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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢷⠂⠔⠋⢀⣀⣀⣀⡀⠈⠉⠉⠛
⣿⣻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⢺⣰⠄⢸⣇⡼⠋⠉⠁⠀⢠⡄⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠛⠛⠿⣿⣿⣿⡟⠃⣀⣤⠴⣌⣁⣀⠀⠀⡜⣢⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⠻⣽⠎⠙⢿⡀⠜⠛⠙⠛⠂⢙⣍⣱⣾
⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⢠⡇⠐⠿⠿⢿⣟⣕⣿⡝
⣿⣿⣿⣿⣿⣿⣯⡄⠾⢢⣌⣛⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢲⠀⠴⠀⠀⠀⠲⠤⠀⠈⠁⠈⢣⣄⠠⢐⣛⣾⣿⣼⢿
⣿⣿⣿⣿⣿⣿⣿⣵⣦⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣀⠀⠛⢧⢰⡇⣠⡄⠀⠀⠀⠀⠊⠉⠉⠉⠁⠈⠀⡉⠛⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⣿⣿⣿⠟⠂⠀⠀⠘⠃⣴⣄⡁⠄⢠⣿⣶⣟⠓⠀⣠⣄⠀⠀⠀⢰⡾⣿⣾⣶⣿
⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⠃⠀⠀⢀⡀⠀⠀⢹⣿⠥⠤⢾⠀⠀⡛⢂⡄⣝⠛⠶⠴⢤⣾⣿⣿⣏⣙⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⡉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡾⠛⠐⠀⠀⠀⠀⠘⡿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠺⠿⠛⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡴⣟⠉⣟⣷⠀⠀⠀⠀⠀⠀⣷⣿⣭⣝
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⣠⣀⣸⡄⠈⠀⠀⠀⠀⠀⠀⠐⡻⣾⡟⠛⠇⠰⡿⠻⣧⣤⣄⣄⠀⠀⣽⣿⣿⣮
⣿⣿⠏⡵⠒⡙⣿⣿⣿⣿⣿⣹⣷⣿⠹⢿⡟⠀⠀⢠⠴⣶⠄⠀⠀⠀⠀⠙⣽⣿⣶⣾⠷⠀⠈⣉⣋⣭⣟⣟⣋⣉⣋⠉
⣿⣿⡄⢢⠀⢄⣻⣿⣿⡿⢿⣿⣤⠄⠀⠘⠃⠀⠀⢸⣀⠀⠀⢀⣠⡤⠁⢠⣿⣿⣥⣴⠂⠠⠀⠠⠸⠉⠾⢽⣿⣿⣿⣿
⣿⣿⣿⣶⢆⣘⣻⣉⣋⠙⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⢸⣽⣿⡿⠿⠤⠀⡀⢀⠰⠶⡖⣦⢬⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⠂⣴⠖⢲⡀⠀⠀⠠⣦⣀⢀⣀⣀⡀⠠⠄⣼⣿⣿⡆⠼⠂⢠⣦⣼⣶⣄⣡⣾⣶⣰⣶⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣾⡿⠿⠶⠶⠽⠿⠿⠿⢷⣤⣶⣾⣿⣿⣿⣇⠠⠀⢸⣟⠉⣿⣿⡝⣿⣿⣡⣄⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⠿⣿⠘⡇⠘⠛⢆⣹⣍⣿⣾⣿⠁⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄⣀⡀⠀⠠⡄⠀⠀⠀⠉⠉⠋⠛⠛⠀⣼⣟
⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠄⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿
⣿⣿⠋⣀⣲⣾⣗⣮⡿⣟⢛⣿⣿⣿⣿⣦⣤⣄⣀⠀⠀⠀⠻⣿⡿⢿⣿⣿⣿⠋⢰⡿⢿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿
⣿⣿⠀⠛⢻⣿⡍⣹⠟⠫⠾⣿⡿⣿⣽⡿⣿⠛⢛⣻⣿⣻⣷⣾⣙⣴⣉⣿⣿⠘⠶⣇⣼⠃⠀⣤⣤⣤⣴⣾⣿⠏⠘⣿
⣿⣿⣆⠉⠙⣿⠟⣁⣼⡇⠷⣦⣍⡻⣿⡯⣍⣾⣿⣿⣟⡹⢫⣿⣝⣻⠟⣩⣤⣲⠾⠄⢧⣀⡴⣿⣿⣿⣟⣻⣧⡀⠀⠈
⣿⣿⣿⣷⣶⣶⣾⢿⣟⠍⠀⠀⠆⠀⣴⠀⣤⡄⢠⡄⢩⣤⠀⣤⡄⢀⣴⡄⠀⠀⠀⢀⠸⡟⠀⠛⠛⠳⣯⣭⣵⣷⣶⣶
⡿⢋⣠⣠⣦⡙⠋⠛⢻⢧⣀⡀⣶⡿⢣⣄⠀⠀⡤⣇⣈⠣⣤⣄⣤⣬⣧⣦⣤⣲⣤⠶⠀⠁
⠀⢹⡏⢿⡟⠃⠀⡀⠀⠉⠈⠀⠈⣧⠘⡏⠉⣀⡺⠟⠋⠘⠻⠋⠻⢄⣿⠋⠀⠉⠉
⠀⠀⠘⠓⠀⠀⠞⠋⠀⠀⠀⠀⠀⣿⠀⡟⠀⣿⠁⠀⢤⡀
⣀⠀⠀⠀⡖⣳⡶⣧⣦⣠⠜⠃⠜⣿⢸⡇⣰⣿⡆⠀⢺⡧
⣷⡔⠟⢶⣾⣄⣠⣿⣿⣿⣤⣌⣿⠹⣿⠼⢿⣿⣇⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣷⣶⣿⣿⣿⣿⡄⡽⣟⣿⣿⡟⢠⣇⠀⢸⣿⣿⣧⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⣿⣯⣿⣿⣿⡟⣿⣿⣿⣿⣟⠁⠁⠀⢿⢧⣼⣿⣿⣿⣶⠉⠻⠇⠣⣄⡙⠂⣤⣶⡄⠠⠻⣇
⣿⣏⡹⣿⣿⣿⣟⠉⠙⠛⠻⣶⣶⠃⠀⣣⣾⣿⠻⣿⣿⣇⠀⠀⠀⣿⣷⠀⢹⣿⣷⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⡛⣿⡧⣘⣿⣦⣄⣒⣩⣤⣴⣿⡋⠤⠤⣿⣿⣿⣏⣯⣿⣿⣤⣤⠜⠻⢿⡇⣸⣿⣟⣤⣤⡀⠀⠀⢰⣷⣄⠀⠀⠀⠀⢸
⣿⣫⣶⣿⢿⣿⣿⠛⠉⣟⣛⣿⠁⠰⠦⢈⡎⠛⢿⣿⠿⠋⠸⢿⣤⠀⣸⣇⣿⡿⣟⢉⢹⣿⡄⠀⠀⠉⢿⣷⡀⠀⠀⢸
⣻⣿⣿⠿⢆⣽⣿⣿⣿⣿⣛⢻⣄⠀⠐⠛⠁⢠⣿⡿⠠⣤⣤⣤⣤⣶⣿⠿⣿⣧⣙⣚⣽⣿⠇⠀⠀⠀⠈⢿⣷⠀⠀⠘
⣯⠻⣿⣷⣾⣿⣿⣽⡟⣏⣿⣿⡀⠑⠀⠀⠒⢈⠉⠀⠀⠘⣷⣤⣿⡿⠙⠀⠙⡿⠿⠿⠟⠋⢀⠀⠀⠀⠀⠀⠟
⢷⣶⠚⢶⣿⣛⣿⣿⣿⠯⢽⣿⢇⠀⠹⣦⣄⠀⠀⠀⠀⣴⠋⢿⣿⡇⣾⠀⠀⠁⠀⠀⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠈⠑⠒⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⢀⣤⣚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠶⠥⠤⢀⣀⣀⣀⠀⢀⡴⠋⠛⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⣿⡆⢄⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⡠⠄⠠⡴⠁⠀⠈⠖⠉⠃⠀⠈⠁⢀⠱⡀⠀⠀⠈⢻⡿⠖⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠧⠀⠈⢆⡠⠔⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⡠⠂⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡌⠀⠀⠀⡀⢴⣿⡄⠀⠀⠀⠀⢀⡴⠊
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⢴⠊⠀⠀⣿⡇⠀⣀⡤⠚⠁
⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⠀⠑⡄⣘⠤⠊⣄⣰⡇⠀⣀⡿⠗⠋⠁⡆
⡰⠁⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⡌⠀⠁⠀⠀⠎⠛⢻⠉⠀⠀⠀⠀⠀⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠀⠁⠀⠀⠳⠀⠀⢀⠇⠀⠀⠀⠀⡠⠒⠁
⡸⠃⠀⠀⠀⡔⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⡜⠀⠀⣠⠔⠉
⠳⣀⣀⠀⠀⠑⠄⢼⡀⠀⠇⠀⠀⠀⡸⠄⠀⠊⠀⠀⠀⢠⣃⠔⠊
⠀⠁⠈⣆⠀⠀⠀⡄⠀⠀⢀⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⢤⠀⠀⠀⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⣠⡤⣤⣀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⠀⣴⡏⣷⡀⠀⠀⠀⠀⢸⠀⣂⣂⢸⠀⠀⠀⠀⢀⣼⡤⣼⣦⠀⠀⠀⠀⢸⣿⡾⣢⣿⡆⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⢼⣿⣿⣿⣷⠀⠀⠀⠀⢸⣀⣀⣴⠞⠀⠀⠀⠀⢸⣿⣆⣾⣿⠀⠀⠀⠀⠘⢿⣿⣿⠿⠁⠀⠀⠀⠠⠿⢿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⣶⠀⠠⣶⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡆⠀⠀⠀⠈⣠⣶⣶⡌
⠀⠀⠀⠀⠈⢦⡞⠈⢧⡀⠀⠀⠀⠀⠹⣿⡿⠁⠀⠀⠀⠀⠀⣀⣿⣏⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠻⣿⣿⠃⠈
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠃⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠹⠁⠈

⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⡴⠀⠀⡄
⠀⠀⠀⠀⣼⣿⠛⢻⣧⠀⠀⠀⠀⠀⠻⣷⣾⠋⠀⠀⠀⠀⢠⣄⢀⣴⠟⠁⠀⠀⠀⠀⣠⡎⢠⠰⠃⠀⠀⠀⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠸⠿⣷⠿⠇⠀⠀⠀⠀⠀⠾⠋⠻⠇⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠸⣀⡵⠋⠀⠀⠀⠀⠀⠉⣇⣀⣀⡏⠁


⠀⠀⠀⠀⡶⣒⠒⢒⠶⡆⠀⠀⠀⠀⠀⢀⣔⠦⠀⠀⠀⠀⣀⣠⣤⢔⣦⠀⠀⠀⠀⠀⣠⣴⢄⣀⠀⠀⠀⠀⠀⠀⣛⣛⣛
⠀⠀⠀⠀⣇⠄⠑⠁⢄⡇⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠈⢹⡾⢿⡏⠀⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠶⠶⠶
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉
⠀⠀⠀⠀⠀⠀⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⢰⣶⠀⢰⣦⠀⠀⠀⠀⢰⣿⠀⠀⢰⣷⠀⠀⠀⢶⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠈⣿⡀⠘⣿⡀⠀⡁⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣷⢰⢻⣷⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⡼⠈⣿⣆⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠻⠇⠀⠻⠟⠀⠀⠸⠿⠀⠀⠸⠿⠤⠄⠀⠸⠧⠤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⢸⣧⡀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⣆⢀⠀⣸⣿⠀⠀⣻⡆⠀⠀⢠⡄⠀⠀⠀⠻⣷⣄⠀⠀⠀⠨⢿⣦⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⡈⠀⣿⣿⠀⠀⣿⡇⠀⠀⣸⡇⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⢀⠻⣿⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣿⡇⠀⠻⣏⠀⢸⣿⠇⠀⡀⢿⡗⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠀⠀⠀⠈⡀⠀⣿⡿⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣄⠀⠀⠀⣴⡿⠁⠠⡀⠀⢀⣀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⢰⣿⣇⠀⢀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠈⣿⣿⠀⠀⢿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⣿⣿⠀⠀⠈⠉⠀⠀⠀⠀⠘⢿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢀⣠⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⣿⠏⠀⠀⠀⠈⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠻⠦⣄⠀⠀⠀⠀⠀⢄⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⣠⡌⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠡⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾



⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⢀⣀⠀⣀⣀⡀⢀⠀⡀⢀⡀⢀⡀⠀⡀⢀⢀⣀⡀⣀⣀⣀⣀⢀⡀⢀⡀⢀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⣠⣾⣿⡋⢓⢻⣿⡟⠚⠿⠾⣶⠞⡩⢙⣷⡱⢮⣾⣿⣿⢻⣿⡟⢦⣿⣿⠿⡏⣩⡻⢧⣿⣍⣹⣿⡀
⠀⠀⠀⠀⣯⣿⡣⢼⣟⣾⣻⡿⢸⡞⣻⣷⠿⡿⣾⢻⣞⠃⣼⣿⠳⢬⡋⠣⣴⣿⡘⢕⣃⢡⡈⣼⢿⡇⠻⣍⡅
⠀⠀⠀⠀⢺⣥⣌⣟⣋⣯⣿⠳⣷⣿⠉⢀⠀⣷⣿⡟⡻⢯⣿⡟⡗⠉⣬⡙⠓⠲⡄⡲⢤⣌⠙⠀⣏⡿⠿⣾⡇
⠀⠀⠀⠀⠈⠿⣿⣟⣹⡻⢿⣇⣻⡿⢤⡄⠠⡾⠛⣿⡕⢸⣿⣩⣿⣿⣯⠹⣷⣶⣹⣼⣾⢿⡃⡀⢻⣇⣶⠻⡆
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⢿⣿⣏⣡⣀⣀⣤⣴⡇⠀⠲⣾⠿⠟⢀⣼⣽⡇⣴⠋⠙⣿⣽⣿⠏⣡⡀⠙⢿⠷⠁
⠀⠀⠀⠀⠀⢀⣤⣾⠿⣿⣾⣿⣿⣿⣿⣯⣉⣍⡉⠀⠀⣿⣗⣶⣿⡧⠺⢰⣋⠀⠂⠨⣿⣽⡧⢑⡟⣦⠈⢿⡆
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡽⡿⠛⠻⠿⢿⣿⣿⡿⣷⣏⢹⣼⡻⢺⢚⢛⣰⣷⣦⠄⢠⠿⠻⣹⣿⣼⣏⡂⣸⠇
⠀⠀⠀⠠⡿⠗⠾⣿⣻⢯⠿⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢷⣴⡍⠉⢀⠠⠀⠀⣸⠖⠿⣿⣿⣿⣿⢣⡿⡄
⠀⠀⠀⠀⢯⡞⢺⣿⠷⣼⣷⣶⣅⡈⠀⠀⠀⠀⡀⢸⡛⢻⠟⠊⢉⣀⠴⠶⠀⠀⠀⡏⠐⠆⢻⢿⣿⡟⠈⢻⡇
⠀⠀⠀⠀⠾⠛⠁⠙⣂⢥⣴⣖⣿⣿⡏⠑⣾⠒⢳⣿⣿⣟⣿⣹⣟⣷⣶⡶⠀⠀⠊⠃⢰⡀⠈⢦⣩⡇⠂⣿⡁
⠀⠀⠀⠀⠠⢴⣾⣿⡇⢸⣿⣿⣿⣿⣷⢠⠿⢆⣸⠿⠛⣟⠛⠛⠛⠛⠿⡟⠀⠀⠀⠀⢈⣿⣦⣠⢿⣧⠱⢹⡇
