Mem Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⣡⠤⠤⠤⢤⣀⣀⠀⠈⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡤⠒⠀⠀⠠⠤⢤⣀⣀⡀⠀⠈⠉⠓⠦⣄⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⠎⣠⠖⠀⠀⠠⠤⢄⣀⠀⠉⠙⠒⠦⢄⡀⠀⠉⠲⣄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡏⡴⠁⣰⠂⠀⠰⠤⣀⠀⠉⠑⠒⠦⠤⣀⠈⠑⠢⣄⡈⠳⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢰⠃⣰⠃⡸⠉⠓⠒⠦⠭⣛⡒⠦⢤⣀⡀⠉⠑⠲⠄⠙⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠋⡰⠁⣰⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠬⠍⣛⣒⡒⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢁⡜⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠉⠉⡇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢾⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡷⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠸⡄⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢻⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⣙⣋⣓⣂⣀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠛⠛⢛⠛⠛⣛⠛⠛⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢩⣅⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠈⠁⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠙⠛⠛⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡒⠒⠲⡴⠦⠒⠒⣒⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡏⠠⠬⠑⢦⣷⣴⠞⢩⠄⠈⡷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠖⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢧⣀⣉⡥⠞⡿⠿⠦⣔⣂⣠⠇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠲⣤
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠈⠉⠓⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢤⢀⡀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠿⠚⠋⠉⠀⠀⠈⠙⢲⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣠⠾⠞⠉⠀⠀⠀⠀⢀⣤⣴⠶⠶⠖⠀⢹⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣄⣾⣵⠶⠶⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡞⠛⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠊⠷⠿⠀⠉⠀⠀⠘⡤⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢳⠀⠴⠋⠿⠾⠉⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠳⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣷⠀⠀⠀⠀⠀⠀⠹⠦⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⡿⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣠⠤⠄⠐⠒⠒⠒⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠐⠒⠒⠚⠁⠀⠀⠀⠀⡼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠙⠦⣄⠀⠀⠀⠀⢀⣠⠔⠃⠀⠀⢸⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠴⢻⠻⡄⠀⠈⠙⠓⠒⠛⠋⠁⠀⠀⠀⢠⠟⠀⣷⠈⠛⠶⢤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣤⠶⠋⠉⠉⠉⠀⠀⣿⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠶⣄
⢀⡠⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠳⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠠⡤⠄⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⣹⠖⠶⢯⡀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣴⠣⡄⠀⢠⡟⣆⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡸⠃⠀⠹⠀⠘⠇⠈⠳⣄⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⠁⠀⠀⢰⠁⠉⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢲⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢈⣠⠤⢄⠀⠑⠪⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠁⠀⠀⠃⠀⠀⠈⠙⠳⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠶⠀⠴⣤⣤⢀⡀⠀⣀⠁⠉⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠙⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢀⡄⠲⢗⠀⠀⠀⠀⠛⣳⠂⠀⡀⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⢰⠀⢸⡁⠀⠀⠷⠛⠒⠆⠀⣏⠀⠀⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢤⠘⠳⢤⡠⠂⠀⠀⠀⠀⠚⠀⣤⠞⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠉⠀⠀⠀⣠⣀⣀⣀⠀⠉⠉⠀⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⣀⡀⢠⣼⡿⢧⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣼⠛⠒⠶⠶⠶⠶⠖⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣀⠀⠒⠒⠂⠀⣰⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣦⣀⣦⣀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠂⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⢀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠒⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠘⠀⠀⠀⠀⢻⣿⣷⣦⣤⣤⣤⣤⣶⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⣠⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠆⠀⠀⠀⠁⡀⠀⠀⠀⢙⠋⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢀⠀⠀⠐⠀⡀⠀⢂⠙⡄⠠⠀⠶⠋⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣀⠀⠀⠀⣀⠀⢸⣿⣿⣿⣿⣿⠀⣾⠀⠀⡄⣳⢀⡶⡀⢡⡈⠀⠀⠐⢛⣁⣤⡤⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣀⣅⡆⢸⣿⣿⣿⣿⣿⠀⣿⠀⠀⢁⢮⢇⡿⣿⡀⠙⠂⠀⠙⢁⣽⡟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⡀⡇⡆⠀⢢⠀⠀⠑⣿⣷⣄⡀⠀⠀⠈⢁⡀⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⠇⠀⢀⠀⠀⠀⠈⠻⣯⡁⠂⠂⠄⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⠀⠀⠀⠀⠀⠙⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢀⢸⣧⠤⢤⣶⣶⢶⣶⣤⠽⠛⢿⣆⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⠛⣿⣿⣿⣿⣿⣿⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣇⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠛⠿⢿⣿⣿⣿⣿⡿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⡷⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⣿⣿⠇⠀⠀⠉⢀⠙⠁⠀⠀⠀⠛⠒⠒⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⣿⡆⠀⠀⠸⠿⠀⠀⠀⠀⠾⠿⡁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣧⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠷⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡀⠀⠀⠈⠳⢎⣉⢉⣙⠛⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠹⣦⡀⠀⠀⠀⠀⠤⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠀⢠⠀⠀⠈⠻⣷⣤⣄⣀⣀⣠⣴⣏⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠋⡀⠀⠀⠻⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠶⠛⠁⣽⠀⠀⠀⠀⡇⠀⠀⢄⠀⠀⠀⠀⠙⠛⡿⠃⠀⡇⠀⠀⠀⠉⠀⠀⠀⠉⠁⠠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠚⠉⠀⠀⠀⠀⡟⠀⠀⠀⠀⠇⠀⠀⠀⠀⠂⢀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⢀⠶⣎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡄⠀⠀⠀⣠⣦⠉⣷⣆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢿⡄⠀⡰⢿⣿⠂⢀⡙⢦⣠⠞⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⠡⠾⠁⢸⣿⡿⢛⡁⠀⠙⠂⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢧
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠸⠆⠀⠀⠰⠶⠿⣤⣤⣤⡀⠀⠀⠀⡆⠀⠀⠿⠟⠁⢺⣇⠀⠃⠀⠀⠀⣠⣤⣤⠞⠻⡆⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⢀⡷⠋⠀⠀⠀⠀⠸⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⣦⠀⠀⠀⢸⠇⠀⠀⡀⢄⡀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠿⠃⣈⠀⠐⠃⠀⠈⠁⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⡇
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠘⠋⠀⠀⣔⠀⠀⢠⣶⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠰⣶⣿⡇⠀⠀⠀⠀⠈⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠇
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢸
⠀⢠⠈⠀⢀⠔⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸
⠀⠇⠀⡀⠃⠀⠀⠀⠀⠀⠈⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⠇⠀⠀⢸⡄
⠃⣀⠀⠀⠀⠀⠀⠤⠂⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡧⠀⠀⠀⢸⣧
⠘⠃⠀⠀⠠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠾⠃⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠈⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡆⢹⡆⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢻⡄
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⣿⠇⡈⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢱⠀⠀⠀⠀⠀⢈⡏
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠴⠂⠀⢉⡀⠼⠛⠀⣷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠀⠀⠘⡄
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠐⠂⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠙⠓⠶⠶⠶⠶⠘⠛⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠛⠒⠶⠦⢤⣤⣤⣀⠀⠀⠀⠀⠀⢿⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⠄⢀⣀⣀⣀⡀⠤⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠤⠤⠤⣾⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡆⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⣀⣤⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⣠⡴⠞⠉⠀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⢛⠁⡀⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠒⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡐⢈⠋⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠉⠙⠛⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠗⠀⠰⠂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠢⠀⣉⢠⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠤⣦⣀⠀⢨⠄⠀⠁⣰⠆⠀⠙⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⠹⣆⠶⡷⢾⡿⢸⠋⠉⠐⠶⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⡀⠀⠀⠀⠀⠘⢈⣗⢸⡇⠿⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢹⣦⡀⠀⠀⠀⠠⡽⠂⣷⡀⠀⠀⠀⠀⠀⣼⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠁⢻⡄⡀⠀⣎⣿⡊⣿⡇⠄⠀⠀⢀⣰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠀⠀⣇⡹⠆⣿⠀⠀⠀⠀⠀⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡇⠀⠀⠁⠀⠀⣿⡴⣻⡟⠛⠂⠀⢀⢠⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⢶⣿⠀⠀⠀⢠⡄⠀⣿⡛⣾⡃⠰⠆⠀⠀⣾⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠲⠀⠘⡇⠀⣿⣿⢾⡆⠠⡄⠀⢀⣿⠓⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⡿⠀⠀⠀⠈⠓⠀⣿⣿⣿⡇⠀⣄⠀⢸⡅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⢿⣿⡃⢀⣼⠀⣸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠠⠐⠂⢿⠀⠀⠀⠃⠀⢰⣿⣹⣾⣇⢀⣐⠀⠃⣇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠶⠛⠛⠛⡄⠀⠀⠀⠀⠘⡇⠎⡉⢹⣿⣋⡀⢀⣟⣀⣀⡀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠴⣧⣵⣿⣬⣧⣾⣿⣼⠉⡀⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⢀⣸⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⡸⠃⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡧⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠸⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣄⣸⡿⠛⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣶⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠛⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣙⣿⣿⡿⠿⢿⣿⣿⣿⡛⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⠉⠉⠛⠒⠋⠙⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⡀⠀⠀⢻⣿⣷⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣳⠀⠘⣯⡭⣉⠀⠀⠈⢻⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠀⠀⠉⠀⠀⠀⠀⢠⣿⡿⡱⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠘⠋⠁⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠲⠤⠀⠀⠀⠀⢀⢾⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⠁⠀⠀⣀⣤⡖⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⠐⠚⠛⠙⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⢧⢰⡀⠀⠀⡔⠀⣠⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠚⠋⠁⠀⠹⡅⠈⢆⠁⠀⡐⢀⠔⠉⣠⠞⠛⠛⠶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠲⠄⠊⢀⠠⠚⠁⠀⠀⠀⠀⠀⠹⠿⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣴⠆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⣖⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠏⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠑⢌⡲⡀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣳⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⡾⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣉⡵⠛⠛⠛⠛⠛⠛⠷⢦⣤⠤⠤⣤⣐⠒⠋⠙⣧⣀⣀⣠⣶⣿⡏⠀⠀⠀⠀⠀⠀⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠀⠀⢿⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣤⡾⠀⠚⠉⠁⠈⡻⣦⠀⠀⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠁⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢦⣄⣠⣤⣤⣤⠶⠶⢾⣿⣟⣃⣒⣒⣤⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢡⠀⠀⠀⠀⣈⠉⠉⠉⠉⠉⠙⠛⠿⢶⣬⣽⣓⣾⣷⣶⣶⣿⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⣿⣧⣤⣶⣿⠋⠀⢀⣠⣤⡤⣶⣤⣤⠤⠀⠀⠈⠉⠉⣉⣉⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠷⠚⠛⠋⢉⣤⡾⠟⢋⣀⣤⠴⠶⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠀⠂⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠁⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠈⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢁⡰⠢⣄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡆⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⠠⠄⠤⠀⠐⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠲⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⡑⠄⡀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠔⠁⢀⠀⢀⡀⠀⣡⠀⠀⠀⡀⠀⠀⠀⠀⠈⠢⡈⠣⡑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⠉⠉⢀⠈⠉⠛⠀⢐⠀⠌⠀⠔⠀⠈⠉⠒⢄⠈⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠷⠀⡄⠀⠄⣀⠀⠀⠀⠃⠰⠀⡆⠀⠀⠀⠀⠀⠀⣷⡌⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡘⠁⠀⠀⠀⠈⠘⡀⡘⡆⠀⠃⠀⠀⠀⠀⠀⠀⢸⢺⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠃⠀⠀⠀⠀⠀⠀⠒⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣹⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠁⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠇⠐⡇⠀⢀⠀⠀⠀⠀⠀⠈⠰⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⡇⢀⡇⠀⠀⠢⢀⡀⠀⠀⠀⠀⠃⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡙⠀⠋⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠌⠀⠀⠀⠀⡐⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠌⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⠐⠲⠲⠒⠀⠀⠀⠠⠀⠀⠀⢀⡆⠀⡈⠢⠤⣀⡀⠀⠐⠠⠀⠀⠀⠀⠠⠀⠐⠒⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠖⠁⠁⠄⠐⠒⠲⠀⠁⠀⡀⠀⠀⢉⡀⠠⠶⠤⠤⠤⠤⠀⠒⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⡙⠻⣿⣿⠏⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣇⠀⠀⣷⠀⢹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣇⠀⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣇⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣧⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⠟⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣶⣾⣷⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⣸⣿⣿⣿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡅⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠶⠶⠿⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣤⠀⠠⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠛⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣾⣧⡀⠙⠒⠂⣽⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⠀⠙⢿⣿⣷⣤⣴⣿⠟⠈⠙⠛⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢓⠀⠀⡀⠀⠀⠀⣿⠄⢸⣷⠀⠀⠀⡀⠀⡘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⠀⠀⠃⠀⠀⠀⣿⡁⡟⣿⠀⠀⠀⡇⠀⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⣿⡏⣻⣿⡇⠀⠀⠇⠀⣧⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⠀⠀⠀⣿⠂⠀⣿⣱⣿⣽⡇⠀⢰⠀⠀⣿⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣘⠀⠀⠀⢸⡄⠀⣿⣄⣽⡼⣇⠀⠘⠀⢠⡿⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡿⢀⡶⣄⠀⡇⢰⣯⣼⠻⡇⣿⠀⠀⢠⣼⡇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢹⣶⠋⠀⠈⢧⠀⢸⣯⣷⡼⠁⣿⠀⡰⠁⣾⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢿⡍⠁⠀⠀⠀⠁⢸⡧⣼⠃⠀⣿⠤⠀⠀⢸⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⠃⠀⠀⠀⠀⠀⣼⣧⣿⣍⣤⣿⠁⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣄⠐⡇⠁⠒⠶⠶⠏⣿⣿⣿⣿⣿⣿⡘⠛⠉⡿⠒⢰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠿⠼⢿⣿⣿⣶⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠑⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠰⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣯⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⠁⠀⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⢉⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⢻⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋

⠳⣄⠀⠈⢣⡈⠀⠀⠘⡆⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⢰⠃⠐⠀⠀⡴⠋⡰⠋⠀⠀⣠⠞⠑⣩⡾⠋
⢦⡈⠢⠀⠀⠱⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠞⠀⠈⠀⠀⣠⠞⢁⣴⠞⠁⠀⢀⠔
⠀⠙⠦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⣠⠞⢁⠴⠋⠀⠀⠀⣨⡴⢟
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠁⠀⠀⣠⠶⠛⢁⣠⡵
⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⢁⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠀⠀⠈⠀⠀⠀⠀⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⠶⠚⠋⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠋⠛⢿⣿⡟⢿⣿⣟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠤⠈⠉⠀⠀⠁⣐⡄⠈⠉⠛⠛⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⡰⣊⠳⠀⠀⠀⠀⡜⣥⠌⡆⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⢰⡃⠟⡸⢠⠲⠼⠀⢧⣋⡴⠀⠀⠀⠀⢀⠿⣟⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣠⣄⣙⠋⠱⣄⣀⣤⠶⣷⠀⢀⣤⢦⠦⣄⡜⠀⡸⠛
⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣰⠉⠁⠈⠈⠙⢄⠘⠧⣤⠥⠏⡴⠃⠀⠀⠀⠈⢸⣛⠠⢀
⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⡀⠀⠀⠀⣼⣀⣀⣈⣉⣀⣱⣀⠀⠀⢀⣠⣯⠉⢛⢶
⡠⠀⣠⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣾⣿⣦⣄⣨⣿⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⢶⣭
⣠⠞⠁⠀⠀⢀⡜⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠈
⠋⠀⢀⡔⢀⠞⠀⠀⠀⣰⠁⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⠀⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣑⠤
⠀⠀⠋⠀⠋⠀⠀⠈⠀⠃⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠑⠀⠀⠁⠈⠃⠀⠀⠙⠉⠙⠂⠀⠀⠈⠛⠊⠙⠙⠓

⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣬⣿⣿⣄⣠⣿⢁⣀⣼⡿⠁⣰⣿⠇⠀⣠
⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⢋⣹⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⢿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣇⠀⣰⡿
⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣾⣿⣿⠿⢛⣥⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡦⠀⢀⡀⢛⣥⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣠⠟⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣤⣾⣿⡟⢡⣿⣿⠟⠀⠀⠀⢸⡟⣿⣿⣿⣿⣤⠼⠋⠙⠛⠁⠐⠛⠿⢿⣷⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠉⠀⠀⣀⡀⠠⠆⠻⡿⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢟⣽⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⣹⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⢿⣷⣦⣄⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣣⣼⣿⡟⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⡄⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣷⣦⣶⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⢁⣴⣿⣿⣿⠿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⡀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠙⢷⣄⡀⢀⡶⠛⣿⣿⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠘⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⣿⣦⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⠟⢃⣤⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⡃⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⣿⠁⣹⠂⠀⠈⠹⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⢀⣴⠿⢻⣿⣿⠜⣽⣿⣿
⣿⣿⣿⣿⣿⣿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣿⠟⢿⡤⠘⡆⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⡿⠀⠙⠧⠸⣿⣿⣿⢾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡀⣠⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠰⢾⡿⣿⣿⡘⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣏⣡⡶⠀⠀⢀⣀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⡿⢿⠿⠁⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣄⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣺⣷⡀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣇⠀⠈⠛⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⢀⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⣄⣿⣿⣿⣿⠟
⣿⣿⣿⠃⣽⣿⣿⣿⠇⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠚⠛⣿⣿⣿⣿⣿⣿⣿⣧⡀⢻⣿⡟⣰⣾⣿⡼⢿⠟⣿⣿⣿⣿⡁⠀⠀⠀⣿⣦⢿⣿⣯
⣿⣿⠿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠸⢷⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣷⣿⢾⣿⣿⣿⣿⣿⣦⣴⣶⣿⣿⡞⣿⣿⣷
⣿⣿⣦⡈⠻⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠠⡤⠤⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣷⡿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠉⠛⠛⠀⢈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⡏⢰⣄⣤⢿⣿⣿⡿⢿⣿⣿⣿⣿
⠃⠀⢠⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣶⣿⣷⣿⢹⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⠿⠷⠿⠛⠋⠀⡾⣿⣿⣿⣿
⣷⣾⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣯⣉⠙⣿⠛⠛⠛⠛⠛⠛⠋⢹⣏⣀⣠⣿⣿⣿⣿⣿⣿⡷⣼⣿⣿⣿⣿⣿⣿⣿⠏⣿⠿⡿⠛⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⣿
⠟⠿⢿⣿⣿⣿⣿⡟⠀⠀⢈⣿⣿⣿⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢿⡏⢠⣿⠟⠋⠉⠻⣿⠟⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡀⠘
⡅⠀⠀⠙⣿⣿⣿⠁⠰⠷⢸⣿⣿⣯⠀⢠⣎⣿⣿⣿⣿⣿⣿⣿⡟⢁⢼⡀⠀⠀⠘⣷⣾⡿⢀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡀⠀⠀⠐⠘⣿⣿⣷⣶⣾⣿⣿⣿⠇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣸⣧⡀⠀⠘⠋⢿⣷⣮⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢘⣿⣿⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⡄⣼⢸⣿⣿⣿⣿⣿⣿⣧⡉⠛⢿⣿⣿⣿⣦⣀⣀⡀
⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠸⠉⠉⠛⣿⣿⣿⣷⣄⣀⣤⡀
⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣹⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⠷⣶⣄⠀⠀⠉⣻⣿⣿⣿⣿⣿
⡆⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠋⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡀⠀⠉⠁⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡙⠉⠁⠀⠈⠿⠋⠀⠀⠀⠀⠘⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⡇
⣿⣷⣄⠀⠀⠀⠁⢠⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⢻⣿⣿⣿⣿⣷⣝⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠙⠿⠿⡟
⣿⣿⣿⣧⡀⠀⠘⣦⣻⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠉⠁⠙⠛⠛⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠
⣿⣿⣿⣿⣶⠄⠘⠉⠻⢿⣍⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠛⢿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠋⢿⣿⣿⣿⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⡄⠀⠿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣟⣫
⠀⠀⠈⠛⣿⣿⣿⣿⣧⣀⣴⣤⣤⣴⣾⣿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡷⠸⣿⣿⣿
⠀⠀⠀⢘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣄⣀⠀⠀⣼⣿⣿⣿⣿⡇⠀⢻⣿⣿
⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡄⠀⠀⣠⣴⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿

⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠠⢤⣀⠀⢤⣄⠀⠀⠀⠀⠀⡠⠀⠀⠠⠀⠀⢀
⠀⠧⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠀⠀⠉⠐⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨
⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣶⣂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⡒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢛⠀⠤⠐⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠲⢄⡒⠦⣌⣹
⠀⠁⢠⡁⠀⢁⢀⠀⠀⠀⡀⠀⢀⣀⡀⢀⣀⠀⣈⠀⣉⡉⣀⠀⢀⢀⡉⠀⣀⠈⢀⠀⠀⢀⡀⡀⣈⢉⡀⢀⣉
⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢐⢮⡢⡈⢦⡈⢿⡜⣆⠈⢸⡇⠀⡋⠀⡎⠀⢀⡞⢕⢕⠕⢋⡴⠋⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣕⡪⡪⠂⠑⠄⠻⡜⡄⠀⡇⠀⠀⠐⠀⠀⠊⠀⠀⣡⠖⢁⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠲⢄⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⡶⢟⣫⡵⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⡫⠕⠚⠉⠁⠈⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠍⠒⠀⣀⠠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣚⣯⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄⡀⠀⠒⠤⢍⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠋⣡⠔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡽⣄⠀⠀⠀⠈⢓⢤⣀⠀⢬
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠊⡠⡢⡀⡴⠁⠀⠀⠀⠀⠀⡇⡀⢹⡹⣦⠐⢄⠢⣔⢕⢯⡳⢶
⠀⠃⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⢈⡴⣫⢮⢎⡞⠀⠀⡄⠀⠀⠀⠀⣷⢣⠀⢳⡹⣧⡈⢳⣌⠳⣕⢮⡳
⠀⠓⠀⢂⠚⢀⡀⣈⠃⠀⠀⠈⠀⠈⠁⠈⠈⠁⠁⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠤⢤⣤⣶⣦⣤
⠀⣟⣋⣉⡑⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢛⢩⣭⣭⣿⣾
⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⢿
⠀⣿⣿⣿⣿⣿⣶⠒⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢙⣿⣿
⠀⠋⠛⠟⠛⠛⠖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣻

⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣀⢀⣀⠀⣀
⠀⢠⢀⠐⠁⠂⠀⣀⡀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡂⢲⢸⡇⢒⣰⣦⣦⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⠉⠉⢡⡾⠇⣴⠟
⠀⢰⢮⡑⡻⠓⠈⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⡈⠀⠀⠀⠀⢫⣹⣹⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶
⠀⠈⠳⣴⠁⣠⡆⠠⢀⠀⠠⠀⢀⡀⠤⡄⠀⠰⠄⠀⠀⠀⠆⢰⠿⣾⠋⠙⠛⠛⠛⠉⢻⡇⠀⠀⠀⠀⠀⠀⡘⠻⢖⡑⠂⠠⠲⢤⣀⠀⠀⠀⠀⠠
⠀⠀⠀⢣⣀⣀⣠⡴⠆⠒⠒⠒⠚⠠⠐⠇⢰⠀⣰⠀⠀⠀⠁⠈⠉⡇⠀⡀⠀⠀⠀⣀⠀⡇⠀⠀⠀⣀⡀⣀⣙⡛⠺⡏⠀⠀⠀⠀⠀⠉⠑⠢⢄⡠
⠀⠀⠀⢘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠟⢀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁⠀⠃⠀⠰⠃⠀⠀⠈⠉⠉⠙⠻⣟⣶⣄⣀⠀⠀⠀⠀⡀⠛
⠀⠀⠀⠈⠉⠉⢛⡷⣤⢠⣄⡀⣀⠀⠀⣴⠂⠾⡀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠾⡟⢳⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠙⠘⠑⠋⣄⠴⠱⠌⠒⣤⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡜⡄⡿⣆⣐⠦
⠀⠀⠀⢀⡀⠀⢀⠠⡤⠀⠀⠀⠈⠁⢰⢶⣶⣤⣄⢀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠒⠲⢦⡀⠀⣀⣀⣤⠂⡾⢠⠃⡇⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠈⡀⠀⢀⡄⢀⣿⣶⣿⡿⠻⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠿⠃⠾⠃⠘
⠀⠀⣆⠀⡀⠀⠀⠀⠀⠉⠉⣾⠁⠘⢻⣿⡟⢹⠂⣘⠁⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠤⠀⣀⠀⢀
⠀⠀⢸⡆⡀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠘⡟⠓⠓⡈⢁⣨⡾⢇⢈⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠈⠏⠙⢷⣂⣠⠴⢥⣾⡁⠀⠈⠉⠃⠀⠀⠉⠛⣉⣀⣠⣿⢧⣴⣼⠀⢀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣳⠀⣆⢀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣽⣷⣾⡿⠧⠄⢀⠀⣀⣀⢸⣇⠐⠂⠰⠀⠀⠀⠀⠰⠀⢠⡄⠠⠉⣁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠦⣀⢀⣉⣤⡀⠀⠀⠀⠀⠀⠀⠒⠚⡶⠂⠀⠀⠈⠑⠁⠉⠁⠈⣄⣉⡴⠷⠋⠑⠛⠃⠈⠃⠙⠒⠚⠭⡇
⠀⠀⠀⠀⠀⢀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠘⢢⠀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⣄⠀⠠⣠⠀⠀⠠⠀⡐⢠⠤⠒⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠁⡀⠁⠀⠀⠀⠀⢀⢀⠠⢠⡄⡀⠀⠀⠀⣠⠤⠐⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠁⠀⠀⠘⠛⠋⠋
⠀⠀⠀⠀⠄⠞⠁⠀⠄⠠⠀⠐⠉⠀⠀⠀⢴⢃⡈⠂⠸⣄⠀⠀⢈⣷⡤⠄⣼⣏⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣦⣀⠀⡠⠀⠁⠀⠀⠀⠀⠀⡈⠀⠀⠘⣿⣶⠎⠀⠀⢠⠋⠀⠀⠘⠟⠛⠁
⠀⠀⠀⠘⠋⠛⠋⡀⠀⡀⠀⣤⣴⣶⣾⣿⣄⠀⢠⣿⣿⣷⣤⡶⠃⠀⠀⠀⠀⠀⡛⠁⠀⠀⠀⠀⠀⡀⢠⣤⠤⠒⠒⠢
⠀⢠⠀⠀⠢⡄⠐⠀⠀⡄⠀⠉⠋⠉⠉⡉⠁⡀⠀⣠⣤⣤⡾⢀⣀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⣠⡼⠃⣀⣀⣶⣶⣿⣿⣿⣷⣦⣤⡈⠃⢄
⠀⠐⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢟⣁⢴⣶⣋⠜⣀⣤⠀⠀⠀⠀⠀⠀⢀⡴⠇⣠⣾⡿⠉⠉⣁⠀⠀⠢⣌⠉⠻⢿⣃⣀
⠀⠀⠈⠠⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣎⣁⣿⠿⣿⣪⣿⣠⣶⣄⠀⠀⠀⠁⡞⢀⣾⠿⣃⠀⠙⡀⠀⠀⠀⠀⠘⠊⠁⠐⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⡤⠼⣿⣿⣷⣾⣿⣿⠿⡁⠋⠛⠀⠀⠄⠸⠀⣾⣿⠀⢻⣷⣾⣯⣄⠀⢀⠀⠀⠀⠀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡆⣶⣿⢿⣾⣛⣿⣟⣻⠏⠘⠲⢄⢸⠀⡇⢸⣿⠿⡄⠀⣿⢹⣿⣇⣰⣿⡇⣾⣀⣤⢸⡿⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⢿⣿⡿⠀⠃⢸⣿⣽⣿⣿⠀⠀⠀⢸⣿⠀⠇⠸⣿⣧⠀⠀⠈⠀⠉⠉⠉⠙⠳⠏⢹⡇⣿⠃⠠⣿⣿
⠀⠀⠀⠰⣀⠈⠀⠀⠀⠀⠠⠤⠈⠑⠀⠀⠾⣟⠓⠀⠃⢐⡻⢬⣿⣿⠀⠁⠀⢸⠀⠀⠀⠈⠿⠿⠂⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢦⡽⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠂⢀⣄⠔⠉⠙⠾⢿⡄⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠙⢦⣀⣀⣤⡆⠀⢀⣠⣦⡴⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⢄⣉⠙⠻⠶⢋⣬⣬⣿⣿⣟⡀
⠀⠀⠀⠀⡀⢀⡤⠄⠔⠈⠘⠠⠀⠤⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⠀⠉⢉⣿⣛⣯⣍⠳⣆
⠀⠀⠀⣯⣀⠲⢾⠂⠠⠀⠀⠀⠀⠀⠁⠀⢂⡀⠀⠈⠃⠀⠀⠀⠀⠀⠁⣄⠀⢀⠀⠀⠀⡈⠂⢠⡴⢶⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⢾⡇
⠀⠀⠁⣾⣬⡤⢾⡠⢤⠁⠀⠀⠀⠀⠀⠂⠤⠳⡆⠠⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠁⠈⠐⢀⡁⠀⠀⢢⡄⠀⠀⠘⠀⠀⢸⠿⣿⣿⣿⢇⡟
⠀⠀⠀⠃⣤⠐⠨⠄⠈⠓⠶⢆⣀⢀⡀⠈⢤⡆⣅⠠⡇⠀⠀⠀⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠼⠃⡀⠀⠀⠀⠀⠀⣴⣿⣿⣧⡾⠁
⠀⠀⠀⠀⢺⣾⠆⠠⢀⠄⠀⠀⠈⠻⠌⡬⠋⠡⠞⣰⣓⠀⢀⣎⠀⠚⣁⣀⣐⣄⣈⣀⡢⠐⡞⠀⠿⠉⡆⠘⠀⠀⠀⠀⠀⠀⠸⣿⠟⠉⡁⠀⣀⣀
⠀⠀⠀⠀⠘⣿⣷⣵⣦⣤⡆⠑⠀⡐⠠⡛⢻⣠⣾⣽⡟⢀⠷⡦⣿⣬⠉⠀⠀⠀⠀⠀⠈⠉⠑⣚⣤⡾⣷⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⣤⣶⠟⠛⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣄⣿⣾⣿⣿⣿⣿⣿⠁⣾⣿⣟⣿⣧⣶⣆⣴⣆⣴⣤⣠⣀⣶⣿⣿⣹⣿⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⣸⡿⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠟⠛⠢⠆⠤⠄⠀⠤⠾⠿⠿⠿⠿⠿

⠀⠀⠀⠀⢰⣿⠿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⢿⣿⢷⡟⢻⣃⣤⣄⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣋⣴⣟⣵⣾⣿⣷⣶⣄⠀⠈⠀⠀⠹⣷⣄⣶⣤⣾⡇⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢺⣿⣿⣿⣷⡄⠀⣀⣿⣿⠿⠛⢋⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠎⣿⣶⣶⣬⣿⣿⣿⣿⠇⠻⠿⠿⠿⣿⢿⣿⣿⣿⣿⡿⠉⠉⠉⠁⠈⠉⠋⣹⣿⡇⢄⠙⣏⠱⠆⠀⢻⡇
⠀⠀⠀⠀⠀⡿⠙⠿⠿⠿⣿⣿⣿⣿⣿⠇⠿⠟⠛⣋⢉⣥⠠⣶⠀⠺⢆⠘⣳⣿⣼⣯⣭⣵⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣯⡄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠐⠇⠈⠉⠀⠁⠀⠀⠀⠀⠳⠌⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⡍⠁⡰⠀⠀⣿⡇
⠀⠀⠀⠀⠀⡏⠀⠀⢀⣀⣠⠤⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣙⣋⠴⠊⠁⢀⣴⣿⡇
⠀⠀⠀⠀⢀⣤⡈⠛⠉⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⢀⣠⣤⣯⣽⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣦⣶⠶⣣⡄⠸⣿⠃⠀⠀⠀⠛⠋⠛⠁
⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⣀⣠⣤⣯⣿⣦⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⠋⠀⠀⠶⠾⣿⣿⣿⡇
⠀⠀⠀⠀⠰⣿⣷⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠯⠉⠙⠛⠛⠛⠉⠉⠛⠉⠁
⠀⠀⠀⠀⠀⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⡔⠀⢰⣶⣶⣶⣶⣶⡀⠐⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⠶⠴⣶⣶⡶⣶⡖⠀⣶⡆
⠀⠀⠀⠀⠀⢃⣿⠁⠈⣿⣿⣿⣿⣿⣧⠀⣨⠈⠛⠂⢼⠀⣨⣿⣿⡿⠗⠀⠓⠘⣿⣀⡩⠀⢻⣿⡿⢿⣿⣏⠁⠍⢩⣿⣧⣠⣾⣿⣄⢸⣿⣻⣟
⠀⠀⠀⠀⠘⣯⣿⡆⠀⣿⣿⣿⣿⡟⠋⠉⠉⠛⣦⠀⠘⠆⠙⢿⠏⠀⠀⠀⢀⠘⢿⣽⠛⠘⠻⢻⡇⠀⢟⢋⣤⡆⢸⣿⣯⣥⣬⠁⢈⣹⣿⡟⢻⡀
⠀⠀⠀⠀⢰⣿⣿⡇⢰⣿⣿⣿⣿⠀⠀⡀⠀⠀⣿⡆⠀⠁⠀⠸⠆⣴⠀⣀⠈⠈⠀⢹⠀⠀⣠⣶⠆⠀⠚⠹⣏⡀⠀⣿⢿⡎⣻⣅⣤⡘⢿⣧⣾⡇
⠀⠀⠀⠀⢸⣇⣹⣿⣿⣿⡏⠛⣿⡇⠀⠀⠀⠀⢽⡃⠀⠀⠸⣷⠠⣿⠀⢿⡆⠀⠀⠘⠁⣰⡟⠁⠀⠀⠀⠀⢹⠇⠀⢻⣿⠡⣿⣿⣿⡇⠀⢿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣟⣾⣿⠄⠀⣿⣧⠀⠀⠀⢀⣾⠀⠀⠀⢀⠀⠀⢠⣿⣀⣀⡀⠀⣤⣿⣿⣧⠀⠀⠀⠃⠀⠀⠀⠀⢸⣿⡀⢾⣿⣿⣇⠀⣸⣿⡇
⠀⠀⠀⠀⢸⣿⣷⣟⣿⣿⠀⢰⣿⣿⠀⣆⠀⣾⣿⠀⢠⣀⣟⠀⠀⢸⣿⡿⢉⠱⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⡄⢠⣤⣾⣿⣿⣿⣯⣽⣟⡀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣇⣀⣹⣿⢿⡇⣿⠀⡿⣿⡆⢸⣿⣿⠀⣄⢸⣿⠀⣼⠀⠀⣿⠛⣿⣿⡅⣴⡄⠀⠀⣧⣼⣿⣿⣿⠑⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⡾⠉⢉⣉⣉⣩⡴⠛⢡⣿⠀⣧⣿⡃⢸⣿⣿⢐⣿⣾⣯⡀⣿⢠⡀⣿⣇⣿⡿⠘⣿⣧⠄⣰⣿⣿⣿⣿⣿⣶⣿⣿⡿⣿⣿⣦⣾⡇
⠀⠀⠀⠀⠈⣿⡇⠀⣘⠛⠛⢻⡷⠀⠘⠛⠁⠙⠛⠃⠘⠛⡙⠉⠛⠛⡛⠃⠛⠺⠛⠛⠻⠟⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠙⠛⢙⠛⠛⠛⠃
⠀⠀⠀⠀⠰⣿⡶⠢⣼⣿⣿⢿⡷⠀⢾⠟⠛⠛⠉⠹⠿⠿⠿⠿⠿⠿⠟⠛⠻⠩⠿⠿⠿⠿⠿⠿⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋⠉⠻⠿⠿⠇
⠀⠀⠀⠀⠐⣶⣶⣶⡄⠀⢀⣀⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢀⣷⣶⣶⣶⠶⠞⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⡆
⠀⠀⠀⠀⢰⣿⢸⣿⢹⣄⠀⠀⠙⠶⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⠋⠀⠀⠂⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡿⣿⢿⣿⡄⣴⠟⢺⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⡭⠹⠿⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⢼⡿⢸⣿⡇⠘⠤⢀⡏⣿⣿⣿⣿⣿⣿⣿⡷⠈⠁⣠⣴⠘⢋⣴⡆⠀⡠⣶⡐⠀⠀⠘⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣷⣾⣷⣾⡿⠁⠀⠀⣫⣠⣌⡿⣿⣿⣿⣿⣿⡀⠀⠀⢿⣿⠀⣟⢻⠃⣠⣅⠉⠉⠀⢀⡀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⢸⡇⡿⠁⠀⠀⠀⠀⡈⢿⣯⣽⣿⣿⣿⡿⠧⡀⣠⣤⣿⠘⣿⣏⣴⡇⢹⣀⢲⡆⢀⣘⢛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⠈⣿⡇⠀⠀⠀⠀⠀⣗⡀⠉⣩⣄⣿⣿⣧⡀⢁⣿⣿⣿⠠⣿⣿⡿⠛⠘⢽⠿⠋⠈⠋⢽⠟⣟⡛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣰⣟⣧⡀⠀⠀⠀⠀⠻⣿⣿⢻⣿⡿⢿⣿⣧⣸⣿⣿⣿⡀⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠁⠘⠛⠁⣿⣿⣨⡻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣥⡿⣿⡇⠀⠀⠀⣠⣄⣿⣿⠉⠉⠁⢨⣿⣿⠋⢿⡿⠛⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠈⣹⣿⣷⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⢿⣷⡟⢁⠀⠀⠀⢿⣿⠛⠋⠀⠀⡇⠸⣿⣿⢇⠟⠁⠀⠀⠀⠀⠉⠷⠤⠀⠀⠠⠤⠀⠀⠾⠯⠥⠰⠿⠿⠿⠼⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⢸⣿⣾⣿⡿⢀⠀⠀⠀⣿⠛⠃⠀⠀⠀⡇⠀⣿⡏⠎⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠸⣿⣿⣿⡿⠛⠀⠀⢰⡏⠀⠀⠀⠀⠰⠃⠀⣿⡇⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⣤⣦⣼⣧⣀
⠀⠀⠀⠀⠀⣿⣽⠟⠁⢀⡄⠀⠘⠉⠂⠀⠀⣀⣀⣠⠤⠛⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⡼⠇⠀⠀⠀⠀⠀⠀⠘⠛⠉⠹⢆
⠀⠀⠀⠀⢰⡿⠁⠀⢠⠋⢠⣀⡀⠀⠀⠀⠈⡻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠉⠛⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢷⣤⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠂⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠓⠘⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠠⠀⠀⠄⠀⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⡿⣀⠀⣤⣄⣴⣿⣿⣷⣶⣶⣿⣿⣷⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡋⠈⣩⣹⣿⣿⣿⠚⢻⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣆⢹⣿⣿⣿⣿⠀⠄⢹⣿⣿⣿⣿⠋⣁⠄⠈⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⡛⢷⣻⣿⣿⣿⣿⣿⠇⠐⠐⠈⢿⣿⣿⣿⡇⢠⡄⢀⡈⢀⣉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡄⢀⡀⢀⠀⠀⢀⣾⣿⣿⣾⣿⣼⣿⣿⣿⣿⡟⠉⣮⣅⢷⠘⠀⠋⢁⣼⣿⣦⡙⠘⣡⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠀⠠⡮⠓⢻⡆⡁⢄⠈⣿⣿⣿⣿⣿⣿⠟⡛⠉⣿⡀⠀⠙⠟⠁⠀⠀⢠⣾⣿⡿⢽⣥⣾⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠰⠟⠉⢡⣤⡄⣀⠀⠈⡟⢠⣾⣿⣿⡿⠿⢿⡟⠁⠀⢳⡄⢿⣿⡇⢸⠀⠀⢀⠀⣾⣿⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠐⢃⠀⠺⠿⠞⠃⠖⠊⣰⣿⣿⣛⣅⠀⠀⠈⠁⠀⢀⣨⠏⠀⠹⡗⠀⣗⠀⠻⠀⠉⢿⣿⣿⣽⡏⠒
⠀⠀⠀⠀⠀⠀⠠⣌⢧⡀⠀⢀⠀⢄⣴⣿⣿⣿⢿⣿⣳⠄⠀⡴⠀⣿⣻⣂⡀⠀⠀⢾⡟⣄⣰⠖⠂⠀⠈⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣶⣿⢸⣿⡏⢡⣾⣿⠋⢠⣾⡀⠀⠉⢻⣿⡇⠀⠠⣼⣿⣿⠁⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣫⣿⣿⡇⢿⣿⣇⣰⣇⠘⣿⠀⠀⠘⣿⠷⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣿⣿⣧⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠹⠷⠄⠀⠀⠀⢀⣀⣬
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⣀⣄⣬⠱⢨⣽⢾⣿⣟⣿⣿⣟⣫⣅⣠⣤⠴⢲⡃⠀⠀⠘⣀⠹⣿⣿⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⣦⣾⣤⣤⡶⠀⢸⣿⣿⣿⣿⣿⣿⣿⣅⣙⣠⡾⠃⡶⣦⣸⣑⣶⣿⣿⣾⣯⣿⣿⣿⠄
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠙⢋⣉⢻⣿⣷⡂⠘⠃⣿⣿⠉⣛⠙⠿⠿⠻⣿⢤⠖⠣⡽⠿⠗⠿⢹⠏⠁⠱⠿⠯
⠀⠀⠀⠂⡀⠀⠀⠀⣀⣦⣤⣽⣿⣿⣿⣷⠆⠐⡘⢻⣿⣿⣷⣦⠋⢸⡿⣋⠽⢿⢿⣿⣿⠙⢃⣤⣴⣿⣷⠂
⠀⠀⠀⠀⠙⣿⣄⡀⠿⣿⣿⠉⠻⢿⣿⣿⣷⣀⣷⣼⣿⣿⣿⡽⢟⣵⣾⣿⠀⠈⠸⣿⣿⣇⣁⣿⣿⣿⣏⠄⠀⡾⢿⡇
⠀⠀⣰⣶⣦⣽⣿⣿⣿⣿⣿⣶⣶⣾⣿⣯⡙⢻⣿⣿⣿⣿⣫⣾⣿⣿⣿⣿⣧⣀⠠⡜⣟⣿⢹⣿⣿⣿⣿⡀
⠀⢸⣿⣿⣿⣿⣿⠛⠛⠙⣿⣿⣿⡟⣹⢦⠤⠙⣿⣿⣿⣿⣿⣿⣿⡿⣟⠿⢿⡿⠀⠀⠈⠉⠉⠻⠿⣿⠿⠁
⠀⠸⣿⡿⠟⠛⢿⣤⣠⡤⣼⢿⣿⡇⣻⢧⡀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣀⡀⠈⣿⣷⡆⠀⠀⠀⢀⡀⣀⣀⣠⠄
⠀⠠⣤⣤⣾⠀⢀⣹⣧⢀⡈⠁⠛⠀⣷⡉⠀⠀⠙⢻⣛⡛⠋⠈⢿⣿⣿⣿⣷⣾⡿⠛⠀⠀⠳⠞⢡⡟⠳⠎
⠀⠀⣋⠉⠀⡷⢀⣿⣿⠋⠀⠢⣄⠀⢣⡀⠠⠀⠰⣾⠟⠀⠀⠀⢀⣬⡝⠉⠉⠁⠄⠀⠀⠀⠀⠀⠁
⠀⠈⠿⠟⢤⡇⢸⣟⠈⠙⠋⠀⠀⠀⣾⣾⣴⠶⡀⠀⣠⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⢿⣷⠜⢻⡷⢄⣀⠀⣰⣾⠸⢿⠒⠚⢻⡶⢤⣁⡀⣲⣿⣿⣇⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣰⣾⣿⣿⠿⢿⣶⣤
⠀⠠⣦⣤⣀⠀⠀⣀⣠⡈⠛⠛⠛⠁⣀⢾⣿⣆⣸⣷⣴⡍⣰⣿⣿⣿⣿⡿⢤⣤⡾⠀⠈⠒⠚⠁⠀⢀⣿⣿⠿⠋
⠀⠀⠙⠻⣿⠟⠁⠉⣻⢿⣿⣿⣿⣾⣉⠙⣿⣋⣙⣿⣿⣁⣿⣿⣿⣿⣿⣿⡄⠉⠀⠀⠀⠀⠀⡄⢀⣿⡟⠁
⠀⠀⠀⠀⠀⢀⣜⢿⣽⣏⣉⣉⡉⠻⣿⣦⠈⠙⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⠿⠅⠀⠀⠀⠀⠀⠓⠋⠁⠁
⠀⢠⣶⣶⠞⠛⠛⠓⠘⠻⠿⠿⠛⠳⠎⠙⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⡁⠐⢷⡄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡆⠹⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠴⠆
⠀⠠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⡳⠛⠉
⠀⠀⠘⢿⣿⡿⣷⣶⡖⠦⠠⡤⣤⣄⣀⠀⢠⣄⠀⠠⠂⠀⠈⠁
⠀⠀⣦⣀⣤⣦⣼⣷⣿⣿⣿⡿⢋⣽⣿⡀⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠴⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣴⣦⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣠⡤⠤⠤⠂
⠀⠀⠀⠀⢸⣿⣿⣽⣧⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⣡⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠘⢛⣭⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⠛⡩⠭⠭⣙⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠙⢻⡏⠀⠀⠀⠀⠀⠀⠐⢷⣦⣀⠀⢿⣿⣧⣎⡤⠒⠢⣌⠳⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣾⣿⣿⣷⣬⣁⣉⢎⠣⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠉⠉⠀⠀⠀⢠⡾⠋⠁⠀⣈⡙⢿⡉⠙⠛⠿⣷⣵⣄⠙⢦⠙⣿⣄
⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠿⠍⣼⠇⠀⠀⠀⠈⠙⣷⣗⡄⢳⢸⣿
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣾⣸⢸⣿⡄⢀
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡸⣧⠙⠛
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣦⣤⣀⠀⠀⠀⠀⢻⣿⣷⣽⣷⡄
⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣷⠉⠀⠀⠀⡾⠟⠻⣿⣿⠿⠊
⠀⠀⠀⠈⠙⠛⠿⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠿⠆⠀⠀⠀⢀⣼⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣀⣀⣠⣤⠶⠛⠁⠀⢀⣼⡇⠀⠀⠀⢿⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠹⣯⠀⠀⠀⣠⣴⣿⣿⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⣿⡿⠛⠁⣰⠟⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢠⣶⣦⣤⣾⣧⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⢀⣾⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠉⢉⣿⣿⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⠀⣸⡟⠀⠀⠀⠀⣀⣴⡿⠋⠉⠉⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⣉⣛⡓⠒⠒⠶⣆⠀⣠⠤⠦⠴⠶⠶⠶⠲⣿⡃⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⡇⠀⣹⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠀⢀⡟⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⠿⠆⣼⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠴⠞⠛⠛⠿⢶⣦⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠷⠶⠶⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠈⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠠⢟⡂⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⠁⠀⠀⠘⠻⠆⠀⠀⠀⠀⠀⠀⠀⠠⠖⠃⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠚⠀⡄⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⡄⠚⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠶⡀⠀⠀⠠
⠀⠐⢫⡀⠁⠀⠐⠀⠀⠁⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠈⢀⡁⠀⠁⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢉⠠⠄⢀⠀⢀⠀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⠀⠀⠀⠤⠐⠈
⠀⠀⠀⠀⠣⡤⠐⠄⠐⠤⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠈⠀⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠉⠉⠉⠈

⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠉⡓⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠉⠃
⠀⠀⡤⠀⠈⠉⠁⠀⠀⠀⠘⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡑⣢⣤⣤
⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢆⠀⠀⢀⠄⠀⠀⠀⠠⠄⠀⢀⠙⠿⠟⠡⠍⠛⠋⡀⠀⠐⠂
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣀⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠘⠦
⠀⠀⠈⠙⠛⠻⣶⢦⣴⣤⣶⡞⠛⠉⡏⠀⠀⠀⠀⢖⠉⢁⡄⠈⠇⣀⡉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂
⠀⠀⠀⠈⠆⠀⢀⠠⠂⠐⠄⣀⠀⠔⠀⠀⠀⠀⠀⠐⠀⡀⠀⠠⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠛⠁⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠁⠁⠒⠒⠁
⠀⠀⠀⠀⡤⠪⢀⡠⠄⢠⠤⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣀⡀
⠀⠀⠀⠸⣡⠀⠴⠂⠀⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⢠⡀⢤⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⡶⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⡤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣴⠷⠒⣖⣻⡚⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⡀
⠀⣠⠄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⡱⢣⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⠀⠀⠀⠀⠀⠂⣤⠜⡇⠀⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠰⣈⡀⠉⠀⠀⡀⠂⠀⠁⢹⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡆⠃⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⢸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠂⢃⡀⠰⠄⠄⡊⠐⠄⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆
⠠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⣠⠣⡈⠐⠁⢠⡈⠲⠞⠀⠐⠛⢱⠀⡄⠃⠀⠄⠀⠀⠀⠐⠢⡀⠀⠘
⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⠈⠀⡎⠀⠉⠀⠀⠈⣞⠀⡘⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠈⠂⠀⠀⠆
⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠤⠤⠤⢬⠇⣀⠀⠀⠠⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁
⠈⠀⠀⠐⠒⠂⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢣⠀⠀⡆⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠤⠔⠓⡐⣠⠤⠤⢇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣰⣦⣻⣷⣾⣷⣶⣦⣠⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣼⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣵⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⡟⠀⠀⣀⣀⣀⠀⠀⠀⢀⡤⠴⠒⠒⠲⣤⣀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⡏⢠⣶⣿⠟⠉⠉⠉⠑⠋⠁⠀⣀⣀⣀⠀⠈⣿⣧⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡟⢠⣿⡟⠁⣠⡴⠞⠛⠓⢦⣴⠋⠁⠀⠈⠻⣄⠈⠋⠀⠀⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠃⠀⠉⠀⣼⠋⠀⢠⣦⠀⠈⣧⠀⠀⠿⠆⠀⣿⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡿⣦⣀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠘⢷⣤⣤⣤⣾⠏⠀⠈⢻⣿⣿⣛⣁⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣺⠟⢱⠀⠀⠀⠀⢻⣏⡓⠂⠀⠀⠀⠀⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⣴⢺⡇⠀⠀⠀⢸⣿⠹⡆⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⣬⣷⣀⣀⣠⣾⢿⡾⠟⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠈⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠐⡀⠀⠀⠀⠀⠀⠂⠙⠻⠿⣇⠀⠀⠀⡀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣥⣀⣀⣀⣀⠀⠀⠀⠀⠲⠈⣷⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁


⠀⣁⡈

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⠐⡄
⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠥⢇⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣉⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢈⢳⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠄⠠⠤
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣧⣤⣼⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⡾⠋⠙⢷⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠘⣧⡀⢀⣸⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⢸⡇⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣤⠞⠉⢸⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⠓⢲⡖⠒⢲⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⣇⣀⣀⣸⡇
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⠷⠶⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣉⣉⣉⣉⣁⣉⣽⡇
⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢧⡀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣶⣶⣦⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣼⣃⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⣟⣁⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡠⠚⠁⠀⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⣭⣭⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠐⠛⣓⡶⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠉⠉⣽⣣⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠬⠭⣯⡉⠛⠒⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠚⠓⢸⡀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⠶⠚⠒⢆⣠⠞⠋⠉⠉⠙⢧⠤⢖⣒⣋⠭⣹⠇⠀⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⣈⠩⠤⠒⠒⠛⠓⠶⠋
⠀⠀⠀⢠⠞⠉⣉⠒⢤⣀⠞⠀⠀⣈⡓⢧⡀⠀⠀⡀⢰⠇⢀⠀⠀⢀⡴⢻⡀
⠀⠀⠀⢻⡀⠈⠑⠛⠆⠋⠀⣀⣼⠉⠉⢲⠙⢦⠤⠟⠋⠓⠿⠶⣶⣋⠀⠀⡇
⠀⠀⠀⠀⠉⠓⠒⠒⠒⢦⡀⠀⠉⠛⢶⡋⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠲⢤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠸⡟⠲⣦⣤⣀⣀⣀⠀⢀⣀⣰⠚⠛⠛⠓⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣄⠘⠶⠳⣀⡝⠳⠏⡙⢯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⢏⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢿⠀⠀⢀⠏⠹⡄⠀⣳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢶⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⠴⠾⠀⠀⣸⡠⠤⢽⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣲⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣧⢸⡄⠀⠀⢠⠿⡀⠉⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠻⣅⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡤⣼⡈⣧⡤⢤⡞⠀⠉⢹⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠏⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⡧⠞⠀⠈⠙⣦⣄⡈⠉⣩⠝⠁⢠⠏⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠁⠢⣀⠀⠀⣰⠛⢻⠙⠛⠁⠀⢠⠏
⠀⠀⠀⠀⢰⡋⠉⠛⣍⣓⣶⡤⢤⣄⣰⣇⠀⢸⣦⠴⠖⣚⠟⠛⠒⡦
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⠏⠀⠉⠙⠓⠒⠒⠋⠁

