⠀⠀⠀⠛⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⣁⣶⣶⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠂⠈⠛⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣠⢫⣧⣾⣿⡿⣿⢿⡿⢿⡿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠀⠀⠀⠉⠣⢽⣶⣄⠀⠀⠀⢠⢃⣿⠻⣧⣿⡃⣿⣾⣷⣾⣿⣦⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢟⠦⣄⡎⣾⣿⡀⢼⡿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⢻⣷⣿⣏⣿⣿⡏⣷⣾⢤⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⣤⣤⣤⣤⣶⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⣿⠟⠀⠉⣇⡏⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⡀⢀⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⢿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣉⣩⣷⡾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⣀⣀⣀⣀⠀⠠⠀⠀⠀⠈⢸⡟⠉⠈⢿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠉⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣽⡻⢍⡀⠀⠀⢀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣿⢋⣹⠀⠀⠀⠀⠀⠀⢸⣿⡿⣸⣿⡇⣿⣿⢿⣿⣿⣿⣿⣿⣿⣧⠀⠑⠦⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡴⣯⢰⢀⡤⡀⠀⠀⠀⢨⣿⣡⣿⣿⣿⢹⣿⣜⣿⣿⣿⣿⣿⣿⣿⣄⡃⠀⠈⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡷⣿⣾⣾⠛⠁⡀⠀⠀⡼⢧⣼⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡇⡈⠢⡀⢃⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⡿⠋⣀⣬⠝⣡⡾⠟⣛⣶⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⡇⠁⠀⠈⢸⠀⠀⠀⢸⣸⠀⠀⠀⣠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⣿⣿⣿⣷⣾⠟⣡⣾⣓⣋⣉⣉⣽⣭⣿⣿⡻⣿⣿⣿⣿⣷⣻⣿⡇⠀⠀⠀⠘⠀⠀⠀⠀⢹⢦⡀⠀⣹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⢻⣿⣿⣿⡿⠕⠊⠉⠀⠀⠀⠀⠀⠈⠑⢲⡶⠛⠉⠉⠁⠀⠉⠉⠙⠣⢄⡀⠀⠀⠀⠀⠀⢰⢸⠀⠁⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠘⣿⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢂⡀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠘⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢰⠀⠀⠀⠀
⠀⠀⠀⠈⢼⡇⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⢸⠀⠀⠀⠀
⠀⠀⠀⠀⣻⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣿⣼⠀⠀⠀⠀
⠀⠀⠀⠀⣺⠈⠀⡠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣿⠀⠀⠀⠁
⠀⠀⠀⠀⢸⣎⣡⡤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡄⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⠀⠀⠀⠀
⠀⠀⣡⣴⣾⡿⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠐⠀⠀
⠀⠀⣿⣿⡅⠡⠖⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡇⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀
⠀⠀⠿⠟⠅⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⣤⣾⣷⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠘⣿⣿⡿⠋⠁⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢻⣿⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠠⠋⣻⠀⠀⠃⠀
⢀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠈⣿⠀⠀⠄⠀
⠀⡀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀
⠀⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⠠⠤⠐⠾⣷⡀⣀⡀⠀⢀⡀⠀⢸⠈⠙⠟⢻⠀⠠⠀⠀⠀⢀⣀⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⡩⢅⠀⠀⣀⠄⡠⠊⠁⠈⠑⠢⡀⠘⠋⣂⣼⠀⠀⡀⢸⣴⠒⠻⠋⠉⣁⣀⣙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀
⠐⠀⣿⣿⣿⣿⣿⣿⣿⠛⠕⠣⢀⡴⠉⡜⠀⠀⠀⠀⠀⠀⠨⣢⢀⣿⡏⠀⢀⠀⠈⢿⡄⢀⠔⠉⠀⠀⠀⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠄⠀⣿⣿⣿⣿⣿⣿⣿⣧⣅⡴⠛⢢⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣔⣤⣴⣼⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠘⠛⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣷⣤⣠⡀⠀⢤⣨⢄⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⡟⢻⡃⣨⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣥⣌⣮⠀⠀⠀⠀