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⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠻⣿⣆⠀⣰⣿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⠉⠉⠙⠛⠒⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠃⢠⣤⠀⠀⠀⠀⢀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠀⠹⠿⠀⣠⠀⠐⣿⡿⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣦⡀⠀⠘⠙⠓⠀⠀⠀⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡽⠿⣷⠒⠶⣶⠶⣶⠛⠉⢻⣟⣓⣶⣄⠀⠀⠀⠀⠀
⠀⠀⠀⢀⡶⣿⣷⣤⡿⠛⢻⣇⣀⣿⡿⠋⠉⠙⢧⠀⠈⠱⡄⠀⠀⠀
⠀⠀⢀⡞⠀⠈⣹⢋⣤⣆⠈⠛⡿⠋⠀⢀⣤⡀⠈⢧⠀⠀⠹⡄⠀⠀
⠀⠀⣸⠁⠀⢀⣧⣯⣾⣿⡆⢀⡇⠀⢠⢯⣿⣿⣆⢸⠀⠀⠀⡇⠀⠀
⠀⠀⢿⠀⠀⢸⠈⠉⠉⠉⢡⣾⣷⡄⠘⠛⠛⠛⠉⢀⠀⠀⣠⡇⠀⠀
⠀⠀⠸⣆⠐⠺⡄⢰⣶⣆⢈⣽⣯⣥⣄⣰⣿⡇⠀⣼⣐⣋⣸⣧⣄⠀
⠀⣾⡻⢿⣦⡀⢳⡈⠻⢿⣿⣿⣿⣿⣿⡿⠋⢠⡾⣿⢫⢻⡵⣶⣟⠀
⢸⣛⣿⣏⡽⣋⢻⣻⠦⣄⣀⣀⡼⠧⣄⣀⡠⣿⢀⣤⠟⢻⣶⣿⠿⠁
⠀⠉⠁⠙⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠳⢭⣟⣾⠿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣭⣤⣀⠀⠈⠈⠉⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⡿⠷⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣩⣵⢾⣟⠷⣦⣄⡉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠉⢸⣷⣆⠀⢈⣹⣯⣀⡻⠿⠇⠀⢙⣿⠀⠹⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣧⣄⣈⣽⣿⡶⢿⣯⣌⠙⠛⠛⠿⠟⠛⠁⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⣠⣼⡿⠿⠟⠛⠿⣿⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣾⠟⠉⠀⠀⠀⠀⠀⣸⠻⣦⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣦⣻⡄⠘⠲⠶⠶⠒⠋⠉⢠⡿⠀⢀⣼⠏⠀⠀⠀⠀⢀⣤⣴⣶⣶⠖⠒⠶⢶⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣙⠿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣞⣋⣠⣴⠟⠃⠀⠀⢀⣴⣿⣿⣷⣿⣿⣿⣿⣿⣶⣦⣌⠛⢦⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣍⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⢀⣶⣿⣿⣿⠿⠛⠉⠁⠀⠀⠈⠙⠻⣿⣷⡈⢻⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠳⣄⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠻⠛⠉⢉⣁⣀⣀⣈⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⣁⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⢶⣄⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⣸⠇
⠀⠀⠀⠀⠀⣠⡾⠏⢁⣴⣾⣿⣿⡿⠿⠟⠋⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠉⢳⣄⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠿⠷⢤⡟⠀
⠀⢀⡴⠚⠛⠉⠛⠛⠿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠻⣆⠀⠀⠀⠀⠀⢠⡜⠋⠀⠀⠀⠀⠀⠀⠹⡄
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢿⡄⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⣠⠇
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⣇⠀⠀⠀⠀⠀⠈⠓⠶⠦⠤⠤⠴⠚⠁⠀
⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠓⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⡅⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣴⣶⣾⣿⣿⣿⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣝⢷⣄⣀⣀⣀⣀⣀⣠⣤⡤⢤⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⣌⡉⠹⣿⠩⣤⣶⣾⣿⣿⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⣿⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⠟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⣿⠟⣻⠛⣏⡿⠟⢁⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠟⢿⣄⣼⣯⣤⣶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠋⠉⠉⠛⢷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢻⡄⠀⢀⣠⣾⠟⠛⠛⠛⠋⠀⠀⣀⣀⣀⣀⣀⠈⠉⠛⠛⠛⠛⠛⠿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢿⣾⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⣯⣉⠉⠉⠙⠛⠶⣤⣀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⡏⠉⠛⢶⣤⣀⡀⢀⣀⣀⠀⠀⠈⠉⠛⠳⠶⠦⣤⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠶⠶⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⡆⢀⣤⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠶⠶⠶⣼⣿⠿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⣦⡀⠀⠀⠀⠀⠙⠷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣇⠘⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠷⢦⣄⠀⠀⠀⠀⠀
