀⢀⣶⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⣤⣤⠄⣤⢠⣤⡄⣤⣤⣤⠄⢀⣀⡹⠇⠂⠀⠀⠀⠂⠐⠐
⠀⡀⠀⡄⠠⣿⠀⠀⠀⡇⠀⣀⠀⠀⠀⠀⣿⣿⣃⣿⣾⡿⣇⣿⣿⣿⣁⣙⡟⠉⠁⠀⢀⣠⣄⢢⡱⡀
⠉⠁⣖⠀⠀⢼⠂⠀⠀⠱⢆⡈⡁⣠⣶⣾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣷⣆⠹⣽⣿⠆⠃
⠁⠋⡁⢀⢀⡋⢀⣀⡀⠀⠈⢠⣾⣿⣿⠟⠛⠙⠋⢻⣿⣿⣿⣿⣿⡟⠿⠿⠿⣿⣿⣿⣷⣅⠀⠀⠀⠀⠀⠀⠤⠀⢀
⠁⠓⠝⠾⠸⠨⠃⠇⠅⠀⠀⠈⣙⣟⣁⣀⣀⣂⣳⣿⣿⡿⠉⢿⣿⣿⡆⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠔⢠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⣿⣟⢸⡏⢸⣿⣿⡧⠦⠼⣿⣿⣿⠒⠒⠶⠆⠀⠒⠰⠂⠀⠂⠀⠀⠒⢈⠀⠛
⠀⠀⡖⢶⠀⠀⠀⠀⠀⠀⠀⠀⢙⡛⢋⣙⢘⠃⣸⣿⣿⡁⣶⠠⣿⣿⡿⢀⣀⣀⠰⣾⣟⠉⠀⢩⡇
⠀⢠⡗⠞⣤⢄⢀⡀⣀⠀⠀⠀⣈⣭⠈⠉⠸⠤⢼⣿⣿⣷⡛⣼⣿⣿⡇⢘⠛⠛⢛⣽⣽⣷⠀⢘⡇⠀⠀⢠⠁
⠀⠀⠻⠆⣿⢃⠀⢷⢻⡆⠘⢿⣿⣿⣷⣶⣏⣀⣈⣙⣿⣿⣿⣿⣿⣋⣮⣙⣱⣶⣾⣿⣿⠏⠀⣰⡅⠠⠀⢸⠠
⣀⡀⠀⠀⠈⡉⠀⠨⠀⠀⠈⠈⢈⣹⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠡⣉⠈⠀⢸⡄
⠌⠘⠀⠀⠀⣀⣀⣀⠀⠀⣀⡀⠀⠑⣀⠀⠋⢙⣛⣉⠉⢁⠠⣤⣬⠍⡉⢉⡉⠍⢀⣀⣀⣀⡀⢘⣷⠀⠀⠸
⠶⠐⠲⢀⣾⠟⠉⠻⢷⣤⣿⣆⣤⣤⣿⣦⠀⣸⣿⢿⣧⠰⠞⣿⡿⣿⡀⣿⡇⠄⢸⣿⣋⣉⠁⠈⣿⡇⠀⠀⣇⡔⠒
⠀⠀⠀⠸⣿⡉⠀⣠⣴⠃⣿⣿⣟⠛⣿⣿⢠⣿⣿⣾⣿⣧⣁⣻⡏⠙⢿⣿⡇⠃⢸⣿⠟⠟⠀⢀⣿⡇⠀⠀⡇⢑⣚
⣀⡀⠀⠀⢹⠿⠿⡟⠉⢰⣿⣯⣿⣟⠿⠋⠿⣯⣭⢉⣨⡟⠃⠻⠇⠀⠈⠛⢃⣀⠘⠛⠿⠛⠷⠀⠻⠿⠿⠟⡇⢸⣿
⣿⠅⢸⣿⠷⠄⠀⡇⢀⠈⣩⣽⣭⣅⣈⣳⠀⠛⠛⢀⡛⡅⣠⣤⣤⡄⢀⣀⣰⠄⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⡇⢠⡅
⠙⠃⠀⠁⠰⡆⠀⡇⠉⠖⠯⢭⡏⢹⡿⣿⣽⣷⣶⣾⣷⣇⣿⣿⣿⡃⠸⠿⠋⠂⠀⢄⡌⠠⡅⢨⡄⠀⠀⠀⠃⠀⣿
⢱⣧⣦⣦⣭⠁⠀⡇⠠⣄⣻⣉⠄⠀⠀⠿⣿⣿⣿⣿⣿⠉⠀⣠⡀⠁⠀⠀⠘⠿⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠽
⠀⠀⠀⠀⠉⠀⠀⡇⠲⣿⣿⡿⠖⠠⠤⠌⠉⠩⠍⠩⠩⠾⠖⠸⠰⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣀⣀⣀⣤⣀⣀⣀⣇⣘⣉⣏⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡰⡄
⡄⠀⠀⠀⠀⠀⠀⢀⡿⣸⣶⣦⠀⠐⠒⠀⠀⠒⠂⠐⢒⠤⠤⠤⠤⡄⠰⣟⣿⣦⠳⠈⠀⠀⠠
⠅⠀⠰⠶⠶⠶⠶⠿⢾⡯⢭⣭⢥⠄⠀⣀⠤⣶⢾⣟⡛⢲⠀⠖⡦⡘⠆⠙⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠿⠿⠿⠿⢿⠶⣤⡄



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠔⢸⠁⠉⠹⡄⠄⠀⡀⠈⠀⠀⠒⠴⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢂⣄⡀⠀⠀⠀⠰⠄⠀⠀⠶⠐⠐⠲⠰⠒⠶⠆⠀⠀⢀⡀
⠀⠀⠀⠃⢀⠀⠃⠀⠀⠧⠘⠀⠀⠀⠀⠂⠄⠀⠀⠀⠉⡉⠛⠲⠶⢤⡄⠀⠘⠂⡨⠀⠄⠀⠀⠀⠀⠀⠄⠠⠤⠤⠤⠤⢄⠀⠆⠀⠀⠀⠐
⠀⠀⠀⡈⢫⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠈⠠⡒⠄⡐⠃⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠔
⠀⠀⠀⠐⠄⣀⠈⠀⠀⡠⢋⠄⠀⠂⠁⠀⢐⠀⠃⠄⠀⠀⠀⠀⠸⢠⠈⠀⠔⡒⠂⢤⡀⠀⠀⠀⢠⢂⣁⠢⠀⠀⠀⢠⢨⠀⠄⠀⡶⠉⠕
⠀⠀⠀⠀⠀⠬⠁⠀⠡⠔⡊⢰⠈⠐⠢⠄⠀⠀⠈⣀⠁⠌⠀⡀⠀⠀⡐⠁⢀⡀⠐⠀⡀⠀⠀⠀⠐⢌⣋⠔⠀⠀⠀⢐⢸⠀⠂⢀⠑⠤⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠂⠈⡠⠘⡀⠄⢀⣀⠀⠠⠀⠢⡉⠢⡀⠀⠁⢀⠃⣌⣀⡤⠧⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⣸⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⣤⡀⠀⠁⠀⢃⠀⣔⠨⠁⢈⠠⠤⣐⢆⠱⠀⠈⠒⠒⠁⠄⠙⠠⣐⡆⠀⡀⠒⣀⣀⠬⠥⠐⠶⠒⠋⡛⠉⠍⡑⠢⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠀⠀⡾⠾⠒⠉⠀⠠⠄⣤⢂⣁⠈⣷⠁⠱⠀⢀⣀⢠⡀⢐⠀⠀⠐⠀⠹⣄⠁⠄⠐⡂⠈⠤⢠⠶⠮⡀⠀⠀⠀⠀⢑⡢
⠀⠀⠀⠁⢀⡈⠀⠈⢀⠠⠐⠈⠀⠀⠨⡻⡁⠈⢃⢄⠆⠀⠄⠀⢶⡁⣀⣏⠆⠀⠉⠁⣀⠘⢷⣄⠀⠀⠀⠀⢈⡂⠖⣉⡤⠖⠚⠛⠉⠀⢀
⠀⠀⠀⠀⡷⡠⠀⠸⢤⠐⠀⠒⠄⡀⠀⠈⠂⣀⠔⠓⠈⠀⠀⠀⠉⠋⠙⠂⠀⠐⠀⠀⢉⡀⠐⠈⠳⠖⠂⠉⢒⡞⠁⠀⣀⢰⠄⠀⢈⠢⡀⠁
⠀⠀⠀⠀⠣⡀⠀⡀⠈⣄⠀⠀⠀⣈⠶⠐⠈⠀⠀⢰⠑⠄⠀⠎⠡⠊⡇⠀⠀⡐⡩⢁⢢⢀⢎⠭⢱⠀⠔⠉⠀⡤⠤⣄⠈⠀⠊⢆⠀⠀⢡
⠀⠀⠀⠀⠐⠁⡰⠁⠀⡍⠂⠂⠉⡰⠀⣀⠀⠐⠂⠀⡆⡇⢀⠀⠂⠀⠄⠀⠀⠃⠀⠈⠀⡌⠈⠀⠉⠀⢣⠀⠀⠑⠴⠀⠀⣀⠈⢆⢢⡀
⠀⠀⠀⠂⠀⡔⢁⣀⡀⠈⠒⠄⠀⢇⠀⣀⡀⢀⠂⠢⠐⠐⠊⡜⠐⠀⠄⠀⣀⡆⢄⠐⡐⢀⢠⠐⢄⠔⢀⠀⠀⠀⠀⠀⠀⠉⠐⠈⢏⠻⡄
⠀⠀⠀⠀⠈⢰⠁⠄⠈⢂⠀⡘⠀⠀⣱⣁⣸⠊⠀⠀⡆⠑⠊⠀⣀⠑⠒⠒⠂⠈⠒⠊⠀⠀⠒⠀⠊⠴⠃⣠⠴⠚⠒⠛⠛⠲⣅⠀⠈⢆⡏⢀
⠀⠀⠀⠒⢠⠈⢢⡔⢫⡈⢂⠣⠀⠀⠑⠈⠘⢀⣀⣀⡈⠂⠐⠊⠀⠇⠀⠘⠃⠀⠀⠃⠀⠀⠘⢄⠀⠀⡼⢁⠀⡔⢁⡌⠂⠀⠈⢂⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠑⢄⠑⠬⢷⠊⠀⣉⠂⠐⠐⠊⠁⢀⣀⣀⡑⢄⢀⣀⠑⠊⠉⠉⠩⠭⠉⠉⠁⠀⢀⠑⡀⡁⠀⠀⠘⠀⠐⠂⠀⠀⢸⠀⠀⠂⢻
⠀⠀⠀⢀⣀⡠⢀⠈⢑⠄⠓⠤⢈⡀⠤⠐⠂⠉⢀⠀⠀⠈⠢⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠐⡀⠀⠠⢀⠀⠎⠑⠦⣜⠀⢀⠄⠂
⠀⠀⠀⠘⠄⣠⠎⠀⠎⠀⠀⠀⠐⠂⠀⠄⠠⠂⠀⠀⡂⠀⢀⠃⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠹⣤⢃⠀⠆⠰⠀⢀⡍⠀⠾⣀⡆
⠀⠀⠀⠐⠀⢀⣀⢃⠌⢀⠙⠄⠂⠀⠄⠈⠰⠂⠨⠀⢁⠀⠋⢀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⡤⠊⢠⣶⡆⠀⠈⠣⢆⣘⠀⠃⠼⠀⠀⢄⡀
⠀⠀⠀⡤⠰⡁⠀⠒⢄⢀⠀⠀⠀⠀⠄⡀⠐⠀⠈⠀⢀⠀⢣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢉⠀⠀⠀⠄⢠⠀⠀⠀⠈⠂⡀⡀⢀
⠀⠀⠀⠀⢀⠠⠇⠀⠠⡒⠈⠈⠀⡀⠈⠄⢀⡀⢀⠀⢀⡀⠂⡇⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⡄⠀⠠⠀⢻⠀⠄⠠⠀⠀⡆
⠀⠀⠀⡐⠁⡔⢆⠂⠤⢈⡁⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⡠⠊⠀⡠⠐⠒⠒⠊⠁⠀⠂⠀⠀⠀⠀⠰⡗⢆⠀⠀⢸⠀⠁⡄⡆⢸⠀⢠⠀⠰
⠀⠀⠀⡃⠰⠀⠀⠣⡀⠀⠀⠉⢈⠰⡄⠀⠀⠀⠈⠁⡊⠀⠀⠀⡄⠈⠁⣠⢴⠠⡀⠀⠀⠊⠀⠀⢀⠁⡐⠀⠀⠀⠐⠂⠀⢁⠈⠄⡈⠀⠘⡀
⠀⠀⠀⠣⣀⠂⠠⠄⡈⠒⠊⠁⣷⢀⠃⠀⡰⠁⢱⠀⠃⠴⠀⠀⠆⠀⠀⠙⠚⠒⠁⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⡀⠀⢀⠤⠐⠁
⠀⠀⠀⠁⠀⠉⠉⠐⠢⠄⣉⣐⡃⠈⠠⠊⠀⠀⠘⠄⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠤⠐⠊⠀⠀⠃⠀⠘⠄⠸⠀⠐

⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣾⠁⠀⠀⠀⣠⣆⠀⠀⡿⠧⠿⠀⠀⡔⠀⢸⣿⡟⠉⢹⣿⣿⠉⠉⣿⣿⠀⠐⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠇⠇⡆
⠀⠀⠀⠀⣿⠀⠠⡶⠷⠀⠝⠃⠀⡇⢾⠆⠀⠀⡇⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⡇⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⠀⣤⠄⡇
⠀⠀⠀⠀⢿⣀⣀⣀⣠⣄⣀⣀⣀⣧⡆⢦⠀⠀⡇⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⡇⠀⠀⡿⢿⠿⠿⡿⠿⢿⠿⠀⡄⡄⡇
⠀⠀⠀⠀⠇⣶⣶⡶⠶⠶⠶⠶⠶⣚⣽⢝⠀⠀⡇⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⡇⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⠀⠁⠁⠇
⠀⠀⠀⠀⠀⢈⣥⣤⣤⣤⣤⣤⠀⠀⡈⠀⠀⠀⠣⠄⠸⠿⠧⠀⠸⠿⠿⠀⠠⠿⠿⠀⠤⠃⠈⡴⣩⣉⣉⣉⣉⣉⣉⣭⠉⠉⡙⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠸⡿⠀⠀⢀⡄⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⠛⠛⠛⠛⠛⢛⢻⡇⡀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⡶⠲⡄⠀⠓⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠃⣷⠛⠛⠛⠛⠛⠛⠛⠛⠣⢀⠄⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣀⣤⡀⠀⣏⠿⠛⠛⠛⠛⠛⠛⠻⠓⠠⢂⡇
⠀⠀⠀⠀⡀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⢉⣉⣁⣀⣀⣀⣀⣀⠉⠉⠉⡀
⠀⠀⠀⠀⡇⡎⠀⢻⣿⡇⠀⣿⣿⠁⠈⠂⠀⠀⠈⠉⠉⠁⠠⠄⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠀⠀⠉⠉⢱⠐⠂⡇
⠀⠀⠀⠀⡇⡇⠀⣸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣠⣤⣤⣤⡐⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⡄⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⡇⢇⠀⠉⠁⠀⣰⣶⣶⠀⡀⠅⠀⠀⠈⠉⠉⠉⠁⠈⠁⠀⠀⠛⠀⠀⠠⠶⠶⠀⠀⠀⡇⣰⠏⠀⣠⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⡇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣤⡜⠀⠀⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⣀⣀
⠀⠀⠀⡧⠤⠤⢤⡤⠬⢭⣤⣥⣬⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠀⠉⠁⠈⠉⠉⠉⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⡃⠀⠀⠀⢰⣾⣿⡿⠿⠿⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⢙⡋⢛⠷⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣦⣄
⠀⠀⠀⡇⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣀⡄⢠⣅⢈⢀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠋
⠀⠀⠀⡇⠀⠀⠀⠘⣧⡀⠀⠙⠂⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⡂⢘⡇⠘⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢺
⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⢸⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⡇⠀⠀⠀⠀⠀⠠⣶⣶⠶⠶⣶⣿⡆
⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⢸⣿⠛⠉⠉⠉⠉⠉⠛⠛⣿⡇⠀⠀⠀⠀⠀⢨⡍⠉⠉⠉⠉⠉⠁
⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⡀
⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⡇⠀⠀⠀⠀⢸⠿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠛⢻⣿⣿⡇
⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇⠀⢤⡧⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣤⣬⣥⠉⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⢀⣰⣇⣀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁
⠀⠀⠀⡆
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣛⣛⠛⣛⣋⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⢛⠛⠛⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠳⡄⠀⢀⡾⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⣶⣶⣿⣷⡀
⠀⢸⣿⠀⢀⠀⢀⠀⢸⣿⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣦⣤⣤⣤⣤⣾⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⣾⣧⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⡿⠿⠿⠿⣿⡿⠃
⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠉⢻⣿⣿⣷⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⣄⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⣿⣿⡿⠟⠛⢶⣴⠟⠛⠿⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠒⠚⢿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⣿⠛⠛⠛⠳⣆
⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠛⢻⠋⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢸⣿⣾⣻⣽⣿⣿⣿⣿⣤⣤⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⢤⣿⡏⠉⣿⣿⣿⠉⢻⡇
⠀⠘⣿⣿⣿⣹⣿⣹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣿⠿⣿⣶⣿⠃
⠀⠀⠈⠻⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠉⠀⠀⠀⠉⠛⠛⠛⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠥⢠⠀⠌⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⠂⠐⠂⠒⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣁⣈⣁⣣⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⣠⠤⠤⠤⠤⢤⡀
⣿⠀⢀⡤⡤⠤⢤⠤⠤⢤⠤⠀⢸⡇⠀⢸⠀⢰⣤⣦⡖⣶⢶⠲⣆⡆⠀⡇⠀⠀⢸⡇⠀⠐⠂⠂⠒⠒⠂⠀⡇⠀⠀⠀⠀⣿⠀⡀⣀⠀⢸⡇
⣿⠀⠰⠇⠇⠛⠀⠀⠀⠈⠀⠀⢸⡇⠀⢸⠀⠈⠀⠉⠛⠁⠈⠀⠙⠀⠀⡇⠀⠀⢸⡇⠀⠀⠹⠈⠿⠁⠀⠀⡇⠀⠀⠀⠀⣿⠀⡤⢤⡤⢸⡇
