Soy sauce Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠙⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠡⠀⠄⢀⠀⠠⠀⡄⠀⡀⠀⡄⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠁⠀⠈⠀⠀⠁⠈⠀⠀⠀⢁⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠀⠀⠀⠀⠀⠀⠀⠈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠗⠲⡶⠂⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠼⡄⠀⠀⠀⠀⣸⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣶⣄⡐⣁⣀⣈⣿⣶⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠀⠉⠻⠿⠿⠿⠋⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢈⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣸⠖⠒⠒⠒⠒⠒⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⢋⣀⣀⣀⣀⣀⣀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⣿⣛⣛⣛⣛⣛⣛⣛⣟⣛⣛⣛⣻⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠁⢀⣤⣦⣤⣤⣴⣶⣶⣶⣄⡀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠃⢀⣴⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣯⠀⠺⣿⠟⢋⡙⢛⡉⢿⣙⣿⣿⣿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⠻⠂⣨⣄⠛⢃⣼⡧⢀⣿⣿⣿⣿⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡟⠉⣽⣿⠿⠿⣿⣿⣿⡿⣿⡴⢿⣿⠟⠛⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣦⠉⠻⠶⠄⢸⠀⠤⠀⠀⠀⣿⣇⠐⠿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣷⣤⣴⣿⣦⣤⣾⣷⣤⣶⣿⣷⣾⣿⣿⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⡀⠈⠙⠛⠛⠛⠛⠛⠋⠁⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣭⣽⡟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⣿⣭⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠻⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⡄⠀⠀⠀⡴⡄⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠉⠱⡀⠑⡞⠀⠀⠀⠈⠁⡆⡜⠂⢣⠀⠃⠀⠸⢠⠖⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⢤⠱⢄⠀⠔⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠒⠁⠘⢄⣀⠀⡼⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⢀⣠⣤⣴⣆⣤⣤⣴⣶⣶⣶⣾⣿⠿⠿⠶⠤⠀⠀⠀⢀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣦⣤⣬⣀⣀⠀⠉⠻⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣾⡿⠟⠛⠛⠛⠛⠻⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢝⣟⣻⣿⣿⣿⣿⣿⣿⣛⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠙⠏⣿⣽⣼⣿⣾⣿⡿⢋⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣶⣤⢤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣫⣵⣞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡁⠐⠂⠁⠩⠯⠿⣯⣿⣾⣭⣭⣽⣿⣿⡾⠟⢛⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⡀⠀⠀⣸⣿⣿⣿⣭⣭⣴⣿⣿⣥⡾⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠛⠛⢻⣿⣿⠟⣻⣿⣿⣿⣢⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠀⠘⠛⠛⠚⠛⠛⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠛⠋⠉⠙⠛⠓⠲⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⠶⢤⣤⣀⣀⣀⣀⣀⣤⡤⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠛⠒⠒⠒⠒⠚⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣀⡤⠴⣶⣒⣒⣒⣒⣲⠶⢤⣄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⠿⣤⣘⣻⡷⠤⠴⠶⣿⣛⣠⡼⣧⠘⣇
⠀⠀⢀⣀⣤⠴⠶⠒⠒⠛⠛⠛⠓⠒⠲⠶⠤⣤⣀⠀⠀⠀⠀⠀⣼⢀⡟⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠸⡆⢹⡀
⣠⠞⠋⣁⡤⠶⠒⠛⠛⠉⠉⠉⠙⠛⠓⠲⠦⣄⡈⠙⢦⡀⠀⢠⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣇
⣯⠀⡞⠁⢀⣠⡤⠴⠶⠒⠒⠒⠒⠶⠦⣤⣀⠀⠙⣦⢀⡇⠀⣼⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹
⢿⠻⢷⣴⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⠿⢻⡇⠀⡏⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸
⠘⢧⣀⠈⠉⠙⠛⠒⠲⠶⠶⠶⠶⠒⠚⠛⠉⠉⠀⣠⡞⠀⢸⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠈⢿⡻⣶⠶⢤⣤⣤⣀⣀⣀⣀⣤⣤⠤⠶⣾⢫⡟⠀⠀⢸⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠈⠳⣬⡳⢦⣄⣀⣀⠀⠀⣀⣀⣤⠴⣛⡵⠋⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠙⠓⠦⢭⣭⣭⣭⣭⠥⠶⠛⠉⠀⠀⠀⠀⠀⠘⢦⣈⠓⠶⢤⣄⣀⣀⣀⣀⠀⣀⣀⣀⣠⣤⠴⠞⣋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣭⣭⣉⣉⣩⣭⣥⠤⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡭⠭⠭⠭⢭⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠤⣤⡤⠤⠤⠤⠠⠤⠀⠀⠀⢀⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢠⣤⣤⠤⢤⣤⣤⣤⡤⠤⠤⠤⠤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠠⠙⠼⢟⠐⡕⠁⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢉⠑⠓⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠾⠭⢿⣟⣛⣛⣿⠦⠽⠛⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⡂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠀⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⢣⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡼⡀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣿⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠃⠃⠀⠀⣠⢞⡭⠖⠲⢮⣙⢦⡀⠀⠹⢧⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠏⠀⠀⢀⡜⣱⠋⠀⠀⠀⠀⠈⢧⡹⡄⠀⠈⢇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡜⠀⠀⢀⡎⡸⠁⠀⠀⠀⠀⠀⠀⠀⠱⡙⡄⠀⠘⣞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣼⠁⠀⢀⡞⠰⠁⣀⣄⡀⣀⣄⡀⣀⢀⡀⠱⠸⡄⠀⠸⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣷⠀⠀⡼⢠⠃⠀⠻⢦⡄⣏⠈⡇⠹⡾⠁⠀⠁⢳⠀⠀⣷⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣾⡏⠀⢰⠃⠀⠀⠀⢓⣚⣁⣙⣛⣁⣀⣃⣀⠀⠀⠈⡇⠀⢻⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⢿⠁⠀⠸⠀⠀⠀⠠⡀⠀⡄⠀⠀⠠⠀⢄⠀⠀⠀⠀⢹⠀⢸⣷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⢿⠀⠀⢡⠀⠀⠀⠀⠈⠈⠉⠀⠂⠘⠂⠐⠀⠀⠀⠀⡸⠀⢸⢻⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣼⡀⠀⠀⠳⣄⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢀⠞⠁⠀⡾⣼⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢧⢧⠀⠀⠀⠈⠳⣄⡙⠲⢄⣀⢀⣠⠴⢊⣡⠞⠁⠀⠀⢠⢇⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣮⢧⠀⠀⠀⠀⠀⠙⠲⢤⣬⣭⡤⠖⠋⠀⠀⠀⠀⠀⢌⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⠳⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⠳⢤⣄⣀⡀⠠⠤⠤⠤⠤⠄⠀⣀⣠⡤⢞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡓⠶⢤⣬⣉⣉⣉⣉⣉⣉⣉⣩⣭⠴⠖⣫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣬⣏⣩⣭⣍⣛⣣⡤⠤⠖⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠐⠀⠂⠀⠀⠀⠁⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠄⠒⠒⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡔⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠘⡁⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠑⠢⠤⣀⣀⡀⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠄⢀⣤⠀⡄⣰⠀⠀⢀⡶⠆⠀⣤⠀⠀⡄⢠⡄⠀⣤⡄⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠈⡇⢻⡟⠀⠀⠸⡇⠀⠀⣿⡇⠀⡇⢸⡇⢸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣿⠀⣿⢸⡇⠀⠀⠀⢹⡆⢰⡇⣇⠀⡇⢀⡇⢸⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿⠸⡇⠀⠀⠀⠀⡿⢸⠷⢿⠀⡇⢸⡇⢸⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⠷⠃⠀⠇⠀⠀⠐⠛⠁⠉⠀⠘⠀⠙⠚⠁⠘⠷⠆⠸⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠁⠀⠀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠁⠀⣇⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠒⠋⠉⠉⠁⠂⠐⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡔⠁⠠⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⡄⠀⠀⠀⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠤⠤⠀⠤⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⣴⠛⣆⢻⣸⡇⠀⠀⢸⠋⠁⠀⣾⡇⠀⣿⠀⢸⠄⣰⠟⠃⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⠀⣿⠈⣿⠀⠀⠀⠘⣧⠀⠀⡟⣧⠀⣿⠀⢸⡄⣿⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⠀⣿⠀⣿⠀⠀⠀⠀⢹⣇⢰⡇⢻⠀⣿⠀⢸⡇⣿⠀⠀⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢻⡀⣿⠀⣿⠀⠀⠀⠀⢠⡟⣸⠛⢻⡇⢹⡀⣼⠀⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠓⠋⠀⠛⠀⠀⠀⠈⠋⠀⠈⠀⠈⠁⠈⠛⠋⠀⠘⠓⠂⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⡀⣤⣤⠤⠲⠄⣉⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣍⡁⢟⢒⠶⡲⠦⣤⠤⠦⠤⠰⡞⢲⠚⣻⢩⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠿⢾⣿⣶⣷⣼⣷⣾⣦⣾⣶⣷⠿⠾⠿⠛⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣦⢤⣤⣤⢤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠖⢲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠌⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⣾⣿⡛⡋⢉⡩⠽⡛⣿⣦⣽⡿⠟⠋⠙⣉⣿⣛⣯⣿⣿⡄⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⣿⣿⣿⡿⣿⣿⣾⠿⣷⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠱⡄⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣼⣿⣻⣿⡿⣿⣿⣿⣿⢸⣿⣿⡇⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⡄⠀⠀⢰⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⣯⣾⣿⣯⡼⠿⠸⠷⠯⠾⠿⠽⣧⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠘⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣾⣿⣿⣿⠋⠀⠀⣠⣶⣶⡶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⣾⣇⠘⣆⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⣿⡏⠀⠀⣸⣿⣿⠏⠀⠀⠁⠀⠀⣠⣤⢤⡀⠀⣴⡄⠀⢀⣤⠀⠀⢹⡿⣿⣿⣆⢹⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠇⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⣰⣿⠻⣤⣧⣼⣿⠃⣠⣾⣿⠀⠀⠸⣷⣾⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣦⣄⡀⠀⠿⣇⣠⠞⠁⠈⠻⠞⢁⣿⡿⠶⠀⠀⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡽⢿⣿⣿⣦⡀⠈⠁⠀⠀⠀⣠⠞⣻⣿⠁⠀⠀⠀⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠀⠀⠀⠀⣴⠋⠀⠀⢻⣿⣿⣷⠀⠀⠀⢀⣾⠁⣠⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣿⣿⣿⡀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⠄⠀⠀⠀⠛⠶⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⠀⠀⠘⣇⠀⠀⠀⣼⣿⣿⡿⠀⡠⠀⢠⡀⡀⠀⢀⡄⢀⢄⠀⠀⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⣯⣽⣿⣇⠀⠀⠀⠙⢶⣶⣾⣿⣿⠟⠁⠀⠀⠂⠂⠃⠃⠄⠑⠄⠘⠀⠀⠀⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣿⣿⣿⣦⠤⣄⣀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⣿⣿⣿⣿⣿⣯⣟⠈⣿⣿⣿⡻⣷⣶⡖⣶⣶⢶⡲⣿⣻⣏⣿⣿⣿⣿⣿⣿⣿⡿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣏⣿⣿⣸⣿⣿⢻⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⢿⡟⣿⣿⣿⣿⡿⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⡛⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣟⣿⠾⣃⣛⣹⡿⢋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣍⣉⣉⡉⠉⠉⠛⠛⠛⠛⠛⢛⣃⣉⣩⣵⠞⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠿⠯⠿⠟⠛⠛⠉
⣤⡀⠀⠀⠀⠀⢴⡆
⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣛⣿⣻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⣤⣦⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⢀⡉⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠒⠶⠶⠚⠿⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⢋⣉⣭⣤⣤⣤⣤⣌⣉⡙⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣡⣾⢿⣻⣿⣩⣝⣭⣩⣉⡛⠻⣷⣄⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣿⣷⣿⡏⠉⠈⠀⠁⠀⠉⢽⣷⣴⣿⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣌⣿⠇⠀⣠⣶⣿⠷⢦⣄⠀⠨⣿⣡⣾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⠻⡟⣴⣿⡇⠀⠀⠈⢹⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⠻⢿⡿⠟⠋⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡄⣤⠠⣤⠀⡤⣠⡀⣄⣄⣤⣠⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠁⠉⠉⠁⠉⠉⠁⠉⠉⠉⠁⠠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠒⠀⠒⠒⠀⠘⠀⠀⠀⠀⠘⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⠦⠶⠆⠶⠖⠰⠆⠀⠀⠀⣜⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢛⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠚⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡳⢌⡙⠳⠤⠦⠰⠶⠚⠀⣡⠔⣡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣍⣁⡒⠒⠒⠒⣊⣉⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠀⠘⠛⠓⠒⠂⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⠀⠉⡝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠒⠶⠶⠶⠖⠒⠛⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣢⣀⣀⣀⣀⣀⣀⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣶⣶⣶⣶⣶⣶⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣿⣿⣿⣿⣿⣿⡇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣾⣿⣿⣿⣿⣿⣿⣿⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⢉⣉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢿⣍⠃⠀⣠⢤⡀⣄⢠⠀⠀⠀⢡⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⢤⣨⡗⠀⠧⡼⠃⠙⢾⡇⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣠⢦⡀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⢶⡀⣰⣶⡀⣴⢰⡄⣴⠂⣰⣶⡄⠀⠐⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠚⠃⠈⠉⠁⠈⠉⠃⠙⠓⠈⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢲⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣍⣉⣛⡛⠛⠛⠛⠛⢛⣛⣋⣩⠵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠲⠶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⢿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣹⣿⣿⣿⣿⣿⣿⣧⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣶⣦⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠊⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣽⣿⣿⣿⣛⣻⣛⡿⢿⣿⣿⣿⣿⣿⣿⠙⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿⣿⣿⣿⣹⡜⣿⡏⣿⣿⣿⣿⣿⣿⣿⠄⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⠖⠀⠵⣿⣽⣿⣿⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⢆⠠⢥⣯⡿⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⢻⠾⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⢸⣿⣷⡄⢀⣿⣿⣿⣿⣿⣿⠟⣗⠀⢰⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⢿⠛⠑⠀⠂⠈⣁⣠⣇⠠⢄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⡟⢀⣧⡖⣻⣿⣿⣿⣿⣿⣿⣜⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢀⣈⠃⠿⡁⢀⡔⢿⣾⡿⠛⠻⢶⡾⠇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣼⠏⠁⣶⠀⠀⠁⠀⠉⠙⠿⣷⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠌⠀⣠⡾⠋⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠐⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠢⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠶⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠓⠚⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠦⠤⢤⠿⠿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠉⠉⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⣀⣀⣀⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣠⢶⠾⣟⣻⣭⣭⣤⣤⣤⣭⣍⣈⡉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡼⣝⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣈⢧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢧⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⡼
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢆⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⣛⠋⠉⣴⡖⡰⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠉⢔⡠⣈⡕⢌⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣻⠀⠀⠀⠀⠀⠑⢦⣀⡄⣤⣤⣤⣤⣤⣼⣴⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣀⣀⣀⣀⣀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠃⠀⣴⢶⡄⢿⢀⡟⠀⠀⠀⣼⠋⠃⠀⢰⣧⠀⢠⡆⠀⣾⠀⢠⡶⠲⢰⡶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⡇⠈⣷⠸⣿⡇⠀⠀⠀⢿⡀⠀⠀⣾⢿⠀⢸⡇⠀⣿⠀⣾⠃⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡇⠀⣿⠀⣿⠁⠀⠀⠀⠈⢷⡀⠀⡟⢸⡆⢸⡇⠀⣿⡇⣿⠀⠀⠀⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇⠀⣿⠀⣿⡇⠀⠀⠀⠀⢸⡇⢸⣧⣸⡇⠘⣇⠀⣿⠃⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⡟⠈⣿⣀⡿⠀⣿⡇⠀⠀⠀⣄⣸⠇⢸⠃⠀⣿⠀⢿⣀⡿⠀⢻⣇⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠋⠁⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠋⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⠶⠖⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣰⠏⠉⠙⡥⠀⠀⠀⣛⠛⠛⠛⠷⢤⠦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠁⠃⡀⠰⠀⠀⠀⠈⠉⠀⠀⠀⢀⡈⠀⠂⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠛⠉⠀⠠⠀⠈⠀⠀⠀⢀⡄⠀⠀⠒⠀⠀⠘⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠱⠂⠀⢰⠂⠀⠀⠰⠂⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠈⠀⠍⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠚⠀⠀⠞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⠀⡀⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⢀⣤⢿⠄⣴⠀⠀⠀⣠⠀⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠷⣄
⠀⠀⠀⠀⠀⣴⠋⢠⡟⠀⠰⡂⠀⠀⡉⠁⠀⠀⣀⡀⠀⠀⠀⠈⠃⠀⠀⢠⠀⠀⠀⠋⠀⠀⠀⠀⠀⠰⠄⣿⠀⢸⣧
⠀⠀⠀⠀⠀⣿⡄⠘⢧⣉⠀⠀⠀⠘⠒⠀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⡼⢻
⠀⠀⠀⠀⠀⣇⠙⠶⣄⡉⠳⢦⣀⡁⠀⠘⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠂⠀⠀⠔⡆⠀⠀⣀⣠⡾⠋⣠⠞⠁⣼
⠀⠀⠀⠀⠀⢻⡀⠀⠈⠙⠲⢤⣈⡙⢦⡴⢶⣦⣤⣀⣀⣙⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠾⢉⣠⡴⠚⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⠉⠓⠲⠶⠤⣤⣉⣉⣉⡉⠙⠛⠛⠛⠛⢋⣉⣉⣠⠤⠶⠚⠋⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡈⠉⠙⠛⠓⠒⠒⠶⠶⠶⠶⠶⠶⠒⠒⠛⣉⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠦⠤⠤⠤⠤⠤⠴⠶⠖⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⣷⣀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢏⠲⣧⠀⠀⠀⠀⠀⠀⠀⢰⠟⢉⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠀⠀⠸⣄⠉⢀⣤⡄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣆⠀⠀⠀⠀⠀⠀⠀⡀⢻⡀⠀⢠⠀⠀⠀⠀⠠⡖⢀⣹⣆⣏⡜⣾⣡⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢆⡀⠀⠀⠀⠀⠀⠑⣮⣷⡠⡟⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⢀⢴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⡈⠂⣄⠀⠀⠀⠀⢸⠋⠉⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⠸⡿⠞⠛⠦⣷⡀⠀⠀⠀⢸⠆⠉⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⡀⠀⠀⠐⡄⠳⡄⠸⣷⣄⣀⢀⠄⡿⠇⠀⠀⠀⢸⡇⠀⠙⡇⠀⢀⡀⠄⠠⠤⠔⠒⠂⠤⠤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠤⠀⠾⡄⠀⠇⠈⢧⡙⢿⣿⠟⠇⢂⠀⠀⢢⣸⣇⠀⠘⠃⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠀⢰⠀⠀⠀⠈⢆⢠⣴⣖⣚⡗⢦⡀⠻⣻⣶⠏⠀⠀⠀⠈⠉⠓⠒⠂⠀⠀⠀⠀⠀⠂⣉⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⣄⡄⠀⠉⠀⡿⠐⠁⠀⠀⠘⣽⠋⠉⠉⠉⢻⣧⠀⠈⠙⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⢰⡙⢷⣶⠂⣤⣼⠻⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢾⠁
⠀⠀⠀⠀⠠⠔⠰⠉⠢⡀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠈⢁⠀⠀⠀⠀⠻⣊⠙⠛⣋⣥⡤⣬⡢⡀⠀⠀⠀⠢⠄⣒⠂⠀⠀⠀⠀⠀⠀⠈⠈⠃⠴
⠀⢀⠄⠂⠀⠀⢀⡀⠀⠈⠐⠂⠔⠳⡀⠀⣀⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠹⣞⣿⠁⠀⠀⢹⣾⡄⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠈⠲⢄⠐⠀⠀⠀⠀⠀⠀⠁⠀⢠⡭⠿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⣻⣅⣮⣾⠟⡇⠀⠀⠘⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⡉⠀⣈⡣⠤⠤⡀⠀⠀⠀⠀⠀⡠⠤⠌⡑⠊⠑⠢⠥⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠒⠀⡀⠐⠁⠀⠐⡠⠂⠘⠻⠏⣯⣀⡀⣀⡘⠁⠀⠑⠀⠘⢀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠠⠐⠒⢺⡗⠉⠉⠉⠻⡶⠓⠉⠙⡊⢙⢑⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢄⣀⡞⠻⣶⡒⠐⠂⠳⣆⣴⠶⢻⡀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢋⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠶⠶⠖⠚⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⠭⣿⣛⢛⣛⣛⢛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⣒⠋⠭⠒⢂⣈⡠⢤⠄⣀⣀⣀⡤⣤⢤⠄⣀⡀⠉⠉⠛⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢞⣫⡴⠞⠋⠁⣀⠔⣩⣤⠆⣹⢭⣋⡿⢶⣻⣷⣶⣧⣏⠉⠛⠏⡏⠛⡿⡖⣶⣄⠉⢷⣦⣀
⠀⠀⠀⠀⠀⣠⣟⠵⠛⢛⣇⣤⡦⣚⣕⣺⢏⣖⡺⠗⠻⣞⣫⠯⣗⣻⡖⢮⡿⣦⣤⣀⣄⣀⣠⣆⣋⠟⢻⡗⣜⢿⡷⡄
⠀⠀⠀⠀⢠⣿⡏⢀⡰⢯⣿⣤⢙⣖⣩⣿⠿⣿⡾⠿⣷⣿⣻⠿⣟⡼⠿⢏⣸⠯⠘⣿⣿⣿⣶⣒⣻⣍⡽⢫⢸⠘⢉⣧
⠀⠀⠀⠀⢸⡿⣷⢬⡳⣯⣽⡯⣟⠛⢷⡾⣷⠿⣶⡏⡼⢿⣭⡿⢭⣿⣳⣿⣤⠞⠋⠒⣿⡔⣫⠿⠭⣯⣵⣾⣗⡷⣺⣿
⠀⠀⠀⠀⠀⣿⠘⠳⢿⣗⡺⠭⠟⡕⣲⣶⠒⣫⣝⣺⣳⣿⣿⣿⣷⣿⣽⣿⣯⣀⠶⣎⡹⣟⣓⣶⢿⣿⣯⣽⣿⡟⢱⡇
⠀⠀⠀⠀⠀⢸⡆⠀⡈⠀⡏⢛⡲⠤⣝⣚⣣⣏⣸⣟⣿⣙⣍⡁⠻⡩⢟⣿⣞⣯⣶⣾⣼⣷⣾⣿⡿⠿⡟⠋⣧⠇⣾⠃
⠀⠀⠀⠀⠀⠀⢿⠀⢇⠀⠀⠈⠭⣓⣒⠢⠭⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⢋⠩⠭⠁⠀⠀⠈⠁⢸⠋⣸⠏
⠀⠀⠀⠀⠀⠀⠘⣧⠘⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⢚⣩⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⢤⣤⣤⣤⣤⡤⠤⠴⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠐⠢⡀⠀⠀⠀⠀⣀⠠⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡀⠀⢀⡀⠀⠆⠰⠀⠀⠀⠀⣐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡠⠬⡁⠀⠀⠀⠃⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠓⠰⣀⡀⢣⡀⠅⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⢻⡍⠃⠃⠀⠋⠠⠤⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⠁⠠⠀⠐⣖⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠒⠒⠒⠄⢀⡂⢠⠐⡴⡀⠀⢰⠂⠀⢀⠉⡀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠰⠀⠠⠀⣀⠀⠁⠑⡆⠊⠇⠀⠸⠀⠀⠀⡀⠀⢤⣄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠉⠳⠠⣅⣴⠀⠄⣤⣤⡀⠀⠀⠂⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠀⠠⠤⠄⠀⠠⠁⣌⠼⣧⣏⡀⣴⠚⣶⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣤⡤⡦⠜⢀⠀⠞⠀⣀⣿⡇⠐⠹⡄⠈⠳⢄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣈⠀⠀⠀⣠⣾⠏⠀⠀⣶⡏⡇⠀⠀⢐⠂⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⣩⠖⠀⠀⣀⡾⠀⣹⣿⠀⠀⠀⢣⡀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢠⡶⠎⠀⠈⡟⠻⣄⠀⠀⠠⡟⢒⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠞⠁⢀⡴⢿⠐⡄⠈⡇⡀⠀⢙⡑⠿⣦⣤⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠄⠀⠴⠟⠁⢰⡀⠈⠲⠴⠤⣀⠀⠦⡄⠀⠈⠂
⠀⠀⠀⣠⡶⢾⡟⠻⢿⠿⠿⡿⣛⣿⡻⡿⠖⣦⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠐⠀⠹⣀⠀⠈⠂⢀
⠀⠀⠈⠿⣓⣪⣜⣣⣓⣄⣒⣡⠰⣇⣧⣧⣤⡿⠃⠀⠀⠀⠀⠀⠀⠘⠄⡄⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⣾⠟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⢋⣭⡽⠟⢓⣶⠞⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡿⠋⠁⠀⢰⠋⠀⣠⠞⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠀⢀⣀⣀⣀⠘⣤⠞⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢀⣴⡾⠋⠀⠀⢀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠁⠀⡞⠉⠀⠀⠀⣠⠟⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠖⠋⠁⠁⠀⠀⠀⢸⡄⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠿⠟⠛⠋⢀⠴⠒⠊⠛⠲⢄⠀⠀⠳⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⣀⣀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢳⣠⠞⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⠛⠉⠀⠈⠙⢦⠀⢸⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⣀⡴⠋⢠⠏⠀⠀⠀⠀⠀⠀⠈⡇⠸⡀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⣿⠀⠀⡏⠀⠀⠀⠀⠀⢀⣀⣤⠷⠶⠛⠛⠋⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠙⢦⠀⡇⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⢀⣀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠈⢧⡙⢢⡤⠞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣠⡴⠚⠉⠀⣀⣴⡦⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠚⠛⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠶⠖⠛⠋⠉⠀⠀⢀⣠⡤⠶⠶⠶⠤⠤⣤⣤⣄⡀⠀⠈⠲⠦⠤⠴⠶⠚⠉
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠾⠛⠿⢶⠶⠦⠀⠀⠙⣆
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⣤⣤⣤⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢣⠀⠀⠠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡜⠿⠉⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢳⡄⡜⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠜⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣧⣀⠀⠤⡈⡇⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠄⣸⠀⠀⠀⠀⠀⠠⠄⠄⠀⠒⠉⠁⠀⠀⠀⠀⠨⠻⡄⠀⠀⠇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠄⠀⠀⢀⣤⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠀⠀⠀⠸⡒⢢⣤⢠⡤⠞⠉⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢜⡴⠃⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠶⣄⠈⡇⠘⡁⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠗⠀⠀⠀⠀⠀⡠⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠈⡱⣼⡏⠉⠓⢦⣤
