Timeline Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⢀⣠⠥⠄⠐⠒⠈⡄⠀⠀⠀⢀⠰⠶⠤⢠⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣄⡨⢃
⠀⠀⠀⠠⠄⡀⣀⠠⠂⡤⣄⠁⠀⠀⠀⢨⣠⣄⠀⠀⣠⡰⠄⠀⠀⠀⠀⠀⢀⢤⣝⡄⣎⣡⡞
⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠈⠐⠚⠁⠘⠃⠀⠀⠀⠀⠀⠢⣻⡿⠿⠻⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⢴⡅
⠀⠀⠀⠀⠠⠤⠄⠠⠄⠀⠄⠀⠤⠤⠵⠦⠤⠤⠤⠤⠤⠄⠀⠀⠀⠄⠳⠆⠀⠐⠒⠒⠒⠐⠆⠀⠀⠇
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⣶⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠿⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠉⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⣶⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠬⠓⠇⢿⡁⣇⠸⠸⠳⠇⣟⠀⠸⠺⣉⠸⣩⠾⠳⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⣧⣥⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⣿⣿⣷⣶⣾⣷⣷⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⢰⡗⢻⡞⣿⢠⣾⣻⡆⣿⠀⣷⢸⡟⢻⡄⣾⣹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠛⠘⠇⠸⠃⠻⠈⠻⠶⠂⠻⠀⠟⠸⠃⠸⠃⠛⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⣀⣽⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢶⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠁⠀⢀⣠⡶⠶⢟⡻⠶⢦⣄⡀⠀⠀⠀⠹⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡟⣩⡄⠀⠘⠋⠀⢠⣍⢻⣆⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡟⠀⠈⠁⠀⢸⡇⠀⠈⠁⠀⢻⡆⢀⣴⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⣾⡇⠿⠂⠀⠀⠸⣧⡀⠀⠐⠿⢸⡿⠛⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⠏⠀⠘⣷⠀⣠⡀⠀⠀⠈⠁⢀⣄⠀⣼⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠙⢷⣍⡁⠀⢴⡦⠀⢈⣩⡾⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⣦⠀⠀⠀⠀⠉⠛⠷⠶⠶⠾⠛⠉⠀⠀⢀⣤⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠁⢹⡧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠄⠄⠀⠀⠀⠀⠀⠀⢠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢘⢹⠉⡇⢾⠘⢸⡇⡇⡏⢹⢸⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠀⠀⠀⠀⠀⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠐⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠈⠻⣧⣀⣠⡀⢀⠀⠀⠀⠀⠀⠘⢿⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⢻⣦⢀⣄⠀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠒
⠀⠀⠀⢀⢠⡀⠀⠀⢘⠿⡃⠉⠀⠀⠀⠁⢼⣷⡜⠀⠀⠚⠂⠀⠀⠀⠀⠸⣿⡜⠀⠒⠚⠁⠀⠀⠀⠀⠀⢿⣤⡃
⠀⠀⠀⠀⠀⣰⣷⣆⡈⠉⠁⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⢀⠄
⠀⠀⠀⠈⠉⠁⠈⠻⠷⣭⡇⠀⠀⠀⠀⠀⠀⠀⠙⠟⣮⡷⠀⠀⠀⠀⠀⠀⠀⠘⠛⢮⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⠀⠀⠀⠀⠀⠀⠀⠲⡶⠠⠤⠀⠀⠀⠀⠀⠀⠀⠴⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡇⠀⡞⠃⣾⢹⠀⢺⠀⣖⡆⢰⢳⠀⡞⡆⣾⠢⢰⠂⢰⢻⠠⠻⠀⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⣇⣷⠒⣇⡗⢺⣼⠘⣿⢷⣧⣷⢺⣼⡚⣇⣷⡻⡜⢻⣸⣻⣼⣛⣼⣏⢧⣿⠉⠉⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠢⢀⠖⠐⡆⢀⠖⠒⠰⡀⠀⠏⠉⠑⠢⡀⠀⢀⠔⠢⠀⢰⠉⠃⢸⠉⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠀⢠⡡⡄⠀⠃⠨⠀⣄⠀⢡⢰⠀⠘⠀⣠⠂⢀⠇⢰⣊⣀⣸⠀⢐⣚⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣀⡼⠀⢇⣀⡇⠃⣠⢄⠀⠈⣲⡆⠀⣤⣀⣑⡜⢦⣀⣠⠞⠡⠤⠼⠸⠤⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⡀⢀⠀⡀⠀⠤⢄⠠⠠⠀⠤⢠⠀⠀⠂⢠⠀⢲⠠⠄⢠⢀⠠⠄⠠⠤⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠠⠀⠅⠠⠤⠄⠠⠠⠜⠒⠐⠀⠄⠔⠘⠒⠃⠘⠄⠐⠸⠤⠜⠠⠤⠄⠠⡀⠄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⢰⠀⠂⠂⢠⠒⠄⠔⠰⡄⠖⢤⠀⠄⢠⠀⡔⢢⢠⠀⢄⠠⠄⢤⠠⠒⠆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠑⠚⠈⠂⠂⠘⠂⠂⠈⠚⠁⠃⠊⠀⠂⠘⠀⠣⠞⠘⠂⠊⠈⠂⠂⠘⠒⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠒⠀⠒⠒⠄⡊⠱⠀⡎⠂⢘⠉⡆⠐⠉⡆⡒⢒⠀⠒⢲⠀⡒⢢⠀⠒⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠚⠁⠉⠉⠁⠉⠈⠁⠁⠁⠈⠉⠁⠈⠉⠀⠉⠁⠀⠉⠁⠀⠉⠁⠀⠁⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠬⠭⠍⢁⣚⣛⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⠒⠒⠒⠀⠀⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣂⣐⣂⣀⣀⡀⠀⠄⠒⠒⣒⣛⣛⣛⣛⣛⣒⣚⣓⣀⡀⠀⠀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠓⠲⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡄⣠⡀⢠⢠⣤⣤⣠⢠⡄⠀⢀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⠿⡿⡿⢿⣿⣿⠿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠓⠛⠓⠚⠊⠛⠛⠛⠚⠃⠀⠀⠛⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣿⣿⣾⣷⣶⣾⣿⣿⣿⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣏⣈⣷⣀⣼⣉⣙⣆⣀⣾⣁⣹⣆⣠⣏⣉⣳⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⠟⠀⠀⠀⣤⠇⠀⠀⢀⡼⠃⠀⠀⢠⡞⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠈⠁⠀⠈⠁⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠆⠀⠠⠄⠀⠠⠄⠀⠴⠀⠀⠤⠀⢀⣀⣤⣄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⢀⡀⠀⢀⡀⠀⢀⠀⠀⣠⠞⠉⠀⠰⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠈⠁⠀⠈⠀⠀⠈⠀⢰⠃⠀⠀⠀⢰⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠂⠀⠐⠂⠀⠐⠂⠀⠒⠀⣿⠐⠂⠀⢀⡼⠀⠀⠀⠒⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣆⠀⠀⠈⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡘⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⠀⢸⢳⡆⠀⢰⠛⡆⠀⢰⠒⡆⢰⠓⡆⢸⠓⡆⢰⠛⡆⢠⠖⡆⢰⠟⡄⣾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⢼⣈⠤⢼⠰⠧⠄⠸⡁⡷⠐⠾⡀⡳⠰⣂⡗⠺⣀⡗⠺⣀⡷⠲⡀⡗⢺⣀⡓⢻⢘⡛⠋⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠙⠁⠘⠋⠀⠀⠀⠉⠁⠀⠈⠃⠁⠀⠉⠁⠀⠀⠁⠀⠁⠁⠀⠃⠃⠀⠁⠁⠈⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠋⠙⢐⠁⢐⠀⠁⠈⠀⡁⠈⠀⡏⠀⡅⠊⠀⠀⠈⠀⠈⠰⠁⠈⡀⠈⠀⡆⠉⠀⠁⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠉⠉⠀⠀⠀⠁⠈⠉⠁⠈⠁⠀⠈⠁⠀⠀⠈⠈⠀⠙⠒⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠁⡂⠀⠂⠈⠉⠂⡋⠉⠁⠈⠁⠆⢸⠁⠱⠐⠂⠐⠀⠂⠒⠀⢘⠐⠂⠈⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠀⠑⠒⠀⠈⠀⠁⠈⠀⠀⠈⠉⠁⠈⠀⠀⠈⠀⠚⠀⠓⠒⠀⠈⠀⠁⠈⠀⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡖⠀⠆⠘⠀⠐⠰⠉⠉⠀⠏⠙⠀⠉⢹⠀⠋⠉⠃⡊⠉⠀⠘⠉⡀⠐⠂⣂⢰⠀⠒⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠃⠈⠀⠈⠁⠉⠀⠁⠈⠉⠀⠁⠚⠀⠉⠉⠁⠈⠉⠀⠈⠉⠁⠈⠁⠀⠈⠁⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⠄⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⣀⠀⠐⠒⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠴⠴⠦⠤⠭⣭⡭⠭⠥⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠒⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣶⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣿⣶⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣀⣀⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⣀⣸⣇⣀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣯⣤⣶⡿⠿⠿⠿⠿⠿⠿⢿⣶⣤⣽⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠋⣴⣾⠿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠻⣷⣦⠙⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠲⠄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠆⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠓⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣁⡀⠀⠀⠠⠄⠀⠀⢀⣈⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠿⢿⣾⣶⣶⣾⡿⠿⠛⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣾⠀⠀⣷⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⠀⢹⡆
⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⢾⡇⠀⢸⡷⠶⠶⠶⣿⠀⢸⡷⠶⠶⠶⣿⠀⠀⣿⠶⠶⠶⢾⡅⠀⣿⠶⠶⠶⢾⡇⠀⢸⡷⠶⠶⠶⠶⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡇⠀⢸⣇⠀⠀⢀⣿⠀⠸⣧⠀⠀⢀⣿⠀⠀⣿⡀⠀⠀⣸⡃⠀⣿⡀⠀⠀⣸⡇⠀⢸⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⣇⠀⣸⢿⡆⠀⣿⣿⡀⣸⣿⡇⠀⣾⢿⡀⢀⡿⣷⠀⢸⣿⣇⢀⣿⣿⠀⢰⡿⣇⢀⣸⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⢮⣭⣥⠟⠀⠀⠙⢮⣭⣭⠞⠁⠀⠘⠷⣭⣭⠾⠃⠀⠈⠳⣭⣭⡵⠋⠀⠀⠻⣬⣭⡵⠟⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠉⠉⠀⢹⠀⠀⣿⠉⠉⠉⠁⣿⠀⠀⣿⠉⠉⠉⠉⣿⠀⠀⣿⠀⠁⠀⠉⣿⠀⠀⡏⠀⠉⠉⠈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠓⠒⠒⠒⠚⠀⠀⠙⠒⠒⠒⠒⠛⠀⠀⠛⠒⠒⠒⠒⠛⠀⠀⠛⠒⠒⠒⠒⠋⠀⠀⠛⠒⠒⠒⠚⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢰⠛⠛⠛⠛⢻⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⡟⠛⠛⠛⠛⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⣤⣤⣤⣤⣼⠀⠀⢿⣤⣤⣤⣤⣿⠀⠀⣿⣤⣤⣤⣤⣿⠀⠀⣿⣤⣤⣤⣤⡿⠀⠀⣧⣤⣤⣤⣤⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠅⠂⠀⠀⠀⠀⠑⢼⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡤⠾⡀⠀⠀⠀⠀⠀⣀⣈⣷⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣴⡎⠀⠀⣀⣀⣬⣦⠀⠀⠄⣯⡀⠀⠀⠈⠑⠢⡀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣏⢋⣿⡿⣫⡤⡶⢶⣧⣬⣵⣤⣀⡀⢀⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⢟⡻⠙⡉⠉⠀⠰⠉⠉⠻⢿⣿⡟⠉⠛⢍⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡵⠋⠀⠀⠀⠀⠀⣤⠀⠀⠀⠂⠹⡹⡄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡎⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠑⣽⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠓⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⠁⠈⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠀⠀⠀⠀⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⣀⠁⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡼⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⢧⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣬⡼⡃⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⣯⣢⡤⢀⠀⡀⢀⣀⣀⣤⠖⣋⡉⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡾⠋⠁⠉⠛⠻⠿⠿⠯⢭⣭⢄⠈⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠈⠉

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⢤⡴⡄⣬⣤⢭⣧⢤⢤⣤⣤⡤⣤
⠀⠀⠢⠖⠡⠀⠀⢠⠀⠀⠐⠈⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠉⠩⠉⠉⠨⠍⠛⠛⠋⠙⠉⠛⠙⠋⠙⠙⡟
⠀⡁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⡀⠀⠀⠀⠀⡇⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂⠀⠘⠀⠀⡧
⡟⢀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⡇⢀⣈⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⡏⠀⠀⠀⣿⢓⣶⣦⣐⣖⣀⣶⣆⣶⣶⣾⣆⣶⣶⣳⣶⣶⣷
⠰⡔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠁
⠀⠙⢌⠢⠈⠀⠀⠀⠀⠀⠄⡀⢌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠉⠒⠤⢭⣠⣤⠬⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡍⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢔⡢⠖⠊⢹⡄⠉⠛⠷⣠
⠀⠀⠀⠀⠀⠀⢀⡆⠈⠈⢰⠄⠈⠉⣏⣉⠁⠀⢰⡂⠁⠠⡀⠀⠀⠠⡞⠛⠓⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢺⠋⠀⠀⠀⢾⡗⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠘⢲⡀⠖⣠⠔⠉⠀⠠⠀⠀⢉⠂⡝⢶⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢈⠀⠀⣀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡼⠁⢠⣄⠀⠀⠀⢀⡀⠑⠘⡀⠁⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠁⠀⠀⠀⠘⠁⠉⠋⠁⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠉⠳⣦⡞⠋⠀⠠⠀⠁⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⣐⡅⣀⡀⣀⡠⢦⣔⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠠⠀⠀⠄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠿⡟⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠒⠒⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠙⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⣻⠞⠁⠀⠀⠀⠀⢀⠀⠤⠀⠁⠼⠤⠥⢈⣀⣤⣀⡀
⠸⠧⠥⠴⠦⠤⠄⠄⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⠀⠙⢦⣀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠰⠾⠿⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⡄⠀⠀⠀⠀⠀⢻⡇⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠘⣇⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠏⢈⠁⠁⠀⡀⢀⠀⠀⠀⠙⢏⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣶⠄⠀⠀⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡅⠀⠈⠂⠀⢀⢸
⠀⠀⠈⠐⠄⠂⠂⠈⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡧⠐⠾⢿⡗⢦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡀⠀⠀⢀⡀⠀⣀⠀⠠⢸⠈
⠀⠀⠀⡀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⣰⠀⠠⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢸⠸
⠀⠠⢁⠄⠀⠀⠀⠐⠀⠑⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⣔⠁⠀⠼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠠⠄⢀⠠⡀⠀⠀⡀
⠀⢃⠇⠀⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⡏⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠂⠀⡞⢹⡏⠙⠂⠈⠀⠀⡀⡆
⠀⡀⠃⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠇⠉⣠⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠇⡈⠡⠀⠀⠀⠀⠀⠀⠇
⢠⣗⣐⣒⣆⣓⡒⠚⠒⠒⠚⠆⠀⠀⠀⠀⠈⠠⣈⣀⣠⣤⣼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⡟⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⢿⡟⠙⢿⣁⣹⣏⣈⣿⣀⣸⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢤⣜⠋⢾⡋⠙⣏⣉⣿⣅⣈⣿⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢾⣿⣷⣶⣶⣶⡶⢻⣆⣀⡿⣶⠞⢻⡟⠛⣿⠉⣿⠉⠉⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡿⠞⠋⠉⠉⠙⠛⢿⣷⡎⢉⣿⣠⡼⣧⡾⢷⡶⠻⣿⠛⢿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⡷⠋⢀⣤⣶⣦⠤⣤⣤⣿⣧⠟⢿⡁⣻⣀⣸⣧⣼⢧⣤⡿⣦⣼⣿⡟⠀⡆
⠀⠀⠀⠀⢀⣤⣾⣟⣻⡿⠟⣵⢶⣿⠷⠿⠓⠷⠶⡿⠿⠻⢶⠊⢻⣅⣹⣄⣸⣧⣀⣿⣀⣸⣇⣀⡇⣀⠁
⠀⠀⡠⢺⣿⡿⠟⠋⠁⢀⣠⣿⠏⠁⣀⣤⣀⣀⡀⢲⣼⣧⡼⢿⠋⢻⠉⢻⡏⢉⡏⠈⣿⠁⢹⣿⡇⡿⡀
⢀⣾⡿⠛⠉⢀⣠⣶⣿⣿⡿⢟⣥⣿⣿⣿⠿⣽⣿⣿⣯⣸⣦⡼⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠂⡇
⠸⡟⠁⣠⣾⣿⡿⠿⠋⠀⠀⣘⡿⠉⠀⣀⣀⠀⠈⣩⣝⠉⠖⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠤⣶⣶⡆⠀⡇
