⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣽⣄⠀⠀⠀⠀⠀⠉⢳⣦⠀⠀⠀⠀⠀⠀⠀⠘
⠁⠀⠀⠀⠀⣴⣿⣿⣿⠟⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⣿⠟⠁⢀⣠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀
⠀⠀⢀⣾⡿⠋⠤⢴⣚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀
⠀⠀⡾⠋⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣖⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀
⢀⡼⠁⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠓⠺⠿⠓⠒⠒⠶⠶⠤⠀⠀⠀⣀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀
⡿⠁⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⡆⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡫⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠚⠉⠁⠀⠀⣀⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠐⠊⠉⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣲⣶⣶⣦⣔⣒⠤⣄⠀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣷⡿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣇⣀⣀⠴⣇⠀⠀⠀⠀⠀⠀⣠⡾⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⣒⣯⣍⣹⣓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⣯⡙⠋⠹⠃⠀⠈⠉⠉⠉⠛⣿⠿⠒⢻⠂⠀⠀⠀⢀⣾⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣷⣾⣿⣿⣿⠻⠿⠷⣶⡒⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡿⡇⠀⠀⢠⡿⣩⡷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⣤⣾⣿⠟⠻⣟⠛⠛⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠹⡄⠀⣼⣰⡏⢸⣿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠞⠉⠀⠀⠀⢈⣉⠴⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡀⣿⣿⠁⣼⠃⣭
⠀⠀⠀⠂⠀⠀⠀⠀⠛⣻⣷⣟⣡⠀⠀⢀⣰⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢳⣼⣿⣼⢇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢳⠀⡸⢻⣿⣷⣿⣿⣿
⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣄⠀⠸⣄⣷⣿⡿⠟⣿⣿⣿
⠰⣶⡶⠒⠋⠁⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠓⠋⠀⠀⡿⠿⠿⠶⠟⠛⠻⠿
⠀⠀⠙⢦⡀⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀
⢇⡀⠀⠀⠳⣮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⢀⣀⣤⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀
⠀⡓⢤⣀⠀⠀⢹⡷⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢫⣳⡀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢘⠆⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠙⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠛⡏⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣙⣻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⢀⣚⣿⡀⠀⠀⢜⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠓⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠛⠶⠿⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⠀⠀⠀⠀⠹⣷⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀
⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⡄⠀⠀⠀⠀⠹⡇⠀⠀⠀⠘⡆⠀⠀⠀⢀⡞⠁⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠓⠲⠶⢤⠤⢤⣤⣸⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⢹⡀⠀⠀⢠⠇⠀⠀⢠⡞⠀⢀⣼⡿⡁⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡷⡄⠈⠻⣽⡟⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣆⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⡟⠀⣠⠿⠙⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⡇⠀⠀⣾⠀⠀⡾⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠈⠻⣦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⢀⡇⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠙⢿⣿⣷⣦⡀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡇⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⡻⣿⣿⣇⣀⣼⣿⣿⣿⣷⡶⠤⠤⠤⣼⡦⠰⢶⢗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