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⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⡠⠤⣀⣐⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⢤⡀⠤⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣾⠉⠀⠉⢻⡀⠀⠀⠀⠀⣠⣊⠀⠠⠤⠀⢀⣀⣀⣠⣤⡬⠵⠶⠖⠚⠛⢿⡀⡯⠀⢀⠀⠨⣐⠨⡢⡀⠀⣠⣮⠶⡻
⠀⠀⠀⠀⣿⠀⠀⢰⣟⣷⡔⢀⣭⡥⠵⠶⠒⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡅⠐⠂⢅⢢⣡⡴⡾⠋⠉⠁⠁
⠀⠀⠀⠀⠸⡆⢀⠸⠄⢿⢇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠨⡜⠡⠼⠞⢋⠁⠁⠐
⠀⠀⠀⡖⠈⠛⠀⠀⠀⠀⠘⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣟⠢⠐⡡⢢⢀⠆⡄⠄
⡔⠋⠉⢗⣄⠀⠀⠀⠀⠀⠀⠇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣹⠂⠐⣀⠥⢊⢎⠌
⠳⡦⣠⣶⠛⠻⣶⣄⠀⠀⠀⢰⣸⡆⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⠙⣧⢤⠄⡪⠔⠏
⠀⠀⠀⢿⡇⠀⠀⠙⣷⡄⠀⠈⡀⢇⠀⣌⣤⣤⣀⡀⠈⠐⠒⠠⠤⣤⣤⠴⠶⠖⠚⢳⡀⠀⠀⠹⡌⠀⠠⠾⢦⡀
⠀⠀⠀⠘⣷⠀⠀⠀⠈⣿⣄⠀⠃⠸⢤⣿⡇⠉⠉⠙⠛⠷⣶⣦⣤⣄⣀⡈⠀⠒⠠⠶⢧⠀⣀⠀⠹⡀⠀⢠⡳
⠀⠀⠀⠀⣿⡶⢶⣶⣶⣿⣿⣬⣷⣤⣼⣿⣀⣀⣀⣠⠒⠁⣀⡬⠆⠉⠉⠛⠛⠷⠶⣦⣤⣀⣀⠈⠉⢣⠀⢁⠃
⠀⠀⠀⠀⢸⣧⢀⣿⣄⣻⣯⠈⣿⣏⡙⣿⠋⢻⣿⡻⢿⡿⠻⣿⣿⣶⣶⣶⣶⣦⣤⣤⣿⣯⣤⣈⣱⣈⣌⡀
⠀⠀⠀⠀⠀⣿⡝⣿⡀⢸⣿⠉⣻⡿⠟⣿⡏⢻⣿⠖⢺⡏⠀⣿⡀⠀⣿⠁⢸⡇⠀⣿⣿⢻⣿⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⢹⣇⣿⡇⣛⣿⠋⢻⡟⠚⣿⡇⢻⣿⠛⢻⡟⠙⣿⡇⢸⣿⠃⢸⡇⢀⣿⣿⣾⣿⡌⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⡟⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡷⢾⣿⠶⣾⣷⣾⣿⣷⣿⣿⣷⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⣿⡄⢸⣿⠀⢿⡇⢸⣿⠀⢸⡇⠀⣿⣀⢸⡷⠀⣿⣇⣿⣿⣿⣿⣿⠂⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣿⡇⢘⣿⠀⣸⡇⢸⣿⣀⣸⣇⣀⣿⣛⣻⣿⣶⣿⣿⣿⣿⣿⣿⣧⣤⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣿⡿⠻⣿⠟⢿⡟⢻⣿⠛⢻⡟⠛⣿⠿⢿⣿⣿⣿⣏⣹⣿⠛⣿⡏⢹⡿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⣇⣀⣿⣀⣸⣇⢸⣿⡀⣸⣇⣀⣿⡀⣸⣇⣈⣿⣛⣻⣿⣇⣿⣤⣼⣧⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⣿⣶⣿⠷⠾⠿⠿⢿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢺⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡇⡍⠙⠉⡀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣈⣪⣭⣥⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣭⣭⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣼⣿⣷⣶⣶⣤⣄
⠈⠉⠉⠛⠛⠿⠛⠿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⣶⣶⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡏⠉⠉⠛⢻⡟⠻⠿⠷⣶⣶⣦⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢹⡏⠉⠙⠛⢻⡟⠿⠿⠶⣶⣶⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠛⢾⡷⠶⠶⣤⣾⣧⣤⣄⣀⣿⣀⣀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠈⢹⡏⠉⠛⠛⣿⠿⠿⠿⢶⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠈⠉⣿⠉⠉⠛⢻⡿⠻⠶⠶⢾⡷⣤⣤⣤⣿⣄⣀⣀⣸⣇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣸⣇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠉⠉⠉⢹⡟⠛⠛⠛⣿⡷⠶⠶⢾⣷⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠉⢹⡟⠙⠛⠛⢻⡟⠓⠶⠶⣿⠶⠶⢤⣼⣧⣤⣤⣄⣼⣇⣀⣀⣀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠘⣷⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢹⡏⠉⠉⠉⣿⠛⠛⠛⢻⡿⠶⠶⠶⢾⡷⣦⣤⣤⣿⣤⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣼⣧⣀⣀⣀⣸⣇⣀⣀⣀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢸⡏⠈⠉⠉⢹⡏⠉⠉⠙⣿⠛⠛⠛⢻⡿⠳⠶⠶⢾⡷⠶⠶⠶⣿⣤⣤⣤⣼⣧⣤⣤⣀⣼⣇⣀⣀⣀⣿⡀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢹⡏⠉⠉⠉⣿⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢾⣷⠤⢤⣤⣾⣧⣤⣤⣤⣿⣤⣤⣤⣼⣧⣤⣤⣤⣼⣇⣀⣀⣀⣿⣀⣀⣀⣸⣇⣀⣀⣀⣸⣇⣀⡀⢀⣿⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡏⠀⠀⠀⢹⡏⠉⠉⠉⣿⠉⠉⠉⢹⡏⠉⠉⠉⢹⡏⠉⠉⠙⣿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⣻⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣽⣿⣯⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢉⣿⣿⡍⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠻⠿⠟⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡘⠿⠿⠃⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠾⠿⠿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣀⣀⡈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣤⣄⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣭⣧⣯⣿⠞⠉⠻⣿⣿⣿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣨⣿⠀⠀⢺⣇⣨⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣧⠐⠶⠂⣼⣿⣏⠸⠿⠂⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⢀⣸⣿⣿⡇⡀⢀⠀⠀⠀⡀⣼⣿⣿⡀⡀⢀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠉⠉⠉⠉⠉⠙⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⣿⠉⠉⠉⠉⠉⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⣿⣿⣿⣀⣀⣀⣀⣀⣠⣿⣿⣏⣀⣀⣀⣀⣠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠉⠀⠀⠀⠀⢠⠒⢄⠀⠀⠀⠀⠀⢀⠞⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢠⠞⣿⣶⣶⡊⢏⣾⣀⣟⠀⠢⠀⠀⢲⠀⢀⣶⣿⣷⣌⠁⠀⠀⠀⠸⠀⠈⡆⠀⣠⢤⣤⣃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠓⣮⠟⠛⠷⣦⣽⣟⠋⠹⣧⡀⠀⠀⣦⡜⠉⠉⠉⠉⢉⡟⢻⡄⠀⢇⠀⡸⠞⢈⢹⠏⣾⡖⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣌⣓⣳⣘⣿⣆⣀⣙⣓⣠⡶⠟⡇⠀⠀⠀⠀⠀⠁⠈⣇⣰⠛⠚⠛⠛⠒⠛⠒⠓⠇⠾⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡄⠀⣭⣭⣭⣭⡍⣭⣉⣉⣉⣉⣉⣙⡓⡒⢒⠒⠒⠛⠛⠒⠒⠠⠶⠧⠤⠤⠤⠤⠰⠿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢫⠀⢻⣿⡞⠻⣇⠀⠸⡇⠀⠘⡇⠀⢸⣇⣉⢹⡿⠍⠀⡟⡻⣿⡟⠀⠀⣶⢲⣶⣶⠒⠲⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣂⣀⡿⣦⣀⡟⣀⣀⡹⣦⣀⣛⣀⣉⣅⣀⣘⡀⣀⣀⣳⣿⣿⣧⡄⣀⣇⣸⣿⣇⣀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠈⢇⢹⡄⠀⠹⡀⠀⠀⠈⢄⢠⡉⡁⠀⠌⠀⠀⡄⢁⣶⡻⣿⡾⢯⢭⣿⠿⢿⣷⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⢇⡗⣥⡤⢧⠀⠐⠳⢋⣷⠡⣈⣟⢲⡀⣠⢩⠤⠼⡦⢼⡇⠴⠖⡿⠒⠦⡿⡖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠙⡿⠐⠋⠁⠊⠉⠁⢍⠈⠛⠀⢸⡞⣇⠀⣤⣠⢤⡀⠈⣧⡀⢀⣇⣀⣸⣧⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠸⣿⢷⠀⠈⠀⢀⡜⣓⣺⢺⡧⠼⠡⠦⢝⣿⠚⣖⠊⣷⣿⣏⣧⠀⠸⠷⢃⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⢻⡯⡓⠈⣘⠹⣁⡁⠀⣿⣇⠀⠧⠄⠘⣛⣂⣫⣥⠭⠓⠒⠒⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣧⠟⣒⣛⣻⣭⣭⣴⣶⣶⠾⠿⠏⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⣿⠟⠛⣋⢉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡼⠁⠘⠁⠀⠀⠀⠁⠀⢰⣾⡆⣶⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠞⠀⠀⡀⡀⢤⣮⡗⠂⠀⣤⣿⢛⡿⠋⢸⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠊⠀⠀⠀⡇⡇⠀⠀⡂⠀⠀⣿⣗⠸⠃⡀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣽⣿⣿⣿⣿⣿⣾⣿⣿⣯⣽⣭⣯⣭⣭⣭⣭⣭⣭⣽⣽⣯⣯⣏⣉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠏⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣲⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⢻⣿⡁⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⠉⠉⠉⠉⠉⠉⣿⣯⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⢹⣿⠏⠉⠉⠉⠉⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠉⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠉⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉





⠀⠀⠀⠁
⠀⠑⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠀⢠⡀⠀⡤⠀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⡃⠀⡖⠀⢫⠀⠀⢽⠂⠠⣅⠀⠄⣛⡁⢸⠄⠀⣖⠀⠀⡤⠂⢠⡤⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⣸⠆⠸⡀⠀⢸⠀⠀⡟⠀⠀⠁⠀⢸⠁⠀⡇⠀⠉⡆⠁⠀⡗⠀⢿⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢹⠀⠳⡇⠀⢰⠆⠀⡇⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⡈⡀⠘⡄⠀⢸⡀⠀⢸⠀⠀⣄⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣃⠐⣇⠀⢣⡀⠈⡄⠀⢸⠀⠀⣷⠀⢸⠀⠀⡇⠀⠀⡇⠀⢀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⡶⠂⠖⡂⢲⣒⠐⡖⠐⢺⠒⠀⡶⠂⢺⠒⠐⡖⠀⠐⡖⠀⢒⡖⠐⡖⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡃⠀⡏⠀⢣⠀⠘⡄⠀⡇⠀⢸⠀⠀⡇⠀⠀⡇⠀⢈⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡆⠀⡇⠀⠃⠀⣸⠀⠀⡇⠀⠃⠀⢸⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡀⠼⠮⣿⠀⢘⡀⢐⡃⠀⠦⠄⢸⠑⢀⣏⠀⠐⡗⠀⢹⠉⡁⡇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢙⡭⡿⠖⠛⠲⣆⣥⠶⠥⠄⠭⠅⠸⠄⠐⠒⠀⠰⠯⠭⠜⠂⠐⠓⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠧⠤⠤⠤⠴⢏⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠂⡰⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠠⢘⣖⣿⡻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣏⠽⠏⠉⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⣆⣠⣬⣮⣷⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣲⣮⣴⣄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁

⣟⠉⣹⣛⣛⣛⣛⣛⣟⣛⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⢀⠄⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡜⠈⠀⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⢠⠠⡇⠀⠀⠆
⠀⠀⠀⢀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢣⠀⠀⠓⠂
⠀⠀⠘⢀⠰⠀⡔⠀⠀⠀⠀⠀⠂⠀⢤⠀⠀⠀⠀⠀⠀⣀⠀⠀⣁⣀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠐⠓⠃⠀⠑
⠀⠀⢢⡀⠀⠀⠀⠘⡂⠀⠀⢀⢠⢄⣘⠂⠀⢀⡀⠠⣀⠀⠀⠁⠃⠐⠀⢢⠄⠀⠁⠨⡐⠀⠀⠀⠀⢀⠀⠘
⣁⡀⠈⣁⡀⠤⠤⠤⠤⠄⠴⠁⠄⠀⡔⠚⠛⠓⠓⠆⠙⡖⠀⠚⠒⠒⠁⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠃
⢀⢀⠀⢐⠁⡒⠂⡀⣀⠀⡇⠀⡀⢀⡘⢀⡀⣀⢀⣠⣀⠷⢀⠀⣀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠈⡄⡀⡈⠀⠁⠀⠁⡦⢐⡁⠀⠀⣀⢰⠂⠲⠠⠐⠀⠠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⢰⠠⠙⡄⠀⡀⢰⡃⣄⢰⠀⡀⢻⡈⠃⠮⡀⢀⣤⠐⢂⢠⠁⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸
⠀⠀⡄⡀⢡⠀⠀⠀⢧⡀⠀⠀⢸⠐⠀⠀⡠⠁⠈⠈⣆⠈⠀⠩⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢨⣉⠈⠁⠘⠀⠀⠘⠉⠂⠒⠓⠒⠰⠀⣄⠄⠼⠻⠄⢪⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠉⠉⠉⠓⠓⠓⠒⠢⠦⠤⠄⢀⠸⠦⣹⣀⡘⢄⠸⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠉⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣆⡀⠀⠈⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠺⠀⠤⠄⠗⠀⠀⠀⠀⠈⠀⠒⠒⠂⠀⠀⠀⠀⠒⠂⠀⠈⠣⡀
⠀⠀⠀⠀⠀⢤⡈⠣⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠳⠱⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁
⠀⠀⠀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣁⣀⣀⣂⣁⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠉⢳⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠈⠙⣦⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠉⢹⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢳⡀⣃⢘⡀⢸⠀⡆⣸⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⢸⣆⣒⣐⣆⣒⣰⣀⡏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣭⣥⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣯⣭⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠸⣭⣥⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⠸⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣻⣷⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⢖⣫⣴⠟⠋⠁⠀⠘⣷⡀⠀⠀⣼⣷⣶⣶⣶⠦⠤⠴⢶⠟⠙⣛
⠀⠀⠀⣠⡴⣚⣭⡶⠟⠉⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⣿⣏⠀⠀⠀⣀⡴⠚⢁⣤⠀⣿⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠄
⣠⣶⣛⣵⡾⠛⠁⠀⠀⠀⠀⠀⣀⣀⣠⣴⣛⣻⣿⣛⣿⣿⡀⢴⠞⠁⢀⣴⣿⠃⠀⢿⡇⠀⠀⠀⠀⠀⢀⡜⠀⢰⠋
⠛⠟⠛⠻⣶⡀⠀⠀⠀⠀⢀⣴⡿⢿⣿⣿⣿⠏⠉⠉⠉⣿⣿⡀⠀⢀⣿⣿⡇⠀⢀⣾⣷⠶⠶⢦⣤⣴⠟⡡⠀⢸⣧
⠀⠀⠀⠀⢹⣧⠀⣀⣤⡞⠋⢹⣷⢸⣿⣿⣿⠀⠀⣠⣄⢹⣿⣇⢰⣿⣿⣿⣷⠀⠼⢿⣿⣤⣀⠀⢸⡟⢸⡇⠀⣿⣿⠓⣲
⠀⠀⠀⠀⠀⢻⣿⠿⣶⣷⣤⣌⣻⣂⠿⠿⠟⠀⣼⣿⣿⣋⢉⣻⡼⣿⣿⣟⣱⡾⠿⠮⠉⠉⠨⣶⣮⡃⣾⠀⠠⣿⣿⠞⢻
⠀⠀⠀⠀⠀⠈⢿⣆⠀⣿⣯⠉⢹⡟⣛⠛⣷⠶⢶⣦⣯⣭⣉⣉⣀⠻⠿⠋⢿⣿⡿⠛⢳⣴⣶⣦⡈⠓⢟⣤⡾⡟⢹⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⠘⣆⢸⡇⣿⡇⢿⣴⡟⠀⣿⣭⢉⡙⣿⠛⠻⠷⣶⣿⣤⣤⣭⣁⣀⣙⣡⣶⡟⢻⠀⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢹⣼⡏⣿⣇⢸⡟⠁⠀⣿⢿⣿⡇⣿⢸⡟⠀⣿⢠⣤⢸⡏⠉⠙⣿⠋⢸⡇⣿⠀⡇⣸⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠻⡇⢹⣿⣾⡇⢀⡄⣿⠘⣻⡇⣿⢘⣿⡆⣿⢘⡏⢸⡇⣀⡀⣿⠀⢸⡄⣿⠰⣧⣾⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠙⠛⣿⠾⠿⢼⣧⣜⣃⣹⡈⢿⡇⣿⢸⣿⡇⣿⣼⣥⣼⡇⢻⡇⣿⠀⣼⠁⣿⢺⡇⣼⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣿⠀⣷⢸⡇⢀⣉⢹⡟⣛⡓⣿⠶⣯⣥⣿⣸⣿⣟⡇⠈⠀⣿⢀⣿⣟⣿⣿⡇⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⢠⣿⣾⡇⣿⣿⢼⠇⣿⣿⢻⢰⣶⡄⣿⢉⡉⠙⡿⠛⠶⣿⢋⣿⣿⣿⢻⡇⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⣿⡿⡇⣩⣥⢸⡄⣿⣿⢸⠀⣿⣿⣿⢸⣟⡃⣯⣺⡇⣿⣿⣿⡏⡟⢸⣥⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⣛⣃⣇⠈⠙⢸⡇⠿⠋⢸⡇⣿⡇⣿⢸⣿⡇⣿⠹⠇⣿⠉⣿⠀⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡛⠛⠛⠻⠿⠿⢷⣶⣦⣼⣧⣟⣋⣿⠀⠀⠀⣿⠀⠀⣿⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⣶⣶⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⡿⠷⢶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⠀⣠⠞⠉⠉⠛⠛⣿⠷⢶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⠟⠻⠿⠟⠁⣠⠞⠁⠀⠀⢀⣤⠞⠁⠀⠀⠀⣨⠟⠙⠛⠻⠿⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⣿⣿⠟⠁⠀⠀⠀⣠⠞⠁⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣴⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡟⠻⠷⠶⣶⣴⣿⣥⣀⣀⣀⣴⠟⠁⠀⠀⠀⣴⠞⠁⠀⠀⠀⢀⣴⣿⠟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣿⣿⡷⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠶⢶⣶⣿⣥⣄⣀⣀⣀⣴⣿⠟⠁⠀⠘⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣛⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠿⠿⠛⠋⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠙⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠁⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⣠⣼⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣦⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⠓⠄⠀⠂⠸⢉⣡⠆
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡀⠀⠀⣶⣶⣶⠈⠀⠀⠀⠃⠀⠉⠀⠀⠜⠀⠀⠐⠀⠻⠿⣶⣤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠿⠿⠛⠃⠀⣀⠋⠉⠉⢰⡦⠰⣦⡇⠀⢢⠐⠀⡄⠀⠀⠀⠃⠀⠀⠀⡀⢀⣀⣀⣴⣤⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡁⠀⠀⠀⠈⠓⣏⣼⢁⠀⠈⠀⠀⣷⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣞⡉⣻⠋⠀⠀⠀⠀⠟⢥⡼⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣰⡇
⠀⠀⠀⠀⠀⠀⢰⠘⠀⠈⢡⣾⠿⠉⢀⣤⡀⠐⠀⠀⠀⠀⠀⠶⠀⠐⠒⠚⢉⣉⣉⣩⠇
⠀⠀⠀⠀⠀⠀⠘⠄⠤⠤⠴⠛⠛⢚⣋⣀⣁⣩⠤⠤⢤⡖⠒⠒⢺⠉⠉⢹⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠠⠴⠶⠒⠒⣾⢉⢉⣹⣇⣀⣶⣿⣤⣤⣼⣧⣤⡶⣿⠶⠶⢿⡗⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡟⣻⣿⡟⠏⢉⣏⠙⢷⣿⠃⠀⣾⡇⣉⣀⣼⣀⣀⣸⣇⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⡷⢶⣿⣿⠚⣿⣿⡛⠛⢙⡟⠋⠉⣻⠉⠉⢹⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⣿⣵⣧⣿⣼⡿⢿⣦⣤⣴⣷⠶⠶⣿⠶⠶⢾⠷⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣿⣯⣭⣼⣄⣤⣼⣥⣤⣬⣯⣤⣭⣿⣤⣤⣼⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠀⢸⣧⣄⣸⣅⣠⣽⣇⣀⣀⣿⣀⣀⣿⣀⣀⣼⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⣤⠤⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢸⡆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣽⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋

⠻⠿⠿⠿⠿⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣤⣤⣤⣤⣽⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡉⠉⠉⠉⠉⣿⡍⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⢩⣿⠉⠉⠉⠉⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀⣀⣀⣹⣧⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣼⣏⣀⣀⣀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠛⠛⠛⠛⣿⡟⠛⠛⠛⢻⡟⠛⠛⠛⢻⣿⠛⠛⠛⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⢸⣷⠀⠀⠀⢸⡇⠀⠀⠀⣾⡏⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠻⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠻⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃

⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠏⠉⠉⢹⠉⠉⠙⡍⠉⠉⣿⠉⠉⠉⡏⠉⠉⡏⠉⠉⢹⠋⠉⠉⡏⠉⠉⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠒⠒⢺⡓⠒⠚⡗⠒⠒⢻⠒⠒⠚⡗⠒⠒⡟⠒⠒⢺⠓⠒⢺⠗⠒⢺⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡦⠤⠬⡧⠤⠤⣧⠤⠤⢼⠤⠤⢤⡧⠤⢤⡧⠤⠤⣼⠤⠤⣼⠤⠤⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⣀⣀⣇⣀⣀⣷⣀⣀⣸⣀⣀⣀⣇⣀⣀⣇⣀⣀⣏⣀⣀⣞⣀⣰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡍⠉⢹⠉⠉⢹⠉⠉⢹⠉⠉⠉⡏⠉⢉⡏⠉⠉⡏⠉⠉⡏⠉⡍⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠒⢺⡖⠒⢺⠒⠒⢺⠓⠒⠒⡗⠒⢺⠗⠒⢺⡗⠒⢺⠗⢲⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣂⣘⣃⣀⣘⣂⣀⣘⣂⣀⣐⣃⣀⣘⣂⣀⣘⣃⣀⣘⣂⣊⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠧⢤⣴⣶⣶⣦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣶⣶⣦⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢺⣿⣏⣹⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣩⣿⡗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣠⣤⣤⣀⣤⣤⣤⣤⣄⣠⣄⣀⣤⣤⣤⣤⣤⣤⣤⣾⣿⣷⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⣤⣤⣄⡀⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣀⡇⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠠⠄⠀⠄⣿⠤⠠⠤⠤⠄⠄⠠⣿⣿⣿⡇⠤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⠀⣶⣶⣶⣶⣶⡆⢾⣿⣿⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⡇⣸⣿⣿⡇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣀⣀⠀⣿⢀⣀⢀⡀⣀⣀⢀⣽⣿⣿⣇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠁⠂⠀⠀⠀⣿⣶⣤⠀⠘⠓⠰⠆⣹⣿⣿⡇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⡇⠈⠉⠁⠀⠈⠁⠀⡇⣀⠀⠀⣀⠀⠀⠀⣿⠹⠋⠀⣠⣤⣶⣦⣬⡛⠿⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠈⠈⡇⠈⠈⠉⠈⠀⠈⠈⡏⠀⠈⠁⠉⠉⠀⠀⣿⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠻⣿⣿⣿⡿⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⣋⣴⣾⠿⠟⠉⢻⣷⡀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣩⣴⡾⠟⠉⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⢸⣿⣿⡿⠿⠿⠟⠛⠛⠛⢛⣩⠟⠁⢈⣥⡄
⠀⠀⠀⠀⣀⣤⠶⢛⣥⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣤⠞⠋⠀⣠⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⢄
⠀⣠⣴⠾⣋⣵⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⣿⣿⠷⢾⣿⣿⠀⢀⣤⡾⠋⠀⠀⣠⣾⡟⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢰⡞⠁
⢸⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⠿⠷⠿⢿⣿⣶⡌⠉⠀⠀⣰⣿⣿⡟⠀⠀⠀⣿⣿⣠⣀⣀⡀⠀⠀⣠⣶⠟⠀⠀⢾⣇
⠀⠉⠉⠀⠈⢹⣿⡄⠀⠀⠀⠀⢀⣠⣴⣿⣟⠹⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⣠⣿⣿⣿⡇⠀⠀⣸⣿⣿⡍⠉⠛⠛⠛⣿⣿⠃⣼⠀⠀⣾⣿⣆⡀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⢀⣤⣶⡟⠋⠀⢸⣿⡄⣿⣿⣿⣿⡇⠀⠀⣠⣤⡄⢻⣿⣿⡄⢸⣿⣿⣿⣿⣿⠄⠐⠿⢿⣻⣥⣤⣀⠀⢠⣿⡏⢸⡟⠀⠀⣿⣿⡏⢉⣹
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣤⣿⣄⣀⣘⠿⠇⠸⣿⣿⣿⠁⠀⣸⣿⣿⣿⠋⠁⠚⣯⣼⣿⣿⣿⠿⢡⣾⣿⣷⣮⡉⠛⠃⢩⣤⣤⣙⠃⣾⡇⠀⢸⣿⣿⣵⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠈⠉⣿⣿⠛⠛⢿⣿⠿⣶⣶⣤⣤⣤⣉⣋⣹⡟⠛⠿⠟⠋⠉⣿⣿⣿⠿⣿⣿⣴⣶⣶⣤⣀⣀⣀⣙⠉⠻⡆⣿⠀⣀⣼⣿⢻⡯⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⣿⡝⣧⠀⢸⣷⢰⣶⡄⢹⣏⢉⣟⠛⢿⣿⠿⣷⣶⣶⣤⣬⣝⣃⣀⡙⠿⠟⠁⠀⣰⡿⠛⠻⣿⣧⠀⢀⣽⣾⡟⢹⡟⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⢿⡇⠸⡆⢸⣿⢸⣿⡇⢸⣧⣾⠏⠀⢸⣿⣶⢀⣤⢸⣏⠉⠉⠛⢻⣿⠿⠿⣶⣶⣶⣤⣤⣄⣈⣥⣾⣿⡏⢸⡇⢸⡇⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⡇⠀⢻⣌⣿⢸⣿⣿⢸⣿⡟⠀⠀⢸⣿⢿⣿⣿⢸⣿⢸⣿⠃⢸⣿⠀⣤⣤⢸⣿⠉⠙⠛⣿⡟⠁⣿⠂⢸⡇⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⢿⣿⡀⣿⣿⣾⣿⠀⠀⣀⢸⣿⢸⣿⣿⢸⣿⠘⣷⣶⠸⣿⠀⢿⠇⢸⣿⠀⠀⠀⣿⡇⠀⣿⠀⢸⡇⢸⡇⣸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣤⣤⣿⣇⣛⠿⢿⣿⠀⣴⣿⠈⣿⢰⣾⣿⢸⣿⢠⣿⣿⠐⣿⢠⡾⠃⢸⣿⢰⣶⠇⣿⡇⠀⣿⠟⢸⣇⣸⡿⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⣻⡏⠛⡛⠻⣿⠶⢶⣦⣤⣿⣀⣛⡿⢀⣿⠸⣿⣿⠀⣿⢸⣷⣾⣿⣿⠸⣿⡆⣿⡇⠀⣿⢀⣼⣿⣻⡇⣼⠇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⢹⡇⠀⣿⡇⣿⠀⠀⣀⡈⣿⠉⠛⠛⠻⣿⠷⣶⣦⣴⣿⣜⣻⣿⢻⣿⠀⠀⠀⣿⡇⣠⣿⣋⣿⣿⢹⡇⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⢸⡇⢠⣾⡇⣿⠀⣿⣿⡇⣿⠀⣿⣿⡇⣿⢀⣤⣤⡈⣿⠉⠛⠛⠻⣿⠷⣶⣶⣿⡿⢫⣿⣿⣿⡟⢸⠅⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣸⣇⢻⣿⣇⣿⠀⠿⠛⠛⣿⠀⣿⣿⡷⣿⠈⢿⣿⡇⣿⠀⣷⣶⠆⣿⢀⣤⡄⣿⣷⣿⣿⣿⣿⡇⢸⠀⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢸⣿⣿⢿⡇⢶⣿⡇⣿⡆⣿⣿⡇⣿⡆⣶⣿⡧⣿⢀⣿⣥⡄⣿⢸⣿⡇⣿⣿⣿⣿⠇⣿⠁⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣬⣭⣉⣸⣇⠀⠈⠋⢹⡇⡿⠛⠁⣿⡇⣿⣿⡇⣿⠘⣿⣿⠇⣿⠸⠿⠃⣿⠁⢸⡇⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠛⠛⠿⠿⠿⢿⣷⣿⣷⣤⣤⣤⣿⣇⡛⠻⠇⣿⠀⠉⠀⠀⣿⠀⠀⠀⣿⠀⢸⣧⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠛⠛⠛⠿⠿⠿⣿⣷⣶⣦⣤⣿⣄⣀⣀⣿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣤⣤⣀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣷⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⣹⡿⠛⠻⠿⢿⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⢀⣠⠟⠉⠀⠀⠀⠀⢀⣴⠟⠉⠉⠛⠛⠿⣿⣿⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣠⣶⡿⠋⠙⠛⠛⠋⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠉⠉⠙⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⢸⣿⣿⠿⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡴⠏⠁⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⡿⣿⣶⣦⣤⣤⣄⣴⣟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢻⣿⣿⣿⡇⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⣿⣶⣶⣿⣤⣤⣀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠋⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⣷⣶⣶⣦⣤⣤⣾⣿⡿⠃⠀⠀⠀⠀⠙⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣦⣄⢀⣤⣤⠆⠰⠀⠛⠛⠛⠛⠃⠚⠀⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⡴⢶⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠃⢤⣀
⢠⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣸⣷⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⢓⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣾⠷⠶⠶⠶⠶⠶⣾⡗⠀⣼⠃⣄⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⠉⠉⠉⠉⠉⠁⣈⠀⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉
⢿⡶⠶⠶⠶⠶⠶⢾⡇⢀⣿⠀⠈⣷⣤⠙⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢀⣴⠟⠛⠛⢶⡄⠀⠀⠀⠀⢸⡇
⣾⠷⠶⠶⠶⠶⠶⢾⡇⢸⡟⠛⠛⠿⣬⡿⢿⣼⢿⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⣼⣧⣤⣤⣤⣬⣿⡄⠀⠀⠀⢸⡇⣴
⢿⡟⠛⠛⠛⠛⠛⢿⡟⠛⠻⣶⢦⣤⠀⢹⣦⡿⢾⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⢦⡄⢸⡇⠻
⠈⠛⣿⠟⠛⠛⠛⠛⠛⢻⡟⠋⣰⡏⣠⣿⡟⠁⣾⣁⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢿⣭⣭⣽⠂⠀⠀⢿⣭⣭⣿⢸⡇⢸⡇
⠀⠀⣻⡿⠛⠛⢷⣄⠀⢸⣧⣴⣿⣴⣿⠏⠀⣸⠏⠉⠻⣆⠀⠀⠀⠀⣿⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿⢸⡷⣾⣇⣀⣀
⠀⠀⣿⣤⣤⣤⣤⣿⣤⣼⣯⣤⣤⣤⣤⣤⣴⡟⠛⠛⢻⡟⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣩⡯⢸⡇⢸⡏⣻⠟
⠀⠀⠈⠻⢦⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⣿⠀⠁⠈⠀⠉
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠋
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⣶⠶⠶⣶⠶⠶⠶⠶⠶⣶⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣤⣤⢴⣿⡤⠤⣿⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⡀⠺
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠘⠶⠴⠟⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⡇⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⣿⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠈⢁⡄⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿⠀⠀⠀⠀⠀⠀⠸⣯⣽⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⣾⠁⣿⠀⣿⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⣤⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⣿⠀⣿⠀⣿⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠠⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢰⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣿⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⢠⣤⠤⠄⣤⠤⣤⡀⣠⡤⣤⡀⣤⡤⣤⡀⠀⢠⣤⢤⣄⢠⡤⠤⠄⣤⠀⢠⡄⣤⡀⢠⡄⣤⡤⠤⢠⡤⣤⡠⣤⢀⡤
⠀⠀⠀⢸⡟⠒⠸⣇⣀⣸⠗⣿⣀⣸⡇⣿⣀⣸⡇⠀⢸⣇⣀⣿⢸⣗⣒⠨⣗⣀⢸⡇⠸⣧⣾⠁⣿⣓⡂⢸⡷⣾⠃⠘⣿⠁
⣶⣶⣶⢾⣷⣶⣶⣾⣿⣿⣶⣾⣿⣿⣶⣿⣿⣷⣶⣶⣾⣿⣿⣷⣾⣿⣿⣶⣿⣿⣿⣷⣶⣿⣷⣶⣿⣿⣿⣾⣷⣾⣷⣶⣷⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣠⠴⠒⡒⠒⢤⡀⠀⠀⠀⠀⡤⠼⠥⣄⠀⠀⠀⠀⠀⡟⣟⠀⠀⠀⠀⢀⡤⣒⣶⣿⠶⢝⣆
⠀⠀⢰⡥⠤⠤⠤⠤⠤⣽⠀⠀⠀⢺⡭⠭⠭⣭⠇⠀⠀⠀⢸⣓⠻⡄⠀⠀⢠⢫⣮⣅⡿⡿⣟⡯⡅
⠀⠀⠸⣏⣙⠾⠭⢞⣉⡿⠀⠀⠀⠀⡇⠀⠀⡞⠀⠀⠀⠀⢻⡸⢂⡇⠀⠀⢸⢸⢉⣏⠛⢽⣆⡇⡇
⠀⠀⠀⠓⠚⠢⣠⠞⠓⠃⠀⠀⠀⠀⣇⠀⢀⡇⠀⠀⠀⠀⣸⠀⠸⡇⠀⠀⠀⠣⣙⠭⠾⠶⣊⠜⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢀⡴⡟⣇⡟⡷⣄⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⢀⡠⠤⠦⠤⡀⠀⠀⠀⠀⢀⡴⠋⠻⣷⡄
⠀⠀⠸⡍⠉⠉⠉⠉⣹⠀⠀⠀⠀⢸⠛⠛⠛⢳⢤⡀⠀⠀⢯⠤⡤⠤⡠⢬⠇⠀⠀⠀⡞⠀⠀⢀⡼⠁
⠀⠀⠀⣇⠀⠀⠀⠀⡇⠀⠀⢀⣀⣸⣄⣀⣀⣿⣊⡀⠀⠀⠘⡆⢇⢠⠇⡞⠀⠀⣠⣴⡹⠶⠒⠉
⠀⠀⠀⠘⠒⠒⠒⠚⠁⠀⠀⠀⠓⠶⠶⠶⠶⠶⠚⠁⠀⠀⠀⠙⠚⠚⠚⠁⠀⠀⠙⠚⠀⣠⣄⡀
⠀⠀⠀⠀⠀⣀⠴⡀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠀⢠⣞⣓⣞⣳⡄
⠀⠀⠀⡼⠭⠤⠤⠽⡄⠀⠀⢰⠋⠉⠛⠉⠙⠉⠉⢳⠀⠀⠀⠀⠀⣀⡞⠑⠺⡀⠀⠘⡟⢛⡻⢿⠃
⠀⠀⠀⣟⢲⡚⢲⠛⡇⠀⠀⣿⡯⢽⡯⠟⢭⣿⠽⣿⡇⠀⠀⢀⠴⣅⠈⠓⢼⠁⠀⠀⢸⡥⢒⡟
⠀⠀⠀⢸⡀⡇⡟⣸⠀⠀⠀⠙⠛⠿⠭⠭⠭⠽⠟⠛⠀⠀⠾⣙⣆⣈⣧⠴⠋⠀⠀⠀⠀⢟⣹⠁
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣴⢒⣟⣲⡀⠀⠀⣠⠒⢄⣠⠒⣄⠀⠀⠀⣾⢿⠆⣜⣳⡄⠀⠀⠀⠀⠀⣴⣽⡷⠦⡀
⠀⠀⠀⢀⡴⠋⠀⡯⢽⣻⠃⠀⢸⠁⡸⠈⡇⡹⠈⡇⠀⠀⣿⠸⣤⠏⠘⢦⠀⠀⢀⡤⣪⣾⠞⠀⢀⡗
⠀⠀⢰⠋⠀⣠⠞⣠⠞⠁⠀⠀⠈⢦⠃⡰⢧⠃⡴⠃⠀⢸⠁⠀⢹⠀⠀⢸⠀⠀⢗⡿⠚⠁⠀⡠⠊
⠀⠀⠀⠑⢞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⢸⣀⣀⣸⣀⣀⣸⠀⠀⠈⠳⠤⠴⠊

⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠉⠉⠉⠛⠹⣿⣿⡏⠙⠋⠉⠉⠙⠋⣿⣿⡟⠉⠉⠉⠉⠉⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠉⠉⠉⠉⠹⣿⣿⡏⠉⠉⠉⠉⣿⣿⡏⠉⠉⠉⠉⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣤⣤⣤⣤⣿⣿⣧⣤⣤⣤⣴⣿⣿⣧⣤⣤⣤⣤⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⡔⣢
⠀⠤⢴⠒⣊⠩⠍⠁⠐⠈⣃⡧⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠁⠈⣇⣡⠠⠄⠒⠈⠉⠀⣀⠴⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡉⢉⣉⣉⣉⠉⢈⢀⣤⠟⢻
⠀⠀⢈⠉⠉⠉⠉⠉⠉⠈⠉⠁⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⠤⠤⠤⠴⠶⠶⢤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⢠⡤⠉⠉⢀⣤⡎
⠀⠀⠈⠰⠆⠀⠀⣀⢠⣔⠒⠶⣒⣒⡀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣇⡠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣜⠳⣄⢸⢨⠀⠀⠀⠀⠀⠀⣻⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠙⠋⡋⢻⠙⢛⡿⠛⣉⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠌⢺⠘⠀⠀⠀⡠⡀⣧⢸⡛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⣸⠔⢋⣴⢎⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡆⡄⠠⡢⡨⠂⢹⢸⠉⠓⠢⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⣠⠔⢫⣿⠈⠻⡦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⠃⢱⣣⣀⣘⢈⣀⣗⢰⡿⣆⠹⡆⠀⠀⠀⠀⠀⠤⠀⠀⠄⣨⠼⡆⣸⡏⠀⢀⠸⣞⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢻⣎⣯⡍⠫⡩⢻⡆⢽⡋⢙⣶⣲⠖⣶⠶⢦⠤⣼⣎⠽⣽⠿⣈⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢢⡀⠀⠉⠉⠙⠲⣄⣼⠁⢸⣗⡉⡥⠔⠋⠀⢦⠡⡳⡉⠀⠀⣿⠀⠈⠳⣄⣼⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡁⢕⠤⣀⢘⣀⢀⡤⡙⢠⠏⠀⠀⠀⠀⠀⠀⠀⠳⢌⠪⢔⣠⣯⣀⣀⣤⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠭⣒⣋⠉⣒⣈⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠨⠥⠤⠥⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠛⠓⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣀⠠⠴⠤⠤⢀⣀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣶⣾⣿⡿⠿⠿⠿⠶⣮⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠿⠃⠀⠀⠀⠙⠂⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡄⠰⠶⠀⡤⠀⠛⠂⠀⠉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢧⠀⠐⣖⣒⣒⡽⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢬⣍⣭⣤⣾⣁⣀
⠀⠀⠀⠀⠀⡴⠶⠶⠖⠒⢲⠖⠶⢤⡀⠀⠀⣠⠖⠙⠮⠓⠚⠰⠀⡀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠘⡇⢀⡜⠁⢠⢂⠤⠤⡤⠤⢤⡏⠙⠢⡀⠈⢦
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡌⠃⢀⣀⣇⣾⠀⢠⡇⢸⠒⠀⠑⠂⢩⢹⠀⢀⣸⡀⢸⡀
⠀⠀⠀⠀⠀⡼⠤⠤⠤⠤⠧⢼⡿⠿⡿⠿⠧⣿⡇⠘⢄⣀⠀⢀⠼⣸⣶⣿⢦⡼⡾⣤⡀
⠀⠀⠀⠀⠀⢳⠒⠒⠒⠒⠒⠚⠶⠶⡷⠒⢲⠻⢧⠤⠤⠤⠤⠤⠕⠺⣌⠉⣻⣾⡇⢡⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠘⣄⠀⠀⠐⣶⡀⠀⠈⢳⡇⠉⣷⢬⠟
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣼⠴⠤⠬⠷⣄⠀⠘⣏⡷⠀⣸⢿⡈⢹⡀⣇
⠀⠀⠀⠀⠀⣀⣝⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢺⡏⠀⢰⣿⣄⣰⡟⢠⡇⢀⠏⠉⠉⠓⠦⡀
⠀⠀⠀⠀⢺⣸⣁⣀⡇⡠⠀⠀⠀⠀⠀⠤⡀⠐⠂⠐⢸⣥⣤⣾⣿⣿⣴⠁⡼⠀⢈⡸⡤⠤⠤⡤⠕
⠀⠀⠀⠀⢰⠋⠤⠤⣜⡞⠉⣽⣿⣍⠉⣷⢸⠠⠤⠀⠸⠿⠿⠿⠿⠜⠃⢠⡇⣠⠏⣧⢻⣶⠀⢹
⠀⠀⠀⠀⠀⠉⠉⠉⠉⢇⠀⠻⠿⠟⢀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢣⡀⠻⠿⠟⠀⡸
⠀⠀⠀⠀⠀⠀⠠⠤⣤⣬⣷⣤⣤⠴⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⢦⣤⣤⢾⠥⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⣰⣊⡑⢢⠀⠀⠀⠀⠀⣀⢤⡤⠤⣄⣠⠤⡄
⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⣠⣖⣂⣁⣀⣲⣄⠀⠀⠀⠀⣞⣁⢀⣹⣌⢱⡄⠀⠀⠀⣈⠞⠀⡞⠺⣡⡞⢾⣽
⠀⠀⠀⠀⣢⣤⠤⢤⠤⡤⠤⢤⣤⣼⡀⠀⠀⠀⠀⢧⠒⠒⠒⠒⢲⠞⠀⠀⠀⠘⢧⠶⣖⣒⠶⢲⠋⠀⠀⠀⢸⡒⠼⠼⢄⠿⠼⠚⡏
⠀⠀⠀⠀⢱⣊⣉⡲⠮⠭⢷⠭⣙⣤⠃⠀⠀⠀⠀⠈⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⣎⢉⣱⡎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠣⠤⠍⠮⡇⢙⡴⠵⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣦⠙⡾⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⠔⢋⠤⡤⠤⠤⣉⢆⣴⠒⣆⠀⠀⠀⠀⣠⠔⠊⠻⡍⠙⢏⠐⠢⡀⠀⠀⡤⠒⠉⠉⠉⠙⠒⢄⠀⠀⠀⠀⠀⡴⠊⠉⠉⠲⣄
⠀⠀⠀⠀⠎⡞⢙⣚⣁⣡⢞⣕⣋⡀⣇⢸⠀⠀⠀⣜⠁⠀⠀⠀⠘⣄⣸⠦⠤⠜⠀⠘⡤⠤⠤⢤⠤⠤⠤⡞⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡆⡘⠨⠈⠁⠹⢉⠟⢲⢲⠉⠉⠀⠀⢠⠋⠣⡀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠸⡀⢇⠀⡇⢸⢠⠇⠀⠀⠀⠀⠇⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠰⡌⠳⡉⣉⣢⣀⡴⢁⡞⠀⠀⠀⠀⢸⠲⢄⣨⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⣘⡀⢃⣁⡜⠀⠀⠀⠀⣀⠇⣹⠦⠔⠋
⠀⠀⠀⠀⠀⠈⠓⠢⠤⠭⠔⠒⠉⠀⠀⠀⠀⠀⠸⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠯⣅⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡆⢠⣆⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢠⣴⡒⢒⣖⠲⠦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣧⣃⣀⣀⠀⠀⠀⠀⠀⠀⢩⠩⣇⡼⠼⡄⠀⠀⠀⠀⢀⠴⢄⡀⣠⠔⢄⡀
⠀⠀⠀⠀⢰⢛⡥⠞⠓⠦⠥⠭⡻⢄⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣾⡻⣹⠀⠀⠀⠀⡈⠀⣍⠖⠒⢦⡀⠀⠀⡰⠋⠈⡆⠙⢇⢸⡀⠙⢦
⠀⠀⠀⠰⣏⡜⠀⠀⠀⠀⠀⠀⠘⣾⡄⠀⠀⣀⢀⣱⣄⣀⣀⣤⠞⡛⢩⡁⠀⠀⠀⡞⠁⠨⠁⠀⠀⠀⢹⠀⠀⡇⠀⢠⠁⠀⢸⠀⡧⠀⢸
⠀⠀⠀⢠⣋⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⠀⠀⠘⠢⢤⣤⣄⣀⣤⠤⠴⠊⠀⠀⠀⠀⡇⠀⢀⡇⠀⠀⠀⢸⠀⠀⠱⣄⠸⠀⣀⢮⡀⠇⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⡇⠀⠀⠀⢈⠀⠀⠀⠀⠑⠎⠁⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣿⡒⢢⠀⠀⠀⠀⠀⢸⢾⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⢱⢦⡀
⠀⠀⠀⠀⠀⣀⡠⠤⣐⣵⢿⡿⠊⠉⢣⠀⠀⠀⠀⡜⣈⢘⣄⠀⠀⠀⠀⠀⠀⣠⠊⠀⣹⣳⣲⡹⠀⠀⠀⡴⠲⡤⠒⠒⡤⠐⠲⡤⠤⢤
⠀⠀⠀⢰⡯⣒⡲⣩⣓⠵⠊⠀⠀⢀⡜⠀⠀⠀⠸⡩⢍⠉⠈⡆⠀⠀⠀⠀⡰⠗⠒⠉⠁⠀⣠⠇⠀⠀⢸⣄⣈⣁⠠⠄⣁⣀⢀⣁⣀⣠⡇
⠀⠀⠀⠺⣈⠞⠉⠁⠀⠀⠀⠀⡡⠊⠀⠀⠀⠀⠀⢱⢀⠇⠸⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢮⣽⣚⡈⣉⣐⣒⢒⠮⣉⣳⢯⡽
⠀⠀⠀⠀⠘⢆⡀⠀⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠸⡄⠀⠀⠑⠤⣄⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠢⠤⠒⠒⠠⠤⠁⠂⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⠤⠃⠀⠀⠀⠀⠈⠳⠴⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠁⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⣀⠤⢒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠤⡴⠎⠭⠔⠊⢡⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣉⣎⣄⡇⠀⠐⡢⠊⠉⠉⡆
⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀⣀⣀⣠⣀⣀⡀⠣⢈⣎⢷⢤⠊⠀⠀⢀⠴⠁
⠀⠀⠀⢸⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⢘⠀⠀⣹⠳⣽⣓⣶⣿⠁⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠸⡖⠛⣿⡟⠛⠛⠛⠚⣿⡟⠃⡞⠀⢰⢹⠀⠀⠑⢄⣻⢦⣀⣀⣀⣀⣀⣀⣸⣷⣀
⠀⠀⠀⢀⡆⠀⣿⡇⣠⠤⠤⠤⣿⡧⡀⡇⠀⡆⠈⢦⡀⠀⠀⠹⠗⠒⠂⠉⢹⣵⣺⡋⠹⠀⠈⠙⡦
⠀⠀⠀⢸⡇⠀⣿⡇⡇⠀⠀⠀⣿⡇⢰⡇⠀⡇⠀⠀⠉⠂⠤⢀⣠⠤⠔⠊⠉⠀⠀⠈⢢⠀⠀⢀⡇
⠀⠀⠀⠘⡇⠀⣿⡇⣇⠀⠀⠀⣿⣇⡜⡇⠀⢥⠀⠀⠀⠀⣀⠠⠗⠒⠀⠒⢤⠀⠀⠀⠈⡇⠀⢻
⠀⠀⠀⠶⠳⠒⠛⠓⠊⢙⠉⠙⠻⣯⡐⠃⠰⠼⢏⡉⠉⠀⠀⠀⠀⠀⢀⠄⠀⡇⠀⠀⠀⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⡤⠬⠷⣤⠤⠤⠤⠬⠵⠶⠶⠶⠶⢭⡇⠀⢰⡇⠀⠀⠀⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡸⠿⠀⠒⠒⠒⠒⠒⠤⢤⡀⠀⠀⠀⡞⡅⠀⢸⢸⠀⠀⢸⠀⠀⠯⠆⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢠⠁⡇⠀⣈⢀⡇⠀⡬⠀⢀⠔⠋⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣗⡲⠯⣜⣳⡶⠁⢠⢋⠴⠒⠂⠛⢇⠚
⠀⠀⠀⠀⠀⠀⠀⠘⢣⠈⠭⠭⠭⠭⠭⠄⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠻⠟⠿⠟⠛⠁⠀⡟⢡⠊⣉⡉⠳⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡵⠦⡴⠶⠴⡥⠤⡤⠤⠤⠤⠔⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇⠸⡀⠙⠃⡰⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣈⠉⢉⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣈⠉⢉⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣀⡀⠀⡀⠀⠀⠀⠀⢀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⣄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠦⠀⠀⢄⠈⠰⠤⠀⠒⢐⠂⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠸⡁⢀⠄⠀⢀⠀⠀⠀⢀⠐⠀⠀⠀⡐⠊⠀⠈⠸⡀⠀⠀⠀⠀⠐⠀⠈⡀⠄
⠰⡀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠈⠃⠀⠋⠀⡎⣉⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠑
⠀⠀⠐⠒⠒⠚⠁⠀⠀⠀⠀⠀⠊⢲⠂⢑⠤⠘⠁
⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⣀⠀⣀⡀⣀⡀⢀⡀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⢰⣷⠘⣇⠁⢹⠀⣇⡀⡏⢹⢸⢹⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠼⠻⠴⣬⠇⢸⠀⡇⠀⢧⡼⠸⣼⠃⣧
⠀⠀⢠⡀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠌⠸⣧
⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠷⠴⠆⢁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠻⠁⠁⠀⠉⠈⢲⡀⠀⠀⠀⠘⠟⠛⠲⢴⠿⣠⡒⠶⢅⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠦⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⡎⠱⣄⡀⢐⡀⠕⠒⢬⡧⠃⠀⠀⠀⠀⠀⠀⠀⢄⡐⠢⠀⣀
⠀⠀⢛⠀⠀⠀⠠⠈⠀⠈⠀⡀⡉⠀⠀⠀⠀⣤⠀⠐⠃⠐⠓⠒⢬⡀⢂⣤⠈⣠⠀⠀⠐⠀⠀⠀⡀⠄⠉
⠀⠀⠀⠈⠀⢁⠂⠀⠐⡀⠖⠃⠀⠀⠀⠀⠃⠠⠤⠦⣀⠐⢁⠝⢊⠀⠀⠈⠈⣀⣢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢉⣉⣉⣉⣉⣉⣉⡉⠉⠉⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣠⠞⠛⠛⠛⠛⠻⠿⠛⣿⠀⣿⣇⡇⠀⠀⠀⣴⣄⠀⣴⠦⠀⡆⠀⢠⠀⡄⢀⡄⡶⠦⢰⣦⡠⡄⢠⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣰⣯⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡇⠀⠀⠀⣿⢘⡇⣿⡄⠀⡇⠀⢸⠀⣿⢸⠇⣧⠄⢸⡾⠃⢻⣾⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⡇⠀⠀⠀⣿⡼⠃⣿⣀⠈⣇⡀⢸⠀⢸⡿⠀⣇⣀⣾⠻⡄⢨⡇⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢷⣿⣤⣤⣤⣤⣤⣤⣤⣤⣽⠀⣿⡇⡇⠀⠀⠀⠈⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠭⠭⠿⠀⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠉⣿⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣒⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⢀⣿⠀⠀⢀⣴⣿⠟⠛⠉⠉⠙⠻⢿⣗⢄⠀⣿⣧⡧⠤⠤⠤⢤⣤⣤⣤⣤⡴⣿⠟⠋⠉⣉⡉⠛⠻⣮⡳⣀⣀⣀⣀⣀⣀⣸⡇
⠀⡀⠀⠀⣿⣸⣿⣀⣀⣞⣿⠃⠀⣔⣭⠿⣷⣆⠀⢹⣿⣆⣻⣇⣇⠀⠀⠀⠀⠀⠀⠀⣘⣼⡏⠀⡔⣩⠶⢯⡷⡄⠘⣿⣇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⢻⣟⣛⣛⣛⣻⣿⠀⠸⣿⣇⠀⣸⢿⠀⢸⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⡄⠀⣇⣧⣀⣠⠇⡇⠀⣿⣟⣛⣛⣛⣛⣿⣿⣿
⠀⡀⠀⠀⠸⣷⣶⣶⣶⣶⣿⣆⠀⠙⠛⠿⠗⠁⠀⣼⣯⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣷⡀⠈⠛⠿⠷⠊⠀⣰⣿⣭⣭⣭⣥⣤⣤⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣴⡾⠟⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣼⣿⣶⣶⣦⣤⣀
⠀⠈⠉⠙⠛⠛⠛⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠛⠛⠻⣿⠷⠶⢶⣶⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠈⠉⢹⠋⠛⠛⢻⡿⠿⠶⢶⣶⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠶⢤⣿⣤⣤⣀⣿⣀⣀⡀⢸⡀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠉⣿⠉⠙⠛⣿⠛⠿⠷⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣿⠀⠀⠈⣿⠉⠉⠉⢻⡟⠛⠓⢾⡷⠶⠦⢤⣿⣤⣤⣀⣿⣀⣀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀⣀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠈⣿⠉⠉⠙⣿⠛⠛⠒⢿⠶⠶⠦⣿⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⣿⠉⠙⠛⣿⠛⠓⠲⢾⠷⠶⠶⢼⣧⣤⣤⣤⣿⣀⣀⣀⣿⣀⠀⠀⣿⠀⠀⠀⢸⡀⠀⠀⢸⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠄⠀⠀⢸⡇⠀⠀⠀⣿⠈⠉⠉⣿⠉⠉⠛⣿⠛⠛⠒⢿⡷⠶⠶⢾⡧⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠶⣿⠤⠤⣤⣿⣤⣤⣤⣼⣄⣀⣀⣸⣇⣀⣀⣀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠅⠈⠉⢹⡏⠉⠉⠙⣿⠙⠛⠛⣿⠛⠛⠓⣿⠶⠶⠶⢾⡷⠦⠤⣼⣧⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣿⣀⣀⣀⣿⡀⠀⠀⣸⡂⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣿⠉⠉⠉⣿⠉⠉⠉⢻⠛⠛⠛⢻⡟⠛⠛⠛⣿⠛⠛⠛⣿⠒⠒⠒⣿⠶⠶⠶⢾⡷⠶⠶⢾⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⠶⠶⠶⠿⠶⠶⠶⠾⠷⠶⠶⠾⠷⠶⠶⠶⠿⠶⠶⠶⠿⠶⠶⠶⠿⠶⠶⠶⠾⠷⠶⠶⠾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣩⣿⣿⣿⣿⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⣿⣿⣏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣾⣿⣧⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢠⣿⣿⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣉⣛⣡⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣙⣋⣤⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣸⣿⠀⠀⠀⡀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣿⣻⠀⠠⡄⢏⢧⠐⡀⠙⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠠⠀⠀⣶⣿⣿⡇⢀⡀⠀⠀⠀⠣⠀⢸⡄⠈⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⢹⣿⣿⣿⣿⣇⠁⠀⠒⠒⠒⠒⠒⠛⠉⣁⡤⠒⠀⠈⠉⠉⢀⠐⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⢳⠒⠓⠂⠀⡀⠀⡸⢦⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⡀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠈⣦⣤⣄⠀⠀⢀⡏⣠⠈⢙⡳⢦⡤⣄⣀⣀⣀⣤⡤⠤⠆⠒⠃⡀
⠀⠀⠀⠀⠀⠀⡯⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣶⣤⣾⣧⡇⢸⠸⣡⣼⢇⡐⠂⠐⠀⠀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠙⠿⠿⠿⣿⠻⠿⠿⣿⣿⣼⣿⣇⣀⡀⣠⢼⣋⣴⡡⠇
⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⠀⠀⠠⡀⠀⠈⠛⠪⠿⠭⠬⠭⠭⠿⠖⠋⠁⠀⡇
⠀⠀⠀⠀⠀⠈⠓⠲⠶⠾⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠉⠐⠒⠒⠂⠘⠑⠒⠀⠤⠤⠤⠤⠤⠤⠀⠤⠠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣥⡶⠃

