⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣾⣶⣥⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠛⠁⠀⠀⠀⠈⠉⠻⣿⣶⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣶⣾⣭⣀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠟⠛⠉⠡⢀⠐⡀⢂⠐⡈⠄⠡⠈⠄⠡⠈⠄⠡⠈⠄⠂⠄⡀⢈⠉⠛⠻⠿⣿⣿⣶⣥⡀⢄⠀⠀⠀⠀⠀⣿⣿⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⡀⠄⠂⠌⠠⢁⠂⡐⢀⠂⡐⠠⠈⠄⠡⠈⠄⠡⠈⠄⠡⢈⠐⠠⠐⠠⢈⠐⡀⠂⠄⡉⠛⠿⣿⣷⣮⡢⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠠⠐⠠⢁⠐⡈⠐⡈⠐⡀⢂⠐⡀⠂⠄⠡⠈⠄⠡⠈⠄⠡⠈⠄⡁⠂⠌⠠⢁⠂⠄⠂⠄⡁⠂⠄⡁⠂⠌⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠋⢀⠐⡈⠄⡁⣂⣄⣂⣄⣡⣀⠡⠐⡀⠂⠄⠡⠈⠄⠡⠈⠄⠡⠈⠄⡁⠂⠄⠡⢈⠐⠠⠈⠄⡁⠂⠄⡁⠂⠄⠡⢈⠐⠠⢈⠛⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⢁⠠⠐⡀⣂⣴⣾⣿⣿⠿⠿⠿⠿⣿⣿⣶⣤⡁⠌⠠⠁⠌⠠⠁⠌⠠⢁⠂⠄⠡⢈⠐⠠⢈⣄⣥⣦⣴⣵⣤⣤⣁⣌⠐⠠⢈⠐⡀⠂⠌⠹⣿⣷⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⡐⢀⠂⣡⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣦⠁⠌⠠⠁⠌⠠⢁⠂⠌⠠⢁⠂⣬⣾⣿⡿⠿⠛⠛⠛⠛⠿⠿⢿⣿⣦⣆⠐⠠⠁⠌⡐⠈⢻⣿⣎⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣼⣿⠟⠁⠄⠡⠐⣀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠈⠄⠡⢈⠐⠠⠈⠄⡁⣢⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄⢁⠂⠄⡁⢂⠹⣿⣧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣳⣿⡿⠋⢀⠡⠈⠄⣁⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⣇⢈⠐⠠⠈⠄⡁⢂⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠈⡐⢀⠂⠄⡘⣿⣧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣵⣿⡟⠁⡐⢀⠂⡁⢂⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⠀⢹⣿⡆⠈⠄⡁⢂⠐⣾⣿⠏⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠐⠠⢈⠐⡀⠘⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⣷⡑⡀⠀⠀⠀⠀⠀⠀⣠⣵⣿⠟⢀⠐⡀⢂⠐⡀⠂⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡇⠌⡐⢀⠂⢰⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠈⡐⢀⠂⠄⠡⢹⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣣⣿⣿⠀⠄⠠⠀⠄⠠⢀⠃⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⡇⠠⠀⠄⡘⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢀⠠⢀⠘⡀⢃⠀⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢹⣿⣆⢂⠀⠀⠀⠀⢰⣾⣿⠇⠠⠈⠄⠡⢈⠐⡀⢂⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠃⠀⠀⢀⣾⣿⠇⠠⢁⠂⠄⠹⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢀⠂⠄⢂⠐⡀⢂⢹⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠂⠀⢿⣿⡼⡀⠀⠀⠀⠸⣿⣿⠀⢂⠡⢈⠐⡀⢂⠐⠠⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⢀⠁⠂⠌⠠⠁⢿⣿⣆⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢀⠂⠌⡐⢀⠂⡐⠠⢸⣿⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠠⠀⢸⣿⡷⣇⠀⠀⠀⣧⣿⣟⠀⢂⠐⡀⢂⠐⠠⠈⠄⡁⢂⠹⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠃⡐⠠⢈⠐⡈⠄⠡⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⢀⠂⠌⡐⢀⠂⡐⠠⢁⢰⣿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⡇⠀⠀⠄⠀⣿⣿⠘⠀⠀⠀⡿⣿⣷⠈⡀⢂⠐⠠⠈⠄⡁⢂⠐⡀⠂⠄⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢀⠐⠠⢁⠂⡐⠠⠈⠄⠡⢈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⡿⠃⡀⠂⠌⡐⢀⠂⡐⠠⢁⠂⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣇⠀⠠⠀⠀⢿⣿⡼⠄⠀⠀⢉⣿⣿⠀⡐⠠⠈⠄⡁⢂⠐⡀⠂⠄⡁⠂⠄⠡⠀⠄⠠⠉⡉⢉⠉⡁⠄⠠⢀⠂⠌⠠⢈⠐⡀⠂⠄⠡⠈⠄⡁⠂⠄⡈⠻⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⡿⠟⠡⢀⠂⠄⡁⢂⠐⡀⠂⠄⡁⠂⠌⣸⣿⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠂⢸⣿⡇⠀⠀⠀⠈⠘⣿⣷⠀⠡⢈⠐⡀⢂⠐⠠⢁⠂⠄⠡⠈⠄⠡⠈⠄⠡⠐⡀⢂⠐⡈⠐⠠⠈⠄⡁⢂⠐⠠⠁⠌⠠⢁⠂⠄⡁⠂⠄⠡⢀⠉⡉⢉⣉⣉⣭⣭⣭⣦⣤⠀⡁⠂⠌⡐⢀⠂⡐⠠⢁⠂⠄⡁⢂⣿⣿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠠⠀⢸⣿⡇⠀⠀⠀⠀⠀⠜⣿⣿⣄⠂⡐⢀⠂⠌⡐⠠⠈⠄⠡⠈⠄⠡⢈⢸⣿⣿⣷⣶⣦⣤⣥⣦⣥⣆⣐⣠⣈⣄⣡⣈⣐⣠⣈⣴⣤⣥⣬⣴⣶⣾⣿⣿⣿⠿⠿⠛⠛⣿⣿⠀⠄⡁⢂⠐⡀⠂⠄⡁⠂⠌⡐⠠⢸⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣆⠀⠀⢸⣿⣏⡇⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡂⠌⠐⠠⠁⠌⠠⠁⠌⠠⢁⠂⠄⡉⣙⣿⣿⡛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢰⣿⡿⠈⡐⢀⠂⡐⠠⢁⠂⠄⡁⠂⠄⣁⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣤⣼⣿⣿⣭⣭⣧⣐⠄⠀⠀⠀⠈⠝⠿⣿⣷⣮⣤⣁⣌⣠⣁⣌⣐⣠⣼⣶⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⢂⠐⡀⠂⠄⡁⠂⠌⡐⠠⢁⠂⣾⣿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣀⣤⣴⣶⣶⣶⣶⣿⣿⠉⠉⠉⠉⠉⣩⣽⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠻⣿⣷⡱⡀⠀⠀⠀⠈⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⠟⢁⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠏⠠⠈⠄⠂⠄⡁⠂⠄⡁⠂⠄⡁⠂⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢠⣿⣿⠟⠛⠉⠉⠉⠙⠛⠋⠀⠀⢀⣴⣿⡿⠟⠋⠁⠀⠀⢀⠀⠀⢀⠀⠀⡀⢘⣿⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⡐⠠⠈⠄⡉⠻⢿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡿⠋⠁⠄⠡⠈⠄⡁⠂⠄⡁⠂⠄⡁⠂⢄⣿⣿⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⠁⠀⠀⠠⠐⠀⢀⠀⠀⠠⢀⣿⣿⣋⣀⣀⣤⣴⣶⣿⣿⠃⠀⡀⠠⠀⠀⣼⣿⢏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣿⡿⠋⡀⠄⠂⠄⠡⢈⠐⠠⠁⠄⡈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⢁⠠⢀⠡⠈⠄⡁⠂⠄⡁⠂⠄⡁⠂⠄⣡⣾⡿⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢻⣿⣆⠀⠈⠀⢀⠠⠀⠀⠈⢀⠀⠻⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⢀⠀⠀⣀⣾⣿⢫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠡⢀⠂⡐⢈⠐⡈⠐⠠⠈⠄⡁⠂⠄⠡⢈⠐⡀⠂⠄⠠⠀⠄⠠⠀⠄⠠⠀⠄⠂⠌⠠⢁⠂⡐⢀⠂⡁⠂⠄⡁⠂⠄⡁⠂⠄⢡⣼⣿⢟⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠈⢻⣿⣆⠀⠁⠀⠀⠀⣠⣷⣷⣶⣶⣿⣿⠟⠁⠀⠀⠀⡀⠀⠂⠀⣀⣶⣿⡿⠹⠁⠀⠀⠀⠀⠀⠀⢀⢀⣠⣾⡿⠟⠁⡀⠂⠄⢂⠐⡀⠂⠄⡁⢂⠁⠂⠄⠡⢈⠐⡀⠂⠄⠡⠈⠄⠡⠈⠄⠡⠈⠄⠡⢈⠐⡈⠐⡀⢂⠐⡀⠂⠄⡁⠂⠄⡁⠂⠄⠡⣼⣿⠟⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠻⣿⣷⣄⣀⣡⣾⣿⡟⠋⠉⠉⠉⠀⠀⠀⡀⠄⠁⠀⢠⣴⣿⣿⢿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⡀⢂⠐⠠⠁⠌⡀⠂⠄⡁⢂⠐⠠⢈⠐⡈⠐⡀⠂⠄⠡⠈⠄⠡⠈⠄⠡⠈⠄⠡⢈⠐⡀⠂⠄⠡⠐⡀⠂⠄⡁⠂⠄⡁⠂⠄⢡⣼⣿⠿⠫⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠈⠻⠿⡿⡿⣿⣯⣤⣤⣤⣤⣀⠀⠂⠀⠀⢀⠠⢸⣿⡿⡫⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⡀⢂⠐⠠⠈⠄⡁⠂⠄⡁⢂⠐⠠⢈⠐⡀⠂⠄⠡⢀⠡⠈⠄⠡⠈⠄⠡⠈⠄⠡⢈⠐⡀⠂⠄⠡⠈⠄⠡⢀⠁⠂⠄⡁⠂⠄⣡⣼⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠿⡿⣿⣿⡿⠀⠀⠁⢀⠀⢠⣿⡿⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⣢⣾⡿⠋⡀⢂⠐⠠⠈⠄⡁⠂⠄⡁⢂⠐⠠⢈⠐⡀⠂⠄⠡⢈⠐⡀⢂⠡⠈⠄⠡⠈⠄⠡⢈⠐⡀⠂⠄⠡⠈⠄⠡⢈⠐⠠⢈⠐⠠⢀⣡⣾⣿⠟⠃⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⢿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⣿⠟⠀⠀⠐⠈⠀⢀⣿⣿⡱⠁⠀⠀⠀⠀⠀⠀⠀⡰⣵⣿⠟⢁⠐⡀⠂⠌⠠⢁⠂⠄⡁⢂⠐⠠⢈⠐⡀⠂⠄⠡⢈⠐⡀⢂⠐⡀⢂⠡⠈⠄⠡⢈⠐⡀⠂⠄⠡⠈⠄⠡⢈⠐⠠⢈⠐⠠⣨⣴⣿⡿⠋⠁⠀⠀⠠⠀⠂⠀⡀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣷⣦⣄⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