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⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠐⠀⡆⠀⠀⢠⠠⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢛⠩⡑⢾⣩⡝⢲⣄⢈⡄⡀⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⠊⠭⣁⠳⣌⣻⣦⢻⣴⡃⢀⠂⡥⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⢌⣁⠂⠄⡀⠄⠀⠄⡁⠒⢨⠙⡜⡻⢾⣿⣻⢽⣆⢳⡘⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢠⠂⣞⡬⢆⢏⡼⠀⠄⠂⠀⠄⡁⠢⠸⢰⢹⡻⢷⡿⣻⣎⢷⡹⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡔⡙⢬⡓⢯⡒⢆⡉⠄⠈⠐⠀⠀⢀⠁⠆⣎⠽⣫⣞⡷⡿⢮⡗⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⣈⠳⣏⠳⣭⠒⡌⠐⠀⠀⠀⠀⠀⡈⠒⡈⢞⡱⣎⢿⡹⢯⡽⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡏⡴⢎⠱⡑⠊⠄⠁⠌⠀⠄⠀⠐⠀⠡⡙⢦⢛⡼⣣⢟⢮⡙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠿⡴⢪⠑⠌⡡⠈⠀⠀⠈⠀⠈⣠⠂⠔⡐⣄⢋⠶⣙⠮⣖⡭⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠡⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠾⠁⠈⠀⢎⠲⡝⣚⠤⢳⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇⠠⠀⠀⡗⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠁⠌⢣⠘⠤⠙⠂⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠼⠀⠀⠀⠀⡐⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠄⠁⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣞⡿⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢀⠀⠠⢀⢈⡇⠀⠀⠀⠀⠀⠄⠁⢂⠁⡐⠀⠄⡀⠄⠀⠀⡀⠀⢀⠰⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣾⣳⢿⣽⣻⣽⣻⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠄⢬⣱⠾⡈⠠⠂⠁⠠⠈⠄⠂⢁⠀⢂⠐⡈⠐⠠⠐⠀⠀⠀⠐⡀⠂⡉⠄⡐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣻⣞⡷⣯⣟⣾⣷⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠣⠘⡈⠂⠭⠚⠀⠄⠂⢁⢀⠢⢜⡡⢂⠌⡐⠠⠄⡉⠐⠈⠀⠀⠀⢀⠂⢡⢂⠐⡀⢀⠂⠐⠀⡁⠂⠠⠀⠀⠀⡐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⣿⣻⢾⣻⢷⣿⣽⣻⢷⣻⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⢠⡅⠂⠡⠀⠁⠐⠀⠀⠐⢀⠢⢘⢦⠱⡈⠤⡁⢃⠒⡀⢂⠀⠄⠀⠀⡀⠂⢄⠂⠂⠄⠂⠠⢃⠆⢄⡁⢂⠡⡀⠄⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣾⣿⣿⡽⣯⡟⣿⣞⣯⣟⣯⢷⣯⣟⡿⣿⣿⣿⣷⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠏⡟⠈⣼⠆⣹⠖⠁⠀⠀⠀⡄⠢⠄⡌⡳⢃⢣⠘⡰⢁⠎⡌⠐⠀⠂⠠⠀⢀⠐⠠⠂⡄⢣⡔⢨⠐⠆⠈⠒⣌⣇⠒⡄⠂⢀⠂⠌⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣻⣵⣻⢧⣟⡾⣽⣞⡿⢾⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠞⠁⠞⠁⠀⠩⠀⡁⠐⠠⡐⠤⡑⢈⢔⠡⢎⠢⡉⠔⡡⠂⠄⠁⠀⠁⠀⠠⠀⠠⠀⠱⣜⠞⠄⠀⠀⠠⠄⠀⠘⠌⠑⠀⠉⠀⠊⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣞⣵⢾⣽⣳⢾⣽⣳⢯⣟⡿⣞⡷⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⣡⣰⡦⢶⢲⢒⣿⠄⠠⣝⠀⡡⢙⣆⠱⡈⠆⡈⠄⡑⠈⠆⡑⠌⢠⠐⠂⠀⠀⠀⠁⠤⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣻⣞⡿⣾⣽⣻⢾⣽⣟⣾⣽⣳⡽⣳⢏⡾⣝⢯⡿⣽⢯⣟⣯⡿⣟⣿⣳⣟⣿⣿⣿⣣⠿⣭⣛⢾⡃⢞⣮⡱⣌⡛⣢⠱⡰⢋⠜⡠⠑⡈⠐⠀⠀⢁⠂⠌⠨⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣄⠉⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⣿⣳⢷⣻⢾⣽⡷⣿⣻⣿⣻⣾⣟⣾⣳⣟⡷⣯⢷⣛⣮⣟⣾⣟⣾⣳⡿⣟⣷⣻⡼⡿⢟⠿⣽⢿⡲⣟⣧⡽⣿⣣⠷⣌⢯⡱⢣⡑⢍⠒⠤⠑⠈⠀⠀⠀⠀⠀⠌⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