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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣄⢀⡀⢀⠄⣤⡀⣀⡀⣀⢀⠀⣤⣀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠚⠑⠊⠚⠛⠛⠐⠃⠙⠒⠀⠛⠋⠒⠉⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠉⠉⠳⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⠄⠀⠀⢄⣀⣠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠣⢀⡞⠙⢆⡀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣎⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢅⠀⠘⠧⠒⢦⠉⢙⠎⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡀⠉⠐⠒⠒⠒⠉⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠑⠢⠤⣤⣬⠶⠻⡄
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢰⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢸⣠⡠⠀⠀⡇⠈⠑⠒⢤⡘⠄⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣀⠴⠖⠁⢸⠀⠀⠠⢀⠂⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠹⡙⡆⢀⠁⠀⠀⠀⠀⠈⠉⠉⢒⠒⠋⠀⠀⠀⠀⠀⠀⠀⢐⠅⢒⡊⠉⠉⠀⠀⠂⠀⠈⠦⠤⠖⠾⣀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣌⠊⠀⠀⠐⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⡄⠀⠀⠀⢀⠀⠀⠀⠀⣠⠊⠀⠞⠳⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⢵⣆⢄⣀⣈⢒⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠷⠄⣀⠐⠃⠀⡀⠄⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡁⣈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡍⠒⠊⠁⠀⢹⠄
⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠄⠚⠀⠀⠈⡗⢄⡀⠀⠴⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⣚⠠⠁⠀⠀⠀⠐⠻⠀⠀⠀⠀⡼⠠⣰⠇
⠀⠀⠀⠀⠀⠀⢀⠴⣶⠀⣤⠀⠘⠀⠀⠁⠀⠁⠀⠀⠤⠀⠀⣧⡍⠲⣤⣤⣤⠄⢡⣀⣈⣀⠀⠀⠀⡤⠤⠁⠀⠀⠀⠤⠚⣆⢀
⠀⢀⠀⠀⠈⠀⠘⢦⣼⠖⠁⠀⠒⠒⠀⢰⠀⠀⣶⣈⠁⠀⡬⠃⠐⠃⠀⢀⡁⠀⠃⢈⡿⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢧
⡀⢠⠀⠀⠀⠀⠐⠠⡉⠀⠀⠀⣠⠀⠀⠀⡄⠀⢱⡈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠄⣤⠂⠀⢠⣀⠀⠀⠀⢀⣄⣤⣤⣺⣀⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠍⠠⠄⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣀⠉⠉⠉⠉⣗⠀⠤⠤⠤⠀⣂
⣇⠀⠀⠀⠀⠀⠐⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣄⣀⡀⠀⢀⣀⣀⡤⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠋
⡟⠶⠦⠤⣤⣄⣀⡀⠀⡐⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡤⠛⠛⠲⠶⠭⠭⠭⠵⠖⠒⠋⠀⠀⠀⠀⠀⠀⠰⠦⠤⠤⠶⠩⡀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠉⠓⠚⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
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⡿⠿⢿⠿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠲⠶⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢁⡀⢻⣦⣀⢰⣦⣀⣀⣀⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠍⠉⠉⠉⠉⠉⠉⠉⢸⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠓⢒⡖⠀⢲⣒⠛⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣿⠀⠙⢻⠀⡟⠋⠁⣻⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⣤⣬⣤⣥⣤⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠢⡀⠀⠤⠀⢀⠔⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⢤⣀⢀⡗⠀⢺⡅⣀⡤⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡞⠁⠀⠀⠀⠀⡀⠉⢃⡶⡘⠉⠳⠤⡀⠀⠀⠉⢳⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠡⠀⠀⠀⠀⠐⠖⢒⣉⠀⠈⠒⠲⠂⡼⠀⠀⠀⠆⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆⠀⠀⠐⡺⠁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⠤⠔⠁⠀⠈⡙⣆⣱⠁⠀⠀⣠⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣠⠔⠊⠉⠀⠀⠀⠀⣀⣀⠀⠀⡾⠧⣤⣴⠚⠃⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⡞⠁⠀⠀⠀⠀⠀⢀⡔⠉⠉⠓⠊⠀⠀⠀⠈⠳⣄⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⡇⠀⠀⠀⠀⢀⡠⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢘⣦⣤⡶⠓⠉⠀⠀⠀⠉⠓⠦⠤⣄⣀⣀⣰⠦⠤⠤⠷⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⠤⠂⠢⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⢠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣘⡍⠀⠀⠀⠀⠀⠀⠀⡦⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠤⠤⠄⠀⠤⠄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⠀⢐⠦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠘⠁⠑⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠶⣄⡠⠏⣀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠃⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣀⠴⢉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠀⠀⠘⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠄⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠈⠒⡤⠀⢀⡉⠄⠂⠛⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢁⠀⠀⡆⠀⣀⡠⡚⠉⡉⠑⢆⡀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⠀⡸⠀⢃⠀⠀⠀⠀⣆⢸⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠁⠀⠙⡆⠣⠤⠔⠁⢉⠋⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠳⢅⠀⠈⠁⠒⠒⠂⠁⠀⣷⠀⠀⠈⡆⠰⠲⠶⠶⠶⢌⠉⠉⠉⠉⠀⡸⠀⠀⢠⡀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠱⠤⡀⠀⠀⣀⡀⢤⠞⣧⡀⠀⠸⠀⠀⠀⣠⡶⠚⠛⠛⠛⠋⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡀⠀⢦⣧⢔⡩⢄⠖⠃⣠⠟⠁⠀⢰⠀⠀⢘⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⢋⣼⠿⠓⠁⢀⡠⠋⠀⠀⡴⠃⠀⣭⠾⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠓⠤⣄⡀⠿⠁⠀⢀⡤⠊⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠀⠀⠀⠀⠈⠀⠐⠂⠉⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⠇⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⢀⠏⠈⠈⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠈⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠃⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⡀⠀⠀⢀⣀⣠⠇⠀⠀⠀⠀⠸⠤⣤⣀⣀⠤⠤⠜⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⠚⠁
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠉⠛⢶⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠀⠀⣴⠶⠄⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢠⠟
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠁⢀⣤⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣧⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢤⣄⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠉⠀⠀⠀⠽⠁⠀⠀⠀⠀⠀⠀⠀⢴⣦⣠⣴⣚⣋⣁⣀⣀⣀⣀⣈⣉⣉⣉⣶⣤⣠⣶⡶
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⠭⠭⠭⠭⢭⣽⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⡀⠀⠀⠀⠀⠀⠻⠿⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⣉⣀⣀⡀
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠙⠻⠿⣭⣍⣉⣉⣭⣭⠿⠿⢟⣛⣭⣍⣩⠤⢬⡿⠛⠉
⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠻⣧⠀⠀⠀⠀⠈⠙⣿⡿⠟⠛⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⣤⡞⠁⣽⠏⠁⠀⠀⠛⠋⠩⣭⣥⡶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡴⠶⠶⠶⠶⢾⣷⣀⣤⣦⡄⢀⡀⠀⠀⠀⠀⠀⣰⡿⠀⣾⠃⠀⠀⣠⣴⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣤⣀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢠⣿⠃⣸⡇⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠉⠉⠛⠻⠿⣿⣯⣄⡀⠀⢠⣾⠇⢠⡿⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠉⠛⢿⣷⣿⠏⢀⣿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⠇⠀⠀⢠⣿⠇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⣼⡿
