High five Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⠀⠀⠀⠐⠦⢄⡀⠀⠀⠀⢀⠁⠀⠀⢹⡄⠀⠀⠀⠀⠒⠒⠉⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠒⠀⠀⠀⠀⠀⠘⠀⠀⢸⠈⡧⠀⠀⠀⠈⠉⠉⢠⠖⠉⠉⠁⠀⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡆⠀⡆⠈⠀⢿⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⣀⠀⠀⠀⠀⠀⠼⡇⠈⠁⠀⢀⠀⣀⠓⠰⠇⠀⠀⢸⠀⠀⠀⠀⡠⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠁⠐⠋⠉⠁⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⡀⠈⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠑⠀⠀⠀⠉⠉⠙⠿⠲⡀⠀⢠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢿⣿⠀⠀⠂⠉⠂⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⡆⠀⠰⣄⣀⠀⢠⡀⠀⢀⠀⣄⠋⠑⠀⠀⠀⠈⠉⠑⠀⠀⠡⠀⠀⠑⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢱⠈⡠⠀⢀⠜⠁⡴⠁⠈⣇⠀⢣⠀⠢⠀⡇⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢠⠂⢸⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢄⠀⠀⠀⠛⠒⢒⠒⠒⠒⡸⠁⢀⠂⠀⠀⡠⠋⢀⡜⠁⠀⠀⢸⡄⠀⢇⠀⠀⠘⡀⠀⢦⠒⠂⢈⠁⠉⠁⠀⢈⣂⠐⡁⢀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢤⢄⠀⠀⠀⠀⠉⠉⠉⠀⢠⠏⠀⢀⠜⠀⢀⡞⠀⠀⠀⠀⠀⢳⠀⠘⣆⠀⠀⢰⡄⠀⠉⠉⠀⠀⠀⡠⠛⠀⠀⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⢀⡀⠀⠀⠀⠃⠀⡠⠎⠀⢀⠎⠀⠘⢠⠀⠀⠀⠈⠇⠀⠘⢄⠀⠈⡈⠓⠶⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠃⠀⠀⠀⠀⡄⠀⢀⣠⢤⠔⠋⠀⢀⠔⠋⠀⠀⠀⠈⠀⠀⠀⠀⠘⢆⠀⠈⠓⢤⣀⡀⠀⢤⠜⠀⠀⠀⢀⠏⠀⠰⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⢄⠀⠀⠰⠀⠉⠀⠀⠈⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡌⠁⠁⠀⠀⢂⣀⡀⠔⠁⠀⠀⠀⠰⣠⠃
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⢢⠀⠆⠀⠀⠀⠀⢸⣉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢣
⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⠀⠀⠀⠀⠉⠀⠀⢰⠖⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⢳
⠀⠀⠀⠀⣸⠛⠒⢢⡴⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡄
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⡇
⠀⠀⠀⢀⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣀⡇⠀⡇
⠀⠀⠀⢸⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢰
⠀⠀⠀⠈⣆⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡼
⠀⠀⠀⠀⠸⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⠀⠀⠀⠀⢀⣠⡇⠀⡇
⠀⠀⠀⠀⠀⠱⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠆⠠⠠⠄⠀⠐⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠅⠍⠩⠉⠙⠈⢽⢿⡒⠒⠶⢤⡀⢀⣀⣀⣀⣀⣀⣤⠤⠴⠚⠛⠛⡒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⡇⠀⠀⠀⡟⠉⠁⠩⢉⠟⠡⠌⠀⠀⡀⣤⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠂⠀⠀⠀⠀⠀⠂⠄⣇⠀⠀⡄⢃⠀⠀⢴⡧⠀⠀⠀⢼⠀⠀⠀⡁⣩⢰⠞⠀⠀⠀⠀⠀⡙⢦
⠀⠀⠀⠒⠒⠛⠛⠙⠉⠊⠁⠀⠀⠀⠀⠀⠀⠀⠰⢰⡟⢆⠀⠁⢨⠖⠀⠀⣣⣄⣀⠀⢘⡀⢀⠐⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢈⡳⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠀⠁⠒⠈⠀⣠⠊⠀⠀⠀⠀⢸⡇⠀⠀⢀⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠚⣄⠀⠀⠀⠀⠈⠣⡄⠴⣿⢷⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠒⠀⠀⠀⠠⢄⣴⠟⠉⠁⠐⠀⠀⠀⠀⣸⢿⢮⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⡀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠇⣿⢻⡅
⠀⠀⠀⢀⣀⣠⣬⡥⠽⠶⣄⣀⠀⠀⠀⠀⠄⠀⣀⣀⠀⠣⣍⣀⣠⡤⠋⣹⠆⠀⠀⠀⠀⢀⣺⠁⠀⠓
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢿⡂⠠⠄⠀⣀⣤⡞⠀⡀⠠⠠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠦⠥⠤⢤⣤⣥⣤⣄⣤⣤⠤⠶⠒⠛⠓⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⣿⣿⣿⡇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣿⣿⣿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣤⣤⣿⠁⠈⢻⡿⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣹⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⠀⠘⣿⠀⠀⢸⡇⠀⠀⢻⡶⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⠀⠀⢹⡇⠀⠘⣇⠀⠀⢸⠁⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣀⣤⣶⣶⣶⡄
⠀⠀⠀⠀⣿⡄⠀⠈⣷⠀⠀⢿⠀⠀⢸⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣇⠀⠀⢻⡀⠀⢸⠀⠀⢸⡇⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣭⣭⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠸⡇⠀⠘⠀⠀⠀⡇⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⣭⣭⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠘⣯⡈⠉⠛⢦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠈⢷⡀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃
⠀⠀⠈⢷⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣫⣾⣿⣆
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠋
⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠎⠡⠔⣋⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠂⢠⡄⠀⠀⡀⠐⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢠⣿⡿⠀⠺⠄⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢉⠸⠿⢿⠰⠀⠀⠀⠀⠀⠀⡾⠟⠛⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠐⢉⠃⠀⠀⠀⠀⠀⠀⣄⠁⣀⠀⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠟⡁⠀⠀⠀⠀⠀⢀⠋⠀⡇⢰⠂⢳⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠤⠄⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡟⠁⠀⠒⠈⢂⠀⠀⠀⠀⡈⠀⢀⠁⢸⠀⠠⠀⠀⠀⠀⠀⡀⠀⢢⡠⠤⠊⠀⠀⠛⠁⢙⠁
⠀⠀⠀⠀⠀⠀⠸⣿⡇⢈⠀⠀⢖⠒⠒⢈⠀⠀⠀⢠⠁⠀⡈⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠈⠂⠄⠀⠀⠀⡶⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⠈⢓⠠⠂⢀⡀⠀⡄⠀⢀⠁⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⣀⡠⠤⠐⡞⠀⠀⡠⡏⠐⠂⢄⣀
⠀⠀⠀⠀⠀⠀⢀⠊⠙⢅⡂⢤⠊⠀⠀⣀⡀⠠⠴⠀⠀⠰⠀⠀⠀⡇⠀⢠⣀⠠⠄⠂⠉⠀⠀⠀⢰⡇⠀⡰⢁⣷⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⢀⠊⠀⠄⠒⠚⠈⠁⠉⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠴⡴⠅⠞⢹⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠄⠀⠸⡀⢃⡀⠌⡠⠊⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣀⣠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠺⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠉⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠸⣿⠟⠁⠀⣀⣀⣠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⠁⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠛⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⣿⣿⠇⠀⢀⣴⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⢰⣿⣿⠉⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠁⠀⣿⣿⠀⠀⣾⣿⠏⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⣿⣿⠸⣿⣿⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⣿⣿⠀⢰⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⣇⢿⣿⡆⠀⠀⣿⣿⣿⣶⣄⠀⠀⠀⠀⣸⣿⡏⢰⣿⣿⠀⠀⣿⣿⠀⢸⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠈⣿⣿⡘⣿⣿⡀⢰⣿⣿⠉⢻⣿⣧⠀⠀⠀⣿⣿⠀⢸⣿⡟⠀⢠⣿⣿⠀⢸⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢹⣿⡇⢹⣿⣧⠀⣿⣿⡀⠀⢻⣿⣇⠀⢸⣿⡿⠀⢸⣿⡇⠀⢸⣿⡟⠀⠀⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠘⣿⣿⠀⢿⣿⡄⢸⣿⣧⠀⠈⣿⣿⡄⢸⣿⡇⠀⣸⣿⡇⠀⣸⣿⡇⠀⢸⣿⡿⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⣿⣿⡆⠸⣿⣧⠀⣿⣿⡄⠀⠸⣿⣷⣸⣿⡇⠀⣿⣿⠁⠀⣿⣿⠁⠀⢸⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⣠⣤⣤⣄⡀⠀⣿⣿⡀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⠀⢻⣿⣿⣿⠃⠀⠉⠉⠀⠀⠈⠁⠀⠀⠈⠉⠁⠀⠀⢸⣿⡇⠀⠀⣠⣴⣶⣿⣿⣶⣄
⠀⠀⠀⣿⣿⠻⢿⣿⣷⣼⣿⡇⢸⣿⣧⠀⠻⠿⠀⠸⣿⡿⠀⠀⠈⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣠⣾⣿⠟⠋⠉⠉⢻⣿⣧
⠀⠀⠀⢻⣿⣦⠀⠈⠻⣿⣿⣿⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⠁⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠻⣿⣷⡀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡟⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠀⠀⠀⢠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⠋⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⣀⡀⠀⣀⡀⢀⣀⠆⣠⠀⢀⣆⡀⢀⣀⠀⠀⣔⡀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠸⠗⠧⠐⠃⠃⠿⠚⠆⠩⠆⠞⠁⠃⠯⠺⠄⠞⠛⠀⠿⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⢀⢤⠇⢹⠛⣷⠢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢸⠀⡬⠀⡟⢦⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⢸⠀⠟⠀⡇⢸⢸⡄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠁⠀⠀⠀⠀⠇⢸⡀⣇⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠄⠀⠀⢸⣇⢻⠀⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠆⢀⡞⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⣠⡟⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠉⢠⣟⠦⠂⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⠏⠈⢆⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠆⠀⠀⠀⠰⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⢠⡀⠀⢀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢲⠃⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⣦⠠⠤⠤⠤⠒⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⡀⠀⠺⠃⠀⢠⠈⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙⠢⠄⡄⠔⠁⠉⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡤⠄⡀⠀⠃⠱⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠤⠔⠂