⣰⣾⢯⡿⣽⣯⡿⣯⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣷⣿⠃⢭⡘⠭⣷⣹⢶⢇⣟⡥⣿⠞⣭⢣⣿⡧⣜⢢⠉⠄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦⡄⣠⣾⣿⣯⢿⣽⣿⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣞⡿⣽⡾⣿⣿⠈⠌⣼⣷⡧⣻⣧⠌⡁⠒⡙⠾⡔⣫⡽⣹⢁⠣⡌⠰⡀⠡⠂⠀⠀⠀⠀⢀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣻⣿⣾⣿⣿⣾⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣻⣽⣿⣘⣬⡝⣿⣽⣿⣟⠤⣁⠂⡘⢂⠜⡠⠑⡀⠈⠀⣍⠑⣈⠡⡄⠀⡀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠽⢭⣛⣿⢟⢯⠲⣅⡚⢤⢣⢘⡔⢡⢂⣳⠄⠐⣦⢀⢲⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣛⡟⡿⢿⡿⣿⣿⢿⠫⡝⣎⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢭⣗⡫⢱⣌⢽⢢⣏⣮⢻⡷⢌⠧⣊⢞⣘⡦⠓⠋⠀⠀⠈⠙⠒⢧⣘⣠⢳⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣷⢫⠞⣵⢋⠶⡱⢌⠣⡝⡼⣍⣞⠲⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠏⢟⠧⠿⢼⠇⣰⡟⡽⣎⣻⡼⢭⣻⡽⢈⠐⢀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣯⠯⡙⢤⠋⢆⠱⣈⢧⡹⡱⢎⡼⣛⠶⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣾⡷⣯⢾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠀⠀⠈⠁⢰⣿⢻⡜⡿⡱⢟⢳⢋⡔⠂⡅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡷⢩⠜⡂⢍⢢⢓⡜⢦⡛⣕⢫⠶⣭⢻⣕⢫⢻⣿⢿⡿⣿⣿⣿⣿⣿⣻⣿⡿⢿⣯⢳⡽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣡⠇⢳⠙⠠⠈⠠⢈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣏⠒⢬⡘⢬⢒⡫⣜⢣⡝⢤⠓⣎⢷⣫⢞⣧⡓⢞⡿⣿⣽⠿⣟⠻⣧⣍⣃⠻⣶⣬⣍⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠇⡈⣍⣻⣷⣤⠁⡆⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡙⢢⢜⢢⠫⠴⣉⠖⡘⢦⡙⡌⠶⣹⢯⣶⣛⣧⡾⡟⣥⠛⣬⢳⣮⢿⣮⣷⢦⣙⢻⡇⡠⢂⠊⠍⢛⠻⠿⣿⣿⣿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠆⠒⠻⣗⠿⡯⠀⠐⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡷⣘⠧⢎⢎⠵⣃⠇⣎⠱⣢⠱⣌⢣⣽⠾⣚⡝⢎⡵⢺⣴⣿⡼⣷⣻⣿⣿⣿⣿⣾⣿⣳⠱⡀⠬⠧⢉⠳⠜⢦⠤⠌⢛⠻⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠮⠄⠐⣭⣿⡆⡐⠄⢁⠂⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡷⣩⢎⡭⢊⠖⣡⠚⡤⢓⣥⡷⢞⣫⠶⣏⠵⣊⠷⣌⠧⣌⢍⡛⡙⠟⡿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣑⣚⠒⠓⡐⠦⣭⠠⣊⢃⠄⣚⢮⣍⣳⡒⣖⣤⣄⣀⡀⠀⠐⡌⣽⣻⡂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣟⡴⣋⠴⣫⠜⡤⢋⣴⡟⡱⢎⠯⣑⠯⣜⢯⣻⡽⣮⣗⢾⣘⠶⣉⢞⡰⣉⠟⣹⢿⣿⣿⣿⣿⣿⣟⡿⣾⢷⣷⢤⣈⠓⠦⣬⢥⣑⣤⠞⣙⠿⣒⣞⣺⣿⣿⣿⣶⣾⣽⡿⠿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⠧⢷⢛⢻⡙⢦⢩⡕⣫⠞⣥⢛⡌⣏⢳⣛⢷⣯⢿⣭⣿⣹⢦⡳⣜⢬⢣⠞⣬⢛⡻⢿⣳⢯⡽⣞⡽⣞⣻⢾⣽⣲⠬⣍⡓⠦⣏⣳⣏⣫⡿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⠏⡜⢢⡉⢦⡑⢎⡱⡘⢥⢛⡴⢫⡜⢦⡳⣌⢞⡸⢋⡟⡳⡟⣯⢷⡹⢎⢧⡚⣤⢣⠜⡈⢿⣣⠿⣜⣯⢟⣽⡻⢾⣽⣻⢷⣯⣝⡲⢦⣼⣡⡟⣿⣿⣿⣿⣿⣿⣿⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⡘⣌⢣⠜⣢⠙⢦⠱⣉⠶⣩⢖⡣⢞⡥⠳⣍⢞⡱⣏⡴⢣⡙⡔⡣⡝⣭⢲⡹⣤⢣⢞⣰⣎⣯⢻⡵⣞⢯⢾⣽⢻⡶⢯⣻⢾⡽⣿⣷⣦⣭⣛⠻⢿⣿⣿⣿⣿⣿⣟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣟⠴⣊⢦⡛⡴⣋⢦⡛⣌⠧⣓⠮⣱⢋⡜⠳⣌⠮⡱⢎⡵⢫⡵⣣⢧⣙⠖⣏⠷⣭⣛⣮⢿⣼⢇⡳⢞⡹⢮⣳⢞⡯⡽⣏⣯⢿⣽⣻⢿⣿⣿⣿⣿⣷⣾⡍⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣏⡞⡱⢎⡵⢣⡝⢦⡝⡜⢮⡱⣋⠶⣭⢞⡳⣎⢷⡹⣎⡞⣧⠿⣵⣻⣞⢿⡞⣿⢷⣿⣾⡿⢣⠜⡌⢧⣋⠷⣹⢮⣝⣳⣛⣾⣻⢾⡽⣯⡿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