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                             ``~~

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢷⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡆⠙⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣇⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⠀⢸⠀⠀⠀⠀⠙⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢿⣦⡅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠻⢷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣠⣤⣦⠴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠐⠂⠈⠉⠁⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡏⢀⣀⣀⣠⣤⡤⠤⠔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣾⠿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣠⣤⣴⣶⣶⣶⣾⣿⣿⢿⠿⠿⡟⢿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣷⣿⣿⡿⣿⠿⢿⣻⡟⣻⢻⡹⣍⢧⡙⢦⡓⢾⣮⡙⢧⡙⢦⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣷⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣦⣶⣶⣶⣾⣿⣿⡿⡿⢿⡛⢿⡛⣏⢻⡍⢯⡱⣹⣼⣶⡿⠿⣿⣷⣿⣭⣳⢃⠷⡌⣶⢩⢖⡹⢎⣽⣏⠶⣙⢎⡳⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⣛⣽⡿⢿⠿⠿⣿⠿⡿⠿⡿⣟⢛⣛⣿⣛⣯⡿⢭⣯⡵⣯⠶⣧⠷⣘⠵⢣⡙⢦⡛⡽⣦⡙⣶⣿⣻⣷⣿⣿⣿⣿⣶⣯⣟⢿⣯⣞⡱⣚⢿⡦⣽⡪⢜⣿⢮⡱⣎⡵⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⡟⢩⢹⣧⣾⡿⣷⠥⣟⢻⣏⢟⡹⠿⣌⠻⠻⠿⣏⢰⣿⠯⣿⣁⣾⢻⣧⠍⡞⣥⢛⡦⡝⡜⣧⣳⣿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⡴⡯⣞⡛⠼⣿⡬⡙⢯⣷⠽⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⣟⣋⣿⣷⣿⢿⣿⣧⣿⡥⢫⡜⣏⢣⠻⣬⢳⡹⢼⡛⣏