Exoskeleton Emojis & Symbols

⠀⠀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠠⣄⡀
⠀⠈⠠⡬⣤⠈⠀⠀⠀⠀⠄⠢⢀⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⡁⠀⠀⠀⢠⠀⠀⢀
⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠐⢂⠌⠁⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠙⡦⢴⠛
⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⠨⠀⠀⠀⠀⠀⠀⠃⠑⠃⠙⠀⠀⠀⠀⠠⢐⢄⠀⠀⠀⠀⠀⠈⠉⠀⠙⡇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠈⢲⠉⠁⠀⠀⠀⢠⠐⢢⠀⠜⠁
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠃⠀⠂⠘⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⡀
⠀⠀⠈⠰⠄⠀⠀⠀⠀⣠⠦⠀⠀⠆⡀⠀⠀⠀⢰⢀⡀⡄⠀⠀⠀⠀⠲⣤⠐⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠈⡐⠀⠀⠀⠀⠀⠿⠀⡔⢢⠀⠆⠀⠀⠀⠘⣂⣐⠃⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠈⠡⡉
⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠐⠂⠘⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠛⠑⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⣠⣀⢄⡠⢀⣀⠀⠀⠠⠤
⠀⢐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠠⠀⢸⡇⠀⠄⠀⠀⢐⢨⠀⢔⢀
⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌⠠⠀⠁⠀⠀⠀⠀⢀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⣉⠁⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠋
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠒⠀⠐⠂⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠐⠀⠂⠒⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⣹⣷⣤⣄⣀⠀⢸⡟⠛⠛⠛⠛⣿⠀⢀⣀⣤⡴⣿⡁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⠀⠀⠈⠉⠛⠿⠷⢶⡶⣶⠶⠿⠟⠋⠉⠀⠀⢸⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢀⣀⣰⣶⣶⣾⡇⣿⣶⣶⣶⣀⣀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣀⣀⢸⣯⣭⣿⠏⢹⡇⣿⠉⢿⣯⣭⣿⢀⣀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⠀⠀⢀⣀⣀⣸⡇⣿⣀⣀⣀⠀⠀⢸⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⣿⣿⣤⣤⣼⣯⣭⣽⡇⣿⣭⣭⣿⣤⣤⣼⣿⡟⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠃⢸⡇⣿⠀⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⢀⡾⠛⢻⡟⠛⠛⠛⠛⣿⠛⠻⣆⠘⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⠶⠞⠁⠀⠀⠀⠀⠙⠷⢾⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣺⡅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠰⠶⢶⡆⠀⠀⣶⠶⠶⠘⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠙⠛⠛⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠫⡏⠀⠀⠀⠠⣏⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠄⠀⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠀⠀⠀⢀⡠⢴⠃⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢙⠀⠈⢀⠀⠀⢰⠆⠹⣗⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢨⠀⠀⡀⠀⠀⢸⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣀⠀⠀⡠⠀⠀⠀⣀⡀⠀⡜⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠥⡐⢖⡂⠀⠀⠀⠁⠀⠀⡅⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠗⠹⠟⡛⠛⢳⣦⣦⡶⠘⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆⡈⢹⡃⠀⠀⠀⠀⠉⠀⢀⠏⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠰⣖⠀⣤⠀⠀⠉⠁⠒⠂⠀⠒⠀⠀⠈⠀⠁⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⣥⡔⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠃⣲⢴⠇⠐⠀⠀⠀⠀⠀⠀⡀⠀⠲⠀⠀⠀⠀⢠⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⠛⠀⡄⠀⠀⠀⠐⠀⠀⠐⠀⠀⣧⡀⠀⡀⠙⢀⢞⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠐⠃⠀⠀⠀⣀⡀⠀⠀⠀⠀⠳⣄⠈⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠅⠀⠀⢰⠇⠀⠈⡁⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡟⠶⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⠀⠘⠇⠀⠠⠎⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⢀⠘⠃⢀⣤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠃⠀⠀⠀⠉⠙⠁⠀⠈⠑⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⣧⣠⠾⢧⣤⣤⡼⢷⣄⣼⠗⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⡿⣅⠀⠀⢸⡇⠀⠀⣸⢷⡄⠀⢹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⢶⡏⠀⠈⠳⠶⣛⡛⠶⠞⠁⠀⢹⡶⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⣧⠀⠀⠀⢛⣩⣙⡛⠀⠀⠀⣼⠁⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠓⢶⡾⠛⢷⡄⠀⠋⠁⠈⠙⠀⢠⡾⠛⢷⡶⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡇⠀⠘⡇⠀⠀⢠⡄⠀⠀⣼⠁⠀⢸⡇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⣴⠟⠀⠀⢠⡿⣶⡶⠶⠶⢶⣶⢿⡄⠀⠀⠻⣦⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⠟⠀⠀⠀⢀⡾⠁⠈⢻⡄⢠⡟⠁⠈⢷⡀⠀⠀⠀⠻⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢻⡟⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣿⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠹⣿⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⣿⣿⣇⣀⣀⣉⣉⣉⣉⣀⣀⣼⣿⣿⣿⣇⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢹⣿⣿⣿⣿⣿⣿⠁⣠⣶⣶⣄⠈⣿⣿⣿⣿⣿⣿⡏⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣿⣿⣿⢻⣿⣧⠀⢿⣿⣿⡿⠀⣼⣿⡟⣿⣿⣿⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣠⣿⣿⣿⠇⢸⣿⣿⣧⣄⣉⢉⣠⣼⣿⣿⡇⠸⣿⣿⣿⣄⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⠁⠈⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠁⠈⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡏⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⡏⠀⢹⣿⣿⡇⢸⣿⣿⡏⠀⢹⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠘⠿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠟⠃⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⣀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣀⣠⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢻⣿⣿⣿⣿⡟⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣦⣤⣍⠛⠛⣩⣤⣴⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢰⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⡆⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠈⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠀⠢⠠⠠⠀⠀⠆⠀⠀⠀⠄



⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣿⣿⣻⣿

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠘⣿⡷⢶⣶⣶⣶⣿⣿⠀⠀⠀⠀⣰⡟⠛⢷⣀⡄
⠀⠀⠀⠀⠛⣧⢼⣿⣿⡧⣿⠃⠀⠀⠀⠀⡟⠀⠀⠈⣿⠇
⠀⠀⠀⠀⠀⣷⠈⣿⣿⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⢠⣼⣤⣿⣿⣼⣧⡀⡀⠀⠀⢀⣧⣀⣀⣠⣇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⣻⡀⠀⠀⠘⠛⣻⣟⠛⠛⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⢧⡀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣇⣠⣤⡄⣤⣄⣀⡟⠀⠀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⢹⠉⠉⠁⠉⠉⢉⠇⠀⠀⠀⠀⠰⠾⡷⠆⠀⠀⠀⠀⣿⡿⠿⠿⠘⠿⠿⢿⡇⠀⠀⠀⢸⣿⣿⠂
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⡷⠶⠆⠲⠶⡆⠀⠀⠀⠀⠀⠀⣴⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⢹⡄⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡟⠛⠃⠀⠛⠛⢻⡇⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⡧⠂⡀⢀⠒⣧⠀⠀⠀⠀⠀⢠⢠⡇⡀⠀⠀⠀⠀⢸⣇⣄⠀⠀⢀⣄⣸⠃⠀⠀⢀⡀⣿⡟
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⣰⣤⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠈⣿⡟⢻⣿⣿⡟⣿⡿⠀⠀⠀⠀⣼⠋⠉⢻⣤⡆
⠀⠀⠀⠀⠈⡗⢺⣿⣿⠗⣿⠁⠀⠀⠀⠀⡏⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢿⠀⣿⣿⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡿
⠀⠀⠀⢠⣰⣾⣶⣿⣿⣾⣷⣄⠀⠀⠀⢠⣷⣶⣶⣶⣧⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡀⠀⠀⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⣸⠄⠀⠀⠈⠉⣽⣯⡉⠉⠀⠀⠀⣰⡟⠛⠿⠿⠿⠟⢳⡄⠀⠀⠻⠿⣿⣿⠿⠇
⠀⠀⠀⠀⢧⣴⣶⡆⣶⣶⣦⡎⠀⠀⠀⠀⠀⣙⣟⠁⠀⠀⠀⠀⣿⣇⣤⣄⢀⣤⣄⣸⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠘⠛⡟⠃⠀⠀⠀⠀⢿⡟⠛⠛⠈⠛⠛⢻⡇⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠘⡇⢀⡀⢀⡀⡿⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⡟⠛⠃⠙⠛⡇⠀⠀⠀⠀⠀⠀⠾⡿⠀⠀⠀⠀⠀⢸⣇⣤⡄⠀⣠⣀⣸⡀⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡏⠉⠁⠀⠉⠉⢹⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⠁⠄⠠⠁⡷⠀⠀⠀⠀⠀⠠⣰⡇⡄⠀⠀⠀⠀⠸⣧⣦⠀⠀⢠⣦⡼⠁⠀⠀⢠⡆⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢿⠷⠀⠀⠀⠠⠞⡿⠅⠀⠀⠀⢀⣀⡀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⢢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠉⠁⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⡄⠀⠀⡀⠐⠀⠀⠀⠉
⠀⠀⠄⣒⣺⠻⣿⣷⣦⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠋
⠀⠀⠔⠉⠻⠿⢿⡿⠿⠿⠂⠀⠸⢉⣛⣿⡷⠦⣤⣀⣀⣀⠀⠈⠲⣼⡧⠐⠀⣀⣀⣀⣀⣠⠴⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠈⠙⠉⠑⠈⢻⡛⢾⣿⣿⠚⠛⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⣆
⠀⠀⠀⠀⠀⠀⢰⡀⢀⣄⠀⠀⠀⣀⣤⣶⡖⠒⡶⢶⣶⠿⣻⣿⢿⣭⡟⠿⠿⣿⣶⠶⠖⠀⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠲⣮⣿⠈⣿⠀⠀⠀⠉⠠⣶⣆⠉⠄⠀⠀⠀⠈⠁⠀⣽⡇⠈⠆⠀⢂⠀⠂⠀⠀⠀⠀⠀⠁⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⡷⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⡔⠂⡟⠛⢿⠇⢿⠀⠸⣇⠘⠁⠟⢴⣦⣄⡀⠀⠀⠀⠾⣻⠟⠀⠀⠀⣼⢸⣷
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⣿⣽⣷⣶⣾⣧⡊⠀⣻⠀⠀⠘⣧⠂⣵⣾⣿⣿⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢼⠪⣿
⠀⠀⠀⣰⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠎⢹⣿⣧⣿⠀⠀⠀⠀⣿⠀⠀⠀⠸⣦⡸⠿⣿⢫⠍⢹⡆⠀⠀⠀⠡⠂⠀⠀⠈⠦⠟
⠀⠀⠰⠯⢥⠁⠀⠀⢠⠀⠀⠀⠀⠀⣬⢦⣉⡩⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠳⣭⣶⣂⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠊⠀⠀⠀⠄⠀⠀⠳⣄⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⣀⠀⠀⠀⠈⠉⠉⠀⣰⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠹⣄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠈⠙⠅⡒⠒⣶⠲⠂⣡⠀⠀⠀⣄⣀⣀⣀⣀⠤
⠀⠀⠀⠘⠚⠾⠿⡿⣷⠷⢼⡿⠿⠬⠷⠾⠶⠶⠶⠒⠖⠂⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠽⠯⠿⠯⠵⣿⣿⣯⢯⣿⣯⣧⣬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣤⣸⣿⣿⣿⣿⣿⣇⣠⣤⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠇⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣿⣤⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠿⠿⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡿⠿⠿⣿⣿⣿⢿⣿⣿⣿⠿⠿⣿⣿⣇⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⢀⣴⣿⡿⠈⢿⣿⣄⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢹⣿⣿⣭⣿⣿⡿⠁⠀⠈⢻⣿⣿⣯⣿⣿⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠉⠙⣿⡟⠁⠀⠀⠀⠀⠻⣿⡟⠉⠹⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢿⡦⠀⠈⣧⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⣿⣇⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣵⣾⢷⣷⣝⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢸⣇⣿⡿⣫⣬⢻⣿⡎⣿⢿⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⡶⣶⣷⣝⡛⣼⣿⡇⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⡇⠿⠿⠇⡇⠿⠿⢇⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣿⡇⡇⣿⡇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣃⣿⡇⣿⡇⡇⣿⡇⣟⣛⣛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣼⣷⣿⡇⣇⣿⢇⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣘⡅⡿⣾⣿⣗⣭⣟⣛⣀⠀⠀⠀⢀⣀⢀⣠⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣭⣭⡱⣶⢸⡿⣼⣿⣿⣿⣿⣿⣿⣷⢸⣿⢨⣭⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⡴⢶⡿⣑⣛⣛⣛⣻⣿⣭⣭⣭⣜⠛⠘⠶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣃⣛⣛⣛⣛⣛⣛⣿⣿⣿⣭⣭⣭⣭⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⣴⣥⣤⡶⠂⠀⠀⠀⠀⠀⠀⠲⢦⣤⣽⣾⡶⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠰⠜⠋⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠈⠙⢾⡅⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⢻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣠⣄⠼⠋⠀⠀⠀⠀⠀⣌⣱⣤⣔⣉⣀⠀⠀⠀⠀⠀⠉⠳⡴⢶⠴⠧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⣿⠃
⠀⠀⠀⠀⢿⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢰⠀⢱⡝⣀⡤⠖⠂⠄⠠⠉⠔⠉⡁⠈⠩⠸⠣⢞⡒⠲⠦⢤⡀⣇⡇⠀⡀⠀⠀⠀⠀⠀⠀⣠⣾⠃⣠⠇
⠀⠀⠀⠀⠀⠻⣤⣌⣻⣆⠀⠀⠀⠀⠀⠘⢶⢿⡟⠁⠀⠀⠠⠄⠀⠀⠀⢄⠀⠀⢀⠀⠀⣀⢿⠈⠀⠎⠙⣿⣷⡿⠁⠀⠀⠀⠀⢀⡼⣎⣴⠟⠁
⠳⣄⠀⠀⠀⠀⠀⠻⣍⡛⠛⣷⣄⠀⠀⠀⠀⢿⠀⠠⠀⠠⠀⠁⠀⡰⠀⠈⠀⠀⠀⠀⡀⠬⠈⠋⠀⠒⠂⢘⡟⠁⣀⣀⣀⡴⠒⠛⣉⠟⠁⠀⠀⠀⠀⣀⡴
⠀⠈⠓⢦⣤⣀⠀⠀⠀⠉⠓⠙⠃⣿⣭⣿⣮⣇⠀⠀⠀⠀⠄⡄⠀⠀⢶⠂⠲⠂⠡⠀⠁⢀⠀⠄⠀⡠⢈⡈⣽⣇⣏⣅⡽⠒⠚⠋⠁⠀⠀⣀⣤⡶⠛⠉
⠀⠀⠀⠀⠉⠛⠿⢷⣦⣄⣀⣠⠤⣀⠀⠈⠉⠉⢆⠂⠀⠈⠀⢐⡀⠴⠀⢿⣄⠀⠀⠀⠀⠀⠰⠀⢈⠠⢬⡔⠉⠉⢁⣀⣠⠤⣤⣄⣄⣼⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠶⠮⠽⢟⣶⢶⣦⣷⣄⡀⣁⢠⠀⢆⠂⡄⢀⡀⢠⣈⡵⠤⠓⢀⣢⣶⣿⢿⢽⡩⠿⠷⠓⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣔⣀⣶⣶⡐⣴⣭⣬⣭⣉⣉⣻⣦⣂⡀⠀⠙⠃⠀⠙⠀⢉⣩⣿⣿⢋⣩⣵⣾⣵⢖⣒⣷⢆⣠⣶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠉⠉⠁⢈⢀⣩⣾⠺⣿⡿⣻⣞⣿⣿⣿⣦⣖⣦⣤⣀⣾⣿⣿⢿⣿⢿⣛⣿⠛⣻⣯⠄⢀⠀⠀⠀⠙⠛⠿⠿⣦⣀
⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⣠⣶⣭⠿⠎⠉⠉⠑⠊⠓⠟⠓⠃⠈⠁⠈⠉⠉⠉⠉⠉⠉⠒⠛⠚⠓⠋⠉⠁⠈⠛⢿⢿⣦⠀⠀⠀⠀⠀⠈⠛⠷⢄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣴⡟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣤⣾⣧⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣞⣂⠰⣶⣷⣼⣯⣥⣴⣿⣿⣙⢻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡤⣹⣿⣿⠛⠀⠀⢻⣾⣿⣿⣿⣿⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣷⣿⣟⠁⠀⠀⣴⣿⡇⣿⠗⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⠁⠀⠰⣾⣾⣿⣿⣷⣿⡖⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⠤⠽⣿⣿⣿⣇⣠⣿⢄⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⠛⢻⣿⣿⣳⣦⠻⣿⣿⣛⣿⣟⣻⣷⣠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠛⣱⣿⣿⣿⣦⣽⡿⠟⢛⠹⣇⠙⠷⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢿⣿⡿⡏⠙⣮⣿⣥⡀⠙⠀⠀⠈⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣀⠀⠀⢹⣿⣿⣧⡶⠛⠆⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⣿⣇⠀⠀⠈⢿⣿⢯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣾⣿⣿⣿⡷⠂⢠⣿⡟⣉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠿⠟⠀⢰⣾⣯⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⣿⠿⠋⠀⠀⠀⣀⣹⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⠃⠀⠀⠀⠀⠿⡿⣿⣖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⣘⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠉⡟⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⠞⠉⠁⡾⠘⠲⣄⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡞⠀⠀⠀⣠⣇⠀⠀⠈⢷⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢰⡇⠀⢹⡀⣘⣀⣀⣠⡏⠁⠈⢳⠀⠀⢸⡇⣀⣀⣀⣀⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡏⢩⠁⠀⠀⢳⣤⣤⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⢸⠀⠀⠀⢸⡇⢸⠀⠀⠀⢸⡇⠤⠤⠤⠤⠤⣤⠏