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠻⣷⣤⡀⠀⠀⠀⢠⣾⠇⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠀⢠⣾⠏⠙⠿⣶⣄⣠⣿⠋⠀⠀⠀⠀⣿⡏
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⣰⡿⠁⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⣼⣿⠅
⠀⠀⢄⡀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠘⣿⣿⡛⢿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠠⠀⠀⠀⠐⠀⢀⣼⠛⠛⠿⢿⣶⣤⣄⡈⢿⣷⡀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⢀⣼⣿⠃
⠀⠐⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⠀⠀⠀⠀⠈⠙⠻⣿⣶⣦⣾⡟⠁⠠
⠀⠀⠈⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣀⣠⣤⠴⣶⣶⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠐⠀⠀⠸⣿⣇⣀⣤⣶⣿⣯⣵⠶⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⣿⡉⠁⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢤⢶⡴⠆⢠⠆⡴⠀⠄⡇⠸⠰⠇⠰⠀⠀⡶⠄⡶⠶⢶⠄⣰⡠⠠⢆⣴⠀⡶⢠⠶⠂⠠⠀⠠⠶⠰
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⣀⣀⣀⠀⢀⠀⣀⣀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⣄
⠀⠀⠀⠀⠀⠓⠓⠋⠐⠀⠛⠓⠛⠘⠃⠀⠛⠁⠊⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠊⠀⠀⠀⠀⠈⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡗⠄⠀⠀⠠⣀⣀⣀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣞⠀⠀⠀⠀⠀⠀⠉⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠑⠢⠤⣄⣀⣠⠤⠜⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠖⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠒⠒⠦⠤⠒⠚⠓⠉⠀⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣀⢠⣤⣤⠄⠀⠝⡂⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠄⠀⠀⡀⢀⢂⡀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡤⢯⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠳⠋⠱⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠲⠒⠁⠀⠀⠀⣀⠤⣺⠁
⠀⠐⠀⠀⠀⠀⠀⠈⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⣰⡢⢄⠀⠀⢀⡽⡶⠶⠶⣶⠊⠙⢊⠉
⡇⢹⠁⠀⠀⠀⠀⠈⡇⠀⠀⢨⣀⠀⠀⠀⠀⠀⣀⠀⠨⠁⠀⠀⠱⣻⡆⠙⠀⠟⠬⠈⠁⠐⡇⠀⠀⠀⠁
⡇⠸⡀⠀⠀⠀⠀⢀⠇⠀⢀⡘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡤⠶⡊⠑⠀⠂⠀⠑⠒⠠⢀⡑⠠⢤⠜
⡇⠀⠙⠒⠒⠒⠒⠋⠂⠀⠀⠆⠀⠀⠀⢈⡀⠈⠁⠲⠄⠀⠈⢆⠀⠀⠀⠀⠀⡀⠀⠤⠒⠋⠁⡠⢷⠞⢣
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠸⠀⠳⠆⠈⠀⣄⠀⠀⠠⣛⠄⠀⡠⠔⠊⠀⢸⡫⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠁⠀⠀⠀⠀⠀⣧⡅
⡇⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⢱⠒⠒⠀⠐⠀⡆
⡇⠀⠀⠀⢀⠈⠉⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠓⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠈⠉⠒⣏⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠀⠰⡉⠉⠒⠂⠦⠤⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠐⠒⠁
⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠚⠛⠉⠁⠉⠉⠉⠑⠒⠠⠄⣀⣀⣀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣇⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢹⣿⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣄⡈⣿⣇⠐⣄⠀⡀⠀⠀⠈⠀⢀⠀⢀⠀⠀⢰⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣝⠿⣿⣿⣦⡈⠳⡌⢄⠀⠀⠀⠀⡇⠘⡀⠀⢧⠤⠤⠿⠦⠄⠀⠀⠀⠀⢀⡀⠃⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⣿⣿⣆⠸⠈⢠⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⢀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡴⠞⢋⠁⠀⠤⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⡔⢠⠿⠿⣷⣦⡀⠀⡀⠀⠀⠀⣠⣴⣶⡶⣦⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⠐⠋⠀⠀⠀⠈⠙⠛⣤⠓⠀⠁⠾⠟⠁⣀⠀⠈⠀⠀⣿⠛⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡙⢿⡿⢿⠃⠀⠀⢀⣄⠀⠀⣼⡆⠀⠘⠀⡄⣡⡈⢳⣄⠂⠀⠈⢴⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢁⣸⣧⡄⣩⠄⠀⢌⣉⠅⠈⣿⡁⠀⠀⠀⠤⠉⠉⠄⠈⠁⠀⢤⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠀⠀⠀⠀⠀⠀⠀⠸⣧⢋⢻⣿⣿⣤⠰⠀⠀⣀⢀⡞⣿⠋⠀⠁⠀⠀⣀⣠⠀⠀⠀⠀⠻⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡴⠳⠄⠞⠉⠁⣼⠆⢀⣀⡀⠀⠀⠀⠀⢻⠈⢸⣿⢿⣿⠖⢾⡿⢯⣌⣿⣿⠷⠆⠘⠈⢹⣿⣿⡇⠀⣆⠀⠈⠟⠁⠲⣄
⠀⠀⠀⠀⠀⡠⠊⠀⣸⠀⠀⡺⠉⢀⣀⣈⡨⠃⠀⠀⠀⠀⣸⣦⣼⡇⠈⡟⠀⠈⠧⠠⣬⣉⠻⠶⢰⣾⣛⣧⠄⠋⠁⠀⢸⠀⠀⢰⠀⠀⠘⢷⣤⣄
⠀⠀⠀⠀⢰⠁⠀⠞⠁⣠⡞⠀⢀⡘⠉⠀⠀⡔⠳⣀⣠⣾⣿⡿⢿⣇⠀⢿⠀⠀⠀⠀⠈⠻⣶⣤⣤⣴⠞⠁⠀⠀⠀⠀⡈⠀⠀⢸⠀⠀⠀⠀⠀⡟⢦
⣠⣄⡀⠀⡌⠀⢀⡄⠀⠁⢸⠀⣩⣅⣠⣤⣶⡀⠀⠙⡏⢸⣿⠁⠸⣿⣆⠘⢧⡀⠀⠀⠀⠳⣤⣉⣁⡅⠀⠀⠀⠀⠀⡰⠁⠀⢠⠎⠀⠀⠀⠀⠘
⢣⡀⠘⢾⡇⠉⠉⢱⣄⡠⠀⠉⠁⠸⢿⣿⡿⡿⠦⠀⣧⠸⣿⡆⠀⠻⣿⣄⠀⠙⢦⣀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣠⠞⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢫⡀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⢄⡀⠟⠀⠻⢧⠀⠀⠈⠻⣶⣦⡀⠉⠳⢦⣀⠀⠀⠀⣀⠔⠋⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠈⢳⡄⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⡆⢠⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⣤⣙⢛⡛⢛⣁⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠎
⠀⠀⠀⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠾⠿⠷⠶⠦⡤⢀⠀⠈⢫⡉⣉⠅⠈⠀⠀⠀⠀⠀⢀⡀⠀⢀⠔⠁⡴⠃
⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣌⣀⣀⣉⢀⣀⣀⣤⣤⣴⠶⠞⢋⣠⠖⢃⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⠛⠿⠶⢶⣶⣤⣤⠶⠶⠖⣿⡟⠙⠻⠿⣯⣭⣭⣤⣴⡶⣛⣩⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⢠⣤⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⡰⢉⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡘⠇⢷⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢹⣞⢳⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠆⣎⣱⠟⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠸⣾⠟⠀⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡍⡉⠀⠀⣼⡰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢰⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⢿⣁⠀⣀⠠⠛⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢠⠇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠫⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡴⠃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣠⠞⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⠁⠀⢰⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⣧⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠻⠿⠛⠛⠋⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠤⠤⠶⠦⠤⢤⣄⡀⠊⠀⠀⠈⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠬⢭⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠠⠂⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡇⠀⠒⡠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣉⡉⠉⢉⣉⣉⠙⢶⣄⢸⡧⢬⣭⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢴⡦⠀⠀⠠⢥⠄⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣯⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢜⣉⣀⣀⡈⠱⡄⠀⠀⠀⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠉⠀⠀⠉⢙⡲⢤⡀⠀⠀⠀⠈⣧⠈⠙⠿⠶⠞⠉⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡝⣆⠀⠀⠀⠈⣧⡀⠉⠁⠀⠀⠀⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⡆⠀⠀⣴⣯⠉⠉⠉⠀⠀⠀⢸⣿⠟⡇