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⢸⣶⣶⣶⣴⣦⣦⣶⣶⣶⣶⣴⡇⠀⠀⠘⢷⢶⣶⢶⠶⣶⡶⢶⣶⠇⠀⠀⠀⠀⣿⠀⠁⠈⠁⢸⡇
⠉⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⠉⠁⠀⠀⠀⣠⣤⣤⣴⣿⣦⣤⣤⣄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣦⠀⠀⠀⢿⣿⣿⣹⣿⣿⠇

⢀⣶⣶⠶⠶⠶⠶⠶⠶⠶⢶⣶⡄⠀⠀⠀⣶⡶⠶⠶⠶⠶⠶⠶⠶⢶⣶⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡄⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⢶
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠈⠛⢻⣿⣿⡿⠿⠿⣿⣿⣿⠛⠋⠀⠀⠀⠿⠿⠿⠟⠿⠿⠿⠿⠾⠿⠿⠀⠀⠀⢛⣛⣛⡻⠿⡿⡿⢟⣛⣛⡛⠁⠀⠙⠛⠛⠛⣛⣿⡛⠛⠛⠛
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠚⠛⠉⠉⠙⠛⠂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠐⠢⣀⣀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣤⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡔⠊⠀⠀⠀⠀⠀⠀⢀⣀⣈⣲⣴⣎⣀⣀
⠀⣶⡶⠶⠶⠶⠿⠷⠶⢶⣶⢶⡆⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⠛⢿⣦⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⢶⣶⣦⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠉⣧
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣷⣄⣀⣀⣀⣀⣀⣤⣿
⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⢿⣗⣶⣶⣶⣶⣶⣾⡟⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣼⣛⣿⠀⠀⠀⠿⣤⣿⣤⣤⣼⣧⡼⠏
⠀⠈⠹⠿⠿⠿⠿⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠈⠉⠿⠋⠉⠉⠉⠉⠿⠏⠁⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠈⠁⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠓⢄⠀⠀⢀⠴⠊⠁
⠀⠀⣠⣤⣾⣯⣤⣤⣤⣤⣤⡀⠀⠀⠀⢀⣀⣀⣀⣒⣉⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣑⣒⣚⣁⣀⡀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣶⣀
⠀⢸⣿⡿⠉⠉⠉⠉⠉⠻⣿⣿⠀⠀⠀⢸⣿⠟⠛⠛⠉⠙⠛⠛⣿⣏⣎⡇⠀⠀⢰⡿⠋⠉⠉⠉⠉⠉⠉⢻⣇⠀⠀⢰⡟⠉⠀⠀⠀⠀⠈⠉⢷
⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢹⣯⣪⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⣷⣤⣤⣤⣤⣤⣼⣿⣿⡄⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠻⣯⣿⣿⣿⣿⣿⣿⣿⣯⡿⠁⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠈⢻⡿⠶⠶⠶⠶⠶⠾⣿⠃⠀⠀⠀⠉⣛⠓⠶⠶⠖⠛⣟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠗⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⣠⣾⡿⠃
⢠⡴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⢦
⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣰⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢷
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⠞⠛⢷⡄
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣀⣸⠇
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠁
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⠓⢶⡀
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡀⠀⣸⠇
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠋⠁
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠻⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤
⠀⠀⠉⠉⠉⢩⡏⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣏⠉⠙⡏⠉⠉⠉
⠀⠀⠀⠀⠀⣸⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⠠⠄⠀⣀⣀⣀⣀⣀⣐⣀⣀⣒⣒⣒⣒⣒⣒⣂⣀⣐⣒⠂⢒⡢⢀⣀⣀⣀⣀⣠⢸
⠀⠀⠀⠀⢸⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠈⣿⣿⣿⣿⣿⠺
⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⠈
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠀⣿⣿⣿⣾⣿⢨
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠿⠿⣿⠿⠈
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢐⢢⡾⢖⣷⠚
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⢨⡁⠙⠍⠃⡌
⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢐⢲⣻⣟⣷⣆
⠀⠀⠀⠀⢸⠠⡐⠈⠒⠦⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠒⠒⠒⠂⠈⠁⠐⢠⠀⣾⣧⣬⣭⣥⡎
⠀⠀⠀⠀⠀⢄⣀⣀⣀⣀⣉⡉⠉⠉⠉⠉⠁⠄⠤⠤⠤⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠄⠤⠄⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⢀⣀⣀⣀⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠭⢿⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⠿⠶⠶⠾⠷⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⠤⠤⠶⠖⠶⠒⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⣤⣤⡀
⠀⠀⠀⠀⠀⢠⡞⠉⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⣿⠀⠀⢠⡤⣤⣄⡈⠹⡆
⠀⠀⠀⠀⠀⣾⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿⠀⠀⡇⢾⣉⣷⡇⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠈⢳⣿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⢀⣴⠚⡓⣤⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⢸⠄⣏⣙⣦⡇⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⠀⠑⣲⣷⠋⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡇⠀⠀⢠⠖⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠈⢷⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢰⣉⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⡇⢀⣀⣐⣛⣀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣸⣹⡀⣇⣸⣉⡇⢸⠁
⠀⠀⠀⠀⠀⣿⠀⠈⠷⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠏⠀⢸⡇⡏⢹⠉⡏⢹⠉⡇⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣤⣤⣬⣥⣤⣭⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⠇⣏⣹⣹⣗⣻⣺⠁⣿
⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣿⠒⢺⠓⠒⢺⡗⠯⢿⠂⠀⠀⠀⠀⠀⠀⠀⠸⣯⠭⣯⠭⠭⣯⠭⣿⡀⠀⢸⠀⠯⣿⣹⣇⡽⠝⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣯⢭⣭⣉⣉⣛⣛⣛⣛⣛⣛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠷⠶⠾⠷⠶⠿⠶⠶⠶⠶⠶⣶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⢤⢤⣤⣤⣤⣤⠤⢬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣋⣛⣋⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⡴⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⢠⠤⠴⠋⠳⠤⢤⠀⠀⠀⠀⠀⣀⡀⢳⣠⠇⣀⡀
⠀⠀⢧⡀⠀⣠⠇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼⠀⡴⠒⠲⡄⠸⣄⠀⠀⠀⠀⢡⠟⠻⡽⠛⢏⠁
⠀⡠⠦⣍⣉⡥⠦⣄⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠘⢦⠀⢧⣀⣠⠇⢠⠞⠀⠀⠀⠞⢿⢠⣆⡇⡦⢸⠛
⠸⣄⠀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠈⠳⣄⠀⠀⠀⠀⠀⠸⠤⢤⡀⣠⠤⠼⠀⠀⠀⠀⠀⡼⠦⣌⣁⡤⢿⡄
⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢰⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣻⠆⢸⡇⢸⣹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡞⠛⠛⠛⣷
⠀⠀⠀⠀⠘⢲⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⠃⢀⠀⠀⣿
⠀⢰⡏⠉⠉⢻⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠿⠀⠸⠇⠸⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢴⠂⡾⢐⡄⣿
⢠⡞⢳⡆⢰⠚⢲⠀⡖⠓⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⣤⠀⡠⣄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠇⠈⠀⣿
⠈⠓⠚⠃⠘⠒⠚⠀⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠛⠀⠓⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠋

⠀⢀⣤⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⣴⢯⠿⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⢸⠰⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⠀⢤⠀⠀⠀⡇
⡇⢰⡿⠻⡿⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠟⠈⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⣼⠀⠀⠀⡇
⡇⠘⠿⣲⠽⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢠⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠤⠤⠴⠶⠾⠟⠒⠒⠚⠒⠒⠶⠤⠤⠤⠤⢄⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣤⡴⠖⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⣄
⠀⠀⠀⡴⠋⣸⠁⠀⢀⡤⣄⡤⠤⠴⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⠤⢤⣄⣢⣄⠀⠀⠀⣀⡀⢤⡀⠸⡆
⠀⣠⢴⠟⣠⡏⠀⠀⡼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠔⡄⢹⣾⡆⠀⡾⢣⡱⡄⢹⠄⢿
⢰⡣⣾⢎⢼⡇⠀⢠⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠀⣿⣧⠀⠳⣄⡙⣎⠜⠀⠸
⣟⠅⣿⢢⢾⠀⠀⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢿⣻⠀⠀⢀⡬⢤⡀⠀⢰
⣯⢊⣿⠁⢺⠀⠀⡈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⡜⠳⣜⣄⢹⠆⢸
⣿⡳⣻⡠⢻⠀⠀⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠾⠀⠙⢦⣹⡾⠞⠀⢸
⢿⡔⣹⠁⢸⡀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⡀⠀⢸
⢸⡐⢹⣡⢺⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠀⠭⠿⠟⢻⠆⢸
⠸⡷⢸⣧⠚⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠈⠉⠥⠄⢸⠀⣾
⠀⣷⠣⣧⡞⣷⠀⠀⢸⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠀⢸⢈⣙⡷⡗⢼⢀⡇
⠀⢹⡡⢻⡜⣹⠄⠀⠈⣇⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢿⠀⢀⡀⠀⡞⣸⠁
⠀⠘⣧⢻⡏⣸⡇⠀⢀⡹⣄⠚⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡅⠀⠘⠦⠤⠤⠔⢡⡏
