Flatulence Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠊⠲⡄⠀⠀⠀⣠⠔⠒⠲⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⠀⠀⠀⠘⡄⠀⡐⠁⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⠀⠀⠀⠀⢳⢰⠁⠀⠀⠀⢠⠁⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢸⡇⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢠⡤⡀⠀⠀⡰⡆⢂⢀⠀⢸⠀⠀⠀⢀⠎
⠀⠀⠀⠀⢀⣠⠤⠭⠞⢦⠤⠇⠧⠤⠦⠀⢜⠀⠀⠀⠼⠤⠤⠒⢄⡀
⠀⠀⠀⡴⠋⠀⢀⡀⠀⠀⠀⠀⡀⠀⠠⢀⠀⠠⡠⠂⠀⠀⠠⠀⢀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠑⢢
⠀⠀⡸⠀⠔⢤⢠⠌⠆⠀⠀⠘⠁⠒⠋⠈⡄⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡰⠉⠁⠀⠀⠀⠈⠉⠙⠲
⠀⠀⢸⠀⠢⠀⠐⢒⠆⠀⠀⠘⢍⣀⣀⠌⠁⠀⠀⠀⠀⠐⠂⠀⠀⢠⠶⠾⠓⠒⠶⠒⠒⣒⠊⠁⣀⡀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠈⢳⠀⠈⣉⠄⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⡞⠒⠶⠤⠄⠀⠀⠠⠄⠂⠀⠠⠠⠀⠀⠀⠀⠈⠛⢦
⠀⠀⢀⠔⠋⢁⣀⡤⠤⠤⠊⠙⠀⢀⡠⠴⣀⡀⠀⠀⢀⡤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠙⠓⢲⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡸⠀⡜⠁⠀⠉⠒⣆⠤⢤⠎⢏⢄⡀⡇⡇⡇⣷⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊
⠀⢠⠃⣰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⡇⣇⣧⡝⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⠀⠀⠀⢸
⢞⡉⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠈⠙⠋⠁⠀⠀⠉⠳⣄⣠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠘⣆
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠶⠒⠂
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠄
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠴⣶⠶⠶⠶⠶⣶⣶⣶⣶⠶⠶⠶⠶⠶
⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣀⡤⠤⡄⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣄⣠⡤⠔⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠶⢦⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠚⠛⠒⠦⡀⠜⠐⠀⠉⠀⠀⠀⠈⢳
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠉⠁⢈⡠⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢄⠑⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢈⠒⠦⢤⣀⣀⣀⡤⠖⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⡀⠀⠀⠈⡆⠜⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⠁⠐
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠉
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣻⣿⣿⣿⣻⣿⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠛⠶⣄⣴⠋⠁⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣿⡴⠞⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⣠⠞⠁⠀⠀⢀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠟⠁⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⢁⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠋⠁⠰⠚⠉⣠⣤⣤⠤⠤⠶⠶⠶⠶⠶⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢉⡿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡶⣤⣀⣀⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣤⣴⠾⠛




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣭⣧⣬⣩⣯⣽⣟⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠏⠀⠀⠀⠀⠀⠀⠀⢸⡆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠘⠋⠁⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠰⠇⠀⢈⣇⣀⡀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠈⠁⠈⠉⠳⡄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⢦⡴⠶⢶⣄⠀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠠⣾⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣹⡾⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⡅⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠀⠀⠀⠘⢦⡞⠉⠀⣀⠈⠙⠳⠶⠦⣤⠀⠀⠀⠀⣀⣀⡽⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣷⠰⠊⠁⢀⣠⡤⠆⠀⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠿⣤⣀⠘⠉⠁⠀⣀⣤⠾⠷⠶⠶⡆⠀⠀⠀⠀⠀⡞⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢛⣉⠙⠷⣤⣄⣀⣠⠀⠀⠀⠙⢧⡤⠶⠛⢯⣅⠀⠀⠀⠀⠙⢦⣤⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠫⠭⠴⠒⠋⠉⠰⢒⣋⣀⣤⡤⣾⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠒⠒⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠸⣧⠀⠀⠀⠀⠀⢀⡀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣠⡴⠏⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠛⠛⠛⠓⠶⣄⡀⢠⡾⠉⠀⠀⠀⠀⠀⢄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⣧
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠻⣶⡟⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⣾⠃⡘⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠂⠀⡀⠀⠀⡀⠄⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⢰⡇⠀⠁⠈⠀⠀⠀⠀⠠⠆⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠠⠿⠛⠶⣤⡀
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣄⠀⢸⠀⠀⣀⠀⠀⠀⢸⢠⢠⠀⠀⠀⠀⠀⠀⣀⠀⠈⢷⡀
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠁⠀⢸⢇⠀⠀⠀⠀⠀⡼⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇
⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠎⢧⡀⠀⠀⠀⠀⣠⠎⠈⠓⠦⠤⠴⠚⠀⠀⠀⠀⠀⢲⠆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠙⠲⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⢰⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢿⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⢦⣄⣀⣠⡤⠚⠉⠉⠑⠒⠦⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢷⣄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠸⡀⠀⠚⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸
⠀⠀⠙⠻⣶⡄⠀⠀⠀⠀⠀⣀⠀⠙⠢⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⠁⢠⡟
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣀⡀⠀⣀⣠⡴⠏
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠐⠀⠒⠉⢀⣴⠞⠉⠻⣷⣄⣀⠀⢀⣀⣤⠾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣒⣒⣶⣶⣟⣋⣁⣀⣀⣀⣈⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢛⣉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣢⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⠉⠁⣉⣽⣿⡿⠟⠋⢩⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠙⣻⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣶⣿⣶⡆⠀⣼⣿⣿⣿⣿⣟⢀⢀⠄⠢⠤⠤⡀⠀⠀⠀⢾⠿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⠐⠼⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡆⠣⡀⠀⢀⡤⠂⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠟⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢰⡄⢀⡀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣼⡟⠀⢈⠈⠁⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣠⣤⣼⣿⣿⣿⣿⠁⣀⡀⢡⡉⢃
⠀⢀⣠⣤⣤⣤⣴⣶⣶⣚⣛⣉⣭⡍⠉⠉⠻⣿⣿⣷⣖⠒⠒⠲⢦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢿⣻⡄⠇⢀
⢠⣿⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠙⢿⣿⣶⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠛⠀⠐⡏
⣾⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⡄⠀⠀⢀⣼⣿⣿⡀⠀⠀⣠⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⡇
⢹⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⡏⠁⠀⢀⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡄⠀⢀⣙
⠀⢻⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢾⣿⣿⠇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠐⠶⠔⠛
⠀⠈⣧⢻⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢹⣥⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⣿⠘⣿⣷⣌⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠋
⠀⢀⡟⠀⢿⣿⣿⡀⠙⣿⣿⣿⣿⣆⠀⠀⢀⡤⠀⠀⡿⠁⠀⠀⠀⠀⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⡿⢀⡾⠁
⢀⣿⡇⠀⢸⣿⡟⠁⠀⠋⠹⣿⣿⣿⡇⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢠⠀⢻⣿⣿⡿⠛⠉⠀⠀⡼⠁
⢸⣿⠁⠀⠀⣿⡷⠀⠀⠀⠀⠹⣿⣿⣿⣿⡏⠲⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠛⢧⠸⡿⠋⣴⡃⠀⠀⡰⠁
⠈⣿⠀⠀⣰⣿⣄⠀⡄⠀⠀⠀⣿⣿⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠉⢹⠒⠚⠁
⠀⡿⠀⠀⣿⡿⠉⣸⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇⠀⠸⡇
⠀⠁⠀⠀⢻⠀⢠⠇⠀⠀⠀⠀⢻⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⡀⠀⡇
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠈⢇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⢠⠃⣸⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⡟⢀⡏
⠀⠀⠀⠀⢸⠀⠸⡆⠀⠀⠀⠀⠀⠀⠓⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⡼
⠀⠀⠀⠀⣼⡄⠀⠁⠀⠀⠀⠀⠀⠀⢸⣄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠘⡇
⠀⠀⠀⠀⠀⢹⣶⣶⣦⠀⠀⠀⠀⠀⠀⠘⣷⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⠖⣏
⠀⠀⠀⢰⠶⠶⡶⣄⣀⣀⣠⣀⣠⣤⣤⣴⣶⣶⣶⣿⢿⣦⣤⡾⢿⣶⣦⣄
⠀⠀⠀⠀⠑⠒⠂⠀⠀⢀⠀⢀⣘⡷⠶⠶⠖⢻⢸⡇⢸⡇⢸⡇⢸⡇⣷⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢻⣿⣷⢸⡇⢸⡇⣼⣿⣿⣝⣦⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡇⢸⡇⣿⣿⡏⠻⡾⠋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⢸⡇⣿⣿⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣇⣸⣇⣿⣿⠀⠀⢹⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣻⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠀⠀⠿⠿⣿⡷

⠀⣀⣀⠀⢀⠋⢆⠀⢀⡀⠸⠳⡀⣸⣿⢆
⠀⣿⣮⣑⡾⠤⠞⠁⠀⢿⣷⣤⣿⣿⡏⢸⠄
⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⢉⠻⢿⣿⠀⡼⠀⢀⣀⣤⣤⣴⣶⣶⣶⠒⠒⠒⠒⠒⠠⣤⣤⣄⣀⡀
⠀⠀⢹⠟⡡⣂⠐⠖⠉⣉⢳⡀⠀⠉⡞⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣄
⠀⠀⠸⣀⣄⡽⢁⡠⠤⠬⢴⣃⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠁⠀⠙⢆
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⠀⠀⢀⠀⢡
⠀⠀⡇⠀⣶⡄⠀⠀⠀⠰⣿⠆⠀⢸⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠈⢱⡄⢣
⠀⠀⠱⡀⠉⠁⠀⠀⠀⠀⠀⢶⠂⡜⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠋⠀⠀⠀⠀⠀⠈⠉⠁⠉⠛⠋⠀⠀⠀⢰⢻⢸⡆
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⣠⣋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢆⡇
⠀⠀⠀⠀⠀⠉⠓⠒⡲⠚⠙⡇⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠰⡄⠀⠀⠀⠀⡀⠀⡜⠀⢠⣾⢿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⢀⣸⠀⠀⠀⠀⠀⢀⠁⠒⠤⢤⣶⣾⣿⣿⣿⣤⠤⠖⠋⠙⢦⡄⠀⢸⡥⠊⠀⠀⡏⠃⢸⠸⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢠⡄⠀⠰⡀⠀⠘⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠘⢄⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⠉⠒⡇⠀⠀⠀⢷⠂⠤⠄⢀⡤⢇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⢱⠀⠀⠀⢸⠀⠀⠀⢸⠀⠘⢴⠛⠚⠉⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⣤⣴⠀⠀⠈⣧⣄⣀⣼⠀⠀⠀⠘⣦⣴⣾⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠀⠀⠀⠻⠿⠿⠟⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈



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

⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⣴⣶⣶⣶⣶⣶⣶⣶⣶
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⠿⠿⠿⠿⢟⡛⠛⠿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣤⣤⣴⠶⠛⠛⠻⠁⠀⠀⢤⣽⣿⣿⣿⡟⣰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠾⣿⣿⣁⡀⠉⢿⣿⡇⡀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣬⣭⣤⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⣠⠿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿
⠀⠋⠙⠛⠛⠳⠶⠄⠁⢠⣬⣭⣄⣀⢀⡙⢛⣙⡿⣿⣿⣿⣿⠏⡈⠙⠛⠛⠻⠿⠿⢿⢿⣿⣷⣶⣶⣶⣦⡤⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠛⠀⠀⡄⠈⠀⠀⠀⣤⣀⢀⡌⠛⠋⠀⣮⠻⠿⠬⢿⢿⠏⠈⠻⠀⠒⠀⠰⠀⠐⠀⠃⠀⠀⠉⠍⠉⠙⠡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡷⠤⢠⠤⣄⣀⠈⠹⠟⢃⣁⡄⠀⢀⠀⠀⠈⢀⡀⠀⠀⢰⣀⣠⡄⣀⡀⠀⠀⢀⢀⡀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⣧⣞⣗⣀⠀⠉⣤⡴⠄⠋⠁⠘⠿⠿⠷⠂⠀⠀⠁⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠈⠈⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿
⠀⣿⣿⣿⡿⠟⠀⠀⢀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣿⣷⢿⣿⣿⣿⣷⣶⣶⣶⣶⡆⠀⠀⣺⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠛⠉⠀⠀⠀⣠⢔⡡⠶⠞⠃⠀⠀⠀⠀⢀⣠⡞⠀⠿⠿⠿⣿⣯⡆⠉⠉⠉⢿⣿⣿⡿⠟⠁⠀⣼⢻⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⢿⣽
⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢀⣦⣾⣿⣿⣷⣿⣿⡆⠀⠀⢺⡿⠯⠁⠀⠀⣠⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠞⢀⣿⣶⣤
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⣿⣿⣶⠦⠀⣰⣿⣿⣿⣿⣿⣿⣷⣦⣄⣠⡄⢠⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠾⠟⣴⣶⠂⠀⠀⠀⠀⠀⠠⠂⠀⣾⣿⣿⣿⣿⡿⠿⠟⠁⠞⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣿⠿⣿⣿⣿⣿
⠀⣄⠀⠀⠀⢸⣿⠃⠀⣠⠀⣠⡆⠀⠀⠀⠾⠟⠛⠋⠉⠁⠀⠀⣰⡇⠀⠇⢶⡈⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⣿⣧⡀⠀⠸⠛⡀⠐⠋⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⡆⠈⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⣀⣀⣴⠿⠋⠀⢀⣾⣶⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠏⣱⣾⡷⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠛⠉⠁⠀⠀⠀⢀⣾⣿⣿⡆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⡴⢋⣾⣿⡆⢀⣶⣾⠟⣹⡟⠀⠀⠀⣀⣀⠀⠠⢆⢄⣀⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿
⠀⠵⠿⠿⠾⠶⠿⠷⠷⠶⠟⠀⠀⠀⠉⠀⠠⡴⠾⠿⠟⠋⠀⠂⠀⠀⠀⠀⣼⢿⡿⢿⠿⣿⡷⠄⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡧⠸⠿⠿
⠀⣶⡲⢶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣀⡀
⠀⠛⢿⣶⡌⠙⣿⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠱⡌⢻⣷⡀⢠⡀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⡀⠙⠁⢠⣽⣿⣿⣛⣛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣌⣿⢛⣧⡇⣶⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂
⠀⣿⠿⠀⢰⣾⠏⠙⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢡⠛⢸⡟⠃⢟⡼⣣⡦⠁⣀
⠀⠀⠚⠳⢾⣿⣧⠀⠸⣿⣿⣿⡿⠓⠀⠀⠀⠀⣸⠛⢻⣟⠓⢆⠀⠀⠘⠁⠀⠈⠰⢋⣴⠇⠈⠀⠀⠀⠀⣠⣤⣴⣶⣶⣦⡴⠦⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⠿⣦⣄⡀⠀⠀⣰⣿⣄⠀⠀⢀⣈⡓⠀⠀⠐⠄⠀⠸⠏⠀⢀⡴⠀⠀⠀⠀⠻⣿⡿⠫⠟⠿⠿⠧⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣆⢺⣿⣿⣿⣷⣦⣄⡀⠀⠀⣀⠱⡸⣇⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣄⣀⠐
⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠿⠿⠿⠿⠿⠿⠇⠀⠙⠃⠁⠙⡼⡷⠀⠀⠀⠀⠁⠀⠀⠀⠀⣨⣿⡏⢰⣶⣿⣶
⠀⠠⠿⠿⠛⠓⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠈⠙⠛⠿
⠀⠀⠀⠈⠉⠁⠈⠀⢈⡉⢉⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⣀⣄⣀⣀⣀⡀⣠⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣾⢾⣷⣾⡿⠶⠆⠠⣦⣤⠀⢲⣦⡄⠀⣴⣶⣶⣿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠾⠿⠿⠖⠚⠙⠉⠛⢛⠛⠿⠿⢷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⢠⡏⡄⠎⡜⠁⠁⠀⢠⡄⢹⣿⠀⠈⠻⠇⠀⠈⢡⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣶⣤⣤⡍⠉⠙
⠀⣿⣶⣶⠆⠸⠀⠃⠀⠂⠀⠀⠀⠀⠳⣈⠋⠀⠀⠀⠀⠀⣺⣇⣿⡀⠀⠷⠧⣤⠀⢀⡀⢠⡤⠘⠿⠉⠀⠰⠄⠀⢀⠀⠀⠐⠆⠀⠀⡀⠂⠈
⠀⣿⣿⣿⡆⠀⠁⠀⢠⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢲⡆⠀⢶⡶⣿⠂⠀⠀⠄⠈⠀⠀⠁⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣉⠀⠀⠄
⠀⠿⣿⣿⣷⡄⠐⠂⠠⠶⠆⠀⢀⠀⠈⠂⠀⠀⠀⢸⠇⠀⣸⢉⣿⠀⠀⠀⠂⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠏⠀⡀⠄
⠀⠀⠛⠛⠛⠉⢀⡐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣿⣶⣿⠀⠀⠀⠁⠀⢠⠀⠀⢀⡀⠄⣠⣿⠄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⣤⣄⣈⠙⠛⠷⠂⠀⠀⠀⠀⠀⠀⠀⠆⢰⣄⠀⣿⢿⣿⠀⠀⠀⠠⠇⠀⠀⠀⢀⣷⣾⣿⡿⠶⠶⠦⠀⠀⢴⣶⡶⣶⣴⡟⠀⠋
⠀⢀⣤⣼⣯⣽⣟⣿⣷⣶⣄⣀⠁⠀⠀⢀⡇⠐⢀⣼⡟⠀⣿⣆⣀⣤⠄⢠⠁⠀⠀⠀⠀⠙⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠐⠒⠀⠠
⠀⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠷⠀⠀⠿⠷⠀⠘⠏⠀⠀⠿⠿⠿⠿⠞⠃⠀⠸⠂⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠆⠒⠐⣶⣶⣤⣤⣤⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢤⡤⣰⢆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⢶⣿⣿⣿⣿⣿⣽⡄⠸⠐⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣴⡏⠀⠀⣻⣭⣦⡝⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠇⠀⠀⣽⣿⠿⣿⣿⣿⠟⠀⠀⠀⠻⠏⠉⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡀⠀⣿⣿⣿⣿⠎⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⡿⠁⣀⠔⠀⠀⠀⠀⣿⠏⠉⠀⡎⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣦⣴⣻⣿⣿⣿⠉⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⢀⡞⢉⡙⠶⢦⣤⣄⣀⣠⣤⣴⣇⠀⠀⣠⠀⢶⣿⣿⣿⣷⣄⡄⠀⠀⠀⠀⠀⢿⠿⣿⡻⣿⡧⢛⣥⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢡⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠆⠀⠛⠳⠶⠨⣅⢈⣿⣩⠡⠀⠀⠀⠀⠀⠀⠲⣀⡇⠉⢀⡘⢳⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⠤⠻⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠠⣤⠀⠀⣿⡀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⢲⡔⣄⢠⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠈⢘⢁
⠀⢀⣤⣶⣶⣶⠲⠶⠦⣤⣄⣀⣀⣀⣠⣤⡤⠤⠠⣤⣄⣀⣀⣀⣀⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⠛⠶⠞⠋
⠀⢸⣿⣿⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⢰⣾⣿⣿⣿⣿⣿⠿⣟⡧⠀⠀⠀⠀⠀⠀⢦⡠⠠⠢⡔⣀
⠀⢸⣿⠛⣿⢱⡆⠀⠀⣼⣿⣿⣿⣿⣿⠿⠛⠁⠈⠁⢘⣿⣿⣿⣀⣄⡻⢿⣆⡄⠀⠀⠘⠷⢿⡄⠃⢠⣿⣿⣿⣦⠐⠢⣶⣤⣤⣶⣶⣶⣶⣶⣿⢉⡙⢿⣷⣶⡄
⢀⣾⡟⠁⠀⢈⡴⠁⠘⠿⠁⠀⠻⠿⠋⢀⡀⠀⠠⡆⠈⠉⡹⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⣧⣀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣧⡸⠧⠸⢿⣿⣿
⣸⣧⡿⢂⡞⠋⠛⢦⣄⣀⣀⣀⣠⣴⠶⠿⣧⠀⢸⣀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣭⣿⣿⣿⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⠉⠉⢀⠀⠀⢾⣿⣿
⠀⢹⢃⡾⠙⣤⠤⠚⠁⠉⠉⠉⠁⠀⠀⠀⠘⡇⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠆⣴⣶⠐⡌⠉⠉⠁⠉⠁⠀⠀⠘⢣⠀⠀⠘⡇
⠀⢰⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⠀⠉⣻⢇⣷⣄⣀⣀⣠⣤⡶⠺⠏⣈⡱⠀⠀⣧
⠀⠘⠶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⠀⢠⠎⠀⠈⠉⠉⠉⠁⠀⠀⠃⢨⡆⢃⠈⡿
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⢤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡈⠀⠀⡀⡇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⢄⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡐⠀⠀⠀⣇⡇
⠀⠀⠀⠀⢹⣿⣿⣿⡛⢛⡃⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠸⣿⣿⣟⠁⠘⠁⠀⣸⣿⣿⣿⣿⡟⠿⣿⠃⠀⢺⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠐⣦⣀
⠀⠀⠀⠀⠀⣿⢻⣯⠀⠀⣼⠀⠈⠛⠉⠙⠿⢰⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠏⢀⠤⣿⣭⣽⣲⡶⠒⢀⣶⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢸⣷⣿⠈⠁⢀⡞⠻⠷⢦⣤⣤⡤⠴⠿⡆⠀⣶⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⢠⣤⣾⠟⣡⣿⣿⡿⠛⠀⢻⣿⣿⣿⣿⣿⣿⣿⠛⠀⠿⢲⣤⣀
⠀⠀⠀⠀⠏⡇⢿⠀⡜⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⣼⡿⣏⠰⢈⡟⠛⠋⠁⠀⠀⠀⠈⠓⠀⠈⠹⣿⡟⠷⠀⣶⣿⡿⠿⢿⣿⣿⠍⠀⠀⡄⠀⠻⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠈⢂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢶⡀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣋⡴⠖⠀⠀⠀⢡⣂⡀⠙⠋⠀⠀⠰⡇⢰⡿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠱⠶⠈⢷⡷⠀⠀⠀⠀⠀⠀⠀⠘⣦⡄⠀⠻⠿⠘⢧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢦⣄⣠⠀⢸⡿⠿⠿⠿⢿⣷⣦⠘⣁⠀⠠⠟⠛⡟⢁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠆⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄⠀⠀⠀⠀⠀⠱⣇⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⡾⣦⣄⡀⠀⠀⢀⣿⡛⠛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⢷⣬⣙⢻⣿⣿⠻⣷⣷⣤⣾⢟⣿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣭⣛⢿⣧⣿⡇⢹⡇⢸⡷⢸⣿⢸⣯⡛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⣷⡼⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⢸⡇⢸⡇⢸⣿⢸⠿⣿⡀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢸⡇⢸⣿⣾⠀⠈⠙⠓⢶⣶⣀⢈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣸⡇⢸⣇⢸⣿⣿⠀⠀⠀⠀⠀⠹⢝⠗⠦⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣧⣼⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⢹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠿⠿⠟⠀⠀⠻⠿⠿⣿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠒⠒⠤⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣤⣠⠡⠤⠀⠀⠠⠐⠁
⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⠿⠋⠀⠀⠀⠻⠿⠟⢿⣿⣿⣿⣿⣶⣾⣄⡙⢍⠀⠈⠁⠈⠛⠉⠗
⠀⠀⡔⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⢾⣝⠻⠿⠿⢿⣿⣿⣿⡟⠀⠛⠀⠀⢨
⠀⠀⢱⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠻⠿⠛⠃⠀⠀⠀⣀⠀⢂
⠀⠀⠀⡸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣿⣿⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡱⠂⠀⢁⣄⡸⠃
⠀⠀⠀⠀⢻⣧⠀⠀⠀⣠⡀⢢⣸⣿⣿⣿⣿⣿⣿⣩⡄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡜⣿⡀⠀⠀⣿⡷⠀⠈⠻⢿⣿⣿⣿⣿⣿⣧⣬⡉⠀⡊⠀⠀⠀⠰⠁⠀⠠
⠀⠀⠀⢸⡇⣿⠇⠀⠸⣿⠁⣠⠈⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠈⡡⠀⠀⢀⠃⠁
⠀⠀⠀⣾⢱⣿⠀⢀⣼⠗⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡀⠀⠈
⠀⠀⠀⠋⢸⠇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡐
⠀⠀⢐⣢⡿
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠋⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠠⠤⠀⠀⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠑⡀⠀⠠⠫⠹⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡎⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⡙⢿⣿⠀⠀⠀⢀⣀⠰⠄⠚⠻⡃⠀⠈⠁⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⠀⡀⢀⠄⠀⢾⡿⠢⡟⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠚⠀⠀⠈⠀⢰⡇⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⠀⢀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⢨⠓⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠄⠁⠀⠀⠱⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⡠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠔⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡧

⠀⣼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠄⠀⢀⡀⢀⠀⡀⠀⠐⠂⢀⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠃⠐⠀⠀⠀⠀⠀⠙⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡀⠀⠀⠂⠘⠂⠒⠒⠊⠓⠘⠀⠃⠚⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣦⠀⠠⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡃⠀⠀⠀⠀⠀⠿⠇⠠⠭⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⢶⢤⣤⣿⣼⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⢀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠠⣦⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠆⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⠖⠒⠒⠂⠒⠆⠀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⠠⠋⢻⠀⠀⠀⠀⠀⡒⠺⠋⠂⠰⣶⢤⣀⣶⣶⢚⣶⣛⣦⣤⣄⡄⠀⠙⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠄⠰⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣤⣀⡴⠋⢳⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣼⣇⠴⠋⠛⠉⠀⠀⠀⠑⠾⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⢻⠀⠀⡖⠉⠈⠉⠑⠂⠀⠀⠀⠂⠉⠁⠀⠈⢳⠀⡏
⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢸⡆⠀⠀⠀⠘⡆⠀⡇⠀⢀⣠⣤⡀⠀⠀⠀⢤⣴⣀⠀⠀⢸⣠⠇
⠀⠀⠀⡠⠚⠁⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠘⡦⠇⠀⠀⠘⠉⢀⡴⢀⣀⡠⠥⠤⠤⠤⠼⢯⡴⠲⠤⠤⣄⣀⡀
⠀⠀⠊⠀⠀⢀⡴⠊⠉⢐⡄⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⣴⡞⢉⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠉⠉⠲⣄
⠀⠀⠀⢀⠴⠉⠀⠀⠒⠊⢀⣀⠀⡀⠀⠀⠀⠀⠀⠑⠤⣾⠟⠋⣩⠤⠀⢀⣀⠀⢀⡤⠖⠷⠤⢄⣀⣀⠀⠀⠀⠀⠀⢘⡆
⠀⠐⠚⠁⠀⠀⠀⣀⠤⠚⠁⢀⢀⡸⠀⠀⠀⠀⠀⢀⡠⠞⠰⣏⣠⠖⠋⠁⢏⣠⠎⠀⠀⢀⡤⠚⠁⠈⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢸⡦⠠⢄⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠒⠓⢀⠠⣿⣿⡥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡧⠀⢶⣟⣵⣯⣶⣶⣶⣦⣤⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢷⣷⠿⠿⣿⡿⠻⢿⣿⣿⣿⣿⠁⠀⢙⣯⣉⣿⣿⣿⣿⠎⠁⢼⣷⠀⠘⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠤⠄⠠⠤⡀⠀⣀⣀⡀⠀⠙⠻⠛⠁⠀⢠⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⣿⣿⣶⣦⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⢴⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢹⣿⣿⣿⡂⠀⠀⠀⠀⠙⣿⣿⡿⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⠛⠙⢿⡆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣴⡇⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣵⠀⠀⣸⣷⡆⠤⠴⠆⠀⠀⠀⠀⠨⢔⠀⠀⠀⠀⢘⣦⡀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⢺⣿⠟⢆⠈⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡏⠀⠀⠀⠀⠰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠄⠀⠀⣀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠋

⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⠀⠀⠀⠀⠈⢻⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⠁⠀⠀⠀⠀⠀⠀⢹⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣦⣄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⣉⠁⠀⠀⠀⠀⠘⣷⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⡾⠃⠀⣐⣶⣖⣒⣚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠋⠀⠀⠀⢀⣠⣴⡟⠁⠀⠐⠾⠭⢭⣭⣉⣉⠛⠋⠀⠀⠀⠀⠀⠀⠀⣼
⣿⡇⠀⠀⠀⠀⣀⣀⣠⣤⣴⠾⢻⡟⠛⠛⠛⠛⠛⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠆⠀⠀⠀⠀⠀⠀⣿⠋
⢹⡇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⣤⣤⣤⡾⠋
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢈⣁⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠛⠙⠀⠀⠀⠀⠀⢲⡠⠶⠞
⠀⠀⠀⠀⠀⠐⠻⣥⡴⠶⠟⠛⠛⠀⢀⡐⠲⢮⣁⠀⠀⣀⠀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠂⠾⣋⠉⠁⡀⠘⠛⠂⠀⠈⢳⣆⣜⢲⣾⣿⠇
⠀⠀⠀⠀⠀⠀⡀⣒⠀⠀⠅⢸⠷⠆⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⢩⠹⡤⠀⢾⠀⠀⣸⠁⠀⠀⠀⠀⠀⠙⣿⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⡘⠃⠘⢟⠙⣄⡤⠒⠻⡅⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠍⠀⠀⠀⠈⢣⡀⠀⠀⠄⠀⢰⣆⠀⠀⠀⠀⠀⠀⢠⡤⣘⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡀⠓⣀⣌⢠⡼⢷⣸⡀⠀⠀⣘⠳⠁⠀⠠⣀⡀⠀⠀⠀⡿⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠆⠉⠁⠠⠓⠈⠁⠰⠀⡾⢻⡆⠀⠀⠀⠘⢿⣷⡶⠶⠶⠤⣇⣠⠤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠒⠀⠓⠂⢰⠂⡀⠀⠀⠀⡀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠶⢤⣤⠙⠢⣘⣧⡀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣇⠀⠀⠀⣧⣿⠀⠀⠀⠀⢸⡟⠀⠠⣠⠀⠀⠀⠀⠘⠻⣿⣾⣿⣧⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠐⠀⣠⠈⣯⠀⠁⠀⢀⣿⠏⠀⠀⢀⠀⢸⡇⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠘⢧⣛⣿⣿⣿⣧⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠹⠂⠈⢂⣄⡀⠊⠀⣤⠀⠘⠘⠂⠈⡇⠀⠀⢰⢀⡴⠀⠐⠢⠀⠀⠀⠀⡈⠙⠻⣿⣿⠇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⡄⠀⠀⢠⡏⠀⢸⣾⠀⢿⣄⣾⣷⣆⠀⠃⠀⠀⠀⢿⣿⣶⡄⢀⡷⠀⠀⠀⠀⠀⠀⠉⢻⣿⡀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⢀⠀⠰⠿⣇⠰⠋⡹⠀⠈⣿⣿⢿⡀⠁⠀⠀⠀⠀⡻⣿⣿⡧⠏⢁⠀⠤⢂⠀⠀⠀⠀⠀⢿⠆⣸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠒⢿⣄⠟⣷⡄⢸⣯⠐⠂⠀⠀⠀⣻⣿⣿⣇⠀⠀⠀⠀⠀⢢⡌⠉⠁⣠⡀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠂⡎⠉⠹⡷
⠀⠀⠀⠀⠀⢰⣖⣿⡆⠰⡀⢿⣿⣆⠈⠓⠀⢻⣧⠀⠀⠀⣴⣿⣿⠀⠉⠀⠀⠀⠀⠀⠈⢿⠀⢴⣷⠀⠹⣿⣿⡿⠇⠀⠀⠀⠑⠀⢡
⠀⠀⠀⠀⠀⠀⠙⠿⠇⠂⠁⠟⣿⣿⣧⠀⠀⠀⡿⡆⢀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠈⠻⣶⣌⠒⠛⠁⠀⠀⠀⠐⢶⣶⣶⣤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠸⣿⣿⣷⡀⠀⠀⠈⠀⠐⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣰⣶⣾⣿⣧⠈⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠸⠀⢀⣹⣝⡿⣿⣄⠀⢀⠀⠀⢹⡧⠀⠐⠄⠀⠀⠀⠀⡀⠀⠀⠠⣬⣽⣿⣿⣿⣿⣇⣠⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠡⡄⠀⠀⠀⠙⢿⣾⣿⢷⡄⢣⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡞⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡨⣿⣿⣿⣧⠀⢷⣄⠀⠈⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠰⢀⡀⠀⠰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠹⣝⡿⠋⢹⣇⢸⣿⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠙⠦⠀⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠙⠋⠀⠂⠈⠀⣿⡀⠀⠀⠀⣷⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⢰⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢻⠳⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⠙⠄⠰⠀⠀⠀⢠⣿⣿⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉

⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛
⣤⠤⠦⠤⢤⡀⠸⡧⣾⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⢸⡇⠀⠸⡳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢹⡄
⣿⠀⠀⠀⠀⠙⣦⡟⢳⣠⠞⠁⣤⡏⢀⡔⠋⠁⢸⣷⠶⣦⣰⠲⣆⣞⡉⢩⡿⢾⠈⣛⣷⣰⠋⢻⣤⠖⢶⠖⠒⢦⠀⣠⠴⠟⠉⢻⠀⡇⢰
⣿⠀⠸⠟⠀⢠⠙⠃⢸⠛⠦⠤⡄⠈⣾⡀⠻⠃⠀⣿⡀⠉⠁⠀⡏⡴⠚⠀⢙⠸⡄⠳⠶⣤⡀⠈⡇⡄⢰⢲⡀⠸⡏⢧⠀⠛⠀⢸⣠⣟⣿
⢿⣀⣀⣀⡤⢋⠀⠧⠼⢃⡏⠉⠁⣀⢇⠙⠚⡆⠀⡇⠙⠦⡤⠞⠀⠓⠒⠒⢋⡀⠙⠒⠚⠃⠙⠉⣀⠉⢉⣈⠙⢛⣁⣌⠉⡇⠀⠘⡏⢣⡼
⡼⠮⠭⠤⠔⠋⠲⠶⠞⠦⠭⠭⠭⠕⠋⠙⠒⣇⠀⢻⠓⠦⠤⠴⠋⠑⠒⠒⠋⠙⠒⠖⠊⠉⠙⠋⠉⠋⠋⠈⠉⠉⠀⢸⠀⡇⠀⠀⡟⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠤⣼⣁⡇⠀⢸⠃
⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠶⠋⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡾⣏⠀⠈⠉⠀⢀⣾⡀
⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⢠⠟⣗⠦⠤⠤⢞⣿⡀⣠⢄⠀⠈⢠⣄⠀⢀⠀⠀⢀⢦⣀⠀⡄⠉⢀⣀⣈⣃⣈⣉⣃⣒⣾⠿⢹⡇
⢸⡀⡠⣄⠀⢸⠁⡇⠀⢰⠻⡄⢸⠀⠀⠉⡟⠋⢉⡇⠓⠃⠸⡆⠄⡏⠈⠓⠋⠹⡆⢸⠀⠀⠀⣿⠀⢸⠉⠈⠉⢸⡇⠀⣿⢹⡀⢨⠇
⠀⠉⠀⢸⡀⣸⠀⢹⣀⡟⠀⠳⠎⠀⠀⠀⢳⠀⢰⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡀⡸⠀⠀⠀⠘⣇⠀⡏⠀⠉⠉
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠎⠀⠀⠀⠀⠳⡴⠁⠀⠀⠀⠀⠧⠏⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⣦⣄⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠉⠻⠋⠉⠀⠀⠉⠛⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠿⠟⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣾⠋⠀⠀⠈⢻⡆⠀⠀⠀⠀⢀⣴⠟⠀⠈⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⠀⠠⣶⡶⠶⠶⠟⠛⠁⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⠈⢿⡄⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⠀⠘⡛⠛⠛⠻⠶⣦⣄⠀⠀⠀⠀⠈⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠉⢻⣄⣀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢠⡿⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⣿⠛⠛⠀⣼⢷⠀⠀⣿⠛⢻⡖⠛⣿⠛⠃⠀⠀⠀⢈⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⣿⠛⠃⣸⠧⠼⣧⠀⣿⠛⣯⡀⠀⣿⠀⠀⠀⠀⢠⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠋⠀⠀⠉⠀⠀⠙⠀⠋⠀⠈⠃⠀⠙⠀⢠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣶⡶⠾⠛⠛⠷⠶⠾⠛⠛⠷⠶⠾⠛⠋







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

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⢻
⣀⣠⣤⣤⣀⡀⠀⢸⣇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⢸⠧⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡆⠀⢸
⢿⠀⠀⠀⠀⠉⢳⣈⣷⣾⡀⣠⠞⠋⠙⡗⠀⢀⠴⠞⠛⢳⣀⣤⣄⢠⣤⣤⢀⡖⠛⠛⣶⣸⠇⠻⢤⡄⢸⠷⢿⡀⣠⣤⣄⡤⢤⣀⠀⠀⢀⣠⡖⠚⣷⠈⡇⠀⣿
⢸⠀⠀⣖⡄⠀⠈⠿⡇⠈⡷⣇⠀⠘⠛⠳⣤⡏⢠⣤⠀⢸⣿⠈⠻⠼⠀⢸⠞⣓⣢⠈⠳⡸⡇⢴⣒⡓⠛⠀⠈⡏⡇⠈⢁⡀⠀⢹⡶⢾⠉⠡⡆⠀⢻⠀⣧⣤⡏
⢸⠀⠀⠉⠀⠀⡼⠀⡇⠀⡇⢈⣍⣩⠆⠀⡟⢧⣀⣁⠀⢸⡹⣆⠀⠀⠀⡜⢸⣁⡀⣀⡠⠃⢣⡀⠉⣹⠿⣄⣀⠇⠧⢤⠇⢹⣀⣈⡇⠘⠦⣤⠀⠀⢸⣶⡧⢾⡅
⢸⡤⣤⠤⠴⠋⣠⠀⠙⠛⡁⢾⣀⣀⣀⡴⢣⣄⠀⢹⠀⠈⣇⡈⠛⠛⠋⢀⣀⡈⠉⠉⣠⣄⠀⠉⠉⣩⣀⣀⣀⣤⣀⣀⡰⢦⣀⣠⠤⠶⡄⢸⠀⠀⢸⣇⣨⠟⠁
⡞⠓⠲⠶⠒⠋⠈⠓⠚⠛⠓⠶⠶⠶⠖⠚⠁⠈⠙⢻⡆⠀⣿⠙⠲⠶⠖⠋⠀⠉⠉⠉⠁⠈⠙⠓⠋⠁⠈⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡏⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⡇⡼⠀⠀⢸⠁
⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠶⠾⠁⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢠⢞⡏⠀⠀⠉⠀⠀⢀⣿
⢻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⢠⠿⡷⢤⣀⣀⣠⢞⣏⠀⢀⣄⠀⠀⠃⢀⡀⠀⠀⠁⠀⠀⣠⡄⢠⢤⡀⠐⠀⡀⢀⡘⢦⣙⠒⠦⠤⠤⣒⡿⢹⡄
⢸⠀⠀⡀⠀⠀⢰⠛⡆⠀⠀⡴⣄⠀⢸⠀⠉⠒⣦⠥⠶⢻⠉⢧⡞⠈⡆⡀⢰⠏⠙⣤⡼⠶⡆⠀⡟⠈⠁⠀⢷⠀⠀⡟⠚⠙⠚⢹⡏⠛⢻⠻⡇⠀⢸⡇
⠘⠧⠞⠉⡆⠀⣟⠀⢧⠀⢸⠀⢸⣀⡿⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⠃⠙⠦⠼⠁
⠀⠀⠀⠀⢳⣀⡇⠀⠘⠳⠞⠀⠀⠉⠁⠀⠀⠀⠘⣇⠀⡾⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⣟⢀⡇⠀⠀⠀⠘⣦⡼⠁⠀⠀⠀⠀⠳⣤⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣧⣀⠀⠿⠋
⠀⠀⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⡄⠀⠀⠀⢠⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣟⢻⣄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠖⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠻⠇⠀⠀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡷⠀⢠⣬⣿⣿⣿⣿⣿⣿⣿⠁⡴⠁⠉⠉⠁⠙⠐⠚⠁
⠀⠀⠀⢸⡏⣿⣿⡿⠉⠛⢤⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⠟⠃⠜
⠀⠀⠀⠀⠀⠹⢻⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠋⣿⣿⠁⠀⠀⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠄⡆⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠐⠟⠃⠀⠈⠁⠀⠀⣸⣿⣿⠏⣸⡿
⠀⠀⠀⠀⠀⠀⡃⠇⠀⠀⢠⠊⠉⠉⠢⡄⠀⠀⠀⠀⡀⠀⣀⢀⣀⣂⣤⣅⠿⠀⠀⣃⡄
⠀⠀⠀⠀⠀⣰⠅⠀⣠⠞⠁⠀⠀⠀⠔⠛⠛⠿⠆⠚⠨⠬⠋⠉⠉⠉⣿⣿⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠋⠀⠀⢹⣧⣤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠷⢀⡆⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠲⠆
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⢠⣤⣄⢀⣀⡀⣀⠀⢀⠀⡀⢀⠀⣄⠀⠀⣀⡀
⠀⠀⠀⠙⠓⠀⠺⠟⠙⠛⠛⠒⠛⠋⠱⠛⠫⠽⠖⠙⠒⠛⠚⠛⠛⠛⠛⠛⠛⠲⠾⠓⠒⠚⠓⠐⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⣀⣤⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡁⠀⠀⣻⡥⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡞⠁⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣤⣼⣣⡟⣀⣀⠀⣧⣠⣤⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⢉⡽⠋⠁⠀⡈⠛⠋⠉⠉⠙⠻⣅⠀⠉⠙⢯⡉⠉⠛⠳⣤⡀
⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⣴⠏⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠹⣦⠀⠀⠈⠻⣆
⠀⠀⠀⠀⣼⠃⠀⠀⢀⡾⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠈⢷⡀⠀⠀⠘⣧
⠀⠀⠀⣸⠃⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠘⣇
⠀⠀⢠⡏⠀⠀⠀⢸⠇⠀⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠸⡇⠀⠀⠀⢹⡄
⠀⠀⢸⠃⠀⠀⠀⣿⠀⠀⠈⢩⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠉⠀⠀⠀⣿⠀⠀⠀⠘⡇
⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⣸⠇⠀⠀⠀⡿⠀⠀⠀⠀⡇
⠀⠀⢸⡆⠀⠀⠀⢿⠀⠀⠀⠈⠛⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠛⠋⠀⠀⠀⢰⡇⠀⠀⠀⢰⡇
⠀⠀⠀⢷⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢧⣠⠶⢶⣄⣴⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡾
⠀⠀⠀⠈⠳⣄⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠉⠳⣤⣀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⠋⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠾⠷⣄⡀⠀⠀⠀⠀⢀⣠⠾⠷⠶⠞⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡟⣽⢧
⠀⠀⠀⣿⣿⣿⣿⣿⠉⠻⣿⡇⠀⢸⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡾⢰⡏⣠⡀
⠀⠀⠀⢻⣿⣿⣿⣿⡸⣆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠃⡟⢰⢻⡇
⠀⠀⠀⠈⣿⣿⣿⡿⠁⠲⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠉⣼⠁
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⡇⠀⠀⠀⠀⠀⠀⠸⡀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣰⠀⡇⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⢀⠆⠀⠒⠛⠲⢤⣙⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⠃⣰⠇⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡴⠋⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⠂⠤⠤⣉⡒⠒⠶⣤⣀⠀⠀⢰⣧⠞⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡀
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣔⡋⠀⠉⢲⡿⠁⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢠⠇
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠢⣠⡞⠀⠀⠀⠀⢀⣿⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠄⠀⠒⠒
⠀⠀⡇⠀⠀⠀⣀⣠⣤⣤⡴⢤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠤⣤⣀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠙⠒⠚⠁⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠁⠈⠀⠉⠀⠁⠸⠿⠛⠛
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠃


⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠫⣄⡁⢄⡰⠲⡄⡖⢦⠠⢒⡄⡇⠀⢸⢢⡔⢦⢠⠀⡄⡇⠀⠀⢠⢠⡖⡦⢲⢠⠰⡇⠔⢲⠰⠆⣔⣢⠀⠀⡖⢢⢰⠀⡆⣔⠢⢸⠖
⠣⠤⠇⠂⠣⢴⡇⠇⠸⠸⠩⠇⠇⠀⠸⠐⠦⠜⠸⠤⠇⠇⠀⠀⠘⠀⠇⠀⠸⠨⠀⠇⠏⠹⠀⠆⠯⠭⠀⠀⡧⠼⠸⠤⠇⠬⠽⠸
⣤⣤⣤⣤⣼⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣬⣯⣤⣭⣽⣥⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⡿⢿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣰⣿⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢿⣷⡀
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠃⠀⠀⠀⠀⠀⠈⣿⣧
⣸⣿⠁⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠙⠛⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣄⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣶⣿⠟⢿⣷⣾⡿⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡈⡱⠆⣀⣀⠀⠀⢞⣵⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢷⣱⡇⣠⡈⠃⠐⠁⣽⣿⣿⣇⣀⢰⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢡⡜⢮⣏⢛⡁⣀⠀⣴⣯⠹⣿⠝⣡⠘⠓⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⣧⠀⡀⠂⠻⣾⠟⠉⠀⠈⠀⠀⠀⠀⠠⢾⢿⣄⠀⠀⠗⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣔⣀⣠⡅⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠀⠀⠅⠀⠸⠀⢀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣿⣿⣿⡟⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣆⣰⣿⡿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣾⣿⡟⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡿⢫⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣸⣿⡿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠻⣟⠻⣷⣆⢄⠀⠙⢮⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⣿⣿⣿⣿⣟⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⡾⡀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢿⡟⠋⢁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡏⠀⢄⣀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⡆⠾⣟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠒⢠⡀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡬⣿⡄⠀⠀⠸⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢶⡟⡟⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⢸⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠳⣆⣸⣇⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⡐⢿⠁⠋⣠⠏⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠈⠰⣴⠏⣾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰⣦⣼⠋⠻⡆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠚⣇⡁⠀⠢⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⢸⡟⢿⡝⢷⣆⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠉⣉⡀⡀⠀⢀⣤⡆⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣄⠾⣯⣌⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⣝⠿⠟⠃⢸⣯⡤⢤⠄⠑⢢⢀⡀⠀⠀⠀⠀⢠⣶⣦⡀⣿⣇⣽⣿⣿⡓⡄⠀⠈⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡄⠁⠒⠀⠀⠀⠀⠀⠈⢧⣁⣀⠀⠀⠀⠘⠿⠊⠁⢻⣿⡿⠿⢿⣿⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠠⣦⣒⢿⣿⣷⣎⠉⠁⠀⣴⣦⣀⡰⢶⣦⡀⠀⠀⣴⣻⠥⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠟⣛⣹⠀⠈⠀⠀⢀⣭⠍⣋⡩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⡐⠈⠀⠑⠠⠀⠀⠀⠀⠠⡄⣀⠀⠈⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠒⢄
⠀⠀⠀⠀⠁⠀⢈⠀⠀⠀⢂⠀⠰⠀⠀⠀⠀⠀⠁⡠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠢⠀⠊⠀⠀⠀⠀⡄⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠐⠄⠊⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⡸⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠂⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠘⠐⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠂⠒⠢
⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⡄⠀⠀⡀⠄⠀⡀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⡠⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡀⠉⢀⠀⠐⠁⠀⠀⠀⠀⠀⠠⠁⠀⠀⡀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠴⠒⣀⣠⣶⣶⣶⣶⣤⣶⣶⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄⠀⠀⢰⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢻⣿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⣿⣟⡁⠀⢠⣤⡀⠀⢿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣼⣿⣿⣿⣿⣿⡆⠀⢹⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠙⠋⠘⢿⡿⠁⠀⢸⣿⣷⡄⠀⠻⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠁⢿⣿⣿⣿⢿⣯⡻⢿⣿⠿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣀⣀⣠⣴⣿⣿⣿⣿⡄⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠛⠿⣿⡇⠙⠇⠀⠀⠀⠈⠻⢿⣿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⢰⡁
⠀⠀⠀⠀⠀⣀⣤⣄⣰⣿⣦⣾⣿⣿⣿⣿⣿⣿⣟⠉⠁⠀⠰⡄⠀⢰⣦⣁⠃⠀⠀⠀⢀⣀⣀⣀⡀⠀⣤⣾⣿⣿⣿⣿⣿⣿⠟⠛⢫⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⣄⠀⠀⠉⠀⢠⣾⠏⠐⠉⠛⠛⠻⠿⠿⣿⣿⣷⣦⣍⣉⡻⢿⡿⠟⠁⠀⠀⠀⠁⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠛⠁⠀⠈⠁⠀⠀⠙⠋⠀⠀⠀⠀⣾⡏⠀⢢⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⣄⣀⣀⠀⠰⠇⠀⠀⠀⣽⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡿⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡀⠀⠀⠱⡀⠀⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠐⠃⠀⠀⠀⠀⠀⠀⣠⡀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣎⡀⠺⠀⠀⣠⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠒⠂⠀⠀⠐⠒⠒⠒⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⣾
⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⢠⡟
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⣸⠃
⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⢶⣤⣤⣀⣀⣀⣀⣀⣤⡴⠶⠶⠖⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠇⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣠⣴⠶⠶⠚⠛⠛⠛⠛⠛⠛⠶⢶⣤⡀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣾⣋⣁⣀⣀⣤⣤⣴⠶⠶⠶⣤⣤⣀⣀⡀⠙⣷⣄⣤⡾⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣓⣒⣦⣔⣶⣤⣴⣶⣞⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠘⢷⣄⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣤⣄⠈⣿⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠻⣦⡀⠙⠗⠀⠀⣴⠶⠿⠶⠶⣦⠀⠀⠀⠀⢀⡉⠀⣉⠀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠛⠀⠀⢠⣤⣿⣤⣤⣤⣤⣿⣤⡀⠀⠀⠘⠁⠀⠿⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⣿⣿⣿⡄⠀⠐⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⣿⠀⠀⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⠁⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣏⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠟⠋⢧⣄⡀
⠀⠀⠀⠐⣟⠛⠛⠟⠢⠤⠴⠤⠶⠤⣶⠶⠾⠿⠿⢿⣿⣿⡟⣿⡶⣾⡟⢻⡟⣿⠿⠿⠿⠶⢶⡦⡤⠤⠶⠦⠴⠞⠿⠛⢛⡁
⠀⠀⠀⠀⠀⠉⠑⠒⠂⠀⠄⢀⣀⣀⣹⣛⣛⣛⣛⣻⣿⣿⡇⣿⠇⢿⡇⢸⡇⣿⣛⣛⣛⣛⣻⣅⣀⣀⠠⠄⠀⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠛⢻⣿⣿⡇⣿⠀⢸⡇⣸⡇⣿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣿⠀⢸⡇⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⢿⡇⢻⡇⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢿⠀⢸⡇⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠀⢸⣿⣿⣿⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣛⣋⣀⣈⣛⣻⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔
⣀⣛⠃⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⣤⠤
⣿⣿⣂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣤⣦⡀
⣡⠀⠛⠁⠂⠀⡄⠀⠰⣄⠀⠐⠛⠠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⢛⣅
⡿⣀⠀⠀⠆⠀⢰⠀⢭⠘⠃⢀⡀⠀⠀⡀⢀⠀⠀⠃⠀⠀⠠⡀⢀⡚⠠⡖⢠⣀⣀⠀⠀⠀⠀⠂⣶⣼⣿
⣰⣿⣿⣆⣠⡄⠐⠀⠀⠈⠙⠂⣼⣤⠞⢠⡟⠀⠰⢿⣄⠀⣦⢘⣾⡤⡷⢀⠀⢩⣥⣀⠀⠀⠀⢰⣿⢿⡯
⣿⣿⣿⡽⣿⣦⣴⡇⣾⢪⣌⡘⠻⠉⢰⢯⡾⠀⢴⣷⠛⠲⠆⡙⣯⡿⠚⠁⢱⣷⠸⡯⢰⡄⣴⣿⡻⣼⠿
⢻⣿⣷⣾⠿⣿⡿⠹⣵⠫⢻⠇⢀⣄⠄⡌⠻⣧⢈⣭⣯⢠⣶⣀⣶⠃⠀⠀⢨⣍⣷⡿⠶⠛⢿⡿⠏⠟
⠀⠼⣟⠀⠀⡈⠁⠰⣿⡦⠀⠀⣸⠏⠀⡄⠀⠿⣿⡄⣾⣷⣿⠏⠀⠁⢠⣷⣖⢨⡏⠋⢰⠷⣞⠛⠆⠘⢷

⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⣜⣿⣶⣔⠆⠀⠁⠉⠀⠀⣠⣤⣶⣦⣄⠀⠀⠀⠀⢀⡀
⠀⠈⠃⣻⠏⡀⠀⠀⠀⣀⠀⢽⣟⣻⠟⠈⠑⠶⢶⡦⠈⠀⠀⠈⠉⠩⣥⡬⢿⣿⣭⡧⣦⠀⠀⠀⠀⠠⠀⠤⠀⠀⣰⣠⣌⣄⣀⠠⢤⣦⣽⣥⣔⠄
⠀⠀⠀⠈⡷⠀⠀⠀⠀⠉⠑⠀⣿⡏⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠁⠉⠑⠈⠀⠀⠀⠉⠙⠿⣷⣶
⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⣸⣇⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢻⣷⡆
⠀⠀⢸⣦⣶⡖⠀⢀⣠⣾⣿⡿⢻⣿⣿⣿⣿⣿⣴⡶⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣄⠱⣲⣄⠀⡠⡇⠀⢻⡇
⠀⠀⠈⠛⠛⠛⠛⠻⣿⣿⣿⢷⠋⠸⣿⣿⣿⣿⣿⠀⠀⢀⠄⠉⣷⣄⢠⣄⠀⠀⢀⠄⠐⣀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣿⣿⡿⡣⡚⠑⣿⣿⢽⡧⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣂⣠⠙⣿⣿⣿⡗⣂⣷⣾⡆⠀⠈⠻⣿⣿⣶⣄⣸⣿⣧⣿⣼⣴⣿⣇⣼⣡⠄⣴⣿⣿⣿⣿⣔⡡⣀⠀⡁⠀⠀⢈⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⠁⠙⠻⢿⡟⣿⣿⣿⣿⣷⣿⣶⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣤⢂⣤⠈⣄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢀⡘⣿⣿⣿⣿⣏⠉⠿⣿⢿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠉⠿⢿⣿⣿⣿⣿⣽⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⡀⠈⣿⣿⡏⢿⣿⣷⣿⠃⢈⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠛⡟⠿⡶⠶⣿⠿⡟⢲⣭⣿⣿⣿⣿⣿⡕⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣽⠾⢎⣻⣿⣾⣿⡯⢿⠶⠚⠺⠿⠿⣿⣿⣿⣿⣿⣿⣿⣃⠧⠀⠚⠉⠀⠀⢻⣿⣷⣶⡟⠌⢻⡆⠀⢼⣿⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣽⣿⣿⣿⣿⣿⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡄⠀⠹⣿⣿⣿⢱⣾⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣢⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠆⠀⠀⠘⣿⣿⣀⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⣿⡏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⠄⠀⠀⠀⠀⠘⣿⣟⠳⣇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⢹⠛⠄⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣏⡷⣿⡅⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢃⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⢶⣮⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⡶⢤⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠛⠛⠛⠳⣦⡀⢀⡾⠉⠀⠀⠀⠒⠒⠤⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⣤⣀⡀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠙⣆
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠻⣾⠁⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⡿⠀⡎⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡼⠡⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠠⢵⠀⠀⠀⠀⠀⠀⠘⠋⠙⠳⣦⡀
⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠶⠆⠀⡅⠄⠶⠈⠀⠀⢰⢠⠐⠀⠀⠀⠀⠀⠀⡦⠀⠈⣷
⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢇⠀⠀⠀⠀⠀⠀⢠⠳⣄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⢹⣷⠄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⢦⣀⠀⠀⢀⡤⠃⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣄
⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠓⣄⠀⠀⠀⠀⠀⠙⢷⣄
⢐⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀⣠⣤⡊⠀⢀⡹⠀⠀⠀⠀⠀⠀⠀⠹
⣼⠁⢘⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠑⢄⠀⠀⠀⠀⣀⠤⠦⠤⣀⡀⠀⣠⠜⠀⠀⠀⠀⠀⠈⠉
⢹⣇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⠒⠒⠚⠛⠒⠊⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠙⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⠏⣴⠀⣰
⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠈⢀⣴⠏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⡶⠶⠾⠛⠁
⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⠻⢦⣤⣤⣤⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠤⠤⠒⠁⣀⡴⠋⢻⣦⣀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⡴⠶⠟⠋⠀⠀⠀⠈⠛⠻⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⣀⠀⣠⢄⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠟⠭⠿⠿⠸⠿⠟⠀⠧⠴⠸⠿⠹⠇⠀⠯⠅⠯⡇⠧⠇⠿⠟⠭⠿⠭⠺⠹