⠀⡀⠣⡀⠘⢄⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⢶⠈⣿⣿⣷⠂⠂⠀⡖⠈⠉⠉⠁
⢇⠀⠁⠻⠆⠈⠁⡨⠂⠀⡘⠀⠀⠀⠀⠀⢸⠀⣿⣿⠃⠀⠀⠀⡇
⢀⠩⠶⠂⠀⠀⠈⠀⢀⡌⠀⠀⠀⠀⠀⠀⠘⡇⠙⠅⠱⠤⠀⠀⠙⠒⠐⠈⠁⠒⠒⠢⢄⡀
⠘⠒⠂⠉⠳⢆⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⣀⡠⢤⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠴⠖⠚⠙⣄⣀⣀⣀⣤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠙⣲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠉⣉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⠤⠇⠀⣀⠀⢠⡀⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠘⠤⠔⠊⠉⡇⠀⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠙⠢⢤⣀⣀⣀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⡀⠀⠀⣀⣀⡠⢤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠁⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣦⣄⣧⠀⣌⣠⣴⣶⣾⣿⣿⣶⣶⣤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡎⠙⡆⠀⠀⡇⠀⢸⠀⠀⠀⡴⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣦⣶⣦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠹⠀⠀⡇⠀⠈⡄⠀⢠⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣤⡿⠟⠉⠀⢀⡸⢿⣿⠇⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢇⠀⡇⠀⠀⡇⠀⡞⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠾⣆⠀⠀⠀⠀⠙⠛⠉⠀⠀⢀⠃⡀
⠀⠀⢸⠉⠡⠀⠀⠀⠃⠀⠘⡄⣿⠀⠀⡇⢰⠅⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠦⠤⠤⠤⠖⠚⠋⠉
⠀⠀⠈⢧⡀⠑⢄⠀⠰⠀⠀⢇⡿⠀⠀⣇⠇⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢳⡀⠀⠁⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⣉⠐⢄
⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠋⠁⠀⢠⠛⢦
⠀⠀⠀⠀⠀⠀⠸⡆⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢣⡀⢀⣠⠔⠀⠀⠀⠢⠀⠀⠀⠀⢀⡀⠀⣀⡤⢄⡀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠠⠈⠀⠀⢀⠖⠁⢀⣀⡀⣀⠀⠈⠢⠀⠀⠘⢄⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠠⠃⢀⣴⣅⣀⣼⣤⣭⣶⣄⠀⠀⠀⠈⠳⣄⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠉⠉⠉⠛⠉⠉⠙⢻⣷⠄⠀⠀⠀⠈⠳⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣀⠀⢀⠀⣀⡀⠐⠁⡠⠀⠀⠀⠀⠀⠈⠢⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⠤⠒⠚⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠀⠀⡠⡈⢧⠀⢀⣰⣀⠀⠀⣠⣆⡀⠀⣀⣤⣀⠀⢀⣰⣄⠀⠀⣀⣆⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠑⠀⠠⠃⣸⠀⢹⡿⣿⠁⠈⣿⢿⡏⠀⢹⣿⡏⠀⢙⡿⣿⠁⠈⣿⢿⡏⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣠⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⡀⠀⣀⠙⢦⠀⠀⣠⡀⠀⠀⢀⡄⠀⠀⠀⣤⠀⠀⠀⢠⡀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠐⠗⠀⠄⠇⣸⠀⢻⣿⣿⠃⠘⣿⣿⡏⠈⢻⣿⡟⠁⢹⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⣀⣴⣃⣀⣈⣁⣉⣀⣀⣁⣈⣁⣀⣈⣀⣁⣀⣈⣁⣈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡀⠀⢈⠙⣆⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡃⠀⣁⠇⢸⠀⢲⣿⣷⠂⠰⣾⣿⠖⠐⢾⣿⡶⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢀⣠⠏⠀⠈⠉⠙⠀⠀⠉⠉⠃⠀⠈⠉⢁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠉⠉⠉⠻⡍⠉⠉⢉⠉⠉⠉⠉⡉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠔⠀⡊⡆⢹⠀⢶⣿⣶⠆⠰⣾⣷⡦⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠀⠀⠈⣠⠟⠀⠘⠋⠛⠀⠀⠛⠙⢂⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⢿⡉⠉⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⠂⠀⠒⡄⢹⠀⣤⣾⣤⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠁⢀⡞⠀⠸⠛⠷⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠿⠶⠶⠶⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⠀⠃⣅⠀⡀⠀⠀⠀⡉⠀⠀⠀⠀⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⠀⠀⠀⠀⠃⢈⠭⠍⡃⠀⠇⠀⠀⢀⠀⡀⠒⢲⠶⠒⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⢀⡄⠀⠈⢙⣷⣌⣿⣿⡆⢼⡆⠠⣄⠀⠀⣀⡤⣀⢀⣀⡠⠀⠈⢁⡇⠀⡿⠀⠸⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣴⣿⣿⣶⡤⢊⣬⣻⢟⣿⠁⢹⠃⠀⠀⢣⡿⣿⣶⣿⣦⢹⠃⠀⠀⣾⠁⠀⠀⠀⠂⠰⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢿⣛⣿⣿⣟⡛⣿⠃⠀⠀⣧⠀⠀⣼⣿⣿⣯⣿⣿⡄⠀⠀⠀⢻⡇⢸⡀⠠⡀⠐⣻⠧⠀⠀⠀⠐⠢⣦
⠀⠀⠀⠀⠀⠀⠻⠶⠀⠳⠄⠀⠀⠀⠈⠿⢿⣿⣿⣿⣻⡄⢀⠀⣿⣇⠀⣿⣿⢿⣿⣿⣿⠃⠀⠀⠀⠘⣿⣆⣵⣄⠱⡶⣻⡾⠠⠘⠀⠀⢂⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠉⣿⣇⣽⣿⣿⣼⣿⣯⢨⣿⣿⣿⣇⠀⠀⠀⠀⢰⢿⣿⠿⣱⣾⠟⡀⠠⢀⠰⠖⠁⠂⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣶⠄⠠⣶⣁⣠⣬⣿⣿⣿⣿⣿⠀⠀⠈⣿⣿⡏⢺⣿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢑⠀⠃⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠈⣶⡆⠀⠻⣿⡏⣠⡄⠛⠛⠋⠉⢿⣿⠀⢻⡏⠀⠀⠀⣹⣿⡇⢘⣿⣿⡇⠀⠐⠂⠀⠀⢀⠀⠀⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠿⣟⠁⠺⠃⠀⠀⢠⣯⣽⣷⡼⢿⡿⠇⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⢤⠁⠀⠠⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣦⣝⣤⣮⣬⣷⣥⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣠⡴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡈⠙⢧⡼⠃⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠘⠃⠀⣸⠁⠀⠀⠀⠀⠀⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠾⡇⠀⠀⠀⠀⠀⢿⠀⠰⠷⠀⠀⢠⠏⠉⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢳⡀⠀⠀⠀⢀⡞⢦⣀⣀⣠⡴⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠲⠶⠚⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠻⣦⡀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠈⠛⠶⣦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⣶⣶⣶⠞⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⡍⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠉⠉⠉⠛⠒⠺⠇⠹⣄⠹⠴⠶⠒⠚⠋⠉⠙⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠟⠲⠶⠤⠤⢤⣤⣤⣤⣤⠤⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⣿⣿⣿⣿⣶⣶⣤⣤⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣴⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁


⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢤⣼⢡⡆⣿⠀⡿⠀⡇⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣦⢹⢸⡇⣿⢠⠁⡀⡇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣴⣿⣾⣶⣷⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣤⠙⠁⣤⡏⢠⠙⡇⣤⢹⢸⣿⡇⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣩⣴⠀⣼⡇⢸⠀⡇⣩⣼⢸⣿⡇⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣿⣿⣤⣭⣷⣬⣼⣧⣿⣿⣤⣭⣧⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢹⡏⣿⢹⢩⣍⣯⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⢸⣦⡙⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣼⣧⣡⣾⣬⣥⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢻⡛⡏⣉⣽⢉⣩⡏⣉⠻⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⡌⠀⡇⣉⣿⢈⣹⡇⣿⠀⣿⡇⡶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣿⣠⣇⣙⣻⣈⣛⣇⣙⣰⣿⣇⣶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⣿⠻⡟⣿⢛⠻⡟⣿⠻⣿⣿⠟⣛⡟⢟⢻⡟⣿⢛⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠛⠀⡇⡇⢸⠲⡇⠛⠠⣟⢻⡄⢛⡇⢸⠸⠇⡇⠘⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⣀⡇⣧⡘⢡⡇⣿⢀⣿⣿⡄⣿⣇⣸⡆⢰⣇⠘⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢿⡟⣿⣿⠿⣻⡿⣻⠿⡿⠿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣀⢸⣏⣿⣤⡀⢽⣇⣽⣀⡃⢟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢿⡿⡿⠿⡿⣿⡿⣿⡿⠿⣿⣿⢿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⡇⣗⣿⣅⣿⢁⢻⡸⠗⣅⣗⡺⢸⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⣉⠛⣿⠛⣉⡙⢿⡏⠙⣿⠋⠙⣿⢉⣉⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣌⠛⠿⣿⢸⣿⡇⢸⡃⣆⠉⢠⠀⣿⠘⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠙⠿⠂⣿⠘⠿⠇⣸⠆⣿⣤⣿⠀⡿⠸⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣤⡌⢻⠁⣤⣤⣼⠋⣤⣌⢹⡇⢠⣤⡙⣿⢸⣿⣿⡏⢠⣤⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣛⣃⣼⠀⣉⣉⣿⠀⣿⣿⢸⡇⢘⣛⣠⣿⢸⣿⣿⡇⢈⣉⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢀⣿⣿⣿⣀⣛⣛⣻⣄⡛⢋⣸⣇⣸⣿⣿⣿⣘⣛⣛⣇⣘⣛⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣿⢿⣿⡿⢿⣿⠿⢿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠀⣿⢰⣿⡇⢸⠁⢶⣦⣽⣶⡆⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