⠀⠀⠀⠀⠀⣀⣀⡀⢠⣠⣤⣤⣄⣀⣀⠀⢀⣤⣤⠀⣠⣤⠀⣠⢤⠀⠀⠀⡀⢠⣤⣤⣤⡀⢠⣤⣄⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠰⠦⠒⠈⠛⠈⢠⡜⣿⠷⣾⣏⣙⠛⡄⢸⣿⠃⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡄⠹⣿⡄⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣶⠇⣤⡏⠈⠃⠀⠋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⢹⡿⠀⠻⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⢀⠀⢤⠶⠐⠉⠁⠙⠀⠛⠀⠉⠀⠀⠀⠀⠀⢸⣿⡇⢠⣄⠀⠀⠀⠀⠀⠀⢀⠰⣿⡇⠀⠀⠀⠀⠀⣠⠤⠀⡀⠀⠠⠔⠉⠉⠲⠀⠀⢦⡀⣤
⠀⠈⠀⠻⠷⠄⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⡀⢠⠀⢿⣷⠈⣿⣆⠀⢠⣤⣴⢦⣤⡴⠚⠀⠀⠀⣤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢹⣹⡇⠀⠛
⠀⠀⠀⠒⠰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠳⢾⣿⣧⠼⢿⣷⣤⣈⠁⠀⠉⠠⠊⢀⠤⠘⣀⡀⠀⠀⠀⢀⣀⢀⡴⣦⣼⣿⡏⣿⠸⠙⠇⣸
⠀⢀⡀⠆⢠⡄⢠⡄⢠⡀⡀⠀⣴⠟⠛⢷⣿⡀⠀⢨⣿⣇⠀⢀⣿⣋⡁⠀⠀⠀⠀⠀⠘⠀⣿⡿⣷⣤⣾⣷⢾⣽⠋⠷⠋⠈⠁⠁⢀⣀⣠⡿⠁⣠⠀⠀⠄
⠀⢈⡱⠎⠁⠀⠀⠐⢀⠀⢙⢢⣿⣄⠀⠘⠋⠁⠀⠀⠈⠁⠐⢿⣿⠉⠙⣷⠀⠀⠀⢰⡇⠀⢹⣷⠟⠙⠛⠁⠈⠀⢀⣀⠀⣠⣿⣷⠟⠉⠉
⠀⢨⡁⠀⠀⠀⠀⠀⡈⢶⡟⠛⠻⡿⠁⠀⠀⠀⣀⠀⠀⠦⠶⠴⢨⣦⡔⠃⠀⠀⠀⠈⠛⢦⡀⠀⢀⣤⣀⣤⣾⣷⣿⣿⡿⠟⠁
⠀⢨⠾⣅⠶⡀⠞⠋⡀⢻⣄⠀⣀⠀⡀⠦⠀⠀⠉⣀⠃⣀⠀⢸⣼⠏⠈⣿⠀⠀⠀⠀⠀⠀⣻⣿⢻⡏⠻⣿⣿⣿⡿⠛
⠀⠸⢷⡽⢞⡡⢞⣡⡜⢠⣼⠿⠟⠀⠈⠀⠀⣸⣤⣤⣦⣿⣿⣧⢿⡳⠶⠛⠀⠀⠀⠀⠀⢀⣿⠉⠀⠣⠀⠈⠉⠁⠀⠀⠀⠀⢠⣄
⠀⠘⣯⡽⢎⠱⣎⡡⠘⣴⣇⠀⠀⢀⣄⣸⣶⠿⠻⣿⡟⣡⣎⠁⠀⣹⠒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⢸⣯⠁⣤⠀⠉⠱⡛⠄⠻⡗⢲⣿⡏⠈⠁⢰⣷⠈⠇⢨⡿⠗⠚⠳⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣷⣶⣶⠀⠀⢀⠀⠀⠀⣿⣿⡇
⠀⢸⣷⠖⣀⡈⠠⠖⡀⠦⡀⠀⠈⠐⠻⠦⣴⠮⡘⠷⠖⠞⠀⠀⠁⠀⠀⠀⢀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣷⣆⠀⣿⣿⣇⣤
⠀⢸⣿⣿⣿⣷⠼⠿⣧⣴⡅⢀⡐⠛⣿⠆⣉⠤⠹⠄⠃⠐⠀⡁⠐⠀⣀⡴⠛⣤⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣯⣀⣿⣿⣿⣿⠆
⠀⠈⠛⠻⠁⣀⠀⠀⠀⠀⠈⠻⡤⠿⠁⠶⠡⢦⠱⡄⢃⡀⢀⡁⠀⠀⣹⠃⠾⢀⡐⠀⠀⠈⠉⠉⠙⠿⠙⠻⣿⣿⣟⣿⣿⣿⣿⠻⣿⠿⠾⠃
⠀⠀⠘⠀⠀⠛⠻⠳⡶⣠⣄⣀⠀⠂⠀⠀⠀⠀⣄⠰⣄⢠⡄⣴⠀⠀⠉⠲⣀⠘⠱⢆⣠⡀⢀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠋⢀⡤⠤⠄⠀⣄
⠀⠀⠀⠀⢤⣀⡀⠀⠀⠈⠉⠉⢀⡀⠀⠀⠀⠀⣍⡳⢌⡣⡕⠜⣦⠀⠀⠘⠟⠲⠆⢺⢟⡵⢋⡤⢄⠀⠘⠃⠀⠀⠀⣀⠀⠀⢸⣏⣠⠸⣾⣶⢻⠆
⠀⠀⠀⠀⠀⠉⠙⠛⠛⢰⡞⠻⠎⠁⠀⠀⠀⠰⠶⢝⡫⢗⡘⡞⣿⣄⡀⠻⢦⣤⠴⢦⠫⣝⡢⠶⡛⠂⠀⣴⠟⠛⢿⣉⡷⢰⢶⡼⠟⣿⣿⡿⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠂⠀⠀⣨⣩⡥⠶⠋⢸⡾⠋⠀⢹⡿⢾⡏⣴⡆⠛⢶⣭⡅⠈⠀⠀⣿⣄⣀⣼⡏⠳⠟⠉⣁⣠⣿⡿⠧⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣖⠒⠾⣷⣿⠃⠀⠀⡜⠁⢸⣧⣤⣹⠷⠚⠛⣒⡚⡏⠀⠈⠙⠛⠉⢀⠤⠽⣿⡿⢿⣿⠀⠀⠨⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣉⡛⠃⠉⠀⠀⠀⠀⢀⣾⣣⡴⢛⡶⢚⣉⠭⠤⣤⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢀⠈⠃⠀⠐⠀⡂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣩⠌⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⠳⠎⢾⣹⠖⠛⠃⠺⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠉⢨⣨⡅⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠚⠉⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⡆⠂
⠀⠀⠀⠀⠀⡦⠠⠈⡶⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⡤⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢶⠎⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠒⠗⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣀⠀⠢⠤⡀⣾⣿⣿⣦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⣠⠴⠶⢴⠏⠀⠀⠘⠛⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢀⣀⣤⣤⣤⣴⣦⣀⠸⠟⣋⣥⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣄⣸⡁⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠐⠀⠀⠐⠂⠈
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠆⢸⣿⠉⠈⢻⣆⣀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⡘⠁⠀⢁⠙⡉⢧⣠⣦⠠⢀⡀
⠀⢸⣿⣿⣿⣿⣿⡟⠙⠃⠀⢻⠀⠀⠀⢠⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠆⠀⢀⡛⠉⡱⠀⠄⠘⠀⢰⠀⢻⣼⣇⡄
⠀⢸⣏⣽⣿⣿⠛⡋⠈⠁⠀⣦⠠⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢤⡈⠑⢄⠀⠦⠀⠐⢁⣈⣤⣾⢿⡝⠁⠀⠂
⠀⢸⡏⣿⠉⡋⢀⡁⠀⠀⢀⠙⠋⠁⠀⣀⡀⣲⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠳⡞⠒⠙⠦⢾⣿⣴⣦⣶⣾⣩⡿⠟⠃⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠠⡤⠾⠋⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⢆⡛⠛⠛⢛⣛⠉⠁⣠⠂⠐⠂⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⢀⠀⡀⢴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⢬⡷⢶⡶⣿⣻⣿⠉⠉⠀⠀⢀⠀⣀⣸⡧⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⡜⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠠⠀⡳⢾⣽⣿⠿⡿⠀⠀⠀⠀⠀⠙⠿⠿⣷⣾⣿⣿⣶⣠
⠀⠀⠀⠀⠀⠀⢀⡠⠠⢦⣶⣾⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⠲⢆⠀⠈⠉⠉⠋⠉⠁⢀
⠀⠀⠀⠀⠀⠀⢠⠘⢻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣶⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠰⠈⠛⠀⠒⠀⠐⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡶⣻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠂⠐⠀⠙
⠀⠀⠀⠀⠐⠀⠒⠒⠒⠒⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠛⠒⠛⠛⠛⠟⠃
⣶⣶⣶⣾⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢟⢻⡿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡂⠀⡦⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠂⠐⠲⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⢀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠁⠈⠉⠉⠐⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣄
⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠠⠀⠁⠀⡈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠇
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠂⠅

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒⢀⡀⠀⠀⠄⠀⠀⠀⠀⢀⡔⠉⠁⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⢄⡀⠀⠁⢦⡀⠀⢀⡤⠶⢯⡀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣀⠀⢀⠀⠈⠙⠶⡀⠙⠞⠁⠀⠀⠈⣄⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠈⠁⠈⠙⠶⣴⢆⠀⠈⠿⣀⡀⠀⠀⠀⠀⠑⠆⠰⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⢀⠴⢄⠀⠀⠈⠓⣀⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⠶⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢮⠻⢃⠔⠀⠀⠀⠀⠀⠀⠐⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⢘⣧⠾⢡⠑⠒⢠⣀⣀⠀⠀⠀⢠⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣷⠀⠀⡆⠀⠀⠉⠻⠃⡀⠀⠀⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢣⣤⣤⡤⠞⠛⠠⠲⢶⣅⠀⢸⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⣿⡀⠘⣧⠖⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠻⠟⠉⠀⠀⢿⡆⠀⠀⠈⠱⣄
⠀⣤⠋⠉⠉⠁⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠀⡀⠀⠀⠈⢆
⢸⠇⠀⠀⣧⠘⣩⡶⠾⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⡴⠶⠶⠶⢛⡈⡇⠀⠀⢀⠈⢃
⣾⠀⠀⠀⣿⠀⢻⣤⡴⠂⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⢠⡔⠀⠀⠀⢀⠞⠁⡇⠀⠀⢸⠀⠘⡄
⢻⡀⠀⢀⠏⠀⠀⠀⠰⡆⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⢀⡜⢀⡷⠉⠀⠀⢀⠖⠁⠀⠀⢨⠀⠀⠀⡄⠀⢹⡄
⠘⢧⣀⣿⠀⠀⠀⠀⠀⡄⠀⡰⠋⣀⡤⢀⡀⢀⣠⡴⠁⠀⠘⠁⠀⠀⠀⠙⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠈⣧
⠀⠀⢙⡇⠀⠀⡇⠀⠀⣥⡌⠓⢤⡜⠰⠋⠉⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⣀⠀⠀⠀⢀⡟⠀⠀⠀⡧⠀⠀⢸
⣄⠒⠋⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠰⢆⠀⠀⠀⠀⠀⢀⡴⢿⣥⣴⠖⢟⠛⠀⠀⠀⢸⡇⠀⠀⠀⢣⠀⠀⠈
⠀⠐⠀⠀⠀⢰⠀⢸⠀⢸⠀⠀⠀⠀⠀⠑⠦⣤⠴⠞⠁⢠⠁⡼⠇⠘⠀⠁⠀⠀⠀⠀⣷⠀⠀⠀⢸⡄
⠀⠀⠀⠱⡀⢸⡀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠸⡇⠀⢰
⠀⠀⠀⠀⠹⠟⠉⢻⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢄⠀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣢⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣤⠉⡑⠒⠒⠀⠐⠒⠒⠻⣆⠀⠀⢀⣸⠏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠁⠀⠈⠉⠉⠉⠉⠉⠀⠀⠈⢳⠒⢋⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣶⡦⣤⣤⣤⡤⡶⠂⠀⠈⢋⠙⣛⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⡾⠃⠀⠀⠀⠀⣰⠀⠀⢀⡰⢍⠓⠋⠁⢁⠀⠀⡄⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⣠⠞⠁⠀⠀⠀⠀⣸⠇⠀⢀⡞⠀⠀⢡⠀⠀⠈⠙⠣⣄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠚⣧⠀⠀⠀⠀⢀⡴⠁⠀⠀⢸⠀⠀⠀⠈⣦⠀⠀⠀⠀⠙⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠻⡆⠀⠀⠀⢸⡃⠀⠀⠸⣇⠀⠀⠀⠀⠈⠳⠦⣤⣀⣀⣀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣏⠀⠀⠀⠈⢳⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⡏⠀⠀⠀⠀⠀⠙⣆⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠘⣦⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⣸⡂⢠⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠉⠙⠋⠁⢸⠀⠀⠀⠀⠀⣠⣸⣇⣀⣸⡃⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣾⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇⠉⠉⠉⠉⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣀⣼⠇⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣨⠿⠚⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣤⠀⠀⠀⠀⠀⠠⢤⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠓⠺⣇⣀⣀⣀⣀⣀⣠⣤⣤⣤⡒⠶⠶⠏⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⢿⣿⡿⠟⢿⣿⣿⣿⣿⠿⢿⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣥⣦⣼⣧⣠⣾⣠⣴⣿⣼⣴⣿⣥⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣤⠀⠀⢀⡠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣄⠠⠤⠐⠉⠀⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣐⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠀⠀⡀⠃⢀⠀⢲⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠐⠎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢠⠆⡀⠀⠑⠠⢀⠴⠢⢄⠀⠈⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠴⠒⠂⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠄⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢊⡀⠀⠀⠀⠀⠀⠀⢄⡠⠛⠉⠀⠀⠀⠀⠈⠉⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⣀⡉⠊⠀⠀⠣⠤⠤⠀⣂⣀⡀⠠⠤⠤⠤⠤⠤⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢⡤⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠤⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠠
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⢀⢀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠄
⠀⠀⠈⠢⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠂⠒⠒⠂⠒⠂⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⣉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠏⢹⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠋⢹⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠓⡂⠠⢄⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⠈⠰⠃⠀⠀⠀⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠲⠄⠀⠀⠀⠀⠴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⣿⠀⠀⡷⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠐⠢⠀⠀⠀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢺⠀⠀⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⣠⡞⢅⡀
⠀⠀⠀⠀⢀⠄⠒⢉⠀⠀⠚⠓⠒⠻⢦⠀⠀⠀⢦⡄⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⢩⢉⡏⢀⡇⠀⠙⢦⡀
⠀⠀⠀⠀⠠⠊⠀⠁⠀⢀⡀⠀⠀⠀⠀⠠⣄⡀⠤⣌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠛⣄⡌⢠⠎⠀⠀⣄⠀⠑⢄
⠀⠀⠀⠀⠰⠀⡐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠐⡄⠈⡇⠀⠘⠗⠁⠀⠀⠀⢏⢱⡄
⠀⠀⠀⣂⡀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢘⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⣠
⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢺⠀⡀⡈⠀⠀⠈⠛⠄⠀⠀⠀⠀⠀⠀⠲⠾⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠇
⠀⠀⠀⠀⢢⠀⠀⢰⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢰⡄⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢺⡏⠁
⠀⠀⠀⠀⠠⠀⠀⠘⠀⠀⠀⢻⠉⡗⠐⠴⠀⠀⠀⠀⠀⠠⢫⡇⠀⠀⠀⠀⠀⠀⠀⢉⠉⠛⢛⡓⠒⠒⠒⢾⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⡎⣧⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⢳⡄⠀⠀⢸
⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⢸⡄⡏⠀⠀⢈⡇
⠀⠄⠈⠀⠀⠀⢀⠀⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠖⠀⠀⠀⠀⢀⠴⠒⠉⠉⢉⡙⣧⠷⠦⠴⠖⢋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣣⣾⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣛⣛⡛⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⢹⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣷⢹⣿⣿⠿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣸⣿⣿⡇⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⣿⣿⢳⣿⣿⣿⡷⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⣿⣿⢸⣿⣿⡿⠁⠀⠀⣿⡇⣿⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢸⣿⣿⣇⣿⣿⣸⣿⣿⡇⠀⠀⡀⣿⣧⣿⣧⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣛⣛⣛⣛⡛⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣻⣿⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉

⠀⠀⠀⢀⡀⠄⢠⣤⣤⣤⢤⡄⢀⣤⠄⣤⡤⠀⠠⠀⠀⠀⠤⢤⣤⡄⢤⣤⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠉⠁⠁⠀⠀⣞⢳⠧⢍⡙⠲⠸⡏⢰⡏⠀⠀⠀⠀⠀⠀⠀⠠⡈⢿⡄⢻⡇⠹⠃⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠠⠐⠊⠉⠉⠈⢀⠀⠀⠀⠀⣾⡇⢠⡀⠀⠀⠀⠀⠀⣴⡇⠈⠁⠀⠀⡀⠄⡀⠄⠀⠒⠢⠀⠀⢄⣀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣷⠘⣷⡀⠲⣶⠶⣶⠖⠁⠀⠠⠂⠊⠀⠀⠀⠀⠀⢀⣀⣤⢸⢸⠀⠈
⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⠉⠙⣿⡟⠙⣿⠷⠀⠀⠀⠀⠀⠂⣤⣦⢀⣤⣠⢶⢴⡿⠸⠹⠘⠈⢨⡔⢃
⠀⣘⠔⠊⠀⠁⠈⠐⢦⣿⠀⠀⠟⠂⠈⠉⠁⠲⣿⠒⢦⠀⠀⠀⡄⠀⣿⡽⠻⠟⠈⠉⢀⠀⣀⣶⡴⠚⠋⠁⠈
⠀⢭⠀⠀⠀⢀⠄⡾⠉⠙⠋⠀⠀⠀⠀⠄⠢⠐⢠⡖⣊⡀⠀⠀⠙⠢⣀⣀⣤⣤⣾⣷⣿⡿⠛
⠀⣪⠷⡡⣜⠄⠠⠳⣤⣤⠀⠀⠀⠀⠀⣀⣤⣄⣺⣁⣸⠄⠀⠀⠀⠀⡿⠋⡂⠉⠻⠛⠋⠀⠀⢀
⠀⢶⡻⢞⣬⠂⠑⣹⠁⠀⢀⢀⣴⠾⢿⡿⣩⡛⠉⡲⠀⠀⠀⠀⠀⣶⣷⣤⡄⠀⠀⠀⠀⠀⠀⢸⡆
⠀⣻⠁⠀⠈⠁⠙⠘⠳⢒⢿⠁⠁⡶⡆⠁⣸⠛⠚⠃⠀⠀⠀⠀⠀⣩⣿⣿⣷⣶⡆⠀⣄⠀⠀⣿⡇
⠀⣽⣏⣶⡄⣤⡄⡀⠀⠀⣈⡋⠯⠁⠝⠉⢃⠀⠀⠀⠀⣠⠀⠀⣶⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⣿⣷⣶
⠀⠻⠿⠋⡉⠀⠉⠋⠲⡠⠖⠡⠢⡔⣈⢀⠀⠐⠀⠀⡿⢠⠁⠄⠈⠛⠛⠛⠿⠻⢷⣶⣿⣿⣾⡿⢿⡿⠴
⠀⠀⠁⠈⠛⠓⠖⣤⡄⠀⠀⠀⠀⠀⣄⢤⢤⢴⠀⠈⠑⣄⠈⠲⣠⡄⣀⠀⠀⠀⠘⠛⠛⠛⠉⢁⡀⠀⠀⢠⡀
⠀⠀⠀⠂⠴⢄⠄⣀⣤⡰⠄⠀⠀⢐⡮⣓⠕⢅⢷⠀⢠⡉⠉⡀⣫⢎⢑⣲⠀⠀⣀⣤⡤⣄⢀⣛⣶⣌⣾⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣐⡏⠯⣊⢱⠟⠓⣶⣼⡏⠠⡒⠝⣏⠀⠀⢸⡇⠀⣻⠻⡼⠛⢁⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⣍⣥⡏⠀⠀⠃⠈⡀⠂⣈⡭⠭⠚⠂⠀⠉⠛⣁⠀⠒⠶⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠭⠎⠀⠀⠀⠀⠀⡾⠢⡌⢐⢬⣭⣔⣒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⡅⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠅
⠀⠀⠀⠀⠤⠀⡶⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠀⢀⣠⣤⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⡠⠤⠀⡉⠀⠀⠁
⠀⢀⣐⣃⣂⣉⡀⢿⠿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠞⠁⠉⠁
⠀⣿⣿⣿⣿⣿⠿⠀⣿⠉⠙⢦⢀⡀⠀⠀⠀⠘⠀⠀⠀⢠⠃⠈⠀⢀⠂⠀⠈⠀⠩⢤⡠⠄⡀
⠀⣿⢿⣿⣿⠿⠉⠀⠘⠂⠀⠀⠾⠟⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢌⠀⠁⠀⠀⠀⠆⢸⣷⠄
⠀⢻⡟⡟⠃⠀⠀⠀⠸⠞⠂⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠐⢄⡡⣀⣀⣠⣆⣼⡯⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠒⣋⡀⠀⠀⠀⠀⠀⠀⠀⠣⡀⣠⡐⠙⢛⣋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣁⠻⣹⣽⡏⠀⠀⠀⠀⣤⣾⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠜⠛⠉⠁⠀⠀⠀⠲⢩⠙⠙⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠐⠾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠳⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣽⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠉⠀⠀⠀⠀⠡⡇⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡨⠝⠛⠛⠂⠀⠀⠀⡀⠁⠀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠔⠁⠈⠑⢄⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠐⠁⠀⠀⠄⠀⠀⠀⠀⠀⠤⠱⡀⠀⠳⡄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⡀⠈⠏⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⢁⠨⠀⠀⠀⢀⣀⠀⠀⢀⠃⠀⢀⠀⠀⠀⠀⠀⠈⠈⠢⢤⠤⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢍⣁⡐⠀⠤⣧⡇⠁⠀⠀⠘⠁⠀⠀⠁⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠘⢳⠖⢀⠔⠚⡇⠀⠡
