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⠀⠀⠀⠀⠀⡀⠔⠒⠈⠁⠈⠉⠉⢉⡒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⡀
⠀⠀⢀⣔⣉⠀⠀⠀⣀⣤⣴⠟⣛⣋⡰⣾⣿⣿⣦⡀⠀⠀⠀⠀⡠⠔⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠢⣄
⠀⣴⣿⣿⣿⣷⣖⣦⣬⣭⣤⣼⠟⠙⠓⠀⠻⣿⣿⠟⡄⠀⡠⠊⠀⢀⠰⠂⡸⠀⢲⠀⡆⠠⠀⠒⠀⠀⢢
⣸⣿⣿⣿⣟⣽⣉⣌⣿⣯⣀⠀⣀⣽⣿⣷⢦⠹⠅⠀⠘⡈⠢⡀⠀⣈⠃⡀⡇⢤⠈⣇⠀⠀⠀⠂
⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢣⠀⢰⠀⠈⠉⠀⠉⢨⠁⠈⠁⠀⠉⠉
⡇⠀⠀⠈⣿⣿⣿⣿⣿⠟⢉⣉⣁⣤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⡇⠈⡀⢲⠀⠀⠀⠀⠀⠀⠐
⢳⠀⠀⠀⢿⣿⣿⣿⣦⣤⣥⠀⢡⣤⣠⣤⣤⣀⠀⠀⠀⡜⠀⠀⠐⢄⠀⠀⠧⠛⠧⠸⠤⠆⠀⡀⠔
⠈⣆⠀⠀⠀⠉⠋⠉⠉⠉⠉⢠⣾⣿⣿⣿⣿⣿⣿⠇⡰⠃⠀⠀⠀⠀⠑⠂⠀⠀⠀⠂⠂
⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⡿⠏⠁⡴⠁
⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢀⠤⠊
⣀⣀⣀⣀⣀⣈⣑⣒⣤⣤⣤⣤⣤⣤⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⢉⣭⣭⣉⠻⣿⣿⠛⠻⣿⣦⠀⠀⠀⠀⢀⣶⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢰⣿⡿⠟⣛⠀⣙⣛⠁⠀⠸⠿⣷⣶⣶⣶⡾⠋⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠈⢿⣄⣘⣋⣴⣿⣿⠀⢠⣿⣶⣶⣦⣤⣤⣴⣶⣿⣿⣏⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣦⡀⠙⢿⣿⣿⣿⠇⠀⣾⣿⠏⢀⣴⡌⠁⣼⡟⠀⣽⣿⠏⢠⣿⣿⡷⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣤⣤⡄⠙⣿⡄⠈⢻⣿⡿⠀⣸⣿⠇⠀⣾⡿⠃⢸⣿⠁⢰⣿⠟⠀⣾⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿⣃⣴⣿⣿⠀⢸⣿⡇⠀⠛⢁⡄⠀⠛⣡⠀⠘⢃⠀⠘⡋⠀⣼⣿⣿⡟⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠘⢿⣿⣿⣿⠟⠃⣠⣾⡿⠿⠷⠾⢿⣿⣶⡾⠿⠷⠶⠿⠷⠿⠃⠰⠿⠛⣋⣤⣾⣿⣿⣥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⣀⣤⣴⣾⠟⢁⣴⣾⣿⡿⢀⣿⡃⣶⣿⣿⣿⡿⠟⣠⣾⣿⣿⣿⣿⣿⡟⠉⣉⣉⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠿⠛⢛⣛⠋⢀⣘⣩⣭⣥⣶⣿⣿⣷⣦⣬⣭⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣧⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣰⣾⣿⡏⠀⡼⢋⣉⠉⢻⡏⠡⢴⣤⡉⢿⠏⠀⠙⣉⠀⠙⢋⡉⠉⣿⡟⠋⣩⡍⣿⣿⣿⠆⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⣸⣿⣿⠁⠀⣰⣿⠏⠀⣿⠃⢀⣉⣉⠁⣴⡇⠀⣸⡿⠀⣰⡿⠁⣰⡟⠀⢘⣋⣴⣿⣿⡿⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⠀⢠⣿⡟⠀⣼⠏⠀⠸⠿⠋⣰⣿⠀⢠⣿⠃⢰⣿⠁⣰⡿⢁⠀⢿⣿⣿⠿⢋⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣼⣿⣧⣄⣥⣾⣷⣶⣶⣾⣿⣿⣦⣾⣿⣦⣾⣿⣄⣨⣴⣿⣿⣶⠶⢶⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⣛⡛⠻⣿⣿⡿⠛⠛⠻⠿⠛⣻⣿⠛⠛⢿⡏⢡⡌⢻⣿⣿⠿⠟⢿⣷⣦⡀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⠀⢻⣧⣾⡿⠋⢀⣼⣿⡧⠀⢰⣿⣿⠁⠀⣼⣷⣌⠃⢼⠏⢀⣴⠿⢠⣿⣿⡇⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣉⣭⣥⠀⢠⣶⡆⠀⢸⣿⡿⠁⠀⣿⣿⠏⠀⢠⣿⡿⠋⣴⣶⠀⢠⣴⣾⣿⣿⠟⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⡙⠛⣁⣠⣼⠿⠧⣀⣈⣉⣴⣧⣀⢈⣁⣴⣄⣀⣉⣠⣾⡿⠿⣦⣀⣙⣛⣛⣡⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠟⢁⣶⠂⣿⣿⠿⣿⣿⣿⣿⣿⡿⢛⡛⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿⢋⣉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⣾⣿⠏⢀⡾⢋⣼⣿⣅⣠⣾⣿⣿⣿⣿⣷⣌⠃⠸⠿⠟⣋⣴⣿⣿⣿⣿⣷⣿⣿⣷⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣈⡉⠀⣠⣶⣿⣿⡏⠀⢹⣿⣿⠁⠀⢸⣿⣿⠇⢸⡿⠋⣠⡌⢻⡇⠀⢾⣿⣿⡿⢃⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⠀⣿⣿⣿⡿⠁⢀⣿⣿⠏⠀⠀⣿⣿⡟⢀⡟⠀⠘⢋⣴⡟⣰⣦⠈⠛⣋⣴⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⢿⡿⠟⡁⠀⢸⡿⢋⣼⡀⠀⣿⠟⢀⣾⡇⠀⣿⣿⡟⣰⢋⣵⡦⠈⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣠⣤⣾⣷⣄⣠⣴⣿⣿⣧⣄⣠⣴⣿⣿⣷⣤⣈⣡⣾⣿⣬⣉⣡⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠋⠙⠛⠿⠛⠋⠀⠙⠻⠿⠛⠋⠀⠈⠛⠿⠿⠛⠉⠛⠻⠿⠛⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡆⠀⠀⠀⠀⠀⠠⣶⠀⠀⢀⡀⣼⡿⠀⠂⠀⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢠⡿⢁⠔⠀⠀⠀⠀⣠⡄⠀⠈⠉⠁⠀⠀⣿⡇⠀⣠⡖⡲⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⣿⠃⢡⠔⠄⠀⣤⡰⢡⡟⢠⣴⠛⣀⠀⢸⣿⠀⢸⣿⠒⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⡏⠀⣼⠇⡸⢰⡿⢁⣿⠃⣿⠃⢸⡟⢀⠺⡧⠔⠙⠿⠄⣠⠖⠋⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣿⡇⣸⣿⡠⠁⣾⠃⣸⡏⠀⠛⠀⠘⠗⠁⠀⠀⠀⠀⣠⡞⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠁⠈⠉⠀⠈⠀⠀⣩⢤⠻⠧⠃⢀⡔⠐⠄⠀⠀⠀⠀⣰⡟⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣤⠀⠀⢠⡾⠁⡐⠀⠀⠀⣾⠀⠀⢠⠀⣠⣴⢶⣿⡁⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⠦⢤⣿⠃⠈⣼⡁⢱⠀⣼⠦⢀⡄⣴⣿⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡔⣶⡄⣿⡇⠀⢰⡿⠁⡜⢹⡏⠀⠜⠀⠿⡧⠜⠘⢷⣀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣠⡿⠁⣿⣀⠀⠛⠷⠔⠀⠈⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠐⠋⠀⠀⣠⡤⡀⠀⠀⠀⠐⣶⡄⠀⣴⡆⠀⣠⡶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠃⢀⣤⠶⠀⢸⣷⠀⢹⠇⢰⣿⠇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠊⣿⡇⢰⣿⢃⣾⡇⣸⡿⢀⠆⠀⠘⠿⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⠔⣿⠇⠘⠛⠀⢾⡧⠙⠛⠁⣠⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠛⢉⡴⠒⠒⢆⢀⡀⠀⠀⣼⠏⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡿⠁⠀⠀⡞⠺⠇⢀⣼⣏⠔⠁⣠⡶⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠃⢀⡠⠚⢰⣶⠀⢻⣿⣇⣀⣰⣿⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣾⡿⠂⠁⠀⠠⣾⡏⡠⣾⡏⡇⠀⠈⠻⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣀⠔⠁⠛⠓⠁⣿⢹
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⡜⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⠻⠤⣄⣼⠁⠀⢀⡞⠀⠙⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⡖⡼⠀⠀⣰⠋⠙⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠒⠒⢶⣊⡼⠁⠀⣰⠃⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠳⡍⠓⠶⣇⠀⡰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⢁⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠠⠤⢶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⠀⠀⣠⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠙⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠀⠀⢀⣴⣶⣦⣄⠀⠀⢀⣤⣶⣶⣦⡀⠀⠈⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡀⠀⢈⣹⠿⠋⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡶⠟⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢀⣤⠞⠋⠀⠙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠟⠋⠁⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠴⢯⡅⢨⣴⢶⠟⢻⣿⡿⠛⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢆⠔⣰⡇⢈⣆⣉⡆⠘⣮⠆⢠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢏⣧⣤⡞⣿⡟⣾⠋⠙⠻⣼⣇⢀⠾⡟⢉⢙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢋⣴⡿⢛⣉⡁⣈⡁⣾⡕⠖⣼⠉⡀⢁⡆⣅⠰⢊⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢁⣾⣿⣿⣿⣿⣇⡿⢇⣻⠟⠉⢻⡟⢿⣿⡷⣿⡿⠋⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠘⢉⣁⡀⠀⢙⡟⢱⣿⠏⢀⠀⢸⡇⢸⣿⢡⡿⠀⣰⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⡴⠀⣼⠃⣾⠏⢠⣿⠀⣸⡇⠈⠇⣸⡆⠀⣤⣾⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠼⠁⣰⡿⢆⣀⣴⣿⣿⠀⠟⣡⠀⢠⣿⣷⠀⠙⠃⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣀⣀⣴⣿⡇⢸⣿⣿⡇⣿⡀⣰⢿⣿⢡⡌⣿⣶⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⢼⣿⣿⠃⣿⠁⠿⢸⣯⠰⣾⠏⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣾⣿⠀⣿⡇⠀⣼⣿⣦⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⡤⠿⠟⠛⠉




⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢽⣷⢼⣿⢸⡝⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣯⣿⢸⡟⡿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠿⣾⣿⣷⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⢸⣿⣾⣹⣇⣿⠁⡇⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⢸⢻⢻⢹⣌⣧⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣷⢿⣿⣿⣷⢿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣸⣏⣿⡗⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣏⡇⣿⢸⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⢻⡿⣿⣿⢻⣟⣿⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣷⣿⣿⣻⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣘⣿⣨⣿⢿⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⢭⣂⣀⡀⠀⢀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠲⠶⠾⡏⠁⠀⠀⠈⢉⣉⣁⣀⣀⣔⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⡶⠴⠒⠂⠀⠀⠈⢻⣦⣤⣴⠟⠁⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣚⠋⠀⠀⠀⠑⠐⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣈⠉⠀⠀⠉⠉⠉⠀⠐⣀⣴⣿⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣭⣉⣀⣀⣀⣀⣀⣀⣤⣠⣾⡇⠀⠄⢈⠀⠀⡀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⣧⡐⠰⢀⣄⠀⢍⠂⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠳⣤⠟⠀⠈⢢⠑⢡⣠⠔⢿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⢐⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠁⡀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠄⣾
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢡⡶⠞⠻⠷⢆⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠐⠀⠀⠀⠀⢠⢄⡃⠀⠀⣀⡰⡀⠈⠙⠉⢠⡴⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⣴⠋⠁⢹⠛⡆⠀⠀⡾⠋⢳⡘⢦⡇⠀⠸⡷⠁⣈⣑⣒⣒⣉⣄
⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⢀⡀⠀⢻⣄⡀⠈⠋⠁⠀⢸⣇⣀⣈⡇⠘⣇⠀⢸⡇⠀⣟⠈⠉⠉⠙⠊
⠀⠀⠀⠀⠀⠈⠉⠉⠙⢻⡏⠀⢰⠲⡍⠉⠙⠛⢦⡀⣾⠋⠉⢹⡇⠀⢻⠀⣾⠀⣸⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠻⣄⠀⠳⢷⣤⣤⣤⠾⠁⣉⠆⠀⣟⣳⠀⠘⠿⠃⠸⣧⣤⣔⣒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠹⡆⠀⠀⢠⡄⠀⣠⠀⢀⡄⠤⢬⡷⠖⠀⡶⠦⠀⣶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⣿⠙⣆⠀⣿⢀⡇⠀⡞⣷⠀⠈⡇⠀⠀⣷⡤⠀⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⡁⢀⠀⠀⠀⡿⠀⠘⣦⡟⣿⠇⣸⠋⢹⡄⠀⡧⠀⢸⣇⡀⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣨⡾⠁⠀⠀⠈⠃⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠡⠀⠀⠐⠀⠂⠀⠀⠀⢀⠎⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠤⠀⠡⠀⠀⠀⠀⠀⠀⠰⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢀⠂⠀⠀⠀⢂⠀⠠⢀⢀⠀⠀⠠⠈⠐⠂⠀⢀⣐⠂⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠒⠀⠈⡏⠑⠀⠈⠉⡏⠒⠀⠉⡎⠑⠈⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠿⠀⠀⠀⠀⠀⠀⢰⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠟⠀⠀⠀⠀⢀⣾⠑⣆⠀⠀⠀⠀⠀⣠⡶⠲⣶
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢀⣤⣄⠀⠀⡀⠙⢦⡇⣀⡤⣄⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⠉⢷⣼⡀⣀⣠⡀⠀⢿⡄⠀⠁
⠀⠀⠀⠀⠀⣠⡄⠈⢷⡀⢠⡿⠁⢠⡄⢸⡟⠀⡸⢰⡟⢀⡿⠃⠀⠀⢸⡇⠀⠀⠀⣿⡅⢀⣼⠃⢀⡍⣭⡏⢁⣽⠀⠈⢻⣦
⠀⠀⠀⠀⠰⡟⠀⠀⣸⣇⣼⡇⠀⣼⠃⢻⡇⡠⠃⣾⠛⠉⠀⠀⢀⣠⣿⡀⠀⠀⢠⡿⠀⢸⡏⢀⡜⠀⣿⠐⠛⢡⠢⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠒⠚⠋⠀⠈⠓⠉⢿⡤⠞⠛⠁⠀⠘⠓⠚⠃⠀⠈⠈⠉⠙⠓⠂⠙⠷⠤⠊⠉⠉⠀⠀⠹⢤⠠⠘⠓⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠰⠄⠠⢽⣂⣀⡀⠀⢀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠂⠂⠲⠶⠾⡏⠁⠀⠀⢈⣉⣉⣁⣀⣤⣖⣔⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠶⠖⠒⠀⠈⠉⢻⣦⣤⣴⠟⠁⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣚⡛⠀⠀⠀⠛⠒⣂⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣈⠉⠀⠈⠉⠉⠉⠁⢐⣀⣴⣿⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣭⣉⣅⣀⣀⣀⣉⣠⣤⣦⣾⡏⠀⠔⢈⠄⠀⡠⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⣧⡜⠰⢃⣄⠀⢍⢂⡐⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⢢⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⢜⠷⣤⠟⠀⡈⢢⠑⣥⣠⠔⢿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⢐⠥⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠌⠄⢃⠋⡇⠀⢸⠆
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⡄⢉⠆⣾
⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢘⢡⡶⡞⠻⠷⢖⡆⠈⣸⠇
⠀⠀⠀⠀⢀⢰⢨⠀⠀⠀⠀⠀⡠⠒⢀⣀⣀⡀⠁⠐⠀⠀⠠⠤⣠⢌⡃⠉⠀⣀⡳⡀⠈⠙⠋⣠⡴⠋
⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠈⠀⣴⠋⠁⢹⠛⡆⠀⠀⡾⠋⢳⡘⢦⡇⠀⠸⡷⠁⣈⣓⣒⣒⣉⣄
⠀⠀⠀⠀⠀⣦⣀⡀⠀⠠⢀⡀⠀⢻⣄⡀⠈⠋⠁⠀⢸⣇⣀⣘⡇⠘⣇⠀⢸⡇⠀⣟⠈⠉⠉⠙⠊
⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⡏⠀⢰⠶⡍⠉⠙⠛⢦⡀⣾⠋⠉⢹⡇⠀⢻⡀⣾⠀⣸⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡻⣄⠀⠳⢷⣤⣤⣤⠾⠃⣙⠇⠀⣟⣳⠀⠘⠿⠃⠸⣧⣤⣔⣒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠠⠹⡆⠀⠀⢠⡄⠀⣠⠀⢀⡄⠤⢬⡷⠖⠀⡶⠦⠀⣶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠄⠀⠀⠃⣿⠙⣆⠈⣿⢀⡏⢀⡞⣿⠀⠈⡇⠀⠀⣷⡤⠀⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⡃⢀⠄⠀⡄⡿⠀⠘⣦⡟⣿⠇⣸⠋⢹⡄⠀⡧⠀⢸⣇⡀⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣨⡾⠃⠀⠀⠈⠃⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣤⢀⣤⠀⠀⠀⢀⣤⡶⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠛⠳⠒⠻⠋⠁⠀⠀⠰⢏⣀⣾⣿⣴⢿⣯⠁⢰⣰⣰⢂⣾⣏⡏⣰⡏⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢫⣀⣀⣀⣀⠶⣄⠀⠀⠙⠚⠋⠘⠀⠃⠋⠁⠀⠈⠉⠁⠋⠘⠘⡅⠋⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢻⣿⢏⡀⠉⢹⡧⠛⠷⠂⠀⠀⠀⠀⠀⠠⣞⠂⣴⣷⣠⣾⣖⠃⡼⢡⢇⣶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢋⡌⠀⢲⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠟⠙⠉⠏⠚⠋⢠⠁⠞⣘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠁⠂⣛⡋⢹⡃⠀⠘⠛⠋⠁⠀⠁⣠⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠀⠀⠸⠋⢹⣯⠹⣦⣀⠀⣀⡴⠞⣹⡿⠋⠈⢙⡂⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣷⡔⢤⢧⡘⢿⡷⣦⣼⣦⣤⠶⠟⠋⠀⠀⠀⠀⣡⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣻⡄⠻⢿⣿⣿⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⣷⣶⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠊⠀⢀⣛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⣤⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣝⣩⣭⣩⣭⣩⣟⣹⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⡼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣴⠦⣭⠟⠉⠂⠀⠀⣍⡉⠿⠿⠃⠀⠀⠀⠀⠀⠀⣤⡶⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⢀⣠⠖⠛⠉⠻⠇⠀⠀⠀⠀⢀⣀⡴⡶⣴⣯⠿⣥⣤⠌⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⣿⡄⡆⠀⠤⠉⠀⠀⠀⢸⣯⠉⠙⣿⣿⣿⠀⣾⢻⣆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠈⠉⡇⢠⠤⡴⠁⠀⠀⣴⢻⣿⠀⠀⠈⠉⠙⠿⢿⣧⢹⣆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⡘⣧⠀⠀⠀⠀⠀⡰⠃⣾⠇⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈⣦⢀⣀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠰⠚⠀⣿⡳⢴⡄⠐⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢧⠸⣯⢀⣤⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣧⠀⠀⠾⡉⠹⣿⣶⣶⠾⢛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣷⣷⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢙⢦⣀⣉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⢦⣌⡑⠐⠚⢛⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠛⠻⣿⣍⣳⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡃⠘⣻⣿⡆⣿⡄⣾⢷⣿⣽⣿⠸⠑⣶⠿⠿⢇⠀⠀⠀⢰⠋⠡⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⣿⠿⢾⣯⣿⡇⠸⣿⡟⠈⢿⣿⠦⢾⠧⣶⡞⢻⣻⢶⠀⠀⢸⣤⡄⠀⠀⢮⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣶⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠀⠀⠀⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠰⣶⣶⡦⢠⣶⡆⢰⣦⣰⡆⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⡆⢸⣿⠀⣾⣽⣿⡀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠟⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠈⠉⠈⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣤⡄⣤⡄⣠⣤⣤⡄⣤⣤⡀⣤⣤⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡷⣿⡇⣿⡇⣹⡿⣿⣿⣿⣿⣿⢸⣿⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠃⠈⠛⠛⠁⠙⠉⠛⠉⠛⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⣤⣹⣿⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣿⣭⣄⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢸⡟⣿⣿⣿⣿⡟⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⡿⣿⡄⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⣿⣿⣿⣿⡇⣿⡇⠀⠀⣾⡦⠀⠀⠀⠀⢾⡷⠀⠀⣸⡟⢸⣿⣿⣿⣧⠸⣧⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⣿⣿⣿⣿⡇⣿⡇⢠⣶⣾⣷⣦⡀⣰⢾⣿⣿⡶⡆⠿⢁⣿⣿⣿⣿⣿⡄⠿⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠈⠁⣿⣿⢹⣿⡇⠉⠀⣾⣹⣿⣿⡞⣇⢸⢸⣿⣿⡇⡇⠀⣼⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⢸⣿⡇⠀⠀⢁⣿⣿⣿⣷⠉⠙⢸⡿⢿⡇⠃⠀⠀⠈⣿⡇⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⢹⡏⣿⠉⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣈⣁⣀⣉⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣁⣈⣀⣀⣀⣀⣀⣈⣀⣈⣀⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠙⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣆⠀⠀⢀⡞⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡉⠻⡄⠀⠊⠀⠀⠀⠀⢀⣴⠟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣁⠀⠹⠀⣰⢧⡀⢀⣴⠟⠁⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⠿⠛⢡⢤⠤⡤⢠⡀⣀⡌⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠸⢄⠀⡇⣼⡇⢻⠀⠀⠀⠈⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠀⢀⠀⣀⠀⠀⠐⢞⣀⠃⠋⠓⢘⠀⠀⠀⠀⠀⣀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣇⣸⢸⠉⡁⣷⣿⢸⡀⠀⠀⠀⣏⠁⣼⡘⡇⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠏⢸⠸⣀⠷⡇⢹⢸⣀⢀⠀⠀⣌⠇⠷⠇⢻⠃⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⢸⡆⠀⣠⣤⡶⣤⠀⠀⣠⡞⢙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢉⡀⠀⠀⣶⣶⡀⠀⢸⡇⢰⣿⡏⢀⣸⠇⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣿⣿⠀⠀⣿⣿⡇⠀⢸⡇⢸⣿⡏⠉⠁⠀⠀⡾⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⣿⠀⢠⡟⣿⣿⠀⣿⠁⠀⠻⣿⣤⣀⣠⠞⠁⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠹⣿⣤⠏⠀⠈⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢾⡿⠓⠶⢤⡀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠶⠶⢶⣦⣤⣀⠈⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠈⠳⢄⠀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠈⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡄⠀⢀⠀⢆⡀⡼⡀⠀⢰⡀⠀⠰⣀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢐⢮⠄⠎⡙⣷⢣⢌⣁⣞⡁⢆⡅⢈⠋⢹⡅⠩⡍⠺⡤⢠⡃⡄⠶⣀⠶
⠀⠀⠀⠀⠀⠀⠥⡤⠛⣬⣱⡊⡭⢁⡀⠚⢠⢞⣔⣐⡢⠔⢬⠙⠂⣼⣃⠈⢥⠒⠳⡀⢌⡉⠬⣚⢀⣀⡄
⠀⠸⣀⠀⢨⡧⡎⣅⡌⠑⣂⢑⠂⠀⢷⡶⡀⢠⢈⠔⠆⠉⣤⢰⡃⠲⠍⠃⢳⡅⡙⠘⡉⣥⠛⢭⠎⠱⣗⢒⢀⡸
⠀⢛⡵⡼⡥⠈⠁⠃⡀⡧⣆⠄⠈⠧⢸⡍⣃⡶⣉⠀⢀⢳⡐⡂⣺⢀⡓⠉⠤⣀⠐⠂⠁⣬⠲⢈⠡⠄⠠⡽⡴⣍⠃
⠦⣬⠤⠇⢚⡈⢩⡉⠁⠤⢥⢉⠚⢀⡬⠱⠦⠭⠅⠐⠎⠵⠁⣍⢖⠾⠷⢉⡥⢈⡐⣂⠉⢀⡘⢐⢒⠀⣘⠂⠧⢬⡤
⠀⠒⣛⡿⢎⡴⠀⣼⢱⢀⠛⢘⣰⢄⠐⠂⡉⠀⢹⠠⢘⠁⠀⠲⠨⠏⠁⣌⠴⠀⣲⠀⠀⢎⢹⢨⡍⠰⣜⢪⣍⡃⠓
⠀⠀⠀⠀⠎⣡⠅⠆⢺⠈⡵⠀⡄⠼⡤⢀⠀⠀⠢⣈⠹⣆⠀⡴⠀⢀⡤⠂⣜⠓⠇⠀⠈⡁⢰⠄⣆⣧⠙⠆
⠀⠀⠀⠀⠀⠠⠎⠱⢆⢣⡈⠦⣐⢆⡇⠌⠓⠶⣶⣿⣷⣿⣿⣷⣾⠿⠖⠊⠅⣇⢖⡠⠎⡬⢣⠶⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠘⠁⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠁⠛⠐⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣿⣿⣿⣟⣻⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣻⣶⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠁⠀⢹⡄⠀⠀⠀⢀⡟⠉⢑⡄
⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⣧⢠⠄⠀⢧⠀⠀⠀⠸⡇⡾⠻⣸⡄
⠀⠀⠀⠀⠸⡁⠘⠏⢢⠀⠀⠀⠀⠸⣿⠛⣠⣼⡀⠀⠀⠀⡇⠀⠾⣻⡇
⠀⠀⠀⠀⠀⢡⠀⠀⢀⣣⠀⠀⠀⠀⢧⠘⠁⠈⢡⠀⠀⠀⣷⣆⢰⡿⢿
⠀⠀⠀⠀⠀⠀⢣⡻⠀⠁⢱⡀⠀⠀⠘⠀⠀⠀⠀⢇⠀⠀⠋⢀⣼⡋⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠱⡄⠀⠀⡄⠀⠔⠈⢹⡀⠀⡆⢘⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⡠⡙⢆⠀⢹⡐⠀⠀⠀⡇⠀⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⡴⠓⠲⢄⠀⠀⠀⠀⠀⢇⠏⢁⠈⠀⣆⠀⠇⠑⠤⠤⠾⡤⠃⠔⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⡠⢀
⢳⣠⣠⣤⣷⣄⠀⠀⠀⠈⣦⠀⠀⠀⠘⠖⠀⠀⢀⠀⠀⠀⠀⠀⠹⣆⠀⣷⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⣠⢄⡜
⠀⠙⢆⡉⠉⠉⠑⢦⣄⠀⠈⣧⠀⢾⣤⠀⠀⠈⠟⢀⢀⠀⢤⡀⣔⣈⡶⠿⡀⠀⠀⠀⠀⠀⠼⠹⢻⡾⠿⡞
⠀⠀⠀⠉⠢⡄⣸⡯⠈⠵⠒⠋⠐⡄⠀⢀⠀⢀⣴⣿⠿⠷⠛⠊⣿⠛⡀⠐⡿⢆⠀⠀⠀⣼⠒⡛⢐⠄⡼⠁
⠀⠀⠀⠀⠀⠈⢧⡄⠨⠐⠢⠚⠀⠀⠐⠇⠤⠛⠉⠶⢶⣶⣟⡄⣳⣇⡛⣰⡏⢙⡳⠶⠾⡅⣐⢴⠌⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⣄⠠⠀⠀⠀⠀⠀⠄⠀⣄⠊⠁⣫⡷⢗⡰⢈⡇⡿⣌⢁⢀⣀⠀⢄⡿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡜⠷⠆⠀⠤⠢⠈⠀⠈⢻⣷⣦⡉⠀⢀⣴⡄⠀⠥⢟⣼⠽⠋⠃⢼⣁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣤⣷⣿⠶⠀⠀⠐⠃⢹⡷⠀⠀⠈⠁⠨⢉⡙⠈⡿⡷⠀⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡿⣿⠻⠆⢰⠀⠀⠀⠀⠀⠀⠒⢘⡟⢦⡿⠂⠀⣤⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢈⠠⠀⠈⢆⠀⠀⠐⢦⣤⡠⠀⠀⠊⡗⢰⣿⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣉⠀⢀⣀⣀⠀⠤⠦⠴⠄⠀⠐⠀⠐⠻⠶⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⣹⣞⡉⠉⠉⣤⣤⣿⠀⠀⠛⠒⠒⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣤⡀⠡⢤⠄⠀⢹⡿⠓⠤⠤⠄⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⡡⠐⣰⠶⠶⠶⠀⠀⠘⠀⠛⣹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡛⢳⡏⡄⣰⠆⢜⡇⡚⡄⣠⡏⢸⡆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⢼⡿⢿⡣⠀⠀⣠⣤⣤⠀⠀⢇⠄⠀⠶⡄⠂
⠀⠀⠀⠀⠀⠀⠒⡀⠐⠀⠀⠀⠂⠐⠀⠀⠘⠁⠀⠉⠉⠀⠀⠀⢠⠰⠆⠉⠀⠀⡐⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠗⠈⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠜⣏⠁⢄⡀
⠀⠀⠀⠀⠢⣤⠠⡄⠾⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⠀⠀⢀⠖⡾⠂⠀⣀⠇⠀⠀⠳⣆⠀⠀⠘⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠓⠀⠀⠀⠀⠀⠀⠙⠰⠃⠀⠀⠀⢣⠀⠀⢁⠀⠀⠀⠀⠈⠻⠄⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠖⠋⠀⡠⠄⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢃⢀⡤⠈⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠈⠀⠑⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⢀⡄⠀⠀⢀⠀⠀⢀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠠⠶⠤⠉⠓⠚⢤⣀⣀⣀⣀⠤⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⡤⢶⡶⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠿⢦⠄⢀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠄⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⢴⡒⠛⠉⢙⣉⣡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡁⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣏⠀⢦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠓





⠀⠀⠀⠀⠀⠀⠀⠠⢠⣤⣤⡀⠀⣦⣄⣀⣠⣤⠀⢠⣄⣀⣠⣠⡄⢠⣀⣄⣀⠀⣤⣤⣠⣀⡄⢠⣄⣀⣄⣀⣤⣤⣀⣠⡀
⠀⢀⣀⣀⣀⠀⣀⣀⣀⣀⣁⣀⡀⢁⣀⣁⣀⡀⣉⣀⣀⣡⣀⡀⢀⣈⡀⢀⣈⡀⣤⣀⣀⣀⡁⣀⢀⣀⡁⢀⣁⣀⣀⣩⣁⣀⣀⣤⠤⠄
⠀⠈⠉⠉⠉⠀⠉⠉⠉⠈⠉⠈⠁⠈⠀⠁⠈⠀⠉⠉⠉⠉⠉⠁⠈⠉⠁⠈⠉⠀⠉⠉⠉⠉⠀⠈⠈⠉⠀⠀⠁⠈⠀⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⢈⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠶⠶⠦⣄⠀⠀⠀⢀⡴⠞⠋⠛⢦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠞⠋⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠀⠀⠒⠐⣄⠀⠀⠀⠐⠃⣠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠻⠷⣶
⠀⠀⠀⠀⣠⡶⠶⠚⠛⠳⠐⠒⠒⠦⠀⠀⠀⠀⠀⠀⠠⠁⠂⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠲⠤⠠⠄⠀⣤⣦⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣦⣦⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠿⠁⣠⣆⣀⢀⣠⠼⠶⠄⠀⠀⠀⠀⠈⣿
⠀⠀⣹⡶⠀⠀⠀⠀⠀⢀⡀⠀⠀⢉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⣀⣤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠱⣀⠀⠀⠀⠀⣠⣤⡿
⠀⠈⢷⣤⣤⣤⣄⡀⠐⠦⠀⠀⠀⠐⠀⠀⢠⠀⠀⠖⠻⣿⠋⠈⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠉⣹⡂⠀⠉⠙⠛⣷
⠀⠀⠀⠀⠀⠀⠙⣿⠄⣀⠀⠀⠒⠦⠶⣴⡶⣶⡤⣶⣶⣿⣤⣀⡀⣀⣀⣀⠀⢀⡐⠂⠺⣶⡊⠁⠀⠀⠠⠀⢠⡿
⠀⠀⠀⠀⠀⢀⣴⠏⠈⠁⠀⠀⠀⢀⢀⡈⠑⠈⡻⡟⠚⠋⠈⠿⠉⠛⠱⠿⣿⡙⠒⠀⠀⠈⠓⠲⢿⣶⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠿⣤⣬⡴⠶⠶⢶⡥⠈⠀⠀⠑⠀⠀⠈⣠⣤⣀⣀⠀⠀⢺⡷⢦⣀⠀⠀⢀⣀⣀⣀⣠⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠞⠛⠛⠛⢿⣿⣝⠛⢊⣹⡷⠾⠋⠁⠀⠉⠛⠳⠾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠐⡠⠀⠘⠉⠀⠀⠀⠈⠀⠀⡀⠀⢀⠄⠔⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⠄⠀⠠⡀⠀⠁⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⡀
⢰⢦⣂⣙⣙⣉⣉⣉⣉⣉⣉⣉⣉⣢⣄⠀⢀⡤⠚⠉⢀⡠⠤⠰⠄⠀⠈⢢⢀⡴⠋⠁⣠⠤⠖⠒⠛⠛⠛⠛⠷⣤⡀
⢘⠀⣹⠉⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠊⢄⡴⠊⠁⠀⠀⠀⠀⠀⠀⢸⡋⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⡛⠀⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⠇⢀⡞⠀⠀⠀⠀⢀⡠⠤⠤⢠⠼⠂⣸⠁⠀⠀⠀⣠⠔⠐⠒⢦⣄⠀⠀⠀⠀⠹
⡇⠀⡏⠀⠀⠀⠀⠙⠛⠛⠛⡄⠀⠀⡎⠀⣸⠀⠀⠀⠀⡔⠁⠀⠀⠀⣼⠀⢀⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⡏⢳⡀
⡇⢀⡇⠀⠀⠀⢀⡀⣀⣀⡠⠇⠀⠀⣯⠀⣟⠀⠀⠀⠰⣇⡀⠀⠀⠀⢻⠀⢸⡅⠀⠀⠀⢸⠀⠀⠀⢀⡼⠁⢸⡇
⡇⠨⡇⠀⠀⠀⢸⠖⣒⡒⠚⠛⢒⡓⢿⡀⢳⠀⠀⠀⠀⠫⣌⣉⣉⡉⠙⣆⠀⣇⠀⠀⠀⠈⠻⣒⢊⠉⣀⣠⠟⠀⠀⠀⢀
⣗⢠⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹⣜⣆⠀⠀⠀⠀⠀⠍⠉⠉⠉⠙⣦⡘⢦⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢠⠞
⠙⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⢶⣄⣀⡀⠀⠀⠀⠀⠀⠀⣇⠙⠶⣝⣦⣄⡀⠀⠀⠀⠀⠀⠀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣤⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢶⡛⢣⢬⢤⠛⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢛⠩⢙⠒⡄⠀⣴⣋⡰⠤⠴⣈⣻⡄⠐⠲⣖⡶⣒⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠁⠀⠈⢱⢹⠆⠀⠉⣧⢾⢦⡏⠁⠀⠠⠾⢽⣗⢅⠜⣼⣆
⠀⠀⠀⠀⠀⢠⣶⣦⠘⢧⡕⢠⡔⢣⡼⢀⣶⣦⡀⢸⠀⢠⣶⣦⠀⠀⠸⡏⢹⡏⢩⠛⣆
⠀⠀⠀⠀⠀⣿⢽⡽⡇⠀⠈⠹⡏⠉⠀⣿⢽⡿⡇⢸⠀⡿⣽⢽⡇⠀⠀⣇⠀⢧⠀⠆⠘⣆
⠀⠀⠀⠀⠀⠘⢿⣿⠃⠀⠀⠀⡇⠀⠀⠙⣿⡿⠃⢸⠀⠻⣿⡿⠁⠀⠀⢈⡽⢺⠗⠚⠒⠚⠂
⠀⠀⠀⠀⠀⠀⡬⡧⣧⣷⢼⠤⠧⠼⠴⠦⠼⠧⠧⠼⠤⢧⣾⠤⠾⠼⠤⠾⢴⢧⡤⡧⡾⣤
⠀⠀⠀⠀⠀⠀⣇⢷⢹⢸⡘⡆⢸⡃⢰⡶⡲⢄⠀⣀⣴⢿⣿⠷⣄⡀⢎⠆⡼⣸⢱⢣⡇⡏
⠀⠀⠀⠀⠀⠀⠸⡜⣎⢧⢳⡹⡄⢜⡆⣹⢾⣾⣿⡱⢸⢾⢈⣙⢮⣿⠆⡴⣣⢃⢏⡞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣌⢮⡳⡙⢌⠢⣐⠷⠵⠊⡠⡉⠻⣾⣿⡞⢋⡡⢞⡴⡡⢋⢞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢬⡲⣝⡪⢝⣲⠤⠵⣥⡤⠤⢵⣚⡭⣚⡥⣊⠕⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢭⣒⠭⣗⣺⠭⠭⠭⠭⠭⢽⣒⡯⢕⣪⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠭⠭⠭⠭⠭⠽⠒⠊⠉

⣏⣈⣿⣽⣿⣯⣯⣽⣽⣿⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣷⠄⢿⠆⢸⠿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠆⢰⣦⢰⣄⢀⣶⠄⣶⠄⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢾⣦⠁⢈⡋⡇⣩⡀⠁⣶⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⣤⡸⡿⠟⣿⠻⣳⣡⣤⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⢉⣅⣻⣼⣿⣰⣏⣩⢁⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢛⡛⣿⣿⣿⣙⡋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⣿⡟⠟⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣴⣶⣤⣾⣿⣷⣴⣾⣿⣿⣷⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⠀⠴⣴⠦⠀⠀⠀⠀⠀⢀⣤⢤⡄⠀⠀⠀⠀⠀⢠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣳⠄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣧⣠⡟⠀⠀⠀⠀⠀⢸⡷⠞⠁





⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣶⠀⠀⡄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⢹⣇⠀⠀⣧⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡄⠀⣿⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣶⣦⣿⣧⣾⣿⣴⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⢼⣿⣿⡧⠤⠤⠤⠤⠤⡤⠖⠉⠹⠧⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⠎⠀⠀⠀⠀⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠶⢄⡴⠁⠀⠀⣀⣤⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠁⠲⣎⡁⠀⣰⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⣨⢨⡇⠀⢠⠀⠀⠀⠀⣠⢄
⠀⠀⠀⢠⡄⣏⠸⡇⠀⠀⣹⡆⠀⠀⠧⣏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢸⣁⡀⠀⣤⠀⠀⣩⢄⠀⣀⠀⡀⢸⡏⠀⣉⠀⣼⠑⢀⡀⢀⠀⣸⡙⠀⠀⠀⢀⣀⣿⠀⢀⣄⡀⣀⠀⡀
⠀⠀⠀⢸⠉⣷⢸⡗⠀⣰⡏⣾⢠⣿⢸⡇⣸⡇⢠⡇⢠⡧⢀⣸⡇⣿⢀⣿⠀⡀⠀⠀⣿⢠⡇⢠⡟⣸⠃⣿⢠⡇
⠀⠀⠀⠈⠐⠋⠀⠙⠈⠈⠁⢛⠀⠉⠈⠓⠘⠓⠉⠓⠁⡟⡇⠈⠁⠙⠂⠙⠊⠀⠀⠀⠈⠉⠓⠁⠉⠘⠊⠈⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⠀⠀⡀⠀⠀⡳⡃⣀⠀⢀⡄⠀⠀⠀⢰⡃⢘⢀⡀⠀⢠⡄⢀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠠⣏⢩⠄⠀⠀⠙⣦⣾⢱⡇⢿⡄⢸⡃⠠⠀⠀⣿⠀⣿⠀⣷⠀⢿⠀⠘⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠀⠀⠀⠀⠘⠋⠀⠈⠁⠈⠀⠀⠁⠀⠀⠀⠈⠀⠉⠀⠈⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡄⢀⣒⠒⠦⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣙⣤⡌⢻⣷⡆⠀⠀⠀⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠉⢶⣾⣿⠁⠀⠀⣀⣀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠈⠀⢀⣈⣿⣿⣷⣾⣿⣿⣟⣻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⢀⣤⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡮⠽⠿⢤⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⣡⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⣿⣧⡔⠶
⠀⠀⠀⠀⣀⡀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠘⣿⣆⠀⠘⣿⣿⣿⣿⡆⠀⠀⠀⠈⢻⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⢠⠀⢐⡄⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣦⣀⠘⢿⣿⣿⠃⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣰⣾⡟⠁⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡈⠻⣏⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣀⣼⣿⡿⠋⢀⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣤⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣋⣤⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡄⠈⠉⠛⠛⠛⠛⠛⠋⠉⠁⢠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢄⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⢒⢹⠱⡀⠀⠀⠀⠀⠀⢄⡿⡪⡜⢸⡃⡂⡶⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⢏⠢⢆⠀⠚⠉⠁⠀⢀⡟⠦⣀⣀⡀⠀⠀⠉⠙⠓⠘⡐⡉⢊⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠍⣆⠃⠁⠀⣀⠤⠐⠊⠉⠉⠀⠀⠉⠳⡂⠭⢋⡖⠦⣤⣀⠀⠀⠁⡁⠥⠪⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠈⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠆⢠⠀⠠⠪⠈⠱⢦⡀⠀⠉⢎⡴⡋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢌⣛⠆⠀⣠⠊⠁⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣠⠞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⡄⠀⠙⠇⡠⢠
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠎⡄⠀⢀⠔⠁⠀⣀⠤⠒⠈⠉⠀⠀⠀⠀⡧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡌⢢⠀⠘⠣⢐⣘⡄
⠀⠀⠀⠀⠀⠀⢀⠄⡀⡢⠀⢀⠎⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡒⠂⠀⠀⠐⠦⠤⢄⣀⠀⠀⢀⡉⢷⠀⢳⠀⠈⣿⡁⣢⡀
⠀⠀⠀⠀⠀⠀⡀⢁⣹⠀⠀⡎⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⣆⣀⣀⠐⠒⠺⣝⣁⠀⠀⠀⠸⠀⠀⢣⠀⠈⣲⡭⢐⡄
⠀⠀⠀⠀⠀⠀⠍⠥⡇⠀⢸⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠘⡄⠀⠀⠀⢸⠀⠀⠈⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢰⣟⣿⣆⡜⠀⢧⡶⣷⣆⢀⠇⠀⠀⠀⡸⠀⠀⠀⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⢀⣀⣈⠻⠿⠷⠊⠀⠀⠀⢠⠃⠀⠀⠀⠀⠘⠀⢅⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠢⡀⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡁⠀⠀⡕⠀⠀⠀⢀⡀⢀⠎⠀⠀⠀⠀⡆⡜⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⣀⣀⣀⡼⠀⢰⠸⡀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠉⠒⠉⠀⠀⠀⠀⠘⡈⠉⠀⠀⠀⠀⢀⠁⡇⠀⠀⠀⠠⢤⣀⠸
⠀⠀⠀⣏⣀⠀⠀⠀⠀⠈⢆⢣⠀⢸⠀⠈⠉⠒⠢⠤⣀⠀⠀⠀⠈⠫⣑⠢⠦⠤⢀⡀⠠⠤⡲⠃⠀⢇⠀⠀⠀⠀⠀⠸⣀⠈⠁⠐⠂⠤⠤⠜
⠀⠀⠰⣁⣀⡤⠄⠀⠀⡀⠜⠈⢢⡆⠀⠀⠀⠀⠀⢎⠁⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠤⠒⠋⠀⠀⠀⠸⡀⣀⡠⠤⡔⠒⠉⠁
⠀⠀⠀⠀⠣⠤⠔⠊⠁⠀⠀⠀⠸⢀⡀⠀⠀⠀⠀⠈⠢⡀⠀⠠⠤⡀⠀⠸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⠢⡀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠈⠓⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢄⣀⣀⠀⠉⠁⣒⣒⠲⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄⡈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⢀⣠⣄
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣆
⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠋⢀⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠟
⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇

⠀⠀⠀⠀⠀⠀⠄⠆⠄⠀⠀⠀⠀⠀⢶⣿⢏⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣺⣀⠀⠀⠀⠀⠀⢰⡶⠶⠶⢶⡆
⠀⠀⠀⠀⠠⠅⣾⣿⠎⠥⠀⠀⠀⠀⡏⢱⣿⡆⢹⠀⠀⠀⠀⠀⣻⡥⣩⡇⠀⠀⠀⠀⠱⡶⢵⣛⣦⢴⠎⠀⠀⠀⢸⡇⠂⠐⢸⡇
⠀⠀⠀⠀⠀⠁⠆⡅⠎⠁⠀⠀⠀⠀⠳⣄⣀⣠⠞⠀⠀⠀⠀⢠⡿⠿⠿⢧⠄⠀⠀⠀⠀⠸⠿⠃⠯⠏⠀⠀⠀⠀⠘⠿⠶⠶⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠠⡰⢠⣄⢆⣤⠶⡀⠀⠀⠀⣀⡴⠒⢦⡀⠀⠀⠀⠀⢠⢾⡷⠉⢳⡄⠀⠀⠀⢀⣴⡿⣛⣿⣦⡀⠀⠀⠀⢀⢻⠽⡮⡿⡀
⠀⠀⠀⠀⣹⡇⢸⣯⢀⣿⡁⠀⠀⢰⠃⠀⠀⠀⢉⡇⠀⠀⠀⢳⠈⢰⣦⢀⡇⠀⠀⠀⠀⣿⠲⠻⣌⡇⠀⠀⠀⠀⠠⠽⢭⡯⠯⠅
⠀⠀⠀⠀⠃⠘⠋⠘⠚⠀⠃⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠑⠚⠓⠊⠀⠀⠀⠀⠀⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠚⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⡤⠤⡀
⠀⠀⠀⠀⠀⠰⢻⣷⣀⠀⠀⠀⠀⠀⠀⢴⡞⠦⠀⠀⠀⠀⠀⢿⣆⣀⣀⣸⠷⠀⠀⠀⢰⣿⡋⠉⢹⣿⡀⠀⠀⠀⠀⣼⣀⣀⣵
⠀⠀⠀⠀⠀⠀⢨⢳⡀⠀⠀⠀⠀⣞⠉⣧⠹⣞⠉⣣⠀⠀⠀⢘⣯⣿⣿⣽⠀⠀⠀⠀⠘⢿⣿⣿⣽⡿⠀⠀⠀⠀⠀⣯⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠓⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⢠⣤⣼⣽⣤⣤⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⡄⠀⠀⠀⠀⠀⢀⣤⢶⡄
⠀⠀⠀⠀⠀⣿⢏⣍⢻⡇⠀⠀⠀⢀⢿⣶⠦⣶⣷⠀⠀⠀⠀⢠⣀⣿⣿⣠⡆⠀⠀⠀⢰⣿⠇⠉⢻⣷⠀⠀⠀⠀⢠⡮⠁⠈⢫⣄
⠀⠀⠀⠀⠀⢻⣌⣫⣾⠁⠀⠀⠀⠨⡿⠷⠲⠿⢿⠀⠀⠀⠀⠼⠥⠼⠧⠬⠯⠀⠀⠀⠀⠉⠇⠀⠟⠉⠀⠀⠀⠀⠙⠵⠆⠴⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡸⣆⣇⡇⠀⠀⠀⠀⠀⢀⡀⠀⣤⣤⡄⠀⠀⠀⠰⣆⠐⠿⠀⠀⠀⠀⠀⠴⢤⠤⡒⠤⠴⠄⠀⠀⠀⠀⠰⢶⡶⠆
⠀⠀⠀⠀⠀⢿⡿⠻⣧⡴⠃⠀⠀⠈⠻⠿⠖⠛⠋⠀⠀⠀⠀⠀⠉⢻⣿⣿⡆⠀⠀⠀⠰⣨⣍⠉⣩⣡⠆⠀⠀⠀⢸⣶⣾⣷⣶⡄
⠀⠀⠀⠀⠀⠘⢧⢀⠟⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁⠀⠀⠀⠀⠉⠛⠐⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣦⢀⡀⠀⣀⢀⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⣰⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣻⠏⠀⢿⣿⣽⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡽⠛⠉⢉⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣶⣁⣀⣀⣀⣹⣶⣎⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣽⣿⣿⣿⣿⣟⣯⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡆⢸⣿⣟⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣛⣿⣟⣾⣿⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⠛⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⣀⣤⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣼⢿⡦⣶⠀⢶⣦⡿⣷⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣸⣤⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠁⠀⠈⢻⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⢰⣿⣿⡿⣿⡿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣾⣿⣿⣿⣿⣿⣳⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⡿⠿⠻⡿⣶⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣾⣿⣿⣿⣷⣿⣷⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠺⠿⠿⠾⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⣿⣾⡿⣿⡿⠿⠿⠿⠿⠯





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⣦⠴⠋⠁⠤⣤⣠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠣⢀⡠⠶⠳⠎⠁⠀⠀⢀⣠⡤⣿⡽⣋⡡⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢾⣇⠀⠐⡀⠀⠀⣸⡏⠉⢿⣿⣇⣞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⣬⠀⠘⠁⠀⡸⣹⠇⠀⠀⠀⠉⠹⣧⠳⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠚⣸⡦⠤⠀⣀⡰⠋⠀⠀⠀⠀⠀⠀⠹⢦⢳⢉⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠻⡈⠻⢷⣟⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⣀⡚⠳⠾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠯⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⢀⣀⡀⡀⠀⠀⡀⠀⠀⠀⡜⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣅⣹⣿⡾⣧⡾⢿⣿⠇⣃⣚⣛⣣⡀⠀⢸⠁⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠛⠁⠙⠁⠈⠉⠁⠉⠉⠁⠉⠁⠀⠘⢿⣤⣤⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠢⣶⠛⠉⠉⠉⠉⠙⠓⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡰⡶⠟⢣⠀⠠⠁⢀⠀⠀⢀⢡⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠞⠫⠿⠝⢐⠁⠀⠐⠐⠁⠀⠀⠀⠀⠀⠠⠀⠀⡁⢠⠈⠠⣌
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠰⣀⠄⠁⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡦⠐⠙⠂⠊⠁⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⡨⠟⢠⠲⠀⠀⠄
⠀⠀⠀⠀⠄⠀⠀⡀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠀⠄⠠⠀⣨
⠀⠀⠀⠀⡄⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢂⡄
⠀⠀⠀⢀⠃⠠⡀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸⠀⠁⠦⢤⠐⠒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂
⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⡀⠠⠘
⠀⢀⠀⠀⠀⠡⠀⡀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠊⠠⠐⡀⠀⠀⢂
⠀⠐⠀⠠⡀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠄
⠆⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠠⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⢠⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢄⠀⠀⠀⠢⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⡆⠀⣄⣠⠄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣷⠀⠸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠟⠋⠀⠰⠬⡵⠂⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⡿⠀⢠⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠴⠋⠀⠀⠀⠀⠀⠀⡠⢠⠀⠈⠃⢀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠒⠂⠽⠆
⠀⠀⠐⢰⠃⡠⠃⠀⠀⠀⠀⠀⠑⠸⡾⠶⠀⠀⠀⠀⠀⠀⢀⣴⣊⡡⠀⠠⠚⠹⠿⣿⣿⣿⣿⠤⠄⠀⠀⡔⠂⠀⠀⣤⣄⣴⡀
⠀⠀⠀⢿⠎⠀⣿⠟⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⡄⠀⠈⠉⠛⠇⠁⠀⠈⢀⣀⠀⢀⣈⡛⢿⠃
⠀⢀⣶⡄⠀⣤⠄⠀⠀⠀⢌⡁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠈⠙⠇⠈⣀
⠀⠘⠏⠀⠀⠑⠀⠀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠙⠀⠀⠀⠀⠼⣴⡶⠀⠀⠀⠀⢈⣿⣦⣄⠀⠀⠀⢰⠁⠐⠹⠄
⠀⣬⠄⢀⠀⠀⠀⣼⠁⠐⠀⡄⢀⡖⡄⢶⠦⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠉⠉⢰⠀⠀⠈⠉⠑⠈⠟⠀⠀⠀⠘⢀⢰⠢
⠀⠦⠆⠀⠐⠊⠩⠃⠀⠀⠈⠄⢂⡧⠃⠀⡀⠀⠀⠀⠀⠚⣢⣤⠢⠤⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠴⠶⢈⠘⢷⡟
⠀⠓⠟⠀⠀⢰⡇⣀⠀⠴⡲⠖⠼⠃⠀⠀⠀⠀⠀⠀⡼⣮⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠘⠃
⠀⢈⡛⠃⠀⠀⠑⢧⣁⣀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠁⠀⢹⡄⠘⠿⠂⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠂⠀⠠⠄⠀⠺⠶⠟⠃
⠀⠨⡭⠀⠀⠀⢈⠈⠉⡁⣠⢘⡀⠀⠀⣀⠀⠀⠐⠀⠀⠀⢆⡿⠂⠐⡀⠐⠊⠀⢀⠔⠡⠀⠅⠑⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢸⣄
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢐⢻⠆⣰⣄⡀⠀⠀⠈⠾⡀⠀⠁⠀⠙⠉⢁⣠⣖⡭⠴⠋⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⣜⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠐⠿⠿⠁⠃⠀⠀⠈⠚⠉⠀⠀⠀⡴⢋⠭⠖⠒⠒⢲⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠠⠤⠤⠼⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⢶⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⣠⠟⠁⠀⢸⠀⠀⠙⢦⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⠲⣴⣇⠀⠀⠀⢸⠀⠀⠀⢈⣧⡴⠋⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡏⠈⠙⠛⠒⢿⠒⠚⠛⠉⠘⡇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⣇⣀⣠⡤⠤⠾⠤⠤⣤⣀⣠⡇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣠⠴⠻⣏⠀⠀⠀⢸⠀⠀⠀⢈⡟⠳⢤⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠙⢦⡀⠀⢸⠀⠀⣠⠞⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠾⠶⠾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠐⢈⣠⠤⠴⠖⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠐⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣀⣀⣀⣀⣀⣀⡀⠤⠤⠄⠀⠀⠀⠃


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

⠀⠠⠠⠀⠠⠄⠠⠠⠄⠀⠄⠄⢠⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠄⠄⠤⠤⠀⠤⠄⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠟⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⣴⠞⠛⣿⠟⠛⠿⢷⣶⣦⣤⣀⣈⣉⣁⣴⠟⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⢿⣁⣠⠾⠋⠀⠀⠀⠀⠈⠉⠛⠛⢛⡛⠉⠀⠀⣿⡟⢀⣴⣿⠿⣦⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠉⠁⠀⠀⠀⣠⣿⡿⢳⣿⡶⠀⣾⣿⠂⠀⣼⡿⢁⣾⣿⠏⣰⣿⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⣼⣿⡏⠀⣼⣿⠃⣸⣿⡏⠀⣰⡿⠁⣼⣿⣇⡴⠟⠁⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⢻⣿⣿⣷⡀⠀⣸⣿⡟⢀⣼⣿⡏⢀⣿⣿⠀⣼⠟⠀⠀⣿⡿⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠈⣿⣿⣿⡇⠀⣿⣿⢁⠞⣿⡿⠀⠘⣿⣧⡾⠋⠀⠀⠀⠙⠛⠛⠋⠁⣠⣶⣦⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠙⠛⠋⠀⣿⡇⠀⠀⠀⠀⣀⠀⠀⠀⢠⣿⣦⠀⠀⣼⣿⠃⢘⣿
⠀⠀⠀⠀⠈⢿⣿⣷⣶⣶⣿⠿⣶⣿⣿⡿⢁⣴⡶⠶⣶⡶⠿⣿⠶⣶⢶⣾⡯⢤⡀⠀⣼⣿⠃⠀⣼⣿⠏⢀⣼⡿
⠀⠀⠀⠀⣠⣾⣿⡿⢿⣷⣶⡾⢿⣿⡋⠀⢾⠏⠀⢀⡿⢀⣼⣇⣼⠏⢾⣭⣄⣬⣥⣼⣿⣯⣤⣴⣿⣿⣾⡿⠛⠁
⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣰⢆⣾⠁⣸⡏⢸⡏⣠⣾⢋⣡⠌⢩⣿⡿⠉⢉⣿⣿⢁⣀
⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⣴⣾⡿⠏⠀⠀⠀⠈⠉⠁⠀⠉⠀⠈⠉⠀⢉⣭⡀⢀⣿⣿⠃⠀⣼⣿⣷⠋⣿⣷
⠀⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⢦⣶⣦⠀⣸⣿⣦⢏⣿⠇⣼⣿⠇⠀⢰⣿⡿⠁⣸⣿⡏
⠀⠀⠘⠻⢿⣶⣤⣴⣶⣶⣶⣄⠀⠀⠀⣴⣿⡟⠁⣼⣿⠇⢠⣿⣿⠃⠘⠋⢠⣿⡟⠀⢀⣿⣿⠁⢀⣿⡿
⠀⠀⠀⢀⣴⣾⣿⣿⠟⠉⠁⠀⠀⠀⢰⣿⡟⠀⣼⣿⡏⢀⣼⣿⠃⠀⠀⠀⣾⣿⢁⡴⢻⣿⠇⠀⣼⣿⠇
⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⡜⣿⡿⢠⠾⣿⡏⠀⠀⠀⠀⠛⠷⠚⠁⠀⠉⣀⣠⣽⣿⡶⠶⠶⣦⣄
⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⡾⠻⠷⠞⠀⢙⣛⣁⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠉⠀⠙⠿⠶⠤⠼⠋
⠀⣿⣿⣿⠁⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⣾⠟⠉⠻⣿⡷⠀⠀⣀⣤⣶⣾⣿⠿⠛⠁
⠀⠘⠻⠿⠷⠶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣶⣶⣶⣾⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠻⠉⠙⠀⠁⢀⣀⠀⠉⣟⡖⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⣉⡀⠀⠀⠀⠉⠀⠁⠀⠈⢠⠇⢀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡥⠒⢢⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⢙⢛⣊⠁⠈⡧⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢠⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⢞⡇⢀⣇⡜⠀⡘⢢
⠀⠀⠀⠀⠀⠀⢠⠏⡠⠤⡶⠛⣹⠀⠀⣠⠀⠀⠀⠀⣀⣀⣀⡤⠎⠀⢸⡎⠀⠈⠉⠀⠐⠁⣼⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠛⠳⠁⠀⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⢀⡼⠤⣀⠀⠀⠀⠀⠘⢁⣌⠀⠀⠈
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠞⠒⣌⠳⠀⠀⢀⡤⢪⡇⠀⠀⠀⠘
⠀⠀⠀⠀⢠⡏⠉⠲⢄⣀⣀⣀⡇⠣⣀⠙⠒⢤⡀⣀⣀⡤⠴⠚⠉⠀⠀⣀⣼⣀⠗⠚⠉⣠⠏⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢣⣀⡀⠀⢘⣤⣀⣈⠉⠉⠀⠀⠀⠟⢛⣆⠀⢀⣀⣠⡴⠾⠛⠉⠀⠀⣠⠞⠹⣠⠇⠀⠀⢸⣏⡃
⠀⠀⠀⠀⠀⠘⡆⢹⠉⡳⢄⣀⠈⠀⠀⠀⠀⢸⠀⡜⠉⠉⠉⠀⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⡼⠀⠀⣞⠉⢳⡄
⠀⠀⠀⠀⣀⠀⠹⡤⠹⠙⢼⠓⠛⠁⠀⢀⣀⡼⣄⡱⠒⠒⠒⠒⠒⠋⠉⠁⠀⠀⠀⣀⡀⡰⠃⠀⠀⠙⠛⠛⠁
⠀⢠⡖⠋⠉⠀⠀⠘⢦⠀⡀⠉⠉⠉⠉⠉⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠋⠉⢉⡝⠁⣀⠀⣀⣾⢃⡄
⠀⠀⠉⢢⠴⣦⠀⠀⠀⠑⢷⡤⠤⠤⢄⡀⠀⠹⡀⠀⠀⠀⠀⢀⣀⠞⠁⠀⠀⣀⠴⠋⠀⠀⠉⢤⡳⡌⠉
⠀⠀⠀⠘⢋⣵⣣⢄⡀⠀⠀⠙⠲⣄⣀⠈⠙⠒⠚⠛⠀⠀⡔⠋⠀⣀⡤⠔⠊⠁⠀⣤⠀⣠⢽⠚⠁
⠀⠀⠀⠀⠀⠀⢰⠿⠽⠀⠀⡀⠀⠀⠉⠙⠓⠒⠒⠲⠒⠚⠛⠋⠉⠁⠀⠀⡀⠐⠒⠾⢆⡳⠞
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡞⠃⡼⣥⢀⣀⠀⢀⡀⠀⠀⡀⠀⠀⢀⡀⣦⠀⡽⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠱⠏⢾⡛⠀⠈⣧⠶⣼⠃⣏⡵⢻⠁⠘⠆⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠁⣀⣀⣛⡆⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠠⠤⢤⣤⣴⣿⠒⠀⠈⠀⡘⠛⠛⠋⠉⢁⣠⢔⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣤⣤⣤⠤⠤⠐⠒⠺⣦⣀⣀⣠⣾⠋⠉⠉⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣠⣴⢟⠛⠛⠉⢦⡤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢙⠿⠖⠀⠠⠤⠤⠤⠔⠂⠉⠙⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠦⠤⠤⠀⠀⠠⠤⠄⠀⢒⠟⣿⠇⠀⢀⠤⠐⠒⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣦⣤⣤⣤⣤⣶⣿⠷⠟⣿⠀⡐⢀⢔⠠⠤⢮⡀⠀⠫⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡁⢚⣵⠮⢅⠢⡑⢔⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡜⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢔⣝⣻⡾⠃⠐⠄⠱⡈⢞⡤⠖⠙⣧
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠠⠄⠒⠥⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠄⢱⠸⢠⡀⠀⣿
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⢤⡘⡧⢸⡇
⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣋⣥⠶⡞⢿⠶⣮⡳⠀⢀⡿
⠀⠀⠀⠀⠀⡀⡄⠇⠀⠀⠀⠀⠀⡀⠌⠓⢀⣈⠉⠉⠐⠒⠠⠤⠀⡀⢀⣴⡚⠂⠉⠀⣈⢷⠁⠈⠹⠟⢗⣠⠞⠁
⠀⠀⠀⠀⠀⠃⠃⡀⠀⠀⠀⠀⠘⠁⢠⡾⠋⠉⡿⢧⡀⠀⠀⣴⠞⢶⡘⢧⣹⠀⠀⢼⡽⠃⠑⠶⠤⠤⠚⣃
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⠀⠀⠀⢿⡀⠀⠀⠳⠜⠁⠀⢸⠇⠀⠈⣷⠈⣿⠀⠀⣼⠃⢀⡟⠛⠛⠛⠻⠭⠃
⠀⠀⠀⠀⠀⠈⠻⠷⠶⠶⢶⠶⠃⢀⣈⡛⠶⠶⠶⣤⡀⠀⣿⠷⠶⠾⣿⠀⢸⡆⢀⡿⠀⣸⠷⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣧⠀⠘⣧⣻⡀⠀⠀⢀⣿⢀⢿⡀⠀⣸⢿⠀⠈⣷⣼⠃⣰⠟⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠌⢷⡀⠈⠙⠛⠛⠛⠛⠁⠈⠚⠁⠀⠹⠞⣀⡀⢈⣁⠀⢙⣛⡛⠓⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⡌⣷⢠⡀⠀⣾⡄⢠⡆⠀⣰⣆⠛⠛⣟⠉⠀⢻⠉⠁⠘⣯⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢊⠀⠀⠁⣿⠀⢻⡄⢻⣧⢸⠁⢰⣯⣿⠀⠀⣿⠀⠀⣼⠷⠂⠀⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡡⠀⢀⢆⡿⠀⠀⠹⣾⢹⡿⢀⡿⠀⠹⡆⠀⢿⠀⠠⣿⡤⠀⠰⡇⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣠⣤⠾⠁⠀⠀⠀⠉⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣻⣟⣓⠀⢤⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡀⢀⣿⡩⠽⠷⠮⣕⠢⡀⡞
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠹⢉⣻⣟⠈⠀⠀⢉⡗⣜⢇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⡬⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠉⠙⣿⣇⣀⣠⡞⠈⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣎⣤⣤⡒⢠⣀⠀⠉⠀⡀⠙⠓⠀⠀⢤⢤⣴⣋⢙⣿⠟⠃⠙⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢳⣀⡸⣿⣿⣿⡿⠷⠈⠁⠀⠀⣃⣀⠀⠀⠀⠀⠀⠈⠉⣿⣿⡇⠀⠐⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⣡⠀⢀⢀⣀⠀⢉⣿⣟⠛⡃⠀⠀⠔⡁⠀⠐
⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠛⣿⠿⠻⠟⠀⠀⣿⠀⠘⠉⠀⠠⠤⣄
⠀⠀⠶⠀⠀⠀⠀⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠀⠀⠀⠐⠈⣹⣿⣿⣶⡶⠁⠀⠈⠁
⠀⠀⠀⠑⢤⣼⡞⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⡠⣄⣀⠴⣞⡯⠞⠋⠁
⠀⠐⢆⠀⠀⠈⠁⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠛⣿⠏⠞⠋⠰⠊⠁⣀⠠⣤⣶⣾⡦⠔⠋⠄
⠀⠀⠈⣆⠑⣤⣀⠈⠻⣧⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠹⠷⠄⠀⠀⠰⣮⣴⣶⣿⣿⣿⠟⠂
⠀⠀⠀⠘⢉⡙⠉⠢⣾⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠈⠀⠀⢰⢃⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⡀⠀⣀⡀⢀⡰⠁
⠀⠀⢀⣤⡄⠀⢀⡏⢨⡤⠤⠤⠀⠂⣴⠀⠀⣐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠉⣰⡿⠀⠃
⠀⠀⠀⠀⠐⣁⠨⢯⣁⣈⣁⠄⠀⠈⠙⠛⠷⠿⠷⡀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⢀⣠⣞⠉⠈⢁⡧⠦⠄
⠀⠀⠀⠀⠤⢌⠒⠀⠀⠀⠀⡤⠴⠤⠀⠀⠀⠀⠀⠀⠘⠒⠾⠿⠿⠟⠛⠉⠀⢀⣤⠤⠖⠋⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠿⠤⠀⣀⣤⣤⣤⣤⣤⠤⠤⠼⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠶⠦⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢾⣁⣀⡠⠤⠖⠒⠲⠦⠤⠴⠒⠛⠛⠓⠲⠤⠤⠤⠴⠒⠲⠼⠷⣄
⠀⠀⠀⠀⠀⣠⠿⠤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄
⠀⠀⠀⢀⡼⠁⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⠒⠦⠤⠤⠔⠚⠓⠒⠒⠋⠁⠈⢧⡀
⠀⠀⢀⡾⠁⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⡾⠁⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⣸⠇⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣇
⠀⣿⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⡇⠀⠀⡞⠹⡄⢻
⢰⡏⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⢠⠞⠀⠀⠳⣀⡏⠀⠀⡇⢸
⢸⡇⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠖⠁⠀⠈⣇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⠁⠀⢠⠇⢸
⠀⣷⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣼
⠀⢹⡄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡏
⠀⠀⢷⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⡼⠀⡾
⠀⠀⠈⢧⡀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⢀⡰⠃⡼⠁
⠀⠀⠀⠈⢳⡄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠙⢦⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠤⠖⠋⠓⠶⠖⠒⢲⡄⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣴⣒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡒⠀⢈⠁⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⣠⣰⣴⠤⠓⠂⠉⠁⠀⠈⠉⠒⡈⢠⠞⢡⠼⢪⠀⠀⠀⠀⠀⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠐⠀⣄⣀⡼⡲⡽⠋⠁⠀⠀⠀⠐⠀⠀⠀⠀⠸⠀⡡⠀⡎⢸⡈⣄⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠨⠙⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠂⠎⢽⣄⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⡠⢒⠑⠁⠀⠀⠀⠀⠀⠀⣈⣁⣠⣀⠐⢃⠋⢾⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢘⢵⡈⣽⢱⡼⠀⠀⠀⠀⠀⠈⣧⣺⣭⠫⡑⠀⢊⣠⣀⠀⠀⠀⢠⡧
⠀⠀⠀⠀⠀⢀⠀⡆⡆⠀⠀⠀⠇⢘⣤⠝⠻⠁⠀⠀⠀⠀⠀⠀⠉⢿⢭⣤⠔⠀⠈⣣⠌⠃
⠀⠀⠀⠀⣧⣿⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⠼⠄
⠀⠀⠀⠀⠣⣤⢠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⡆
⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡘
⠀⠀⠐⠀⠁⠀⠓⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠊⠎
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢁⠠⡺⠞⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⢀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠑⠀⡀⢠⡖⠀⠀⠀⠀⠀⡤⠤⠌⠛⠃⡎⢀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⢀⡴⣺⠀⠀⠠⠄⣀⡀⢉⣑⡒⣲⣄⡶⠈⡼⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠠⣼⣦⣄⠞⠀⢠⠖⠛⡅⡐⣹⢧⣄⠘⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢰⠄⠀⢠⡀⠀⠛⠶⠆⠀⠀⢀⢸⣶⣶⣿⣀⣎⠉⠹⣷⣾⣥⣄⣤⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠸⠶⣄⣴⣿⡀⠀⢀⣴⣤⣿⡛⢻⡏⡿⣿⠿⣿⣤⡄⣠⡤⢡⡍⣳⢀⣼⣀⣤⣤⣴⣦⣤⣤⡀⠲⠛⠉⠉⠀⠀⠉⠙⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠛⠚⠋⠁⠀⢸⣿⡜⢿⣷⣌⣧⣿⣿⠙⠻⣷⣬⣽⢿⡿⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⢦⣤⡄⠀⠉
⠀⠀⠀⠀⢀⣀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣼⣯⡅⣿⣿⣿⣆⠀⠈⠁⠠⠎⠀⠀⣀⠶⠞⠛⠛⠛⠒⠀⠀⢁⣠⣤⣤⣀⠈⠻⣦
⠀⠀⢀⣀⣙⠁⠀⠚⠉⣄⡄⢀⡤⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠨⣿⠛⣿⣿⣿⣻⣿⣿⡆⣀⠀⠀⠀⠀⡜⠉⠀⣠⡤⠤⠄⠀⠀⠀⠈⠵⣴⢄⠙⣷⠀⠈
⠀⠈⠉⠁⡼⠛⠈⠁⠀⢀⣨⣿⠟⠀⠀⠀⠀⣤⣼⣏⣸⡇⣴⢶⠰⠟⠛⠛⣿⣾⣧⣽⣿⣿⡿⠀⠀⠀⠀⠀⠀⠛⠁⣀⡀⠀⠀⠀⠀⢀⣼⣿⠈
⠀⠀⠀⠸⠁⠀⢦⡴⣿⣿⣿⡇⣠⡄⣠⡾⣷⢻⣩⡉⣹⣇⣹⡞⠳⡀⠀⠀⠈⣿⣿⣷⣏⣿⣿⠀⣀⣶⣴⣦⣤⡠⠛⡡⠶⠀⠀⠀⠀⣿⣟⠿⣿⣷⠄
⠀⠀⠀⠀⣀⣾⣨⣾⣧⣦⠛⣧⠿⣝⠛⣶⠹⣏⢹⣿⣿⠟⠿⠻⣥⢷⠠⣤⡀⣿⣟⣻⡿⣿⣿⣯⣿⣿⣿⣋⣸⣇⢠⣾⠿⣶⣦⣤⣰⣿⣿⣿⣿⣿
⠀⠀⠀⣙⡟⠙⠿⣿⣿⣝⠳⣿⣇⣿⠀⣿⠀⣾⣿⢯⣤⣶⣦⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⢷⣽⡷⠶⠟⢸⣧⣄⠀⣨⣽⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣹⣤⠖⠃⠀⠹⣿⢸⣿⣿⣿⣾⠏⢀⡿⣻⡼⢻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⣺⣶⡛⠻⣯⣿⣿⣹⣿⣿⡟
⠀⠰⣾⠏⢉⡆⠀⠄⢀⠸⣷⣿⠟⡘⣿⠀⣼⠟⣻⣿⣷⣿⢿⡏⢙⣿⠹⣿⣿⣿⠿⠿⠿⢿⣿⣿⣷⣶⣤⡴⠂⠀⣿⣿⣿⣿⣿⡟⢸⡿⢿⣿⡄
⠀⠀⣤⢀⡾⢁⣀⣾⡏⠁⢺⣿⣄⣵⣹⣧⣿⣿⣿⣿⣭⢽⣾⣿⣾⣧⣼⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣭⣿⠆⣠⣿⣿⣷⣿⣿⢣⡿⢤⡘⣿⣷⡀
⠀⠀⠘⣧⡁⣨⡏⢹⣿⠛⠁⠘⣷⢾⣏⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣭⣍⣿⣯⣾⡽⠿⣤⡿⣿⣷⡄
⠠⡀⢤⣾⣇⠙⣷⣼⠏⠀⠀⠀⢹⣷⣟⢦⢼⡝⢛⣿⣿⡿⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡜⢻⣽⣿⣿⣄⡾⡿⢾⣿⣿⡄
⠀⣿⡌⣷⠻⣷⢿⡟⣶⡏⠀⠀⠈⣿⣽⡼⣣⣾⣿⣿⣏⡄⢿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢫⣿⣿⣿⣿⣿⣿⣦⠹⣿⣯⠛⠿⣿⣶⡿⠿⠃
⠐⣾⣷⣿⡄⢹⣎⠷⣾⢡⣤⣤⣀⣿⣿⣷⣿⡿⣷⠘⠿⣿⣇⣿⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣵⠿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣷⡝⠛⣿⠟⠋
⠀⠈⢻⣟⢿⣷⡻⣦⡹⢿⡿⢁⠃⣿⡿⣿⡿⡉⠻⣧⠀⠀⠉⠛⠁⠀⠀⠛⢿⣿⣮⣽⠿⠋⠀⠀⢀⠉⠉⠉⠀⠹⣿⣿⣿⣿⣿⣿⣄⡏
⠈⠻⣦⣽⣆⠻⣷⣜⣧⠰⠿⣆⣸⣿⣿⣟⣷⣰⣆⢨⡇⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠂⠈⠀⠀⠀⠀⡀⡀⠀⠀⠙⢿⣿⣿⣟⣿⣿⣷⣄⡀
⠀⠀⢿⡻⣿⡓⠌⠛⠿⣿⡾⠏⣿⣿⡿⠿⢿⣽⣿⡿⣟⠀⠐⣁⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣄⠂⠰⠄⢶⣾⣾⣿⣿⣿⣿⠿
⠘⣤⣾⣿⣿⣷⣀⠀⢀⣙⣻⣾⠋⣿⠀⢰⡦⠈⠻⡿⠃⢀⣾⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⠻⣷⣄⠀⠀⣿⣿⣿⣿⣿⣿⣦
⢀⣿⣿⣿⣝⠻⣿⡳⣿⣯⠉⠀⢸⣇⠀⢿⡷⠀⠀⠀⣰⠟⣡⢎⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣍⢳⠈⠛⢷⣌⣾⣿⣿⣿⣿⣿⡏
⠸⣿⣿⢿⣿⣷⣬⣿⣶⣟⠓⠀⢸⣿⣤⣦⡄⠀⠀⠞⠁⢸⣧⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠐⠀⢸⣿⣿⣿⣻⣿⡿⠃
⡴⢿⣷⣀⣈⢻⣾⣿⡿⢉⠉⠓⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠉⠉⠋⣾⣿⣿⣿⣦
⢳⣌⣿⣿⣿⣿⣟⢿⡛⢈⡀⠀⠀⠘⣿⣿⣿⣷⣦⡀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢻⣿⣿⠻⠋
⢾⠏⠁⠀⠀⠉⢙⣿⣿⣤⠝⢠⡀⠀⠙⣿⣿⣿⣿⣷⡄⠀⢀⡄⠀⠀⠀⠀⠀⣴⡀⠀⠀⠘⣷⡀⠀⠀⠀⢠⣦⡀⠀⠀⢴⣟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣦⣝⣻⣿⣷⡄⠀⠀⠠⣿⣿⣿⣿⡧⡇⠀⠺⠿⠀⠀⠀⠀⠘⠿⣧⣤⣤⣼⡟⠁⠀⠀⠀⠻⡾⠃⠀⠀⢠⣿⡿⣁
⠀⠀⠀⠀⠀⠀⢸⣿⢟⡁⠸⣇⠀⠀⠀⣽⡄⠻⣿⣷⣿⡀⣠⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣀⠀⠀⠀⠀⣶⠄⢠⣶⡿⠋⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡄⠀⠉⠀⠀⠈⠟⠃⠀⠈⠙⢿⣧⣿⡇⠀⠀⢀⡴⠞⠉⢉⣤⣤⣤⡈⠉⠻⠦⠀⣠⣽⡾⠟⠋⠀⠀⢾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⡶⠟⠛⣽⣿⡄⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠉⠁⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣤⡶⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣠⢶⣄⠀⣀⣀⣶⣄⠀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣤⡀⢠⣤⠀⠀⠀⣀⣤⣤⣼⠟⢙⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡏⠀⢸⡏⠀⠙⣿⡏⠙⣯⡛⢸⣿⡦⠀⠀⠀⠀⢠⣿⣿⡟⠙⣿⣿⣿⡄⠀⠀⠀⠘⠛⣿⠈⣡⣶⣦⣼⠟⠁⢸⡏⠀⢸⡟⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⢹⣇⠀⢸⡇⠀⠀⢸⡇⠀⠸⣧⠘⠋⠁⣴⣶⠆⠀⠘⣿⣿⣷⣄⣿⣿⣿⠀⠀⠀⠀⢶⣶⣄⠀⣿⠋⣾⠃⠀⢀⣾⡇⢀⣿⠁⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠠⣿⡀⠘⣷⠀⠀⠸⣧⠀⠀⠹⣧⣠⣄⢿⣯⡾⠛⢻⡞⠿⠛⣿⣿⣿⠃⠀⣠⣴⣶⣬⡙⢻⣧⣿⣿⠃⠂⢀⣾⠋⢄⣼⠇⡀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢹⣇⠀⢹⣇⠐⠀⢻⣆⠀⠀⠹⢿⡿⢈⡿⠟⠛⠋⠁⠀⢠⣿⣿⠁⠀⢀⠿⡵⠟⣿⣯⣿⡿⠟⠁⢀⢉⡾⠃⠘⣼⠟⠀⢁⣾⢏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⢻⣆⠀⠹⣧⡀⠀⠻⣦⡐⠄⠈⠻⣦⣜⣛⠰⣶⣦⠀⠀⠻⠿⠀⠀⠀⢀⣴⡾⠛⠉⠁⠀⠀⠀⠀⠩⠔⠃⠾⡏⠂⣰⡿⠃⣁⣾
⠀⠀⠀⠀⠀⠀⣀⠰⣀⣽⣧⡀⠻⣦⡀⠈⢿⡄⠠⠙⠿⠆⠀⠀⠀⠈⠉⠛⢶⣍⡀⠀⣸⣷⡄⢀⣴⠟⠁⡁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡸⠯⣙⣦⡾⢣
⠀⠀⠀⠀⠀⠒⣠⡿⠋⡀⠈⢿⣤⡉⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠐⠘⠿⠃⠸⠃⠀⠀⠀⠀⢠⣤⣤⣤⣽⣿⣿⣿⣿⣯⣤⣵⣾⣾⢟⡛⢿
⠀⠀⠀⠀⠒⢋⣘⠿⠶⢶⣦⣦⣈⠿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣼⣿⣿⣽⣽⣿⣿⣽⣿⣯⣿⣾⣿⣷⣾⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠁⠀⠀⠈⠙⠋⠉⠀⠀⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟
⠀⣀⣤⡀⠀⣀⣀⣀⣀⡀⢀⡀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⣠⣀⠀⠀⣀⠀⠀⠀⢀⠀⠀⣤⣤
⣾⠁⠀⢹⡄⠀⢸⡇⠀⠀⣾⣧⠈⢷⡀⠀⢠⡏⠀⠀⠀⠀⠀⡇⠀⣿⠀⡞⠁⠈⣧⠀⣿⡄⠀⠀⣿⡇⠀⡇
⢿⡀⠀⠀⠀⠀⠀⡇⠀⢠⡇⢹⡄⠈⢷⣠⡏⠀⠀⠀⠀⠀⠀⡇⠀⢿⢠⡇⠀⠀⣻⠀⣿⢻⠀⢸⢿⡇⠀⡇
⠈⠛⢦⣄⠀⠀⠀⡇⠀⣸⠗⠚⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡷⠒⢿⠈⡇⠀⠀⣿⠀⣿⠘⣇⡿⢸⡇⠀⡗⠒
⠐⢧⣀⡿⠀⠀⠀⡇⠀⡟⠀⠀⠸⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢻⣄⡴⠃⠀⣿⠀⢹⡇⠈⡇⠀⣧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠰⢁⠠⠤⠤⢄⡀⢀⣀⠀⠀⣀⢀⣄⠀⠀⠀⡇⠀⢀⠀⡀⢠⠀⣠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⡇⠦⠯⠜⠚⠈⠻⠥⠂⠀⢰⠀⠀⠢⠜⠗⠁⠙⠥⡾⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⠤⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠑⡆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⡐⠁⠀⠀⠤⢆⠀⡔⠈⠀⠀⠪⠉⠆⡀⠀⣠⢈⣑⡈⠁⠁⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⠤⠤⠚⠣⠀⠀⠢⡀⡄⠙⢂⠀⢀⠄⠀⠀⡼⠦⣩⠂⠀⠀⠈⠑⠠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁⠀⡀⠀⠀⠀⠱⡀⠀⠀⢉⡈⠸⡒⣉⠀⠀⡐⠀⠀⡄⢀⠂⠐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⣔⠙⢯⣿⡁⠴⠦⠺⢀⠆⠰⠀⡶⢀⠼⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠄⠘⠓⠒⢲⣆⠀⣇⠠⠄⠐⠈⡉⠤⠋⠑⠈⣏⣀⡀⠀⠀⠈⢠⠂⠀⡜⠓⠢⠠⠤
⠀⠀⠎⠩⢃⠤⠒⢢⠀⡊⠀⢈⣦⠤⢐⢀⠛⡄⠀⠀⠅⡀⠀⠀⠀⡈⠐⠒⠒⠐⠑⢎⣀⡀⠀⠀⡔⠂⠒⠈⠁⠂
⠀⠀⠀⠀⠈⠁⠊⡡⠊⠉⠋⡔⢿⠀⠣⡀⡠⠂⠀⠀⠀⠀⢂⠀⡌⠀⠀⠀⠀⠘⠓⣉⣁⣵⠂⠀⠡⠠⡀⢀⢀⠇
⠀⠀⠀⠀⡠⠤⠘⠄⣀⣀⡠⠋⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡌⠀⢂⡀⡠⠀⠠⠤⠊⢡⠊⠉⠣⡀
⠀⠀⠀⡅⠀⠖⠦⠵⡧⠞⠀⠀⠀⡠⠂⠉⠉⠐⠂⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠱
⠀⠀⠈⠢⠃⠀⠀⠀⠁⠀⢀⠤⠐⠀⠠⠘⠈⠁⠓⢆⠀⠀⠀⠀⠠⠐⠁⡉⠐⢤⠀⠔⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠀⢠⠁⣰⣿⣿⣷⡀⠁⠀⠀⠠⠁⣰⣿⣿⣷⡄⠁⠈⠂⠄⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⠑⢄⠁⠂⠀⠐⡄⠙⢿⣿⠿⠁⠄⡠⠠⡐⡄⠙⢿⣿⠿⠁⠜⠒⠄⡀⡠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢨⠑⢄⡁⠀⠀⡤⠈⠂⠤⡄⠔⠊⠀⠀⠀⠀⢈⠖⠄⠀⠔⠊⠀⢤⢄⡀⣀⠔⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠉⠉⠀⠈⠥⣃⠀⠈⠒⠢⢄⠀⠀⠀⠈⠀⠀⠀⠀⠘⡫⠄⠀⠀⡀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠱⣄⣮⠽⠁⠀⠀⠀⠀⠀⢡⠀⠀⠰⡀⠀⠀⠀⠈⠹⠋⡦⠊⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠒⠠⠀⢲⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠈⡄⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠄⠊⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⢴⡀⠀⠀⠀⣀⠬⠄⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠦⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠒⠲⢶⣄⢀⡀⠈⢳⣽⠀⠀⢀⡴⠞⠉⠉⠉⠙⠲⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠠⠄⠀⠒⠒⠶⣬⣷⡈⠳⣌⣿⠀⣴⣫⠴⠒⠛⠛⠛⠒⠂⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⢿⣦⣸⣿⣼⠿⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠉⠉⠉⠉⠉⠛⠛⠓⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣁⣀⣀⣠⡤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣚⠒⠀⢀⠄⣀⣀⣀⢈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠁⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⢀⡀⠀⠀⠉⠙⠲⡼⠒⢿⠽⣟⢿⡍⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠈⠁⠻⠟⠃⠀⠀⠀⠀⠷⣶⡝⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣽⣶⡿⠂⠦⢦⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⡄⠀⣠⡀⠀⣤⠶⣞⠛⢠⡾⠁⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⣀⣼⢯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⡾⠁⠹⠼⠁⠀⠩⡷⠛⠀⠀⠀⠀⢀⣀⠀⣸⠃⠀⠀⠀⠀⣠⡴⠿⢧⣿⠗⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣷⣿⠇⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⢠⠏⠉⠉⠁⠀⠀⣠⠴⠾⠃⠀⠀⠈⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⣹⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⠼⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⡟⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠀⣾⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢄⡏⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⣀⡀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠻⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⣄⣉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣌⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢺⣿⣿⣿⣿⠉⣹⣿⣿⣿⣿⣿⣍⠲⢄⠀⠀⠀⠀⠀⠹⣷⣄⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⣴⣾⣿⣿⣡⣶⣿⣿⣿⡏⣬⠋⠀⠀⠀⠑⣄⠀⠀⠀⠀⠈⢿⣦⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠸⠋⠈⣿⣿⣿⣿⣿⣿⣿⣷⠾⠇⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢻⣧⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣄⠀⣶⡤⠀⠀⠈⢇⠀⠀⠀⠀⠀⢿⡆⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢹⡇⣠⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠾⠿⣿⣿⣷⡆⠀⣀⠀⡾⠙⣆⠀⠀⠀⠘⣿⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠁⠀⠸⣆⠉⣿⡇⠀⡟⡆⠀⠀⠀⣿⡀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡿⠀⢠⣧⠇⠀⠀⠀⢸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠘⣿⡿⠈⠉⠻⣿⣿⣿⣿⣿⣿⣄⡀⣀⣀⠀⠀⠀⠀⢀⣾⠇⠀⢸⡟⠀⠀⠀⠀⢸⠀⣿⣿⡇
⠀⠀⠈⠣⡀⠉⠉⠉⠒⠢⢤⡀⠀⠀⠀⠸⣿⠀⠀⢹⣿⡄⠀⠀⠘⠻⢿⠏⠘⣿⡏⠹⣿⣿⣇⠀⠀⠀⣾⡏⠀⠀⣼⠃⠀⠀⠀⠀⡞⢸⣿⣿
⠀⠀⠀⠀⢱⠀⠀⠀⠀⢄⠀⠈⠢⡀⠀⠀⠙⡆⠀⠀⢿⣿⣦⠀⠀⠀⣸⠀⢠⣿⡇⠀⢿⣿⡏⠀⢀⣼⡿⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⣄⠀⠱⡀⠀⠀⢻⡀⠀⠀⠉⠛⠣⣄⡼⠋⠀⢸⣿⡇⠀⠘⢿⣧⠴⠛⠉⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⢧⡀⡧⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠘⢷⡇⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢰⠇⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠦⢤⣀⣠⠤⠚⢿⣦⡀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⣠⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡇⠀⢠⡇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⡇⠀⠀⠀⠀⠀⡇⠀⢸⠃⠀⠀⠀⠀⣸⣧⣴⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣶⣶⣤⣷⣤⣾⣶⣶⣶⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣒⡀⠈⠩⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣂⣀⠀⠠⠔⢹⣧⣤⣤⣤⠤⠤⠐⠒⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠉⠉⠉⠉⠐⠒⠒⠶⠿⠟⢿⠁⠀⠀⠀⠰⠂⠉⣀⣀⣀⣀⣠⣤⣶⡡⢂⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣴⣶⣶⣤⣴⠦⠐⠂⠉⠉⠻⣷⣄⣀⣀⣴⣿⠏⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⢀⣀⣤⣿⠛⠛⠛⠙⣧⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢿⣯⣭⡀⣁⣀⣀⣀⣀⣁⣠⡤⠟⠳⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠘⣀⣠⡴⣿⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣛⣛⠛⠛⠀⠀⠀⠀⠀⠐⠂⣀⣉⠭⢉⣵⡟⠀⠀⡠⠒⠉⠉⠉⡉⡉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠻⢿⣿⣷⣶⣶⣶⣶⣶⠶⠾⠿⠿⠟⢹⡇⢀⡎⢀⢔⡩⠤⠤⢷⡔⠄⠀⠻⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡌⣿⣼⠁⠝⢋⣔⣈⡑⠢⡈⢦⠪⠀⠁⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⢆⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰⡘⠿⣦⣤⡿⠁⢄⠉⢢⡈⢢⢣⢀⣠⠴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣡⢊⠎⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠑⠒⠃⠀⠀⢢⠣⠀⢳⠀⡎⡯⠐⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠀⠈⠉⠁⠈⠉⠁⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢃⠈⡇⠀⠵⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡞⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠘⠀⡇⡆⡂⣹⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡨⠓⣩⣴⡶⠶⢶⢶⣶⡼⡔⠀⠁⣼⡇
⠀⠀⠀⠀⠀⠀⢸⢠⠀⡇⠀⠀⠀⠀⠀⠀⢀⠤⣓⡀⠭⠍⠐⠛⠩⠤⣀⣀⠀⠀⠤⠄⠀⠀⠤⣰⡪⠥⠒⠁⠙⣵⡘⠂⠪⢲⢶⢟⠄⢀⣴⠟
⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠀⠀⠀⠀⠀⢀⡔⠉⠀⣠⣤⠶⣶⣤⡀⠀⠀⠀⠈⢉⣉⣉⠁⣾⠛⢦⠀⠀⠀⣠⠶⡌⢮⡠⡀⠀⠁⢂⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠸⠘⠀⡆⠀⠀⠀⠀⡄⠸⠀⠀⣼⠋⠀⠀⢸⠏⢻⡄⠀⠀⢠⡿⠉⠙⣷⠈⢷⣼⡇⠀⠀⣿⡞⠁⢀⣌⣙⡛⠒⣉⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⢀⡀⡅⠄⠀⠈⣿⡀⠀⠀⠘⠓⠚⠁⠀⠀⣿⠁⠀⠀⢹⡇⠈⣿⡀⠀⠀⣿⠃⠀⣸⠏⠉⠉⠛⠛⠳⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠻⠿⢶⣶⣦⣤⣶⡶⠇⠀⢀⡈⠻⠶⢶⣦⣤⣄⡀⠀⢰⣿⠷⠶⠶⢿⡇⠀⢸⣧⠀⢠⡿⠀⢀⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢿⠙⣧⠀⠀⠀⠀⠉⢿⡄⣸⣿⠀⠀⢀⣿⡇⠀⠀⣿⡀⣼⠃⢀⣼⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⢻⣆⠀⠈⠳⣿⣦⣤⣤⣤⣴⠿⠁⣏⣹⠀⠀⢸⡇⣳⠀⠀⠘⠿⠋⠀⢾⣧⣤⣤⣖⢚⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠄⢻⣇⠀⠀⠀⠀⢉⠉⠉⠀⠀⠀⠈⠁⠀⣀⣀⣙⣩⣤⠄⢠⣤⣤⡀⢀⣤⣤⣌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠄⠀⠘⡄⣿⡄⣤⡀⠀⢸⣷⠀⢰⡟⠀⢀⣿⡆⠉⠉⢻⡏⠀⠀⢸⣏⠀⠀⠀⣿⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡕⠁⠀⠀⢱⢸⡇⠈⢷⡄⢸⣿⡆⣼⠃⠀⣾⣉⣷⠀⠀⢸⡇⠀⠀⢸⡿⠶⠆⠀⣿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠔⡡⠀⠀⡌⣸⡇⠀⠈⢿⣼⡟⣷⣿⠀⣸⡟⠉⢿⡄⠀⢸⡇⠀⠀⣼⡇⡀⠀⠀⣿⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⠰⠀⠐⣡⡿⠁⠀⠀⠀⠻⠇⠹⠏⠀⠿⠀⠀⠈⠁⠀⠈⠃⠀⠀⠙⠛⠁⠀⠀⠋⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⡶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠈⠁⠀⠉⠈⠒⠉⠱⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠠⠤⣄⣀⡠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡠⢤⠤⠤⠎⠉⠉⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⢠⡴⠠⠤⠔⠖⢂⣀⡀⠀⠀⠀⡔⠲⠖⠒⠒⣄⠀⠀⠀⠀⠀⠀⠀⠓⠠
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⡠⠟⠀⠀⠀⠀⠀⠀⠙⣾⣿⢿⠒⠁⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠓⡄
⠀⠀⠸⣁⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣀⣀⣄⠀⠀⢻⠉⡨⠧⠤⢤⣤⡀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠘⢄⠀⠀⠀⠀⣰⣾⣯⣀⢀⡼⠛⠿⢡⠊⠈⢹⠀⠀⠀⠀⢨⠻⣛⠀⠀⢀⡼⠻⠇⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡎⠀⠀⠀⡤⠋⠁⠉⢏⠁⠣⠀⣘⠤⠦⠐⠓⠒⠚⠂⠴⠥⣎⠀⣡⠒⢊⠄⠢⢤⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠈⠉⠑⠐⠁⠀⠀⠀⠀⣣⠔⠩⢔⠒⠒⠂⣰⠂⠀⠀⠀⠀⠀⠉⠣⢤⠃⠀⠀⠀⠑⠢⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣿⡿⠀⢈⡠⠒⠈⠁⠀⠀⢠⠔⣭⣈⠻⠒⠊⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠷⠃⣈⣱⠀⠀⠀⠀⠀⠐⠲⣷⠜⠛⠛⠁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⡤⠚⠋⠀⠀⠀⠀⠀⠀⠀⢲⣣⢒⡺⠦⣤⡟⠛⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠒⠓⠺⡇⠀⢠⣀⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠶⠤⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⠤⠃⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠚⠓⠲⢄⡀⠀⠀⠀⠀⠀⠢⣀⡀⢀⠀⠀⠀⠀⠀⠈⢩⠂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠓⠂⢤⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⡔⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠅⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢀⠆⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⠃⠀⠀⠎⠀⡀⠀⠀⠀⠀⠠⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠈⠀⠍⠉⠉⠛⠓⠴⢮⣀⣀⡎⠀⡔⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠙⠳⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡄⠀⠉⠢⡄⠀⠀⠀⡠
⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⣠⠖⠚⠲⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⢦⡠⠊⠀⠀⠈⢀
⠀⠀⠀⢈⠀⠀⠀⠀⠐⡁⣀⠂⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡠⠶⠶⠦⡄⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⡜⠀⠀⢀⠀⠘⠳⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠒⠶⢤⣤⣄⣀⣀⡐⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠢⢀⣸⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠱⣄⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠐⠒⠂⠒⠒⠊⠁⠀⠀⠀⠀⢙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠁⠂⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⢦⡀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠔⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⠖⠁⠀⠀⠀⠠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠂⠀⠀⠀⡉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢢⡀⠀⠀⣰⡧⠖⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⡑⢄⠈⠙⠛⠲⠶⢦⣤⣤⣿⣤⡄⠠⠙⠛⠉⠁⠀⠀⡀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠂⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠈⠉⠁⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⢀⣀⣴⠶⠶⠞⠁⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠈⠉⠉⠀⠀⠉⠑⠒⠢⢀
⠀⠀⠀⠀⠀⠀⣠⣾⣷⣿⣦⣤⣄⠘⠻⠛⠁⣶⡾⠿⣧⠈⢻⣿⣿⣿⣆⠀⠀⢀⡤⠊⠀⠀⣀⠄⡄⠀⢤⠀⣠⠀⡄⢀⡀⠀⠑⢄
⠀⠀⠀⠀⠀⣼⣟⣿⣿⣿⣿⣿⣞⣿⣿⠟⠛⠉⠀⢶⣶⣤⣀⠸⣿⠋⠀⢣⠰⣋⠀⠀⠀⢢⡀⠀⠇⡀⠘⡆⠃⠀⠃⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣟⣷⣿⣿⣿⣷⣶⣠⣶⣿⣿⠟⠋⠃⠙⠀⠀⠀⡆⠈⢂⠀⠐⠦⠤⠸⠄⠹⠄⠳⠄⠀⠣⠤⠄⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠙⠃⠈⢿⣿⡿⣟⢿⣿⣾⣿⠿⢿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⡄⠀⠀⠀⢰⠀⠴⠀⢰⠀⠤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⢁⠤⠤⠤⠶⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠀⠀⠀⠸⠀⣀⡀⢸⠀⠄⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣆⠀⢴⣶⣴⣶⣤⣤⣀⡀⠀⠀⡆⠀⠀⠀⠡⡀⠀⠀⠒⠉⠒⠘⠒⠒⠀⠠⠐⠊
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠈⠓⠈⠀⠈⠉⠉⢠⣾⣿⣿⣿⢿⣿⣿⣿⠁⡼⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣿⠿⠉⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⢄⣀⣀⣀⣀⣀⣀⠬⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠀⢠⣼⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠈⠀⠠⣬⣾⣿⣿⣛⣿⣽⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⣼⢉⣈⠉⢉⣩⣿⣿⣯⣷⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⣵⣿⣿⣿⣿⣿⣿⣷⣍⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠹⣿⣿⣿⣿⡷⣰⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⢻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡾⣿⣦⡀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠋⠀⠀⠀⠀⠀⣠⣾⡿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⣿⣦⣀⠀⠀⠈⠛⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡈⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣴⣿⣿⣿⣩⣩⣛⣻⣹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣯⣭⣿⣯⡽⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⡿⠿⠀⠸⠿⣿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣶⣽⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⡙⢿⣿⣟⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣀⢸⣎⠿⠻⢏⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⢠⡛⡉⠁⣶⡶⣄⡀⠒⠾⢼⣧⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⣻⣿⡇⢀⠀⠘⠄⠈⠀⢶⣶⣿⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢫⡿⢃⣿⣇⠀⠀⠀⠀⣠⣧⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠁⠀⡜⠙⠃⢺⣄⣉⣀⣀⣈⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣤⣤⣶⣶⡿⣻⣿⣿⣭⡙⠛⠂
⠀⠀⠀⠀⠆⠁⠠⣤⣤⣤⡶⠾⠿⠻⢻⣻⣿⣵⣾⣾⣏⣾⣿⣷⣶⣄⢠⣴⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⣖⠈⢻⠹⣛⣲⣷⣿⣿⣿⣿⠿⠟⠿⠛⠏⠛⣀⠙⠻⣅⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⠀⠀⠀⠀⠀⠐⡵⠈⡠⠤⠀⠀⠁⠈⠁⠀⣠⢄⣄⢰⣠⡃⠀⠀⠈⠳⣄⠀⠀⠀⠉⡒⠀⡀⠀⠀⠀⠀⠉
⠀⠀⠈⠀⠀⠀⠀⠀⠘⢀⠈⠀⠢⡀⠒⣻⣶⣿⣿⣷⣯⡽⠿⠃⢱⠀⠀⠀⠀⠈⠛⣤⡀⠀⠰⡄⠈⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡀⠀⣐⠢⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢛⣤⡙⢿⣶⡄⠀⠈⠉⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠛⣤⣾⣿⠃⠀⠀⠀⠀⢀⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⣿⣧⣶⣿⣿⣟⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠄⠀⠀⠀⠀⠀⣀⣤⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀⣶⣦⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣄⣰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣮⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠏⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠺⢿⣴⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠙⠻⣷⠄⠁
⠀⠀⠀⠀⠸⣶⡄⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢠⣶⠇
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠹⣿⣿⣿⣿⠂⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⣶⠀⠀⠀⠀⠀⣃⣤⡾⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣶⣄⠘⢿⣿⡏⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣠⣴⣿⠟⠁⣠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠙⢦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣶⣾⣿⠟⠁⣠⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣬⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣤⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠱⠀⠪⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠁⠘⠓⡣⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠤⠂⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡞⠢⡀
⠀⠀⠀⠀⡔⠊⠀⠉⠉⠂⢄⣄⡀⠀⠀⠀⠈⠒⠄⠀⠀⠠⠀⠔⠀⠀⠀⠀⢀⠤⠊⠄⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⡀⣠⠤⠀⠔⠙⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢈⠀⠈⠉⠰⣀
⠀⠔⠀⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠉⠂⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢃⠀⠠⠐⠀⠀⠀⠁
⠀⠀⠈⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠠⠎⢠⠤⢀⣃⠚⠀⠀⠀⡀⠀⠉⠀⠠⡠⠀⠀⠸⣉⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⠀⠀⠴⠂⠂⠋⠀⠀⠀⠢⠀⠐⠀⠈⠁⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⢀⣠⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠅⣀⠸⠃⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠠⠀⣠⠄⢁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠐⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠁⠀⠀⠃⠀⠀⠐⡀⠤⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⢠⠒⢢
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠀⠀⠨⣌⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠑⠨⠀⠀⡜⠀⠁⠀⢀⡠⠔⠐⠀⠈⠒⠂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⠃⠀⠐⠊⠁
⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠒⠚⠄⠀⠀⠀⠾⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⡈⠀⠀⠀⠀⢀⡀⠑⠂⠀⠀⠀⠀⢀⠤⠐⠀⠀⠒⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠂⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⢀⠆⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠩⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⡀⠀⠀⠀⠈⠁⣸⠁⣌⠀⠀⠀⠀⠈⠇⢱⣠⣄⣀⠀⠈⠆⠂⠁⠀⢠⠀⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠰⠙⠀⠛⠀⠈⠁⠁⠀⠀⠀⠀⠀⠃⠁⠁⠀⠜⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⡆
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠉⠀⠀⡀⠀⠀⠀⠀⢀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠰


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠆⠀⠀⠰⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠖⡂⠀⡀⠀⠀⠀⠀⠀⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⣠⣤⣿⣆⡀⣶⣆⡀⠀⠀⢀⡇⠀⠀⠈⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣾⡿⣻⣿⣿⣿⣿⡿⠿⣛⣿⣛⠛⠻⠷⢦⣥⣠⠋⠀⠀⠀⠀⠚⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠊⢁⠀⢙⣿⣿⣿⣿⣛⠛⢻⣿⠿⠛⠋⠀⠀⠀⠀⣨⣝⣷⣌⣾⠁⢀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⠗⣹⣯⡀⠈⠁⠀⠀⠀⠀⢴⣄⣹⣿⣿⣿⣿⣿⣷⡔⠀⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣷⡿⠍⠃⠀⠀⠀⠀⠀⠀⣄⡿⠿⡿⣿⡟⠿⣿⣿⣟⢶⠆
⠀⠀⠀⠀⠀⠀⢰⣿⡧⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣷⣶⣀⣉⠙⣻⣿⣽⣿⡎⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⢹⠛⢉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠁⠀⠁⢲⡀
⠀⠀⠀⠀⠀⠀⣸⣟⢼⣿⠘⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⠻⠿⠿⠿⠿⠓⠃
⠀⠀⠀⠀⠀⠀⢩⡏⠀⣿⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠈⠻⠿⠛⢿⣿⣿⣿⣿⡟⠀⢸⡏
⠀⠀⠀⠀⠀⣴⣿⠇⠀⡸⣇⠀⠸⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡏⠀⢀⣿⢡⢇
⠀⠀⠀⠀⠀⠉⠉⣠⣾⣿⣿⣧⡀⠙⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢣⣦⡾⣡⠟⠀⢣
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣷⣄⣻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⢁⣴⢟⣴⠋⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠈⠋⠕⣉⢿⠿⠓⣸⠻⢷⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⡡⠺⠥⠤⠆⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⢠⠇⠔⣠⠞⠉⣛⣛⣻⣟⣛⡛⠋⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢌⡴⠁⠀⠀⢸⡿⠿⢿⣿⠃⠀⢠⠋⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠦⠋⠀⠀⠀⠀⠐⠂⠤⠴⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠧⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣰⡇⣴⣼⡧⣴⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣾⠻⠿⠞⣿⢏⠙⢫⣾⣿⠰⢟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡊⣙⢫⡈⢿⣿⣶⣿⣷⡎⢫⣵⣠⣿⡷⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⢻⣽⣧⠿⢘⣿⢰⡏⣛⡝⣞⠓⣻⣏⡍⠉⣥⣼⡯
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣶⡁⣝⠛⢳⡟⢻⣟⣶⡿⡁⣿⡾⠷⣾⣧⣠⢿⡏⢡⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⢻⣶⡾⢻⡞⠛⠶⣯⣽⣷⣽⣿⡶⠛⠉⠻⠅⢿⣧⡈⠻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣞⣭⣿⣿⢛⡓⠲⢤⣶⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣡⣼⣿⣿⣿⡿⢎⣉⣀⠀⠀⠈⠻⣿⠏⠡⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣿⣿⣿⡿⠏⠈⠉⠙⠛⠛⢀⠀⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢢⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⣿⣿⣿⣿⣿⣦⣴⣦⡄⠀⠀⠀⢀⣼⣿⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢸⣿⣿⡇⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⣯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠹⠿⣿⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠹⠿⣿⣿⡿⠿⢿⡿⣿⡿⣿⣿⣿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣶⣶⣶⣶⠟⠉⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠉⣯⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣤⠀⠀⠲⣆⠀⠀⠀⠀⢀⣠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⡏⠀⠀⢿⡄⠾⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⣷⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠉⠛⢦⣤⠀⠀⠀⠸⠧⠀⠰⠟⠀⠀⠀⠀⠘⠦⢀⣀⡼⠟⠀⣴⠟⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⡻⠷⣦⡄⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠐⣏⠀⠀⠀⣠⠛⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⣏⡀⠀⠁⢀⡴⠞⠒⣹⡿⠆⠒⠀⠉⣩⣭⣥⣀⠒⠛⠿⣄⠀⢀⡀⠀⠀⠑⠂⠚⠁⠀⢀⣼⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣠⠶⠻⡄⠊⠀⢠⠜⣋⠀⢤⠔⠁⠀⣤⣀⡀⡀⠛⣟⡁⠛⠻⢿⣶⣶⣔⡇⠈⠂⠀⠀⠀⠀⠠⣴⠟⢡⡀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠁⣀⠔⠀⠀⡴⠛⠉⠁⠀⣰⣤⣴⣶⣬⣿⣷⣾⢦⡞⠙⣷⠀⠈⣿⣿⣿⣧⡀⠀⠀⠈⢦⠀⠀⠀⢀⡼⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢠⠞⠀⠀⢀⡤⠞⠻⣿⣿⣿⣿⣿⣿⣯⣀⠀⣾⣦⣄⠀⠈⠛⠉⠈⠙⠠⠄⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣿⣿⡛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡠⠆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⢀⡄⠐⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠻⠷⠴⡄⠀⠙⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣿⣿⣷⣤⡀⠀⠆⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠸⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣹⠁⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠨⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠁⢀⠃⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣤⣀⣀⣀⣀⣟⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⣿⡿⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣿⡅⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢤⡀⠀⠀⠀⠀⠀⠀⣼⢿⡄⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠐⠁⠀⠀⠀⠀⠀⠀⠏⠘⣷⠄⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⢦⣄⠀⣶⠆⠀⠀⠀⠀⣤⠀⠀⠰⣦⡀⠀⣶⠄⢴⡦⠀⢦⠀⡶⢴⡦⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠴⠟⠀⣿⠀⠀⠀⠀⣜⢹⣇⠀⠀⡏⢻⣄⣺⠀⢸⡧⣾⠈⠀⠁⢸⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⣿⡀⢤⣯⣼⡅⠀⣿⡄⢠⣧⠀⠙⠿⠀⢸⣇⣈⣼⠁⠀⢸⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⣤⢀⢠⡸⠀⠀⠀⠴⠀⠀⠀⡆⡀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠀⢤⠋⠰⠧⡶⡀⣰⠀⢤⠏⢠⡕⠋⠀⠲⠞⠱⠂⠘⣒⡃⣼⢀⠐⡆⢃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⢠⡄⠲⡜⡁⡬⠴⠀⠙⢓⣃⢬⢱⢃⠛⣂⡘⠋⢠⣆⣄⠉⠁⡄⠈⠥⠄⢋⡰⢌⡭⠀⠀⠐⠦⣩
⠀⠀⠀⠀⠀⠀⠀⣄⢈⠩⡀⣡⢣⠜⣅⠒⠃⠀⣄⠈⠀⠚⠩⠜⡄⠨⠙⣑⠒⠂⠀⡀⠳⢡⢄⠀⠳⣩⠘⢎⡴⠀⡄⢈⡓⣥⠄⣤⠰⠃
⠀⠀⠀⢀⡀⠀⠐⠚⣘⢳⡄⠆⠒⣁⡐⡫⡀⠀⢛⡺⢃⡀⠰⣄⡰⠐⠆⠀⢠⡤⠬⠁⠐⠦⠘⠀⢛⡅⢘⢢⠖⣄⠵⠂⠡⢖⡗⠀⠫⠜⣁⠀⣌⠃
⠰⡀⠀⠶⣡⠄⠀⠀⠀⠐⢃⡤⢛⢰⢈⠀⢉⣤⠨⡍⠘⢀⠲⠛⠀⠀⠀⠰⠤⠶⠀⢖⠀⠰⢘⠉⣤⠀⢈⢣⠖⠉⣰⡳⠦⡈⢉⠀⠀⢠⣹⠴⡎⠛⠁
⠒⢶⠆⠓⢃⠶⠀⠵⠖⣃⣤⠀⠉⠘⠉⠒⣠⠀⢀⠋⣀⠛⠲⠚⠁⠀⠘⢃⡋⠈⢶⡉⣀⢳⡉⠒⣀⠀⣈⠳⠢⠖⠀⠀⣆⠯⠄⠀⠀⢰⡁⢷⡘⢋⢀
⠀⢛⣢⠈⣁⠘⣁⠀⠈⠡⠔⠸⢀⡀⠀⢆⣠⠘⠛⣀⠠⣜⠈⢩⡀⠀⣄⡬⠀⠘⠚⢰⠨⠤⠀⠦⠉⠀⢈⡐⠚⠀⡀⡦⠇⢶⠼⠀⠀⢷⠤⠍⡅⣸⡈
⠀⠀⠀⠚⢡⠞⠐⠆⡆⣤⢀⠋⠘⠃⡄⢠⠍⠰⢀⠉⠤⠀⡀⠠⠇⠃⣡⠂⠀⠈⢓⠀⠛⠀⠐⠆⠋⠀⢈⠍⠀⣀⢳⡄⠇⠒⠂⢀⢹⡌⡵⠠⣥⠉⠁
⢀⡲⢄⡓⣨⢓⡌⠣⠆⠰⠎⡧⢑⡼⢀⠰⠐⠒⣸⠠⡄⠀⠀⠲⣄⠈⢹⣆⠀⢠⡆⠀⢀⣠⠔⠡⣄⠋⠛⠂⠀⠈⢃⠈⣹⠀⠴⢼⡄⠐⠒
⠈⠁⠈⠀⠁⠀⢀⡀⣈⠁⣻⠰⡆⢦⠤⡄⡒⡄⠀⠈⠓⠲⢦⣤⣼⣿⣶⣿⣷⣿⣤⣶⣿⡥⠶⠒⡀⢤⢠⠄⢀⡝⠃⡙⠀⢶⠚⢢⡀
⠀⠀⠀⠀⠀⠀⠈⠱⠊⠵⢠⠆⡆⠼⠘⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠈⠁⠹⠀⢦⠚⢥⡋⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠓⠉⠀⣿⣿⣟⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠰⣏⣤⣉⣥⣿⣿⣿⡌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣾⣦
⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠉⠉⠀⠘⢿⣿⣿⡷⠄⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣤⠀⠀⠈⢿⣷⣦⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁⠀⠀⠀⢠⣤⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⡀⠀⠙⠿⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⣸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⢿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠤⢤⡤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠁⠈⣤⠄⠀⠙⠀⡀⡀⠂⠆⠀⠜⠃⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠈⠀⢀⣀⡨⠡⡤⠰⠀⠄⠁⠀⢀⠀⠀⣈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡁⠀⠀⠀⠀⠀⠁⠈⡁⠀⠤⡔⠀⠀⠀⠀⠓⠐⠀⠂⠁⠀⠲⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣰⡆⡠⠄⣀⠀⠀⠀⠈⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢀⠧
⠀⠀⠀⠀⠀⢠⣀⡀⠀⠐⢀⡜⠀⣁⡀⠀⠈⡂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠃⠀⢀⢀⣳⣾⣧⠀⠠⠤⠄⠒⠒
⠀⠀⠀⠀⠀⠀⡀⢌⠁⠒⠒⠧⢀⡙⠛⠮⡀⠑⠀⠀⠠⢀⠀⡂⢄⠀⡀⠀⠔⠀⡠⠂⢀⠴⠓⠉⠀⠀⠀⠀⠀⠀⠊⠘
⠀⠀⠀⠀⠀⠀⡇⢀⡱⡀⠀⠀⠀⠀⠉⠒⠽⣇⢜⠚⡀⢁⠈⠂⠕⠈⡀⠐⠐⣡⢾⠖⠁⠀⢀⠀⠀⠀⠀⠀⡐⠁⡊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠈⠑⢮⡂⠑⠈⡂⠀⠄⠦⠰⣋⠖⠁⠀⠀⡎⠀⡑⠀⠀⢀⠊⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⢈⠢⡀⠀⠀⠀⠀⠀⠀⠙⢄⠀⡪⡰⢘⢅⡢⠉⠠⡀⠀⠘⡄⠀⢃⣄⠔⠁⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠙⢀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢢⠞⠀⠇⢀⡜⠀⠀⠈⢛⠟⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠍⠀⠀⠀⣑⢄⠀⠀⠀⠀⠀⠀⢹⣴⠋⠀⠀⠀⠉⠀⠀⠠⠂⠁⠀⠀⠀⠄⣀⡠⠃⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠠⠀⠀⠂⢀⡹⠦⡀⠀⢀⠀⠀⡇⠀⠀⠀⠀⣠⠐⠁⠀⠀⠀⡀⠄⡥⠖⠉⠀⠀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡢⠚⠉⠉⠉⠉⠙⠓⠒⠂⠤⠤⠄⠤⠖⠛⠈⠉⢙⢂⣀⡤⠖⠉⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠇⡠⡸⠁⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⡀⠀⠀⠕⠊⠀⠀⠀⠂⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠁⠀⠐⠀⠐⠒⠀⠀⠀⠀⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⢐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⢠⠞⠁⢸⠉⠉⠉⣉⣉⠉⠉⢉⣉⣉⣉⣉⣉⡉⠉⠉⢉⡉⠉⢉⡉⠉⠉⠉⠉⠉⣉⢻⠈⠳⠄
⠀⠀⠀⠀⠀⠀⢸⠀⣰⡿⠛⠛⠻⣦⠙⠛⠛⣿⠛⠛⠛⠀⢀⣿⣿⡀⠈⢿⣆⠀⠀⢀⣾⠏⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠸⣧⣄⣀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⣼⠏⠹⣧⠀⠀⠻⣧⣠⡿⠃⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠛⠿⣦⡀⠀⠀⣿⠀⠀⠀⣸⣟⣀⣀⣻⣇⠀⠀⢹⣿⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣆⣀⣀⣠⣿⠃⠀⠀⣿⠀⠀⢠⡿⠉⠉⠉⠉⢿⡆⠀⢸⣯⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠛⠀⠀⠛⠁⠀⠀⠀⠀⠈⠛⠀⠈⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⢰⠀⠀⣠⠔⠒⠲⢤⡀⠀⢰⣄⠀⠀⠀⣰⡆⠀⡖⠒⠒⠂⢸
⠀⠀⠀⠀⠀⠀⢸⠀⣇⣀⣀⣀⣸⠀⣸⠁⠀⠀⠀⠀⢱⠀⢸⠘⢆⠀⡼⠁⡇⠀⣇⣀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⢸⠀⢹⡀⠀⠀⠀⠀⡼⠀⢸⠀⠈⠛⠀⠀⡇⠀⡋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠘⠀⠀⠙⠲⠦⠖⠊⠁⠀⠘⠀⠀⠀⠀⠀⠃⠀⠳⠶⠶⠆⠸
⢀⠤⠤⣄⠀⠀⢀⡄⠀⠠⡀⠀⠀⠀⡄⢠⠤⠤⠄⠀⠀⠀⡀⠀⠀⠀⢠⠀⡄⠀⠀⠀⡄⢠⠤⠤⠤⠀⢠⠤⠤
⠻⣀⡀⠈⠀⠀⡞⠘⡄⠀⢳⠀⠀⡼⠀⢸⣀⣀⡀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠹⡀⠀⣰⠁⢸⣀⣀⡀⠀⠣⣀
⡀⠀⠉⢳⠀⡼⠤⠤⢵⡀⠈⢇⢰⠁⠀⢸⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢣⢠⠇⠀⢸⠉⠉⠁⠀⡀⠀⠉
⠉⠒⠒⠊⠐⠁⠀⠀⠀⠃⠀⠘⠃⠀⠀⠘⠒⠒⠒⠀⠀⠀⠙⠒⠒⠂⠘⠀⠀⠈⠋⠀⠀⠘⠒⠒⠒⠀⠙⠒⠒
⣀⣀⣀⣀⣀⣀⣰⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣔⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡮⠷⡤⠴⠞⠋⠁⠀⢰⡂⣠⣶⣤⣾⡖⠀⣤⣄⠄⣴⣒⡏⣵⢒⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢃⣀⣀⣀⡶⣄⠀⠀⠓⠛⠋⠁⠋⠛⠁⠀⠛⠋⠚⠙⢘⡐⠋⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⢁⠉⠙⡷⠘⠷⠀⠀⠀⠀⠀⠠⣞⢀⣾⣦⣼⣷⠁⡜⢰⣰⣦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⡽⠡⣄⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠋⠉⠘⠘⠉⣐⣁⠋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢠⣤⢯⣼⣇⡄⢈⠉⠀⣀⣤⣶⡷⣦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⢤⣿⠸⣿⣟⣳⣾⠐⢚⣩⡾⠋⠀⠀⠸⠸⢿⠆
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣌⠫⣦⣼⣯⣭⡝⠛⠉⠁⠀⠀⠀⠀⠈⠛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣽⣦⣬⣽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⠀⠀⣿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⣀⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⣠⠃⠰⡧⠒⠐⣭⣸⣛⢟⡛⣧⣄⡶⠒⠲⢄⠀⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠊⠁⡠⠟⠀⠀⡰⠉⠀⢹⡟⠉⠛⢿⣷⡄⠀⠀⠑⠷⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠊⠀⠀⠀⠜⠀⠀⠀⠀⠼⢄⠀⠀⠈⠻⣧⡀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⠀⠀⡰⠊⠀⠀⢀⠎⢀⠀⠀⠙⢆⠀⠀⠈⠙⠷⠶⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠀⠐⢊⣄⠀⠀⡔⠁⠀⠀⠑⢦⠀⠀⠑⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣡⠤⢞⡂⠀⠠⠗⠀⢀⣁⣤⣤⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⢄⡤⠠⠎⠉⠦⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠁⠀⠈⠀⠀⠀⠀⠀⠐⠓⢢⠤⠄
⠀⠀⠀⠀⣠⠒⠖⠉⢲⠒⠚⠉⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢦⠤⢄
⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⣁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠘⠋⠉⠚⠑⣄
⢀⣾⠉⠁⠀⠀⠀⠀⠀⢀⣀⡜⠉⠁⠃⠉⠹⣄⣠⡄⡀⣢⠴⠉⠉⠙⠙⠉⠛⡆⠀⠀⠀⠀⠀⠀⢀⡕
⢏⡀⠀⠀⠀⠀⠀⠀⠠⡚⠀⠀⠀⠀⠀⠀⠀⢙⡿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠀⠀⠀⠀⠀⠀⠈⠦
⠀⠁⠀⠀⠀⠀⠀⣀⠠⠁⠀⠀⢀⣀⣾⡿⠚⠒⠑⠀⢉⠀⢀⠐⣶⣤⣤⡀⠀⢠⡇⣀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠦⠀⠀⠀⠉⠛⢧⠀⠀⢠⠀⢸⠀⠀⠀⢸⠏⠉⠳⢦⣶⣿⣿⣷⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⡠⠤⠓⠂⠉⠉⠉⠙⠒⠂⠾⣀⣀⠀⢠⠟⠀⠀⠈⠓⡀⢀⣀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⠚⠁⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢉⡶⠏⣠⣇⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠉⠑⠲⢤⠇⠘⣿⠿⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠉⠑⠾⡟⠒⠀⠀⠀⠀⠀⢠⡄⠀⠈⠾⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢻⣦⣴⣟⣒⣄⢑⠂⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⢨⠟⠋⠒⠋⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡆
⠀⠀⠀⠀⠀⠀⠘⠉⠻⣍⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠋⠀⠀⠀⠀⠀⢀⡀⠀⢀⠟⠀⠀⠀⠀⠀⢀⣠⠴⠲⠶⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⢀⠉⠈⠉⠁⠀⠀⠀⢀⡠⠴⠿⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣒⣀⣤⣠⣔⣒⣊⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