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⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢦⣄⣀⠀⡠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⠻⣏⠀⠀⠙⢋⠀⢀⣀⣠⣴⠶⠤⠤⣄⠙⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠈⠛⠟⠛⠉⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⢀⡙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⣀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡗⠦⣤⣶⣿⣾⣿⣿⣿⣿⣿⣶⣄⠰⠦⡄⠀⠐⠀⠀⣴⣼⣿⣿⣿⣧⣿⣿⣿⣧⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⡿⢦⠀⠀⠀⠶⠀⠄⢈⣉⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣇⡀⠐⠒⠶⢦⣤⣤⡉⠁⠀⣉⠁⠀⠙⠆⠀⠀⠀⠀⠀⠈⠙⠋⠋⠹⠿⠀⠙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢻⣶⣤⣀⡀⠀⠀⠓⠒⠒⠈⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⣶⡀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠙⣁⣢⣤⠀⠀⠀⠀⠀⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠞⢹⣿⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷⣤⡀⠀⠀⠀⠀⠈⠒⢶⣤⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣶⣤⣄⡈⠙⠲⡄⠐⠙⠂⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣧⣀⡑⠆⠀⠀⠀⠀⢠⡀⠀⠈⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⣿⡿⣿⣦⠀⠀⠀⢻⣧⠀⠀⢻⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡀⠻⣧⡀⠀⠀⢿⡆⠀⠀⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠘⣧⣈⠀⢻⣷⢘⠆⠘⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠷⠶⠾⠾⠧⠴⠾⠷⠴⠶⠶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢾⠁⠀⡀⠀⠂
⠀⠀⠀⠀⣀⡄⠐⠂⢀⣼⣿⡄⠀⠀⢰⠃⡀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⣥⠿⢛⠁⣀⢀⡤⠈⠠⠖
⠀⠀⠀⠀⠀⠀⣰⠿⢷⣶⣿⣛⣃⣁⡀⠐⠉⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠜⣡⡀
⠀⠀⠸⠀⠀⠀⠀⠀⠸⣹⡿⠧⠀⠉⠁⠈⠩⢷⢤⡎⠉⣉⣭⣭⡭⠿⠷⢒⣂⣠⣤⢤⡶⢶⡎⠁
⠀⠀⠀⠈⠲⡶⠖⠂⠈⠉⠀⠀⠀⠀⠀⠀⢸⠈⣉⡀⢨⡍⣶⠶⠒⣿⠀⣿⢹⣏⣀⢸⣇⣼⠇
⠀⠀⠀⠀⠀⠈⠀⢀⡼⣿⣄⣀⣀⣀⠤⠒⠛⠀⣿⣿⣸⡇⣿⣴⠄⢹⣷⡟⢸⣏⣩⣸⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡀⣀⣴⠀⣿⠘⣿⡇⣿⣶⢆⢘⣿⣇⠘⣿⠉⣶⣾⡛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠂⠀⠈⠉⠀⠀⣿⠀⣹⢶⣬⣷⢿⡄⣿⢹⣏⡀⠀⣿⠀⣿⣿⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⣿⠄⣿⠰⣶⣿⠸⣷⡏⢸⣯⣥⠀⠻⠶⣋⣻⣧⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⠒⠒⠒⠛⠉⢹⡄⠿⠶⠟⠛⣀⣛⡥⠬⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠦⠤⠤⠤⠄⠒⠋⡷⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣴⣶⣤⣴⣾⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠙⠀⠀⠀⠀⠉⠙⢧⠀⠆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢸⣿⣿⡆
⠀⠀⠀⠀⢸⡟⠙⠁⠀⠀⢿⠆⠀⠀⠿⠁⠀⠀⠀⠀⠀⢾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠁
⠀⠀⠀⠀⢸⣷⣆⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⣄⣀⣸⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠛⠛⠛⠃⣀⠀⠀⠉⠙⠁⣀⠀⠀⠀⠀⠛⠉⢹⣿⡏⠉⠉⠀⠀⠀⠀⠀⢠⣾⣦⣄⡀⠉⠉⠉⠀⢀⣠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣿⣿⣿⣷⡦⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⡄⢠⣴⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠺⠟⣿⣿⣿⣿⣿⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢰⣿⣶⣤⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⠿⣿⡇⠀⠘⠿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠿⠇⠸⠿⢿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠙⠒⠦⡀⠀⠀⢀⠴⠒⠋⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡴⠃
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⡵⢄⡀⠀⠀⠈⠿⣦⣤⣤⣴⠿⠁⠀⠀⢀⡠⣾⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣖⡳⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢹⠛⠀⠀⠙⠦⣀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠴⠛⠛⠓⢤⠀⠀⣀⡠⢔⣺⠭⠒⠋⠁⢱⣱⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⢀⣴⣾⣥⣤⣤⣤⣤⠾⣒⡯⠕⠊⡩⢢⠐⠢⠀⠀⠀⢳⢳
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⢈⣎⣇⠀⠀⠀⢇⠈⡷⢀⠇⢀⣀⠤⣃⡧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⡼⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠻⡍⠉⠉⠉⠘⣞⣆⠀⠀⠀⣁⡨⢕⣺⠽⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⢀⡴⠺⡜⠦⣒⡯⠗⠚⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠘⠉⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⢠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⣀⠴⣉⠇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⡢⠀⠀⠀⠀⣀⢼⠀⠳⡮⠕⠒⠰⢏⣆⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡴⢋⠔⢹⢄⠘⠍⠉⠓⢄⡀⠉⢦⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡼⠀⡇⠀⡏⠀⠀⠀⠀⠀⠂⠉⠢⢄⣙⠓⢆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣇⠀⢳⡀⢳⠀⠀⠀⢀⣀⣀⠀⠀⠠⡇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⠀⠀⢀⢾⠀⣼⡀⢠⠈⣾⠀⠀⠀⠀⠉⠁⠀⠀⠀⠑⣄⠀⠀⠘⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⢧⢱⠳⡘⡆⠉⠀⠀⠀⠀⠀⢠⡀⠀⠀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢨⣦⠃⠱⣽⠀⠀⠀⠀⣐⣦⣤⠭⠷⢲⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠘⠋⠁⠀⠀⠁⠀⠀⡄⡞⠈⠀⠀⣄⠀⠀⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠼⣔⡃⠀⠀⠀⢀⡩⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠙⠲⢄⡈⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⡄⠝⠗⠀⠀⠀⢳⡁
⠀⠀⠀⠀⠀⠀⣠⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡄⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⢰⠃⡇⡇⢣⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠋⠀⢇⠀⠀⠀⠀⠀⠠⡾⣶⢦⣤⣀⠄
⠀⢀⡠⠀⢀⠎⢰⠁⡇⠀⠱⣄⠀⠀⠀⠀⠀⢸⠀⢀⡠⢺⠁⠀⠀⠈⢡⠀⢀⣀⠤⠚⢄⠀⠑⠮⠝⠁
⠀⢸⣗⣒⣩⠔⠁⡰⠁⠀⠀⠈⠣⡀⠀⠀⠀⠸⡞⠁⠀⡏⠀⠀⠀⠀⠘⡆⠘⡌⠒⠤⠤⠿
⠀⠀⠑⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⣇⢀⣸⠁⠀⠀⠀⠀⠀⠘⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠿⠏⠛⠛

⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣰⣿⣹⣿⣿⡟⠀⠉⠛⠇
⠀⠀⠀⠀⠀⣿⣿⢻⠿⡿⠃⠀⡄⢀⡏
⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⣠⠖⢻⠀⠀⠀⠀⣤⣨⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣄⡀
⠀⢀⠴⣿⣿⣆⠦⢣⣴⣄⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣍⠆⠀⠿⢿⣿⣻
⣰⠁⠀⢻⣿⣿⣦⡠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣿⡏⠘⠃⠀⠈⣿⣿
⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠓⠚⠙⣢⡀⠀⠀⠀⠀⠀⡿⠷⠖⠀⢀⣘⣻⡿
⡏⠀⠀⣿⣿⣿⣿⠟⠋⠊⢐⣆⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⣴⣾⠟⠻⢄⣂⣤⣄⣀⡘⣲⣖⣀⡞⠉
⣷⢀⣀⣿⣿⡿⠻⡀⣀⡤⢞⣝⡆⠀⠀⠀⠀⠀⠈⠲⢤⣇⣹⣷⣴⡖⠢⢤⣀⠈⢹⠏⠉⠉⣩⢷⡀
⣿⢏⡎⠉⠀⠈⣳⣾⣿⣷⡋⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⠛⠛⠋⠁⠐⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣭⣭⣭⣭⣥⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⢀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⡀⣀⣤⣶⣿⣿⣿⣿⣷⠄⠀⠈⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⣥⣅⡠⠂⢀⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠙⠿⢿⡗⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠱⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⡨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣵⠄⠙⠦⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⢦⡈⠢⠀⠀⠀⠀⠀⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡛⠫⠤⠤⠤⠤⠤⠤⠬⠭⠝⠻⠿⠟⠛⠛⠓⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠻⣿⣿⣿⣿⡍⠙⠿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⡿⣻⣶⣵⣿⣿⣿⠀⠠⠤⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⡯⢻⠟⠁⠀⠤⡆⢀⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⣏
⠀⠀⠀⠀⠀⠀⢀⡬⣿⠋⠀⠀⠀⠀⢀⠀⠘⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⢀⣤⣶⡏⢂⠘⣆⠀⠀⠀⠀⢿⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡛⠟⢻⣿⣿⣿⣶⣄
⠀⢀⡴⢿⣿⣿⣿⣄⠳⠜⣦⣷⠦⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⠇⢀⠐⠟⢿⣿⣮⣿
⢠⡏⠀⠀⢻⣿⣿⣿⣮⣀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣻⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡁⠈⠛⠀⠀⠈⣿⣿⣿
⣾⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠧⠷⢭⡽⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣄⠀⠀⠀⢠⠿⣻⣾
⣿⠀⠁⡀⣠⣿⣿⣿⣿⣿⣣⡾⣻⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣤⠤⠐⣐⣶⣿⣄⡀⠀⠀⠀⠀⠀⠸⣌⠉⢀⡀⠀⣸⡷⠾⠛⠋
⡿⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁⠀⠁⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢠⣞⣿⠟⠈⠓⠦⢭⣩⣖⣶⣤⣄⣈⣿⣋⣠⣼⠁
⣿⡂⢀⣸⣿⣿⣿⡿⠳⡀⠀⢀⣤⠞⣯⣹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣟⠀⢹⣿⣤⣶⠖⠦⣤⣀⠀⠀⢈⡿⠀⠀⠀⠀⣸⢷
⣿⣵⢋⡝⠛⠋⠁⠰⣆⣨⣾⣿⣷⠞⠁⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠙⠳⢾⣷⣴⠶⠶⠛⠉⢸⡇
⣿⣧⣾⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣦⣀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠤⠤⠤⠤⠤⠬⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣻⣷⣂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠩⢉⠉⠡⠈⣿⣇
⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠛⠛⡿⠾⣳⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢸⠀⠀⠘⠀⠀⠀⢀⠽
⠀⣠⣾⣿⣿⣿⣿⣭⣦⣷⠀⢀⣤⣤⡀⢼⣥⠊⠙⢋⡉⠁⠠⠄⡟⠀⠀⠀⠀⠀⠀⢠⠞⠛⠋⠁⠀⣄⠀⢢⠀⠌⠤⠀⡴⠁
⣼⣿⠇⠀⢐⡋⠉⠉⠭⢸⣾⣿⠿⠿⠿⠻⠻⢿⠖⠀⠀⠔⢂⠴⠀⠀⠀⠀⠀⠀⠀⣞⡳⣄⢩⠄⠼⠛⡀⢀⡵⣤⣤⣎
⠻⣿⣆⡀⠀⠀⢤⠴⢀⣿⢿⣧⠀⠤⠄⠀⠐⠚⡏⢁⠦⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⡁⠀⠀⡈⠠⡎⠀⢻⠾⠄⠷⡀
⠀⠈⠉⠉⠁⢢⣀⣰⠉⠙⠾⣿⡀⠀⠐⠤⠃⡠⢡⡿⠴⢻⡿⠲⡀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢠⡏⠰⠀⢰⡋⡆⠘⠀⠈⢢
⠀⠀⠀⣰⣿⠕⣦⡀⢈⡗⠀⠉⠁⣱⡀⣔⠁⢀⠉⡍⡰⠈⣠⣶⠐⡀⠀⢆⠈⠀⠐⢀⠈⠉⢉⣆⠀⠈⡄⠀⠘⣏⠁⠈⠄⠀⢸
⠀⠀⣼⠋⠛⠠⡀⣿⠘⡟⠈⣦⣄⣉⣉⡑⢤⠈⠀⢀⠁⡠⢿⡏⣃⠱⠄⢢⣿⠟⠅⢋⡀⠈⢻⠃⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠸
⠀⢰⡇⠀⠀⠀⣴⢡⠂⠀⣿⠀⠀⠀⠀⠀⢀⢃⠀⢸⢸⠀⢼⠉⠉⠁⢰⠋⠒⠒⠀⠀⠇⠀⢘⠆⠀⠀⠀⠀⢤⡃⢴⠀⠐⠀⡇
⠀⣾⠀⣀⡤⡪⠊⢋⡄⢰⡇⠀⠀⠀⠀⠀⠀⠁⠂⡨⠙⢀⠼⠀⠀⠀⢸⠀⠀⠀⠀⠀⣁⡀⠺⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢰
⠀⣿⢀⡧⠈⠑⠢⣉⡌⣟⠀⡔⠀⠀⠀⠀⡰⠣⡀⠉⠀⠷⡀⢲⡶⠖⡞⠢⠀⠀⠆⠘⡀⢈⠃⠀⠰⠀⠀⠀⠀⠈⢢⡄⠀⢾
⠀⠙⠂⠀⢀⣤⡶⠿⠟⢿⡔⠉⢦⡀⠀⠰⠁⡐⠉⠢⠴⠀⠘⡀⢻⣃⠠⠤⡁⠒⠤⠀⠀⡏⠀⢀⠃⠀⠀⠀⠀⠀⠀⠳⠄⢸
⠀⠀⠀⣠⣾⠓⠠⠄⣀⣸⠣⡀⢀⣭⣿⣧⢠⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⣸⠁⣎⡀⣀⡀⠀⢨⠀⣱⡿⣿⣿⢿⣿⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⠑⠒⠐⠀⢸⠖⠀⠀⡠⠔⠈⠂⠀⠘⡄
⠀⠀⠛⠦⣄⣈⠛⠑⢢⠎⢰⣿⣿⣿⠹⢿⡿⣧⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡄⠈⠄⠀⠀⠀⠁⠀⠈
⠀⠀⠀⢠⠇⠀⠈⠉⠁⠀⣿⣿⣏⡿⠃⠏⣸⢿⣇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠐⡀⠈⢄⣀⠠⠀⢀⠠
⠀⠀⠀⣾⠀⠀⠀⠀⠀⣸⢭⣿⡇⣷⢠⠐⢉⡾⢿⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⢄⣀⠠⠔⠈
⠀⠀⢀⡟⠀⠀⠀⠀⠀⣿⡎⡻⢻⠏⡌⢠⠞⢡⠈⢻⡀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⢸⡇⠀⠀⠀⠀⢸⣯⠃⢿⠸⠀⣁⡞⠀⠈⡆⠈⢿⡀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⢸⠁⠀⠀⠀⠀⣿⢹⡀⢸⠇⠀⠏⠀⠄⠠⠙⠀⠘⣷⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡇
⠀⠀⣼⠀⠀⠀⠀⢈⠶⡆⢀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⡸⣧⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇
⠀⠀⡟⠀⠀⠀⠀⡿⠆⠃⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠛⡆⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⢸
⠀⢐⡇⠀⠀⠀⢰⣧⠀⠘⠀⡀⢀⠀⠀⡐⠀⠃⠀⠀⠀⢀⠜⣡⠆⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⢸
⠀⠈⠋⠉⠧⣤⣾⠈⣦⠀⠄⡅⣸⠀⠰⠀⡞⠐⠀⠂⡀⠈⡘⠁⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⡆⠙⠀⠀⠁⡉⠀⠂⡘⠀⠆⠀⠐⠀⡀⠀⠀⠀⠂⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣏⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠁⡆⠀⡆⠀⡀⡀⠀⠀⠰⠀⡇⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⣻⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⢠⠃⠀⠰⠡⠀⠀⠀⢃⠀⠀⠀⠀⢰⡆⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠅⣣⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠃⡘⠀⠀⠐⠔⠛⠶⠦⠈⣠⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣨⠀⠀⠀⡇