⠀⢡⣰⣿⠟⢁⣀⣸⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣷⣾⣿⢐⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠃⠀⣿
⠀⢸⣟⣡⣴⣿⣿⣿⡍⠀⠉⢽⡿⠋⢁⣀⠀⠈⠛⢛⣣⡏⠴⢃⣀⣤⣤⣤⣤⣤⣤⡤⠀⣤⣤⣶⡆
⠀⠀⢻⡟⠋⢀⣐⣿⣶⣿⣷⣿⣟⣻⣿⢭⣷⣶⣤⣸⣿⣷⣀⠻⠿⠿⠿⠿⠻⠟⠛⠓⠂⠛⠛⠛⡇⡀⣬
⠀⠀⣨⣴⣶⣿⣿⡉⠉⠉⢩⣽⣏⠁⠀⠈⠉⠙⠛⣛⣿⠋⣅⠀⣀⣀⣀⣠⣤⣤⣀⣠⣤⣤⣵⣴⡇⠁⣷
⠀⣼⠟⠋⣀⣺⣿⣿⣿⣶⣶⣿⣾⣿⣿⣿⣿⣶⣬⣿⣿⣶⣺⣿⣿⣿⣿⠿⠿⠿⠿⠿⠶⠿⠿⠿⡂⠀⡗
⠀⣧⣶⣿⡿⣿⣧⡀⠈⢉⣽⠟⣡⡀⠀⠈⠙⠛⣻⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠁
⠀⣿⢁⣀⣼⣿⠿⣿⣿⣾⣿⣶⣿⣿⣿⣷⣶⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢠
⢠⣿⠿⠿⣿⣟⡀⠀⣽⣿⠟⠉⠀⠀⠉⠛⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢸
⢸⣁⣴⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⡿⠿⠿⠇⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⠛⠋⠙⠛⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣀⠀⠀⠰⠆⠀⠀⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠈⠁⠀⢸⡇⠀⠀⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⢀⣴⠟⠁⠀⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠈⠁⠀⠀⠀⠀⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⢰⡆⠀⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⡴⠞⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⡴⠶⢶⣴⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⣶⣾⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⠀⣉⣉⣉⣽⣿⡟⠛⠁⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠠⠔⣊⣉⣩⣭⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠶⠛⠉⣉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣋⣠⠴⠒⠋⣉⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣋⣁⣤⣴⡾⢿⣟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⣋⣡⣴⣿⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣵⣶⣿⣿⣿⣛⣉⣡⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣶⡶⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡙⢏⣿⣿⡛⠖⠛⠋⠀⠀⠀⠀⠀⢀⡶⢛⣡⣾⣿⣿⠃⠐⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡘⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠛⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠙⠋⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣾⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣠⣶⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⢺⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣼⠿⡯⠋⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣤⡀⠀⠀⢠⣾⢛⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⢏⠙⠌⠋⠻⣯⡀⠀⠀⠀⠈⠙⢷⣤⣀⠀⣀⠀⡀⢤⣀⠀⠀⠹⣿⣦⡀⠹⣷⠀⠀⣹⡯⠀⠀⢹⠶⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡣⠕⠨⠢⠁⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⠪⡦⡆⣽⣿⣿⣿⣿⣿⣷⡀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⣾⣿⢿⡳⢝⠢⡈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠨⣟⢿⣶⣶⣿⡟⠛⠉⠉⠈⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡿⣙⠦⡉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⡠⢋⣴⣿⠿⣿⡇⠀⠀⠀⠀⣿⣿⡷⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣸⡟⠑⢮⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄⠀⠀⣰⣿⣿⠗⣋⣽⡇⠀⠀⠀⠀⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣭⣟⠿⣿⣾⣿⣿⣿⣯⣵⣾⠃⠀⠀⠀⠀⠸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠉⠀⠈⠛⣿⣿⡿⠟⢻⡇⠀⠀⠀⢠⣶⣾⣿⣿⡀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢋⣠⠖⠛⣧⠀⠀⢰⣿⣿⣿⠿⠿⠿⠶⣶⣤⣄⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠀⠀⠀⠹⡄⢰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⣿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⠋⠉⠁⠀⠀⠀⠈⠉⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢃⣄⠀⣜⡇⠀⠰⠆⠀⠀⠀⠀⠀⠰⡄⠀⣿⡀⠖⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⡽⠞⠋⠉⠭⢝⠢⡀⠀⣖⢲⡤⣄⡤⠒⠒⠛⢧⣞⣳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⡤⢒⡶⢋⣀⣈⣽⣖⣧⡀⠀⠀⠀⠀⠉⠛⢄⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⢺⢀⡤⣚⣩⠿⠛⠉⠉⠉⠉⠉⠙⠳⢤⣈⢳⣄⡀⠀⠀⢸⡌⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢋⡽⠋⣀⣴⠖⠛⠉⠉⢹⠛⠒⠤⡀⠈⠻⣟⠛⠲⢤⣀⣡⠇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⠏⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢣⡀⢠⠙⣅⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢳⡀⢇⠸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣇⢸⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠂⠀⠀⠀⠀⠸⡊⢂⠀⠀⠀⠀⠀⠀⣀⣧⠀⢹⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡄⡇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⡼⢸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢇⡏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⣜⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣼⢞⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠓⢍⣒⠤⠀⠰⠀⠀⠀⣀⡴⠊⠁⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣲⢤⣈⠉⠛⠒⠛⠋⠉⢁⣀⡤⢞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⢧⡀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣰⠟⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠑⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡰⠋⠀⢀⡴⠟⠉
⠀⠀⠀⠀⠘⠛⠷⠶⢤⣄⠀⢀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠉⠀⡀⢀⡀⠀⠄⠀⡀⠀⡀⠈⠂⠀⠘⡄⢰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⡀⠀⢀⠜⠋⠁⠀⠀⠀⠠⠀⠉⠃⢆⠀⠄⢀⠇⡎⠀⡰⠛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡴⠁⠀⠀⠰⠆⠈⠊⠀⠀⠀⠀⠀⠳⣄⣀⡀⠀⠀⠊⠉⠁
⠀⠀⠀⠀⠠⠠⠠⠀⣀⠀⠀⠤⠾⠁⡀⠘⠓⠀⢄⠀⠀⠀⠀⠀⠐⠀⠀⢃⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠢⢄⠀⠂⠈⡆⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠄⠀⠀⠀⠀⠆⠆⢸⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⠀⠰⠀⠂⠀⡄⠀⠀⡞⠀⠀⠀⠀⣀⠀⠀⡈⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠡⡀⠀⠀⠠⠀⠀⡀⠀⢠⠀⠀⠀⡰⠁⠀⠀⠀⡤⠒⠡⠤⠶⠆
⠀⠀⠀⠀⠀⠤⠶⢚⡅⢀⠞⠀⠀⠀⠀⢦⡀⠀⠀⠀⠁⠀⠀⢀⡠⠎⠀⠀⠀⠀⠘⢇⠠⣘⠒⠲
⠀⠀⠀⠀⠀⠠⠴⠋⠀⠃⠀⠀⠲⡄⠐⠀⠀⠉⠀⠒⠒⠂⠈⠁⠀⠀⠀⢀⡄⠂⠀⠀⠐⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡆⠃⡀⠈⠙⢦⠀⣠⠀⢒⣉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢲⡄⠀⢸⡇⠀⢸⡇⣸⢡⠀⡄⠀⣳⢛⠀⢉⡁⠀⢰⠀⠀⡄⢀⡀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢻⡄⠘⠇⣦⠀⢣⠇⠼⡄⠀⠈⠁⢿⡀⠉⢁⠀⢸⠀⠀⡇⢸⠃⢸⠆⣸
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠹⣆⣘⠖⠀⠛⣞⣸⣧⣈⡉⠉⠉⠀⠸⣄⣐⡃⣾⠀⠚⢊⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⠓
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠷⡦⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠐⠂⢄⣀⠀⢀
⠀⠀⠀⢀⠠⠤⡎⠠⠁⠀⠀⠀⠀⠐⢢⠤⠒⠒⣦⣄⠀⠀⢰⠁⠀⠀⠀⠀⠀⡔⠨⠢⠔⠀⠈⢂
⢀⠔⠈⠠⠄⢰⡇⠀⠀⠀⠀⡄⢀⠏⡁⠀⠀⠀⠈⠉⡂⠀⠘⠄⠀⠀⠀⠀⠀⢆⠀⠀⡀⠀⠀⠀⠑⡀
⢸⠀⠠⠄⠀⢸⠀⡀⠀⠀⠰⠀⠎⠈⠀⠀⡠⠂⠀⢠⠁⠀⠀⡠⠓⠒⠈⠀⠀⠬⢦⠀⠁⠀⠀⠀⠘⠌⠄
⠈⠤⠀⠤⠄⠸⡀⠀⠀⠀⠀⠼⠀⠀⠀⡀⠤⠀⡠⠃⠀⠀⠘⠀⠀⢀⠀⠀⠈⠁⠀⠡⡀⠀⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠑⡀⠀⣀⠌⠀⠑⠠⠄⠠⠄⠊⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⡕⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠐⠉⡲⠈⠂⠄⢀⡀⠜
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠄⠜⠊
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡐⠂⠈
⡇⠀⠐⠀⠀⠀⠀⡠⡊⠀⠀⠀⠀⠀⠀⠀⠂⠔⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠑⢀⠀⠀⠀⡠⠊⠀⠀⠁⠂⠠⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠀⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⡄⠀⢤⣤⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⣁⣀⣸⣀⡙⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⣩⣷⡾⠟⠛⠛⠛⠛⠛⠿⢶⣿⣙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠀⠈⠙⢢⣦⡄⠀⠤⠤⠄⢤⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠷⣶⣶⣶⣶⡶⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣤⣤⣄⣀⣤⣤⣤⣀⣠⣖⣦⣄⣠⣤⣤⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⣯⢽⣦
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇⢱⢻⡇
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⠀⣉⡀⢀⣀⡀⣀⡀⢀⠀⡀⢀⣠⡀⢀⣀⢀⣉⡉⠉⠉⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢿⡼⠀⣷⡀⠸⣤⠀⢿⡄⢸⠟⢿⢸⣗⡇⢸⣂⢸⠷⡏⠀⠀⠀⡏⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢰⢒⣶⢲⠤⣤⠤⡤⠤⠤⣤⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣷⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠒⢺⠒⠒⢲⠗⠛⠏⡦⠬⠭⠤⡇⢐⣒⣸⣀⣒⣈⡋⠯⢹⠇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢀⣀⣜⣀⡀⢸⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⡆⠀⠀⠸⠀⠈⣹⠀⢸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⡇⠀⠀⡏⠉⠉⢸⠓⠒⠂⡞⠒⠒⢲⠇⠤⠤⡧⠤⢤⡇⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡧⢄⣰⣁⣀⣀⠃⠀⠀⡇⠀⠀⢠⠃⠀⠀⢸⠀⠀⢀⠃⠀⠀⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⠃⠀⣸⠀⠀⡎⠀⠀⡜⠉⠉⢉⠋⠉⠉⡻⠀⠒⠒⠗⠤⠤⢼⠤⠤⢤⠇⢰⣿⠃
⠀⠀⠀⠀⠀⠀⡞⠀⠀⡗⠂⢰⠧⠤⠤⡧⠤⠤⣼⣀⣀⣀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⢸⣿
⠀⠀⠀⠀⠀⣸⠁⠀⢸⠀⠀⠸⠀⠀⢰⠀⠀⠀⡇⠀⠀⢸⠉⠉⠉⡝⠉⠁⢸⠓⠒⠲⡇⠀⣿⡏
⠀⠀⠀⠀⢀⡇⠀⢀⡏⠉⠉⡏⠀⠀⡞⠒⠒⢲⠧⠤⠤⡦⠄⣀⣠⣇⣀⠀⡘⠀⠀⢸⠁⢠⣿⡇
⠀⠀⠀⠀⢸⠀⠀⣸⠀⠀⢸⠀⠀⢰⠁⠀⠀⡜⠀⠀⢠⠃⠀⠀⢸⠀⠀⠈⡏⠉⠉⡟⠀⣸⣿
⠀⠀⠀⠀⢿⣄⣀⡈⠉⠉⠛⠒⠒⠚⠒⠒⠒⠧⠤⠤⠼⠤⣄⣀⣌⣀⣀⣸⣀⣀⡰⢃⡤⣿⡏
⠀⠀⠀⠀⠀⠳⢶⣿⣭⣭⣽⣶⣒⣒⣒⣒⣒⣒⣲⣦⣤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠔⠋⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠭⠭⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡤⠴⠒⢚⣉⣉⣉⣉⣉⣉⣉⡙⠒⠲⠤⣄⡈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡠⠖⢋⡡⠔⠚⠉⠉⠀⠀⡄⠀⠤⣄⠀⠈⠉⠙⠒⠤⣉⠓⠦⣀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣀⠔⢉⡴⠊⡅⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢠⡄⠉⠢⣌⠑⢤⡈⠳⣄
⠀⠀⠀⠀⣠⠋⢀⠜⢁⠔⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢄⠙⢆⠈⠣⡀
⠀⠀⠀⡼⠁⡰⠃⡴⠁⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠱⣄⠳⡀⠙⡄
⠀⠀⡜⠀⡼⢁⠞⢠⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⢆⠹⡄⠘⡄
⠀⡼⠀⡼⢁⠎⠀⢸⠀⣯⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆⠹⡄⠸⡄
⢠⠃⢠⠃⡞⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⡆⢳⠀⢳
⡼⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢳⠘⡆⠸
⡇⠀⡇⢸⠀⢀⠄⢦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣀⣀⣤⠤⠤⠶⠒⠒⠻⠖⠉⠀⠐⠦⠀⢸⠀⡇
⡇⠀⡇⢸⠀⠈⢃⠟⠛⠧⠼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢄⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢘⠀⡇
⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇
⢫⠀⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⢰
⠘⡄⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⡼⠀⡜
⠀⢳⠀⢱⡈⢇⠀⠀⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠗⢀⠏⣰⠃⢰⠃
⠀⠀⢣⠀⠳⡈⢦⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠃⢠⠃
⠀⠀⠀⢳⡀⠱⣄⠳⡄⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⡴⢃⡴⠁⣰⠃
⠀⠀⠀⠀⠙⣄⠈⠢⡈⠢⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⢀⡠⠊⡠⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢⣌⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠑⠢⣌⡑⠲⠤⣄⣀⡀⠀⠀⠣⠘⠀⠀⣀⣀⡤⠴⠒⣉⡤⠖⠉⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡉⠑⠒⠤⠤⣍⣉⣉⣉⣉⣉⣉⡥⠤⠔⠒⠉⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣒⣦⣰⣋⣳⣤⡮⠭⡦⠾⠭⡷⠿⢿⡷⠿⢿⡿⠚⠓⣺⠗⠓⢲⠞⠛⢒⡾⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠩⠞⠀⠀⠚⠀⠀⠚⠁⠀⠉⣀⡀⢉⠀⡀⣨⣄⠀⣤⡀⡤⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⠀⠀⠀⠀⢰⠋⢠⡿⡄⣇⡀⢿⡄⢹⠓⠇⠳⠜⠐⠛⠃⠛⠙⠀⠀⠀⣀⣀⠈⡇
⠀⠀⠀⠀⠀⠀⡏⠀⢻⣿⡇⠀⢀⣀⣀⣩⢤⡤⠤⠤⠤⢤⠒⠒⠒⢲⡒⢒⣚⣻⣉⠉⢽⡭⠭⢽⠭⠭⡇⢹
⠀⠀⠀⠀⠀⣸⠁⠀⠸⣿⣿⠀⠸⡖⠒⠒⠒⠗⠒⠈⠉⢹⣉⣉⣉⣉⣇⣀⡠⢸⡤⠤⠤⡦⠤⠤⡦⠤⢿⠈⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⣿⡇⠀⢷⠒⠊⠉⢹⠉⠉⠉⢀⣇⣀⣀⡀⢥⠤⠤⠤⡧⠤⠐⢷⠒⠒⢻⠊⠙⡆⢹
⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⣿⣿⠀⠸⡖⠒⠈⠉⡏⠉⢀⣀⣹⣀⠠⠤⠼⡤⠤⠄⢲⠒⠒⢺⡂⣈⣙⣏⣉⣱⠈⡇
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⣿⣿⡆⠀⢷⠒⠒⠈⢹⠉⣁⣀⡬⡦⠤⠔⠒⢗⠒⠒⠚⡏⠉⠉⣏⢉⣀⣣⣀⠬⡇⢸⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢻⣿⣷⠀⢸⡊⠉⠉⢉⣅⣀⣀⣀⢸⠤⠤⠤⢺⠒⠒⠒⢻⠉⣉⣹⣀⡠⠼⠤⠤⠞⠀⣇
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠏⠉⠉⠁⣣⣀⣀⣠⠤⠦⠴⠖⠚⠓⠋⠉⠉⠉⢀⣀⣀⣠⡤⠤⠤⠴⢶⠛
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⣇⠀⠘⠚⠉⠉⠁⠀⢀⣀⣀⣀⡤⠤⠴⠖⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⢰⠃⣀⣀⠤⠤⠖⠒⠒⠙⣿⡼⠤⠶⠖⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠖⠒⠒⠋⠉⠉
⠀⠀⠈⠙⠓⠶⠤⣄⣀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠒⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠾⠦⠖⠒⠚⠉⠉⠁


⣶⣶⠶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠬⠀⠀⠀⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠩⠍⠐⡒⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠢⡀⠀⠀⠀⣠⢤⣤⡤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⣠⠿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠛⠃
⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣻⣏⣙⢂⠀⠀⠀⠠⠤⠤⠀⠒⠂⠁⠀⠀⠀⣠⡤⠤⣤
⢸⠀⠐⠀⠒⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⣀⡀⠀⠀⠐⢻⡅⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣴⠀⠀⠀⠀⢀⡴⠒⢲⡇⠀⠐⠒⠒⠒⡖⠛⠛⠛⠁⢀⡟⠉⠉⠻⣦⡀⠀⠀⢿⣀⡾⠁⠀⠀⠀⠀⠀⠀⢠⡏
⠘⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠛⠛⠛⠛⠛⠀⠀⠠⡕
⡄⠀⢠⡿⠒⠒⠲⢾⡇⠀⠀⠀⣿⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠈
⡇⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣴⠀⠀⠀⢙⡷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢿⣄⠀⢀⡾⠁⠀⠀⠀⠀⠛⠀⠀⠀⡠⠂
⡇⠀⠸⠃⠀⠀⠀⢸⠇⠀⠀⢰⡿⠀⠀⠀⠿⢶⣤⣤⠞⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⣀⡀⠠⠄⠂⠉
⠧⠀⠐⢆⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣤⠤⠤⠤⠀⠐⠒⠒⠒⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⠡⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢸⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⠀⢸⠀⠀⠔⠜⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⢀⠀⡇⠀⠀⠀⢸⠠⠀⠀⠀⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠂⠸⠧⢤⡬⠐⢠⠄⠉⠁⠂⠨⠁
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠒⠒⣚⣛⣛⣛⣛⣛⣛⣛⣛⣓⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢠⠚⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢳⡄⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠳⣞⣴⣣⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣝⣮⣳⠚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⡫⠁⠀⠀⠀⠀⠈⣓⡄⠀⠀⠀⠀⠀⠀⣴⣊⠉⠈⠀⠀⠀⠊⠝⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠀⠀⠀⣠⠴⢚⣉⠥⠔⠒⠒⠒⠒⠒⠒⠦⢬⣉⠓⠦⣄⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡴⢋⣱⠖⢉⡄⠒⠈⠁⢠⠀⠀⠀⠀⠁⠐⠤⡙⠲⣎⡙⢦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡴⠋⣠⠊⡠⠊⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠑⢌⠱⣄⠙⢆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡜⢁⠚⠄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⡈⢧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠁⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠂⡠⠄⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠜⠀⠀⠀⠀⠀⠀⠀⢨⡏⢱⠄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢄⠄⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⠄⡀⠀⡀⠀⠀⠀⠀⠒⢐⠀⠀⠀⠀⢑⠄⢊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢲⣌⡘⠀⠄⠀⠀⡀⠀⠀⠀⠠⠄⢂⣡⡖⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣠⠎⠀⠉⠙⠒⠶⠤⠤⠤⠤⠖⠒⠋⠁⠀⠻⣦⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠿⠛⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠿⠿⠟⠛⠻⠿⣿⣷⣶⣶⣦⣼⡇⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠇⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠏⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠉⠁⠀⢀⣴⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠟⠉⠀⠻⠋
⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠉⠈⠻⣿⣿⣦⣀⣠⣴⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣀⣠⣾⣿⡿⠋⣠⠽⠻⢿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣦⣀⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠉⠉⠉⠁
⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣷⣶⣤⣤⠀⠀⠀⣀⣀⣠⡀⠀⠀⢠⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⡿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⢿⡟⠏⠉⣻⠉⠳⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆⠀⠀⢀⣠⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢠⣿⣿⢟⣷⣿⣿⣿⣿⣿⡄⠀⠈⠙⠧⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⡇⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣯⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣟⠿⠛⠋⣿⣿⣿⣿⠀⠀⠀⠈⢹⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣄⠀⣼⣿⣿⣿⣿⠀⠀⢀⠀⠈⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣦⣻⣿⣿⣿⣿⢿⣿⣯⣭⢻⢿⣿⣿⣿⢛⡟⣿⡄⠘⢿⣿⣿⣿⠁⠀⠈⠃⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠾⣽⣿⣿⣿⣿⠟⠁⠈⢻⣿⣿⣦⣤⣤⣿⠏⢸⠃⣿⣿⣤⣿⡿⣿⣿⣧⣀⡀⢀⣠⣴⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡘⠿⠶⢿⣷⣿⡇⠀⠀⠀⠉⢻⣿⣿⣿⡟⠀⡾⢀⡇⢿⣿⣿⢃⣿⡧⠀⣿⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣷⣶⣿⣿⡗⠀⠀⠀⠀⣼⣿⣿⣿⠃⣼⠁⢸⠓⢸⣿⣿⣿⣿⢃⣨⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡹⣿⣿⢿⢿⣿⠀⠀⠀⢠⡿⢹⣿⡏⠀⠁⠀⠀⠀⣼⣻⣿⣿⢡⡿⠋⢀⣤⣶⡀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡘⢿⣾⡛⣿⡆⠀⠀⠸⠅⣼⡿⠀⠀⠀⠀⠁⣰⢯⣿⡿⠃⣾⣾⣟⢃⣶⣄⠃⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣎⠻⠇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣾⣿⣇⡸⣿⠁⠀⠈⢉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠻⣏⢿⣧⡄⢸⡇⠀⢀⡀⠀⠀⠀⠀⠀⡀⣰⣿⣶⣿⣷⣿⡗⣶⡛⠛⠢⠞⢴⠶⢴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠙⠳⠿⠇⢸⣷⣾⣿⣟⣿⡏⠀⠀⣸⣆⣻⣷⣿⣿⣿⣿⣿⣘⡇⠈⠁⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣶⣄⣀⣿⠏⠙⢿⣿⣿⣷⣀⣰⣿⣿⢿⡋⠉⠈⠿⠿⢻⣯⣿⣤⣤⣄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠉⠛⠿⣿⡿⣻⣷⢟⣽⢟⡗⠘⠻⠻⠿⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣷⣬⣴⣾⣿⣿⣿⡆⠀⠀⠀⠉⠉⠛⣿⡿⠟⠀⠈⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⣴⣿⣿⣯⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠀⢤⣄⠄⠀⠛⠻⠿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠄⣾⢻⠎⠒⠁⠀⠀⢻⣷⣶⠆
⠀⠀⣠⣴⣶⣾⣿⣏⣻⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡄⠀⠚⠛⠛⠛⠚⠷⠆⣀⢰⣿⠟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⣿⠛⠛⠛⢁⣬⡠⣦⠀⠀⠨⠾⠿⠚⠛⠛⠉
⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⡽⠶⣔⠺⣿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡉⠓⠀⠂⠘⢆⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠣⠀⠄⠀⡀⠈⠀⠀⠀⠐⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⡁⠐⠉⠂⠀⠀⠀⠄⠀⠀⠀⢀⣴⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣆⠀⠠⠀⠀⠀⡈⠀⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠉⣬⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠁⠛⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢴⠄⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠁⠀⠈⣣⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠔⡴⠀⠈⡑⠀⠀⡠⠖⠒⢤⠀⠀⠀⣏⣁⠀⡂⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠴⠁⠃⡠⠤⠁⠀⠀⡇⠰⠇⠀⡇⠀⠀⠀⡤⢉⢔⠡⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢠⡎⠻⠄⠀⠀⣠⡤⠈⠒⡆⠊⠀⠀⠀⠀⠰⢋⠂⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠁⠀⠀⠀⠰⡄⣰⠀⠀⡇⠀⠀⠀⠀⡠⡇⠀⠀⠀⠄⡠⠀⢳
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡾⠀⠀⠳⡀⠀⢅⡰⢡⠃⠀⠀⣀⠇⠀⠀⠀⡰⠀⠛⠇⠀⠌⣼⠔⠃⢸⡆
⠀⠀⠀⠀⠸⡟⡄⠀⠀⢰⠇⠀⠀⣠⠽⢦⡀⠡⡎⢀⣠⣘⠀⠀⡞⠀⡔⢁⠰⠒⢂⢈⠞⡁⠀⠀⠸⡇⠀⠀⠀⡰⣷
⠀⠀⠀⠀⠀⣷⠸⡀⠀⣾⣀⣀⠞⠀⠀⠀⠙⢦⡀⡌⣻⣭⣿⡁⢴⠣⢷⡈⢦⠀⠾⠋⠀⠘⠲⠄⠀⡇⠀⠀⡜⢱⡇
⠀⠀⠀⠀⠀⢹⡆⢡⢀⡏⠀⠈⢆⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⢱⠐⣄⠀⢱⠀⡰⢁⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⣾⡥⣶⢾⡽⠐⠒⠋⠉⠉⠉⠉⠁⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⠈⣦⠘⣴⢁⣾⠇
⠀⠀⠀⠀⠀⠀⢹⣷⡏⢸⢫⡟⠀⠀⢀⡠⢖⣩⠔⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠰⠸⡀⠀⠀⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣄⣤⣴⣾⣛⣩⡵⡇⢀⠇⠀⢸⠀⢣⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⢣⡴⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡿⢋⡾⢁⡎⠀⠀⠀⠇⠈⢧⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⣿⠟⠉⣠⡟⠀⡜⠀⠀⠀⠀⠨⠀⠈⢦⢀⠀⠀⠀⢸⠀⠀⠀⠘⡄⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⡹⢁⡠⠞⢁⡀⠀⠛⠀⣜⣀⠀⠀⠀⠀⠐⡄⠀⣈⡴⠃⠀⠤⠤⢤⠤⠶⢖⠇⠀⣟⠛⣟⡀
⠀⠀⠀⠀⠀⢀⣯⠖⠉⠀⢰⠋⠙⡆⠀⠀⢿⠋⠑⢄⠀⠀⠀⠁⢏⣁⣀⠀⠀⠀⠀⠀⠀⠀⢯⣤⡤⠼⠀⢸⡇
⠀⠀⠀⡠⣖⣩⣀⡀⠀⠀⠘⡄⠀⠸⡀⠀⠀⢑⡄⠈⠳⣄⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠘⠁⠀⠈⠙⠷⣤⣤⡞⠀⠀⠈⠙⠉⠉⠀⠀⠀⠀⠙⠲⠤⠤⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⠁⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠉⠒⠢⠴⠾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠠⠂⠀⠀⢠⠀⣤⢤⣀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠐⠐⠂⡰⠁⠀⠀⠀⠀⠠⠞⠀⠡⠴⡶⠶⠶⠶⠶⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠢⠤⡾⠃⠀⠀⠀⣠⠃⣀⠴⢂⡂⡉⠉⠈⡀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⢀⣀⣐⣂⠀⠤⢀
⠀⠀⠀⠀⠀⠀⡈⠀⢀⡠⣠⠞⡵⢊⠀⣪⡥⠂⠒⠒⠒⠂⠶⢄⡀⠌⠂⠀⠀⠀⠀⠘⠣⣷⠈⠙⠛⠻⠿⠿⠿⠷⠮⠶⡑
⠀⠀⠀⠀⠀⢰⡇⠎⣩⠞⢡⠊⠐⣡⠎⠁⠀⠀⡁⠀⠠⠀⢀⠀⠉⠣⡔⠌⢢⡀⠄⣈⠁⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠇
⠀⠀⠀⠇⠰⡀⢓⠜⠁⢀⠇⠀⡼⠁⠀⠁⠀⠀⢂⠀⠀⠀⠀⠐⠀⠀⠸⡆⠀⢣⠊⠀⠈⠀⠈⢦⡀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⢠⠇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⢻⠀⠘⠀⠀⠀⠀⠀⠀⠑⠤⠌⠁⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⢸⠀⠀⠀⠀⠀⠀⠀⡾⠂⠁⠀⠀⠀⢂⠀⢸⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢇⠀⠀⠀⠀⠀⢻⠿⠀⠀⠀⠀⠀⠀⠀⡜⠐⢰⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠑⣆⠀⠀⠄⠀⠨⠀⠀⠀⠀⠀⠀⢀⡼⡁⢠⣧⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠈⡳⢆⡀⠀⠀⠀⠀⠀⠀⡠⡴⠋⠀⡰⠋⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡲⢌⡀⠈⠙⠻⠯⠼⠾⠾⠚⣩⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠙⠒⡶⠒⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠴⠶⠞⠛⠛⠛⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠒⣶⠒⠦⣤⣀⠀⠀⢰⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⣀⡔⠾⣀⠀⠉⠀⠀⠀⠙⢷⣄⣈⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣀⣬⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠹⣿⠂⠀⠀⡱⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⢡⢴⣿⣶⠐⠃⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣠⣠⣶⣿⢏⣿⣿⣿⡆⢀⡀⠀⢀⠠⠤⠾⣞⠁⠀⠀⢿⡅⠆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⡓⠀⠀⠀⠀⠀⢫⠀⠀⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣀⣠⣿⣖⣻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠘⠄⠀⢢⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠸⣶⠀⢸⣦⡾⣿⣤⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠛⠛⠋⠀⠈⠉⠉⠉⣉⣉⣭⣽⣿⣬⣽⠙⠓⠒⠒⠛⠛⠁⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠄⠛⠉⠉⠀⠀⠀⠀⠘⠛⣿⣦⡌⠙⠶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⣁⠙⣿⣔⡐⠚⠛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣀⣀⠀⠤⠤⠤⠶⢋⡡⡢⢝⣿⣤⣀⣈⡁⠚⣦⡀⠀⠀⠀⠀⠀⣠⠞
⠀⠜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⡴⠁⠀⡀⠐⠂⠉⣀⣀⠀⠁⣒⣒⣂⣭⣉⣷⣾⣿⣭⣤⣤⡄⣀⠈⢳⡄⠀⠀⣠⠔⠁⠠
⠈⠈⠌⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠟⠀⠀⠀⠀⠀⢈⡁⢠⣤⣴⣶⣿⠿⠟⠛⠉⠉⠙⠻⢿⣿⣧⠄⠀⠁⠀⠹⣤⠊⠉
⠀⠀⡐⠀⠀⢉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢰⠏⠀⠀⠀⡀⠔⠃⠈⣡⡴⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⢱⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠀⠀⠀⠀⣠⠇⢀⠃⠀⠀⢀⠂⡠⢔⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢇
⠀⠠⠠⠀⠄⠀⠄⠀⠀⠡⠠⠀⠌⡼⠉⠉⢣⣠⡏⠀⠀⢀⠀⣰⡗⣩⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⢂⠸⡧
⠀⢡⢡⠉⡌⠈⠌⠀⠀⠁⠠⠉⡜⡵⡤⠠⢼⣿⠁⠀⠀⡌⢠⣟⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⢢⠈⡄⣿⠀⠀⠈
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠀⠀⣾⣭⣄⡀⢰⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣆⢃⣰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡿⣅⢀⡸⡀⣿⣿⠃⣤⠶⠶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣠⠴⠾⠛⠛⠉⠛⠒⠀⠀⠀⠀⢸⣿⣿⡎⢹⣾⡇
⠀⠌⠌⠀⠡⠡⠡⠀⠀⠌⠌⠤⠡⠡⠀⢸⣷⠀⠀⠀⢱⣿⣿⡀⠀⣠⣴⣦⣤⡢⠀⠀⠀⠀⠀⠀⠀⣠⡶⡶⠿⣷⣦⡀⠀⠀⠀⠀⠸⠹⣿⣷⠈⣿⡉
⠀⡈⡈⠀⠀⠀⠁⠀⠀⠀⠀⠀⢡⠀⠀⢠⣼⡆⠀⠀⠸⢹⣿⡇⠘⠛⠯⡭⠀⠉⠶⠀⠀⠀⠀⠀⠘⠙⠲⠂⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡀⣿⠟
⠀⠐⠐⠀⠂⠀⠂⠀⠀⠀⠀⠀⢂⠂⠀⠀⣸⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣾⣿⣧⣸⡇
⠠⠡⠡⠈⠄⠄⠌⠀⠀⠠⠠⠀⠌⡄⠄⣀⣸⣇⡿⢀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠀⠀⠀⠹⣿⣷⣶
⠀⢁⢃⠈⠀⠀⠀⠀⠀⠀⠂⣀⣼⣾⣿⣿⣿⣿⡇⠰⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⢰⠀⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠉⠻⣿⣿⣷⣦⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠐⠂⠀⠀⢸⠀⠀⣿⣿
⠀⠄⠄⠠⠀⠠⠠⠀⢀⣿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⣿⣿⣿⣧⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠃⠀⡀⠀⠀⠸⠀⠀⠘⠀⢰⣿⡇
⠀⠈⠈⠀⠁⠀⠁⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⢹⣿⣿⣿⣧⡀⠀⠀⠀⠈⠛⠿⠷⠾⠋⠁⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡆⠀⡀⠀⣸⣿⡇⣾
⠀⡀⡐⣀⠀⢀⢂⠀⣿⣿⡇⠀⠀⣄⠘⣆⠀⠀⢻⣿⣿⣇⡈⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠸⢀⣥⣾⣿⣿⣰⡿
⠀⡸⡀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠸⣦⠘⣧⠀⠀⢿⣿⣿⡠⠀⠀⠈⡹⢿⣿⣿⣷⡀⠀⣀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡗⠂⠸⣿⣿⣿⣿⡇
⠀⠽⠡⠀⠄⠄⠀⢸⣿⣿⡇⠀⠀⠀⢹⣧⡹⣷⡀⠸⣿⣿⣧⢂⠀⠀⠈⠙⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⡟⣿⡟⡀⠘⠀⢿⣿⣿⣿⣷
⠀⢱⢣⠉⠈⠈⡌⢸⣿⣿⡇⠀⣤⠀⠀⢻⣷⣿⣷⠀⣿⣿⣿⣟⣦⠀⠀⠀⠸⡞⣹⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⢠⣾⣇⣀⠀⠂⢻⣧⠇⠀⠁⠈⠹⣿⣿⣿
⠀⢃⢂⠀⠀⠐⡀⢸⣿⣿⡇⠀⣿⣿⣄⠈⢿⣿⣿⣧⢸⣿⣿⣿⣿⣧⢆⠐⡀⢻⣣⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣼⣰⠃⠸⣧⠈⠀⠀⠀⠀⠹⣿⣿
⠀⢚⠆⠀⠀⠀⠀⢸⣿⣿⡇⣀⢹⣿⣿⣷⣜⣿⣿⣿⣿⣿⠙⣿⣿⣿⣾⣦⢳⡈⣿⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣼⠀⣿⣆⠐⠀⡄⠀⠀⢿⣿
⠀⠜⠄⠀⠠⠠⠠⢸⣿⣿⡇⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⣿⣿⣿⣿⡎⣷⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡞⣼⣿⣆⢠⣷⠀⡆⢸⣿
⠀⣼⡌⠀⠀⠠⠀⢸⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡆⢻⣿⢿⣿⣷⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣿⣶⣧⣼⣿
⠐⡸⡐⠂⠂⠂⠂⢺⣿⣿⣧⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃⢠⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⡐⣄⠊⢧⡲⡀
⠀⠀⠀⡠⠔⠉⠀⠀⠀⠙⢦⣠⠵⠌⠃⠀⠁⠘⠄⠀⠀⢀⣀⣀⡀
⠀⣠⠎⠀⠀⠀⠀⠀⠀⢠⠎⠁⢀⣀⡀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣰⠁⠀⠀⠀⠀⠀⡴⠋⠈⠆⠴⠛⠛⠃⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣄
⡇⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢣⡀
⠹⢄⡀⢀⡠⠞⠀⠀⠀⡀⠀⣀⠀⡱⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢳⠀⠰⠀⢢
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣣⡘⠿⢷⠀⢀⣤⡀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠇⠀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣽⣿⣿⡆⠌⣛⠕⠁⠀⠀⢻⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡀⠀⡇
⠀⡎⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠑⣄⠀⠀⠘⡗⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⣸⠁⠠⠁⢀
⠀⢣⡀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠄⠀⠀⠀⠀⠘⡄⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠒⠒⠒⠶⠊⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠁⠒⠖⢲⣶⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠉⠒⢤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠘⡿⠟⠛⠛⠿⠶⣶⣶⣾⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠤⠜⠛⠁⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢇
⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⠀⢀⣀⡠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡌⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⢦⡀⢀⡴⠁⠀⠀⢳⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠹⣄⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡗⠲⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈⠃⠀⠀⡙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠠⠚⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠄⢀⠀⠀⠀⠀⠀⡿⢤⠄⠋⠁⠀⠀⠀⠉⠒⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡇⠀⠀⠀⣿⣿⣿⡆
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢿⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣀⣉⣀⡀⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⡿⠟⠋⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⢀⣤⣤⣴⣶⣶⣶⣾⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠠⣿⣿⣿⡇⠀⢠⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡆⠀⠀⣷⣿⣿⡃⠀⢸⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠸⣿⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⠸⣿⣿⣿⠇⠀⠀⣿⣿⣿⠇⠀⠀⣿⣿⣿⠇
⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⣉⣤⣤⣤⣤⣬⣍⡛⠶⠀⠀⣀⣤⠶⠛⠛⠛⠛⠛⠷⣦⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⡤⠶⠶⠶⣦⣬⡁⠀⠀⠘⠉⣠⠴⠞⠛⠛⠛⠻⠶⣦⣅⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢠⡆⣰⠇⢀⠔⠉⠀⠀⠀⠀⠀⠀⠉⠻⣦⡠⠞⠻⢷⣆⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀⠈⣷
⠀⠀⠀⠀⢠⡟⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣷⣄⣠⣠⠿⠒⠛⠶⠶⣤⣀⠀⠀⠈⢷⠀⢸⡇
⠀⠀⠀⠀⢸⠇⣿⠀⡇⠀⠀⠀⠀⣀⣠⡴⠚⠀⠀⠀⠀⣟⣹⣽⣁⣸⣿⣿⣿⣶⣄⠙⢷⡀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⢸⡆⣿⠀⣷⠘⠛⠛⠋⢩⡀⢶⣿⣷⡀⠀⡀⠉⠉⠙⠉⣉⣉⣉⣙⡛⠛⠷⠦⣿⣄⡈⠀⠺⣦⣀
⠀⠀⠀⠀⠈⠃⠹⣦⠘⢷⣄⠀⠀⠀⠻⡌⠻⢟⡥⣊⣤⡴⠾⣻⣿⣿⡿⠿⢿⣿⣿⣿⣶⣶⣭⣙⠷⣦⣀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⡠⠚⠙⠷⢶⣤⡤⠶⢚⣥⣞⣉⡠⠖⠋⠉⠀⢀⡄⢠⣤⡄⠀⠉⠙⠻⣿⣿⣷⣦⡙⢷⣄⠙⢷⡀
⠀⠀⠀⣀⡀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠔⠛⠒⡴⠋⠀⠰⡇⠰⡇⠀⡇⣼⣷⡇⠀⠀⢴⡆⠀⠙⢿⣿⣿⣦⡙⢷⡈⢷⡀
⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⡇⠀⠇⡇⢿⡶⠃⠀⢸⡇⠀⢀⡀⠙⢷⡝⣿⡌⢿⡄
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡴⢳⡄⠃⠀⠃⠀⢸⠇⢸⡇⠀⠀⠘⠀⠀⠟⢹⠀⠈⢿⡌⢿⡌⢷⡀
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢸⠀⣇⢀⡇⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⣴⣏⣴⠀⠈⢿⡘⣿⠘⣇
⣿⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡜⠀⠀⠘⠀⠙⠛⠁⠀⠀⠀⠀⢸⠇⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠸⣇⢹⡇⢹
⠿⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡇⠀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⠀⣿⠈⣇⢸
⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣻⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡇⠀⢈⡽⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠶⠋⠀⡿⠀⡿
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠛⠁⢀⣴⣆⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⡴⣿⠀⠀⢀⠇⠀⡇⢸
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢹⣇⠀⠀⢸⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣧⣿⡄⠀⠈⠀⠸⠀⠄
⠀⠀⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⢻⣧⠀⢻⣆⠀⢸⡁⣿⠀⠀⠲⠶⣶⠂⠀⣠⠆⠀⠀⣠⠛⠁⠀⠀⣿⠀⠀⠀⠀⠁
⠀⠀⢀⡀⢹⣦⣤⣤⣤⣤⣤⣤⡄⠹⣷⡄⠻⣧⡈⠛⠃⠀⠀⠀⢰⠇⠀⢰⣏⡀⠀⠀⠹⣦⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠈⢿⣄⠙⢷⣤⠤⠤⠤⠤⠤⠤⠜⢿⣦⣙⢿⣦⡀⠀⠀⢀⡟⠀⠀⣾⠙⣧⠀⢠⣴⠏⠀⢀⡤⠚⣲⠎⡀
⠀⠀⠀⠀⠙⢦⡀⠙⢷⣶⠶⠶⠛⠛⠻⣶⡟⠻⣷⣽⣿⣷⣤⣀⡀⠀⠀⠘⠛⠃⠀⢀⣀⣤⠖⣋⡴⢞⣡⠊
⠀⠀⠀⠀⠀⠀⠙⢷⡄⠙⠻⠃⠀⠀⠀⠈⢻⣄⣀⡙⠛⠿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⠿⠞⢛⣩⠖⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣿⣶⣶⠶⠶⠆⠀⠀⠉⠉⠉⠉⢉⣉⣡⡤⠶⠚⠉⠛⠛⠻⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠖⠋⠁⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠣⣉⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣉⡡⠃⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠖⠶⠲⡆⠀⢠⠖⠶⠖⢦⠀⠀⡄⠲⠖⢦⠀⠀⡴⠲⠒⠲⡄⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠃⠀⠘⠓⠒⠒⠛⠀⠀⠓⠒⠒⠚⠀⠀⠛⠒⠒⠚⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣠⠤⠤⢤⡄⠀⢠⠤⠤⠤⡄⠀⢀⡤⠤⠤⣄⠀⠀⡤⠤⠤⢤⠀⠀⣠⠤⠤⢤⡀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣇⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠓⠖⠶⠒⠃⠀⠘⠶⠶⠶⠃⠀⠘⠒⠶⠖⠞⠀⠀⠳⠶⠶⠞⠀⠀⠓⠲⠖⠖⠃⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠷⠤⠤⠤⠇⠀⠸⠤⠤⠤⠇⠀⠸⠤⠤⠤⠼⠀⠀⠧⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⡀⠔⢮⢴⠠⡰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢽⣹⠞⡁⢸⡘⣿⠚⡵⢟⣿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣟⣓⣇⠈⠸⣦⠀⢰⡄⢤⠈⠀⣍⠉⠭⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡀⠄⠀⡉⠖⠁⠀⠘⠈⠈⢁⠘⣆⠁⠸⣀⣋⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠄⡐⡘⠂⠉⠀⣘⠈⠀⠃⠈⠀⠉⠀⠀⠀⡐⡂⠀⣈⢇⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠌⠠⢀⠆⡥⠑⠀⠁⢲⣽⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠘⣫⠀⡀⠀⠀⠁⠸⠀⠊⠈⠈⠈⣑⢠⡔⣆⠘⣼⢻⣶⡂⠗⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠷⠌⠉⢀⡠⣤⠠⣴⠞⣰⢿⢶⡾⠲⠎⠀⢩⠟⠝⣧⡑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣞⢂⡇⠧⢪⣀⣾⣾⣼⣪⣷⠟⣦⠀⢩⠆⠲⣆⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⠆⡳⣯⡯⣞⢿⣻⠽⡖⠟⢏⠱⢧⣘⣧⣼⡛⠓⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠀⠀⠐⠳⠛⣯⢎⠰⢇⢃⡴⡴⣚⣷⣯⠌⡁⠀⠀⠀⠀⠑⠪⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⡍⢁⣏⣈⢡⣞⣿⡵⣶⣮⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣭⢭⣽⠙⣝⠌⡶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⢏⢴⡥⡔⢦⣴⣚⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣚⣻⡯⠄⠈⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⣉⣁⣤⠴⠖⢋⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣉⣁⣤⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⣏⡛⠻⠿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠛⠃⠀⠀⠀⠈⠉⠛⠿⢿⣿⣷⣶⣤⣄⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠘⡶⠦⠀⣀⠀⠀⠀⠉⠛⠻⣿⣿⣟⣷⣷⣤⣀
⠀⠀⠀⠀⠀⢀⠏⢀⣤⣀⠀⠀⠀⠐⠒⠃⢠⠾⣽⠀⣰⣆⠀⡀⠀⢹⡦⠈⠉⠛⠻⢿⣿⣆
⠀⠀⠀⠀⠀⡞⠀⡜⠙⢺⠯⣷⣲⠤⣄⡀⠀⠀⠈⠀⠃⠘⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠏⢸
⠀⠀⠀⠀⢰⠃⢰⠉⢑⠗⠤⣠⡏⠙⠳⣼⣉⣖⢢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠈⣇
⠀⠀⠀⢀⡏⢀⣏⡉⡾⠤⢄⣜⠈⠙⢲⠧⢄⣈⠟⠚⠦⣽⣓⡶⢤⣀⡀⠀⠀⢰⣿⠁⠀⠀⢹
⠀⠀⠀⡼⠀⡼⣄⣸⠓⠲⣼⣉⠉⢒⠧⢤⣀⡎⠉⠒⣴⣅⡈⠉⡞⠺⢭⡇⢀⣿⠏⠀⠀⠀⠈⡇
⠀⠀⢰⠃⢰⠧⢄⣏⠙⢲⠧⢄⣉⡟⠢⢤⣜⠉⠑⢲⢧⣀⠈⡝⠓⠤⡼⠀⣾⡟⠀⠀⠀⠀⠀⢹
⠀⠀⣞⠀⠘⠒⠼⣁⣉⠟⠢⢄⣼⠉⠒⡾⢥⣉⢓⠾⢤⣀⣹⠓⠦⣤⠃⢸⣿⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠻⢽⣓⠦⢄⣀⠉⠑⠲⠤⣇⠉⢲⠧⢤⣀⡝⠓⠢⢤⣏⠑⢲⡎⢠⣿⣏⣀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠉⠓⠺⢽⣒⠦⣤⣀⠉⠛⠢⢄⣸⠉⠒⢢⠯⣀⠉⡞⠀⣾⣿⡀⠈⠉⠒⠦⢄⣀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠯⢽⣳⣦⣤⣀⠉⠓⠺⠤⣀⣹⠁⣼⣿⡇⠈⠑⠒⢤⣀⣠⠤⠭⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⢽⣓⠦⣤⣀⠀⢠⣿⡿⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⢽⣛⣻⣧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣄⠀⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠁⠉⠙⠓⢦⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢤⡦⠟⡶⠄⠙⠒⠤⣆⡀⣰⠀⠀⠀⣠
⠸⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⣳⣌⠀⠉⠘⡀⠀⠀⠉⠉⠹⠔⡦⢊⡏
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠖⢦⠀⠀⠀⠀⠀⠀⠀⠴⠶⢄⠈⠁⠀⢀⣄⣤⣾⣄⣄⣀⠀⠀⡰⡇
⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡴⠃⡜⠁⢠⠎⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⡏⡷⠤⠚⠉⠀⠀⠈⠉⣻⣦⡄⠸⢤⡄
⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠊⠉⠀⠀⠣⠔⠁⠀⡌⠀⡜⢀⠊⡇⠠⡲⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣉⡁⠀⠧⣀
⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣠⡶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠈⠊⠀⠀⠀⡰⠋⠀⠀⠤⣀⠀⠀⠀⠀⢀⣿⡟⠛⠂⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡞⠰⢦⣀⡈⡙⠀⠀⢀⣸⠋⠁⢰⡚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠸⢮⣁⡬⠃⠀⡴⣋⣉⣠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣩⡀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣡⣦⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠈⠀⠠⡀⠀⠀⡠⠤⠴⠃⠁⠈⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⢄⠀⠀⠐⡀⠀⡀⠘⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣄⠀⠀⠉⠉⠓⠃⠀⣧⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠉⠉⠓⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⣀⠴⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠉⠁⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⢾⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⡉⠁⠀⠀⡼⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠹⢄⣀⣠⠾⡁⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⢠⠜⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣐⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠈⠉⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⡀⠄⠒⠀⠀⠀⠀⠠⣤⠀⠠⠄⢦⡄⠀⠈⠀⠃⠠⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⠄⠈⠀⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⢀⣀⣋⣀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⢤⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠄⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣦
⠀⠀⠀⢠⣾⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⡀
⠀⠀⢠⣿⠏⠀⠦⠀⢠⣤⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠆⠀⢻⣷⡀
⠀⢀⣿⡏⠀⡠⠀⠀⢸⡏⠀⣸⡃⠀⢸⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠀⢻⣷
⠀⣼⡟⠀⢠⠀⠀⠀⢸⣇⠀⢹⡇⠀⢸⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡀⠈⣿⣇
⢰⣿⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠛⠀⠀⠀⢀⠀⢸⣿
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⣠⠂⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃⠀⢿
⣿⡇⠀⠆⠀⠙⠶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣬⡀⠀⠀⠶⠀⢹
⣿⡇⠀⢀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣕⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⡄⠀⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠐⣿⣿⣿⡟⠁⢀⠀⠀⣿
⠸⣿⡄⠀⠀⠀⠀⠀⢀⡀⢀⡐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣾⣿⣿⣿⣦⣈⠀⢸⣿
⠀⢻⣷⠀⠙⡀⠀⠀⠸⣆⣠⠃⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏
⠀⠈⢿⣇⠀⠐⠀⠀⣴⠉⠙⣷⣰⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣦⠀⠖⠀⠈⠓⠚⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠈⢿⣧⡀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠲⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠃⢀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⢠⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠁⢾⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⡇⠈⢻⡆⠀⠀⠀⠀⠀⠘⠂⠈⠁⢠⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⢀⠀⠤⠐⠈⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣧⠁⠉⠀⠘⠂⠀⠉⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⣼⣿⣿⣿⢿⣿⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⣷⣶⣶⣶⣶⣾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣏⣻⣿⣦⣈⣏⣙⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣑⣾⣿⣛⣿⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⢁⣤⣶⠾⠟⠛⠛⠛⠛⠻⠿⠶⣶⣤⣀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿⠛⠿⢶⣦⣄⠀⠀⡟⠉⢹⡇⠀⢀⣤⡶⠿⠛⠛⠛⠻⢶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣀⣠⡾⠃⠀⠀⠀⠈⠹⣷⠀⣧⠀⢸⡇⢠⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⡂⣻⣀⣸⡇⣺⡏⠉⠛⠛⠷⢶⣤⣀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠉⠛⢛⣷⣦⣄⡀⠀⠀⠈⠙⠷⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⣁⣤⣴⠶⠾⠛⠛⠛⠛⠛⠛⠻⢷⣤⡀⠀⠀⠈⢻⣆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣴⠶⠿⠛⣋⣥⡶⠟⢋⣁⣤⣴⡶⠶⠶⠶⢶⣦⣤⣀⠀⠙⠻⣷⣄⠀⠀⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⣠⡾⠛⢁⣤⠾⠛⠉⠀⠀⣀⣄⡀⠀⠀⠀⠉⠛⠿⣶⣄⠈⠻⣷⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣰⡿⠋⢀⣴⠟⠁⠀⠀⠀⠀⠸⣧⣤⠗⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠟⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠏⢀⣾⠋⠀⠀⢀⣴⠞⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣾⡏⢀⣾⠁⠀⠀⠀⠻⣤⡀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢰⡿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡘⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⡇⠀⣿⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠀⣹⡆⠀⠀⠀⠀⠀⠐⢯⣨⠇⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠸⣿⠀⠿⣬⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣏⠻⣦⡀⠀⠀⠀⠀⠀⠁⠀⢀⣿⠇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢸⣇⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠳⣄⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⢿⡀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣙⣷⠀⠀⠀⢀⣼⠏⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢷⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠋⠁⠀⠀⣠⡾⠃⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠙⠄⠈⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠺⣭⡿⠀⠀⠀⠀⢀⣠⡾⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣄⣀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⠾⠟⠉⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣀⣠⣤⣶⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠿⠷⠶⢶⣶⣦⣤⣤⣴⣶⣶⠶⠶⠿⠛⠻⣍⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⠋⠀⠀⠀⢉⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⣷⠦⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠁⣠⠞⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢏⣴⣤⣾⣿⣷⡶⣌⢹⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣷⡀⠀⠀⣴⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢿⣿⣿⣿⣿⣿⠟⠻⢦⡀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⢹⢿⣿⣿⠁⠀⠀⠀⡇⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣇⡙⢿⣷⣄⠀⠀⡷⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣙⢻⣿⣿⡄⢻⣿⣶⣤⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⠈⡎⢿⣿⣷⠘⣿⣿⠿⠿⣷⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⡀⠀⢣⠸⣿⣿⡆⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣌⣦⡙⢿⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣿⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣤⣀⠉⠉⠉⠛⠛⠿⣷⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⣻⡿⠿⠛⠻⣶⣥⡀⠀⠀⠉⠻⣷⣄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠉⠀⠉⠁⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣛⣛⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣒⡒⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠖⢾⢳⠟⠓⠚⠓⠛⠒⠒⠒⠒⠒⠒⢶⡏⣴⠿⠿⠿⠦⠽⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣛⣛⣛⣛⣛⣛⣛⣛⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⠤⣤⣤⣤⣄⣀⣠⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡧⠤⠴⠶⠶⠤⠬⠭⠽⠿⠿⠿⠒⠶⠶⠾⠿⢿⣿⣛⣛⣓⣲⣶⣤⣤⣠⠞⣿⡿⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣿⣇⡇⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣇⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⢻⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡏⡿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢱⢇⡟
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⡏⣸⠁
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⣽⢁⡏
⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⢳⠇⣸⠁
⠀⠀⠀⢠⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⡟⢀⡏
⠀⠀⠀⠀⠈⠉⠉⢛⡷⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡟⢸⠃⡾
⠀⠀⠀⠀⠀⠀⠀⣾⣤⣤⣄⣈⣉⠓⠶⠤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⢁⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠦⠭⣭⣿⣿⣶⣶⣶⣶⣾⡿⠿⠿⠯⢽⣿⣶⣶⣤⣤⣠⠞⠁⢠⡇⣼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⢭⣽⣛⣛⣿⣿⣿⣒⣶⣶⠾⢡⠇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⠿⠿⠴⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠟⡿⠀⠈⠉⠉⠉⠛⠛⠓⠲⠶⠶⠶⠦⢤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡏⠀⢠⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⣧⣾⣿⡷⠶⠤⠭⠭⠭⣍⣉⣙⣒⣒⣒⣒⣒⣒⣒⠲⣶⣤⣤⣿⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠋⢷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣼⠿⣷⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⢿⠀⠀⠀⠀⢀⣤⣶⠿⢻⡿⠀⠀⠀⠉⠀⠹⣷⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣇⠀⠀⠀⠀⠁⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣼⣿⠾⠛⠛⢷⣄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢿⡆⠀⠀⠀⠈⠻⡧⠀⠀⠀⠀⠀⠈⠛⠶⠶⡶
⠀⠀⠀⠀⠀⣿⢹⡟⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⡀⠀⠀⠀⣿⣿⣷⢏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⣤⣤⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⣿⡟⣿⣹⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⣿⣧⢻⡜⣿⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⢹⣼⢹⢱⢇⢧⣷⣬⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠘⣇⣎⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠙⢷⣄⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡄⠀⠀⠀⡼⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⣀⣀⣁⣀⣀⡀⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠈⢧⠀⠀⡤⠊⠉⠁⡄⠈⠉⠙⠛⢔⠄⠈⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⢠⣄⣼⣃⡉⠀⠀⢃⠀⠰⠃⠀⢸⡆⢀⡀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⢀⣴⣿⣿⣿⣿⣶⡱⠀⠫⣯⡶⢪⣷⣾⣿⣿⣏⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠈⠿⣛⣿⣿⣿⣿⣶⣾⣿⣟⣛⢾⣿⣿⣿⣿⣿⢧⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⠈⢊⡿⣫⡵⠞⠛⠉⠹⠛⠻⢿⡻⣧⣽⣿⣝⣛⡘⠀⢀⣀⣤⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣰⢫⣾⠋⢀⡀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣯⠻⣿⡄⠈⠁
⠀⠀⠀⠀⠶⠶⠶⠶⠶⠆⠀⢰⢇⣏⠇⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⢷⢹⣇⢻⡿⠀⠀⠰⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⠀⠀⠀⠈⡖⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠶⠒⠋⠉⠀⢸⡸⡼⡀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢹⠇⡿⢸⣧⠁⠈⠉⠓⠲⠦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢧⢳⡱⡀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢀⡾⣸⢇⣿⠇⠠⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⡈⢳⣵⣌⡢⣀⠀⠀⠀⠀⠀⢀⣠⢞⣴⣫⡿⠃⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⢿⣿⣖⣶⣦⣴⣶⣶⣿⣷⡿⠛⠁⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢉⠉⠛⠛⠋⠉⠁⠘⣳⡀⠈⢆⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⡈⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⠇⠀⠀⠀⠁⠀⠀⠀⢳⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣈⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠙⠻⢷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡘⠀⠀⠀⡰⢻⣆⣌⢛⣦⠀⠘⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣧⣶⣠⣦⣷⣿⣼⣿⣿⣿⠀⢈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣦⠀⢸⡏⠸⡏⠀⢹⣿⣿⡇⠐⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⡇⠀⢸⣿⣿⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣅⣤⣡⣤⣬⣭⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠋⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠷⠒⠒⠛⠛⠛⠛⠿⠦⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠴⠊⠀⠀⠀⠀⣠⠔⠚⡛⠥⠐⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⢐⠵⠁⠀⠀⠀⡀⠐⢉⠤⠂⠁⠐⠂⠀⠀⠀⠀⡶⠀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣀⡀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⣠⡖⠡⠀⠐⣄⢳⡀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠈⢿⣆
⠀⠀⠀⡰⠁⠀⠀⠀⣴⢃⡴⠃⠈⠀⠀⠀⠈⠳⣳⡀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡙⢶⣿⣦⡀⠄⠈⢿⣆
⠀⠀⡴⠀⠀⠀⢀⡾⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠔⠈⠀⠈⢻⣿⣿⣶⣆⠀⢿⣆
⠀⢠⠃⠀⠀⠀⠞⠠⠀⠀⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⣴⠿⠛⠀⠀⢻⣿⣿⣿⣶⠘⣿⡄
⠀⡄⠀⠀⠀⡘⢀⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢻⣧
⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⣿
⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⣿⡇⢿⡇⠐⣿
⠀⠀⠀⠐⣼⡇⠰⠀⠬⠍⠉⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠿⢀⡀⣿⣟⢸⣷⡀⣿
⠀⠀⠀⠘⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⣌⠧⠀⣿
⢠⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⣿⡇⢟⠁⢀⣿
⠈⡤⠀⠀⢹⡿⢣⠀⠀⢀⣄⢖⣫⢱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣄⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣦⡀⠀⢠⣿⡻⠼⠞⠀⣸⡏
⠀⠐⢀⠀⠘⣷⡉⣧⡀⡈⠻⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉⠛⠶⠙⠀⢠⡿⡷⢡⠚⠀⣸⡟
⠀⠀⠀⢂⠀⢻⣷⡹⢿⡇⠀⠋⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⢘⣧⡀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡟⣡⠎⠀⣰⡟⠁
⠀⠀⠀⠀⡣⡈⣿⣿⣌⠢⣾⣦⠀⠀⠀⢀⣤⠏⠗⠀⠀⠘⠃⠈⠛⠉⠀⠀⠀⠈⠀⠻⣄⠀⠀⠠⠄⣠⣾⣯⢏⣴⠁⠀⣰⠟
⠀⠀⠀⠀⠘⢷⠈⢿⣿⣷⣝⠿⣷⣄⠀⠈⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠂⢀⣤⣾⣯⠟⣡⠟⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢂⠉⢿⣿⣿⣮⡝⢷⣦⢄⡀⡀⠀⠀⠀⠘⠂⠰⡆⡀⠀⠠⠀⡀⣠⡴⣞⠿⠟⢋⣴⡟⠉⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⢻⣿⣿⣿⣷⣮⣘⣤⠉⡀⠐⠒⠂⠤⣤⣤⣶⣶⣶⣿⣿⠿⢞⣋⣴⣶⡿⠗⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣍⣉⣁⣀⣠⣤⣤⣶⣶⣿⢿⢛⣉⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠠⣦⠀⢠⡄⠀⣴⠀⠀⣦⠀⢠⡆⠀⢰⡄⠀⣤⠀⠠⣦⠀⠀⠀⠘⢿⣿⣆
⠀⠀⢀⣾⣿⠏⠀⠀⠀⣾⣾⣿⣶⣾⣷⣶⣿⣶⣶⣿⣶⣾⣷⣶⣿⣷⣷⣿⣶⣾⣿⣾⡆⠀⠀⠈⢿⣿⣆
⠀⠀⣼⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⢻⡟⠛⠛⡟⠛⠛⡟⠛⠛⣿⠛⠛⢻⠛⠛⢻⡇⠀⠀⠀⠈⣿⣿⡄
⠀⢰⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣀⣠⣇⣀⣠⣧⣀⣠⣧⣀⣀⣧⣀⣀⣼⣀⣀⣼⡇⠀⠀⠀⠀⢸⣿⣷
⠀⢸⣿⡏⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⡇⠀⠀⣿⠀⠀⢹⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⠖⠒⢺⠖⠒⢺⡗⠒⠲⡗⠒⠒⡷⠒⠒⣿⠒⠒⢾⠒⠒⢾⡇⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣧⠀⠀⠀⠀⠀⣿⣀⣀⣸⣀⣀⣸⣇⣀⣀⣇⣀⣀⣇⣀⣀⣿⣀⣀⣸⣀⣀⣸⡇⠀⠀⠀⠀⢠⣿⣿
⠀⠘⣿⣿⡀⠀⠀⠀⠀⣿⠉⠉⢹⠉⠉⢉⡏⠉⠉⡏⠉⠉⡏⠉⠉⣏⠉⠉⢹⠉⠉⢹⡇⠀⠀⠀⠀⣸⣿⡟
⠀⠀⢻⣿⣧⠀⠀⠀⠀⣿⠤⠤⢼⡦⠤⢼⡧⠤⢤⡧⠤⠤⣧⣤⣤⣿⣤⣤⣾⣤⣤⣼⡇⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⢿⣿⣧⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠈⢻⣿⣷⡀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣾⣆
⠀⠀⠀⠀⡾⢹⡟⠆
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠁⠀⠀⣰⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⠟⠀⠀⢀⡴⠋⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡾⠋⠀⠀⠁⠀⠀⠀⢠⣿⣷⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠘⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢋⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣋⣥⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠦
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢞⣫⣥⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣠⡴⠀⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣬⣭⡷⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢿⣧⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣠⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠛⢁⡀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠀⠀⠀⠀⢠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢈⠛⠦⡄⠀⠀⣿⠀⠀⠀⠀⠀⠖⠋⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡿⠃⠙⣿⣆⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠤⠶⠋⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠴⠞⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣤⠖⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⡴⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢀⠴⠚⡁⠀⠀⠀⠀⠀⣀⣤⠀⡉⠻⢦⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⡆⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣴⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠉⣠⠀⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⢁⡤⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⣠⠀⣿
⠀⠀⠀⠀⠀⢹⠀⠀⠀⢠⣿⠀⠀⠀⠀⠐⠛⢁⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠚⠋⣁⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠐⠛⢁⢀⣿
⠀⠁⠀⠀⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠐⠖⠋⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠰⠚⠁⣾⡇⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠖⠋⣸⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠛⠛⠛⠟⠛⠛⠿⠛⠛⠛⠀⠀⠀⠀⠙⠛⠟⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠛⠛⠛⠛⠛⠟⠻⠟⠛⠛
⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣦⡄
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠠⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢂⠠⠂⢂⠀⠀⠆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠉⠀⢇⠈⠄⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢉⣎⠢⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠂⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠉⠰⠛⡏⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⢁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠠⠤⢧⡀⠀⠲⣄⠘⡀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⣴⠃⠀⠀⠀⠈⠢⢀⠈⠳⠵⣄⣀⠀⠘⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠑⡀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠒⢀⡀⠀⠀⠑⠦⡀⠀⠀⠀⠀⣰⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠔⠈⣿⣿⣿⣿⣿⣷⣤⣴⡿⠁⠀⠢⣀⠀⠈⠢⡀⠀⣰⣿⡟⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠹⠀⠀⡌⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠔⠂⠁⢸⠀⠀⠀⠀⠉⠙⡇⠀⠀⠀⠀⠀⠁⠀⡐⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠐⠁⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⡗⠈⠈⠁⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡋⠁⠐⠒⠒⠒⠚⢁⡀⠀⠔⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡟⠉⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⢼⣁⠀⠙⡙⠋⠙⠉⠉⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⡠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⡤⠤⢀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢀⡲⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠛⠀⠀⠛⠀⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣾⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣷⣾⣿⣿⣾⣭⣍⠛⣦⠰⣧⣤⣼⠇⢾⣁⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⣿⣄⣀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡿⠀⡾⠀⣿⣿⣿⣿⣿⡟⣉⣤⠶⣛⣫⣭⣭⣿⣿⣿⣿⣿⣷⣶⣿⡟⠻⣿⣿⣿⡇⢸⠄⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⡿⢋⡽⠛⣩⣶⣿⠿⠛⠛⢉⣉⠉⠉⠛⠻⠿⣿⣿⣿⣷⣄⠉⠛⠧⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⣤⠞⠁⣰⠋⣠⣾⡿⢋⡅⠀⠀⠀⢸⣿⠀⠀⠀⠀⠖⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠴⠶⠀⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⢸⣿⠁⢠⡏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇⠀⠹⣿⣿⣧⠀⡤⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⢡⣿⠏⢀⡾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣦⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠞⣱⣿⠋⢠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢳⡀⠈⢻⣿⣿⣷⣤⣘⠃⠀⠀⠀⣿⠄⠀⠀⣀⣤⠞⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⣶⣶⠶⣖⣛⣭⣴⡿⢿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠋⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠉⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠒⠋⠉⠉⠉⠀⠀⠀⠉⠉⠑⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⣀⣀⣠⡤⣶⠦⠛⡻⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⣀⣀⠀⠠⡶⠶⠞⠛⠋⠙⢉⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⠀⣒⠛⠚⠋⠑⠻⠉⠁⠀⠀⠀⠀⢠⡤⠀⠀⠀⠂⠳⢄⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠍⠉⠓⠀⠀⡀⠀⠈⠂⢀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⢀⣈⠦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠳⣒⠒⠓⠂⠈⠂⢠⡏⣀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡠⠤⢄⡀⢤⣌⠑⢀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡀⠊⠀⠀⠀⠀⠈⠀⠈⠛⢦⢌⠣⢀⡐⠆⠀⠀⠀⠀⠀⣠⠴⠩⠀⠐⠂⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⣴⠀⣀⠀⣈⠳⡴⢢⡑⢤⡤⠤⡔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣠⢊⠽⠋⠀⠀⠀⠈⠰⡄⠈⠂⠐⣤⠠⡀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠕⠉⠀⠀⠀⠀⠀⠀⠠⢻⠀⠀⠀⡈⠣⣀⢀⣀⣀⠠⠤⠒⠁⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡠⠈⡠⠁⠀⠀⠀⠀⠀⠀⢀⠴⢷⣘⠗⠁⠈⠛⠑⢌⠃⠀⢀⡤⠐⠂⠉⠀⠁⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠀⠀⠀⠀⠀⠀⢀⠔⠓⠚⢲⠏⠀⠀⠀⢠⠀⠀⠣⡠⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⡀⠔⠈⠀⠀⠀⣠⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠬⡛⢤⠤⣤⡀⠠⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠀⠀⠔⠒⠈⠒⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡄⠀⢠⠂⠀⠀⠀⠀⡆⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠔⠀⠰⠁⠀⠀⠀⠀⢐⠁⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⣀⠀⠀⠀⠀⠀⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢸⡇⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⢠⢞⠤⣀⠀⢲⡟⠂⡠⠄⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠐⠁⠧⠀⠇⠀⠻⠀⠘⠁⠐⠲⠏⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⡴⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⡤⠶⠶⠶⢤⡞⠋⠉⣉⣙⣯⡤⠴⠒⢻⠏⢧
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⣹⣿⣠⣤⣤⣶⠾⣿⠛⠋⠉⢁⡿⠀⠀⠰⡿⠀⠈⢧
⠀⠀⠀⢀⣠⠼⠷⢒⣚⣛⣭⠭⠞⣟⠉⠁⠀⣀⡟⠀⠀⠀⠘⠃⠀⠀⢀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⣸⢠⠖⢻⠟⢻⡅⠀⠀⣴⣏⠀⠀⠀⠛⠁⠀⢀⣤⡀⠀⣷⢦⣸⣆⠀⠀⠀⠈⢷⡀
⠀⠀⠀⡏⢸⠀⢸⠀⠀⡇⠀⠀⠉⠁⠀⠀⠴⠶⣿⠋⢸⣯⡿⣄⠹⡆⠉⠛⠀⠀⢀⣀⡈⢳⡀
⠀⠀⢰⡇⢸⡇⠸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⣹⠀⠸⠇⠀⠉⢀⣀⣤⡴⠒⣻⡩⠄⢹⡄⠻⡄
⠀⠀⢸⡇⠸⡇⠀⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⡤⢶⠚⣙⣅⠄⢳⣈⠥⠿⣎⠉⠙⣆⠙⣆
⠀⠀⢸⡇⠀⡇⠀⢸⡀⠀⢹⡀⠀⠀⣀⡤⠶⡖⢋⡉⢧⠒⣊⣧⠴⠚⣏⠁⠙⣆⣀⠼⢶⡊⠉⢧⡈⢧⡀
⠀⠀⢸⡇⠀⣇⠀⠀⣇⠀⠀⢷⠀⠈⢇⠰⠒⣱⣬⠴⠚⢏⠁⠈⢧⣀⠼⢶⡊⠉⠣⡀⢀⣱⣤⠒⠛⣄⠙⢦
⠀⠀⢸⡇⠀⢻⠀⠀⢹⡄⠀⠈⢧⠀⠘⣞⠉⠁⠹⡄⣀⡬⢶⠚⠉⠳⡀⠀⣱⣤⠖⠛⢏⠀⠈⢣⣀⡬⠷⡀⠳⣄
⠀⠀⠘⡇⠀⢸⡄⠀⠀⢷⠀⠀⠈⢧⠀⠘⢦⠴⠚⠹⡅⠀⠈⢣⣠⠤⠛⢏⠁⠈⢢⣀⡨⠷⣞⠉⠙⢦⡀⢙⣦⡈⠳⣄
⠀⠀⠀⡇⠀⠀⣇⠀⠀⠈⣧⠀⠀⠈⢧⡀⠈⢧⡀⣀⡼⢶⠚⠉⠙⣄⠀⣈⡷⡔⠋⠙⢦⡀⠈⣳⣤⠔⠛⢯⡀⠙⣦⠈⠑⢄
⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠘⣆⠀⠀⠈⢳⡀⠀⠻⡄⠀⠈⢣⣠⠴⠚⠻⡁⠀⠈⢲⣄⠤⠛⢯⡁⠈⢲⣤⠴⠛⠋⠁⠀⢀⣠⠗
⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠘⣆⠀⠀⠀⠹⣄⠀⠙⢶⠊⠉⠙⢦⡀⣀⡼⠶⡚⠉⠈⠳⣄⡠⠽⠚⠉⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠘⡇⠀⠀⢳⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⢦⡀⠀⠳⣄⡤⠖⠛⢧⡀⠀⣈⡷⠖⠋⠁⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⣷⠀⠀⠘⣇⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⣦⠀⠈⠑⢦⣠⠴⠚⠉⠁⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⢸⡄⠀⠀⢹⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⣧⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣹⢦⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣣⠀⠀⠀⣀⡤⠴⠚⠳⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠳⠒⠛⠉⠉⠳⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠋⠻⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠻⠛⠃⠈⠛⠟⠁⣽⣿⣿⣿⡇⠀⣠⣶⣤⠀⠨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠁⠀⢀⣀⣀⡀⠀⠈⠻⠿⢿⣿⡆⠀⠻⣿⡿⠀⠠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣴⣾⣿⣷⣆⣠⡀⣀⣰⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣷⡄⠀⠀⠉⠉⠀⠀⢠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣶⣶⡄⢠⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠓⢶⢛⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⣛⣿⣿⡛⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢤⣤⣬⡶⠶⠎⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡃⠀⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⢠⣤⡘⣦⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠿⠙⠁⠀⢘⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠓⠀⠘⡻⣼⣯⡏⣿⢿⡄
⢰⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣍⢷⣤⠀⠀⠀⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠀⠀⢙⡿⣋⣴⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⢧⡀⢻⢆⠀⡑⠀⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡠⠂⡀⠀⠈⠟⣠⠏⠐⠤⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⢸⣿⣿⣿⡟⠀⠀⠀⣠⠞⠁⢀⠀⠀⠒⠒⠦⣷⠈⣬⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢀⣎⡔⠁⠀⠀⠀⠈⠳⣄⠀⠀⢸⣿⣿⣿⣿
⢸⣿⣿⣿⠇⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠂⠢⠀⠀⢸⠟⠀⠉⠈⢲⡀⠀⠀⠈⠳⡀⢸⣿⣿⣿⣿
⢸⣿⣿⡿⠀⣰⠏⠀⠀⠀⠐⠶⠾⠿⠿⢿⣂⣀⣈⠹⢝⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠴⠀⢇⠀⡼⠰⠒⠚⣋⠄⢣⠀⠀⠀⠀⠘⢆⢻⣿⣿⣿
⢸⣿⣿⠁⡴⠃⠀⠀⢀⡢⣀⠀⠀⠒⠛⠛⠛⠛⠂⠀⠀⣷⣡⣱⣢⠰⠒⠆⠀⠀⠀⠀⠀⠀⠀⢀⡘⠈⠒⢄⢈⢆⡷⠶⠛⠉⠀⢀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⢸⣿⣿⠘⠁⠤⠶⠞⣻⣿⣿⣟⣶⢤⣀⠀⠀⢀⣤⡤⠴⠀⠉⠉⠁⣂⢀⣄⠈⠁⠀⠀⠀⠀⠀⡇⢢⣀⢰⠈⢧⡼⠁⠀⠀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠉⠁⣶⠿⠋⢁⠞⠉⣿⢶⣄⡀⠀⠀⠀⠀⠀⠀⢿⢀⡇⢰⠀⠀⠀⠀⠀⠀⢳⣳⠀⡎⢠⠎⠀⠀⠀⣰⠛⠲⠶⣖⣒⣠⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠸⣿⠎⠀⠉⠓⠢⢤⣄⣀⣠⠟⢸⣷⠸⠀⠀⠀⠀⠀⠀⢸⠿⠀⠃⢸⠀⠀⢀⡴⠋⠀⠀⠀⢷⠀⠀⢔⡀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⣀⣀⣀⣤⣤⣀⣄⡭⠖⢁⣤⠞⠁⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⣼⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠉⠉⠉⠉⠉⠉⢀⣠⣠⢞⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠺⣦⣝⢲⠦⢤⣤⣤⣤⣼⣿⣿
⢸⣿⣿⠀⢀⣀⣤⣴⡾⠛⠛⣫⢾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣻⠿⣆⣟⠛⢷⣍⠛⣶⠶⠄⢸⣿⣿
⢸⣿⣿⡀⢛⣛⣉⣤⣴⠂⠘⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⢿⣦⣍⣶⣤⣽⣦⣼⣿⣿
⢸⣿⣿⣿⣿⠿⣻⠿⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣽⣿⣿⡏⠙⣧⠀⣿⡇⣼⡿⠛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣷⣶⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠡⠾⠟⢹⡿⠟⠀⠀⠈⠀⠛⠁⠈⠀⠀⠈⠛⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⣤⡴⠶⠔⠿⠃⡀⢠⢠⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⡀⠀⢠⡄⠀⢦⣠⣆⠀⠀⠰⠁⠀⠀⠶⠋⢁⡀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠸⠟⡃⠄⢰⣶⠿⢈⣉⣭⣤⣤⣦⣴⣤⣿⣧⣀⡺⠻⣦⣤⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡈⠁⢀⣠⣴⣾⠛⠛⣻⣿⣿⣿⣿⠛⠛⠛⣻⣿⣿⣦⣄⡀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠶⠀⠈⣁⣿⣿⣿⣿⣧⣄⡨⣽⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⡦⢀⣶⣿⣿⣿⣿⣿⣹⣿⡇⠋⠈⠛⠛⠋⠙⠛⠻⣿⣿⣼⣿⣿⣿⣿⣿⣷⣠⣬⠁⠀⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣿⣿⣿⣿⣿⣿⡄⢘⡁⠀⠀⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠀⠀⠺⢦⣾⣿⣿⣿⣿⣿⣿⣿⢸⣟⠠⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠙⢿⣿⣿⣿⣿⡿⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡻⠇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⡄⠀⠙⢿⠿⠉⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡎⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡹⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣙⣿⣿⠀⠀⠈⠀⠀⢠⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢠⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣷⠀⠀⠀⠠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣟⣈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢋⡈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠛⢶⠄⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣴⣦⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⢋⡄⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠐⢤⡄⠈⢻⣿⣿⣿⣿⣿⢿⣿⣧⣀⣴⣿⣿⡿⢿⣦⣠⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋⢘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠙⠿⣿⣿⣿⣾⠟⠿⢧⣶⣿⣿⣷⠒⠿⠿⢿⣿⣾⣿⣿⣿⡿⠋⣄⡀⢊⡙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⢴⣌⣻⢿⣿⣤⣈⣿⣿⣿⣿⣿⣭⣥⣤⣿⣿⣿⡿⠟⠉⠠⣶⡈⡁⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢨⡿⠻⢂⣤⠉⡙⢻⢻⠿⠿⡿⡿⠟⠛⣛⠉⢡⡄⢀⠛⠀⠂⢀⣄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⢰⣶⡄⠻⠆⠛⢃⣈⡁⠀⠀⣿⣿⠶⣿⠁⠈⠁⣿⠁⠛⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠀⢱⡜⢂⣀⣀⣠⣄⠱⣆⠀⣄⠀⣰⠃⠀⡙⠁⠾⠂⠠⠚⠛⠋⢰⣿⠀⠁⢴⡷⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⢀⣡⣼⣿⣿⣿⡏⣩⣦⣦⣶⣿⣿⡟⠀⠢⠉⠀⠀⠀⠀⠀⡀⠀⠀⠁⡄

⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠶⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⣠⣾⠟⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠙⠻⢷⣦⡀
⠀⢀⣾⠟⠁⠀⢀⣴⠞⠉⠀⠀⣤⡤⠴⠶⢿⣿⣿⣶⣤⡀⠀⣤⣤⣶⣶⣆⠀⠀⠀⠀⣀⣤⡶⠚⠋⠁⠒⠠⣴⡆⠀⠉⠓⢦⣀⠀⠀⠙⢿⣆
⠀⢸⠇⠀⠀⣰⠟⠁⠀⠀⠀⡔⠉⠀⠀⢀⣼⣿⣿⣿⡿⠛⠀⠙⠉⣿⠉⠀⠀⠀⠀⠘⠻⣿⣷⣦⣄⣀⠀⠀⠀⠱⣄⠀⠀⠀⠙⣧⡀⠀⠈⢿
⠀⠸⠀⠀⢰⡟⠀⠀⠀⠀⡌⠀⠀⢀⣴⣿⣿⣿⠟⠉⠀⣀⡠⠤⠴⠛⠒⠒⠶⠦⢤⣄⣀⠈⠙⠿⣿⣿⣿⣦⣄⣀⣿⣧⠀⠀⠀⠘⣷⠀⠀⢸
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢰⣇⢀⣴⣿⣿⠿⣫⣤⡠⠖⣋⣥⣶⣾⡿⠿⠿⠿⠿⣿⣿⣶⣯⣿⣦⣄⣬⡙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⣿⣿⡿⠋⠁⠀⡼⢋⣴⣾⠿⠛⠉⠀⠀⠀⣾⡆⠀⢀⠀⢉⠛⠿⣿⣿⣿⣄⠀⠈⠛⢿⣿⡿⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢠⠎⣴⣿⠟⠁⠀⠹⠂⠀⠀⠀⠉⠁⢀⣿⠀⠚⠁⠀⠈⠻⣿⣿⣷⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⡿⠃⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣾⣿⠁⠈⠛⠀⠀⠀⠙⢦⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⠉⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠀⢸⣼⣿⡇⣴⣶⡆⠀⠀⠀⠀⠀⠀⠈⣶⣿⡄⠀⠀⠀⠀⠀⠀⠸⢿⠷⠀⣿⣿⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⠀⢸⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⡄⠘⣾⣿⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⢰⣿⣿⠁⢠⠇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⠀⠹⣿⣿⣧⡀⠋⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⡡⢣⣿⣿⠇⠠⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⢀⣄⠀⠀⠀⢋⡀⠀⠀⠀⢠⡄⠀⢀⠔⣡⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⡈⠁⠀⠀⠀⢸⣿⠀⠀⠀⢀⠡⢂⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣭⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠀⠘⠛⠁

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠿⠷⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠦⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣆
⠀⠀⠀⣸⡟⠀⣀⡀⠀⠀⢀⣴⠟⠁⣿⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⣼⣏⣴⣾⠏⠀⠀⢀⡾⠃⠀⠀⠸⣇⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣿⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠰⠿⠿⠋⢹⡟⠀⠀⢀⡿⠁⠀⠀⠀⠀⠹⣦⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣼⡇⣀⠀⣾⠃⠀⠀⠀⣤⠾⠷⠙⢷⣄⠀⠘⢷⣄⠐⠛⢿⣷⡀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⣿⡿⢿⣷⡏⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠻⠶⣦⣽⣿⣶⣾⡟⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠻⠇⢸⣿⡇⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⢻⣷⠟⢻⣆⣰⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠈⠻⣦⣀⠀⠀⠀⠀⠀⢤⠤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠙⠛⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⢿⡁⠀⠀⠀⣀⣀⣤⣶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢹⡟⠳⠶⠶⠾⠛⠋⠁⠀⠀⠘⠷⣶⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠘⠷⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠁⠀⠈⠉⠉⣹⡟⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢀⣼⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⣤⣤⣾⡃⠀⠀⢀⣿⠛⠷⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠉⠉⠻⣷⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣻⣶⣶⣶⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⢀⠤⡀
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆
⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠠⡆⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠔⠀⠈⠣⠀⢀⣀⣀⣀⡀⠤⢷⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡄⠀⠀⠈⠁⢀⠀⠀⢀⠄⠙⡂
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠀⠀⠀⠀⢀⣦⠶⠒⠃
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠐⢾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢦⡠⠖⠤⠀⠢⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⢠⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⣄⠰⠀⠀⠐⠒⠄⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠁⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢐⠀⠀⠸⠁⠨⠑⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢸⠄⠒⠶⠂⢖⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠈⠀⢈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⠤⠠⠤⠤⠤⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠐⠀⠀⣀⢀⣀⡀
⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠄⠘⡇
⠀⠀⠀⡟⣆⠀⠁⠒⠒⠒⠂⠐⠂⠤⠄⢀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠄⠀⢀⣀⣀⠀⠀⠀⣀⠠⠐⠊⠁⠀⠀⣀⣀⣀⣠⠃
⠀⠀⠀⢣⣈⠓⠠⠤⠤⠤⠄⠤⠄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠁⠀⠀⢀⠎
⠀⠀⠀⠈⢦⡣⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠉⠲⣕⠄⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠜
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠾⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠟⠛⠉⠉⠀⠀⣤⣤⣤⣬⡄
⠀⠀⠀⠀⠀⠀⢠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠐⡿⠛⢻⣧⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣀⣀⣀⡀⠀⠀⢸⡇⠀⢸⡿⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⢀⣠⡾⠁⠀⠀⢸⡟⠛⢻⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡟⠛⢻⡇⠀⠀⠀⠙⠻⠋⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠐⠶⣿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣤⡤⢤⣤⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⠶⣿⠀⠀⠀⢸⠇⠀⣸⠀⠀⠀⠀⡏⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡀⢀⣿⠀⠀⠀⠀⣧⣀⣠⣧⠀⠀⠀⢸⣧⣀⣸⡇⠀⠀⢸⣇⣀⣀⡇⠀⠀⠀⠀⣿⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠈⠛⠛⠛⣿⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⢻⡿⠛⠛⠛⠛⠛⠛⠛⣿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⠛⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⡄⣤⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠋⡼⠁⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⢰⠃⣀⡀⢸⠃⠀⣠⠴⠶⠋⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠎⠳⣄⠀⠀⠀⠁⠀⣴⢤⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⣴⣦⠀⠀⠙⠋⠁⠙⠋⣤⣾⠵⢦⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣠⡼⠗⠀⠀⠀⠀⠈⠉⠀⠙⣤⡇⠀⠀⠀⢠⢦⣴⢦⡀⠀⠀⠙⠃⠉⠁⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⣠⠔⠚⠉⠛⠲⣌⠁⢀⡴⠖⠚⠀⠀⠀⣉⣙⣦⠀⢀⡴⠛⠉⠉⠉⠙⢦⡀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⣷⠀⠛⠋⢳⠀⠀⢰⡇⠈⠁⠀⢾⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⠧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠼⠃⠀⠀⢾⠀⠀⠀⢀⣠⡤⠖⠋⠀⠀⣀⣸⠀⠀⣸⣇⣀⣀⣀⡈⢙⡶⠠⣤⣀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠛⢯⠀⠱⣶⠚⠋⠉⠹⣭⠄⣴⣾⣅⣀⠼⠉⢩⠿⠁⢰⣇⡈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡇⢀⣠⠆⠀⣀⣀⡀⠀⠀⠀⠀⣀⢀⣀⣀⡳⠄⠨⠂⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠉⣁⠤⠒⠉⠀⠀⠉⠀⠒⠒⠚⠁⠀⠀⠀⠈⠉⠒⠀⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡠⠊⠁⠀⢠⡀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠜⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⡄⠀⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠎⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⡄⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⠇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠔⠻⣙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠲⢄⠀⠀⠀⡀⠀⠀⡜⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠈⢢⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠥⠀⠀⠋⢀⠞⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠦⣀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠈⠑⠢⣤⣀⡀⠀⠀⠀⡶⠀⠀⠀⠀⢀⣉⠤⠒⠁⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣤⣀⡤⣄⣀⠈⠉⠙⠓⠒⠒⢒⣒⣈⣭⣤⣄⢀⡀⢴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡞⠀⠀⠀⠀⠉⠉⠛⠓⠚⠛⠋⠉⠉⠀⠀⠀⠈⠉⢳⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⢀⡀⠤⠤⢠⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⡠⠀⢄⠸⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⣰⠃⠀⠀⠀⠀⠰⡅⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠉⠀⠁⠸⠀⠀⠀⢠⠄⠀⠢⡀⠀⠀⢧⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⡶⠒⢲⠶⡶⠒⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⡄⠀⠠⠁⠀⠀⠸⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠟⠛⠛⠛⠒⠛⠚⠟⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠗⠶⠶⠶⠴⠴⣿⡩⢽⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣽⠀⠀⠀⠠⠤⠄⠀⢰⠀⠀⠀⢈⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⠀⠀⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣏⢼⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠒⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣀⠀⢀⡀⣀⣀⠠⠔⠊⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⡏⠈⠈⠉⡉⡉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣩⣩⡩⡍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣓⣓⣂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡼⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣰⡻⢤⣀⣀⣀⢀⣀⣀⣀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢹⡇
⠀⠀⢀⣀⣀⣀⣤⣤⣼⣧⣿⣯⣼⣯⣥⣴⣮⣽⣯⣽⣫⣋⣝⣩⣉⣙⣽⣫⣪⣗⣡⣠⣟⣕⣽⣟⢙⢁⡽⡭⣫⠿⡿⣩⠞⠕⣽⠁
⠀⠀⠸⣆⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠂
⠀⠀⠀⠈⠉⠉⠉⠉⠙⠚⠛⠛⠓⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠚⠚⠓⠚⠛⠛⠒⠛⠛⠓⠛⠛⠓⠚⠒⠒⠒⠲⠶⠶⠴⠶⠴⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⣠⡤⣄⠀⠀⠀⢀⣤⠤⣄⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣷⣶⠶⠿⠶⠶⣿⠶⠶⠾⠷⠶⠶⢾⠿⠶⠾⠷⠶⠶⣿⠷⠶⢾⣷⣶⣶⣿⣥⣤⣽⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⢷⠄⠀⠀⠀⠀⠹⠦⠀⠀⠀⠀⠀⠘⢷⠄⠀⠀⠀⠀⠙⢶⠀⠀⠀⠀⠀⠳⣤⠀⠀⠉⢹⡿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⣄⢠⠤⠤⠤⡄⡤⠤⠤⢤⢀⠤⠤⠤⢄⡠⠤⠤⠤⣀⠤⠤⠤⡄⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡿⢤⠀⠀⢀⡇⣇⠀⠀⢸⠠⡀⠀⠀⡸⢇⠀⠀⢀⢿⡀⠀⠀⡧⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡎⠉⠀⠐⣴⠉⠉⠉⡇⠘⠉⠉⠙⡆⡎⠉⠉⢹⢠⠋⠉⠉⢱⡖⠉⠉⠙⣶⠋⠉⠉⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠧⠤⠤⠤⠛⠤⠤⠤⠇⠰⠤⠤⠤⠃⠣⠤⠤⠼⠀⠦⠤⠤⠜⠣⠤⠤⠤⠛⠤⠤⠤⠇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡏⠀⠀⠈⣾⠉⠉⠀⡇⠈⠉⠉⠈⠃⡁⠈⠉⢹⢰⠁⠉⠉⢱⡏⠉⠉⠈⣾⠉⠉⠉⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡵⠶⠶⣖⡹⠶⠶⢶⢃⠴⠶⠶⡔⢁⠧⠴⠶⡞⣨⠦⠶⢶⠊⡱⠶⠶⡔⡹⠦⠶⢴⠃⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⡀⠀⠀⣿⡇⠀⠀⢸⢸⠀⠀⠀⣸⢠⠀⠀⠀⡇⣇⠀⠀⢰⢸⠀⠀⠀⡯⡇⠀⠀⠸⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢠⠊⠙⠋⡋⡔⠛⠛⢻⢥⠒⠛⠛⢫⡔⠒⠛⠛⡯⠔⠒⠛⢳⠁⠀⠉⠉⠉⠀⠉⠉⠉⠁⠀⣼⢹⣿⣿
⠀⠀⠀⠀⢠⣼⠏⠀⠀⠳⠤⠤⠤⠃⠧⠤⠄⠼⠸⠤⠀⢀⠜⠅⠀⣀⡠⠇⠄⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣿⣿
⠀⠀⠀⠀⠈⠉⠛⠛⢳⡶⠲⠶⠶⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣭⣭⣭⣭⣭⣭⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣆⣳⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠏⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⢠⣂⣀⣤⣿⣻⣍⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠁⠀⠀⠐⣦⢿⣿⣶⡦⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃⠀⠀⢱⣯⡵⣻⡏⢻⣿⡚⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠳⡆⠀⠈⠌⡼⠯⢶⣒⡚⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢠⣦⣬⢿⣣⣨⠛⠃
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⡀⠐⠐⠌⠉⠀⠀⠀⣉⡀⠀⠀⠀⢀⣴⣿⡳⡀⠀⠀⣾⣻⣷⠀⠀⡠⢚⣩⣿⡿⣛⠳⣄
⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠙⢬⠋⠀⠀⠀⢘⣓⡒⠀⠀⠀⢠⠞⠁⠛⠃⠀⠀⠀⢹⣿⡟⠀⠀⡉⢭⡤⢁⣿⣦⡉⠁
⠀⠀⠀⠀⠠⠁⠂⠀⢀⠹⣮⣗⣲⠤⠴⠂⠀⠀⠀⠀⠀⢠⠏⢀⢀⡀⣀⣀⠀⠀⠀⠙⠁⠀⠀⠉⠛⢣⣾⡶⢵⡛
⠀⠀⠀⠀⠀⢤⣀⠀⠀⠈⣾⡍⠁⠀⠀⠀⠀⠉⠀⠀⣠⢏⠔⠣⠺⣿⠂⠸⡒⣠⠀⠀⠀⠀⠀⢠⠀⣡⢛⡼⢦⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⡇⠒⠀⠐⢦⣀⡀⠀⠀⢐⡦⡦⠖⡇⡇⣻⡥⣣⢦⡮⢿⠄⠀⠀⢀⣿⣛⠽⣻⡾⣖⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠐⣲⣾⣄⠐⠲⠤⠤⠬⠍⠀⠀⣸⣷⡭⡾⡏⣧⣮⡶⣽⡻⣶⣿⣄⡄⠀⠀⠈⢀⠊⣿⣿⣮⣳⡄
⠀⠀⠀⠀⠀⠐⠀⠀⢻⡏⠀⠀⠀⠠⠀⠂⠀⢀⢠⣿⢺⢰⣻⡞⣕⡾⡻⣵⢶⢪⡟⡃⣦⠀⠀⠀⣆⡼⠹⡆⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠪⢕⠄⡀⢄⠂⠀⠀⠀⡴⣻⣵⣙⣾⡈⡇⡯⡝⡯⣺⣟⣼⣇⢷⡄⠀⠀⠉⠁⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢛⠒⠀⠀⠀⢸⢧⡞⢛⣞⢮⡓⡆⣵⣧⣆⠿⡽⣲⢭⢽⡵⡄⠀⣰⣞⢳⡀⣰⣶⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠟⢓⣸⣽⡚⡇⠏⡽⣆⣿⣜⡻⢤⠋⠑⠃⠀⣿⣿⣿⠃⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠂⠓⠁⠀⠁⠉⠋⠉⠀⠀⠀⠀⠀⠀⠸⢿⡿⠀⠹⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⡛⠋⠉⠉⠉⠉⠛⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⠿⠛⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⡓⠦⣀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣍⠂⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣷⣦⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣌⠻⣿⡿⣿⣦⣀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠿⠿⠗⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣌⠻⣮⡻⣿⣦⡀⠀⠀⠀⢹⣟⣭⣵⣶⣾⠿⠿⣏⣙⢦⡀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣎⢻⣷⠈⣷⣿⣿⣷⡀⠀⠀⠀⢻⣿⡿⠿⣶⣖⡒⠾⣿⣷⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠀⢋⣿⣿⣿⣷⡀⠀⢤⣌⣿⡄⠀⠘⢢⣝⣻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⢿⣿⣿⣷⢸⡘⣿⣿⣿⢦⡀⠀⠀⣠⠍⡿⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⠿⢿⣏⢷⣿⣿⣿⣎⢿⡆⠘⠁⠀⢀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠻⣿⣿⣿⡿⣟⣾⣷⡄⠀⠐⠻⣤⣈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠻⣿⡃⠹⣿⡏⣰⣂⣀⣐⣿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⢿⣯⣉⣝⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠘⢷⡀⠹⣿⠇⢉⣉⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣦⣝⢿⣶⣶⣬⡻⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣷⡀⣹⣧⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣽⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣌⢿⡻⢿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣍⠳⣍⡻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢽⡻⣿⣿⣿⣷⣌⠻⣦⣝⠿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢝⠿⣿⣿⣷⣌⠻⣷⣬⡻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠙⠻⣿⣷⣌⢻⣿⣮⣽⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣝⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠛⢿⣿⣿⣦⣤⣾⣿⣿⠟⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠀⠀⠀⠙⢿⣿⣿⡿⠟⠁⠀⢰⠆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⢠⡄⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋⠁⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⡀⣀⣄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣤⣠⣶⠋⠀⠀⢀⣴⡤⣤⠘⠿⢿⡧⢘⣟⡃⠀⣀⠀⠀⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠈⣛⣃⣈⢁⠀⠀⠀⢘⣻⣿⣾⠟⠀⠀⠛⠴
⠀⠀⠀⠀⠀⠀⠀⠾⠀⢈⠉⠉⠙⠃⢴⣶⣦⠀⠀⠿⣿⡿⢸⣿⣷⠀⠉⠉⠉⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣾⢻⣿⠀⠀⠀⠀⠀⢸⡏⣿⡀⠀⠀⠀⠀⠀⠀⢹⡿⣿
⠀⠀⠀⠀⠀⠀⢠⢿⡇⠀⠀⠀⠀⢠⡏⠘⡟⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠀⠀⠀⠀⠀⣾⠁⢿⡀
⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠙⠀⠈⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⠀⢀⣤⣤⣄⠀⣠⠤⠤⣄⣠⠤⠤⣄⢠⠶⠶⢦⡀⢠⠖⠛⠲⣄⣰⠚⣛⣳⣄⣞⣋⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣉⣉⣙⣦⣾⡥⠤⢼⡷⠿⠶⠶⣾⠷⠛⢒⣚⡟⠛⠛⣛⡟⠋⠉⣩⡿⠋⠉⠉⢉⣽⠋⠉⠉⢩⡟⠉⠉⢉⣽⠋⣧
