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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠂⠀⢰⡆⢠⠀⢂⠀⡂⠀⢀⠀⢲⠀⠀⠀⠐⢎⠀⡔⠀⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢊⡀⡘⠀⠀⠀⠈⠀⢸⠃⠐⠂⠀⠀⠀⢀⡁⠄⢹⠀⢈⠠⠈⠀⠂⠀⠀⠀⠐⣁⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠈⠀⠸⠆⠠⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠀⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠸⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠂⠈⠀⠀⠀⠀⠀⠀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⡀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣮⠀⡈⠀⢈⠒⢄⣀⡄⠀⠀⠀⠀⡀⠈⠁⠉⡂⠀⠀⢀⠀⢈⠂⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠉⠉⠦⣾⣷⡌⠐⠂⠈⠀⠲⠀⡀⢈⡆⠀⠈⠸⠁⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠠⠤⣄⠀⠀⠀⠀⠀⠀⢠⡘⠐⠠⠀⠂⠙⠁⠀⡀⠁⢂⢀⡴⠃⠀⠀⠀⢀⠀⣀⠠⠄⡀⡀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠉⠂⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⣀⠀⠔⠀⠠⠁⠀⢀⡄⠀⢺⠁⠀⠈⠁⠀⠀⠈⠁⠀⠈⠠⣀⠐⠁⠀⠀⠑⢀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠠⢆⠄⠀⠈⠀⠀⢰⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠊⠁⠀⠐⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⡔⠀⠀⠀⠐⢄⠉⢢⡔⠂⢶⡁⢉⠀⠘⠁⠱⠂⠄⠀⠐⠀⠀⠀⠀⠀⠀⠈⡀⡀⠠⠘⠃⠉⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠔⠀⠈⠁⠈⠐⠀⠀⠀⠀⠀⠃⣤⠁⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠠⡄⠂⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⢌⠋⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠁⠀⠀⠂⠀⠀⠀⡈⠉⡡
⠀⠀⠀⠀⠀⠉⠢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡀⠐⠒⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠈⠡⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠀⠀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢬⠆⠀⣀⡀⠱⠂⠄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠀⠀⠀⠀⠀⢂⠀⠀⢄⠀⠀⡀⠐⠦⠔⠀⠀⠈⠀⠀⠀⠀⠁⠐⠒⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠑⠠⠈⠣⠀⠐⠀⠙⠂⡐⠣⡀⠀⠀⠀⠀⢀⠔⠉⠀⠆⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠠⠆⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠄⠀⠀⠈⠂⡄⡀⠀⠈⠓⠀⢤⡀⠈⢂⠈⠖⢐⠐⠈⠀⠀⡀⠈⠀⠀⠀⠀⠲⠀⠀⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠈⠉⠢⡄⡠⠐⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⣀⣀⠘⠀⠀⠰⠀⠐⠀⠀⠀⢐⠶⠒⠀⠀⢀⣈⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⡆⠀⠈⢁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠆⠀⠀⠀⠱⠁⠂⠈⡸⢗⡂⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠶⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠆⠔⠂⡌⢀⠀⠂⠀⠀⠠⢀⠤⢤⠒⢂⡄⢀⠖⠀⠀⠤⠴⠀⠊⢀⠀⠀⠀⠀⢲⡄⠂⢸⠉⣤⠐⢂⠠⢄⠠⠤⣀⠈⢀⡘
⠀⠀⠀⠉⠁⠀⡇⠀⠀⠀⠓⠀⢸⠃⠀⠀⠀⠀⠀⢀⠸⠀⢸⠀⠀⠠⠁⠀⠀⠠⠈⠂⡄⠀⠀⢀⠀⢰⡄⠀⠇⠀⡄⠀⠏⠀⢸⠀⠀⠳⠀⣏⠐⠀⢄
⠀⠀⠀⠀⢀⠀⡇⠀⠀⠐⢐⠀⠘⠀⢐⠘⠀⢀⡄⢈⢃⠀⢸⠁⠘⠂⠀⠐⠘⡀⠁⠠⠙⠄⠀⠀⠠⠂⢀⠀⠀⠀⠁⠠⠀⡄⠈⠀⢸⠄⠀⠉⢄⠀⠀⡀⠂
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉
⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢾⠀⣻⠌⢳⢲⡜⠀⢰⠃⠀⠀⢦⡠⠞⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠈⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⣌⡀⡏⢸⠃⠀⡇⠀⠘⢦⣀⣀⣀⣀⣤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠘⠀⠃⠃⠘⠀⠀⠃⠀⢲⠀⠀⠉⠀⠀⢀⡀⠀⠼⠃⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣧⡇⠰⢳⡄⣾⢹⡇⡇⡾⢠⢤⡌⡄⢀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡸⣦⠀⠀⠀⠀⡀⡐⣦⠿⣯⣿⠣⠐⠼⠋⠙⢸⠇⠉⠳⠋⠺⠾⠛⡽⣡⢖⣆⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣷⡽⣧⡸⣧⣸⣧⡿⠏⠁⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠉⢛⣾⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠋⠳⣌⢳⣏⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⡆⠀⡇⢳⡞⠰⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡴⠒⠒⢄⠀⠈⠋⠉⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢰⣰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡒⠤⠚⠀⠀⢀⡠⠒⠉⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠀⠛⠁⠃⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⣐⣮⣤⣶⣶⣶⣶⣤⣤⣀⠀⠘⣾⡟⢁⣠⣤⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢀⣉⣥⣭⣭⣭⣭⣝⣛⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⢏⣴⣿⣿⣿⣮⡻⣿⣿⣿⣿⣿⣶⡝⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣜⢿⣿⣿⣿⣿⣿⡎⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡹⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⢇⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣝⢿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⢏⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠿⠝⢿⣿⣿⣿⣿⣿⡿⢣⣿⣿⡿⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠰⠀⠀⠆⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣟⢹⡇⡸⢄⠠⡿⠧⠀⠀⠁⠀⠀⠀⠀⡀⠀⢠⠀⡷⠲⣄
⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠂⠀⠈⠫⠟⠉⠁⠀⠘⡒⠠⠀⠀⡄⠬⠂⠀⠄⠀⠐⠐⠀⠀⠙⠿⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠀⠆
⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⡠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠠⢤⡀⠁⠙⠰
⠀⠀⠀⠄⠀⠂⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⡁⠀⡘
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⢈⡄⡀⠀⠀⠀⠀⡔⠃⠐⠊⠀⠀⡀⠱
⠀⠀⣀⡄⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢘⡔⠀⠀⠀⠁⠀⠂⠀⢁⡄⠐⡐⢠⠘
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠢⠠⠤⠀⠀⣠⢀⠊⠂⠀⠀⠀⠘⠳⡄⣀⠀⠀⠀⠉⠠⠈⣇⠉⠀⠁
⣴⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠍⢀⠀⠠⠸⡷⠔⠁⢚⡐⠡⠀⠀⠈⠈⠐⠄⠀⠻⣀⢀⠄⡞⣠⠊
⠙⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠯⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠉⠻⡦⠥⠏
⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠂⠱⡿⢀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢻⢿⣿⣖⣤⡠⢠⣴⢀⡀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣶⣦⣀⣹⣇⣸⠿⢨⣿⠿⣯⣿⡻⣿⠁⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠟⢿⢟⣦⠁⡙⣏⠙⠜⠃⣙⣛⡡⠏⢩⢣⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡈⠒⢤⣰⡀⠘⠷⢬⣙⢇⣤⣄⣩⣀⡾⣷⣬⣽⠿⠓⠉⢰⠀⠦
⠀⠀⠀⠀⡀⢨⠟⣀⣤⣤⣤⣤⣶⣾⢋⣷⣶⣾⡟⠱⠛⠛⠋⠀⠀⠀⠈
⠀⠀⠀⠠⠀⣐⠀⠸⣿⣿⣿⠿⠄⣟⢾⣿⣿⣿⡿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠋⠁⢰⠀⡁
⠀⠀⠀⠀⠀⠙⠄⢀⠁⣀⣈⣱⣤⠝⢮⣙⣫⣵⡤⠾⣿⣄⡆⠀⠀⠀⠀⠀⣠⣶⡾⠁⢀⡴⠂⠉⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⢹⣦⠐⣠⠌⠉⠐⠗⠀⠀⠉⠻⡀⠀⠀⣾⠋⠭⠁⠸⠂⠬⠁⠀⠀⡦⠀⠘
⠀⠀⠀⠠⣤⠀⢀⠀⠀⠀⣄⠀⠙⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠊⣉⡄⠀⠀⢀⠀⠀⠾⠁
⠀⠀⠀⠀⠓⢴⣿⣄⠀⠀⢸⡆⠀⠀⢦⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⢠⠀⡇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⣆⣠⡀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⡇⡁⠀⠄⢐⢈⠀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠀⠀⣁⢀⡀⠀⠀⠃⠉⠉⠉⠁⢘⣀⣀⢀⡀⡀⠀⠀⠤⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠆⠈⡐⡡⠄⠙⠝⡀⠀⠡⢉⠤⡈⢂⠁⠰⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠁⠤⠄⢲⣁⢤⠞⡐⠁⠂⠾⡔⡀⠬⠤⠈⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠔⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠂⢠⠂⠀⢀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠁⠀⠀⠰⢄⠀⠀⡠⠂⣀⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡐⠠⢑⡈⠄⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠂⠐⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠔⠁⠀⠀⠈⠢⠀⠀⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢫⡇⢸⠀⣄⠀⣄⣰⣿⠂⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣇⣼⣄⣿⣤⣿⠙⣷⣤⣠⣤⡴⠞⠋⠁⠀⠀⣠⣄⠀⠀⠉⢀⣀⣤⠤⠶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠻⢾⣿⠁⠉⠈⠙⠁⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢠⣿⢘⠀⠀⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⣾⣿⠀⢀⣿⣤⣼⠉⠀⠀⢀⣤⣄⠀⢀⣴⡄⢸⣿⣼⡀⠰⣿⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠚⠛⠛⠀⠀⣼⢣⡿⠀⢩⣿⠀⢸⡔⣀⠀⣸⡇⣿⠀⢸⡏⢸⠀⣿⣷⡃⢀⢿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠹⠟⠁⠀⢸⣿⠀⡸⣟⢹⡆⣹⣧⠞⣧⠿⡷⠈⠒⠹⠋⠙⢡⣿⡇⠀⠀⢀⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠁⠻⠀⠹⠅⠀⠀⠀⠤⢄⣀⠀⠀⠀⠀⠀⠉⠀⠀⠰⡇⠀⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣤⣈⢂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⣀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⢲⠀⠀⠀⠀⠀⠀⣀⣀⡙⢦⣀⣠⠴⠒⠒⠚⢛⡛⠛⠛⠓⠲⢤⣀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠈⠁⠀⠀⣠⠴⠚⠉⠁⠀⣀⠄⠁⠒⢤⡀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⣶⣄⠀⠒⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⢼⢳⠚⠩⣝⢦⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠾⠥⠤⢤⣄⠓⠄⠐⠁
⠀⠀⡔⢰⠃⠀⣧⠀⠈⣆⠇⡞⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⠇⠀⠀⠀⢸⡇⠀⢀⠔⡉⠠⣄⠀⠑⠢⡈⢳⣄
⠀⠿⠀⢸⡄⠀⠇⣀⡠⠋⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⢀⡾⠁⠀⡏⢸⡇⠀⢹⡆⠀⠀⠱⠀⣿⡄
⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⣇⠸⣧⠀⣼⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠹⣆⠹⢰⣿⢇⣀⣀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢀⣠⠴⠶⢤⣀⣀⡀⠀⣀⣠⣟⣁⣀⣤⡶⠛⠁⠀⠀⠀⣀⠤⠴⠖⠚⠓⠘⠡⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠈⠉⠹⠙⠓⠒⠛⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⢮⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠫⠅⠀⡨⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠒⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⠛⠛⠛⠛⡓⠶⣼⡇⡇⠸⠸⣇⣠⡴⠶⠒⠛⠛⠛⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠓⠒⠶⢤⣄⡀⠠⡀⠑⣼⠇⡇⠀⠀⢻⡄⡠⡨⠐⢈⣠⡤⠶⠚⠛⠛⠛⠒⠶⢤⣄