⠀⠀⠀⣠⡾⠓⠊⠀⠀⠀⠀⣄⠆⠀⠀⠀⠀⠀⠀⠀⢀⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠏⡀⠀⠀⠀⠀⠀⠀⠀⠘⡦⡘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠄⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡬⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢨⠿⠛⣋⡉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣄⣂⣉⣉⣁⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠒⠒⠀⠀⠀⠈⠇⠛
⠀⠀⠀⠀⠀⠀⠀⢴⠋⠉⠁⠀⣀⠀⠈⠁⠁⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠠⡀⠀⠀⠠⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⠉⠈⢠⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⡖⠀⠀⠃⠀⠀⠀⠐⠀⡁⣠⠄⠄⢄⠨⢐⡄⠀⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠆⠀⠠⠀⠀⠀⠀⠨⠈⠓⠤⠠⠐⠴⢋⢈⠗⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠢⢄⡈⠉⠀⠀⠄⠀⠄⠀⠁⠂⠠⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠄
⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠂⠢

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢁⣀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠑⠠⠇⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡀⠀⣀⠀⣀⣀⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡋⠀⣁⠀⠀⡈⠀⠈⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⣅⡀⠙⠂⠈⠛⡐⣀⣠⣀⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠄⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣈⣁⣄⣀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠞⠁⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠘⠀⠀⠛⠛⠛⠻⠋⠀⠀⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⠀⠾⠆⠀⠠⣶⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤
⠀⠀⠀⠀⠀⠀⢀⠀⠈⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⢐⠖⠒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠘⠓⠖⠀⡠⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢡⠀⢃
⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠂⠀⠈⠐⠐⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠈⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠂⠀⠁⠀⠄⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠀⠃⠈⠀⠀⠀⠀⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠴⢆⣀⡀⢀⡊⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⢀⡠⢂⠚⠃⠒⠃⡀⠖⠉⠁⠀⠀⢣⠀⠀⠀⡀⠀⠀⠀⡠⠁⠀⡐⠡
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠡⠀⠀⠀⢠⣀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⢉⠙⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠓⡆⠀⠈⠆⠒⠈⢀⠠⠊⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢈⠀⠈⠸⠀⠀⠐⠀⠀⠀⢀⡰⠂⠀⠠⠀⠀⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⣀⠠⢚⢂⠁⡀⠈⡠⠀⠠⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠠⠀⠠⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠉⠰⠀⠰⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⡀⣀⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⠸⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠲⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣐⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣒⣤⣤⣤⣤⣤⣤⣴⣃⣀⣀⣀⣈⣒⣂⣠⣤⣤⣤⣤⣤⣔⣁
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢁⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⢀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⠰⢛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣻⣇⠀⡘⠀⣠⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⡇⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠠⠤⢤⣧⠴⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⣀⠀⠀⠀⠀⠉⠐⠒⠂⠤
⣿⣿⣿⣶⣶⣾⡆⠀⠁⠀⠀⠀⠀⠐⠖⠀⢀⡀⠤⡤⠀⠀⠀⠤⠴⠒⠉⣩⡷⢶⣾⡛⠛⠻⣅⣀⡀⠀⢀
⠀⠀⣈⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⢘⠛⠤⠄⣀⣠⠄⠀⠀⠀⠀⠀⠀⠉⠉⣠⡿⢠⡄⠀⠀⠀⠉⣁⠄⠠⣬⠄⣀
⣿⣾⣿⣿⣶⡇⢸⠀⠀⠀⠀⠠⡄⠉⠓⠒⠂⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣰⣻⠦⠀⠐⠂⠉⠁⠀⠀⢸⡀⠀⠉
⣿⣿⣿⣿⣿⡇⡏⠀⠈⠁⠒⠒⠛⠒⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣣⡀
⠉⠛⠛⠛⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠁⠀⠀⠀⠀⠀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠑⢄⡠⠂⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⣀⠀⠀⢘⡄⢀⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⡄⠀⠀⣿⠀⠀⠨⡇⠸⠇⠀⠀⢀⡏⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡸⠣⡀⠀⠀⣀⠎⠀⠀⠀⠠⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢶⠀⠀⠀⠈⠒⠠⠐⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⡰⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠁⢲⡤⠤⢀⡀⠀⢀⠤⠀⢲⠈⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠆⠚⠀⠀⠐⠀⠠⠴⠤⠤⠠⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠠⠄⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠂⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠙⠄
⠀⠀⠀⠀⠀⠀⣸⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣀
⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢰⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⢲⡖⠒⠒⠒⠒⠲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡶⠒⠒⠒⠒⠒⠒⠒⢲⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡟⠉⠁⠈
⠀⠀⠀⠀⠀⣠⣴⢿⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠷⠶⢶⣶⣤⣤⣤⣤⣤⣴⣶⠀⠀⡀⢀⠉⠒⠒⡞
⠀⠀⢀⣴⡾⠋⠁⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠓⠐⠚
⠀⣴⡟⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠹⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣿⣄⣀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣜⠩⠉⠛⢾⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡅
⣀⣼⡹⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇
⢇⠈⢡⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠈⠓⠌⣀⢀⣀⣤⢾⡇⢀⣤⠶⠋⠁⠀⠙⢦⡀⠀⠀⠀⣀⡴⠊⠁⠀⠙⢦⡀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠉⠀⠀⠘⢿⡉⠀⠀⠀⠀⠀⠀⠀⠉⠲⡴⠚⠁⠀⠀⠀⠀⠀⠀⠙⢦⡤⠖⢉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⢶⣶⣶⣶⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠒⣀⣠⣤⣤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠙⠚⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠘⢲⡻⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣟⠳⠀⣉⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠸⡓⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢃⠀⠀⠝⠀⢐⣤⠌⠀⠀⠀⠀⠀⠀⠈⠄⢤⢀⠙⠛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠐⠂⠀⠘⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠂⣬⡟⡀⢁⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠦⠰⣪⠄⠀⠀⠀⠀⠀⠱⠇⢰⢹⠻⡆⠀⠀⠐⠄
⠀⠀⠀⠀⠀⢀⠔⠁⠢⣄⠀⣆⠀⠀⣤⠀⠀⠉⠉⠍⠀⠀⠀⡀⠀⢀⡅⠈⠀⠀⠈⢁⢠⡀
⠀⠀⠀⣀⡔⠁⠀⣀⠤⠘⠂⠨⡓⠀⠘⠃⠀⠀⠀⠁⠐⠒⠫⠄⠀⢈⡇⠰⡀⠀⠀⢘⣀⠱⣄
⢀⡴⣿⣿⡧⠐⠉⠀⠀⠀⠀⠰⠵⠀⠀⠀⠀⠀⠈⠀⠈⠂⠈⠀⠀⠈⢻⠀⠁⠀⠀⠀⡅⠀⠈⠂⠀⠀⠀⠀⢈
⣮⡆⡙⣻⣄⠀⠀⠀⠀⠀⠠⡀⢀⠠⠀⠀⠠⠄⠈⠯⠄⠈⠃⠤⠪⠀⠘⠦⣄⠀⢀⣀⠈⠓⠒⠀⠀⢀⠆⢀⣈
⠙⠻⣂⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⡐⠠⠄
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠖⠛⠛⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠓⢦
⡾⠁⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡅⠀⠀⠀⠀⠀⠀⢀⡼
⠈⠳⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠻⡄⠀⠀⠀⠀⣠⠞
⠀⠀⠈⠓⠦⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⢷⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠙⠒⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⡞⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⡴⠞⠋⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠛⠒⠦⢤⣄⣀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣄⡀⠈⢳⡀
⠀⠀⣰⠊⢁⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⠲⢤⡀⢀⡴⠚⠉⠉⠛⠳⢤⣀⡀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢻⡀
⠀⡾⠡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠙⣾⣀⣀⣄⡀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠉⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠛⠦⣀⣠⡴⠖⠋⠉⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⢤⣤⣟⡛⠲⠦⣤⣀⡀⠀⠀⠀⣠⠞
⠘⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⣧⣀⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⠏⠉⠛⠛⠁
⠀⠀⠙⠲⠶⠤⠶⠖⠛⠉⢹⠂⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠙⠛⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⢘⡧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⠞⠛⠳⢦⣤⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠛⠋⠀⠀⠀⢀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠖⠛⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣉⣀⣀⣀⣤⣤⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠋⠉⠀⠀⠀⠀⠀⠉⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠤⡀⠀⠀⠀⢀⣀⣀⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠞⠁⡠⠖⠚⠛⠋⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠊⠁⠀⠀⠈⠉⠒⠒⠐⠓⠢⠤⠒⡖⠋⢰⠁⠀⠈⠀⠀⠀⢀⡴⠁⡄⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡎⠀⠀⣀⠤⡄⢀⠏⣀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠚⣅⠀⢇⠘⡰⠉⢆⢳⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢸⠀⢀⢳
⠀⠀⠀⠀⠀⠘⡄⠀⢰⠁⠠⠏⠋⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢰⠀⡄⠀⢹⠁⠀⡅⠀⠘⢤⠁⠀⠘⢦⡁⠀⠀⠀⠀⠀⠀⠈⢧⠀⡜⣠⠇⡌
⠀⠀⠀⠀⠀⠀⠘⡄⠈⢢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠉⢠⠃⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⣀⣠⣈⣀⡐⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣋⠀⠀⠀⠀⠀⢐⠀⠘⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢃⢷⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠉⠉⠽⠀⢸⠁⡑⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠦⠄⣀⡀⠀⠊⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢠⠇⠸⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠻⣦⣀⡀⠀⡹⡌⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢄⠀⠀⠋⡸⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠋⡄⠀⠀⠈⠙⠛⠛⠋⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣚⢁⣀⠉⠑⡲⠤⠤⡤⠤⠄⢶⡒⣩⠋⠀⠀⢀⠀⡇
⠀⠀⠀⠀⡜⠀⠀⢸⠀⠀⠀⡠⠀⠄⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠊⠁⠀⡜⠑⠤⠴⠚⢫⡀⠀⣀⠤⢃⠜
⠀⠀⠀⣸⢄⠀⢠⠀⠀⠀⠠⠠⠀⠀⠀⣸⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢤⠴⠒⢲⣦⣄⣀⣠⢎⡀⢀⡀⣠⠋⢱⡈⢻⠃⠉⠁
⠀⠀⠀⠇⠀⢹⢛⠀⠀⠀⠀⠀⠀⢀⠀⢻⠀⠙⠒⠳⡄⠀⠀⠀⠀⠀⢸⠬⡑⢠⡄⢐⠋⠀⣀⢀⢉⠤⢷⣀⠀⠁⣀⡸⠁⠘⡆
⠀⠀⣸⠀⠀⠇⢸⡀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠑⠂⠈⠀⠀⠀⠀⠀⠀⢸⠀⠂⠰⠮⣒⣂⠤⠬⠀⡲⠗⠋⢁⡠⠔⠋⠁⠀⣀⣽⡀
⠀⠘⢤⣀⠘⠄⠘⡛⠭⠭⠭⠭⠭⠭⢭⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠀⠘⠐⠀⠀⠀⠀⠀⠓⠀⢹⡏⠉⠉⠈⠋⠉⠑⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⢀⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣘⠀⠄⠠⠀⠁⠀⣀⣀⡿⠀⠘⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⢸⡗⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠔⠒⠊⠉⠁⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⠾⣀⠠⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠤⠤⡤⡤⠤⢴⠊⠁
