Promising Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠀⠹⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⢠⣾⣿⣿⢟⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣇⡀⠈⢹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣯⣶⡄⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠰⣷⣾⣿⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣀⣤⣿⣿⣯⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉


⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢀⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⢀⣽⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣶⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⢾⣿⣿⣿⣿⣿⣎⣿⣧⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣤⣾⣿⡿⣿⣷⣄⣸⣿⣭⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠙⢿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡜⠉⠲⢔⠎⠈⢲
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣖⡤⠈⠢⣀⡀⠀⠀⠀⢇
⠀⠀⠀⠀⡠⠒⠈⠁⢀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢃⠉⠉⠳⡄⠀⠈⢹⡆⠀⡀⠘⡀
⠀⠀⠀⢰⠁⠀⣠⠞⠛⠁⠀⠠⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠄⠐⠿⣄⠀⠀⠘⢦⡀⠈⢯⠉⠀⠀⢷
⠀⠀⢠⠃⠀⢰⠃⠀⠀⠀⢀⠀⠀⠀⢡⠀⢀⠠⠄⠒⠲⠾⢎⠀⠀⠀⠀⠘⢆⠀⠀⠈⢳⡤⢌⠉⠀⠀⠈⡇
⠀⢠⠏⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠇⠀⠀⠀⡇
⠀⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠶⠤⡄⠐⣀⣦⣀⠀⠀⠀⢹⠀⡠⠒⢁⠀⠀⠀⠀⡇
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡔⠛⣷⡀⠀⠈⠣⡰⡀⠘⠀⠀⠀⢀⠇
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠁⢀⠔⠁⠀⠀⠈⠙⠢⠤⠾⠢⣀⠀⠀⠀⠀⠸
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠳⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠒⠂⠀⣲⡀
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠄⠊⢀⠟⣦
⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡠⠂⠀⠀⠀⠀⠜⠅⠁⠀⠂⠀⠹
⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣆⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⠀⠀⠀⠀⠀⠀⠀⣄⣤⣾⡿⠁
⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⣤⣶⣿⣿⡿⠁
⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁
⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃
⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣆⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⢸⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⠀⠀⠼⠿⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⠿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⢠⣼⣿⢿⡏⠁⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠃⠀⠀⠀⠀⢀⣾⣿⡁⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠂⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢸⣶⣾⣿⣿⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠈⠲⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣸⣿⣿⡆⢀⣾⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⠀⠀⠘⢟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣤⣤⡴⠋⡸⢻⣻⣾⣧⡀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢹⣿⣯⠿⣦⣤⣀⣀⣨⣿⣿⣿⣿⢿⣿⣿⠛⣀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡤⠀⣀⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣸⣿⣿⣿⡟⠉⠙⣿⠇⠀⣿⣷⣾⣿⣦⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣮⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⡟⠂⠀⣿⡿⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⢠⣿⡃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⠏⠀⠀⠀⡸⠀⠀⢰⡿⠃⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠟⣿⣷⣾⠟⣿⡆⠀⢸⠋⠉⣿⡿⠲⠿⣿⡟⠉⠁⢤⡶⣿⠇⠀⢀⣤⡾⠃⠀⢠⡞⠁⠀⢠⣿⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠛⢻⣏⠀⠹⡇⠀⢨⠀⠀⢹⣿⣶⣄⢹⠇⣠⠴⠟⣼⣿⠀⢠⣾⡿⠃⢀⣴⣿⠀⠀⢀⣿⣿⡁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⣿⠀⠀⣇⠀⢸⡆⠀⠈⣿⠟⠇⠸⠰⠇⠀⠀⡽⠻⣿⣿⡟⠁⠀⣾⣿⠏⢀⣴⡿⠿⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢻⠀⠘⡃⠀⠀⠋⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠘⢦⣀⣼⣿⠉⠀⣾⡟⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣀⠘⠢⡀⠈⠲⢤⣤⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣯⣀⣼⣿⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣽⣷⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣄⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣷⣶⣶⣤⣤⣄⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣾⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠋⢀⣾⡿⠁⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣾⡿⠁⠀⢰⣿⣿⣿⣿⠋⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⢀⣾⡿⠁⠀⢠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⢀⣾⡿⠁⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣆⠈⢻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃⠀⢺⣿⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠻⣿⣿⣿⠟⠀⠀⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠉⠻⣿⣦⣀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣤⣤⡀⠀⣀⡀⠀⠈⠻⢿⣦⣄⠀⠀⠙⢿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣀⡙⢿⣶⣄⠀⠀⠙⠻⣷⣦⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣿⣦⣄⣀⣼⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠸⣿⣿⠟⢿⣶⣴⡿⠉⠛⠛⠋

⠀⠀⠀⢀⣴⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⡰⠋⣨⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠟⠳⡄
⠀⠀⠁⢲⠧⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⡀⠀⠀⠉⠲⣤⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⡿⢿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠙⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠖⢀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⠇⠀⠀⠀⠀⠀⢰⠀⠒⠀⠀⣿⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢾⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⢸⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠉⡦⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢲⡴⠃⠀⠀⠀⠀⠀⣿⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢸⠀⠀⠀⠀⠀⠀⢇⣠⠀⠀⠀⠀⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠐⠒⠒⠀⠀⠒⠒⠒⠒⠂⢸⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⣹⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣬⣑⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⣉⣉⣉⣉⣉⣉⣉⣉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⢚⣉⣥⠤⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣍⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠘⢆⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢉⣉⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠐⣆⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣠⠔⡻⠛⢳⣀⠀⣀⡀⠇⠀⠀⠀⠀⠀⠀⢠⢿⠁⠀⢸⠀⠀⠀⡆⢱
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⣆⠀⠀⠀⠳⣀⣠⠤⠴⠒⠊⠉⠹⣯⠀⢸⢁⢦⡀⠉⢹⡟⠉⠉⠓⠲⠤⣤⣀⣰⠃⡞⠀⠀⢸⠀⠀⠀⡇⠘⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢈⢉⠀⠱⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⡇⠀⢇
