⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⠓⠦⣦⡒⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠤⣤⣀⣀⡴⠛⠀⢸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⢦⠀⠀⠀⠀⠈⠉⠓⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠛⠛⠉⠉⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⣇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡤⠖⠒⠒⢤⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡟⠀⠀⠀⠀⠀⡾⠁⢠⣶⣄⠈⢳⠀⠀⠀⠀⠀⠀⡇⠀⠀⡼⠁⣤⣤⡀⠙⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡆⠀⠀⠀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠻⡀⠘⠛⠋⢀⡞⠀⠀⠀⠀⠀⢠⠇⠀⠀⢳⡀⠙⠛⠁⣰⠃⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉⠀⠀⠀⠀⠀⠀⡾⣆⠀⠀⠀⠉⠒⠒⠚⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⡠⠴⠶⠶⢬⡑⠲⠤⠄⠀⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠤⠤⠤⠶⠒⢉⡡⠟⠁⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠲⠤⢄⣀⣀⣀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣉⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀
⠀⠀⠀⠀⣸⠤⠒⠚⠋⠉⠉⠛⠓⠒⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠉⠙⢧⡀⠀⠀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡤⠔⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠢⢤⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢳⠀⠀
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡏⠀⠀⠀⠀⠀⠀ ⠈⡇⠀
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠀⠀⡾⠀⠀⢧⠀⠀⡞⠀⠀⢿⠀⢠⡟⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀
⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠘⢦⡼⠁⠀⠀⠈⠧⠜⠁⠀⠀⠘⠦⠋⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀ ⢸⠀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀ ⠸⠀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇
⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣾⠀⠀⠀⠀⢷⡀⡞⠀⠀⠘⡇⣰⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠾⠃⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀ ⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⢸⠀⠀⠀⡇
⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⠓⠦⣦⡒⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠤⣤⣀⣀⡴⠛⠀⢸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⢦⠀⠀⠀⠀⠈⠉⠓⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠛⠛⠉⠉⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⣇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡤⠖⠒⠒⢤⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡟⠀⠀⠀⠀⠀⡾⠁⢠⣶⣄⠈⢳⠀⠀⠀⠀⠀⠀⡇⠀⠀⡼⠁⣤⣤⡀⠙⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡆⠀⠀⠀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠻⡀⠘⠛⠋⢀⡞⠀⠀⠀⠀⠀⢠⠇⠀⠀⢳⡀⠙⠛⠁⣰⠃⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉⠀⠀⠀⠀⠀⠀⡾⣆⠀⠀⠀⠉⠒⠒⠚⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⡠⠴⠶⠶⢬⡑⠲⠤⠄⠀⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠤⠤⠤⠶⠒⢉⡡⠟⠁⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠲⠤⢄⣀⣀⣀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣉⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀
⠀⠀⠀⠀⣸⠤⠒⠚⠋⠉⠉⠛⠓⠒⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠉⠙⢧⡀⠀⠀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡤⠔⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠢⢤⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢳⠀⠀
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡏⠀⠀⠀⠀⠀⠀ ⠈⡇⠀
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠀⠀⡾⠀⠀⢧⠀⠀⡞⠀⠀⢿⠀⢠⡟⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀
⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠘⢦⡼⠁⠀⠀⠈⠧⠜⠁⠀⠀⠘⠦⠋⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀ ⢸⠀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀ ⠸⠀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇
⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣾⠀⠀⠀⠀⢷⡀⡞⠀⠀⠘⡇⣰⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠾⠃⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀ ⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⢸⠀⠀⠀⡇
⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⠀⠀⡇