⣿⡟⠀⠀⡀⠂⠀⢀⣿⣿⡱⠁⠀⠀⠀⠀⠀⠀⢀⠜⣾⣿⠋⡀⢂⠐⠠⠁⠌⡐⠠⢈⠐⡀⠂⠌⡐⢀⠂⠄⠡⢈⠐⡀⢂⠐⡀⢂⠐⡀⢂⠡⢈⠐⡀⠂⠄⠡⠈⠄⠡⢈⠐⠠⢈⠐⠠⣼⣿⣿⣿⣿⣤⣤⣤⣤⣆⣀⣀⠄⠀⠀⠄⠀⡀⠀⡀⠁⢀⠈⠀⠠⠀⠀⠄⠀⠀⠉⠻⢿⣷⣄ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⡀⠀⢀⢀⣿⣿⠣⠁⠀⠀⠀⠀⠀⢀⣤⣷⣾⣿⠃⡐⢀⠂⠌⠠⢁⠂⠄⡁⢂⠐⠠⢁⠂⡐⠠⠈⠄⡁⢂⠐⡀⢂⠐⡀⢂⠐⡀⢂⠐⡀⠂⠄⠡⠈⠄⠡⢈⠐⠠⢈⠐⠠⠈⠄⡉⢉⠉⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣶⣤⣀⠀⢀⠀⠄⠀⠀⠂⢀⠀⠁⠀⡀⠂⢀⠀⠈⢻⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⡀⠀⠄⠀⣼⣿⢧⠀⠀⠀⣀⣤⣴⣿⠿⠛⠋⣿⣿⠀⡐⠠⠈⠄⡁⠂⠌⡐⢀⠂⠌⡐⢀⠂⠄⠡⢈⠐⡀⢂⠐⡀⢂⠐⡀⢂⠐⡀⢂⠐⠠⠁⠌⠠⠁⠌⡐⠠⢈⠐⠠⠈⠄⡁⢂⠐⠠⠈⠄⠡⠈⠄⡁⢂⠐⠠⠀⠄⡉⠛⠻⣿⣷⣦⣤⣤⣤⣦⣤⡀⠀⠂⠁⠀⢀⠀⠀⠄⢸⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠄⠀⢠⣿⣿⣼⣶⣾⣿⠿⠛⠉⠀⠀⠀⢀⣿⣿⠀⠄⠡⢈⠐⠠⢁⠂⡐⠠⢈⠐⡀⠂⠌⠠⢁⠂⡐⢀⠂⡐⢀⠂⡐⢀⠂⡐⢀⠂⠌⠠⠁⠌⠠⢁⠂⠄⡁⠂⠌⠠⢁⠂⡐⠠⠈⠄⠡⠈⠄⡁⢂⠐⠠⠈⠄⡁⠂⠄⡁⠂⠌⠻⣿⣿⣿⣿⡿⠟⠃⠀⠠⠀⠂⠀⠀⠄⣠⣾⣿ ⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⢀⠈⠀⠀⠄⠀⠙⠿⠛⠋⠉⠀⠀⠀⠀⠄⠂⠁⠠⣿⣿⠀⠌⡐⠠⢈⠐⡀⠂⠄⡁⢂⠐⠠⠁⠌⡐⢀⠂⡐⢀⠂⡐⢀⠂⡐⢀⠂⡐⠠⠈⠄⠡⠈⠄⡁⠂⠌⡐⠠⠁⠌⡐⢀⠂⠄⠡⠈⠄⠡⢈⠐⡀⠂⠌⠠⢁⠂⠄⡁⠂⠄⡁⠂⠄⢻⣿⡏⠀⠀⠀⢀⠐⠀⠀⠄⢂⣤⣾⣿⡿⠋ ⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⡀⠀⠁⢀⠀⠂⠀⠀⢀⠀⠄⠐⠈⠀⠀⡀⠄⠀⣿⣿⡌⠐⠠⢁⠂⡐⠠⢁⠂⡐⠠⠈⠄⡁⢂⠐⡀⢂⠐⡀⢂⠐⡀⢂⠐⡀⠂⠄⠡⠈⠄⠡⢈⠐⠠⢁⠂⠄⠡⢈⠐⡀⠂⠌⠠⠁⠌⠠⢁⠂⡐⠠⠁⠌⡐⠠⢈⠐⠠⢁⠂⠄⠡⢈⢸⣿⡇⠀⠠⠈⠀⠀⣀⣤⣾⣿⣿⠟⠉⠀⠀ ⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠠⠀⠀⠐⠈⠀⠀⠠⠀⠁⠀⠀⡀⠠⠀⠂⢁⣀⣤⣶⣿⣿⣷⠁⠂⠄⠂⠄⡁⢂⠐⠠⠁⠌⡐⢀⠂⡐⢀⠂⡐⢀⠂⡐⢀⠂⡐⠠⠁⠌⠠⠁⠌⡐⠠⢈⠐⠠⠈⠄⡁⢂⠐⠠⠁⠌⠠⠁⠌⡐⢀⠂⠄⠡⢈⠐⠠⢁⠂⠌⡐⠠⠈⠄⡁⠂⣼⣿⡇⠀⢀⣠⣴⣾⣿⣿⣿⣽⣃⣀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠐⠈⠀⢀⠐⠀⠀⠐⠀⠁⢀⣀⣤⣶⣾⣿⡿⠛⠋⠁⢻⣿⣆⠡⢈⠐⠠⠐⠠⠈⠄⡁⢂⠐⡀⢂⠐⡀⢂⠐⡀⢂⠐⡀⠂⠄⠡⠈⠄⠡⢈⠐⠠⢁⠂⠌⠠⢁⠂⡐⠠⠈⠄⠡⠈⠄⡁⢂⠐⠠⠈⠄⡁⠂⠌⡐⠠⢈⠐⠠⠁⠌⡐⠠⢡⣿⡟⠀⣴⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⡧⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⡈⠀⠀⠐⠀⣀⣠⣡⣴⣶⣿⣿⠿⡟⠻⠉⠈⠀⠀⠀⠀⠀⣻⣿⣆⠄⡈⠄⡁⢂⠁⢂⠐⡀⢂⠐⡀⢂⠐⡀⢂⠐⡀⠂⠄⠡⠈⠄⠡⢈⠐⠠⢈⠐⠠⠈⠄⡁⢂⠐⠠⠁⠌⠠⠁⠌⡐⢀⠂⠌⠠⢁⠂⠄⡁⠂⠄⡁⠂⠌⠠⢁⠂⠄⣡⣿⡿⠁⠀⠈⠉⠁⠀⠀⢀⣀⣴⣾⣿⠟⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣧⠐⢠⠐⢠⠈⡄⠂⠐⠀⠂⠐⠀⠂⠐⠀⠂⡄⢡⠈⡄⢡⠈⠐⢠⠈⠐⢠⠈⡄⠁⠂⠐⢠⠈⡄⢡⠈⡄⠁⠂⠐⢠⠈⡄⠁⠂⡌⠐⢠⠁⠂⡄⢡⠈⠐⢠⠈⢰⣿⣿⠁⠀⠀⠂⢠⣴⣶⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢿⣧⡄⠂⠄⢂⠐⡈⠄⠡⠈⠄⠡⠈⠄⡁⢂⠐⡀⢂⠐⠠⢈⠐⠠⢈⠐⡀⠂⠄⠡⢈⠐⡀⢂⠐⡀⠂⠄⠡⢈⠐⡀⠂⠄⡁⠂⠄⡁⠂⠌⡐⢀⠂⠌⡐⠠⣸⣿⡿⠁⠀⠀⠂⠀⠈⠻⠿⢿⣿⣿⣶⣶⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣾⣶⣦⣤⣬⣤⣥⣬⣤⣅⣂⣐⣀⣂⡐⢠⠈⡐⠠⢈⠐⡀⠂⠄⠡⢈⠐⡀⢂⠐⡀⠂⠄⠡⢈⠐⡀⠂⠄⡁⠂⠄⡁⠂⠄⡁⢂⠐⠠⢈⠐⢀⣶⣿⠟⠀⠀⢠⣦⣤⣤⣤⣤⣤⣄⣀⣉⣉⣻⣿⡷⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣦⣴⣤⣦⣴⣤⣥⣬⣴⣤⣆⣄⣁⣂⣄⣁⣂⣄⣁⣂⣐⣠⣈⣐⣤⣼⣿⡿⠋⠀⠀⠠⠘⢿⣿⣿⡿⡿⣿⣿⣿⣿⣿⢿⠿⠅⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡧⠀⠀⠀⠀⠀⠀⢸⣿⡏⠈⠉⠉⠉⠉⠛⠛⠛⠛⢻⣿⣿⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⢻⢿⣿⣿⣶⣤⣐⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⡀⠂⠀⠄⠀⣿⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠈⠺⠝⣿⢿⣿⣶⣦⣤⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⠀⠄⠀⡀⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠈⢀⠀⠁⠀⣿⣿⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⡿⠿⠿⠟⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⡀⠠⠀⠀⣿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣇⠀⠐⠀⠀⠐⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⠀⠀⠄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⠀⠀⠐⠈⠀⠀⠸⣿⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠂⠀⠀⢹⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠂⠀⠄⠁⠀⠻⢿⣿⣿⣿⣿⣶⣶⣶⣦⣯⣥⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣼⣾⣿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⠐⠈⠀⢸⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⢿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠁⠀⠀⠠⠀⢀⠐⠀⠀⠂⠀⠁⠀⠁⠀⠀⠄⠀⠠⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣷⠀⠀⠄⠁⠠⠀⠁⠀⠂⠀⠐⠀⠀⠂⠀⡀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠂⠀⠄⠀⠀⠄⠂⠀⠈⠀⠐⠈⠀⠠⠀⠂⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⡀⠄⠀⠀⠂⠁⠀⠈⢀⠀⠁⠀⠄⠀⠀⠁⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡻⣿⣷⣦⣀⣀⣀⣂⣀⣀⣀⣈⣀⣐⣀⣀⣂⣠⣤⣴⣶⣷⣾⣿⢷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣷⣶⣴⣤⣤⣥⣀⣀⣈⣀⣀⣀⣐⣀⣀⣀⣡⣶⣿⣿⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢝⣻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠝⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