⠀⠀⠈⠲⢻⣀⣷⠀⠀⠑⠬⣉⡉⣛⣓⠒⠒⠒⠒⠒⠒⠒⠄⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠋⣱⠃⠀⢀⡀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠙⠺⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠤⠤⣤⣄⣀⣠⣀⣀⣄⣀⣀⣀⣀⣤⣄⣤⣤⠤⠤⠶⠒⠚⠋⠉⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⣄⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⢀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠟⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢠⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠤⠴⠶⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣆
⠀⠀⠀⠀⠀⢰⢋⢼⡏⠀⠀⣠⣒⠴⠶⠶⠖⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠶⠦⢔⡦⡄⠀⠀⢀⢤⡴⠤⣄⠸⡆
⠀⠀⠀⢀⡴⡿⡡⣾⠁⠀⢠⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠴⡰⠈⢿⣿⠀⢰⠛⢦⡱⡄⢹⠀⣿
⠀⠀⢠⣏⢼⡟⡕⣿⠀⠀⡸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢊⢸⣿⡆⠈⠳⣄⣛⡯⠏⠀⢸⡄
⠀⠀⢸⡵⣸⡏⠔⡏⠀⠀⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣟⡇⠀⢀⣠⡤⢤⡀⠀⢸⡇
⠀⠀⢸⣜⣽⡏⠐⡇⠀⢀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡇⠀⡎⢳⡱⡄⢹⠆⢸⡇
⠀⠀⢸⣮⢾⣇⠜⡇⠀⠈⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠱⢌⣹⠿⠏⠀⢸⡇
⠀⠀⢸⣧⢻⡏⠀⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢀⣀⣀⣀⣀⡀⠀⢸⡇
⠀⠀⠘⣧⢎⣏⡔⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠬⠭⠭⠽⢽⠀⣸
⠀⠀⠀⣿⠊⣿⡔⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡷⠰⠶⠶⠆⢸⠀⣿
⠀⠀⠀⢻⡜⣽⣠⢻⡆⠀⠀⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣏⢘⣫⠦⠤⣿⢠⡇
⠀⠀⠀⠸⡟⡸⣧⢊⣷⠀⠀⢸⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⣗⣂⣁⠂⠀⡏⣸⠁
⠀⠀⠀⠀⢿⡜⣷⠃⣿⠀⠀⠀⢧⡠⠗⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠷⣀⣐⣀⡼⢡⡏
⠀⠀⠀⠀⠈⠳⣽⡞⢸⡆⠀⠘⠦⣙⠒⠒⠒⠲⠶⠦⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠶⠛⡼⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠙⠯⣷⡀⠀⠀⠀⠈⠉⠉⠙⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠉⠀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⠤⣤⣤⣀⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣠⡤⠤⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣀⣠⡤⠄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⠟⠉⠀⢰⣶⣶⣶⣶⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣷⣤⡤⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⡿⠛⠛⣿⡇⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⠃⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⢦⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣡⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡳⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢦⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠱⣄⠀⠀⠀⠀⠀⢀⣴⢫⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠗⠛⠛⠋⠉⠉⠉⠙⠛⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣯⣤⣤⣤⣤⣤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠲⠶⠦⢤⣄⡀
⢰⠖⠚⣛⣛⡙⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣩⣭⣭⣭⣍⣉⠁⢸⡆⠀⠀⠀⠉⠛⠲⠦⣄⡀
⢸⠘⠖⢒⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠾⣩⠟⠛⠙⢻⣦⡈⢻⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⢸⠀⣴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠳⣦⡀⠀⠀⡗⡏⠁⠀⠀⠀⣿⡇⣸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣼⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⡇⠱⣄⣀⣤⣞⣵⠇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⡇⠀⠀⠉⠉⠉⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡀
⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣤⠞⣿⢿⣶⣄⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠳⡄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢰⡇⠀⠁⠀⢹⡽⣧⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢷⣀⠀⢀⡾⢡⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⢰⣏⣉⣉⣹⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠀⠙⠛⠛⠛⠋⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⠰⣧⣀
⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⠀⠀⢀⠤⣤⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⣤⣤
⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢧⣀⣼⡿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢙⠛⠛
⠃⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠉⠁⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⢺⠃⢿⣿⣿
⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⢠⠊⠙⣷⡀⠀⢸⠃⡿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⢻⠅⡴⠤⣴
⠻⠤⣀⠀⠀⠉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⠀⠈⠲⠾⠛⠁⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠉⠁⢸
⠀⠀⠀⠉⠓⠶⣤⣀⡴⣾⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡇⠀⠀⠀⢀⡀⠀⠀⢸⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣍⡻⢿⣿⡏⠛⠶⠶⢦⣤⣤⣤⠶⠾⠋⠀⠀⣧⡀⠀⠦⠤⠟⠀⠀⣸⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠶⠶⣤⠤⠴⠞⢻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⣀⠀⠀⠀⠀⠀⡷⣏⣉⠙⠛⠛⠛⡇⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣄⡓⠤⣄⣉⠑⠒⠾⠇⡀⢸⠀⠀⠀⠀⢀⣀⣴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠮⠍⢑⡒⠤⠇⡀⠀⣀⡤⠞⠋⠁

⠀⠀⣀⣤⣴⣶⣿⣿⠉⠋⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠋⠙⠉⠉⠙⠙⠙⠋⠛⠉⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠟⠿⢿
⢠⣾⣿⣿⣿⣿⣿⣿⠀⠀⢠⣆⣀⡀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠚⠓⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄⠉
⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣟⣙⡇⡄⠀⡇⠀⠀⢀⡾⠆⠴⠆⢠⣄⣴⣦⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣞⡙⡇⡇⠀⡇⡆⢠⡏⠔⠺⢶⠀⠉⠁⠰⠆⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣹⣞⡃⡇⢀⡇⡇⣾⣰⢤⣐⡿⠉⢟⡋⠴⠤⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣯⣠⡇⡷⠈⡇⡄⠘⢇⡲⠋⠀⠈⠛⠂⡾⡽⠉⠀⠀⠀⠀⠀⠤⠤⣠⠤⢄⡀⠀⠀⠀⡇⠈
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠉⠹⡇⡇⠀⡇⡇⠀⠈⠻⠢⡀⠐⢦⡤⡠⠀⠀⠀⢀⠀⡀⢀⡀⠀⠀⠉⠢⣙⣆⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⠐⠁⡗⠀⡇⣇⣀⡔⠁⠀⠈⡑⠒⠊⠀⠀⠀⠀⠈⣞⠀⠀⠀⠈⠲⢤⡀⠈⠻⠀⠀⠇
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠄⠂⠀⡇⠀⡇⡏⠃⠃⠀⠁⠀⠉⡿⡖⣄⠀⠀⠀⠀⢛⠀⠺⠄⠀⢠⣀⠑⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣷⣦⣆⡇⠀⡇⣷⠀⢡⣀⣀⣀⣀⣇⣇⠸⡀⠀⠀⠀⢶⣠⠀⠀⠀⠚⢛⣷⣿⣀⡆
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠁⢸⡇⠀⡇⡏⣠⠶⠭⡭⢩⣅⣖⣿⣆⡇⠀⠀⠀⢦⣤⣤⡤⠀⢠⣷⡟⣉⠁⠁⠀⠆
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⢻⡿⠏⡇⠀⡇⠗⠒⡎⢩⠛⠛⠁⠋⢙⡾⠘⠀⠀⢀⣙⣤⠏⣀⠨⣌⠁⣠⣿⠢⠂⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⢧⣄⡇⠀⡇⠉⠈⠁⠉⠉⠀⢈⣉⣉⣉⣉⣈⣉⣀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠞⠁
⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠒⠒⠛⠁⠀⠙⠚⠙⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈
⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠛⠋⠉⠉⢹
⠀⠙⠻⣿⣿⣿⣿⣿⣶⡞⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠠⠤⠤⠜
⠀⠀⠀⠈⠉⠛⠛⠿⢿⣇⣀⣀⣀⢀⡀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠐⠒⠒⠂⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢘⣛⣛⣛⣛⣛⣋⠀⠀⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢨⣭⣭⣭⣭⣭⣭⠀⠀⡄⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣛⡋⠀⠀⡆⠀⠀⠀⠀⠀⣠⣄⣀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣀⢄⣀⣀⣀⣠⣤⣬⣭⣭⣍⠀⠀⠆⠀⠀⢠⠊⠉⢙⣯⠟⠛⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧⠖⠋⠒⠒⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⣺⣥⣭⣴⣷⠒⠒⠄⠈⣿⣧⣀
⠀⠀⠀⠀⠀⣴⣤⣤⣾⣿⣸⠀⠈⠛⠛⠛⠛⠢⣀⣀⣀⣀⣀⣀⠀⠈⢆⠀⠘⠛⡯⠄⠒⠺⠃⠀⣀⣴⣿⠟⠋
⠀⠀⠀⠀⣠⣿⡿⠛⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠸⠅⠀⠀⢁⠀⠀⠀⠃⠀⠿⠿⣿
⠀⠀⠀⠛⣿⣽⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠘⣰⠀⢰⢀
⠀⠀⠀⠀⣸⣟⣧⣄⣀⣠⢾⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⡼⠃⠀⠈⢧⡀⢃⡇⠈⡞⡀
⠀⠀⠀⠀⠉⠉⠛⣿⡿⠟⢻⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠹⣦⢳⠀⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢀⡼⠼⡀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⠞⠁⠁⠃⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠟⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢰⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠄⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃


⡟⠉⣻⣛⣛⣛⣛⣟⣻⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣠⣼⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡹⡄⠀⠀⠀⠀⠀⠀⠀⣀⢔⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀⡀⠀⠀⠀⠈⣧⢹⡀⠀⠀⠀⢀⡤⣪⠵⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠣⠻⣟⠛⢛⡵⠚⠛⠻⠟⠛⠛⠶⠶⠶⠶⠶⣶⣦⣤⣤⡀
