⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣠⡾⠿⠹⠟⣲⡿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⣿⠇⠩⣘⡟⣀⢀⠛⠀⠉⢉⡽⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣟⣛⣷⣾⣤⣦⡴⣿⣽⣴⣾⣤⣤⣵⣮⣛⡂⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠤⢼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣼⢻⠛⣿⣿⡟⣿⣦⢰⣵⣯⣤⣤⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⠝⣽⠝⠋⢰⡿⣿⣿⣦⣏⣷⣿⡞⠋⠈⠀⠌⠐⡖⠁⢰⠻⠽⣤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⣠⣶⣾⣯⠹⡦⠑⡆⠀⣀⠐⠛⣿⢿⢇⣽⣿⣿⣶⣾⠀⢀⣬⣇⣆⣀⡀⣶⣳⡆⠀⠀⠀⠀⠀⠐⠿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⡰⢶⣦⢀⣤⣈⣀⠉⣾⡏⠿⣿⢯⣿⣿⣿⡟⠙⢀⡟⠛⠿⣷⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡻⣻⣿⣿⣿⣤⣿⣿⣾⣹⣿⣿⢟⠻⠿⠀⠟⢀⡀⣸⣯⡄⠲⠀⠀⠀⡐⢉⠙⣻⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠻⠿⣟⣻⣿⣧⢯⣽⣿⢫⣶⡾⢿⡧⣠⣮⡴⣽⣿⣋⣁⠔⠉⠣⢨⠣⣲⡿⢿⠿⠟⠃⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀
⣦⡄⠀⠀⠀⠀⠠⣾⣿⣷⣿⣶⣶⣿⣿⢊⠹⠉⣻⡿⣭⣿⠷⠀⣻⣷⣮⣾⠿⣟⡿⢿⣶⣺⣤⣠⣷⣷⣿⣦⣿⠇⣲⣶⣄⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⣿⣿⢿⣿⣿⣟⣙⣩⣯⣿⣿⣿⣿⣧⣿⣶⣿⣿⠿⠙⠈⠚⠀⢀⠀⠉⢟⠻⣍⠋⠐⡉⠉⠁⠄⢙⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠻⠻⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣴⠠⠐⣀⣠⡙⣿⡶⠚⡄⣸⢋⡀⠄⠀⢌⢡⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⠿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣯⣻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣰⣬⣿⣿⣤⣐⣍⣻⡾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⠉⢻⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠟⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢩⢿⡟⠻⠛⠃⠛⠟⠃⠘⣿⣏⡗⣴⡿⠁⠀⠛⠛⡟⠟⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣿⡯⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣼⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣦⣾⣶⣶⣶⣿⣿⣿⣿⢿⢸⠾⣿⣶⣶⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣿⡟⣷⢹⣿⣿⣿⣿⣿⣧⣦⣤⣄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⣄⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⡿⣿⣿⣟⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⢦⣦⣿⡿⠿⡿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