Lowercase Emojis & Symbols

⢀⠀⡀⠀⢀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⠴⠉⢀⠀⡀⠀⠀⠀⡄⠀⣀⢀⠀⠀⡀⠀⠀⢀⢀⠀⠀⠀⣀⠤⣀⢄
⢇⣀⠅⠀⢔⣈⡇⠀⢧⣀⠀⠀⠀⣀⡀⠘⣀⣠⠀⠸⣐⡂⠀⠀⠀⢂⣀⠄⠀⢸⠆⠊⠀⠀⠀⠇⠀⠄⠀⠀⠰⠁⠄⠀⠀⠅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⡇⠀⡆⠰⠁⢈⠀⢈⠀⢰⠀⢄⠀⢙⠀⠸⠀⠀⢘⠂⠄⠀⠀⠀⢄⢀⡄⠀⢂⡐⠀⠐⡐⠡⡜⠀⢀⠶⡀⠀⢣⡐⠀⢀⠀⡄⠀⡠⠊
⣀⣀⣀⣀⣈⣁⣀⣀⣈⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣜⣀⣀⣀⣤⣆⣀⣀⣀
⣿⣿⣉⢉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣀⣀⣤⠤⠤⠤⣄⣀⣀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠩⠔⠚⠛⠳⠦⢤⣀⣀⡀⠀⠀⠈⠙⠲⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡉⠉⠙⠃⠀⠀⠈⠀⠀⠀⠀⠐⣶⣲⣄⡀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠐⠒⠐⠶⣦⡀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢸⡟⠉⣻⣦⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣴⠁⣠⠖⠋⣼⢀⣤⠠⡄⠀⢹⠓⠋⠀⠀⠀⠀⠀⠀⠰⠤⠴⠚⠁⡷⢸⣿⡿⠿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⡧⠊⠁⠀⠀⢿⠋⢺⡀⠈⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⣿⣄⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⡟⠁⢀⣴⡀⠀⢸⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡰⢋⣾⠟⠉⠛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡼⢹⠀⢸⡇⢀⣾⣿⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⢈⣧⠴⣊⣴⣿⣿⣄⠀⣀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠘⡆⠘⡇⢸⣿⣿⠃⣰⠇⠀⢀⣤⡤⠚⠁⠀⠀⠀⠀⠀⢹⣦⣤⣶⣿⣿⡏⠉⣿⣿⣿⡿⠋⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢦⣹⣼⣿⣣⡴⠧⠤⢄⣿⣿⡇⠀⠀⠀⠀⠰⡤⣴⡿⢿⣿⣏⣹⣿⣿⣾⣿⡿⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢹⣴⣶⣦⣄⠉⠻⢷⡀⠀⠀⠀⠀⠀⠙⠋⠻⠿⠿⢛⣻⠿⠛⠁⢀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠦⣄⣤⣞⠿⣿⣿⡿⡷⣀⠀⠈⠀⠀⠀⢠⣤⠀⠀⠄⠀⠉⠩⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣧⠈⠠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠈⠑⡄⠀⠂⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠘⡇⢸⡆⠀⣯⣿⡏⠀⢀⡀⢠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢳⡀⠈⠿⠀⠀⡼⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣀⠀⠀⠀⣀⡤⠖⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠙⠈⣇⠁⡀⠀⢀⡼⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠑⠄⠐⢁⡴⡻⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⠀⠙⣄⠋⣇⠀⠀⠸⡐⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⢸⣰⣿⣿⣇⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢦⠀⠀⠈⠦⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠳⠂⡄⠀⠀⠀⢸⠟⢋⡁⣀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠀⡀⠀⠀⠙⠏⠉⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⢢⣤⠀⢀⡇⠁⠀⠀⢀⡐⠈⠉⠉⠁⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⠂⢀⣀⠀⠈⠓⠦⢬⣑⣒⠲⠶⠤⠶⠒⣋⣡⣴⠟⠃⠀⠀⣠⠎⣁⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠈⠙⠒⠲⠤⠤⠤⣭⣭⡭⠭⠭⠿⠛⠉⠀⠀⢀⣤⣾⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢹⣿⡇⠀⣿⣿⠈⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⣿⡇⢠⣿⣿⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣿⡇⣿⠀⠹⣆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⣿⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡟⠛⣿⡧⠀⢸⣿⡇⠛⣿⣿⠛⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⠤⢿⣧⠀⢸⣿⡇⢰⡿⠿⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⣿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣦⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢶⡄⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡿⠾⣧⢾⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣷⢸⣇⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠆⣿⣨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠆⣴⠆
⠀⠀⠀⠀⠀⠀⢸⣇⣠⡟⢾⣉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⢸⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⡴⠲⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⣠⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠸⣇⣘⡷⢰⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⣿⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡴⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣄⡿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣧⠀⡿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⡀⠀⣀
⠀⠀⠀⠀⠀⠰⣿⣿⢿⢰⡏⣻⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⠁⡏⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣸⠇⣿⣹⡇
⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⣴⢶⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢶⡄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠚⠁⣿⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣽⠃⣿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢾⡁⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣭⣛⣧⢘⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣸⠇⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢹⣶⠏⠸⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⢻⡿⠘⣶⢿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⣤⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠻⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣃⠀⣹⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢟⣉⣻⡿⠿⠿⢿⠿⠿⢿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣮⣈⣿⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⣿⣿⡇⠸⡟⠩⢹⡉⠉⣹⠉⢙⡏⣝⠩⠽⢉⢹⡏⠙⡋⡉⣻⡇⠸⣟⠉⢹⡀⣥⢸⠋⡉⡏⣩⠉⢹⣿⡁⠶⡇⢸⢸⠩⠍⣿⡏⠀⢿⢩⡅⢼⣿
⣿⣿⣿⣶⣿⣶⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣾⣾⣷⣿⣿⣾⣿⣿⣶⣿⣷⣿⣷⣿⣾⣿⣾⣿⣿⣷⣾⣿⣿⣾⣿⣿⣿⣴⣾⣿⣿⣿⣾⣾⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢸⠅⢿⡇⠈⠍⠏⢙⠫⡏⠹⣿⠃⠙⣿⡌⡈⣅⠉⡏⡅⡟⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⢀⡀
⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠀⠀⠀⠈⠁⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⢀⢸⠀⢄⡆⠀⠀⠀⠀⠀⠀⣀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⢀
⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇
⡇⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠘⠂⠀⠈⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠
⣷⡄⢠⣤⠠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡄⣤⡄⢠⣤
⣯⡽⢯⣨⠿⣭⡇⠀⠀⠀⣿⠿⢲⡖⣶⣦⣶⣶⣶⣴⣲⢼⣗⡇⣶⣦⣶⡾⠀⣿⣯⣴⣰⢴⣖⣴⣖⣴⣶⣴⣶⣰⣶⣸⠃⠀⠀⠀⡿⣏⡽⢯⣨
⣿⣷⣾⣹⣶⣛⣧⣤⣤⣤⣬⣤⣬⣥⣬⣥⣬⣭⣬⣬⣭⣬⣭⣥⣥⣥⣿⣿⣤⣭⣥⣬⣭⣤⣭⣬⣭⣬⣭⣤⣭⣤⣭⣬⣤⣤⣤⣴⣷⣏⣳⣾⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠛⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣠⠇⠀⣷⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣤⡄⠹⣆⠀⠀⡇⠀⠉⠒⣄⠀⠀⢠⠋⢠⣤⣽⡆⠀⢠⠋⢰⣦⠀⣿⠀⠀⢰⠋⣤⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢴⡦⠀⣿⠀⠀⢂⠀⠿⠃⣸⡇⠀⠸⡀⠸⡧⢄⡀⠀⠈⢆⠘⠏⠀⣿⠀⠀⢡⠀⣤⠴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠿⠀⠀⠙⠶⠶⠾⠟⠀⠀⠀⠘⠲⠶⠾⠃⠀⠀⠈⠛⠲⠾⠟⠀⠀⠀⠓⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⡄⠀⠀⠀⠀⣰⠒⣄⠀⠀⠀⢀⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡾⠇⠀⢀⠔⠒⠢⡀⠀⠀⢀⠀⢸⣇⣀⠀⠀⠀⣸⠒⣿⠀⠀⠀⢈⠖⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣼⠆⠀⡏⠀⣿⡆⢸⡆⠀⠀⠀⢀⡀⠈⣧⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡇⠀⠀⠱⣄⣛⠁⢸⡇⠀⠀⡆⢸⡇⢀⡟⠀⠀⢿⠀⣿⠀⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⢯⣉⣀⣾⠇⠀⠀⠈⠛⠉⠛⠁⠀⠀⠈⠙⠋⠀⠀⢯⣉⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⣦⠀⠀⠀⠀⡖⠒⣦⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠟⣩⣦⠀⠀⡇⠀⣿⠀⠀⢸⠉⠙⠓⠂⠒⢢⡀⠀⢸⠉⠙⠒⢢⡀⠀⢀⠖⢉⡙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⡁⠀⠀⡇⠀⣿⠀⠀⢼⠀⣶⠀⢰⠀⢈⡇⠀⢸⠀⢰⡆⢸⡇⠀⢸⠀⣿⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣦⣤⣷⠀⠀⢇⣀⣹⡄⠀⠸⣀⣿⣀⣼⣀⣸⡇⠀⠸⣀⣸⣅⣸⡇⠀⠈⠢⣬⣡⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠈⠉⠈⠉⠀⠀⠀⠈⠉⠉⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣰⠁⢸⣆⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣈⠉⢦⠀⢀⠞⢁⣀⠀⣷⠀⠀⡇⠘⠋⣱⡄⠀⡜⠁⣤⣩⣦⠀⠘⢤⠀⢠⣼⠇⠀⡇⢸⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢀⣾⠃⠘⣄⠘⠛⠀⣿⠀⠀⡇⠀⣾⠟⠁⠀⣈⢶⡄⠙⣆⠀⠀⢀⠀⢸⡇⠀⠀⣇⠸⡿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠛⠁⠀⠀⠈⠛⠆⠀⣿⠀⠀⠣⣴⠿⠀⠀⠀⠘⠲⢤⡾⠟⠀⠀⠈⠦⣼⠇⠀⠀⠈⠢⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠟⠀⠀⠀⠀⠀⠀⠙⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣀⠤⢤⠀⢀⡤⢤⡤⣤⠤⣄⠀⠀⡤⢤⡤⢤⡀⠀⡤⢤⡠⠤⣄⠀⢠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⡟⠀⣿⠁⠈⡀⠸⠀⠹⠀⣿⠁⠀⢣⠀⢀⣾⠃⠀⢱⠈⠃⢰⡟⠀⠈⢲⠂⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣸⡏⠀⠀⠇⠀⣠⡀⢀⣿⠀⠀⡜⢀⠀⢻⡀⠀⠈⡇⠀⣼⠇⠀⢠⠃⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠈⠛⠛⠙⠛⠋⠀⠀⠙⠛⠛⠛⠃⠀⢠⠁⣰⡟⠀⠀⠈⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠉⢻⣿⣷⣴⣾⣿⣿⠉⠉⣿⣿⠉⠉⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⡿⠛⢹⣿⣿⠀⠀⣿⣿⠀⠀⢻⣿⡏⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣏⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⡟⠀⢸⣿⣿⠀⠀⣿⣿⡆⠀⢸⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣇⢠⣼⡿⠿⠿⠿⣿⣿⡃⠀⢸⣿⣿⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⢻⣷⠀⢸⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⡄⢸⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣧⣤⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⣿⣿⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣠⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐
⣿⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⡃⠀⢸⣇⣤⣄⠀⠀⢸⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⢀⣠⣄⣿⠁⠀⠀⠀⠀⢀⣀⣀
⣿⠀⠀⠀⠀⣾⠏⢹⣿⠀⠀⡇⠀⢸⡏⠉⢹⣧⠀⢸⠀⠀⣾⠋⠉⠉⠀⠀⠀⠀⣿⠋⠉⣿⠀⠀⠀⠀⢰⣿⣭⡿⠃
⣿⠀⠀⠀⠀⠻⣦⣴⣿⡄⠀⡃⠀⢸⣷⣶⡾⠃⠀⢸⠀⠀⠿⣦⣴⠶⠀⠀⠀⠀⠻⢶⡶⠿⠀⠀⠀⠀⠘⠿⣶⡶⠆⠀⠨
⣿⠀⠐⠒⠒⠒⠒⣒⣒⠐⠐⠖⠒⠒⠒⠒⠒⠒⠂⢺⠒⠒⣒⠂⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠂⠖⠐⠒⠒⠒⠂⠒⠒⠺
⣿⠀⠀⠀⠀⠀⣸⣟⡉⠀⠀⡇⠀⢀⣴⠶⢶⡄⠀⢸⠀⠀⣿⢀⣀⠀⠀⠀⠀⠀⠀⠘⡛⠀⠀⠀⠀⠀⠀⠀⢈⣇
⣿⠀⠀⠀⠀⠛⣿⡟⠋⠀⠀⡀⠀⢸⣿⢀⣼⡇⠀⢸⠀⠀⣿⠟⢻⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠀⠀⡙⢛⣿⠇⠀⢸⠀⠠⣿⠀⢸⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣀⣿⠀⠀⠐
⣿⠀⠀⠤⠤⠤⠭⠥⠤⠤⠤⡦⠤⠬⠿⠿⠭⠤⠤⢼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠿⠯⠤⠤⢬
⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠂⠀⠀⠀⣿⠀⠀⠀⢸⠀⣀⢀⣀⣀⣀
⣿⠀⠀⠀⠀⣿⣧⣾⠟⠀⠀⡃⠀⠀⠀⣿⠀⠀⠀⢸⠀⣿⡟⣿⡟⢻⡇⠀⠀⠀⣺⣷⠿⣷⠀⠀⠀⠀⢀⣶⠿⣷⡄
⣿⠀⠀⠀⠀⣿⡏⢿⣦⠀⠀⠂⠀⠀⠀⣿⠀⠀⠀⢸⠀⣿⠀⢸⡇⠸⣿⠀⠀⠀⣿⡇⠈⣿⠀⠀⠀⠀⠸⣧⣀⣾⠇⠀⠐
⣿⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠅⠀⠀⠀⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠁⠀⠉⠀⠀⠤⠀⠀⠈⠉⠁⠀⠀⠠
⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠉⠉⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⢄⠀⠒⢀⠬⠏
⠀⠀⢤⡆⠀⢈⠀⠀⠀⢁⠀⠀⠱⠀⠀⠀⠒⠖⠀⠀⠀⠐⠀⠀⠀⠀⠌⠀⠘⠀⠀⠆⠀⠀⢀⠄⠈
⠀⠀⠀⠰⠃⠈⠂⠀⠀⠡⠀⠀⡠⠀⢀⡀⠀⣀⠆⠑⠤⠤⢄⡰⠀⠤⡀⠀⠐⠀⣀⣀⠀⠠⡀⠀⠀⢣
⠀⠀⠈⠒⠉⠑⠃⠀⠒⠒⠒⠊⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠊⠁
⠀⠀⠀⢃⠀⠀⠀⢀⣀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⠀⠰⡀⠀⠐⠂⠀⠢
⠀⠀⠀⠘⢀⠈⠂⠀⠈⠃⠀⠀⠈⡇⠀⠀⠀⡁⠀⠜⠀⠀⠠⠀⠀⠀⢀⠄⠀⠠⠀⠁⠀⠀⠊⠀⢀⠀⠀⠴⠀⢣
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡄⠂⠀⠀⠀⠐⠀⡀⠀⢘⠀⣀⡇⠀⠀⠀⣀⡀⢀⣸⠠⣀⠉
⠀⠀⠈⠉⠀⠉⠁⠐⠀⠂⠀⠁⠀⠀⠀⠀⠄⠆⠀⠒⠒⠐⠒⠂⠀⠈⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠠⠠⡀⢀⣈⡂⠀⠀⠀⠀⠆⠀⠐⡀⠀⡰⠀⠀⢠⠀⢀⡀⠀⠄
⠀⠀⠀⠀⡌⠀⠘⠀⠀⠐⠀⠰⠀⠀⠀⠒⠰⠀⠠⣀⠊⠰⠀⠀⢸⠀⠀⠀⠀⠄⠀⡅⠀⠀⠇⠀⠀⠀⠀⠊⠀⠘
⠀⠀⠀⠀⠤⠀⠀⠐⢤⣀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠑⠀⠰⠀⠀⠀⡁⠠⠤⠤⠤⠤⠠⣤⣤⡄⠀⢠⣠⣤⣆⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠉⠉⠀⠀⠀⠘⠂⠂⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠁
⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠐⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠠⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠄⠀⠈⠀⣀⢀⠀⠀⡠⠀⠀⠀⠀⢀⢀⠂⠈⠃⠀⠐⠖⠐⠶⡁⠀⠲⣌⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠇⠀⠀⠰⠁⠀⠁⠀⠀⠈⠄⠀⠈
⠀⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠰⠄⠀⠀⠀⠠⡈⠀⣄⡀⠂⠈⠒⠐⠒⠀⠀⠀⠢⠄⠤⠘⠂
⠀⠀⠁⠀⠁⠈⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡄⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⣠⠀⠀⠀⠀⢠⠀⠀⠀⢀⡄⠀⠀⠀⠀⡄⠀⡀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣸⠃⠀⠃⡟⣰⡃⠀⡾⠀⠀⠀⢁⡟⠘⣆⠀⣏⠀⠀⠀⠀⡆⠀⠀⠀⢸⠁⣰⠀⠀⢠⡇⣼⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢠⠇⡁⡸⢰⡇⠀⠀⠀⢸⠃⠀⡟⣰⠉⠃⠀⠀⢰⡗⠂⠀⠀⠉⠠⡇⠀⠀⡸⢹⠃⠀⢀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠋⠀⠐⠁⠀⠁⠈⠀⠐⠋⠀⠈⠀⠈⠀⠀⠀⠀⠊⠀⠀⠀⠀⠑⡜⠀⠀⠐⠁⠘⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⢀⡤⢀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢠⡄⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⡀⠀⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧⢀⡇⠀⠀⠀⠀⢻⡀⠀⠂⣼⠁⠀⠀⠀⣿⢀⡿⢠⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⡞⢸⡆⡀⠀⠀⢰⡇⠀⢠⠀⠠⠃⣼⠁⣧⠀⠀⠀⠀⢨⡇⠀⠀⡇⠀⡀⠀⢀⡟⠈⠀⠘⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣦⠀⠀⢔⠂⠂⠀⢰⠀⡄⡀⢰⡄⢠⠀⠀⢰⠀⠠⢲⠀⠐⣆⢀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢲⠃⠈⠀⢺⠀⠀⠀⢸⠀⠀⠀⣿⢁⡏⠀⠀⢸⠀⠀⠁⠀⠀⢻⡸⣧⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠸⣆⠀⠀⡼⠁⠀⡀⡼⠀⠀⠀⣇⠘⢇⠄⠀⢸⡇⠀⠀⠀⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢱⡄⠠⠂⠌⢠⡇⡄⠐⠠⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⣿⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠷⠀⠀⠀⠀⣠⠏⠠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣿⠀⣠⠄⢀⡤⡾⠁⢀⡤⠀⢀⡟⠀⢀⠄⠀⠀⣏⣀⠀⣠⠀⠀⣀⠀⢸⠃⢀⠀⣸⠁⠀⢰⣦⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠰⠷⠆⠀⠯⠀⠿⠀⠸⠅⠇⠀⠸⠇⠀⣸⡇⠀⠺⢻⠀⠠⠇⡏⠀⠿⠀⠀⡟⠀⢸⣷⠁⠀⠻⠀⠀⠸⠿⠯⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠄⣺⠃⠀⠀⠀⠀⠀⠠⢐⠏⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⣰⡄
⠀⠀⠀⠀⠀⢠⣄⣠⠀⠀⣠⡀⠀⣄⢀⡀⠀⢀⢤⡀⠀⢠⡀⠀⢀⠀⠀⡟⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠸⠏⣏⠀⠸⡅⠃⢠⡿⠴⠃⠀⠧⣿⠀⠀⢾⠁⠀⢸⠇⠸⡇⠀⢸⡧⡟⠀⢸⡃⠀⢺⢵⡆⠀⣠⠿⡁⠀⠿⢸⠀⠞⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠠⢠⡟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠊

⠀⠀⠀⢀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⡿⠋⣿⡄⠀⣇⣠⠇⠀⢸⠋⠀⠿⠂⠀⠉⣿⡉⠉⠙⢿⡆⠀⣤⠤⢤⡄⠙⣿⡟⠛⠛⠛⣿⡆
⠀⠀⠀⣾⣁⣀⣸⡇⠀⡇⠀⢱⠀⢸⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⠀⢺⠀⣀⠁⠀⣿⡇⠀⠀⠀⠈
⠀⠀⢸⡟⠛⠛⠻⣇⠀⠿⠤⠜⠀⢸⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢸⠉⠋⠀⠀⣿⣧⣤⣼⡇
⠀⠀⣸⠃⠀⠀⠀⣿⠀⠐⡆⣠⠆⠸⣇⠀⣲⠆⠀⠀⣿⡃⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⣿⡇⠀⠙⠃
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⡁⠀⠀⠉⠙⠉⠀⠀⠀⣿⡆⠀⠀⠀⣿⠀⠺⠶⠶⠃⠀⣿⡇⠀⢤⠄⠀⢤
⠀⣤⣿⡄⠀⠀⢠⣼⣧⠀⡇⢳⡀⠐⢿⣆⠀⢶⠂⠀⣿⡇⠀⠀⣠⡟⠀⣠⠶⠶⣦⠀⣿⡇⠀⢸⣇⣀⣸
⠀⢀⣀⠀⠀⢀⡀⢀⣀⠀⠁⠀⠁⠀⢸⠈⠳⣼⠀⠰⠿⠷⠶⠿⠛⠁⠀⣿⠀⠀⠉⠁⣿⡇⠀⢸⡏⠉⢹
⠀⠈⣯⠀⠀⢸⡇⠀⡏⠀⢰⢀⢴⠀⢸⠀⠀⢸⠀⣠⣤⣤⣤⣀⠀⠀⠀⣿⠀⠀⠀⠘⠛⠛⠃⢸⡇⠀⢸
⠀⠀⣿⠀⠀⢸⡇⠀⡇⠀⢸⠀⢸⠀⢸⠀⠀⢸⠀⠀⣿⠀⠈⠙⣷⠀⠀⣿⢀⣄⣀⠀⣿⠛⢢⠘⠃⠀⠚
⠀⠀⣿⠀⡄⢸⡇⠀⣇⠀⠈⠀⠈⠀⠚⠂⠀⠚⠂⠀⣿⣤⣤⡾⠏⠀⠀⢻⡈⠁⣿⠀⢿⡖⠋⠀⠀⣀⣀
⠀⠀⣯⠀⠳⠞⠀⢀⣿⣀⣀⡆⠀⠀⢠⡏⠉⢷⠀⠀⣿⠀⠀⢰⠋⠉⣇⠈⠙⠛⠉⠀⣿⠹⡄⠀⠀⣇⣈
⠀⢀⣿⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠛⠀⠙⠂⢀⡀⠈
⠀⠈⠉⠀⠀⠀⠈⣿⠋⠀⠀⢹⣿⠁⠈⡇⠀⣸⠀⠀⣿⠀⠀⢸⡄⠀⣿⠀⠈⠻⣟⠀⠀⢸⡿⠃⠈⠧⠴
⠰⡟⠻⡿⠛⠷⠀⣿⠀⠀⠀⢸⣿⠀⠀⠳⠤⠟⠀⠙⠛⠋⠀⠀⠳⠾⠯⠀⠀⠀⠹⣧⢠⡟⠁⣀⣀⣀⣀
⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠘⡟⠀⢸⡟⣀⠀⠀⠀⠀⣀⠰⣶⠀⢰⠖⠀⢹⡟⠀⠀⠋⠉⢩⡏
⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⢻⠀⢸⠁⢸⡄⠀⠀⢠⠇⠀⠹⣤⡏⠀⠀⢸⡇⠀⠀⠀⢀⡞
⠀⠀⠀⡇⠀⠀⠀⣿⡆⠀⠀⢸⣿⠀⠀⢸⡆⡿⠀⠀⡇⣸⡇⢸⠀⠀⠀⣿⡀⠀⠀⢸⡇⠀⠀⠀⡼⠁
⠀⠀⠀⣇⠀⠀⠀⢹⣇⠀⠀⢸⣿⠀⠀⠀⣷⡇⠀⠀⢹⡇⢸⡇⠀⠀⣼⠁⢷⠀⠀⠸⡇⠀⠀⢸⣃⣀⣠
⠀⠀⠘⠛⠀⠀⠀⠀⠛⠿⠿⠟⠃⠀⠀⠀⠛⠂⠀⠀⠈⠁⠈⠁⠀⠰⠇⠀⠸⠗⠀⠚⠛⠂⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⣿⣧⣽⠻⢿⠏⢽⣿⣶⢶⣤⠀⠀⠀⢸⣻⣿⣴⣿⣷⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣽⣣⣭⢻⠿⠃⠘⠴⣿⡯⣿⠄⣿⣧⡀⠀⢹⣿⣻⠈⠀⠙⣞⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⠀⠘⠃⠀⠀⠀⠀⣸⣹⣿⣡⣿⢿⡄⠀⢸⣿⣽⣄⡀⣰⣯⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⣤⣴⣶⣾⣿⣿⢈⢯⣛⢷⣞⠃⠀⠸⠿⠿⠻⠿⢷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⣿⣽⣿⠿⠏⠛⠋⢹⣾⣿⣿⡠⣿⡇⠀⠀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣯⡝⠀⠀⠀⠀⠀⣸⣿⢿⣿⣿⣿⠇⠀⣼⣷⣻⠛⠙⠻⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡭⣯⣿⡇⢜⣦⣀⣀⣤⣎⡹⣿⣧⣿⣿⣿⡇⠀⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣿⣟⢴⣿⣦⣷⡿⠛⠁⢿⣿⣋⣉⢻⣷⠀⠛⣿⣳⣦⣤⠾⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉

⠀⠀⠀⠀⣀⡀⣀⡀⢨⣇⣀⣀⠀⢀⣠⣀⡀⢀⣀⣸⣯⠀⣀⣤⣀⠀⣸⣯⡁⣀⣀⣀⡀⢸⣏⣀⣀⠀⣉⡁⢈⡁
⠀⠀⠀⠀⣤⠔⢺⡇⢸⣏⠀⣿⠀⣿⠀⢈⡁⢸⡇⢈⣿⠰⣿⠂⠚⠃⢸⡏⠀⢿⣀⣹⡇⢸⡧⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⡀⠉⠉⠉⢁⠈⠉⠉⠉⠀⠈⠉⠉⠀⠈⠉⠉⠉⠀⠉⠉⠉⠀⠈⠉⠀⢬⣍⣼⠇⠈⠉⠀⠉⠀⠉⢠⣼⠇
⠀⠀⢸⡿⢠⡤⠀⣿⡇⢠⣄⣤⣄⣤⣄⠀⣤⣤⣤⡀⢀⣤⢤⡄⠀⣤⣠⣤⠀⢠⣤⣠⡄⢀⣤⣤⢀⡤⠤⡄⢠⣾⣤
⠀⠀⠘⣿⠻⣧⠀⢿⡂⢸⡇⢸⡏⠀⣿⠀⣿⠀⢸⡇⢸⣇⣀⣿⠀⣿⣀⣸⠇⢻⣀⣸⣇⢸⣿⠀⢈⡙⢒⣶⠀⣿⡀
⠀⠀⠈⠁⠀⠉⠁⠈⠀⠈⠀⠀⠁⠀⠉⠀⠉⠀⠈⠁⠀⠈⠉⠀⠀⠿⠈⠀⠀⠀⠉⠸⡇⠀⠉⠀⠀⠉⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣶⠐⣦⠀⣶⠀⣦⠀⣶⡄⢠⡆⠰⣦⣰⠖⠐⣦⠀⣴⠀⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡤⣿⠀⠹⣾⠃⠀⠘⣾⠃⢻⡞⠀⢠⡟⢻⣆⠀⠹⣾⠇⠠⣾⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃
⠀⠀⠀⠀⠀⠀⠈⠈⠂⡄⠀⠀⠀⠀⠀⠨⣧⡤⡀⠀⠀⠀⠀⠀⠀⠂⢤⠀⠀⠀⠀⠀⠀⢠⡄⢸⡇⠀⠀⠀⠀⠀⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢘⠀⠀⠀⠀⠀⠠⡆⠀⠙⡇⠀⠀⠀⠀⣴⠀⠈⠀⠀⠀⠀⠀⢠⠄⠀⠘⡇⠀⠀⠀⠀⠛⠖⢈⡇
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠸⡄⠀⠀⠀⠀⠘⠓⠒⠘⠁⠀⠀⠀⠀⠙⠒⠋⠛⠀⠀⠀⠀⠘⠓⠂⠰⠇⠀⠀⠀⠀⠙⠂⠉⠇

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⢁⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠧⢤⡄⠀⠀⠀⠀⠤⠴⠤⡆⠀⠀⠀⠀⠐⠛⠒⣆⠀⠀⠀⠀⠐⣞⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⠀⣄⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⢠⡀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠘⢳⠀⠀⠀⠀⠀⡀⠣⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠠⠃⠀⠐⠁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠔⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠐⠷⠒⢒⡚⠐⢆⠀⠀⠀⠀⠘⠒⠐⠲⠀⠀⠀⠀⠀⠀⠚⠛⠳⣆⠀⠀⠀⠀⠠⠖⠲⢦⡀
⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠦⠀⠨⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠰⣏⠀⠀⢻⠀⠀⠀⠀⠐⣏⣀⠀⡧
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠟⠀⡈⠃⠀⠸⠀⠀⠀⠀⠀⠇⠀⠀⠰⠀⠀⠀⠀⠀⠙⠚⠀⠉⠀⠀⠀⠀⢀⡌⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢤⣀⣠⣤⠀⠀⠀⠀⠠⠤⣤⠀⠀⠀⠀⢀⡄⢸⣧⡀⠀⠀⠀⠀⠀⣀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢈⠀⠀⣷⠀⠀⠀⠀⠀⢈⠁⠀⠊⠀⠀⠀⠀⠀⢶⣈⠀⠀⠀⠀⠀⠀⢠⠜⠁⠀⠀⠀⠀⠀⢻⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠃⣯⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⣤⡀⠙⡇⠀⠀⠀⠀⠀⠸⣅⠀⠦⠀⠀⠀⠀⢀⡀⡠⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠈⠛⠃⠀⠀⡿⠁⠀⠀⠀⠛⢆⠀⣀⠀⠀⡻⠀⠀⠀⠐⢻⡦⢐⠏⠀⠀⠀⠀⣤⡀⠀⣤⠄⠀⠀⠀⠀⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠳⢀⠜⠀⠀⠀⠀⠀⠀⠘⢀⠉⠀⡰⠁⠀⠀⠀⠀⠀⠁⣎⠀⠀⠀⠀⠀⠈⢧⣀⠘⡇
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠖⠀⠀⠀⠀⠀⠀⠠⠂⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⢠⠇


⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⡴⠶⢶⣴⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⣶⣾⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⠀⣉⣉⣉⣽⣿⡟⠛⠁⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠠⠔⣊⣉⣩⣭⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠶⠛⠉⣉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣋⣠⠴⠒⠋⣉⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣋⣁⣤⣴⡾⢿⣟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⣋⣡⣴⣿⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣵⣶⣿⣿⣿⣛⣉⣡⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣶⡶⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡙⢏⣿⣿⡛⠖⠛⠋⠀⠀⠀⠀⠀⢀⡶⢛⣡⣾⣿⣿⠃⠐⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡘⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠛⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠙⠋⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣾⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣠⣶⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⢺⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣼⠿⡯⠋⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣤⡀⠀⠀⢠⣾⢛⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⢏⠙⠌⠋⠻⣯⡀⠀⠀⠀⠈⠙⢷⣤⣀⠀⣀⠀⡀⢤⣀⠀⠀⠹⣿⣦⡀⠹⣷⠀⠀⣹⡯⠀⠀⢹⠶⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡣⠕⠨⠢⠁⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⠪⡦⡆⣽⣿⣿⣿⣿⣿⣷⡀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⣾⣿⢿⡳⢝⠢⡈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠨⣟⢿⣶⣶⣿⡟⠛⠉⠉⠈⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡿⣙⠦⡉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⡠⢋⣴⣿⠿⣿⡇⠀⠀⠀⠀⣿⣿⡷⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣸⡟⠑⢮⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄⠀⠀⣰⣿⣿⠗⣋⣽⡇⠀⠀⠀⠀⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣭⣟⠿⣿⣾⣿⣿⣿⣯⣵⣾⠃⠀⠀⠀⠀⠸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠉⠀⠈⠛⣿⣿⡿⠟⢻⡇⠀⠀⠀⢠⣶⣾⣿⣿⡀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢋⣠⠖⠛⣧⠀⠀⢰⣿⣿⣿⠿⠿⠿⠶⣶⣤⣄⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠀⠀⠀⠹⡄⢰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⣿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣱⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢆⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠢⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⡀⠀⠀⠀⠀⠀⠈⡀⠀⢰⣳⠁⠀⠀⠀⠀⠀⠈⠉⠙⠟⠤⠀⠂⠈
