Diamon Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠦⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠉⠉⠓⢲⠖⠒⠒⠒⠒⠒⠒⠒⠒⢲⡒⠋⠉⠉⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⢄⣀⣀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀⣀⣀⡬⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⠉⠉⠑⠒⢲⠧⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢿⠒⠒⠉⠉⠉⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⣇⠀⠀⠀⠀⠀⠀⡏⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠸⡄⠀⠀⠀⠀⡸⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢳⠀⠀⠀⢠⠃⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⣆⠀⠀⡎⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡀⣸⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢲⣶⣶⣶⣶⡶⢠⡀⢶⣶⣶⣶⣶⠖⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣆⠹⣿⣿⠏⣰⣿⣿⣄⠻⣿⣿⠋⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠧⠈⠃⠼⠿⠿⠿⠿⠦⠘⠁⠾⠿⠿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣶⣶⣶⣶⠐⣶⣶⣶⣶⣶⣶⠂⣶⣶⣶⣶⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡆⢻⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⠘⣿⣿⣿⣿⠃⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⢻⣿⣿⡟⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠈⣿⣿⠃⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⢹⡟⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠁⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢂
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢢⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠘⠌⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠜⠁
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠈⡀⠀⠀⡈⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠠⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⡄⡄⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡴⠶⠶⠚⠛⠛⣻⡿⣉⠉⠉⠉⠉⠉⢹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠚⠛⠋⢉⢽⡿⣇⠀⠀⠀⠀⠀⠀⣰⠉⣧⠈⠢⡀⠀⠀⠀⠘⡀⠙⢦⡀⠀⠀⠀⢀⣴⠞⠁
⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡏⠀⠀⠀⡰⠋⢸⠇⠸⡀⠀⠀⠀⠀⢠⠃⠀⢹⡄⠀⠈⠢⡀⠀⠀⡇⠀⠀⠙⢦⡀⠀⠉
⠀⠉⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢰⠁⠀⣠⠎⠀⠀⡿⠀⠀⢳⠀⠀⠀⢠⠃⠀⠀⠘⣇⠀⠀⠀⠈⠢⡀⢸⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢀⠇⢀⠔⠁⠀⠀⢸⠇⠀⠀⠈⢇⠀⢠⠏⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢈⣲⣦⣤⠶⠖⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣠⠞⠁⠀⠀⣸⡰⠋⠀⠀⠀⠀⡿⠀⠀⠀⠀⠸⣄⠎⠀⠀⢀⣀⣀⣸⣧⣤⠶⠾⠛⠋⡹⠁⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣤⣤⣤⣤⣯⣤⣤⣤⣤⡤⣼⡧⠴⠶⠶⠶⠒⢻⠛⠛⠋⠉⠉⠉⢹⡇⠀⠀⠀⠀⡰⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠈⠢⡀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣰⠁⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠘⢆⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⡟⠀⠀⠀⢠⠃⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠑⣄⠀⠀⠀⢻⡄⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⢠⠇⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠣⡀⠀⠀⢻⡄⠀⠀⠀⡆⠀⠀⠀⣾⠀⠀⢠⠏⠀⣴⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⢦⠀⠀⢻⡄⠀⠀⢱⠀⠀⢀⡟⠀⢀⠎⢀⡾⠁⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠑⣄⠀⢻⡄⠀⠸⡀⠀⢸⠇⠀⡞⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⠳⡀⢻⡄⠀⡇⠀⣿⠀⡜⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣦⡹⡄⢰⢠⡇⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣎⣾⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⢤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠙⢦⣠⠴⠖⠋⠉⠀⠀⠉⠙⠲⠦⣄⡴⠋⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⠀⠀⠀⣠⠴⢻⡟⠲⣄⠀⠀⠀⠀⠀⠀⣠⠖⢻⡟⠦⣄⠀⠀⠀⡏⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢸⢀⡤⠞⠁⠀⢸⠇⠀⠈⠳⢄⡀⢀⡠⠞⠁⠀⠸⡇⠀⠈⠳⢤⡀⡇⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢿⡒⠒⠒⠺⣿⠒⠒⠒⠒⢺⡖⠒⠒⠒⠒⢻⡟⠒⠒⠒⠒⢲⡗⠒⠒⠒⠒⣿⠗⠒⠒⢒⡿
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠹⣧⡀⠀⠀⠀⢧⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⡼⠀⠀⠀⢀⣼⠏⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠹⡝⣆⠀⠀⠘⡇⠀⠀⠀⡏⢻⠀⠀⠀⢸⠃⠀⠀⣰⢫⠏⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢻⡌⢳⡀⠀⢻⡀⠀⢠⡇⢸⡄⠀⢀⡟⠀⢀⡞⢡⡏⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣳⡄⠙⣆⠈⣧⠀⢸⠁⠈⡇⠀⣼⠁⣰⠋⢀⡟⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠈⢳⡸⡆⡿⠀⠀⢿⢰⢇⡞⠁⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠙⣿⡇⠀⠀⢸⣿⠋⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⣇⠀⠀⣸⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠸⡄⢠⠇⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠖⢺⡿⠋⠻⣿⠷⠲⠶⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣺⣯⣟⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣶⠶⣿⣿⡿⠿⠿⠿⠾⠿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⠤⠔⠚⠉⢉⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⣾⣛⢦⡀
⠀⠀⠀⠀⣠⣴⠿⢿⣿⠀⠈⠓⢄⡀⠀⠀⠀⠀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠹⢸⠳⣴⡹⣦⡀
⠀⠀⣠⠞⠉⠀⠀⠀⡟⠀⠀⠀⠀⢹⣶⡆⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⣯⣄⣁⠀⠸⡇⠀⠙⠈⢻⢶⣀
⣠⠎⠁⠀⠀⠀⠀⢤⡇⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠈⠉⠉⢙⣻⣶⣾⣿⣿⣿⠛⠟⡛⣿⣿⡿⠟⣟⡽⠟⢻⠷⣧⠀⠀⠀⠈⣣⠙⢳
⡇⢀⣀⠤⠂⠁⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣍⢀⡼⠟⠀⢉⠄⠀⠀⠀⠁⡇⢡⡀⠐⠢⢤⣷⢸
⢿⠉⠀⠀⠀⠰⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡰⢺⠛⡅⠀⠰⡿⠿⠟⠛⡟⠉⣻⣿⣿⣿⣧⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡣⡀⠀⠔⠉⣿
⠈⠳⣌⠠⡄⠀⠀⠀⢻⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⡓⣠⣾⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢠⠄⠀⠐⢿⣤⣦⡾⠋
⠀⠀⠈⠱⣯⣀⡀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠈⢿⡁⠀⠁⡿⣠⣾⣿⡿⡿⡿⣟⣯⣼⣻⣿⣷⣦⣀⠸⠀⣚⣳⣬⡾⠋
⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣦⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠈⢻⣦⣀⢷⣿⡿⠃⠐⠊⠉⠚⣐⣉⣻⣤⣽⣿⣿⣷⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣯⡋⠻⣿⣿⢿⣿⣶⣶⣾⣤⣤⣤⣤⣤⣽⣷⣿⣯⣭⣬⣯⣿⣭⣶⣽⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡙⢌⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢎⡳⡌⠨⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠂⠹⢿⣌⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠻⣿⣻⣿⣿⣿⣿⣿⣿⡿⣻⣿⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠈⢻⣿⣿⣿⣿⣿⣟⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣈⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⢟⡛⢻⡻⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠶⠶⠒⠲⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣀⡀⠀⢠⣴⠶⣶⣶⣶⣶⣶⣶⣶⠀⢀⣀⠀⠠⣤⡶⢶⣶⣶⣶⣶⣶⣶⡄⢀⣈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣴⠟⢻⣦⠀⠻⣧⡀⠀⣠⣶⡿⠟⣩⡴⠋⠙⢷⣤⡈⠻⣦⣄⠀⢀⣼⣿⠏⣠⡟⠙⣷⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣰⡿⠃⠀⠀⠹⣷⡀⠘⣿⣿⠟⢋⣴⡾⠋⠀⠀⠀⠀⠙⢿⣦⡈⠻⣷⣿⠟⢡⣾⠏⠀⠀⠈⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣾⣿⣥⣤⣤⣤⣤⣼⣿⣦⠀⢀⣾⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣦⠀⠀⣴⣿⣯⣤⣤⣤⣤⣤⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠉⠉⠉⢉⣉⣉⣉⣉⣉⡁⠀⠀⣈⣉⡉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣁⡀⠀⠈⠉⠉⠉⣉⣉⣉⣉⣉⣉⡀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡈⢿⣍⠉⠉⠉⠛⠛⠛⠛⣿⡄⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢛⣿⠇⢀⡿⠉⠉⠉⠉⠙⠛⠛⣻⣿⣯⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡙⢷⣄⠀⠀⠀⠀⠀⠸⣷⡀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⣼⠃⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣎⠻⣷⡀⠀⠀⠀⠀⢿⣧⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⣸⡏⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡌⢻⣦⡀⠀⠀⠘⣿⣇⠈⣿⣆⠀⠀⠀⠀⠀⠀⣼⣿⠃⢠⡟⠀⠀⠀⢀⣾⡿⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢿⣦⠀⠀⢻⣿⡄⢹⣿⡄⠀⠀⠀⠀⣸⣿⠇⢠⣿⠃⠀⢀⣴⣿⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣎⠻⣷⣄⠈⣿⣿⡈⣿⣷⡀⠀⠀⢰⣿⡟⢠⣿⡏⠀⣴⣿⢟⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣝⢿⣷⣜⣿⣧⠘⣿⣧⠀⢀⣿⡿⢀⣾⡟⣠⣾⣿⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡻⣿⣿⣿⣇⢹⣿⣆⣼⣿⠃⣼⣿⣿⡿⣫⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣮⢻⣿⣿⡄⢻⣿⣿⡏⢰⣿⣿⣫⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣝⠿⠇⠈⠻⠟⠀⠙⢛⣵⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣋⣭⠽⠟⠛⠉⠻⢭⡉⠉⠉⣉⣩⡭⠟⠛⠉⠉⠙⠻⠯⣭⣉⡉⠉⠉⣩⠝⠋⠙⠛⠯⢭⣍⣳⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠉⠁⠀⠀⠀⠀⠀⢀⣠⣽⢾⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⢾⣥⣀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣦⣀
⠀⠀⠀⠀⣀⣴⠟⠁⢸⠀⠀⠀⠀⢀⣠⠴⠚⠉⢠⡏⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠘⣇⠈⠛⠶⣤⡀⠀⠀⠀⠀⣸⠀⠹⣷⣄
⠀⢀⣠⢞⡽⠋⠀⠀⢸⡆⣀⡴⠞⠋⠁⠀⠀⢀⣞⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⣠⠶⠋⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠉⠓⠦⣄⡀⣿⠀⠀⠈⠻⣝⢦⡀
⠀⠸⣟⠿⣿⠛⠛⠛⠿⢿⡿⠛⠛⠓⠛⠛⠿⠿⣿⠿⠛⠛⠛⠛⠛⠛⠿⠿⡽⠿⠟⠛⠛⠛⠛⠛⠿⢿⡽⠿⠟⠛⠚⠛⠛⢻⣿⠿⠟⠛⠛⢻⡿⢛⡿
⠀⠀⠈⠓⢮⡳⣄⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣤⡟⠁⠀⠀⣀⡴⣫⠖⠋
⠀⠀⠀⠀⠀⠙⢮⡑⢦⡀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣸⢻⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣾⠋⠀⠀⣠⢞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⢦⡀⠀⠙⣟⢦⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡟⢸⡇⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣠⢞⡟⠁⠀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡲⣄⠈⢷⡙⢦⡀⠀⠀⢳⡀⠀⠀⠀⢰⡇⠀⣧⠀⠀⠀⠀⣸⠃⠀⠀⣠⠞⣱⠋⢀⡴⣫⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠹⣆⠙⢦⡀⠀⢷⠀⠀⠀⣸⠁⠀⢻⠀⠀⠀⣰⠃⠀⣠⠞⢁⡞⢁⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣷⢬⣣⡀⠙⢦⡈⢧⠀⠀⡿⠀⠀⢸⡇⠀⢰⠇⣠⠞⠁⣰⣫⢖⡽⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣿⣄⠀⠙⢮⣧⢠⡇⠀⠀⠀⣇⢠⣯⠞⠁⢀⣾⣿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠻⣾⠁⠀⠀⠀⢻⡟⠁⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠘⣆⠀⠀⢀⡞⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡸⡆⢀⡞⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠯⠙⠋⠻⣉⡉⠭⠛⠉⠉⠛⠭⣉⣉⠝⠙⠛⠽⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣀⡴⡾⠋⡄⠀⢀⠤⠐⢊⠏⠑⠤⡀⠀⠀⢀⠤⠒⠙⡗⠢⠤⡀⠀⢰⠙⢷⢦⣀