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢀⣀⡠⢚⡡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢶⣄⣀⣤⠞⠁⣸⡇⠀⠀⠀⠘⢦⣤⢀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⣀⣀⣤⣡⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⢧⠀⠀⠀⠀⠀⠙⠀⠙⠀⠐⡖⠀⢂
⠀⠀⠀⠀⠀⠘⠉⠘⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠙⠲⠦⠤⠴⠛⠉⠀⠑⠤⣍⠀⣀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢻⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠿⣟⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠇⠀⠀⢸⣿⠋⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⠀⢀⡀⢰⡙⢄⡀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⣿⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠁⠀⠀⠀⠉⠂⠱⣄⠀⠀⠀⠀⠀⠀⠈⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣋⠁⠀⢲⡌⠓⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠁⠀⠀⢀⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡯⢄⡀⠀⠀⠀⡀⠤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠧⣢⡀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡲⡆⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠁⠀⠈⠀⠀⠐⡟⢛⣻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠄⡀⢀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠻⡀⠻⣄⠀⠀⣀⣀⣼⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠉⠛⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠻⠿⠿⠿⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠚⢛⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣠⣄⡀⠀⠀⢀⣠⠴⢚⣋⣭⠤⠶⠖⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡛⢭⢼⠖⠗⢋⢻⡶⠞⣫⡵⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⠾⡖⠂⣍⣨⠷⢊⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣞⣉⣭⣥⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠤⠶⠶⠦⠤⠤⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠫⢅⡶⠬⠖⡀⠵⢠⡤⢈⡓⠦⠄⣉⣙⠓⠲⢤⣄
⠀⠀⠀⠀⠀⢀⣠⢞⠚⠣⡼⢑⣩⣄⢘⡀⠱⡶⠀⠆⡈⢁⡤⠤⡄⠨⢙⠊⢂⡉⡳⣆⡀
⠀⠀⢰⡶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠓⠒⠒⠶⠶⠶⠶⠾⠿⢶⡆
⠀⠀⠀⠻⣔⠂⠒⠀⠒⠀⠂⠐⠂⠒⠂⠒⠀⠐⠀⠒⠀⠒⠂⠒⠀⠒⠂⠒⠀⠒⠐⠂⢒⡾⠁
⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠾⠛⢛⣋⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢟⡛⢛⡯⡽⣧⣄⣀⣤⣴⠶⢟⣛⣋⣭⣥⡷⠾⠟⠛⢛⣉⣉⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠋⣧⠅⠺⣥⢙⡈⢿⣿⠷⠾⣛⣛⣩⣭⣤⡶⠶⣾⣿⣛⣫⣭⣭⡽⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠊⣥⠤⠅⣲⣀⣩⡥⠾⢿⣛⣛⣯⣭⣽⠶⠶⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣴⣯⣭⡿⠷⠞⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣼⡟⠛⠿⠛⠉⢁⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠾⠛⣋⡭⢀⠈⠋⣤⡤⠘⠃⠉⣠⡙⠷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⡶⠿⢋⣀⡴⠎⡉⢀⡘⠁⠐⠒⠢⣈⣋⡀⠐⠷⣤⡜⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠆⢉⡤⠿⢉⠀⠛⣀⣥⠀⠻⠿⠋⢙⣉⠉⠷⢤⡃⠈⢛⣀⠈⠻⠶⣦⡀
⠀⠀⠀⠀⣀⣀⣾⣏⣁⣊⣁⣀⣈⣉⣥⣤⣬⣥⣤⣈⣳⣦⣬⣥⣤⣤⣤⣤⣄⣀⣀⣀⣈⣃⣹⣷⣀⣀
⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⣿⣀⣤⠀⢀⣀⠀⣀⡀⢀⣀⠀⣀⡀⢀⣀⡀⢀⣀⠀⢀⠀⠀⢀⡀⠀⢀⣀⠀⢀⣀⢀⣀⣿
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠉⠁⠈⠉⠁⠈⠉⠁⠈⠉⠀⠈⠉⠀⠈⠉⠀⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠻⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿
⠀⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡆⠰
⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⣦⠘⡄⠀⠀⠀⠀⠀⠲⢆⣤⡀⣀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠈⠳⡐⢠⢀⣂⡆⡔⠀⡆
⠀⠀⠀⠀⠈⠀⣀⠀⠐⠂⠀⠀⠀⣀⡡⠊⠀⠀⠀⠀⠈⣴⠀⠉⠉⠉⠀⠀⣠⡤⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡏⠽⠉⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣄⠀⠀⠠⠯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠈⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠩⠅⠀⠀⣀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢠
⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠒⠂⢀⠢⠍⠀⠀⠀⠀⠈⠰⠄⠀⠀⠀⠀⠀⢀⠀⠲⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠛⢅⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡌⠉⠀⠒⠀⠀⠶⠚⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⠀⠀⠀⠀⠀⠀⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠠⠤⣤⣾⠿⠋
⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⣤⣰⡶⠒⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⢿⣤⣀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⡈⠈⠁⠀⠀⠃⠀⠴⠂⡀⣠⠞⠀⠀⠀⣰⡀⠀⠀⠀⠤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠀⠀⠀⢶⡟⠀⠀⠀⠀⠀⠀⠪⠥⠀⣒⣀⣀⣀⣀⡼⠟⠁⠀⠀⠀⠀⠠⣔⡠⠷⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣠⡤⠟⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⣀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢻⡻⣛⣛⢻⣿⣻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣁⡤⠄⠀⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡠⠊⢉⢠⡄⣀⠔⠁⠀⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⡶⠿⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢠⣤⢴⡾⣿⠶⠚⠛⠶⣶⣤⣄⣀⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡴⢋⠔⠁⠀⣰⠤⠿⠛⠛⠛⠛⠛⠓⠒⠛⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⠟⠡⠃⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠠⣊⠔⠁
⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠤⣄⣒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠂⠐⠶⠤⠤⠤⠤⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠐⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠐⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠼⠀⠩⠁⠀⠀⠉⠁⠐⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠦⣀⠄⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⡋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠⠀⠠⠊⠤⡨⠂⠀⡔⢀⠱⠱⡜⢠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⠅⢒⠀⠴⠄⠄⠀⠀⠀⠀⠀⠄⠂⠁⣿⠀⠐⠢⢅⠒⠄⠀⠣⠖⢀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠤⠀⠅⡨⠈⠂⠀⠊⠀⡨⠶⠃⠄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠁⠀⠂⡀⡇⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⡄⢸⠄⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⢄⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⡂⢈⠄⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠄⠀⢢⠀⠉⢀⡀⠐⠂⠉⠀⢀⠉⡑⠀⠎⠉⣠⠜⠓⣆⠀⠂⠈⠁⣉⠀⠉⠛⡅⠀⣅⠀⠀⠈⠀⣠⡂⠀⠂
⠀⠀⠀⠀⠀⠣⣀⡀⠃⢈⠁⢉⠀⢀⠤⠄⠀⠁⠄⠢⠄⠠⠐⠆⠀⠘⢣⠔⠀⠖⠠⠈⠂⠠⡤⠤⠾⠀⠒⠑⡂⠈⠀⠑⣠⠤
⠀⠀⠀⠀⠀⡈⡀⠉⠁⠀⠔⠀⠀⠈⠀⠀⠀⠢⠠⠐⠀⠐⠀⢃⡘⠛⡀⠀⠔⠃⠀⠖⠀⠀⣀⠀⠀⠓⠀⠊⠑⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⢀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⡀⣤⠀⢀⣤⠠⢆⠨⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠔⠀⣀⠠⣤⣤⢝⣠⣮⠐⣣⣰⣆⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⢀⣠⣤⠾⠀⠐⠒⠃⠉⠁⣀⡀⣠⣤⡤⡶⣎⣩⡴⣀⠀⠀⠀⠐⠲⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠠⠖⠋⠉⣀⡤⡤⣴⣶⣶⢿⠿⠟⠻⠍⠃⠴⢲⣄⠁⣉⡉⢂⠲⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠⠂⠀⣀⣤⠔⠚⠋⠅⠒⠂⠉⠁⠈⠨⡷⡤⡄⠀⠐⠂⡔⢈⠭⠀⠈⣯⠀⠁⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠐⠞⢱⡆⠀⠀⠀⠂⠈⢿⣷⣀⣤⣤⢠⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠐⢉⣠⡖⢕⡊⢎⡑⠎⠰⡌⢛⠒⠋⢀⡀⠀⠀⠀⠀⠐⠀⠂⠠⠀⠀⢁⠤⠀⠈⠅⠄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡯⠖⠁⠀⠒⠉⠃⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠛⠏⠿⠉⢿⠛⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠯⡁⠀⠁⠀⠃⠔⠄⠜⠡⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠋⠰⠠⠋⠔⠂⡰⢣⠠⣿⣿
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⣧⡨⢀⠐⠃⢠⡚⡉⠻⠔⠠⢈⡖⣤⣿
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠈⡈⠠⠂⣀⠢⡤⠃⠀⠺⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣦⡀⢀⠀⠌⠨⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣿⣶⣷⣾⣿⡿⠟⠁⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡧⡃⡆⣴⠂⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠶⠴⠾⠏⢈⡍⠁⠈⣭⢁⣗⠺⠦⢆⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠮⠟⠀⢀⠀⢴⠀⣘⣋⡾⠐⠀⠀⠉⣀⣀⢀⣌⠓⠲⠦⢤⡀