⠀⠀⠀⠀⠀⠀⠀⡔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣳⡤⣏⠻⣄⡀⣀⣠⠴⠖⠋⠁⣸⠷⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⣌⣙⠒⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡺⣽⣶⡾⠀⠻⣄⠀⢹⡇⠀⠀⠀⣠⡾⠃⠀⠀⠈⠉⠉⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣶⠶⣤⣤⣤⣤⠬⠽⢿⣶⣶⠟⠋⠁⠀⢀⠠⠬⢷⣬⣧⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢻⠈⢿⣄⣀⡾⢰⡇⠀⠀⠀⠀⡾⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⣄⠀⠈⠉⠀⠘⢧⣀⡀⠀⢰⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⣈⡽⠛⣦⡸⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠖⠚⠉⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠈⠀⠀⠈⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⠂⠀⠀⠀⢀⣀⡀⠈⠳⣦⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⠤⠀⠘⢁⠴⠶⠀⠀⠸⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠴⠶⠈⠋⠉⠀⠀⢀⡜⣧⡀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⣠⠎⠀⡈⠑⡀⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡎⠓⢄⠠⠄⠊⡏⠀⠀⠀⠀⠈⠃⠀⡷⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠀⠀⠀⠀⠠⠀⠐⠂⠳⠀⠐⠂⠖⠞⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠡⠀⠃⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⠀⢈⠀⠐⢄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣠⡖⠀⠀⠀⠢⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣿⠃⠀⠀⠀⠀⠘⠤⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⣀⣀⣠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠡
⠀⡀⠔⠂⢉⣠⣴⡶⠄⡉⠁⠀⠀⠀⠈⠀⠁⠈⠉⢁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⢇⠘⠄⠈⠙⠛⠀⢨⠓⣶⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣙⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠁⠂⠤⢀⠺⠀⠃⠾⢃⣠⣼⣿⡏⠀⠠⠀⠀⠀⠀⠈⠉⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆⢤⣬⠉⢉⣈⣋⣀⡀⣀⣀⣀⣠⡤⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠓⠀⠀⠀⠀⠀⡌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢺⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠳⠶⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⣀⡮⠕⠀⠀⠀⠀⢀⣴⣁⣤⡂⠀⠀⠀⣧⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠻⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⢸⣄⠄⠀⠀⠉⣻⡉⠉⠀⠀⠀⠀⠀⣀⠤⠄⠒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀⣀⣀⡴⢦⡀
⠀⠀⡟⠀⠀⠀⠐⠁⡽⣄⠀⠀⢀⠴⢋⠀⠀⠀⣀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⠦⢤⡴⠋⠁⣀
⠀⠀⠳⡀⠀⠀⠀⠀⢀⣾⣦⣴⣧⡀⠀⠶⠂⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣠⢾⠆⠀⢉⣢⡴⣣⠷
⠀⠀⢀⣿⣷⣶⣶⣶⣿⣿⣿⡟⡛⠇⠀⠀⠀⠐⠘⣁⢶⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠠⡏⢰⠋⠀⡼⠉⠀⢀⡇
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠂⠀⠘⠣⡄⠭⠀⠸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠓⠈⠀⠀⠁⠀⠀⣸⠇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⣸⡄⠹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠭⠥⠬⠙⠳⠶⠦⠶⠮⠛⠿⠥⠤⠉⢹
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣧⠀⠈⠓⠒⠒⠊⢹⣥⠖⠓⠋⢹⠀⠈⠉⠛⠿⣿⣿⣿⣿⣶⣶⣒⣒⡒⠲⢶⠒⠒⠒⢚⡠⠛⠋⠉
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠇⠻⢤⣀⣀⢀⣠⠖⠻⣤⠤⢂⣠⠟⢦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢰⣶⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀⠀⠀⠀⠀⠉⢹⠉⡙⢦⣸⠏⠉⢠⠒⠋⠀⠀⠀⢠⠀⠀⠀⠉⢿⣿⣿⠏⠙⠷⢶⣶⣾⡿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠙⠒⢒⡶⢶⡚⣆⠳⠈⣽⠀⠀⠀⠳⢄⠀⠀⠀⢈⣇⠀⠀⢱⠀⠙⡟⠀⠀⠀⠀⠀⠀⢀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡏⢳⠀⠎⠻⣆⡸⣹⠀⠀⠀⢀⠜⠁⠀⠀⠈⢿⡇⠀⠀⠇⣸⠇⠀⠀⠀⠀⠀⠀⣸⣿⡇
⢹⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣠⠀⣸⠀⢾⠲⢤⣀⠉⢣⡏⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠘⣿⣿⣿⣿⣿⠃⡰⠋⠀⠀⠀⣹⠋⠙⠀⠓⢤⠀⠈⠉⢻⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⠀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⢹⣿⣿⣿⣇⡞⠁⡤⠖⠒⣾⠁⠀⠀⠑⠢⣨⠇⠀⠀⢸⡊⡠⠔⠒⠀⠀⠀⢴⡦⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⢻⣿⣿⡟⠀⠀⠀⢠⠞⠃⠉⠀⠀⠂⢄⡾⠀⠀⠀⠀⠻⡄⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣀⣀⠀⢀⣠⣾⣿⣿⠏
⠀⠀⠀⠻⣿⡁⠀⠀⠀⣾⠀⠀⠀⣀⠉⢲⠞⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⣇⠀⠀⠀⢹⠀⠀⢺⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠾⠆⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢳⣤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠲⣦⣶⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠾⠇⠀⠀⠀⠁⠀⠀⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣤⣤⣀⣀⠀⠀⠀⢸⠀⠀⣀⣠⣤⠶⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠⠖⢀⠀⢠⣌⡑⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠐⠘⠛⣩⣿⣿⣿⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠉⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⡟⠁⠀⠀⠀⠀⠀⣔⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣄⢦⢾⣿⣷⡀⠀⠀⠀⠀⣀⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠿⠿⠧⠀⠀⣴⣾⠉⠉⠉⠉⠉⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣆⣻⣿⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠁⠀⠐⢦⡤⠀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠁⠀⠀⠉⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠐⠒⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⠣⠀⠀⠀⠉⠐⠊⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣽⣿⣦⡀⠀⠀⠀⣀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⢤⡙⢿⣿⣿⣿⣦⣄⡛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠳⣄⠀⠙⢦⡙⠛⠿⠿⠿⣿⣿⣯⡹⡓⣄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠈⢷⠀⠀⠙⣆⠀⠀⠀⣿⣿⣿⣷⣧⠈⡄
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣧⠀⠀⠻⣿⣿⣿⣿⣷⣵⠀⠀⠀⠀⠀⠈⣽⡂
⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⢧⡤⠤⢿⣿⣏⢻⣿⡇⣇⠀⠀⠀⠀⠈⣽⠃
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠙⠃⠹⡆⠀⠀⢠⠌⠁⠀⣀
⠀⠀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠐⣿⠈⡆⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢹⠀⢠⠏⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⢦⡏⠀⠀⣄⢀⠰⣿⡗⠈
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣾⣇⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⠚⢉⣁⠀⠀⠿⠶⠦⡞
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠔⠈⣀⣷⡟⠁⣀⣀⡠⠤⠔⠒⠚⢛⠀⠀⠀⠀⠀⢠⡆⣥⠀⠀⠂⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⢱⠀⠀⠀⢀⠔⣋⠔⡹⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣧⡿⠄⠂⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠘⡆⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠋⠁⠘⡇⠲⠄⣿⣟⣉⠉⡉⠉⣉⣉⣉⠁⣤
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣠⣤⣤⣤⠴⠲⢖⠀⡀⠈⡿⣱⣄⣀⣄⣀⣷⣀⣀⣀⣟⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⡀
⠀⠀⠀⠀⠀⠀⢀⡉⠀⠉⠉⠃⠀⠒⠀⠈⠘⠀⠉⠀⣏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⢀⡀⣀⣀⠠⠒⠂⠀⠐⡆
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠘⠎
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠠⠔⠈⠀⢈⣃⠡⠠
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠆⠀⠀⣠⠖⡠⠚⠉⠈⠀⠀⠉⠑⠇
⠀⠀⠀⠀⠰⡀⠀⠀⠈⠀⠀⠰⠁⠆⠁⢀⡄⠀⠀⠰⢰⡄⡆
⠀⠀⠀⠀⠀⠈⠒⠆⣈⠀⡈⠉⡄⢂⠀⠘⠃⠀⠀⠀⠋⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⡀⠀⠀⠀⢀⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡀⠀⠀⠈⠳⠄⡍⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠑⠢⣀⠀⠁⠒⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠤⡘⢄⠀⠀⠑⠒⠒⠒⠺⠠⢀