⣿⢸⣿⡇⢸⢷⣦⡌⢻⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣬⣉⣴⣿⣦⣉⣥⣾⣦⣉⣡⣾⣿⣇⣿⣿⡇⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⢿⡟⢿⡟⠛⠛⢻⡟⠛⠛⢻⡿⠛⠻⢿⣿⣾⣿⡿⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⡇⢸⡇⠿⠿⣿⡇⠸⠿⢿⡇⢸⣿⠀⣿⣿⣿⠀⡄⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⣇⠁⢸⡇⣶⣶⣿⡇⢲⣶⣾⡇⢸⣿⠀⣿⣿⡏⠘⠃⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⣶⣾⣧⣴⣦⣼⣷⣶⣴⣼⣷⣤⣶⣾⣿⣿⣧⣾⣿⣦⣿⣿⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠙⣿⡟⢻⡟⣿⠟⣉⡛⢿⠛⣿⣿⢹⣿⣿⣿⡟⢉⣉⣙⡟⢻⠛⣿⡿⢹⡟⢉⣉⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠛⠃⢸⡇⣿⢸⡿⠿⢿⠀⠛⠛⢸⣿⣿⣿⡇⠘⠛⢻⡇⢸⡇⢸⡇⣸⡇⠘⠛⢻⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⡇⢸⡇⣿⠘⠿⠆⣸⠀⣿⣿⢸⣧⣤⣼⡇⢸⣿⣿⡇⢸⣿⠈⢁⣿⡇⠸⠿⢿⡿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡿⣿⣷⢿⡿⣶⣿⣿⢿⣿⣿⣾⣿⠿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣾⣿⣷⣶⣶⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⣷⣤⣿⣷⣧⣥⣿⣽⣿⣿⣿⡁⠀⠈⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣴⣷⣯⣽⣥⣿⣿⣤⣧⣧⣤⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡄⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣩⡤⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠙⢦⡀⠀⣠⡶⠛⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠋⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣤⣄⠀⠀⠈⣷⢀⣤⡄⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⣾⣉⣿⡆⠀⠀⣿⣾⣸⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣹⡄⠀⠀⠀⢻⣽⣿⠃⠀⢀⡿⠙⠿⠟⠀⠀⠀⠀⠀⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢠⣿⠋⠙⠛⢉⠉⢷⡀⠀⠀⠀⠉⠁⠀⢀⡼⢷⣄⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⢸⡿⠀⣴⠿⣏⠀⠀⠙⢦⣄⣀⢀⣀⣴⠟⠁⠀⠙⠷⠦⣤⠴⠾⠋⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢸⡇⠈⠀⠀⠈⠻⣶⣤⣀⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣠⠂⠀⠀⠀⠉⣀⣠⡼⠟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠷⠶⣤⣄⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠻⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠷⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡏⠉⠛⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⠶⢶⣤⣤⣤⣤⣤⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⢿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡶⠶⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⣾⠁⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠹⡆⠀⠀⣿⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣿⠀⠀⢻⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠘⡇⠤⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⢇⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡇
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⢸⠀⠀⢻⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⡇⣀⠀⠀⠀⢸⡇
⠀⣾⠉⠉⠉⢳⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⢳⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⡄⢸⠠⠤⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇
⠀⠸⡆⠀⠀⠀⠈⢦⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣷⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⢻⠀⠀⠀⠀⠘⣇⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠇⠐⠀⠀⠀⠀⡜
⠀⠀⠀⢇⠈⠀⠀⠀⠙⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢃⠀⠀⠀⠀⠀⢸⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠘⡆⠀⠸⡇⠀⠀⠀⠀⠀⢳⢸⡂⠀⠀⠀⠀⢐⡄⠀⢀⠇⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠹⡆⢀⠀⠀⠀⢱⠀⠀⢳⠀⠀⠀⠀⠀⠸⣴⠟⠀⠀⠀⠀⠘⠓⠰⠛⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢆⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⢷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡠⠖⠉⠂⠀⠀⠀⠀⠀⠀⣰⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣢⣷⣤⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣶⡸⣏⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣴⠶⠶⣤⣻⣷⣄⣸⡿⣧⢻⣏⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣯⣭⣀⢻⣅⠀⠀⣸⡏⢿⡍⠻⠃⠙⠃⢉⡁⠀⢰⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡽⠇⠙⠛⠛⠋⢀⣈⠀⢤⣴⠶⢶⡜⣷⡀⠀⢻⣟⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⣤⠶⠆⢠⡾⠋⠉⠻⣮⢿⣤⠾⠃⠘⣷⡴⠎⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣹⡗⢹⣷⠶⠘⢷⣄⣀⣴⠟⠘⠿⠀⠀⢀⣠⣤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠁⠀⠀⠻⠓⠛⠀⠀⠉⠀⢠⣄⣰⡞⠛⠈⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⢻⡌⠙⠛⣳⡆⠈⠿⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠻⢦⣤⠾⠃⠘⠛⠉⡀⠀⠀⠀⠀⢸⣟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢁⣀⣠⡀⣤⠶⠆⢻⣟⠉⠻⣦⠀⠀⢸⣿⠉⠙⠗⠀⢀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠈⣿⡼⣯⣁⡄⢹⣷⠶⠀⢿⣆⣤⠟⠀⠀⠈⢁⣀⠀⠀⣿⠈⣿⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠻⣾⣷⡹⣧⣤⠤⠻⠗⠛⠈⠉⠀⣤⡶⠖⠸⣧⠙⢷⣄⣿⠀⠘⣷⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠉⣀⣤⣤⠠⣦⠀⠈⢿⡄⠀⠘⣷⡶⠂⢻⣇⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠈⣿⡜⣷⢸⣟⠠⣤⣤⢹⣷⠛⠛⣿⡀⠀⠹⠇⠀⠀⠉⢀⢀⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠙⣷⡸⣧⠙⠷⠶⠾⠃⠛⠂⠀⠈⠁⣀⣠⠤⠴⠄⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⣀⡀⢀⣀⣤⠀⠄⠶⠢⠛⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡏⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠛⢉⡿⠀⠀⢻⡄⢰⡇⠀⣠⠃⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣼⠃⠀⢀⣀⢳⣼⣧⠞⠁⠀⠀⠀⢸⡉⠙⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⢦⡀⠀⠀⠀⠀⠈⠁⠀⢰⡟⠉⠉⠹⠿⠶⣆⠀⠀⠀⠈⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣤⠿⠂⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠒⠂⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢀⣠⡤⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⠠⠤⠤⣿⠀⠀⠀⠀⢀⣠⣤⡀⠀⢀⡴⠛⣉⠉⣷
⢀⡴⠋⣡⣤⣤⠀⠉⣠⣤⣈⡙⡷⠀⠀⠀⢀⣀⡀⣼⡃⠀⠀⠀⠀⢀⣟⣀⠀⠀⠀⠘⢧⡈⠳⢶⠟⠀⠚⢁⣴⠋
⢿⣤⡾⠉⡴⢁⡴⠀⡏⠀⠈⠙⠁⠀⢠⠞⠉⠁⠀⠈⠙⢦⡄⣰⠞⠉⠉⠉⠙⢦⡀⠀⠀⠙⠳⠶⣄⠀⠶⣯⠤⠶⠛
⠀⠀⠀⠰⡇⠘⢁⡼⠁⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⠳⠦⣤⡤⠶⠚
⠀⠀⠀⠀⠙⠶⠛⠁⠀⠀⣴⠶⠶⢾⡀⠀⠀⠀⠀⠸⠇⠀⢸⡄⠘⠛⠀⠀⠀⢀⡿⠶⠶⠶⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣠⡞⠳⣄⡀⠀⢀⣠⠞⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣀⠀⠈⠳⢦⣤⣠⡤⠾⠋⠀⠀⠈⠙⠛⠋⠁⠀⢀⣤⠀⠀⠀⣿⡇⠀⢠⣞⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⣿⡇⠀⠀⠈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⠤⠤⠤⠤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣦⣄
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣀
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠈⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⣿⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠋
⠸⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⡿⠿⠛⠋⠁
⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡿⠛⠛⣿⠟⠛⠿⡿⠿⢿⠿⢿⡿⠿⠿⣿⡿⠿⠿⣿⠿⠿⢿⠿⠛⠿⡿⠿⠿⢿⠿⢿⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣇⣀⠙⠏⠀⣷⠀⡇⠀⠈⠀⠘⡇⢈⣹⣿⡇⠘⠀⣸⠀⣉⣿⠀⣿⠀⣿⠀⠃⢸⠀⢸⣿⠀⣉⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣧⣤⣤⣿⣦⣤⣴⣧⣼⣦⣷⣴⣧⣤⣤⣿⣧⣼⣿⣿⣤⣤⣼⣦⣤⣴⣿⣴⣿⣿⣦⣤⣼⣤⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣧⠀⣿⠀⣿⠁⢹⠁⠠⢬⣧⡄⢀⣼⣿⠁⠙⠇⢸⡇⠀⣤⣽⠀⢠⣬⡇⢀⣄⠉⣿⣿⠁⠀⠹⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠟⠀⣿⠀⠻⠀⣸⠶⠄⠀⣿⡇⢸⣿⣿⠀⢠⠀⢸⡇⠀⠶⢿⠀⠰⠾⡇⠘⠟⠀⣿⡏⠀⣁⠀⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠶⠾⠿⠷⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⣤⡄⣤⣤⢀⣤⣶⣦⠀⣤⡄⢠⣤⠀⠀⠀⢠⣤⣤⡄⣤⡄⣤⣤⢀⣤⡄⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⡇⣿⣿⢸⣿⢠⣤⡄⣿⣷⣾⣿⢠⣤⣤⢸⣿⣶⡄⣿⡇⢸⣿⣸⣿⠀⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠿⠇⠿⠿⠘⠿⢶⠿⠇⠿⠇⠸⠿⠀⠀⠀⠸⠿⠀⠀⠿⠇⠀⠿⠿⠏⠀⠿⠷⠶⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣘⢀⢈⣷⢰⢈⣀⠀⢾⢀⡏⡀⡃⢾⠀⠿⣿⠈⠀⡆⣆⣶⢈⢸⡇⡈⣏⠰⢈⡁⢃⡘⢈⢀⢸⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣖⣲⡒⠒⠒⢒⡒⠒⠒⠒⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⣖⠒⠒⠒⢒⡒⠒⠒⠒⣲
⠀⢸⡃⠈⣦⡠⠏⠈⠓⢄⠄⢸⠀⢀⡤⠤⠤⠤⠤⠤⠤⡀⠀⠀⢀⠤⠤⠤⠤⠤⠤⢤⡀⠀⣟⠀⢹⣄⡼⠅⠉⠢⡠⠀⡇
⠀⢸⠫⣄⡍⠈⣢⠀⡨⠓⡆⢸⠀⡼⠓⠛⠛⠛⠛⠛⠛⢣⠀⢀⡜⠛⠛⠛⠛⠛⠛⠛⢧⠀⡟⢥⣈⠉⠑⡄⢀⠜⠣⠀⡇
⠀⠸⡇⠃⠉⠍⠸⢏⠐⠀⡘⡏⠀⡇⢀⠖⠡⢶⣌⠑⡄⢸⡇⢸⡇⠀⡞⠢⡀⠀⠀⠀⢸⠀⢹⠸⠉⠩⠁⡾⡉⠂⢀⢱⠇
⠀⠀⡏⢑⠖⠠⣇⠎⠉⠚⢡⡇⠀⡇⠃⠀⠰⣾⡟⠀⠈⣼⡇⢸⡇⠀⡄⠀⠿⠦⡀⠀⢸⠀⢸⠈⡲⠂⢴⡠⠁⠁⠛⣽
⠀⠀⡿⡀⠌⠀⢨⣀⡀⢀⢺⠇⠀⡿⠠⠶⠲⠾⠶⠖⠄⢸⡃⠸⡇⠀⠈⠲⠥⠦⠚⠀⢸⠀⠸⣏⠀⠁⠀⣧⣀⠀⡰⢹
⠀⠀⢻⠈⠢⡞⠋⠁⡸⡁⢸⠀⠀⡇⠔⠲⠶⠒⠶⠶⢠⢻⡆⢸⡇⣖⣲⢶⣰⠀⡶⡆⢸⠀⠀⡇⠑⢴⠚⠁⢀⢫⠁⡟
⠀⠀⢸⠀⠀⠸⡀⢀⠠⠇⢸⠀⠀⡇⠈⠢⣀⡛⣀⠔⠁⢸⡇⢸⡇⠀⠀⠐⠒⠒⠀⠀⢸⠀⠀⡇⠁⠀⢧⠀⡀⠤⠂⡇