⠀⠀⠀⠀⡤⢚⣡⡟⠋⠛⢠⠀⠀⠑⠛⠀⡠⠁⠀⢠⢤⡠⠞⠃⢀⣌⠈⠋⠈⢆⣀⡀⠙⢷⡢⡀
⠀⠀⢀⡎⣰⠟⠋⠁⠀⢈⡼⠃⡀⠘⠂⢂⣺⢆⢤⠦⠀⠀⢀⣤⠀⠉⠀⢠⣄⠀⣈⠂⠘⠈⢻⣎⣦
⠀⠀⠸⡇⣿⣄⠺⠤⣀⠬⠀⢈⣉⣀⡥⠈⠉⠛⠀⢆⣴⣀⠨⠈⠀⠀⠒⠋⠀⠀⠀⣬⠀⠀⢰⡟⢸
⠀⠀⠀⣇⠬⡻⢦⣤⠸⠍⠀⠈⣀⠐⠆⠀⠀⠁⢒⠙⣦⡴⠐⢒⠀⡤⠀⠀⠀⢰⠤⢤⣤⡴⢋⡠⣿
⠀⠀⠀⢹⡀⠈⠲⢌⡛⠶⣦⣬⣑⣀⡂⠐⠶⠟⢆⠟⠏⡛⠮⠙⢃⣄⣚⣹⡦⢶⣚⣯⠵⠚⠁⢠⡟
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⠀⠒⠺⠿⠭⠽⠽⣿⣿⣿⣿⣿⣿⠭⠭⠿⠿⠒⠛⠉⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⣄⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣠⠤⣶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⠉⠉⠉⠉⠉⠉⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠚⠋⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⠀⠰⡌⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⡀⠀⠀⢀⣀⡠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⢻⡏⠀⠀⠀⣠⡑⠀⡄⠀⠀⢀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡼⠃⠀⠀⢀⠉⡍⠆⠹⣶⣤⠃⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡠⠞⠁⠀⠀⠀⠆⠀⠅⡰⠀⠈⢻⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠔⢉⡴⠀⢠⣥⠀⠀⠀⠀⠀⢪⠀⠤⣀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠆⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠈⠛⣆⡘⢁⣠⠖⠛⠃⠠⠄⠀⠀⠈⠑⠄⠀⠠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠻⡇⠀⠀⢀⡤⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠜⠋⠀⠠⡀⠀⠀⢐⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣤⡴⠞⠁⠀⠀⠙⠄⠀⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠐⠂⠉⠀⠀⠀⠀⠀⠁⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢆⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠖⠶⠖⠀⠁⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡿⢠⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣁⡀
⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⢙⣶⢇⠸⣀⡆⠀⠀⢀⠀⠀⠀⠈⠲⣄⣀⠀⠀⣀⣄⣀⢀⣀⡤⠞⠀⠀⠉⠉⠉⠁⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠿⠿⠁⢀⡑⣝⠓⠳⠖⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢢⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢸⠇⠉⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣿⠇⠀⠀⠀⠳⣄⣈⢉⣡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡤⢯⡀⠀⠀⠀⣹⡄⠀⠀⢠⡞⢹⡭⠂⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢬⠙⠒⠐⠚⣡⣈⠁⠀⠁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢳⠀⠀⠀⠀⠀⡼⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣⠀⠀⠀⡜⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⡆⣠⠎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⣅⣀⡀⠤⠼⢿⠥⠤⣤⣤⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⣙⣯⡛⠒⠛⠉⠉⠁⠀⠘⣒⠤⣭⣉⡐⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣁⡠⠔⠋⠁⠀⠉⠲⢤⣀⡀⠀⠀⢀⠀⠈⠉⠙⠓⠶⢦⣍⣮⠐⠢⢄⡀
⠀⠀⠀⠀⠀⣤⠤⠤⠤⠖⠛⠚⠉⠁⠀⠀⢀⣀⡀⣀⡴⠶⠋⠭⢍⢛⠶⠂⠒⠢⢤⣄⠀⠀⠈⠙⠓⢦⣄⣉⣒⣄
⠀⠀⠀⢀⣀⠬⠓⠦⣄⡀⠀⠀⢀⣀⣤⣶⡶⠞⡻⠉⠀⢀⣀⠤⠤⠿⡀⠺⠷⠆⠀⣱⠖⠋⠈⠑⠢⡄⠀⠉⠒⢽⡕⢄
⠀⠀⠈⠓⠦⠤⣀⣀⣈⣹⠟⠃⠉⠛⠫⡁⠀⠀⠘⢶⡶⠉⣴⣲⠆⠀⢈⡷⠒⠈⠉⠛⠢⡀⠀⠀⠀⠘⣆⡀⣀⠀⠈⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢄⣀⡀⢀⡟⠀⠀⠀⡠⠗⠒⠒⠒⢤⡴⢿⡀⢠⣤⣄⠀⢀⡟⠶⣦⡤⠞⠉⠙⠓⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠉⠀⠀⠀⢸⠁⣠⢤⣤⠀⠀⡓⠀⡹⠒⠂⠈⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣉⣐⠢⠔⠁⢸⡀⠀⠀⢀⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠿⠿⠗⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣍⡑⢰⠋⠉⢳⠈⢣⡠⠃⠀⣯⠈⡗⠈⡏⣭⠃⠀⡸⢳⡀⠀⡟⢦⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠚⠈⠒⠐⠊⠀⠐⠃⠀⠐⠓⠒⠊⠐⠓⠒⠚⠐⠃⠈⠓⠀⠓⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣤⣤⡶⠶⠶⠶⣤⣴⣦⡾⢿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⠞⠋⣠⡙⠒⣠⣄⠐⣃⠀⢀⠀⠰⢦⣀⣉⡁⠙⠛⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡿⢈⡄⠉⣥⠏⠉⣉⠁⠤⠀⠉⠁⠛⠙⠠⢶⠀⠈⠃⠴⠰⠶⣤⣀⢭⡛⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢛⡙⡫⠶⡞⢃⣁⣴⢲⡀⢉⡀⠀⣛⡛⠂⠘⢂⡀⢀⡤⠤⠴⢦⡀⠤⠌⣉⠠⠅⣤⠨⡝⣿⣄
⠀⠀⠀⣤⣤⣤⣤⣿⣯⣥⣭⣤⣥⣬⣥⣤⣤⣤⣬⣦⣾⡦⠶⠾⠷⣶⣴⣾⠶⢶⠶⠶⠶⣾⣶⣶⣦⣥⣤⣤⣤⣤⣤⣤⣽⣷⣤⣤⡄
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣠⡟⢡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠉⠛⠛⠋⠉⠉⠳⣄⠀⠀⣴⠏⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣉⣤⠶
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠋⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⢁⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⢋⣡⡴⠟⠉
⠀⠀⠀⢀⣠⣤⢶⣿⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠀⠀⣰⠟⣰⡟⠉⠙⠛⢶⡄⠀⠀⠀⠀⠀⢀⣠⡶⠛⣉⣤⠾⠋⠁
⠀⢀⡴⠛⣁⣤⠾⣗⠶⠀⠀⣤⠶⠄⠀⠀⠀⠀⠿⠀⠀⢀⡾⢋⣼⠋⠀⠀⠀⠀⠀⡿⠳⣦⣀⣴⠞⢋⣡⡴⠛⠉
⣰⡏⣠⡾⢯⡜⠓⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣡⡿⠁⠀⠀⠀⣀⠀⢘⣻⡶⠟⣉⣴⠞⠋⠁
⣿⠀⣿⣹⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⠏⠀⠀⠀⠀⣀⣼⠷⢋⣥⡶⢻⡏⢸⡇
⢻⣦⡟⠷⠀⠀⠀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⡴⠞⣩⡴⠞⠋⠁⠀⢸⡇⢸⡇
⠈⣿⣿⣗⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⠶⠛⠉⠀⠀⠀⠀⠀⢈⣹⣿⠁
⠀⠘⣿⡙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⣿⠃
⠀⠀⠘⢷⡝⠶⠍⠛⠻⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⠶⠟⠛⠉⠀⣼⠏
⠀⠀⠀⠈⠻⣾⠛⠦⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠙⠷⣝⡛⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣤⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢤⡀⢀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⡀
⠀⠀⠀⠀⠱⢤⡀⠀⣠⠞⡀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⣟⣠⣕⠀⠀⠀⠀⡠⠄⠀⠀⡀⠀⠀⠒⠁
⠀⠀⠀⠈⢙⠀⠤⠀⠍⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠆⠀⠀⠈⠈⠈⠁⠀⠀⠂
⠀⠀⠀⢀⢆⡣⠔⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢀⠀⠆
⠀⠀⠀⠈⠀⡀⠄⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠠⠅⠀⡾⠀⠀⠀⠀⠀⠁⣀⠄⡀⠀⠀⠀⠀⠑⠠⠼⠁⣴
⠀⠀⠀⠀⠀⠨⠒⠐⠄⠶⠀⢡⠀⠀⠀⠀⣀⢀⡀⠀⢸⠁⠀⠀⠀⣾⠘⠁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⡀⠌⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣉⠈⠙⠃⠀⢸⠄⠀⠠⠈⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠂⠘⠀⠀⠀⣠⣈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠈⢀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⠀⡐⠲⠖⢂⠏⠀⢀⠀⠐⠺⠀⠀⠐⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠘⡧⠈⠁⠀⠀⠈⢀⢀⡠⠐⠀⠀⠁⠀⡀⠘⡠⠂
⠀⠀⠀⠐⠂⠈⠑⠀⢈⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⣸⠎⠉⠀⠐⠁⠀⢀⢈⡁⠘⠁
⠀⠀⠀⠀⠠⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠟⠛⠂
⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⢄⡉⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢦⠵⠝⣴⣞⣿⣶⣵⣄⣀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠸⡌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠈⣉⠉⠛⠓⠛⠚⠛⡽
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⠤⠤⠀⡀⡄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠁
⠀⠀⣿⣿⠿⡿⣿⢿⠿⡿⣿⣿⡇⠀⠠⣛⣴⣺⡯⡿⢷⣦⣷⣮⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠘⣿⣷⣷⣿⣾⣷⣿⣿⡟⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⢙⡿⠁⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠉⠀⡤
⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⠤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣉⣩⠭⠤⠤⡴⠒⣚⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣧⣾⣿⣿⢰⣿⣷⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠭⠵⠿⠞⠛⠛⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⠀⠀⠀⠀⠀⠀⡼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣹⡎⠀⠀⠀⠀⠀⠀⢁⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢠⠏⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠖⠒⠒⠋⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠑⠺⡗⠒⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡶⠀⢧⠀⠀⣀⡤⢄⠀⠀⡤⠤⣄⠀⠀⡴⠒⠢⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣔⠀⢸⠀⠈⡇⠀⣸⡀⠀⢧⢀⡾⢤⡀⢧⠀⣀⡅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢫⠀⠈⡆⠀⢃⠈⠉⠉⡷⢸⡀⢀⣠⠟⠈⠦⠼⢧⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⣇⠀⠈⠛⠋⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⢄⣀⣸⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠴⠒⠚⠉⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣤⣴⣤⣤⣄⣀⣀⣀⣀⣠⣤⣶⡶⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⣤⣶⣏⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⣿⣿⣿⢿⣄⠀⠀⠀⣠⡿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⡟⣧⡆⣴⡏⡷⣿⡿⣿⡏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⡽⢿⣿⣿⡟⣷⢿⣸⣿⣿⣿⢟⡴⣿⠁⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣙⡾⠿⠿⡇⢹⣷⣿⣿⡭⣿⡟⢰⣟⣠⡴⠞⠋⠉⠀⠀⠉⠛⠛⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⣛⣷⣦⣙⡌⢿⡇⠀⣸⣿⠶⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠠⠄⠀⠀⠉⠳⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣙⣻⣿⣿⣿⠛⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⣀⣌⣩⠿⠛⠁⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠋⠀⠀⠀⠀⠀⠀⠐⠂⣉⡿⠿⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠩⡴⠞⠁⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢀⡀⠀⠠⠀⠀⣀⡴⠖⠛⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⢳⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⢀⠐⠂⠀⠀⠐⠂⠀⠁⠀⠀⣠⣴⣛⠁⠀⠀⠀⠀⣀⡴⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠈⠙⠓⠶⠤⢤⣤⣤⣄⣀⣀⣀⠀⢀⣀⣠⡤⠴⠶⠶⠶⠖⠒⠛⠋⠉⠉⠀⠀⢀⣠⠶⢋⡉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠛⠋⠐⠚⠉⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⠛⠶⠦⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠶⠶⠦⠤⠤⡤⣄⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠾⠿⠴⠶⠒⠒⠒⠶⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠶⠶⢤⣤⣤⡤⠴⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠒⠚⣶⣖⡆⢨⡇⡀⠀⠀⠀⠀⠀⠀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠈⠉⠉⠷⠌⡁⢇⣤⣆⠈⠙⠐⠋⠈⠀⠒⠂⢄⡀⠀⠀⣤⣤⣴⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢠⣞⠻⣏⢡⣶⣾⣿⡅⠀⠀⠀⠀⣠⣶⣛⠗⢊⡆⠏⣴⣿⠿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣼⣾⠷⣿⣿⣿⡖⠀⢀⢾⣯⣀⠜⣓⡈⣷⢴⣇⣀⢀⡀⢹⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣠⠞⠛⠁⠀⢈⣻⣴⡟⢿⡷⢤⢨⣾⣷⣵⡏⢠⡟⠁⠀⠈⠙⠇⠀⠀⠻⣧⣹⣦
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⡼⠁⠀⠀⠀⠒⠀⠀⠀⠉⠉⠉⠛⠳⣽⣿⣿⣶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⣿⡟
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠹⣿⣧⣶⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⣷⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢤⣣⣿⡟
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣿⣶⠀⠀⠀⠀⠀⢤⣁⣨⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠖⠖⠺⠛⠉⠉⠉⠉⠉⠉⠉⠉⠘⠛⡲⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⣍⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠠⡠⣫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣶⡄⠀⠈⠑⠂⠤⠀⠀⠈⠉⠒⠶⣤⣄⣀⡀⡀⢀⡀⠀⠀⠀⣀⣀⣀⣴⠰⠾⠱⠛⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢸⣿⣽⣅⠀⠀⠀⠀⠀⠀⠈⠀⠓⠄⠀⠀⠀⠀⠉⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣟⣿⣿⡶⣦⣤⣄⣤⡴⠟⣳⡄⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⠻⠰⣽⣤⡾⢋⡀⢠⡿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠙⠛⠺⠿⢷⣤⣴⠿⠋⢀⠔⠃⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣀⣠⣤⣔⣾⣣⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠄⠀⠀⠀⠀⠀⢀⢠⣴⣶⣶⣶⢶⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣒⢒⠒⣒⣒⣒⣒⣒⣲⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠠⠤⠴⠒⠒⣚⢒⢒⣐⣐⡇⠀⠠⠤⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠂⠠⠬⠭⠭⠭⠭⠍⠉⣹⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠒⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣉⣛⣟⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⠉⠉⠉⠉⢉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⢤⡴⠶⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⣀⣤⢶⣫⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠏⠀⠁⠀⠈⠀⡀⠈⠈⠷⠶⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠉⠀⠀⠀⠀⢀⣀⡤⠶⣒⣫⠽⠚⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡞⡛⠠⠂⠠⠆⠐⠀⠈⠀⠀⠀⠈⠃⠈⠉⠳⢦⣀⣤⠶⠚⠉⠀⠀⠀⢀⣠⠴⠖⣋⣭⠴⠒⠋⠁
⠀⠀⠀⠀⠀⢀⡼⠁⠀⡤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠉⠀⢈⢷⠀⠀⠀⠀⣀⡴⢚⣩⠴⠚⠋⠁
⠀⠀⠀⠀⡶⠋⠀⠀⠀⠈⠀⡀⢀⡀⠲⠀⣀⠀⠀⠀⠀⣀⠀⠀⠘⣿⠀⢀⡤⠞⣡⠖⠋
⠀⠀⠀⠀⣷⠰⠀⠲⠂⠀⠀⠀⠉⠋⠀⠀⠁⠀⠀⠀⠘⠁⣀⢀⡼⢃⡴⠋⣠⠞⠁
⠀⠀⢀⣴⣿⠂⢤⠀⠀⢠⠄⠀⠀⠀⠀⠠⡖⠀⠘⠁⢀⡤⠞⣩⠞⠋⢀⡾⠁
⠀⠀⢸⣿⣇⠡⡀⠀⠠⣄⢀⣀⡘⠂⠴⠂⢀⣠⠴⢚⣩⠴⠋⠁⠀⣰⠋
⠀⠀⢸⠙⣮⡛⠶⣤⣄⣀⣤⣬⡤⠴⠒⢋⣩⡤⠖⠋⠁⠀⠀⣠⠞⠁
⠀⠀⠘⢧⡀⠙⠓⠶⠬⠭⠤⠤⠴⠖⠛⠉⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠶⠒⠚⠋⠉⠁

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⣈⢱⠀⠀⠀⠞⠛⠒⠲⠚⠛⢻
⠀⠀⠀⠀⠀⠀⢸⠓⠚⠛⠛⣋⠀⠀⠀⢣⢤⣤⣶⣶⡶⣾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣾⠀⠀⠀⡇⠀⠀⠀⠀⠀⡌⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡏⡆⠀⢸⠀⠀⡴⠉⠂⠀⢷⣇
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠹⡵⡀⡞⠀⠀⡇⠀⠀⡇⢸⢸
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⠱⡇⠀⢸⠃⠀⠀⢹⠈⡏⡆
⠀⠀⠀⠀⡇⠐⠒⠲⠶⠶⠶⠒⠒⠀⡇⠀⢸⠀⠀⠀⠸⠀⣇⡇
⠀⠀⠀⠀⡇⠀⠀⢀⡤⠤⣢⡀⠀⣴⡇⠀⡟⠀⠀⠀⠀⡇⢹⢰⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠢⢄
⠀⠀⠀⠀⡇⠀⠀⡞⠀⠀⠙⢷⠀⣿⡇⠀⠁⠀⠀⠀⠀⢷⢸⣼⠀⠀⠀⠀⠀⢀⡠⠊⣁⣤⡦⠊
⠀⠀⠀⠀⡇⠀⢠⢳⠀⠀⠀⡼⠀⣿⡇⠐⢄⣀⣀⣀⣠⠞⢸⢿⠀⠀⠀⡠⠔⣩⣴⠾⠋⠁
⠀⠀⠀⠀⡇⠀⠈⠣⣅⣀⡴⠃⠀⠙⣇⠀⠀⠀⠀⣀⣀⣠⣯⣾⡐⠒⣩⡴⠿⠋⠁
⠀⠀⠀⠀⡇⠢⢤⣀⣀⣀⣀⣤⠤⢂⡯⡷⠟⠛⢛⣩⡿⡻⠇⠀⢸⢜⡟⠷⣶⢄
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀⣠⣿⣜⢴⡊⠉⠁⣿⣜⣀⣀⣴⡁⠜⠉⣒⡤⣣⠇
⠀⠀⠀⠀⠀⠙⠒⠻⠿⠿⠿⠛⠊⠉⢆⠉⠚⠛⠯⠵⠶⠶⠶⠶⠬⠝⠛⢛⠍⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⢀⣀⡤⢖⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣭⣭⣭⣭⡴⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠂⡀⠀⠀⠀⠒⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢂⠤⠂⠀⠀⡌⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠄⢂⠉⢢⡀⠂⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⠐⠻⠀⠎⠄⡀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠈⠀⠀⠐⣆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⡔⢠⠀⣴⡀⠀⡎⠀⠠⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠀⠐⠀⢄⣀⠈⠂⠁⠃⠀⠀⠀⠀⠀⠐⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠈⠑⠒⠺⣀⣰⠶⠂⡀⡀⠀⠉⠒⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢈⡄⠀⡠⠃⢉⣿⠂⠸⡇⠺⢆⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⠉⠁⣰⡖⠀⠀⣽⢻⠀⠀⢡⠄⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⡩⠁⠀⣀⣾⠛⣽⡆⠀⠀⢆⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⢀⡴⠞⠀⢸⡟⢧⡀⠀⠠⡟⠒⠀⠘⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢠⡞⡇⠰⡀⢈⠄⠀⢩⠝⠳⠦⡤
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠁⠀⠐⠋⠀⢱⠀⠈⠉⠓⢤⠀⠓⢄⠀⠡
⠀⠀⢰⣿⡙⢯⡍⡟⣋⡹⢙⣿⠉⠏⢹⣷⠄⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠢⢀⠀⠁⠂
⠀⠀⠀⠁⠒⠷⠾⠶⠷⠶⠷⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡠⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡜⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⢎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡰⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⡜⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⢎⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⢀⡠⣔⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣠⠴⣚⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⡿⠃⠀⠀⠀⠀⢀⣤⠞⣋⠤⣚⡭⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⢤⣀⠀⢀⣤⠴⠶⣤⡀⢀⡞⠁⣴⠟⠀⠀⠀⢀⣠⠞⡫⢔⣫⠵⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⢀⡴⠞⠛⠋⠀⠀⠀⠀⠈⠳⠋⠀⠀⠀⠀⢙⠟⢀⣼⣋⠀⢀⣠⢖⣫⣷⡽⠚⠉
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢎⣠⠞⢀⣩⣗⣯⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠿⠛⠁⢻⣤⣤⡀
⠀⠀⠀⠀⣠⠖⠛⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⣠⣤⣤⡏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⡛⠛⠛⣷
⣾⠁⠀⠐⣏⠀⠀⠀⠀⠀⠀⠀⠰⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢠⣤⠟