⠘⢿⠖⡇⠀⣯⣟⣹⡜⢩⣟⣴⢣⢏⣶⡝⣼⡯⣽⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⡗⣧⢹⡳⣬⣿⢿⠛⠛⠿⡿⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣷⣾⣿⣿⣿⣟⡿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⣯⠙⠷⠶⡿⣿⣟⣟⣿⣿⣿⠯⡿⣉⢶⡾⠛⢿⣔⡣⣽⠺⣷⡘⢿⣆⢹⣷⣤⡍⣥⡜⢋⣼⡇⣸⡿⢶⣾⢻⣖⣷⣹⣷⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⡛⡿⣐⡫⣵⡿⠁⠠⠂⠀⠀⠀⡀⠀⠀⣸⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⣶⣾⣿⣿⣻⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠚⣿⠈⣴⢫⡔⣧⡃⢭⡙⣭⣙⣥⣾⣷⡘⣿⣶⢮⡴⢹⣧⢣⣻⠴⣟⠦⡟⣷⣦⣴⡎⣿⣐⠎⣽⠡⣽⠭⠙⡏⣾⡟⣿⣧⣻⣿⣮⣛⡻⢿⡿⣿⡿⣿⣯⣾⣿⢫⡜⣱⢣⡓⣽⣷⣤⣀⣀⣠⣴⣾⣷⣶⣾⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣿⣽⣿⢿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣠⣀⣀⣀⠀⠀⠀⠀⣺⣿⠿⢈⡙⠛⢇⣂⢣⡙⢓⣯⣿⣄⣿⣿⡟⡿⠟⢠⡶⢾⣥⢈⡿⠟⢻⡜⣬⠳⡽⠳⢿⢿⣆⡻⡟⢧⣼⠷⣼⣇⠘⠿⠿⢧⡟⢹⣞⠶⣽⣻⠿⡿⢿⡿⠿⣟⡻⠕⣎⠧⡜⣥⠳⡜⣢⠝⣛⢿⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣻⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣟⡿⣿⣾⣟⣿⣿⣿⣿⣿⣿⣿⡛⠛⣻⣿⣿⣿⣶⣤⣼⡿⢻⡖⡀⠒⡉⣌⣗⡐⢨⠠⣿⣷⣟⡻⣿⣿⡿⠆⣹⡷⢈⣹⣯⣼⡏⡘⣯⣶⢟⠋⣴⠟⡰⢿⣿⢻⢷⡶⣬⣏⣾⢻⡱⠺⢏⡔⢺⡉⣮⢻⡏⣿⣹⣷⢚⡝⢦⡙⣭⢲⡙⡞⣴⢣⠳⣡⢛⡬⢲⡙⣧⢹⠿⣿⣿⣿⣿⣿⣿⣿⢪⣟⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⢻⣿⣿⣿⣿⣯⣿⣻⣿⣿⡿⣿⢿⣿⣿⣷⣭⣟⣿⣾⣿⣿⣿⡿⢁⡄⣽⣿⣿⣿⣿⣿⣿⣿⢰⣿⣗⣞⠡⣸⡿⢁⣧⠈⣶⠛⣯⣹⡗⣈⡡⢰⡾⣌⠛⢿⣿⣡⡺⣧⣼⢻⣇⡼⢞⡷⢬⡶⢯⠷⢻⡞⣼⠱⢮⣙⠧⣽⣙⠞⡶⣭⣷⣶⡿⣷⣬⣥⣛⠻⡞⢦⡙⢦⢣⡝⠶⡡⢏⡚⣥⢫⢜⡣⡝⡴⢛⡼⣡⢏⡹⢻⢿⣿⣿⡻⡞⢯⣭⣟⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢻⣿⣿⣿⣿⡼⣷⣻⣿⡗⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⢃⢒⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡝⢻⣯⡕⣿⣧⣒⣃⡖⣸⣧⠺⢿⣷⣿⣧⡌⠻⢷⣌⢺⣿⡿⢿⣿⣾⡱⣎⣴⡿⢋⡞⣇⠢⣄⢻⡇⢼⡟⣭⠹⢾⡴⣩⡟⢺⣼⢬⠗⣾⡿⢷⣯⣿