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠦⢀⣇⠀⠁⠈⠋⠒⠤⠤⢀⡀⠠⠀⠀⠀⠀⠀⠀⢰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠐⢇⣀⠎⠀⠀⠰⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠐⠓⠒⠓⠒⠒⣾⣧⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⡄⠸⢀⡠⠔⠢⣄⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢺⡁⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⢾⠟⠋⠉⠉⠉⢀⠀⠀⠀⠀⠀⢀⠇⠖⠁⠤⠴⠀⡟⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠤⡕⡀⠀⠀⠀⠀⠀⠈⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⡜⢠⡄⠀⠀⢀⠔⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢆⠀⣸⡌⢢⡀⠀⠀⠀⠀⣅⠀⠀⠀⠠⠐⠐⠀⠤⠖⠑⠂⠒⢒⡉⠟⢠⠎⣈⠟⠋⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠩⠥⠼⣄⠙⠢⣀⠠⡤⠘⠟⠀⠂⠀⠀⠀⠀⠀⠀⠍⠀⡀⠈⠀⣤⡣⠊⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⠐⠂⠀⠀⠈⠁⠀⠀⠈⠀⠐⠀⠀⠀⡠⠔⠊⠀⠀⠀⠉⠀⠀⠁⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⡆⢣⣀⠀⠇⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⣠⠆⠀⠀⠀⢸⢷⡷⠜⠀⠹⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠌⠀⠀⠀⠀⢸⠸⠁⠀⢀⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣤⠀⠀⠌⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠺⠀⠀⠀⠀⢀⠄⠁⡆⢿⣿⡏⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠖⠀⠉⠁⠀⠀⠹⡀⠳⠀⢀⠔⠁⠀⠀⠹⠇⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⢱⠴⢇⠀⠀⠀⠀⠀⠀⠈⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣤⣀⣤⣴⣾⠥⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢫⣍⢳⠖⠯⠭⣽⣿⣛⣛⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡓⠀⠸⠆⠄⠀⠀⠀⠀⢐⠒⠊⠉⠉⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠒⠂⠐⠀⠂⠆⠂⠒⠀⠀⠂⠀⠐⠐⠐⠐⠂⠀⠀⠀⠀⠐⠀⠃⠃⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⡛⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⣈⣀⣈⣀⣈⡉⠳⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠉⠉⠉⠉⠉⠉⠛⠲⢤⣀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢸⠦⣀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⣄⠈⠉⠛⠛⠛⠛⠉⠉⢀⣼⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⠃⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⡾⠥⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣁⡀⠀⠀⠈⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠁⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠈⠙⠛⠳⠶⠶⠞⢛⡿⠋⠙⠓⢦⡀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⣠⠤⠶⠒⠲⠶⠤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠁⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠈⢻⣄
⢀⡀⢀⠔⠉⡀⠀⠐⣆⠀⠀⢢⡀⠈⠙⠛⠓⠒⠛⠛⠋⠉⢸⠃⣀⣀⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⢀⡠⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢨⠿⣆
⡟⠉⠿⠴⢾⣇⣀⣠⡟⢷⠶⢿⣥⣀⡀⢸⡟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⡤⠖⠋⠀⠀⠀⠀⠈⣷⠀⠀⠀⠒⠒⠁⠀⢻⡄
⠷⠤⠤⢤⣤⣭⣭⣉⣀⣀⣀⡀⠀⠈⠉⠛⢿⣄⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠟⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢳⠀⠢⡤⣙⠳⣤⣠⣶⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠉⠓⠧⣈⠳⣤⣤⣤⣤⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⢻⡄⢹⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿⠀⠀⠀⠀⠈⣹⠦⣀⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⢀⣀⡤⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⢿⠀⠈⠛⠞⠁⠀⠀⣀⣀⣠⣀⣤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣿⠃⢸⠁⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡿⠀⡿⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⢀⣾⠁⣸⠇⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠷⣦⣀⠀⠈⠙⠛⠒⠒⠒⠒⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⣠⠾⠛⢶⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠁⠀⠀⠀⠈⠉⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⠟⠛⠛⠛⠛⠛⠛⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡇⢸⠁⢀⣠⣤⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⣄⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢻⡈⠉⠉⠉⠉⡷⠋⠁⠀⠈⢻⣇⣀⡀⣤⠀⠀⠀⢰⡀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⡦⢷⣄⠀⠀⡼⠁⢀⡴⢶⡄⠈⣯⠉⠙⠛⠓⠶⠶⠾⠟⠛⠛⠉⠛⠻⠶⠶⣦⣤⣤⣤⠖⠛⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠛⠛⢻⠳⢶⡇⠀⢸⡀⣸⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣠⠶⣄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣠⡴⠏⠀⠘⡇⠀⠈⠛⠁⠀⢰⡟⠛⠓⠒⠶⠶⠶⠤⠤⠤⣤⣤⣤⣄⣀⣀⣀⣸⡇⠀⣧⣠⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠉⠹⣇⠀⠈⠁⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠿⠳⣦⣤⣴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠂⠈⠉⠉⠉⠀⠒⠢⢄⡀
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⣀⣤⣤⣀⠀⡶⠤⣀⠀⠈⠲⣄
⠀⠀⡠⠊⠀⠀⠀⣠⠔⠛⠉⠉⠉⠉⠙⠳⢯⣊⣓⣄⠀⠈⠢⡀
⠀⡜⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠙⢯⣾⢧⠀⠀⠑⡀
⣸⠃⠀⠀⢀⠃⠀⠀⠀⠀⠀⡠⢊⢔⡡⠀⠀⠀⠀⠙⣯⣣⠀⠀⢷
⣯⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠔⠁⠉⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⡆
⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠀⠄⠀⠀⣧⡇⠀⢸⡇
⢸⣥⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢊⠄⡠⠂⠀⢠⣷⠃⠀⢸
⠀⢳⡅⠀⠀⠐⡀⠀⠀⠀⠀⡠⠊⠊⠐⡡⠊⠀⠀⠀⣾⠋⠀⢀⡟
⠀⠀⠳⡵⣄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠞⠁⠀⢀⣾⣷⢄⡀
⠀⠀⠀⠙⢧⡧⡄⠀⠀⠑⠂⠤⠄⠤⠤⠤⠐⠊⠁⠀⣀⣴⣾⣿⣿⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠙⠻⢥⣅⣄⡦⠀⠀⡄⢀⠀⣤⣤⣶⡿⠟⠉⣟⣫⣃⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠚⠚⠛⠉⠁⠀⠀⠀⠘⢗⢵⠤⠄⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣖⠒⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⡫⡀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡴⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⡣⡂⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡢⣂⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡷⣠⡄⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠒⠀⣀⠉⠁⠠⠄⠀⠀⠀⠀⠀⣀⣠⠛⠋⠈⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠢⡄⠀⠀⠀⢀⣠⣤⣮⠀⠀⠀⠀⠀⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡳⠆⠉⠀⠀⠈⠉⠲⣤⡀⠈⢢⣤⠖⠉⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⠠⠶⠶⢆⣀⢀⡀⠀⠈⢣⣰⠇⠀⠀⠀⠀⠀⠋⢦⡀⠀⠀⣤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠈⠻⣟⠦⠀⠀⠙⢧⡤⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⢾⠃⠀⠀⠐⠒⢦⡀⠀⠀⠈⠒⠂⣄⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣸⣧⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠶⠂⣦⠀⠀⠀⠀⠀⠈⠉⠀⠤⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠖⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⣦⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠚⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢀⡄⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⣸⣷⠛⠱⡎⠉⠁⠶⣤⣤⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣷⠀⠀⢸⡄⠀⠀⠈⠛⠛⠀⠀⠀⠀⡧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠸⣧⣤⣤⣤⣬⣍⡁⠀⢠⣶⡃⠀⠈⣦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⡀⠹⣟⡁⠹⣧⡀⠶⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⣧⡼⠏⠀⠀⢹⡇⠀⠀⠈⠙⢦⡀
⠀⠀⣠⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠹⣄
⠀⡼⠃⠀⢠⡀⢀⡴⠛⠳⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⠀⠀⠀⠀⠸⣆
⢸⡇⠀⠀⠸⣦⠟⢰⡟⠛⠛⠋⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠤⠖⡿⠟⠛⠛⠛⠋⣤⠋⠀⠀⠀⢾⠀⠸⣆
⣿⠀⠀⠀⢰⡟⠀⠸⣦⣴⡖⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠇⢀⡴⠋⠀⠀⠀⢀⡴⠃⠀⠂⠀⠀⠀⡆⠀⢻⡀
⢻⡄⠀⠀⢸⠇⠀⠀⠀⠀⢰⠀⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡿⠋⠀⠀⠀⢀⡾⠋⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⢿
⠘⢧⡀⢰⡟⠀⠀⠀⠀⠀⢸⠃⠀⢰⡿⡀⢀⣤⠀⠀⠀⠀⣠⠞⠃⠀⢰⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠸⡇
⠀⠈⠙⢾⠇⠀⠀⡀⠀⠀⢸⡀⠸⢧⡀⠙⡏⣰⠿⠶⠶⠎⠁⠀⠀⠀⢺⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⢱
⠀⢀⣠⡿⠀⠀⢠⡇⠀⡀⢸⡟⠃⠀⠘⢿⣠⠏⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⣀⣠⣶⣦⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠶⠀⠉⠀⠀⠀⢸⡇⠀⡇⢸⡇⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⢀⣤⡴⢻⡝⣿⣿⠋⢩⢷⡈⠀⠀⠀⠀⢹⡇⠀⠀⠀⢸⡏⠀⠀⢸⡇
⠀⠈⠳⣄⡀⠀⣼⠁⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠉⠲⠷⠶⠛⠉⠀⣸⠂⢼⡟⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣷⠀⠀⢸⡇
⠀⠀⠀⠈⢣⡀⣿⣀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢿⡄⠀⢸⠃
⠀⠀⠀⠀⠈⠹⡿⠈⢻⠃⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢷⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢂⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡉⠂⠾⠆⠀⢀⣀⣀⣀⣀⠸⣷⡀⠀⠀⠀⢀⣼⠷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠛⠓⠂⠀⠀⠠⠍⠉⠉⠐⠛⢷⣄⣀⣤⣾⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⣤⡐⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣹⠟⠓⠶⠶⠶⠚⣿⠃⠀⠀⠀⣰⡄⢻⣀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⠞⢡⠟⠋⠁⠀⣏⠀⠀⠤⠄⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⣰⠇⠀⠀⢰⠏⠀⠀⢶⠀⠀⠀⠈⠉⠿⣤⠄⢿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠠⢾⡁⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢠⡏⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠻⠦⠾⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠸⣿⡄⠀⠀⠀⠀⡼⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⠘⢷⣤⣀⡀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠘⣧⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⠶⠶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⡇⠀⠀⢸⠇⢄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠛⠛⠛⠛⠓⠀⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇⠘⠛⠶⠾⠟⠂⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠚⠛⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣠⣴⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠤⠤⠄⣀⣀⣀⣀⣀⣀⢠⣤⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢤⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠛⠛⢿⣄⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣞⠻⠷⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠛⠛⠛⠒⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣦⣶⣶⣶⣶⣶⣾⣿⡏⣴⠟⢷⡌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣍⣭⣍⣭⣭⣽⣿⣿⣿⣧⡘⠳⠛⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠻⣿⠟⢛⣿⣿⡿⠟⢋⣼⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠰⣿⣭⣤⣴⣾⠟⢛⢛⢛⣛⠋⣉⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣴⣿⣷⣬⣿⣿⠛⣉⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣤⠀⠀⠀⢠⣤⡄⠀⠉⠙⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣿⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⢶⠋⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢸⡇⢀⡇⠀⠀⠀⠀⠀⠀⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠁⠁⢼⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⠈⠀⠀⠀⠀⢸⣿⠙⢿⣿⣦⣄⠀⢸⠀⠈⠀⣠⣾⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠻⢿⣿⣿⣶⣶⡿⠿⢿⣷⣦⣤⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⢠⣿⡏⠀⠀⣶⠚⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠾⠿⠷⣄⣀⣸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢳⣠⡞⠁⠀⣀⣀⡀⠀⠀⣹⠟⣧⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠀⢀⣼⣿⣀⣀⣀⠀⣶⠞⢧⡀⠀⠀⠀⡟⠀⠀⢿⣮⢠⡾⢟⣛⣿⠿⠾⠟⢲⣿⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢶⣶⣶⡾⠛⠉⠀⠀⠀⠉⠓⢿⣿⡧⠹⣦⠀⢰⡇⠀⠀⠈⣿⣿⡁⠘⣧⠀⠀⠀⠀⢸⣿⣇⣀⣸⣏⠒
⠀⠀⠰⡦⠞⡆⠀⠀⠀⠀⠀⠀⢹⣿⡠⠀⣠⣤⣦⣤⡀⠀⠀⢻⡏⢄⠈⠳⣼⠁⠀⠀⠀⠘⣿⠳⣤⣿⡾⠟⠀⠀⣼⠙⣏⠁⠀⢿⣤⣤⣀⣀
⠀⠀⠰⠧⢄⡶⠀⠀⠀⠀⠀⠀⣿⣿⣟⣾⠏⠁⠀⠉⢻⡆⠀⠀⢿⣡⡄⠀⠉⠀⠀⣿⡀⠀⠸⣇⠀⠀⠀⢀⣠⡾⣫⡴⣿⣶⠶⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣯⣿⠀⠀⠀⠀⠀⡿⠀⠀⣸⡟⣿⣦⡀⠀⢸⡟⣷⣀⣀⣽⡿⠿⠿⠟⠋⢰⡟
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣠⠞⠁⠀⢀⣿⣿⣿⠙⢿⣤⣾⠀⠘⠛⠛⠉⠁⠀⢄⡀⠀⠀⢳⡀⠀⠀⠀⠀⣨⠁⠸⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡤⠊⢻⣿⣿⣷⠀⠀⠀⠀⠀⣠⣾⡿⠿⠟⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠵⣎⠹⣍⠉⠉⠉⠁⠀⠀⠀⠀⠱⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠈⠛⣿⣷⣶⣶⡶⠟⠛⠻⣷⡄⠀⠀⠀⠀⢀⣠⣤⢤⣄⡀⢸⠀⠈⢧⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠟⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢀⡴⠋⠁⠀⣀⡀⠈⠲⡄⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠘⡇⢀⡞⠁⢠⡞⠋⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠋⠀⢲⠅⠀⠀⠀⠀⠀⣿⡞⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⢤⣰⠇⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠿⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⢆⡜⠱⠔⠶⠀⠀⠐⣿⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠉⠀⠒⠀⣠⠞⠁⡴⡒⠢⠀⠀⠀⠇⣠⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠐⣫⣀⠀⡦
⠀⠀⠀⠀⡠⠛⠲⠎⠁⠀⠀⠣⢀⠔⠀⠀⠀⠀⠁⠀⣠⠀⠀⠀⡴⠊⠑⡤⠀⡀⢀⡀⠀⠀⠀⠀⡠⢀⣟⣊⡁
⠀⢧⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠲⠛⠀⠀⢸⠇⠈⢰⠇⠈⠀⡼⢸⢃⡎⠀⣴⣷⣯⠛⠁
⠀⠀⠁⠀⠀⢤⡄⢀⠀⢼⠀⡖⠋⠸⡆⠳⠀⠀⠀⠾⡠⠀⠀⠘⢦⠀⣾⢰⡄⢀⠧⣿⢼⡀⠀⠙⠛⠁⠀⠀⠀⠋⠐⠆
⠀⠘⢶⡄⠋⠁⡇⠈⢣⢸⡄⠒⠃⢢⣇⡐⠒⠖⠃⠀⠀⠀⠀⡀⣀⠀⠀⠉⠉⠙⠚⠁⠦⠀⢠⡖⢙⠒⠦⠀⠀⠈⠉
⠀⠀⠀⡇⠀⠀⢹⠐⣈⠔⠧⠖⠚⠁⠈⠀⣠⣤⣄⠀⠀⠀⠘⠆⠤⠃⠀⠀⣠⢦⣄⣠⠦⡄⢾⡀⠄⠁⡸⠀⠀⠀⢀⡄⡀
⠀⠀⣀⣃⠤⠞⠘⠋⠁⠀⠀⣀⣀⡀⠀⠀⠛⠟⠛⣠⢶⣶⣦⡀⠀⠀⢀⣤⠯⢄⣠⣄⠀⠑⢤⡑⠒⠚⠀⠀⠠⢊⡅⢠⠟
⠀⠀⠉⠀⠀⠀⠀⠀⢀⠔⠈⠁⠈⢻⡿⠦⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⢇⠀⢸⡀⢀⡇⠀⡼⠇⠀⠀⡰⢾⠶⣫⠖⠁
⠀⠀⣰⡆⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⠛⠛⠛⠁⣀⡀⠀⠈⣗⠀⢉⣉⠀⢠⠁⠀⠀⠀⠛⠾⠋⠁
⠀⠀⠈⠿⣷⡄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢰⡟⠉⠛⣦⠀⠈⠓⠁⢈⡙⡁⠀⠀⠀⠆⠀⠀⠰
⠀⠀⠀⠀⠉⠀⠀⠀⠈⢣⡀⠀⡆⠀⠀⡜⠁⠀⠀⠀⠀⠀⠈⠷⣤⣄⣼⠀⠀⠀⠀⠀⠀⠰⢐⡖⠂⠰⠐⡔⣦⣼⡃
⠀⢠⢤⡔⣖⡤⢠⣤⠄⠀⢧⠀⠳⢀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣀⣀⡀⠀⠈⠀⠉⠈⢀⠈⠀⠁
⠀⠘⣎⢱⡇⡀⠨⡥⠀⠀⣾⡘⢸⣾⠁⠀⢀⣾⡿⠿⢿⣶⡄⠀⠀⣰⠿⠛⠛⠻⠿⠆⠀⠀⠀⢸⣦⣴⣴⣶⣷⣷⣶⣿⡦
⠀⢀⢘⠈⣇⠃⢀⠈⠃⠀⢹⣯⣿⣿⠃⠀⢸⣿⠀⠀⠀⢹⣿⡀⠀⢿⣤⣀⣀⡀⣸⡟⠀⠀⣀⣈⡉⢉⣡⣀⣀
⠀⠈⠁⠀⠉⠁⠈⠉⠀⠀⠀⠉⠉⠁⠀⠀⠈⠻⣷⣄⡀⠀⣿⡇⠀⠀⠙⠛⠛⣿⡿⢀⣴⠀⢇⢈⡩⠿⠁⣁⠘⠢⡀
⠀⢀⡰⢂⣻⠃⠀⠀⢀⣀⠀⠀⠀⡖⠀⠀⠀⢀⠈⠛⠿⢷⢿⠿⠗⠂⣀⡀⠀⠙⠀⣨⠐⠠⣨⠀⠀⣤⠲⠀⠀⠐⢙⡆
⠀⠹⠟⠉⢡⠄⣤⠤⠿⢨⠀⠀⠸⡧⠈⠒⡆⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃⠐⠦⠀⢰⣇⡀⢀⢸⠁⣸⢡⡈⣨⠃⣨⡜⢰⡄
⠀⠀⠠⠠⠌⠀⠀⠀⠐⠀⠀⠀⠀⢳⡀⢰⡃⠀⠀⠀⡀⣠⢤⠀⠀⡧⠖⠀⠲⠀⠀⣄⠄⠈⢺⣤⢀⢫⠇⡀⠘⠀⠀⣜⠁
⠀⠀⢸⡀⠀⡆⣆⣆⢠⡀⠀⠀⠀⣎⠀⠠⢥⠀⠀⢈⠎⠁⠐⠖⠀⠁⠰⠀⠀⠀⠀⢹⡀⠐⠉⡀⠒⣻⡞⠁⠟⠀⡄⢸
⠀⠀⠨⡿⠠⡿⠽⣿⢼⣧⠀⠀⠀⠈⠉⠁⡾⠀⠀⠈⠐⢋⡠⠤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢏⡙⣍⠪⠇⠱⣇⣠⣊⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢶⣤⣀⠀⠀⢀⣤⣤⣤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠈⠙⠷⣴⡟⠁⠀⠀⠙⢿⣮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⠶⠶⠶⠶⠶⣾⣷⣀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢞⠋⠁⠀⠀⠻⢷⣦⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⢰⠟⢉⡙⢷⣴⠞⠛⠳⣌⠁⠀⠀⠀⠀⠀⠹⣧
⠀⣀⣤⣀⠀⢀⡞⢉⡗⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢹⣄⠘⢋⣼⣧⠀⠻⢇⡿⠀⠀⡄⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠲⣆
⠀⠻⣄⣈⡙⠾⡄⠾⢦⣄⣠⣤⣤⣤⣄⣸⡿⠀⠀⠀⠀⠀⠀⠙⠋⠉⣹⠏⠻⠟⠛⠁⠀⣸⣧⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣿
⡴⠒⠒⣺⡇⠀⣇⠀⠀⢉⣀⣠⡤⠤⠤⢼⡇⠀⠀⠀⠀⣴⠟⠛⠛⠛⠁⠀⡀⠀⠀⠀⣴⠋⣿⠀⠀⠀⠀⠀⣿⣟⣛⣛⣛⠛⠒⠶⡞⠋⠛⣰⣇⣀⣀⡀
⠉⢛⡩⠽⢃⣤⣀⣀⡴⠋⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣠⡿⠀⣠⡾⠁⠀⣿⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⠓⣦⠀⠀⠀⣿⠀⢦⣈⠉⠳