⠀⠀⠀⠀⢀⣤⠞⠉⠀⢀⠠⠒⠠⢀⣀⣴⣶⣮⣿⣶⣌⡾⠉⠀⠀⠀⠀⠀⡙⢧⣠⣴⣯⡥⣤⣤⣀⡀⠀⠀⠀⠀⠠⠈⠛⢦⣀
⠀⠀⠀⢠⠞⠁⠀⠀⠊⠀⢀⡴⠞⠋⠁⡀⠠⠤⠭⠿⢿⣿⡄⣤⠴⢾⡶⠶⠶⣞⠉⠀⠀⠠⠄⠀⠈⠙⠳⣄⠀⠀⠀⠀⠁⠀⠙⢧⡀
⠀⠀⢠⡏⠀⠀⠀⠀⠀⣰⠏⠀⢀⠔⠁⠀⠀⡀⠐⠈⠁⢈⡟⠁⠠⠠⠀⠀⠡⠈⢷⡀⠈⠠⠀⠀⠈⠂⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⣾⠀⠀⠀⠀⠀⣼⠃⠀⡐⠁⠀⠀⠀⠊⠀⠀⠀⠀⡾⠀⠀⠁⠀⠀⠀⠀⠁⠈⢷⡀⠀⠐⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢷
⠀⢠⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢸⡆
⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⢸⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡇
⠀⢸⡇⢠⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⢸⠇
⠀⠘⣇⢸⠀⠸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠠⠀⠀⡄⡾
⠀⠀⢻⡘⡇⠸⢀⢻⡀⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⠐⢸⡏⠀⢀⠐⣸⠃
⠀⠀⠈⢷⣹⡀⢇⠈⣧⠠⠠⠘⡄⢁⠀⢀⠀⠀⢂⠀⢻⡀⠀⢰⠆⡆⠀⠀⡀⠀⡆⡇⠀⡟⢀⢀⠂⠀⢢⠀⡈⠀⣠⣿⢁⠆⢆⣵⠏
⠀⠀⠀⠈⠻⣷⡌⢆⠺⣷⡡⠡⡱⡀⢂⠈⠄⢃⠈⠄⢘⣧⢠⠠⢀⡇⠀⠀⡇⠀⡇⡇⢸⠇⠄⡈⡀⢠⠇⢀⡟⢰⣿⠃⢌⣼⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣦⡣⡘⢿⣷⡑⠔⡄⠢⡐⠌⢆⠈⢌⡼⣆⠂⡌⡇⢠⠀⡅⢀⠀⠀⡾⠠⠠⠁⢀⡞⢀⡾⣰⡿⢃⢬⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣽⣿⣦⡙⢦⡑⢌⢂⠣⣀⠹⢿⣆⡴⣻⢘⠀⡃⠨⡌⣼⡓⣡⠕⣠⠏⢀⣾⣴⣯⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣶⣬⣽⣦⡽⠶⠛⠻⣿⣷⣄⣀⣷⠾⣿⣾⣥⣾⣥⠶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⢀⠄⠀⠰⡀⠀⢀⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⠐⠐⠀⢄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⡀⠁⠀⠀⠤⠖⠁⠀⢀⠀⣱⠂⢨⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⢰⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠐⢄⠀⠀⠀⠀⢲⡌⠀⠁⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠀⠉⠉⠁⠀⠀⠂⠐⣋⠀⢀⣈⢥⡍⠤⠓⠉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⠀⠀⠀⠀⠈⠒⠤⢀⢀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢀⢀⡠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤⠞⠁⠀⠀⠀⠀⠀⠊⢢⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠚⠃⠘⠛⠛⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⣀⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠑⠒⠊⠉⠁⠀⠀⠀⠀⠀⢀⡤⠒⠈⡁⠀⠁⠀⠠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠠⢼⡄⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠔⠀⢁⠠⠀⠀⠃⠠⠀⠀⡀⠂⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢸⠓
⠀⠀⠀⠠⠀⠀⠈⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣾⠔⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⡄⢡⠀⠀⠀⠀⠠⠁⠰⠀⠀⠀⡸⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡜⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠
⠀⠀⠀⠀⠠⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢰⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠎⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⣌⠀⠀⠀⡄⠀⠠⠆⡘⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢰⠐⠂⢲⡁⣀⡐⠀⠀⠀⡂⠀⠀⠀⠀⠳⣄⠀⣇⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⢀⡼⠃⠀⠀⠐⢁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⡷⣀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠳⢌⣦⡀⠀⠀⢳⡀⠀⠀⠀⠀⢀⠎⠀⠀⣀⣌⠴⠋⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠈⠒⠖⠃
⠀⠀⠀⠀⠀⠱⣌⠑⠀⠀⣀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠵⢤⣀⣀⠴⠏⠤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠤⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⡨⠊⡑⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠉⠤⠉⡑⢂⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠑⠄⣀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠀⠉⠑⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢄⣀⡀⠀⠀⠀⢀⣀⣠⣐⣀⣀⣀⣁⣂⣤⣤⡤⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢉⠄⡑⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣄⠰⢇⡀⠅⠖⣀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⢐⠠⠐⠀⠀⠉⠀⣓⣉⣁⣀⠐⠂⢬⡂⢰⡖⢶⠂⠈⠤
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢐⠔⣡⠮⠐⠈⠄⠒⠲⠿⠽⠌⠦⠓⠛⠁⠀⠀⠙⠆⠀⠑⠀⠁⠢⡀
⠀⠀⠀⠀⢀⠀⠁⡄⠀⡯⠊⡁⠄⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢠⡀⠀⠀⠈⢂
⠀⠀⠀⠠⠀⣄⠐⢀⠊⠀⡌⠠⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⣂⠀⠄⠄
⠀⠀⠀⠀⠀⠈⠀⠈⠠⠊⠁⠀⠀⠀⠐⠤⠤⡧⠚⢀⡀⣀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠑⡐⡆⠀⡆⢢⡄
⠀⠀⠀⢀⠐⠰⠀⡠⣁⠀⡀⠀⠀⠀⣀⠠⠠⠖⠨⡉⠓⡏⠃⠘⠓⠄⠀⠀⠀⠀⠀⠀⠐⠼⠀⠉⠨⡅
⠀⠀⠰⢹⠀⠀⠠⠁⡣⢪⠀⠀⠀⢏⣀⠀⢸⢄⠀⠀⠀⠐⠣⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⡄⢊⠂⠅
⠀⠀⠀⠈⡀⡆⠀⢸⠇⠀⠀⠀⠀⠀⠀⠄⠐⢾⢶⢀⣄⢀⣀⣸⡆⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠰⡄⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢜⠈⠋⠉⠈⠙⠼⠙⠉⠄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄⢀⢀⢀⣀⢠⣠⠀⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⢑⣚⡂⠈⠁⠑⡀⠊⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⡀⠰⠄⠠
⠀⠀⠀⠀⠀⠀⢠⣌⠀⠀⠀⠀⣆⠀⠀⠀⠀⠸⡀⠀⠈⠳⡄⠀⡀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥
⠀⠀⠀⠀⠤⠀⣧⠙⠀⣍⠈⣛⠂⢡⠀⠀⠀⠀⠈⠉⠀⠀⠙⠮⣀⢁⠜⠀⠀⠀⠀⠀⢠⢀⡀⢀⠸⠀⠰
⠀⠀⠀⠀⠀⢋⢉⠀⣀⡀⠢⢈⢳⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⠃⠉⠠⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠈⠐⢠⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⢀⠅⢁⡄⣫⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⢀⡁⠐⠀⠀⠀⠄⣄⢙⡋⠙⠃⠁⢀⠞⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠢⢍⡴⠦⣒⡚⠀⡀⠠⠀⠄⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⡀⠀⠀⠀⠀⡰⠂⠉⠙⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⠀⢙⠏⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠡⡈⠀⠀⢀⠠⢺⢣⢀⡀⠀⠘⠀⠀⠀⠤⢒⣤⠄⢤⡀⢀⢀⠀⠀⠀⠀⠀⢀⡀⠀⣩⢷⡿⣿⡄⠑⢠⢂⡴⣞⣳
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠥⠈⠀⠱⣄⣚⣁⣀⣘⠀⣇⠀⢀⡆⣠⢉⢽⡀⠰⠇⢌⡲⢟⣂⠁⠀⠀⠀⠀⠒⠙⢃⠘⡧⣿⣿⣽⠉⣂⡾⣟⣽⣲⠅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢹⠀⡦⢠⠄⠀⠈⠢⡀⠀⢸⢀⢿⠀⢸⢹⡝⡇⢸⠀⠂⢀⠈⣈⢊⣻⣅⣀⣀⠀⢀⣀⣀⢈⡤⠞⠓⡯⠇⠄⣯⡛⢞⣿⠏⣐⡏
⠀⠀⠀⠀⠀⠀⠀⢘⢀⣸⡄⠷⠔⠒⠉⣋⣑⠤⣁⣈⣆⠀⡟⠛⠂⠘⡖⠊⠙⠆⠸⠁⢹⠇⠀⢸⡠⠃⢀⠌⣍⠀⢰⠋⡡⡳⢂⣿⣪⣷⠇⢜⠏
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⣀⣀⠉⢸⠀⡄⠀⠀⢰⠁⢘⣹⠀⡄⠀⠀⠸⣤⠀⢸⡖⠀⠈⠀⠰⡃⠀⠈⢢⠀⠙⠻⣠⠞⠉⠤⡴⣡⡊⣀⡀
⠀⠀⠀⠀⢠⡀⠀⣤⠀⠒⠂⠀⢸⠀⠁⠀⠘⠋⠁⠀⢀⡄⠀⣈⣁⠀⢠⠀⢺⡀⠀⠀⢸⠉⠀⠸⠦⡀⠈⢙⡏⠉⠀⠀⣠⠏⡴⠶⠭⢄⡥⠤⠤⡌⢆
⠀⠀⠀⣓⢤⡉⢂⡎⢣⡀⠄⠀⢸⠀⡀⠀⠰⠚⢳⠀⢸⠀⢠⠁⠘⣇⣸⣄⠼⠘⠦⠔⠚⠛⠒⠚⠀⣈⣲⣧⠓⢲⠚⢻⡭⠋⢀⡀⢤⢾⠀⠀⠀⠀⠘
⠀⢰⠠⠽⡄⠉⢤⠸⠦⡘⡴⢶⡚⢦⠳⠦⠾⣀⠼⠶⠮⢍⡉⠀⣠⠔⢦⡔⠚⢃⠀⢀⠋⠉⡧⠒⢦⡇⠀⠈⢆⢸⠀⢸⠁⢰⢡⡤⠶⠚⠃⠙⡃⢈⡇
⠀⠀⡷⣅⠘⢆⡀⠳⡀⠃⠳⣈⢥⡀⡣⢤⠞⠁⠀⣀⣄⡠⠋⠀⢱⠀⠈⡱⡀⠘⡆⠸⠀⢸⡇⠀⢸⡇⠀⠀⠈⢾⠀⢸⡀⠘⣄⣧⡶⠀⠐⡋⠁⠘
⠀⠀⠣⡙⢬⣆⠱⣆⢠⣼⣼⠁⡐⠉⢇⡈⠀⠀⠚⣤⠤⠓⠒⠉⠉⡃⠀⢧⠱⡀⠈⠆⠀⢈⠁⠀⠈⡇⠀⠻⡄⠀⠀⠈⠳⣄⠀⠀⠀⡀⢀⠧⣀
⠀⠀⠀⠈⠦⣈⣰⡚⢿⠍⣸⣭⠙⠃⠦⣥⠀⠀⢆⢰⣤⡄⠀⠰⠂⢇⠀⠸⡀⠱⡀⠀⣠⢦⠛⠊⡤⢿⣀⡼⠘⣦⠖⣾⢳⣎⣟⢻⣍⡙⠀⠀⠀⢃
⠀⠀⠀⠀⡠⠊⡥⠒⠙⡾⠌⠧⡈⠉⠢⡈⢧⡀⠀⠉⠉⠀⣀⠀⢆⢸⠆⣉⠉⠛⠻⣉⣜⠠⢄⠒⡤⣈⠈⢡⠴⢉⡟⠯⣹⠾⡠⠄⣰⠃⠀⠀⠀⣼
⠀⠀⠀⢀⠁⡀⡈⠀⠈⠶⠀⠀⠸⠀⠀⢡⠈⣏⠁⣖⠚⠉⢉⢉⡽⡇⠀⢋⡀⠤⢀⣟⠀⢞⡊⣁⠀⢁⡔⡉⢑⠋⠍⠑⢺⣂⠬⠛⢁⠂⠀⠀⡰⡎
⠀⠀⠀⠸⠀⡇⠃⠀⠀⣧⠀⠀⠀⡇⠀⢸⠀⡹⣰⠗⠊⣱⢤⡸⢶⣿⣁⠍⡢⢄⠀⢹⠋⣥⠀⠀⠙⠙⢅⠀⠄⠀⡀⢀⠊⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠃⢁⠀⢸⡄⠀⠀⠀⡆⠇⠀⢪⡞⣰⡘⠂⠸⠁⠀⢱⣠⠽⡡⠀⠉⠢⡑⠾⡈⠋⠀⠀⢇⣀⠘⠂⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⡔⡰⠁
⠀⠀⠀⠀⠈⠺⢦⡀⠛⡄⠀⢠⠃⠀⢃⣈⢇⠈⠀⠀⠀⣀⢤⡆⠘⠀⠈⠒⠬⡀⠉⠳⢞⠓⠂⠀⢈⡘⠚⠂⠀⢀⡀⠾⣜⠢⠤⠤⢔⡩⠞
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠋⠊⠉⠀⠀⠑⠢⠤⠤⠒⠁⠈⠑⠋⠑⠊⠤⣀⡙⠒⠠⢵⠒⠒⠉⠈⠂⣂⠀⠥⠄⠚⠁⠉⠑⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣀⠀⢀⣀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠸⣄⣀⡧⢤⣀⡼⠀⠀⠀⠀⠀⠀⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡀⠀⠀⠀⠀⣀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⢀⠀⠐⠐⢉⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠘⠛⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⡀⠀⠀⢠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣬⣥⣭⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠒⠉⠉⠉⠉⠉⠒⢤⣸⡇
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⣾⡏⡙⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⢶⡆⢠⠄⢀⡤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡀⠀⠀⢀⡀⠀⠤⠤⠤⠤⣤⣄⠀⣿⠀⢸⣿⠘⣰⡏⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠸⣿⣷⠘⢡⠀⠀⠀⢀⠀⠀⠀⢀⡀⢠⣿⡏⠀⣿⠀⠀⣿⠇⢿⣇⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠈⠒⠀⠠⡠⣄⠀⡀⡞⠀⣿⡟⠀⠐⠿⠀⠀⠹⠀⠈⠛⠷⠤⠤⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠙⠿⣷⣤⣀⢱⡽⠎⡁⢹⢥⡞⠋⠀⠀⣀⢀⣀⣀⡀⣀
⠀⠀⠀⢱⠀⠀⠀⢂⠀⠈⠉⠙⡛⡠⠧⠄⠄⢠⠀⠀⠈⠀⡇⠀⣿⣿⠀⢸
⠀⠀⠀⠈⠁⠀⠀⠤⡴⢳⡜⠒⢳⣠⠊⢦⣐⠀⠄⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠐⠢⡠⠇⡠⠶⠹⠀⠀⠀⢢⠠⣔⡀⢙⡀⠀⣀⠀⠀⠀⣿⣿⠀⠀⢸⣇⣠⡀⠀⠀⣀⠠⢤⣀⠀⣀⣀⣠⡀⢸⣿⠀⢀⠀⠀⣀
⠀⠀⠀⠀⡔⠀⠊⠀⠃⠀⠀⠀⠀⠑⠛⠀⢵⣾⣃⠖⠲⣠⠀⠀⣿⣿⠀⠀⢸⡏⢸⡿⠀⣀⡥⠤⠄⣿⡆⣿⡏⢸⡇⢸⣿⣤⡀⠀⢸⣧⠈
⠀⠀⠦⠀⠟⠶⡀⡀⠆⠀⠀⠀⠀⠄⠀⠀⠟⠉⠻⠀⠀⠁⠀⠀⣿⣿⠀⠀⢸⡇⢸⡟⣼⡏⠀⠀⠀⣿⡇⣿⡇⢸⡇⢸⣿⠹⣷⠀⠀⠙⣷
⣀⣀⢄⠀⠀⡀⠂⢸⡠⡀⠀⠀⡀⠱⡞⠁⠀⠈⠀⠠⠼⠂⠠⣀⣿⣿⠀⠀⢸⡇⢸⣷⠘⢷⣄⣀⡠⣿⡇⣿⡇⢸⡇⢸⣿⠀⢹⣧⠰⡀⡸
⠀⠀⢠⡀⠁⠁⠆⡶⠁⠀⠀⠀⠀⢢⠄⠀⠀⠈⠀⠀⠒⠀⢀⠄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠸⡾⣑⠀⠲⠇⠀⠄⢒⡀⣠⠋⠉⠰⡜⢤⠀⣄⠀⢈⣀⡈⡉⣭⣅⣀⠀⢀⣀⠀⠀⣀⠀⠀⢀⡀⠠⢀⣾⣿⣟⣀⣀⢤⣤⣤⢤⠤
⠀⠀⠀⠳⠀⠠⡦⠤⠖⡊⠀⠉⠁⠀⠀⠀⡏⢡⠘⢉⣿⠋⠀⠀⢰⠃⠈⠊⠱⣶⣶⡶⠶⢤⣬⡭⢛⡛⠿⢛⣉⣉⡛⠋⠉⡩⠕⠋
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠄⠀⠀⠀⢀⠼⢃⣹⡃⣉⠉⣨⡗⠀⢳⣀⢀⢰⣰⡏⠛⠶⣬⡛⠞⠻⠿⣟⣵⣏⡏⣽⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠴⠦⡉⣶⡉⠟⠛⠀⠉⠛⠋⠉⠉⠈⠁⠘⣽⣦⣤⡎⠁⠀⠀⡹⠓⠒⠿⠓⠺⠳⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⢎⠀⠀⢀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⠋⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠁
⠀⠀⠀⠀⠀⣀⡤⠴⠋⢩⡉⠀⠀⠈⢫⡝⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⡆
⠀⠀⠀⢠⠞⠁⠀⡀⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢠⠂⠀⣠⡛⠒⠒⠀⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠄⠒⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢠⠖⣞⣹⡖⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢲⣎⣸⣅⡿⠀⠀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣨⣧⠴⢧⡨⣿⡗⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠣⠀⠀⠁⠘⢳⣿⣤⡟⠀⠀⠮⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⠀⢀⠀⠉⠁⠀⢉⡇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠈⢳⡤⠊⠉⠀⠉⠃⡀⠜⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⢀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⠃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠘⡄⠀⠀⠰⠄⠀⣤⣄⢠⣎⣹⣯⣠⠖⠒⠒⠳⣄⠀⠀⠀⠘⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠈
⠀⠀⠀⠀⠀⠈⠒⠠⠤⠤⠚⠻⣠⢯⡬⠷⠼⠁⠀⠀⠀⠀⠀⠉⠒⠒⢾⠀⣠⡌⠁⢠⣀⡀⢀⡀⠤⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠎⠁⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣦⡀⢀⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⣍⠐⠁⠉⠉⠉⠋⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠁⠀⠀⢀⠛⠘⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⠐⢀⠀⢸
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⢀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡀⠤⠤⣅⠀⠈⠪⣀⠈⠐⡀⠀⠀⠀⠀⠠⠀⡇⡇⠀⠈⠒⠚⠁⠀⠀⠀⠀⠀⠀⠐⠀⠠
⢀⢣⣀⣀⠈⠙⠦⡀⠘⢧⠀⢡⡖⢂⠀⠀⠀⠀⡇⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠒⠂⠤⡀
⢇⠀⠀⠉⠙⠒⠄⡈⠢⡈⢇⠀⣇⢀⡄⠀⠀⢰⡇⠘⣄⠀⠀⠀⢤⡀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠔⠀⠈⠁⠀⡀
⠰⠁⠉⠉⠐⠒⠠⠌⡑⢌⢎⠀⢿⢸⠁⢀⣢⠞⠃⠀⠱⡡⣎⢎⠄⠉⣺⡂⠐⠒⠒⠒⠂⠒⠒⠂⠐⠊⢜⣀⠤
⠀⠑⡶⠒⠒⠂⠠⠤⠤⣑⣻⠧⠼⠚⠊⠉⠂⠒⣆⢤⣐⠐⠙⡊⠀⠈⠟⠒⠒⠚⠂⠀⠀⠀⠠⠀⢐⡒⠚⠡⠒
⠀⠀⠑⠤⠤⠶⠒⠒⠒⣲⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣑⠰⠀⠱⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠂⠤⠒⠉⠈⢣⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡔⠁⠱⠀⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢐⢄⠀⠀⠀⠀⠒⠀⠀⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠄⢺⡶⠂⠀⠤⠄⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠢⢔⣶⡒⢆

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢨⣿⠀⢰⡄⠀⠀⣰⠖⠀⠀⠀⢀⠀⠀⠀⣿⠂⠀⣾⡇⠀⢲⡀⠀⢸⣟⠀⢰⣦⠀⢠⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣯⠀⣸⡇⠀⢰⣿⠀⣸⡇⠀⢸⣧⠀⢰⡿⠀⠀⣿⡇⣶⠟⠀⠀⢸⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⡇⢀⠘⠿⠀⣿⠇⠀⠘⠿⠄⢹⡇⠀⠘⠿⠀⣾⡇⠀⠀⠻⠇⢿⡄⠀⠀⢸⡇⠀⠈⠿⠀⠘⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠙⠀⠀⠈⠇⠀⠀⠀⠀⠀⠈⠀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⢿⡆




⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡰⡉⠀⠀⠀⡀⠐⠈⠀⡀⠘⠁⠐⠃⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠘⠀⠁⠁⠂⢸⠀⠀⠊⠐⠠⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⠁⠔⠨⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⠀⡇⠀⢋⠀⡇⠐⠀⡥⠂⢈⠉⠀⠊⠀⠀⠈⠀⠇⠀⠀⡀⠀⠸⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠇⠚⠘⠀⠀⠇⠀⠇⠀⠀⡀⠀⡀⠚⡢⢀⡀⢠⠄⠀⣀⠀⠀⢰⠀⠢⠀⠜⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠂⠀⠀⠀⠑⠄⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⢂⡂⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠢⢀⡁⠀⠄⠐⠒⠚⠗⠁⠀⠀⠀⠀⠈⠃⠁⠂⢄⠀⢘⡀⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢛⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠄⢀⠀⠀⠁⡜⠁⠐⠄⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡐⠁⡠⠐⠊⠒⠎⠁⠀⠀⢸⠁⠀⠀⠀⠀⠃⠀⠀⠈⢂⢀⠁⠀⠀⠀⠈⢀⡌⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠂⠀⠀⡐⠉⠉⠞⠀⣾⣿⠑⠀⠱⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠐⠠⡀⠀⠀⠜⠫⠉⠐⡄⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠰⡇⠂⠈⠛⠈⠀⠀⠆⢄⡠⠓⠐⠂⣀⡆⠀⣀⠀⠀⠂⠀⠀⠀⠐⣤⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠠⡀⠁⢠⡄⢐⠃⠀⠀⠀⠀⠀⠀⡆⡀⠁⡐⠤⡀⠉⠣⡀⠀⠘⣅⣀⡀⠀⠀⠊⠀⠁⠀⠀⠀⢠⠂⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡠⠒⠘⠓⠠⢀⣸⠀⠐⠀⠀⠀⠀⢨⠀⡠⠛⠠⡀⡈⢦⠒⠘⠣⠀⠙⠄⠀⠱⡀⠀⠀⠀⣀⢀⠔⠃⠂⠄⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢰⠀⠀⠁⠀⠀⠂⠀⠢⠐⠀⠀⠀⠀⢠⠘⠂⣠⠀⠛⠄⢀⣡⠀⠐⠇⠀⠺⠀⠐⠇⠀⠦⠘⠃⠀⠀⠀⡄⠈⡀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢐⠂⠈⠁⠀⠁⠚⠡⢀⣀⡀⠼⠐⠒⠇⠀⠚⠀⠲⠆⢀⠀⠀⠀⢀⠐⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠠⠚⢲⡄⠀⠶⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠐⢤⠤⠰⠄⠀⠀⠙⠆⠄⣠⠐⠚⠀⠘⠃⠠⣄⠈⠤⠀⠒⠛⢄⠀⠀⠀⣿⣿⣿
⠀⠀⠘⠀⠀⠂⢠⠟⠘⠛⡀⠀⠀⠀⠀⠀⠀⠆⠑⡀⡰⠀⠀⠀⠠⠀⠐⠢⡀⠈⠲⠒⠚⠂⠰⡄⠂⠚⠠⠀⣀⠀⠀⠆⠀⠀⣿⣿⣿
⠀⠀⠈⠀⠆⠐⢸⠀⣠⠤⢲⠠⡀⠀⠠⠔⢄⠀⡏⠀⠈⠦⡀⠀⢠⠂⢄⠀⠈⠲⠚⠐⠂⠤⣴⠁⣀⡀⠀⠀⢈⠀⢀⠆⠀⠀⣿⡏⢹
⠀⠀⠀⠈⠗⠀⣺⠀⠀⠀⠀⡀⣤⠀⡘⠀⠀⠹⢶⡀⠀⠐⠀⠈⢃⠀⠀⠑⢤⠠⠄⠀⠀⠀⠀⢑⠀⠀⠀⠉⠑⠖⠁⠀⠀⠀⣿⡇⢸
⠀⠀⠀⢠⠁⠀⣮⡀⠠⠖⠊⠁⣄⠸⠃⠄⣠⠁⠀⠓⠠⠘⠄⠀⠀⠡⢀⣀⡸⠀⠀⠀⠈⠐⢆⠉⠐⠄⣀⣀⠠⠂⠀⠀⠀⠀⣿⣟⣺
⠀⠀⠀⠀⢀⠀⠇⡀⠀⣀⠄⠘⠻⠃⠀⠀⠉⠀⠀⠶⠀⠀⠈⠦⠀⣀⠀⠀⠑⢄⡀⠀⠀⠀⢀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⣿⣇⣸
⠀⠀⠀⠐⢫⠀⠸⠋⠁⠁⠀⢀⣶⡰⣤⡄⠀⠀⡐⠀⠡⡀⠀⠀⠀⠀⢠⡀⠀⠀⡁⠈⠉⠁⠀⠀⠀⠤⡤⠄⠀⠀⠀⠀⠀⠀⣿⣇⢾
⠀⠀⠀⠀⠀⠀⠀⢻⠒⠘⡇⡠⠁⠑⣨⠄⠀⠖⠠⣤⣊⡨⢆⠁⠀⠀⠀⠅⠀⠈⠠⡄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⡇⢺
⠀⠀⠀⠀⠘⠀⠠⠇⠀⢲⡘⣄⠰⠊⠑⣄⠹⢦⠐⠒⠈⠤⡄⠃⢄⢀⣀⠄⠀⠀⠀⠀⠄⠀⠩⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣺
⠀⠀⠀⠀⠀⠁⠁⠐⠐⠀⠈⠠⣄⡰⠔⠉⢄⢀⠱⢆⠁⠀⢈⠴⠆⠉⠉⢐⠤⠄⣀⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠦⠊⠡⡀⣀⠳⣊⠀⠀⠀⡴⠾⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⢝
⠀⠀⠀⠀⠀⠀⡀⠀⠄⢐⡀⠀⠤⠄⠂⠀⠀⡀⠀⠀⠠⢄⠂⠀⠀⠒⣰⠐⠀⠜⠠⠠⠄
⠀⠀⠀⠀⠀⢀⠀⢀⠈⠲⢠⡀⠈⠀⠀⠀⠑⠀⡂⠀⠀⢸⠀⡀⢀⠃⠟⢈⠱⠀⠀⢆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠠⣈⠀⡈⠀⠀⠀⠁⠄⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⡄⠂⠠⠀⡀
⠀⠀⠄⠁⢰⠂⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⢀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠁⡀⠁
⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠃⠈⢧⠉⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠧⢀⠀⣀⠀⢔⢇
⠀⠠⡔⠀⠀⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠅⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢸⠏⠀⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡶⣶⣴⣤⣤⡤⠀⠀⠀⠘⠼⠀⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢆⣶⣞⣥⣤⣴⣾⣠⣥⣬⣥⣤⣤⡄⠀⠁⠗⠀⠀⠠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠊⠀⠫⠀⠀⠀⠄⡰⠉⠡⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⢀⠋⠀⠀⠈⠀⠀⠀⠟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠡⠈⠒⢄⠈⠀⠀⠲⠀⠀⢀⣀⡖⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡓⢶⠀⠀⠀⢄⡘⠀⡄⢒⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡀⢰⠏⠀⠀⠀⠀⠀⠥⠤⢴⡡⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠈⠀⠈⠁⠀⠐⠀⠡
⠀⠄⡀⢀⠀⠊⠀⢀⣤⠀⠇⠀⠀⠀⠚⠂⣀⠀⠀⠉⠆⠀⠠⣀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚
⠀⠀⠁⠀⠀⠀⠀⠀⠆⢸⡄⠀⡀⠀⠀⠀⡀⣀⡀⠀⠠⠀⡀⣀⢠⠀⠤⠀⡀⠈⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡩
⠀⠀⠀⠀⠀⠐⠉⢀⠀⠀⠰⣌⠇⠀⠀⠀⠀⠈⠀⠀⢀⠁⡀⣀⠀⢀⡑⣨⠄⡉⣉⣉⣩⢉⣉⣉⣻⣉⣹⡿⢿⣉⡻⡧⣱⠤⠎
⡀⠀⠈⠁⡀⠀⠀⢸⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⡘⢚⠇⣆⠘⠲⠶⠀⠙⠦⠤⢤⢥⠤⠤⠥⠤⠝⠽⠛⠋⠛⠒⠊⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠤⠤⢀⢀⣀⡇⠈⠀⡘⡀⠑⢠⢇⠆⢃⡻⠓⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⢀⡀⠘⠀⣠⠂⠒⢲⠤⠠⠒⠔⠈⠡⡍⡘⡳⠊⠀⠈⠀⠀⠄⠀⠀⠀⠠⠄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢐⠀⡖⠉⠾⠂⠀⣰⠀⠀⢠⠟⠦⡈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠷⠀⢀⠠⠘⠀⠀⡈⣞⣓⢚⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠀⢰⡏⢸⡏⣀⢀⣀⡀⠀⢀⣤⢸⣗⡀⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢺⡇⡸⠇⣏⠾⢽⣧⣿⢻⣏⠼⣿⡁⣼⡇⢾⣼⠃⢿⠀⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠉⠀⠈⠁⠀⠈⠉⠀⠀⠛⠙⠷⠀⠈⠋⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠆⠈⠉⠰⠄⠀⢰⠂⠈⠉⡏⠁⠋⠁⠁⠁⠀⠁⠋⠀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡦⠀⣀⣠⠀⢀⠀⡀⠀⠀⡇⠀⡄⠒⠀⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠧⠠⠄⠀⠄⠌⠀⠀⠄⠀⠀⠀⠀⠄⠄⠥⠀⠀⠀⠦⠄⠀⠠⠄⠠
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡄⣤⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡀⢈⠀⠃⠃⠛⠃⠋⡁⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣙⠋⠀⡐⢐⣒⢠⡀⠙⠉⠀⠍⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠒⠒⠁⠀⠀⠀⠀⢠⣟⡅⠀⣼⠋⠀⠀⠈⢆⠈⢓⣵⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠈⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢀⠄⠹⣟⠠⣀⣿⡎⢰⡖⢰⣧⣰⡆⠈⣴⠂⢠⡖⣿⠀⠀⠠⡀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡏⠸⣇⠜⠗⠃⣼⡇⣿⠁⠙⠖⠉⠉⢻⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠀⠀⢠⠀⡠⠀⠠⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⡞⠄⡗⠁⡗⠂⠸⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡴⣰⠆⠀⢀⣀⣀⡀⢠⠖⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣀⣤⣄⡰
⠀⠀⠀⠀⢠⡶⠚⣫⠉⣹⠁⠀⠀⠀⠉⠀⡀⢠⡇⢠⢰⡂⡴⢒⡄⢐⠀⡜⠀⠀⡉⠀⠀⠀⠉
⠀⠀⠀⠀⠺⠁⢰⣿⠀⡏⠀⢠⠖⣢⣶⢠⠏⣼⢂⠁⢀⠇⠁⢘⡇⡞⣰⠇⠀⡴⢠⠀⢀⡀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⣀⠀⢿⢀⡇⣽⠈⠀⠷⠋⢿⠀⠁⠈⠳⠤⠤⠴⠃⡇⠀⠃⠀⠈⠠⠞⣠⢯⠖⠂⡄⡸⠁