⠀⠀⠀⣀⣤⣾⣯⣭⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⠶⠟⠛⣿
⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⣶⣶⣶⣤⡶⠾⠛⠋⠉⠉⠀⠀⠀⠀⠀⢰⡿
⠀⠀⢸⡇⠀⡖⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣸⠀⠀⡇⠸⢉⢉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡆⠀⢀⠀⠀⠀⢀⢀⣀⠀⠀⢸⡇
⠀⠀⢸⡆⠀⡇⡇⠞⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠿⣶⡀⣿⠀⠀⢿⢸⢰⣾⢸⣸⣿⠀⠀⢸⡇
⠀⠀⢸⡇⠀⡇⢧⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⢧⣤⡿⠃⣿⠀⠀⣾⢸⢸⠿⠸⠛⠛⠀⠀⢸⡇
⠀⠀⢸⡇⠀⣷⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⣿⠀⠀⠙⠈⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣤⣤⡀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⠀⢈⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠳⠟⠃⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠶⢶⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⣿⠀⠀⠘⠓⠶⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠀⢸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢿⣤⣀⣀⣀⡀⠀⠀⠀⠉⠉⠉⠛⠛⠓⠒⠲⠶⠶⠶⠶⢤⣤⣤⣤⠞⠀⢧⣤⣾⢠⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⠇
⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠷⠶⠶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠷⠶⣶⣶⣾⣷⠶⠿⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣀⣤⣤⣤⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⢿⣛⣛⣛⣻⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣟⣛⣛⣻⠿⢿⣿⣿⣿⢛⣭⢫⣭⡻⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⠟⢛⣛⣉⣭⣭⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠻⣿⡇⣿⣿⢣⣿⠏⠈⢻⣿⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡏⣿⡏⢰⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣏⠻⣷⢰⣿⢟⣼⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⣿⣿⣷⣶⣷⣶⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣧⣿⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣿⡟⣡⣶⢰⣶⣝⢿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡹⣿⡅⠀⣸⣿⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣷⣙⠿⡸⠿⣣⣾⣿⡇
⠀⠀⠀⠀⢸⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠛⢿⣿⡿⠛⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⣴⣾⣿⣷⣶⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢸⣿⢸⣿⣒⣒⣒⣒⣒⣒⣒⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢛⣃⣿⡟⣿⣿⣖⡖⣶⣶⢶⡶⣲⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣮⣟⣛⣛⣛⠿⠷⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠿⣛⣛⣛⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢀⢤⡄⠀⠁⠂⢀⠀⠘⠁⡠⠄⠈⠀⠠⠔⠒⠀⠀⣀⡄⡰⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢳⡠⠀⠀⠶⢶⠄⡀⠀⣀⣴⡀⣾⢒⡐⣄⢤⣤⣔⡟⣿⡜⠁⣀⣀⡀⠸⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣵⡄⠀⣀⣄⢠⢻⣶⣁⠖⠁⣩⣆⣿⣧⢿⣿⡁⢤⡝⠀⠈⠒⣒⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄⠰⡖⠞⣿⣟⠟⠭⠱⢾⢋⡟⣳⣇⡈⣟⡿⡟⠀⠘⣿⣿⣑⣦⣷⠷⣁⣴⢉⣙⡠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠁⣶⢴⡾⣏⣣⣾⣶⠿⠯⠿⠵⠿⠽⠿⠿⠧⡤⣶⣻⣶⢀⣹⣟⣷⣯⣷⡦⠖⢖⠵⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣖⣾⣞⣓⣛⣗⣒⣒⣒⣚⣛⣛⣛⣛⡛⠛⠛⠻⠿⠷⠾⠧⠤⠤⠭⠭⠿⠿⠿⠤⠾⠮⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣟⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣓⣓⠒⢃⡘⠩⠭⠭⢉⣉⡍⢩⣅⣝⣿⣶⣶⣶⣶⡄⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⡉⠙⢻⣿⠿⠧⠶⠶⠶⠶⠶⠤⠤⠤⣤⣬⣉⣝⣿⣻⠟⢺⣷⠀⢁⣽⡭⠤⣄⡉⢳⡃⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⢹⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⠀⡿⣿⠀⣿⡏⢀⣴⠟⢳⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣟⣿⠀⢻⣿⡟⠁⢀⡼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠿⣿⠀⠢⡙⠻⣿⣋⢀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇⠠⣿⡗⣦⣰⢦⣆⡦⣶⢲⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢸⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇⢠⣿⡇⠛⠙⠚⠋⠓⠛⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡗⣿⡌⡇⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢰⣿⡇⠐⠷⠺⠇⠿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⢳⠀⢻⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⢸⡇⢐⣒⣒⣒⣲⣆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠁⠘⡄⠘⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠀⠘⢸⡇⢨⣭⣭⣩⡉⠗⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡄⠳⠶⠭⠽⠺⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠋⡀⡀⢸⡇⣠⡄⠉⣼⠦⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣯⣷⣶⣶⣖⣦⣤⡴⣤⣤⣤⣀⣠⣀⣀⣠⣼⣿⣿⣶⡶⣦⠆⢸⡧⡠⠠⣤⣹⠏⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣉⣧⣽⣘⣬⢃⣬⣝⣀⡒⣀⣰⣯⡇⣾⣥⣸⣭⣄⠀⢀⣸⣦⣉⣈⣁⣀⣀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣖⣒⣒⣀⣐⠒⠒⠒⠂⠀⢀⣐⠒⠚⠋⠛⠛⠛⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⡇⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⢹⣏⣻⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣯⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠟⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⡓⢿⡍⣭⣭⣭⡍⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣼⣿⣿⣿⣿⣿⠟⠉⠉⠉⢉⣽⣿⣿⡿⣻⣿⡆⠁⣿⣫⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⡿⣫⣾⣿⣿⣧⠀⣿⣿⣿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⠀⣿⣥⣾⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡃⠀⠀⠀⣴⣿⣿⡿⣫⣾⣿⣿⣿⡿⠋⣼⣿⠀⠛⣛⡛⠃⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⡇⠀⣠⣾⣿⡿⣫⣾⣿⣿⣿⡿⠋⣠⣾⣿⡏⠀⣾⣿⣿⡆⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⢹⣧⣾⣿⣿⣯⣾⣿⣿⣿⣿⣉⣤⣾⣿⣿⣿⢃⡄⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣓⣀⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣊⣀⣒⣛⣁⣈⣛⣋⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠿⠶⠶⠿⠷⠶⠾⠿⠿⠷⠶⠾⠿⠶⠶⠿⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠶⠿⠶⠾⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⢿⡏⠉⣿⠀⠀⢀⣤⡤⠤⠶⠶⠖⠒⠒⢒⣒⣒⣒⣒⣒⡒⠒⠒⢒⣒⣒⣒⣒⣒⣒⣀⣀⣈⣛⠓⢶⡄⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇⣠⡾⠛⠋⠉⠉⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⢸⡇⢠⣾⠋⠈⠱⡄⠀⠀⡇
⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠘⣿⣀⡀⣠⠇⠀⠀⡇
⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠉⠉⠁⠀⠀⠀⣷
⠀⠀⠀⢸⡇⠀⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⡾⠛⠻⣦⠀⠀⣿
⠀⠀⠀⢸⡇⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⡀⠀⣠⠟⠀⠀⣿
⠀⠀⠀⢸⡇⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠉⠁⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣀⡀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣼⠟⠉⠙⣆⠀⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣿⠀⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⠀⠹⣧⣀⣤⠟⠀⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡇⠀⢀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣿⠀⢸⡏⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠾⠁⢸⡇⠠⡟⠉⠉⠙⡧⠀⠀⣿
⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⣇⣀⣉⣙⣓⣒⣓⣒⣓⣚⣛⣛⣛⣛⣛⣛⣛⣫⣿⣯⣭⣭⣭⡽⠯⠿⠶⠶⠛⠛⠃⠀⠙⠶⠴⠚⠁⠀⢀⣿
⠀⠀⠀⠸⣧⣀⣠⣿⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣾⠿
⠀⠀⠀⠀⠈⣿⣿⢿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⠏⠉⢹⣿⣿⠛⡏
⠀⠀⠀⠀⠀⢽⠉⠘⠇⢘⣻⣦⣴⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠘⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠀⠀⠀⠀⠈⠉⠙⠳⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⢡⣏⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⡟⠛⢻⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡶⠴⠋⡉⣻⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣧⡙⠄⠻⣄⣧⠰⣆⠠⡄⣀⠀⡀⠀⡆⢸⠀⠀⠀⡜⣡⠏⠀⠀⠀⢴⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠈⠻⠶⠶⠋⠙⢷⡜⢦⡓⠸⠄⡇⠰⠁⣋⣠⣦⣠⠾⠋⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⢠⣤⣤⣤⠾⠾⣍⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⡁⢶⣄
⠀⠀⠀⠀⠀⣠⠞⠋⣿⠁⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠹⣧
⠀⠀⠀⠀⢰⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣇⠀⠀⠀⠀⠀⠻⣤⣄⣤⡤⠶⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⢀⣼⠃
⠀⠀⠀⠀⠘⠷⠤⠄⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⠶⠾⢶⣄⠀⠀⡏⠀⠀⡆⠀⠘⠦⣤⣄⣀⣤⣤⣤⣤⣤⣤⡤⠾⣏⠉⠉