⠀⠀⢸⡃⠉⠀⠉⠉⠀⠁⡟⠀⠀⢣⠤⠤⠤⠤⠤⠤⠤⡜⠁⠈⢣⠤⠤⠤⠤⠤⠤⢤⡞⠀⠀⢻⠈⠀⠈⠉⠁⠈⢘⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠒⢤⡀⠀⠀⢀⡠⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠱⣄⡴⠁⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣿⢢⠀⠀⠀⢸⠇⣾⢃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⡴⠶⢿⠀⠀⠘⣿⡟⠀⠀⠀⢸⡃⠻⠟⠀⠀⠀⠀⢠⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣆
⠀⠀⠀⠀⠘⡇⠀⠈⢣⡀⠀⠀⠀⠀⠀⢠⠏⠱⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣧⠀⠀⣀⠙⠢⠄⣀⡠⠔⠁⠀⠀⠀⠁⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠰⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⠀⠀⢸⠀⠉⢱⠖⠒⠒⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡄⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣦⠼⠒⢦⡴⣦⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠲⠒⠉⠀⠁⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠦⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⣿⠀⠀⢀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠈⠛⢷⠀⠀⠉⠀⠀⠸⠋⠀⠀⣀⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⡾⠛⠁⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠀⣰⣿⣦⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠏⠀⠀⢀⣴⣾⣿⣿⣿⣿⠟⠀⣰⣿⣿⣿⣿⣦⣀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⣉⣀⣤⣴⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣦⡀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⠁

⠀⠀⠀⣀⣤⣶⡆⠀⠶⠶⠖⣤⣾⣿⣿⡆⢀⡀⠀⡄
⠀⠀⢸⣿⠛⠉⢀⡆⠀⠀⢰⣿⢹⣿⣿⣷⣿⣿⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠸⣿⣷⣶⣾⣿⣀⣠⣿⣷⣼⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠸⣽⠃
⠀⠀⢰⣿⡟⠁⠉⠛⠛⠋⠉⢹⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⡄⠀⠀⠀⠀⣶⣶⠿⠛⢣⡄⠀⠀⠀⠀⠃
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡏⠁⠀⠀⠸⠟⢲⡆⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣷⣤⣤⣬⡶⠀⢶⣶⡀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⢸⡏⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢃⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⢶⠶⠀⠀⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢠⠁⣔⢶⣃⡠⣆⣤⣠⠞⠀⠀⢀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢿⣐⡀⢀⡘⠃
⠀⠰⡇⠀⠀⣠⠇⠀⠝⡯⣛⠿⠈⠛⠟⡀⠄⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠈⡿⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢺⡁⠀⢀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠘⠒⠐⠀⠒⠒⠓⠀⠀⠐⠃⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠅⠀⠀⠛⠟⠛⠀⠉⠙⠒⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⡠⢀⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠙⠛⠛⣿⣌⠻⣆⡀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠆⠀⠀⣀⣉⣤⡾⢿⣿⣻⠃⣤⠀⠀⠀⠀⠀⠀⣄⢙⣿⣿⡳⡌⠳⢦⣄⣤⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⠉⠀⢀⣸⣿⡿⡦⠙⠀⠀⠛⠛⠓⠐⢿⠋⠻⣿⠁⠘⠳⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠟⡉⠳⣷⣾⡞⢶⣤⣤⣤⣴⣎⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⠈⠉⣻⠟⠿⠟⢿⡉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣿⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⢀⠀⠄⠉⠀⠄⢘⡉⠀⡄⠠⠀⠀⠀⣿⣿⣇⡀⠀⠀⢠⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠁⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⢠⡾⢫⡤⠃⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⠰⠶⢄⠀⠀⠀⠀⠀⢠⣤⠄⠀⠘⡆⢀⠙⡆⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠠⠀⠓⣾⠠⢶⣾⡆⠀⠀⣰⠖⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠈⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠐⡗⠰⡿⠄⠈⠀⠀⠀⣀⠀⣀⠄⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⡇⠀⢴⣤⡅⠀⠀⡔⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣦⠀⢿⣶⣶⣶⠆⠀⠀⠀⠙⠀⠀⠀⠀⠙⣿⣤⡀⠀⠀⣤⣤⠃⠀⣶⣿⠋⢠⠀⠁⢹⡀⠀⢠⡄⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⡆⠀⣀⣭⡀⠀⠀⠀⢰⠟⠁⠀⠀⠀⢀⠼⣿⣷⡀⣀⣸⣧⡀⣨⣿⠇⠀⠺⡀⢸⣾⣧⠀⠀⠑⠀⠀⠐⣀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠉⠉⠉⠉⣆⢀⡄⠀⠀⠀⢠⣄⠀⠠⠹⣿⣷⡝⠃⠈⣵⣿⣇⢙⣄⠀⢱⣤⠛⢿⠆⠀⠀⠀⢀⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣷⣴⡁⢀⣶⡏⠘⠃⠀⠀⠀⣼⠃⠀⠀⠀⢙⣿⣅⠒⢀⣽⠛⠆⠈⣿⡀⠈⠡⠾⠋⠀⠀⠀⢀⣿⣿⣯⣿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡻⣿⣿⠟⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡿⠟⠛⠛⠋⠡⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠤⠘⠿⠿⠛⠛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠘⠻⠀⢠⠀⠀⠀⠀⠀⠸⣧⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⠻⠀⠀⠀⠀⠀⠀⠁⢀⡿⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⢠⠀⠀⠀⠀⠀⣺⠀⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⠀⠀⠀⠀⠀⢸⡀⠘⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⡼⠀⠀⠀⡿⠄⠀⠀⠀⢁⡾⠟⠀⠀⠀⠠⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢾⣿⣅⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠢⠤⣀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠢⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠘⢻⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡠⠋⠀⠀⠀⠀⠈⣿⣤⠴⠶⠶⠶⠚⠛⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢰⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡏
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⣿⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⢤⣿⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠒⠋⠀⠙⢦⣀⠀⠀⢀⡠⠞⠀⠀⠘⢦⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠔⠈⠻⣷⣦⣄
⠀⢀⠎⠀⠀⠀⠀⠀⠙⠳⣝⡦⡀
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡢⡀
⡆⠀⣀⣠⡤⡆⢔⠀⠀⠒⡢⠀⠈⢻⣝⣄
⡷⢄⣓⠈⡪⡰⠅⠄⠠⡊⠊⠀⢨⢆⠹⣎⢆
⢱⠀⠀⢉⠘⠢⣕⡡⡠⠂⠀⡑⡣⡪⡂⠹⣏⢆
⠀⡇⠀⢘⣰⠚⠒⠾⣶⣌⠚⠪⡪⢊⠌⠄⢻⣟⢆
⠀⠸⠀⡒⢹⠀⠀⠀⠈⢫⣳⡈⢐⠅⠑⢑⡄⢿⣜⢆
⠀⠀⡆⠙⢺⠀⠀⠀⠀⠀⠹⡿⡄⢡⡅⡠⠟⠌⣷⣙⣄
⠀⠀⢁⢠⢸⠀⠀⠀⠀⠀⠀⠙⡿⡄⠺⢶⡣⡢⡸⣧⣎⠢⡀
⠀⠀⢸⠠⢼⠀⠀⠀⠀⠀⠀⠀⢹⣽⡀⠀⠈⠘⠶⠙⣧⡇⣌⠲⠤⣀
⠀⠀⢸⢨⢾⠀⠀⠀⠀⠀⠀⠀⠈⢏⣧⢑⠄⢄⡠⠪⠈⠙⠲⢯⣵⣀⡭⣉⣒⡦⢤⣀⡀
⠀⠀⢀⢘⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⡆⣀⣉⢄⢔⢕⠁⣲⢒⣄⢌⢉⣉⠛⠻⠦⠦⠞⠁⠐⠢⢄⡀
⠀⠀⠸⢈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢷⡨⣐⢕⢕⣡⡀⠁⢈⠁⢀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⢠⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⢣⡩⣑⡝⢗⢕⢅⣀⣵⠕⡕⠀⠄⠀⡰⡀⡀⡁⡑⡡⡐⡐⡐⣤⣡
⠀⠀⢘⠰⣾⠀⠀⠀⣀⡀⠀⣀⠀⠀⠄⢹⣗⢙⠪⢔⣕⡕⠽⠗⡡⠞⢕⠄⢴⠀⠊⣎⣊⣈⣊⣂⡐⢎⣴⢛⣿⠀⢠⣠⣤⠠⡠⣀
⠀⠈⠙⠛⠏⠀⠭⠉⠅⠤⡈⠀⠄⡠⢀⠥⣿⢷⣕⣤⣤⢌⢉⢝⢍⣄⣈⣝⣝⣍⣴⣔⣔⣔⣔⣔⣔⣝⣭⣿⣿⠶⡣⠢⠵⠜⠚⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⠶⠶⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⡛⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣴⠸⠃⠰⠴⠢⢌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠆⠐⠇⠀⡈⠏⠘⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⡂⠀⠀⠀⠁⡀⢈⠔⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡀⠀⠉⠉⢀⡜⣄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢄⠀⢹⠷⠖⠒⠉⡀⣸⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢽⠀⠀⠀⠀⡇⠥⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠰⠱⢃⣩⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠐⢅⠀⠀⠀⠀⡨⠂⠁⠀⠈⠉⠳⡄⠀⠀⠀⠀⠀⢀⠆⢰⠁⠉⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠂⢀⡈⠉⠉⣿⣧⢀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠀⠘⠄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠘⠟⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⡘⠀⠀⠈⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⢤⡃⠀⠀⡐⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢹⠀⠀⠀⠀⠀⣲⣦⠀⢠⠁⠀⠀⢠⠴⠊⠀⠀⢣⠀⡐⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡞⠀⠀⢀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⢈⠏⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⠀⣀⠀⠀⠀⣿⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠃⠀⠀⠈⡀⠛⠁⠀⠀⡟⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠀⠀⢠⠀⡇⠀⠀⠀⢸⡇⠀⠈⢢⠀⠀⢣⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢈⠀⡠⠁⠀⢷⠒⠂⠉⠉⡇⠀⠀⠀⠑⢤⡀⢧⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⠴⠊⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡀⠀⠀⠀⠀⠸⡇⠧⡄⠀⢰⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠖⠚⠛⠛⠲⠃⠀⢰⠀⠀⠛⠛⠛⠛⠛⢶⣶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠻

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣉⢉⢠
