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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡾⠿⠿⠟⠛⣛⣛⣻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⠿⠟⠛⠋⠉⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⠿⠿⠛⠛⠉⠁⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡿⠿⠛⠋⠉⠁⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣟⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⠿⢿⡟⠉⠉⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⣉⡀⠀⠀⠈⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⠛⠋⠉⠁⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⣀⣀⣤⣤⣶⠟⠋⠁⢿⣦⡀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠛⠛⠉⣀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠀⣀⣤⣶⡾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠛⠋⢉⣀⣠⣤⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⢀⣀⣀⣴⣾⠿⠛⠉⠀⠀⠀⢠⣆⣤⣴⣶⠄⠀⠀⠀⠀⢹⣧⡀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠟⠛⠉⢁⣀⣭⣵⣶⣿⡿⠟⠛⠉⢩⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⠿⠋⠉⠁⠀⢀⣠⣤⣶⣶⠿⠛⠛⠉⠉⠀⠀⠀⢀⣀⣀⣀⠸⣿⡟⠋⣉⡀⠀⠀⠀⠀⠀⢹⣧⡀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⡿⠿⠛⠉⠁⢀⣀⣤⣴⣾⢿⣿⣿⠉⢻⣿⢇⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⣀⣠⣴⣶⡿⠟⠋⠙⠁⠀⣀⠀⠀⠺⠀⣼⣧⢰⣿⡿⢿⠿⠀⣿⣷⣾⠿⠟⠀⠀⠀⠀⠀⠀⢹⣷⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠟⠋⠉⠁⢀⣀⣤⣤⣶⣿⣿⡟⣿⣯⢻⣮⣿⣿⡞⢿⣿⣿⣧⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠿⠛⠋⠁⠀⠀⢀⠀⢀⣠⣤⡍⣾⣿⣤⡄⢻⣿⡼⣿⣦⣤⣀⣀⡘⣿⣧⣠⣾⣿⠀⠀⠀⠀⠀⠀⢿⣦⠀⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⠿⠛⠉⠀⢀⣠⣤⣶⣿⢿⣿⢻⣯⣈⣻⣿⣅⣿⡿⣷⠿⣏⣿⣥⣶⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⡴⠿⠛⠛⠉⠁⠀⠀⢀⣀⣀⣀⡀⣿⣿⡿⠛⣋⡁⠘⣿⣿⣿⣶⣿⣷⢈⣉⣙⣻⣿⣿⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣿⡿⠋⠀⣀⣤⣴⣾⣿⣿⣏⢹⣇⣨⣿⡇⠉⢻⡇⠛⣛⣩⣥⣶⣿⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣼⣿⢿⣿⡷⢹⣿⣶⣿⠿⠃⠀⠸⣿⣎⠛⠿⠿⠘⠛⠛⠛⠋⠁⠀⠀⠀⠀⢀⡀⠀⢀⣤⣀⠀⠀⠘⣿⡄⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣰⣿⣿⠃⣿⣿⢿⣿⣿⣦⣸⣏⣹⣿⣿⡾⢇⣛⣠⣴⣿⡿⠿⢛⣉⣥⣶⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢿⣧⣀⣸⣿⡛⠀⠀⠀⠀⢻⣿⡀⣀⣶⣶⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣄⣿⣿⣿⡆⠀⠀⠘⣿⡄⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢿⣿⡿⡀⠸⣿⣾⣿⣄⣿⡏⠛⣛⣯⣶⣿⠿⠟⣛⣭⣵⣶⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠸⣿⡆⠀⠀⠀⠀⠸⣿⣇⠈⣿⣷⣆⠀⣠⣶⡌⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣾⣷⠀⠀⣿⣿⣋⣫⡿⣿⣇⠀⠀⠀⠸⣷⡀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣧⠁⠀⠹⣿⣯⣽⣿⣿⡿⢟⣛⣭⣴⣶⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢻⣿⣂⠀⠀⠀⠀⢻⣿⣤⠈⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣦⣹⣿⡟⣋⣦⡄⠀⠈⠙⠿⣿⣿⣿⠟⠀⠀⠀⠀⢻⣇⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀
⠀⢹⣿⣿⣄⠀⠀⠹⣿⣛⣭⣴⣾⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠈⢿⣿⠀⢀⣠⣦⡈⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⣿⣿⢿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣿⡆⠛⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠸⣿⢿⣿⣀⡀⠉⢻⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣄⡀⠀⠀⠀⠀⠘⣿⡄⠀⠘⣿⣷⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣍⣀⣀⡀⠀⠀⠀⠀⠈⢿⣷⣾⣿⢿⣿⣿⣿⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠘⣿⣇⠀⠀⠀⠀⠀