⡾⡟⢳⡝⢮⡱⢎⡳⣍⢧⡙⣆⢏⡲⣱⢹⣌⠳⡜⡥⢎⡵⢋⡖⡿⢩⢿⡙⢦⣳⣼⣿⡾⣿⣦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠹⣿⣿⣿⣷⢿⣟⣿⣿⣻⣿⣿⡟⠗⣺⣿⣿⣿⣿⣿⣿⡌⠢⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠱⣿⣷⡈⠟⠿⠻⢡⡆⠉⠝⠯⠝⣁⣿⣥⣵⡾⣇⣿⣙⠛⠏⣹⡦⢉⣯⡐⣤⠏⣯⢇⡿⢯⠘⣧⣟⡱⣮⡙⣦⠹⣾⣳⠛⠾⠳⣜⣰⢧⡂⠥⢙⣿⣯⣶⣿⢶⡝⣪⣕⠾⣦⡝⣮⢎⡝⢦⡫⣔⠫⡵⣙⣾⣜⣣⡝⣰⣋⢦⣙⢣⢿⣦⣙⠃⡾⠾⢿⣦⡁⠀⠀⠀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⢿⣾⣿⣟⣿⡇⠧⣈⠘⣿⣿⣿⣿⣿⣿⣿⠳⢠⠻⣿⣿⣿⣿⣿⣿⣿⣇⢊⣡⣼⣿⠟⠠⢡⢈⣿⡄⣍⠘⡄⢺⢼⣿⣌⠗⡐⠌⣽⠧⡈⣱⣾⢧⣦⢽⣻⣇⣼⣇⠢⣐⣾⡇⢼⣇⡘⠡⢌⡸⢗⡋⢽⡛⢶⡟⠛⣧⣈⡿⠲⠏⡄⠛⣿⣧⣾⣌⠳⢾⣿⠿⡹⣜⢫⡜⣳⢛⡾⢷⠟⣌⢳⣌⡳⢬⡱⡜⢦⣋⣎⠶⣩⣟⠷⣶⢻⣾⡽⢿⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⢿⣞⣿⣿⣷⠡⣐⢾⣿⣿⣿⣿⣿⣿⠻⠌⣁⠆⣹⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣮⣶⠟⢩⠙⣯⣴⡈⣼⡏⣹⣿⣴⡈⠆⢿⣦⣵⢾⣗⣢⣴⡾⣛⣧⣬⣽⠟⡍⣦⣹⣩⡟⠷⡦⢿⡅⢊⠍⣹⣿⡌⢻⣮⡽⢣⢙⣦⠑⡌⢡⠣⣿⣿⣿⣮⡜⣟⠳⣿⣎⠷⣼⡑⡞⣜⢣⣿⠸⣇⡝⢿⣡⣇⢿⢻⡼⢬⢷⣧⢚⣳⢶⣎⢷⡞⣯⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣾⢿⣿⣿⣿⡥⠨⢽⣿⣿⣿⣿⣿⣿⣏⠡⠌⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⡿⣿⣿⠥⢺⡇⢌⣰⠄⣌⡌⢻⣷⣇⣲⠴⡇⣸⡗⠠⠆⠌⣉⠍⡑⢢⢁⠋⡍⠦⡑⡆⣸⣷⢨⡟⠶⣥⡻⢷⣬⣾⠛⣴⣹⢣⡽⢷⡟⢢⠌⣋⠜⡠⠱⡬⠿⣿⣿⣡⣿⡶⣿⢯⡙⢦⣙⢾⡈⢷⠸⣟⡥⢿⡎⢷⣛⣾⣌⡿⢎⠶⣌⡳⢼⡏⣽⣎⣿⡺⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠩⢃⠡⠌⢿⣿⣿⣿⣿⣿⣿⡧⠌⠜⣿⣿⣿⣿⣿⣿⣿⣿⡇⡬⠑⣿⣿⣿⣥⠿⢶⡞⢨⣿⠱⡈⠭⢙⠻⢏⡴⣹⡣⢑⡨⢸⣿⢄⡉⢢⠡⠎⢴⡟⡄⡧⢘⢿⣴⣿⡖⣹⣧⣉⣙⣤⡍⢿⣵⣂⣶⡿⢆⣹⡗⠦⠟⣣⠡⡙⢴⠿⣿⣿⠇⢴⠿⢶⣯⢹⡎⣞⢳⡜⣮⡇⣟⠼⡿⣼⡇⢺⠹⣿⣎⢳⡜⣿⢣⡝⣼⢻⡼⡷⣽⣧⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠌⣰⠁⠮⢉⣼⣿⣿⣿⣿⣿⣿⡀⠌⠠⢿⣿⣿⣿⣿⣿⣿⣿⡿⣎⠡⣙⢻⣿⣿⡿⢈⣷⡘⠻⠶⢷⡋⢰⢈⣲⠄⡹⢷⣬⣵⠌⡛⠾⢛⠟⢃⡚⣸⣿⣰⣉⣲⡆⢰⡸⣏⢹⣎⣭⡉⣼⡔⢢⡘⠭⣍⡑⠢⠼⣿⣶⣿⠿⢁⠼⢸⣷⣿⣿⡗⣶⣄⣾⣿⢺⣕⣻⣦⡟⡼⡷⣹⣶⢻⡟⣧⣿⡖⡽⣧⢾⣼⣘⠳⣼⡿⣽⢛⣷⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⣀⠇⢨⣿⣿⣿⣿⣿⣿⣿⣿⣌⠆⠡⢀⢹⣿⣿⣿⣿⣿⣿⠁⠞⡰⢠⣾⣿⡿⠅⢆⢡⠌⣿⢒⠸⢦⣝⡠⢿⣰⡐⣶⠈⡄⠚⣠⠕⡈⠜⡰⣤