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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠉⢀⡀⠀⡇⠀⠀⢀⢴⠀⡘⠯⠀⠙⡆⠀⡠⠊⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠐⡫⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠂⠁⠀⠀⠀⢤⢰⣎⡇⠀⠇⠀⠀⠀⠸⠰⠀⠒⡈⠴⠁⢠⡃⠀⣼⣧⡾⣸⢿⠋⣴⠂⠀⡴⠀⠀⣠⣤⠀⣀⠣⠀⠈⠁
⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⡄⡾⢳⡜⠿⠉⢁⢴⠃⠀⠀⠀⢐⠂⠐⠐⢀⡜⠀⢸⠳⡀⠉⠀⠁⠙⠛⠚⠷⠶⠸⠧⠤⢾⣥⣽⢐⠃⠀⠀⢄
⠀⠀⠀⢀⠀⡀⠀⢰⡄⣧⣿⠙⠋⠀⢀⠔⠁⠀⠀⠀⠀⠈⠓⠂⠮⠕⠁⠀⠀⠀⠑⢌⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢄⡀⠀⢢
⠀⠀⠀⠈⠄⠰⠀⠀⠛⠁⠀⠀⢀⡴⠃⡠⠊⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠬⣉⡒⠒⠀⠲⠖⠀⠒⠐⣒⡨⠅⠢⣌⢦⡀⢡
⠀⠀⠀⠀⠐⠀⢃⠀⠀⠀⠀⡠⢊⠔⠊⠀⠀⢀⠔⡒⠒⠐⠊⠀⠀⠈⠐⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠢⠱⡀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⢊⠔⠁⠀⠀⡠⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⠀⠀⠡⠀⠐⠐⠂⠀⠈⠉⠀⠀⡀⠀⠀⠀⠀⠀⠱⣱⡀
⠀⠀⠀⠀⠀⠘⡀⢁⠀⠗⠁⠀⠀⠀⢀⠇⠀⠀⣀⢤⡀⢤⡀⢶⡀⢰⢲⠀⢹⠂⠀⠀⡡⠀⠀⠀⠀⠠⡄⢦⢰⣖⠐⣶⠂⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠡⠀⠄⠀⠀⠀⠀⢠⠁⠀⠀⠀⣿⠀⡇⢠⡗⣦⠳⠸⠏⠇⢰⡄⠀⠐⡅⠀⠀⠀⠇⠀⡅⠸⠀⠻⠄⠘⠀⠸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠑⠲⠀⠀⠈⠘⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠂⠀⠀⠄⣀⠀⠄⣀
⠀⠀⠀⣀⠀⠀⠀⠰⠀⠳⠀⠀⠠⠀⠀⠡⢈⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠠⡀⡂⠀⠀⠀⠀⢀⠀⠀⠀⡌⠀⣤⢠⡄⠀⠀⠀⠀⡑⠠⠁
⠀⠀⠀⣀⠈⠢⡀⠀⠡⠁⠀⠀⠈⠐⠁⠀⠀⠀⠀⠉⠂⠂⠁⠂⠐⠉⠈⠉⠁⠀⠀⠀⠀⠙⠈⠀⠀⠀⠚⠁⠀⠀⢠⠁⢀⠌⣸⢇⠗⡷⠀⡐⠀⠁
⠀⠀⠠⢀⡈⠀⠐⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢄⠀⠀⠀⣀⠄⠀⠄⡀⠀⠀⠁⠈⠘⠁⢠
⠀⠀⠐⠀⠀⠙⠂⠀⠀⡠⠈⠀⣀⠀⢀⡄⣠⡠⣤⡤⢤⠀⢠⣄⣤⣠⠄⢀⡤⣠⢄⡄⠀⠠⠀⠀⡇⠀⠈⠔⣢⢀⠀⠀⠈⠀⠀⠄⠄⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢄⣀⠀⠀⠈⠁⠉⠀⠉⠁⠈⠀⠀⠁⠉⠉⠁⠀⠉⢉⣀⣀⣀⠠⠤⠔⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⡀⣀⠤⠒⠒⠒⠒⠒⠠⢄⠀⠀⠀⠀⠀⠀⡀⠤⠁⠀⠐⠒⠂⠈⠉⠉⠨⠀⠀⠀⠀⠀⠀⠀⠠⠔⣊⠜⠉⠢⠤⠤⠰
⠀⠀⠀⠀⡚⣡⠊⠁⠀⠀⠀⣀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠇⣠⠀⢀⡀⣀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠌⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢐⡃⠀⠀⡖⢺⠋⠙⣺⠁⠀⠠⢆⠀⠀⠀⠀⢰⠀⠥⡖⠫⠚⠥⠓⡜⠀⡘⠀⠀⠀⠀⠀⡄⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠆
⠀⠀⠀⠀⡼⠁⠀⢀⣁⡈⢀⢀⢁⣀⠀⠀⠀⢳⠀⠀⠀⡈⠀⠂⢰⢰⠶⢱⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠡⠄⠒⢀⠀⣀⠀⠀⡎
⠀⠀⠀⠰⠁⠀⠀⠾⠂⠇⠾⠾⠿⠧⠀⠀⠀⡰⠀⠀⠀⠇⠀⠓⠙⠃⠘⠘⠓⠈⠀⠐⢄⠀⠀⠀⠠⡁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠠⡀
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⣀⡀⠀⠀⠤⠤⠄⠀⠐⠁⠑⠄⠆⠀⠀⠀⠢⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠒⠉⠁
⠀⠀⠀⠀⠀⠉⠒⠶⢤⠄⢀⠰⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⡘⠀⠀⠀⠀⠀⠁⠀⠐⢠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⡤⠒⠂⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠐⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢶⣤⣴⠢⢔⣄⡠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⡸⠆⡼⠙⢄⣠⢴⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⢿⠾⠃⠁⠈⠁⠀⠀⠉⠙⠻⢀⢀
⠀⠀⠀⠀⠀⠈⢢⣄⣀⠟⢄⠴⠋⠒⠁⠀⠀⣾⣋⣀⢸⣼⣆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠉⠀⠀⡀⠀⠠⠈⠂⠐⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢀⡅⠀⢀⣴⠶⠿⣟⣻⡆⣶⣤⡽⢿⡾⣿⠿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠁⠀⢤⠐⠦⠀⣀⠱⠀⣠⠈⣅⠃⠀⢼⣄
⠀⠀⠀⠀⠀⠈⠉⠲⠀⢸⣇⠀⢀⣿⠛⢶⡟⣙⢆⣘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡆⠀⠐⢦⡀⠑⡄⣀⡠⠧⠈⣀⠄⢀⡃⣾⡉⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡠⠤⡝⠛⠛⡍⡠⣨⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡅⠀⠓⠄⠔⠒⠡⠀⠠⠈⠘⠁⢀⡀⠛⠛⠓⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠋⠱⢿⠟⠠⠀⠹⠇⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣳⡦⡀⢀⣀⡀⠀⢀⣤⡤⡀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠊⠩⣝⠞⠋⠀⠐⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⡀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣸⠋⠉⢲⣆⠆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣼⡃⠀⠈⠿⠀⠀⠘⠇⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠳⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⡴⠛⠛⠷⠀⠀⠤⠠⠒⠔⠦⣠⡌⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢱⣟⣷⠄⠀⠀⠀⠀⠀⠀⠙⠆⠀⠻⡄⠀⠀⠀⠸⢧⡀⡀⠀⠐⡄⠀⠐⠊⠂⠙⠉⡇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡔⠈⠉⠱⢞⡴⣀⡀⢀⡀⡀⠀⠈⢦⠀⠀⠀⠀⠀⠀⡼⠛⠃⠀⠁⠀⢰⣄⣠⣄⣦⣤⢋⠉⠀⢀
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠐⠚⣷⣛⡾⣳⢱⡆⣦⢈⠀⠀⠀⠀⠀⠀⢧⡀⢈⣤⢨⣷⠿⠿⣿⡟⣉⠻⠀⢡⣆⠈⠀⠈
⠀⠀⠿⣇⡀⠀⠈⢱⡀⠀⠀⠀⠁⠈⠁⠉⠘⠀⢣⡏⠀⠀⠀⠀⠀⠀⠀⠉⠋⣏⠀⠁⣾⡇⠙⠀⣿⢆⠀⣆
⠀⠀⠀⠙⠷⠀⠀⠀⠛⠦⣄⢄⠀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠙⡟⢒⢚⠑⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠒⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⠆⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⣀⡐⠀⠀⠀⠀⠀⠰⠶⢐⡄⢀⣠⠞⠛⠉⠉⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⠟⠉⠉⠈⠉⠙⠆⠀⠀⠀⠀⠀⠀⠈⢀⣿⠁⠀⠀⠀⠀⠀⣀⢀⠀⠙⢠
⠀⢄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡽⠃⠀⠀⠀⣠⣄⣦⣶⡳⠀⣿⠤⠀⠀⠀⠀⠀⠘⡄⠀⠀⣤⣲⣀⡤⠀⠀⠂⠈⢱⡃
⠀⠘⠀⠀⠘⡈⣿⣷⣶⣄⠀⠀⣈⠤⠀⠀⠀⠀⠀⠀⠈⠇⢰⣶⡾⠽⠛⠋⠈⠀⢀⡶⠯⠤⠄⠀⠀⠀⠀⠀⠳⣀⠁⠈⢀⡠⢤⠀⠀⡶⠒⢋
⠀⠀⠀⠀⠐⠀⠙⠿⣿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠁⠀⠀⡀⢀⣴⠴⠚⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠁⠀⠀⠃⣄⣧⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠶⠞⠃⠏⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠂⠀⠀⠀⠰⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣠⣴⡾⠋⠉⠉⠉⠙⢿⠶⢦⣄⠀⠀⠀⠀⣠⣴⣾⡟⠋⠉⠉⠉⠉⠉⠉⠛⣲⣤⡀
⠀⠀⠀⢀⣾⠟⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⣸⡟⠈⢻⣄⣷⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿⠇⠀⢻⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠋⠀⠀⠈⠻⢶⣶⠶⠾⠿⢿⣦⠀⣰⠟⠛⠋⠁
⠀⠀⠀⠀⠙⠷⢶⣶⠒⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⣁⣴⠏
⠀⠀⠀⠀⠀⢀⣸⣿⠿⠿⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠛⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠿⠿⠿⢶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⢶⣦⣄
⠀⠀⠀⢠⣾⣯⡅⠀⠀⠀⠀⠀⠀⠀⡈⢻⣿⣿⣦⠀⠀⠀⠀⠀⢠⣿⣯⣀⡀⠀⠈⠹⣿⣶⣦⣄
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠇⠀⢻⣇⠀⠀⢠⣾⠟⠉⠉⠈⠉⠀⠀⢀⡿⠀⠈⢿⡆
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⡿⠀⠀⣿⡇⠀⣀⣀⠄⠀⠀⠀⠈⠀⢀⠀⣸⡇
⠀⠀⠀⠀⠙⠿⣶⣶⡶⡞⠀⢸⣷⣦⣄⣠⣴⣦⣾⠃⠀⠀⠘⢿⣾⡟⠀⠀⠀⠀⠀⠀⢀⣿⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⣾⠃⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠹⣿⣤⣀⢀⣀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠒⠒⠂⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣭⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠤⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⢦⣄⡀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣴⣶⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢿⡀
⠀⡀⢀⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠃⠎⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣼⣿⠁⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀
⠀⠀⢠⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⣼
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⢃
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠖⠁
⠀⠐⡀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⢣⡑⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢋⡀⣠
⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋⠁⠂⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠦⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠛⠛⢻⣦⣀⠀⠀⠀⢻⣆⢆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⢷⣦⣀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠻⢷⣦⣙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠘⢦⠎⠀⠀⠀⡷⠒⠖⠲⠗⠤⠤⠤⠤⠤⠤⠤⣀⢀⣭⡉⢸⠀⣸⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢀⡠⠤⠧⣼
⣤⣄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⡟⠖⠁⢸⢸⠹⡀⠀⠀⠀⠈⠓⠤⢤⣤⣴⠃⠀⠀⠀⠈⢧⣠⣤
⠀⠀⡇⠀⠀⠀⣀⠞⠀⢀⠔⠊⠉⠉⠙⠒⠚⢡⠇⠀⠀⡇⠀⠀⢸⢸⠀⠉⠉⠉⡏⠉⢳⢰⣋⣁⣸⠀⠀⠀⠀⠀⣸⡇
⠀⠀⡧⠤⡀⢎⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⢸⠾⣄⣀⣀⣀⣧⠖⠹⡎⠀⠀⢸⡣⢄⣀⣠⠔⠁⡇
⠀⠀⢧⣄⠉⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡧⣤⠤⠞⠀⠀⢧⣈⠁⡇⠀⠀⠁⠀⠀⠈⠀⠀⠀⢸⠀⠀⡇
⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠧⣇⠀⠀⠀⠀⡔⡩⠖⠚⠦⣄⡀⠀⠀⠀⠀⠀⡴⠓⢦⠀⡇
⠀⠀⡧⠤⠤⠤⠤⠤⠵⠤⣤⣀⣀⡀⠀⠀⢀⠞⠀⠀⠀⣠⡴⠃⠀⠲⢖⡛⠁⠀⠀⠀⢀⣉⣉⣉⠉⠉⠒⡡⢄⢀⠧⡇
⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠚⠁⠀⠀⠀⢠⠁⡆⠀⠀⠀⠀⡟⡆⠀⠀⠀⡏⠀⠀⠀⢹⡀⠀⠑⠋⡇⠀⣇⣀
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠤⠤⠤⠤⠔⡣⠊⣧⠀⠀⠀⠀⡿⡇⠀⢲⠲⠃⠀⠀⠀⢸⡇⠀⠀⠀⡄⠀⠀⠈
⠀⣠⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⠤⠎⣠⡀⠰⣶⠲⠒⠁⠀⠻⠤⠤⠤⠤⠇⡇⠀⠀⠙⡆⠀⠀⠀⢸⠇⢠⠚⠉⠳⠤⣤⢤
⠔⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣀⡰⠃⠙⣄⢈⣙⣢⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢠
⠤⠤⣄⡀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⢆⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⡇⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣄⡀⠀⠀⠀⣠⠔⠓⠒⠺⡇⠀⠀⠀⢸⠤⠧⣄⠀⠀⡠⠚⠁
⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠀⠀⠀⣠⠔⣇⠀⠀⠀⡸⠀⠀⢸⠀⡜⠀⣀⣠
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠰⠁⠀⠀⠉⠉⠉⢣⠀⠀⢸⠀⡅⠀⡇
⠤⠤⠤⠤⢤⠭⠟⡹⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢣⡀⠳⠤
⣀⣀⡤⠖⠁⠀⠀⢇⠀⠀⠀⠀⠘⠤⠤⠤⠤⠤⠤⢤⠤⡤⠤⢄⠀⡤⠤⠜⠀⠀⠱⢄⡀⠀⣀⡠⠋⠀⠀⠸⣄⠀⠑⢤
⠛⠒⠲⠤⣔⠒⠋⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠲⡇⠀⠀⠻⠀⢠⠖⠒⠒⠒⠒⢧⠟⠳⢦⡀⠀⠀⢀⡤⠛⢀⣼⣀
⠀⠀⠀⠀⠈⢣⠀⠀⠈⠓⠦⠤⠤⡠⠤⢄⠀⡞⠁⠀⠀⣇⠀⠀⣀⡤⠼⠶⠶⠦⣄⠀⠈⠀⠀⠀⡇⠀⠀⡸⠀⠀⡇⠘⠁
⣀⣀⣀⣠⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠂⠀⠀⡨⠖⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡗⠒⡎⠀⠀⠀⠙⣆⣀
⠀⠀⠀⠀⠙⠒⠒⢲⠄⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⡲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣀⣀⣠⠗⣏⣁⠀⠀⠀⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠺⢲⠁⠀⠀⡰⠒⠒⠒⠒⠒⠚⡃⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⡴⠋⠁⠀⠀⠀⠀⠉⢢
⠤⠤⣄⠀⠀⡤⠤⠴⡊⠘⠯⣉⠁⠀⠀⠀⠀⠀⠀⠙⠦⢤⡀⢀⣤⠔⢺⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠈⠑⢄⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠐⢄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡠⠤⡄⠀⢣⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠥⠀⠀⠬⠆⠈⡄⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⣍⠀⠂⠀⠄⠂⡘⠀⡇⠀⠀⠀⢸⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠈⢢⡀⠫⢃⡠⠃⡠⠃⠀⠀⠀⠗⠊⠁⠀⠑⢄⣀⠀⣀⡠⠞⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣤⠞⠀⠉⠹⠦⣒⠛⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⠂⠤⠤⠤⠒⠁⠉⠢⡀⠀⠀⢀⠛⠉⢙⠏
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⣤⡤⡤⡄⠀⠀⠀⠘⡄⢀⠼⣀⡠⠃
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡆⢀⠎⠀⠈⣿⠇⠀⢰⠀⠀⢈⠎⠀⠀⡏
⠀⠀⠀⠀⠀⠀⡘⠀⠀⡠⠋⠁⠑⢢⠎⠁⠀⠀⢸⡄⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠇⠀⠊⠀⠀⠀⡠⠊⠀⠀⠀⠀⢸⠈⠒⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡴⠊⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠒⠒⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠟⠉⠉⠉⠉⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠷⠤⣤⣀⣀⣀⡠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⣀⣠⢊⣽⠥⡿⠭⠧⣀⡀⠀⠀⠀⠀⢠⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⡞⠁⠐⡍⠓⠲⠤⠤⠤⠤⠤⠤⠒⠚⠉⠁
⠀⠀⠀⠀⢀⣰⡅⠚⡀⠙⠄⠓⠀⠀⠀⠈⠑⢦⡀⠀⠀⢓⠶⠚⠁⠀⠀⠀⠀⠀⠀⢙⠒⠒⠁
⠀⠀⠀⠀⣷⠁⠑⠄⠈⠀⠀⢀⡀⠀⠀⡀⠀⠀⠙⢆⠀⠓⠐⠁⠀⠀⠀⠀⠀⠀⠀⠐⠔⠀⠀⣦⣔⣪⣅⡠⠈⠀⡐⠤⣀⠀⠀⠔⠀⠉⠒⡄
⠀⠀⠀⢰⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠈⡎⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡔⠁⠀⠈⠁⠀⠀⠀⡌⡆⠈⠳⣍⣿⢍⠙⡄⢸
⠀⠀⠀⢸⢠⠤⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⠀⢀⡀⠀⠱⡘⣄⠀⣈⢧⠈⠀⣇⡇
⠀⠀⠀⠘⣌⠉⠀⠀⠀⠀⠀⠈⠢⡄⠠⠈⠀⠀⠀⣠⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠐⠇⠀⠈⠀⠀⠀⠘⢮⡓⠬⡎⡄⠀⡟
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⣀⠔⠁⠀⢈⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠈⠁⠐⡇⠀⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢄⢀⣀⠠⠤⡔⠊⠁⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠉⠒⠛⠁⠀⠀⠀⠀⠀⠀⠬⠇⡸⢌⠂⠈⡃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠙⠖⠃⡇⠳⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢓⠀⠀⠀⠉⠢⢀⠀⠀⠀⢀⣀⠀⣐⡠⠊⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡿⠀⠐⠒⠃⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡴⠐⢄⠜⢹⠏⠣⡉⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠁⢸⡇⠀⠈⠉⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⣤⣀⠔⠋⠀⠀⠀⢹⡏⠉⠉⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⢀⡞
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⡿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠋⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠳⡎⠙⢞⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠀⠀⢹⡀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣠⠇⠀⠀⠉⢿⠄
⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠲⣤⣽⠆⠀⠀⠒⠲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠙⠋⠁⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠛⠲⢦⣄⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⡀⠀⠀⢀⣀⡤⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⢤⣠⣇⡴⠋⠉⠉⠉⠉⠁⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⠿⢿⣿⣷⣠⠶⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠙⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣭⣭⣭⣭⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠬⠭⠭⠥⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿

⠀⠀⠀⠀⠀⠀⡇⠀⣾⠀⠀⠀⠀⠀⠀⡗⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⢰⡆⠀⠀⣗⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⠀⢸⡆⠀⠀⣇⠀⣿
⠀⡀⠀⠀⠀⠀⠙⠒⣿⠀⠀⢸⡇⠀⠀⣿⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡇⠀⠀⢸⡇⠀⠀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⣸⣇⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣼⣧⣤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡋⢙⣿⠉⢹⣿⠉⢹⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣶⡦⠀⣶⡆⠠⣶⡆⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⢦⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡹⣿⣿⣿
⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⢈⡇⠀⠀⢸⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡷⣹⣿⣿
⠀⠀⠀⢀⡴⠿⠋⠁⠐⢤⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢸⣿⣿⣿⣿⢋⣠⣴⣿⣏⡻⣿⣿⣿⣿⣿⠟⣡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠦⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣭⣧⣙⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣦⣀⣴⠾⠛⠟⢷⣦⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⣾⠋⠀⠀⠀⠀⠀⠈⠛⠁⠀⠐⠠⡱⡙⣷⣤⣤⣤⣤⣀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠻⠹⠱⡱⡹⢙⢷⣄⠈⠆
⠀⢀⡤⠖⠂⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠡⠁⠁⢃⠜⣻⡆
⠰⠁⣠⡶⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠛⠟⡷⣦⡀
⠁⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡐⠠⠊⣻
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣽
⠀⠀⠻⣦⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⢠⡟⠁⢠⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠸⣧⣸⣟⣗⣴⣼⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠙⠿⣾⣮⣿⣧⣿⣮⡵⣡⡖⡠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣵⣟⣼⣱⣳⣷⣆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠁⢻⡆⠀⠀⠀⠀⣰⠟⠳⠶⠶⠶⠞⠉⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋⢀⠴⠤⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣤⡾⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠁⣀⠔⠉
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠽⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⠕⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⡀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⢻⡀⠀⠀⢠⠃⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣴⣠⡀⠳⡀⠀⠸⡀⠀⠀⠀⠀⡠⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢳⣷⠀⣠⢛⡩⠭⠭⣻⡶⣚⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠙⠤⠥⣰⠞⠁⠈⠉⠒⠦⣀⡀⠀⠀⠀⠀⢀⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣊⡛⠀⠀⠀⠁⠀⠘⣆⠀⠀⠀⠀⠉⠑⠢⢄⡀⢜⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠁⣩⡄⠀⠀⠀⠀⠀⠊⠳⣄⠀⠀⠀⠀⠀⠀⠉⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡇⠀⠀⠀⠀⠀⠐⠀⠈⠑⠦⣀⠀⢀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⢒⡶⢶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⢀⡠⢮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣍⣇⠀⠀⠀⠐⠉⠀⠀⡇⢀⠀⢀⢧⠋⣠⣾⢧⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠌⣑⠲⠤⣄⣴⠤⣤⢿⠋⡇⠾⠞⡄⠉⠀⠸⠒⠒⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠉⠲⣤⣈⢣⢀⠗⢧⣧⠀⢀⠟⢆⡀⢚⠉⠉⠲⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⠓⢬⡑⠿⣞⠤⠀⡛⣷⠾⡀⢒⣩⣣⠬⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠉⠢⣈⠙⠶⠗⠁⣠⠞⢢⢇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠳⣄⣠⣞⠙⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡟⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠒⠶⠤⢄⣀⣀⡀⠀⠀⠀⣼⣄⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⠔⡡⠎⢉⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⣠⠤⣶⡥⠎⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⣼⣁⣀⣀⣀⣠⣎⣀⢠⣾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠛⠓⠈⠉⠉⠀⠀⠈⠉⠈⠛⠐⠀⠬⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠐⣀⣀⡠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠂⠉⠁⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⣤⣤⠶⠶⠓⡿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⢀⡀⠀⠨⠐⠀⠀⠀⠀⠉⠙⢋⣉⡠⠤⠴⠒⠛⠁⡀⠀⠀⠐⣀⠀⠀⢣⡄⠀⠀⠸⡀
⠀⠀⠀⠀⠁⠄⠀⠀⢠⡤⠶⠚⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠖⠁⢀⠀⠂⡐⣦⡈⢦⢨⣷⡀⡐⠀⢱
⠀⠀⠀⠀⠐⠈⣠⠠⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠄⢦⠔⢆⣤⠡⠙⣉⢤⠞⠛⠉⠀⠀⠀⠘⡂
⠀⠀⠀⠀⠀⣯⣆⢀⢾⡇⠀⠀⠤⠒⠋⡁⠀⢀⢋⠀⠩⣑⠘⠚⣀⠤⠒⠋⠁⠀⠀⠀⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⢈⢿⣆⠀⠫⠀⠀⡀⠂⠰⠀⠃⠀⠐⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠉⢻⡦⢼⡄⠀⠘⡂⣀⡁⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠠⢳⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠔⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠋⠉⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠛⠿⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡐⡄⣤⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢻⣄⢀⣹⣶⣽⣿⣿⣧⣿⣿⣤⣴⣶⣶⣶⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢢⡈⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢄⠱⣾⣿⣦⣙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠬⢦⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⡉⠙⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠈⠻⢿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠴⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠿⣿⣿⣿⣷⣤⡈⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⣒⣒⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣟⣿⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣯⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠥⢞⣻⣿⣿⣿⣿⣿⣿⣏⢿⣿⣧⢻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢊⡿⢟⣻⣿⢿⣿⣿⣿⡙⢾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠩⠄⠊⠩⠚⢡⣾⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⢿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡞⠁⠁⡾⠋⠹⡟⠁⠉⣽⡟⠉⠹⠿⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠀⠀⠉⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢂⠀⠔⠂⠀⠐⠒⠁⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠎⢐⠢⢀⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⡀⠀⠀⠀⠰⢄⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⠀⠀⡂⡄⠀⠀⠀⠀⠔⠀⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠌⠚⠐⠠⡷⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠡⡟⠀⠂⡤⢤⠃⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠀⠀⠉⠐⠒⠃⠀⢅⢀⠀⠀⣸⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠦⠤⡬⠃⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠂⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠂⠁⠈⢂
⠀⠀⠀⠀⠀⠎⠈⠁⢀⠀⠀⠀⠡⠀⠀⠀⠈⠐⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠠⠐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⠀⢰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⡠⢰⡯⡔⠁⠀⠀⠠⠹⣂⠀⠀⠀⠀⠀⠀⡆⠀⠁
⠀⠀⠀⠀⢠⠰⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠔⠁⢀⡩⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⢃⠀⠀⠀⠀⣀⠤⠐⠂⠈⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠑⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⠀⠀⢆⠀⠀⠐⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠢⠤⠓⠂⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⡀⠀⠀⠀⣀⣀⣀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠉⠂⠈⠁⠀⠈⠙⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠲⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢠⠑⠄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠾⠃⢤⡌⠁⠘⠀⡀⢄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠒⠐⠁⡶⡰⢤⣚⠉⠀⠀⠈⠃⠀⠁⡀⠀⠁⢠⣀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠲⠄⠄⢸⣏⠀⠑⠎⠡⡀⠀⠀⠀⠈⠀⠀⠉⢀⠀⢀⡀⠉⠄⠀⠅⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡀⠀⢀⣄⢠⡈⠉⠈⣀⠀⠀⠘⠃⠀⡘⠀⠀⢀⠀⠈⠓⢂⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣠⠀⡀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠐⠀⠀⠙⢋⣀⣼⢶⡀⠀⡲
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣤⣀⡟⡳⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠿⣀⡆⠉⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠙⡏⡯⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣀⠀⠀⠀⠈⠙⣉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⢘⣎⣤⠀⠀⠀⠀⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⠀⠀⡄⠀⠀⢇⡿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⣀⠀⣸⢠⡤
⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⡄⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢻⡉⠓⡦⢀⠐⠃⠀⢀
⠀⠀⠀⠀⠀⣠⣁⡴⠁⠈⠉⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣰⠐⡠⣄⡀⣀⡈⠗⡎⠠⠍
⠀⠀⠀⠀⠴⠟⠻⢶⡀⠑⠈⠀⠀⠐⠁⠈⠁⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⣴⡾⠖⡉⢻⣿⣧⢶⢷⣦
⠀⠀⠀⠈⠴⠀⢤⣼⢽⠀⠀⠀⢀⢀⡀⡄⠄⠀⢀⢸⠄⠀⠀⠀⠀⠀⠀⣜⠁⠀⢘⡗⠀⠀⠙⠻⠂⠋⡀⠎⠁
⠀⠀⠀⠀⠀⠀⡝⠁⠈⠀⠁⠤⠬⣈⠄⢀⠔⠀⣮⣼⠦⠀⠀⠀⠀⠀⠀⣝⠉⠉⠉⢳⣴⡆⠀⠉⠀⠰⠗⠂⣉⠁
⠀⠀⠀⠀⡄⠀⠂⠀⠀⢠⠖⠀⠀⠀⠀⢀⠀⠨⢿⣿⠑⠂⠀⠀⠀⠀⠀⠈⢆⠠⣴⢀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⣣⡀⠄
⠀⠀⠀⠁⠀⠀⠀⢁⠀⠾⠀⠀⠀⠀⠀⠘⡀⠠⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡥⠀⠀⠀⠀⢀⣰⡃⠙⠷⠇⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠉⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡛⣻⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠙⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠋⢉⣀⣀⣤⣤⣄⣀⡀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣀⠈⠣⡀⠀⠀⠀⣀⡠⠤⠤⠤⠤⣄⠀⠀⠙⢦⡀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⠎⠈⠳⣄⠈⠢⣠⠎⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠱⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡏⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢹⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋⠀⠀⢤⣄⣀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠠⣤⣬⣭⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⣆⠀⣤⠶⠞⠋⠁⢹⣀⣀⡀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⡰⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⢆⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⣀⣀⣀⣠⣄⠈⠢⡀⡰⠁⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⢀⠤⠊⢁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣈⠉⠉⠀⠀⠈⠉⣀⣠⠤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣄⣀⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣿
⠀⠀⠀⠀⣠⣾⡿⢻⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿
⠀⠀⠀⣰⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠈⠁⢈⣍⡙⢻⣿⣿⣿⡇⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠈⣿⣿⣿⠃⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣜⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⠙⣷⠀⠿⣿⡟⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠈⣿⠿⠻⢿⣿⡇⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⢈⣸⠇⠀⢸⡇⣠⠞⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡇⠈⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⢸⠋⠀⠀⠈⠛⠁
⠀⠀⠀⠀⢙⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⠀⠀⢸⣷⣶⣦⣤⣤⣀
⠀⠀⣠⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣤⣤⢤⡤⠴⠚⠁⠀⠀⣾⣿⣿⣿⠃⠀⣽
⠀⣼⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠛⠛⣿⠀⠀⠀⠀⠀⣸⣿⣿⠛⠳⣤⠴⢧⡀
⢺⣿⣿⣿⣿⣿⣿⣿⠏⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⣠⡾⠛⢿⡇⠰⢤⡏⢀⠞⢻⡆
⠙⠿⢿⣿⡿⠿⠛⠁⠀⠀⠀⠙⣳⡦⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠺⣧⡀⠀⠈⣷⠀⠈⠙⠻⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠁⠀⠀⠀⠉⠛⠻⠿⠿⠛⠁⠀⢀⠀⢙⣦⡞⠁⠀⢠⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠹⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠟⠀⠀⠀⣰⠏
⠀⠀⣀⣤⣤⣠⡤⠞⠁⠀⢀⡴⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣤⠞⠁
