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⠀⠀⠀⢟⠉⢇⠀⣰⣊⠍⠍⢍⠢⡕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢄⡀
⠀⠀⠀⠘⡄⠈⢦⡁⠁⠠⠀⠀⠗⢻⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣼⡷⠲⠀⠲⣤⣌⡆
⠀⠀⠀⠀⠱⡀⠈⣇⠀⠀⠀⠀⣰⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣏⠀⠀⠁⠀⠀⢈⣺⡀
⠀⠀⠀⠀⠀⠱⡀⠈⠓⠶⠶⠊⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⡤⡄⠀⠙⡄⠀⣀⠔⠋⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠰⡖⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠯⣤⡩⠿⡈⠚⡏⢉⣉⣁⣤⣺⠥⠖⠛⠊⠑⠒⠲⢤⡀
⠀⠀⠀⠀⠀⣀⡠⠧⠖⠒⠛⠉⠉⠉⠉⠓⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡩⠽⠛⠉⠉
⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⢤⡬⠚⢛⡆
⠀⠀⠀⠀⠀⢠⠚⠢⡽⠁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⢀⡏⢉⠏
⠀⠀⠀⠀⠀⡎⠀⢰⣡⠎⢁⡆⠀⣨⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⣸⠀⢸⠛⣾
⠀⠀⠀⠀⠀⡇⠀⠘⣿⠲⠈⠀⠌⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠈⡇⠀⡟⠀⢸
⠀⠀⠀⠀⠀⢱⠀⠀⢻⡀⠀⡔⠀⠢⠋⣩⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢰⠃⠀⣇⢀⢘⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠣⡙⡦⣄⣤⠮⠞⠛⠁⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠀⠀⣯⡬⠞
⠀⠀⠀⠀⠀⠀⡇⠀⠐⡅⡷⡱⠁⣀⡤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠀⠀⢀⠇
⠀⢀⣠⠤⠒⠒⠛⠐⠒⠒⠓⠤⢤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣀⠀⠀⠀⡞
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⣄⡀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠞⠁⠀⠈⠉⠓⠾⣅⡀⣠⠔⠒⠛⠛⠓⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⢄⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣦⠀⠀⠀⠀⢀⣀⡀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⢯⣭⣁⣀⣈⣉⣙

⠀⠀⠀⠀⠀⣠⠞⠛⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠃⠀⢄
⠀⠀⠀⠀⠰⣿⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠛⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠙⢿⣷⣾⠿⠂⠀⠀⢀⠉⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠀⠈⠻⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢦⠔⠉⠀⠓⠤⠔⠈⣽⣿⡿⠊⠁⠀⠀⠀⠀⠉⠀⠘⣿⣿⣿⠄⠀⠀⠀⠀⠈⣾⠏⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢘⠈⡀⠀⠀⢀⣀⠀⠘⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣀⠈⠀⠚⠀⠉⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠸⣿⣿⣿⣿⣷⣄⣼⣿⣧⠀⠀⢷⣶⣶⠇⠀⠀⣸⣿⣿⡿⠛⠉⢀⣠⣴⡦⠀⠐⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⣿⣿⣿⡿⠿⠧⡾⢋⣷⡄⡀⠈⠁⠀⢀⣴⣟⠛⠯⠀⢠⣴⣿⣿⣿⠃⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠋⠉⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠈⠙⠛⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣾⣷⣿⣶⣀⣀⣀⣀⣀
⣴⣿⠁⠀⠈⢢⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣤⣀⣠⣾⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠛⢿⣿⡿⠟⢀⠀⠀⢀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⢸⣿⠿⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢻⣦⠀⠀⢂⡐⠁⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣧⠀⠈⠄⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⣰⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡿⠀⠀⣸⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⠀⠀⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⡇⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⡼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⢺⢸⡜⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠤⠖⠀⣀⠀⠀⠀⠀⠐⢌⠃⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠆⠁⠀⠀⠀⠀⠀⠉⡴⠃⠀⠀⢠⠋⠀⠉⠐⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠀⠀⠀⠀⠀⠀⣄⠀⣆⠨⡄⠀⠀⡸⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁⠀⠀⢀⡀⠔⢂⠜⠃⠁⠀⣘⣄⠴⠁⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⡄⠋⠀⠀⠀⠀⠀⠀⠸⠄⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢢⡀⣀⡋⠀⠀⠀⠀⠀⠀⢀⢠⠈⡄⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠁⠉⡰⠂⢤⠀⠀⠀⠀⠀⣠⠃⠀⡆⠀⠉⠉⠐⠊⠉⠉⠉⠐⠒⠒⠨⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠈⠀⡇⣀⡔⠃⠀⠀⠙⠂⠒⠂⠉⠈⠆⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⣠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠢⢄⡈⢹⠀⠈⣦⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢇⠀⠀⢈⣖⣁⣀⠀⠀⣀⠂⠴⠚⢉⡤⠊⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⢂⠤⠒⠀⠫⠀⠀⢠⠄⠀⠀⠀⢣⠟⢢⡠⠓⡄⠀⠀⠀⠀⠀⠀⢱⠊⠁⠀⠁⡀⠉⠙⣤⠔⠉⠉⢀⡠⠶⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠔⠁⠀⠀⠀⠀⣀⠔⣁⠴⠆⡐⠒⠉⠉⠛⢄⣀⡎⢀⣤⠤⠤⠤⠐⡣⠀⠀⠀⠀⠈⢣⠀⠷⢀⡤⠒⡏⢀⠇
⠀⠀⠀⠀⢀⣠⡖⠁⢀⢀⣀⣀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⠊⠀⠸⡴⠀⠀⠀⠀⠀⠉⡆⠛⠁⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⡌⢀⠟⠁⠀⠀⢇⠼
⠀⡴⠶⠭⠤⠤⠬⠭⣥⣀⣀⣀⣀⣈⡉⠺⡒⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠞⠒⠒⢘⠆⠀⠀⢰⠁⡸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡠⠔⠋⠀⠀⠀⢀⡜⠀⠀⠀⢸⠀⣇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠒⠒⠊⠉⠀⡅⠀⠀⠀⣀⠴⠊⢡⠇⠀⠀⠀⡼⠰⡏
⠀⠓⢲⡖⠒⢦⠤⠤⠤⠤⠤⣤⣄⣀⣀⣀⣸⡄⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠔⠋⠁⠀⠀⡸⠲⠤⠤⠜⡇⠀⠑⣠
⠀⠀⢸⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⠞⠁⠉⠉⠉⠉⠉⢹⠉⠉⢻⠈⠉⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠘⡔⠤⠰⠏
⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠉⡐⡄
⠀⠀⠀⣇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠔⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣠⠅

⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⡠⠀⠒⠀⠉⠀⠀⠀⠀⠀⠠⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠂⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢂⠈⠆⠀⠸⠀⢀⠄⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠢⠄⠁⠀⠀⠀⠀⠀⠔⠋⠀⠀⢠⣀⡄⠐⢶⡶⣄⢸⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠁⠒⠀⢀⣤⠘⠂⠀⠀⠀⠀⠀⠀⠈⠁⢀⡤⠀⠺⠆⣀⡀⠀⠀⠱⠀⠀⢀⣿⣆⣾⣿⣳⣶⣦
⠀⠀⣀⡠⠀⠀⠚⠃⠠⠀⠀⠀⠀⠀⠄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠾⣿⠛⠿⣿⠟
⠀⠀⠁⠀⢠⣾⡿⠂⠀⢠⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⠀⠀⠀⠀⠀⠠⠤⠖⠒⠒⠛⠃⠂⠘⠀⠤⠤⠤⠤⠄
⠀⠀⠀⢀⣇⣠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢤⡀
⠀⠀⠀⠘⣿⣿⠀⠀⠀⢈⠀⠀⠀⢀⠀⢠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠚⠁
⠀⠀⠀⠀⠙⢝⠢⡀⠀⠘⠁⠀⣠⠀⠀⠀⠠⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠛⠋⠑⠂⠀⠈⠈⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⣶⣿⣅⣀⠀⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠛⠛⠛⠛⢦⣄⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠍⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠡⡄⠀⠀⠀⠀⠀⠀⠀⠏⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠈⠉⠁⠁⢀⡀⠀⠀⠀⠀⡠⠀⠂⠀⠀⢀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢀⡀⠿⠞⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠁⠀⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠈⠃⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠇⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠃⠀⠀⠀⠒⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣀⠤⡄⠀⠀⠀⠀⠀⠀⢀⠄⠠⠀⠀⠀⡐⠀⠀⠀⠠⠄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣶⠛⠛⠛⡗⣄⠀⠀⡀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠄⠀⣽⠆⠄⠀⡀⠀⡀⠀⢀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠂⠀⠘⠑⠒⠛⠁⠀⠀⡨⠄⠀⠀⠀⠈⠁⠀⠘⠂⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠂⢀⠡⠀⠁
⠀⠀⠀⠀⠀⢨⠀⠐⠠⡤⠤⠀⠀⠀⡀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⢰⢐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠁⠀⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⢀⠅⠀⠁⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠒⠨⠭⠄⢀⣒⠒⠀⠤⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠫⠭⠭⢍⣒⡒⠠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⡤⠒⠢⣄⠠⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠉⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⢀⠀⡆⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡀⡸⠀⡆⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⢀⡀⢀⠦⣠⠁⠈⢧⡀⠀⠀⠀⠘⡦⢤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠒⠛⠁⠀⢇⠠⡀⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⣄⠊⠓⠁⠀⠀⠀⡀⠈⢷⠀⠀⠀⠀⠉⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⢆⠷⠨⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠂⠀⠀⠰⠀⠀⠆⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⢸
⠀⠀⠀⠀⠠⠔⠪⠕⢂⠿⡀⠀⠀⠀⠀⡤⠒⠂⡗⠀⢀⠆⠀⠀⠀⡇⡅⠀⢸⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠⢸
⠀⠰⠪⠐⠈⠀⠀⠀⠀⢑⡛⠦⣀⡀⠀⠈⠒⣚⡤⠖⠉⢄⠀⠀⠀⣇⠀⢀⢸⠃⠒⠠⠬⢉⣒⠢⠤⢄⣀⡀⠀⠀⠀⠀⠀⠰⢰⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⡈⠉⠉⠉⠩⢉⠐⠀⠀⠀⠱⣄⢀⠀⠙⠉⠙⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠬⠭⣒⡒⠦⢤⡆⠇⠀⡆⣿
⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠠⠀⡠⠄⠳⡀⠈⢢⡉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣰⠀⠀⠇⣿
⠀⠀⠀⠈⡄⠀⠀⠀⠈⠂⠀⡈⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠈⢀⠀⠙⠄⠀⠁⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠁
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡇⠀⠀⡁⡀⠀⠐⠀⠀⠀⡠⠐⠁⠑⠖⠈⠂⣀⠀⠀⠀⠀⠉⠁⠐⠀⠤⣀⡀
⠀⢀⢀⡀⡔⠀⠀⠀⠀⠀⡠⣄⡠⠊⠀⠑⡀⠀⠈⢢⠊⠀⠀⠀⠀⠈⠑⠛⠁⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠈⠁⠒⠠⠀⠀⣀
⠀⠈⢸⠀⠣⠀⠤⢤⡒⠉⢱⠊⠀⠀⠀⠀⢈⠦⡤⠚⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢢
⠀⠀⢸⠀⠀⠀⠀⠀⠈⢡⠃⠀⠀⠀⠀⢠⠋⠀⠈⠤⠴⠇⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢈⡇⠀⠀⠀⠀⠀⠀⣠⠔⠁⠈⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡇⠈⠁⠐⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠈⠉⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠃
⠀⠀⠘⠒⠢⠤⢀⡀⠀⠇⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⣠
⠀⠰⠴⠄⢀⣀⠀⠀⠉⠑⠲⠤⢀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢤⣀⡀⠀⠀⠔⠁⠀⠀⠀⠀⢀⡴⠋⣸