⠀⠀⠀⠀⠀⠺⣟⠚⢗⠚⢳⡛⠓⠒⠒⢻⠒⠒⠒⠚⠳⠞⠛⠒⠒⠒⡻⠒⠒⠚⢻⠞⠓⡺⠓⣻⠗
⠀⠀⠀⠀⠀⠀⠈⠙⢦⡑⢄⠙⣦⡀⠀⠀⢣⠀⠀⠀⠐⠃⠀⠀⠀⡰⠁⠀⢀⣴⠋⡠⢊⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣷⣌⢎⠢⡀⠈⢆⠀⠀⡇⢸⠀⠀⢠⠁⢀⠔⡕⣡⣪⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣱⡙⢄⠈⡆⠀⠃⠘⠀⢠⠃⡠⢃⣎⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄⠑⢼⣘⠀⠀⣃⡧⠊⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠹⡀⢀⠏⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⡼⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣤⣴⣶⣄⠻⣿⣿⠿⠟⣋⣤⣴⣾⣶⣤⣌⡛⠿⢿⣿⡿⢃⣴⣶⣤⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⣿⣿⠗⠀⠐⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⠀⠐⢿⣿⣿⣿⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⢸⣿⣿⡿⢋⣵⣾⠁⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⠟⣡⣾⡇⢹⣶⣍⠻⣿⣿⣿⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⢸⠟⣩⣶⣿⣿⣿⢸⣿⣿⣿⣷⣌⠻⣿⡿⢋⣴⣿⣿⣿⣿⢸⣿⣿⣷⣮⡙⢿⠀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣍⣭⣥⠀⣬⣭⣭⣭⣭⣭⠈⣭⣭⣭⣭⣭⣥⡀⣠⣭⣭⣭⣭⣭⡍⢨⣭⣭⣭⣭⣭⡄⢀⣭⣍⣉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠈⢻⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⠋⢠⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⢠⡙⣿⣿⣿⣿⡌⣿⣿⣿⣿⡿⢸⠸⣿⣿⣿⣿⡏⣸⣿⣿⣿⡟⣡⢠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⢻⣌⢻⣿⣿⣧⠸⣿⣿⣿⠇⣼⡄⣿⣿⣿⡿⢠⣿⣿⣿⢋⣴⢃⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⢻⣦⡙⣿⣿⡆⢿⣿⣿⠀⣿⡇⢸⣿⣿⠇⣾⣿⡟⣡⣾⢃⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⣌⠻⣿⡘⣿⣿⢸⣿⣿⢸⣿⡟⣸⣿⢋⣼⣿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣿⣧⡙⣧⢹⡇⣸⣿⣿⡀⣿⢡⡟⣡⣿⣿⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣌⠀⠃⣿⣿⣿⡇⠃⢈⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠐⣿⣿⣿⡗⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⢻⣿⣿⢃⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡌⣿⡏⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⠛⠻⠛⠻⣻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠶⠶⢶⣶⡶⠤⢤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⡟⠀⠀⠀⠀⢀⡠⠚⣹⠏⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⠙⣄⠀⠀⠀⠀⠉⠉⢻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⢀⡜⠀⠀⠀⣀⠴⠋⠀⣰⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣿⠀⠈⢦⠀⠀⠀⠀⠀⢸⠈⢷⡄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⡞⠀⠀⡠⠞⠁⠀⠀⢰⡟⠀⠀⠈⡇⠀⠀⠀⠀⢀⡜⠁⠀⠀⣿⠀⠀⠀⠳⡄⠀⠀⠀⢸⠀⠀⠻⣦
⠀⠀⣠⡾⠋⠁⠀⠀⢀⡞⢀⡴⠋⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣇⠀⠀⠀⢠⠎⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢆⠀⠀⢸⠀⠀⠀⠘⢷⡀
⢴⣾⣏⣀⣀⣀⠀⢀⣾⠖⠁⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⢸⠀⠀⡰⠃⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠈⢣⡀⢸⠀⠀⠀⠀⠈⠻⣆
⠀⠻⣯⡉⠙⠛⠛⠻⣿⠶⠶⣶⣤⣤⣤⣠⣾⣁⡀⠀⠀⠀⠀⠘⣆⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠱⣼⣀⣀⣀⣀⣀⣠⣽⣷
⠀⠀⠈⢻⣆⠀⠀⠀⠈⢦⠀⠀⠀⠀⠈⠙⣿⠉⠛⠛⠛⠛⠛⠛⣿⠛⠷⠶⠶⠶⠶⠶⠶⣾⡷⠶⠶⠾⠿⠛⠛⢛⠟⠛⠋⠉⠉⠉⣡⡿⠋
⠀⠀⠀⠀⠙⢷⡀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⢳⡀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠱⡄⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢠⠞⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠙⣄⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡰⠃⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠘⢆⠀⠀⠈⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⡜⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠈⢦⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠁⠀⠀⢠⠎⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⢣⡀⠀⢻⡄⠀⠀⣿⠀⠀⠀⣼⠇⠀⠀⡴⠃⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠳⡀⠘⣷⠀⠀⣿⠀⠀⢰⡏⠀⢀⠞⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡹⡄⢹⣇⠀⣿⠀⢠⡿⠀⣠⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣜⣄⢿⡀⣿⠀⣾⠁⣴⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣧⣼⣼⢇⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡛⠻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣻⣿⣿⣿⣛⣿⣿⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⣫⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣿⣿⡀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⣿⣿⣿⣿⣿⣯⢸⣿⣿⣿⣿⣿⣿⣿⢗⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡝⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡛⢿⣿⣶⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣜⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠷⣶⡄⠈⣶⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠈⠀⢀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⠟⢿⣦⡀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠙⠻⣿⣿⣿⣿⣿⣿⣧⢹⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⣷⣾⣿⣿⣿⣛⠋⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣿⢹⡿⠋⢀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⡎⢰⣷⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡍⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡘⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠛⠛⣻⣿⣛⡻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⣀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣬⡉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣷⡄⢀⣤⣄⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣰⣿⠋⠉⢹⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⢰⣿⠁⠀⠀⢸⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⣿⡇⠀⠀⢠⠇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⢻⣧⣀⡴⠋⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣤⣴⣾⣏⠉⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣾⣿⣿⣿⣿⣿⡿⠇⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠿⠿⠟⠋⠀⠀⠁⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣅⡋⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⢸⠗⠀⠀⠀⣀⠤⠶⡾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡺⣿⣿⡟⠉⠉⠓⠒⠒⠢⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⢙⣿⡿⠁⡀⠁⠀⠀⠀⠀⣠⣴⣿⣿⣦⣄⠀⠀⠀⠀⠌⠀⠀⠀⢸⡇⢀⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣴⡿⠁⡀⠀⠀⠀⠀⣠⣾⡿⢻⣿⣿⣿⣿⣿⣦⣀⡼⠀⠀⠀⠄⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠟⠉⠉⠉⠉⠉⠉⠙⠛⠻⣿⣿⠃⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠐⠀⠀⠀⠀⢀⠀⠰⢁⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⢶⣄⣀⠀⠋⣠⠂⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⣷⣻⣀⠈⡅⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣜⣿⣶⣆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠎⠙⠛⠉⠁⠠⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠒⠒⠒⠺⠇
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀
⠀⠀⠀⠀⡀⢤⠴⢧⠤⠤⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠐⠒⠚⠓⠀⠉⠀⠉⠐⠓⠦⣄
⠀⣠⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⡘⠛⠿⠿⠿⠟⠛⣡⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞
⠀⠉⠘⠓⣢⣤⣤⡄⠀⠀⠀⣀⠀⠀⠀⣠⣔⣢⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣡⡒⢄⡀⠠⣤⠤⠤⠐⠾⡇⠈⢹⡄
⠀⠀⠀⠀⣿⣤⠋⠀⠀⠈⠉⠉⠉⢉⡾⣯⠚⠁⠀⠒⠒⠂⢰⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⢎⡹⠀⠀⠀⠀⠀⠀⠈⠑⣾⡏⡄
⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⣎⠻⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢡⠛⣄⠀⠀⠀⠀⠀⠀⠈⢠⢇⡀
⠀⠀⠚⠀⡄⠀⠀⠀⠀⠀⠀⠀⣷⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡀⠀⠀⠀⠀⠀⠀⢺⠀⢣
⠀⠀⢷⢀⡇⠀⠀⠀⠀⠀⠀⣴⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡇⠀⠀⠀⠀⠀⠀⢨⣠⠆
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡈⠃⠀⠀⠀⠀⠀⢀⡙
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠇⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢋⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠋⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠛⢿⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠃⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠙⠻⣶⣦⣀⣀⠀⠀⠀⠀⣀⣀⣠⣶⠿⠋⠁⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠈⠉⠉⢻⡟⠛⠛⢿⠉⠉⠀⠀⠀⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠂⢤⡤⠄⠀⣸⠀⠀⠀⣿⠃⠠⣤⡤⠐⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡀⠀⠀⠀⠀⢹⠁⠀⠀⢻⠁⠀⠀⠀⠀⣼⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣟⠁⠀⠈⠉⠙⠋⢻⡇⠀⠀⢹⣋⡉⠉⠉⠁⠈⢻⡄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡿⠛⢦⡀⠀⠀⠀⠘⣃⠀⠀⢻⠉⢃⠀⠀⣠⠔⠿⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣀⠁⠀⠀⠙⢢⡄⢀⡼⠛⠷⣤⠆⠀⠘⣷⢾⣁⠀⢐⡇⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢀⣽⣀⣀⡀⠴⠧⣤⡼⠳⠦⠀⠀⠀⠀⢰⠆⠀⠀⠈⠛⠛⠦⠄⣀⡀⠀⠀⠷⡀
⠀⠀⠀⠀⠀⢀⡮⠁⠀⠀⠀⠘⠁⠈⠉⠈⠁⠀⠀⠀⢠⣶⣤⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠻⡄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣀⣀⣀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠁⠈⠙⠿⢲⡤⠤⣤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠐⠒⢶⡶⠶⠶⠶⠶⣶⠶⢶⣶⠶⠷⠶⢶⡶⠒⠒⢉⡴⢦⣤⣄⣈⣽⠋⢻⡷⠒⠒⠒⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢧⡸⠦⠤⠤⣾⠀⣄⣀⣀⣘⠛⠁⠀⢛⣋⣉⣥⠀⣇⢀⠀⢼⣆⢨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⣄⢀⣶⣄⠈⠛⠁⠀⠁⠙⣷⢤⣤⣾⠉⠀⠉⠚⠃⠀⠀⠀⠈⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣼⣿⠖⠺⣿⠉⢹⡿⠳⣄⠀⠀⠀⣾⠇⣸⣇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢙⣁⣀⠤⠴⡿⠄⢀⣭⠀⠀⢠⡏⢠⠛⠻⣧⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣗⠚⣉⡁⣤⠤⣶⠦⠤⣤⣏⠀⠀⡶⠀⣻⠀⠄⠈⣦⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣉⣉⣀⡀⢰⡏⠀⠀⣸⣧⡄⠸⣄⢰⡋⢹⡷⠀⠀⣦⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡘⣿⠛⠋⠁⠀⠛⠃⠉⠛⠋⠀⠙⣂⠀⠙⠃⠙⡇⠀⠀⢀⠛⠀⠉⠛⠋⠀⠀⠀⠀⠀⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣦
⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⣰⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢠⡇⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠙⣤⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠈⢁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠧⠤⠤⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠤⢚⣿⣿⣿⡿⠟⣛⣛⣯⣭⠉⣿⡟⢩⣽⣿⣛⡛⠛⢿⡿⠿⣿⡓⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⣠⣿⣷⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠃⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣡⠤⢀⣀⣀⢀⣀⠠⣦⣉⠷⠋⠱⠞⣡⡤⠄⣀⣀⣀⠀⢠⢤⣜⠟⠁

⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⠄
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠹⣏
⠀⠀⠀⠈⠸⠄⠀⠀⠀⠂⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⣀⣤⣀⡀⢀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠈
⠀⢠⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣾⣿⣿⣿⣿⣶⣤⣾⣿⣿⣷⠟⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠁⠈⠀⠀⠀⠀⢿⠀⠀⠀⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢤⢁⢀⣀⠀⠈⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐
⠀⠀⠠⣄⠀⠠⡄⠀⣼⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥
⠀⠀⠀⢫⣆⡸⣌⠀⠈⠁⠀⠢⠀⠀⢘⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⠙⢿⡟
⠀⣠⠀⣨⣝⠃⢹⣧⡀⠀⠀⡀⠀⠀⢬⣙⣻⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⡡⠀⠀⠀⠀⠀⡒⠀⠙⠀⠀⠈⠉⣅⣈⠃
⠀⢻⣿⣿⣿⣷⣄⢁⢻⡈⠉⣿⡆⠉⠀⠝⠻⠿⠿⠿⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣬⣿⣧⠄
⠀⠀⠉⢻⣿⣿⣿⣷⣿⣀⠀⠈⢣⣄⠀⠀⠀⠈⣠⢦⡄⠀⠒⣺⣿⣿⣿⣿⣿⣿⣿⣿⡙⠀⠀⠃⠀⠀⠀⠀⢨⣥⠈⣴⣿⠿
⠀⠀⠀⢬⣿⣿⣿⣬⣿⢯⣷⡀⠠⢿⣷⣲⠀⣦⣥⠞⣠⣶⣾⣟⣥⣿⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣠⠞⢀⣀⠁⢠⣤⣿⠀⠴
⣶⣦⠀⠺⣿⣿⡿⢿⣿⣾⣿⣿⣧⣀⢀⣑⣶⣾⣧⣿⣿⣿⣽⡟⠿⡆⣤⣶⣿⣿⣿⣿⣿⣷⣤⠠⣶⠋⣠⡾⢛⣥⠀⠉⠉⡀
⠉⢁⠀⠘⣻⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⢻⣿⣟⣙⣿⣿⣿⣿⣾⣷⣿⡾⣿⣿⣿⣿⣿⣿⣯⡶⣴⣿⣏⣴⣿⣵⣶⣿⠈⣋
⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⡿⡿⠛⠟⠫⣿⢿⠿⢿⡿⣿⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠷
⠀⠀⣘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣶⣤⣾⣟⣾⣷⣯⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿⣯⣯⣁⣀
⠀⠀⠈⠓⠢⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠘
⠀⠀⣀⡀⠀⢭⣡⣾⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣶
⠒⠶⠶⢶⣄⣨⠍⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣼⡿
⠀⠀⠀⠻⢿⣌⣀⡠⣀⣠⣶⣿⣿⣭⣟⣩⠗⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣴⡆⠛⠁
⠀⠀⠘⢃⣀⠈⠹⣶⣉⠉⠹⠋⠁⠈⠛⠁⢀⣸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢟⠻⣿⢿⣤⡰⢒⣾⣿⣿⠂
⠀⠀⠀⠈⠙⠋⠉⠉⠉⠖⠋⠉⠁⠀⠀⠀⣬⣽⣿⣿⣿⠛⣟⠛⠸⣿⣿⣿⣿⣷⣮⣟⠡⣬⠉⠉⠁⡈⢳⣀⡀⠈⠋⠙⠛⠀⣧
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠿⠀⠙⠙⢯⣍⡙⠎⠉⢀⣼⣙⣳⠀⠈⠙⠉⢻⣤⠀⢀⠐⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠈⠃⡀⠂⠀⠀⠙⠛⣿⠃⢠⣁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⣶⡄⠀⠊⢻⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠈⡵⠾⠟⠋
⠀⠀⠓⠶⠶⠦⠄⠀⠙⠛⠶⠾⠿⣶⣦⣽⣿⡿⡷⣶⣤⠸⠶⠄⠀⢀⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⢦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄⢸⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠇⠀⠀⠀⡀⠈⠀⣇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢀⡼⣇⠀⠀⠀⠀⠀⠁⣸⡾⢿⢠⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠸⢱⡟⣤⡀⠀⠀⠀⠐⢿⡇⠛⣾⡷⢦⣤⠠⢿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄⡡⢾⡷⢹⣇⠀⠀⠀⠀⢸⠁⠀⠉⣗⣲⠀⢸⠶⣧⡈⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⡋⠀⠀⠀⠀⢾⣤⠀⢰⣿⡿⠓⢨⠠⣿⠁⡀⢠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⣶⣶⣶⣶⣶⡆⠀⠀⣀⠁⣿⠀⠀⠀⠀⣼⠁⠀⠀⢿⠛⠀⠀⡴⠋⠀⢹⣾⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⢻⣿⣿⡇⠀⠀⣿⠀⣸⣧⠀⠀⠀⢸⠠⠀⠀⣿⣔⠂⠀⠀⠀⠀⢸⠿⠛⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⣿⣷⣜⣿⣿⡇⣴⢰⡾⣻⣿⣿⡃⢀⡇⢸⡀⠀⢸⠛⡭⠭⢁⣀⠀⠀⠦⠀⠢⡈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠀⣾⣷⣿⣿⡿⠋⢀⣾⣻⢷⠟⠉⢀⡙⣧⣾⣇⠀⠈⠀⠀⡄⢠⢀⠀⠈⠙⡇⠀⢱⡘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⣄⣿⡿⠿⠿⠲⠶⠁⠠⠀⢺⣀⣷⣬⣷⡾⠻⠦⡠⢾⡀⡀⠃⠘⡈⠀⠀⢀⡇⢀⢞⡁⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⢀⠀⠀⠀⠀⠚⢃⣴⣿⠩⣿⣧⣷⣖⣾⢁⢘⣞⡷⠚⢀⣳⣿⣦⡸⣷⠃⢸⡗⠊⢻⡀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠀⠀⠀⠀⠠⠾⣿⡟⠛⣿⣿⡿⣿⡁⠀⣿⠾⣿⠟⢈⣤⣽⣿⣿⠁⡟⢠⣿⣥⣤⡉⠆⠀⠟⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠂⠀⠀⠀⠈⠀⢰⣿⣿⠐⠿⢛⣴⡏⠀⠘⢷⣿⣿⣿⣿⣿⣉⣵⠟⣧⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⡸⠂⠀⠀⣿⣿⡦⠤⣜⠿⢧⠀⡀⠈⣿⣿⠁⢹⣿⢉⣁⣾⠋⠉⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣶⣀⠀⠀⠀⠀⠈⠁⢀⣼⣿⡇⠀⣄⢀⣠⡍⢀⣿⣄⢠⠛⢄⣈⡉⠛⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁⣄⠀⠀⠀⢀⠾⠿⠛⢷⣷⣃⡚⣋⡴⠋⢹⡇⠀⠀⠀⣽⠗⡼⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠀⠀⣠⡾⠀⠀⠀⠐⠿⠉⠉⢿⠁⠀⣸⠁⠀⡀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠉⠁⠀⠀⠀⠀⠀⢀⢀⠟⠀⠀⠃⠀⣰⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠘⡄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠘⡇⠰⢆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⢇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠐
⠀⠠
⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⣤⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣶⣶⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠈⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡬⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠈⢹⣇⠀⠀⠀⠀⠀⠈⠙⠻⣶⡶⠞⠻⣿⣦⡀⣠⠄⠀⠀⠀⠀⠐⠻⣿⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡷⠀⡽⠋⠉⠙⠉⠉⠉⠉⠉⠙⠛⠒⠶⡖⠚⠻⡟⢛⠛⠒⠂⠠⢄⡀⠀⠈⠿⠛⠉
⠀⠀⠀⢰⠀⣀⣀⢠⠀⠀⠀⢰⣶⣟⠛⠻⢿⣿⠿⣿⣯⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁
⠀⠀⢰⢎⢰⣿⣿⣼⣷⡶⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒
⠀⠀⢸⣼⣿⣿⣿⣿⡿⣁⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑
⠀⠀⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡄⠀⠠⢴⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⠀⠀⣧⣀⡴⠸⠶⣶⣶⣤⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⠀⠀⠉⠉⠐⠂⢀⣩⢿⣿⣶⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣶⣶⣤⣾⣬⣦⣽⣿⣿⣿⣿⣶⣆
⠀⠀⠘⠋⢹⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠐⠂⠀⠈⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠙⠛⠟⠻⣿⣿⣿⣿⣿⣿⠻⣿⣿⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠁⠀⠀⠉⠓⣶⣥⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣀⣀
⠀⠀⢰⣮⣼⠏⢠⣄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⠛⡃⠀⢻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠘⠓⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣫⣵⠶⠿⠛⠛⠛⠛⠛⠛⠷⢦⣍⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣥⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣩⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣵⠛⣁⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠙⣆⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣭⣶⡿⠿⠿⠿⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣦⡀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⣿⣿⢻⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡝⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡇⠀⠀⠀⢹⣷⣼⣿⡇⠿⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⣛⡛⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⣭⠄⠀⠀⠀⠀⠀⠈⣿⣯⣹⡿⠚⠛⢳⣝⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⢸⣧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠉⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⡣⡎⢠⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣿⣿⠀⣶⣿⣿⣿⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡇⠈⣾⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡅⣿⠀⠉⠋⠈⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡸⠂⠈⢠⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡆⢸⠀⠀⠀⠀⢋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠁⠀⢀⡾⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⣼⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⣿⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡇⠀⠀⠀⠀⢀⡀⣿⠤⠤⠤⣤⠶⠖⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠛⠒⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⡀⢹⢸⣇⠀⠀⠀⠐⢏⠁⠀⠀⠀⠀⠀⠀⠀⠱⡌⠳⣄⠀⠀⠀⢀⠤⠤⠚⠛⠛⠛⠛⢻⣶⢽⡇
⠀⠀⠀⠀⠀⣠⠾⣫⣽⣶⣶⢮⣍⡻⣇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⢦⡔⠚⢻⡄⠀⠤⠤⠤⠤⠄⢸⣿⢸⡇
⠀⠀⠀⠀⣴⢫⡞⢡⠞⠀⠲⢀⡉⢻⣦⢠⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠳⣄⠀⠛⠆⣼⡄⠀⠀⠀⠀⠀⠀⢸⡿⣸
⠀⠀⠀⢰⡏⣾⠀⢈⠀⠀⠀⠀⢳⠀⣿⡌⣷⣌⢻⣆⡀⠀⠀⠀⠀⠀⠀⢶⡋⠁⠀⠀⠹⢦⡀⠀⠀⠳⡄⠀⠀⠀⠀⢀⣾⣧⡝⣦
⠀⠀⠀⠸⣇⢿⠀⠘⣄⠀⠀⢀⡌⠀⣿⢣⡟⠈⣣⠝⣿⣦⣀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢷⠀⠙⠦⠤⣀⣀⣀⣀⣀⡤⠛⢀⣼⠟⣼⠁
⠀⠀⠀⠀⠹⣎⠷⣄⡈⣛⡿⠋⣡⡾⢋⡾⠁⢰⡋⣾⣿⣿⣿⣷⣦⣤⣄⣀⣀⣹⣦⠀⠸⣇⡀⠀⠀⠀⠉⠉⠉⠉⠀⢠⠏⠙⢷⡝⣧
⠀⠀⠀⠀⠀⢈⣷⠎⣙⣿⣰⣿⠻⡆⣿⠀⠀⠈⠛⠶⢮⣍⡛⢻⣿⣿⣿⣿⣿⣿⠙⠀⠸⣇⠙⠂⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢻⡌⢧
⠀⠀⠀⠀⠀⡟⣵⠟⠋⢙⡄⣟⠀⢿⡹⣄⠀⣀⣠⣤⠖⣫⣴⣿⣿⣿⣿⣿⣿⣿⣾⠄⠀⢹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢻⡜⣧
⠀⠀⠀⠀⠀⣷⣸⡧⠚⢛⠄⢛⠀⠈⢻⡌⢛⣭⣷⣶⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠻⣅⠀⠀⠀⠀⠀⠀⠘⠃⠸⠂⠀⠀⠈⢿⡸⣇
⠀⠀⠀⠀⠀⢳⡙⣷⠞⢛⡆⢠⠀⠀⢸⣇⣼⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡹⡆⠲⣀⠀⠈⠳⠶⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣧⢹⡀
⠀⠀⠀⠀⠀⢸⡄⢿⡤⠛⣹⡷⠂⠀⠀⢀⠉⠙⠻⠏⠀⠀⠉⠛⠿⠿⠟⠛⠁⢰⡆⠳⣄⣀⣀⡄⠈⡀⠀⠈⠉⠙⠓⠶⠶⡇⠀⠀⠀⣼⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠹⣮⡛⠶⠿⣿⠉⠐⠦⣼⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⢿⣿⣿⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⣿⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣠⡞⠁⣀⣠⠞⠁⣾⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⡆⠀⡴⠃⠀⠀⠀⠀⠀⣠⠟⡌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⡁⠀⠀⣿⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠻⠿⠷⠶⣶⣶⣶⡖⠚⠁⣰⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢰⣿⣿⣿⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢹⡇⣾⣿⣿⣿⠀⠄⠀⠈⠁⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⣋⣾⣿⣿⣿⣿⠐⠶⠀⠀⠀⣤⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⣾⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢩⣶⣿⣿⣿⣿⣿⣿⣅⣼⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠛⠀⠀⠐⠒⢛⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢿⣿⣿⣿⡿⢟⣭⢸⣇⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣝⣛⣭⠶⠛⠳⣮⡻⢦⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣝⡓⠶⣤⣤⣄⣀⣀⡀⠀⠀⠙⠒⠀⠠⠤⠤⠤⠒⠊⠁⣀⣀⣠⣼⣿⣿⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠭⢭⣍⣛⣛⣛⣛⣛⣛⣒⣒⣚⣛⣛⣛⣛⣫⡭⢭⡜⣿⣿⠟⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠻⣮⣡⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⢛⡿⠟⠛⠋⠉⢉⣿⠟⠛⠛⠛⠛⠛⠋⠻⣿⡋⠉⠉⠉⠛⢿⣝⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⣿⢀⠉⠀⠀⠀⠀⢠⣾⠋⣆⠀⠀⠀⠀⠀⠀⡴⠘⢿⣄⠀⠀⠀⠀⠈⡀⢿⣟⠳⣄⡀
