Haircutting Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣐⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠑⠶⢄⡀
⠀⠀⠀⠀⠀⢀⣬⠉⠀⠛⠛⠉⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣁⣲⣶
⠀⠀⠀⠀⠀⢉⣤⣤⣴⣵⣤⣴⣶⣶⢾⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⡭⢾⡟⠋⠉
⠀⠀⠀⠀⠀⠸⣇⠆⢹⡄⠀⠙⠉⠋⡀⠇⠀⠀⠀⠀⠀⠀⢀⡄⣬⠀⠀⠀⠀⠘⠁⡴
⠀⠀⠀⠀⠀⠀⠉⠒⢶⣧⣀⣀⢀⣽⣽⠆⠀⠀⠀⠀⠀⠀⣸⡧⢧⢠⠖⡆⢶⣼⢰⡗⡶⣰⡯⠓⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣟⢟⡿⣶⣿⡆⠀⠀⠀⠀⠀⠀⣿⢀⠃⠺⠼⠇⢀⠟⠘⠛⠁⠛⠀⠀⠘⠃
⠀⠀⠀⠀⠀⣶⠃⠀⠀⠸⣿⣿⣷⣿⡿⠁⡀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢾⡁⢠⡿⠀⢠⠤⠀⡤⢤
⠀⠀⢂⣀⣾⣿⡀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣷⣽⢿⡇⣿⡴⢸⡧⠋
⠀⠀⣿⣿⣿⣿⡇⠐⠠⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⣤⠀⠀⠀⠀⠀⠀⢎⡴⠏⠉⠈⠀⠀⠀⣿
⠀⠀⢻⣿⣿⣿⣷⠀⠀⠐⠻⠻⣿⣿⣿⣿⡏⠀⠐⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⣀⣼⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⢠⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⡟⠀⠐⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢿⣧⠀⣸⣿⣿⡄⠄⠀⠀⣠⣿⠃⠀⢀⣀⣠⣴⣦⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⣇⠁⣿⡿⣿⡭⢀⣠⡿⣁⡠⢲⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣻⣷⣾⣿⣿⡟⠀⠀⠈⣿⣿⡏⠙⠻⣷⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣾⣷⣶⣤⣄⣈⢿⣷⣿⣧⣿⡯⣤⣀⡀⠀⣿⣿⡇⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣠⣄⣤⣧⣀⣠⣄⣤⣤⣤⣤⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡉⣙⣿⣿⣿⣿⣿⣿⣭⣭⣽⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠙⡻⢫⣴⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣯⣻⣿⣿⣿⠟⠿⡿⣿⠿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣦⣤⣴⣇⠸⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⡿⣿⣿⣿⣿⣶⣶⣮⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠿⢻⣿⣿⣿⣿⣿⢏⣿⣿⣟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⠙⠟⠺⠾⠿⠇⠈⠉⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣍⣸⣿⣿⣿⣿⣿⡟⡃⠒⠛⠁⢶⠿⢿⣿⣯⣿⣿⣿⣿⣿⣿⣶⣶⡶⠶⠿⢲⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⠛⠛⠉⠀⣟⠀⣿⡄⠠⠠⠾⠽⠛⠛⠏⣿⡏⠩⡏⠁⠘⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠋
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣼⠀⢀⠀⠀⠀⠀⢠⡆⣿⣇⠀⠁⠀⢸⠀⠀⠀⣮⣀⠀⠀⣇⠀⠀⡀⠰
⠀⠀⠲⠦⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠇⠀⠙⠀⠙⠀⠀⠀⠀⠘⠁⠈⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⢀⣀⢤⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢴⠇⠀⢠⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠒⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢸⠀⠀⠀⠒⠐⠀⠀⠄⠄⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠤⠀⡸⠀⠀⠀⠀⠀⠀⢀⣄⢠⠆⠀⠀⠀⠀⠀⢘⠀⠀⠀⡚⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⡄⠀⡇⠀⠀⠀⠀⢀⠔⠙⠁⠞⠰⠧⣦⠀⠀⠀⡨⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⡠⠋⠀⡀⠀⠀⠀⠀⢂⡄⠀⢠⣧⣄⣀⢀⡇⠂⠀⠐⠀⢠⠀⡀
⠀⠀⢀⠀⠠⠄⠀⣀⣀⠀⠀⠀⢰⠀⢃⠀⠀⠀⡇⡰⠊⠉⠀⠈⠛⢤⢨⠃⡄⠘⠦⣅⡀⢦⡇
⠀⠀⠁⠀⠀⠜⠉⠀⠀⠙⡄⠀⢸⠀⢸⠀⠀⢐⠱⡁⠀⠀⠀⠀⠀⠀⡿⠄⢇⠲⠀⢨⢁⡇⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⢉
⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠸⠀⢸⠀⠀⠀⠻⠁⣖⣒⡒⣒⣢⠌⠋⠀⠀⠣⡑⣇⠞⠀⠀⠁⠁⠀⠀⠀⠀⢠⡃⠀⡄
⠀⠠⠀⣦⣼⡀⠀⠀⠀⢀⠆⠀⠀⡆⢸⠀⠀⠀⠀⠑⢮⠁⠀⠀⡠⠊⠀⠀⠀⣠⢉⢌⡢⢢⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⢀⠀⡁⢄⣀⡤⡎⠀⠀⠀⠿⣿⠤⡀⠀⢀⢀⡄⠁⠀⠈⠢⠄⡀⠀⢞⣁⣽⠓⢶⠋⠀⠈⠉⠛⠒⠤⡴⢉⡁⠈
⠀⠀⠀⠀⠀⠀⣼⠀⢀⠇⢀⣀⠀⢷⠠⣏⢾⣶⣁⠀⠀⠚⠀⠄⠀⠀⠀⢢⡴⠁⢘⠄⠛⠉⠓⠢⢄⡀⠀⠀⣆⢁⣀⠈⠁
⠒⠀⠀⢀⢓⣋⠠⠬⠥⠤⠤⠈⠕⠣⠀⠙⠳⡌⠛⡓⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡰⡮⠃⠸⡂
⠉⠉⠠⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠎⡀⡉⢷⠀⠀⠀⠀⠀⢠⡗⣤⢠⡜⠀⠀⠁⠀⠆⠀⠀⠀⢰⠀⠀⠀⠐⢹⡶
⠤⠤⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⠒⠒⠒⢺⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢄⠀⠀⢠⠊
⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢅⡀⠠⠤⢸⡀⠀⠈⠀⠀⡸⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠴⠢⠄⢀⡠⠃⠠
⠀⢀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢈⠀⠀⠀⡸⠀⠀⠁⠁⠄⢾⠀⠀⠐⠀⠀⠉⠒⠦⢀⡀⠀⠀⠀⠘⠁
⡀⠈⢦⠙⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠉⠀⠀⡀⠀⠀⠹⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⡁⠆
⠀⠀⠈⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠂⠀⠀⢀⠀⠀⠐⠂⠀⠀⡠⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠌⠀⠀⢀⠄⠀⠀⣿⠀⠀⠀⡦
⠰⠀⠀⠀⠈⠁⠱⠲⣀⠉⠉⢉⡼⢹⠀⠀⠀⠀⣨⠀⠀⠀⡻⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⡁⢀⡀⣘⠁⢀⣀⠤⢼⠀⠀⠀⡇⠇⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣈⡶⠉⢁⠸⠀⢹⡍⡀⠁⠀⡆⠀⠀⠀⠇⢸⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠒⠒⠚⠉⠙⢉⠁⡴⠊⠄⠸⠀⠆⠊⠀⠈⡂⠀⠃⠀⠀⢸⠂⠈⡇⠀⠀⢃⠀⠀⠀⠀⠀⠁⠀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⠀⠀⣀⡠⠞⣠⠊⠉⠉⠁⢻⠀⢀⡼⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠘⠛⠛⠛⠋⠀⠘⠧⠤⠤⠦⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣃⡄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢀⣀
⠀⠀⠀⠀⠀⠀⠨⢤⡿⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡠⠂⣠⡋⠁⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⡤⡔⠁⠀⠀⠀⠀⠊⡔⠁⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈⠉⠁⠀⠈⡇⠀⢀⣴⡮⠀⠀⠀⢠⢧⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠞⠑⠒⠋⠃⠇⠀⢀⡴⠀⠀⠑⢼⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠂⢸⠀⠊⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡠⠐⠚⠊⠀⠈⠀⠠⠔⠚⠛⠲⣤⡇⠀⠀⠀⠹⡀⡀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠙⣄⠀⡠⠂⠀⠀⢠⠘⡄⠀⠀⢹⡇⠀⠀⠀⠀⠱⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠀⡇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⠀⢰⠇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠠⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⢸⠀⢸⡀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢻⠓⠒⠒⠒⠲⠁⠀⠀⠀⠀⢁⢸⠀⠀⠱⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢓⣒⠒⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⢀⠣⣀⣸⣏⣉⣉⣲⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⡟⠉⠉⠉⠉⠈⠉⠢⣄⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠥⢀⣀⣀⡀⠀⠀⠀⡌⠙⠦⣄⡀⠀⢹⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢿⣿⡿⡟⠋⠉⠉⠉⠙⠢⣼⠀⠀⠀⠀⢸⠀⢂
⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⠀⢸⣤⠊⡀
⠀⠀⠀⠀⠀⠀⠀⡷⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢸⠉⠻⣮⡑⠤⡀
⠀⠀⠀⠀⠀⠀⣾⢧⡤⠯⣦⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⢼⠀⠀⢀⡇⢀⣀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣮⣤⣤⣾⠿⢷⠆
⠀⠀⠀⠀⠀⠀⠹⢒⠲⠤⣌⣉⣙⡆⠀⠀⠀⠀⢠⡊⠁⠀⠀⠀⠀⠈⠙⠉⠉⠀⠀⠀⠉⠑⢢⡈⠲⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠤⠤⣤⠤⠤⢤⡤⠤⠤⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣡⣾⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠼⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⡀⠠⠴⠂⠀⠸⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⢄⣀⡀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠐⡞⠀⠀⠀⠀⠀⠀⣀⠔⠁⡀⢀⠘⠸⣧⠄⠀⠀⠀⠀⡠⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⡟⠀⠘⣿⣇⣀⣀⣠⠞⠁⠈⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠒⣂⣠⣤⣤⣤⣀⠀⠀⢀⡇⠀⠀⠟⠙⢻⡩⢶⠀⠠⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣴⣿⣿⣿⣮⣽⣿⣿⣦⡴⠋⠀⠐⣤⠀⠀⠈⠀⠸⡄⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⣿⡀⠉⠉⠉⠉⠉⠀⣿⡇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢺⡟⢠⣄⡀⠀⢀⣀⠀⢻⡧⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠚⠋⠀⠈⠛⠂⢸⠃⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⢀⡀⠀⠀⠸⠊⢒⠘⠐⠀⠀⠀⠀⠀⠀⠀⠃⡼⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠤⠉⠤⠔⢀⣼⣧⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢱⢦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⠀⠀⣰⠊⣿⣿⣾⡸⣧⣀⠀⠙⠒⠐⠀⠀⠀⠿⢋⡤⠤⣤⣀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡻⠿⠿⠟⠁⢠⣿⣿⢿⣷⢻⣿⣻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣷⠘⣯⠐⣐⠀⢠⡟⡟⠈⢸⡟⠿⣿⡿⡀⠽⠛⢷⣦⣤⣀⠀⠒⠲⢤⣀⣠⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣾⠋⣿⠸⡆⠘⣆⢠⡴⠏⠐⠁⠀⡸⠃⠀⠈⠻⢧⣶⣺⠆⠙⠧⡟⠉⠓⠲⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⡇⠀⢸⠀⢳⡀⠈⠀⠀⠀⢠⠄⢀⠁⠘⠀⠀⠀⠀⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡧⠀⠈⠃⠀⠀⠀⢀⣠⠖⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢻⡄⠀⠤⣀⠀⠒⠋⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠀⠉⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠡⠀⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢶⣄⡀
⠀⠀⠀⠀⠠⠐⠚⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⡀⠂⢉⠀⠀⠀⡲⡀⠀⠀⠀⠀⠀⠠⠆⢀⡦⣤⣴⣴⢊⣶⣤⡾⡀⠀⠀⠀⠀⠀⡠⢦⡴⠤⠄⠀⠛⢿⣦⡀
⠀⠀⠀⣰⣦⣦⣀⣤⣠⣄⣄⣄⡀⠀⠀⠀⠀⠁⠖⠚⠉⠓⠛⠁⠈⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⣸⠉⡜⠂⠠⠤⠀⠡⡞⣙⠇
⠀⠀⠀⣡⡉⠈⠉⠉⠈⠉⠈⠀⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠀⢹⡕⠀⠀⠀⠀⠀⠀⠘⢫
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠒⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⣀
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⡄⣈⡆⠀⠀⠀⠀⠀⠀⠀⠉⣉⡁⠘⢉⢣⡀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠠⢾⡿⣷⡄
⠀⠀⠀⡆⡟⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⣌⠗⠋⠈⠛⠋⠛⠏⣹⠀⠀⠀⠀⠀⠀⣾⣷⠶⠛⠷⠗⠚⠷⣇⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⢐⠏⡁⢿⡇
⠀⠀⠀⢿⠂⠀⠀⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⢸⡿⠟⠓⠚⠛⠋⠁⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠷⢆⣠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠐⠈⠉⠀⠀⠀⢀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⣷⠆⠤⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⢀⢊⢠
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⠦
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠂⠀⠀⠐⠂⠡⠤⠀⠣⠐⠀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠕⢾⣿⣿⡿⣫⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢰⣤⣄⣤⣴⣾⣿⣿⣿⣿⢿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡍⠙⠛⠛⠋⣈⠉⠉⠉⠉⣹⢿⣿⡿⠃⠀⢻⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠉⠁⢰⠶⠶⣦⢶⣮⣿⣯⡟⠈⢻⣿⣦⣄⣈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠀⠀⠟⣿⣿⣿⣿⡇⠀⠀⣿⡿⢟⣉⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠸⡟⠛⠋⠍⠛⠲⠆⡤⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠶⠏⠀⠀⠀⢋⡀⣠⠀⠀⠐⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⡀⠄⠀⠀⠀⠀⠉⢀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢁⠀⡶⠀⣶⣶⠀⢺⠃⢰⡆⠀⣾⠀⢰⣴⡆⢰⡇⠀⣶⡆⠈⠀⢾⡀
⠀⠀⠀⠀⢀⣤⣤⣤⣄⠈⠉⢠⣭⣤⡅⠉⠀⣬⣤⣤⣍⠀⠉⣥⣬⣤⣥⡈⠉⣤⣥⣥⣤⡀⢩⣤⣤⣤⣀
⠀⠀⠀⠀⠀⣿⡿⠿⣿⡇⠀⢸⣿⣿⡇⠀⠀⣿⡟⠛⢿⣇⠀⢸⣿⠉⢻⣿⠀⣿⣿⢉⡉⠁⢸⣿⠋⠙⣿⡆
⠀⠀⠀⠀⠀⣿⣇⣀⣿⡇⠀⣸⣿⢹⣷⠀⠀⣿⣇⣀⣾⡟⠀⢸⣿⣿⣿⣿⡀⣿⣿⠿⠏⠀⢸⣿⣶⣾⡿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⣿⣏⣸⣿⡀⠀⣿⡿⢿⣿⠀⠀⢸⣿⢀⣸⣿⠇⣿⣿⣤⣤⡄⢸⣿⠀⢻⣷⡀
⠀⠀⠀⠀⠀⣿⡏⠉⢻⣿⢠⣿⡿⠿⣿⡇⠀⣿⡇⠈⣿⣧⠀⠼⠿⠿⠟⢋⣀⣽⣿⣿⣭⣥⣈⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣿⣇⣠⣾⡏⢸⣿⠁⠀⠿⠗⠀⠛⠁⡀⡈⢠⡴⠶⠶⣶⡄⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡿⠟⠀⠉⠉⣀⣀⣐⣚⣛⣛⣚⣙⣓⣘⣃⣤⠴⠛⠁⢻⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠈⠉⠀⣠⢶⠟⢿⣿⠉⣩⡍⠉⠉⣉⣉⠉⣴⠞⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⡞⠁⠸⢦⡾⠟⢠⣿⣶⣆⣾⢁⣿⣸⡏⣸⠟⢠⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⣀⣤⣶⡄⣸⡇⣿⢹⣯⡼⢃⣿⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣷⣿⠿⠛⣿⠃⣿⠘⠛⠀⣀⣀⣬⣥⣤⣤⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢛⠋⠁⠀⣰⠃⠈⠥⠖⠚⢫⣭⣀⣀⣀⣼⡟⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⢀⣠⠞⠁⠀⠀⠀⠛⣛⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠹⠟⠋⠁⠀⣀⡤⠶⣚⣫⡭⠆⠀⠠⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠐⡇
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣚⡻⠖⠀⣠⣶⠚⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⢻⣿⣿⣿⣿⣿⣿⡇⢀⣄⢡⠇
⠀⠀⠀⠀⠀⢰⡟⠀⠘⠉⢩⡟⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⡟⠛⠓⠀⣿⡧
⠀⠀⠀⠀⠀⠘⠿⣶⠶⠞⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⣿⣿⣿⣿⡯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⡧⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠐⣆⢲⠂⠀⠈⠫⣭⣙⣛⣛⣛⣛⠛⠛⠛⠛⠛⠻⠿⠿⠛⠟⣡⣿⣿⣿⠟⢻⠂
⠀⠀⠀⠀⠀⠀⣤⣨⢤⣁⡬⣄⣤⠀⠀⢀⣤⣯⣾⣤⣤⣟⣙⣉⣉⣉⣻⡗⠃⠶⢿⠿⠛⠋⠓⠒⠸⡅
⠀⠀⠀⠀⠀⠀⠿⢹⡿⠋⣿⠟⠼⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡾⠀⢸⢾⣰⣿⣸⣸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⡇⠀⢨⠶⡤⡬⢶⠦⣴
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⢰⠇⠀⠈⠂⠙⠃⠘⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠡⢓⣴⠊⠉⢛⣿⣁⠸⠃⠒⠒⠖⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⣤⣭⡉⠉⠻⠖⠀⠀⣿⠁⠉⠀⢶⣦⣐⣀⠀⣀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢶⣤⣄⣀⣐⣿⡇⠀⠀⠰⠿⠿⠿⢟⣘⣃⣠⣤⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⣒⣲⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠒⣒⣲⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣶⣒⡋⠉⠛⠿⣯⣟⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⢾⣵⣒⠋⠉⠛⠻⢿⣛⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠉⠙⠷⣦⡰⣌⢻⣷⡜⠞⡆⠀⠀⠀⠀⠀⢠⠏⠈⠛⠛⢿⣶⣤⡰⣄⠹⣷⡘⠺⡇
⠀⠀⠀⠀⠀⠀⠀⢞⣤⡄⠀⠀⢰⣟⣀⣙⣿⣧⢻⣿⠀⡗⠀⠀⠀⠀⠀⢘⣀⡀⠀⠀⢼⣟⣠⣙⣿⣧⢹⣷⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠀⠀⢀⣿⠟⣛⣻⣿⣾⣿⢠⡇⠀⠀⠀⠀⢀⠌⠙⠀⠀⠀⢸⣧⠶⣛⣻⣿⣼⣿⢀⡏
⠀⠀⠀⠀⠀⠠⢇⠀⠀⠀⠀⠀⠀⠁⠊⣵⠏⣿⣿⣿⡼⠀⠀⠀⠀⠀⣆⠀⠀⣀⣤⡐⣿⠁⠘⣡⠟⣿⣿⣿⣼⠁
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣘⢿⡿⠁⠀⠀⠀⠀⠀⢸⣭⣟⣻⣿⣷⣽⣷⡆⠸⣆⣙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⣀⡠⠎⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢻⢹⣿⡿⢟⡿⠁⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⣿⣤⣤⣤⣤⢤⡶⠟⠁⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣭⣁⣀⡴⠶⢾⣿⣿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣴⣫⣍⣀⣠⠶⠶⣿⣿⡿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠛⠛⠲⢶⣤⡀⢈⠻⣷⣌⠳⣳⡄⠀⠀⠀⠀⠀⣹⠛⠛⠓⠶⣦⣄⠀⡙⢿⣦⡙⢟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⣿⣿⣮⣷⡜⣿⣧⠀⣇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣸⣿⣷⣽⣦⢻⣿⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠉⠀⠀⠈⣿⣿⡿⣿⣿⣸⣿⠀⡇⠀⠀⠀⠀⠘⢸⡏⠀⠀⠀⢻⣿⣿⢿⣿⣇⣿⡇⢸
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠐⠛⠡⢞⣾⣿⣿⣿⣸⠃⠀⠀⠀⡠⠁⠀⠀⠀⠀⢀⡿⠋⠶⣱⣿⣿⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⣦⣀⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠀⠀⠀⡠⠀⠈⠙⢿⠛⠁⠀⠀⠀⠀⠀⠀⠋⠈⠉⠀⠀⠀⢀⠀⠀⠉⠻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⢶⠚⠉⠁⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠑⠠⠴⠲⡖⠋⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠖⠀⡀⠐⢂⡀⠉⠏⠉⠃⠀⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠒⠄⠀⠀⠀⡠⠄⠁⠘⠁⢀⠚⠡⠂⠊⠀⠀⢠⠘⠀⢡⠀⠀⠀⠙⠲⢄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠁⠁⠀⠀⠠⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠀⢲⣄
⠀⠀⣴⢷⣦⣤⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⢀⠌⠀⢣
⠀⠀⠃⢤⡌⠙⠻⢶⣼⣄⠀⠀⠀⠀⠀⠀⣠⠄⠃⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠘⡆
⠀⠀⠀⠀⡀⠉⠂⠄⣈⠛⠷⢶⣆⠀⠀⢻⡇⠄⠀⠀⢰⡀⠘⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠣⠄⡉⢧⠀⢸⣷⡈⠆⠀⠀⢃⠀⠘⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⡀⠀⠀⢹
⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⢇⢸⡞⡖⢸⡀⠀⠀⣇⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡠⠂⠉⠉⠙⠂⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡈⣇⢹⣼⠀⠄⠀⠸⢤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠱⡅⠨⢇⠀⠘⣜⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠶⣤⣸⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢨⠠⠤⠉⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⡎⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⢀⠀⠀⠀⠀⢨⠀⢀⠆⠀⠸⢡⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠳⠀⠀⠀⢘⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠖⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⣲⠶⠒⠀⠈⠁⠀⠀⠀⣀⠴⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⢀⡠⠖⠁⠒⠊⠁⠀⠀⠀⠀⠀⠀⠔⠈⠁⠀⡸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⠒⠒⠒⠀⠀⠒⠊
⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣀⠔⡊⠁⠀⠀⠐⠒⠒⠂⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⡏
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⡜
⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠐⢄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⣨⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⣯⣤⡀⠀⠀⠀⠀⢻⣓⡀⠀⡠⣤⣍⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣆
⠀⠀⠀⡟⠤⢤⡄⠀⠀⠀⣸⣿⠋⢠⡖⠂⢹⡆⠘⣿⣿⣿⣿⣿⠟⠀⠀⠈⣿⡆
⠀⠀⡼⠀⠈⠁⠀⠀⠀⠀⣿⣷⣄⣸⡆⠀⡼⠁⠀⣿⣿⣿⣿⢿⣦⡀⠀⠀⢻⣿
⠀⠸⢅⣀⡲⠀⠀⠀⠀⢠⣿⣿⣿⠋⢀⠞⠁⠀⢰⣿⣿⠟⠀⠀⠙⣿⣦⡀⠸⣿⣇
⠀⠀⢰⣿⣿⣶⡄⠀⠀⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠈⠻⣿⣦⣿⣿⡄
⠀⠀⠀⠁⢹⣿⣿⡆⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡧
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣿⣦⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠻⣦⣀⡶⠒⠶⣄
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀⡤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢻⡇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⢿⣆⠀⠀⠀⠻⢦⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⣿⣼⣛⡷⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠉⠁⠀⢘⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠆⠀⠀⠀⠀⠀⠘⣦⠀⣦⠄⠀⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠎⠀⠀⠀⠀⠀⠀⠖⠁⠀⠈⠀⠀⠘⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢠⡏⠀⠀⠀⠘⢦⡀⠀⠘⣗⡲⠄⠀⠀⠀⠰⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢹⡿⡍⡄⠀⠀⠀⠈⢳⣄⠀⢆⣀⣀⣤⣴⠼⠚⠁⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠊⠉⢁⣹⣷⣶⣶⣖⢢⣤⣿⣚⠲⣏⡟⣉⡠⠤⠒⢉⡴⠐⡩⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⢫⡦⣀⣤⣩⣍⡁⢍⣽⡂⠅⠈⢃⠀⠴⠋⠀⠀⠀⠘⠁⡠⠔⠋⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠶⠸⣷⣍⠀⠀⠀⢆⠀⠀⠀⠀⠀⡼⢁⣤⣖⣢⣤⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡀⠀⢒⡒⠀⠨⣗⠵⢻⠳⣄⠀⠈⠑⢤⡀⠀⡇⡇⠁⠈⠡⡀⠀⡠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢊⠸⠁⠈⢧⡀⠀⠀⠈⠑⠦⣉⣀⡀⠀⠀⠁⠘⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠑⠀⠀⠀⠀⠀⠈⠹⣤⡁⠀⠀⠙⢦⡀⠀⠀⠀⣾⠘⢝⠂⠀⢠⠞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠐⠋⠀⢀⣴⡥⢜⡝⠱⣀⠀⠀⠀⠙⠷⣤⣀⡟⠀⠀⠁⠀⢸⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⣶⣶⣾⣻⢯⣠⢾⠀⡴⡩⢋⠓⠶⣄⠀⠈⣿⣇⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠉⣙⣹⡷⢡⡿⢠⢁⠎⡸⡅⢡⢂⠔⡈⡻⣦⠀⠉⠙⠳⠶⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠌⠀⠌⡼⡕⡴⠫⢠⢁⠊⠌⠔⣱⣡⢋⢌⠜⡔⠘⣳⣤⠴⠒⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⣰⢫⢌⢂⢂⢎⢂⠄⢜⠌⡔⡀⡠⢁⢂⢎⠌⡴⢡⢃⢂⠎⠜⠄⠜⡴⣹⣧⡀⠼⠒⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠲⠲⠔⠆⠢⠲⠐⠖
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣄⡀⠤⣀⣀⠀⠀⡠
⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⡿⠿⠿⠿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣽⠛⠂
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠛⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠁⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣀⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣤⣧⣄⣖⣢⣤⣆⣄⡄
⠀⠀⠀⠀⠀⢿⣿⠿⠛⠻⠿⠛⠻⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⢸⣿⡿⠟⠻⢿⣿⠾⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡦

⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣤⣷⣤⣭⣷⣶⢤⣬⡖
⠀⠀⠀⠀⠀⢿⡿⠋⠙⠛⠛⠛⠛⠿⠀⠀⠀⠀⢿⣿⣿⡿⠿⢿⣿⠟⠛⠃⠀⠀⠀⠀⢸⠿⠋⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠐⠈⠀⠠⠊⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠜⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠴⠂⠀⠀⠀⠀⠐⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⠀⠁⠰⠚⠓⠀⠀⠀⠀⠠⠦⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡐⠡⢌⢀⡇⠀⠐⢁⣌⣂⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⢀⠛⠀⠀⠀⠐⠿⠿⠋⠀⢰⡰⣿⡿⡎⠀⠀⠀⠀⣶⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠤⡽⠀⢠⠁⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠎⠰⡟⠁⠑⡀⠀⠀⠀⠐⠂⢤⠆⠀⢀⠎⠀⠀⠀⠀⠶⣒⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⡄⠀⢀⠀⠀⠈⣦⢄⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⢫⡥⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⣀⢀⡀⠱⠀⠀⢢⣀⣀⡌⠀⠏⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⠋⠀⠀⢱⠀⢂⡔⠒⠦⠴⠚⡳⠚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⠁⠀⠀⠀⠀⡇⠀⠰⠀⠀⠀⠀⡇⠀⠀⠈⠢⡀⠀⢀⠀⠀⠀⠠⢂⡆⠀⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⢡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠌⢡⠟⢉⠠⠄⠱⢪⡤⡬⠉⠩⠃⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠈⡄⠀⠀⠂⠀⠂⢠⠀⠀⢀⠈⢠⡏⠀⠈⣓⣂⣱⠋⠠⠒⢻⣶⣦⢲⠄⠨⢝⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠈⢒⡥⠔⠋⠀⠀⠨⡷⣿⢃⡢⢣⠞⠀⡟⠇⠉⠳⣕⠄⡙⣧
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢃⠀⠀⢀⠠⠀⢀⣱⠀⠀⢀⣠⠦⠚⠉⠀⠀⢀⠔⠛⠃⢰⣧⣯⠖⣁⠀⠀⣿⢲⠀⠀⢨⡳⠤⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢄⠔⠁⡠⠂⠁⠈⠓⠚⠉⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢘⢩⡇⠀⣴⣶⢠⠃⢸⢀⣴⡶⠄⠀⣢⣄⣄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢢⠞⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⢂⠛⣧⠀⠛⠛⠀⢀⠌⠈⢛⠃⠄⠀⣿⣇⢺
⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⢀⠔⠁⠀⠀⠙⠦⢄⣀⡠⠴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣿⡀⠀⠀⠀⠁⢀⠀⠀⠀⠀⢀⢻⣸⢨
⠀⠀⠀⠀⠀⠀⢺⣿⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡿⡉⠢⣀⠀⠀⠁⠀⢀⠀⢄⠉⠀⡾⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠄⠡⠀⠀⠀⠀⠀⠀⢀⡤⣿⢣⣷⠈⠻⠛⠯⣉⣩⡏⠀⠘⠷⢄⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢤⠑⡄⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡐⠡⣶⡄⠀⠀⠈⠛⠃⠀⣄⡀⠀⢣⣏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠌⣀⣀⠉⠁⠠⣶⡆⠀⠀⠈⠟⠃⠀⣠⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀⣈⣂⣀⣀⣀⣀⣀⣁⣂⣀⣂⣈⣀⣀⣀⣀⡀⠀⠀⠀⠀⠌⠈⠿⠃⠀⠀⠀⠉⠀⠀⣴⡦⠀⠀⠀⠛⠃⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠒⣒⣒⣒⠒⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢀⡄⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣀⡀⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿⣿⣿⣿⣿⡸⣿⣿⣿⡿⣿⣿⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⣾⡿⠿⢿⠛⠧⠈⠻⠿⠷⠘⠋⠘⢋⣵⡿⣟⡅⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠉⠉⠙⢲⡶⠲⣶⠋⠁⠀⠀⠙⢿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⠀⠀⠀⠀⢀⡇⠀⢷⠀⠀⠀⠀⠀⣸⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡙⢦⣀⣀⡠⠊⠀⠀⠈⠳⠤⠠⠤⠞⣱⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣰⣿⣿⢁⡀⠀⠀⠀⢀⣶⣿⠶⢤⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢧⠀⠀⠀⠠⣿⣿⣷⣀⣤⣴⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⠋⠀⠀⠀⢀⣾⡿⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⡀⠀⠀⠘⣷⡀⠀⠀⢾⣿⣿⣿⣿⠛⠛⠋⠀⠉⠉⣽⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⡿⠁⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡓⠶⣦⣼⣷⡀⠀⢸⣿⣿⣿⣿⣷⣤⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⡄⠀⢠⣿⡟⠁⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠉⠙⣟⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⡟⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠳⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠁⠀⠈⠋⠻⠿⠻⠿⠿⠿⠟⠛⠿⠋⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣽⣿⣾⣶⡶⢮⣭⣛⠲⣌⡳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣭⣝⣛⡻⢿⣷⣯⡛⢦⡙⢎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠉⠛⠿⣷⣮⣻⣿⣶⡝⡈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢸⣦⡙⣿⣿⣷⢿⡜⣄⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡘⣿⣿⢸⣿⢸⣿⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⣿⡏⣾⣿⢸⣿⢃⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣼⣿⣯⣿⢏⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣷⡟⢡⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢏⣾⣿⠇⣴⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡏⣿⣼⣿⠁⠐⣂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠸⣯⡙⣿⣿⣄⢠⣿⣟⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠳⡙⠿⣦⣙⢿⡷⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠡⣿⣿⣦⣈⣾⡝⢆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣭⣾⡿⠃⠙⣫⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠻⠉⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣏⠤⣔⠴⣶⢴⣶⣜⠵⡶⠀⠠⢶⠴⡶⠴⡞⠀⢠⡧⢶⣦⡦⠾⡧⡮⡶⠦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⢂⡰⠀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠘⣾⡿⠀⠀⠙⠋
⠀⠈⠁⠂⠀⠀⠰⣼⣿⠀⠀⢴⣯⠇⠀⠀⠀⡀⠀⠀⠀⠘⠛⠀⠀⠀⠾⣷⠄⠀⠀⠀⠙⡇⠀⠀⢀⣿
⠀⡴⣶⡄⠀⠀⠀⠻⡇⠀⠀⠀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣾⡇⠀⠀⢊⡴⡃⠀⠀⠀⠀⠃⠀⠀⢠⣿
⠀⢞⣵⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⡠⠁⡀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣫⢾⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠨⠮⡈⠀⠀⢠⣸⣿⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⣵⣻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⢿⡟⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⡆
⠀⠈⠉⠀⠀⠀⠀⠆⠀⠀⠀⠈⣉⡀⠀⠀⢋⠿⡃⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡐⠁⠀⠀⠀⠀⢹⡇
⠀⢸⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢿⡇⠀⠀⡡⢚⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠊⠩⢎⡁⠀⠀⢠⣸⣧
⠀⢸⣿⠀⠀⡐⢓⡜⠂⠀⠀⠀⢸⣷⠀⠀⠔⣫⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠃⠀⠀⠸⣼⣿⠀⠀⠺⠷⠃⠀⠀⠀⡄⠀⠀⠀⠈⣉⡀⠀⠀⢚⡿⠂⠀⠀⠀⠈⡇
⠀⠈⠋⠀⠀⠀⠴⣷⠄⠀⠀⠀⠹⡟⠀⠀⠀⣤⠀⠀⠀⢀⠆⠀⠀⠀⠀⣿⡇⠀⠀⣩⢞⡁
⢀⠀⣿⡇⠀⠀⢚⡵⠂⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⠬⡠⠄⠀⠀⠀⢸⣇⠀⠀⠴⣫⡄
⠀⡀⢿⡇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠑⠄⠀⠀⠸⣼⣿⠀⠀⠺⡷⠃⠀⠀⠀⡄
⠀⢠⣸⡇⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣠⣦⡀⠀⠀⠀⠻⡇⠀⠀⠀⣤⠀⠀⠀⠀⠆
⠀⠈⢿⡇⠀⠀⠀⣁⠀⠀⠀⠐⠂⠀⠀⠀⠀⣤⡄⠀⠀⡴⣫⡄⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⢌⠠⠄
⠀⠀⠈⠇⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⠐⠀⢻⣇⠀⠀⢞⡵⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠐⠑⠄
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠔⢣⡜⠂⠀⠀⢀⢸⣿⠀⠀⣫⣞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠇⠀⠀⠈⣾⣿⠀⠀⠈⠋⠀⠀⠀⠀⠆⠀⠀⠀⠀⣤⡄⠀⠀⡵⣫⡅
⠀⠀⠄⠀⠀⠀⠈⣉⡀⠀⠀⠾⣿⠆⠀⠀⠀⠘⡇⠀⠀⠀⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢻⡇⠀⠀⢞⡵⠆
⠀⠰⠂⠀⠀⠀⠀⣿⡇⠀⠀⢚⡵⡇⠀⠀⠀⠀⠁⠀⠀⢠⣿⠀⠀⠔⠓⡜⠂⠀⠀⠂⢸⣧⠀⠀⣫⣞⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⢁⣀⣀⡀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠞⠂⠈⢁⡤⠀⠘⣿⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠚⣰⠀⢀⠀⢀⠠⡉⠐⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢓⡶⢽⡲⡶⠑⠘⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣴⣶⣶⣤⣤⡀⠀⠘⢦⡓⠀⠚⢁⠀⠀⠀⠀⠀⢀⡷⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠋⠉⠀⠀⠉⠙⠛⠻⣷⣦⡀⠙⢦⡁⠁⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠴⠆⠀⠀⠤⣄⠀⠀⢿⣿⣿⢀⡤⠟⠲⢤⣤⠴⠚⠁⠀⠀⠀⠀⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢴⢀⠀⠀⣄⠈⠀⠀⠸⣿⣿⢯⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣓⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠾⣡⢤⠀⠁⠀⠀⠀⠀⠙⠙⣷⢍⣻⠅⡉⠉⠒⠲⠤⣄⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠘⠿⣖⣩⠿⠂⠀⠀⠀⣼⡛⠛⠁⠉⠓⠶⠔⠲⣞⠒⠶⢴⡇⠈⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⢋⠲⣦⣤⡴⠒⠒⠻⢤⣉⡽⠃⠀⠀⠀⢠⠔⠚⣦⡀⠈⠉⠛⠒⠲⠦⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⡀⠙⠒⠀⠒⠈⣀⠄⠀⠪⠭⣓⢦⡄⠀⠉⠀⠈⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠧⣄⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠉⠒⠢⠤⠤
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠱⡐⠒⠢⠤⠤⠤⠤⠶⠒⠋⠁⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠺⢧⣄⡀⠀⠀⠈⠑⠦⢤⣀⡀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠈⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣠⣤⡀⠀⢤⣤⣤⠄⠠⣤⣤⣄⠀⢀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠘⣿⣛⣷⠀⢸⣯⣽⣧⠀⢸⡷⠆⠀⢸⣿⣶⡟⠀⢿⣭⡉⠀⣿⣃⣀⣿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡶⢿⣦⠀⠀⣼⣿⣧⠀⠀⣿⠉⣿⡀⢸⣷⣶⠿⠁⠸⠷⠶⠇⠸⠇⠸⠷⠀⢶⣼⠟⢀⣿⠙⢻⡇⢰⡟⠁⢹⡇⢀⣿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣻⡇⠀⠿⠉⠙⠃⠀⠉⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡁⠀⠙⠁⠈⠻⠶⠟⠁⣸⡟⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⣀⣠⠤⠴⠖⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠶⠦⢤⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⡀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣿⣿⡟⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠁⠀⠀⠑⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠀⠈⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠫⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠝⠁⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠑⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠨⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠈⠢⠄⠈⢠⠀⠀⡠⠗⠁⠀⠀⠀⢐⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠕⠀⠠⠪⠂⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⡀⣀⡠⠬⠭⣥⡄⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠠⠼⠓⡉⢠⡆⠀⠀⠀⠙⣴⠟⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⣀⣄⣦⠸⡧⠈⣻⠂⠀⠀⢀⣇⢐⠆⠈⠀⢠⣴⠘⢧⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠾⠟⠁⢀⡠⡄⠀⣿⣝⢿⠀⠛⠁⣠⡶⠶⠶⠟⠋⠀⣀⢰⡸⡇⠹⡄⠘⠁⠀⣠⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠠⣿⡘⣗⢷⠀⠋⢉⡤⠐⣫⡾⠋⠀⠀⠀⠀⠀⠐⣇⡈⠳⠛⠀⢀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠸⠃⠉⢀⣠⠔⣉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠠⢄⠉⠢⣑⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⢴⣾⠟⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡙⠳⣿⡷⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⣈⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠂⠂⠀⠐⠂⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤
⠀⠀⠀⣠⣀⠀⠀⠀⠁⠀⠀⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡃⠜⢛⣉⡉⠉⢁⣀⣠⠤⠤⠤⢼⠄
⠀⢰⣖⠛⢛⣼⣷⠀⡷⠈⡄⠈⡇⢰⡀⣼⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⠃⠀⠈⠁⠀⢀⡀⢀⠀⠀⠀⠀⠀⡀⠀⡆
⠀⠀⠹⡇⠿⠟⠁⠀⡇⢀⡇⠀⡇⠀⠀⢠⡇⠀⠀⠀⢀⠈⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣤⠾⣣⣮⣿⡅⠀⢀⡀⡇⠀⠃
⠀⠀⠀⠰⠀⠀⠀⠀⣿⠈⠀⠀⡇⠀⠀⠀⡇⠀⠀⠆⢠⣀⡀⠀⠦⠠⣁⣀⣰⣦⣤⣄⣤⣴⣿⣿⣿⣤⣶⡤⣇⠸⠟⢸⣿⠇⠀⠀⠀⡇⠀⡄
⠀⠀⠀⢀⠀⠀⠀⠀⣿⢀⠀⠀⡇⠀⠀⢰⡇⠀⠀⠀⠀⠁⠠⠠⠶⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⣼⡑⢂⣼⣯⠀⠀⠀⢾⡇⠀⡇
⠀⠀⠀⠘⡇⠀⠀⠀⣿⠘⠀⠠⡇⠀⠀⢸⡯⣄⠚⠟⠻⣦⣀⡄⠀⠀⠕⠉⠉⡭⠉⠭⢉⠁⣽⣿⣿⣿⣿⣴⡯⢰⣿⣿⣿⣷⠀⠀⢸⡇⠀⠁
⠀⠀⠀⠐⠇⠀⠀⠀⣿⠀⠀⠀⡇⠀⢀⣘⡇⠀⠀⠀⢰⠂⠙⠁⠀⠐⡇⠀⠤⠁⢄⠀⢸⡅⣿⣾⣿⣿⣷⣤⣵⣿⣿⣿⣿⣿⠀⠀⢸⡇⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣤⣂⣱⣮⣿⣿⣿⡆⠀⢀⡐⢠⠈⠀⠀⠀⠀⣀⣤⢀⠀⠠⣾⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⡿⡿⠀⠀⣸⠀⠀⡇