⠀⠀⠀⠀⠀⢿⡀⡸⠀⠋⢿⠀⠀⠀⠀⢀⡴⠒⣉⠉⡄⢀⡀⠀⡧⠠⢤⣤⣤⡴⠃⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠧⠓⠙⢘⡠⠞⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠉⠓⠒⠐⠈⠁⠀⠀⠀⠀⠀⠀⡀⡄⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠞⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊⠂⠓⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠤⣴⡟⣚⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠀⢤⡀⠀⠀⣰⡟⣰⢿⡾⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡄⣸⢇⣠⣀⠀⠀⠀⣀⣁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⢡⣾⠵⠋⠀⠀⡎⢀⡾⠀⢠⡟⣸⢀⣴⢋⡞⠀⠀⣠⠞⣱
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢰⡿⠓⢲⡿⠁⠀⠁⠀⠀⠳⢏⠀⣠⣿⠔⢁⣾⠧⠊⠀⠀⣴⠯⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⡿⠁⢠⡿⣱⣆⣴⢂⡖⣰⠆⣤⢠⡿⣳⢀⣾⠁⣴⢠⡖⣰⠏
⠀⠀⠀⠀⠀⠸⠀⠄⠀⠀⠻⠇⠐⠟⠘⠋⠺⠇⠿⠐⠛⠸⠃⠟⠻⢁⡾⢻⠘⠃⠛⠐⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣶⣦⣾⣧⣴⣶⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣼⣿⢿⣿⡏⣵⣟⣿⢫⣽⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⢾⣾⣿⣿⣧⡻⣿⣷⣌⣙⣻⣬⣙⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡿⣫⣽⡝⣿⣿⣿⡿⠿⣿⣻⡿⣾⣿⣿⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⢁⣿⠳⢟⣫⣼⣿⣿⣿⢻⣇⡿⣻⣿⣿⠏⣾⣷⡦⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⢰⣾⣿⣟⢰⡇⢸⣻⡿⢿⡏⢻⢻⠀⡛⡇⣹⢿⢿⠀⣿⣿⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⢠⣾⣿⡇⢸⡇⢾⠸⢸⡟⡇⢸⠘⢠⣹⡃⢫⣘⣘⡃⣿⡿⣿⣿⣿⣷⣦⡘⣿⣿⣿⣿⡿⠉⣸⡇
⠀⠀⠀⠀⠀⣽⣿⣧⢸⣇⣼⣿⣦⣦⣽⣿⣾⢟⣿⡇⢻⣿⠏⣿⣮⣿⣛⣿⣿⡟⣭⣷⣸⣿⣿⣿⡁⠀⣿⣇
⠀⠀⠀⠀⠀⠉⣽⣿⣌⣻⣿⣿⣿⣿⢋⣵⣿⡼⢿⡿⢖⣿⢠⣿⣿⡇⣹⣿⣼⣇⢹⣿⢟⣻⣿⣿⣇⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡘⢿⢹⣿⢃⡇⠿⢻⢸⡇⣶⡅⣭⣇⢿⠸⡌⢿⣶⣯⣿⣿⣿⣆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢀⣠⣿⢟⣯⢰⡿⢨⣵⣿⣶⣬⣷⣬⡇⢿⣿⢖⣟⢻⣷⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣿⣿⣿⡿⡿⢃⡾⢸⢇⣿⢡⣦⡿⢛⡟⣻⣬⣿⣜⣿⠘⣟⣿⣿⣿⣿⣻⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡾⡏⢰⢟⢿⡘⣇⢺⠃⣶⡟⣽⢸⣿⣿⣿⣿⣭⣾⣿⣿
⠀⠀⠀⠀⠀⢨⣶⣿⣿⣿⢡⣶⣿⢿⣿⣿⡇⣼⠟⡄⣩⠸⣟⣷⢸⡇⣷⣽⣤⣿⡌⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣛⣿⣿⣿⣮⣟⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣋⣸⣵⣿⡿⢛⣿⣿⣿⡏⣾⢿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢯⣶⣿⣷⣶⣬⣭⣙⣛⠻⠿⠿⠋⠴⠿⢟⣱⣿⣿⣶⣾⠟⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣮⣯⣿⣿⣿⡿⣯⣭⣭⠝⣃⣐⠶⠿⠿⢿⣿⠿⠿⢛⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣭⣿⣿⣿⡿⠟⠛⠻⢿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠈⠉⠉⠁⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⣿⣇⠀⠀⢀⣤⡾⢿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠳⠀⠈⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠛⠋⠉⠛⠁⠀⠀⠀⠀⠈⢷⡾⠋⠉⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣀⣿⠀⠀⢀⣀⣀⠀⢀⣤⣶⣶⣤⠀⠈⠀⣀⣀⠀⠀⠀⠹⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⣿⣿⣿⠀⣼⣿⠿⣿⣷⡀⣿⠟⠛⢿⡇⣠⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⣽⡁⠗⠀⣸⣿⣿⣆⠛⢀⣤⣌⠻⡗⢡⣴⣦⡈⠣⠛⣉⡉⠙⣿⠋⣴⣾⣷⣦⡄⠀⢀⣯⣀
⠀⠀⠀⠀⠀⠀⣴⢛⡁⠀⣀⡌⣶⣾⣿⣿⣿⣿⡄⣿⣿⢁⣀⢰⡟⣉⠹⣟⣴⣿⣿⣿⡀⠯⠾⠻⢿⣿⣿⡿⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠈⠛⠉⡿⢸⣿⡇⢽⣿⣿⣿⣿⣿⡇⣬⡛⣼⣿⡗⣼⣿⣇⠟⣁⡀⢹⢟⣥⣶⣶⣦⠀⣟⣡⣶⣶⣶⡄⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⣿⣿⣿⣿⣏⠁⠉⠁⣹⣿⣿⣿⣿⣣⣾⠿⠛⠀⠁⠀⢸⣿⡟⣰⣿⣿⣿⣿⣿⣿⠄⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣟⡇⠀⣿⣿⣿⣿⣿⡄⠀⣴⡟⣭⣭⣿⠷⠋⠁⠀⠀⠀⠀⠠⡿⠿⡤⣵⣶⣤⡀⠙⠿⢋⣤⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⠟⣵⣤⣿⣿⣿⣿⣿⣿⣦⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⣰⣿⣿⣿⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⢀⠇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡛⠟⣥⣾⣿⡿⠟⣋⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⣄⠀⠇⣄⠀⠀⠀⠀⠀⠀⠛⣽⠟⣠⡿⠿⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠤⠤⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡄⠏⠀⠀⠀⠀⠀⣠⡶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠒⠂⠈⠁⠤⠄⠀⠀⠀⠀⠀⠀⠀⡔⠃⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠐⢒⣀⡐⠒⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⢀⣠⣤⣦⣄⠀⠀⣀⣤⣤⣀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⡿⠳⣾⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠈⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣧⠀⢀⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⠿⠃⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⠇⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣼⡟⠀⢠⠊⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⠁⡰⠁⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠀⠀⢰⣿⠀⠀⠀⣸⡿⠊⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢸⡟⠀⠀⠠⣿⡇⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⣾⡧⠀⠁⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⣿⡇⠌⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣀⣶⡀⢠⡞⣡⡄⠐⢹⡖⣽⡇⢀⣿⡟⣰⠟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡏⣿⡇⣿⣃⣿⠇⠂⣿⠁⣿⠁⣼⣿⣿⡅⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠉⣿⠁⢻⡷⠀⠁⠻⠎⠀⠉⠀⣿⢰⢹⣿⢻⣇⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠜⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⠃⠸⠿⠈⠻⣄⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢳⣦⠀⣰⡄⢾⠀⣴⡌⡆⠀⣠⣶⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣿⡀⣹⣇⠈⢲⣿⠄⢼⠀⣿⡇⢠⣿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣿⡇⡟⣿⡄⠹⣿⠀⡆⠀⣿⠇⣸⣿⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢁⣿⣇⠀⠙⠊⠀⠀⠈⠛⠈⣿⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡼⠁



⣶⡶⢶⢶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⡶⢶⢶⣶⣶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⡇⢁⡘⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠐⢀⠄⠐⠁⠀⠀⠈⠀⠀⠁⠀⡐
⠀⠀⠐⠀⡀⠀⠒⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠀⢠⠊⠲⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⡄
⠀⢠⠄⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠟⢀⡮⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠶⡶⣦⠦⡦⠀⠀⠀⠹⢁⠀⠀⠐⠇
⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠘⠾⠯⠶⠦⢶⡧⡶⠶⠶⠦⣶⠀⠀⡜⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠀⡀⠘⠐⠁⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⡀⡀⠀⠀⠀⠃⠐⢀
⠀⠀⠂⠀⠀⠀⠈⠀⢀⡄⠀⠀⠀⠀⠀⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠘⢀⠱⠀⠀⠀⠠⠀⠉⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠐⠀⠀⠀⠀⠀⡉⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⡀⢉⢀⠀⡀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠀⠈⢅⡹⠀⠀⠀⠀⠉⠀⠀⠈⡀⠀⠀⢈⠀⡥⠀⠀⠀⠀⢀⠀⡀
⡀⠀⠉⠠⠀⠀⠨⠀⡀⠀⠿⠀⡀⠀⠀⠀⠀⢀⠓⡇⠀⠘⠲⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠠⠠⠰⠀⠀⠤⠇⡈⣸⢄⠊⠌⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠠⠴⠀⠐⠀⠀⣴⢁⠉⠙⠠⠊⠉⢀⡀⠢⡬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠄⣼⠀⠉⠀⠠⠃⠀⢈⢇⢩⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈⠀⠀⠀⠈⠒⠐⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢷⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣩⡍⣰⠋⠑⠀⠀⢀⠀⠀⠀⣴⢶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢶⣟⡀⠀⠹⡄⠀⡀⠃⠀⠀⠀⠀⠈⢻⣿⡄⠀⢀⡎⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣯⠍⣂⠀⠠⠐⠰⠂⠘⡻⢍⠙⠤⠤⢭⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠗⠀⠀⠨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣼⣿⣧⠔⠁⠀⠀⠀⠀⠀⠀⣀⠀⠶⠀⠀⠀⠈⠴⣆⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠍⡿⡿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⢝⡋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠈⣠⣬⠄⢊⡁⢤⠆
⠀⠀⠀⠀⠀⠀⠀⣀⡐⡞⠁⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⠀⠀⠀⠒⠁⣠⠥⠧⡠⠚⢶⢁⡖⠊
⠀⠀⠀⠀⠠⠴⡚⡁⠄⡃⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢀⠀⢀⠜⠁⠀⠀⡹⡄⠨
⠀⠀⢀⣴⡋⡅⡂⠀⢁⣘⣄⠀⠀⢄⠀⠀⢀⠤⢣⡀⠀⠀⠀⠀⣰⠁⠀⠁⠀⠀⠀⡰⠁
⠀⠀⠀⠈⠛⠉⠱⠛⠛⠋⠉⠁⠒⠺⠁⠒⠡⢄⠀⠑⠢⠤⠄⠞⠁⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣠⠤⣄⡀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢴⡷⣼⠕⠁⠀⠀⠀⠠⠀⠈⠑⢲⡿⠐⠀⢠⠞⡁⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⢄⠀⢸⡇⢿⡐⣶⠀⡴⢛⡀⣷⠎⡿⢸⡅⡸⢃⣟⠀⣴⢀⡄⣴⢃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠊⠀⢸⡇⠸⡇⢿⡐⠃⠺⠆⠻⠐⠧⠘⠛⠇⢸⡗⠈⠧⢾⣱⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠼⠁⠀⠀⠀⠀⠀⠀⢠⣸⠂⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢅⡐⠈⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠓⠔⠛⠋⠀⠀⣠⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠉⣄⠀⠰⣄⡀⣸⠧⠖⠒⢶⡇⢀⠀⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣃⡿⠀⣹⠆⣴⢛⠂⣾⠀⢰⡿⢸⡇⠸⣄⢧⡹⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠞⠁⠀⠻⠀⠁⠛⠀⢿⡠⠙⠒⠸⡏⢂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢰⡀⠀⢘⠁⠀⣔⠂⢠⡔⠒⠒⢀⠀⠀⢰⠒⢚⠀⢉⢀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⡠⠶⠈⠀⠀⠀⠀⠀⠁⠀⠸⡀⠠⠀⠀⠉⠀⠀⠜⢠⠀⠀⠼⠁⠁⠀⢀⠈⠀⠀⠳⠄⠀⡀⠀⢀⠀⠀⢴⠘⠀⢠⣚⠂
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠁⢄⠉⠀⠐⠈⠀⠠⠃⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠉⠠⠽⠊⠀⠀⠀⠘⠁
⠀⠀⡈⠰⠋⠀⠀⠀⢀⠠⢀⣀⠈⠀⠀⠀⠀⠀⠀⠁⠒⠒⠀⠈
⠀⠐⠀⠆⠀⠀⢄⡄⠀⠄⢀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠠⠀⠀⠀⠀⠈⠃⢰⠀⠀⠠⠃⠤⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠂⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢨⡄⢉⣧⣤⡾
⠀⠀⠐⠀⠀⠐⠀⠀⠐⠄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠛⠀⠞⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠢⠀⠀⠀⠈⣰⠁