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⠘⠦⡤⠤⢼⡇⢸⠦⠤⣤⠞⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠀⠘⡇⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣸⠀⠀⣿⠀⠀⡇⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⢸⡄⠀⢹⠀⠀⡇⢸⡄⠀⣿⠈⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⠞⠁⠀⢸⠀⠀⡇⢸⣇⣠⠷⢄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⣤⠴⡇⢸⡏⠀⠀⢀⡷⠤⠤⣤⣈⣃⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠖⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢐⣇⣀⣠⠗⠺⣇⠀⠀⡾⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⢾⠉⠉⣿⣀⣠⡤⠤⠶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⢹⣄⣀⡼⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣄⣀⣀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⣏⣀⠀⠀⣰⣧⠤⠤⠤⠤⠶⠶⠶⠒⠒⠚⠛⠛⢦⡤⠿⣄⣀⠈⠉⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡞⢁⣀⣉⣉⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠠⠤⡇⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣋⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⡆⣲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡖
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⢀⡀⣀⣀⢀⡀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⡇⢘⢨⠀⠃⢠⠹⡀⠇⠇⠀⠀⡷⠀⠀⠳⢄⠈⠁⠨⢜⢘⠀⢱⢰⢎⠸⠄
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣶⣶⣶⣶⣶⣷⣷⣾⣶⣾⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣷⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠊⠉⠉⠉⠙⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⡛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⢖⣛⣫⣭⣭⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⠶⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⣿⣟⠛⠛⢫⡏⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡞⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⢀⣠⣞⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠐⠉⠉⠉⠉⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⢀⣀⣾⣿⣿⠿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢉⡞⢉⡟⠁⠀⠀⢱⣾⣿⣷⠀⢀⡤⠶⡋⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡜⠀⠀⠀⠀⠀⠙⠿⠿⠚⠉⠀⠀⢧⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣦⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣴⣶⣦⣤⡽⠄
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠛⣿⣧⡀⡇⠀⣤⣶⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠉⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣄⣤⠴⠒⠋⠁⠀⠀⠀⠀⠀⣘⣿⣿⡷⠞⢛⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⣀⣤⣤⣤⣴⣿⣿⣿⣿⣷⠀⠀⢀⣀⣤⣴⡶⠿⣋⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣠⣶⣋⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠉⢛⣻⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠏⠉⠉⠉⠉⢻⣆⠀⢀⣤⣊⡽
⠀⠀⠀⠺⠿⠟⢋⣽⣿⠿⠿⠛⠉⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡋⢻⣿⣿⣿⣆⠀⠀⢀⣠⠼⠿⡴⠋⢉⡏
⠀⠀⠀⠀⠀⠴⠛⠛⠉⠀⠀⠀⢿⡿⠿⠿⡿⠟⠛⠻⠟⠋⢁⡴⠟⠛⠿⣿⣿⣿⣿⣿⡷⠖⣩⣴⣶⠶⣶⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⣾⠋⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡞⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⣦⣄⠀⡞⠀⠀⠀⠀⢠⣇⣀⣀⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⠹⡟⠀⠀⠀⠀⠀⠀⡿⠋⠉⠛⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⠶⢶⡤⠇⠀⠀⠀⠀⠀⢿⣀⣤⣤⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⠀⠀⠀⢳⠀⠀⠀⢀⣠⡤⠞⠋⠁⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⠤⠤⠴⠒⠋⠉⠈⠳⠤⢤⣀⣀⣀⣀⡤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠛⠁⠀⠀⠀⠀⠀⠈⠻⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦⣶⡖⠀⠀⠀⠀⠐⣦⣤⣴⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠿⣿⣟⣉⣀⠀⠀⠀⠀⠀⠀⢠⡀⣤⣤⣰⣤⣆⣴⣀⣰⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⠿⠃⠀⠀⠀⠀⠀⣠⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⠀⠀⢿⣿⣿⣿⣿⣶⣤⣴⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⣤⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣠⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣠⣴⣾⣿⡿⢏⣁⣀⣀⣀⣀
⠀⠤⠔⠶⠶⠿⠿⠿⠿⠿⢿⣷⣾⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠿⠿⠿⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣛⣛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣛⣛⣉⣉⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣷⣦⣄⡀
⠀⢀⡤⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣶⣶⣶⣶⡄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠲⣄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠻⣿⣿⣿⣿⣿⡿⠋⠉⠛⠹⠋⠿⠻⡏⠙⠋⠙⠋⠉⠻⣿⣿⣿⣿⣿⠟⠋⠘⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣩⣙⣉⣝⣉⣭⣉⣭⣋⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⢰⣾⣿⣫⣥⣠⡝⠉⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⠆⠈⣣⣿⣿⣿⢳⣿⢧⡀⣦⢤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠀⢰⣿⣿⠟⠁⠠⡹⡿⣿⣷⣼⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⣲⠆⢒⣿⡏⠀⠀⣴⣿⣷⣏⣿⡿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⠂⠀⣺⠿⣿⣟⣘⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⡀⢠⡿⠷⢿⣏⣟⣽⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⣿⡆⣸⣿⣟⠻⣻⣿⠟⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢮⣙⣯⠉⠙⠉⡛⠛⣲⣾⡯⠠⠀⢈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣸⣿⣇⠈⠿⢃⡞⠹⣿⠀⢀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢽⡿⠋⠁⡼⠀⠀⣯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⡴⠋⠀⠀⠀⢱⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⠐⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠺⡿⠁⠀⠀⠋⠠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢻⣀⣀⣀⣅⣄⠘⢓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⢠⡄⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠐⢣⠀⠀⠎⠆⢀⡄⠀⢸⢸⠀⡇⠃⠁⠀⡇⠀⠀⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⢸⠀⠀⠄⡄⡀⢠⠀⠈⢸⢀⠀⠇⠀⠀⠇⠀⣠⠀⠀⠀⠀⡆⠠⢢
⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠈⠘⠀⣀⠀⡇⡇⢸⠀⡀⠐⠘⢸⢸⠀⠀⠀⡞⠀⡇⢰⠀⢀⢀⠃⡀
⠀⠀⠀⠀⠀⠇⢁⠀⠀⢸⠀⠀⢰⠀⠀⠀⡇⠸⠀⠀⠃⠀⠌⠘⠀⠀⡿⠀⠀⠃⢸⠀⠘⠀⠀⠅
⠀⠀⠀⠀⠀⡆⠀⡆⢡⠀⡇⠀⠈⠀⠀⢰⠀⠀⢀⢸⠀⠀⡇⡀⠃⢀⠃⠀⢰⠀⡆⠀⡄⠀⡘⠠⠓⢀⠎⡁
⠀⠀⠀⠀⠀⠸⡀⢱⠘⡄⢱⠸⠆⠀⢇⠸⡄⢀⡈⠸⠀⠰⣮⢹⠂⢰⢰⠀⡈⢀⠇⢰⠘⢠⢃⠀⠀⡌⠐
⠀⠀⠀⠀⠀⠀⢃⠀⢇⠱⠘⡀⢀⠀⢸⠀⣷⢸⡇⢠⡔⡀⣿⢰⢁⢸⠸⠀⡇⠸⠀⡆⠆⡆⡌⡜⠰⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠂⢥⣌⠷⠚⠁⠉⠉⠉⠉⠓⢾⣇⢸⠸⢸⠀⢰⠀⠇⢰⠸⠸⡰⠰⢡⢡⡢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⠰⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣴⠀⣆⡺⡘⠀⣼⢃⢣⠃⠡⠣⠃⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠿⠲⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣻⠾⠧⠀⣿⣸⡆⡴⢁⡷
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠈⠂⠬⠍⠈⠉⠉⠩⠵⠶⠶⠦⠒⠉⠀⠀⠀⠀⠀⠀⡟⠓⢾⣡⠁
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡰⠁⠀⠀⠈⠁⠢⡄
⠀⠀⠀⠀⢀⠎⠉⢆⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⠤⢊⠥⠊⠀⠀⠀⣀⣠⠔⠀⡸
⠀⠀⠀⠀⠸⡀⠀⠀⠑⠢⠭⠭⠀⠠⠤⠤⠔⠒⠒⠈⠉⠉⠑⠒⠒⠒⠒⠉⠁⠀⢀⡤⠖⢉⡀⠤⠖⠚⠁
⠀⠀⠀⠀⠀⠉⠲⠤⠄⠀⢀⣀⣀⣀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠤⠴⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣾⣿⠿⠛⠛⢛⣛⠿⣿⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⢿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠳⡌⠿⠋⠁⠀⠈⠉⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠏⠀⠀⠀⠙⢦⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⣠⣶⣿⠿⠿⠛⢿⠏⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡄
⠀⠀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡟⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣿⣿
⢰⣿⣇⡏⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿
⢸⣿⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⢸⣿⣿
⢸⣿⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡿⠘⣿⣿⡀
⢸⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡇⠀⠸⣿⣧
⠸⣿⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠁⠀⠀⢿⣿
⠀⠹⣿⣦⣤⣀⣀⣠⣿⣆⡠⣀⣀⡤⠔⣸⣷⡀⠀⠀⠀⢀⣀⣤⣾⣿⣷⣶⣶⠤⠀⠀⢀⣀⣾⠿⠳⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠛⠿⢿⡿⠿⠿⢿⣿⣷⣶⣶⣿⣿⠟⣻⣿⣛⠛⠛⠛⠉⠁⠀⠉⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⢰⠃⠀⠈⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⣤⢤⣤⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣾⣧⣀⢠⡄⠀⠀⡼⠀⠀⠀⢀⣮⣵⣾⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣷⣤⣄⣥⣤⣶⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⣿⠿⠿⠿⡿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣻⣟⣿⣿⣿⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⡀⠀⢀⣤⠤⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡀
⠀⠀⠀⠈⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠈⢷⣤⣼⣿⣿⣿⣿⣧⣴⡟⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⡿⠿⠓⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⣿⠿⣿⠻⣷⣍⣉⡀⠀⠀⠀⠀⠀⠈⠉⢉⣿⡿⣿⣦⡀⠀⠀⠀⠀⠉⠀⢀⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢠⠔⠁⠀⠈⠙⣶⡍⠉⠙⣷⠀⠀⠀⠀⠀⠀⣼⡟⠁⠈⠛⠿⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣄⠀⢀⣶⣦⢀⣶⣦⡀⠀⢀⣤⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣍⠀⠀⠀⠹⣶⣿⡇⠈⠻⣟⣈⣻⡟⠀⠀⣿⣷⡾⠁⠀⠀⠙⠻⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⡿⠛⢻⠛⠛⠿⣿⣷⡄⠀⠀⠀⠁⠈⣶⣿⣿⣿⣿⣿⣷⡎⠀⠀⠀⠀⢀⣴⠖⢶⡀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⡀⠋⣠⣾⣿⠿⠇
⠀⠀⠀⠀⢰⠿⠿⠿⡿⠛⠲⣶⠄⠀⠿⠛⠃⠀⠀⠀⠋⠉⠉⠉⠀⠛⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠐⡀⣠⣴⣶⣿⣷⣤⡀⣠⡔⠀⠀⠀⠀⠤⢞⣄⠀⠀⠀⠀⢠⣤⣠⣿⣿⣿⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠶⠂⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣸⢶⡝⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠟⠁⠙⠧⠀⠀⢒⣯⠅⠀⠀⠀⠀⠀⠀⠈⠻⢛⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠩⠷⣟⡄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⡏⢀⡄⠀⠀⠀⠀⠀⠀⠈⠱⢿⠧⠀⠀⣴⣿⣿⣷⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⣧⣾⠀⠀⠀⠀⠀⠀⠀⣬⡿⡟⠃⠀⠀⠙⠿⢿⣿⣶⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣸⣿⠄⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⢿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⡠⠤⡀
⠀⠀⠀⠀⠀⣈⣁⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⡘⣆⢠⠎⣠⢄⢡
⠀⠀⠀⠀⠈⠀⠀⠈⠁⠒⠌⡑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣜⣀⡮⢼⠞⠁⡴⠒⠚⢆⢳⠒⠀⠤⠤⠤⠤⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢽⣲⣤⡴⠒⢒⡖⠉⠁⠀⠀⠀⡠⠛⠒⡠⠖⠁⢀⡎⠀⠀⠀⠈⢆⢣⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⢷⠁⢧⠀⠀⢀⣀⠔⣠⠔⠋⠀⠀⢀⠎⠉⠉⠁⠒⠒⠚⢦⢣⣀⣀⣀⠠⣤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⡀⠣⢴⢿⢆⡘⡍⣾⡇⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⠣⡤⠤⠔⠚
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠑⡶⢊⠾⣈⠧⢶⠛⠉⠉⠉⠁⠉⠙⠓⠒⠒⠒⠂⠈⠛⠟⠛⠓⢔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣔⠕⠁⠀⠀⠀⠈⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠶⠒⠛⠁⠀⠀⠀⠀⠀⠀⠙⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣿⣆⣸⣿⣤⣾⣇⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⣿⣷⣿⡟⣿⣿⣿⣿⣿⢿⣷⣴⡿⠁⠀⠀⠀⢀⠀⢠⡀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣹⣿⣿⣿⠟⢣⢻⣉⣩⣽⣧⣿⣿⣿⣿⣾⡟⠰⣦⣼⣷⣾⣷⣿⣶⣿⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣛⠿⣅⣤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣉⠉⢻⡟⠁⠈⢻⣿⣧⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣾⣿⡿⠟⠛⠉⠹⡍⠉⠉⠛⠻⣿⣿⣿⣯⣷⣾⣿⣦⣤⣾⠿⣿⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣰⣄⣶⣲⢽⣿⣫⣿⣿⠟⠉⢦⣀⣠⠤⢶⣷⠶⠦⣤⣞⠀⠈⠛⢿⣿⣾⣿⣿⡿⠀⠀⠈⣿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠃⠀⣠⠞⠻⣀⠀⠠⡿⠄⠠⣯⠟⠓⣤⢀⣨⢿⣿⣿⣻⡾⢿⡟⢻⣜⠛⠂
⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠲⣴⣇⢀⡀⠋⠀⣀⣅⣄⠴⠉⠢⡀⠘⢻⡀⠀⢻⣿⣿⣷⣾⢷⡞⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⡇⠀⢰⡏⠈⠛⠀⢠⣿⣿⡿⣇⣾⣦⠀⠀⠀⠈⣧⠀⠀⣿⣿⣿⣿⣿⣙⣿⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⢼⣗⣲⡆⠀⣿⣿⣿⣅⣿⣿⣿⠆⠀⠓⠂⣿⠉⠉⣿⣿⣿⡿⡜⡟⠛⠒⡾⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⡇⠀⣠⡀⠘⢿⣿⣾⣿⣿⠟⠀⣰⢦⣀⡟⠀⢀⠿⣿⡋⡠⢟⣙⣤⣘⡳⣇⠑⢄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⡔⠋⠻⢭⣻⣿⣿⣿⡄⠀⣹⣾⠟⠓⢀⡀⠈⠉⠉⠀⣠⡄⠀⢉⡟⠓⠦⣼⣿⠒⠽⣱⡟⠉⠉⠉⢻⣎⢶⡩⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣧⡀⢀⣸⣿⣿⣽⣿⣿⣎⠁⠈⠳⣤⣼⠋⠀⠘⠇⠀⠈⣭⡶⠋⠀⠀⣜⢫⣏⣀⡇⢿⡀⠀⠿⠀⢠⡿⢘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⠛⠉⢻⣿⣿⣿⣿⣿⣿⢿⣿⣷⣄⠀⡰⠉⠛⠒⠒⡗⠚⠉⠁⠘⢠⣬⡷⠿⢿⣭⣵⠛⢌⠻⣦⣤⣴⠟⡡⠯⣝⡳⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣄⣀⣰⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣦⣤⣄⣀⣧⣤⣤⣶⠔⣶⣿⣇⣀⣸⣧⣈⣣⣴⢫⠖⠒⢶⢻⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣹⣿⣿⠏⠀⠉⣿⣿⣾⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⢛⣿⡿⢧⢿⣿⠟⠻⣎⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣼⠿⣷⣤⣴⣿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⢸⣿⣿⣿⣷⡒⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣛⣿⣾⠂⠉⠉⠙⣿⣿⣿⠉⢹⣿⣿⡇⠈⠉⠒⢦⣄⡀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⣿⣄⠀⢀⣰⣿⣿⣿⣷⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣷⣦⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢿⣿⣿⣿⡟⠋⠉⠻⣿⣿⠏⠁⠈⠙⣿⣿⣿⣿⣌⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣿⣿⡀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣿⣿⣿⠉⠙⠁⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⡿⠿⣿⣦⣴⣾⣿⣿⣿⣷⣾⣿⣿⡉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡿⠉⣿⡿⠻⣿⡟⢻⣿⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠻⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⣀⡄⠀⠄⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⢀⣀
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠘⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂⢄⠀⠀⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠒⠤⠤⠠⠀⠂⠀⠀⠎⠀⢄⠆⠀⠀⡁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠋⠀⠀⠀⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠩⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠒⠠⠄⣀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠐⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠐⡄
⠀⠀⠀⠈⠁⣀⠀⠈⠁⠒⠀⠀⠀⠈⠢⣀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⢁⠀⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡎
⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠒⠀⠀⠈⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠌⠢⠄⣀⡀⠀⠀⠀⠠⠄⠐⠈⠁

⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢀⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠉⡀⠊⣠⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⣠⡗⠀⢀⠆⠀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢐⣤⠀⢠⣆⣀⣬⡤⢐⣊⠭⣶⣚⠋⠀⠀⠀⠀⠀⠴⣾⡟⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣈⠛⠖⠄⣼⠟⠁⠛⠋⠉⠉⠉⠉⠛⠙⠃⢀⣀⣀⣄⣀⡌⣆⠀⠀⠀⠠⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⡰⠊⢐⠁⠀⠀⡠⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢹⡷⠤⠤⠼⠀⠠⠄⠀⣈⡁⠢⡀
⠀⠀⠀⠀⠀⠀⡻⠂⠠⠞⠛⠒⣶⣇⣀⣠⣤⣶⣶⡦⠀⠀⠀⠀⠰⣿⣬⢕⣂⠀⠀⠀⠀⠀⠈⠑⢌
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⢁⣨⣿⣿⣄⣼⠟⠀⠀⠀⠀⠀⠀⠨⣷⣤⡉⠓⠀⠀⠠⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣤⡭⠤⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⠳⠑⠂
⠀⠀⠀⡄⠀⠘⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡀
⠀⠀⠀⢛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⢀⣰⣴⣶⡈⠀⠀⠀⠀⣠⠃⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠙⠿⣷⢤⣤⣤⣤⣤⡖⠁⠠⠐⢡⡾⣿⣿⣷⠀⠀⠀⠠⠃⠘⠃⢦⡄⠀⠰⡄⠀⠀⡘⠀⠈
⠀⠀⠀⠀⠀⠀⠰⠄⠀⢀⡀⠈⡇⠀⠀⠀⡋⠀⠘⣿⣿⡇⠀⠀⢀⠊⠀⠀⠈⠳⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢰⣷⠀⢰⠲⠇⢠⠂⣼⢿⠇⠀⢠⠏⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⡆⠾⠤⠌⠀⢸⠟⠀⢀⡏⠀⠀⡀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⣴⡶⠀⠀⢠⡂⡀⠁⢀⠀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠐⢋⡴⠯⡇⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⢢⡀⠀⠀⠀⠀⣠⠴⢦⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠿⣦⣠⣠⡿⢿⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⣿⡏⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠙⠀⣀⣠⡷⠀⠀⠈⠣⢶⣒⠀⠁⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠈⠢⣀⠀⠄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠀⠀⣀⡀⠐⣠⠄⣀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠃⢤⡇⠔⠊
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢨⠐
⠀⠀⢀⡞⠉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⡀⣼⣅⡝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠑⠬⠉⠐⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠖⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠖⠚⠉⢹⠇⠀⠀⠀⠀⠀⠈⠓⢦⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⢠⣿⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠉⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⡰⠛⠉⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⢷⡄⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠹⡄
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠅⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠨⣅⣀⣀⣈⣿⣿⣦⠀⠀⠀⣀⣴⣶⣿⣿⣿⣦⣀⠀⠀⠀⠐⠚⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⠿⠛⠁⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⠋⠉⠙⠛⠛⠛⠛⠋⠁⠈⠛⠻⠷⠶⠶⠶⠾⠿⠿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⣀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⡼⠁⠀⠀⠀⠀⠙⠛⠂⠀⠀⠀⠀⡆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠦⠤⠴⠶⠶⠦⠄⠒⠒⠒⠶⠶⠒⠊⠁



⡿⢿⠿⡿⠿⢿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣟⣻⣿⣟⣿⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣶⠍⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠁⢪⡢⣀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⠔⢢⠀⠀⠐⢤⡀⠀⠀⠀⠀⠉⢎⢂⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢠⣾⡀⠠⡧⠀⠀⠀⢧⡄⠀⠀⠀⠀⠀⠓⡡⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡟⡛⠀⠀⣃⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠒⡥⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠉⠁⠀⠠⠇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⡁⠀⢀⠜⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠱⠚⠁⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠒⠒⠂⠐⠊⠉⠉⠀⠜⡉⠠
⠀⠀⠀⠀⢣⡀⠀⢀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠤⠂⢂⠩⠂⠁
⠀⠀⠀⠀⠠⡉⠻⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠁⠠⢍⠂⠠⠀⣀⡀⡀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠄⠀⡀⠀⡀⠀⠀⠀⢀⠀⣀⡀⠠⠄⠀⠀⠒⠂⠈⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣾⢁⣤⣶⣿⣯⣶⠟⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⣐⣚⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣓⣀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣞⠛⢿⣿⣷⣾⣿⠛⠉⠉⠳⠙⣿⢿⡿⣿⠋⠞⠉⠉⠛⣿⣷⣾⣿⡿⠛⣱⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠉⠛⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠛⠛⣿⡿⠛⠛⠉⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣶⣾⣿⣀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⣿⣷⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⣿⣷⠟⠀⠀⠀⠀⠀⢄⠀⠀⡠⠀⠀⠀⠀⠀⠲⣄⠈⢿⣿⣿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⣿⣿⣧⡀⢿⣿⣶⣦⣄⠀⠀⠈⢦⣦⣴⡴⠁⠀⠀⣠⣴⣶⣿⡷⢀⣼⣿⣟⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣭⠀⠐⣿⣿⠁⢘⣳⣄⢀⡀⢸⡇⢀⡀⣠⣞⡇⠈⣿⣿⠂⠠⣭⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⡿⡋⠀⠀⠀⠙⠻⢷⣤⣥⡾⠟⠛⠟⠻⠛⠻⢷⣤⣤⡾⠟⠋⠀⠀⠀⠙⢿⣿⣿⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣵⠇⠀⠀⠀⠀⣀⡀⠙⢷⣿⣷⣦⣴⣾⣿⡾⠋⢀⣀⠀⠀⠀⠀⠸⣦⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⣞⠏⡹⠳⣄⠉⠙⠿⠿⠋⠉⣠⠞⣏⠹⣳⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⢸⣿⣿⣿⣿⣿⣧⡅⢤⠘⢷⣇⢠⡍⣙⢲⢶⡶⡞⣋⢩⡀⣹⡾⠃⣤⢠⣼⣿⣿⣿⣿⣿⡏⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⠏⣿⣿⣿⣿⡇⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠸⣿⣿⣿⣿⠹⣿⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠰⣿⡟⣿⣿⢰⡆⠀⢸⡇⠹⣿⠿⡿⡿⢿⢿⠿⣿⠏⢸⡇⠀⢰⡆⣿⣿⢻⣿⠇⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢻⣿⣿⡇⡄⠈⠷⣿⡴⠦⠞⠓⠚⠳⠴⢦⣿⠾⠃⢀⢸⣿⣿⡟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿⣿⣷⣷⣄⠀⢀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢿⣏⢻⣷⣿⣀⣆⣿⡄⢀⣿⣸⢠⣴⣾⡟⣹⡿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢿⣿⣿⣿⣷⣼⣿⣿⣿⡿⠋⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣿⣿⣿⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠉

⠀⠀⠀⠀⠀⠀⠀⡠⣤⠶⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠢⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢴⡒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠲⡤⢤
⠀⠀⠀⠀⠀⡉⠙⠀⢸⠀⢀⠄⢤⠀⠀⠀⠀⢠⠞⠣⣐⣄⠀⠁⠢⣄⠀⠀⠀⡶⠛⠃⠉⠉⠙⢻⡷⠴⠄⠀⠀⠀⠀⣀⡤⠞⢦⠀⣄⣡⡔⠛⣆
⠀⠀⠀⢠⠃⣹⢄⡁⣺⣟⣁⣠⠀⠣⡀⠀⠠⣫⣄⠀⠘⢿⡷⠎⢀⡼⠀⠀⠀⡝⠒⠚⠓⠒⠓⢚⣥⡀⡼⠀⠀⢠⡏⣑⣤⡀⠸⡆⠓⢤⣤⠀⢸⡄
⠀⠀⠀⠈⣆⠀⠉⠓⠋⢋⠝⠋⠛⠄⢀⠀⠀⠀⣿⠀⠀⠈⠁⠀⡸⠁⠀⠀⠀⢳⠖⠚⠓⠲⠒⢼⡿⠙⠰⠀⠀⠀⣷⡀⠙⢯⠞⢛⡷⠟⠛⠓⠀⣷
⠀⠀⠀⠀⢻⠀⠀⠈⠃⠀⠀⠀⠀⠀⠊⠀⠀⢠⡇⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡄⠠⠵⣆⠀⡀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠘⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢸⡃⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠈⠉⢷⡴⠀⠀⠀⠀⠀⢳⠀⠀⠀⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠋⡅⠀⠀⠀⠀⠀⠈⢷⠐⠚⠃⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣠⠀⣠⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⣠⣻⠂⠀⢈⠁⠰⠦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠺⠐⠩⠙⠀⠈⢦⣤⣅⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⡀⠀⠀⣀⡀⠀⠉⡿⠡⣤⡒⡆⠀⢀⠀⠄⠐
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠩⡐⠐⠲⣋⡦⠴⡤⡒⠐⠀⠙⣦⠀⠀⠠⠐⠈
⠀⠀⠐⢋⣠⣀⠀⠁⠀⠀⢐⣶⠦⠄⠠⡄⠀⠠⠀⡀⡿⠀⠀⠈⠉⡔⠸⡁⠦⠔⠆⢱⡖⠀⠄⠀⠐
⠀⠀⠀⠉⠩⠙⠛⢡⡾⠋⣡⠦⢄⠀⠀⠈⠁⠀⠀⠈⢷⠄⠀⠠⠦⣤⠓⠉⣀⡙⠉⠉⠂⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢀⣬⣾⣝⡛⠛⠦⠄⠀⠀⡀⠀⠀⠀⢈⣀⠏⠙⠋⠃⠀⠀⠀⣀⠀⡀⣌⡉⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣼⣿⠋⢻⣿⣿⣶⣤⡨⢒⡠⠑⠶⡶⠦⠴⠮⣿⡾⢍⡿⠋⢷⢿⣦⡀⠉
⠀⠀⠀⠀⢠⡾⢈⠁⡤⣋⡽⣿⣿⣿⣿⣿⣟⡿⣤⣥⢈⡵⢀⠀⠈⠳⣦⡀⠨⠑⢭⣟⠛⠂⡀
⠀⠀⠀⠀⣸⠀⡈⢼⡁⠯⠀⣼⣟⠛⢿⣿⣿⣿⣿⣿⣿⡙⢧⡑⢄⠀⠈⠻⣷⣦⠀⢿⠳⢄⠈
⠀⠀⠀⢀⣿⣧⠈⣆⢱⡀⠘⣿⢻⡄⠀⠹⣿⡏⠃⣿⡻⠿⣶⠁⢄⠀⠢⠄⠀⢻⣷⣀⢀⠀⣽⣆⣦
⠀⠀⠀⠀⠡⢾⡇⣘⣷⣟⡷⣬⣍⣷⣤⣀⢿⣿⠈⣼⣷⡆⢸⣧⡀⢀⠀⠈⢶⠀⠜⢏⠘⡇⠈⠋⢧
⠀⠀⠀⠀⢸⠔⢳⣼⠣⣽⡭⢚⠙⠛⠻⠿⣿⣬⠜⣽⠙⠃⠀⠀⠉⢦⠐⠨⡎⢶⢀⣿⣤⢹⣦⠀⢸⡆
⠀⠀⠀⠀⠈⣤⡀⣿⣷⣿⣝⡾⢿⣭⣿⣽⣿⣷⣾⡝⣡⡄⠀⠀⠀⠈⣁⠀⣽⡌⢠⡝⣾⣬⣽⣧⡿⠃
⠀⠀⠀⠀⡀⢽⣦⢻⣿⣧⢿⣷⣿⣿⢴⣷⣶⣾⣿⡟⠉⠀⠀⠀⠀⢠⠉⠁⣈⣿⣶⣙⢿⣿⣿⠟
⠀⠀⠀⠒⢀⡿⣿⠾⠿⣿⡏⠀⠀⠉⠙⠿⢽⣿⣿⣿⣤⣄⠀⠀⣠⣾⡷⢤⣛⣻⣿⣿⠻⣿⠏
⠀⠀⢠⣾⣿⣾⠟⣿⣦⣝⣷⡀⠀⠀⠀⠀⠀⠙⢻⣷⣬⣙⠷⣸⣿⠏⠀⠀⢸⣿⣷⣿⣷⠇
⠀⠀⠾⠿⠿⣧⣄⠹⠿⣯⠻⠷⡄⠀⠀⠀⢀⡴⠞⣹⢿⠟⣵⡟⣾⠀⠀⠆⢺⣹⣿⡿⠋
⠀⠀⠀⢀⡤⠀⠈⠙⠳⠈⢮⠀⠀⠀⠀⠈⠜⠡⠚⢉⣳⣿⠫⡼⡃⠁⠀⠈⣩⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢈⠟⠧⣾⢽⠃⡄⢀⣣⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠘⣶⢻⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⣀⣀
⠀⠀⠀⠠⣶⣧⣗⣼⣖⡖⠈⠷⡝⣿⣶⣄⡀
⠀⠀⡰⠅⢶⠰⠟⠛⡶⠙⣷⢱⣇⣈⣇⡼⣿⠻⡿⣶⣦⣄
⠀⠀⠁⢀⠈⣿⣾⠿⣆⠖⠹⢸⣭⡙⠁⠏⡻⢀⠇⣿⢸⣿⣿⠲⣄
⠀⠀⠐⠒⠵⠃⣉⣴⡟⡠⣴⡿⠀⠀⠀⠀⡇⣿⠸⣳⣫⢿⡿⢶⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠐⠿⠛⠈⠀⠈⣿⠧⠵⣖⡒⠿⢻⠠⣿⣿⣾⣁⣴⡾⣯⢳⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠒⠂⢢⣳⣼⡷⣷⢸⡹⣿⡋⢠⣶⢬⡻⣿⣟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⣛⣻⣠⠿⣕⠉⠛⢤⣽⣾⣿⣮⣦⢞⣽⢽⠛⢖⣿⢧⡀
⠀⠀⠀⠄⠠⠔⠀⠐⠂⠤⠀⠟⠬⣍⠃⢠⣾⣿⣦⣘⣻⣿⣛⣾⣟⠾⢡⣾⠛⣒⣿⡶⣷
⠀⠀⠐⡂⣡⡤⢰⢀⠀⠀⠀⠀⠀⠪⠀⠸⡿⠯⢝⠺⢍⡛⢝⡯⡙⣷⣼⡣⡞⢋⣭⣵⣿⣌⣦
⠀⠀⠁⡁⠟⠱⠉⠉⠀⠀⠠⢀⡀⠀⠀⠀⡇⠀⠀⠙⠂⠉⠢⣷⣈⠋⠳⡷⣷⣽⢁⠴⢬⣆⣧
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⢧⠹⣦⡟⢧⡙⡏⠠⡞⠉⠉⠑⠒⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠙⠙⢿⡃⣄⢐⠀⠀⠀⠀⠀⠘⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣿⠘⡏⢆⡀⢠⢄⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡟⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⡀
⠀⠀⠀⢀⡠⠤⢤⠾⠒⠢⣄⠀⠀⠙⡆
⠀⠀⢰⡯⠒⠒⢮⢠⣖⣄⠘⡆⠀⠀⣇
⠀⠀⢺⠀⢰⣷⠈⢦⣙⣃⣰⡧⠦⣄⠀⠙⠢⣄
⠀⠀⠀⠳⣄⣉⣠⠏⠀⠘⠲⢄⡀⠈⠳⣄⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠢⡀⠀⠉⠲⣄⠈⢣⡀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⢢⡀⢳⡀⠀⠀⠘⢦⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠷⠴⠇⠀⠀⠀⠈⠓⢦⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠲⠤⣀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⡴⠊⠁⠀⠀⠊⠹⡆⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⢢⠀⢸⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣀⡴⠊⠀⡞⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠙⠉⢿⠛⠙⠊⢹⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⡴⠁⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣄⠸⡄⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠋⠉⠀⠀⢠⠟⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⢍⡤⠃⡰⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠒⠖⠊⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⢀⡤⠞⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⣀⣀⣀⣀⣠⠤⠶⠋⠉⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠖⠒⠲⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡤⠤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡠⣤⡀
⠀⣀⣀⣀⣶⣶⣤⣤⣽⣶⣶⣦⣦⡀⠀⠀⠀⠀⢠⣶⠶⠶⠞⠀⠀⠀⠀⠀⠀⣷⡆⣘⠓⠀⠉⠉⠀⣼⠀⠀⠀⠀⠀⠀⠀⣷⡄⢛⡃⠉⠁⠀⢠⡄
⠀⠉⠉⠉⢿⠿⠛⠻⣿⠟⠻⠛⠋⠁⠀⠀⠀⠀⢾⣧⣤⣄⡠⣤⣤⠀⠀⠀⠀⠉⠀⠈⠁⠈⠳⠖⠀⠀⠒⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠛⠓⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆
⠀⢤⡀⠀⢀⠀⢀⣶⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣤⣶⣆⣲⣂⡀⠀⠀⠀⠀⠀⢀⠀⢘⣿⡃⠀⡀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣤⣠⣤⣠⣴⡀
⠀⣸⡿⠿⢶⣦⣒⣾⣿⣾⣿⡿⣷⣶⠀⠀⠨⠿⠁⣀⣀⣉⠉⠉⠩⠿⢄⠀⠉⠓⠀⠀⠀⠀⢩⠿⠟⡿⠅⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠛⠘⠛⠛⠿⠁
⠀⠉⠀⠀⠘⠁⠀⠞⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠂⠀⠀⠄⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢰⡟⠧⠀⠀⠽⢷⡄⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⡴⠿⠧⠄⠀⠀⠀⢠⣤⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⣠⣶⢾⣝⣿⣷⡀
⠀⡤⢾⣷⣿⣿⣿⣷⣿⠶⣄⠀⠀⠻⠿⠿⠿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣭⠉⠉⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣿⡿⠟⢀
⠀⠰⠩⠻⠛⠛⠛⠻⠻⠹⠈⠀⠀⠀⠀⠀⠀⠋⠁⠁⠉⢿⣯⡁⠀⠀⠀⠘⠿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠉
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈
⠀⠀⠀⠀⢀⣠⡴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣶⣛⣟⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡠⠖⠛⠛⠃
⠀⣷⢦⣤⣿⣾⣿⣿⣿⣿⣟⣶⣤⠀⠀⣾⣶⣤⣾⣿⣿⣿⡿⠿⣿⣥⡄⠀⠀⢙⣶⠤⣜⡷⣶⣶⣿⣿⣿⣯⣿⣷⣶⡦⠀⠀⠀⡽⠇⢠⣀⠀⠰⡦⠄
⠀⠛⠃⠀⠿⣟⣿⣿⡿⠷⠿⠛⠉⠀⠀⠉⠉⠀⠲⢟⣟⣿⣿⠿⡟⠉⠁⠀⠀⠈⠀⠀⠁⠀⠛⠁⠀⠀⠁⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠚
⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⢀⣠⡦⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣖⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣷⡦⢤⠶⠶⠶⡿⢿⠿⡶⢶⡤⣤⠀⠀⠀⢳⣀⣠⣤⣶⣷⣿⠿⢿⣦⣤⣄⠀⠀⢀⣿⠶⢿⣿⣿⣿⣿⣯⣿⡿⠇⠀⠀⢰⡆⠐⠒⠚⠛⠛⠛⠛⠛⠓
⠀⠟⠃⠀⠶⠀⢤⡄⠀⠀⠐⠂⠂⠈⠀⠀⠀⠞⠀⠀⠉⣹⠟⠿⢿⡿⠟⠛⠋⠀⠀⠈⠀⠀⠀⠉⠛⠿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⠀⡀⣠⣤⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣤⠄⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⣀⣀
⠀⣹⠖⠂⠈⡉⠉⠉⠁⠈⠭⠉⠀⠤⠀⠀⢀⣿⡇⠀⣀⠀⠀⠀⠠⡤⠀⡁⢠⠀⠀⠀⠤⠤⠤⠤⢴⣿⣿⣿⣿⣿⡧⠀⠀⠀⢨⡟⢐⣚⠛⠉⠉⠉⢉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠰⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠖⠀⠉