⢡⠉⡍⡍⡘⢛⠛⡡⣸⢁⠆⣌⠙⣿⡰⣠⣵⢂⢻⣄⣣⣦⠟⣉⢃⠒⠌⢒⠠⢻⣿⡽⡷⢭⣹⣄⠓⢹⣷⡎⢿⠻⡴⣧⡙⢿⣏⢿⡸⣿⡹⢾⢽⡏⢿⣌⠳⣿⣱⡿⣯⢼⣩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡈⢰⠠⣿⣿⣿⣿⣿⣿⣿⣿⠿⢂⠁⢢⣿⣿⣿⣿⣿⣿⣿⠈⢆⠱⢸⣿⣿⣦⣛⣡⢆⢺⣟⠈⢳⠌⣷⣶⠾⢇⣸⢃⠡⢼⡧⢸⣦⡕⠧⡘⣿⣦⣑⣤⣿⠈⢆⠱⡐⣿⡌⣶⠠⡉⠻⢋⠝⣹⣤⠢⣍⢉⡄⠚⡄⠻⣌⠘⡄⢃⠂⣿⣷⣣⠹⣯⢹⣛⢭⣫⢼⢫⣧⣱⣈⣿⡣⣿⡾⣷⡹⣗⡏⣼⣠⢾⣯⣹⡗⢾⡗⣯⡟⡴⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⣹⣹⣿⣿⣿⣿⣿⣿⣏⡆⢂⠡⢺⣿⣿⣿⣿⣿⣿⣿⡎⣄⠃⠔⣈⡝⣿⣿⢟⣜⡆⠛⠿⠟⡛⢋⠔⢢⡖⢻⣤⣾⢿⣗⠠⠡⢴⢸⡇⠤⡉⢍⣣⠄⢃⠎⡰⢁⡚⢹⡟⠺⠐⡱⠨⡄⣉⢹⣟⣳⢾⣇⣡⠘⠤⢈⠲⣀⢃⢂⡙⣿⣯⢗⣶⣬⣍⢺⠹⣾⣧⢿⢹⣷⢺⣱⣧⢿⣭⠳⣽⣯⡙⢦⣯⣿⣲⣏⢾⡟⣼⡟⡖⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠑⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐⠌⡰⣰⢸⣿⣿⣿⣿⣿⣿⣿⣦⣓⢬⢀⠆⡙⣿⣿⣋⣟⡘⠤⢃⡌⢡⡊⢼⡇⠲⠠⢄⡞⠻⠿⠟⠫⠘⣷⣦⡅⢌⣿⡐⠌⢒⡁⠦⠌⣻⡄⢣⣱⢖⠡⠐⡄⠢⡈⠍⡉⢩⢉⠣⠜⣀⠃⢌⢂⠂⡔⣻⣿⡜⡶⣧⢛⡞⠾⣇⠿⣻⢮⣽⣼⣷⢻⣆⣻⡷⣘⢿⡝⡶⣏⢶⢻⡜⡾⣹⡿⣸⢣⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠃⡌⡐⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢎⡰⠁⠆⣻⣟⢠⡆⡑⠢⢌⣿⠃⢌⠛⢛⠲⢨⣟⠀⢆⠉⣣⣱⡤⢀⣓⡌⡿⠷⣼⣶⠿⠐⢂⣉⠻⢋⠅⠺⠶⢓⠢⠁⠆⡥⠘⡄⠣⡐⠦⢠⠉⡔⢂⠡⡀⢻⣿⣇⣟⣧⢛⣎⡓⣮⢜⣱⣳⢌⡿⡜⣧⢚⡭⢳⡹⡖⣻⣖⣹⢎⣶⡿⣱⡟⣧⡕⣎⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣇⣢⡌⠐⣈⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡖⢀⠃⠜⡀⣿⡟⣱⢃⠱⢸⣿⡁⠎⣌⠂⣿⣀⠛⢿⡶⢟⠣⠈⢷⠦⡴⠤⣿⡐⠤⣀⣢⡟⠠⣉⡐⢩⠌⡑⠌⠢⢌⠡⣿⢁⡖⢈⢡⠰⠌⢂⡑⣆⢉⣴⡐⣨⣿⡧⡷⣟⢬⢲⢩⢳⣎⢶⣹⢺⣇⠯⣧⢛⡯⢧⡱⣿⢹⡧⣞⢣⣾⠧⣽⢚⣿⡱⢺⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡟⡑⣡⠡⢀⠎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣘⠄⠽⣠⢂⡹⢿⠯⣡⠃⡌⢻⢿⣶⣤⡃⠜⡛⠞⣫⡕⠚⠶⠟⢻⡆⡑⠆⡙⠷⠷⠞⢋⡙⢣⠡⡘⡎⡐⢒⢈⢡⠢⠘⠿⣦⠴⡌⢺⡅⠸⠤⡈⢝⡛⠡⣐⢰⣿⡏⣷⡟⡜⢮⢥⢫⣿⢚⡷⢭⢿⡘⣿⡹⣿⢢⣓⠿⣓⡷⣩⢾⣟⢳⢬⠻⣥⣋⣿⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡰⠄⢂⡁⠢⢈