⠀⠀⠀⠀⠀⠀⠀⡐⠉⠄⠀⠈⡀⠀⠀⢈⠐⠲⡌⠀⢃⠓⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠁⠀⡐⡎⠀⡎⠓⠲⠃⡇⠀⠀⠀⣀⠢⢀⡀⠠⣴⡶⠁⠀⠀⠁⠀⠑⠮⠟⠀⣀⠀⢝⣳⠐⠀⠀⠀⢀
⠀⠀⠀⡈⠈⢁⡂⢀⠀⠀⠀⣇⠠⠤⢇⣃⠀⠀⢡⠀⠀⠠⠀⢀⡄⠁⠐⡡⢄⠀⢀⣀⠀⠂⠒⠃⠀⣀⣀⣠⠀⠀⠀⢀⠠⢤⡀⢐⡆⠀⢳
⡔⠡⠂⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⢹⠆⠀⡸⠀⠈⠢⠤⠼⣧⠀⠀⠒⣉⡐⠥⠠⠆⢀⡠⠀⠀⠠⠄⠈⠤⢀⡠⠜⠀⡨⠜⠋⠁⢀⢮
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠣⠴⠚⠀⠀⠀⠀⠀⠀⢁⡕⠦⢄⣀⠁⠐⠒⠒⠒⠂⠦⠼⠓⠛⠖⠒⠒⠁⠉⢁⣀⡤⢴⡺⠕⠁
⢠⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⡎⠤⠤⠤⠤⠤⠤⠄⠤⠌⠈⠪⠧⠼⠭⠭⠶⠶⡒⢲⣶⣶⣶⣖⣒⣶⣶⡾⠯⠟⠒⠋⠁
⠀⠉⠒⠉⠩⠀⠒⢄⡀⠀⠀⠀⠠⠄⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠚⠂⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣲⣤⣤⣖⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠋⠁⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠷⠧⠦⠧⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠄⣀⣀⣀⠠⠄⢂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠂⠤⠤⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢀⠉⠄⢀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⢀⠄⠠⠀⠐⡒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⠖⠁⠀⣤⡌⠣⡀⠀⠀⠀⠀⠈⠀⠐⣊⡅⠀⠀⠀⠐⢈⡑⠤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡰⠐⠋⢁⠀⠀⠀⢀⣠⡄⠀⠀⢀⠀⠀⠀⢀⡀⠀⠑⣀⠀⠀⠀⠀⢀⠉⠈⠗⠁⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⢀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠁⠑⠀⠀⠛⠀⡠⠀⠀⠛⠁⠀⠠⠒⠀⠀⠀⠀⠐⡀
⠀⡐⠐⠤⡈⠒⠀⠀⠀⠀⠠⠙⠀⠀⠀⠀⠀⣄⣀⣠⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⣠⣬⣆⣤⣤⢤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⣹⡜⣸⣆⣂⡼⠦⠀⠀⠀⠀⠀⠀⠀⢀⢁⡒⠀⢂⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⡐⠠⠠⠄⣀⣡⡄⠀⠒⠀⢀⡀⠀⠀⠀⠀⠀⢸⡀
⠂⠠⣤⣤⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠉⠀⠀⠈⠀⠀⠀⠒⠀⠀⣒⠠⡀⠀⠁⠈⠰⠀⡄⠂⢰⠀⠀⡀⠀⠀⠀⠀⣧⠇
⠀⠀⠈⠉⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠐⠉⠛⠀⠈⠁⠀⠚⠀⢶⠓⠋⠐⠀⠁⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠠⠄⠠⠈⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠡⢀⡀⠀⠀⠀⠀⠀⠀⢰⠈⣅⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⣉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⢊⣁⠀⠀⠀⠀⠀⠀⠰⡄⠠⠤⠦⠀⠋⠑⠉⠑⠉⠉⠁⠁⠀⠀⠀⡠⡆⠀⠀⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠳⠤⠔⠃⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢷⠀⢸⠃⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⡀⢘⡦⠶⠒⠺⢄⠊⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⡄⠀⠀⠀⣀⣠⠄⠾⠀⠀⠀⠀⠈⠇⢡⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⣀⡿⠀⠀⠀⣀⠀⢀⢀⣶⠀⣀⠀⢀⢀⣤
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠁⠀⠀⣷⢰⣿⠀⣾⡾⢸⠌⣿⢠⣿⡟⣾⢔⡿⢀⣶⠷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⠛⠶⣌⠙⠃⠘⠛⠉⢁⠀⠀⠙⠋⠈⠀⠀⠘⠷⠻⣇⣠⡾⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⡿⠀⠀⣆⢰⡇⣴⠀⣿⣽⠀⣾⡒⢠⡞⠦⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢻⣼⠀⣿⠀⣿⢸⢷⡇⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠙⠂⠛⠋⠈⠓⠂⠘⠾⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢸⠀⢰⠇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠤⢀⡵⠒⠋⠓⠢⡉⡠⠤⠖
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢶⡄⠀⠀⠠⠴⠖⠘⠁⠀⠀⠀⠀⠛⠐⠶⠆
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠠⠼⠃⢀⡀⣠⡀⢠⣆⣾⣇⣤⠀⢠⣤⢾⢂⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⡀⠼⣧⠿⣇⡼⣿⠘⠋⢾⣰⠿⠏⠻⢫⡏⣰⠟⠋⣠⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠉⢻⡆⠀⠀⠈⣀⠀⠆⣠⢤⡀⣀⣀⠀⣈⡉⠙⠲⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣤⠴⠁⠀⢸⣤⡇⢸⡇⣿⠛⡄⡿⠀⠸⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⡇⢿⡴⠃⣷⣀⢠⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣧⣶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠁⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡾⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠖⢶⡀⠰⢖⡓⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠉⠉⠀⠀⠈⠁⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣀⠙⠃⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢄⠀⠈⠙⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣾⠐⣤⣀⣀⡠⠀⢸⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠤⠞⢿⣟⠢⡐⠉⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠈⠛⠂⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠒⠉⠻⣇⢰⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣖⣈⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⢿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⢿⣂⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⠟⠋⠁⠑⢆⢻⣷⡂
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⢻⡆⠀⠀⠀⠘⠁⠀⠀⠀⠀⢈⣠⡏⠉⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⡄⢀⣀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠉⠉⠉⠛⠶⣄⠄⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡿⣱⠞⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⣼⡏⠀⠀⠀⠀⠀⠀⢀⡠⠰⠿⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠄⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡏⠐⢁⢀⣰⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠘⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣠⠀⢀⣰⣇⣴⣽⣦⣌⡖
⠀⠀⠀⠀⠀⠀⠀⣀⠙⢠⠻⠿⠋⣿⠀⢰⡇⠀⠀⠀⠀⣀⣰⣇⠱⡀⢀⡖⣯⣸⣷⣄⡠⢻⡄⠀⠀⠀⠀⠹⢄⡹⣧⣾⠏⢿⢋⢹⡁⡙⠁
⠀⠀⠀⠀⠀⠀⠀⣏⢀⠀⢠⠤⣖⢤⣠⢾⠁⠀⠀⠀⠘⠉⠀⠙⢆⠓⡏⠃⣹⠁⢹⡉⠃⡸⢁⠒⠖⠁⠀⠀⠀⠓⠀⣥⠘⣏⢾⢀⡜⣧
⠀⠀⠀⠀⠀⠀⠀⣿⢯⢸⡌⢳⡘⡏⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠆⠈⣦⠙⠛⠾⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠘⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⢰⡄⡆⣆⠰⢠⡆⠀⠀⠀⠀⠀⠀⠀⢁⠶⠙⢒⠸⠚⠆⠁⡀⠀⠀⠀⠀⢠⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⢠⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⠀⢱⡇⡈⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⡌⡈⠀⠀⢀⠰⡄⡈⠀⠀⠀⠀⠀⠀⢣⢨⢸⢸⢰⠀⢡⠐⢰⠀⠈⣦⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠄⣤⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠏⠒⠀⡀⠮⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁⢀⡀⢀⡁⣀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡿⡀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠠⠀⢀⠅⣠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⠂⠘⡅⡷⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠚⠃⠁⠙⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠃⣘⠀⢰⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠋⠁⠓⠁⠃⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣿⠄⢈⡀⠀⠀⠀⠀⠀⢀⡴⢢⡄⠀⠀⣈⣉⣉⣉⣉⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣦⣤⢦⠂⡀⢀⠀⠀⠀⠀⠸⣧⡜⠁⣀⢀⣤⣤⣠⣤⣤⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠯⢅⣿⣴⣟⠛⣇
⠀⠀⠀⠀⠀⠀⠀⡠⡀⡀⠀⠁⣡⣌⠀⠁⣸⠀⠀⠀⠀⠀⠀⠙⣷⠯⠻⠿⠘⠙⠋⠘⠹⠛⠄⢿⡹⠃⠀⠀⠀⠀⠛⠋⠈⠃⠈⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠃⣿⠁⡜⠧⠻⡌⠟⠿⢯⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣒⣒⣒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⢻⠀⣯⠁⣯⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠓⠀⠛⠙⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿
⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⣿⣿⣧⡙⣯⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⡇⣿⣷⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣿⡇⣿⣿⣻⢴⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⢿⣿⣿⢹⣶⠹⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣸⣿⣿⡜⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⣿⣿⡿⢿⣿⣿⡷⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠧⡆⠀⠀⠐
⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣠⣀⡀⢀⣠⡀⠐⠒⠶⣖⠲⢶⣄⣒⡒⠲⠄⣀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⠛⠛⠩⣭⣵⠾⠿⣟⣓⣒⣂⣉⡐⠂⠀⠉⠉⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠉⠒⠀⡠⣄⣉⣉⣉⣍⣓⣶⣶⣶⣾⣋⠉⠩⠍⠭⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠋⠁

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠉⢷
⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣱⣄⣤⣤⣤
⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⣤⠀⠀⣀⠀⢀⡄⠀⢀⢴⡄⠀⢀⣴⠶⣆⠀⠀⡇⠀⣰⡟
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣤⡄⠀⠀⣤⠀⠀⣼⡇⡴⢹⣇⠔⢡⡟⠀⢰⡏⡰⣹⢇⡜⢡⡿⠀⢰⡟⢁⠠⠃⠀⡴⠁⣼⠏
⠀⠀⠀⠀⠀⠙⣷⡄⢀⠔⢉⣿⠁⣠⢺⡏⠀⠀⣾⡟⠀⣿⠏⢠⡿⠀⠀⣼⡟⠀⠿⠋⢀⣾⠁⠀⣿⡇⠀⠀⣠⠎⠀⣸⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢸⣿⣴⢃⣿⠀⠌⠀⠛⠁⠀⠋⢀⣾⠁⡔⠀⠿⠁⠀⠀⠀⣸⠇⢀⠂⠹⣧⡴⠚⠁⠀⠀⠹⢧⠤⠚⠁
⠀⠀⠀⠀⠀⢀⠎⠘⣿⠀⠈⠛⠁⢸⡇⡸⠀⠀⠀⠀⠀⠀⣸⢇⡜⠀⠀⠀⠀⢀⣀⠀⣟⡠⠃
⠀⠀⠀⠀⢠⠃⠀⠀⣿⠇⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣼⠋⢹⠀⠉⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠴⠀⢀⣾⠃⢠⠋⠀⢀⣠⡤⣄⢀⣾⠋⣀⠟
⠀⠀⠀⠀⢧⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢏⣸⠁⢰⡖⠀⣼⣇⠔⠁⠀⢠⡿⠁⢀⠜⢸⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠇⢀⣿⠁⢰⡿⢁⠖⢄⢀⣿⠃⠉⠁⣠⠃⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⡞⠀⣸⡇⠀⣼⡇⠀⠓⣺⠁⢿⣄⣀⡴⢁⣀⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡟⠀⠀⢹⣇⡴⢻⡇⠀⣠⠃⠀⠀⠉⠁⠀⡎⠀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠀⠀⠀⠀⠳⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⡐⠉⠁⠀⠀⠀⠀⠀⠱⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢀⠄⠒⠀⠚⡆⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠤⠄⠀⠀⢀⡄⢠⠤⠄⠒⠀⠘⠛⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⡰⢋⡄⠃⠀⡐⠀⠀⢠⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡀⠐⠈⠔⢣⡘⡀⠀⠸⠇⠀⠀⠘⢞⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠀⠀⡇⠉⠀⢀⠀⠀⠀⠀⠀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠀⠀⠈⠲⣆⠀⠐⠂⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠄⡀⠀⠐⠡⠅⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠑⠤⡀⠀⣠⢔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠈⠀⠀⠀⢠⣄⠀⠐⠤⢌⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⢀⠀⠀⡟⠀⡴⣇⣀⠘⡄⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣀⠤⠒⠂⠉⠉⠀⠀⠉⠉⠙⠒⠠⢄⡃⠀⠀⠀⠃⠀⠈⠑⠢⡀⠀⠀⠀⠘⠀⠀⡐⠊⠀⡠⠚⡀
⠀⠀⠀⠀⠀⡠⠐⠂⢒⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡾⠀⠀⠀⠐⠉⢀⣄⠠⠀⠀⠀⢐⠒⠈⣀⠤⢑
⠀⠀⠀⢀⠌⡀⠈⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡇⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡊⣉⡀⠄⠊⠄
⠀⠀⠀⢀⠈⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣤⠀⠄⠊
⠀⠀⢀⡼⠞⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠚⢆⠀⠀⠀⠀⠁
⠀⠀⢎⠀⠀⢪⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠐⠂⠀⡀⠔⣸⠃⠀⢠⠃
⠀⠀⠈⠳⢤⣀⠑⠢⠤⣀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⣐⣠⠥⠖⠋⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⠭⢭⣽⣿⣟⣛⣛⣛⣛⣻⣿⡭⠭⠭⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⣄⠀⠀⢠⣶⠂⠀⢲⣦⠀⠀⢀⣤⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⣖⠀⠉⣧⠀⠈⡟⢧⣀⡿⢹⠀⠀⢸⠻⣇⣠⠟⡿⠀⠀⣿⠛⢻⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣙⣃⡀⢻⡄⠀⢸⠆⢀⣿⡄⠻⠁⠼⠷⠠⠾⠄⠹⠋⣰⣇⠀⣰⠏⠻⠁⠀⢠⡟⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⢈⡷⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⢀⡔⣹⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠃⣠⡿⢦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠁⠀⠀⠀⠀⠀⠀⣀⡤⣴⣯⡿⠱⢿⣷⠶⢢⡄⣂⣁⠀⠠⣤⣄⡀⠈⠁⠸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⢶⠋⣽⢣⡇⣼⢡⡾⢰⠇⣸⠿⢤⠍⠱⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⢠⣾⣿⡇⠀⠀⢰⢀⡡⠻⢈⢡⣬⣴⣿⠦⠔⢚⣒⣮⣅⣤⣄⣀⣹⣿⣿⣆⠀⣀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⢀⣾⣿⣿⣠⣤⣤⣤⣤⣤⡀⠀⠀⠈⠉⠀⠀⠘⠋⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⠋⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⠃⡨⠋⠉⠱⠦⢀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⠦⠄⠀⡀⠇⠈⠀⢀⠀⡆⠀⠀⠀⢈⣡⣿⣿⡿⠛⣿⡿⠙⢻⡿⣿⣿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢊⣿⣿⠟⠹⠃⠘⢿⠋⠙⠛⠿⡷⣄⠀⠙⠊⠐⠊⠁⠀⠐⠐⠃⠀⠛⠉⠁⠀⠀⠀⠛⠀⠀⢡⠀⠈⠋⠻⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠐⠂⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡟⠛⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⢶⣶⠶⠂⠀⠉⠁⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠂⡆⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠠⠒⠒⠣⠤⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠧⠤⠚⠒⠒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠚⠛⠛⠛⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠉⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠠⠤⠒⠊⠉⠀⠀⠉⠉⠓⠒⠒⠒⠒⠘⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠤⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠄⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣔⣂⡀⠀⠀⠀⢀⠤⠈⠉⠑⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡡⠀⠀⡞⠁⠙⡀⠀⠀⠐⢋⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣍⠤⡃⠀⠀⢣⡔⡄⠀⠀⠀⠀⠋⠂⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢀⢀⡇⠀⢀⠞⠁⠀⠀⠀⠉⠀⠀⠀⠀⠠⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢣⠀⠀⣀⣀⠀⣀⡠⠀⠈⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠲⡀⢠⠓⠤⣀⠀⠣⡀⠣⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣧⠱⡼⣄⠀⠈⠳⡄⠙⠦⡀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠒⠓⠢⡈⢳⡀⢀⠋⠀⢀⡜⡖⠀⣀⡴⠊⢣⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⢀⣠⡤⠚⢦⡴⢀⡌⠀⠀⠈⢧⠘⢾⡉⢀⡠⠃⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠴⢮⠠⡜⠀⠀⠀⠀⠀⠉⠀⠉⢟⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠤⠤⢤⠴⠚⣄⢣⠀⠀⠀⠀⠀⢀⣀⣠⠦⢖⠴⠂⣸⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⣸⠋⡆⢣⣀⡤⠖⠚⠉⢁⡠⠖⠉⠀⠀⢻⣇⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡴⠞⡄⠁⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⢸⡟⢦⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⡴⠒⣬⣇⡼⠋⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠎⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠋⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠇⠀⠈⠳⡄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠚⠋⠉⠀⠀⠀⠀⣠⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡈⠙⠒⠶⠦⠤⠴⠿⠶⢶⣶⠶⢒⣀⣠⠄⣠⠞⠉⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠉⠉⣉⣉⡳⠿⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣷⣶⣶⣾⣶⡿⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣾⣷⣿⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣷⡾⢿⠿⠿⠿⠾⣷⡶⠶⠶⠶⠶⢿