⠀⠀⠀⠀⠀⠀⠘⠦⣤⣔⡏⠀⠈⠙⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡒⠒⣇⢡⣀⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⢨⣇⣀⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣶⣲⡀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢶⣟⠊⠁⠈⠿⡧⠀⠄⠂⢠⣄⣀⠀⠀⢀⣦⣷⡌⣿⣷⣄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠇⠀⠸⣿⡄⠀⠀⠀⠀⠀⣀⡶⣿⣷⣿⡆⠀⠙⢼⣿⣦⢶⣀⣹⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⣼⣷⣴⣾⣷⡀⣿⣿⢠⠐⢈⠟⡃⠀⠼⠠⡌⢤⣒⠻⢿⠋⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣢⢤⣾⣆⣿⠛⢿⣿⣿⡗⠌⢻⣿⠟⠫⠀⠀⠀⢈⠂⣠⠞⠻⠷⣄⠲⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠿⢿⣧⡄⠙⠛⠛⠀⠀⠈⠹⢃⡀⠘⠆⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣀⢈⠑⣿⣿⢫⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠈⠻⣷⠀⠀⠀⣦⢠⣄⣋⣙⣓⣶⣤⠀⠀⠀⡀⠸⣿⣿⣶⣿⠿⣫⠜⠛⢈⣀⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⡀⠀⠀⠀⠸⣧⠀⠘⠡⣿⣿⠍⠉⢿⣯⣨⡗⣶⢠⡇⠀⣹⠿⠛⠓⢈⠅⡤⠞⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢸⠄⢀⠰⣿⣿⣔⣫⣾⢿⠁⠃⢿⣾⡇⢰⣿⠣⣀⣀⢾⡌⣀⠀⠀⠀⠀⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠈⢑⡀⠘⠙⠛⠛⠛⠛⢩⣾⡄⣿⣿⣤⡟⣡⣿⡿⠁⠠⣼⣍⣥⣤⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⣀⠉⠓⠌⠳⢮⣿⢺⣎⢿⣇⣿⠟⢹⣣⣿⢫⠖⢁⣾⣇⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⣻⢿⣿⣶⣌⢤⣌⢿⢘⣿⡞⣿⠟⠀⣿⡿⣷⣷⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣮⣝⣿⣧⢹⡆⢛⣿⣷⣿⠀⠈⢻⣿⡿⢋⡽⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣭⣟⠻⣷⣧⣿⡟⣿⡇⠀⢠⣾⣯⠀⠀⣀⣶⣶⡿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣝⡿⣿⣿⣿⣿⣰⣿⣿⣷⡿⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣧⣿⣿⣿⣿⣿⣿⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠙⠿⣿⣀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡦⣄⡒⣠⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠺⣿⣿⡦⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⠀⠆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⠶⠚⠆⠁
⠀⠀⠀⠀⠀⠀⢠⠄⢦⣴⣿⣿⣿⠿⠿⢿⠏⡿⣿⣿⠀⠀⠀⡀⠀⣀⣴⡿⡿⣹⣿⢿⣶⢚⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡀⣠⢀⣴⣤⣿⣿⠀⢸⣋⣤⡴⠌⢁⣾⢏⣇⣿⣼⣿⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⣿⣿⣿⣾⣿⣿⣿⣿⠀⠘⣿⠛⠒⠒⠂⠊⠘⠋⠹⠋⠋
⠀⠀⠀⠀⠀⠀⠀⢀⢰⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠉⠛⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠉⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠉⠉⠉⠉⠒⠢⢄⡀
⠀⠀⡠⠊⠀⠀⡠⠔⠒⠉⠉⠉⠉⠐⠢⢌⡒⢄⠀⠀⠀⢀⠔⣊⠤⠒⠊⠉⠉⠉⠑⠒⠤⡀⠀⠉⠢⡀
⠀⡜⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡱⣄⡴⡡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠘
⡸⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⢰
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⢸
⢡⠀⠸⡀⠀⠀⠀⠀⠀⠁⢠⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⠀⠁⠀⠀⠀⠀⠀⡸
⠈⡄⠀⢇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠁⠙⡇⠀⠀⠀⠀⠀⠀⢾⠉⠈⢹⣿⣿⠆⠀⠀⠀⠀⠀⢀⠇
⠀⠰⡀⠘⡄⠀⠀⠀⠀⠀⠀⠻⢿⣷⡤⠔⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⡜⠀⡰
⠀⠀⠱⡀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡰⠁
⠀⠀⠀⠘⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠜
⠀⠀⠀⠀⠀⠳⣄⠘⢦⡀⠀⠀⠀⠀⠀⠀⠲⣤⣄⣀⣀⣠⡤⠖⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡠⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡈⠐⠤⣀⠀⠀⠀⢀⡠⠔⠈⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠑⠒⠒⠁⣠⠔⠊⠁
⣀⣄⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣦⣖⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢰⠀⣶⠶⠀⣶⢠⡆⢰⠶⠂⢰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⣿⠃⠀⢸⣾⠁⢸⡛⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠉⠀⠉⠉⣤⣤⣭⣤⠈⢹⣥⣬⣤⣭
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡄⢠⡤⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠃⠓⣤⠧⠙⢬⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡟⠛⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⡄⠀⣿⣧⣤⣤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⠋⢤⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⢰⣷⣾⡄⠀⢰⣶⡶⣾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⠂⣼⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣿⣿⡇⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢏⠸⠁⠻⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣦⣿⢹⣿⡇⠀⢸⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠿⠃⠿⠇⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡏⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⠃⠸⠿⠇⠀⠸⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠆⣾⢹⡆⢸⣼⣇⡿⠀⣿⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠃⠿⣼⠇⠈⠿⠹⠇⠀⠟⠻⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⣿⣿⣷⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢠⢮⡾⠛⠉⢟⡻⠉⠛⢷⡷⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣦⣤⡶⠶⠚⠻⠿⣧⣠⢔⡾⢷⡢⣄⣼⠿⠟⠒⠶⢦⣤⣴⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣶⣄⠁⠀⠀⡸⡳⠁⣨⣅⠈⢞⣇⠀⠀⠈⣠⡲⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠄⠙⡻⢎⣤⡶⢽⡷⢋⣉⣉⡙⢾⡯⠶⢤⣱⢟⠋⠠⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣿⡆⠀⠀⣠⡯⡮⣿⡟⡔⣵⢚⡓⣮⢧⢻⣿⣽⢿⡄⠀⠀⢰⣿⣲⢄
⠀⠀⠀⠀⠀⠀⢀⣞⣾⠋⠉⣻⢶⡪⢷⡷⢛⡭⡁⠀⣑⡃⢘⣊⠁⢈⠭⡛⢾⡾⢗⡶⣟⠉⠙⣷⣳⡀
⠀⠀⠀⣀⣀⣀⣾⣾⣧⢄⣐⡵⡁⣠⡏⢰⡱⢛⠵⠼⢿⣷⣾⡿⠇⠮⡛⣎⡆⢹⣄⢈⢮⣂⡠⣼⣷⣷⣀⣀⣀
⠀⠀⠀⠉⠉⠉⢿⢿⡟⠊⠹⡳⡁⠙⣇⠸⡹⣬⡲⢰⣾⡿⢿⣷⡆⢖⣥⢏⠇⣸⠋⢈⢾⠍⠑⢻⣿⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢯⢿⣄⣀⣽⠾⢵⡾⡷⣭⣲⡁⢀⡩⡅⢨⣍⡀⢈⣖⣭⣾⠷⡮⠷⣯⣀⣠⡿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⣿⠇⠀⠀⠘⣇⣗⣿⣧⠧⡻⢬⡥⢟⡜⣼⣾⣻⣺⠃⠀⠀⠸⣿⠽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠂⣠⣵⢎⠛⠷⣺⡷⣌⣉⣉⣡⢾⡟⠶⠚⡳⣮⣄⠐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠾⠋⡀⠀⠀⢱⡵⡀⢙⡋⢀⢮⡎⠀⠀⢀⠙⠷⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠟⠛⠷⠶⢤⣴⣶⡟⠙⠪⢳⡾⠗⠋⢻⣶⣦⠤⠶⠾⠛⠻⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠘⢞⢷⣤⣀⣮⣵⣀⣤⡾⡳⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢽⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠒⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢴⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠸⠟⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠻⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⣀⣀⡀⠠⠀⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠐⠀⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣄⣄⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠘⠋⠛⠋⠋⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠦⠀⠀⠐⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⣄⠠⣄⠄⠒⠠⠔⠈⠩⠭⠀⠒⢄
⠀⠀⠀⠀⠀⡠⠊⡉⠀⠈⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⡐⡡⠋⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⢀⢠⠁
⠀⠀⠀⢠⠁⠁⢀⠔⠋⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢀⣀⡠⠔⡢⡢⠃
⠀⠀⠀⢸⢆⡀⠀⠀⠀⣀⠠⠔⠒⠂⠀⠀⠀⣀⠀⣀⣀⡠⠼⣿⡇
⠀⠀⠀⢸⢸⠁⡙⠶⠒⠂⠒⠒⠈⠉⠀⠈⠁⠀⠀⠀⠹⣁⠀⡿⡆
⠀⠀⠀⠘⡈⣴⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢁⠁
⠀⠀⢠⠐⡇⢃⠃⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⣸⠢⡄
⠀⠀⡇⠊⢡⠜⠀⠀⠈⠀⠉⠀⠀⠀⡄⠀⠀⠉⠀⠁⠀⠀⠈⠋⠙⢸
⠀⠀⠙⠦⢴⠀⠀⠀⠀⠀⣴⠋⡹⡿⠛⠉⠳⢦⠀⠀⠀⠀⠀⡶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠦⢄⡀
⠀⠀⠀⠀⠈⠆⠀⠘⣷⠾⠇⠠⠇⡠⢄⠈⠢⠀⢑⣸⠇⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠐⠂⠀⠛⠁⠈⠳⡀⠀⠘
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠑⠒⠓⠉⠁⠐⠋⠙⠛⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠔⠳⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣤⠒⠀⠀⠀⣀⠈⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣧⡘⠁⠐⠄⢀⠀⠀⠖⠄⠸⣼⢻
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢚⠀⢠⠀⠀⡁⠈⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡁⠀⠀⣠⠤⠤⢄⠀⠀⠀⡝⠋
⠀⠀⠀⠀⠀⣠⠊⠁⢤⠀⠘⡀⣠⠗⠺⣄⠀⠆⠀⠀⢳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠤⣀⠀⠀⠀⣀⠔⠒⠦⠀⠀⠀⠑⢄⡀⠈⠂⠐⠁⠀⣠⠞
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠱⡄⠈⢰⠧⣼⣾⠉⠀⠀⠀⠘⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⡟⠉⢹⠀⠀⠀⢤⣀⡀⠀⠀⠀⠉⠢⡀⠀⡐⠊⠑⡆⠀⢳
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢦⡸⠁⢀⣀⠀⠀⠀⠀⡄⠀⠀⠈⠑⠤⣀⠀⠀⡶⠭⢀⡇⠀⠀⣀⢧⠤⣼⢀⡀⠀⢰⡦⠚⢧⣀⡤⠤⡶⠁⠀⢳⡆⠠⢜⠂⠊
⠀⠀⠀⠀⢰⠀⠂⡀⠀⠀⠀⠀⠻⣎⠥⠺⡇⠀⠀⠀⠗⠤⢀⠀⠀⠀⠀⠈⠑⢣⠄⣀⣑⠈⡿⠁⡼⠠⠼⡆⠹⣍⣫⠤⠰⠏⠀⠀⠀⠷⡤⠶⠋⠔⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠙⢤⡀⠀⠀⠀⠈⢦⡠⣆⠀⠀⠀⡇⠀⠀⠉⠐⠢⠄⣀⠀⢰⠀⠀⠉⠙⣦⡼⠤⣦⠤⠒⣤⠾⠏⠀⠀⡦⣤⠀⠀⠀⢀⡰⢇⡀⢀⠀⡁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠻⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠀⠂⠀⡏⠀⠀⡇⠀⠀⠀⠀⢢⡠⢤⡄⣐⡀⠤⠒⠁⢀⡠⠛⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢘⡷⢤⣀⠀⠀⠀⠉⠛⠳⠤⠄⣀⢀⡰⠉⠆⠒⡮⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠸⢅⠀⡁⠀⠠⠤⠐⠊⠉
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⣅⠔⡩⠛⡶⠤⣄⣀⠀⠀⠀⡄⠈⡇⠀⠀⠘⡷⠢⠤⣄⡀⡆⠀⠀⡇⠀⠀⢀⡀⠤⠀⠈⠀⣠⠀⠠⠤⠤⠤⠤⠤⠔⠺
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠜⠀⠀⠀⠀⠛⠉⠘⠃⠒⠛⢦⣘⠋⠀⠀⠀⠀⠀⠙⠐⠂⠧⠐⠊⠉⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⢀⡰⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢱
⠀⠀⠀⢸⠀⠀⠀⣀⡠⠴⠚⠉⠀⠀⠉⠉⠀⠐⠛⠣⢄⡀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣅⣠⡇⠀⢀⣨⡄⣀⠀⣀⣤⡄⣠
⠀⠀⠀⠈⢲⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡉⠀⠃⠀⢹⠀⠀⢠
⣠⠶⠞⠛⠛⡳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠀⠀⡘
⠑⠤⠤⠶⠄⠧⡀⠀⠉⠐⠒⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠀⠀⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠀⣀⠀⠀⠈⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⢸⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠠⠤⠄⢀⡉⠢⣄⡀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠋⣉⠃⠀⠎⠉⠑⠢⡄
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣷⣶
⣿⣿⣟⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠉⠉⠉⠉⠒⢤⣄⡀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠉⠉⠉⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⠄⠀⠀⠀⠀⠀⠈⠙⠒⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡎⠲⡆⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣥⢤⡤⠀⠀⢀⣴⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠉⠋⡀⢠⢀⡽⢀⣙⣡⣦⣀⠺⠃⠀⢄⡀⣀⢀⣤⢀⠀⣀⠀⠉⠉⠛⠿⣦
⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠸⣇⠀⠰⣄⣷⡌⠁⣴⠛⣧⣼⣿⣿⣵⠟⢷⣾⣿⡿⣿⡇⣾⣀⡀⠀⠀⢀⣴⠶⠋
⠀⠀⠀⠀⠸⡃⠀⠀⠐⠛⠶⢄⡜⠁⣦⣸⡇⠈⠛⠛⠶⠟⠻⠋⠉⠀⠀⠈⠛⠾⠾⣿⣿⣿⣷⣿⣷⣄⡀⠙⢷⡀⢀
⠀⠀⠀⠀⠀⠛⢦⣄⠠⣦⣄⣤⣤⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⡉⢻⣿⣏⡙⣧⡈⢻⠋⢻⣆
⠀⠀⠀⠀⠀⣠⡄⠚⠃⠀⠈⠉⠻⣿⠉⠀⠀⣠⠰⠖⠉⠉⠓⢦⡄⠀⠀⣠⣤⡶⠶⠶⣦⣀⠈⢨⣿⣿⣏⡰⠿⠖⣤⡞⠁
⠀⠀⠀⠀⣞⠁⠀⠀⠀⠖⣠⣶⣆⣸⠄⠀⠞⠁⠀⠀⠀⠀⠀⠀⢿⠀⣼⠟⠁⠀⠀⠀⠀⢻⡇⠀⣿⣿⣿⣿⣦⣀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠙⠛⠛⠻⠿⢻⣿⢿⢭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⣿⣟⡙⠿⠿⢧⡄⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢤⣀⠀⠁⠀⠀⠀⠀⠈⠻⠟⠛⠒⠀⠀⢸⡇⠠⠤⣤⣤⠤⠄⠀⠁⠀⣿⣯⣿⡟⠓⠢⠴⠤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢉⣀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠇⠀⠁⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⣧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠦⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠛⠛⠁⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠢⣤⣤⣈⣉⣁⣀⣀⣠⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠈⠉⠉⢉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢂⠀⠀⠀⠀⠀⠀⠀⣼⡀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣦




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣷⣆
⠀⠀⣠⣤⣄⣀⢀⣄⣀⠀⣾⣿⣿⣿⣿⣿⣦⡀⠀⣠⣤⣶⣿⣤
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⣿⣿⣿⣿⣷⠶⣽⡾⠛⠿⠇
⠀⠙⣿⣿⣿⣿⠿⠿⣿⣿⣷⣹⣿⡟⠉⣿⣿⡟⣰⡿⣥⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⣴⣿⣿⣿⣿⣆⠀⠈⢻⣿⣿⣿⠀⢠⣿⣟⣴⢏⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠘⣿⣿⣿⣿⣿⣷⣦⡀⢻⣿⠇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⣀⣬⣭⣭⣭⣽⣿⣿⡄⣴⣾⡿⠿⠛⡉⠉⠉⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣄⡀
⣰⣿⣿⣿⣿⣿⠋⠉⠁⢀⣴⣄⠰⢾⣿⠿⣿⣿⣶⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣦⡀
⢻⣿⣿⣿⣿⡁⠀⣀⣤⣿⣿⣿⣧⠀⠙⣷⣦⣙⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣇⠀⣸⣿⣿⣆⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠈⠙⠛⠋⣹⣿⣿⡟⠁⢰⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧
⠀⠀⠀⣰⣿⣿⡟⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠀⠀⣰⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠈⠉⠁⠉⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆
⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠈⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠⡶⣶⠀⠀⠀⠀⣼⡿⢁⣠⣤⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣼⢿⡆⠀⢸⣷⠏⠀⠺⣇⢸⠏⣴⢟⣁⡼⠇⢿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⢸⣇⢀⣾⡏⣠⡜⠀⢻⡟⢸⣟⠉⠁⢀⡤⠞⢛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢿⡄⠈⠿⠿⠉⠛⠉⣠⡤⠀⠀⠀⠙⠛⠛⠉⢠⡾⢋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⣴⠶⣦⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⣤⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⢀⣀⡙⢁⣄⠀⣤⠀⠀⣶⣄⣿⢱⡿⠶⠛⠋⠀⠀⠀⣤⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⣷⠘⠯⠿⣯⣿⡄⣿⡀⣸⡟⣿⡇⢸⣧⣠⠴⠛⠁⠀⣸⠏⠀⢹⡇⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣻⡟⠗⠘⠻⠛⠀⠀⠀⠀⣴⠀⠀⣤⠀⢠⣯⡶⠋⣼⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣴⠿⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠀⢰⣿⣠⣿⣿⠷⠾⠟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⣶⡄⠀⢀⣀⠀⠉⠀⠀⠻⠶⠟⠉⠋⢸⣿⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢋⣿⠀⣼⡇⠀⠙⠛⠀⠀⠀⢠⣤⡀⠀⠀⠀⠈⢩⣄⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠀⣸⣯⣾⡿⣶⣄⠀⢀⣀⣀⡀⣾⢸⡇⠛⠀⣾⢻⡆⢻⣦⣿⢣⡾⢛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣿⠋⠉⠀⠈⣿⢠⣿⢋⣽⠇⣿⡿⢀⣷⠀⣿⠞⣁⠘⣿⠇⣿⡷⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣾⡟⠀⠀⢀⣰⣿⢹⣿⠛⣁⡀⣿⠁⢸⣧⣠⡿⠿⠇⠀⠉⠀⠿⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠰⣶⡿⠟⠁⠈⠿⠿⠟⠀⣿⣴⡿⠻⠛⠁⠄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢠⢃⣶⢿⡀⣼⠛⡇⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣸⡇⢠⣿⡇⠀⠀⠀⠀⠀⢰⣟⡗⠀⠀⣠⣤⡈⠛⠁⠈⠁⣿⢠⠆⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣧⣿⢻⡗⣴⠶⣄⣴⠀⡈⠛⠷⣶⠀⣿⣸⠃⢠⡖⣶⠀⣿⡟⠘⠻⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⢸⡇⢸⠛⣿⣿⣰⡇⠀⢸⣇⣤⠘⣧⡀⣸⡶⠋⣠⡟⣠⡆⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠸⠾⠃⠛⠛⠿⠿⠘⠛⣳⣶⠞⣷⢹⣤⡾⠛⠿⠟⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⣬⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢴⣷⣶⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠰⣿⣿⣿⢇⣌⣉⣅⠀⢤⡀
⠀⠀⠠⢶⣶⡖⣶⠶⠶⢶⠶⠶⣆⣶⠴⠦⣀⠀⠀⡐⠀⠀⠈⠀⠘⢿⣿⠿⡛⠉⠀⠀⠀⠀⠀⠀⠐⢄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡼⠁⢀⡎⠉⠉⠀⡄⠀⠀⠀⠨⣿⣿⣃⢀⣄⢀⠀⠀⠀⢀⢰⣤⢀⣄
⣠⣬⣄⣶⣶⣶⣿⣾⣷⣾⣶⣿⠃⠀⡏⠀⠀⠀⠀⣷⢀⣁⣠⣼⣾⣿⣿⣿⣿⣾⣴⣄⡾⡌⣼⣏⣿⡿⣶
⣿⢻⣧⣻⣛⣿⣿⣇⣽⣧⣿⢯⠀⠀⣷⢴⣴⣀⡌⣿⣿⣿⣾⣾⡿⠿⠿⠿⡶⠾⠿⣤⣿⣷⣥⣤⣴⣶⣜
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢻⠁⠁⠛⢣⠸⠿⣿⣿⣿⣇⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣄⣠⣀⢰⣀⠀⠀⠸⣹⣿⡻⢿⣶⠀⡀⠀⣠⠀⣴⣶⣴⣼⣿⣧
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢹⣟⣛⣿⣿⣶⣌⠃⢿⣆⠙⢿⣶⣶⣄⣰⣋⣴⣻⣿⣭⡍⡥⠈
⠀⢿⣿⣿⣿⣿⡿⠿⢿⡿⢟⡛⢿⣧⠆⢀⡟⠛⠊⠈⠻⣿⣦⢿⣿⣷⡀⠀⠉⠉⠉⠉⠉⠁⢀⣀⠀⠁
⠀⠸⣿⣿⣿⣿⣀⠂⢄⣷⡀⠂⣸⢻⣶⢾⠀⠀⠀⠀⠀⠙⢿⣧⡹⢯⣿⣶⣶⣶⣤⣴⣶⣶⣿⡟⠘
⠀⠀⠻⠟⠛⠙⣛⣛⣋⣉⣙⣋⣉⣠⣿⡔⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠙⠛⠿⠷⠿⠟⠛⠋⣡
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⡶⠴⠦⠒⠚⠜⠁
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠀⠶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠻⠄⠀⠀⠀⠀⠀⢠⣿⠿⣿⣾⠿⣿⣆⠀⠀⠀⠰⡧⠀⠀⠀⡀⠀⠀⠰⣶⣶
⠀⠀⠀⠀⠈⠻⠿⣿⣿⠄⠘⠋⠁⢈⣥⣤⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣅⠀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⣀⠃⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠐⠙⠀⠁⠀⠀⠀⠀⠀⢀⠹⠦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⢤⡀⠀⠙⢿⡇⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡈⠀⠁⠀⠀⠀⠀⠀⠀⠀⡼⢡⣶⣶⣆⢹⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠃⠀⠛⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⡀⠀⠰⢞⡻⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣳
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠺⠂⠀⠀⠀⠠⢄⠀⢀⡀⠀⠀⢲⡄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠄⠀⠀⢧⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡗⣕⡀⠀⠀⠀⠀⠀⠂⠀⠀⡈⠁⡒⠀⢠⡣⠀⣀⠀⠄⠀⣀⠀⠀⠀⣀⣤⢤⣀⠀⠀⠀⠈⠂⠀⠨⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠿⠁⠀⠀⠀⠀⠀⠀⠋⢀⡄⠘⠤⠤⠊⠃⠁⢈⣡⣤⣻⡋⠀⢀⡞⠁⡀⠀⡈⢳⡀⠀⠀⠀⠈⣁⣀⡀
⠀⠀⠀⠀⠀⢶⡶⠀⣶⡦⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠘⡇⠘⠃⠘⠃⢸⠇⠀⠀⠀⡋⢰⡦⢰
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠆⠀⠀⠀⠀⠐⠐⠁⠀⠁⠄⠀⠄⠠⠤⠸⠀⠀⠀⠀⠀⠙⠦⢤⡤⠔⠋⠀⠀⠀⠀⠘⢖⢒⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢀⣠⢠⡤⠤
⠀⠀⠀⠀⠀⠀⢀⠠⣿⠶⣆⠀⠤⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢩⡤⠨⠀⣰⠞
⠀⠀⠀⠸⡇⠀⢇⡅⢻⣠⡿⠀⢀⣀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣀⠀⢤⠛⠁⠁⠀⢠⣾
⠀⠀⠀⠀⠙⣷⣄⠀⠈⠳⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠶⠷⠈⠀⠀⠀⠀⠀⠀⠐⠆⠛⠀⠀⠀⠀⠈⠈⠀⠀⣄⣀⣀⠀⠾⠛
⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣦⣭⣴⣦⣬⣦⣵⣾⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣀⣤⡤⠤⠄⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣀⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⢀⠀⠀⠈⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣶⡿⠛⠛⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠐⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠂⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣤⣤⣄⣀⣠⣤⣤⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠶⠶⠶⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⣠⣴⡟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣿⠟⠋⠀⠀⠀⠀⢀⣿⣷⣶⣦⡞⣻⡀⣠⣶⣾⣿⣿⣿⣷⠆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠁⠀⢂⢀⣔⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡁⠀⣠⣿⣿⣿⣅⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⡲⠄⢃⢣⠀⣬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡧⠰⣿⡿⠃⣹⣿⣿⣿⣿⣷⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣉⠁⢀⣤⡀⠀⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⠀⣹⣷⣄⠀⣠⣼⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡀⢀⣼⣿⡆⣲⡏⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡁⣍⣻⣿⡿⣟⣻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣿⣏⣴⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣃⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⢿⣿⢧⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠌⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣞⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣤⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣮⠉⠡⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⣿⣿⡿⠛⠿⠿⠋⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⣿⡟⢥⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⡿⢿⢿⣿⠿⣯⣤⣿⣟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠈⠉⠁⠺⡿⠡⣾⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠃⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠯⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢰⣦⣾⣶⡖⢸⠂⢰⢸⡃⣟⠀⢺⣆⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⣿⣿⣿⣶⣶⣶⣿⣿⣿⠿⠿⢿⣿⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠘⡿⠿⠿⠇⠸⠁⠸⠼⠅⠇⠀⠸⠫⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢻⣿⣿⣥⣴⣴⣆⣉⣀⣠⣤⣨⣿⣿⠇⠀⠀⠀⠀⠀⠰⣦⣴⣤⣤⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⣤⣠⡄⣄⣤⣠⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠻⠏⠛⠋⠛⣡⡿⠃⠀⠀⠀⠀⠀⠀⠘⢻⢻⡛⣿⠹⡇⠿⠿⠇⠸⠿⠿⠁⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠰⠛⠒⠚⠐⠚⠛⠓

⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢿⣿⣿⣿⣿⣿⠛⠛⠂⠀⠀⠀⠀⠀⠀⠐⠛⠛⣿⣿⣿⣿⣿⡿⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠓⠒⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠴⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠉⠉⠉⠉⠉⠙⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⣄
⠀⡎⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⡆
⠀⢱⡀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡞⠉⠀⠀⠉⢦⡀⠀⢀⡴⠒⠒⠲⢄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢰⠃
⠀⠀⠳⢦⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⡞⠀⣀⢴⣦⡀⠀⢳⠀⡜⠀⡀⣤⣄⠀⢳⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢠⡴⠃
⠀⠀⠀⠀⣱⠀⠀⠀⠀⠉⠲⡄⠀⢠⠇⠀⠀⠀⠀⡇⠀⢻⣿⣿⠇⠀⢸⠇⡇⠀⠿⡿⠟⠀⠈⡇⠀⠀⠈⢧⠀⠀⡠⠚⠉⠀⠀⠀⢰⠋
⢀⡤⠒⠉⠉⣀⣀⠀⠀⠀⠀⠘⡆⡼⠀⠀⠀⠀⠀⠹⣄⠀⠈⠁⠀⢀⡜⠀⠣⡀⠀⠀⠀⢀⡼⠀⣀⡀⠀⢸⡄⡜⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⡀
⡏⠀⠀⠉⠙⠀⠀⢱⡀⠀⠀⠀⣧⡇⠰⠿⣅⡀⠀⠀⠈⠑⠒⠒⠚⠉⠀⠀⠀⠈⠉⠒⠊⠉⣀⡴⠚⠹⠀⠀⣧⡃⠀⠀⠀⣰⠉⠀⠙⠂⠀⠀⠈
⠳⣀⠀⠀⠀⠀⢀⡼⢦⡀⠀⠀⡇⡇⠀⠀⠈⠙⢷⡶⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⠋⡽⠁⠀⠀⠀⢀⡇⡇⠀⠀⢠⠼⣄⠀⠀⠀⠀⠀⣠
⠀⢜⠙⠒⠒⠊⠉⠀⣠⢳⡀⠀⡄⢸⡀⠀⠀⠀⠀⠙⢶⣤⣉⣉⠉⠉⠉⠉⠉⠁⢀⣠⣴⣿⠁⠀⠀⠀⠀⣸⠀⡀⠀⣨⢧⠀⠈⠑⠒⠒⠒⢛⠄
⠀⠈⢳⠦⠤⠤⠴⠚⠁⡸⠙⡞⠁⠀⢧⠀⠀⠀⠀⠀⠈⠻⣿⠿⠛⠛⠋⠉⠙⠛⠻⢿⣿⠃⠀⠀⠀⠀⢠⠇⠀⠑⡖⢳⡀⠓⠦⢄⣀⠠⢴⠊
⠀⠀⠈⠢⢄⣀⡤⠴⠚⣁⠜⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⢀⢀⣀⣤⡼⠃⠀⠀⠀⠀⣠⠋⠀⠀⠀⠹⣄⠙⠦⣄⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠉⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣤⠴⠋⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣳⣶⣶⣶⣶⣶⣶⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣮⡻⣿⣷⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠛⢿⣮⣿⣿⡄⢳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣧⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣶⣔⣍⠻⣿⣿⡜⣿⣿⣿⣿⣟⠻⢿⣿⡟⣿⣿⣿⢀⢸⣿⣿⣿⣿⣿⡄
⢹⣿⣿⣿⣿⣧⠀⠀⠉⢻⡝⠛⢿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣸⡌⠻⣿⣿⠀⠀⠻⡇⠈⢿⣿⡌⡆⣿⣿⣿⣿⣿⣿
⣇⢿⣿⣿⢿⣿⡀⠀⠀⠀⠙⠀⠀⠈⠙⢿⣿⣿⣿⣿⣾⣿⣧⣷⠀⠘⢿⠀⠀⣀⣤⠴⠞⢿⣇⢳⠸⣿⣿⣿⣿⣿⡆
⣿⢸⣿⣿⣈⣿⣷⠶⠶⠚⠛⠓⠒⠒⠒⠀⠙⢿⣿⣿⣿⣿⣿⣿⡄⠀⢈⡴⠚⠉⠀⠀⠀⠀⢻⡌⢧⢻⣿⣿⣿⣿⡇
⣿⡇⣿⡟⠋⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣿⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢷⡻⣿⣿⡏⢿⡀
⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⣂⡒⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⡀⢿⣆⠻⣿⣿⣷⡈⠃
⣿⣇⣿⣇⠀⠀⠀⢀⠤⠊⢁⣴⠾⢿⣿⣿⣿⡶⣄⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⢀⣤⠾⠿⠿⠋⠀⠘⣿⣧⡙⣿⣿⣷⡄
⣿⣿⣿⣿⠀⠀⠀⠁⣤⣿⢿⠃⠀⠀⣹⣿⣿⣷⠈⠄⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡌⢿⣿⣿⣦
⣿⣿⣿⣿⡆⠀⠀⠙⠛⣿⢸⣷⣶⣶⣷⡅⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡌⢿⣿⣿⣧
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⠈⡇⠘⠛⡿⠃⣠⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡘⣿⣿⣿⡆
⡙⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢋⠜⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⢻⣿⣿⡇