⠀⠘⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⣥⡴⣾⠋⠀⣠⡞⢹⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠘⣦⡀⠐⠋⠀⣼⠙⠻⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⠉⠀⣼⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⢧⠙⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠋⢠⣇⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣀⡆⢠⠄⠀⢀⠀⢀⠀⢀⣐⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⣿⠻⠶⢿⡄⠸⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢿⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠾⠦⠸⣇⠀⢸⡇⠸⠿⠶⠶⢶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣛⣋⣉⣁⣀⣀⣀⣀⣀⣤⠟⠀⠸⢧⣤⣤⣄⣀⣀⣀⣀⣉⣹⣷⡄
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡿⠛⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠀⣠⣾⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⠀⢠⣄⠀⠙⢿⣿⣿⣆⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣷⠀⠀⢻⣿⣿⣆⠀⢻⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢿⣿⣿⡀⠘⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠇⠀⣿⣿
⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡀⠀⢀⣿⣿⡿⠀⢰⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢻⣿⣿⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠟⠁⢠⣾⣿⣿⠃⠀⠾⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠹⣿⣿⣷⣤⡀⠉⠉⠉⠉⠉⠉⠀⠈⠀⣠⣴⣿⣿⣿⣿⣿⣦⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣭⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣇⣀⠀⢀⣴⣿⣿⣿⣷⣄⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣤⠀⠀⠀⠀⠀⢀⣰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⣿⣿⣿⣿⣿⣿⣿⣷⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣷⠀⠀⢙⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⢩⣿⣿⢿⡿⣿⢿⣿⣮⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠰⢻⡟⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣚⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⣤⡄⠀⠀⠀⠀⠀⢸⣿⣿⠸⣿⣿⣼⣧⣿⣽⣿⡿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠭⣭⣭⣭⡭⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⢰⣶⣶⢳⡞⣶⢳⣶⣶⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⡄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⣿⡇⣿⢟⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣧⡰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣶⠀⠀⠸⠁⠀⣘⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢘⢿⣿⣿⣿⣮⡛⠃⠀⠀⠀⠀⠀⠀⣾⢫⣭⣽⣿⡇⠿⠦⠤⠤⢤⠄⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣷⣭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢸⣿⡿⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣄⣀⠜⠛⠋⠉⣉⣉⢑⣠⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠞⠋⣿⠟⠉⠀⠀⠀⣀⣤⣴⣖⣚⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⡷⣄⣀⣀⡼⠁⠀⠀⣰⠏⠀⠀⠈⠁⠀⠀⠀⠀⠊⠉⣽⠟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠀⠀⠀⠀⣰⠇⠀⡀⠀⣀⣀⣠⣤⣤⣤⣤⣼⣧⣄⣀⠀⠀⠀⠀⡀
⠀⠀⣿⠙⠲⣤⡀⠀⠀⠀⣤⣄⠀⣀⣿⡤⣶⣿⡟⠉⠛⠁⠀⠀⠉⠀⠀⠈⠀⠉⢳⠀⠀⠀⢈
⠀⠀⢸⠀⠀⠠⡙⣦⠀⣰⠃⢀⣿⣿⣿⠀⠘⣿⡇⠀⣿⣿⣷⠀⠀⠈⠍⠒⣦⣴⠋⠳⡀⠀⠈⠓⡆
⠀⠀⠈⢧⡀⠀⢜⡾⢷⡗⢲⣿⠀⠻⡿⠀⠀⠀⠀⢰⣿⣷⣾⠀⠀⠀⠀⠈⠁⠀⠀⠀⠹⡄⠀⠀⢱
⠀⠀⠀⢀⡗⠀⢀⡇⠈⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⣿⣓⣤⣀⣀⣀⣀⣀⣀⣠⡴⠃⠀⠀⢸⡄
⠀⠀⠀⣾⠀⢀⣡⠟⠛⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⠛⠻⣿⢿⡤⠋⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⡿⠖⠋⠁⠀⠀⢷⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⢸⣾⢳⣤⠖⡆⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⢆⠀⠀⢯⣑⠢⢄⠀⠈⠑⠢⠤⣤⠜⠉⠁⠀⠀⡇⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⢧⡀⠹⡏⠁⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠐⠒⡅⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠢⠤⠀⠠⠄⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⢄⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠢⣄⠑⠄⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⣀⣠⡤⣶⠟⠉⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠘⣅⣤⣭⡅⢀⡴⠃⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⣀⣀⣀⣀⣈⣉⣯⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣠⡤⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⣰⣿⡯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣧⠹⠀⡀⠐⠣⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡿⠶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡈⠀⢀⢀⡋⣓⣋⣋⡀⣀⠀⢀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠈⠉⠉⠉⠀⠈⠉⠉⠉⠉⠀⠈⠉⡉⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠒⠲⡀⠀⢪⡀⢸
⠀⠀⠀⡀⠀⠀⠈⢦⣀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠂⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠁⡀⣈⡀⠀⠀⠀⡀⠀⠐⠳⣄⣴⣦⡀
⠀⠀⠀⠀⠀⣄⣤⡀⠂⡁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡟⡟⠉⢰⡶⠀⠀⠀⠀⠀⠘⠿⢿⠃
⠀⠀⠀⠀⠈⠛⢻⠁⡆⢀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠈⠀⠀⣄⣀⢀⣤⠤⠆⠄⠖
⠀⠉⠉⠛⢦⡀⠈⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⡀⠘⠿⢿⠋⠀⠀⠂⠀⢘
⠀⠀⠀⠀⠀⠙⠀⠀⢐⠀⠓⠀⠘⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠐⠀⠄⠀⢇⡘
⠀⠀⠀⠀⠀⠀⠀⠠⡈⠂⠤⠠⠃⢀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⢀⡀⠀⠀⠸⣿⠗⠀⠂⠀⢘⡶⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠑⠁⠁⠄⠀⠠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠠⠆⠀⠀⠀⠐⠀⠀⠀⠀⠈

⣿⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔
⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⠀⠁⠀⠁⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢠⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⠾⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠄⠘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠁⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤
⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣶⠞⠳⢶⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⢾⡇⠀⠀⠀⡿⠶⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⠀⣴⠛⢁⣀⣀⣀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠿⢦⡴⠾⢧⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⢸⠇⠀⠈⠁⠀⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠆⠸⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⠹⣦⣀⡀⠀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠉⠉⠉⠉⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡞⠉⠀⠀⠈⢳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠟⠻⠛⠟⠿⠻⠟⠷⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⡴⠟⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣠⣤⣤⣤⣤⣤⣀⣄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⡖⣦⣰⡆⣟⢦⡄⣲⠆⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠉⠁⠁⠘⠁⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠊⠀⢀⡀⣣⠴⠂⠉⠁⠺⠎⢅⠀⠀⠈⠢⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠚⠉⠆⣥⡄⣤⡀⠀⠀⠀⢎⠲⠤⠤⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠃⠦⡟⠁⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣤⠀⠀⠀⠀⢰⠆⠀⣠⠞⠁⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠳⢼⣄⣀⣀⣠⣦⠴⠚⠁⠀⠀⠀⠀⠀⠱⣄⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠐⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠁⠀⠀⠉⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢳⡀⠀⠹⣄⠀⠀⠀⠀⠀⠸⣄⠀⢀⠀⠀⠀⠀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⡀⠀⠈⢧⡀⠀⠀⠀⣰⠋⠁⠀⠁⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠈⢳⠀⠀⠀⠹⣄⣠⠞⠁⠀⣠⠟⠢⢄⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠑⣤⠞⢀⡿⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡠⠖⠋⠀⠀⠉⠀⠀⠈⢀⡴⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⣤⠎⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⠒⠒⠉⠓⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠞⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢠⠚⠒⠒⠦⠤⢄⣀⣀⡇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⢦⣀⡀⣀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⡴⠓⠦⠤⠤⠴⠒⠚⠉⠀⠀⠀⡼
⢌⣀⠀⢀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠖⠉⠀⠀⠀⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⢸⣉⣉⣉⣰⠒⠒⠒⠒⠒⠒⠛⠓⠇⠀⡆⠀⠀⠀⡆⠀⡿⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠤⢾⠀⡆⠀⠀⢀⠀⠀⡀⡋⠛⠒⠒⠶⠶⠶⠒⠛⠉
⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⢇⣀⣀⣤⠗⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢧⣄⣀⣴⠤⠤⠋⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣽⣾⣿⢷⣶⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⢹⠛⠙⡛⠀⠀⠈⢻⢇⣑⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢪⠀⣹⠂⠀⠀⠈⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠐⢲⠲⠤⠀⠀⠀⠀⠀⠀⠉⠙⠈⠋⠀⠀⠀⢀⡳⠀⠀⠑⡄
⠀⠀⢀⣀⣀⡠⠚⢭⣉⠀⠂⠀⠀⢆⣀⠴⠃⠀⠀⠤⠄⠂⠁⠀⠀⠀⠀⠈⢆
⠀⠀⠉⢙⣿⠃⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡤⠒⠉⡅⢉⠐⣄
⠀⠉⠂⠈⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠤⠠⠇⠀⠀⢠⠒⠀⠀⠀⠀⢀⣀⠠⠤⢲⠧⡌⢀⠌⠃⡐⢡⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⣡⠁⠈⠀⡀⠀⠀⠈⠉⠉⠀⠀⣀⡤⠾⡁⠀⣇⠀⣆⡨⠣⠀⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢁⠆⠀⠀⠀⠈⠁⠒⠒⠒⠒⠋⠉⠙⢆⠀⠙⡄⠈⢳⡚⣶⣖⡶⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠠⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣱⠀⠀⡤⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠹⠵⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⠀⠀⠀⠀⠈⢆⠉⠉⠉⠁⠀⠐⠀⠐⠒⠒⠒⠒⠒⠒⠒⡲⠋⠉
⠀⠀⠀⠀⠀⠔⡯⡴⣯⢓⠄⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠲⡶⠤⠤⠤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠉⠒⠢⢤⠤⢤⠵⠀⠀⠂⠠⡔⢶⠶⠶⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠀⠀⠀⠀⠀⢱⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠂⠈⠐⠢⠴⠊⢚⣠⠀⣀⠀⢔⠵⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠋⠉⠉⠀⢨⠊⠀⣀⠤⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣐⣄⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠛⠛⠛⠛⣿⡏
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣂⣤⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⠇⠀⣀⠔⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢛⠋⡀⢀⣬⣦⡀⠀⠀⣟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠀⠒⠂⠈⠀⠀⣷⡿⠋⠄⠀⠀⠀⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠉⠁⠀⠀⠀⠀⠐⠈⠀⠀⢈⡄⠀⣀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣷⠀⢀⠀⣶⣦⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣜⣹⣿⢻⡇⠀⠀⠀⠀⣿⡿⠛⠻⣿⠀⠀⠀⠈⠁⠀⣦⣦⣾⡇
⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣼⡇
⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⣁⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⠄⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣀
⠀⠘⠿⠛⠛⠛⠃⠀⠀⠀⣿⣿⣿⣿⣿⣟⠁⠀⠠⠇⢲⣾⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣙⣛⣛⣛⢸⣿⡇⠀⠀⠀⣿⣿⣿⡇⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⠸⠿⠃⠀⠀⠀⣿⣿⣿⣧⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⠇⣿⣿⣿⡇⢀⣿⣿⠋⠉⠙⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⣿⣿⠴⣿⣿⣿⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⠀⣿⣿⣷⣶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠲⠏⠉⠉⠍⠐⢢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠁⡀⠀⠀⠗⠀⠀⠂⠀⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⣀⣠⣀⣰⣆⣀⣰⣆⣨⡀⡇⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡊⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠑⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠈⠳⣄⠀⠉⠀⠀⠀⠀⠀⠈⠉⢀⣠⠞⠑⢆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠈⠳⣤⣀⠀⢀⣀⣤⠴⠚⠋⠀⠀⠀⠸⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠉⢙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⣠⠞⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢸⠀⠀⢸⠀⢀⠞⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⡘⢠⣿⣿⣿⣿⣿⣿⣿⣆⠰⡄⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⡇⢸⣿⡿⠛⠛⠋⠙⢿⣿⠀⡇⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⢻⠸⡿⠶⠶⠶⠶⠶⠾⡿⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠘⡄⠻⣄⠀⠀⠀⠀⡴⠃⡀⠀⠀⠘⡆⠀⠀⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠗⠋⠀⠀⠀⢀⠇⠀⢠⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣤⡴⠃⠀⠀⠚⠀⠀⡾⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠉⠉⠈⠳⣄⠀⠀⠀⢀⡼⠃⠀⠀⠀⡇⠀⠀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡼⢆⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠙⠶⠤⠶⢻⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠈⠙⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡄⠀⢀⡞⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠸⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣧⠔⠉⠀⢀⡞
⠀⡠⠒⠂⠒⠒⠒⠀⠀⢼⠁⠀⠀⠀⠀⣇⢸⠀⠀⢰⡄⠀⠀⠠⠶⠂⠀⠀⠘⡄⠀⠀⣧⠀⠀⠀⠀⠀⠸⡦⠤⠖⠛⠉⣉⠱⡄
⡜⢀⠎⠀⡰⠒⠁⠀⢠⡏⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠹⡄⠀⡟⠀⠀⠀⠀⠀⠀⠘⢦⣀⠐⡄⠈⢧⡱
⠙⠺⢤⣤⣇⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⣆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢾⡀
⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢀⣠⠤⠤⠏⠀⠀⠀⠀⢸⣦⣤⣤⣤⣷⠤⠀⠀⠀⠀⠈⠛⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠸⣄⠀⠀⠀⠀⠀⣀⡴⠋⡀⠀⠀⢀⠀⠀⠀⢀⠈⠳⡄⠀⡞⠁⠀⢰⠂⠀⠀⠀⡀⠀⠀⡄⠘⣆⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠈⠛⠛⠚⠛⠛⠙⠦⢼⣃⠀⠀⢸⡀⠀⠀⢸⣄⣀⣿⠀⠳⠤⠴⢿⣄⣀⣀⣀⣇⣀⣀⡽⠒⠋⠙⠛⠛⠛⠛⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢰⡇⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣾⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⣡⣾⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣄⢠⢣⣿⣿⣿⣿⠟⠋
⠀⣸⣦⡄⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡤⢊⣡⣴⣶⣿⣿⣿⣿⣶⣝⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⣿⣷⣦⣭⣑⣒⣲⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⣤⣥⣵⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠉⠛⠙⠛⠉⠉⠀⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢔⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣿⣿⠏⠀⣿⣿⣿⣝⣿⣿⡟⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣇⠀⠐⣿⣿⡿⢿⣿⣿⣿⣝⠏⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣿⣿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣯⠝⠋⡀⠀⠹⣎⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢸⠁⣽⣿⣿⣿⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠸⠛⣿⡿⠇⠀⠐⠀⠀⠀⠹⠿⠟⠻⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⡟⠀⢱⣿⠻⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⡌⢠⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡸⠁⠀⠀⢳⡀⠙⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠻⠿⠿⠃⢀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡟⠁⠀⠀⠀⠀⠙⢦⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠲⡟⠲⡞⢇⢀⣼⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡄⠀⡘⣦⡙⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⡿⣉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⡼⣿⣿⣿⣿⣿⣧⢻⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⣴⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣎⠭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠘⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⠿⢿⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⣄⣀⡰⠤⠜