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡬⠁⢌⠢⣁⠢⣬⣡⠃⠬⠔⣨⠐⠸⣇⣹⠐⠃⣼⡇⢩⣂⣩⡿⢃⠰⡉⢄⢣⡘⢌⠢⢄⠃⢃⠌⠳⠾⠴⠋⣸⣇⣁⠒⡀⠆⠄⡉⠛⡅⢂⠱⠘⣧⢁⡒⣸⣿⡱⣯⢳⡟⡜⢮⡱⢎⣧⢺⡜⣲⡟⣼⠗⣿⢣⡻⡜⣽⣓⣧⡿⢬⡗⣎⢷⢆⣳⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠇⢂⠤⠑⡠⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⢲⠠⢀⠂⣴⡷⢉⠐⡌⠠⡘⠂⢼⠉⣿⠴⠚⠿⠶⢋⢹⣷⠈⠰⠔⡈⠼⣄⣢⡴⢈⢲⠈⢄⡓⠐⢂⠣⡀⢌⡁⠆⢌⡐⠹⡆⢡⠐⡡⢚⠠⢉⠛⣤⣿⣿⡔⣣⢏⡵⣙⢦⡝⣺⡧⣛⢮⢽⡏⣷⢛⣯⢳⠳⡭⣿⡜⡿⡱⣿⠹⣼⠧⣾⣿⠃⠂⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡬⣄⠂⠥⢠⠁⠂⠌⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⠯⢂⠂⢾⣿⡤⢇⣂⠄⢃⠘⠠⢹⣦⢅⣼⠇⢈⠒⡀⢊⠻⠿⠷⠟⢃⡶⠀⠆⡐⢢⢀⡉⠤⣧⣘⢀⢂⠁⣦⠐⡈⠄⢂⠑⣧⡂⡼⣧⠂⣅⢊⡔⣾⡿⡜⡵⢣⠞⡼⡱⢎⡼⣡⠗⣭⢚⣼⣚⡿⣎⡽⣧⢻⣜⣹⡜⡵⣹⢆⡻⣬⣷⣿⠏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⡷⣅⠂⢡⠁⢒⠠⠉⣼⣿⣿⣿⣿⣿⣿⣿⢲⡌⠄⡙⠿⣿⣷⣾⠐⣌⠂⡑⠠⡀⢬⣿⠀⢂⠡⡈⢼⡆⠐⡄⠌⠸⣇⣱⡜⠀⠆⡀⠲⠠⠄⣉⠉⢂⠌⣿⡀⠘⢄⣃⡼⢀⠙⡡⠘⠻⢼⢠⣼⣿⢯⡵⣍⢧⡛⡴⣙⢎⡶⢥⠻⡔⣯⡟⡴⢟⡴⣋⠶⣯⢲⣹⡮⡱⢿⣌⡳⣼⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣭⡻⣄⡌⢁⠢⠡⢻⣿⣿⣿⣿⣿⣿⣿⡌⠡⢘⠠⠑⠬⠤⣿⣃⣖⠠⠑⡐⠠⣿⡹⠷⠶⡖⣧⡘⢧⣂⡐⠘⢂⠉⡉⠻⠶⣦⠁⠢⣁⠂⠄⠃⠆⡌⠙⠷⠿⠞⢋⠐⡈⠔⠤⢉⠢⢸⣾⣿⢯⣏⠷⣸⢦⡹⡱⣍⠞⣼⢣⢏⡳⣥⢻⣜⢣⢞⣌⢻⣏⡇⡞⣝⢳⣙⡦⣿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢿⣾⣝⠦⣆⠡⢉⠛⠿⣿⣿⣿⣿⣿⣧⡐⢢⡌⣐⢀⠂⢛⠛⣿⣦⣥⣼⣅⠻⣷⡭⣹⢱⣙⣷⡼⣋⡡⢁⢂⠘⠠⢑⠠⢿⣤⣥⠶⢹⣤⠙⢧⡐⢉⡶⠁⢎⠡⢊⠔⡡⠂⡅⣲⣿⣿⢯⢏⣬⠳⡱⢎⡵⢳⡜⣏⠶⡹⢎⡵⢪⢵⣏⠞⡼⣨⢛⣧⣞⣼⣌⠳⣭⣿⠟⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣟⡶⡬⢙⠊⣿⣿⣿⣿⣿⣿⣷⣶⣿⠠⠚⣀⠎⣼⣿⣿⣿⠏⡰⢀⡑⠻⣾⠿⢋⠡⠌⢰⣶⠏⡀⢃⠂⢆⠉⡉⠤⡁⠆⡐⠌⡌⡁⠎⣇⣉⢆⡘⡄⢊⠴⣱⣼⣿⣟⡟⡜⣎⠶⣩⠽⣿⡔⣣⢞⡥⢫⡕⣫⢞⡱⢾⣋⡞⡱⣷⢍⡞⣭⢲⠬⣷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣥⣚⢈⡙⠛⣿⣿⣿⣿⣿⣿⡐⢡⡄⢂⣽⣿⣿⣿⣦⣱⣠⢁⣣⡄⠒⣈⡉⠄⢿⣇⡂⠳⠌⡌⡐⢸⡇⠒⡤⠑⡨⠐⡄⡉⢆⠩⡍⠡⢆⢉⣦⣿⣿⡿⣏⠶⡹⡜⣬⢳⡱⢏⡽⣿⢶⡻⣜⣳⡼⢱⢎⡵⢻⡧⢞⡱⣛⠼⣘⢦⢫⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣥⣻⣿⣿⣿⣿⣿⣿