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⡒⣄⠀⣸⣹⡆⠀⠀⢠⠊⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢷⡜⣧⢻⡄⣷⢺⡇⠀⠀⠠⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠙⣇⠀⠀⠀⠀⠀⢀⣤⣤⣝⣦⡹⠞⠊⠉⠉⠉⣽⢳⠀⣂⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⠠⢈⢽⡀⠀⠀⠀⣾⠞⢁⡽⠉⢀⠀⡄⣸⠤⡷⣿⢸⠇⠨⠂⠐⠂⠔⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠲⢶⡀⠀⠀⠀⠙⠲⣄⣀⣈⣳⡞⠀⠀⣄⢷⣼⡶⣛⣿⠿⠿⠶⡄⢧⣀⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⣌⡂⠄⠉⠈⣩⠋⠹⡀⢤⣝⣿⡋⠘⠻⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢉⡦⠍⠢⠀⠀⠀⠈⡉⠐⢻⣔⠗⠁⠀⠀⢠⣤⡥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠉⠀⠀⣎⡠⣀⣄⣀⣦⠐⢾⣎⣉⡟⠀⠀⠀⠀⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡞⢁⠴⠐⠦⡉⢳⡟⣫⡅⠠⠞⠀⡀⡤⣼⠙⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⣼⠀⠡⠁⠀⠁⢸⠀⣽⣿⡷⢀⠀⠀⠭⠁⠛⠀⠐⣿⣲⣲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢀⠼⣆⠢⢅⣀⡠⠎⣰⣾⢛⡀⠨⣿⠛⠓⠲⣦⠠⢤⠋⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣢⠏⠀⢌⢷⣦⢤⣴⣶⡿⠃⢄⠐⠀⠈⠣⡀⡐⠉⢛⠨⡆⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⣷⠃⢀⣡⣴⠎⠉⡛⠉⠁⠀⠀⣠⣷⣄⠂⢰⠌⠿⠷⢊⣼⠉⠺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣏⡟⠀⠠⢬⡿⠖⠦⢤⣤⠶⠓⠊⠉⠉⠀⢱⣬⠾⣚⣿⠉⠀⠒⡖⢌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣇⣀⣀⡸⠂⢀⡋⠒⠠⠷⣶⡴⣤⠆⢚⣛⣦⠰⣿⡿⠃⢀⢠⣉⢦⢚⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣻⠉⠉⠉⣗⣠⣁⣰⣼⣷⣦⡀⢿⡤⡆⠄⣼⢀⣠⠇⠐⡠⢄⠓⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣳⣷⠇⠀⢀⡔⠉⠉⠉⠁⡉⠱⣏⣀⣙⠛⢛⣁⣈⡇⠀⠠⢀⡟⠑⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣫⣴⠛⢦⣇⠈⠟⠀⠀⠀⠀⡰⠀⠁⠘⠒⠚⠈⠉⠳⡀⠒⣄⠑⣄⡧⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡼⠛⢄⠀⠂⢝⡴⠱⠆⠀⠀⠀⠀⠀⢀⠛⡘⡞⠥⡇⢸⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣝⢦⣴⠋⠓⠦⣆⡈⠃⠀⠀⠉⠁⠀⠀⠠⠄⡡⠛⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⡼⠓⠃⠀⠀⠀⠀⠙⠳⠦⢴⣤⣤⣤⡤⠶⠛⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⡍⠀⢸⣿⠋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣸⡇⠁⢀⣾⣿⣄⠿⣿⣏⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣽⣟⣴⣿⣿⣿⣿⣶⣻⣯⣟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⣸⣿⡿⠇⠸⠿⠿⣿⣯⣯⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣾⣽⡇⠐⣦⣀⣁⣴⠊⠈⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣾⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣇⣀⣈⣻⣿⣁⣀⣸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣦⠀⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣠⣬⣥⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⣹⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⣟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣟⣛⡛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⢻⣿⣿⠿⠿⠿⠿⠀⠰⢤⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠟⠒⠀⠀⠠⣤⣄⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶⣶⣶⣶⣦⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣥⣤⣤⣤⣤⣤⣤⣤⣙⣿⣿⣷⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠫⣄⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⢻⡄⣸⠾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⡔⠉⣽⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢀⣠⣿⠋⢀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡾⠧⠤⣿⣄⣀⠀⠉⠻⣦⡀⠀⠀⠀⠀⣰⡟⢠⣶⠟⠉⠈⢹⡋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣤⣤⠟⠛⠻⢷⣦⣌⡻⣦⠀⣀⢠⣟⣴⠟⠁⣀⣤⣴⡋⣹⡟
⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡼⠦⣤⣀⣀⣈⣙⣛⣾⣷⣿⣾⣿⣵⣶⣿⡿⠿⠿⢿⣍
⠀⠀⠀⠀⣼⣏⠉⠻⡟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣿⣿⠿⠿⣿⣿⠿⣿⣧⣼⣿⡟⠛⠛⠒⠲⢤⡴⠿⢷⡄
⠀⠀⠀⢰⡿⠿⣆⠀⠘⢦⡈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣀⡴⠋⠉⠀⣸⣿⠉⣧⠙⣟⢦⡀⠀⠀⠀⢱⣦⣴⣧
⠀⠀⠀⣿⡧⣀⠈⠳⣄⠀⢻⣦⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡋⢉⡟⠀⠀⠀⢰⠇⡏⠀⠻⠀⠘⣆⠙⢦⣄⣤⣾⣄⣸⡅
⠀⠀⠀⢸⡇⠈⠻⣶⡼⠿⢦⣿⡍⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠋⢙⣦⣄⣴⠏⢰⡇⠀⠀⠀⠀⠘⢧⣀⡽⢧⣀⣽⣏⡀
⠀⠀⠀⢸⣿⠖⢶⣿⠀⠀⠀⠈⠻⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣞⠁⣸⠟⣤⡾⠀⠀⠀⠀⠀⠀⠀⠙⠳⡼⠿⠋⠉⢻⡗⢦
⠀⠀⠀⢸⣇⠀⠀⠙⢧⡀⠀⠀⠀⠙⡄⠹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⡿⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠘⢿⣦⣀⠀⠀⠙⢦⡀⠀⠀⢹⡀⣿⠀⠀⠀⠀⠀⠀⢰⣛⡟
⠀⠀⠀⠀⠀⠉⣛⣿⣶⣤⡀⠙⢦⡀⠀⢧⢿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣴⣾⣿⡿⠿⠿⠿⠿⠿⣷⣤⡙⢦⡘⣾⡇⣠⣄⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠠⣿⠀⣏⠀⠀⠀⠀⠀⠀⣀⣉⡙⢿⣿⣿⣿⣿⣋⣤⣶⠾⣛⣛⣉⣁⣀⣀⣀⣀⣈⣛⣿⣏⣙⠛⠛⠷⣦⣀
⠀⠀⣿⣖⠻⢦⣤⣤⡶⠿⠛⠛⠛⢉⣩⣿⡟⣡⣿⣿⣿⣉⣉⠉⠀⠀⠀⠀⠀⠀⠈⠙⢿⣍⣉⠉⠉⠓⠒⠿⢿⣷⡀
⠀⠀⠨⣿⣤⣾⠋⠁⠀⠀⢀⣴⣾⠟⢉⣼⡟⣻⡟⣿⠻⢾⣍⡛⠳⠶⣤⣀⠀⠀⠀⠀⠀⣻⣍⠛⠲⢶⣤⣤⣴⡟
⠀⠀⠀⠈⢿⡍⢳⣄⣤⣴⡿⠟⠁⢠⣾⠏⠀⣿⠇⢸⣧⠀⠙⢿⣷⣤⡀⠉⠙⠛⠲⠶⢿⡉⠉⠓⢤⣀⣸⡿⠋
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⢠⣿⠋⠀⣸⣿⠀⠀⣿⡇⠀⠀⢻⣿⡿⣶⣄⠀⠀⠀⢀⣿⣦⡶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢀⣿⡇⠀⠀⢸⡇⠀⠀⠀⣿⣷⠈⠛⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⣸⣿⣄⣀⣀⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣙⣻⣟⣀⣙⣯⣥⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠛⠁
⣿⣏⢹⣟⠻⣿⢛⢟⣻⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡭⠶⠤⣀⣀⡤⡴⢯⣕⣆⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠸⡀⣠⣖⣿⣿⣽⡗⠫⣭⣿⣿⣷⣦⡄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣴⢻⡿⠛⢍⣻⠺⢿⣿⣴⣿⢿⣦⣧⣿⢢⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⣇⠘⢧⡀⠀⠙⣿⣏⡼⣓⣼⣧⡿⢷⡯⠠⠃⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠤⣍⣹⠟⠷⢿⡋⠩⠭⠭⠗⢮⡇⠐⠦⣤⣀⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠊⠛⠒⠒⠒⠋⠀⢀⡴⠶⡎⠱⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⣰⣫⣭⡳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠘⣦⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣿⡇⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠀⠀⣤⣶⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠈⢧⠠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⢀⣤⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠧⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠒⠚⠁⠀⠀⠉⠀⠀⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⡀⠀⠀⠀⠒⠒⠶⠶⢾⣿⡆⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠐⢦⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠙⢾⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⢀
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢻⣷⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢠⠞⠁⢀⣴⣿⣷⡄
⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⠀⢻⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢀⡟⠀⡰⠋⠀⣰⣿⣿⠟⠋
⠀⠀⠘⠦⠀⠈⠻⣷⡀⠀⣤⠀⢻⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⣠⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⣒⠒⢶⣤⣤⣤⣤⠤⠶⠒⣶⠶⠦⣤⡀⠀⠀⠀⠾⠋⠁⠀⢀⣠⡴⠷
⠀⠀⠀⣶⣤⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠔⢫⠭⠉⠀⢻⡄⠀⠀⠀⠀⠈⠡⠖⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠓⠂⠀⠀⠀⠀⠀⠀⣀⠴⠒⠻⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢠⣤⣴⡶⠿
⠀⠀⠀⠶⠒⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑⡫⠁⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⣆
⠀⠀⠀⠀⠀⠀⠀⡿⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢼⣆⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⡠⠖⠒⠛⠒⠀⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣀⣀⠄⠀⢸⣿⡄⠀⠠⠤⠶⠚⣡⣼⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠈⣉⣁⣠⠔⠀⣸⣿⣿⣷⣶⣶⣶⣿⠟⠋⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠤⠤⠖⠀⢻⣆⠀⠒⠒⠒⢉⠀⢸⣿⣦⣤⣌⣩⣤⣴⣾⣿⣿⣿⠿⠏⠁⠀⢠⣠⣠⡿
⠀⠀⢤⠤⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣬⣭⣴⡿⠿⢷⣤⣉⣉⣡⣤⣾⠿⣿⣿⣿⠯⠭⠭⠶⣦⡀⠀⠀⠀⠀⠀⠀⣷⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⡏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣀⣴⡿⠃
⠀⠀⠀⢶⡾⠟⠋⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⠤⠴⠶⠚⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⡀⠀⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢴⣄⠀⠈⠳⣤⣀
⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁⠀⣠⡟⠀⠀⢰⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⢱⡀⠀⠈⢻⣷⣄⠀⠈⠻⠃
⠀⠀⠀⣠⣾⣿⣿⠋⠀⠀⢰⣿⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠹⣿⣷⣄
⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⣤⠀⣰⣴⡆⢠⣶⡄⢠⣤⣴⡄⣴⣶⡄⢀⣀⠀⠀⡀⢠⢠⡄⠀⣠⣤⣶⣶⣶⣄⠀⢠⣦⣀⠀⠀⠀⢠
⢰⣿⣦⢻⡿⢣⡌⠟⣡⡘⣿⡿⠀⣿⣿⠇⢸⡿⠀⢾⡏⢿⢈⡃⠉⣭⣥⣄⠈⢉⢉⠀⣿⠟⠋⡶⠁⣣⣴
⠘⡿⠛⠀⠰⠆⠁⣤⡌⣰⣮⠀⣛⣈⢩⣤⣤⠉⣤⣤⡄⠴⠞⢡⣄⡱⠄⠻⠀⠚⠈⣤⣍⠴⠂⢄⠐⣂⠈
⢠⣶⣿⣷⡆⢾⡷⠶⠗⠙⣛⡁⡾⠉⣤⣭⠹⠶⠨⠉⣥⢼⡷⠀⣠⣤⡘⠀⠛⠂⠩⠉⣒⣋⡁⠘⠰⢟⡃
⢀⣩⣼⣿⣤⣄⠛⡿⠿⢢⠙⡁⡇⣴⡎⠁⠘⠛⠓⠒⠒⠀⣦⣾⣿⣿⣿⡆⠉⠻⠿⣶⠈⠀⠔⠚⣁⣞⠓
⠘⠻⠿⠿⠿⠛⠘⠛⠈⡻⢿⢠⠃⠈⠁⠸⢿⣷⡷⠿⠇⢀⣹⣅⣈⣙⡁⠀⠒⠲⢤⠐⢊⣡⡄⠉⢁⣽⠔
⠀⣴⣿⣷⡄⢠⣶⣴⡥⠀⣴⡜⠀⠈⠀⠴⠖⠠⣴⣦⣄⢘⣽⣿⣿⠿⢏⠢⢤⣾⣾⣶⠛⠡⠖⠀⠛⢁⣀
⠀⠻⣿⠿⢃⡘⠿⠿⠋⠀⠿⠃⠀⣄⣀⡀⠀⢠⣦⡀⠀⠘⠋⠋⠙⠂⠀⠀⠘⠙⠛⢉⣀⢿⣂⠈⣋⠘⠛
⠀⣶⣶⣦⠙⢿⡆⢰⣿⡇⠀⣀⠀⠽⠿⠷⠄⠀⠉⠀⣥⡁⠀⠀⣤⠀⠀⢀⠀⠘⠀⠐⠉⠻⠃⠘⠛⣋⠄
⠀⠀⠀⠐⠾⠷⠂⢀⣤⣶⣿⠟⢠⣤⡤⢄⡠⠀⠀⠈⡁⠀⣤⡆⠰⠖⠀⠙⠃⠘⠻⠆⢸⠆⠐⠋⠀⠟
⠀⣿⣇⠈⠉⠋⣾⣿⡟⠋⠤⠀⠀⣰⠀⠈⣤⠀⠲⠾⡇⠀⣈⣀⠀⠀⠛⠁⣿⠀⠰⠆⣤⡀⠀⡀⠀⠙⠁
⠀⠈⠁⣾⣷⣦⢀⠀⢿⡀⠀⣸⠇⢹⣤⣤⠏⠠⣄⡀⠘⠀⢻⡿⢀⠀⢀⡀⢀⡄⠀⣷⠟⠛⠈⠉⢁⣀⣀
⠀⢐⠽⣿⣿⣿⣟⠣⠨⣷⣷⡯⠀⠈⢻⡟⢁⡘⢿⣇⠀⠀⡆⠀⠉⠁⠊⠁⠈⠃⠀⠀⠤⠄⠶⠒⠀⠉⠉
⠀⢀⣈⣈⠛⣩⠶⢦⡄⢹⣿⠁⣀⡀⠘⠁⣈⠱⣶⡦⠀⢠⡤⠀⠀⡀⠀⠠⠶⠶⢶⠀⠓⠂⠀⠉⠛⠚⠒
⠀⠺⠋⠉⠀⠠⠴⠶⠖⠀⢰⠟⠉⠁⠀⠟⠛⠷⠀⠶⣶⠈⢱⣦⠀⠀⠲⠤⡄⠀⠀⠛⠓⠲⢄⠴⠶⠤
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠀⠂⢀⠀⠁⠀⠀⣀
⠀⠀⠀⠈⠀⠀⡀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠉
⠀⠠⠈⠀⢀⡀⠀⠀⠀⠀⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠦⠁
⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠰⠁⢉⠄⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠒⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠈⠱⣫⠆⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⡄⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠂⠀⠀⢀⡀
⡔⠐⠂⠀⠀⠀⠠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⢀⠁⠀⢀⠀⠀⡀
⣑⣐⣁⣀⣠⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣘⣰⣄⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣰⣾⣷⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⠟⠙⠿⠁⠁⠈⠈⠁⠉⠙⢻⡿⣖⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⠿⣿⣿⡆⠻⠀⠀⠰⣶⢤⣤⡤⠶⠄⠁⣿⢸⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⣀⠀⢰⣿⣿⠃⠀⠀⡀⠀⠂⠀⠀⠀⠂⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠈⠁⠀⢠⠟⢻⡆⠀⠀⣡⠐⠀⠀⠀⠀⠀⠋⠁⣸⣿⣿⠿⠿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢹⠀⠀⠀⢀⠀⠈⢻⡄⠀⠉⠀⢎⡽⣭⣌⠀⢸⣦⡿⠛⠀⠀⠀⠈⠉⠻⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⡈⠸⠀⡆⢸⠁⠀⠀⠙⣶⣾⡓⠸⠀⣉⠻⠀⣾⠉⠀⠀⠀⢰⡆⢫⣀⠀⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠃⢠⡾⠁⢼⡀⠀⠀⠀⠈⢿⣷⣦⣀⣾⣤⣼⠏⠀⠀⠀⠀⢸⡇⠀⠻⣦⡀⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠟⠁⢀⣼⣿⡄⠀⠀⠀⠀⠙⠿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠙⠀⠀⠻⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⢀⣿⣿⣿⣿⣶⣦⡐⠲⢤⡶⠿⠿⠷⠀⠀⣀⣤⣴⣾⣿⣿⣿⢿⣶⡄⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⡟⠂⠀⢠⣾⢿⣿⣿⣿⣮⡉⠁⠀⠀⠀⣠⡀⠀⠀⠀⠈⠻⢛⣿⣿⣿⣿⠀⠙⢿⣆⠒⠀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠸⠇⠀⣿⣿⣿⠉⠛⠒⠀⠀⠚⣭⡛⠓⠂⠀⠐⠚⣻⣿⣿⣿⠇⠀⠀⠀⠻⣧⡀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣴⠀⠀⢹⣿⣿⣷⠈⠂⠀⠀⠘⣿⠛⠗⠀⠀⠶⢶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⣷⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣯⠀⠀⢨⣿⣿⣿⣶⠆⠐⠲⠶⠾⡿⠦⠀⠀⣀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠛⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⢹⣇⣠⣼⣿⣿⣿⣿⣿⣷⣤⣶⣴⣾⣦⣤⣦⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⠆⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⢠⡀⠀⢽⣿⠋⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢹⡇⣇⠀⠹⣦⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢄⣙⠀⠘⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠸⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠌⠣⠀⠈⢣⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠛⠿⠿⠿⠿⣿⣿⣦⠀⠘⣿⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⣿⣅⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠈⣻⣿⣧⠀⣿⡇⣤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⡿⣏⣿⡦⠀⠀⢼⣾⣏⠉⠀⠀⠀⠀⠉⠁⠀⠈⠙⠻⠿⣿⣿⣷⣷⠀⠀⠀⢀⣽⣿⣿⣧⣿⠓⣏⠇⢀⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣇⠀⠀⠉⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣄⣤⣴⣾⣿⣿⣿⣿⠏⠀⠙⣦⡈⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠘⠢⠀⠀⠙⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⡴⢂⣤⠘⣽⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣰⣇⣴⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠮⠶⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⡠⢦⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢩⣷⠋⠀⠀⠉⠀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⢀⡀⢤⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣁⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠞⢉⡋⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠶⣿⡽⢣⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⠿⡤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠼⣿⡝⣧⣴⠖⠁⢁⡀⠀⣶⣦⡀⠀⠀⠀⠀⠾⣊⠤⢟⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⢠⠀⠈⠈⠛⠛⠚⡉⠛⠾⢯⣭⣍⡀⠀⢲⠆⠸⡏⢝⡈⠁⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠈⢶⡋⠐⠘⢷⡀⣀⠀⠀⣠⡆⣴⣲⣶⢦⠈⠉⠛⠛⢷⣄⣬⣷⡃⠛⠀⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⢷⣄⠀⠈⠙⠛⠛⢛⡉⡁⣿⣿⣷⣿⠇⠀⠀⠀⠀⢹⣉⣿⣿⠳⢀⣀⡆⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⢲⣦⣀⣤⣤⠴⠟⠁⠹⢯⣍⠹⣄⣀⠀⠀⠀⠈⠙⠛⠿⠋⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠒⢺⠿⠿⣷⡶⠶⠀⠀⠀⠀⠈⠳⣾⣿⣿⣦⠀⠀⠀⠀⠀⠘⢣⣴⣷⣶⣶⡂⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡉⠁⠠⠾⢿⣿⣶⠀⠀⠀⠀⠀⠀⢺⣿⣿⠏⠀⠀⠀⠀⠀⠀⡤⠄⠈⠉⢻⠿⠴⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⣯⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢠⡤⡒⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡞⠀⠀⡠⠔⠊⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠁⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠴⠿⠂⣀⠀⢀⡀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠄⣀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠃⠀⠀⠈⠢⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠀⠠⠟⠃⢀⠚⠃

⠐⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠋
⡄⠀⠀⠀⠀⠀⠀⠐⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠂⠄⠀⠀⠀⠀⠰⣍⡀⢀⠠⣈⠩⠍
⡗⠀⠀⠀⠀⠀⠀⠀⠢⠀⠑⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣃⣦⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠿⢯⣼⠏⢹⣮⣤⣤⡄
⣷⣤⣤⣦⡄⠀⠀⠒⠀⠀⠀⡀⢀⡀⠀⠀⠻⣝⢶⣤⣦⡴⠞⡽⠃⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠋⠉⠛⠧
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⣿⣿⣷⣾⣧⠀⠀⢀⡀⣈⣈⡈⠐⣽⠉⣀⣋⣁⣹
⣿⣿⡿⠟⠒⠒⠂⠀⠃⠐⠃⠀⠊⠋⠀⠠⠌⠁⠀⠉⠀⠐⠀⠃⠀⠀⠤⠄⠆⠤⠀⠤⠥⠰⠦⢠⠤⠈
⣿⣿⠇⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠩⠠⠭⠅⠄⠭⠭⠽
⣿⣿⠃⠄⠀⠀⠀⠨⠉⠋⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⢀⣥⠀⠀⠀⢀⣸⣄⠀⠀⠀⠀⠀⠉⠀⠀⠃
⣿⡿⠀⠀⠀⠀⠀⣀⡄⣺⣶⠿⢛⣻⣿⣿⣿⣿⣿⣯⣽⣿⣿⣶⣶⣶⣿⣿⣿⡇
⠋⠁⠀⠀⠀⠀⠼⣽⡿⠛⠓⣶⣾⣿⣿⣿⣿⠟⢻⣿⣿⡿⠿⠿⢻⣿⣿⣯⠟⠁
⣿⠂⠀⠀⠀⠐⢤⡯⠤⡤⠤⣤⣬⣥⣤⣤⣤⣭⣬⣥⣤⣤⣴⣶⣼⠿⢿⢸⡆
⣟⣀⠀⠀⠀⠀⠋⠀⠀⢀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉
⡏⠁⠀⠀⠀⠀⠀⠀⠓⠉⠙⠁⠘⠐⠃⠊⠀⠀⠂⠂⠀⠒⠒⠀⠁
⣯⣤⣤⣤⣤⣤⣽⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠈⠤⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣤⠀⠌⡀⠀⠠⠀⡀⠤⠀⠄
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐⠢⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠀⠀⠒⠀⠈⠉⠀⠀⠈⢳⠀⠄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠨⠀⠀⠝⢤⡀⣀⠀⢀⣀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢈⡁⠤
⠀⠀⠀⠀⠰⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⡅⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠠⠰⠀⠠⢄
⠀⠀⠀⠠⣎⠁⠹⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠩⠀⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⣶⡀⡡
⠀⠀⠀⠐⠀⠀⠐⠐⠀⠄⡈⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠶⠝⠤
⠀⠀⢀⠁⠀⠀⠀⠀⡁⠀⠢⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢠⠀⠀⠀⠁⢀⡀
⠀⠀⢨⠀⠀⠀⠐⠤⢢⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠐⠁⠀⠂⠀⠂⠔⠂