⠀⠀⢪⡀⠀⠀⠀⣀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠛⠉⢩⢤⡍⠘⠩⠞⠁
⠀⠀⠀⠈⠉⠓⠚⠁⠰⡀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠼⠤⠂
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠒⠒⠒⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⣠⡼⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⢤⣄⡀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠶⠞⠋⠉⠀⠀⢀⣀⣀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠲⠟⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⡰⠞⢋⣉⣤⠴⠞⠛
⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠋⠁
⠒⠋⠉⠉⠉⠁⠀⠀⠉⠙⠒⠶⠋⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠶⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣁⡈⠁⠀⠀⠀⢀⣐⣀⣼
⠀⢀⣀⣤⣤⠤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⢿⠟⣿⠛⠛⢛⣉⣉⣉⣉⣙⣑⣦
⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠶⠶⠚⠋⠉⠉⣉⣩⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣾⠂⠈⠙⣧⢹⡉⠉⠙⠛⢿⡙⣿
⠀⢀⣀⣠⣤⠴⠶⠞⠋⠁⠀⢀⣀⣠⣤⠴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡖⢿⡀⠀⠀⠸⣆⠙⢦⣦⠀⢸⡇⢸⡇
⠛⠋⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣷⠛⠉⣷⠀⠀⠙⢦⣀⢻⣆⣀⣷⢸⡇
⣠⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣴⢲⣖⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣼⠃⢠⡶⠛⠀⠀⠀⠀⠉⠛⢿⠉⠹⣾⣷⣆
⠁⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⢛⣩⣭⠵⢿⡄⣏⡉⠉⣹⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠘⠁⢀⡿⠀⠀⠀⠀⠀⢀⣠⠄⢸⣇⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣟⣩⠴⡄⢸⡃⢀⣼⡄⣿⠈⣿⠉⢻⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡾⠁⠀⣀⣤⡶⠞⠋⠁⠀⠀⣿⡄⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠋⠀⣠⠿⢿⡟⠋⢼⡟⠛⠲⣾⣤⣿⣦⡀⠀⠀⠀⠀⠀⢺⡄⠀⣼⡇⠀⠘⠋⠁⠀⠀⠀⠀⠀⢸⣧⡿⣄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢷⠞⢁⣤⢶⠟⠀⠈⣷⠑⣦⣄⠙⠷⠈⠻⣄⠀⠀⠀⠀⠈⠳⠞⠉⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡇⠀⠈⠓⢄⠀⠑⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣴⡏⢸⣼⡇⠀⠀⠘⢷⣌⣉⣷⣦⣀⢀⣹⠂⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⡇⠀⠀⠀⠀⠑⢦⠨⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⡟⣿⡿⠾⠋⠀⠀⣀⡤⠀⠀⠀⢻⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢠⣶⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠳⡷⡿
⠀⠀⠀⠀⢀⣤⠶⠦⣄⡈⠀⠻⣿⠃⠀⣤⠶⠛⠉⠀⢀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⢸⡇⣿⠀⠀⣼⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⢀⣤⠾⠋⠀⠀⠀⠀⢉⠀⠀⣿⣀⣤⣤⣴⡶⠾⠿⠛⠛⠋⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢻⡄⢀⡇⠘⡇⠀⠀⠀⠀⠀⢸
⠶⠟⠁⢀⢄⠀⢀⡜⠛⠛⠀⡴⠋⠉⠉⠀⢀⣀⡀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⢸⡇⢸⡇⠘⠇⠀⣿⠀⠀⠀⠀⠀⡏
⠀⠐⠒⠁⠀⠉⠉⠀⠀⠀⡼⢡⡆⠀⢀⡴⠉⢉⡇⠀⠀⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⢠⣄⡴⠄⠀⠀⠀⡴⠁⣼⠁⢀⡾⠁⠀⢸⡇⠀⠀⣾⠁⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢻⣄⣀⣀⣿⠀⠀⠀⠀⢸
⡴⠢⠀⠈⢀⣤⠀⠀⡼⠁⢰⡏⢠⡟⠀⠀⠀⢸⡇⠀⣸⡏⠀⢸⡇⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⡆⠀⠀⠀⣸
⠐⠀⠏⡅⠈⠀⠀⢸⡀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡇⠀⠉⠀⠀⢸⡇⠀⠀⠀⠈⠛⠦⠀⠀⠀⠀⠀⣰⣏⠉⠉⢹⡇⠀⠀⣿⠀⢉⣩⣷⡀⠀⠀⡏
⠀⠀⠀⠋⠙⠆⠀⢸⡇⠀⠀⠀⣸⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⠛⠀⠀⢸⡇⠺⠆⠘⣦⣈⠳⣾⣷⡀⠀⡇
⠀⣠⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⡀⠀⣠⣾⣿⠿⣆⠀⠀⠀⠀⠀⠀⠀⣯⣈⢁⣀⡤⠶⠞⠁⠀⠀⠀⠉⠙⢻⣦⣼⡇⡀⠛⠿⠿⠿⠿⠿⠿
⠸⠃⠀⢣⡖⢲⡆⢸⡇⠀⠀⠀⣿⠀⡰⠒⢦⡀⠀⠈⠳⣾⡿⠛⠁⠀⠹⣆⠀⣀⠀⢀⡴⠛⠉⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠙⠳⣦⡀⠀⣀⣀⡀
⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⠛⠂⠀⠙⣦⠀⠀⠀⣸⠟⡀⢀⣴⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠳⡜⠿⠿⣿
⠶⠶⠶⠖⠲⠖⠀⢈⡏⠀⠀⠀⢸⡀⠤⣤⣤⣤⣄⣀⣀⠀⡸⠿⢃⡴⠛⣀⡥⠞⠉⠉⠉⠳⣦⡴⠶⠶⠶⣤⡀⠀⠀⢀⡴⠞⠛⠉⠉⠙⠛⢶⣄⠀⠀⠙⣤⣀⣈
⣀⣀⣀⣀⠀⠀⡾⣿⢄⣀⣠⣤⡾⠁⢀⣀⣀⡀⠀⠈⠉⠀⠙⠒⢋⣵⠞⠉⠀⢀⣀⣀⣀⣀⡈⠀⠀⠀⠀⠀⠉⠛⠛⠻⠶⢶⡶⢶⣶⣶⡤⠤⠼⠓
⠛⠛⠛⣻⡃⠀⠙⠛⠋⠙⠁⠀⠁⠀⠈⠛⠛⠛⢀⣀⣀⣉⣀⣠⠟⠁⣀⠐⠛⠻⣥⠤⠬⠍⠉⠓⠲⣤⣠⣤⠶⠶⠶⠶⠶⢤⣤⣀⠀⠀⠴⠶⠶⠶⢦
⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠒⠒⠖⠛⠩⠷⠎⠿⠿⠯⠾⠯⠭⠽⠦⠀⠄⠰⠤⠤⣤⠀⠀⠀⠙⣿⣦⡀⠀⢀⣀⣀⣀⣉⣹⣆⣤⣀⣶⣶⠛⠉⠉⠛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠉⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢡⠀⠠⢄⣀⣂⡠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠁⠂⠀⠀⠠⡀⠀⠀⠈⢢⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠐⠂⠀⠂⠂⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠇⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣎⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⣠⠃⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⠀⠒⠒⢒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠑⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠃⠤⣀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⠃
⠀⠀⠀⠁⠙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠉⠈⠀⠈⠁⠀⠀⠀⠁⠢⠀⠀⣀⡠⠤⠒⠋⠁⠀⠈⠈⠈
⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐⠀⢀⠔⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣶⣶⣶⣶⣶⣦⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠪⠭⠩⠍⠉⠉⢉⣒⣒⣒⣒⠒⠒⠤⠤⠤⠤⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⣷⡾⠯⠍⢉⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⣁⣐⡒⢒⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠲⠦⠤⠤⠤⠤⣤⣉⣉⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠐⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠷⠟⢿⡀⢠⠞⠉⣉⡻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇
⠀⠀⢔⢢⡀⢀⠖⠆⠀⠀⠀⠀⢸⠀⢾⣿⠇⢸⠀⢸⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⡇
⢯⡙⠺⠆⠱⣼⠐⢦⠀⠀⠀⠀⢀⡟⠚⠉⠀⠀⠑⠲⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⡇
⠪⠭⣭⠄⠀⠧⠀⣘⠤⠤⠔⣂⠭⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⡇
⠀⠰⠥⠊⠑⠒⠚⠁⠉⠉⠉⠀⠀⡇⠀⢢⡀⠀⠀⠀⣈⢷⢦⠀⠀⠀⠀⠀⠀⢣⠘⢄⠀⣿⠀⠀⡇⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢏⠂⠉⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⡑⠿⣤⣠⠗⢁⡜⣀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠼⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⢦⠀⠀⠀⡇⠀⠒⠚⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⢳⣄⣀⣄⠠⣍⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣎⡇⠀⠈⠦⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠉⡟⠙⠛⠓⠒⠒⢶⠲⡶⠶⠦⠤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡙⣄⠀⠀⠀⠀⢰⠃⡜⠀⠀⠀⠀⠀⠀⠀⢣⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠓⠤⣀⣠⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⠇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣄⡖⣀⣰⣷⡆
⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡷⠲⣾⣧⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⣿⣷⣷⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣷⣤⣿⡟⠛⠙⢿⣿⣿⣿⣿⣻⠇⢀⠸⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠈⡿⢿⣿⠈⣿⠀⠈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⡁
⠀⠀⠀⠀⠈⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠈⢻⡷⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⡀⠀⠀⠀⡈⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣤⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⢀⠈⠅⢼⠀⣰⠦⠈⠁⠀⢸⣿⣿⣿⣿⣿⣿⠉
⣠⣴⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⡏⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⢩⡇
⣿⣿⡟⡔⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆⠀⠈⣷⣼⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⠀⠈⡁
⣿⣿⡀⣇⡜⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢄⣀⣀⣤⣤⣶⡶⣶⣶⣶⣶⣶⣶⣾⣿⣷⡆⢹⠀⠀⠇
⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⠿⠏⠀⠀⠈⢹⡿⠿⠿⠿⠟⣿⡿⠳⠘⠳⠶⠂
⣿⣿⣿⣿⣧⢺⣦⣦⣇⠀⡀⢿⠀⠀⠀⠀⠀⠻⣧⣀⡿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠑⣄⣸⣷⣦⣤⠤⠯⢥⡄⠀⠀⠀⠄
⠛⠛⠛⣿⣿⣾⣿⢧⣿⠀⣷⣘⢧⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣦⣀⣦⣀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣁⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢿⣿⣿⣿⣷⣿⡆⣷⣧⠘⣄⠀⠀⠀⠀⠘⠛⠉⢉⣿⣿⣿⣿⣿⣿⣶⣆⠀⣠⠂⣠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣤⣤⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠙⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠿⠻⡿⠛⢻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛⠛⠛⠿⠛⠛⠉⠉⠉⠉⠙⠉⠉⠁

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠠⠶⠾⠿⠿⠿⣿⣿⣿⡟⠀⠈⠙⢛⡺⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⢄⡀⣄⣮⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣵⣿⠎⠙⠋⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣄⡀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡶⠋⠉⠉⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠐⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣦⣤⣤⡎⠹⠿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠒⠛⠉⠀⣶⣿⣦⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⣛⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣏⠉⢉⡴⠋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣷⣿⡽⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁⠀⠀⢻⣿⣿⣿⡿⠛⢻⣿⣿⡇⠈⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⠚⢛⡿⣿⡇⠀⠀⠙⢿⣿⣦⠀⠀⢀⣴⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⣿⣿⣿⣇⠀⠀⠀⠈⠻⣿⣷⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡉⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠻⣄⠀⢹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⡟⡿⠀⠹⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⢀⡄⣠⠀⢠⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒⣦⣴⣷⣿⣿⢿⡿⢿⣿⣷⣶⣦⣄⡀⠀⠀⠀⣧⢸⡄⠘⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⢟⣄⣀⣁⣠⣼⣿⣿⣿⣿⣿⡆⠀⠀⣿⠘⣇⠀⢿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢸⡄⣿⠀⠀⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢸⣧⣿⡇⠐⣦⣤⣠⣴⣆⢶⣤⣤⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠈⠁⠀⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⣀⠀⠀⠀⠀⠀⡰⣻⡳⣀⠳⣦⣄⠀⢀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠊⠁⠈⠉⠓⠒⠒⠚⠣⢿⢫⣴⣵⠘⠀⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⡗⠪⠒⠛⠋⠀⢀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⡀⢤⣾⠇⠀⠀⠀⠀⠀⠈⡗⠒⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠒⠒⣒⠮⢡⠎⠀⠀⢀⡰⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠛⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⡐⠑⣂⡥⠚⠒⢚⡿⠀⠀⠀⠀⣿
⠀⠀⠀⡄⠰⢆⠀⢸⣀⡀⠀⢠⠃⡤⠤⢀⡀⠀⠈⠒⠒⢊⠠⠒⠉⠉⡧⠖⣋⠀⠀⠀⠀⠀⡿
⠀⠀⡜⠻⠁⠘⠋⠉⠈⠉⠑⡇⢘⠄⠀⠀⢳⠀⢀⠔⠊⠁⣠⠔⢊⠼⠚⠉⡀⢱⠀⠀⠀⠀⡇
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠑⣭⢞⣀⡠⠁⣸⠐⣕⡀⢰⡟⡡⠚⠁⠀⣠⠜⣱⠃⠀⠀⠀⢰⠃
⣤⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡤⠞⠁⠀⠀⠉⢹⣞⢁⠄⢠⠚⠁⠔⠁⠀⠀⠀⠀⡞
⡍⠇⠀⠀⠀⠀⠀⡴⢋⠀⢀⣴⠋⢳⣀⡀⠀⠀⢀⡤⠋⠈⠘⢲⠋⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠇⠀⠀⠀⠀⠀⠀⠈⠛⡤⠄⠈⠆⢸⠀⠙⣶⠊⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠸⡀⠀⠀⠀⢀⠀⡸⠳⠇⠀⠀⢸⠂⠀⠀⡽⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠱⣄⠀⠀⠘⡗⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡠⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⣠⣤⣤⣤⣤⣤⣄⣀⣠⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣀⣄⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣸⡿⣱⣿⣿⠃⣴⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣷⣿⡁⢿⣿⠏⢰⣿⣿⣿⠟⠁
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠋⠹⣿⡿⠀⠀⠙⠛⠛⠁⢠⣾⡇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣾⣿⣀⡀⢀⣴⣷⡶⠀⠈⠟⠀⠘
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⢁⠸⡛⠿⢛⡻⢿⠁⠀⠀⠀⢀⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠊⢿⢀⢿⣿⡽⠘⣶⣆⣀⣴⣿⣿⣷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⣠⡄⠈⠁⠚⢀⣩⠙⣿⣿⣿⣿⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⣠⣤⣴⣤⣤⣄⠳⠀⠐⣿⡎⠩⡅⢰⣿⣿⡿⠟⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣶⣿⣿⠟⠙⠛⠻⠿⡏⠄⠀⠘⢫⠖⠀⣌⠙⠏⢀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⢁⡸⣄⠁⠀⡀⠀⡍⠻⡀⠙
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⡿⠗⢀⠀⠀⠀⠀⠈⠁⢚⡁⣈⢄⡰⢃⠚⡇⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣹⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣼⡿⠉⠳⡫⢄⠺⣄⡟⠛⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡿⠳⣟⠂⠀⠀⠙⠄⢀⢠⣄⡴⢟⣴⡻⢼⡟⡃⡠⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢻⣿⣿⡁⢀⣼⣷⣆⣀⣀⣠⣿⡈⢿⡀⢁⡀⠈⢦⢰⡛⣴⡋
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠏⠘⢧⣭⠯⣼⡿⠈⠙⠻⡾⣅⠈⠁⣠⣟⠚⡑⢯⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣷⣄⠀⠀⠀⢀⣿⣧⠀⠈⠐⠳⣤⣤⡉⠈⢫⣰⢱⣎⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡀⠈⢿⣿⣷⡀⠔⣹⣇⣄⡢⠤⣶⢆⡋⡄⢀⢢⣿⡟⢀⣿⣿⣏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⡟⢀⡘⣿⠟⠙⠑⡖⠀⣼⣾⣾⣿⣿⣿⣭⢹⣿⣿⠿⠀⠀⠀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⠀⢪⠓⢿⣿⣿⣾⣴⣿⢿⣭⣷⣿⣿⣿⠳⢰⠏⠀⠀⢠⣾⣿⡟
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣿⣧⢹⣿⣦⣠⣤⠿⠛⠟⡿⢿⣿⣿⣿⣿⣿⡿⣎⡽⣇⠀⠀⢺⣿⠏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢈⣛⠹⠻⣷⡙⢿⣿⡇⠀⠀⠀⠉⣶⣮⢻⣿⡿⣩⣿⢥⡞⣠⠀⠀⠀⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠈⢿⣿⣶⣮⣿⣦⣙⢷⡄⠀⠀⣤⠉⡛⢃⣴⣿⣿⣿⣿⠃⣟⡛⠲⠦⢤⣤⣄⣀⣀⣀
⠀⣿⣿⣿⠿⣟⣛⣩⣵⠾⠏⠀⣼⣿⣿⣿⣿⣿⣿⢻⣶⣶⠦⠟⠀⢻⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⣦⣤⣤⣤⣄⣤⡀
⠀⣾⣿⣿⣿⣿⣭⣷⣶⣿⣿⣦⡘⢿⣿⣿⣿⣿⣿⢸⡿⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⢀⣹⣿⣿⢿⣿⣿⣿⠿⣿⣿⡯
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⡸⣿⠀⠀⠀⠀⢰⣿⣿⡿⣟⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣒⣒⣲⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡄⢤⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⠀⣻⡦⢱⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠠⢢⠄⠀⡀⠀⠀⠀⠀⠀⠈⢧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢀⠀⠀⠐⣦⣤⣤⣵⠄⡒⠈⠋⠀⠀⠀⠀⠀⡅⢰⠀⢠⠘⠆⠸⠃⠀⠰⠊⣠⠸⡲⠀⡁⠔
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠒⢀⣠⣀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⢀⣅⢠⠀⢏⠀⠀⠀⠰⢟⣁⢰⠇⠀⡃⠂
⠀⠀⠠⠂⡀⡀⣷⣿⣂⠀⠀⠆⠂⠈⠀⠀⠀⠀⠀⠀⢀⠙⠻⣭⣀⠆⠀⠀⠈⠀⠀⠔⠀⠀⣿⣿⡶⠡⠈⣀⠀⣵⣿⣾⣷⣆⣤⡁⠠
⠄⠀⠀⠀⠈⠀⡏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣦⢨⠏⢀⡤⡶⠊⠀⠀⠀⡀⠀⠈⢟⠁⠀⣤⡟⠛⢍⠉⠉⠈⠑⠉⠿⠃
⠀⢀⠀⠀⠀⠀⣗⠠⠊⢀⠀⡀⠀⡴⠂⠖⠒⢀⣐⡻⢤⡸⠛⠈⠀⠀⠀⠈⠀⠐⢛⡛⠛⠠⠎⠚⢐⣵⣿⣿⣾⠆⠙⠲⢤⣀
⠠⢁⠈⠀⠔⠀⠈⠦⠙⠁⠀⠀⠄⠀⠀⠀⢙⡎⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⡓⠀⠶⠶⠶⠋⠀⠀⠀⠀⠀⠀⣠⣶⣨⡓⢤⡀