⠀⠀⠛⠻⠿⠿⠿⢿⡏⣫⡽⠾⣾⣋⣫⣿⣿⠁⢈⡬⠀⢉⢀⣀⣒⣴⣶⣿⣷⡄⠛⠀⠀⠈⠛⠡⣻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣄⢸⢃⡿⢻⢹⣿⡇⣁⣳⢀⠀⠈⠱⣿⣿⣿⡟⠁⠀⠀⠀⠈⠀⠁⠀⠿⣿⠿⣿⣿⣿⣿⣿⢇⠘⠀⢰⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⢸⣇⠰⣾⣿⣇⣷⡇⢠⠙⣢⢐⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠸⠿⣿⡿⠟⠋⢀⣀⠀⢰⣶⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠈⣿⡆⢸⣿⣷⣿⣸⣿⣿⣿⡈⣿⣿⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡇⠀⢼⣷⡟⠓⠒⠋⠀⠚⠿⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣷⠀⣿⣿⡿⠈⣏⣿⣿⣿⣷⣿⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠐⣿⣿⡥⠬⠤⠤⠤⠶⠤⠥⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡰⠀⡇⠞⢿⣽⡇⠠⠹⠟⣟⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⢻⣿⡧⠀⠀⠀⠓⠀⠇⢰⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠉⢀⣯⢴⣟⣽⣇⠀⣲⠫⠝⣻⣭⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠛⠛⠛⢻⣿⣶⣄⡤⣄⠀⠘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠀⠈⢇⣨⣿⣿⣿⣿⢯⡴⣂⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⠖⠐⠀⢀⡀⠸⢻⣷⢰⡆⣑⠀⣧⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⡿⠿⠟⠻⠉⠉⢀⡴⠃⠐⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠘⣠⣾⣻⣿⣦⡀⠉⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠸⠟⠉⠓⠛⠁⠀⢀⡠⠋⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⢀⣠⣾⣷⡿⢿⣿⣿⢀⠀⢈⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠰⣆⣄⣼⢿⣦⣤⣤⡀⠀⠋⠀⠀⢀⣴⣾⣯⣻⣿⠃⣾⣿⣷⣰⣿⣿⣷⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣾⣿⢸⡿⠟⣿⣿⣿⣿⣿⡟⣿⣿⣽⣿⣿⡥⣾⣿⢿⣿⣿⣿⣿⡗⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⢺⡟⡃⣿⣿⣏⣱⣿⣿⡋⣿⣿⡿⢀⣼⣿⣿⣷⡟⠻⢿⣿⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠀⠀⢀⣸⠞⢁⣿⠀⠘⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⡿⠟⢑⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠠⠞⠋⠉⠀⠉⢿⢽⡶⠿⣿⣿⠿⠟⠛⢉⢁⠐⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⣩⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣽⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⠉⠛⠛⠛⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⢿⣿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠧⢽⣦⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣦⠀⢴⠶⠒⠀⠸⠿⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⠒⠒⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣄⢀⡇⠈⠀⠈⠑⠂⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠬⣽⠇⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⣌⡔⢸⢿⣷⣲⠐⠀⠀⠀⠀⠀⡸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⠀⡇⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⢠⡾⠁⢿⣿⣿⠉⠁⠀⠀⣠⠞⢁⠏⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠰⣀⣀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⢿⡿⣷⣶⠶⣟⡡⠐⠀⠠⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣠⣥⣴⣶⣾⣷⣶⣦⡀⠀⢀⡼⠃⠀⠀⠈⠁⠈⢳⠊⢻⠀⣠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣾⠿⣿⣷⣿⣦⣿⣿⡿⢿⣶⠟⠁⠀⠘⣷⠀⠀⠀⠀⠀⠘⣦⠠⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⢸⣿⠀⠀⠈⠉⠉⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠀⢻⡟⢀⣀⠀⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠬⠌⠂⠀⠙⠂⠤⠀⠚⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣷⣆⡠⠌⡒⠀⠂⠀⠀⠀⠀⠀⢸⡙⢷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⠀⠲⠴⠊⢀⠀⢠⣧⠁⢺⣊⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿⣈⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠁⠐⢿⠀⠛⠉⢀⣾⣿⣿⣈⣹⡇⠀⠀⢯⣀⠀⠀⠀⠀⠠⢠⣿⣧⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣀⡀⠀⣀⣴⠏⣿⣿⣿⣿⡽⣷⣄⣀⠀⠉⠉⠉⠈⠓⠐⠚⠿⠤⠶⠶⢦⣤⣀⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⣟⢿⠿⠿⠟⠁⢀⣿⢻⡟⣿⣿⠻⣿⣿⡿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠘⣿⡟⠒⠒⢀⣀⣽⢿⠀⠃⢸⡿⢿⣿⣿⣷⠨⡟⠿⣷⣦⣄⣀⠀⠰⠶⢦⣤⠀⠀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡾⠻⣿⢸⡆⠘⣧⠈⠉⣡⡾⢣⠏⠀⠀⣾⡇⠀⠈⢿⣾⣴⣧⣷⣮⡿⣿⣿⡟⠲⢤⣤⣨⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⡍⢸⡇⠀⣿⠀⣿⡀⠘⠧⠈⠉⠠⠋⢀⠀⢠⠃⡇⠀⠀⠀⠀⠙⠻⣷⣿⢡⠋⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠘⠀⢸⠀⠀⠹⡄⠈⠇⠀⠀⠀⠀⢀⡴⠃⢀⠏⠀⠇⠀⡄⠀⠀⠀⠀⠈⠻⣾⣵⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⠀⠀⢀⣤⡴⠋⢀⠠⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠈⣷⡀⠑⠲⢤⡀⠈⠉⣁⡤⠈⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠁⠉⠁⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣠⡴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇⠀⡠⢤⠀⢺⢄⡀⢠⠖⠂⠀⡔⢰⠈⠧⣄⠀⣀⢀⠈⠆⠀⡴⢤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠒⠚⢄⢀⠀⡸⠀⡄⢸⠀⡌⢠⡇⣼⠉⢁⠎⢀⠃⣼⠀⠀⠁⠀⡘⠁⡐⢀⠎⢀⡄⣠⠋⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠶⠐⢀⡘⠘⠐⠓⠊⠀⠈⠉⠉⠁⠀⠀⠉⠁⠀⠀⠉⠀⠑⠒⠊⠉⠀⠀⠓⠛⠄⠚⠶⣁⠼⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡷⠶⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣇⣀⡀⣴⡆
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠁⠀⠀⠀⠈⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡖⠤⢀⣤⣶⣶⣾⣿⠷⠶⢾⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⣠⢾⣿⣋⡁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡀⠀⠀⠀⠀⠀⠀⡿⠸⠇⠘⣸⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⣿⢸⡟⠛⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⣇⢀⠀⠀⠁⢞⣿⣿⠟
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠙⠎⢧⠀⠀⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠻⣿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠜⡄⠀⢻⣿⣿⣿⠿⣿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠀⠈⢿⣿⣷⠀⠀⠉⠻⣷⣄⠀⣼⣿⠃⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠐⡀⠀⠘⢿⣿⣆⠀⠀⠀⣾⣻⣿⣿⠋⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠈⢆⠀⠈⠻⣿⣿⣶⣶⣿⣿⡿⠻⣦⡀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠙⠆⠐⢻⣿⣿⣿⠟⠋⠀⠀⠘⢷⡄⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⣀⠀⠹⢿⣧⡀⠀⢀⣀⣤⣾⣿⣿⠟⠻⣿⣄⠀⠀⣠⢖⡈⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⣿⠿⠿⠛⠛⠋⠉⠙⢶⣤⣼⣿⣷⣄⢁⣿⣿⠟⣱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⣀⣹⣿⣿⣿⣿⣿⡿⣷⣿⠟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⢿⣿⡿⠿⣿⡜⡇⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠒⢄⠀⡰⠁⢀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠻⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠜⠰⠣⠃⢆⡜⠜⢱⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⢟⣵⡿⢏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⡔⠁⣄⠀⡠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⣄⠀⠀⠼⣟⣵⣿⠏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠞⠐⢖⡁⡜⢀⡀⠀⣀⢠⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣟⣿⣿⠏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠋⠀⠀⠰⠓⠁⠎⠸⠱⠁⠏⠸⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣾⣿⣿⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣛⣻⣿⣟⣛⣛⣓⠾⢭⣅⡙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⢯⣭⣭⣭⣭⣭⣟⣻⢶⣬⡙⢶⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣶⣬⣭⣙⠻⢿⣷⣌⠛⢦⡙⢦⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠂⠀⠀⠉⠛⠿⣿⣦⣹⣿⣯⢦⡉⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⠀⠀⠀⠀⣷⣌⠻⣿⣿⣿⡎⣷⡹⡄⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠘⣿⣿⡇⣿⣧⢹⣽⡜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⡆⢿⣿⢃⣿⣿⢸⣿⡇⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⡟⣼⣿⡿⣼⣿⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣾⣿⡟⣽⡿⢃⣼⢏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢫⣾⠟⢠⠞⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣵⡿⣵⠃⣴⢣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠰⣿⢹⣿⢸⣿⠘⠀⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠠⣟⠌⣿⣿⣿⡀⢄⣒⣞⣵⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢻⣆⠻⣷⣌⡻⢿⣷⣤⣺⣿⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢷⣝⠣⠈⡛⢿⣦⡙⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⣸⡿⣷⣆⠠⣷⣎⠳⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣩⣾⣿⡿⠀⠑⣣⣷⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠐⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣲⠳⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⠄⠀⠀⠀⢀⡴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠥⠖⠎⠵⠻⠧⠿⠎⠮⠸⣷⠃⠀⠎⠿⠴⠿⠱⠿⠇⠀⠰⡧⠎⠯⢦⠇⠿⠇⠮⠾⠿⠾⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡶⣶⢶⠶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⠶⣶⣶⣶⠶⡶⢶⣶⣶

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⣤⣄⠀⠀⢀⢀⡀⢀⡤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠄
⠀⠀⠀⢰⠟⢋⣩⠀⠀⡇⠀⡱⢮⠀⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢋⣡
⠀⠀⠀⢰⡾⠟⢛⠀⠀⠀⢀⠀⠀⠡⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢰⡾⠟⢋
⠀⠀⠀⢨⣴⡾⢿⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⠢⠀⠀⠀⠌⠀⠀⠀⢠⠁⠈⠀⠀⠀⣚⣿⠀⠀⢸⣴⡾⠟
⠀⠀⠀⢘⣡⣴⠿⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠊⠉⠱⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣒⣾⠀⠀⢘⣡⣴⠿
⠀⠀⠀⠸⣋⣥⣾⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠤⣾⠀⠀⠸⢋⣥⣶
⠀⠀⠀⠸⠛⣉⣴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠠⠭⣽⠀⠀⠸⠟⣉⣴
⠀⠀⠀⠁⠈⠉⠀⠁⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⣀⣀⣀⣀⣐⣂⣩⠭⠀⠀⠀⠀⠀⠉⠭⠀⠀⠁⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⢀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⢰⡾⠀⠀⠀⠀⠀⢷⠀⠄⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⢠⠀⣦⠀⠀⠀⠀⠀⠀⠀⣾⡘⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⠀⠀⢀⠔⢋⣉⡙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠆⠀⠀⠈⢼⣷⠄⠀⠀⠀⠄⣿⠇⠁⠀⠀⠀⠀⣠⠀⣾⣉⡷⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢶⣄⠀⣤⠎⠡⠀⠀⠀⠀⠶⠁⠘⠦⡈⡉⢡⠴⠁⠨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⠴⣼⣿⣿⣿⡷⢶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠠⣿⣿⣿⣟⠁⠀⠈⠙⠛⠻⠋⠉⠑⠛⠓⠒⠒⠶⠶⠆⠀⠀⣠
⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⢤⡄⣀⣒⠒⠶⠆⣿⣿⣿⣿⠁⠀⣀⣠⣶⣄⠀⠀⠀⠀⠀⠀⣀⠀⡼⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠚⠛⣯⡄⠀⠀⡀⠀⢈⢳⡇⠀⠀⡀⠀⠀⣻⣿⣿⡿⢀⣈⣿⣿⣿⠿⠷⢦⣄⡀⠀⣧⣿⢠⡇⠀⠀⡘
⠀⠀⠀⣾⡆⣶⣼⣿⣿⠠⠄⠀⠁⠀⠰⠆⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⠿⠿⣿⡶⣤⣹⣿⣦⡛⣿⣿⣄⠄⢠⢁⣆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣤⣼⠆⠀⠠⠤⠀⠀⣿⣿⣿⣿⣿⣿⡤⠀⠀⠀⣰⠟⢻⣿⠃⡋⢻⣿⣿⣦⣧⣼⡿
⠀⠀⠀⣿⣿⣿⡟⢿⢻⢀⠠⣶⣿⣿⠉⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿⡿⣻⡄⠀⠀⣼⣿⢴⡿⡟⠀⣲⡏⠘⣿⣿⣿⣿⣷
⠀⠀⠀⢿⣿⣿⣧⢾⣿⣿⣷⣿⣿⢿⢿⠉⠒⠲⠒⠒⠐⣿⣾⠋⢻⣧⣻⣯⣤⣾⣻⣿⣾⡟⠁⢀⣞⠉⢠⠋⠛⠛⠻⢿⠇
⠀⠀⠀⠀⣿⣿⣿⣾⣾⣿⣟⣛⣛⡥⡀⢸⠀⠀⠒⠲⢴⣿⡿⣧⣾⡿⣿⣿⣿⣿⠿⠿⠿⠃⠀⣾⠉⠀⠀⠀⠈
⠀⠀⠀⣤⣿⣿⣿⣿⡟⢻⣿⣿⣿⠛⠿⠀⠀⢀⣀⡀⢸⣿⡋⢻⣿⣾⣾⣿⣿⣿⣷⣂⣷⡇⣼⣹⠀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⠀⣤⣸⣇⣿⣿⣷⣼⣿⡿⠺⠿⡿⢻⡿⠿⠿⠿⠿⢇⣼⣁⣿⡄⠀⠃⠀⢀⡀
⠀⠀⠀⢿⡿⠿⣿⣿⠟⠉⠉⠛⢿⡆⠀⠈⠙⠛⠿⢿⣿⣷⣿⣿⣷⣯⣀⠇⠀⠀⠀⠀⠀⠉⠛⠿⠿⠦⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣤⡶⠶⢶⣶⣶⣶⠶⠦⣄⣄⣀⣀⣀⠄⠒⠐⠂⣄
⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠿⠿⣯⡴⠿⠿⠿⠷⢶⣤⡈⠙⣷⣆⠀⢻⣷⣸⣄⣀⣀⣠⣴⣿⣧⣀⡀
⢠⣴⣿⡿⠟⠛⠉⠁⣀⠀⠰⠚⠁⢴⣄⠐⣦⡀⣠⡙⣿⣆⠘⣿⡆⠿⢿⣿⣿⠠⠤⠧⢼⣿⣿⠻⣿⣷⣤
⢸⣿⡇⠀⠀⠀⠀⠀⠁⠐⢩⠾⣦⢐⣹⣦⡘⣿⣿⣧⡘⣿⠰⣿⠇⣠⣾⣿⣿⠀⣶⣶⢸⣿⡇⠀⠀⣿⣿
⢸⣿⡇⡠⠀⠀⢠⠂⠀⢂⠀⢼⣿⣦⡈⠉⠙⠻⢿⡧⠷⠋⠐⠙⠛⢭⣿⣧⣏⠀⠛⠋⠈⠸⠀⠀⠀⣿⣿
⠸⣿⡇⠁⠀⢀⠃⠱⢰⠃⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠃⠀⣀⣤⢠⣾⡇⠀⠀⣿⡿
⠀⣿⡇⠀⠀⠙⡄⠀⣾⠀⠀⢸⣿⠃⠶⠶⠒⣲⡀⠀⠀⠔⠶⠶⠶⡌⢿⣿⣼⡇⣿⣿⢸⠏⠁⠀⢀⢿⡇
⠀⣿⢸⠀⠀⢀⡜⡶⢻⠀⠀⢸⡏⠀⡠⢶⡗⠯⠃⠀⠀⠰⢶⡽⢎⠛⠀⣿⠟⠃⠉⠀⠀⣇⡀⠀⢸⢸⡇
⠀⢹⡏⠀⠀⠈⢤⡜⠸⡀⠈⣿⠇⠀⠀⠈⠁⠀⠀⠄⠀⠀⠀⠉⠁⠀⢀⢦⣞⠡⣴⣶⣾⣿⡇⠀⡸⣾⠁
⠀⠸⣧⠀⠀⠀⠸⡇⠀⡇⠀⠱⣇⠀⠀⠀⢀⣀⣤⣤⣤⣴⣀⠀⠀⠀⠀⡟⠁⡾⠿⠿⠛⡇⠀⠀⣧⡿
⠀⠀⢿⣷⠀⠀⠀⡇⠠⠃⠀⠈⢿⣷⣤⢖⣩⣭⣥⢶⣾⣭⣬⣙⠦⣠⣼⡟⠀⠀⠀⠀⣀⣧⡆⢸⣿⠇
⠀⠀⠸⣿⣇⠀⠃⢧⠀⢠⠀⠐⣾⣿⣾⣿⣿⡉⢹⣶⣿⡉⣹⣿⣿⣿⣟⣿⣀⢰⣿⣿⣿⡟⣧⣿⡿
⠀⠀⠀⢿⣿⡄⠈⠃⢦⢸⠀⠀⢼⣧⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡇⣷⠾⠭⡥⢤⣷⣾⣼⠇
⠀⠀⠀⠈⢿⣳⠀⠀⠀⣨⡄⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣼⣿⣤⠤⢧⣼⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣧⠀⠀⢹⠀⡂⠨⡏⠀⠈⣷⠀⡂⢸⠂⢐⠀⡇⢐⡦⣖⠀⡂⢸⡀⠀⢈⢛⡝
⠀⠀⠀⠀⠀⠘⢿⣳⡀⠻⠤⠴⠖⡴⠾⠤⠶⠤⠷⢶⠤⠶⢴⡷⠶⠦⡶⠶⠦⠼⠃⣰⣳⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⡧⠄⠘⡇⠠⠄⢸⠀⠿⢀⡇⠨⡤⠆⠀⠀⢀⣴⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠛⠒⠛⠛⠚⣷⣾⣿⣶⠛⠓⠛⠃⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⠀⠛⠋⢁⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠂⠒⠶⠶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⡿⢹⣿⣿⣦⢀⣀⣠⣤⣶⣦⣤⣶⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⢸⣿⣿⢽⡿⢀⣈⡻⣿⣿⣿⣿⢿⡛⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠿⠀⢸⣿⣧⣀⣙⣻⣛⢳⡎⠙⣿⡏⠀⢠⣴⣾⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠧⠀⣿⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠙⣴⣿⠃⠒⠋⠨⣽⣻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡿⠋⠠⠀⢲⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠘⠛⠀⡠⠔⠁⣴⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⡿⠀⠀⠀⡠⠊⠁⠀⠀⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢀⠔⠁⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⡄⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠊⠀⢀⣴⣿⣿⢿⣿⣿⣿⣿⡜⠿⣿⣦⡀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⣴⠃⢀⣴⠿⠋⠡⣾⠘⣧⣿⣿⣿⣿⡄⣈⠙⢳⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⠉⢿⡿⠁⠀⠖⣴⣿⣦⣝⣿⢻⣿⣏⣿⡏⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⣸⣷⣄⣠⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⠀⣿⣿⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⠀⢀⣧⣼⣿⠀⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⡆⣼⠀⠀⠀⣰⠞⣿⣿⠛⠟⠃⠸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⠀⢀⠀⠀⠸⠋⡿⠀⢀⣷⢶⣶⠄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣤⢸⡇⠀⠀⠀⣠⣶⣿⠇⢾⡀⣔⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣧⠀⠀⠘⣿⠟⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠓⠶⡜⣃⣤⣤⠤⢄⣀⣠⡞⣫⣥⣶⣦⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⢀⡈⠀⢰⣾⡟⡿⢁⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⣠⣾⣿⣿⣿⡘⢿⣧⢧⠸⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⣰⣿⣿⣿⣟⢿⣿⣦⣙⢿⣷⣝⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⣿⣿⣿⣿⢿⣷⣝⣿⣿⣿⣾⣿⣿⣿⠿⠿⣿⣿⣿⣟⣋⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣆⠀⣿⣿⣿⣻⣷⣽⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠈⠉⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⡀⢹⣿⣿⣿⠿⢿⣿⠟⠉⠉⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⢿⣧⠀⢻⣿⡆⠀⠈⠁⠀⣀⣄⣀⣀⣀⠉⠻⠶⠀⠀⢀⣀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⢸⣿⡆⠈⢿⣿⡀⠀⠀⠀⠘⢿⣟⡻⠿⣿⠀⠀⢀⣀⣉⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣇⣿⣿⣧⣿⡇⢸⡎⣿⣿⣆⠀⠀⠀⠀⠈⠋⠋⠀⠀⠀⠞⣻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⣿⣿⡇⣼⣧⣿⣿⣿⢧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⡤⠤⡙⣿⡿⠛⠓⢿⣿⣿⢹⣿⠀⠙⢦⣄⠀⠀⠙⠷⠒⢀⣴⣯⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣴⡏⣾⣷⡇⡸⢰⣿⣿⡌⣿⡿⣾⣿⡀⠀⠀⠙⢷⣄⣀⣡⣶⣿⣿⣿⢟⣩⣭⣀
⠀⠀⠀⠀⢰⣿⣷⣙⣛⡡⢣⣿⣿⡿⢠⣿⣿⣿⡟⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⣟⣵⣿⣿⡿⠋⠀⠈⠆⠀⠀⠈⠻⢿⣿⣿⣿⡿⣿⣶⡾⣋⡾⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠁
⠀⠀⢀⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⠀⢀⡀⣀⣀⣀⣀⠀⣀⣀⢀⡀⣀⣀⣀⠀⣀⡀⢀⡀⣀⠀⣀⣀⣀⡀
⠀⠀⢸⡿⢼⣯⡼⢿⢿⡿⠹⡏⠹⡏⠀⡿⡧⡿⡿⢿⡿⠀⣿⡿⡾⠷⣿⢿⢿⠀⢽⡷⡿⣧⣿⡶⣿⠿⡿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⡀⠠⢄⡀
⠀⠀⠀⠀⠀⠐⠒⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢨⠋⠀⠀⠀⠈⠆⠀⠠⠚⠁⠀⠀⡄
⠀⠀⠀⠐⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠄⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⡄⠈⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⣑⣾⣿⠿⠗⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣿⡄⡀⢠⣤⠔⠊⠁
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠲⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⣸⣧⠀⣾⠵⠆
