Hospitable Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⣀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⢋⣘⡙⠖⠒⠶⠶⢜⣛⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⡙⣶⠋⢰⠋⣤⡌⡆⠀⢀⣠⣤⣉⠀⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⡃⠀⠈⠲⢤⠔⠁⠠⡏⠠⠶⣸⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⣧⣄⣀⣀⣭⡟⠓⠀⠙⠓⠚⠁⠀⠀⢸⠛⠓⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠃⠀⡇⠀⠀⠀⠀⠳⣄⣀⠀⣀⣴⡖⠤⢄⡬⠤⣄⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⣤⣤⣤⡴⠉⠀⣀⣠⣤⣽⣦⣤⣤⣄⣠⣤⣭⣭⣭⣵⣬⣵⠏⠀⠀⣨⠧⠤⠤⢤⡀⠀⠈⢧⡀
⠀⠀⠀⢸⠀⠀⡠⠇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠠⡇⠀⠀⠀⢸⡇⠀⠀⠀⢳⡀
⠀⠀⠀⢸⠀⠘⠶⣎⣀⠦⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣇⣰⠶⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡇
⠀⠀⠀⣿⠀⢸⠀⢠⡄⢀⡄⣾⣉⡄⠀⡿⠀⠀⢠⠖⠂⠀⡤⠲⡄⣸⢦⡼⡇⢰⣋⠁⠀⠀⡇⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣿⠀⠈⣧⡏⢧⡼⠁⠿⣤⠄⠀⠷⠔⠀⠸⠤⠤⠀⠙⠚⠁⠛⠀⠀⠃⠸⠧⠤⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠶⣤⣀⣇⠀⣀⠀⠀⠀⡟
⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⡏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠤⠶⠚⠉⠁

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⢿⠿⣿⠿⣿⣿⡿⠿⠿⠛⠛⠉⠉⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠐⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⣴⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠀⠀⠀⠙⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠤⠀⠀⣤⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⣿⠀⠈⡇⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⣿⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠈⠀⣿⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠋⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⢶⣶⣤⣤⣀⣀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉


⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠿⣾⣿⠿⠿⢿⢿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⢦⡀⠀⢸⡇⠀⢀⡴⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠞⠃⢠⡾⢷⡄⠘⠓⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣯⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣀⡈⣛⠉⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⣍⣙⠻⢶⣟⣉⣹⣆⠀⣤⣤⡄⢀⣤⠾⣟⢻⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⣼⡋⠛⠶⠟⠉⠉⠉⠀⣧⣠⡇⢸⡷⣤⠘⠋⢠⡶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢶⣄⠈⠻⣦⡀⣴⠶⠶⠶⠶⠾⠷⠶⠾⠷⠿⠶⠶⠾⠷⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣸⠋⠳⣆⠈⣷⠻⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡟⠀⠀⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣀⣸⠃⠀⠀⢿⡀⣹⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠙⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡤⢔⣤⣤⣤⣤⣽⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⣣⡾⠟⠉⠉⠉⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⠀⠀⠀⠀⠀⢀⠀⠀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡋⢿⠀⠀⠒⠀⡀⠀⠛⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢇⠀⠀⠀⠀⠤⠄⠄⠀⠀⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠓⠒⠚⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀⡠⣤⠤⢠⠎⠰⠒⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⢿⣎⠁⠙⠻⣿⣟⠉⠀⠀⣧⢠⡐⣤⡶⠶⠚⠓⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠁⠀⠀⠀⣿⣆⣀⣴⠇⠈⢷⠦⡾⣿⡎⣿⣿⣄⠙⣛⠛⠛⢒⣂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢤⡘⣿⣦⠀⡠⢸⡏⡻⢿⠖⠀⠐⠄⠪⠳⢿⣾⣿⡗⠀⠈⠩⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢙⡎⢿⣇⠈⠂⣷⠀⠀⠀⠀⠂⠀⠀⠀⢀⣿⡿⠃⠀⠀⠀⢺⣿⣿⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠈⣿⣇⠀⢸⡖⡖⠒⢚⠋⡋⢉⢩⣻⠋⠀⠀⠀⠀⠶⣿⣿⣿⡧⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢄⠀⠀⠀⣿⣿⣧⣸⠀⠀⣄⢆⣦⣆⣴⣸⡇⠀⠀⢀⠤⣒⣿⡿⠁⠀⠀⠐⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⡲⢄⡀⠀⢻⡯⢳⣿⣴⣿⣿⣾⣿⣿⣿⣿⡧⠤⡤⢄⣩⠽⠿⣷⣄⡠⣤⠅⢀⣼⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⡌⠀⢀⡤⣾⣷⡾⠿⠿⠟⠛⠛⠙⠋⡉⠀⠀⠀⠠⠄⣹⣿⣴⣿⣿⣦⡖⣺⣿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢽⣂⡀⡰⠟⠉⠀⠀⠀⠀⠀⠀⢷⠰⡄⣱⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⠏⣰⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⣿⠀⠁⠀⡀⠀⠠⢔⠠⣀⠄⠀⢀⣼⣾⣿⣦⣄⠀⢀⣀⢀⣀⡶⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡘⠀⠀⠀⠂⢐⠨⢔⣪⣴⡦⣞⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣠⣬⣭⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⢿⡿⠿⠿⠟⠛⠛⠛⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⢀⣴⡶⠟⠛⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠐⠟⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⣼⣏⣀⣀⣠⣴⡿⠂⠀⣤⡀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠁⠀⣠⣾⠟⢿⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠿⠿⠿⠛⠋⠁⠀⠀⠙⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣶⣄⠈⠉⢀⣴⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡀⠀⠙⣷⣶⡟⠁⠀⣹⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠈⢿⣴⡾⠋⠉⠻⣦⣾⠏⠈⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⣠⣾⠟⠁⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⣀⠀⠀⠀⠀⢀⣠⣦⣀⣠⣤⣄⣤⣄⡴⠶⠆⠴⠰⠂⠀⣀⠂⠀⠄⠀⡀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠈⣃⠀⣠⣾⣿⣿⣿⡟⠿⠆⠙⢹⠿⠂⠀⢁⠠⣀⢠⣰⣿⣷⣤⣶⠄⠀⠀⠈⠂⠀⠀⠉⠀⠀⠀⠀⠀⢠⡆⠀⠀⠈
⠀⢸⡲⠄⠐⠿⠿⢿⣿⣿⣟⠓⠆⠀⣣⣤⣤⣶⡴⠞⠛⢻⣿⠟⢛⣿⣿⠳⠤⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣌⣷⣄⡀
⠀⢸⠂⣴⣀⠀⠶⣿⣿⣿⣿⣶⣶⣾⠛⠋⠉⢿⣥⡄⠀⠨⣿⣯⣾⣿⡭⠤⢶⣦⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡿⠏⠁⣹⣷⡶⡅⠄⠀⠀⠄
⠀⠸⡀⣿⣿⣿⣶⣿⣿⣿⣿⣻⡟⢁⢁⡴⡆⠀⣿⣥⠄⣠⣿⣿⡿⠋⢨⣵⣾⣿⣷⣼⣇⢀⣠⡀⠀⡀⠀⠀⠂⠠⢾⣭⢭⡙⢷⠀⣆⠀⠀⠀⠁⠠
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⡟⠃⠙⢦⣿⣿⣿⣿⣿⠟⠄⠘⠉⠉⠃⢀⣼⡟⣿⣟⠋⢁⠄⠀⣉⠀⠀⡄⢉⡳⠊⠌⢸⣦⠏⢛⡗⠂
⠀⢰⠀⣿⣿⣿⣿⡟⠦⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⡟⠛⡏⠀⠀⠀⠀⠀⠀⠿⠛⠷⠬⠉⠋⢠⡦⠀⠀⣟⢁⡈⢾⡏⠀⠀⠀⢹⠀⠁⢰⡥
⠀⢸⠀⡿⢩⡉⣿⡷⣾⣿⣿⣿⣿⡟⠠⡀⠀⣿⣿⣿⣷⢼⢃⠌⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⣍⠈⠳⠾⢢⡤⠠⠔⢨⣄⠀⠃
⠀⢸⠀⣇⣸⡴⠟⣠⣿⣯⡹⢸⣿⡇⢰⡵⣷⣿⣿⣟⠉⠘⣿⡴⢁⡄⠀⢀⣀⡄⠀⡾⢶⡖⠀⡜⢰⠄⠀⡏⣠⡌⢀⠀⠁⡀⢀⠀⠈⣴⠃
⠀⢸⠀⣿⡄⣴⣾⣿⠛⣿⣿⣛⠛⠛⢷⣹⡟⣿⣿⢣⢄⢨⣿⣦⣿⣧⠀⡸⢀⣿⣼⡇⣾⣏⠀⢐⠁⢣⣸⣿⠙⠀⢸⠀⠀⢿⡬⠧⠤⣿
⠀⢸⠀⣿⣿⡈⢿⣿⣷⣜⣻⠿⠀⠀⠉⣿⣩⠼⠛⠟⠀⠛⠻⠛⠛⠃⢠⠁⠺⢿⣿⠟⠛⠻⠁⠛⢠⠚⠉⢙⣷⣶⡌⠃⠀⢁⠀⢠⠄⡄
⠀⢈⠀⣿⡻⠁⣠⣿⣿⣿⣿⡇⠀⢄⣣⣽⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⢀⡆⢀⡿⠀⢀⠶⠦⢐⡁⠀⠔⠋⢸⣿⣿⠷⠶⢋⡀⣀⢠⡇⠀⡀
⠀⢸⠀⡏⠉⢻⡿⣿⣿⣿⣿⡷⠀⢺⣿⠏⡀⡄⠀⠀⡀⠀⢠⡴⠛⢠⣂⣈⡇⢈⡃⣀⣀⣆⡀⠀⢠⣀⣀⣀⣸⣿⠀⠀⠀⠘⣿⡿⣮⣇⠀⠀⠂
⠀⢰⠄⡇⠀⣾⠁⢹⡿⣿⣿⡗⠐⣊⠀⠀⠐⡇⢠⠀⠐⠒⣾⡦⠄⠊⠉⠙⠛⠚⠛⠛⠁⠀⠀⠀⠈⠈⠁⢩⣿⡇⠀⠁⠀⠀⡟⠓⠘⠋⢠⡀
⠀⢸⠀⣿⠂⢿⣦⣾⣿⣿⣿⣿⣁⣿⣧⡀⢿⡰⣠⠘⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⣇⢀⠀⠀⠀⣧⠐⢧⡄⠀⠉
⠀⢸⠀⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡧⢠⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⠀⣟⢸⠀⠀⠀⣻⣶⣦⣽⣿⢗⠉
⠀⠍⠀⣻⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠀⠄⡼⢛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠟⠙⡂⠀⢠⣿⣿⣿⣿⣿⣿⣷
⠀⣾⠀⣿⣿⣏⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⠞⢗⣫⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣀⡀⠚⠃⢀⠉⠉⠁⠏
⠠⠏⣠⠟⠉⠉⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣰⣿⣿⣯⣍⢃⠃⠀⠄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠠⡇⠀⠀⣿⠙⠠⠀⠏⢈⡀⠀⣁
⠀⣤⠕⠢⠅⠄⠐⠄⢐⠄⠀⠀⠀⠉⢛⢿⣿⣿⣿⣿⣿⣻⡏⡑⢲⠀⠠⠉⡉⠉⠉⠉⠙⠛⠛⠿⡯⠉⣋⡸⠧⢤⣀⡟⠀⠀⠁⣨⣏⣁⣤⣬⡤
⠀⠘⠳⠶⢆⠖⢂⠤⠰⠢⡀⠀⡄⡁⡁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⡁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⢀⣈⢉⣁⣰⣿⣿⣿⣿⣿⣿⡙
⠀⠛⠓⠀⠀⠈⠙⠋⠈⠉⠀⠘⠓⠓⠀⠀⠁⠚⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠦⠀⠀⠁⠒⠓⠚⠛⠛⠛⠛⠛⠒⠛⠙⠛⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣊⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⣩⣤⣬⣛⠒⢀⣈⠳⡀
⠀⠀⠀⠀⢀⡤⠚⠒⠳⠏⠐⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣶⡷⠞⠋⣟⡝⡗⡄⠓⡲⠘⢆
⠀⠀⡀⠀⠱⡩⣄⠀⡮⡵⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⠁⣄⡀⠀⠀⠘⣼⡘⣾⠈⣓⡶⠈⡆
⠀⡐⠁⠀⠀⢇⠟⠀⢉⡅⠀⢰⡲⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢤⠈⠱⠆⠀⠦⠬⠛⢻⠀⣩⠭⠆⢃
⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡳⢙⡳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡈⢾⡀⢄⠀⠀⢀⢀⠔⣄⡤⢩⠉⡂⡀
⣀⡇⢠⣒⠆⠀⠀⠀⠀⡀⠀⠀⣠⢣⠉⠀⠀⠀⠀⠀⡠⠒⣲⣶⣿⣯⢿⣛⢿⣮⣑⠂⠁⠈⠓⠒⣉⡹⡖⠲⠃
⠳⢛⡖⠶⡳⠰⠤⠤⠔⠁⡠⠂⢻⠁⠀⠀⠀⠀⠀⢾⣶⣞⠻⠣⢴⠦⢵⠒⠾⢷⣙⠛⣛⣭⠟⢯⠘⣽⡇
⠀⠘⠒⠲⣓⡦⠤⠔⠂⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠈⢹⠒⠒⠶⣖⡒⠊⠁⡀⠉⣿⣿⢧⡜⠀⢻⣷⣉⡆
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⣶⣾⣿⠀⠀⠀⠙⠻⢷⠁⠀⠀⡇⠛⠉⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⢤⣎⣀⡤⡤⣤⡀⠀⠀⢸⣷⠿⠛⠀⠀⠀⠀⠀⣼⣠⣴⡶⡇⠀⠀⠀⠀⠀⣘⣄
⠀⠀⠀⠀⠀⠀⢸⡿⠤⠔⣿⣇⣀⠵⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢿⢟⣽⣿⡗⢲⣖⣶⠝⡿⠯⠝⢦
⠀⠀⠀⠀⠀⠀⢸⣀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢟⣉⠀⠉⠉⡉⢉⣉⣀⠤⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡴⠛⠈⠨⠥⠦⠦⠮⠤⠍⠈⠳⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⡀⡄⠆⠀⠰⠒⠂⠂⠐⠒⠂⠆⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⡇⠀⠀⠀⡴⠫⢤⣤⡀⠀⢀⣠⡤⠴⡄⠀⠀⠀⠀⣿⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢦⣿⡇⠀⣰⣿⡆⠀⠉⠉⠁⠀⠈⠉⠉⠀⢘⣆⠀⠀⠘⣿⡖⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⢋⠙⠛⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⣿⠀⠀⢱⣿⣧⠀⠀⠀⣀⣀⡀⠀⠀⢀⣿⡎⠀⠀⠀⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡶⣰⣿⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⡇⠀⠀⢰⣿⢟⣧⢰⠟⠍⠉⠩⠛⣆⣾⣼⣗⠀⠀⠀⢸⡿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡴⣻⢫⣿⣷⢶⣏⣶⡶⣠⣶⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡿⣿⡙⢿⢆⣬⢿⢧⣰⡟⢋⣿⢘⠷⢦⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠛⠻⠟⠷⠛⠟⠛⠛⢿⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢡⡘⡠⡂⢻⣿⡿⣷⣷⢿⣼⣿⢿⠁⡦⠀⢁⣃⡀
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠾⠚⠲⠞⠿⠾⢲⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⡄⣻⠁⠀⢠⡟⣿⣿⣏⣿⣿⡯⢻⡄⠱⠀⣮⠳⡝⠀⢀
⢴⡖⢂⣐⡈⢉⣉⠀⣀⠀⠈⢁⣈⠉⣉⣉⠉⣉⣀⡉⢉⡀⢀⣀⠒⣶⠄⠀⣼⠇⢸⣿⡯⠁⠀⢸⡤⢸⣽⡟⣽⠟⣿⡁⢹⢪⠁⣀⠀⡽⠆⢻⡇⢸⣷⡀
⢸⣿⣟⠛⢻⣿⢻⣷⣿⡇⣼⣿⢻⣷⣿⣿⡷⣿⣿⣽⣿⣷⣿⣿⣯⣿⠀⠈⣿⠀⣟⠜⠀⠸⠂⡈⢧⡀⢧⠀⠱⣇⡑⡴⢀⡈⡀⡈⠂⠀⡄⠞⡇⢸⣫⡇
⢸⣏⣻⣿⣟⣻⣿⣋⣿⣿⣏⣿⣿⣋⣿⣛⣷⣿⣻⣏⣳⣏⣻⡿⣿⣿⠀⠈⠿⠀⣛⠀⠀⠀⠀⠃⠸⠀⣾⣏⣀⢹⣮⡆⠀⡀⠐⠀⠀⠠⠁⠘⢿⠀⠏⠁
⠀⠙⠈⢹⡇⠀⠀⡯⣿⡷⢰⣯⣆⢰⡙⣻⣇⡿⣟⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠘⠇⠀⠈⠠⠄⠀⠀⠀⠹⢏⣿⣚⣛⣏⠀⠃⠈⠀⠀⠆⠀⣤⢸⠀⡆
⠄⠀⠀⢸⡇⠀⠀⠿⠇⠀⠷⠚⢺⠎⠾⠯⠞⠯⠷⠀⠀⠀⣿⠀⠀⢠⠀⡀⡀⠀⣀⠀⠀⡀⠀⠀⠀⠀⣾⡟⡞⣿⣃⣿⡀⠰⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠁⢀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢸⢸⡆⢉⣀⡁⣰⠀⡇⠀⠀⠀⠀⢢⢌⠇⣱⡨⠽⠀⠰⠌⠀⠀⠀⡄⠀⣰⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠼⡿⠿⠾⠿⠿⠷⠿⠸⠹⠿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣧⣶⣼⣿⠆⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⢀
⠀⠀⠀⠀⡄⡀⠀⠀⠀⣘⡀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠋⣼⣿⣿⡻⠃⠘⠄⠀⠀⠀⠀⢐⠐⣂⠂⣀⠀⡀⢂⠒⠒⡒⡐⠐⣀⠂⠒⣚⢺⢦
⠀⠀⠀⠀⠈⠛⠛⠛⠀⠻⠙⠃⠘⠛⠛⠁⠙⠛⠛⠃⠘⠋⠀⠀⠀⢸⣜⠛⣧⣿⡏⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠛⠛⢀⠋⠛⠋⢛⡛⠛⠛⠛⠛⠂⠛⣸⣴
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡿⣇⠐⣛⣛⡁⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⢀⡴⠋⣠⡤⢥⣉⠉⠉⠉⠉⠙⠛⢦⣉⣤⠴⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⣄⡈⠑⣴⠋⠀⡾⠁⣠⣤⠘⡇⠀⠀⠀⣀⣀⣀⡀⠀⢠⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣿⠅⠁⠀⠻⣄⣈⣉⡰⠃⠀⢠⠞⠁⣤⣌⢹⡆⠉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢿⡀⠀⠀⠐⠀⣭⣥⣤⣄⠀⠹⣦⣀⣉⣡⠞⠁⠀⠀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢨⡿⢦⣤⣤⣤⡴⢾⡇⠛⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢿⠀⠀⠉⠉⠙⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⢸⡇⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⢀⣶⠷⠤⢄⣀⣈⣤⣄⡀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠁⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠖⠛⠻⡍⠛⠦⣴⠏⠁⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⢰⡖⠛⠛⠛⢻⡇⠀⠀⢠⡶⠚⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠛⠛⠛⠛⠛⠛⠛⠿⠛⡟⠁⠀⠀⢰⡟⠙⠛⠛⠛⠙⣷⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⢸⡇⠀⠀⣠⠼⠀⠀⠠⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⠀⠀⠠⢾⡅⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢸⡇⠀⠘⢧⣴⡇⠀⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠃⢀⣶⠴⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢾⠀⠀⢹⡄⠀⣰⡆⠀⢰⠄⢸⣅⣠⡄⠀⢸⡇⠀⠀⠀⣰⠞⠓⠀⠀⣠⠖⢶⡀⢸⡿⣄⡞⣷⠀⣼⡋⡉⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢸⠀⠀⠘⣧⢸⡇⣧⢀⡿⠀⢸⣁⣀⡀⠀⢸⣇⣠⠄⠀⣿⣀⣀⡀⠀⠻⠦⠞⠀⢸⠀⠉⠀⠙⠀⢻⣉⣁⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠘⠿⠀⠘⠛⠁⠀⠈⠉⠉⠁⠀⠈⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠳⢦⣄⡀⢸⠀⠀⢀⡀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠸⠷⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣤⣤⣤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣌⠀⠀⠀⠀⠄⠈⠁⡘⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠣⡁⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡗⠀⠀⠀⠀⠐⠢⢄⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠑⢄⡀⠀⠀⠀⠀⠀⠉⠀⠒⢀⠀⡴⠋⢐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡠⠚⠉⠀⡈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠙⢦
⠀⣠⡖⠛⠓⠂⠀⢉⡁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⠉⠓⠠⠄⢸⡟⣴⠋⠀⠀⠀⢀⣴⠶⠆
⠈⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠒⡤⢤⣄⣀⠀⠀⠀⠘⠃⠀⠀⠘⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠑⠒⠴⠂⠄⢀⡔⠒⠋⣰⠋⢀⡠⠔⢉⠀⡖⣀
⠀⠀⠀⠀⠀⠙⠓⠒⠂⠰⠶⠶⢦⡜⠁⣾⠀⠀⠀⠉⠓⢦⡀⠀⠉⠀⠀⠀⡄⠀⠠⠀⠱⣄⠀⠀⠀⠀⠀⡀⠀⣠⠎⠀⠀⠙⠁⡰⢃⠔⠋⢀⠔⢁⡴⢃⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠟⡀⠀⠈⡀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠈⠑⠦⠀⢘⣿⢛⣀⣁⣤⣤⡾⠋⢰⠃⠈⠀⠘⣡⠞⡋⠀⠈⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠘⣄⠀⠀⠀⠀⣙⠻⢶⣤⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠓⠋⠽⠓⠛⠸⢦⣼⡀⠀⣠⠎⠀⠠⡇⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢯⠀⠀⠀⠀⠘⡄⠀⠀⢹⣆⣀⢀⠀⢀⣠⣤⣤⣼⣿⣛⣁⡀⠀⠀⠀⠀⢹⣇⡸⠁⠀⠀⠀⠻⣄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠑⠀⠀⠀⠀⠘⢆⠀⠀⢷⡀⠈⡿⠿⣿⣿⠿⠿⢿⣿⣿⣿⣷⣦⣄⡀⠀⢿⣇⠀⡀⠄⠀⠀⠈⠁⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡈⠢⠀⠀⠀⠈⠢⡀⠈⢷⡀⠹⡄⠀⠉⠒⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⠚⠛⣧⡐⠈⠁⠐⠀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⠶⣤⣀⠀⠀⠑⠄⠀⠑⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⠛⠛⠐⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠻⢄⡀⠀⠈⣽⣷⣶⣦⣤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⣦⠀⠀⠉⠢⢠⣿⣿⣿⣿⣿⣧⡙⢏⠐⠢⣤⠀⣀⣀⣴⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀⠈⢷⣄⠑⠦⣄⠉⠛⠿⣿⣿⣿⣿⣦⣵⣦⣤⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣶⣄⡙⠳⠄⠈⠙⠶⢤⣄⣉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠉⠙⠻⠶⣤⣤⣀⣀⡀⠀⠉⠙⠒⠒⠲⠟⠛⡿⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢇⠀⠀⠑⢄⠀⠐⢄⠀⠀⠀⠀⠀⡀⠠⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠸⡀⠀⠀⠀⠑⠴⢄⣀⡤⠠⠶⠀⠀⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠀⠀⠰⠀⢇⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⢑⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⠀⠀⢸⠀⠸⡀⠀⠠⠀⠀⠐⠀⢀⠀⠀⠐⠒⠚⠋⠁⠀⣹⡀
⠀⠀⢀⡴⢀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠀⠀⢸⠀⠀⢧⠀⠀⠁⠀⠀⠀⠈⠙⠓⠢⠴⠶⠶⠚⠛⠉⠷⢄
⠀⠀⢸⡇⠠⡅⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢸⠀⠀⠸⡇⠀⠸⣆⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⢠⡎⠈⠓⠀⢱⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⠀⠀⠀⣧⠀⠀⢹⡄⠀⠀⠘⢦⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡽⢦⡀
⡸⠀⡈⠀⠀⠈⢦⡀⠀⠀⠾⢥⣄⣀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢷⡀⠀⠀⠀⠳⣄⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⠀⣀⣠⣽⣦⣀
⡳⠠⡄⠀⠀⠀⠀⠉⠓⠢⠴⠊⠀⠘⣷⡀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢧⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⣀⡤⠒⠛⢿⣉⠎⠀⢈⡉⠉⠉⢢⡄
⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠊⠁⠀⠀⠀⠀⢉⠀⠀⠋⢀⣠⣴⡿⠁
⠀⠀⠈⠙⠦⣄⡀⠀⠀⠠⢄⠇⢸⡀⠀⠀⠈⠛⢦⡀⠀⠀⢠⡀⠀⠀⢀⣀⣤⠤⠤⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠒⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠲⣤⡀⠀⠀⡧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⢀⣠⣄⣠⠞⠁⠀⠀⠠⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠉⠉⠀⠀⠀⢈⣿⣲⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠍⠂⠀⠀⠀⠀⠀⢀⠤⠂⠉⠉⠉⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠂⠒⠒⠢⢄
⠀⢠⠄⢀⡂⠬⠐⠀⠉⠀⠀⡀⠈⠠⠀⠀⠀⠀⠠⠁⡠⠰⠎⠉⣉⢦⣠⠌⡄⠀⠀⠀⡀⠀⢀⡀⠠⠄⠐⠀⠉⠑⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠂⠁⠀⠀⠀⠀⢉⣿⣾⣥⡄⠄⠆⠀⠀⠀⠀⠀⡀⢀⠜⠀⠱⠃⠋⡁⢰⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠉⠟⢻⡀⠀⠰⠘⡀⠀⠀⠆⡀⠰⣁⠀⠀⠀⡱⡄⠇⠄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⠀⢤⠚⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠇⠃⠀⠀⢀⡀⠤⠽⠳⠀⠀⢃⢁⠀⠀⢸⣡⢆⠀⠉⠀⠈⠀⢠⠘⠀⠀⠀⠀⠃⢀⡀⠀⠤⠉⠀⠀⠀⠈⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠸⠈⡄⠊⠁⢀⡀⠠⠀⠐⠂⣉⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠑⠒⠉⠀⠀⠠⠔⠂⠉⠁⠀⢀⠤⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠲⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠞
⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⡔⠒⠈⠉⠁⠀⠀⠀⠀⠀⡨⠂⠀⠀⢠⠖⢉⠇⠀⠀⠀⠀⣰⠖⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣠⡤⠤⠄⠒⠊⠁⠀⠀⢰⠃⠀⠈⠲⡄⠀⠀⠘⣉⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠀⠛⠘⠓⠈⠉⢡⠀⠀⠀⠀⠟⡄⠀⠀⠀⣇⠀⠀⠐⢓⣙⢒⡷⠀⢺⠀⠀⡔⠤⠄⠀⠀⠒⠒⢐
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠈⠑⢄⠀⠰⡁⠀⠀⠀⢀⣀⣠⠤⠰⠀⠀⠀⠀⠂⣹⢾⣭⣭⡁⠀⠀⠘⣇⠀⠀⠀⡠⢿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠈⡀⠀⠀⠁⠈⠉⡋⠀⠄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⠑⣆⡀⢰⢡⠂⠀⠀⣇⣀⡀⠀⠀⠀⢀⣸⡀
⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠘⡀⠀⢀⣀⠀⠀⣠⠗⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⡠⠖⠛⢉⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⢡⢀⡤⠶⠂⡡⣀⡴⡊⠁⠀⠐⡀⠀⠀⠀⡠⠶⠀⠐⠃⠄⢀⣰⠀⠀⠀⠀⠀⠀⡠⣀⠤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⢸⠻⣓⠴⢾⣿⣿⣿⣦⠀⠀⠀⢣⠀⠐⠊⠂⠈⠀⠰⠦⢀⣄⠀⠉⠀⠒⠂⠀⠠⠖⠁⡠⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢏⠀⠀⠐⣝⣛⡿⠟⢀⡠⠐⠁⠀⠐⠢⠤⣀⠀⠔⠒⠒⠠⡀⠛⠀⠓⠀⠀⠀⣰⡉⡄⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠈⢢⡔⠔⠊⡁⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⢀⠧⠀⠐⠐⠒⠒⠉⠁⠈⠁⠀⠀⠀⠀⠀⠈⠂⠤⠤⠤⠤⠤⠤⠤⠤⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠤⠤⠬⢤⠄⡁
⠀⠀⠀⠀⠀⠀⠀⣇⠤⠔⣀⠔⠐⠀⢲⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠠⠤⢀⡀⠀⠀⠑⢄⠀⠀⢀⠔⠋⠊
⠀⠀⢠⠀⠀⡠⠁⠉⠢⣀⠋⠀⠀⠀⠈⠀⠀⠀⣠⡄⠼⠏⠁⠀⠚⠁⠉⠉⠉⢆⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠴⠦⠵⠶
⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⡀⠔⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠠⠤⠄⠐⠒⠀⠁⠁⠀⠀⠀⠀⠙⠛⠳⠶⣶⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡿⠘⠀⠀⢻⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣔⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⣠⡠⠀⠠⢤⠴⠄⠀⠀⢀⣀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄
⡀⠈⠛⠙⠚⣰⢊⡋⡛⠟⠧⠿⠎⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠤⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠁⠐⠒⠒⠚⠃⠀⠉⠈⠀⠰⠷⠿⢕⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⠀⠀⠀⠸⣉⡀⠂⠄⢠⣢⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠂⠀⠀⢸⡹⣆⡄⠄⢀⠀⠀⠀⠀⡀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠳⠈⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠘⡩⡺⢠⠀⠀⠀⠀⡀⠀⠐⢀⣁⠀⠀⠸
⠀⠄⠀⠠⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠒⠀⠂⠀⠁⣀⡀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠑⠁⠀⠀⠀⡀⠀⠠⠀⠀⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠤⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠁⠈⠀⡖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠇⠄⠀⠀⠀⠀⠀⢦⡰⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⡀⣀⣀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⣰⣀⣠⠤⠤⠠⠯⠿⠯⠿⠗⠀⠀⠀⡇⠀⠉⠉⠉⠛⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢁⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢤⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠒⠀⠄⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⡐⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠠⠠⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠆⠰⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣀⠀⢀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠐⠉⠀⠈⠋⡀
⠀⠀⠐⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠀⠀⠀⠁⠁⠒⠦⠴⢤⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠐⢿⣾⣿⣿⣿⣿⣿⣿⣾⣿⡶⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢰⣤⢬⣉⡅⡉⠽⠉⠿⠯⠶⠷⠳⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠘⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⠽⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠄⠀⠀⢤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢄⠀⠁⡀⢠
⠀⠀⠀⠐⢪⣷⣻⠯⡑⠀⠳⠟⣓⡃⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⠹⠧⠃⡁⠤⠚⡤
⠀⠀⠀⠀⢸⡋⠚⠧⢵⠒⢲⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡜⣒⠙⠉⣀⢐⠸⡆⣀⡚⡀
⠀⠀⠀⠀⠀⣷⠋⠠⢈⣵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡟⠻⡶⢄⠻⣯⠑⣾⡟⢘⡟⣾
⠀⠀⠀⠀⠀⠘⣆⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢛⢯⠟⣾⣠⣿⣯⣽⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠚⠾⠉⠒