⠀⠀⣿⣷⣿⣏⣁⣀⡈⠻⢷⣆⢰⣦⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠙⣷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠳⣾⡿⠛⢻⣿⣆⠀⠀⠀⠀⠀⢻⣿⣤⣼⣿⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣽⣿⡄⠀⠀⠀⠀
⠀⠀⠸⣿⣿⣿⣉⣝⡛⣷⡄⢻⣧⢹⣧⠀⣾⠟⠉⣥⣭⣛⣷⣄⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠄⠀⣠⣿⣧⠀⠀⠀⠀⠀⠀⣿⠀⠘⢷⡿⢿⣿⡅⠀⠰⣿⡇⠀⠀⣹⣿⠀⠀⠀⠀⠀⠈⢻⣿⠋⠁⠀⠀⠀⢀⣤⠀⠀⠀⠚⠃⢰⣆⠀⠀⠀⠀⠀⠀⠈⢻⣥⣬⣿⣇⠀⠀⠀⠀
⠀⠀⠀⠹⣿⣿⣧⡝⢿⣎⢿⡄⢻⡇⣿⠸⣿⣸⣇⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⢰⡿⠁⢿⣇⠀⠀⠀⣠⠀⣹⣶⣦⡀⠀⠈⣿⣧⠀⠀⠻⣿⣦⣶⡿⠃⠀⠀⠀⠀⠀⠀⣀⠀⢴⡀⠀⠀⠀⣾⠇⠸⣇⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢹⣿⡆⠀⠀⠀
⠀⠀⠀⠀⢻⣿⣿⣷⣾⣿⠈⠁⢸⡇⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣛⠛⠿⠿⣿⣿⣿⣿⣿⣿⡶⠀⢸⣇⢸⣎⢿⡆⠀⣿⣿⣷⣿⠙⣿⡇⠀⠀⠈⣿⣷⡄⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠈⣿⡄⠀⠈⢻⣿⡀⠙⠻⣿⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡈⣿⣷⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⡿⣁⢀⣤⣾⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠸⣿⣶⣤⣾⣎⡙⢿⣿⣿⣿⣷⡄⠀⢻⣮⡻⣮⣿⡆⠻⢿⣾⣿⣦⣿⣷⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢋⣻⠀⠀⢿⣿⣤⣀⠀⢿⣿⡄⠀⠀⠈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢟⣿⡆⠀⠀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣬⣍⣭⠄⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏⢈⣿⡇⠰⣦⠰⣿⣭⣭⣾⡷⠀⢻⣿⣿⣿⣿⡄⠀⠙⠿⣶⣽⣷⠀⠀⠉⠛⠻⠛⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣷⠀⠈⢿⣿⣿⣷⣮⣿⣷⡆⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⢀⣴⢶⣦⣀⣾⢿⣿⡀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⣠⣾⡟⣿⡟⢡⢶⣿⣯⡉⠁⠀⠀⠈⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣦⢠⡄⣶⣾⣿⣿⣶⣬⢽⣿⣿⣧⣤⣄⡈⢿⣷⣌⠙⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡎⣿⣷⠀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠉⣡⣾⡿⠃⣸⣾⠿⣻⣿⣷⣦⡀⢰⣿⣿⣿⣿⣿⣿⡤⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢠⣼⣿⠀⠀⠻⣿⣇⠀⢹⣿⠃⠈⠙⣿⡟⢹⣿⣿⠛⠻⠏⠈⠻⠟⠁⠀⠀⠀⣀⣀⣠⣴⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠟⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀⠉⠉⠀⢰⣿⣷⣿⣿⡿⠿⠿⢣⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠈⠙⣿⣷⣴⣿⣿⣿⣆⠛⢿⣦⣜⣠⣿⡇⠈⠛⠛⠀⠀⠀⠀⠀⠂⣀⣠⣴⣶⣿⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣄⠀⠀⢸⣿⣿⠟⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠙⣿⣿⡋⠀⢻⣿⣧⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⣀⣤⣶⢾⣿⣯⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⢀⣤⣴⣿⣿⣿⣿⣶⣀⣀⣀⣰⣶⡈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠿⠛⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠿⠛⠉⢀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣫⣿⣿⣿⣿⣿⠇⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⣀⢀⣰⣦⣾⣿⣿⣿⡇⠘⣿⣿⣟⣭⢹⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣿⣿⣿⣿⡟⠿⣿⣏⣩⣦⣿⡍⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣩⡉⢸⣿⣿⣿⣿⣿⣿⠛⢻⣧⣶⣾⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⡞⠻⠿⣿⣯⡁⠀⣾⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⣿⣧⣴⣾⣿⣿⣧⠀⠙⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⠀⠀⠀⠀⠀⠻⡿⠉⠁⠀⠀⢸⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧⣤⣤⣤⣴⣶⣶⣾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣩⣭⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣀⠀⣀⡈⠙⢢⡀⠀⠒⠒⠂⣀⡤⠖⠒⠒⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢰⣿⣿⡿⠛⠃⠀⠀⣹⠀⠀⠀⠸⡇⠀