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⢠⠉⠁⠒⠤⠄⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠠⠤⠒⠉⢡
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⢱
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⡜⠀⢀⣀⠠⠤⠤⠤⠤⠄⢀⡀⡈⡈⠀⢇
⢠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠈⡆⠀⠀⠀⠀⠀⡠⠺⢍⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡵⢤⡀⠀⠀⠀⠀⠀⡆
⠀⠘⢄⣀⡀⠔⠊⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠈⠢⢄⡀⢀⠜
⠀⠀⠀⠀⠀⠀⠒⢲⠓⠂⠤⢀⡀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠘⡆
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠑⠤⣀⠀⢈⣛⣀⣛⣁⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⣘⣀⣀⣀⣀⣀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠉⠓⢄⢀⣀⣀⡀⠀⠀⠀⢀⣣⣀
⠀⠀⠀⠈⠉⢠⠃⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢹⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊

⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⢸⡇⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⣿⡇⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⣿⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⢀⣀
⢹⡇⠀⠀⠀⢸⡿⣦⡀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⢀⣴⡿⣿
⢸⣷⠀⠀⠀⢸⣧⠈⠻⣦⡀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣠⡾⠋⢰⣿⠀⠀⠀⢸
⠀⣿⡀⠀⠀⠸⣿⠀⠀⠈⢿⣄⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢀⣾⠏⠀⠀⢸⡟⠀⠀⠀⣼
⠀⢻⣇⠀⠀⠀⣿⡄⠀⠀⠀⠹⣧⡀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣠⡿⠁⠀⠀⠀⣼⡇⠀⠀⢀⣿
⠀⠈⣿⡀⠀⠀⢹⣇⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣴⡟⠀⠀⠀⠀⢠⣿⠀⠀⠀⣼⡏
⠀⠀⠹⣧⠀⠀⠀⢿⡄⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣼⠇⠀⠀⢰⡿
⠀⠀⠀⢻⣆⠀⠀⠘⣷⡀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⣿⠁
⠀⠀⠀⠀⢻⣆⠀⠀⠹⣷⡀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢠⡿⠀⠀⠀⠀⠀⣰⡿⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠹⣷⡀⠀⠀⠀⠀⢿⡄⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣾⠇⠀⠀⠀⠀⣰⡟⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠘⢿⣄⠀⠀⠀⢸⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢰⣿⠀⠀⠀⢀⣼⠏⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠻⣷⡀⠀⠈⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣴⡿⠃⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⣿⡆⢀⣀⣽⣧⣤⣴⣶⣶⣤⣤⣿⣄⣀⠀⣿⡇⣠⡾⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠈⣻⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⡋⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠠⣶⡶⠶⠿⠟⠛⠛⢛⣿⠿⠿⠶⢶⣶⣤⣤⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠾⠿⢿⣿⡛⠛⠛⠿⠷⠶⣶
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣿⡋⠉⠉⠉⠉⠉⠉⣻⣧⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣾⣿⣦⡀⠀⠀⣠⣾⣿⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⣾⣇⣠⣴⠶⠟⠛⠋⠉⠉⠀⠀⠀⠀⣀⣤⣴⠶⠿⢷⣤⣴⠿⠷⢶⣤⣄⡀⠀⠀⠀⠈⠉⠉⠛⠛⠷⣶⣤⣀⠀⠘⣷
⠀⠀⠀⣠⣴⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠋⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣿⡀
⠀⠀⠘⠋⠀⢻⣧⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠠⣶⣤⣄⣤⡾⠛⠿⣦⣄⣤⣴⡦⠀⠀⠀⠉⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⣴⡟⠛
⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢾⠏⠀⠈⠙⠻⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠛⠋⠉⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠷⠶⢶⣶⣤⣤⣤⣤⣤⣴⣶⠶⠾⠿⠛⠛⠉⠉

⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⣿⡂⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢋⠿⡗
⠀⠀⠀⠀⣏⠁⠀⠀⠙⠂⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠀⠊⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠉⠐⠢⢄⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠂⠉⠀⠀⠀⠀⠀⣴⠇
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠺⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠂⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⢀⠠⣄⠀⠀⡈⠃⠀⠀⠐⠀⢧⣀⠀⠀⠀⠀⣠⣼⣷⠀⠀⠀⠘⠁⠀⠀⠀⠀⡠⠀⣨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡌⢳⡄⢣⡀⢷⠀⠀⠀⠀⠀⣌⣿⣿⣷⣶⣿⣾⣿⣧⡀⠀⠀⠀⡿⠀⠌⢠⠟⣡⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣷⡀⢻⣎⠀⠀⠀⠀⣸⡿⠋⠉⣍⢋⠙⡟⢿⣟⡇⣀⠀⠀⣳⠏⢠⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢷⣌⠹⠃⢤⡎⠉⠀⠀⠀⠀⠀⠀⠑⠁⠀⠈⠛⢿⡆⢾⠏⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⠟⠀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠆⣀⠈⢿⣧⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⠋⠀⠠⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢠⣶⣦⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡴⣷⡀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡿⠁⢈⣷⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡁⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠟⠀⠀⠹⣿⠇⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠈⠿⠏⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠇⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢳⡟⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⣧⡄⠀⠀⠒⠂⠀⠀⠀⠤⠄⠸⢿⠿⠇⠠⠤⠀⠀⠀⠘⠀⠀⠀⠐⣟⣻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡗⢿⣷⠀⠀⣘⡁⠀⠀⠈⠀⢦⣄⢺⡆⣠⣤⠀⠀⠀⠀⢘⣇⠀⠀⣰⢧⢺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠹⡿⡃⠀⠀⢆⠀⠀⠺⣀⣤⣉⣿⣿⣁⣀⣠⠖⠀⢀⡴⠆⠁⢲⣿⠏⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠦⠀⠛⢻⡶⣀⡀⠀⠀⠀⠉⠙⠛⠛⠛⠟⠋⠉⠀⠀⢉⠀⣤⢾⡟⠃⠀⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠺⠀⠀⠈⠛⠹⠦⢺⡇⠄⠀⠀⠠⠀⠠⠄⠀⠀⠰⢾⡟⠶⠏⠞⠁⠀⢸⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⡶⠛⠛⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠳⣦⡀
⠀⠀⣼⠋⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠻⣆
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢠⡟⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⢳
⣸⠇⠀⠀⣰⠟⠉⠀⠉⠻⣦⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⡶⠋⠁⠀⠉⢷⡄⠀⠀⢸⡆
⣿⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⢳⡄⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣰⠏⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⡇
⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢀⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⢹⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣾
⠀⣿⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡟
⠀⢹⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⣧⣤⣤⡤⠶⠶⠶⢤⣤⣤⣤⣽⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠃
⠀⠀⢿⡀⢻⡀⠀⠀⠀⠀⢀⣠⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⢰⠇⣰⠏
⠀⠀⠈⢷⡈⢷⡀⢀⣤⠞⠋⠁⠀⠀⠀⢠⠀⠀⠀⣿⠀⢠⡀⢻⡀⠀⣄⠸⠆⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⣰⡟⢠⡟
⠀⠀⠀⠈⢷⡈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠀⠘⣃⠀⠀⠀⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠏⣠⠟
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠿⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠛⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⣀⣤⡀
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠛⠋⠉⣼⠃
⠀⠀⠀⠀⠀⠈⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠉⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢿⡀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⣿⠀⣀⣤⡶⠛⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠉⠛⠛⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠛⠛⠉⠁⠀⠀⠀⣰⡟⠛⠉⠁⠀⠀⠀⠀⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉⠙⠳⠶⠶⠶⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⡀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢸⡷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾
⣾⠷⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠾
⠸⡆⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⢰
⠀⣷⠀⠀⠸⡟⠶⢤⣀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⣀⡤⠶⣻⠃⠀⠀⡾
⠀⢸⡆⠀⠀⢻⡄⠀⠈⠛⠦⣄⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⢀⣠⠶⠛⠁⠀⢠⡏⠀⠀⢰⡇
⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⠾⢋⠔⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⡿
⠀⠀⢸⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⡼⠃⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡇
⠀⠀⠘⣇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣸⠁
⠀⠀⠀⢿⡀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⡿
⠀⠀⠀⠘⣇⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⢢⡀⠑⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣿⣀⣤⣤⣤⣤⣤⣤⣀⣿⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠙⠦⣄⠀⠀⠰⣶⡖⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⢲⣶⠆⠀⠀⣠⠖⠋⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠈⠙⠶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠶⠋⠁⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⣠⣤⠴⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣤⣄⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠏⡽⠁⢀⣾⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⣇⠘⢯⠳⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⠃⢠⣿⣿⡦⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡶⠾⡇⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢿⠀⠸⣿⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⣿⣿⠃⠀⡗⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣀⠀⠀⠀⠀⠸⣆⠀⠈⠁⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠈⠁⠀⣰⠇⠀⠀⠀⢀⣀⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠙⠳⢤⡀⠀⠉⠛⠒⠛⠉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠛⠒⠛⠉⠀⢀⡠⠚⠉⠉⠉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠈⠉⠙⢶⡄⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⢤⣞⠶⠀⠀⡞⣳⡴⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢲⡖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠹⡿⣆⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⢀⣠⢿⡿⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠚⡟⠁⠸⠇⠈⢻⠓⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠐⠲⠤⠤⠖⠂⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⠶⠛⠉⠉⠁
⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠙⣆⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠋⠀⢷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⣸⢁⠇⠀⠀⣿⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠸⡀⠀⠀⢧⢸⡀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢹⠀⠀⠈⡇⢿⠀⠀⠀⠀⠀⢸⢰⠁⠀⠀⡜⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⡆⠀⠀⡇⢸⠀⠀⠀⠀⠀⡇⢸⠀⠀⢠⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢱⠀⠀⢱⢸⠀⠀⠀⠀⠀⡇⣜⠀⠀⡾⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡆⠀⢸⢸⠀⠀⠀⠀⠀⡇⡏⠀⢠⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⠀⢸⠀⡀⠀⠀⠀⢠⠃⡇⠀⡜⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⡆⢸⣀⣇⣀⣀⣀⣘⡀⡇⢰⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠞⠉⠁⠀⠀⠀⠀⠈⠉⠓⠮⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣼⡷⡄⠀⠀⠀⣰⣿⣆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠘⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠉⠑⠲⢤⣀⠀⠀⣠⠴⠦⢤⠀⠀⢀⣠⠴⠒⠉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀⠠⢀⣀⣀⠑⢦⣱⡀⠀⡰⣠⠖⢉⣀⣀⡀⢀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠻⢧⡀⠀⠀⠀⠀⠀⠉⠱⣟⠛⠋⢚⡗⠉⠁⠀⠀⠀⠀⢀⡴⠟⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡄⢀⢀⡀⢱⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡎⢀⠀⡀⢠⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠼⣇⣼⣠⣧⠼⠉⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠜⠧⣸⣄⣧⡸⠧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠉⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠂⠈⣧⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣸⠁⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣧⣀⣤⣤⣤⣄⣀⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⢀⡤⠞⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⠀⢠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢻⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠃⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠸⡇⠀⠀⠀⠀⠀⢀⡴⠋⢀⡞⠁⡀⠀⢠⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣦⠀⠀⣀⠙⣆⠈⠳⣄⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠻⣄⣀⣀⣠⠴⠋⠀⢠⡟⠀⠀⠙⠷⠀⠇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠇⠰⠞⠁⠀⠘⣧⠀⠈⠳⢤⣀⣀⣀⡼⠃
⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡴⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠶⣄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠾⠁⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠸⠆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣤⡤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣏⣀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⣈⣇⡀
⠀⠀⠀⠚⠛⠛⠉⠉⣿⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢹⡏⠉⠙⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⣼⠁
⠀⠀⠀⠀⠀⣀⡤⠖⠋⢷⡀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣰⠏⠛⠦⣄⡀