⠀⢀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠤⠄⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⠄
⠀⠘⠀⠀⠀⠀⠀⠈⢄⠀⠰⠁⢀⠀⠉⠆⠀⠀⠌⠠⠊⠠⠀⠟⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠈⠀⠡⣈⢀⡄⠐⠰⣲
⠀⠀⠐⠀⠐⠀⠀⢉⠣⠀⠀⠣⡀⠈⠀⠈⠆⠈⢠⠴⡀⠊⠀⠀⠀⠂⠀⠠⠀⠄⠀⠁⠀⠀⡠⠁⠀⠒⠄⠀⠉⠉⠀⢐
⠀⢀⠀⣋⡀⠀⠀⠀⠈⣀⢀⢀⠈⠒⠈⠄⢀⡴⠀⡯⢚⠁⠀⠀⢠⠐⠂⡐⠠⢀⢀⠀⢀⠜⠀⢣⢄⣀⡀⠀⠀⠀⠤
⠀⠘⠀⠀⡆⠀⡠⠊⡁⠀⠃⠂⢌⡑⢄⠀⠸⠋⠀⢸⡵⠀⠀⢸⠃⠀⢀⡅⡀⠘⠄⠐⠁⢀⠀⠠⣀⡀⠀⡀⠠
⠀⢠⠖⢦⠀⢄⣠⠒⡂⢜⣊⠼⠓⢊⠁⢠⠒⠒⠤⡀⠃⠀⠀⠀⠁⠨⠉⠁⠀⠀⠀⠀⠠⠂⢤⠀⠠⠀⠀⠁⠀⠈
⠀⠀⡆⣈⢂⠀⠀⠀⠀⠒⠒⠒⠈⠁⠀⡌⠀⠀⢰⠀⠰⢠⠀⠀⠂⠆⠀⢀⣴⣶⢻⣟⣳⡆⠀⡀⠀⢀⠀⢀⡰⠉⡘
⠀⠀⡄⠂⢠⠀⠀⠀⢖⠒⠀⠠⠀⢲⠆⠑⠀⠲⠈⠀⠀⠁⠃⠀⠀⠀⠀⢀⢀⠽⠉⠉⠉⠉⠀⠀⡀⠐⠀⠀⠉⠉
⠀⠀⠈⠉⠁⠔⢄⠀⡄⠉⠀⠒⠀⠁⢀⠀⣴⡢⡄⠀⠠⠀⠀⠀⢀⠖⡆⠀⠀⠀⠀⠍⠀⠁⠀⡀⠃⠀⠀⡔⠉⢆⠁⢠
⠀⠦⠀⠀⠈⠀⠀⠀⠀⢀⠀⢈⠈⠡⠏⠓⢉⠠⠣⢶⠠⢅⢤⡤⠀⠀⠀⡀⠈⣁⠀⠈⠈⠀⠂⠀⠀⠀⠨⠀⡠⠊⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠂⠈⢀⣐⣃⡠⠤⠗⢉⠀⠀⠀⠀⠰⠀⠀⡀⠼⢑⠀⠉⡂⠲⢀⠑⢀⡱
⠀⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠈⠉⣤⠀⠀⢘⠭⠁⣀⠀⠐⣁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠎
⠀⠐⠂⠀⠀⡠⠄⠀⠀⠊⢟⠀⠀⢠⠶⠛⠉⠐⢹⣶⠄⠈⡠⠀⠀⠀⠀⠈⠀⢠⠄⠀⠀⠀⠀⠀⠀⣡⠀⠀⠘⠀⠂⠄
⠀⠰⠠⠍⢀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢈⠠⠒⠀⠀⠀⠀⠉⠉⠉⠈⠐⠡⠄⢀⣀⠀⠂
⠀⢠⠀⠀⡀⠄⠈⡆⡕⠀⠈⠀⠐⠀⠀⠠⠒⠃⠀⠀⠀⢰⠀⠀⠀⠌⠀⠀⠄⠠⠏⠀⡀⠀⢀⠠⠀⠀⠙⠇
⠀⠀⠀⡀⢀⡄⠀⢡⠁⢀⡀⠌⠀⡄⠂⠂⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠒⢅⡀⢀⠰
⠀⠠⠐⠊⠸⠈⠆⠈⢀⠡⠀⠀⠦⠠⢖⠀⠀⠀⠈⠁⠀⠈⠀⠐⠀⡄⠀⢠⠂⠉⠀⠀⠀⠈⠀⢀⠜⡀⠀⠀⢻⠧⠂⠁
⠀⠀⠠⠨⠄⠀⠂⠀⠀⠒⠈⠀⠀⠀⠀⠄⠀⠀⠀⠠⠒⠃⠀⠀⣀⠀⢌⢀⠢⠤⠬⠔⡀⠤⠔⢊⠈⠃⡀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠠⠂⠀⣀⡀⡔⠒⠒⡆⠀⢢⠀⠐⠀⠀⠁⠀⠈⠀⠠⡈⡀⢄⠀⠠⠀⠈⠀⠀⠀⠀⠐⠁
⠀⢀⡀⠐⠀⠀⠀⠀⠂⠄⠁⠀⠀⠘⠄⠂⢘⣀⣸⠀⠀⡀⠀⠐⠄⢤⠆⠀⠀⠈⠚⠁⠡⠊⠈⠠⠁⠀⠤⠀⠐⠀⠒⢤
⠀⠀⣠⡶⢀⡶⠀⢀⠀⠃⡀⠂⠈⠠⠀⠀⡀⠄⠠⣀⠀⠀⠘⠀⢀⠀⠀⠀⠃⠀⠠⣀⣠⣤⠄⠀⠀⠀⢄⠀⠀⠀⠀⠔
⠀⢸⣆⠅⠊⠀⡀⠁⠀⡂⠀⠀⡀⢊⠀⠌⡠⠊⢉⡉⠠⠀⠈⠠⠊⠀⠀⠀⠁⠀⢀⠜⠈⠡⡀⠀⠀⠀⡈⠀⠈⠁⠁⢠
⠀⠀⠁⠂⠀⠀⠃⠀⠀⠈⠀⠉⠀⠈⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣀⡀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻⡟⠉⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣀⠀⠀⣿⢿⡇⠀⠀⠠⠤⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⣯⣤⣿⡄⠀⠀⠀⠀⣾⣿⣿⣇⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢳⣶⣦⣀⠀⠀⣀⣤⡀⠀⣰⡿⠛⡙⠛⣷⡄⢀⣴⣾⣿⣿⣿⡋⣴⣿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠘⣿⣿⣿⡶⣟⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣅⣶⣿⣿⡇⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⣸⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⢀⡀⡄⣆⢳⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⢀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⢺⡜⡎⣧⢳⢹⡜⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠧⠼⠘⠈⠁⠀⠀⣶⡄
⠀⠀⠀⠀⢀⣤⣤⣴⡖⠒⣶⣶⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣤⡀⠀⠀⠀⣀⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠷⠤⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢽⣽⣧⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣾⣿⣿⠀⠀⢸⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡷⠿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠚⠋⠁⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⢱⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⠴⠦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢳⣀⣀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠛⠙⠯⣭⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠋⠍⠅⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⠿⡀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⣴⣿⡿⡟⠻⠆⣽⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⡿⠋⡀⠀⢠⣼⡿⠃⢸⣿⣿⣿⣿⡏⠉⢻⡟⠀⢹⣿⡄⠁⠉⠻⠿⡉⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠙⠀⣸⡿⠁⠀⢸⣿⣿⣼⣯⡇⠀⠘⡁⠀⢀⣽⣿⠖⠒⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⢸⣿⠏⢹⣿⠇⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠈⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠦⠀⠀⢘⣺⣄⡠⢀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠄⣠⡁⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠋⠀⠀⠐⠀⠀⠀⠀⠀⢀⠘⠛⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⢀⠀⠀⡆⠀⠠⠘⢥⠽⠀⠀⠀⡀
⠀⡀⠀⠐⠀⢠⠀⠒⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠠⠄⠀⠀⠀⠀⢀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠰⣷⢄⠀⠀⠂⠀⠁⠀⢠⡇
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣊⢄⠀⠀⠀⠐⢜⢃⡀⢀⣀⣐⣤⡤⢊
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠑⠈⠉⣓⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠄⠀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⢊⠀⠀⠀⠀⡊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⡠⠀⠀⠀⠉⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠐⡀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠠⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡉⢉⡒⡄⠀⠀⠀⠐⠠⠀⠀⠀⠀⣄⡠⠉
⠀⠀⠀⠀⠀⠤⢤⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠀⠀⠈⢶⠀⠀⠈⢱⠂⠐⠒⠒⠋⠁⠄⠀⡇
⠀⠀⠀⠀⠀⢠⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⣄⡀⠐⢆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⡠⡧⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠉⠘⣆⡀⠀⢀⠀⠲⡄⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠄⠼⠀⠀⠀⠀⠀⠀⠀⢌⠢⣄⠐⠀⢀⡴⠤⢛⢽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠐⠒⠂⠈⡼⠀⠀⠀⠀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣠⠀⠑⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠞⠁
⠀⠃⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠛⠫⠄⠀⠀⠁⠀⠐⠤⢀⣀⣀⡠⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠀⠀⠘⠋⠀⠠⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⣰⠀⠀⠀⠀⣠⠄
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⣧⠶⠛⠛⠉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⡅⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⢻⠛⡟⢻⢻⠛⠙⢹⡇⢀⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡶⡟⣿⢷⣤⡾⣿⠀⡇⢸⠘⡀⡇⢸⣈⣀⣠⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣴⢿⠉⡏⢻⡇⣧⢸⢸⡈⡇⢸⣀⣷⡾⠟⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠾⠿⠟⠛⠁⠀⠀⠀⠘⠃⠀⠀⢠⣴⣿⣾⡄⢇⢸⣧⣼⣼⣤⣇⣧⡾⠛⠁
⠀⠀⢠⡶⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⢸⣿⣧⣼⣼⣿⠁⠈⠙⠛⠋
⠀⢰⡟⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠙⢿⡜⢿⡏⠀⠈⠉
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣠⢾⣷⠟⠋
⠀⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣼⣷⢸⣿
⠀⢸⣷⢹⠛⡿⢿⡿⡷⢶⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⡿
⠀⠀⠻⣿⣄⣧⠘⡇⡇⢸⢼⣧⠀⠀⠀⠻⠇⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠈⠙⠛⠿⠿⠿⠾⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⡿⣶⣤⣤⣀⣤⣴⡿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡇⠀⡏⢹⠉⢻⠈⡇⢣⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣷⣬⣦⣼⣤⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⡋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣧⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠄⠀⠀⠀⠀⠀⠠⠡⠊⠀⠄⠀⠑⠀⠀⠀⠀⠀⢀⠤⠐⡶⢐⡟⠉⠓⣄
⠀⠀⠀⠁⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠈⠢⣌⠥⣝⡭⢯⠓⢺
⠀⠀⠅⠌⣉⠫⠀⠀⠀⠀⠀⠀⠀⠀⠠⡩⢀⠘⣦⠈⡱⠀⠀⠀⠀⠀⠈⢌⠠⠀⠄⠠⢈⠔⠀⠀⠀⠀⠀⠀⠀⠓⢥⠂⠋⠁⠁
⠀⠀⠐⠒⠛⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⣶⠠⠐⣀⠐⢀⠠⢢⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠢⡄⠀⠀⠀⠀⠀⢀⣤⣄⠁⠀⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡃
⠀⡎⢀⡀⠈⡕⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⢔⡛⣆⠀⠀⠀⠀⠀⠀⠀⠿⠝⠠⡚⠙⢲⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠅⣈⠀⠅
⠐⡆⠢⡙⠚⠟⠡⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣠⠺⣠⣀⠇⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶


⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⠐⠤⠖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠀⠁⠒⠫⡭⠂⠁⠀⠀⠀⠀⠀⠠⢀⠀⠄⠀⢁⣔⡤⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⣲⣄⠀⠶⠶⠖⠇
⢀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⠀⢐⣂⠀⡄⠀⠰⠍⡀⠀⠀⠀⠀⠀⠀⡈⠀⠓⠀⡀⠈⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⢀⣀⠀⠂
⠀⠀⠐⠐⠦⠄⠀⠀⠀⠀⠀⠀⠀⠻⠩⠈⠒⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠙⠠⠀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢀
⠀⣠⠄⣀⣐⠠⡌⠀⢢⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠶⣶⠀⠀⠀⠀⠀⢀⠴⠶⠲⡊⠨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⣠⠔⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⠈⠁⠃⡀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠓⠲⠒⢄⠀⠀⠀⠀⠀⠀⠈⠀⠛⠁⠈⠆⡄
⠁⠀⠀⠐⠁⠚⠁⡀⡀⠀⠀⠀⠀⠀⠸⣟⢲⡶⠦⢬⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠠⢲⣀⠀⢀⡐⠋
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠛⠟⠲⠺⠟⠅⠀⠀⠀⠀⠀⠀⠀⣄⣣⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⠋⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⣰⡖⢥⣃⡂⠄⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⡁⢀⣴⠾⣧⣼⢟⠘⢍⠀⠉⢳⢌⠢⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠊⠀⠈⠀⠉⠑⣊⡟⡄⣡⢀⠀⢀⡼⠀⠑⠌⢂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂⠡⠀⠀⠀⠀⠘⢄⡌⠙⢂⠀⠒⠁⠀⠀⠀⢈⡆⢃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⡀⠸⠙⠂⠀⠀⠀⠀⠘⡷⢛⡀⡀⢀
⠀⠀⠀⠀⠀⠠⠂⣀⣀⣐⠂⠄⠀⠐⣄⠀⢀⠤⢤⠘⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠚⠇⠁
⠀⠀⠀⢀⠤⠤⠤⠤⠤⠬⠧⠦⢤⡀⠈⠑⠴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠃⠆⢠
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠀⠀⠀⠀⠙⠉⠁⠁⠢⠄⡀⠀⠀⠀⠈⠀⡙⠠⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢁⢠⡄⠀⠀⠀⢠⠀⡄⠀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠗⠛⠉⠉⢝⡭⢭⣍⢂⠀⠀⠀⠀⡰
⠀⠀⠸⠀⠀⠸⠇⠀⠀⠀⢹⠋⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣤⠂⣿⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠄⡀⠀⠀⢃⡈⠀⢀⣘⠂⠄⠀⢀⡀⢸⠄⠄⠊⢂⠈⠉⠁⠰⠀⠀⠀⠀⠠⠂
⡔⠐⡇⠀⠊⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⠉⢳⠂⠀⠈⠉⠁⠀⠐⠊⠒⠀⠈⠁⠀⠒⡀⠀⠀⠀⠣⠀⠀⡄⠀⢀⡀⠤⠐⡞
⡯⢁⠀⢠⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠍⠁⠀⠒⠀⠀⠒⠀⠉⡀⠀⠀⠀⠀⠀⠛⠀⠀⢀⢔⣑⡀⠃⠀⠀⠀⠀⠀⠐
⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠇⢈⢀⠀⣄⠀⠀⣠⢀⠈⢠⠀⠀⠀⠀⠀⠘⠈⡠⠢⠈⢹⡟⢻⠀⠀⠀⠀⠀⠰
⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠠⠀⠀⠀⠀⠀⠀⠃⡀⣄⣀⣀⣠⠀⠐⣋⢀⣀⣀⣤⠤⠼⠤⠤⠤⠤⠤⠴
⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣴⣶⠆⣴⠟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⢀⡴⣿⡿⠁⠀⠲⡔⠚⢛⣋⠀⢰⡏⣠⠖⣒⠉⠉⢛⣒⠒⠒⣒⠒⡚⠛⠛⠛⠉⢁⡀
⠀⠀⠀⠀⢠⣿⢰⣿⠁⢠⣶⠀⢡⣰⡏⡹⢠⣿⠘⢂⣾⠋⠃⣠⡾⣉⡧⢾⣯⢺⣿⡔⣿⠇⣼⡟⡹
⠀⠀⠀⠀⢿⣿⢸⡇⠀⢸⡏⠀⡎⠘⠧⠔⠁⠻⠤⠊⠳⠤⠔⠹⠤⠜⠁⠼⠃⠸⠟⠀⠻⠤⠜⠯⠤⠆
⠀⠀⠀⠀⠀⠀⠘⣇⣠⠃⠳⠜⠀⣀⣠⣤⣴⣶⠶⠶⠞⠛⠛⠓⠒⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠰⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠴⠶⠋⠉⠙⠶⠦⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⣹⠟⢉⡟⠉⠉⠉⠉⠉⠉⠻⡉⠛⣏⠙⠻⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣧⣤⣼⡄⠀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣧⡀⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠿⣿⣿⢿⠗⠒⠒⠒⠒⠒⠒⠺⢿⣿⣿⣿⠇⠚⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡇⠹⣆⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢠⣞⣣⠶⣤⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣤⡄⠐⡄⣤⢰⠀⠀⠁⠶⠀⠂⣈⠀⠀⠀⠀⣧⠀⣷⠀⠙⡆⠀⠐⠉⢃⠀⡀⠈⠀⠀⠀⣴⣻⢞⡽⣻⠎⠀⠀⠀⠀⠀⠀⣤⠚⣶
⠀⠀⠀⠀⠀⠀⠓⠃⠘⠀⠀⠈⠛⠋⠛⠉⠀⠀⠀⠀⢹⣁⠀⢂⡼⠃⠀⠀⠒⠊⠉⠀⡂⠀⠀⠈⠐⠫⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢀⣤⣤⣠⣦⣤⠀⠀⠀⣀⠐⠄⣈⠀⠀⠀⠀⠀⢠⣄⣠⣄⣤
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡒⠦⠀⠀⠀⢠⣤⠙⠃⠀⡼⠂⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⣴⠆⠀⠑⢿⡄⠀⠀⠘⠛⢿⣿⠋⠁
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣤⠐⠂⠛⠁⠠⠀⠀⠀⢀⠈⠀⠔⠊⠀⠀⠀⠀⠁⠀⠀⡀⠀⣦⠀⠠⠟⠃⠐⠛⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠛⠛⠛⠛⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠐⣶⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠚⠀⠁⠒⠀⠀⠀⢤⡀⠈⠹⠇⢤⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡀⠀⠈⠿⠀⠀⠀⠀⠀⠠⠁⠀⠀⡙⣷
⠀⠀⠀⠀⠠⣶⣤⣤⠀⠀⢾⡀⠁⠀⠀⠀⠰⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⠀⠀⠀⠠⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁
⠀⠀⠀⠀⠀⠿⣮⠗⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢠⠦⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠉⠛⡂⠀⠀⠀⠀⢰⠀⢃⣠⠖⠀⡀⠀⠀⠄⠀⠰⠀⠀⠀⣴⣶⣶⣶⡶⢤⣀
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⡣⠀⠠⢤⡄⣤⠖⠉⠀⠀⠀⣅⢠⠄⠀⣀⡀⠀⠀⠀⠳⢄⠀⠀⡴⠖⠋
⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠹⠁⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠡⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⢾⡿⠁⠀⢀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠐⣄⠁⠠⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⠀⠐⣿⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠁⠀⠀⠀⠀⢀⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⣷⢄⡇
⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠄⠀⠉⠑⠀⠄⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠰⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀
⠀⢆⣀⣀⠀⠀⣀⣠⠀⠀⠸⣿⣍⣶⣍⣻⡀⠀⠀⠀⠰⠒⠆⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⠀⢀⡜⠁⣴⣴⣀⡶⢤⡀⢠⣾⣛⢛⣿
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢶⣤⣄⣤⠶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠠⠀⠃⣜⢁⣸⡧⡴⠃⠀⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣶
⠀⠀⠀⠘⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢤⣤⠀⠀⠀⠀⠘⢿⡿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⡄⠀⠀⠀⠀⠀⠂⢠⠁⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠄⣀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠙⠀⠀⠠⠀⠀⠀⠘⠳⠶⠶⠟⠉
⠀⠀⠀⠀⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢀⠒⠀⢀⡇⠀⠀⠀⠀⠀⠤⢤⠐⠂⠀⠀⠀⣶⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠾⠷⠂⠐⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⢠⡜⠛⣅⠀⠀⠉⠉⠉⠀⠀⠃⠀⠀⠀⣷⡾⡿⠻⣆⠀⠀⠀⠀⠀⣾⢀⡀⠟⠀⠀⠀⣤⣤⠀⠀⢀⠀⠀⠀⢀⠸⠋⠙⠉⠙⠁
⠀⠀⠀⠉⠀⠉⠂⠉⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠳⠊⠐⠓⠀⠛⠁⠀⠀⠠⣅⠈⢨⠅⠀⠀⠀⠀⠉⠠⣦⠆⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠂
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⢀⠤⠀⠀⣤⠀⠀⠀⠀⠀⣤⣤⡀⠀⡀⠀⠀⠀⠿⡇⠀⢀⣠⠤⢄
⠀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠠⠀⠀⠘⠇⠀⠀⠀⣴⠾⠟⠉⠁⠀⠐⠀⠀⠁⠀⣘⣦⡄⠀⠀⠀⠀⠀⠀⢠⡀⣠⠡⠀⠀⠀⠁⠐⠀⠀⠿⠁⠟
⠀⠀⠀⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠟⠁⠀⠀⢈⣿⡷⠒⠾⠟⠂⠀⠀⠀⠀⠀⡞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠛⠛⣿⠋⢻⠛⢛⡛⠟⣛⣿⢛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣾⣿⣶⣾⣿⣷⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠀⣿⣿⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⠂⠀⠠⠈⠀⣼⣿⡟⢋⣁⡈⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠙⣏⠀⠀⠆⠀⠀⢈⣭⡖⠚⠿⠃⢨⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣄⠀⠀⢸⣿⠃⢸⡿⠀⠀⠃⠀⠀⠙⢿⡿⢣⣶⣾⡝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⠉⠙⣆⠀⠈⢿⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡂⠜⠀⠀⠀⠀⠘⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢺⣿⣿⣿⣠⣶⣦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⢄⣁⠈⠹⢿⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡧⠀⠾⣿⡗⠀⣻⣦⣄⠀⠀⠀⠀⡍⠀⠀⠀⠋⠙⢋⣈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⡔⢀⣀⣤⣿⣿⡟⠀⠀⠀⢠⠉⠀⠀⠚⠀⠲⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢿⣿⣿⡷⠀⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡜⠀⢠⠆⠀⠀⣼⣷⣿⣷⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡠⠃⣴⣶⣤⣹⠿⠿⢻⡥⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠄⠀⢾⣿⣿⡿⠈⢣⣄⣌⠀⠙⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠐⠀⠀⢉⣩⡖⠒⠺⠿⣷⠀⢉⣽⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠰⣿⣿⣿⣆⣴⢀⣀⣰⢼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⠀⠀⠀⡀⠂⠘⠿⠟⠉⢼⣿⣿⣿⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠔⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⣶⣄⠀⡘⡏⠀⠀⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣄⣀⣈⡀⠙⠛⢠⡿⠁⠀⠀⢀⠖⠉⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠿⠷⠒⠐⢮⠃⡄⠀⠀⠉⠀⠐⢶⣶⡄⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡥⠀⠀⠀⠀⠀⠀⠈⠻⣆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⣄⠀⠙⠳⠤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠿⠿⠿⠷⢶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⢀⣀⠀⢀⣀⣀⡀⣾⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣿⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⣧⠈⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣤⣤⣀⣀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢶⣾⣿⣤⣄⣸⣿⡀
⠀⢀⣤⣶⣾⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣿⣿⣷⡀
⣰⡿⠟⠛⠛⣿⡟⠛⠻⣶⠶⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⠶⠛⠉⠀⣿⣿⣇
⣿⡇⠀⠀⠀⡿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠉⠉⢻⡟⠷⠶⢦⣤⣀⠀⠀⣀⣀⣤⣤⡶⠾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿⢻⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⠹⣿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡇⠀⠀⠈⣿⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆
⣿⡇⠀⠀⠀⣿⣀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢠⡿⣻⣿⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢸⡇⢿⣹⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⣿⡇⠀⠀⢸⣇⠘⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠘⡯⢈⣉⡉⠉⢹⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⣿⡇⠀⠀⠀⢻⣶⢶⣶⡾⠃⠀⠀⠀⠀⠀⠀⢸⡇⣿⣉⣿⡄⠘⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢶⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟
⣿⡇⠀⠀⠀⠠⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣍⣉⣀⣀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⠀⠀⢰⣤⣦⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢙⣿⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠛⠻⣷
⠀⠀⠀⠻⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡟⠀⣀⠀⢹
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⣠⣤⣶⣾⣿⡿⠿⠛⠋⠉⣿⣿⡇⠀⠿⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣤⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣶⣶⣶⣤⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢬⠉⠙⠒⠀⠀⠐⡆⠀⠐⠊⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣼⠀⢀⣀⣀⠀⠀⢇⣀⣀⣀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠒⠊⠉⢀⠚⠛⠉⠀⠀⠀⡈⠋⠋⠉⠉⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⢄⣀⠀⠀⢠⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠨⢆⠈⠀⠀⡸⠐⠒⠒⠒⠒⠒⠲⡀⠄⠀⠀⠄⠀⠉⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢰⡐⠄⠀⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠒⠒⠈⣀⠠⠘⠋⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠔⠈⠀⠉⡠⠂⠀⠀⠀⠀⠀⠠⠐⠀⡡⠔⠋⠙⠉⠻⠲⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠔⠁⠀⡠⠊⠀⠀⠌⠀⠀⠀⠄⠈⢢⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⢰⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣠⠀⣤⣤⣶⣆⣀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡲⠄⢀⣀⡀⠉⠁⡒⠒⠒⠂⠤⠀⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⠧⡌⣯⠁⣹⠽⢿⣷⣧⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠁⠀⠀⣕⠂⡤⢠⠄⠀⠤⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠃⣿⣷⣸⡦⠀⣿⠁⠀⠀⠁⠐⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⢀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡀⡀⠘⣆⣀⣀⣽⣿⣿⣷⣄⠀⢈⠂⠄⡀⠀⠀⠈⢄⢀⡴⠃⠀⠀⠀⠀⠀⢆⠠⠺⢉⡗⢒⠤⣠⢲
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠠⠀⠀⠒⠁⠀⠀⠐⡄⠰⢊⡴⢁⡞⠀⣠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠃⠈⠀⠀⠀⢠⣄⠀⢀⡜⠵⣋⣴⢏⣠⠞⠀⠏
⠀⠀⠀⠀⠀⠀⡘⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠄⠀⠀⠀⠀⠀⢈⣕⡶⢭⡋⢥⣠⠯⣣⡀
⠀⠀⠀⠀⠀⠀⠠⠰⠄⠤⠠⣤⠚⠓⢲⡴⠒⠒⢦⡀⣀⣀⣀⣀⣀⣇⠘⠁⠀⠀⠀⠀⠀⠀⠉⠐⠠⣈⡑⠪⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠙⠷⠔⠚⠛⠢⠴⠟⠉⠛⠃⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠀⠀⠀⠀⠑⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠂⢤⡀⠀⠀⠀⠀⢀⡠⠔⠂⢉⠀⠀⠉⠑⢢⡀⠀⠀⠀⠀⠀⢀⠔⠊⠉⡁⠀⠡⡐⢀⠀⠀⢀⡀⠀⠀⢀⠔⠈⠣⠄⠠⠜⠁⠢⡀
⠀⠀⠣⡀⠀⠈⠐⠂⢀⠍⢠⠀⢁⠚⠥⠈⠀⠁⠈⠁⠀⠕⠀⠀⠀⢀⠃⠀⠀⠀⠁⠀⠀⠀⠑⢆⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⢀⠜⠂⣈⠕⠀⠈⠀⡡⠐⠒⠢⢄⠀⠀⠀⠀⡀⣀⠜⠁⠀⠀⢸⠠⠈⢈⡈⡀⠀⣠⠀⠔⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠘⠒⠫⠤⠄⠄⡤⠊⢀⠀⢀⡠⠚⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⢳⠀⠀⡆⠁⢀⠀⡘⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠮⠤⠴⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣢⡈⢲⡰⠁⠀⢘⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⣀⡀⡠⠔⠢⢄⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⣀⠀⠀⠀⣀⠀⠀⠈⠒⣄⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠇⢯⡱⢸⠒⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⡤⠴⠎⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⢀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠛⠁⠀⣠⠄⠒⠒⡤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁⣠⡀⡄⠀⢡⣧⠀⠀⠀⢨⠉⠉⡇
⠀⠀⠀⡀⠀⠐⠚⠀⠀⠐⠻⣄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⢰⠃⠀⣙⡷⠃⠀⣿⠇⠀⠀⠀⠈⠳⠞
⠀⠀⢰⠁⠀⠀⠀⠀⠠⡄⢠⠀⠃⠀⠑⠐⠊⠉⠉⠐⠢⠤⠀⠤⠔⠊⠀⠀⢀⠀⠀⠀⠇⠀⠐⠇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⢸⠂⣠⣀⡀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⣀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠲⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈⠈⠁⢐⡏⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡧⠀⠐⠒⠒⠀⢤⢄⠈⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠂⠚⠉⠢⣄⠀⠀⠀⠀⠀⠙⠶⠶⢥⣤⣤⡾⠁⠀⠀⠀⠀⡚⠓
⠀⠀⣇⠀⠀⠈⠀⠂⠈⣾⠘⠃⡅⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣄⣴⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⡀⣠⣖⠠⠀⠐⠀⣷
⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⠐⡀⡇⠀⠀⠀⠊⠀⠀⠀⠀⢰⠃⠙⠋⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⢠⡄⠀⠀⠀⠀⠄⠀⢀⢸⣿⣿⡇⠀⠀⢸⡅
⠀⠀⠀⠲⠶⢤⠤⠤⡾⠡⡾⠐⠁⠀⠀⠀⠀⠀⠀⣀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠈⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠛⠶⠶⠶⢴⣄⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⣉⠀⢀⡂⠀⠀⠀⠁
⠀⠀⡀⠀⣀⠠⠖⠒⠊⠱⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠁⠀⠠⠔⠒⠂⡄⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣤⣀⣀⣀⡀⢈⠀⠀⠀⠀⠀⠀⠀⢀⢄⠤⠄⠀⠀⠄⠀⠀⠰⠂
⠀⠀⢸⡀⠀⠀⠒⠡⡐⠀⠰⡀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⡄⠕⠁⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠈⡃⠀⠐⠂⠚⠁⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⡈⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⡀⠀⠀⠀⡧⡀
⠀⠀⠀⢣⠀⣤⠴⠖⠚⠀⠀⡁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⢣⣤⠇⠀⠀⠀⠀⠀⡀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠂⠀⠒⠊⠁⠈⠒⢄
⠀⠀⠀⠈⠆⡀⠀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⠋⠘⠻⠂⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠉⠉⠐⠲⠀⡀⠤⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠚⣿⣿⠲⡄⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⢀⣀⣀⣀⣀⣛⣛⣚⡇⠀⠀⠀⠀⠘⣤⣤⣄⡀⠀⠀⠀⠀⣠⣤⣥⣬⣬⣏⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⡏⢹⣿⣟⣿⠃⠀⠀⠀⠀⠀⣿⣿⣷⠙⢦⣠⡤⣾⡿⣿⣿⣾⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣻⣇⠀⢸⣿⣿⡁⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⣼⠀⠀⠀⠀⠙⡇⠿⢿⣿⣿⣿⣿⣧⢩⡬⠁⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠘⣿⣿⠂⠀⣾⣿⡟⠀⠀⠀⠀⠀⠘⠉⠉⠉⠠⠞⡅⠀⠉⠉⠈⠛⠏⢽⢻⣿⣿⣏⠟⠒⣦⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⢹⡗⠀⠸⣾⣿⠃⠀⠀⠀⠀⠀⢸⣷⡒⡞⡛⠹⠂⢂⢤⡖⢻⣿⣿⣿⣿⡟⠛⠁⠀⠀⡿⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠅⠻⠇⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⠃⡟⠀⢰⠀⠸⠠⡇⣋⣿⣿⣿⣿⣷⠀⠀⠀⢀⡟⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⢾⡇⠀⢸⠀⠀⠀⣿⢀⢿⣿⣿⣿⣿⠀⠀⠀⣸⢱⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⢹⡷⠀⢸⠀⠀⠀⣿⠀⢼⣿⣿⣿⣿⠀⠀⢠⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣟⣿⢺⠃⠀⠘⠀⠀⢁⣿⠐⡾⣿⣿⣿⣿⡀⢠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⢸⣿⣆⣾⣿⣷⠀⠀⠀⠀⠀⢀⠅⡉⠉⠉⠁⠈⢂⡔⡄⠞⠂⠛⢿⡽⣭⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⢸⣾⣋⣵⢻⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠛⢲⠒⠒⢓⣷⢲⣿⣿⣿⣿⣿⡏⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣃⣀⣰⣴⣾⣿⠿⣿⠀⠀⠀⠀⠀⢀⢨⠉⠉⠙⠒⠚⢒⡢⠄⠙⠘⠛⠛⠛⢛⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠓⠒⠒⠒⠒⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠛⠒⠒⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠛⣻⡇⠀⢠⣧⣤⣄⣀⣉⣉⣙⠛⡦⠀⠀⡾
⠀⠀⠀⠀⠀⢀⡤⠚⠫⠤⢤⡤⠖⠉⣀⣇⣀⣸⠁⠀⣠⠖⠛⠛⣿⠙⣇⠀⢸⡧⣤⣀⣀⣀⡀
⠀⠀⠀⣠⠖⢫⠀⠀⣀⠔⠋⢀⡴⠚⠧⠬⣭⣁⣀⠀⠛⠉⠉⠙⠃⠀⣇⠀⣼⠀⣠⠞⠉⢉⡽⠛⠓⠲⢶⣤⣀
⠀⠀⢻⠛⢲⠶⠷⣾⢤⣤⣞⣁⣀⣀⡀⠀⠀⠀⠈⠉⣩⠿⣖⣒⠒⠶⠬⢭⡵⠊⢀⣠⠞⠉⠀⠀⠀⢀⣸⡶⢿
⠀⠀⢸⡴⠋⠀⠀⡇⠀⢸⠁⠀⠉⠉⠙⠛⠛⠓⠶⢾⠧⣤⣤⣏⣀⣀⡴⠋⢀⡴⠋⠈⢉⣑⣒⣤⡴⠋⠁⠑⣾
⠀⠀⢸⡇⠀⠀⠀⠁⠀⢸⠀⣴⠒⠋⢻⠀⠀⠀⠀⠘⣆⠀⢹⡀⠈⢹⠉⠛⡿⠶⡿⠿⣭⡽⢻⠃⠁⠀⠀⠀⣾
⠀⠀⠘⣧⠤⡄⠀⠀⠀⢸⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠸⠤⠶⠃⠀⢸⠀⠀⡏⠀⠱⡀⠀⣇⡞⠀⢸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⣧⠀⢸⠀⢸⡇⢸⡇⢀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠙⢦⣿⠃⠀⠸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠶⠒⠦⣀⠀⣸⠀⠀⡇⣤⠖⢲⠀⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣿⠤⠼⠀⢸⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠀⢦⠈⣇⣿⠀⠀⢷⢹⠀⢸⡄⣿⠀⠀⢠⠀⠀⠀⢸⡀
⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⣿⠉⠉⠉⠉⠉⠉⡇⠘⣄⠢⠤⠜⢀⡏⣿⠀⠀⢸⠈⠷⠴⠇⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⢹⠒⠒⠒⠒⠒⠚⠁⠀⠈⠓⠒⠚⠋⠀⣿⠀⠀⢸⠑⠲⡄⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣧⡀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⡄⢸⠀⢻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⣧⠈⢢⡀⢻⠀⢸⡄⠀⠀⡏⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⣀⡤⠋⠀⢸⠀⠀⠀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠙⠓⠲⠷⢼⣤⣤⣇⣀⠀⠻⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⡇⠀⢀⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠶⠤⣤⣀⡀⠀⠀⠀⣾⠀⠀⢸⠀⠀⠀⢀⣼⣇⠀⠀⡇⢀⣾⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣿⣀⡀⢸⡆⢀⡴⠋⢸⡇⢳⣤⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠶⠦⠼⠗⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠘⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣶⣦⣼⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣄⣠⣤⣤⣴⣶⣾⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⢰⣖⣴⢄⡀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⣙⣻⣷⣎⡼⠆⠀⢀⡄⢠⣄⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⢩⣭⠿⣿⡴⢎⠾⣌⣷⣾⡿⢭⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠷⣿⡿⣯⣙⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠰⡷⡄
⠀⠀⠀⣀⢶⣘⣷⢰⡀⠀⠀⠘⠳⠆⠉⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠒⠒⠀⠛⠛⠃
⠀⠀⠈⠫⡟⣻⡿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠀⠀⠀⢺⣷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡮⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠉⠉⠀⠀⠁⠀⠒⠂⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠃⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⣀⣀⣠⡶⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣉⣀⣀⣀⣠⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⠉⠝⢻⡇⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠙⠛⢶⣄⠀⠀⠈⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡇⠀⠀⠠⠙⠿⠷⢶⣄⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⠶⠃⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠾⠶⠶⣶⣶⡶⠶⠶⠟⠛⠉⠉⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠁⠀⠁⢰⣿
⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠤⡦
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣖⣤⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠦⠀⠀⠀⡄⠀⠀⢀⡎⠀⠁⢀⡖⠀⠀⠀⠀⢸⠃
⠀⠀⢸⣶⣤⡀⠀⠀⠀⠀⠀⠉⠛⠒⠳⠶⢢⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠄⠀⠉⠀⠀⠀⣀⡤⠖⠉⠀⢀⣤⢸⡂⠀⠀⡘⠋⠀⠀⠀⠐⣿
⠀⢀⠉⠉⠛⠛⠽⠆⣶⣶⠀⢸⡗⠒⠛⡿⠟⠁⠀⠀⠀⢀⣀⣀⠈⠉⠑⠂⠀⠀⠀⢀⣀⠦⠚⠉⠀⠀⠀⠀⢰⠃⠉⠀⡻⢣⡞⠁⠀⠀⠀⠀⠜⠁
⠀⠈⠻⢿⣶⣤⡄⢀⣀⠀⠀⣾⠂⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠈⠉⠀⠀⣀⡠⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⢁⡌⠁⠀⠀⠀⢶⣾
⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠀⠁⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⢀⠠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣇⡄⣴⠉⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⣀⣤⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣿⣤⠍⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠀⠉⠉⠓⠶⠤⣀⠀⠀⠀⠀⠀⠀⡾⣿⡁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠛⠒⠒⠚⠛⠋
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⡄⡀⠀⠀⠀⠸⣿⣿⣿⢷⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠤⠦⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠉⠛⠿⣿⣮⣭⡙⠳⢦⣤⠔⠒⠀⠀⢠⣼⠃⠀⠀⠀⠀⠀⠠⡶⠋⠉⠋⠉⠓⣆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠉⠙⠟⢿⣷⣀⣠⡦⠔⠋⠉⠀⠀⠀⠀⠠⠎⠈⠐⢦⣄⣀⣀⠀⠀⣸⠷⣦⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠰⠛⠙⢸⡟⢛⣏⠀⠀⠘⡇
⠀⠀⠀⠑⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡄⣠⠀⠀⠀⠀⠈⡿⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠈⠙⠿⠿⢿⣏⣀⣚⢛⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⡀⠑⢤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠲⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠤⠤⠤⢀⡈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠤⣤⢤⣤⣠⣀⣀⣀⣀⢀⠂⣐⣁⣀⠀⠀⠀⠈⡄⠰
⠀⠀⡠⠒⠚⠛⠙⠛⠒⠚⠛⠻⠯⠿⠲⠶⠤⠤⠤⠤⣛⣲⣿⣃⠉⡉⣉⢉⣉⣷⣼⡷⠓⠒⠒⠒⠶⠴⠀⠒⠖⡦⠤⠤⠤⠤⢤⣤⣀⠀⠀⡀
⠀⠘⠐⠈⠉⠉⠁⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠤⠤⠤⠤⠀⠀⣀⢀⣉⣉⣁⠉⠈⠁⠉⠉⠃⠓⠒⠚⠛⠿⠿⠻⠘⢛⠀⢤⣀⡠⠤⠔⠈⠀⢮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠂⠐⠒⠒⠀⢄⠀⠈⢢⠈⠆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⢸⠀⠀⠀⠀⠀⡇
⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢺⠀⠀⠀⠀⠀⠠
⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⠀⠴⠀⠀⠀⠀⠀⣷
⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡄⠀⠀⠀⠀⢀⢸
⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⢇⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⢸⠸
⠀⠀⠁⠐⠨⠭⠍⠅⣐⣒⠂⠤⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⣸⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠒⠒⠬⠭⠄⣁⢐⣒⠀⠤⠤⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⡇⢀⣠⠴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠒⠂⠨⠭⠤⢀⣀⣒⠀⠠⠤⠀⠀⠴⠀⢀⣄⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠒⠀⠉

