⠀⣀⣀⠀⣀⣀⠀⢀⣀⡀⢀⣀⡀⠀⣀⣀⠀⣀⣀⠀
⠀⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀
⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀
⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀
⠀⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀
⠀⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠈⠉⠀⠈⠉⠁⢰⠀
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀
⠀⠧⠤⠀⠤⠤⠀⠠⠤⠄⠠⠤⠄⠀⠤⠤⠀⠤⠼⠀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣾⠿⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠻⠛⠁⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀
⠀⣀⣀⠀⢀⣀⠀⢀⣀⠀⢀⣀⡀⠀⣀⡀⠀⣀⣀⠀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀
⠀⠁⢠⣤⡄⠀⣤⣤⠀⣤⣤⠀⣤⣤⠀⢠⣤⡄⠈⠀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀
⠀⠃⠠⠤⠄⠀⠤⠤⠀⠤⠤⠀⠤⠤⠀⠠⠤⠄⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀
⠀⠧⠤⠀⠠⠤⠀⠠⠤⠀⠠⠤⠄⠀⠤⠄⠀⠤⠼⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇⠀
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀
⠀⠀⠛⠃⠀⠀⢰⣦⠀⠀⠀⠀⢠⣴⠀⠀⠀⠛⠃⠀
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀
⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀
⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀
⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀
⠀⣀⣀⠀⣀⣀⠀⢀⣀⡀⢀⣀⡀⠀⣀⣀⠀⣀⣀⠀
⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⠀
⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀
⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀
⠀⡄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠀
⠀⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠀
⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀
⠀⠧⠤⠀⠤⠤⠀⠠⠤⠄⠠⠤⠄⠀⠤⠤⠀⠤⠼⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⢰⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀
⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀
⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠛⠀⠀⠀⠀
⠀⣀⣀⠀⢀⣀⠀⢀⣀⠀⢀⣀⡀⠀⣀⡀⠀⣀⣀⠀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀
⠀⡄⠘⠛⠃⠀⠛⠛⠀⠛⠛⠀⠛⠛⠀⠘⠛⠃⢠⠀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀
⠀⠧⠤⠀⠠⠤⠀⠠⠤⠀⠠⠤⠄⠀⠤⠄⠀⠤⠼⠀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀
⠀⠀⢠⣿⠁⢀⠀⠀⠛⠛⠛⠛⠀⠀⣸⡟⠀⡀⠀⠀
⠀⢶⣾⠃⢠⡿⠁⠶⠶⠶⠶⠶⠶⣶⡟⠀⣼⡟⠀⠀
⠀⠈⠻⣿⡿⠁⠀⢀⣀⣀⣀⣀⠀⠘⢿⣿⠏⠀⠀⠀
⠀⠀⣰⠟⠉⢿⡄⠈⠉⠉⠉⠉⠀⢀⣼⠋⠉⣿⡄⠀
⠘⠿⠿⢿⡟⠛⣿⠛⠛⠛⠛⠛⠻⠿⠿⣿⡟⠛⣿⠀
⠀⢰⡖⢺⡇⢰⡆⢰⣶⠶⠶⣶⠀⣶⣶⣿⣷⣶⡀⠀
⠀⣼⡇⢸⡇⠈⣿⣿⣿⠀⠀⣿⠀⣿⠀⣿⡇⢸⣧⠀
⠠⠿⠀⢸⡇⠀⠉⢹⣿⣤⣤⣿⠿⡏⠀⣿⡇⠀⠿⠀
⠀⠀⠀⠸⠇⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀
⠀⣀⣀⠀⣀⣀⠀⢀⣀⡀⢀⣀⡀⠀⣀⣀⠀⣀⣀⠀
⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⠀
⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀
⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀
⠀⡄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠀
⠀⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⠀
⠀⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠀
⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀
⠀⠧⠤⠀⠤⠤⠀⠠⠤⠄⠠⠤⠄⠀⠤⠤⠀⠤⠼⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀
⠀⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣷⡀⠀⠀⢀⣤⣾⠆⠀⠀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⢿⣶⡾⠟⠉⠀⠀⠀⠀
⠀⠀⠀⣀⣸⣿⣤⣤⣤⣴⣶⡆⠙⠿⣶⣤⣀⡀⠀⠀
⠀⠀⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠇⠀⠀
⠀⠀⠀⢸⣿⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀
⠀⠀⠀⢸⣿⠉⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⠁⠀⠀⠀
⠀⠀⠀⢸⣿⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⡄⠀⠀⠀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇⠀
⠀⠀⢀⣷⠂⠀⣶⡆⠀⠰⣦⠀⠀⢻⣆⠀⠀⣿⡇⠀
⠀⢀⣾⡟⠀⠀⢸⣧⠀⠀⢻⣇⠀⠀⠛⠁⢰⣿⠁⠀
⠀⠛⠋⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠸⠿⠿⠿⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀
⠀⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣷⡀⠀⠀⢀⣤⣾⠆⠀⠀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⢿⣶⡾⠟⠉⠀⠀⠀⠀
⠀⠀⠀⣀⣸⣿⣤⣤⣤⣴⣶⡆⠙⠿⣶⣤⣀⡀⠀⠀
⠀⠀⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠚⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⣤⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠰⣿⣄⠀⠀⠀
⠀⠀⣸⡿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀
⠀⢠⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀
⢀⣾⡟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣷⡆⠀⠸⡿⠄
⠀⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