⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⠀⠀⢀⣀⣠⣶⣶⡆⠀⠀⢸⣿⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⡿⢠⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣥⣤⣿⣿⣿⣿⡿⠋⣀⣀⠀⠈⢸⣷⠀⢻⣦⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠀⠸⢇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣶⣶⣾⡿⠀⠀⠈⢹⣦⣀⣹⠿⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠟⣤⠀⢸⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢸⡿⠋⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠛⣿⣿⣿⣏⠀⠀⠀⠀⣤⣸⣿⣿⣿⠛⠘⣿⣿⠟⠛⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⢠⣾⣦⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣷⣦⠀⠀⣿⣿⡿⡏⠀⠀⠀⢀⣀⡀⠀⠉⠹⣿⣄⠀⠀⣰⡆⠀⠐⣾⠃⠀⠀⠀⣾⡏⠉⠉⠿⠶⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣤⠐⢛⣻⠛⠛⣦⢈⣏⡀⠛⣦⠀⠀⣿⣿⢀⣤⣿⡇⠀⣿⠟⠀⠀⠠⣴⡏⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣎⡹⢇⠀⠙⣟⠻⣇⣴⠿⢰⡆⠾⣿⣿⣿⣿⠷⠟⠁⠀⠀⠠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⠉⠀⠈⠉⣹⡾⡇⠀⠺⣿⠟⠁⠀⠀⠀⠀⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠿⣶⢶⣶⣀⣀⡀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⡸⠟⣿⡏⢐⣠⡿⠇⠀⠀⠀⠀⢀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠶⠶⠾⠿⠟⠛⠋⠀⠀⠀
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣮⣁⣀⣿⠉⢸⣯⡁⠀⠀⠀⠀⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠖⠿⠟⠉⠉⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣤⣾⣧⡼⠛⣿⢠⣼⣿⣿⣤⣤⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡛⠋⣸⣿⣿⣿⣿⣿⣹⡷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣛⣴⠀⠀⠀⠀⢠⠘⠣⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⡏⠙⠸⣿⠿⠿⠋⠉⠉⠉⠉⢹⡏⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇⠉⡿⠷⢶⡶⢦⣤⣤⠀⠀⠀⠀⣰⣶⡶⠶⠾⠾⠟⠉⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠈⠲⡀⠀⠀⠀
⠀⠀⠀⠀⣿⣟⠣⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠻⣿⣿⣿⣿⢿⡇⠀⢻⡄⠀⠘⢣⣄⠘⠛⠛⠷⠤⢤⣀⣙⣧⠀⠀⠀⠀⠀⡤⠒⠲⠶⠶⠶⠏⠀⠀⠀⠀⠻⣇⠀⠀
⠀⠀⠀⠀⠉⠙⡇⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⣿⠧⢸⡇⠀⢸⡇⠀⠀⠈⢿⣶⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⢹⣿⣿⣷⣄⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣄⢸⡇⠀⢸⡇⠀⠀⠀⠀⠙⢷⣄⠀⠀⣀⠴⠶⠶⡶⠶⠴⠞⠁⠀⠀⠀⠀⠀⠐⠻⣿⡟⠛⠛⠛⠛⠚⠛
⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⢸⡟⠓⣿⣿⣷⣼⣇⣀⣀⠀⠀⠀⠀⠀⠻⢼⡇⠀⠈⢳⡦⠀⠀⠀⠀⠀⠙⠿⣅⣽⠀⠀⠀⢻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⡇⠀⣿⡙⢿⣭⣍⠉⠛⠒⠒⣶⣶⣧⣤⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠘⣏⠉⠉⠓⠒⠢⣤⣠⣀⠀⠀⠀⠀⠛⣶⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣿⡟⠃⠀⠈⣻⣟⠘⠛⢿⣦⡀⠀⢻⣷⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⠛⠤⠤⠄⢀⠀⢳⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⠀⡇⢀⣀⣀⣿⣿⡀⠀⠀⠀⠹⣦⣸⣿⣧⣀⣼⡷⠶⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣍⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡿⠋⠉⣿⣿⣿⣧⠀⠀⠀⢀⣼⣿⣿⣿⡟⢻⡇⠀⠉⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠸⣇⠀⠀⠘⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣇⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣿⣿⣿⣿⣧⡄⠀⢸⣿⣿⣿⣿⣿⡄⠘⣶⣄⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣷⠴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⠿⢿⣷⠀⢹⡿⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣼⣿⣤⢸⡇⠀⠙⣯⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⡏⠿⢸⡇⠀⠀⣿⡿⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⢸⡇⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⢸⡇⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣾⠇⠀⠀⠀⢸⣇⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