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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⢠⠲⡄⠀⠐⠀⢀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢃⠀⣄⠀⠀⠌⠀⠀⠆⠀⠀⠉⠀⠀⠀⠀⡄⠀⠰⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠈⠆⠊⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⠄⠑⡀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⠀⠀⠀⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⡀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠀⡰⠄⠐⠁⠀⠁⠀⠀⢀⠀⠀⠁⠀⠀⢀⣀⠀⢀⠀⠼⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⠀⠀⠄⠀⠗⠀⢀⣤⣤⡶⠒⠐⠈⠀⠀⠱⢆⠙⠧⠴⠄⠀⠀⠁⠚⠲⣥⠀⡀⠀⠀⠐⠀⠈⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠰⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⡚⠀⠀⠉⠃⠘⠀⣰⠒⠀⠀⠀⠀⠀⠑⢆⠀⠀⠂⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⢠⣍⠁⠘⣠⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠏⢀⣀⡀⠀⠀⠀⠀⠀⠡⡐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠛⠒⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⠞⠹⠁⠀⠀⠀⠀⠀⠀⠹⡤⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⣢⠄⠄⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡟⠢⠄⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠷⠀⣠⠈⠀⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣏⢁⣄⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠄⠀⠢⢺⠇⠀⢀⡥⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠡⠀⠀⠰⡄⢀⠀⠁⠀⠀⠍⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠤⣥⡌⠀⡀⡿⠃⠀⠀⠀⠀⠀⠁⠌⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠋⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡴⣤⡂⡹⠋⢁⣘⣊⡥⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠐⠈⡄⠀⠀⠀⠀⡘⠀⠂⠀⡀⠀⡄⠁⠀⠉⠰⡀⠒⠤⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⡒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡉⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠂⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⣠⢆⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣄⠀⠀⠀⠀⠀⠺⣄⡀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⡀⠔⠈


⡿⢿⡿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡀⠀⠀⣀⣤⡴⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠿⠋⠁⣠⡾⠋⠀⠀⢀⡄⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣴⠟⠛⠷⠄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣤⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠻⢶⡶⠟⠁⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠈⠳⣤⣀⠀⠈⠁⠀⢀⣠⡾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣴⠞⠛⠛⠻⠦⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢰⡏⠀⣤⣼⣧⣤⣤⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠲⢿⡶⠶⠶⠶⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠳⣤⣀⣀⣠⡤⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠈⠉⠉⠁⠀⣠⡾⠋⢀⣠⣶⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠞⠛⠉⠀⠀⠈⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣟⣻⣿⣛⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠿⠿⢿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠆
⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⣄⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣆⠀⠀⠻⣿⣿⡿⠿⠿⠿⠿⠿⢿⣷⣄
⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠛⠛⠻⢿⣷⣄⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⡇⠀⠀⠀⠀⣶⣶⣿⣿⣶⣶⣶⣶⡆⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠘⣿⡇⠀⠀⠀⠀⣠⣤⣿⣯⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⣇⠀⠀⠀⠀⠙⠛⣿⣿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠛⠛⠛⠛⠛⠻⠿⢿⣶⣤⣀⠀⠈⠁
⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣶⡶⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣷⣶⣦⣀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠈⠹⣿⡆⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣄⣀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠉⠙⠛⠛⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣀⣀⣀⣀⣼⣿⠃⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣤⣤⣤⣤⣼⣿⣷⣄⠀⠈⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣹⣿⡿⠋⠀⠀⠀⠈⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣶⣶⣦⣤⣴⣶⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠓⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠄⢠⢠⠄⢤⠆⢠⠀⢠⠠⡀⡄⠀⡤⠀⠀⢠⠆⠀⡤⠀⠄⢄⢠⠐⢄⢠⢠⠄⡤
⠀⠀⠀⠀⠀⠀⠀⢸⣁⢸⣀⡰⠰⠀⢸⢀⠃⢣⢸⣉⠆⣇⡀⣏⣁⠀⢈⡱⠀⡇⠀⠃⢣⠣⣀⡸⢸⠱⡀⣏⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠻⣿⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠙⢦⡀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⣀⣀⣀⣉⣦⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢹⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢷⣄⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣉⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⡴⠞⠃⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⣠⡶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣄⣀⣾⣉⣀⣼⣧⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⣿⣤⡾⠛⠛⠛⠛⢻⣶⡟⠛⠛⠛⠛⠛⠛⢷⣤⡀⢰⡟⠛⠛⠛⠛⠉⠻⠀⣸⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠈⠇⠀⠀⠀⠀⠸⡇⣽⠀⠀⠀⠀⠀⠀⡄⠘⣧⣾⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠇⠀⣇⠀⠀⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⣠⡆⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⡇⠺⠶⠶⠶⢶⠀⠀⠀⠀⢀⣀⣄⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠻⢤⡀⣿⣻⠁⠀⠀⠀⢞⡧⠀⠀⠀⠰⣄⣀⣀⡀⢸⠀⠀⠀⠀⢻⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣼⠃⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⣷⢸⠀⠀⠀⠀⢸⠌⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⣼⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠇⠀⠃⣸⠀⠀⠀⠀⢸⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢀⡇⠀⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠏⠉⠹⣆⠀⠀⢀⣸⣧⡀⣿⣤⣤⣤⣤⣼⠂⠀⠀⢹⡇⢀⣤