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣀⠀⠀⠉⠑⠒⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠀⠀⠀⠀⢀⠔⠉⡠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠁⠒⠢⠤⣀⡀⠀⠈⠁⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣠⠔⢁⡴⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠙⢦⣅⣤⠒⢄⠀⠀⠈⠉⠒⠲⠤⢀⡀⠀⠈⠉⠒⠒⠤⢄⣀⠀⠀⠀⠀⠸⠀⡠⠊⢁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⣀⡎⠀⠀⠑⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⢀⡀⠀⠀⠉⠑⠒⠤⡔⠋⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠂⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠤⢄⣀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠤⡀
⠸⠀⠀⢈⠭⠀⠀⠀⠄⠀⠀⠀⠈⢀⡀⠂⠀⢠⡖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠠⡔⢠⠀⠀⠀⠀⠀⢸⡏⠉⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢨⣤⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠈⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠃⠀⠀⣿⠁⠀⠀⠁⢀⡄⠑⠀⠀⠀⠀⣤⠂⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢐⡢⠄⡀⠁⠃⢸⠀⠀⠉⡀⠀⠀⡄⢀⢀⠹⠄⣠
⣠⠂⠀⠁⠀⠀⠀⠈⠈⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠀⠀⠂⠀⠀⠀⠀⠀⣿⣡⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠈⢀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠈⠀⠀⠐⠈⠀⠀⠀⠀⠀⠈⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸
⠀⢀⠀⠙⡀⠀⠠⠂⠀⠀⠘⠀⠈⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠒⠨⠷⢀⣀⠤
⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁
⠸⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⡴⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠐⠀⠀⢀⠎⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠆⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⡒⡄⠀⠀⠀⠀⠀⠀⠀⣴⠊⠓⠼⠋⠛⣦⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣚⣢⡻⠒⢄⡀⠀⠀⠀⢰⠏⢀⡠⣄⣀⠀⠠⠗⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⡀⠈⠲⢤⡀⢸⡌⣉⢁⣈⠙⢦⣄⡦⠀⢀⡠⠔⠊⡍⣏⠐⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠳⣏⣸⢄⠄⠀⡐⢻⠥⠊⠁⢀⡠⠔⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢠⣞⣌⣁⡤⡾⢳⠁⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠹⠆⢃⡴⢁⠇⣀⡴⠁⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢤⠉⠉⠀⠀⢹⠁⠀⠉⠲⢕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠉⠁⠀⠀⠈⠢⠶⠆⠀⢤⠀⠀⠀⠀⡞⠀⠀⠀⢀⡴⢳⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⡤⠀⠀⠀⠸⡷⢄⠀⣰⡷⠋⡼⠁⠀⢙⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠚⠉⠉⠩⠟⠁⠀⠀⠀⢀⠠⠐⠂⠀⠀⠀⠀⠈⠢⢤⣀⡁⠒⠤⣀⣰⠁⠀⠉⠛⠶⠊⢁⡠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⢍⡃⠀⠉⠑⠒⢆⠀⡤⠚⠁⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡇⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡇⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⢄⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣀⠤⠚⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠧⢤⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡀⠀⠀⡠⠄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠠⣀
⠸⠀⠂⢀⡉⡁⠀⠀⢠⠀⠀⠀⠀⠈⣀⠀⠀⠀⣀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⡀
⠀⠀⢈⡡⢀⠀⠀⠀⠀⠀⠀⣾⠉⠻⠏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂
⠀⠀⠀⠑⣀⠁⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡙⠃
⠀⠀⠀⠀⠟⠀⠀⢰⡖⠀⠀⠀⠃⠈⡁⠆⠀⠀⠀⠀⢀⠠⠀⠀⠈⠃
⠀⠀⠀⠀⣄⠀⠀⠘⠣⠀⠀⠀⠠⠶⠓⠀⠀⡀⠀⠀⢹⠂⠀⡀⠀⠘
⢀⠀⠀⠔⠁⠉⠐⠂⠀⢤⡙⠀⠀⠈⡂⠀⠀⠧⠞⠴⠄⢀⡘⠁
⠸⠀⠀⡀⠀⠀⠀⠀⠀⢸⡷⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡈⠒⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠂⠜⠀⠀⠀⠀⢠⠄⠀⠀⡉⠄⠀⠀⠂⠙⠓⢤⡀
⠀⠀⠑⢤⡀⠀⠐⠀⠀⠀⠀⡈⠀⡀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⣀⢀⣀⣀⠀⡠⣄⠀⡒⠄
⠀⠠⠂⠀⠠⢀⠀⠃⠀⠀⠀⠃⡀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡏⡁⠤⠒
⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘
⠘⠀⠀⠀⢀⠄⠀⠀⠀⠠⠀⠀⠀⣠⠊⠛⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠐
⠈⠀⠀⠀⠎⠀⠀⠁⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠀⠀⠀⠠⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢳⡔⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣋⣉⣉⣉⠓⢦⡀⠀⢻⡉⠀⠀⣉⠿⠀⠀⠀⠀⢴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠖⣹⡋⠁⢀⡀⠉⠓⠦⣉⠳⣼⠋⠉⣏⠁⠀⠀⠀⠀⠀⠈⣹⣧⠶⢶⡀⣴⠲⡄
⠀⠀⠒⠒⠒⡆⠀⠀⠀⠀⠀⠰⡏⠉⠙⡆⠀⣀⣠⡇⡎⢸⠙⠦⠼⠏⠉⠛⠒⢮⣙⡇⠀⠀⢸⠋⠉⠉⠉⠛⠲⢤⡟⠘⣆⣸⠷⠚⡟⠁
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡗⠀⠀⡷⠋⠀⢸⡄⣠⠿⠄⠀⠀⠀⠘⠛⠀⠘⣾⡇⠀⠀⢸⡦⠤⢤⣤⠔⡷⠸⣅⡀⠈⠀⠀⢸⡧⠤⢤⣯
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣇⠀⣀⣷⠒⣫⠉⢧⠹⡇⠀⠚⠁⠀⢁⡀⠀⠀⣿⠀⢀⠂⢸⡇⠀⠀⠀⠸⡄⠀⠀⢝⠓⢦⣄⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⠒⠻⠞⣭⡮⠓⢁⣧⣼⣀⠹⡀⠀⠀⠀⣿⣷⠀⢀⡿⡄⠀⠈⢀⡷⠶⢤⠀⠀⡇⠀⠀⢸⠀⠸⡇⠉⠒⠃
⠀⠀⣀⣀⣀⣀⣸⠄⠀⠀⠈⠉⣶⣀⠽⡆⠀⡎⠀⡀⠈⠉⠓⠤⣄⣀⠈⢁⣠⢾⡀⣹⠀⠀⠈⣧⠿⠋⠉⠉⠓⢦⣞⣝⣆⠀⡇
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣷⢻⠉⡧⠊⠳⢤⣀⡁⠀⠀⠀⠀⢉⠉⠻⡀⢸⠍⠁⢀⠠⠀⡏⠀⢀⣘⠰⠀⠀⣿⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠈⣇⣠⣀⣀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠈⠉⠓⢦⡀⠀⠀⠓⠁⠈⠁⠀⠀⠀⠈⣠⠷⡤⢤⣩⣄⢀⣠⠏⠘⡇⠀⢸
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⠢⡀⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡀⠁⠀⡼⠁⠀⠳⠞⢧⡟⠉⠙⣦⠀⡇⠀⢸⡄
⠀⠀⣒⣒⠯⠥⠤⠤⠤⠤⢤⠇⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣘⣇⣀⣈⣀⠀⠀⠀⠀⢸⣟⣋⡷⠤⠶⠶⠶⠤⢄⣼⠀⣿⠀⢸⡇
⠀⠀⣤⣤⣤⣤⣤⣤⡀⠀⢸⠀⡿⠀⡞⣠⣖⣶⡤⠴⠒⠋⠉⠁⠀⠀⠀⠈⠉⠑⠒⠉⠉⠉⠀⢀⣀⣠⣤⣄⣀⠀⠈⠓⠿⣀⣸⡇
⠀⠀⠠⡄⠀⠀⠀⢸⡇⠀⢸⠀⡇⢀⡿⠋⠁⠀⠀⣀⣠⡤⠤⠤⠤⣤⣄⣀⣀⣀⡤⠤⠴⠒⠋⠉⠀⠀⠀⠀⠈⠉⠳⢤⣀⡀⠈⠉⠉⠙⠲⣄
⠀⠐⠒⠒⠒⠒⠒⠚⠁⠀⠸⠀⠧⠏⠀⠀⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠤⠄⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣰⡦⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣠⡄⠀⠀⢱
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡠⠲⠔⢁⡁⢣⣀⡸
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⠁⠀⠺⠟⣶⠂
⠀⢠⠇⠀⠼⠧⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⠷⡀⢸⣲⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣾⢤⡆⢀⣼⢥⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠈⠻⣄⣹⣖⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠠⣶⠍
⠀⠀⣾⣤⠀⢠⢾⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢤⠈⠁⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠋⠙⠙⣄⠴⠒⠀⣤⠄
⠀⠀⢠⣿⡆⠐⠋⠀⠀⠀⠀⡴⡆⢀⣤⢀⣤⣆⣠⠄⢸⠀⢧⡶⠛⢇⠀⢸⣧⠞⠙⢆⠀⠀⠀⠐⡒⠂⠀⠀⠀⠀⠀⠀⡴⢲⡞⠁⠀⠀⠀⡀⠈⠙⡇⢀⣛⠃
⠀⠀⠀⣠⢖⣾⣿⢿⣲⡀⠞⡀⠟⠁⠟⠉⠚⠋⠀⠀⡇⠀⠀⠳⡀⠈⣧⠎⠈⠳⣄⠈⠓⠂⠤⠤⠀⣤⣤⠤⠔⠒⠂⣾⠀⢠⠓⠠⣌⠙⠻⢇⡀⢸⠃⠺⠋
⠀⠀⠀⣧⣾⣏⠼⢠⣧⡇⠘⣧⠔⣿⡤⢶⣆⡤⠄⢀⠃⠀⠀⠀⠈⠲⣌⠉⠒⠤⣀⣉⢭⡴⠶⠒⠾⠁⠀⠀⣠⠞⠀⠉⢆⢠⠀⣄⣀⠖⠓⢂⣧⠞⠢⣄
⠀⡞⣿⠿⠿⠿⠛⢉⣚⣛⣻⠁⠀⠉⠀⠋⠁⠀⠀⣰⠦⢄⣀⣀⣐⡾⢀⡽⠒⠢⠬⡀⠈⣱⠂⠀⠀⠀⠀⡜⠁⠀⠀⠀⠸⡜⢆⠀⠚⢉⡠⢫⡵⠆⠀⣨⣧⡀
⠀⠉⠙⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠠⣄⠉⡷⠋⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠑⣚⣩⠖⠉⠀⣀⢔⡩⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠀⠒⠒⠚⠃⠉⠉⠉⢉⡝⠁⠀⣀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣗⠈⢢⠴⠾⠓⠉⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢯⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠠⠄⠒⠀⠈⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠀⠀⠀⢀⠤⠔⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⡠⠔⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠒⠉⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⣛⢻⣏⡛⠛⠛⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡿⢛⡽⠂⣩⣿⠟⢿⡳⠯⢷⣦⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⣠⠞⠉⠀⠀⠀⠙⢧⣀⡈⠙⢛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇⠰⢻⣄⠀⢀⣤⣤⣄⠈⢻⣿⣷⠀⣿⣏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⡏⠀⡀⣿⠉⠀⠀⠠⠤⣄⠀⢸⣿⠿⢷⣽⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⠇⢼⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠀⣼⣿⣷⣶⣦⡀⠀⠀⠀⢀⣿⣿⣏⠀⠀⡿⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣧⣼⣿⡇⠛⠟⠛⠁⠀⠀⢀⣾⠟⣸⣿⡄⠀⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢻⣿⣿⣿⣦⣤⣤⣤⠶⢾⡿⢡⣾⣹⣿⠃⡜⣰⢠⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢠⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣵⣥⣾⣿⣿⣿⣿⣄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⣿⣿⢻⣿⣿⠿⠏⠀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠀⠀⠀⣾⣟⣵⣶⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣽⡷⢀⣀⣰⢠⣿⣿⣿⣿⣿⠛⣿⣿⣶⣿⣿⣿⣿⢿⣧⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣠⣉⣽⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⣿⡟⣸⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣖⣾⣿⡿⢡⡯⣹⣧⠿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠀⣼⡴⣥⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡗⣿
⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡟⠀⠀⣿⣷⢛⣿⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⢿⣼⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⠟⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣭⣿⣿⣿⣷⣿⠆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣘⠿⣋⠹⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣻⣵⣿⣿⣷⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣉⠙⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣓⠶⢽⣆⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣌⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠬⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⣋⣥⠤⠤⣄⡀⢀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢠⣖⠉⠁⠀⠀⠀⠀⢻⠀⠓⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣁⣀⠀⢰⠏⠀⠑⣄⠀⠀⠀⠀⡿⠀⠀⠀⢿⣄⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠘⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢱⠁⠀⠈⣇⢸⡀⠀⠀⠈⢆⣀⡤⠞⠓⠈⠉⠑⢦⠈⠓⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡄⠀⠣⠀⠀⠘⠀⠓⠤⢤⣤⡜⢧⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠹⠤⣤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠳⡄⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢾⡉⢹⡖⢦⠀⢠⠃⠀⢳⠀⠀⠀⠀⠀⠘⢦⣀⣀⣳⡤⠖⠋⠀⠀⠀⠀⠴⣖⠓⠈⠙⠛⠀⠀⠀⠀⢰⣇⡤⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⢱⢀⡃⢸⢀⡏⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡈⠳⠤⡎⠀⠀⢹⠞⢢⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠋
⠀⠀⠀⠀⠀⠀⠀⣾⢵⣀⣀⣈⣉⣤⠋⠀⠈⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠉⠁⠈⠣⣍⣈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠙⠲⣄⠀⠀⠀⠀⢀⣼⠋⢣⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠈⡻⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠙⠚⠷⠀⣸⣄⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⣸⠃⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⡏⠛⠲⠶⠖⠚⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡤⣀⡀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠉⠙⠒⠠⠤⠌⢿⠒⣖⠒⢲⡒⠛⡷⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⢙⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣃⢻⣄⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⣯⣠⢀⢍⡍⠙⠓⠒⠶⠶⠦⢤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠓⠐⠲⠤⠤⠤⠤⠤⠤⠤⠤⠔⠛

⠀⠀⠀⠀⣴⠋⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⡄
⠀⠀⠀⠀⢿⣄⣀⣼⠀⠀⠀⣀⠀⠀⠀⢀⣼⣿⡿⠟⠉⠛⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣹⣧⣀⣠⠇
⠀⠀⠀⠀⠀⠹⣟⠉⠀⣀⠠⠀⠄⣀⠀⢺⣿⡿⠄⠀⠀⠀⠠⠙⢻⣿⣿⡀⠀⠀⠀⠠⣴⠏⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣆⢂⢁⠀⠐⢀⡀⠐⢿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡀⠀⠐⠀⠋⠀⡀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⣿⣿⣿⣦⡼⣿⡆⠀⠳⣶⡾⠀⠀⣼⣿⡿⠋⠁⣠⣴⡦⠀⣀⡰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸⣿⠿⠟⠛⠙⠋⠼⠗⠄⠀⠀⠠⢾⣏⠉⠀⠐⠿⠿⠟⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢀⣠⡤⢄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠴⣿⣾⣷⣤⣤⣤⣄
⣾⡇⠀⠀⣡⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢈⣿⣿⣿⣿⣿⣿
⠹⢿⣶⣾⡷⠀⠀⠀⠀⠿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢸⣿⣿⣷⣶⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠻⣏⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣆⠀⠑⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⢰⡏⠀⢰⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡾⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⠀⢰⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢀⡇
⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠻⠄⣹⠔⠶⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡽⢡⢧⠤⡀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡀⠀⠀⢉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠾⣁⡀⠈⠑⠤⡀⠀⠀⣧⠴⠖⠋⠓⠲⡀⠀⢱⠀⠀⠀⣀⡠⠤⢲⠖⢽⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠈⠑⠤⣿⡖⡇⠐⡉⠀⠉⢽⠎⡠⠔⠊⠁⢀⣨⠧⠖⠬⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢈⣇⠫⠚⠁⣠⣆⠜⠉⠀⠀⡠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠺⣼⢲⡖⢋⠇⡜⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⣴⡫⠔⠃⣴⡞⠀⠬⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⢦⡀⢀⠀⠀⠀⠖⠀⠀⠀⠀⢸⠀⠀⠀⠈⠓⢍⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠈⠊⠛⠀⠀⠆⠀⠀⠀⢀⡞⠀⠀⠀⣀⣼⡛⣷⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠠⡠⠆⠀⠀⠀⢊⡗⠦⡀⢰⡷⠋⣰⠇⠀⢀⡹⢦
⠀⠀⢀⡤⠖⠒⠒⠚⠉⠉⠁⠀⠚⠁⠀⠀⠀⠀⣀⠤⠒⠂⠀⠀⠀⠀⠀⠙⠢⠤⣀⣉⠒⠢⢄⣸⠁⠀⠈⠉⠓⠚⢁⣠⠔⠋⠀⢸⡆
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠬⢍⠁⠀⠈⠉⠛⢢⠀⠠⠖⠉⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⣷
⠀⢸⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⢀⡠⡄⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠤⠚⠉⠀⠙⠉
⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⢀⡀⠠⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠉⠁
⠀⠀⠀⠈⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠐⣾⠅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⡖⠒⠒⠉⠉⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠣⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠀⠀⠀⠀⣠⢤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⢀⡄⠀⠀⠀⠀⡀⠀⢀⠤⢄⠀⠀⠀⠠⡋⠀⠀⡔⠋⠑⣆⢹⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠎⢧⠀⠀⢀⣼⡀⠀⡼⠀⠈⡇⠀⠀⢀⣣⠀⠀⢖⠯⣴⠞⢋⣼⡉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⢲⢼⡶⠚⠒⠒⠉⠀⠓⠢⠼⠦⠀⣙⣤⢀⢸⠟⠀⠀⠈⠳⣅⠀⠀⠀⠱⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⡤⠞⠛⠛⠓⠀⠀⠚⠛⠛⠳⠄⠈⣟⡜⡿⠀⠀⠀⠀⠀⠀⠙⠢⣤⠴⠚⠁⠑⣄
⠀⠀⠀⣀⡤⢤⠔⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠣⣷⠋⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣿⠵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⢣⡀
⠀⠀⢸⡇⡇⢸⣀⣚⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⣀⣙⣋⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠱⡄
⠀⠀⠘⡿⢯⣉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠘⡇⢀⣉⡛⡇⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⢇⠀⡇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠙⠣⠤⠝⠁⣠⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠸
⠀⠀⠀⣸⠄⢀⠠⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠙⠒⠒⠒⠋⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠘⠂
⠀⠀⡴⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠲⣅⡀⠀⠀⠀⠀⠀⠀⣀⠽⠒⠶⢶⡒⠒⠒⠒⠢⠤⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡰⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠴⠚⢩⠞⠁⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⢠⠇⠀⠀⠀⠀⢰⠃⠀⢀⡤⠚⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⣸⠀⠀⠀⠀⢀⡾⠖⠋⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⠤⠖⠊⠁
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⠤⠖⠒⠉⠉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠚⠉⠁⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠙⠒⠒⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣶⣖⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⣘⡿⠿⠟⠿⠿⣿⣿⣻⡏⠁
⠀⠀⢠⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠞⡡⢋⣄⡠⠙⢄⠈⢣⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡜⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡀⠡⠆⣐⡀⠀⠀⠣⡏⠀⢑⡿⠋⢤⣿⠟⠀⠀⡧⢄⠀⠀⠀⠀⠀⠀⠐⡀⠂
⠀⢰⠃⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⣀⠀⠀⠀⠀⠠⡇⢠⡈⠙⠁⣀⣼⠒⠀⢀⡗⠋⠀⠀⠐⠒⢈⠒⠄⢠⢂
⠀⡏⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠄⠀⠀⠀⢇⠀⠙⣉⠉⣵⠃⠀⠀⡼⠁⠀⠀⠀⠀⣀⠼⡤⠔⠃
⢸⠁⠀⣀⣤⠀⡶⠶⠖⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡂⠄⠈⢦⣀⠈⠭⠄⢀⣠⠞⢁⠀⠠⢄⡂⠭⠚⠃
⠿⠒⠋⠁⣿⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠬⠙⡹⠋⠉⠡⠀⠀⠒⠈⠁
⠀⠀⠀⠀⢸⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⡇
⠀⠀⠀⠀⠘⡇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠐⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⠁⢠⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡇⠈⠦⠊⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠐⡉⡬⠁⠀⠠⠤⠴⣄⡉⠒⠤⠤⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⢀⡇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⡇⠀⠀⠀⠀⡠⠔⢻⠤⢄⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⢍⣙⣀⣈⡨⢥⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠹⣄⠠⠐⢉⣠⠖⠋⠀⠀⠑⡄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⢤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣤⣀⣀⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡒⣦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣻⣿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡉⢻⣿⠚⠛⣷⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⡀⠀⠐⠟⠻⠿⠛⡿⠓⣟⠻⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄⠰⡆⠀⠂⠠⠁⡤⢤⠀⢠⠅⣼⠉⠁⣤⡬⠃⠀⠀⠁⠲⠀⢉⠄⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣬⠴⠞⠻⣿⣶⣶⣤⣶⣾⣧⣹⣅⡶⢴⠆⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠛⠳⠦⠴⠿⡇⠀⠀⡈⠀⢰⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⣲⠿⢳⣤⣠⠄⣺⠿⣽⣦⣠⣠⣤⢖⣰⣤⡀⠐⠀⠞⠀⠀⢴⡃⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⢀⢀⣀⣉⣉⣩⣤⣤⣤⣭⣿⣯⣤⣤⣼⢧⣔⠀⠀⠀⠐⠛⠘⣷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣻⡛⣿⣭⠀⠀⠀⢀⣸⡇⠀⣿⡇⠀⡏⠀⣀⡐⠀⠀⠀⠀⠤⠛⠳⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠀⠉⠀⠀⠈⠛⢻⣇⠀⢨⣿⡇⠀⣤⡄⢰⡇⢠⣤⡌⠀⠉⠀⠀⠁⠐⢤⢰⠦⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⡴⠂⠉⠈⢀⡀⠰⠿⣀⡀⠈⠉⢣⡔⠛⠃⢘⡃⠀⠀⣸⡇⠀⡷⠀⠤⢤⠀⠀⣴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠈⠀⢿⡅⢰⠰⣿⡇⠀⢀⡿⠁⣼⡗⠙⠀⢠⡀⢻⡇⠸⠃⣸⠀⢸⠀⣀⢸⠀⠧
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢼⠀⠸⡧⣤⠀⠁⢀⣠⣿⠀⠀⣾⠇⢠⣿⠁⡠⠀⠈⢁⣼⠇⠀⢰⡏⠀⣿⠀⡟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣀⠈⣠⡀⠀⣀⠸⠇⠀⠀⠛⠀⠀⣿⡀⠘⠃⢰⡇⠀⡁⢸⣿⠀⣴⠸⡇⢠⡟⠀⣀⠘⠋⠄
⠀⠀⠀⠀⠀⠀⡀⠸⡏⠉⢙⡟⠉⠉⠛⡶⢶⡟⠛⠛⣶⣬⣿⣶⣾⣬⠷⠳⠿⣦⣿⣦⣿⣦⣷⡾⡱⠢⣍⣛
⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠉⣤⠀⢀⣤⣀⠍⠉⠩⠃⠐⠀⠀⠀⠚⣒⠒⠂⢁⣠⠀⣠⣤⡽⠒⠒⠐⠛⠋
⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⣀⠃⠀⠀⠁⡀⠰⡄⡀⣇⠀⠆⢶⣠⡆⠲⡄⣲⠀⡤⢸⡀⣉⢃⠀⠀⠀⠤
⠀⠀⠀⠀⢠⣼⡮⠁⠶⠴⠀⠉⠐⠀⠀⠀⠀⠛⠿⠤⡆⡟⠤⢶⠹⡜⠖⠗⠇⠻⠀⠋⠳⡆⢫
⠀⠀⠀⠀⠄⠈⠁⠠⠀⠀⠠⠜⠂⠀⠀⢀⠀⢠⣧⠀⠻⣧⣀⠲⠾⡥⠃⠐⠦⠀⠀⠀⠀⠧⣠⢄⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠰⣆⣀⣐⣠⡕⢈⣗⡂⣀⠇⡷⡘⣆⡇⡇⠷⣈⠐⣺⡓⣓⢸⣄⡳⣇⠦⣹⣏⢡⠧⡀⣠⣾
⠀⠀⠀⠀⣶⡆⢀⠻⠋⣽⢿⡁⠚⠦⠈⢲⣘⠲⠧⠻⠼⠇⠙⠂⠳⠬⠹⠗⠙⠞⠆⣱⡍⠣⠅⠹⠘⠆⠀⠁⣱⡀
⠀⠀⢠⣿⣯⢀⠙⢃⡠⠑⠚⠷⠦⣴⠔⠛⠛⠷⡀⢀⡀⠀⣠⡾⠿⢦⣤⣄⣀⣠⠿⠻⢷⣦⡄⣠⡶⡛⠓⠾⢻⣧⣀
⠀⠀⠀⢁⡉⢤⡄⠈⠁⠐⠷⠂⠀⣦⣠⡤⠐⣀⠈⣉⡉⢉⡁⢀⠀⣀⠀⣉⠩⢀⠀⣀⢀⣈⡉⣀⣠⠀⢴⡶⠈⠛⣉⠛
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠉⠉⠉⠋⠠⠉⠄⠘⠛⠛⠛⠛⠛⠻⠛⠛⠀⠈⠀⠛⠛⠛⠛⠛⠃⠐⠀⠂⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠊⠁

⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣀⢀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡤⠀⠀⠐⠒⠒⠒⠒⠒⠒⠲⠆⠀⢸⠀⠀⡇⡇⠀⠀⡅⢤⠤⠤⣤⣤⣤⣤⡄⠤⠤⠤⠤⠀⠠⢀⠀⠁⠀⠃
⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠅⠀⠘⠀⠀⡇⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀
⠀⠀⠀⠘⠀⠀⠐⠐⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠁⡀⠸
⠀⠀⠀⠀⠀⠀⠘⡁⠀⠠⡀⠠⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠂⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠆⠀⠜⠁⠀⠈⠈⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⢹⠀⠀⠀⠀⢀⠿⢜⡅⠀⠀⠀⠀⢀⠀⠇
⠀⠀⠀⠀⠐⠒⡲⠒⣒⢤⡁⢀⠨⠀⠂⠀⠈⠁⠒⠢⢄⠀⠀⣇⣽⣀⠀⠀⠀⠀⢾⢩⠀⠀⠀⠀⠀⠸⢠
⠀⠀⠀⠀⠀⢘⠀⠠⠔⢒⠟⡀⠈⠗⢤⡀⠀⠀⠀⠀⠀⠉⢶⡉⡀⠀⠘⠀⠀⠀⡿⠘⠀⠀⠀⠀⠀⡆⠘
⠀⠀⠀⠀⠀⠈⡰⠁⠀⠃⢰⠀⠰⠌⠀⠈⠓⢄⡀⠀⠀⠀⠀⢁⢹⢢⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⢀⠃⠈⠒⢄⠀⠀⠈⠐⠐⠀⠀⠘⣿⠀⡶⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⡠⠒⠀⡇⠀⡠⠊⠀⠀⠀⠀⠈⠈⠐⠒⠒⠒⢄⠀⠋⠐⢄⣀⣀⣀⡇⡀⠀⠀⠀⠀⠉⠀⠃
⠀⠀⠀⠀⠀⠃⢰⢒⣇⡈⠀⠀⣰⡆⠀⠀⡀⠀⠀⣴⡆⠀⠈⡤⣄⠀⢸⠀⠀⠀⠀⠈⠁⡈⡉⠈⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠀⠀⠀⢠⠦⠁⠩⠶
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢀⠜⠀⠆⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⣏⢑⡤⠤⣀⡁⠂⢤⣀⣀⣀⣀⡤⠔⣊⡉⣤⢤⡊⢰⡇⡖⠯⣁⠐⢂⡘⠖⢄
⠀⠀⠀⠀⠀⠀⢉⠍⡧⢀⡀⢀⠀⡌⠙⠒⠒⠋⢀⣀⣀⣀⣀⣨⠉⡅⠃⢳⠀⠹⠿⠟⡀⢀⡈⣦⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡤⠤⠤⠤⢼⠖⠀⠉⠉⠉⢹⠥⠴⠶⠂⠄⣀⡏⠉⠉⠁⠀⠉⢫⠲⡌⠀⠀⠀⠈⠀⠀⠀⡦⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⡸⠀⠐⠒⠒⠂⡡⠐⠐⠒⡂⠐⠠⢉⡐⠒⠒⠒⠐⠐⢶⡇⠀⠀⠀⠂⠀⠀⠠⠤⠤⢤