⠈⢻⡆⠀⠀⠉⠑⠒⠀⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠠⠤⠔⠒⠉⠀⠀⠀⣼
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡆⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠋⠉⠉⠉⠉⠉⠙⠻⠟⢛⡟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣾⠛⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠾⡏⢉⣛⣿⡏⢩⣿⣿⣿⠳⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⣁⣠⣄⣉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠋⠉⠀⠘⠃⠰⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠷⠂⢀⡀⠀⠀⠀⢀⡀⠀⠀⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⢶⣦⡈⠉⠛⠛⠛⣉⣡⣶⠾⠷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣶⠉⡿⢿⣿⠿⠿⠿⠋⠁⣀⡆⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠀⡀⠘⣣⠀⠀⠀⡅⠈⢻⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⢀⣠⡷⠀⠘⣾⠇⠀⠀⠀⠋⠀⠈⠀⠈⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢳⡄⠀⢿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡜⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⠂⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⡄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡄⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠙⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⡘⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣤⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⣠⠓⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⡇⠀⠀⠀⡄⠀⠀⣸⠁⠀⢷⠀⠀⠀⠀⠀⠾⠏⠀⠀⢸⠇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠛⠂⢠⣾⡀⢰⣿⣇⠀⠀⢀⣄⢿⣄⡀⢠⣄⡀⢾⡆⠀⠀⠀⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣈⣉⣤⣀⣉⣁⣀⣤⣈⣥⣼⣄⣁⣈⣁⣤⣼⣧⣄⣠⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢁⡀⢀⣀⣀⣀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⠠⠄⠀⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡄⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠀⠀⢸⣏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡃⠀⠀⠄⠀⠀⠀⠀⣀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠀⠀⠀⣟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢷⣦⣀⠀⠀⠀⠀⢰⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢠⠄⠀⠀⠀⠠⠀⠀⠈⠀⠀⠰⠦⡄⠀⠈⠻⣧⠀⠀⠀⠀⣾⡿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⡄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⢻⡄⠀⢀⡀⢿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⢻⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠈⠇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⣰⠃⠀⠀⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠈⣿⡆⠀⠀⠀⠐⠀⠀⠀⠒⠀⠈⠁⢀⣁⡤⠁⠀⠀⠀⠀⢸⡇⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⢻⡇⠀⠀⠀⠀⠈⠁⠰⠦⣀⣀⠴⠞⠉⠀⠀⠀⠀⠀⢀⣼⣀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡯⣽⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡀⣿⡄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿⡷⢻⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣿⡇⠀⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠀⠁⠀⢩⠉⠉⠙⠛⠿⠿⠿⠟⠻⠛⠛⠛⠻⠟⠻⠿⠛⠙⠋⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠘⠀⠀⠇⠀⠇⠀⠿⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡘⠤⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣬⣥⣴⠂⡀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣦⣤⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠿⠻⠶⠶⠾⠟⠛⠛⠛⠛⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠾⠛⠻⢶⣤⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣤⠿⠁⠈⠒⠖⠉⠁⠀⠀⠀⠀⠀⠉⠻⣦⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣋⠄⠀⢁⠈⠉⠉⢉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⠇⠊⠀⠒⠚⠉⠛⠋⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡟⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁⠀⠀⠐⠒⠒⠄⠸⣧⡀
⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠉⠉⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠂⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣀⡿⠀⠀⢀⣰⠉⠉⠀⠀⠀⣀⠤⠒⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡠⠖⢋⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠀⠀⠈⢻⣦⡀
⢠⠞⠀⢀⠎⢠⠏⠀⠠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⢰⡇⠙⢦
⣿⣄⠀⠸⣄⠀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⡠⠤⠀⠀⠀⣠⠞⠁⢀⣼
⣯⠈⠳⣄⡈⠓⢤⣄⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠒⠛⢀⣀⣤⠶⠛⢁⣠⠴⠋⢰
⣿⡀⠀⠈⠙⠳⢦⣄⣉⠙⠳⠶⢤⣬⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠶⠖⠛⠋⣉⣤⠴⠞⠋⠁⠀⠀⢸
⢻⡙⠦⣄⠀⠀⠀⠀⠉⠛⠳⠶⢤⣄⣀⣀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣉⣉⣁⣤⣤⡤⠶⠖⠛⠉⠀⠀⠀⠀⢀⣠⠴⡿
⠘⣇⠀⣬⣝⡲⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣺⣭⡀⢰⠇
⠀⢻⡀⣞⣂⡇⢠⠬⣍⠓⠲⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠚⠩⠟⡓⣦⢰⣦⢷⡟
⠀⠘⣷⢮⣉⠀⢰⣗⡼⠃⣚⡛⡆⠠⠭⢭⡉⠙⣛⣒⡒⠒⠒⣒⣒⠒⠒⠒⣒⣒⡒⠛⣩⡭⢭⡀⠐⣋⡉⣧⠀⠿⠵⣏⡤⠖⡿
⠀⠀⠸⣆⠈⠙⠲⠦⣄⣈⠳⠞⠁⢰⣖⣨⠇⠈⣥⡄⢹⠀⢘⣩⡍⢳⠀⠘⣡⠄⢹⡄⢠⡖⣠⡇⠀⢻⣖⣫⠤⠖⠋⠁⠀⡼⠁
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠉⠓⠒⠦⢭⣥⣀⣀⣓⠒⠋⠀⠘⠷⠶⠛⠀⠀⠛⢶⣛⣀⣤⡭⠵⠖⠛⠉⠁⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠻⠶⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠖⠛⠉⠉⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠐⠒⠤⠤⣀⠀⠀⢀⠞⠁⠀⠀⠀⢠⠀⠀⠄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣀⡤⢆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠈⠙⢦⣇⠀⠀⠰⡇⢀⠴⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠰⠏⠁⠀⠀⣿⠛⠾⣝⠳⢦⣄⡀⢀⠀⠀⠈⢧⡀⠀⠀⠀⠂⢿⢀⣘⣄⣞⢕⡭⢖⡀⠀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠉⠊⣍⡚⠲⠶⠤⣤⣉⠳⡄⠀⣀⣼⣽⡿⣂⢁⡏⠀⠈⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣔⣤⣠⣤⣤⣤⣈⣉⣉⣉⠠⠤⠬⠽⡿⠽⠟⠃⠘⠋⣿⣟⠁⢀⣀⣠⠞⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣖⠋⠉⠉⠉⠉⠙⠛⠂⠐⠶⠖⠛⠋⠀⠀⠀⠀⠀⠀⢀⠤⡝⠛⠛⢋⠉⠠⠀⣀⣀⣄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⡠⠄⣁⡾⣁⣆⣴⠛⠀⠐⠃⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠈⣥⣴⡄⢀⠶⣒⠳⠤⣀⣀⣀⣀⠤⠀⣠⠝⠺⠷⠿⠫⣚⠽⠋⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡷⠊⠀⠀⠁⢸⠁⠀⠙⠿⠆⠈⠨⠭⠴⠖⠋⠉⠙⠛⠛⠛⠋⠁⠐⠀⠄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠐⠚⢦⡀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠒⠚⠛⠛⠛⠛⠒⠒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢀⣀⡤⠤⠤⠤⠶⠶⠶⠶⠤⠤⣄⣀⡀⠀⠀⠉⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⡠⠔⠊⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠔⠉⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠖⠒⠒⠒⠒⠶⠤⢤⣀⡀⠀⠙⠳⢄⡀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⠔⠁⠀⠀⢀⣠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠉⠙⠲⢤⣀⠙⠶⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢀⠎⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠀⠀⡈⠉⠑⠒⢄⠀⠀⠀⠈⠙⠶⣜⡆⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⠖⠉⠉⠉⡱⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⠤⠖⠁⠀⢠⠃⠀⠀⠀⠀⡞⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⣀⣀⠤⠤⠔⠚⠁⠀⠀⠀⢀⡴⠋⠀⢠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⡏⠳⣄⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⣀⠴⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠈⡗⢤⡀⠈⠙⠒⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠒⠊⠉⠀⣀⡤⠚⢹⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡅⠀⢷⠀⠈⢱⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⣻⠁⠀⠀⡿⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡇⠀⢸⠀⠀⠸⡄⠀⠀⢹⠙⠓⠒⢲⠖⠲⠒⢲⠚⠛⠍⠀⡇⠀⠀⡿⠀⠀⢀⠇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢱⠀⠈⡇⠀⠀⣇⠀⠀⢸⡆⠀⠀⢸⠃⠀⡀⣼⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⣸⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