⠀⠐⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠀⡠⠀⠻⣿⣁⠉⠐⡄⠀⠠⠽⠿⠟⠛⠛⠛⠂⠀⠀⠀⢀⠠⠐⡾⠿⠿⠛⠯⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣃⠀⠓⠥⣈⡀⠉⠍⠐⠀⠀⠀⠀⠀⢀⠀⠀⠦⠀⠒⠀⠀⠀⠀⡀⠀⠄⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠑⠛⠓⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⡶⠶⠀⠀⢀⣤⣄⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠞⢩⣿⣦⡔⢹⣿⣆⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠇⠀⠀⠘⣇⠘⢻⡿⢧⡈⣋⡝⠀⠀⠀⣀⣿⣤⣶⡶⠶⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢇⠀⠀⠀⠀⠉⠛⠉⠀⠀⢉⣡⣤⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⠂⠀⠀⢤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣾⠀⠈⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣄⣀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡇⠀⠀⠀⠉⠙⠛⣿⡟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⠀⠀⠹⣦⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣷⠉⠳⢦⣄⣀⣄⣉⣉⡅⠀⠀⠀⣀⣠⣴⡾⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣠⠞⣿⠀⠀⠀⠈⠉⠛⠛⠻⠶⠾⠛⠛⠛⠉⠁⠀⠀⣿⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢁⣤⠞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡌⠻⣦⡀
⠀⠀⠘⢍⠓⢤⣠⠴⠶⠶⠋⢀⡴⠋⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦⡈⠛⢦⡀
⢰⣚⣛⣛⣛⠂⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⡀⠙⢦⣀⠀⠀⢀⠔⣶⣀⡀
⠯⣀⣀⠠⠤⠀⠀⣀⣤⡀⢀⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢦⡀⠋⠉⠙⠃⢚⣩⣔⣁
⠀⠯⠤⠤⠖⠚⠉⠀⠀⢹⡀⠳⣄⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠤⠔⠋⣠⣄⡀⠀⢀⣬⣥⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⠶⠞⠁⠀⠙⠶⣄⣈⠙⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⢷⡀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠉⠛⠛⠛⢻⡟⢻⠉⠉⠀⠀⠀⠀⠀⣠⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⢿⠳⠶⠶⠶⠶⢶⣼⡇⢸⡀⠀⠀⠀⠀⢀⡇⢸⣀⣀⣀⣀⣀⣤⡞⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢷⡀⠀⠀⢰⡟⠁⠉⠉⠀⠀⠀⠀⠈⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⣶⣦⣤⣴⡾⠃⠀⠀⠘⠷⢶⡶⠶⠶⠶⠶⠶⠶⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡂⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⠀⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⣶⣶⡄⢸⣿⣿⣿⠀⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⢰⣶⣶⡦⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⡇⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠘⣿⣿⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⡟⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡈⠻⡯⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠿⠋⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠛⠻⠿⠃⠸⠿⠛⢋⣀⣴⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⠿⠛⠉⠛⢁⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠈⠉⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠦⣀⠐⣋⠁⠀⠀⠀⣆
⠀⠀⢀⠠⡄⠤⠀⠀⠀⠌⠀⠀⠀⠀⠀⠂⠀⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠻⠋⠻⣏⡀⠁⠀⠀⠀⠘⡆
⡔⠋⠀⠀⠀⠢⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⣠⠠⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡅
⠑⣄⠀⠀⠀⠀⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠈⠀⠀⠀⠈⢶⣶⣿⠌⠆⠀⠀⡜⠀⠒
⠀⠈⢂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠘⠅⠀⠋⠀⠀⠐⠀⠱⠀⠀⠀⠐
⠀⠀⠀⢃⠀⠀⠀⠀⣯⣀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠈⠄⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠚⠙⠂⠀⠀⠀⢸⠀⠀⢹⠠⡀⢀
⠀⠀⠀⠀⡂⠀⣠⠂⠁⠀⠀⠀⠀⠀⠂⠠⠒⠄⠀⠀⠠⠀⠀⠀⢀⠄⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⣹⠈⠃⠀⠨
⠀⠀⠀⠀⠰⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠐⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠨⠒⡄⠀⠀⠀⠀⠲⠴⠬⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠈⠁⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠜
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢤⣀⣀⡠⠤⡤⠖⠚⠅⠤⠀⡀
⠀⠀⠀⠀⠀⠈⠒⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⡀⠀⠠⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣶⣶⣶⣶⣦⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⣤⣤⣤⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⢁⣀⣤⣤⣤⣀⡈⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⠟⠋⠉⠀⠀⠀⠉⠙⠳⣦⡀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⣿⠏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⢻⡄⠀⠀⠀⠀⠀⠀⢶⡟⠛⠉⠁⠀⠀⢀⣼⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠀⣸⠇⠀⠀⣀⡴⢶⣤⣀⠀⠀⠀⠀⠀⠀⠹⣇⠀⢿⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣠⡾⠋⠀⠀⠀⠉⠛⠷⢦⣤⣀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣀⣴⠞⠛⢷⡄⢠⡟⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⡿⠀⣸⠇⠀⢀⣴⠾⠋⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣿⡀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡿⠛⠾⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⠉⢷⡀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢠⣾⣅
⠀⠀⠀⠀⣿⠀⠀⣠⡶⠋⠁⠀⠀⠀⠻⣦⡀⠉⠻⠶⣤⣤⣀⣤⣤⠶⠟⠉⣀⣴⠟⠀⠙⢷⣤⣄
⠀⠀⠀⠀⣿⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⣀⣀⣀⣤⡴⠾⢿⡄⠀⢀⣴⠟⠁⠙⣷
⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⣶⠟⠁⠀⣠⡾⠛⢷⣄
⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣤⣾⠋⠀⠀⠀⠙⢷⣄⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⣄⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡼⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠉⠉⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣁⡤⠤⠤⠤⢤⣀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠞⣆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣸
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠘⠀⡜⠀⢻⡀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠀⠒⠲⢤⡞⠀⣠⠏
⠀⡀⢀⡠⠴⠒⠚⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⡀⠀⢀⠃⠀⠈⢧⡀⠀⠀⠘⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⢊⣀⠀⢀⠤⡈⠀⠈⢀⠀⠀⠀⠛⢦⣤⡴⠃⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡰⠁⠀⡇⡇⠀⠘⣆⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⢧⠁⡷⡷⠀⢸⠀⠀⠘⢄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠦⠤⠋⠀⠹⣄⣀⡜⠀⠀⠑⡌⠢⣄⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⠊⠉⠉⠉⠈⠉⠉⠙⢢⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣇⠀⠲⠀⠀⢀⡾⠆⠀⡼⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠘⠦⣜⣀⣆⣨⠧⡤⠞⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇⠹⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⡤⠤⡞⠉⠳⣶⢶⡖⠛⡉⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⠁⠀⠹⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠛⠛⠉⠀⢹⠀⠈⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣸⠃⠀⠀⡼⠂⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⡴⠯⠚⢀⡴⠁⠀⠀⣰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡄⠀⠀⠀⠠⡜⠲⠖⠁⠀⠀⡀⠀⣀⡴⢚⠁⢀⠄⠀⣼⠁⠀⠀⢰⠃⠀⢠⠏
⠀⢀⡠⢤⣀⣀⡀⠀⠀⣀⣀⣠⠴⠞⠁⠀⠀⡟⠙⠓⠒⠒⠘⠲⣄⠀⠀⠀⠹⣎⠀⠈⠛⠖⠾⠴⠖⡵⠋⠉⠉⠉⠀⢠⠏
⣰⠋⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣤⠼⠗⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠘⠧⡜⠀⠀⡆⠀⢹
⣇⣠⡇⢠⠀⠀⢀⠀⢀⣀⣀⣤⠴⠞⢋⡥⠖⠒⠊⠉⠉⠉⠀⠀⠀⢀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠒⠚
⠀⠉⠛⠾⢦⣤⡼⠏⠉⠉⠀⠀⠀⠀⡟⢠⠂⠀⡖⠀⠀⣴⠶⠚⠋⠉⠉
⡀⢀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠓⣻⣤⣴⣧⣤⡾⡁⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀⣀⢀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣤⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⠀⢀⣤⡄⠀⣤⣤⠀⢀⣤⣤⣤⠀⠀⠀⣤⣤⣄⡀⢀⣤⣄⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠻⣽⡏⢡⠀⣾⣿⠁⣼⣿⠁⣴⣿⠟⢁⣤⣴⣶⣤⡄⠙⠙⠻⣄⠙⣿⣆⠘⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⠇
⠀⠀⠀⣯⢹⢀⠰⡄⣤⢤⠀⡀⠀⠈⡇⠛⠁⠀⠀⢈⣿⡇⢰⣿⠇⢠⣿⠟⢉⣉⠙⢁⡀⠀⠀⣹⡄⢹⣿⡀⠙⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠘⠲⠰⠄⠂
⠀⠀⠀⠁⠈⠈⠉⠁⠉⠘⠘⢥⠇⠚⠙⠆⠀⠀⠤⠈⢻⠀⢸⣿⣶⡿⣿⣿⡞⢿⠇⢀⣴⡆⠐⣿⡇⢸⣿⡇⠐⠀⠀⠀⠀⡀⣀⠘⢦⠀⠐⠀⡀⠒⢀⡿
⠀⠰⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣄⠀⢠⠿⡽⠛⠛⠙⠳⣎⣤⣾⡿⠁⣼⣿⠁⣾⣟⠴⠀⠀⠀⣆⠀⠁⠀⠀⣈⠧⠀⡸⢣⣀⡼⠆
⠀⠀⠀⠀⠨⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢻⣿⡆⣾⠀⠀⢰⡀⠀⢸⣟⢉⣠⣾⡿⠃⣰⡿⠃⠀⠛⢀⡴⠋⠀⠀⠀⠆⠉⢀⠀⠀⡄⠤
⠀⠀⠀⣤⠀⡀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠻⣯⣿⡀⠀⠈⠁⠀⣸⣿⡿⠟⠋⣠⣾⡿⠁⠘⠀⠀⢈⡄⠀⠀⠀⣀⠀⣠⣧⣾⣿⡆
⠀⠘⢣⡔⠂⡁⣤⣥⣴⠟⠛⠳⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⣤⣀⡴⠶⢯⠀⠀⠰⠻⠿⠋⠀⠀⠀⢰⡂⢉⠀⠒⠀⠀⠀⠀⠀⢺⣿⣿⣿⡆
⠀⠘⢣⣸⡷⠟⠋⠀⠀⠀⠀⠀⠀⠈⠳⠶⢦⡀⠀⠀⠀⠀⠀⢚⠁⠈⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⣀⠀⠀⠀⢠⣿⡿⠋
⠀⢘⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠈⠀⢠⣀⠀⠀⠀⠈⠻⢆⠀⠄⢀⣦⠀⠀⠰⠞⠓⠄⠀⠀⡽⣿⣏⠀⢰⡟⠉
⠀⢸⡇⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⡻⠀⠂⠀⠀⠈⠋⢀⣴⣶⣦⠀⠀⠀⢠⡾⡿⢧⣎⣟⣻⣟⣛⡃⠀⠀⡃⠀⠀⠆
⠀⢸⡄⠀⠉⠁⠀⢠⠀⢰⠀⢠⠁⢠⡄⠀⢤⠉⠉⡃⢀⡀⠀⠀⠀⣀⣈⡻⢿⠇⠐⠀⠀⢤⣶⣿⣴⣏⣟⣻⣏⣙⡃⠀⢘⠙⠢⠀⠀⡀
⠀⢸⣷⣄⣀⣀⣤⠞⠳⠊⠛⠂⠛⠃⠙⠶⡤⢦⣄⣀⣸⣁⣀⢠⡿⠋⠉⣿⠆⠐⣀⣶⣤⡀⠈⣧⠈⣿⣟⣻⣟⣿⡧⠀⢘⠀⠀⢀⣶⣹⡇
⠀⢰⣟⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⡄⠀⠀⠰⠏⠀⢸⣷⣿⣯⡁⠴⠦⢀⣿⡟⢻⡟⣿⣇⠀⠸⠀⠀⠸⡟⢉⣁⡀
⠀⢸⣿⣿⢿⣦⣀⣤⡄⠀⠀⠀⠀⠀⠀⢄⣠⣴⠿⠛⢃⡀⠋⠀⠀⣠⢄⣤⠀⠸⡏⠉⠉⠀⠀⠀⠀⠙⠿⣿⡟⣿⡇⠀⢀⠀⠀⠘⡟⢿⣿⣿⣿⠆⢄
⠀⢸⣿⣾⣿⡽⢿⡹⢿⣳⡖⢳⣤⣤⡶⢏⡱⠎⠡⡈⠘⢧⡱⠆⠘⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡷⠀⢸⠀⠀⠘⢿⣿⣿⣿⣿⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠨⠐⣶⠈⠉⠁⠈⠉⠁⠈⠁⠀⠀⠈⣤⣤⡉⠛⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠈⠃⠀⠀⠁⢀⠘⢟⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⠅⣌⠁⠀⡀⠀⠀⠀⠐⠀⠀⢀⡔⠈⠛⠁⠓⠸⣲⠖⣲⠀⣾⠀⣀⢀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠐⠃⠜⠋⡤⢀⣤⡀⠀⠀⠀⠀⡀⠈⠛⠦⠟⠀⠀⠰⣋⢴⣿⣧⣌⡳⣍⢊⢟⡱⢟⣡⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⠉⠙⠀⠀⠀⠀⠀⡿⠷⣤⡤⢄⡶⢛⣠⡞⢩⣴⣿⣿⠶⣬⡳⣝⡛⠶⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⠀⠀⠀⠀⠉⠳⣿⣻⣥⡴⠟⢶⣟⣾⠋⠀⠸⣿⣿⣶⣬⣿⣦⣍⣛⣭⠂⢐⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⠈⠀⠁
⠀⠀⠀⠀⠀⠠⠤⠆⠈⠀⠀⢀⡀⠀⠄⠀⠀⠰⣷⡞⢿⣥⣆⢨⣭⡗⠀⠄⠀⢹⣿⣿⣿⠆⡷⠶⢒⣺⠀⠈⡄⠻⡃⢴⡦⢘⡁⢰⡦⠈⠇⠈⠗⠀⢲⠖⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠂⠀⠀⠀⠀⠸⡿⠛⠃⢙⣵⣇⡀⠀⠀⠀⠀⣰⣿⣿⣿⠷⣚⠫⠭⣧⣤⣤⡾⠻⢦⡴⠞⠳⣦⣤⣾⣤⣆⣰⣀⣿⢤⡘⡃⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠓⠀⠁⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⠻⠋⠘⣶⣾⣿⡿⠾⠋⢡⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢷⠶⠆⠆
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠿⠸⡟⠁⠀⠀⠈⠀⠀⠃⢀⠄⠀⠀⠀⠀⠀⠀⠀⣌⡉⣙⢘⣋⡃⠃
⠀⠀⠀⠀⠀⠀⠁⠖⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡄⠘⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠠⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⢰⣤⡀⠀⠀⠀⢀⣴⠆⠉⠀⠀⠠⠄⠀⠀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⣠⠤⢤⣤⠜⠚⠻⠇⠰⠻⢏⠉⠀⠀⠈⠉⠉⠀⢀⡤⠂⢀⡄⢒⠀⢀⠀⣀⣠⠀⣤⣴⠆⣤⠈⡿⠂⠀⠀⠐⢀
⠀⢀⣤⣤⣤⣤⣶⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘⠁⠘⠁⢾⠀⣉⣩⠀⣉⣠⠀⢉⣀⠟⢀⡀⠀⠀⢰⣦
⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⢀⡄⢀⡀⢳⣄⠀⠀⠀⠀⠀⣿⠀⠀⢾⡄⠚⠃⣸⠂⠉⢹⡆⠉⠨⡆⣻⠁⣴⠛⠀⠀⠀⣀⠁
⠀⢸⣿⣿⣿⡇⠰⠇⠀⣀⠸⡆⠠⡄⢠⡄⠁⢸⠀⣊⡻⢛⠇⠈⠀⠀⠀⠀⠘⠷⣦⣈⣙⣿⣟⣛⣟⠻⣿⣿⣿⣟⣋⣁⣴⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠
⠀⢸⣿⣿⣿⣿⡀⠛⣀⣷⣴⠗⢦⡷⠦⠙⢦⠝⢦⣈⣡⡾⣶⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠛⠿⣿⠟⠀⠀⠙⠏⠉⠛⠋⠁
⠀⢸⣿⣿⡋⣻⣟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⣻⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠡⠔⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⣩⣿⡄⠈⠙⠛⢿⡀⠈⣵⠦⣤⠴⠀⢀⡰⠊⣴⠶⠛⢉⡴⠚⠋⠀⠀⠀⠀⠉⠲⠶⣀⠘⢄⡀⠀⠀⠀⠀⠀⢲⣆⠠⠤⠀⠀⠾⠶⠶⠦⠴⠀⠆
⠀⠀⢸⣿⣇⠀⠀⠀⠀⠙⠋⠁⠀⠈⠉⢻⡯⠉⠛⠁⠀⠀⣸⡀⣀⢠⣤⣄⣤⠀⣤⠀⣠⣦⡙⣾⡛⣦⡀⠀⢨⡛⢄⣙⡛⠻⣷⡄
⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⢿⡿⢿⢸⣯⣠⣿⡀⣿⣀⣿⣸⡗⣿⡟⣾⡟⢳⣤⣹⣾⣿⣏⠰⣿⡛
⠀⢸⣿⣿⣿⣿⣷⣶⣦⠀⠀⣠⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢷⡀⠀⠉⠁⠉⠉⠉⠋⠉⠛⠁⠹⣿⣼⣿⣾⣿⣿⣯⡛⣳⣶⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣷⣆⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣠⣄⡀⠀⠀⠀⣠⣾⢿⣿⣿⡏⠹⢷⠽⠿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠹⢆⡹⣛⢟⣹⣿⣿⣿⠀⠀⠀⠀⠈⠁⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠼⣽⠿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠰⠷⣅⠄⠀⢠⣤⠤⠠⠶⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣿⣼⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡴⠞⢤⡀⣡⢦⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⡆⠀⠸⡄⢡⠘⣣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠉⠙⠒⠂⠴⠤⠤⠴⣯⡴⠖⠒⢚⣛⠒⡖⠿⣼⣉⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠤⠤⠤⠤⠒⢚⣯⠄⡇⠀⠀⢀⡤⠴⣇⡿⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡇⠦⢀⠎⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⣠⣚⡛⢢⣼⠀⠁⠀⠀⢸⠀⠀⠀⢀⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣸⡀⠀⠱⡀⠀⢸⡁⠀⠀⠘⣹⠀⠀⠀⣀⠞⠀⠀⠀⡎⠀⠀⡸⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠀⠀⠙⣌⢂⠀⡇⡠⢄⡉⠉⢺⠟⠀⠉⠈⠉⡇⢀⠤⢄⠀⠸⡀⠞⡱⠁⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣚⣉⢑⡈⠋⠀⣿⣀⡠⣈⠖⠁⠀⠀⠀⠀⠀⡞⢁⠶⣎⠀⡠⠃⠛⢡⣋⡩⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⣩⠀⠀⠀⠉⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⡇⢜⡲⠮⡉⠃⠀⠀⡠⢤⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠔⠋⠀⠦⡄⣠⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⣄⣅⠀⠀⠀⢘⠄⠉⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡇⠀⠸⣄⠀⠀⠀⠀⠀⠀⣀⠞⠀⢸⠃⠈⡏⢱⢰⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠈⠑⠒⢢⠴⠖⠊⠁⠀⢀⡇⠀⡰⠃⡌⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠸⠀⠀⢀⡤⠒⠒⠚⣦⣀⡔⠳⣄⡜⢠⠞⠁⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢇⠀⠀⡏⠀⠀⠠⠒⢩⠉⠀⠀⢀⡴⠁⠀⠀⢀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢌⠳⡀⠳⠤⠔⠁⠀⣾⠀⠀⢠⠎⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⡶⢄⠀⠀⠀⡇⠀⡰⠁⠀⣠⣄⣠⠤⣫⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠑⠒⡞⠀⢸⠦⠲⠚⢁⣡⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠀⠀⠀⠀⠠⣇⣀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⢋⣭⣀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡤⣗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢆⡁⠀⠳⣄⠀⠀⡜⠀⠈⠑⢤⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡙⠲⡉⢻⣣⠶⠁⠀⠀⠀⠀⠹⠙⠢⢄⡀⠀⠀⠀⠠⢤⢀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⠪⡃⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⢳⠒⠒⠒⠒⠋⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⣠⣿⣦⡄⠀⠈⡶⢄⠀⠀⢠⠁⡇⠀⠀⠀⠈⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⢏⠣⣙⢦⡄⢳⠀⠙⠻⡾⠶⠤⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠱⣌⢲⡿⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⣀⣀⣤⣀⣀⣈⣉⣀⣀⣀⡀⠀⣿⣴⣤⣤⣤⣤⣤⠤⠤⠿⣾⠿⠻⠿⡍⠉⠩⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠊⠉⠁⠉⠉⠀⠉⠋⠉⠋⠉⠉⠀⠀⠈⠑⠒⠂⠠⠬⠭⠍⠉⠉⠉⠂⠘⠬⠉⠩⠭⠵⠋

⠀⠀⠀⢠⢶⠈⠄
⣠⡷⡀⠀⢙⠀⠼⠄
⠘⢶⡥⠊⢱⠆⠂⠀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠘⢇⠀⠉⢀⡴⡿⠈⠙⠀⠠⠤⠐⡒⠀⠀⠀⠴⠦⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⡶⣊⠐⣤⣐⣈⣉⣤⠴⠶⠓⢐⠦⢤⣤⣤⡄⢀⣀⡠⠄
⠀⢄⠚⢧⣴⢯⣠⠀⠐⠦⠀⠀⠺⡱⣤⠢⡀⠀⠀⠀⠐⠀⢢⣀⡤⢄⠒⠀⠁⠀⠀⠀⠀⠁⢠⡷⠄⣘⣛⠛⠛⡀⢈⠁⠀⣴⠱⡄⠀⠀⠀⡈⢋
⠀⠸⡎⢔⠉⢽⣕⠀⠀⠀⠠⢤⡀⠈⠪⣷⡈⡀⠀⠀⠀⠀⠀⠱⡊⡁⠀⠀⡠⠖⠀⢥⣭⡈⠈⢕⢿⣿⣟⠛⠛⠛⠓⠂⠀⠀⢀⣿⡌⠛⠁⢠⠈⣆
⠀⠀⢓⠁⠐⠕⠈⠂⠀⠰⣤⡀⠛⢧⣄⡈⠻⣰⠀⠀⠀⠀⠀⠀⠱⡀⠁⠠⠵⠒⠒⠓⣆⠤⢠⠀⠀⢹⣿⣥⡅⠉⠌⠀⠐⠐⢻⣿⢃⣄⣀⢠⠀⡾⠅
⠀⠀⢸⠀⠀⠀⠈⠀⠙⢦⡀⠙⣦⣀⢈⣳⣄⡠⣢⠀⠀⠀⠀⠀⠀⠁⠀⠄⠢⠀⠀⠀⢹⡉⠄⠀⠀⠀⠹⣿⣻⡚⠒⠀⢰⣶⠞⣷⠚⠛⠛⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠉⠻⡢⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠈⠩⠙⠗⠐⢾⣛⣿⣿⣄
⠀⠀⠀⢻⠀⠀⠠⠈⢻⢿⣶⣶⡀⠘⣟⣛⣛⣀⣤⠑⠧⠀⠀⠀⢀⠀⠀⢂⡠⠄⠀⠂⠀⠀⠀⢀⣠⣄⡀⠀⠀⠠⢐⣀⡀⣹⣼⣿⣿⣶⣡⡀
⠀⠀⠀⢸⡄⠀⠁⠁⠀⠋⠣⠶⡇⠀⢋⡭⢭⠭⢤⣶⢀⡆⠀⠀⠀⠃⠀⠈⢖⢀⣢⠖⠩⠂⠀⠀⠘⠋⠉⣠⣤⣶⣾⣿⣿⠿⠿⠖⠛⠉⠁
⠀⠀⠀⠀⣧⠀⠈⠢⠰⠀⠁⠐⠁⠀⠘⠋⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣠⣤⡴⠶⠚⠛⠉⠙⠉⠉⠉⠁
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡸⠁
⠀⠀⠀⠀⢸⡻⣄⣀⣠⣤⣤⣤⣤⠤⠤⡤⠤⠴⠶⠒⠖⠒⠛⠛⠉⡉⠡⠀⠀⠈
⠀⠀⠀⠀⠀⡏⠁⠉⠀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢿⡆⢀⠈⠩⠀⠂