⠀⠀⠀⠀⠀⠠⠀⠈⢶⠤⠖⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠰⠐⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⠐⠀⠤
⠀⠀⠀⠀⠀⠀⠠⠄⠢⣀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠶⠟⣉⠩⠅⠀⠦⠄⠶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡎⠀⠀⡠⠅⠐⠂⢁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠙⠒⠒⠰⢦⡀⠀⠁⠀⣀⠂
⠀⠀⠀⠔⠋⠁⠂⣀⣀⣀⣀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠤⠔⠊
⠀⠀⢀⡀⠀⣸⠁⠉⠀⠊⣁⣀⡀⠓⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠸⡇⠀⠸⠦⢄⠐⠤⠤⠠⣈⣀⡀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠙⠆⠀⠀⠀⠻⡀⠀⠀⣀⣀⡤⠊⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⠠⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠓⠖⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠌⠈⠀⢀⡅⠀⠀⠀⠀⢀⣀⣀⣬⣀⡠⠄⠀⠑⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠈⣦⠘⠃⠀⠀⠀⠀⠈⢄⠈⢂⠉⠐⢠⠗⠈⢠⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠘⠒⢌⠂⠀⡀⠀⠈⢀⢡⡀⠑⢄⠀⠀⠀⠂⠠⠚⡀⢄⡀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠙⣄⠑⠄⠀⠀⠁⡈⠀⢀⠀⠀⠉⠀⠀⠀⡗⠀⠉
⣿⣿⣤⣤⣼⣧⣤⣾⣧⣷⣥⣥⣤⣤⣴⣬⣴⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠑⡀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠔⠚⠀⣠⠜
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠄⡀⣀⠀⠀⠀⠀⠤⠄⠠⠂

⡿⢿⠿⢿⡿⠻⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡅⡟⣥⡄⡄⣤⡄⣤⢠⡄⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠒⣦⠀⠃⠏⠉⠙⡋⢹⠋⠀⣰⠃⣤⠲⣄⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢠⣿⠀⢠⡏⢀⡼⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⠀⢸⡇⣠⢤⡀⠀⢀⣘⡅⢀⡄⢀⣀⠀⢸⣇⢴⡄⣤⡄⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣷⠃⣿⡇⣰⡟⣹⡇⢠⣿⠃⣿⠀⢸⣷⠋⢀⣿⣀⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⢿⡏⠀⠻⠶⠿⡴⠙⣇⠸⠟⠀⣿⣀⣾⠙⠧⢺⢾⡏⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣘⠁⠰⡶⠀⠀⣀⣀⠀⣦⠀⠀⠀⠁⠘⠀⠀⣈⠉⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠁⠀⠁⢀⡾⠙⣧⢰⡆⠀⣶⠙⢾⢰⡧⠀⣿⡔⣷⡄⢀⡞⠉⣇⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⣿⢸⡇⢠⣿⢀⡇⠸⡇⢠⣿⠁⣿⠃⢸⣇⡼⣿⢀⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠤⣿⠄⠙⠁⣈⣋⣀⣴⠟⠉⠉⠀⠙⠒⠋⠉⢀⣿⠋⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡟⠀⠀⠀⠀⠀⠀⣤⠀⡦⢀⡀⠀⢀⠀⠀⣯⠟⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⠀⠀⠀⠀⠐⠁⠐⣇⡰⠻⡿⡇⣧⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒





⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣶⣥⡀
⠀⠀⠀⠀⠉⣁⠍⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠤⠴⣶⠃
⠀⠀⠀⠼⠋⠁⠀⠀⠲⠀⠀⠀⠀⢠⣴⡆⠁⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠭⠭⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣿⡯⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⠼⣃⣴⠆⣠⠀⠀⢀⣀⣐⣐⣒⣒⣒⣒⣚⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠠⡄⢀⠀⠀⠀⠠⢀⡀⠀⠒⢒⣒⣛⣋⣭⣤⠶⠋⣀⣿⠂⠒⠒⠲⠦⠀⠀⣀⣐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⠗⠂⣠⣭⣭⠭⠭⠭⠵⠶⠾⢛⣫⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠩⠁⠤⠴⠾⠲⠒⠒⠚⣛⣛⣋⣭⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⢀⣀⣀⣉⣉⣉⣭⣭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠠⡂⢠⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠉⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡲⢠⠀⠀⢊⠐⠶⠄⠀⠀⠠⡶⠀⡁⠂⠀⡄⢒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢖⠐⢀⣠⠗⠢⡅⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠒⠨⠔⠺⣄⡀⠂⡰⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⣠⠖⡃⠐⠀⠀⠀⠀⢀⣤⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⠢⣄⠙⠦⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡴⠷⠒⠋⠁⠀⢀⡀⢀⣀⠀⢰⡿⠁⠀⢛⣃⡀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠈⠙⠒⠾⢧⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠋⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⣧⣿⡇⠀⠀⣿⣿⣇⢸⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡃⠒⠀⠀⠀⠀⠀⢸⣟⢸⣿⢿⣿⡇⠀⢰⣿⣾⣿⢸⣿⣄⣀⢸⣿⣀⣀⠀⠀⠀⠀⠀⠒⢘⠊⢧
⠀⠀⠰⢀⡂⣠⠀⣼⢁⡞⡁⣃⠀⠀⠀⠀⠀⠘⠋⠘⠛⠈⠛⠃⠀⠘⠛⠀⢿⣎⢛⣛⠛⠘⠛⠛⠋⠀⠀⠀⠀⠀⡘⢈⢳⡈⣧⠀⣄⢐⡀⠆
⠀⠐⢨⢩⠑⢠⢰⠇⡾⣠⠱⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠈⠻⠾⠟⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠡⠌⣄⢷⠸⡆⡤⠊⡍⡅⠂
⠀⠀⠀⢸⢰⡆⡾⢠⣷⠋⢃⣀⣀⡀⣀⡀⣀⡀⣀⠀⣀⡀⢠⡿⠁⣐⡟⠀⣀⣀⠀⠀⠀⣀⣘⠟⣹⣧⡀⣀⡀⣀⣀⡘⠘⣾⡄⣷⢰⡆⡇
⠀⠀⠀⠸⣾⡇⣇⡼⠁⠀⠛⢻⡟⠃⣿⡇⣿⡇⣿⡇⣿⣧⣾⡇⣾⡟⠿⠐⣿⣿⠇⠀⣾⡟⠿⠀⣿⢻⣧⣿⠃⣿⣏⡉⠀⠈⢧⣸⢸⣿⠇
⠀⠀⠀⠀⠹⣷⡏⠀⠀⠀⠀⣸⡇⠀⣿⡟⣿⡇⣿⡇⣿⡿⣿⡇⢿⣧⣾⡶⣮⣿⡇⠀⢿⣧⣼⡆⣿⠀⣿⣿⠀⣿⣯⣥⠀⠀⠀⢹⣾⠏
⠀⠀⠀⠀⠀⣧⡧⢄⠀⠀⠀⠈⠁⠀⠈⠁⢹⣇⢩⡄⠉⠀⠉⠁⠀⠉⠉⠀⠈⠁⠀⢀⠀⠈⠉⠁⠉⠀⠉⠁⠀⠉⠉⠁⠀⠀⡠⢼⣼
⠀⠀⠀⠀⠀⠸⣷⢨⡕⠰⢐⠂⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠆⢪⡅⣼⠇
⠀⠀⠀⠀⢀⡠⠼⡆⣧⢹⠰⠀⠀⠀⢠⣶⣶⣶⢰⣶⢠⣶⠀⣴⣶⡄⢰⣦⠀⣿⢱⣶⣴⡶⣰⣶⣦⠀⠀⠀⠀⠀⠀⠖⡏⣼⢰⠧⢄⡀
⠀⠀⠀⠀⠀⠆⢧⣹⡸⣾⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⣼⣿⢀⣿⣻⡇⢸⣿⣿⣿⢸⣿⣿⠃⢘⣿⣏⠀⠀⠀⠀⠀⠀⠀⣷⢇⣏⡼⠰
⠀⠀⠀⠀⠀⠀⠀⢙⢳⡙⣆⠀⠀⡀⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⠻⣿⢸⡿⢻⣿⢸⣿⢻⣆⠻⣿⡿⠀⠀⠀⢀⠀⠀⣰⢋⡞⠋
⠀⠀⠀⠀⠀⠀⣒⠨⣔⠻⣼⡆⣤⡕⢄⢃⢂⡀⠀⠀⠀⣿⡤⣶⡀⠻⣦⣰⡆⠀⠀⠀⠀⠻⣦⣶⢀⡐⡚⣄⢪⣤⢰⣧⠟⣢⠅⣒
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠳⣌⢳⡀⠙⢆⡈⢳⡀⠀⠀⠀⠈⠛⠛⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⢀⡞⢁⡴⠋⢀⡞⣡⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⠳⣄⠙⠶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠴⠋⣠⠞⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⠽⠦⣄⡉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢉⣠⠴⠯⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠧⢤⣤⣀⣀⣀⣀⣀⣀⣤⡤⠼⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣟⢽⠘⢈⠇⢊⡟⡑⢪⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡯⢣⣈⣾⣒⢾⣀⠎⠀⡰⠃⢌⣿⡢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠰⠄⠀⠂⠀⠀⠀⠀⢀⣀⣤⣤⣾⣾⣿⣿⣾⣿⣿⣿⣦⡵⣎⠀⠐⠋⣒⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠂⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣗⡂⠉⣁⢈⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠛⠛⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡏⠀⠀⣈⢈⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⣴⣇⠀⠀⠀⡄⢀⡨⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⢟⣤⠄⢀⡁⣨⣫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⣼⣿⣿⣷⣤⡴⢺⠁⠄⠄⠆⠀⠀⢀⠀⠸⢿⣿⣿⣿⢟⣏⠀⢀⡜⠳⣯⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠉⡙⡿⢷⡄⠠⠰⠠⠄⡀⢀⣠⣴⡆⣠⣸⣿⡟⠫⣀⠱⢮⣀⣛⠎⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⡀⢀⠀⠇⠀⠀⠹⣦⣁⣠⣴⣾⣿⣿⣿⣿⣲⣿⣿⣿⡾⠬⠗⠺⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢂⠀⠁⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠄⠀⠉⠰⠀⠀⠀⠰⡟⣿⣿⣿⣟⣽⣿⣿⢎⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⡒⠀⢀⢀⡂⠚⠜⠀⢸⣿⣿⣿⣿⣿⣵⣟⣿⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⣠⡄⠠⢀⢠⣿⣿⣿⣿⡿⣿⣿⢷⣙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡶⡾⣾⣿⣿⣿⣿⣟⡷⣽⣮⡋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡝⢀⡟⡆⢻⣿⣿⠿⣯⣾⣯⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢊⢸⣶⡯⠤⢯⠀⢿⣿⣯⢿⣷⣻⣏⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠼⠙⠊


⣿⣿⠿⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡆⠀⠀⣠⠞⠁⠀⠀⠀⢀⡔⣿⢳⡄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠸⣗⣾⣸⡟⠀⡀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡀⠀⢰⡟
⠀⠀⠀⠀⠀⣾⡇⠀⣿⠁⢠⡄⠀⢀⡴⠀⠈⠉⠁⠀⠀⣿⠀⠀⣿⠀⢠⡀⠀⣾⡁⠀⣄⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠁⠀⣿⡀⢸⡇⠀⣾⠃⢸⡇⠀⣿⡄⢰⡿⠀⢀⣿⢶⡟⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⣼⡇
⠀⠀⠀⠀⠀⢹⡆⠀⠙⠀⢿⠁⠀⠙⠂⠸⡇⠀⠙⠃⢸⡇⠀⠀⠛⠸⣇⠀⠀⣿⠀⠀⠛⠀⠸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠋⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⣄⡀⠀⠀⠀⣀⡀⢀⠀⣀⡀⠘⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⣿⠀⡇⠀⢠⡞⢹⡇⠸⣿⠉⣿⠀⣿⡼⠃⠀⠀⠀⠀⠀⠀⠘⣧⢰⡇⣴⠛⢶⣸⡇⠀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⣿⠀⣇⣠⣾⣀⡿⣧⣀⡇⠀⣿⢀⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⣷⣀⡼⠻⣇⡼⣷⡀⢘⡁
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠉⠁⠈⠉⠀⠈⠉⠁⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠐⠀⠘⠀⣸⠇⠈⠉⠁⠀⠀⠀⠈
⠀⠀⢸⡙⠳⣯⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⣀⣠⡴⠏
⠀⠀⣠⠽⠦⣝⡾⠛⠛⠓⢦⡀
⠀⠀⠻⠤⠶⢺⡇⠐⠀⠒⠈⡇
⠀⠀⠀⠀⠀⠀⢧⠀⠘⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣷⣀
⠀⠀⠀⣰⠞⠉⡟⢧⡴⠋⠁⠀⢠⠞⠋⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢼⣧⣟⣹⣇⠀⠀⠀⢐⡶⠶⢧⣀⠀⠀⠀⠀⣠⠿⠦⣤⠄⠀⠀⢠⠰⣇⡆
⠀⠀⠀⣇⠘⢀⠀⢸⡇⠀⠀⢠⡏⠀⠃⢀⠀⣽⠀⠀⠀⡾⠛⣆⢰⠇⡟⢸⡆⢹⡟⠒⠛⠀⠀⠘⡏⠰⠀⡌⣧⠀⠀⢀⣼⠃⠆⢠⠘⡷⠀⠀⠈⠻⡟⠀⣼⣡⠄
⠀⠀⠀⠘⢦⣀⣀⣼⣧⣤⡀⢸⠁⠐⠃⣠⠼⠁⠀⠀⠀⣇⠀⢹⣼⠀⣇⢸⡇⠈⡇⠀⡤⢤⠀⠀⠹⣤⡀⢠⣧⣤⣶⡀⣿⠀⠲⠀⣸⠃⠀⢀⠀⢀⡷⠚⠛⠳
⠀⠀⠀⠀⣠⠟⠁⢀⡀⠀⠙⢾⡤⠖⠛⠁⠀⠀⠀⠀⠀⠈⠳⣼⣿⠀⣿⢸⡅⢠⡇⢸⠃⠸⡇⠀⠀⢦⡽⠋⠀⢠⡀⠹⣿⢀⡤⠞⢁⠀⠀⣸⢧⣸⠁
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡄⢸⡆⠀⠀⢀⡀⢠⣄⣤⣯⣀⡄⠀⢹⠀⣿⠈⣇⢸⠃⡟⠀⠀⡇⠀⠀⢾⠃⠈⠀⠀⠀⡀⢿⠋⠀⢀⡏⢳⣠⡏⠈⣿⡶⡄
⠀⠀⠀⠸⡇⠀⠛⠀⠀⡀⠀⠀⡇⢀⡿⠛⡿⠻⣟⠳⡄⠹⣦⠤⢸⡆⣷⠀⡏⣾⡼⠁⣀⡼⠃⢀⡄⢸⡄⠀⠈⠁⠀⠁⣿⠀⣤⣜⡇⠀⠹⡆⠀⣿⠀⣷
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠁⠀⢸⡇⢹⡇⠀⡇⠀⢹⢰⡇⢠⣧⠀⠘⡇⢹⢸⣧⡟⠛⠉⠁⢨⠟⢋⠙⢓⢷⡈⠀⠀⠆⢠⡇⠀⣧⠈⠳⡄⠀⢻⠀⡟⢠⡟⠋⡷
⠀⠀⠀⠀⠈⢧⠈⠀⠀⢠⠀⣼⠀⠸⡇⠀⢻⠀⡿⡼⢀⡞⠀⠀⠀⣇⣿⢸⣿⠀⠀⠀⠀⢻⣄⠈⠤⠈⠀⣳⣄⠀⠀⣼⠀⠀⠘⢧⡀⢹⠀⣾⢸⢃⡾⣠⠞⠁