⠀⠀⠀⠀⠀⠐⠓⠘⠋⠹⢉⣬⠀⠀⡱⢒⣂⢆⡐⠀⠈⠉⠩⠉⡉⠑⠚⢷⠖⠒⠒⠒⠒⢲⡞⣀⣀⡀⡄
⠀⠀⠀⠀⢀⡎⠂⢜⡒⠀⡬⠖⡆⠀⠀⠈⠉⠁⠁⣶⠲⠦⠰⠀⠅⠂⣢⡈⣇⠀⠀⡀⠤⠤⠤⡀⠀⠀⠐⡀
⠀⠀⠀⠀⡼⠤⠁⠖⠃⠹⣀⡠⠉⠉⡑⢒⣒⣛⣈⣀⣂⣀⡉⠐⠒⠈⠀⠉⠘⢆⠈⢒⠴⠶⠂⠡⡀⠀⠀⠡⠀⠀⠂
⠀⠀⢠⠒⠃⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠂⠈⠀⠐⠀⠐⠁⠀⠀⠀⠄⠀⠆
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⡀⡄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠤⠤⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠬⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⣶⣷⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠔⠀⠈⢐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡀⡈⢢⠔⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠈⠀⢐⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠛⠻⠿⠻⢻⣿⣷⡀⠀⠀⠀⠀⡐⠀⠈⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠑⠤⠠⠀⠉⠈⠂⠀⠀⡀⠀⠀⠀⢘⣿⣿⠋⠤⠀⠀⠀⠀⠄⠀⢻⣿⣄⠀⠀⠀⠔⠀⠀⠊⠀⠀⠁⠠⠤⠒⠁⠀⠔⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠂⡀⠀⠀⠐⠄⠐⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠑⠠⠌⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡀⠀⠀⠠⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⡰⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈⠢⡀⠀⠑⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢀⠀⠁⠁⠀⠀⠌⠀⠀⠀⠄⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠄⠐⠿⣿⣿⣷⣄⣀⣀⣀
⡄⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠁⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛
⢁⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠
⠀⠈⢂⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⡀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠁⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⢠⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠻⣿⣷⣦⣤⡀
⠀⠀⠀⢀⣤⠶⢻⢟⣯⡀⣒⠻⠿⠏⠛⢷⡀
⠀⠀⣰⢃⣴⣃⡴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠏⣾⣿⣿⣂⡄⠀⠀⠀⠀⠀⠀⣀⣠⠀⢰⢷⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣵⣶⡤⣄⣀
⠀⢰⠸⢻⠻⣿⡟⣅⠀⠀⠀⢀⡴⢛⣉⡑⢀⠫⡠⢼⠄⠀⠀⠀⠀⠀⠀⠤⢈⡉⠘⢶⠀⠀⣀⠀⠀⠀⢸⠃⡎⠙⠿⣿⣷⣿⣿⣶⣤
⠀⢸⣆⠸⣇⡾⢺⣿⣧⠀⠀⠈⠠⠚⠛⠀⠀⠁⠈⢐⡐⠄⠀⠀⠀⢀⠄⠠⠤⢀⠝⢎⠑⠂⠀⢽⡄⠴⢿⢀⡇⢦⡀⠐⢬⡉⠉⠛⠿
⠀⠀⢻⣦⠻⣷⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠘⠁⢸⠀⠀⠀⠀⠤⠀⠄⢀⢱⠘⠀⠀⠀⠘⠃⠀⡞⣸⣷⣿⣿⣿⣮⣻⣷⣶⣄
⠀⢄⣄⣻⣿⣽⣤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⢠⣾⣿⠇⠀⠀⠀⠀⠀⠤⣄⠀⠸⡱⠚⠉⠀⣠⣀⣀⡼⣱⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⢈⡍⡛⣿⡿⡄⢦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⠟⣫⡿⠀⡳⡔⢦⣤⡄⠀⠀⠀⠀⠀⠀⠉⠛⣻⡥⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿
⠐⠓⡤⠞⢠⠊⠀⠘⡄⠙⢮⡢⢤⣀⡀⠀⠀⠀⠀⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣾
⠠⠬⠤⠐⠁⠀⠀⠀⣱⠄⣀⣙⣦⣀⠀⠛⠷⠖⠶⣶⢭⣁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⢻⣦⡀⠀⢀⣀⣀⣠⡞⣛⣭⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣄⡀⠉⠉⠉⠉⠉⠉⢡⣦⡙⢿⡒⠶⣾⣭⣭⣭⣭⣰⣶⣷⣷⣿⣿⢛⡋⢠⣿⣿⣾⣿⡿⣫⣿⣿⣿⣿
⢸⣿⣷⣶⣶⣛⣭⣉⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢸⡿⢿⢂⣻⣷⣄⠀⠀⢉⣿⣿⣿⣿⣿⣿⣅⣾⣿⣿⣿⣿⡟⢫⣾⣿⣿⣿⣿⡏
⢸⣿⣿⣿⣿⣯⣍⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠸⡇⢸⣿⠭⣽⣿⣷⣤⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣰⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⢻⣿⣷⣄⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣷⣄⠸⣿⡶⠶⠬⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⣰⣿⣿⣿⡿⠟⠛⠋⠁
⠘⠛⠙⠟⠛⠃⠙⠙⠛⠷⠄⠀⠀⠀⠀⠈⠙⠻⠿⣿⡿⠷⢿⠛⠒⠶⠬⣿⡿⡆⠀⠻⠋⠙⠿⠟⠋⠀⠀⠀⠿⠋⠉⠁
⣶⢶⣶⢶⢶⢶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠐⠂⠤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠐⢀⠀⡰⠂⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁
⠀⠀⠀⠀⡂⠈⣀⠇⠀⠀⠀⡀⠚⠀⠁⢺⠟⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠈⠈⠁⠀⠀⣠⣎⠄⠊⠅⣂⣒⣀⣰⣀⡀⠀⠀⠀⠀⠀⠀⢠⢲⠐⠤⣤⣶⣶⣶⣤⡡⣂⠄
⠀⠀⠀⠀⢀⡇⡴⣪⠟⡕⠡⣢⠞⠉⠀⠀⠀⠀⠀⠉⠲⣄⠈⢂⠀⠀⠀⢹⡆⠀⠀⠀⠈⠉⠉⢉⣷
⠀⠀⠰⠀⡈⢣⠜⠁⡜⠀⡼⠁⠐⠀⠀⢁⠀⠀⠀⠀⠀⠈⢧⠀⢣⠀⠁⠀⠉⢆⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⠃⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠘⡏⠈⠀⠀⠀⠀⠀⠑⠠⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⡀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢷⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⢀⡼⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠳⢄⠀⠀⠈⠀⠀⠀⢀⡠⠞⢈⠔⠋⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠉⠓⠲⣶⣶⠾⠟⣃⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠓⠖⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠔⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣾⣿⡀⢰⢁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆⠀⣰⣾⣿⣳⣿⣿⣿⣿⣿⣿⣻⣿⣉⣻⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣚⡂⠠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⠉⢹⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⣈⡉⠁⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⢡⡄⠀⢀⣀⣠⣤⣶⣾⣿⡿⢱⠄⢀⠀⣾⠠⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡄⠀⠀⢀⡇⠀⠰⠀⣀⠀⠀⠒⣀⠲⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⡇⠀⢠⣿⡀⠚⡍⠛⠢
⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⣴⣷⠔⠉⢛⡥⡤⣀⠀⣛⠒⠋⠉⡜⠛⣒⠿⠛⠋⠉⠌⡁⠩⠩⣠⠄⡈⠉⡓⠴⠸⢿⡇⠛⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⠧⠘⠉⠁⢀⠔⡅⢀⠈⠁⠀⠁⠀⠸⠓⠒⠉⠀⢀⠄⠘⠆⢀⣿⠀⠀⡀⠉⠈⠑⡷⢤⠀⠀⠈⡖⠉
⠀⠀⠀⠀⠀⠄⠀⠙⠤⠊⠁⠀⡀⠂⠀⣼⢰⡄⠘⠃⠀⠀⠀⢠⡀⠂⢀⠀⠀⣿⢸⣿⡄⣀⣀⠀⢀⠀⠂⠀⡆⠀⠀⠇
⠀⠀⡆⠀⠂⠀⠀⠠⢀⣀⠀⢀⣤⣤⡤⣭⡁⠀⠰⠆⠀⣠⡄⣾⡇⠲⠵⠄⠜⠛⠛⠉⠉⠀⠄⠈⠀⠀⠈⠀⡇
⠀⠀⠁⠨⠁⠠⢈⠀⢀⠀⣀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠀⠒⠠⠅⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⠿⡷⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠃⢂⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⠿⠿⣋⡵⠋⠉⠉⣛⠛⣿⣿⣿⣷⣼⠒⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠁⠀⠈⣿⡟⠛⡉⠀⠀⠀⠀⠄⠀⣿⣿⣿⣿⡟⠀⢨⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠈⠂⣠⣿⡇⠀⠀⠀⣀⡀⠀⠀⠀⢸⡸⣿⡿⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡴⣿⣿⡇⠀⠀⢠⣚⣢⡀⠀⠀⢸⣷⣿⠃⠀⢠⣄⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣧⠀⠀⠸⡛⠡⠇⠀⠀⢸⠿⠉⠢⠤⢼⣿⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⣿⣿⣿⣷⣤⢤⣠⡤⠤⠤⢖⣞⠈⡄⠁⠄⢸⠋⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣿⠿⠿⠋⢖⣠⣦⣴⣧⣨⣄⣠⣄⣒⠐⠎⡼⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠿⠶⠛⠛⢿⣤⡀⠀⣀⣀⣀⣈⣻⣿⠗⠧⡠⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢊⣡⣤⣤⣤⣶⣶⣿⣿⣷⣿⣿⣿⣶⣿⣽⣻⡷⣄⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡠
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⢛⣛⣛⣋⣙⠛⠿⠿⠿⠿⠶⠶⠶⠦⣤⣄⡈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠁

⣐⢦⣲⣄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⢀⣾⣵⣗
⢩⣿⣟⠁⠉⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⣈⡿⠷
⠀⠀⠀⠉⠙⢄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡜⠉
⠀⠀⠀⠀⠀⠈⢆⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠒⠢⡄⠀⠀⠀⠀⠀⠀⢀⠜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠑⢄⠀⠀⠀⠀⠀⡌⠀⠀⣤⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⠊⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠈⠣⡀⠀⠀⠀⣃⡤⢊⠀⠑⠢⠤⣠⠀⠀⠀⠀⡴⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠙⢄⠀⠸⣹⠀⠐⠀⠀⠐⠀⣟⠇⠀⢀⠎⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠜⠑⢤⠋⢇⠀⢠⠬⠄⢠⢫⢀⠔⠛⢦⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣧⠀⠀⠀⠣⡀⢱⠌⡩⢴⢁⠴⠏⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⡌⠱⡀⠀⠀⠈⠹⡀⠀⡸⠁⠀⡀⠀⣠⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠘⡄⠀⠀⠀⠈⠊⠀⠀⠀⢣⡔⢣⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣼⡁⠀⠀⠀⠀⠀⢸⡄⠀⣀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠛⠻⠶⣦⣤⣤⣼⣷⠾⠛⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠞⠛⠛⠛⠛⠛⠻⠿⠶⣶⣤⣤⣀
⠀⠀⠀⢀⣠⣴⣶⠿⠛⠋⠉⠁⠀⠀⣀⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦
⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⣀⣴⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠛⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢻
⣼⡿⠁⠀⠀⠀⠀⠀⢀⣴⡾⠋⢡⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⢀⡆⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠙⠿⣶⣄⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⢀⣴⠿⠋⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⣄⣀⣼⣇⣤⣴⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⢿⣇⠀⠀⠀⣾
⠸⣷⡀⠀⠀⠀⣿⠃⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠈⠉⠉⠉⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⢸⣿⠀⢀⣾⠏
⠀⠙⣷⣄⠀⠘⣿⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢀⣴⣾⡷⠀⣾⣿⣦⡀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠘⣿⣶⠟⠁
⠀⠀⠈⠻⢷⣤⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢰⣿⣿⣿⡁⠀⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⣼⠿⢿⣿⣿⡿⠿⠿⢿⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⣿⣡⡼⠋⠀⠀⢀⡿⠉⠀⠀⠀⠀⠻⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡄⢰⣿⠋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣿⡇⢸⣿⡋⠉⢹⠋⠛⠛⠛⢶⣦⣤⣀⠀⣰⡿⠁⣠⣤⣤⣶⡶⠿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⣿⡇⠀⢿⣧⡀⢸⠃⠀⠀⠀⣾⠀⠀⣩⣿⠟⠁⢀⣿⠏⠀⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠹⣷⡀⠀⠙⠿⣿⣶⣤⣤⣴⣷⣶⠿⠛⠁⢀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⡀⠈⠉⠉⠉⠀⣀⣀⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⢀⡔⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠟⠿⠿⠿⠿⠄⠀⠀⠀⢀⣤⣾⣷⣶⣦⡄⠀⠀⠀⠀⠀⠀⣴⣶⠶⠿⠿⠟⠛⠓⠀⠀⢠⣴⣿⣷⣶⣶⣤⣤⡀⢸⣿
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣷⣸⣿
⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢰⣿⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣿⡃⠀⠀⠀⠀⠀⠀⣿⣄⣤⣤⣶⣿⣧⣤⣤⣤⣤⣬⣿⣦⣤⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠶⣶⣶⣶⡶⠟⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣴⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⢀⣀⡠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀
⠀⠀⠀⠀⠀⣼⣿⣧⠀⢠⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⢠⣤⣤⡄⣿⡇⠀⠀⣀⡴⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣧⣾⣿⢃⡰⠆⠉⢁⣴⠏⠀⠀⠈⢻⡇⠀⠀⠀⠐⣷⣬⣁⣀⡀⣿⠇⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠘⣿⠟⣻⣯⠙⠛⠉⢁⣾⣿⠋⠀⠀⠀⠀⢾⣧⣀⠀⠀⠀⢉⣷⣯⠀⠘⠋⠻⣿⡿⠏