⠀⠢⠀⠰⠄⠀⡀⠔⠁⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢠
⠀⠈⠦⠂⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠁⠠⡀⠀⠀⢀⠂⠃⠀⠀⠀⠀⠀⠐⢇⠀⠀⠀⠀⡌⠀⠶⡄⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢳⠀⢀⡀⠀⠀⠀⠐⡀⠀⠀⢠⢀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⣌⡀⠀⣀⠀⠠⠠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠠⠝⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠈⠊⠋⠲⠸⠀⠀⠀⠀⠀⠀⢁⡌⡽⣈⠁⠀⣥⠻⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⠇⢂⠀⠸⢀⢏⣇⠀⠩
⠆⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠔⠁⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠐⣬⢃⠄⠂⠙⢀⢛⠂⢁⠂
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⣠⡆⠀⠀⠡⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⣀⢀⣀⠀⠀⠀⠀⠀⢰⠃⠖⡤⠃⠀⠈⠄⠀⠀⠙
⢀⠀⠀⠀⠀⠘⢁⣤⡆⠀⢫⡇⠀⠆⠘⢀⣒⠀⠀⠀⠀⠀⠀⡀⣀⡨⠥⠥⢴⣚⢀⡣⠀⠀⠀⠀⠌⢸⠀⠀⡃⠊⡄⡀⠀⠀⠀⠀⠄
⠈⠙⣷⣄⣀⡀⢚⣀⡀⠀⠀⠀⠀⠣⢔⡥⠒⢋⡀⣀⠀⠀⣠⡥⠠⠈⣍⠑⢦⢁⠈⠀⠀⠀⠀⠀⠀⡀⠫⡄⠉⠀⠃⠁
⠀⠀⠀⠉⠉⠀⠈⠫⠇⠀⠀⠢⣤⣴⠾⠃⠀⠀⠀⠀⠀⠀⠐⠙⠀⠁⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢹⠤⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠲⠦⢤⣤⠿⠤⡤⢤⡀⣿⠔⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸⠃⡿⠀⠀⠀⢿⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠠⣄⠀⠀⢀⣀⣤⠾⠃⢰⡇⠀⠀⠀⡟⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⢹⣿⣿⠏⠀⠀⠀⣼⠁⠀⠀⢸⠇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⡰⠞⠛⠻⣿⣄⣀⠀⢶⡿⠀⠀⠀⡿⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠀⠀⠀⠈⠙⠻⣿⣿⣏⢀⣀⣀⣧⢀⠀⠀⣼⡏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣭⡿⠿⠦⠿⠿⠋⠉⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⢀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣀⣀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⠉⠀⠀⠀⠀⠀⠀⢁⣀⣀⣈⣙⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠅⠉⠡⣀⡠⠤⠀⠉⠭⢍⣉⠙⠿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠂⠛⠋⠉⠉⢁⣀⣀⡀⢀⣤⣄⡄⢠⣤⢀⣤⣀⣀⣀⡀⠉⠉⠙⠛⠑⢲⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠺⣿⣙⠃⢻⡇⣿⡇⢨⣧⢸⡿⢶⣿⠘⢩⣿⠹⢻⡷⠀⠀⠀⠀⠀⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⡿⠦⠀⠚⠓⠈⠛⠛⠋⠘⠛⠘⠛⠀⠺⠿⠀⠾⠅⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠙⠉⢧⡀⠀⣀⣀⣀⣀⣀⣀⠶⠶⠆⠀⢠⠴⠤⠀⡄⠰⠶⠶⣄⣀⣀⣀⣀⣀⠀⠀⣼⠋⠙⢧⣀
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠘⣿⣿⡏⠉⠀⢸⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⠀⠉⢹⣿⣿⠇⠀⠀⠀⠈⢣
⠀⠀⠀⠀⡸⠃⠀⠀⡀⠀⣠⠤⠿⠿⠀⠀⣾⡇⠀⠀⠀⠀⠀⣾⠀⠀⢸⠇⠀⠀⠀⠀⢻⣿⠀⠀⠾⣿⣤⣤⠀⠀⠀⠀⠀⢧
⠀⠀⠀⡔⢰⡤⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⣦⢡⡀
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⠀⠀⠀⠀⠀⠀⣠⠞⡆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢣⡀⠀⠀⠀⣰⠁⠀⢹⡀⠀⠀⠀⣠⠞⠉⡇
⠀⠀⠀⣤⣄⡀⠀⠀⠀⢰⠇⠀⠀⠱⡄⠀⢰⠃⠀⠀⠀⣧⠀⢀⡼⠁⠀⠀⣷
⠀⠀⠀⠘⡄⠙⠲⣄⡀⣼⠀⠀⠀⠀⠙⣦⡟⠀⠀⠀⠀⠘⣦⠞⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠖⠋⣧
⠀⠀⠀⠀⠘⡄⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡇⢀⡴⠋⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠹⡄⢀⣀⡀⠀⠀⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⠴⠊⢹⠇
⠀⠀⠀⠀⢠⡖⠋⠉⠀⠘⠊⠁⠀⠀⠀⠈⠳⡤⠖⠈⠉⠉⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠴⠊⠁⠀⠀⡼
⣀⡀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠱⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⣠⡤⡄
⣿⠉⠲⢤⡄⢻⡀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡀⠀⢀⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠘⠛⠉⠁⢀⡼⠁
⠛⠚⠛⠛⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠰⣟⡷⠀⠀⠀⠀⢸⠀⠀⠷⠚⠉⢧⢀⠴⠋⠀⠙⢄⠀⠀⠀⠀⣤⢄⣀⡞
⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⢰⠦⣤⣤⣴⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⣢⠀⢀⡴⠁⠀⠈⣱⠆
⠀⠀⣀⠴⠚⠁⢹⠀⠀⠀⢸⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⢀⡞⠀⠀⢠⠞⠁⢀⡠
⢤⣮⣥⠤⡤⣄⠘⣧⣤⣴⠾⠀⠀⠀⠀⠀⠙⠶⢤⣤⣤⣤⠴⠞⡇⠀⠀⠀⢀⣠⡀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡎⠀⠀⣰⠋⠠⣶⠋⢀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠠⠀⢄⠀⠀⣀⠤⠀⠤⢀⠀⠀⠘⣦⡴⠞⠉⠀⠹⣄⣠⠴⠋⠀⠀⠀⠀⠈⠓⠦⠼⡇⠀⠀⠀⠙⠚
⠀⠀⠀⢠⠏⠀⣀⡠⠤⠖⡺⠃⠀⠀⠀⣡⠊⠀⠀⠀⠄⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⡴⠒⣦⠀⠀⢹⡀
⠀⠀⠰⠷⠖⠋⠁⠀⠀⡔⠀⠀⠀⢀⡔⠁⠀⠀⠁⢀⡌⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃⢀⣤⣀⢳
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢣⠀⣠⠞⢧⠀⠀⠀⠀⢀⣤⠀⠀⠀⣸⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⡀⠀⠀⠀⠀⡟⠁⠀⠸⡆⠀⠀⢀⠞⠈⢧⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠃⠀⠀⠀⠹⡄⢀⠞⠀⠀⠈⢳⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠹⠞⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠈
⠀⠀⠀⠀⠀⡠⡍⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⢠⠌⠀⠀⠂⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠂⢄⢀⡀⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣈⣑⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⢀⣠⣴⡶⠶⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣻⣿⣿⣿⣭⣽⣿⣿⣿⣻⡶⣤⡙⢳⡀
⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⢠⣴⢿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⡼⣿⣿⡶⣅
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠈⠻⣿⣿⣧⡀
⠀⠀⣿⡇⠀⣐⣶⣂⣀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢂⠌⢻⣿⢳⡀
⠀⠀⠸⣧⠀⠹⣿⠿⣯⣀⣴⣶⡿⣠⡿⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢻⣿⣷⠐⠢⣄
⠀⠀⠀⠙⢷⣄⡀⠀⠈⠉⠉⣉⣵⠟⠁⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣜⣿⣿⡆⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⢸⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡧⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠈⠱⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⢿⣿⢀⡴⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⡄⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⣿⣾⣣⠟⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠈⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣒⣂⣸⢇⣿⣿⣏⣳⣛⣡⣤⣶⡶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠡⠀⠚⠁⠀⠀⠀⠀⣀⣙⣾⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢸⣵⣛⣽⣿⣿⡷⣻⣿⣯⣽⣿⣿⢿⠿⡞⠁
⠀⠀⠀⠀⢀⡠⠒⠉⠁⠀⠀⡸⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣞⣿⣻⣿⣿⣻⣿⢟⣟⡁⣩⣤⣴⠾⠋⠉⠡⡄
⠀⠀⠀⢠⠋⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⡓⠛⠛⢛⣭⣵⣿⣿⣿⣿⣿⣯⣝⣾⡉⢠⠖⠒⠒⠚⠁
⠀⠀⠀⢸⡀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣶⣟⣿⣿⣿⡿⠛⠛⠺⠝⣦⠙⡇⠘⠦⠄
⠀⠀⠀⠀⣿⣄⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢹⠇⢐⠀⠀⠘⡄
⠀⠀⠀⠀⠙⣌⠳⣤⣿⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠉⠙⠉⠉⠀⠀⣀⣀⣀⠀⢀⣀⣴⠟⠀⠀⠙⠒⠚⠁
⠀⠀⢀⣤⣤⣈⡻⣮⢻⣄⠀⢸⠀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠈⠉⠉⠁⢀⣀⣀⣀⡀
⠀⠀⠈⢛⡿⣿⣿⣿⣽⣿⣦⠟⣿⡟⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠋⠉⠙⠛⢿⣦⡀
⠀⠀⡄⠈⠙⠷⣦⣤⣌⣹⣛⣶⣿⣧⠀⠻⠃⠀⠀⠀⠀⢠⣶⣶⣶⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠉⠉⠉⢲⠀⣼⠻⣿⣿⣿⣿⣿⣷⣶⣄⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠉⠀⠃⣾⠛⠉⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠸⢄⡠⠔⠀⢿⡀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠟⠛⠛⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠛⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣄⣀⣤⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣿⢿⡿⠿⢿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠒⣍⡀⠱⠀⠀⡈⠙⠲⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠉⠀⠀⠀⠀⠀⣏⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢣
⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢇
⠀⠀⠀⠀⣸⡇⠀⠀⢣⣶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⢸⡄⠀⠀⠀⠀⠀⠀⠐⢤⣽⠈⡆
⠀⠀⠀⠀⠇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠠⣿⠃⠀⠀⠀⠀⠀⢀⢼⡟⠀⠀⠀⠀⠀⠀⠀⠈⢮⣿⢠⢿
⠀⠀⠀⢸⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⡑⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠌⣸
⠀⠀⠀⢸⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠨⣿⠀⠀⠀⠀⠀⠀⠀⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⢝⣯⡴⠃
⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠠⡽⠀⠀⠀⠀⠀⠀⠀⠀⢨⣢⣿⠁⡆
⠀⠀⠀⠀⣷⠄⠀⠀⣿⣀⠀⠀⠀⠀⠀⣇⠀⣀⡀⠀⠀⣠⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣻⡇⢰⠁
⠀⠀⠀⠀⠸⣷⡄⢐⣿⣦⣀⠀⠀⢀⣀⣿⡂⠀⠈⠁⠀⠀⣽⣷⡄⠀⣀⠀⠀⠀⣠⣤⣽⣿⠃⣾⡀
⠀⠀⠀⠀⠀⠙⠲⠶⠟⠻⢿⣶⣤⣶⡶⠿⠿⣶⣿⣋⣰⣶⣿⣿⣗⣀⣀⡉⠉⣃⣼⣿⡟⠉⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠋⠁⣈⠙⡉⠉⠋⠙⠛⠿⣿⠛⠏⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡖⠁⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⠤⣤⣤⣮⠀⠀⠀⠀⢠⠀⠀⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠧⡶⢸⠀⡁⠀⡄⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣼⣥⣶⡿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⢾⠦⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠉⢾⣳⣿⣷⡖⠈⠁⠀⢰⠓⡄⢰⢠⢴⣶⣴⢂⣿⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠙⢷⣍⠀⠜⡰⠀⣼⣶⣥⡾⠿⠼⢿⣟⢯⣽⡂⠙⢦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤⣄⠘⣛⡙⢆⠉⣦⣿⣿⡟⣿⣟⣿⣿⣯⣿⢟⣷⣴⣶⣿⢾⡗⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⣦⢹⡿⣮⢻⣧⣿⢫⣉⣭⠑⢙⢿⣿⣿⣿⣿⣿⣿⣏⣈⡄⢀⣀⠤⠒⠺⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡈⠻⠇⢀⣜⣿⣿⣿⣿⣯⡳⠌⣡⣭⡉⢻⡿⢻⣿⣮⣼⣦⣻⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣼⡛⠏⣨⡿⣿⠿⣿⣻⣿⣏⡁⢸⢱⣿⣿⣾⡗⣿⣿⣃⡝⣧⣽⢯⣜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢿⡇⠘⢿⡻⣶⣝⣿⢿⣧⣙⠃⠘⡎⢿⣿⡿⢡⣿⣿⣿⣿⣻⠟⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡒⠒⣚⣻⡄⠺⣿⣿⣿⠻⡷⣽⠇⣠⣛⣿⣦⡿⣶⣿⣿⣿⣿⣾⢻⠃⠀⠑⢲⠫⢭⣓⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣹⣷⠶⣾⢯⣭⣄⢹⣿⢟⣷⠅⣉⣉⡙⢿⣿⣿⢂⣼⣿⣷⣍⣡⠄⠉⠀⣀⠈⢦⣻⣥⣍⣁⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠉⡟⠚⣾⡷⠜⣫⣿⣁⣼⣟⠙⢿⠀⣛⢁⣾⣿⣿⣿⣩⡇⠸⣄⠰⠐⠠⠚⠉⢠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⡐⢉⣰⢎⣼⣿⣿⣿⣿⣤⠟⠠⠿⢸⣿⣿⣿⣿⣻⠃⠀⠛⣷⡌⠄⠠⣤⡐⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡸⠟⠋⢜⣿⣿⢟⣿⣿⣻⡟⢽⣷⣾⣷⣷⢾⣿⣿⣽⠁⢂⢰⡀⠈⠙⢦⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡤⠠⣯⠟⢃⡈⢉⣃⠘⣄⡈⢙⢷⡟⠻⣦⠙⠻⠈⣳⣧⡀⠑⠀⠀⠀⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠄⠠⠔⠉⠁⢻⡉⠂⠿⡞⣻⢶⣟⢰⣻⣦⠀⠀⠙⢾⡓⠒⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⢂⡀⠀⠈⠇⠄⢸⡿⠏⠻⠇⠀⠀⠸⠖⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠃⠐⠃⠀⠀⠴⣫⠄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠐⠁⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠂⠈⠀⠠⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠀⠀⢀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⢤⣠⠤⠒⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⢇⠀⠀⠀⣀⣀⡤⠤⠴⠖⠒⠚⠉⠉⠁⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠴⠒⠛⠒⠒⠲⠶⠦⠤⢤⣀⣀⡀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣀⢦⡰⠢⠞⠢⠚⠤⠚⡤⠢⣠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠢⠎⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠋⠳⡴⢄⢀
⠀⠀⠀⠀⠀⣀⡰⢦⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠣⣀⡄
⠀⠀⠀⠀⡴⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡄
⠀⠀⠀⡸⠁⠀⠀⢀⢀⡠⠤⠀⠐⠂⠀⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠰⡀
⢀⠤⠠⣇⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⡠⠴⠒⠂⠙⠉⠁⠀⠀⠀⠀⠀⠉⠑⠲⣄⠀⢱⡤⠤⡄
⠈⠰⠲⡌⡞⠀⠀⠐⠦⣤⡤⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡠⢤⠤⣤⠶⠂⠀⠩⢱⠎⣠⢤⠘
⠀⢸⢠⣼⠃⠀⠀⠐⢿⣞⠛⠒⢻⣷⡤⣀⣠⡎⠀⠀⠀⢊⠀⠀⠀⢀⣆⠤⢞⢋⡿⣷⠚⢻⡄⠀⠀⠀⠈⣞⡁⢸
⡀⡄⣿⡟⠀⠀⠀⠀⠀⠈⠓⠮⣶⣯⠥⠤⠟⡿⠂⠀⠀⠀⠀⠀⠀⠙⣏⠒⠮⢿⡟⠓⠊⠁⠀⠀⠀⠀⠀⢿⢹⢸
⡇⡇⢹⠁⠀⠀⠀⠀⠀⠀⠐⠚⠫⠤⠤⣄⡞⢀⡤⠖⠊⠉⠙⠒⠢⢄⠈⣆⡜⠓⠚⠓⠂⠀⠀⠀⠀⠀⠀⠘⡞
⢣⠈⠃⠀⠀⢳⢄⠀⠀⠀⠀⠀⠀⠀⡠⣿⡮⠕⠒⠉⠉⠉⠉⠉⠁⠒⠳⢽⢧⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢃⠜⢠
⠀⢹⠀⠀⠀⠈⡆⠁⣀⠀⠀⠀⡤⢪⠏⠙⢄⣀⣄⡀⠀⠀⠀⢀⣀⡀⡠⠎⠳⡝⠢⣀⠀⠀⢀⡴⡛⠀⠀⠀⠘⠖⠁
⠀⠀⢦⡀⠀⠀⠰⡄⠈⠲⡤⠊⢀⠎⠀⠀⠀⠀⢶⡁⠀⠀⠀⠸⡄⠈⠁⠀⠀⠹⡀⠈⠳⠖⠁⢀⠇⠀⠀⠀⢰⠇
⠀⠀⠈⢮⡀⠀⠀⠱⡀⠐⠀⢀⡟⠀⠀⠀⠀⠀⠀⠈⠢⠤⠒⠉⠀⠀⠀⠀⠀⠀⢡⠀⠄⠀⢀⡜⠀⠀⠠⢂⡟
⠀⠀⠀⠀⠙⢤⡈⡨⠷⡄⢂⡌⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣠⠾⠣⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢣⠀⠔⢘⣼⢱⠀⠀⣠⡔⠊⢹⢈⠣⠜⠦⠁⠌⠤⢙⠀⢐⡤⢄⡀⢸⢚⣇⠁⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠢⠘⢸⢠⡏⠠⡘⢄⢸⠀⠀⠀⠀⠀⠀⠀⠸⡠⢋⡂⠀⠙⠪⢈⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠈⡀⠘⠃⢣⠀⡇⠀⡛⠀⠀⠤⠤⠤⠄⠀⣘⠁⠸⡛⠀⡌⠀⢸⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠃⠀⠀⠈⡆⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳⠂⣰⠀⠀⠸⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠸⣼⡄⢦⠀⠀⠀⠀⠀⠀⠀⠰⣄⡎⢠⠃⠀⠀⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢹⡟⠈⢦⢓⡔⣣⢓⡜⢡⠁⢹⣰⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠳⢊⣉⣀⣀⣀⢀⡈⠩⠦⠜⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠆⠀⠀⠀⠀⢀⡀⢀⣠⠆⢀⣠⡴⠀⠀⠀⠀⠀⠀⠐⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢹⠁⠀⠀⠀⠂⢠⣿⣠⣿⣿⣾⣿⠏⣀⣤⣤⡤⠄⠀⠀⠀⠀⢳⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣄⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⠀⢀⡞⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠈⠙⢛⣽⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⢁⠀⠀⠙⣿⣿⠹⣿⣿⡋⠁⠀⠀⠈⢭⡙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠀⠁⡀⢀⠀⠈⠻⠄⠈⠻⢧⠀⠀⢀⠠⢀⠙⠆⢿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⣿⠇⠀⢘⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡁⠈⢸⣟⠛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⠟⢡⣿⠀⠀⠊⠛⣧⠙⠶⣄⣠⡄⣀⣄⣠⠾⢁⡿⠉⠁⠀⠀⣿⡌⠻⣿⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠸⣿⣿⠀⠀⠀⠀⠈⠛⣻⣏⣉⣙⣃⣈⣹⣛⠛⠀⠀⠀⠀⢀⣿⣿⠃⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣀⠀⠀⣀⢹⣿⡀⠀⣀⡤⣶⠶⢦⣭⠍⣿⣿⢭⡥⠶⣶⡶⣤⣀⠀⢸⣿⡏⣀⠀⠀⣀⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠈⢻⣿⢧⠀⣿⣇⣸⡀⣸⡁⠀⢸⠁⠀⡇⠀⣹⣀⣾⣿⠀⠿⣿⡟⠁⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣀⠀⠀⢀⠈⠻⡄⠃⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡏⠘⣰⠟⠁⡀⠀⠀⢀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢰⡏⠀⢰⣷⡀⠈⢧⢻⣧⣽⣻⣿⣿⣿⣿⣋⠤⡟⣸⠟⠀⢰⣿⡆⠀⢹⡆⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣿⣄⣼⣷⣀⣼⣿⣧⡀⠀⠙⠢⢼⣇⣀⣿⣀⣈⣧⠤⠛⠁⡠⢀⣾⣿⣧⣀⣾⣧⣠⣿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⡿⠿⠛⣿⣿⣷⣄⠂⢑⠲⢤⣭⣭⣭⡥⠴⠒⡠⠊⣠⣿⣿⣿⠛⠿⢿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣴⠁⠉⠙⢿⣷⣦⣄⣀⣀⡀⣀⣀⣀⣤⣴⣿⠟⠋⠁⠈⣆⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣷⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆⡀⣸⣿⠻⠿⢿⣿⣿⡿⠿⠟⣿⣇⠀⣀⣤⠞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢏⣩⣍⣽⣿⣤⣤⣤⡤⠤⢤⣤⣴⣿⡏⢩⣍⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⣾⠟⠿⢦⡀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢀⡴⠿⠻⣷
⠀⠀⠙⢷⣦⡈⢻⣦⠀⠀⠀⠘⣿⠀⢰⡿⢷⡆⠀⠀⠀⠀⢰⡿⢿⡆⢀⣿⠁⠀⠀⠀⣴⡟⢁⣴⡾⠋
⠀⠀⠀⠀⠙⣷⠀⢻⡇⠀⠀⠀⢹⣧⠘⠻⠟⠃⠀⠀⠀⠀⠘⠻⠟⠃⣼⠏⠀⠀⠀⢸⡟⠀⣾⠋
⠀⠀⠀⠀⢀⣿⠀⣸⡇⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⢸⣇⠀⣿⡀
⠀⠀⠀⠀⣾⠏⢠⡿⠁⠀⠀⠀⣸⡏⢈⣿⣶⣄⣀⣀⣀⣀⣤⣶⣿⡅⢹⣇⠀⠀⠀⠈⢿⡄⠹⣷
⠀⠀⠀⢰⣿⠀⣿⡇⠀⠀⠀⢠⡿⠀⣾⠇⣼⡏⢹⡿⢿⡏⢹⣧⠸⣷⠀⢿⡄⠀⠀⠀⢸⣿⠀⣿⡆
⠀⠀⠀⠀⢿⣆⠘⢷⣤⣤⣴⡿⠃⣸⡟⢀⣿⠁⣼⡇⢸⣧⠈⣿⡀⢻⣇⠘⠿⣦⣀⣤⡾⠃⣰⡿
⠀⠀⠀⠀⠈⠻⣷⣤⣈⣉⣁⣤⡾⠟⢀⣾⠃⣰⡿⠀⠀⢿⣆⠘⣷⡀⠻⢷⣤⣀⣈⣁⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢀⣴⡿⠁⣴⡟⠁⠀⠀⠈⢻⣦⠈⢿⣦⡀⠈⠉⠉⠉⠉
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣴⡾⠟⢉⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⣄⡉⠻⢷⣦⣤⣀⣀⣀⣀⣀⣀
⠀⠀⣾⡛⠛⠋⠉⠉⢉⣉⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣉⡉⠉⠉⠙⠛⢛⣷
⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣖⡩⠔⠊⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣿⡯⠵⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣒⣒⠂⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⣤⣤⣤⣤⣤⣐⣀⡠⢄⡀⣓⠘⡧⠼⣆⢢⠈⠿⣿⣿⢿⣷⠧⣭⡉⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠈⠩⢥⣿⣿⣿⣯⣵⣊⣝⠉⠩⡤⢼⣶⡷⣞⢿⣯⡑⠦⠉⡑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠈⡿⠐⠩⠞⣡⣿⢿⣿⣶⠀⢧⣀⠟⠈⠈⢣⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠁⠁⠈⡘⢿⢿⡳⠂⣠⠁⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠴⠀⠀⣩⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠛⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⢐⡆⢄⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠛⠒⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⢀⡀⠄⠀⠀⠀⠁⠀⠀⠐⢦⡤⠾⠋⠀⠘⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠂⠀⠀⣼⠫⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⣠⠌⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠶⠤⠀⢄⣴⠋⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠈⢡⡄⠀⠀⠀⠀⠛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠀⠄⣼⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠿⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠈⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠲⣌⠁⡰⠛⠋⠉⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣦⣀⠀⠈⠀⠀⣼⠀⢄⡀⠀⠀⣀⠀⠐⢦⣶⡾⠋⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣧⡧⢾⣦⣼⣶⠿⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡆⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠀⠀⠀⠀⢺⣿⠓⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠂⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⣿⠏⠁⣤⠆⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣛⣿⡿⠀⠀⣾⠠⣤⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣤⣀⠀⠒⠀⠀⠔⠀⠀⠀⠳⠒⠶⠟⠷⢄⡀⣹⠀⠉⠀⠰⠄⠀⢀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⣽⣽⣿⣷⣿⣤⣽⣤⣴⣀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠁⠙⠛⠖⠺⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠒⡀⠐⠒⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣌⡙⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉⠉⠉⠁
⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣾⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠙⠲⠤⠤⣤⣤⣤⣤⣤⢶⡾⠿⠟⠿⢷⣶⣶⠿⠿⠻⠿⢷⣶⣤⣤⣤⣤⣤⠤⠤⠖⠋⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⢿⣦⡀⠀⠴⠛⠋⠁⡐⣉⠤⠤⠶⠖⣚⡻⠶⠿⠿⠶⢟⣓⠲⠶⠤⠤⡉⢂⠈⠙⠛⠦⠀⢀⣴⡿⢿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡦⠀⠀⠉⠂⠀⢀⣀⣾⢫⡵⠖⠚⠛⠉⢉⣩⣿⣷⣾⣿⣍⡉⠉⠛⠓⠢⢮⡝⢷⢀⡀⠀⠐⠉⠀⠀⠴⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⡀⢠⡄⠀⠀⠉⠀⣘⣡⠖⠊⠀⣴⡿⠛⠉⠉⠈⠁⠉⠉⠛⢿⣆⠀⠈⠲⣌⡂⠀⠉⠀⠀⢠⡄⠀⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠖⣲⣶⠖⠀⣠⣦⡿⠁⠀⠠⢚⣩⣴⣾⣧⣾⡄⢰⣷⣼⣷⣦⣌⡓⠄⠀⠈⢿⣴⣄⠀⠲⣶⣖⠲⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠀⣠⡾⢟⣛⠁⠀⠀⠚⣉⣉⣭⠄⠀⣈⠁⠈⣁⠀⠠⣍⣉⣉⠓⠄⠀⠈⣻⡿⢷⣄⠀⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠃⢠⡾⠁⠀⢾⣿⡇⢠⣄⢸⡟⢋⡔⢠⡾⠋⠀⠀⠙⢿⡄⢠⡉⢻⡇⣠⡄⢸⣿⡷⠀⠈⢷⣄⠘⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠚⢠⣿⠀⢰⡇⢀⣼⡇⠻⠿⠿⣇⠟⠀⠿⣾⣡⠀⠀⣌⣶⠿⠀⠻⣸⡿⠿⠟⢸⣧⡀⣸⡆⠀⣿⡄⠓⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠆⢀⣿⡟⢀⡿⠃⠾⣉⣽⡛⣾⠻⣦⡁⠀⠀⠟⠛⠀⠀⠛⠻⠀⠀⢈⣷⠟⣷⢛⣯⣉⠷⠘⢿⡀⢻⣿⡀⠰⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠏⠀⠀⣾⡟⠸⡇⠀⠘⢯⣀⠙⠉⠢⠴⢿⡦⠐⠍⠀⠀⠀⠀⠹⠂⣴⡿⠦⠔⠳⠋⣀⡽⠇⠀⢸⠇⢹⣷⠀⠀⠹⡻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠆⠈⣽⡿⠀⢿⣧⠀⠠⠤⢠⣄⡀⣀⡀⠙⢦⠦⡀⠀⠀⢀⠴⡶⠋⢀⣀⣀⣠⡄⠤⠄⠀⣼⡿⠀⢿⣧⠁⠸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⡆⠈⢠⡷⠀⠈⠙⣧⣀⠀⣼⠛⠛⠛⠲⢌⠒⠀⠀⠀⠀⠀⠀⠚⡡⠖⠛⠋⠛⣇⠀⣀⣼⠋⠁⠀⣾⡄⠁⢰⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⠾⣿⣶⡄⣄⠹⢿⣿⣿⣄⠰⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠆⣠⣿⣿⣿⠏⣠⢠⣶⣿⠃⢰⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡇⣠⠛⢹⠿⣿⣷⣮⣿⣿⡏⠳⣌⠓⠀⢀⣠⣄⠁⠀⣠⣄⡀⠀⠚⣡⠞⢹⣿⣿⣵⣾⣿⠿⡏⠙⣆⢸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢾⡏⠀⢃⠻⠻⣿⢿⣿⣦⡙⠳⣄⠀⢠⡌⠓⠚⢡⠄⠁⣠⠞⢋⣴⣿⣿⣿⠟⠗⡜⠀⢱⡶⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢢⡿⣧⣰⣙⠆⠻⣷⣭⡑⠀⠀⠀⠻⣄⣠⠟⠀⠀⠀⢈⣭⣾⠟⠐⣋⣆⣼⢾⡔⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠣⣵⣽⡿⣿⠲⣶⣤⣶⣿⣿⣦⣤⣶⠖⣿⣿⣯⣯⠜⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⡈⣦⠙⡟⠋⠉⠉⠉⢻⠏⣸⢁⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀⣄⠀⡀⠀⠀⢀⠀⣠⢀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠷⠌⣙⡛⠉⠉⢛⣋⠡⠾⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⠀⠀⠈⠁⠈⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠦⠴⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⢿⣦⠀⠀⠀⠀⢀⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠹⣷⣄⠀⣠⡿⠋⠀⢿⡆⠀⠀⠀⠀⣀⣠⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣀⣀⣿⡇⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠘⣿⣀⣤⣶⠿⠛⠋⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢀⣤⣦⣤⣴⡾⠟⠻⣷⣤⣤⣤⣤⣄⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠲⣶⣶⠿⠿⠿⠿⠿⠧⠀⣠⣶⢶⣶⠿⠋⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠉⢻⣧⠀⠀⠘⠛⠛⠿⠿⠶⣶⣶⡤
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣿⡇⠀⠀⢰⡆⠀⠀⠀⣿⡆⠀⠀⠀⣿⡀⠀⠀⠀⣿⡇⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡇⠀⠀⣼⡇⠀⠀⠀⣿⠇⠀⠀⠀⣿⠃⠀⠀⠀⣿⡇⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⣴⣿⠋⠁⠀⠀⠀⠀⢿⡇⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠈⠙⣿⣦⣄
⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⡀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣇⠀⠀⣠⣴⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡗⠀⠀⢿⣦⣤⣼⣿⣄⣀⣠⣿⣇⣀⣀⣠⣿⣷⣦⣴⡾⠏⢸⡿⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠉⢿⡟⠛⠋⠁⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⣀⣠⣤⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⡶⠶⠶⠿⠿⠿⠋⠀⣀⣀⡀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⢀⣼⣿⠿⠟⠛⠋⠉⠀⣿⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠻⠿⠿⠿⣷⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⢀⣠⣴⡾⠻⣷⡀⠀⠀⠀⠀⣴⣷⣦⣀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠟⠋⠁⠀⠀⠹⣷⡀⠀⢀⣾⠟⠀⠉⠻⢷⣦⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠹⣷⣠⣿⠋⠀⠀⠀⠀⠀⠉⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠁

⠑⠀⠈⢠⡀⣧⠛⠾⡓⠀⠀⠀⠀⠀⠀⢀⠸⡃⠀⢀⠀⡀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠐⣠⠄⠀⣠⠊
⢀⣶⣕⠌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠁⠀⠐⠁⠘⢆
⡉⠀⠁⢠⠶⠈⠀⠀⠀⡀⠆⠘⠖⠒⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠈
⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⠓⠀⠀⠀⢀⠐⠑⠆⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠐⠂⠁⢀⠀⣀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠀⠘⠉⠁⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠋
⠀⢀⡄⠀⠂⢘⠀⠀⠀⠀⠏⠈⠀⠀⠢⠀⠀⠀⠁⠐⠀⠀⠀⠸⠂⢈⠄⠋⠈⠁⡆⠓⡀⠀⢄⠀⠀⠀⠐
⡈⠀⠀⠀⠀⠠⠛⠈⠁⢐⠀⠊⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠥⠀⠀⠰
⠉⠁⠀⠀⢀⠤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠂⠠⡄⠀⠀⢨⠥
⠓⠠⠄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰


⣉⣉⣀⠤
⣉⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣉⣤⡴⠖⡊
⠉⣥⠶⢃⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⢿⣶
⣈⣴⣮⣅⣤⣴⣎⣠⣠⣄⣤⣄⣠⣇⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣀⣤⣤⣈⣤⣌⣵⣦⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢨⠀⠀⡔⢁⣇⡤⠒⠉⠀⡜⠀⠀⢀⣠⠤⠔⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⠩⣁⡌⣴⢸⠁⠀⣀⣀⠴⠔⠒⠉⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢕⡢⢄⣠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡸⢒⠏⣸⡄⡟⠒⠉⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡁⢾⠆⢻⡑⠢⠤⠄⣀⣀⣼⠶⠕⢲⠋⢠⢃⢱⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⢣⢴⣷⣦⠀⠀⣠⣾⠉⠉⡹⠃⠀⡞⢿⡏⠉⠉⠉⠉⠀⠉⠉⠉⠉⠐⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠛⢳⢞⡜⣙⣳⢟⡹⣰⣀⠏⠀⠀⡜⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠛⠁⠀⡴⡣⠊⠈⠁⠙⠉⠀⢸⡍⠳⠤⠺⢧⣰⣟⣠⣄⣀⣤⣀⠜⠂⠚⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⠀⠀⠙⠳⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⢉⡤⠍⠛⠒⠤⢄⣀⣀⠤⠤⠐⠒⠒⠠⠤⢄⣉⠉⠛⠭⣕⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⢔⠢⡜⢱⠀⠙⠢⡀⠀⠙⢢⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢠⣚⢎⠈⡧⡭⣃⠀⠈⢦⠀⠀⢱⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠤⠏⠣⠛⣦⣟⣁⠣⢢⠀⢇⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡠⢊⡥⣌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢖⢊⠔⠉⠉⠑⡄⠮⡄⣱⠸⡄⢸⠀⢠⠃⢸
⠀⠀⠀⠀⠀⣠⠊⠀⢏⣵⡜⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠎⠀⣎⡸⡀⠀⠀⠀⡼⠰⣛⣁⡏⠁⢸⠋⠁⢀⠎
⠀⠀⠀⢀⡾⠤⢃⡐⠠⠤⠤⠀⠀⠈⠑⠒⠒⠒⠒⠉⠀⠀⠀⠰⠥⡌⢒⠒⣊⣐⡏⠁⠈⠀⠀⡼⠤⠒⠁
⠀⠀⠀⢸⣀⣀⣀⣈⣉⣒⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣑⣃⡀⣀⢀⣀⣀⠜⠁
⠀⠀⠀⠀⠓⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⠚⠂⠠⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠂⠀⠈⠀⠀⠳⡀⠀⠀⠀⠂⠘⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠥⠀⠐⠀⢀⣀⠓⠮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠖⠃⣀⠀⠐⠂⠀⠅
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⡵⠖⠂⠀⠀⡙⢾⠀⠀⠀⠀⣠⡶⠂⠀⠒⠶⢅⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠂⠀⠀⠀⠈⠢⡈⢍⠁⠀⠐⢀⢄⡠⠂⠀⠀⠀⠀⢂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠧⢀⠀⠀⠀⠀⠙⢇⠀⠀⠉⠀⠞⠁⠀⠀⠀⣀⠀⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢹⣷⣄⠀⠀⡀⠀⠀⢀⠀⡀⠀⢀⣴⣾⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠟⢰⡿⢧⣠⠿⢄⣼⣈⡹⢧⣤⠿⣿⠈⠏⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠤⠤⠀⠞⠉⠁⠀⠀⠀⠀⠉⠙⠇⠠⠤⠤⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠠⣄⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⠀⡠⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⢈⣦⣡⣠⠏⡀⠄⠀⠀⠀⠀⠀⣀⢻⣥⢀⣞⡠⠔⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢀⠈⢉⠋⠈⠀⠀⠀⠀⠠⣀⠀⠠⣉⣣⠈⢀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢌⡝⠁⠀⠀⢀⡀⠀⠀⠀⠀⣀⡒⠀⠉⠙⣏⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⡀⠰⠇⢀⣀⢀⣀⡀⠀⠿⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢟⠋⠙⠛⠋⠉⠙⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠢⢄⣀⣀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⢤⣤⣤⠤⠞⠁⠀⠀⣀⣀⣠⡞⠍⠉⠲⠋⠀⠂⠘⡶⠚⠉⠲⡀⠀⠀⠀⠉⠁⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢠⠏⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣧⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢰⡇⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢇
⠀⠀⠀⣠⣴⡋⠀⠀⠀⠀⠸⡄⠀⠀⢸⡄⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⣿⡆⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠈⣉⠓⠶⣄⠀⠀⠳⣀⣀⣼⣇⠀⠀⣸⣇⡀⠀⢠⣇⠀⠀⣰⠃⡇⣠⠤⠶⠤⠤⢷
⠀⠀⠀⢀⠄⠀⠀⠀⠊⢳⡀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⣹⢋⠉⠉⠉⠁⢀⡿⠁
⠀⠀⠀⠘⠄⠀⠀⠀⠀⢘⡧⠄⠀⣀⣀⣀⡀⠀⠀⠀⠀⣋⣤⣴⣒⡒⠒⠊⡇
⠀⠀⠀⠀⠐⠂⠐⠂⠀⣼⡤⠖⠋⠁⠀⠀⠙⣆⠀⠀⢠⠞⠁⠀⠀⠉⠢⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠄⠀⠠⠔⠀⠀⠀⠘⡆⢠⠏⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠻⣷⡄⠀⠀⠹⣿⣆⠀⠀⠻⣷⡀⠹⡄⠀⠀⠘⣇⠀⠀⣿⡆⠈⠀⠀⢿⠀⠀⠀⡿⠀⢠⠇⠀⠀⢀⠏⠀⢀⡾⠁⠀⠀⠀⢠⠞⠀⠀⢠⠞⢁⣴
⢤⡈⠻⣦⡀⠀⠈⠻⣧⡀⠀⠙⢷⡀⠹⡀⠀⠀⢹⡄⠀⢸⣇⠀⠀⠀⢸⠀⠀⢰⡇⠀⠘⠀⠀⠀⠈⠀⢀⡞⠁⠀⠀⠀⠐⠁⠀⠀⠐⢁⣴⡿⣋
⠢⠙⠢⣈⠻⣦⡀⠀⠈⢿⣄⠀⠈⢳⡀⠁⠀⠀⠀⢧⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸⠃⠀⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠊⠁
⠀⠀⠀⠈⠑⠈⠻⣆⠀⠀⠙⢦⡀⠀⠱⠀⠀⠀⠀⠈⠄⠀⢸⡄⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⣦⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⣠⠖
⠀⠙⠳⢦⡀⠀⠀⠀⠀⠈⠑⢄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⢀⣠
⢦⣀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣀⣀⣤⣶⣶⣿⣿⣷⣦⣀⢀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⡡
⢤⣌⡛⠢⣄⡀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⡤⠖⠋
⠀⠀⠉⠁⠀⠈⠐⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠚⠉⠀⠀⠀⠀⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠂⠉⠁
⠻⠶⠶⠦⢤⣤⣄⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠉⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⡀
⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠙
⠶⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠻⢿⣿⣿⣿⡿⠗⠸⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢈⡛⠿⠿⠿⢛⣡⣶⣬⣛⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣦⠀⠀⠀⠀⠀⠠⢀⡉⠑⠲⠤
⠀⣀⡠⠄⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤
⣋⣥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠈⠙⠶
⠀⠀⠀⢀⠄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠱⣤⡀
⣀⡤⠚⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⣰⠀⠀⡏⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⢿⣦⡀⠢⣀
⠉⠀⡠⠂⢀⣴⠟⠁⠀⠀⠀⢀⠎⠀⣰⠃⠀⠀⠀⢠⡇⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢻⡄⠀⠀⠀⠈⢢⡀⠙⣿⣦⡈⠳⣄
⡴⠊⢀⣴⠟⠁⠀⠀⠀⠀⣰⠋⠀⣰⠃⠀⠀⠀⠀⣼⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⡆⠀⠀⠀⢸⡆⠀⠀⠈⣿⡄⠀⠀⠀⠀⠱⡄⠈⠻⣿⣦⡈