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⣹⢾⣁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣹⢾⡁⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠙⠛⠒⠲⠒⠒⠛⠉⠙⠓⠶⠖⠛⠉⠙⠓⠒⠶⠒⠒⠛⠉⠁⠀⠙⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⢠⠃⡆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⡆⢣⠀⠀⠀⠀⠀⡎⢰⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢇⠘⡆⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠸⡄⡇⠀⠀⠀⡏⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀⠀⠀⠀⠀⡇⣧⠀⠀⠀⡇⠀⢸⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢡⠀⠀⠀⠀⣇⢻⠀⠀⢰⡇⠀⢸⡄⠀⢀⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⣯⢸⠀⠀⢸⣿⣢⣸⡇⣄⡾⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣟⣿⣛⡛⠲⠷⣿⣛⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⡋⠩⠋⠋⠄⠀⠉⠈⠢⡙⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠎⠀⠀⡀⠀⠀⠀⠀⠀⠐⣴⣦⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⣿⠨⠂⠀⠀⣤⠀⠀⠀⠘⢼⣿⡾⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠓⠬⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⠀⠀⣃⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠒⡀⡀⡖⠠⠀⠙⠮⣍⡴⠟⠀⠀⢠⣄⡀⠠⠤⢴⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢿⣿⣶⣶⣿⠐⠒⠄⠀⣈⡀⠀⠀⠀⠘⠂⠁⠍⢭⡯⠉⡉⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠙⠻⢿⣿⣷⣖⠒⠊⠀⠈⠙⠒⣋⣾⣯⡾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣋⠉⠉⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⢠⣄⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⣿⣿⣦⡀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣿⣿⣿⣿⣄⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠁⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣿⣿⣿⣿⣆⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⣾⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⣯⢿⣿⣿⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠇⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠘⣎⢿⣿⣿⣇⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⣿⢫⢷⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠙⣦⢻⣏⣿⠀⣸⠀⡐⢛⠀⢲⢂⡏⢰⣿⣿⣷⠋⠀⢹⣶⣄⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢮⣳⣿⡿⠀⠛⠋⠁⠀⠀⠉⠉⠓⠺⠿⠛⢁⣠⠾⠋⠀⠈⠛⠳⠶⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡞⣻⣧⠀⠀⠀⠀⣔⢹⣷⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⡀⣿⣿⡟⠀⢀⡀⠀⢻⣿⣿⣀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠉⠁⠀⠀⠀⢴⣿⣷⣾⣿⡆⠀⠀⠀⠈⠉⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢀⣴⠒⠀⠀⠀⠈⠻⢿⡿⠟⠀⠀⠀⠀⠒⣦⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠈⢧⣀⠀⠀⠀⣠⢾⣷⣄⠀⠀⠀⣠⡾⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠙⠛⠛⢿⡁⢸⠇⢨⡟⠛⠛⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣼⠁⠀⠀⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠄⠄⠀⠄⠀⠤⢠⡄⠠⠠⢠⠤⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡤⣄⣠⣄⢠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢶⣙⣧⠞⠵⡏⢬⣸⠄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⡀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⡀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢏⠉⠑⠦⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⠔⠉⡽⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣑⠦⣀⠈⢢⡀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡰⠃⢀⠞⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢮⡓⢄⠙⢆⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡜⠁⡰⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠱⡌⢣⠹⡀⠀⠀⠀⠀⠀⠀⢠⢏⡞⢀⡞⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⣆⢣⢳⡀⠀⠀⠀⠀⠀⣞⡞⢠⢎⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡜⡆⢧⣧⣀⣀⡀⠀⢸⡿⢠⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠊⠉⠁⠀⠉⠉⠉⠓⠮⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣠⠶⢄⠀⠀⠀⠀⢀⡤⣀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠻⣿⠿⠀⠀⠀⠀⢺⣾⣾⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⡠⠒⠛⠒⡄⠀⠀⣀⣤⣄⠀⠉⢀⠤⠖⠦⣄⠀⠉⠉⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠳⠤⠤⠔⠃⠀⠀⠳⢤⠞⠀⠀⠘⢤⣀⣀⠜⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠉⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⣸⣏⣀⣀⣀⣀⣀⣀⣹⣇⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠠⠿⠿⠟⠛⠛⠛⠛⠻⠿⠿⠄⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢶⣤⡀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⢀⣤⡶⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠂⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠐⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⠀⢀⡾⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡍⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠑⠛⡇⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠐⠶⣦⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠈⣿⡤⠔⠀⣈⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠠⢶⣾⣯⠙⠿⠿⠒⠢⢤⣤⣽⣿⣿⣾⣛⣿⡦⢀⣼⣿⡆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣂⠈⢀⣩⣿⣿⣿⣿⣿⡍⠙⢿⣿⣿⣿⣇⡔⠀⢀⠠⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡟⢻⣿⣿⣿⠏⠀⢀⣾⣿⣿⣿⣿⣿⡟⢛⠤⣀⣄⣈⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣅⣈⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⠻⠿⠟⠀⠀⠘⢿⣿⣿⣿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠪⠻⣧⠠⠀⠀⠀⠠⢆⣈⣿⣿⣿⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⠄⠀⡜⢢⠀⠀⠀⠙⠻⢩⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠄⠀⠘⠁⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⡃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⣿⠀⠁⠘⣷⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⣠⣿⣿⡿⠇⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⠁⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠙⡄⠀⠿⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⡏⡂⠀⠀⠈⠻⢇⡀⠘⢀⡿⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠢⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠑⢤⣼⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⡳⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢊⣿⣿⣿⠷⢶⣦⣀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢱⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⣾⠟⠋⠁⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢷⣄⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢷⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⢾⡿⠿⢿⣿⣟⢿⣿⣿⣿⡟⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⢍⠙⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠓

⠀⠀⠀⡄⠘⢣⠋⡀⡇⡀⠀⠀⠀⠀⠀⠠⢊⡥⠦⡄⠁⠠⢀⠀⠀⠀⠀⣠⠴⢄⡋⠙⣆⠤⣄⠀⠀⠀⠀⠀⢀⢄⠸⠑⠘⠂⣠⡄
⠀⠀⠓⣒⠌⣁⣄⡀⠡⢀⠤⠀⠀⠀⠄⠀⣸⣜⣱⣼⡡⡙⣆⠂⠀⠀⢀⡹⣄⣀⣣⣴⠃⣀⣼⡀⠀⠀⠀⠐⠍⠀⠱⣄⣤⠎⠁⠱⠄
⠀⠀⣫⠄⡀⢿⣼⡇⠨⠍⠒⠀⠀⠀⠀⡞⡩⢹⣿⣿⢶⡵⠋⡀⠀⠀⠸⣀⣀⠬⢯⣿⠯⣀⡀⣸⠀⠀⠐⣇⠀⠚⢻⣿⢿⡋⠐⢒⠦
⠀⠈⠁⡌⠴⡤⢠⠠⢙⡘⠃⠀⠀⠀⢁⡳⢬⠽⣏⢤⣦⠆⡀⠀⠀⠀⠀⢞⠁⢀⡎⠘⣆⠀⣹⠀⠀⠀⠀⠛⠀⡠⠀⢸⠀⠙⡀⠼
⠀⠀⠀⠀⠋⠀⠂⠱⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠈⠉⠁⠊⠀⠀⠀⠀⠀⠀⠉⠁⠣⠤⠇⠉⠁⠀⠀⠀⠀⠀⠀⠓⠳⠎⠒⠋⠁
⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢠⣾⣶⠻⣿⢷⣾⣲⠀⠀⠀⠀⠀⡖⡃⡆⠆⠆⡥⡓⡆⠀⠀⠀⠀⢀⡴⠋⠉⠉⠉⠳⢆⠀⠀⠀⠀⠀⠀⠀⡏⢠⠙⡆⢀⣀
⠀⢀⣼⣏⣻⡶⡻⣾⣋⣽⣦⠀⠀⠀⡸⣰⠉⣞⣿⣿⣮⢐⡚⡇⠀⠀⠀⡾⠀⠀⢀⢤⠀⠀⠈⣧⠀⠀⠀⡎⣉⡒⢇⠉⣆⠗⣋⠍⡇
⠀⠘⠾⣟⣹⣿⣵⢿⣉⣻⠞⠁⠀⠀⢇⠼⡊⢿⣿⣿⡿⡒⢇⡇⠀⠀⠐⣗⠀⠀⠈⠚⠀⠀⠠⡟⠀⠀⠀⠑⠌⢙⢳⣶⣷⢚⡁⠔⠁
⠀⠀⠸⣿⡿⣴⣷⣬⣿⡿⠃⠀⠀⠀⠘⢪⣙⢰⢩⠱⣔⣩⠜⠀⠀⠀⠀⠘⢧⣄⣀⢀⢀⣤⠟⠁⠀⠀⠀⠀⢾⠠⠾⣻⡙⠳⢈⠆
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⣀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⢞⠁⠈⡷⠁⠀⡱⠀⠀⠀⠀⠀⣾⠟⠻⡇⢠⡟⠻⣳⠀⠀⠀⠀⣯⡖⠓⣿⣴⣿⠚⣲⣽⠀⠀⠀⠠⣎⠙⢿⡄⢸⠟⠉⡦
⠀⢠⠞⠛⠃⣴⣶⡄⠚⠉⢳⠀⠀⠀⣴⡺⠳⠦⣽⣾⡴⠞⠻⣦⠀⠀⠀⠙⠯⣶⣳⢿⣜⣶⠿⠃⠀⠀⠀⡶⠛⠳⢬⣷⣯⠶⠛⠓⡄
⠀⠘⠧⣤⠖⠛⣿⠋⠲⡤⠞⠀⠀⠀⠘⣳⠖⠋⡿⢻⡙⠳⣶⠛⠀⠀⠀⣴⡿⠻⣟⣿⣹⠟⣿⣆⠀⠀⠀⠙⡶⠛⢋⡟⣯⠙⠶⡖⠃
⠀⠀⠀⢧⣀⡠⢻⣀⣠⠇⠀⠀⠀⠀⠀⠳⠷⠾⡇⡨⡷⠼⠞⠀⠀⠀⠀⣯⣳⣖⡿⠙⢿⣲⣞⡽⠀⠀⠀⠈⠳⠴⢻⡇⢸⠷⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀
⠀⠀⠠⠀⣘⠃⢠⠛⢂⠡⠄⠀⠀⠀⢀⡐⡏⠈⣦⠀⢈⠀⠦⡄⠀⠀⠀⠀⢶⠄⣀⠋⣀⠤⠖⠀⠀⠀⠀⠀⣶⣫⠽⠞⠶⣭⢳⡆
⠀⠠⠘⢁⠈⣦⣷⡔⠊⠝⢠⠀⠀⠀⣏⠀⠈⢢⣼⣶⡁⠀⣠⠃⠀⠀⠀⣀⠤⠷⣼⣶⣧⠤⠤⣀⠀⠀⠀⣼⢽⠃⢀⣤⡄⠈⢻⢽⡄
⠀⠀⢠⣄⡠⠝⡿⢇⠀⢐⠔⠀⠀⠀⣤⠀⠀⢹⠿⠿⢏⠠⠈⡧⠀⠀⠀⠈⢉⠝⢻⠿⢟⠫⠀⠀⠀⠀⠀⢿⡽⣄⠈⠛⠃⢀⣸⣹⠇
⠀⠀⠐⠀⢐⡠⠁⢄⠇⡔⠂⠀⠀⠀⠙⠦⢔⠁⠀⠄⠄⡡⠒⠁⠀⠀⠀⠀⠚⠀⠈⡀⠀⠁⠒⠀⠀⠀⠀⠈⠻⣽⣓⡶⣖⣯⡷⠋
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⡿⢿⣿⣿⡟⠿⣿⡿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠁⠀⠀⠙⠻⣿⣷⣦⣍⣙⡒⠾⢭⣤⣈⡙⠲⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣝⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣬⣍⣙⠓⠲⢮⣭⣳⢮⣷⣵⡝⣿⣿⣿⣿⣿⣿⣿⣿⣦⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⢷⣒⣶⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⠓⠂⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣶⠈⠀⠹⣿⣿⣿⣿⣿⢧⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠰⠿⠋⢁⠀⣀⣀⣀⡀⢀⡠⠤⠒⠉⠁⠙⣿⣿⣿⣿⣧⡐⢀⣿⣿⣿⣿⣿⣷⡁⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢰⡶⠿⠗⠀⣠⠊⠉⠀⠉⠻⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠸⡟⠘⣿⡞⣿⣾⣿⣿⣿⣿⣿⣿⣷⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣷⠬⢄⣀⢠⣀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣀⡼⡟⢺⢋⠳⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⢃⠀⡇⠀⠙⠢⢄⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣆⠀⠈⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡇⡿⣸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢠⠉⠉⠑⡄⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡇⠁⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣏⠄⠀⠀⠈⢆⢈⡤⠴⠒⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡟⣦⠀⠀⠀⠀⠡⡄⠐⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡏⢿⣿⣿⣿⣿⡟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⠘⣷⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⢸⠻⣿⣿⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡇⠈⢿⣄⠀⠀⠀⠀⠈⠂⠀⠒⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣾⣿⣦⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡆⠉⠻⢷⣄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⢋⣽⢿⣻⣿⠟⠛⣿⠿⢿⣄⣄⠀⠈⣷⡄⠀⠀⠀⠀⢹⣷⣀⠀⠀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⢯⣿⣿⣿⢿⣿⡉⠉⠁⠀⢻⣷⢫⠶⣧⡀⠈⢻⡀⠀⠀⠀⠀⠘⣿⣶⢾⣯⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⣮⡙⣏⣧⣦⣼⣧⡀⠀⠀⢸⡿⠃⠀⣿⣿⡀⠈⣀⣀⣄⡀⠀⠀⢹⣿⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣾⣿⣿⣿⣿⡿⣧⡀⣠⣿⡀⠀⠀⣸⠟⠀⠀⢸⣿⡿⠁⢠⠀⠀⠙⠿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣴⣿⣿⣿⣿⣄⠹⡿⣿⢿⣷⠿⣷⣿⠜⠁⠀⠀⢀⣰⠇⠀⠈⡿⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠙⣿⢱⣿⣿⣿⣷⣿⣤⡾⢛⣷⢿⡿⠀⣰⡆⠀⣿⡏⠀⠀⣼⠅⠀⣸⠅
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢹⣿⢋⣈⡙⣿⣿⣍⣿⣯⣵⣿⣧⣼⠏⠀⣠⣿⣷⣤⣶⠟⠀⢰⣿⣴⡀⢘⠂⠀⠁
⠀⠀⠀⠀⠀⢀⣀⠀⢠⣤⣀⠀⠀⡉⠈⢻⠿⠷⣟⡻⣟⠋⠙⣿⣿⡏⣵⣾⣿⣿⠉⠁⠀⠀⠀⠀⣸⣿⣿⣿⡋
⠀⠀⠀⠀⡖⢀⣭⡆⢈⡛⢝⠀⠀⠀⠀⠉⠙⠻⣮⡟⠿⢿⡛⠉⣩⡶⡟⠉⠿⢿⡄⠀⢀⠀⢀⠴⠛⢿⣿⣿⠛⠓
⠀⠀⠀⠘⢇⠘⠾⠛⢀⠃⠀⠐⠄⠀⠀⠈⠛⠀⢸⣿⠷⠢⠷⠾⠿⠒⠀⠀⠀⠸⡦⢤⣴⡟⠁⠀⠀⢸⣿⣿⡇⢰⠀⢀⡠⠀⢀
⠀⠀⠀⠐⠚⠃⢰⣴⣿⣤⣿⣿⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠀⠀⠀⠀⣍⠛⣦⡸⠀⣉⠳⣔⣀⣱⣄
⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⡟⠀⠀⠀⠀⢠⣾⣿⣤⣿⣦⡀⠈⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁⠀⠁⠀⢸⣁⣷⠔⠀⢈⡤⣼⡙⡿⠿
⠀⠀⠀⠀⠀⠀⠀⡀⠙⢻⠏⠀⠀⡀⠄⠰⢿⣿⡀⠉⠉⢻⣿⣶⡀⠀⠀⠀⠀⢠⣾⣻⣾⡉⠀⠀⠀⠀⠞⠋⠛⠳⠀⠲⠄⠀⠁⠁
⠀⠀⠀⠀⠀⠊⠠⠍⣈⣧⠀⠀⣄⠀⢀⡠⢀⠉⠁⠀⠀⣀⠀⠉⣿⠀⠀⠀⠀⡺⠛⠃⠛⠁⠀⠀⠀⠀⠀⡇⠈⡻⠁⠠⠄⢠⣄⢛⣿
⠀⠀⠀⠀⠐⢀⡀⠀⠿⠟⠀⢰⡷⡀⠀⣤⣷⡀⠀⠀⡜⠗⠀⣿⣿⣶⡜⣣⣠⡗⣶⣴⡟⠀⠀⠀⠀⣤⣶⣿⡋⠑⠄⠠⡦⠈⣻⠛⠛⢠⣤
⠀⠀⠀⠀⣾⡯⢴⣿⠶⠂⠀⠀⠛⢿⣿⠝⣧⣈⣿⣶⡷⠛⡤⢞⡹⠑⠄⢹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⣼⠦⠀⠀⠇⠠⢠⠈⠉
⠀⠀⠀⠠⠿⣷⠿⠿⠄⡀⠀⠀⠀⠀⡄⠀⠉⣿⠏⣯⣡⣿⡀⠌⢐⢀⣄⣼⡏⢀⡀⠘⣿⡆⠐⣦⡀⠀⠘⣾⣯⣀⣈⠳⠶⠂⠀⢠⡴
⠀⠀⠀⢠⣴⣿⣿⣠⠎⣰⠀⣴⣷⣬⣁⢸⠏⡟⠠⢃⣽⣿⣧⣴⣾⣿⣿⣿⣷⣿⣿⣦⡉⠓⠸⠉⠿⠶⠿⠿⢿⣿⣷⣄⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠉⠉⠁⠘⠃⠘⠿⠏⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠁








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⣀⣤⣖⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣷⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣌⣿⣯⣟⣹⣬⣿⣷⣿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿⣿⠿⠛⠻⠻⠿⠿⠿⠿⣿⣿⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡧⢶⣶⣦⣀⣀⣠⣶⣶⣖⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣈⣻⣿⣿⡏⣒⡀⣿⣿⣿⣿⣿⣿⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡹⠟⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠦⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⣅⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣍⣁⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣆⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣿⣦⣀⠀⣰⣟⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠛⠛⠛⠿⣿⣦⡀⢀⣴⣿⠟⢛⣛⡉⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⢀⣾⣿⡿⠋⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢰⡇⠀⠀⠀⠀⠙⢿⣾⠟⣡⣾⡿⠋⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢸⣿⠟⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⣿⡄⠀⠀⠀⠀⠈⠟⢰⣿⠟⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢸⣿⡟⣹⣿⡇⠘⣡⣴⣿⣿⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⠻⣿⣄⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⢸⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣦⣈⠻⢷⣦⣀⠀⢸⡏⠀⠀⠀⠀⢀⣴⣿⣧⣤⣀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢟⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠉⠛⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣭⣙⢿⣷⣄⠀⣼⣿⠿⠛⠉⣠⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠹⣧⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⡹⣿⣆⢉⣤⣴⣶⣿⣿⣿⣿⡟⠁