⠀⠀⠀⢀⣶⠋⢀⣿⣿⠻⢀⡀⠀⠸⠋⠁⠀⠀⠀⠀⠀⡨⣄⠈⠁⠰⠊⠉⠉⣳⠀⠀⠀⠑⠆⡀
⠀⠀⠀⣼⠇⠀⣼⣿⢯⡁⠈⠀⠐⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣰⠯⠀⠀⢈⠟⠶⣦⣤⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⣠⠀⠀⠎⠁
⠀⠀⠙⠈⠀⢀⠎⠀⢀⡇⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠇
⠀⠀⠀⠀⣠⡏⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣞
⠀⠀⢀⡼⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠣⣄
⠀⢀⡾⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠀⠀⠈⠳⠄
⠀⠸⠅⠀⠀⠈⢁

⠋⢀⠞⠀⠀⠺⣿⡷⠿⠿⠃⠀⠀⠀⠾⡁⠀⠀⢀⣀⣇⠀⠀⢿⠿⠇⠀⠀⠀⠶⡁⠀⠀⣠
⣶⡏⣀⠓⠄⡀⢉⠩⠀⠀⠀⠀⢠⡀⠈⢉⠀⢸⣿⣏⣽⣷⡮⠀⠀⠀⠀⢠⡀⠈⢉⠁⠀⠉⠁⠀⣴⡆
⣷⣿⡂⠤⣍⠱⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢁⣿⣿⣿⣿⣿⡷⢀⣄⠰⠆⠉⠀⠐⠢⠄⠀⠐⠀⠀⠀⠀⣀
⠋⠀⠐⠿⢩⣤⣴⣾⣿⣷⣶⣄⠀⠠⠀⠀⠀⢹⣿⣿⣿⣟⠀⠟⠻⠷⢿⣶⣦⣄⠀⠀⠈⠀⠀⠄⠀⠀⠙⠈
⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣸⡞⢡⢃⠈⣿⣿⣿⣿⣷⣦⣤⣀⠀⠈⠉⠙⠛⠦⢠⠄⠈⣤⣤⠂⠀⢠⣶⣶⣄
⠃⡄⢰⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿⠄⢀⣿⠿⣿⣯⣈⣉⡙⠟⠷⢶⣦⣤⣀⡀⣰⠀⠀⢛⣫⣄⠀⠻⢿⣿⣿
⠐⠈⠀⢈⣛⣿⣿⣿⠛⡟⠛⠿⠿⠿⠿⢃⣷⣿⣿⡿⠀⠠⠈⠙⣆⡀⠀⠀⡁⠈⣿⡿⠙⠒⠈⢮⡹⣧⣀⣿⡯⠁⢀
⡀⠓⠠⣿⠟⠉⠻⠿⠀⠙⢀⡵⢶⣾⣿⣿⣿⣿⣿⣿⣦⣄⡀⠈⠉⠉⠀⠀⠉⠛⠛⢠⣿⠈⢐⣾⣿⣾⣯⣷⠆⠀⠠
⠏⠘⠂⠀⠀⢀⠀⣠⠶⠒⢺⢣⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⣶⣿⣡⣾⡿⢟⡝⡿⠹⣿⣎⠁⣠
⣀⢀⡀⠀⢃⣸⣿⣿⣄⣀⢆⠚⣁⠈⠰⠈⠹⢿⣿⠉⠋⠙⠛⠿⣿⡿⢿⡿⣷⣾⣿⣿⡬⡆⢠⠉⣴⣷⣦⠙⢿⡆⢻
⠛⠀⢁⣆⠃⠘⢿⡿⣩⣴⡶⠆⠉⠀⠀⠀⠀⣾⣿⣷⣦⣤⣤⣀⠀⠀⣿⣷⣾⣿⣿⣿⣧⡀⠀⡾⠙⢟⣏⣶⠒⣤⣼
⠀⠀⠈⣿⠀⠈⠐⠹⣿⠛⠁⣀⠸⠁⡠⠓⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢂⠎⢀⡈⠂⠻⠿⠉⠋⠉
⣶⣿⣿⣿⣿⣶⣾⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⠙⡿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣆⡠⡀⠈⠓⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐⠀⠠⠓
⠀⠀⠀⠀⠀⡔⡏⠂⠀⠂⠀⠀⢰⣿⣿⣿⡿⣿⡿⠉⠙⠋⠁⠹⣧⠀⠉⢀
⠀⠀⠀⠀⠐⠈⡲⠇⠀⠀⣀⠀⣿⣿⣟⠉⠐⠀⠀⠀⠀⠀⠀⠀⠑⠮⠄⠀⢠
⠀⠀⠀⠀⠀⠉⠠⣄⠀⡠⠀⠣⣹⣿⣿⡗⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⠒⠀⠈⢻⣍⠋⠀⠀⠀⠀⠶⠾⠟⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡼⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⡠⠚⡄⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣣⡀⣀⡠⠀⠀⠙⠳⡖⠤⠤⣔⠂⠱⠃⠤⢇
⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠛⣦⡄⠀⠀⠆⠠⠈⠦⠴⠁⠈⠣⣀⠔⠁⢀⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⢌⣻⣆⠀⠠⠂⠀⠀⠀⠈⠉⠑⣑⣄⣀⠁⠤⠤⠀⠒⠀⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⢻⡛⣻⣆⠀⣀⠄⢀⠤⠒⠈⠁⠀⢀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢱⠈⣻⣆⡡⠊⠁⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⢘⣾⡿⠋⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠉⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⡀⠀⢸⠃⠀⡄⠀⢐
⠀⠀⠀⠀⣀⡀⠐⠂⢀⣼⣿⡆⠈⠀⢠⠏⣀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⣥⠾⢛⠁⣠⢀⡬⠈⠁⠖
⠀⠀⠀⠈⠀⠀⣠⡿⢷⣶⣿⣝⣋⣉⣀⠀⠚⠁⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⢋⣄
⠀⠀⠸⠀⠀⠀⠀⠀⠸⢻⡿⠷⠀⠉⠁⠈⠉⢷⢄⡆⠙⢉⣭⣭⡭⠽⠿⢓⣒⣠⣤⡄⣶⠶⣦⠉
⠀⠀⠀⠈⠲⣶⠖⠂⠈⠉⠀⠀⠀⠀⠀⠀⢈⠃⢋⣀⢠⡍⣴⡶⠖⢻⡆⣸⡏⣿⣀⡀⣿⣠⡿
⠀⠀⠀⠀⠀⠈⠁⢀⡼⣿⣅⣀⣀⣠⠤⠐⠚⠀⢸⣿⣼⡇⢸⣧⡆⠸⣷⣿⠁⣿⣉⡅⣿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⣀⣀⣰⠀⢾⡏⢿⡇⢸⣧⢶⣀⣻⣯⠀⢻⡏⢱⣾⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠒⠀⠉⠈⠀⠀⢿⠀⢸⡷⣮⣵⢾⡇⢸⠟⣯⣁⠀⢸⡇⢸⣿⣷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⣸⡄⣿⠰⣶⣿⠈⣿⣿⠀⣿⣭⡄⠘⠷⢟⣘⣧⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠒⠒⠒⠚⠋⢹⡇⠻⡶⠟⠻⣂⣛⣫⠤⠅⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠲⠤⠤⠤⠄⠒⠊⢳⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⣀⣀⠀⠉⠉⠁⠀⠀⠀⠀⡉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠒⢤⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣠⠊⠀⠀⡠⠐⠀⠀⠀⠀⠀⠢⢀⠀⠈⠉⠒⠢⢄⡈⠢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡰⠁⠀⡴⠉⠀⠀⠀⠀⠀⣀⠀⠂⢄⡈⠐⠢⠄⣀⣀⠈⠢⠈⠈⠆
⠀⠀⠀⠀⣀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠰⠁⠀⠊⢀⠄⢀⠔⠁⡠⠊⠀⠑⠢⣀⠈⠐⠢⠄⣀⠀⠈⠁⠈⠂⠸
⠀⠀⢠⠤⣣⠹⡄⢂⡐⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠐⢊⡡⠖⢁⡠⠊⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⣀⡉⠉⠉⠁⠀⢀
⠀⠀⢀⡵⡌⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣊⡩⠄⠒⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠉⡋⡙⠏⡀
⠀⠀⠀⠑⠤⣀⠀⣀⠔⠉⠑⢄⠀⠀⠀⠀⠰⠹⡨⢁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠴⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡧⠊⢀⡇
⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⣄⠈⠚⢆⠀⠀⡔⠒⡄⠀⠀⠐⠤⣠⠆⠀⠀⠀⡆⢐⠀⠀⠀⡌⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠑⠢⡜⠀⠐⣠⡨⠢⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠞⠂⡅⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⠲⢅⣀⠀⠈⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠁⠀⠀⠘⢮⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠈⠁⠒⠒⠀⠀⠀⠒⠂⠉⠀⢠⠑⢄⠀⠀⠀⠸⡀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣬⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⢀⣀⠠⢤⠤⣀⣀⣀⠠⠔⠁⠀⠀⠑⢄⠀⠀⢈⡖⣈⣽⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⠏⡿⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⡟⡏⢇⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠑⠤⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠣⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⡣⠀⠀⡀⠔⠊⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢠⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡝⣠⡏⡀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣤⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢟⣻⣷
⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠀⠈⠙⠻⠛⠁⠀⠄⡉
⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⡀⠀⠀⠀⡀⠁
⠀⠀⠘⣿⣧⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⠒
⠀⠀⠀⠈⠻⢿⣷⣤⣀⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⡇⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠹⠿⠟
⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⣿⣿⣇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢿⣿⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢠⣿⣿⡟⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠘⢿⠋⠀⠀⠈⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠉⠈⠁⠈⠁⠀⠈⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠊⠔⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠀⠡⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢈⠀⠀⢀⡠⠒⠁⠑⢄⡐⢄⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢡⡀⡈⠅⠠⠐⠊⠁⠀⠀⠀⠀⠀⡉⠢⠌⠐⡀⠁⢀⠀⠀⠀⠠⢄⡌⠿⣸⠰
⠀⠀⠀⠀⠀⠀⡳⡇⠏⡆⣀⠀⠀⠀⠀⡊⢅⢇⠀⠀⠀⠘⠃⠀⠀⠤⠀⠀⠉⠀⠀⠀⠀⣿⠌⠃⠀⠀⣸⡀⠀⠀⠐⡳
⠀⠀⠀⠀⠀⠘⡉⠀⠀⠈⠐⠀⠀⠀⠀⢸⠀⠉⠄⠀⠰⣱⠀⠀⠠⠤⠂⠀⠀⠆⠔⢀⡌⠀⠘⢄⡠⠊⠀⠈⢳⠔⠉
⠀⠀⠀⠀⠀⠀⠈⠢⡔⠈⠑⠄⡀⠀⠀⡆⠀⡰⠛⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠦⣬⠵⢄⠀⢢⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠁⠚⠀⡸⠀⠎⠹⡀⠁⠲⡄⢄⠀⣰⠀⡚⠁⢀⡁⠈⠂⠀⠣⡐⡡⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠰⠑⢴⠁⠀⠀⠀⠈⠃⢲⠑⡜⠀⢡⠌⡐⠐⠁⠁⠀⠀⠀⡀⠺⡀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⢀⢌⣀⠀⡀⠀⢀⠀⠀⠑⠌⢤⢂⠔⠁⠀⠀⢸⠀⠒⠈⠀⠀⠈⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁⠀⡌⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡐⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠰⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⠣⠄⠀⠀⠀⠀⠀⠧⠀⠀⠀⠤⠤⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠓⠄⣀⣀⠀⠀⠀⠚⠂⠀⢀⢀⡀⡤⠄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠌⢠⠁⣰⠁⡉⠉⠉⠈⠁⠁⠘⠀⠘⠈⢄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠡⠂⡰⠁⡠⠁⢠⠁⢸⠀⢰⠀⠰⠀⡆⠀⢦⠈⣦⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠔⠁⡔⠁⡰⠁⠀⡎⠀⢸⠀⢸⠀⢠⠀⠠⠀⠸⡆⠈⠧⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣎⢀⠜⠀⠀⡸⠀⠀⡟⠀⡈⠀⢸⠀⠰⡆⠀⢲⠀⠀⢲⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠈⠁⢒⠴⠀⣀⣀⡃⠀⡇⠀⣸⢀⣀⠧⡤⠐⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠋⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠈⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⡤⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⡀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠨⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣌⠻⡄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠙⣆⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⡾⠋⠀⠀⢀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠂⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡞⠁⠀⢀⡴⠟⠋
⠀⠀⠀⠀⠀⠘⠿⠿⠶⣦⣤⣀⠀⢀⠄⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠀⠠⣀⡀⠀⠀⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢀⠴⠋⠁⠀⡀⠀⠁⠀⡀⠀⠀⢄⠀⠈⠙⠢⡀⠀⢦⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⡀⠀⠀⢐⡧⠒⠉⠉⡉⠉⠉⠉⠒⠬⣂⡄⠀⠈⠀⢸⠂⡿⠀⠀⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡆⠘⠀⠐⠀⢠⠔⠁⠀⢀⣀⠁⢠⣠⠀⠁⠀⠄⠈⠳⡄⠀⠀⠎⠐⠁⠠⢚⣥⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣵⠃⠀⢂⢀⠈⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠗⠦⠤⠤⠤⠤⠤⠀⡤
⠀⠀⠀⠀⠀⠐⠐⠘⡀⠠⡄⠀⠤⠐⠚⠇⠀⠁⠈⠉⠀⠑⢄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⡀⠀⠀⠀⢀⠄⠀⢀⠠
⠀⠀⠀⠀⠀⠦⢄⡠⠘⠀⢸⡆⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠆⡆⠀⡇⠀⠀⡴⠁⠀⠀⠦⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠜⠁⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀⠀⠀⢀⠃⠀⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⡀⠀⠀⠰⡀⠀⠀⠋⠀⡀⠈⠀⠀⠀⠀⡀⠀⠁⠈⢀⠎⠀⠀⠀⠀⢀⡠⠄⠀⣁⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⣹⠀⠀⠀⠙⢄⠀⠀⠀⠃⠀⠐⠆⠀⠀⠃⠀⠀⡠⠊⠀⠀⠀⠀⢰⡏⠀⠈⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠘⠋⣡⡴⠁⡰⠃⠀⠀⠀⠀⡈⡑⠢⢄⣀⠀⠀⠀⠀⢁⡠⠴⢎⠡⠀⠀⠀⠀⠀⠀⠳⠈⠲⣤⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠾⠏⠀⠈⠁⠀⠀⠉⢳⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠁⠁⠀⡴⠊⠁⠀⠀⠈⠀⠈⠙⢷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⡖⠈⠀⡀⣉⠉⠳⡄⠀⣠⠐⢚⣉⡅⠉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⡆⠈⠂⣿⠀⠀⢠⣿⠀⡿⢀⠀⣤⠄⠜⡴⠿⠀⠉⢉⠁⠀⣠⠀⠀⣠⠁⢀⡀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢿⠀⣄⠀⠻⢸⠃⣙⡀⠘⠀⠾⢁⣿⠀⢚⠛⠀⠀⣿⠀⠀⣿⠀⣾⠀⢴⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠈⠀⣿⡄⢀⣏⡔⠘⣧⣀⡄⣶⡀⠻⠶⠶⠶⠆⠀⣿⠀⠀⡿⢀⡏⠀⣬⣤⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣀⠌⠿⠶⠚⠀⠀⠀⠿⠶⠛⠷⠶⠆⠀⠀⠀⠀⠙⠷⠶⠖⢸⣧⡠⠀⢠⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠰⣿⣷⣶⣴⣶⣤⣀⡀