⠀⠰⡴⡀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠐⠲⣶⢄⠀⠀⠀⠀⠁⠀⠀⢱⢧⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠘⢿⢆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡯⡆⠀⠀⠀⠀⠀⠀⠀⣮⡄⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠈⢎⢆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠔⠡⠁⠀⠀⠀⠀⠀⠀⠀⠘⡾⡀⠀⠀⠀⠀⠀⠐⠸⢰
⠀⠀⠀⠀⠈⡍⢆⠀⠀⠀⠀⠀⠘⠀⠂⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⢧⠀⠀⠀⠀⠀⠀⠀⣟⡄⠀⠀⠀⠀⠁⠀⣀⣄⡀⠀⠠⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠘⣮⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⠀⣾⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸⣱⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸⡓
⠀⠀⠀⠸⠃⠀⡆⠸⣳⡀⠀⠀⠀⠀⠀⡀⠠⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠈⢟⡃⠀⠀⠀⠀⠀⠀⣟⡀⠀⠀⠀⠠⠀⢸⣇⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠐⠑⠀⠀⢷⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣞⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡟⠇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⡆
⠀⠀⠢⢀⡙⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠙⠀⠀⠀⠀⠀⠀⠈⣇⡇⠀⠀⠀⠀⠀⠸⣳⠀⠀⠀⠀⢀⠀⠀⠀⡠⡾⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠀⠀⠀⠀⠀⠓⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠑⠀⠐⠈⠀⠀⠀⠀⠀⡀⠁
⠀⠀⢤⣤⢠⠄⠀⠀⠀⠀⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⡟⡆
⠀⠀⡸⠁⠟⠢⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣁⣀⢀⣀⣟⣇⣀⣀⣀⣀⣀⣄⠹⣜⣦⣄⣀⣀⡀⢀⣀⣀⣀⣠⣆⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣦⣤⣄⠀⠀⢀⣤⣄⢠⠞⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣧⢹⣷⡴⠁⠈⣿⡏⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡟⠘⣿⡇⠀⠀⣿⡇⠀⢸⡇⠀⠀⢀⡤⠄⠠⣿⡑⢀⣴⠂⣾⣡⣤⡿⠁⠀⣴⡶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⡇⠀⠀⣿⣇⠆⠀⡹⠓⣿⠃⠀⠘⢡⡿⠁⡠⣺⡿⠁⡰⣿⠇
⠀⠀⠀⠀⠀⠀⣠⡶⠛⠳⡀⠀⢰⡖⢆⠀⠀⣿⡇⠀⠀⢿⡇⠀⢸⣇⠀⠀⢻⣧⣠⠜⠁⠸⣏⡠⠊⠀⠸⡧⠔⠀⢿⡥⠊⣼⠏⠀⠀⢀⡷
⠀⠀⠀⠀⠀⢰⣿⠃⠀⢀⡇⠀⢿⣇⢸⠀⠀⣿⡇⠀⠀⠉⠁⠀⠀⠻⠦⠴⠃⠀⠠⠤⠶⣤⣤⣴⣆⣀⣠⠤⠒⣀⠀⢠⣾⠯⠤⠤⠶⠟⠁
⠀⠀⠀⠀⠀⣸⣿⠈⠷⠾⠁⠀⠸⣿⣸⠀⠀⠀⠀⢀⣀⡀⢠⣾⠀⠀⠰⠆⠀⣰⡟⡃⠀⠀⣹⡟⠉⠉⠀⢠⡊⣀⣤⠿⠃
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⣾⠁⣧⢾⣿⠀⠀⣶⡆⠀⢻⣿⠀⠀⢠⣿⠁⠀⣠⠀⣈⠉⢉⠁⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠈⣿⣆⠐⠙⣷⡘⠷⣿⢸⣿⠀⢀⣿⠃⠀⠌⢿⣇⡀⢼⡇⠀⢀⣿⠃⣼⠆⢹⡇⠀⣴⡟⢩⣿⠁⢸⣧⠆⠀⢦⡀
⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢹⣇⠀⢸⠈⣿⣀⠜⣿⣇⠜⣀⠼⣿⡆⢾⡇⣠⣾⡇⠀⣿⠀⣾⠁⢸⡿⠀⣼⡇⠀⡸⣿⡄⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⣿⣷⠀⠈⣿⡀⡜⠀⠀⠀⠀⢈⠽⠊⠁⠀⣿⣇⠈⠉⠁⠈⠀⠈⠃⠀⢿⠔⠻⠷⠊⠻⠦⢊⡠⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⡏⢿⠇⠀⠈⠋⠁⠀⠀⠀⡴⠁⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡀⢀⣶⣶⣶⣶⣶⣶⠒⠒⠒⠒⠒⠋⠀⠀⠙⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⣠⣀⡀⢸⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⡀⠀⠀⢠⡞⠉⠙⠓⠚⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡿⠁⠀⠀⠘⢧⣀⣠⠴⢦⡄
⠀⠀⠀⠀⠈⠉⠉⠁⢸⣿⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⡄⠀⠀⢠⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⣿⡇⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠖⠋⠉⢳⡄⠀⠀⢀⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣄⣀⡼⠃⠀⠀⠈⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⡟⠉⠉⢻⣿⣿⣿⣿⣿⡇⠈⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⣄⠀⠀⣠⠤⠤⠤⠤⠤⠿⠿⠿⠿⠿⠿⠁⠈⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⡰⣶⣶⡀⢶⢂⣤⣶⢆⣴⢠⡆⣰⡆⢰⠀⣶⠀⣶⣶⡶⠂⣴⣆⠀⣴⠆⣴⡄⣶⠶⣶⠀⣒⡶⠦⢰⣦⢰⣶⠶⠀⠶⣒⡀⢰⣦⠠⣶⢰⣶
⣿⣌⢿⣿⠆⣾⠟⠁⣾⣿⢸⡇⠹⣷⣶⣾⠟⢠⣮⡍⢡⣬⣿⣿⣾⠟⠘⠿⠛⣰⡆⢡⣿⠟⠻⣷⠘⠟⠸⣿⢀⡲⠄⣿⡷⢸⣟⢤⣁⢼⡿
⣿⣿⢃⣴⡇⣿⠘⢷⣦⣝⣸⡇⣶⡶⢶⣶⣶⣦⣤⣴⣶⣦⣭⣴⡄⣶⠷⣘⡧⢸⣿⠸⠇⣾⢇⣿⢠⡾⠷⣆⢺⠿⠂⢀⣤⠸⡿⢈⣿⣷⣴
⣋⣥⣶⣶⣤⡁⣶⡀⣰⡎⣩⣥⡄⢾⡷⢶⡀⣴⣄⣶⠀⣶⡀⣶⡄⠟⠘⣹⡆⢠⣉⣾⡇⣉⣈⣋⡘⠳⠞⣃⠰⡿⠛⠈⢿⡇⠿⣠⡀⢦⣤
⡿⢩⣤⡉⢻⣷⢸⣷⡟⠀⠿⠿⠏⠸⣧⠀⢷⡘⢛⣋⣰⣿⠇⠛⠛⣿⡟⣛⣁⣙⡛⡛⢃⣩⡍⣿⢡⡟⠿⠻⣧⠙⣼⠃⣛⣛⠘⠿⠿⢂⣉
⠁⠘⠛⠃⠀⠛⠀⠛⠃⠛⠛⠛⠛⠂⠛⠀⠘⠃⠘⠛⠋⠁⠘⠛⠃⠙⠃⠉⠉⠙⠃⠙⠘⠛⠁⠋⠘⠛⠘⠛⠘⠃⠙⠀⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠣⠀⠄⠀⡀⠈⠀⠀⠀⠐⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⡁⠐⠉⠂⠀⠀⠀⠄⠀⠀⠀⢀⣴⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣆⠀⠠⠀⠀⠀⡈⠀⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠄⠛⠉⠉⠀⠀⠀⠀⠘⠛⣿⣦⡌⠙⠶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⣁⠙⣿⣔⡐⠚⠛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣀⣀⠀⠤⠤⠤⠶⢋⡡⡢⢝⣿⣤⣀⣈⡁⠚⣦⡀⠀⠀⠀⠀⠀⣠⠞
⠀⠜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⡴⠁⠀⡀⠐⠂⠉⣀⣀⠀⠁⣒⣒⣂⣭⣉⣷⣾⣿⣭⣤⣤⡄⣀⠈⢳⡄⠀⠀⣠⠔⠁⠠
⠈⠈⠌⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠟⠀⠀⠀⠀⠀⢈⡁⢠⣤⣴⣶⣿⠿⠟⠛⠉⠉⠙⠻⢿⣿⣧⠄⠀⠁⠀⠹⣤⠊⠉
⠀⠀⡐⠀⠀⢉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢰⠏⠀⠀⠀⡀⠔⠃⠈⣡⡴⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⢱⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠀⠀⠀⠀⣠⠇⢀⠃⠀⠀⢀⠂⡠⢔⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢇
⠀⠠⠠⠀⠄⠀⠄⠀⠀⠡⠠⠀⠌⡼⠉⠉⢣⣠⡏⠀⠀⢀⠀⣰⡗⣩⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⢂⠸⡧
⠀⢡⢡⠉⡌⠈⠌⠀⠀⠁⠠⠉⡜⡵⡤⠠⢼⣿⠁⠀⠀⡌⢠⣟⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⢢⠈⡄⣿⠀⠀⠈
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠀⠀⣾⣭⣄⡀⢰⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣆⢃⣰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡿⣅⢀⡸⡀⣿⣿⠃⣤⠶⠶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣠⠴⠾⠛⠛⠉⠛⠒⠀⠀⠀⠀⢸⣿⣿⡎⢹⣾⡇
⠀⠌⠌⠀⠡⠡⠡⠀⠀⠌⠌⠤⠡⠡⠀⢸⣷⠀⠀⠀⢱⣿⣿⡀⠀⣠⣴⣦⣤⡢⠀⠀⠀⠀⠀⠀⠀⣠⡶⡶⠿⣷⣦⡀⠀⠀⠀⠀⠸⠹⣿⣷⠈⣿⡉
⠀⡈⡈⠀⠀⠀⠁⠀⠀⠀⠀⠀⢡⠀⠀⢠⣼⡆⠀⠀⠸⢹⣿⡇⠘⠛⠯⡭⠀⠉⠶⠀⠀⠀⠀⠀⠘⠙⠲⠂⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡀⣿⠟
⠀⠐⠐⠀⠂⠀⠂⠀⠀⠀⠀⠀⢂⠂⠀⠀⣸⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣾⣿⣧⣸⡇
⠠⠡⠡⠈⠄⠄⠌⠀⠀⠠⠠⠀⠌⡄⠄⣀⣸⣇⡿⢀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠀⠀⠀⠹⣿⣷⣶
⠀⢁⢃⠈⠀⠀⠀⠀⠀⠀⠂⣀⣼⣾⣿⣿⣿⣿⡇⠰⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⢰⠀⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠉⠻⣿⣿⣷⣦⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠐⠂⠀⠀⢸⠀⠀⣿⣿
⠀⠄⠄⠠⠀⠠⠠⠀⢀⣿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⣿⣿⣿⣧⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠃⠀⡀⠀⠀⠸⠀⠀⠘⠀⢰⣿⡇
⠀⠈⠈⠀⠁⠀⠁⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⢹⣿⣿⣿⣧⡀⠀⠀⠀⠈⠛⠿⠷⠾⠋⠁⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡆⠀⡀⠀⣸⣿⡇⣾
⠀⡀⡐⣀⠀⢀⢂⠀⣿⣿⡇⠀⠀⣄⠘⣆⠀⠀⢻⣿⣿⣇⡈⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠸⢀⣥⣾⣿⣿⣰⡿
⠀⡸⡀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠸⣦⠘⣧⠀⠀⢿⣿⣿⡠⠀⠀⠈⡹⢿⣿⣿⣷⡀⠀⣀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡗⠂⠸⣿⣿⣿⣿⡇
⠀⠽⠡⠀⠄⠄⠀⢸⣿⣿⡇⠀⠀⠀⢹⣧⡹⣷⡀⠸⣿⣿⣧⢂⠀⠀⠈⠙⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⡟⣿⡟⡀⠘⠀⢿⣿⣿⣿⣷
⠀⢱⢣⠉⠈⠈⡌⢸⣿⣿⡇⠀⣤⠀⠀⢻⣷⣿⣷⠀⣿⣿⣿⣟⣦⠀⠀⠀⠸⡞⣹⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⢠⣾⣇⣀⠀⠂⢻⣧⠇⠀⠁⠈⠹⣿⣿⣿
⠀⢃⢂⠀⠀⠐⡀⢸⣿⣿⡇⠀⣿⣿⣄⠈⢿⣿⣿⣧⢸⣿⣿⣿⣿⣧⢆⠐⡀⢻⣣⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣼⣰⠃⠸⣧⠈⠀⠀⠀⠀⠹⣿⣿
⠀⢚⠆⠀⠀⠀⠀⢸⣿⣿⡇⣀⢹⣿⣿⣷⣜⣿⣿⣿⣿⣿⠙⣿⣿⣿⣾⣦⢳⡈⣿⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣼⠀⣿⣆⠐⠀⡄⠀⠀⢿⣿
⠀⠜⠄⠀⠠⠠⠠⢸⣿⣿⡇⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⣿⣿⣿⣿⡎⣷⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡞⣼⣿⣆⢠⣷⠀⡆⢸⣿
⠀⣼⡌⠀⠀⠠⠀⢸⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡆⢻⣿⢿⣿⣷⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣿⣶⣧⣼⣿
⠐⡸⡐⠂⠂⠂⠂⢺⣿⣿⣧⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃⢠⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⡼⠛⠃⠀⠀⠀⠀⠤⢶⡀⡤⠴⡳⠊⠂⠀⠀⠀⠀⠀⠀⠈⠝⢆⡀⠀⠀⠀⢀⡀⢀⣔⠔⠛⠃⠀⠀⠀⢀⠠⣤
⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠋⠉⠉⠀⠀⠀⠀⠀⣨⡞⠁
⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⣴⠗⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠏
⠀⠀⢤⠆⠀⠀⠀⠀⠛⢶⢄⠄⠀⠀⠀⠀⠈⠙⠈⠈
⠀⠀⣻⠃⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠷⣄⠀⠀⠀⠀⠀⢀⣀⠀⣄
⠀⠈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠊⠻⢆⠀⠀⠀⠀⠉⠉⠁
⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢲⢤⡠⠀⠀⠀⠀⠀⠀⠉⠷⠂⠀⠀⠀⠀⠀⠺⡀
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠩⢷⠂
⢠⣄⣠⢜⠎⠉⠁⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠙⠛⠓⠦⣄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣎⡀⠄⠦⡁⡄⣠⠖⠄⠂⡂⡄⡄⡎⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⡀
⡤⣄⣄⠀⠀⠀⠀⠀⠈⠛⠒⢶⠄⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡃
⠀⠈⠉⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠀⠀⠀⠀⠶⠗
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠐⡶
⠀⣠⡴⡓⠉⠀⠀⠀⠀⠀⠘⠟⢄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠲⠖
⠙⠈⠀⠀⠀⠀⢤⡆⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⢀⡥⠀⠀⠀⠀⠀⠘⠷⣤⡀
⠀⠀⠀⠀⠀⠀⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣦⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶
⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⢀⣠⢞⠟⠀⠀⠀⠀⠀⠁⢳⡄
⣴⠀⠀⠀⠀⠛⠊⠀⠀⠀⠂⠛⠂⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⣀
⠩⣦⡂⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⣴⡉⠁⠀⠀⠀⢀⣤⡜
