⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠞⠑⠈⠀⠀⠀⠀⢄⠂⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠿⢿⣧⡢⡀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⠫⣻⣀⠀⠀⠀⠀⠀⠀⡰⢩⠆⡾⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⡁⠀⠀⠀⠙⣿⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡰⢡⣾⣿⣿⣇⠀⠀⠀⠀⠈⠡⡾⢋⠁⠀⢀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣄⠀⠀⣠⣏⣿⣿⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡼⢱⣿⠋⠀⢹⣿⡇⠀⠀⠀⠀⠔⠀⢒⣞⠻⠁⠀⠀⠀⠀⠀⡀⠛⢿⣿⣿⡿⣯⣽⣾⠟⠁⣀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⢡⣿⠋⠀⠀⣼⡿⠀⠀⠀⠀⠀⢠⡀⢠⠻⣶⣀⣀⣀⢀⠀⠈⠊⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡇⣾⣷⣀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠈⠁⢃⣼⡟⠁⠑⡍⠀⠐⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢐⠇⢿⣿⣷⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣺⣷⣶⣶⣷⣶⣶⣦⣀⡄⠀⠀⣸⠊⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⢱⡶⣯⡁⠠⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢻⡿⣏⡿⢻⣹⠿⣟⡿⢿⣻⣿⣿⣧⣀⠀⠀⠀⠀⣀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠃⠚⢃⣁⣶⣀⠀⠀⢀⣼⣿⣿⠿⣾⠉⢿⠀⠘⡇⠀⢻⠀⢸⣇⣈⣏⣸⣡⣏⣿⡛⣛⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣧⢧⣹⡞⡹⣻⣏⣷⣾⡿⣿⠉⢿⠀⣹⣀⢼⠶⠖⢻⠋⢻⠉⠹⣟⠻⣿⣻⣯⣧⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⡿⣧⡋⣷⢟⣹⣿⣯⣧⡼⣖⠛⡏⠉⡇⠈⣇⠀⢸⣄⢠⢧⣰⠛⢿⣿⠿⣻⢿⡿⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⡶⢝⠻⣿⣿⣿⣿⣽⣄⣧⡿⣤⣿⢤⠟⠤⠃⠓⢋⣈⡩⣤⣴⡶⠯⠒⠋⢁⣾⢤⣤⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢣⠀⠀⠳⠈⠙⢿⣿⣭⣟⣒⣒⣚⠒⠒⠒⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣄⣯⡄⠀⠀⠀⠀⠀⢀⣴⠞⡗⣄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⡁⠄⠐⠤⢰⣄⣠⣻⠛⡫⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣯⣿⣿⢿⣧⣤⣶⣦⠤⠚⠁⠀⢧⣿⢋⣇⣀⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠳⡀⠀⠀⢙⢟⡷⣿⣦⣶⣬⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⣇⣧⡿⠿⡟⣿⣿⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠐⠈⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠦⣀⣀⣀⡠⠴⠶⠲⠤⣄⠀⠀⠀⠀⣀⣼⣿⣿⠿⣿⣦⠀⠀⢀⣠⡶⠿⡋⢰⠚⡵⢛⡾⢷⡬⢿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⣷⡄⣀⣸⣿⣿⣿⣿⣿⣿⠛⠙⢯⠵⠾⣯⣿⡇⠀⠀⢄⡀⣏⣷⡴⢚⣍⡤⣶⡿⣼⡇⠀⢀⣴⣾⡟⠋⠀⡀⠱⣌⢧⣴⠟⢇⠀⡉⠙⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣯⡿⣷⣶⣤⡼⣿⢿⣿⣿⣿⡥⠀⠰⣧⣿⣿⡿⢀⣠⣾⡿⢹⣿⣿⣤⣶⣿⣶⡋⣬⢟⣥⣾⣿⠟⠃⠀⠀⠀⠀⢀⣨⠿⣟⣄⣾⣴⣿⢾⣏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡏⢣⣼⣠⣿⠻⡷⢿⣹⣿⠋⠀⠀⠌⠁⠝⣿⡶⣾⠿⢯⡦⣸⣿⣿⣿⣿⣿⣋⣷⣿⣿⠟⠋⠀⠀⠀⠂⠀⣠⣶⣿⣯⣴⣿⣿⢿⣏⣿⣦⡙⠳⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣇⠏⣶⡏⣿⣴⡗⢋⡀⣿⠀⠀⢈⡴⢿⣷⣿⣇⡄⣠⡴⠿⣿⡯⣿⣿⣾⣿⣿⣿⣟⠁⠀⠀⠀⢀⣀⣀⣰⣿⣿⢿⢟⠃⣼