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣤⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠛⠛⠿⠿⠟⣠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡅⠀⠐⡇
⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠐⠀⡀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢙⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠀⠀⠀⠙⠂⡇⠀⠀⠨⠈⠇⠀⠀⠈⡃⠀⠀⡀⢘⠄⠼⡇⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡁⠔⠀⠀⠀⠀⠤⠤⠤⠄⠄⠂⠀⠀⠚⠒⠉⠈⠀⠀⠁⠀⠀⠀⢨⣀⣀⣤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰
⠀⠀⡴⠒⠒⠒⠒⠂⠐⠀⠀⠀⠀⠤⠤⢤⠤⢤⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⠀⠀⠀⠀⢠⠞⠉⢳⡆⠀⠀⠀⠀⠸
⠀⠀⡇⢀⠖⠉⠉⠙⠢⡀⢰⠉⠉⡆⡎⠉⠙⣼⠀⠀⠈⠄⠀⠀⡇⠀⡇⠀⠀⢸⠀⠀⠀⠀⠘⢄⡀⠞⢃⠀⢀⠀⠀⠀⢀
⠀⠀⡇⡎⠀⠀⠀⠀⠀⢣⠈⣢⣖⠁⢙⣶⣊⢸⠀⠀⠀⠀⠀⠀⣃⠀⣁⠀⠀⠠⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠻⡄
⠀⠀⡇⠱⡀⠀⠀⠀⠀⡜⢸⠀⠀⡇⣇⠀⢈⢺⠀⠀⠀⠀⠀⠀⡥⠤⠬⠀⠀⢀⣀⣀⡀⠀⢰⠁⢰⠀⠀⠀⠀⠳⢤⠄⠌
⠀⠀⣇⠀⠑⢢⣤⡤⠊⡀⠈⠁⠉⠀⠈⠉⠁⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠄⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠻⠇
⠀⠀⠀⠈⠀⠀⡀⠀⠀⢘⠌⡆⠀⠀⠀⡇⠸⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⣈⠂⠀⠲⠰⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠃⠀⠀⠁⠀⠀⢾⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⡟⢿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⢀⠠⢄⡀
⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄⠀⠀⠀⡧⣗⠊⢹⣭⣾⢿⠀⢀⡔⠛⠛⠲⡄⠀⠀⠀⠀⡇⠀⢰⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢻⡒⣠⣻⣽⣳⡿⠀⢸⡀⠀⠀⠀⡹⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠒⠈⠀⢀⠴⠢⡉⠛⠛⠛⠉⠀⠀⠈⠓⠦⠤⠞⠁⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⡄⠀⠀⠰⠀⠀⠐⠓⢰⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⢠⠀⠀⡀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠲⠀⠘⠀⠀⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠄⠀⠈⡂⠀⡀⠀⠀⠣⡀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡸⠸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠀⠀⠀⠁⠀⠀⠐⢀⣤⣄⢀⠀⠐⠂⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠛⠈⠁⠀⠀⢀⠀⠀⣏⣿⣿⡏⠄⠀⣮⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡈⠀⣀⣰⠀⠀⠀⡎⠻⣧⣙⣏⠝⠛⡌⠻⣟⣋⡡⠟⢩
⠀⠀⠀⠀⠀⢀⡀⣤⣀⠀⠈⠑⠂⢄⡀⠀⢻⣃⣼⣵⡟⠀⠀⠈⠑⠀⠀⠀⡄⣰⠂⠉⠁⢠⣧⡾⢋⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⢤⣀⠀⠀⠀⠐⢤⠛⠛⡍⠂⠀⠀⠀⠀⣀⣀⣀⣈⣁⣀⣀⠀⢨⣧⠶⡛
⠀⠀⠀⠀⠀⠀⠺⠉⠳⢤⣀⢿⡙⠒⣶⣄⠐⡄⠀⡇⠀⠀⡀⠀⠀⠉⠉⠈⠙⠋⠉⠉⢀⣼⣯⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢿⣍⢛⠿⣟⣿⣶⡱⠂⠡⢤⠾⠿⠦⠄⠀⠀⠀⠀⠀⠀⠀⣚⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣓⣶⣤⣭⠷⣯⣸⠏⠀⠄⠀⠁⠢⢀⣀⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠦⢼⣉⣡⠃⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⣤⢾⡡⠀⠀⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠛⠏⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⡀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡦⢝⠃⢀⠠⠄⠈⠁⠀⠀⠀⣀⡀⠀⠀⡴⣶⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢦⡀⠀⠀⠀⠠⠀⠀⠀⢀⣺⣧⣾⢴⠥⠞⠀⠀⢀⠀⢰⡎⠧⡇⠀⠀⡇⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠑⢄⣀⠀⡀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣹⠀⡎⠁⠀⡇⠀⠀⡇⢤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠐⠀⠁⠈⠁⠁⠉⠙⠛⠛⠒⠊⠀⠐⠒⠓⠊⠰⠀⡇⣐⣧⠏⠀⠀⣇⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡄⠀⠀⠀⠀⠀⠀⡇⢻⡆⣇⣀⠀⡢⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠓⠀⠉⠘⠛⠛⠛⠒⠒⠚⢠⣠⣀⢀⡠⣄⣄⠀⣇⡿⠷⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢀⡀⠀⠀⡀⣀⡄⠀⠀⠀⣄⣠⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠋⣀⢸⣭⠄⠲⠟⡇⢄⠉⠀⠉⠀⠛⠷⡈⠃⢀⡄⠸⢻⡟⢨⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈









⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⡿⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⣀⠤⠀⠠⢠⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⠀⢄⡇⠀⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⣸⠁⠈⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠋⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠳⣄⠂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠆⠀⡦⠛⠉⠓⠂⢀⠀⣙⣿⣿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⠀⠀⠈⠛⢿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⢲⠀⠀⣀⠀⠀⠄⢀⡤⠋⠀⠀⠀⠑⢦⠦⠆⠼⠀⢀⣀⢐⠂⠒⠀⡅⠀⠆⠀⠀⠀⠀⠈⠈⢹⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⠀⡀⣀⠀⠀⠸⠁⡀⠀⠀⠀⠄⠨⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠸⠤⠆⠀⠀⠀⠋⠀⠀⠀⠀⠈⠂⠀⠀⠀⡰⠠⡇⠀⠀⠀⠤⠃⠀⠦⠤⠤⡤⠮⠤⡼⢁⠓⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠐⠀⠢⣄⠀⢆⠀⢦⢨⠄⢀⠄⠀⢤⠔⢀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢌⣡⢾⡿⠿⢾⣦⣈⠆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠈⠢⢈⠐⠂⠀⠢⠹⠂⠀⠐⠀⣠⠐⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⠁⠐⡄⠀⠙⣯⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠤⠠⠄⠰⠀⠈⡆⠀⠀⠀⠀⠀⠈⠁⠠⠄⠤⠄⠇⣀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡈⠉⠁⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⣀⠀⠠⠄⠀⠇⠀⠀⠀⠀⠀⠤⠀⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⡟⣶⣶⣞⣷⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢠⠥⡐⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⡠⢤⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠒⠀⠀⢸⠀⡇⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠠⠠⠤⠠⠀⠄⠤⠠⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠠⠤⠬⠥⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢘⠀⢇⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠆⡑⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣠⠔⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠁⠀⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠟⠻⣿⠿⢿⣿⣿⡄⠀⠀⠀⠀⢀⡂⠀⠓⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠑⢤⡀⠀⠈⢁⠀⠀⠈⢄⠀⠀⠀⠀⢸⣿⣿⠛⡁⠀⠀⠀⠀⣀⠀⠹⣿⣇⠀⠀⠀⢀⠎⠀⠠⠈⠀⠈⠀⠀⢠⠔⠁⠀⡠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⢂⡠⠔⠈⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣏⠑⠠⡠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠢⠀⠀⠀⠀⡄⠂⠘⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠋⠀⠄⠀⠀⠀⠀⠂⠂⠀⠀⠠⠁⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠊⠠⡀⠀⠈⢢⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⡠⠉⠀⠒⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⢀⠂⠀⠀⠀⡀⠀⠀⡌⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠀⠀⠀⠀⠲⠀⠀⠀⠰⠆⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⡀⡠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣴⣦⡀
⡠⠊⠀⠀⠁⢂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠁⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿
⠇⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀
⠘⠤⣀⡀⠀⠈⠄⠀⠀⠀⢀⠼⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⠀⠸⠀⠈⠀⠀⣀⡀⠔⠁
⠀⠀⠐⠄⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⠄⠀⠀⠀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠄⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡤⠶⠤⢤⣄⡲⣤⣄⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠉⠙⠓⠉⠳⢌⣧⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢖⣒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠉⠃⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢶⣦⣙⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⢀⠀⠀⠀⠐⠾⠧⣀⠴⣊⡍⠉⠑⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠐⠀⠀⢁⡂⢂⡤⠶⠶⣄⡀⠀⠀⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠁⡈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⠀⢄⢀⠈⠀⡎⠀⠀⠀⠀⠹⡀⠀⠀⠐⡄⠀⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⢠⠃⠀⠸⡀⠀⠀⠀⠀⠀⢡⠀⠒⠒⢼⡀⠀⠀⠈⣉⠲⠄⠀⠀⠀⠀⠀⠀⣈⢀⠄⠀⠐⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⢏⠞⠛⠲⣇⠀⠀⠀⠀⠤⠉⠂⠀⣀⡐⠁⠀⠀⠉⢉⡴⠖⠀⠀⠀⠀⠀⣐⡀⠙⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢸⠀⠣⢄⡀⠋⠳⠉⠀⠀⡄⠛⠀⠀⡼⠁⠀⠀⠀⠰⠏⢲⠆⠀⠀⢠⠖⠒⢇⣠⠢⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢸⠀⠀⣤⢀⡀⠀⠀⠀⠀⢓⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⢸⠤⠔⠒⠒⢧⠐⠼⢥⡌⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠙⠺⠭⣗⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠳⡴⠊⣠⠄⣠⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠑⠟⣌⠉⢙⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠄⢼⠿⠆⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⣀⠴⠻⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠐⠀⠀⢀⠞⠀⠀⠀⠀⠰⠊⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⣤⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠈⢁⡰⠖⠒⠒⠀⠊⠁⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠂⠈⠀⠀⠀⠀⠈⠃⠄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⡠⠁⡰⠋⡍
⠀⠀⠀⢰⠋⠉⠉⠉⠁⠀⠀⡠⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠤⠒⠁⠀⣠⠞⡡⠞⠀⢠⠃
⠀⠀⠀⢸⠀⠀⠢⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⣸⠁⠀⢀⡴⠺⠱⣐⣴⠴⢋⣵⡎⠀⡀⠀⠸
⠀⠀⠀⢸⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠙⠢⣤⠇⠀⠀⠉⠉⠉⠉⣉⠤⠚⠛⠺⢤⠀⠇⠀⡇
⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠈⠗⠇⠀⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣴
⠀⠀⡟⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠓⠾⠴⣤⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠧⠴⠶⠒⠒⠒⠒⠒⠂⠉⡏⠉⡩⠀⡸
⠀⠀⠀⠀⠀⠑⠂⠠⠤⢤⠤⠗⡖⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⣇⣀⣀⡀⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠲⠤⡀⢀⠄⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸⣇⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠭⠽⡁⠀⡀⣀⠈⢳⣄⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠉⠋⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠐⠂⠀⠀⠀⠂⠂⠂⠀⠃⠂⠁⠀⠀⠐⠀⠐⠀⠀⠀⠀⠈⠀⠃⠀⠈⠀⠘⠈⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠈⠉⠉⠑⡦⠤⢲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠁⠀⠀⠀⠀⠀⣇⠀⠀⢙⡶
⠀⠀⠀⠀⠀⠀⢠⣴⢲⢤⣠⠀⠀⠀⠠⣞⣁⡤⠂⢀⡄⢀⡴⡉⢷⢬⣿⠄⠀⠀⠀⠀⠀⣀⢤⢶⢦⣄
⠀⠀⠀⠀⠀⠀⣿⢿⣎⡿⢹⠀⠀⠀⢠⣎⠘⢳⠏⣡⠷⠋⠀⠐⠢⢸⣈⣦⠀⠀⠀⠀⢰⡷⣄⣧⡿⢿