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠊⢹⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⢁⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣏⣥⣽⠄⢤⡄⣀⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⠿⠛⠛⠉⠁⣀⣤⣤⣦⣤⡀⢈⠻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣡⣶⡟⢀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣦⣤⣤⣤⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣟⡥
⠀⠀⠀⠀⢈⠓⠶⠶⠾⠿⠿⠭⠭⠭⣭⣭⣭⣉⣙⣛⣛⣛⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣛⣛⣻⣿⣿⠿⠿⠿⠗⠛⠉⠔⠃
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⠀⠀⢰⣾⡆⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠙⢷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⢳⡄⢸⠁⠀⢸⡇⠀⡠⠚⠛⣦
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢈⣿⣤⡄⠚⠃⠀⠀⢀⣴⣄⢿⣿⣴⣿⣦⠀⠀⠀⠀⠘⡇⠀⠀⣷⢸⠀⠀⢸⡇⣸⠁⠀⢀⡟
⠀⠀⠀⠀⢠⣴⣿⣿⠛⠋⣿⡇⢀⣶⡄⢠⡶⠚⢿⣿⢾⣿⠃⣿⣿⠀⠀⠀⠀⠀⢻⡀⠀⢸⣸⠀⠀⢸⢡⠃⠀⢀⡾⢀⡴⠛⢻⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⡇⢸⣿⡇⣿⡅⠀⢸⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠈⣇⠀⠀⣿⣀⠀⣼⣎⠀⢀⣜⡵⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣧⢸⣿⡇⠻⣿⡿⢿⣿⡇⠁⠀⣿⡿⠀⣀⠤⠤⣄⡀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠺⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠁⠈⠿⣃⡀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠰⢅⣀⠀⠀⠙⢾⠧⠴⠦⠤⣄⠀⠀⠀⠀⠀⣴⠊⠁
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣴⣿⡿⠿⠿⠋⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⡄⠀⠀⠶⣂⣼⠟⠀⠀⢀⠀⣠⣶⣦⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢾⣿⣷⣶⡶⣿⣏⢵⣿⡄⢠⡿⢰⣿⣁⣿⠃⠀⠀⠀⠀⢨⣷⢄⠀⠀⠀⠀⢀⣀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠐⣿⣿⠘⣿⣿⣼⠇⢸⣿⣉⣁⡦⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⣤⡤⠞
⠀⠀⢀⣀⣤⣤⣤⠀⠀⠀⠀⢸⣿⣇⠀⣿⣿⠀⠹⣿⡿⠀⠈⠻⠿⠛⠁⠀⠀⠀⢰⣾⡿
⠀⠀⠸⢿⣿⣟⠃⣀⣤⠀⠀⠈⢿⠿⠀⠘⠛⠀⢀⣨⣅⠀⣴⣦⣀⣤⡀⢀⣤⣤⡈⣭⣦⣰⣶⣆⣠⣄⠀⠀⢀⣠⣤⣿⣿⡆
⠀⠀⠀⢸⣿⣿⠀⣿⣿⣶⠿⣷⡄⢀⣾⠛⠓⢠⣿⠉⠙⣷⣿⣿⠟⢻⣷⡟⠉⣿⣿⣿⣿⣸⣿⣿⠛⣿⣧⢠⣿⡟⠉⢻⣿⡇
⠀⠀⠀⠘⣿⣿⠀⢻⣿⡇⠀⢹⣿⣸⣿⠀⠀⢸⣿⣄⣠⡟⣿⣿⠀⢸⣿⡇⠀⢿⣿⡾⣿⣿⣿⡏⠀⢸⣿⠸⣿⣷⣤⣼⣿⡇
⠀⠀⠀⣠⣿⣿⣷⢸⣿⡇⠀⢸⣿⠸⣿⣷⣿⠇⠙⠛⠉⠀⢻⣿⠀⢸⣿⠇⠀⢸⣿⠇⣿⠟⣿⠇⠀⢸⡿⠃⠙⠛⠛⢋⣿⠇
⠀⠀⠀⠙⠛⠉⠁⠈⠛⠁⠀⢸⣿⠃⠀⠉⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠿⠋
⠀⠀⠀⠀⠰⠿⠛⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠶⠶⠶⠿⠿⠿⠟⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠂⣠⠖⢲⣄⣶⢠⣆⡶⠒⠂⠀⣰⠒⣶⣴⠒⠒⢀⡴⠒⣦⡶⠒⣦⡶⠀⢠⡶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⣈⣉⡽⢿⣁⣀⡼⡿⠹⢫⣿⣋⣉⡀⢀⡟⠊⢱⣏⣉⣡⣏⣀⣠⢿⡟⠋⣼⣁⣠⣼⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣉⡁⠀⠁⠀⠀⣉⡉⣀⣀⣀⠀⠀⠈⠉⡁⠀⣈⡉⠀⣈⡀⢀⣈⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡏⣸⠁⢠⡿⠯⢭⡍⢩⡏⠉⠀⣸⣧⢰⣿⡯⠭⢁⣯⠭⠁⡞⠉⡷⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠾⠁⠧⠴⠾⠲⠴⠞⠁⠼⠀⠀⠀⠟⠘⠿⠿⠤⠶⠾⠤⠶⠾⠷⠚⠁⠠⠋⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⣤⣤⠤⠀⣠⠤⢀⡄⢠⡄⠀⢠⡤⠤⣤⣤⠤⡄⠀⣠⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠒⡟⢀⡏⢠⠎⢴⣶⣾⠓⣿⣀⣀⡾⠒⠂⢠⡏⠀⣷⣰⢫⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠓⠛⠋⠉⠛⠋⠀⠋⠀⠃⠀⠀⠁⠀⠐⠛⠋⠁⠘⠁⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠐⠰⠆⠐⠰⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠂⠐⠒⠂⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⢸⣿⠀⠀⠀⣰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠹⣿⣆⠀⢸⣿⠀⠀⣼⡿⠃⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠘⠛⠀⠀⠉⠀⠈⠛⠁⢀⣠⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠸⣿⣿⣧⠀⣴⣶⣆⢰⣶⣦⠀⣼⣿⣿⠇⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠸⣿⣿⣧⢹⣿⣿⣇⢻⣿⣿⣎⢿⡟⣸⣿⣿⠏⣼⣿⣿⠏⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠹⣿⣿⣆⢻⣿⣿⡆⢻⣿⣿⡌⢰⣿⣿⡟⣸⣿⣿⡏⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⢻⣿⣿⡆⢿⣿⣿⡌⢿⣿⣿⡸⣿⡿⢰⣿⣿⡟⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⢿⣿⣿⡌⢿⣿⣿⣾⣿⣿⣷⡙⢡⣿⣿⡿⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣡⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⡿⠀⢀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⢁⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⠿⠟⣭⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⣿⣿⣿⣿⡶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⢿⣿⣿⡿⠉⢻⣿⣿⠟⢻⣿⣿⡟⠛⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣧⠀⢸⣿⣿⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠉⠉⠀⢸⣿⣿⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⠀⢸⣿⣿⣷⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇⣠⣿⣿⣿⣧⣴⣿⣿⣤⣼⣿⣿⣏⢈⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⢀⣼⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠙⢧⡀⠀⠀⢸⡏⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣀⠀⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣶⣾⡆⠀⢀⣼⠏⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢶⣶⣶⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⠃⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡤⠶⠒⠛⠋⠉⠉⠉⠙⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠁⠀⠀⡇⠀⠀⠈⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠠⠤⠁⠀⢠⡿⠀⠀⢀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢸⣇⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠢⠤⣄⣀⣠⠖⠋⠉⠀⠀⠀⠈⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣢⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢷⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡏⢠⠰⣷⣦⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⡷⠀⠀⠀⠀⠀⠀⢸⣶⠟⣀⣈⣉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⡀⡇⠀⠀⠀⠀⠀⠀⢸⣷⠿⠟⠛⠉⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⠙⣒⠶⣦⣀⠀⠀⣻⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣹⡄⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠻⣿⡾⠃⠀⠀⠀⠀⠀⠀⢠⠇⠐⠂⠤
⠀⠀⠀⠀⠀⠀⠰⠶⠀⢀⣴⠾⠟⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠱⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠈⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣦⡀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣴⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⢸⣿⡇⠀⢀⣴⡿⠋⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠸⣿⠇⠀⠘⠛⠁⣠⣶⣾⡿⠛⢻⣷
⠀⠀⠀⠀⠀⠀⣰⣿⠛⣿⣷⣤⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣾⡟⢹⣿⠁⢀⣼⣿⣶⣄
⠀⠀⠀⠀⠀⣾⡿⣿⡄⣿⡏⠙⣿⡆⠀⠀⠀⠀⠀⠀⣠⣴⣿⠁⢸⣿⠀⣼⡟⠀⠘⣿⡀
⠀⠀⠀⠀⠀⣿⡇⢿⣧⣿⣧⠀⢸⣿⣦⣄⠀⠀⠀⣼⡿⢹⣿⠀⢸⣿⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⣇⠸⣿⡜⣿⡄⢸⣿⠉⢿⣧⠀⢠⣿⠇⢸⣿⠀⣸⣿⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⢿⣇⢹⣷⠘⣿⡆⠈⣿⡇⢸⣿⠀⣼⡟⠀⣿⡇⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿⠈⣿⡇⢻⣷⠀⠸⣿⣼⣿⠀⣿⡇⠀⣿⠇⠀⣿⡇⠀⢸⣿
⠀⠀⣴⣶⣶⣤⣹⣿⠘⣿⡆⢿⡷⠸⣿⡄⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣤⣾⠿⠿⢿⣦
⠀⠀⢻⣧⡈⠛⢿⣿⡆⠛⠃⠀⠀⠀⠉⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁⠀⢀⣼⡿
⠀⠀⠀⠻⣷⣄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠏⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠘⣿⣆⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢾⡿⠛⠉⠁⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠆⠀⠀⠀⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠄⠤⡀⠀⠠⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⢃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⣤⠀⠘⠰⠆⠀⠀⠓⠂⠈⠀⠉⠉⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠢⣀⠀⠀⡠⠣⠤⠤⠜⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡀⢀⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠃⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠤
⢀⡀⢢⠀⠡⡀⠀⠀⡀⡀⠀⠀⠀⠀⢸⠀⣿⣿⡗⠀⠀⠒⠈⠉⠉
⢇⡀⠈⠓⠀⠁⡠⠋⢀⠀⠀⠀⠀⠀⠈⡄⣿⣿⠀⠀⠀⢸⠀⠀⢀⡀
⢠⠀⢉⣄⡀⡀⠀⡠⠃⠀⠀⠀⠀⠀⠀⡇⠈⠂⠓⠂⠀⠈⠉⠀⠀⠀⠉⠉⠒⠤⣀