⠀⠀⠀⠀⠀⠀⢠⡶⠾⠶⠶⢤⡤⠤⠴⠶⠀⢿⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⢧⡀⢦⣹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠤⠼⣧⣴⠾⠃⠀⠀⢨⡽⠿⠿⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⡠⠤⠤⠤⠙⠷⢶⣦⣤⣤⣤⣤⣤⡴⠀⣤⣶⢸⣆⣷⠽⠀⠀⠀⠉⠉⢡⣤⣤⣤⣬⣍⡛⠻⠶⠶⠶⠶⠶⠶⠿⠟⠃⠒⠚⠷⠶
⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠈⠉⠉⠉⠭⠡⠍⠙⠚⠁⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⣶⠶⠶⠦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠭⠭⠩⠽⠿⠧⠴
⠀⠀⠀⠀⠀⠈⠉⠉⠩⠥⠄⠤⠤⠤⠤⠤⠤⠴⠦⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣒⡒⠒⠒⣒⣒⣒⣒⣀⣀⣀⣀⣀⣈⣁⡉⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣩⠉⠉⠉⣩⢉⡭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠌⠉⠀⠀⠠⠠⠉
⠀⠀⠀⠀⠀⠀⠀⡐⡉⠀⠀⠀⠀⠀⡔⠂
⠀⠀⠀⠀⠀⡠⣏⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠠⠘⣆⠀⠠⠈⡠⠊
⠀⡠⠈⠀⠀⠀⡠⠈⡤⠊
⢇⠐⠀⠀⠀⠨⡦⠂
⡇⠀⢠⠀⡀
⡧⠐⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀
⡧⠂⢠⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢩⣿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣾⣿⠆
⠀⠀⠘⣷⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣸⣷⣿⣿⣿⣷⣦⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⡿⢿⣿⣿⠟⠁
⠀⠀⠀⠹⣿⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿⣷⣝⠳⢶⠆⠀⠀⠀⡞⠀⠁
⠀⠀⠀⠀⠘⢛⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⢶⣿⣿⣿⣿⣿⣿⣿⣯⢽⠿⠋⠁⠀⢙⣿⣿⢦⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣤⣄⣀⡀⢉⣻⡿⠿⠛⠻⠏⠹⣟⣿⣿⣿⣷⣻⣿⣿⣿⣯⡟⠟⠈⠔⢠⡄⢀⣴⣿⣿⠿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣭⣩⣭⡇⣄⡀⠀⠲⢲⣾⢻⠍⣹⣿⠿⢛⠟⠐⠉⠟⠑⠀⠀⠺⠆⠀⢿⣿⠋⠁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠷⠂⠀⠀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⣀⣼⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⠀⣀⣠⠄⠀⠀⠀⣠⣾⣿⡅
⠀⠰⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠁⣭⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢻⣿⣿⢿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡙⢲⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠑⣠⡇
⠀⠀⠀⠀⠀⠀⠘⠲⠚⡛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣁⠌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢣⡀⠀⠀⠀⠀⠀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠽⠶⠶⠶⠶⠶⠿⠿⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣴⣊⣥⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠽⠷⠤⠦⠶⠶⠦⢤⣀⡀
⣰⢛⠛⢯⡍⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣬⣭⣭⡥⢄⡁⢸⠀⠀⠀⠉⠛⠒⠤⢀
⢹⠉⣩⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣄⡀⠀⠀⢘⡱⠏⠁⠈⢿⣦⢡⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲
⡄⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⢸⠡⡀⠀⣀⡾⡿⣼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⠉⠙⠛⠋⠀⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣠⠖⡷⢷⣦⡀⡇⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠢⡀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠀⠀⠀⡿⣷⡇⣿⠀⠀⠀⠀⠀⠀⣠⣀⡀⣿⠀⠀⢹
⡟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠘⠦⣤⣾⡧⢿⡇⡿⠀⠀⠀⠀⠀⠀⠓⠦⠤⣿⢀⡄⢸
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⠀⣀⡀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣉⣻
⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⢬⣀⣿⠆⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠛⢻
⠁⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡇⠀⢀⣉⡀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢰⣟⡒⠒⡇⠻⠿⢿
⣀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇⠀⣅⣸⡷⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⠓⢺
⠈⠑⠂⠤⣀⡈⢉⣻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠈⠉⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣸
⠀⠀⠀⠀⠀⠉⠓⠯⣭⣛⢿⣿⡗⠶⢤⣤⣤⣤⣤⡤⠞⠁⢸⡇⠀⢠⣨⠆⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⢉⣲⣶⣦⣶⠖⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⡀⠀⢰⠻⠭⢉⣉⡉⣿⠸⢸⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠭⣔⡢⠤⢉⡉⣠⠈⠀⠀⣀⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣤⢀⡴⡄⣄⢠⢠⡆⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⣸⢷⢿⢸⡇⡇⢻⡏⠐⡇⣿⠄⠺⠇⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢀⠌⠀⢅⠀⠀⡄⠀⠀⡀⠀⠈⠀⠂⠂⡄
⠀⢀⠠⢀⣀⣀⡠⣀⠉⠁⠀⠀⠻⠀⠘⠈⠳⠃⠘⠃⠰⠇⠻⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠖⠐⠀⡠⢀⠑⠀⠀⠀⢀⠆⠀⠆⠀⢀⠀⡀⢀⡀⠀⡀⡄
⠀⡐⠺⠿⠟⠛⠋⠘⠀⠀⠑⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⠀⢀⡣⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄
⠀⠀⠀⢴⣶⣶⠀⠀⣂⡀⠄⠀⠀⠀⢀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠠⠈⠀⠄⠘⡁⠀⠀⠀⠰⣀⠁⠠⠒⢄⠀⠀⠀⡠⠒⠀⠀⠀⠐⠢
⠀⠐⢄⠈⠻⠋⡀⠌⠀⠀⢀⣀⣀⣀⣁⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡀⠀⠀⡰⡒⠀⠀⠂⢀⠁⠀⠈⢱⠀⠈⠀⠀⠀⠀⠀⠁⠀⡃
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡀⠀⠀⣦⢰⢠⠠⠀⠀⢀⠀⠀⢨⠅⡀⣇⢀⠠⡄⢠⠄⢀⠀⠀⠄⠘⠆⣀⠀⠀⡁⠀⠂
⠀⡠⠀⠈⠀⠑⢄⠀⢠⠀⣿⣿⣷⣿⢷⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠈⠈⠀⠈⠀⢀⠄⠀⠀⡤⠀⠀⠀⠠⠂⠀⡆⠀⠀⠐⠀⠀⠀⠀⣀⠄⠀⠀⠀⡀
⠐⡀⠀⠀⠀⠀⢀⠄⠈⠀⣿⣟⣥⣯⣷⣯⣽⣿⠀⠀⠀⠈⠀⠰⠎⠀⠀⠀⠀⠉⠠⠤⠅⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠰⡀⠀⡐⠀⠀⠀⠊⠛⢷⢄
⠀⢳⡆⣤⠀⡤⠂⠀⠀⢄⣉⣉⣉⣉⣉⢉⣉⢉⣀⣀⡀⠄⠀⢀⠠⠄⠰⠂⠀⠀⠀⠀⠂⢀⡀⠀⠘⠤⠀⠤⠂⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⣿⣤⣀⠈
⠀⢸⠇⠉⠘⡇⠀⠀⠀⠉⢀⣀⠀⠠⠤⠄⠀⠄⠀⠈⠃⠠⠂⠡⠒⠲⠤⠜⠢⠀⢀⠔⠐⠀⡆⢑⠢⡀⠀⠠⠊⠀⠀⠰⠆⠀⠀⠑⠀⠀⠁⠀⠀⠚⠋⠁
⠀⢸⡇⣶⢸⡇⠀⠀⠀⠀⢠⠀⠀⠀⡤⢖⠀⠘⠀⢀⠴⠁⠀⢠⠀⠀⡀⠀⢀⠀⢘⡀⢀⠠⠁⠂⠂⠐⢄⠀⠀⠀⠉⠇⠿⠀⠀⠀⡄
⠀⠈⠃⠿⠘⠃⠀⠀⠀⠀⠈⠀⠀⠠⠖⠉⠀⠀⠀⠈⠢⡀⠀⠁⠑⠂⠠⡐⣉⠔⠁⠁⠀⡄⢊⠀⠃⡀⠈⠀⠂⠀⠀⠀⠀⠀⠀⢂⣀⣀⣀⣀⣀⣀⣀⣁⡀
⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⡁⠀⠐⠖⠀⠂⠀⡀⠀⠀⠈⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣉⢟⠙⣿⢹⣿⢉⠁⣿
⠀⠀⣰⡿⣫⡵⣶⣦⣝⢿⡄⠐⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄⠤⠄⠠⠀⠀⠀⠰⠀⠀⠀⠐⡀⠸⠛⠛⠋⠀⠀⠙⠛⠛⠛
⠀⠀⠿⡼⠁⠀⠈⠉⠛⡾⡇⠀⢠⣤⣶⣶⣶⣶⣶⣶⣄⠀⠉⢅⠜⠁⠀⡠⡀⠀⠰⠋⣄⠀⠀⠀⠛⠐⠊⠄⠀⢘⠀⠀⠀⠈⠇⠀⠀⠀⠂⠈⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠋⠹⠉⣿⢡⠈⣿⠀⣀⣠⣤⠀⠊⠀⠁⠀⠀⠘⠻⠃⡠⠐⠤⡀⠀⡀⠀⠐⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⢀
⠀⠀⠙⣷⣄⠀⠀⢀⣼⡏⠀⠀⢸⡇⠀⡄⡀⢸⠠⡴⣿⠀⠙⠁⠈⣀⡜⠻⠂⠀⠘⠢⠤⠎⠀⠀⠀⠈⠈⠀⠀⠀⠢⢠⡤⠄⠈⢸⠀⠀⠀⠀⠀⠀⠄⠠
⠀⢀⠰⡿⠻⡇⠀⢿⠛⠗⠀⠀⢸⣷⣴⣇⣷⣨⣄⣠⣿⡇⠀⠀⠀⠈⠙⠟⠀⠀⠀⠀⠀⠀⠰⡎⠀⠀⠀⠈⠁⣠⡀⠀⣆⠀⠀⢈⣀⠠⠰⠀⠀⠠⠤⠄
⠀⠀⠀⠴⠂⠀⠀⠀⠉⢂⠀⠀⢸⣿⠩⡏⢽⠙⢹⠩⣿⡇⠀⢰⣨⣥⣬⣥⣠⡀⠀⠀⠀⢀⠠⠀⠀⢄⣠⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⠁⠁⠈⠉⠉⠉⠉⠁
⠀⠀⣼⣶⠀⠀⢠⣤⣴⣚⡆⠀⢻⣿⣨⣧⣈⣴⣼⣐⣿⡇⠀⡀⠀⠀⠀⠀⠈⠁⠈⠀⠀⢸⡤⢤⠀⢈⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰
⠀⠀⠀⠁⠀⠂⠙⠛⠉⠁⠀⠀⠀⠙⠛⠉⠉⠉⠉⠉⠉⠀⠀⡀⠀⠀⠁⠀⠀⡀⠀⠀⠠⡏⠀⠈⠁⠈⠀⠰⠇⠈⢧⠀⡀⠀⢠⠂⠀⠀⠀⠄⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠁⠢⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⡸⠃⠁⠀⠀⠣⡀⡰⠀⠀⠀⢀⡀⢀⡜⢀⠀⠀⠀⣴⣶⣦⣄⠀⠀⠀⠀⠁
⠀⡀⠐⠐⠒⠒⠂⢠⠀⠀⠀⢀⡆⣀⣀⡀⣀⣁⠀⠀⡄⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠘⢝⡿⠿⠛⠀⠠⠂⠀⠈⠂
⠀⠁⠀⠀⠀⠀⠀⢈⠆⠀⠀⠁⠀⢠⣴⣀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠁⠀⠀⠓⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠛⠻⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⡠⠖⡈⠒⠁⠀⠀⠀⠀⢀⣀⠂⡄⠀⠐⢄⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠻⠏⠀⠀⡅⠀⠀⠀⠀⠤⡀⡠⠄⡄⠈⠀⠘⠀⠀⠀⣠⢊⠔⠊⠀⠀⠀⠀⠀⠀⢡⡿⢛⣷⢈⠀⠠⠂⠀⠑⠄⠀⠎⡘⢋⡠⠀⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠛⠃⠂⠃⠀⠃⠀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢆⠀⠈⠻⠟⠁⠀⠀⢃⠀⠀⠀⡀⠀⠀⠀⢠⣶⠀⠀⢠
⠀⠘⠒⠛⠉⠉⠉⠉⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠑⠀⠒⠀⠤⠂⠀⠀⠀⠀⠑⠠⠔⠁⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡟⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⣤⣶⣿⣶⣶⣤⣤⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠛⠉⠁⠀⠀⠈⠉⣙⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⢿⣶⣦⣤⣠⣴⣾⠟⠋⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯⣤⣤⣤⣤⣄⠀⠀⠀⣿⡗⠀⠀⠀⠀⠉⢹⣿⠉⠀⠀⠀⠀⣿⡇⠀⠀⢀⣤⣤⣤⣤⣬⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⠉⠉⠉⠉⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠉⠉⠉⢉⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣤⣀⣸⣿⣠⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠙⠛⠿⠿⠟⠉⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣿⡇⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁



⠀⠀⠀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡏⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⠀⡴⠶⠶⠶⠶⠶⢦⡈⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⠀⡇⠀⡴⠒⢦⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠂⠒⠒⠒⠀⠀⠀⢸⠀⡇⢀⣷⣤⣼⡀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠉⠀⠉⠉⢉⠉⠉⠉⠁⢸⠀⣇⣯⣀⣀⣀⣹⣼⠇⡇
⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⠀⣤⣀⠀⠀⠀⠀⢀⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠛⠀⠋⠁⠀⠐⠛⢻⡭⢥⣀⡬⢭⠟⢳
⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⢯⠉⠉⠉⠉⠉⠉⠁⢸⡧⠤⡤⠤⢼⠀⠯⠇
⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠶⠶⠶⠶⢶⠶⠾⠿⠶⢶⠶⠶⠶⠶⠆⠘⢇⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣉⣉⣉⣉⠉⠉⢱⡦⠤⠾⠦⠄⠀⠀⢀⢤⡾⣟⠛⣹⢶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠤⠤⢸⡟⠒⠤⣄⡀⢀⠜⠁⠀⠑⣌⠉⡡⠋⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠶⠶⢦⡀⠀⢸⡧⣄⠀⠀⠉⠁⠀⢰⡆⠀⢸⠙⡇⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⣀⠰⡃⣎⢙⡆⡷⢀⡸⠃⠀⠀⠲⢄⣀⡤⢺⡇⠀⠀⠁⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢮⣭⠖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⣾⢸⡶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢷⠀⠀⠀⠀⠀⢀⡠⣺⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠶⠶⠶⠶⢤⣤⣤⣤⣼⣾⣧⣤⣤⣤⢔⣽⣮⣥⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠀⠐⠁⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣉⣻⣿⣿⡆
⠀⠀⣰⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⠶⠶⠶⠶⣶⣶⡶⠶⠶⢶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣤⡴⠶⠞⠛⠛⠋⠉⠁⠀⢸⡇
⠀⠀⣿⠀⢀⡤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠉⠉⠉⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⣿⠀⢸⠀⡖⡀⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⢸⢰⠰⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⢶⣄⢸⡏⠀⢸⢳⣶⣾⢸⣾⣷⠀⠀⣿
⠀⠀⣿⠀⢸⡟⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⢦⣤⠟⢸⡇⠀⢸⣼⣿⣿⠸⠟⠋⠀⠀⣿
⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡇⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡴⢦⡀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡀⣸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣙⡉⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⡿⠀⠘⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⣿⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⡇⠀⠀⠈⠉⠉⠛⠒⠶⠦⢤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣰⠃⡇⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠙⠓⠲⠶⠶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠁⠀⠙⠛⠋⣸⡇⠀⠀⠀⠀⣀⣠⣴⡶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠷⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣀⣀⣽⣇⣠⣴⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠛⠋⠙⠋⠀⢀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠻⢿⣦⡙⠿⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⡿⠻⣿⣧⣤⣶⣶⣶⣦⣽⣦⡀⢰⣿⣿⢿⡇⠘⠿⠀⡀⠈⠉⠀⠈⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣼⣿⣿⡿⠀⠀⡀⠀⠀⠀⠷⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠃⡸⠻⣿⣿⠇
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠋⠀⠀⠤⠀⠀⢠⣿⣿⣿⡿⡿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣆⠀⣀⠀⠀⠈⠃⡀⠀⠀⠀⠌⠀⠀⠀⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠏⠉⠛⢿⣿⣿⣤⠀⠀⢐⣐⠡⡖⣿⣿⡏⢿⠑⠇⠘⠧⠙⠂⠻⠇⣿⣿⣿⣿⣏⠀⠉⠀⢠⠊⠁⡸⠀⣠⠊
⠀⠀⢶⣿⣿⣿⣿⠏⢉⠉⠀⠀⠀⢀⡈⢹⣿⡅⠉⠈⠉⠉⠀⡿⣿⠀⠀⠋⠀⠀⠀⠀⠀⠁⠀⠈⢹⠿⣿⠏⠑⠤⠀⠀⠁⣩⠔⢈⠍⢸⠉⠒⢄
⠀⠀⢸⣿⣿⡿⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⢿⡃⠀⠀⠀⠀⠘⣄⡙⠀⠀⢸⡆⠀⡀⠀⢰⡇⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⡐⠱⠤⠓⠔⠁⠀⠀⠀⠑⡄
⠀⠀⠈⢏⠘⠃⠀⠀⠀⠶⠀⠀⡀⠿⠀⠀⡠⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠃⡀⠀⠀⢀⢀⣀⠠⠄⠀⠘
⠀⠀⠀⠀⠉⠱⣄⠀⠀⠠⠤⠀⠀⠀⢀⠼⡃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠐⠪⡁⠀⡇⠀⠀⢆⠀⣡⠀⣀⣀⡼
⠀⠀⠀⠀⣠⠊⠉⠙⠒⠒⠒⠒⠒⠛⠙⢦⡃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠑⠄⠠⠓⠊⡇⠀⠀⠀⠉⠁⠀⠀⠈⡀
⠀⠀⠀⢰⠃⠀⠀⡤⠀⠀⠀⠀⠀⢀⡀⠈⣇⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢻⠀⠀⠈⠛⠲⢦⡤⣴⠖⠛⠁⠀⡿⠀⠤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠑⠢⢄⣀⣀⣸⠁⠀⢱⠤⠔⠊⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⣠⣾⠟⠁
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣷⣄⣠⣾⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⣰⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡴⠖⢶⡄
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣇⠀⣀⡟
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡶⠛⠳⡄
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢧⣀⣠⠏
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠃
⢳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠈⠉⠉⠉⣿⠉⢉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⡏⠉⢻⠉⠉⠉⠁
⠀⠀⠀⠀⢠⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡇
⠀⠀⠀⠀⢸⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷

⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⡯⡯⡯⡯⡅⠀⠀⠀⠀⠀⠐⠫⣗⣺⠝⠂⠀⠀⠀⠀⠀⣴⣾⣷⡄⢀⠆⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢹⠯⣿⡆
⠀⠀⠀⠑⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⣇⣄⡸⠀⠀⠀⠀⠀⠀⠀⠙⠾⠷⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⣴⡾⠛⣧⡀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣦⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⣼⣧⣿⠄⠀⠀⠀⠀⠀⠀⣀⣠⣿⣀
⠀⠀⠀⠉⠃⣀⠏⠃⠀⠀⠀⠀⠀⠈⠉⡃⢰⠉⠁⠀⠀⠀⠀⠀⣘⣒⠒⢒⣚⡀⠀⠀⠀⠀⠀⠰⣾⢾⠷⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉


⠀⠀⠀⣤⡀⠠⠒⡂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⣰⠇⠀⠀⠀⠀⠀⢀⡀⣀⣠⡄⠀⠀⠀⠀⠀⠠⣶⣶⣤⣤⡄
⠀⠀⠀⠹⣦⡈⠊⠁⠀⠀⠀⠀⠀⢸⡯⣿⠉⠙⡇⠀⠀⠀⠀⠀⣶⣻⣰⣯⡟⠀⠀⠀⡄⠀⠀⠿⠿⠿⣿⡗⠀⠠⣦⣤⠀⢰⣶⣶⣶⣶⡇⠀⢠
⠀⠀⠀⠀⠈⠛⠻⠂⠀⠀⠀⠀⠀⠈⠓⠚⠓⠚⠃⠀⠀⠀⠀⠀⠸⠿⠋⠉⠁⠀⠀⠀⡗⠀⠀⠐⠂⠀⠈⠁⠀⠀⣿⣿⠀⠘⠛⠛⠛⠛⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⢶⣿⣿⡶⡄⠀⠀⠀⠀⠀⡟⠯⣉⠝⣿⠀⠀⠀⠀⠀⢰⠛⢛⢚⠫⠅
⠀⠀⠈⠲⠶⡶⠶⠞⠀⠀⠀⠀⠀⠀⠁⢀⡀⠈⠁⠀⠀⠀⠀⠀⠑⣄⡛⠉⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠟⠀⠀⠀⠀⠀⠸⠄⣀⡂⠬⠂
⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡒⠿⢟⣲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠸⠀⠀⠏⢸⢢⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⣼⠭⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⡤⢾⠉⠉⢉⣉⣉⣉⣉⣉⡉⠉⠉⠋⠛⠛⠒⠒⠒⠒⠶⢤⣤⠖⠤⠤⠤⠤⠤⣀⣀⣀⣀⡀⣀⢴⡿⢐⣩⡼⠥⠔⠖⠶⠤⠤⣄⡀
⠀⠀⢸⢀⣨⡇⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡦⠀⡜⢸⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⡽⠃⠋⠀⣈⢡⣄⣀⣰⣶⣶⣶⣶⣶⠻⢶⡄
⠀⠀⡞⢻⡻⢳⡄⢹⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⠇⡌⠀⠰⣉⣁⣀⣀⣀⣲⡖⠛⠠⠐⠊⠁⠀⠀⠀⢩⠀⠀⠀⠀⠉⠉⠁
⠀⢰⠃⠈⠁⣼⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⣼⠀⡇⠀⠰⠤⠄⢀⣀⣀⣿⠖⠀⠀⡇⠀⠀⡀⢀⡴⠋
⠴⣏⣀⠀⣠⢻⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⢠⡇⠀⠃⢰⡂⠀⠀⠀⠀⠠⢹⣞⠀⠀⢿⠀⠈⠀⡞⠀⠂⠂
⠀⠀⠈⢹⡁⢸⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⠃⣸⠁⢸⠀⠀⠁⠀⠈⠉⠉⠈⠙⠉⡟⣤⣼⠀⠀⡠⠛⢄⡠⠤⡀
⠀⠀⠀⠀⣷⣿⣅⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠀⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢺⣻⡀⢼⠀⠀⢸⡗⠂⢙⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠘⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡆⢸⠃⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠺⢿⠇⠈⣓⣲⠞⠉⠛⠋⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠛⢦
⠀⠀⠀⠀⡄⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⢸⠀⠀⡇⠀⠀⠀⠀⢀⡤⠖⠒⠒⢻⠀⠗⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⣃⣀⣀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡇⡌⠀⢀⠇⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠳⣤⡤⠺⡄⠀⠀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜
⠀⠀⠀⠀⢸⡯⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠃⡇⠀⢸⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠙⢌⠉⠓⠒⠶⠤⠤⠤⠤⠤⠖⠒⠈⠁
⠀⠀⠀⠀⠸⣷⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠀⡇⠀⢸⠀⠀⠀⠀⡇⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠙⠙⢻⠉⠉⠉⠉⠉⠉⠛⠿⠛⠿⠽⢻⣟⡃⠀⢸⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⠓⢦⠀⠘⢆
⢠⣤⠤⠤⠤⡾⣖⠚⢖⣲⣒⣶⣲⣄⢤⣄⢤⡤⣤⣾⢯⡻⣦⣽⠀⣶⡄⢶⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠢⡰⠗⠒⠉
⠘⢮⣷⣄⠀⠈⠪⣦⡬⠵⠮⠷⠿⠮⠷⠽⢶⡿⣿⠉⠳⣝⣦⣿⣄⣀⣠⠴⠚⣉⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣖⠛⠋⠲⡄
⠀⠀⠘⢿⣷⠤⠤⠬⠭⠭⠭⠩⠉⠉⠉⠩⢹⡅⠈⣆⠀⠀⠹⣿⣿⣾⠷⠚⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⢀⡏⡗⡾⠀⢀⡬⠿⠏⠸⠧⡄
⠀⠀⠀⠀⠙⣶⣶⠒⠒⠒⠒⠒⠒⠒⠒⣠⣸⡍⠒⡇⠀⠀⠀⢹⢸⠡⠤⠀⠀⠀⠀⠇⠀⠀⢀⣀⣀⣤⡎⠑⣷⠃⠀⣰⠋⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⣉⣶⠋⠀⠀⠀⢸⢸⠀⠀⢀⠀⠀⢀⠀⠀⣠⠎⠀⠁⢰⠁⣀⢳⣤⢾⣭⣳⡛⢤⣀
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡝⠀⠀⠀⠀⠘⢸⠀⠀⠀⡀⡀⢸⠋⠉⢰⡎⠀⠀⠘⠋⠉⢦⡱⡿⠁⠀⠣⣼⡽⠦⠼⣹⠆