⠀⢰⣶⣤⣀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠏⠉⠁⠀⢀⡠⠞⠋⠀⠀⠀⠀⠻⣄⠀⠀⠨⢿⣿⡷⠄⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠿⡀⠈⠀⢀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠹⣷⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⠝⠋⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⢠⢏⣴⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠙⠢⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⢀⡏⡞⢸⡄⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠸⡇⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢳⡄⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⡇⣷⠈⠉⢷⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡧⣾⠃⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⢸⡘⡆⠀⠙⣆⠀⠈⠑⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⢰⣿⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⣳⣳⠀⠀⠘⣆⠀⠀⠀⠀⠉⠙⠛⠿⠶⢤⣤⠤⠤⠤⠤⠶⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⢀⣠⠴⠚⢉⣡⣡⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠇⢀⡄⠀⠀⠀⠀⠀
⣇⡤⠖⣉⣀⣠⣴⠞⠋⠁⣷⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⠗⠁⣠⠟⠀⠀⠀⠀⠀⠀
⡧⣚⠿⠛⠉⠊⠁⠀⠀⠀⣿⡄⢈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠂⠁⠀⠈⠁⣀⠔⠁⠀⠀⠀⠀
⡏⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠳⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⡀⢀⠀⠰⢾⣿⡀⠀⠙⣆⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⠇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⡠⠊⠁⠀⠀⠘⠈⣿⣧⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡇⠀⠀⠀⠸⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠘⡆⠀⠀⠀⠀⠘⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⢄⠀⣧⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⢀⣀⡤⠤⠴⢲⠃⠀⠀⠈⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠈⢧⢹⡄⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠈⠉⠉⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠒⠒⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠀⢷⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣠⠴⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣷⣤⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⠞⠁⠀⡇⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⣾⡟⠛⠉⠙⠳⢦⣄⣧⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⣼⠃⠀⠀⢀⡴⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡟⠸⡇⠀⠀⢸⠛⣧⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⡇⠀⠀⠀⣸⠃⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣼⠁⢰⢿⠀⠀⡏⠀⠘⣷⡄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠸⡏⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⣰⠃⡀⡀⢸⡇⠀⡇⠀⠀⣸⣷⡀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⢰⠃⠀⠀⠀⢸⡇⢸⢃⣤⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠏⠀⠀⠀⠀⢹⡇⣼⣿⣿⡿⠉⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣰⠋⠀⠀⠀⠀⠀⠈⢡⣿⡿⠟⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢘⣿⠧⠀⠀⠀⠀⠀⠀⠒⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢀⣸⣥⡤⠶⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣭⣥⣽⣶⣿⣿⣷⣦⣤⣤⣤⣤⣤⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