⠀⠀⠀⠀⠀⢀⣼⠋⢳⣶⡶⠒⠈⢁⣴⣿⠒⠒⠒⠒⠖⠊⣠⠷⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠒⠋⢀⣤⣄⠉⠉⠉⠉⠉⢉⣿⣿⡀⠀⠀⢠⡏⠀⠀⠀⠀⢻⠟⠙
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠛⠛⠋⠉⠁⠘⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠙⣿⠛⠛⠛⠛⢻⠉⢻⠛⠛⠛⠋⠁
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡰⠋⠀⠘
⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣠⠔⠋⠀⠀⠀⠀⢺
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠻⠶⣄⣀⠀⠀⠀⠈⠓⣤⣤⣤⣀⠀⣀⣠⠶⠶⠶⢶⣄⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⣠⠶⠶⠦⣴⡏⠀⠀⣀⡴⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣛⣛⣋⣀⣀⣈⣉⣛⣉⣁⣀⣀⣀⣀⣉⣻⣶⣶⣾⣛⣉⣁⣀⣈⣉⣻⣶⣴⣛⣁⣀⣀⣀⣈⣙⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⢀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡠⡀⣄⡀⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠓⠋⠋⠃⠛⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠿⠰⣉⠸⠺⠸⠧⠟⠾⢮⡥⢿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠦⡀⠀⠀⠀⡖⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠶⣣⠄⠀⢤⡘⢢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⣹⢁⠉⠈⠃⠹⢎⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁⢸⠸⠘⠶⣓⣒⡳⢚⠥⡀⠀⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⢦⣉⡆⠀⠀⠀⢇⣱⡸⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠆⠀⠀⠀⣀⠀⠀⠀⡀⠀⣠⣾⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷⠂⡐⣞⢂⠀⠀⠋⠨⠀⠁⣁⣥⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠌⠩⠾⠈⡑⠄⠁⢀⡀⢸⣿⣸⣿⣸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⠀⢆⡠⢧⠀⡧⠤⠥⠬⠭⠭⠭⠭⠥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠉⠀⠀⢰⠀⢸⡀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠴⠤⠴⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡅⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⣀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠐⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣠⣾⣿⣿⡿⣿⣿⣿⣦⣌⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣴⣿⣿⣿⢿⣿⣿⣷⣤⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣴⣿⣿⣿⣿⣍⡁⢉⣽⣿⡿⣿⣷⡌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⣿⣏⠃⠸⣿⣿⡇⣿⣿⣿⠆⢋⣽⣿⡄⢻⡄⠀⠀⠀⠀⠀⠀⢀⡟⢠⣿⣯⡙⠻⣿⠟⠛⠛⠻⢿⡿⢛⣹⣿⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣧⣶⣶⣬⡙⠃⠛⣉⣴⣶⣦⣽⣿⣷⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣷⠃⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡛⠻⠟⢋⣡⡄⣤⣍⠛⠿⠟⣻⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⡿⣆⠀⠀⠀⠀⢀⡿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠹⣿⣏⡄⢸⣿⣿⡇⣿⣿⣿⡆⣬⣽⣿⠃⣾⠁⠀⠀⠀⠀⠀⠀⠈⣷⠘⣿⣯⣥⣶⣿⣶⣤⢤⣴⣿⣷⣦⣹⣿⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠙⢿⣿⣿⣿⣋⡄⣈⣹⣿⣿⣿⡿⢁⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡙⠻⠿⣿⣷⣿⡿⠿⠛⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣈⠛⠿⢿⣿⣾⡿⠿⠟⣋⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣯⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠛⢿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⠟⠉⠁⣀⣀⠈⠉⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⢴⣾⣶⣤⣿⣿⡏⣸⣿⣿⣿⣿⣿⡿⠁⠠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠙⠻⢿⣿⣿⠁⣿⣿⣿⣿⣿⣤⡄⠀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢹⣿⡀⣿⣿⣿⣿⣿⠛⠀⠀⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠿⣿⠿⠀⣸⣿⣇⢹⣿⣿⣿⣿⣿⣷⡀⠘⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣤⣾⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠉⠉⠀⣀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣀⠀⠀⢀⣤⡴⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠿⠛⠁⣠⡾⠋⠀⠀⣀⣀⡀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⣾⠋⠉⠻⠇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣿⣄⣀⣀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣹⡏⠉⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣠⣿⠶⣦⣤⣴⠄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⣀⣠⠾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⣠⡶⠋⠉⠀⢀⡀⠀⠉⠻⢦⡀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣴⠿⠷⣦⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⢷⣤⣴⠟⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠘⠁⠀⢀⣠⡾⠋⢀⣤⣶⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠚⠋⠁⠀⠀⠈⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⣠⠞⣉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⡙⢦⡀
⠀⠀⠀⣿⢸⠁⢹⠀⡇⡒⠒⠀⠀⠀⠀⠀⢸⠁⡏⠈⡇⢹⠠⠤⠄⠀⠀⠀⠀⠀⡇⢸⠁⣿⢸⡇⡒⠒
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠂⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣄⣿⢸⡇⠂
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠥⠤⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢐⣀⡀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠥⠤
⠀⠀⠀⡟⢸⠀⢸⠀⡇⡁⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⡁
⠀⠀⠀⣷⢸⠀⢸⠀⡇⣂⣀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⣂⣀
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠆⠀⠀⠀⠀⠀⠀⢸⠀⣷⣾⡇⢸⢈⠉⠁⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠆
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠅
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡋⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠰⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡋⠉
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡂⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡂
⠀⠀⠀⣿⢸⣷⣿⠀⡇⠍⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢘⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠍⠉
⠀⠀⠀⣿⢸⣿⣿⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠅
⠀⠀⠀⡏⢸⣿⣿⠀⡇⠒⠒⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠤⠄⠀⠀⠀⠀⠀⡇⢸⣿⣿⠐⡇⠒⠒
⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄⠀⠀⠀⠀⠀⢠⠞⣠⣿⣿⣧⡘⢦⠀⠀⠀⠀⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄
⠀⢸⡇⣾⣿⣿⣿⣿⡆⢻⠀⠀⠀⠀⠀⡟⢰⣿⣿⣿⣿⣷⢘⡇⠀⠀⠀⠀⢸⠃⣿⣿⣿⣿⣿⡆⣿
⠀⠈⢧⠙⢿⣿⣿⠟⢡⠟⠀⠀⠀⠀⠀⠹⡌⠻⣿⣿⡿⠋⡼⠁⠀⠀⠀⠀⠈⢧⠙⢿⣿⣿⠟⢡⠏
⠀⠀⠀⠙⠶⠤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⣧⡀⠀⢸⣷⡀⠀⣄⢸⡇⠀⠀⡀⠀⠀⣴⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢹⣷⣄⣨⣿⣷⣶⣿⣿⣷⣤⣾⣁⣠⣾⠇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⠟⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⢸⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣏⠀⠀⢀⣀⣠⢤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠋⢠⣿⣿⣯⣻⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣤⣠⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡻⢯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⣿⡻⠋⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠿⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠛⠉⠙⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡝⠘⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⣸⣿⠟⢻⡿⢿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⠎⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢠⡟⠁⠀⠚⠁⠀⣿⠟⡏⢀⡹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣠⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠈⠉⠉⠉⠉⠉⠰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⡄⢶⠋⠉⠉⣙⢻⠋⠑⠶⣤⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠋⠹⣄⢳⡀⠀⣿⢸⠀⠀⠀⣠⠟⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⠏⠀⠀⠀⠙⣆⠙⣦⡁⠸⠀⡴⠚⣡⠞⠁⢻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠄⠀⠲⠶⠒⠒⠒⠒⠒⠒⠒⠲⠂⠀⠀⠄⢠⡏⠀⠷⠠⠤⢤⣤⣤⣬⣧⠘⠁⠘⢁⣀⠊⠁⠀⠀⠀⢳
⢀⣠⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠗⠒⠲⠢⠤⣤⣤⡀⠀⡀⠀⠈⣉⠉⠐⠒⠒⠂⣾
⣿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣴⣋⣤⡾⢻⣆⠙⣏⠉⠉⠉⠉⠀⢶
⣿⠀⠀⠀⠉⠙⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣵⣤⣵⡦⢲⡶⠛⡩⠛⠉⡄⢸⡟⢶⡈⢷⡀⠀⠀⢰⠇⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⢂⡀⠈⠀⠀⠀⡇⢸⡇⠀⠙⣎⡱⣄⣰⠋⢠⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⠤⣀⡀⠀⣧⢸⡇⠀⣀⣨⡷⠉⢁⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠉⠁⠘⢻⡟⠈⣉⣁⣤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢺⠟⠀⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⣷⢠⡞⠙⠛⠦⠴⠦⠤⠖⠛⠛⠉⠉⢉⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠋⠉⠙⠛⠶⣄⠀⢠⡏
⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠈⢣⠉⠛⠳⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇
⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⣠⠂⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇
⠀⢸⡆⢿⠀⠀⠀⠀⠀⣾⠀⢀⣀⠴⠚⠉⢣⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⡇⢸⡄⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⣷⠈⡇⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⠶⠖⠒⠐⠀⠒⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⢹⠀⣧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⣀⠴⠞⠁⠀⡧⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠑⠶⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢠⡏
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⢟⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⡏⢸⡇
⠀⠀⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠳⢮⣵⣏⠀⠀⠀⠀⠀⠀⠀⢸⠅⡇⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠘⡆⢿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢐⣇⡀⠀⠀⠀⢹⡗⠶⠠⠤⠤⠀⠀⠀⠶⠁⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⠀⠀⢰⠋⠀⠀⢠⠏⠀⠉⠳⢦⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⢃⣦⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⡄⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢻⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⢦⣀⠈⠙⠛⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁⣀⡤⠾⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠛⠳⠶⠦⢤⣤⣀⣀⣀⣀⣤⠤⠴⠶⠛⠛⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢀⣀⣀⣀⣀⣀⣀⣀⡤⠶⠶⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣠⡀⠦⢀⣠⣤⣤⣄⡀⠴⢀⡄⠀⠀⠀⠀⣼⡄⠀⣀⣤⣤⣄⡀⠀⣼⡄⠀⠀⠀⠀⡀⣆⠀⢀⣠⣤⣤⣀⠀⢠⡆⡀⠀⠀⠀⠀⣠⠄⣠⣴⣶⣶⣦⣄⠠⣄
⣤⠍⣰⠋⠉⠛⠛⠉⠙⣆⠩⡤⠀⠀⠀⠀⡉⢠⠞⠉⠙⠛⠛⠉⢦⠈⡁⠀⠀⠀⠸⣇⢛⡼⠉⡉⠛⢛⡉⠳⡜⢠⡇⠀⠀⠀⢈⠇⡾⠁⣤⡉⢉⣤⠈⢧⠸
⠀⠀⡇⠀⠘⠆⠰⠧⠀⢼⡀⠀⠀⠀⢘⡿⠀⣿⠀⠨⠶⠠⠮⠀⢸⡇⢨⣿⠂⠀⠀⠟⢸⡅⢮⢾⠄⢾⣿⠀⠇⠘⠀⠀⠀⠀⠀⣠⠗⠘⠿⢃⡐⠿⠃⠺⡄
⠀⡞⠀⢠⢖⣂⣐⡢⡄⠀⣳⠀⠀⠀⠀⠀⢸⠁⢀⡴⣖⣖⡲⢄⠀⢹⠀⠀⠀⠀⠀⢠⡇⠀⡠⣲⣤⣔⣦⡀⠈⡆⠀⠀⠀⠀⠀⣯⠀⢰⡫⠒⠒⢝⡆⠀⣽
⠀⠻⣄⡑⢾⠈⠁⡷⢂⣠⠋⠀⠀⠀⠀⠀⠘⢦⡘⠣⠓⠒⠣⠟⣀⠞⠀⠀⠀⠀⠀⠀⢳⣀⠳⠍⠛⠛⠕⢃⡴⠃⠀⠀⠀⠀⠀⠘⠦⣬⢁⠁⠐⡈⣧⠔⠁
⠀⠀⠀⠉⠛⠓⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠛⠉⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠐⡀⠻⠟⣀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣰⠞⠳⣆⢠⣾⠁⣿⠆⢰⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣾⠉⢹⡆⠀⠀⠀⠀⠀⠀⢠⡏⡉⠳⠆⢠⠵⢞⣂⣠
⠀⠈⠁⠀⠿⣿⣧⠀⠁⣶⠀⠀⠀⠀⠀⠀⠘⠳⠞⠃⠀⠉⣻⣶⡶⢶⡅⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⣿⢸⣞⡇⠀⠀⠀⠀⠀⠀⢸⡇⣿⢠⣄⠐⠶⠭⡭⠉
⠀⠀⣷⠀⣷⢿⣿⣧⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢦⠑⣦⢀⣿⣿⠀⠸⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⢺⡿⡇⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⠀⠀⠊⠓
⠀⢸⡏⠀⢡⡾⠁⢻⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⣆⢀⣾⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣼⢱⣿⣳⣹⡆⠀⠀⠀⠀⠀⡎⣰⣍⡈⣷
⠀⠈⠃⠀⢸⣇⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠹⣮⣛⣋⡼⠁⠀⠀⠀⠀⠀⢳⣝⣛⣡⠟
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠲⠟⠳⠖⠂⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠻⣶⣶⣾⠛⠲⣤⡀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⣉⡉⢹⡇⠀⠀⠀⠀⠀⠀⠀⣠⠶⣺⡿⢿⡗⢶⣄
⠀⠀⣴⠾⢷⣦⣴⣾⣷⣦⠀⠀⠀⠀⠀⢺⡍⠀⣀⠀⢈⣉⡀⢀⡀⠀⣹⠆⠀⠀⠀⠀⠀⣧⢰⡶⢿⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⡼⠃⣰⣿⠃⠸⣿⣆⠘⣧
⠀⠀⣿⣦⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠛⣿⠰⣏⣉⡷⢸⡟⠛⠃⠀⠀⠀⠀⠀⠀⣿⠘⣿⣾⢻⡇⣸⠀⠀⠀⠀⠀⠀⠘⠓⠒⠟⠛⢶⡶⠛⠛⠒⠛
⠰⢤⠈⢿⣿⣿⣿⣿⠟⢁⡤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⢉⣿⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠘⠒⣄⠙⠿⠟⢋⣰⠾⠃⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⣤⣄⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣙⣛⣛⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⣤⠴⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣀
⠀⠀⢰⠏⢀⠀⣲⠀⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⢛⣫⣤⣌⡛⠂⣀⠀⠀⠀⠀⠀⠀⢠⠞⣉⡀⠀⣴⡄⣩⠷⡄
⠀⠀⣼⠀⡞⠸⡏⢰⡏⢸⡇⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣾⣧⠀⢸⡇⠀⠀⠀⠀⠀⠘⠿⢡⡟⠁⢰⠀⠙⣆⠻⠏⠀⠀⠀⠀⢠⡏⠀⠙⣡⡶⣢⠞⢡⡄⢹
⠀⠰⢧⡀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣭⣥⣴⡿⠁⠀⠀⠀⠀⠀⢀⣤⠸⡇⠀⠈⠉⢀⡟⣠⡀⠀⠀⠀⠀⠸⡇⠴⡆⣫⢞⣵⡟⡈⠀⢸
⠀⠀⠘⢧⢤⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠃⣙⠳⠶⠶⢋⠀⠛⠋⠀⠀⠀⠀⠀⠳⣤⠞⢱⡦⠀⠘⠛⣠⠟
⠀⠀⠀⠀⠸⣧⡤⠤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠶⠚⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠢⠤⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣫⣭⣭⣭⣝⣛⠿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣘⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⠷⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠤⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⡟⣭⣭⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠠⣽⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣍⣉⣁⠀⠀⠀⠀⠀⠀⠀⣸⣁⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠏⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⡻⣿⣛⣿⣿⣿⣿⣿⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢧⠛⠤⣀⡀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣙⣥⣛⠿⢿⣿⣿⣿⡿⠿⣛⣵⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠠⠀⠀⠀⠀⢠⠀⠘⠩⠴
⠀⠀⠀⠀⠀⠀⠠⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⢀⠤⠀⠀⠀⠀⠀⠀⠓⠊⠀⢈
⠀⠀⠀⠀⠀⠀⠀⣭⡐⠆⢄⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⢠⢀⠠⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢩⠣⠈⡷⠀⠀⠘⡆⠀⠀⢤⣈⣰⢄⣢⢰⣄⠀⢘⠀⢠⡀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠰⠃⠀⠀⠀⢳⠀⠈⠠⢡⠀⢀⠞⠍⠁⠠⠮⠷⡜⢷⢾⣶⣤⡴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⢚⢻⠘⡆⣄⡄⠀⡼⡀⠀⢀⢸⠀⡀⡆⣀⣆⣂⣲⣀⣈⢀⢀⡉⢋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠠⢄⢸⣬⣼⣴⡇⣶⢅⠀⣐⡇⠀⡈⡰⠃⠂⡆⠁⠓⠂⢸⠐⠘⠐⢨⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠤⠁⠃⠘⠈⠀⠈⢇⠀⠀⠈⠗⠒⠧⠂⢨⣴⠉⠴⠘⠄⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⣄⡀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣰⠉⠉⠓⠲⢤⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⢢⠀⠘⢶⡖⠢⢄⣀⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣰⢿⣦⣠⠤⠬⣃⣀⡜⠀⣔⣉⡴⠷⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣳⠦⠤⠬⣿⡆⠀⠀⠀⣀⠝⠀⢰⣷⣏⣀⣀
⠀⠀⠀⠀⡜⣱⠋⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⠠⡤⠈⠓⠒⠒⠉⠁⠀⠀⡷⢋⣡⡤⡌⢳⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⠋⠠⢁⠃
⠀⠀⠀⠀⠀⠉⠀⠑⠦⣥⣈⣁⣈⣀⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠁⠇⠀⣀⠤⠒⠊⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⢹⠛⢲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠖⣉⠀⠄⢀⣀⣸⡯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡰⠧⠤⢄⣀⣙⣢⣾⣠⣏⣙⣆⣀⣀⣠⣤⠤⠤⠤⠞⣆⠿⢋⡀⠒⠒⠢⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢙⡇⠠⡄⡈⠑⣄⢀⣴⠿⣦
⠀⠀⠀⠀⡰⢃⠖⣠⠞⣁⣸⠧⠤⠿⠭⠭⠭⠭⢉⣉⡉⢉⡩⠟⠉⠀⠀⠀⣠⠞⡇⠰⠁⠇⠀⠘⢟⠗⠕⠀⠁
⠀⠀⠀⠀⠃⣏⢰⡁⢊⡔⠀⣀⣀⣠⣤⠤⠶⠒⠒⣲⡞⠋⠀⠀⠀⠀⣠⠞⠁⠀⢿⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⢹⠀⠈⠓⠦⣀⣰⠁⡇⠀⢀⣠⠴⠋⠁⠀⠀⠀⢸⡀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⡼⣸⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣜⠀⢇⠻⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⢘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡿⠿⢷⡶⠒⠒⠒⠒⠲⣶⡶⠶⣶⠟⠋⠉⠉⠁⠀⠈⠉⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢾⠀⠀⠀⠀⠀⠀⣧⡤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠭⠿⠀⢯⠀⠀⠀⢀⣔⠾⣕⡶⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠛⠒⠩⠥⠤⠬⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⡿⢿⣿⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣄⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠛⠛⠛⠛⠿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠛⠛⠛⠿⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⢀⣤⣤⣤⣤⡀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⣶⠀⣶⠀⠀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⢀⣾⡟⠉⠉⠙⣿⡆⠀⠀⠈⢿⣧⠀⠀⠀⠀⣼⡿⠁⠀⠸⠿⣿⣿⠿⠿⣿⣶⡄⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢸⣿⠃⠀⠀⠀⠀⣿⣿⠀⠀⣸⣿⡇⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠿⢿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠿⠿⢿⣯⣄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢸⣿⡄⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡆⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠠⣾⣿⣿⣶⣶⣶⣶⣦⠀⠀⢀⣾⡟⠀⠀⠀⠀⢻⣷⡀⠀⢰⣶⣿⣿⣶⣶⣾⠿⠛⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠈⠀⠀⠀⠈⠉⠉⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠿⠀⠿⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣶⣤⣤⣤⣤⣶⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣤⣤⣤⣶⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣶⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠙⠻⢿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣷⣾⣿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⢉⣉⣠⣄⣉⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢠⣶⣴⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠘⠟⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⡀⢲⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣈⣉⠙⢋⣉⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠉


⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣷⣄
⠀⣴⣿⣿⣿⣿⡟⢉⣠⣤⣤⣄⡉⠻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣄⠹⣿⣿⣿⣿⣿⣿⠏⣠⣿⣿⣿⣦
⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⠘⢿⣿⣿⡿⠃⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡈⢻⡟⢁⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣤⣿⡇⢠⣤⣤⣿⣿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣦⣈⠙⠛⠛⠋⣁⣴⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠿⠇⠸⠿⠿⣿⣿⣿⣿⣿⠟
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⡆⢰⣶⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⣦⣤⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡗⠒⠀⠀⠀⠀⢰⡋⠉⡆⠀⣠⠞⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠚⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡏⠉⠉⠉
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⣇⣀⣀⣀⣀
⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡧⠤⠤⠤
⠀⠃⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢸⠀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⣴⣶⣿⣷⣶⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⣿⡟⠉⢻⣿⣿⣿⣷⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⣿⣿⣶⣿⣿⣿⣿⣿⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⠛⠿⠟⠛⠁⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣤⡤⠔⠚⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠌⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣌⢸⡇⢀⣀⠠⠤⠐⠒⠒⠒⠒⠲⠤⢤⣀⠀⣿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣃⡼⠛⠉⠀⠀⢀⣠⠀⠁⠀⠀⠀⢀⣀⠈⠙⢻⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠿⣄⡀⠀⠀⠀⢋⠄⠒⠒⢄⡀⠒⠒⠠⡹⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⢤⡀⠀⠀⠀⡞⠀⠀⠀⠈⠷⠤⠘⠐⠁⠀⣤⡀⠀⠀⣠⣄⠀⠐⠰⡦⠏⠻⡀
⣦⣄⡀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠉⠢⡀⢀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠐⢀⠉⢀⠀⠃⠚⢅⡠⠂⠀⠀⠄⠀⡇
⢹⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠙⠦⣀⣠⡴⠧⠙⠢⠄⠀⢀⣀⡀⠀⠀⠀⠀⣇⣠⡴⢦⡀⢀⠇
⢸⡆⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢛⠛⠛⢿⡀⠀⣀⠤⢾⠁⠀⠀⠐⠠⠀⡈⠙⠻⠶⠶⠖⠊⠀⡀⠠⠀⣻⠁
⠀⣇⠀⣀⡠⣶⡿⠛⠁⡠⠀⠀⠀⠀⠀⣠⠝⠋⠀⠀⣸⣇⠠⠀⡀⠀⠀⠀⠈⠉⠁⠈⠉⠁⢀⠠⠐⠈⣸⠳⢤⡀
⠀⠉⠁⠀⠀⠘⡆⡠⠊⠀⠀⠀⢀⡤⠊⠁⠀⠀⢀⡾⠁⠈⠲⣄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢧⡀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⡖⠒⠲⠒⠚⠉⠉⠀⠀⠈⣷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠶⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡾⠓⠤⣀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠙⠻⠿⠿⢷⡚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣶⠿⠋⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢠⡇⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣿⣷⡈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠋⣷⡞⠉⠀⠀⡶⢤⣀
⠀⠀⠀⠀⠓⠶⠶⠖⠋⠀⠀⢻⡀⢀⣠⠇⠈⢳⣦⣀⡀⠀⠀⠉⠙⠛⠲⠲⠶⠶⠶⠶⠶⠶⠒⠛⠉⠉⠀⠀⣰⠟⢧⠀⠀⢀⡇⠀⠈⠓⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⡇⠈⠙⠲⠦⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠋⡇⠀⠈⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⡓⠒⠈⠉⠉⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⢳⠆⠀⠀⠀⠀⠠⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠒⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣁⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠃⠂⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠨⠈⠏⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧

⡏⠛⠛⠉⠉⠉⠉⠉⢉⡉⢉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⡀⣀⠀⠀⠀⠀⢰⡆⢰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄
⡇⠈⠒⠃⠚⠀⠈⠉⠉⢸⡄⢸⠀⠉⠉⠁⠀⠐⠂⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠁⠀⠀⠉⠒⢦⡀
⡇⠐⠂⠃⠜⠀⠁⠈⠉⢰⠆⢸⠀⠉⠉⠀⠀⠐⠀⠟⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡐⣁⡀⣠⢤⠤⡄⡄⣤⢠⣄⢙⣆
⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡔⠋⠈⠉⠁⠀⢀⣀⣀⣀⡀⣀⣈⣀⡉⣣
⡇⠘⠓⠃⠛⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⠘⠃⠛⠀⡇⠀⠀⠀⠀⠀⠀⣥⠐⠐⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⡇⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐
⡇⠀⠈⠿⠃⠀⠀⠐⠀⢨⠀⢸⠀⠐⠒⠀⠀⠻⠌⠂⠀⡇⠀⠀⠀⠀⠀⡀⢰⠀⠈⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⠀⠳⡄⠀⢀⣀⠠⠤⠐⠀⠀⠀⠀⢀
⡇⠀⠼⠣⠇⠀⠒⠒⠒⢈⠀⢸⠀⠒⠒⠂⠀⠿⠧⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠈⠓⢤⣀⣀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⣤
⡇⠀⠀⠛⠃⠀⠒⠒⠒⠀⠀⢸⠀⠒⠒⠂⠀⠙⠛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⣀⣤⣲⣄⣭⡟⠁⠉⠻⠀⠀⠁⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠄⢸⢀⡤⢄⠀⡇⠀⠀⢸
⡇⠀⢴⠴⠇⠀⠒⠒⠒⢈⠀⢸⠐⠒⠒⠂⠀⣤⠼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠌⠳⣄⡼⠉⠐⢦⠇⡇⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡘⠀⠘⡇⠀⣄⠀⠀⢣⠀⣀⣴⣷⡄⠂
⡇⠀⠸⠿⠇⠀⠒⠒⠒⢀⠀⢸⠐⠒⠒⠂⠀⠷⠽⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠀⠈⠻⡀⠏⠀⠀⠈⠻⣿⣿⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢈⠀⢱⣎⠀⠰⠀⠀⠈
⡇⠀⢤⣴⠇⠀⠒⠒⠒⠀⣶⣾⠐⠒⠒⠂⠠⣤⡄⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⠀⠀⢻⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⠀⠀⠀⠁
⡇⠀⢰⢴⠆⠀⠒⠒⠒⠀⣿⣿⠀⠒⠒⠂⠀⡤⠤⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃
⡇⠀⣰⠄⠆⠀⠀⠀⠒⠀⣿⣿⠀⠂⠐⠀⢀⡆⡖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠰⢄⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠠⠀⡶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⢀⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢸⡅⠐
⡇⠀⠀⢴⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠄⠀⠀⣤⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⢀⣨⠟⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⡇⠀⠀⠀⠀⠀⠀⣤⣟⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠿⠭⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⠴
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣭⣥⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣾⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢻⣿⡿⢿⣿⠿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢾⣿⠀⣿⡏⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡈⢿⣦⠙⣿⡄⢻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠙⣧⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢃⣾⡟⢠⣿⠇⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠹⠿⠿⠿⠿⠿⠿⠿⠿⠋⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣶⣀⣀⣀⣀⣀⣀⣀⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉



⣶⣶⡶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣷⣶⣶⣷⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣦⣄⠘⢻⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⣠⣴⣶⣶⣤⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⢀⣠⣾⣟⣁⣀⣀⡀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢈⣿⣿⣛⣛⣛⣛⡃⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⠀⠀⠀⠀⠀⣼⣿⡿⠀⠘⠛⣿⣟⡛⠛⠛⠃⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢾⣿⣶⣶⣶⣾⣿⠟⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠈⠛⠿⠿⠿⠟⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠉⠙⠿⠟⠋⠁⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠁⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠛⠁⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠘⣿⣿⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠙⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⠞⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡀⠀⣰⠇⣰⡇⠀⣰⣆⠀⠀⠀⣠⡶⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢺⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⡏⠀⢠⡇⡜⢱⣣⠎⠿⡟⠀⡠⠊⢰⠇⡴⠚⡇⡴⣿⠀⣠⡆⠀⠀⠀⠀⢳⡀⠘⡦⢘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠇⠀⢸⠏⠀⠸⠃⠀⠀⣷⠊⠀⠀⢸⠏⠀⠀⠟⠀⣿⠞⢹⢃⣤⠀⠀⠀⠈⡇⠘⣇⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣸⠀⠀⣾⠀⠀⠀⢀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠏⢹⢀⢼⡇⠀⣿⠀⢇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡼⠁⠀⢀⣿⣤⣴⠾⠟⠋⠉⠛⣆⠀⠀⠀⠀⣀⣽⣦⣄⡀⠀⠀⠀⠸⠋⢸⠃⠀⣿⠀⠘⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠟⢸⠁⠀⠀⠀⠀⢀⢸⡆⠀⠀⡜⠁⠀⠈⠳⣝⢦⡀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢻⠀⠀⠀⠀⠰⠿⣸⠁⠀⣸⠀⠀⠀⠀⠀⣈⠙⡿⠒⠀⠀⢸⣠⠼⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⡼⠁⠀⠀⠀⠈⠳⣤⣄⣀⡤⠞⢁⡀⠀⢿⡀⠀⠀⠀⠰⠟⣠⠇⠀⠀⠀⣿⣣⢚⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠋⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⣭⠀⠀⠀⠉⠓⢦⣤⡤⠞⠁⠀⠀⠀⢸⠛⢋⡽⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣗⠺⠭⣄⠤⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠦⣄⡀⠀⠸⣆⠀⣠⣄⣤⡴⠶⢻⠖⢦⠞⣯⠤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢯⡀⠀⠀⠀⠀⣀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠶⠶⠯⢤⡶⠀⠈⡻⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣤⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢽⣭⢭⡭⠭⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠠⠮⠤⠖⠊⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣃⣠⠴⠋⠁⠀⣀⠀⢦⣤⣉⣓⣦⣤⠀⣀
⠀⠀⠀⠀⠰⡇⠀⠀⣻⠄⠀⠀⠀⠀⠀⣀⣀⣀⡈⠳⣄⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⢒⣒⡿⠿⣍⠹⡄⢠⣄⠀⢹⠓⢬⣦⡀⠉⣩⠗⢴⠋⢠⡄
⠀⠀⢀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠀⠚⠯⢤⣉⡉⠙⠛⠒⢶⣶⡴⢲⣄⡤⣄⠀⣜⠉⠀⠀⠀⠈⠳⡝⣼⠀⠳⣼⡀⠀⠈⠉⠀⣳⢀⡼⠁⣾
⠀⠀⠈⢣⡀⠈⣢⠀⠀⠷⢤⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠈⡇⢧⠀⠀⠀⠙⠉⠙⠖⠚⠳⠒⠲⠖⠿⠈⠀⠀⠈⠁⠀⠀⠀⢼⠁⣨⠇⢶⠁
⠀⠀⠀⠀⡇⠀⠙⢦⠀⠀⠈⠳⢦⣤⡤⠶⡄⠀⢀⡀⠈⢦⣤⣇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠈⣧⠀⠼
⠀⠀⠀⠀⠘⡆⠀⢎⠀⠀⠀⠀⠘⣧⠀⠀⠙⠋⠉⠓⠒⠋⠉⠉⠓⢦⢤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠋⠀⡴⠁
⠀⠀⠀⠀⢀⠃⠀⠈⣷⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠙⠚⠙⠲⠛⠶⠶⡤⢦⠖⢶⡿⠙⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⡖⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣟⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠉⠉⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⢩⣄⡀⠀⣼⡇⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⢸⡆⢿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⣾⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠁⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠛⠳⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⢀⣶⠀⣼⣧⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣀⣀⣀⣠⣴⠟⠁⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⢸⣿⡏⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠿⠇



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠸⣿⡇⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⢀⣤⣤⣤⣄⠺⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⣿⣿⣿⣿⣿⣷⠀⣠⣤⣄
⠀⠀⠀⠀⠰⠶⠳⣦⠀⠀⠘⢷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⠃⠸⣿⣿⣿⣿⣿⡿⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠙⠻⠿⠿⠛⢡⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣷⡀⠀⢿⣅⡀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠺⡿⠋⠀⠀⣶⣆⠀⠈⠻⠿⠃
⠀⣤⡶⢷⡄⠀⠀⠀⠀⠀⢻⣧⣄⠀⠉⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠿⠿
⠀⠁⠀⠈⢿⣤⣴⠶⣦⡀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠃⠀⠀⢸⣿⣿⡏⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣤⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣾⣿⣿⣿⡿⢿⣿⣿⠟⠁⠀⠀⠹⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡆⢀⠾⠶⠶⣤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⢀⣠⣿⡿⠿⠋⠁⠰⠿⠟⠁⡀⢰⠀⡀⠀⡄⠙⠿⢿⣿⡇⠘⣿⣿⣿⡇⠀⢀⣤⡶⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀⣀⡼⣧⡤⠞⠀⠆⠀⠀⠀⡀⢸⠀⠻⢤⣥⣬⠅⠀⠉⢳⣾⣿⣿⣿⠃⠀⠀⠀⠙⢳⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢈⣉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠆⠀⠂⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡼⡏⠀⠠⠤⢤⣤⠀⠀⠀⠀⠀⠀⢀⣤⡤⠦⠄⠀⠀⡀⠀⢸⠟⠋⠛⢧⡀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⠃⡇⠀⠠⠞⠉⠀⠀⢀⡖⠀⠀⠀⠀⠀⠉⠙⠆⠀⣸⢳⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⢷⣀⡴⠃⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⢀⠏⠀⠀⢠⡏⠸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⡜⠁
⠀⠀⠀⢀⡞⠀⠀⠀⠈⠳⡾⠃⠀⠀⠀⠀⢧⠀⠀⠀⣴⠒⠒⠒⠒⠒⠒⠚⠀⢶⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⣼⢁⡾⡆⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⠣⡄⠀⠙⠶⠒⠚⠉⠉⠒⠶⠒⠋⠀⠀⠀⠀⣠⠎
⠀⠀⢰⠇⣾⠁⢹⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿
⠀⠀⡾⠀⠻⠶⠟⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⢤⡤⠤⠦⠒⠛⠉⠁⠀⢹⡄
⠀⢰⠃⠀⠀⠀⠀⠀⣇⣠⣤⣤⣤⣶⣶⣶⡶⣾⡿⢿⣿⣿⣿⣻⣿⡁⠀⠀⠀⠀⠀⠀⠀⣠⣟⣛⣿⣿⡶⣦⣤⣀