⠀⠀⠀⠀⠀⠀⣠⣴⡟⣿⡉⢡⡴⠃⠀⠀⠰⠟⠀⠀⠀⠛⠃⠀⠀⠘⠁⠀⠀⠀⠉⠀⠀⠀⣈⣀⡀⠀⣤⡀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢀⡏⠸⣧⣿⣇⠀⠀⠀⠀⠀⠀⣠⠴⠆⢀⣶⡄⢰⡆⠀⢸⣯⠅⠀⣿⢦⣹⠀⣯⢈⡷⢰⡧⢿⠀⣷⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠇⠀⣿⣾⣿⠀⠀⠀⠀⠀⠀⢿⣀⡄⢸⠛⠷⠈⠧⠤⠘⠓⠒⠀⠛⠀⠙⠀⠙⠋⠁⠈⠀⠈⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡀⢸⡆
⠀⠀⠀⠀⠀⡿⠀⠀⢹⣿⡟⡇⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠴⠶⡶⠖⠒⠒⠛⠛⣿⠛⠛⢉⣉⣹⣉⣉⣉⣭⣯⠤⠤⠤⢽⠤⠤⠼⡇⠀⣷
⠀⠀⠀⠀⢰⠇⠀⠀⠘⣿⣷⢻⠀⠀⢸⣍⣡⣤⡤⠤⠼⡦⠤⠴⠶⠒⠒⢻⠒⠒⠒⠒⠛⢻⠉⠉⠉⠉⠉⡇⠀⠀⠀⣸⣀⡀⣀⣘⣆⣀⣀⣻⠀⢸⡄
⠀⠀⠀⠀⡿⠀⠀⠀⠀⢿⣿⡿⡇⠀⠀⣧⠀⣀⣀⣀⣀⣷⣀⣤⣤⠤⠤⠼⡧⠤⠶⠒⠒⠚⡗⠒⠛⠋⠋⢻⠉⠉⠉⠉⡏⠉⠉⠉⣿⣉⣁⣘⡆⠀⣧
⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⣿⣿⢿⠀⠀⢸⡍⠉⠁⠀⠀⠸⡆⠀⠀⠀⣀⣀⣷⣀⣀⣠⡤⠤⢿⠤⠴⠶⠖⠺⡖⠒⠒⠛⢻⠉⠉⠉⠹⡍⠁⠀⢻⠀⢸⡄
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⣿⡿⡇⠀⠀⣷⠴⠒⠒⠒⠚⢿⠉⠉⠉⠉⠉⠸⡀⠀⠀⠀⢀⣸⣄⣀⣠⣤⠤⣧⠤⠤⠴⠾⡖⠒⠒⠚⢿⠉⠉⠙⣇⠀⣷
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢹⣿⣧⣷⠀⠀⢹⣀⣀⣀⣀⣠⢼⡤⠴⠒⠒⠒⠛⣟⠉⠉⠉⠉⠀⣇⣀⣀⣀⣀⣸⣤⠤⠤⠤⢷⠒⠒⠒⢻⡖⠛⠋⢻⡀⠸⡆
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠸⣿⡿⣾⡆⠀⠈⣏⠉⠀⠀⢀⣀⣧⣀⣠⠤⠤⠶⠺⡖⠚⠉⠉⠉⢹⡉⠀⠀⠀⠈⣇⠀⠀⣀⣸⣀⣠⣤⠤⣧⠴⠶⠞⣇⠀⢷
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣇⣧⠀⠀⢹⡛⠉⠉⠉⠁⢹⠀⠀⠀⠀⢀⣀⣧⣀⣤⠤⠤⠼⠷⠲⠒⠒⠚⢻⡋⠉⠁⢸⣇⣀⣀⣠⣽⣤⣤⣤⠿⠄⢸
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⡄⠀⠈⡷⠶⠶⠖⠒⠚⡏⠉⠉⠉⠉⠀⢹⡀⠀⣀⣀⣀⠶⠤⠴⠶⠶⠾⠓⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⣤
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡎⣇⠀⠀⢻⠀⢀⣀⣀⣤⡽⠤⠴⠶⠒⠒⠛⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠶⠶⠶⠚⠛⠛⠋⠉⠉⠉⣧
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢻⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠶⠖⠒⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠁⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠖⢿⣿⡼⣇⣀⣀⣤⣤⡤⠤⠶⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠶⠖⠚⠛⠉
⣿⣤⣶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠸⡟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠶⠖⠒⠛⠋⠉⠉
⠀⠀⠉⠙⠓⠲⠦⣤⣀⣀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠶⠖⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠤⣼⣧⡤⠤⠶⠖⠚⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠋⠉⠙⠛⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢿⣿⣿⣿⡿⠃⠀⠀⡀⠉⠉⠛⠷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣧⣤⣀⠘⢿⣦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⢟⣭⣭⡙⢻⣷⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣯⣿⠀⠀⠀⠀⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⡿⠋⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠙⣿⣿⣧⣀⢀⣠⠤⡀⠀⢀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣿⠇⡀⠙⢿⣿⣿⣿⣧⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡾⢳⣌⢢⠀⣿⣿⣿⣿⠿⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠛⠛⢟⣡⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣄⡀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠄⠀⠤⠄⣿⣶⡝⢿⣿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠁⠀⠈⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠻⣷⣬⡻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠙⠛⠀⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⡀⠀⠀⣢⠠⢄⡀⠀⠀⠠⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠠⢠⣀⠀⢀⣤⢠⡄⠆⢄⡀⡨⠃⢠⡰⠌⠖⠆⢀⠀⢀⡃⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⡀⠀⠈⠐⠀⡠⢊⣭⡭⣒⠏⣤⣤⢓⣬⡆⣷⡂⣦⠄⡠⣸⠤⠼⡂⠈⠈⢊⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣀⣦⣯⡷⡱⣧⠾⣭⣓⡻⠎⣓⣛⡵⣧⢞⡕⣹⢫⡞⣅⢢⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠙⠛⠛⠙⠓⠚⠛⠲⠻⠻⠙⠛⠚⠛⠳⠿⣯⣶⠿⠿⠟⠓⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⢂⠀⠀⠀⠀⠂⢀⠁⠀⢄⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠄⠀⠐
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⡆⠀⠈⠁⠀⠀⢀⠎⠀⠀⠀⡀⠊⠁
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡅⠦⠠⡀⢠⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠒⠶⠒⢊⡁⠴⡀⠀⠀⠙⠂⠦⠈⠀⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠀⣤⠄⠐⠠⠊⠀⠸⠴⠦⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠂⠐⠂⠁⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠁⢀⢂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢤⣿⣿⣧⣄⡷⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⣿⠑⠈⠉⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠚⠀⢾⣿⣟⣷⡀⢀⣴⣿⣿⣿⠀⠑⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⣉⢀⠰⢚⣩⣿⣿⣿⣿⣿⣿⣍⡗⠂⡀⣅⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⡾⠶⠌⠈⢯⣿⣍⣹⣯⡼⠋⠡⠶⢷⣿⣧⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣷⣌⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣡⣾⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠅⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠚⠛⡿⡿⣿⣿⣿⣦⣶⠛⠓⠒⠟⢳⣴⡿⣿⣿⢿⡿⠋⠃⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠀⠀⢹⡇⠈⣼⣿⣿⣿⠆⢠⡤⠰⣿⣿⣿⣧⠁⢸⡟⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⡀⠈⣅⣸⣿⣮⡙⠛⢷⠦⠤⡾⠛⢋⣱⣿⣇⣸⠁⢀⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣮⣠⣽⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣯⣄⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣍⣤⢿⡻⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⣤⣉⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣽⣿⣆⠁⠹⢿⡅⠂⠐⢨⣿⠏⠉⣠⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠙⠁⠀⠀⠙⠄⠀⠞⠁⠀⠀⠋⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⡷⠶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⣀⣀⡽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣀⡤⠴⣖⣺⣭⠷⠞⠛⢋⠁⠀⠀⣴⠶⢦⣤⠀⠀⠉⠋⣀⣀⣀⣀⠐⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⡧⢖⣺⣭⠶⠞⣿⣽⠷⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢶⣿⣀⠀⠀⢿⡝⣏⠉⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠛⠋⣁⣤⠶⠛⠉⠀⢀⣀⡤⠤⠴⠒⠶⠦⠤⢤⣀⠀⠀⠉⠻⢦⣄⠻⣜⢦⠀⠀⠀⠙⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡾⠋⠀⢀⡠⠖⣊⣭⠶⠞⠛⠛⠛⠛⠛⠶⠶⣬⣉⠲⣄⠀⠀⠙⢷⣝⣯⢳⡀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⣠⢔⣯⠶⠛⢩⡀⠀⠀⠀⠸⠇⠀⢀⡾⠰⢶⡍⠻⣦⡱⣄⠀⠀⠹⣿⣷⡱⣄⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⢞⣵⠟⣁⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⣀⣄⠛⠿⠁⠀⠈⠻⣎⢳⡀⠀⠘⣦⠻⣜⢆⠀⢰⡇⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡰⢣⡾⠁⢿⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣿⠀⠀⠀⠀⢀⣀⠙⣧⠹⡄⠀⠸⣇⠹⣮⣷⡟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣰⢡⡟⠀⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⠁⠀⠀⠀⠀⠈⠁⠀⠸⣇⢳⠀⠀⢿⡀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⡿⠀⢰⢣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⡇⢀⣀⠀⠀⠀⠀⠀⣠⠞⠛⠓⣶⣟⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⡷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡸⡇⠈⠉⠀⠀⠀⠀⠀⠙⢶⠶⠛⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣼⢃⡇⠀⣸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢧⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⢀⡿⠀⠟
⠀⠀⠀⠀⠀⠸⡄⠘⣧⠀⠈⣏⣷⡀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣼⣷⢠⡟⡴⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠁⠀⢹⣆⠀⠘⢎⢷⡄⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠳⠟⣰⢏⡞⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆⠀⠈⠳⣝⢦⡀⠀⢸⡿⣮⠁⠀⢠⡄⠀⠀⠰⡆⠀⠀⢀⣠⠞⡡⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⣌⣷⣄⠀⠈⠳⢽⡷⣬⣍⣁⣀⣀⠈⠀⠀⢀⣀⣤⡴⠞⣫⠵⠋⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡴⠟⠁⠈⠻⣦⣀⠀⠉⠓⠮⠭⠭⢭⣍⣛⣛⣉⡭⠴⠒⠉⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡞⠋⠁⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠙⣿⠚⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠶⠟

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠒⠠⡀
⠀⠀⠀⠀⠀⣀⢀⡮⣤⠄⣀⣠⣄⠰⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⠀⠘⠀⠠⠤⠤⠠⠄⠤⢀
⠀⠀⡔⠊⠉⠉⠈⡇⠀⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠈⠀⠀⠈⠉⠉⠐⡀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠈⡗⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠥⠄⡃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡄⠀⠈⠉⠉⠁⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠀⠀⠀⠀⢠⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡅
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡰⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠐⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⠐⠒⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢀⠀⠀⠀⠘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠸⠀⠀⠀⠀⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠜
⠀⠀⠈⠦⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠐⠒⠒⠒⠒⠒⠂⠐⠃⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⢻⢉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠢⠽⠞⠾⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠰⡉⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣠⠷⡄⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⠉⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠉⠂⠘⣾⡀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⢀⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠹⣑⣲⡚⣧⣤⠀⢀⠔⠉⠀⠀⠀⠀⠀⠠⡊⠉⠉⢹⡆⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢿⡍⠉⠁⢸⡴⠃⠀⠀⢀⠁⠀⠀⠀⠀⡀⠒⢒⣋⣤⣄⣀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣧⣸⠟⠳⣀⠞⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠁⡴⠋⠀⠀⠀⠈⠑⢆⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠉⢫⡀⠀⠀⠀⢻⠀⠀⠀⠀⢰⡄⠤⠤⡀⠀⠀⢰⠁⠀⢀⣄⠀⠀⠀⠈⢧⠈⣧
⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢧⠀⠀⠀⠐⠀⠀⠀⡇⠘⡦⠘⠓⠢⣼⠛⠇⠀⠀⠀⠈⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠘⡆⠀⠀⠸⠀⠀⡸⠀⠀⡇⠀⠀⠀⠈⢣⠀⠀⠀⠀⢀⡏⠀⢠⠀⠀⠄⠀⠀⣠⠖⠉
⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⠆⠀⠀⢣⠐⠁⠀⢠⡁⠀⠀⠀⠶⢼⠒⠒⠒⣒⣉⡴⠃⢨⠎⠀⢀⡴⠚⠁⠀⠀⢰
⠀⠀⢸⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⣄⠀⠈⢧⣀⠀⠈⠙⠦⠄⠀⠠⠎⠀⠀⠀⠉⠁⠀⡐⠁⠀⠘⠉⠀⠀⠀⠀⢀⡟
⠀⠀⠸⡄⠀⠀⣯⠙⡶⠶⠤⠤⢄⣀⣀⣀⣀⣰⠀⠀⣠⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠶⠶⠂⢀⡔⢹⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⡇⠀⠀⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠃⠀⢀⣶⠀⡖⢄⠀⣀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⢁⠀⠀⠲⠷⣴⣤⠄⠀⠀⠀⠀⠀⠀⡇⠀⠟⠀⠀⣟⠸⠀⠇⠀⠑⠏⠱⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⢀⣠⡼⢦⣤⣄⣀⢷⣄⣀⣀⣀⡀⣀⣠⣀⢦⣀⣧⠀⠀⠋⠀⢠⠃⠀⠀⠀⠀⢷⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⡜⠉⠀⣠⠁⠨⠿⠓⢀⠿⣆⠀⣸⡧⢨⠔⢣⠀⠈⠹⠑⠦⠄⢰⠇⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⢳⣤⣤⣼⣖⡢⠴⠂⠀⠀⠀⠈⠛⠳⠶⢶⠒⣠⣤⡾⠀⠀⠐⠷⠤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣥⠄
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠦⢤⣄⣀⣀⣀⣀⣀⣠⣤⠤⠖⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠂⠃⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⠆⣿⣿⣿⣿⡏⠉⠁⠉⠁⠈⠀⠉⠉⠉⠁⠉⠁⠉⢹⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⢿⢧⣿⣟⣛⡿⣧⣤⡤⠤⠤⠤⠤⠤⠴⠤⣤⣄⣠⣤⣼⣿⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⢸⠷⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠿⠙⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡈⠋⠀⠀⠀⠀⠀⢶⣾⠡⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢄⠀⠀⠀⠀⢀⡾⠁⠁⠀⢀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⣿⠀⠀⠀⠀⠀⢄⠀⢀⡮⠊⠀⠀⠀⠈⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢿⣷⠀⠀⠀⠀⡀⠈⢸⠀⠑⠀⠀⠀⠀⠀⡾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠈⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠑⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡇⣸⣦⢤⣤⡤⠀⠀⠀⠀⠀⠠⠧⣴⣃⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣾⣾⡞⣿⣯⣽⣷⡟⠛⠳⠒⠒⠒⠒⠲⠖⠒⠛⠋⠑⠛⢻⣷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⣿⢆⣿⣿⣿⣿⣇⡀⠀⠀⡀⢀⠀⠀⢀⡀⣀⣀⣀⣀⣸⣿⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠂⠈⠑⠀⠀⠀⠢⠀⠉⠙⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢰⠰⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢄⠀⠀⠀⢀⠀⠀⠠⢀⠀⠀⠀⡀⠀⠀⠄⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠈⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢀⠀⠀⡂⣀⠀⠀⢂⠀⠀⠐⢠⠀⣀⡐⣀⠀⠀⢆⠀⠀⠀⢀⡀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⢀⠀⠀⠂⣀⢀⣐⡨⠤⡀⢠⠄⠀⠀⠤⡄⠀⠠⢌⠀⠀⠀⠤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⡄⢈⢁⠴⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⢰⡇
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⢀⠀⠀⠒⡀⠀⠀⢊⡀⠀⣀⣚⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⡀⢸⡇
⠀⠀⠀⠀⠀⠰⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⢈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