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢤⣀⠀⠀⡆⠸⠀⠀⠀⠀⠁⢸⡶⢒⡟⢀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢶⢤⣈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣇⣇⡤⠶⠒⠋⠉⠉⣱⣎⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠙⣇⠀⠀⡇
⠀⠰⡿⢗⢦⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣤⠴⠚⠉⠈⠙⠛⠛⠓⠲⠶⢶⠦⠤⠖⠒⠛⠛⠁⠉⠉⠉⠳⡆
⠀⠀⠈⠢⣝⢷⡜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣆⣘⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠈⠳⣍⠢⡍⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣝⣛⣉⣉⣉⣙⣯⢭⡀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣼⣆⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣜⣷⣤⣤⣄⣀⣀⣀⡀⠉⢯⣽⣼⣹⡀⣰⠋⠀⠈⠀⠀⠀⠀⣐⣻⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠤⠤⠤⠤⠬⠿⠭⠛⠛⠒⠒⠒⠒⠒⠒⢂⣷⣠⠗⣞⣃⣤⣬⣤⣴⣖⣚⣉⣿⠁⠤⠤⠤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣀⣀⣀⣀⣀⣼⣁
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠦⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡖⠒⠂⠐⠀⠀⠀⠀⠂⠐⠀⠀⢀⡀⠈⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠁⠈⣍⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⠀⠀⠀⠀⡀⠀⣀⠀⠀⠽⠠⣾⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⢔⠀⠀⠀⠀⠀⢠⢠⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠈⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢀⠀⠀⠙⣅⠀⠀⠀⠀⡰⠟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣎⢆⢤⣿⠟⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠐⠋⠈⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⣦⢹⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡸⠟⠿⣯⣋⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⡇
⠀⠀⠀⠀⠀⢀⡴⠋⡠⠘⠆⠈⢿⣶⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⣸⣾⡗
⠀⠀⠀⢀⠔⢫⢄⣬⡔⠁⠈⢡⣿⣿⠿⠿⠷⠶⠶⠶⠶⠶⠶⣶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠿⠿⠗
⠀⠀⡖⠁⡠⠣⡧⣯⢇⣠⡶⠟⠁⠀⠀⠀⣀⣀⣤⠔⠒⢊⣉⣀⣤⡟⠙⠒⠦⣤⣄⣀⡀
⠀⠀⢳⣄⠐⠀⣀⣴⣿⠋⠀⢠⣴⣶⡬⠭⠴⠞⠓⡛⢛⣉⠉⠀⠤⠄⠉⠉⠙⠒⠚⠭⣿⣿⣶⣦⣤⣄⠀⢀
⠀⠀⠀⠙⠓⠛⠛⠋⠉⠻⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢩⡄⠀⠀⢀⡍⠛⢷⣄⠀⠀⠀⠀⠀⣿⣤⣿⣤⣿⣧⣼⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⣦⣤⣤⣤⣤⣤⣤⣤⡿⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⠀⣼⡟⣻⡆⢰⡆⢠⣿⢻⣷⠀⣿⠉⣿⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⣿⠀⠈⠛⠛⠀⠘⠃⠀⠙⠛⠃⠀⣿⠶⠿⠀⠀⣀⣀⠀⠀⠀⠀⣈⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡾⠟⠂⠘⠃⠐⠃⠀⠛⢷⣤⣿⠀⢀⣴⠟⠋⠙⠻⣶⣴⠟⠛⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣅⠀⢰⡶⠶⠶⢶⣦⠀⢨⡿⠃⠀⣾⠃⠀⠀⢀⣤⣬⡁⠀⣤⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣇⣀⣀⣸⣷⣾⠛⠁⠀⠀⠹⣧⡀⠀⢸⣧⣼⡇⠀⣿⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣼⡇⠉⠉⠉⠉⢉⣿⣤⣤⣤⣄⡀⠈⠻⣦⡘⠃⠘⠃⠈⠛⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠈⠀⠉⢿⣦⣦⣤⣴⡾⣋⣈⡈⠈⠉⠻⣦⠀⠉⣿⣦⡀⠀⣠⡾⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣴⣿⣩⣿⠶⠶⠶⣿⡷⠾⠏⠉⢻⡾⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⢶⣶⣄⠨⣿⣤⣤⣤⣤⣤⣤⣽⡏⢡⣶⠶⠶⠿⠇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣼⠏⠻⣦⣬⣨⣄⣤⣤⣠⣤⣌⣴⠿⣧⣄⡀⠀⠀⠀⠀⢀⣼⣷⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⢹⡏⠉⢹⡏⠉⠙⠿⣧⣄⠀⠉⠛⠛⠛⠛⠛⠛⢷⣼⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⣷⣴⠶⠾⠟⠃⢰⡟⢻⡆⠀⠀⠈⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢸
⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸
⠀⠀⠀⠀⠙⠂⠐⠒⠒⠒⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⣀⣀⡀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠀⢀⡦⠀⢠⣄⠤⠤⠤⡤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠥⠤⢤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣂⣤⣴⣴⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠋⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠀⠀⠐⠒⠲⣔⢲⢰⠁⡀⢀⣤⡤⠤⣤⣤⠤⢤⣤⡀⢀⠈⡆⢠⣤⣤⣤⣤⣤⣤⢠⣤
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣤⡘⠳⠆⠐⢨⡅⢸⢸⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⡇⡇⢘⣻⣻⣻⣻⣛⣟⠈⣈
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣄⣀⣀⣸⡤⢤⢸⢸⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⡇⡇⠸⢿⢿⢿⠿⠿⠿⢠⣡
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠖⠒⠒⠒⠲⢭⠍⢸⢸⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⡇⡗⣈⢉⢉⣉⣉⣉⣉⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⠀⣤⠀⢸⠀⠓⠘⠛⠃⢀⣛⣛⡀⠘⠛⠃⠊⠁⣷⣵⣤⣤⣤⣤⣤⣴⣠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢀⣉⡀⠸⠼⠄⠀⠀⠀⠀⠀⠀⠤⠤⠤⠴⠤⠤⢿⡷⠶⠶⠶⠶⠶⢶⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⠀⢸⠳⠶⠶⠶⠶⠶⠶⠂⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠀⠀⢁⣀⣀⣀⣀⣀⣀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠃⢀⣛⡃⠀⠀⠀⠀⠶⠿⠶⠆⠤⠀⠀⡼⠁⠀⣀⣀⠀⠀⡇⢀⡔⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣉⣁⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢧⣈⡁⠀⠁⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢳⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣎⢧⠀⠀⠀⠀⣀⢔⡽⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠞⢿⡛⠳⣪⠷⠿⠶⠶⠶⠶⠶⠶⢶⣦⣤⣤⣄⣀
⠀⠀⢀⣤⣶⣯⣭⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠞⠛⣿
⠀⠀⣾⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⠶⠶⠶⠶⡶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⣘⢉⡩⠭⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢻⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⣿⠀⣿⠸⢰⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⣀⠀⣿⠀⢠⣄⣀⣠⢰⢠⡆⠀⠀⡇
⠀⠀⣿⠀⣿⢰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣏⣁⣹⠂⣿⠀⢸⣿⣿⣿⢸⠾⡇⠀⢰⡇
⠀⠀⣿⠀⢹⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠉⠉⢸⡇⠀⠘⠟⠋⠉⠀⠀⠀⠀⢸⡇
⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣾⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⡇⠀⣿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⡿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢻⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠈⠉⠙⠓⠶⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡼⢹⠀⢸⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠈⠙⠛⠲⠶⠶⠦⣤⣤⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠋⠀⠈⠛⠛⢹⣿⠀⠀⠀⢀⣀⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠷⠶⢶⣶⣦⣤⣤⣤⣤⣼⣏⣤⡴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣿⡏⠉⣽⣿⣿⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⡉⢹⣿⣿⡍⠉⣿⡏⠀⠀⠀⠀⠀⢰⡟⠛⠛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠛⠛⢷⡄
⠀⠀⣿⡇⠀⣿⣿⣿⠀⣿⡏⠉⠉⠉⠉⠉⠉⢹⡇⠰⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⡇
⠀⠀⣿⡇⠀⣿⣿⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⣿⡇⠀⣿⣿⣿⠀⣿⡆⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⣿⡇⠀⣿⣿⣿⠀⠻⠷⠶⠶⠶⠶⠶⠶⠿⠇⢸⣿⣿⡃⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⣿⡇⠀⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⡅⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇
⠀⠀⣿⣇⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⣀⣿⣧⠀⠀⠀⠀⠀⢸⣧⣤⣤⣬⣭⣍⣭⣭⣭⣭⣭⣭⣭⣤⠤⡤⣼⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⢸⡇⢰⣶⡄⠀⣿⣿⣿⣿⣿⣿⡇⠀⢰⣶⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣭⣤⣤⣽⣭⣼⣯⣿⣿⣥⣤⣤⣭⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⣤⣤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣿⢸⣿⣿⣟⣿⣿⡇
⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣾⣷⣾⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣿⣷⣿⣿⣿⡇
⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡿⣷⢿⡿⣿⡇
⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠟⡛⢿⣯⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠭⡭⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏⠄⡇⡌⡏⡇
⠀⠀⠀⠸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣶⣦⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣷⠃⡇⢳⣷⡇
⠀⠀⠀⠀⢰⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣟⣿⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣘⣿⣷⣷⣿⣟⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣶⠶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣾⣮⣾⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢟⢿⣿⡱⣿⣭⣭⣭⣭⣭⣿⣭⣿⣿⢎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱⣶⣶⣾⣿⣦⢃⣿⡱⣿⣿⢫⣿⣷⢎⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣛⢫⣾⣿⣿⡑⣿⣿⢣⣿⡿⣪⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⡻⣟⣤⣭⣭⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡫⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣧⠀⠀⠀⣿⣿⣿⠘⡆