⠀⠀⠀⠀⠀⢀⠤⠒⠂⠄⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡠⣤⢤⠀⡄⣤⡤⣠⡀⣄⠀⢠⣤⢀⡤⡠
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠉⠉⠀⠀⠁⠀⠈⠁⠀⠀⠁
⠀⠀⡰⠉⠀⠀⠀⢠⠒⠢⣀⣀⣀⣜
⠀⠀⠣⣀⣤⣀⡀⢸⠀⡠⠤⡄⠀⢸
⠀⠀⠀⠀⢈⠆⠀⠀⠉⠀⠀⠘⡶⠋⠑⠲⢄
⠀⠀⠀⢠⠚⠒⠤⠤⠤⠤⠒⠉⠀⠀⠀⠀⠀⠈⠖⣆
⠀⠀⢀⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⢳⠤⣀⡡⠦⣀⣀⣠⠤⢤⡤⠊
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠐⠂⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣀⣀⢠⠀⠀⠀⠀⠀⠂⠌⠠
⠀⠀⠸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠺⠅⠠⠤⠤⠤⢀⡸⢑
⠀⠀⣠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠤⠚⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡑⠂⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠓⠤⡀⢀⠀⠀⢸⠓⢖⠲⡶⢒⡾⡛⠋⠀⠀⠀⡤⡂⠀⠀⠀⠀⣀⠀⠀⠀⠐⠰⠁⠀⠀⠠⠀⠀⠀⠀⠈⠀⠄
⠃⠀⠀⠀⠉⠉⠉⠲⠊⠀⠀⡇⠒⠈⢡⠼⠀⠀⠁⣞⠉⡁⠀⠀⠈⠤⠀⠀⠤⠀⠨⡇⠀⠀⠀⢸⠂⠀⠀⠀⠀⠐⡇
⠁⠀⠀⠀⠀⠀⡤⣼⣀⣀⠤⣫⠤⠐⠒⠒⠒⠀⠐⠓⠶⠤⡁⠀⠀⠀⠀⠀⠀⠁⠸⠇⢀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠃⢼⠋⢃⠀⠉⠂⠁⢒⢀⠀⠀⠐⠒⠂⢀⠁⠀⠀⣄⣀⣀⡠⠀⠂⠀⠨⠀⠀⠈⢷⣦⣤⣤⣴⡾
⡆⠀⠀⠀⠀⠀⡞⡸⠀⠀⠁⠂⠄⠀⠀⢒⡄⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠺⣅⡀⠤⠀⢀⠀⠈⢩⣩⡽⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢭⠋⠁⢀⡏⠉⠏
⠀⠀⠀⠀⠀⢠⠖⠃⣈⠓⠤⣎⢉⣀⠹⣄
⠂⠀⠀⠀⠀⡇⠀⠀⠩⡉⢈⣇⠀⠑⠀⠈⢱
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠁⢸⣄⡀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠔⠋
⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠒⠂⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⡾⢺⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠒⠀⠀⠀⠀⠀⠀⠀⠙⢄⢠⣟⡾⣿⡀⠀⠀⠀⠀⠀⣠⡾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑
⠀⠀⠀⠀⠀⣀⠄⠒⠈⠁⠀⠀⠀⠈⠁⠀⠠⢀⠀⠀⠀⠀⠈⢺⣯⠈⠌⠛⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢂
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠄⠀⡀⠀⠀⠀⠀⠐⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡤
⠀⠛⠤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠒⠉⠱⠀⠀⠀⠀⠉⠀⠀⣤⢺⠈⠲⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠐⠀⠀⠀⠰⠐⠋
⠀⠀⠀⠀⠈⠈⠙⢛⠲⣴⠢⣄⡤⢀⡀⠀⣀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠀⠐⠘⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠄⢀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠓⠈⡍⢀⠐⠟⡙⠛⠛⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢬⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠓⠀⢿⠗⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⢳⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠈⠄⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠁⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⣓⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠄⠀⠀⠐⠉⣇⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢹⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡐⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠊
⣿⣿⡟⠻⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡞⢹⠧⣄⠀⠀⠀⢀⡤⠼⡎⢳⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠗⠀⠀⠀⠺⣍⠀⣠⣄⠈⡽⠗⠀⠀⠀⠲⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢺⢃⡞⢳⡹⡟⣸⠁⠘⡞⣻⢋⡞⢳⡘⡗⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⣇⠸⡇⢰⠀⣿⠃⡼⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣤⡴⣄⣤⣄⣠⡶⢻⣄⣧⣸⣠⠟⢶⣧⣄⣠⡴⠦⣴⢖⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡁⢺⣧⣾⠀⣿⣧⢸⠙⢠⠀⠋⠀⡟⠛⣄⠘⠃⢸⡇⢐⣓⣸⣮⢁⡍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢲⡌⢻⡀⣿⡇⣼⠀⣾⡆⣸⠀⣷⠀⣿⢀⣿⢸⡇⠸⡿⢻⣿⣼⡇⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠛⢳⡬⢶⣻⡇⣿⣿⠟⠀⠿⣴⡿⠛⠿⣼⠟⣓⢶⡏⠀⠈⠳⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣿⢹⡇⣾⡂⣿⠇⣿⠀⣻⣥⣇⠘⢯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢁⣿⡇⡿⠁⣤⡘⣿⡀⠿⢏⣿⡟⠶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠓⠚⠷⠶⠞⠋⠙⠛⠋⠀⠙⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⢶⣤⠙⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣶⣬⣿⠟⠦⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⣼⠋⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⠁⠀⠀⠀⠀⠀⠀⠀⢿⣆⠆⣠⡅⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠐⣋⣽⡇⠀⢾⣭⡉⢸⡿⢤⠖⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠀⠀⠉⠀⠀⠀⠀⠸⠿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⠀⠓⠒⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⢓⠂⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⢀⣴⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⢿⣿⣿⣿⠋⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣍⡋⠁⠀⣠⠎⠀⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣌⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⣤⠴⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣢⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⡰⡁⠀⠀⠀⡀⠐⠀⠀⢸⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠁⢀⡸⠇⢀⠀⠀⠀⠸⡆
⠀⠀⡼⠁⠈⠀⠀⡌⠀⠀⠀⡀⣼⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣤⣾⣿⣿⡟⢻⡗⣦⠙⢷⣌⠀⡀⠀⠀⣿
⠀⢠⡇⠤⠒⠘⣼⡃⠀⠀⠀⢥⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠋⣾⣿⣿⣿⣧⣸⣧⠀⠂⠀⠻⣷⡘⣆⠀⢸⡀
⠀⢸⡇⠀⠀⠀⠹⣁⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠃⠠⣿⣿⣿⣿⣿⣷⠟⠂⠀⠀⠀⠻⣖⠘⡆⢸⡇
⠀⣜⠁⠀⠀⠀⢀⣻⣴⣶⡞⠁⣇⠀⠀⠀⠀⠀⠀⢸⡇⢰⣴⣿⣿⣿⣿⣿⣿⣶⡶⠀⠀⠀⠀⠀⢹⡇⣷⢸⡇
⠀⠘⣦⡀⠀⠀⠸⣴⣿⡟⠀⠀⠏⠀⠀⠀⠀⠀⠀⠘⠣⣾⣿⣿⣿⣿⣿⣿⣿⡉⢻⠀⠀⠀⢔⣀⢴⠂⠀⠸
⠀⠀⠈⠻⣄⣀⣀⣻⠟⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠦⠶⣄⡐⠀⠒⠀⢁⡾⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠷⠲⠾⣽⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢸⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠟⠃⠠⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠂
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⡆
⠀⠀⠀⢸⠟⠛⠋⠙⢿⣿⣿⣿⣿⣿⡄⠀⡀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣴⢸⣦⠀⣻⡀
⠀⠀⠀⠀⠐⠄⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⠘⣧⢚⣩⡿
⠀⠀⠀⠀⠀⢈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⡏⢹⣇
⠀⠀⠀⠀⠀⢸⣷⡄⣿⣿⣿⣿⣿⣿⣿⣧⣸⣟⢳⣾⣿
⠀⠀⠀⠀⣠⣾⣃⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⢙⣿⡿⣆
⠀⠀⠀⢸⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣷⢻⣿⣧⣿⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡇⣷⠘⡆
⠀⠀⠀⣿⣿⣿⢻⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠠
⠀⠀⠀⣿⣿⣾⣿⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⣿⣿⡇⠀⠀⠠⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠉⣿⡿⣿⠋⠁⣸⠇
⠀⠀⣰⣿⣿⣿⣧⡀⠀⠀⠀⠀⢻⠀⠀⠀⢠⣿
⣠⣾⣿⣿⣿⣿⣿⣿⣷⣾⣷⣄⠀⠀⣀⣀⣾⡿⠀⠀⠀⢀⠆
⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⣿⣿⣿⣿⣿⣿⡿⠓⠤⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡧⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⠂⠐⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠤⠤⠀⠺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣨⠟⣒⣀⡤⢎⠐⠄⠀⠀⠀⠀⠀⢠⡞⠛⠁⠀⠀⡀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠐⠉⠦⠀⠑⠀⠀⠀⠀⠈⠓⣶⠀⠘⠋⠁⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠾⠛⣶⡄⠘⣄⠀⠀⠀⢠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⢦⡀⠀⠲⠌⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⢮⡀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⠦⣴⡾⠀⠀⠀⠀⠀⠀⠀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⠀⠤⣶⣶⣶⣶⣤⣄⣠⡤⠴⠂⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢀⡼⢻⠇
⠀⠀⠀⠀⠀⣴⣾⣿⣿⠀⣽⣿⢹⣦⡀⠀⠿⢟⢛⣉⠀⣤⡶⣦⡤⠀⡀⠀⣼⢹⣧⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢀⣴⡿⢁⡟
⠀⠀⠀⠀⠀⠩⠙⣿⣿⣿⣿⣷⣶⠖⠀⣆⠈⣿⡏⠻⠀⠈⠻⣮⡵⣦⠀⠀⡏⠘⢻⣿⣆⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠳⣌⠿⣿⣿⣿⣿⡥⢂⡀⠨⠿⠇⠀⠀⠀⠀⠈⣿⡏⣠⣼⠇⠀⠀⢿⣿⣷⣄⢀⣠⣴⣾⣿⡿⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣰⣿⣿⠀⠀⠀⠀⠘⠻⢿⣿⣿⠟⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣍⡿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⣭⣽⣿⣿⣿⣛⡳⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠉⠁⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⢰⠚⢶⠚⠙⡶⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠀⠀⠀⠙⢿⣶⣤⣤⡴⣦⠀⠀⠀⠀⠀⠀⢹⡄⢸⣇
⢴⣦⣤⣤⣤⣤⣤⡏⢹⠀⢸⠀⢀⡇⠀⣿⣭⣭⣧⣤⣤⣤⣶⣶⠀⠀⠀⠀⠀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣄⠀⠀⢸⡇⢸⡿
⢸⣿⠀⠀⠀⠀⠀⢣⡼⠧⠞⢦⠞⢧⠔⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠛⠶⠟⢁⣾⠃
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢠⠤⠄⠀⠈⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠙⠻⢷⣶⡶⠾⠛⠁
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⡀⠀⠀⠀⣼⠃
⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⣤⠤⠄⠀⠀⠀⠀⠀⠀⢀⠠⡄⠈⠻⣿⣦⣀⣼⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠉⠀⠀⠈⠻⣿⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⣆⡀⠀⠀⠀⣠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⢸⡗
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⠿⠷⠾⢯⣁⡀⠀⢀⣠⡞⠃⠀⠀⠀⠀⠤⠶⠛⠁
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠤⠬⠽⠿⠛⠉
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠔⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠟⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢧⣀⣨⠗⠒⠒⣒⣛⣻⣿⣯⣭⣭⠤⠤⠀⠀⠀⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠈⠙⠷⡶⠤⣜⣟⣠⠤⢶⠶⠋⠁⠀⠀⠀⠀⢸⡇⠀⠀⢰⣆⠈⣤⣀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠻⠿⠷⢶⣶⣦⣤⣌⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟⠘⣇⠘⣟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣄
⠀⠀⠀⠀⢀⡀⠠⡤⢤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇⢀⡼⠁⠀⢿⡄⠸⡆⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⠇⠀⢸⣷⡀⢻⡀⠘⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡴⠋⠀⠀⠀⠹⣷⣄⡙⠂⠈⠛⢿⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡟⣀⡤⠎⠀⠀⠀⠈⠙⠻⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸
⠀⠀⠀⠀⠀⣀⡠⠤⠤⡇⡧⠤⠤⢄⣀
⢀⠴⠊⠉⠉⠑⢢⡤⠒⠉⠉⠐⢤⡠⠒⠉⠉⠑⠢⡄
⡎⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⡆⢀⣀⣀
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠉⠢⡀
⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡠⠒⢩⣿⢿⠀⠀⢠⡀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⢢
⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠎⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡼⠉⠉
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠓⠀⠀⠀⠘⠃⠀⠰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠏⢈⡾⢥⣠⠎
⠀⢃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢆⠀⠙⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⡠⠧⠤⣀⠀⠀⣰⠏⢀⠞⢠⠾⡅
⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠈⢢⡀⠀⠓⠀⠀⡀⢀⡠⠤⠒⠉⠸⣆⠀⠈⢣⡾⠁⠀⠀⢀⡵⡞⠓⡄
⠀⠘⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⣰⣉⠦⠤⠒⠚⣯⠁⠀⠀⢀⡀⠀⠹⠆⠀⠀⣧⣤⣤⡬⠵⠖⠓⠉
⠀⠀⠱⣄⠀⠀⠱⣄⠀⠀⠀⡠⠒⠉⠉⠳⠀⠀⢀⠀⠀⠘⣧⠀⠀⠈⢷⡀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠉⠒⠊⠉⠉⠒⠺⣀⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣀⣘⣃⠠⠤⠒⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⡷⠀⠀⠀⢰⠇⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠸⣷⢀⣠⠤⠒⠂⠈⠉⠁⠒⠲⢤⡀⠰⠋⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢲⠄⠀⠀⠀⠘⠀⠀⠀⠀⢀⣤⡀⣠⣄⠀⢀⠀⢀⡀⢀⣤⡀⠀⠀⢀⡀⠻⢼⢤⣄
⠀⠀⠀⠀⠀⡀⠀⢀⠽⣿⣇⣠⠞⣿⠀⢠⣶⠀⠀⣷⠎⢸⣷⢣⡿⠀⣿⡠⠋⣷⠎⣸⡇⠀⣼⣯⠇⢠⠏⣼⡟
⠀⠀⠀⣰⠁⠀⠀⠀⠀⣸⣿⡇⠀⣿⣀⠎⣿⠀⡘⣿⠀⢸⡇⣸⡇⡼⣿⠃⠀⠟⠀⣿⠃⣰⣿⠁⣠⠋⢸⣿⠀⣰⠂
⠀⠀⠀⠑⠀⠀⠀⢀⡾⢁⣿⡇⠀⠈⠋⠀⣿⡸⠀⠀⠀⠀⠀⣿⡿⠁⠈⠀⠀⠀⠀⢿⡴⠋⠙⠋⠁⠀⠘⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢧⠾⠟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠃⠀⣾⠀⡟⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢰⠗⢺⡇⢀⣧⣾⢤⣄⣴⣾⠹⠆⣰⣿⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡼⠀⢸⡇⣸⡿⢁⣴⡏⢺⡇⠀⣠⠇⡼⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠈⠛⠋⠀⣎⠼⠁⠈⠛⠚⠁⠈⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠢⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡌⢿⠴⣷⢏⣭⠽⠄⠀⠀⠀⠁⠀⡀⠈⠳⠶⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡺⣠⠄⣙⠁⢠⠚⠀⠠⠀⠀⡿⡅⠉⠀⡀⢴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣯⣀⣤⠟⡻⠊⠀⢰⠞⠀⠀⣧⠁⠨⠟⠋⠋
⠀⠀⠀⠰⣦⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣩⣍⣿⣇⢡⠏⠀⠀⠀⡀⠈⠀⠘⠁⠀⠀⢠⣴⠞⡀
⠀⠀⠀⢴⣿⣿⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣝⣿⣷⣫⡟⠀⠀⠀⡈⣙⠀⠀⠁⠀⢠⠀⠀⣽
⠀⠀⠀⣾⣿⣿⣋⡿⢄⡄⣰⠀⠈⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣢⣶⣾⣿⣿⣿⣿⡰⡀⢀⣄⠉⡉⠀⠀⠀⢠⣾⠀⢠⣿⡀⣒
⠀⠀⠀⢾⣿⣿⣯⣥⣬⡷⣿⣇⣰⣇⣸⣧⣠⣆⣠⣄⢀⣀⣤⣤⣴⣾⡿⣶⠈⠀⠙⢚⣿⣿⣿⡃⣼⣿⣦⣿⠄⠀⠐⣽⣿⡇⢼⣿⢷⣧
⠀⠀⠀⢲⠹⡏⠻⢿⠿⣼⣿⣉⣿⣏⣙⠻⣟⣭⠭⢽⠷⠾⠟⡛⠉⣱⠒⠀⠀⠀⣼⣻⣿⣿⡟⣿⣿⣿⢿⡏⠀⠠⢠⣾⣫⠋⠀⢀⣼⣽
⠀⠀⠀⣚⢀⠀⠐⠀⠉⡌⡉⢻⠻⢿⢿⣿⠝⡁⠀⢀⣀⡛⠈⠙⠀⢡⣤⣤⣄⣤⣿⠿⢿⣿⣷⢁⣲⠞⣯⡇⠀⠒⣸⣮⣶⡆⠀⢸⣿⣿⡁
⠀⠀⠀⠸⠀⣁⡀⠀⠺⠀⠉⠠⠀⢀⠸⡃⠀⠀⠐⠚⠛⠻⡀⣠⣶⣿⢹⣏⢙⡝⠁⠀⠀⢹⡏⢶⣿⣿⣿⡇⢀⠠⢹⣿⣿⠄⠀⢸⠛⡿⠁
⠀⠀⠀⡀⢡⠀⠀⠀⠁⠀⠈⠀⠀⣰⣴⣿⠗⣬⠄⠀⢚⡉⠉⣙⣛⡛⠊⣛⣉⣿⣧⠀⠠⡀⠉⢆⠿⠛⢛⠃⠸⠀⠘⣟⠙⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣙⠧⠌⠋⠁⢘⢒⢹⣾⠿⠛⠛⠛⣿⣿⣿⣿⣯⣤⣶⡾⠋⡏⠐⠀⣾⡄⠀⠀⠀⠊⠀⢰⣤⠀⠃⣗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢸⠞⠀⠀⠀⠈⠁⠀⠠⣺⣿⣿⣿⣿⡆⠀⢧⠀⠀⢢⠇⠀⠀⠀⠀⠀⠰⢼⠀⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⠚⠈⠀⠆⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⢠⢀⣴⣿⣿⣿⣿⣿⡏⠁⣾⠀⠀⣸⠀⠀⠐⠀⠀⠀⠀⢸⡐⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠒⢄⠀⠀⠀⠀⠀⠀⠀⡔⠈⠁⠀⠈⠁⠈⠈⠑⡂
⠀⠀⠀⠀⠀⠀⠀⡖⢤⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠔⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠢⡎⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠌⠀⣀⠀⠀⠤⠘⠒⠀⠈⠒⡄
⠀⠀⠀⠀⠀⡠⠄⠒⠚⢂⠤⠒⠉⠉⠉⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⠀⠀⡴⠁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀⠄⡒⠂⠈⠉⠀⢐⠪⣄⠤⠄⠈
⠀⠀⠀⠀⠀⠀⡣⣼⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠣⡀
⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠈⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠡⠜⠀⠀⠀⠀⠠⠀⡟
