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