⠀⠀⠀⠀⠀⢀⠄⠀⢄
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠨⢦⠀⠐⠒⠒⠒⠀⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠒⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣡⡀⢈⡖⠉⠁⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠠⠤⠄⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠹⡍⠉⡍⠉⠉⠉⠹⡉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⢱⣀⣘⣄⢀⡀⣀⡘⢤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠀⢀⣀⠀⡆
⠀⠀⠀⠀⡠⠔⠒⠒⠒⠤⠤⠀⢰⣊⣍⠀⢱⡄⠃⠀⠀⠀⠀⠀⡑⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠤⠤⠤⠀⠒⢸⠁
⠀⠀⠠⠊⢀⡤⠒⠒⠤⠤⢄⠸⠀⢀⠃⠀⠀⠹⡀⢡⠀⠀⠀⢀⡠⠛⠉⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠃⡆⡌⠀⠀⠀⠀⠱⠀⢢⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢡⠀⠑⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠈⢢⠀⠑⣄⠀⠀⠀⠀⠀⠋⡄⠀⠀⠀⠀⠀⠀⡄⠀⠈⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠑⠤⣀⠑⠂⢤⠄⠐⠦⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠡⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⣀⣉⢰⠊⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⡄⢰⠘⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠈⠉⠉⠉⠀⠂⡄
⠀⠀⠀⠀⡊⠁⠀⠀⢠⠀⠀⠀⢰⠃⠇⠀⠀⠀⠀⠀⠀⠑⡀⡄⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⣀⠀⣤⠖⠁
⠀⠀⠀⠀⠀⠉⠛⠫⢉⠁⠀⠀⠛⠃⠘⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠚⠁⠉⠉⠉⠉⠉⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⡐⠒⠂⠠⠤⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠄⠐⢲⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣠⡾⠀⠀⠀⢀⡀⠳⣄⠹⡄⠀⠀⠴⢾⣭⠽⠃⠀⣠⡴⠞⢉⡴⠺⡇
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠉⠀⣀⠀⠀⠟⠀⢤⡀⠀⢹⠀⠶⣦⠄⢀⣠⠶⠋⠁⠀⠀⢾⣀⣀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠰⠃⠀⠀⠸⠆⠒⠠⣀⢸⣤⣄⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⡼⢃⡼⠁
⠀⠀⠀⠀⠀⠀⢠⠘⠟⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠉⢠⢄⠎⠀⡃⠀⠀⠒⠶⠶⠶⠒⠂⠸⡄⠸⡀
⠀⠀⠀⠀⣀⠄⢸⣌⠃⠀⠾⢸⢡⡾⠿⠿⠟⠻⣿⠆⠀⣼⡼⠀⠀⠳⣄⡀⠀⠉⠛⠒⠆⣠⣴⢷
⠀⠀⣠⢾⡏⠀⠀⠙⠶⢶⡀⠈⠀⠓⠒⢢⣭⠉⠉⠀⣠⠇⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⢾⣧⣴⣾
⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⢀⣀⡴⠋⢀⣀⣀⠀⠀⠀⠀⠀⠀⢉⣷⣌⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣮⡍⠛⢛⠛⢻⣧⣴⣾⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠀⢸⣇⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣾⣿⣿⣿⣿⣇⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠹⣟⠶⣄
⠀⢀⠀⠀⠛⠻⢿⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⢦⣈⣧
⠰⣌⣓⠋⠉⠑⠢⣼⠁⢀⣿⣿⣿⣿⣿⡿⠁⠈⠛⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢀⠜⣁⠈⠀⠀⠇⠀⣸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣄⡀
⢰⣯⢀⠌⡚⡆⡟⢶⣤⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠈⠛⠀⠛⠃⡴⠃⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⢤⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⣸⡆⠲⢤⣥⡤⠀⡞⢀⣠⡤⠖⠛⠉⠁⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⣀⣚⡥⠐⡆⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣼⠔⠋⠀⠀⣀⢠⡠⣄⡘⠟⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⢻⣤⣤⣤⠶⢊⡁⠀⠀⠘⠟⠋⠃⠁⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣧⣾⠟⠻⠿⠀⠀⠀⠀⠀⢹⢡⡶⡌⢶⣉⡤⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣀⣀⣀⣀⣀⠀⢀⣤⠶⠞⠂⢀⠼⣸⠷⠚⣛⣛⡛⢛⠋⠉⠁
⠀⠀⠀⠀⢀⣴⣒⣊⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠙⠚⠛⠋⠉⠉⠁⠀⠚⠓⠒⠛⠛⠛⠋⠉⣙⣦
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⡶⠾⠟⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡔⣀⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⢀⡠⢊⣀⡠⠂⡀⠀⠀⣚⡫⣇⣤⣠⣄⠨⠄⠀⣃⡀⠀⢀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⢦⡐⠖⡉⠛⣷⠀⠀⠻⣅⠈⠩⢅⠈⠓⠲⠌⠉⠀⠈⠙⠀⣞⠁⠀⠂⠀⢍⠲⠭⠉⠉⠉⠩⢏⠂⠋⡇
⠀⠀⠀⠀⠀⢀⠉⠀⠉⠐⠒⠁⡀⠀⠘⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠃⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠄⢁⠀⢘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠈⠀⠠⣀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠬⢥⠴⠤⠤⠤⠮⠤⠤⠴⡿⠄⠀⣶⣾⡆⠀⠀⠤⠤⢒⠒⡶⢐⠿⣄⠀⠀⠙⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠀⡐⠀⠀⠀⠠⠔⠉⠐⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠿⠿⠿⠿⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⢉⠀⠤⠐⠆⠘⠀⠒⠀⠆⢠⡌⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢁⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢠⠉⠀⠀⠀⢠⣤⡀⢠⣤⢠⣤⣤⣤⠠⣤⡄⣤⡄⢠⣤⠀⠀⠀⠀⠒⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠤⠀⠀⠀⠀⢸⣿⣷⣼⡏⢸⣿⣤⡄⠀⢿⣷⣿⣿⣾⡟⠀⠀⠀⠀⠈⡁⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠆⠀⠀⠀⠀⢸⣿⠹⣿⡯⢸⣿⣉⣁⠀⠸⣿⡟⢿⣿⠃⠀⠀⠀⠀⠀⣉⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⠂⠀⠀⠀⠀⠈⠉⠀⠉⠁⠈⠉⠉⠉⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠃⠀⠀⠀⠀⢠⣟⡇⢿⣽⢾⣿⠗⡇⢻⣼⢳⣿⡄⣿⠀⠀⠀⠀⠀⠠⠄⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠈⣀⠀⠀⠀⠘⠉⠛⠚⠛⠘⠛⠂⠃⠘⠋⠛⠉⠓⠛⠂⠀⠀⠀⠀⠶⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣈⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠘⢃⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢿⣷⣦⣌⣀⠃⠐⠂⠰⠀⠶⠀⠓⢈⣁⣤⣶⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡞⣼⣻⢿⡿⣿⣿⣷⣶⣶⣶⣿⣿⣿⡿⡟⣏⢷⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⢱⢧⡏⡾⣹⢳⢇⡟⠀⠀⢻⡹⡜⣏⢧⢻⡼⡎⣧⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⢇⡟⡾⢰⢣⣏⡟⣼⠁⠀⠀⠈⣧⢳⡹⡜⣇⢷⢻⡸⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡟⣼⢹⢣⡟⡾⣸⢱⠇⠀⠀⠀⠀⠸⡎⣧⢳⣹⡜⠇⢷⢹⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⠁⠁⠀⠀⠸⢳⢇⡟⠀⠀⠀⠀⠀⠀⢹⡜⡎⡧⠁⠀⠈⠀⠓⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣼⠁⠀⠀⠀⠀⠀⠀⠀⢷⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡟⢉⣥⠉⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣤⣶⣾⣿⠛⠛⢿⣿⣿⣷⡈⣿⡇⢸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⢛⣉⣥⣴⣶⠿⠿⣿⡏⠤⢙⣿⣤⣤⣿⣿⣯⣉⣁⠀⠴⠾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⣿⣿⠏⡉⢻⣿⣈⣀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣏⠀⠀⣿⣷⣤⣞⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠷⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣩⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣡⣴⣶⣿⣿⣿⣿⣷⡶⠂⠀⢰⣄⣴
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⡿⠟⢛⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠸⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣥⣾⣿⣶⣾⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠾⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠞⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣘⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶

⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⣤⣤⡀⣤⣤⡀⣠⢠⡄⠀⣤⠀⣤⡀⢀⣄⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣇⢸⡷⣾⠁⣿⢾⡇⣾⠀⢿⣸⡇⣸⣯⣧⢰⣏⠀⠈⠳⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣯⣤⣿⣼⣧⣼⣧⣿⣬⣷⣿⣤⣼⣿⣤⣿⣤⣽⣦⣿⣿⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠂⠈⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠔⠒⠒⠒⠒⠦⣄⠕
⠀⠀⠀⠀⠀⣀⣀⢠⠴⠶⣒⡶⠶⣬⣱⡿⠖⢂⣰⣿⣦⠀⢀⣀⣤⣤⣿⣤⣤⡴⠦⣀⣒⠶⠦⠤⠤⠴⠤⠄⢄
⠀⠀⠀⠀⠀⡁⠀⢀⡠⠎⠉⠀⠘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠀⢂⠀⠀⢀⠁⠆⡁⢸
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠐⠀⠐⠃⠀⠀⠈⢄⠉⠁⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⢛⠀⠀⠀⠑⢤⠚⠁⢰⡇⢸
⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⠉⠉⡇⠀⠀⠠⠀⠀⠠⠉⠀⠄⠀⠀⠀⠀⡈⠁⠀⣿⠀⠀⠀⠀⠀⡑⠄⠜⢣⠸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⡀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠑⠤⠊⠀⠀⠀⠀⠠⢰⡌⠁⢰⢸⠀⡇
⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠆⢸⠘⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⠙⠂⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢀⠀⠀⠵⡠⠀⡉⠏⠀⡇⢸
⠀⠀⠀⠀⠀⠧⠀⠀⠀⠠⡀⠀⠀⡇⠀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠁⠛⠀⠀⠁⢸
⠀⠀⠀⠀⠀⡄⢈⠢⠐⠈⠁⠀⠀⢡⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢂⢀⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠢⡇⢸
⠀⠀⠀⠀⠀⠧⠬⠤⣰⠤⠤⠄⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⡇⠀⢸⠂⠀⠀⠀⠀⠀⠀⢀⣧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠓⠒⠒⠤⠤⢭⡤⠎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣤⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢿⣿⠀⢸⡇⠀⣠⣶⣿⡆⠀⠀⣴⡄⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠉⠹⡆⠀⠀⢉⣠⠟⠀⣼⣿⡟⣰⠇⠀⠀⠈⠀⠀⠀⠀⢸⣿⡆⢹
⠀⠀⠀⠀⠀⣴⡋⢹⣿⣷⡀⠀⠀⠙⠳⠦⣿⠴⠖⢋⡁⠀⣼⣿⡟⣰⠋⣀⣀⠁⢀⡶⠋⠉⠳⡄⠀⢀⡾⠀⢈⣠⣴⣦⢀⣤⢶⣶⣦⠀⠐
⠀⠀⠀⠄⠀⣿⣷⠈⣿⣿⡇⠀⣶⣦⡀⠀⣿⢠⣾⣟⣹⢷⣿⣿⡟⢡⣾⣿⠋⣷⡜⣷⣾⣷⢀⣻⡴⢻⣦⣴⠏⣹⣿⣿⠋⢀⣾⣿⡏⣠⣴⣦⡀
⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣷⠀⢸⣿⡇⢠⡇⢿⣿⡏⠁⣸⣿⡿⠀⣿⣿⡇⠘⠛⣡⣿⣿⡏⠉⣿⠀⣼⣿⠃⢠⣿⡿⠁⢠⣿⣿⡟⣸⣿⡿⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⣸⣿⣧⣾⠁⠈⠻⠷⠞⢻⣿⡇⠀⠻⣿⣷⡶⠞⠹⣿⣿⠀⣰⠇⠀⢿⣿⠀⠀⠙⠃⢠⣿⣿⠟⠀⢿⣿⠀⣴⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠋⠻⠿⠃⠀⠀⠀⠀⠄⠸⣿⣇⠀⣤⣦⡀⠀⠀⠀⠉⠛⠛⣡⠴⠶⢦⠀⠀⠀⠁⠀⣸⣿⡿⠒⢦⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣶⠄⠀⠀⠀⠹⣿⣄⢉⣰⠇⠀⢀⡀⠀⠁⢸⠇⢰⣾⣿⠃⢀⣠⠴⠛⠁⠙⠓⠒⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠉⠀⠀⠀⠈⠁⠀⠀⠘⢧⣤⣭⡤⠖⠋⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣿⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣷⣶⣷⣿⣿⣷⣿⣿⣷⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣾⣹⡁⢀⡇⢷⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠴⠂⢳⡶⠋⠉⠉⠉⢀⣈⠉⢦⢶⡩⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠐⢺⠃⠀⠀⠀⠀⠘⠋⢰⣿⣧⠠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⡀⠀⢠⡄⣴⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡝⠃⠀⠈⠁⠈⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢟⡠⠄⠀⠤⠤⠤⠤⣅⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⢠⣲⠀⢄⣦⠀⠀⠀⢹⡄⢠⢲⣭⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠁⠀⢈⡡⠤⠒⠒⢹⠇⠘⣯⠿⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠘⠄⣤⠺⠁⠀⠀⠀⣠⠜⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡷⠛⠖⠒⢒⣒⠛⠛⠉⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠐⠉⠛⠛⠀⠀⠀⠀⣿⡿⠿⠀⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠰⢾⠗⠰⠶⠳⢶⡖⠺⠿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⠣⣀⠀⠶⠆⠀⢀⡠⡟⢠⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⡇⣴⡌⠁⠀⠉⠙⢛⡇⢸⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣧⠀⠀⣀⣠⣤⡤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢶⡄⠀⢸⡿⠀⢀⣴⡇⠀⠀⠀⣠⠋⠛⢢⠀⠀⠀⡇⢰⣷⠸⣶⡏⠤⢴⡒⠂⠒⠒⠒⣠⣄⢵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠄⠀⠀⠀⠀⢿⣴⣶⣾⣿⣧⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⡇⠈⠛⠀⢧⣷⠿⠃⠀⠀⠀⠸⡿⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢺⡆⣀⣰⠙⢿⣿⣿⣿⡄⠀⠀⢇⠀⡠⠂⡇⠀⠀⣇⠀⠀⠀⢸⢿⠳⣄⠐⠿⠀⠀⢀⣠⣮⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⡇⣾⣿⣿⣿⣷⠀⠀⢸⡜⠀⠀⡇⠀⢠⢹⠀⠀⠀⢸⢸⡀⣶⠍⠉⠉⢨⣅⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠙⠛⠛⠿⠛⢆⠀⠈⡇⠀⢰⠁⢀⡞⣸⡄⠀⠀⠘⢄⣇⠀⠀⠀⠀⠈⠁⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠤⢄⣀⣈⣶⣶⣻⡀⣼⣿⣿⣗⡥⠤⠖⠒⢄⠀⠹⣟⠀⠰⠷⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀⣠⠊⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⢣⣾⣿⣿⡦⠤⠴⠿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⡹⠁⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠠⠀⠈⡟⠁⡀⠐⡤⠀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣉⠀⡌⠁⠀⢳⡥⡞⠀⡥⢠⠆⠀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠠⡱⣇⣅⣂⢴⠡⣀⢄⣠⠄⠐⢈⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠅⣠⠖⠉⠀⠀⢹⠟⠉⠉⠙⡄⢈⠋⠠⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠁⡀⢀⣡⡄⠀⠀⡞⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠙⡅⠰⠂⠀⠠⠀⠀⢤⢀⠀⠄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠛⠁⠀⠀⣰⠷⠓⠂⢄⡀⠀⢸⠀⠀⡀⢠⣀⠴⠁⠀⠁⢂⠀⣰⠰⠟⠓⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠃⡠⠤⠀⡜⠁⠀⠀⠀⠀⣤⢖⠟⠙⠛⠫⡙⢥⣀⠀⠀⠀⠘⡌⠉⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢀⡄⠄⠀⠐⢉⠁⠀⡀⠀⠀⢸⣷⠁⡠⠀⡀⠀⠀⢂⢳⠠⣀⣠⠴⢧⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠘⠓⠚⠀⠀⠀⢼⣀⠀⠐⠂⠈⠐⢀⣼⣿⠚⠁⠀⠀⡼⠀⡐⠈⠐⠲⢤⣂⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢀⡲⢆⠀⠀⠀⠎⢿⣶⡤⣄⢠⣤⡾⠏⠀⠈⠐⠒⣹⠰⣂⡀⢀⠂⠀⢢⠀⠙⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣤⣄⣆⢴⠁⠀⢀⡥⣖⠀⠀⠀⠀⠩⠋⠙⠈⠀⠀⠀⢹⠒⠊⠪⠼⣆⣲⠋⣘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠜⠃⠚⢆⠀⠈⠀⠈⢄⡀⢀⣠⠂⠀⠀⠀⢀⣀⣠⣼⠳⣷⣞⠁⠈⠇⠋⠱⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⠀⠁⠀⠄⠐⢩⡆⠠⠀⠐⠂⠀⠈⠁⠁⢠⡖⠒⢲⠉⢽⠅⡛⠀⣿⢫⠩⠁⠶⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⠀⡤⠈⠀⢀⠣⣌⡾⢀⠤⢀⡠⡀⠀⠑⣷⡖⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠐⠀⢀⡀⠀⠁⠀⠛⢻⠀⠀⠀⠀⠇⠒⠀⠀⣳⣀⠠⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠈⠛⠂⠀⠀⠀⠐⠀⠀⠀⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠂⠀⠀⠀⠀⠀⠈⠈⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⢽⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⢿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⢀⡠⠐⢊⣱⡀⠀⠀⠀⠀⣀⡾⠤⣄⠀⠀⠻⠋⠀⠀⠀⠀⠀⣠⠖⠈⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⡛⠀⠤⠊⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠠⠂⠉⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠸⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡿⠇⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⠀⡠⠴⣚⣵⡀⠀⠀⠀⢀⣠⡾⠤⣄⠀⠀⠳⠏⠀⠀⠀⠀⠀⣠⠔⢘⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⠛⡁⠐⠂⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠁⠤⠊⠉⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢲⣵⣤⠠⠦⠶⠀⠀⠈⢉⣓⠂⠤⢀⡀⠘⠤⠊⢀⠤⠤⠤⠤⠤⠤⡀⠀⢠⠒⠒⡄⠀⠀⠀⣳⣀⣀⣐⣎
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣲⣥⠄⢀⠄⢀⠄⠀⠀⢀⠄⢠⡌⠭⠐⠒⠈⢹⠀⠀⠀⡜⢂⢤⣄⠀⠒⢰⢇⠀⠀⠑⠞⠁⢀⠴⠊⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⢀⣁⣀⣬⠭⠑⠒⠀⠉⠙⠠⠚⠁⣠⡀⠀⠀⢸⠀⢀⡎⠀⠡⣠⡾⠀⣠⠋⢀⢂⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⢠⡴⠭⣔⠂⣸⢍⠉⠀⢠⠋⠁⣱⠒⢳⡀⠀⠀⠀⠀⠀⠈⠁⠀⣤⣬⠀⢸⠉⠉⠉⠉⠈⠉⡏⠉⠉⠉⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⣀⠀⢣
⠀⠀⠀⡄⡀⠈⡋⠀⣠⠂⠀⠀⠙⠒⢹⠆⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⠀⡆
⠀⠀⠀⠃⢁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠐⠦⠞⠀⠔⣴⣲⣦⠤⠔⠂⠉⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠸⠨⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠒⠒⠚⠉⠁⠀⠀⠀⡔⠂⠒⠈⠲⡀⢈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢧⣇⠤⠠⠄⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠜⠀⢨⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠤⢄⣀⠀⠙⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡼⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠈⠉⢉⣛⠒⠶⠐⠐⠒⠀⠒⠒⠀⠉⠉⠁⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡤⣤⣤⣤⣤⣬⣥⣁⡤⣉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣞⠏⠩⣩⣫⡇⠀⠀⠀⠀⠀⠀⠀⢸⠢⣀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⣠⢀⡄⣀⣠⠀⢠⣀⠄⢘⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠙⠒⠙⠳⡃⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠓⠠⡀⢨⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠿⠛⠇⠿⠻⠤⠼⠻⠄⠀⡇⠀⡇
⠀⠀⠀⠀⠀⢀⠧⠄⣀⣀⠀⢀⡀⠀⠀⠈⠢⣀⠀⠀⣀⣀⠤⢼⠞⠀⠀⢠⠃⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡄⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠁⡰⠁
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢩⠏⠉⠉⠉⠉⠉⠛⢄⠀⠀⢠⡎⡀⠀⢠⠃⠀⠘⠄⠀⠤⠄⠤⠄⠷⠀⠄⠔⠀⠀⠈⠒⠂⠄⠠⠠⠤⠠⠤⠀⠤⠀⠀⠚⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣽⣥⣈⣶⠁⠀⠀⠀⠀⠀⢠⠤⠤⠤⢤⡄⠀⠀⢀⡤⠤⠤⠤⣤⠀⠀⠀⣤⠤⠤⠤⢤
⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⢆⠀⢰⠒⡲⠀⢨⣥⣤⣤⣼⠀⠀⠀⠀⣯⣠⣤⣬⡇⠀⠀⠀⢸⣥⣤⣄⣏
⠀⠀⢰⣅⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡄⠀⠉⠀⠀⢰⣤⠀⢀⢨⡀⠀⠀⠀⣧⡀⠀⠀⡇⠀⠀⠀⣠⣄⢀⠀⢠
⠀⠀⢸⠀⠈⠑⠂⠤⢤⣇⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠤⠐⠊⠁⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠀⠸⠀⠀⠀⠆⠀⠀⠀⠈⡄
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣰⠀⠀⠀⠀⠀⠸⣀⣀⠇⠀⠀⠀⠀⠀⢇⣀⡘⠀⠀⠀⠀⠀⢰⣀⣀
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⢨⢀⠃⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢨⢸⠀⠀⠀⠀⠀⠀⠀⡃⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡧⠤⠧⠤⠤⠤⠤⠤⠤⠴⠶⠤⠤⠤⠤⠠⠤⠼⠾⠤⠤⠤⠤⠤⠤⠤⠶⠤⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠐⡆⠂⠀⠀⠀⠀⠀⠲⠒⠒⠀⠀⠀⠀⠀⠰⢲⠒⠂⠀⠀⠀⠀⠀⠖⠀⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠀⠀⣠⠠⠇⠠⠤⠀⠀⠀⠀⠼⠤⠠⠤⠀⠄⠠⠄⠸⠼⠄⠤⠄⠄⠀⠀⠀⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠀⠠⠤⠤⠤⠤⠄⠀⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⣤⡤⠤⠤⠶⢶⣦⣤⣶⣤⣤⣤⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣶⣯⣽⣽⣃⣀⠀⢀⣀⣉⠶⠖⠒⠈⠷⠿⠾⢭⣤⣞⣻⠿⢷⣶⣦⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⣸⡍⠉⢻⣿⣿⠷⠶⢦⢤⡤⢤⣀⣀⠀⠀⠈⠉⠙⠒⠢⢍⣟⣻⣿⠦⢀⡀⠐⠀⠲⠄⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠎⠯⠀⢸⢯⠀⢠⣴⣖⣶⣿⣿⠋⠙⠉⠺⠶⢦⣀⢀⣀⠄⢘⠩⠐⠒⠛⠀⠠⢄⡀⠀⠀⡀⢭⣽⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢹⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡾⣁⣀⢴⣮⣭⣤⣭⣶⣾⢿⣵⡚⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠄⡐⡬⢖⣦⡀⣼⣬⡀⠈⠁⠀⠀⠀⠀⠀⠀⣴⣿⢯⠡⠀⠸⠿⢧⣿⣿⠈⢿⣿⣿⣿⣿⠃⢸⡿⠃⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣄⢛⣩⣥⠄⠿⢿⠇⠀⠀⠀⠀⠀⢠⡄⠀⠰⣼⣧⠜⠀⢸⠀⠄⠈⠃⠀⠀⢹⢻⣿⡿⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢨⠈⠘⠛⠙⠀⢠⣤⠂⠀⠀⠀⠀⠀⢳⣬⡀⠀⠀⠀⠀⠀⣻⣿⡇⠀⠀⠀⠀⢼⢸⣿⠁⠀⠸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠸⡟⡳⡄⠀⠀⠀⠀⢈⠙⡃⠀⡶⠟⠀⠘⢸⣷⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⢀⣴⡇⠀⣬⠀⠀⠀⠀⠀⠀⠀⢹⡯⠖⠀⠀⠀⠀⢸⣾⠃⠠⢿⢟⠠⠀⢸⡏⠁⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢐⠞⡿⣿⡇⠀⣟⡇⢠⣶⣄⠀⠀⠀⣸⣔⠑⠾⠆⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⢸⡧⠄⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠿⠙⠗⠀⣏⡷⣿⡬⢉⢿⣦⡀⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⣿⠄⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⡀⠀⠀⠀⣧⡇⠈⠑⠿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣵⣀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢲⠦⡄⠸⠀⠀⢾⢿⣦⣄⠰⢸⣏⡀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣄⣀⠀⠀⠀⠀⠀⣆⣻⡄⣼⠀⢠⠀⠀⡴⣿⣄⣠⠞⠀⠀⣿⣗⠂⠀⣿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⣤⢀⣀⠀⠈⠉⠉⠉⠛⠀⡄⡇⠀⠈⠻⠀⠀⠀⣯⣅⠀⠀⡏⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠒⠆⠤⣄⡀⢀⣄⡇⠀⠀⠀⠀⡠⣶⣿⣬⡤⢠⡏⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠢⢤⣠⣎⣀⣿⣿⣿⣷⣼⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⣳⠘⠁⠀⠀⠠⠂⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠑⠀⠀⠀⣷⣶⣾⡓⠀⠀⢡
⠀⠀⡀⠄⠀⠀⠛⢀⡌⠀⡴⢁⠔⠉⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⢂⡀⠀⠴⠖⠈⠈⢹⡿⠏
⠀⠀⠀⠀⣘⣿⣀⠀⠀⠈⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣿⠟⠻⡁⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠈⡇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣐⣿⣧
⡀⠀⢀⣌⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙
⠿⣿⣿⣍⠀⠀⠘
⠀⢨⡿⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠤⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠑⠲⣀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⡂⢇
⠀⣴⣄⢀⡠⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈
⠉⢹⣿⣿⠉⠀⢤⠀⠀⠀⣀⠔⢁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⣶⣤⣴⠯
⣠⡾⠋⠁⠀⠀⠐⠁⢀⡔⠡⠠⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠙⣿⣿⠆
⠁⠀⠀⠀⠀⠂⣺⣶⣾⡋⠁⠉⢖⠒⠂⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠉⠁⠀⢹⡟⠁⠀⠀⡀⠀⢀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣶⣶⡆⢀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⠀⠀⠀⡀⢀⠀⠈⣛⣿⣿⣿⣿⣿⣟⠋⣀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠙⠛⠛⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡹⠉⡿⠛⡿⠿⢿⣿⣶⣶⣶⣾⣿⠿⠿⡟⠛⡏⠹⣁⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠻⢿⣿⣿⠿⠃⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣉⣀⣀⣀⣈⣁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣼⣿⣿⣿⠀⠘⣿⣿⣿⡆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣼⣿⣿⣿⡟⠉⠉⣿⣿⣿⣿⡆⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⣼⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡄⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⣼⣿⣿⣿⣿⣿⡷⣶⡶⣿⣿⣿⣿⣿⣿⡄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠹⣿⣆