⠀⠀⣾⣿⣷⣄
⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠿⠻⠛⠿⠿⠟⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠀⠀⠉⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏⠁⠀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⣤⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣦⣤⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⣤⡦⣰⡄⠀⠀⢰⡦⢦⣰⣤⣤⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⣿⠀⣶⠓⣦⠀⠀⠀⠀⠀⣤⡀⢀⣠⡀⢠⣄⡀⣀⡀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠀⠙⠒⣿⡄⠀⠀⣿⠀⠑⠾⠟⠛⠃⠚⠻⠟⠘⠛⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⢀⡿⠁⢈⠀⣯⠀⢰⠀⣤⢀⠀⡾⢀⣄⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠟⠀⡇⠀⠋⠀⢿⢹⢠⡇⢸⡇⠀⡁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠑⠈⠁⠁⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢓⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠻⠂
⠀⠀⠀⠀⠀⠀⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⢀
⠀⢀⠤⠠⣤⣠⣇⡔⠇⠀⣀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠤⠤⠤⣄⡀⠀⠀⢀⡠⠤⠀⠒⠂⠀⠀⠉⠀⣠⡴⢿⡀
⣤⣯⣤⡚⠀⠀⠈⡟⠉⠲⡀⠀⠀⢱⠀⠀⠀⣀⠤⢄⠀⠀⠀⢠⠖⠊⠉⠁⠀⠀⠀⠉⠘⠒⠢⣄⡙⣦⠀⡸⢤⣂⡀⠀⠀⠀⠀⣤⡼⢟⣴⣾⠇
⡇⠠⠀⠙⠶⢤⠴⠖⠒⠲⠿⣤⣤⣞⣳⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠒⠄⠀⠀⢀⡐⠀⡂⡀⠂⠘⠏⢹⡆⢰⠆⢤⣿⣋⠐⠒⢚⣩⡴⠟⠋
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢻⠶⠇⠀⠀⠀⠀⠀⡀⠀⠀⣸⣦⣤⣄⣄⣠⣄⣠⣦⣠⣦⣶⣶⣞⣿⠀⠀⠒⢄⠉⣁⠴⠟⠋⠁
⢸⠀⡆⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⣧⠀⠀⠨⣀⠀⠀⡇⠀⢾⢀⡙⠓⠛⠳⠶⠛⠷⠞⢛⣋⡭⠖⣻⠏⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠛⢆⡀
⠘⡄⠁⠀⠀⠀⠀⠀⡔⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢥⣭⡭⣭⣭⣭⣭⣭⠭⠴⠖⠋⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠸⠀⢀⠨⡸⡄
⠀⠉⠛⠛⠛⠓⠒⠺⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣈⠀⠀⠈⠇⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡘⠲⠂⠠⢀⠉⠛⢦⣄⠀⠀⠀⠀⢷⠀⡀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣀⣠⣴⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠰⠞⠃⠀⠘⠀⠀⠀⠀⠀⠀⢀⡞⠱⠀⠀⠀⠀⠐⠢⡀⠈⠳⣄⠀⠀⠘⠇⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⢋⠉⢊⠈⢉⡏
⠀⠀⠀⠀⠀⢀⣘⠃⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⡀⠀⠀⠀⠀⠈⠢⡀⠈⣆⠀⠀⠀⠃⠀⢀⡀⠸⠀⠀⠀⠀⠀⠠⡐⠠⠑⢨⠅
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢠⡟⠀⠀⠀⠀⠁⠀⠀⢀⣠⣶⠶⠛⠚⠋⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡏⠰⠀⠀⠀⢠⣀⣴⠾⠿⠋⠁⠀⠀⠀⠀⠀⢰⠀⠀⡈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠰⢀⠢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠂⠀⠀⠀⢀⡿⠀⡀⢈⣤⡶⠛⠋⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⢰⢬⠁
⠀⠀⠀⠀⠀⠀⠀⡆⢉⣻⣦⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⢸⣃⣴⠾⠛⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠷⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⡠⠦⠴⢤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡤⠶⠛⠿⢫⡥⠅⠀⠦⢤⡀⠀⠀⠀⠀⢀⣤⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⢀
⠀⠘⠀⢤⠀⠈⡇⠀⠀⠀⠀⠀⣀⠠⣤⡄⢠⠶⠤⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⠚⠀⠀⠀⠀⣀⠀⢀⣿⠀⠀⣠⣤⢴⣤⣤⣤⢴⡄⢠⠀⠀⠀⠈⢳⣿⣄⠼⠶⣤
⠀⡆⠘⡆⠀⠀⣿⠀⠀⠀⣠⣾⠟⢁⡽⠓⠉⠉⠛⠛⢴⣄⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣀⣤⣞⣫⣽⠀⠀⠈⣽⡇⠉⢉⢻⡸⡇⣸⣤⠀⠀⠀⠀⠋⠀⠈
⠀⠃⣴⠃⠀⠀⣿⠀⠀⢰⠻⠏⡠⠊⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⣿⡀⠀⠁⣀⡠⢤⣶⠿⡿⣋⠁⣴⠼⠃⣤⠀⢸⣏⣤⣸⣄⣧⣷⣿⠃⠀⠀⠀⠀⠘
⠀⢸⠋⠆⠀⠀⡟⠀⠀⠘⢾⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠻⠿⡏⠉⢀⣬⠥⠌⣻⣯⣭⠶⡞⢻⠃⠀⠘⡆⠀⠉⠉⠉⠀⣼⠁⠀⠀⠀⠀⠀⢢⠤⠬⠉
⠀⢸⣠⡇⠀⢰⡇⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠀⠒⠒⠒⠁⠀⠀⠀⣳⣟⠦⠴⣤⣾⡿⠛⠋⠀⢀⡷⠋⠀⠀⠀⢷⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⡂⣀⠰⠊
⠀⠈⣇⠀⠀⣼⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⣄⣀⣀⣉⣠⠤⠔⠚⠉⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠸⣆⣰⠇⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠐⠒⠐⠒⠋⠀⠀⠀⠀⡀⠔⠄
⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠿⠯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢠⡄⠲⡀
⠀⠀⠀⢈⢧⠀⠀⠀⠠⡤⠤⠤⠤⢤⡄⣀⣀⣀⣀⣈⣐⣄⠀⠀⠀⠀⡤⢀⣤⣤⣴⠾⠿⠛⠉⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣣⠀⠈⣢⡀
⠀⠀⠀⠘⡜⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢉⡀⠀⠙⡀⠀⠀⠔⠉⠝⠉⠁⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⢀⣀⣀⠤⠼⢧⡽⢤⣾⠿⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠐⢦⠀⠀⠀⠀⣀⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⠀⢈⡟⣻⣳⡄⣀⠀⠀⠀⠸⣇⠈⣿⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢹⠈⠙⠚⠷⣿⡿⢿⣹⣇⠀⢀⡤⢿⡄⣹⠷⡞⣛⣓⢶⠀⢀⣾⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⢸⣿⣀⣳⠈⠉⠙⢦⣀⣿⣿⣠⣇⣉⡛⠈⣷⠋⣠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⣠⠴⠒⣻⣿⡿⠿⣄⠀⢠⠞⠁⣀⠜⠉⠙⢦⣿⠀⢸⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣯⣳⣾⣁⢿⢷⢠⠞⡶⠋⣠⣞⣁⣠⠤⢴⣖⣛⣭⣭⠽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣬⣟⠺⣿⣸⡮⢿⣒⣚⣭⠭⢶⡞⢻⣉⣨⡧⠴⡿⠒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡧⣿⡛⣿⣿⣏⣽⡧⠴⢾⠚⢻⡏⣹⣁⣸⣧⣤⡷⢾⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⢾⣍⣻⢾⣿⠶⣷⠒⢻⠛⣻⣏⣹⣉⣼⣤⣼⣧⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⣿⣷⣽⡟⣿⡗⢻⠋⢻⠉⢹⠁⣿⠀⢸⠀⣸⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⠉⠓⠿⣿⣷⣚⣒⣚⣒⣛⣚⣛⣿⣭⠭⠭⣽⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣶⣖⠶⠾⣿⡷⠓⠛⠓⠒⠒⢺⢿⠧⣄⣀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⡈⠳⢦⣿⣧⣀⣤⠤⠤⠶⢾⢿⠓⠚⢻⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣇⠀⠀⢸⣿⠀⠀⢀⡶⡿⢿⠿⣄⣀⡾⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣟⡭⣝⣦⢼⡿⣆⡀⢸⣿⡇⢯⣽⣻⢛⣧⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠿⣗⣻⡇⡞⣟⢻⢹⡆⠙⠛⠲⠾⣿⣘⣚⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠮⠭⠞⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣰⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣠⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠲⡏⣯⡀⢀⠖⠁⡠⠀⠈⡑⠢⡀
⠀⠀⠸⡨⠼⢈⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠓⣦⣷⠇⠮⠬⢠⠁⡇⡇⡦⢷⢱
⠀⠀⠀⠀⠀⠈⠙⠳⣌⠳⡀⠀⠀⣀⣀⠠⣖⣲⣶⣶⡦⠀⠀⢀⣾⠂⡎⢲⣼⠉⠉⠉⠛⠙⢛⣇⣧⡧⡏⡿⠸⢀⠤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢸⡀⠀⣥⣿⢠⡋⠀⡑⢩⠳⠀⢠⡚⡒⢛⡶⢟⡽⠀⣀⣇⣼⠧⢽⣿⠙⠺⠿⠷⢛⠡⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣧⠀⠃⣻⣘⠑⠀⡇⡖⠛⠉⠉⠄⣃⣯⠉⢛⣹⡏⣀⠀⠀⡨⠦⠛⠛⠓⠚⠚⠒⠚⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠹⢼⣸⣉⣉⣉⣿⣭⣧⣀⣀⣀⣜⣯⣿⣷⣯⣿⣧⡤⢿⠶⠇⠀⠀⠤⠴⠥⠿⠵⠶⠾⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣶⢶⣶⠶⠶⣶⣶⣶⣲⣶⣶⡶⣦⣶⠖⢲⣶⡶⢶⣶⣶⡖⢶⡖⢒⢿⢾⡺⡚⣿⡟⢺⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣏⢹⣇⠇⢪⣷⠯⠥⠿⡍⣿⠉⠛⢋⠉⣿⢸⡇⣑⡆⣗⣇⡀⡇⡤⢻⠸⠚⢣⢺⡟⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⣶⣛⣒⣴⣿⣾⣤⣿⣷⣟⡳⣾⣷⣷⣿⣾⣿⣛⣛⣛⣛⣛⣛⣃⣌⡭⢬⢭⣭⣭⣭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣿⣿⡟⡧⡟⠿⠀⢕⣬⢻⣇⡋⡇⠃⢻⢹⡥⣿⡏⣐⣷⡏⣿⠯⠏⣇⠐⠼⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⡿⣿⣛⣛⣚⣓⣛⣛⣛⣛⣛⣓⣶⣛⣩⣭⢭⣭⠭⡭⣭⡭⠭⣭⣭⢭⣭⣭⣭⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⢿⣿⣿⡎⡹⣷⠛⡛⣋⢹⣗⡒⠚⢹⣛⡂⠉⡨⢨⢸⠓⣲⢧⢻⢾⡹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣭⣙⢓⢛⠛⠛⠓⠛⢛⠚⠚⠚⠛⠛⠛⠛⠒⠛⠚⠒⠛⠚⠛⠓⠓⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⢿⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠓⠓⠓⠛⣛⣶⢺⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⢫⣸⠎⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢾⣸⢾⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠄⠠⢂⠔⠠⠳⢤⣉⣩⢴⠎⡠⢀⠄⡀⢄⠠⡠⢀⠔⡠⡀⢄⠠⡀⢀⠤⡠⣀⢈⢦⣑⣒⣧⡮⡠⡠⡠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢩⠓
⠀⠀⠀⠀⢀⡤⠊⠁⠀⠠⢄⠘⢶⠂⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣫⣴⠾⠃⣤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠀⢠⣾⡄
⠀⠀⠀⠀⢸⡀⡀⠀⠓⠒⡢⠤⠤⠾⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠾⡋⢻⡴⠏⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢿⡾⠇⠿⣤⣤⣦⠉
⠀⠀⠀⠀⢸⣿⠩⣀⡴⠊⠀⠀⠀⠀⠀⢮⣀⡀⠀⠀⠀⠀⣤⠂⣰⣟⢻⣅⡦⠟⠉⠀⢀⣰⡿⠃⠀⠀⠀⠀⠀⣆⠚⢧⣼⡇⠨⣶⡄⠒⠉⠉⠛
⠀⠀⠀⠀⠀⠙⠢⠿⠇⠀⠀⠘⠷⠜⠀⠈⣬⡃⠀⠀⠀⠀⠙⢀⣷⡜⠋⠁⢀⢀⣶⣿⡶⠛⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠹⣿⣯⡀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣀⠀⡀⠀⠤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠻⣀⣴⣷⣉⣴⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⡉⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠶⠶⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠂⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⢾⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠦⠀⠀⠃⡀⠂⣨⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡆⠀⠀⢾⠆⡠
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠀⢀⣀⣐⢃⣠⣴⠾⢃⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⢿⡀⢻⣇⠀⠀⠰⢺⠃⣾⣷
⠀⠀⠀⠀⠀⣾⠁⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢶⣮⣽⣿⣿⣧⣬⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⣿⡖⠏⠉⠓⠀⠀⢼⣯⣥⠂
⠀⠀⠀⠀⠀⢉⠀⠀⠠⣿⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣿⣏⣉⣙⣛⣭⣼⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠠⠟⠋
⠀⠀⠀⠀⠀⠈⠳⢀⣰⡙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⡤⣤⡿⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣠⣴⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠠⡄⠀⡀
⠀⠀⠀⠀⠀⢠⣧⣤⣴⠶⢠⣾⣷⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠲⠦⠟⢿⣻⡆⠀⠀⠀⠀⠀⠀⠀⢠⡶⣾⣷⣶⡿⣿⣿⣿⣗⠂⡄⠐⡀⠁
⠀⠀⠀⠀⠀⠘⢩⠟⠳⠶⠷⣬⣭⣵⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠁⠒⠂⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣹⣯⣷⣿⣛⣻⣿⣷⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⣴⣶⡖⢢⣾⡏⠀⠀⠀⠀⠀⠀⠀⠛⢶⣌⠙⢳⣤⣿⣿⠿⣋⣽⠃⣸⢳
⠀⠀⠀⠀⠀⠀⠀⣧⡐⢿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⠬⠁⠼⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⣿⠸⣦⣾⣿⡇⠀⡿⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠈⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢷⠀⠉⠀⣉⠹⠟⠃⠘⠃⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠛⠶⠾⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⣶⣬⠓⣯⣾⡻⠛⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣹⡧⠀⠀⢀⣼⣅⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢹⢻⡄
⠀⠀⠀⠀⠀⣠⠊⣿⣶⣶⡉⣭⢳⣾⣿⣖⣠⡇⠀⠀⠀⠀⠀⠀⠀⣿⡗⢩⣭⡧⠀⢠⣿⠧⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⣮⣤⡿
⠀⠀⠀⠀⠀⠛⢶⣿⣛⠻⣿⣶⣿⣤⢒⣽⡿⢋⣆⠀⠀⠀⠀⠀⠀⣶⠀⠀⢸⡇⠀⠈⡇⠀⠀⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⢃⠲⣼⠷⠆
⠀⠀⠀⠀⠀⠤⢂⠈⠙⠻⢿⣍⣙⠿⠼⢿⣤⡾⠛⠀⠀⠀⠀⠀⠀⣿⠀⣤⣼⡇⠀⠀⡟⢰⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⡘⣟⣓⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣀⡠⠈⠙⠿⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠉⢹⣷⠀⠀⣷⠈⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢫⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⣸⣯⠀⠀⢹⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣦⡜⣷⣷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲⠤
⠀⠀⠀⠀⠀⠀⢻⣹⣿⣿⣿⢡⣷⢸⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠂⡀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣇⣸⡟⢸⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⣡⠔⠃⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠚⠛⠻⠿⣿⣿⣴⣶⡀
⠀⠀⠀⠀⠀⠐⣷⠿⠿⠿⠿⠿⢧⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⠐⠊⠁⢀⣴⣿⣿⣿⡟⣻⠀⠀⠀⠀⠀⠀⠰⢖⣀⣀⠐⠠⠔⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⢀⡐⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣟⡁⠤⣳⡿⢻⣿⣽⡿⠽⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢷⣦⣄⡀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣷⠾⣿⡿⠶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡻⢶⣬⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠈⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⢂⠔⣈⡔⠊⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⣪⢚⠥⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡤⠔⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠰⠑⠁⢼⠀⠀⠀⠀⠀⠀⠂⢀⣶⠶⠞⠺⢿⡖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠤⠄⢀⠃⠀⠀⢸⠀⠀⣤⡤⠴⢖⠛⠙⡇⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠀⠈⠰⠠⠆⠼⠶⢯⡇⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠤⠖⢈⣀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠉⠀⠀⡘⠀⠀⠐⠗⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠸⠀⢸⡇⠀⠀⢸⠀⢠⣷⠦⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠠⠀⠠⠀⠀⠀⢠⠰⠀⠀⠃⠀⠀⠀⠀⠐⠀⣹⡇⠀⠀⣸⣿⣻⡿⢶⣾⣯⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠂⠐⠨⠀⠀⠠⢰⡈⠋⠉⠙⢹⠀⢹⡗⠀⢨⣿⣿⠿⣧⣾⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠸⡇⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⣖⣊⡇⣁⣾⣷⣸⢶⣷⡿⢛⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡇⠀⠈⠉⠛⡟⡟⠛⠊⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠉⠐⠀⠄⡀⡗⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠛⠉⠉⠉⠉⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠤⠤⠀⠀⠠⠀⠄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠰⠆⠀⠀⠀⠶⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣤⠀⠀⢀⡴⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⣀⣀⣀⠔⠋⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡎⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠁⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⢄⠠⠀⠈⠂⠀⠀⣠⠞⠀⡐⣾⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠳⠀⠀⠀⠢⠌⠁⠀⡦⢱⣿⣿⣿⣿⣿⣶⣿⣿⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣤⠤⠼⠤⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣦⣤⣤⣤⣄⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣇⣰⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣤⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢡
⠀⠀⠀⠀⠀⣰⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢇
⠀⠀⠀⠀⢠⠏⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠆
⠀⠀⠀⠀⡞⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢷
⠀⠀⠀⢰⡆⠂⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄
⠀⠀⢠⡉⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⣸⠃⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠛⠀⠀⠀⢰⡇
⠀⠀⡇⠀⢠⡞⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠑⠂⢀⠈⠢⣼⠃
⠀⠀⡀⢾⠀⠀⡈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠀⠈⣁⠀⣿
⠀⠀⠑⢾⡃⠀⠀⠉⠀⢀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡉⠁⠀⢀⡟⣠⠏
⠀⠀⠀⠈⢧⡀⠈⠛⠓⠒⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠁⣄⠙⠓⠒⢒⣶⠶⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣒⠚⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣍⣿⣟⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣹⣿⣿⣿⣿⣷⣶⣶⣶⣿⣾⣿⣿⣶⣶⣶⣶⣶⣿⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠎⠀⠀⡃⠀⠀⠀⡆⠀⠀⢨⠀⠀⠀⢐⠀⠀⠱⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⣠⣮⣤⣤⣤⣮⣤⣤⣤⣧⣤⣤⣤⣥⣤⣤⣬⣤⣤⣤⣼⣤⣤⣤⣵⣤⣤⣤⣵⣄
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡸⢇⠀⠀⡨⢇⠀⠀⢨⢇⠀⠀⡸⢅⠀⠀⡸⢇⠀⠀⠨⢇⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠀⣠⣱⣪⣥⣤⣵⣪⣥⣤⣭⣨⣤⣬⣅⣬⣤⣤⣵⣊⣅⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⡟⠛⠛⠛⢻⡟⠛⠛⠛⢻⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣧⣠⣤⣤⣼⣧⣤⣤⣄⣼⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⡿⠿⠿⠿⢿⡿⠿⠿⠿⢿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢀⡀⣰⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣆⢀⡀
⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠻⠿⠿⡿⠿⠿⠿⠿⠿⠋⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠹⡀⠘⣿⠉⠉⠉⢿⠉⠉⠉⠹⡏⠉⠉⢹⡏⠉⠉⢻⠉⠉⠉⢉⡟⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢹⡖⠒⠒⢺⡗⠒⠒⠺⡷⠒⠒⢺⡗⠒⠒⣿⠒⠒⠒⢺⡗⠒⠒⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠼⠀⠈⣧⣀⣀⣈⣇⣀⣀⣀⣿⣀⣀⣸⣇⣀⣀⣿⣀⣀⣀⣼⣀⣀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠉⠉⠉⢣⠀⢸⡏⠉⠉⢿⠉⠉⠉⣿⠉⠉⢹⡏⠉⠉⡏⠉⠉⠉⡿⠉⠉⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣷⡤⠤⢼⣧⠤⠤⣾⡤⠤⣼⡧⠤⢴⣧⠤⠤⢴⡧⠤⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸⡆⠀⠀⣧⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⢸⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠻⠲⠒⠛⠒⠒⠚⠓⠒⠚⠓⠲⠚⠓⠲⠒⠟⠒⠒⠚⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣿⡟⠉⢹⣿⣏⣛⣋⣉⣛⣛⣛⣛⣋⣻⣿⠉⠙⣿⣟⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⠟⠁