⠀⠀⠀⠀⠈⠣⣤⠀⣀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠈⢡⣾⠇⢀⡶
⠀⠀⠀⠀⠀⡰⣏⣄⢹⡿⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⡼⠁⣼⠃
⠀⠀⠀⠀⠀⠉⠉⢻⠈⢀⠴⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠟⠉⣈⡷⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⢁⠀⠀⠀⠻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡇⠺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠈⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠴⠿⠉⣶⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠟⢻⡿⠛⠘⠁⠀⠀⠀⠀⠀⠸⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢂⣸⣿⣿⡇⠀⠀⠀⠀⠙⠛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⠟⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠁⢀⡏⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠓⠁⠈⢿⣶⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠶⠶⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡈⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢐⠂⢈⡠⣌⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢅⠘⠿⠿⣦⣕⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⣬⡐⠤⣀⠈⠻⢶⢿⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢸⣿⣄⠀⠁⠐⠠⠠⠔⠹⣟⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠺⣿⡄⠀⠀⠀⠀⠀⠀⠀⠑⠲⣽⢷⣦⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⢀⡀⠤⢐⣢⣍⣤⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄⣀⣠⡤⠵⠚⠛⠛⠠⣭⣤⣶⣾⣿⣿⠿⠛⠋⠉⠉⠋⠙⠖⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠀⠀⢀⣠⣉⣠⣤⣼⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣤⡾⠶⠶⠶⠿⠛⠛⠛⢫⡀⠀⣺⣿⣿⠆⠀⠀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡼⠉⠁⠀⠀⣼⠏⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⣸⣿⣠⠊⠀⠀⠀⣠⣴⡿⢳⡄⣼⡿⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣼⢁⠀⠀⢀⣤⡤⢼⣿⠀⣴⣿⢁⣿⡏⠀⣿⣿⠟⢉⣿⠀⢰⣿⠟⣠⠞⠀⡿⠷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠞⠋⢻⣇⢠⣿⡏⠀⣿⣿⠀⣸⠿⢻⣿⠁⢀⣿⡿⣦⣼⡏⢀⣿⣿⠋⠁⣠⠞⠀⢰⣿⠇⡠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣠⡿⠁⢸⣿⡇⢠⣿⡏⡠⠁⠀⣾⣿⣠⠚⣿⣇⡼⠋⠀⠈⠿⣿⠤⠞⠁⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠞⣸⠃⠀⠸⣿⠷⠋⠿⠋⠀⠀⠀⠙⠋⠁⠀⠉⠁⢀⣀⠀⠀⠀⢀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠁⠀⣴⣿⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⣸⣿⣯⠔⠁⠀⠀⣴⣾⠷⣆⠀⢠⣿⡖⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣀⠴⣿⣿⠅⣴⡦⠀⣸⣿⡇⠀⣿⠀⣼⣿⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡅⢸⣿⡟⢸⣿⠁⣠⣿⣟⢷⣠⡧⠈⣿⡟⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⣿⡇⣼⣿⠇⣾⣿⠜⠁⠻⢿⡶⠟⠀⢸⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣼⡿⠃⠻⠟⠀⠀⠁⠀⠀⠀⠀⠀⠐⠊⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⢿⣟⡙⠿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣻⣿⣿⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣴⣷⣭⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⣾⣿⣿⣟⡷⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣾⣿⣿⡳⢯⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⣿⣿⣿⢿⣝⡳⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⣿⣿⡿⣯⣛⠮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⢷⡀⠀⠀⠀⠀⠀⢠⠋⣰⣿⣿⣿⣟⡷⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠻⣄⠀⠀⠀⡰⠃⣼⣿⣿⣿⡳⢮⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠙⣆⠀⡰⠁⣼⣿⣿⣿⠿⣝⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡖⣮⢿⡁⠘⣿⣿⡷⣯⣛⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠁⠀⠳⣄⣉⣻⣽⡶⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣄⣀⠀⠙⢆⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡰⠻⡈⢹⠲⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡴⠁⠀⢳⢸⡄⠀⠑⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠜⠀⠀⠀⠈⢇⢇⠀⠀⠀⠳⡌⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠊⠀⠀⠀⠀⠀⢸⡌⡆⠀⠀⠀⠘⡄⠈⣩⠭⣽⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡡⠃⠀⠀⠀⠀⠀⠀⠀⡇⠘⢆⠀⠀⠀⡘⣼⠁⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡴⠁⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⣬⣕⡢⡀⢰⣹⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡼⠁⠀⠀⠙⣾⣧⠷⣝⢦⣀⣀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢳⠀⠀⠀⠀⢸⣿⠀⠈⠙⠒⠓⠛⢬⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡳⣄⣀⣀⡼⡿⠀⠀⠀⠀⠀⠀⠀⠳⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠺⠗⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠠⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣀⢤⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠇⠄⢨⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠐⡉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠖⠀⠀⠀⠤⠄⠆
⢀⣀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠘⠄⠇⡸⠀⠀⠀⠀⠀⠀⢀⡀⢠⠄⠀⠀⠀⠀⠠⢘⠀⠀⠀⡛⠠⠀⠀⠀⠀⠀⠀⠄
⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠁⠈⡆⠀⡇⠀⠀⠀⠀⠀⡔⠙⠁⠞⠸⠳⣆⠀⠀⠐⠨⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⠀⡠⠈⠀⡀⠀⠀⠀⠀⢂⡄⠀⢨⣧⣄⢀⢀⡀
⠀⠀⢀⠀⠀⠀⡀⣀⢀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⡰⠊⠉⠂⠈⠫⠤⡈⠃⡄⠘⠦⣅⡀⢦⠗
⠀⠀⠀⠀⠀⠜⠁⠀⠀⠑⡀⠀⢸⠀⢸⠀⠀⢀⠲⡀⠀⠀⠀⠀⠀⠀⠷⡄⢌⢲⠀⢈⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢀⠀⠀⢘⠀⠀⠀⠀⠀⠡⠀⢸⡀⢸⠀⠀⠀⠙⠀⣆⡀⠈⢀⣰⠂⠇⠀⠈⢢⡹⣏⠎⠀⠈⠀⠀⠀⠀⠀⠀⢀⠃⠀⡅
⠀⠀⠀⣤⣽⡀⠀⠀⠀⢀⠗⠀⠸⠇⢸⠀⠀⠀⠀⠑⠮⠁⠀⠀⡠⠈⠀⠀⠀⣠⢉⢈⡢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⢀⡀⡁⢀⣀⡠⠆⠀⠀⢀⠻⣿⠤⠀⠀⢀⣀⡐⠁⠀⠈⠢⣀⡀⠀⢸⣁⣽⠓⢶⠋⠀⠈⠉⠛⠒⠤⡴⢉⡁⠀⠠
⠀⠀⠀⠂⠀⠀⣸⢀⡀⠤⢀⣀⠈⣧⡠⣃⢾⣶⣁⠀⠁⠸⠈⠀⠀⠀⠀⢦⡴⠁⢐⠖⠚⠑⠂⠄⡤⢀⠀⠀⡀⢀⣀⡈
⠒⠒⠐⠒⠀⣋⣀⡬⠡⠤⠤⠬⠑⢆⠀⠙⠳⡌⠛⡃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⣰⡬⠃⠹⡦⠐
⠈⠉⠉⡽⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⠀⠉⢇⠀⠀⠀⠀⠀⠀⡗⣄⢠⡜⠀⠀⠀⠠⠄⠀⠀⠀⢰⠀⠁⠀⠀⢛⡶
⠤⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠒⠒⠒⢺⠀⠀⠀⠀⠀⢸⠘⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢢⢄⠀⠀⠐⠈
⠀⠐⣀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢗⢀⡀⠠⢼⡀⠀⠈⠀⠀⠸⠀⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⢰⠤⠀⢀⡠⠃⢀⡀
⠀⢀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⠀⠀⠀⠸⠀⠈⠁⠂⠀⢣⠀⠀⠀⠀⠈⠙⠒⠦⢀⡀⠀⠀⠀⠘⠉
⣀⠀⢦⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⠉⠁⠑⠁⠀⣀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⣂⠆
⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠀⠀⢠⠀⠀⠀⣾⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠒⠲⡐⠀⠀⢉⡝⢻⠀⠀⠈⠀⢸⠀⠀⠀⡹⠄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢅⠀⠉⡀⢈⣀⣘⠀⢀⣀⠤⡴⠀⠀⠀⡇⠆⠀⠀⠸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣈⣴⠉⠃⠈⡀⢹⡉⠀⠁⢀⡇⠀⠀⠀⠇⢰⠀⠀⠀⡆
⠶⠒⠒⠚⠛⠉⠁⡠⠒⡄⠀⠀⠧⠎⠀⠈⡆⠀⡀⠀⠀⢰⠂⠘⡇⠀⠀⢱⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠀⠀⢀⡠⠔⣀⠌⠁⠉⠉⢻⠀⢀⡴⠒⠒⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠙⠚⠛⠛⠑⠋⠀⠸⠧⠠⠄⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣶⣶⣶⣶⣶⡿⢷⣶⣷⣷⣷⣿⣾⣷⣾⣷⣿⣾⣷⣶⣶⣶⣿⣿⣶⣿⣾⣿⣷⣾⣶⣾⣷⣾⢷⡶⢾⠷⢷⣿⡾⢷⣾⠾⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡛⢛⣻⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣽⡿⠿⠇⣿⡿⠟⠻⠿⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⣀⣀⠀⣠⠲⣤⣦⣤⡀⠀⠀⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠘⠋⠴⢋⡤⣛⣷⣿⣿⣿⣦⡄⠀⠀⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠋⣴⡟⢙⣫⣿⣿⣿⣿⡄⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣀⠐⣷⠾⣋⣼⡿⣿⡿⠛⣽⣿⡷⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣡⠶⢁⣤⣻⣯⣼⠏⣰⠞⢹⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠉⠴⢟⡵⠋⠁⠀⠀⠀⠰⣟⣿⣿⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠊⢁⣤⣤⣶⣶⣤⣤⣄⠈⠉⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣠⡾⠟⠋⠉⠉⠉⠛⢩⣿⣷⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⣰⣿⣷⠓⠶⠀⠀⠀⠐⠉⢉⣿⣿⡄⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⣴⢏⣿⡟⠀⢴⠆⠀⠀⠀⠶⠘⣿⣿⣇⢀⣀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠊⠙⣧⡀⠀⠀⠀⠀⠀⠀⣤⡾⢿⣭⣯⡽⠋⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠤⣀⠀⠀⠀⠈⢛⡀⠀⢴⣶⠆⢀⡟⠀⠀⠀⣀⡤⠚⢻⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠻⠂⠈⡻⣦⡀⠀⢸⡇⠐⢀⣤⣤⣿⠀⠀⣠⠞⢋⣤⠀⠀⣁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣻⣀⣈⣋⠀⠙⣦⡈⣣⡀⠀⠀⢀⡙⠀⣿⡇⢀⣈⣁⣀⣈⣛⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⠉⠛⢻⣷⣿⠟⣹⡟⠉⠲⠶⠛⢁⣴⡏⠀⠈⣉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠛⠁⠉⠛⠛⠙⠿⠇⣸⠋⢹⣶⡿⠆⠚⣛⠋⢀⣾⠀⣤⡌⠛⠳⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⢰⣦⠀⠀⠘⠛⠀⠀⠀⠀⢻⡄⢀⠙⠿⣷⡆⠉⣠⣾⡿⠀⠈⠁⠀⣀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠿⠂⠈⢿⡄⠰⣶⠤⠴⠟⠛⠋⠠⡶⠀⠀⠘⠛⠀⠀⠈⠁⠘⠋⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⡀⠀⠘⠟⠀⢀⣀⠀⠀⢀⠈⣿⡄⠀⠀⠀⣶⠄⠀⠀⠀⣀⣠⣤⣤⣀⡀⠐⠷⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⠛⠀⠀⠀⠀⠘⠋⠀⠀⠉⠀⠼⠿⣿⣦⡄⢀⣠⣴⠾⠟⠋⠉⠉⠀⠨⡇⢀⣀⠀⠰⣶⠀⢿⡄
⠀⠀⠀⠀⠀⢸⣿⡀⠘⠃⠀⠀⠀⠀⠿⠂⠠⡄⠀⠀⣀⠀⠀⠀⠀⠈⢁⠈⣁⠀⠀⠀⠀⠀⠀⠀⠐⡇⠘⠋⠀⠀⠀⢀⣼⣿⡀
⠀⠀⠀⠀⣠⣿⡿⠀⣀⠀⠀⢶⡦⠀⣀⡀⣀⢠⣜⠷⣉⠀⠁⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠛⢂⠤⠠⣷⣄⠠⣶⠀⣴⠟⠁⠈⠛⢦⣀
⠀⠀⠀⠀⠘⠿⢧⣄⠉⠀⠀⠀⠀⢀⡙⠃⠀⠀⢹⣶⣫⡴⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⠿⣿⠀⠀⠈⠉⠳⣤⠀⠙⣦⡀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢀⡴⠀⠀⠀⠙⠧⢀⣀⣰⡶⠂⠀⠀⣀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠉⠰⢶⣤⣤⣀⡀⠙⠓⠴⠿⡶⠦⠤⠄⠀⠙⢷⣄⠀⠀⠀⠀⠈⠁
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣤⠶⠟⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠸⠃
⣸⠇⠀⠀⠀⠀⠀⢀⣀⡠⠶⢾⣿⣷⣴⣆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠀⢀⣠⠖⠻⠇⠀⢀⣬⠛⢿⣦⣄⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⡄
⣿⣦⣤⣤⣴⣶⣶⠟⠉⠑⠶⣤⡈⢻⣿⠁⢀⠘⠳⠆⠀⢀⣀⡀⠀⠀⠀⠁⠀⠃⠀⠀⣠⣄⠒⠉⠁⠀⠀⠀⠀⠀⣉⠑⢀⡈⠉⠙⣿⠶⣶⣶⠿⠋
⠈⠛⠛⠛⠋⠁⢸⣄⣈⣁⣤⡉⠀⠈⣿⣧⠈⠁⢀⡀⠀⠀⣀⠈⠛⢛⣶⣶⣶⣶⣶⣄⠀⠙⠛⠿⣶⣤⡀⢀⠀⡀⠀⠀⠈⢉⡉⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⣿⠀⢀⣿⡿⠀⠀⠘⠛⠀⠀⢈⣤⣾⣿⣿⣿⡿⠛⠉⠉⠳⣄⠀⢀⠈⠙⠇⠈⣾⠃⠀⣴⠿⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢸⣿⠀⢀⣤⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢹⡆⠸⠷⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⠏⣾⣿⠁⠰⣾⣿⣿⣿⣿⡿⠛⠻⣇⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢴⣿⣷⡀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣤⠀⣄⣼⣿⠃⠀⣰⣿⣿⣿⣿⣿⠁⠀⠀⠈⢷⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠙⢷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣠⢿⣦⢻⣿⡿⠷⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⣿⡄⢸⣆⠀⠀⠀⠙⠦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡼⢿⣼⢿⣿⣿⡇⠀⠰⣿⣿⣿⣿⣿⡇⠸⣧⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⣷⠀⠉⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠛⡇⢸⡿⠀⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠇⠀⠁⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠹⠷⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⢿⡆⢠⣿⠃⢠⣿⣿⣷⣶⠄⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⡇⠀⠻⣿⣿⣧⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⡀⢰⣶⠀⠀⠀⢰⣶⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⢿⣿⣿⣀⡀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⣷⡀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡧⠀⠀⡀⠘⣿⣿⣿⠟⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⡇⠀⠀⣠⣄⠀⠀⠀⠀⣤⣄⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠸⣿⠀⠈⠛⠛⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⡇⠀⠀⠈⠉⠀⠀⠀⠀⠙⠋⠀⠀⢹⣿⡇⠀⠀⢠⡄
⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⣧⠀⣤⡄⠀⠀⢰⡆⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⢴⡆⠀⠀⠀⠈⠁⢀⣰⠿⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢹⣿⣿⡀⠈⠀⠀⠀⠈⠁⠀⠀⣠⣀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⣠⠟⠁⠀⠻⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠸⠟⠻⣷⣶⠖⠒⠲⡦⠠⣄⠀⠙⠛⠀⠀⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠘⢿⣧⠀⢶⠆⠀⠀⢛⣴⠞⡏⠀⠀⠀⠀⠀⠉⣿⠀⠀⠸⠉⠻⣿⣿⣿⣿⣿⣿⣿⡀⠀⢰⠀⠈⣿⠀⠀⢰⡇⠀⠀⠻⣦⡀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⣴⣾⡁⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⢠⡄⠀⠀⠙⠻⠿⠿⠟⠁⠀⣿⠀⠈⢀⠀⣿⠀⠀⣤⠁⠀⠀⠀⢸⣿⡄⢰⡿⠃⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⢹⣧⠀⣄⠀⠀⠀⠀⠀⠀⣿⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢾⣿⠀⣴⠁⠀⠀⠀⠀⠈⠿⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠘⢇⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⠇⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡿⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⢀⡜⠁⠀⠀⣠⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠙⠉⠀⠀⠀⡴⠁⠀⠀⠀⠁⢀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣵⣶⡶⠂⠤⣄⣠⡼⠁⣠⠞⢁⠀⠀⡈⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣿⡟⠀⠀⡀⠀⢉⣴⡖⠁⠊⠉⣀⠀⢉⢀⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⣠⣾⢃⣠⡾⢹⡇⠀⠈⠉⠉⠀⠈⠛⢿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣷⠿⢻⣿⣿⠋⠀⠈⢀⣀⠀⠀⠀⣠⠖⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⡙⣅⡀⠸⠘⣿⢠⡎⠀⠻⠿⠗⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡌⢳⡘⢿⣷⣄⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣦⠃⢸⢿⣿⣵⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⣴⣡⣾⠟⠁⢸⡟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⣿⠀⢿⡀⠀⣸⡿⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⢀⣽⣟⠧⣄⢵⣏⠁⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠋⠀⠈⢣⢹⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢰⠀⠀⣼⠉⢀⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠆⠁⠀⢼⣏⠀⠾⡆⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⠳⠀⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣆⠀⠀⢸⣿⣿⣇⠀⡀⠀⢿⣇⠀⠘⣷⠀⢀⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡹⣿⣿⣆⠀⢿⣿⣿⣿⣀⣇⠀⠈⣿⣆⠈⣿⠀⠘⢿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⣷⣹⡿⠿⠂⠸⣿⣿⣿⣿⡿⠀⠀⠺⢿⡄⠈⠀⡶⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⣁⣀⠀⠀⡉⢀⡀⠀⠀⠻⣿⣿⣿⠇⠀⠀⠀⠀⠹⡀⠸⠃⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣴⣿⡿⠀⠘⠂⠠⠤⠕⠀⠀⠸⣿⠿⠇⠀⠀⠂⠀⠀⠳⠀⠀⠀⠀⠀⠀⣶⣿⣿⠿⠿⠟⠋⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⣾⣿⠏⠀⢀⣠⣤⠶⠊⠁⠀⠀⣠⡤⣤⡀⠀⠐⢦⡀⠀⠀⠣⠀⡇⠀⣠⣿⣿⣿⠏⠀⠀⠶⠟⣛⣉⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣴⣿⡿⠁⠀⠾⡻⠋⠀⠀⣠⣴⢶⠀⣿⣎⣻⣷⡄⠀⠀⢻⣆⠀⠀⢰⡇⢀⣿⣿⣿⠃⠀⣠⣶⣾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣠⠟⠋⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠈⢿⣤⠿⣿⡄⠀⠀⠋⠁⠀⢸⡇⢸⣿⠟⠁⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣿⠁⢀⣴⠋⠀⠀⠀⠀⠐⣾⡿⠟⠂⠀⠀⠀⣀⡀⠀⠈⠿⣷⣿⡧⠀⠀⠀⠀⠀⣼⡇⠙⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣀⠎⢸⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣠⣾⣿⣞⠀⠀⠀⠉⠻⣧⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠊⢙⣠⠛⠄⠀⢠⣿⠇⠀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⡭⠀⠀⠀⠀⠹⣧⡀⠀⠀⢠⣿⣷⠀⠀⠀⣇⠀⠿⠶⠦⠌⠉⠛⠛⠛⠉⣠⡄
⠀⠀⠀⠁⠀⠰⡄⠀⢠⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⡄⠉⠀⠐⣿⣿⣧⠀⠙⣷⠀⠠⢼⣿⣿⣿⠂⠸⠿⠀⠀⢀⣠⠄⣠⡄⠀⢀⣼⡿⢃⣴⡶
⠀⠀⠀⠀⠀⠀⠁⢠⣿⠏⠀⢹⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⡾⠋⠀⠀⠀⠻⣴⡾⠀⠀⠈⡡⠀⡀⠀⠙⠁⠀⠀⢀⣤⠖⠋⠀⢀⣿⣿⢀⣾⣿⣵⣿⢟⣵⣶
⠀⠀⠀⠀⠀⡴⢻⣿⡿⠄⠀⠀⠙⠻⠻⣿⣿⣦⠀⠀⠀⡼⠁⠀⠀⣀⠀⠀⠘⣷⠀⠀⠀⠀⠂⠐⠁⣀⣤⣶⠞⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣛⣥
⠀⠀⠀⠀⠘⠁⣻⡿⠁⠀⠀⡀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠁⠀⠀⣼⣿⡟⠂⠀⠹⡄⠀⠠⠋⢘⠟⠁⣡⠞⠁⠀⡀⣼⣇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⢠⣶⣿⠁⠀⣠⠞⠀⠀⠀⡀⠀⢻⣿⣿⡀⠀⠀⢀⣼⣿⠟⠀⢀⠀⠀⢇⠀⠈⠁⡝⠒⢺⠁⠀⣀⠀⠙⣾⣿⡀⢾⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⢠⣽⣿⠃⢀⣴⠏⠀⠀⢀⣤⠐⠀⠀⠻⣿⡇⠀⠀⣾⣿⡏⠀⢰⣟⡿⠀⠈⠀⠀⠤⠃⠀⡁⠀⠀⣿⣇⠀⠀⢻⡇⠀⠈⠻⣿⣿⠟⠁⣴⣿⠦⣄