⡄⠀⢣⠀⠀⢻⠀⠀⠘⡇⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⢀⡇⠀⢀⡇⠀⠀⡇⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⠆⠀⠸⡄⠀⠀⡇⠀⠀⠈⡆⠀⠀⢸⠀⠀⠀⢸⠀⠀⠸⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⣃⠀⠀⠃⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢘⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣴⣶⣦⣤⣤⣤⣿⣿⣿⣷⣤⣴⡶⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢈⣉⣭⣭⣁⣉⣁⣌⣤⣀⡀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠋⠉⢉⣿⠛⢻⣿⡉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⡛⠿⠿⠿⠟⠛⠛⠻⢶⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣴⠾⠛⠛⢷⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡉⢻⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⢻⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⣼⣶⣿⣡⡟⠀⠀⠀⠀⠀⠀⢀⣴⡾⠷⠾⠿⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⣿⣍⡙⠛⠛⠛⠛⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⡉⢿⣦⣀⢀⣀⣀⣀⣿⠟⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠙⠶⢭⣙⣛⣛⣛⣩⡥⠞⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⣤⣬⣭⣭⣥⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣦⠈⠉⠉⠉⠉⠉⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⠀⠀⡄⠀⠀⠀⣿⡇⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡧⠀⠀⠀⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⢰⡇⠀⠀⠀⢸⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⠀⢸⡇⠀⢠⡄⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠃⠀⢸⠇⢸⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠀⠀⢠⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡟⠀⣀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⡏⢸⠀⠀⠀⢸⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⠀⡇⠘⠀⠀⠀⢸⣇⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⡟⢠⠇⠀⢀⡀⠀⢸⣿⠀⠀⢰⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣀⡀⠀⢸⡇⠀⠀⠀⢸⠃⠀⢸⡇⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣴⡟⠛⠻⠇⠸⡿⠀⠀⠀⠘⠀⠀⠘⣧⠀⠘⠋⠉⣠⣄⠀⠀⠘⠋⠉⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠇⠀⢀⣤⡄⠀⠀⠀⠀⢀⡀⠀⠞⠁⠀⠲⣦⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠸⣤⠀⠀⢰⡟⠁⠈⠙⠃⠠⣇⠀⠀⠀⠀⠐⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣠⡄⠀⠈⠀⣀⠀⠀⣀⠀⠈⢉⣤⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠿⠛⢿⣤⣀⣠⡿⠃⠀⠻⣶⣶⠾⠛⠿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡔⣿⣴⣷⣿⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠛⣉⣡⣤⣠⣆⣸⠉⠓⠻⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⣤⣿⠾⣯⣿⢧⠶⣿⣿⠛⢻⣶⢦⡘⠿⣿⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡞⣿⡶⠋⠛⠀⠉⠀⠀⠉⠙⠛⢻⣶⣮⣿⠶⡮⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⣀⣿⣿⡿⠉⠁⠀⠀⠀⣀⠔⠒⠀⠀⠀⠀⠀⠈⣧⣤⣿⣦⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠏⢀⣠⣴⣷⣾⣿⣿⣗⣿⣤⣄⣀⠀⠀⠀⠀⠘⢧⣘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣷⡿⠛⠉⢨⠁⠀⠉⠉⠹⠿⣿⣹⣧⣤⡀⠀⠀⠘⣿⣿⣞⣻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣿⣻⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣷⣦⣴⣶⣿⣿⣿⣯⣿⣶⡇⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠤⠈⠀⢷⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣽⣿⣦⣿⣍⢿⣷⣌⣿⣿⡿⢿⠶⣶⠐⠒
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⢿⣿⠷⠞⢷⡙⢿⣻⣿⣩⢿⣷⡚⠛⣿⣻⣆⡀
⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⡇⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠏⣹⠏⠀⠀⠀⠀⣠⡾⠋⠀⣽⣼⡿⠻⣤⠀⢹⣟⣿⡁
⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⠀⡇⠀⢠⡿⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣾⣿⣷⣾⣦⣤⣾⡏⠀⠀⠀⠀⠻⠳⣦⡿⠿⣶⠹⣿⣿⣤⡀
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⡇⠀⣼⠇⢸⣷⠀⠀⠀⠀⠀⢀⠀⠺⠋⣼⣿⠇⠀⢈⣿⠿⣿⡳⣦⡄⠀⠀⠀⠀⠻⢿⣷⣽⣃⠉⣿⠻⣧⡀
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠇⢠⣿⠀⢸⣿⡄⠀⠀⠀⢀⠌⠀⠀⣼⣿⡟⠀⣴⡿⠁⠀⠀⠙⣿⣻⣇⠀⠀⠀⠀⢀⣿⠋⢹⡟⢙⣧⠙⠁
⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⢹⣿⠀⢠⣼⡇⠀⢸⣿⣇⠀⠀⠀⡌⠀⠀⣰⣿⣿⣣⣾⠟⠀⠀⠀⠀⠀⠉⢿⣹⣷⠀⠀⠀⠈⠛⣿⢿⣧⣜⣻⡶
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠆⠘⣿⡇⠀⠘⣿⣿⠀⠀⡄⠀⠀⢠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢽⣟⣷⠄⠀⠀⠀⣻⡀⣾⣯⣭⣷⠄
⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⢸⣿⡆⢰⣿⠃⠀⢸⣿⣿⠀⠰⠀⠀⠀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣦⡄⠀⠀⣹⣿⡛⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⢸⣿⡇⣼⣿⠀⠀⢸⣿⣿⠀⠇⠀⠀⣰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠀⠀⠀⢉⣿⣿⣛⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⢸⣿⣿⣼⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠀⠀⠀⠀⢽⣿⣋⣽⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⢸⣿⣿⣿⣿⢷⠀⢸⣿⣿⡇⠀⣴⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣟⢁⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠘⣿⣿⣿⣿⢸⡄⢸⣿⣿⠃⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣄⢨⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⣿⣿⣿⣿⠈⡇⢸⣿⡿⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⣿⣿⣿⣿⠀⣷⢸⣿⣷⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⣺⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢻⣿⣿⣇⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣜⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿

⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣝⣤⣙⣅⣼⣡⣼⣧⣡⣾⣋⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠁⠐⠐⠀⢄⠀⠀⡐⠋⢀⠂⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠓⡜⠁⠀⠲⢀⡀⠠⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⢰⡀⠀⠐⣠⠧⢂⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⣀⠀⢸⠁⣠⠇⠀⢸⡌⠐⠇⢀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⣀⣀⡈⠶⡄⠰⠟⠒⠒⡜⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠉⠄⠀⠀⡤⠓⡄⠀⠐⠁⠀⠃⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡶⠀⠀⠈⡄⠀⠀⠀⢠⠋⠈⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⢸⠀⠀⢀⣧⠎⢸⡇⠨⡄⠀⠊⠁⠀⠉⠑⠲⠆⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠀⢠⠀⠀⠀⠘⠀⠀⠸⠀⠒⡜⠀⠈⠀⢀⡄⠀⠀⠀⠢⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⣀⣤⠤⠒⠀⠀⠀⢀⠀⡀⠀⠀⢤⣀⢁⠀⡼⠛⠁⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⣀⠉⠉⣇⣠⣦⠑⠉⢣⠀⠠⢀⡀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⠀⣿⠁⠛⢾⣧⠈⠘⢣⡎⠀⠀⣤⠊⢉⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠀⠀⡰⠃⠀⠀⢸⣟⠶⠀⠀⠁⣀⣄⠉⠙⠛⠒⠶⠀⠀⠄
⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⢈⡝⠀⠀⠀⡙⢹⣿⡄⠀⠀⠸⠿⢦⣀⠀⠀⠀⢬⣴⡤⠄
⠀⠀⠀⢰⠃⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣠⡜⠀⠀⠀⢀⣿⠀⣿⢸⣀⡀⠀⠀⠐⢬⡉⠙⠉⠋⠁
⠀⠀⠀⠘⣆⠻⡄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⠀⠀⠤⠶⢛⡀⠀⠀⠀⣸⠃⠀⣿⠀⠀⠸⣦⡀⠀⠀⠳⣄
⠀⠀⠀⠀⠘⠢⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⣴⠂⠀⠊⠀⠀⢰⡎⠁⠀⠀⣤⠀⠀⠀⠈⠓⠠⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠛⠁⠀⠠⠁⠀⠠⣻⠃⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠚⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢧⠀⠀⠀⣼⠏⠀⠀⠀⠀⡴⣯⠅⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⢀⣥⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠓⠒⠚⠁⠀⠀⠀⠀⢸⠀⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢐⡥⠂⠈⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⣀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠀⠀⠤⠚⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⣉⣦⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⣤⠶⠛⠉⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⣠⡴⠞⠋⠉⠀⠀⢀⣀⣤⠶⡞⠋⢉⣹⠆
⠀⠀⠀⠀⠀⠀⢰⡶⢤⣤⣤⣄⠀⠀⣀⣤⠶⠛⣁⡤⠶⠛⠉⠀⠀⢀⣀⣤⠶⠚⠋⣁⣸⡤⠷⠛⠋⠁
⠀⠀⠀⢠⡾⢋⢻⢥⣂⣰⡄⢸⣷⣛⣩⡴⠖⠋⠉⠀⣀⣠⡤⠶⠛⠋⣁⣤⡴⠞⠋⠉
⠀⠀⣰⠏⣤⡍⢊⠀⣉⡙⠙⡄⢸⡏⠁⢀⣠⡤⠶⠛⢉⣡⣤⠶⠚⠋⠉
⠀⠈⢻⡆⢔⣩⠖⢢⡄⠰⣌⣡⡼⠷⠚⢋⣡⣤⠶⠚⠋⠁
⠀⠀⠈⣷⠰⢍⣡⡼⠶⠛⣉⣡⡴⠶⠛⠉⠁
⠀⠀⠀⠈⢻⣏⣥⣶⠾⠟⠋