⠀⠀⢀⣀⣀⣠⣤⣄⣰⣿⣿⠁⢀⣀⣀⣠⣤⣴⣾⠿⠋⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⡆⠉⣹⣿⣿⣿⡿⣿⣿⡄
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣉⣛⠛⢛⣋⣡⣴⠂⠀⠀⠀⢸⣿⡄⢶⣤⣀⣀⣀⣀⣠⣴⣾⣿⠛⢉⣾⡿⢋⣥⡶⠀⠈⢻⣿
⠀⠀⠀⠈⠻⣿⣷⣦⣄⣉⡛⠿⠿⠛⠛⠛⠟⠛⢿⣿⠀⠀⠀⠀⢸⣿⡇⢸⡏⢻⣿⡿⠛⠛⠛⠛⠿⣷⣾⡿⢠⣿⡟⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣷⣶⣶⣤⡀⠸⣿⡆⠀⠀⠀⢸⣿⢃⣿⡇⢸⣿⡀⣤⠀⠀⠀⠀⠈⠙⡇⢸⡿⠁⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠟⠋⠉⣀⣤⠀⠹⣿⣆⡀⠀⠿⣋⣾⡟⠀⠈⢿⣇⢻⣷⣄⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⣾⡿⢶⣶⣶⣤⣀
⠀⠀⠀⣠⣾⣿⣿⣿⡿⠟⠉⢀⣀⣤⣴⣾⣿⠏⢀⡀⠈⠻⣿⣶⣾⡿⠋⠀⠀⠀⠀⠹⣦⡙⠿⣷⣦⣄⡀⠀⢸⡇⠀⣀⣤⣤⣶⣶⣶⣦⣍⡻⣿⣷⣄
⠀⣠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⡿⠋⣴⡿⠿⣷⣤⡈⠻⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⠷⠆⠈⠉⠛⠛⣿⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠓⠘⣿⣿
⠰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⣼⡟⣱⠂⠀⠉⠻⣆⠀⠀⠀⢀⣀⠀⣴⣾⠟⠁⠀⠀⠀⠀⢀⣼⠏⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⢹⣧⡶⢛⣛⣛⣿⣿⠃⠀⠀⠀⠀⣠⣴⡿⠁⠀⠀⠈⣿⣦⠰⣤⣤⣤⣤⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣽⣇⢻⣷⠀⠀⠀⠘⣡⣾⡿⠟⠋⣿⣿⠀⣠⣤⣶⣿⠿⢋⣴⡆⠀⠀⠀⠘⣿⡆⢻⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠀⠀⠉⠀⠹⣧⠀⢀⣴⠟⠁⠀⠀⠀⢿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣇⠀⠀⠀⠀⢿⣿⢸⣿
⠀⠀⠀⠀⠸⣿⣿⣷⣶⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣿⣃⠀⠀⢀⣠⣴⡾⠋⠁⠀⢠⣄⡉⠙⠛⠿⣿⣦⡀⠀⠀⠼⢃⣿⡏
⠀⠀⠀⠀⠀⠹⣿⣮⡙⠿⣿⣿⣮⡻⠿⣷⣶⠶⠶⠟⠛⢹⡏⠉⠛⠷⣶⣬⡙⢷⣄⠀⠀⠘⣿⣿⣷⣶⣤⣌⡙⣿⣷⣦⣴⣿⡟
⠀⠀⠀⠀⠀⣀⣨⣿⣿⣷⣦⣬⡛⠿⢷⣶⣶⡞⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢿⡆⢿⡆⠀⢠⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⣠⣶⣿⣿⣿⣿⠿⠿⠿⠛⠛⠓⠀⢀⣿⠃⠀⠀⠀⣾⡇⣶⣄⠀⠀⠀⠀⠙⢸⣿⠀⣿⣷⠈⣿⣿⠙⠛⠻⠿⠿⠿⠿⠿⠿
⠰⣿⣿⣿⣿⣿⣷⣤⣤⣴⣶⣶⣾⠟⠀⠘⣿⡀⠀⠀⣼⡿⢁⣿⠻⢿⣶⣶⣶⣶⣿⣿⠀⢹⣿⣧⣽⣿
⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⢻⣷⣀⣘⣭⣴⡿⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⣦⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡦⠀⢀⣠⡾⠛⠛⠒⠒⠒⠚⡛⣦⣄⠲⢄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠒⠿⢶⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣿⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⢠⠚⠁⡄⠸⣇⠀⡀⠀⠀⠀⠀⠀⡠⠀⠐⠋⠀⠘⠁⠚⠿⢿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡄
⠀⠀⠀⠀⠀⠈⠄⡄⠰⠀⢛⣼⠀⠀⠀⠤⠂⠉⠀⢀⡀⢀⣀⣐⣄⣠⠀⠀⠈⠉⠻⠷⠀⠀⠀⢀⣀⣀⣀⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠾⣦⡳⡀⡟⠀⠀⠀⠀⢀⢴⠀⡀⠙⠿⢿⣿⣿⣽⣝⣢⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠙⠿
⠀⠀⠀⠀⠀⠀⢀⠠⠟⠿⣿⣿⠀⣴⠋⠀⡂⠀⠈⠉⠲⣶⣤⡝⠋⠃⠀⠹⠗⠒⠢⡀⠀⣀⣤⡶⢦⣤⣞
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠹⣇⢸⠇⣴⡹⠁⠄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣇⣀⣉⡿⠛⠐⣼⣿⠛⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣠⠎⠉⠉⢿⠀⠀⠀⠉⣤⣷⠀⠈
⠀⠀⠀⠀⠀⠰⡀⢆⠀⠀⢀⠀⡋⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣀⡴⠞⠡⠄⠀⠀⠠⠀⠀⠀⠀⣠⢏
⠀⠀⠀⠀⠀⠀⠙⢦⣥⣄⣈⢳⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠞⠁⢸⠂
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠈⠉⢿⡏⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢿⡆⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠶⠐⠆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡟⠀⢑⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠀⠀⡀⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢴⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡦⣤⣰⡿⠀⠀⠀⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡀⢁⠬⠁⠌⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣁⢰⣧⡞⠂⢤⠀⠈⠀⠀⠀⠀⠐⠖⠆⢸⡷⠋⣠⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢯⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢲⣄⡈⠳⣏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢷⠀⠀⢰⡇⠀⠀⠀⠀⠀⣾⢉⢻⣶⠿⠀⣀⣷
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠲⢿⡇⡀⠘⡟⠀⠀⠀⠀⠀⢹⣧⣾⠁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⡒⣾⣿⣀⣶⡇⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠻⠿⠋⡕⠀⠀⠀⠀⢀⠐⠉⡒⠖⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⢲⣆⠁⠑⣄⠀⠀⣾⡠⢠⣇⠨⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⠆⠀⢻⡆⠀⣿⢃⣾⠃⢸⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠰⡸⡄⠈⠟⠀⢹⣾⡏⢠⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣀⢷⡁⠀⢸⠀⠈⣽⡁⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⠏⠿⡀⢸⠀⠀⠙⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣙⣛⣛⣹⣯⣩⣉⣿⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣽⣭⣽⣿⣯⣭⣭⣭⣭⣿⣭⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠁⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠙⠀⣰⠉⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠆⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⢀⠉⠙⣗⠉⢀⡆⢳⣤⣠⣌⡴⡀⠘⡿⠎⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⡀⠀⡀⢴⢦⣿⡀⢴⡇⠈⣿⣿⣿⢁⣧⠐⣹⡴⠈⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⢢⣿⣼⣏⠈⣻⣿⣷⡀⢘⣿⡟⣿⡯⡶⠉⢀⣾⣿⣦⠔⢐⡊⠀⠈⠣⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠨⣓⣚⣿⣿⣿⣶⣭⡿⣿⣷⢴⠿⢯⣿⣿⢁⣴⣿⣿⣿⣟⠛⠁⠀⠀⠀⢀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢐⢤⡈⠉⠙⠻⢿⡿⡫⠚⠉⠈⠑⢌⢳⡿⡛⣹⡏⣙⣿⣿⡷⠄⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⡀⠪⠁⡀⣉⣶⣤⣄⣼⠕⠀⠐⠈⠀⢢⠀⣾⣿⣷⣿⣿⣉⣀⣹⣆⠘⣁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡋⢀⣶⡾⢿⣿⣻⣿⣿⣥⠀⠓⡔⡤⠞⢀⣿⣏⠀⠈⠈⠉⠙⠻⠍⠈⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣼⠀⠀⠀⢻⣻⣿⣾⣿⡿⠛⢹⣷⡵⣲⣌⣠⣤⣿⣿⡿⣿⣿⣷⣿⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠸⣀⠀⡀⠽⢿⣿⡿⠋⣀⣰⣯⣵⣾⣿⡇⠈⢿⣿⣷⠇⠽⣿⣿⣿⡙⠢⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠝⠒⣠⡞⠛⢩⢾⡿⣿⣿⣷⠀⢸⡟⠻⣿⣦⡼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠠⡟⠀⣁⣀⣼⣿⣿⣿⣿⣿⣦⣸⡿⣄⣸⣿⣍⠛⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⢒⠀⢸⡇⢆⠈⢹⠋⠼⡻⠻⡟⣿⣿⠏⠁⠀⢤⠔⠙⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⡆⠁⣺⠄⠈⠛⠒⠀⠀⠀⠀⠀⠀⡅⠀⠃⠀⠀⠘⠖⠀⣀⠠⠳⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠠⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⡠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡼⠈⢣⣸⠁⢸⡄⢀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⢤⣘⡌⠳⡇⠀⡎⡇⠀⠀⡷⠃⡼⢀⡤⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⢳⡀⠘⣴⡁⢰⠀⢸⠃⢠⠟⠉⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⢦⡀⠙⣄⠘⣇⢸⣀⡎⡰⠃⠀⣠⡮⠤⠴⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡶⡎⠙⠢⣄⣀⠙⠢⣌⣶⣿⣿⣿⣿⣷⡴⠊⠁⣀⣤⣾⢥⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⡀⡞⠀⢷⠚⠉⢉⣭⢿⠶⣾⣿⣿⣿⣿⣿⣿⣿⡂⠉⠁⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⢶⠲⢤⡤⣄⣇⠀⢿⠃⠀⢀⣤⡞⠋⢀⡾⢴⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡉⠛⠛⠲⠤⣀
⠀⠀⢠⡤⠤⣌⣧⡀⠙⢦⣻⡄⠘⡆⠀⣼⠏⢀⡠⠞⠉⢁⣠⡾⢿⣿⣿⣿⡟⢧⡈⠑⢯⡙⠳⢞⡋⠉
⠀⠀⠀⠳⢄⡀⠈⠙⠦⡀⠙⣿⣶⣷⣶⣿⣴⠋⢀⣠⣴⡫⠵⠷⢾⣼⡇⢹⡟⢆⠈⠢⣄⠈⢢⡤⠽⠂
⠀⠀⠀⠀⠀⠙⢲⣤⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⢀⣠⣴⢋⡟⡇⠀⡿⣌⠳⣄⠈⠳⣤⣹⡄
⠀⠀⣀⡤⠖⠛⠛⠉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⣉⠉⠉⠉⢉⣙⣷⠀⡇⠘⢆⠈⢳⠦⢼⣆⠁
⠀⠈⠙⠒⢲⡶⠖⠋⢁⡤⠚⣻⣿⣿⣿⣿⢟⠓⢄⡙⠪⣽⢿⢉⢯⢹⠻⣄⡇⠀⠀⠳⣼
⠀⠀⢀⣴⣋⣠⣤⠞⠁⢀⠜⠁⡌⠀⡇⠘⡌⢦⡀⠙⣤⣈⠙⢿⡚⠛⠀⠛
⠀⠀⠀⠀⠀⣴⢃⣠⠴⡏⠀⣼⠀⠀⡇⠀⢹⡄⠙⢦⡘⡏⢻⡉⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⣇⠜⣿⠀⢠⣇⠀⢸⠙⢦⡀⠙⡟⠶⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠏⢹⡄⡾⠀⠀⠙⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠚⣭⣶⣿⣷⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⣴⣾⣿⣷⡀⠈⣿⣿⣿⡇⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣛⢉⣩⣭⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⢛⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣰⣿⣤⠀⢀⣠⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠛⢛⣉⣉⣭⣬⣽⣿⣦⣀⣀⣀⣴⣿⣿⡿⣣⣾⣿⡿⢋⣿⣟⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣟⣵⣿⣿⡟⢠⣿⣿⣿⡏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣬⡙⢿⣿⣿⣿⣿⠿⠛⢛⣉⣉⣉⣩⣭⣥⣤⣶⣿⣿⣿⣿⣿⠟⢁⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢛⣉⣥⣴⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⢀⣶⣦⣬⡙⠿⣿⣿⣿⣿⣦⡹⣿⠿⠛⠻⠛⠛⠛⢛⣋⣥⣶⣿⣿⣿⣿⣿⣿⣷⣤⣬⣉⠁⣁⣴⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⣛⣉⠽⣟⣫⣴⣾⣿⠋⣾⣿⣿⡇⣴⣄
⠀⠀⢀⣴⣦⣄⣈⡙⠻⢿⣿⣿⣿⣦⡙⢿⣿⣟⠻⠛⠻⠿⠿⠿⢛⣫⣴⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⢸⣿⣿⡟⢠⣿⣿
⠀⢠⣾⣿⣿⣿⣿⣿⣶⣤⡙⠿⣿⡿⢻⣾⣷⡾⢿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠟⣛⣯⣥⣤⣴⣷⣿⣟⣋⣠⣴⣿⣿⣿⢁⣿⣿⡿
⢀⣙⣙⠛⠛⠛⢿⣿⣿⣿⢿⣷⣾⡆⣷⣿⣿⣴⣷⣿⠛⣿⣿⣿⣎⢻⣿⣿⣦⠀⠉⠉⠉⠉⠛⠛⠻⣿⣿⣿⣿⠟⢡⣾⣿⣿⠇
⣿⣿⡿⠿⠛⢛⣛⠟⣽⢃⣿⣿⣿⡇⢹⣿⣿⣷⣹⣿⡇⢹⣿⣿⣿⣆⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣙⡻⢧⣴⣿⣿⡿⠋
⠋⣡⣴⣾⣿⣿⢋⣾⡇⢸⣿⣿⣿⡇⢸⣿⡟⢿⣿⣿⣿⡈⢿⣿⣿⣿⣇⢹⣿⣿⡇⠀⠀⠀⠀⠠⡄⠘⠛⠁
⣿⣿⣿⣿⠟⣵⣿⣿⡇⢸⣿⣿⣿⠇⣾⣿⣇⠸⣿⣿⣿⣷⠈⣿⣿⠿⠟⠀⠉
⣿⣿⠟⢡⣾⣿⣿⡿⢀⣿⣿⡿⢣⣾⣿⣿⣿⠀⠿⠿⠟⠛⠁
⠟⢁⣴⣿⣿⡿⠏⣴⣿⣿⣿⠇⠘⠟⠛⠉⠁

⢠⣤⡀⠀⣠⣤⣄⡀⠀⢀⣄⠀⠀⢠⠀⠀⠐⢊⣍⠶⠀⠀⡂⠀⠀⠀⢀⣀⠀⠀⢠⣦⡀⠀⣠⣤⣤⡀⠀⢀⣀⠀⠀⢠⠀⠀⠰⢎⣍⠶⠀⠀⡆⡀⠀⠀⢀⣀
⠈⠋⢀⡞⠁⢸⠀⠙⡆⠘⠛⠁⠀⠀⣶⣶⣶⣦⣤⣤⡄⠀⠘⠫⠀⣤⣿⡟⠀⠀⠈⠋⠀⡞⠁⢸⠀⠙⡆⠘⠛⠁⠀⠀⣶⣶⣶⣦⣤⣤⡄⠂⠜⠫⢀⣤⣿⡏
⣀⠀⠘⢧⡔⢻⠳⣤⠇⠀⠀⠀⠀⢸⣿⣯⣿⣿⣽⣿⠃⠀⣀⠀⠀⢹⣿⢿⡷⠆⣀⠀⠈⢧⡔⢻⠓⣤⠇⠠⠀⠀⡃⢰⣿⣯⣿⣿⣽⣿⠇⠀⣀⠀⠀⢩⣿⢿⡷
⣿⡗⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠈⣛⠛⠛⠛⠛⠛⠀⢠⣿⣿⣇⠀⢀⠀⠀⢲⣿⡗⠀⠈⠙⠛⠛⠁⠀⠀⠈⠀⠀⠈⣛⠛⠛⠛⠛⠛⠀⣠⣿⣿⣇⠀⢀⡀
⠀⢁⡴⠖⠢⢄⠀⢀⣶⣷⣄⣠⡀⠀⠉⢀⡶⣟⠛⠳⣄⠀⠘⠋⠁⠀⡾⣿⠃⠈⠁⢁⡴⠒⠦⣄⠀⢀⣶⣷⣄⣠⡀⠀⠙⢠⡴⣟⠛⠳⣄⠀⠛⠋⠉⠀⡾⣿⠃
⠀⡏⣴⠛⢷⡈⣇⠈⢿⣿⣿⣿⠇⠀⠀⣾⠀⢸⣶⠤⣽⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡏⣴⠛⢷⡈⡇⠀⢿⣿⣿⣿⠇⠀⠀⣿⠀⢹⣶⠤⣿⠀⠀⠀⠀⠀⠀⡀
⠀⢣⡙⠄⠚⣠⠏⠀⠈⠻⠛⠁⠀⠻⠃⠘⠷⣿⣼⡶⠋⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⢣⡙⠤⠚⣠⠇⠀⠈⠻⠛⠁⠀⠿⠃⠘⠷⣿⣬⡷⠋⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠉⠛⠛⠁⠀⠙⣀⣤⣴⠶⢿⡀⠀⠀⠀⠀⠀⣰⣧⣤⡀⠀⡶⠆⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠙⣀⣠⣴⠶⢿⡀⠀⠀⠀⠀⠀⣰⣷⣤⡀⠀⡶⠆
⠀⠀⠀⠀⠀⣶⠞⣛⣩⣥⣶⣲⠈⢷⠀⠀⠀⢀⠘⠻⢿⣯⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠞⢛⣭⣥⣶⣲⠀⢧⠀⠀⠀⢀⠘⠛⣿⣏⠀⠀⢠⣤⡀
⠀⠀⠀⠀⣠⠹⡄⠳⢟⣛⣭⣵⣾⠿⠃⣿⣷⣿⡇⠀⣀⢀⡀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⣠⢹⡄⠳⢞⣛⣭⣴⣶⠿⠂⣿⣷⣿⡇⠀⣀⢀⡀⠀⠻⠿⠟
⠀⠀⡠⠶⣄⠀⢳⡾⠟⠛⠋⠁⠀⢠⣷⠎⠙⠛⠀⢠⣽⣿⣷⣟⠀⠀⠀⠀⠀⠀⠀⠀⡠⠲⣄⠀⢿⡾⠿⠛⠋⠁⠀⢠⣷⠊⠙⠛⠀⢠⣽⣿⣷⣟⠀⠀⠀⠀⢀
⣠⡊⠀⢄⣈⠳⣄⡀⠠⠄⢰⣿⣇⣠⡀⠀⠀⢀⠂⠀⠘⠛⠟⠉⠀⠰⠷⠀⠉⠀⣠⡊⠀⣄⣈⠳⣄⠀⠠⠄⢰⣿⣇⣤⠀⠀⠀⢀⠀⠀⠘⠛⠟⠉⠀⠐⠿⠀⠉
⠙⢿⣦⡘⠻⡷⠈⠻⣦⠀⠘⣿⣿⠿⠃⠀⠀⠈⠄⠀⢀⡀⠀⣰⣦⠀⠀⠀⠲⠀⠙⠿⣦⡘⠻⡷⡌⠳⣤⠀⠘⣿⣿⠿⠁⠀⠀⠈⠀⠀⢀⡀⠀⣰⣦⠀⠀⠀⠲
⠀⣀⠈⠻⣦⡈⢀⡴⠋⣤⣤⡁⠀⣤⣤⡄⠀⠀⠀⠃⠙⢿⣿⣿⣇⠀⢀⣀⡄⠀⠀⢀⠉⠻⣦⡀⢀⡴⠃⣤⣤⡉⠀⣠⣤⡄⠀⠀⠀⠃⠸⢿⣿⣿⡇⠀⠠⣀⡄
⠀⠛⠁⠴⠈⠻⠋⠀⠘⠻⠿⠀⠀⠙⠛⠃⠐⠉⡉⠑⠀⠈⠛⠈⠁⠀⠁⠋⠓⠀⠀⠛⠁⠴⠈⠻⠋⠀⠘⠻⠿⠀⠀⠙⠛⠋⠐⠁⡏⠑⡀⠈⠛⠈⠁⠀⠁⠏⠛
⢠⣶⡀⠀⣠⣤⣤⡀⠀⢀⣄⠀⠀⢠⠄⠀⠐⠬⡍⠶⠀⠀⡀⡀⠀⠀⢀⣀⠀⠀⢠⣶⡀⠀⣠⣤⣤⡀⠀⢀⣀⠀⠀⢠⠄⠀⠐⠌⣍⠶⠀⠀⠂⡀⠀⠀⢀⣄
⠈⠉⢀⡞⠀⢸⠀⠙⡆⠘⠛⠀⠀⢀⣶⣶⣶⣦⣤⣤⡄⠀⢜⠃⢀⣤⣿⡟⠀⠀⠈⠉⠀⡞⠀⢸⠀⠙⡆⠘⠛⠀⠀⠀⣶⣶⣶⣦⣤⣤⡄⠂⠙⠫⠀⣤⣿⡏
⣀⠀⠈⢧⡞⢻⠓⣦⠇⠠⠀⠀⠁⢸⣿⣿⣿⣿⣽⣿⠁⠀⣀⠀⠀⢹⡿⠿⡷⠆⣀⠀⠀⢧⡖⢻⠓⣤⠇⠀⠀⠀⡃⢸⣿⣷⣿⣿⣽⣿⠃⠀⣀⠀⠀⢩⡿⠿⡷
⣿⡗⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠈⣉⠛⠛⠛⠛⠛⠀⣤⣿⣿⣇⠀⢀⡀⠀⢺⣿⡗⠀⠀⠙⠛⠋⠁⠀⠀⠈⠀⠁⠈⣉⠙⠛⠛⠛⠛⠀⢤⣿⣿⣇⠀⢀⡀
⠀⢁⡴⠒⠲⢄⠀⢀⣾⣷⣄⣤⡀⠀⠉⢠⠶⣏⠙⠳⣄⠀⠘⠋⠁⠀⠾⣿⠃⠈⠀⢁⠴⠒⠢⢄⠀⢠⣾⣿⣄⣤⡄⠀⠉⢠⡶⣟⠙⠳⣄⠀⠙⠋⠁⠀⠾⣿⠃
⠀⡏⣴⠛⢷⡈⡇⠈⢿⣿⣿⣿⠇⠀⠀⢿⠀⢸⣶⠦⣾⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡏⡴⠛⢷⡌⡇⠀⢿⣿⣿⣿⠇⠀⠀⣿⠀⢸⣶⠦⣾⠀⠀⠀⠀⠀⢀⣀
⠀⢣⡙⠠⠚⣠⠇⠀⠀⠛⠛⠁⠀⠻⠃⠈⠳⣿⣼⠷⠋⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⢣⡙⠀⠚⣠⠇⠀⠈⠻⠋⠁⠀⠻⠃⠈⠳⣿⣼⠷⠋⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠉⠛⠋⠁⠀⢉⣀⣤⡴⠶⢻⡄⠀⠀⠀⠀⠀⣴⣧⣤⡀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⢉⣀⣤⡴⠶⢿⡀⠀⠀⠀⠀⠀⣼⣧⣤⡀⠀⠿⠆
⠀⠀⠀⠀⠀⣶⠞⢛⣭⣵⣶⣱⠈⢷⠀⠀⠀⢀⠘⠻⢿⠧⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⣛⣩⣵⣶⣹⠌⣧⠀⠀⠀⢀⠘⠛⢿⡧⠀⠀⣠⣤⡀
⠀⠀⠀⠀⣤⠸⡄⠁⢛⣋⣭⣴⣾⠿⠃⣿⣷⣿⡇⠀⣀⢀⡄⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⣠⠸⡄⠳⢛⣋⣭⣴⣶⠿⠃⢿⣷⣿⡇⠀⣀⢀⡄⠀⠻⠿⠟
⠀⠀⡠⠶⣄⠀⢳⡾⠟⠛⠉⠁⠀⢠⣷⠊⠙⠛⠀⢠⣽⣿⣷⣟⠀⠀⠀⠀⢀⠀⠀⠀⡰⠶⣄⠀⢳⡿⠟⠛⠉⠁⠀⢠⣷⠂⠙⠛⠀⢠⣽⣿⣷⣟⠀⠀⠀⠀⢀
⣠⡊⠀⢀⣈⠳⣤⡀⠰⠄⢸⣿⣧⣤⡀⠀⠀⢀⠀⠀⠘⠛⠟⠉⠀⠰⠿⠀⠉⠀⣠⡊⠀⢀⣈⠳⣦⡀⠰⠄⢸⣿⣧⣤⡀⠀⠀⢀⠀⠀⠘⠛⠟⠉⠀⠐⠿⠀⠉
⠙⠻⣦⡈⠻⡷⠈⠻⣦⠀⠘⣿⣿⠟⠁⠀⠀⠈⠔⠀⢀⡀⠀⣴⣦⠀⠀⠀⠺⠀⠙⠿⣦⡘⠻⡷⠌⠛⣦⠀⠘⣿⣿⠟⠁⠀⠀⠈⠄⠀⢀⣀⠀⣴⣦⠀⠀⠀⠲
⠀⣀⠈⠻⣦⡀⢀⠔⠁⣴⣤⠀⠀⣤⣦⡄⠀⠀⡐⠃⠘⢿⣿⣿⣇⠀⠰⢀⡤⠀⠀⣀⡈⠻⣦⣄⢀⡴⠁⣶⣤⡀⠀⣤⣦⡄⠀⠀⡐⠃⠘⢿⣿⣿⡇⠀⢐⢠⡄
⠀⠛⠁⠶⠈⠛⠋⠀⠘⠻⠟⠀⠀⠙⠛⠃⠐⠁⡍⠑⠀⠈⠛⠀⠁⠀⠀⠈⠓⠀⠀⠛⠁⠶⠈⠛⠋⠀⠘⠻⠟⠀⠀⠙⠛⠋⠔⠁⡇⠑⠄⠈⠛⠀⠁⠀⠁⠎⠛
⠀⠉⠁⠁
⠀⠐⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠑⠄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⠀⡀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⢱⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣄⠩⣽⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠃⠠⣤
⠀⠀⠀⠀⠢⠘⠀⠀⠀⠸⡀⠀⠀⠏⠖⠒⣀⣀⣤⣤⣤⣴⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠖⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠆⣀⣀⣀⣺⣿⣿⣿⣿⣿⠝⡛⢯⡬⣬⣭⡛⢿⣿⡏⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⠀⠄⠃⠰⠃⢠⠒⠛⠋⣽⣿⣿⣿⣿⡿⠴⠷⣶⠀⠀⢳⣿⣿⣿⡆⢹⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠤⠶⠂
⠀⠀⠀⠀⠄⠀⠰⡄⠀⠃⠀⠀⢸⣿⣿⣿⣿⣾⣿⣄⣸⡫⠀⢀⢘⣄⢠⣿⠃⡎⠀⡠⠀⡥⠊⠀⠐⠦⠄⡰⠈⠙⠒⠂
⠀⠀⠀⠀⢀⠈⠀⠑⡀⠰⠀⢀⣿⣿⣿⣿⡿⠟⠛⠻⣿⣷⣾⡇⣞⣿⣷⠎⣸⣠⠎⢀⠀⠠⠖⠋⠁⠀⡐⠀⢀⡈⠴⠊⠁
⠀⠀⠀⠀⠀⠑⠤⠀⠁⠀⠃⢸⣿⣿⠟⣁⣤⣾⣿⣦⠈⢻⣯⣄⡠⠁⠀⢰⡟⠁⠠⠁⡀⠴⣊⡤⠔⣉⠀⠀⠉⣀⣄⡀
⠀⠀⠀⠀⠐⠀⠐⠋⠵⢶⣄⣿⣿⣡⣾⣿⣿⣿⣿⣿⠇⣼⣿⣿⡅⠀⠀⣿⡇⠀⠠⠊⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡿⠟⠋⠉⢟⣿⡟⣰⣿⣿⣿⢧⠀⠀⣿⡇⠀⠀⠀⢀⠔⠀⠈⠒⢀⠒⠦⠖
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣼⣿⠟⠉⠁⢠⠄⠀⠰⠉⢁⣿⣿⡿⠁⢸⠀⠀⡟⠀⠀⠈⠀⠈⠷⡙⠦⢀⡀⠁⢄⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⢴⣦⣤⣴⡆⠀⠀⣾⣿⣻⣶⢦⠶⣧⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣷⣷⣿⣾⣿⣿⣿⠇⠂⡸⠿⠿⠿⠶⠶⠞⠛⣄⠀⠁⠒⢂⣴⣶⣶⣤⣶⣆⢀⢀
⠀⠀⠀⠀⢸⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⡁⠀⢀⣀⣀⣀⣀⣤⣤⣄⣀⡬⠳⠆⠚⠘⠟⠻⠛⠟⠉⠁⠁⠙⠛⠑
⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠈⠉⠙⠋⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡐⠐⢶⠄⠦⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠈⠡⡗⢤⢳⠄⣤⣱⠉⢨⠕⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⢹⣲⠇⠊⠀⠀⣼⣷⣦⣚⣙⡁⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⡖⣸⣤⡾⠎⠀⠀⠀⠙⢯⣟⡟⣷⣿⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠨⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠈⠉⠁⣶⠆⠀⠀⣶⡆⠁⢸⣼⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠠⢀⠄⠛⠳⣤⠴⡋⠗⢀⣼⡿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠗⡄⠇⠀⠀⠘⠀⠀⠁⠀⠘⢿⣷⣜⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠁⡄⢸⠀⠀⠀⠀⠀⢰⠀⡄⣎⣿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣏⠀⢄⠀⣸⠀⠀⠀⠀⠀⠸⡁⠀⠏⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠎⢈⠁⠀⠄⠀⠀⢀⠀⠉⠀⢲⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⠛⠃⠀⠂⠀⠀⠐⠀⠄⠀⠀⠘⣯⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠣⢀⣀⣀⣀⠀⠠⠤⠔⠓⠒⠒⠐⠒⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⢀⡀⣠⣄⣀⣀⠀⠀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⢋⣵⡉⢹⠈⢉⣹⠋⢉⣿⢉⠉⣿⡁⠀⠄⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⠀⠀⠀⠀⢸⡇⠀⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠛⡆⠀⠸⠇⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⡀⠀⠀⢙⡆⠀⢰⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⣾⣇⢀⣼⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠉⠉⠉⠁⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠀⠀⠀⠀⢀⣤⣄⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⣴⣿⣿⣿⢢⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠛⠟⠛⢁⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣀⣠⣤⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣴⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡈⠀⠀⠈⠉⠉⠛⠛⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠅⠀⠀⠀⠀⢀⣀⣀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠘⠃⠀⣀⠖⠊⠉⠉⢠⣿⣱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠋⠉⠛⣦⣤⡾⠂⢰⡇⠀⢀⡶⠾⢿⣧⣼⣿⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢹⣇⣀⣀⣈⣿⣷⡟⠀⠚⢾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢈⣻⣿⣿⣿⣿⣿⡃⣨⣿⣿⣿⣿⣿⣿⣷⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠀⠢⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠶⠶⠒⣾⣉⢁⣍⣽⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣿⣿⠸⡟⢿⣯⡠⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⡿⣿⣿⣿⡗⠲⢾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢂⣻⠟⣛⣈⣀⣀⡀⠀⠀⠹⡟⠟⣟⠱⠈⠙⠂⠙⣲⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠈⣮⣼⡟⠛⣿⣿⣿⣿⣧⠀⠘⣇⠀⣧⠀⠀⠀⡀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⢶⢶⣿⣿⠃⣻⣿⣿⣿⣿⣿⠀⢼⣾⠇⠃⠀⣠⣿⠚⠛
⠀⠀⠀⠀⠀⣤⣄⣰⣿⣶⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢾⡿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠹⣿⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣿⣿⣿⣿⡿⠿⢿⣿⣿⡷⠄⠀⠙⠟⠓⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣴⠿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠙⣿⣯⡀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠘⣿⣷⡄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⣿⣷⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡏⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢰⣿⠀⣿⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⢸⡇⠀⠸⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠈⠁⠀⠀⠈⠙⠛⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠐⢷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⢶⢶⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⡟⠀⠀⠀⠀⢀⣴⡾⠋⠁
⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢸⣧⠀⠀⠀⢀⣾⠃
⣾⡟⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠘⣿⠀⠀⠀⢸⡇
⠘⢿⣄⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⣿⡄⠀⠀⢸⡇
⠀⠀⢻⣧⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢹⣇⠀⠀⢸⡇
⠀⠀⠀⠙⢷⣦⣌⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⣤⣤⣤⣄⠘⣿⠀⠀⠸⣷⣶⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠠⣤⣤⣤⣤⣶⡶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣭⣭⣭⣭⣽⡇⠙⠷⠶⠶⠶⠾⠃
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣤⣄⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣰⢛⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣤⣝⢦⡀⠀⠀⣸⢣⡏⡿⢸⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⣠⠖⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢳⡈⢧⠳⡄⢀⡏⡾⢰⢇⡟⠀⠀⠀⠀⠀⣾⢡⠶⣝⢦⠀⣸⢱⢻⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢮⠳⣌⣧⢻⡾⣸⢣⢏⡞⠀⠀⠀⠀⠀⠀⠘⣎⢧⡸⡜⣇⡟⡞⣸⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⣨⠗⠈⠁⠈⠁⠛⠋⢯⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣧⣷⣿⢿⢃⣏⡟
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠁⡾⠁⠀⠀⡀⠀⢀⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣠⠴⠚⠈⠋⠉⠈⠋⠻⣄⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⢰⡇⠀⠀⠄⠉⠀⠆⠘⣀⠀⠀⢸⡇⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢸⡗⠆⠀⠀⠀⠀⠀⠀⠤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣷⡄⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠋⠉⠛⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢰⠋⢡⣠⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠉⠛⠋⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠄⠐⢤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠑⢤⠊⠁⠀⠀⠀⠈⠑⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀⠀⠀⠀⡅⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⡃⠀⠀⠀⠀⡄⠀⡄⢀⣇⣀⢰⡀⣰⠀⠀⠀⠰⠃⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⣣⡀⠀⠀⢄⡨⠂⣉⡁⠀⠀⠀⢉⣉⡑⠢⣴⠁⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠉⢲⠎⠀⠙⠛⠁⠀⠀⠀⠈⢙⠉⠀⠈⢣⠒⠁⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠠⡌⠀⡔⠉⠉⠱⡀⠀⠀⡎⠁⠈⢱⡀⠀⠓⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠐⡇⠀⡍⢰⣶⣿⠇⡀⠘⣹⣿⡯⢁⠃⠀⠷⠤⠄⣀⠤⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠒⢃⠀⠈⠂⠙⠋⠀⠧⠀⠈⠛⠒⢁⠀⢠⠇⠀⠀⠈⢢⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⡀⢀⠤⠂⠙⡄⠀⠑⠤⣀⡀⠀⣀⣀⡠⠔⠁⢠⡊⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢸⠁⠀⠀⠐⠚⣢⣀⠀⠀⠀⠀⠁⠀⠀⣀⡔⠁⠈⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠈⠁⠌⠉⠲⠒⠒⡖⠒⡏⠁⠈⢆⡀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠙⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠠⡤⠤⠔⢳⠀⠀⠀⠀⠀⠀⠀⠀⡸⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠘⣄⡀⠀⠀⢀⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢉⣀⣉⠀⠈⡏⢹⠁⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠉⠠⣀⣀⡀⣠⡬⢱⡇⢸⡮⠥⠤⣤⡀⠀⡀⠈⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⠀⠀⠀⠒⠈⠁⠀⠒⣁⠔⠊⡇⠀⠈⠢⢄⣀⠈⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠒⠒⠈⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡂⠀⢀⣀⠠⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠂⣈⣉⣉⣉⣉⠥⢶⡇⣷⠊⠤⣤⡤⢤⣔⡂⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠠⠐⠊⠉⠚⢁⠴⠊⡇⠆⠑⠤⡀⠉⠀⠀⠀⠉⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⣀⡀⠀⠀⠀⣀⠠⠔⠊⠁⠀⠀⡇⠀⠀⠀⠈⠁⠒⠀⠐⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢔⣠⣾⣿⣿⣿⣿⣿⣧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣁⣥⠖⠛⠛⠛⠿⣿⣿⣿⣿⣿⣷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡗⣒⣂⡀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⣹⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⣀⡀⠀⠀⠸⣿⣿⣿⣿⣆⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⢰⣯⣥⣾⠃⠀⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡿⣿⣿⣿⣿⣧⡀⠀⠈⠈⠉⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠹⣿⣿⣿⣿⣿⣷⡶⠆⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣹⡄⣈⣛⣛⣛⣛⠻⢫⡄⠀⠼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠟⠉⠙⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣧⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⣿⣿⣿⣿⣿⣆⠈⣻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣾⠟⠛⠟⠚⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠊⠁⠀⠀⠀⠀⠺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠫⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣄⠀⢠⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠈⢿⣦⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣷⡄⠀⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠻⣿⣿⣿⣿⡿⠋⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠃⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠠⣤⡄⢀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠸⠁⠘⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠀⢉⡿⠿⢻⡅⠀⣠⣄⡀⡁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⣀⣀⣀⡀⠀⣀⣠⣔⢂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣻⣿⣿⣿⣿⣿⣿⢿⣿⡤⢐⡀⣀⠀⠀⠀⠄⠈⢷⡀⠈⢮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⣼⣿⢸⣿⠶⠟⠛⠋⠀⠀⠀⠀⠀⠀⠙⢷⣾⣧⣠⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣚⣿⡿⣻⣿⣧⡈⢷⣿⣷⣔⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣽⣸⡿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣿⠿⠉⠛⢿⣿⣾⢿⠉⠋⠛⠿⠿⠤⠀⠀⠀⠀⠀⠑⢹⣿⣿⣿⣿⣿⡿⠛⠛⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢛⣹⣿⣿⣧⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣠⣾⣷⡿⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢄⠀⠀⠀⠀⠀⠀⠀⠈⣉⣽⠿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⣀⣀⠀⠀⠈⠀⣈⡴⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢿⡏⠛⠑⣲⠚⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⡘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠁⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⣿⣼⡇⢸⣿⣧⠙⣿⠃⢸⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⢹⡇⢸⣯⣿⠀⣿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠛⠘⠃⠙⠁⠛⠁⠋⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⣤⡴⠆⣶⢸⡆⢿⣟⠀⣿⢹⡇⢹⣿⡏⢸⡏⣿⢸⣷⣾⢰⡶⠆⠀⢀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣇⠠⠈⣿⠀⢹⣿⡇⢸⡏⠀⣿⢿⡇⠀⣿⠁⢸⣇⣿⢸⣿⣿⣸⠋⠀⠀⠛⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠨⡇⠀⠙⠶⠀⠻⠃⠘⠛⠃⠛⠈⠋⠀⠙⠀⠈⠙⠋⠘⠋⠃⠛⠃⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠦⠒⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⣖⠀⠀⠀⠀⣀⠀⠀⡀⢐⠀⠥⠒⠉⠀⠀⢀⡀⢠⠜⢲⡀

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡷⠀⠀⠀⠀⠂⠀⢀⡀
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠄⠀⠀⠄⠤⠀⠠⠤⠤⠤⠄⠀⣠
⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠘⠀⠢⠅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣈⡐⠒⠒⢦⠒⠖⠶⡀⠀⠀⠛
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣐⡄⣀⣀⡬⡀⠤⠤⠦⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡤⠤⠤⠤⣶⣶⠶⢤⣠⣄⣤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠾⠓⠀⢰⠯⠠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠯⣭⠥⠠⠀⠸⣟⣧⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡔⠋⢹⡤⣀⠀⠀⠠
⠀⠀⠀⠀⠡⠀⠇⠀⠀⢀⠀⠀⠀⢀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡼⡟⡂⢠⡁⡅
⠀⠀⠀⠀⢀⠀⠸⠀⢠⠃⢠⣶⣴⣦⠆⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⠚⡏⢌⢀⡴⠃⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠘⢿⡿⢏⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢉⠂⠝⡊⢀⣀⣤⣶⣿⣿⣿⣿⣿⣷
⠀⠠⠀⠂⠀⠀⠀⠀⠀⠈⠓⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⣻⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣸⣭⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⠃⣿⡆⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠉⠉⠛⢀⡿⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡀⠀⠀⠀⠀⠈⢈⡀⣾⠁
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢰⣿⣦⠀⠀⠠⠁⠀⠻⣄
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⢖⠹⠛⠍⠉⠉⠉⠍⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⠗⠋⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠐⠃⠰⠀⠀⠁⡀⠀⠂⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠐⠈⠀⠀⡀⠀⠀⠐⠁⠠⠄⠠⢤⡀⠀⠀⠀⢀⠀⠀⠀⠐⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣴⠏⠰⢀⠀⠑⠀⠀⠁⠀⡄⠀⠂⠀⠀⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠲⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠈⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠑⠄⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠠⠀⠀⠀⢀⣿⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠄⢀⡼⢷⡤⢑⡄⠳⠀⠀⠀⠀⠀⠀⠀⠢⠐⠐⠀⠀⠀⠀⠀⠀⣼⣷⣿
⠀⠀⢀⢀⠀⠀⠀⠀⢐⣼⣾⣴⣯⣾⣽⣿⠶⢛⡛⠂⣠⡀⢁⡀⢰⣮⠉⣠⣄⡀⠀⠀⡀⣀⣸⣟⣿⠟⣥⢀⣀⠀⠀⠀⣠⠄
⠀⠒⠓⠚⠃⣐⠚⠚⢟⠋⠙⠛⠛⣻⣿⠿⢝⣿⣿⣿⣿⣳⣛⣶⣾⡷⠾⢿⡾⣿⡾⠾⠏⠷⢿⢿⡧⣞⡥⠖⠐⢛⡛⠛⠃⠀⠒⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣐⣁⣀⣈⣂⣉⣀⣁⣀⣀⣀⣙⣀⣀⣀⣄⣀⣀⣀⣁⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⣶⠠⠀⠰⠰⣦⡆⠰⠂⠀⢤⣆⠀⡤⠶⠀⢰⣶⡆
⠀⠀⠀⠀⠀⠞⠀⠀⠚⠲⠾⠄⠀⠀⠠⠿⠼⠂⠀⠈⠗⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀⠀⢀⠀⣀⢀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠈⠀⠈⠛⠁⠈⠉⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣿⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠸⠃⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠈⢧⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢉⡴⠃⠀⠀⢝⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣱⡀⠀⢈⡃⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣵⣌⠸⢶⡖⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⠟⠛⢿⡿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠫⡿⣶⣼⣿⣷⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⠗⠀⢺⣷⣯⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠘⠛⢈⣿⢷⣌⡁⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⠀⠀⡀⠶
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⡎⠀⢠⣿⠏⢴⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⡀⠀⣰⡆⠀⠀⠀⠀⣿⡆⠀⠀⠀⠇⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠳⠤⣿⣿⣀⡖⠁⣀⢙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣤⣾⡷⠾⠿⠿⠿⠷⣶⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⡈⠙⠉⠀⠇⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢍⠀⠀⠀⠀⢀⡄⢀⡤⠉⠡⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠠⣦⠀⠀⢀⡄⠀⣷⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⠀⢃⠀⠀⠀⢀⡀⠈⣣⠀⠀⡄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⢷⡀⠀⠁⢰⡟⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⠀⠀⠀⢘⣷⣄⠀⠀⠃⠀⣿⡇⠀⠐⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⠃⠀⠈⢿⡄⠀⣾⠇⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣟⠀⠀⠀⠈⠀⠹⣦⠀⠀⣸⡏⠁⠀⠀⠈⠻⠷
⠀⠀⠀⠀⠀⠀⠀⢹⠿⣄⠃⠀⠀⠀⢸⣿⣾⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⢹⣧⣴⣿⠀⠰⠀⠀⠀⠀⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⣹⡆⠀⠀⠀⢠⡿⠋⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠸⣿⣿⡿⠀⠀⠆⠀⠀⠀⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣟⣷⣦⠀⠀⢸⡇⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⡄⢀⡀⠀⠀⣿⠟⠁⠀⢠⣵⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢋⣿⠀⡴⠀⢰⠀⢰⣿⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠀⡀⠋⠀⠀⣸⡿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠨⣿⡆⠀⠀⠘⢀⣾⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⣿⠀⠀⠀⠇⣰⡿⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⠇⠀⠀⠀⡿⠋⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⢻⡇⠀⠀⢴⡟⠁⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⢸⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢸⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘
⣀⣀⣀⣀⣀⣠⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⢀⣀⡀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠱⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣇⡀⠀⠀⠤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⡆⠀⢿⡿⠈⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⠄⡘⠀⢀⡀⠤⡶⡈⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⡿⠆⠀⢀⡨⡅⠀⠀⠙⠛⠙⠋⠙⠋⠛⠛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⡦⠀⠂⠈⣉⣃⣀⣈⣣⣀⢤⣀⣀⣄⣈⣀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠛⠛⠉⠙⠒⠀⠀⠀⠘⠛⠛⠁⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠄⠁⠀⠀⠀⢌⣥⡉⣡⠟⢿⠼⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠛⠛⠉⠙⠟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡄⠀⢀⠉⠀⢠⠀⡍⢁⠉⠉⢠⣿⣿⣿⣿
⠀⣀⣀⣠⣠⣤⣤⣤⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢤⠤⠤⣤⣤⢠⣤⣤⣴⣤⣬⣤⣦⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⡽⠳⣶⣦⠤
⠀⠀⠀⠀⢾⠃⢠⡞⠁⠂⡄⡠⣦⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⣵⣿⡆⢰⣿⣿⡷⠶⠶⠄⣀⠀⠀⠀⡀
⠀⠈⢶⠀⢾⠀⠼⠁⣆⣸⠇⠂⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠙⠒⠉⠀⠀⢀⡀⠀⠀⢀⠀⠛⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠁⠀⢧⣤⣀⣄⡀⠀⣀⣤⣤⣾⣿⣆⡀⠈⠓⠀⠀⣼⣿⣿⣧⠀⣠⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣧⠀⢠⠄⢻⡿⠿⠿⠿⠿⠛⠉⣿⠛⢿⢿⣿⣿⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠠⣴⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⢰⠀⠀⠀⣰⠄⠀⠀⠀⠁⠀⠀⠀⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⠖⠒⠾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢘⣫⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣌⠁⣠⠀⠀⠸⣆⠀⠙⡀⠈⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⢱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣧⠞⠁⣠⣴⡆⠙⠦⠊⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢿⣿⠿⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡰⠋⢠⣾⡿⠋⠀⠀⣨⣿⣿⣿⡟⠋⠁⠈⠙⠻⣿⣾⣿⣿⢿⣷⣿⣿⣾⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⢹⡇⠀⠈⠁
⠀⠀⠀⠀⢠⣤⣾⣇⠀⢾⣿⣷⣀⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⡿⠋⠁⢀⣼⠛⠋⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠈⡿⢿⣟⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡞⠋⠀⣠⣶⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣤⣸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣠⣿⣆⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠁⣠⣾⡿⠟⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠏⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣴⣿⡿⠋⠀⢀⡬⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢘⣿⡿⠟⠛⣱⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠀⢀⡿⠋⠀⢀⣴⣿⣷⣀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠙⠉⠙⣿⠿⠻⠏⠁
⠀⠀⣨⣿⡃⠀⢰⣿⣿⣿⡿⠛⠁⢌⢿⣿⣿⣿⣿⣿⡿⠟⠀⢀⣤⣿⡿⠋⡛⠛⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⣾⣿⣿⣷⣤⣼⣿⣿⡏⠀⠀⠀⠈⣷⣿⣿⣿⣿⡉⠁⠀⢠⣿⣿⠟⠁⠀⣻⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⡀
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣠⣴⣿⣿⣿⣿⣿⣿⣆⣠⣾⡿⠃⠀⣠⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣷⣄
⠀⠈⠟⠁⠉⢻⣟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⡾⢿⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣦⣶⣿⣿⠶⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡁⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣄⣠⣿⣿⣄⠀⠀⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠈⠑⣿⢻⡽⣿⣿⣿⣧⠿⠛⠻⠿⣿⠿⠛⠣⠲⣶⢦⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⢾⢳⢻⣿⣿⣿⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠛⠁⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠄⠉⠻⣿⣿⠇⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠱⠏⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣿⣿⡿⣿⣿⠟⠉⠀⠀⣾⣿⡏⠱⠄⢤⠀⠀⢈⠃⠀⢀⣠⡤⠄⠐⠂⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⢀⠀⠀⠀⠀⣾⣿⣆⡀⢲⠄⠀⠀⣈⣿⣏⡁⠀⠀⠀⢀⣴⠀⠀⠙⢿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⣸⡟⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣤⣾⣿⠀⠀⢀⣾⣿⣿⣿⣿⣾⡄⠀⠰⣯⣿⣛⣇⣀⠀⣴⡿⠋⠀⠀⠀⢘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣾⠟⢁⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠟⠋⠁⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠟⠵⠀⠀⢠⣶⣿⣿⣽⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⡀⢀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠸⢿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣼⣦⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠂⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣾⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⢀⣠⣴⠶⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⡿⠋⠀⢀⡤⠆⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⣀⡤⠒⠉⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠙⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣠⣀⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣀⣄⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠋⠀⠀⠙⢦⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡴⠋⠀⠀⠙⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠳⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢱⡈⣇⠀⠀⠀⠀⠀⠀⣸⢁⡎⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⢃⠸⡄⠀⠀⠀⠀⢰⠇⡘⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⠀⠀⠀⣼⠀⠁⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠐⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣸⡀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡴⠚⠉⠀⠀⠀⠀⠀⠈⠉⠲⢄⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠞⠳⣄⠀⠀⠀⠀⠀⢀⡼⠚⢦⡀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣟⢙⣿⣾⠀⠀⠀⠀⠀⢸⡗⢻⣿⣷⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡄⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⢠⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⢹⡏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠳⢤⣀⣀⡴⢻⡟⢦⣀⣀⡠⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠈⠧⠼⠧⠼⠁⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣄⣀⣀⣠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⢀⡖⢀⣠⣶⣄⡀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣷⠀⢰⠄⢿⢧⡈⠀⠓⠂
⠀⠀⠀⠀⠀⣼⣿⠰⠎⠉⠀⠈⠴⠺⣧⡀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⡧⠀⠀⠀⠀⡠⠔⠁⠁⡧⣴⣿⠀⢹⣶⡽⡀
⠀⠀⠀⠀⠀⣿⡇⣠⣖⠢⠶⠜⢧⣄⢻⣷⡀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠓⠀⠀⠀⠉⠁⠀⠀⡔⠁⠉⠉⠀⠘⠋⠉⠓⡄
⠀⠀⠀⠀⢸⣿⢃⡘⣦⠈⠟⢩⣼⡿⣸⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠳⣆⠀⠀⢠⠀⠀⣠⡼⡃
⠀⠀⠀⠀⠀⡿⠟⠛⠻⡿⠿⠿⠟⡱⢿⠸⣷⠀⠀⠀⠀⠀⢠⠼⠧⡀⠀⠈⠀⠀⠀⠀⠀⢨⠆⠈⡑⠀⠀⠀⠚⠉⠀⠑
⠀⠀⠀⠰⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠛⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠈⠉⠋⠉⠉
⡇⠀⢀⣀⣨⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠻⣿⣿⡉⠉⠉⠉⠉⠉⠀⠀⣾⣿⣿⣿⣍⡀⠀⠘⣿⣿⣿⣶⣾⣿⡇⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣴⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⡀⢿⣿⣿⣿⠿⠟⠃⠀⠀⠘⠻⠗⠒⢾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⢉⠉⠿
⣿⣿⣿⣟⣿⣿⣥⣤⣄⣠⣤⠀⠀⠀⠀⠀⠀⠈⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣀⣠⣤⣤⣤⣶⣶⣾⣿⣤⣤⣄⣀⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡭⣽⡷⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠈⢻⣿⣿⣿⣿⢁⣿⡏⠀⠀⠀⠀⠀⢠⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣤⣴⣦⡀⠀⠀⠉
⣿⣿⣿⣿⣿⠟⠻⠋⢿⣿⣯⣿⣿⣿⣿⡿⠋⠉⠉⢨⣿⣷⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣯⣶⡄⢠⣾⣿⣿⣿⣿⣿⣿⣧⣴⣄⡀⠀⢿⡿⠿⠟⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠛⠛⠹⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣉⠁⣀⠀⠀⣀⣤⣄⡀
⣿⣿⣿⣿⣿⣿⣷⣬⡍⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⣾⣧⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⡟⣿⢟⣿⣿⣷⠀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠑⣤⡔⣠⠀⠁⢉⡁⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠁⠀⡸⠿⠏⠃⠀⠀⠠⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⣄⠙⠒⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⢙⠿⣿⣿⣿⠛⠁⠀⣀⣀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⣠⣤⣆⠀⢀⠀⠀⠀⠀⠂⠐⠚⢉⢣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠀⠀⠀⠑⣸⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡆⠇⣀⡀⢼⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⢀⠄⣄⢀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢣⡆⠉⠀⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⣰⣿⢦⠀⢠⣤⠀⡔⠀⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⠀⠀⠸⠿⠆⠀⣸⢿⣆⢦⣄⣀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠠⣶⠖⢀⡜⢀⣭⣬⣟⠻⠃⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣟⣿⣳⣶⣶⣾⣿⢿⣿⣾⣯⣷⠙⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣿⣿⣯⣿⣟⣿⣿⣿⣿⣿⣿⣿⣻⡾⣿⠁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠛⢛⠉⢤⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠈⠁⠀⠐⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣿⣯⣽⣻⣿⣿⣯⣯⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡎⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠉⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠿⣿⣿⡟⠋⠙⠛⣛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠒⠂

⠀⠀⣠⠴⠚⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠓⠦⣄
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠗⠚⠛⠋⠉⠉⠉⠉⠉⠛⠓⠺⠧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠻⣄⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠙⢦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⡰⢆⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⢟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⣀⣹⣄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣏⣀⡀
⠀⠀⠀⠀⠀⢠⡏⠁⠀⠈⠓⠦⣤⣀⣀⣀⡾⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠴⠛⠁⠀⠈⢹⡄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⢀⣠⠴⠤⠤⠤⠤⠤⠤⠦⣄⡀⠀⢀⣀⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣻⡿⣿⣿⣿⣿⣡⣴⣿⣭⣭⡟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠿⠛⠉⠙⠿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⡹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⢶⠀⠀⠰⠾⠛⠁⠀⠀⠀⠈⣿⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⡇⢰⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⣿⣿⣯⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢈⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣝⢻⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣿⣻⣿⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣻⣿⣿⣿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢻⣿⣧⡉⠻⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠀⠀⠀⠀⠀⠀⣀⡶⣶⠄⠀⠆⠀⠀⠀⠀⠈⡎⣹⣿⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⣆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⣿⣆⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⢸⡀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣠⣀⣀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢣⡆⠀⠀⠀⣾⡟⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠈⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⠸⠂⠙⠻⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⢀⣴⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣠⠀⡀⠀⡀⠰⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢠⡾⠃⠈⠙⢦⣤⡀⠀⠀⠀⠀⢠⡾⠃⠀⢻⣄⠀⠀⠀⠜⡁⣄⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠛⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡟⠠⣾⡄⠈⠎⠲⠖⠂⠘⢹⡇⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠃⠘⣆⡙⠀⡘⠻⡄⠀⠀⣀⠀⣠⣾⣿⣿⡟⠀⠀⠹⣿⣄⠀⠘⠶⠃⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠞⠁⠀⣴⠁⠐⡌⠉⣀⢀⡇⠁⠀⣀⠉⠀⠉⠋⠁⠘⠁⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠔⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⣀⡀⢻⣀⣀⡘⠂⠀⠈⠴⠊⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣄⣀⠀⠀⠀⠀⣠⡴⠞⠿⠉⠉
⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⢠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠈⠀⠀⠀⠀⠉⠉⠁⣠⠈⠁
⠀⠀⠀⠀⠀⣼⠁⢠⡟⠁⢰⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠈⠓⢤⣿⡿⠿⣿⠟⠃⠀⠀⠀⠀⢠⡀⠀⠀⢀⡠⣶⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⠀⠀⡀
⠀⠀⠀⠀⠀⢧⡀⠆⠠⠾⢿⣿⣦⣄⣀⣀⣄⣀⣤⠔⠻⠗⠓⣉⡁⠰⣿⣆⠀⠀⠃⢠⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠙⣦⢦⠀⣦⣱⡿⣫⠌⢉⡁⠤⠀⠀⠘⠉⠁⣀⠀⠀⡄⢹⡶⢦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠶⢒
⠀⠀⠀⠀⠀⠀⠀⢹⣖⣲⣝⡿⢾⠉⢹⣄⡀⠠⠤⠀⡏⠉⢉⠀⢰⠀⡃⠘⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⠺⡄⠈⢻⡤⠤⠖⠂⡏⠉⠠⠀⡅⣰⠤⢄⠙⣦⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣄⠀⠻⠀⠈⡟⠂⠒⠠⠉⠀⢀⠀⠀⠉⠀⠀⠉⢺⡆⠔⠂⠀⣀⡤⠾⠛⠊⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠛⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠰⣿⡀⡷⡴⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠖⠀⠀⠀⠀⠀⢀⠈⠳⡄⠀⠀⠀⡞⢀⠀⠀⠀⢀⡤⣿⣍⡁⠘⠒⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠈⡖⢲⣦⡥⠈⢧⡆⠀⠃⠔⠛⠋⠉⠁⠀⠀⠀⠙⡄
⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠇⠀⢻⣼⡄⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⣦⣀⡼⠳⣿⡀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠈⠑⠲⠦⣤⣀⡀⠀⠀⠀⠸
⠀⠀⠀⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰
⠀⠀⠀⡿⠇⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⣦⣤⣤⣀⣀⠠⡇⠈
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢔⡴⠃⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠤⠶⠞⠉⠁⣀⠤
⠀⠀⠀⢹⡇⠀⠀⠀⢀⡔⠋⠁⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⠀⣴⠂⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⣧⠈⠶⠆⠀⠀⢀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡀⠈
⠀⠀⠀⠀⠘⢷⣤⡞⠁⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣠⠶⠤
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢤⠈⢢⣀
⠀⠀⠀⠀⢀⣾⡉⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠈⠓⠈⠁
⠀⠀⠀⢠⡞⢻⡟⢛⡃⠱
⠀⠀⠀⠻⠁⢿⠀⢼⣇⡀

⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⢀⣂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠚⠩⠝⢛⡆⣤⣈⡐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⣀⣷⠀⠀⠀⠙⠛⠒⢿⣿⣷⣄⡈⠢⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠈⣉⣛⣿⣤⠈⢢
⠀⠀⠀⠀⠐⡄⠀⠀⠀⢘⣧⡀⠀⠀⠀⠀⠀⠁⠉⠻⢿⣶⡀⠑⣄
⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠭⣛⣤⡀⠀⠀⠀⢰⣄⣠⣼⣿⣷⡀⠈⠹⣿⣿⣶⣶⠦⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠐⣋⣰⣿⣦⣄⡀⠘⢇⡻⠿⢿⡿⠳⠀⠀⠜⠛⠙⠂⠀⠀⠈⠙⠻⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠯⣟⣿⣿⣿⣿⣷⣶⣤⣤⣤⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠉⠀⠀⠀⠮⣱⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢒⡶⢤⣆⣤⣤⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⣿⡄⠀⠀⠈⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣿⣿⣿⣉⣿⣷⠀⣰⣿⣄⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣈⣿⣿⣿⣿⣿⣿⣿⣷⣴⡀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠠⢤⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠀⠀⠀⢉⡙⣿⣿⣄⣄⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣃⠀⠈⠁⠀⠀⠈⠓⠙⠿⣿⢆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⡀⠀⢰⣏⡀⢀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⣻⣧⣤⣼⣿⠀⠀⠐⢠⡀⠂⠀⠂⠀⠰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣌⠹⣿⣿⣿⣷⡟⢶⣰⣠⣬⣝⣃⡤⣤⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢊⣿⡴⣟⣿⠿⠛⠉⠉⠛⠻⠿⠿⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣋⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠙⢶⡀
⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⣷
⠀⣾⠋⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⡠⠦⡀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⢠⠞⠀⠀⡇⠀⠀⣿
⣾⠁⠀⢰⠋⠳⡄⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢠⠋⠀⠀⠀⡇⠀⠀⡿
⣿⠀⠀⡆⠀⠀⠹⡄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠⠃⠀⢰⠇
⣿⠀⠀⡇⠀⠀⠀⠹⡀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡸⠀⢀⡿
⣿⠀⠀⡇⠀⠀⠀⠀⢳⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⢠⠃⠀⣼⠃
⢸⡇⠀⢸⠀⠀⠀⠀⠈⣧⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠁⠀⠀⠀⢠⡏⠀⣼⠃
⠈⣷⠀⠈⢇⠀⠀⠀⠀⠸⡆⠀⠀⠀⣿⣠⣤⠶⠖⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⢮⣀⡀⠀⣠⠏⢀⡼⠃
⠀⠘⣧⡀⠈⢧⡀⠀⠀⠀⠸⡀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣴⠟⠁
⠀⠀⠈⢳⣄⠀⠳⣄⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠙⠷⣄⣈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢉⡀⠀⢀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠀⠀⠠⠤⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠞⠋⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠶⠶⠶⣶⠶⠶⠶⠶⠒⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠠⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠳⣀⡀⠀⠀⠀⢀⡞⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣼⠇⠀⠀⠀⠀⠳⢦⡀⠀⠀⠀⡀⣠⠞⠁⠀⠈⠙⠛⠳⠘⠋⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⣰⡋⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⢀⠀⢀⣀⣠⠟⠛⠻⣤⣀⣀⣀⣀⢀⣀⣀⣼⠁

⠀⠀⢠⡖⠉⠙⣦⡠⣤⣀
⢀⡤⠼⣦⡀⢀⡏⠀⠀⢹
⣿⠀⠀⢘⡟⠉⠙⣶⠶⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣀⣀⡀
⠈⢻⠚⠛⢧⡄⢤⣇⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣇⡀⢹⡏⠀⣹⣤⣄
⠀⠹⣄⣠⣞⠁⠀⢹⡟⠚⠃⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣘⡿⣪⣧⡿⣏⠀⣸
⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢸⠉⠈⣩⡻⠿⢧⡛⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠛⣿⠋⠙⣿⠈⢱⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⠉⢿⣿⣿⣿⣀⣼⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠙⠲⠾⠉⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡟⠉⠁⠀⠉⠻⣿⠋⠛⣛⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢋⣁⣀⣼⣷⣄⡀⠀⣀⣴⡛⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⣦⡛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠘⠿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢰⡇⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣾⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⣡⣾⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣄⢠⢣⣿⣿⣿⣿⠟⠋
⠀⣸⣦⡄⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡤⢊⣡⣴⣶⣿⣿⣿⣿⣶⣝⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⣿⣷⣦⣭⣑⣒⣲⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⣤⣥⣵⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠉⠛⠙⠛⠉⠉⠀⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢔⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣿⣿⠏⠀⣿⣿⣿⣝⣿⣿⡟⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣇⠀⠐⣿⣿⡿⢿⣿⣿⣿⣝⠏⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣿⣿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣯⠝⠋⡀⠀⠹⣎⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢸⠁⣽⣿⣿⣿⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠸⠛⣿⡿⠇⠀⠐⠀⠀⠀⠹⠿⠟⠻⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⡟⠀⢱⣿⠻⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⡌⢠⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡸⠁⠀⠀⢳⡀⠙⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠻⠿⠿⠃⢀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡟⠁⠀⠀⠀⠀⠙⢦⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠲⡟⠲⡞⢇⢀⣼⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡄⠀⡘⣦⡙⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⡿⣉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⡼⣿⣿⣿⣿⣿⣧⢻⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⣴⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣎⠭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠘⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⠿⢿⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⣄⣀⡰⠤⠜
⠀⠀⢪⡀⠀⠀⠀⣀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠛⠉⢩⢤⡍⠘⠩⠞⠁
⠀⠀⠀⠈⠉⠓⠚⠁⠰⡀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠼⠤⠂
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠛⠉⠙⠛⠿⠟⢻⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠁⡀⠀⠂⠀⠀⠐⠃⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢠⠢⢈⣶⣤⣤⣤⣦⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠚⠀⠀⢀⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢪⡘⢊⠕⠀⠀⠀⠀⢏⠑⣶⠏⣴⠀⠀⠀⠀⠀⡄⠔⡂⠀⡜⠀⡘⠀⠀⡀⠀⠀⢘⡹⣦⡀⢰⣏⣿⠀⠀⢀⢤⣄⠀⣴⣲⠀⠰⠂⡠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⠚⠉⠚⠩⡀⠀⠀⠀⠀⠉⢚⡠⠔⠀⣴⣗⠎⡱⣪⠂⠀⡀⢻⢬⠳⢾⣼⠃⠀⠀⠈⢷⣽⣶⣇⡿⠀⠀⠈
⠀⠀⠀⠐⡵⢦⡀⠰⠙⣳⠀⠀⠀⡇⠐⠀⠀⠀⢱⠀⠀⠀⠀⢠⠛⠀⢀⠀⠈⡟⠋⠉⠀⠀⢠⢡⠊⠀⠀⠀⠈⠆⠀⠀⢠⠜⠛⠋⠛⢇
⠀⠀⠀⠘⢶⣌⢟⣶⣆⡝⠀⠀⠀⠱⠀⢼⢻⠄⠱⠀⠀⠀⠀⠀⡧⢄⣀⢠⠜⠀⠀⠀⠀⠀⠀⣨⠄⠀⠀⠀⢠⠃⠀⢠⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠋⠉⠀⠁⠑⡄⠀⠀⢰⠀⠙⠼⠁⢀⣳⠀⠀⠀⠸⡀⡠⣄⢀⠇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢰⠃⠀⠀⢳⠄⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠠⡀⠀⠂⠀⠀⢄⠀⠀⠀⠙⠈⢩⣁⠉⠾⣷⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠔⠢⠔⠁⠀⠀⠀⠈⡇⠀⠀⡀⢣
⠀⠀⠀⠀⠀⡇⠀⠈⠛⠀⠠⡆⠀⠀⠀⠀⠀⣿⡆⢀⣏⡀⢀⡀⡀⣠⠐⡆⣠⠐⣦⢠⡠⣆⠀⣔⣢⣀⡤⡀⠀⠀⠀⠀⠀⠑⠤⠺⠔⠛
⠀⠀⠀⠀⠈⠳⠤⠄⣰⠖⠁⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣾⡧⡧⠋⠀⡇⠈⠀⣿⠈⠡⡏⠀⠿⠼⢿⡿⠁⠀⠀⠀⠀⠀⣠⣄⡀⢀⣴⢶⡄
⠀⠀⠠⡤⠀⢄⠀⠀⠀⠀⠀⠀⠀⢖⡹⠀⠀⢸⡇⡔⢁⣿⠇⣴⠂⢀⡞⠁⠀⠀⠀⡌⣸⠇⠀⠀⣾⠀⠀⠀⠰⡙⠆⡄⠀⢻⡌⢳⣾⢃⢺
⠀⠈⠊⢷⣦⠛⠉⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢷⠾⠋⢸⣿⠀⢸⠇⢰⡀⡀⢳⢀⣅⡠⣀⡀⡏⢀⠀⠀⠀⠉⠉⠀⠀⠀⡟⠋⠈⠁⠳⡄
⠀⠀⠝⠥⢤⠃⠀⠀⢀⡉⠣⢀⠀⠀⡔⣝⢆⠀⠀⣠⣄⠀⠘⣿⠀⠸⣧⠙⠋⡟⠋⠘⠛⠃⣿⢸⣇⠄⠀⡀⠀⢀⣤⡀⠀⢠⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡎⠀⠀⢧⣿⢸⣠⡾⣻⡺⠀⠀⠌⠉⡀⡄⠀⠄⠁⢠⢀⠀⠀⠈⢀⠠⠀⢸⡞⢷⣜⡏⣿⠀⠘⠤⣤⢲⡄⠰⣊
⠀⠀⠀⠀⠀⠀⠉⠒⢦⠄⠂⠁⠀⠀⡸⠋⠋⠙⠳⡟⠁⡤⣄⠀⠀⠀⠀⠀⠀⠀⢀⢔⠢⠀⠀⠀⠀⠀⠀⠙⣦⠿⠿⠿⡀⠀⢠⣋⣈⣄⡠⠟⠁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠐⠀⢸⠀⠉⠃⠀⠰⣱⣶⣄⠀⡰⠕⣹⠝⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢱
⠀⠀⠀⠀⢇⠘⣋⠽⠀⠀⠀⠀⠀⢼⠂⠀⠀⠀⠀⠾⡀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠉⡟⠠⣄⠀⠀⠀⠀⠀⢇⡀⠀⣨⠀⢀⠆⠀⠀⠀⡞⢳
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠈⠦⡀⠀⠀⠀⢸⠁⠬⣽⢿⠁⠀⠀⠠⣎⣉⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠒⠤⠞⠋⠀⠀⠀⠀⠀⠀⠪⠵⠒⠦⠤⠤⠤⠤⠋⠀⠀⠀⠀⠈⠐⠘⠒⠘⠂⠀⠀⠀⠀⠁

⠀⠀⢀⡤⠖⠋⠉⠉⠓⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠉⠉⠙⠲⢤⡀
⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠘⣦⡀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⡠⠜⠳⠤⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣳⠶⠦⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠴⠶⣞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠾⢷⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡾⠷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠙⠲⠦⠤⣤⡼⠃⠀⠀⠀⠀⠀⠀⠘⢦⣤⡤⠴⠖⠋⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣠⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⣀⠀⠀⠀⠀⠀⡰⣻⡳⣀⠳⣦⣄⠀⢀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠊⠁⠈⠉⠓⠒⠒⠚⠣⢿⢫⣴⣵⠘⠀⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⡗⠪⠒⠛⠋⠀⢀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⡀⢤⣾⠇⠀⠀⠀⠀⠀⠈⡗⠒⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠒⠒⣒⠮⢡⠎⠀⠀⢀⡰⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠛⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⡐⠑⣂⡥⠚⠒⢚⡿⠀⠀⠀⠀⣿
⠀⠀⠀⡄⠰⢆⠀⢸⣀⡀⠀⢠⠃⡤⠤⢀⡀⠀⠈⠒⠒⢊⠠⠒⠉⠉⡧⠖⣋⠀⠀⠀⠀⠀⡿
⠀⠀⡜⠻⠁⠘⠋⠉⠈⠉⠑⡇⢘⠄⠀⠀⢳⠀⢀⠔⠊⠁⣠⠔⢊⠼⠚⠉⡀⢱⠀⠀⠀⠀⡇
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠑⣭⢞⣀⡠⠁⣸⠐⣕⡀⢰⡟⡡⠚⠁⠀⣠⠜⣱⠃⠀⠀⠀⢰⠃
⣤⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡤⠞⠁⠀⠀⠉⢹⣞⢁⠄⢠⠚⠁⠔⠁⠀⠀⠀⠀⡞
⡍⠇⠀⠀⠀⠀⠀⡴⢋⠀⢀⣴⠋⢳⣀⡀⠀⠀⢀⡤⠋⠈⠘⢲⠋⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠇⠀⠀⠀⠀⠀⠀⠈⠛⡤⠄⠈⠆⢸⠀⠙⣶⠊⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠸⡀⠀⠀⠀⢀⠀⡸⠳⠇⠀⠀⢸⠂⠀⠀⡽⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠱⣄⠀⠀⠘⡗⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡠⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣷⣦⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠈⢻⣷⡀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⣿⣷⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⣿⣿⣿⣿⣝⠓⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⢿⣿⡤⠀⠀⠀⢻⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⠛⠛⠉⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⣶⣶⣶⠖⢲⠚⢲⠂⢲⠒⢶⡖⠒⡖⠒⠲⢒⢲⡖⢲⣶⠒⢲⠒⠒⠖⠒⢲⣶⡖⢲⣶⠒⢲⠒⠒⠒⣶⣶⠒⣶⡖⠐⢒⠒⠒⠂⣶⣶
⣿⣿⣿⣇⣾⣤⣸⣀⣰⣀⣽⣯⣀⣼⣤⣷⣌⣴⣇⣼⣿⣤⣾⣴⣄⣧⣁⣼⣿⣇⣼⣿⣄⣸⣴⣧⣠⣽⣿⣠⣿⣧⣀⣬⣤⣥⣰⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣛⡻⣿⣭⣯⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡏⠁⠀⠀⠀⠈⠉⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣗⣿⠁⠀⠀⠀⠀⢀⡀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣿⣿⣿⢿⣟⣒⣂⠀⠀⣫⣭⣥⣄⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⢻⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⣿⡿⣹⣿⣿⠟⠀⠀⠙⠿⠿⠋⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣰⣿⣟⣿⣧⡀⠀⠲⠶⠂⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⣾⣵⣟⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢳⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣾⢿⢻⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠸⡿⠓⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⠃⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠿⢶⢤⣦⣤⣄⣄⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠉⠁

⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⣻⣿⣿⣿⣿⠋⢻⡿⢻⣿⣿⣿⣿⣿⠋⣩⡉⠻⡏⠛⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⣀⣀⣀⣰⣿⠟⢋⣽⣿⡇⠈⢤⣾⣿⡏⣉⣹⡇⠰⣿⣿⠀⣇⣸⣇⣸⣿⣿⣿⣿⣿⣿⣿⠿⠇⠙⣋⠘⢛⣻⣿⣿⣿⣿
⣿⡿⠁⠀⣴⣾⣿⡿⣿⣿⡭⣿⡀⠿⠿⢿⣷⣀⣶⣤⣿⣿⣿⣿⣷⣀⠻⠟⠀⠉⠉⠈⠉⡻⣿⣿⣿⣿⣿⣿⡿⢂⡈⠛⢠⣉⣻⣿⣿⣿⣿
⣿⡇⠀⢸⣿⣿⣿⣇⠘⢻⣧⠸⣷⣶⡶⠋⠁⠀⠀⠀⠉⢙⠻⣿⣿⠸⢿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣬⣿⣿⣿⣿⣿⣿
⣿⣷⡀⠘⢿⣿⣿⣿⡄⠟⣹⣦⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡙⠀⠀⠀⠀⢀⠀⠀⠀⢰⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿
⣿⣿⣿⣶⣦⣄⠉⠙⢷⣶⣿⣿⣿⣧⢠⠀⠀⠈⠐⢀⠄⡀⠀⠀⠀⠀⢰⡆⢠⠎⠀⣠⠄⡨⠊⠼⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⣿⣿⣿⡧⠀⢈⣿⣿⣿⣿⣿⣦⡡⠀⠀⠀⠀⠁⠠⢦⣄⠀⠈⣼⣇⡟⣠⣖⠃⠉⠐⠀⠀⠀⠐⠍⣿⡟⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿
⣿⣿⣿⣿⡿⠟⠁⢀⣾⠟⠁⠙⠻⢿⣿⣿⣦⣑⠠⢀⣀⠀⠀⠈⠑⡮⠁⠍⢙⣇⣡⣄⠀⠠⠤⠤⠀⠀⢸⣸⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠈⢹⣯⣽⣥⠄⠉⠭⠵⠛⢛⡻⢤⣤⠼⣥⡀⠀⠀⠀⠀⠀⠀⡠⣢⣿⣿⣷⣶⣤⣀⠙⠻⢿⣿⣿⣿
⣿⣿⣿⣿⣷⣿⡟⠀⠀⠀⢀⣄⣀⠀⢠⣿⡿⠋⠄⠀⠀⠀⢤⠾⠋⠀⡴⠹⡆⠌⠀⠀⢡⣔⣂⣂⣭⣶⠟⠛⠛⠙⠻⣿⣿⣿⡄⠀⠈⠻⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠁⠁⠀⡀⠀⠀⠁⠀⢀⢘⡇⠠⢩⠈⠐⣦⣤⣹⣿⣿⣿⣿⠀⠀⢠⡀⠀⠈⣿⣿⡿⠀⠀⠀⢹
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠄⠉⠘⠁⠀⠀⠀⠀⠹⠟⠛⠉⠁⠈⣿⡀⠀⠸⠃⠀⢠⣿⡟⠁⠀⠀⠀⣼
⣿⣿⣿⡟⠀⠀⠀⠀⢀⣄⣀⠀⣼⠁⠉⢹⣦⣀⣀⠀⢠⣤⣶⣇⠑⠓⠀⠀⠀⠇⠀⠀⠀⠀⢰⣶⡿⣿⡇⠀⠀⠀⢰⣿⣿⣇⠀⠀⠀⣼⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣼⣿⠋⠁⠀⠀⠀⠹⡇⠀⠀⣧⡢⠀⠀⠠⠠⢦⠀⠀⠀⠀⣻⡇⠀⠀⣄⠀⠙⣿⣿⡀⣠⣾⣿⡿
⣿⣿⣿⣦⣄⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⣿⡇⠀⠀⢴⡇⠀⠀⡇⠀⠀⠋⠉⢰⡆⠈⠀⣄⠀⠀⢰⣿⣿⢷⣀⣀⣿⣦⣤⣾⣿⣿⣿⣿⠟
⣿⣻⡿⣛⣿⣿⣶⣤⣿⣿⣿⡿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠘⠁⠀⢰⣿⠀⠀⠀⠁⠀⢘⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣰⣿⣷⣦⣤⣤⣴⣿⣇⠀⠀⣼⡀⠀⠀⢀⣿⣿⣦⣴⣶⣶⣿⣿⣿⣿⣿⣿⠿⣋⠁
⣿⣿⣿⠿⡛⡍⠐⣶⣴⣖⡆⡄⢀⠈⣉⢙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⡛⣩⣴⡈⢻⡄⠀⠀⠠⡄
⣿⣿⠃⢰⠛⠃⠀⠁⠀⠀⠀⠉⠈⠃⠛⠛⠂⠿⠇⡖⠀⢰⣾⢩⡍⢙⡛⡛⢛⣛⢛⣛⡛⢩⣭⢡⣶⢠⡎⢸⣷⡘⠏⠓⠀⢀⠠⠻⠇
⣿⡇⠀⠀⠀⠀⠂⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠸⠧⠸⣧⡟⢿⣬⠿⡷⣏⢸⣿⠄⠽⠃⠙⡁⢡⡀⠄⠸⠬⠊⠋⠀⠀⠀⢀
⣿⠀⠢⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠑⠚⠈⠀⠀⠀⠀⠀⠀⢀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠄⠄⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿
⣿⣧⡀⠀⠀⠀⠀⣤⣰⢠⣤⣤⣤⠀⠀⣄⣀⣀⣀⡀⢀⡀⢠⣀⡀⡀⠀⢠⣤⣀⢠⠀⠀⡀⣀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⣿⣿⣷⣤⡀⠀⢠⡄⠈⣬⢈⣉⡈⡄⠀⢡⡍⡈⡉⢉⠉⠀⡌⠉⡉⠁⠀⡈⠉⢁⡘⢀⠀⠓⢙⠀⣀⠂⡁⠙⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣤⣥⣀⣉⣉⣈⣁⣁⣀⣈⣉⣉⣉⣉⣀⣀⣁⣋⣁⣉⣈⣁⣀⣈⣉⣈⣀⣀⣘⣂⣛⣀⣃⣘⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣧⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⣠⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⣧⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⡴⠁⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢼⣧⠀⠈⣧⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⣿⡒⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠐⣿⣄⠀⢸⡆⠀⠀⠀⠀⠀⣼⠁⢰⣿⡏⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠹⣿⡆⠈⣷⠀⠀⠀⠀⢀⡿⠀⣿⡿⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢺⣿⠀⣿⠀⠀⠀⠀⢸⡇⠀⢩⠂⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠈⢻⡆⢸⣦⣤⡤⢤⡿⠃⠀⢚⣰⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣼⣷⡄⠸⡇⠸⣇⠀⠀⢸⠇⠀⡴⢻⡗⠈⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣾⣿⠉⠀⠛⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⣅⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠁⠐⡀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⣽⡆⠀⠀⠀⠀⠀⠀⠀⣾⣶⡇⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠄⠀⠀⢜⣿⣇⠀⠀⠀⠀⠀⢸⠀⣿⡏⠀⠀⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⣿⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⢸⡇⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠰⣿⡀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⣸⣧⠀⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⢀⠠⢿⣷⣖⣲⣿⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⣹⣀⣻⣿⢏⣩⠄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣰⠟⠀⣷⡄⡠⠔⠚⣉⠟⢻⢿⣤⡀⠀⠉⢳⡞⠁⠀⣠⣼⣷⡿⠫⢙⡩⢐⠊⠄⠀⠹⣇
⠀⠀⠀⠀⠀⣸⠃⠀⣀⠉⠇⠀⡠⠊⡡⠊⠁⠀⠉⠉⠛⠻⠮⠽⠞⠛⠉⡉⠁⢤⠀⡄⠈⠐⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⢠⡟⠀⢰⢋⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠘⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢠⠁⢸⡇
⠀⠀⠀⠀⠸⣧⠀⠀⢧⣮⡠⡄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⣠⣾⠀⣿
⠀⠀⠀⠀⠀⢿⡆⠀⠘⢿⣿⡇⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠰⠏⠘⠋⠀⠀⠀⢠⣾⡟⠁⢸⡇
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⢿⣿⣼⠀⠀⠀⠀⠀⠙⠋⣦⡀⢀⣀⣤⣶⠃⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⣧⣠⠀⠀⠙⢿⡆⠀⠀⠀⠀⢀⠀⠈⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣄⣀⣤⣿⣦⡀⣀⠀⠈⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢀⡾⠯⢆⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡏⠙⣽⡿⠃⠋⠀⠀⠈⠀⠘⣿⡏⠀⠈⠁⠀⠘⠃⠚⣷⠀⠘⢡⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⢰⣹⣧⠀⢀⣆⠀⣙⡀⠀⣿⣀⡀⣜⠀⢀⡇⠀⣰⣏⣦⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⠀⡃⢀⢮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⣄⡇⢰⠔⠩⠴⣧⣸⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⠓⠀⠀⠀⠀⠈⠟⢀⡨⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠇⢀⠀⠸⣹⣦⡄⠀⣴⣿⢀⠀⡀⣇⣄⠉⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠘⠛⠛⠀⠔⠟⣿⣺⡶⢿⡟⢉⣀⡌⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣢⣀⡀⠀⠓⠐⣂⣴⣿⣾⠀⠀⠑⢦⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⣿⡿⣯⣿⣛⣻⣿⣿⣿⠿⣾⣷⣄⠤⠄⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣭⡿⡼⣿⣿⡿⠿⠛⠛⠛⢻⠛⠛⢿⡀⢀⣵⡏⢀⣢⣄⡀⢾⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠈⠉⠉⠻⣾⠟⠃⠄⠀⢀⠤⠤⠎⠠⢤⡜⢿⠁⣦⠛⠀⠉⠉⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡀⡴⢏⡀⠀⠀⠐⠀⠀⠀⠰⠀⢠⡇⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠀⠀⠸⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡐⢲⣶⣇⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣽⣼⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣤⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠻⠿⢿⠏⠉⠀⠉⠩⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠉⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠑⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠤⠚⠛⠋⠉⠉⠉⠉⠉⠙⠓⠻⠤⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠆⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀
⠘⠦⣄⣀⣀⣠⠤⠚⠉⠀⢸⠀⠀⠀⠀⠀⢰⣿⣁⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣈⣿⡄⠀⠀⠀⠀⢸⠀⠀⠀⠉⠳⠦⣄⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠠⠒⢄⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠙⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠰⡀⠀⠀⠀⠀⠀⠘⡄⠀⢸⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⢀⠃⠀⢸⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣠⠖⠋⠉⠒⢌⠑⠲⠤⠤⠎⠀⠀⠀⠳⠤⠤⠴⠊⡥⠚⠉⠙⠲⣄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠸⠁⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⠀⠀⠀⣀⣠⠼⠤⠤⣤⣤⣤⡤⡤⢧⣀⠀⠀⠀⠀⢀⣠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⠯⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣥⡍⠀⢉⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣡⣶⣌⣻⣧⠀⠀⠀⠀⠀⣀⠰⣿⣿⡆⢀
⠀⠀⠀⠀⠀⣶⣿⣷⢙⡻⢣⣾⣷⠀⠀⠀⠀⢀⠀⡀⠀⠈⠛⠛⠉⠛⠛⠁⠀⠀⠀⠀⣾⣿⡷⣨⣟⢱⣿⣷
⠀⠀⠀⠀⠀⠛⣫⣥⡻⢟⡸⠿⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣵⣦⡛⣋⣜⣛⠋
⠀⠀⠀⠀⠀⠀⣿⣿⡗⢿⣿⡿⠂⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠈⢿⣿⠏⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢠⣶⡦⢙⣙⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⡇⠀⠀⠈⣡⡛⢛⣬⡉⠀⠀⠀⣾⣦⡀⠀⠀⠀⠀⠀⣴⣶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡁⠀⠀⠘⠿⠃⠈⠻⠃⠀⠀⠀⠸⣿⣷⠀⠀⠀⢠⣼⣿⡿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⡄⢀⠀⠈⠻⢿⣿⣿⣷⣦⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⠀⣰⣶⣶⡄⢀⣀⠀⠀⠀⠀⢿⣿⣿⣅⠀⣸⣿⣯
⠀⠀⠀⢰⣿⡿⠛⠿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟
⠀⠀⠀⣿⣿⡀⠀⠀⣽⣿⠇⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠄⠠⠀⠀⠀⣠⣽⣿⡿⠉⠀⠀⠈⠻⣿⣯⣤⡀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠐⣿⣿⣶⣾⢿⠗⠀⠀⠀⠀⠀⢀⣠⡤⣤⣀⠀⠒⠐⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡤⠖⠻⠒⠢⡤⠈⠳⠦⢄⡀⠀⠀⠉⣹⣿⣷⣄⣀⣀⣠⣾⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣤⣿⣶⣿⠶⢶⣶⣿⣦⠴⣷⠀⠀⠀⠻⣿⡿⢻⣿⣿⣿⠿⣿⠿⠁⠀⢀⣤⣀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠛⠾⠯⢿⣿⡆⠀⣿⠿⠿⠛⠁⠀⠀⣀⡀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠀⠀⠺⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⣿⡟⠀⠀⠀⠀⣿⡇⠀⣇⠀⠀⠀⠀⢀⣾⣿⠃⠀⢀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⢸⣿⠇⠀⢸⠀⠀⠀⠀⠘⠛⠋⠀⠀⠘⠺⠇⢹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠛⠶⠶⠞⠁⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠄⢼⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⣦⠀⠀⠀⠀⠀⣠⣤⣼⣿⢹⣷⣤⣄⠀⠀⠀⠀⠀⠀⢀⣼⡟⠉⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣤⣿⡧⠀⠀⠀⠀⠀⢿⣯⣍⠁⠈⣉⣽⣿⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⣾⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣿⣏⣴⣦⣹⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⠛⠋

⣀⣀⣀⣀⠀⠀⣀⣤⣶⣶⣶⣶⣦⡀
⠀⠉⠙⠻⣿⣾⣿⣿⣗⣼⣿⣯⣭⡝⢷⡄
⠀⠀⢀⣴⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠿⣦⡀
⠀⣰⣿⣟⣿⣿⣦⣴⣿⣿⣿⣿⣿⣽⠻⣽⣾⣻⣆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣿⣾⣷⢮⡻⣿⣿⣿⠶⣄⡀
⠀⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⡽⣿⣿⣿⣻⡷⠙⠉⠓⢿⣬⣻⣄
⠀⠈⣿⣿⣿⣿⣿⣿⡹⣟⣿⣿⣿⡿⠃⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⢿⣿⣿⣿⣿⣷⡝⢮⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠹⡌⢻⣷⡀
⠀⠀⠸⣿⣽⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⣽⣿⣿⣡
⠀⠀⠀⢿⢻⣿⢿⣽⢿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠘⣯⣿⡎⢿⣯⢻⣿⡀⠀⢀⣤⠶⠒⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠘⣿⣿⢸⣿⠘⣿⣿⣄⠀⢀⣤⣴⣾⣷⣿⣿⠉⠛⠛⠋⠉⣿⣷⡀⠀⠀⢁
⠀⠀⠀⠀⠀⠹⢿⣧⣻⣆⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡿⠀⢀⣔⠂⢸⣿⣿⣿⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣯⠙⣾⣻⣿⣿⠿⠟⠉⠀⠀⠁⠀⢀⣼⣿⠘⣿⣿⣦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣈⠧⠻⣟⡷⣄⠀⠀⠀⠀⠀⣢⡿⣽⡇⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣻⣻⣟⠴⠗⠋⠃⠛⠃⠀⠀⠀⢀⡿⢃⣿⣷⢸⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡿⠋⠀⠀⠀⠀⡄⡀⠀⠀⠀⠈⠁⢾⣿⡿⢯⣿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠶⣃⣄⣴⠄⠀⠈⠿⣿⣸⠟⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠡⠟⠐⠲⠶⣤⠄⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⣿⣷⠀⠀⠈⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢸⡏⠀⠀⠈⠓⠠⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡇⠀⠀⢠⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠸⡎⠀⠀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⡀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠄⠨⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢢⣥⠘⠁⠘⠙⢷⡇⠀⠀⠀⠀⠀⠀⠀⢀⠄⡴⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠲⠼⠆⠉⢠⣦⣄⠀⠀⣡⠀⡞⠋⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠄⠀⠀⢀⠴⠋⠀⠀⣀⣀⣚⠀⠀⠸⣿⡯⠁⣼⣿⣶⣧⣦⣤⡴⠆⠙⢆
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠙⣏⠽⠛⠿⣄⡀⠀⠐⠆⠠⢄⣠⢥⡝⠉⠙⢢⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠈⠷⠖⠒⠲⠯⠗⠲⢦⡀⠀⠈⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠧⣤⣄⣤⣄⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢘⠀⠀⣥⣀⣠⠔⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⡏⠉⠈⠢⢀⣤⠤⠒⠋⠀⢤⡩⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠐⠀⠀⠀⠒⠀⢘⠇⠀⠀⠀⠈⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠠⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣽⣿⣄⡀⠰⠀⠀⠀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠿⠿⠿⠟⠛⠛⢛⣛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠚⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠲⣤⡀
⠀⠀⢀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⡇⠈⠳⡀⠉⠳⣄
⠀⠀⠀⠑⠀⠀⠈⠉⠓⠒⠦⣤⣀⢹⡄⠀⠙⣄⠀⠙⢧
⠀⠀⠀⠀⠀⠱⡀⠀⠁⠢⢄⠀⠈⠛⠿⣄⠀⠈⠆⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠂⠀⠉⠢⣄⠀⠈⠛⢦⣄⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠳⣄⠀⠀⠙⣦⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⣈⠢⠀⠀⠈⠣⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠈⠂⠀⢈⣉⡑⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⢀⣤⠀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠈⠀⠁⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠓⠈⠢⢀⡀⠀⢀⠰⠄⡀⠀⠀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠑⣄⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢄⡀⠀⠀⠈⢷⡀⠀⠙⣆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠈⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡜⠀⠀⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡠⠐⠁⠀⠀⣠⠏⠀⠀⣠⠞
⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠤⠴⠚⠁⠀⡀⠈⠁
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠶⠤⣤⣀⣀⣀⣀⠒⠀⠤⠄⠀⠁⡀⠀⡡⠀⡌
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠳⢦⣀⠀⠀⠀⡄⠻⢯⡁⠀⠀⠐⢄⠀⠈
⠀⢸⡇⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠈⠓⢦⣄⠁⠀⠀⠙⢦⡀⠀⠀⠳
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡀
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⠂⠀⠀⠀⠀⢸⡇⠀⠀⢠⠏⠀⠀⠀⠘⠀⡠
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⠞⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉
⠀⠀⢀⣀⣈⡻⣦⣤⣀⣀⣀⣀⣤⡾⢃⠀⠀⠀⠀⣰⠃⠀⢀⡞⠁
⠀⠔⠉⠀⠀⠀⠀⠙⠛⠿⠿⠛⠉⠀⢸⠀⠀⠀⣸⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢀⡟⠀⠀⠀⠻⠀⠀⠀⠙⠓⠖⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠠⢄⣀
⠈⠀⠀⠀⠀⠐⠒⠚⠋⠀⠀⠀⠀⠘⠀⠀⠀⡀⢀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠆⡀⢉⡂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣀⣛⣋⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣤⣀⣰⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣤⣄⣀
⣿⣿⣿⡿⣿⣶⣦⣤⣀⡀⠀⠀⠀⣀⣠⣤⢶⣶⢿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⣶⣿⣭⣿⠩⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⡏⢽⠮⢩⣴⣶⣶⣶⣶⣶⣿⡿⣿⣿⣶⣤⣀
⣍⣛⡛⣿⣿⠿⣻⢻⣿⣿⣿⣿⣿⠽⢿⣏⣫⣗⣤⢤⡾⠻⠾⣿⣿⣿⣿⣼⣿⣿⣷⠘⡿⣿⣿⣿⣷⣤
⣿⣿⣿⣿⣿⣿⡾⣿⢻⣿⣿⣻⠧⢉⡥⢞⣝⣔⡿⣿⣶⣶⡜⢿⣯⣟⠿⠿⠛⠍⠛⠤⣼⣿⣿⣿⣿⣿⣷⣤⡀
⣩⡭⠶⠿⠒⢺⣵⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣻⣿⣿⣿⣿⣯⣐⣍⣉⣋⣇⣀⣴⣷⣿⣿⣻⣻⣻⡟⣻⡿⢟⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣾⡻⡿⢿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡗⢠⠙⠻⣿⡿
⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣽⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣏⠞⡠⠰⠀⠀⡸⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⣿⣿⣿⣻⡟⠉⢉⡁⣠⣴⡟⣿⣷⣉⡯⣴⣠⢃⣂⡌⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣋⣞⠀⠈⠁⠉⠛⠷⠒⣟⢰⠟⠚⠹⣻⣿⣿⣿⣿⣾⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⠺⢶⠄⡐⠁⠀⠀⠀⠀⠀⣠⡷⠏⠁⠁⡈⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣶⣳⢺⣶⢦⣴⣾⡷⠟⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠌⠿⣿⣿⣿⡿⡿⣣⠟⠛⠿⡏⣿
⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢸⣅⣙⣮⠑⠒⠺⡁⠁⢀⡙⣏
⣭⣿⣿⣶⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠁⠈⠤⠄⠀⣀⣵⡇
⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠐⣳⠀⠀⠀⠀⠀⣌⡻⠁
⣿⣷⣿⢙⡟⣿⠿⠛⠻⠟⣿⢯⡻⣿⣿⣿⣟⡷⣮⢆⡀⠀⠀⣾⡿⠃
⣿⣿⣏⣞⣩⠿⢦⢭⡄⣡⠀⢀⣿⠑⢿⣿⣿⣧⠀⢦⠄⠆⣾⣿
⣿⣿⣿⢻⢜⢷⡄⠆⠂⢈⡘⣆⠉⠃⠸⣿⣿⡟⡛⠃⢀⣴⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠉⠉⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣁⡤⠤⠤⠤⢤⣀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠞⣆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣸
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠘⠀⡜⠀⢻⡀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠀⠒⠲⢤⡞⠀⣠⠏
⠀⡀⢀⡠⠴⠒⠚⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⡀⠀⢀⠃⠀⠈⢧⡀⠀⠀⠘⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⢊⣀⠀⢀⠤⡈⠀⠈⢀⠀⠀⠀⠛⢦⣤⡴⠃⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡰⠁⠀⡇⡇⠀⠘⣆⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⢧⠁⡷⡷⠀⢸⠀⠀⠘⢄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠦⠤⠋⠀⠹⣄⣀⡜⠀⠀⠑⡌⠢⣄⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⠊⠉⠉⠉⠈⠉⠉⠙⢢⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣇⠀⠲⠀⠀⢀⡾⠆⠀⡼⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠘⠦⣜⣀⣆⣨⠧⡤⠞⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇⠹⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⡤⠤⡞⠉⠳⣶⢶⡖⠛⡉⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⠁⠀⠹⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠛⠛⠉⠀⢹⠀⠈⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣸⠃⠀⠀⡼⠂⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⡴⠯⠚⢀⡴⠁⠀⠀⣰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡄⠀⠀⠀⠠⡜⠲⠖⠁⠀⠀⡀⠀⣀⡴⢚⠁⢀⠄⠀⣼⠁⠀⠀⢰⠃⠀⢠⠏
⠀⢀⡠⢤⣀⣀⡀⠀⠀⣀⣀⣠⠴⠞⠁⠀⠀⡟⠙⠓⠒⠒⠘⠲⣄⠀⠀⠀⠹⣎⠀⠈⠛⠖⠾⠴⠖⡵⠋⠉⠉⠉⠀⢠⠏
⣰⠋⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣤⠼⠗⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠘⠧⡜⠀⠀⡆⠀⢹
⣇⣠⡇⢠⠀⠀⢀⠀⢀⣀⣀⣤⠴⠞⢋⡥⠖⠒⠊⠉⠉⠉⠀⠀⠀⢀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠒⠚
⠀⠉⠛⠾⢦⣤⡼⠏⠉⠉⠀⠀⠀⠀⡟⢠⠂⠀⡖⠀⠀⣴⠶⠚⠋⠉⠉
⡀⢀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠓⣻⣤⣴⣧⣤⡾⡁⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀⣀⢀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢸⡇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣥⡜⠀⠀⡴⠞⠛⠻⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⣸⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⠠⠾⠟⠛⣸⠀⠀⣿⡗⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⢀⣴⡷⠿⠷⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠻⠀⠀⠈⠉⠉⠀⠀⠐⠑⠛⠀⠀⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⣼⡇⠘⢹⣿
⠀⠀⠀⠀⠀⠀⢀⣿⢸⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⡇⡅⢸⣿⠄
⠀⠀⠀⠀⠀⠀⡸⡏⢸⣧⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠈⠀⠀⣿⡇⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠱⠇⢀⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⢰⠀⠀⢠⣿⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⢸⣾⣇⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⠈⠀⠀⢸⡸⠆⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠁⠀⢰⢸⢸⢻⣿⣆⠀⠃⠀⠀⠀⠀⠀⠐⠀⠘⠋⠀⠀⠄⠀⠀⢠⠀⠈⡇⠛⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢸⢸⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠈⠀⢰⠁⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⢸⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢸⠀⣠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣾⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣸⠀⡏⢸⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠟⠀⡇⢸
⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⢠⠀⠀⢻⢸⣆
⠀⠀⠀⠀⠀⠀⢁⠀⠀⣸⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣸⠀⢄⡿⢸⠌⡆
⠀⠀⠀⠀⠀⠀⡘⡆⠀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⣰⠀⠀⠀⡇⠀⢸⡇⠎⢦⣙⣶⣦⣤⣤⣀⡀
⣀⣠⣤⣤⣤⣤⣴⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢙⠇⠀⢰⠁⠆⠀⣿⣷⣮⣭⣭⣿⣿⣿⣿⣿⣷⣤⣀⡀
⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⢸⠀⠀⢸⠀⢠⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⠀⠀⠀⠀⠘⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⣴⣿⢠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⢸⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣤⣄⣀⣠⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⣀⣀⣀⣾⣿⣿⣿⣼⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⡿⣿⡿⣿⡟⣿⣽⣿⡟⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⣿⣿⠿⠛⠛⠛⣿⣿⣿⣿⣿⣿⠀⠀⡟⠀⠀⢿⣿⣿⣿⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠄⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⡏⠙⡟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢃⠀⠀⠀⠀⢸⠀⡇⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡄⠀⠀⣸⠄⢸⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂⡄⢠⢄⣿⡤⠥⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣇⠸⡇⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠒⠀⠂⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠉⠟⠀⠃⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠀⠀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠚⡄⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠔⠀⠡⠀⠀⡃⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⠀⡠⠛⢀⠇⠨⢹⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣤⡶⠒⠆⠀⡇⠻⠿⠀⡄⠀⠀⠀⠀⣀⣀⡀⠀⠑⢆⠀⢞⠁⢀⡈⠂⢱⠞
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠤⠤⠤⣤⠤⠀⠠⡀⠀⠥⠀⠀⠁⠙⠓⠫⡍⠁⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⣱⣶⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠠⠺⠀⠀⡂⢁⠛⠀⠀⠀⠀⠙⡒⠶⠤⠤⠀⠀⠀⠀⠀⡬⠀⠀⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⢻⡂⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⡄⠀⢿⡃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠒⠤⠦⠐⢀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡔⣹⠃⠀⠀⡜⠀⠀⠈⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠂⠐⠀⠏⠁⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠎⢀⠋⠀⠀⡼⠁⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⣀⣴⠶⣱⣆⠀⠀⠀⠀⠀⢠⡊⠤⠀⠀⠊⠉⡃⠀⠀⠀⡄⢤⠒⠈⠀⠀⠀⠈⠀⡠⢚⠁⠀⠀⠀⠀⡹⢆
⠀⠀⠀⠀⠀⢣⣏⣡⡾⣟⡅⠀⠀⢀⡀⣀⣨⡀⠀⠀⠀⡀⠴⠃⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠈⣠⠂⠀⠀⠀⠀⣤⠦⠼
⠀⠀⠀⠀⠀⠀⠙⠋⠊⠉⠁⡠⠊⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠤⠐⠚⠉⠀⠀⠀⠀⠀⢰⢻⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⢰⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢈⡀⠘⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⢀⠀⠀⠀⠀⡼⠀⢰
⠀⠀⠀⠀⠀⠀⢠⠎⠀⡀⡄⡐⠀⠃⠐⠒⢶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠁⠈⢜⡓⠲⠶⠶⠇⠀⠘⠃
⠀⠀⠀⠀⠀⠀⡎⠀⢈⠀⠘⠓⠢⡀⠀⠀⡴⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠔⢉⠔⢊⠅⠒⠒⣠⠤⠊⠑⢠⠆⢈⡀⢠
⠀⠀⠀⠀⠀⠀⡇⠀⠊⠁⠀⠄⢤⠁⠀⢸⠀⢠⠃⠀⠀⠀⠀⠴⠀⠀⠃⠀⡠⠊⡑⠂⢁⡐⢪⠕⠁⠀⠀⠰⠥⠒⡉⠡⠂⡁⠈
⠀⠀⠀⠀⠀⠀⠈⠢⡑⠤⡄⣠⠘⠀⠀⣈⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠒⠊⢀⠔⢁⠔⠀⠀⡼⠃⠀⠔⡇⣴⢃⡀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡠⣯⠉⣙⣉⡀⠔⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢧⡲⠀⠪⠠⠎⠀⠀⢸⢸⠀⠀⢠⢱⡙⠈⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠈⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⢠⠀⠀⠀⠈⠉⠀⠈⠃⠀⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢣⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⣶⣶⣦⣶⡆⣴⠆⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢼⢯⢶⡶⣽⣝⣿⡟⣭⣻⣿⣷⠘⣋⡼⡱⠓⠺⣄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢿⠿⢺⠿⣗⣯⣿⠵⠟⢻⣿⣯⣾⠿⠟⠻⠿⠷⣹⢶⠼⣥⢿⣿⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡳⢟⣗⣶⣿⠣⠛⠁⠀⠀⠀⠨⠿⣞⡛⢿⠇⠀⠀⠀⠉⠙⠹⣟⣷⣼⠨⣿⢷⡆
⠀⠀⠀⠀⠀⠀⢠⣼⣖⢿⣦⣴⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠒⠇⠀⠀⠀⠀⠀⠀⠀⠙⢫⣭⣟⢿⣩⡇⢄
⠀⠀⠀⠀⠀⣠⣿⠛⢹⢺⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣟⠓⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⣻⡞⢻⣿⣤
⠀⠀⠀⠀⣰⢮⣯⣾⣌⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡃⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⡦⣂
⠀⠀⠀⢀⣛⢻⡿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢯⣝⣛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣭⣔⠿⣸⡂
⠀⠀⠀⢿⣽⠧⣾⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣯⢺⣿⣿
⠀⠀⠀⣀⣺⣥⢡⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⠿⡿⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠳⣮⣾⡄
⠀⠀⠠⠟⣻⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠟⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡹⣜⣫⡅
⠀⠀⢰⣷⠙⣡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⣿⣿⣿⣿⣧⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠿⠿⣧⡆
⠀⠀⠀⣟⣲⢽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣿⣿⣽⣿⣿⣿⡿⢣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢶⣶⣿⡇
⠀⠀⠀⢿⡭⣿⣻⣇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣹⣳⣿⣶⣿⣿⠿⣿⣿⡆⢶⡞⣧⢼⣍⣄⠀⠀⠀⠀⠀⠀⠀⣴⣷⡎⣽⡄
⠀⠀⠀⠸⡶⠘⡷⢞⣧⡀⠀⠀⠀⣀⣶⣽⢍⣲⣸⠳⠋⠀⣿⡿⣬⡿⡇⠀⠶⣴⡾⣶⢭⣴⣄⠀⠀⠀⠀⣴⣮⢉⣼⡛⠋
⠀⠀⠀⠀⠠⣾⣿⢨⣛⠡⠀⣠⣼⡻⠻⣿⣧⠋⠓⠀⠀⠰⣶⢄⡼⢤⠀⠀⠀⠛⠿⡛⣝⡻⢫⣷⡄⠀⣐⣩⣟⢿⣿⡿
⠀⠀⠀⠀⠀⠈⣷⠻⣟⡲⣿⡿⡻⣿⡷⡹⠛⠁⠀⠀⠀⢠⣶⣎⠳⢟⠁⠀⠀⠀⠀⠙⠿⣓⣾⢛⣧⣕⣺⣿⡿⣅⠂
⠀⠀⠀⠀⠀⠀⠘⠿⣬⣽⢿⣿⣶⣿⡿⠀⠀⠀⠀⠀⠀⢀⣭⡊⣚⡾⠆⠀⠀⠀⠀⠀⠀⢙⣿⡿⣿⣿⣤⡃⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣶⣾⣭⣴⢿⣠⡀⠀⠀⠀⠀⢀⡿⡍⣛⣣⡁⠀⠀⠀⣀⣤⡲⢿⠿⣿⣿⣻⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠵⢭⣇⢄⣮⢷⣿⣻⡖⢶⣞⣿⠿⣿⣷⢧⣦⣺⡗⣿⣭⡿⢰⣶⡟⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠅⣈⠉⣯⣷⣻⠿⢿⣾⣿⣿⣻⢴⣟⣻⡇⣩⣰⠎⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠑⠟⠺⠽⠳⠾⠽⠿⠛⠛⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⡦⠀⠀⡀⠀⣠⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⡴⠁⠀⣢⣴⣿⣿⢿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣾⣁⠈⠀⠀⣾⣿⣧⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠇⠀⠀⠀⠀⡡⠿⠈⠗⠰⣿⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠀⠀⠀⠀⢹⠔⠀⠀⣐⠂⣼⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠘⠓⡂⠀⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢶⠶⠶⣶⡆⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠁⠀⠀⠀⢲⣼⡛⠛⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢄⣀⣀⡀⠀⠀⡸⠏⠉⠁⠙⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠔⠁⠀⠀⠀⠀⠀⠈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⡀⠀⠀⠀⠠⢇⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠂⠤⠴⠲⢦⣭⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡿⢿⣯⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⣀⠈⣠⣾⣷⠉⢿⣿⣬⣈⡀
⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠄⠀⠈⠁⣠⣾⠋⠁⢹⡝⠙⢷⣶⣤⣆⠁⠀⡀
⠀⠀⠂⠠⠼⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡘⠋⣿⠀⠀⣸⠇⠀⠀⠉⠱⠄⡀⠀⠠⣀⣀⣄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣠⠀⠀⣀⣼⣷⡾⠟⠓⠒⣿⠀⣀⠀⠈⠃⢧⣽⣾⡉⠉⠙⠻
⠀⠀⠀⠀⠈⠷⢶⣿⡶⠾⠛⠉⠉⢀⠟⠋⠁⠀⠀⠀⠀⠈⢀⠘⠄⠀⠀⠈⢛⠃⠈⠲⠄⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠴⢦⠀⠀⠹⢄⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠛