⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ㅤㅤㅤ ⠀⠀⠀⠀⠀⠀ㅤ⠀⢀⠀⠀⠀⠀ ⡀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀ ⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⠦⡀⠀⠀⠀⠀ㅤㅤ⠀ ⠀ ⠀⠀⣀⣴⠟⠁⠀⠀ㅤ⠀⠀⢀⣀⠤⡾⠃⠀⠀⣀⡼⠁⠀⠀⣠⣾⣣⣴⡾⡾⠀⡼⠀ ⡾ ㅤㅤ ㅤ ⡼⢹
⠀⠀⠈⡄⠙⡄⠀⠀⠀⢀ ⠀⣰⠃⠀⣠⠞⢻⠃⣠⠴⠒⠛⠉⠉⠉⠀⣠⣮⠤⠴⠒⠋⣟⣁⡤⠶⢿⣿⡿⣿⡿⣟⠀⣠⣾⣦ ㅤ⠀ㅤ⣰⠃⡼
⠀⠀⠀⠱⡀⠘⣆⠀⠀⣸⠀⢰⠇⢀⣼⣏⣢⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⢍⣹⣁⣴⣾⢿⣯⣾⣿ ㅤ ⢀⡜⠁ ⣸⠀
⠀⠀⠀⠀⢳⢰⠸⡄⢠⣿⠀⣿⢠⡿⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠁⢩⣟⢿⠇ㅤ⢠⠎⢀⠔⡟⠁⠀
⠀⠀⠀⠀⠈⣿⠀⢡⣼⣿⣔⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⣠⠃⣠⠏⡼⠋⠀
⠀⠀⠀⠀⠀⣿⡄⠈⣿⣿⣷⣯⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣷⠃⡴⠃⡜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣄⣿⡟⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡜⠀⠀⡜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣯⡿⠇⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠎⠀⠀⡼⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣴⣾⣷⡌⠰⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠞⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⡰⢋⡥⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏⠀⠀⢠⢃⣠⠀⠀⠀⠀⠀⠀
⠀⠀⡵⠋⣀⢤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢡⡏⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀
⠀⣜⡴⢋⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⣿⠃⠀⠀⡾⣣⡴⠃⠀⠀⠀⠀⠀⠀
⢠⡞⢀⠞⢀⣠⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡀⠀⠀⣼⠀⣿⠀⠀⠀⡇⠀⠀⣤⠀⠀⠀⠀⠀⡸⠃⣼⡏⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠀⡞⡴⠋⡼⠁⡄⠀⠀⡔⠀⠀⢀⡆⠀⢀⡾⠀⠀⡆⠀⢰⡇⠀⢠⢿⠀⡏⣆⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⢠⠃⣸⣿⠃⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⡽⠁⣰⠃⣼⠃⠀⣼⠀⠀⢀⣾⠀⠀⡜⡇⠀⣸⠁⠀⡾⡇⠀⠸⠈⠇⡇⠘⡄⠀⣯⣇⠀⢹⢆⠀⠀⢀⡏⢰⣿⣿⠀⠀⣸⣿⠇⠀⣀⡀⠀⠀⠀⠀⠀⠀
⠀⢸⡇⠀⡇⡸⢹⠀⢸⡇⠀⠀⡼⢸⠀⡸⠁⡇⠀⣿⠘⠀⡇⡇⠀⡆⠀⠘⣷⢀⣼⣦⢹⡈⢆⢸⠈⢦⡀⡼⠀⣾⣿⡏⠀ ⢰⣷⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢧⡇⢸⠀⡏⢱⠀⠀⡇⣦⣦⡇⠀⢳⠀⡟⡆⠀⡇⢸⡀⣇⣀⣤⣿⣿⣿⣶⣿⣗⣮⡷⣇⠀⡏⡇⢰⣿⣿⠇ ⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡇⠀⡇⡇⠘⡀⠀⡄⠀⢻⣻⣶⣬⣇⣷⣷⡀⡇⠀⠻⣾⣿⣿⣿⡿⠿⠻⠿⠿⡟⠃⠈⠀⣧⠁⢸⣿⡏⢀⣾⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠁⠀⠘⣷⠀⠱⡀⣷⣶⣿⣿⣿⣿⣿⣿⡃⠱⣧⠀⠀⠈⣱⣿⡟⠀⠀⣀⢀⣴⠇⠆⠀⠀⢹⠀⢠⠏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⣷⣻⡉⣷⠄⠈⠉⠙⣿⡄⠘⠁⠀⠀⠃⠧⠳⠶⠭⠤⠏⠀⠀⠀⠀⠀⠘⣆⣡⣾⡿⠿⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⡷⠾⠷⡒⡲⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡔⠉⠛⢧⠀ㅤ ⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⢹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋ ⣷⡄⠀ ⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⢶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀ ⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠳⣄⠀⠀⠀⠤⠔⢒⣶⡖⠀⠀⠐⠀⠀⠀⠀⣠⠞⠁⠀ ⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀ ⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡵⠦⠤⠤⠤⣤⣔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⡤⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠑⠄⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠹⢄⠀⠀⠠⢄⠈⢣⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠈⠳⣄⡀⠀⠳⢤⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢷⣄⠈⠻⣦⣴⣤⠛⢦⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣿⣦⣄⠻⣿⣿⣦⣙⢦⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠙⢟⣿⢿⣷⣤⡙⢿⣿⣷⣌⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢦⡻⣷⣿⣿⣿⣦⣝⢿⣯⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⠢⠙⢿⡻⣿⣿⣿⣿⣷⣝⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠂⠡⠀⠙⢾⣝⢻⣿⣿⣿⣷



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