⠀⠉⠀⠀⠙⠲⣖⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⣀⣀⡤⠼⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⠂⠉⠉⠑⠒⠒⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠓⠂⠉⠉⠉⠀⠀⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢠⠏⠀⠉⠒⠖⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠇⠀⠀⠀⣀⠚⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠘⠤⠄⠊⠉⠀⢨⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠐⠤⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠙⠢⠤⢤⣄⡤⠤⠊⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⣶⠐⠒⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣶⣤⣧⠀⣧⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣄⠀⠘⣿⣿⣆⠀⠀⢀⣤⠀⠀⠀⣿⣿⡇⠀⣰⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⣿⣿⣦⡀⠘⣿⣿⣆⠀⢸⣿⣧⠀⢸⣿⣿⠁⢀⣿⣿⡟⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡀⠘⢿⣿⣷⡄⠹⣿⣿⣆⠘⣿⣿⡆⣾⣿⣿⠀⣼⣿⣿⠁⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆⠈⢿⣿⣿⣆⢻⣿⣿⣦⣿⣿⣷⡸⣿⡇⣸⣿⣿⡇⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣤⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣯⣀⣠⣴⣾⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣱⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠛⠉⠐⠿⠿⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠶⠞⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⢠⠤⡀⠀⣄⠀⠀⢠⡄⢠⢤⠀⠀⣤⣄⢀⡤⡄⢠⠤⡀⠀⣤⡀⢠⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣿⠀⢸⠀⡿⡆⠀⣾⡇⢸⠀⠀⠀⣿⢸⠨⡇⠀⣏⠀⢹⠀⡇⡇⢸⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣿⠀⢸⠀⡇⢻⣸⠋⡇⢸⠉⠀⠀⣿⠉⢰⡏⠀⣏⠀⢸⢐⡟⠁⢸⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢻⡀⣼⢠⡇⠸⠏⠀⡇⢸⠀⠀⠀⣿⠀⠸⡇⠀⢻⡀⡼⠸⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠀⠀⠉⠁⠈⠁⠀⠀⠀⠃⠈⠉⠀⠀⠉⠀⠀⠉⠁⠀⠉⠁⠀⠁⠀⠈⠉⠁⠉⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢤⠄⢠⠀⢠⡄⡤⠄⠠⡤⠄⠀⢠⡄⠀⢠⠀⡤⡄⢠⠤⠀⡤⡀⠈⢉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⣇⠀⠀⣇⠀⠀⢸⢻⡀⢸⠀⣇⠀⢸⠀⠀⡇⠸⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⠘⡆⠀⢻⠀⠀⢸⠀⢇⢸⠀⡏⠁⢸⠉⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣀⡸⠀⢀⡟⠀⢸⠀⠀⢸⠀⠘⣾⠀⣇⠀⢸⠀⠀⣇⣸⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠈⠉⠀⠉⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡄⢠⠀⢤⠤⠀⡤⠄⠀⣤⢠⡄⠀⠀⠀⠀⢠⠤⠠⢤⠄⢠⠀⠀⡄⣠⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⡇⢸⠀⢸⠀⢸⡇⠀⠀⣿⠸⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠘⡇⢰⠁⣽⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡇⠀⠀⡗⢺⠀⢸⠀⢸⡅⢤⡄⣿⢺⡇⠀⠉⠉⠀⢼⠃⠀⢸⠀⠀⢳⣸⠀⣿⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣷⠀⠀⡇⢸⠀⣸⣀⠸⣇⣸⡇⣿⠸⡇⠀⠀⠀⠀⣿⠀⠀⣸⣀⠀⢸⡏⠀⢿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⢀⡀⠈⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠄⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠂⠀⠅⠀⠀⠂⠨⠀⠀⠇⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⣤⡀⠀⠀⠀⠙⢷⡄⠀⠀⠘⠃⠀⠀⢠⡾⠋⠀⠀⠀⢀⣤⠶⠶⠶⢶⣄
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠁⠀⣀⠀⠀⣀⠀⠈⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⣠⡾⣿⠀⠀⣿⢷⣄⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣼⠃⠀⠀⣠⡾⠋⠀⣿⠀⠀⣿⠀⠙⢷⣄⠀⠀⠘⣧⡀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠙⠷⠶⠴⠶⠛⠁⠀⣠⡾⠋⠀⢀⡾⠃⠀⠀⠘⢷⡀⠀⠙⢷⣄⠀⠈⠛⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⠶⠟⠉⠀⠀⣠⡟⠁⠀⠀⠀⠀⠈⢻⣄⠀⠀⠉⠻⠶⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠻⣻⢛⠟⠿⠛⡟⠿⣻⠛⡟⣿⡿⠟⢻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⠟⢻⢻⣿⠟⣻⣛⠟⢿⣿⢿
⣿⣾⣾⣶⣷⣶⣾⣷⣶⣾⣶⣷⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣾⣿⣷⣿⣶⣿⣾⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠀⠀⠀⠀⠷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⠠⢄⡀⢀⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠙⠆⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠸⠰⠄⠀⠀⠷⠒⠒⠒⠈⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠱⣄⠀⠉⠀⠰⢦⣀⡀⠔⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠸⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇
⠀⠀⢇⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣦⡤⠤⠀⡤⠔⠒⠒⠂⠈⠉⠉⠁
⡆⠁⠊⢦⠀⠓⢀⡡⠊⢙⠀⠀⠀⠀⠀⢣⢸⣿⡿⠀⠀⠀⢠
⠈⡲⠴⠀⠀⠀⠊⠀⠰⠁⠀⠀⠀⠀⠀⢸⠈⠯⣙⡤⠀⠀⠘⠖⠒⠒⠒⠲⠦⢤⡀
⠘⠀⠂⠙⠶⣂⣀⠖⠁⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡤⠤⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠴⠒⠚⠉⢹⣀⣀⠤⠤⠤⠖⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⢙⠖⠢⠄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠚⢁⡠⢄⠀⡦⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠈⠃⠉⠀⠸⠀⠰⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠦⣀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠈⠉⠓⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣟⠈⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣶⣦⣧⠀⣧⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⢤⠄⠀⠄⠀⠀⠒⠦⢀⡀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠀⠀⠀⢄⣃⣀⠀⠀⠀⠀⠀⡤⡄⠠⠀⠀⠉⠁⠀⠠⠀⠐⠀⡀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠀⠇⠐⠛⠛⠉⠃⠀⠀⠀⠈⠑⠏⠀⠠⠄⠀⠈⠀⣠⠀⠀⠈⢍⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⢀⢀⠭⢭⠀⠠⡀⣠⣄⣀⡠⠀⢀⡒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣴⣿⣿⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠘⢆⡁⠀⣑⠀⠀⠉⡀⠁⣭⡇⠁⠀⠀⠺⠅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠉⠁⠀⠿⠧⠦⠀⠀⠀⠐⢁⠃⠀⠂⠀⠀⢠⣦⣤⡝⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠀⠀⠀⣤⠒⣄⠀⠠⡄⡄⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⣀
⠀⠤⠀⠀⠀⠀⠢⠀⣰⣿⠄⠀⠀⠀⠀⠠⣤⠘⣿⣿⡆⠴⣻⣷⡀⠀⠠⡠⠐⠀⠀⢠⠶⠀⠀⢀⣴⡿⠣
⠀⠉⠀⠀⠀⠀⠀⠀⠙⠁⠑⢠⡀⠀⠀⡑⠂⠀⠈⠋⠁⠸⠿⠋⠀⠢⠀⠐⠀⠄⠢⡀⠄⠐⠂⠻⣛⠱⠀⠀⠀⣴⣤⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠋⠉⠉⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⣤⡄⠀⠀⢠⣾⣷⠀⠀⣠⡄⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⠀⠿⠇⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣿⣧⠀⠀⢸⣿⣿⠀⢠⣶⣦⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉⢹⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⣿⣿⠇⠘⣿⡏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠻⠛⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠶⠦⣤⡀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⢶⣤⣤⡶⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠈⠹⣆⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠉⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣴⡆⠀⠈⠀⠀⠀⠀⢀⠀⠀⣿⠗⠀⠀⠀⠀⢀⡀⠁⠀⠑⠀⠰⠀⠀⠀⠐⠠
⠀⠀⢀⠀⠀⠊⠀⠀⠈⠀⠀⠀⠀⠈⠿⠃⠀⠀⠁⠀⠈⠁⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠔⠀⠐⠀⠀⠀⠀⠀⣀⠀⠔⠳⠀⠀⠀⠀⠀⠒⠀⠠⠄⠀⠀⠀⠀⠀⠀⠄⣀⠀⡀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠃⠹⠀⠐⠄⠈⠢⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠢⠀⡀⠀⠀⠀⠀⡀⠚⠀⠀⠀⠤⠀⢐⠀⢀⢀⠀⠀⠀⠀⠂⢘⡀⠀⠁⠀⠀⢀⡠⠄⠀⡀⢀⡀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠔⠀⠂⠉⠀⠀⠈⡀⠀⠂⠀⠀⠀⠀⠀⢀⠀⢀⠔⠃⠀⢠⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠘⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠈⠀⠠⣀⠵⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠠⠂⠀⠀⠀⠀⠀⡐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠐⠋
⠀⠀⠀⠀⠀⠰⠯⠦⣽⠓⠁⢀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢢⣀⠀⠀⠀⢀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠒⢀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⡠⡁⢬⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠈⠀⠀⠠⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠀⠀⠀⠀⠒⠛⢲⣄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⡠⠒⣉⣉⡉⠓⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠈⠳⡀⢳⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣭⠭⣭⣄⡀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⢳⠀⡇⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⢻⣝⠿⢖⣿⠃⣙⡷⠤
⠀⠀⠀⠀⠙⠒⠤⠤⢤⠀⢻⡄⡇⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠿⠛⠛⠉
⠀⠀⠀⠀⣤⡄⠀⠀⠀⡇⠈⢇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⠀⠀⠀⠀⣿⠁⣀⠤⠊⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠳⠊⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠋⠈⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠢⣀⣀⠔⠛⠛⢓⡀⠔⠁⠀⢸⠀⠸⠀⢸⠀⠀⡇⠀⡜⠀⢹
⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⡆⢸⠀⠀⡇⠀⠇⠀⠸
⠀⠀⠀⠀⢀⣠⣄⣀⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢣⠘⡀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇
⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⡄⡇⠀⢠⢸⠀⠀⡇⡆⠀⢀⠃
⠀⠀⠒⠚⠉⠉⠙⠛⠋⠉⠙⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠇⠀⢸⢸⠀⠀⣧⠁⠀⢸
⠀⠀⠈⠒⠤⠤⠤⠤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⡶⠀⠈⠁⠀⠀⠉⠀⠀⡈⠀⠀⠀⢀⠤⠤⠤