⠀⠀⠀⡿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠶⢦⣤⣄⠀⠀⠀⠀⠀⢶⣶⡛⠲⠶⢤⣤⣀⠀⠀⠀⢰⣶⡖⠲⣦⣄⡀⠈⠁⠀⠀⠀⠀
⠀⠀⠀⡇⢦⡀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣀⠀⠀⠀⠉⠛⠶⣦⣤⣈⣉⠳⣦⡄⠀⠈⠙⠓⠾⢿⠷⠀⠀⠀⠀⠀⠀
⠀⠲⠶⠷⠦⣭⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠳⢦⣄⡀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⡆⢿⡏⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⡤⠶⠖⢻⠋⠉
⠒⠚⠛⠳⠾⢿⣏⣁⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣳⠶⠛⠉⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀⢰⡶⣾⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⢂⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡗⠋⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣋⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣦⠀⠀⠐⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⢀
⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣾⣷⡟⣦⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠛⠛⠛⢛⣿⣿⠛⠛⠛⠻⡲⠶⠤⣄⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⠷⠖⠚⠛
⢬⡿⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠛⠛⠁⠈⠳⠶⠖⠚⠛⠉⣿⢞⣧⠀⢀⣀⣀⣀⣙⣿⣛⣀⠀⠀⠀⠀⣗⢛⡟⠉⠛⠲⣄⣀⣀⣀⣤⡤⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⡞⠁⠀⠈⠙⠲⠦⢤⣤⣄⣀⣀⣠⣤⡴⠟⠀⠀⢠⣤⣶⡄⠀⠀⠀⢀⣼⡿⠗⠛⠋⠉⠛⠋⠉⠉⠉⠙⠫⢿⣶⠶⣽⣿⠁⠀⠀⠀⢈⣹⣉⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⢲⣤⣤⠤⢤⣤⣀⠀⠀⡶⢶⡀⠀⠀⠀⠀⠐⢻⣿⡿⠇⠀⢀⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣶⡀⢰⣿⣡⡟⠀⠀⢘⣧⠀⠀⣀⣀⣠⣤⣤⠶⠚⠛
⢰⠟⠁⠀⠀⠀⠀⠈⠉⠛⠳⠾⠧⢴⣶⣶⡶⠶⠾⣍⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⢀⣶⠖⠛⠉⠉⠙⠛⠿⢭⣭⣿⡛⡇⠀⠀
⢸⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠓⢈⣳⣄⠀⠀⢙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤
⠈⠻⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠾⠃⠀⠀⠰⢝⢦⣲⣏⣹⠟⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡅⠀⠀⢀⡾⠋⠛⢷⣶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠙⠒⠒⠲⠒⠒⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⢍⠻⣥⡀⠀⠀⠉⠛⠓⠶⠤⣤⠤⠤⠶⠖⠚⠋⠁⢻⣍⣽⣠⣴⠟⠁⠀⠀⠀⠉⠳⣤⢨⣿⡟⠛⠛⠒⠒⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⣿⣩⡇⠀⠀⠀⠀⢀⣀⣤⠶⠟⠙⢿⡷⢦⣀⠀⠀⠀⠀⠈⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⢉⡟⠉⠉⠉⠙⠓⣶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠒⠚⠋⠉⠻⡄⠀⠀⠀⠈⢷⣼⣿⣿⣦⣄⣀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⣸⠃⠀⣼⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠘⣿⣟⣿⡟⠄⣻⡟⠶⢤⣄⣘⣧⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⢤⠇⠀⣼⠁⠀⠀⠀⠈⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢸⣿⡟⠁⣴⡿⠃⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⣤⣀⢰⠏⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠉⢣⠀⠀⠀⠀⢠⣾⠇⠙⠳⢦⣼⡏⢠⣾⡟⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⢦⣀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⡾⠀⠀⠀⣴⡿⠃⠀⢠⣶⣶⠉⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⠤⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⡾⠁⠀⠀⣸⡟⠀⠀⣀⣾⣿⡿⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠚⠋⠀⠀⠀⢀⣿⣠⡴⠛⠁⠈⠻⣆⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