⣧⠸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⣑⡀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣼⣿⣿⣧
⣿⡇⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⢠⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⡘⣿⣿⣷⣌⠙⠛⠿⠷⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣤⣤⣶⣿⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⢹⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠙⠻⠏⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠳⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠚⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡉⠀⠀⢀⡀⡀⠀⠀⠀⢠⣤⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡏⠰⠚⣡⡏⢠⠆⠀⠀⠛⠁⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⠁⠰⣤⠇⡾⣰⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡤⠶⢄⣠⠴⢞⣻⣶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⣴⠋⣁⡀⣀⣀⣤⣬⣉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠗⠛⠉⠁⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⢰⡇⣠⠿⡝⢿⠋⠛⠛⠃⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠈⣧⢸⣦⠀⠀⢠⡄⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠘⣧
⠀⠀⠀⠀⠀⢸⣟⢃⠀⠀⠀⠀⣀⣉⡀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠉⠻⣦⢦⣀⠀⠉⢉⡼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢉⣻⣉⡥⠤⡄⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢻⣆
⠀⠀⠀⠀⣀⣤⣶⣛⣉⣷⣤⣿⡁⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⡇⠹⣦⠀⠀⠀⢿⡄
⠀⠀⢠⣞⣛⡂⠀⣤⡭⢭⣤⣼⣇⠰⢦⣀⢻⣶⡄⠀⠀⠀⣸⠇⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠘⣧⠀⠀⠸⣧
⠀⠀⣼⣧⠬⡍⢠⡏⠓⠊⠉⣿⢿⣧⠀⠉⠙⠀⠙⠦⣤⣀⡿⠀⠀⠀⢠⡏⠙⠳⣄⠀⠀⠀⠀⠀⣸⠃⠀⣾⠀⠀⠀⠸⣇⠀⠀⣿
⠀⢰⠏⠉⠉⣠⠞⠁⠀⠀⢠⡷⠾⣏⠳⣤⣀⣀⣀⠀⣿⠁⠀⠀⠀⠀⡾⠁⠀⠀⠙⣧⣀⣀⣀⣀⣀⣠⣴⣛⢤⡀⠀⠀⢻⣶⣤⣿⡇
⠀⡿⢠⠴⠋⠁⠀⣴⠖⠒⠋⠛⠛⢛⣦⣤⡼⢻⡏⠻⣿⣆⡀⠀⠀⣼⠁⠀⠀⢀⡼⠋⣹⣿⠛⠛⠻⣟⣧⠀⠈⢻⣄⠀⠘⣧⠀⢹⡇
⢰⡇⠀⠀⠀⢀⣾⢋⠉⠉⠉⠉⠉⣉⣉⣸⡇⢸⡇⠀⠀⠉⠛⠻⠞⠁⠀⠀⢀⡾⠁⢰⣏⡇⠀⠀⠀⢹⡌⢇⠀⠀⢹⣆⠀⢻⡀⠘⣿
⠈⢷⣀⣠⠶⠋⣽⣭⣍⣩⣭⣭⣭⣭⣭⣽⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⣾⢸⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⡆⠘⡇⠀⢹
⠀⠀⠀⠀⠀⠀⢸⣓⣲⣤⣤⠴⠖⢒⣒⣻⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣇⢀⢸
⠀⠀⠀⠀⠀⠀⣸⣷⣮⣭⣭⠭⣽⣾⣛⣽⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠹⣿⢸
⠀⠀⠀⠀⠀⢰⡟⠺⠿⠯⠭⠭⠭⠖⠛⠚⣷⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⠟
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠇

⠛⠁⢀⡞⢇⠣⡐⢺⣿⣿⣷⣿⡿⣿⠅⠀⠀⠀⠀⡤⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⡿⡿⠀⠀⠀⠀⠠⣴⡃⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠂
⣄⣠⠏⠀⣸⡀⠈⢂⠻⠛⠶⠈⠁⠀⠀⠀⠀⠀⠀⠠⠽⠶⠀⠀⣰⣶⡾⠿⢷⣤⡀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠠⠧⠦⠀⠀⠾⠧⠀⠀⠀⣀⡀
⣿⢿⡄⠦⠁⢘⣰⡄⡁⠄⠒⠀⠀⠀⠀⠀⠀⠵⠄⠀⠀⠠⡀⢠⣿⣿⣷⣶⣿⣿⣗⠂⠀⠀⠀⡀⠀⢠⡷⠀⠀⡠⠄⠀⠀⠀⠀⠀⠐⢾⢿⠁
⣿⣿⣏⡑⡤⢤⡉⠸⠃⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠁⣼⣿⣿⣿⣻⣿⣿⡿⠖⢒⡎⠀⠾⠂⣉⠀⠀⠐⠲⢤⡀⠀⡀⠂⠀⠀⠂⠀⠀⢀⠀⠀⡀
⠟⠉⠈⠠⣷⡖⢃⢉⢄⣦⣶⣿⣾⣤⣦⠀⠀⠀⢀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⢸⠿⢿⣷⣴⣶⣤⣤⣀⡀⠀⠀⠁⠖⠀⠀⠀⠁⠀⠀⠀⡻⠃⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⢠⣀⡤⢠⡀⠀⢿⣿⣿⣿⣿⣧⣀⠃⠀⠀⠈⠉⠙⠻⢿⡿⣿⣤⡀⠀⠀⠀⠀⢈⠀⠀⢀⠄⠀⠀⢢
⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣼⠿⠀⣾⠠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠉⠙⠂⢰⠞⠀⢠⣶⣶⡎⠀⠀⠀⡾⣿⣿⣿⡇
⠃⢠⠀⢴⣾⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣧⠄⠀⢁⣿⡿⣿⣿⣏⡀⠉⠈⠙⠿⠿⢶⣶⣶⣤⣄⡀⠀⠀⡰⠃⠠⠘⠿⣛⣁⣀⠂⠸⣷⣿⣿⣿⡾
⢀⡈⠐⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠇⣤⣴⣿⣿⣶⡏⠉⠉⠙⠷⢿⠁⠀⠀⠀⠈⠛⠉⠻⣿⣿⣶⢻⣤⠄⢢⡘⠻⣿⠇⠀⣾⣦⡝⠛⠉⠱
⠀⢐⡀⠀⣠⣶⣾⣿⢿⣿⣿⠀⢙⣀⠀⣭⠉⠉⠉⣬⣡⣴⣿⣿⣿⣿⣿⡀⠀⠉⠀⠀⢤⣷⣦⣀⠀⠀⣶⣄⣀⣽⡿⠁⣤⢠⡄⠀⢽⣦⡘⣿⣵⡿⠟⠋⠁⠀⠂
⣀⠘⠃⠰⢿⡿⡋⠀⠚⠻⠛⠀⠈⠁⢀⣴⠟⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠉⠙⠁⣴⣿⡔⠈⣩⣾⣿⣿⣿⣷⣾⡛⠂⠄⠀⠦
⡟⠀⠶⠀⠀⠀⠤⠐⠀⠀⣠⣶⡶⠲⣾⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠁⠀⣶⣾⣟⢉⣴⣿⣿⡿⢋⢾⡿⢹⣿⣿⡍⠉⠁
⠀⠀⠈⠤⠀⢰⡄⢰⣦⣼⣿⠀⠈⢢⠉⣀⡟⠋⠙⢀⢉⠻⣿⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣰⣿⣿⣇⠚⠛⠋⠁⠔⠋⣼⣍⡁⠹⣿⣿⡄⢸⣿
⣶⡈⢡⡄⠀⠀⠰⢶⣿⣿⣿⣷⣤⠜⠣⠉⢤⣄⠀⠀⠋⠀⠀⠛⣿⣿⢄⠀⠂⠀⠈⠉⠙⢿⠿⠋⣿⡟⠻⣿⣿⣿⣿⣿⡟⠻⡀⠰⠂⣸⣿⣿⣿⡗⠈⠻⠷⠈⣿
⠉⠁⠀⢀⣶⠘⠃⠘⠿⣿⠿⢁⣴⣶⠶⠶⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣶⣤⣤⣀⡀⠀⠀⢰⣿⣷⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⡽⢏⠻⡛⣯⢼⣖⠋⣡⣶⡝
⠀⠀⠀⠈⢿⡇⠀⠘⢃⠞⢳⣿⠿⠿⢃⠀⠐⣶⠂⠀⢰⣆⣀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⠞⢑⢄⠣⡈⢿⣿⠟⠛⠜⠛⠃
⣀⣀⣀⣀⣸⣹⣀⣀⣀⣀⣘⣿⣧⣀⣰⣶⣆⣐⣀⣚⣁⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣏⣀⣰⣲⣁⣈⣂⣉⣙⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠀⠀⣀⡤⠖⠒⠂⠐⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⡴⡫⠄⠀⢉⠛⠳⣄⢠⠞⠁⠐⢉⡁⠀⡉⢉⡂⠈⢣⠀⢀⡴⠚⠛⠒⢄
⠀⠀⠀⠀⠀⠀⡘⠀⣮⡞⢿⠖⢳⠀⠘⡏⠀⢀⡎⠀⠈⣾⠁⠀⠹⡄⠀⣧⡟⠶⢿⡿⢶⠄⢇
⠀⠀⠀⠀⠀⠀⡆⠀⠑⠶⣋⣽⣧⣤⢠⡃⠀⠈⣇⠀⢃⡿⣘⠀⣰⢇⠀⢸⠀⠈⡥⠌⣡⠴⠚⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢗⠀⠀⠸⣿⣿⢿⠏⠙⡛⠳⡔⢌⡙⠉⠀⠈⢉⡡⣮⠴⠞⢲⣦⡀⡜⢡⣘⣇⣀⣀⢹
⠀⠀⠀⠀⣀⡤⠾⠖⠶⢤⡀⡏⠠⡩⢹⡶⣧⠹⡄⠉⠻⠿⠿⠟⢸⠁⡩⢤⡐⢤⢹⡀⣼⢿⣻⣷⡈⠙
⠀⠀⢀⡞⡡⠒⠀⢉⡭⠀⠹⣷⠀⠓⢺⣭⠁⠀⢡⣀⣠⣤⣤⣀⢠⠀⢣⡾⠳⣾⠀⡗⢭⣄⣈⡝⠀⡸
⠀⠀⢸⠁⡞⠉⠀⣾⡠⣀⣧⣽⣦⣀⠸⠛⡄⢠⠞⢁⡄⠀⠠⢌⠻⣄⠀⢼⣿⣦⣀⣃⡀⢈⡭⠛⣭⡉⠛⠲⡄
⠀⠀⢸⠀⢧⣀⢁⣯⡟⡿⢆⡟⡳⡌⠱⡌⠁⠏⠀⡠⠤⠄⠠⠤⢀⢹⡄⠠⠚⠉⡂⠈⣭⡻⣄⣞⠫⣡⠤⡄⢹
⠀⠀⠸⡇⠀⠉⢉⢾⠀⢧⢖⣻⣭⣍⠀⠘⣌⠀⠀⣡⣄⣀⡀⢀⡀⠀⣷⠃⣼⠯⢽⡿⢽⡆⢹⣯⠶⣿⣷⠂⡼
⠀⠀⠀⡇⠀⠀⠈⢸⠀⠁⣯⣿⣿⣿⣇⠀⢹⠀⠀⢿⡿⠿⢷⣿⡇⠀⢹⠀⢘⠶⠞⠳⠴⠁⢸⠳⣄⠈⠁⣰⠃
⠀⠀⠀⠸⠀⠀⠀⠀⡄⠀⢽⣿⣿⡿⢟⠛⢉⣉⣛⢦⡉⠲⠞⠉⢀⣠⠼⣄⡈⠑⠶⠖⠉⠀⡞⠀⠀⠀⡀⠃
⠀⠀⠀⠀⣣⠴⠚⠋⣙⣓⠦⡁⣽⠐⢁⣀⠀⣴⠤⠄⢳⠀⢀⡔⣀⡀⠀⣀⡙⣆⡠⠖⠋⠉⠉⠙⠓⠶⣅
⠀⠀⢀⡞⠑⠉⠂⠈⠀⣈⡄⠈⡇⠐⣭⢷⣢⣴⣒⡗⠀⠃⡼⢰⢉⡝⣴⣋⢳⠸⡄⣠⣞⣛⠲⡄⠀⠀⠈⢳⡀
⠀⠀⡾⠀⡖⠲⠞⢻⡙⠃⣸⣀⣸⣀⡈⠻⠿⠿⢟⣁⣤⣧⡇⠈⠉⡉⢈⡋⢉⣀⣳⡧⠤⢮⣉⣷⢾⠿⡆⠈⡇
⠀⠀⡇⠀⠑⠤⠴⠋⣹⢟⠍⢀⣠⠀⠙⢆⢠⠾⢭⡀⠀⣀⠬⠁⣄⠉⡴⠛⠉⠉⢉⡙⢦⣀⠘⢫⡉⠹⠃⠀⡟
⠀⠀⠇⠀⠀⠀⣆⢺⡇⢃⠆⠰⡇⡌⡆⢨⡏⡤⣤⣈⣀⢤⣤⢄⠘⡾⠁⣤⣀⠀⡵⢿⡀⢷⠤⠶⠶⠦⠀⢠⠇
⠀⠀⢰⠀⠀⠀⠙⠢⢧⠈⠧⣭⠟⠚⠁⣸⠀⠳⠬⠵⠙⠦⢥⡊⠀⣷⠀⣗⣠⠟⡗⠀⠇⢸⠀⠀⠀⠀⠀⡼
⠀⠀⠘⡇⠀⠀⠀⠀⠘⠀⠐⠲⣾⠿⡇⢿⠠⣿⣿⣀⣒⡊⣽⣿⠀⢹⠀⠈⣵⡖⡍⠉⠀⡀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠰⡄⠀⠈⠚⠃⢸⠀⠙⠛⠒⠶⠶⠚⠋⠀⠸⠀⠀⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄⣤⣤⣤⢠⣤⣤⡄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣯⠀⣿⠶⠆⢸⡷⠶⠆⣿⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠃⠛⠓⠒⠘⠿⠿⠃⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢶⠀⢠⣾⡄⢰⡇⠀⠀⣿⡄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠂⣿⠛⣷⢸⣇⣀⣸⣿⢿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡀⠀⣀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣧⢸⣿⣴⣿⢸⡇⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⠚⠃⠙⠛⠘⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢠⣆⢰⣶⠶⠶⢰⡆⠀⣰⢤⣶⠶⠶⠀⣶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⡇⢸⡿⠶⠖⠈⢿⣠⡟⢸⣿⠶⠖⠰⣿⢶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠊⠻⠃⠘⠿⠶⠶⠀⠘⠿⠁⠘⠿⠶⠶⠂⠟⠀⠻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣤⣤⣤⠀⠴⣶⡶⠆⠀⢀⣤⡀⢀⣤⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⠶⠶⠀⠀⣿⡇⠀⠀⢸⣿⣧⣾⣿⠐⣿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠀⠿⠶⠶⠄⠀⠿⠃⠀⠀⠸⠇⠹⠃⠿⠀⠿⠶⠶

⢠⣤⣤⣄⠀⢀⣤⣤⣄⠀⣠⡀⣠⡄⢠⡄⣀⠀⢀⡀⠀⠀⠀⢀⣤⠀⠀⢀⣤⣤⣀⠀⢀⣤⡀⠀⣄⠀⣀⠀⢀
⢸⡏⠈⣿⡇⣿⡇⠀⢻⡇⢹⣧⡿⣷⣼⠇⣿⣧⣸⡏⠀⠀⢀⣾⣛⣧⠀⣿⠁⣤⣭⠀⣾⣿⣇⠀⣿⢰⣿⣧⣸
⠸⠷⠶⠟⠁⠻⠷⠤⠞⠁⠈⠿⠃⠹⠿⠀⠿⠙⠿⠇⠀⠀⠼⠏⠉⠻⠇⠻⠶⠶⠟⠸⠏⠉⠿⠄⠿⠘⠿⠙⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠀⠈⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠒⢄⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠑⡄⢇⠀⢰⢀⠆⠀⢀⠜⡠⠔⠉⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡌⡆⠘⠘⠀⣠⡪⠊⠀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠑⠄⠀⡸⠚⠒⠂⢴⠏⠀⠔⠉⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠉⠁⡎⠀⠋⢀⠛⠀⣇⠀⠈⠁⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⣉⠩⠭⠛⠳⢄⠘⠊⢀⡠⠫⣍⠑⠢⠤⠠⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⢀⠔⡀⠀⢰⣿⠉⠁⢀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠁⠰⠁⢀⣿⡏⠀⢰⠀⠣⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡰⠁⠈⠑⢦⡀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⢏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⡠⠤⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⢄⠀⠀⢀⠌⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⡠⠒⢁⡠⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠉⠀⠉⠀⠸⡄⠈⢆⠀⠀⢀⡠⠔⠊⠀⣠⠊⢀⠔⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠢⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠉⠉⠀⠀⠀⢀⠜⢁⠔⠁⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⢦⡀⠈⢢⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠃⠔⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠙⠦⡀⠑⢄⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⢀⠜⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠳⣄⠀⠀⠀⡇⡆⠀⠀⢀⠎⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠳⣄⠀⢰⢸⠀⢀⠆⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⡑⢄⡆⢧⠎⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢽⣈⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡋⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠹⠈⠉
⡇⠀⠀⠀⠀⠀⠀⢠⡼⢰⠀⡀⠀⠀⠀⠐⠒⠈⡇⡀
⡇⠀⠀⠀⢠⠀⠀⢸⠁⣸⠛⢁⠀⠀⠀⠀⢩⣿⠀⢨
⡇⠀⠀⠀⠈⠄⠀⠏⡰⠃⡞⢩⣶⡄⠀⠀⠘⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⡍⠑⡀
⡇⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠛⠋⠀⠀⡀⠀⠀⠐⠃⠸⠀⡀⠀⢀⣀⠆⠒⠔
⡇⠀⠀⠀⠠⡀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡀⢁⡀⠀⠐⠙⠻⣩⠟
⡇⠀⠀⠀⠀⠀⠡⣀⠤⠀⣀⠒⢶⣀⢀⣨⠇⠛⣤⢀⡛⡀⣀⢀
⡇⠀⠀⠀⠀⠀⢦⡡⣠⡏⠉⣡⡀⠈⠉⠁⠀⠀⢀⡻⢗⢁⠉⠒⠉⠐⠂⠀⠀⠀⠀⠀⣀⠀⢤⡀
⡇⠀⠀⠀⠀⠀⣈⡞⢤⠘⠃⣀⠀⠀⠀⠠⠀⠀⠀⠈⠛⠂⠉⠀⣀⠀⠀⠈⠀⠀⠀⠈⠉
⡇⠀⠀⢀⡴⠿⠃⠁⠦⠰⠂⣩⡀⠀⠀⠐⡀⠋⡒⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠚⠛⠃⠊⢙⠉⠁⠄
⡇⠀⠀⠀⠙⠂⠀⠀⢀⠀⠉⠉⠁⠂⠐⠀⠑⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠁⠁⢸⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠨⢒⢀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⡂⠔⠹⢐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠁⠀⠀⠀⠀⠉⠢⡀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢡⣴⣆⣴⡀⠀⢁
⡇⠀⠀⠀⠀⠀⠀⠀⡴⠋⣰⡗⣄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢐⡀⠀⠈⠀⠒⠓⠶⠷⠀⠀⠀⠐⠂
⡇⠀⠀⠀⠀⠀⠀⢸⠃⠺⢻⡇⠈⢦⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢘⣒⣒⣒⣒⠛⠀⠸⠿⠒⠶⠶⠇
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠓⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠁
⡇
⡃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⢿⡷⠀⠸⠿⠄⠘⢿⠆⠀⠿⠇⠀⠸⠿⠀⠸⠿⠆⠰⠿⠇⠀⠿⠗⠀⠸⠿⠀⠸⠿⠂⠐⠿⠇⠀⠿⠗⠀⠸⠟⠀⠘⠟⠂⠐⠿⠃⠀⠻⠇⠀⠻⠗⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⠀⠀⠀⠀⠀⣀⠚⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠈⢿⠇⠀⣤⣤⣥⠶⠚⠛⢻⣷⣆⣀⣠⣤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⢀⡴⠿⣛⡫⠉⠀⡀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⣠⢀⣤⢀⣤⣴⣏⠈⠁⠀⠹⡄⠻⠆⢱
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠁⠀⠀⣀⣈⣵⠷⠞⢻⡄⠀⣿⣿⣄⠀⠀⡄⡇⠀⣶⡛⠂⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠩⣴⣤⣤⠴⠲⠄⠀⠐⢬⠍⣴⠄⠀⠈⡇⠀⣷⠏⢹⠀⠐⠀⣧⠀⠈⠻⠄⠸⡀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠑⢶⣄⠀⠰⣦⠀⠸⣧⠀⠸⣼⠃⢠⠄⠸⡇⢰⡟⣠⢿⡀⢠⣶⠋⠀⣀⣀⣼⣡⣇⠀⠀⠀⠀⠀⠀⢡⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⢸⡆⢰⡙⢷⡌⡇⠀⢀⣠⡴⠃⢸⣤⡏⠘⢧⠀⢀⣠⣾⣩⣭⡉⢡⣶⣶⣶⡤⣄⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⡇⠘⡛⢦⡁⠘⣄⠘⠋⢀⣴⣿⣷⠞⠐⠛⠟⠋⠘⠛⢿⣿⣿⡟⠉⠁⠈⠀⠈⢤⠀⠙⣄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⣈⡇⠀⢡⡄⣷⣞⣻⣗⣾⣷⠞⠁⢻⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠀⠀⠀⠀⠐⠻⢦⡀⠀⣤⡌
⡄⠀⠀⠀⠀⠀⢸⣇⠐⠀⠌⡹⢶⡿⠋⠁⠉⢉⣿⡟⠃⡤⠀⠀⠀⣠⠀⠀⠀⠀⠒⢫⡄⣀⣠⣼⠂⠀⠀⡠⠁⠀⡁⠀⢹⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⢰⣾⣀⣠⡄⣤⣿⡏⠁⠀⠀⡠⠀⠹⡇⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠛⢲⡆⠀⠀⠰⢧⠀⠀⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠹⣶⢦⠉⠁⠀⠀⢷⠀⠘⣳⢮⠞⠁⡤⠁⠀⠀⢀⠀⠤⠀⠤⠬⣀⣀⣀⣀⣤⡼⠃⠀⠀⠀⠈⠃⠀⠷
⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠃⣀⣴⡆⣿⣄⡀⣇⠀⠀⠀⡀⡀⠀⠈⠹⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⠀⠀⠀⠀⠀⡴⠂⢠⠛⠻⠂⠀⠂