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⣀⣠⠤⠴⠖⠚⠋⠉⠉⠋⠛⠙⠁⠉⠓⠒⠢⠤⢄⣀⣀⠀⠀⣼⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡇⠀⠀⠀⡏⠀⠀⢀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠶⠶⠤⠤⠤⠤⠶⠦⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁⠀⠀⡜⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⡇⠀⠀⣰⠶⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⢿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣒⠢⡤⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣩⢉⡷⠛⣄⣇⣀⣠⣀⣾⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡉⣑⠴⠋⡹⢀⣽⣧⣟⠟⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠉⠉⠸⢭⠝⠣⠬⠏⠁⠀⠀⠀⠀⠀⢀⡀⢀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⢀⣀⠈⠀⠀⠀⠀⠀⠀⡐⠁⠘⠂⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠠⢄⣀⠀⡀⠀⢰⠉⣀⠀⠀⢳⡀⠀⠀⠀⠀⠔⠀⡀⠀⢃⠀⢀⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣐⠀⠀⠀⠈⠙⠓⠷⠷⣄⣈⣉⣳⡞⢠⠤⡄⠔⠀⠀⢰⢧⠖⠾⢣⡸⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢶⡤⠤⠤⠤⠒⠀⠈⠁⠀⠓⠀⠂⠀⠀⠀⣰⣻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡫⠥⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠩⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠃⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠓⠊⢳⡀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⠷⡀⠀⠀⣀⠤⠊⣡⠖⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠳⣽⠖⢉⠁⢀⠔⠁⠀⢠⠘⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⡖⠕⠀⠀⠙⠁⠉⡁⠀⠀⡀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⣤⠚⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢠⢠⠄⠀⠀⠤⠤⠸⡇⡄⠐⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢀⢸⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢾⣤⣄⠀⠀⠀⠀⠀⠠⠔⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢻⣕⠦⢄⣨⣉⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠛⠛⠹⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠠⠀⠁⠀⠀⠀⠠⠒⠒⠄⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⡀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⡀⠀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠄⠀⠳⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢛⣿⣿⣷⣀⣠
⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⣠⣦⠀⠐⠂⠠⢤⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⡅⠀⢠⣄⠀⣴⣶⣤⡀
⠀⠀⠐⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⢾⡗⠀⠀⠀⠂⠈⢿⣂⡀⡤⢐⣟⣷⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄⢉⣯⡑⣿⣿⣿⣿⡄
⠀⠀⠨⡀⠀⠀⠀⠬⠀⢺⠀⠀⢄⠀⠁⠀⠀⠀⠀⠀⠸⣿⣇⠑⢴⣿⡽⠆⣀⠀⠀⠈⣿⣿⣿⣿⣿⠮⠛⠑⠟⢻⣛⣛⣃
⠀⠀⠀⠨⠄⠀⠀⠀⠀⠘⠉⠃⠈⠁⠀⠀⠃⠐⠁⠐⠀⠹⡙⡇⠀⠈⠓⠈⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⡀⠀⠀⢤⣽⣿⠿
⠀⠀⠀⠀⠆⠀⢸⡆⠣⠸⡀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠉⠉⠒⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠈⢰⠀⢸⣷⣄⠇⠁⠀⠀⠀⠈⢆⠀⠀⠰⢁⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢹⠳⢜⣿⣿⠆⠀⠀⠀⠚⠀⠘⠊⠦⠄⠀⠃⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠈⢁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠱⡆⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠈⠉⠀⠀⠉⠁
⣀⣀⣤⣬⣤⣤⣤⣤⣤⣤⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠈⣿⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣠⠞⢻⣤⣼⠛⠚⣿⣿⢛⣿⣿⣧⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡹⠻⢿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⣹⡇⠘⠁⠀⠀⠈⠁⠚⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣒⠒⠀⠀⠀⠀⣰⣾⣫⣿
⠂⠀⠀⠀⠀⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⠏⡛⠃⢻⠉⠓⠄⠀⢠⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠤⠄⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡾⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⣈⡾⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠿⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠀⠀⣤⣀⡀⠀⣚⣻⣿⣿⣿⣿⣿⣿
⠀⠀⢠⠀⣿⣿⣿⣿⣿⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⡟⠛⣿⠀⣉⣿⣯⣿⣿⣿⣿⣿⣿⣿
⣿⣬⡰⢤⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠻⣿⣏⡇⠀⣿⣘⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿
⢀⣤⣶⣒⣿⣿⣿⣿⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣛⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⣻⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣬⣿⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢰⡄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢸⡇⠀⣾⠃⢀⡾⠃⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣿⣶⣿⣷⣾⣇⣠⡾⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⣦⣼⣿⡿⠛⣉⣿⠿⣿⣿⣤⣴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣤⣼⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣤⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⠻⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⢠⡟⠛⣿⠛⢿⡅⠀⠻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⣉⣴⠃⣼⠇⡀⢹⡆⠀⢻⡆⢠⣌⣡⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣤⣡⣾⣇⠚⢁⣷⣌⣡⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠂⠀⠀⠁⠀⠀⠀⠀⠀⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠒⠋⠁
⠀⠀⠀⡀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠊⠊⠀⠀⠀⠘⠦⡀⠀⠀⢀⠔⡁⠁⠀⠀⡀⠀⠀⠠⢊⠔⠁
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡐⠀⠀⠀⠀⠀⠀⠀⠌⠢⠔⢁⠀⠀⠀⠀⠀⠁⡀⠐⠠
⠀⠀⠄⠄⠌⠃⠀⠀⠠⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡠⠀⠀⠀⠀⠀⠀⠀⠐⢤
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠄⠄⠊⠉⠁⠀⠀⠀⠀⠀⠀⣰⠲⠞⠙⠴⢲⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⣀⣠⠤⠀⢰⠒⠒⠒⠒⠒⠒⠒⣲⡤
⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⡏⠁⠀⠀⠀⠀⣨⢉⡇⠀⡰⠁⠀⠀⠀⡠⡢⠙⡆⠀⠀⢸⠀⠀⣀⠀⢀⢔⠈⢻
⠀⠀⠀⢀⠠⣀⠄⡀⠀⠀⡀⠺⡁⠀⠄⢀⠀⡀⠁⢀⡹⠀⡇⠀⠁⡀⠊⠄⠀⠀⢀⠀⠀⢸⠀⠪⠀⠄⡁⠁⠀⢸
⠐⡈⠄⠀⠑⠠⠂⠀⠀⡰⠁⠠⣧⣄⠀⠀⠀⠀⢀⣬⠇⠀⠘⣄⠀⠀⠆⠀⠀⡠⠃⠀⠀⢸⠀⠀⠐⠄⠀⠀⠀⢸
⠀⠈⠢⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⠖⢢⡰⠒⠞⠀⠀⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄⡀⠀⠀⠀⠀⠠
⠀⠐⡁⠀⠀⠀⠀⠀⠢⠁⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠊⠀⠀⠀⠀⠀⢀⠼⡀⠀⠀⠁⠀⠀⠀⠀⠀⡐⠁⡄
⠀⢠⠀⡠⠀⢀⠀⡀⠁⠀⠀⢠⢀⠀⠑⠀⠀⡠⠂⠀⠀⠀⠘⠀⢠⡀⠀⢀⠐⠋⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈
⠀⠈⠀⠀⠀⡡⠂⠀⠀⠄⠀⠈⠀⠀⠀⠡⠈⡀⠁⠀⡰⠀⠸⠀⠀⠩⠒⡁⠁⠀⠀⠜⠀⠀⠀⠂⡀⠂⠁⠈
⠀⠐⢄⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡀⠀⠈⠀⠀⠞⠁⠀⠀⠁⢀⠀⠉⠀⠀⡀⠞⠁⠀⠀⠀⠠⠈⠀⠀⠀⢐⠄
⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠒⠂⠀⠀⠀⢀
⠈⠠⠀⠀⠀⠀⠀⠀⠖⢁⠀⠀⠁⠄⠀⠀⠀⠀⠀⠔⠁⠀⠀⠂⠄⠀⠀⠀⠀⠠⠖⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢀⢀⡠⢀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡀⢄⠀⠀⡠⠀⠔⠁⠀⠀⢀⡀⠀⠀⠀⠠⠂⣠⠀⠀⠀⠀⠀⠄⠀⠀⠔⢁⠀⡄
⠰⠨⠑⠄⠑⠊⠠⠀⠀⠀⠀⠀⠀⠁⠀⡀⠊⡠⠂⠈⠀⠀⠀⠈⠁⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⡇
⠠⠨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠀⠀⠀⠀⠈⠀⠀⠠⠑⠔⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⡀⠀⠐⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⠿⠿⠟⠻⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠢⣤⣀⣄⣀⣠⣇⣄⣀⣀⣀⣴⣄⣀⣄⣄⢄⣤⣤⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⢿⠿⠻⠏⠁⠀⢀⣰⣶⣤⣠⣤⣤⣔⡀⡀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣾⣿⣯⠙⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣫⣤⣴⣿⣿⠿⣿⣿⣿⣿⣿⣛⣤⣦⡀⡀⠀⠀⣀⠀⠀⠀⠉⠙⣿⣷⡀⠘⣿⡻⣯⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⠉⠛⠛⠛⢛⣽⣃⣄⣸⣿⣿⣿⣿⣿⣿⣻⣟⣮⡺⣞⢀⣠⡀⣀⣠⢄⣾⣿⣧⠀⢸⣿⣷⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⠀⡀⣀⣀⣠⣤⣼⣿⡿⣏⡙⣿⣿⣿⣿⣮⡿⣿⣿⣿⣿⣿⣏⣯⣿⡨⣊⣿⣿⣿⣿⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⠓⠿⠿⠻⠟⠿⣻⣿⣿⣾⣷⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⡀⣀⣴⣿⣿⡿⢛⢩⣷⣧⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⢀⢸⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠟⠛⠉⠀⣴⣷⣯⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣕⢙⡧⣪⣿⣿⣿⣿⡏⠀⠸⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠇⠀⠐⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣿⣿⠿⠟⠛⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⢀⡀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠋⠹⣯⢸⣧⣿⡽⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣭⣭⣵⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠒⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠒⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠒⠶⠦⣄⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⡖⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⢠⡾⠋⠀⢀⡄⠀⡄⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⣠⠞⠉⠀⢀⡀⢠⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢠⠏⠀⠀⠀⠀⢰⣷⣄⠀⠀⠀⠀⠀⠙⣆⠀⠀⢰⠏⠀⠀⠀⠈⣇⣸⣇⠀⠀⠀⠀⠀⠹⣆⠀⠀⣰⠋⠀⠀⠀⢸⡇⣿⣼⠀⠀⠀⠀⠀⠹⡄
⠀⠀⡿⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⢹⡄⠀⡟⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⠀⢠⡏⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⢿