⠀⠀⠀⠀⢀⣴⠞⠁⢠⣿⠇⠈⣧⠀⠀⠀⣰⡿⠁⠀⠘⣆⠀⠀⠀⢀⡼⠁⠀⠈⢻⣦⠀⠀⠀⣰⠃⠘⣿⡆⠈⠻⣦⡀
⠀⠀⣀⡴⠛⠁⠀⠀⣾⠋⠀⠀⠸⣇⢀⣼⠟⠀⠀⠀⠀⠘⣧⠀⠀⡾⠁⠀⠀⠀⠀⠹⣷⡀⣰⠇⠀⠀⠘⢿⠀⠀⠈⠙⢦⣄
⣠⡾⢋⣀⣀⣀⣀⠀⣁⣀⣀⣀⣀⣹⣿⣋⣀⣀⣀⣀⣀⣀⣈⡀⢈⣁⣀⣀⣀⣀⣀⣀⣘⣿⣏⣀⣀⣀⣀⣈⠁⣀⣀⣀⣀⡙⢷
⠈⢳⣌⠻⣿⣿⣿⡇⠹⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠏⢰⣿⣿⣿⠟⣡⠞
⠀⠀⠙⢷⡜⢿⣿⣿⡀⠹⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⠏⢀⣿⣿⡿⣡⡾⠋
⠀⠀⠀⠀⠙⢦⡹⣿⣷⡀⠹⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⠏⢀⣾⣿⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠙⣮⣻⣧⠀⠹⣿⣿⣿⡄⠈⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡿⠁⢸⣿⣿⣿⠋⢀⣾⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠹⣿⣿⣷⠀⠘⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠃⢀⣿⣿⣿⠃⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠙⣿⣿⣇⠀⢹⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⠇⠀⣼⣿⣿⠃⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠘⣿⣿⡄⠀⢻⣿⣿⠀⠀⠀⠘⣿⣿⡏⠀⢰⣿⣿⠃⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⢿⣷⠀⠈⣿⣿⠀⠀⠀⠀⣿⡿⠀⢀⣿⡿⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣿⣧⠀⠸⣷⠀⠀⠀⠀⣿⠁⠀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⢹⡇⠀⠀⣸⠇⠀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⢿⠀⢠⡟⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣇⣾⣵⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⣉⣀⣀⡀⠀⠀⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠋⠙⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠀⠀⠙⠛⠛⠉⠹⡟⠛⠛⠋⠁⢀⣾⣿⠟⢩⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⣠⣿⡿⠁⠀⣾⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠚⢧⡀⠀⠰⣄⠀⠀⠀⠐⢲⡾⠉⠁⠀⠀⢠⣿⡿⠁⠀⣼⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⡄⠈⠳⡄⠀⢸⣿⣶⣤⣤⣾⣅⡀⠀⠀⠀⣾⣿⠁⢀⣼⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣹⣶⣾⠉⠁⣸⣿⣿⣿⣿⣷⣦⣄⣿⣿⢠⢞⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠻⣿⡟⠧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣇⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⣀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⠟⠉⠀⠀⠀⠀⠀⣄⠙⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⡀⠀⠀⠀⠀⠀⠀⠲⣌⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢷⣄⣴⣿⣿⣋⣿⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⡅⠱⠀⠀⠀⠀⠀⠀⠙⢟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢹⡄⠓⠂⣀⠀⠀⠀⠀⠀⢷⠨⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠤⣿⡈⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠋⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⢈⡬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠐⠂⠀⢠⠀⠀⠰⣿⠀⠖⠀⠀⠀⢠⠀⠀⠀⣼⠀⠀⣁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⡾⠄⢀⠇⠀⠐⢰⡇⣀⡀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⢩⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢛⠛⠛⠓⠒⠚⠓⠒⠃⣼⠀⡍⠀⠀⠀⢀⡇⠀⡌⢸⡇⠀⠁⠸⠂⢻
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣭⣵⣶⣅⣀⣤⣨⡥⠴⠖⠚⠋⠻⣿⣿⣿⣿⡿⢾⣷⣶⣷⣿⣯⡉⠝⠀⢡⢸⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⡟⣿⣿⣂⠀⠀⠀⠀⠀⠈⢻⣷⣯⠿⠚⠻⣾⣿⢿⣿⣿⣦⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠻⣇⠈⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣌⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣼⣿⣿⣿⣿⣿⣿⡟⠁⠉⠛⠶⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⠉⢉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣦⠤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⣰⣿⠿⠛⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⢀⡠⠴⠒⠉⠁⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢰⡇⠀⠀⠀⢀⢀⣀⣀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⣠⢤⣿⣿⣿⣿⣿⡷⣤
⠀⠀⢀⣀⣤⣤⣤⠀⢀⣾⠏⠀⠀⠀⠀⠀⠳⢤⣴⣾⣿⡿⣿⣿⡇⡜⣈⡧⠀⠀⣀⣤⡖⠋⢠⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠡⡉⠻⡿⠴⣊⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡝⡓⢸⣿⣿⣭⢤⣴⣺⡽⠛⠉⠀⣴⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢳⡀⣠⢞⠟⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠷⠃⣏⠉⠉⠉⢩⣾⠉⠁⠀⠀⣠⣾⣿⡿⡿⠿⠿⢿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⡴⢻⣿⣿⠟⠁⠀⠀⠘⡄⠀⠀⣾⡏⠀⠀⠀⠘⠋⠉⠉⠀⠀⠠⢀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡿⠛⠁⠀⠀⠀⠀⠀⣷⠀⠴⢿⣇⠀⠀⠔⢉⢀⣦⣔⠶⠋⠚⠂⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢻⡄⢆⠤⠊⠈⠀⢠⡄⢀⠆⡀⢐⣤⣤⡴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠜⠀⡏⠐⠘⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣦⣴⣶⣦⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⠋⠉⠈⠓⢄⣀⣀⡤⠶⠚⠉⠁⠀⠉⠙⠒⠦⣤⣀⣀⠴⠋⠈⠉⠓⠲⠬⣷⣤⡀
⠀⠀⠀⠀⢀⣠⠞⣯⠀⠀⠀⠀⠀⢀⣠⠴⢺⠛⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢻⡒⢤⣀⠀⠀⠀⠀⠀⢈⡟⢦⣄
⠀⠀⣀⢴⡿⠃⠀⢻⠀⢀⣠⠴⠚⠉⠀⢠⠏⠀⠀⠈⠙⠦⣀⠀⠀⢀⡠⠞⠉⠀⠀⠀⢧⠀⠈⠙⠲⢤⣀⠀⢸⠃⠀⠹⣷⢄⡀
⠀⢾⡷⣿⠶⠶⠶⢾⣾⠿⠶⠦⠴⠶⢶⣿⡶⠶⠶⠦⠶⠶⠾⣷⣶⡿⠶⠶⠶⠶⠶⠶⣾⣷⠶⠶⠤⠶⠾⢿⣾⠶⠶⠶⢾⡷⣿⠆
⠀⠀⠙⢮⡳⣄⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⣴⠟⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠀⠀⠉⠢⣕⢦⡀⠀⠹⣷⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣰⡿⠃⠀⣀⢔⡫⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡀⠈⢯⡳⡄⠀⠀⠘⡆⠀⠀⠀⠀⡾⠸⡆⠀⠀⠀⠀⡼⠁⠀⠀⡠⢪⠟⠀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡢⣄⠱⣌⠢⡀⠀⠹⡄⠀⠀⠀⡇⠀⣇⠀⠀⠀⡼⠁⠀⣠⠞⡴⢃⡠⣾⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣷⣌⢦⡙⢦⡀⠹⡄⠀⢸⠃⠀⢹⠀⠀⣸⠃⣠⠞⢡⢞⣴⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣷⣄⠙⢦⣳⡀⣼⠀⠀⠸⡆⣰⣣⠞⠁⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠙⢷⡇⠀⠀⠀⣷⠟⠁⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⢧⠀⠀⢠⠏⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣧⢠⣯⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣷⣼⢟⣩⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣀⣼⢿⣿⣿⣿⡿⡿⢗⣻⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡈⠉⠁⣉⣽⡿⠿⠟⠃⢬⣛⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢙⣿⣇
⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⢀⣠⡤⠤⣌⡓⢀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢘⣿⣿⠁⢀⣀⣀⣀⡀⢀⣹⣧⠀⠁⠀⠀⢹⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢻⣶⣦⣿⣿⣿⣿⡻⡌⠉⠀⣷⠀⠀⢠⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣩⣴⣤⣤⣿⠀⢀⡎⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣷⡧⣤⣄⠀⠹⠀⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⣠⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣤⡴⠞⠛⠻⣾⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠻⡆⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⢿⠻⣿⣿⣿⣿⣿⣿⡿⡇⠸⢿⣿⣿⣿⣿⣿⣿⢿⡿⢧⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⡈⠉⠛⠛⠛⠋⠀⡇⢸⠂⠉⠛⠛⠛⠉⢁⢨⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡙⠇⠀⠀⠀⠀⢶⣀⠃⠈⣀⡶⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣹⣟⣟⣉⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠁⠙⠛⠛⠻⠟⠛⠛⠃⠘⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢦⡀⠀⠀⠈⢹⠿⠿⠋⠁⠀⠀⣠⣾⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢮⣿⣦⣆⡀⠀⠀⠀⠀⢰⣦⢞⠕⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠛⠿⢿⣶⣶⠶⠞⠫⠐⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣼⣷⣻⣆⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡏⢷⡀⠀⠀⢀⣾⣿⣿⣿⣻⣆⠀⠸⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠾⠿⠿⠿⠿⡷⠛⠂⠰⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣶⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣆⠙⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⠋⣰⣿⣿⣿⣦⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡈⢿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⡿⢁⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⡀⢀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⡀⢀⣉⣉⣉⣉⣉⣉⣉⣉⣉
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡈⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡆⢹⣿⣿⣿⣿⡏⢰⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⢿⣿⣿⡿⢀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠘⣿⣿⠃⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢹⡏⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢿⡍⠉⠉⢩⣿⠉⠉⠉⠉⠉⠉⣿⡍⠉⠉⢩⡿⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⢿⣆⢀⣾⠃⠀⠀⠀⠀⠀⠀⠘⣷⡀⣰⡿⠁⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣁⣀⣀⣀⣀⣈⣿⣾⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷⣿⣁⣀⣀⣀⣀⣈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⠉⠉⠉⠉⠉⠉⢿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡿⠉⠉⠉⠉⠉⠉⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠹⣧⠀⠀⠀⠀⣼⠏⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⢻⣇⠀⠀⣸⡟⢀⣰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣿⡆⢰⣿⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⢀⡀⢠⣤⡀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠊⠡⠀⠀⢈⠁⠀⠀⡆
⠀⠀⢠⡠⠶⢤⣀⣤⡕⠦⠔⣉⣠⠀⡀⠾⠱⡄⡀⡀⠀⠘⢀⣀⢀⡰⠛⠦⠤⠊⠑⠒⢻⠁⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠠⠀⡀⠀⠄⡀
⠀⠀⣷⡁⠂⠉⠙⠹⠇⢤⠤⠽⠋⠀⠰⠦⢈⣱⠀⠘⢦⡔⠂⠄⠁⠀⠘⠀⠀⣠⡤⣶⣿⣀⣀⠀⠀⢀⣴⣦⡤⠠⠆⢀⣄⣤⡀⠠⡦⢀⣬⠦⡀
⠀⠀⠻⣤⡠⢤⣶⢧⣤⣀⣰⣤⡞⣷⠶⣆⣤⠛⠀⠀⠈⣷⣶⣟⡷⢖⣚⣋⠉⢉⢉⣅⠀⢈⠉⣆⠐⣏⠀⠈⠀⠀⠐⠛⠋⠙⠓⠀⠀⠀⠈⠀⣸⠆
⠀⠀⠀⠘⠅⠀⠈⠰⠉⠚⠊⠛⠃⠀⠀⠨⠃⠀⠀⠀⠀⠘⠉⠘⣥⣬⣩⣤⣽⣭⣬⣍⣤⡍⠁⠉⠂⠈⠢⣀⣄⠀⠀⠀⠀⡄⠀⠀⠀⣠⠀⡐⠁
⠀⠀⠀⠀⠈⠢⡀⠐⣄⠀⠀⣀⣦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢶⡎⠟⠛⠉⠃⠉⠙⠉⠏⣡⡀⠀⠀⠀⠀⠙⢿⡷⢶⣶⣾⣧⣼⣿⣶⣿⠊
