⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⣰⠟⠀⠀⠉⠛⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⠶⢦⡀⠀⢀⡴⠊⠁⣠⠞⠁⠀⠀⠈⠹⣆⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣷⠞⠉⡰⠂⠀⠀⠀⢘⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⢸⡅⠀⠀⠹⣄⠞⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠻⣿⣿⣿⣿⠿⠧⠶⠊⠀⠀⠀⢀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠹⣄⠀⠀⠹⡄⠀⣰⠃⠀⠀⢀⡤⠒⠢⣄⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣨⣽⣿⣃⠀⠀⠀⠀⠀⠀⣠⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢦⠀⠀⢻⠀⡏⠀⠀⢠⠏⠉⢲⠛⠻⣷⡀⠸⡆⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢀⡴⠞⠟⠋⠀⠀⠀⠈⠻⣷⣤⡀⢠⣾⣵⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠓⠒⠉⣸⠀⠀⢀⠏⠀⠀⡎⠀⢠⡏⢧⠀⢱⠀⠀⠀⠀⠐⣿⣿⣿⣀⣀⡀⣠⣿⠃⣠⣤⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡾⠀⠀⠀⡏⠀⠀⡼⠀⠀⢰⠇⠀⣸⠀⠸⡆⠘⡇⢀⣀⣀⠀⠙⢿⣿⣷⠉⠙⢿⡋⣼⣿⠇⠀⠀⠀⠀⢀⣼⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡇⠀⠀⢠⠃⠀⢀⣇⠀⠀⢸⠀⢀⣿⡀⠀⡇⠀⣿⣿⣿⣿⣿⣦⡸⡌⠁⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠔⣩⠤⢤⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠙⣦⠀⡎⠀⡸⠙⡇⠀⢸⠀⢸⣿⣿⣿⣿⣿⣷⣙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠇⠀⠃⠀⠀⢹⠏⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⠀⠀⠀⣼⠀⠀⡿⡄⠈⢧⠧⣤⣧⣀⡧⢤⣸⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣶⣭⣹⢳⠒⠶⢶⡶⠾⠿⣿⠏⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡿⠀⠀⠀⡟⠀⠀⣧⣿⣶⣾⣄⡏⠀⢣⠀⢀⣿⡇⢸⠿⠿⠿⠿⠋⠁⠀⠉⠙⠻⢿⡜⢦⣀⣀⣀⡴⢞⠁⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣷⣿⣾⢲⠉⢘⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣭⣭⣥⣶⣯⣤⣶⣶⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣇⣀⡀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣾⣷⣾⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠴⠚⠉⠀⠀⠈⡆⡇⠀⠀⣿⣿⣿⣿⣿⣿⣉⡟⡟⠛⢻⠛⢻⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣇⠀⠀⠀⣀⡤⠞⠁⣷⠀⠀⡏⢹⠟⠋⡟⣏⠉⢳⡇⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠋⠁⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠓⠒⠊⢹⠀⠀⠀⢹⠀⠀⣿⡟⠀⢰⠃⠸⣄⠀⠱⡄⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠤⣄⠀⠀⢧⣀⣠⡇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⡄⠀⠀⢸⠀⠀⢸⠁⠀⡾⠀⠀⠸⣆⣠⣷⠇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⡀⣸⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⣇⢠⠇⠀⠀⢀⡏⠀⡜⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⠀⠀⠘⣾⠀⠀⢀⡞⠀⡼⠁⢠⡏⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⣧⠀⠀⠈⢶⣶⢋⣠⠞⠁⠀⣼⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠶⣄⣈⣒⣶⠤⡤⠀⠀⠀⠀⠀⢹⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠘⣆⠀⠀⠀⠈⠉⠁⠀⠀⡼⠉⠳⣤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠉⠀⠀⠀⠉⠋⠁⠀⠳⠦⢤⣤⣴⠖⠛⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣳⠋⠙⣿⢦⡀⠀⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⡿⠤⠽⠶⠤⠤⠞⠋⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠛⠲⠶⣾⠫⣿⣿⡟⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⢹⡆⠀⠀⠀⣰⠞⣻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢭⣝⣒⣒⣶⣶⣶⣶⡶⠶⢒⣾⠁⠀⠀⢰⠟⠒⠈⠉⠓⠛⠒⠒⠋⠒⣦⠀⠀⠙⢦⡀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢿⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠙⠲⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣶⢤⣄⣀⣀⣤⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