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠃⠀⠀⠃⠀⠈⠓⠛⠃⠘⠀⠀⠀⠀⠹⠘⠟⠃⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠙⠂⠐⠃⠀⠀⠀⠀⠙⠊⠀⠃⠛⠚⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠖⣟⣋⣛⣛⣛⣙⣋⣛⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠉⠉⠁⡂⠈⠀⠈⢀⠀⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⡴⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠉⢀⡈⠈⠣⣤⠀⢰⡄⠀⠀⠀⢠⡀⠂⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠞⠁⠀⠀⡄⢯⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠐⠀⠀⢀⠒⠋⡀⠀⠀⠀⠃⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠄⠐⠒⠂⠈⠁⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠴⠋⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠄⠊⠀⠀⠀⢀⠖⠀⠀⠀⡀⠀⠀⢀⠀⢀⠀⠀⠠⠉⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠐⠐⠂⠠⢤⡀⠄⡀⠀⠀⠑⢆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⠖⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠉⠦⠒⠠⡄⠡⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠲⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣀⠀⣀⡀⠂⠐⠀⠀⠈⠉⠙⠛⠛⠻⣥
⠀⠀⠀⠄⠀⠀⠀⠀⠈⡶⠀⠀⢣⠀⣠⢦⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⡔⠈⠀⠀⠀⣀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡤⠄⠀⠀⠀⡠⠊⠁⠀⠠⠤⣄⡀⢀⠀⡰
⢇⡀⠀⠉⠉⠑⠂⢄⠸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠈⠐⠠⢀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⣠⠚⡇
⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠀⠀⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠐⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠶⡁
⠀⠀⠀⠀⠀⠠⠃⡌⢰⠁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢦⠐⣍⠢⢀⣉⣉⣉⠔
⠀⠀⠀⠀⠀⢁⠞⢠⠇⡰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠆⠱⠠⠤⠜
⠀⠀⠀⠀⠀⠀⠐⠏⠀⠁⠀⠀⠑⢄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⡀⠀⠀⠀⠀⣀⣀⠠⠄⠒⠉⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⠁⠉⠙⠉⣋⠀⠀⠀⠀⠀⠀⢸⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠸⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⡀⠀⠀⠐⢐⠁⠀⠀⠀⠀⠀⠀⢰⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⢁⣀⣀⠠⠤⠀⣀⠠⢌⣉⠢⠀⠀⠀⠀⠀⠆⡦⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠉⣁⠀⠄⠐⠈⠁⠀⠀⠀⠀⢈⡀⠠⠀⠒⡉⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠄⠒⠀⡀⠋⡤⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠊⣠⠊⢀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠞⠀⠀⡌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠰⣀⠃
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣠⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢖⠀⠤⠤⠤⠀⠒⠂⢡⠃⢀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠉⠀⠚⠓⠀⠀⢸⠐⠒⠂⠁⠀⠐⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠐⠂⠉⠑⠂⠄⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠢
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡠⠒⢠⠘⢄⡀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠱⠀⠀⠀⢀⡀
⠀⢐⢉⡌⠶⠊⠁⠀⠉⢲⠗⠤⡀⠀⡔⢁⠦⣨⣀⡀⠀⣇⣹⡁⣂⠀⣈⠰⢆⢻⡶⠮⢅⠀⡀⠤⠤⢼⣀⡠⡊⣠⣈⡆
⠀⠈⢂⠘⠻⡢⠤⠄⣠⡥⠐⠒⢺⡏⢄⡟⢠⠏⠀⠈⠑⡗⠈⣧⢨⠋⠀⠀⠀⢏⡠⠂⠉⢫⡀⠀⠀⠀⠙⢦⠞⢂⠌
⠀⠀⠀⠡⡌⠱⠄⠄⠴⠷⠀⠀⠼⠶⠎⠨⠴⠦⠶⠒⠶⠻⠲⠶⠲⠟⠶⠒⠦⠼⠷⠀⠀⠴⠏⠑⠶⠤⠄⠼⢡⠖
⠀⠀⠀⠀⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁

⠀⠀⠀⠀⢿⣿⣿⣿⠇⢸⣿⡇⢸⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣧⠀⣿⣿⠀⣿⣿⠀⢠⣿⡟⠀⣿⣿⠿⠀⢻⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢠⣿⣿⣿⡆⠀⢸⣿⣿⡀⣾⣿⠀⣿⣿⢀⣾⡟⠀⠀⣿⣿⠀⠀⣸⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⠈⣿⡇⠀⢸⣿⣿⣇⢿⣿⠀⣿⣿⣾⡟⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣷⣾⣿⡇⠀⢸⣿⠀⣿⡇⠀⢸⣿⣿⣿⣾⣿⠀⣿⣿⡟⠀⠀⠀⠀⣿⣿⣶⠀⣽⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡏⢹⣿⡇⠀⣾⡟⠀⣿⣿⠀⢸⣿⡟⣿⣿⣿⠀⣿⣿⣷⡀⠀⠀⠀⣿⣿⠉⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⣿⣷⣦⣿⣿⠀⢸⣿⡇⢿⣿⣿⠀⣿⣿⢻⣷⡀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇⠀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⢠⣿⡏⠉⢹⣿⡆⢸⣿⡇⣾⣿⣿⠀⣿⣿⠈⢿⣷⡀⠀⣿⣿⠀⠀⢹⣿⡀⣸⣿⡇⢸⣿⣧⣾⡟⠙⡇
⠀⠀⠀⠀⠀⠸⠿⠇⠀⠸⠿⠇⠸⠿⠇⠸⠿⠁⠀⠸⠿⢧⡼⠿⢷⣿⡿⠿⠀⠿⠿⠀⠈⠿⠷⠀⢿⣿⡆⠀⠈⠻⠿⠿⠋⠀⠸⢿⣿⡿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⡟⠛⠛⠇⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⠏
⠀⢀⣠⣤⣤⣀⡀⠀⠀⣴⡿⠀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣄⡀⠀⢀⣿⡿⠀⠀⣀⣀⠀⠀⣀⠀⠀⣸⣿⡷⠋
⣴⣿⠏⠀⠀⣿⣿⠦⠀⣹⠷⢻⣿⡿⠀⠀⣴⣿⠋⢨⣿⣿⠀⠀⣿⣿⠀⠀⠀⢀⣾⣿⠉⣹⡄⢸⣿⡇⠀⢠⣿⡿⠀⣼⣿⡇⠀⢹⣿⡇
⣿⣿⠀⠀⣸⣿⣿⠀⢠⡟⠀⢸⣿⡇⠀⣼⣿⡇⢀⣾⣿⣿⠀⢠⣿⣿⠀⠀⠀⣸⣿⡏⠉⠉⢠⣾⣿⣅⣤⢾⣿⡇⢠⣿⣿⡇⠀⣸⣿⡇⠀⠀⣼
⠘⠿⣦⠾⠋⣿⣿⣠⡞⠀⠀⢸⣿⣦⡾⠻⣿⣧⠞⠁⢿⣿⣴⠏⠹⣿⣦⣤⡶⠛⣿⣧⣠⡴⠋⣿⣿⡅⠀⠸⣿⣷⠟⠸⣿⣧⣴⠏⢻⣷⣤⡾⠁
⠀⠀⢀⣠⢾⣿⡟⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠁⠀⠀⣿⣿⢻⡀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⣰⠟⠁⣸⣿⣇⣀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⣿⣿⣈⣧⣀⠀⢀⣀⣀⣀⡀
⢸⡇⠀⢀⣿⡟⣿⣿⡿⣿⣿⡄⠀⢸⣿⡇⠀⠀⣿⣿⣿⠅⢀⣾⣿⠿⣿⣦⠀⠀⣴⣿⡿⣿⣷⣿⣿⣿⣿⡿⠀⢸⣿⡿⣿⣿⣷⡀
⠘⣇⣀⣼⠟⠀⣿⣿⠀⠈⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇⢰⣿⡏⠀⢸⣿⡟⢿⣿⡿⠁⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠘⣿⣧⠀⠸⠿⠇⠈⣿⣷⠀⠸⠿⠇⢠⣿⣧⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⠇⠀⢸⣿⡇⠀⠀⣿⣿⣤⡀⠀⠹⣿⣷⡀⠀⠀⠀⠙⣿⣷⡀⠀⠀⢸⣿⣿⣤⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠛⠻⣿⣦⠀⢸⣿⡇⠀⠀⣿⣿⠛⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠘⢿⣿⡄⠀⢸⣿⣿⠛⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢀⣀⠀⠈⢿⣿⠀⠀⣀⡀⠈⢿⣿⡄⢸⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣄⣤⣿⣿⠀⢸⣿⣇⡄⠀⣿⣿⣤⠀⠘⣿⣧⣄⣼⣿⠃⠈⣿⣷⣄⣼⣿⠇⢸⣿⣯⣀⠀⢸⣿⣇⣠⣾⣿⠇
⣤⣤⣠⣤⣤⣤⣿⣿⣿⣿⣿⣡⣤⣼⣿⣿⣧⣠⣿⣿⣿⣄⣤⣌⣿⣿⣿⣡⣤⣤⣌⣿⣿⣿⣩⣤⣸⣿⣿⣿⣤⣼⣿⣿⣿⣿⣫⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣩⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠃⠀⠀⠀⢀⣤⠢⠄⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢀⣴⡄⠀⡀⢁⡇⠀⢺⣧⡙⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣴⣿⣮⣅⢀⡇⢸⡇⠀⣿⣿⣤⣬⢥⣞⠀⠀⡀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⣤⣟⣿⠯⣼⣾⠅⡘⠀⢰⣿⣻⠻⣟⣤⣤⣾⣟⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃⠩⢷⣗⢀⢻⡟⢰⠏⣠⠛⢙⡏⠔⠂⠉⠉⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣀⢀⡈⠻⠯⢽⣥⡿⠂⠁⠀⠠⠁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠋⠀⠏⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⡀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠠⠀⠄⠀⢠⠀⢀⠐⠀⠄⠙⠐⠀⡀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⠀⠐⠈⠁⠈⠀⠂⠈⠂⠀⠀⠀⠀⠐⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢀⣀⣀
⠀⣀⣀⠀⠀⢠⡄⡄⢀⡀⢀⡀⢀⡀⠀⢀⠀⠀⠐⡖⣠⠀⢀⡀⠀⠀⢸⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠰⠒⠋⠉⠁
⠀⠀⣀⣠⡄⢸⡿⡇⣼⣧⣯⠇⣿⡗⠾⣾⠀⠀⢐⡇⣿⡀⣼⡇⣿⢂⣾⢇⢘⡄⢿⣷⢰⠠⡟⠷⡆⢾⢂⠼⣿⠄⠀⠒⠂
⠚⠋⠉⣀⠄⠘⠀⠁⠁⠈⡏⠀⣯⠀⠀⡹⠀⠀⠀⠃⠃⠀⠁⠀⠀⠀⠈⠀⠈⠀⡔⡿⠈⠀⠉⠀⠁⠀⠀⢠⢹⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠐⠢⢄⠀⠀⠀⠀⠀⠀⠣⠤⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣈⡙⢦⣀⡤⠴⠒⠒⢒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⡠⠀⠁⠒⢄⠀⠀⠀⠉⠲⣄⠀⠈⠙⢦
⠀⠀⠀⢻⣦⣀⣥⡀⠀⠀⢀⠔⠉⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⣧⠀⠀⠈⢷⠀⠀⠀⠀⡰⣦⠄⠀⠀⡀
⠀⠀⠀⡴⠘⡄⠐⠌⡆⣰⠃⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⢀⠤⠚⠉⠉⣙⠳⣌⠉
⠀⠀⠀⡇⠀⠇⠀⡰⠁⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠀⠀⠀⣼⠁⢰⢃⡆⠈⣷⠀⠀⠣⠸⣧
⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡾⠁⠀⢸⠘⣧⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣀⡴⠋⠀⠀⠀⠈⢧⡙⢸⡿⣀⣀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⣀⣀⣀⡤⠚⠙⠲⠤⢤⣤⡤⠼⠧⠤⠶⠛⠉⠀⠀⢠⠔⠒⠀⠉⠉⠁⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠄
⣿⣿⣿⣻⣛⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣿⣿⣿⣛⣿⣻⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠐⠂⣀⠀⠀⠀⠀⠀⠤⠤⠤⠠⠀⠀⠀⣠⠊⠁⠀⠀⠂⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⡀⠀⠀⣤⠀⠠⢤⠀⠤⡄⢀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢶⢀⠟⡀⠒⠊⢰⠒⠁⢸⠁⠀⠀⢀⠀⠀⠀⠀⠠⠐⠐⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠂⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠈⠈⠀⠀⠀⡀⠈⠂
⠀⠀⠀⠀⢀⠀⠐⠀⠀⢀⡤⢤⠀⠀⠂⠂⠀⠀⠉⠀⣠⠄⠀⠀⠈⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⣀⣀⡤⠤⢤⡞⠐⠋⠀⠀⠤⠖⠒⠒⢠⡞⠀⠀⣀⠀⠀⡀⠀⠀⡀⠀⡄⢀⠀⠤⠖⠛⠛⠳⢆⠁
⠀⠀⢠⡿⠋⢠⡆⠀⣼⠀⠀⠀⠀⠀⠀⠀⢠⡀⣼⢀⡴⠂⣇⢰⠋⢹⡇⢰⠆⢲⠀⠀⠀⠆⠀⠀⠀⠀⠀⣀
⠀⠀⠘⠀⠀⢾⡇⠀⡟⢀⠀⡴⠋⡖⣿⢀⡟⢠⡏⢦⠀⢀⡟⠀⠀⣸⡇⠞⢠⡟⠀⢀⡾⢁⡄⠀⣤⠀⠀⠈⢻⡆
⠀⠀⠀⢀⡀⠘⣇⢠⡧⢸⠇⠃⠀⡇⠋⢼⠁⠈⠁⠈⠳⣦⣤⡤⠞⠁⡇⠀⠈⠀⠀⠉⠠⠞⢀⡾⣡⠖⠒⡤⢠⠏
⠀⠀⠀⢾⡅⠀⣿⠘⠟⢸⠀⠀⠀⠀⠀⠀⣀⡤⠒⠒⠒⣒⢀⠀⠀⠀⡧⢀⣀⣀⣀⣀⣤⠇⠊⠀⠁⠀⣼⠃⠁