⠀⠐⠝⣵⠀⠀⠀⠀⠀⢽⠅⠀⠀⠀⠀⠐⠺⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣀⡾⠅⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⠁⠀⠀⠀⣿⣿⠀⠀⣴⣶⠄⠀⠀⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠈⠙⢒⡒⢺⣿⣿⠒⢒⣿⣿⠛⠒⢲⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠐⠿⠟⢸⣿⡏⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣶⡆⠈⣿⣿⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣇⠀⠹⠿⠂⠀⣿⣿⠀⢀⠈⣿⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠉⠉⣠⣄⠀⣠⣄⠹⣿⣧⠎⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⡰⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⡟⢀⣀⣀⣀⣀⣀⠀⠀⠀⢳⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠕⠫⠋⣉⣭⣉⠉⣉⠉⢉⡀⠹⣿⡟⢀⡀⠀⢀⡀⠀⣀⠀⢀⡀⢹⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠸⡿⠀⠉⢀⠈⠁⣀⠈⠀⠈⢁⣀⡈⠁⣀⡉⠀⢈⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢿⡇⣾⠋⠇⠀⠈⢹⡏⠁⣾⠋⠁⢀⣿⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢸⡇⠹⣆⠀⠀⠀⢸⡇⠀⣿⠛⠁⢸⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢸⡇⠀⠙⣷⡀⠀⢸⡇⠀⣿⠀⠀⢸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢳⡄⠀⢹⡇⢸⠇⠸⣇⣸⡇⠀⠸⠇⠀⣿⣤⠄⢸⡇⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠘⠃⠀⣸⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⣽⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣇⣀⣷⣶⣿⣿⠛⠛⣻⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⣿⡙⠀⠀⠈⠁⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠉⠽⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠸⠿⠿⣿⣇⠀⣤⡄⠀⢠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣦⣾⣿⡇⠀⠀⠀⠀⠀⢻⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣿⣀⣾⣶⣿⣿⠿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⠛⠻⠿⠿⠉⠿⠿⣿⡏⠁⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣇⣠⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⠷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡷⠀⣀⡀⠀⠀⣤⣿⣿⣿⣿⣿⡏⠉⠉⣿⠏⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠇⠈⢱⣶⣿⣷⣶⣶⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠐⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣶⡆⠀⣸⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⡀⠔⢮⢴⠠⡰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢽⣹⠞⡁⢸⡘⣿⠚⡵⢟⣿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣟⣓⣇⠈⠸⣦⠀⢰⡄⢤⠈⠀⣍⠉⠭⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡀⠄⠀⡉⠖⠁⠀⠘⠈⠈⢁⠘⣆⠁⠸⣀⣋⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠄⡐⡘⠂⠉⠀⣘⠈⠀⠃⠈⠀⠉⠀⠀⠀⡐⡂⠀⣈⢇⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠌⠠⢀⠆⡥⠑⠀⠁⢲⣽⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠘⣫⠀⡀⠀⠀⠁⠸⠀⠊⠈⠈⠈⣑⢠⡔⣆⠘⣼⢻⣶⡂⠗⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠷⠌⠉⢀⡠⣤⠠⣴⠞⣰⢿⢶⡾⠲⠎⠀⢩⠟⠝⣧⡑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣞⢂⡇⠧⢪⣀⣾⣾⣼⣪⣷⠟⣦⠀⢩⠆⠲⣆⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⠆⡳⣯⡯⣞⢿⣻⠽⡖⠟⢏⠱⢧⣘⣧⣼⡛⠓⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠀⠀⠐⠳⠛⣯⢎⠰⢇⢃⡴⡴⣚⣷⣯⠌⡁⠀⠀⠀⠀⠑⠪⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⡍⢁⣏⣈⢡⣞⣿⡵⣶⣮⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣭⢭⣽⠙⣝⠌⡶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⢏⢴⡥⡔⢦⣴⣚⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣚⣻⡯⠄⠈⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⣉⣁⣤⠴⠖⢋⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣉⣁⣤⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢄⠀⠀⠀⠀⠀⠀⡔⠉⢣⠀⢠⠇⠀⡤⠚⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⠀⠀⠀⠂⠄⢤⠙⠁⠀⠘⢆⠀⠀⡨⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠋⠀⠀⠑⠒⠒⠊⢳⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠉⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⣀⡀⠀⠀⠀⠀⢀⠤⢄⠈⡆⠀⡠⠊⠀⢀⠤⢤⡀⠀⠀⠀⠀⠘⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⡆⠀⠀⠀⡇⠀⠀⠉⠀⠀⠳⣄⠀⡊⠀⠀⢘⠆⠀⣀⠤⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⣀⡰⠀⠀⠀⠀⠑⠤⠒⠉⡆⠀⠀⢀⣉⡁⢠⡒⠉⠀⡰⠁⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣅⠀⡸⠀⠈⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠉⠉⡇⠀⠀⢏⠉⠉⢉⠧⣀⣀⣀⡸⠐⣅⣀⣀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡀⠀⠀⣀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠖⠢⠤⠞⠀⠀⠈⠉⠉⠀⠀⠀⠸⢄⠠⠲⡄⠀⠀⠀⠰⠁⠉⠒⠎
⠀⠀⠀⠀⠀⠀⠸⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⢀⡇⠀⠀⠀⠰⡀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢱⠀⠀⡴⠊⠉⠐⠄⠀⢈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⢱⠄⠀⢠⠃⠀⢸⡤⠤⠒⠦⠀⠀⡤⠒⠒⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⢠⠃⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠔⠁



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

⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣄⠀⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠁⠉⠙⠓⢦⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢤⡦⠟⡶⠄⠙⠒⠤⣆⡀⣰⠀⠀⠀⣠
⠸⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⣳⣌⠀⠉⠘⡀⠀⠀⠉⠉⠹⠔⡦⢊⡏
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠖⢦⠀⠀⠀⠀⠀⠀⠀⠴⠶⢄⠈⠁⠀⢀⣄⣤⣾⣄⣄⣀⠀⠀⡰⡇
⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡴⠃⡜⠁⢠⠎⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⡏⡷⠤⠚⠉⠀⠀⠈⠉⣻⣦⡄⠸⢤⡄
⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠊⠉⠀⠀⠣⠔⠁⠀⡌⠀⡜⢀⠊⡇⠠⡲⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣉⡁⠀⠧⣀
⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣠⡶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠈⠊⠀⠀⠀⡰⠋⠀⠀⠤⣀⠀⠀⠀⠀⢀⣿⡟⠛⠂⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡞⠰⢦⣀⡈⡙⠀⠀⢀⣸⠋⠁⢰⡚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠸⢮⣁⡬⠃⠀⡴⣋⣉⣠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣩⡀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣡⣦⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠈⠀⠠⡀⠀⠀⡠⠤⠴⠃⠁⠈⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⢄⠀⠀⠐⡀⠀⡀⠘⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣄⠀⠀⠉⠉⠓⠃⠀⣧⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠉⠉⠓⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⣀⠴⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠉⠁⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⢾⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⡉⠁⠀⠀⡼⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠹⢄⣀⣠⠾⡁⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⢠⠜⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣐⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⡠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠈⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠙⠟⠂
⠀⠘⢷⢄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⢻⣦⠀⠀⠀⠀⠀⠀⠘⡺⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢯⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠀⠀⠀⠀⠀⠀⠀⠹⣡⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠈⢯⢆⠀⠀⠀⠀⠀⢡⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡆⠀⠀⠀⠀⠀⠀⢡⡇
⠀⠀⠀⠀⠈⢏⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠀⠀⠈⣽⡀⠀⠀⠀⠀⠀⠘⣶⠀⠀⠀⠀⢀⠀⢺⡏⠁⠀⠈⠉⠉⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠈⢟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢹⣇⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠰⠀⠠⠘⡾⡄⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠁⠔⠄⠀⢸⣥⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠈⠁⠀⠈⢸⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠣⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠆⣈⠒⠀⠀⠀⠘⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⢀⠰⠟⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡄⠀⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠑
⠀⠠⢠⡄⡦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠜⠘⠙⠀⠀⠀⠀⠸⢇⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⣠⣀⣀⢀⡤⢿⣆⣠⣤⣄⣀⣠⠈⠾⣵⣦⣤⣀⣀⣀⣀⣤⠤⠶

⠀⠀⠀⠀⠀⣠⡴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠓⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣀
⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⢀⣸⣇⢀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿
⠀⠀⣭⣽⣯⣽⣿⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿
⠀⠀⠀⢸⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠶⢾⡷⢶⣾⣿⡆⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⣿⣿
⠀⠀⠛⢻⡟⠙⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣀⣸⣇⣀⣤⣤⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⣿⣿
⠀⠀⠭⢿⡯⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿
⠀⠀⣶⣾⣷⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠉⢹⡏⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⣀⣸⣇⣠⣤⣤⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⣿⡇
⠀⠀⠶⢾⡿⠾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⡇⠀⢀⡀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⢸⣿⡇
⠀⠀⣛⣿⣟⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠉⢹⡏⠉⠛⠛⠁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⣼⣿⠇
⠀⠀⣤⣼⣧⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠚⢿⡟⠻⠿⠿⠇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⡇⠀⠉⠉⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡄⠀⠀⣿⣿
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠈⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣞⣯⡾⠁
⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⣭⣭⣭⣭⣭⣭⡭⠿⠿⠽⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣦⣤⣄⣀⣀⣀⡀⣀⣀⣀⣤⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⢘⣿⣷⡍⠶⡌⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣀⡴⠊⠝⢊⡡⠔⠚⠉⠈⠙⢾⠿⣨⣟⢢⣌⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠴⠊⠁⠀⠀⠉⠁⠀⠀⠀⣠⣴⣀⣀⢻⣙⢫⣶⣯⠱⣦⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠓⠦⣜⣫⣶⣮⠸⠿⣇⣼⡷⢀⣈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡠⣄⠙⣋⡺⣿⢗⣝⡛⣥⣮⡛⢡⣄⡉⠻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⡿⣗⣉⣋⣾⣿⢝⠻⢟⣦⣽⣿⣧⣮⣩⣾⣷⢏⡱⣶⡄⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠋⠁⢀⣠⣼⣕⡿⠟⠋⠀⠀⢈⡿⠛⠁⠙⢧⠻⡿⣇⣜⡷⢆⡈⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⠋⢹⡿⠋⠀⠀⠀⠀⠒⠉⠀⠀⣠⠖⠛⠻⣆⣙⠿⣣⣮⣟⠗
⠀⠀⠀⠠⢄⣠⣤⠴⠶⠒⠛⠁⠀⢈⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⡿⡿⣟⣭⡛⢵⣶⣦
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢛⠿⣿⣃⡽⢿⢟⣤⣍⠁
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠋⠉⢁⣤⣾⣮⡛⠋
⠀⢦⣤⡈⠳⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣿⠇⣙⠿⠋
⠀⠀⠙⢿⣦⠈⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣀⣴⣄⡻⣿⣷⣦⡀
⠀⠀⠀⠀⠹⣷⡄⠙⠲⠂⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⠿⢟⡁⠈⠻⠿⠋
⠀⠀⠀⠀⠀⠈⠳⣄⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⢿⠶
⠀⠀⠀⠀⠀⠀⠀⢈⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢠⣾⠵⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈
⠀⠀⠀⠀⣠⠞⣩⠏⠀⠀⠀⠀⣴⣿⡅⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⢀⡴⢋⡿⠁⠀⠀⠀⠀⢠⣾⡥⠞⠀⠀⠀⠀⠀⣀⣼⡁
⠀⠀⠀⢰⡧⢾⣧⠄⠀⠀⠀⣾⡭⠞⠁⠀⠀⠀⠀⣾⣤⠴⠀⠀⠀⠀⣾⠶⢻⣧⠄⠀⠀⠀⠠⣯⣤⠄⠀⠀⠀⠀⠀⢀⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡴⣫⠟⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⢀⣴⠷⠋⠀⠀⠀⠀⢠⢞⡿
⠀⠀⠀⠀⢀⡴⣫⡽⠃⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⣶⠞⠀⠀⠀⠀⠀⢀⣤⢴⡞⠁⠀⠀⠀⠀⣶⢻⠇⠀⠀⠀⠀⢀⣠⠟⠋
⠀⠀⠀⠀⠚⣫⣿⡅⠀⠀⢀⣴⠏⣵⠋⠀⠀⠀⠀⢠⣾⣁⡀⠀⠀⠀⠀⠀⠀⣰⣯⠤⠀⠀⠀⢀⡴⢻⣃⣀⠀⠀⠀⢀⣼⣁⣠
⠀⠀⠀⡖⡻⠋⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⢞⡿⠁⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠞⣻⢛⡟⠀⠀⠀⠀⣠⠞⣻⠇⠀⠀⠀⢀⡴⠚⢙⡇⠀⠀⠀⠀⠀⣴⠋⣻⠀⠀⠀⠀⢀⡴⢋⡿⠂⠀⠀⠀⠚⣻⠷⠋
⠀⠀⠀⣴⢃⡼⢱⣏⡄⠀⠀⠀⡼⠃⣴⣏⡄⠀⠀⠀⣾⣁⣤⠞⠀⠀⠀⠀⢀⡾⠷⠖⠃⠀⠀⠀⠀⠿⣶⠟⠁⠀⠀⠀⣰⣎⣥⡤
⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⡥⠖⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠤⠶⣾⠁⠀⠀⠀⠀⢛⡿⠛⠉⠁⠀⠀⠀⠀⢀⡼⠁⣠⠞⠀⠀⠀⠀⠀⢘⡿⠁⡾⠛⠉⠀⠀⠀⠀⢀⡴⠋⣠⡞⢡⡟⠉
⠀⠀⠀⢠⣀⣠⡟⠀⠀⠀⠀⢰⣯⣤⡤⠀⠀⠀⠀⠀⢠⣾⡴⣾⣧⣤⠀⠀⠀⠀⢠⣯⡤⠞⠁⠀⠀⠀⠀⠀⠀⣾⣤⢾⣏⣠⠞
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⡀⠀⠀⠀⠀⠀⢀⡄⠀⡤⠀⠀⠀⠀⢀⣤⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⡾⠉⠉⠀⠀⠀⠀⠀⢠⣎⣤⠞⠀⠀⠀⠀⠀⣀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠻⠗⠃⠀⠀⠀⠀⠀⠀⣉⣽⠅⠀⠀⠀⠀⠀⠞⠻⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⢇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⢀⣠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠘⡽⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢧⢧⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡼⡜⠀⠀⠀⠀⠀⠀⠐⠛⠻⣗⣄⡀⠠⠀⠂⠁
⠀⢀⣪⡂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢤⣤⡀⠀⠀⠀⠀⠑⠀⠀⠈⣎⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠙⣕⣄⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠙⡝⡆⠀⠀⠀⠀⠀⠀⠀⢸⣱⠀⠀⠀⠀⠀⠀⠀⠂⢰⣻
⠀⠀⠀⠘⡍⢆⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⣣⠀⠀⠀⠀⠀⠀⠈⡈⡏⡄
⠀⠀⠀⠀⠘⡭⢆⠀⠀⠀⠀⠀⠀⢧⠀⣐⠥⠊⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⣏⡆⠀⠀⠀⠀⠀⠀⠀⢡⣇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠸⡌⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠄⢀⠀⠀⠸⣳⡀⠀⠀⠀⠀⠀⠀⠘⣾⡀⠀⠀⠀⠀⢀⠀⢰⣶⠒⠂⠀⠉⠉⠉⠉⠁⠉⠉⢣
⠀⠀⠀⠀⠀⠀⠸⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢧⣇⠀⠀⠀⠀⠀⠀⠀⣟⡆⠀⠀⠀⠀⠈⠀⢸⣻
⠀⠀⠀⠀⡀⠀⠀⠱⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡼⡄⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⢸⣼
⠀⠀⠀⠠⠁⠀⠸⠀⢫⢧⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠲⣦⣄⠀⠀⠀⠀⠀⠀⠀⢻⢥⠀⠀⠀⠀⠀⠀⠸⣽⠀⠀⠀⠀⠘⠀⠸⠿⠀⠀⠀⢀⣠
⠀⠀⢠⠁⠠⠁⠁⠀⠈⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⠀⠀⠀⠀⠀⡀
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠸⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠹⠀⠀⠀⠀⠀⠀⢸⡿⠄⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠇
⠀⠀⠚⢄⡈⠁⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡩⠃⠀⠀⠀⠀⠀⠀⠈⣗⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠀⠀⠀⡠⢾⠏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⠀⠀⠀⠀⠀⠓⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠂⡗⡇⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠈⠑⠂⠀⠂⠁⠀⠀⠀⠀⠀⡐
⠀⠀⠄⣄⢠⠠⠀⠀⢀⠀⠨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⢸⡻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⣠⠟⡜⠂⡈⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠈⠀⠀⠀⡼⡇⠀⠀⠀⠀⠀⢀⡀⣧⣓⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠳⠶⠶⠶⠶⠼⠻⠛⠛⠉⠝⠛⠋⠉⠭⠉⠭⠹⠿⠯⠭⠭⠽⠟⠤⠄⠉⠍⠉⠙⠛⣉⣉⢉⣉⣉⠉⠍⠩⠥⣀⡤⢠⡤

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣂⣰⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⢠⠗⠈⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠤⠤⠤⠤⠀⠐⠒⠲⢲⡀⢀⠀⠄⠂⠀⢸⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠸⡀⣷⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⡜
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠐⠐⠈⠀⠀⠀⠙⢦⣀⡠⠤⠒⠁⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⡆⠀⠀⠀⠀⣀⢀⠀⣀⠀⠀⠀⠀⠀⠀⢏⡇⠀⠊⠁⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠖⠚⠉⠙⡄
⠀⠀⠀⢡⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⢸⢹⠀⠸⡏⠙⠒⠦⢌⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡆⢲⡒⠒⠀⠀⠠⣭⣉⠓⠢⢤⡠⠤⠤⠔⠒⠊⢉⠏⠸⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢇⢬⡡⠀⠀⠀⡇⡇⠈⠙⠢⢄⣀⠀⠀⠀⠀⡴⠁⠀⠀⢣
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠐⠂⠀⠀⠀⠳⠃⠀⠀⠀⠀⠈⠉⠓⠦⠊⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠸⠄⣠⣤⠄⡀⠒⠒⠒⢒⣒⡒⠒⠒⠒⠒⡟⠒⠒⠒⠒⠒⠛⢀⠐⢢⣤⡄⠀⠠⡄⠈⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠐⠒⠚⠓⠛⠛⠒⠓⠓⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠛⠒⠚⠛⠓⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⡦⠀⠀⠀⡷⠒⠲⡄⠀⠀⢀⡴⠒⠢⠀⠀⠀⢀⠔⠒⢾⠀⠀⠀⣠⠖⠲⡄⠀⠀⠀⣸⣀⠀⠀⠀⢀⡴⠒⢶
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⠀⠀⠀⡧⠤⠤⠖⠀⠀⠀⢸⠁⠀⠀⠀⣼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⣅⠀⠀⠀⣇⠀⢀⡜⠀⠀⠐⣀⠀⢀⠀⠀⠀⠱⡀⠀⣸⠀⠀⠀⢡⡀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠰⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠊
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣇⣀⢀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⡠⠂⠀⠀⠀⡇⠀⠀⠀⣀⡠⣀⢀⡠⣄⠀⠀⠀⠀⣀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢱⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⡷⠺⢤⡀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢫⠀⠀⢱⠀⠀⠀⠋⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⢱⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠘⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⢠⠊⠉⠳⠀⠀⠀⠀⡗⠉⠑⡄⠀⠀⠀⡔⠋⠑⠷⠀⠀⠀⠰⠋⠉⠙⠀⠀⠠⡎⠉⠑⠀⠀⠀⠁⢽⠉⠀⠀⢸⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡁⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠒⢤⡀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢄⣀⠼⠁⠀⠀⠀⣇⣀⡠⠎⠀⠀⠈⠣⣀⣀⡧⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢄⣀⡰⠃⠀⠀⠀⢿⠀⠀⠀⠘⢄⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⢢⠀⠀⢠⠀⢠⠀⠀⠀⠀⢠⡀⢠⠂⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠐⠒⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡸⠀⠀⠀⠀⠈⡆⠀⡇⢀⡏⠀⠀⠀⠀⠀⣱⡃⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠃⠀⠀⠀⠀⠀⠸⡼⢣⡜⠀⠀⠀⠀⠀⡰⠃⠱⡄⠀⠀⠀⠈⣄⠀⣀⡇⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⡈⠉⠀⡇⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠀⠀⠉⠉⠉⠉⠀⠀⠀⠐⡄⠀⠀⠈⠉⠀⠐⠀⠐⠒⠒⠒⠒⠁⠀⠀⠐⠒⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢲⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠈⠑
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠣⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠸⠀⠐⠒⠒⠒⠒⠄⠀⠀⢀⠤⠤⠤⠤⠄⣀⢐⠀⠀⠠⠤⠤⠤⢄⠀⠀⠀⠠⠄⠄⠤⠤⠄⢘⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠒⠂⢄⠀⠘⣄⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠉⠒
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠒⠄⠔⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠒⠤⠀⠀⢡
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠢⠀⠀⠒⠀⠀⠀⠓⠠⠤⠀⣀⡀⠀⠇⠀⠀⠰⠀⣀⣀⣀⣀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣤⣤⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠋⠻⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠻⠛⠃⠈⠛⠟⠁⣽⣿⣿⣿⡇⠀⣠⣶⣤⠀⠨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠁⠀⢀⣀⣀⡀⠀⠈⠻⠿⢿⣿⡆⠀⠻⣿⡿⠀⠠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣴⣾⣿⣷⣆⣠⡀⣀⣰⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣷⡄⠀⠀⠉⠉⠀⠀⢠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣶⣶⡄⢠⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠃

⠀⠀⠀⠀⠀⠀⣾⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢛⣶⠀⣿⡆⠀⠀⠀⠀⠀⢠⣾⠟⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠲⣤⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⣸⣯⣴⡿⠀⢀⣤⣄⠀⠀⠀⠀⢸⡿⢿⣽⣅⡀⣿⣧⣤⣀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠀⣠⣤⣀⠀⠀⠀⣿⡇⠀⠀⢸⡇⢀⣀⣠⣿
⠀⠀⠀⠀⢠⣿⠉⠛⠛⣶⣾⡇⢸⣿⣧⠀⠀⢸⡇⠀⠀⢈⣷⣿⡇⠀⣹⡇⠀⠀⠹⣇⠀⠀⢀⣶⢾⡏⠀⣉⠀⠀⠀⣻⡇⠀⢀⣼⡷⣿⠉⠉⣿
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙⠋⠛⠟⠉⠛⠀⠀⠘⠃⠐⠚⠛⠃⠘⠓⠚⠛⠁⠀⠀⠀⠉⠛⠺⠛⠉⠀⠙⠺⠛⠁⠀⠀⠙⠃⠛⠛⠋⠀⠙⠛⠚⠋

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠁⣾⡋⠉⠁⠀⢀⣼⠟⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠐⢿⣷⢸⣿
⠀⠀⠀⠀⢀⣭⣽⡷⠀⠀⣠⣤⣤⡀⠀⠀⢀⣭⣽⡷⠀⠀⣨⣽⡷⠀⠀⢸⡏⠀⠀⣀⣀⠀⠀⣤⣤⣤⡀⠀⠀⠀⣿⣶⠟⢳⣾⣿⢸⣟⣠⣤⣤⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠿⢛⠁⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢿⡀⠀⢈⣽⡗⢸⡏⠀⢸⣿⠀⠀⠀⣿⣿⠀⢸⣿⣿⢸⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠈⠙⠛⠁⠀⠀⠘⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⡙⠛⣻⣿⠀⠀⠀⠙⠛⠀⠈⠙⠋⠀⠙⠋⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣷⠀⠀⡀⠀⠀⠀⠀⣿⣿⢀⣴⣿⣶⣾⣿⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣽⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣍⠀⠀⠀⠀⣿⠻⢿⣿⠟⠁⣿⣿⣧⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡄⠀⠀⠀⣤⡀⠀⣠⡿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⣿⡇⢹⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠙⠁⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⣿⠀⠀⠀⠀⠛⠀⠀⠈⠙⠀⠙⠃⠀⠙⠀⠀⠀⠀⠉⠛⠛⠛⠂⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡿
⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣧⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⡀⠀⣽⡇⠀⠀⠀⠀⠀⠀⣰⡿⠉⠙⣦⡀⠀⠀⠀⠀⠀⢸⡟⠉⢹⣧
⠀⠀⠀⠀⢸⡇⠻⣾⡏⣿⡇⢠⣶⢦⣤⡶⣤⠀⠀⢸⡇⠘⣷⠀⣿⡇⠀⣴⢶⡄⠀⠀⣿⠀⠀⠀⢸⡇⢠⡶⣤⡀⠀⢸⡷⠤⠾⠏⢠⣶⠶⢦⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⢸⡇⢸⣿⡇⣿⡇⠀⢸⡇⠀⠘⣧⣿⡇⠀⣿⢸⣿⠀⠀⠹⣦⣀⣠⡿⠃⢻⣇⣼⡇⠀⢸⡇⠀⠀⠀⢸⣇⣀⣸⡗
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠈⠉⠀⠀⠉⠀⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⢸⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠁⠀⢹⣿⣿⡟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣤⡴⠋⠉⠘⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠅⠀⠀⠀⠀⣿⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢃⡀⠀⠀⠸⠁⢀⠔⠀⣰⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠘⢓⠖⠁⢀⣴⣿⣻⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣰⡿⠛⠉⠀⠀⠙⢷⣼⣦
⠀⠀⠀⠀⢠⣾⣿⣿⣷⣦⣲⣾⣿⣯⣷⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣽⡄⠀⠀⠈⣓⠂⠁⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⢠⡀⣿⣿⣿⡆
⠀⠀⠀⠀⡏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣠⠏⢀⠀⠀⠱⠀⢀⡄⣠⣀⡀⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⣿⡇⡟⣿⣿⣇
⠀⠀⠀⠀⢷⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⡻⣮⣿⣶⡣⠄⣤⠒⠺⡙⠈⠠⠴⠤⣶⡄⠹⡇⠀⠀⠀⠀⠀⣰⡟⣹⢳⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡵⠀⡇⠀⠀⠀⠀⠒⠤⣄⣘⡆⣰⠁⠀⠀⠀⠀⢠⠋⣰⢯⣿⣿⣿⣿