⠀⠀⠀⠀⠀⠀⠙⢾⡿⣷⣿⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠘⣆⠀⠀⠀⠀⡴⠁⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠇⠘⡿⠁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⡟⠟⠀⠀⣀⣠⣤⡤⠤⠄⣀⠀⠀⠀⠀⠁⣹⠀⣼⣦⠚⣷⠈⠀⣀⡠⢤⠄⠤⠄⣀⠀⠀⠢⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⡠⣺⠏⠃⠞⠁⠀⠀⠀⠀⠙⣶⣠⡀⠀⠙⣆⢻⡏⡼⠃⡄⠊⢯⣀⠻⢦⣄⡠⠉⠻⣦⡀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣁⠀⠐⠀⠓⠦⠰⡶⢤⢴⣿⡿⢛⣆⠀⠈⢮⡗⢁⡎⠀⢀⠀⠛⠁⠈⠈⠀⠀⠀⠘⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠴⠄⢦⡄⠀⠀⠀⠁⠉⠘⢁⠋⠈⢦⠀⠀⠀⢸⣵⣆⠘⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠑⠛⢢⠰⢄⣀⠶⢆⠀⠀⣻⠸⠇⠀⠀⠢⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠸⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⡀⠀⠘⠉⠈⢹⠯⠒⠉⠀⠀⠀⠀⣤⣼⠇⠠⠀⠀⠀⠀⠀⣀⡆⢴⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⢱⣟⣵⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⣷⡤⠼⠁⠈⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠒⠉⠀⠀⢀⣀⠀⠀⡀⢀⠀⠀⠀⢀⠀⠀⠙⠛⠀⠻⡀⡀⠴⠆⠂⠁
⠀⠀⠀⠀⠀⢀⡤⠤⠤⣤⠤⠤⠤⡤⢤⡀⠀⠀⢀⠀⠞⠉⣩⠉⣁⠀⣨⡷⢀⡘⠋⠀⠀⢄⠀⠀⠀⠀⣀⣤⣶⠦⠤⠤⠤⣀⡀⡀
⠀⠀⠀⠀⣴⣷⠂⠀⢠⣾⡆⢀⣍⡠⡴⡉⣦⠂⣿⣥⡄⠠⠄⠀⠛⠃⢡⣤⣿⡍⠑⠠⠔⠺⠓⠂⢶⣉⠁⠈⠉⠙⢠⣤⣤⣤⣿⢿⣗⣢⣄
⠀⠀⠀⠀⣤⡿⠂⠛⡛⠉⠙⠺⣽⣣⡄⢀⣾⠀⠈⠂⡁⠀⠀⣀⣀⣴⣾⠟⢙⣥⠀⣠⡤⠂⠀⠀⠘⡌⠉⠲⡔⣄⣀⡀⠀⠈⠙⠇⠛⠙⠛⢣⡀
⠀⠀⠀⠀⣿⠃⣀⠐⠀⠀⠀⠀⠈⠻⡶⠘⣿⠀⠀⠀⠀⠀⢤⣿⡻⠋⠉⠀⣸⣁⠜⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠁⠀⠈⠁⠀⠲⢄⡤⠤⠀⣛⣱⠄
⠀⠀⠀⠀⣽⠀⠨⠀⠀⠀⠀⠀⠀⢸⢄⣄⢿⠀⠀⠀⠀⠀⠀⠙⢿⣄⣀⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⣇⠀⠘⢀⠀⠀⠀⠀⠀⡌⠈⡛⢨⠀⠀⠀⠀⠀⢀⣀⣀⡉⢋⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢹⣷⣴⣶⢊⣶⣴⡤⠚⠁
⠀⠀⠀⠀⣿⣧⠀⣈⠦⠀⢀⣀⣾⣇⢠⣧⣘⠀⠀⠀⠀⠀⠛⠉⢨⣄⣬⣅⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⡿⣣⣿⡿⠋
⠀⠀⠀⠀⢿⠟⠴⠈⣙⠃⠉⠘⣁⢀⡋⢩⡿⠀⠀⠀⠀⠘⣿⣿⡁⠀⠀⠈⠾⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡿⣱⡿⠋
⠀⠀⠀⠀⠀⠹⠶⠬⠭⠤⠴⠶⠿⠯⠷⠊⠀⠀⠀⠀⠀⠀⠈⠀⠑⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠻⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿⠏⠀⠀⠘⢿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁

⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠆⡀
⠉⠈⠀⠀⣤⣤⠤⠤⠤⠤⠠⠠⢄⣄⣀⡀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢴⢦⡈
⠀⠀⠀⠀⡟⡌⠑⠢⢄⡀⠀⠀⢀⡴⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡟⢟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⣀⢏⣀⠽⣦⡀
⠀⠀⠀⠀⡇⠘⡄⠀⠀⠈⢑⣶⣏⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢟⡺⢗⡺⣗⡟⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⢸⠀⢠⠟⢧⡀⢰⠁⣙⣆
⣖⣲⠀⠀⡇⠀⠘⡄⢀⡴⠋⠈⡆⠈⠑⢺⠅⢐⠊⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⢠⡀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⢸⢠⠏⠀⣀⡽⢿⡋⠁⡆
⠈⠁⠀⠀⡇⠀⢀⡼⡯⠤⢄⣀⢸⣀⠔⠁⠐⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣖⣋⣁⣀⣀⣙⣦⠃
⠀⠀⠀⠀⣇⡴⠋⠀⠘⡄⠀⢀⠝⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠒⠒⠒⠤⠾⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⢀⠴⠦⡀⠀⠀⠀⠀⠀⢠⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⣯⠟⠗⠪⣦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢁⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⢧⡟⣟⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡏⠳⢄⡀
⠀⠀⡀⢆⡀⠀⠀⠀⠀⠀⡰⠛⡆⡸⠙⡄⡸⠱⣄⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⡇⠀⢀⣉⣦⠀⠀⠀⡴
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⡗⠒⢺⠓⠒⢺⠗⠒⡺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⡠⠴⡷⢍⡁⠀⠰⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢇⠀⡎⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⣴⡟⠒⠋⠉⠀⠀⠀⡇⠀⠉⠓⣴
⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠈⠺⣶⠞⠁⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠓⠲⠤⢄⣀⣇⡤⠖⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⡠⠚⢏⠉⠒⠒⠒⠤⣤⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⡏⠐⠲⠤⣬⠄⡐⠂
⠀⢰⡶⣒⢒⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠈⢦⠀⠀⣠⠞⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡇⢀⡠⠞⠁⢠⡉⠁
⠀⢸⢱⣴⢻⣲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡠⠊⢻⠉⠙⠒⠲⠬⣧⠞⠁⠀⠀⡇⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠙⠗⠉⠀⠀⠀⠀⠁
⠀⠸⠯⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣑⠀⢠⣎⠀⠀⠈⡇⠀⠀⣠⠞⠉⢆⠀⠀⢀⡇⠀⠀⠀⠙⠊
⠀⠀⠀⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡀⠉⠒⢤⣸⣠⠞⠁⠀⠀⠈⢧⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⠉⠒⠤⣀⠀⠀⠈⢇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠔⠀⠀⠀⠀⢰⡷⠾⣑⡷⡆
⢠⠶⡄⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢛⣠⠞⠁⠀⠀⠀⠀⠈⠉⠒⠤⣈⢦⡇⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⢸⣏⣶⣛⣯⠇
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠒⠒⠒⠒⠒⠚⠛⠃⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠈⠛⠸⠞⢁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣖⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠔⡹⠙⡝⢦⡀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⢠⡏⢦⣀⣠⠴⠚⡽⡇⠀⠀⠀⠈⠁⠀⠀⠀⢤⠦⠆⠀⠀⠀⠀⠀⠈⠠⠈⠀⠀⠀⢀⠔⠁⢠⠇⠀⢱⠀⠙⢦⡀
⠀⣤⠀⢸⡤⠒⡟⢢⡀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣯⠟⣽⠳⡀⠀⠀⠀⠀⠀⠀⠰⡇⠤⠤⣾⠠⠤⢰⣷⠠⠄⣿
⠀⠐⢣⠀⠙⢾⡤⠒⡿⢧⡀⡇⠀⠀⢈⡁⡋⠀⠀⠀⠈⢧⡹⡁⣏⡽⠃⠀⠀⠀⠀⠀⠀⠀⠙⢆⡸⠀⢣⣀⠎⠘⣄⡔⠁⠀⠀⠀⢰⢱⠄
⠀⠀⠀⠀⠀⠀⠙⠾⠥⠐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣼⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠑



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠⣴⡦⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡶⡞⠛⠛⢲⣶⠟⢿⡽⣶⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⡿⠁⠈⠀⢀⡰⢿⡏⠘⣡⣾⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠋⠀⡜⠁⠀⠀⢶⠂⠁⠈⠀⠾⠿⠿⠷⠿⠿⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠞⣿⠀⠀⠙⠀⠀⣀⠤⣼⡃⠀⠀⠶⡶⢶⢶⠶⡖⠖⣶⠒⠒⠈⠉⠙⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣼⡶⠀⣷⠀⠀⠀⣺⢿⢡⣆⣄⡻⣿⠛⢿⣿⣇⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠘⠉⠁⠀⢹⠀⡠⠚⢡⠇⣽⣿⣿⠛⠿⣶⣷⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠁⠀⠀⠀⠀⠀⠀⠸⠊⠀⢱⣾⣦⡎⢀⣹⣦⣴⣿⣿⣿⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣄⣀⣀⣀⠤⣶⣿⡷⠦⢶⣿⣿⣿⡿⠋⢹⣿⣿⠟⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣍⡻⣿⡛⠉⠁⢠⣾⢋⡁⣀⣀⣴⣿⡋⠉⢷⣴⣾⡿⠛⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⠉⢀⣸⡏⠐⢴⣿⠟⠁⠁⢀⣼⡏⣿⠛⢃⣟⣿⣿⠀⠀⠀⢿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢂⠀⣉⣀⣹⣠⠚⠉⠀⠀⢀⣾⣿⣿⣷⣖⢡⣬⣤⣈⠳⣄⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⢘⣿⣿⣽⣋⠀⠀⣀⡿⣯⣿⡟⠋⠰⠟⠀⣨⣥⣤⣤⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣍⣙⣿⣿⣿⣷⣿⣿⣿⣷⡶⠶⡾⠿⠟⠟⠛⠛⠛⠙⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠓⠒⠒⠦⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠴⠖⠒⠚⠉⠉⠙⢦⡀
⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠻⡗⠲⠦⠤⣤⣀⣀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⣀⣀⣠⠤⠴⠖⢺
⠀⠙⣆⠀⠀⠀⠀⠈⠉⠉⠓⢿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡿⠚⠉⠉⠁⠀⠀⠀⠀⣠⠏
⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠸⡄⠀⠀⠀⠀⢠⠇⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢳⠀⠀⠀⠀⡞⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⣇⠀⠀⣸⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⡄⢠⣇⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣶⡷⠶⢶⣶⣦⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⢿⣿⣿⠓⠒⠀⠀⠈⠉⠛⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠻⠃⠀⠨⠽⢶⡆⣤⠠⠿⠿⠿⢓⣴⡮⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⠀⠀⠬⠇⠂⠀⠀⠀⠠⡫⠋⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⡞⠉⠁⢿⣶⠀⠀⠀⠠⠠⠔⠡⠀⠀⠀⣿⠃⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⢰⠁⠀⡆⠈⢿⣦⠀⠀⠀⣹⣦⣄⡀⠀⠀⢩⣘⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡀⠀⠘⣤⠀⢰⠀⢠⣿⣿⢿⣿⣿⣿⡿⠋⠉⢉⡉⠛⡙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱⠀⠸⡁⠀⠘⣆⠏⣼⣿⣤⣿⣿⠟⠀⠀⠀⢄⡈⢢⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠙⣄⠀⢻⣼⣿⣿⣿⠟⠁⠀⠀⠀⠐⢤⣹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠆⢸⡿⠛⠋⢁⠀⠀⢀⣀⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠴⠊⠁⠀⠀⢠⠈⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠟⠻⡛⣛⡻⠟⠛⠛⠿⣟⣛⢛⡿⠛⠿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⡏⠀⢀⣠⠤⠒⡟⠓⢤⡀⠀⠀⢀⡠⠞⠻⡲⠤⣄⡀⠀⢹⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⠒⠻⣾⠿⠷⠶⢾⡒⠒⠒⠚⢳⡞⠛⠒⠒⠒⡿⠶⠾⠿⣷⠟⢒⡿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣕⢄⠙⣧⡀⠀⠀⢳⠀⠀⠀⡼⢣⠀⠀⠀⡼⠁⠀⢠⡾⠋⡠⣫⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣷⣌⢿⡢⡀⠈⢇⠀⠀⡇⢸⠀⠀⣰⠃⢀⢔⡟⣡⣾⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣵⡝⢦⡈⣆⠀⡇⢸⠀⢰⢃⡴⣣⣮⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠙⢼⣼⠇⠸⣧⡷⠋⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠳⣦⡹⡄⢀⢏⣤⠞⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⢈⣻⣷⣾⣟⡁⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⣤⡶⠞⠛⠋⠉⠉⠉⠉⠙⠛⠳⢶⣤⡀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣋⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⣛⢶⣤⣄⣀⠀⠀⠀⠀⣀⣀⣤⠶⣛⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠿⣿⣿⣿⣿⣿⣿⠿⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⣣⡴⠎⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⢀⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣤⡴⣧⣤⣤⣤⣤⣤⣤⣤⣾⡧⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⢰⡅⣀⣀⣤⡄⠀⠀⠈⢉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠸⢄⣘⣿⡷⠃⠀⠠⢄⣀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⢸⠇⠀⠲⠖⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠸⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⢀⡇⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠷⠶⠶⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣼⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⣨⡟⢻⣏⠉⢹⡇⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠁⢹⡏⠙⢧⣈⡇⡤⠀⠀⠀⠀⢰⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡻⡇⠀⢀⠸⡇⠀⠀⢹⡟⠀⠀⠀⠀⠀⠈⢣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⡸⠁⡇⠀⠸⠀⡇⠀⠀⣾⠀⠀⠀⠀⢀⠀⠀⢸⠀⠸⡀
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⠇⠀⡇⠀⡄⠀⡇⠀⢰⠁⠀⠀⠀⠀⠸⠀⠀⠈⣇⠀⣿
⠀⠀⢀⣀⣹⣿⠿⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣿⠀⠀⡇⠀⠇⠀⡇⠀⡜⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠀⢸⡆
⠀⣴⣶⣿⡇⢉⣀⣀⠘⣆⢰⠛⢦⡀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣠⢺⡇⠀⠀⡇⢀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠈⣧
⣷⢿⡟⠁⠀⠻⣄⡈⠓⢾⡿⠀⠠⡈⠳⡶⠖⠋⠁⠀⢀⡴⠊⠁⣼⠁⠀⠀⡇⢸⠀⠀⡇⣸⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⠀⠀⢸
⠙⢦⡀⠀⠀⠀⢠⡙⠆⣼⠃⠀⢀⡇⠀⢇⡀⣠⡴⠖⠉⠀⠀⠀⡿⠀⠀⠀⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠘⡄
⠀⠀⠹⣦⠀⠖⣩⠁⠀⢥⣤⡴⣏⠑⢦⣜⠉⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⡏⠀⠀⠀⣇
⠀⠀⠀⠈⢣⡄⠁⠀⠀⠀⠀⡤⠈⠙⢦⣈⠑⠦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠻⣄⠠⢄⠀⠈⠀⠀⠀⠀⢻⡟⠃⠸⠷⠀⠀⢠⡏⠀⠀⠀⠀⡟⢧⣄⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡜⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠈⠙⡟⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⡇⠠⢤⣀⣀⡞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⢨⡉⠛⠳⠦⢤⣼⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⠀⠈⠓⠶⢀⣀⠀⠀⠉⠛⠛⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⠴⠖⠚⢙⡇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠤⠤⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠏⠀⠉⠙⠛⠓⠶⠶⠦⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠶⠶⠶⠖⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠶⠖⠀⠉⠉⠁⠀⠀⠀⣠⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠋⠀⢘⣧⡿⠷⠶⠶⠶⢀⣤⡤⠤⠤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠒⠒⠉⣉⣉⠁⢀⣠⣤⡤⠶⠚⠉⠹⣤⣤⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠿⠿⠿⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⢲⡄⡀⠀⢀⡈⠉⠙⠛⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⠅⢀⡤⠏⠀⠀⠸⡇⠀⡴⠊⠀⠀⠀⠁⠈⠉⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠘⡡⠞⣁⡴⠟⠀⠀⠀⡇⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠏⠁⠀⢀⣠⠞⠉⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⣥⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⠁⡠⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡍⠀⠑⣄⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠖⠁⢀⡄⠀⠀⢀⠎⠀⠀⠀⠀⢃⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠋⠀⠀⠀⠀⡴⠋⠀⣀⠔⠁⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠥⠄⠀⠀⠘⠦⡀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⣀⣤⠶⠊⠉⠀⢠⡍⠒⠀⠖⠀⠤⣍⡱⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢻⣦
⠀⠀⣠⡿⣿⠇⠀⠀⠀⢀⠰⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠹⣷⡄
⠀⢰⡿⢰⣃⠆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢿⡄⠘⣿
⢀⣿⢃⡾⣉⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠦⢀⣀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⠣⢸⣿⠀⣿
⣼⡇⢎⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⢀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣿
⢻⡇⠈⡁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⠖⡀⠄⣀⡀⠀⠉⢠⣾⠋
⠀⠹⣦⠠⡄⢀⡀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠡⣀⠀⢿⠄⣠⡿⠃
⠀⠀⠘⢷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣤⠰⣾⡟⠁
⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠒⠀⠞⢀⣼⠏
⠀⠀⠀⠀⠀⠘⢿⣆⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⡤⠠⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠿⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⣀⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⠁⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠂⠀⢀⡉⢁⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠉⠀⠀⣠⠟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⡀⠲⠊⠉⠑⠒⠰⠶⣂⣀⣤⡤⠤⠀⢈⣁⠀⠀⢀⡠⠴⠞⠁⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⠠⡀⠀⠀⠀⠀⠉⠉⠐⠒⢦⡈⠀⠀⠠⠄⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣬⡳⢦⣄⠀⠀⠀⠀⠀⠀⠹⣶⠂⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣍⡛⠻⠶⣤⣤⣄⣀⡰⣤⣤⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠈⠉⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠾⠟⠁







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢈⡇⡌⢁⢤⡄⣠⣤⣠⡄⡠⡄⢀⣤⡄⢀⢠⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⣞⣡⣅⣽⣼⣴⣋⣌⣸⣄⣧⣇⣼⣹⣄⣧⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠈⠀⠉⠀⠀⠈⢁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢩⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠉⠀⠀⠈⠁⠘



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠢⠄⣀⠀⠀⢸
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠘⠂
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣤⣀⠀⠉⠉⣿⣿⣶⣴⣤⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⢹⣿⣿⣿⣿⣿⡏⠔⠈⠙⡄⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠛⢿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣮⣴⡇⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠱⠟⠀⠀⢸⠉⠉⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢸⣿⣿⣿⣿⣿⡄⠘⡏⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠙⠻⢾⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠛⠿⣟⡛⠁⠀⢹⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠈⠙⢦⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠂⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣤⣤⣤⠤⣤⢤⠤⠤⣤⣤⣤⣤⣤⠤⣤⢤⡤⡤⣤⣤⣤⡤⢤⡀
⠀⠀⠀⠀⠀⢀⣠⠾⠗⠚⠉⠁⠀⠙⢢⣤⠤⠒⠋⠉⠀⠀⠈⠉⠓⠲⢤⣤⡞⠁⠀⠉⠙⠒⠿⢦⣄
⠀⠀⠀⣀⣴⠏⢹⠀⠀⠀⢀⡠⠔⠊⡽⠉⠓⢦⡀⠀⠀⠀⠀⠀⣠⠴⠋⠙⡏⠓⠦⣄⡀⠀⠀⢀⡏⠳⣤⡀
⠀⣠⣾⣟⣁⣀⣸⣤⣴⣚⣉⣀⣀⣰⣇⣀⣀⣀⣈⣓⣤⣀⣴⣊⣁⣀⣀⣀⣹⣄⣀⣀⣉⣒⣤⣸⣁⣀⣈⣿⣢⡄
⠀⠙⢯⡫⣍⠉⠉⠻⣍⠉⠉⠉⠉⠙⡏⠉⠉⠉⠉⠉⠉⣏⠉⠉⠉⠉⠉⠉⡹⠉⠉⠉⠉⢉⡽⠋⠉⠉⡩⣟⠝⠃
⠀⠀⠀⠙⠪⣳⣄⠀⠘⢷⣄⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⢿⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣴⠟⠁⢀⡴⣪⠞⠁
⠀⠀⠀⠀⠀⠈⠳⣕⢤⡀⠻⡷⣄⠀⠀⠹⡀⠀⠀⠀⢸⠸⡆⠀⠀⠀⣰⠃⠀⢀⡴⡻⠋⢀⢴⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⢮⠳⣄⠀⢳⡀⠀⠀⡏⠀⡇⠀⠀⢠⠃⠀⡰⢋⠞⣡⣶⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣵⡌⠳⣄⢳⡀⢠⠇⠀⢹⠀⢠⠏⡠⠊⣰⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠈⠳⣧⣸⠀⠀⠸⣄⡿⠊⢀⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢯⠀⠀⢀⡟⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⣇⢀⣞⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡾⠋

⠀⠀⢀⡴⣟⡻⠻⣟⡷⢆⠀⠀⠀⣴⣺⢞⣶⣾⠿⣤⡀⠀⠀⣠⡶⢿⣿⣿⡿⣶⡄⠀⠀⣠⢖⣉⣉⣉⣉⡲⣄
⠀⠀⠸⣏⠹⡋⠉⢹⠋⡽⠃⠀⠐⢿⣦⣾⣾⣾⣲⣿⠃⠀⠈⢿⡛⣿⣿⣿⣟⣻⠏⠀⠀⢳⡶⡧⠤⠤⢼⢦⡞
⠀⠀⠀⠈⢦⣳⢀⣯⠞⠀⠀⠀⠀⠀⠻⡾⣿⣯⡟⠁⠀⠀⠀⠀⠻⣽⣿⣿⡷⠋⠀⠀⠀⠀⠙⢿⡄⢀⡾⠊
⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠛⢻⣟⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⠈⠡⠃⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠧⠤⡤⣼⣈⣷⢦⠦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⡴⢦⣬⠶⣾⠶⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠋⠀⠀⠉⠀⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡵⠀⠀⠀⠜⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣠⠄⠀⡠⠚⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⡛⠉⢹⠉⢳⡤⠊⠀⡸⠀⠈⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣄⡀⠀⠀⠀⡜⠀⡀⢸⠠⣿⡿⠦⢠⠇⠀⠀⣇⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣾⠇⡰⠦⠾⣄⠔⠁⢸⣷⣾⣧⠐⡎⠀⠇⠀⢸⠀⢀⠀⢠⡄⠂⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⣿⡟⣻⡜⠀⠀⠈⠛⠋⠉⡿⠀⢸⡇⠀⡀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⠀⢿⡆⠀⠀⠀⠀⠸⣋⣼⣿⠁⠀⠀⠀⠀⢀⣼⢁⣀⣾⡀⠀⢷⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠀⢸⣷⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢀⡞⣿⢛⣋⡟⠀⠀⢸⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡏⢠⡞⠀⣿⠀⢸⡟⡆⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⡠⠻⡭⡤⠤⢸⡇⢀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠈⠀⠀⣿⠀⠈⠁⢹⡀⠀⠀⠀⡜⠀⠀⠀⢀⡄⠀⠀⢠⠄⠀⣼⣷⠈⢇⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣠⣤⣄⡀⣿⠀⠀⠀⠀⢳⠀⠀⡜⠀⠀⠀⣠⡞⠀⠀⢠⡟⠀⠀⣿⣿⠀⠘⢧⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣅⠀⢤⣬⡁⣿⡀⠀⠀⣤⠘⣧⡼⠁⠀⢠⣾⠋⠀⣀⠀⡞⠀⠀⢸⣿⣯⣤⡶⠦⣅⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡉⠓⠀⠀⠀⣿⡇⢸⠀⠉⠀⣹⠁⣀⢶⡿⠁⠀⠀⠛⠀⢰⠀⠀⣿⣿⣇⣀⡰⠖⠋⢡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠛⠛⢀⣿⡇⢸⠀⠀⠀⢣⠞⠡⠋⣀⣀⣄⠀⠀⠀⢸⠆⢸⣿⡿⢋⡉⠀⢠⠾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡀⠀⠘⣶⣿⠐⠚⠛⠉⠁⣤⠀⠀⢠⠟⠀⣿⠟⠀⠉⠁⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⡏⣇⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣶⡴⠋⠀⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠐⢾⣿⣄⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣿⠁⠀⠀⠔⢋⣵⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⡇⠀⠀⠀⢸⣿⠉⠉⠀⢰⡧⡤⠀⠀⠀⠀⢠⣤⠀⡀⠚⠿⠿⣦⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢷⣶⠟⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠞⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⢸⣁⢿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣤⢸⡷⢿⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡿⣦⣿⣤⣤⣄⣼⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠀⠆⢸⢹⡉⠉⡹⢿⠉⠉⣭⡿⠋⠉⠉⠉⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢸⠸⡇⠀⣷⠘⡆⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣘⠂⢳⠀⣿⠀⢣⠀⠀⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣾⣧⣤⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣾⡿⣷⣷⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣩⡖⠋⠁⠀⠘⣧⢳⡀
⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠙⠛⠃⠒⣂⠤⠤⣄⣹⣼⡇
⠀⠀⢰⣿⡏⢹⠿⣦⠀⠀⠀⠀⠀⠀⠀⠺⠤⣀⣀⠀⡠⢤⢄⠊⠙⠳⣤⡀⠀⠉⠉⢢⣀
⠀⠀⢸⣿⣇⠛⠀⠘⢷⡀⠀⠀⠀⠀⡠⠀⠈⠁⠀⠀⢰⣶⠀⠇⠀⠀⠘⠉⠑⡶⣶⠒⠚⠃
⠀⠀⠈⣿⣷⠀⠀⣶⢈⣷⡀⠀⠀⠺⢤⣀⣤⣤⣤⢤⣸⡁⠀⠀⠀⠀⠀⣀⡠⠋⣸⣿⣶⠶
⠀⠀⠀⠹⣿⣧⢠⡇⡀⠸⣇⠀⠀⠀⠀⠀⠀⡇⠀⠂⠙⠤⣤⠖⠀⠀⠀⠀⠀⣸⣿⣿⠁⣠⠞⠃⣠⣄
⠀⠀⠀⠀⠹⣿⣿⣡⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠐⠂⠂⠚⢣⣀⠀⡠⠔⠁⣼⣿⣿⡷⠀⠘⢀⡈⠘⠉⢪⣦⣄⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣧⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠦⣄⣼⣿⠿⣿⢳⠀⠐⡿⣿⠢⣤⡈⠉⠈⣙⣢