⡏⠀⠀⠀⠀⠀⠢⢦⣈⣿⠁⢰⡿⠈⠀⣿⠀⠀⠙⣿⢴⠉⠀⠀⠀⠀⠀⡀⠰⠶⠤⠀⢀⣀⣀⣤⡶⠀⠀⠀⠀⢦⠀⠀⢸⡿
⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠈⣣⣤⣦⣽⣦⣤⣀⣷⡀⠀⠀⣤⠐⠒⣟⠛⠉⣉⠉⠈⠋⠉⠛⢫⠀⠀⠀⠀⢀⠀⠀⠀⢸⣧⡀⠄
⣇⠀⠀⠀⠀⠀⠀⠛⢯⣿⣴⣿⣿⣿⣙⣿⡏⢈⡻⣿⣿⡛⠀⠤⠀⣀⡼⠛⠶⣤⡤⠶⢤⣤⠾⠃⠀⠀⠀⣠⡁⠀⠀⠀⡀
⣿⠀⠀⠀⠀⠀⠰⢶⣿⡿⠿⣿⣿⠃⡀⠲⢾⣿⣿⣾⡿⠳⢴⣾⣭⡀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠠⣀⣼⠁⢠⠀⠀⠚⠃⠀⠀⠀⠀⡀
⣿⠀⠀⠀⠀⠀⢠⣤⣽⣷⣤⣸⣿⣿⡦⠁⣀⣵⡿⠛⠁⠀⠀⠙⠻⡿⣦⣄⠀⠀⡠⠀⠢⠀⠀⠀⣾⠋⣉⠀⡀⠙⠉⠉⢀⠀⡤⠴⠊
⣿⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⠙⢻⣿⣧⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠙⢎⠻⣶⣼⠁⠀⠀⢸⣤⡾⠙⡈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣄⡀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⢘⠻⣿⣾⣿⣿⣯⡅⠀⠈⠛⢦⠀⠀⠀⠀⠀⠀⠀⢣⠈⣧⡏⠀⠀⠘⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⡿⠟⠋⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⡾⢋⠕⠀⢀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⢀⣿⣽⣥⢀⠀⠀⣆⢀⡀⠴⠊⠉⠀⢀⡀⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⣾⣷⡖⠋⠩⠅⠲⠖⠺⠁⠀⢦⠀⠀⢠⡄⢀⠀⠻⣷⣀⣴⣿⡿⠿⠂⣸⡇⢀⠿⠀⡀⠀⣠⡄⢢⠀⡿⠙⢠
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠋⢻⣷⣤⣒⠀⠀⠀⣠⠤⠘⢧⡀⠸⣽⣿⣧⣀⠙⢿⣇⡈⠵⠤⣤⣭⠄⢸⡆⢿⠃⢸⡏⠑⢺⡀⠓⠈⠑⢇⡄
⣿⡇⠀⠀⠀⠀⠀⣠⡾⢏⡁⢠⣶⣾⣿⡛⠿⣿⣶⣿⠁⠺⣿⠀⡳⣄⠀⢹⣿⣼⠆⠀⡾⠓⠆⠀⣼⠇⠀⢸⠃⢸⡀⠺⠀⡤⠄⠀⢠⡀⠚⠋
⣿⡇⠀⠀⠀⠀⣾⣿⣧⣌⣷⣿⣿⡿⣿⠳⠄⣹⠏⢹⣿⣗⣿⢾⣷⢋⣀⣄⢠⠀⠀⢰⣦⡀⠀⢰⠿⠶⢶⣦⡄⠀⣀⠄⠐⡇⠀⠀⡌⠁
⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⢻⣴⣾⡟⠀⠸⣿⣿⣿⣿⣿⣟⠡⢿⡆⠀⠀⠀⢸⡇⠀⠀⠆⠀⣾⠟⠀⠀⠹⠓⢰⣇⡠⡘⠁⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⡿⢻⡿⠂⠘⣿⣿⣷⣶⣾⣿⢽⣉⣉⣯⣿⠀⠀⡇⣆⠁⢀⣾⠁⠀⣰⠀⠀⢻⡀⠀⣴⠃⢀⠞⢏⠙⡁⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣿⣿⣿⢻⣩⠹⢿⣽⣿⠐⢸⣿⠘⣿⡂⠀⣿⠃⡀⢸⠇⠀⣴⡟⠀⢰⡈⠧⢰⡃⢀⡸⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⣿⣿⣷⠀⣁⣀⠀⠀⠀⢸⣇⠙⣿⣿⡿⣿⠉⢷⣄⠀⠙⠳⢶⢿⣷⡈⠃⣜⠃⢠⣏⠀⢀⢰⣿⡃⠀⠻⢷⡄⢸⠰⠯⡤⣶⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⣿⣿⣯⣿⣿⣿⣿⣿⣮⠸⣿⡀⢈⡹⣷⣿⡆⠘⣿⣿⡦⣤⢿⡃⠙⠟⠀⢈⣵⡿⣿⣣⣬⣬⣿⣷⣶⣶⣸⠷⠿⠷⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣇⠀⣄⢻⣿⣷⣀⠍⠉⠐⣻⣿⣧⣤⣤⣴⠿⠿⠛⠉⢋⣉⣉⣈⣉⣉⡛⠻⣿⣿⣶⣤⣸⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣴⠆⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣯⣴⣶⣶⣿⣿⣿⣿⣿⠿⠿⠉⢻⡉⠉⠛⠛⠟⠛⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⡅⠉⠉⢁⠻⣿⣿⣿⣿⣿⡿⠋⣭⣿⣿⡿⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏
⣿⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠘⢦⡈⠃⡽⠛⣠⣼⣿⠀⣠⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏
⣿⣿⣿⣮⣥⣮⣽⣿⣿⣿⣿⣿⣿⣯⣭⣭⣤⣤⣤⣥⣬⣥⣤⣤⣭⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣾⣿⣿⣿⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠙⠛⠦⣄⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢱⡀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣿
⠀⠀⠀⠻⠿⠛⠛⠀⠀⠀⠀⣇⠀⠀⠀⢺⣄⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⢦⣉⠓⠒⠶⠶⢒⡿⠋⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠈⠙⠒⠒⠋⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠚⠉⠁⠀⠀⠀⠀⠀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠈⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠦⠤⠤⣄⡀
⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⣤⣄⡀
⠀⠙⠦⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠈⠓⠾⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡹⠓⡶⢤⣀⣤⡤⠀⠠⢤⡀⠀⠀⠀⠀⠹⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠤⠤⢤⣤⣀⠀⠀⠀⡗⠸⠁⠀⡩⠿⣦⣀⣀⡀⠉⠓⠶⠶⢆⡀⠑⢌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣄⡀⠘⢄⠀⢈⣿⢤⣉⠓⠦⠤⢤⡀⠙⠒⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠛⠛⠉⠓⠮⠭⠿⠦⣄⡉⢲⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠀⠀⠀⠜⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡒⠀⠁⠀⡠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡳⡈⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⢄⡉⠓⢴⣖⣒⠦⣄⣠⣤⡤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠣⢤⡀⠙⠒⠒⠦⣌⡙⢺⣏⠀⠈⢢⠀⠉⠙⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠢⡀⠉⠶⠶⠦⣄⠀⠉⠈⠙⢷⣖⡁⠀⣰⠠⡇⠀⠀⠈⠙⠛⠒⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⡀⠀⠀⠀⠀⠙⠒⠀⠐⠛⠋⠙⠲⠧⣼⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤⣄⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡿⠟⠁⠀⠀⠴⠒⠒⠦⠄⠀⠀⠀⠀⠀⠠⠴⠒⠒⠦⠄⠀⠀⠀⠀⠀⠀⢹⣿⣿⠲⣄
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢀⠈⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢁⡼⠃
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣠⣀⣀⡀⠀⠀⠉⠛⠋⠁⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢻⣿⣦⣭⣍⣙⣛⣒⣒⣛⣩⣭⣽⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⠀⠀⠈⠻⢿⣉⡉⠉⢉⣉⡿⠛⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⣟
⣿
⢿
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⡠⠒⠒⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠄⢀⡀⠀⠀⠀⠀⠀⠑⠈
⠀⢀⡀⣀⡠⠒⠂⡄⠀⠀⠀⠀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠀⠀⢀⠄⠒⠢⣀⡀⡀
⠀⠁⠀⠀⠀⠀⠀⠸⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⢨⠀⠀⠀⠀⠀⠈⠄
⠀⠱⣀⠀⠀⠀⡀⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠂⠀⠀⠀⢀⠔
⠀⠀⠀⠉⠒⢐⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⠐⠊⠁
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠁⠢⠄⠀⠀⠀⠀⠀⢀⠤⠂⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠀⠉⠐⠊⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠒⠈⠁⠈⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⣀⠤⠊
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐⠐⠈
⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢁⡀⠀⠂⠈⠉⠉⠉⠐⠂⣄⡙⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣴⣶⣶⣶⣶⣶⣶⣦⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠄⡄⣤⠠⠤⠤⠤⠤⠤⠄⡀⠠⡤⠀⠀⠤⠤⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠠⠄⠀⠀⠀⢤⣀⠠⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠠⠤⠤⣤
⠀⠀⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣀⢀⣀⡄⡄⡄⠀⠸⡇⠀⠀⠀⠂⠶⠀⠶⠒⠂⠀⠀⢸
⡠⡄⠀⠀⡇⡇⣀⣀⡈⢁⣉⡉⢁⣀⡀⠀⠇⠀⠀⣀⢈⣀⢀⣀⡀⡀⠀⠀⠀⠀⠈⠛⠈⠙⠛⠓⠋⠀⠀⠁⠀⠀⡛⠐⠚⠓⠓⠚⠚⠃⠀⢸
⠈⠁⠀⠀⡇⡇⠀⢉⣠⣬⡍⡄⣨⡅⠀⠀⠀⠀⠄⠤⠀⠠⠀⠀⠠⠤⠀⠀⡀⠀⢰⣰⣤⣆⡆⠋⠀⠀⠀⠀⠀⠀⢉⣀⡘⢋⡉⡛⣀⠃⠀⢸
⢀⣀⡄⠀⡁⡅⠀⠤⠤⠀⠴⠶⠤⠆⠀⠀⠂⠀⠒⠐⠂⠀⠀⠂⠀⠒⠒⠀⢘⠀⠀⠁⠀⠈⠀⠀⠀⢀⡰⠀⠀⢀⡌⠭⣉⠉⢉⣨⣩⢁⠀⢸
⣀⣙⡧⠀⠇⡇⠀⠀⠠⠒⠒⠂⠀⠀⠀⢠⠀⠀⠀⠉⠁⠁⠀⠉⠀⠀⠀⠀⠘⠀⠒⠢⠤⡀⠀⠀⠠⠈⢆⠀⠀⠀⠀⢠⣤⢤⣮⠄⠀⠀⠀⢸
⠀⠋⡀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠛⠀⠹⠀⠀⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠹⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠐⠠⠊⢀⣈⣵⣶⣶⡒⠲⢦⢴⣾⣿⣿
⢢⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣾⡿⣫⣷⠏⢻
⡀⠁⠀⠀⠀⠇⠀⠀⠞⠃⠘⠃⠁⠐⠞⠂⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣾⠿⢿⣿⣿⣿⡟⣿⣿⣿⣿⡟⠀⢸
⣧⠀⠀⠀⠀⠂⠐⠛⠛⠛⠀⠛⠐⠋⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠼⠋⠀⣀⣴⣿⣿⣿⣇⣿⣿⣿⠀⠀⠀⢸
⣿⡆⠻⣀⠀⠂⠀⠛⢚⠙⠛⢛⠀⠛⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⣽⣿⠟⢶⡄⠀⢸
⡉⢧⠠⢹⠀⠀⠀⠀⠀⠓⠃⠈⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⡀⠀⢹⠀⣽
⣧⠸⣷⣌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠰⢧⣢⠄⣾⣿⣿⣿⣿⣿⡟⣸⣄⠀⢸⡇⠀⢸⠀⣿
⣿⡆⢹⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⠀⠀⠈⠁⠀⣼⡙⠃⣿⣿⣿⣿⣿⡟⢀⡇⠹⣧⠀⡇⠀⣾⢀⣿
⠛⢃⠘⢁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠄⡀⠀⠀⠀⠀⣹⡦⣤⡤⣶⣾⠉⠟⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠀⣟⠛⡏⢻⣿
⣀⡀⠀⣰⠀⠁⠀⠀⠀⠀⠰⠶⠂⠔⠀⠀⠀⠀⠀⢧⣤⡀⠀⠀⠀⢘⢇⣿⠀⣿⡇⢿⣿⠃⢀⣿⣿⣿⠋⢻⣿⠀⠀⠀⠀⠀⠀⣿⢊⠉⣺⢸
⠛⠓⠀⢸⠀⠀⠀⢀⠘⣛⠀⡑⢋⢂⠛⡙⢀⠀⠀⢸⣏⣥⠀⡈⠀⢸⢸⣿⠀⢻⣷⠀⢿⠁⣼⣿⠟⠁⠀⢸⣿⢠⠀⠀⠀⠀⠀⢹⡚⢳⡏⢸
⠀⠀⠀⢸⠀⠀⠀⢈⣭⡅⠨⡉⣤⠬⠉⡡⠠⠀⠀⢸⠟⠁⡀⠛⠀⢸⢸⣿⠀⠘⠋⠀⠀⣰⣿⡏⠀⠀⣄⣸⣿⡟⠀⠀⠀⠀⠀⠈⣧⡟⢀⢾
⠀⠀⠀⢸⠀⡀⠀⠀⠒⠤⠴⠂⠶⠄⠴⠖⠀⠀⠀⢆⠀⠀⠀⠀⠀⢸⣦⠋⠀⠀⠀⢀⣴⣿⡿⢀⣠⣀⠈⣿⡟⠀⠀⠀⠀⠀⠀⢀⣜⣁⣤⣺
⠀⣠⣠⢿⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣿⠆⠁⣀⠀⠀⢸⣗⠂⠀⠀⢠⣾⣿⣿⢷⣭⣙⣻⣷⣿⣷⣆⠀⠀⠰⣎⠀⠺⣌⠹⠓⢻
⠀⠈⠉⢡⠀⣭⣭⣭⣵⠒⠠⠤⠤⠤⡤⢀⠀⢀⠀⢀⡀⡀⠿⡗⠂⠀⣿⣷⣄⣶⣿⣿⣟⠀⠘⢦⠉⠙⡿⠉⠀⠤⠰⢶⣶⡬⠉⢿⠉⡀⠤⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀
⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀
⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀
⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⠛⠿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠳⣄⡀⢀⣠⣾⣟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣼⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣴⣏⢈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣰⣿⣿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⢰⣿⣿⠁⠀⢹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⠁⠀⣸⣿⡿⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⢿⣿⣷⣤⣿⣿⣇⣀⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠘⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⢿⣷⣽⣿⣷⡀⢙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣦⡀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⣿⡌⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣆⠈⠛⢿⣿⣿⣿⣿⣿⣿⣮⣦⡙⣿⣿⡸⡿⣿⣿⣯⠙⢿⣿⠙⣿⣿⡇⠘⣿⣿⣿⣿⣷
⡌⣿⣿⣿⣿⡄⠀⠀⠙⠆⠉⠛⢿⣿⣿⣿⣿⣮⣿⣧⣧⠈⠻⣿⠀⠀⠙⠀⣘⣿⡇⣇⢿⣿⣿⣿⣿⡇
⣷⢻⣿⡏⢻⣇⣀⣠⣤⣤⣤⣀⣀⡉⠻⣿⣿⣿⣿⣿⣿⡆⠀⠹⣀⡤⠖⠋⠉⠈⢷⡸⡜⣿⣿⣿⣿⣧
⣿⣼⣿⡷⠛⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⠀⠔⠁⠀⠀⠀⠀⠀⠈⢷⡹⣜⣿⣿⣟⢿⡀
⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠄⠀⠀⠃⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡄⣀⡈⣷⡘⢿⣿⣿⡈⠣
⣿⡇⣿⡇⠀⠀⠀⡠⠂⢁⡴⠾⣿⣿⣷⡦⡄⠀⠀⠈⢻⣿⠀⠀⠀⠀⣀⣤⠾⠿⠟⠁⠸⣿⣌⢻⣿⣷⡄
⣿⣷⣿⡇⠀⠀⠈⣤⣾⢿⠀⠀⢸⣿⣿⣷⠘⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠻⣿⣿⣦
⣿⣿⣿⣿⠀⠀⠈⠙⣇⢸⡷⣶⣿⣦⠀⠈⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⢹⣿⣿⣧
⣿⣿⣿⣿⣧⠀⠀⠀⠈⠘⡀⠉⠙⠃⣰⢄⠁⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⢿⣿⣿⡄
⡌⢿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠤⡀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢸⣿⣿⡃
⣿⡘⣿⣿⣿⣿⣷⣄⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣇⢹⣿⣿⡛⠿⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⣀⣀⣠⣾⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⡜⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠻⢿⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣜⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⠣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠿⠿⣿⣿⣿⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠷⢄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣽⣭⣯⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡂⠀⠐⠢⢄⢀⠔⠊⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠊⠀⠀⠀⠀⠑⡍⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢱⠀⡴⣦⠀⠀⢸⡇⣀⡤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⠀⠀⠀⠀⠠⠀⠸⠀⠹⠿⠀⠀⡸⠁⠈⠁⠈⠩⠀⠙⢦
⠀⠀⠀⠀⢀⡤⢶⣿⠏⠶⠁⠢⡀⠀⠀⠀⢀⠆⠈⠂⠠⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠐⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢰⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⣸
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠉⠀⠀⠀⠀⠀⢠⠃⣀⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢢⠀⠈⠑⡖⠂⢲⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢸⠀⠈⡇
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠲⡀⠀⣁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡁⢠⡃⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠁⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⡜⠀⢠⡋⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡲⠦⢤⣄⣀⣀⣀⣀⣀⣠⠤⠔⠲⣏⡁⠀⠀⠀⠀⠀⠀⡁⠀⢀⡟⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠄⣊⣽⢱⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⡑⠦⠤⠖⠊⣉⡠⡀⡄⠉⠒⠒⡏
⠀⠀⠀⠀⢠⠒⣩⠒⠉⢰⣿⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠐⠒⠋⠉⠀⠀⠈⠉⠙⠛⠉
⠀⠀⠀⠀⠈⠢⢈⡑⠢⢼⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⡠⠔⠌⡳⠘⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⢐⠦⠄⠀⠀⠹⢄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠣⣀⠄⠀⡠⠃⠀⠉⠒⠤⢄⣀⡀⢀⣀⣀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠹⠄⠊⠀⠀⠀⠀⠀⠀⢘⠸⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⠀⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⠀⠀⡇⡇
⠀⠀⠀⡀⠤⠄⠐⠒⠒⠒⠒⠂⠠⢤⠇⠀⠀⠀⠀⡇⠠⠔⠒⠂⠉⠉⠉⠉⢉⡉⠠⢶⣤⡀
⡰⠟⠋⡡⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⡞⠁⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠑⡄⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⣄⢀⡄⠀⠀⠀⠀⠀⡀⢀⡀⠀⡆⠀⠀⢠⠀⠰⠒⠀⠂⠀⠀⠀⡄⢠⠀⠀⡄⠀⠀⠐⠆⠐⠂
⠀⠀⠠⡀⠀⠀⠀⡠⠂⠀⠀⠀⠁⠀⠁⡠⠁⠀⠀⠀⠤⢶⣶⡱⠁⠀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠲⡀⠀⣠⠂⠀⠀⠀⠒⠀⡀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣰⠦⠀⠀⠴⣄⠀⠀⠀⣀⠀⠀⣀⠄
⠀⠀⠀⠀⠤⠀⠄⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠰⠀⠀⡁⠀⠁⠠⠄⠰⠀⠀⠀⠀⠈⢠⠄⢠⠈⢈
⠀⠀⠀⢰⡆⠀⢸⡄⠁⠀⠀⠀⢉⡽⠗⠀⠀⠀⠀⢀⠂⠐⣀⠔⠀⠀⠠⣀⠀⠀⡀⠄⠀⠀⠂⡀⠲⠂⣀⠌
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠒⠦⠀⠤⠄⠀⠀⠰⠃⡀⠀⡈⠢⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠔⢡⡄⢠⡌⠢⠀⠀⠐⢁⠀⢀⡑⠆
⠀⠀⠀⠘⠲⠶⠖⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⡀⠈⠁⠐⠟⠀⠀⠀⠀⢀⡤⣤⡀⠀⠂⠀⠀⢈⣀⣈
⠀⠀⠈⠦⡴⡿⠟⠗⠀⠀⠐⠼⠏⠉⠿⠀⠀⠀⠐⠄⡘⠛⠀⠀⠀⠀⠑⠌⣀⣀⠥⠂⠀⠀⠀⠀⡼⠷⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠐⠁⣀⠀⢀⠑⠀⠀⠐⠁⣀⢀⡀⠑⠀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠉⠓
⠀⠀⠀⠀⣨⢤⡀⠀⠀⠀⠀⠀⣱⢶⡁⠀⠀⠀⡀⠐⠂⠐⠂⠀⠀⠀⠀⠈⣁⣘⠄⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠈⠂⠄⠀⠡⠊⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠐⠠⠀⠀⠤⠊⠀⠀⠈⠀⠁⠈⠐⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠄⠀⠀⠠⡀⠀⠀⢀⣤⠀⢀⣄
⠀⠀⠸⠁⣄⠀⠴⠀⠀⠀⠀⠀⡀⠀⡀⠈⠄⠀⠀⠀⠀⠀⡀⠈⠀⠀⠰⣐⠂⠐⣂⠜⡀⠀⠋⠰⠀⠀⠆⠣
⠀⠀⢀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠃⠈⠃⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢼⠃⡖⢲⠘⡧⠀⠀⡀⠀⣴⣦⠀⠐
⠀⠀⠀⠑⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠈⠀⠀⠀⠈⠂⠀⠀⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠙⠋⠉⠉⠛⠛⠛⠿⠻⢿⠉⠛⠛⠉⠉⠙⠛⠛⠛⢛⡟⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠘⠁⠀⠃⠀⠀⣾
⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⡇
⣿
⣿⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠑⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡅
⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡿⠁
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⠀⠀⠀⠀⠀⠀⠠⠀⠠
⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⢀⡀⣀⢀⣀⣠⣤⣶⣦⣠⣶⣿⣿

⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠉⠉⠉⠋⠙⠿⢭⣭⣉⣉⣛⣛⣿⣿⣿⣿⢿⠟⠛⣿⣷⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣟⣻⡿⠵⠾⣿⠿⠟⠛⠉⠶⠿⠿⢶⣤⣽⣄⡤⠉⠉
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠁⢤⡀⠤⠶⠤⢤⣤⠈⠉⠙⠛⠫⠄⢈
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠖⠲⠶⠢⣶⠎
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣀⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣤⠶⠋⠉⣩⡟⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⡾⠋⣀⠀⠀⠀⢀⣀⡀⣠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠁⣀⣤⠾⢋⣴⠟⢉⣴⡶⠀⢀⣩⡿⠟⠷⣦⠀⡀⠐⠒⠀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠙⠛⠛⠉⠁⠀⠘⠷⠶⣿⣡⣤⣶⠟⠋⠀⠀⠀⠀⠠⡾⠛⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠒⠋⠁⠀⠀⠠⠀⢀⠔⠂⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣠⣶⠟⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢼⣧⣽⠿⠛⣲⣶⠿⠷⣶⡆⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠇⠀⠀⢠⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏
⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⢁⣴⠿⠋⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠰⠂⣠⣾⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠏⠁⢰⡄⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠞⠛⠉⠀⠀⠀⠀⠀⠈⢀⠀⢀⣤⠙⠐⡀⠭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶
⠀⠀⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⡟⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢁⣼⠏⠀⠀⠀⠀⠀⠀⠀⢿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣿⣟⣁⣤⠴⠆⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⢛⣟
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠘⠯
⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⢸⠇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠋⠀⠒⣀⢀⡀
⠁⠒⠀⣤⣀⠀⣠⣾⣟⣩⣿⠟⢀⠀⠀⠀⠀⢀⠎⠙⠋⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡵⠂⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⣀⡀⠀⠀⠉⠛⠛⠊
⠀⠀⠀⠈⣻⣿⠿⠿⠛⠋⠁⠰⠿⠃⠀⠀⠀⢈⡚⠃⠀⣶⠏⠀⠀⠀⠀⠀⡀⠀⣀⡘⣾⣽⡏⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣄⠙⢿⣿⠃
⠀⢀⣠⣾⠟⠁⠀⠀⣀⣴⡦⠀⠀⠀⠀⠀⠀⠈⠙⠛⠆⠀⠀⠀⠀⠀⠀⣀⠀⠈⢹⣗⡅⠈⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⠦⠤⠤⠶⠤⠤
⣴⠿⠋⠁⠀⠀⢀⣼⡿⠋⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⡩⠃⠉⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣏⠉⠋⠉
⣿⣄⠀⠀⠀⢠⣿⣋⣀⣠⣴⣾⣿⠿⠿⣿⡧⢀⣀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣏⣉
⠻⣿⣿⠟⠀⠈⠛⠛⢩⣿⡿⠋⠀⣠⣾⡿⠟⢻⣿⡷⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⣾⠟⠉⠀⣴⣿⠟⠁⠀⣴⣿⡿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣾⣟⣁⣀⣤⣾⡟⠋⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⣽⡟⠀⣠⣴⡿⠛⠛⠻⠗⠂⠀⠀⢀⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣤⣤⠺⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡏⠀⠀⠀⠀⣠⡀⠀⠚⣀⡘⣶⣦⣼⣿⣿⣿⣿⣿
⣴⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄⣀⢀⠐⠶⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡀
⠘⠻⢠⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⠈⢙⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠓
⠀⠀⠉⢼⣋⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠈⠉⢿⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠋⠛⠋⠁⠉⠉⠉⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣴⠛⠛⠳⠟⠉⠛⠶⠚⠛⠳⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣀⣠⡶⠦⠤⠶⠛⠙⠓⠚⠙⠷⣤⣤⣤⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⣀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⠴⢦⣀⠀⠀⠀⣸⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢻⣏⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⣷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠓⠀⠀⠀⠉⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣯⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⠀⠀⢰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⢶⣾⣿⣿⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⣿⣭⣛⣛⣛⡭⢾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⡏⠿⠿⠾⠶⠶⠟⡇⠀⠀⢀⣀⣤⣴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣟⠛⠒⢷⡀⠀⠀⠀⠀⣰⠷⠖⠛⠉⠉⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢹⡏⠛⠳⠶⢿⣦⡀⣠⣾⣥⣤⣤⡶⠶⠛⠋⢻⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⢰⡇⠀⠀⠀⣀⣠⣼⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠐⠶⣾⣤⣴⠞⠛⠉⠁⢹⣇⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⣉⣹⣇⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠛⢉⣩⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠿⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠶⠶⠾⠷⣤⡶⠶⠶⣴⢾⣇⠀⠀⢈⣷⢤⣤⣤⣤⣤⣤⣀⠈⣧⢀⣴⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣰⡿⠶⣤⣯⡤⠶⣶⡞⢿⡀⠀⠀⣿⠀⠙⠳⠶⠛⠁⠀⡾⠋⠉⠻⡆⠉⠻⣿⢿⣅⣰⣏⣸⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⣉⣽⠿⠷⣾⠿⣭⣷⠟⠋⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣠⡟⠀⠀⢹⡿⠛⣿⣟⠛⠿⣭⣉⠉⠉⠉⠹⣇
⠀⠀⠀⠀⠀⣼⠃⠀⣠⡾⠋⠀⠀⣤⣼⣷⠟⢻⣄⣀⣤⡶⢷⡄⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⢉⣭⣅⣀⣠⡶⠛⠁⣴⣯⡿⠶⣦⠀⠙⢷⡄⠀⠀⢹⣆
⠀⠀⠀⠀⣸⠇⠀⢠⡏⠀⠀⠀⠀⣿⠻⢯⣛⠿⠯⣭⣤⣶⣾⣧⣀⣤⣾⣏⣩⣿⣿⠶⣤⠶⠛⠉⠉⠉⢋⣠⡴⠛⣹⣯⣀⣴⠟⠀⠀⠀⢻⠀⠀⠀⢻⡄
⠀⠀⠀⣰⠏⠀⠀⠀⢻⣄⠀⠀⠀⠙⢦⣄⠉⠛⠲⠦⣤⣬⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣠⣤⣤⠶⠶⠛⠋⢁⣠⠾⢿⣌⣿⠀⠀⠀⠀⣰⠟⠀⠀⠀⠈⢿⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠉⠛⠶⢦⣤⣄⣀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠉⠉⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠈⣷
⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣤⣤⣄⣈⣉⡙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⢀⣀⣀⣠⣤⣤⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠔⠊⠉⠀⠀⠀⠀⠀⠀⠂⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢁⠀⠀⣀⣀⡀⠉⠀⠉⢁⠀⡔⢹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡉⠉⠀⠀⠉⠉⠀⠀⠀⠀⠃⠨⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠋⠅⠒⠈⢁⠢⠠⠂⠌⠐⠒⠉⢢⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⡱⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠳⠠⠀⠀⠈⠀⡘⠛⠀⠈⠀⠀⠀⡸⠃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠄⡠⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠀⠀⠐⠀⠀⠂⠐⠊⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⢀⡌⢀⡠⡠⡢⡄⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠈⠉⠀⠀⠀⠺⢦⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠠⠀⠈⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢄⠘⠉⠛⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⡁⢂⣀⠀⠱⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠑⠤⠤⠔⠁⢈⡀⠀⠀⠀⠈⠂⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⡀⠡⢮⢹⡣⠆⠀⠀⠃⢄⠀⠀⠀⠀⡰⠈⡍⠀⣀⢡⠠⡄⣀⠀⡧⠐⣂⠠⠔⠊⠀⠈⡂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡚⠉⣁⠀⠀⢰⠀⠀⠑⠄⡀⠀⠐⠂⢸⢈⠋⡘⡆⢣⠈⣬⠂⠘⢇⢀⠀⠈⠁⢌⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠱⡄⢐⠀⠀⢸⠀⠀⢀⠀⠀⠁⠁⠀⠀⠈⠏⠈⡏⠀⢙⠈⡤⢬⣉⡨⠄⠂⠁⡠⠛⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⢄⠀⠀⠈⠺⣄⠀⢸⠀⠀⠀⠈⠐⠂⠆⠀⠀⠀⡇⠀⡇⠀⠀⠀⠣⡠⠄⠤⠀⠂⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⣄⡀⠀⠉⠚⠤⠀⠀⡀⠒⠄⠌⠁⠚⠠⠇⢀⣇⠀⠴⠐⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠌⡉⠓⠂⠤⠀⢰⠀⣄⠀⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠤⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⢀⡀⢀⡠⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠄⡀⢃⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠇⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠐⠂⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠠⠀⠀⠁⠂⢀⡀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠒⠆⠰⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⢀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣉⠁⠀⠀⠉⠉⠛⢿⣦⣄⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠈⣿⣷⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣤⣴⣶⣶⣦⣤⣄⡀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡟⠁⠀⠀⠀⠈⠉⠛⢿⣷⣶⣤⣼⣿⠏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⣿⠟⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠋⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠾⠛⠋⠉⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⣠⣿⡿⠁