⠀⠀⠀⠀⠀⠀⠈⡻⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠟⠀⠀⠀⠿⠿⠟⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⣿⣿⠟⣿⣿⣿⣿⣿⡿⢿⡿⣿⠛⡟⢿⠟⡿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⡟⠛⠛⠿⠛⣛⡻⢻⡟⠛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠿⣿⠀⡟⢉⠉⠡⡌⠐⠎⢇⣦⣴⢁⢠⣶⠁⡈⠃⠀⠀⠈⡃⠀⠈⠉⢁⡀⠘⠉⠀⢸⡏⠀⠰⠆⠉⢨⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣇⠘⠂⣷⠀⡈⠃⠃⣠⣿⠾⠾⠿⠿⢾⣿⣠⣿⡆⠚⠁⠀⣻⣽⣶⡄⢐⡀⠀⢀⣁⢸⡇⠀⠀⣀⠁⠸⠤⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣿⣆⠀⠸⠖⠋⠉⠠⠀⠀⠐⠒⠒⠚⢻⠿⣭⡽⣃⡂⠊⠀⠀⠿⣷⣳⠂⣧⠁⠀⠘⢹⣿⢄⠀⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄
⣿⠿⠒⠃⣀⠀⠀⠀⢀⣴⣶⣾⣟⣹⣧⣾⡟⣷⣟⣿⡠⠄⠀⣀⡼⠋⠐⠛⣆⠀⢉⣹⣟⠀⠀⠀⠀⣀⠀⢸⡇⣶⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣀⣠⣤
⣷⣚⣿⣿⣿⣿⣯⣤⣿⣯⣿⡏⠑⠚⠛⠋⠓⠛⠛⠛⠀⠀⣾⣏⣀⢀⡠⠀⠀⠀⠀⢿⡧⠤⠤⠀⠀⠀⠀⢸⡇⢫⠁⠀⠀⠀⠀⠀⠁⠹⠋⠈⠘⠛⠛⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠐⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡤⠰⠆⢠⣤⣴⣶⡀⠀⠀⢀⣾⣿
⣿⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠿⠿⢿⣿⣿
⣿⠀⠀⠀⢀⠀⣀⠀⠀⠞⠿⠀⠀⠀⣿⡿⣿⣿⡿⣿⣿⣿⢿⢶⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⢀⣀⣐⣀⣸⣿⣿
⣿⠀⠀⠀⢈⣁⠌⠁⠀⢸⠀⠀⠀⠀⣿⣷⣿⢿⡗⢙⢻⣯⣤⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⢹⣦⡔⠤⢿⠅⠀⠞⣀⡟⡟⣧⣌⠓⣄⡈⡿⣦⡿⣿⣷⠼⣿⣿⣿⣿⣿⢿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣽⢿⢙⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠋⠙⣿⣿⣿⣏⡍⣨⣤⠋⠙⠧⡄⢿⣿⣿⣿⠟⠈⣷⠚⠟⠛⠛⠛⠛⠛⠛⠻⠟⠛⠿⠛⣿⡾⠿⠿⠿⢿⣿⣿⣿⣿
⣿⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣶⠀⠀⠀⡟⠋⠙⠛⠟⠿⢿⠀⣿⣶⣧⣴⣿⣿⣿⣿⣀⣮⣶⣾⣦⡚⣳⣶⡖⠒⢾⣷⣶⣦⡀⣾⠧⠄⣀⣀⠀⢀⣀⣀⣄
⣿⣬⣥⣄⣀⠀⢀⣤⣤⣤⠤⣤⢋⣷⡄⣤⡟⠀⢀⠀⠀⠀⠀⠀⠈⣙⢻⣿⣿⠿⢿⢿⣿⣿⣿⣿⣦⣠⣯⣽⣿⠀⢀⣧⣏⣏⡆⠙⠀⡁⡁⢙⡆⠈⠛⡛⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣶⣾⣷⣄⢾⣾⣿⡿⠞⣛⢛⡛⣛⢓⡛⡋⠉⠉⠉⢻⡇⢸⡟⠛⢛⡇⠸⠻⠿⠳⠛⠃⠘⠷⠷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣽⣷⣿⣿⣿⡟⠒⠒⠲⣿⠇⠘⠃⡀⢹⡇⢸⡀⠀⠂⠺⠃⠀⠂⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢀⣤⡄⠀⠉⠂⢀⠀⠸⠁⡠⠀⠘⠀⢠⠖⠒⠀⠀⣀⡄⡴⠁⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢳⡠⠀⠀⠶⢶⠄⡀⠀⣀⣴⡀⣾⢒⡰⣄⢤⣤⣴⣟⣿⡜⠁⣀⣀⡀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣵⡄⠀⣀⣄⢠⢹⣶⡱⠞⠁⣩⣦⣿⣯⢿⣿⡁⢤⡟⠀⠈⠒⣳⠛⢳⣄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠰⡖⠞⣿⣟⠯⠭⠱⢾⣻⡟⣳⣇⡜⣿⣿⡟⠀⠙⣿⣿⣱⣦⣷⡷⣁⣴⣉⣛⡠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⢑⣶⠽⣾⣏⣳⣿⣷⡿⠯⠿⠵⠿⠽⠿⠿⢧⣤⣶⣻⣶⣠⣹⣿⣷⣿⣷⡶⠶⢾⠵⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣾⣟⣓⣛⣗⣒⣒⣒⣚⣛⣛⣛⣛⡛⠛⠛⠻⠿⠿⠿⠧⠤⠤⠭⠭⠿⠿⠿⠤⠾⠮⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣟⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠚⠛⠘⠫⠭⠭⢉⣉⡍⢩⣭⣽⣿⣷⣶⣶⣶⡄⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠙⢻⣿⠿⠧⠶⠶⠶⠶⠶⠤⠤⠤⢤⣬⣍⣝⣿⣿⠟⢺⣷⠀⢁⣽⡯⠤⣤⡉⢳⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⢹⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⠀⣿⣿⠀⣿⡋⢀⡴⠟⢳⢸⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⠀⢻⣿⡟⠁⢀⡼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠿⣿⠠⠂⡙⠻⣿⣋⢀⡾⡇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇⢰⣿⡗⣶⣲⢦⣆⡦⣶⢲⣇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⢨⣿⡏⠛⠙⠚⠋⠓⠛⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡈⡇⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢸⣿⡇⢰⠷⠺⠇⠿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⢳⠀⢻⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢸⢸⡇⢐⣒⣒⣒⣶⡆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⠁⠘⡄⠘⣿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠀⠈⢸⡇⢨⣭⣭⣭⡭⠇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡄⠳⠶⠯⠽⢻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠋⡀⡀⢸⡇⣨⡅⠉⣼⠦⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣯⣷⣶⣶⣶⣶⣦⡶⣤⣤⣤⣤⣤⣠⣀⣠⣿⣷⣿⣶⡶⣧⠇⢸⣧⡠⠠⣤⣹⠏⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠿⣿⣍⣧⣽⣸⣬⠃⣬⣽⣀⢖⣀⣰⣯⣇⣾⣥⣼⣭⣃⣀⣀⣸⣧⣉⣉⣁⣀⣀⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣾⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣖⣒⣒⣁⣒⢒⢒⠒⠒⢈⡂⣒⠒⠛⠛⠛⠛⠛⠛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣉⡇⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⢹⣏⣻⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣷⣦⣤⣤⣤⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣄⡀⠀⢠⠖⠒⠒⠒⠒⢦⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡄⢸⠐⣹⡱⢦⠡⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡙⠶⣛⠐⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡠⢀⡄⡀⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⣼⡋⢻⠐⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠨⠞⠫⠊⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣖⣒⣒⣒⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠠⣿⣿⣿⣯⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣟⣒⣒⣒⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢨⣿⣿⣿⣿⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠙⢷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠟⠀⠸⣦⣤⣤⣤⣤⡞⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣭⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣛⣛⠿⠿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡟⠟⢻⢛⣛⡛⠛⣛⡛⣛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠏⣴⣶⣶⣶⣿⣿⣿⣿⠟⠿⠿⠻⠿⢿⣿⣿⣷⣶⣶⣦⡙⡇⣿⣿⢿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿⢟⣵⣿⣿⡇⡇⣿⣷⣮⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿⢟⣵⣿⣿⣿⣿⣧⡇⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡟⠁⠀⠀⠀⢀⣴⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⣱⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠋⠀⠀⠀⠀⣴⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⡿⠁⣾⣿⡇⡛⠛⠛⠛⢃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣿⠀⠀⠀⣠⣾⣿⣿⢟⣵⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⡟⡇⢠⣾⣿⣷⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⡆⣠⣾⣿⣿⢟⣵⣿⣿⣿⣿⣿⠿⠋⣠⣾⣿⣿⣿⡇⡇⣿⣿⣿⣿⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣆⠻⠿⠿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⠿⠿⠿⠟⣠⡇⠹⣿⣿⣿⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣭⣭⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣭⣭⣥⣥⣬⣭⣥⣬⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣶⣾⣿⣷⣶⣶⣿⣿⣿⣷⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⠼⠿⢦⣤⡦⠀⠀⠀⠀⠀⠀⠀⢲⡶⠖⠛⠲⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠁⠀⠀⠀⠹⣇⣀⠀⠀⠀⠀⠀⢠⣼⠁⠀⣤⣤⣬⠁⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⠀⠀⠀⢀⣤⣿⣶⣶⣶⣶⠀⠀⠀⢸⣦⡀⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠳⢶⡄⡿⣛⣛⣛⣻⣿⡋⡄⠀⠀⠉⠁⠁⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢱⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡏⣽⢏⣿⢝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣾⣾⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣜⣛⣋⣶⣶⣶⡄⠀⠀⠀⠀⠴⢧⢦⠀⢽⢶⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢫⢻⣿⣿⣿⣿⣿⣷⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇⣷⡀⠀⣿⣿⣿⡇⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢋⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⢻⡇⠀⣿⣿⣿⡇⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⣿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣭⣻⡟⣭⣻⡟⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣧⡻⠿⣳⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣾⣾⣿⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠻⢿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⣦⡻⡿⣥⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠚⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠀⢀⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢠⠖⢋⣉⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢠⣾⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇⣴⠁⠀⠀⣩⣿⡿⠃⠀⠀⠀⠀⠀⣬⡻⣿⣿⣿⣿⣞⢿⣯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡄⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣛⣛⣛⣫⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