⠀⠀⠀⠀⠀⠈⠿⠛⠓⢚⣩⡿⠀⣤⣯⣾⠇⢨⣹⡒⢀⣼⣄⠀⠙⠛⠟⠀⠀⡰⠀⠀⢠⢧⣈⡉⠀⣹⣰⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠄⠊⠉⠉⠙⠒⠧⢤⣄⠀⡈⠛⢃⡘⣠⠈⠃⠀⡰⠀⣠⠂⠈⠀⠀⠄⠀⠈⠿⣌⢹⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢌⣾⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠚⢏⣤⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⣤⡜⠀⠀⠨⢈⢄⣩⣿⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣺⣦⣿⣿⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠐⣿⠿⣿⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⠆⠐⢒⣶⢒⣻⣿⣿⣫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡄⣐⠀⠋⣀⣸⣿⣟⣋⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢨⣬⠿⣿⣿⣳⣾⣻⣿⠏⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣿⣤⣳⣿⣿⣿⣿⠿⠂⠲
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣁⣸⢀⠔⠃⠀⠤⠤⢀⣀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠄⠦⡀⢇⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠬⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠑⠀⡀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡔⡩⠀⣀⠔⠈⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢀
⠀⠀⠈⠐⠉⠈⠄⠀⠀⠀⢀⣠⠤⡍⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠠⠂⠈⠢⠄⠊
⠀⠀⠀⠀⠀⠀⠈⢢⡠⠂⠙⠥⣐⢗⠖⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⠄⠊⠀⠈⣀⠕⡟⢉⡷⠀⠀⠀⠀⠈⠢⢀⠀⠀⠈⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⡠⠊⠀⠀⢈⢵⠟⣻⠆⠑⢄⡀⠀⠀⠑⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣠⠂⠁⠀⡩⠪⡀⠀⠀⠈⠢⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠪⢀⠀⠀⡳⠂⠒⠁
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢰⡶⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣼⣃⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡾⠀⠀⠀⣸⠏
⠀⠀⠀⠀⢰⠏⠁⠀⣤⠶⣶⠀⢠⡴⠖⠆⣤⢲⠄⡴⢀⠐⠲⡄⢠⠶⡆⢀⡿⠀⣾⠁⠀⠀⠀⠻
⠀⠀⠀⠀⡿⠀⠀⠀⠳⠞⠛⠒⠛⠃⠀⠈⠻⠭⠀⠷⠻⣤⣤⠇⠘⢿⡥⠞⠧⠄⠻⣤⠄⠀⢰⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣤⣤⠴⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠰⠐⠀⠓⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠘⠃⠃⠈⠓⠃⠀⠂
⠐⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠂⠀⠠⠀⠐⠀⠂⠆
⠀⠀⣏⠀⠀⠀⠐⢄⣰⢢⢖⢲⡖⣖⣶⢂⢶⠇⠀⠀⠈
⠀⣀⠀⢀⠀⠀⢀⠁⠉⢘⠀⢁⡴⢦⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⡆⢰
⠀⠀⠓⠃⠀⠐⠛⠀⠈⠚⠃⢸⡇⠀⠃⣴⠒⣦⢠⠖⢦⡀⡴⠲⡇⢸⠖⣦⢰⡄⣴⢠⠖⢶⢠⠀⠀⡀⠀⠀⡀
⠀⢨⠁⣩⠈⠈⠉⠀⠀⠀⠀⢸⡇⢹⡇⣿⠀⣿⢸⠀⢸⡇⡇⠀⡇⢸⠀⣿⠈⣧⡇⢸⠓⢚⠙⣀⠘⠀⠁⠘⠁
⠀⠼⠾⡹⠿⡹⢩⠀⠀⠀⠀⠈⠳⠚⠃⠙⠒⠋⠈⠳⠚⠁⠛⠚⠃⠘⠒⠋⠀⣹⠁⠈⠓⠚⠀⡇⣱⣰⣖⡲
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠂⠂⠀⠀⠠⠀⠂⠃⠀⠑⠁⠀⠀⠀⠀⠀⠀⢠⠘⠋⡀⣀⡀⣀⣀⡈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠐⠐⠀⠐⠂⠀⠐⠀⠀⠂⠂⠈⠓⠀⠀⠀⠉⠐⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⡉⠀⠀⠐⠘⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠆⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠓⠂⠈⠉⠃⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡀⢠⣤⣀⣤⣄⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠶⠞⠛⣉⣉⣡⠀⣤⣤⣤⣤⣤⣤⣀⣘⠓⠿⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠛⣉⣀⠐⠾⠛⠋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠷⣶⣬⣙⠺⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠛⣡⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣙⠳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢋⣴⡿⠋⠀⠀⠀⠀⢠⣷⣆⠀⠀⠀⠀⠀⠀⠘⠛⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⣷⣌⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⣀⣠⡤⡤⠤⢤⠤⢀⣀⡀⠀⠀⠀⠠⢿⡿⠀⠀⠀⠈⢻⣷⡌⢷⡦
⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⡶⠟⠀⠀⠀⡀⠀⠀⠀⠀⣀⡴⠞⠛⢸⠀⠃⡇⡇⠸⢰⢰⠀⡏⡟⡶⢄⡀⠈⠀⣀⣠⣴⣾⣿⣿⣿⣆⠻⣶
⠀⠀⠀⠀⠀⠀⣀⡼⢁⣾⠃⠀⠀⢘⣾⡧⠄⠀⣠⠾⢩⠀⡇⡄⡘⢸⢸⠁⠇⡆⡾⢸⢰⠀⠁⣇⣼⣽⣶⣿⣟⠉⢿⣿⣿⣿⣿⣿⣧⠹⣦
⠀⠀⠀⠀⠀⠀⣹⠁⣾⠁⠀⠀⠀⠀⠈⠁⢀⡞⡇⠸⢸⢀⠁⡇⡇⡘⢸⢰⠀⠇⡇⣆⣸⣴⡿⠟⢿⡉⠙⣿⣿⡆⠈⢿⣿⣿⣿⣿⣿⣆⢹⣤
⠀⠀⠀⠀⠀⢸⡇⣸⡏⠀⠀⠀⠀⠀⠀⢠⢾⠀⠇⡆⡌⢸⢰⠁⠃⡇⣚⣸⣼⣶⣿⡟⢻⡅⠀⣤⣾⣷⡀⠹⣿⣿⡄⠈⢿⣿⣿⣿⣿⣿⡀⢿⡀
⠀⠀⠀⠀⠐⣿⢀⣿⠁⠀⠀⠀⠀⠀⠀⡞⠸⢰⠀⡇⡃⡈⢸⣸⣶⡿⢿⠋⢻⡄⠈⢧⠀⢷⡀⠘⠋⠹⣧⠀⠹⣿⣿⡄⠘⠋⠉⣿⣿⣿⣧⢸⡇
⠀⠀⠀⠀⢰⡟⢸⡿⠀⠀⢠⣷⡶⠀⢰⠃⡇⢘⣠⣤⣶⣿⣿⡋⠁⣀⣼⣧⠀⠳⠀⠈⠂⠘⣷⡀⠰⣿⣿⣧⠀⠉⠁⣹⣤⣶⠿⠛⠋⠉⣿⠈⡟
⠀⠀⠀⠀⢠⡇⢸⡇⠀⠀⠀⠋⠉⢀⣸⣤⡿⣿⣿⠉⠀⣀⠈⢷⡀⠸⠟⠻⣧⡀⠀⠀⠀⠀⢹⣧⠀⠉⢀⣸⣷⡶⣿⠛⠉⠀⢀⠀⠀⠀⣿⠀⡿⠂
⠀⠀⠀⠀⢈⣧⢸⣷⠀⣀⣤⣶⠾⠟⢿⡇⠀⠈⢿⣧⠀⠻⠄⠸⣧⠀⠠⣶⣿⣷⡀⠀⠰⡀⣀⣿⣷⡾⠟⢻⠉⠃⣿⠀⠀⠽⡿⠆⠀⢠⣿⠀⣷
⠀⠀⠀⠀⠈⣻⠘⣿⣿⣿⣿⡀⠀⣤⣾⣷⠀⢀⠀⢻⣆⠀⢠⡄⠈⣧⠀⠙⠉⢘⣿⣤⣾⡿⢹⢉⠁⡇⡇⠘⢸⢠⡏⠀⠀⠀⠀⠀⠀⢸⡟⢸⣧
⠀⠀⠀⠀⠀⠻⡆⢻⣿⣿⣿⣷⡀⠘⠛⢻⣇⠈⠇⠀⠹⣆⠀⢻⡄⣈⣧⣶⡾⢿⢻⠁⡇⡄⡈⢸⢀⠁⠃⡇⢸⡼⠀⠀⠀⠀⠀⠀⠀⣾⠇⣾⡁
⠀⠀⠀⠀⠀⠘⢷⡈⣿⣿⣿⣿⣷⠀⠰⣿⣿⡀⠠⣶⣀⣸⣦⣾⢿⠛⠋⡇⡆⠸⢸⢠⠀⡇⡇⠌⢸⢸⠀⢃⡟⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⡋
⠀⠀⠀⠀⠀⠀⠸⢧⠘⣿⣿⠿⣿⣧⠀⢹⣿⣧⣴⣿⢿⠋⡇⡇⡘⢸⠸⠃⡇⡇⣼⢸⢠⢁⠇⡇⡘⢨⡼⠋⠀⠠⢾⣶⠁⠀⠀⣰⡟⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡈⢯⣆⠉⣸⣷⠾⠛⠉⠀⠀⠐⠴⣅⡆⡇⠘⢸⢰⠀⠇⡇⡄⢸⢸⣀⡥⠟⠋⠀⠀⠀⠀⠈⠀⠀⢀⣼⡟⣠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠻⣧⡈⠀⠀⠀⠰⣾⡶⠂⠀⠀⠉⠑⠓⠾⠼⠴⠤⠧⠗⠚⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢋⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡘⠻⣦⣀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠂⠀⠀⠀⠀⣀⣠⠈⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣌⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠐⠺⡿⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣴⠿⢋⣥⡞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⣦⣌⠛⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⢋⣠⢶⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠓⣦⣬⣉⣛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⣋⣩⣤⣶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠙⠿⠛⠳⠖⠛⠿⠛⠻⠋⠉⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⡀⠀⢀⡀⠀⣀⣀⣀⡀⠀⣀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⠿⢲⣿⣿⢻⣿⡇⢛⡋⢸⣿⡿⠟⠿⢸⣿⣿⡄⣿⡇⢸⣿⣿⣿⣦⢰⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡦⠀⢸⣿⣿⣿⡟⠁⣿⣷⣸⣿⡷⠾⠶⢸⣿⡿⣿⣿⡇⣾⣿⠀⣹⣿⠈⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⠏⢻⣷⡌⣿⣿⢸⣿⣷⣶⣶⠼⣿⡇⢻⣿⡇⣿⣿⣿⣿⠋⢰⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠉⠀⠈⠁⠈⠉⠉⠉⠁⠀⠈⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀⢀⣀⣀⣀⣀⢀⣀⠀⠀⣀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⡀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣈⣿⡇⣿⣿⠟⠛⠛⢸⣿⡀⢸⣿⡇⣿⣿⠟⠛⠋⢸⣿⣟⣻⣿⡇⠀⠀⠀⠠⣿⣿⠛⠟⠀⣼⡿⢿⡞⣿⣇⢰⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡇⣿⣿⠿⠿⠟⠸⣿⣇⣼⣿⠃⣿⣿⠿⠟⠃⢸⣿⣿⣿⡋⠀⠀⠀⠀⠀⠈⠻⣷⣦⢀⣿⣷⣾⣿⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⠇⢻⣿⡇⣿⣿⣿⣿⣷⠀⠹⣿⣿⠏⠀⣿⣿⣿⣿⡷⢸⣿⠈⢻⣷⠀⠀⠀⠀⠠⣶⣾⣿⠿⢹⣿⠇⠹⣿⡇⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⢀⣤⣄⠀⠀⠀⣠⣤⡀⠀⢠⣤⣄⣀⠀⢀⣠⣤⣤⣀⠀⣤⡀⠀⣠⡀⢀⣤⣤⣤⣄⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠛⠃⣰⣿⠿⣿⣷⢀⣾⡿⢻⣿⡆⣸⣿⠿⣿⣷⢸⣿⣏⣽⣿⠇⢿⣧⣰⣿⠇⣿⣿⡟⣛⡋⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣶⡦⢿⣿⡀⣿⣿⠻⣿⣇⣨⣿⡏⣿⣿⣀⣾⣿⢸⣿⡿⣿⣧⡀⠀⣿⣿⠇⠀⢿⣿⠟⠛⠛⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠃⠈⠿⣿⠿⠋⠀⠹⢿⣿⠟⠁⢿⣿⡿⠟⠁⠸⠿⣷⣿⠿⠃⠀⠸⡿⠀⠀⢾⣿⣿⣿⡿⠀⢴⡶







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

⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⠶⠶⣯⢭⣭⠤⣤⡐⠒⢒⣒⣀⣁⣶⣶⣤⣤⡤⠶⠦⠴⠶⣒⣂⣀⣀⡀⢠⣤⠤⠤⠄
⠀⠀⠀⠀⣾⣧⣸⡄⠀⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠲⠶⣶⡂
⠀⠀⠀⢀⣿⢱⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣧⡄
⠀⠀⠀⠘⣿⣿⢹⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣴⡆⠀⠀⠀⠀⠀⠀⣰⠀⣶⣾⠏⠀⢰⡄⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⡸⠁⠀⠀⠀⣤⣴⡿⠁⠀⣸⣷⠀⢸⣷⢻⡇⢸⡿⠋⠀⡇⠀⠀⣾⠀⠀⣿⠠⣿⠁⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⣿⠉⠀⠀⢀⣿⣿⡀⢸⡏⢸⠇⢸⡇⢀⠀⢿⠀⢠⣿⣇⢰⣿⠀⣿⣴⠆⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣾
⠀⠀⠀⠀⣿⡇⡟⠀⠀⠀⠀⣿⣶⠆⠀⢸⡟⢻⣇⢸⣧⡞⠀⣼⡿⠋⠀⠸⡇⢸⣿⣿⣸⣿⠸⣿⠁⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⣿⡇⣷⠀⠀⠀⠀⣿⠃⠀⠀⣼⣿⢿⣿⢸⣿⣇⠀⢹⡇⠀⠀⠀⢧⣸⡇⢻⣿⡏⠀⣿⢀⣀⣤⢸⡇⣠⣶⡆⣿⣠⣶⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠁⠘⣿⢸⡇⢿⡆⢸⣧⣶⣾⠁⠘⣿⠁⠘⣿⠇⠀⣿⣿⠟⠃⢸⣿⡿⠋⠀⣿⡿⠋⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⡿⠀⠀⠀⠀⠻⠀⠀⠀⠏⠀⠀⣿⠀⠇⠈⠃⠸⠟⠋⠁⠀⠀⠁⠀⠀⠈⠀⠀⠉⠀⠀⠀⠈⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠁⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀⣤⣤⡤⠤⢔⡟
⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠶⠾⠛⣻⣭⠽⠫⠭⢭⣝⣋⡉⠴⠄⠘⠛
⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣴⣶⡶⠶⠶⠿⠷⠶⢤⣄⡀⠀⠀⣮⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠗⠒⠒⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢷⠀⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣇⡇⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠘⣿⣿⣿⣿⣿⣿⠇⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠀⣤⢿⣿⣿⣿⣿⣿⢠⠈⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⢰⣿⢸⣿⣿⣿⣿⡏⣾⡇⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢳⣿⡇⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠙⠛⠛⠛⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣤⣤⣤⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣦⣤⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣶⣶⣦⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠠⠺⡁⠡⠀⠀⠀⠀⠔⠂⢀
⠀⠀⠀⠀⠀⠀⠰⣆⢌⡄⠁⣴⠉⠲⡄⠀⡜⠀⠀⠀⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠄⣰⠗⠊⠁⠀⠈⠉⠁⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣐⡋⡠⢤⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠘⡹⠄⠀⠀⠀⠀⠀⠁⢲⠄⠄⠠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠴⢈⠁⢀⣀⠁⠀⡠⠀⠒⠢
⠀⠀⠀⠀⠀⠐⢀⡄⡾⠀⠀⠐⡨⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠰⠞⠰⠁
⠀⠀⠀⠀⠀⠀⠘⣍⠁⠀⢀⠀⠘⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠆⠄⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠔⡀⠈⠀⠠⠈⠁⠂⠤⠀⡀⠀⠀⡇⠀⠀⠀⠀⢀⣔⠒⠈⠀⠀⠓⠃⠟⠤⢒⠀⡄⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢰⡉⢜⡀⢘⠃⠀⠀⠀⠀⠄⡀⢀⠙⣄⣿⠂⠐⡀⠀⠈⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢾⠀⢙⢻⠉⠠⠠⠄⢀⣀⡀⠀⠁⠀⠈⠁⡦⠜⠁⠀⠊⠸⠤⠤⠤⠄⢰⠒⠚⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⢠⢾⡌⠆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠙⢉⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡈
⠀⠁⠣⠌⠙⢄⠀⠀⠀⠀⠀⠠⠤⣤⠤⢮⣦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠒⠒⠉⠁⠐⠒⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠐⠂⠤⣸⡀⠀⠠⠠⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣲⣍⢃⡠⠤⠒⢉⠞⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢿⣿⣿⣿⣿⣿⣷⠦⢀
⠀⠀⠀⠀⢀⣴⢂⣵⡏⡼⠸⣉⢁⣾⠀⡰⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⠛⠉⠀⣬⣷⣄
⠀⠀⠀⣠⠋⡿⣻⡈⣕⡸⢣⡇⠰⣋⡞⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣭⣼⣿⠿⠛⢹⣿⠅⠀⠀⣸⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⡀
⣀⣠⣜⣡⢠⠏⡸⣸⠅⣀⣬⣶⡏⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠻⣷⣄⣶⣽⣿⣿⣿⣿⣷⡛⠋⠀⠀⢀⡴⠃
⣾⡏⢹⠏⢞⣊⣵⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠘⠛⠙⠋
⣿⠧⠾⠾⠿⠿⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣿⣿⠀⠉⠛⠛⠛⣓
⠀⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⡟⠛⠻⠶⠶⣤⣤⣤⣿⣿⣟⣀⣀⣀⣀⠀⠀⠀⣠⠤⠖⠒⠤⣄⠀⢿⣯⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿
⣄⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣠⣼⣿⣿⣿⣏⣤⢶⣿⣿⣧⠀⣼⣿⢶⣶⣶⣦⠀⠐⠸⣿⡼⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⣉⣿⣿⣿⢁⠀⠨⠀⠙⣿⣿⣿⣿⣿⣋⠛⢻⣿⢿⣿⣿⡟⠛⠚⠋⠝⠟⠀⠁⠙⠈⠉⠉⠁⠀⠀⠀⣿⡇⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣾⠇⠀⣶⣾⣿⣿⣿⣿⣿⠿⣀⣾⣿⣿⣿⣿⡇⠀⠀⢀⣠⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿
⣿⡟⡙⠛⡻⠀⡆⢻⡏⡟⠙⣿⣿⣿⣯⣧⢸⣿⣿⣿⣿⣷⣤⡶⠋⠙⠲⠶⠛⠚⠉⠙⠻⣶⠿⠶⠀⣿⣇⣿⣿⣿⣿⠠⠜⠛⠛⠛⠛
⣿⣷⣤⣷⣾⣾⣶⣾⣷⣷⣦⣿⣿⣿⣿⣿⢸⣿⡿⢿⣿⣿⣿⠛⠻⣟⠓⠒⠒⠒⢛⡉⠛⠛⠀⠀⣰⣿⣿⣿⣿⣿⡇
⣛⣿⢿⣿⣿⣭⣭⡟⠛⠿⢿⣧⠖⣂⣬⣭⣬⣁⠀⠀⣿⣿⣿⣷⣄⠈⠯⠶⠶⠟⠋⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⠃
⣿⡇⠈⣿⣿⣿⠏⠀⠀⠀⢠⢋⣾⣿⣿⣿⣿⠃⠀⠘⠸⣿⣿⣿⣿⣦⡀⠀⠈⠁⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠻
⣿⡇⠀⣿⣿⠏⠀⠀⠀⢠⣎⢸⣿⣿⣿⠟⠁⠀⣠⡇⠀⢹⣽⣿⣿⣿⣿⣶⣶⣾⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⡏
⣿⢡⣤⠉⠁⠤⠤⠀⠀⠈⠁⠈⢛⣋⣥⣤⣶⣾⡿⠱⠇⠀⢿⢻⣿⣿⣿⠏⠉⠁⣙⢃⡟⣼⣿⣿⣿⠿⣿⣿⣡⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⠁⠀⣰⣾⣄⣆⠀⠈⠁⠀⠀⠉⠛⠛⢻⡭⠀⠀⠀⠀⠘⡎⣿⣿⣿⠀⠀⢰⢋⣾⣿⣿⣿⠟⠁⠀⢸⣿⢼⣿⣿⣿⡟⣿⣿⣿⣿
⣿⡄⠀⠀⠠⠠⡄⠀⠄⠀⢠⠀⡀⣀⣤⡀⠀⠀⣀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⣿⣸⣿⣿⣿⠁⠀⠀⠀⠈⡇⢸⣿⣿⣿⣇⣼⣿⣿⣿
⠉⠿⠿⠿⠿⠶⠗⠶⠿⠿⠿⠀⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣭⣉⣿⣿⣿
⣤⡤⡄⢠⣴⣧⣷⡼⢷⣄⣄⣼⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⢇⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣄⡁⠙⣽⢳⡿⢿⣷⣾⡿⠛⢻⣿⣿⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢾⣏⣿⣟⢻⡇⢿⡆⠀⡄⠀⠀⠀⠈⠀⠁⠉⠀⠀⠁⠀⠁
⠛⠀⠀⠈⠳⠶⠧⠭⠷⠄⠴⠀⠀⠹⣿⣿⣿⢦⠀⠀⢀⣤⣀⡄⣰⡿⣀⠈⣿⣿⣿⣿⣷⡌⢿⠔⠛⣚⣛⣉⣁⣀⡀
⠀⠀⠀⠀⠀⣀⣠⠴⢖⣒⡶⠖⠈⠑⠚⢿⣿⣾⣷⣟⣜⣿⠿⠛⠹⣷⣾⣧⣤⣤⣤⣤⣤⣤⣤⣶⡁⠌⢽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠃⢤⣼⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣶⣦⣄⡀⠀⠉⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⣷⣿⡏⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠈⠙⠻⣿⣦⡀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠙⢿⣷⣄⠀⠀⠈⠻⣿⣦⡀⠀⠀⣼⣿⠛⠛⣿⣧⠀⠀⠀⠈⢻⣿⡄⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠈⠻⣿⣦⡀⣿⣧⠀⠀⠸⣿⡆⠀⠀⠀⠀⣿⣷⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠈⠻⣿⣾⣿⡄⠀⠀⢻⣿⡀⠀⠀⠀⠘⠟⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠈⠻⣿⣿⡀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠈⠻⣿⣦⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠙⢿⣦⡀⠀⠀⠙⢿⣷⣄⠀⠀⠙⠻⠃⠀⠀⠀⠈⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠙⢿⣦⡀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠙⢿⣦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⣾⣷⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢹⣿⡀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠈⠛⠿⢿⣿⣷⠄⠀⠀⠀⠈⠻⣿⣦⣄⡀⠀⠀⠀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣷⣶⠆

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠰⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠴⠶⠋⠉⠙⠶⠦⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⣹⠟⢉⡟⠉⠉⠉⠉⠉⠉⠻⡉⠛⣏⠙⠻⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣧⣤⣼⡄⠀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣧⡀⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠿⣿⣿⢿⠗⠒⠒⠒⠒⠒⠒⠺⢿⣿⣿⣿⠇⠚⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠉⠁⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡀⠀⢀⢀⠀⢀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⡎⠉⠉⠉⠉⠉⠑⣲⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠁⠠⠔⠋⠑⠄⠠⠌⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿
⣿⠿⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠛⠿⣿⡷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿
⡅⠀⠀⢻⣿⣿⣿⣿⣿⡦⠀⠂⠀⠀⠀⠀⠀⠈⢘⠀⠂⠀⠀⠀⠀⠀⠀⠀⢁⠀⢱⠍⠙⠻⢿⣿⠟⠀⠀⠈⠀⠀⠀⠈⠻
⣷⠀⠀⢸⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡄⠀⠀⠀⠈⠉⠁⠂⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣧⠀⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⣿⣿⣧⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⣿⣿⣿⡇⠀⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⣇⠀⠈⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣘⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣄⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠐⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢰⢖⠤⢠⡤⠄⠀⣀⡢⠉⠕⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠆⢺⠎⠑⠤⢀⣀⣈⣀⢱⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⢹⡠⠄⡀⠀⠀⠀⠈⡀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢆⡀⠀⠀⠆⠀⠈⡇⢈⡆⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠠⠝⠀⡀⠐⠀⠀⣫⠖⠀⠀⠈⠀⠀⢐⠁⠀⠈⠒⠤⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠘⠀⠀⠀⢀⣠⠀⠘⠀⢅⠀⠀⠂⠀⠂⠀⠃
⠀⠀⠀⠐⠀⢀⢀⡄⠀⠔⠏⡀⠀⠹⢶⡆⢀⠐⡀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠘⠄⢂⠀⠈⠂⠒⠀⢼⡯⠃⠀⢀⠄⠅⠃⠀⡀⠂⡀
⠀⠀⠀⠀⠀⠈⢁⠈⠢⢄⠉⠁⠢⠄⡈⣉⣁⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢀⣀⣠⡄⠀⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠑⠒⠂⠺⣁⣀⡗⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⢤⡤⠋⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⢰⠀⠀⠀⠀⠀⣄⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠀⢸⠁⢀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠃⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠁⠀⢀⣐⡤⠆⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⠀⠀⢘⡰⠀⠀⠀⠀⠋⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⠀⠀⠀⠀⠀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠉⠋⠉⠉⠋⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢒⢲⠒⠐⠲⠒⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⠀⠐⠀⢀⠄⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⢄⡀⠀⡆⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠆⠀⠐⠒⠛⠷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠾⠿⠶⠃⠀⠠⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠿⠿⠿⠷⢶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⢀⣀⠀⢀⣀⣀⡀⣾⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣿⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⣧⠈⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣤⣤⣀⣀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢶⣾⣿⣤⣄⣸⣿⡀
⠀⢀⣤⣶⣾⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣿⣿⣷⡀
⣰⡿⠟⠛⠛⣿⡟⠛⠻⣶⠶⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⠶⠛⠉⠀⣿⣿⣇
⣿⡇⠀⠀⠀⡿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠉⠉⢻⡟⠷⠶⢦⣤⣀⠀⠀⣀⣀⣤⣤⡶⠾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿⢻⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⠹⣿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡇⠀⠀⠈⣿⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆
⣿⡇⠀⠀⠀⣿⣀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢠⡿⣻⣿⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢸⡇⢿⣹⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⣿⡇⠀⠀⢸⣇⠘⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠘⡯⢈⣉⡉⠉⢹⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⣿⡇⠀⠀⠀⢻⣶⢶⣶⡾⠃⠀⠀⠀⠀⠀⠀⢸⡇⣿⣉⣿⡄⠘⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢶⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟
⣿⡇⠀⠀⠀⠠⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣍⣉⣀⣀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⠀⠀⢰⣤⣦⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢙⣿⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠛⠻⣷
⠀⠀⠀⠻⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡟⠀⣀⠀⢹
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⣠⣤⣶⣾⣿⡿⠿⠛⠋⠉⣿⣿⡇⠀⠿⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣤⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣶⣶⣶⣤⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠛⠒⠒⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠛⣻⡇⠀⢠⣧⣤⣄⣀⣉⣉⣙⠛⡦⠀⠀⡾
⠀⠀⠀⠀⠀⢀⡤⠚⠫⠤⢤⡤⠖⠉⣀⣇⣀⣸⠁⠀⣠⠖⠛⠛⣿⠙⣇⠀⢸⡧⣤⣀⣀⣀⡀
⠀⠀⠀⣠⠖⢫⠀⠀⣀⠔⠋⢀⡴⠚⠧⠬⣭⣁⣀⠀⠛⠉⠉⠙⠃⠀⣇⠀⣼⠀⣠⠞⠉⢉⡽⠛⠓⠲⢶⣤⣀
⠀⠀⢻⠛⢲⠶⠷⣾⢤⣤⣞⣁⣀⣀⡀⠀⠀⠀⠈⠉⣩⠿⣖⣒⠒⠶⠬⢭⡵⠊⢀⣠⠞⠉⠀⠀⠀⢀⣸⡶⢿
⠀⠀⢸⡴⠋⠀⠀⡇⠀⢸⠁⠀⠉⠉⠙⠛⠛⠓⠶⢾⠧⣤⣤⣏⣀⣀⡴⠋⢀⡴⠋⠈⢉⣑⣒⣤⡴⠋⠁⠑⣾
⠀⠀⢸⡇⠀⠀⠀⠁⠀⢸⠀⣴⠒⠋⢻⠀⠀⠀⠀⠘⣆⠀⢹⡀⠈⢹⠉⠛⡿⠶⡿⠿⣭⡽⢻⠃⠁⠀⠀⠀⣾
⠀⠀⠘⣧⠤⡄⠀⠀⠀⢸⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠸⠤⠶⠃⠀⢸⠀⠀⡏⠀⠱⡀⠀⣇⡞⠀⢸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⣧⠀⢸⠀⢸⡇⢸⡇⢀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠙⢦⣿⠃⠀⠸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠶⠒⠦⣀⠀⣸⠀⠀⡇⣤⠖⢲⠀⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣿⠤⠼⠀⢸⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠀⢦⠈⣇⣿⠀⠀⢷⢹⠀⢸⡄⣿⠀⠀⢠⠀⠀⠀⢸⡀
⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⣿⠉⠉⠉⠉⠉⠉⡇⠘⣄⠢⠤⠜⢀⡏⣿⠀⠀⢸⠈⠷⠴⠇⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⢹⠒⠒⠒⠒⠒⠚⠁⠀⠈⠓⠒⠚⠋⠀⣿⠀⠀⢸⠑⠲⡄⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣧⡀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⡄⢸⠀⢻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⣧⠈⢢⡀⢻⠀⢸⡄⠀⠀⡏⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⣀⡤⠋⠀⢸⠀⠀⠀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠙⠓⠲⠷⢼⣤⣤⣇⣀⠀⠻⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⡇⠀⢀⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠶⠤⣤⣀⡀⠀⠀⠀⣾⠀⠀⢸⠀⠀⠀⢀⣼⣇⠀⠀⡇⢀⣾⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣿⣀⡀⢸⡆⢀⡴⠋⢸⡇⢳⣤⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠶⠦⠼⠗⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⢿⣷⣦⣄⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⢰⣿⠋⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠁⠈⠙⢿⣿⣿⡄⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠙⢿⣿⣦⡄⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣾⡟⠉⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣶⡿⠟⠻⢿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣦⡀⢿⣇⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⢿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣮⡿⣷⣄⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠈⠹⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣯⡻⣷⣄⠀⠀⠀⠀⠈⠻⣷⣀
⠀⠀⠀⢀⣤⣶⣾⣿⣿⣦⡀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣾⡟⠁⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢿⣧⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠈⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠈⠻⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⣠⡄⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢻⣿⡄⠀⠀⠘⢿⣷⣤⡀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣿⣄⠀⠀⠀⠉⠻⢿⣶⣤⣀⡀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠉⠛⠻⠿⠷⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣤⣀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁
⡀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠙⠻⣿⣿⣿⣿⣿⠟⣋⠁⣀⣀⢀⣀⡀⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠤⠤⠤⢀⡈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠤⣤⢤⣤⣠⣀⣀⣀⣀⢀⠂⣐⣁⣀⠀⠀⠀⠈⡄⠰
⠀⠀⡠⠒⠚⠛⠙⠛⠒⠚⠛⠻⠯⠿⠲⠶⠤⠤⠤⠤⣛⣲⣿⣃⠉⡉⣉⢉⣉⣷⣼⡷⠓⠒⠒⠒⠶⠴⠀⠒⠖⡦⠤⠤⠤⠤⢤⣤⣀⠀⠀⡀
⠀⠘⠐⠈⠉⠉⠁⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠤⠤⠤⠤⠀⠀⣀⢀⣉⣉⣁⠉⠈⠁⠉⠉⠃⠓⠒⠚⠛⠿⠿⠻⠘⢛⠀⢤⣀⡠⠤⠔⠈⠀⢮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠂⠐⠒⠒⠀⢄⠀⠈⢢⠈⠆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⢸⠀⠀⠀⠀⠀⡇
⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢺⠀⠀⠀⠀⠀⠠
⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⠀⠴⠀⠀⠀⠀⠀⣷
⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡄⠀⠀⠀⠀⢀⢸
⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⢇⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⢸⠸
⠀⠀⠁⠐⠨⠭⠍⠅⣐⣒⠂⠤⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⣸⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠒⠒⠬⠭⠄⣁⢐⣒⠀⠤⠤⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⡇⢀⣠⠴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠒⠂⠨⠭⠤⢀⣀⣒⠀⠠⠤⠀⠀⠴⠀⢀⣄⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⡤⠐⠚⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠋⠀⠀⠒⣒⠤⣀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⠉⠢⡀⢀⣄⡀⠙⠀⠀⠀⠈⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⣠⠀⠘⢫⠄⠈⠒⠓⡄⠀⠀⠀⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢷⡐⠃⢱⠀⠀⠀⠀⡎⢀⠀⢠⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢷⡀⢈⠀⠔⠁⢀⠇⢸⣦⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢆⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣤⣹⣦⣀⣁⠠⢾⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡝⢇⣷⢡⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣡⢤⠶⠇⠀⠀⠈⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡹⡌⠈⠀⡏⠘⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣹⣱⣃⡘⡄⣀⣀⡠⡼⠃⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⡔⠁⢸⠀⠀⠀⠀⠀⠀⡜⠁⢀⠎⠉⠀⣐⢟⣿⠂⠀⡰⠁⡰⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⢪⡀⠀⠀⠀⢀⡞⠀⣀⡎⠀⠀⣸⣿⣟⠉⠀⠰⠁⣠⠇⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠒⠊⠱⡀⠀⢀⠎⠀⢻⣿⣇⣀⡠⠿⣿⣿⡄⢀⠏⢻⣡⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠱⣀⠞⠀⠀⢀⢻⣿⡹⠀⠀⠘⣿⣷⡎⡴⢫⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠹⠀⠀⠀⠘⠈⢿⡇⠀⠀⠀⠘⣿⣿⡠⠃⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢃⠀⠀⡗⠀⠈⣿⡀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣼⡇⠀⠀⠟⣧⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⠞⠁⡂⠀⢸⠀⠹⣧⡀⠀⠀⠀⠀⠀⢀⡤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⢸⠁⠀⣨⣀⣀⠙⠉⣲⢤⣤⣤⡶⠋⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠈⢹⠀⠈⠉⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣀⠀⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣠⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠛⠇⠀⠀⠀⠀⠘⠶⠶⠾⠿⡿⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠃⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢸⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡸⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠃⠺⠿⠿⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠳

⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⣤⡤⠤⠤⠶⢶⣦⣤⣶⣤⣤⣤⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣶⣯⣽⣽⣃⣀⠀⢀⣀⣉⠶⠖⠒⠈⠷⠿⠾⢭⣤⣞⣻⠿⢷⣶⣦⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⣸⡍⠉⢻⣿⣿⠷⠶⢦⢤⡤⢤⣀⣀⠀⠀⠈⠉⠙⠒⠢⢍⣟⣻⣿⠦⢀⡀⠐⠀⠲⠄⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠎⠯⠀⢸⢯⠀⢠⣴⣖⣶⣿⣿⠋⠙⠉⠺⠶⢦⣀⢀⣀⠄⢘⠩⠐⠒⠛⠀⠠⢄⡀⠀⠀⡀⢭⣽⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢹⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡾⣁⣀⢴⣮⣭⣤⣭⣶⣾⢿⣵⡚⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠄⡐⡬⢖⣦⡀⣼⣬⡀⠈⠁⠀⠀⠀⠀⠀⠀⣴⣿⢯⠡⠀⠸⠿⢧⣿⣿⠈⢿⣿⣿⣿⣿⠃⢸⡿⠃⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣄⢛⣩⣥⠄⠿⢿⠇⠀⠀⠀⠀⠀⢠⡄⠀⠰⣼⣧⠜⠀⢸⠀⠄⠈⠃⠀⠀⢹⢻⣿⡿⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢨⠈⠘⠛⠙⠀⢠⣤⠂⠀⠀⠀⠀⠀⢳⣬⡀⠀⠀⠀⠀⠀⣻⣿⡇⠀⠀⠀⠀⢼⢸⣿⠁⠀⠸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠸⡟⡳⡄⠀⠀⠀⠀⢈⠙⡃⠀⡶⠟⠀⠘⢸⣷⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⢀⣴⡇⠀⣬⠀⠀⠀⠀⠀⠀⠀⢹⡯⠖⠀⠀⠀⠀⢸⣾⠃⠠⢿⢟⠠⠀⢸⡏⠁⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢐⠞⡿⣿⡇⠀⣟⡇⢠⣶⣄⠀⠀⠀⣸⣔⠑⠾⠆⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⢸⡧⠄⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠿⠙⠗⠀⣏⡷⣿⡬⢉⢿⣦⡀⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⣿⠄⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⡀⠀⠀⠀⣧⡇⠈⠑⠿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣵⣀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢲⠦⡄⠸⠀⠀⢾⢿⣦⣄⠰⢸⣏⡀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣄⣀⠀⠀⠀⠀⠀⣆⣻⡄⣼⠀⢠⠀⠀⡴⣿⣄⣠⠞⠀⠀⣿⣗⠂⠀⣿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⣤⢀⣀⠀⠈⠉⠉⠉⠛⠀⡄⡇⠀⠈⠻⠀⠀⠀⣯⣅⠀⠀⡏⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠒⠆⠤⣄⡀⢀⣄⡇⠀⠀⠀⠀⡠⣶⣿⣬⡤⢠⡏⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠢⢤⣠⣎⣀⣿⣿⣿⣷⣼⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠙⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣦⠀⠀⠀⠀⢠⣦⡄⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⣀⣀⣀⣀⣰⠀⠀⠀⠘⡇
⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠆⠀⠀⠙⠋⠉⠙⠁⠀⠀⠀⠀⡇
⣴⢯⡻⡌⢦⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣽⢮⡳⠜⠎⠛⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠆⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣧⣀
⠙⠳⣅⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠘⠷⣄⡀⠄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠳⣄⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠙⢦⡀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠹⠟⠀⠀⠀⠀⢀⡴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣠⠞⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣤⠾⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡄⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠰⠁⠀⠀⠀⠀⢠⠊⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠠⠁⠀⠀⠀⢀⠔⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠀⠒⠒⠀⠈⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠖⠉⠐⠈⠃⠀⠀⠀⠽⠀⠪⠂⠪⠂⠸⠇⠐⠕⠀⠀⠀⠀⠀⢀⣀⠀⢎⡁
⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠤⠤⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠂⠉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⡴⠶⡄⠀⢀⡀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠳⡄⢳⡀⠹⡄⠀⠹⡆⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣄⢳⡄⠙⣆⠳⡄⠙⣆⠀⠹⠀⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⢦⡙⣆⠈⢧⡹⡄⠘⣆⠀⠀⠀⢠⠞⠉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⡌⢧⡀⠳⣽⠄⠘⢧⣀⣀⡟⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠹⣆⠳⡄⠙⢦⣡⠀⠀⠀⠀⠀⠉⠉⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢷⡀⠈⢧⡈⢳⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠳⣄⠀⠹⣄⠙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⡤⠴⠚⠁




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

⠀⠈⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⠠⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠈⠂⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠈⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠠⠤⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⡀⠤⠐⠊⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠔⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠖⠉⠀⡠⠂⠀⠀⢀⠞⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⡤⠚⠁⠀⢀⠔⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣷⣶⣾⣾⡿⣾⣶⣶⣾⣿⣾⣾⣶⣶⣶⣶⣿⣶⣾⣷⣶⣾⣶⣶⣿⣶⣷⣾⣷⣿⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣾⣷⡷⣷⡾⢾⣾⢷⣶⣷⠾⠶⡿⠶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⠤⠤⠤⢄⣈⢑
⠀⠀⠀⠀⠀⣀⢀⣤⣤⢶⡤⢔⣠⣶⠤⢤⣼⣧⠀⢀⣀⣀⣨⣧⣀⣤⣄⣀⣤⣤⡤⣤⣤⣤⠄⡀
⠀⠀⠀⠀⣎⠀⢀⡠⠊⠁⠀⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣴⠋⠁⣄⠀⠀⠀⠆⡇⢱
⠀⠀⠀⠀⡁⡔⠁⠀⠀⣂⣀⣓⠀⠀⠈⢄⠁⠀⠀⠀⠀⠀⠀⠰⣂⡀⣹⠀⠀⠈⠑⢔⠁⢸⡇⢛
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠨⡂⠊⡇⢸
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠓⠀⠘⠀⢸
⠀⠀⠀⠀⡇⠀⠈⠚⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠩⢄⠄⠐⠞⠀⠀⢸
⠀⠀⠀⠀⠸⢄⠀⠀⢂⠀⠀⢸⠀⠀⠀⠀⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠑⠄⠀⠈⠚⠀⠀⠀⣆
⠀⠀⠀⠀⡆⢀⠠⡀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠄⠀⢠⡀⢠⡄⠀⢲⠀⢠⠀⠀⠀⠀⠀⠂⢱⡄⣾
⠀⠀⠀⠀⠑⠒⠒⠞⠒⠒⠀⠘⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣀⠀⡀⠀⠀⠄⣨⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠉⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⢊⣩⠴⣶⡿⢿⣿⣶⣶⣶⣶⣭⣭⣭⣭⣖⣒⣒⣠⣤⡤⠤⠄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠧⢄⣉⣀⡈⢠⡎⠁⠊⠡⡆⠩⠚⠝⠛⠝⠻⠛⠿⢟⢿⠿⢿⢿⣿⣿⣶⣶⣶⣶⠬⠇
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠉⣧⡏⠉⠉⣷⢒⠒⠒⠒⣚⢧⣶⠒⣺⢇⠦⢌⠝⠉⠁⠀⠁⣾⣷⡀⡠⡄
⠀⠀⠀⢀⠔⣿⣿⡷⡀⠀⠀⠀⠀⢻⢰⠀⣀⣽⡜⣒⠿⠟⣾⣿⡿⡟⠁⠀⠁⡘⠀⠀⠀⣸⠀⣇⣜⡛⣞⠁
⠀⠀⠀⠀⡆⠸⣿⣿⣎⢦⠀⣀⡠⠼⡏⡟⠛⠀⣇⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣷⣴⣾⣿⡿⢂⣥⣺⢿⠁
⠀⠀⠀⠀⡇⠀⠻⠿⠿⠓⠉⠁⠀⠀⢻⢠⣀⣀⣹⣰⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡀⠛⡜
⠀⢠⡒⠩⠥⠶⠶⠶⠂⢀⣀⣠⠴⢚⠭⣭⢶⣶⣾⣭⣭⣭⣿⣟⣛⣛⠿⣻⡏⣋⣉⣙⣤⠤⠒⠊⠉
⠀⠀⠀⠉⠉⠓⠚⣿⢿⠿⠿⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠓⠒⠛⠛⣛⠿⢱⠁⠘⣿⣸⣤⣄
⠀⠀⠀⠀⠀⠀⣟⣵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⢦⣧⣀⣼⣷⡋⢻⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠂⣸⡿⠉⠉⠷⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡖⠒⠒⠒⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠶⠶⡶⠂
⠀⠀⠀⠀⠔⠁⢸⠀⠸⠉⠁⠀⠀⠈⠉⢙⠀⢁⠀⠀⠂⠀⠀⠀⠀⡰⡆⠀⠒⠒⠛⠓⠒⠛⠛⠛⠛⠃⠘⠓
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡆⠀⠀⠀⠉⢢⣠⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠁⠓⠒⠀⠀⠐⠂⠐⠀⠀⠀⠒⠚⠉⠈⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣾⡿⡿⣷⣆⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣶⣶⣶⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠊⠉⠁⠉⠐⠉⠈⠁⠈⠁⠈⠀⠈⠫⡉⢻⡇⠀⠀⠀⢠⡟⠉⠉⠉⣿⠛⠉⠉⠉⠙⠋⣿⠋⠋⠉⠂
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⢀⣸⡗⠈⠀⠈⣉⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠙
⠀⠀⠀⠀⡇⠀⠀⡟⡷⠀⠀⠀⠀⠀⢟⢻⠀⠀⠀⢠⠄⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⣤⣤⠀⢠⣤⣤⣤⣤⣤⡄⢀⣤⣤⣤⢠⠀⠀⠀⠘⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢨⡆⠀⠀⠀⠘⡧⠁⠀⠀⣉⡀⠀⠀⠀⠀⠀⣉⡁⠀⠠⢤
⠀⠀⠀⠀⠄⠙⠿⠇⠸⠿⠿⠿⠿⠿⠇⢸⠿⢟⣁⠝⠁⠀⠀⠀⠀⠑⠀⠀⠀⠿⠒⠂⠀⠀⠀⠀⠿⠗⠂

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣯⠊⢻⣿⣿⣿⠸⠀⡇⢹⢸⣿⣿⣿⠟⠀⢰⢿⣿⣿⠟⣴⡿⣫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⢿⣷⡀⠹⣟⣿⠀⠁⠁⢸⢸⣿⣿⡏⠀⢠⠏⣾⡿⢣⡾⢋⣾⢟⣫⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣆⠙⢿⣄⠙⡞⣇⠀⠀⠀⢸⣿⣿⠀⠀⠈⢠⠋⡴⢋⣵⢟⣵⢟⢽⡯
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣙⢷⣄⠙⢦⡈⠘⠄⠀⠀⠘⢻⠇⠀⠀⠀⠀⠊⡠⢋⠴⠋⠀⠐⠝⣋
⠀⠀⠀⠀⠀⠀⠀⢨⣁⠉⠛⠿⠿⣇⡈⠳⢄⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣬⣍⣒⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣁⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠨⡥⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⠠⣦⠀⠀⠀⠀⠀⠐⠒⠭⠽⢷⣞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠶⠶⠶⠤⠀⠀⠀⠀⠈⠟⠋⠙⣈⠀⣡⠀⠀⠀⠀⠀⠀⠨⠭⠽⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠯⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⠸⠟⠟⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠭⣙⡓⠶⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡙⢦⣈⢷⣦⣄⡀⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠐⣋⢀⣠⡶⣀⠔⡀⢀⡴⠁⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠹⣄⠙⢷⣽⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⢟⣭⢞⣵⣞⣵⠟⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠘⣎⣆⠘⢷⣄⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⡷⣛⣵⣿⣿⡿⠃⣴⠃⡼⠀⠀⡿⠀⣾⢠⠀⡄⠀⢹⣞⣧⠈⢻⣷⣌⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⢁⣼⣟⣼⢁⡾⢸⡇⣿⣿⢸⠀⣷⢀⢸⣿⣿⣷⡀⡹⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⡿⣫⣍⠻⠿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⡴⠒⢦⣀⣀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⡿⢡⣿⠟⣿⠃⣾⡆⣿⣿⢰⣿⡆⠿⢿⣿⡟⢻⣧⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⠃⢠⡄⢰⡏⠉⡇⠀⢸⠁⠈⣇⣀⠀⠀⣤⠈⢻
⣿⣿⣿⣿⣿⣿⡇⢸⣟⢸⣿⠀⣿⡇⣿⣿⢸⣿⡇⣾⣷⢹⣿⠀⣿⡆⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⠀⡇⠀⢸⠀⠀⡏⠈⣷⠀⢸⡇⢸
⣿⣿⣿⣿⣿⣿⣧⠘⣿⣌⣿⠀⣿⡇⣿⣿⢸⣿⡇⣿⣿⢸⣏⣼⡿⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⡄⠘⠇⢸⡇⠀⡇⠀⢸⠀⠀⡇⠀⣿⠀⠟⠀⣼
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⠀⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⡇⠀⡇⠀⠸⠀⠀⡇⠀⣿⠀⠀⠘⠃
⣿⣿⣿⣿⣿⣿⢡⣶⣮⣝⠻⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⠋⠙⠲⢤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣮⡻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⢦⡄⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⢀⡴⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⣛⣛⣛⣛⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⡤⠤⠖⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⣠⡤⡄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁⢢
⠀⠀⠀⠀⠀⠀⡸⠵⠖⠒⠀⠐⠀⠠⠦⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠐⠁
⠀⠀⢀⠔⠁⠀⠂⡀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠄⠈⠀⠀⠀⡠⠊
⠀⠀⢠⠀⠀⠀⠀⠀⠁⠂⠄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⣀⠀⣀⠀⠠⠄⠂⠁⠀⠀⠀⢀⠔
⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⡀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⢠⣀⡀⠀⡀⠀⠀⠁⠐⠠⢠⡊
⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⡠⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢌⠐⠠⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⡀⠢⡀⠀⠀⠀⠁⠐⠤⡀⠀⢀⠀⠠⠤⠔⠒⠂⠑⠀⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡑⢤⡀⠀⠈⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠌⡀⠀⠑⠠⡀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡂⠁⠀⠀⡀⠊⠑⠠⢀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⠶⠀⣠⠒

⣿⣭⣭⣍⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⠿
⣿⣿⣿⣿⣿⣿⠷⠶⠶⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣶⣶⣶
⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣭⣭⣭⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿
⣯⣭⣭⣭⣭⣭⣍⣉⣉⣉⣉⣉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠛⠛⠛⠛⠉⠙⠉⠙⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐
⣟⡛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣒⣲⣶⣾
⣿⣿⣿⣛⣛⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣻⣿⣿⣿
⣛⣛⣉⣉⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠿⠿⠿⣿
⣿⣿⣿⣿⣿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶
⣿⣿⣿⣛⣛⣛⣛⣉⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣚⣛⣿⣿⣿⣿
⣷⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛
⣿⣟⣋⣉⣉⣉⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣉⣩⣽⣿⣿
⣿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣩
⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶
⣿⣿⣿⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿
⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿
⣿⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠻⠿⠿⠿
⣿⣯⣭⣭⣭⣭⣭⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣬⣭
⣓⣒⣒⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠶⠾⠿
⣿⣿⣿⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣼
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⣧⣀⠠⠄⠰⣖⢆⢀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠓⠦⢄⣸⣿⣷⢤⣀⣀⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠈⠳⣄⠀⠈⠉⠁⠀⠨⣝⣓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠒⠛⠉⠉⠉⠈⠉⠉⢙
⠀⠀⠀⠀⠀⣀⠤⠒⠻⣝⡻⣶⣦⣄⣤⣴⣾⠇⠀⠀⡀⠀⠀⠀⠉⠻⢶⣮⣍⣒⠢⠤⣀⡀⢀⣀⣠⡤⠶⠖⠚⠋⠉⠁⠀⠀⠀⠀⣠⠚⠭⠈⠛⣦⣠⣾⠟
⠀⠀⠀⠀⠀⠧⠤⠴⠶⠞⠛⠛⠉⠉⠁⠘⣧⡀⠀⠸⣙⠷⣤⣀⠀⠀⠀⠈⠙⠻⣿⣷⣦⣍⣙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣥⡎⢀⣠⣿⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠳⣎⠙⠷⢦⣀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣮⣍⡓⠦⣄⠀⠀⠀⠀⠀⣇⣽⣶⣿⡿⠉⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠷⣦⣌⠙⢧⣸⠉⢳⢦⣀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠓⠈⠙⢦⣄⣠⣶⣿⡿⢏⠀⠀⢸⣫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠉⠻⢦⣌⠳⢾⡀⢹⠓⢦⣀⠀⠀⠀⢀⣠⡤⣤⣀⡀⢰⠋⠹⡿⠋⠉⢦⣑⠀⢩⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⢀⡤⠤⣄⡀⠀⠀⠀⠀⠉⠻⢦⣍⠺⣄⠀⠘⣧⠀⠀⡟⠀⠀⠀⠀⠹⡜⣆⣄⣹⡄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⢀⢞⢕⢰⠂⢐⢝⢦⠀⠀⠀⠀⠀⠀⠈⣻⣾⣷⢦⣽⠀⠀⢻⣠⡄⣀⠀⠀⢹⣜⣟⣣⣧⣤⣄⡀
⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⢯⣿⡏⠀⠀⠀⢳⣈⣧⣠⣤⣶⣶⣿⣿⣿⣿⣿⣧⣀⣠⣄⡈⢿⠻⠿⣀⡴⠛⣩⢞⢕⠸⣦⡀⢙⢦⡀
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⣿⣥⣴⣟⣾⣿⣿⣿⠿⠿⠟⠛⠛⠉⠻⣧⡉⠙⠛⠁⠉⢻⣬⣷⡼⠋⠀⣰⢣⣳⠓⣈⣬⣏⢢⡍⢷
⠸⣿⣦⣄⣀⣀⣠⣤⣤⣶⣶⣾⣿⢿⢿⢿⡟⠛⠛⠉⣽⣯⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⠳⣤⣀⠀⣀⣴⠀⠀⠀⢠⡇⣿⣿⡽⢿⣿⣿⣾⣾⢸
⠀⠈⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠈⢧⠳⣍⠂⢰⠐⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣦⡄⠉⠛⠷⢾⣷⣄⣀⢸⡇⣿⣿⣧⣤⣼⣿⢿⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠈⣿⡀⠀⠀⠀⠀⠉⢿⣿⣟⢻⠟⢏⠉⠉⠀⠿⢅⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⢦⠁⢸⡈⣢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠖⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠈⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠒⠈⠉⠈⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⢑⠤⣀
⠀⠀⠀⡀⠤⠐⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⠃⠀⠀⠉⠑⠢⢄
⠠⠂⠁⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⡀⡀⠀⠀⢀⡴⠒⠲⡄⠀⡏⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁
⠘⡄⡠⠀⠀⠀⡰⠀⠀⠀⠑⢄⢀⡠⠊⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⣧⠀⠸⣇⠀⢀⡇⠀⠳⢤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠜⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣤⠏⠀⣀⡈⠩⠭⠄⠀⠖⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⡤⠊
⠂⠀⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠁
⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣧⠀⠀⠀⠀⠀⢀⣀⠠⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⢴⡁⠀⢸⡤⠄⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡠⠒⠁⣀⡹⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠑⠄⢀⣀⡀⠠⠐⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠳⡀⠀⠀⠀⡏⠀⠀⠘⠀⠀⣠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⠀⠀⢸⠃⠀⠀⡇⠀⣰⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠀⠀⡟⠀⠀⢰⠁⢰⠃⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⢸⠀⢰⠃⠀⠀⡸⢀⡏⠀⠀⡸⠀⠀⢀⡴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⣼⠀⠀⢀⡇⡼⠀⠀⣠⠃⠀⡰⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢷⠏⠀⠀⢸⣵⠁⠀⢠⡏⢀⠞⠁⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⠠⢄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠋⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢳⡀⣸⠃⠀⠀⠀⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠓⢿⣀⡀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠉⠳⣄⠀⠀⣀⡤⠒⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘⠀⠐⠁⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⠀⠀⣄⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠻⠶⠶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⢀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⠠⠲⠆⠀⠀⠐⠶⠤⠤⠀⠀⠀⠀⠠⠤⠤⠦⠀⠀⠠⠶⢤⢠⢀
⠀⠀⠈⠀⠐⠐⠀⡀⠀⠀⠂⠠⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⣀⢁⣭⠉⠉⢉⣥⢉⡈⡂⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⢒⢒⠠⠀⢂⠀⣒⡒⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠉⠉⠉⠉⠁⠈⠀⠅⠀⠀⠀⠁⠉⠀⠀⠈⠉⠁
⠀⠀⢀⠀⠈⠀⠀⠠⠀⠠⠈⠀⠀⢀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠄⠦⡆⠀⠀⠀⢠⠄⢠⡄⠠⠀⠠⠀⠦⠀⠀⠀⢢⠠⠄
⠀⠀⠀⠰⠤⠰⠀⠀⠀⠈⠇⠤⠤⠀⠀⠀⠘⠀⠤⠀⠀⠀⠀⠀⠠⠰⠠⠁⠀⠀⠀⠄⠀⠀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠜⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠐⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢀⡇⠀⠀⠀⠀⠀⠀⢀⢀⡇⠀⢀⣇⠀⡀
⠀⠀⠀⠤⠐⠦⠂⠀⠀⠀⠦⠆⠠⠄⠘⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠡⠔⠀⠀⠐⠦⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠤⢶⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠠⠀⠈⠆
⠀⠀⠀⠀⢠⠤⠤⡾⠟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠰⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠘⠈⠂⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠛⣝⢛⣉⡛⣻⣛⣹⡟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣷⡝⣆⠹⣿⣿⣿⣿⣿⣿⠀⣷⠀⢻⡆⢸⣿⢸⣿⣿⣿⣿⣿⣿⣼⠁⠏⢠⣿⡟⣾⣿⣿⣿⠟⢀⣾⣿⡿⢋⣾⣿
⣿⣿⣿⣿⠋⠻⣿⣿⣿⣌⠣⠘⣿⣿⢿⣿⡿⡆⠘⡆⠘⡇⠀⣿⢸⣿⣿⣿⣿⣿⠇⠇⠈⠀⣾⠏⣼⣿⣿⣿⠋⣰⣿⣿⢟⣵⣿⣿⠿
⣿⣿⣿⣿⣷⣄⠈⠻⣿⣿⣦⠀⠈⢿⣯⢻⣷⠑⠀⢁⠀⠣⠀⢿⠸⣿⣿⣿⣿⡟⠈⠀⠀⣸⠏⡼⣻⣿⡿⢁⣾⣿⠟⣡⣿⣿⠟⣡⣾
⣿⣿⣿⣿⣿⣿⣷⡄⠈⢿⣿⣷⡀⠀⠻⣆⢻⣇⠀⠀⠀⠀⠀⢸⠀⣿⢻⣿⣿⠁⠀⠀⢠⠏⠘⣱⡟⠁⣠⣿⠟⣡⣾⡿⠋⣡⣾⠿⣫
⣿⣿⣿⣷⣝⠻⣿⣿⣦⡀⠙⢿⣿⣄⠀⠙⣆⢻⡆⠀⠀⠀⠀⠘⠀⣿⢸⣿⡏⠀⠀⠀⠈⠀⣰⠋⢀⡼⠋⢀⣼⠿⢋⣴⡿⢛⡵⢚⣥
⣿⣿⣿⣿⣿⣷⣮⡙⢿⣿⣦⡀⠙⢿⣦⡀⠘⢆⢻⡀⠀⠀⠀⠀⠀⣿⣾⣿⠀⠀⠀⠀⠀⠐⠁⡠⠋⢀⣴⠟⢁⣴⠟⠩⢒⡡⢞⣫⠴
⢽⣛⠿⣿⣿⣿⣿⣿⣦⡈⠻⢷⣄⠀⠙⢷⣄⠀⠀⠳⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⡠⠊⢁⠴⠋⠀⠀⠐⠩⠐⠉⣀⡤
⣤⣈⠉⠀⠉⠛⠿⣿⢷⣽⡢⠀⠙⠳⣄⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢒⣩⣵⣾
⣉⠙⠛⠶⢤⣀⡀⠀⠀⠈⠉⠓⠄⠀⠈⠑⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⠁
⣻⠿⠷⣶⣤⣄⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣠⣤⣶⡾⣿
⣉⣉⡁⠀⠀⠈⠉⠙⠓⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠄⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⣚⣋⣩⣭⣴⣶⠿⠿
⣿⣭⣭⣭⣍⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡿⢻⡽⣿⢂⢿⣂⢸⡼⠃⠀⠀⠀⠀⠀⠀⠐⣀⣉⣉⣩⣥⣤⣴⣶⣶⣾⣿
⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠖⠒⠒⠀⠀⠀⠀⠀⠀⠹⠟⠘⠀⠈⢁⣈⡅⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠲⠿⠿⣿
⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⣿⣾⢻⣂⢙⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⣓⣶⣶⣿
⡿⠶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠙⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⣀⡀⠀⠀⢈⣉
⣤⣴⣶⠆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠐⠒⠤⣄⣉⠙⠛⠷⢶⣶
⠛⠉⢀⣀⣤⡴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⢦⡀⠀⠐⢤⣀⠀⠀⠀⠈⠙⠓⠶⣤⣄
⣶⡿⠟⣋⠥⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠙⢷⣌⡓⣮⣛⢶⣦⣤⣀⠀⠀⣀⠉
⡵⠖⠋⣀⠄⢀⡤⣒⠅⢀⠴⠊⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢷⣄⠀⠻⣿⣮⣝⢿⣿⣿⣿⣷⣦⣄⣉
⣠⠴⣊⣵⣞⡽⢊⣡⠞⢁⣴⡄⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠀⠀⠀⠀⠀⠈⢧⠰⡄⠀⠻⣷⣄⠈⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿
⣵⣿⣿⠞⣡⣴⠟⣡⣴⡿⣫⣾⡿⠃⠀⠀⠀⢠⠀⠀⠀⢸⡇⠀⢸⠀⡆⠀⡀⠀⠀⠘⣧⠹⣆⠀⠙⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿
⡿⣋⣴⣿⠟⣡⣾⣿⢟⣵⣿⡟⠁⣠⠆⠀⣰⠇⠀⠀⢀⣿⠀⡄⣼⡀⡇⠀⣤⠀⠀⠀⢹⣧⡹⣦⠀⠈⢿⣿⣦⡐⢌⠻⣿⣿⣿⣿⣿
⣾⡿⢋⣵⣾⣿⢟⣵⣿⣿⠏⠀⣰⣯⠎⢰⡏⠀⠀⡆⣸⡟⢠⡀⣿⡇⣧⠀⢹⡆⠀⠀⢀⢿⣷⣹⣷⡀⠀⠻⣿⣿⣦⡳⣝⢿⣿⣿⣿
⣫⣴⣿⣿⣿⣿⣿⣿⡿⠃⢀⣼⣿⡟⣰⡿⠀⣸⡿⢀⣿⡇⢸⣿⣿⡇⣿⠀⢸⣷⠀⠄⠘⣾⣿⣷⣿⣿⣄⢢⡙⣿⣿⣿⣮⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⢠⣾⣿⡿⣱⣿⠇⣰⣿⠇⣸⣿⠁⣿⣿⣿⡇⣿⠀⠘⣿⣇⢸⡀⣿⣿⣿⣿⣿⣿⣦⠹⣌⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣄⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠈⠁⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠛⠋⠀⠀⠀⠀⠛⠃⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠘⠶⠶⠶⠞⠀⠀⣠⠟
⠀⠀⠀⠀⢀⣶⢾⡹⣆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⣤⠞⠋
⠀⠀⠀⠀⢻⣍⢧⡹⡜⠿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⡇⠀⠀⠀⠀⠟⠳⣤⣄⣀
⠀⠀⠀⠀⠹⢮⡓⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠀⠀⠀⠉⠀⠤⠄⠀⠁⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠙⣦⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠳⣄⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠘⢷⠟⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣠⠞⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡼⠃⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣸⣇⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣆⠀⠀⠐⠶⢄⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⡀⠘⢿⣿⣿⣷⡀⠀⠀⠀⠹⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⡀⠻⣿⣿⣿⣦⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡙⢿⣿⣿⣿⣦⡘⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣝⢿⣿⣿⣿⣦⣻⣿⣿⣿⣦⣄⠀⠀⠀⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣌⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠘⢿⣿⣿⣿⣦⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢧⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠙⠲⠆⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⠿⠟⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢄⠐⡢⠖⠢⡂⢄⡄⡆⢔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠁⠀⣠⠴⢊⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣾⣿⣭⣤⣤⣤⣤⣴⣋⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⢿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⠶⣶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⣡⠔⠋⠀⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣶⣦⣤⣾⠟⠁⣠⠎⠀⠀⠀⣼⡟
⠀⠀⠀⢠⣾⣟⠁⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣧⣦⣤⣤⣴⣿⡁
⠀⠀⢀⣿⠋⠙⣿⠲⢦⣤⣤⣍⣻⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣿⡇
⠀⠀⠀⢿⣆⠀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣈⣿⣿⣀⣘⣿⣏⣀⡻⠿⠁⠀⠻⠿⠁⠹⣿⣏⣉⣿⣏⣉⣉⣿⡿⢿⡿⠿⠿⠾⣷⣿⡇
⠀⠀⠀⠈⠻⣿⣇⡀⠉⠉⠉⠉⠉⠉⠉⠓⠒⣾⡿⠛⠛⠛⠛⠋⠉⠉⠛⠛⢿⡛⠿⠿⣿⣭⡉⣉⣉⣉⣩⣭⣭⣭⣭⣿⣿⣿⣶⣬⣓⠦⣀⣀⣙⣿⣦
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣦⣤⣤⣀⣀⣀⣀⣙⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠙⠿⣿⣟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠉⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣬⣽⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠙⠻⠿⣶⣦⣤⣀⣀⡀⠀⠀⠀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠷⣦⡀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣴⠟⠛⢷⣄⠀⠈⠻⣦⡀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⠻⣦⡀⠀⠙⢷⣄⠀⠈⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣄⠀⠀⠀⣼⠏⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣄⠀⠹⣇⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣄⠀⠈⠻⣦⡀⠀⠈⠻⡆⠀⠀⠙⢷⣄⢻⡆⠀⠀⢻⡄
⠀⠀⠀⠀⣴⡀⠀⣀⠀⠀⣾⡋⠈⠙⢷⣄⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⠀⠈⢿⡄
⠀⠀⠀⠀⠘⣷⠀⠸⣧⡀⠘⢷⣄⠀⠀⠙⢷⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠘⣷⡀⠘⢷⣄⠀⠙⢷⣄⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠙⢷⠄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡿⠁
⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠉⠉⢙⠉⠙⡛⠛⠻⠿⠿⠿⢿⣿⣿⠛⠻⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠠⠿⣦⣴⣉⡻⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠿⠷⠌⣽⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣶⣶⣶⣶⣶⣶⣶⡦⠀⣀⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣤⠾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠙⠛⠋⠁⠀⠀⠙⠛⠋⠁⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⣭⣿⡟⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡯⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡺⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠤⠴⣶⣶⣶⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⢿⣿⣍⠁⠀⠀⠀⠀⠀⠐⠒⠲⠶⠿⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣥⣿⠖⠂⠀⠀⠀⠀⠀⠀⠚⣓⣐⣪⣬⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⠆⠀⠀⠀⠀⠀⠠⠤⠶⠶⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣆⠀⠀⠀⠀⠀⠀⢐⣺⣿⣿⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⢙⣻⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣟⣁⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠴⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣮⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣿⣿⠒⠀⠀⠀⠀⠀⠀⡀⢺⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⣿⣿⣿⡽⢂⣀⠀⠀⠀⠀⠘⣫⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠿⠟⠛⠉⠀⠀⠀⠐⡚⢿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣙⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⡀⠤⠴⠒⠚⠙⡆⠘⠛⠶⠶⢤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡴⠞⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⢦⠀⠄⢠⣤⣤⣤⢤⡤⠾⠟⢱⣟⡷⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠐⠦⢈⡀⠕⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣦⣴⠇⠀⠀⠀⠀⠀⢂⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⣴⣧⡤⠖⣛⢑⡀⠀⠈⠉⠉⠛⠛⠛⠚⠒⠛⠋
⠀⠠⠄⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠨⠉⠀⢿⣛⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡂⠄⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠨⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠃⠀⠀⠀⠀⠀⠐⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢤⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠠⠴⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠉

⠀⠀⠀⠀⠀⠠⠤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡷⠀⠀⠀⠀⠀⠈⢹⣿⡟⠉⠀⠀⠀⠀⠀⢀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠒⢲⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉

⠀⠀⠀⠀⠈⠐⢤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣇⠀⠀⠀⠀⠀⠀⠠⠴⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠁⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠭⠿⠿⠿⠒⠂⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠴⠂⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⣀⠤⠴⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠺⠿⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⠒⠀⠀⠀⠀⠀⢀⣠⣶⡾⠋⠀⠀⠀⠀⢀⣺⠒⠒⠂⣀⣠⡕⠒⠀⠀⠀⠀⠀⢨⡁⠐⢾⠖⠲
⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠠⢬⣭⠙⡒⠂⠀⠀⠀⠀⠀⠀⡀⣭⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡆
⠀⠀⠀⠀⠀⠒⠚⠨⠭⠥⠀⠀⠀⠀⠀⠀⠀⠬⡝⠓⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⣿⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠀⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣐⣶⡦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡆
⠀⠀⠀⠀⠀⠠⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⢿⣓⣒⣀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⡟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⣠⣤⣈⣉⠛⠲⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣄⡉⠙⠳⠦⣤⣀⡉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠈⠙⠒⠦⣤⣈⠉⠓⠦⠤⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠉⡿⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⣸⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⡶⠖⠒⠲⠶⠶⣟⠙⢧⣼⡀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠈⠙⠳⢤⣀⠀⠀⠈⠳⢦⣈⠙⠳⢦⣀⠀⠀⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠈⠙⠢⢄⣉⠛⠶⢿⣼⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠉⠛⠶⣤⣀⣽⠂
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⠳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⠉⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⡶⠚⠛⠋⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⠶⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⣆⣀⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⣤⣿⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣴⠒⠲⠶⣶⠀⠀⠀⣿⠀⠀⠀⠀⣿⢁⣀⣽⡆⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⣻⡆⣀⣀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠸⢧⣤⣹⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⡀⠰⢯⣀⠉⠻⣷⣜⡳⠶⠶⠛⠀⠀⠀⣿⠀⠀⠀⣼⢛⢳⡄⢸⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠉⠉⠉⣇⠀⠀⠈⠓⠶⣬⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⢽⢸⡇⢸⣿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣌⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣿⢟⣦⣾⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⢤⡀⠀⠀⠀⠀⠀⢰⠛⢉⠙⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡿⣏⣁⣾⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⠷⢦⣀⠀⠀⠘⣧⣍⣠⠟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣧⣬⡽⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢘⣇⣠⢶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣄⠀⠀⣿⠀⠀⢀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⠿⠶⠚⠉