⠀⠀⠀⠈⠛⠁⢠⣾⠏⠀⣠⣶⣿⣏⠀⠀⠀⠀⠙⠃⠀⣰⠟⠉⠁⢠⣿⠟⢁⡀⠀⠀⠀⢰⣤⡈⠫⣦⡀⠸⣿⡆⠀⠘⣷⠀⠀⠀⠀⠁⢀⣾⣿⢫⠊
⠀⠀⠀⠀⠀⣰⣿⣿⠀⠼⣯⣽⠟⠛⠳⢤⠀⢀⣴⠆⣰⠏⠀⠀⠀⡿⠋⠀⠘⢛⠐⡀⠀⠈⠻⣷⣄⠈⠙⠀⢹⣿⡀⠀⢹⠀⠀⠉⠂⢀⣾⣿⡳⠃⠂
⠀⠀⠀⠀⠀⠉⠁⠋⠀⠀⢸⣟⣷⡀⠀⠀⡴⠟⠁⡰⠃⠀⠿⠶⣤⡀⠀⠀⠱⠈⠀⠠⠀⠀⠀⠙⢿⣷⡀⠀⠀⢿⣧⠀⠘⠀⢸⢀⠃⣼⣿⠟⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣷⣦⠀⠀⠀⠁⣤⣄⠀⠀⠀⠈⠁⠀⠀⢠⠈⡀⠀⠀⠀⠠⠀⠹⣿⣦⡀⠘⣿⣇⠀⠀⠈⠁⣰⡿⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢿⣦⣜⢿⣷⡀⠀⣶⡆⠀⠀⠀⠀⣤⣴⡀⠀⠀⠀⠀⠈⠻⣷⡄⠈⣿⡄⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠟⠁⢾⣦⣝⣿⣿⣿⣿⣤⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠢⣄⠀⠀⠙⢿⣄⠸⣷⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠘⣿⣿⣇⠀⠀⠈⠻⣦⡀⠀⠹⣗⡭⣃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣰⡟⠸⣿⣿⡄⠀⠀⠀⠘⡿⣷⡄⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠟⠁⢠⢿⡇⠀⢻⣿⣿⡄⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠺⠷⠁⠀⠈⠻⠛⠃

⣶⣶⣶⣶⣾⡾⢷⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⡶⠿⠷⠿⠿⣶⡶⠶⠶⠶⠶⠾⢷⡶⠿⠶⠶⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⡴⣾⣿⣟⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣹⣿⣦⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⣿⣿⡗⡆
⠀⠀⠀⠀⠀⣸⠱⠏⠀⠀⢰⣾⠄⠀⠀⠀⠀⠀⢸⣿⡏⠁⠀⠀⢻⣧⠀⠀⠀⠀⠀⠘⢧⠸⠋⠙⢻⢣⣇
⠀⠀⠀⠀⠀⣿⣄⠂⠀⠀⢀⣿⡆⠀⠀⠀⠀⣤⣾⣧⢷⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠰⣿⣩⡦⣘⣣⢾⣿⡷⠀⠀⠀⠀⠘⢿⣿⣶⡃⠶⢢⣿⠏⠀⠀⠀⠀⠀⠀⠀⣄⠖⡠⢸⠏⢸⡇
⠀⠀⠀⢀⡴⣯⢱⡆⢀⣅⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⣯⣀⣾⡇
⠀⠀⠀⠘⢧⣬⡟⠀⢸⡟⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⡇
⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠈⢸⠛⠁
⠀⠀⠀⠀⠀⢀⣤⠶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢷⣺⣯⣶⣄
⠀⠀⠀⠀⢠⣿⠛⣾⠿⠟⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣧⢱⣿
⠀⠀⠀⢠⣾⢿⠗⠝⠛⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⡄⡀⡀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⠏
⠀⠀⠀⠘⠿⢿⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢍⡁⠀⠉
⠀⠀⠀⠀⠀⠊⠀⠀⠂⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢢⡄
⠀⠀⠀⠀⠀⣠⠞⠋⠙⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠿⠟⠉⢻⣿
⠀⠀⠀⠀⣠⣿⠀⣠⠔⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⣿⣽⢧⡷⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⢠⠻⠃⠠⠟⠻⣝⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⢠⣾⣿⣇
⠀⠀⠀⠈⣿⠱⢾⣦⠑⣣⣿⣄⠀⠀⠀⠀⠀⢠⣿⣠⠂⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠞⠟⣿⡆
⠀⠀⠀⠘⣟⢣⡞⠋⠀⣿⣤⠟⠀⠀⠀⠀⠀⣸⣌⠻⡀⢤⠄⡰⡿⠿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣴⣶⣾⠋⠉⠁
⠀⠀⠀⢰⣿⡿⠋⠀⠀⠘⢻⣷⠀⠀⠀⠀⢸⡏⢁⣤⡯⠐⠊⢷⣉⢻⡿⠀⠀⠀⠀⠀⠀⠀⢿⠋⣋⢸⠄
⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠁⠻⠟⠁⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠰⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣄⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣻⣟⣷⣦⣀⣄⣄⠀⣠
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⡉⠙⣿⣿⣿⠋⠀⢻⣿⡿⠋⣽⣿⣿⠘⠀⠙⠻⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣷⣄⠈⠛⠁⠀⠀⠈⣿⠁⣼⣿⣿⣿⢸⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⡋⠉⠉⠉⠳⣄⠐⢾⡴⣼⢣⣾⠋⢛⣿⡟⢸⣷⣦⣀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠃⢨⣄⣵⣿⣿⠰⣿⣿⡇⢸⣻⣿⣿⣷⣦⣄⡀⠀⠀⠉⠻⠿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⢾⣷⣾⡟⣛⣿⠀⠋⠉⡇⢸⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣤⣛⣿⣿⢡⣀⣧⣰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⢾⣯⣿⣿⣿⣿⡿⣾⣿⣿⡇⢸⣿⣿⣿⣿⣿⠟⠙⠻⣿⣿⣿⣿⣶
⢤⢠⣔⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⠀⣾⠀⠀⠀⣿⣦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡿⠟⢉⣾⣷⠀⣼⣿⣽⣿⣿⣿
⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⢠⣿⡇⠀⠀⣿⣿⣿⣷⣼⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣡⣴⣿⡿⠥⠴⠿⠿⠿⠿⠿⠿
⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠒⠚⠛⠓⠒⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣤⣦⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢀⠁⣸⣿⣿⠇⠀⢀⣶⣿⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠨⣿⠿⠃⢀⣰⣿⡻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⣀⣀⠀⠐⠄⣠⣶⡟⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣣⠾⢿⣯⣎⢰⡏⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⣄⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢴⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⡀⠀⠀⠀⠀⠀⢀⡄⠚⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠉⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⡀⣠⣟⡿⠶⠶⣤⣀⣀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠐⢦⡤⠀⠰⡿⣿⣿⠆⠀⠀⠀⠀⠀⣾⣴⣿⣿⣿⣓⣀⣨⣍⣛⢿⣗⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⣦⡀⠸⢿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⢿⡋⠙⣿⣿⣆⢻⣿⣿⣶⡀⠀⠀⠀⠀⠀⢰⠀⢨⣿⢿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢛⣿⣿⣿⣿⣿⡻⡄⠀⠀⠀⠀⠈⠀⠘⠻⣪⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢾⡇⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣸⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⢠⣜⣻⣿⣿⠀⠀⠀⠀⠉⠐⣥⣀⡀⠆⠀⣹⣯⠙⣿⡿⢋⣿⣟⣽⣧⣿⡿⠀⠀⠀⠀⣃⠀⢠⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢸⣿⣷⣆⠘⠀⠀⠀⠀⠀⠠⠜⠁⠁⠀⠀⠘⢿⣦⠈⢿⢸⠌⣿⣼⢃⣿⠀⠀⠀⠀⠀⣷⡀⢸⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⢀⠒⣀⠀⠀⠀⠀⠀⠘⣳⠀⠀⠀⡼⢿⣿⣾⠟⠀⠀⠀⠀⠀⣿⡇⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⢻⣿⣿⠀⠀⠀⠀⢺⣗⠚⢧⣄⠀⢀⣠⣼⡉⠀⠘⣿⣧⣞⢋⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⣶⣿⡟⢻⠀⠀⠀⠀⢘⡻⣿⠿⠟⠀⢾⣟⠟⠃⠀⢸⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⡧⠀⢠⣿⣿⡟⢹
⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁⠀⢻⣿⣿⣿⡇⠀⠀⠀⢀⡛⠁⠀⠀⣾⡿⠗⠀⠀⢀⣼⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠉⢹⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⡿⠁⠀⠀⠀⠈⢧⣶⠀⣤⣭⢴⣦⣶⣿⠿⢿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠘⠂⠒⠲⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⢰⣿⣿⣷⣤⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠁⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣤⣀⡀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣦⡀⠀⠀⠀⠀⣀⣤⣴⣆⠈⠀⠠⣤⣶⣤⣤⡿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣜⢿⣿⣼⡇⢠⣴⣶⣿⣿⣿⣿⣿⣷⣮⣤⣈⣹⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣾⢟⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠉⠀⠙⢿⠗⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠹⢟⣵⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠘⣿⡧⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⡾⠋⣴⣶⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢴⣶⡘⢷⣄⠀⠀⠀⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠰⡽⠇⣼⡟⢠⣿⡇⠀⢠⣶⠄⣤⠀⣤⡄⢤⣤⠀⠀⣾⣧⠸⡦⡈⢿⡷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠾⠿⠀⠀⢸⡇⢰⣟⠀⢽⡇⠀⣷⠀⠀⢹⡿⠆⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣭⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠿⠿⠿⠿⠿⢭⣽⣓⠂⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣾⣛⣛⣛⣛⣛⡿⢶⣦⣍⡛⠶⣄⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣭⣙⣛⡛⠻⠿⣿⣿⣶⣝⠻⢷⣌⠻⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠉⠛⠿⣿⣷⣦⣍⠻⣿⣷⡤⡝⠳⡌⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⢉⠻⢿⣷⣼⣿⣿⡹⣆⢢⡀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠸⣷⣄⠙⣿⣿⣿⡇⢿⣆⢳⡰⡰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣦⠸⣿⣿⡇⢸⣿⡄⣷⣷⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡿⡄⣿⣿⠇⣿⣿⡇⣿⣿⡇⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡟⣸⣿⣿⢣⣿⡿⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣼⣿⣿⢫⣿⡿⢁⣾⢏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⣵⡿⠋⣠⠟⢡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢫⣾⢟⡔⢈⣵⢢⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢡⣿⣏⣾⠃⣾⡇⡟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⢿⡘⣿⣦⣿⡇⠁⠀⣩⣰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⣯⣁⠹⣿⣿⣷⡀⢒⡬⠥⢞⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢦⠻⣷⡙⢿⣷⣌⡛⢿⣿⣶⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠈⣷⣮⡓⠀⢌⡛⢿⣦⠹⣯⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢜⣼⡿⣿⣦⡠⣺⢷⠙⡧⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠙⢛⣭⣾⣿⡿⠃⠐⢚⣼⡇⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠀⠀⠀⠰⡟⠋⠉⠁⠀⠀⠀⠴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠞⣄⣡⠤⡠⣴⢀⣄⡤⡰⢓⣄⡤⠀⢀⣤⡤⣠⣤⢄⣤⡞⠁⠀⣠⢋⣤⡤⡠⡆⣰⣫⣠⢌⣤⣆⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠙⠊⠁⠙⠈⠉⠃⠛⠈⠽⠃⠀⠈⠈⠃⠉⠙⠈⠉⠃⠀⢠⡻⠉⠉⠋⠙⠉⠉⠉⠙⠈⠋⠉⠉⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡀⣀⠄⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⣒⠟⢓⢦⡹⡆⢄⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⡀⠀⠀⠐⢤⡀
⠀⠀⡠⠤⠦⠞⢁⡤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣰⣫⠖⠒⠂⠀⠑⢣⠞⠀⠀⠀⠀⠀⣰⠋⢀⡴⠋⣀⡠⠆⠈⢢⠱⡄
⠀⢸⣷⠄⠀⣀⢧⡀⢈⡀⠀⣀⠀⠔⢦⠀⠀⠠⡇⠏⠀⣀⠀⠀⠀⠈⣙⠀⠀⠀⠀⢀⡏⢀⡞⠀⠈⠁⠀⠤⢠⠈⡆⢳
⠀⠈⠦⠄⠸⣤⡵⠃⠈⠁⠀⠋⠀⡤⠊⠀⠀⠀⢇⠀⠁⠀⠀⠀⠀⠖⠘⠀⠀⠀⠀⢸⡇⢸⠀⠑⠀⠀⠀⠀⡠⠀⣇⢸⡆
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⢦⡀⠀⠀⠀⠀⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠁⠀⠀⠀⠀⠐⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⡀
⠀⠀⢀⡖⣫⠽⠂⠲⠶⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣩⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⣈⠉
⠀⠀⢸⡈⠃⠁⠲⣶⠛⠉⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠋⠁⢀⡞⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁⠀⠉
⠀⠀⠀⠑⠀⢉⡋⠁⠀⠀⠄⢠⠀⠀⠀⠊⠀⠀⣀⣀⣤⠴⠋⢰⠇⠴⠀⢠⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠂
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠈⠁⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠁⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⡒⢀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠐⢆⠠⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠈⠀⠘⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡐⢀⠃⠰⠀⠈⠀⠈⠀⡆⠀⠀⠀⠀⠀⢌⠀⠀⡀⠆⡀⡄⡀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠤⠃⠀⠀⢠⠀⡀⠀⢈
⠀⠀⠀⠀⢀⠃⣸⠀⡇⠘⠂⠀⠒⣳⠃⠀⠀⠀⠀⠀⠁⢰⠈⠀⠀⠀⢡⠐⢀⠄⠀⠀⠀⠀⠈⠂⠸⡖⠀⠐⠃⠈⢲⠌⠂⠂
⠀⠀⠀⠀⠌⠀⣿⠀⠇⠀⠀⠤⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⢰⣃⠰⠄⠀⠀⡹⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⢶⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⡀⠀⠀⠀⢠⣶⣾⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣠⠂
⠀⠀⠀⠘⣿⣿⣿⣿⣧⢠⣿⠀⡀⠀⣼⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡞⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣼⣿⢠⣿⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢸⣿⣾⣿⣿⣿⣿⠃⠀⠀⠀⢀⡴⢋⠏⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠇⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⢹
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡃⠀⣿⣿⣿⣿⣿⡟⠀⠀⡰⠊⡛⢀⣼⠀⠀⠀⠀⠀⠀⡔⠘⠋⣀⡎⠀⠀⠀⠀⠀⢀⡃⢈⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠇⠀⢸⣿⣿⣿⠟⠀⠀⠀⠑⠒⢹⠁⢸⠂⠀⠀⠀⠀⠀⠓⠂⢛⠁⡹⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⡟⠀⠀⠀⠀⠀⠀⢀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⡈⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⡫⠊⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢏⠂⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣽⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣿⠿⠿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏
⠀⠀⠀⢠⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣝⢽⠋
⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⠟⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣄⠀⠀⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠛⠿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⡄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣰⣾⠀⠀⠀⣼⣵⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠙⢿⡄⠀⠀⠙⠿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⣼⡿⠀⠀⢀⣾⡟⠁⠀⠀⣴⠟⠁⠀⢼⣿⡷⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣰⣿⠋⠀⠀⠀⠈⠀⠀⠀⢀⣿⣾⣷⣇
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣵⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠟⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⠥⠵⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣘⢤⣬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠘⠛⠁⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠁⢂⣠⡔⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⡿⢷⡤⢀⣀⡴⠂⠀⠉⠓⠠⢄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⢵⣶⣿⠥⠤⠀⠀⢄⠀⠀⠈⠉⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣽⠉⠀⠀⠺⠂⠆⡄⠈⠀⢁⠀⠀⠀⠐⣀⢀⠐⠁⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢉⠐⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠈⠢⠄⠀⠀⠙⢄⠀⠀⣴⢳⣀
⠀⠀⠀⠀⠀⢠⠖⡙⠀⠤⢀⠀⠐⡳⠀⢧⠀⠀⠀⠀⢀⡀⠈⠢⡀⠀⠉⠲⡤⡄⢩⣆
⠀⢀⣀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠉⠛⢦⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⢃⠘⠄⢻⡆
⠀⠈⡿⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠖⣮⠐⣤⠼⡧
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠚⢻⡟⠁⠀⠁⡼⠁
⠀⢸⠀⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡄⠀⠀⠀⢠⠟⠀⡀⡠⠂
⠀⣾⠀⡥⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⢄⡀⠀⢰⠃⣠⠖⡹⣧
⢀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠱⠤⠁⠀⡀⢘⠃⢹⣆⡀
⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠄⠁⠀⠀⢛⡀⠘⠳⡈⠁⠀⠈⡁⠈⠉⠙⢦
⣼⡇⠰⠀⡆⣠⠤⠤⠤⠤⠄⠀⠀⡀⠤⠴⠖⢩⠀⠀⣀⣀⠴⠛⢧⣀⡰⠚⠀⠉⠉⠀⠀⠀⠀⠀⠳⣦⡀
⠀⢅⣀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠉⠁⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡀
⠀⠀⠸⠋⢲⡷⢿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⢟⡰⢰⣻⣿⣿⣿⣿⣿⠏⣴⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠹⣟⣿⣧⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⡟⢲⢠⣤⣄⡀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣶⣶⣯⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡏⠛⠛⠁⠛⠛⠛⠛⠛⠛⠛⠛⢳⡆⠛⠛⠛⠛⠋⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⢸⣿⡇⢀⡾⢧⣄⠀⠀⠀⠀⠀⠀⠀⠈⠃⡆⠀⠀⠀⠀⠀⣀⣀⣀⣿⡏⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠸⠧⠒⠛⠓⠒⣶⣶⣶⣶⣶⣶⣄⣻⣲⠖⠿⢯⣿⣟⣿⣛⣿⠀⠀⠙⢿⣿⣿⣷⣀⡴⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⢩⢿⡇⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⡿⠆
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠉⢀⣠⠖
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢑⠂⠉⠉⠉⠹⣃⣹⣿⣿⣿⣿⣿⣿⣿⣏⣿⠀⠀⠀⠀⠈⣿⠿⠛⠉⣱⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⠿⠟⠛⣉⡠⢿⣶⣶⣠⣤⠚⠁⡄⢀⣼⣿⠿⠏
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⣿⣇⡠⡄⠒⠈⠋⠘⠄⠒⠈⠀⠍⢀⠰⡟⠉⠉⠁⡌⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣶⣞⣿⠀⠀⠀⠀⠀⢐⢸⢇⡐⠈⠆⠂⠈⢀⡠⠀⢀⣀⣤⣾⡷⢇⣒⣂⠠⠚
⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡠⠽⠟⠛⠟⠀⠀⠀⠀⠀⠀⢂⣁⣌⣹⡟⣓⣦⡀⠀⢀⠠⠄⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠈⠉⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣄⣈⡉⠃⠀⠀⠀⠀⠐⠂⠀⠀⠈⠉⢀⣀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣷⣶⣾⣿⣿⣿⡿⠿⠿⠛⠛⠁