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⢤⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⢤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠴⠟⠘⣧⣸⠃⠀⢀⣛⣛⡃⠀⢰⡟⢻⡆⣿⠶⣾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠃⠛⠀⠘⠃⠀⠛

⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⡄⡀⡀⣠⢀⠀⡤⢀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠘⠒⠒⠒⠒⠒⠀⠒⠁⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠆⠀⠰⠤⠤⠀⠄⠤⠤⠤⠤⠤⠄⠰⠄⠀⠆⠄⠠⠶⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⡄⠀⠠⢀⣀⣀⣀⡀⢀⡀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⡠⠚⣯⣤⡔⠊⠛⢦⡀⠀⠀⠀⢀⢴⠊⠁⠀⠀⠈⠳⢄⠀⠀⠀⠀⡠⠊⠉⠒⢄⡈⠳⡦⡀
⠀⠀⣰⡡⠚⢿⠿⠓⠤⣀⣸⠘⡄⠀⢀⠎⠸⡀⠀⠀⠀⠀⢠⠀⢣⠀⠀⣜⠌⠉⡱⢄⠀⠑⡀⢐⡵⡀
⠀⠀⣿⡀⠀⢸⠀⠀⠀⣸⣿⡗⡇⠀⢸⠀⠀⠑⡀⠀⠀⠰⠁⠀⢸⠀⠀⡏⠀⠎⠀⠠⠒⠐⠓⠉⢀⡇
⠀⠀⠸⡈⢲⣿⣷⡖⠉⠀⢹⡰⠁⠀⠀⢇⠀⢀⠃⠀⠀⢠⠀⢀⠌⠀⠀⠱⡸⠀⠀⠃⠤⠤⠄⠂⡹
⠀⠀⠀⠈⠲⢏⣉⣑⣤⡶⠛⠁⠀⠀⠀⠀⠑⠧⣀⣀⣀⡠⠕⠋⠀⠀⠀⠀⠙⠣⢄⣶⣠⣤⠴⠊

⠀⠀⠀⢀⠴⠚⢭⣙⣍⡱⠦⡀⠀⠀⠀⠀⠀⠀⡠⡔⢺⣿⣿⣍⢱⠀⠀⠀⢠⡔⠈⠉⠉⠉⠒⣄
⠀⠀⢰⡥⠊⠉⢀⠇⢩⠹⡑⠳⡄⠀⠀⠀⡠⣫⣼⠟⠉⠀⠙⣿⣿⠀⠀⡰⠁⠙⡆⠀⠀⠀⡞⠁⠱
⠀⠀⡏⠀⠀⢠⠎⠀⢸⠀⠱⡀⡇⠀⠀⣜⢱⠟⠁⠀⠀⠀⠀⢈⠇⠀⠀⡇⠀⠀⣷⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠱⡀⢠⠃⠀⠀⢸⠀⠀⣧⠃⠀⢰⣿⣯⡀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠱⡀⣰⠇⠀⠀⠀⢧⡀⡰
⠀⠀⠀⠑⠾⣀⠀⠀⣜⣠⠔⠁⠀⠀⠸⡈⠻⣿⣦⡤⠔⠊⠀⠀⠀⠀⠀⠀⠘⠣⣀⣀⣀⣀⡨⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⣤⡔⠢⠀⠀⠀⠒⠴⡤⢀⠀⠀⠄⠀⠀⠄⠀⠀⡄⠂⠐⠀⠀⠀⠀⢀⡲⢶⡀⠀⠠⡀⠀⢠
⠀⠰⠀⣉⠀⢾⠆⠀⠁⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡅⢠⡼⠋⠀⠀⠃⠀⠀⡁⠀⡇
⠀⠀⢠⣟⣣⡴⠀⠀⠐⣤⣠⣤⠈⠀⠀⣤⣀⣠⠄⠀⠀⢢⣤⣤⡄⠀⠈⢻⣦⣤⡀⠀⠀⠐⣥⣤⣼

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠠⡄⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢧
⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⠟⠛⠁⠀⠐⠀⣄
⠀⠀⠀⠀⠐⠈⠀⠀⠉⠀⠀⠀⠀⠃⠀⠈⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⣴⡦⠀⢪⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⡆⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠺⢿⣷⠀⠀⠈⠪⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⠋⢻⣿⡟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⢀⢀⠀⣀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢄⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⢉⠀⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠠⠀⠀⠠⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠠⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠤⠤⠤⠶⠤⠤⠀⠤⠆⠀⠀⡄⣀⣀⣀⣀⣀⡀⢀⣠⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡀⡀⢀⡀⣀⠀⠀⠀⠀⣤⣄⣄⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣰⣃⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⣸⠃⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡤⢤⢠⡄⢠⡄⠀⣀⣀⠀⠀⠈⡀⢠⡤⢤⡠⢤⠤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡗⠚⠀⢷⡾⠀⠀⠿⠿⠀⢸⡏⢹⢸⡗⢿⡀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⡼⠥⠤⠤⠤⠤⠤⠤⠤⠤⢤⡼⠥⠤⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠲⠶⠤⠒⠢⠆⠀⠀⠀⢆⠤⡤⡄⢐⡢⡤⠤⢤⠤⡤⠤⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢭⡙⠲⣾⠀⠀⢸⡀⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡘⠓⠶⣧⣦⠙⠆⢦⠈⢷⣌⠳⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣁⠀⠀⠀⠀⠛⠂⠈⠳⢤⣈⡉⠀⠀⠙⠷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⢀⡴⠋⠁⠠⠒⠉⠉⠉⠁⠀⣀⡤⠖⢒⣠⡤⠤⠄⠀⠀⠉⢙⡛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠙⠛⠓⠂⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⡰⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠲⣄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⣸⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⡇
⠀⠀⠀⠀⠀⣿⡄⠀⢸⠁⠀⠀⠀⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⢀⡼⣡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⡀⣀⢸⡇
⠀⠀⠀⠀⠀⣿⠃⠀⢸⡆⠀⠀⠀⢀⣤⣤⣤⣤⣤⠴⠖⠒⢾⡆⠉⠛⠋⠛⠁⠀⠀⣄⣀⣠⡴⢻⣿⣷⣦⣄⣀⠀⢹⡏⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠠⠟⠀⠀⠠⠤⠤⠔⡂⠔⠈⠁⠀⠀⡀⢠⡀⠀⠀⠈⣹⠟⠓⠾⠿⣿⣧⣄⣉⣷⠘⡇⢰⠇
⠀⣠⡴⠶⠶⣿⣤⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⢠⠊⠀⠲⢤⣤⣤⣤⡄⠀⠀⠀⢸⣇⡾
⡾⠁⣀⣀⣀⠀⠈⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⠀⠠⠋⠀⠀⠀⠸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⠛⠳⢦
⠇⠚⠉⠉⠙⢷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡶⠶⣤
⠀⡀⠀⠀⠘⠻⣿⢀⠀⠀⢀⠄⠀⠀⠀⠀⣰⠟⠁⠸⣆⠘⠿⢷⡄⠀⠀⠀⠀⠀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠂⠀⠀⢀
⠀⠹⣆⢀⠀⠀⢿⢸⡀⢠⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀⠚⠋⠉⠉⠘⣿⠀⠀⠀⠀⠀⢸⠀⢀⡼⠀⠀⣀⠔⠁
⠀⠀⠈⠻⣆⠀⠼⢸⡇⣾⠀⠀⠀⠐⠁⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⡇⠀⠋⠀⠀⠉
⠀⠀⠀⠀⢹⡄⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡴⠟⠛⠹⣄⣴⠉⠳⣄⡙⠒⠤⠤⡀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣾⠃⢀⡤
⠀⠀⠀⠀⠀⠙⠶⠶⢻⡇⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⡴⠿⣄⠀⠸⡟⠒⠤⣤⣤⣀⠀⠀⢸⠇⠀⠀⠀⣾⡟⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡞⠁⠀⠀⡴⠛⠉⠀⠀⠀⠀⠉⠀⠃⠀⠞⠙⢿⣿⡆⢀⡟⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡿⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢴⣻⠖⡼⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠈⠁⢀⡼⠁⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠞⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣤⠶⠛⠉
⣿⣿⣿⣿⣷⡾⢿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣷⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⡿⡿⡿⠿⠿⣿⣷⠿⠿⣿⠿⢿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣣⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠁⠀⠀⠀⠀⠀⢠⣿⠉⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠀⠀⠨⣽⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⣣⣀⠀⠀⠀⠀⢻⣤⡤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣯⣙⠻⣶⣶⡶⣿⣇⣀⣼⢶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣟⡷⣿⣿⡇⢸⡏⢹⡇⢸⡇⣿⡻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡇⢸⡇⢸⡇⢸⡇⣿⣿⣎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⢸⡇⢸⡇⣸⣧⡟⢿⣿⡷⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⡇⢸⡇⣿⡿⡇⠀⢹⠄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢸⡇⢸⡇⢼⣥⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣾⣷⣿⣿⠀⠀⢨⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠿⠿⠀⠀⠿⢿⣿⡿⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢿⡗⠀⠀⠀⠀⠀⠀⠶⣶⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⠀⠀⡀⠰⣿⠆⠉⠀⠀⠀⠀⢸⣿⣀
⠀⠀⢠⠴⣿⣿⢦⠀⠀⠀⢀⠄⡾⠿⠟⠦⠀⠀⠀⠀⡴⢿⡿⢦⡀⠀⠀⠀⠙⢺⣿⡗⠚⠀⠀⠀⠀⣿⠿⠿⠆
⠀⠀⠀⢰⡏⣹⠆⠀⠀⠀⡟⠀⢾⡄⣾⠂⠀⠀⠀⠀⠰⣆⢰⡆⠀⠀⠀⠀⢰⡖⠀⣶⠄⠀⠀⠀⠀⠀⣶⢸⠇
⠀⠀⠀⡀⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⡦⠀⠀⠀⠀⢀⣤⡀⡀⣀⣤⡾⠁⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⢆⠀⡁⠀⠀⣀⣀
⠀⠀⢀⡴⣿⡷⢦⠀⠀⠀⣨⣿⣷⣅⠉⠀⠀⠀⠀⠀⣠⣽⣥⡀⠀⠀⠀⠀⠀⣻⣟⠐⡄⠁⠀⠀⢀⣿⣿⡇
⠀⠀⢀⣴⢦⠰⣦⠀⠀⠀⢡⡌⢡⡌⠁⠀⠀⠀⠀⠈⣡⡉⣩⡉⠀⠀⠀⠀⠞⠻⠿⠳⠄⠀⠀⠀⠀⣭⠉⣅
⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠈⠃⠘⠁⠀⠀⠀⠀⠀⠀⠘⠃⠛⠁⠀⠀⠀⠀⠘⡧⢸⠇⠀⠀⠀⠀⠀⠛⠀⠛
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠁⠉⠉⠈⠁⠀⠀⠀⠀⠁⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⢘⣿⣇⠀⠀⠀⠀⠀⠀⢈⣽⣷⡄⠀⠀⠀⠀⢀⣽⣧⡀⠀⠀⠀⠀⢀⣽⣯⡀
⠀⠀⠀⢘⡍⢹⡁⠀⠀⠀⠐⢋⡛⢋⠓⠀⠀⠀⠀⠀⢀⡌⢩⣄⠀⠀⠀⠈⢩⡙⢩⡙⠀⠀⠀⠀⢋⡉⢋⡉
⠀⠀⠀⠘⠇⠛⠀⠀⠀⠀⠀⠹⠇⠾⠁⠀⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠘⠇⠺⠁⠀⠀⠀⠀⠈⠗⠸⠃
⠀⠀⠀⠐⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⠐⠐⠒⠀⠀⠀⠀⠐⠀⠀⠂⠂⠀⠀⠀⠂⠒⠐⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠐⠀⠀⠀⠀⠁
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⡶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠁⠀⠀⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⡷⢦⣤⣤⣤⣤⣀⣴⠏
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⣠⣾⡇⠀⠈⠁⠈⡇⠀⣐⣯⠀⠘⣧⡀⠀⠀⠀⠀⠀⠈⠙⠷⣄⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠛⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠙⣶⠖⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠙⠶⣤⠀⠀⢀⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣷⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢰⡟⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⣀⢀⣀⠀⠈⠛⠿⣦⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠁⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣶⣶⣶⣦⣄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠿⠿⠿⢟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠣⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣀⣠⣿⣿⣿⣿⠀⠀⣠⣤⣴⣤⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣠⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⢿⣿⠛⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠛⠷⢶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠛⠿⠿⠿⠿⠿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⠀⠀⠀⠀⠀⠀⣸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠳



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⣛⠦⢋⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⡴⠟⠃⠀⠀⠀⢰⣿⡿⠿⣛⣫⣅⢭⣥
⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣥⣾⣿⣿⢏⣵⢨⡻
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣤⣤⣄⠙⠿⠿⣣⣿⡏⠞⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣴⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣤
⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠙⠛⠛⠋⢁⣤⣾⣿⣿⣿⡿⢟⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣫⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁⣀⣤⣄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢰⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⠉⠉⢀⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣶⣄⡀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡿⠛⠁⠈⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣝⠻⠿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠉⠙⠻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠀⠀⠘⢿⣿⣿⣿⣷⡄⠀⠀⠀⡰⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠀⢀⡴⠁⠀⠀⠀⣴⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠴⠋⠀⠀⠀⠀⠸⣿⣿⣿⣿⡋⠀⠀⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡠⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠒⠲⠶⠴⠶⠚⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡄⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠘⢿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⠿⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠸⡿⠃⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿
⠀⠀⠀⠀⢠⣄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣄⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠸
⠀⠀⠀⠀⠘⠋⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⢠⣾⣦⣀⠀⠉⠁⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⢠⡄⠈⠁⠀⢸⣿⠉⣿⣿⠀⠀⠀⠀⠺⣿⣤⣄⠀⠀⠀⠀⠀⠀⠰⢿⠿⠁
⣦⡄⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣸⣶⠋⠉⠀⠀⠀⠀⢠⣾⡿⠋
⣿⣿⠆⠀⠀⠀⠹⣿⣿⠀⠀⠰⣿⠋⠀⠀⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠
⢿⣿⣄⠀⠀⠀⠀⠉⠁⠀⠀⢠⡇⠀⠀⠀⠀⢀⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠙⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⣤⠀⠀⠀⠀⠀⠀⠀⣴⣷
⠀⠀⠀⢴⣾⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠸⣿⠃⠀⢻⣧⡄⠀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠈⠛⠀⠀⠀⢀⣴⣿⣿⡆⠀⠀⠀⡀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⠛⠁⠀⠀⠀⠿⠷⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢠⡆⠀⠀⢰⣿⡆
⠀⠀⠀⠀⢠⣦⠁⠀⣼⠇⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢈⡇⠀⠀⠘⠛⢃
⠀⠀⠀⠀⠀⠉⣷⣶⣶⣦⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⢾⡇⠀⣼⡆⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⣿⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠈⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⣶⠀⢠⣤⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠆⠀⠀⠀⢿⣦⣿⣿⣄⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠙⣧⣤⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠙⠃⠀⠀⠀⠀⠀⠀⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⣸⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢈⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣴⣿⠁⠀⢠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣴⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡀⠛⢀⣀⠛⠿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⢱⣶⣿⣿⡦⠀⠈⠁
⠀⠀⠀⠀⠀⠈⢿⣄⣴⠟⠁⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠷⠆⠀⠀⢠⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠤⠰⣿⠇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢻⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣶⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⢠⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⣤⡀⠀⠀⠀⢀⣄⣹⣦⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠛⠛⠀⠀⠀⠈⢻⣿⠏⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⠀⠀⠀⠞⠉⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠿⠃⠀⠀⠀⠚⠀⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠰⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠋⠀⠀⠘⠇⠀⠀⠀⠀⠈⠻⣧⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⡿⣷⣦⠀⠀⠀⠀⠀⠀⣿⣇⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⣴⡟⠁⠀⠠⡄⠀⠀⠸⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⣴⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⣤⠀⠀⠀⠀⠀⢠⣤⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣄⣀⣽⣷⠀⠀⠀⠀⠘⣿⡁⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⡿⠀⠀⡄⠀⠀⠀⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠁⠀⠀⠁⠀⠀⠀⣨⣯⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⡆⠀⠀⠀⠀⠹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⣿⣿⠀⠙⢣⡄⠀⢠⣶⡇⠀⠀⣶⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⠃⠀⠀⠘⠃⠀⠘⢿⣧⠀⠀⢸⣦⠀⠀⣤⠀⠀⣤⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣤⠀⠀⠀⢰⣶⡀⠀⠀⠉⠀⠀⢸⡟⠀⠀⣿⠁⠀⠈⢠⠀⠀⣴⣷⣦⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⣸⡿⠆⢀⣶⠀⠀⠿⠀⠀⠀⢿⠃⠀⠀⠘⢀⡀⠘⠻⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡀⠀⠀⠀⠀⠀⠸⣿⠀⣼⣿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⡀⣼⣿⡀⢸⡿⠀⠀⠀⠟
⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣿⡿⠀⢀⠀⠀⠀⣄⡘⠿⠋⠉⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠁⣀⠀⠀⢴⡄
⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⡀⠀⠿⠇⠀⠀⠹⣷⣀⠀⠻⣿⡿⠛⢀⣤⡄⠈⠀⠀⠀⠙⠟⠀⠀⠿⠃⠀⠘⠇
⠀⠀⠈⢃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⡀⠀⠀⠀⠈⠿⠀⢸⣿⡇⢀⣜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣶⠆
⠀⠀⠀⠘⠿⣧⣀⠀⠀⠀⠸⣷⠀⠀⢘⣿⠄⠾⠟⠓⣶⡦⠀⣀⣀⠈⠉⠀⠸⠏⠀⢸⣿⠀⠀⠀⠟⠀⠀⣰⠿⠋
⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⢠⣿⡆⠀⢸⣿⠀⢸⣷⣶⡿⠁⠀⢿⡏⠀⡄⠀⠀⠀⠀⢸⡏⠀⢠⡄⠀⢠⡞⠉
⠀⠀⠀⠀⠠⠶⠿⣿⣧⠀⢸⣿⡇⠀⠘⣿⡷⠀⢹⣿⠃⠀⠀⠈⠀⢸⣿⠀⠀⣿⣦⣾⡇⠀⠀⠱⠆⠈
⠀⢰⣦⣄⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⣸⣿⡁⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⡆⠀⢻⣿⡿⠛⣶⡦
⠀⠀⠙⠻⠧⠀⠀⢴⣶⡄⠙⢻⣷⠀⠸⣿⣷⠀⠀⢈⠁⠀⠀⠀⠀⠀⢿⣿⠀⠀⣿⡇⠀⣸
⠀⠀⠀⠀⠀⠸⢿⣀⣠⡇⠀⠀⣿⣦⡀⠘⠛⠀⠀⢸⣿⡆⠀⠀⠀⣶⣿⠀⠀⠀⢿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⣤⣤⠛⢿⣿⠀⠀⠉⠙⢻⡀⠀⠀⠀⠘⢿⣿⡄⠀⢰⣿⠟⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⣧⣶⡄⠀⢺⡇⠀⠘⠃⠀⣼⠏⠀⠀⠈⠋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⣿⡆⠀⠀⣾⣿⡇⠀⠀⠀⣼⣦⠀⠀⠀⣠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠘⢻⣷⡞⠉⣀⠀⠀⢻⣿⡇⠀⠀⢰⣟⠉⠠⣄⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢉⡀⣼⣿⠆⠀⠈⠻⠷⠀⠀⠀⠿⢿⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣅⣤⡀⠀⣴⡆⠀⣶⣦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠉⠁⠈⠛⠛⠿⠿⠆⠘⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣦⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠛⠿⠿⠿⠿⠟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⣀⣤⣄⣀⣀⣀⣄⣀⣀⣤⣄⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡦⣠⢀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣴⣿⣟⣿⡿⣿⣿⣿⣿⡣⣫⣾⣿⠿⣃⡄
⠀⠀⠀⡰⠋⣩⣿⣿⢿⣿⣿⣽⣿⣿⣿⣾⠟⠉⣴⣾⣟⣯⡷⢟⣋⣥⡄
⠀⠀⠀⠁⣼⣽⢋⡶⢦⡈⣿⡿⣿⡗⡾⣹⣝⡶⣤⣔⣒⣒⣛⣛⣩⣥⣶⠆
⠀⢀⣤⣼⣽⣿⣎⠓⠛⢁⢿⣿⣫⡿⣿⢷⢯⣯⣿⣿⡿⣿⣿⣿⣿⢲⣿⣆
⠀⠘⠿⣷⣿⣷⣭⢿⣞⣿⣿⣿⣞⣯⣷⣋⣯⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣄⠀⠀⢀⣀⣤⣤⣤⣀
⠀⠰⣿⣟⢿⣻⣿⢿⣯⣿⣻⣿⣽⣿⣟⣘⣫⣝⣿⣿⣯⣿⣿⣿⣿⣿⣿⢿⡷⣼⠇⠈⠩⣟⡳⢮⣝⡻⠗
⠀⠀⣷⡻⣽⢿⢾⣽⣕⣿⣝⣾⣿⣾⣮⣯⠐⣿⣘⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢐⣾⠷⣄⠹⣦⡙⠿⠇
⠀⠀⠯⣽⣽⣸⢯⢻⡝⣛⣽⣿⣞⣽⢷⣿⣟⣘⣿⣊⣿⣼⣿⣿⣿⣿⣿⣿⡃⣷⡳⡝⢶⣘⢷⣜⣿⣿⠆
⠀⠀⢸⣾⡷⣹⡿⣯⣷⣝⣿⣿⣻⣾⣿⣷⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⣿⣧⠻⣤⣙⠿⣿⣿⡿⠆
⠀⠀⠀⢷⡿⡿⣮⣷⣽⣾⣿⣻⣿⣿⡻⣿⡿⢾⡭⣛⣿⣿⣿⣿⣿⣻⣾⣾⠁⢸⣟⣷⣌⠻⢿⣮⡛⠃
⠀⠀⠀⠘⣝⣷⢾⢿⣽⣿⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣻⡿⣿⣿⡷⠃⠀⠀⢻⣮⣻⣿⡮⠙⠁
⠀⠀⠀⠀⠈⢿⡗⣿⣿⣻⣿⣿⣿⢷⣿⡿⣛⡿⢿⣵⣿⣿⡝⣿⣿⠿⠁⠀⠀⠀⠀⠙⠻⠃
⠀⠀⠀⠀⠀⠈⠙⢿⣿⢿⣿⣿⣯⣿⣿⣿⣻⣿⡿⣧⢶⡟⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣿⣾⣿⣿⣿⣿⢷⣿⣷⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠠⠄⢀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠃⡖⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢱⢀⠞⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠱⠒⠉⢱⡨⠀⠀⠀⠀⣠⡀⠀⠘⡆⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠰⠄⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠠⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⢀⠀⠂⠀⠤⢀⣀⡄⢄⣸⠛⣯⡲⢂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠔⡁⠀⠉⠱⠇⠀⢀⣀⠈⠀⠀⠈⠓⠈⠁⠁⡀⠈⣀⠀⠈⡀⠠⠠⠀⡄
⠀⠀⠀⠀⠀⢸⠀⢠⡰⠀⠀⠀⠂⠄⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠙⢠⡄⠁⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠱⡌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠔⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠂⢀⠒⠁⠀⠀⠀⠀⢠⠀⠀⢀⠀⢤⢀⢲⠀⠠⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠰⢸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⣀⡀⢙⢚⢀⣀⢘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢙⢛⡟⠛⠛⠿⣛⠛⠻⢿⣉⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣶⣶⣟⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⢒⣶⣶⣦⣤⣤⣤⣤⣴⡶⠒⠚⠛⠛⢻⣿⣿⠛⠋⣿⣿⣿⣿⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⡾⠁⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠈⠛⠋⠀⠀⢻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣤⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⢻⣿⣿⣀⠀⠀⣰⣶⣶⣤⣾⠟⠉⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⣿⣿⣿⣿⡉⠁⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⡀⣠⣿⣿⣁⠀⢈⣹⠶⣤⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠸⣿⡇⢹⣿⠁⠙⠾⣾⡅⠀⠀⠀⠉⠉⠉⠁⣸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠟⠀⢸⣿⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⢻⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠹⢿⣄⡀⠀⠀⠀⠀⣾⡇⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⡰⠋⠁⠀⠀⠉⢀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣯⣶⡖⡁⠀⠀⠀⠀⠀⣈⣷⡶⣫⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⢰⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠓⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣷⣦⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠇⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣷⡘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡾⣶⣶⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⠀⠉⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠈⠿⠿⠙⠿⠿⠁⣿⣿⣿⣿⣿⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠻⡌⠟⠁⠉⢻⠛⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢿⣿⣿⣿⡿⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡇⠀⠀⠀⣿⢠⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣤⣀⣠⣾⣷⣤⣄⠛⢿⣿⡇⠀⠀⢾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣤⣼⣿⣿⣿⡟⠀⠉⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⡀⠹⣆⠀⠈⠁⠀⠀⠈⠻⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⢸⣧⠀⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⡟⠉⠙⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠀⠈⢧⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⡀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠋⣹⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⣿⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣶⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣯⠐⠀⠀⠀⣾⣷⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢀⠇⠀⠀⠀⠀⠀⠀⢠⢖⠟⠉⠲⡄⠀⠀⠀⠀⠀⠀⠸⡎⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⡀⠀⠀⠀⠀⠀⢀⣾⡃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡏⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⢡⠀⠀⠀⠀⠀⠀⢽⡇⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣼⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣤⡀⠀⠀⠀⠀⠈⣿⣤⣶⢼⠁⠀⠀⠀⠀⢀⣤⣟⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⣿⣍⡻⣶⣤⣠⣤⣾⣇⠂⠀⣸⣷⣤⣄⣤⣶⢿⣫⣿⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⡙⠻⣦⣿⢹⣿⠈⣿⡿⢿⣿⠁⣿⡏⣿⣷⠟⢋⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣌⣻⣿⣿⠀⣿⡇⢸⣿⠀⣿⠇⣟⣡⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⠀⣿⡇⢸⣿⠀⣿⢠⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡄⣿⡇⢸⣿⢠⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⢸⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⣿⡇⢸⣿⢸⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣧⣼⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠟⠀⠀⠻⠿⠿⢿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠖⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣻⣿⡾⣁⡀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠟⠁⢻⣿⣿⡄
⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⣿⠶⠰⢾⣿⣿⣯
⠸⢿⣿⣷⣆⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⢳⣤⣤⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠚⠻⣿⣿⣿⣇⣠⣾⠀⣀⡤⠀⠀⣀⠉⢉⣉⣿⠿⣿⣿⡿⠿⠆⠀⠀⠀⠀⡀⠀⡄⣶⠀⣸⣦⠀⣼⣧⣴⣷⣶
⠀⠀⠀⠽⠟⢹⣿⣿⣿⣾⣿⣵⡖⢀⣿⣷⡛⠋⣽⣿⣿⣿⣿⣶⡶⣦⡄⠀⣸⣿⣰⣿⣿⣷⣿⣿⣿⡿⢿⣿⠛⠉
⠀⠀⠀⠀⠀⠼⠃⢸⣿⢻⣿⢿⣧⣿⣏⡛⣷⡤⣾⣿⣿⣯⠁⢸⣷⣿⣿⣾⣿⢿⣿⣿⣿⡿⣿⡇⠙⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠷⠾⣷⠾⣷⣿⣿⣿⣿⣿⣼⣿⢛⣽⣷⣿⠛⢿⡆⣸⡇⠀⠙⢷⠙⠃⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡇⣸⣏⣟⣿⣿⣿⣿⣯⣤⣶⣿⣆⢀⣉⠋
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣾⣿⣿⠻⣏⡉⢧⣴⣏⠀⠿⣏⣻⠿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠛⠃⠡⣄⠀⢻⣾⣯⣁⡀⠒⠛⣿⣟⣽⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢸⣿⠿⠋⠀⠀⡰⣄⠙⢷⣼⢿⣄⠈⠁⠀⠾⢛⣿⣿⣿⣿⠿⣶⠶⠄⠀⠀⠈⠉
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣝⣿⣿⣧⠀⠀⠀⠀⠀⠠⠾⢛⣿⣿⣋⣀⡟⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣼⣿⠿⢿⣿⣧⠀⠀⠀⠀⠀⠐⠚⠛⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠲⣶⣿⣿⡿⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠠⣤⣿⣿⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢐⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠿⣿⣿
⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣄⠀⣶⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⢠⣿⠀⣀⣀⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣿⡈⠿⠟⠋⠉⠙⠻⣦⡀⠀⢻⣆⠀⠀⣴⣶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⣷⡀⠀⢿⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢸⡇⠀⠘⣷⠀⠸⠛⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣠⣤⣄⣀⣿⠀⠀⠀⠀⣸⡇⠀⠀⣿⠀⢀⣤⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⡏⠀⠈⠉⠁⠀⠀⢀⣴⠟⠀⠀⠀⣿⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠈⢻⡄⠀⣷⠶⠶⠖⠋⠁⠀⠀⠀⢀⣿⠀⠰⠶⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⣸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠉⠻⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠁⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣦⠀⠀⢰⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣼⣿⣿⡟⠋⠉⠻⣿⣿⣿⣿⣿⡇⠀⠙⠛⠛⠛⠛⡋⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⣿⣿⣿⣿⡿⠟⠉⠉⢻⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⢿⣇⠀⠀⠀⠀⠀⠀⢿⠄⢰⡟⠀⠀⠀⠀⠀⢀⣠⡟⠉⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣯⠀⠀⣀⣀⣀⡀⠀⠀⠀⠉⣿⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣿⠉⠀⠀⠀⣀⣀⣀⣀⠀⠀⣽⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠷⠟⣦⡀⠙⣄⠀⠀⠀⠀⠀⠀⣀⡞⢀⣾⠻⠾⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⠟⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠈⣷⠀⠹⡄⢴⢤⡧⢶⠤⢿⠇⢸⡏⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠙⣿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣇⠀⠀⠐⠀⠆⠸⠂⠋⢀⣾⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣏⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉

⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣤⣀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠒⠦⡄⠎⠐⠀⠃⠀⠀⠈⢳
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠘⠆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠌
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡰⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠐⢌⡉⠙⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢀⠇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠠⠊
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠂⠀⠀⠀⠀⠉
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⢤⣤⡀⠀⠀⣠⡴⠋⠀⠀⠀⣀⣀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⣷⣴⠋⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠛⠳⢦⣀⡞⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠟⠛⠓⠀⠀⠀⠀⠀⠀⠙⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣼⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡇⡀⢀⠀⠀⠀⠀⠉⠀⠈⠉⠳⣄
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠤⠐⠈⠛⠀⠀⡄⠀⠘⠃⠂⠄⣀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢹⡄
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡸⡅⠀⠀⠀⠀⠀⠀⠠⠉⢦⣀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢹⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠙⢦⣀⠀⠀⣀⠔⠁⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣄⡀
⠀⠀⣠⡶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠂⢄⡀⠤⠐⠊⠙⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢠⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠿⣦⣄⡀⠀⠀⠈⠁⠀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠀⠀⣾
⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠙⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣦⣤⣤⣤⠶⠟⠁
⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠸⣦⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡀⠈⠉⢉⣀⣤⡶⠛⠁⠈⠻⣷⣤⣤⣤⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂

⠀⠀⠀⢀⣀⣀⣀⣀⣐⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢿⣍⠫⢍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠭⠍⣭⣍⣉⠉⠉⢉⣉⠉⣿
⠀⠀⠀⣿⠀⠁⢀⠑⠦⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠒⠀⠀⠀⠀⢀⣴⣿⣣⠥⡘⠷⠀⠺⡳⣄⠀⠀⠀⣿
⠀⠀⠀⣿⡄⢀⡀⠙⠢⣈⡒⣄⠀⠠⠀⠀⠄⠀⠀⠀⠀⡄⠀⣠⣾⣿⠟⢡⠊⡐⢰⠀⢃⠱⡌⢷⡀⠀⣿
⠀⠀⠀⢼⠄⠨⢿⡷⠂⠈⠢⣀⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡗⡰⠃⢠⠁⢸⠀⠸⠀⠱⡀⠃⠀⣿
⠀⠀⠀⢼⠀⣤⠈⠉⠂⠐⠂⠈⠉⠀⠀⠀⠐⡄⠀⠀⢀⢢⣿⣿⡟⣰⠁⠀⡆⠀⢸⠤⡄⡄⡀⣃⠈⣠⣿
⠀⠀⠀⣸⠀⠀⠒⠛⠳⠶⠶⠀⠀⠀⠀⠀⠀⢻⠢⡄⠀⣾⣿⣿⠖⠯⡁⢠⠀⠒⡚⠀⠀⣇⡡⠽⠊⡇⣿
⠀⠀⠀⢸⠀⠀⢀⣠⣦⣄⣀⣭⠷⠒⠀⠀⠀⢸⠐⠈⢱⣿⣿⣿⠀⠀⠀⣙⠓⠒⡗⠒⠚⡇⠀⠀⠀⡇⣿
⠀⠀⠀⢻⠀⠤⠘⠛⠛⠻⠁⣸⠀⠀⠀⠀⢀⠎⠑⢄⣾⣿⣿⣿⠀⠤⢀⡇⠀⢠⠀⠀⠀⡇⠠⢰⢀⡗⣿
⠀⠀⠀⢾⠀⠀⣴⣶⣿⣿⣯⡇⠀⠀⠀⡠⠊⠒⠂⢈⣿⣿⣿⣷⠀⠀⢸⠃⠀⢘⠀⠀⠀⠇⠀⢸⢸⡇⣿
⠀⠀⠀⢺⠀⠀⣭⣭⠉⣻⠀⡗⠀⠀⠆⠀⢠⠀⠄⢹⣿⣿⣿⡹⠀⠀⡼⠀⠀⡈⠀⠀⠀⠂⠀⢸⢸⠀⣿
⠀⠀⠀⣹⠀⠀⠐⠛⠠⠶⠀⢣⠀⠀⠀⠀⣻⠒⠀⢹⣿⣿⣿⡇⡄⢀⠃⠀⠀⠇⠀⠂⢰⠀⠀⣀⡜⠀⣿
⠀⠀⠀⣽⠀⠓⠤⣶⣶⣶⠆⠈⠃⢀⠆⣀⠈⠈⠁⢸⣿⣿⣿⡆⠇⢸⠀⠀⢸⠀⠀⠀⠸⠀⢀⣿⠷⡄⣿
⠀⠀⠀⢸⠀⠤⠤⠤⢬⠿⣶⠀⠀⡀⡆⠀⠐⣦⣤⣼⣿⣿⣿⠇⠸⡾⠀⠁⡄⠀⠀⠀⡆⠀⠸⡿⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⢉⡀⠀⠁⡇⢀⣾⣁⠈⣿⣿⣿⣿⠶⠲⡇⠀⠀⠃⠀⠀⢠⠁⠀⣽⡇⠈⠀⣿
⠀⠀⠀⢿⠋⠹⠿⣿⣧⡤⠀⠀⠀⠀⡇⡟⠳⡋⡃⣿⣿⣿⣿⢠⢸⡷⠀⢸⠀⠀⠀⡌⠀⣴⣿⠢⡀⠀⣿
⠀⠀⠀⢸⠀⠘⣛⣻⡛⠁⣤⣤⠀⠚⠁⠘⠒⢋⡄⢻⣿⣿⣿⢸⢸⠀⠣⡌⠀⠀⢠⠁⢠⣿⠇⠀⠈⠓⣿
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠉⠹⠿⠿⢷⠀⣠⢎⡔⢹⣿⣿⣿⣼⢸⢀⣤⡓⢄⢀⡇⢀⣿⡏⠀⠀⠀⠀⣿
⠀⠀⠀⣽⠀⠛⠋⠉⠉⠉⠉⠉⠛⢻⠀⢷⡥⠊⠀⢘⣿⣿⣿⣿⡟⠀⢠⢠⠄⡽⠁⣿⡿⠀⠀⠈⠁⠀⣿
⠀⠀⠀⢹⠀⢀⣀⣈⣉⠹⠀⣀⣀⣀⡀⠞⠁⡠⠒⠉⣿⣿⣿⣿⡇⠀⡘⠀⢰⣁⡀⠛⠁⠀⣠⠀⠀⠀⣿
⠀⠀⠀⣸⠀⣀⣀⣘⣻⣿⣿⣿⡿⠈⢀⡒⠒⢲⣀⣀⢘⣿⣿⣿⡇⠀⠇⢠⠏⠈⡠⠀⢛⣰⠃⠀⠀⠀⣿
⠀⠀⠀⣸⠀⠈⠉⠉⠉⢰⣶⣌⢀⠀⠀⠈⠐⠂⠭⣏⠈⠹⣿⣿⡇⢰⠀⡞⣠⡶⠛⠒⠞⠃⠀⢀⡄⠀⣿
⠀⠀⠀⢼⠀⠤⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠺⢿⣧⢿⣼⠿⠋⠁⠒⠂⠀⠤⢀⡀⠀⠀⣿
⠀⠀⠀⣸⠆⠀⡖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢹⢘⠃⠤⣀⠀⠀⠀⠀⠀⢀⠀⠉⠁⣿
⠀⠀⠀⢺⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠈⠑⠢⠄⡀⠀⠈⠀⠀⣿⠀⡄
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣀⣺⡟⠛⠿⠇⠉⠸⢿⣿⠿⠃⠺⣤⣤⡶⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠀⠀⢠⣶⣶⣶⣤⣤⡘⠫⡑⠀⢰⣆⠀⠁⢀⣄⠀⠘⡏⢖⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣥⠀⠀⠀⣀⣠⣀⠈⠙⢻⣿⣿⠟⠁⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠗⠉
⠀⠀⠀⠀⠀⠀⡠⠐⠀⠘⠛⠀⠀⣾⣿⣿⣿⡇⠀⠀⠈⠁⣀⣀⠀⠞⣙⡂⠀⠀⠀⠀⢀⡤⡀⠄
⠀⠀⠀⠀⠀⣠⠔⠒⡀⠀⠀⠀⢸⣿⣿⡿⠟⠁⠀⠀⠀⣾⣿⣿⡄⠀⠙⠃⠀⠀⠀⠀⠾⠃
⠀⠀⢀⣤⠊⠀⠀⠀⢱⠀⠀⠀⠘⠿⠋⠀⠀⣀⣀⢀⣰⣿⣿⣿⣧⠀⠐⠢⡤⣄⣀⡀⠠⠆
⠀⠀⠊⢉⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠀⠁⡒⠊⠀⢀⡠⠔⠁
⠀⠀⠡⠌⠀⠀⠀⠀⢴⣿⣷⠆⢄⡀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠙⠛⠉⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠇⠀⡀⢠⠀⡔⢲⠖⠀⠀⠀⠀⢿⣿⠀⢰⠄⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠐⠑⠂⠘⠐⠂⠈⠀⠀⠀⠀⠸⠿⠃⢨⠠⠀⢙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⢠⠀⠀⠀⠀⠠⠀⠒⢄⠀⠀⠀⠀⢆⠀⢄⠀⠯⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠤⠔⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣀⡴
⠀⢰⣟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⣖⠁⠀⠁
⠀⠸⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠉
⠀⠀⣈⡅⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣈⣤⡔
⠀⠈⠻⢿⣼⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢀⣼⣿⡿⠏
⠀⠀⠀⢻⣿⣆⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢠⣀⠀⣆⢀⣾⡿⣿⠏
⠀⠀⠀⠀⠙⣿⣧⣻⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠰⣿⣷⣼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣇⡰⠀⡇⠀⠀⠠⡇⠀⠀⣒⠀⢲⣾⠟⣶⣍⡷⠟⠁⢀⡞⡂
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣿⠀⡠⢾⡇⠀⠀⠋⠑⠈⣿⠀⡸⠁⠀⢀⣴⠏⠈⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠻⣿⣿⣶⣗⣼⡇⠰⢠⡏⠻⡐⡈⢰⠇⠀⢠⡿⠉⡀⠀⠀⡠⠛⠁⠀⠀⠀⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠛⣿⣿⣿⡀⢾⡇⡐⢦⠑⠘⠀⠀⣾⠂⠁⠸⣶⣤⣤⣤⢀⣠⣠⠾⠛⠁
⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⡙⢿⣿⣾⡡⠟⢸⠀⠀⠀⡸⠁⠀⠀⠀⠈⠙⠛⠋⠉
⠀⢀⠀⠘⢦⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠨⣻⢿⣶⣿⠀⠀⢀⡇
⠀⠸⡆⠈⠙⠛⠛⠛⠿⡿⠛⠓⠁⠦⠴⣄⠀⠀⠒⢝⣿⡆⠀⣸⠁
⠀⠀⡰⠀⠀⢠⣤⣧⣤⣤⡀⠀⠀⢹⣶⡶⢦⣄⡀⠈⠈⢧⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣤⣠⣤⣶⣠⣤⡤⡄
⠀⢸⣃⠀⠀⢀⡾⠿⢿⣿⣿⣿⣯⣿⠟⠁⡠⢾⣿⣦⠤⠛⠛⡁⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣟⠛⠛⠛
⠀⠀⠉⢁⡾⡟⠁⠐⠚⠋⠉⠀⠀⠀⠀⢀⠀⠚⠟⠁⠀⠀⠀⠈⠀⠀⠀⢠⣾⣿⣿⡟⠿⠯⠍⠉⠠
⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠫⠛⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢠⡖⠀⠀⣼⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢀⠌⠀⣸⠇⠀⣸⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠊⠀⣀⣤⣿⡠⠖⠛⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⢡⠟⠟⢂⠀⠀⠀⠙⠅⢀⣀⣀⡈⠳⠀⠀⠀⣀⣠⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠂⠀⠀⠀⢲⠀⠀⠙⡁⠀⣠⣾⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⡆⠀⠀⠀⢸⡇⠀⠀⠇⠀⣿⣷⣠⣼⣿⣿⣿⣆⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⡇⠀⠀⢰⡄⠹⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠚⠁⠀⠀⠸⡇⠀⠈⠙⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠸⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛
⣀⣤⡤⢤⣄⡀⠀⢻⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⠧⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻
⣿⠀⠀⠀⠀⠉⢧⣨⣶⣇⠀⡴⠋⠉⣹⠂⢀⡤⠞⠋⠹⣆⣤⣤⠀⣤⣤⡄⡾⠛⠛⣷⣼⡇⠻⠶⡄⢸⠟⢿⣀⣤⣤⣤⠤⢤⡀⠀⢀⣀⡴⠚⢻⡇⢸⠀⢸
⣿⠀⢠⣶⠀⠀⢸⢿⠀⢸⠾⣁⣀⡉⠙⢦⣼⠀⣴⡄⠀⣿⡇⠘⠶⠃⢨⡟⣙⣲⠀⠳⡄⡇⢰⣚⣛⢻⠀⠘⡏⡇⠈⣡⡀⠀⢹⠖⡟⠁⠴⠄⠀⡇⢸⠦⣼
⣿⠀⠀⠀⠀⢀⠎⢸⠀⢸⡄⡨⠤⠽⠀⢸⠛⢤⣠⡀⠀⡏⢷⣀⠀⠀⡸⠀⣇⣀⣀⡴⠃⠹⣄⣈⣹⠟⢤⡤⠇⠷⠤⠇⠹⣤⡼⠀⠙⠤⡆⠀⠀⣷⢾⡖⣧
⢹⠦⠤⠔⠚⢁⣤⠈⠉⢉⠐⠧⣄⣀⡤⢊⡤⣄⣀⡇⠀⢳⣀⠉⠛⠋⢀⣤⣀⣀⣀⣠⢦⣀⠀⢉⡠⣤⣀⣀⢶⣀⣀⡖⢦⣠⠴⠶⢲⠀⣧⠀⠀⢹⣀⡽⠃
⠀⠙⠓⠚⠋⠁⠈⠉⠉⠉⠙⠒⠒⠒⠊⠁⠀⠈⠉⣷⠀⢸⡇⠙⠒⠒⠉⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⢾⣣⠇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠒⠃⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⠤⢤⡤⢞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣌⣙⡒⢒⣒⡽⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠉⠛⢶⣄⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⣼⣧⣄⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠠⠐⠂⠁⠉⠀⠀⠀⠀⠉⠈⠐⢂⡎⠂⠁⠀⠀⠀⠀⠀⠛⠀⠀⢸
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠌⠜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢄
⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⢸⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⢸⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠻⢷⣄⣀⠀⠀⠈⠀⢀⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⠇
⠀⠀⠀⠀⠉⡙⠁⠀⠀⠀⠈⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡠⠋
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣶⠤⠾⠛⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠰⢦⢠⣤⡤⢴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣄⠀⠀⠀⠀⠀⠊⢁⣠⡾⠋⢿⣦⣀⡀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠲⠶⠞⠛⠁⠀⠀⠀⠈⠙⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠀⢈⠀⡆⠀⠀⠐⠀⠘⠀⠀⢠⠤⡄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠚⠀⡤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠃⠈⡄⠀⠀⠂⠀⠀⠀⠀⠀⢁⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠼⠁⠲⣤⡦⢄⣤⠀⠀⡖⠸⠿⠃⠀⠀⠀⠀⠀⠀⢸⡟⠛⢷⡄
