⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠒⠊⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡖⠂⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⢀⣀⣀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣶⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠇⠀⠀⣴⣿⡿⠋⣤⣴⣶⣿⣿⠀⢸⣿⡿⠀⠀⠀⠀⢈⣿⡏⣿⣿⣿⡇⢰⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⢰⣿⡟⠁⠀⣿⡇⠉⠉⠙⠂⢸⣿⡇⠀⠀⠀⠀⢸⣿⢁⣿⣿⡿⠀⣼⡿⠿⢿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⠀⢀⣿⡿⠀⠀⠀⣿⣿⣿⣿⡄⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣸⣿⣿⠇⠀⣿⡇⠀⢀⣿⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠃⠀⣀⣤⠀⣾⣿⣿⡇⠀⠀⠀⣿⣷⣶⣾⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠛⠛⠉⠁⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⠀⣿⣿⣶⣿⡿⠿⠀⣿⣿⡿⠃⠐⣄⠀⠉⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠠⠄⠄⠀⠀⠀⢸⣿⣿⢿⣿⡄⠀⠀⠀⢸⣿⡿⠿⠟⠻⠀⠋⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠤⠤⠤⠤⠄⠀⠀⠀⢸⣿⡇⠘⣿⣿⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣤⣤⣤⡄⠀⠀⠀⢸⣿⡇⠀⢹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⡀⠀⠀⠀⠸⠛⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⠶⠶⠤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡀⠀⠘⠛⠛⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠁⠈⠀⢰⡆⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣡⣤⣶⣇⠀⠀⠀⢠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣾⣿⣿⠟⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣰⣿⣏⠉⠻⠿⠿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢀⣿⣿⣿⣷⣷⣶⣿⣿⣿⡿⠁⠀⢀⣠⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡿⠟⠋⢁⣤⣶⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⣿⠁⠀⣠⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠘⣿⣿⣿⡏⢀⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⣿⣿⣿⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⡰⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠉⠉⠉⠉⣁⡀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠤⠤⠀⠀⠐⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀⢸⠀⠀⠀⣀⣀⠤⠤⠀⠀⠐⠒⠂⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