⡄⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⣴⣿⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⢀⣾⣿⠏
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⣠⣿⡿⠃
⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢠⣾⣿⠟
⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠁
⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠋
⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠁
⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⣠⠞⠉⠁⠀⠉⠙⢧⡀⠀⠀⢀⡼⠋⠉⠀⠈⠉⠳⣄⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠳⢦⣀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⠆⠀⠈⢳⡀⠀⠀⠀⠀⠀⣀⡴⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⠶⣄⡀⢠⠟⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠻⡄⢀⣠⠶⠋⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠓⠶⠆⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠖⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉



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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠹⠩⠛⠟⣛⠙⠛⠛⠛⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣬⣦⣽⣬⣬⣤⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⠀⢀⣀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠐⠒⠀⠐⠀⠐⠒⠀⠠⠤⢉⡀⠂⠠⢀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠠⠄⠀⠄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠒⠀⠀⠀⠀⠀⠀⠒⠂⠠⠬⢁⡐⠂⠤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠒⠓⠀⠀⠀⠀⠀⠀⣤⣄⡀⠈⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣦⠀⠠⠀⠀⠀⣠⢤⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⣿⠟⠀⠄⠀⠀⢰⣧⣼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡌⠁
⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣄
⠀⠀⠐⠀⠀⠂⠒⠠⣄⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⡀⠀⠀⠁⠀⠀⠉⠄
⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡄⠤⠤⠤⠄⢀⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠀⠈⡄
⠀⠀⠔⢁⡠⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃
⠀⠀⡤⠁⡠⠂⢀⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠘⠂
⠀⠀⠑⠺⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⡀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣽⣭⣭⣭⣥⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣷⣾⣿⣷⣶⣿⣿⣿⣿⣿⣾

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠙⠳⠷⠀⠀⠉⠻⣿⣿⣿⣿⣿⣧⡰⣶⣛⣿⣭⡛⢛⠻⣽⣿⠟⠛⢻⣧⢀⣠⡀⠀⠀⠉⢶⡄⢲⡄⢈⣵⠟⠛⣻⢏⣠⠄⠀⠀⢀⠘⠻⣷
⣤⠤⠟⠛⠉⠀⠀⠀⠙⢿⣿⣿⣿⣿⣜⢛⠀⡛⠁⣈⡬⠻⣿⣧⠀⢈⣉⠈⣿⠇⠀⠀⠀⠀⠭⣆⣴⡞⣡⣦⡸⠞⠉⣝⣃⣀⣰⣾⣧⣀⡈
⠀⣄⣀⣀⣀⡀⢰⣶⣦⣤⣬⣿⣿⣟⠉⠙⣦⡀⣴⠤⠆⠀⠈⠀⡀⠀⠀⠀⠉⠀⣀⣀⣀⢷⠀⡁⢸⠟⠟⠋⣿⣠⣾⣿⣿⣿⣿⣯⡝⠘⢁
⣀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⡗⠀⠙⠶⠂⠀⠙⠃⢀⣀⡤⠤⠀⠹⣿⣿⣾⣶⣤⡈⠰⠇⣀⠛⢛⠿⠿⢿⣿⣿⣿⣷⣤⣼
⢻⡅⠀⠉⠉⠙⢸⣿⣿⣿⣿⣿⣿⢷⡌⠁⢈⠁⠀⠀⠀⣶⡄⣰⣯⠉⠄⠠⣤⡄⠀⢻⣿⣿⢿⣿⣧⣀⡀⠈⣠⢰⠇⣀⣠⠖⣺⣿⣿⣿⣿
⣤⡁⠀⠀⠀⠀⠘⠟⠛⠋⠉⠀⢀⣾⢣⣴⡟⠀⣰⣷⠶⣯⡰⠋⠀⠀⢰⣤⡿⠙⠰⣾⣿⣿⠀⠀⣽⣿⠇⠀⠙⢷⠔⠋⠀⣼⣿⣿⣿⠿⠿
⡲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⣾⣿⢃⡼⠟⠁⢀⣿⡁⢀⠴⣲⣭⠞⠁⠀⠀⢩⡉⠁⠀⠘⠻⠛⠀⢀⣀⠀⠀⠀⢸⣿⣦⣤⣤⣀⣀
⢱⣄⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⣰⡞⢛⡿⠋⠀⠀⠀⠘⠘⢿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⡀⣠⠀⢀⡄⠈⠁⠒⠂⣀⣸⢿⡿⠿⠿⠿⠿
⠸⣿⠀⠀⠉⠉⢸⣄⣦⣀⠀⠀⢹⣿⣿⠇⢠⣴⣆⡤⠀⠠⠀⠸⣷⢤⣤⠀⣠⠤⠮⣀⡀⠀⣽⠞⢒⡴⠂⠀⠀⠀⠸⢷⣶⢶⣶⣶⣶⡶⢶
⢀⣾⠀⠀⠀⢀⣀⣿⣿⣿⣿⣦⣾⣿⣿⠀⠂⠉⠀⠀⠀⠀⠀⢀⡏⠀⡀⢨⣤⠀⠀⠈⠁⠀⠀⣰⣋⣁⣀⠀⠀⢸⡇⠞⠱⠛⠉⠓⠉⠹⠿
⢎⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣧⢠⣶⣶⣶⣶⣿⣿⣶⣾⠯⣭⣁⣴⣿⡄⠀⠠⠆⠐⠀⠉⠉⣡⠅⢠⣤⣌⠁⠀⠀⠐⠒⠒⠒⣻⣯
⣼⠀⢒⡒⠒⠉⠉⢉⠉⠻⠛⠉⣰⣿⣩⣦⣤⣈⡉⠙⢿⣧⠹⣿⣧⠈⠻⠛⠩⣶⣾⣇⢻⣿⣶⠶⠒⢶⣦⣮⣤⣤⣤⠀⠠⠤⠶⠶⠶⠶⠶
⠏⠀⠨⣿⣮⠢⡀⠈⢳⡄⠀⠀⠉⠉⠙⠛⠛⠿⢿⣶⣄⠹⣇⠈⠿⠀⠀⠀⢀⠏⠈⡿⢸⠏⠁⠀⠀⠀⠀⠀⠉⠉⠉
⢀⡀⡘⠑⠁⣠⣾⢷⡯⠁⠀⢤⣤⣀⢀⠀⠀⠀⢸⣿⣿⣷⣮⡂⠀⠀⠀⠀⣼⠀⣸⢡⠏⠀⠀⠀⠸⠿⠀⠀⠶⠶⠶⠶⢆⣤⣤⣤⣤⣤⣤
⠆⠠⠐⠂⣠⠋⠀⢿⡇⢀⣤⢸⣿⢇⣿⣿⣷⣶⣼⣿⣿⣿⣿⣿⣶⣄⣀⣼⣿⣰⠃⠛⣤⣀⠠⣄⣲⣤⣀⣀⡇⠀⠀⢀⡎⠉⢁⣼⡿⣿⣿
⡤⠀⠀⠈⠘⠀⠀⠸⠁⣄⢿⢸⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣸⣿⡿⠃⠀⠉⠛⢿⣍⠁⠀⣠⣿⡇⠀⠈⠁⢾⣿⣿
⠾⠦⠚⠃⠀⣴⣦⡀⠀⠙⢾⡏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠛⣻⣿⣿⡀⣰⠟⠉⠀⠀⠀⢀⠀⠀⡬⠀⠀⠈⠉⢹⣿⠟⠀⠀⠙⠿
⣀⣁⣀⡇⠘⢿⡟⠀⡀⢀⣤⣷⣿⢿⢿⣿⣿⢿⣿⣿⣿⣿⡟⠀⠀⢠⣿⠿⣻⠁⢀⡀⠀⢀⣠⡀⢙⠂⡆⠀⠀⣄⣴⠀⠸⡋
⡿⠟⣹⡗⠛⠋⣛⠉⡉⠈⠛⢧⡟⢓⣰⡾⣧⣄⡀⠒⢾⡛⠣⠀⠀⣾⠏⢰⠃⢀⣾⠃⠀⣸⣿⣿⣿⡷⠄⠀⠀⠀⠤⠴⠝⡪⠐⣃⣀⢀

⢀⡴⠛⠉⠉⠓⠢⡄⠀⠀⠀⠀⠀⢠⠔⠛⠉⠉⠓⠦⡀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠒⢤
⡎⠀⠶⠀⠀⠰⠂⠘⡆⠀⠀⠀⠀⡏⠰⠆⠀⣀⠰⠂⢹⡄⠀⠀⠀⠀⣼⠀⠰⠂⠀⠰⠆⠈⡇⠀⠀⠀⠀⢰⠁⠀⡲⠀⠰⣇
⢧⠀⠑⠦⠤⠖⠁⢠⠇⠀⠀⠀⠀⢇⠀⠞⠉⠉⠉⠂⣸⠃⠀⠀⠀⠀⢹⡀⠈⠓⠒⠚⠁⢀⡇⠀⠀⠀⠀⠘⡄⢲⣤⡤⠤⠴⠋
⠀⠙⠲⠤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠈⠛⠭⠤⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣄⣀⢒⣶⠄⠛⠃⠀⠀⠀⣀⡤⠤⢤⣀
⡰⠋⣀⠀⠀⢀⡉⢢⡀⠀⠀⠀⠀⡴⣭⣠⡄⣤⣤⣽⢦⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠈⠳⡉⠁⠀⠀⠀⢀⣾⣥⣤⣤⣤⣤⣽
⡇⠠⡤⠤⠤⠤⣤⠀⡇⠀⠀⠀⠀⡇⠙⠿⢃⢈⠿⠋⢸⡇⠀⠀⠀⠀⢟⠀⠙⠚⠀⠙⠋⠀⡇⠀⠀⠀⠀⢸⢿⣿⣿⠟⢿⣿⣿
⠙⢄⡙⠦⠤⠔⣃⡜⠁⠀⠀⠀⠀⠱⢄⡀⠈⠉⠀⣀⠞⠀⠀⠀⠀⠀⠘⢦⣀⠑⠒⠀⢀⡴⠃⠀⠀⠀⠀⠀⠣⣌⠁⠤⠖⢉⣠
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⢲⣤
⠀⣠⠴⠶⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⠿⠿⢭⡁
⡼⠁⣄⡀⠀⣠⠌⠳⡀⠀⠀⠀⠀⡞⠑⣋⠀⠀⣉⡌⢳⡀⠀⠀⠀⠀⣰⠃⢀⡀⠀⢀⡀⠹⡄⠀⠀⠀⠀⢠⠏⠀⣀⠀⢀⣀⠙
⣇⠀⠈⢁⣀⣉⠀⢀⡇⠀⠀⠀⠀⡇⠐⠋⢀⣄⡉⠀⢸⠇⠀⠀⠀⠀⢯⠀⠈⢁⣀⡈⣥⠀⡇⠀⠀⠀⠀⢸⡀⠀⣄
⠘⠦⣄⣈⣀⣀⡠⠞⠀⠀⠀⠀⠀⠘⠦⣀⣙⣚⣁⡠⠞⠀⠀⠀⠀⠀⠈⠢⣄⣉⣀⣀⣩⠜⠁⠀⠀⠀⠀⠀⠑⢤⣈⣉⣉⣀⡴
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣄
⠀⣠⠶⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠦⣄⠀⠀⠀⠀⠀⠰⡉⣧⠴⠒⠒⠲⢼⣉⡇⠀⠀⠀⠀⠀⢀⡴⣖⠒⠲⠤⣀
⡞⢀⢤⣄⢀⡴⢦⠹⡄⠀⠀⠀⠀⡞⠀⣠⡀⠀⣠⡀⢳⡀⠀⠀⠀⠀⣸⠁⢤⣀⠀⣠⠄⠙⡆⠀⠀⠀⠀⢠⠏⢤⣀⠙⣶⣶⡌
⣇⠈⠓⣊⣀⡓⠚⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢠⢤⣴⢨⠇⠀⠀⠀⠀⢻⠀⣦⠥⠤⠾⣶⠂⡇⠀⠀⠀⠀⠸⡀⠀⡁⠀⠙⢛⡙
⠈⠲⢤⣙⣛⣉⡤⠊⠀⠀⠀⠀⠀⠈⠢⢤⣀⣈⣀⠵⠋⠀⠀⠀⠀⠀⠈⠓⠬⣓⣲⣒⡡⠞⠀⠀⠀⠀⠀⠀⠙⠦⣌⣑⣚⣩⠔


⠀⠀⠀⠀⣾⡿⣰⣶⣧⣾⣻⡞⣿⠓⣷⣾⣟⣶⣟⣷⣾⣶⣶⣿⡆⠀⢸⣿⣶⣷⢀⣾⣦⣾⣟⣲⣷⣆⠀⠐⠗⠗⢲⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠟⠽⠟⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⠀⢀⡀⠄⠒⠀⠐⠒⠒⠒⠤⢤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣦⣀⡀⠀⠀⠀⠀⠀⠙⠛⢟⠓⢊⣁⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⡆⣨⣭⡉⠛⠓⣶⠾⣿⡟⣿⡛⠉⠉⠀⣀⡀⣤
⠀⠀⡠⠔⠊⠉⠉⠉⠒⠲⠤⠤⣤⣠⡤⡤⠾⠓⡶⣿⠙⡡⠃⠀⠀⠷⠴⠛⠉⣿⠀⡀⢴⠺⠉⢁⣬⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⢀⣈⡥⠘⣔⣡⠚⠉⣩⠃⠀⠀⠀⠈⠓⢺⣁⠠⠋⠁⠁⠚⠀⠀⠀⠀⠀⢀⡤⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠙⡏⠁⡀⠤⠞⠉⢀⡀⠚⢓⡄⠀⠀⠀⠀⠀⠀⠙⠙⢄⣀⣀⠀⢀⣀⡠⠔⠊⠁
⠀⠀⣠⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣀⠀⠀⠀⠀⣀⣀⣀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⠛⠛⠿⠿⣿⣷⣦⣀⠀⠀⠀⣠⣶⣿⡿⠿⠟⠛⠻⠿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⢠⡞⣫⣽⣿⣷⡄⠀⠀⠀⠀⠀⣴⢛⣭⣿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠸⣿⣯⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⡀⢀⣿⠀⠀⠀⠀⢰⡇⣾⣿⣇⠀⣸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠳⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡦⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠙⠿⣿⣶⣦⣤⣤⣤⣤⣤⣶⣾⡿⠟⠁⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⡀⠀⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣤⣍⣝⣻⣿⣿⣷⣶⣿⣷⣏⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⠋⣉⣿⣿⣿⣿⣿⡟⢻⣦⣀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⡏⠀⠀⢈⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢣⠶⡜⣿⠀⣻⣿⣿⣿⠁⠐⠃⣀⢀⡀⠁⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢻⣿⡟⢠⠀⡇⢻⡾⠋⢿⣿⡟⠀⢄⠀⣿⣿⡇⢡⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠙⠷⣷⣿⣷⣜⣈⣷⢟⠇⠀⠈⡿⠇⠀⣿⣷⣿⣿⣧⣼⣦⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣤⣄⠈⠙⠇⢉⣉⣉⣠⠀⠀⠀⣠⣄⡀⣈⡝⠛⠛⠛⠛⢡⣄⣠⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣀⠀⢀⣹⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠴⠛⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠙⠳⠄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡠⠐⠀⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠈⠁⠂⢄⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠐⠀⠀⠔⣈⣉⠢⡀⠀⠂⠀⠀⠀⠀⠠⠁⠀⠔⣉⣁⠢⠀⠀⠂⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⣸⣿⣿⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⣇⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠐⣄⣠⠽⢿⣿⡯⠀⡠⠂⠀⠀⠀⠀⠈⢄⠀⢽⣿⡿⠯⣄⣠⠂⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠎⠀⠀⠀⠀⠀⠀⠀⠀⠱⠊⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡤⠖⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠲⢤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠁⢰⠁⠁⢢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡔⠈⠈⡆⠀⠹⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡇⠀⢨⠖⠀⠄⠀⢀⡀⠀⠠⠀⠲⡅⠀⢸⡀⠀⢱
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢳⣇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠨⡃⠀⠀⠀⠀⡇⠀⠀⡇⠀⣸⡞⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠄⠑⢾⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡷⠊⠠⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠈⢄⡼⠉⠉⠚⢳⠤⠤⠤⢤⡤⠤⠤⠤⡞⠛⠉⠉⢱⡤⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠑⢄⠀⠀⡞⠀⠀⠀⠀⠄⠀⠀⠀⢸⡀⠀⡠⠊⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠈⠐⠣⠀⢀⠀⢀⡃⢀⠀⠀⠄⠃⠁⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠄⠀⠠⠀⠠⠤⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡰⠁⢀⠈⠓⣤⠊⢀⢄⠀⠑⡄⠀⠀⠀⠀⠎⢁⢀⠈⠢⣠⢋⠠⡠⠈⠂⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠄⡂⢀⠠⢂⠅⠂⢄⠃⠀⠀⠀⠰⠀⠂⢕⠀⠀⠁⠔⡨⡂⠅⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠑⢄⠨⢐⠅⡊⠐⠄⢂⠂⠀⠀⠀⠀⠀⢆⠀⠔⠠⡢⢐⠅⡪⠐⠠⠁⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠪⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠣⡀⠠⡂⢐⠅⢊⠔⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠀⠀⠐⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠀⠠⠤⠄⠀⠒⠒⠂⠉⠀⠀⠀⠀⠀⡀⢃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⡀⠀⠀⡀⠀⢀⢀⠄⠄⡀⠀⠀⠀⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⠥⠈⣀⣀⣀⣀⡈⠀⠤⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⡀⠔⡀⠄⡀⠀⡠⢐⠠⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡨⠁⢀⠡⠀⢂⠅⠐⡈⠔⡈⠄⢂⠡⠒⡡⠐⡊⠄⠂⠅⣂⠅⢂⠡⠐⠡⠒⡡⠐⠈⠔⠈⠒⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠁⠐⠁⠐⠉⠀⠈⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⠞⢶⠀⠀⣴⠓⢆⠀⠱⡀⠀⠀⡼⠁⠀⢀⠀⠀⠀⠀⡀⠀⠈⢧⠀⠀⢀⠎⠀⠔⠓⠆⠀⠰⠚⠢⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡤⠄⡈⠁⠀⠁⡀⠠⡄⠀⡇⠀⢰⠁⠀⠈⠉⠑⠀⠀⠚⠉⠁⠀⠈⡆⠀⠸⠀⠀⣀⡀⠀⠤⠀⠀⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢹⣦⣤⣀⣀⣀⣤⣶⠇⠀⡆⠀⠘⡄⠘⢗⡂⠠⠤⠤⠀⢒⣨⠃⢠⠃⠀⢸⠀⠈⢿⣿⣶⣶⣶⣾⣿⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠙⢿⣿⣿⣿⡿⠏⢀⡜⠀⠀⠀⠱⡄⠈⠛⢿⣶⣶⡿⠟⠁⣠⠏⠀⠀⠈⢧⡀⠀⠻⡿⠛⠻⠿⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣉⣁⡠⠔⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⣠⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠉⠑⠢⣄⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢀⡀⠀⠀⠀⣀⠀⠈⢳⡀⠀⠀⡰⠃⢠⠶⠆⠀⠰⠲⡄⠀⠘⣆⠀⠀⢀⡞⠁⢀⡀⠀⠀⢀⡴⠤⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠋⠉⠀⠀⠈⠉⠑⠀⠀⠇⠀⢠⠁⠀⢀⠀⠀⠀⠀⠀⠢⡀⠀⠘⡄⠀⢸⠀⠀⠁⠙⠀⠀⠘⢀⠠⣶⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢖⣒⠒⠒⠒⠒⠒⣒⡆⠀⡇⠀⠸⡀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⠃⠀⢸⠀⢺⣆⡀⠀⠈⢀⣠⣾⠏⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠙⢿⣿⣿⣿⡿⠋⠀⡰⠁⠀⠀⢳⡀⠘⢯⣙⠻⢟⣫⠽⠁⢀⠞⠀⠀⠈⢆⠀⠻⢿⣿⣿⣿⠿⠋⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠈⠉⠁⣀⡤⠚⠀⠀⠀⠀⠀⠙⠦⣄⡀⠉⠉⢀⣠⠔⠋⠀⠀⠀⠀⠀⠓⢤⣀⡀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