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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠐⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣤⠼⠗⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠔⢺⡇⢠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠿⣭⣉⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⡇⢸⠀⠀⢀⣀⣀⣤⣤⡴⠶⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣶⡇⢸⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣤⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾⣿⠀⠘⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠾⠛⠋⠉
⣀⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣿⣧⣶⣶⠶⠟⠛⠛⠉⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡶⠿⠛⠋⠉
⠙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠦⠤⣄⡀⠀⠀⢸⣿⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⠿⠟⣿⡏
⠀⠀⠀⠀⠉⠉⣛⡛⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠀⠈⢉⣑⣺⣟⣤⣶⣾⡿⠿⠟⠛⠉⢹⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠸




⢀⢀⢀⠀⡀⠀⡀⢀⣀⡀⢀⡀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⣀⢀⠀⣀⡀⢀⡀⣀⡀⠀⡀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⡀⠀⢺⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡈⠈⢽⡟⠋⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠿⠿⠿⠿⣿⣿⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠓⢦⣀⠀⠀⠀⠀⣠⠞⠙⡇⠀⠿⠿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢈⡷⢦⣀⠔⠧⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣔⣋⣀⣀⣀⣀⣀⣀⣁⣷⠿⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⣡⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠋⣡⠔⠋⢸⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣛⣡⠴⠋⠀⠀⠀⢸⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⣛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢟⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⡾⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢋⣉⣉⣉⡉⢻⣧⣿⡏⠉⠉⠭⠭⠭⠭⠭⠭⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⣽⠀⠒⠂⠀⠀⣼⢻⣿⠃⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠭⠭⠍⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢤⡿⢓⡿⣿⡇⠈⠉⠁⠀⡿⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠊⡰⠋⡜⣹⡇⠀⠀⠀⢸⡇⣽⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⡉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣠⣾⣥⡾⢳⣿⡃⢀⠊⣄⣼⣵⣿⣇⠈⠄⠐⡰⢠⢀⢀⠀⡀⢀⠂⠀⣀⣀⣀⣄⣀⣄⣀⣀⡀⠀⡀⢠⠤⢤⠤⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⣰⢋⣾⣧⢂⠜⣿⣿⣿⣿⣿⠂⠈⡴⡱⢣⠫⠂⠔⢠⠃⡜⢠⠋⠀⠌⠀⠬⢤⠦⡴⠄⠴⠰⠂⢰⠒⠲⠓⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⢡⠎⣼⡇⠈⡰⣿⣿⣿⣿⣿⠃⡜⡰⡱⢡⠃⠌⠠⢃⠜⢠⠃⢀⠎⢀⠀⠀⠁⠌⢀⠌⡐⠉⡘⠉⣞⣁⡞⣓⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠰⢣⠞⣹⡇⠀⠀⠈⠉⠉⠉⠀⢘⠞⡐⠡⢁⠌⡰⢁⠎⠠⠁⢀⠊⢀⠃⡰⢁⠊⠠⠂⠄⢀⠌⢀⠂⢠⠮⢤⢬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠠⠋⡰⢻⡇⠀⠀⢀⣀⣀⣀⣀⣈⣀⣀⡁⣈⠀⡀⠀⠀⠁⠀⠂⠠⠃⠜⡠⠃⡰⢁⠊⠀⠂⠀⠁⠐⠁⣠⢊⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡼⢁⣾⡇⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠈⠀⠁⠀⡠⠁⡐⠁⠀⢀⠊⢠⠖⣽⣻⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠾⣆⣠⠃⣿⡿⢋⣽⣿⡆⠐⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠴⠯⡽⢧⢯⠼⢥⠏⣹
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣚⣫⣭⡤⠶⠶⣛⣛⣻⣭⣤⣾⣿⣿⡷⠶⠶⠿⠿⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣯⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣷⣶⠿
⣀⣤⣤⣶⣾⡿⠿⠟⣛⣛⣭⣭⠵⠶⠖⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣧⢻⠎⡿⢹⣿
⢿⣯⡥⠴⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣥⣇⢿⡜⣷⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣏⣾⡰⣧⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡋⢸⢡⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡇⣸⢁⡟⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⡇⣹⢊⡟⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⡇⢸⠈⡗⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⢰⠊⡷⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⡇⢰⠞⣥⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡃⡇⢀⣸⠃⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⡇⢸⢸⠏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⡴⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢇⢰⢸⣡⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⢸⢁⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⢸⠏⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠁⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣈⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⣤⠀⢀⡠⠤⠭⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⠔⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠖⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠏⢰⠁⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡎⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⡊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠷⠶⠶⠭⠭⠤⠤⠤⠤⠤⠤⠀⠨⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⠤⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠰⣅⠅⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠰⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠐⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⢤⡤⡀⢠⢄⢠⠂⣄⠤⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠀⠈⠁⠃⠘⠜⡀⡀⠁⡀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠤⠉⠾⣆⢶⣰⣃⣓⢇⣉⡡⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠉⠉⠉⠀⠀⡄⢀⣀⣠⠤⠴⠖⠒⢻⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⡇⢸⠁⠀⠀⠀⠀⠀⠸⣿⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠧⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠇⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠒⠒⠒⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠁⠂⠠⠊⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⡘⣇⠀⣀⣀⣠⡇⠀⠀⠀⠀⣀⠄⠊⠁⠀⠈⡆⠀⠀⢀⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⠤⠚⢀⣹⠋⠉⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⢀⡤⠔⠛⠀⢀⡆⠀⢄⠀⢱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠚⠁⠀⠀⠀⠀⠀⠀⠑⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠙⢬⡉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⢖⣂⢈⠭⡝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠸⠀⠀⠀⠀⣀⡤⠚⠉⢀⣿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠞⠁⠐⠄⢉⡼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢺⠁⣀⠀⢀⠞⠸⡏⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣄⢀⣄⠀⡶⡄⠐⠀⡇⠀⠀⠀⠸⠀⢰⠀⠀⠈⢇⠈⠉⢉⣠⡴⠛⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠉⢹⢠⡥⢧⠀⡆⣇⣀⡦⠀⠀⡇⢸⠀⠀⠀⠈⠳⠚⠉⠀⠳⠶⠖⠲⠶⠞
⠀⠀⠀⠀⠀⡇⠤⠤⠤⠀⢀⠀⣀⡀⢸⡇⠸⠀⠀⠘⠏⠀⠈⠁⠁⠉⠀⠀⠀⢀⣃⠜
⠀⠀⠀⠀⠀⡇⠦⠤⠄⠀⢀⠀⠀⠇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠈⠁⠀⢀⡠⠤⠔⠒⠒⢹⢿
⠀⠀⠀⠀⠀⠻⠥⠥⢤⣴⣆⣤⡬⠀⣸⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣀⣠⣔⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⢧⠀⠀⠀⢀⡇⢈⣀⣀⣀⣉⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⢊⣨⠇⠀⠀⠈⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠖⠒⠒⠒⢚⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢄⡀⢀⣠⠒⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠋⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⣀⣀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡄⠀⠘⠀⢠⠀⠔⠁⣠⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣬⣡⡄⠀⠀⠀⠀⠈⢡⠴⠂⠁⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢀⣰⠗⠒⠒⠒⠚⠃⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡉⠉⢦⣤⡀⠀⢸⣿⡏⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠳⡱⡀⢸⣿⡷⠤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢣⢣⠈⡟⡀⠀⠀⠀⠈⢳⣶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢯⡉⠉⠙⠛⠛⠛⠛⠿⠿⠿⢿⠀⠀⠸⠸⠀⠀⠂⠀⠀⠀⠀⠀⢳⢳⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠢⣄⠀⠀⠀⠀⠐⠛⣿⠂⠄⠀⠀⡄⣴⣿⣭⢀⠀⠀⠀⠀⠈⡇⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠳⢤⡀⠀⠀⠀⠉⢀⡇⠀⠀⠀⠈⠻⠤⠊⠀⠀⠀⠀⠀⡇⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠉⠲⣤⣀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡤⠊⠀⠀⠀⠀⠹⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠏⣀⣀⡀⠀⠀⠀⣀⣡⡼⠷⢿⠿⠿⠷⠶⠶⠶⠶⠶⠦⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣶⣶⣶⣶⠶⠶⠒⠋⠉⠀⣀⣤⣼⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⠤⠤⠤
⠀⠀⠀⠀⠀⡠⠔⠚⣉⣡⠤⠜⢛⣛⠿⠿⠛⠛⠉⠁⠀⠀⡀⠠⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠘⡗⠒⠚⠉⠉⠉⠉⠀⠀⠀⣀⣀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏

⣶⣶⠿⣷⠶⢶⣶⠶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⣀⠀⢸⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡈⠀⢸⡇⠀⠀⠀⠀⠀⠄⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠸⢤⡾⠃⠴⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⠀⠀⢠⢠⣰⢠⣀⢀⣆⣠⢀⣄⣠⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠹⣿⣿⣿⡏⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⣿⣿⠀⠉⠠⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣷⠊⢹⣿⠀⢠⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢈⠇⢼⣿⠀⢨⠀⠀⢀⣥⢐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡧⣠⣿⢷⣿⣿⠀⢸⡤⣴⠿⠀⠒⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣱⣿⣿⣷⣿⣿⡇⢸⣿⣧⠐⢦⡄⣤⠈⠀⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡾⢛⠁⠐⣾⣿⠿⠇⢸⣿⡿⠿⠋⠀⠀⣀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠆⣿⠇⠀⠀⠸⣯⣖⠄⠀⠀⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠭⢟⣺⠦⢹⡀⠀⠀⠈⠩⣵⣾⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠓⠀⠀⠀⠀⠈⠒⠾⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠾⠿⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣌⣿⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠠⠶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⡄⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⢤⠄⠀⠀⠀⣿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠷⣤⡀⠀⠒⠒⠂⠀⠒⠲⠶⠶⠶⠶⠶⠶⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⠷⣍⡉⢉⣛⣋⣛⣃⣀⣀⣀⡀⠀⣀⣤⠾⠛⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⢶⣌⡉⠉⠉⠉⠉⣉⣵⠾⠋⠁⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⣐⣾⣿⣿⡩⢽⣓⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠐⠂⠈⠁⠠⠒⠒⠉⠋⠉⠀⠁⠀⠁⠀⠁⠀⠉⠲⢄⠈⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⢀⣤⠖⣒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⡀⢠⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣊⣑⡉⢉⡈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠂⠈⠙⡎⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⢏⣵⣛⣫⣍⠽⠗⣎⣶⣤⡀⠑⢄⠀⠀⠀⠀⠀⢸⠁⠀⢀⡄⠀⠀⠀⠀⠀⠐⠓⠂⡀⠌⠈⠀⠀⠀⠀⢀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡯⡄⣿⠏⣓⣀⢄⣑⣢⣆⡛⡎⡆⠈⠆⠀⠀⠀⡀⢸⠄⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⠰⡘⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡥⣐⣟⠛⢭⡀⠀⠀⠢⡢⢂⢦⡇⠀⢸⠀⠀⠀⢐⣼⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢁⢃⠢⢁⢡⣇⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣗⢀⣿⠀⠀⠉⠑⠦⣄⣁⡑⣿⠃⠀⠀⠀⠀⠀⢀⢽⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⡈⠈⡄⠌⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣧⣨⣿⣆⠠⠤⠀⠒⠉⠅⡐⠚⠀⠀⠀⠄⠀⠀⢐⡾⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠠⠑⠸⢠⢰⢇⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠐⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠨⢽⢠⢐⠀⡆⠀⠀⠠⣇⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣨⠃⠆⠂⢅⢃⠂⢳⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠉⠑⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⢽⢨⠐⠂⡇⠀⠀⠀⢿⡇⠸⠒⠢⡄⠀⠀⠀⠀⠀⠀⠠⠑⡀⠂⡈⡘⠌⠄⡈⢌⢌⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⠙⠓⠲⢤⣤⣀⣘⣼⠠⠄⡇⠀⠀⠀⠽⣇⠀⠮⠀⡘⠀⠀⠀⠀⢀⢀⢃⢓⠰⠰⠱⠠⡘⠐⠈⠉⠼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠘⣿⢀⡂⠀⠀⠀⠀⢨⡿⢦⣤⡴⡑⡀⠀⠀⠀⠌⠆⢄⢢⢁⢀⢣⢡⢠⠣⠹⠰⢸⣃⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⣿⡐⠂⢰⠀⠀⠀⢜⠌⢼⠋⠑⡐⡐⡜⡌⡌⡌⠌⠆⠆⠂⢀⣀⠡⠤⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣼⢿⣿⣆⠀⠀⠀⢀⠎⠀⠀⠀⠬⡀⢿⡇⡁⠸⠀⢀⣀⢂⢊⢂⠠⠰⠐⢐⣈⢀⡤⠄⠐⠂⠉⠉
⠀⠀⠀⢀⣀⣤⣤⡶⡞⡟⠛⡏⣉⣠⣤⣼⠾⢧⡀⢠⠎⠀⢀⣐⣢⡵⢶⣿⠏⡯⠻⣈⣎⠣⠤⡧⢖⠖⡚⡩⡣⠡⢪⡆
⢠⢔⡾⠟⠛⠁⣀⣧⣧⠷⣖⡛⠟⠉⠁⠐⠀⠀⠑⠷⠶⣚⠛⣉⣧⣿⡭⠟⢣⢙⡾⠁⠀⠀⠀⡗⡡⡪⢊⢎⠔⡕⡡⡇
⡇⣏⢀⡴⠿⠛⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⡇⣇⡧⠚⠉⠀⠀⠀⠀⠀⡯⣪⢔⡵⡫⢊⢔⠼⠁
⢛⠪⡺⢷⣷⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣒⢭⠗⠋⠃⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⢚⠔⡡⡢⢺
⠘⢦⣈⠉⠐⠛⠿⡭⢶⢶⢶⠦⠿⠻⠛⠉⢡⢀⣸⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠐⡡⢊⢔⢽
⠀⠀⠀⠉⠒⠒⠢⠧⠴⠼⠼⠴⠴⠒⠚⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢒⠵⡫⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠵⡪⢊⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣯⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⣭⣶⣾⣿⣿⣦⣄⠀⠀⢨⡟⠦⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣾⣽⡿⣿⣿⣿⣿⣿⣿⣧⠀⠰⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⡇⢹⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠉⠉⠛⠻⢿⣿⣿⢸⡇⡆⢸⡇⣿⣷⣶⣤⣤⣀⣀⠈⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⢹⢸⡧⣇⣸⠇⣿⢿⣿⣿⣿⣿⣿⣷⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠸⡆⢱⢀⠀⠀⠈⠂⡀⢸⡀⣿⣿⣿⠀⣿⠘⠋⢻⠛⠛⢻⡏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢷⠸⡜⡆⠀⠀⠈⠙⠋⠁⣿⣿⣿⠀⣟⣤⢀⢀⠀⠀⢀⣃⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢷⣸⣤⣧⣷⣄⣀⣀⣒⣒⣂⣿⣿⣿⣠⣽⣿⣨⣾⣷⣾⣩⣯⣾⣴⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⡿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⠃⠀⣿⡿⡇
⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠃⠀⠀⠉
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠁
⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠞
⠀⠀⠀⠀⠀⠀⠑⠶⢤⣄⣀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠶⠶⣶⡖⠒⠲⠶⠤⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣡⠴⠒⠒⠒⠓⠲⢤⡉⠳⡀⢄⢤⡀⢀⠦⠠⣍⡳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⣆⠊⠙⠋⠁⠐⠀⠿⠌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣬⣤⡤⠤⠶⠖⠒⠒⠒⠒⠓⣆⢇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡗⠛⠛⠋⣉⣉⣉⣉⣩⣭⡿⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠀⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢰⠖⢤⢰⠛⢾⡙⠀⠉⡇⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠄⢸⠆⠚⢸⠀⡇⠓⢆⠀⡇⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⡀⠀⠘⠤⠃⠒⠛⠀⢷⠀⢸⡇⡿⠀⠀⠀⠀⣀⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢸⠀⣀⣀⣀⣠⣤⡤⠤⠤⠤⣤⣤⣾⢀⡇⣶⣴⡾⠿⣛⡽⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣼⡯⠟⢒⣒⣒⣛⣛⣉⣉⣉⣉⣩⣭⣽⣿⣿⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢸⡇⠉⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⣤⣿⡀⢸⢸⣄⠀⡴⠀⣦⡀⣤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢿⠷⡼⠳⣿⠻⡟⠳⡟⣾⠉⢷⠠⢻⠇⠙⣿⠳⡏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠸⢿⣿⣿⣿⣿⠵⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠉⢹⣿⠀⠀⠀⠀⠉⠉⠛⠒⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣦⠀⠀⣿⢳⡀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠷⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢸⡇⠀⡟⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⡀⠀⠀⠀⠀⠀⠙⠦⠴⠟⠀⠀⣿⠀⠀⠙⣆⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣀⡴⠞⢧⣀⡤⣾⠋⠁⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠈⠉⠀⠘⣆⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣀⠀⠀⠀⢿⣷⣦⣄⠀⠀⠀⠀⠘⣆⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠳⢦⣀⠸⣿⣿⣿⣿⡶⣤⡀⠀⠘⣆⠀⣸⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠉⠓⢿⠛⠉⠁⠀⠀⠉⠳⢦⣹⣦⡏⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣠⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢦⡀⣀⡤⠖⠉⠀⠀⡈⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠘⠀⠙⠁⠀⠀⠀⠀⢠⣀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⡆⠀⠀⠀⣀⢀⠀⢀⠄⠀⠀⠈⢁⡰⠇
⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠠⠁⠈⠁⢐⠞⠀⠇⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠹⡀⠉⠉⠀⠀⣨⠁⡄⠀⢠⢶⠰⠆⡀⢠⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⢀⣹⠄⠀⠀⠀⠀⡐⢃⠀⡀⠀⠣⠀⠀⠈⠁⠛⠲⠤⣄⣀
⠀⢀⡤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⡀⠀⠀⠀⣀⠄⠂⣿
⢎⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠡⠤⠒⠈⠑⠤⠀⠪⠀⠀⢠⠇⠀⠀⠀⢀⡤⠦⣄
⠀⠩⡀⠀⠄⠀⠀⠀⠀⡐⠉⠁⠀⠀⢀⣐⠇⠀⠀⠀⠀⠀⠀⠱⡀⡼⠀⠀⢀⠔⠋⠀⠁⠀⠑⢄⡀
⠀⠀⠹⣄⠀⢀⠁⠀⠀⠀⣀⡤⠖⠊⠉⠉⠙⠛⠒⠲⠶⠤⢤⣤⡼⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠘⢦⣂⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠘⠈⠀⠀⢀⣼⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣀⡀⠀⠤⠄⠐⠁⠀⢀⣰⠋⠀⠀⢀⣀⣀⣀⠀⠀⢀⣀⠉⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠉⠉⠉⠉⠀⢀⡔⠉⡤⠊⠁⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠜⠳⣄⠀⠅⢀⡴⠋⠀⠀⠀⡀⠀⠀⢠⠏⢠⠊⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡎⠀⠀⠀⠙⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⣀⣀⠤⠼⠦⠤⠤⠤⠤⠤⠤⠤⠌⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠤⠤⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡂⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠸⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠤⠤⠄⠀⠀⠀⠀⠀⠠⠤⠬⠭⠭⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⢳⡞⡀⢀⠒⡄⢯⢩⠖⠒⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⣀⠁⠸⠐⢡⠈⠈⠀⠀⣆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠤⠉⠸⢧⡜⣄⣿⣀⣑⠦⠀⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠃⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠉⡏⠉⠉⠙⠛⠛⠓⠒⠲⠶⠶⠤⠤⣤⠀⠀⠀⣾⠋⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢘⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⢸⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣤⣧⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⠶⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠶⠶⠶⠾⠿⠷⠶⠶⠶⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣴⠟⠉⠛⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⢷⣦⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡄⠀⢀⣿⡀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠈⢿⣄⡀⠉⠛⠋⠉⣉⣭⠁⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⠀⠀⠀⠀⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠈⠛⢷⣤⣀⣠⣴⠟⠋⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠇





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶⣄⠀⣀⠴⠚⠁⠀⠀⡈⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠃⠈⠟⠁⠀⠀⠀⠀⢠⣁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⢰⡀⠀⠀⠂⡀⠀⠀⠀⠌⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⢠⠏⠀⡅⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⡇⢀⠈⠀⡰⠆⠴⣁⠀⡀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⢹⣄⠠⠐⠂⠁⢀⠾⠀⢀⠁⠐⠀⠀⠈⠉⠒⠺⠥⣀⡀
⠀⠀⣀⠤⠐⠊⠉⠀⠀⠀⠀⢀⠎⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⣻
⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠠⠐⠚⠢⡀⠀⠂⠁⠀⠀⠇⠀⠀⠀⠀⢀⣤⣀
⠘⢃⠂⠠⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠒⠀⣠⠄⠀⠀⠀⠀⠀⠀⢠⠀⡸⠀⠀⠀⣠⠞⠁⠁⠈⠓⢄
⠀⠀⢣⡀⠀⠀⠒⠀⠐⠁⠀⢀⣠⠴⠖⠛⠓⠶⠦⠤⣤⣄⢀⣀⣀⣳⠃⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠱⡄⠈⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⠞⠁⠀⠀⠀⠀⠰⠤⠀⠀⠀⢘⡇⠀⣀⣀⡀
⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⠂⠀⠀⢀⡴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠒⠈⠈⠁⠸⠀⠀⠀⠀⡠⠛⠒⠒⠒⠒⠒⠒⡰⠃⣀⠔⠒⠁⠈⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡱⢄⠀⠀⡄⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⢠⠊⢠⠚⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡰⠀⠀⠑⠢⣧⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠃⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⡀⠀⡆⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⢶⡶⢤⣬⣭⣬⣥⣀⣀⣀⣀⣀⣀⣉⣈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⢂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠦⠼⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠤⠤⠤⠤⠤⠤⠄⠀⠤⠤⠤⢤⡤⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⠀⡀⣀⠀⠀⢀⠀⣀⠀⡀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠹⠁⠆⢀⠉⡆⠙⠠⠏⠈⠙⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⠸⢀⣤⣰⢸⢈⠑⢀⠐⡀⠀⠐⣀⡀⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠐⠚⠀⠙⠻⠎⠣⡿⠤⠬⠗⠤⠤⠄⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡏⠀⠉⠉⠉⠉⠙⠛⠛⠓⠒⠲⠶⠶⡄⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⢸⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣘⣠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣼⣶⠋
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣤⠤⣍⠉⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠸⣇⣀⡼⢁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⣤⠿⠛⢋⣉⣭⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⢀⣠⡴⢖⣋⣭⣶⠾⡿⠿⠉⢿⣵⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠷⠟⠋⠙⠋⣭⠀⢀⣶⠤⡄⢿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠀⠀⠀⠐⠚⠉⠃⠀⣠⠴⠃⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣟⠀⠀⠀⢀⣴⠋⠀⠀⢀⡔⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⢓⡗⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠈⠳⠾⣧⠀⠀⠈⠁⠳⠶⠶⢏⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣾⣛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠁⠈⣿⠻⡄⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣨⣷⡀⠀⠀⠀⠀⢀⣠⡤⢾⠁⠈⣹⣥⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠚⠛⠛⣷⣦⡤⠿⠚⠛⣇⠀⣷⠀⠀⠀⠀⢀⡤⠚⣁⣤⡶⠛⣿⡼⢷⠶⠶⠖⠛⠉⠀⠀⣿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣏⡉⢹⡇⠀⠀⢹⣧⠴⠶⢾⡄⢿⣀⣿⠶⠦⣤⣤⣾⠗⠋⣽⢿⣿⠲⣿⢦⡌⣧⣠⣤⠴⠶⠚⠛⠉
⠀⠀⠀⠀⠀⢶⡛⠉⢳⡘⡟⠛⣷⢸⣇⣀⣤⣼⡷⠟⠉⠉⠛⠲⠦⣤⣭⣽⣾⣇⠀⢹⣆⠘⣆⡇⢸⡅
⠀⠀⠀⠀⠀⠀⠹⣆⠀⢳⣿⣠⡿⠞⠛⣩⠿⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠳⢤⣘⣷⡈⣿⣆⠹⣷⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠛⠉⠀⢀⡴⠞⢁⣠⣀⣠⣤⠶⢿⣟⠋⠀⠀⠀⠀⠀⠈⠙⢧⠸⣿⣆⢻⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⣇⠀⠀⠹⣟⠉⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⣿⣤⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠳⣦⠖⠛⢳⣄⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣋⣩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⢴⡇⡀⡄⢺⠁⢹⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠛⠉⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⡧⠀⣧⣳⣹⣌⣧⣾⢀⣿⠀⠀⠶⠖⠛⠋⠀⠀⠀⣿⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠻⠚⠋⠉⠉⣉⣥⣶⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣼⣏⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠉⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣀⣠⡶⠛⠉⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣿⣤⣴⣶⣛⣉⣁⣀⣤⡴⣶⡆⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣌⡉⢁⣼⠟⠛⢻⡿⠉⠀⠀⣼⠃⢀⣾⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⡟⠉⠙⠛⠁⠀⣠⡟⠁⠀⠀⣼⣯⣤⡾⠇⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠙⠻⢶⣤⣤⣴⠾⠋⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁

⠀⠀⢀⡤⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢄⡀
⡔⠒⠋⠀⢈⡁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢙⠆⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠁⠢⢄⡀
⠈⢂⠀⠂⠀⠆⠐⠃⠀⠀⠂⠐⠀⠀⠊⠀⠐⠘⢠⠎⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⠀⠀⠀⠀⠈⢰
⠀⠸⣀⡀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⡜⠀⠀⠀⠀⣀⢀⡀⠈⠑⢄⡀⠀⠀⠀⠂⠀⠀⠌
⠀⠀⠈⡇⠀⠐⠒⠒⠒⠒⠒⠒⠀⠂⠒⠒⠂⢀⡇⠤⠒⠉⠀⠀⠀⠑⠀⠀⠀⠀⠑⠢⢄⡀⢀⡔
⠀⠀⠀⡇⠀⠀⠀⠀⢸⠋⠉⠉⠛⠫⠁⠀⢸⠡⠀⡀⠀⠀⠀⠤⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠲⢄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠒⠒⠒⠀⠉⠈⠀⠈⢃⠀⠀⠈⠀⠀⠀⢀⡀⠜⠁⠀⠀⠀⠀⣀⡠⠔⠚⠉⠀⠀⢀⠀⠈⢆
⠀⠀⠀⢸⠀⠀⠀⠀⡤⠠⠤⠀⠀⠀⠄⠀⠀⠀⡦⡄⠤⠄⠒⠉⠀⠀⠀⠀⠀⠀⢔⣉⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠈
⠀⠀⠀⢸⠀⢀⠀⠀⠀⠈⠀⠀⠀⠈⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠉⠀⠰⠀⠤⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⡐⠀⠀⠀⠀⠀⠔⠒⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠀⠀⡀⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠤⠖⠒⠉
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⢀⠤⠤⡠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢉⠤⡤⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇
⠀⠀⠀⠀⡔⠃⠐⠒⠐⠒⠒⠒⠒⠒⠒⠒⠘⠒⠳⡀
⠀⠀⠀⠀⡇⠠⡀⢄⠠⠄⠠⢄⡠⠄⠠⠄⠠⠦⠄⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠄⠠⠀⠠⠀⠠⠀⠠⠀⠠⠂⠂⠀⡇
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⣶⣶⡶⣶⢾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠔⠒⠒⠒⠈⢉⠛⠀⠀⠀⢀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠉⢀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠛⠋⠉⠉⠉⠉⠀⡠⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠢⠄⡀⢀⠔⠃⢀⠀⢀⠀⠀⠀⠀⢀⠊⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢁⡰⠀⠀⣠⡀⠀⣑⠾⠁⠄⠀⠔⠉⠒⢄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡃⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⡂⠠⠂⠐⠈⠣⡀⢀⠌⠀⠀⠀⠀⠀⣀⣈⠴⠃
⠀⠀⠀⣀⡤⠴⠒⠒⠊⠉⣉⡉⠭⠤⠈⠛⠛⠛⢳⠀⠀⠀⡀⠀⠀⠀⣠⠊⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠓⣀⡤⠔⠒⠉⠉
⢀⡴⠋⠁⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⣸⠀⡠⠀⠉⠒⠐⠁⠀⠁⠂⠀⠀⠠⠄⠤⠠⡄⠀⠀⠀⠀⠑⢤⡀
⡎⠀⠀⠀⠀⠀⡆⠀⣀⠤⠄⠒⠀⠘⠋⠀⠀⠀⠀⠳⠤⡤⠤⡤⠤⠤⠤⠤⠤⠤⣌⠖⠀⠀⠀⠐⠄⢠⠐⠂⠂⢀⡿
⡇⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡇⠀⢡⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⣠⠎
⡇⠀⠀⠀⠀⠀⢸⠀⡀⢀⣀⡀⠠⠤⢆⣀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⣧⠜⠁⠀⠀⠀⣀⣠⣤
⡇⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⣀⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⠤⠖⠊⠉⠀⢀⠂⢳⡀
⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠒⠀⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠆⠀⠀⢣
⡇⠀⠀⠀⠀⠀⠀⡇⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠒⠀⠄⢀⠀⠀⢀⠜⠂⠀⠤
⡇⠀⠀⠀⠀⠀⠀⡇⢀⡠⠤⠤⣀⡤⠒⠤⣰⠊⠩⠿⢄⢠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⣀⠠
⢧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡿⠀⠀⠀⣯⠟⠃⠸⠠⣨⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠂⠀⢀⣀⡤⠴⠚⠉
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠊⠀⠀⠢⠄⠴⠁⠒⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠉
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⠁
⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⢀⣀⣀⠤⣴⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⢸⡀⠀⢻⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠻⠙⠧⠌⠀⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⠴⠶⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠄⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣾⠛⠛⠛⠶⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠃⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣼⠁⡾⠶⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣸⣶⣶⡒⠚⠛⠛⠉⠉⠉⠀⠀⠏⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⣤⠞⢛⡇⠀⡇⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠸⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⣰⠟⠁⠀⠀⠻⠟⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠂⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢰⠏⢠⡤⠤⠤⠤⠤⠤⠤⠤⠄⠀⢸⡇⠀⠀⠠⠒⢲⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠐⠶⠶⠶⢶⣤⡤⠤⠤⠤⠀⠀⣿⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⣤⡴⠞⠁⠀⠀⠀⡀⠀⠀⠀⠠⡞⢠⠀⢸⣡⢶⣦⡿⠀⠀⠀⠀⣸
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⣠⣤⡀⠀⠀⢀⣿⠀⠀⣠⠀⠀⢸⢀⡞⠘⠏⠙⠂⠀⠀⠀⢀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣻⠈⡷⢀⣤⣸⠿⣆⠿⠩⡄⠀⠈⠋⠀⠀⠀⠀⢀⣠⣤⣶⠋⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢦⣠⠿⡼⠉⠀⠉⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⢀⡟
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⡴⠛⢿⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⠀⢀⣠⡴⠞⠁
⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠃⠀⣼⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠓⣛⣥⠶⠛⠋⠁⠀⠀⣠⡴⣶⡟⠉
⠀⠀⠀⠈⠉⣻⠳⠶⢦⣤⣤⣀⣀⡀⠀⣼⣀⣤⡴⠖⠛⠉⠀⠀⢀⣠⡤⠖⠋⢻⣄⣿
⠀⠀⠀⠀⠀⠙⠶⢶⣤⣄⣀⠉⠉⠛⠛⠛⠋⡉⠀⠀⢀⣠⡴⠞⣿⠋⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⢦⣤⣀⣧⡴⣾⡏⠁⢀⡾⠁⣰⠞⢦⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢹⡇⠀⡌⠀⢠⠁⣀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⡆⠀⡄⡰⠛⢧⢰⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠘⠁⠀⠃⢁⣤⠘⣸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢸⡄⠀⣻⡼⠃⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⡯⠀⣇⠀⠛⢿⠀⣰⠙⢸⡿⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⢸⡷⠀⢁⡀⠀⢡⡘⠁⡄⢸⣧⡴⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡗⠀⠈⢃⠘⡀⠛⠛⠁⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⣿⠀⠀⠙⡄⠘⠶⣤⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠷⣾⡏⢳⡀⠀⠱⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⣷⠀⠀⢳⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠸⡇⠀⠘⣇⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⣿⠀⠀⢹⡀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⣿⠀⠀⢸⡇⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢠⡟⠀⠀⣸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⡟⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠈⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⢀⡠⠤⠒⠒⠒⠤⢄⡀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⡜⠁⠀⠀⢀⡴⠊⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⡇⠀⣠⠔⠁⠀⠀⠀⠀⡸⠘⣄⠀⢠⣴⢦⣄⠀⠀⣿⠀⠀⠀⠀⠈⠢⣄⠀⠀⠆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣧⠞⡇⠀⠀⠀⠀⠀⠀⣇⡀⠈⠳⠿⣁⢀⣿⣦⢞⠇⠀⠀⠀⠀⠀⠀⢸⠳⣄⠂⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠁⠈⠢⣀⠀⠈⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠉⠑⣢⣤⣀⣤⣔⠊⠉⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢲
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣒⣉⣁⣀⣀⣀⣈⣉⣒⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢺
⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⠶⠶⠶⠾⠿⠶⠶⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠶⠂⠀⠀⠀⣿
⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠻⣶⣄⠀⠘⣛⣛⣃⢀⠛⠛⠛⠛⠒⠒⠶⠶⠆⠀⠀⠀⣠⣴⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠻⣮⣉⠉⣉⣉⣉⣙⣁⣀⣀⣀⣀⠀⢀⣠⣶⠟⠋⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⠳⣮⣉⠉⠉⠉⠉⠉⢉⣩⡶⠟⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢉⠉⠉⡡⠀⡄⠈⠈⠉⠹⠡⣈⡙⣇⠀⠀⠀⠀⠃⠒⠠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠒⣀⣤⡤
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠙⠲⠤⠀⠀⠀⠠⢍⣉⣒⣀⣀⣀⠀⠀⠀⠀⡈⡀⠀⢐⠀⠀⠀⠀⢸⠇⠈⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠐⠒⠀⠤⠤⠀⠈⡉⠙⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠤⠠⠄⠒⠒⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠐⠐⢄⠀⠀⣀⠔⢱⠧⠅
⠀⠀⠀⠀⠆⠀⢀⠀⡀⢰⠁⡀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣤⠄⠀⠀⠠⠈⠒⠀⠀⠀⠈⠁⠀⠀⠀⢰⠶⣟⠁⣃⠈⣦⠊⠁⠀⡞⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡄⣀⡀⠀⠇⠀⠨⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣾⢷⡜⠀⢹⠀⠀⢠⣇⡀⠀⠃
⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠛⠁⠀⢸⠀⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⡄⢈⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠒⠲⠂⠂⠉⠉⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡀⠠⠦⠄⢖⠐⠂⠄⠒⠂⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢁⣉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⢩⡃⠀⠆⠄⠀⠀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠄⠤⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⢀⣀⣀⣀⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠖⠒⠛⠛⠓⢦⣄⡀
⠛⠛⠛⠳⠶⢤⣤⣤⡤⠤⠤⠴⠶⠶⠖⠒⠒⠛⠛⠛⠉⢹⡟⠿⣗⠲⠶⣤⡀
⠀⠀⢀⡤⠞⢋⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⢦⣀⠙⠳⣤⡀
⠀⠀⠙⢶⣌⠘⠋⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠳⢤⡀⠙⠳⢤⡀
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠳⣄⠀⠀⢀⡀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⠀⠈⠳⣄⠩⠾⠃⠀⢀⣀⣸⣀⣀⣀⣤⣤⡤⠤⠴⠶⠶⠚⢳⡶⠶⠤⠽⣶
⠀⠀⠀⠀⠀⠀⢸⡇⠈⡷⣄⠀⠀⣀⣼⣿⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⢞⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠸⣇⠃⢇⠈⠳⠞⠋⢸⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⣀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇⠘⠦⠶⠶⠒⠒⠒⠛⠛⠛⠛⠋⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡜⠄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣤⣀⣀⣠⣤⣤⣤⡤⠤⠤⠴⠶⠶⠶⠖⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀⢀⡀⠀⠈⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠉⠉⢃⣿⡿⠭⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⢰⣿⡇⠀⠀⠈⠉⠉⠒⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⠠⠤⠤⠿⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠟⢿⡇⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠙⢦⣄⠀⠀⠀⠀⣠⠞⠁⠀⢸⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢀⡼⠛⠶⣤⠜⠛⠲⢄⡀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡇⡟⣦⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣼⣷⠶⠶⢶⣶⣶⣶⣶⣤⣤⣶⡧
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣽⣟⣯⣾⣿⣿⣿⣿⣿⣿⠿⠋⢁⡤⢾⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣤⠞⠋⠀⢸⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣛⠏⢁⣠⠖⠋⠀⠀⠀⠀⢸⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠿⢿⣿⣿⣿⣿⣶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⠋⠙⠙⠛⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⣤⣤⣦⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣾⡏⠁⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⢋⠏⢉⢉⡿⠀⢹⣿⡛⠻⠛⡟⠻⠛⠻⠛⠿⠟⢿⢿⠿⠿⢷⢶⡶⣶⢶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⣝⣿⠀⢀⠈⠈⡇⠀⢨⣿⠀⠀⠀⠰⡀⠀⠀⠀⠀⢡⠀⠂⠆⠀⠄⡄⠑⠈⠄⠳⡘⡌⢢⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠻⣯⣿⡇⠀⠀⢰⡇⢀⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠘⠀⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡳⣣⢿⡇⠀⠀⢸⣀⣈⣿⡇⠠⠤⠤⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⣯⢿⡇⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡷⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣾⣸⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣼⣸⣿⣏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡼⣹⣿⡃⠀⠀⠄⠀⡠⢄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠚⢿⣿⣿⣿⣿⣃⠀⣀⢔⡫⢒⡡⠖⣡⠴⣋⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⢸⡇
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠛⠋⣉⣥⣴⠶⠾⣛⠿⣛⣿⣿⣿⣷⣮⣴⣩⣴⣩⣴⣯⣵⣊⣤⣄⣡⣴⣠⣤⣀⣤⣤⣤⣤⣨⣴⣨⣄⣠⣴⣠⣄⣥⣆⣼⡇
⠀⠀⠀⡴⠞⢛⣉⣁⣤⡴⠶⠛⠋⠍⡲⠞⠛⠉⣒⣩⣄⣤⡤⢴⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠹⠋⠉⣻⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⣿⠾⠿⣯⡥⣤⢤⠶⠶⠶⠶⢶⢛⣛⣫⣭⣭⡷⠶⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡐⢹⡇
⠀⠀⠀⠈⠛⠓⠓⠒⠒⠒⠓⠚⠚⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⡎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠠⢆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠀⠀⣻⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⣩⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠈⡷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⣟⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⡫⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠕⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣾⣋⣁⣀⣀⣈⣙⣷⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⠀⠀⠀⠀⠀⠀⣴⠛⢉⣭⣍⣉⠙⣦⠀⠀⠀⠀⠀⠀⣿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣀⣿⠀⠀⠀⠀⠀⢰⡇⠀⡏⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⣀⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠿⣤⡀⠀⠀⠀⠀⢷⣄⠙⠒⠚⠛⢒⡛⠀⠀⠀⠀⢀⣤⠿⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢶⣄⠀⠀⣠⡶⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠛⠛⠛⠒⢶⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⡼⠤⠤⠤⠒⠒⠋⠉⠀⠉⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠄⠐⡇⡴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠙⠳⣄⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⢷⡀⠀⡇⢈⣀⣀⣠⡤⠤⠶⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣷⠚⠇⠸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⢀⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⢀⣀⣀⣤⡤⠴⠶⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠉⠉
⠾⣭⣉⡉⠉⠉⠉⠉⠙⠛⠻⠭⢤⣀⡀⠀⠀⣿⠀⠀⠀⣀⣠⣤⣴⡶⣿⡟⠋⠁
⠀⠀⠈⠉⠙⠓⣒⣒⣒⣒⣒⣀⣀⣀⣀⣉⣰⣿⣴⠾⠿⠛⠋⢹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣼⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠀⠉⠀⠠⠀⠀⠀⠐⠐⠀⢠⡠⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣺⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣀⣴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣽⣮⢠⡤⠄⠀⠀⠀⠀⠐⢃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⠉⠉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⣿⡿⣿⡆⠀⠃⠀⠀⠀⠀⢰⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣿⣷⣬⣅⠰⣄⡀⠀⠀⠀⢸⣻⣾⡀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠘⠉⠛
⠀⠀⠀⠀⠀⠰⢤⣴⠂⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣽⣿⣿⣿⣷⡌⣿⣶⣦⣤⣼⣿⣺⣴⣿⣭⣽⣤⣼⣽⣽⣿⣿⣴⡄⠄⠀⣤⠀⠀⢶⡦
⠀⠀⠀⠀⠀⢁⠞⠀⠀⠀⠀⠀⠀⠀⠀⢦⡇⠀⠘⡽⣿⣿⣯⣽⣧⠈⠻⣿⣿⣿⣿⡿⠿⡿⢿⡿⠄⣾⣿⣿⡿⡿⣿⣧⠀⢠⣿⠀⠀⠈⠋
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠞⠝⣿⣿⣿⣿⣯⠀⠀⠹⠿⣷⣿⣧⠿⠂⠿⢻⣦⡿⠟⠹⠇⣣⡿⢹⡿⠟⠟
⠀⠀⠀⠀⡄⣸⠃⠀⠀⢠⠂⠀⠀⠀⠀⠀⢽⡇⠀⣾⣿⣿⣿⣿⣿⣷⡦⢀⣠⢠⠟⠁⠀⠀⠀⠀⠉⣁⣨⣶⣿⣿⣤⣉⢀⣀⣀⣼⠀⠀⣠⠄
⠀⠀⠀⢰⢡⡏⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢸⣷⠀⢻⣿⣿⣿⣿⡿⠿⠗⠙⠡⢾⣯⡞⠀⢣⠀⠠⣿⣿⡻⢷⣾⠛⠙⢻⣿⣿⣿⣥⣴⠋⢑⠂
⠀⠀⠀⠀⢸⣷⣶⣶⣾⣶⣦⣤⣀⣀⠀⠀⣾⣿⡤⣼⣿⣿⣿⣷⣀⢄⣤⣖⣀⣀⣷⡀⠀⠀⠀⢀⣿⣿⣿⣶⣿⣿⣿⣫⢍⣉⠀⠀⠀⣀⡈
⠀⠀⠀⠀⣆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣾⣿⣿⣿⢿⣷⣿⣿⣿⣿⣿⢾⣇⡀⠀⠀⠀⠹⠿⠛⠛⢉⡵⣻⠓⠚⣿⠿⠅⠠⣹⡇
⠀⠀⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣟⣿⢇⣿⣿⣿⣿⣿⣷⣾⣿⢿⣿⣶⡿⣿⣿⣿⣿⣿⣻⠀⠀⣀⠀⠁⠀⠀⠄⢀⡟⠉⠸⣿⣷⠀⠤⢿⡇
⠀⠀⠀⠀⠍⢸⡯⣿⣷⡇⣿⡿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢺⡁⠀⢀⡀⠀⢀⡞⠁⠰⠀⠛⠁⠀⠀⢼⡇
⠀⠀⠀⠀⠈⠘⢿⠿⢿⡇⣹⣧⠀⢰⢿⣯⣽⣿⣿⡿⣻⣿⣿⠟⣿⣿⣿⣿⣿⡿⢻⣿⣿⣁⣤⠾⠾⠋⠁⠀⡼⠁⠀⠠⠄⠉⠓⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠀⠀⢸⡿⢿⣿⣿⣿⠟⠋⠁⣰⣿⣿⣿⣿⣿⣿⣶⡯⠟⠋⠁⠀⠀⠀⠖⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⡿⠟⠻⠟⠉⠛⠓⠒⠶⠯⠦⠄⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠰⠒⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⡀⠀⠀⠀⠒⠀⠂⠂⠐⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⢈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣅⣤⣀⣠⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠴⠶⠶⠒⠚⠛⢻⠛⠲⠶⣤⣄⣀
⠀⠀⢀⣤⣤⠤⠶⠶⠶⠒⠚⠛⠛⠋⠉⠉⠉⠉⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠙⠛⠲⢶⣤⣄⣀
⠀⠀⡾⡉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠘⠉⠙⠛⠲⠦⣤⣄⣀
⠀⡼⠱⠋⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠿⠛⢳⡄
⣼⠋⠢⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠖⠛⠋⠉⠉⠀⠀⠀⠀⢀⠠⠠⡹⣆
⠈⠛⢦⣀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠾⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠑⠘⢧⡀
⠀⠀⠀⠉⢳⣤⡈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠘⢳
⠀⠀⠀⠀⢸⡇⠙⢶⣄⠙⠢⡀⠀⠀⠀⠀⢄⡴⢡⠟⣿⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠒⠈⠀⣀⣠⣤⠴⠶⠚
⠀⠀⠀⠀⠈⣷⢀⠀⠈⠻⢦⡈⠑⢄⡀⠀⠀⣱⠏⠀⣿⡆⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠄⠒⠈⣁⣀⣤⡤⠶⠖⠛⣻⠉
⠀⠀⠀⠀⠀⠙⢦⣕⠤⡀⠀⠙⠳⣄⡈⠢⣴⠏⠀⠀⡟⣧⠰⠀⠀⠀⢀⡀⠀⠠⠄⠒⠀⢈⣁⣠⡤⠴⠖⠛⠋⠉⠁⠀⠀⠀⠂⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⠢⣀⠀⠈⠙⠶⠏⠀⠀⡇⡇⢻⠄⠒⠀⠉⣀⣀⣤⡴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⠀⠠⠐⠂⢀⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡑⢄⠀⠀⠀⠀⡀⡇⡇⠸⠷⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠄⠀⠀⢁⣀⣤⡴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣌⠢⡀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠂⠉⣀⣠⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡑⢄⠀⡇⠀⢀⡀⠄⠐⠊⠁⣀⣠⡤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣙⡟⢉⣀⣤⡴⠶⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠔⠒⠒⠒⠊⢉⠙⡆⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠓⠉⠉⠉⠉⠀⠀⢀⠌⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠉⠐⢠⠄⠀⢀⡐⠣⢀⠀⠰⠀⠀⠀⠀⠀⠐⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠐⠃⡢⠀⠀⣀⡀⠀⢁⡤⠀⠀⠀⠠⠊⠐⠄⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⢀⠔⠉⠙⠊⡀⠀⡐⠁⠀⠀⠀⠀⠀⢈⣠⡼
⠀⠀⠀⠀⣀⣠⡤⠴⠒⠒⠚⢛⣋⠉⠭⠽⠛⠿⠿⡀⠈⠢⡀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠲⣀⣀⡤⠴⠒⠘⠉⠁
⠀⣠⠖⠋⠉⠀⠀⣠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠘⠐⠤⠔⠈⠉⠒⠀⠀⠀⢀⢀⠀⣀⡀⠀⠀⠀⠈⠳⣄
⡼⠁⠀⠀⠀⠀⡘⠁⠀⢀⣀⠤⠀⠀⠶⠀⠀⠀⠉⢳⣌⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡁⡮⠀⠀⠀⠡⠀⠀⡀⠀⠀⠈⢳⠄
⡇⠀⠀⠀⠀⠀⢰⠀⡛⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡅⠈⡅⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⡅⠀⠀⡰⠋
⡇⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⣠⠞
⡇⠀⠀⠀⠀⠀⠈⠀⢰⠄⠚⠓⠀⠉⠉⠙⠂⠦⢀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠁⠀⣀⡤⠶⠚⠋⠙⠳⡀
⡇⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠵⠾⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠒⠉⠀⠀⠀⠀⢀⠃⠀⠱⡀
⡇⠀⠀⠀⠀⠀⠀⡇⠸⠄⠤⠄⠐⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢁⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠣
⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠤⢄⡔⠊⢉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠉⠀⠐⠦⠄⠤⠂
⡇⠀⠀⠀⠀⠀⠀⡇⠀⡆⠉⠉⢡⡎⠀⠈⢳⠦⣄⠉⡏⡆⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠁
⢹⠀⠀⠀⠀⠀⠀⡇⠀⡇⢠⠤⠚⡅⠀⠀⡀⢏⣁⡠⠃⠑⠂⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣃⣀⠤⠔⠒⠉
⢸⠀⠀⠀⠀⠀⠀⡇⠀⢇⡰⠀⠀⠈⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⡁
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟
⠀⠉⠓⢦⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠈⠙⢦⣈⣀⣀⣀⣠⡤⠔⢲⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣇⠀⠸⣗⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⡟⠷⣠⠁⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⠟
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣄⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣬⣭⣥⣬⣭⣥⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠃⣿⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⢸⣿⠘⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣅⠀⠀⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⠀⠀⣨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠛⢿⣦⣝⠃⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠘⣫⣴⡿⠛⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠉⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⢀⣤⣤⡀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣶⡿⠋⠙⢿⣶⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⢠⡄
⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠰⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠤⢄
⠌⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢳
⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣄⣁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠞⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠈⠀⣀⠜⠁⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠤⠚⠀⠙⠓⠒⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠠⡅⢀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠠⠀⠓⠘⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡄⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠂⠀⠠⠀⠀⠀⠀⢀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⡀⠐⠀⠀⠄⠀⠁⢀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠁⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠘⠠⠀⠀⢀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠔⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡰⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⡀⠄⣀⠀⠺⣦⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠠⠐⠀⠀⠠⢐⠠⢴⣄⣉⠿⢚⣅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣤⡀⢙⡶⢔⡋⠑⢷⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⣦⡤⠟⠹⣦⣈⡽⠛⣯⣀⣠⡤⠄⠀⠀⠀⠄⠒⠀⢠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⠀⢶⣄⣩⠟⢫⣅⣈⣳⠶⣿⠉⠻⣦⠴⠋⠉⠀⠀⠀⣠⠄⡀⠰⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣙⡷⢚⡁⠘⣷⣤⣾⠙⢷⣄⡬⠟⠋⢤⡀⠀⢀⣠⡴⠎⠋⠀⡁⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠐⠆⠀⠀⠙⠦⢴⠋⠰⣦⣠⡽⠫⣯⣀⣹⠗⠊⠁⠀⠀⠀⠀⠉⠓⠍⠁⠀⠀⠐⠀⠁⢠⠐
⠀⠀⠀⠀⠈⠀⠀⠈⠐⠀⠀⠈⠙⠒⣏⠈⢻⡦⠖⠉⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠂⠀⠀⠀⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⡀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡀⠀⠄⠈⢦⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣴⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣄⣀⣀⡀⢀⠀⠀⠀⠀⠀⣿⣿⣟⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣯⡉⠙⢶⣤⠀⠀⣿⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢯⢧⠀⣿⠟⠒⠦⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣯⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠘⡞⡄⠿⡄⠀⠀⠀⠀⠻⡿⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⡀⠀⠉⠉⠉⠉⢉⣻⡟⠃⠀⠀⠇⡇⣠⣁⠀⠀⠀⠀⠀⢱⢱⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠙⢦⣀⠀⠀⠀⠀⠸⡏⠈⠀⠀⠀⠸⣿⠲⢰⠀⠀⠀⠀⠈⡆⡇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠓⢤⡀⠀⠀⣀⣶⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⡇⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠉⠳⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡴⠋⠀⠀⠀⠀⠀⠸⠋⣿⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣈⣉⣁⣀⣀⣀⡠⠴⠒⠋⠉⢉⣧⣬⣍⣉⣉⣉⣉⡉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⣉⣻⠿⢿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠐⠚⢿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢰⠭⠤⠶⠿⠍⠐⠒⠂⠉⠀⠀⢀⣀⠠⠄⠂⠈⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠤⠖⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣜⣛⣤⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢟⣽⠿⣽⢿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⡀⠀⠀⠀⠀⢀⢺⣏⣿⣿⣾⠹⡟⠾⣸⣿⡇⣻⢂⠀⠀⠀⠀⠀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣷
⠀⣿⣶⣝⡻⣿⣿⣿⣿⣿⡿⢟⣭⣾⣿⠀⠀⠀⠀⢸⣷⣦⣝⢿⣮⣟⣻⣿⣿⢟⣣⣶⣿⠀⠀⠀⠀⣿⣮⣝⢿⣿⣿⣿⣿⣿⡿⣛⣽⣿⡿⠿⠃
⠀⣿⣿⣿⣿⣮⠝⠿⠟⢫⣾⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⢎⣝⣛⣍⠾⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣮⠛⠿⠟⣽⣾⣿⣿⣿⡇
⠀⣿⣿⡿⣫⣵⣿⣿⣿⣷⣯⡻⢿⣿⣿⠀⠀⠀⠀⢸⣿⡿⣫⣶⣿⣿⣿⣿⣿⣮⡻⢿⣿⠀⠀⠀⠀⣿⣿⢿⣫⣶⣿⣿⣿⣷⣭⡻⣿⣿⡇
⠀⠛⠵⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠯⠛⠀⠀⠀⠀⠘⠩⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠝⠀⠀⠀⠀⠫⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠝⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠚⠋⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⡴⠒⢲⠦⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⢒⡶⠂⣠⠴⠒⠉⠉⠙⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡰⠋⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠊⠁⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡜⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠂⠀⠀⡼⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣀⣀⡤⠤⠖⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠃⠀⡎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠦⠤⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡇⠀⠀⠀⠀⠀⠖⠰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠃⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠾⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡀⢠⣄⢤⠀⠀⣤⡀⣤⠤⢤⢠⢤⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠘⠀⡇⢸⢂⢂⠈⠀⡏⢸⢸⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢰⠶⣰⣸⢨⡈⣄⠆⢇⣸⠘⠦⠄⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠘⠃⠀⠉⠛⠎⠳⠼⠤⠤⠟⠒⠒⠚⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠻⠤⠤⢤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠚⠛⣿⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⢸⡇⠀⠈⠉⠉⠉⠉⠙⠛⠓⠒⠒⠲⠶⣆⠀⢘⡄⢸⡉⠀⠀⠀⠀⠀⠀⠀⢻⣰⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣷⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⣤⣧⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠰⠿⣷⣆⠀⠀⠀⠀⠀⠀⢰⣿⢿⣷⡀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣆⠀⠀⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠘⣿⣆⠀⠀⢀⣀⣀⣘⣿⣄⣿⣷⡿⠋⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣄⣼⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣶⣶⠿⠛⣛⣛⣿⣿⡿⠿⣶⣿⣟⣃⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣈⣻⣿⠿⠿⠿⣿⣿⣛⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠀⠀⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⣿⣤⣀⣠⣾⣿⣿⡇⠀⠀⠀⣿⣿⡟⠛⠻⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢀⣤⠀⠀⠉⢉⠉⠻⠿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣦⣀⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⠀⠀⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣟⠉⠛⠉⢹⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⣀⠀⠀⣀⠉⢉⣉⠉
⠀⠀⠀⠀⠀⠀⠿⠿⣿⣷⣤⣴⣿⡇⠀⠀⠀⢙⣿⣿⣿⣅⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢿⣿⡿⠟⠿⠂
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⣀⠀⠀⣀⣀⣠⣶⣶
⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠿⢿⣿⡇⠀⠀⠀⣿⣿⣇⠀⠋⢈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠿⠿⠀⠀⠿⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⣿⣦⡀⠁⢀⣼⣿⡇⠀⠀⠀⢹⣿⣿⣷⣾⣿⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣦⠀⠀⣠⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠷⠿⠿⠿⠇⠀⠀⠀⠾⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠛⠛⠛⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣦⢀⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣮⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣙⣿⣿⣍⣿⣏⣽⣿⠇
⠀⠀⢤⣀⠀⠀⣴⡿⣫⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⡿⢿⣿⣿⣿⣿⣅⡀
⠀⢸⣧⡹⣷⣼⣿⣿⣿⣇⢲⣶⣦⠰⣶⣶⣦⠲⣶⣶⣶⡄⢶⣶⣶⣄⢲⣶⣶⡄⠀⠀⣠⣴⡜⠇⠀⠀⠠⠾⠛⣹⣷⣾⣿⣿⣿⡎⠻⢿⣦⡀
⠀⠀⠹⣷⡙⣿⣿⣿⠻⣿⣦⠹⣿⣷⡜⢿⣿⣷⣜⢿⣿⣿⣎⢻⣿⣿⣧⡙⣿⣿⣄⢠⡹⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣿⣿⣦
⠀⠀⠀⢿⣧⢹⣿⣿⣷⡹⣿⣧⡙⣿⣿⣦⠻⣿⣿⣦⡹⣿⣿⣧⡹⣿⣿⣿⣌⢻⣿⡘⣷⣌⡻⢿⡇⠀⠀⢰⣿⣏⣹⣿⣿⣶⣾⣿⣿⡷
⠀⠀⠸⣦⠻⡆⠛⢹⣿⣷⡘⣿⣷⡜⢿⣿⣷⡜⢿⣿⣷⣌⢻⣿⣷⡜⢿⣿⣿⣷⣿⣧⢹⣿⣿⣶⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⣼⡇⠁⠀⠀⠙⠛⠓⠘⠛⠛⠈⠛⠛⠛⠈⠛⠛⠛⠃⠙⠛⠛⠂⠛⠛⣿⡿⣿⡄⠛⢿⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠘⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⡠⢘⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⡆⠀⣀⣀⠄⠀⠀⠀⣀⠔⠊⠁⠱⡀⠀⢀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠐⣀⠼⠁⠀⠀⠀⠀⠀⠈⢅⠀⣠⠔⠒⠁⠀⠄⡐⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠠⡄⠤⣈⠉⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⠀⠀⠀⣀⡠⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠆⠀⠀⠀⢀⡠⠒⢹⠆
⠀⠀⠀⠀⢠⠊⠀⠅⠁⠇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⠀⠀⢾⠁⠀⣀⠆⢷⠈⠉⠁⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡶⢀⡆⢠⢣⠀⠀⠀⠀⠀⢸⠀⡄⠀⠈⡄⢀⡠⠴⠚⠁⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡜⠘⠄⠀⠖⠚⠀⠸⢀⠇⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⡁⠀⠀⠀⠉⠈⠒⢸⠀⠁⠀⠀⠀⠀⠀⠀⡀⠤⠀⠘⠉⣀⣀⠤⠤⠄⣶
⠀⠀⠀⠀⠳⠤⠶⠤⢤⣤⠇⣘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⢠⡀⠤⡊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⢀⠔⠊⠙⡀⠀⠀⣧⠠⠤⠤⠤⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢄⠀⡠⠒⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠤⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠄⠀⠀⢀⢄⡊⠀⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⢄⠀⠀⠀⡀⠐⣁⣀⣔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁
⣿⣿⠻⠿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣿⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣯⣀⠀⣿⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⢸⡇⢀⣨⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⠈⠙⠿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⠉⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⣠⣤⣴⣤⣤⡀⠀⠀⣀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣶⡾⠋⠁⠀⠀⠉⠻⣶⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣼⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣠⡴⢶⠛⠛⠛⢶⡄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠙⠛⠓⠛⠋⠁⠀⢸⠀⣴⠞⠛⠛⠛⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⣤⣤⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠛⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣯⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢹⠀⢹⡇⠀⠀⠀⠀⠀⠀⣠⡶⠛⠶⣦⣄⡀⢀⣸⡷⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠘⡆⠘⣧⠀⠀⢸⡆⠀⠀⣿⠀⣠⠄⠀⠉⣿⠋⠁⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢧⠀⣿⠀⠀⠈⡇⠀⠀⣿⠀⣠⠖⠀⠀⡏⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠸⡄⢸⡇⠀⠀⢷⠀⠀⢿⠀⠀⠞⠁⠀⡇⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⣷⠀⠀⢸⡀⠀⢸⡇⠀⠀⠀⢰⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⠛⠛⢷⡀⠀⠀⠀⢹⡆⠀⠀⡇⠀⢸⣇⠀⠀⠀⣼⠀⠀⢰⣿⣴⡶⢶⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⣷⠀⠀⢹⠀⠀⠿⠀⠀⠀⠁⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⣶
⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⢠⡶⠛⠛⣷⡀⠀⠀⠘⠃⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋
⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉
⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋
⣿⠓⠚⠲⠶⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠞⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢻⡅⠀⠀⠙⠒
⣿⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠓
⣿⠀⠀⠀⠴⠋⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠂
⣿⠀⠀⠀⠐⠋⠁⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢠⡇⢠⠄⠀⢸⡇⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠀⠋⠀⠀⢸⡇⠀⠀⣠⡾⠋⠀⠀⠀⣠⡾⠋
⢿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠟⠉⠀⠀⠀⣠⡾⠋
⢸⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⠀⠀⢀⣴⠟⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⡴⠛⠁
⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠃⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠳⣤⣀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠳⣦⣀⣠⡶⠟⠉⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣴⣦
⠀⠀⠀⠀⠀⢀⣤⠶⠶⢦⣀⣼⠏⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⡿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠈⠙⣷⣤⡀⠀⠀⠀⢀⣾⢿⡟⠛⠷⠶⠚⠒⠒⠒⠒⠒⣀⣶⣶⣾⣿⣿⣷⣶⣶⣤⡀
⠀⠀⠀⢠⡾⠓⠀⠉⠉⠉⠛⠳⣤⣴⣿⠁⢸⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣹⣿⡿⢫⣽⢿⣿⣿⣿⣶⡄
⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡈⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣿⣿⠟⠛⣶⣿⣿⠂⠙⢿⣿⡻⣿⡀
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣣⡴⠟⢷⠈⠉⠀⠀⠀⠙⢿⣿⣿⠀⠀⣠⡄
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⢋⣁⣤⣾⠁⣧⣤⠴⠖⠺⡇⠀⠙⢫⣶⣀⣠⣄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠙⣷⠿⠀⠹⣷⣶⣾⠀⢁⠀⠀⠀⣿⣯⡙⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⢀⣿⠤⠴⠆⠛⠉⠛⡀⠀⠀⠙⡿⠃⠀⢸⡥⠞⠉⠀⠈⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⣿⠀⠀⠀⠀⠀⠸⣿⣤⣤⣤⣶⡆⠀⡇⠀⠀⠀⢠⠀⠀⠸⠁
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠿⣧⡀⠀⠀⠀⠀⢰⣷⠈⠻⣿⣿⠁⠀⢡⠀⠀⠀⠀⢡
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠛⠁⠀⠀⠀⠀⢨⣿⠀⠀⠈⠁⠀⠀⠘⣃⣀⡠⠤⠻⠂⠀⡀
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢻⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠴⠇
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠠⡤⠀⠰⠶⠻⠇⠀⠀⠀⠀⠀⠰⣶⣶⣿⣿
⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣷⣶⣶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠺⠟⠉⠉⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠾⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠌⣀⠶⠁
⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⡀⢀⡞⠁⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⢰⡟⠉⠙⠛⠀⠀⣿⢁⡾⠛⣿⡄⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠛⢶⣄⣀⣀⣠⡏⢸⣇⢠⡿⠁⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠉⠉⠙⢧⡈⣯⣾⠃⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣴⠶⣦⣄⣙⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠈⠻⢶⣤⣬⣭⣿⣧⣴⠀⠈⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣷⢶⣶⣄
⠀⠀⠀⠀⠀⢰⡟⠻⣦⠀⢰⠀⠐⢩⣶⠄⠀⠀⣿⣿⣿⡆⠀⠀⣠⣤⡀⠀⠀⠘⣿⣾⣿⣿⣿⣽⡿
⠀⠀⢀⣤⣤⣾⡇⠀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠀⣿⣿⣿⡇⠀⠀⣯⠉⣷⠀⣠⠾⠛⠛⠉⠉⠉⠁
⠀⠀⣿⠉⠉⠉⠁⠀⢸⠀⠀⠀⣠⣄⠀⠘⣧⠀⣿⣿⣿⡇⠀⠀⢿⣤⣿⡶⠋⠀⠀⣀
⠀⠀⠀⠙⠓⠶⠶⠶⠿⢷⡄⢸⡏⣿⠀⠀⠸⣷⣿⣿⣿⡇⠀⠀⠈⢻⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣰⠟⠁⠀⠀⠀⠀⡐⠀⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⣿⣿⣿⡇⠀⡰⢁⣀⣤⣄⠀⠀⠀⠈⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠹⣦⠀⢢⡄⣿⣿⣿⡇⣴⣷⣎⣉⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡦⠀⠀⠘⣧⠈⢿⣿⡏⣿⡿⠉⠉⠉⠉⠁
⠀⠀⢀⡶⠖⠰⢦⡀⠀⠀⠀⠀⠀⣶⣄⠈⠷⣸⢿⣶⣿⡇⠀⢀⣤
⠀⠀⠈⠳⠾⠃⠀⢹⡆⠀⢰⣄⣀⠈⢿⣷⣬⣿⡞⣿⣿⡇⣴⣿⡏⢀⣤⣶⡆
⠀⠀⠀⠀⠀⢀⠄⠙⢻⠢⢤⣿⣝⢻⣾⣧⠙⣿⡇⣿⣿⡿⠣⣾⣴⡿⣡⣇⣀⡴⣦⣀⡤⠤⠶⣤⡀
⠀⠀⠀⠀⠀⠸⡀⢀⡜⠀⠀⠙⢿⡆⣻⣿⡆⣿⡟⣿⣿⡁⢰⣿⠏⠀⣿⠟⠁⢀⠜⣿⠀⠀⣤⢈⡿
⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠈⠁⠻⢿⣿⣯⣿⣿⣽⣿⡤⠟⠂⠀⠈⠀⠀⢿⣤⠟⠀⠀⠉⠙⠁

⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠉⠉⠋⣆⡎⣿⣿⣿⣿⣿⣿⡇⣿⡇⠉⠀⠀⢸⠁⢹⠩
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡇⡿⢿⠿⣿⣿⣿⡇⣿⡇⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠁⠆⠀⠙⠛⢿⡇⣽⠁⠀⠀⠀⢸⠀⠙⠠
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠇⠀⠀⠀⢸⡇⠘⠀⡀⠀⠀⢸⢰⣤⣴⣔⠀⣲⣿⣿⣶⣀⣒⠀⠀⣂
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⣀⣀⣈⠸⠤⠀⠁⠉⠁⢹⡏⠉⠙⠁
⠀⠀⣠⣴⣶⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡇⠀⣾⠟⠁⠀⠀⠀⠀⠀⢀⣿⣾⡗⣄⠲⣄
⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⡟⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⣤⠉⠉⠁⢠⣄
⡏⠀⠻⣿⠿⠟⢡⠀⠀⠀⠀⣰⠞⠋⢹⡏⠀⠀⡄⡄⠀⠀⣀⣠⣧⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠀⣤⣴⢿⡿
⣿⣦⠀⠀⢶⣶⣦⣄⣀⠀⠀⠉⠀⠀⡼⢀⣄⣄⣦⡱⢊⡍⢡⢺⣟⠁⡄⠀⠀⠀⠀⠠⣾⣿⣷⣧⣏⣤⣶⢿⣅⣼⣿⡤
⣿⣟⠀⠀⠘⣿⣿⣿⣿⢳⡀⠀⠀⢸⠃⡄⢓⡆⣿⠀⠿⣿⣽⣿⣧⡴⡗⡀⠀⠀⢀⣴⣿⣿⣿⣿⢟⣱⡾⠛⣬⣿⡟⠁
⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣦⣠⣀⣼⣿⣼⡇⡿⠖⣻⣅⣠⣽⣷⣾⡭⠁⠀⠀⠀⠁⣹⡿⣿⠣⠞⠛⠋⠀⠉⣛⠗
⣿⣿⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⠁⠀⢾⠟⣿⣏⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡶⠈⠻⣦⣤⣠⠉
⣿⣿⠄⠀⠀⠀⠹⣿⣿⣿⣿⠟⠋⠉⠈⢻⣷⣜⠀⠀⢀⣠⣿⠿⠤⠄⢀⡀⠁⠤⢴⣿⣿⣿⣹⠀⠀⠀⠀⠙⢿⠶⠤⠄
⣿⣿⠢⠛⠀⠀⠀⠿⠟⠋⠁⠀⠀⠀⣠⣿⣿⣿⣶⣾⣿⣿⣿⡀⢠⣶⣿⣿⣿⣿⠿⠛⢿⡿⣿⠀⠀⠀⠀⠀⢿⣿⡤⠄
⡿⠃⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠛⠛⠛⠻⣩⣤⡀⢀⣾⠿⡏⠀⠀⠀⠀⠀⠘⠉
⣷⠆⠀⠀⠀⠀⠀⠀⠄⠐⠛⠁⠈⢿⢹⣿⣿⣿⣿⣟⠉⣉⡁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣇
⣿⣿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣥⣤⣴⣄⣀⣄⠄
⣿⣯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⠟⠙⠉⠉
⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠋⠁
⣿⣯⣷⣄⣀⣀⣀⣀⣀⣀⣤
⣿⣛⣻⣿⣿⣿⣿⣿⣿⠛⠁⡀⠀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⣀⣠⣀⣀⠀⠀⢀⣠⣠⣄⣀⣀⣀⡀⠂⢀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡽⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣟⠛⠛⠛⣿⡟⠙⠛⠛⠛⠛⠋⠉⠙⠋⠉⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡼⡷⣿⠐⠀⢸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⢀⣜⣤⣿⠀⢀⣼⣿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠉⢉⢞⣿⠀⢻⣿⣿⡇⢠⢃⡞⠁⠀⡐⠘⠁⠉⠡⠦⠂⢲⢂⢒⣚⡂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⡷⣿⠀⠙⠛⠛⠁⠎⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢎⣔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡔⣿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣦⡯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣖⣧⣎⡿⣫⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠐⠈⣘⣛⣋⣃⡿
⠀⠀⠀⣠⣤⣴⣶⣾⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⢹⣯⡿⡟⡟⣿⠉⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠙⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢛⣇⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡋⡷⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⣗⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⡴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢞⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢿⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠘⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠘⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠈⠓⢤⣀⠔⢶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠑⢍⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠑⢮⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠢⡀
⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⢢⡀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⢼⡁
⠀⠀⢰⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡴⠃⠀⣱⡄
⠀⠀⢸⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡠⠊⣀⠔⠊⠁⡇
⠀⠀⡎⠀⠀⠀⠀⠉⠲⢄⡀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣼⠕⠊⠀⠀⠀⠀⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⠔⠋⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡢⠔⠋⠉⠒⠶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⡇
⠀⠀⠀⡆⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⡇
⠀⠀⠀⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⠉⠁⠀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⢖⡉⠀⠁⠢⢄⡀⠀⠀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢒⡤⣀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⢰⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢅⡀⠀⠑⠢⢄⡀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⢸⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⢈⡲⠤⡀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⡀⠈⠑⠢⢄⢀⣀⢤⡴⠞⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣤⠀⣀⠤⠒⠁⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⣠⡪⠔⠊⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⢸⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⢀⠕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠁⠢⣤⢄⡀⠀⠈⡄⣀⣸⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣘⣗⢌⣆⠀⡇⡴⡿⢿⣷⠖⠢⠌⠳⡄
⠀⠀⠀⠀⠀⠀⢀⢩⠐⠒⢻⣀⠀⠀⠀⣠⡿⢸⣿⣾⠿⢼⣾⣤⡶⠊⢹⠦⠤⠄⢨⡇
⡤⠤⠤⠤⠤⠤⠼⠶⠴⠶⠶⠿⠟⠲⠶⠿⠿⢷⠟⠷⠶⠾⠻⠛⠿⠲⡟⠿⠿⠿⠾⠤⠦⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⡇⠀⠀⠀⠀⠀⠀⢰⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⠉⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣶⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣖⢤⢤⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⢸⣟⠀⠀⠀⠀⠐⠒⠀⠄⠠⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡆⢡⠘⣽⡾⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠘⠘⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢿⣿⣿⣷⠀⠀⠀⡇⢀⡄⠠⠄⠀⠀⠠⢠⠀⠀⠀⠀⠠⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠓⠚⠛⠛⢻⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡗⠀⡇⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣉⡀⣀⣾⣿⡇⠀⠁⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢻⡿⣿⠁⢸⣿⡆⠀⢀⠀⠈⠀⢤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⢈⡧⢼⣿⣇⠀⢸⡀⠀⠀⢂⣠⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠄⣰⣾⣦⣾⣿⡽⠀⢸⠇⣠⣴⣟⠉⠡⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢰⠇⣼⣻⣯⣌⣿⣿⣿⠀⢸⣿⣾⠋⣄⣀⠀⠀⠘⠲⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣈⣶⣿⠿⠿⣿⣿⢿⣿⠀⠸⣿⣿⣂⣀⡻⠏⠺⠆⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠻⡿⢫⡀⠔⢉⣿⣿⠿⠻⠂⢼⣿⡿⠟⠛⠁⠀⠐⠄⣀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠉⢺⠄⣿⡏⠀⠀⠀⠱⣯⢶⣣⠄⠀⠀⠄⠱⠤⠄⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣚⣛⠶⠶⢾⡄⢿⡁⠀⠀⠀⠘⢛⣋⣤⠞⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠒⠋⠉⠡⠌⣧⠐⠀⠀⠀⠀⠙⢛⣉⠥⢆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠻⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠘⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠐⣀⣀⣄⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣀⣼⠟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡶⠶⢶⠶⠶⡇⠀⣿⣿⠶⢶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⢾⢿⡇⠉⠀⠆⢳⡇⠀⢺⡟⠂⠈⠀⢣⠀⠡⠀⠑⠀⠱⠈⢃⠹⡈⠅⠹⠙⢏⠻⡙⢍⠻⡙⢟⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣻⣾⡇⠀⠁⠀⢸⠀⢀⣾⡇⠈⠀⠀⠀⠆⠀⠁⠀⠄⠀⠇⠈⠀⠁⠘⠀⢃⠈⠆⢱⠈⡆⠰⠈⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢼⡹⣿⣷⠀⠀⠀⣼⠀⠜⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢌⡷⣯⣿⠀⠀⠀⣷⣶⣾⣿⠁⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡮⣵⣻⡿⠀⠀⠀⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⡱⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡾⡤⣿⣏⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢼⣥⣿⣯⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⣸⣡⣿⣏⡠⠊⠠⠊⣀⠴⣂⠄⣠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠖⠛⣻⣿⣿⣿⣿⣏⠀⢀⡤⢚⠕⢊⠴⢊⣡⠖⣋⠔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⠄⢀⣿
⠀⠀⠀⢀⣀⣤⣴⠶⠞⠛⢉⣉⣤⣴⠶⢞⡻⣿⢿⣿⣿⣿⣿⣶⣥⣎⣡⣞⣥⣶⣫⣴⣋⣤⣄⣁⣴⣠⣤⣀⣠⣤⣤⣤⣄⣡⣴⣡⣄⣡⣴⣡⣔⣡⣆⣩⡿
⣠⠶⠛⠋⣉⣠⣠⡴⠶⠛⡛⠭⣴⠶⠛⠫⢙⣊⣉⣰⣬⠤⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⢻⠉⠹⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠁
⣿⣶⣿⣿⣭⣭⣤⣤⣤⣴⢶⠶⠶⠶⢛⣛⣛⣩⣭⣿⡾⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠘⢘⠜⣿
⠘⠳⠶⠦⠦⠴⠴⠴⠶⠶⠞⠒⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⣨⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⣽⢂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢡⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠰⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠼⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢴⡳⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢈⠀⢺⡝⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠐⠀⢹⠎⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡃⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢘
⣿⣿⡿⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠖⠚⠛⠉⠉⢀⣀⡀⠠⣤⣌⡩⠍⠛⠓⠲⠶⠤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣤⣤⠤⠶⠶⠶⠶⠖⠚⠛⠉⢩⣅⣀⣀⠤⠤⠔⠒⠒⠉⠉⠓⠢⠤⣀⡀⠙⠛⠒⠶⠤⣤⣀⣐⡂⠀⠀⠈⠉⠉⠉⠛⠛⠛⠒⠲⢦⣄
⠀⠀⠀⠀⢿⢿⠛⠫⢭⣉⠓⠒⠒⠒⠛⠻⢿⣯⣀⠀⠉⠲⠤⣀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠒⠦⠤⣐⡂⠀⠀⠉⢉⣑⣲⠶⠦⠤⠙⠒⢒⣚⣿⢿⣿⡇
⠀⠀⠀⠀⢸⠀⢳⡀⠀⠈⠉⠓⠦⢄⣀⠀⠀⠀⠉⠛⠲⢤⣄⡀⠉⠑⠢⢄⣀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠜⠓⠊⠉⠁⣀⣀⡠⠤⠒⠚⠉⠉⠁⠀⡜⢸⠃
⠀⠀⠀⠀⢸⠃⠀⢱⡤⠤⢤⣐⠀⠀⠈⠉⠓⠦⢄⣀⠀⠀⠀⠉⠓⠢⢄⣀⠬⠽⠖⠒⠋⠉⠉⠀⠀⢀⣀⡤⠶⠚⠋⠉⠀⠀⠀⠈⠀⠤⠀⢀⡞⠀⢸
⠀⠀⠀⠀⢸⡄⠀⠀⡇⠀⠀⠈⠙⠖⠦⣄⣀⠠⠀⠈⠉⠓⠶⢤⣄⣀⠀⠹⡀⠀⢀⣀⣤⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀⣤⣤⠞⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀⠀⠀⠀⠉⠙⢛⣷⣞⠉⠁⠀⠀⠀⢀⣀⣀⠤⠤⢤⠖⠒⠒⠉⠉⢹⠁⠀⠀⢸⡀⠀⠀⣾
⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠛⡶⠤⣤⡴⠋⢸⠀⠑⣤⠴⠒⠋⡏⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⢸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⢹⠀⠃⢸⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣗⠀⠀⠀⡇⠀⠸⡆⠀⠀⠀⠀⠹⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⢸⠁⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠈⡆⠀⢡⠀⢠⠀⡄⠈⠀⠀⡟⠀⠀⡀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⠀⠀⠀⡇⠀⠈⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠃⡇⠀⠀⡇⠀⠀⡄⠀⣤⠀⠇⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢸⡇⠤⠀⣧⠀⠀⠀⢸⠀⠀⢰⠀⠀⠇⠀⠀⠀⡇⠀⠀⠇⠀⠀⡇⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⢸⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⢀⡔⠛⢄⠀⢸⠀⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⡆⢻⠀⠀⠀⠀⡀⠀⢀⠀⠀⡄⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⢦⠀⡏⠀⢸⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣡⠞⠳⢤⣀⠀⡇⠀⠀⠀⠀⣇⣀⠤⠲⡇⠀⢸⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢛⣧⠤⠾⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢹⡔⠁⠀⠀⠀⠈⠙⠳⠤⣀⠀⠀⣿⠈⡇⠰⡾⠄⢸⠀⠀⢸⠇⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣀⣸⡤⠤⠖⠚⠋⣉⠀⠀⠀⠈⠳⣄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠄⠀⠀⠀⠀⠉⠒⠾⣄⡉⠛⡇⠀⢸⠀⠀⢸⠀⠀⠈⠀⢀⣀⣤⠷⠖⠚⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢈⣳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠐⠂⠀⠀⠉⠑⠷⣀⣸⠀⠀⢸⠀⠀⣸⠋⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠤⠔⠂⠀⠀⢀⣀⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⢳⡀⢸⢀⠇⠀⠀⠀⠀⠀⣀⣤⡴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠱⣸⡏⢀⣠⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠛⠟⠛⠻⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠻⠛⢛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠻⢦⣄⠀⠀⠀⠀⣠⡴⠟⠉⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠉⠻⢦⡴⠟⠉⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣴⣼⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