⠀⠀⠀⠈⠙⠊⠁⠀⠀⠈⠁⠀⠀⠀⠀⢰⡟⠀⠀⣉⢦⠛⡀⠋⢀⡼⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣀⣁⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⡀⡄⢂⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⣠⢀⡂⡄⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⣴⣦⣤⣴⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣨⣷⢿⢻⣯⣯⣉⡙⠻⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⣫⣶⠞⠛⠻⠛⣿⣧⣄⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢈⡀⠀⠀⠀⠀⠀⣨⣾⣋⣶⣿⡇⠀⠀⠀⠀⢘⢿⣿⡆⠀⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢺⠟⢁⡤⠴⠖⠒⠒⢾⠻⣯⢿⡿⢂⠀⠀⠀⠀⡨⣾⣿⣇⠀⣸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⣴⠋⠁⠀⠀⠀⠈⠀⠁⢠⠰⣼⣷⡄⣠⣤⢼⣦⣿⣯⠥⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⣴⣿⠀⠤⣤⣢⢀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣡⣾⣶⣿⣯⠬⣼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠾⠟⠁⠀⠀⢨⢿⣾⣦⣔⠤⢴⡤⠟⠛⠛⢿⣿⣿⣿⣿⡿⢏⣡⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⠀⠀⠁⠀⠀⠘⢿⠯⠻⡿⢿⡿⠁⠨⣀⠘⣛⣿⣿⡿⣣⡾⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡥⢀⠔⠁⡀⠀⠀⠀⢠⡶⢠⣾⠟⠂⠥⠀⢉⣓⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣿⣶⣞⡀⣈⢐⠷⣿⣷⡀⢔⡪⡲⢞⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⣿⡶⠿⢿⠏⠐⠍⣮⡰⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠴⢧⠀⢀⣤⣜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠀⢀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢰⠀⣄⡱⠀⠈⠠⡢⠐⠘⠀⣲⠀⠣⡀⠚⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠸⠀⠉⠁⡆⠐⠀⠄⠊⢸⠀⠄⠘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠡⡀⢀⠘⢆⡠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⡆⢰⠀⠸⠇⢰⡆⠀⠀⣀⠀⠐⠐⠀⠔⠉⠀⠈⠁⠀⠀⡈⢈⠀⠀⡷⢰⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢾⢠⠗⡇⠈⢡⠀⠘⢇⠀⡈⠂⠀⠀⠤⢠⠀⠀⠀⡀⠀⢸⠀⠀⠠⡄⠠⡀⢂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈⠁⠀⠑⠒⠁⠈⠁⠀⠁⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠐⠢⡀⠀⠀⣀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠢⡀⡌⠀⠀⠀⠀⠀⠘⣀⠊⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠸⠀⠀⠀⣀⣀⣸⡃⠄⠒⠀⠀⠀⠒⡧⠤⣀⠀⠀⠀⠀⣀⠄⠒⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡉⢶⡄⠊⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠢⣀⠶⠁⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡠⠋⠀⣀⣧⣀⡠⠐⠂⠐⢣⠔⠂⠀⠀⠂⡼⠁⠉⠢⡀⠀⠹⠀⠀⠈⠑⢄⠀⠄
⠀⠀⠀⠀⠀⢀⠄⠒⠒⡲⠷⣄⡴⠉⣤⡤⡀⠑⠄⠀⠀⠸⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢦⠇⡀⠀⠀⠀⠈⣦⠤⠤⣀
⠀⠀⠀⠀⢀⠁⠀⠀⠌⠀⠀⣸⠀⠸⣿⣇⠘⠀⠀⠀⠀⠀⣆⢀⡀⠀⡘⠀⠀⠀⠀⢠⠂⠀⠀⠢⡀⢀⠔⠀⠣⡀⠀⠂
⠀⠀⠀⠀⠈⡄⠀⠀⠀⢀⠐⣹⠈⠂⠀⠀⠀⠀⢘⡁⠱⠮⢀⠀⢈⣢⡗⠒⠒⢄⡐⠁⠀⠀⠀⢀⠝⠃⠀⠀⠀⢡⠀⡄
⠀⠀⠀⠀⠀⠘⢂⣄⠐⠳⠤⡌⠀⠀⠀⠀⠀⠀⠸⠣⠄⣀⠄⠛⠤⠀⠈⢢⡀⠄⢧⠤⠤⣠⠔⠁⠀⠀⢡⠀⠀⢀⡞
⠀⠀⠀⠀⠔⠉⠨⠉⠁⢀⠖⠇⠀⠀⠀⠀⠀⠀⠲⣀⠂⠉⠑⣤⠀⠛⡉⠀⠀⠂⢀⢑⣀⡀⣀⠀⠀⠀⣠⠄⠐⠉⢂⠁⢄
⠀⠀⠀⠸⠀⠀⡆⠀⢠⠋⠀⠈⠂⠁⠀⠀⠀⠀⠖⢉⣑⡴⠉⡈⢃⡠⠼⠈⠉⠋⠀⠈⢃⠀⢙⠀⠙⠇⠈⠂⠀⠀⠀⡆
⠀⠀⠀⠀⠢⡀⡇⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⡉⠑⠢⣨⠄⠘⢇⠈⠻⡀⠈⣃⠉⠻⠂⣤⠀⠀⠀⡧⠊
⠀⠀⠀⡔⠊⢙⠧⣀⠘⡇⡌⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⢲⠐⠘⢂⠀⠀⠀⠓⠂⢴⡂⠈⠁⠀⠙⠀⢲⠄⠸⠂⠀⢫⠑⢄
⠀⠀⠈⠀⢠⠁⢠⡞⠁⠋⢃⠀⠀⠀⠀⠀⠀⡰⢃⢡⡀⡜⠀⠀⠀⠱⠆⠒⠑⢄⠀⠉⡏⠉⠉⠐⢳⠘⠈⠓⠒⠠⣄⠀⠀⠂
⠀⠀⠈⢄⡂⡰⢡⡇⢠⡦⠐⡗⢤⠀⠀⠴⠚⢄⢨⡅⠀⠈⠦⠄⣀⡠⠓⠤⡀⠀⠳⠂⠁⠀⠀⠴⡎⠀⡄⠀⢀⣀⠨⠀⡠⠃
⠀⠀⠀⠈⠳⠁⢸⠁⠀⠀⠀⢡⣠⡄⢀⠃⠀⠀⡛⢧⡀⠀⢰⠀⠈⠣⠀⠀⠈⢦⠤⠧⢀⡀⠀⠀⠀⡂⠀⠀⠀⠉⠑⡎
⠀⠀⠀⢠⠁⠀⣾⡄⠠⡖⠒⠉⢄⡀⠟⠢⢀⣰⠀⠈⡑⠤⠚⢄⠀⠀⠱⢄⣀⡰⠀⠀⠀⠈⠁⠲⡈⠁⠂⢄⣀⡀⠔⠁
⠀⠀⠀⠀⢀⠌⢸⠀⠀⢠⡠⠔⠛⠟⠀⠀⠠⠁⠀⠐⢧⠀⠀⠘⠦⢀⣀⠆⠀⠁⠢⣀⠀⠀⠀⢀⡄⠀⠀⠀⠈⢢
⠀⠀⠀⠐⢺⠀⠈⡟⠉⠈⡀⠀⣰⣆⣦⣤⠁⠀⠀⠜⠀⢣⡀⠀⠇⠀⠈⢢⡀⠀⠀⢨⠀⠉⠉⠁⠀⠑⠢⣤⠤⠐
⠀⠀⠀⠀⢸⠀⠀⢸⡦⠄⢳⠈⣝⠉⢆⣣⡈⢲⠶⠀⣠⠊⣹⢖⠁⠀⠀⠀⢉⠀⠈⠁⣄⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠸⡀⠀⡾⠈⢲⡞⢃⡠⠶⠈⢃⠈⡣⣄⠰⠃⠙⣤⡜⠂⢀⠀⠀⡠⠀⠀⠀⠈⢂⠀⢁⠖⠂⠀⠈
⠀⠀⠀⠀⠀⠑⠊⠀⠄⠊⠈⠫⣥⠀⣢⠔⠋⡀⠈⡢⡔⠈⠀⠉⣢⡄⠋⠙⠂⣄⠀⢀⠀⡰⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⢦⣤⠚⣇⠀⢈⠦⠒⠉⠀⢀⣱⣶⠉⠠⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢈⠇⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⣄⠀⢀⠀⠀⠆⠐⠂⠀⡤⢤⢀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⢸⡆⢀⡀⣴⢂⡀⢹⠂⡿⢸⣅⠁⢻⣰⡏⣰⢸⠇⢸⠀⢪⡆⢈⣅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣧⠀⣷⢸⡇⣿⣸⣇⠾⠘⠇⠘⠈⠆⢸⡇⢁⣿⠘⠊⠘⠂⢾⢀⡾⢁⡾⣡⠖⡂
⠀⠀⠀⠀⠀⠀⡀⠀⠹⣆⠸⠆⠃⠈⠁⠀⠀⠀⠀⠀⠃⠀⠼⠁⣸⠇⠀⠀⠀⠀⣄⠀⠀⡈⠃⠛⣴⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⡀⠀⢠⣶⣦⠰⣶⣦⡀⠀⣶⣶⠈⠀⢾⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⣿⡟⢿⣿⠀⣼⣿⣿⣧⠘⢿⣷⣼⣿⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⣿⣷⣸⣿⢰⣿⣟⣹⣿⣧⠀⢻⣿⣿⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢋⣿⣿⠿⠿⢿⣿⣧⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢈⣉⣁⣠⠆⠀⠉⠩⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢉⢉⠝⡛⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡂




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣀⣋⡤⠈⠖⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠀⡠⠂⠉⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎⠄⠀⠀⠀⠀⠀⠘⡄⣀⡈⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⢀⠀⣠⢁⠠⠀⠀⠠⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠈⡁⣰⠀⠐⢁⠊⠁⠀⠀⠀⠂⡈⠉⢐⠘⠀⢐⣄⣨⠍⠰⢄⡀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠔⠈⣠⠔⠉⠁⠚⠉⠀⠀⠀⠈⢢⡀⠀⠀⠈⠁⠂⠈⠑⠢⡀⠉⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡤⢀⠁⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠂⠜⠀⠈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⠨⠀⠀⠘⠀⠌⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠢⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠁⡠⠁⠙⠀⠈⠀⠀⣀⠀⠀⠀⠀⠤⠃⠀⠤⠐⠁⠀⠀⠀⠑⠠⠀⠀⠓⠄⠀⠀⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣫⣛⣉⣝⣻⣋⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠶⠒⢦⢀⡴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣔⣋⣲⣾⣥⣽⣷⣿⣿⣾⣷⣶⣤⡀⢻⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠛⠛⡿⢫⢩⡏⢀⡄⡼⠉⠙⠻⢿⣿⣾⣥⣄⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠋⠙⢳⢤⡀⠀⠀⣴⣶⠟⢹⢸⠃⡼⢸⠃⡎⢰⠀⡜⡼⠁⠀⡴⠀⢀⠟⠻⢿⣿⣷⡴⠖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠴⡁⠘⠃⠀⠀⠀⠈⣷⠀⣼⠋⣽⣤⠿⣼⠚⠛⡟⠛⢻⡿⠛⢷⣥⠦⣾⡁⢠⠏⠀⡠⠂⢈⣿⣷⣦⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠈⠀⢀⠀⠀⠀⠀⠐⠀⣼⢸⡯⡞⣻⣧⣤⣷⣿⣿⣷⡾⣾⣤⣖⣟⡁⡀⠈⣿⠷⢦⣞⠁⣠⠏⠀⡹⣿⣷⣸⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢛⡟⠓⣾⠷⢶⠎⠀⢀⣲⠏⢱⣶⣿⣿⣿⣿⣿⣿⣝⣿⣿⢿⣷⣿⣿⣿⣤⡼⠁⡀⠀⣿⣴⣅⣠⠞⠀⣿⣿⣅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠟⠚⠁⠈⣔⣼⣧⣶⣿⣿⣿⣿⣯⢿⣿⣾⣿⣿⣻⣿⣿⣿⣿⣿⣟⣿⣞⣠⠞⠁⠀⢘⡇⣠⠔⢉⢻⣿⡴⠯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⠀⠀⠀⠀⣀⣼⣿⣿⣮⣿⣿⣟⢿⣯⣿⣷⣿⣿⢿⣿⣿⣿⣿⢿⣟⣿⣿⣿⣻⡖⣁⠴⠛⠛⣧⠞⢉⡴⢿⣷⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⢾⡇⣰⣄⣸⣏⢙⣿⣾⡯⣿⣟⣿⣿⡿⣿⣿⣮⣿⣾⢿⣿⣷⣽⣿⣿⣿⣿⣿⢿⣿⡤⠔⢀⣠⣿⠚⣉⠄⢈⣿⡟⢧
⠀⠀⠀⠀⠀⠀⠀⢰⢇⡼⠹⢑⢹⣣⣭⣷⣝⣎⠹⣯⣿⣿⣿⣿⡟⣿⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣻⣿⣷⠿⣗⣚⡉⠀⢹⣏⡤⠞⠋⢻⣿⣸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢯⡐⣢⠤⣍⠕⡻⣻⣌⣣⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⠉⣽⢹⣼⣿⠤⠄⠘⠋⣸⡇⠚⣹⠇⣶⣒⠛⣷⠔⠒⢁⣼⣟⠛⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣲⢬⣉⣒⣲⡥⣾⣑⣖⣾⣀⣿⣽⣯⣿⣿⣿⣿⢋⡾⣏⡽⡽⢻⢉⣿⣦⠤⠀⢴⢩⣿⣯⠀⠘⢩⡿⣤⣻⠴⠞⠋⢹⣿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣙⠻⠯⠥⠔⣾⠗⡒⣿⠙⣿⣿⣿⣿⣿⣟⢣⠟⣶⡿⣹⣟⣿⣾⣿⠄⠀⢠⠔⣻⡯⠤⠒⠈⠁⡠⠟⡟⠊⢉⣀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣅⣀⣙⢬⢴⣫⡾⢃⣿⣿⣿⣿⣿⣿⣟⣟⠞⣉⡟⣻⣿⣿⣿⠛⠳⠤⠤⢾⣯⣤⠴⠤⠈⡀⠠⠰⡁⠉⠀⣼⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⢛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡙⢿⠻⢷⣈⠓⠤⢠⣾⠟⠒⠒⠂⠈⠁⡀⠠⢾⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠍⢻⡷⡦⠭⠽⣿⠯⠭⣉⣛⣛⡋⠉⠥⠐⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣽⣿⣤⣀⣚⣻⡿⠶⠶⢶⠤⠤⡒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⡻⣿⣿⣿⣿⣿⣿⡿⢿⡟⢿⣛⢦⢯⡿⢯⡑⠦⡀⠁⠢⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠡⡹⢰⡅⣷⢠⡿⣦⣽⢿⠻⣿⢶⡙⢆⠈⠢⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢿⡏⢘⠗⠋⢘⠃⢋⡹⡌⣷⠈⢆⢃⠀⢠⠀⢉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣒⠊⠆⠀⢧⣤⣠⠞⠀⡋⠀⠘⠏⢇⢸⢆⣇⢸⢆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒⣀⠖⡠⢊⣓⡩⢭⠆⡠⠆⠽⠴⠄⠈⠞⠘⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠊⠀⠈⠀⢰⠸⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠈⠉⠉⠉⠉⠐⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡠⠒⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⠒⠒⠒⠠⠆⠒⠒⠉⠁⡰⠁⢀⠮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠉⠉⠉⠉⠁⠚⠓⠢⠤⠤⢼⠄⠒⠒⠈⠁⠀⠘⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⡠⠤⠔⠚⠢⠄⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠙⠄
⠀⠀⢀⣀⣀⠀⠀⣀⡇⢀⣀⠀⣀⠀⠀⠀⠀⠀⢠⠄⠢⡄⠀⠀⠀⢠⠊⣉⡱⡄⠙⠦⡀⠀⠀⢀⡀⠀⠀⣠⠔⠁⠀⠀⠀⢸⠀⠀⠀⢀⠤⢄⡀
⠀⠠⠃⣠⡎⠗⠉⠀⠀⠀⠈⣱⠟⠢⢤⠀⠀⡔⡃⡳⠆⡸⣀⣀⠀⢸⢎⣹⡠⢱⠀⢀⡈⠵⢖⠚⡷⠖⠋⢤⠀⣀⠤⠤⠤⠼⣀⣀⠖⡃⣲⣄⣱
⠀⠀⠱⡄⠼⢿⢢⢄⣀⣀⣰⣁⠤⠒⠀⣷⠉⡁⡸⠓⣢⠏⠀⠀⠙⢺⠆⣈⣇⠄⡝⠉⠀⠀⠀⢱⠁⡠⠒⠉⠻⡁⠀⠀⠀⠀⠈⢢⣠⠟⠆⡜⠁
⠀⠀⠀⠉⣆⣐⣻⣀⣀⣀⣹⣇⣀⣀⣀⣮⣠⣜⣁⣀⣼⣀⣠⣀⣤⣜⣢⣠⣽⣔⣱⣶⣀⣠⣄⣈⣾⣁⣀⣀⣀⣿⣓⣦⣄⣀⣀⣀⣗⣂⡘
⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⣠⣴⣷⠟⠋⠛⠋⢶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢋⡁⢸⡏⡿⢶⣦⣷⡀
⠀⠀⠀⠀⠠⠅⣰⣿⡿⠈⣿⣯⠸⡗⠋⣻⠿⣿⡇
⠀⠀⠀⠐⠀⢀⣌⢋⣴⣳⡜⡟⣀⠀⠀⢀⠈⡙⡇
⠀⠀⠀⠈⢀⣿⢏⣰⣝⠏⢉⠉⠁⠠⠴⣀⠐⣎⠃
⠀⠀⠀⠢⢼⡿⢨⣻⡟⢄⠎⡀⢀⡄⠀⠙⠂⠲⠴⠄
⠀⠀⠀⠀⣾⡆⠈⣯⠀⢊⡼⠑⡩⠂⣦⠀⠀⠀⠠⣟⣓⣤⡀
⠀⠀⠘⠀⢽⡧⠿⠿⠃⠟⢠⠄⢣⢞⡕⠐⣋⢈⣨⣼⢻⣿⣿⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢿⠇⣥⣸⢆⡄⠋⢠⠋⢀⠌⠉⠕⡽⢿⣿⢿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⣀⠀⠀⠀⠀⠀⠐⠈⢻⡀
⠀⠀⠀⠀⢸⣿⣬⢡⠈⠀⡆⣀⣄⢰⡇⡴⣼⢶⠟⢿⣟⡡⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠋⠍⡀
⠀⠀⠀⢀⡸⠿⠇⠾⠀⢸⢣⡏⠛⢨⠃⢀⣿⣀⢀⠈⣿⡽⣵⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣄⠀⠀⠀⢱⣾⡇⢀
⠀⠀⠀⠀⣰⠄⣀⠻⠧⡜⡀⠃⡒⡘⠞⣿⣿⠁⢸⠀⠈⢿⠹⠅⣿⣹⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢻⡇⠘
⠀⠈⡁⠀⠀⠀⠈⠳⡀⠃⢻⣤⠀⣀⣠⠋⢈⠤⠎⠠⠦⡄⠇⠀⠀⠙⢻⡿⣧⣿⣽⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⡷
⢀⢀⡀⠶⠄⠙⠀⠀⠈⠂⠀⠹⠀⠙⠟⣀⢂⡀⡄⢠⡓⠀⠘⠁⠀⠸⠀⠁⢺⣿⣿⣯⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢾⣆
⠈⠐⠯⠀⡄⣐⢤⠠⢌⡓⠀⣰⡶⠿⠞⢋⡈⡹⢞⡿⠃⠀⢠⠀⡇⡀⡐⠈⢉⡘⣿⣿⢟⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⢾⣿
⠀⠀⠀⠀⠸⠍⠀⠁⠠⢄⣀⣹⡽⠶⠟⣛⣉⣅⠁⢘⢻⣄⣌⣦⣠⣇⡃⡄⠸⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣡⡴⣺⠅⠃⡁⠁⣤⣧⠀⠀⠈⢡⡇⡁⠹⣧⠰⣾⣷⣿⣎⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣦⣰⣿
⠀⠀⠀⠀⠀⠀⠀⢈⡡⠴⠈⡁⡥⠶⠆⣿⠓⠀⢐⡛⣿⣴⡀⣸⠁⠠⠁⡸⣄⣀⣟⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⣀⡥⢆⠧⠷⡚⣎⣠⢩⠆⠀⠉⠐⠛⠿⣧⡿⣠⣶⣦⣁⢋⣍⢧⢹⣟⣿⣿⣿⣙⣟⣻⣿⠿⣻⣿⢛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣧⠶⠟⢋⣡⣭⣶⢆⠀⠀⠀⠀⢀⡠⠐⠋⢁⠼⠛⣹⣻⣤⣶⣎⣿⣍⡿⢫⡿⣭⢹⢻⣟⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠁⠛⣉⣀⣤⣼⠿⠿⡾⠷⣾⠆⠀⠠⠊⠁⠀⠀⠀⠀⡠⠔⠋⣩⣿⣿⣛⣛⣾⣿⣦⣿⠒⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠳⣟⣋⣅⣁⣅⣤⠿⢟⢻⢋⣿⢿⣤⣤⡄⠀⠠⠀⠈⠀⣤⣶⢿⣿⣵⣞⠏⣷⡯⣭⣎⣸⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠐⠚⢛⣋⡋⠏⢉⡁⣑⡾⠦⠻⡄⢭⡏⣟⣿⠿⣿⣿⣱⣶⣟⢷⣹⣿⣿⣿⣿⣯⣉⣾⣫⣽⣿⣻⣿⠿⢻⡿⢆