⠀⠀⡇⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⢸⡇⠐⡇⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⠂⢸⡇⠀⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⢻⡀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣼⠀⠀⢷⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⡾⠀⠀⣧⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠈⢷⡀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⡼⠃⠀⠀⠈⢧⡀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⢀⡼⠁⠀⠀⠘⢧⡀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠙⠶⣄⣀⣾⣿⣿⣇⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⢸⣿⣿⣿⣆⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠘⣿⣿⣿⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⣛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⡛⠉⠉
⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠉⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠉⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠉⠛⠶⣄⡀
⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣴⠋⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⡴⠋⣤⣆⢠⡀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⣰⠃⠀⠀⠀⣶⢰⣧⡆⣀⠀⠀⠀⠀⠈⢧⠀⠀⣼⠃⠀⠀⠀⢸⡇⣿⡿⢠⠄⠀⠀⠀⠘⣧⠀⠀⡼⠑⣆⢻⣿⣼⡇⠀⣀⣄⠀⠀⠀⠀⠘⣆
⠀⠀⡟⠀⠀⠀⠀⢹⣾⣿⣷⠏⠀⠀⠀⠀⠀⢸⡆⢀⡇⠀⠀⠺⣦⣸⣿⣿⣷⡏⠀⠀⠀⠀⠀⢸⡀⢰⡇⠀⢸⣿⣿⣿⣷⡾⠁⢻⣾⣿⣷⣄⠀⢻
⠀⠀⣧⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠇⠈⡇⠀⠀⠀⠹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⠁⠸⡇⠀⠘⣿⣿⣿⡿⠁⠀⠘⣿⣿⣿⡟⠀⣼
⠀⠀⠹⡄⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⢀⡞⠀⠀⢻⡄⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⢠⡟⠀⠀⢳⡀⢀⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⠀⢠⠏
⠀⠀⠀⠙⣦⡀⠀⠀⢻⣿⣿⡄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠻⣄⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⣠⠟⠀⠀⠀⠈⠳⣾⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣴⠋
⠀⠀⠀⠀⠈⠙⠶⣤⣸⣿⣿⣷⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣸⣿⣿⣿⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣥⣀⣀⣀⣀⡼⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⢀⡾⠁⢠⣶⢠⡆⠀⠀⢰⡆⢀⡆⠈⢳⡄⠀⠀⢀⡞⠁⣤⣦⢰⡄⠀⠀⢸⢀⡇⡤⠈⢳⡀⠀⠀⢠⡞⠁⣤⣆⢰⡄⠀⠐⡇⣸⣰⠀⡈⢧⡀
⠀⠀⣼⠁⢷⣸⣿⣾⡇⠀⣀⠘⣧⣼⣅⡀⠀⢻⠀⠀⡾⠐⣦⣹⣿⣼⡇⢀⣀⢸⣾⣿⣧⠀⠀⢷⠀⠀⡟⠘⣦⣹⣿⣼⡇⢀⣀⣿⣿⣿⣼⠁⠈⣧
⠀⠀⡇⠀⠈⣿⣿⣿⣷⣾⠃⢰⣿⣿⣿⡇⠀⢸⡇⢠⡇⠀⢸⣿⣿⣿⣷⡿⠁⢸⣿⣿⣿⠀⠀⢸⠆⢸⡇⠀⢸⣿⣿⣿⣷⡿⠁⣿⣿⣿⡿⠀⠀⢸
⠀⠀⣷⠀⠀⢻⣿⣿⡿⠁⠀⠘⣿⣿⣿⠃⠀⣸⠃⠀⣇⠀⠀⣿⣿⣿⡟⠁⠀⠸⣿⣿⡏⠀⠀⣼⠀⠘⣇⠀⠈⣿⣿⣿⡟⠁⠀⢹⣿⣿⡇⠀⠀⣾
⠀⠀⠘⣆⠀⣾⣿⣿⠁⠀⠀⠀⢿⣿⣿⡀⣠⠏⠀⠀⠹⣆⢀⣿⣿⡿⠀⠀⠀⠀⣿⣿⣷⠀⣰⠇⠀⠀⠹⣄⢀⣿⣿⡿⠀⠀⠀⠀⣿⣿⣷⠀⣰⠃
⠀⠀⠀⠈⠻⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⡷⠋⠀⠀⠀⠀⠈⠿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠿⠃⠀⠀⠀⠀⠙⢾⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠤⠤⠶⠞⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⣼⣿⣿⣿⠏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣻⣿⡿⠋⠁⠀⢀⣤⣿⣿⣿⡿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⢠⣼⣿⡿⠀⠀⠀⣰⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠉
⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣴⣿⣿⣯⡀⠀⠀⠸⢄⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉
⣀⣼⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⠀⢹⣿⣿⣿⠋⠀⠀⠀⠀⠀⠙⠛⠁⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠟⠃
⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣷⣶⣾⠉
⣿⣇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡦
⣿⣿⣷⣾⣿⣉⣻⣿⣿⣿⣿⣋⣉⣿⣶⣶⣶⣶⣤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡈⠉⠛⠿⠿⢿⣃⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⢸⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠉⠀⠀⠉⢉⣹⣿⣿⣿⠋⢙⣻⣭⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠶⠾⠷⠶⠦⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿
⣿⣏⣿⣉⣉⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⠀⢴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⢠⡾⠛⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀
⣿⠃⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⡿⠟⠛⠛⠻⠿⣦⣄
⣿⡆⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⣀⣰⣿⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⣿⠛⠁⠀⠀⠀⠀⠀⠀⠙⣿⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡇
⠀⠙⣧⡀⢀⣀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢸⣷⡤⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠁
⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⠃
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿
⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⣿
⠀⠀⠀⠀⠀⠀⠹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢀⣤⣤⣤⣄⣀⠀⠀⠀⢠⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠛⠛⠛⠳⢦⣤⣉⡙⠛⢀⣠⡿⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢠⠂⠀⠀⠀⠀⠀⣠⣿⠋⠙⠛⠙⠻⠿⢷⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠲⠒⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢹⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⢿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣂⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠘⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣥⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡓⠖⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⣤⣄⣤⡔⠐⠂⣄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣹⣿⡿⣿⠋⠉⣑⢸⡆⣀⣤⢀⠀⠀⣸⣾⠟⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⢺⢉⣍⣿⣿⣧⣶⣿⢸⣿⠿⢻⣾⡓⣴⣿⡟⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⣌⣟⢻⣿⡻⠟⠏⣸⣿⠀⠈⣿⣤⠛⠿⢧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣟⣿⢿⣿⣷⣿⠝⠉⠙⣷⣼⣿⣏⡀⣠⡶⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣿⣿⣣⣾⠟⠁⢫⠉⣶⣿⢞⣿⡿⠫⣿⣿⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⠏⠀⠐⡀⠀⢑⣾⢟⣽⢻⡟⠺⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⢸⣦⣀⣴⣾⡶⣋⣴⣿⡏⣿⠸⣷⣄⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣭⣉⠭⣹⣿⣿⡟⢧⢸⣥⣿⣾⡏⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢠⠀⠀⠀⢀⠄⣿⣾⣿⣇⣼⢷⣿⣿⠿⡉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣄⠀⣈⣟⣽⣟⣿⣿⣿⣾⣜⡿⠛⠤⡈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠉⠛⣿⣿⠿⡏⢻⡟⣿⣿⡟⠇⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠙⣿⣿⣿⣿⣮⢾⣻⣥⣤⣶⠿⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠉⠛⣿⣿⣷⣶⣿⣿⣷⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢾⣿⣿⠿⢿⡿⡋⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢻⣿⠀⠛⠉⠁⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢂

⠀⠀⠀⠀⠀⣾⠛⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢿⣦
⠀⠀⠀⠀⠀⠈⣷⠀⢨⡇⠀⠀⠀⠀⣀⡤⡶⠾⠋⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⡀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣿⠇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⣀⣀⣤⠾⠋⠁⠁⠀⠈⠻⣧⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠉⠃⠀⠀⠀⠀⢠⡟⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡾⠄⡀⠀⠀⠀⠸⢫⣄⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠘⣷
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠰⣏⠛⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠲⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣦
⠀⠸⣧⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠹⣷⠀⠀⠀⣀⡴⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠟⠻⣦⣀⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠙⣶⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⢀⡾⠃
⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣤⣤⣤
⣿⣿⣉⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⠁⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⡐⠅⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠈⠠⢀⠀⠀⠀⠌⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠠
⡥⠀⢲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠑⡀⠀⡐⠀⠀⠎⠀⠀⡄⠀⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠌⠢⡀⠀⠄⠀⠠⠇⠀⢨⡔⠀⡆
⠁⠀⠌⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠘⠔⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠌⠀⠀⠀⠀⠀⠘⠀⠠⠁
⠀⠀⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣤⡀⠀⠛⠘⠀⣴⣆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⠀⠀⡇⢳⠀⡖⡄⠀⠀⠀⠀⢀
⠀⠀⠠⠀⠈⢃⢡⠀⠀⠀⠁⠂⠘⠀⣀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠘⡂⠡⠀⠇⠈⠀⠆⠀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠠⠁⠀⠀⢁⠀⡆⠸⠐⢂⡄⡠⢰⢻⠃⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢂⠀⡄⠐⠐⡆⡀⡄⢘⠙⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠈
⠄⠁⠀⠀⠀⠀⠇⠈⠤⠀⠸⠀⢀⡠⢤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠐⠀⠀⠀⠀⠈⡁⠐⠄⠀⠧⠀⢀⠤⢤⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠄⠈
⠀⢠⡀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠰⢀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣀⠈⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠏⡗⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡰⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⢀⡀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡈⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⡀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠆
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⠁⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠰
⠀⠀⠀⡐⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠈⡠⢀⠀⠀⠀⠈⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠁⢂⠠
⡥⠂⢢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠠⡀⠀⡐⠀⠀⠎⠀⠀⡄⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠌⠢⡀⠀⠌⠀⠠⠇⠀⢈⡔⠀⡄
⠁⠀⠌⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠔⠀⠀⠀⠀⠀⠘⠁⠀⠇⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠌⠀⠀⠀⠀⠀⠘⠀⠠⠁
⠀⠀⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣤⡀⠀⠛⠘⠀⣴⣆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣄⠀⠀⡇⢳⠀⡶⡄⠀⠀⠀⠀⢀
⠀⠀⠠⠀⠈⠃⢓⠀⠁⠀⠅⠄⠸⠀⢀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡊⡡⠀⡇⠈⠀⠇⠀⠀⡀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠠⠁⠀⠀⢁⠀⣄⢸⠐⢂⡄⡠⢰⢻⠃⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⡄⠰⠐⡇⡀⡄⢘⢹⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠔
⠄⠀⠀⠀⠀⠀⠇⠈⠤⠀⠸⠀⢀⡠⢨⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⠀⠔⠀⠀⠀⠀⠈⡁⠐⠄⠀⠧⠀⢀⠦⣤⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⠤⠄
⠀⢀⡄⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠰⠀⡈⢄⡘⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⣀⡀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⣀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡫⠂
⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠳⣤⠤⠤⠤⠤⠐⠒⠚⠉
⠀⠀⠈⠑⠶⠶⠤⠤⠶⠶⠤⣤⡎⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡳⠦⠤⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠤⠰⠀⠄⠀⠆⢀⠀⢄⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡀⠐⠁⡀⠀⠀⢀⡀⠄⣠⣤⡤⡴⠀⠀⠀⠐⠀⠀⠀⠀⢀⢄⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢄⡠⠞⠴⠋⠑⠈⠁⠀⠘⠧⠤⠤⠤⠄⠒⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡞⢻⠉⣿⠉⡏⢹⣆⠀⠀⠀⠀⠀⠀⠀⡞⢻⡏⢹⠀⡇⢀⣇⠀⠀⠀⠀⠀⠀⢰⠛⡏⢹⡏⢹⠀⣿⠀⣠⠴⣄
⠀⠀⠀⠀⠀⠀⠐⣧⡼⣄⣿⣠⢷⡇⠈⢧⠀⠀⠀⠀⠀⠀⣧⡼⣇⡼⠶⠃⣏⠀⢳⡀⠀⠀⠀⠀⢸⣤⢯⣸⣇⡼⠀⣿⠜⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠤⠿⠀⠈⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠴⠼⠆⠀⢧⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣀⣠⠤⠎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣠⠤⠞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⡍⠁⠀⠀⠀⠀⠀⠀⠀⠈⣭⣭⣭⣭⣭⡍⠁
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣦⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⢳⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡞⢳⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⡏⢸⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⣠⣤⡴⢾⠀⣿⠀⡇⠀⠀⣀⠀⠀⠀⠀⢀⣤⡇⢸⡇⢸⠀⡇⠀⠀⣀⠀⠀⠀⠀⢸⠀⡇⢸⡇⢸⠀⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⠀⠋⠀⡇⡰⠋⠉⡇⠀⠀⠀⣿⠀⡇⠘⠁⠘⠀⡇⣠⠎⠉⡷⠀⠀⠀⢸⠀⠃⠈⠃⠘⠀⣿⣠⠞⠉⣱
⠀⠀⠀⠀⠀⠀⡟⠋⠑⠋⠀⠀⠀⠛⠁⣠⠞⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠛⠁⢠⠞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠛⠁⢀⡜⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠎
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣤⣤⣶⡶⠶⠿⠿⠿⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⢛⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢿⣦⣤⣠⣤⣶⠾⠟⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠉⠻⢷⣄⡀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠙⠿⣦⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠈⠛⢿⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⢿⣦⣀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⣶⣿⠛⢿⣦⣀⠀⠀⠀⠈⠻⢷⣦⡀⠀⠀⠀⠉⠻⣷⣤⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⣿⣄⣀⣨⣿⣷⣤⡀⠀⠀⠀⠙⢻⣷⣄⡀⠀⣠⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⢿⣯⣻⣶⣄⡀⢀⣼⡟⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⣴⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⣠⡞⠁⢀⣤⣀⣤⡼⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢠⠆⠀⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⢄⢀⣴⡟⣠⣶⣿⣿⡿⠋
⢠⣄⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋
⠀⠙⣦⡀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⣀⠀⠀⠙⢿⠟⠋
⠀⠀⠈⢿⣄⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⡟⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠀⠀⠘⣇
⠀⠀⠀⠈⢿⣷⡄⠙⠻⢆⠀⠀⠀⠀⢀⡞⠉⠛⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣿⡿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠏⢧⣀⠀⠀⠁⣀⠀⡠⠋⠀⣤⡾⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣴⣿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⠀⠈⣿⣷⣦⣤⠛⣶⣅⣀⣠⣿⣷⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⠀⠀⣠⠀⢀⡟
⠀⠀⠀⠀⠈⣇⣈⣿⢻⣿⣿⣿⡿⣿⠟⢁⠋⣸⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⡀⣰⡏⠀⣸⣇
⠀⠀⠀⠀⠀⠉⢫⠙⠀⠘⠗⠙⠈⠻⠴⣋⡶⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⢡⣿⣷⣾⣛⣿⠂
⠀⠀⠀⠀⠀⠀⠈⠢⡔⠐⠂⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠊⣠⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠈⠉⠉⠁⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠁⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠧⠀⠈⠀⠀⡎⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠀⡰⠂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⠀⠀⠀⠀⣥⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⡔⠂⠀⢠⠐⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡄⣦⠀⠈⡀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠇⠀⠈⠃⡎⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⠀⢠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠁⢠⠋⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⢀⣴⣿⣿⣿⣿⣿⣿⣻⣿⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⣠⣿⣿⣿⣿⣿⣿⣿⡿⢯⡉⠳⠶⠎⠉⠛⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⢀⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⣼⠋⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⣠⣿⡄⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡌⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⠆⠀⠀⠀⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣮⡳⣄⠀⠀⠀⠀⢀⣤⣞⣵⠏⠀⠀⠀⣼⣿⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣏⣵⡀⠀⢀⣼⡿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠉⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠴⣶⣤⣀⣀⣀⣀⡀⠔⠊⢀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣭⣍⣥⣤⣵⠶⠶⠛⠿⢷⠖⣴⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣶⣾⣶⡷⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁


⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⢀⡠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⢠⡾⣽⠿⠛⠙⢻⡄⠀⠀⠀⢠⣿⣿⡿⠿⠿⣿⣆⠀⠀⠀⣴⢫⣶⠞⠋⠙⣷⡀⠀⠀⢠⢞⣯⠿⠛⣿⢷⡄
⠀⠀⠀⠀⢸⣿⡄⠀⠀⢸⠀⣿⠀⠀⠀⢿⣿⡁⠀⠀⢠⣿⣿⡆⠀⠀⣿⣌⡁⠀⠀⠀⢹⡇⠀⢀⣿⣼⠃⠀⠀⣰⠈⣷
⠀⠀⠀⠀⠈⠻⣿⣶⣤⣭⡾⢻⠀⠀⠀⠘⢿⣿⣶⣶⣿⣿⣿⠀⠀⠀⠹⢿⣿⣦⣤⣶⠀⠀⠀⠀⢿⣿⣷⣶⣞⣡⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣸⠃⡼⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠉⠉⠉⣿⢣⠃
⠀⠀⠀⠀⠰⣶⣤⣄⣚⣡⠞⠁⠀⠀⠐⢦⣄⣤⡰⠛⠉⠉⠀⠀⠀⠀⠀⣴⣄⠀⣂⠄⠁⠀⠀⠀⠘⣤⣀⣠⣾⡵⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢖⣊⡉⠉⠑⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡈⠩⠥⣀⢀⣀⠀⢈⣵⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡣⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⠀⡔⢀⠀⢲⢄⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⡇⡌⠉⠙⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡰⠀⡜⠀⠀⣿⣇⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢒⣞⠁⠈⡞⠀⠀⠀⢿⢡⠟⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢹⠉⠢⡔⠃⢠⢤⣤⡀⡤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⢛⠒⠊⠀⢀⣎⡘⠻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁⠀⢹⠀⠀⠀⡼⣀⢗⠇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠁⠀⠀⠀⠀⣰⠃⠀⢀⡾⣀⡜⡞⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⣀⡤⠊⣵⠀⢀⠞⢤⡜⡻⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⣀⣀⠀⠀⠀⠀⠀⠈⠒⠚⠒⠒⠈⠁⠀⠀⠻⢤⠋⣢⠊⣱⠁⠀⠀⠈⣅⡠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣰⢄⣈⣉⣉⡒⠂⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡱⠁⠀⠀⠀⠀⣧⠞⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣿⡆⠀⠀⠀⠁⠘⠀⠸⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠀⠀⠀⠀⣠⠖⠁⣨⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚⢤⣼⠇⠀⠀⠀⢀⣾⣏⣙⠢⡄⣀⠀⠀⠀⠀⣧⠀⠀⠀⣀⡤⠒⠉⢀⡤⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠊⠉⣳⠀⠀⠀⠀⢸⣧⣤⡌⠁⡞⣿⠉⠓⠢⣶⠽⡖⠊⠉⠀⠀⡠⠖⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠸⠷⠒⡆⠘⠧⠇⠀⢤⡔⣧⣿⠛⠒⠒⠒⠚⠁⠀⠀⠀⡬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠤⢄⣀⠀⠀⢸⠋⣉⢭⣽⣖⣒⣶⢲⡾⠀⠀⠀⣀⣠⡤⠀⠀⠀⠐⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠃⠠⡟⣉⢀⣀⣀⣸⢸⣧⠤⠒⠋⠁⠀⠉⠀⠀⠀⣀⠤⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⣉⣉⣹⣤⢤⠤⠤⠤⠤⠄⠶⠒⠊⠉⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢊⡏⣹⡟⠛⠈⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⣷⠈⢦⡀⠰⣼⣤⣀⣀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⡀⠤⢴⠧⢤⣤⡧⣀⣀⣀⣀⣀⣀⣀⣠⣧⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠟

⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⣠⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⣠⠏⠀⣿⢠⠞⡥⠀⠀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⣄⠀⠀⢰⠃⠀⡇
⠀⠀⠀⡼⠁⠀⢀⣏⣼⠞⠁⠀⢰⠇⢸⡇⠀⢀⡤⢤⣀⡤⣤⠀⠀⠀⣸⣇⣀⠀⠙⣦⠀⢸⠀⠀⢻⡀
⠀⠀⡼⠁⠀⣠⠞⠉⠀⠀⠀⠀⠟⠀⣸⠳⡆⢸⡄⠀⡿⠀⣸⠀⠀⣸⣁⡀⠉⠙⢷⡈⢧⡈⣧⠀⠀⢧⡀
⠀⡼⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⡇⠀⢳⡴⠇⠀⣿⠀⡏⠁⠈⠉⠛⢦⠈⣧⣠⣧⣿⠀⠀⠈⢷⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⡴⠋⠀⢀⡼⠉⠳⣿⡉⠛⢶⣀⠈⢳⡘⡇⠀⢹⡀⠀⠀⠈⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⡴⠛⢷⡀⠀⠈⠙⠀⠀⡹⢧⣀⡷⠋⠀⠀⠷⠀⠀⠀⢹⡀
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠙⣦⡀⠀⠀⠊⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⣷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠉⠛⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⢀⣠⣤⣤⣌⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠚⠷⠞⠛⠋⠙⠉⢉⡀⠀⠀⠉⣧⠀⠉⠛⠲⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⣈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⢾⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠶⠶⠶⠞⠀⢀⡴⠿⠔⠁⠀⠀⠀⠀⠀⢀⣀⡩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠃⠀⠀⠐⠒⠀⠀⡈⠙⠲⠦⠤⠤⠄⠐⢚⣋⡤⠶⠀⠀⠀⠀⠀⠘⠫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠀⠀⣰⠃⠈⠻⣄
⠀⣠⠋⠀⠐⠠⣀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡐⠁⠀⠀⠀⠸
⣴⣁⣀⣀⣤⣤⣀⣽⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣠⣄⡀
⠛⠋⠉⠉⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠈⠙⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠁⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠉⠓⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠉⠀⠀⠀⠀⠈⠉⠛⠲⠦⠤⢤⣤⠤⠴⠒⠊⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠠⢤⢄⣼⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄
⢀⢀⠀⢀⡀⠀⡀⠐⠀⠠⠴⠶⠶⠶⠶⠶⠆⠂⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠶⠶⠶⢤⣈⣳⣄⠀⡀⠀⠀⠀⠀⢀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣦⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣲⣶⣶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠤⣤⣄⣤⣦⣄⣀⣠⣤⣠⣤⣤⣤⣤⡶⠋⠉⠉⠀⠀⠀⠀⠀⠉⠓⢶⣤⣀⠀⢠⣴⣾⣿⣿⣿⣿⡅
⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠙⠋⣛⣉⡡⠀⣠⣾⣷⣶⣶⣶⡦⣀⠀⠀⠀⠈⠙⠛⠿⣷⡌⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠙⠛⠻⠛⣻⣏⣹⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⢀⡀⣹⣿⡄⢹⣾⣝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⣄⣤⣴⣶⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡇⠘⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠉⠉⠉⣹⣿⡿⠟⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋⢀⣶⣯⣿⣿⣿⣿⣿⣿⣿⣟⣩⣿⣿⣿⣿⠇⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⣤⣶⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⢻⡹⣷⣷⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠷⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠀⠘⣿⡿⠿⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡀⠀⠀⣀⢠⣤⡄⠀⣶⣆⣀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠂
⠀⠀⠀⢀⠀⠀⣸⠷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣽⣯⠀⠀⡈⠛⣿⣦⠀⢹⣿⣿⡿⠶⠖⢠⣀⠀⠀⠁
⠀⠀⠀⠈⠀⠀⠉⢀⡦⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠻⠿⠇⠘⠉⢢⣭⠛⠂⠂⣿⣿⡇⡄⢀⣼⣿⣿⣦
⠀⠀⠀⠀⠀⠐⠶⣾⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢶⣿⢟⡃⠀⠀⠀⠈⠻⠟⢾⣧⠈⠉⠁⠻⣿⣿⡿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠃⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⡧⠄⠀⠀⠀⠀⠀⠤⣥⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢄⣤⡀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⡄⠀⡀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣾⡿⣷⡶⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⠤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣟⣽⣷⣳⣿⣇⠀⠀⠀⠀⢠⣤⣸⣿⣿⣿⣶⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⢿⣟⣁⠀⢀⡖⠀⣾⣿⣷⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⠛⠁⡀⢸⡇⡀⠈⠻⠿⢿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛

⠀⠀⠀⢺⢳⣴⣄⡀⢸⣦⣄⣠⣤⡆⠰⣰⡄⢰⣦⡄⢠⣦⣤⣄⣏⣄⣤⣶⣠⠀⣠⣴⣦⣴⣤⣰⡆⢰⢳⣤⣤⣤⡄⢰⡶⣶⢠⡄⣤⢠⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠈⠀⠁⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⣠⢠⠐⡇⢠⡄⢸⠆⣤⢠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⡴⠋⠉⠛⠛⠒⢂⣠⣴⣶⣿⣿⣷⣶⣶⣶⣶⣷⣌⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⣠⠏⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡼⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠰⠷⢀⡞⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠉⠻⣯⣛⡛⢛⣋⣽⣿⣧⠹⣏⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠳⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠇⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠙⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣫⣵⣾⣿⣿⡿⢇⣤⡀⠀⠀⠀⠠⣄⠀⠀⠈⠳⣄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢛⣩⣶⣿⣿⠟⠀⠀⢀⠀⠈⠳⢦⡀⠀⢈⡷⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣫⣵⣿⣿⠆⠈⠓⣦⣀⣠⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠓⠒⠛⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠛⠟⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠘⣧
⠀⠀⣤⡄⠀⣀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⡀⠀⠀⢰⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣿⡄
⠀⠰⣼⡀⠀⢹⠄⠇⠀⠈⣿⠁⠸⠀⡏⠘⠿⠌⠀⠸⣷⠀⠜⠩⢍⣿⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⣤⣾⣿⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠏⠀⠀⠀⠀⣹⣿⠏⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠸⣴⣷⠀⢠⠆⠀⢦⣾⡃⠀⠀⠀⣴⠃⠀⢉⣷
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⡿⢻⡖⠁⠀⠀⡆⠀⢻⠀⢠⠆⠾⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⢳⡾⠂⠁⣼⠇⠀⠀⢀⣇⣄⠈⡄⠘⣧⣠⡯⠀⢀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠇⠀⠀⠤⠄⣀⠓⠀⠶⠉⠀⠀⠔⠀⠘⠀⠀⠀⢀⠘⠋⠀⠀⠈⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⡀⠀⠀⢲⠀⠀⠀⠀⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠰⠆⠈⠷⡦⠀⠀⠈⠃⠀⠁⠄⠈⠷⠃⠀⣤⠘⠻⠤⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠛⠶⠶⣦⣄⡤⠀⠀⠀⠳⠆⠰⣼⡷⠀⠉⠁⠀⠀⠀⠀⠀⡰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣦⣤⣿⣷⣴⣤⡀⢢⠀⠀⠃⠀⠀⠀⠀⣤⣤⣴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠍⣉⣀⠀⣉⡟⠋⢁⣤⡴⣾⣶⣦⣄⣀⣀⣛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡴⠞⠛⠛⠿⣯⡁⠀⠉⠀⠙⡏⠀⠀⠙⠉⢹⢽⡿⠏⠿⠷⠾⠿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢴⣦⢉⡄⠀⠀⠀⠀⠀⣾⣷⠄⠀⠐⡀⠀⠀⠈⠙⠛⠻⠷⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⡄⢙⡿⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈⠉⡀⠀⣄⢀⣤⣅⡀⠀⠸⣆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⢀⣀⣐⣾⣿⣯⣇⠀⢁⣈⠉⠂⠀⠀⠉⢻⡆⠹⣧⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠀⠀⠀⠀⠘⣿⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⢿⣿⡌⢿⣄⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣈⣀⠈⠿⠆⡀⠀⠀⠀⠙⢟⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⠿⣾⣿⣄⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢹⡧⠄⠀⠀⢉⡇⠀⠀⠀⠀⠈⠁⠈⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⢙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣤⣶⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠿⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣯⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⣠⣴⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣉⡉⢠⣿⣷
⣤⡀⠀⠒⠒⠚⠉⠉⠁⠈⠂⠀⠀⠘⢟⣛⣛⡛⠛⠻⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⣿⣅⣼⣿⣿⣄⣀⠀⠀⠀⣤⡀
⣿⣿⣷⣶⣿⣶⣂⣀⣠⣤⠀⠀⢦⣤⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠌⠳⣶⢦⣄⠀⠀⠀⢻⣿⣿⣿⣿⢛⡿⠃⠀⠀⠘⣿⣶⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⣀⢀⣰⠶⣀⣀⣀⣀⣠⣤⣄⣀⠰⠀⠴⠖⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣀⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⣠⣾⡿⠟⠁⣠
⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣋⣉⣀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣷⠄⠢⠀⠀⠀⠀⠀⠸⣷⣶⣶⣟⣻⡟⠁⢿⣿⣿⣿⡇⠀⠀⠀⠊⠁⢈⣀⠴⢻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠶⠒⠒⣒⠚⣻⣿⣿⣿⣿⣿⣷⣶⠶⠖⠀⠀⠀⠀⠀⠻⣿⣿⡿⢿⣷⣧⠈⣿⣿⣿⠀⢀⠀⠀⢨⣿⣏⣵⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡤⠤⡤⡤⠄⠀⠀⠙⢧⣤⣶⣿⣿⣤⣿⣿⣿⡇⠀⢠⣶⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣭⠿⣿⣿⣿⣛⣛⣀⠐⠂⠀⣀⣀⣀⣄⡀⠉⠿⠋⢻⡿⣿⣿⣿⣷⣦⣾⣿⡄⠀⠙⡟⠻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣋⠛⠁⠙⠛⠀⠀⠈⠷⠈⡙⠛⠛⠛⠋⠉⠁⠀⠀⡁⠀⠉⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣼⣿⣿⣿⣿⣷⣤⣤⣶⣤⣄⣀⣄⣀⢠⡌⠃⠀⠀⠀⠀⠀⠀⣴⡋⠀⠀⢠

⠀⠀⣀⣤⣔⠀⢀⣤⣀
⠀⡞⠙⠿⢿⠲⡜⠿⠏⠳
⠀⢷⣦⠀⢸⣆⡉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⢀⣀⡀
⠀⠈⠳⠄⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢀⡄⠀⢠⡾⠃⠀⠀⠀⠀⠀⡴⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠈⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⣀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣦⣤⣤⠀⠀⠀⠂⢠⣶⣿⣷⣦⣀⠀⠐⢄⠱⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⢿⣿⣿⣿⣿⠇⠀⠀⠀⢿⣿⣿⣿⣿⠿⣦⠀⡰⠂⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠻⡛⠿⠟⠃⠔⠉⣻⠇⠨⠟⢋⠿⠏⠀⢹⣇⢃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢸⡿⠀⠁
⠀⢆⠂⠤⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣆⠀⠀⠀⢀⣶⣦⡀⢀⡆⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⠘⣿⣿⣿⣷⡿⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠊⠩⠄⠀⠐⠉⠀⠀⠀⠈⠻⣧⣄⣀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢛⣿⠀⠀⠀⣀⣀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⣤⣽⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣿⠇⠀⠀⠀⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀⣀⣠⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠸⡏⠛⠋⠁⣿⠀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⡤⠢⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⣀⡀⠀⡇⠀⠘⡀⠀⠀⠜⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇⠀⠀⡰⠁⠀⢸
⠀⠀⠀⢈⠀⠘⡄⢰⠀⠀⢃⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡇⠀⢠⠃⠀⠀⡎
⠀⠀⠀⠘⠀⠀⢡⠀⡆⠀⠘⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠁⠀⡎⠀⠀⠰
⢠⠒⢢⠀⢃⠀⠀⠆⢱⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣸⠀⡸⠀⠀⢠⠃
⠈⢆⠀⠱⡘⡄⠀⠸⡀⡆⠀⠸⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡆⢠⠃⠀⠀⣎⣀
⠀⠈⢣⠀⠘⣼⠀⠀⢇⠰⠀⠀⠳⠃⠀⠀⡇⠀⠀⠀⠀⡠⠒⠒⡄⠀⠀⠀⠀⡎⠀⠙⢦⠀⠀⠀⠀⢸⠀⠀⠀⣧⠏⠀⠀⡜⠀⠀⠑⡀
⠀⠀⠀⠳⡀⠘⣧⠀⠘⠒⠁⠀⠀⠀⠀⠀⡇⠀⠀⡠⠊⠀⠀⢀⠇⠀⠀⠀⠀⢣⠀⠀⠀⠑⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⡼⠠⢄
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠊⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠢⠜⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠞⠀⠀⣸
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⠎⠀⠀⡰⠁
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⡎⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⢋⠎
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠤⠤⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⢄⣀⠀⠀⠀⠀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠴⠒⠒⠒⠒⠒⣛⣛⠉⣧
⠀⠀⠀⠀⠀⠀⠀⣰⠶⠭⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠱⢶⡀⢻⠔⣠
⠀⠀⠀⠀⠀⠀⢸⠻⣮⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠸⣞⡵⢋
⢀⣀⣀⣀⣀⣤⣏⡓⠀⠈⣹⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡴⢃
⣿⠉⠉⠁⣴⣾⠳⠁⠀⡰⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⢋⡄
⢸⠀⠀⠀⢶⣧⠉⠀⣴⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢋⡄⠀⠀⠀⠀⠀⣠⢊⣝⠒⣷⠓⢲⡄
⠸⡇⠀⢈⣿⡂⢀⡾⠷⠆⠀⠀⣀⣀⣀⣀⣤⠤⠤⠴⠀⠀⣀⣀⣀⣀⣠⡤⠀⠀⠀⣧⢋⡤⠀⠀⠀⢀⡜⡵⢻⡇⢀⡼⣻⢿⡟
⠀⡇⠀⢒⡧⢀⣞⡛⠃⢈⣭⠭⠭⠤⠴⠶⠒⠒⠚⠋⣉⣉⣉⣡⡤⠤⠤⠄⠀⠀⠀⢷⢎⡴⠀⠀⣠⣾⠞⣠⠛⠙⠯⣞⣱⣿⠋
⠀⡇⠀⢽⣇⡾⠄⠀⠀⠠⠤⠤⠶⠒⠒⠒⠋⣉⣉⣉⣩⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⢸⠎⡴⢃⡴⢻⣣⠞⢁⡰⢌⣓⡞⠉
⠀⣧⠀⣻⣿⠂⠀⠀⠀⠀⠶⠶⠒⠒⠛⠉⠉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣰⣏⣠⠿⠁⢀⠲⣝⣲⠟
⠀⢸⠀⢾⡏⠁⠀⠀⠀⠀⠤⠴⠶⠒⠚⠋⠉⠉⠉⠀⢀⣀⣀⣀⣀⣤⠤⠄⠀⠀⠀⠀⢿⣔⡴⠉⠀⣜⠦⣙⠞⠁
⠀⢸⠀⠨⡇⠀⠀⠀⠀⠀⠤⠤⠴⠒⠒⠒⠛⣋⣉⣉⣩⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⢠⡓⠬⡽⠋
⠀⢸⡆⠨⣷⠀⠀⠀⠀⠀⠒⠒⠒⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡠⣌⠲⣭⠟⠁
⠀⠘⡇⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⢠⡲⢽⣪⠟⡁
⠀⠀⡇⠀⢺⡆⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⠋⢨⣒⡬⢿⣧⠎⡀
⠀⠀⢧⠀⢩⡇⠀⠀⠀⠀⠀⠀⢀⡞⢁⡶⢀⠆⣠⢀⡀⠈⡉⠉⠉⢁⡀⣠⣯⣦⠵⡋⢁⡈⣧⢎⡄
⠀⠀⢸⠀⠐⣧⠀⠀⠀⠀⠀⢠⡞⠀⡼⣡⢏⡴⢧⢾⣠⠞⠥⠔⠲⠞⠉⠉⠑⠋⣚⣥⣠⠖⣵⠾⢀⡄⣠⢀⡀
⠀⠀⠘⡇⠈⢿⠀⠀⠀⠀⠀⣙⣤⣞⣉⣁⣤⣤⠤⠤⠤⠶⠶⠒⠒⠚⠛⠋⠉⠉⠉⠉⠀⠈⢻⣵⢋⡞⢡⠞⣡⠋⡰⢂⡄⡀
⠀⠀⠀⣧⠀⢽⡄⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠏⡴⢁⡴⢁⠞⣡⢋⠜⣱⢋⡔⣠⢂⡀
⠀⠀⠀⢸⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⣁⠀⠀⠀⠀⠁⠋⠘⠁⠚⠰⠣⠞⠼
⠀⠀⠀⠘⡇⠐⣿⠘⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⢯⡜⢁
⠀⠀⠀⠀⣧⠀⣿⡄⠋⠉⠁⠁⣀⣀⣀⣀⣀⣀⡤⠤⠤⢶⣒⣒⣚⣿⠭⠭⠿⠷⠒⠚⣧⠞⣰⠋⡴⠃
⠀⠀⠀⠀⢹⠀⢨⡿⠯⠭⠭⠽⠿⠛⠛⠓⠒⠛⠉⢉⣉⣉⣉⣠⣤⢤⡤⢤⡴⣶⠖⣲⢋⡴⣡
⠀⠀⠀⠀⠸⣦⣤⣤⠤⡴⢶⡶⣶⠒⡶⢛⡟⢉⡝⣩⢋⡼⢁⡞⢡⠞⢠⠏⠰⠃⠐⠁⠘⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀⢰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⢀⣾⣿⣟⣠⣄⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⡀⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣦⠀⢀⡀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣿⣿⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡿⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠰⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢁⢸⣭⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠉⠀⢀⣀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣏⡁⠀⢰⣅⢄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠴⠀⠈⠁⠀⠸⢮⣜⣧⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠁⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣀⣾⠿⠿⠓⠦⠼⠾⣷⣷⣷⣾⣾⣴⣠⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣠⠴⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠟⠁⠀⠀⠐⠂⠀⠀⠀⣠⣿⣦⠀⣤⣄⢤⣀⣀⣀⣤⣤⣴⣦⣤⣾⠟
⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⢉⠁⠠⠂⠀⠀⠀⠀⢀⠤⠂⠛⠛⠛⠙⠋⠙⠋⠉⠋⠉⠉⠁⠀⠀⠀⡀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⣀⣤⣴⣶⣦⣠⣴⡤⢰⣐⣀⣠⣦⣥⣴⠦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠛⠐⠉⠉⠉⠉⠙⠛⠛⠋⡹⠶⠛⠉⠣⠍⣛⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷⣿⣾⣿⣾⣶⣼⣶⣴⣦⣀⠀⣀⡄⠀⣤⣀⣄⢀⢴⢀⠉⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⣯⡶⠛⠉⠙⠻⢿⣿⣿⣿⣿⣿⠿⠿⠿⠷⠿⠛⠛⠻⠾⠿⠾⢦⣤⣤⣀⡈⢙⣦
⠀⠀⠀⠀⠀⢀⣴⣶⡿⣿⣻⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢾⡿⢟⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠁⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡔⠢⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣿⢫⣝⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠠⡉⢳⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣟⢶⡯⣻⡿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢌⡩⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣷⢾⣻⣵⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⣤⣤⠤⠤⠤⡤⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣻⣛⡟⢿⣛⣛⣛⣟⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⡠⠤⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⡶⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⢆⣀⣀⠀⠀⠀⠀⠀⠙⠦⣀⡰⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣹⡿⢿⣿⣿⣿⣿⣿⣮⣝⢿⣏⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢾⣡⠜⢲⣄⠀⠀⠀⡀⠢⣌⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡧⠾⣣⡍⠻⣿⣿⣿⢿⣝⠷⢼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠓⢶⣁⡴⢳⡔⢦⣈⡶⠊⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣮⡉⠾⣫⣬⣿⣝⠷⢉⣵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣶⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⢀⡠⠖⠛⢉⠉⠙⠓⠦⣄⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠙⠒⢦⡀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠛⠲⢄⡀
⠀⠀⣠⠋⠀⠀⠀⣾⡀⠀⠀⠀⠈⢳⡀⠀⠀⡴⠋⠀⠀⣿⢠⡇⠀⠀⠀⠙⢦⠀⠀⢀⡞⠁⠀⠀⡆⡶⡄⠀⠀⠀⠙⣄
⠀⢠⠇⠀⠀⠀⢰⣿⣿⣦⠀⠀⠀⠀⢳⠀⢸⠁⠀⠀⠀⣽⣿⣷⡆⠀⠀⠀⠈⡇⠀⡞⠀⠀⠀⠀⣷⣿⣧⠀⠀⠀⠀⠸⡄
⠀⢸⡀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢀⡇
⠀⠈⣇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⡞⠀⠸⡄⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢠⠇⠀⢳⡀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⣸⠁
⠀⠀⠈⢧⡀⠀⢀⣿⣿⡇⠀⠀⣠⠞⠀⠀⠀⠙⢦⡀⠀⠈⣿⣿⣆⠀⢀⡴⠋⠀⠀⠀⠳⣄⠀⠀⢸⣿⣿⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠓⠺⠿⠿⠷⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠶⠿⠿⠿⠒⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠿⠿⠗⠚⠁
⠀⠀⠀⠀⣠⠴⠒⠚⠛⠒⠲⢤⡀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠒⠶⣄⡀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠛⠓⠒⠦⣄
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⣰⠋⠀⠀⣀⢠⡀⠀⠀⠀⠙⢆⠀⠀⢀⡴⢁⣴⢠⡄⠀⠀⠀⠀⠈⠳⡄
⠀⢠⡏⠀⠀⠀⣇⣿⡇⡤⠀⠀⠀⠀⢷⠀⢰⠃⠀⣀⠀⣿⣿⣏⡴⠀⠀⠀⠘⡇⠀⡾⠘⣦⣿⣿⡇⣠⠴⣄⣠⣀⠀⢹⡄
⠀⢸⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⢸⠀⣿⠀⠀⠙⣷⣿⣿⣿⠁⠀⠀⠀⠀⣿⠀⡇⠀⢸⣿⣿⣿⠏⠀⣿⣿⣿⡧⠀⡇
⠀⠘⣇⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⡾⠀⠸⡄⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⢠⡇⠀⢷⠀⢨⣿⣿⠃⠀⠀⢸⣿⣿⠁⣸⠃
⠀⠀⠘⢦⡀⠀⠘⣿⣿⡄⠀⠀⣀⠞⠁⠀⠀⠹⣄⠀⠀⢸⣿⣿⡄⠀⠀⣠⠏⠀⠀⠈⠳⣾⣿⡟⠀⠀⠀⢸⣿⣿⡴⠃
⠀⠀⠀⠀⠙⠲⠤⢿⣿⠷⠴⠚⠁⠀⠀⠀⠀⠀⠈⠙⠶⠤⢿⡿⠷⠶⠋⠁⠀⠀⠀⠀⠀⠈⠛⠧⠤⣤⡤⠼⠟⠋
⠀⠀⠀⠀⢀⡤⠴⠒⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠲⠤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠦⢤⡀
⠀⠀⢀⡞⢁⡀⢀⠀⠀⢀⡀⢀⠙⢦⠀⠀⠀⣠⠞⣁⡀⡀⠀⠀⣄⢠⢈⠳⣄⠀⠀⠀⡴⠋⣀⢀⡀⠀⢠⢀⣤⡈⢳⡀
⠀⢀⡏⢤⣸⣿⣼⠀⢀⢸⣇⣿⡀⠈⢧⠀⢰⠃⣄⣿⣧⡇⠀⡀⣿⣿⣿⠀⠘⡆⠀⡼⠱⡌⣿⣾⡇⢀⣸⣾⣿⣴⠃⢹⡀
⠀⢸⠀⠀⣿⣿⣿⣶⠋⢸⣿⣿⡟⠀⢸⠀⣾⠀⢸⣿⣿⣿⡾⠁⣿⣿⣿⠀⠀⣷⠀⡇⠀⢹⣿⣿⣷⡟⢹⣿⣿⡷⠀⠈⡇
⠀⠘⡇⠀⢹⣿⣿⠃⠀⠘⣿⣿⡇⠀⡼⠀⢸⡀⠀⣿⣿⡟⠁⠀⢹⣿⣯⠀⢀⡇⠀⢧⠀⠘⣿⣿⠟⠀⠀⢿⣿⡇⠀⢸⠃
⠀⠀⠙⣦⣿⣿⡇⠀⠀⠀⣿⣿⣇⡼⠁⠀⠀⠳⣴⣿⣿⠁⠀⠀⠈⣿⣿⣤⠞⠀⠀⠈⢧⣼⣿⣿⠀⠀⠀⢸⣿⣿⣴⠋
⠀⠀⠀⠈⠙⠿⣥⣀⣀⣠⡽⠟⠋⠀⠀⠀⠀⠀⠈⠛⠿⣄⣀⣀⣠⠿⠛⠁⠀⠀⠀⠀⠀⠙⠻⢧⣀⣀⣀⣬⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣾⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢠⠏⢰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠃⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡀⠀⠀⠀⢸⡟⣽⠁⠀⠀⠀⢀⣠⡴⣞⡯⠗⠋⠁⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⠃⠀⠀⣠⠋⣰⡿⡆⢀⣤⠞⠋⠁⠊⠁⠀⣀⣤⣾⣿⣾⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡏⠀⠀⠐⠁⠰⢋⣼⡿⠋⠀⠀⠀⠀⠀⣐⣯⠽⠟⠋⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⣤⣀⠀⠀⠀⢀⡼⣣⠟⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠊⠉⠁⣀⣤⣴⣿⣭⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠠⡄⠀⠀⠉⠓⢦⣶⣯⠞⠁⠀⠀⠀⠀⣪⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⠭⠷⠖⠛⣋⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣿⡄⠀⠀⢠⠀⠀⠀⠀⠉⠃⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣠⣤⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠓⠺⣿⢶⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠛⠛⠛⠛⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠒⠬⣝⣻⣿⣿⣛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣝⣶⣶⡶⠭⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡄⠀⠀⠀⠀⠈⢦⡀⢄⡀⠘⢦⣄⣀⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡱⡀⠀⠀⠀⠀⠀⠙⢶⣝⣶⣤⣙⣿⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠀⠀⠀⠀⠀⠀⠀⢻⡙⣏⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢳⢸⡛⠛⠓⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠛⠻⢿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠓⠂⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠈⠉⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠈⠛⠦⣄⡲⢤⣄⣀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣆⡀⢰⡄⠱⣄⢄⠀⠈⠲⣤⣄⡀⠉⠳⢭⣻⣏⡛⠲⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⣿⣷⡀⢻⣆⠈⢮⣷⣤⡀⠈⢿⣿⣷⠦⣤⣉⡻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢻⣷⡘⡏⢷⡀⠻⣿⡙⢦⣄⠙⢿⣷⡀⠉⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠻⣧⣿⠀⠙⢦⡹⣷⠀⠈⠛⠶⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠙⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⢿⣷⡀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⢸⣿
⢀⣴⡿⠟⠿⢿⣦⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⣾⡏⠀⠀⠀⠈⣿⡇⠀⠀⠈⣿⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⢻⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⠈⣿⡆⠀⠀⢻⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠈⣿⡆⠀⠀⠀⠀⢻⣧⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⢻⣷⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠸⣿⡀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣾⠟⠉⠉⠙⢻⣷
⠀⠀⣿⡇⠀⠀⠀⠀⢻⣧⠀⠀⣿⡇⠀⠀⠀⠀⣿⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢻⣧⠀⠀⠀⠀⢸⣿⡀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⡀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⣼⡟⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣿⡀⠘⣿⡆⠀⠀⠀⠀⣿⣧⣤⣤⣼⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣿⡇⠀⣿⡅⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⠀⢸⣿⢠⣦⣠⣶⢶⣦⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣷⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⡟⠘⣿⣿⡇⢀⣿⣿⣿⣤⣄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠇⠀⠈⠛⢿⣿⣿⠉⠻⣿⠻⣷⡄
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣏⠻⣷⣾⠟⠀⢹⣿⡀
⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣿⠟⣿⣆⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡗⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⣿⡄⢸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⣸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣾⣷⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⢂⣂⣀⣀⣁⣀⡀⢸⡇⠰⣣⡴⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⡇