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡦⣤⠖⠆⠀⠀⠀⠀⠀⡰⠛⣿⣿⡿⣿⣾⣤⠤⡷⠺⠀⠐⡖⣶⠖⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣲⣡⡆⠸⣿⣿⣿⡖⢚⣣⡎⠉⣸⡆⢺⠭⠀⠀⠤⠄⠀⠀⣀⣀⣀⡎⠀⠉⢙⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠉⠀⢰⣿⣿⣿⠋⠈⠋⣠⠞⠉⡇⣴⣶⠀⠀⢴⢠⠀⠀⠠⡤⠄⠀⢨⡯⠏⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠤⠠⠊⠀⠀⠀⡇⠀⠉⠀⠀⠈⠈⠀⠀⠀⠉⠁⠀⠈⠉⠁⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣶⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠋⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠌⢤⣄⣀⡀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡖⠒⠓⣷⣶⣾⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣾⣿⣿⣧⡀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠛⠻⣿⣧⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⠿⠿⠿⠟⠛⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠲⠶⠾⢿⠿⢟⡿⠿⠏⠽⠿⣟⠿⢿⠿⠶⠶⢒⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⠏⢩⠛⠉⣁⠔⡾⡒⠒⠛⢳⠛⠃⠒⡺⡖⢄⡉⠙⢯⠉⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⠀⡇⣧⡃⠔⠈⠓⡀⠇⠈⡲⢄⢸⢈⠴⡊⠀⠇⡰⠋⠐⠄⣢⡆⡇⢈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡫⠁⠸⠸⠠⡟⠄⠀⢀⠀⠣⡀⡔⢀⠤⡿⡧⢄⠐⡄⡠⠃⢀⠀⠀⠜⡣⠰⢸⠀⠩⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢶⡤⣰⣞⠀⠷⠚⢢⠸⠀⡀⠿⢋⠁⠰⠀⠰⠀⢉⠺⠇⡀⠸⢠⠚⠲⠧⣐⣶⡠⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣍⠉⠓⠒⢻⡞⠦⠤⠮⠶⠧⢤⠤⠷⠮⠦⠤⠾⣾⠓⠚⠋⢉⡽⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠀⠱⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠁⠀⢀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣳⡄⠀⠹⡄⠀⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⣴⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠘⢆⠀⠀⢸⠀⠀⢀⠞⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⢦⢀⢸⠀⢠⠎⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣣⣸⢠⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠟⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣛⠿⣿⣦⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡿⠟⠛⠻⢿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣦⣀⠀⠙⠁⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠿⠛⠿⣿⣦⠀⢠⣿⡿⠁⠀⠀⠉⠻⢿⣷⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⢹⣿⡇⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠸⣿⠟⠁⣠⣴⣶⣶⣶⣤⡀⠀⢸⣿⡇⠈⠁⠀⠀⣀⣠⣤⣤⣄⡀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠈⠙⠻⣿⣦⡈⠋⠀⠀⢀⣴⣾⡿⠟⠛⠛⠻⣿⣦⡀⠙⠛⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣤⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⣴⣿⣿⣶⡄⠀⣿⣿⠀⢠⣾⣿⣿⣷⠀⠀⠀⢰⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⣿⣿⣿⣿⣿⢰⣿⣿⡇⢸⣿⣿⣿⣿⠄⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠈⠻⠿⠿⣳⣿⡟⢿⣿⡌⠿⠿⠟⠁⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣶⣿⠿⠋⠀⠈⠻⣿⣶⣶⣾⣿⠿⢋⡁
⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⣀⣤⣭⣍⠀⠀⠀⠀⠀⠀⠀⢉⣭⣭⣴⣿⡿⢿⣷⣄⣠⣤⣤⣤⣀
⣠⣶⣾⣿⣿⣿⡏⠁⠙⣿⣿⡿⠛⣿⣷⠀⠀⠀⠀⠀⢰⣿⡟⠻⣿⣿⠀⠀⣹⣿⣟⠛⠛⢻⣿⡆
⢿⣿⣤⣤⣼⣿⣿⣶⣾⣿⢿⣷⣶⣿⠟⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⣿⣶⣾⣿⠿⠿⢿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠙⠉⠉⠠⠦⠄⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠉⠙⠉⠛⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠰⠧⠤⠄⣀
⠀⠀⢀⣠⡄⠒⠛⠓⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠑⢢
⣀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣌⠳⠆⠈⠉⠉⠉⠉⠠⠾⢋⡴⢆⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠖⠉⠁
⠀⠀⠀⠀⣿⠛⡟⠀⠠⠤⠤⠤⠤⢀⡴⣛⡾⠁⣀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠠⠴⠶⢌⡷⡄⠚⠛⠋⠁⠈⠻⢄⣘⡷
⠀⠀⠀⡌⢿⠏⠀⠀⠀⠀⠀⠀⢀⢮⡙⠁⠀⠀⠀⠀⠀⢰⠄⠀⢴⠄⠀⠀⠀⠀⠀⠈⠳⣡⡄⠀⠀⠀⠀⠀⠀⠹⠇⣴
⠀⠀⣴⠃⠘⠀⠀⠀⠀⠀⠀⢠⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⡸⠂⠀⠀⠀⠀⠀⠀⢸⠉⡆
⠀⠀⣅⢠⡆⠀⠀⠀⠀⠀⠀⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣴⠀⠀⠀⠀⠀⠀⢸⠀⡘
⠀⠀⠈⠲⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠻⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢴⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠃⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠓⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣀⠀⠀⠀⠀⠙⠿⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⣴⠷⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀⠀⠀⠀⠈⠛⠛⢷⡾⠷⠾⠟⠛⠉⠀⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⢤⣤⡄⠀⢸⡂⠀⠀⣿⠂⢠⣤⡤⠰⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢹⠃⠀⠀⣟⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣼⡁⠀⠀⠉⠉⠉⢹⡇⠀⠀⣿⣉⡉⠉⠉⠀⠙⣧⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢸⠉⠳⣄⠀⠀⠀⠘⣇⠀⠀⠟⠈⢡⠀⢀⡴⠛⠋⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡌⠀⠀⠈⢳⠄⢀⡜⠛⠳⠤⠀⠀⠈⠾⠯⠤⢀⣸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⣧⣤⡤⠰⠛⠲⠞⠙⠲⠀⠀⠀⠀⠸⠀⠀⠀⠈⠉⠑⠒⢠⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠷⠶⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⠤⠤⠀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠈⢤⠤⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⠀⡀⠀⠀⣇⣀⣀⣀⡈⠉⠿⢂⠒⠒⠀⢤⣤⣄⣀⡀⡀⢀⡀⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠉⣟⠉⠉⢻⠋⠹⠟⠛⢩⣽⠛⠏⣭⡅⠴⡉⢹⠖⠒⠐⡟⠈⢿⡁⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠋⢉⠉⠹⢇⠿⠲⠶⢶⡄⠀⠀⣰⠴⠶⢧⣸⠇⠀⠈⠙⠆⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣄⣠⣾⡦⢾⣿⣦⡀⠀⠀⠀⣠⡟⢹⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠠⣿⡟⠛⠿⠀⠙⠁⣠⡛⠀⠱⠀⠀⠀⡛⠁⡾⢷⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣥⠄⠒⠒⠈⢀⣍⠉⠀⢿⠀⠀⣠⠃⢸⠀⠀⠳⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠂⠈⠉⣹⠏⠉⢹⣿⠀⢠⡋⠀⠿⢢⡄⠀⢣⣀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠛⠉⠠⠷⠤⠰⠾⠛⠳⡄⠙⠾⠆⠾⠇⠀⠈⠁⠖⠁⠶⠶⠄⠀⠀⠀⠀⠠⠶⣷
⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠀⠀⢀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄
⠀⠀⠀⠀⠀⡌⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⡀
⠀⠀⠀⠀⠀⠏⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⢃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠧⠤⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠐⠀⣠⣄⣠⣀⣀⣀⣀⣀⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⢞⣛⡿⢿⠿⣛⣛⣫⣿⣬⡁⣿⡏⣨⣿⣯⣛⣛⣛⠿⠿⢿⣛⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠶⢂⣀⣀⣀⣀⡰⠶⠬⠓⠉⠘⠫⡴⠖⣀⣀⣀⣀⣀⠶⢦⠝⠁

⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄
⠀⠀⢀⣴⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠉
⠀⠀⠀⠉⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢿⣿⣶⣦⠀⠀⠀⠀⢀⡀⢰⣿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⠉⠈⠙⠀⠀⢀⣀⣀⡽⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠂⠀⠀⠀⡼⠛⠋⢁⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠊⠷⢸⣿⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠉⣿⣿⠿⠿⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠋⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀⣀⣀⣤⣤⡤⠿⣿⠀⠀⠀⠀⠀⠀⣰⣿⠋⣽⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠚⠛⢛⣿⣿⣻⣿⣿⣿⣿⣿⣶⠂
⠀⠀⠀⠈⠀⠀⢀⡈⠉⠀⠀⣀⡀⠀⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⢛⡿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⠉⠉⠉⠉⠛⠛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⡀
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠃⣡⠞⠋⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢷
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⣰⠃⣀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢠⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠆⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⢊⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡖⢉⣥⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠠⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢡⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣘⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⠐⠒⠈⠉⢁⣀⣉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠛⠋⠉⠁⠁⢉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠳⣄⣀⣀⣀⣤⠤⠤⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⠉⠀⠀⠀⠠⠄⠀⠉⠳⣤
⠀⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠉⠙⠓⠒⠶⠶⠶⠖⠒⠚⢻⣄⡤⠤⠶⠒⣒⣈⣩⣿⡃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠳⣄
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠀⠀⠀⢀⡠⠤⠛⠓⠚⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠀⠀⠀⠀⠈⠓⠀⢀⡀
⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠯⣍⣀⡀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠛⠛⠛⠛⠋⠙⠛⠛⢦⡀⠀⠀⠀⠙⠲⣄⠀⠀⠈⠛⠛⠲⠶⠶⠦⢤⣤⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠑⣤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠚⠛⠓⠦⣠
⠀⠂⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣤⠤⣤⣄⣀⣀⣀⣀⣀⡠⠔⠉
⢠⠀⠀⠀⠀⠀⠻⣤⣀⣀⣀⡈⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣤⣤⣀⣀⣀⣠⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⣀⣤⠶⠶⠦⢤⣤⠴⠖⠋⠁
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠉⠉
⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⣀⣠⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠂⠀⠀⠀⣀⡀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠤⠤⠤⠀⠀⢀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠩⠉⢉⣁⣀⠀⠠⢼⡯⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠉⠐⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣒⡒⠒⠆⠀⠀⡜⠧⡀⠰⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠄⠀⠠⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠀⠤⠭⠭⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠁⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⠉⢁⣐⡒⠢⠠⠭⣁⣺⢀⣀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⡒⠒⠢⠬⠍⣉⣒⠒⠼⣿⠉⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠤⠤⠤⠭⠍⣁⣒⠂⠠⠬⢹⠿⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠙⠒⠂⠤⢤⡨⣄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠘⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠲⠈⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠔⠉⠉⠲⡀⠈⠀⠀⠀⠀⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠂⠴⣷⠈⡄⠠⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡈⢁⠔⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠘⠈⠐⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⡀⠀⠁⠁⠀⠀⠀⠁⢀⠂⠀⠀⠀⠀⠀⠌⠁⠀⠀⢀⠂⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⡀⠀⡀⠀⠂⠀⠀⠄⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠈⡁⠂⠀⠁⠄⠀⡀⢢⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠐⠄⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠤⠀⠀⠀⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢠⠀⠀⠂⠀⠠⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠍⠉⠀⠀⠀⠘⢄⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠘⠂⠀⡀⠄⠒⠒⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠁⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠠⢴⠈⠐⠀⠀⠀⠁
⠀⠀⠁⠀⠀⠀⠄⠠⠄⠀⠰⠀⠀⠀⠀⠀⠳⠀⠀⠀⢀⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠧⠀⠀⠀⢸⢔⠀⠐⠀⠢⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠔⠠⠄⠀⠁⠀⠀⠀⠀⠀⠠⣾⠀⠀⠀⢎⡠⠒⠀⠀⠀⢀⠀⠀⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠈⠀⠐⠌⡒⠨⠅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣠⠇⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠁⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠙⠈⠀⠠⠐⠂⠀⠀⠈⠠⠃⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠑⠒⠆⠴⠢⠖⠋⠀⠀⠀⠠⡤⠀⢤⢁⠄⠰⠐⡄⠀⣤⠤⠀⢠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡠⠀⠐⠤⡀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⠢⣠⣿⣌⠶⡡⣴⠁⠀⠄⣊⣄⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⡀⠀⠀⠀⠀⠠⠀⠂⠀⡀⢥⠀⠀⢰⠀⢇⠠⢀⠀⠀⠑⠐⠟⠚⠄⢘⠤⢙⣋⠁⠴⠛⠗⠃⠀⠉⠣⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠂⠠⡈⠉⠒⠤⢍⡀⠀⠁⡘⠡⠀⠀⢀⠔⠄⡐⠤⡀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⠂⠀⠀⠒⠒⠢⠀⠁⠀⠀⠔⠀⣄⣀⠀⠈⠀⠈⠀⠁⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠆⠀⠠⠐⠦⠀⠀⠠⠀⠀⠀⠄⣀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁⠀⠉⠁⠈⠣⡀⠈⠉⢙⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢶⡻⠿⠟⠛⠛⠛⠲⣄⡀
⠀⠀⠀⠀⠀⢀⣠⣾⣿⠀⠢⡀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⠾⠷⣽⢦⡀
⠀⠀⠀⢀⠴⠋⠁⠀⡅⠀⠀⠈⢶⣄⠀⠀⣀⣀⣀⡀⢀⣀⣀⣀⣤⣤⣴⣾⣯⣅⣀⠷⠈⠁⠹⣦⡀
⠀⠀⢰⠃⠀⠀⠀⠐⡇⠀⠀⠀⠀⠙⠄⠀⢀⣠⣴⣿⣿⣿⣿⣭⠉⣻⠟⠋⠛⠉⠙⢿⠂⠀⠀⠳⡘⡆
⠀⠀⢸⡎⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀⠴⠶⠉⠩⡿⠿⠿⠛⢻⣿⣷⣧⡀⠀⠀⠀⠀⠀⠈⢆⠀⠙⢳⡇
⠀⠀⠀⠙⢦⣠⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⣤⠀⠀⢺⢠⣾⣿⣿⣿⡿⣷⣄⡀⠀⠀⢀⡈⢷⡶⠋
⠀⠀⠀⠀⠀⠙⢶⣶⣦⣤⣀⠀⠀⠀⡀⠀⠀⠘⢦⡀⣭⣿⠟⠉⠛⠻⣛⣿⣛⣿⣷⣥⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢿⣿⣷⣶⣷⣶⣤⣤⣬⣿⣾⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡪⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⡈⠉⠙⢯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⢿⣿⣿⣿⣿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡘⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠾⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠟⠑⠐⠉⡁⣁⣁⢩⡑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡠⠀⠔⢐⠁⢨⠀⠰⠰⠄⠠⠐⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠨⠀⢰⡅⢈⠀⢀⠀⠠⠆⠲⠀⠂⠲⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⠤⢰⣟⡀⢉⠀⠀⠄⢀⣒⡐⠂⠀⠀⠀⠉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⢓⣋⣥⣶⣶⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⣁⣤⣶⣾⣿⣿⣿⣿⣿⠃⢟⣧⢻⣿⠈⠹⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣿⣏⣸⣿⣿⠟⢇⠘⠷⠺⠿⠇⡴⣿⣁⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⢅⠿⢿⡿⠋⠁⠀⡂⠰⣴⠇⠀⠀⠀⢉⣻⠏⠀⠀⠀⠀⠀⠀⠀⢀⢠⡼⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⡀⠁⠀⡀⢰⠇⢸⡇⠀⠀⠀⠀⢻⣧⡄⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢂⠘
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠶⠛⠃⠀⡀⠀⠀⠀⠀⠘⡄⠀⠈⢾⠀⣼⣷⣄⣀⣀⣀⣀⣿⣷⠀⠀⠀⢀⡀⠀⠀⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠁⠢⣀⠈⢢⡀⠀⠀⠀⠸⣆⠀⠈⢷⣿⣟⠁⠉⠉⠉⠉⠁⠀⢀⡟⠀⣸⠷⠂⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠑⡀⠀⠀⠀⢹⣦⠘⠘⣿⣿⣧⠀⠀⠀⠀⠀⠐⢌⣀⣴⠯⠀⠀⣀⠀⠀⢈⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣷⡀⠀⠸⡄⢻⣧⡀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⣿⣿⡷⠀⠀⠓⠈⣿⣷⠄⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⢀⣿⡇⢀⣀⠀⠠⢸⣇⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡿⢹⣿⠸⠟⠋⢳⣈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣼⡇⢸⣿⡇⠠⣄⠈⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠐⢁⡠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⣿⡷⣦⢻⣧⣿⣿⣿⣿⣿⣿⣿⠀⡀⠐⠁⠀⠀⠈⠉⠁⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣿⣿⡇⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠐⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣚⣒⠛⣿⣿⣿⣇⣿⣿⣿⣿⡿⠟⠻⡁⠀⠀⠀⢠⠄⠀⠀⢚⠰⠛⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢟⡽⣱⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⡘⢦⡀⡀⠀⠀⠀⠀⠈⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⣿⣿⠟⠉⠀⠀⠖⢋⠉⠁⣀⢐⡈⢳⣿⡦⠀⠀⠼⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⡀⠉⠹⠀⠙⣶⣿⣤⣿⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢀⡀⣄⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⢿⡻⡿⠟⢻⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠐⠒⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⣠⣴⣶⣿⡉⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠄⡀⢀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠙
⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢼⣿⣿⣿⡿⣿⣿⣿⣦⣤⢺⣿⣿⣿⣿⣿⢗⡴⠂⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⡾⠛⠀⠀⠀⠀⣄
⠀⣿⣿⣿⣿⡿⠋⢸⣿⣿⡏⢸⣿⣿⣿⣿⣡⠞⢁⣼⣿⣿⣿⣿⣿⡗⣼⠧⣀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⠀⠀⠀⠀⡷
⠀⢻⣿⠟⠁⢀⣤⣮⣭⣥⣶⣾⣿⣿⣿⣿⣿⣴⣿⣿⢿⣿⣿⣿⡟⣼⡏⠀⠀⣿⣆⠀⠀⠀⠀⣈⠁
⠀⠈⢻⣶⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⡉⠀⠀⠀⣿⣿⣾⣿⣀⣤⡤⠿⠛⠐⣧⠀⠘⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠉⢟⢦⣹⣿⣿⡿⠿⠛⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠑⠹⣿⣦⣀⠀⠀⠹⣿⣿⣿⠃⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⢿⣷⣶⣬⣿⡏⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⢀⣠⡂⠀⠀⠀⠀⠀⠀⢠⠿⠀⣀⣠⠴⣂⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡻⣿⣿⣷⢶⣶⣶⣦⣤⣤⣤⣍⣉⡃⣘⣋⣩⣤⠤⠄⠀⠀⠀⢀⠀⠋⠁⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣝⣿⣿⣿⣿⣯⣭⣈⠁⡋⠉⣀⣀⡀⢀⡀⡄⠂⠹⠠⣪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠟⣩⣭⡅⡿⠿⢿⣿⣶⠟⠁⢀⣴⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣜⢿⣿⡱⢀⢀⡔⠀⠀⠀⢠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠛⢿⣦⣅⠟⠈⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣻⡻⠒⡡⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡤⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣨⣿⠀⢀⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢿⡞⣿⡧⣤⣤⣄⣀⣀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣟⣀⣙⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⢫⠏⠀⠀⠈⠹⣷⣿⣿⣿⢟⡽⠋⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣱⠋⠀⠀⠠⢄⠀⠹⣿⣻⣿⣿⠠⣄⣀⠀⢹⡿⠷
⠀⠀⠀⠀⠠⠀⢀⣿⣿⣿⣿⣏⣼⠃⠀⢸⠀⠨⢈⡄⠀⠙⣿⣿⡿⠀⡀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠲⣼⣿⣿⣿⣿⣿⡏⠀⠀⣸⣀⣀⣀⡇⠀⠀⡏⠹⠇⢀⣀⣁⠀⢸⠁
⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⡇⠀⠀⣤⣄⣀⣉⡀⠀⠀⣷⣤⣦⣤⣤⣤⣤⣤⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣠⣾⣿⣿⣿⣿⣷⣤⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡤⠤
⠀⠀⠀⠀⣸⡄⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⠙⠛⠛⡉⠛⢁⣿⣿⣿⠛⢻⡍⠻⣿⠟⠉⡉⠉⠉⡅
⠀⠀⠀⠀⣿⣷⢸⣿⣿⣿⣿⣿⡇⠹⣿⣿⠀⠀⠀⡇⠄⢸⣿⠏⠁⠀⠼⠅⠁⣿⠀⠀⠁⠀⠀⠁
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⠤⠀⣇⣠⠘⣿⣤⣤⠄⠰⡆⠂⣿⠀⠀⡆⠀⠀⠄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠉⢻⠛⠛⢛⠛⠁⠀⣿⣿⣷⣤⢰⡇⠤⠿⠤⠤⡧⠤⠤⣷⡄
⠀⠀⠠⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠾⠶⠶⣿⣿⡾⠿⠿⠿⠿⠿⠶⠷⠒⠀⠀⠀⠘⠻⠷⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡉⠭⠍⣁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠁⠀⠀⠀⠀⠉⢢⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⣂⠉⠉⠉⠉⠉⠉⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⢌⠢⢄⣀⣀⠤⢊⠜⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣁⠂⠀⠀⢂⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⠉⡉⠉⢹⡁⠀⣀⠀⠀⠀⠀⢸⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠘⠋⠀⠀⠈⠁⠢⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠂⢀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠱⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⣀⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡎⢐⠄⢀⠠⡀⠀⠀⠀⠘⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⢀⢀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠠⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠉⠰⠒⠉⠑⠂⠀⠁⢠⡍⠉⠒⠊⠉⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢎⠈⡳⠸⠏⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠋⡀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠋⠑⢌⣉⡥⠚⠀⠤⢄⡀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠘⠤⢂⢌⠔⠀⢸⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⠌⠀⠀⠀⠀⠀⠈⠑⢠⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠢⣑⠤⣀⠀⠀⢎⡰⠀⢀⡠⢐⠥⠊⠀⠀⢸⠀⠀⠀⠀⢠⣎⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⡀
⠀⠀⠀⠀⠀⠘⡤⠤⠤⠤⣬⠃⠀⠀⠀⠁⠒⠨⠭⠬⠭⠭⠐⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠪⣓⢦⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠇⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠈⢁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀
⠀⠀⠀⠀⠀⡰⠡⢤⠬⠥⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠂⡶⡰⣒⠂⠲⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⡜
⠀⠀⠀⠀⡰⠱⠁⠀⠠⡄⠀⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠃⡣⡘⣢⡀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠤⣀⠀⢀⠔⠁⢀⠜
⠀⠀⠀⠰⠡⠃⢀⡀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⢊⡁⠈⠓⠤⣀⠜⠁⠀⠀⠀⠀⠀⠀⠑⡁⠀⡠⠊
⠀⠀⢠⢁⠂⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠲⢄⡀⠸⠀⠀⠀⠔⠁⢑⠀⠀⠀⢰⠈
⠀⢀⢂⠂⠀⠀⠀⠀⠰⣉⡁⠀⠀⠑⡐⡤⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠸⠤⠤⠀⠤⠄⢤⠀⠀⠀⢙⠀⠀⠀⠈⠐⠈⠀⠀⠀⢸
⢀⢂⠂⠀⠘⠃⠀⠀⠀⠀⠀⠰⠉⠆⠱⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣆⣘⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣁⣵⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