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⢀⣀⠀⡀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢆⢸⠤⡃⠿⢼⠀⡇⢱⡜⢠⠿⡆⢸⡀


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠛⠻⠟⠛⠻⠟⠛⠛⠛⠛⠛⠛⠿⠿⠻⠛⠛⠻⠛⠻
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠁
⣿⡄
⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣏⠁⠁⠀⠈⣿⣿⡏⠛⢶⣄⡀
⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹⠋⡀⠀⠀⠀⣀⠀⠀⠀⢬⣿⡟⢶⣄⣀⣀⣤⣄⣀⣀⣀⣀⣀
⣿⡇⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⡼⠀⠀⠀⣀⣦⣠⡢⣷⣧⠀⢵⣾⣁⡠⠤⠴⠖⠒⠒⠒⠀⠀⠈⠉⠒⠢⠄⠀⠀⠠
⣿⠃⠀⠀⣰⠏⢠⢆⠄⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⠇⠛⠗⠀⢙⣻⡷⠒⠒⠒⠒⠒⣒⣂⡠⠔⠛⠛⠉⠉⠁
⣿⡀⠀⢰⠏⠀⠠⣫⠂⠀⠀⠀⠀⠀⠀⠀⠸⠶⠂⠥⠠⣴⣿⣿⢏⣅⣠⠴⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⣾⡄⠀⠀⠁⢀⠀⠀⠀⠀⢀⣠⠖⢀⣀⣄⣀⡴⠉⠉⠁⢞⣭⠇
⣿⡄⢸⢿⡟⠀⠀⣴⣯⡖⠀⠀⠐⠋⠁⠉⢉⣝⣽⠟⡡⡂⠀⣠⣾⣿
⣿⡇⣾⢀⡄⡀⠀⠃⢟⣴⣷⢆⡄⠀⠊⣀⢄⡀⠀⡸⡑⡰⣴⣿⣟⠓
⣿⠀⣿⣼⣸⡟⠀⠀⠘⠹⠋⢸⢳⡇⡄⡀⠀⠠⠇⠀⢱⢰⣿⡟
⣿⠠⠃⠛⢿⡇⡀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣷⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡆⠀⠀⠀⠐⡝⡰⢰⢇⡄⠀⠀⠀⠀⠀⠋⠃⠐⣖⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠇⠀⠀⠀⠀⠈⠠⠃⡜⡼⠀⣷⣦⢰⡀⠀⠀⠘⡎⠈⠀⢷
⣿⠀⣤⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠉⠘⠙⣾⡆⣦⠀⠀⠀⣌⣇
⣯⣾⣿⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠘⠀⢳⣤⣈⠻⣆
⣿⣿⣿⢀⣿⣿⣿⣿⣦⠀⢤⣀⠀⠀⠀⣀⡀⠀⠀⠀⠈⢿⣿⣷⣦⣄
⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣼⣿⣿⣶⣤⣌⣿⣿⣶⣦⣤⣘⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴
⣿⣿⠿⢿⢻⠿⡿⢿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣶⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⠁⣀⡀⡱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢹⡧⣟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣷⠋⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡚⢛⡛⠻⠟⠛⢓⠔⢻⠀⢸⣿⣿⠿⢟⢃⣤⣾⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠰⣇⠎⡠⡀⢃⡄⠀⠈⠀⠈⣃⠈⠅⠀⠀⠠⡏⠁⠀⠀⣤⣴⣶⡶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠀⡀⠐⢻⠀⠏⠦⠬⠤⠿⠧⠶⠉⠁⣦⡏⠦⠤⠦⣠⣤⣥⣤⣄⠐⣨⠒⠠⠦⢠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠑⠀⣠⣶⣿⣿⣿⣷⣦⠀⠀⢀⣀⣀⡷⣧⣀⠔⡿⠚⣿⢄⠀⠊⡗⠐⠒⠀⠠⣼⠐⣅⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⣿⠀⣠⣵⣿⣿⣿⣷⡜⣿⣴⣾⣿⣿⣿⣿⢩⠏⢰⠁⠀⠈⣿⡂⣤⢣⠶⠿⠄⠀⣿⣏⢉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡛⠛⠉⠻⢿⣿⣿⡿⣿⠀⠀⠁⢸⣿⣿⠿⠘⢄⣿⡀⠀⠀⣼⣇⣏⣨⠃⠒⠒⢰⡿⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⠀⠘⠒⠒⠒⠛⠛⠀⠀⠀⠀⠈⠉⠉⠉⢉⣀⣈⠩⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣺⣿⣿⡏⢻⠿⣶⣶⡖⣶⣖⣶⡾⢿⢻⣿⣷⣿⣶⣶⠶⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⡟⢹⠀⠈⠉⢓⢺⡀⠀⢹⣯⣿⣿⣃⡒⠠⢸⡇⠀⠉⠙⢏⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠚⠓⢺⡒⠦⠠⠾⠎⠉⠁⠈⠀⣌⡇⠀⠉⣉⣉⣑⠒⠠⠸⠟⠛⠿⠋⢩⣉⣉⣭⣽⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠇⠀⠉⢘⠛⢛⣀⠀⠀⠄⢀⣀⡈⠓⢛⡿⣿⣿⣿⣿⠛⢷⣶⠛⠻⠿⣿⠿⠟⢻⣷⣾⠧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⡀⠀⠀⢨⡾⠥⣀⣀⠀⠀⢀⣀⣨⡆⣸⣼⣿⣿⣿⣿⣷⠈⣿⡃⠀⠀⠠⠀⣀⣇⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠭⢍⣛⣉⠓⠤⢀⣀⠈⢉⣁⣀⡽⠿⠛⢿⣿⣿⣿⣿⣿⣾⣋⡈⣁⣂⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠲⠶⠶⠶⣦⣤⠦⠤⠉⠑⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡾⠵⠥⠤⠴⢲⡎⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⠿⣤⣤⣤⣤⣤⣀⠢⠭⢛⣩⣿⣷⣦⣄⣀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠀⠐⠒⠒⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡴⣀⣤⣤⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠑⣀⡁⠶⡉⣣⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣟⡄⡸⣾⠀⠀⠀⠐⠠⠀⡐⠂⡂⢢⡀⠀⠀⢀⠰⣄⠩⢤⠘⢣⣄⠹⣆⡙⣥⠚⡾⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⠛⠘⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡈⢤⡈⢤⡀⠻⡤⠑⢢⡈⢵⠆⠓⠀⠈⠈⠀⠸⢡
⠀⠀⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠱⠦⠉⠠⠚⠓⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃
⠀⠀⠀⠀⠛⠃⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢠⡤⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⣀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⡤⠧⠀⢀⡤⠀⠀⠀⠀⠀⠀⡤⢀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⡀⠛⠀⣠⡶⠀⠀⠀⠠⠇⠉⠉⠉⠉⠛⠉⠉⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⢀⡤⠖⠀⠀⣀⣠⢠⣤⣤⠶⠶⠴⠁⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⣀⡀⠰⠁⠀⡀
⠀⠀⠀⠀⠀⢈⠛⠃⠀⠀⠀⠀⢲⣶⣾⡦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠘⠲⠆⠒⠊⠀⠀⠀⠀⠈⠉⠠⠈⠀⠁⠉⡗⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓⠁⣀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠄⢤⠀⠀⠧⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠹⠁⠠⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠈⡦⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠀⠘⠀⠀⠃⠀⠅⠀⠐⠀⠀⠀⠀⡶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣐⢁⠺⠁
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠁⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠀⠐⠀⠀⠀⢄⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⠤⠊
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠠⢤⠄⠈⠀⠀⠀⠈⡀
⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⢆⠴⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠉⠁⠉⠀⠀⠀⠂⠀⠠⠤⠄⠤⠔⠈
⠀⠀⠀⢀⡀⡧⠀⠴⠆⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠓⡼⠆⠀⣀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠐⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠐⠒⢤⡄⠀⠈⠁⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠈
⠀⠀⠀⢀⡀⠀⠀⠟⠀⠀⠴⠀⢁⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣚⠁
⠀⠀⠀⠀⠁⠀⠷⠀⢰⡀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠐⠆⠀⠀⠌⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⡶⢠⡀⢒⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠠⠈⠃⠀⠀⠀⣀⠀⠀⠀⠈⠇⡈⠕⠉⠀⠉⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢁⠀⠃⠠⠄⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⠴⢶⠄⣼⠇⣾⠔⢚⡉⠉⠉⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⡏⠀⣠⡤⠀⣿⠘⠁⣼⡏⠉⢠⣾⠙⡂⢠⡬⠙⠻⢶⣦⣤⣀⣴⡄
⠀⠀⠀⠀⠀⢀⣤⣄⠀⢻⣧⡀⠹⡄⣿⡇⠀⣿⡄⠀⣿⣇⠀⣾⡇⢠⢁⣿⠗⣹⡇⢴⡎⠉⠛⡋⠁
⠀⠀⠀⠀⠀⢸⡀⠹⣷⡀⢻⣷⣄⢱⠻⣷⣠⠛⠧⠔⠈⢙⡋⠈⠓⠃⢾⠏⣰⡟⢡⡿⢡⣶⣟⣉⠇
⠀⠀⠀⠀⠀⠈⠛⠀⠘⣿⣌⡏⠙⠋⠀⠀⢀⣀⡤⠖⠊⠉⠈⠙⠒⠦⢤⣀⡈⠀⠿⠵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢡⣄⠀⢠⣾⠛⢛⡶⣒⠒⣀⡀⠻⠟⢡⣄⢪⣿⠿⡄⠀⠈⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⢠⣿⠀⢸⡆⣿⣇⠀⠀⣾⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⢸⣿⢸⣿⠀⢸⡇⣿⢻⣆⢸⢸⣿⢸⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠸⣿⠀⢸⠇⢸⠀⢻⠇⢸⣿⢸⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣤⣬⣭⣤⣬⣭⣥⣤⣭⣤⣤⣤⣬⣭⣭⣭⣭⣍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣧⣠⣶⡂⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣀⠞⡡⠂⠪⡉⠉⠉⠉⠲⠔⢒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣑⣾⠃⠐⢧⣘⣁⠇⢀⡔⢉⡙⡆⠊⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣀⠀⠀⢘⡶⠦⠈⠳⠬⠔⠁⠀⢸⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⢸⠉⠉⠉⠉⢳⣀⠀⠀⢀⣤⠤⢀⣈⣀⡀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⣀⣀⣀⣤⠊⠉⢀⣀⣈⣧⣄⣀⣀⣀⣈⣉⣋⣙⣮⣑⡾⠁⠀⣨⣦⣤⡄⣀⠀⠈⢧⡀
⠀⠀⠀⡇⠀⣀⠟⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠠⣇⠀⠀⠀⢿⠀⠀⠀⠳⡄
⠀⠀⠀⡇⠀⠓⢾⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⣠⠗⠋⠀⠀⠀⢸⠀⠀⠀⠀⢳
⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⡀⠀⢸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⡇⠈⡇⢰⣇⢠⠆⡷⠒⠀⢸⠀⠀⢰⠋⠁⠠⣎⡳⢰⠛⠞⣇⢸⠅⠀⠀⢸⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡇⠀⠙⠏⠘⠚⠀⠙⠒⠀⠘⠉⠀⠈⠓⠈⠀⠈⠀⠈⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⠙⠺⠒⠒⠀⠀⣰⠃
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣤⠤⠴⠚⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠟⠉⠉⠉⠉⠉⠻⣿⠛⠛⠛⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⣶⡆⢰⠀⢠⡆⣠⠶⠄⣴⣀⣦⠶⡶⠆⠀⢠⣦⠀⣶⣶⢠⣶⣦⢠⡄⣦⢠⠄⣴⡄⢰⡄⠀⣴⡦⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠟⠁⠸⠶⠖⠇⠻⠾⠇⠻⠉⠟⠀⠇⠀⠀⠟⠛⠇⠏⠻⠘⠏⠻⠘⠇⠙⠟⠸⠛⠻⠸⠷⠖⠾⠟⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿

⠀⠀⠀⠀⠀⠀⣤⠄⣀⣀⣀⣀⡀⣀⠀⣠⠄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠛⠀⠳⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⢸⠀⣿⢦⡾⣇⠀⣧⢀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆⢹⠀⠀⢻⡆⣿⠉
⠀⠀⠀⠀⠀⠐⠛⠗⠀⠀⠉⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃⠘⠃⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠕⠈⠉⠉⠓⢌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡤⠤⡤⠤⡮⠀⠀⠀⠠⠀⠬⣮⣇⣤⡤⢄⡠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⠔⠁⢀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢮⠥⡽⠀⠀⠀⠀⠀⠀⠀⠀⠮⡅⡽⠂⠈⠢⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⣄⢾⣼⡷⠀⠀⠀⠀⠀⠀⠀⠰⣞⣐⡳⠀⠠⠔⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⡎⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠀⡰⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠛⠛⠒⠒⠒⠂⠀⠀⠀⠐⠓⠓⠒⠓⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⣿⠙⢱⡎⢹⡆⠀⠀⠀⠀⠐⠒⡶⠒⠂⢶⢲⠀⠀⢀⣶⡀⢰⡀⢰⡇⢺⠉⠀⣴
⠀⠀⠀⠀⠀⠀⢿⠀⠸⣇⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⣄⠀⢼⠗⢷⠀⢳⣼⠁⢸⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠈⠀⠛⠀⠈⠀⠀⠋⠀⠘⠋⠁⠉⠛⠁