⠀⠀⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆⠃⢄⣀⠀⠀⠀⠀⠀⠈⠁⢝⢦⡑⠀⠀⣠⣿⣞⣱⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠁⠀⠀⠈⠙⢦⡀⠀⠀⠀⢰⠌⠧⠁⠀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⢀⡜⣿⣿⣿⣿⠟⠿⡟⠈⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢠⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣸⠇⢿⡀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⡿⠀⠘⣇⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣼⠃⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠋⠉⠉⠉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠚⠛⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠛⠉⠉⠉⠉⠉⢳⡄⠀⠀⠀⠈⣷
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠛⠛⠃⠀⠀⠀⠀⢹
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣴⠟⠋⠉⠉⣿⠀⠀⠀⠀⢸
⠀⢀⡏⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡏⠀⠀⠀⠀⢸
⠀⣾⠁⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢻⡄⠀⠀⠀⠀⠙⠶⠶⠞⠋⠀⠀⠀⠀⠀⢸
⣸⠇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⢀⣠⣶⡀⠀⠀⠀⢸
⣙⣛⣛⣛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣈⣙⣳⣶⣶⣶⣞⣋⣁⣘⣛⣛⣛⣛⣛
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠣⠤⠴⡆⠀⠀⣀⢀⡀⠀⠀⠶⠤⠤⠃⠀⠘⠤⠤⢶⠀⠀⢀⡀⣀⠀⠀⣰⠦⠤⠜⠀⠀⠣⠤⠴⡆⠀⠀⣀⢀⡀⠀⠀⡶⠤⠤⠃
⠀⠀⠀⠀⠇⠀⠘⡇⢀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⣸⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⢀⠇⠀⠀⠇
⢸⡉⠉⢙⠋⠐⠒⠋⠈⠓⠒⠈⢋⠉⠁⡹⠀⢏⠈⠉⠻⠁⠒⠚⠁⠙⠒⠂⠹⠉⠉⢉⠇⠸⡁⠉⢩⠉⠐⠒⠋⠈⠓⠒⠈⢋⠉⠁⡹
⠔⠃⠀⢸⣀⠤⢄⠀⠀⢀⠤⣀⣸⠀⠀⠓⠠⠚⠀⠀⣇⡠⠤⡀⠀⠀⡠⢄⣀⡆⠀⠘⠢⠔⠁⠀⢸⣀⡤⢄⠀⠀⢀⠤⣀⣸⠀⠀⠓
⡀⠀⠀⢠⠤⣄⡰⠀⠀⠸⣀⡤⢤⠀⠀⢀⣀⠀⠀⠀⡤⢤⣀⠅⠀⠀⢄⣠⠤⡀⠀⠀⢀⡀⠀⠀⢠⠤⣄⡸⠀⠀⠠⣀⡤⢤
⢨⠆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢏⠀⡹⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠸⡁⢈⠆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢏
⠘⠂⠒⠒⡓⠈⠉⡗⠰⡋⠉⠐⡗⠒⠀⠚⠀⠓⠐⠒⢺⠀⠉⢹⠆⢾⠉⠁⢺⠒⠒⠐⠃⠘⠂⠒⠒⡗⠈⠉⡷⠰⡏⠉⠐⡗⠒⠀⠚
⠀⢀⣀⡀⡇⠀⠘⠧⠠⠇⠀⠀⣁⣀⣀⠀⠀⠀⣀⣀⢸⠀⠀⠸⠄⠼⠀⠀⢼⢀⣀⡀⠀⠀⢀⣀⣀⡇⠀⠀⠧⠠⠃⠀⠀⡇⣀⣀
⠀⢇⣀⣨⡅⠀⠀⠀⡀⠀⠀⠀⣭⣀⣀⠇⠀⠸⣀⣀⣭⠀⠀⠀⢀⠀⠀⠀⢨⣅⣀⡸⠀⠀⢇⣀⣨⡅⠀⠀⠀⡀⠀⠀⠀⣭⣀⣀⠇
⠀⠀⠁⠀⡇⠀⠠⡏⠉⡆⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⢸⠉⢱⠀⠀⢙⠀⠈⠀⠀⠀⠀⠁⠀⡇⠀⠀⡏⠉⡇⠀⠀⡇⠀⠁
⢰⠒⠒⠲⠆⠦⠤⠟⠘⠦⠤⠆⠗⠒⠒⢲⠀⡖⠒⠒⠾⠰⠠⠼⠃⠻⠤⠴⢸⠒⠒⠒⡆⢰⠒⠒⠒⠧⠦⠤⠟⠘⠧⠤⠖⠗⠒⠒⢲
⡨⠇⠀⢸⠀⣀⡀⠀⠀⠀⣀⠀⢸⠀⠀⢧⣀⠽⠀⠀⡆⢀⣀⠀⠀⠀⢀⡀⠈⡄⠀⠸⣄⡠⠇⠀⢸⠀⣀⡀⠀⠀⠀⣀⠀⢸⠀⠀⢧
⠀⠀⠀⢈⣉⡁⢘⠀⠀⢸⠀⣉⣉⠀⠀⠀⠀⠀⠀⠀⣉⣉⠈⡇⠀⠀⡁⢈⣉⠁⠀⠀⠀⠀⠀⠀⢈⣉⡁⢸⠀⠀⢌⠀⣉⣉
⢑⡆⠀⢸⠀⠉⠁⠀⠀⠀⠉⠁⢸⠀⠀⡖⠉⢲⠀⠀⠇⠈⠉⠀⠀⠀⠈⠉⠀⠃⠀⢰⠊⠑⡆⠀⢸⠀⠉⠁⠀⠀⠀⠉⠁⢹⠀⠀⡖
⠸⠤⠤⠤⠆⠖⠒⡦⢠⠖⠒⠆⡦⠤⠤⠼⠀⠧⠠⠤⢴⠰⠒⢲⡄⣴⠒⠲⢸⠤⠤⠤⠇⠸⠤⠤⠬⡆⠆⠐⣦⢠⠖⠒⠆⡦⠤⠤⠼
⠀⠀⠀⠀⡇⠀⠰⣇⣀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢸⣀⣸⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣇⣀⡇⠀⠀⡇
⠀⡎⠉⢑⠃⠀⠀⠀⠁⠀⠀⠀⣓⠉⠉⡄⠀⢰⠉⠉⣚⠀⠀⠀⠈⠀⠀⠀⢙⡊⠉⢱⠀⠀⡎⠉⢑⡃⠀⠀⠀⠁⠀⠀⠀⣓⠉⠉⡆
⠀⠈⠊⠁⡇⠀⢀⡔⠒⡄⠀⠀⡋⠉⠋⠀⠀⠀⠙⠉⢹⠀⠀⢰⠒⢢⠀⠀⠺⠉⠙⠁⠀⠀⠈⠋⠉⡇⠀⠀⡖⠒⡄⠀⠀⡏⠉⠋
⢠⠤⠤⢤⡅⣀⣀⡧⠰⣅⣀⡀⡧⠤⠤⢤⠀⡤⠤⠤⢬⢀⣀⣸⠆⢾⣀⣀⢸⠤⠤⠤⡄⢠⠤⠤⠤⡇⣀⣀⡷⠰⣇⣀⡀⡧⠤⠤⢤
⢘⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣎⠀⣳⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⡁⢘⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣎
⠉⠀⠀⠘⠒⠊⢑⠀⠀⢰⠉⠒⠚⠀⠀⠈⠉⠀⠀⠀⠓⠒⠉⠆⠀⠀⠊⠑⠒⠃⠀⠀⠈⠉⠀⠀⠘⠒⠊⢱⠀⠀⠐⠉⠒⠚⠀⠀⠈
⠢⡀⠀⢸⠉⠒⠊⠀⠀⠈⠒⠋⢹⠀⠀⡤⠒⢤⠀⠀⡏⠑⠒⠁⠀⠀⠑⠚⠉⡆⠀⢠⠔⠢⡀⠀⢸⠉⠒⠊⠀⠀⠈⠒⠊⢻⠀⠀⡠
⢰⣁⣀⣨⣄⡀⠤⣄⢀⡤⠄⣀⣌⣀⣀⣱⠀⣎⣀⣀⣠⡀⠠⢤⡀⣠⠤⢀⣰⣀⣀⣈⡆⢰⣁⣀⣘⣄⡀⠤⣄⢀⡤⠄⣀⣌⣀⣀⣱
⠀⠁⠁⠀⡆⠀⢠⡇⠈⡆⠀⠀⡇⠀⠀⠁⠀⠈⠀⠀⢸⠀⠀⢸⠁⢹⠀⠀⢸⠀⠈⠈⠀⠀⠁⠁⠀⡇⠀⠀⡏⠈⡆⠀⠀⡇⠀⠈⠁
⠀⡔⠒⠶⠇⠀⠀⠉⠊⠁⠀⠀⠶⠒⠒⠄⠀⠠⠒⠒⠾⠀⠀⠈⠑⠉⠀⠀⠹⠔⠒⠢⠀⠀⠔⠒⠶⠇⠀⠀⠉⠊⠁⠀⠀⠷⠒⠒⠄
⠀⡀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣀⣠⣤⣶⡆
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠻⠁⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠿⠷⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠦⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣆
⠀⠀⠀⣸⡟⠀⣀⡀⠀⠀⢀⣴⠟⠁⣿⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⣼⣏⣴⣾⠏⠀⠀⢀⡾⠃⠀⠀⠸⣇⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣿⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠰⠿⠿⠋⢹⡟⠀⠀⢀⡿⠁⠀⠀⠀⠀⠹⣦⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣼⡇⣀⠀⣾⠃⠀⠀⠀⣤⠾⠷⠙⢷⣄⠀⠘⢷⣄⠐⠛⢿⣷⡀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⣿⡿⢿⣷⡏⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠻⠶⣦⣽⣿⣶⣾⡟⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠻⠇⢸⣿⡇⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⢻⣷⠟⢻⣆⣰⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠈⠻⣦⣀⠀⠀⠀⠀⠀⢤⠤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠙⠛⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⢿⡁⠀⠀⠀⣀⣀⣤⣶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢹⡟⠳⠶⠶⠾⠛⠋⠁⠀⠀⠘⠷⣶⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠘⠷⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠁⠀⠈⠉⠉⣹⡟⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢀⣼⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⣤⣤⣾⡃⠀⠀⢀⣿⠛⠷⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠉⠉⠻⣷⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣻⣶⣶⣶⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⢀⡀⠤⠤⢠⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⡠⠀⢄⠸⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⣰⠃⠀⠀⠀⠀⠰⡅⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠉⠀⠁⠸⠀⠀⠀⢠⠄⠀⠢⡀⠀⠀⢧⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⡶⠒⢲⠶⡶⠒⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⡄⠀⠠⠁⠀⠀⠸⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠟⠛⠛⠛⠒⠛⠚⠟⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠗⠶⠶⠶⠴⠴⣿⡩⢽⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣽⠀⠀⠀⠠⠤⠄⠀⢰⠀⠀⠀⢈⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⠀⠀⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣏⢼⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠒⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣀⠀⢀⡀⣀⣀⠠⠔⠊⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⡏⠈⠈⠉⡉⡉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣩⣩⡩⡍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣓⣓⣂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡼⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣰⡻⢤⣀⣀⣀⢀⣀⣀⣀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢹⡇
⠀⠀⢀⣀⣀⣀⣤⣤⣼⣧⣿⣯⣼⣯⣥⣴⣮⣽⣯⣽⣫⣋⣝⣩⣉⣙⣽⣫⣪⣗⣡⣠⣟⣕⣽⣟⢙⢁⡽⡭⣫⠿⡿⣩⠞⠕⣽⠁
⠀⠀⠸⣆⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠂
⠀⠀⠀⠈⠉⠉⠉⠉⠙⠚⠛⠛⠓⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠚⠚⠓⠚⠛⠛⠒⠛⠛⠓⠛⠛⠓⠚⠒⠒⠒⠲⠶⠶⠴⠶⠴⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣤⣶⣤⣶⣧⣴⣦⣴⣦⣤⣶⣤⣶⣤⣵⣦⣴⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⢻⣿⣾⣿⣾⣿⣶⣿⣷⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣠⡀⢧⡤⠒⠉⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⣀⡠⠔⠒⠒⢦⡀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢙⣦⡀⠀⠸⡤⠤⣤⣀⡀
⠀⡔⠃⠓⣠⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢠⠔⠂⠀⠒⠢⡀⢀⠤⠐⠒⠀⠐⢢⠔⢋⣠⠽⠆⠀⠀⠀⠀⢀⣨⡆
⠀⠈⠓⢋⠠⠒⠢⣠⠖⠲⠀⠀⠀⠀⠀⠙⢆⠀⠀⣠⡀⡘⢁⠀⠀⠀⢀⡴⠇⢠⡏⣄⢤⠤⠤⠶⠾⢭⣁⣀⣀⠀⠀⠐⠂⠤⢄⣀
⠀⡠⠀⠁⢔⠈⠉⠒⠃⠐⣆⠀⠒⠒⠒⠒⠒⠛⠓⡋⠉⠈⠛⠃⠀⠒⠁⡶⠀⠀⠘⣡⠋⠀⢀⠴⢄⠀⠀⠈⠉⠉⠙⢲⠆⠀⠀⢈⣱
⠈⠁⠒⠂⠀⢬⠐⣠⠀⠀⠀⠙⠒⢤⡤⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠎⠉⠈⠒⠚⠁⡠⣈⢢⠀⣴⠀⠀⡰⠃⠀⠀⣠⡞⠁
⠀⠀⠤⣀⣉⡐⠮⢀⠀⠀⠡⠀⠀⠀⠉⠒⢤⣀⠀⠀⣀⣀⠤⠔⠊⠀⠀⠀⣸⠖⢤⣀⣀⠤⠊⠀⠈⢦⣩⠞⢢⠎⠀⠀⢀⣼⠋
⠀⠀⠀⣀⡤⢄⡋⠔⠁⢔⡩⡹⠦⣄⠀⠘⣿⣧⣌⠉⡰⠒⡶⠠⠤⣤⠤⠊⠀⣠⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⣠⡿⠁
⣤⠊⠉⠀⢾⡁⠀⠀⠀⣜⡩⢔⢀⠦⢒⠥⠊⠉⠁⠛⢁⠈⠲⢶⣉⠀⠀⢀⣼⡟⠳⠭⣉⠒⢄⠀⠀⠀⡔⠁⠀⠀⣜⠟
⠀⠈⠉⠐⢒⢤⣀⡀⠂⠉⢠⠸⠄⡎⢠⢲⣊⣍⠦⡀⢀⠠⢚⡿⠟⠻⠶⠛⠛⠤⢀⣀⠀⠈⠈⠓⡀⠜⠀⠀⢀⣾⠎
⣴⠦⣒⠪⢽⣼⣶⣗⣦⣅⠂⠝⡒⠁⠈⡭⡌⡗⠂⢋⠅⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠒⢢⣜⣊⠀⠀⠀⢼⡎
⠀⠉⠂⠍⡒⠭⡛⠿⣿⣽⡯⢶⣦⣈⡐⠧⠍⠴⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠵⣀⠀⡿⠈⠉⠓⠺⠞
⠀⠀⠀⠀⠀⠑⠢⢕⡢⢍⠛⢽⡯⢷⠷⣒⣄⣀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⢑⠢⣀⠀⣀⣤⢔⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢍⡲⢌⠛⠉⢋⠉⣤⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣽⡪⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢠⡆⢰⡄⠀⠀⡶⠂⡴⠂⠀⡴⠂⣰⠂⡖⠀⢠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⠃⠀⢰⡇⠀⡧⠀⠀⡇⠠⡏⠀⡇⠐⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠸⠄⠸⠄⠀⠘⠇⠀⠿⠀⠀⠳⠄⠣⠀⠷⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄
⠀⠀⠀⠀⠀⢀⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⡟⠗
⠀⠀⠀⠀⠀⣠⣾⣄⡀⠀⢀⣤⠴⠀⡄⣤⣆⠤⠐⣦⡄⡠⠰⣤⡀⢀⣤⠆⠠⠀⣠⡴⠀⠀⠀⣀⣴⣦⣀
⠀⠀⠀⠀⠀⠸⠿⠯⠀⠀⢿⡇⠀⠀⡠⣿⡧⣀⠒⠋⢰⣇⣀⣼⡇⢸⣇⠀⢀⠠⣿⡆⠀⠀⠀⠈⠿⠿⠁
⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠈⠁⠀⠉⠁⠁⠀⠈⠉⠈⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⢖⣒⣛⡿⠭⠍⠁⠀⠀⠀⠀⠈⠉⠽⣿⣛⣒⡲⠶⠶⠶⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠛⠋⠉⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣷⢰⡆⠀⠀⡾⠶⢼⡆⣶⢀⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠉⠀⠀⠉⠀⠈⠁⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠐⠒⡶⠒⠐⣶⠒⠀⣶⡀⢰⠀⢲⡄⢠⡆⠀⠀⠀⣶⠒⠂⠀⣶⡆⠀⢠⡖⠲⠂⢰⡖⠒
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢺⠀⠀⡇⢳⣸⡀⠀⢹⡟⠀⠀⠀⠀⢹⠶⠆⢠⣏⣻⡀⢸⠀⠀⠀⢸⡧⠤
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠠⣼⡄⠀⡇⠈⢿⠇⠀⠸⡇⠀⠀⠀⠀⢸⠅⠀⢸⠉⠙⠇⠸⢦⡴⠂⠸⣧⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⡀⠀⠀⣢⠠⢄⡀⠀⠀⠠⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠠⢠⣀⠀⢀⣤⢠⡄⠆⢄⡀⡨⠃⢠⡰⠌⠖⠆⢀⠀⢀⡃⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⡀⠀⠈⠐⠀⡠⢊⣭⡭⣒⠏⣤⣤⢓⣬⡆⣷⡂⣦⠄⡠⣸⠤⠼⡂⠈⠈⢊⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣀⣦⣯⡷⡱⣧⠾⣭⣓⡻⠎⣓⣛⡵⣧⢞⡕⣹⢫⡞⣅⢢⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠙⠛⠛⠙⠓⠚⠛⠲⠻⠻⠙⠛⠚⠛⠳⠿⣯⣶⠿⠿⠟⠓⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⢂⠀⠀⠀⠀⠂⢀⠁⠀⢄⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠄⠀⠐
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⡆⠀⠈⠁⠀⠀⢀⠎⠀⠀⠀⡀⠊⠁
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡅⠦⠠⡀⢠⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠒⠶⠒⢊⡁⠴⡀⠀⠀⠙⠂⠦⠈⠀⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠀⣤⠄⠐⠠⠊⠀⠸⠴⠦⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠂⠐⠂⠁⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠂⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢲⠀⠀⡰⢳⡹⣦⡀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⡴⠊⠁⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠁⠀⠀⠀⠀⡜⠀⡐⠻⣶⣱⡑⡙⣧⠀⠀⠀⠀⠀⠀⠘⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣆⠀⠀⠀⠀⠀⠀⠀⢇⠈⠩⠭⠭⢻⣷⣠⡟⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢈⠎⢇⠀⠀⠀⠀⠀⠀⠈⠢⢤⠀⠭⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢀⣀⡼⠁