⣰⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶⣤⡀
⢿⣇⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠹⣿⡄
⠈⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢺⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢻⣧
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠿⠿⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠿⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⣤⣶⣶⣤⡈⠻⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿⠉⣠⣴⣶⣦⣄⠈⢻⣷⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢰⣿⠋⠀⠈⢹⣷⠀⢹⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣸⡿⠉⠀⠈⢻⣧⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠸⣿⣄⠀⢀⣸⡿⠀⣸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡀⢹⣷⣀⠀⢀⣼⡟⠀⣾⡇
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠛⠿⠟⠋⢁⣴⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⣄⠙⠻⠿⠟⠋⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣶⡾⠟⠋

⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⡀⢀⡁⠀⠀⠀⠀⠀⢀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠲⢤⣀⡐⠒⠂⠀⡀⠀⠄⠒
⠀⠀⠀⠑⣂⣾⢛⣻⠗⠶⣽⢳⣖⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠎⣰⣌⣩⢥⡓⢒⡦⣟⣩⠇⢀
⠀⠀⢰⣿⢻⣿⠿⠿⢿⣿⣿⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⢳⣶⣩⠵⠲⢿⣿⣶⣾⣷⣆⣻⠇
⠀⠀⠀⣁⠩⢵⣶⡆⠈⠁⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣄⣀⣀⣂⣤⠀⠀⠀⣠⣠⣤⡤
⠀⠀⠀⠉⠉⠉⠛⠛⠛⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⢤⣖⠚⢿⣤⠿⣿⣶⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⣞⠃⢿⠿⡇⢡⣜⠉⢀⣼⣿⠜⠥⣆⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⣉⡀⠴⠰⠀⠀⣐⣚⣙⣀⣐⣔⣀⣀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠠⢀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⢫⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠩⠻⡁⠈⠀⠀⠀⠀⢀⢠⠄⢒⡒⠒⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠊⠃⠀⠤⠼⡀⠜⠊⣥⠀⠀⢹⣧⠈⠐⢄⠀⠀⠀⠠⡂⢀⣄⣀⠐⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢐⡦⠀⠉⠐⢳⠄⠘⣿⠤⠄⠀⠂⡀⢰⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⠁⢀⡈⡶⢤⣀⣥⠄⠊⠙⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠈⠩⠀⠅⠀⠀⠀⠀⠈⠏⠂⠴⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⣀⠂⠀⠰⠀⠀⠀⠀⠐⠂⠀⠀⢀⣈
⠀⠀⠀⠀⢠⣴⣾⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢠⡆⠀⣮⠲⠦⠖⠀⠂⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠁⢂⡀⠀⠀⠀⡠⠃⠄⡉⠃⢀⠒
⠀⠀⠀⠀⢿⣿⣿⠛⠟⠀⠀⡠⡅⠀⠀⠔⠀⠀⠠⠀⢠⠂
⠀⠀⠀⠀⠀⠙⠻⢶⠀⠀⠀⣄⡃⠠⠈⠀⠀⡠⢁⠄⠁
⠀⠀⠀⠀⠀⠀⡀⠜⠂⠀⠀⠀⠀⠀⠀⠀⢌⠕⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠂⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⠶⠿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⠶⠶⠿⣼⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣽⣷⣤⣄⢀⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠁⠀⠀⣷⠢⠎⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢬⡆⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢏⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⣾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⡇⠀⢠⠂⠀⠀⠀⠀⠀⠀⡏⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢐⠄⡸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⠀⠀⠀⡔⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢜⠄⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠊⡀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⢎⠎⣽
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠞⣸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⡐⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠊⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠦⣄⣀⣀⣠⡴⡟⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢥⣾⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢪⢺⡆
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⠆⢀⠔⠀⠀⢀⠔⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡱⢣⢾⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⡰⢣⠞⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⡡⠃⣧
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡱⠃⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠱⢁⢺⡇
⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠳⠦⢼⠇
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠻⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⠄⠴⡉⠉⢣⠀⡝⠛⢲⡀
⠀⠀⠀⢀⡤⢧⡀⣠⣿⣶⣿⣿⣿⣿⣿⣷⣤⡤⠤⣄
⠀⠀⢰⡋⠀⢀⣾⣿⣿⠿⠿⣿⠛⠛⢯⠀⠙⡆⣀⣠⠃
⠀⠀⢠⠟⠒⣿⣿⣿⡇⠀⠀⠈⡆⠀⣼⡤⠞⠫⡀⠀⢱
⠀⠀⠱⡄⠀⣿⣿⣿⠗⠤⣀⣠⠞⠉⠀⣇⠀⣠⠗⢴⠋
⠀⠀⠀⠙⠂⢿⣿⡿⠀⡀⠀⠑⠤⠤⠜⠁⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠉⢆
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠉⠙⠂⠀⠀⢄⢤⣀⠴⢂⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢠⠈⢁⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣀⠤⠤⣀⣠⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠰⠀⣠⠀⡄⢸⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠝⠉⠙⠶⠤⣤⣤⠤⠚⣦⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠁⠀⠀⠀⠀⠈⠲⠤⣠⣀⣀⣀⣀⠴⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⢰⠓⠦⡤⠶⠦⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠊⠀⠀⠀⠀⠀⠀⠉⠉⠀⢩⠀⢸⡉⠑⠦⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠈⠙⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠇
⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠒⠒⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠋⠁
⠀⠀⠀⠀⡠⠊⠡⠀⠀⠀⠄⠀⠀⠒⠀⠠⡈⢳⡀
⠀⠀⠀⠀⣇⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⢀⣠⡇
⠀⠀⠀⠀⣧⠉⠒⠲⠦⡤⠤⠄⠔⢶⠒⢋⣽⡞⣽⡄
⠀⠀⠀⠀⣿⣙⣓⣚⣺⣦⣤⠤⢶⡧⢾⠛⠒⢏⡹⡀
⠀⠀⠀⠀⢼⠀⣉⢹⠀⢸⠉⠉⡉⠚⠐⢅⠠⢃⡵⢇
⠀⠀⠀⠠⡛⢦⣄⣙⣤⣼⣉⣉⣥⡤⡴⠖⠚⣹⣤⡏
⠀⠀⠀⠀⠸⡖⠢⠤⢤⣌⣀⣀⣠⣤⣏⡴⢾⠉⢠⠃
⠀⠀⠀⠀⠀⠱⢄⣁⠀⠀⠉⠁⠀⠠⠂⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠄⡀⠀⣀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠔⠁⢠⠆⡐⠫⠄⠀⣀⢡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣸⡱⠂⢠⠀⣎⠁⠒⠧⠛⡓⠈⠀⢀⠈⣇⣰⢶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡿⢿⣶⣾⣞⠒⡲⣶⣿⣟⣳⣾⣷⣶⣚⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋⠿⣿⡗⠞⠏⣿⣯⣹⡛⢩⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠿⢿⣤⣴⣢⡏⠁⠉⣶⠯⣽⣯⢷⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠧⣤⣟⣀⢸⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠁⠀⠑⠀⠀⠀⠀⠉⢹⠛⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠿⠈⣁⣿⣧⣤⠤⠴⠶⠒⢻⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣉⣉⣛⣀⣀⣀⣀⣀⣀⣀⣸⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⠛⠛⠋⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣄⡀⠀⢠⣾⣯⣤⣤⣤⣿⣧⣤⣤⣬⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⢸⣿⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣧⡀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⠀⠙⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⡏⠀⠀⠀⠀⠈⠙⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃

⣿⣿⠟⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠉⠉⢻⣿⣟⠉⠉⠉⠉⢹⣿⣿⠋⠉⠉⠉⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠶⣶⣾⠿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠉⢻⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡟⠋⠉⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣤⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⠤⠀⠀⣴⣥⡀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠞⠁⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⢠⡄⠀⠀⡄⠀⠀⠀⠁⢶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠂
⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠰⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⢠⣴⠢⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠘⠠⣠⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠄⡀⠀⠀⠀⡠⠢⠤⠁⠀⠀⠀⠒⠰⠀⠀⠀⠀⠄⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠊⠂⠀⡀⢈⠔⠀⠈⠢⡑⠢⠐⠍⠙⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠑⠁⠁⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠏⡅⡊⠉
⠀⠀⠑⠂⠠⠤⠄⠀⠀⠀⢀⡀⠀⠀⠀⣀⣼⣠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠇⣇⡇⣀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠳⠿⠆⠆⠢⠄⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡎⠀⠀⠀⠀⠀⢀⡄⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⡶⣶⡶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠙⠻⣿⣦
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣴⣿⠿⠛⠻⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣾⡿⠟⠛⠿⣿⣦⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣤⣤⣼⣿⠏⠀⠀⠀⠀⢹⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣴⣿⡟⠀⠀⠀⠀⠘⣿⣧⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⣇⠀⠀⠀⠀⣼⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣧⡀⠀⠀⠀⣰⣿⡟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁





⡿⢿⠿⡿⡿⣻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢈⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠁⠀⠀⠀⠀⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣋⣿⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⢶⣶⣾⡧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣩⣿⣿⣥⣶⠶⠶⠾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡶⢧⡀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⡾⣷⣦⣞⣱⠃⠈⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣿⣟⡿⣿⣟⣿⣿⣶⣇⣀⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⢻⣿⣿⣿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡛⣾⣿⣳⢿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⡟⢃⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣻⢡⣿⣗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠉⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣛⣛⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢿⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠙⠻⣿⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠟⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁


⠀⠠⠂

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠤⢤⣤⣤⣤⣤⣤⣤⣄⡀⡀
⠀⠀⣀⣴⣥⣼⣿⣿⣿⣿⣏⡀⠀⠀⢀⡠⣊⣹⣿⣿⣿⣿⣷⠶⠆⠦⠀⢀⣠⣮⣬⣿⣤⣀⠀⡀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣶⠂
⠀⠘⢻⡻⠻⠿⠿⣟⡿⠛⠛⠉⠀⠺⢿⢻⣿⣿⣿⢻⣿⠿⠟⠉⠈⠁⠀⠛⣟⠟⠻⠿⠷⠷⠈⠀⠀⢻⣛⡿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠤⠁⢸⠛⢲⡀⠀⠀⠀⢀⣤⣠⣤⣤⣤⣤⡄⣀⣀⡀⠀⠀⠀⠀⠀⢐⣾⣂⠀⠐⣶⠀⠀⠀⠀⠀⠀⢰⣿⣿⣴⣿⠤⡄
⠀⠀⢰⣾⣶⣿⣿⣿⣿⣷⡄⠀⢠⣶⣾⣿⣿⣿⣿⣿⡇⣿⠟⠀⠠⠄⠴⣶⣿⣿⣿⣿⡟⠀⣼⣧⡄⠀⠐⢒⡶⣾⣿⣿⣿⣹⢃⣿⣤
⠀⠀⠘⠯⠟⠛⠛⠛⠯⠟⠃⠀⠈⠯⠟⠛⠛⠛⠛⠭⠃⠃⠀⠀⠀⠀⠀⠀⠿⠭⠟⠛⠛⠛⠉⢯⠽⠀⠀⠀⠁⠙⠶⠛⠛⠛⠋⠘⠶

⠀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣶⣶⣶⣦⣤⣤⡀
⠀⢀⣬⣅⣷⣾⣍⣿⠓⢶⡦⢠⣀⡀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀⠘⣇⠀⠐⠲⠶⣶⣾⣿⣿⣿⠉⢉⣿⣷
⠀⢸⣿⢿⣇⢿⣏⣹⠷⢶⣷⡊⠉⠀⠐⠶⣶⣿⣿⣿⣿⣿⣷⠀⠤⠈⠼⢿⣿⣿⣿⣿⣁⣠⣿⣷⡄⠈⠉⠉⠙⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠑⠞⠁⠀⠁⠀⠀⠘⠖⠃⠀⠀⠉⠙⠛⠳⠟⠉⠉⠧⠃⠀⠀⠀⠀⠐⠿⠼⠛⠉⠉⠉⠘⠧⠇⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣴⠷⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⣤⠀⠀⢀⣰⣿⣿⣿⣿⣷⣴⣿⣶⠄⠀⣼⣤⣤⣥
⠠⠨⢽⣿⣿⡇⠀⠆⠈⣷⣶⣦⠀⠀⠀⣀⣤⣴⣿⣏⣽⣏⡹⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣼⣋⣳⣄
⠀⠉⠀⠘⢋⣷⣄⣀⣴⣿⡿⠃⣀⡠⢽⣾⣿⣿⣿⣯⡴⠋⠀⠀⠐⠒⠛⠿⢿⣿⣿⠿⢿⣿⣿⣿⠀⠀⡸⢿⣿⢿⡿⡿⣿⣿⡿⢿⡿
⠀⠀⠀⠀⠘⠔⠀⠉⠘⠔⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠖⠛⠛⠉⠁⠀⠀⠙⠊⠙⠚⠀⠀⠀⠀⠙⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠐⢲⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⡶⠤⡤⣤⣀⡀⠀⠀⠀⠀⠀⡎⠄⠂⠀⠀⠀⢸⠀⠠⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠚⠄⣇⠈⠈⠠⠀⠉⠝⠝⠒⠦⢤⡇⠀⠀⠀⠀⠀⣬⡀⢄⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢀⣠⠠⠤⠖⠚⠚⠙⡟⠾⣅⣀⠀⠀⠀⠀⣏⠠⢐⠇⠀⠀⡀⠤⠀⠀⢀⣤⢄
⠀⠀⠀⠀⠀⠀⡠⣤⡤⡄⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠨⠇⠈⠰⢹⠦⠀⠀⠀⡗⠈⣺⠀⠀⠎⣠⠴⠶⠶⠼⠶⠟
⠀⠀⠀⠀⠀⡀⠀⢸⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠐⠌⣿⠅⠁⠀⢐⡃⡁⡇⢀⠌⡰⠁
⠀⠀⠀⠀⡠⠴⠀⠀⠓⠛⠀⠂⣀⣒⣒⠒⠒⣂⣐⣂⠀⣀⣈⣩⣉⠉⣭⡭⡭⡉⢉⣭⣭⣍⠉⡰⠁
⠀⠀⠀⠀⠄⢎⡋⠉⠹⢷⠸⡍⠀⠉⠀⠁⢸⡀⠀⠀⠠⡇⢼⠂⡜⢰⣟⡠⢀⠀⣿⠂⣸⠆⡰⠁
⠀⠀⠀⠀⠈⡄⢻⡀⡀⢟⣧⢱⡀⠀⠀⠀⢸⠃⠀⢸⠀⡁⣼⣺⠃⣼⡯⢤⠖⠠⣇⢄⠟⡰⠁
⠀⠀⠀⠀⠀⠘⡄⢉⣉⣉⣉⠀⣁⣉⡀⣁⢈⡩⠭⢅⢀⠭⣤⢤⢀⣴⡖⢖⠄⡴⠖⡆⢠⠃
⠀⠀⠀⠀⠀⠀⠘⡌⢻⠿⠏⢇⢷⡀⠀⢰⢸⠀⠀⠸⢸⠀⡟⣻⢸⢽⠃⡴⢠⣻⣱⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠣⣘⣸⠈⠧⠀⠹⠈⠇⠨⠄⠸⠥⠷⠃⠘⠛⣓⣃⣘⡛⢡⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠉⠈⠉⠉⠁⠀⠀⠉⠁⢠⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⡀⣀⣀⣀⠀⠀⡁⢘⡄
⠀⠀⠀⠀⠀⠀⠀⣥⡤⠤⠀⠀⠒⠒⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠤⠄⣄⣠⠃
⠀⠀⠀⠀⠀⠀⠈⡐⢸⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⢲⣆⡀
⠀⠀⠀⠀⢀⠀⠀⢽⣿⢟⣬⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⠀⠀⠀⠀⣨⡳⠷⢟⣇⡀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠂⠁⠈⠀⠁⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠉⠁⠉⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⢿⣿⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⢀⣀⣀⣀⣀⠜⠀⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠔⠉⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠈⠑⠄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠸⡀⠀⠀⠀⢨⠂⠂⠀⠀⠀⠀⠀⠀⠀⠂⢺⠀⠀⠀⠀⠸⠂⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢟⣿⣿⠿⠿⠿⠿⢿⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⣫⣷⣶⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣶⣾⣝⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠳⣿⣿⣿⣿⡷⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢺⣿⣿⣿⣿⠏⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⡯⠀⠀⠀⠀⠀⠈⢿⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠟⢛⣛⣁⣀⣀⠀⠀⠀⠉⠉⠘⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣿⣿⣿⡒⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⡀⠀⠤⠀⢰⠂⠀⠀⣀⢀⡀⠀⢀⠀⠀⠀⢸⡏⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠁⣀⠀⠀⣿⡀⠏⢢⡿⣿⡟⢠⡎⠀⠀⠀⠘⣷⢹⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⠈⣿⡆⢇⢹⠀⠀⣿⣿⠀⠀⠀⢸⡇⠨⠀⠀⠀⠀⠀⣿⠠⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣷⣄⢹⣇⣾⡄⡅⡀⢹⡏⠀⢸⣷⣿⠁⢠⡀⠀⠀⠀⠀⢻⡗⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢿⡸⣿⣿⣻⣧⡀⣷⢸⠇⠀⣼⣿⠇⠀⣼⣧⣤⠄⢰⡄⠚⠓⠼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣼⠷⠛⠛⠉⢻⣿⣿⣾⡰⢠⣿⣿⣶⣦⣿⣿⣿⣴⣀⣤⣐⣴⣦⡀⠘⠿⡃⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⠻⠿⡿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋⠛⠉⠉⠁⠀⠀⠀⠉⠉⠙⠟⣿⣿⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⠸⣿⠉⠁⣀⡤⠴⠶⠶⣶⠶⡶⠖⠲⠦⠤⠤⠴⠃⠈⠂⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⢦⡨⢛⣩⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⠐⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⡈⠉⠘⡿⠿⣛⣋⣻⣍⣉⡋⢉⡉⠙⣛⣻⣛⡛⠛⠃⠲⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠉⢁⣶⣾⣿⣿⣿⣿⣿⠿⠿⠷⢶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣠⣈⠻⢿⣍⣍⡀⠀⡀⠆⠁⠈⠀⠀⠀⠈⠈⠉⠑⣢⣦⣠⡴⠏
⠀⠀⠀⠀⠤⠤⠶⣶⣾⣿⣿⣤⣤⣤⣤⣤⣼⣇⣤⣤⣤⣤⣴⠿⠶⠞⢛⣟⣓⣛⣛⣻⣟⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣻⣟⣿⣿⣿⣿⣷⣶⣶⣶⣷⣿⣾⣿⠿⠿⠿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠙⠛⠿⠿⠿⠿⠟⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠸⠿⠿⠿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⣿⣧⠀⠀⠀⠀⣼⣿⡟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠇⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉⠦⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠡⠀⠈⠢⡀⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠂⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⢀⣀⠀⡆⠀⠀⠀⠀⠀⣀⢀⣀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠄⢤⠒⠈⠇⠋⠀⠃⠋⠑⠢⠄⠠⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⣴⣦⠀⠀⠀⣾⡆⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠁⠀⣇⠀⠙⠁⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⠀⠠⠈
⠀⠀⠀⡈⠧⠀⠀⠀⠌⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠀⡀⠑⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠁⠀⠈⢓⡀⢈⠀⡀⢁⡘⢄
⠀⠏⠓⣇⡄⠐⠀⠘⠑⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣠⣨⠀⠒⠒⠂⢸⠅⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⠤⠀⠐⠀⠀⠙⡌⠠⠉⢺⣅
⠀⡄⠙⣯⡄⠀⠀⠄⠙⢿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠉⠀⠀⢄⠀⠀⢀⠌⠀⠀⠀⠠⡀⠀⠀⠀⠳⢢⡰⠁⠚⠕⠒⠠⠈⠀⠀⠈⠻⣥⠄⢰
⠐⡀⠛⡷⠆⡘⠀⠀⠙⢺⠖⡈⠀⣶⣀⣀⣈⣀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⣰⡆⠀⠉⠂⠀⠋⠀⠀⠀⠀⢀⠈⢛⠶⢀
⠀⠀⠀⠇⠁⠀⠀⠀⠀⢸⠁⠀⢀⣿⣶⣦⣦⣶⡶⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠄⠀⠠⢹⡧⠀⠀⢠⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠌⠀⠈⠂⡔⠃⠈⠓⢤⠀⠀⠀⢈⠠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠌⠀⡅⠶⠀⠀⠠⠄⠠⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⠂⠁⠙⠃⠒⠂⠐⠒⠐⠂⠀⠈
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠘⠀⠀⠠⢦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠖⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⡀⠀⠀⠀⠈⣄⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⡰⠐⠀⠂⠀⠀⠀⠀⠀⠀⣀⡤⠤⠀⠀⠀⡀⠀⠈⠈⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠍⠀⠘⠀⠒⠀⠀⣀⠀⡀⠀⠀⡀⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣤⣤⡀
⠀⠀⠻⣿⡿⢻⣿⡀
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢻⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡋⠉⠉⠉⣿⠉⠉⠉⠉⢹⡟⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠙⣿⠉⠉⠉⠉⢹⡏⠉⠉⠉⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢺⣿⠛⠛⠛⣿⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⢻⡟⠛⠛⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠶⠶⣿⠲⠶⠶⠶⢾⡿⠶⠶⠶⠖⢺⡗⠒⠒⠶⠶⠾⣿⠶⠶⠶⠶⢾⡿⠶⠶⢾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⠶⣿⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⢶⣿⣶⣶⣶⣶⣾⣷⠶⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⢸⣇⢀⣀⣀⣠⣼⣷⣶⠶⠶⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣿⣤⡶⠶⠾⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣴⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠓⠻⣭⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠤⠤⢤⡤⠶⠶⡶⠶⠶⡾
⠀⠀⠀⠀⢴⠛⠛⠛⠉⣯⠉⠉⠀⢻⡀⠀⠀⠀⢻⠀⠀⠀⣾⠀⠀⢰⠇⠀⢰⠇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠸⡄⠀⠀⠈⣧⣀⣀⣀⣸⣄⣀⣠⣧⣤⣤⣾⣤⣤⡟
⠀⠀⠀⠀⠀⢹⡖⠒⠒⠚⢻⡛⠉⠉⢻⠉⠉⠁⢹⠁⠀⢸⠃⠀⣼⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢧⠀⠀⠘⣇⠀⣀⣸⣀⣀⣸⣀⣰⣇⣀⣸⠃
⠀⠀⠀⠀⠀⠀⢸⡶⠒⠒⠒⠛⣟⠋⠉⢻⠉⠉⢹⠉⠉⣿⠁⣹⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣹⣄⣀⣘⣇⣀⣸⣄⣠⣯⣠⣯⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠶⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡄⠀⠀⠀⠀⠀⠀⣰⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⣀⡟⠀⠀⠀⠀⠀⠀⠹⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⡗⠒⠠⠤⠤⠤⠄⢀⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⣷⣾⣅⣉⣉⣉⣉⠙⠛⠒⠒⠂⠠⠤⠭⠭⠭⣭⣭⣛⣒⣲⠖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⠃⡶⠲⠖⣤⠤⠤⢤⡤⣉⣉⣁⡘⠒⠐⠒⠲⠤⠄⣠⠋⠙⣄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⡏⢩⢰⠀⡇⢸⠀⡟⢰⠀⠸⠀⡇⢸⠈⡍⢸⠋⡗⠒⠒⣰⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠏⡇⣅⣤⣶⣿⢻⡟⠀⠁⠀⢠⡇⢸⠀⡇⢰⠀⡆⣸⣁⡶⠴⠗⡞⢹⢃⣃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⢺⠉⣹⣤⡶⠟⡟⠩⡇⣿⣿⣈⡇⢸⠀⡆⠸⠁⣈⢠⠧⠾⠊⡏⢸⠃⣦⢸⣠⡧⡞⠛⢩
⠀⢀⣀⣀⡤⡴⡞⢹⠉⢀⣀⣭⢶⠟⢻⠀⠀⠀⠃⡄⢳⣯⣿⣾⠃⣘⡠⠷⠞⠛⠉⢸⠀⣄⣀⡇⢴⠒⡏⡉⢰⣸⠇⡇⡘
⡎⠀⢼⣭⣇⣇⣥⣼⠚⠋⠉⠀⢸⠇⢸⠀⣠⣤⡶⢟⣿⡭⡷⠛⡏⠀⡇⢠⡂⣠⠤⡾⠈⡏⠈⠇⠘⢸⠇⡇⣸⣿⢴⠛⠇
⠑⣲⠚⠀⠈⠀⠁⢨⠉⠛⠓⡖⢶⡟⠙⠛⠛⠋⠙⡋⠉⠀⡇⠀⣧⣠⡗⢺⠉⣿⠀⡇⢀⠃⢸⣀⡤⠾⢺⢋⡏⢸⡺⢸
⠀⢸⠉⠛⢚⠲⡦⢼⠦⣼⢈⣇⢘⡓⠤⠤⡤⢶⢚⡇⠴⠒⡛⠉⡇⠀⡇⢸⠀⣿⢀⡧⢼⠃⢹⠀⠁⠛⢸⣾⠇⡌⡇⡇
⠀⢸⡄⢎⢸⠁⡄⢸⠀⠘⠀⡏⣈⡅⡿⠐⡗⠛⠉⠃⢨⠀⠀⢀⣇⣠⠤⠾⠘⠋⠀⠀⠘⠀⠘⢰⠀⠇⣾⣸⣤⡷⢢⠁
⠀⠘⠁⠛⢹⠓⣗⢺⠦⢾⢼⣇⣈⡀⡇⠀⡇⡇⢀⡄⡴⢐⡏⠈⠁⢰⠀⡇⠀⡆⢸⠀⡆⢀⡦⣼⠲⢻⡏⡈⢠⢸⡎
⠀⢀⡦⣴⢸⠃⡇⢸⠀⠀⢸⡇⠈⠉⣏⠙⡗⡏⠙⡇⣿⠀⠈⢸⠀⢼⢀⣇⣀⣧⢼⠒⡏⠀⠇⡸⢰⣿⡃⠇⠘⠘⠇
⠀⠈⡏⠙⢹⠓⠷⢾⠄⣤⢰⡃⢚⡈⡇⠀⡇⡇⠀⡇⣿⢠⡤⢼⠖⢻⠉⡇⢰⡇⠸⠁⠛⢰⠂⡇⣼⣭⠶⠓⢿⢹
⠀⠀⡇⠶⢺⠀⡇⢸⠀⢸⠈⠁⢹⠉⣟⠛⣿⡷⠚⡇⣿⢨⡄⢸⠀⣘⢀⠇⣀⡠⣤⠴⠖⢺⠚⡏⡿⠸⢸⢀⡟⡇
⠀⠀⡅⠙⠛⠗⡷⢺⡤⣼⢤⣔⣸⡃⡏⠀⡇⡇⠀⡇⣿⣐⡀⢬⠔⢿⠙⡏⠉⠃⡶⠠⠂⡌⢰⡏⠀⡤⣘⣼⣷⠁
⠀⠀⠃⠰⠛⠂⡏⢸⠀⢸⠀⡀⢸⠁⡟⠉⠛⡗⠚⡏⣿⢨⠁⢸⠀⠘⠀⡇⢸⠀⡇⢐⠀⣧⣼⡡⠞⠉⠉⢙⠃
⠀⠀⠘⠚⠲⠧⠧⣨⣀⣸⢐⡇⠀⠀⡇⠀⠀⠇⠠⡇⢿⢸⠀⢸⠀⢀⣀⣧⠼⠴⠗⠚⠒⢣⣀⠀⠀⡈⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠚⢰⠤⢴⡄⣦⡤⠥⠶⠰⠖⠚⡛⢏⠁⠀⠀⠀⠀⠀⣠⠞⢁⡠⠊⠉⠓⢦⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢳⡀⠀⢀⡴⠊⣠⠔⠋⠀⠀⠀⠀⠀⠳⡄⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠋⣩⠖⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢦
⠀⠀⠀⠀⠀⠀⢀⣮⠀⢴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠾⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆⠀⢢
⠀⠀⠀⠀⠀⢀⠏⡔⠢⠤⢄⣈⡑⠒⠤⠄⣀⡀⠀⠀⣠⠖⢋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⣤
⠀⠀⠀⠀⠀⡜⠸⠀⠀⠀⠀⠀⠈⠉⠒⠒⠠⠄⢀⠞⢁⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢎⢺
⠀⠀⠀⠀⡠⠇⠡⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛
⠀⠀⠀⠀⠓⣤⠀⠀⡷⡄⠀⠀⠀⠀⠀⠀⡜⠀⡆
⠀⠀⠀⠀⢸⠴⡳⣤⡇⡙⠀⠀⠀⠀⠀⢰⠣⠠⠑⣄
⠀⠀⠀⠀⠈⠳⠌⠩⠔⠁⠀⠀⠀⠀⠀⠈⡵⢢⡀⣾⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⠱⡿⡸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣟⣿⢻⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠊⠇⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣉⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣇⠀⡠⠼⠔⢣⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⣠⠔⠚⢆⢫⠀⠀⠘⡆⢰⠀⠀⠸⠀⡎⠉⠑⢢⢀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢣⠀⠀⣈⡮⣆⠀⠀⠱⠘⠀⠀⠀⡿⠀⠀⢀⣃⠏⠀⠀⢹⠂
⠀⠀⠀⠀⢀⣀⠤⣤⣴⠦⠤⠶⢾⠀⠷⠶⠦⢴⣦⣤⠧⣀⣧⠀⠸⡀⠀⠀⢻⡂⠀⢀⠇⠀⠀⡘⠉⢓⠄⢠⠃
⠀⠀⠀⢰⣅⠰⢍⣀⠀⠀⠀⠀⠘⠖⠁⠀⠀⠀⠀⣀⡩⠆⣨⡄⠀⠇⠀⠀⢸⡇⠀⠸⠀⠀⠀⡇⠀⡞⠀⡏
⠀⠀⠀⠘⡓⠮⠭⠭⢛⣛⣛⣓⣒⣓⣚⣛⣛⣛⡛⠭⠭⠵⢺⠹⡀⢸⠀⠀⢸⠇⠀⡇⠀⠀⢰⠀⢠⢁⣼
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⢇⡀⡆⠀⢸⠀⠀⡇⠀⠀⢸⢀⡼⠋⡌
⠀⠀⠀⠀⢰⠀⠀⠀⢀⡾⣧⠀⠀⠀⠀⠀⣼⢧⡀⠀⠀⠀⣶⠀⠀⠈⠑⠒⠚⠤⠴⠇⠐⠒⠉⠁⠀⢠⠃
⠀⠀⠀⠀⠘⡀⠀⠀⠚⠁⠘⢃⣤⣤⣤⠘⠃⠈⠃⠀⠀⢀⠿⠀⠀⡴⠛⢶⠄⠀⠀⠀⢠⠞⠳⡦⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⡿⢿⠇⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠻⠦⠞⠀⠀⠀⠀⠀⠀⡌⠀⢡⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣅⡢⡖⢈⡉⠭⠶⣞⡢⢤⣤⣤⣤⣈⡉⣀⡀⠀⠀⠠⠼⡤⢤⣐⡲⠶⠍⣉⣑⠦⠄⣤⣄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠱⣹⣉⣐⣲⣖⣂⡉⠉⣉⣒⠢⣍⡙⠛⠛⢛⣛⡻⠟⣋⣒⡢⢤⣴⣶⣊⡕⠊⠉⠁⠈⠆
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠈⢇⡀⣠⠞⠉⠀⠶⠀⠬⠍⣁⠙⣃⡐⢛⣒⠛⠓⠒⠚⣛⠚⠋⢻⠉⢉⣍⢀⣄⠀⠈⠙⢦⡀⢀⠆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠋⠀⠀⠀⠙⠂⠈⠶⠀⠁⠗⠈⠃⠠⡏⠰⠃⠘⠃⠴⠂⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠤⠤⢀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⠠⠤⠤⠤⠄⠒⠒⠊⠁



⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣠⡤⢤⣤⣤⣤⣤⣤⣤⡄⠠⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⡼⠋⠉⠉⠉⠉⠉⠉⠉⢹⡇⢨⣿⢸⡇⠀⠀⠀⣿⢷⡄⣾⠛⠀⣾⠀⠀⣷⢸⡆⢰⡆⡟⠋⢸⣿⡆⢷⢀⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢠⡾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⢸⡇⠀⠀⠀⣿⠀⡇⣽⠖⠀⣿⠀⠀⣿⠀⣧⣾⠀⡷⠂⢸⣿⠃⠘⣿⠃⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⢸⡇⠀⠀⠀⣿⡼⠃⢿⣤⡀⢿⣤⠀⡿⠀⢹⡏⠀⣧⣤⢸⠙⣧⠀⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠘⠿⢯⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡄⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⣭⣽⠇⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣼⡏⢹⡇⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣶⣶⣶⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⢀⣴⣿⡿⠟⠛⠛⠛⠿⣿⣷⢦⠀⢸⣿⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⢞⣷⠿⠛⠛⠛⠛⠻⢷⣽⣦⡀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⣾⣿⠋⠀⡠⣒⣶⣦⣄⠀⠹⣿⣷⠈⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⢏⣿⠃⢀⡴⣪⣭⣗⢦⡀⠙⣷⣿⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⢹⣿⠿⠷⠾⠿⢿⡏⠀⢸⣱⡏⠀⢹⡟⡇⠀⣿⡿⠾⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠾⣾⡏⠀⢸⢸⡏⠀⢸⡇⡇⠀⣿⣿⣶⣶⣶⣶⣶⣶⣿⣦
⠀⠀⠀⠀⠈⣿⣉⣉⣉⣉⣹⣷⠀⠈⠻⣷⣶⣾⠽⠁⢀⣿⡓⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⣿⡀⠈⠫⣿⣶⣛⡵⠁⢀⣿⠓⠒⠒⠒⠒⠒⢒⣺⡿
⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠛⠻⣷⣄⡀⠀⠀⠀⢀⣠⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⣄⣀⠀⠀⠀⢀⣠⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣿⣳⣦⡀⠀⣀⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⢠⣤⣄⡀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠹⡄⠀⢀⣠⣤⣽⣿⣉⠉⠉⠻⣭⡾⠻⣏⠀⠀⠀⣠⡴⠶⠚⠷⢦⣄⠀⢹⣦⡼⠟⣧⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠶⠆⢻⣴⠿⣥⣴⠋⢀⡿⠁⢀⡼⠋⠀⠀⠈⣻⣦⣾⠃⠀⠀⠀⠀⠀⠙⢷⣟⣿⣠⡼⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠰⠶⠶⣾⠋⢀⡾⠃⢠⡟⢁⣴⠏⣀⡀⠀⣴⠟⠉⠉⠉⠳⣄⡀⠀⠀⠀⠀⢸⣷⠟⠁⠀⠈⣷
⠀⠀⠀⣠⠤⣤⣤⣤⣄⠀⠸⣷⠾⠁⠀⢀⣾⠁⣠⡿⠛⠲⠏⣠⠞⠁⠸⣏⡹⠀⢻⣦⣄⣀⠀⠀⣿⠛⠶⣤⣀⣀⣸⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠻⣤⣼⣿⣿⣿⣷⣄⣻⣀⣀⣀⣞⣉⣛⣿⣀⣀⣀⣾⣋⣀⣀⣀⣀⣀⣀⣸⣃⣈⣉⣹⣛⣁⣀⣀⣸⣋⣉⣉⣳⣄⣀⣀⣼⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡛⠛⠛⢻⡟⠛⠛⢻⡟⠙⠛⢻⣿⠛⠛⠛⣿⠛⠛⠛⣿⠛⠛⠛⣿⡟⠛⠛⢻⡟⠛⠙⢻⡟⠙⠙⢻⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢸⣇⠀⠀⢸⣷⠀⠀⠘⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣼⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠸⣿⠀⠀⠈⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠇⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣠⣿⣄⣀⣠⣿⣄⣀⣀⣿⣄⣀⣀⣿⣄⣀⣀⣿⣄⣀⣀⣿⣄⣀⣀⣿⣇⣀⣀⣿⣇⣀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡋⠉⠉⣿⡋⠉⠉⣿⡏⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢻⡇⠀⠀⣿⡇⠀⠀⣿⡄⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢸⡇⠀⠀⢹⡇⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢰⣿⠀⠀⢸⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣸⣧⣀⣀⣸⣇⣀⣀⣹⣇⣀⣀⣿⣀⣀⣀⣿⣀⣀⣀⣿⣀⣀⣸⣟⣀⣀⣸⣇⣀⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣽⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣽⣷⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣯⣿⣿⣮⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⣿⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡾⠟⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⠶⠶⠛⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠟⠉⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⠀⠈⠉⠉⠉⠉⣉⣁⡀⠀⣀⢀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⠀⢽⣿⡇⠀⠀⠀⠀⢰⣹⣿⣸⠀⠻⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⣽⣿⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟
⠀⠀⠀⠀⠀⠀⠸⡄⡞⠉⣭⣭⣭⣄⢸⠀⡏⠀⠐⠀⠐⢸⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⡇⣸⣿⣿⣿⣿⠘⡀⡇⠢⠀⢴⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠻⠿⢿⣿⠿⠆⡇⢇⠀⠀⠀⠀⠄⠿⡿⢻⡟⠛⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⢈⣒⣒⣲⠶⠎⢍⣁⣨⠭⠭⠽⠿⠟⠛⢇⠸⡇⠀⠀⠈⠹⣿⣿⣿⣷⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⢠⣀⠤⡀⢸⡇⠀⠐⣄⠐⠆⠀⢸⠀⣷⠀⠀⠰⣶⣿⣿⣿⣿⣤⣤⠄⠀⠀⣼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡇⢀⡠⠂⠈⠐⠀⢸⣷⠀⠀⠀⠗⠦⠄⠸⠄⠛⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⢠⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢻⡇⠀⠶⠇⠀⠈⣿⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⣸⣿⣿⣿⡿⢃⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠘⠦⣤⣤⣤⣶⣖⣚⢉⣉⣉⣩⣭⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⣿⠟⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⢀⡤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⠐⠂⠀⠐⠂⢸⢸⠈⠁⢸⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠻⠃⠀⢸⠈⡠⠦⠆⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣅⡘⣒⣒⣒⣒⣊⣉⣀⣓⣀⣭⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣮⡻⢿⠿⢟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⡟⠉⠛⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠒⠀⠀⠀⠐⠚⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⡿⠟

⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⠲⡄
⠀⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡗⠀⠤⠤⠄⣀⣀
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡏⠛⠛⢶⡀⢸⢣
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⠈⢷⠈⡆⢣
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⡇⠀⠀⠀⠘⣇⠸⡀⢣
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠓⠒⠒⠒⠚⠋⠀⠓⠒⠣⣀
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡗⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄
⠀⠀⠀⣾⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣆⠘⢧
⠀⠀⠀⣿⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁⡇⠀⠀⠀⠀⠀⢸⡟⢉⣤⣤⣉⠻⣿⣏⣉
⠀⠀⠀⠈⠒⠒⠒⠒⠒⠛⡇⣿⠁⠀⢘⡇⢹⠛⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⢾⠀⡏⠀⠀⢹⡆⠻⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡘⠳⠶⠞⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠶⠶⠛⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣽⣽⣭⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡄⠀⣄⠀⣀
⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⡰⠟⡗⠚⠎⡛⡹⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⢀⣄⢠
⠀⠀⠒⠚⠋⠉⠉⠀⠀⢠⡌⠈⠀⠀⠀⠐⠀⠈⠃⠀⠀⣠⣴⡶⠂⠀⠀⣀⡠⠂⣑⣀⠔⠡⣬⠀⢇⠎
⢠⣀⢄⠈⢉⣆⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠠⢾⡿⠗⠁⠀⠀⡘⠲⡇⠀⣝⡏⠀⡰⡾
⠈⠲⠃⠀⠈⢀⣀⡀⠀⠠⠀⠀⠀⠀⠃⣀⢀⡀⠀⠀⠀⠀⠀⠰⠀⢠⣅⣙⡃⣰⢾⠁⢠⣿⠁⠠⠂
⠀⠀⢀⢄⠀⡆⠄⠔⠁⠀⢠⣍⡆⠀⢘⢖⠉⠃⣠⠠⠻⣀⠠⠀⠀⢸⠃⠀⠉⠈⠉⠋⠒⡇⢀⢄⣠⡄
⠀⠒⢁⢈⠀⠃⡡⠴⠀⢀⢃⠭⡸⠀⠈⣲⠍⡆⠀⡙⣎⡁⢠⠶⣦⢸⠘⠂⠉⠐⠥⠀⣾⠀⠀⠀⠕⡁
⠀⠀⠀⢻⠈⠂⠃⠀⠀⠘⠊⠀⠓⠃⠀⠑⠒⠁⠀⠑⠐⠃⠈⠋⠁⠈⠓⠒⠒⠂⠲⠀⠃⠀⠀⢠⢰⠟
⠀⠀⠠⠒⢣⡀⠃⡑⠀⠀⠠⠫⡷⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⠞⠁
⠀⠀⠀⠀⠁⠈⢙⢤⡄⠀⠀⠁⣂⡀⡀⠀⠀⠹⠒⠇⠀⠀⠠⠀⠀⢠⡔⠀⠀⣀⡀⠠⠀⠀⠰⣿⠴⠂
⠀⠀⠀⠀⠀⠀⠣⠜⠁⠀⠀⢸⠔⠒⠊⢀⢎⢭⢳⠀⡔⡒⠄⠀⠀⠠⡀⢀⠞⠉⠙⢦⠀⢠⠶⢦⡀
⠀⠀⣼⣷⠀⠀⠠⠀⠀⠀⠀⠘⢨⠴⠂⠸⡘⠐⡸⠰⠐⠝⡘⢰⠸⠵⡘⠘⡀⠀⠀⢸⠀⠃⠀⢸⠃⠁
⠀⠀⢳⣿⡇⠀⠀⢀⣀⠠⢀⠘⠚⠀⠀⠀⠉⠉⣠⠤⢵⡮⣀⠘⠒⠈⠠⠀⠘⠢⡤⡎⠀⣀⠔⠁⠀⠸
⠀⠀⠀⠉⣴⣖⠈⠙⢷⣄⠈⣆⢤⠀⠐⠈⢀⠼⠢⡖⠚⠳⡤⣇⠠⢰⡅⠀⠀⠀⢧⠥⠀⠿⠀⠠⡂⠢
⠀⠀⠀⠀⡍⢮⣦⣠⠾⣿⠊⠡⠀⢠⣤⠀⢸⠑⠾⠛⠓⠛⠓⠺⠀⠈⠀⠀⠀⠠⡀⡄⠀⠀⡄⠀⠜⠤
⠀⠀⠀⠘⡄⠀⠈⠀⠀⣿⠀⠀⡇⠀⠉⠀⠀⡀⡆⠀⠉⠊⠀⠸⠀⠀⠀⠀⠀⣀⣥⣤⣠⡀⠁⠀⢱⡎
⠀⢀⣀⡀⠙⠂⠂⠒⠘⠛⠂⠉⠀⠀⠀⠀⠀⢟⠿⠚⡟⠛⠛⡇⠦⠀⠀⠀⠀⢠⣉⣉⣉⡁⠀⠀⢅⡸
⠀⢠⡸⣈⡵⡆⢠⣶⡐⡎⢱⠀⢔⣉⡱⠀⠀⡘⠲⠐⠓⠒⠒⠃⠀⢰⣉⡕⠀⠸⠶⠴⠴⠇⠀⠀⡔⢢
⠀⠠⠳⣦⣄⣈⣋⣀⣉⡥⠋⠀⠀⠀⠀⠀⢼⣽⠰⣿⠠⠴⠦⠀⠀⠀⠀⠀⠀⠀⠁⠒⠚⠁⠀⠀⠘⠇
⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⠿⠿⣿⡇⠀⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣷⣶⣿⣷⣿⠛⣉⠙⣿⣿⣿⠟⢋⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣿⠇⠀⠀⣿⣄⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣆⠠⠶⠄⣰⣿⣿⡀⠻⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣆⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠞⠁⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⣟⣲⡲⣄⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣗⠒⠒⠒⠒⠒⠒⢲⢎⡷⠿⠧⠤⢬⣽⣷⣄⣀⢈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣋⣀⣀⣀⣀⣀⣠⣿⢟⣄⣄⠀⡄⠀⠀⢘⣿⠉⠉⠉⠉⠉⠛⠓⠒⣒⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢽⡿⣿⡏⠩⠉⢹⠿⣿⣿⠟⠻⠿⠻⠷⠻⠿⠾⢿⡦⣿⣷⣧⣧⣤⣶⣸⣏⠩⡪⢂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠚⠋⢠⡶⣤⠀⠉⡸⠇⠂⠀⠀⡞⣋⡟⠀⠀⢸⣷⢿⡆⢀⠀⢸⣏⢙⣿⡆⢘⢕⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡠⣤⠀⠀⠒⣫⣤⢄⣈⠁⠀⢀⡂⠑⠔⠊⠀⠀⠓⠃⠀⠀⠀⠠⠈⠓⠃⣾⣇⠀⡩⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠧⡾⠀⢀⣾⢋⣥⡈⡙⢷⡰⣷⡄⡆⠀⠀⢀⣀⠀⠀⠠⢊⠀⡐⠄⠀⠁⣿⣿⡀⠪⢀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠓⣲⣾⣷⣾⡷⠾⠷⠾⠿⠶⠾⠥⣤⣤⣯⣴⣆⠀⢀⠠⡑⣣⠃⠀⠺⣿⣘⣇⠀⢕⡹⣇
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⡋⠀⠀⠀⣀⣾⠋⢴⡄⠈⢻⣆⠀⣤⠀⠀⠀⠀⠀⢀⣠⠔⠻⡇⠀⠑⠒⠉⠀⠀⣀⣸⠀⢿⡀⠩⢆⢿⡄
⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠺⠶⢿⣯⣤⣤⣤⣧⣧⣿⣤⣀⣦⣄⣤⣴⣸⣏⠁⠠⡈⣧⠀⠀⠀⠈⠀⢸⢿⣿⠄⠸⣇⠐⠕⠸⣧
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⢹⠀⠈⡀⢹⣀⣤⠄⢤⡀⠌⠒⢻⠅⠀⢿⡀⠠⠂⢿⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣔⢍⣿⠻⠲⢻⠇⠀⠀⢸⠇⡀⠸⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⡷⠒⢲⣤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠔⡡⣷⠨⠉⠡⠀⠀⠀⢸⡇⠂⠀⣷⠀⢠⠶⠸⣇
⠀⠀⠀⠀⠀⣾⡇⠗⡅⠀⣰⡖⠀⠑⣤⠊⠙⢄⡠⠋⠡⣈⡩⠋⠫⣉⡝⠛⣍⣹⣷⠀⠪⠀⢸⣠⣄⡀⠀⡔⠒⢌⣇⠁⠀⢸⡀⠐⠩⣂⢹⡄
⠀⠀⠀⠀⠀⣿⣅⠀⢈⠾⡁⠈⡢⡊⠀⠑⣴⠋⠑⢄⡠⠊⠢⣀⠴⠋⢄⡠⠚⢻⣿⡆⢀⠀⢸⡟⠙⠁⠀⠷⣷⠞⣿⠀⠀⣼⢟⠻⣻⣿⠿⠃
⠀⠀⠀⠀⠀⣿⢈⠶⡁⠀⡸⢎⠀⠈⣦⠋⠈⠣⣠⠊⠣⣀⡰⠋⢢⡠⠊⠑⢄⢼⣿⣷⠀⠄⡨⣇⣀⠀⠀⠀⠀⠀⣿⠀⣼⣏⣵⡾⠛⠁
⠀⠀⠀⠀⢠⡟⢄⠀⡸⢎⠀⠀⡱⡎⠀⠑⣤⠊⠈⢢⡔⠉⢢⡠⠊⠑⢤⠔⠉⢺⣿⢿⡄⠘⠄⣿⣧⡷⠀⠀⣰⣏⣻⣼⡿⠟⠁
⠀⠀⠀⠀⢸⣇⡠⠛⣄⠀⡱⢎⠀⠈⡦⡊⠀⢱⡔⠁⠑⣤⠊⠑⢤⠔⠁⠑⣤⢺⡿⣽⣧⣴⣇⣼⠛⠙⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⢸⡋⠢⣀⡠⠫⡀⠈⡷⡊⠀⢙⢶⠁⠈⡲⡊⠀⢳⡔⠁⠑⣤⠎⠀⢹⣿⣿⠣⢈⣽⠟
⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠶⠾⠶⢧⣤⣵⣎⣀⡈⣤⠃⠈⢢⡔⠉⠣⡤⢺⣿⢯⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠳⠷⠤⣮⣌⣾⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠉⠉⠀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡤⠤⡌⡹⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡇⠀⣶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢀⢀⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢂⠀⡑⠦⣠⠄⠈⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠠⠀⠀⠀⢀⠾
⠀⡀⠀⠀⠀⠀⠠⠤⠤⢀⠠⠀⠀⡀⠄⠀⠒⠈⠁⠀⠀⠀⠀⠈⠒⡦⠾⠓⠚⠀⠀⠀⠀⠀⠀⢀⠄⠄
⠀⠃⢰⠒⠚⠂⠀⢀⠄⢠⠛⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⡃⠒⠀⠤⢍⠀⠤⢤⣖⠁⠀⠀⡄
⠀⠀⢸⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⠁⢀⡜⠉⠉⠁⠀⠐⠤⢔⠁⠀⠀⠀⠀⠀⠈⢔⡀⠔⣿⠲⠢⠄⠂
⠀⠀⠸⢀⠐⢁⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⠀⠀⠀⠀⠈⠀⠂⠐⠄
⠀⢠⡐⢁⡔⢁⣀⣀⣀⢀⠀⡆⠠⠒⠓⠤⠤⠄⠀⠀⠀⠠⠤⡤⠄⠀⡈⠐⠅⠢⡀⠀⠀⠀⠈⡄⠈⢄
⠤⠴⠴⠦⠤⠄⠀⠀⠀⠀⠒⣃⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⢠⠀⠘⠀⠀⠀⠀⠀⠌⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠀⠄⢀⠈⠀⠂⠀⠀⡈⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠪⠐⠀⠁⠉⠁⠐⠠⡀⠀⠀⢀⢄⠐⠠⡤⠃⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠦⠀⠈⠁⠦⠈⠰⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠈⠀⠀⠀⠀⠀⣀⠴⡀⢄⠈⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠏⠋⠙⡉⠉⠉⠀⡙⠉⠋⠏⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠄⠀⡅⠠⣐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠐⠠⠤⠐⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢁⣒⣀⣂⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣂⣀⣂⠁⠊
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⣒⡺⠭⢿⠛⠭⠽⢷⣖⣶⣶⠶⠦⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⣚⡩⠥⠐⠊⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠑⠒⣒⡭⢿⠆
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣒⡚⠭⠕⠒⠊⠉⠀⠀⠀⢀⣀⠤⠔⠒⠂⠉⠀⠒⠒⠒⠦⠤⢤⣀⣀⡤⠔⣒⡭⠕⠚⣹
⠀⠀⠀⠀⠀⠀⠀⢳⠤⠭⢙⣓⠲⠤⠤⣤⣴⣒⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⢚⡩⠔⠒⠉⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠒⠒⠦⠬⠭⣉⣙⠓⠒⠲⠶⠔⠊⣁⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⡏⢉⣭⡓⢒⣶⡄⠈⠉⠙⠓⢺⡏⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⢶⢚⠟⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⢟⠟⢲⡾⢃⣔⣓⠚⠻⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⠤⠖⠋⠁⠈⠢⡏⡌⡞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡤⢊⡴⠥⣀⣉⢙⠓⡶⠤⣤⣄⣀⣈⣇⣠⠴⣚⡉⠀⠀⠀⠀⠀⠀⠀⢷⣇⢇⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⠋⣰⡻⠋⠉⠓⠒⠿⠵⣽⡶⠤⣄⣈⢫⠉⣶⢋⠟⣙⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠞⢡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠻⢷⡇⡏⡾⣈⡿⡇
⠀⠀⢀⠴⣒⠤⢴⣋⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢧⠣⡩⢣⡇
⠀⣴⠡⡎⠉⠙⠓⠒⢮⠉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠚⠋
⠀⠈⠙⠚⠯⢵⣂⣖⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠯⣷⡄⠀⠀⠀⠀⠀⠀⠀⡞⡾⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠘⡇⠀⠀⢀⣶⣶⣶⣖⠛⡗⣒⣛⣿⣫⡽⠿⣤⠀⣀⣤⡀
⠀⠀⢸⣿⣏⣉⡉⡏⠉⠉⠉⠉⡏⠉⣹⣟⣿⡿⠛⠻⣿⠋⠉⠁⠉⠀⣰⠦⣿⣺⣧⠀⢹
⠀⠀⢸⠉⠉⠉⠉⠋⠉⠉⠉⢩⣯⣭⣭⣽⣿⣤⣤⡤⢼⠤⠤⠤⢿⣯⣽⣰⡇⠈⢹⡗⠾⣄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⡇⠀⠀⠀⣸⣀⣀⣀⣸⣾⡀⠙⠷⢤⣸⠁⠀⢀⣩⣷⣦
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠏⠉⠉⠉⠛⠉⠉⠉⠉⠉⡇⠀⠀⠀⠙⠓⢤⡼⡇⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠢⣄⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠦⣄⣿⠗⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠀⠀⠀⠰⠠⠐⠀⠀⠐⠀⢺⠄⠐⠀⠠⡂⠐⠀⠉⠉⠀⡀⠀⠰⡍⠕⡀⢤⡀⠀⠀⠀⢳⠂⠀⠀⠀⠣⠘⠀⠶⠀⠀⠀⠀⢀⠀⠠⢀⡀⠘
⠀⠐⡀⠀⠀⡀⠅⠀⠀⢠⠠⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡠⣌⠉⠃⠊⢀
⠀⠀⠀⠀⠐⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠐⠏⠝⢀⡄⢲⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠎⠙⠃⠀⠀⠀⠺⠆⠁
⠀⠀⠀⠀⠄⠈⠁⠀⡠⠚⢁⠤⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠎⠕⠀⠀⠀⠉⠁⠀⠀⠠⠀⠀⠀⠂⡀⠐⢠⣀⠀⠀⠀⢀⠀⠀⢀⡠⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠈⠣⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠘⠀⠀⠀⣀⠀⠀⠂⠈⠙⠃⠀⠂
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠇⠀⠐⡔⢣⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠂⠰⠂⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠑⢰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠠⠀⠚⠀⣄⠀⠂⠀⢀⠀⠀⠀⠉⠀⠀⠀⡋⡜⠃⠀⠀⠂⠀⠐⠀⣤⣄⣁⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠁⠈⠀⠀⠢
⠀⠗⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠃⠀⠄⠀⢀⠧⡀⡮⠸⠃⠀⠀⠀⠀⠤⡁⠉⠛⠃⠀⠒⠶⠃⠀⠀⠀⠀⠢⠲⠁⠲
⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠆⢀⠀⠐⠃⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⢀⢼⠁⢀⠎⡁⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⡀⠀⠠⠈
⠀⣀⠀⠠⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢀⠘⠄⠄⠀⣤⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠁⠀⠀⠀⠀⠂
⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠆⠀⠀⠀⠘⠀⠀⠄⠀⠀⠀⠀⠀⠚⠠⠉⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⡂⠠⠄⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠦⠤⢤⡤⠤⠖⠒⢲⠖⠒⠒⠒⠒⡶⠒⠒⠒⠒⡶⠶⠶⢴⡦⠦⠴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡼⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢳⠀⠀⠀⢧⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡰⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⢣⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠘⡆⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠤⠤⠤⠤⠾⠤⠤⠤⠤⠤⠶⠺⠒⠒⠒⠒⠒⠒⠺⠒⠲⠶⠶⠶⢿⠶⠶⠒⢠⠛⠒⠒⡒⠶⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⣼⡇⠀⢀⣶⠀⠀⣸⡀⠀⠀⣸⡀⠀⠀⣸⡀⠀⠀⣸⣀⠀⠀⡸⢇⣀⣀⡼⣄⣀⣀⣷⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠋⣿⠋⠉⠀⠉⠉⣹⣉⣉⣉⣀⣉⣉⣉⣀⣈⣒⣚⣁⣈⣉⣉⣀⣸⡁⠀⠀⠀⠈⣷⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⡿⠤⣀⠀⠀⢸⠉⠏⠉⠈⠀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠉⣉⠙⠉⠉⡇⠀⠀⡏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠉⣉⣄⢀⣄⣸⡄⠀⢸⠐⡮⠀⣇⣀⣎⡳⢳⠞⠦⠃⢿⡦⠸⠻⠌⢓⣦⠀⢠⠇⣰⠚⠋⠛⢿⢤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣵⢰⣾⠁⠈⠉⢈⠿⠼⠀⢸⣄⡥⣤⡭⠤⠤⠴⠶⠲⠶⠶⠖⠒⠒⠒⠒⠚⠒⠒⠾⢘⣏⢠⣤⡴⣄⣤⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡶⠸⣄⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢰⠛⡄⠀⠀⣸⡇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⢁⣀⠀⢠⢦⡀⠀⠀⠀⢀⣠⣤⣀⡤⢤⣄⠀⠀⠀⠀⠀⠀⢀⢶⡀⠸⣼⠀⢻⢲⡖⠋⢇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠛⢊⣙⠒⠓⣤⠿⡴⢲⣀⡀⡧⠴⣏⢀⡤⣀⠹⣄⣀⣀⡴⠶⢋⣦⢷⠴⣄⣴⢺⠜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠑⡦⢣⡠⠇⠐⢇⡤⢤⠈⢁⣽⣅⢖⡌⣻⢶⡚⢀⡏⢀⡟⢥⠀⣬⣹⣄⠀⠁⠸⡄⠀⠡⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠙⠣⠤⣶⠦⠴⠋⣛⠉⣆⢈⣿⠮⣉⡴⣇⣠⠟⡟⣉⠙⠧⣏⠛⠁⠀⣉⢧⣤⡖⠳⠼⠶⣏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡸⠽⢀⠈⢳⠀⠸⠭⢇⠈⣻⠣⢴⠡⡏⣝⠹⣆⠙⠋⢀⡶⣏⣰⣀⣶⡼⠀⠀⡳⢀⠀⡘⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠚⣇⠀⣸⡟⢛⣶⠞⣆⡸⠉⢳⡠⠾⣆⡑⢚⣡⠋⡵⢚⡛⣿⠀⠀⢠⠿⢶⠀⡀⣱⠊⠒⠉⠑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⡁⣽⠋⠁⠻⢤⠃⢸⡃⢠⣙⣇⢯⡀⣤⠉⢻⣆⠑⠔⠋⡸⠛⣤⣾⣋⡈⠋⠉⣯⣤⠀⡀⢀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⣸⢯⠿⣤⠜⠳⠔⠓⠶⠋⠈⠒⠬⠤⠠⠜⠀⠉⠉⠉⠀⠀⠀⠀⠙⠦⠚⠓⠁⠘⠖⢹⠇⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⢀⢠⡤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⢤⡤⠤⠤⠤⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⠀⠀⠸⡆⠀⠀⢡⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⢠⡇⠀⠀⡄⢀
⠀⠀⠀⠒⠒⠉⠁⠉⠙⠁⠓⠒⠚⠓⠦⠤⠼⠆⠒⠒⠚⠓⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠛⠒⠒⠒⠚⠓⠂⠉⠙⠹⠦⠤⢤