⠀⠀⠀⠀⠀⠀⢳⠀⠉⡤⠋⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⠀⣔⠁⠀⠈⣿⠂⠀⠀⠀⠀⠀⠹⣄⡉⠀⢸⡂
⠀⠀⠀⠀⠀⠀⣸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠚⠿⡀⠀⠀⡾⠀⣥⠀⢠⣷⡃⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠱⡄
⠀⠀⠀⠀⠀⢀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠛⠚⢁⣤⡋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⠀⣀⡤⠴⠶⡞⡇⠉⠉⠉⠉⣸⠉⣦⠒⠒⠶⢤⣀⠀⠀⠀⠀⢸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇⣀⣤⠴⠋⠁⠀⠀⠈⣇⠳⠤⡀⠀⢰⣇⠀⡇⠀⠀⠀⠀⠈⠉⢛⡦⣤⣸⡇⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠈⠉⠉⣇⠀⠀⠀⠀⠀⠀⠈⣷⠖⠙⣆⠎⠀⠙⠛⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠘⢦⣀⣀⠀⢀⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠛⠒⠦⠤⢤⣤⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⢳⣷⣦⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⠀⢽⡏
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣷⠙⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣼⡧⠤⠤⠴⠶⠖⠒⠒⠒⠒⠲⠶⠦⠤⠤⠤⠤
⠒⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣨⡷⠄⠀⠀⠀⠀⢿⠗⠀⠀⠀⠀⠀⢸⠓⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠋⠉⠁⣀⣧⠤⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠒⠺⣄⡀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⣠⠴⠚⠉⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⣄⠉⠓⢦⣀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢀⣀⡿⠞⠉⠀⠀⣠⠴⠚⠉⠉⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⠀⠈⠙⠦⣄⠈⠛⠲⠦⠷⠤⢤⠤⠤⠤⠤⠤⢤
⢠⣤⠤⠴⠖⠛⠉⠁⠀⠀⢀⡴⠊⠁⣀⡴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⢤⡈⠙⠲⢤⣀⣀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⢀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡙⠓⠒⠒⠒⠒⠒⠒⠒⠒
⠤⠤⠶⠖⠒⠛⠉⠉⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⢗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠈⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠰⠦⣀⢨⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠈⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡄⠛⠒⠣⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⣠⡀⡾⠀⠀⠀⠀⠀⠀⠊⠁⠀⣒⠀⠀⠠⠖⠀⣠⠯⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⡗⢒⡦⠭⠛⠛⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⢋⡀⠤⠢⡀⠀⠅⠠⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠥⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠡⠿⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡠⠐⠂⠉⠀⠀⠁⡇⠀⠠⠀⠀⠰⠂⢤⠧⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣲⣀⣿⣴⣍⣹⠂⢄⠀⠀⠀⠀⠀⢀⣠⠔⢃⠜⠁⠀⠈⡐⠰⢀⣀⡀⠤⠀⠛⠉⠉⠀⠒⠲⠦⠤⢤⡤⣤⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠋⠁⡇⠉⢿⠳⠤⢀⡁⣀⠤⠒⠈⠀⠁⠀⠀⠆⢄⠀⠀⠀⠱⠀⠀⡠⠐⠀⠉⠁⠉⠀⠀⠀⠠⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⠀⡇⠀⣼⣷⠀⠤⠀⣀⡀⡀⡉⠀⠀⡠⡆⠀⠀⢑⡄⠀⡘⠠⠊⠀⢀⠠⠐⠈⠉⠀⠈⡀⠂⠀⠈⠰⠇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⣠⠄⠒⠹⠶⠦⠤⠶⠟⠁⣀⠤⠐⢙⠀⠀⠁⠀⠀⠃⠃⠀⠀⡦⡆⠀⢀⠃⠀⣰⠡⠤⠠⠤⢀⣀⠀⠀⢀⡤⠠⠔⢁⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠂⠤⠌⢁⡐⠀⠠⠔⠂⡁⠄⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⡐⠀⠈⠉⠀⠒⠂⠒⠈⠭⠅⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢷⠐⡈⠄⠛⡞⠛⠁⠒⠀⠤⢼⣀⡄⣀⡰⠄⠨⠇⢠⠀⠀⡄⠈⠄⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠠⠐⣈⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠠⠊⡱⠊⠀⡸⠣⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⣀⠔⠈
⠀⣋⠠⠐⠉⠀⠈⠉⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠂⠠⠤⠀⠀⢀⣄⠔⠊
⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠻⢓⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⢀⣀⠤⠖⠚⠋⠉⢹⡄
⠀⠀⠀⠀⣀⣤⡴⠖⠊⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠠⠤⠤⠤⠤⠤⢼⠅⠀⠀⠀⠀⢀⠤⢤
⠀⠀⣴⠟⠩⠴⠒⣲⠀⠀⠠⣤⣴⠾⠋⠀⢰⣏⣼⡇⠀⢀⣀⣀⣀⡐⠒⠦⢤⣀⢀⠏⠀⠀⠀⠀⢰⠁⠀⣾
⠀⠀⠈⠛⠲⠤⣄⢁⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⡗⠊⠉⠽⠁⠉⢻⠀⠀⢀⡼⠛⢦⡀⠀⠀⠀⢾⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⡠⠀⠀⢀⣴⣯⠀⣠⠴⠶⠤⢀⠜⠀⡠⠊⠀⠀⠈⢷⡀⠀⠀⠸⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡴⠋⡜⢠⠀⣾⠁⠉⠞⠀⠀⢀⡴⠊⣠⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⢻⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⡠⠋⠀⣸⠁⢸⠀⠹⠤⠀⠤⠐⠊⣁⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⣧⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⣄⣠⠃⠀⠸⡀⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⡿⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡛⠛⢻⠛⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠉⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠠⣿⡿⡿⠀⠀⠄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣴⣿⣷⣶⣤⣴⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣋⣿⣿⣷⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣴⡦⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣄⣠⣀⠀⠃⠀⠀⠀⢠⡞⣿⣿⣿⣿⣧⣺⣼⣟⢵⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣤⣤⣶⣾⣿⣿⣟⣫⣿⣿⣿⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠗⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣘⠛⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣯⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⡇⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣶⡄⣰⣦⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⡀⣰⣦⢠⣶⡄
⠀⠀⢸⣿⡇⣿⣿⠀⣿⡇⢲⡄⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣦⢸⣿⡇⣿⣿⢸⣿⡇
⠀⠀⢀⣉⣤⣌⣁⣤⣉⣠⣿⠃⠀⠀⠀⠀⠀⠀⠿⢛⣉⣉⣉⣉⣉⣤⣶⣷⣤⣉⠛⠿⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠘⣿⣄⣉⣤⣌⣡⣤⣉⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⢙⣉⣋⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⡉⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣥⣤⣬⣿⣿⣿⣿⣿⣿⣡⣤⣬⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⢛⣉⣉⡙⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠘⠿⠿⠃⣸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⠻⠿⠶⠶⠿⠿⢛⣁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟
⠀⠀⠀⣠⡀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠚⠻⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⠟⠓⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⠻⠿⠿⠿⠿⠟⢋⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⣠⣤⣍⡙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣩⣤⣄⠀⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣶⣿⣿⣿⣿⣿⣷⡄
⢦⣄⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣠
⠀⠈⠙⠻⠆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠰⠞⠋⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⣋⣉⣉⣉⣉⣉⣉⣉⣁⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣈⣉⣉⣉⣉⣉⣉⣉⣙⠛⠿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⡿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⡇
⠀⠀⠀⠀⠈⠉⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⢀⡀⢀⡀⢀⡀⢀⡀⠀⣀⠀⣀⢀⡀⣀⠀⣀⠀⢀⣀⡀⠀⢀⡀⢀⡀⢀⣀⡀⠀⢀⣀⣀⠀⣀⡀⠀⣀⣀⠀⣀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⣿⣼⣧⣼⢀⣟⣿⠀⣿⣾⡁⢸⡇⣿⢧⣿⢰⡏⢡⣅⠀⢸⡇⢸⡇⢹⣬⠿⠀⢸⡧⠄⢠⣟⣷⠀⣿⣼⡇⣿⠀⠈⢷⠟
⣤⣤⣤⣤⣤⣼⣧⣼⣧⣼⣭⣽⣧⣿⣬⣷⣼⣧⣿⣬⣿⣬⣿⣶⣯⣤⣬⣷⣾⣥⣽⣤⣤⣤⣼⣷⣶⣼⣭⣽⣦⣿⣬⣧⣿⣶⣤⣼⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡤⢖⣂⣀⣀⣀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣏⣶⣟⣿⣯⣽⣛⡋⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⡻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣿⣿⢸⣿⡿⢿⣯⢻⣿⣆⠙⢿⣿⣿⠘⣿⣿⣿⣿⣿⣴⡎⢽⣧⡇
⠀⠀⠀⠀⢰⡽⠷⠉⡇⠀⣿⣿⣿⣿⡷⠖⠛⠳⠍⠻⠦⠀⠈⣉⠉⢹⣿⣿⣿⣿⡿⠁⣠⠊
⠀⠀⠀⠀⠀⠑⠰⡄⠙⢆⢻⣿⣿⣿⠀⢰⠿⠆⠀⠀⠀⠀⠸⠛⠓⠸⡿⠻⣿⠟⠀⣰⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⠻⣿⠰⠙⠂⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣡⠏⢀⣴⣟⣛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠈⣷⣤⣦⡀⠀⠀⢻⣿⣿⣿⠃⠀⢀⣼⣿⣿⣴⣿⣛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⢿⣿⣿⣿⣦⣤⣄⣉⣉⡡⢤⣶⣿⣿⡿⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⠀⢠⣄⢀⣸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⠛⠻⠿⠛⠛⢛⠛⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣰⣃⣀⠀⠈⢧⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡟⠁⣿⣾⡇⠀⢸⣄⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢛⣿⡇⠀⠘⣾⠟⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⢹⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⢉⡉⠉⠁⠉⠉⠀⠀⠀⡉⠑⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠔⠁⠀⢀⠠⠀⠀⠀⠀⠀⠠⢀⠀⠉⠁⠒⠤⢄⡀⠑⢦⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠋⠀⢠⠖⠁⠀⠀⠀⠀⢀⠀⠠⢄⠉⠒⠤⢀⡀⠀⠈⠓⢄⡑⠘⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠃⠀⡴⠁⢀⠄⢀⠎⠀⡠⠊⠑⢤⡀⠉⠒⠤⣀⠈⠉⠁⠒⠂⠉⠄⠸
⠀⠀⢀⡀⢊⢢⡏⢣⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠤⠊⣡⠔⢁⡠⠊⠀⠀⠀⠀⠈⠑⠠⢄⣀⠉⠑⠒⠢⠤⠄⠀⠀⠇
⠀⠀⠈⠪⡙⠆⠉⠀⠚⠐⡀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠁⠤⠒⢋⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⡤⠤⡜⡀
⠀⠀⠀⠫⡉⠀⠀⠀⠀⡠⠓⢄⠀⠀⠀⠀⠀⠸⠦⠀⠖⠒⠉⠁⠀⠀⠘⠟⠀⠀⠀⣀⡀⠀⠀⠐⠿⠀⠀⠀⠀⠀⠀⡇⡣⠃⡇
⠀⠀⠀⠀⠈⠑⢢⠒⠉⠀⠀⠀⠳⣄⠀⠀⠀⠀⠈⠪⣱⡀⠀⠀⢀⣀⡀⠀⠀⠀⡀⠈⢁⡀⠀⠀⢀⡠⢄⠀⠀⠀⣰⠉⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠈⠳⢄⠀⣰⠓⢀⠀⠱⡄⠀⠣⠠⠇⠀⠀⠀⠈⠙⠉⠀⠀⠀⠘⠤⠜⠀⠀⣠⡃⡔⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠙⠢⣄⢠⠾⠊⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⢇⣠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⢄⡏⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⣀⠤⠔⢺⣄⠀⠀⠀⠀⢹⠀⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠀⠀⢀⡾⠈⠳⣄⠀⠀⠘⢿⡆⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢒⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠐⠂⡖⠲⠀⠐⠂⠒⠉⠀⠀⠀⠈⢣⡀⠀⠰⢯⣵⣾⢿⡓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡜⢸⠹⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠏⡏⠹⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣧⣠⣀⣀⡀⠀⠀⠀⠀⠀⣇⠀⠀⣀⡀⠀⣀⣀⡠⠇⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠩⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠉⠉⡟⡄⢀⡠⠔⠉⠉⠉⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥
⠀⠀⠀⠀⠠⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠤⠮⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠰⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⢹⠈⠉⠈⡇⡞⠀⠈⠁⢻⠀⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⠈⠁⠂
⠀⠀⢰⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢹⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠘⣷⠛⠓⠒⠒⠒⠒⠒⠲⣿⢸⠛⠛⠉⠉⠛⠀⠀⠐⢰⠄⡎
⠀⠀⢸⠀⢠⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⡆
⠀⠀⢸⠀⠸⡆⠀⠀⠀⠄⢰⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠄⠀⢀⠀⠀⠀⢸⡇⢸⠈⠰⠀⠀⠀⠀⠰⠁⡎⢰⠇
⠀⠀⢸⠀⠀⡆⠀⠀⠀⡤⢄⠀⠀⠀⠈⠁⠁⠀⠀⣾⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠠⠄⢰⡇⣈⠀⢀⠀⠀⠀⠀⢀⠀⠁⠸
⠀⠀⢸⠀⠀⠛⠀⠀⠀⡇⡄⠀⠉⠶⢂⡁⡃⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠈⠀⠐⠤⠤⠄⠊⢰⠁⣶
⠀⠀⢸⠀⠀⢸⡄⠀⠠⠃⠉⠀⠉⠉⠈⠀⠀⠐⢄⣹⠀⠀⠀⡆⠀⠀⡇⢸⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣁⡽⠤⢣⠀⠀⠀⠀⠂⠀⠀⡘⠀⡍
⠀⠀⢸⠀⠀⠀⣧⠌⢤⣠⣤⡤⢔⠒⠒⠂⠉⣋⠉⡥⠄⠀⠰⠇⠤⠄⣇⣸⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠈⠡⣴⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⢀⠁
⠀⠀⢸⠀⠀⠀⠀⠀⡞⠁⠠⠒⠋⢓⣤⠔⠋⢥⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠀⣼⡾⠛⡛⠲⡄⠀⠀⠀⠀⣼⠀⠇⠀⠀⠀⠀⠀⠀⢠⠁⢸⠃
⠀⠀⢸⠀⠀⠀⠀⢸⠀⢀⠔⠊⠉⡝⠁⡄⠀⠐⠉⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⢠⣀⠀⠀⠸⠀⠀⠀⠀⣿⠀⡆⠐⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⢸⠀⠀⠀⠀⠸⠀⡎⠀⠀⡜⠀⡸⠀⠀⣶⡀⠂⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠙⡀⣿⠱⡀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠄⡆⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠷⠀⠀⢰⠀⠀⠃⠀⢸⠀⠙⢤⡀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⣿⣿⠀⢃⡸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⠀⢠⠃⠀⠀⠀⠈⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠃⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠑⠀⢸⠀⢰
⠀⠀⢸⠀⠀⠀⠀⡴⠊⠉⠈⡇⠀⢀⡴⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⢢⡀⢸⠉⠉⠓⣤⣠⣀⣤⣖⡇⢸⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⢸⠀⠀⠀⠈⠃⠀⡶⠰⣇⢠⠋⠀⠀⠔⣴⡀⠀⠀⠀⠀⠀⠀⢀⢠⣶⠀⠀⠀⢁⣸⡀⠀⠀⣏⠀⢀⠀⠀⠁⠈⢈⣉⣉⣁⠙⠚⠃⠀⡄
⠀⠀⢸⠀⠀⠀⠀⢡⠀⢱⡞⠉⢿⠀⠀⠀⠘⠋⠁⠀⠀⠐⠂⠀⠀⠀⠙⠋⠀⠀⠀⠙⠉⢓⠀⠀⡇⠀⠀⢀⡤⠬⠭⠭⣍⠉⠉⠉⠉⠉⠉⠁
⠀⠀⢸⠀⠀⠀⠀⠘⠀⠈⠳⠄⢄⠁⠀⠀⠀⠀⠀⠀⡒⠀⠒⡂⠀⠀⠀⠀⠀⠀⠀⡠⢤⠊⠀⠀⠇⠀⠀⠏⠑⠒⠒⠊⠙
⠀⠀⢸⠀⠀⠀⠀⠀⡆⠁⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⢁⡀⣀⠂⠀⠀⠀⠀⠀⣀⠞⠁⠀⡇⠀⢠⠀⠀⡸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⣷⠛⢢⣀⣀⠀⠈⠑⠦⢄⣀⠀⠀⠈⠀⠀⠀⣀⡠⠴⠚⠉⢀⣀⣠⠟⠲⣼⠀⡤⢏⠐⠂⠠⠤⠀⠲⡧⣄
⠀⠀⠸⠀⠀⠀⠀⠀⢹⢶⠶⡏⠈⠉⠙⠐⠒⠖⠾⢿⣏⣉⢉⣻⠟⠏⠒⠊⠉⠉⠙⠁⢹⠶⡖⢹⠀⣇⠀⠹⣶⣴⣴⡖⠉⠀⠈⢦
⠀⠀⢈⠀⠀⠀⠀⠀⢀⡼⠀⠷⢂⣄⣀⣠⣄⢰⡆⠀⠀⠉⠁⠀⠀⢰⣤⣤⣤⣤⣤⣤⠎⠀⣣⡘⢸⢸⠂⠠⠘⠛⠋⡠⠄⠤⠤⠤⠷
⠀⠀⢠⣄⣀⣀⣀⣀⣠⣿⠊⠉⠉⠉⠉⠉⠉⣿⣀⠤⠶⠶⠖⣀⡀⡀⡇⠀⠀⠀⠀⠀⠀⠉⢡⡕⢺⡈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⢠⠀⠀⠀⠀⠀⢀⣇⣀⠠⠤⠤⠤⠤⠔⢉⡀⠤⠤⠤⠤⢀⣈⠉⠛⠤⠤⠤⠤⠆⠀⠄⠌⠿⡟⠛⡌⠉⠉⢸⣿⣷⠀⠒⢳⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⠀⣘⣀⣀⣤⢤⡤⠤⠤⠔⠁⠀⢠⠀⢤⠒⢀⣀⡉⠐⠲⠤⠤⠤⡦⠤⠤⠤⠀⣿⠆⠃⠀⠀⠀⠚⢀⣀⣀⠸⡟⠉⠉⠉⢿⡀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⢠⠉⡀⡈⠁⠀⠃⣠⠶⡂⠠⠀⠤⣋⣙⣌⡱⣄⠊⠀⠀⡀⢀⡔⠿⢶⡷⢤⠀⠉⢶⡦⣬⠭⠤⠭⠭⠽⠽⠿⠥⠤⠤⠤⠼⡧⠀⠀⠀⠀⠿
⠀⠀⢸⠀⠀⠀⢀⡏⡟⢀⠜⠒⢆⢁⡌⡶⢶⠀⠀⠀⠀⡀⠀⡉⠘⠀⣴⠶⠶⣆⢘⡂⢸⡅⠾⡐⣂⣈⢇⠀⠀⠀⢀⡀⣤⠤⠤⣄⡀⠀⠀⠀⠰⠄
⠀⠀⢸⠀⠀⠀⡾⠁⠼⣀⣘⣏⠱⡎⠀⠀⡞⠲⠶⠦⣠⣥⣤⣥⠤⠔⠛⣄⠀⠈⠷⠤⢤⣉⠤⠋⡙⠳⠛⣆⠀⠐⢯⡀⢀⣄⣀⠴⠃⠀⠀⠀⠀⢳
⠀⠀⢸⠀⠀⣘⠓⠛⠀⢉⣁⣀⣀⡱⡴⠞⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠆⠤⠠⠀⢀⣈⣉⣉⣉⡉⣦⠀⢀⡔⠋⠙⠀⠈⠐⢢⠀⠀⠀⠈⢇
⠀⠀⠸⢰⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢊⠁⠉⠉⠆⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠑⠒⠒⠓⠒⠊⠁⠀⠀⠀⠀⠸⡄
⠀⠀⢠⠀⣄⠀⠠⠒⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠂⠀⠀⡾⠠⠄⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⢶⠀⠁
⠀⠀⢸⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⢸⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢈⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏
⠀⠀⢈⡀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠹⣆⠀⠀⢺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣾⠏⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡘⢧⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢠⡾⠃⠀⠀⠀⣠⣾⠿⠂
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢠⡟⠁⠀⠀⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠶⢦⣤⣀⠀⣠⠀⠀⠀⢀⣀⠴⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠦⣠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⠴⠋⠁⠀⢀⠀⠊⠀⠒⠂⠀⠑⠄⡀⠀⠈⠑⠦⡀⠀⢣⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠁⠀⡀⠀⣲⠤⠒⠊⠉⠉⠉⠉⠑⠒⠬⣶⢤⠀⠀⠠⠀⢸⡇⢸⠃⠀⠀⣴⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠑⠄⣇⠀⠃⠀⠄⢐⠉⡠⠊⢁⠀⠐⠈⠀⣀⢀⠀⠁⠂⠀⠈⠙⢆⠀⠊⠀⡼⠀⠏⠀⠠⠞⣁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⡄⠀⠀⠀⣆⠞⠀⠀⠀⠀⠋⠃⠀⠈⠁⠀⠀⠰⠀⠀⠀⠀⠳⣤⣈⣀⠀⠀⠀⠐⠉⠉
⠀⠀⠀⠀⠀⠀⣄⣀⢠⠀⠀⠀⠀⠀⠀⣀⣤⡏⠀⡠⠐⠶⠆⠢⡀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⢣⠀⠉⠉⠉⠉⠉⡉⠉⠁
⠀⠀⠀⠀⠀⢀⠇⠀⠘⢆⠀⠻⡍⠉⠉⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠘⡀⠀⠀⠀⣠⠊⠀⠀⡂⠘⢄
⠀⠀⠀⠀⠀⠉⠐⢢⠎⠉⠀⠀⣿⠀⠀⠀⢸⠀⠘⠀⠘⠁⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠘⠀⡇⠀⠇⠀⠀⢰⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠈⡄⠀⠂⠀⣠⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⠀⠐⠀⢸⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⡀⠀⠀⠱⡄⠀⠀⠉⠀⢀⡀⠁⠀⠀⠀⠀⡀⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠀⣠⠴⠒⢈⣁⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⢀⡇⠀⠀⠀⠘⢆⠀⠀⢀⠐⠀⠀⠰⠇⠀⠀⠙⡠⠀⠀⡴⠃⠀⠀⠀⠀⠀⣾⠁⠀⢈⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠛⢉⣤⠊⢀⠞⠀⠀⠀⠀⠀⡠⢑⠦⢄⡀⠀⠁⠒⠀⠈⠀⢁⡠⠴⡛⠄⠀⠀⠀⠀⠀⠀⠙⢄⠐⢦⣀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠋⠀⠀⠀⠓⣆⠀⠒⠀⠀⠀⠉⠁⠒⠒⠒⠒⠈⠁⠀⠀⠈⠚⠀⠀⣠⠖⠒⠒⠂⠀⠐⠀⠙⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠀⠀⠐⠒⠦⡀⠀⠀⠀⣀⣤⠖⠒⠒⠒⠦⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⢰⡎⠁⠁⢸⡇⠀⣤⠂⠐⡀⠀⠙⡄⢀⣿⠀⠴⠶⠞⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡴⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣼⡇⠀⠀⢸⡇⢠⡏⣸⡀⠸⡏⠉⣹⠋⣽⠀⣀⣀⠄⠀⢰⡇⠈⠀⣾⠀⣦⡆⠀⠀⠄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠸⡇⢰⡄⠀⠃⣼⠀⢶⣆⠀⠃⠀⠃⢸⣿⠀⠨⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⠀⢸⡿⠀⢹⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢸⣷⡀⠰⣧⡄⠀⢿⣤⡴⠀⣧⠀⠻⠛⠛⠛⠋⠀⢸⣇⠀⠀⠿⢀⡟⠀⣤⡤⠴⣿⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⡤⠘⠷⠖⠋⠀⠀⠀⠸⠷⠖⠙⠷⠶⠃⠀⠀⠀⠀⠀⠛⠷⠶⠖⢹⣧⣤⠂⠀⣴⣁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄
⠀⠀⡄⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡼⡟⠁⠀⠀⠀⠀⠀⠀⠀⡀⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛
⠀⠀⣇⣛⣛⣃⣙⣀⣀⣀⣀⣀⠀⠀⠀⣷⣙⣛⣃⣙⣀⣀⣀⣀⣀⡀⠀⠀⣧⣿⣟⣃⣀⣀⣀⣀⣀⣀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣶⠘⢿⢲⡆
⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠇⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡾⠘⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠄⠀⠀⠸⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠹⠄
⠀⠀⠀⡀⣠⡐⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠋⠉⠙⣦
⠀⠀⠀⣇⣉⣙⣋⣀⣀⣀⣀⣀⣀⠀⠀⠀⣷⣙⣙⣃⣙⣀⣀⣀⣀⣀⠀⠀⠀⣤⣙⣃⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣇⣀⣀⣀⣤⣤⣙⣀⣀⣀
⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠄⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠇⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠰⠿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠸⠄⡀⣀
⠀⠀⠀⠀⠀⣀⣤⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣷⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⢸⡄⠀⢹⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⢾⣾⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡷⠶⠶⢿⣿⡷⠶⠶⠶⣶⠀⠀⠀⡷⠶⠶⢿⣿⣷⠶⠶⠶⢾⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⠶⠶⢾⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠈⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠃⠀⠀⠈⠁⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣷⣄⠀⠈⢛⡜⢦⡛⠓⢚⣣⡆⠍⣳⡽⢿⣧⣟⣿⣽⡽⣿⢉⣻⣯⣭⡝⠃⠀⠸⡟⠁⣦⠈⢻⡇⠙⠿⠈⠟⠿⣷⣶⢠⠟⠀⠱⡿⠁⠀⡛
⣿⣿⣿⣿⣿⣿⣭⣀⣘⠛⠃⠀⠚⠶⡄⠈⡷⢴⣿⢣⣿⢾⣾⡽⢿⢻⣫⡯⣧⣰⠘⠂⣶⠀⠀⠄⢤⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⣆⠈⠃⢀⡰⢿
⣿⣿⣿⣿⣿⣟⠻⣿⣌⢱⣶⡄⣀⡀⠀⣀⠃⣠⣌⡷⠶⣾⣿⠧⣾⢴⣽⣝⠣⠍⢀⣄⣿⣟⣻⣼⡀⠀⢀⡀⠀⠀⡀⠘⠇⠀⠤⠀⠘⣇⠉⣠⣶
⣿⣿⣟⡉⢉⣹⠟⣿⡟⢈⣏⡹⠋⠁⡀⠀⠀⢾⡍⠙⠂⠀⠀⠀⠀⠈⠈⠉⡀⠀⠘⠇⠀⢹⣏⡜⠴⠀⡌⠡⠆⡀⠉⠃⠐⠀⠀⢠⣶⠿⠋⠛⠁
⣿⣿⣿⣿⣿⣿⣴⣿⠿⠟⣛⢥⣶⢾⣄⣀⣛⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⡞⠀⠋⠈⠁⠠⣶⠀⣄⠀⠀⠆⣾⠄⠠⠄⠀⠟⠁⠁⠠⢶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣲⠸⠋⠉⠁⠺⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⡓⣸⡀⡼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣼⢀⣀⡀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣀⣼⢟⠏⣽⣷⡷⡀⠀⠀⠘⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣾⣿⠃⣀⠀⣀⠀⠀⠀⠀⢀⣠⡤⠤⡀⠀⠀⠀⠀⠀⣤⠀⠀⠻⣃⡹⣸⠿⢫⣏⣿⡅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⡀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠆⠀⠀⠀⣄⠉⠉⠀⢸⠇⢁⢸⡍⠁⢠⠀⠀⠀⢀⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⢛⣋⣚⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠐⣀⢀⣄⠛⠋⠾⠀⠀⠀⠀⠀⠀⠀⣙⣷⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠟⣫⠅⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⢶⢀⡀⢀⠤⠀⠀⠀⠀⠀⠉⢠⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡾⣫⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠙⠁⢠⡀⠠⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣣⢜⡵⠀⠀⠀⠀⠐⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⣀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣇⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⢴⣿⣅⣀⣣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢿⢈⡿⠋⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣀⠀⢀⠐⠀⠀⠀⠀⠀⠰⠞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢏⠉⠂⣦⣤⠠⠀⠀⠀⣞⢶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⣿⣷⡤⠤⢹⠍⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠰⡶⣿⠒⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢀⠀⠀⠀⣿⡏⠰⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⣇⣠⣾⡷⣃⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡡⠒⢦⣤⣿⣻⣄⣽⣿⠖⠢
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢢⢵⣾⡻⣋⣽⣽⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠐⢿⠀⠘⢗⢾⡿⠁⠉
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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