⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⠶⠒⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⢿⢟⣵⠃⠀⠀⢠⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⣴⣷⢫⣾⣞⠀⠀⠀⠀⢧⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⡆⡘⡄⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⣏⣹⡖⡄⡶⣾⣧⠇⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣋⢿⣿⠲⠿⠊⢿⡽⣿⣽⠿⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢫⠿⣾⡟⠀⠀⠀⠈⠙⠂⠁⢰⣿⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠻⣣⡀⠀⠀⠀⠀⠀⢠⡟⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠦⡀⠀⡠⣶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣼⣶⣿⣿⣿⠉⢷⣶⣶⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣞⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣹⣮⣿⡏⣿⠿⣿⣿⣧⡀⠀⠀⠀⠀⠀
⠀⠀⣀⣴⣾⢟⡞⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣿⣠⡿⣛⢸⣷⡄⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⣿⠸⣿⣿⣿⣿⡿⠿⣷⠀⠀⠀⠀
⠀⢸⣿⣿⣿⡏⠰⢶⣿⣿⣿⣿⣿⣿⣷⢄⠉⠑⠈⠙⣿⣿⣷⣿⣿⣿⡿⠀⣀⣼⣧⡄⠀⠀
⠀⢸⡿⣿⣿⣧⠀⠙⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠈⢻⠻⣿⠛⠻⢿⣾⣯⣿⢿⣿⣦⠀
⠀⢸⣿⣿⣯⣿⡆⢀⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⡼⠀⣿⠀⣄⠀⠻⣿⣿⣦⢻⣿⠀
⠀⢸⣿⣿⣿⢹⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠓⣼⣿⣿⣿⣦⡀⡈⢷⣝⣇⣿⣇
⠀⢸⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⢹⣿⣿⣿
⠀⢸⣿⡘⣿⣇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣷⣿⣿⡿
⠀⢸⣿⣟⣿⣿⣤⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⠀
⠀⠘⣿⣿⣾⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢻⣧⣽⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠁⠉⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢹⣿⣿⣻⡄⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⡄⠀⠀⠀⠀
⠀⠘⣏⣽⠿⠻⣿⣿⣿⠛⠿⠿⠿⠛⠛⠛⠛⠛⠋⠹⣿⣿⣿⣫⠽⢟⣽⠟⠽⠿⠀⠀⠀⠀
⠀⠀⢷⠃⢀⣤⣿⣿⣿⡀⠉⠲⣄⡀⠀⠐⠒⢶⣦⣤⣿⣿⣿⡷⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡀⣎⣘⣻⣿⣿⠈⠳⢄⡀⠉⠲⣦⣄⣀⡀⠀⠘⢋⡁⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠹⣜⣿⢼⣿⡇⠀⠀⠀⠉⠳⢦⣄⣙⢻⠿⠛⠋⠉⣀⣐⣚⣩⠼⡅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠙⣾⡇⠀⠀⠀⠀⠀⠀⢹⣿⢹⡗⠦⣤⣀⣀⣀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⡌⠛⢮⠉⠉⠛⠻⣯⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢸⠀⢿⣷⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟⠀⢸⣿⠀⠀⠀⠀⠀⠈⣼⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠘⣿⠀⠀⢹⣧⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⠤⢛⡓⢦⡸⡀⠀⠈⣿⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⢲⣭⡼⡇⢁⡇⠀⣰⣿⡿⣌⣻⣭⡭⢿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠠⢂⣴⠗⢻⢀⡼⡟⠈⢻⡄⢭⣍⣩⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠋⠀⠀⣾⣿⠅⠃⠀⠀⢿⣌⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣿⠀⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠒⣿⣾⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠀⢿⠇⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣴⣿⣤⣤⡄⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠹⣿⢏⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣿⣸⣭⢯⣠⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢼⣻⡏⠉⡏⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⢳⡇⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣠⣼⡇⠀⡇⠀⣳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠓⠋⠉⠑⠋⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢾⣿⣿⣿⣿⡿⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢔⣒⠂⣀⣀⣤⣄⣀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⠋⢠⣟⡼⣷⠼⣆⣼⢇⣿⣄⠱⣄
⠀⠀⠀⠀⠀⠀⠹⣿⡀⣆⠙⠢⠐⠉⠉⣴⣾⣽⢟⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠤⢴⣿⠿⢋⣴⡏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⠻⣿⣶⣦⣭⣉⠁⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⠉⠉⠉⠉⠇⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣘⣦⣀⠀⠀⣀⡴⠊⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⣿⣿⣿⣿⠻⣧⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣿⠉⠻⣇⠘⠓⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀
⢶⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠹⠻⢿⣿⣿⠿⠛⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