⠀⠀⠀⠀⢠⢋⢉⢛⡛⡛⣯⣟⣯⡷⡛⣄⣤⣦⡿⣯⡟⣧⣙⣿⡟⣧⣽⣾⣿⣿⣿⣿⣷⣿⡿⠿⠿⠛⠋⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠿⢻⣿⣿⣥⣭⡥⠴⠖⣇⢋⣤⣟⡮⣝⣶⣻⠦⢉⣻⠿⠿⠛⠛⠉⠀⠈⠀⠈⠋⠁⠀⠀⣰⡃
⠀⠀⠀⠸⡙⡈⢸⠊⡏⢰⣘⡇⡝⢻⡽⣷⡾⡇⠘⠚⠁⠀⠸⡎⡄⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠀⢀⣿⣷
⠀⠀⠀⠐⠁⢷⣞⢻⠹⣉⣭⡷⡟⢏⠣⠜⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⣼⣿⣿⣄⣠⡀
⠀⠀⠀⠀⠀⡾⢼⣾⣿⣭⣮⣷⡿⠿⣿⣿⣿⣿⣿⣩⣭⣽⣾⣶⣶⣤⣀⣛⠾⣷⣶⣾⣷⣶⣾⣾⣿⣿⠿⠿⣿⣿⣶⠶⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠟⠉⠀⠉⠀⠀⠐⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⣿⣿⣷⣼⣿⣿⣿⠿⣿⣿⣟⠛⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⡤⠀⠰⣞⢲⡄⣷⡇⣠⠖⡶⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣾⠷⢤⣿⣇⣀⣀⠈⠙⠛⠈⠙⠓⠋⠀⣀⡤⠞⢠⣿⠃⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢠⣿⠀⣼⠇⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⣀⠀⣼⠇⣀⠘⢬
⠀⠀⠀⠀⠀⠀⡴⠚⡁⢸⡏⠀⣿⠔⣹⡆⠀⣠⡶⠋⣷⡟⠀⣴⣦⢺⡿⢀⣿⠊⣨⡇
⠀⠀⠀⠀⠀⠀⡷⠊⠁⠸⣧⠀⠛⠀⣿⠁⢰⣏⠴⠊⣿⢀⠴⣿⠃⣿⢁⠝⠻⣿⠉⠀⢠⠃
⠀⠀⠀⠀⠀⣤⣀⢤⡰⣄⠙⠦⠀⠀⠻⠒⠁⠀⠀⠀⠈⠁⠀⠉⠀⠈⠁⠀⠀⠈⠓⠒⠁⠲⣄
⠀⠀⠀⠀⠀⠀⢈⠗⡿⢺⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⢿⡦
⠀⠀⠀⠀⠀⠘⠁⠈⣁⡋⠀⠀⠀⢰⣿⣆⠀⢀⣠⣤⣤⠀⠀⠀⣠⣤⡄⠀⠀⠀⢀⣤⣄⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⣠⣿⣿⣿⢠⣿⣿⠏⠀⢿⣏⠉⣿⣿⡇⠀⠀⠀⢸⣿⣿⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣤⠴⠋⢸⣿⡿⣿⣿⣿⠀⠀⢸⣿⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣼⣿⠇⢹⣿⣿⡀⠀⢸⣿⠀⢹⣿⣷⠀⠀⣼⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⢀⣽⣿⠿⠄⠀⠙⠿⠷⠶⠟⠁⠀⠀⠙⠻⠷⠞⠁⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠦⠤⠤⣀
⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⢠⣀⣀⠤⠶⠤⡌⡑⡄
⠀⠀⠀⠀⣠⠎⠀⠀⢀⡰⠋⠁⠀⠀⣠⠄⣸⠀⡇
⠀⠀⠀⡼⠃⠀⠀⠀⢘⣧⠀⠀⠀⠀⠻⠿⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣦
⠀⠀⣼⠃⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢪⡟⠃
⠀⢸⠏⠀⠀⠀⠀⠀⡀⠈⠉⠓⠲⠤⢄⣤⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠿⡄
⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣡⣤⣤⣤⣭⣉⣉⠓⠲⢤⣀⣠⣤⠤⣴⡇⢰⣧⠤⢤⣀⡀
⠀⣿⢡⠏⠠⠀⠠⢀⠀⠀⠀⢀⣴⣾⠿⣿⣿⣿⣿⣿⠟⠛⠋⢙⡶⠚⠉⠀⠀⠚⠙⠟⠉⠉⠁⠀⠀⠉⠓⢄
⠀⣿⡃⣠⠀⠀⠀⠀⠂⢀⡔⠉⠉⢱⠛⢉⠢⢢⣞⡁⡀⠐⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠱⡀
⠀⢸⣥⡟⢀⠀⠀⡀⠀⡾⠀⢠⣶⣷⣤⣴⠿⠿⠯⢭⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠈⣿⣧⡌⠀⢀⡆⢸⡇⠀⣈⣿⣷⡶⣿⣛⠲⠦⣤⣌⣽⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⡞⢻⣿⣄⡀
⠀⠀⢻⣏⣧⣧⠰⡆⠸⣟⡄⢹⣿⣿⣷⡀⠙⠷⣦⣌⣻⣿⣿⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⠉⠉⠋⠀⠀⢹⠁
⠀⠀⠀⠻⣿⣿⣁⢿⣄⢻⡄⠀⠀⠙⣿⣿⣿⣾⣿⣿⣅⠀⡩⠗⠒⠛⠢⣄⠀⣈⣀⣀⣀⣀⣰⠁⠀⠀⠀⡀⢸⣧⡄⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠻⢿⣿⣿⣷⣽⡟⠑⠲⠎⣹⡟⠝⢻⠏⠁⠈⠻⠆⠅⢀⠀⠀⠘⠋⠀⠀⠀⡁⠀⠸⠾⠽⠧⠥⢬⣔⣉⡀⠀⠀⠄⣪⠏
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢡⠀⠀⠀⡏⢧⠀⠀⠀⠀⠔⠁⠀⠀⠀⢣⣀⣼⣤⣶⣦⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⡎⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⣢⠄⣀⡼⢌⡢⣄⠀⢸⡄⠀⠀⠀⠀⠈⠀⠈⣿⠀⠱⠿⣷⣶⣄⣀⣀⣀⣄⣠⣤⣤⡴⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠀⠙⠓⣄⡰⠚⠤⠤⢄⣿⣷⣶⠾⠫⠿⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠐⠋⠀⠸⠆⠈⠀⠂⢲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⠂⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠈⠼⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⡴⠒⢒⡶⡤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠘⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠀⠀⠀⠀⠳⠄⡸⣰⠏⠀⠀⠀⠀⠀⢰⡏⢠⡟⠀⠀⣰⠟⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠨⠀⠀⠀⠀⣀⣠⢧⡟⣇⢼⣿⣱⢟⣧⡿⣁⡿⢰⠷⣷⣏⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠒⠂⠤⠄⠒⠒⠀⠇⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⡏⠓⣿⢳⢀⣿⠈⡏⠁⡏⢸⡏⢸⠀⡇⡇⠀⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⢳⢹⣏⢸⢻⡆⡇⠀⡏⢸⡏⢸⡀⡇⡇⠀⠀⠀⠀⠀⠘⣠⡄
⠀⠀⠀⠀⠀⠀⠘⠁⡉⣀⠀⠀⠀⠀⠀⠀⠙⢋⢌⠈⠘⣀⡃⠁⠀⠛⠈⠁⠀⠙⡡⠟⠃⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠀⠀⠀⠀⠀⠀⠀⣎⣜⠀⡜⡡⠁⠖⠁⠐⠢⢄⣀⣞⠠⠂⠀⠀⠀⠀⠀⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠂⠀⠀⠀⠀⢸⡁⢨⣾⠊⣱⢀⣖⣠⣒⡶⣠⣶⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡔⠀⠀⠀⠀⠣⠤⠊⠘⠖⠉⠁⠋⠘⠁⠈⠁⠈⠒⠁⠀⠀⠀⣀⠠⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠤⢄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠆⠁⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠀⠠⡄⠠⠤⣀⠁⠀⠠⡔⠉⠁⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠆


⠀⠁
⠀⠅

⠠⡆⠈⠑⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠂⠒⠂⠊⠇⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠊⠀⠀⣀⡈⣐⡀⠤⠀⠤⠀⡀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠄⣀⡀⠀⡀⠀⣀⠀⡀⠄⠊⠉⠉⠁⠑⢄
⢪⣴⣤⡊⣠⡀⠀⡌⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠑⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⣤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠈⠓⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜
⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡠⠠⠄⠠⡄
⠀⠀⠀⠉⠉⠟⠂⢄⠀⠀⠀⠒⠒⠠⢄⠀⠀⠀⠀⢀⡤⠒⠤⠠⢲⠀⠒⠤⠔⠂⠐⠤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠉⠀⠀⠀⠀⠁⠢⠤⠐⠉⠀⠀⠀⠀⣄⠞
⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠀⠀⠀⠐⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⡰⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡈⢈⠁⠀⠐⡈⠁⠀⠀⠃⢠⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⡠⢄⡀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠉
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠣
⠀⢠⠄⠠⣀⠄⠒⠒⠤⣰⡄⠀⠀⠀⡐⣉⠐⠀⠀⠀⠂⢤⠈⡀⠁⠢⢄⣀⣤⣀⠄⠊⠀⠀⠀⠀⠇⠀⣠⢒⠂⠄
⠀⠐⡌⢣⡥⣀⠀⠀⠰⢁⠀⠑⡤⠊⢰⠍⡰⠊⠐⠠⡧⠒⡃⠐⠔⠊⠀⠀⠌⢀⠄⠳⡂⠁⠀⠈⠑⣌⠠⠎⠱⠂
⠀⠀⠈⣆⢘⣃⣀⣉⣹⣀⣀⣀⣏⣬⣓⣀⣇⣀⣀⣤⣣⣠⣩⣌⣦⣄⣠⣀⣰⣁⣀⣀⣽⣂⣤⣀⣀⣈⣓⣰⠁
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁






⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⢰⠖⠒⠛⠀⢂⠠⡀
⠀⠀⠀⠀⠀⣴⠚⠁⢈⣤⣤⢤⣤⢉⣉⠀⠗⠄
⠀⠀⠀⢠⠈⠁⢤⣾⣯⡻⣿⣟⣿⣿⣿⣷⣄⢈⢄
⠀⠀⠀⠿⠀⢠⣿⣿⣿⣿⣿⡿⢛⣩⣿⣿⣿⣦⣼⣤⣴⣤⣾⣤⣴⣦⣴⡀⣀
⠀⠀⠀⡃⢰⣿⣿⣿⣾⡿⢋⣿⣿⡟⣿⢟⣼⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⡠⠒⠒⠒⢄
⠀⠀⠸⠁⢸⣿⣿⣿⣿⣧⣿⠟⣿⣸⣿⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⡇⠀⠀⠲⠘⣄
⠀⠀⢀⠆⢼⣿⣿⣿⡏⣿⣿⢸⡟⣿⣿⢻⣿⣿⣿⣏⢽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⠀⠣⡀⠰⣄⣄⡿⠆
⠀⠀⢬⢃⡸⣿⣿⣿⣷⣿⣿⣾⣇⢻⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣏⣾⣿⣾⣿⣿⣿⣿⣷⣄⢱⡀⠀⢸⠇
⠀⠀⢰⣣⠐⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠘⡄⠆
⠀⠀⠈⢿⠄⠙⣿⣿⣾⣿⣿⣻⣿⣿⣯⣿⣿⡻⣿⣿⣿⣿⣿⡙⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⢸
⠀⠀⠀⢪⢰⡀⢿⣿⣿⣷⣿⡿⡟⢿⣿⣿⣻⣷⣿⣿⣯⣹⣿⣿⣏⠹⣿⣼⡿⣿⢻⡿⣿⣿⣿⣿⣻⣿⣿⣿⣏⠀⡸
⠀⠀⠀⠀⠓⢅⡀⣻⣿⢿⣿⣿⣿⣟⣿⣿⣟⣿⣯⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⡿⣿⣧⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠙⢏⣠⡿⠫⣹⣴⣯⣿⡽⣿⣿⢿⡻⣿⣿⡿⣷⣹⡷⣿⣿⣿⣧⢻⣏⣿⣸⣿⣿⣿⣟⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣾⣟⣿⡿⢿⡿⣿⢻⣯⣿⡟⣏⢿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣻⣷⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢸⡃⠰⢀⣧⠼⢷⡋⣽⡶⣏⠿⣮⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⢹⣅⠾⣤⠛⣆⡹⣧⠚⣻⣴⡛⣿⢿⣿⡿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⣁⣆⣼⠤⣿⠊⣻⣤⠊⣿⣼⣿⣿⠿⢎⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣭⣯⣠⣷⣶⣉⣿⣤⣻⣾⣿⣿⣿⣁⣀⣜⣈⣓⣒⣒⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢠⡄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣞⢷⢿⣼⡟⡷⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢷⣰⣌⣿⣷⣿⡟⠈⠧⠀⢛⡱⣖⣤⣰⢿⣵⡶⣶⣴⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠻⡜⠏⠁⣶⢁⢠⣶⢼⣽⡇⢿⢹⡿⠸⠛⠳⢧⡝⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣾⣿⣤⠽⢷⣏⡿⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⡠⢿⣮⣳⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣙⢷⣿⣾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠟⢦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⢲⣦⠀⠀⠀⢠⢶⣦⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠙⠿⠟⠀⠄⠤⠘⠿⠟⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠈⠂⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⡟⣢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡏⠐⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⢶⣿⣗⣖⣾⢲⡖⠲⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠂⢀⢀⣀⣀⣈⢛⠻⣿⡟⢀⡠⠛⠁⠀⠛⠓⠒⢤⣤⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠮⠀⠏⠀⢉⣿⣧⡿⠗⣿⣿⣿⠀⠀⢸⣷⡦⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠒⠸⡟⠋⠅⠨⠭⢾⠿⣿⣿⠀⠀⢸⣿⣿⠿⣿⣿⣿⠃⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⣤⣿⣷⡆⢀⠄⢲⢲⠋⠀⠐⠀⠀⠀⠀⠀⢘⠋⣿⠇⠀⠀⠐⢠⣀⠀⠴⣿⣦⣄⡀⣼⣿⣧⣤⡤⡎⠉⠀⠀⠀⠀⠀⠈⠲⣶⣶
⠀⠀⠀⠀⠀⣶⠋⠁⠀⢰⣾⣿⠋⠀⠀⠀⠘⣾⣶⠂⠒⢲⡀⠘⠒⠀⣬⣵⡇⢀⣐⠀⢀⠘⠛⣷⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⡿⡿⢻⠉⣿⡟⡟⢻
⠀⠀⠀⠀⠐⣿⠀⠀⣠⡼⢿⣿⠀⠀⠀⠀⠀⢍⣉⣐⡀⠶⣷⣖⣒⣖⣚⣿⣇⣻⡇⠀⣤⣦⡀⣿⣷⣿⠯⠌⡻⣏⣀⣀⣉⣿⡀⣶⣰⡄⣦⢀⣤⣗⣧⣧⣸
⠀⠀⠀⠀⠀⣿⣶⠶⢻⣧⣿⣃⣀⣀⣀⣵⣶⣶⣾⣿⡉⣽⣿⣷⣠⣾⣿⣟⣸⣿⣷⡄⠛⢿⣿⣾⣿⣯⣤⣤⣄⣉⣹⡀⣺⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⠿⠯⠿⠯⠿⠯⠿⠿⣿⣭⣿⣽⡏⠘⡿⣿⢹⣿⣿⣇⣿⣿⣿⣾⣿⣿⣯⣿⣿⣻⣿⣶⣶⣾⢹⣿⡏⣿⢸⣧⣷⣿⣶⡟⣹⣿⣟⣿⣻⣟⣟⣻⣿⣿⣿⣿
⠀⠐⠀⠀⠀⠀⠀⠈⠀⠁⠉⠁⠁⠀⠀⠀⠁⠀⠋⠀⠀⠈⠉⠉⠉⠉⠁⠈⠉⠁⠈⠈⠈⢀⠀⠈⠉⠁⠈⠈⠁⠁⠉⠈⠁⠀⣸⠈⢈⠀⡀⢀⢀⣇⣀⣀
⣤⣭⣭⣤⣬⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