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⡄⠀⣶⡀⠀⢠⡤⢀⣤⣶⣦⡀⠀⣴⣤⡆⢀⣴⣶⣤⡀⠀⣠⣶⣦⣤⡄⠀⣠⡶⣶⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⣸⣿⡀⠸⣷⣀⣿⠃⣼⣯⣤⣼⡿⠀⣿⡋⠀⢈⣥⡤⣿⡇⢸⣿⠀⠀⣿⡇⢸⣿⣤⣬⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠏⠙⠋⠻⣷⠀⠹⡿⠇⠀⠙⢷⣤⡶⠆⠀⢿⠀⠀⠻⣧⡴⢾⡇⠈⠿⣶⠶⣿⡇⠈⠿⣦⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣴⡿⠁
⠆
⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢸⣿⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⣿⡿⠛⢿⣶⠛⢿⣆⠀⣾⡆⢸⣿⠀⣿⡃⣴⡿⠃⠀⠀⢀⣶⠟⠛⣧⡀⢸⣿⠛⢿⣦⠀⣾⡆⢠⣾⠟⠻⣦⡀⢿⡄⠀⣼⠇⣴⡟⠛⣷⣄⠰⣿
⣿⡇⠀⢸⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡿⢿⣆⠀⠀⠀⢸⣿⠛⠛⣻⡁⢸⣇⠀⢸⣿⠀⣿⡇⢸⣇⠀⢀⣿⠇⠘⣷⣼⡟⠀⣿⡟⠛⣻⡋⢠⣿
⠟⠃⠀⠘⠛⠀⠘⠛⠀⠛⠃⠘⠛⠀⠛⠁⠀⠛⠃⠀⠀⠀⠙⠛⠛⠋⠀⠘⠋⠀⠈⠛⠀⣿⡇⠀⠛⠛⠛⠉⠀⠀⢸⡿⠀⠀⠈⠛⠛⠛⠁⠈⠛
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣴⣶⣶⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⣿⣯⣿⡿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⡏⠁⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣶⣶⣶⡆⢠⣤⣀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢻⣇
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠻⢿⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣀⣠⣴⣶⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⡿⠁⠀⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢧⣼⣿⡷⠶⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣽⣿⣿⠟⠙⢿⣿⣷⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⡿⣟⣀⡉⣠⡌⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠻⣿⣿⠁⠗⠀⠀⠀⠀⠀⢀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠛⠁⠛⢛⣿⡶⢻⡟⢀⣿⣦⣄⣀⣠⣶⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⢨⠉⣁⣾⣿⠿⠟⠛⠛⠛⠛⠻⠛⠛⠛⠻⠛⠻⠟⠛⠛⠒⠒⠖⠒⠒⠶⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⡀⠀⢈⣨⡻⣿⣿⠿⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⢀⡎⢠⣾⣿⣿⣦⣰⣶⣤⠛⢷⠄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢿⣿⣿⣿⣠⡟⢠⣿⣿⣿⣷⣄⠈⠉⠁⡘⠷⣤⠀⠀⠀⠀⠀⠀⠀⡿⢋⡈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠛⠁⢸⣿⣿⣿⣿⡇⣼⣿⣿⣿⣏⡉⠛⠀⠘⠛⠿⣆⡀⠀⠀⠀⠀⠀⣾⠄⠸⡇⠀⠈⠛
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠀⠀⠀⠀⢸⡿⠿⣿⣿⣷⣿⣿⣿⣟⡉⠑⠊⣁⠶⢷⣶⣌⡛⠀⠀⠀⠀⠀⠈⠀⠀⢰⡀
⠀⠀⠀⠀⢀⣠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣯⣉⠙⠛⠣⠀⠀⠶⣦⣄⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⢠⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠿⣽⣿⣿⣿⣯⡈⠑⠒⢀⡀⠉⠐⢈⡉⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢛⣭⣿⣿⣿⣿⣿⣶⣄⡀⠀⠙⠄⢀⣶⡦⠈⢿⠇⠀⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⠘⢿⣿⡶⠂⠀⠀⠀⠀⠀⠀⠀⠉⠙⣏⣿⣿⠿⣿⣿⣏⣽⠓⣂⡀⠛⠁⢄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⡿
⡆⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠶⢺⣿⣿⣿⣁⣤⣈⡹⢿⣟⡋⠀⠀⠠⠀⠠⠄⠀⠠⣤⣶⡤
⠁⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣟⠛⢿⡷⠞⠋⠙⠒⠀⠀⠀⠐⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢿⡁⠆⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⠀⠀⠀⠀⠀⠾⠟⣩⣹⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⡀⠀⣄⢠⡀⣤⠀⣄⠀⡤⢢⡀⣠⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢠⣶⣿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠐⠀⠘⢇⣀⡿⠸⠀⣿⠀⢸⡟⣀⣸⠇⢹⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⢡⡄⠀⠀⢀⣁⣸⣿⣿⣷⣅⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠿⠖⠛⠋⠃⠐⠀⠉⠀⠈⠀⠉⠉⠀⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⠋⠙⢿⣻⡿⠋⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⠀⠀⠀⢀⣿⠃⠀⠀⠈⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣿⣿⣿⣿⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠆⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⣿⣿⣿⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⣾⣿⣿⡏⠙⣿⣿⡀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⢶⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣤⣿⣿⣦⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⡟⢹⠏⣉⣥⠈⠉⠉⣤⠈⠀⣰⡦⠀⠀⣴⡄⠀⠀⣴⠆⠈⢿⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣩⣿⣿⠿⠚⢩⣿⣄⣀⢼⣿⡄⣤⠿⠓⠠⣤⣿⣧⣤⢼⣿⣦⣤⡄⢨⣷⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠸⠟⠈⠁⠀⠀⠘⠿⢋⣿⣌⠋⣡⣷⡄⠀⠀⣿⠉⢡⣿⡆⠉⣵⣿⡅⠙⠛⣿⡿⠏⢉⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠙⢛⣤⡆⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⢙⣛⣃⣬⣿⣯⣤⣬⣽⣇⣀⠙⠛⠀⠀⣾⣿⣾⣞⢙⣿⣟⣿⣿⣄
⠀⠀⠀⠀⠀⣴⡿⢋⣴⢟⣵⣶⡾⠇⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⡿⢿⡿⠿⠿⠿⠟⠻⠿⠿⣿⣿⣶⣾⣭⠿⢿⣿⣿⣻⣿⣧⣝⣿⣷⣤
⠀⠀⠀⣠⣿⣛⣴⣿⣻⡏⣿⣿⡆⢠⡄⠀⣠⣾⣿⣿⣿⣛⣋⣉⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣷⡌⠋⣨⣿⣿⣽⣿⡷⠹⣿⣷⣆
⠀⢀⣼⡿⢻⣎⢹⣿⣿⣧⣼⣿⡿⡿⠋⣼⣿⣯⣿⢿⣿⣿⣿⣿⣿⣿⣭⣿⣻⣶⣛⣓⣶⣛⣟⣻⣿⣿⣟⣻⣿⣆⠈⠛⠟⣋⣴⣿⢾⣇⣽⣿⣿
⠀⣿⣿⣦⣸⡾⢾⣯⣴⣾⣻⡍⣾⢶⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣋⣭⣽⣿⣿⣿⡇⠀⣴⣟⣿⣿⣷⡆⣹⣯⣿⣏
⠀⣿⡿⣷⣝⣦⣴⣽⣿⣾⡿⠀⣿⣤⣶⣾⠿⠿⠿⠿⠿⠿⠿⠿⢛⣟⠛⠛⢛⣛⡛⠛⢓⣛⠛⠛⢛⣙⡛⠛⠛⠁⣴⣿⡿⣻⣵⣾⡿⣿⢏⣿⡗
⠀⣿⠻⣿⣟⣷⣷⣌⠴⣟⢵⡪⣥⠿⢋⣐⢿⡿⠀⣀⠀⣾⣿⠆⠻⠿⠟⣀⠛⠿⠟⠀⣻⣷⡿⢀⣹⠿⠟⣠⣤⡚⠛⠉⣾⡿⢛⣼⣽⣧⣿⣿
⠀⣿⣷⣜⢿⣧⠙⢮⣳⡔⠙⣤⣖⠀⢻⣿⠷⠀⠿⣿⡗⢀⠀⠷⡾⠇⢾⣿⡿⠀⠸⣿⣿⣀⡐⠿⠿⣇⠸⡿⠿⢃⣼⠿⣭⢦⣾⣿⣃⣿⣿⣿
⠀⢿⣿⣬⢷⣜⢷⣄⠻⣿⢶⡉⢹⢟⡾⠃⠰⣿⡾⠂⢀⣿⡿⠇⠰⣬⣿⠄⣶⣿⡷⠀⠻⠷⢿⣀⡀⠿⠷⠋⣠⣼⣧⠶⢫⣫⣿⠻⣻⣿⡿⠃
⠀⠀⣿⣿⣶⣿⣢⡙⢦⡙⢷⣝⣄⠁⠚⢁⣀⣤⡤⠞⠿⠿⠃⣴⣶⣮⠀⣤⣴⣦⠀⣶⣶⡄⠻⠷⠟⣠⣴⣿⡿⢿⣿⣤⣷⣏⣿⣾⣿⣿⡇
⠀⠀⣿⣿⣿⣽⢏⡝⢦⡹⣎⡻⣿⡳⣶⠿⠛⣷⠰⣶⣤⡶⠞⠉⠉⠀⠀⢈⣉⠁⠀⡈⠉⠁⣠⡴⣿⢨⣹⡎⠙⠛⣻⣽⡿⢣⡌⣽⣿⣿⡇
⠀⠀⣼⣿⣿⣿⣦⡵⠄⠑⣌⠳⣤⢛⣦⡂⠤⠽⠧⢤⠤⠤⢤⠤⠤⠴⡦⠚⢻⠷⠶⣆⣶⣚⣛⠓⡚⣒⣛⣓⣒⢒⣿⣿⣻⡛⣀⣿⣷⣿⡇
⠀⠀⣿⣿⣿⡿⣿⣿⣦⡀⠈⠳⣜⢿⣿⡿⠭⠿⠭⠿⠽⠧⠼⠿⠿⠿⠭⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠞⢿⣿⣿⡿⣥⡟⣼⣿⣿⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣟⢳⣦⡒⢘⣷⡙⠦⠤⠤⠦⠤⢦⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣿⣟⢿⣿⢯⣿⣟⣿⣿
⠀⠀⢸⣿⣿⣿⢿⡿⣾⣾⣽⣿⣌⣷⣽⣶⣶⡶⢶⢲⣶⡒⠶⣆⠐⢒⣒⣶⣶⡶⠒⠒⠓⠲⠒⠒⣲⠖⠚⠓⠺⣧⢿⣿⣡⣿⣿⢿⣿⣿⣄
⠀⢀⣼⣿⣿⣾⣿⣽⢿⣿⣿⣻⣽⣿⣶⡟⠲⡝⠀⠠⣶⠏⠠⠴⠆⠀⠀⠄⠚⠃⡀⠛⠷⠐⠇⢨⠉⠀⠀⠺⠠⣾⡟⣤⣿⠷⣷⣿⠿⣏⣿
⠀⢸⣿⣝⠻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠟⣲⣻⣿⣖⣚⣚⣛⡒⢒⡙⢲⠐⢺⡚⣿⠁⣩⠱⡝⠿⣿⢿⣿⡍⢉⣻⣾⡿⢧⣾⣿⠋⡈⢹⣿
⠀⢸⣿⣯⣿⣦⡙⢿⣿⣿⣿⣼⡿⣿⣿⣦⣿⣧⣽⣯⣽⣿⣿⣥⣭⣇⣬⣽⣿⢿⣭⣥⣯⡄⢽⣿⣿⣼⣍⣽⠏⣹⣿⣴⣿⠋⢁⠐⢇⣼⣿⡁
⠀⢸⣿⣿⣽⣻⣟⡶⢌⠻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⢿⣿⢷⠿⣷⠾⠷⣻⣿⣟⣿⣾⣟⣿⣿⣿⣿⣿⣿⣿⡿⣋⣾⠟⡉⣠⣤⣤⣟⡽⢿⡅
⠀⣾⣿⡿⢿⣿⣿⣝⠺⣧⣤⣙⢿⣿⣿⠟⣿⣿⣿⣿⣛⣿⣛⣿⣿⣞⣚⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣯⣿⣯⣵⡿⢡⣾⢉⡡⢞⣿⠏⢀⣾⡇
⠀⣾⣿⣿⣶⣝⠿⣿⣷⣯⣿⣿⣿⣍⣻⡆⢛⣻⣟⠻⠟⠛⣿⣿⡟⢿⡏⠻⠉⢭⠙⣻⠿⢟⣉⠩⢉⡟⠛⠋⣼⣦⣿⣵⠟⣷⠟⢁⣵⣿⡟
⠀⠈⠙⠿⣿⣿⣷⣮⣻⣿⣿⣿⣷⣽⣿⡿⢷⠴⠿⣶⣿⠟⠁⣏⣱⠿⣶⡿⢠⣴⡛⠟⣶⢋⡙⠠⢟⣈⠙⠆⣿⣿⣿⣿⠟⣁⣴⣿⡿⠋
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣝⠻⣿⣿⣿⣧⣾⣶⣶⣦⣬⣿⣤⣭⣵⣶⣴⣿⣤⣤⣵⣦⣬⣼⣿⣷⣾⣿⣶⣶⣿⣯⣾⣿⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⣯⡿⣿⣯⣽⣿⣭⣭⣭⣭⣭⣯⣭⣭⣭⣭⣭⣭⣿⣭⣭⣭⣭⣭⣭⣽⣽⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣾⣟⣿⣯⣉⣫⣼⣧⣤⡤⢀⣥⣬⣄⣀⡉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀⡀⠈⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠈⡃⣴⠃⣦⠀⣿⠙⡇⢠⡖⠶⢤⣖⣰⡄⣴⠐⠂⣿⠃⢸⡇⢰⡶⢲⡆⢠⣖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠾⠋⠻⠶⠟⠰⠿⠾⠿⠘⠷⠴⠘⠿⠴⠇⠿⠆⠀⠿⠾⠾⠧⠼⡷⢼⡧⠸⡯⢿⠗⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠳⣆⢤⣠⠤⢔⢂⣰⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠠⠀⡀⢀⠠⣄⣠⣤⣿⡚⡟⢁⡰⡎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡡⠒⠉⢈⠹⠎⠉⠀⠀⠙⢾⣥⣹⡀
⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢢⣴⠒⠒⠿⠋⠑⠄⠂⠙⣿⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤⠄⢢⡀⠇⠀⡰⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠉⡆
⠀⠀⠀⡀⢠⠔⠚⠒⠒⠒⠒⠻⠟⠀⠘⠷⠯⠤⢒⠷⢾⣶⣦⠢⣑⡀⠀⠴⠀⠐⠄⠀⠀⠈⣴⡠
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠄⠀⢈⠏⠀⠀⠀⠀⠀⠐⣄⠀⡔⠜⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢸⡄⠀⠀⠐⠒⠖⠙⠐⡥⠤⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢸⠄⠀⠀⠀⠀⠀⠀⢆⠸⡐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠸⠀⡀⢀⢀⠀⢀⠠⢸⢐⠌
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⣷⠶⠤⠤⠬⠠⠄⣊⡜⠋
⠀⠀⠀⠿⠦⢤⠤⣄⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⢀⢒⠂⢄⣔⣵⣷⣭⠨⠤⠴⠚⠉⠀⣀⣠⡠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠙⠃⠒⠒⠻⠚⠛⠋⠋⠁⠀⠠⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠈⠈⠉

⠀⠀⠀⢀⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⢀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⢠⡟
⠀⡦⠶⠾⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡘⠁⠀⠀⢠⣌⠆⠀⠁⠀⠀⠀⠖⠈⠙⠉⠉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⢠⡈⠀⣀⠁⠀⠀⠀⠀⠈⠂⠀⠈⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⠀⠀⠀⠀⠀⠀⠄⠃

⠀⠀⠀⠀⠠
⠀⠀⠀⠈⢀⠀⠀⠀⠀⢰⠉⠁⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠤⠖⣆⠀⠀⠀⠑⠀⢂⠐⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⠄⠀⠀⠀⠀⠈⠄⠀⠠⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠆⠈⣁⣈⡉⠂⠀⠀⠐⣶⡀⢱⣮⠄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠐⠈⠠⢻⠍⠀⠀⠀⠀⠂⠀⠨⠙⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣊⣄⡤⠄
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣸⠀⠀⠀⠀⠀⠀⠠⢀⢀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⠀⢀⡴⠆⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠊⢁⠀⡀⠁⡄
⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠉⠰

⠀⠀⢀⢀⡀
⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶
⠀⢐⠀⢫⠀⢀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠛⠁⠀⠀⢠⠀⠀⠀⢠⡀
⠀⠸⠀⠸⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⣀⠀⠀⠶⠞
⠀⠀⠄⠘⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⢤

⠀⠀⠀⢨⠉⣙⠉⠉⢭⠉⠉⠉⠉⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠇⠈⢷⣀⠉⡿⢁⢀⠄
⠈⠀⠀⢼⣀⠛⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣿⠀⢐⠾⠀⣀
⠀⠈⢦⣄⠉⠶⠄⢢⣹⡉⣦⡔⠁⣼⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⢁⣉⠂⡽⢃
⠀⠶⠀⠙⢷⡄⠘⣪⠀⠇⠨⠀⠀⣿⠀⠀⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡤⠀⠉⠀⢀⠈⠸⠅⣢
⠀⠰⢆⠀⠀⠹⡆⠁⠈⠀⠄⠀⠀⠈⠃⠠⢻⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡘⣭⠢⡷⡸⠃⡠⠀⠀⠄⠀⠀⠋⢀
⠀⠂⠶⢠⠶⡆⠀⠀⢀⢀⠀⡀⠃⠰⢯⢃⠯⠀⡁⣀⠓⢰⠀⠷⡀⠀⠀⠀⠀⠋⠀⠄⠠⢇⢏⠀⠙⣦⣀⠀⠈⠀⠀⢀⠀⢸⡿
⠈⠓⢦⣄⣀⠀⡆⠀⠀⠀⠈⠠⠟⠁⠲⡍⠀⠠⠋⠁⠀⣿⠀⠀⠁⠀⠀⣄⠈⢳⡄⠀⢀⠀⢹⡀⢠⠇⠀⣠⣤⠀⠡⣤⡆⠀⢀
⠑⢂⡄⢈⠙⠛⠛⢄⠀⢀⡀⠓⠲⢆⠀⢡⠀⠀⢀⠄⠐⠙⡷⠀⠁⠀⠈⠻⣶⣈⡧⢀⡈⠑⣠⠷⢂⡤⠾⠛⠀⠃⢐⠈⢿⠀⠈
⠀⠀⠀⢹⠐⠀⢋⢊⡤⠶⢛⡢⠤⠤⢴⠋⢺⣖⣡⠀⠀⠻⣧⠀⠀⡄⠀⢠⡿⠟⠃⠉⠉⠉⣩⠤⣾⠓⠦⣤⡴⠆⠶⠾⡲⢀
⠀⠀⠀⢸⠀⠀⠞⠃⣤⣶⠇⢀⣠⡄⠊⠗⢛⠉⠇⠀⠀⢀⠘⠂⢀⡀⠀⠊⣀⠀⠀⢈⣋⡀⠀⢰⠀⢡⣄⠀⢡⣤⠀⡈⣤⡉⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠄⠔⠋⠉⠁⠀⠸⠀⠘⢧⡀⠐⠀⠀⠀⠀⢄⠀⠀⢀⡀⣀⣀⣈⡁⠁⠀⠞⠂⠀⣙⠣⣌⢢⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠚⠾⠂⠀⠐⠀⢠⣤⡇⠀⠀⠈⠀⠀⠀⠀⠄⠰⠀⠉⠉⠙⠙⠻⠶⣄⠈⠃⠘⠋⠉⠉⠓⠇⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠠⠤⠴⠛⠛⠉⠒⢀⡀⢠⠂⠀⠁⠀⠈⡄⠀⡀⢀⡀⠀⠀⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀⠀⠤⠀⠴⡃⠋⠦⡀⠀⡀⢤⠉⠏⠈⢢⠘⡇⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⣟⠀⠈⠀⠘⢧⣸⠀⠘⠒⡠⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢰⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⣽⠏⣰⡗⠀⣼⢃⡄⢀⣾⠁⣠⡀⢀⣄⡀⢀⠀⣄⠀⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠋⠀⢰⡏⢰⣿⢁⡼⢱⣟⠼⣾⠃⣾⠋⢳⡟⢠⢧⡿⢡⡿⢰⡏⣼⢏⠟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠸⠷⠋⠀⠛⠀⠘⠛⠘⠇⠀⠛⠐⠋⠘⠁⠘⠀⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⢀⣀⡀⢀⠀⠀⡀⠀⠀⡀⣀⡀⠀⣀⡀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⡄⢀⠀⢀⡀⢀⢀⡀⠀⡀⣠⡄⡀⢀⠀⣠⡀⠀⠀⢠⠀⠀⠀⣀⢀⠀⣠⣀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢸⠀⠀⠀⠀⠀⠄⠠⠤⠀⠄⠀⠤⠄⠀⠤⠤⠀⠠⠤⠀⠤⠄⠤⠀⠀⠠⠤⠤⠀⠤⠄⠤⠀⠤⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠀⠀⠐⠀⠒⠀⠐⠂⠐⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠄⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠠⠀⠀⣠⣤⣄⡀⢠⠔⠆⠀⣀⣀⠰⣇⠆⢰⠀⣦⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢰⣛⣯⣍⢹⣿⠷⠤⠞⠿⠷⠥⠤⠇⠀⠘⠶⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣰⠛⠂⢀⣄⣜⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠎⠛⠛⠛⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⢀⡀⡀⠀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡄⠀⠀⠀⣧⣉⣭⣿⣟⣿⣉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⠉⣉⡉⣙⣻⠀⠀⡁⠀⠀⠀⠀⣁⠀⠀⠀⢰⣶⣶⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⠶⠓⠀⢰⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡖⣴⠾⣾⠏⣽⠐⣮⡀⣞⠚⠛⣧⣟⣶⡄⠀⠎⠉⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣇⣴⣖⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠚⣿⣇⣼⣄⠁⠈⠅⢈⡀⣁⠀⢄⡀⢀⣤⣤⣤⣤⣤⣤⣴⣧⣿⠷⣶⠀⠀⠀⣤⣀⣼⣯⣈⣉⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⡘⣿⣿⣿⣿⣿⣾⣿⣿⣤⣼⣿⣿⣿⣽⣽⣿⣿⣷⣤⣤⣄⣀⣠⣾⣥⣤⣿⣿⣭⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠀⠀⠀⠀⠀⡄⠀⢂⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠛⠀⢈⠉⠁⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⡇⠀⣀⣠⠀⠑⠀⠂⠀⠰⠂⠀⡀⣉⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠉⠁⠈⠀⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠠⠤⠀⠀⢠⣄⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⣶⣶⠶⢶⣶⡶⠶⡶⢶⢲⣶⣦⣶⢶⣶⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢶⡆
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣤⣼⣿⣿⠀⠀⠀⠀⠛⠳⠄⠘⣀⣤⡀⠀⠈⠀⠁⠀⠁⠤⣿⠀⠀⠀⠀⠀⢀⡛⣿⣿⣿⠃⢀⢠⣴⣶⣦⣀⢲⡇
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⣴⠾⣿⣿⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣼⣿⡿⣿⣷⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⠏⠀⠠⠋⠀⣄⢘⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣾⣿⠃⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⢀⡿⠟⠀⠢⠹⣿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠘⠟⡋⠈⣶⡏⠀⠀⠀⣼⡿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡖⠲⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣤⣿⡄⠀⠀⢰⣿⣰⡦⠀⠀⢰⣾⣧⠀⠀⠀⠀⡄⢢⣩⠟⠛⠛⢻⣣⣤⣤⠾⠛⠾⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣩⡄⢾⣧⡀⠉⣿⣿⣿⣿⣿⠀⠀⠀⠀⣤⣴⣿⡏⠀⠀⢸⣿⣿⡇⠀⠀⣼⣿⣿⠀⠀⠀⢰⣷⣶⣯⢀⣢⣀⣰⣿⠷⠁⠀⢐⣶⣿⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣾⣿⣷⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣾⣿⣿⡿⣤⣼⣿⣿⣿⠀⠀⠀⢸⣿⣿⣋⣲⣿⣿⣉⣠⣤⣐⡉⣼⣿⣿⡇
⠀⠀⠀⠀⠛⠛⠛⠛⠻⠟⠻⠛⠛⠟⠛⠻⠿⠿⠛⠀⠀⠀⠀⠿⠿⠿⠿⠀⠾⠿⠿⠿⠷⠤⠽⠿⠿⠟⠀⠀⠀⠸⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠿⠉⠉⠙⢹⢿⢿⠻⠟⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠛⠋⠁⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⣦⣤⣄⣾⣄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⣀⣀⣄⣸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠘⠋⠙⣿⣿⠻⠇⠀⠀⠀⢸⣿⣿⣿⡯⠌⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣠⣄⣠⠉⠅⠀⠀⠀⠀⠀⢸⡟⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠖⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⡽⢁⣾⡿⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⡝⠄⠈⡄⠀⠀⠀⠀⠇⢀⠨⠀⠁⡘⠀⠀⠀⠠⠃⠀⡠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠜⢄⠘⢌⡄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⡔⡡⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠑⢄⠈⠒⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⢖⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⠢⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠈⠁⠀⠓⢁⠔⡊⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠆⠜⠊⡠⠐⠫⠃
⠀⠀⠀⠀⠀⠓⢄⡀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⡀⠀⢀⣀⣀⣀⣤⣾⠆⠀⠀⠀⠀⠀⠈⢀⠠⠔⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⢒⣦⠙⠖⣫⡾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠔⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠈⠑⡀⠀⠀⠈⣅⢀⣉⡇⠀⡙⢇⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⡄⠀⠀⠀⢀⢀⠕⠀⠀⠀⠈⠛⠁⢁⠈⠀⠈⠢⣜⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢟⣏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣳⣤⠀⡀⠉⠻⣷⣄⣀⠀⠀⠀⠀⢴⣶⡿⢟⣻⠟⠹⢻⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡀⣿⣶⠠⡈⠻⡛⠋⠀⡀⣾⠼⠶⣚⡭⠋⣀⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⡐⡆⠹⣷⣌⠂⠈⢀⡌⠷⠶⢚⡻⢋⡴⠟⣋⣵⢾⡗⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠳⣕⣀⠈⠻⣷⣤⣌⠿⣛⠿⢋⣴⡵⠾⣛⡭⢶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⢈⢶⣀⣘⡟⢸⠟⣡⣚⣩⣥⠾⢛⣩⠶⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⡀⠀⠀⠀⠀⣶⣦⠩⣻⠉⠠⠾⠿⠛⣉⣤⠾⣛⣵⢞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠁⢀⠀⠀⠉⡈⠂⠘⠒⢣⡴⠾⢛⠋⢰⡿⣋⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⡀⠀⠀⠐⠠⠀⠈⠡⠀⠀⠀⠀⠐⢒⡵⢺⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⣧⢔⣫⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡼⠋⠋⠁⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⡜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣆⠀⢸⠁⠘⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⣨⠇⠀⠄