⣠⣶⡿⠃⠀⠀⠀⠀⠀⡼⠃⠀⣼⠃⠀⠀⠀⠀⢰⠇⠀⠀⣿⠃⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠘⢦⠀⠘⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠀⠰⣦⡐⠂⢶⠀⣶⠤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠉⠀⠀⠁⠀⡀⠀⠉⠐⠡⠄⠛⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠐⠠⠿⠃⠁⢸⡆⠀⠀⠀⠀⠠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⠀⠂⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⡀⣄⠀⠀⠀⠘⢢⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠛⠂⠒⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠒⠀⠄⡀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀⡀⣤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⡀⣀⡀⠀⢀⠆⠀⡀⠀⠤⠰⠆⠀⠄⠀⣀⡀⠀⠀⠀⠈⠄⠀⠀⡀⠖⠉⠀⠙⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠⠌⠀⠈⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⢂⠀⠀⣀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠂⠂⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢡⠬⠀⠀⠀⠀⣤⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇⠘⠆⠰⠀⠃⢠⠄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠙⢤⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⡄⠐⠤⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠐⠀⠠⡀⢱⠀⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⣹⠀⠄⠀⠀⠉⠱⠦⢄⣅⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠲⣸⠀⠀⡁⠀⠀⢀⢀⠐⠢⠄⠀⠀⠀⣧⢄⠀⠀⠀⡠⠣⠀⠀⠀⠆⡼⠀⠀⠁⠀⠀⠀⢀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⣀⠈⠠⡄⠈⠳⡀⠰⠀⠀⠀⠈⠀⠀⢠⠀⠂⠚⠈⢡⡤⠶⠹⣀⠀⠄⡻⣠⠔⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠐⡀⠀⠐⠦⢍⠀⠈⢱⠠⣐⡄⠤⠀⠀⠀⠀⠀⠀⠀⠶⢀⡾⠰⠏⠀⠲⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠚⠆⠓⠒⡤⠀⠠⠀⠀⠀⠀⠀⠠⠎⣠⣴⢩⠀⣀⠀⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠆⠀⠀⠀⠀⢈⠇⠀⠀⠀⠀⠠⠔⠛⠋⠁⠂⠒⠦⡄⠀⠈⡀⣘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠣⣀⡄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠴⡀⠀⠄⠀⠀⢀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⡛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠬⠒⠃⠀⠀⠀⠈⠀⠄⠀⠀⠂⠀⠀⢀⠀⠘⠦⠀⠀⡀⠀⠅⠀⠁⢧⡀⢹⠋⢻⣬⢺⣣⣼⣼⡿⠉⡽⠁⡨⢋⠁⠃⠼⢁⠞⠹⠃⢈⡼⠋
⣅⡀⠀⠀⠀⠀⠀⠈⠀⠈⢢⣠⣤⣄⣤⣀⡀⠀⠠⠀⠀⠀⠀⣠⠖⠾⣇⡀⢱⢋⡷⠻⠋⠀⠈⣷⡼⢁⣀⠀⡈⠀⠀⢀⡀⠀⣀⠐⠁⠀⠈⠀⠀⢀⡴
⢤⢉⠁⠤⣀⠀⠀⠀⠀⠀⣾⠉⠀⡀⢀⣉⣻⡖⡦⠥⡀⣀⣴⠧⠁⠀⠙⠉⢨⠮⣤⡀⠀⠀⠰⠟⠗⠛⠉⠙⠉⣹⠋⠁⠹⠏⢉⠷⡄⠀⡀⠔⠊⠉⠉
⣓⠀⠀⡀⠀⠐⠀⠀⠀⠀⢻⡍⠀⠈⠯⡉⠁⠀⠁⠀⠀⠉⠉⠗⠲⠄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠉⠀⠀⠸⠀⠀⣈⡗⠂⠁⠀⠀⠠⠐
⠦⣠⠀⠁⠉⠒⠀⠀⠀⠀⠻⣆⡄⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⣲⢤⠤⢤⠀⠀⠀⠀⢲⡋⠀⠀⠀⠀⠠⣄⠼
⠂⠀⠅⠀⠂⠀⠀⡀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⢀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⡀⠀⠠⠆⠀⠈⠹⠀⢀⣸⠃⠀⠀⣀⣠⠇⠀⠀⠀⠀⠀⠀⢀
⠙⠃⠀⠀⠀⠀⢀⢀⣠⣔⡓⢺⡋⣷⠒⠀⠀⠀⠀⠀⠀⠈⠁⠉⠠⠄⠀⠀⠱⠀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⠞⠁⣴⣃⣠⡀⠀⠀⠀⠀⠄
⠚⠛⠓⠲⠆⠀⢰⣏⡀⠉⠀⠀⠃⠈⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠹⠻⡟⢲⡄⣀
⣛⠖⠃⠀⠀⢀⣿⠉⠀⣀⠴⠚⠁⠀⠙⠂⠀⠀⠀⠒⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠀⠀⠀⠀⠁⠀⠀⢀⡼⢋⣑
⠻⠍⣠⣶⠋⠙⠇⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶
⠉⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠀⠀⠀⢠⣤⣄
⠤⠼⢿⣓⣤⣰⠯⠀⠸⣧⣀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠂⠐⠀⠀⠀⠀⠀⠐⠂⢠⠀⢀⡀⠀⠀⠀⠀⠁⠀⠉⠉
⠃⠉⠀⠀⠀⠉⢳⣀⠀⠀⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠘⠆⠀⠀⠁⠢⠄⠀⠀⠀⠀⠀⠀⠀⠈⣠⣸⠇⠀⣠⡼⠟⠂⠒⠠⠤
⣴⡖⠊⠀⠀⠀⠀⠈⠛⠷⠖⢡⡀⠀⠀⠀⠀⠺⢤⠴⠦⠄⠀⠀⠀⠀⠘⡄⠀⡀⠀⠀⠀⠠⠀⠀⠠⠨⠆⠀⢀⠀⠀⠀⣹⡝⠀⢙⠀⠀⢄⣀⠀⠘⠑
⡩⠀⠀⠀⠀⠀⣀⠄⠈⠁⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⢆⣀⡔⠀⠀⠰⣤⠤⡀⠙⠆⠀⠀⢀⣄⠀⠀⡀⠘⠒⠋⢹⣷⣛⠁⢠⣴⡏⠀⠀⠀⠀⠉⠀⠈
⠂⠀⢀⡀⠄⠊⠀⠀⣀⡤⡾⠊⠳⠴⠤⠴⠻⣄⣀⣰⣯⠉⣅⡀⡐⢀⣽⡆⠁⠘⠲⢤⢶⡞⠉⠓⠚⠻⣤⣤⡽⠛⠀⠉⠙⠋⠁⠀⠀⠀⠀⠑⠢⢈⡐
⣪⠕⠉⠀⠀⣀⠔⠛⡉⠈⠁⠀⠀⠀⠀⢀⠀⠀⠁⡰⠃⢠⠻⣤⠶⣞⠈⡻⣆⠀⡀⣀⡾⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠁⠀⢂⡠⠋⣡⡴⡋⢀⣄⣠⠊⡰⢂⠔⣉⡴⠀⣰⠃⣰⣦⣢⠘⣶⡔⣾⣧⣨⣃⢹⡉⡀⠀⡀⠀⠪⠀⠰⣄⠀⠐⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠐⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⣶⣶⡆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣠⣼⣿⣿⣧⣄⣀⣠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣠⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣷⣿⣿⡿⢛⣩⣵⣶⣶⣾⣿⣶⣶⣮⣭⣛⠿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⣡⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣮⡙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣾⣿⡿⣡⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣮⠻⣿⣿⣦⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⣱⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⢹⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡜⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠠⣴⣾⣿⣿⣷⡹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠟⢿⣿⣷⡝⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠿⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡙⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⡿⣿⣿⣷⣬⣙⡻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠙⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⠏⠉⠙⢻⣿⣿⡟⠋⠉⠙⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠉⠁

⠀⠀⢠⣶⣟⣻⢷⣄⠀⠀⠀⠀⠀⠀⢠⣠⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣆⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⢀⣤⡄⠀⠀⠀⠀⠀⢰⣾⡦⢨⣾⣷⡤
⠀⠀⣿⡱⢽⣷⡥⣿⡆⠀⠀⠀⠀⠈⠋⠀⠉⠫⣿⡤⠀⠀⠀⠀⠀⠀⢨⣿⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⣡⣄⡘⠻⠃⠀⠀⠀⠀⠀⠀⠉⣁⣀⠹⠚⠉
⠀⠀⠙⢿⣟⣹⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠐⢿⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣯⣽⠗⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠆
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⡄⠄⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠶⢀⡀
⠀⠀⣼⣿⣿⡱⢎⣷⠀⠀⠀⠀⠀⠀⠌⣀⡀⢈⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣔⡿⣿⣮⠁⠀⠀⠀⠀⠀⣠⡾⢟⣛⡛⢷⣄
⠀⠀⢽⣿⡹⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣽⠟⠶⠀⠀⠀⠀⠀⠻⣧⣘⣛⣃⣴⡟
⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖⠐⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁


⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣼⠛⣉⢿⡀⠀⠀⠀⠀⠀⠀⢠⠖⠊⢉⠲⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣟⣿⣿⠀⠀⠀⠀⠀⢠⡏⠉⠙⠆⠴⠏⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘⣦
⠀⠀⠀⣾⠠⠠⢸⠇⠀⠀⠀⠀⠀⠀⢿⠈⠐⡀⡒⡯⠀⠀⠀⠀⠀⢠⡿⣯⣟⣏⣿⠇⠀⠀⠀⠀⠀⠈⠳⢦⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡿
⠀⠀⠀⠙⠷⠶⠟⠁⠀⠀⠀⠀⠀⠀⠉⠳⠥⠤⠚⠁⠀⠀⠀⠀⠀⠈⠻⣷⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠈⠻⡤



⠀⠀⠀⠀⣶⠐⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣏⣶⣦⡀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡸⠂
⠀⠀⢀⣴⡟⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣞⢙⣏⣸⣷⠀⠀⠀⠀⠀⢸⡀⠂
⠀⠀⣾⡏⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣻⣏⣹⣿⠇⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⡀
⠀⠀⠘⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣤⣤⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠈⠉⠻⣷⣄⣴⠟⠋⠉⠙⠛⠛⠿⣶⡟⠋⠙⠛⠛⠻⢷⣆
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠻⣷⡾⠛⠛⠿⠿⢷⣤⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⣿⡇⢦⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⣿⣇⣤⣝⣷⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣿⡀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⣰⡟⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⣷⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃⠀⠀⠀⢠⣤⣤⣿⠇⠀⠀⠀⠠⣤⣼⡿⠀⠀⠀⠀⠰⠶⣿⡏
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⠷⢶⣦⣤⣤⡴⠶⠾⠛⢋⣿⣯⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⠶⠿⠛⠉⠹⣷⣄⣀⣀⣀⣤⣿⣧⠀⠀⠀⠀⢀⣾⢷⣤⣤⣤⣤⣴⠿⠃⣿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⢹⣷⣶⣶⡶⠟⠃⠀⠀⠉⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠄⠀⠀⠀⠀⠀⠀⠉⠛⠛⢛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⠛⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢀⣀⣤⣶⣾⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠴⠾⠛⠛⠛⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⠿⠿⠛⠻⣧⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣸⣿⣿⣤⣴⣶⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢿⡋⠉⣿⣿⠛⠻⠿⢿⣿⣿⡟⣿⣿⠿⠿⣿⠛⠻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⣼⠃⠹⣇⠀⠀⠙⠦⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠈⠛⢷⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⡞⠛⠛⠋⠉⠉⠉⠉⣻⠛⠛⠳⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢹⣷⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⡏⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⣀⣿⣀⣀⣠⣤⣤⣤⣤⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠷⠶⠶⠾⠟⠚⠛⠛⣿⠛⠛⠛⠛⠛⠛⠉⠉⠉⢹⡏⠉⠉⠁⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⡀⠀⢸⣿⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣀⣀⣀⣀⣀⣀⣀⣸⣧⣀⣀⣀⣀⡀⠀⠀⣿⠇
⠀⠀⠀⠀⢰⡅⠀⢸⣧⠀⠈⢻⣄⢀⣀⣤⣤⡶⠶⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠶⠦⣤⡀
⠀⠀⠀⠀⠘⣇⠀⠀⠻⢷⣤⡶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠙⢷⣄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠿⠽⠿⠿⠿⠿⣿⠿⣹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡤
⠀⠀⠀⠘⠢⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠮⣀
⠀⠀⠀⠩⠨⠀⠀⠀⠀⣸⠆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⠀⠀⡐⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠁⠒⢦⡴⠋⣴⠏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠂⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠦⢀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠑⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠐⠖⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⠑⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⡶⠀⢀
⠀⠀⠀⠀⡀⠀⣀⠜⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⡄
⠀⠀⠀⠀⠨⡉⠁⣠⠄⠀⡠⠀⣰⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠢⠁⠈⠂⠚⠉⠋⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣾⣷⣶
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣽⣿⣽⣿⣭⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⡉⠁⠀⠊⠉⠉⠋⠉⢙⡳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠞⡁⠀⢀⠢⠔⠶⠆⠤⣤⣤⡀⠠⠀⠀⠀⠠⠄⠙⠲⡀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⡴⠁⢤⠴⠊⠁⡀⣀⣀⡠⢄⡀⠉⠛⢷⣄⡐⠃⢀⡠⠒⢂⡙⢦⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⡜⠠⡴⢁⣤⠞⠋⠁⠀⠉⠙⠺⣍⠷⣆⡀⠹⣷⣔⠁⢰⡤⠐⠓⠈⢳⡀⠀⠀⠀⢸⠘⡀⠀⠀⠄⣿
⠀⠀⠀⢰⠃⢸⠁⣼⠃⢀⠖⠀⠀⠙⢷⣄⠈⢏⠻⣷⡀⠈⣿⣟⠁⠀⠀⠀⣴⠄⢱⠀⠀⠀⢸⡄⠉⠀⠀⠀⠛⢧⡀
⠀⠀⠀⢸⡑⣿⠀⣿⠀⢾⠐⡁⢢⢑⠀⣿⡇⡈⣷⣿⡇⠀⣿⡏⠴⠋⢶⡀⠀⢤⡘⣇⠀⢀⠎⠀⠀⠐⣀⠀⠀⢸⣷⡀
⠀⠀⠀⠸⡇⢹⡈⣿⣆⠸⢧⣍⣡⡞⢀⣿⠁⠀⣿⣿⡇⠀⣿⠇⠚⣁⣴⡄⠀⠀⣀⣻⣴⠃⠀⣴⣷⠾⠇⠀⢠⣾⡟⠉
⠀⠀⠀⠀⢧⡐⢷⡸⣿⣦⣀⡈⠀⣠⣾⠏⠀⣸⡿⠙⠀⢰⣯⣧⣴⣿⣭⣶⣶⣾⠿⠿⠃⢀⣾⣿⠟⣭⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠈⢳⣦⠙⢦⣙⠙⠿⠟⠋⢃⣠⣶⣻⠁⠀⣠⡿⣼⣷⣿⠿⣿⣿⣤⣴⣤⡶⢾⣩⣿⡟⣳⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠙⢷⣀⠈⠙⢛⠻⠛⠿⠿⠋⠁⣠⡾⢋⣼⡿⠟⠤⠀⠀⠄⠄⠤⠤⣖⣩⣪⣿⢨⣿⣿⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣬⡤⣤⡤⢶⣚⣫⣽⠾⠛⣁⢄⣄⣤⣶⣷⣴⣪⡿⣾⣽⣿⠟⢣⣿⣿⡱⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣚⣉⣭⣿⣽⣿⣿⣿⠿⠿⢶⣵⣷⣿⠟⣿⣿⣎⢨⡉⡻⠛⠛⠁⣠⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣛⣫⣭⣼⣶⣯⣉⣈⣵⣿⣿⣿⣾⡒⠘⣈⣡⣿⣿⣿⣷⣥⣤⣤⡶⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣭⣍⣹⣩⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠣⠸⣄⠀⠀⠀⠀⢀⠟⡄⠀⠀⠀⠀⡠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡏⠣⡀⠀⠀⡜⠀⢱⠀⠀⢀⠜⡇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠢⠔⠁⠤⠀⠓⠒⠁⠀⡇⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠙⢍⠒⠤⠀⠀⠀⠼⠐⠒⠐⠂⠀⠀⠀⠀⡀⠁⠀⠗⠐⠫⠞⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠠⠀⠠⠐⠀⢚⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠠⠤⠀⡀⢀⣀⣀⢀⡴⠁⠀⠀⠀⠀⣤⡇⠀⠀⠀⢐⠀⠀⠀⢘⠀⠀⠀⢨⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⠳⣄⢀⣀⣀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠈⢀⠷⠰⠀⠀⢸⠂⠀⠒⢌⠔⠂⠀⢺⠀⠀⠀⡰⣇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⢸⣀⠈⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠠⠐⠛⡁⠀⠀⠀⠀⠀⠀⠐⣄⠄⠉⠀⠀⠀⠀⠀⠀⡙⠗⠦⠀⠀⠠⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢰⠇⠀⠀⠀⠀⠀⢼⠉⠂⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠊⢹⠄⠀⠀⠀⠀⠀⢧⠀⣀⣀⡀
⠀⠀⠀⠒⠀⠁⠉⠀⠀⠀⠀⠉⠣⢄⡀⠀⠀⠀⠈⠦⣄⠀⠀⠀⠀⡈⠀⠀⠀⠈⠀⠀⠀⠀⢀⡤⠎⠀⠀⠀⠀⣀⠤⠋⠉⠀⠀⠀⠈⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠖⠒⠀⠀⡤⠀⠒⠂⠂⡀⠒⢀⠀⠀⠊⠀⠒⠊⠀⠀⠀⢖⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⢃⠴⠊⠱⡀⠐⠀⣠⢄⡈⠒⠀⡜⠁⠀⠉⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⠁⠀⠀⠀⡇⡠⠊⠀⠀⠙⣄⠀⡇⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣰⠀⠛⠀⠀⠀⠀⠀⠘⢦⡇⠀⠰⠀⢦⡈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⠤⠤⢤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⢶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⠛⠛⢉⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⠉⠛⠛⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢤⣀⡞⠀⠀⠊⠁⠈⠘⠛⠲⠤⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠠⠤⠚⠛⠁⠀⠉⠂⠀⠀⢳⣀⡤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⢻⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡟⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⢸⡇⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢀⣤⠤⢤⣴⠖⣲⡦⣄⡀⠀⢠⣿⣿⣿⣯⡄⠀⣀⣠⢤⣖⠲⣶⡤⠤⣤⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⣦⠠⡹⣄⣿⣿⠞⠁⢀⣹⣿⡿⢿⣿⣏⡀⠈⠳⣿⣿⣠⢏⠀⣴⠀⠀⠀⠀⠀⢻⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⡆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠀⣷⡇
⠀⠀⠀⠀⠶⣤⡤⣤⣀⣸⣿⣼⠀⢤⡀⠀⠀⠀⠀⠀⢀⠴⢊⠁⠀⢈⣹⣿⣿⣿⣿⣿⣿⣏⡁⠀⠉⡙⠢⡀⠀⠀⠀⠀⠀⢀⡤⠀⣧⣿⡃⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠛⢷⣶⣼⣿⣿⠀⠀⠙⡦⠀⠀⢀⡖⢁⠖⠁⠀⠈⠉⠘⠿⢿⣿⣿⡿⠿⠁⠉⠁⠀⠈⠳⡈⢲⡀⠀⠀⢰⠎⠀⠀⣿⣿⣃⣉⣭⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⣏⢧⠀⠀⠀⠀⡠⠛⠀⠀⠠⠀⠀⠀⢂⣒⣠⣿⣿⣿⣿⡄⣐⡐⠀⠀⠂⠀⠀⠀⡛⢄⠀⠀⠀⠀⡼⣹⢿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠈⠊⠃⠀⢀⡞⠁⡜⠀⠀⠠⠔⠚⣉⣡⣶⣾⣿⡿⢿⣿⣷⣦⣌⣉⠓⠢⠄⠀⠀⢣⠈⢳⡀⠀⠘⠐⠁⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡄⢠⠀⠀⠀⠀⢼⢁⠎⠀⣠⣾⣿⣿⣿⣿⡷⠶⠶⠶⠶⣿⣿⣿⣿⣿⣷⣄⠀⢱⡈⡧⠀⠀⠀⠀⡄⢀⡏⣿
⠀⠀⣀⣀⡤⠴⠖⠒⠛⣿⡄⢳⠀⠀⠀⠀⠀⢸⠀⠀⣼⣿⠿⢿⣧⢯⡤⠶⡦⢠⡄⢴⠶⢤⡽⣼⡿⠿⣿⣧⠀⠀⡇⠀⠀⠀⠀⠀⡞⢠⣿⠙⠒⠲⠦⢤⣄⣀
⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣧⢀⠣⠀⠀⠀⠀⢾⡀⠀⣿⣷⠀⠘⡇⢀⣇⣀⣄⣠⣄⣠⣀⣨⡀⢸⡇⠀⣾⣿⠀⢀⣗⠀⠀⠀⠀⠘⡀⣼⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⢆⠀⠀⠀⠀⢹⡇⠀⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⣿⠀⢸⡏⠀⠀⠀⠀⣰⣱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣎⠆⠀⠀⠀⢸⡇⠀⣿⣿⠿⣾⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣷⡿⣿⣿⠀⢸⡇⠀⠀⠀⠰⣱⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣻⣿⣿⣆⠀⠀⠀⠈⣇⠀⢻⣿⡄⢽⣷⡀⣻⣿⣿⣿⣿⣿⣿⣟⢀⣾⡿⠀⣼⡟⠀⣸⠁⠀⠀⠀⣰⣿⣿⡿⢦⣀