⣷⣇⠂⠜⠻⢿⣿⡿⣿⣿⣛⡋⡄⠡⢄⡰⢉⠌⠻⠷⠷⠶⠤⢁⠺⢷⣥⣔⠡⡐⢡⠒⠅⢊⠔⣈⣥⣾⣿⣿⣟⣧⢳⡙⡮⣕⢯⡜⣣⢞⠳⣜⣱⡎⢶⣱⢚⠿⣧⡚⣜⣳⡛⣬⠳⣌⡳⢭⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣟⠻⣿⣿⣿⣿⣿⣿⣿⣶⣧⣓⢢⠐⡄⢻⣿⣿⡞⢢⡑⠢⡐⠡⢊⡑⠚⢤⠑⠌⣂⠓⢢⠐⡌⠒⠡⢆⣉⣸⣬⣶⣿⣿⠟⠋⠈⠀⠐⣣⣝⢲⣿⡜⢮⡱⢎⡟⡴⢣⢿⣧⠲⢭⡚⢥⣛⡴⢋⠷⣆⠯⣹⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡙⠛⠻⢿⣿⣿⣿⣿⣶⣁⣂⡥⢉⠛⢿⣾⣃⠱⢈⡁⠆⡌⢡⢌⣂⣉⣴⣬⣤⣧⣴⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⡸⢿⣺⣥⡟⡼⣸⣱⢋⡖⣻⡟⢶⣙⠲⣱⠘⣍⡳⣮⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣄⣀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢖⡥⢫⡝⣷⣵⡨⢏⡜⣡⢛⡦⡹⢗⡌⣯⣴⣷⣿⠿⠛⠛⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⡵⢳⠼⡱⢎⣽⣦⢹⡰⢩⡕⣭⣶⣾⣿⣿⢛⠙⠿⢶⣯⣤⣤⣭⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⢞⡡⢿⠰⣃⢦⣩⣭⣼⣿⣿⣿⣛⠯⢡⠙⠻⣿⣶⣄⡉⠛⢿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⣩⣽⣾⣷⣿⠿⠟⠛⡏⢡⡿⠈⠙⠷⣮⣍⣰⠈⠹⢻⣿⣮⣄⣈⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠰⢿⠿⠁⠀⠀⠀⠀⠉⠙⠛⠷⣦⣤⣝⣿⢟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡏⣤⣶⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣦⣄⣠⡄⠀⠀⡀⢀⡀⠀⣤⠀⠀⠀⠀⢠⣤⢾⠛⠋⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⢋⣡⣭⡿⠿⣶⠿⠻⢾⣿⠛⠻⠶⣶⣶⠛⠛⢿⣦⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁⠀⠀⠛⣷⡄⢂⠹⣿⣆⡐⠘⢿⣿⣤⣂⠈⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠂⠽⣿⣷⣆⠠⠙⢿⣝⠻⠶⢦⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠁⢼⣿⠻⣿⣦⣩⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣿⡆⠂⠙⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