⠀⠀⠀⠀⢠⠆⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⡏⠀⠀⢸⠀⠀⠀⠀⠀⣄⠀⠀⢸⡇⠠⣤⠤⠀⠀⠀⠒⠶⠶⠆⠀⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠
⠀⠀⠀⣸⠀⠀⠀⢸⠀⠀⠀⠀⢠⡈⠳⠀⠘⡇⡘⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣸
⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠛⠊⠀⠀⠀⠀⠀⠀⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢠⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⣠⠋
⠀⠀⠀⠹⡀⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⣀⡀⠀⠀⢀⣀⡴⠋⢉⣹⠶⢤⣤⣤⣤⣤⠤⠤⠴⠖⠚⠋⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⡟⠁⠀⡴⠋⢀⣄⠀⠀⠈⣧⢱⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⡇⣇⣀⣸⣅⡴⠋⠈⢳⡀⠀⣸⢸⢀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠻⣏⣡⠴⡃⠀⠀⠀⣄⠹⡖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣖⣚⡽⣄⣸⠦⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣉⠀⠠⠀⠀⠄⢀⡁⠈⠈⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠈⠈⠀⠐⠈⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠐⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⢸⠒⢤⢠⡀⢠⢢⠀⠠⠀⢠⢰⠀⠀⠀⠀⠀⠀⠈⡝⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈⠐⠀⠐⠂⠀⠀⠀⠀⠀⠄⠁⡀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠰⠂⠀⠡⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠁⠀⢀⠀⠀⠀⠁⠁⠁⠀⠀⠉⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠁⠈⠚⠀⠘⠀⠘⠀⠀⠁⠁⠃⠈⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠀⠀⠀⠀⠐⣶⠂⠀⠀⢀⠤⠒⠒⠲⢶⣤⡀⠀⠀⢲⡖⠒⠲⢦⡄⠀⢲⡖⠒⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⡐⠁⠀⠀⠀⠀⠀⠈⢿⡆⠀⢸⡇⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠠⠤⠠⣿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⢸⡇⠀⠀⠞⠁⠀⢸⣧⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠘⠿⣦⣀⡀⢀⣀⠔⠁⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠈⠁⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣄⠠⠂⠄⠀⠀⠠⠠⠀⠀⠤⡀⠀⠀⠀⡀⢆⣱⣄⢂⣸⠀⠐⡀⠆⢉⢰⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⢀⢠⠀⠀⠀⠀⡆⢀⠀⢀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠂⠂⠀⠀⠰⠃⠐⠐⠸⠘⠀⠆⠚⠆⠟⠧⠀⢐⠀⠀⢀⠀⠈⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠠⠀⠄⠄⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣡⠀⠄⠀⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⢄⠀⠀⠀⠀⡀⢀⢠⢠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⢀⢀⠀⡀⣀⠀⡀⢀⢀⠀⢈⠁⠀⡀⡀⡀⣀⠀⠘⠀⠐⠆⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢠⠀⠁⠀⠁⠈⠈⠘⠉⠀⢀⠁⠁⠃⠀⠀⠨⠀⠠⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠸⠈⠢⠽⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠐⠀⠰⠀⠀⠀⠀⠰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠠⢠⠀⠢⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠈⠉⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡎⠈⢂⠀⠀⠀⠀⡇⠀⡅⠀⠀⠀⠎⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠒⠀⠀⠀⠀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⡄⠀⠀⠀⠀⠀⡅⠀⠀⠠⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⡀⠀⠃⠀⢠⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⡀⠀⠀⠀⠀⠇⠀⠀⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡀⢴⠀⢰⠀⠈⡄⠀⡀⣀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠰⡀⠀⡄⠀⢠⠀⠰⠐⠄⠄⠀⠀⠀⠀⡀⠔⠢⠀⠀⠀⠀⢡⠀⠀⢁⠈⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠱⡀⡃⠀⠀⣀⠄⠀⢠⠀⠀⠀⢀⠔⠡⢀⠄⠀⠀⠀⠀⠀⢂⠀⠀⠦⠀⠀⠐⠜⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠡⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀
⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡠⠂⠠⠀⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢈
⠀⠀⠠⡀⠠⠊⠁⠢⡀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠂⢄⠀⠀⠈⠢⠠⠌⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⡠⢚
⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠁⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠒⠋
⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠂⠐⠂⠒⠂⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⠀⠘⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢀⠈⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣻⣿⣿⣿⣿⣷⠀⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠊⣠⣁⣠⠖⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠀⠀⣹⡏⠁⠀⠀⠀⠉⡀⠀⣠⣄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠡⠶⠄⠀⠀⡡⠂⠀⢹⣿⣆⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠊⠠⠐⠂⠈⢿⣿⣷⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⠟⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⡙⢶⠲⣄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⣂⡼⠋⢧⣸⡤⢄⣤⣀⡿⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡁⣙⡦⠞⣈⡇⣀⣯⣷⡟⠟⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⡉⠉⠁⢮⣩⠽⢧⣬⠟⠁⠀⠀⠀⠀⠀⠀⢀⡀⢀⡴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇⠀⠀⠀⢀⣀⠒⠒⠀⠀⠀⠀⠀⠀⠀⡴⠁⢹⡎⠀⠀⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⢤⣀⡀⠀⠀⡇⣣⢀⡴⠋⠉⠉⠙⢦⢀⡠⡰⠀⠀⢀⡞⠁⢀⣈⡇⠀⠀⠙⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡀⠀⠀⠉⠓⠦⢧⣸⣸⡉⠁⠒⠢⣤⣼⣁⠀⡿⢀⡴⠋⠀⣰⠏⣀⣰⢆⠘⡗⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢭⣀⣀⠀⠀⠀⠀⣉⡙⠒⠲⠒⠚⣇⣘⣻⠛⠉⠀⠀⣠⢯⠞⠀⠉⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠑⠍⠀⠀⠀⠀⣰⠷⠿⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⢛⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡸⣍⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢻⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠒⣤⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠦⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣣⢾⢕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠘⣆⠀⠀⠀⠀⢀⡠⠞⢁⡴⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣾⣌⢧⣀⠤⠞⠉⢀⡴⠋⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣨⢟⠁⠐⢦⡴⠫⣤⠀⠀⢹⡀⠱⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠩⠇⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢻⣿⠁⠀⣀⡤⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠉⠁⡶⢲⠒⠒⠒⠒⠒⢒⣺⣸⡏⠉⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢇⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣒⢾⣤⣅⡀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣐⣻⣳⣤⣀⣨⡙⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⠴⠖⠶⠮⠭⣇⡀⠨⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡀
⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠠⢤⠄
⠀⠀⠀⠀⠀⠀⣹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⣷⣄⠠⡀⠖
⠀⠀⠀⠀⠀⠀⠉⢁⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠒⠀⠀⠂⠐⠀⠀⠀⠀⡀⠀⠘⢈⠀⠀⣿⣿⣶⣿⣧⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂⠄⠀⣀⠤⠀⠉⠀⠠⠀⠀⠀⠀⣰⡀⠀⠀⡀⠄⠐⠈⠀⠀⠀⠀⠉⠀⠈⠉⠉⠛⠛⠻⠄⠐⠎⢠
⠀⠀⠀⠀⠀⠀⡀⢸⠀⣀⡀⡴⠷⠈⠓⠀⢀⣦⡤⠾⣷⣄⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⠂⠀⠀⠀⡀⠀⣀⠔⢃⣠
⠀⠀⠀⠀⡔⠈⠁⠀⢁⢻⣅⠀⠀⠄⠀⠠⠾⣿⠿⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⢤⠆⠠⠀⠀⠀⡿⡽⡏
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠻⡄⠀⢆⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⡀⢠⣿⠙⠁
⠀⠀⠀⠀⠀⢣⣀⣀⠠⠐⠈⠀⢠⠼⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠐⡸⠐⠃⢸⡥⡸
⠀⠀⠀⢀⡠⠂⠉⠀⠀⠀⠀⡰⠀⠀⠀⠀⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⡲⠖⠀⠀⠀⠀⠁⠀⠆⠀⠀⠸⡋
⠀⠀⠀⠐⠒⢠⣤⣄⣀⣀⢐⠁⠀⠀⠀⠀⠁⠀⠈⢲⡄⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠠⠤⠖⠁⠀⠀⠚⣲⠀⠀⠈⠀⠧⣰⢾⠿⢗
⠀⠀⠀⠀⠀⠀⠀⢦⠒⠉⠐⠀⠠⠀⣀⡀⠀⡨⠴⢾⣂⣀⣀⣀⡀⠀⣀⣤⡆⠀⠀⠀⠀⢀⡇⠀⢠⠀⠀⠀⡶⡠⠊⠀⠓⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⠑⡢⠦⠀⠀⠀⠉⠀⠈⠁⠄⠀⠀⠀⢔⠀⢸⠀⢀⠠⠀⠀⡠⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢀⠔⢋⡴⡲⠙⠂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢘⠀⠀⠀⠀⠀⠀⠀⠑⢂⣀⠀⡤⣤⣟⠁⠀⠹⠣⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠡⢠⡾⠀⠀⠐⠊⠠⠶⠚⡷⣊⠵⠉⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠟⠀⠋⠛⠙⠃⠘⠙⠃⠘⠒⠛⠛⠛⠛⠿⠛⠃