⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣶⡴⠓⠋⠉⠉⠉⢻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡟⣿⣿⠀⠀⠀⠀⠀⠘⠛⠈⣾⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠯⠅⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⢀⢔⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⢰⣿⢿⣿⣦⡀⠀⠀⠀⢘⣇⠀⠀⠀⠀⠀⣠⣳⣗⣁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡍⠀⠀⠀⠀⠀⣾⣿⣿⢿⡇⠀⠀⠀⠙⠿⣿⣿⡗⠀⠀⠀⢸⣾⣄⣤⠴⠶⠛⠛⠉⠙⡙⢫⣑⠂⠭⠢⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⢀⣴⣿⣷⡄⠀⠀⠀⠀⠐⢿⣿⡿⢏⠀⢀⣀⣀⣤⡄⠈⢥⣀⣤⢴⣶⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠈⠢⡉⠲⡁⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⢄⠀⠀⠀⠀⠀⠁⠸⣇⠀⠙⠻⠿⣿⠃⢀⣽⣭⣶⣿⣿⣻⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠈⠐⠨⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢄⠀⠀⠀⢊⡄⡆⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣤⣴⣾⣿⣷⣾⡦⠚⠩⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠫⡢⡀⠀⢱⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠑⢄⠰⠀⠡⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢹⡽⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⢀⣴⠟⠀⠀⠈⠛⠘⠛⠯⠿⠛⠿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠒⢤⢄⣀⢠⠰⣆⠸⠒⠀⠀⠈⠁⠀⠉⠉⠙⠛⠻⠿⡿⠿⣿⠿⣿⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⡠⠙⡝⠳⠤⡀⣀⡀⡀⠠⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠟⢿⣿⣿⣿⣿⣿⣿⡿⠒⠀⠀⠀⠑⡱⣰⠀⠀⠈⠛⠻⢏⣛⡞⡿⡛⠛⠒⠚⠁⠀⠀⠀⠀⠀⠀⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⢁⢠⡢⠀⠻⣇⣽⡏⠉⠀⠀⠀⠀⠀⠀⠈⠘⡇⠀⠀⠀⠀⠀⠈⠉⠒⠻⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣼⣶⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢠⡃⣠⣤⣔⣦⣤⣦⣦⣦⣾⣷⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢇⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⠟⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣵⣦⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣧⠀⠀⠀⣼⡟⢻⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠳⢖⠢⠴⠲⠣⢚⣽⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⢀⣾⠟⠀⠀⢻⡿⢿⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣯⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⣿⣿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣯⣤⣤⣤⠯⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⠶⠟⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