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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡾⠿⠿⠟⠛⣛⣛⣻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⠿⠟⠛⠋⠉⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⠿⠿⠛⠛⠉⠁⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡿⠿⠛⠋⠉⠁⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣟⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⠿⢿⡟⠉⠉⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⣉⡀⠀⠀⠈⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⠛⠋⠉⠁⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⣀⣀⣤⣤⣶⠟⠋⠁⢿⣦⡀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠛⠛⠉⣀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠀⣀⣤⣶⡾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠛⠋⢉⣀⣠⣤⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⢀⣀⣀⣴⣾⠿⠛⠉⠀⠀⠀⢠⣆⣤⣴⣶⠄⠀⠀⠀⠀⢹⣧⡀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠟⠛⠉⢁⣀⣭⣵⣶⣿⡿⠟⠛⠉⢩⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⠿⠋⠉⠁⠀⢀⣠⣤⣶⣶⠿⠛⠛⠉⠉⠀⠀⠀⢀⣀⣀⣀⠸⣿⡟⠋⣉⡀⠀⠀⠀⠀⠀⢹⣧⡀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⡿⠿⠛⠉⠁⢀⣀⣤⣴⣾⢿⣿⣿⠉⢻⣿⢇⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⣀⣠⣴⣶⡿⠟⠋⠙⠁⠀⣀⠀⠀⠺⠀⣼⣧⢰⣿⡿⢿⠿⠀⣿⣷⣾⠿⠟⠀⠀⠀⠀⠀⠀⢹⣷⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠟⠋⠉⠁⢀⣀⣤⣤⣶⣿⣿⡟⣿⣯⢻⣮⣿⣿⡞⢿⣿⣿⣧⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠿⠛⠋⠁⠀⠀⢀⠀⢀⣠⣤⡍⣾⣿⣤⡄⢻⣿⡼⣿⣦⣤⣀⣀⡘⣿⣧⣠⣾⣿⠀⠀⠀⠀⠀⠀⢿⣦⠀⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⠿⠛⠉⠀⢀⣠⣤⣶⣿⢿⣿⢻⣯⣈⣻⣿⣅⣿⡿⣷⠿⣏⣿⣥⣶⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⡴⠿⠛⠛⠉⠁⠀⠀⢀⣀⣀⣀⡀⣿⣿⡿⠛⣋⡁⠘⣿⣿⣿⣶⣿⣷⢈⣉⣙⣻⣿⣿⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⡿⠋⠀⣀⣤⣴⣾⣿⣿⣏⢹⣇⣨⣿⡇⠉⢻⡇⠛⣛⣩⣥⣶⣿⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣼⣿⢿⣿⡷⢹⣿⣶⣿⠿⠃⠀⠸⣿⣎⠛⠿⠿⠘⠛⠛⠛⠋⠁⠀⠀⠀⠀⢀⡀⠀⢀⣤⣀⠀⠀⠘⣿⡄⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣰⣿⣿⠃⣿⣿⢿⣿⣿⣦⣸⣏⣹⣿⣿⡾⢇⣛⣠⣴⣿⡿⠿⢛⣉⣥⣶⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢿⣧⣀⣸⣿⡛⠀⠀⠀⠀⢻⣿⡀⣀⣶⣶⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣄⣿⣿⣿⡆⠀⠀⠘⣿⡄⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢿⣿⡿⡀⠸⣿⣾⣿⣄⣿⡏⠛⣛⣯⣶⣿⠿⠟⣛⣭⣵⣶⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠸⣿⡆⠀⠀⠀⠀⠸⣿⣇⠈⣿⣷⣆⠀⣠⣶⡌⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣾⣷⠀⠀⣿⣿⣋⣫⡿⣿⣇⠀⠀⠀⠸⣷⡀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣧⠁⠀⠹⣿⣯⣽⣿⣿⡿⢟⣛⣭⣴⣶⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢻⣿⣂⠀⠀⠀⠀⢻⣿⣤⠈⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣦⣹⣿⡟⣋⣦⡄⠀⠈⠙⠿⣿⣿⣿⠟⠀⠀⠀⠀⢻⣇⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀
⠀⢹⣿⣿⣄⠀⠀⠹⣿⣛⣭⣴⣾⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠈⢿⣿⠀⢀⣠⣦⡈⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⣿⣿⢿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣿⡆⠛⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠸⣿⢿⣿⣀⡀⠉⢻⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣄⡀⠀⠀⠀⠀⠘⣿⡄⠀⠘⣿⣷⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣍⣀⣀⡀⠀⠀⠀⠀⠈⢿⣷⣾⣿⢿⣿⣿⣿⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠘⣿⣇⠀⠀⠀⠀⠀
⠀⠀⣿⣷⣿⣏⣁⣀⡈⠻⢷⣆⢰⣦⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠙⣷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠳⣾⡿⠛⢻⣿⣆⠀⠀⠀⠀⠀⢻⣿⣤⣼⣿⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣽⣿⡄⠀⠀⠀⠀
⠀⠀⠸⣿⣿⣿⣉⣝⡛⣷⡄⢻⣧⢹⣧⠀⣾⠟⠉⣥⣭⣛⣷⣄⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠄⠀⣠⣿⣧⠀⠀⠀⠀⠀⠀⣿⠀⠘⢷⡿⢿⣿⡅⠀⠰⣿⡇⠀⠀⣹⣿⠀⠀⠀⠀⠀⠈⢻⣿⠋⠁⠀⠀⠀⢀⣤⠀⠀⠀⠚⠃⢰⣆⠀⠀⠀⠀⠀⠀⠈⢻⣥⣬⣿⣇⠀⠀⠀⠀
⠀⠀⠀⠹⣿⣿⣧⡝⢿⣎⢿⡄⢻⡇⣿⠸⣿⣸⣇⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⢰⡿⠁⢿⣇⠀⠀⠀⣠⠀⣹⣶⣦⡀⠀⠈⣿⣧⠀⠀⠻⣿⣦⣶⡿⠃⠀⠀⠀⠀⠀⠀⣀⠀⢴⡀⠀⠀⠀⣾⠇⠸⣇⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢹⣿⡆⠀⠀⠀
⠀⠀⠀⠀⢻⣿⣿⣷⣾⣿⠈⠁⢸⡇⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣛⠛⠿⠿⣿⣿⣿⣿⣿⣿⡶⠀⢸⣇⢸⣎⢿⡆⠀⣿⣿⣷⣿⠙⣿⡇⠀⠀⠈⣿⣷⡄⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠈⣿⡄⠀⠈⢻⣿⡀⠙⠻⣿⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡈⣿⣷⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⡿⣁⢀⣤⣾⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠸⣿⣶⣤⣾⣎⡙⢿⣿⣿⣿⣷⡄⠀⢻⣮⡻⣮⣿⡆⠻⢿⣾⣿⣦⣿⣷⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢋⣻⠀⠀⢿⣿⣤⣀⠀⢿⣿⡄⠀⠀⠈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢟⣿⡆⠀⠀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣬⣍⣭⠄⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏⢈⣿⡇⠰⣦⠰⣿⣭⣭⣾⡷⠀⢻⣿⣿⣿⣿⡄⠀⠙⠿⣶⣽⣷⠀⠀⠉⠛⠻⠛⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣷⠀⠈⢿⣿⣿⣷⣮⣿⣷⡆⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⢀⣴⢶⣦⣀⣾⢿⣿⡀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⣠⣾⡟⣿⡟⢡⢶⣿⣯⡉⠁⠀⠀⠈⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣦⢠⡄⣶⣾⣿⣿⣶⣬⢽⣿⣿⣧⣤⣄⡈⢿⣷⣌⠙⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡎⣿⣷⠀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠉⣡⣾⡿⠃⣸⣾⠿⣻⣿⣷⣦⡀⢰⣿⣿⣿⣿⣿⣿⡤⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢠⣼⣿⠀⠀⠻⣿⣇⠀⢹⣿⠃⠈⠙⣿⡟⢹⣿⣿⠛⠻⠏⠈⠻⠟⠁⠀⠀⠀⣀⣀⣠⣴⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠟⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀⠉⠉⠀⢰⣿⣷⣿⣿⡿⠿⠿⢣⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠈⠙⣿⣷⣴⣿⣿⣿⣆⠛⢿⣦⣜⣠⣿⡇⠈⠛⠛⠀⠀⠀⠀⠀⠂⣀⣠⣴⣶⣿⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣄⠀⠀⢸⣿⣿⠟⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠙⣿⣿⡋⠀⢻⣿⣧⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⣀⣤⣶⢾⣿⣯⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⢀⣤⣴⣿⣿⣿⣿⣶⣀⣀⣀⣰⣶⡈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠿⠛⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠿⠛⠉⢀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣫⣿⣿⣿⣿⣿⠇⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⣀⢀⣰⣦⣾⣿⣿⣿⡇⠘⣿⣿⣟⣭⢹⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣿⣿⣿⣿⡟⠿⣿⣏⣩⣦⣿⡍⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣩⡉⢸⣿⣿⣿⣿⣿⣿⠛⢻⣧⣶⣾⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⡞⠻⠿⣿⣯⡁⠀⣾⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⣿⣧⣴⣾⣿⣿⣧⠀⠙⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⠀⠀⠀⠀⠀⠻⡿⠉⠁⠀⠀⢸⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧⣤⣤⣤⣴⣶⣶⣾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
¯\_(ツ)_/¯🤷🤷♂️🤷♀️
⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠭⠛⠩⢍⠉⠉⠉⠉⢉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣶⡶⠒⠀⣱⠀⠀⢞⠁⠠⣤⣈⠲⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠘⠁⢀⠄⠊⠁⠀⠀⠈⠢⡀⠙⢿⡀⣁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⢔⠈⢀⠜⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⣀⡋⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⡎⢾⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄⠀⠀⠀⠀
⡇⠀⠀⠀⣿⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣰⠀⠀⠀⠀
⡇⠀⠀⠀⢰⢃⠘⢇⠈⠒⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⡿⠀⠀⠀⠀
⡇⠀⢀⡠⠐⣋⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀
⣇⣪⡥⡴⠊⠁⡇⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⠋⠔⢁⠀⠀⠀
⠇⠁⠀⠀⠀⠀⣿⠐⢕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⠀⢀⠠⠂⠁⠀⠀⠀
⠀⠀⢀⠀⠀⠠⢿⡆⠈⢎⠀⠀⠀⠀⠀⠀⠀⠀⡠⢫⠃⠀⢰⡅⠀⠀⠀⠀⠀⠀
⠀⠐⠀⠀⠀⠀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⠁⢸⠀⠀⠀⠺⡀⠀⠀⠀⠀⠀
⠁⠀⠀⣀⠠⢤⠀⠀⠣⢄⣀⠀⠀⠀⠀⠀⠸⠁⠀⠆⡇⠀⠀⠀⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⡄⠀⠀⠀⠀
⠀⠀⠀⠀⡜⠀⠀⠀⡠⢤⠀⠀⠀⢀⠀⠀⠀⣿⣦⣀⠀⢣⠀⠀⠀⢠⠀⠀⠀⠀
⠀⠀⠀⡘⠀⠀⡠⠊⠀⢸⠀⠀⠀⡜⣇⠀⢰⢿⠁⠈⠉⠺⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⠁⡠⠊⠀⠀⠀⢸⠀⠀⢠⠁⢺⠀⡆⠈⢧⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢩⠊⠀⠀⠀⠀⠀⢸⠀⢀⠃⠀⢸⡆⣇⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠊⠀⠀⠘⢳⣿⠟⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣠⣴⣶⣏⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