⠀⠸⣇⠀⠀⠀⠀⠀⠀⡴⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠸⣟⣛⣻⢿⡄⠀⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠀⠀⠙⠛⠋⠀⠀⢻⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣶⡖⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠚⠛⢻⡇⡟⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡼⢧⣤⣤⣼⠀⣷⣤⣤⣴⢧⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣙⣃⣛⣁⣀⣀⣸⣆⣇⣀⣀⣈⣛⣘⣋⣉⣳⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣶⣶⣶⣴⣶⣶⣾⣋⣷⣶⣶⣶⣶⣶⣶⣶⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⢀⣠⣾⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⣷⣄
⠠⠞⠉⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣉⠉⠙⠛⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠛⠋⠉⣉⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠉⠳
⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣆⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⡦
⠀⠀⠸⢿⣿⡿⠟⠋⠀⠀⢸⡼⡿⢻⣿⣿⡟⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠟⠁⢿⣿⣿⡟⢿⣥⡗⠀⠀⠙⠻⢿⣿⡿⠇
⠀⠀⠀⠀⠘⠿⢷⣤⡀⠄⡈⠓⠘⢪⠥⢚⣁⣠⣀⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢉⣀⣄⣈⡳⢌⣕⠃⠚⢁⠠⢐⣴⡾⠿⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⠶⠿⠾⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠾⠶⠾⠛⠋⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠟⢿⣀⡀⢀⡀⢀⡠⠆⠸⡇⠀⠀⠀⢸⢶⣠⣄⠀⠀⢄⣷⣴⡆⠇⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡞⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡰⢾⣳⠀⠀⢴⣿⣿⣜⢿⣧⡰⡖⠂⠸⣦⣠⠀⠁⠘⢿⡟⢀⣶⡿⢿⠿⠇⠸⠄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⢆⢁⠤⠠⢈⠁
⠀⠀⠀⠀⡀⣀⣾⣧⡌⠉⠀⠀⠨⠿⠋⠛⠟⠛⢁⣿⡶⠆⠈⣷⢷⠿⠋⠁⣠⢿⠁⣀⣠⣄⠁⠀⠀⠈⠓⠂⠂⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⣨⠏
⠀⠀⠠⠤⠿⠿⡟⠉⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢂⡄⠦⢱⣀⠠⢉⣯⣼⣿⣛⣃⠀⠀⣴⡀⣀⣀⠀⡀⢀⡉⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠆⠀⠀⠀⡀⢈⡠⣦⡀⠀⠀⠀⠐⣾⣿⣊⡀⠐⣼⣦⠠⠼⢿⣿⣿⣿⣿⣿⣿⣿⡤⠀⠺⣿⣿⠿⢓⣀⣸⣤⡃⠀⠀⠤⣸⠝⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠑⠀⠁⠠⣍⡓⠸⣿⣧⢤⠀⠀⠠⠈⠛⣿⣿⣿⠏⢡⡀⣌⣶⣦⣿⣿⣿⢿⣹⣻⣄⣤⣿⡇⣠⢬⡙⠿⠧⡴⢠⢠⣀⠀⠀⠴⢦⡀⠐⠀⠀⠄
⠀⠀⠀⠀⣀⡀⢠⣴⠦⠝⠃⠀⣥⡇⠘⣡⠄⢀⣈⠀⢨⡁⢌⢓⣺⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⢈⡛⢁⣠⣈⣻⣿⣿⣦⣶⣲⡛⡁⢀⡂⢢
⠀⠀⠀⠀⠈⣷⠀⠀⠤⠤⡄⠀⠀⣿⣻⣿⣿⣿⣾⡄⣠⣼⡅⣶⢽⣿⣿⣿⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠘⣿⡾⢿⠿⠽⣿⣿⡿⣼⣻⣴⠁⡸⣹⡏⡄⠂
⠀⠀⠀⠀⠘⠋⠀⡇⠀⠀⠄⠀⠀⣿⠁⠸⣿⣿⣿⠇⠋⠿⠿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣷⠀⣾⡷⠱⠜⢾⣿⣿⣿⠿⢿⠟⢿⣥⡌⣁⡀
⠀⠀⣧⠀⠐⠆⢰⡇⠀⠀⠀⠀⣶⣿⡦⠨⣯⠍⢈⣡⣔⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣸⣿⡗⠀⠀⣸⣟⣿⣯⣀⣾⣞⣯⣭⡽⢉⠁⠀⡀
⠀⠀⣾⠀⠲⠛⢸⡷⠀⢀⣤⢠⣽⣿⡗⠒⣷⣰⣿⣿⠟⣿⡶⢾⣿⣿⣿⣿⣻⢟⣩⣿⣿⣿⣿⣼⡻⣿⣿⣿⣾⣿⣷⣴⣿⣿⣿⠟⣻⣿⣿⣿⣿⠿⠃⠀⠀⠐
⠀⠀⣿⠀⠸⠗⢸⣿⣿⣿⠋⠿⠿⣿⣇⢦⣿⠿⣿⣿⡷⢟⡓⠞⣿⣿⣿⣿⣿⣾⣿⣫⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⢿⡟⢛⠛⢡
⠀⠀⣿⣠⡄⠀⢸⣇⠀⠀⠐⠶⠇⣿⣧⠄⣿⣳⣿⣩⣯⣶⣴⣙⣻⣿⣿⣿⠟⢻⣿⣿⠀⠀⢀⠙⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣟⣠⣿⡴⠋⠩⠓
⠀⠳⢻⣿⡇⡀⠘⣿⠀⠀⠀⠀⣤⣻⡿⣀⣿⠈⢀⢨⣟⠛⠉⠉⠛⣻⣾⣿⣿⣛⣻⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠤⠄
⠐⢤⣌⣿⡤⠋⠘⡝⠀⠀⠠⣄⠘⢻⣿⣻⣿⠁⠉⢹⡏⠀⢀⣀⣠⠉⢉⣈⣻⣿⣿⣍⡉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠙⣏⣠⡔⠠⠀⠀⠀⠀⠈⢡⣼⣿⣿⡿⠁⠀⢠⣿⠉⠉⠉⠀⠀⠀⠀⠉⠩⠟⠛⣛⣏⣉⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡀
⡀⣦⣸⣤⠋⠀⠀⢀⠀⠀⠀⠒⢾⣿⣋⡄⠀⡀⠀⡐⣿⡄⠀⠀⠀⠀⠀⠀⣀⣐⠲⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠈⠦
⠁⣈⣽⣏⠁⠀⡀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠐⠀⢀⠀⡁⡀⠀⠀⠀⠀⣲⣶⡟⠉⠋⠙⠛⠻⠛⠻⠿⠿⠟⢻⠿⡿⠋⠻⠿⣿⣿⣇⠀⡄⠀⠀⠀⠀⠶⠰⢶⣦
⠈⢁⣀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⠁⠂⢄⠀⠋⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠁⢈⠒⠀⠀⠀⠐⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡀⠀⠀⠀⠀⠀⠀⢐⠈⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢇⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠒⠒⠃⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣦⡀⠀⠀⣉⠐⠄⠀⠀⡀⠀⠌⣠⠀⠀⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠠⠉⠀⠂⠀⠀⢄⠙⠋⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣄⣠
⠀⠀⠀⣀⡴⠶⣤⡀⣀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠵⠋⣠⠇
⠀⠀⠀⠈⢮⠂⠀⠈⠙⠻⢶⣠⣀⢀⠀⠀⠉⢁⣀⣲⣶⢶⣖⣒⣐⣒⠊⠀⠀⢀⣀⠔⠉⠀⠀⡘
⠀⠀⠀⠀⠀⠑⢕⠄⠀⠀⠀⠈⢹⠋⢤⡄⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠙⢺⡋⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠸⠀⣖⠄⢸⠀⠀⠀⠀⠠⠀⠀⢨⣍⣩⡅⠀⠁⢀⡠⡢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⢐⡀⢸⠀⠠⠤⠦⡠⠅⡀⠐⢒⡔⠂⠀⡏⢣⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢻⠀⢔⡀⠀⢉⣉⣉⣈⡀⠈⠁⠨⡵⠮⡅⠀⡇⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡸⠀⠀⠀⠀⠈⠁⠛⢸⠁⠀⠀⠐⠪⠭⠂⠀⣌⠛⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢡⡀⠀⠀⠀⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⠛⠛⠯⠭⠭⠭⠿⣩⡟⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠸⢄⡀⠀⠀⠀⠀⠀⡀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠀⠄⡄⢹⠄⠀⠀⢀⡈⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠈⠒⠲⠶⠖⠒⠉⠀⠀⠀⠀⠁⠲⠆⠤⡤⡤⠪⠀⠀⠀⠀⠂⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡤⣐⣬⣶⣿⣿⣿⣿⣻⣷⢦⣖⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣬⣷⣿⣿⣿⣿⠱⣿⣿⡿⣟⣻⣿⣿⣛⣛⣿⣾⣽⣮⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣷⢟⣵⣿⡿⣟⣯⣿⣟⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⡄⠢
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣾⢯⣿⡿⣫⣾⣿⣿⡿⠟⠿⣭⣉⣭⡴⢶⣛⣛⣻⣷⣶⣯⣟⡿⣿⣆⢡
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢿⣏⣿⣟⣽⣿⣿⣟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣶⣬⡻⣿⣮⣻⣆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⢻⡟⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣎⢿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠈⡿⢹⣾⢡⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡎⣿⠿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⢀⢂⠈⠳⣤⡀⠀⠀⠀⠀⠀⡇⢸⣿⣸⣿⠇⠀⠀⠀⣠⣴⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢸⣿⡇⢿⠀⡆⠀⠀⠀⠀⢀⡠⠄⠘⠃
⠀⠀⠂⠂⠀⠀⠈⠑⠒⠤⢀⢀⣇⢸⠛⠻⣿⡀⠀⠀⠀⣀⣀⣄⣈⣉⠛⠀⠀⠀⠰⠛⠛⠉⠉⠁⣿⣿⡇⡞⢰⠃⠂⠂⠀⠀⠀⢀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢾⢸⣸⡄⠀⡜⠃⠀⠀⠈⠉⣹⣿⡏⠻⠃⠀⠀⠀⠼⣿⣿⠛⡦⠀⣿⡟⢠⠃⠸⠄⠀⠀⠀⠠⡏⠀⠈⣱
⠀⠌⠌⠄⠄⠄⠠⠠⠀⠀⠌⠌⠎⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⢁⢎⣶⢸⠌⠄⢤⠤⢤⠹⠶⠶⠡⠌⠄⠄
⠀⠈⠈⠈⠀⠀⠁⠀⠀⠀⠈⠈⠜⡹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡎⣾⣿⡎⡌⠈⠱⠄⠴⠃⠀⠀⠉⠈⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡙⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠈⠀⠀⠀⠀⠀⣰⣟⢴⣿⣿⣿⣜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⠳⡞⣧⠀⠀⠀⠀⠀⠀⢴⣤⣤⣤⣤⢀⣤⠄⠀⠀⢰⣿⣟⢾⣿⣿⣿⣹⣦⠶
⠀⠠⠡⠁⠀⠀⠀⠀⠀⠀⠀⣡⠅⣁⣹⠀⠡⢹⣷⡄⠀⠀⠀⠀⠀⠙⢿⠈⢿⠟⠋⠀⠀⣠⣿⡯⠿⠆⠙⣿⣿⣷⢿⡞⡁
⠀⠀⠡⠡⠀⠀⠀⠀⠀⠀⢠⠱⣏⠿⣫⡇⠐⡇⣿⠎⠀⢀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡼⡿⣫⣾⣿⣿⣶⢪⣿⣿⡘⣱⠁⠀⠀⠀⠀⠀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⣽⣽⣫⡞⣴⢧⡿⠀⠀⠀⠙⠲⣤⣀⣀⡀⠀⢠⣶⠋⣶⡁⢿⣿⡿⣿⠿⣨⠟⢋⣰⠁
⠀⢀⢂⢀⢀⠀⠀⠀⠀⠀⢀⢋⢢⣉⣹⣞⠋⠔⠁⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⢿⣧⠻⣿⣌⡉⢁⣐⣚⣭⠶⣻⢃⠀⠀⠀⡀⡀⡀⣀⢀⢀
⠀⠂⢂⠂⠀⣂⡠⠤⣤⣶⡖⠲⡎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣽⢻⣮⣽⠛⣓⣦⠐⠚⠂⢂⠂⠂⠀⠀⠐⠀⠐⠀⠂⠂⠂
⠀⠀⣠⠔⠋⠁⠀⠀⣾⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⢈⣿⣿⣿⣿⣶⠂⢤⣄⡀
⢀⡞⠁⠀⠀⠀⠀⠀⣿⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣤⣼⡟⠀⠀⠀⠉⢢⡀
⡜⠀⠀⠀⠀⠀⠀⢠⡛⣿⡟⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⣟⡇⠀⠀⠀⠀⠀⢑
⠃⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠀⢀⡀⠀⠀⠀⠀⠀⢠⡾⢿⣿⡿⢿⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠐⠄⠀⠆⠀⠀⠀⠀⣷⣼⣿⣧⣾⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠃⠀⠀⠀⠀⢸⣿⣉⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢿⣿⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠌⠡⠡⠁⠀⠈
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣶⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢰⣿⣯⣠⣿⣧⣨⣿⣅⣼⣿⣄⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣅⣤⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⣾⡛⢻⣿⠛⢻⣿⠛⣿⡟⠛⣿⡟⢛⣿⡧⡀⠀⠀⠀⢀⡔⣻⡟⠛⣿⡟⢃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⡿⢿⣿⡿⢿⣿⡿⣿⣿⠿⣿⣿⢦⡀⣠⢿⣿⡿⢿⣿⡿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡷⠠⠀⠄⠄⠀⠀⠠
⠀⠀⠀⠀⠀⢷⣿⣷⣴⣿⣧⣴⣿⣦⣼⣿⣤⣾⣿⣤⣿⣷⣤⣿⣧⣴⣿⣧⣼⣿⣦⣼⣿⣤⣶⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣷⠀⠁⠈⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⢉⣁⣤⣤⣤⣤⣤⣤⣈⡉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣤⣖⣛⡉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣤⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠋⠁⠉⠉⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⢷⡀⠹⣄
⠀⠀⠀⠀⠀⠀⣰⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠻⡄⠹⣆
⠀⠀⠀⠀⠀⢠⡏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⢻⡀⢹⡄
⠀⠀⠀⠀⠀⣸⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣠⠞⠁⠀⠀⢀⡀⠀⠀⠀⠈⣧⠈⣇
⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣾⠁⠀⣀⠀⢶⠀⢳⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⢘⡇⠀⢸⡆⠸⡇⠸⡇⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⢹⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⣰⡏⠉⠁⠀⢀⡇⢀⡇⢀⡇⠀⠀⢀⡟⢀⡏
⠀⠀⠀⠀⠀⠘⣇⠈⣧⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠘⢯⡉⠀⠀⠀⠀⡼⠁⣸⠃⣸⠃⠀⠀⣼⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠘⣆⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠀⢠⠟⠀⠀⠀⠈⠁⣴⠃⣠⠏⠀⠀⣰⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢷⡴⠋⣠⠞⠁⠀⣤⢤⣤⣀⡼⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⡾⠁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢾⣅⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠛⠦⣤⣿⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠛⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣈⡉⠙⠛⠛⠛⠛⠛⢉⣁⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠇⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠒⣛⠒⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⢀⡀⠀⠘⠀⢀⡄⠀⠀⠱⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠐⠀⠀⠈⠁⠀⠜⠀⠈⠁⠀⢀⠆⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠲⠂⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠑⢼⡀⠀⢹⠀⠀⢸⠀⠀⢸⠅⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠠⠼⠄⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡯⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⢹⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢣⠀⣀⡄⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⡇⠀⠀⠀⠀⢀⡇⢸⣀⠀⡝⠒⢄
⠀⠀⠀⠀⠀⠀⢀⠚⢄⣠⠞⠉⠀⠘⠺⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⡇⠀⠀⠀⠀⢸⠗⠃⠀⠉⠢⣄⡠⠓⡀
⠀⠀⠀⠀⠀⢠⣁⠀⡜⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⢧⠀⣀⡄
⠀⠀⠀⠀⢀⠃⠀⡝⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢏⠁⠐⡀
⠀⠀⢰⠊⡉⢉⡘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⣉⠉⠑⡄
⠀⠠⠊⠈⠁⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣤⠀⢠⠄⠀⠀⠀⠀⠀⢠⡄⠀⠤⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠁⠀⠈⠉⠓⢄
⠀⠀⠀⠂⠁⠉⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⡇⠀⠀⠀⠀⠀⢸⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠉⠈
⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠚⡇⠀⠀⠀⠀⠀⢸⠓⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠲⠲⡇⠀⠀⠀⠀⠀⢸⠖⠒⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣗⢒⠲⡇⠀⠀⠀⠀⠀⢸⠖⢒⣺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠀⠀⠀⠈⠙⢆⠀⠀⠀⡰⠋⠉⠀⠀⠀⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣔⢾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠈⠻⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⣿⡟⠁⠀⣴⣷⣌⠻⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⢰⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⣿⣿⣿⣷⣝⢿⣿⣿⡄⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⣝⢿⣿⣄⠀⠀⠀⣤⣈⠛⣦⡀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⣝⠟⠀⠀⠰⣤⠙⣧⠘⣧⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⣄⠀⠀⣸⡇⣹⠆⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⡜⠋⣠⠟⢠⡟⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣦⡁⢴⠟⠀⠀⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠃⠙⢿⣿⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣦⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠉⠑⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠼⠟⠓⠒⠒⠿⠤⠤⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣼⡐⠀⠀⠊⡉⢳⣶⠪⡉⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠀⠈⡞⠀⠳⠽⠊⠍⠳⠿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣶⣿⠀⠀⢠⣤⣄⣄⣔⣲⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡏⡞⠀⡀⠸⠝⠉⠋⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣤⠤⠬⠸⣿⣿⣿⣿⣟⣭⢓⣂⠀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡉⠁⠉⠂⠈⡃⠁⢀⢢⠀⠀⠀⠀⠀⠀⢄⠆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠠⡘⠀⣀⣠⠀⠇⠀⢈⣀⣤⡤⣴⣤⡤⠶⢖⠆⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠘⢦⣤⣾⣿⡇⡿⠀⠀⢽⠀⠴⠊⡉⠁⠙⢲⢘⠅⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⣰⣿⣿⣿⢳⡇⡆⠀⣻⠀⠓⠀⠁⠀⠀⡡⢸⡇⢠⡀⢘⡀