⠀⠀⠀⠀⠀⢠⡃⠙⣦⡄⣈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⡰⠅⠑⠩⢉⢉⡿
⠀⠀⠀⠀⠀⣦⠴⠒⠚⠛⠶⠮⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⢓⣧⣷⣷⣾⡇
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠈⠳⣴⢶⣚⡤⠃
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⡝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠉⠉⠙⢻⣦⠀⠀⠀⠀⠀⠀⠈⠫⡀
⠀⠀⣤⢧⢹⢦⡀⠀⠀⠀⣀⡴⣏⡎⠘⡜⢳⡀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣼⡠⣄
⠠⠄⢹⣆⢊⢊⠋⠛⠹⢉⠁⢈⡿⠿⡾⡀⢨⢳⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠑⢄
⢹⡃⠀⠛⣮⣇⢊⠁⢡⣛⣱⣟⣛⣓⣿⡰⢃⠚⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⡀⠀⠘⣿⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠒⠠⠤⠈⠙⠻⠿⠿⠿⠷⠶⡴⠒⢻⡇⠀⠌⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠈⠀⠐⠀⠭⠤⠛⠉⠀⣀⠀⢸⣧⠈⢈⣾⠷⢶⣶⣶⡶⠾⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢋⠙⡷⠶⠤⡴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣐⣿⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢾⣮⣎⡣⠀⡑⣼⠃⡴⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠐⣻⡿⣹⡇⠀⠀⠀⠀⠀⢀⡤⢴⠶⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢭⣔⣟⣻⣿⣾⡯⣼⠃⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣷⠆⣻⡀⠀⠀⠀⢰⠟⡅⢄⠂⠀⠀⠘⢳⡀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠓⠒⠊⣰⠋⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢀⠈⣷⡂⢀⠿⣄⠀⠀⣾⠿⠛⣻⣿⣷⣟⢐⢎⡗⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢂⣘⣷⢃⢈⠊⢳⡾⠉⠉⠛⠉⠍⠭⢿⣎⢄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠍⠠⠬⠳⣦⠽⠟⠀⡠⠤⠤⠀⣰⣶⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠥⣄⣀⠁⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⢒⠚⠛⠙⡉⠈⠌⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⠄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠉⠓⠳⠦⣦⣴⣤⠴⠾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢻⡆⡔⠱⠳⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠏⢙⡿⠀⠀⠀⠀⠀⠀⠀⠉⢳⣅⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣁⠀⠹⣾⡕⡐⠙⢦⣄⠀⠀⣠⠶⠛⡀⡀⣤⣾⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠬⢿⣄⠄⢌⢋⠛⠛⠅⢀⣰⣴⠛⣛⣻⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⢙⣮⣧⣢⣀⣤⣶⠛⣉⢭⠭⠕⣿⠀⡟⣦⣄⣀⣀⣀⣠⡴⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⡀⢀⠛⠋⠉⠉⠋⠉⠀⠈⠙⢋⡻⣾⣜⣆⢌⡍⡌⢁⣣⡾⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠲⠚⠉⠁⠀⠀⠀⠀⠀⣐⣚⣙⣏⣿⡿⠾⠋⠭⠭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⢒⣒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⣁⠠⠄⠀⠀⠂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⠉⠡⠤⠔⢒⣀⠀⠀⠍⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⣒⠀⠁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠥⠀⠂⠀⢡⠤⠀⠐⠀⢈⠉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢉⡁⠤⠄⢀⠐⢃⣈⡡⠄⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠃⣀⣀⡀⠀⠀⠀⠨⠤⠔⠒⢂⣀⠈⠀⠀⠀⠀⠀⢠⠂⠰⠋⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠠⢐⠂⢶⡇⠀⣴⠒⢅⠀⢀⣀⡤⡖⢢⣤⣤⣴⣦⣕⣒⣚⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡇⣸⣿⢤⣴⣥⣾⣿⡟⠻⡁⢴⢊⣧⣴⢿⡯⢹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⣿⢿⣿⣿⣿⣧⡺⢷⣤⣾⣿⣯⡾⢃⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⠘⣿⣿⣟⣉⣽⣿⣷⣯⣭⣭⣥⣶⣿⣿⣿⡿⢿⣟⣿⣙⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡹⣿⣿⣿⣿⣿⡿⢿⡛⣿⢩⠟⣼⣛⡾⢩⡟⠨⠃⡚⢹⣏⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣜⠿⣿⡓⠾⢄⣛⠸⢅⡿⢼⣟⣾⡥⠶⢜⣓⣪⣭⡼⢫⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣮⡻⢷⣤⣖⣁⣚⡛⣭⣤⢶⣶⣿⣿⣿⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣽⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⡴⠚⠋⠀⠀⠀⠀⠛⠠⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠖⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⡀⢀⡀⠀⠀⠀⠀⢀⠀⠐⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⡿⠿⠿⠿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⡙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣶⣤⣤⣶⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠁⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡉⢻⣿⣿⠏⠀⠀⠀⢀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣷⡄⢹⡿⠀⠀⠀⠀⠈⣀⡀⠉⠛⠋⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠈⠁⠀⠀⠀⢠⠟⠻⠟⠳⣦⣦⡄⠀⠀⣰⡿⠉⠉⠉⠛⠂⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⣿⠻⠿⠟⢳⡄⠀⣨⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠂⢀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣇⣀⠀⠀⠀⣠⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠓⠤⠞⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⡴⠶⠶⠶⠶⢦⣄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣤⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣨⠛⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠈⠑⠶⠶⠶⠶⢤⣶⡿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡾⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⣀⡠⠴⠞⠋⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠁⠀⢻⡀⠀⠀⠀⠀⢉⣷⣶⣶⣿⡇⠉⠁⠀⠀⠀⠀⢀⡿⠉⢻⣄⡀
⠀⠀⠀⢀⣤⠖⠀⠁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠁⠉⠻⠶⣤⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢠⡟⠛⢿⣿⣿⣿⠟⢿⡄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠃⠀⣸⣿⣿⣿⠀⠀⠠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠙⢿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠳⣄⠀⠀⡘⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠢⡀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⣎
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⣀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⡀⣀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⢣⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⠔⠢⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⣠⣄⠀⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠹⣄⠀⣀⠆⠀⠀⠈⠢⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣼⣿⣿⣷⣄⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠒⠢⢄⠀⠀⠀⠀⠀⡀⠀⠤⠴⠆⠀⣀⠀⠀⣠⠾⢿⣾⡿⢷⡶
⠀⠀⠀⢀⢠⠋⢀⣃⣆⣤⠀⣀⡄⡠⠀⡃⡠⠀⢠⣏⠄⠒⠀⠰⣢⠔⠉⠁⠀⠘⠿⣷⣶⣾⡿⠋
⠀⠀⠐⢸⢸⡀⠼⠘⠉⢒⡠⣽⠋⠀⣠⠃⠁⠀⠸⠁⠀⢀⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠳⢄⣀⣀⠘⠋⡁⣀⢼⠁⠀⠀⠀⠀⢀⡞⠁⠀⠠⠄⢱⡀⠀⠈⠀⠀⡄⢀⡄
⠀⠀⠀⣦⠀⠀⠀⠀⠀⠉⠁⠀⠀⣉⠁⠀⠀⡶⠀⣸⢴⡠⣄⠠⣄⣾⡇⠀⠀⢣⢠⢃⠌⠄
⠀⠀⠀⢸⣷⣄⠀⠐⠒⢒⠒⡒⠒⣿⠒⣄⠀⡇⠀⣿⠉⠉⠙⠯⠭⠓⡇⠀⠀⠀⠋⠈⢰⠤⠤⠄
⠀⠀⠀⠀⣧⠙⣧⢰⡒⠛⢰⣿⣆⢿⠀⣿⠒⣧⠀⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⣄⡀⡀
⠀⠀⠀⠀⢿⠀⡸⢸⠯⣥⠌⠀⢿⠸⠀⠛⠀⠻⠀⠘⢆⡄⣀⣀⣄⢼⠁⠀⡆⢆⡞⢣⣇⢰⡯⢷⢷
⠀⠀⠀⠈⠛⢁⣀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠀⠸⡼⠂⣀⣛⣹⡕⠊⠈⠋⠘⠈⠈
⠀⠀⠀⣠⠊⠁⣼⠀⠉⢢⡀⠀⠀⣀⠀⢀⠀⠀⠀⡶⠂⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⢠⡏⠀⢰⣷⡀⠀⠀⣧⠀⣴⣷⠆⡷⢊⠾⡇⠇⣸⢗⠠⠀⠁⠅⡀⠠⠁⠀⠙⠘⠤⠲⡈⠊
⠀⠀⠘⣧⡴⢛⡏⠛⢦⣀⡟⠀⡴⠋⢰⠋⠆⠀⠁⢀⡁⠀⠀⠀⠀⠐⣀⡷⢤⣀⣤⣰⢆⣸⡠⠢⣡⡀
⠀⠀⠀⠈⠓⠾⠷⠶⠞⠋⠀⠚⠁⠀⠀⠀⠀⠀⠀⣘⣳⣤⣤⠄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠄⠀⠒⢆⠘⡄⠀⠐⢯⣉⠁⣨⣟⠁⠐⠂⡄⠀⡀⢀⡀⢀⠀⣀⡀⡠⡆⠉
⠀⠀⠀⠀⠀⣎⠄⠀⢰⠀⠀⡰⢸⠀⠧⢀⠀⠀⢘⡿⣅⠸⡄⠀⠀⡇⣸⣟⢻⢬⣿⡛⡿⢹⣧⣷
⠀⠀⠀⠀⠀⠀⠰⠁⠁⠒⠀⠐⠛⠀⠐⠚⠁⠀⠚⠀⠈⠑⢷⠀⠀⠛⠉⠀⠀⡭⢴⣒⠒⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡉⠑⠦⡤⠋⠈⢱⣆⡤⠒⠚⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢵⠀⢠⣀⢀⣶⣌⣁⡴⢀⣾⠟⠉⢀⣀⢄⣀
⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⣦⣸⠀⠀⣸⡆⠘⠿⣛⣟⡈⠹⣡⣿⠋⣠⠞⠁⠀⣀⡄⠙⣦
⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⣿⠇⠀⢠⣿⡿⠶⣿⠁⡀⠉⣶⠽⢯⡰⠁⠀⢀⣾⣿⠇⠀⣼⣇
⠀⠀⠀⡜⠀⠀⣴⠀⠀⡼⠋⢀⣴⠿⠇⠀⢰⣿⣾⠃⠀⣠⠀⣸⠁⠀⢠⣿⣏⠘⢦⣤⣾⠟⠁⡰⠋⣷⡄
⠀⠀⢸⠁⠀⣼⡏⠀⠀⠁⢀⣠⠄⠀⢀⡀⠘⢛⡅⠀⣰⣿⡀⠋⠀⠀⡀⠀⣬⡍⠙⣧⣁⡤⠊⠀⣼⡿
⠀⠀⠈⡇⣸⡿⡇⠀⠀⣾⣿⠏⠀⠀⢞⡻⠿⠛⠛⠒⠛⠓⠛⠂⠀⠀⣿⣦⣌⠀⢀⣉⣀⣀⣴⣾⠟
⠀⠀⠀⠘⠿⠃⢣⠀⠀⠸⣿⣀⣀⣴⠏⠀⠀⣀⣠⣤⣴⣶⣶⣶⡀⠀⣿⠿⠁⠀⣼⡟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢹⡍⠛⢧⣶⣾⣟⣋⣵⣾⠿⠛⠉⠀⠑⢤⣀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⢀⣤⡶⣒⣒⣒⣒⣒⣲⣶⣒⣒⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⠒⠒⠒⢶⣦
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹
⠀⠀⠀⠀⠀⡏⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⢲⡆⣿⢸⡆
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠃
⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣼
⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣿
⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿
⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⢹⠀⠻⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⣧⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠤⠤⠤⠄⡆⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⣛⣛⣛⣉⣉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⣀
⠀⠄⠀⠀⠀⠀⠀⠀⣰⢏⡭⣭⢭⡭⢭⡭⢭⠭⢭⠭⢭⡭⢭⡭⢭⡉⢩⠭⣽⠿⡿⢿⠻⣆⠀⠀⠀⣾⣡⠜⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢭⠯⢽⠭⢽⠭⢽⠭⢽⡭⢽⡯⠭⡯⠭⡇⠘⣟⣻⣛⣻⣚⣧⠹⣆⠀⠀⠹⣅⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣰⣏⣈⣓⣛⣒⣛⣒⣚⣒⣚⣒⣒⣓⣚⣓⣒⣓⣒⣛⣀⣱⣤⣷⣤⣷⣤⣇⣹⣆⠀⠀⠘⢧⡀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠻⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠛⠶⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠘⣟⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣤⡿⣄⣴⢿⡇⢀⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣀⠀⢀⣤⠀⢸⠀⠈⠀⠉⠁⠸⠷⠋⡟⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠛⢀⣸⣏⣀⠀⠀⣀⠀⠀⢀⡀⠀⠀⡹⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠉⢻⡏⠉⠀⢠⣿⠀⠀⣼⡇⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠘⢻⡟⠛⠀⣿⠀⠀⠀⣰⡾⠻⣦⢠⡄
⠀⠀⠀⠀⠀⣠⣶⡿⠿⣿⣦⡀⠰⢶⡿⠶⠶⠶⠄⠿⠀⠈⠻⠶⠂⢸⡇⠀⠀⣿⡀⠀⠀⢿⡛⠛⠋⢸⡇
⠀⠀⠀⠀⣼⡿⠋⡀⠀⠀⠹⣿⡆⠀⣠⣴⣶⣶⣦⡀⠀⢠⣦⠀⠀⠈⢷⣤⡦⠸⣧⣀⣠⡘⢷⣤⣴⠟⠁
⠀⠀⠀⠸⡿⠃⢸⣿⠀⠀⠀⢹⣿⣾⡟⠁⠀⠀⢹⣷⠀⠘⠟⠀⢰⣦⠀⠀⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠁⢰⣷⠀⠀⣸⡿⠀⢰⣶⠀⢼⣿⣴⡿⢿⣦⠀⠀⢠⣾⠿⣿⡆⠀⢀⣴⡿⠿⣷⡄⢀⣤
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣼⡿⠀⢸⣿⣠⣾⠟⠁⠀⢸⣿⠀⣾⣿⡟⠀⠘⣿⡆⢀⣿⠏⠀⢿⡗⠀⣿⣿⣤⣤⣿⠟⠈⣿⡇
⠀⠀⠀⠀⠀⠿⣿⣿⣤⣴⣾⠿⠁⠀⢸⣿⣿⡁⠀⠀⠀⢸⣿⠀⣿⡿⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⢿⣯⠉⠉⠁⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠁⠀⠀⠀⢸⣿⠙⢿⣦⡀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿⡇⠸⣿⣄⣠⣴⣾⠆⠈⢿⣧⣄⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠙⠿⣷⣜⣋⠀⠙⢋⡀⠀⠀⠙⠁⠀⠉⠛⠛⠉⠀⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣀⣠⣤⣴⣶⡶⠶⠶⠾⠷⠮⠙⠻⠿⠿⠟⠃⠾⠿⠿⠿⠷⠶⠶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛




⡶⣶⢶⠶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡠
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⡠⠀⠀⠀⠀⢠⢈⠀⡀
⡎⠀⠈⠠⠀⠀⠀⡇⠀⠒⢢⠀⠀⠈⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠁⠐⠀⠐⠀⠀⢹⠀⠁
⠀⠀⠀⠠⠄⠀⠀⠧⠤⠂⠤⠄⠀⠐⠂⠀⠤⠀⠀⠁⠄⠠⢤⠄⠀⠀⠀⠠⠄⠀⠀⢸⠤

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⣄⡈⠁⠀⠀⠀⢀⠀⠀⠀⢈⠀⠀⠀⢠
⢈⠀⠈⠠⠀⠀⠀⡇⠀⠈⠰⠀⠀⠈⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠉⢱⠀⠈⡆
⠀⣀⣀⣠⠀⠀⠀⣇⡀⠀⢸⡀⠀⢰⣀⢀⡀⠀⠀⡃⠀⠀⢀⡄⠀⢠⣀⠀⠈⢀⣀⠀⢘⡀⠀⢸⡀⠀⣇
⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⡀
⠏⠀⠒⠀⠀⠀⠀⠀⠒⡄⠀⠀⡅⠀⠀⢠⠀⠀⡔⠀⠒⡄⠀⠀⢀⡀⢀⣄⠂⢘⣀⠘⠂⠂⠀⡇
⢁⠀⠀⠀⠀⠀⢠⡀⠀⢁⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⢀⡀⠀⠁⢈⣁⡀⠀⠀⢀⣉⠻⢃⡌⠀⣿⡘⢢
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠄⠈⠁⠁⠀⠀⠈⠉⠉⠉⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡆⠀⠀⡁⠀⠀⢇⠀⡔⠀⠀⠀⠀⠀⠰⠀⠀⠜⠀⠀⠀⠀⠀⠁⠀⠀⠐⢀⠀⠰⠁⠀⠀⠆⠀⡠
⠀⠀⠀⠃⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⠜⢀⢄⠀⠀⠀⠀⡢⠀⠀⠀⠀⢀⠊
⠀⠒⠐⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠊⠀⠀⠓⠒⠀⠀⠠⠁⠀⠀⠀⠀⠉⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠒⠛⠋⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⠀⢹⡆
⠀⠀⠀⣼⣧⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇
⠀⠀⠀⢻⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⡏⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠘⡇⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⡇⢿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢿⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡧⠀⡇
⠀⠀⠀⠀⢸⡘⡇⠈⣧⣤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣀⣀⣇
⠀⠀⠀⠀⠈⣇⣧⣤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣁⣂⣀⣀⡀⠘⣆
⠀⠀⠀⠀⠀⢿⠹⡄⠀⢀⡤⠤⡤⠄⢤⠒⡖⠒⡖⠘⣏⢹⣉⣙⣍⣉⣅⠨⣦⠽⡤⢧⠼⡄⠘⣆
⠀⠀⠀⠀⠀⠘⣆⢳⡀⠈⢧⠄⠺⡖⠚⡖⠙⡉⢹⣉⣹⣄⣵⡤⠴⡤⠼⡆⠲⡒⢳⠚⡗⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠈⢷⠒⠳⠀⠙⡉⠹⡀⢣⠀⢱⣀⣣⡀⢵⡤⠼⡤⢷⠐⢳⠚⡂⠹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠘⣷⠒⢳⠒⠹⡉⢻⣉⣇⣀⣣⣀⣧⣠⣥⠤⠵⠴⠷⠚⠓⠛⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠘⠒⠚⠛⠋⠉⠉⠉⠉⣭⡤⠤⠤⠴⠶⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣀⡀⢀⣀⣀⡀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⢚⣒⣂⣈⣉⣤⡤⢤⣴⣶⣶⣾⠯⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡷⠤⣤⣤⣶⣖⣒⣚⣻⣿⣭⠭⠭⠽⠿⠓⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢡⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠈⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠘⠛⢿⠟⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡅⠀⠀⠀⠀⢰⢾⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠱⡴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⠖⠙⠛⠧⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠖⠋⢻⡤⠀⠀⡠⠀⠀⠀⠈⠢⡀
⠀⠀⠈⠉⠉⠁⣀⣠⡴⠞⠉⠀⠀⢀⣠⣴⣾⡿⠧⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠒⠈⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⣀⠤⠒⢩⣏⣁⠀⠀⢀⣤⣶⡿⠛⣭⡷⡙⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢿⣷⣦⣿⣿⣿⣷⣾⡿⠋⣑⡬⠛⠉⠟⢽⣮⣍⠃⠀⠀⠀⠀⠀⠐⠒⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠘⠿⢿⣿⣿⡟⠛⣻⡷⣿⣥⣍⠃⠀⠀⠘⡿⠟⠻⠿⢿⣿⣴⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠦⠀⠉⠛⣿⣿⣿⠀⠀⠀⠃⠀⠀⣠⡞⠋⠙⠀⠒⠢⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠑⠤⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣰⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠘⠀⠀⠈⠀⠀⡠⠋⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⡠⠊⠁⠀⠀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠤⠤⠶⠶⠒⠒⢒⣒⣋⣉⣩⠭⠤⠤⠤⠤⠄⠐⠒⠒⠒⠒⠀⠤⡄⠀⡸
⠀⠀⠀⠀⠹⡀⠀⡠⠤⠖⠒⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⠃
⠀⠀⠀⠀⠀⠱⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡏
⠀⠀⠀⠀⠀⠀⢳⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸
⠀⠀⠀⠀⠀⠀⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠱⢤⣀⣀⣀⣤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠒⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⢚⣉⣉⣉⣭⣩⠭⣍⣭⢭⣉⣭⣭⢭⣭⡍⣽⡍⢹⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠙⠒⡊⢉⣉⣈⣭⣭⣨⠭⣤⢤⣤⣠⣴⡆⣖⡶⢸⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⢸⣭⣿⢟⡜⡱⢋⣽⣭⠿⡽⣿⡌⢯⢻⠢⡴⢲⠀⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠃⣹⣴⠃⠘⠰⢡⢏⢞⣩⠯⡵⢌⢻⣦⠃⠃⠙⡄⡆⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢳⠇⠀⠀⠀⢠⣣⢋⠞⢉⢅⣾⣧⠉⠀⠀⠀⢸⠃⢇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⠀⢸⠀⠀⠀⠀⠈⢁⣞⣤⡬⢵⠦⣄⠀⠀⠀⠀⠀⡇⢸⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣀⡎⠀⠀⠀⠀⢀⠮⠬⠭⠭⠭⠤⠬⢆⠀⠀⠀⠀⢧⠼⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⠀⣎⣀⣀⣀⣀⣀⣀⣀⣸⡀⠀⠀⠀⠘⣤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡽⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿⣿⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠟⠀⠻⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣄⠀⣀⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡠⠒⠒⢦⡼⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠻⠋⠉⠉⢿⣿⣿⣿⣿⠟⠉⠉⠻⠇
⠀⠀⠀⣇⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣼⣿⣿⣿⣿⣄⠀⠀⣠⡄
⠀⠀⠀⠈⠉⠉⠁⢸⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣾⣿⣦⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠻⣿⣿⠟⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⣇⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠙⣿⣿⣿⣿⡟⠁⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣀⣴⣿⣿⣿⣿⣷⣀⣀⣠⣦⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡾⠳⢤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⠋⠀⠉⣦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣦⠀⣴⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⣻⣿⡛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢷⣆⠀⠀⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣾⡏⠀⣰⠏⠀⠀⠀⠀⠐⠛⠉⢀⣴⣶⣦⠀⠀⠀⣀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠋⠁⡀⠀⠀⣀⠀⢀⣶⠀⢸⣯⡆⢹⡇⠀⣼⡟⠀⠀⡰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣧⡄⢰⣿⢀⢾⣿⠇⢸⣿⠀⣸⣿⣥⣼⠷⢺⣿⠁⣠⠞⣰⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠁⠀⠀⠀⠀⢀⣼⣿⠎⣼⣟⣴⢻⣿⠞⣿⣇⣴⠇⠀⠸⠷⠞⣵⢿⣿⠋⠉⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⣠⠎⢿⠟⠀⠘⠛⡱⣿⡏⠀⠘⠛⠁⠀⠀⠀⠀⠸⠷⠟⢁⣴⠀⠸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢠⠀⠀⠷⠟⠀⣰⢀⠀⣴⣴⠆⡖⣰⡲⠀⠘⣯⣤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣴⠾⠋⠀⣠⣤⣤⡄⡟⣤⢰⡶⠀⠀⡏⣰⢰⣧⡟⣸⡡⣟⡵⠀⠀⠓⠀⣰⣿⠟⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠴⠀⠸⣯⡤⢠⣧⣧⣟⣿⡡⠀⠘⠇⠛⠘⠃⠉⠈⠀⠀⠀⣀⠀⠀⢠⣿⡯⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀⢀⠀⠀⠀⢈⡇⠁⠈⠀⠀⢀⡾⠗⠀⠀⠀⢀⣀⠀⣰⣿⠟⣷⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠋⠉⠉⠛⠻⣿⡿⠛⠀⣼⡟⠀⣠⡄⠀⣤⠀⢀⣶⡆⡰⣿⡿⣰⣿⠃⣼⣿⢀⡀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⢀⡀⠀⠀⢠⣿⡇⠀⢰⣿⠇⣰⣿⡇⣸⡟⠀⣼⣿⡱⣱⣿⢣⣿⣣⠞⣿⣿⢿⡀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⣸⣿⠀⠀⣼⣿⠁⢠⣿⡟⣰⣿⡟⢠⣿⢃⡾⣿⣿⠁⣿⣷⠟⠙⣁⣼⣿⠃⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣶⡿⠃⠀⠀⣿⡟⢀⢾⣿⢡⠃⣿⣡⡞⠿⠏⠀⠛⠁⠀⠈⠁⢠⡞⢋⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠰⣿⣣⠞⠘⠿⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣶⡶⠒⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣷⣶⣶⣶⣶⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠠⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠂
⠀⠀⠘⠀⠀⠀⠆⢘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠆⢀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠀⠀⠀⠀⠐⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⢀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢤⠀⡀⠀⠀⠤⠄⠄⠀⠠⠤⠀⠀⠀⠀⠤⠀⠀⠰⠤⠤⠄⠀⡀⠠⠄⠄⠀⠀⠤⠄⡄⠀⠀⠠⠄⠀⠀⠀⠤⠀⠀⢠⠄⠤⠤⠀⠰⠤⠄⠠
⠀⠀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠀⠀⠀⠃⠈⠀⠀⠀⠀⠁⠀⠃⠈⠀⠁⠀⠀⠀⠈⢉⠉⠀⠀⠈⠁⠀⠀⠀⠈⠀⠐⠀⠃⠈⠀⠀⠀⠀⠈⠀⠃⠈⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⠀⠀⠀⠠⠈⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠂⠰⠀⠀⠠⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠆⠀⠀⠐⠀⠆⠠⠀⡀⠐⠀⠘⠀⠂⠐⠀⡀⠀⠀⠰⠀⠀⠀⠁⠒⠀⠀⠀⠀⠂⠀⠃⠐⠀⠇
⠀⠀⠈⠀⠀⠀⠄⠂⠀⠀⠀⠀⠁
⠀⡀⢈⠀⡀⠀⡈⣁⠀⠈⠠⠀⣀⠀⠀⠀⢉⡀⢀⠀⢀⣀⣁⣀⠀⠀⢁⡀⠀⠀⢀⣈⢀⠀⠀⢈⣁⠀⠀⠡⠀⣀⠀⠀⠀⣉⣀⡀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠡⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠘⠀⠆⠰⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠐⠀⠆⠐⠂⠈⠀⠀⠘⠀⠂⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠂⠀⠁⠈⠀⠃⠐
⠀⠀⢁⠀⠀⠀⠄⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣁⠀⠀⠀⠀⠉⠀⠀⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⢀
⠀⠄⠠⠀⠀⠀⠀⠠⠀⠀⠰⠀⠤⠀⡀⠀⠀⡄⠠⠀⠀⠀⠄⠀⠀⠀⠠⠄⠀⠀⠄⠠⠀⢀⠀⡀⢠⠀⠄⠰⠀⠠⠀⠀⠀⠀⠄⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐⢀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠐⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⡐⠂⠀⡐
⠀⠀⣈⠀⠀⡰⠁⠉⠀⠁⠀⢁⡉⠈⢀⠀⠀⡉⠀⠀⠈⣉⠉⠀⠀⠊⠈⠁⠉⠀⠀⡈⠈⠁⡀⠁⢈⠀⠁⠀⢁⡉⠀⢀⠀⠀⠉⠉⠀⠈⢀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠄⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠆⠠⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⡀⠠⠀⠂⠘⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⢀⠀⠀⠀⠐⢂⠀⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⠀⠀⠀⠒⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⡀⢀⠀⠀⠀⠀⢀⠁⡀⠀⠀⡀⠀⠀⠀⠀⡈⠀⡀⢀⠀⡀⠀⠀⠀⠀⡉⢠⠀⡀⢀⠀⠀⠀⠀⢀⠁⡀⠀⡄⢀⠀⠀⠀⠀⡀⠀⡄⢀⠀⡄
⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠁⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠄⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠁⠀⠀⠀⠀⠀⠄
⠀⠂⠐⠀⠀⠀⠀⠒⠐⠀⠈⠀⠒⠀⠀⠠⠀⠒⠐⠀⠐⠐⠒⠒⠀⠀⠐⠂⠂⠀⠒⠐⠂⠀⠀⠀⠒⠐⠀⠈⠀⠐⠀⠄⠀⠀⠒⠒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢐⠁⠀⠀⠀⢀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⠀⠀⠀⠀⡈⠀⠀⠀⢀⡀⠀⠀⠀⢀⠉⡀⠀⠀⢀⠀⠀⠀⠂⢈⠁⠀⠀⠀⡀⠀⠄⠠⠀⡍⢀
⠀⠀⠄⠀⠁⠈⠀⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠈⠀⠈⠀⠄⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠘⠀⠁⠈⠀⠈⠀⠄⠀⠀⠁⠀⠀⠀⠀⠃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠱⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⡀⠀⢈⠀⠀⠀⠀⠀⠐
⠀⠀⠠⠀⠀⢀⠀⠤⠀⠄⠀⠀⠤⠀⠀⠀⠠⠤⠀⠀⠠⠀⠄⠀⠀⡀⠤⠄⠐⠀⠀⠠⠀⡀⠀⠀⢠⠀⠀⠀⠄⠤⠀⠀⠀⠀⠤⠀⠂⠠⠀⠄
⠀⠀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠐⠂