⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⢿⣾⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠚⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣤⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⢀⣀⠀⠀⠉⠛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠲⡄⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⣠⣴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣶⠛⠙⣿⣿⣿⣿⣷⣄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠈⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⣀⠀⠀⠀⠀⠈⠙⠉⠀⠹⣿⣿⣷⣿⣿⣿⣿⣷⣶⣴⠶⣶⣾⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⢰⠀⢸⣿⢀⣤⣄⠀⣀⠀⠀⠀⢉⣿⣿⣿⣿⡿⠛⠉⠀⠀⠈⠻⢿⡿⠋⢹⣦⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⣿⣄⠀⣿⣿⣿⠛⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⣿⣿⣦⣀⠀⠀⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠛⣁⣾⣿⣿⠀⣿⣿⣶⣶⣶⠒⠀⢀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠾⣿⣿⣿⣿⣿⣿⠉⠀⠛⠻⠛⠉⠀⠾⠛⠛⠋⠁⢀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣿⠈⣷⣀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣷⣶⣽⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣴⣶⣶⣶⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢀
⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠈⠉⠛⠛⠻⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣶⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣀⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠉⣀⣤⣶⠉⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣶⣿⣿⣶⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣾⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⣿⠁⠀⠀⠀⠀⠀⢀⣤⠀⢸⣦⡀⠀⠈⠛⣿⣿⣿⣿⠿⠿⠿⠟⠛⠻⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣀⣴⠶⠉⡿⣠⣶⠋⠙⠻⢿⣤⣤⣀⣀⣀⠀⠀⢠⣤⣤⣤⣴⣶⣶⣶⣶⠁⠀⣀⠀⠀⠀⣀⣀⣀⣀⣤⣶⡿⢻⣿⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⣷⣦⣄⣀⠈⢛⣿⣷⠀⠀⠀⣿⡈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠀⠈⠉⠛⠷⣦⣤⣤⣤⣴⣿⡿⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣄⠀⠀⠉⠉⠉⠉⢀⣀⣀⣀⡙⣿⣤
⢰⡶⢶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠀⠀⠀⠀⠀⣶⣶⣤⣀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠟⠛⠉⠉⠉⠀⠀⠀⠉⠙⠻⣿
⠀⣿⠀⠒⠠⣄⣀⠉⠙⠓⠒⢤⣤⣄⣴⠿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣬⠁⠀⢠⣶⣴⣶⣾⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠒⠀⠀⠀⠀⠀⠀⠀⢻⣀⠈⢿⣶⣶⣭⣓⠲⣤⣤⣀⠀⠉⠙⠒⠒⢤⣤⣀⣀⠀⣿⣿⠛⠀⠀⣰⣬⣙⢿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⡿⠻⢿⣿⣿⠃⠀⠀⣚⣻⣿⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢲⣄⠀⠙⢿⣿⣿⣿⣷⣶⣭⣙⠛⠶⣶⣤⣄⣀⠀⠉⠉⠛⠒⠶⣴⣿⣿⣿⣿⣾⣿⣿⣏⠰⠀⢸⠿⠛⠉⢀⣤⣶⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠛⣶⣀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣍⣛⠛⢶⣶⣤⣀⣀⠀⠈⠙⠛⠛⠛⠉⠉⠀⣀⣤⣶⣿⣿⣿⠿⢿⣿⣿⣿⠛⣻⣿⣶⡀⠀⠀⠐⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣉⣛⠓⢄⠀⠛⠛⢛⣠⠿⠟⠛⠉⠀⠤⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⣤⡀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠈⠙⠶⣤⣀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⠄⠀⠀⠒⠒⢤⠀⠀⣴⠈⠀⣠⣿⣿⣿⠋⢠⣿⣿⣿⣿⣿⣷⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⠀⠀⠀⠀⠀⣽⡿⠲⣶⣤⣀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣿⣶⣤⠀⠸⠀⢀⣾⣿⣿⠉⠀⣴⣿⣿⡿⠉⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠉⠛⠻⣶⣦⣤⣈⠉⠛⠻⣿⣿⣿⣿⠀⢀⣿⣿⠿⠛⠁⠀⠀⠀⠘⣿⠛⠀⣠⣿⣿⡿⠉⢀⣾⣿⣿⣿⠿⠉⣠
⠀⠀⠀⠀⣠⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣷⣶⣤⣄⣈⠀⣾⡟⠀⠀⠀⠀⠀⠙⣦⠀⠉⠀⠀⠻⣿⠋⠀⣴⣿⣿⣿⠛⠀⡤⣿⣨⣨
⠀⠀⡤⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⢿⢰⣶⡶⠀⠀⠀⠀⠉⠛⣿⣷⣴⣄⠀⠀⠉⠙⠀
⣠⣾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