⠀⠀⠀⠀⠀⠀⢠⠞⠒⠒⢲⡖⠒⠒⠒⢲⠒⠒⠒⠒⢲⠒⠒⠒⠒⢲⠒⠒⠒⠒⡞⠒⠒⠒⠒⡖⠒⠒⠒⠒⡖⠒⠒⠒⠲⡒⠒⠒⠒⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠀⠀⠀⠀⡜⠀⠀⠀⠀⡜⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⠀⠀⠀⠀⢣⠀⠀⠀⠀⢷⠀⠀⠀⠀⢣⠀⠀⠀⠈⢆
⠀⠀⠀⠀⣴⣁⣀⣀⣀⣰⣁⣀⣀⣀⣠⣇⣀⣀⣀⣀⣇⣀⣀⣀⣀⣯⣀⣀⣀⣀⣼⣀⣀⣀⣀⣸⣄⣀⣀⣀⣸⣆⣀⣀⣀⣈⣇⣀⣀⣀⣈⣦
⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢻⣄⠀⠀⣠⠞⣄⠀⠀⢀⡜⢧⡀⠀⢀⡼⢣⡀⠀⠀⣠⠻⣄⠀⠀⣠⠾⣄⠀⠀⢀⡜⢦⡀⠀⢀⡼⢣⡀⠀⠀⣠⠻⣄⠀⠀⣠⡎
⠀⠀⠀⠀⢸⠈⡏⠉⣁⣀⣀⣉⣉⣁⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⢀⠤⠬⢭⡉⠁⠀⠀⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠈⠉⢹⡇⡇
⠀⠀⠀⠀⢸⡆⡇⠀⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠒⠒⠒⣻⠄⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⢒⣒⣒⡒⠴⢤⡀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠘⡇⡇⠀⠉⡏⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠙⠉⡽⠀⠀⠀⠀⠀⠀⠀⢰⠡⡎⢉⣀⣉⣁⣈⡉⢳⢹⠀⠀⢸⢸⠁
⠀⠀⠀⠀⠀⡇⢱⡖⠒⠓⠒⠒⠒⢲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆⡇⠠⠶⠖⠶⠶⠀⡎⣸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⡇⢸⣇⣀⣀⣀⣀⣀⣸⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⡶⣍⣿⣩⣦⢤⠀⠀⠀⠀⠀⠀⠀⡇⡇⠐⣒⣂⣒⣒⠀⡇⡿⠀⠀⡇⢸
⠀⠀⠀⠀⠀⢣⢸⠈⡇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⡇⠀⡇⠈⡆⠀⠀⠀⠀⠀⠀⠇⢻⠀⠭⠭⠭⠅⠀⠇⡇⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡚⠚⠓⠒⠒⠒⠒⠒⠚⠓⠒⠂⠒⠒⠒⠒⠛⠒⠓⠒⠛⠒⠛⠂⠓⠒⠐⠒⠒⠒⠒⠛⠚⠒⠐⠒⠒⠒⠚⠓⠓⠒⠚⠓⡇
⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⣹⠉⠉⠁⠉⣹⠀⠁⠁⠈⣩⠀⠉⠉⠁⣽⠉⠉⠉⠀⣹⠀⠀⠀⠉⣿⠈⠁⠀⠉⣿⠈⠉⠉⠉⣏⠉⠉⠉⠉⡷
⠀⠀⠀⠀⠀⠈⡷⢤⣠⠴⠋⠣⢄⡠⠴⠋⠢⢄⡠⠔⠉⠢⢄⡠⠔⠉⠢⢄⡠⠔⠉⠢⢄⡠⠔⠉⠢⢄⡠⠔⠉⠢⢤⡠⠔⠉⠢⢄⡠⢴⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠉⢲⣤⡖⠉⠀⠈⠑⣦⣴⠊⠁⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⣤⠒⠉⠀⠈⠓⣤⡔⠊⠁⠀⠙⢢⣤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠙⠢⣤⡔⠋⠀⠈⠓⢤⣤⠚⠉⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢇⡠⢖⣒⣁⣒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠙⠂⠀⠀⠈⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⣭⣍⡒⢤⣸
⠀⠀⠀⠀⠀⠀⢀⡟⡴⠋⠀⠀⠈⠱⡌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠋⠀⠀⠀⠈⢢⠹⡀
⠀⠀⠀⠀⠀⠀⢸⡁⡇⠀⠀⠀⠀⠀⡅⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣇⢳⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⡱⢄⡀⠀⣀⡴⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠢⣄⣀⣀⡤⢋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⠭⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠬⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⣉⠉⠁⠀⠀⢀⡏⢝⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⣩⠘⡯⢫⢟⠱⡲⢶⢤⡤⣀⡀⠀⠀⠀⡇⠉⠁⠀⠀⠀⠀⢸⡮⢓⠭⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠘⠂⡯⢔⠥⡪⢖⢕⡠⠪⡪⢙⢝⢗⢲⡇⠀⠀⠀⠀⠀⠀⢸⡪⢐⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢗⠕⣪⣆⠵⠵⠒⠚⠚⠉⢹⢟⡿⢦⣀⠀⠀⠀⠀⠀⣿⡪⠕⣺⠁⠀⠀⣀⠤⠤⠀⠀⠒⣲⡤⢄
⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣀⡇⠀⠀⠀⠀⡖⡉⠉⠄⠀⠐⠀⠀⠀⠀⠀⣾⠕⡊⢔⠮⣷⢄⠀⠀⠀⡭⣒⠅⡞⠀⠀⡜⢁⣠⡤⢴⣴⣼⣧⠾⠏
⠀⠀⠀⠀⠀⢠⠔⠱⠌⣗⢔⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢍⡪⢔⢺⣇⢕⠀⠀⠀⡯⢒⢭⠇⠀⡔⢁⡞
⠀⠀⠀⠀⠀⠀⢣⠀⠀⢻⣂⣺⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠿⠥⠮⠷⠽⠷⠵⠤⠤⠼⠯⠶⠼⠤⠔⢁⡎
⠀⠀⠀⠀⢠⠊⣉⣍⢭⣍⣩⡉⢉⡥⠤⠤⡤⠤⡄⢀⡴⠖⠒⠲⠀⡖⠒⡶⢖⣶⠀⣶⠚⢛⢛⠀⣰⠞⡺⣶⡲⢀⡞
⠀⠀⠀⠀⠈⠆⠳⣡⠀⠀⢻⢻⡈⣇⠁⠀⡇⠁⠁⢸⣥⠄⠀⢸⠀⡇⢰⡯⣐⡏⢸⣿⢌⠢⡸⠀⣿⠅⣪⡿⢀⡜
⠀⠀⠀⠀⠀⠘⡄⢹⣆⢠⡘⣟⣷⠸⡆⠀⡇⠀⢸⠘⡾⠀⠀⢸⠀⡏⢸⣏⣾⠀⣼⣿⣵⣭⠇⠰⣇⣅⡾⠁⡜
⠀⠀⠀⠀⠀⠀⠘⡄⠙⠛⠓⠛⠛⠁⠙⠒⠓⣒⡁⠀⢉⣉⣉⡁⠀⣉⣭⣭⣁⢀⣤⡤⢤⢦⢀⣤⡶⢦⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣯⣟⣿⢣⢸⡍⡏⠀⢸⠀⡏⠀⠀⢸⢸⡇⢸⡳⡟⢸⢕⡏⡢⡝⢸⢻⢢⡏⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⡄⢙⢘⡆⣷⠇⠀⠌⠀⡇⠀⠄⡈⢈⠆⣿⣲⠇⣸⣹⣗⣼⠁⣏⣾⡞⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠒⠂⠃⠘⠓⠒⠚⠀⢈⣀⣉⣁⣈⣙⣉⣉⣄⠬⠭⠥⠤⠤⢤⣤⠠⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣂⣀⣀⢀⣀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⡉⠉⢉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡞⣛⣏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠹⠛⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣥⣕⡼⣷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⡽⣦⡆
⠀⠀⠀⠀⠀⢤⡠⢀⠱⣿⣿⣋⢴⠠⢄⠠⣀⠠⢀⠠⢄⠠⢄⠠⠀⣀⠀⣀⠀⡀⢄⡠⢄⠠⡀⠄⡀⢄⡠⣏⡻⠿⢗⡵⡠⣀⠤⣄⣤⡠⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠁⠈⠀⠈⠀⠈⠀⠈⠀⠈⠀⠈⠐⠀⠁⠊⠁⠂⠁⠊⠁⠊⠑⠉⠐⠁⠊⠁⠈⠀⠉⠉⠉⠁⠉⠀⠉⠈⠁⠉⠑⠉⠈⠀⠁

⠀⠀⠀⠀⣀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⢀⠤⠀⠠⢤⡈⣧⡀⠀⣀
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠲⡀⠀⢀⠀⡴⡾⢉⣧⣟⢵⡱⡤⣾⠾⣀⠀⠀⠀⠀⣀⡔⠂⣄⣶
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⠑⢿⢴⡞⡼⡏⣼⡸⡜⡟⢸⠟⠁⠀⠀⣈⠀⢈⢶⣿⡿⢄⡀
⠀⠱⢅⣀⣀⣀⣀⣀⣀⠠⣥⠜⡶⠀⠀⠈⠉⠁⠛⠘⠿⠇⠓⠊⠹⠀⠀⠔⠁⠀⠉⠀⠤⠭⠤⠤⠤⠙⢲
⠀⣿⣷⣾⣶⢾⣯⣿⠶⣲⣗⡿⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢘⡉⠒⠡⠌⠣⠎⠳⠉⠞⢳⡻
⠀⠳⡿⠓⠛⠪⠭⠿⠿⠿⣿⠇⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠏⠛⠛⠛⠛⠭⠭⠭⠽⠯⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢢⠅⠉⠐⠢⠤⠤⠒⡩⠡⢀⠄⢱⣀⣀
⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⢎⠉⢇⠀⠀⠤⠀⠀⠁⢉⢀⣀⠁⠤⠂⣠⠞⣠⠛
⠀⠀⣒⣓⣒⣒⣀⣀⡈⠉⢭⠭⢭⣭⣭⣭⣭⢭⠄⠀⠀⣝⣚⡳⢶⣖⣒⣒⣒⣒⣲⣦⣶⣶⢟⣒⡽⡄
⠀⠀⠖⢒⣟⣛⣿⡛⠻⠒⠛⠲⠒⠋⠍⠍⠁⠬⠇⠀⠀⠀⠣⠌⣉⣛⣁⣵⣓⣚⣛⣻⣛⠫⠽⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡰⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡀⠀⠀⠀⠀⠀⣠⠒⢚⢙⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢹
⠀⠰⠊⢰⡓⢤⡀⠀⠀⠀⠀⡠⣬⡄⠠⡊⡞⠀⠀⢀⣀⣀⢠⡝⣿⣥⣄⡀⠀⠀⢀⠤⣄⠀⡴⢾⣿⡷⠾
⠠⣝⡺⠒⠺⣛⡽⢄⠀⡰⠂⠓⠐⠓⢢⡸⡇⠀⠀⣿⠦⠤⠤⠩⠥⠤⠤⢿⠇⠀⣵⣁⠙⡇⢣⣷⡟⠠⠄
⠣⣤⣀⣉⢉⣵⡤⠎⠀⢗⠚⠂⠑⢊⣾⣠⢹⡆⠀⠈⣥⠔⢲⣐⠒⠂⣌⡏⠀⠀⠀⢀⣴⣡⡞⣁⠁
⠀⠰⣞⡡⢏⡞⣹⠁⠀⠈⢍⠁⠛⠉⠈⡠⠊⠀⠀⠀⢫⠌⢹⠉⢙⠀⣸⠀⠀⢸⠶⠈⠛⠏⠅⢚⡷⠐⠄
⠀⠀⠹⣿⣷⠾⡇⠀⠀⠀⠀⠈⠈⣉⡁⠀⠀⠀⠀⠀⠘⠀⣕⠈⠄⠀⠇⠀⠀⠈⠇⠒⠒⠒⠒⡹⠷⠤⠃
⠀⠀⠀⠡⡀⡸⠁⠀⠀⠀⠠⡉⠀⠈⠉⡹⠀⠀⠀⠀⠀⢠⣆⡖⢺⡞⠀⠀⠀⣤⠚⠓⠆⠲⠚⠙⢓⡆