⠀⠀⠀⠀⢀⣴⣺⣥⣶⣿⣿⣿⣿⣿⣆⠀⡄⠂⢹⡀⢸⣿⣄⢀⣿⣿⣿⠿⠿⢿⡿⠿⠿⢿⣿⣿⣇⣀⣿⡇⢀⡏⠐⢀⠀⣰⣿⣿⣿⣷⣦⣌⡓⢤⡀
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢻⣿⣿⣦⡸⡌⠈⣧⠘⣿⣷⣄⢸⣿⠁⠀⠀⢛⠀⠀⠀⠀⣿⣟⢀⣼⣿⠃⣼⠁⢡⢇⣼⣿⣿⡟⠉⠉⠉⠙⠛⠛⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣽⡄⢸⡀⢿⣿⣆⠀⣿⠀⠀⠀⠸⠀⠀⠀⢀⣿⡇⢠⣿⡟⢀⡏⢠⢯⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣨⣿⣿⣿⣿⡌⣧⡞⣿⣷⣀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡿⣀⣼⣿⢣⣾⢡⣿⣿⣿⣿⣅⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⣾⡿⠛⠉⠙⠻⣿⣼⣿⣹⣿⣿⣦⢿⣄⠀⠀⠀⠀⢀⡾⣧⣾⣿⣏⣿⣧⣿⠿⠋⠉⠛⠿⣷⣜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷⣻⣿⡇⢸⣿⣿⣶⣶⣾⣿⡇⢸⣿⣟⣾⣿⣿⠁⠀⠀⠀⠀⠀⠈⠙⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⣿⣿⣿⣿⣤⣼⣙⠛⠛⠛⠋⣉⣧⣠⣿⣿⣿⣟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠋⠻⣿⣿⣿⣿⣭⣛⠛⠛⣋⣭⣿⣿⣿⣿⠟⢿⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠈⠉⠛⠛⠛⠻⣿⣿⠟⠛⠛⠛⠉⠁⠀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠁⠀⢀⡀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡠⠂⠀⣀⣀⡈⠑⢄⠀⠙⢆
⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⢠⠋⢔⣀⠀⢳⠈⢣⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠠⠄⡀⠀⣀⢀⣮⢠⠀⠀⡀⠀⢸⠈⡈⠉⠕⡸⠀⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣐⠀⠀⠀⠀⠀⠀⠀⠈⠦⡑⠄⠃⠀⠀⠑⠄⣀⡉⣀⠔⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣶⣦⣤⡤⣀⣀⣀⣈⠑⣺⣦⣀⠀⠀⠀⠀⠀⠀⣠⠞⠓⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⢿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠿⠿⠟⠛⠛⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⣄⠀⢀⡀
⠀⠀⠀⢠⡄⠀⠀⠀⠀⢲⣿⣿⡖⠀⠀⠈⠻⠷⣿⣿⣤⣤⣤⠄⠀⠀⢴⣦⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠰⣶⣿⣷⣦⣄⠀⠀⠀⠀⠀⢂⣀⣀⠄
⠀⠀⢀⣿⣗⠀⠀⠀⠀⢈⣿⣿⣏⠁⠀⠀⠀⠸⣿⠇⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣿⣿⡂
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠁
⠀⠀⠻⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⢠⣤⣀⡀⠀⠀⡄⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠒⠒⢴⣬⣔⡀⠀⠀⠀⠀⢠⣶⣶⣦⡀⠀⠀⠀⠀⢀⠀⠠⠀⣀
⠀⠀⠈⠻⣿⣿⣷⣶⣼⣿⣷⣶⣶⣿⣿⡿⠋⠀⠀⠠⣠⡾⠋⠁⢈⣿⠀⠀⠀⠀⠀⠀⢸⣿⣟⠛⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠰⠷⠂⠿⠾⠾⠿⣅
⠀⢠⡀⠀⢈⡙⠛⠉⣿⣿⡟⡏⠛⠋⠉⠀⠀⢠⣀⣼⣿⣁⡄⠀⠀⡀⠀⢠⠀⠀⠐⠒⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠸⣿⡄⢾⣷⠀⠀⢸⣿⡇⠁⠀⠀⠀⠀⡤⣤⣿⣿⡯⣤⠀⠑⢄⠘⣀⣇⣠⠔⠊⠁
⠀⠀⢻⣿⣼⣿⡇⠀⠈⠿⠃⠀⠀⠀⣴⢷⣿⣿⣿⣿⢥⡀⠀⠐⢾⣿⣿⣿⡇⣛⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⣶⣶⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠲⢿⣿⣏⣽⡀⠀⠀⠄⠒⠙⠉⠝⠣⠀⠀⠀⠀⠀⠀⠀⠋⠿⣿⣿⡿⠀⠀⣶⠀⠀⠀⠀⠀⠉⠁⠁
⠰⣿⢿⣿⣿⡷⠶⠶⠶⠶⠶⠶⠦⠀⠀⠀⠀⠙⠶⠾⠁⠀⠀⠀⠀⠈⠀⠘⠂⠀⠈⢉⣄⠀⠀⢀⡀⠀⠘⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣴⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⢻⣇⠀⣾⠇⠀⠀⠀⠀⠀⢺⣿⡗⠀⠀⠀⠐⠾⢿⣿⣿⡿⠂
⠀⠀⣼⣿⢹⣿⡇⠀⠀⠀⠀⠙⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣦⡄⠀⠈⢿⢸⠏⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠈⢀
⠀⣼⡿⠃⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠋⠀⠀⠀⢺⢂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣀⣴⣤⣤⣅
⠀⠈⠀⠀⠀⠀⠀⢀⣤⣶⠿⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⣀⠀⠀⠀⣾⠘⣧⡀⢀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⣀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⡀⠀⣠⣤⣾⣿⣿⣷⣿⣶⣏⠀⠀⣸⡟⠀⢻⣧⠈⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠐⠀⠝⡙⠫⠅
⠀⠀⠀⠀⠛⠛⠋⠀⣀⣤⣤⡀⢠⣀⣴⣿⣿⡿⠛⠇⠀⠈⢙⣛⡿⡿⠟⠿⠻⡇⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⠵⠛⠁
⢀⡀⠀⠀⠀⣀⣀⣰⣿⣿⣿⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⣤⣶⡤
⠸⣷⡀⠀⠈⠛⠛⠟⠛⠛⠛⠙⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣿⣿⣶⣶⠄⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣤⣀⠀⠀⠀⣿⣿⣿
⠀⠙⠻⠆⠀⠀⠤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠙⠛⠛⠛⠿⠛⠛⠛⠫⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣉⠁⠀⠀⠀⢻⣿⡏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⣤⣤⣬⣭⣥⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠛⢻⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡀⢀⠀⢈⠀⠀⠀⢀⠊⠀⡠⠀⢄⠀⠠⠀⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠁⠀⠀⠀⡠⠁⢠⡶⠶⠦⣤⠊⠀⠀⠀⠙⠀⠀⠀⠙⣣⣤⣄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢼⠈⢑⣀⢀⢀⠜⠀⠀⢀⠌⠠⠆⠀⡀⠀⢴⠄⠠⠀⡤⠀⢠⠉⠙⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠡⡀⡱⣏⠀⠀⢱⠀⠀⠀⡼⠀⠀⠀⠀⠈⠢⠤⠐⠉⠂⠤⠐⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠖⠎⠈⢈⠟⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠠⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠉⠂⡤⠂⠒⡄⠀⠀⠀⠀⢁⠀⠘⠟⠛⠶⠶⠾⣶⣶⠆⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢠⡀⠀⣣⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠔⠁⠀⠀⠐⡈⠀⠀⢠⠂⠑⠤⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡠⠊⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠄⠀⠀⠈⠐⠤⡀⠀⠀⢀⡀⢀⠄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⡀⠀⠔⠊⠀⠀⠀⢀
⠀⡀⠀⠂⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢨⠇⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⡌⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⢁⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠠⣳⡀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠀⠀⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⠀⠀⠀⢀⣀⠀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠎⡀⠀⠈⢋⢲⠞⠋⠈⠉⠋⠉⠛⠁⠈⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⡞⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣢⠖⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡉⠡⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⠀⠀⣨⠀⠀⠀⠀⠀⠠⠀⢤⡠⠞⡀⠀⡀⢀⠖⠓⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⣗⠀⠐⠠⠀⠀⠀⠀⢠⣾⡶⣿⣆⠀⠀⠀⠀⠀⠀⠈⢹⣿⣧⣦⣤⡄⠀⠀⠀⠀⠀⠩⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠃⠀⠠⠀⣀⡀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠠⠘⠋⠻⠟⠁⡀⣀⠀⣄⣀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢇⠆⢀⣤⠂⠀⠀⢀⡇⠂⠀⠀⠀⠀⠀⡄⠀⢀⣀⡵⣅⣩⠴⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠓⠋⠛⠉⢷⠀⠀⠀⠀⠀⢸⠛⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣄⡴⠊⣿⡁⠀⠀⠀⠀⣿⠦⢤⡤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⠀⠀⠐⡂⢹⡤⢄⡀⠤⣀⣿⢐⠤⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠆⢀⠀⣀⣄⠁⠐⠀⠀⣀⣀⣀⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⡏⠈⠛⠦⢾⡶⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⢧⡀⠀⢀⣠⠊⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠐⠒⢴⡉⠉⠋⡇⠀⣀⢀⣿⡁⠈⠉⠡⠀⠀⠀⠳⢦⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⢀⠐⠋⠁⢀⠀⠊⠙⠞⢄⠁⠀⠁⠀⠁⠀⠀⠀⠈⠁⢀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡃⠆⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠰⢎
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠛⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠐⠀⠀⠉⠑⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠛⠃⠉⠀⠀⠀⠀⠁⠀⠐⠐⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⠉⠁⠈⠉⠙⠓⠊⠉⠛⠋

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⡖⠒⠄⠀⠀⠀⠀⠀⠀⣀⣤⣴⣲⣖⡆
⠀⠀⠀⠀⠔⠌⠀⢀⠘⢀⠀⠀⠀⠀⡀⢠⠀⠄⣸⣤⠔⠘⢆⠀⠀⠀⢟⣉⡀⠀⣠⣹⠟⠙⡀
⠀⠀⠀⠀⠀⠘⠀⢀⠀⢠⠀⠀⠀⠀⠇⠐⠷⣦⠿⠷⡶⠧⡌⠣⠀⠀⢇⡀⠤⢀⡤⢰⣿⠞⡁
⠀⠀⠀⠀⠀⠠⢦⣼⡤⠶⠶⠃⠀⠀⠸⠀⠀⠐⠄⠈⠀⠀⠈⠰⠀⠀⠈⡀⠐⣦⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠚⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠱⡄⠀⠈⠉⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠊⠁⠀⣀⣀⡉⠳⣄
⠀⠀⠀⠀⠀⠀⡀⣪⠔⠛⠉⠉⠉⠙⠲⢬⡢⡀⠀⠀⠀⠀⠁⣰⠃⠀⢀⡜⠁⠀⠈⢢⠀⠳⠀⠠
⠀⠀⠀⠀⠀⡌⡾⠁⡀⠐⠈⠀⡀⡀⠀⢀⡙⢮⣦⠀⠀⢰⠛⠛⠀⠘⠛⢢⠀⠀⠀⢠⢠⠀⡇
⠀⠀⠀⠀⢠⢸⠁⠈⠀⢐⡾⠛⠛⠷⣆⡀⢱⣎⠁⠀⡆⠈⡄⢰⡃⡖⠢⠆⠀⠀⡠⢂⠈⢠⠀⠸
⠀⠀⠀⠀⠸⡘⡀⠀⠀⢠⡀⠀⠀⠀⡸⠟⠛⠛⢻⠀⠈⠀⠁⠀⠙⠁⠀⣶⣦⠥⠒⠁⠀⠌⢀⠆
⠀⠀⠀⠀⠀⠃⠁⠀⠀⠾⠇⠀⠀⠀⠢⣠⠐⠒⡛⠀⠀⠀⣀⣀⣄⡀⠀⠹⠂⠐⡄⠠⣮⣴⠞
⠀⠀⠀⠀⠀⠀⠳⣕⢦⣀⠀⠀⠀⠀⠀⢈⡧⠊⠀⣠⡶⠛⠉⠁⠀⣀⣤⣤⣄⣠⣷⣦⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢭⡖⣂⡤⢤⡞⠁⠀⠀⠁⠀⠀⠀⠐⠉⠁⠀⠙⠛⡟⠻⡋⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠈⢹⣿⠮⡵⢄⠀⠀⣠⠆⡜⠰⠤⠒⣾⢀⡛⠧⠼⠓⠤⣘⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢡⡀⠀⠘⣿⠀⠙⠉⣴⠋⠀⠀⡇⠀⠀⠀⢸⢸⡟⠁⠳⣄⣠⠾⣧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⠹⠃⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀⣴⠿⠃⠀⠀⠉⠁⠀⢻⠠
⠀⠀⠀⠀⠀⠀⠀⠘⡅⠇⢡⡄⠀⠀⠈⣷⡄⠀⠀⠀⠀⠊⠀⠀⠀⠈⢣⡤⠒⠒⠉⣸⠀⢀⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡈⡤⠀⠀⢠⠟⠳⡄⠀⠀⢈⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⢃⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⡀⢿⠶⠒⠛⢦⡀⠀⠀⠀⠀⠀⣰⣯⣬⠽⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠘⢆⠀⠀⠀⠉⠲⢤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠬⠐⣂⣐⠠⠤⠒⠒⢉⣭⣬⣍⣑⣂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠛⠛⠛⠛⠻⣿⣿⣿⣿⠍


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣶⣋⣀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⣌⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⢀⡀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠀⠀⠀⠀⠉⠉⠈⠀⠈⠉⠁⠀⠀⠒⠠⠤⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡳⣿⠖⠀⠀⠀⠀⠀⡤⣤⣤⣄⠐⠙⠓⠦⠴⠖⣒⠐⠀⣄⡤⢠⡀⠀⢀⠀⠀⢀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⠥⠙⢓⠤⠀⠈⠀⠘⣣⡏⠘⠁⠠⠤⠄⠲⠖⣠⠬⠴⠦⠐⠛⠛⠋⢲⡾⠿⠂⠁⠈⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡀⠀⠀⠀⠈⡛⠂⡆⢤⠀⠀⠀⠈⠂⠀⠀⢠⢀⡄⠀⠭⠬⢤⣴⣶⠆⢀⡃⠀⠀⡰⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠙⢲⣤⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠊⠁⢀⠐⢒⢤⣀⣲⠆⠀⣀⣀⠀⣰⣟⠁⠀⣆⡔⠂
⠀⠀⠀⠀⠀⢻⣷⣆⡀⠀⢀⡐⢸⠀⠀⣠⣿⣿⣿⣇⡀⠀⣰⠶⣦⣤⢤⢤⣬⡁⠀⢬⠉⠈⠉⠁⠀⠀⠐⢛⣻⣿⣾⣽⣿⣿⡄⢰⣷⣖⠒⣒⣒⡢⠄
⠐⢿⠓⠒⠒⠾⠿⣿⣥⠶⠋⠀⠈⠀⠞⣿⣿⣿⣿⡇⠍⢀⡿⣿⣿⣈⠛⢦⢧⠀⠀⢻⢇⠀⠀⠀⠀⢤⣰⣾⣿⣿⣿⢿⣿⣿⠃⠀⠈⠻⢦⠾⣏⡀⠀⢀⢲⣬⡦
⠀⢨⣿⡶⠶⠤⠩⣭⡬⣄⡀⡀⠀⠀⠀⠀⡼⣿⡿⣿⢣⡞⡠⢋⣿⣿⡷⣆⡀⠀⡆⢰⣾⠀⠀⠐⠈⣰⠿⡏⠏⢸⡏⢞⡿⢋⠀⠄⠀⠠⣛⣴⣿⠉⠉⢤⣼⣯⠄
⠀⠀⠈⢿⣶⡶⠒⢂⣤⡈⡋⠀⠀⠐⠐⣪⡴⠳⠔⣟⣿⣿⢁⣟⠛⠅⠀⠀⠙⣦⠃⠘⢉⠀⣠⣾⠜⠛⠀⠁⠜⠈⠉⣼⣾⣆⣀⠀⣠⣤⣼⣟⠿⡥⠤⠀⢠⡼⠉
⠀⠀⠀⠀⠹⠿⡤⠸⣿⣯⣍⠀⠈⠁⠀⠹⣷⣤⢿⣿⣿⣏⡉⢉⡉⠍⠙⣓⡀⢤⣄⠐⣡⣄⣂⠁⠀⢀⠀⠠⣶⡦⢄⠙⣶⣻⠀⠈⠁⠴⠖⠽⠛⢿⡀⠴⠊
⠀⠀⠀⠀⠸⠀⡧⠄⠃⠹⢿⣂⠀⠀⡶⠋⠺⣿⣏⠉⠓⣶⣥⠀⠀⠀⠤⠉⣗⣠⢂⡀⢹⣤⣐⣖⠀⠈⠴⢆⣀⣄⣾⢟⠘⠻⠿⠀⠀⣠⡖⠀⠦⢸⡇
⠀⠀⠀⠀⠠⢀⣇⠀⠀⠀⠀⠉⡟⠁⠀⠀⠀⠉⠙⢀⡀⠈⠻⣿⡄⢀⣈⣾⣿⣿⣿⠇⢸⣿⡻⣏⠉⠀⠀⠠⢋⣤⠗⠁⠀⠀⠈⡎⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠛⠛⠀⠀⡏⠈⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠐⠀⠲⣄⣄⡹⣿⠟⣋⡥⠴⣿⣤⣮⣿⡇⠀⠈⡟⢁⡴⠋⠁⠀⠀⠀⠀⠀⡷⠀⠀⠀⠐⠛⠺⡧⠈⠅⠂
⠀⠀⠀⣀⣠⡐⠇⣤⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠳⡶⠶⠶⠶⢶⠶⠶⠾⠿⢃⡀⠤⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⢀⡀⠁⠤⡄
⠀⠀⠀⠉⠋⠁⠀⠐⠒⠁⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠶⣄⠐⣛⣓⠄⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⢤⡶⠃⠀⢚⣉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠆⠀⢂⣉⡱⠀⠀⠀⠀⠀⠀⠬⡾⠃⠀⣂⣉⡉⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣥⡄⠺⣿⡀⠀⠀⠀⠀⢀⣛⠗⢠⣬⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠀⣹⡇⠀⠀⠀⠀⢸⣏⠀⠿⠿⠁
⠀⠀⠀⠀⢠⣤⣶⣶⣆⣶⣶⣶⣶⢤⣤⣤⡀⠀⠀⠀⠀⠀⠐⣶⣆⣿⠁⢠⣤⣴⡄⠈⣟⣰⣷⠂⠀⠀⠀⠀⠀⢀⣤⣤⠄⣶⣶⣶⣶⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⠛⠛⠻⢿⣦⣄⠀⠀⠀⢻⡿⣿⣧⣼⣏⣹⣧⣴⣿⢿⡟⠀⠀⠀⣠⣴⣿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⣀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠛⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠛⠿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢙⣻⣽⣿⣿⣯⣟⡋⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠛⠉⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠄⣤⣄⡀
⠀⠀⠀⠀⢀⡰⠿⣃⢠⠃⡔⠁⠈⠤⣉⢳
⠀⠀⢠⠔⠉⠃⢈⡩⠴⠀⠈⠀⠀⢠⠼⠛
⠀⠀⢈⣶⠶⢠⣿⣵⡏⠀⢠⡾⠻⢮⢆⣼⣄
⠀⢠⣿⢹⡄⠸⠻⢿⣇⣆⡈⠉⢇⠈⠘⠁
⠀⣾⡟⢎⢯⡀⠐⢤⡈⠱⠂⣠⠄⠀⠒
⠀⣏⠳⢄⠀⢁⡤⣀⠉⠋⢼⣇
⠀⣟⢶⣄⡳⠿⠀⠏⠀⠀⡈⢈⡽⢓⣢⣄
⠀⣿⡦⠿⣄⠀⠀⠀⠈⠀⡀⢠⡞⡽⢋⣵⡞⣦⡀
⠀⡝⢻⣆⡀⠀⠀⠀⠀⠈⠀⡾⡍⢰⠏⠏⣾⣁⢙⣦⡀
⢸⡷⠿⢦⣭⣁⠀⠀⠀⠀⠀⠐⠄⢸⡆⣦⢿⡉⠨⡾⢿⣷⣄
⠈⣿⠟⣿⡓⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⡜⠈⠐⠓⠠⠍⠤⠈⠀⠁⠂⠒⣀
⠀⢸⣿⣭⢝⡛⠂⠀⠐⠀⠀⠀⠈⠉⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠈⢧⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣤⡦⣤⣶⡲⡟⣆⠀⠢
⠀⠀⠈⢷⣒⠀⠑⠀⠀⡀⠀⠦⣀⠀⠀⠀⠀⠀⠡⠟⢠⣿⠿⡿⢁⣿⢳⣷⠄⡀⠀⠠⠀⠀⠉⡀⡀
⠀⠀⠀⠈⣷⣿⠇⠑⡄⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠏⠘⠁⣾⢁⢞⡼⣧⠈⠁⠐⠛⠓⠘⠳⠇
⠀⠀⠀⠀⠈⠳⣾⠀⡸⡄⠀⠀⣄⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⣘⠣⣥⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢇⡈⢀⠀⠀⢟⣛⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣈⣈⣽⣷⠟⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣀⡀⠈⢯⣍⡂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣭⠆⣠⠶⠷⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡑⡩⠞⠽⣦⠳⣦⠄⠀⠀⠀⠀⠀⠀⠀⠘⢾⡇⡼⢃⡴⠛⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⢦⠀⠀⠂⠘⠀⠀⠀⠡⠄⠐⠢⢄⠀⠘⡁⠷⣌⡤⢘⡙⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡚⠀⠀⠀⡌⠁⠀⠀⠀⠂⠀⠀⠁⠛⠂⠌⢢⣾⣡⡟⡉⠉⣿⣷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠔⠃⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡏⢸⣇⣠⡌⣛⣸⣿⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣼⡁⠙⠙⠣⣤⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠘⢷⡞⠷⣀⢩⣁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⠀⣿⣿⡀⠀⣹⣿⡇⠐⣦⣤⠄⣸⣆⡈⠿⡞⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠃⠉⠁⠀⠀⠈⠉⠛⠂⠴⢿⣟⢀⣿⣏⡠⠄⠰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠯⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢱⣤⡄⠀⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠾⡿⠁⠠⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⡤⢀⡰⠄⠴
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡉⣽⠟⠻⠿⠿⠛⠟⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠁⠀⠀⠀⠀⠈⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡎⠀⠀⠀⠀⠀⠀⢱⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⡇⠈⠀⠀⠀⠀⠁⢸⡀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣸⡇⡀⠀⠀⠀⠀⢠⢸⣇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⡇⠀⠀⠀⠀⠀⠀⢸⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠢⠄⠀⠀⠠⠔⢀⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⠠⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠻⢱⠀⠀⠀⠠⡌⠏⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠰⠄⠀⠀⠀⠀⠀⠀⡔⠁⢠⠂⠀⠀⠰⡄⠈⢂⠀⠀⠀⠀⠀⠀⠠⠆⠀⠰⠁
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠈⢠⠀⠀⡄⠁⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡀⠄⠊⠀⠈⠀⠀⠁⠀⠑⠠⢀⠀⠀⡠⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠁⠒⠀⠂⠀⣠⠖⠊⠑⠲⣄⠀⠐⠐⠒⠈⣠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠒⠖⠒⠋⠁⠀⠀⠀⠀⠈⠉⠒⠲⠒⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣿⣻⣛⣟⣟⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