⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⣄⣀⣀⡀⠀⣠⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⢴⠕⠋⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣽⡷⡄⢻⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣿⣶⠾⢿⣀⡄⠀⠠⠤⡰⢃⠀⠀⠀⠀⠀⠀⠤⠂⣼⣏⡏⡼⢻⣗⡟⢦⡹⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⢀⠄⠀⠀⠀⠀⠀⠈⠉⠉⠂⠀⠀⠀⠁⠀⢲⡀⠹⣿⣿⣦⣌⢻⡱⡀⠱⠜⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢠⣾⠋⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡿⣿⣿⣿⣇⢱⡀⠀⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡴⠁⢀⣧⠛⠳⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣠⠞⠻⢿⣿⡔⣧⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣠⢢⠎⠀⠀⢸⠙⡆⠀⠀⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠴⣀⣉⣿⠸⢧⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⣳⠃⠀⠀⠀⠈⢀⣸⡄⠀⠸⡀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⣷⡏⣏⡇⠀⢣⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡼⣽⣻⠀⠀⠀⠀⡇⡸⠘⣝⣆⢢⢳⡰⣎⠉⠦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣿⡇⣿⣿⠀⠈⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢡⠃⡏⠀⠀⠀⠀⣏⡇⠢⠈⠻⣦⡓⢇⠈⢳⣶⢿⣦⡀⡀⠀⠀⠀⢈⣿⣿⣻⣿⢹⡇⡏⣿⣇⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⡟⠀⢳⢠⠀⢸⠀⣿⣤⣤⣀⠀⠀⠙⢌⢆⠘⠃⠋⣏⠳⣜⢾⡀⡆⢸⡟⠉⡿⣿⣸⣿⣧⡏⢿⡀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡼⣼⠁⠀⠸⠈⣆⠸⡄⣿⡰⣿⣿⣇⠀⠀⠀⠙⠧⠀⠘⠽⠿⢿⣯⠁⡇⣾⣷⣶⣷⣿⡿⢻⠟⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢳⡇⠀⠀⢠⢇⠸⡄⢣⠰⢷⣻⡿⠿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⣿⣿⣿⢿⣏⣠⠋⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⡇⠀⠀⡎⠸⣆⢱⡀⠀⠘⣧⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢈⢸⢿⡏⣾⣷⣫⠟⠁⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠃⠀⢰⠃⡆⠘⢆⠳⣄⠀⢿⣄⡀⠀⠀⠥⡤⢔⡋⠀⠀⠀⠀⡸⡸⣼⢿⡀⢯⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢠⡅⠀⡘⢷⣹⢦⢸⣯⠻⣖⡀⠀⠉⠉⠀⠀⣀⢔⠊⣽⡷⠁⢸⢱⣌⢳⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡇⠀⠹⣌⢻⠹⣷⣦⡉⠑⠢⠤⠶⡏⣡⠋⣰⠟⣀⠤⢺⠛⡍⠳⢿⣄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡇⠀⠀⠈⢿⠀⠘⠛⠛⠀⢠⣤⡴⠿⣧⡶⠛⠉⠀⢀⠃⣰⣿⣦⣀⡉⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣇⠘⠁⠀⠀⠀⠀⠃⣰⡿⠛⠋⠉⠈⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣀⡀⣤⣾⣩⡿⠛⠛⣿⡀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⠖⠋⠉⠀⣼⣳⣿⣿⣷⣶⣾⣽⣇⢠⠖⢙⣄⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡇⠀⠀⢠⢂⢿⡟⢹⠏⢹⣿⣵⠟⢿⣆⢓⣵⣿⠏⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢀⠗⠁⠀⢹⠋⣴⣿⠉⠀⠳⣌⠻⣿⣿⡏⠀⢻⠀⠀⡶⣦⣴⡆⠀⢰⡻⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡟⢀⠔⠁⠀⠀⡰⠃⠀⣼⠇⠀⠀⠀⠈⠉⠁⢿⡇⢀⠏⡇⠀⠷⠿⠼⠇⢠⢣⡇⠀⠀⠀⠀⠀⠀⣇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡆⠀⠀⠀⡰⠁⠀⣸⣿⠖⠒⠀⠀⢠⠀⠀⢸⡇⠘⡀⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡏⢻⠀⠀⠀⡼⠁⠀⢠⣿⠹⠿⠷⢶⣶⡾⠀⠀⢸⣷⡠⠂⡇⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⢛⣇⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠁⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣸⠛⠻⢾⢦⣀⡜⠀⠀⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⣧⣴⢻⣰⠀⠀⡀⣀⣠⣏⡇⠀⠀⠀⠀⠀⠀⠀⠸⡼⣽⡀⠀⠀⠀⠀⠀⠀⠀
⡎⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⣾⢸⠟⠀⠀⢀⡟⢻⣧⠀⠀⠀⠀⠀⠠⠚⣁⣴⡟⠛⣿⠉⠛⠛⠛⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠃⠈⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠏⣠⣴⣟⣼⠏⠀⠀⠀⣼⡆⠀⠙⢷⠀⠀⠀⢀⠔⠉⠀⢹⠇⢺⢿⡀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⠸⡄⢱⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣰⠃⣼⣿⣿⡿⠁⠀⠀⠀⣸⣿⡗⢄⠀⠀⠃⠀⠀⠁⠀⠀⡴⠋⡰⡷⢻⣷⣦⡄⠀⠀⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡄⢣⠈⠇⠀⠀⠀⠀
⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⣰⠃⠀⣿⣿⡾⠁⠀⠀⠀⢰⣿⡿⠀⠈⡆⠀⠀⠀⠀⠀⠀⠘⠀⢈⡞⠁⢸⣿⣿⣿⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣃⠈⠆⠀⠀⠀⠀⠀