⠀⠀⠀⠀⡀⠤⣀⣀⣀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⠀⣠⠞
⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠛⠀⢠⠞⠁
⠀⠀⢀⣤⣀⡄⠀⣀⠀⢠⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⣤⣄⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⢦⠦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣰⠋⠀⢸⠀⠈⠳⡄⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⡀⠀⣰⠃⠀⠀⢸⠀⠀⠀⠹⡄⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠉⢓⡿⠶⠀⠀⠘⠀⣀⣀⡤⠵⠖⠚⠋⠉⠙⠓⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠸⠃⢀⣀⠀⠀⠛⠻⠦⢤⣀⣀⠀⠀⠀⠀⠀⣀⣠⡤⠼⠛⠋
⠀⠀⠀⠀⠀⠀⣧⣤⣤⠀⢠⠖⠋⠉⠉⠉⠳⣄⠀⠀⢀⡇⠉⠙⠓⠒⠛⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠏⠀⣤⠀⣤⢐⠀⠈⡇⠀⡸⠛⢒⡶⠒⢶⣒⣒⣒⠚⡃
⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⡄⠀⣿⠛⣿⢸⠀⠀⣾⠀⡇⢀⡟⠠⠤⠀⠤⠄⢩⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠹⣤⠄⠳⣄⠀⠀⠀⠀⢀⣀⣳⡄⢷⢸⢷⡄⠠⣤⡄⢠⢿⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⠙⠒⠒⠚⠉⠉⠉⢠⡟⢸⢠⡙⠓⠶⠚⢉⡈⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⠀⢀⡞⠀⢸⡄⠙⠲⠶⠖⠋⠰⠇⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢠⠏⢳⣄⣸⠃⠀⠀⠀⠀⠀⠘⣇⣠⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠒⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢻⣥⣄⢠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟

⣀⣠⣄⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣿⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠄⠀⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠳⣄⡴⠋⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠸⠇⠀⣸⠁⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠴⡇⠀⠀⠀⠀⠀⡇⠀⢰⡆⠀⠀⠸⠛⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢣⠀⠀⠀⠀⢀⡿⣄⠀⠀⣀⡴⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠑⠤⠴⠖⠋⠀⠈⠉⠉⠁⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠒⠚⠛⠋⠉⠉⠉⠙⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠋⠀⠻⣦⡀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠈⠙⠷⠶⡶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡸⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⢧⠹⡄⢳⡀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡉⠀⠀⠀⠀⠉⠉⠙⠃⠹⣄⠙⠛⠉⠉⠁⠀⠀⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠤⠤⠃⠐⣬⣪⣐⣠⣞⣈⣌⣀⣓⣠⣦⢰⣴⣑⣉⣑⣋⣒⣐⡨⣮⣀⣎⣒⣐⣐⢄⠠⠤
⠀⠀⠀⠀⠒⠒⣶⣶⣇⠀⢯⡿⢧⠿⢿⡿⢯⢿⡽⣯⡇⠸⠿⠏⡿⣯⣿⣿⠯⠀⡿⠯⠿⠽⠿⠹⠉⣲⠖⢲⣶⣶⠆⡰⢐
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠠⡇⠀⠀⢹⠀⣠⠔⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⢱⢈⡇⠀⠀⢸⠀⡇⠀⠀⢸⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡌⡇⠀⠀⢸⢸⠀⠀⠀⡸⡸⠁⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣷⡇⠀⠀⢸⣼⠀⠀⠀⣷⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡇⠀⠀⢸⡇⠀⠀⢰⡟⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠲⡀⠈⡇⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⢸⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠱⡀⡇⠀⠀⠈⠁⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⡀⠀⠀⠀⣀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⠖⠁⢀⡈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠚⢀⠀⣀⠑⠉⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⢠⣀⠹⠄⠑⠒⠄⠀⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡌⠀⠀⢀⠄⢘⢠⠀⣦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣙⣤⡀⠃⣠⣇⠼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⣿⡠⣀⡄⠀⠀⠀⠀⠀⠀⠘⢊⡩⡵⠏⠁⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⠀⠀⠇⡅⠀⠀⠀⠀⢀⠔⢈⡴⠼⢤⣧⡤⠄⠺⢁⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠀⡇⠀⠀⠀⠠⠃⢀⠊⠀⢰⣿⠟⠀⠰⠁⡎⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠄⠚⠄⠀⠠⠁⢸⣿⣀⡀⢿⣿⡆⠠⠓⣌⠅⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⣤⠁⠀⠀⢻⣇⠀⠀⠹⣿⣆⢎⠊⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⠀⠀⣀⠈⢿⡀⠀⠀⠹⣿⠎⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢠⡇⠀⠸⣧⠀⠀⠀⠙⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⡰⠋⡀⠀⡆⠘⣧⡀⠀⠀⠀⢀⡴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡰⠁⢀⡶⠂⠀⠸⡓⠲⠞⠋⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⢸⡇⠀⠀⠀⡁⠀⠀⠀⣀⣠⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠛⠁⠀⠀⠀⠓⠖⠚⢿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠈⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⢰⣿⡇⠀⠀⠀⠀⠀⡄⠀⠘⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠳⣾⠀⠀⠀⠀⢸⣄⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠶⣄⡷⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⣠⠞⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠇⠀⠀⠀⢸⠇⠀⠀⠀⠀⡇⠀⣰⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠏⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠏⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠶⠶⠒⠛⠉



⣞⣳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣛
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣒⢰⠒⠲⡀⡆⢲⠒⢠⢦⠀⣖⡲⢰⠀⠀⣖⡂⠀⢴⡒⠐⢶⠂⣖⢲⢠⠖⠲⡄⣆⠔⢰⣒
⠀⡀⢀⣀⠀⠀⡀⠀⣉⣀⣸⣄⡴⢁⡇⣸⣀⣟⢙⣆⣏⣹⢸⣄⣀⣇⣁⣀⣀⣹⢀⣺⢀⣏⣹⡘⣄⣠⢃⣏⢳⣸⣁⣀⢀⢀⠀⣀⢀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢰⠀⠀⣿⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⢹⡀⠀⠘⡇⠀⣿⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⣠⠞⠁
⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠄
⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡆⢰⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⣀⡀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣇⡇⢸⡿⢹⣶⢦
⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣿⠁⣾⠃⢸⣹⠸⡆⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠃⢸⡇⢀⡟⠀⣯⠟⡆⣧⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡟⠀⠈⠀⠸⠁⢰⡏⢠⠇⢻⠀⢀⣤⡀
⠀⠀⠉⠙⠛⠛⠛⠀⠀⠀⠀⠀⡏⢸⠁⠀⠀⠀⠀⠀⠘⠀⡞⠀⢸⢀⡞⢰⡇
⠀⢀⣠⠤⠤⠤⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠾⠀⢸⠀⠀⠀⠐⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠆⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡴⠁⠀⠀⠲⢤⡀⠀⠀⠒⠲⠤⣀
⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠀⡀⠘⡆⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⠓⢤
⢀⡤⠚⠉⠀⠀⠀⠀⠀⣠⠞⠁⣰⠃⠀⠀⠀⠀⠀⡴⣎⠀⠀⠀⠀⢳⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣰⠃⠀⠀⠀⠀⠀⢸⠁⠸⡄⠀⠀⠀⠀⢳⡀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠓⠲⠤⢤⣀⣀⡏⠀⠀⢣⣀⣠⡤⠴⠒⠛⣆⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠸⡅⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⢧
⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⢰⠏⠀⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣆⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢹⡀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡞⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⣇⠀⠀⢸⠀⠀⠀⢸⠀⠀⣸⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢹⡀⠀⣿⠀⠀⠀⢸⠀⢠⡏⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣧⠀⣿⠀⠀⠀⣿⠀⡾⠀⠀⠀⡾⠀⠀⢀⡴⠋⠉⢳
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢹⡀⠀⠀⠸⣄⡿⠀⠀⠀⢿⣼⠃⠀⠀⢸⠃⠀⣠⠎⠀⠀⢀⡞
⠀⠀⠀⠀⢰⠋⠀⠙⢦⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⠞⠁⠀⠀⣠⠏
⠀⠀⠀⠀⢸⡄⠀⠀⠀⢳⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⣀⢀⣀⣀⣀⣀⣀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣤⣄⠠⠤⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠓⠢⢤⣀⡈⠉⠉⠓⠒⠲⠤⠤⣍⣉⡛⠺⠭⢭⣀⣒⡒⠢⡀
⠀⠀⠀⠀⠀⠀⢘⠙⠲⡤⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠦⠤⣌⣉⡙⠛⡄
⠀⠀⠀⠀⠀⠀⣆⣉⠙⠇⠀⠀⣦⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⡀⠀⠀⢳⠬⢍⣳
⠀⠀⠀⠀⠀⠀⡧⢌⣹⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠙⠿⠇⠀⢀⡇⠀⠀⠀⣗⠒⠺
⠀⠀⠀⠀⠀⠀⠧⠤⣼⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢠⣦⠀⠀⢸⡇⠀⠀⠀⢩⠉⣙⠆
⠀⠀⠀⠀⠀⠀⠓⠲⢾⠀⠀⠀⢸⣿⣧⣴⣶⣦⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⢸⠒⠲⠁
⠀⠀⠀⠀⠀⠀⣉⠓⢺⠀⠀⠀⠈⣿⡟⠉⢻⣿⠀⣿⡿⠀⠀⠘⠃⠀⠀⠀⢸⠭⣹
⠀⠀⠀⠀⠀⠀⣏⣉⣙⡀⠀⠀⠀⣿⡇⠀⠀⡿⠀⣿⠇⠀⠀⠰⡆⠀⠀⠀⡜⠒⠊⠶⢄⡀
⠀⠀⠀⠀⠀⠀⢣⠤⣌⣱⣄⡀⠀⠘⠁⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣰⣋⡙⠒⠒⠦⠽⣄