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣀⡀
⠀⠀⠐⠁⠉⢄⠀⠀⢰⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⣤⣿⣭⠁⠀⠀⠀⣀⣡
⠀⠀⠘⠦⠠⠓⠀⠀⠺⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠁⠀⠀⡄⠀⠀⠀⠀⢹⣝⠋⠀⠀⠀⠀⠉⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠈⠉⢱⣶⡾⠁⠀⠀⠀⠀⢢⣉⠀⠀⠀⠀⠀⠆⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⡷⠆⠰⢾⡄⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠄⢳⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣥⠒⠒⣬⡞⠀⠀⠀⠀⢠⠭⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣷⡆⠸⢛⠀⠀⠀⠀⠀⣴⣶⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣧⡄⠰⢞⠀⠀⠀⠀⠀⠋⣿⣏⠀⠀⠀⠀⠀⠐⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⢀⣿⣿⣿⡟⢛⡆⠀⠀⠀⠀⣇⡄⢰⠾⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡏⡅⢠⣾⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⡴⠏⠁⢨⠴⢦⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠇⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣞⠻⣿⣿⡏⡀⠀⠀⠀⠁⡇⠀⠀⢸⠈⠀⠀⠀⠀⠸⠆⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡧⠀⠀⠀⠀⠀⠀⠸⡿⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⣿⠟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠦
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⡿⠃⠀⡠⠖⠉⠀⢀⠀⣤⣶⣷⡄⠀⢳⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠟⠀⠀⠀⠀⢀⣴⠾⠛⠁⠘⣿⣿⣿⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⠀⠈⠁⠀⠀⠀⠀⠘⠟⠉⠁⠀⠀⠀⡀⠄
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠂⠋⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣆⠀⠀⠀⠠⠇⢀⣶⠄⠰⠁⠒⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⡄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣰⡿⠁⠀⠇⠀⠠⠴⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⡄⠀⠀⠀⠀⠀⠘⠋⣼⡟⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠤⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠁⣀⡄⠀⠀⠀⡜⠁⠀⢀⣀⠀⠈⠀⠰⡄⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄⠀⢷⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⣆⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⣶⠀⣾⢛⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⢿⣄⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢻⡆⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠘⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⡿⠀⠀⠀⠀⠀⠀⣿⠁⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠛⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠹⣤⡀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⠎⠀⣠⡾⢆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢻⣿⠀⠀⠀⠀⠀⠈⢧⡾⠋⢀⠜
⠀⠀⠀⠀⠀⢰⣿⠋⠛⣿⡆⠔⡄⠀⠀⠀⠀⠀⢀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁
⠀⠀⠀⠀⠀⠸⣿⣦⣴⣿⠃⠐⠁⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡀


⠀⠀⠀⠀⠀⣶⣦⣄⣠⣤⣤⣀⠀⠀⠀⠀⠀⡴⢿⣿⣻⢷⣄⠀⠀⠀⠀⠀⠀⢠⠶⠶⣄⣀⠀⠀⠀⠀⠀⠀⣠⣶⡆⡀
⠀⠀⠀⠀⠀⠛⠛⠿⣿⣽⣿⣿⡇⠀⠀⠀⠀⠇⠀⢹⣮⢷⣹⠆⠀⠀⠀⠀⠀⢸⣦⣴⡏⣭⠀⠀⠀⠀⠀⠀⠹⣿⣷⣇⣾⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠋⠀⠀⠀⠀⠀⢰⣖⣸⣿⣾⡿⠀⠀⠀⠀⠀⠀⢸⠁⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠰⠏⠝⠛⠻⠁⢧⠄⠀⠀⠀⠀⠀⢸⣤⣤⡇⠿⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⡿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣠⡄⠀⠀⠀⠀⠀⠈⠋⣠⠄⠀⣤⡉⠁⠀⠀⠀⠀⠀⠀⢞⢕⡇
⠀⠀⠀⠀⠀⠐⠂⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⡫⡂
⠀⠀⠀⠀⠀⠈⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣾⡗
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣰⡲⣦⣍⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⠾⣭⣛⢶⡿⢯⣖⠵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⢷⡀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣰⣏⡻⣶⣽⣯⣛⠷⣭⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠻⣄⠀⠀⠀⠀⢀⠞⠀⠀⣜⡳⣮⣟⣷⣻⢶⣝⡳⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⣦⠀⠀⢀⠎⠀⢀⡾⣭⣛⢶⡽⣫⣟⠷⣮⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⢧⣠⠋⠀⢀⣞⡻⢶⣽⣫⣚⠷⣭⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⣳⡄⢀⡾⣮⣙⡷⣫⢶⣝⡻⢦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡿⠾⠙⣾⣛⢷⣽⣻⣟⡷⣮⣙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⡄⠀⠈⢻⣻⣚⠷⣭⡛⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠹⡐⢢⣄⡙⢽⡻⢦⣝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡠⢳⢸⠦⢭⣮⡻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⠎⠀⠀⢇⢇⠀⠀⠙⢌⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⠇⠀⠀⠀⠸⡘⡄⠀⠀⠈⢇⢀⡭⠿⢵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇⠘⢄⠀⠀⢸⢸⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢰⣣⠖⠒⢾⣦⡼⢾⢆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢩⣇⠀⠀⠀⠹⣾⠉⠪⢝⣶⡾⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠘⣜⢄⠀⠀⢠⣿⠀⠀⠀⠀⠀⠈⢫⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⠿⠝⠋⠀⠀⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡋⡥⣽⣷⣿⣿⡟⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠐⣾⣿⡿⠿⠿⣍⣁⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⢆⡼⠛⠉⠁⠀⠀⠀⠙⢷⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⡎⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⣄⣧⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢻⡿⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⠸⡻⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣜⠿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣯⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣟⠹⣿⣍⣣⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣳⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣟⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⡁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠈⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⢄⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⣤⣸⡅⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠤⣄⣯⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠉⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡏⠑⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⡄⠀⠀⠀⠀⢠⣿⣿⣯⣿⣤⣆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡋⠀⠀⠀⠀⣰⣿⣿⣯⣿⣿⡿⠃⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠛⠷⣿⣿⡿⠁⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣴⣀⡄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁
⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠟

⠀⠀⠀⠠⠆⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⠂⠖⢠
⠀⠀⠀⠖⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡌⡟⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠈⠀⠀⠐⣦
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠁⠀⠀⠀⡴⠡⠊⠀⠀⠀⠀⠀⠀⠀⠶⠱⠀⠰⠀⠀⢈⠁
⠀⠀⠀⠀⠁⠴⡀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡞⠀⠀⢀⠞⡐⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⠀⣠⢋⠜⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢈⠙⠛⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡏⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⢘⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⢈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣉⣙⣏⣩⣏⣩⣍⣩⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣤⣶⠾⠿⢿⣿⣛⠓⠲⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣛⡛⠛⠶⠦⣄⡉⢳⡄⢀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣿⢻⣿⡛⠿⢭⣉⠒⢦⡈⢳⡄⢿⡌⡇⢻⡽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⡟⣆⠙⢿⣕⡲⢬⡻⣆⣷⣹⣷⣿⣿⣿⣾⣷⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣳⣜⢮⣙⢦⣜⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⡋⠁⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠉⠙⠛⠛⠋⠭⠒⠁⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠰⣿⣿⣿⣶⣤⣠⡾⣦⣤⣴⣾⣿⣿⡷⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠻⠿⠿⢿⡿⠟⣤⡜⠿⣿⠿⠿⠿⠃⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⣀⠀⠀⠀⣸⣿⣿⡀⠀⠀⢀⡀⠀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣧⠀⠀⠉⠁⠉⠁⠀⢰⡿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢿⡒⠒⠒⠲⠲⠒⠒⣺⢳⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠘⢯⣭⣤⡀⣠⣬⡽⠟⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⣹⣿⣿⣿⣿⣻⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣽⣿⣿⣿⣿⣿⣿⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⢠⠖⢉⣭⠉⠯⣭⣖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣧⡈⠀⠁⠀⣽⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀⣀⣀⣀⣀⠀⠀⣿⣿⣧⠀⣠⣷⡞⠓⠉⠉⠱⣶⣿⢱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⠧⡅⢹⠀⡏⣿⠀⠀⣿⣿⣿⣶⡷⢟⣿⣉⣩⣭⣭⣯⣩⣚⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠛⠃⠘⠀⠛⠋⠀⠀⣿⣿⡟⠀⠙⣿⢟⣻⣿⣿⡟⠛⠻⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⢸⣿⠟⠋⠀⢰⣾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⠈⢀⣠⣾⡇⠘⠛⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣄⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠲⠶⠿⠿⠿⠿⠿⠷⠶⠞⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⠀⣾⣿⠀⢸⣿⣶⡄⢰⣶⣶⡄⣶⡶⠶⢰⣶⣶⡆⢠⣶⢶⡆⢰⣶⣾⡗⢠⣿⢿⡆⢸⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⠀⣿⣿⡇⢸⣯⣿⠇⢸⣯⣿⠇⣿⣧⡀⢸⣿⣼⡇⠘⢿⣎⠋⢸⣿⣽⡇⢸⣿⢸⡇⢸⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⢀⣿⣽⡇⢸⡟⣿⡇⢸⡟⣿⡇⣿⡏⠁⢸⣿⢹⣗⢀⣌⢿⣧⢸⣿⣻⣇⢸⣿⢸⡇⢸⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⢸⡿⢹⡇⢸⠗⠿⠇⠸⠿⠿⠃⠿⠿⠷⠸⠿⠸⠿⠈⠿⠿⠏⠠⠿⠿⠇⠘⢿⣾⠇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⣀⣀⣀⣤⣤⡤⠤⠤⣶⣶⡶⠶⠶⢶⠶⠶⢶⣶⡶⢶⠶⢶⠦⠤⠤⠤⣤⣤⣄⣀⣀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⢠⣴⣿⡿⠃⠀⣠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⠸⠟⠁⣠⡄⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⡰⣿⣷⣶⣿⣿⣧⣼⣾⣿⣦⠀⠀⠀⣤⢠⢤⡀⢤⡄⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣷⣶⣯⣭⣉⣉⡩⡭⣭⣤⣾⠇⠀⠀⣿⢘⣺⠇⣼⠁⣟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠘⣟⡷⠤⠤⠤⠼⢿⣿⡞⡏⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡟⣿⣿⠀⠀⢄⣀⢙⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠈⠺⢽⣗⣄⣉⡭⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⠖⠐⠼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠎⡀⢀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠆⠸⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⣾⣤⣤⠀⠀⠈⢃⣠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠟⠟⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⠄⠐⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠌⣰⠛⣿⡏⢰⣤⣤⣤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣈⡂⠁⠀⠘⢴⣷⠟⡩⠀⠋⠉⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⡴⠃⡀⠀⠈⠁⠀⠀⠀⡠⣴⢿⠻⠁⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣤⣘⠀⡿⠀⠀⠭⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⢹⣹⡗⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣺⡅⠀⠀⢰⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠈⠑⢂⣴⠟⡡⣪⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣟⣥⣀⣴⠃⠟⣿⡜⡱⡿⠫⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⣻⠝⢁⠈⠀⠀⠀⠐⠉⢠⡄⠈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠋⠀⠈⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠈⠀⠀⠐⢀⠀⠀⠖⢀⠀⠠⠠

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

⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⢀⣠⢤⡀
⠀⠀⣿⠀⠀⢠⡎⠁⠀⠀⠈⠙⡆⠀⣼⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣀⠀⠀⢸⠀⠀⢸⠀⠀⢹⣿⠀⠀⠀⢸⣿⠀⣼⠁⠀⢳
⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠘⠿⢿⣿⣿⠿⠿⠛⠛⠛⠛⠙⠉⠀⠀⠀⢸⠀⠀⢸⠀⠀⠸⣿⠀⠀⠀⢸⣿⠀⣏⠀⠀⢸
⠀⠀⣿⠀⠀⠀⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢾⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⢹⡀⢀⡟
⠀⠀⣿⠀⠀⠀⠀⠉⢲⡶⠋⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⠀⠀⠑⣿⠀⠀⠐⢾⣿⠀⠀⢹⡟
⠀⠀⣿⣦⡀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠘⠀⠀⠀⣸⠀⠀⠀⢸⣿⠀⠀⢸⣇
⠀⢰⣿⡆⢹⡄⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠚⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣞⡿⠀⠀⢸⡿
⠀⠸⣿⡇⠀⠁⠀⠀⠘⠃⠀⠀⠀⢀⠀⠀⢆⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁
⠀⠀⣿⡄⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠑⢄⠸⣆⠀⠀⠐⠋⠁⠀⣠⣀⣤⡄⠀⠀⠀⠀⠀⠀⠰⣶⣶⣦⣶⣶⡶⠀⠀⣰⡾⠿⢿⣦
⠀⢠⣿⡇⠀⠀⡴⠁⠀⠀⠀⠀⠉⢦⠀⠈⠳⣿⡄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⢈⣿⡏⠀⠀⠀⠀⣿⣅⠀⢠⣿⠂
⠀⠘⣿⡇⠀⣴⠃⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠈⢷⣦⡀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠛⠿⠛⠁
⠀⠀⣽⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢰⠏⠑⢦⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡳⢦⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣤⡀
⠀⠀⠛⠀⠀⠀⠫⢗⠤⠤⠤⠴⡺⠏⠀⠀⠀⢀⠼⡀⠀⠀⣍⠈⣢⠀⠀⠀⢻⣻⣿⣿⣿⣿⣷⡄⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⣿⡇
⠀⠀⢀⡀⠀⠀⠀⠀⣬⣽⣮⣥⠀⠀⠀⠀⠀⢆⢀⡼⠀⠀⠀⠈⠉⠲⠀⠀⠀⠁⠙⠛⢿⡿⣿⣿⠆⠀⢸⣿⣿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⣿⣿⣧⠀⢠⣤⣤⣤⡄⠀⠀⢰⣷⠀⠀⠀⠙⣿⣿⣿⡄⠀⠀⠀⠀⣹⣿⣿⣿⣶⢤⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠉
⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⣴⣾⣿⣶⡄⠀⠀⣿⣿⠃⠀⠀⡆⠀⣠⠟⠋⠉⣿⣿⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⡇
⠀⠀⣿⣿⣿⠀⠸⣿⣿⣿⠃⠀⣿⣿⣿⣿⡇⠀⠓⣿⡇⠀⠀⢀⡇⠈⠁⠀⠀⢸⣿⣿⡁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇
⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⣼⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡷
⠀⠀⣿⣿⡿⠀⠀⢽⣛⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⡟⠀⠀⠀⣿⣿⡁⠀⠈⢿⣿⣿⣿⣿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡙⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠤⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠚⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠁⠀⠀⣈⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⡤⣦⡀
⠀⠀⠛⠁⠀⠈⢷⡀⠀⢠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠈⢷⣶⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⢀⣿⣷⣦⣄⡀
⠀⠀⢀⣴⣧⡀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⣴⣿⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠙⣦⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣂⠒⠁
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠘⠿⠇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣩⣶⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠉⠀⠰⠾⠿⣿⡿⠉⡁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢹⣿⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿⠃
⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⣤⠶⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⣠⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⢀⣴⣿⣿⣿⣿⠟⠀⠀⠀⡐⠄⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢰⠈⢂⠀⠀⣠⣾⡟⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⣾⣧⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⢰⡃⠀⠀⢣⣼⣿⡟⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢀⣴⣿⣿⣿⣿⣧⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣇⢀⣴⣿⣿⢏⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠗⠄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠟⠁⠈⠑⢄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠠⣤⣀⠀⠀⠀⣾⣿⣿⡿⣇⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣜⢿⠟⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠉⠛⢳⠀⠀⠀⠈⠳⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⢸⣿⣀⣨⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣥⣿⣷⣦⣀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠙⠃⠈⠻⠿⠿⠿⠷⣦⣄⡀⠀⠈⠙⠒⠲⠤⠼⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣠⠞⠋⠉⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠉⠻⣦⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡾⠁⠀⠀⢀⣼⡇⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣷⣶⣦⣀⠀⠀⠀⢿⣦⡀⠀⠀⠈⣷⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣯⡀⣀⣴⣿⠟⠀⡤⢴⣾⣿⣿⣿⡿⡿⢿⢻⡿⣿⣿⣿⣷⣶⢦⡄⠙⠿⣶⣤⣴⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠋⠀⠀⠀⡇⢸⠈⡇⢿⢸⠀⡇⢸⠘⡇⡇⢸⠀⡇⢻⢸⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠃⠘⠀⠇⠠⠸⠀⠇⢸⠀⠇⡇⠸⠀⠇⠐⠘⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋

⢀⣤⡮⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠠⢿⠏
⠀⠀⠀⠀⠀⠸⠿⢿⣿⡟⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣠⣤⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠚⠙⠓⠶⣤⣄⡀⠀⠀⠀⠀⠀⣇
⢀⣼⣿⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠀⠀⠀⠀⢿
⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠉⢰⠣⠀⠀⠀⠀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠈⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢀
⣄⠀⠀⠀⠀⠀⠀⠀⡜⢄⢁⣄⠀⠀⠀⠀⠐⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠻
⠈⠳⢦⡀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢸⣦
⠀⠀⠀⠉⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⢴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣷⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠿⠇⠀⠀⠀⠀⠀⢓⠊⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⡿⠟⠉⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢻⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⣆
⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠶⠒⠃⠐⠂⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡏⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟⣼⠇⣿⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡿⠻⠿⣿⣿⡿⢿⠟⣱⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣿⢣⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣠⣿⣯⣿⣟⣋⠙⣿⠁⠚⠁⠈⣿⡿⢉⣉⣉⡽⢉⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠙⠿⣍⣀⣀⣿⣿⡇⠙⠀⠀⠀⠀⠀⣰⠭⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠸⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠼⠿⠋⣿⣿⣿⣿⣿⣿⣿⡸⡀
⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⢹⡇⠁
⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢸⡇⠈
⠀⠀⠀⠀⠀⠀⣸⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡞
⠀⠀⠀⠀⠀⢠⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠰⢾⣿⣿⣿⣿⠿⠋⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠤⠤⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇⢹⣿⡟⠿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢠⡟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠿⠛⠛⠋⠙⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⣥⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠘⠷⣬⠝⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠛⠛⠛⠉⠀⠀⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⣬⢐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠻⠿⣦⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠰⣅⠂⢄⡙⠻⠿⠖⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢿⣧⠀⠀⠑⠲⠊⠘⠯⠁⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠠⠟⣛⣿⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⠀⣈⣢⡰⠿⠗⢀⣂⣭⣤⣶⣿⣿⠿⠛⠛⠗⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠀⢉⡉⢁⣀⣀⣀⣤⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠿⠟⠚⠛⠋⠉⢩⡄⠀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⢀⡴⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⠜⠀⠀⠀⣠⡤⢄⢠⣴⢠⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣰⠃⠀⠀⣠⣤⣴⡦⢠⣶⢂⣾⠇⢸⣿⠞⢺⡆⠀⣾⡟⣡⠎⢸⡿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠞⠋⣿⡇⣼⡟⠀⣿⡇⢀⠿⢻⣿⠀⣸⡟⢦⣾⠃⣸⣿⠊⢁⡔⠁⠀⣿⣃⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⢴⡟⠀⣿⣇⣠⣿⡧⠊⠀⢺⣯⠔⢻⡧⠞⠁⠀⠘⠻⠖⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠊⡼⠀⠀⠙⠋⠀⠉⠀⠀⠀⠀⠀⠀⣀⣤⠴⠦⠀⠀⣠⡖⢙⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣯⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⣼⣿⡠⠊⠀⢀⣤⣤⡀⠀⣠⣄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣀⢼⣿⡏⣤⡄⠀⣾⡏⠀⡇⢀⣿⠇⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡅⣼⣿⢠⣿⢁⢼⣿⠻⣰⠗⢸⣿⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣿⡇⣿⡏⠸⠿⠉⠈⠛⠓⠋⣀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣠⣾⠟⠀⠙⠀⠀⠀⠀⠀⠀⠀⠉⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣦⣤⣤⣴⣦⣶⣴⣦⣦⣴⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣴⣤⣶⣶⣴⣦⣴⣶⣶⣤
⣿⣿⣿⣿⣿⣿⣿⣽⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠰⣶⣶⠀⢰⣶⠀⠂⠀⠰⣶⣶⣶⣶⡆⢤⡐⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣖⣺⣿⣇⠀⠀⠀⢻⣿⣿⣿⣿⣤⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣻⠿⢛⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣭⣭⣿⣿⣿⣿⣿⣿⠿⠿⠟⢛⣛⣿⣿⣿⣿⡄⢠⣤⣾⣿⣿⣿⣿⣿⣧⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⢾⢿⣿⡿⡻⢿⣝⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡄⠁⢫⣿⣿⣦⣤⣿⣞⣿⣿⣷⡈⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⢡⣿⣿⣿⣷⣶⣟⣻⡾⠛⣻⣿⣿⣾⠿⣿⣧⢹⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢻⣷⣿⣿⠋⠉⢻⣿⣿⣯⣴⢟⣫⣼⠿⢿⣧⣿⣿⡆⢻⣿⣿⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⣠⣾⣿⡿⠃⠀⠀⣠⣿⣿⣿⣿⣿⣯⡵⠋⣸⣿⣿⣿⣿⣾⣿⡿⢋⡾⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷⣾⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣽⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡼⣿⣿⣧⣄⣰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣍⠻⢿⣿⣿⣿⡏⢉⣉⣋⣩⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣫⣶⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡛⠿⣿⣯⢷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣹⣿⠿⣋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠈⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣛⣷⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⣶⣿⣿⣿⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡀⢻⣿⠋⠛⠁⠹⣿⣿⣿⣿⡟⢫⡛⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣕⣺⣿⣇⠀⠀⠀⢻⣿⣿⣿⣿⣤⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⠿⡛⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣭⣍⣿⣿⣿⣿⣿⣿⡿⠿⠟⢛⣛⣿⣿⣿⣿⡄⢠⣤⣾⣿⣿⣿⣿⣿⣧⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⢶⢿⣿⡿⠻⣿⣝⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡄⠠⢫⣽⣿⣦⣤⣿⣞⣿⣿⣷⡘⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠟⠁⠿⠿⠿⠷⠶⠟⠻⠾⠛⠻⠿⠿⠞⠻⠿⠇⠹⠿⠆⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⠀⠀⣰⣷⣦⡄⠀⠀⠐⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠂⠀⠀⠀⠟⠩⠀⠀⠀⠀⠀⠈⠇⠀⠀⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢄⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁⠖⠂⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⠀⠐⠗⠀⠀⠀⠀⠈⠅⠀⠀⠀⠐⠋⠀⠀⠔
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠸⡐⡿⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⢀⡔
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠂⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢠⠄⡄⣠⠀⣄⡤⡀⢰⠲⡄⣄⢠⠀⢠⢆⡄⡄⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⡇⣏⡆⣿⣅⠃⠘⢄⠀⣹⢸⠀⡗⢸⢻⡇⠀⠀⡀⠀⠈⢢⡅⠀⠈⣀⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇⠿⠁⠣⠗⠙⠆⠰⠬⠟⠉⠟⠦⠰⠆⠂⠃⠀⠞⠀⠀⠐⠀⠀⡀⠀⢿⠟
⠀⠀⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠐⠀⠄⠀⠀⠀⠀⠰⠀⠀⠰⠀⠀⠐⠀⠀⠀⠂⠂⠐⠀⠂⠀⠈⠂⡁⠀⡈⠁⠘⠄⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⣠⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠑⠀⠀⠐⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⢳⡄⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⡴⠃⡘⠃⠀⠱⡄⢿⠀⠀⠀⠀⣠⣼⡏
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⢠⠎⠀⢀⠜⠀⠀⠀⠹⣿⣀⠀⠀⢌⠈⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣥⠀⠀⠀⠀⡠⡇⠈⠹⠒⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⢀⡬⠓
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠑⠀⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠣⠖⠀⠀⠑⠉⣀⡀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢸⠀⠀⡞⠁⠀⠠
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠱⠚⠚⠢⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⣅⡀⠀⠐⡀⠀⠀⣠⠀⠀⢸⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠘⠻⠿⠿⠀⠀⣿⠀⠀⣿⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢉⠀⠠⡤⠄⠀⢸⠀⡈⡇⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠓⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢄⠀⣄⡀⢀⣿⣿⣿⠛⠉⠉⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠸⣿⣿⠋⠠⠤⠴⠤⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⡆
⠀⠀⠔⠁⠀⠀⣠⠼⠋⠀⢰⠛⣏⠀⠀⠙⠋⠀⠀⠉⠛⢣⠀⠀⠐⠐⠀⠓⠒⠒⠒⠚⠒⠒⠓⠒⠓⠒⠊
⠀⠀⠀⠀⠄⢘⣅⠀⠠⠤⠼⡀⢻⣄⠀⠀⠀⠀⡄⣠⡀⢈⠀⠀⠀⢀⣀⣀⣄⢀⣀⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠐⠀⠀⢇⠘⣆⠀⠀⠀⠈⣾⣿⣿⣾⡏⠉⣉⣀⢿⡟⠀⠀⠀⠘⠛⠛⠛⠛⠛⠂⠂⠐⠂⠐⠚⠛⠂⠀⣀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⡄⢸⡀⠀⣀⠄⠉⠛⢿⣿⣷⣦⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢊⣂⠀⢈⡉⠉⠀⡗⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣾⠗⠊⠀⠀⠀⠀⠀⠀⠉⠙⠛⢉⠷⢄⠀⠀⠀⠀⠀⠆⠴⠠⠸⠠⠄⠷⠀⠆⠶⠀⠀⠀⠀⠀⣶⡄⠰
⠒⠒⢲⢲⢲⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠄⠉⠢⣄⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢠⣶⣾⣏⠀⢠⣿⣷⣆
⠴⠤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣅⠀⠀⢄⡠⠀⢀⡀⠠⡤⠠⡄⠘⠿⠟⠀⠒⠄⠈⠉⠿⠓
⠻⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠆⠀⠈⠀⠀⡄⠠⡀⠀⠘
⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⡀⣀⡀⠀⣇⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠘
⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡈⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣀⠀⠀⣀⡀⠀⠈⠁⠉⠉⠉⠉⠉⣉⠉
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠏⠈⠻⠏
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⣄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠟⡃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠵⠦⡄
⡀⠀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⡀⠀⠀⠀⠑⠢⢄
⣥⠐⠊⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⠃⠀⠀⠀⠠⠀⠀⠱⠄⢀
⢹⠀⠀⠔⣠⠼⠒⠁⠒⠒⠒⠤⠤⠤⢄⣀⣀⣀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⣔⠠⠂⣿⣉⣛⣒⠒⠒⠦⠤⠤⢤⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣾⠀⠀⠉⡆⢧⠈⠉⠉⠉⠉⠀⠒⢖⠀⠤⠍⣩⣉⣙⡓⠒⠒⠲⠤⠄⠁⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠐⠀⠠⠄
⠀⠀⠀⠀⢷⠸⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠀⠠⠀⡀⠀⡀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠘⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠

⣀⠀⠀⠀⠀⠀⠈⠇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣽⣆⠀⠀⢻⣆
⣿⣷⡶⠀⠀⠀⠀⠘⢀⣿⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⡊⠈⢢⡴⠊⠓⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣤⣤⣿⣧⠀⠀⠀⠀⢀⣰
⣾⠋⣡⠀⠀⠀⠀⠀⠆⢿⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠑⢶⠃⢱⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣮⡀⠸⣿⣧⠀⠀⠀⠈⠛
⠃⣴⡿⡄⠀⠀⠀⠀⢰⣼⡀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠸⡄⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣹⣿⣇
⣼⠏⢠⣧⠀⠀⠀⠀⢰⠌⡇⠀⠀⠀⠀⣸⡏⣿⣿⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣝⢿⡄
⠃⣰⡿⢻⡄⠀⠀⠀⠈⡅⠁⠀⠀⠀⠀⡟⠙⣿⣿⡇⠀⠀⠀⠠⡖⠀⠀⠀⣹⡇⠀⠋⢁⣭⣿⣿⠗⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣕
⣸⠏⢠⣾⣇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢰⠇⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢰⡾⠛⢉⣠⡄⢠⣶⡆⠀⠀⠀⠀⠀⠀⠙⠻⢯⣷
⠈⣰⣟⣁⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣽⡇⢁⣠⣶⠿⢿⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠚⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠉⣠⣴⣿⣿⠋⢀⠔⠃⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣡⡶⠿⠛⣹⠟⣁⠔⠁⠀⠀⠀⠀⠻⣷⣶⣾⣷⣶⣤⣄⡀
⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⣠⣴⣿⣿⠞⠁⠀⠀⠀⢺⣿⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿
⢿⣿⣿⣿⣦⣀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣦⠀⠀⠀⠀⠶⣶⣿⣋⣁⡿⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠉⠉⠙
⠀⠈⠉⠉⠉⠉⠁⠀⠀⣀⣀⣤⣤⣶⢶⣾⣟⣿⣽⣧⣿⣿⣿⡆⠀⠀⠀⠀⠻⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠰⣾⢿⣛⣯⣿⣼⠷⠿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠀⠀⠀⠘⣿⣇⠀⠀⠀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢻⠛⠋⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⢉⣁⣀⣀⣤⣤⣦⡀⠀⠛⠛⠀⢰⣶⣿⣿⠿⠿⢿⣿
⢀⣿⡗⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣷⣾⣿⣿⠿⠿⠿⠟⠛⠋⠀⠀⢀⣠⣄⠀⢸⠛⠿⣷⣤⡀⠙
⢸⣿⠇⠀⠀⠀⠀⢠⣞⠁⠀⠀⠀⣀⣀⣠⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⢸⣦⣄⠈⠻⣿⣦
⣾⣿⠀⠀⠀⠀⠀⠈⢻⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣷⣾⣶⣤⣄⠀⠀⠀⠒⢆⠀⠀⠀⠀⠒⠂⠹⣿⣿⣿⣿⣿⣷⣄⡈⠙
⣿⣯⡀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣾⠟⠀⠀⠀⠈⠳⡀⠀⠙⣿⣿⣧⡈⠙⠻⣿⣦
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠙⠛⠛⠻⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠘⢦⠀⠹⣿⣿⣿⣶⣄⠈⠻
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠳⣄⡼⣿⡉⠛⢿⣿⣦
⣿⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠳⢄⠀⠚⣄⣀⠖⠠⠄⠀⠀⣴⣾⣿⣶⡄⠀⠀⠂⠀⠙⢧⡀⠀⠀⠀⠀⠘⢷⣼⣿⣦⡀⠈⠻
⠿⠇⠀⠀⣧⣀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⢹⠋⠙⡆⠀⠀⠀⠀⠘⣿⣿⣾⣿⣿⣦⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠹⡿⠿⣿⣦⣀
⣤⡄⢀⠞⠉⠉⠹⣆⣾⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣿⣄⣠⣽⣿
⠀⠀⠀⣧⡀⠀⣠⠏⠙⢦⣀⣠⠞⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⣶⣦⣤⠟⠀⠀⠻⣿⣿⣿⣿⣿⣦⠐⡀⠀⠈⢳⡄⠀⠀⠀⠙⠙⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠛⢿⠀⠀⢸⡏⠀⠀⠀⠀⠙⠾⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣼⡄⠀⠀⢻⣄⠀⠀⠀⠀⠈⠉⠛⠉

⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠀⠶⠂⠄
⠀⠀⠀⠀⠀⣾⣋⡀⢹⣦⠰⡄⢹⡀⡑
⠀⠀⠀⠀⠀⢻⣿⣿⣴⣿⣐⣥⣼⣇⣿⣼⣃⡀⢀
⠀⠀⠀⠀⠀⠈⢿⣿⠿⠿⠛⠛⠉⠋⠉⠉⠁⠀⠀⢳⡀
⠀⠀⠀⢀⡄⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⣷⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⢸⣿⡆⢶⣿⣄⠀⠀⠀⢄⣀⢆⡸⢤⡙⠯⣀⣀⠾⡄
⠀⠀⠀⠈⣯⣧⡌⢟⣿⡄⠀⠀⠀⠉⠲⠁⠒⠈⠓⠈⠁⠀⠐⡀⢀⣀⣀⣤⡤⠤⠶⠶⣖⠒⠂
⠀⠀⠀⠀⢻⢿⠟⢸⣼⡷⠤⠤⠒⠒⠒⠛⣒⠻⠛⠓⠆⠄⠀⢻⣭⣥⣤⡶⠶⠚⠛⠋⠉⠉⠁
⠀⠀⠀⢰⣞⣩⣥⣄⠀⢀⣀⣠⣤⣴⣶⠶⠿⠓⢖⠘⡇⣠⠀⢸⣿⠛⠁
⠀⠀⠀⠸⣿⣭⣾⣿⢿⣿⣿⡟⠉⠁⠀⡆⠀⡆⣼⣸⣇⣿⣤⡾⠇⠠⠤⠤⠤⠤⠤⠄⣠⣄⣀⣀
⠀⠀⠀⠀⠉⢹⡟⣿⠁⣏⣿⣷⠀⠀⣰⣇⣼⣷⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢱
⠀⠀⠀⠀⠀⠀⣧⠀⠀⡟⣿⣾⣷⣿⣏⣿⣤⣤⣤⣤⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠸⡾⣷⣝⢝⠿⣿⣉⣏⣉⣀⣀⣤⠤⠴⠤⢤⡀⠙⣶⠀⠀⠠⠀⡄⠀⠀⣸⣼⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡜⢿⢇⡬⢽⣿⣻⣐⣒⣀⡬⢭⡍⠙⠳⣼⣦⣿⡅⠀⣀⠀⣇⣸⣧⣿⡿⠟⠋⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡛⣛⠦⢶⣿⢣⡤⠄⢠⣴⣷⢷⠞⢋⣿⣿⣇⣠⣿⣾⠿⠛⠋⢄⠀⢠⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣙⣯⢗⢒⡶⣷⣶⣶⡾⠇⢿⣿⣿⣋⠁⠀⠀⠦⠀⠈⠑⠪⢀⠀⢸⠆⠀⠀⢀⡠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣉⣁⣀⡙⠲⠛⢛⣿⢿⣿⡗⠢⠠⣴⣿⣾⢧⠀⠰⠦⣄⠉⠛⡀⢀⣦⠾⣀⡤⠚⠉⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣭⡽⠿⢿⣟⡟⠈⣿⡷⣶⣶⣾⣿⣿⣾⣆⠀⠀⢈⣄⣤⠟⠿⣷⡎⠁⢀⡀⣤⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⠟⠙⠂⠀⠀⠀⠬⠅⢺⣿⣷⣾⣿⣿⣿⡏⠛⣿⣤⡴⠟⢉⠐⣸⡇⢻⡇⢀⣼⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠂⠤⢀⡀⠀⣴⣾⣿⣿⣿⣿⠿⣟⣿⣿⣾⠿⠿⠒⠒⠶⢓⣿⣿⣿⣧⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠤⢶⣿⣿⣋⣙⣷⣚⣉⣉⣤⣼⣿⣷⠿⠟⠋⠀⠈⣿⡆⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣚⣭⣥⣶⣶⣤⣌⡙⠻⢿⣦⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠃⣠⣾⠟⢋⣉⣀⣀⡀⠉⢉⡓⠀⠙⢿⣽⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⡇⣰⡟⢡⣾⣿⣿⣿⣿⣿⣷⣄⠹⣦⡀⢀⢻⡇⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣿⣿⡇⣿⠁⣿⡿⠛⠉⠙⠻⣿⣿⣿⡆⢹⣿⣾⡆⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣜⠿⣿⣧⡘⠆⠛⠀⠀⠀⠀⠀⢹⣿⣿⡿⢸⣿⣿⡗⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠿⠷⠬⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⣼⣿⣿⠃⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⠂⢀⣀⣀⠀⠀⠀⠀⠀⢰⣿⣿⠏⣼⣿⠟⢁⡼⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠀⠀⠉⠙⠛⢷⣤⡀⢠⣿⣿⠋⣼⡟⠁⠀⡩⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⠃⠘⣿⠛⠒⠢⠀⠈⠀⢸⣿⠇⢸⡏⠐⢠⡾⠁⠀⢾⡷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠁⠀⢾⣷⣤⣤⠄⠀⠀⢸⣿⢰⡘⣇⢸⣿⣇⠠⣀⣙⣛⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢻⡌⢿⣬⡂⠹⣿⣧⣈⠉⠉⣉⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡛⢿⣶⣌⡻⣟⠿⠶⣶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⣭⣛⢿⣾⣷⡀⢤⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⣾⣶⠄⠀⠀⣸⣿⣷⡜⣿⡇⢸⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⠀⠀⠈⠛⠛⠛⠋⠀⢷⡾⣟⣿⠟⠡⠟⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣴⣶⠿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠬⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠁⠀⠀⠀⠙⡆
⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠼⠦
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⣀⡠⣀⠀⣠⠞⠉
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⡀⠀⠀⠀⢀⡴⠛⠃⠀⡠⠚⠉⠀⠀⠀⠀⢰⡏
⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣿⣶⠖⠛⠉⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣀⣀⠀⣠⡴⠶⢄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠼⠷⣄⠀⠀⠀⢀⣠⠜⠁
⠀⠀⠀⠀⠀⠈⣻⢶⣀⠉⢫⣿⠛⠉⠀⠀⠀⠈⠢⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠽⠶⠖⠛⠉
⠀⠀⠀⠀⠀⡔⠁⠀⠻⣿⣚⣧⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠱⣀⠀⠀⢩⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⢸⣿⠇⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠑⢦⣆⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣙⢿⡏⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⠏⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠟⠘⢠⣶⣜⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠛⢖⣥⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣩⡟⠁⠀⠀⠀⠙⢿⣿⣯⣻⣿⣿⣿⣷⡀⠀⢀⡠⢊⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠰⣷⣀⣄⠈⠛⢿⣿⣿⣿⣿⣿⣷⡉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣾⣫⣻⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⣿⣿⡘⢿⣿⣿⣿⣿⡿⢋⣵⡿⠋
⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣯⣟⠿⣿⣿⡿⢋⣴⡿⠋
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣾⣯⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⢠⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡄⢿⣇⠀⠘⡇⢸⠄⠸⡇⢸⠃⠀⣸⡷⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠸⣷⡘⠿⠘⠉⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠑⠃⠿⢠⢾⠇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⡿⠻⣦⡀⠀⠀⠀⠀⠀⠉⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⢀⡇⣧⣿⣧⠀⠀⠀⠀⠀⠀⣄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⣸⢁⡎⡼⢠⡏⣿⣿⣷⠀⠀⠀⠀⠀⣽⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⣇⡞⣸⢁⡼⣾⣿⣿⣿⠀⠀⠀⠀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣽⣿⡿⠿⣿⡀⠀⠀⠀⠀⣻⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠈⠙⠟⠋⠀⠀⣽⣷⠀⠀⠀⠀⠈⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢐⠋⠉⣁⣀⡀⠀⢶⠂⠀⣀⡈⠉⠙⠓⢤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⣯⠖⠢⡀⠀⠀⣴⣿⣿⣿⣿⣷⠈⣰⣿⣿⣿⣿⣷⡄⣻⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡰⠻⡀⢀⠇⠀⠀⣿⣿⣿⣿⡿⠋⣸⡝⣿⣿⣿⣿⣿⡷⢀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠈⠁⠀⠀⠀⣈⡛⠿⢋⠀⣴⡿⣷⡌⠙⠻⢛⣛⣡⠖⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣥⣤⣬⣤⣛⣧⡛⣧⣽⣤⣤⣬⣥⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠁⢀⣨⣿⣿⡻⡿⠛⣿⠛⢿⠿⣿⠿⠛⠁⢀⣀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣾⣇⣷⣶⣿⣿⣿⣿⣝⣷⣶⣿⣶⣾⣯⣥⣶⣶⣿⣿⣿⣟⣶⣴⣦⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠛⠛⠙⠛⠛⠛⠛⠋⠀⠉⠛⠋⠉⠙⠛⠉⠋⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠰⠸⡾⠿⠧⠸⠹⠤⠇⠏⣰⢦⢤⣀⡀⠀⠰⣭⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠺⠀⠀⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠯⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟

⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⠀⠈⣹⠿⡏⠀⠀⡠⠤⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⡆
⠀⠀⢩⣯⠽⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃
⠀⠀⢀⣠⣤⣤⣶⠀⠀⠀⠀⣀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣄⠀⠀⠀⠀⠀⡼⠛⠈⠁⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠘⠋⠉⣁⣀⠀⠀⠀⠰⣿⣿⣿⠿⠆⠀⠀⠀⠀⢠⣤⠀⠀⠀⠈⢦⡄⣀⣠⠞⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣷⡿⠀⠀⠀⠀⣠⣴⣿⣿⣿⣦⣄⡀
⠀⠀⠰⠿⠟⠛⠋⠀⠀⠀⠀⣉⣁⣀⣤⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⣼⡽⣃⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣤⣴⣶⠾⠀⠀⠀⠈⠛⠉⣉⣁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⣼⠏⠀⢻⣆⣰⣿⠟⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⢸⠛⡟⡿⢻⢻⢻⢻⣿
⠀⠀⠈⠉⣀⣀⣤⠀⠀⠀⠠⠟⠛⠛⢉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣅⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⣿⠿⢟⠿⡿⡇⠀⢸⠂⡇⡇⠸⢸⢸⢸⡿
⠀⠀⢶⣾⣷⣿⣾⠆⠀⠀⢠⣴⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠻⢷⣄⡴⠋⠘⣆⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⡏⡇⠀⠈⠀⠀⠁⠀⢸⠈⢸⡇
⠀⠀⠈⠛⢿⡿⠋⠀⠀⠀⠀⠙⢻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⢠⠆⠉⠁⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠘⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢆⠀⢀⣄⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣼⠏⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⡁⠀⡱⡁⠀⡀⠀⣿⢒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠊⠰⡖⡁⢀
⠀⠀⠀⠀⠀⠀⠀⢠⢯⣼⠏⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠈⢻⠁⢘⠏⠀⠀⣿⠬⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⠀⠀⠂⢴⠀⣽⠁
⠀⠀⠀⠀⠀⠀⣀⠍⣾⠏⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠸⡆⢸⠀⠀⠀⣿⣚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠰⡇⠀⠀⠀⠀⣇⢺
⠠⣠⡖⠀⢲⣄⣈⣻⣃⣀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⡇⠀⠀⠀⣿⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠇⠀⠀⠀⠀⢸⡟
⠀⠀⠑⠒⠊⠀⣰⠃⠈⢻⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⣸⠃⠀⢀⠏⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡰⠃⢀⠤⢾⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⢤⡇⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠁⠀⠠⣀⡠⠁⠀⠀⠀⠀⠈⠐⢾⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢦⣀⡼⠁⠀⠀⠸⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠸⣿⠀⠀⠘⠃⠀⠀⠀⠆⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⢀⣄⣀⣀⣀⣀⠀⠀⠀⢠⠉⠉⡲⠦⠤⠤⢸⣷⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡖⣀⣀⣀⡤⠞⠧
⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠙⢷⣍⠙⠓⠶⠤⠀⠀⠀⠀⠀⠤⠴⠖⠚⢋⣵⠟⠁⠀⠀⠀⠑⠄
⠀⢀⣾⣾⣿⣿⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣿⡄
⠀⠸⣿⢿⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⣿⠖⠖⡇
⠀⠀⣿⣾⣿⣿⣿⣾⡇⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢩⣙⠀⠀⢸⣿⣋⣉⡇
⠀⠀⢸⣟⣿⣿⣷⣿⠀⠀⢸⢠⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢻⣿⠤⠤⠀⠀⢸⣿⡥⠤⡇
⠀⠀⠈⣿⣿⣿⣿⡏⠀⠀⢸⢸⢀⢠⢸⢸⠀⠀⢰⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⡇⡇⡀⠀⠀⠀⠀⢺⣿⠒⢚⠀⠀⢸⣿⡓⠒⡇
⠀⠀⠀⣧⣿⣿⣿⡇⠀⠀⢸⣸⢸⣸⣸⢸⣸⠀⢸⣸⣠⣀⣀⣸⡀⠀⢀⣄⣀⣇⣇⣆⡀⠀⣇⣄⣇⡇⣇⣇⡀⠀⠀⠀⢹⣿⢩⣩⠀⠀⢸⣿⣭⣍⡅
⠀⠀⠀⣿⣽⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⠰⠶⠀⠀⢸⣿⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⠂⠀⠘⠇⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣯⣨⣿⣿⣅⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⠀⠉⠀⠀⠀⠀⠀⣰⠊⠁⠈⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣾⣿⣿⣿⣿⡿⢫⣾⣿⣷⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡠⠤⣀⡼⠉⢑⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⣛⡻⠟⢱⣷⢹⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⢣⣀⡽⢸⡶⠊⠀⣀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣧⡻⠿⣸⡎⣩⣾⡿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⡀⠈⢳⠘⡇⠀⢀⣿⣿⣷⣶⠂⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣷⢹⡇⣿⣿⠇⠀⠀⠉⢩⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠃⠀⠀⣿⣿⣿⣿⠀⠘⣳⡇⢠⣿⣿⣿⣿⣿⠀⠈⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣴⣿⣿⡇⠀⠀⠀⢸⣿⡧⠃⣿⠏⠀⠀⠀⠀⢸⣿⣦⣾⣿⣿⣿⣿
⠀⠀⠀⢀⣤⠀⠀⢸⢿⣽⣿⡆⠀⣿⡇⢸⣿⣿⣿⣿⡟⠀⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢻⣿⣿⠀⠐⠀⠈⣿⡇⠀⣿⠀⠀⠀⠀⠀⣼⡟⢻⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣥⠴⠆⠘⣮⡛⣿⣷⠀⣿⡇⢸⣿⣿⣿⡟⠐⠦⢤⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⢻⡟⢛⠋⠛⠓⠀⠻⡗⢸⣿⢧⠀⢀⡆⣾⣿⣿⣿⢿⣿⣿⣿⣿
⠀⢀⣀⣿⣆⠀⠀⠀⢸⡿⣿⣿⡄⣿⠇⣾⣿⣿⣿⠀⠀⠀⢀⣿⣅⡀⠀⠀⠀⣿⣿⣿⡿⠗⠀⢻⣿⣿⣛⠛⡛⠂⠘⡟⠘⣿⠂⠀⡰⣼⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠒⠯⣀⠘⣿⣠⡤⠄⠘⡿⣎⣿⣇⢹⠀⣿⣿⣿⠃⠤⠤⣤⣻⡏⢀⡨⠗⠀⠀⣿⣿⣅⡺⢿⡆⠘⠛⣛⣿⡏⠙⠃⠀⣿⢸⡇⠁⠀⢁⣿⣛⡛⠛⠁⣿⠿⣋⣽⣿
⠀⠀⡎⣀⡽⠟⠂⠀⠀⠓⠛⠉⠁⠈⠀⠙⠛⠉⠀⠀⠀⠚⠿⣄⠸⡀⠀⠀⠀⣿⣿⣿⡿⣼⠟⣀⣼⣿⣿⣧⣤⣴⣶⣿⣾⣷⣤⣴⣿⣿⣿⣯⣄⡘⢿⣼⣿⣿⣿
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⣶⡀⠀⠺⣷⠀⢠⡄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⡄⠰⣿⣿⣇⢰⠀⢹⣧⠈⣷⠀⢿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠿⢮⡿⠟⠀⢿⣿⣿⣿⠀⠀⠻⢇⠈⢰⠆⠙⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠉⣠⣦⠀⡄⠒⠠⡀⠈⢻⣿⣏⠀⠄⠀⠈⠆⠊⠀⠀⠀⢈⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣼⡿⠋⠀⣀⣩⠤⠀⠀⢈⣉⡁⠀⠢⡀⠀⠈⠀⡀⠀⢰⣾⣿⠏⠉⣡⠶⠟⠃
⠀⠀⠀⠀⠀⢰⠃⠀⢠⣿⡟⠁⠲⡿⠋⢀⣠⡴⠄⢿⣯⢿⡄⠀⠹⣦⡀⠀⡇⢀⣿⣿⠏⠀⣤⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢀⡟⠀⢠⠞⠉⠀⠀⣀⣤⣾⠛⠁⠀⠀⠈⢿⣮⣿⡆⠀⠈⠀⠀⡇⢸⡿⠁⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠞⡇⡴⠋⠀⠀⠀⠀⠛⠋⠀⠀⢀⣴⣦⠀⠈⠹⢿⡇⠀⠀⠀⢴⡇⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠐⢻⡰⠇⠀⣰⡖⢀⣀⣤⣤⣤⣶⣾⣿⣿⣿⡦⠀⠀⡀⢻⣆⠀⢠⣾⣿⡀⠀⣸⠀⠶⠦⠌⠛⠛⠛⠋⣠
⠀⠀⠈⠀⢰⠀⣰⡟⢠⣿⣿⣿⣿⣿⡿⠟⠉⣠⠄⠀⢸⡿⢷⠀⠹⡆⠐⠿⢿⡿⠀⠛⠀⢀⡤⠂⣴⡄⢀⣾⢏⣴⡗⡀
⠀⠀⠀⠀⣠⣴⣿⠁⠈⠿⣿⣿⣿⡋⠀⠀⣰⠋⠀⠀⠀⠙⣿⠀⠀⠀⠀⠠⠀⢀⣀⡴⠞⠉⠀⢀⣿⣧⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠘⢱⣿⠃⠀⠀⠀⠀⠸⣿⣷⡂⠀⠁⠀⣠⣿⡦⠀⢸⡀⠀⠔⢰⠖⢈⠝⠉⢀⢴⡆⠀⣼⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣰⣾⠇⢀⡴⠁⠀⠀⠀⠹⣿⣧⠀⠀⣰⣿⠟⢀⣀⠀⠃⠀⠀⡝⢘⠃⢀⣄⠈⢺⣷⠐⢿⣿⣿⣿⡿⢛⡁
⠀⠀⠰⠿⠏⢠⡾⠁⢀⣴⡖⠀⠀⠙⢿⠀⢈⡿⠏⢀⣾⡟⠁⠀⠀⢀⡁⢠⠄⠀⢿⡆⠀⢿⡆⠀⠙⠿⠋⣠⣿⣿⡶
⠀⠀⠀⠀⣰⣿⠀⣴⣿⡿⠷⢤⡀⢀⣤⢠⡞⠀⠀⣼⠋⠀⠷⢀⠀⠘⢿⣦⡀⠲⠐⣷⡀⠘⡇⠀⠒⡄⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠉⠈⠀⠀⣿⣷⣀⠀⠴⠛⢡⠏⠀⠳⠦⢄⠀⠈⡌⠀⢂⠀⠀⠙⣷⣄⠀⠹⣧⠀⠁⠸⡠⢡⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡆⠀⣀⠰⣦⡀⠀⢀⠀⠀⠘⡀⠈⠄⠀⠀⠈⠻⣧⡀⢻⣇⠀⠀⣠⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⣤⣽⣷⣽⣿⣄⣿⡇⠀⠀⢸⣿⣷⠀⠠⣀⠀⠘⢷⡄⢻⡄⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢹⣿⣆⠀⠈⠳⣤⡀⠹⣮⣗⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣷⠀⣼⠇⢿⣿⡄⠀⠀⠙⡿⠆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠋⠀⢸⡿⠀⠘⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣋⣥⢶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⠟⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢉⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⣠⡀⠀⢀⣽⣿⣿⡿⠿⠿⠛⠻⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢿⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡿⢛⣿⣿⣿⣿⣿⣿⠿⢿⢿⡿⢿⠟⡟⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣿⣿⠏⠀⢹⡇⠃⠈⠀⠁⠀⠘⠀⠸⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣿⡏⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠹⣧⠙⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣸⣿⢃⢠⠀⣿⠀⡀⢠⢠⠀⡀⠀⣀⡄⡸⣷⡀⣔⣼⣧⡈⢿⣿⣿⣿⣿⣿⣆⠀⠀⡠⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⡀⢀⠀⠀⣿⣿⣾⣾⣶⣿⣴⣿⣼⣼⣆⣷⣧⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⢿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣼⡄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣦⡈⠻⣿⣿⣿⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⠁⠈⠻⣮⣺⣿⣆⠀⠈⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠛⠟⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⡀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣴⡾⠇⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⢽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣶⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢻⣿⣿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢨⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢀⢀⣠⠰⣀⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⢶⣰⢿⣹⡿⢿⣷⠛⣿⣸⢿⣿⠻⢆⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⡰⣽⣶⣇⠺⣇⣿⠸⣮⠅⠏⢷⢫⣌⡧⢻⣾⣡⢦⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡼⢦⡺⠽⣽⠦⠙⠨⠁⠀⠀⣁⣀⣀⣈⡀⠈⠙⠁⡋⣴⢋⡵⢋⡽⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢝⣾⣍⣷⠉⠈⠀⠀⠀⠀⠃⣿⣿⣿⣿⡇⠀⠀⠀⠀⠐⠻⠶⢋⣵⠞⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⡞⠁⠨⠁⠀⠀⠀⠀⢠⣾⣶⣿⣿⣿⣿⣥⣴⣤⠀⠀⠀⠀⠐⠍⠊⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡞⠀⠀⠀⠀⠀⠀⠀⢀⠩⡽⠛⣿⡿⠿⣿⡟⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠁⠀⠀⠀⠀⠀⠀⢄⡀⠑⢧⡀⠀⢀⢀⠀⠀⣠⢆⡠⠀⡀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡶⡷⢿⡴⠲⡄⠀⠀⡀⠀⠀⠬⢑⣤⣷⡖⠛⡛⢒⣾⣷⠅⣒⠡⠄⠀⠀⠀⢠⠖⣶⣾⣻⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣃⣆⣀⣃⣠⠇⠀⠀⠀⠤⠬⠭⠽⢿⡿⢿⣶⣶⣾⢿⣭⡥⠤⠤⠤⠄⠀⠀⠸⣄⣤⣸⣤⣱⣅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢻⠀⠀⠀⠀⠀⡠⠄⠐⠀⠀⠀⠠⠀⠉⠉⠁⠄⠉⠉⠙⠓⠲⢆⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⣇⠀⠀⠀⠈⠀⠀⢀⡏⣉⡁⠀⠀⠀⠀⠀⠀⢀⣉⢹⡄⠀⠀⠱⠀⠀⠀⢰⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⣆⠀⢐⠀⠀⠀⢈⣿⣿⣷⣆⣤⣠⢤⢆⡠⣼⣿⣿⡃⠀⠀⠀⠂⠀⢠⢟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢧⣸⢄⠀⢀⡗⠀⠀⠈⠙⢉⠉⠈⡡⠛⠁⠀⠀⢸⢀⠀⠀⣄⣴⣫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣿⠀⠀⠀⢈⡄⠀⠀⠀⠀⠈⠉⠀⠤⢀⠀⢀⡅⠀⠀⠀⢹⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣄⠈⠉⠛⠒⠀⠐⠀⠀⠀⠀⠀⠀⠈⢁⣴⠚⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⣿⣿⡏⠙⢛⣛⣦⡀⠈⠻⣿⣧⣄⠈⠉⢭⣭⣽⡇⣿⠋⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡏⠀⣿⣿⣷⣄⡀⠙⢿⣿⣦⡀⠈⠻⣿⣷⣄⠀⠙⢿⡇⢿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⢻⣿⡟⢿⣿⣆⠀⠙⠿⣟⣂⡀⠈⠛⢿⣷⣄⢸⡇⣿⠀⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠤⠼⠟⠋⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⠛⠲⠯⠤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠀⠀⠁⠃⠘⠒⠒⠃⠁