⠀⠀⠀⠀⠀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡤⠴⠖⠒⠦⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣁⣠⠤⠦⠤⣤⣀⣙⡆⠪⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⢔⢄⣠⣤⣴⡶⡟⣷⠾⣛⣽⡿⠶⢖⣿⣷⠲⠶⢶⣶⣾⣿⣬⡶⠮⣁⣾⣶⠶⠶⠴⠶⡶⢦⡀⢠
⠀⠀⠀⠀⠀⡁⣿⣀⠀⡠⠜⠉⠀⢸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢰⠏⠁⢂⠀⠈⠉⢐⣐⠀⣷⢀
⠀⠀⠀⠀⠀⠀⣗⢠⠋⠀⠀⢐⣀⣘⡃⠀⠀⠡⣈⠁⠀⠀⠀⠀⠀⠀⠀⣂⣀⣘⡇⠀⠀⠑⢄⠊⠈⣿⠐⡏⢸
⠀⠀⠀⠀⠀⠃⣼⠁⠀⠀⠀⠈⠀⢸⡇⠀⠀⠂⠀⠀⢊⠀⠀⠀⠀⠀⠠⠁⠀⡸⠄⠀⠀⠀⠀⣑⠰⠹⠀⣿⢸
⠀⠀⠀⠀⠀⠄⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢱⠈⠀⠀⠀⠀⠀⠼⠄⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡆⡇⠀⠈⠶⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⢑⡀⢀⠀⢤⠂⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡂⢻⠄⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠂⠀⠈⠑⠲⠆⠀⠀⢀⡇⢸
⠀⠀⠀⠀⠀⡃⣧⠀⠢⡀⠐⠃⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⢀⠀⠄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠈⣷⢸
⠀⠀⠀⠀⠀⠃⢿⣐⣡⣈⣦⣤⣄⣀⣧⣀⡀⠀⠀⠀⠀⠀⠚⠃⠈⠁⠀⢸⠀⠀⠆⠀⠀⠀⠀⢀⠀⣹⣤⡟⢸
⠀⠀⠀⠀⠀⠠⠲⠠⠄⠘⠐⠀⠉⠘⠅⠐⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠺⠦⠴⠗⠒⠶⣤⣤⣥⡼⠋⠁⠖⢸
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠀⠒⠒

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⢴⣾⣿⣿⡄⠈⠂⠙⠛⠀⠀⠐⠀⠀⠀⠀⠎⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣐⣀⣀⠀⠀⠀⠀⠀⠀⠋⡿⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣼
⠀⡠⠒⠃⠀⠒⢀⠀⠀⠀⠀⠀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⢀⣾⣟
⠞⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⠀⠉⠉⠤⠤⠤⠄⢀⡴⠒⠒⠒⠠⡀⠀⠀⠈⠃⡀⠀⠀⠀⠄⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣀⣶⣿⡟⢉
⠂⠒⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⣄⡀⢈⢢⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣷⣿⣿⣷⣾⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠐⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢯⣿⣿⢿⣿⣿⣯
⠷⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠀⡠⠒⠀⠀⠐⢉⣿⡿⠄⠉⠏⠛
⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢵⣤⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⠇
⠀⠀⠀⠀⠈⡟⠒⠠⠀⠀⠀⠣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠉⠉⠙⠧⠀⢀⠀⠀⠀⠒⠤⢤⣿⣿⣿⡇⠀⣿⣇⣠⣴⢅⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠠⣤⡼⣿⣿⣿⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⢈⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠠⠀⠀⠀⠑⠀⢀⡎⠀⠄⠺⠃
⠀⠚⠀⢤⡀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢔⣾⠿⠛⢿⡇⠐⠦⠀⠀⢀⣀⡀⠀⠀⠀⠃⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠈⠐⠛⢧⡀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠄⣀⣀⣀⡀⠤⢰⠞⠋⡠⠀⡀⠀⠙⣄⠀⠀⠀⠀⠀⠉⠁⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠸⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠖⠒⠻⣷⢶⣷⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠔⠋⠀⠀⠀⠀⠀⠀⠀⠂⠣⣄⣀⡄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠀⠀⠁⠉⠉⠉⠚⢆⠀⠀⠀⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠘⠈⠁⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⢸⣇⣸⣀⣠⣾⠿
⠀⠀⠀⢀⡇⠉⠉⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢱⣄⣆⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⣽⣧⣾⣿⣿⡿⠿⠃
⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢿⣿⣆⡀⠀⡜⠁⠀⠀⢀⣴⣞⡥⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⡏⡁⠀⠀⠀⠐
⠀⠀⠀⡀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⢤⣀⠀⠀⠀⠀⢻⣿⣷⣀⣠⡄⠀⣤⣿⣿⠟⢀⡀⠀⠀⠀⠀⠢⡄⢀⠍⠀⠂⠉⠷⣁⠂⠤
⠒⠀⡉⠁⠀⠈⠀⠀⠐⠄⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠈⠙⢿⣿⡛⠃⢰⣿⣿⣩⠆⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠓⠒
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣶⣶⣉⡄⠓⢦⣹⣿⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠒⠋⠉⠉⠈⠞⠛⠙⠋⢻⡿⢿⠅⣠⣆⠋⣼⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣶⣶⣶⣿⣷⣶⣿⣧⣴⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠋⠀⠈⠉⠁⠀⠉⠁⠉⠉⠈⠋⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠸⡿⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡷⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⣦⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠛⢿⡟⠁⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠰⣄⠛⠀⠀⠀⣼⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠸


⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⣤⣀⠀⠈⠀⠒⠀⣀⡘⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠤⠦⠀⠆⠀⠀⠀⠀⠀⠀⠘⠂
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⡔⠀⠀⠀⣠⠄⠀⠠⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠰⠂
⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡁⡀⠶⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣦⣼⣿⣿⡻⠦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⠻⣿⠛⠛⠉⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⢠⠀⠀⠀⠀⠠⠄⠀⠐⠒⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣶⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠿⠛⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣠⣀⣠⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⠛⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣦⣿⣯⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠋⠉⠈⠙⣿⣷⠛⢹⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣧⠸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⢻⣿⣿⣿⡃⣀⣀⣀⡀⢸⣿⡁⠀⠾⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⣿⠇⠀⢾⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀⣀⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⡃⡹⠀⣿⠀⠀⠀⡏⢠⠀⡟⠉⢹⢹⡟⣻⣿⣿⣶⣶⣦⣄⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⠇⢻⡇⡏⠇⢸⢀⠁⣸⢀⡇⠀⠘⠈⢉⢻⣿⡟⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⠇⣷⠀⠈⢸⠀⡇⢸⠀⡇⠀⣇⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣀⣿⢰⡇⡘⢀⡇⣾⢸⠇⣶⢻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣻⣬⣻⣿⣸⣇⣇⣸⣇⣿⣿⣴⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣀⠀⠉⠙⡏⠙⠛⠛⠟⢻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣤⣤⣄⣸⣤⣤⣤⣤⣄⣰⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠈⣰⡚⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⡉⢉⢋⠙⡛⠛⠻⢿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣿⡃⠌⡈⡀⢿⠀⠇⠀⡄⢁⠀⠀⢀⠀⣹⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣴⢁⢁⠘⡇⢠⠀⣇⢸⠀⠀⠀⠀⣿⠉⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡎⡘⡄⢳⠀⡄⢸⠈⠀⠀⠀⢦⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⠑⢠⡇⠸⡆⢁⠈⡆⣦⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣶⠈⡀⠀⢣⠘⠀⠁⠛⠘⢱⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⡏⢿⡤⠃⠖⠘⡄⠇⢠⠠⡇⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣆⠸⣧⠖⡸⠃⠁⠀⠘⠀⡇⣾⣗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⢹⣷⣇⠀⠸⠀⠀⣾⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡾⣿⡿⢰⠀⡇⠁⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢻⣅⢦⣴⢸⠰⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣾⣷⣬⣤⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣩⡏⠉⢹⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⠻⡀⠀⡏⠀⣀⠱⠸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠷⠾⢷⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠠⣒⣺⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁⠘⡦⠀⠦⣤⣔⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡆⠛⠫⣿⢹⣼⣿⠅⠀⠀⠀⠀⠀⡓⠀⠘⢦⣀⡄⢫⠙⠒⠶⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⣐⣀⣦⣤⣈⣡⢿⣬⡀⠀⠀⠀⠀⠃⢐⠛⠘⣿⠋⠙⣳⣤⣿⡥⠐⠚⡆⠀⠀⠀⢤⣀⣀⠔⢞⠀⠁⠀⢙⡆
⠀⠀⠀⠀⠀⠩⡀⠁⠀⠁⡡⠄⠀⠁⠀⠀⠀⠈⠓⠤⢆⣴⣛⠶⠖⠃⠉⠀⠀⠀⠀⡇⠀⠀⢠⢲⠊⡁⠐⠀⣠⡀⢢⣼⡷⣀
⠀⠀⠀⠀⢺⡀⣀⡍⣠⣄⣁⣽⡀⠁⠀⠀⠸⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣨⣛⡁⠀⠈⠋⠀⢁⣈⣤⡴⠿⠂
⠀⠀⠀⠀⠀⣏⠉⠁⠉⠀⠉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠋⠀⠀⠀⢠⠀⠀⠉⠉⠁⠀⠈⠀⠀⢀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣣⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠉⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢰⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣶⣄⣀⣀⣀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢛⣛⡛⠻⢟⣛⠕
⠀⠀⠀⠀⠀⠀⠀⣥⠀⠁⠀⠀⠀⠀⠀⢀⠆⠀⠀⢀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠢⣌⢣⣀⢸⡀⢠⣴⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢙⠀⠀⠉⠉⠁⠉⠉⠙⠧⣀⡠⠊⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⡀⠀⠤⠒⠂⠐⠀⠠⠤⢄⠀⠈⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⢒⢶⣧⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠠⡄⠀⢀⡄⢠⡄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠄⣥⣮⣽⣿⠿⢀⠞⠋⠔⠡
⠀⠀⠀⡆⠠⠀⠚⠄⣸⣦⣾⢤⠖⢀⠆⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠂⠀⣄⡈⠛⠛⠋⠉⢉⡉⠙⣁⣔⣀⣀⣀⡀
⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠉⠙⠋⠁⠉⠉
⠀⠀⠀⠂⠉⠢⠖⠐⠂⠄⠒⠀⠤⣄⠏⡴⠀⠀⠀⠐⠒⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⣀⠀⢀⣠⡄
⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⢀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠈⠛⠿⠛⠺⠽⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠒⠐⠃⠐⠒⠒⠂⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⠤⠤⠤⢄⣈⢑
⠀⠀⠀⠀⠀⣀⢀⣤⣤⢶⡤⢔⣠⣶⠤⢤⣼⣧⠀⢀⣀⣀⣨⣧⣀⣤⣄⣀⣤⣤⡤⣤⣤⣤⠄⡀
⠀⠀⠀⠀⣎⠀⢀⡠⠊⠁⠀⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣴⠋⠁⣄⠀⠀⠀⠆⡇⢱
⠀⠀⠀⠀⡁⡔⠁⠀⠀⣂⣀⣓⠀⠀⠈⢄⠁⠀⠀⠀⠀⠀⠀⠰⣂⡀⣹⠀⠀⠈⠑⢔⠁⢸⡇⢛
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠨⡂⠊⡇⢸
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠓⠀⠘⠀⢸
⠀⠀⠀⠀⡇⠀⠈⠚⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠩⢄⠄⠐⠞⠀⠀⢸
⠀⠀⠀⠀⠸⢄⠀⠀⢂⠀⠀⢸⠀⠀⠀⠀⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠑⠄⠀⠈⠚⠀⠀⠀⣆
⠀⠀⠀⠀⡆⢀⠠⡀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠄⠀⢠⡀⢠⡄⠀⢲⠀⢠⠀⠀⠀⠀⠀⠂⢱⡄⣾
⠀⠀⠀⠀⠑⠒⠒⠞⠒⠒⠀⠘⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣀⠀⡀⠀⠀⠄⣨⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠉⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣃⠖⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡿⠩⠂⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡗⠵⠀⠀⠀⠀⣄⡀⠀⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡄⡠⠀⠀⠀⠀⡄⠈⠆⢸⠛⡡⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠥⡄⠀⠀⣷⣤⠐⢄⣰⢃⡼⠀⠔⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡨⡄⠀⠀⠻⣿⡇⠀⡹⢁⣤⣤⠀⠀⡀⠄⠚⠅⣀⠄⠒⡩⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠣⢄⡀⠛⡄⠀⠀⠀⢠⢿⡂⢅⠀⠀⠠⢌⠣⣀⢣⣿⣿⠏⠀⠌⢡⠀⢰⢊⡀⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠌⣦⡘⡄⢰⣶⠀⣐⣗⠧⠀⠀⠐⡄⢱⣸⠼⠛⠁⠀⡐⠀⢀⡴⠋⠁⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠑⣆⠀⠁⠀⠃⣿⢎⠄⠀⢀⠜⠁⠀⠀⠉⠐⠒⠇⣀⣼⠤⠐⢒⠞⠀⢀⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⠀⣈⣤⡀⠘⡟⠪⣷⣂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡔⠋⠀⢠⠁⢀⠌⢀⡠⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⢧⠀⡷⠊⠀⠇⠀⠀⠀⠀⠀⢀⠄⠒⠒⠄⡀⠀⠀⠈⠐⠤⣴⠖⠃⠒⠮⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠠⠄⡉⠒⠤⠄⡤⠋⠀⠀⠀⡀⠤⠤⠤⠄⣀⡎⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠈⢋⢣⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠥⣦⠊⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣠⠊⠀⠈⠢⠀⠀⡤⢀⡀⠀⠀⠸⠄⣉⣁⡉
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⣀⢰⠖⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡔⠉⠉⠑⠄⠀⠀⡄⢀⡅⠀⠈⠢⡀⣰⠢⡘⠦⠈⠆
⠀⠀⢀⡤⠒⠫⡀⠀⠀⠱⡄⠀⠈⠑⠺⡀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢣⠀⠀⠶⠂⠀⢠⠗⠈⢡⠀⠀⠀⠘⡅⠣⣈⠄
⠀⡴⢋⡠⠄⠒⠒⠓⣒⣦⠬⠶⣄⣀⡀⢡⡀⠀⠀⠈⠄⠀⠀⠀⠀⡠⠥⡀⠀⣀⠔⠁⠀⠀⡘⠀⠀⠀⠀⢰
⣰⣀⠁⠀⢀⡔⠊⠉⠀⠀⢀⠠⠂⠉⠉⠀⡱⣄⠀⠀⠈⡶⠠⠄⠚⢄⠀⠈⠈⠀⠀⠀⠀⡠⠁⠀⠀⠀⢀⠃⢀⣠⡟
⠁⠀⠀⠀⠀⠀⠁⠒⠂⠴⠥⠄⠤⠄⠒⡨⢤⣫⠶⢆⠈⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡤⠊⠁⠀⠀⠀⢀⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣬⢛⡡⠑⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⠀⠐⠒⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡾⢏⠢⠀⠀⢀⠔⠊⠱⢄⠀⠀⠀⢀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⡟⠌⠊⠀⢀⠜⢁⣀⠤⣴⠋⠉⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡸⡗⡌⠀⠀⢀⠊⠉⠀⢀⠔⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣾⠣⠈⠁⠀⠀⣈⠀⡀⢴⠏⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⣆⣽⢣⠉⠀⠀⠀⠀⡏⠁⠀⠎⠀⠀⡄⠔⠁
⠀⠀⠀⠀⠀⠀⠮⣽⣉⡍⠀⠀⠀⠀⠀⢀⠀⠐⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠰⡰⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣟⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣻⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣶⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣷⡀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⢀⣾⡏⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢿⣷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⡿⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢁⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠠⠀⠀⠄⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠂⢀⠂⠂⠀⠆⠆⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⢸⣧⠀⠀⢀⡏⠁⣰⠛⣷⢀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢲⡈⠓⠲⣦⢸⠚⢧⠀⠘⣧⡄⢿⣠⡏⣼⠓⠀⣾⠙⠃⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠛⣆⢳⣀⣷⠘⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠾⠁⢀⡾⠇⠉⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡙⢶⡁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢠⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⢳⣄⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣠⡞⣳
⠀⠀⠀⠀⠀⠀⢠⣤⣄⣙⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡟⣉⣠⣄
⠀⠀⠀⠀⠀⢀⡀⠙⢾⡉⠁⠀⠀⡔⠒⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠒⢂⠀⠀⠘⢛⣿⠟⠁⡀
⠀⠀⠀⠀⠀⠈⠻⣿⡲⠁⠀⠀⠀⢹⠓⢦⠤⢀⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⢤⡶⠚⡏⠀⠀⠀⠈⠷⣞⠋⣻
⠀⠀⠀⠀⠈⠛⠛⠯⠍⠀⠀⠀⠀⠈⠆⠀⠈⣢⣤⡍⠓⢄⠀⠑⡀⠀⠀⢠⠊⠀⡠⠛⢨⣤⣒⠁⠀⡸⠀⠀⠀⠀⠀⠰⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠁⠀⠀⠈⠒⢄⡑⠄⠱⡀⢀⠃⡠⢊⡤⠚⠁⠀⠀⠈⢸⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⣓⣀⣀⣀⠀⠀⠀⠈⠚⢦⣣⣌⡼⠂⠁⠀⠀⠀⣀⡀⣠⣫⠀⠀⠀⠀⠀⠀⠰⠀⠀⣄
⠀⠀⠀⠀⠘⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠙⣝⣀⣀⣠⡤⠀⠊⠋⢉⡁⢹⠉⠉⠐⠂⢤⣀⣀⣠⡣⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⣀⠠⠘⡹⠇⢸⡍⠂⠄⣀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣷⣀⠤⠒⠉⠀⠀⠰⠁⡄⠘⠘⡄⠀⠀⠉⠒⠤⣠⡟⠀⠀⠀⠀⠀⠀⠀⠐⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠴⠭⠤⠀⠀⠀⠀⠀⠀⠀⢫⠙⢆⠐⠒⠀⠴⠥⠀⣅⣆⠀⠼⡄⠐⠒⠂⡰⠋⠏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠣⢌⢂⠀⢠⠁⠀⡔⢹⠏⢆⠀⠘⡄⠀⡔⡡⠊⠀⠀⠀⠀⠀⠀⠀⣠⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠅⠠⠎⠀⠀⠀⠀⠳⠄⠸⠞⠉⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡐⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠤⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⡌⠀⠐⠄⢰⠀⠀⠀⠑⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣀⠀⡴⠋⢳⡀
⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠉⣻⠇⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⢀⠀⣴⣧⡟⠛⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣰⠶⠞⣉⣻⠶⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠓⠦⠴⠖⠋⠀⠀⢹
⠀⣠⠟⣧⣤⣼⣇⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡟⣀⣾⣩⣹⡦⢦⡽⠋⣇⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⢀⡤⠶⠦⣄⠀⠘⣆⡀
⠾⠷⢶⠋⠀⠀⠙⠷⣄⡀⠀⠀⠀⠉⠙⢳⡟⡇⠀⠀⠀⣾⠿⣾⣵⣏⣀⣈⣳⡞⠁⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⢳⠀⢰⡏⠀⠀⠀⠈⣇⠀⠈⠙⢦
⠀⠀⠘⢧⠀⠀⠀⠀⠈⠛⠷⢶⣶⣶⡖⠊⠙⠇⠀⠀⠀⠸⡆⠀⠀⡍⠉⢩⡟⢠⠀⠀⠀⢷⠀⠀⠀⠀⠀⢀⡴⠏⠀⠸⣇⠀⠀⠀⢀⠏⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢙⡟⣻⠀⠀⠀⠀⠀⠹⡄⠀⠀⢠⠏⠀⠀⢀⠀⠀⠘⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠒⠋⢷⡿⠀⠀⠀⠀⠀⠀⠙⣦⣰⠏⠀⠀⠀⠈⠀⠀⠀⢹⡄⠀⠀⠀⢧⣀⠀⢀⣠⡤⠶⠶⠶⠤⣤⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⢠⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠄⠀⠈⠀⠀⠈⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠈⠳⣄⡀⠀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢩⠗⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣼⡇⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⡤⣠⡋⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡄⠀⠀⠀⢀⡾⢀⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠙⣦⡼⠿⢧⣸⢷⡞⢳⡶⢦
⠀⠀⣀⣀⣠⣤⣤⣼⣵⣧⡤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⣴⠿⠤⠦⠿⢶⠀⠀⢸⡇⢿⠀⢀⡀⠀⣠⣯⠟⠀⠀⠀⡶⠶⢼⣇⣠⡞⣿⢠⣷⡀⠹⣏
⠀⠰⢧⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠿⠀⠀⠀⠀⠀⢀⡿⠖⠒⠒⠒⠺⣄⠀⠀⡷⠘⢰⡏⠹⠻⣿⠁⠀⠀⠀⠀⠛⠛⣶⡌⠙⢶⡇⢸⠏⣷⠀⠘⣆⡀
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⣷⠀⣡⠴⢮⣷⣄⣈⡿⠀⠀⠀⠀⠀⠀⣷⠿⠽⢤⣤⣽⣯⡴⠿⠗⠛⣿⠁
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⡀⠀⢹⡏⠉⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡟⠛⠻⣟⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⢦⣼⣧⣴⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣏⠀⠂⠈⠹⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠒⠒⠚⠛⠛⠒⠛
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⣶⣿⠷⣼⣤⠾⠁
⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠓⠲⠦⢤⣄⣀
⠀⠀⠀⢀⣾⣭⠿⢭⣝⡛⢻⣞⣛⣷⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣠⡤⠶⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⡀⠀⠀⣤⠀⠉⠛⠦⣄
⠀⠀⣠⠞⠁⠀⠃⠀⠀⠉⠳⣍⢹⣟⠻⡍⠓⣦⠀⠀⠀⠀⠀⢀⡴⠋⠱⣤⠀⣄⠉⢂⡤⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⣯⣍⠉⠛⠺⢧⣀⢀⡀⠀⠈⠳⣄
⠀⠰⡇⠀⠀⠀⠀⠈⠃⠀⡀⠈⢳⡉⢳⡟⣳⠌⣧⠀⠀⠀⠀⣾⠀⠟⠀⠀⠀⢁⣠⠀⠀⠶⠂⢹⡆⠀⠀⠀⠀⠀⠠⣇⠚⢷⠀⠀⠆⠈⠙⠦⣄⡴⠄⠈⢷
⠀⠀⠻⣄⠁⠀⠰⠀⠀⠀⠀⠀⡤⢻⡌⠙⣏⡧⣿⠀⠀⠀⠀⠈⡿⠲⠶⠶⠶⠶⠶⢶⡶⠶⠶⢶⡇⠀⠀⠀⠀⠀⢸⣯⡵⠏⣠⠖⣋⢩⣽⡲⣌⣳⣤⠶⠚
⠀⠀⠀⠈⠳⢦⣀⠀⠀⠲⠀⠀⠀⠀⢳⢰⣏⣷⡏⠀⠀⠀⠀⢸⡟⠚⠛⢛⣳⣶⣚⣋⣙⠓⢒⣋⣽⠀⠀⠀⠀⠀⢸⡇⠀⠀⣏⢾⣹⣯⠷⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠃⠀⠸⡟⢉⡟⠀⠀⠀⠀⠀⠀⢻⡛⠛⠉⠉⠙⠻⢯⣍⠉⠉⢩⡇⠀⠀⠀⠀⠀⢸⠉⣳⣤⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⣤⡼⠷⠖⠋⠀⠀⠀⠀⠀⠀⠺⠚⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⣿⣿⣿⠛⣿⣿⣿⣁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⡀⠀⠀⢠⡄⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠳⡟⢻⠛⠛⣛⠙⠋⠛⢉⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⡈⠀⠀⢀⣬⣡⣄⢠⣴⡆⣶⣴⣦⣾
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⡀⢸⡄⢀⣸⣄⣠⣄⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡁⠀⠀⠈⣿⢿⣿⢸⣿⡧⠹⠿⠹⠿⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠾⡿⠻⠟⢻⢛⠛⡛⠹⠍⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢁⠀⠀⠀⣁⢀⣤⢠⣤⣤⣤⣆⣶⣰⡀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⣤⣤⣤⣶⣶⠾⣷⣿⣿⢻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠛⢛⠋⠈⠉⣉⡈⠁⠁⣤⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠘⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⡀⠠⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣾⣶⣶⣶⣶⣶⣖⣲⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣒⣶⣶⣶⣶⣶⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣟⡿⣿⣿⠿⠿⠛⠛⠻⠟⠛⠛⠛⠛⠻⠿⠻⢿⢟⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠏⠁⠈⡄⠉⢳⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⢐⡀⠀⠀⣤⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡏⠀⠀⢀⣠⣀⡈⢷⡄⠀⠐⠀⠀⣰⠃⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡃⢉⣿⠀⠙⠷⠶⠶⠚⣡⣏⡹⠈⠛⣦⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⣙⣓⣿⠀⠀⠀⠀⠀⠐⣯⠢⡅⣢⠜⣸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢺⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⡤⡤⠴⠋⠀⠠⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣇⡀⢸⣒⣔⣶⣘⣆⡠⡤⠀⠀⠀⠀⣰⣶⣶⣼⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⣒⣻⣿⣿⣻⣶⣒⣦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠉⠉⠉⠹⠿⣛⢛⡛⠛⢛⣛⡛⣛⣛⣛⣛⣛⠛⠛⢻⠋⠋⠋⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡄⡇⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣉⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⡉⠉⠉⢁⣀⣀⣀⣈⣉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠇
⠀⠀⠀⠀⠐⣶⡀⠀⣹⣦⠀⣴⠁⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡅
⠀⠀⠀⠀⠀⢻⣿⡷⣾⠗⣿⡿⡄⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣷⢸⢹⣿⠃⣇⣀⣀⣀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣮⣼⣿⠀⠠⠀⠀⡆⢰⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢻⣠⣤⣦⣤⣥⡼⠿⠻⠛⠻⢳⣶⣶⣶⣶⣶⣶⠶⠿⠾⠿⠿⢦⣷⣶⣦⣬⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢻⣯⣤⣤⣤⣤⣤⠜⠛⠛⠛⠛⠻⠛⢤⣤⣤⣤⣤⣤⡿⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⢠⠤⠤⢤⣄⡀⢐⠋⠒⠐⢆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡔⢱⠤⣀⡴⡁⠈⠀⠒⡦⠉⠙⢎⣢⠇⠠⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠆⣰⠚⢍⢍⠙⡉⠉⡝⣧⡷⠉⡝
⠀⡹⠐⠉⠉⠐⣼⡦⠀⠉⢠⠀⢺⢻⣅⠄⠒⢊⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠇⠘⠒⢆⡀⢸⡜⠀⣇⠀⡁⣿⡦⣀⠸
⡎⠀⢀⣤⠒⠪⢀⣁⠤⠀⠄⢁⢁⡼⢏⠙⢀⠇⠉⠀⠀⠀⣀⡀⠀⡀⢀⣀⡀⠀⠀⠸⠴⡆⣜⠷⠎⠢⠼⠴⠤⠿⠔⠓⠃⠘⠃
⠈⠐⠉⠀⠱⠤⠜⠣⢇⣒⣱⡥⠚⠀⠀⠉⠉⠀⠀⢀⡒⠃⠄⢈⡶⡶⠭⣄⠬⣙⡆⠀⠘⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠑⠒⠂⠀⠖⠀⠒⠗⠊⢻
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⡇⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⡠⠤⣀
⠀⠀⠀⠀⢀⢠⠔⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠈⠠⢀⡇⣀⣀⣇⣀⣀⡧⠔⠃⠀⠀⠀⠀⠀⠀⠀⢀⠞⠅⠀⠀⢰⣉⡈⠣
⠀⠀⠀⠀⣀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠠⠂⠠⣈⠆⠀⠀⢆⠃
⠀⠀⠀⠀⢘⠢⡀⠑⠤⣀⠀⠀⠀⣠⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⢣⠠⡀⠀⠠⠀⣀⠹⢀⠁
⠀⠀⠀⠀⠈⢦⡈⠑⠂⠤⢈⣉⣉⡤⠊⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⣠⠖⠂⠈⠁⠒⢢⡄⠀⠀⠑⠆⡈⣉⢀⣤⠴⡃⢤⡀
⠀⠀⠀⠀⠀⠀⡝⠒⡤⠤⠠⠴⠊⠀⠀⠀⠀⠀⢀⠤⢦⣊⠀⠀⠈⠀⠱⠂⠀⣀⣀⠀⢀⠄⠂⠁⢰⠀⠀⠀⠀⠀⢠⠩⠤⡰⠹⡉
⠀⠀⠀⠀⠀⢠⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠓⠒⠂⠬⢲⡤⡀⠀⠀⠀⢀⠀⡠⠊⡄⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⡀
⠀⠀⠀⠀⢠⠏⡰⠁⠀⠀⠀⠀⠀⣠⠖⢉⠀⠀⠀⠀⢀⠔⠀⠊⠉⠑⠒⡶⡉⠓⡄⠀⡧⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠱
⠀⠀⠀⢠⠏⢠⠃⠀⠀⠀⠀⠀⠀⢈⠑⠢⢅⠂⠠⢴⠐⠓⢢⠤⠀⠴⡄⠀⠉⠀⡶⠈⠀⠀⠀⢀⡠⠖⠋⠉⠙⠙⠢⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢸⠄⠄⣠⠀⠀⢼⡠⠄⠤⣇⠠⠂⠉⠁⠀⠀⠀⣠⠋⠀⠀⢀⡤⢤⣀⣀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢐⣀⡜⢁⡤⠄⣻⠀⠊⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣧⠈⠦⡀⠀⠀⠀⠀⠀⠈⣆⠀⠸⡀⢀⡦⠔⢋⢆⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⢍⠡⠀⠉⣧⡤⠀⡆⠀⠀⠀⠀⠀⠀⠈⠣⣌⡙⢧⣀⣀⣼⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠒⠘⠃⢉⠢⠀⠀⠀⢃⢁⡠⠛⠳⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠠⡈⢆
⠀⠀⠀⠠⠠⣤⣀⠀⡔⠉⠭⡉⠢⡐⡀⠀⠀⣂⣣⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠳⡀
⠀⠀⠀⢠⢄⡘⢄⣉⡉⡠⠀⠁⠀⣸⠼⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡆
⠀⠀⠀⣼⠀⢠⢦⠤⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⢀⡠⠟⠓⢶⡠⠤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣀⠤⠖⢋⣴⣶⣶⣤⣽⠀⢀⠞⠋⠀⢠⣟⡡⠤⣾⠉⢀⣀⡙⠤⠀⠒⠳⠀⠈⠉
⠀⠀⠀⠀⠈⠻⣄⣶⣶⣦⠌⢉⠉⠛⠿⠿⣄⣀⡙⠂⣈⣛⣆⣀⣤⡔⠊⡉⢀⡄⢸⠉⠀⢰⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣴⠃⠈⢿⠃⡇⢸⣠⣄⣰⢳⡟⠁⢰⠉⠉⡽⠻⠀⡏⢹⠀⡟⠎⣇⢸⢁⠀⢸⢰⣶⣴⣦⢠⠗⠋
⠀⠀⠀⠀⠈⠋⠉⠉⠋⠉⠻⣌⠋⠘⠃⢸⣋⠀⢸⠀⠀⡇⠀⠀⣇⡸⠀⢇⣀⣿⣼⣿⣷⣾⠸⡿⠛⠁⣈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠆⣀⡨⣍⠛⢚⡿⠦⠼⠭⠤⠽⠾⠴⣶⠒⠋⠉⢹⣿⠿⠿⠟⣋⡠⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣠⠖⠹⠧⠤⣤⡽⠷⠤⠤⣭⡷⠒⠀⠸⠤⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡖⠒⠒⠒⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠶⠶⡶⠂
⠀⠀⠀⠀⠔⠁⢸⠀⠸⠉⠁⠀⠀⠈⠉⢙⠀⢁⠀⠀⠂⠀⠀⠀⠀⡰⡆⠀⠒⠒⠛⠓⠒⠛⠛⠛⠛⠃⠘⠓
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡆⠀⠀⠀⠉⢢⣠⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠁⠓⠒⠀⠀⠐⠂⠐⠀⠀⠀⠒⠚⠉⠈⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣾⡿⡿⣷⣆⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣶⣶⣶⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠊⠉⠁⠉⠐⠉⠈⠁⠈⠁⠈⠀⠈⠫⡉⢻⡇⠀⠀⠀⢠⡟⠉⠉⠉⣿⠛⠉⠉⠉⠙⠋⣿⠋⠋⠉⠂
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⢀⣸⡗⠈⠀⠈⣉⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠙
⠀⠀⠀⠀⡇⠀⠀⡟⡷⠀⠀⠀⠀⠀⢟⢻⠀⠀⠀⢠⠄⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⣤⣤⠀⢠⣤⣤⣤⣤⣤⡄⢀⣤⣤⣤⢠⠀⠀⠀⠘⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢨⡆⠀⠀⠀⠘⡧⠁⠀⠀⣉⡀⠀⠀⠀⠀⠀⣉⡁⠀⠠⢤
⠀⠀⠀⠀⠄⠙⠿⠇⠸⠿⠿⠿⠿⠿⠇⢸⠿⢟⣁⠝⠁⠀⠀⠀⠀⠑⠀⠀⠀⠿⠒⠂⠀⠀⠀⠀⠿⠗⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠖⢲⠛⠛⠛⡏⠉⢹⠉⠉⡏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣼⣤⣤⠤⠷⠶⠾⠖⣚⣛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡹⣆⠀⠀⠀⠀⠀⠀⣰⣿⠉⠉⢷⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠔⠋⠀⠈⢷⡄⠀⠀⠀⠀⠻⣽⣤⣠⠞⠀⠀⠀⠀⢙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⢁⣀⣤⠴⠖⠛⠋⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠓⠒⠶⠶⠶⠶⠶⠤⣤⣤⣌⣻⣄⣠⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⢺⡇
⠀⠀⠀⠀⣤⠤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣿⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠛⠶⠦⢤⣼⡿⣦⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠙⠒⣿⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡞⣧⣤⣤⣤⣤⣼⣧⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡀⣴⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣷⣼⣟⣉⣉⣉⣉⣙⣛⣛⣛⣛⣻⡿⢳⡄⠀⠀⠀⠀⣿⠀⣾⠶⠶⡆⣩⠸⡇⠻⠻⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠹⡏⣿⠛⠛⠛⠛⠛⠒⠚⠛⠛⠛⢻⢹⡇⠀⠀⠀⠀⣿⠀⠿⠀⠀⠛⠈⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⢿⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣧⠤⠼⠧⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⡜⣼⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⡀⠀⢠⠞⠱⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡇⣿⣄⡀⠀⠀⢀⣇⡀⠀⠀⡞⠁⠳⠤⠊⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡟⠀⠣⣀⠜⢹⡇⠘⠦⠞⠁⠀⠀⠀⠀⠄⣘⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣸⣇⡀⠀⠀⠀⢸⡇⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢣⡏⠈⠙⠓⠶⢤⣼⡷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢭⣕⣒⣒⣒⣒⣒⣒⣫⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠂
⣀⡀⠀⠈⠀⣀⡰⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡀
⠀⠠⢀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⠤⠖⠀⢤⠀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⢀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠄⠄⠂⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢿⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡐⠀⠀⠀⠀⠀⢄⢰⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢄⡋⠠
⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠰⠂⢀⣀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⣷⠈⠻⠀⠀⠀⠤⠀⣁⠤⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠴⠀⠀⠱⠀⠀⠹⠆⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠊
⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠁⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠐

⠀⠀⠀⠀⠀⢠⢂⡐⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢈⣌⡠⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠎⡀⢀⢔⠜⠀⠀⠀⠀⠀⡰⠃⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠃⠠⣀⡡⠊⠀⠀⠀⠀⠀⠠⠀⠡⠊⠠⠂⠀⠀⠀⠀⠈⠢⡡⠠⠀⠀⠀⠀⠀⠀⠀⠢⠤⠘⠀⠤⠂


⠀⠀⠀⠀⠀⢀⡴⠶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠧⠀⠆⠠⠅⠀⠀⠀⠀⠀⠰⠌⠉⠉⠉⠁⠀⠀⠀⠀⠈⠂⡈⠐⢡⠂⠀⠀⠀⠀⠀⠀⠃⠤⠤⠃
⠀⠀⠀⠀⢐⡄⢀⡄⠀⡈⠀⠀⠀⠀⠀⢠⣡⣕⣢⡖⠀⠀⠀⠀⠀⠠⢔⠁⠒⡈⠄⠀⠀⠀⠀⠀⠀⣢⠤⢤⡄
⠀⠀⠀⠀⠈⠛⠋⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠇⣶⢲⡄⠃⠀⠀⠀⠀⠀⣾⡆⠀⠀⣿⠆⠀⠀⠀⠀⠀⢗⢄⠠⢪⢆⠀⠀⠀⠀⠀⠀⠀⠛⠘⢆
⠀⠀⠀⠀⠀⡇⠙⠚⠁⠀⠀⠀⠀⠀⠀⠈⠑⢲⠐⠙⠀⠀⠀⠀⠀⠀⢀⢕⠕⠈⠁⠀⠀⠀⠀⠀⢖⠁⡲⠀⠛⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠋⠈⠈⠋⠀⠀⠀⠀⠀⠀⠓⠀⠀⠒⠁⠀⠀⠀⠀⠀⠐⠁⠃⠒⠂⠁


⠀⠀⠀⠀⠀⠀⠒⠘⠢⠀⠀⠀⠀⠀⠀⠒⠀⣀⣠⡀⠀⠀⠀⠀⠀⠠⠽⢄⠀⡸⣀⠀⠀⠀⠀⠀⠠⣄⠓⠚⡦⠄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠸⠀⢸⠀⠀⠀⠀⠀⠀⠚⡐⠆⠘
⠀⠀⠀⠀⠀⠠⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠲⠃⠐⠳⠇⠀⠀⠀⠀⠀⣤⠀⠈⢰⠊⠀⠀⠀⠀⠀⠤⠚⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⢿⡿⠟⠋⠁⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⡁⠀⠈⡁⠀⠀⠸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⠃⠀⠀⠀⢹⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠤⡄⠀⠀⠀⠈⡧⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⢀⠀⠠⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⠁⠉⠉⠀⠀⠀⠈⠁⠒⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠢
⣀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠐⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⢃⠀⠑⠂⠠⠤⠋⡆⠀⠀⠀⢀⠒⠢⣀⠀⠀⢠⠇⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡠
⠈⠦⣀⠀⠀⠀⠀⠘⠢⢄⣀⠈⢤⠀⠀⠀⠈⠋⠘⢋⢱⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠈⠉⠁⠀⠒⠤⣀⠀⠉⠉⠓⠦⣤⡄⠀⢀⡀⠉⠓⠲⠦⢀⡀⠀⠀⢀⡔⠃⠀⠣⠀⠀⠀⠀⣠⠊⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠀⠀⣡⠆⠄⣀⠀⠈⠁⠒⠀⠊⠀⠀⠀⡠⠗⢄⠠⠊⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠊⠁⠀⠀⠀⢿⠒⢠⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠶⠾⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠀⠀⠀⠀⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠅
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⣀⡀⠀⠠⠤⠀⠚⠀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠒⠒⠒⢶⠖⠒⠒⠒⠒⠶⠒⠶⠖⠒⢲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡇⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡁⠀⠃⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢣⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣸⠀⠀⡃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⡄
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠒⠒⠂⠉⡁⠀⣀⠝⠐⠂⠠⠄⠰⠊⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⣉⠀⡇
⠀⠀⠀⠀⡄⠋⠀⠀⠀⠀⢀⠤⠈⡠⠚⠁⠀⠀⠀⢀⡀⠰⠀⠀⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠂⠤⠤⠤⠄⠊⠁⠀⠀⠇⠀⠀⡀⠠⠔⠋
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⣄⠄⢒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠉⠀⠀⠈⠉⠀⠈⡑⡤⢤⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡞⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠒⠂⠀⠀⠀⠀⢂⣀⠀⠀⠀⢄⠀⠠⠀⠄⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠲⠤⠤⠤⣤⣀⣀⣐⣁⣐⣀⣀⣀⣄⣧⡤⠤⠴⠤⠖⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠴⢾⠁⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠀⠀⡜⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢙⡗⠒⠒⠲⠶⡤⠤⠤⠤⡤⠤⠒⠲⠒⠒⡟⠉⢉⡏⠀⠀⠀⠀⢰⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡊⠁⠈⠠⠤⠂⠀⠈⠐⠄⠌⠀⠀⠀⠀⠀⠌⠀⠑⢸⡇⠀⢀⡤⠖⡾⡸⠤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠸⡇⠀⠀⠀⠀⣋⣅⡀⠀⠀⠀⠀⣀⣍⡃⠀⠀⠀⠀⣸⣠⠖⢏⠀⠀⠓⠃⠀⢈⡧⣄
⠀⠀⠀⠀⢀⡎⣡⡽⠌⡇⠀⣷⠀⠀⠀⢸⣥⣿⣿⠀⠀⠀⢸⣿⣧⣽⠀⠀⠀⢀⡟⣥⣁⣀⡀⠀⠀⠀⠀⣀⣀⡼
⠀⠀⠀⠀⠈⠣⣄⡀⠀⢧⠀⢸⡄⠀⠀⠘⠻⠛⠋⠀⠀⠀⠈⠛⠛⠛⠀⠀⢠⠚⢶⣼⠄⠉⠉⠉⠉⠉⠉⢉⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠣⣈⡇⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⡔⠀⠀⠀⠀⠓⡔⠉⢺⡆⠀⠀⠠⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠈⢻⡀⠀⠀⠀⠀⠹⢟⣛⣻⠟⠀⠀⠀⠀⢀⠞⣢⡤⠎⣇⠀⠀⣮⣿⡆⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⡿⠀⢀⣽⠀⠀⣈⣉⣀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠉⠘⡆⠀⠈⠋⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣀⠀⠀⠀⠀⣀⣀⣠⡤⠞⠁⠀⠀⠀⠀⠳⣄⣀⣀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠹⡏⠉⠉⠉⢹⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⠇⠀⢇⠀⠀⠀⡞⠀⠸⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⣀⡼⠀⠀⠀⣇⣀⠀⢀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡟⠛⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠉⠑⠄⠀⠀⢀⠀⠀⠁⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡀⠈⢀⡠⠀⢀⡀⠐⠤⠄⠀⡤⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠉⢣
⠀⠀⢀⠀⠴⠷⠿⠿⠟⠛⠶⠆⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⠇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡹⠀⠀⢹⡆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠁
⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠘⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⢀⢈⠃⠀⠀⠀⠀⠀⠘⠻⠿⠟⠀⠀⠸⡆⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠒⢒⡀⠤⠞⠁⠀⠘⠉⢒⣋⠛⠈⠍⠀⢀⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⢉⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡇⠀⠀⡞⠙⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠲⠚⠁
⠀⢠⠀⡠⠊⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⠏⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆
⠀⢸⠀⣠⠎⠀⡠⠀⠀⠀⠀⠀⣹⡿⠀⠀⢰⡇⠀⠀⢄⠂⠀⠀⡠⠀⣰⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡄⠀⠐
⠀⠈⠀⠁⠀⠊⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠠⣛⡚⠲⠋⢝⣲⣄⣀⢀⣼⡍⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠨⠙⠘⠑⠀⠂⢸⢕⡀
⠀⠀⠈⠀⠶⠆⠀⠀⠀⠀⠲⠎⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠷⠦⠟⠀⠀⠀⠔⠛⢒⠒⠒⠢⠀⠔⠒⠒⠒⠛⠢⠀⠀⠀⠀⠀⠁⡀⠀⠀⡸⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠓⣼⢂⡤⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⡜⢁⠎⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⢠⠤⢭⣤⡤⣴⡶⢶⡖⠖⢴⠏⢺⠆⠀⠛⠛⠛⠛⠛⠛⠷⢼⠇
⠀⠀⠐⠂⠆⠀⠀⠀⠀⠀⠌⠀⠀⡜⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠸⡄⣠⠟⠁⠈⠁⠀⠂⠳⠏
⠀⠀⠠⢤⠈⠀⠀⠀⠒⢶⣶⣶⣞⣠⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠑⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⠏⣷
⠀⠀⠀⠀⠀⠠⠂⢠⣀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡄⠰⠁⠛⠉⠀⠀⠀⠢⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢠⣄⠀⠀⠀⠘⡀⢻
⠀⠀⠀⠀⠀⡀⠒⠁⠉⠉⡇⠈⢈⣆⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⢸
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠘⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠋⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣇⡴⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠐⠲⠶⠴⠶⠦⣤⠴⠤⠤⠤⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡍⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⡶⡄⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢠⡶⠛⠛⠉⠉⠉⣼⡟⠁⡀⠀⢻⡁⠀⣠⡴⠄⠀⣠⡿⠃⠀⢀⣠⢠⡀⠀⣿⡃⠈⠂⢀⣤⡀⢒⣷⡠⣶⡄⠀⣠⡶⠛⡳⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⡟⠀⣼⡟⠀⠀⠉⢸⡿⠀⠀⢠⣿⠃⠀⣠⣿⠁⠈⠀⣶⡟⠁⡈⠀⣰⡿⢡⣿⠋⣴⡿⠁⣼⡟⠃⠉⣀⠴⠞⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠄⢿⡇⠀⠀⠀⠈⠉⠀⠀⠸⣯⠄⠀⠈⠛⠀⠀⠀⠻⠤⠊⠀⠀⠉⠁⠀⠁⠀⠛⠀⠀⠘⠃⠒⠈⠁


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