⠀⡸⠀⠀⠀⠀⠀⠀⠀⣸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣤⣤⣴⠶⣶⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣷⣦⡀
⠀⡇⠀⠀⠀⢀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣆
⠀⠱⣄⠀⢠⠟⣇⠀⠀⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣆
⠀⠀⠈⠉⣿⠀⣽⠦⠤⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⠃⠀⠀⢹⣞⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣽⣯⠉⠉⠛⠻⠾⣷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣄⠀⠀⠀⠀⠀⢀⡞⢹⡀⠀⠿⣿⣿⣿⣿⣿⣿⣯⣿⠛⠿⠿⠿⠿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣟⣿⣦⣀⠀⠀⠀⢸⣁⣸⠇⠀⠀⠈⠙⠻⢿⣽⣿⣿⠏⠀⠀⡀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣻⣯⣻⣶⣤⣀⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣸⠹⡄⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣷⣦⡀⢠⡿⡆⠀⠀⠀⠀⠀⠀⣇⠀⡇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠁⢹⡀⠀⠀⠀⠀⠀⠉⢉⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⢿⡻⣷⢦⣤⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣿⣿⣿⣽⣿⣿⣭⣽⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⣾⠟⢹⣿⠙⣿⣦⡀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣾⠟⢹⣿⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠿⣷⡄⠘⠛⠁⠀⢸⣿⠀⠀⠙⠃⢀⣴⡿⠛⠛⢿⣶⡘⠛⠁⠀⢸⣿⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢹⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢠⠞⠓⣦⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⣾⣷⣆⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⢀⣿⡇⢸⠀⠀⣿⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⣿⣿⣿⢸⣿⡀
⠀⠀⠀⠀⣴⡿⠋⠁⢸⠀⠀⣿⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣿⣿⣿⣿⠀⠙⢿⣦
⠀⠀⠀⣸⡿⠁⠀⣠⠞⠀⠀⠛⢦⠀⠀⢻⣷⠀⠀⠀⠀⣼⡟⠀⠀⣠⣿⣿⣿⣿⣦⠀⠀⢻⣧
⠀⠀⠀⣿⡇⠀⢰⡇⠀⠀⠀⠀⠘⡇⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⢹⣷⠀⠀⢳⣄⠀⠀⢀⡼⠃⠀⣸⡿⠀⠀⠀⠀⢻⣧⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⣼⡿
⠀⠀⠀⠀⠻⣷⣄⠀⠈⠉⠋⠉⠀⢀⣴⡿⠁⠀⠀⠀⠀⠈⢻⣧⣀⠀⠉⠙⠋⠉⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣴⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣦⣴⣶⡾⠟⠋



⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣿⣛⣟⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠟⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣤⡶⣶⣦⡾⠟⢿⣦⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣾⠟⠿⣦⣴⠶⣦⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢠⣿⠁⠀⠀⣉⣤⣶⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢷⣦⣌⡁⠀⠈⢻⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠿⠛⢋⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡈⠛⠻⢶⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠘⢷⣄⠀⠀⣿⠁⠀⠀⠀⠀⢀⡀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣴⠟
⠀⠀⣸⡟⠀⢠⣿⠀⠀⠀⠀⠀⣾⠃⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⢻⡆⠀⠀⠀⠀⢸⡗⠀⠘⣿⡀
⢠⣿⠋⠀⠀⢨⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡗⠀⠀⠈⢻⣧
⠈⠻⣦⡀⠀⠈⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢹⡄⠀⢸⡇⠀⢸⡇⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋
⠀⢀⣾⠃⠀⠀⣻⣧⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⣤⣤⡴⠾⠋⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠻⢶⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⣠⣠⣿⡃⠀⠀⢻⣆
⣴⡟⠁⠀⠀⣼⠟⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠙⢿⡄⠀⠀⠙⣷
⠙⢷⣄⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⡇⠀⠀⣴⠟
⠀⣸⡟⠀⠀⢻⣆⠀⠀⠀⢻⡇⠀⣠⠾⠛⢛⣿⠇⠰⣶⣶⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣶⣶⡶⠀⢿⣟⠛⠻⢦⡄⠀⣾⠇⠀⠀⢀⣿⠃⠀⠘⣿⡀
⠀⣿⡁⠀⠀⠀⠻⣦⡀⠀⢸⣧⢸⣇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⣿⠀⠀⣠⡿⠁⠀⠀⠀⣹⡇
⠀⠈⠻⢷⣦⠀⠀⠈⠻⠷⢾⣧⠀⠙⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠁⢈⣿⠾⠟⠋⠁⠀⢠⣶⠟⠛
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⢻⣧⡀⠀⣿⣀⣿⠀⠀⣠⡶⢦⣄⣀⣤⣄⡀⠀⠀⣀⣤⣄⣀⣤⠶⣦⡀⠀⢸⣇⣸⡇⠀⣀⣿⠁⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣦⣌⡉⠁⠀⠀⠉⠀⠀⠙⠋⠁⠈⠛⠛⠛⠋⠀⠉⠙⠁⠀⠈⠁⠀⠀⠉⣉⣤⡾⢿⣥⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠛⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡿⠋⠁⠀⠀⢹⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⢀⣿⠀⠀⠀⢀⣾⣿⡛⠿⠟⠉⢻⡿⠛⠿⠛⢷⣶⣶⠿⠿⠛⠛⢛⣿⠋⠉⠀⠛⣿⣦⠀⠀⠀⢿⣇⠀⠀⢠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣏⠀⠀⠀⣿⣯⠉⠁⠀⢀⣴⡾⠿⠿⠶⠾⣿⣯⣄⣀⠀⠀⠀⣼⡟⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⣷⡀⠘⣿⣉⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣿⣧⣄⣀⠀⣾⡟⠁⠀⠀⣰⡟⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⣿⠁⠀⢼⣯⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠃⠀⠀⠀⣿⠇⠀⠀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣦⡀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠈⠙⠻⠷⢶⣦⣦⣴⣶⠾⠛⠉⠙⠻⠿⠶⣶⣤⣤⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠉⠉⠉⠉⠙⠻⢿⣶⣄⠀⠀⠀⢠⣾⣿⣧⣴⣦⣤⣤⣤⣄⣀
⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠻⣷⡄⠀⠈⠻⣿⣯⡉⠉⠉⠉⠉⠛⠿⣿⣦⡀
⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⣠⣴⣿⣷⣤⡀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣾⡿⠉⠀⠉⠙⠛⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠻⣿⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣷⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢰⣦⣄⣀⣀⣾⡿⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿
⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠙⠻⣿⡟⠋⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣤⣀⣀⣀⣤⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⣴⣾⠿⠿⢿⣷⣄⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⢀⣤⣼⣿⣥⣤⣤⡀⠙⠋⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣙⣿⣟⣙⣙⣋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠘⠿⢿⣿⠿⠟⠟⠃⢀⡀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠠⣶⣄⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠘⢿⣷⣄⣀⣤⣾⠟⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣤⣀⣀⣀⣀⣙⣿⣷⣄⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⣿⣿⡿⠁⠀⠀⠀⠈⠛⠿⣿⣶⣶⣤⣤⣶⣶⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡿⠿⠿⠿⠿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾⣿⣦
⠀⠀⢀⡀⠳⣄⡇⡴⠃⣀⠀⢸⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢿⠀⢀⣠⠀⢸⣿⣿⣿
⠀⠀⢀⣉⣳⣿⣿⣷⣋⣁⡀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⣾⣖⠉⠁⠀⢸⣿⣿⣿
⠀⠀⠀⡴⢚⡿⡿⣟⠲⡄⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⣼⠈⠙⠲⠄⢸⣿⣿⣿
⠀⠀⠀⠀⠚⠀⠇⠘⠂⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣫⠀⠀⣝⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢫⡀⠀⢨⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⣶⣶⡄⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣶⣶⡦⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⣭⡶⠟

⡟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠙⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠡⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠣
⡇⠀⣼⠦⠫⠝⠐⠒⠒⠒⠀⡇⠀⡇⠐⠒⠒⠒⠂⣼⠖⢫⠓⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠄⠀⠀⠀⠀⡌
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣄⣀⣤⡀⠃
⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠂⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠚⠁⠀⠘⠃⠈⢿⠢⢄⡀
⡇⠀⠳⠆⡹⠀⠀⠤⠤⠤⠀⡇⠀⡇⠠⠤⠄⠠⠀⠷⠆⡸⠉⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠘⠀⠀⠙⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢧
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢰⠀⠀⠀⠀⠀⠀⢀⠰⣿⠆⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⡆⠀⠀⠀⠀⠀⢸⠀⠄⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡇⠀⣸⣦⣜⠃⠀⠀⣀⣀⠀⣿⣿⡇⢀⣀⣀⣀⠀⣰⣆⣘⠂⠀⣿⠀⠃⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠀⠙⢦⡀⠀⠂⠶⡶⢦⣄⠀⠀⠀⠀⠀⠀⠄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠑⠆⠁⠀⠀⠀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠀⠈⠉⠙⣀
⡇⠀⠐⣢⡐⠦⢠⣤⣤⣤⠀⣿⣿⡇⢠⣤⣤⢠⠄⢐⣦⡐⡦⠀⣿⠀⠀⠀⠀⠀⠳⣤⣤⣤⠀⠀⠀⠀⠠⣤⣤⣤⠤⠀⠒⠒⢻⡄
⡇⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇
⡇⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠
⡇⠀⠀⠙⠊⠀⠀⠀⠁⠀⠀⣿⣿⡇⠀⠉⠁⠈⠀⠀⠙⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀
⡇⠀⠀⠠⣤⠀⠀⠉⠉⠉⠀⣿⣿⡇⠀⠉⠁⠀⠀⠀⢠⣬⠁⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀
⡇⠐⠀⠠⡼⠀⠐⠒⠒⠒⠀⣿⣿⡇⠐⠒⠒⠒⠀⠀⠀⢠⡷⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⢸⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢸⠀⠀⠀⢰⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠈⢦
⡇⠠⠀⣠⠠⠏⠠⠤⠤⠤⠀⣹⣿⡇⠠⠤⠤⠤⠀⠤⠈⡅⠼⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣶⣧⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢇
⡇⠀⠀⠀⠀⠀⠀⠀⠘⣾⣧⣾⣿⣿⣿⣯⡶⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠘⣽⢿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢱
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠷
⣧⠀⠀⠀⠀⢀⣀⣠⣀⣤⣤⣤⣤⣤⣀⣀⣀⠀⢀⣀⡀⢀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣼⠀⠀⠀⠀⠀⠀⢠⠀⠺
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠛⠛⠛⠛⠿⢿⣿⣿⣷⣄⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢿⣿⣿⣶⣤⣀
⠀⠀⣴⣿⣿⠟⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠛⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣤⡀
⠀⣾⣿⡿⠃⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⣸⣿⣿⠁⠀⠀⠀⠀⢸⣿⡇⣿⡇⠛⠋⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⣿⣿⡏⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢿⣿⡄⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⣤
⢹⣿⣿⡀⠀⠀⠀⠀⢻⣿⣦⣿⣇⣼⣿⠃⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⡟
⠈⢿⣿⣷⡀⠀⠀⠀⠀⠙⠻⣿⡿⠛⠁⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠇
⠀⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢀⣠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠻
⠀⠀⠀⠙⠻⣿⣿⣷⣶⣤⣤⣤⣄⣤⣴⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠉⠙⠻⣿⣿⣷⣄
⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⣰⣿⡿⠟⠛⠛⠇⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⢀⣴⣿⣿⣥⣤⣤⣤⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠈⠁⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⢀⣽⣿⣯⣀⣤⣠⡅⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠈⠙⣿⣿⡍⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠘⠿⣿⣷⣶⣶⣿⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣷⣶⣶⣶⣶⡄⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠿⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣬⣭⣽⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣍⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⠿⠿⠿⠷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⡤⠶⠶⠶⡶⠛⠛⠛⠻⠶⠛⠛⠛⠶⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢳⣄⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣾⡏⠀⡤⠖⠛⠛⠓⠀⠀⠀⠀⠈⠉⠉⠉⠙⠲⠄⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⡿⠀⠘⠶⠶⠶⠚⠁⠀⠀⠀⠀⠘⠓⠲⠒⠛⠀⠘⣧⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⣼⡇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣠⡤⠀⠀⠀⠀⠀⢠⣿⢁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣬⣿⣄⣀⣀⣀⡀⠀⠀⠀⠉⣉⣁⣀⣀⣀⣠⣤⣴⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠉⠉⢙⣷⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠁⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⣠⡴⠟⠋⠁⠀⣦⣤⡾⠿⢿⡟⠷⢶⣶⣶⣶⣿⣿⡟⢛⣛⣩⣥⣤⣴⠶⠶⣤⣄⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⢀⣾⠏⠀⢀⡴⠟⢛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⣆⠀⠈⠉⠛⠛⠉⠁⠀⠸⣇⠀⠀⠀⠈⠙⠳⣦⡀⠹⣿⡄
⠀⠀⠀⠀⣾⣣⣴⠿⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⡀
⠀⠀⢀⣼⡿⠋⠁⠀⠸⣇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣄⣀⣀⣼⡃⣿⣇
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠹⣆⠀⠈⢷⣄⡀⠀⣀⣀⣀⣀⣀⣀⣀⣼⣧⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢷⣼⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⢈⣿⡏⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