⠀⠀⠀⠀⠀⠀⠰⠡⢌⢱⣴⣿⣿⣿⣿⣾⡇⠀⠀⣎⠎⠰⠀⠀⠒⢀⢊⢸⡇⢸⣷⠆⠘⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡉⣹⣿⣿⣿⣿⠁⡆⠀⣜⠐⣀⣃⣡⣤⣤⠴⠶⠃⢸⣿⣴⠤⠄⢀⠖⠻⡉⠁⠈⡰
⠀⠀⠀⠀⠀⠀⠀⠷⠶⠿⠀⢸⣿⣿⣿⡏⡾⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠑⠂⠠⢠⡀⢀⣽⢿⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠘⣣⡀⠀⠀⣿⣿⣿⢳⡇⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⢅⣀⣀⠤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢉⢣⣀⠀⠻⠿⢿⣾⣃⡵⡀⠀⣀⣈⣤⣤⣴⣷⣶⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢄⡈⠟⡉⠠⠀⣿⣿⡿⠟⠛⠋⢹⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢠⠘⣇⠀⢀⠀⠿⣿⠇⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠿⠒⠋⠀⢀⣿⠀⠀⠀⠀⠀⠆⣤⡠⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⡉⢲⡤⠄⠂⠳⢘⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢈⠄⣡⣤⠶⢦⡌⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⡇⠀⠀⢸⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢈⢠⡏⡏⠀⠀⠀⢸⠀⠀⢰
⢠⣴⣶⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣶⣾⣿⣸⢹⠁⠀⠀⠀⢀⣠⡤⠾⠂⠂⠀⢀
⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡍⣀⣀⠀⠀⣀⣼⣿⣶⣄⣠⠴⣒⠻⠆
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣙⣉⠁⠀⢀⣀⣭⣏⠥⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣟⣻⠿⣿⣷⡎⠉⢀⡨⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣶⣿⣶⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡌⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠙⠛⠛⠛⠛⠛⠋⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣶⣶⣶⣶⣶⣶⣾⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢈⣃⣀⣉⣉⣉⣉⣉⣥⣶⣶⣶⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣼⣿⣶⣶⣶⣶⡶⠛⠉⠉⠉⠉⣿⣿⣿⣿
⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠠⣿⣿⣿⡿⣟⣻⣿⣿⡶⠞⣿⣏⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢀⣀⠠⠄⠐⠂⣩⣥⠀⢰⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣺⣿⣻⣧⡵⢿⣛⣿⣭⣷⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠈⠅⠀⠆⠘⣊⡀⠤⠐⠒⠈⠀⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣯⣭⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃⠀⠀⠀⠙⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⣿⣿⡟⣁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⢏⣼⣿⣿⣶⣤⡀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣾⣿⣿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⠀⠀⣠⣿⣿⠿⠛⠛⠿⢋⣾⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠐⠛⠉⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣤⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠿⠶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⣋⣉⣉⣉⠉⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣥⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠞⠉⠀⣾⣿⡿⠋⢻⣿⣿⣿⣿⣿⡈⠙⠻⢷⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⣿⣿⣿⣶⣿⣿⠿⣿⣿⣿⠇⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⠉⠉⠛⠲⢶⣼⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠞⠛⠛⠛⠛⠳⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⣀⣠⣤⡤⠶⠶⠶⠶⢤⣤⣄⣀⠈⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣀⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣦⣀⠈⠛⢦⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠠⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠙⢷⡀
⠀⠀⠀⣰⠏⠀⣠⣦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠻⣆
⠀⠀⣼⠃⢀⣼⠋⠀⠙⢦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠙⣧
⠀⣸⠇⠀⣾⠁⠀⠀⠀⠀⠙⢦⡀⠈⠳⣶⠶⣤⣄⡀⢀⣠⣤⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹⣆
⢠⡟⠀⣸⠇⠀⠀⠀⠀⠀⣠⡾⠛⢦⡀⠈⠳⣄⠈⠻⠟⠁⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠸⣇⠀⢻
⣸⠇⠀⡿⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠙⢦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⣿⠀⠸
⣿⠀⢸⡇⠀⣠⣴⣋⡁⠀⠀⠀⠀⣀⣠⣤⠿⢦⡀⠈⠻⣶⠶⠶⣤⣄⣀⠀⠀⠀⠀⢈⣛⣦⡀⠀⢸⡆
⣿⠀⢸⡇⠀⠉⠻⣯⡉⠛⠛⠛⠛⠿⣥⣄⣀⡀⠙⢦⡀⠈⠳⣄⣠⣬⠿⠛⠛⠛⠛⢉⣽⠟⠁⠀⢸⠃
⢸⡆⠀⣷⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢦⡀⠈⠻⣄⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⣿⠀⢸
⠘⣧⠀⢹⡆⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠳⣤⡾⠋⠀⠀⠀⠀⠀⢰⡏⠀⣼
⠀⢹⡆⠀⢿⡄⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣽⣦⡀⠈⠳⣄⠀⠀⠀⠀⢠⡟⠀⣰⠏
⠀⠀⢻⣄⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠙⢦⡀⠈⠳⣄⠀⣠⡟⠀⣰⠟
⠀⠀⠀⠹⣦⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠻⠋⠀⣴⠏
⠀⠀⠀⠀⠈⢷⣄⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠂⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠉⠙⠛⠛⠒⠶⠶⠒⠛⠛⠋⠉⢀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⢤⣤⣤⡤⠶⠶⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠙⠛⠛⠛⠻⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⢛⣛⡛⠛⠛⠛⠛⠓⠶⠶⠦⣤⣄⣀⡀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣩⣥⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⣀⣤⠟⠋⠉⠐⠒⠒⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⣴⡿⠟⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⣋⡁⠀⠀⣿⣿⣿⣿⣤⣿⢫⡇⠀⢉⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⡄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠓⠦⣜⣿⣭⣟⣩⣵⣿⡴⠾⠛⠉⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠦⠼⠋⠁⠀⠀⠉⣠⡶⠆⠀⠀⠀⠀⠀⠀⢀⡀⣴⣠⠂⠠⠔⢀⣾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣕⣯⢤⣄⠀⠀⠀⠘⠉⣀⠂⠀⡀⠀⠀⠀⡔⠘⣧⡻⠛⠀⠀⠄⠶⠀⣜⢽⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣙⡶⠽⢄⣎⠄⠀⠀⠀⠘⠃⠁⠀⠀⣄⣸⡽⡏⢂⠁⠒⠢⠀⠀⢀⣀⠀⠂⢐⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠈⡗⠉⠀⠀⠀⠀⠀⠀⠠⢈⠃⡆⣿⢯⡀⠸⢯⡔⠀⠀⢀⠤⣤⠀⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⢤⠒⠚⠀⠁⢿⣷⠀⠀⠈⠁⠀⠀⠀⠟⠚⠁⠀⡌⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢴⠀⠠⠈⡉⠀⠀⠀⣀⣀⠀⠀⠀⠀⡬⢠⡞
⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠄⠀⠁⢀⡠⠆⠀⢙⠄⢀⠷⠀⠀⠠⢇⠉
⠀⠀⡠⠖⠁⠀⠀⠀⠄⠀⠀⠀⢂⠀⠤⠊⠉⠉⠣⣀⡀⠀⢀⠃⠀⠀⠀⠈⢂⠀⠉⠁⠀⡀⠔⠣⠜⠀⠀⠄⠀⢀⡀
⠀⠊⠀⠀⠀⠀⠀⠀⠸⠀⠀⠡⠀⢸⣷⡄⠀⠀⡐⠂⠤⠤⠈⡀⣂⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠌⠀⠠⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⢀⣿⣄⢀⠔⠀⠀⠀⠀⠰⠁⠁⠀⠀⠀⣴⣢⣤⠅⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢾⣿⣿⣷⡀⠀⠀⠀⠐⠃⠀⠀⠀⠀⡠⠛⠛⢯⠁⠀⠸⡀⠀⠀⠠⠄⡎⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⢿⣿⣿⠁⠀⠀⠀⢀⠆⠀⡠⠔⠋⠀⠀⠀⢰⠀⠀⠀⠀⠒⠀⠀⠀⠠⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠉⠃⠀⠀⠀⢀⡮⠀⠀⠀⠀⠀⠀⢀⡠⠊⢆⠀⠀⢸⠀⢀⡠⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡠⠔⠉⠀⠀⠀⠣⠀⠘⡰⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡤⠖⠀⢀⠔⢛⡠⠄⠖
⠀⠀⠀⢠⠊⠀⢀⡰⠁⠀⠁⠀⠈⠉⢉⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠐⠒⠀⠐⠒⠒⠤⢄⡀
⠀⠀⠀⠆⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⡀
⠀⠀⢰⠀⠀⠀⠀⠀⣀⠃⢀⣠⠐⠄⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡤⠚⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠮
⠀⠀⢸⠀⠀⠀⠆⠀⠀⠀⡎⠐⠊⠀⠠⠁⠠⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠠⡉⠀⠚⠋⠙⠓⠃⠀⠀⡖
⠑⠢⢄⠀⠀⠀⠄⠀⠀⣺⠡⢾⠇⠦⠴⠺⡇⠈⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠔⠁⠀⢄⠀⡄⠀⢐⡐⢢⠀⠀⠀⠀⠀⠀⠀⠆⠀⠦
⠀⢠⠜⠀⠒⠀⡏⠀⠀⠀⠑⠀⠀⠀⣀⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣷⠀⠀⠀⠌⠤⣃⠘⢌⣁⠌⠀⣤⠄⠀⠀⠀⠀⠐⠀⠐
⠀⡇⠣⠀⠈⠐⠂⠀⠀⠀⠀⠀⡠⠠⠄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⠀⠀⠀⠀⠸⣅⡀⠀⠀⠀⠰⠁⢡⡀⠀⠀⠀⠀⢸⠚
⠀⠈⠒⣎⠀⠀⠀⠀⠀⠀⠀⢸⢿⣶⠆⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣠⡾⡇⠀⠀⠀⠀⡜
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠰⡬⠋⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠮⣤⡾⠋⣠⠅⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠲⡶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠶⠠⠌⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⡠⠐⢠⠒⠁⣀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠂⡉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢡⠀⠀⠀⢀⡘⠀⠠⠊⠀⠐⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠓⠄⢂⠀⠀⠀⠷⠀⠀⠀⠀⠳⠈⠓⠈⠀⠀⠄⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣤⠀⠐⠀⠈⢄⣀⡤⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢆⠀⠀⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠜
⠀⠀⠀⠀⡀⠀⠀⠶⠀⠀⠈⠀⢰⢠⣤⠄⢳⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠊⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⢀⡴⠁
⠀⠀⠀⠀⡇⠀⢀⡀⠤⠐⠀⠀⠸⠀⠰⠘⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠈⠁⠐⠒⠒⠀⠈⠉⠫⣀⠀⡁
⠀⠀⠀⠀⢨⠉⠀⠒⠒⠒⠒⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⢤⠀⠀⢸⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣙⠀⠀⠀⢇⣀⣀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠣⢄⠀⠀⠀⠀⢘⠓⡆
⠀⠀⠀⠀⢠⠟⠒⠚⠛⠥⢀⡠⠂⠘⢍⠄⠀⠀⡙⠋⠈⠙⢦⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⠀⢀⡀⣤⠀⠀⢸⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢧⡈⣀⡀⠀⠀⡘⠀⠀⠀⠈⡟⠠⠤⠖⣀⣀⠀⡠⣇⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠈⠈⠉⠉⠉⠉⠁
⠀⠀⣀⠀⡉⠈⠈⣉⣁⣈⣁⣀⣀⣀⣐⣇⣀⣀⣈⣁⠀⣀⠐⡈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⡀⠀⠀⡀⣀⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⣶⢦⣶⣶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣾⣴⣾⠿⣿⣿⣟⣿⣶⣾⣿⣛⣮⡒⢟⡶⣦⢀⣀
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣛⢟⣅⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠚⠲⠥⠴⢍⠙⠂⣤
⠀⠀⢀⣠⣶⠿⠁⠔⠒⠉⠡⢿⣾⣿⣿⣵⡖⣀⣀⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠈⠀⠢
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣿⣿⣿⣿⡿⠷⠿⠿⠷⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⡛⠩⠓⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠔⠀⣀⣶⣿⣿⣿⣿⣷⣿⣷⣷⣿⣦⣄⠐⢮⣹⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣷⣮⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣾⣿⣿⣿⣿⡿⠋⢹⣿⣿⣿⣿⣿⣿⡟⣋⣀⣽⣿⢿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠠⠎⣴⣿⠿⠟⣹⢫⢻⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠻⠿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠁⣴⠞⢻⣿⠻⣞⠃⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣨⠿⣫⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠙⠛⠦⢄⣀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⠤⢊⣥⠺⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣬⣉⣐⠚⠻⠿⠿⠋⠓⠒⠒⣈⣠⣴⣾⣿⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⣿⣿⠭⡿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣿⣯⣯⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠾⠿⠿⠿⠿⠿⠿⠷⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⢉⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡉⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⢁⣠⣴⡾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢷⣦⣄⡈⠛⢷⣦⡀
⠀⠀⠀⠀⢀⣴⡿⠋⠀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣀⠙⢿⣦⡀
⠀⠀⠀⣠⣾⠋⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣧⡀⠙⣷⣄
⠀⠀⣰⡿⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠘⢿⣆⠈⢿⣆
⠀⣰⡿⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠈⢿⣆⠈⢿⣆
⢀⣿⠃⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠘⣿
⣸⡏⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠙⠻⠷⠿⠟⢋⣿⠿⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢹
⣿⡇⢰⣿⠀⠀⠀⠀⣀⣤⣶⣿⣟⣩⣤⣤⣤⣤⣤⣄⣀⣀⣴⠟⠁⣠⣤⣤⣍⣻⣿⣶⣤⡀⠀⠀⠀⠀⣿⡆⢸
⣿⡇⢸⣿⠀⠀⠀⠀⢻⣿⡿⣿⣿⣭⣭⣍⣉⣉⣉⣉⣿⣟⣁⣀⣀⣈⣩⣭⣭⣿⣿⢿⣿⡟⠀⠀⠀⠀⣿⡇⢸
⣿⡇⠸⣿⠀⠀⠀⠀⠀⠙⠿⣦⣄⠈⠉⠉⠙⢛⣿⠟⠛⠛⠛⠛⠛⠋⠉⠉⢉⣠⣴⠿⠋⠀⠀⠀⠀⠀⣿⠇⢸
⢹⣇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂⢀⣴⣿⣁⣀⣀⣀⣀⣀⣠⣤⣴⡾⠿⠋⠁⠀⠀⠀⠀⠀⠀⢰⣿⠀⣸
⠈⣿⡄⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢠⣿
⠀⠹⣷⡀⠹⣷⡀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⣾⠏
⠀⠀⠹⣷⡀⠹⣷⡄⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣾⠏
⠀⠀⠀⠙⢿⣄⠈⢻⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⡿⠋
⠀⠀⠀⠀⠈⠻⣷⣄⠉⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡈⠙⠻⢷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⠟⠋⢁⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⣁⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⢶⣶⣶⣶⣶⣶⣶⡶⠾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠒⢂⠀⠀⠀⠠⡁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⣶⣵⡄⠀⠀⠀⠀⢉⠂⢠⢦⡤⣠⣴⡀⣀⣀⣆⣬⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠏⢿⠃⠀⠀⠎⢸⠶⠀⠁⣀⣴⠆⠲⣾⠩⡀⠡⠀⠀⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠂⢘⣂⡴⠖⣀⡈⣶⢀⡆⠚⣯⠷⠶⣿⠀⣧⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣠⣯⣉⡙⠾⣏⣼⣧⠎⠱⠊⢿⣤⣼⡿⠀⠘⣧⣷⡄⢶⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡍⡷⠤⡙⢲⣿⠿⠋⠀⠀⠀⠈⠻⢿⠃⠀⠀⠈⢻⡏⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠉⡇⢻⠀⢬⣽⣿⡄⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⢀⡿⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣠⣅⠈⠁⣾⢿⣿⣧⡀⣨⣧⣤⣤⣤⣁⣀⠀⢀⣾⡝⢿
⠀⠀⠀⠀⠀⠀⠀⢉⢁⣿⡿⠉⠹⠀⠋⠀⣈⣿⡙⠛⠩⠉⠭⠬⣽⣿⣿⣿⡿⢅
⠀⠀⠀⠀⠀⠀⠐⣯⢽⠈⡏⠀⠀⠀⢰⣞⣱⠉⡙⠁⠀⠀⠂⠛⠛⢟⠻⣧⡈⠢⣅⢠
⠀⠀⠀⠀⠀⠀⠀⣥⠞⠀⠈⠀⠈⠀⢸⠀⠉⠙⠀⢰⠀⠀⠀⢌⣀⡙⠉⣿⡅⠢⠈⢹⣖⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠁⠀⠀⢸⠀⠀⠀⢂⣸⣷⠀⢠⣾⣧⠁⠀⠘⣧⠀⠀⡼⣻⢋⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡇⠀⣴⣶⣾⠀⠀⠀⢸⣿⠟⠐⢾⣿⣧⠄⠀⠀⢿⣷⣾⣿⠹⢹⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣰⢏⣴⣿⣿⡿⣏⠀⠆⠀⢸⡟⠀⠀⠈⢿⣷⡀⠀⠀⠈⢻⡿⢧⠜⠟
⠀⠀⠀⠀⠀⠀⠈⠱⣯⣾⠛⣿⠛⠀⣿⠀⠀⠀⢺⣇⠀⠀⠀⢀⣇⣿⡈⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢀⣿⠀⠀⢸⠛⠲⣶⣿⣿⠀⠀⣠⢿⠟⡛⣿⠙⠛⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⢸⡏⣱⣿⣿⡟⠀⣶⣿⣊⡔⠁⣮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣠⣴⣇⠙⢹⣾⣿⣶⣯⣝⣽⣅⣴⣶⡒⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣴⣾⣿⣿⣿⡿⣤⡯⠼⣿⣿⣿⡆⠀⠈⠁⠹⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⢤⣖⣹⣿⣿⣿⣿⣿⣿⡿⠁⠋⠀⠀⠸⡟⣿⣧⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣤⣄⣀⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣠⣴⣾⣿⣿⣿⡿⠧⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⢿⣿⣿⣿⣿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉
⣀⠀⠀⠀⠀⠀⣷⢰⠀⠀⢸⡀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣤⣤⡀
⣯⠛⠒⠒⠊⢰⣿⡸⡆⠀⠘⣇⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠁⡙⠷⣌⡀⠀⠉
⢸⡇⠀⠀⠀⢸⠸⠃⢻⣄⠀⢹⣆⠀⠀⠈⠶⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠙⡆
⠈⢧⠀⠀⠀⣸⠀⢄⠀⣿⠀⠈⣯⠀⠀⠀⣼⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠈⡄⠀⠀⣇⠀⢠⡞
⠀⠀⡃⠀⠀⣿⠀⠘⣇⠸⣧⠀⠹⣆⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⡇⠀⠀⣿⠀⠈⢧⠀⡴⢤⣤⣄⣀
⠀⠀⢷⠀⠀⡿⠀⠀⢹⡆⠘⣷⠀⢹⡦⠀⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⠀⠋⠀⠀⢠⠉⣠⠶⣞⣂⡀
⠀⠀⠘⢷⡾⠃⠀⠀⢸⡇⠀⣿⠀⢸⡀⠀⢿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠀⢸⠀⠀⠘⣠⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡏⠀⠸⣇⠀⢸⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠙⡆⠀⠘⠀⠈⠃⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠴⠚⠣⠀⠀⢼⣄⠀⠀⠣⣀
⠀⠀⠀⠀⣀⣀⡴⢟⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠳⣸⠆
⠀⠀⠀⣰⠀⠀⢀⡼⠁⢤⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠙⠓⠶⢤⣀⣀⣀⣀⣀⠀⢿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠈⠛⠲⣝⡛⠛⠛⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢀⣀⣠⠞⠉⠁⠀⠀⠀⠘⢧⡀
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣄⡀
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊⣰⠀⠀⠀⢾⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⡀⠀⠀⠀⢀⡿⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠋⠉⠉⠉⠻⠉⠉⠉⢉⣩⠽⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠭⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⡴⠞⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡖⢺⡇⣴⣶⣦⢤⠀⠀⢀⣤⣴⣶⣄⢸⡇⠀⢀⡞⠉⡟⡱⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠇⠸⣿⡿⢿⠀⠀⢸⢿⣿⣶⡏⢸⡇⠀⠸⡄⠘⡇⢇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠀⠐⢲⠄⠀⠀⠀⠀⠠⣤⡈⠁⠀⡇⠀⠀⠑⠒⢛⡏⠁
⠀⠯⢭⡙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣀⣈⣙⣒⠲⠶⠶⠶⠾⠆⠀⠀⡇⠀⣀⡤⢾⡉
⡜⠳⠴⢧⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠉⠛⠛⠛⠒⠲⠶⠶⠷⠺⡅⠀⠀⠙⢦⡀
⠈⠒⠒⠙⠲⡏⣸⠣⣄⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⣤⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠙⠳⣄⠈⠙⠛⠛⠛⠉⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⡏⠀⢀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠛⢻⡇⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠛⠋⠉⢻⠻⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⢠⣄⣀⣀⣀⣐⡂⠁⠤⠀⠠⠄⠠⡄⠇⢿⠀⣀⡤⢒⣹⡄⢹⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠤⠤⠤⠄⣀⣀⣉⣉⣉⣉⣉⠛⠓⠒⠒⢻⣋⠤⠚⠁⢸⣷⡞⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⠀⠀⠀⠀⢸⣧⡟⣲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣠⡞⡵⠛⢫⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⢲⡶⠶⢶⡤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣿⣠⡴⠛⠁⣇⣧⠀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣸⡁⠀⠀⠀⠀⠀⣼⠉⠉⣿⠉⠉⠁⠀⠀⠀⠈⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠶⢾⣷⡀⠀⠀⠀⠀⣹⣀⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡛⠁⠀⢀⡴⠋⠀⣇⠀⣀⠴⠚⠁⠉⠛⡿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⠭⣭⣭⣽⠴⠖⠋⢱⡻⠷⠤⢤⣄⣀⡞⢁⡼⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣒⣶⣶⣷⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠾⠛⠛⠛⠛⠛⠿⢿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⠀⠈⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⣡⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡀⠀⠀⠀⢸⠃⡘⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⣠⢲⡄⠉⠀⠀⠀⠀⠀⠀⢁⡇⠀⡇⡇⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⡏⠀⡇⠠⠀⠀⠀⠀⠀⢠⣸⡅⢀⢷⠀⢰⡆⡄⢸⡇⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣧⠀⠀⣠⠦⣠⢄⡄⡖⣀⢸⠀⢰⡼⡇⣈⠛⣇⠀⠙⠃⠀⠀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠸⡤⣄⣿⠀⠛⣾⢸⢷⡇⠸⣄⠘⠋⠹⣌⡆⠀⠀⠀⣠⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠰⡄⠙⠾⠀⢀⡀⠉⠁⠘⠃⠀⠀⠀⠉⠀⠈⠀⠀⠀⢸⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢦⣄⣀⣀⣀⠀⠀⠀⠈⢀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⡀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⣸⣥⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠿⠛⠛⠋⠉⠉⠉⠙⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⡞⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣻⡿⠟⢋⡝⠁⢀⣀⡀⠙⢯⠙⠻⢿⣿⣿⣿⣿⣿⡿⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣾⠀⠠⣾⣿⣿⠀⢸⡇⠀⠀⠈⢻⣿⣿⣟⣛⣩
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠹⣄⠀⠉⠋⠁⠀⣸⠃⠀⠀⣠⣾⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠽⢷⣦⣤⣶⣾⣥⣴⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣾⣭⣛⡋⠂⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠉⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡿⠿⣿⣿⠀⣴⡾⣶⣄⣸⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⣠⣿⣿⠈⣿⣤⣼⡿⣿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠆⠈⠉⠉⢠⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣻⢿⠿⣀⡀⠠⠤⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣾⡧⠠⣤⣖⣲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢰⣾⣿⣷⣿⣿⣿⣷⣦⣿⣿⣿⣾⡟⠑⠆⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣤⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠊⠀⢰⣠⣤⣿⡎⠇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⠇⠉⠉⠙⠛⠏⠉⠀⠀⠀⠀⢠⣼⣿⣿⣿⣷⢸⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣧⡶⣖⣶⣶⣶⣲⣖⢦⠀⠀⠀⠙⣿⣿⣿⣿⢸⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡿⡘⢰⣲⠀⠀⣿⣿⣿⡟⢸⡇⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠉⠀⢡⣿⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡿⣿⣿⠏⣰⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠐⠁⠀⠀⢂⣾⣿⣄⠹⣿⣿⢏⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠂⣹⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢷⡀⢀⣾⣿⣿⣏⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠉⠻⠟⠉⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡤⡠⣢⡪⢂⡠⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⡟⡱⣷⣫⠖⣡⣶⠥⠐⠀⠀⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡈⢦⠑⢌⠢⡦⢄
⠀⠀⣴⣶⣿⡱⣹⠟⡡⠞⠉⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡠⠀⠀⢳⡀⠈⡄⠱⡀⠳⡀⠱⡈⢆⢣
⠀⢸⠇⢸⡗⠡⢋⡼⠁⠀⢤⣤⠀⠀⠀⣴⣅⠀⠀⠀⠀⠉⠉⠀⠀⠹⣆⢸⠀⠱⡀⠙⢄⠱⠈⡆⠇
⠀⢸⣾⠿⠒⠀⠊⠁⠀⠘⠉⠛⠀⠀⠀⠈⠀⠠⠀⠀⠀⠘⠛⠀⠀⠀⠸⣿⠂⠀⠙⢄⡀⠂⡁⠰
⠀⠘⢼⠀⢄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠀⠀⣿⠀⢀⢰⠀⠉⠂⣤⠤⡶⡄
⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠀⠀⢀⣠⠤⢤⠀⠃⠆⠀⠘⡆⠀⠀⠀⠀⠀⡿⠌⠁⠂⠀⠑⠓⠀⠀⢉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠞⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠄⠀⡄⢸⣄⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⢸⠀⡿⠀⡰⢀⢿⡏⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢸⣧⠜⣠⢫⠛⠀⠆⠸
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠘⠁⠔⠱⢁⡀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠐⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⣀⠤⡤⠶⠶⢦⠤⣀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⡜⢀⣶⣂⡀⢣⠀⠉⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣱⡀⠙⠋⢀⣎⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⢀⠉⠉⠉⠉⠉⠉⡀⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠶⠶⠶⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣛⣛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⣤⠊⠁⢠⠃⢰⣶⣀⠈⡆⠈⠑⢦⠀⠀⠀⠀⠀⣴⣾⣿⣿⠁⣴⣆⡀⢹⣿⣿⣦⡄⠀⠀⠀⠀⣴⡞⠉⠀⠀⣿⠋⠙⣷⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣈⣦⡈⠛⢁⣰⣃⡤⠞⠁⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄⡈⠋⣀⣾⣿⠟⠋⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠻⠶⠾⠏⢀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠒⢉⠝⣡⠤⣌⠻⡉⠓⢤⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠏⢁⣄⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⣉⣉⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠛⢄⡀⠸⡄⠣⣀⠜⢀⠇⢀⡠⠞⠀⠀⠀⠀⠀⠻⣿⣿⣿⡀⠻⠿⠃⣸⣿⣿⠟⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⡀⠿⠟⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠶⠤⠶⠟⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⣰⣿⠇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⢏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⢀⣿⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⣭⣭⢄⣾⡿⢣⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣾⣿⣿⡿⢣⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⡟⣱⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⠟⣼⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣴⣿⠿⠛⠿⢿⣧⢿⣿⣿⣿⢋⣾⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⠈⣿⣷⠀⠀⠠⠟⠉⠀⠀⠀⠀⠀⠀⠈⢻⡿⢣⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣍⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣄⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢀⡴⠞⠋⠁⠉⠈⠉⠑⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣞⡶⠦⡀⠀⢀
⠀⠀⠀⠋⠀⣠⣶⣿⣿⣷⣶⠄⠀⠀⠀⠀⢀⠤⠒⠒⢳⣾⠚⠓⠛⠟⠉⠀⠀⠀⡰⠉⠣⡸⠿⢓⡶⠘⡶⠊
⠀⠀⠀⠀⠊⠀⠻⢿⡥⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠐⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⣀⣠⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒
⠀⠀⢠⣴⠟⠋⠉⠀⠀⠈⠉⠑⠢⠀⠀⠀⠀⢀⣴⠟⠋⢁⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡠⠂⠀⠀⠀⠉
⠀⠀⠉⢀⠔⢺⣿⣿⠛⠾⢷⠶⠀⠀⠀⠀⣰⠋⣁⣴⣾⣿⠿⠟⠛⠃⠀⠀⠀⢠⠟⣡⢴⢲⣶⠒⠲⠤
⠀⠀⠀⠀⠀⠈⠫⠁⠀⠄⠀⠀⠀⠀⠀⠀⠁⠰⠛⠙⠿⣃⣀⠰⠂⠀⠀⠀⠀⠀⠎⠀⠈⠪⠭⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⣀⣤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣴⠞⠉⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⢀⣶⡾⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠉
⠀⠀⠈⠁⠀⣠⣾⣿⣿⣷⣤⣀⠀⠀⠀⠀⠘⠁⣠⣴⣶⣿⣿⠻⢷⣷⠀⠀⠀⠀⠰⠛⣡⣤⣴⣾⡾⠾⡿
⠀⠀⠀⠀⠋⠙⢿⣿⢇⣩⡿⠉⠀⠀⠀⠀⠀⠼⠋⠘⠿⡿⠟⢀⡴⠅⠀⠀⠀⠀⢀⣞⠉⢩⣿⣿⣧⠴⠟
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣟⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠼⠤⠤⠤⠤⠤⠤⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠤⢀⠀⠀⠀⢀⠠⠤⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⡂⠀⠄⡂⠁⠀⠀⠁⡊⠄⠀⡆⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠐⠠⠩⠔⠁⠀⠀⠑⠬⠡⠐⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⠂⠘⠈⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠈⠀⠂⠃⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠐⠆⠀⠀⠖⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠐⠒⠲⠶⠖⠒⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡴⢧⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠜⠁⡀⢀⣀⠀⠀⣠⢄⣠
⠀⠀⠀⠀⠰⢮⠉⠙⠁⠐⠈⢉⡉⠁⣴⠋⠀⠀⠀⠘⠂⠒⠒⠀⡴⠋⠉⠉⠉⠉⢁⡀⠀⣦⣤⣿⣾⣿⣿⣾⣿⣿⣿⣶⡶⠂
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⡠⠃⠀⠀⣈⣧⡄⠀⠀⠀⠀⠠⢀⠎⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠛⠉⠉⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡜⠀⠀⠀⠀⠀⡎⠀⣤⠞⠉⠁⡈⡇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠋⠖⠉⠙⢯⠀⣿⣿⣿⣿⣿⠟
⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⠘⢄⡈⠀⠉⡁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠠⠀⣸⠀⢻⡏⢋⣿⣇⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄⠀⠀⣀⣀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠁⠀⠀⢁⡀⠀⠀⡸⠋⠀⢹
⠀⠠⢞⠁⢀⠀⠀⠀⠀⠀⠘⡄⠀⢉⡡⠾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢤⣘⣁⡨⠔⠋⡀⠀⠀⡎
⠀⠀⠀⢉⡽⣠⠀⠀⠀⠀⠀⢸⢸⡁⢀⠀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢹⠀⢰⠇
⠀⠀⠀⠉⠊⢭⣀⠴⢳⠀⢠⠏⠉⢿⢏⡉⠀⠀⡴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢸⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠰⡷⠚⣆⡜⠂⠒⠡⣎⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢀⡇⠀⠀⠀⠀⢀⡞⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡆⠘⠦⢾⡉⠀⠒⢢⣀⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡀⠀⠀⠀⠀⢘⣑⠒⠺⡃
⠀⠀⠀⠀⠀⠀⠀⢈⠗⢦⡴⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⢳⠶⠶⠶⠶⢾⣀⣴⠞
⠀⠀⠀⠀⠀⠀⢌⡀⠀⠸⡇⠀⠀⠀⠀⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠲⣆⢀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠉⠚⠓⠛⡦⠒⠒⡻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⡆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣃⠀⠘⡆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢦⣇⣹⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠉⠙⣦⠷⠋⠁⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⠋⠀⠳⡹⣶⡤⣤⡤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣀⣸⣦⣄⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠼⢄⣀⣀⣱⣠⣀⣠⡶⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⠴⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣲⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠁⣀⣠⠼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⢦⡀
⠀⢀⣀⣠⣤⣤⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⣤⣤⣤⣤⢀⣀⣀⠀⠀⠀⠀⠀⠈⢙⣦
⠀⠀⠨⣿⣿⡿⠏⠀⠀⠀⠀⠀⡀⠀⣲⣮⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣦⡤⢄⡀⠀⠛⣷⣄
⠀⠀⢀⠜⠋⠀⠀⠀⠀⠠⣀⣬⣾⣿⡿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⢤⣟⡻⣿⡿⣦⡀⠀⠉⣷⣤⣀⡀
⠀⡚⡉⠀⠀⠀⡀⢀⡴⢪⣋⡃⢸⣿⣧⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠑⠚⢿⣷⣄⠈⢿⣿⣿⣤⡀
⠀⠀⠀⢀⠀⣠⡎⠉⠛⠃⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠬⣿⣟⣆⠀⠙⠟⠋
⠀⢀⣢⡵⠛⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡙⡷⡀
⠚⠛⠛⢿⢷⡤⣀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣯⣿⣿⣿⣿⣿⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠥⠴⠉
⠀⠀⢦⡀⠀⠈⠐⠨⢓⠠⠠⣄⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠁⠒⠂⠍⠀⠐⠒⢂⡉⠋⠉⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⣩⣥⣀⠁⠀⢄⢀⠐
⠀⠀⠀⠀⠉⢿⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢂⣵⢾⠿⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⡐⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡠⠰⣋⣵⠛⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠓⡂⡴⣦⢄⢦⡄⣂⣔⡐⣐⠄⣔⢌⠾⠊⣋⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠈⠈⠀⠀⠁⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠙⢹⡏⠉⠏⠀⠀⠀⠀⠀⠙⠁⢰⣬⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⠃⠊⠉
⠀⣼⠁⠀⢀⡀⠀⠀⠀⠀⠀⢠⡼⠗⣹⣿⣿⣿⣿⣿⣻⣻⣧⣾⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣯⣹⣿⣏⠳⠰⠆
⠀⢹⡀⠀⠘⠙⢿⡶⠄⣴⢰⣯⣦⣿⣋⡼⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠟⢶⣀⠁
⠀⠈⠁⢀⣶⣶⡏⢿⣻⢋⣉⣙⣿⣯⣿⣠⣾⣿⣿⣿⣷⣿⣇⣀⠙⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠂⣿⡍⡆
⠀⠀⠀⠙⢣⡙⠗⢽⡉⠇⠈⢻⠟⠛⣟⡿⡟⠛⠛⠛⠻⣿⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠉⠈⠹⣀⠑
⢀⠀⣍⣽⡇⠀⠀⢠⣉⣓⠶⣶⡶⣟⡽⡀⢀⠀⠀⠀⣤⣧⣀⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠦⣶⣾⡍⠤
⠀⠛⣿⣿⣿⣿⣧⡶⢦⠋⠹⣿⣿⣿⢀⣁⠀⠒⠂⠙⠛⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⢉⣸⡿⠉⠐⡀
⠀⠀⣇⢹⣷⣿⣿⣷⡆⢳⣿⣿⣷⣿⢟⡉⠤⣿⣖⣀⣀⣀⣬⣷⣿⣿⣿⢿⠚⣻⣿⣾⣿⣿⣯⣼⡷⡆⠀⠀⠁
⠀⡄⢽⢹⡇⢻⣿⣿⡑⠀⠑⡻⣿⣿⡟⢱⢴⢾⠿⠧⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠷⠃⡅
⠀⠻⣿⢺⡧⣞⣿⣿⣅⣤⣶⣦⣿⣿⣻⣿⣻⣤⣴⣶⣶⣿⣿⣿⡿⢧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡀⡇
⠀⠀⢿⣿⣇⠈⠋⣿⡛⢋⣀⢀⡩⣍⠛⢟⣛⣿⣿⣥⣿⣿⣿⣋⣵⠾⠟⠛⡉⣛⣽⣿⣿⣿⣿⣽⠃
⠀⠀⠂⣼⣿⣿⢇⣾⣇⠀⠙⠻⠻⠛⠾⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠁⠲⠭⢍⠁⠌⠉⢉⣻⣭⣀⠄
⠀⠀⠀⠈⠋⠁⠀⠁⠈⠀⠄⠀⠀⠀⠀⠁⢠⣿⣿⣿⡿⠀⠀⢀⠀⠀⢀⡈⡉⠛⣏⣠⣄⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠙⠷⠧⢾⣯⣻⣿⡟⣷⢶⡄⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠊⠛⢋⣛⡯⣓⣹⣦⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠉⠛⠛⠻⣯⠁⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⢀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠖⠚⠛⠛⠛⠓⠛⠚⠛⠛⠻⠷⠶⠾⠿⠿⠿⠶⠶⠶⠶⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⠟⣩⣤⣌⠻⣆⠀⠀⠀⠀⠀⣴⠟⢩⣭⣍⠙⣦⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⠃⠰⣿⣿⣿⠀⣿⠀⠀⠀⠀⢰⡏⠀⣿⣿⣿⠀⠈⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡀⠀⠘⠛⠁⠀⣿⠀⠀⠀⠀⢸⡇⠀⠈⠛⠋⠀⣠⠏⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⣄⣀⣀⣠⡼⠃⠀⠀⠀⠀⠀⢿⣄⣀⣀⣠⡾⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢛⣛⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⣀⡀⠀⢀⣶⠖⢿⡀⠈⠛⠉⠉⠉⠉⠁⠀⠀⠀⣠⡶⠶⣦⣤⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠶⠶⠶⠶⠿⠿⠟⢻⣿⠀⢸⣿⣛⣻⣿⣿⣿⣿⣛⣛⠛⠛⣿⡇⠀⣿⠤⠤⠴⠶⠶⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠦⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⡿⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⡏⠉⠁⠀⠀⢀⣠⣿⣤⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣿⣍⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣀⡀⣹⡆⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢻⡏⢉⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡟⣿⠋⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠈⣛⡿⢿⡿⣷⣶⠶⣦⣼⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠀⣀⣀⠀⠀⢈
⣠⡾⠋⢰⣿⣅⠈⠙⢶⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢳⣶⠾⣿⠟⠻⣿⡉
⢿⡀⢼⠛⠁⠈⢻⡦⠀⣿⠀⠀⢿⡤⠤⠶⠶⠶⠶⠶⠒⠂⣿⡇⢸⣟⠛⠛⠛⠛⠛⠛⠛⠋⣉⣉⣹⡇⢐⣿⣛⣛⠛⠒⠲⠶⠶⢄⣸⡇⢰⡟⢁⣸⡿⢶⣄⡈⠛
⢸⡇⠸⡇⠀⠀⢸⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠛⢿⣆⣸⡟⠛⠛⢻⣦⠀⠀⠐⣿⠉⠉⢹⣧⡾⠋⠉⠉⠛⢻⡆⠀⠀⠀⠉⠃⠘⣇⠘⣏⠀⠀⢩⡇⢰
⠈⣷⢀⣿⠀⠀⣸⡃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣼⠁⢸
⠀⠸⠟⠉⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣆⡿⠀⠀⢿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⣶⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠳⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⠀⠀⢀⣀⣤⣤⣤⣤⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠖⣾⣿⣿⣿⣿⣿⣿⣿⡛⠿⠟⠀⠀⠈⠛⢻⡿⠿⠿⠿⠿⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠉⠉⠉⠉⠉⠉⢷⡀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣀⣠⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣄⠀⣾⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⣦

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⡤⠾⠿⠿⠿⣿⣿⣿⣿⣿⣿⣶⣦⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠷⣤⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠹⡿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠔⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠠⣤⣀⣠⣿⣿⠿⣿⡟⠙⣿⣳⣶⢤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⣠⡴⠒⢶⣚⣶⡾⣿⣶⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠟⢻⣦⣀⣿⡈⠛⠻⣿⡏⠀⠉⢢⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⢀⠜⠁⢻⡉⠲⠿⣿⣿⠀⣀⣽⠿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙⠙⠛⠋⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠈⢊⣀⡉⠙⠒⠓⠛⠳⠋⠙⠃




⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣶⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠔⠛⠉⠉⠉⠉⠛⠛⠻⠿⣿⣿⣿⣿⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣶⣿⣿⣿⡿⠿⠿⠛⠋⠉⠉⠉⠙⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣀⡀⠀⠀⠀⠀⠀⣀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠿⠶⣷⣿⣿⡿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠋⠉⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⠖⠒⠒⠒⠲⠦⠤⢤⣀⣀⡀⠀⠉⠉⠙⠓⠲⣤⡀
⠀⣀⣠⢶⡖⢲⠀⣠⡄⠀⠀⠀⠀⠀⠀⡟⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⢤⡀⠈⢻⠶⣄
⢿⡉⢣⠀⣇⣸⠏⣡⡟⠀⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡸⠀⠈⠳⣄
⠀⠙⣆⣳⣯⡧⣴⠋⠀⠀⠀⠀⠀⠀⡞⠀⣸⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡾⢀⠃⠀⠀⠀⠈⡇
⠀⠀⣹⠥⠶⠒⢻⠇⠀⠀⠀⠀⠀⢰⡇⠀⡏⠀⠀⠀⠀⠛⠛⠒⠲⠦⠤⣤⠀⠀⠀⠀⠀⢐⠇⡌⠀⠀⠀⠀⢠⡇
⠀⠀⣟⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣾⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠁⠀⠀⠀⠀⣸⠁
⠀⠀⣿⠤⠤⠀⠈⣧⠀⠀⠀⠀⢠⡏⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠀⠀⠀⠀⠀⡟
⠀⠀⢸⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⡇⠀⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⡇⠀⠀⠀⠀⢸⠃
⠀⠀⠘⣷⠒⠉⠀⠀⠹⡄⠀⠀⠈⠳⣤⣀⡀⠀⠉⠉⠛⠒⠲⠶⢤⣤⣀⡀⠀⠀⠀⢠⠇⢸⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠹⡄⢀⡤⠂⠀⠙⢦⡀⠀⠀⠀⡽⠋⠙⠓⠲⠤⢤⣄⣀⣀⡀⠈⠉⠉⠛⠛⠁⢀⣎⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠹⣏⠀⠀⢀⠀⠀⠙⠳⠶⡟⢃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⠖⠋⠉⠹⡆
⠀⠀⠀⠀⠀⠘⢦⣴⠋⠀⠀⡰⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣀⣤⠿⠶⠶⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣴⣁⣀⡼⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠋⠨⡇⠀⠀⣶⠀⠀⣰⠁⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠟⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⡀⢀⣀⣀⣀⣀⣀⠀⠁⠀⠀⢠⠞⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠽⠉⠁⢸⠁⠉⠉⠛⠶⣄⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠤⠤⠤⠤⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷⡀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣸⠧⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠈⠀⠀⠀⡟⠀⠀⢀⣠⡤⠖⢛⡗⢲⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⠧⠶⠛⡟⠙⢻⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣼⣀⣤⣼⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣧⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⢴⠶⣤⡀⢀⣀⣠⠖⠺⡁⠀⠘⡀⠉⠛⠒⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠶⠶⠶⣶⠒⠋
⠈⢧⠀⠙⢫⡈⠱⡀⠀⠸⠄⠀⠑⠀⠀⢀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⠛⠳⠖⢦⡀
⠀⠈⢧⠀⠀⢣⠀⠳⠀⢀⣀⣀⡴⠓⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡠⠂⠁⠀⢠⠃⠀⣤⠀⡸⠛⡆
⠀⠀⠈⢧⡀⡼⠦⣤⠴⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢠⠃⠀⢀⠃⡜⠀⢀⡇
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠻⠤⠴⠿⢶⡇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠿⠿⠿⠿⠿⢿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⢠⣿⠏⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⠟⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⣼⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣭⣵⡶⣰⣿⢏⣾⣿⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣿⣿⣿⠟⣴⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⢋⣼⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇⣿⣿⣿⡿⢣⣿⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⢻⣿⠀⠀⠠⠾⠛⠉⠀⠀⠈⠁⠘⣿⡿⣱⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣿⡟
⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠉⠙⠛⠛⠿⠿⠿⠿⠟⠻⢿⣿⡁⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣶⣶⣶⣶⣾⠿⠿⠟⠛⠉


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠤⢤⣐⣶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⠿⠛⠒⠀⠈⠁⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⡟⠓⠒⢄⠀⠀⠉⣉⣙⢲⡀⠀⠀⠀⠀⠀⠀⣠⡖⠉⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡎⠀⠠⣄⠈⡋⠉⠁⠀⠀⠀⠀⠀⡎⠁⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡠⢤⡂⠀⢸⠇⠀⡸⢡⠖⠒⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠹⣜⢳⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⡿⠂⠀⠘⣆⣸⠀⢠⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢣⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⡇⠀⠀⠀⠘⡇⠀⣸⣽⠀⠀⠀⠀⢠⠞⠉⠉⠒⢦⢀⡀⠤⠄⢤⣠⠎⠙⣦⣀⠀⢉⣉⣙⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢷⠀⠀⠀⢀⠇⠀⣯⠿⣂⡭⢑⡆⠀⠓⠤⠤⣶⠞⠉⠀⢀⡠⠊⠀⢀⡜⠁⠈⠀⠀⠉⢢⡉
⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠊⠉⣠⠌⠲⣤⡤⠊⠀⠀⠉⠊⣉⣠⠚⠉⠉⠉⠀⠘⣇⣀⡤⠖⠋⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧
⣠⠤⠒⠂⠉⠁⠀⠀⠀⠀⠀⣠⠞⡹⠀⠀⠀⠀⠀⠀⠀⡎⠁⠈⠉⠉⠓⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠘⢫⡭⠭⢑⠆⠀⠀⠀⠀⡇
⣇⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⡗⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠉⠀⠀⠀⠀⡜
⠈⠑⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣔⠲⡖⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢏⠳⡘⠜⣄⠀⠀⠀⠀⢳⢄⣀⠠⡾⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠀⠈⠿⠳⢤⡀⠘⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠀⠀⠀⠀⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣺⡓⠷⠀⢇⡠⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣈⡆⠀⠀⢸⡏⣰⠆⣰⠃⠀⠀⠀⠀⠀⠀⣬⢠⠋⠀⠙⣆⠀⠘⣠⠞⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠃⠀⢀⡸⠟⠁⢺⠀⠀⠀⠀⠀⠀⠀⢀⠇⡜⠀⠀⣜⣁⣳⣞⠁⢠⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⣹⢀⣀⡤⠔⠚⠁⠀⣠⢶⡅⠀⠀⠀⠀⠀⠀⠀⡜⢸⠁⣠⠴⢊⠉⠁⠈⠓⠃⠀⢤⡉⠦⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣹⠏⠈⣆⠀⢀⠴⠊⠀⡀⠃⠀⠀⠀⠀⠀⠀⢰⠳⠟⠉⠀⠔⢺⡀⢆⠀⠀⠀⠀⠀⢻⠢⡈⠢⡀⠀⠠⡇⡇⣰⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠞⡗⠀⠀⠸⠖⠁⢰⠀⠀⢡⠶⢤⠀⠀⠀⠀⠀⠸⠒⠉⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⠆⠈⠢⡘⢦⣸⢇⣥⢃⠜⣠
⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣀⣸⡄⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡏⠙⠈⠁⢙⣊⠴
⠀⠀⠀⠀⠀⠀⢣⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⢿⣀⡠⠖⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠦⢤⣤⣤⣤⣤⣤⡞⣠⠖⡡⠚⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⡗⢲⠶⠦⠶⠖⠒⠒⢛⠇⡎⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠉⡙⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢇⢸⠀⠀⠀⠀⠈⠉⠚⠈⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⣸⣄⣇⠀⠀⠀⠀⣀⣀⠴⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡖⢲⣦⠄⠐⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣬⣀⠈⠙⡛⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⣀⣀⣀⣘⡹⡀⢫⢡
⠀⠀⠀⠀⢎⣁⣀⡈⠿⠓⠋⠉⠉⠒⠢⠤⠤⠤⣤⣤⣴⠾⠓⠒⠢⢄⠀⠀⠀⠀⠙⠠⡤⠒⠒⠚⠛⠛⠓⢷⡒⠇⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠁⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀⠀⠀⠈⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⣀⣤⣶⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣿⣶⣦⣄⠀⠀⠉⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣠⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⠀⠀⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⢾⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⠀⠀⠀⢿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⣠⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠹⣿⣦
⠀⠀⠀⢀⣾⡿⠃⠀⢀⣾⡿⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠘⢿⣷⡀
⠀⠀⢀⣾⡿⠁⠀⣠⣿⡟⠁⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠈⢿⣷⡀
⠀⢀⣾⡿⠁⠀⢠⣿⡟⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣶⣶⣦⣤⣀⡀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠈⢿⣷
⠀⣼⣿⠃⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷⣄⠀⠀⠙⢿⣿⣍⠛⠿⣿⣶⣶⣿⠿⠛⠉⠉⠉⠉⠉⠛⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠘⣿⣇
⢠⣿⡏⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⢹⣿
⢸⣿⠃⠀⢰⣿⡇⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠘⣿
⣿⣿⠀⠀⣸⣿⠁⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠈⣿⣇⠀⠀⣿
⣿⣿⠀⠀⣿⣿⠀⠀⣴⣿⣿⣯⣥⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣽⣿⣷⣄⠀⠀⠙⢿⣷⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣦⠀⠀⣿⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⠀⠀⠻⢿⣿⣿⣟⡛⠛⠛⠛⠁⠀⠈⠉⠉⠉⠉⠛⠛⠿⢿⣷⣄⠀⠀⠙⢿⣿⣟⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⢛⣻⣿⣿⡿⠟⠀⠀⣿⣿⠀⠀⣿
⣿⣿⠀⠀⢹⣿⡀⠀⠀⠀⠈⠙⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⠀⠀⠀⢀⣿⡏⠀⠀⣿
⢸⣿⡄⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⣀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢠⣿
⠘⣿⣇⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⣸⣿
⠀⢹⣿⡄⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢠⣿⡟
⠀⠀⢿⣷⡀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⣼⣿⠃⠀⢀⣾⡿
⠀⠀⠈⢿⣷⡀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣄⠀⢀⣼⣿⠃⠀⢀⣾⡿⠁
⠀⠀⠀⠈⢿⣷⡄⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⢿⣷⣾⡿⠁⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠙⠋⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡷⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠙⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠋⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠙⠻⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⠿⠛⠉⠀⠀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣄⡀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⢀⣠⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⠿⠿⠿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠁⣰⣿⠏⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠉⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⠀⣸⣿⠃⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠈⠙⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⢻⣿⠁⠀⠈⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⠀⠀⠀⠀⢿⣷⡀⠀⠀⠈⠻⣦⣤⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⣰⣿⠇⠀⠀⠀⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣄⡀⠈⢻⣷⡄⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⣴⣿⠏⠀⣀⣤⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣽⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣥⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣿⣿⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠒⠒⠒⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢀⡠⢂⠠⠤⠤⠚⡇⠀⠀⢃⠀⢀⠀⢀⣄⣀⣀⠤⠤⠒⠊⢠⣄⣤⣄⣠⣶⣴⣆⣀
⠀⠀⠀⢈⠗⠀⠀⠀⢠⠒⠀⠙⣄⠀⠀⠀⠉⠀⢠⠋⠀⠀⠀⠀⢰⣤⣾⣿⠿⠿⠿⣿⣿⣿⣿⡯
⠀⠀⠰⠃⠀⠀⠀⡐⠁⣠⠒⠋⠙⡀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠹⠟⠅⠒⠲⠀⣿⣿⣿⣿⠗
⠀⠀⢀⠀⠀⠀⠀⢇⠀⠳⠀⠀⢉⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠂⢀⠇⠻⠏⣹⣯⡀
⠀⠀⡼⠀⠀⠀⠀⠈⢆⠀⢠⠤⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡁⠀⡀⠤⢀⡠⠋⠀⠸
⠀⠈⠂⢰⠀⠀⠀⠀⠈⡆⡔⠊⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⠒⠂⠁⠐⡆⠀⡇
⠀⠀⠐⠓⣋⣀⢴⠀⢠⠗⢳⣞⠊⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠸⡖⠁⠼⠒⢆⡝⠐⠂⠮⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⠇⠀⠀⠀⢠⠃⠈⡄
⠀⠀⠀⠀⠀⠀⢞⡀⢠⠄⠑⠛⡈⠉⠑⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⡀⠀⠀⠪⡍⠘⣅
⠀⠀⠀⠀⠀⠄⠊⢹⡏⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢉⣉⠉⠉⠑⡗⠁
⠀⠀⠀⠀⠀⠑⠂⠢⢧⡤⠤⠔⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡍⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠼⢄⡀⡇⣸⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⡾⢬⣧⡀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠹⣍⠀⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸⠀⠀⢉⠛⠓⢚⡷
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠊⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠦⢤⣤⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠄⠚⠛⠁⠴⢶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠈⠁⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⠀⠀⢠⠔⠀⠀⠀⠀⠒⠒⠄⠀⠀⠉⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠈⢳⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠘⡃⢤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⠀⢱⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⢸⠂⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⠀⠀⣠⠄⠀⠀⠈⢧⡀⠀⠁⠒⠠⠤⠤⠤⠤⠀⠐⠂⠉⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡴⠖⠁⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⠶⢶⠶⠶⠖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠛⠛⠉⠛⠋⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡂⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢣⡈⠙⠳⢤⣄⣀⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠤⠁⠀⠀⠀⠀⠇⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠱⣀⠀⠀⠀⠉⠉⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡤⠊⠀⠀⠀⠀⠀⡀⠀⠟⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠷
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⢰⡆⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠃⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠃⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠤⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣤⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠠
⡼⠓⠒⠒⠲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠂
⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣴⣦⣴⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣏⣹⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠀⢀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⡨⢠⢢⠀⣶⠀⠈⠀⡌⠰⠢⢸⠅⠁⠸⠇⢟⠀⡀⠀⡇⢸⠂⠕⢎⠇⢸⠂⢏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣔⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⢋⣽⣋⣽⣟⣉⣿⣿⠾⠷⣤⣤⣤⡞⠁⠀⠀⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⣀⣤⣶⣾⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣷⢶⣾⣍⣀⡀⣀⡴⠃⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣀⡤⠖⠋⢹⠛⠁⢠⣿⣿⣷⣿⣷⠛⠃⠈⠙⢻⡛⠻⣿⣿⣦⣤⣤⣼⠿⠛⣛⣣⣴⠞⠋⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⡴⠞⠉⠀⠀⠀⢸⠀⠀⠸⣿⣿⣿⣿⣧⣀⡀⠀⠀⢸⡇⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣭⣤⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠋⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠿⣿⣿⣿⠟⠁⠀⢀⡿⠁⠀⠀⠀⠀⡴⢃⣼⢯⣭⣭⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⣀⣴⡟⠁⠀⠀⠀⡠⢚⣵⣿⣿⣛⠳⠦⣍⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⠞⠛⠲⠀⠀⣀⠀⠀⠀⠀⠀⠙⠻⠿⠷⣶⠿⠿⠛⠁⠀⠀⢀⣀⣤⡾⢿⡛⠶⢤⠈⠙⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⢤⣤⣤⣤⣤⣤⣴⡶⠶⢿⡛⠋⠻⣄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠙⠀⠀⠈








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠖⠐⠒⠒⠒⠲⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠁⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠶⠀⠀⠀⢀⠀⠀⠃⢀⠁⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠓⠠⢀⡀⠀⠀⠀⠀⣀⢠⠒⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠀⠀⠀⡇⠀⠀⠀⠀⣠⠴⡄⠀⢰⠓⠲⡄⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠐⠒⠂⠒⠒⠈⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠱⡄⣱⡤⢮⣀⠴⠉⣳⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣀⣀⣀⣀⣀⡀⠤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠹⡖⠊⢁⡷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⡖⠉⠟⠛⠉⠙⣏⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠀⢀⡴⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣻⠁⠀⠈⢦⣀⠀⠀⠈⠀⠀⣠⠎⠀⠈⠱⡤⢤⡀⠀⠀⠀⢀⡴⠉⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡟⠀⢠⠇⠀⢀⣀⠀⣈⠉⠒⠛⠃⠉⠀⠀⠀⠀⠀⠱⡀⠙⡦⠒⠛⠁⢰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠋⢡⡤⢧⠀⡜⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣺⠤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⣼⠀⡴⠋⠀⠀⢹⠃⠀⠈⠻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃
⠀⠀⠀⠀⢀⠴⠧⢼⠁⠀⠀⠀⠸⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⣀⣠⠃⠀⠀⠀⠇⠀⠀⠀⡇⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣔⠋⣡⠞⣦⣀⣀⣠⠛⣄⠀⠀⠓⢤⣄⣁⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠁⠄⣀⣠⠇
⠈⠻⣁⣰⠃⢨⡍⠈⢦⡬⠃⠀⠀⠀⣀⡈⢉⡛⠒⠒⠒⠠⠤⠤⢀⣲⣶⡒⠚⠻⠉
⠀⠀⠀⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⣏⡤⠄⡈⠳⠀⠀⠀⠀⡴⠉⡀⢤⣹⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣸⡤⢇⣀⠀⠀⣧⣮⡀⠀⠹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠑⠓⠲⣅⠀⠀⠀⠀⠀⠀⢸⠷⠒⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⣤⠞⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⣈⡉⠐⢆⠀⠀⠀⠀⠀⠀⢸⠃⠉⣉⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠁⠠⠁⠈⠉⠚⢦⡀⠀⠀⠀⡠⠟⠊⠉⠈⡤⠈⠁⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⣈⣀⣀⣁⣀⣀⣀⣱⠀⠀⡞⣀⡀⣀⢀⣐⣀⣁⣀⣀⣀⡀⣷
⣀⣀⣀⣀⣀⣀⣀⣳⣤⣦⣤⣼⣤⣤⣧⣤⣤⣤⣼⣼⣀⣀⣧⣭⣤⣭⣤⣬⣥⣤⣥⣤⣬⣤⣽⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠮⠵⠦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡋⡔⡶⠆⢠⢲⢦⢙⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣧⣇⣑⣛⣃⣈⣚⣋⣸⣼⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠚⠓⠒⠒⠒⠒⠒⠓⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡞⠇⠀⡏⣏⣹⢉⡋⣏⣹⢙⠀⠸⢳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢴⡛⢜⢎⡇⠀⠈⠈⠁⡽⢯⠉⠉⠁⠀⢸⡱⣣⢛⡦⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠐⡄⢹⠞⠁⠀⠀⣠⣄⠀⣨⣅⠀⣠⣄⠀⢀⠈⠳⡏⠠⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡸⠀⠀⠀⠈⠧⠼⠉⠧⠼⠉⠧⠼⠁⠈⠀⠀⢣⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣺⠀⠀⣗⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣺⠀⠀⣗⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣒⣺⠀⠀⣗⣒⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣗⣒⣺⡀⢀⣗⣒⣺⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⣸⣇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠛⠛⠛⠛⠛⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠈⠳⣄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣤⠶⠚⠛⠛⠛⠓⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⣦
⠀⠀⡼⠁⠀⠀⣰⠛⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢧
⠀⣼⠁⠀⠀⣼⠃⢰⠏⠙⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⣧
⢰⠇⠀⠀⢰⠇⠀⡟⠀⠀⠀⠙⢦⡀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸
⣾⠀⠀⠀⡿⠀⢰⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⢷⣄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⡿⠀⠀⢠⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢷⣄⠀⠹⡄⠀⠀⢀⡙⠲⣄⠀⠀⠀⢸⡀
⣿⠀⠀⠸⡇⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢷⣄⢙⡆⠀⣬⡙⢳⡌⢳⡀⠀⢸⠃
⢿⠀⠀⠀⣧⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢿⣄⠀⠀⢹⡆⣿⠈⡇⠀⣼
⠸⡆⠀⠀⢹⡄⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢷⣄⠞⢁⡟⢰⠇⢠⡇⠀⠀⢰
⠀⢳⠀⠀⠀⢳⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠷⡉⢰⠏⢀⡟⠀⠀⢀⡟
⠀⠀⢧⡀⠀⠀⠻⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⠶⠞⠛⢦⡀⠀⠀⠘⢷⣤⠟⠀⠀⢀⡾⠁
⠀⠀⠈⢳⡄⠀⠀⠙⢦⡀⠠⣇⣀⣀⣀⣀⣀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠙⠓⠦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠚⠋⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⢤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠶⠞⠋⠁
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠋⣽⣿⠏⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⢿⣿⡍⠛⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠉⠀⠀⠀⣼⣿⠃⠀⠀⠀⢀⣴⣿⡿⠿⠿⣿⣷⣄⠀⠀⠀⠀⢻⣿⡄⠀⠀⠈⠙⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⡀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢿⣿⣄⡀⠀⠀⣠⣼⣿⠃⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠙⠿⢿⣿⣿⠿⠟⠁⠀⠀⠀⢀⣾⡿⠁⠀⢀⣠⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣦⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣥⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⡿⠿⠿⢿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⢫⣿⠟⠋⠀⣠⣤⣤⣤⣀⠙⠻⣿⡝⠻⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠋⠀⠀⢠⣿⠏⠀⠀⠀⠉⢿⡏⠙⠻⣷⡀⠹⣿⡄⠀⠀⠙⠻⢿⣦⣀
⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⣼⡟⠀⣾⣷⣤⣤⣿⠇⠀⠀⢹⣷⠀⢻⣧⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⢹⣧⠀⢿⣯⠉⠉⠀⠀⠀⠀⣼⡿⠀⠈⠁⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣀⣀⠘⣿⣆⠈⠻⣷⣤⣀⣀⣤⣾⠟⠁⠀⠛⠀⠀⢀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣷⣬⣿⣷⣄⡈⠉⠛⠛⠉⢁⣠⣾⣧⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣿⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣼⣿⣤⣶⣤⣤⣴⣶⣤⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⣶⠿⠟⠛⠁⣠⣿⣿⠋⠀⠉⢻⣧⠈⠹⣿⣿⣟⠛⠛⠻⢿⣦⡀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⣼⡟⠹⣿⣀⠀⠀⠈⠉⠀⠀⠉⠉⢻⣧⠀⠀⠀⢻⣧
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣷⣀⠙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡟⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣧⣀⣀⠀⠀⠀⠀⣀⡀⠀⢀⣿⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⠛⢿⣧⣀⣀⣰⣿⠿⠿⠟⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠈⠻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⢠⣦⠀⢰⣶⣶⣶⣶⣾⠟⠁


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⢴⣶⣾⠾⣷⡀⠀⠀⠀⠀⢀⡾⠐⣁⠀⡀⢠⢀⡤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡖
⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣿⡤⠺⠡⣶⢿⣟⠒⢿⡟⢾⡟⣀⣶⣠⡄⡠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠚⠿⠌⠁⠙⣁⣘⣈⣢⣇⡘⢌⣧⣽⡶⡟⢿⡀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄⠾⠵⣻⡰⠞⣠⣴⣾⣿⣿⣿⠿⣿⣿⣿⣷⣿⣄⡻⠸⣊⣽⢷⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣆⣖⣽⡉⢩⣾⣿⡿⡾⢫⠇⠻⣤⡾⠘⣧⠙⢟⢿⣿⣷⡜⠛⠈⣹⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠋⡟⡏⢃⣿⢿⡏⠐⣷⠃⢦⣺⡋⡿⣩⠇⣙⠿⠁⢻⣿⣿⡄⠰⠟⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣦⣾⣾⣅⣼⣿⠟⢶⠲⢊⠗⠒⠟⠛⠛⢋⠿⠁⠚⢋⠛⣿⣿⡷⠤⠼⠚⢯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣱⠟⢃⣿⣿⣤⣦⣤⣯⣤⣾⣤⣾⣤⣷⣤⣶⣤⣼⣦⣾⣿⣷⠀⠓⠋⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡍⠘⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⣿⢻⣿⢿⣿⣇⣿⡆⣿⢏⣿⡟⢻⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⢼⣿⡾⣹⣿⣿⡇⣿⡆⣿⠃⣾⣧⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣻⣿⣾⣿⡷⢸⣿⣿⡇⣿⠁⣿⢱⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣼⣿⣿⣿⣟⢻⡇⣿⠃⣿⡇⢹⣆⢹⣯⣿⣿⣿⣿⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⣏⣿⣟⣿⣿⣿⢻⡇⢸⡄⢼⣡⢸⣿⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠘⠃⠀⠀⢸⣿⣿⣿⢿⣿⣿⣎⣿⣜⣿⣾⣿⣿⣿⢘⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⡾⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠛⠛⠉⠉⠉⠉⠙⠋⠉⠉⠉⠛⠙⠋⠉⠉⠛⠛⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄⠀⣠⠀⣤⣤⠀⣠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢸⠗⠀⡿⠀⢻⡋⠀⣽⠂⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⢸⠀⠀⣷⠀⢸⠁⠀⢿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠃⠋⠘⠗⠁⠹⠂⠘⠊⠂⠸⠚⠀⠃⠀⠀⠀⠀⢘⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⡆⢰⣦⡄⢠⡆⡄⢠⡆⠀⣄⢀⡄⠀⠀⠀⠉⠛⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⢸⡏⠀⢸⡇⢿⢸⠃⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠐⣿⠘⡇⠀⢸⠀⡼⢸⡄⠀⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠋⠀⠃⠉⠘⠃⠀⠘⠛⠁⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢚⣩⣥⠴⠶⠒⠒⠒⠒⠒⠲⠶⠤⣭⣙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣡⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣉⡴⠋⠁⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠉⠳⣌⡛⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⢋⡼⠋⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⣀⡀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠈⠻⣌⠳⣄
⠀⠀⠀⠀⠀⢠⠞⣰⠏⠀⠀⠀⠀⣠⠞⠁⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠙⠓⠦⣄⡀⠀⠙⢦⡀⠀⠀⠀⠈⢷⡘⣦
⠀⠀⠀⠀⢠⠏⣴⠃⠀⠀⠀⢠⠞⠁⠀⠀⠺⣅⠀⠀⠀⢀⣀⣤⡤⠤⠤⢤⣄⡀⠙⠲⣄⠀⠙⢦⠀⠀⠀⠀⢳⡈⣧
⠀⠀⠀⢀⡞⣰⠇⠀⠀⠀⣰⠏⠀⢠⠾⢦⡀⠈⠳⣄⡴⠋⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⢦⠀⠈⢧⠀⠀⠀⠀⢳⠘⣇
⠀⠀⠀⢸⢁⡏⠀⠀⠀⢠⡏⠀⢠⠏⠀⠀⠙⢦⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⢷⠀⠈⢧⠀⠀⠀⠈⣇⢹⡀
⠀⠀⠀⡟⢸⠃⠀⠀⠀⣾⠀⠀⡟⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢸⡄⠀⠀⠀⢻⠸⡇
⠀⠀⠀⡇⢸⠀⠀⠀⠀⡏⠀⢰⡇⠀⣀⣀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣷⠀⠈⡇⠀⠀⠀⢸⠀⡇
⠀⠀⠀⡇⢸⠀⠀⠀⠀⣧⠀⠘⡇⠀⣳⣯⣽⣓⡿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡿⠀⢠⡇⠀⠀⠀⢸⠀⡇
⠀⠀⠀⣷⢸⡆⠀⠀⠀⢻⠀⠀⢧⠘⢳⣷⣶⢾⣷⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⢰⠇⠀⢸⠁⠀⠀⠀⣼⢰⡇
⠀⠀⠀⢸⡄⣧⠀⠀⠀⠈⣧⠀⠘⣷⠿⠖⠋⠉⠀⠈⠧⠤⢤⣄⠀⠀⠀⠀⠀⠈⠳⣄⠀⢠⡟⠀⢀⡏⠀⠀⠀⢠⡇⣼⠁
⠀⠀⠀⠀⢧⠸⡆⠀⠀⠀⠘⣦⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠴⢦⡀⠈⠳⠏⠀⢠⠞⠀⠀⠀⠀⡾⢰⠇
⠀⠀⠀⠀⠈⢧⠹⣄⠀⠀⠀⠈⠳⡀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠆⠀⠀⡠⠋⠀⠀⠀⢀⡼⢡⡟
⠀⠀⠀⠀⠀⠈⢧⡙⢦⠀⠀⠀⠀⠙⠲⣄⠀⠈⠙⠲⠦⣤⣤⣤⣤⣤⡤⠶⠚⠉⠀⢀⡴⠚⠁⠀⠀⠀⢠⠞⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⣄⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉⠀⠀⠀⠀⢀⡴⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⢦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣍⡓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⣋⣥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣭⣉⣛⠒⠲⠶⠦⠶⠶⠒⢒⣋⣩⡥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⣡⠬⠐⠂⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠡⢂⢔⠅⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢄⢠⣒⣀⠈⡁⠀⠀⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠡⡠⡒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣆⠩⣤⠤⠖⠀⠀⠁⡱⠀⠀⠀⠰⠹⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠘⠀⠀⠨⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠊⠇⠀⠈⠢⡀⠁⠀⠀⡠⠀⠀⡠⠂⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠠⠂⠀⠀⠈⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠧⠲⢄⣼⢅⡈⠁⢀⠴⠊⠠⠥⠌⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠐⡊⠓⠤⠔⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡄⠈⠐⣄⢧⢉⠈⣅⡰⡼⠻⠦⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⠀⠀⢀⣼⠏⡸⠀⠉⡆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡀⠬⢧⠁⠀⠈⢿⡳⠷⠃⠂⠀⠀⢀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠛⠣⡀⢠⣞⣟⡠⠃⢀⣼⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢨⡄⡊⠣⡀⠀⠀⠀⠀⣀⠔⠀⠿⠷⠔⠖⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣀⠀⠀⠉⠉⠀⣀⢔⡫⠒⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡁⠀⠀⠲⠶⠂⠉⣀⣀⡈⠀⢀⣀⠀⠀⡀⢀⣀⢀⣀⣀⣀⡐⠛⢙⣃⠽⠷⠞⠉⠀⣈⣡⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⢀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠢⡘⠐⠂⣄⣶⣦⣤⣴⡤⠒⠤⠚⠈⠁⠀⠀⠀⠀⠀⠀⠐⠝⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠠⢄⡉⡀⡀⠀⠀⠀⠀⠀⠀⢠⡇⠱⡀⠀⠀⠀⣿⡝⠀⠀⢸⠀⠀⠀⠤⣤⡄⠀⣀⣀⣒⣒⣈⣴⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠓⠦⠤⢤⣀⣀⡀⠀⠀⣠⠀⢄⡆⠀⢇⠀⠀⠀⠇⡇⠀⠀⢸⠀⠀⠀⠔⠀⣀⠀⣗⣈⠈⠀⠁⣀⣀⣀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡶⡖⠒⠒⠒⠠⡔⠰⠃⠐⠙⢀⣼⠐⠀⠘⠀⠀⠀⡂⡇⠀⠀⠈⢇⠀⠀⣀⣤⠿⠀⢻⢟⠀⠀⠀⠀⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⡇⠀⠀⠀⡐⠀⠁⠀⠃⠠⠄⣒⡀⠄⠀⠡⠤⠤⣁⠇⠀⠀⠀⠈⡀⠀⠩⠔⠊⢁⡸⣯⠀⠀⠀⠀⠀⢈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⡄⠀⠀⠐⡀⠀⠀⢀⠀⠀⠐⠒⠂⠲⠄⡀⣀⣀⣈⣢⠀⠀⠀⢠⠃⢀⡀⠲⠀⠂⠱⡿⠀⠀⠀⠀⠀⢨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⢱⡄⠀⠈⡀⠀⠀⠀⣀⢀⣴⣱⣫⢒⣖⡵⣢⣀⡴⣁⠔⠋⡰⠁⠀⣀⡤⡆⢀⠀⠀⠀⠀⢨⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠉⠙⠂⠨⠐⠺⠿⠒⠶⠿⠨⠒⠀⠅⠒⠠⠐⠈⠩⠐⠈⠩⠐⠠⠑⠢⠻⡤⠖⢛⠉⠷⠾⠛⠭⠄⠢⠽⠜⠺⠔⠢⠒⠥⠂⠤

⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⢿⣿⣤⣼⣿⠀⢸⣿⡿⣿⣶⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣿⣦⠀⠀⠀⠈⠙⠛⠛⠁⠀⢸⣿⣇⣀⠀⠉⠉⠛⠛⠿⢿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣤⣤⡀⠀⣴⣿⠟⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠟⠀⠀⠀⠀⠀⠀⢼⡿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣄⠀⣿⡟⢻⣿⡆⠀⠀⠀⠀⢀⣠⣶⣿⢿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣄⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢠⣿⡇⠺⠟⠁⣀⠀⠀⣴⣿⠟⠋⣀⣴⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡏⠛⠛⠻⠿⠇⠀⣼⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠿⠿⠟⠁⣠⣴⣾⣿⣷⡀⠹⣿⣶⣾⠿⠋⢁⣤⣶⣦⡀⠀⠀⠀⢸⣿⣿⣶⣶⣤⣄⠀⢠⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣴⣾⠿⣿⣧⠈⠻⡿⠀⠈⠉⢁⣤⣾⣿⣟⠉⣿⣷⠀⠀⠀⢸⣿⣇⠈⠉⠙⠉⠀⣼⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠻⣿⣆⠈⠁⠀⠀⣀⣤⣀⠀⠻⠟⠉⠙⠿⠿⠿⠃⠀⠀⠀⢸⣿⡿⠿⣿⣶⠀⢠⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⠛⠀⣠⣴⣿⠿⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢁⡀⠸⠿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⡀⢠⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⢀⣠⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⠻⢿⣿⣿⡅
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⢿⣿⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⣄⠈⢻⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⣿⡟⠛⠛⢁⣠⣤⡄⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢀⣠⣶⣿⡟⠻⣿⣆⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠉⠿⠗⠀⠙⠋⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⢿⣷⡀⠀⠈⠿⠃⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⣿⡄⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠁⠀⠉⠁⠀⠀⠀⠀⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠠⠄⢀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠄⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠢⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠁⡒⠀⠂⠃⠈⠀⠀⠈⠀⠀⠀⠀⠁⠈⠀⠀⠦⡀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠂⠢⣀⡀⠀⠘⠒⣴⠅
⠀⠀⠀⠀⠀⠀⢀⠹⠑⠃⣀⣀⠄⢀⠄⠀⣀⢀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠐⠀⠤⠠⠠⠄⠀⠄⠀⠢⠈⠀⠀⠀⢫⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠄⡀⠈⡺⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣂⡠⠚⠁
⠀⠀⠀⠀⠀⠀⢀⠐⠂⠠⠜⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠊⠂⡀
⠀⠀⠀⠀⠀⡴⠁⠀⠁⠀⠀⠀⢙⠢⠀⠀⠀⠀⢀⠼⣷⠴⠃⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡀⡠⠊⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⢄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⡀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠲⢀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢀⠀⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