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•.,¸,.•*`•.,¸¸,.•*¯ ╭━━━━╮
•.,¸,.•*¯`•.,¸,.•*¯.|:::::::::: /___/
•.,¸,.•*¯`•.,¸,.•* <|:::::::::(。 ●ω●。)
•.,¸,.•¯•.,¸,.•╰ * >し------し---J
⠤⠂⠉⠀⠀⠀⠀⠀⠉⠉⠓⠦⣜⡄⠀⠤⠠⠀⠀⠀⠀⠠⠤⠄⢸⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻
⠀⠀⠀⠀⠀⣀⣀⣀⣄⣤⣤⣤⣿⣶⠄⠀⠹⡄⢀⡤⠔⠛⣁⣤⣤⣤⣤⣤⣼⣻
⣤⢴⡶⠛⣋⣉⣵⣤⣤⠤⠤⠼⣿⣿⣟⠳⢶⣗⣤⢶⣞⣏⣉⣀⣤⣄⣼⣿⣿⣿
⠉⠈⠙⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⣾⡿⠟⠉⠉⠀⠀⠀⠀⠀⠈⠙⠛⢻
⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠁⠀⠀⠀⠛⠶⢤⡀⠤⠤⠄⠲⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠻⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼
⣴⠶⠶⠶⠶⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢿
⠀⢹⠴⠶⠦⣤⣄⣀⣀⣈⠉⠉⠛⠛⠓⠒⠦⡤⠤⢤⢤⡤⡴⠶⠚⠛⠉⠁⢀⡽
⣤⣤⣤⣤⣄⣀⣀⡈⠉⠉⠉⠛⠒⠒⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠖⠒⠒⠛⣏⣻
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠒⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣤⣞⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⣿⣿⣿⣿⣿
🖤✔️⚫️▪️
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⠃⠛⠋⠛⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠙⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣷⣀⣹⣷⣶⣶⣦⡌⠳⣄⠀⠀⠀
⠀⠀⠀⢀⣤⣾⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⢷⡀⠀
⠀⠀⢰⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⢻⡄
⠀⠀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠿⠛⠛⢛⣛⣿⢿⣿⣿⣿⣿⣷⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣠⠄⢽⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠈⠻⣟⢿⣿⣏⠀⠘⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿
⢻⣿⡿⠛⠛⠿⢿⣿⣿⣿⡿⠿⠿⣄⡀⠊⠙⣿⣿⡃⠀⠀⢀⣀⣀⢀⣀⣀⣀⣤⠤⠤⠼⣤⠈⠃⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⣽⠀⠁⠀⢀⣀⡀⠀⣀⣀⣀⣲⣾⣿⡿⠀⢹⣿⣷⠚⠉⣹⣿⣿⣹⡍⠉⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠳⣶⠒⣻⣿⣿⣉⠉⠉⠀⠀⠀⠀⣿⡀⢸⣿⡹⣆⡀⠙⣿⣿⣿⠇⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣄⠙⣿⣿⣿⠇⠀⠀⠀⠀⢠⡏⠀⢸⠟⣇⠀⠱⣄⠀⠈⠀⠀⠀⣀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿
⠀⠀⠀⣇⠙⠳⣀⠉⠁⠀⠀⢀⣠⡶⢛⠀⠀⣸⠀⠸⡄⢸⠀⠙⠛⠒⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠈⠑⠒⠶⠒⠛⠉⠐⠛⠀⢀⠃⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠈⠀⠀⠀⠸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠂⡹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣼⣶⣤⣤⣤⣤⣦⣴⡇⠈⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⣿⣄⠀⠀⠀⣀⣺⣆⡌⢠⣿⣯⣏⣩⣿⠿⠛⠉⠁⠀⠈⠉⠛⢳⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⠛⢶⣤⣼⣿⣿⣿⡇⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⣀⣤⣴⠟⠁⠈⠙⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⡿⠋⢆⢀⠀⠀⠀⠀⣤⣤⣶⣶⣾⡿⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⢀⣀⡀⢄⣀⣀⣀⠀⠠⠀⠒⠀⠈⠉⠛⠩⡉⢂⠑⡄⠀⠐⠒⠂⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢢⠀⠀⠀⢀⠄⢊⠜⠋⠉⠁⠀⠀⠀⠀⠀⡠⠒⠉⠀⡠⠃⠑⠺⢄⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⣄⠈⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠀⠀⠘⠰⣆⠀⠀⠀⢢⠀⠀⠀⠂⠤⢄⡀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠊⡰⠃⠀⠀⡆⠀⣏⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠈⠁⠣⣔⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢁⠀⠀⠀⠀⠀⢀⠄⢀⠞⠀⢆⡜⠁⠀⠀⢰⠅⡀⠹⡇⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠦⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⢋⡔⠁⠀⠀⠀⡰⢀⠎⠠⡝⠀⢸⡜⠀⠀⠀⢀⢃⢡⢣⠀⢣⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢐⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⡎⡝⠀⠀⠀⢀⡞⠁⡞⢠⡝⠀⠀⣿⠃⠀⠀⠀⡜⡌⡜⡼⣆⠈⢣⠀⠀⠀⠀⠈⠀⣀⣀⠀⠀⠀⠠⠄⠂⢁⠔⠋⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⡜⡌⠀⠀⠀⢒⠞⢲⠸⢧⣿⠁⠀⢀⡟⠀⢀⠆⢠⠡⡰⡇⡘⢿⣦⡀⠣⡀⠀⠀⠀⡆⢢⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠀⢠⠁⠀⡌⡄⣎⡗⡡⡌⠀⠀⡼⠇⠀⡜⠀⡬⢤⠧⣇⢱⠀⠑⠱⢄⡈⠢⢄⠀⠇⢢⢣⠀⠀⡠⡪⢲⡆⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⢠⡇⠀⡜⡠⠐⢹⡌⠀⡇⠀⢰⢸⠀⢀⠇⡷⡆⠸⠀⡇⢩⠂⢄⠀⠀⠈⠁⠀⠀⠁⡎⡆⡤⢊⠔⠀⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⣞⠁⢰⣯⣤⣤⡘⢅⠀⡇⠀⡄⡀⠀⢸⢠⢃⢀⡇⠀⣠⢨⠀⡇⠁⡆⠀⠀⠀⠀⠀⣇⠏⡗⠁⠀⠀⠀⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢞⢰⢸⠀⡘⢈⣶⣶⣾⣵⠀⢳⡇⡇⡇⠀⢸⡎⣌⠸⠥⡀⠉⡄⠀⡇⠀⠀⠀⠀⠀⠀⠚⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⡏⢸⠀⡀⣇⢸⠉⢻⡽⢟⠁⠘⢷⢻⡅⠀⣺⣿⣭⣭⣿⣰⢤⣱⠀⠃⢀⡀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡏⠀⣇⡿⡦⣃⡘⣁⠎⠀⠀⠈⢻⠱⡀⠘⠛⠿⣟⣻⣿⡷⡝⣷⠀⣸⠀⠀⢸⢀⢲⠀⡆⢰⠀⠀⡇⠀⢠⡇⠀⡸⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣀⡏⣇⢞⢼⠉⠑⠈⠀⠀⠀⠀⠉⠻⢄⣆⠸⣻⠟⠁⠹⠅⣸⢀⡋⠀⠀⡎⡸⡈⠰⠀⡛⠀⢠⠇⢀⢳⠁⢰⢁⢼⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⢠⠎⠉⠐⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⢹⠊⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣘⠢⢤⣠⡔⣎⠀⡇⡌⠃⠀⡸⢠⢣⡇⢆⢠⠁⢠⢻⠀⣌⣸⠀⢃⠎⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⢸⣆⠀⠀⠀⠀⢸⠃⠀⠀⢰⠉⠑⠤⠃⠀⡇⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢲⣿⡟⣾⣾⡦⠱⣸⣿⠘⠀⣰⡡⠁⠸⣇⢀⠇⠀⠎⡌⣼⡪⢷⠸⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠲⠤⠄⠀⠎⠀⠀⠀⠈⢆⢄⠀⠀⢠⠃⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠉⡝⠚⠿⠃⠆⡝⡝⡆⡴⠟⢁⠂⢀⢋⣬⢀⢊⡜⡼⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣣⡘⡀⠀⡎⢀⡧⠂⡏⡰⢱⠊⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠤⠐⠂⠈⠋⡏⢱⢠⣿⡇⣬⣀⠀⢷⠁⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠀⠀⡇⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠁⣠⡔⠛⢆⠰⡼⣣⠀⠀⠀⠀⠀⠀⡠⠊⠀⡇⣿⣾⠛⠳⠇⠶⣭⣺⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠴⠿⡷⡀⠘⡄⢻⡿⠆⠀⠀⡠⠔⠁⠀⠀⣀⣿⣽⠟⠠⠠⠐⠂⠙⠋⣁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠱⡹⡄⢱⠈⣿⠘⡆⠁⠀⠀⣠⡖⠟⠋⣉⠠⠄⡀⠒⠂⠀⠋⠁⠩⣃⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢵⣿⡄⡇⢹⠘⠀⢀⣴⠟⢉⠤⡚⠭⠐⠈⠀⣀⠀⠠⠤⠀⠐⠂⠉⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠸⡀⠀⠀⣸⣿⣿⣤⢸⠀⡴⠋⣠⡾⢛⠭⠔⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠐⠒⠤⢄⡀⠀⠀⠐⡀⡵⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠉⢻⣰⠊⣠⠾⠓⠉⠀⢀⡀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠒⠂⠤⠤⢐⣣⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠸⣿⠡⠀⠀⠉⠉⠉⠉⠉⢩⣿⢼⡟⣅⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠠⠀⠀⠀⠉⢢⡀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⠁⠀⡄⣿⠀⠀⠀⠀⠀⠀⠀⢠⡿⣹⠀⢰⠸⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁⠀⠀⠀⣀⠔⠊⠁⠈⠉⠒⠠⠔⠊⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡇⠀⠀⠀⢸⠀⠀⣧⠁⠀⠀⠀⠀⠀⠀⢀⢧⠃⣿⠀⠈⡀⡇⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡜⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡇⠀⠀⠀⠘⠀⠀⣋⡇⠀⠀⠀⠀⠀⠀⡸⡜⠀⢻⠀⠀⡇⢣⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁⠀⠀⠀⡀⠀⢰⡿⡇⠀⠀⠀⠀⠀⠀⡇⡇⢰⢸⣰⠀⢱⢸⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡇⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⣷⠁⠀⠀⠀⠀⠀⢠⢻⠀⡘⢸⣷⣀⠼⣼⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢐⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⡇⠀⠸⣯⠀⠀⠀⠀⠀⠀⠸⡾⠿⣿⡟⢽⠁⠊⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠤⠤⠼⠀⠀⠀⢸⠠⠤⠤⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡼⡇⢸⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠆⠀⠀⠀⡆⠀⠀⠀⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡜⠀⠀⠀⠀⠸⡀⠀⠀⠀⠣⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣞⡘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢈⠓⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡠⠊⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠠⡀⠑⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠇⠀⠀⣸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⡠⠊⠀⡰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡄⠸⠀⢠⢀⠂⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠃⠀⠀⠀⠀⠀⡠⠊⠀⠀⡰⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠄⠀⢣⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠄⠀⠀⠘⠈⠄⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠜⠀⠀⠀⠀⠀⡠⠊⠀⡠⢀⡌⠀⡠⢠⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⠄⠀⠀⢇⠈⢆⠰⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⠠⠀⢁⠔⡙⠀⠌⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠀⠈⠀⠀⠘⡄⠀⢂⢣⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠂⠁⠀⡠⠂⠰⠁⠌⠀⠀⠀⠀⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⠀⠀⠀⠀⠀⢇⠣⠀⠊⠆⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⠃⡜⠀⠀⠀⠇⢜⠠⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⢸⠘⡀⠡⡀⠨⡄⠀⡘⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⡌⠰⠁⠀⠀⢠⠌⡜⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⢃⠀⠀⠀⠘⡄⢃⢀⠘⢄⠈⢲⠁⠀⠀⠀⠀⠄⠊⠀⠠⠐⠈⠀⡸⢠⠁⠀⠀⡠⠊⠰⠀⠀⠀⠀⡗⡄⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⠈⡄⠀⠀⡀⣇⠘⡀⠑⢄⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⢃⠂⠀⠀⠔⠁⡰⠁⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⢀⠔⠁⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠄⠘⡀⠀⠀⡇⢀⠰⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡠⠊⢀⠔⠁⠀⠀⠀⠀⠀⡇⠀⠘⠄⢀⠔⠁⠀⠀⠀⢈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢂⠀⠀⠀⢸⡄⢂⢁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠊⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⠀⠠⠠⡈⠅⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⢸⢁⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠑⡈⠄⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠺⣏⠳⣄⠀⠀⠀⠀⠀⠀⣘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⠀⡇⠉⠺⣛⢦⠤⠤⢶⣟⣽⡙⡵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠪⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠁⠈⡟⣻⡿⠑⢼⢞⢢⡀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⡂⠄⢀⠀⢀⣼⢰⠋⠀⠀⠀⠑⠍⣚⡷⠶⢲⡶⠶⠶⢿⡛⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢪⠋⠀⠀⢠⣖⣢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢓⠢⠤⠤⣬⢞⢆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠒⠀⣡⣾⠘⣦⣕⠤⡀⠀⠀⠀⣶⡆⠀⣶⡦⠀⠀⠀⠠⢐⣤⢶⢾⢿⡔⠁⠀⠀⣰⡟⠀⠀⠈⠁⠘⠲⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⢰⢰⢡⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡱⢡⠊⠁⠀⠁⡈⢆⠀⢀⣀⡀⠀⠉⠀⠀⠀⣀⡴⠛⢹⠀⠀⠸⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠈⠑⢏⠡⠶⠤⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⠈⣼⢁⠂⠄⢀⠙⢢⣀⣀⣀⣀⣀⠀⠀⠠⠤⣧⠇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⢆⠴⠛⠁⠀⠀⡏⠀⢠⠃⢒⡮⠋⠁⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣿⣰⠂⠄⠎⣠⡏⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠁⢀⠊⠀⠀⠀⠀⢘⡌⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡜⣼⣿⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠁⡴⠃⡀⣀⢄⡐⠠⠅⠒⠒⠂⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠳⣠⠄⡀⠀⠀⠀⣀⠀⡀⠀⠐⠒⠊⢉⣉⣻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊⢁⠩⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⠤⡤⠥⠔⠒⠒⠒⠒⠈⠁⠀⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⢤⡤⢶⠒⣰⣨⣧⡧⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⣦⡷⢟⣫⢷⡝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠄⠂⠀⠀⢀⡠⠜⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢁⣀⣀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⣀⡀⠀⠀⠀⢀⣈⡰⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢠⣶⣾⣿⠀⢸⣿⣷⣶⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⡆⠀⢸⣿⢸⣿⣿⣿⡇⢸⣿⣿⣿⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡀⠀⣿⡇⠀⢸⣿⢸⣿⣿⣿⡇⢸⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⡆⠀⠀⠀
⠀⢀⣾⣿⠀⣿⡇⠀⢸⣿⢸⣿⠟⠛⠃⢸⣿⠻⠿⠀⢸⣿⢿⣿⠀⣿⡇⠀⠀⠀
⠀⣼⣿⡿⠀⣿⡇⠀⢸⣿⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡟⠀⠇⠀⣿⡇⠀⢰⡀
⢠⣿⠉⠃⠀⣿⡇⠀⢸⣿⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣧
⢸⣷⣀⠀⠀⣿⣷⣶⣾⣿⢸⣿⣿⣿⠀⢸⣿⣿⡆⠀⢸⣿⣧⠀⠀⣿⣧⣀⢸⣿
⢸⣿⣿⣆⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⠀⢸⣿⣿⡇⠀⢸⣿⣿⣇⠀⣿⣿⣿⣿⣿
⠀⠻⣿⣿⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⠀⣿⣿⣿⢿⣿
⣀⠀⢹⣿⠀⣿⡟⠛⢻⣿⢸⣿⠛⠛⠀⢸⣿⠛⠃⠀⠀⠘⢿⣿⠀⣿⡇⠀⢸⣿
⢸⣧⣸⣿⠀⣿⡇⠀⢸⣿⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⢸⡟
⠀⢿⣿⣿⠀⣿⡇⠀⢸⣿⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢀⡀⢸⣿⠀⣿⡇⠀⢸⠁
⠀⠘⣿⡿⠀⣿⡇⠀⢸⣿⢸⣿⣦⣤⡄⢸⣿⣤⣶⠀⢸⣷⣿⣿⠀⣿⡇⠀⠀⠀
⠀⠀⠈⠀⠀⣿⡇⠀⢸⣿⢸⣿⣿⣿⡇⢸⣿⣿⣿⠀⣾⣿⣿⡿⠀⣿⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠛⠇⠀⢸⣿⢸⣿⣿⣿⡇⢸⣿⣿⣿⠀⢹⣿⣿⠃⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠿⢿⣿⠇⢸⣿⡿⠿⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