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⢀⡤⡒⢮⡹⡶⢒⣿⣤⡀⡀⡠⡦⡄⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⢷
⠀⠀⠀⠀⠀⠠⠃⠀⣀⡠⠀⢀⠀⠤⠀⠡⠴⠏⠈⣰⡾⠻⡿⠸⢻⡾⣘⣥⠚⠏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⡀⠃⠀⠤⠤⠾⠡⠾⡐⢃⠠⡼⠷⠀⠵⠦⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣿⣟⣻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠋⠁⠀⣠⣴⣦⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠺⣿⣿⣿⣿⣦⣤⣤⣶⣿⣿⡇⠀⠀⠀⠉⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠻⢿⠿⠻⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠿⠿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⣀⠀⠀⠰⠿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⢀⣶⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⢀⣴⣿⡟⠉⠉⢻⣿⣦⡀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣦⣤⣤⣤⣿⣿⣿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣄⡀⠀⠉⠛⠛⠛⠛⠉⠀⢀⣠⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⣿⣷⣦⣄⡀⢀⣠⣴⣾⣿⣿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡿⢿⣿⣷⣯⣝⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣤⣴⣾⣿⡿⠗⠈⠙⠻⢿⣿⣷⣦⣤⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠠⠴⠤⠆⠤⠤⠀⠤⠆⠀⠀⡤⣀⣀⣀⣀⣀⣀⢀⣠⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡀⡀⠀⡀⣀⠀⠀⠀⠀⣠⣀⢀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣰⣃⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⣸⠋⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡤⢤⢠⡄⢠⡄⠀⣀⣀⠀⠀⠈⡀⢠⡤⢤⡠⣤⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡗⠚⠈⢷⡾⠀⠀⠯⠿⠀⢸⡏⣻⢸⡗⢾⡁⣾⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠈⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⡼⠥⠤⠤⠤⠤⠤⠤⠤⠤⢤⡼⠥⠤⠤⠤⠤⠤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⠠⠶⠤⠄⠀⠀⠀⢔⣀⡄⡄⢐⡂⡤⢤⢠⢀⣠⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠁⠀⠀⣀⠀⠀⠸⠖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠎⠉⣳⣀⡀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠉⠉⢳⣤⣧⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠙⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠲⣴⠋⢀⠈⠓⠶⠤⣤⠀⠀⠀⣀⡸⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠉⣠⠴⠒⠀⢿⡀⠀⠀⠀⠀⠀⢀⣀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠻⣄⣀⠀⠀⠈⠙⢦⣀⣠⢴⡞⠉⠉⠉⢳⣀⡀⢀⡼⠁
⠀⠀⠀⠀⢀⣀⡠⠤⠖⠺⠭⠓⠚⢉⣔⣊⣩⡭⣿⠀⠀⠀⠋⠀⠀⠘⣇⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠘⣦⡀⠀⠀⣠⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⣴⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂
⠀⠀⠀⠀⠈⠋⠀⠉⠀⣀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢈⣤⣤⣍⣂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⠒⠳⣌
⠀⠀⠀⠀⠀⠀⢢⣶⣿⣿⣖⣹⡿⠀⠀⠀⠀⠀⠀⢾⢧⣾⡿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠐⡠⢌⠀⣰⠖⡟⡶
⠀⠀⠀⠀⠀⠀⢘⣿⣋⢨⣨⡼⠁⠀⠀⠀⠀⠀⠀⢳⡋⠉⠙⠛⢻⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠤⠘⡠⠁
⠀⠀⠀⠀⠀⠀⠘⠈⠛⠉⠛⠅⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⠒⠲⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢶⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠷⠠⠴⢶⡄
⠀⠀⠀⠀⠀⠀⡘⢠⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⣀⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠘⢾⡭⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣙⣿⡿⢇
⠀⠀⠀⠀⠀⠀⡸⣾⣷⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠉
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠙⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠑⢤⣄
⠀⠀⠀⠀⠀⠘⠉⠀⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠰⣀⠆⣠⢤⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⡀⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⢀⣤⣤⣄⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢠⡏⠀⠀⠈⣷⠀⢸⣧⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢸⠿⡃⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡗
⠘⢷⣀⣀⣠⠟⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡿⠇⠀⠀⠀⠀⢸⣿⡜⣼⠩⡃⣿⡇
⠀⠀⠉⣿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡾⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⡗⠀⠀⠀⠀⠀⠀⢶⠆
⠀⢠⢤⣿⠤⡄⠀⢸⣿⣿⣿⣿⣿⣿⠀⢀⣀⡀⠀⢰⣾⠇⠀⣂⣀⡀⢸⣷⣶⣶⣾⣇⣸⡗
⠀⢸⣼⣿⣤⡧⠀⢸⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⡄⢨⣶⡇⣰⣿⣿⣷⣘⣛⣛⣛⣻⡧⢺⡇⠀⠀⡀
⠀⢸⣿⣿⣿⡧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠘⡜
⠀⢸⣩⣷⣭⣧⠀⢸⣿⣿⣿⣿⣿⣿⡏⢉⣉⠉⠉⠉⠉⠉⣉⡉⠉⠉⢹⣧⣴⣶⣼⣧⣸⣇
⠀⢸⢽⡶⡿⣇⠀⢸⣿⣿⣿⣿⣿⣿⡇⠈⠉⠀⠀⠛⠃⠀⠛⠃⠀⠿⢸⠿⠿⠿⢿⣏⢻⡏
⠀⢸⢿⣼⠟⣇⠀⢸⣿⣿⣿⣿⣿⣿⡧⠶⠀⠛⢃⣄⠀⠿⠂⠘⠿⠀⢸⣿⣿⣿⣿⣯⣾⡏
⠀⠈⢹⣿⡏⠁⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣤⡀⠈⠉⣤⡀⠾⠆⠰⢶⢸⣯⣭⣭⣽⠖⣽⡧
⠀⠀⢸⣿⡧⣤⡤⣼⡿⠿⠿⠿⠿⢿⣷⡄⠉⠁⣶⠄⠉⣡⡄⠰⠖⣤⣼⣿⣿⣿⣿⡟⣿⡗⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠘⠛⠓⠛⠓⢿⣷⣶⣶⣶⣶⣾⡿⢣⡄⣴⡄⠸⠿⠈⠡⣶⢰⡎⢸⣶⣶⣶⣾⣧⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣷⣶⣿⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠀⣀⣬⣭⣍⣉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⢉⣩⣭⣭⣅⡀
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⠀⠀⠀⠀⣠⠀⠀⠀⠀⡇⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡶⠤⠤⠼⠀⠀⠀⠀⡿⠀⠀⠀⠀⠓⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠸⣿⣿⣿⣿⣿⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣹⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢀⠀⢈⠉⡉⢉⠀⡀⠀⠀⠀⢰⣶⣶⣶⣾⣿⣷⣶⣶⣶⡆⠀⠀⠀⢠⠈⡉⢉⣉⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣤⣤⣤⣤⣶⣶⣿⣶⣿⡷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣦⣾⡟⢻⣟⣿⣿⣿⣿⣿⣿⡉⠹⣿⣿⣟⠉⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⢿⣿⣧⠈⢻⡿⠛⠛⠛⠿⠿⠇⠀⢈⣿⣿⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠟⠋⠻⣿⠟⠻⣦⢻⣿⠀⠀⢀⣾⣦⠀⢠⡀⠀⠀⢸⣿⣿⣿⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣶⣤⣦⡀⠀⠈⣿⣶⣿⡇⠀⢸⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⡆⠀⣠⠀⠀⠀⠀⠀⠀⠉⠿⡇⠘⣟⣿⣿⣿⣿⣿⡄⠀⢿⣟⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠛⠁⠉⠀⠀⠀⠀⠀⠀⣄⠀⠇⠀⣿⣿⣿⣿⣟⠃⠇⠀⢹⢿⣿⠃⠀⠀⠙⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡃⠺⠀⠐⠋⠛⠋⠀⠀⢰⣿⣵⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣟⠻⣿⣠⡤⠀⠙⢿⣿⣿⣿⣿
⣭⡿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠛⠻⠀⠀⠼⠿⠏⠘⢿
⣿⣿⣿⣶⣭⡓⠶⣤⣀⠀⠀⢄⢠⠥⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠿⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣿⣿⣿⣿⣿⣿⣶⣌⠙⢿⣧⣄⡄⠀⠀⠀⠤⠖⠁
⢿⣿⣿⣿⣿⡿⠛⠁⠀⣼⣿⣿⣏⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⠟⠀⠀⢀⣰⣿⣿⣟⣭⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣧⣽⠖⣥⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣾⣧⣲⣦⣤⣄⡀
⣿⣿⣿⣿⡗⠀⠀⢸⣿⣿⣯⣷⢟⡹⡇⡩⢬⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠈⠀⠀⠈⠁⠔
⣿⣿⣿⣿⠂⠀⠀⠀⢙⣿⣯⣾⣊⣵⣆⢓⣫⢴⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠶⠂⠀⠀⠀⠀⢀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡞⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠇⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⣨⠋⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠂⢸⡟⠉⠀⠀⢰⠷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣾⣇⢀⣀⣤⣾⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠻⢿⣧⣬⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⣿⣿⣀⠀⡇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠸⠇⠈⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢤⣄⠀⠀⢀⣀⡐⢢⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠟⢻⡷⡐⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠀⠀⢧⢿⣄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⢄⡀⠀⠉⠠⠀⠀⠀⠓⡛⢋⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠑⢄⠄⠀⢀⠀⠀⠀⠒⡽⡿⣷⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣻⠆⠀⠀⠀⣀⣳⣤⣄⣄⠀⣀⣀⡈⣥⠆⣷⣶⣻⣫⣀⣄⣀⣠⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣷⣧⣴⠊⢻⣿⣿⣿⣿⠟⠿⠛⠋⠉⠃⢈⣤⣽⣷⠿⠿⠛⠿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣛⣿⣿⠟⠛⠋⠋⠀⢸⣶⡿⠋⠉⠀⠀⠐⠂⠈⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠓⠸⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