⠀⠀⠀⠀⠀⠀⠘⢖⠤⢬⣉⣙⠓⠶⠦⣤⣤⡤⠤⠀⠀⢀⣤⣤⠤⠔⠺⠥⢤⣍⣉⣙⠒⠒⠲⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠬⠭⠽⢒⣶⣬⣍⣓⡒⠦⠤⢄⣈⠉⠓⠒⠦⣤⠒⢄⣉⠉⠙⠒⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠐⠒⠛⠛⠂⠉⠁⠀⠀⢣⣉⡑⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣉⣙⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⢴⠟⠁⠀⢀⡤⠖⠋⠉⠀⠉⠀⠀⠉⠉⠛⠲⢤⡀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⢰⣦⣴⣾⣭⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠹⣦
⠀⠀⠀⢀⡴⠛⠳⣦⡞⠉⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⣆
⢠⡴⠖⢾⡇⠀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣻⣆
⣿⠀⠀⠘⣧⠀⠀⠀⣷⠀⠀⠀⢷⣤⡤⣤⡀⠀⠀⠀⢸⠇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⢀⣾⠀⢀⡀⢰⣦⢀⣿⠹⣆
⢸⡆⠀⠀⢻⡄⠀⠀⢹⡆⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣞⣁⣀⣠⠞⢻⣀⡾⣧⣸⡏⠛⢻⡆⢹⡄
⠈⣧⠀⠀⠈⣧⠀⠀⠈⣃⠀⠀⠀⢹⠀⠈⡅⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠛⠁⠀⠀⠀⠀⠀⠉⢩⣍⡁⠒⠛⠉⠀⠙⠋⠀⠀⠀⣿⠈⣷
⠀⢻⡀⠀⠀⠿⠀⠀⠀⠈⠀⠀⠀⠈⣧⠀⢹⡆⠀⢘⡇⣀⡴⠟⠁⠀⠀⠠⠶⢟⣛⣻⣭⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿
⠀⠘⡇⠀⠀⢀⣀⣤⣤⠶⠶⣶⠟⠀⠀⠀⠀⢻⡞⢻⡟⠉⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣙⠻⣦⣄⠀⠀⠀⠀⢸⡇⢸
⠀⠀⢿⡀⠀⠉⠉⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⣴⣿⣿⡇⠀⠸⡆⠀⠀⠀⠀⢀⡴⠋⢁⡀⠀⠉⠙⢦⠉⠀⠀⠀⠀⣸⠀⣿
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⣧⣸⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣿⣿⣿⣿⣾⠀⡇⠀⠀⠀⠀⡟⢀⣾⣿⣿⡇⠀⠀⠈⣷⠀⠀⠀⠀⠛⢶⣏
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠇⢹⡇⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣙⠛⠿⠛⢁⡼⠁⠀⠀⠀⠘⡇⠸⣿⣿⣿⣿⡟⠀⠀⣿⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⣀⣽⡶⠶⠒⢲⣏⠁⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⣤⣈⣛⣛⣋⣀⣀⡴⠃⠀⠀⠀⠀⠀⣾⢀
⠀⠀⠀⠀⠘⠳⠶⠚⠻⣏⠉⠀⠀⠀⠀⠀⠙⢷⡀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢡⣤⡀⠀⠀⠀⠀⣰⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠦⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠙⢦⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠋⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣀⣀⠀⠙⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⢼⡟⠛⠷⣶⠦⢤⣤⣤⡤⣶⡶⠖⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⠦⣤⠴⠞⠋⠀⠀⠸⡏⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⣿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢶⢾⣿⣿⣿⣿⠀⠀⠀⢹⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⠉⠁⠘⠋⠉⠀⠀⠀⠸⡇⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣾⣇⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⣤⣀⣀⣀⢀⡼⠋⠉⠉⠈⢻⣄⣀⣀⣀⣀⣬⣿⣤⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⢹⠂⠀⠀⠀⢸⡉⠉⠉⠉⠀⠀⠀⠘⡯⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣤⣤⣄⣀⣀⡿⠀⠀⠀⠀⠈⣧⣀⣠⣤⣤⣤⣤⡴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⡏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⢧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠷⢶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⣿⠿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠏⢩⡏⠀⣁⢀⣠⠸⡇⢀⣿⣿⠠⢀⠋⢠⣍⢉⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠉⢼⣿⡆⢰⣧⣈⣷⣦⣾⣾⣿⣶⣾⣶⣿⣿⣶⣾⣦⣽⣏⣡⡅⢰⡇⠀⣸⠛⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠛⠌⠉⠀⣰⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣇⣀⡔⠁⣭⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠙⠧⣀⣸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⠟⠉⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⢿⣿⠟⣿⠛⠙⢿⣿⠟⠛⣿⡿⣿⡟⢻⣿⣿⣿⡿⠛⠙⣿⠛⠉⠿⢻⣿⢿⠟⠋⢹⣿⣿⣾⣤⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⠿⠰⣷⡄⣾⣿⢃⣴⣤⣿⠀⡿⠃⣿⣿⣿⣿⠀⠶⠾⣧⡄⣾⡇⢸⠇⣾⠠⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⡄⢰⣿⢀⣿⠇⠈⣁⡈⣹⢀⣤⢠⣟⢉⣁⡌⢀⣤⣶⣿⢠⣿⡇⠘⢸⡇⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡇⣾⠿⠘⢻⡄⠛⢋⣴⡇⢸⡿⢸⣿⣿⣿⡇⢸⣿⣿⠿⠘⢻⣿⢀⣿⡇⠻⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣶⣶⣿⣷⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣶⣿⣿⣾⣿⣧⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⡶⠶⠾⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠻⣿⠿⣻⣿⢟⣿⢻⡿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⣤⣤⡄⣿⣿⣿⣿⣿⣿⢣⡙⣿⣿⣟⣉⢩⢽⣿⣹⣟⣃⣿⢯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠉⠉⠁⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢺⡆⢰⣶⣶⣶⡆⣿⣿⣿⠛⢧⣶⣶⠟⣿⡷⢺⣿⣴⡾⢛⠷⣿⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⡇⢸⣿⣿⣿⡇⣿⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⢠⠃⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡿⠒⠒⠶⠀⠀⣿⠀⠀⠀⠀⣦⠀⠀⠀⠀⣴⠞⠲⣤⠀⠀⠀⠀⠀⠹⣆
⠀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⣻⡄
⠀⠀⢸⠀⠀⠀⠀⠀⣿⠶⠤⠤⣾⠀⠀⠀⡿⠛⠛⠛⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢹⡀⠀⠀⠀⠂⢸⢇
⠀⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⢸⠇⠀⠀⠀⠆⣼⣿
⠀⠀⢻⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣧⣀⣀⣀⠀⠀⢿⣀⣀⣀⠀⣿⣀⣀⣀⣀⠹⣦⣀⣠⠟⠀⠀⠀⠀⠀⡏⡇
⠀⠀⠸⡀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⣸⠃
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⠏
⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⠊
⠀⠀⠀⠀⠀⠈⠻⢶⡤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠭⢷⡀⠀⠀⠀⡰⢶⠲⡖⠦⠲⠖⠶⠲⠖⠖⠒⠒⣂⣋⣉⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⠞⣡⠎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀⠀⠀⠐⢶⡀⠀⠀⣠⠶⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⠋⠁⠀⣿⠀⠀⠀⠀⠴⠘⠷⠀⠠⠛⠠⠄⠀⠀⠀⠰⠛⠀⠙⢿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠈⠀⢀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠘⠛⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠂⠀⠀⠀⠉⠀⠀⠀⠀⠠⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣷⣶⡟⠏⠛⠛⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠀⠈⣷⡄⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢣⠀⢸⢿⢄⣀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢇⠸⢲⠀⠀⠉⠉⠒⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠶⣦
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣠⣴⠶⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢰⡟⠀⣠⣴⣤⡀
⠀⠀⢀⣤⣤⣴⡿⠋⠁⠈⠉⠛⠁⠀⣀⠀⠈⠻⣦⡀⢀⣠⣤⣤⣴⣿⣄⣀⣾⣣⣾⠟⠁⠙⣿⡄
⠀⣴⡟⠉⠀⠀⠀⠀⢴⣶⣄⠀⠀⠸⣿⣷⣄⠀⠈⠻⠟⠉⠁⠈⠙⠻⣿⣿⣿⣿⣅⠀⣠⡾⠋
⠀⣿⠀⠀⢾⣷⣄⠀⠈⠻⣿⣷⣄⠀⠈⠻⣿⣷⣄⠀⠀⠀⣶⣶⣄⠀⠈⠻⣿⣿⣿⣿⣯⣶⠶⠿⢻⣆
⠀⣿⡦⠀⠀⠙⠁⠀⣀⠀⠈⠻⣿⣷⣄⠀⠈⠻⣿⣷⣄⠀⠈⠻⣿⣷⣄⠀⠈⠻⣿⣿⣧⠀⠀⣀⣨⣿
⣼⡏⠀⢀⣀⡀⠀⠘⢿⣷⣄⠀⠈⠻⣿⣷⣄⠀⠈⠻⣿⣷⣄⠀⠈⠻⣿⣷⣄⠀⠈⢻⣿⠿⠛⠛⠉⠁
⢿⡇⠀⠘⢿⣿⡆⠀⠀⠙⢿⣷⣄⠀⠈⠻⣿⣷⣄⠀⣨⣿⣿⣷⣄⠀⠈⣿⣿⣷⡀⠀⠙⣷⡀
⢈⣿⠆⠀⠀⠉⠀⢀⣄⡀⠀⠙⢿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣄⠀⠘⣷⡀
⣾⠇⠀⢠⣤⡀⠀⠘⢿⣿⣦⡀⠀⠙⢿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣷
⢿⡆⠀⠘⢿⣿⣦⡀⠀⠙⢿⣿⣦⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢿⣄
⠈⠻⣦⡀⠀⠙⢿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⢷⣄
⠀⠀⠈⠻⣦⡀⠀⠙⢿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⣹
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠙⠿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠙⢿⣿⣦⡀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠙⠻⠿⣶⣦⣤⣀⡀⠀⠀⠉⠻⣿⣿⣿⣿⠟⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⠀⠀⠉⠙⠛⠻⣿⣶⣄⠀⠈⠻⠟⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⢶⣤⣄⠀⠈⠻⠟⠁⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⣠⡾⢷⣾⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣤⣄⢠⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⣶⡀⣿⡎⣿⣁⠀⠀⠀⣿⣇⣿⣸⣏⢡⡿⢻⣆⣶⢶⡄⣀
⠀⠀⠀⠀⣤⢶⣄⣿⡙⣷⣿⣿⣿⣷⢿⡉⠀⠀⠀⢹⠋⠁⣿⠉⣸⠇⣾⣻⣧⡾⢳⡟⠀⣰⣦⣄
⠀⠀⠀⠀⠙⠷⣥⣼⣧⣸⡿⣯⠛⠹⠟⠛⠃⠀⠀⠘⠀⠀⠛⠛⠹⠿⠋⡿⠀⢠⡟⠀⣰⡿⡄
⠀⠀⠀⠀⠀⠐⢦⣿⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢲⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢠⢰⡐⠈⣇⠀⠀⠈⡗⣿⣸⠀⣾⠀⡞⢠⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡼⡞⡇⣈⡇⠹⠀⠀⠀⠃⠛⠿⠀⠇⢰⡇⠎⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣄⣤⢀⣤⡄⣤⢀⡄⠀⠀⣤⣤⣤⣤⢀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⢸⣇⣉⣿⣼⡇⣤⡄⣿⡄⣿⢹⣼⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⢸⡇⣿⣿⢹⡇⠀⠀⣿⠀⣿⢸⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⠛⠈⠛⠋⠛⠘⠃⠀⠀⠛⠀⠛⠀⠛⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⢟⡞⠀⠀⢰⢠⠄⣤⣠⠀⠀⢠⡄⠠⡆⢸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠸⠃⠀⠀⡞⢸⠀⡏⡟⠀⠀⠀⡇⠀⡏⠘⠆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠓⠀⠀⠀⠁