⠀⠀⠀⠀⠀⢠⠀⠚⠗⡒⢄⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⣌⠀⠂⠐⠈⠀⠷⠤⠠⠀⠤⠤⠂⠙⡄⠐⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⡇⠐⠀⠈⠀⠀⢀⠀⢠⠒⠐⠢⠀⠠⡘⠢⡀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠈⠂⠤⠤⠤⠤⢊⡠⠼⠀⠀⠄⠀⠀⢸⣄⠈⠙⡒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠸⠀⠀⠒⠀⠀⠘⣀⡀⠀⡠⠤⠀⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠂⢠⠔⠁⠄⡀⠐⠀⢀⡰⠀⢁⡰⠁⠀⠀⠀⠀⠈⠑⢄⠉⠲⢄
⠀⠀⠀⠀⠀⠀⢀⡊⠀⢈⠀⠀⠠⠄⠀⠀⢧⠀⠀⠸⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠄⠁⠁⠆⠀⠀⣶⠁⠀⠀⠇⠀⠀⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠢⠀⠐⠄⡀⡄⠀⠀⣒⠀⢈⠖⠊⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠉⠁⠉⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⣀⠄⠀⠀⠀⣀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠁⠢⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠒⠀⢀⡴⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠫⢭⡽⣆
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠰⠀⠖⠀⠀⠇⠀⠀⠀⢀⠀⠀⠀⠀⢠⣟⠀⡺⣀⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⢉⡆⢘⣿⡀⠀⠀⠈⠀⡠⠛⠃
⠀⠀⠀⠀⠀⢀⠀⠃⠣⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡹⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠼⠒⠹⠐⠨⣇⠀⠀⠀⡜⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢦⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⢠⡞⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠏⡃⠏⡁⠾⡄⠀⢰⠇⠰⣶⣤⢴⣤⣿⣥⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⠀⠰⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣃⣙⣢⣙⣻⠃⠀⢸⠀⠀⣠⣿⣶⣿⢿⢋⡄
⠀⠀⠀⠀⠀⠀⠢⣀⣤⠶⠊⠁⠀⠀⡼⠀⠇⡾⠀⠀⠀⠀⠀⠀⠈⠛⢦⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡟⡟⠯⠀⠀⠆⠀⠰⡿⣿⣟⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⠞⢀⣼⡾⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⢀⠀⠀⠀⠈⣿⣿⣿⣿⣛⡁
⠀⠀⠀⠀⣤⣤⠤⠤⠴⠶⠶⠛⣁⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠈⢹⣾⣿⠉
⠀⠀⠀⠀⠀⠈⠉⠀⠒⠒⠒⠉⠉⠀⠀⣀⣀⣀⣀⣤⣤⣤⡴⠶⠶⠖⠚⠟⠛⠉⠉⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⢰⠋⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠞⠁⠀⠀⠀⠀⠀⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣬⣥⠜⢻
⠀⠀⠀⠀⡴⢋⡴⠂⠀⠈⠑⠄⠀⠈⠓⠦⣄⣀⣀⣐⣀⣀⣀⣀⣠⣤⡤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢙⠳⢄⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⢄⢘⡉⠺⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢻
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣀⣀⣐⣠⣄⡶⠦⣄⠀⠀⠀⠈⠀⠀⠀⠈⠋⠀⢹⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⣿
⠀⠀⠀⢸⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡞⠁⠀⠙⡇⠀⢹⡆⢹⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⡻
⠀⠀⠀⠀⢳⣜⠦⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠁⠀⠀⠀⡇⠀⣸⢃⣿⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠠⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⢀⠃⠠⣯⠞⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢢⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢰⢣⡇⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠉⢉⠁⠂⠤⠀⠀⠀⠄⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠤⠔⣣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠃⠀⠀⠃⣠⠖⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡄⠀⠀⠀⠉⠛⠛⠓⠋⠁⢀⠠⡔⠒⢤⢀⣤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣠⡤⠶⢤⣄⣀⣠⠼⠃⠀⠀⠀⠀⢠⠁⠀⠀⠀⠤⠤⢍⡉⢩⡄⣾⣹⠀⣠⠖⠀⠀⠀⠀⠀⡰⡄⣨⢃⠫⣜⣻⣍⠛⢞⣷⣦⡀
⠀⠀⠀⡄⠁⠀⠀⠀⣿⠮⣤⣄⠀⢀⣀⣀⣀⡀⣀⣀⠀⠀⣀⠀⠀⠀⠀⢀⣠⡶⠛⠑⣤⠶⠿⣷⠶⠶⣤⡀⠀⠀⠀⠀⢽⣤⡁⠋⡐⡛⢉⣿⣾⣻⡙⢹⡇
⠀⠀⠀⣱⠄⠀⠀⠀⢿⡀⠁⢉⠓⠋⣀⣹⡅⠘⠋⠹⠓⢾⡏⠸⠀⠀⢠⣾⠟⠀⠀⠜⠁⠀⠀⠈⠃⠀⠈⠻⡆⠀⠀⠀⠐⠛⠑⠆⠈⠒⠲⢿⣷⣤⣅⣾⠇
⠀⠀⠘⠇⢶⠀⠀⠀⠘⣧⡀⠀⠀⠸⡏⠉⠐⠀⠀⠁⢈⡼⠁⠆⠀⠀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡼⢂⡼⠋⠉⠁
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠈⠻⣦⣴⡋⠁⠀⠀⠀⣀⡴⠋⠀⠎⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡜⢰⢃⠊
⠀⠀⠀⠀⠉⠒⠤⢤⡀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⡴⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢰⠃⡜⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠃⠀⠀⠀⠀⠐⠒⠂⠐⠢⠒⠁⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⢀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠮⢤⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠄⢫⣱
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠻⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠠⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠀⠤⢀⡐⠆⠠⢆⡄⠠⡥⢀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢤⡰⣶⣒⣤⠂⠁⠀⠀⠀⠚⠐⠒⠢⠌⠢⣘⡸⠃⢠⡃⢠⠊
⠀⠀⠀⠀⠀⠀⢀⣠⣚⣿⣿⣛⣮⣷⣿⣾⣥⣆⣀⣀⣀⡀⣔⣃⣈⣀⢀⣀⣘⠄⠀⡰⡣⠚
⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠤⠀⠀⠀⠤⠠⠠⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠠⠭⢽
⠀⠀⠀⠀⠀⠀⠈⡄⠂⠀⠀⠀⠀⠀⠀⠄⠀⡀⠄⠐⠂⠤⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠃⠀⠀⠀⠄⠀⠠⠂⢀⡤⠐⠒⠤⡀⠀⠀⢤⠀⠀⠀⠐⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠆⠀⠀⡀⠀⠀⠀⡎⢰⢉⣈⡆⡅⢸⠀⢠⠄⠈⠔⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠰⠀⣁⠀⠀⠀⢡⡈⠒⠓⢊⠔⠁⠀⡄⠈⠂⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠁⠂⠀⠀⠠⠌⠉⠉⠀⠀⠀⠒⠈⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣙⣀⣛⣀⠘⠃⣘⣃⣚⣁⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡠⠤⡀⠀⡤⠤⣀
⠀⠀⠀⠀⠀⠀⡼⠀⡈⢢⢀⠎⠀⠀⠀⠈⢢⠀⣀⡀⠀⠀⠀⡤⠒⠦⡀⢰⠁⠀⠹⡄⠀⠑⠉⠁⠀⢨⡀⢠⠤⠤⠄⢀⡀
⠀⠀⠀⠀⡜⠉⠁⠀⠇⠀⠿⡀⠀⠀⠀⢀⠍⡜⠁⠈⠉⠐⢺⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠶⢅
⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠈⠒⢤⣰⠋⢚⠀⠀⠀⣴⠤⠏⠉⠉⠓⡦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣁⣀
⠀⠀⠀⠈⡙⠀⠀⠃⠀⠀⠀⠑⠀⠀⣹⣀⡂⠁⡐⠠⡇⠀⠀⠐⠀⠠⣇⢀⡴⠚⠙⠒⢅⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⢀⠇
⠀⠀⠀⠀⢣⡀⠀⠀⢀⢄⡀⠀⠀⠀⢹⠁⠐⠀⠉⢰⠁⢀⠀⠀⠀⡤⢨⣟⠂⠀⠀⠀⠀⣱⠀⠀⠀⠀⠀⢻⣇⠀⠒⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡎⠁⠀⢈⠥⣤⡁⠀⠈⢻⠶⠗⠒⡲⠂⢸⢰⡯⢦⡀⠀⠩⢿⡃⠀⠀⠀⠀⢠⢷⣀⠆⠀⠀⠀⠢⠀⢣⠀⠀⠀⢰⠴⠃
⠀⠀⠀⠀⣇⠀⠀⡆⠀⠀⠉⠢⣄⡼⠀⣠⣞⠃⠀⢸⡾⠁⠀⠀⡰⠃⡿⡟⠒⠢⠤⣴⠣⣻⡎⠀⠀⠀⠀⢀⢀⡽⠀⠀⠀⠈⠑⡄
⠀⠀⣄⣀⡈⣒⠖⡹⢆⠀⠀⠀⠨⠷⢆⠁⠘⡧⠄⠘⠃⢄⢀⡜⠁⠀⢈⠞⣆⠀⠀⢀⣠⡟⠓⠜⠀⢀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠽
⠀⠀⡼⠃⠀⠀⠉⠚⠈⠳⡄⠀⠀⠀⢀⠆⠀⢃⠀⠀⠀⡄⠊⠀⠀⠀⠊⡜⢨⡖⠈⣌⠀⣼⡀⠤⢱⠉⠐⠄⣀⢅⠀⠀⠀⠀⡤⠃
⠀⠀⠱⣄⠀⠀⠀⣐⠦⠀⠈⠢⣀⠤⡙⢄⠀⢸⠀⠀⢠⡁⠀⠙⠄⠰⢱⠀⡈⠁⡞⠔⢀⣽⠛⠢⣄⠓⠛⠒⢅⠀⠀⠀⠀⢴⠁
⠀⠀⡖⠒⠛⠀⠀⠈⠀⠀⠠⠀⠀⠀⠈⠊⠀⠘⠢⠤⠞⠀⠈⠀⢠⠘⣾⠮⠉⣴⡽⠀⠈⠀⠀⠀⠀⣣⢀⠈⠀⢳⠀⠀⠀⠀⢱
⠀⠀⠈⣢⡴⠆⠀⠀⠀⣀⠀⠀⠀⡰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠀⢠⣟⠀⠊⠀⡠⠀⠀⠴⠁⠀⠀⠈⡀⠇⠐⠠⣔⠁
⠀⠀⠀⡏⠈⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⡠⠒⠉⠁⠐⢦⣰⠉⠉⠁⠛⠀⢰⢄⠀⠀⣠⠊⠀⡀⠀⠀⠀⢀⡏⠀⠀⠀⠱
⠀⠀⠀⠳⡀⠈⡀⠀⠄⠀⠀⢬⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣸⠔⡡⠊⠉⠉⠄⠑⠦⣀⣀⣀⡠⠎⢡⢀⠀⠀⠈⠆
⠀⠀⠀⠀⢨⡖⠛⠁⠀⠀⠀⡀⢇⣀⢀⠠⢤⠞⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡧⠔⠁⠀⠀⠀⠀⠀⠘⠠⠀⠀⠐⠄⢩⠀⡀⠀⡠⠃
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⢘⡤⡄⠀⢀⣀⣀⢴⠋⠀⠀⣸⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⡈⠀⠘⢣⡀
⠀⠀⠀⢼⡄⠀⠄⠀⠀⢀⡘⠙⠦⣀⡠⢶⣯⣞⠏⠂⠁⠀⠀⠸⢤⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠑⠀⠈⠠⣀⠀⠑⡀
⠀⠀⠈⠓⣅⣐⡀⣀⣀⠜⠀⠀⠀⠀⢠⠟⢉⡆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠴⢄⡀⠀⠀⠐⣲⣄⡴⣤⠐⠺⡀⠁⠀⣀⣅⠢⠕
⠀⠀⠀⠀⠀⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠑⠒⠊⠀⠁⠀⠁⠀⠀⠉⠑⠀⠉⠘⠀⠙⠚⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠙⠶⠦⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⢀⣰⡩⠂⠀⠀⠀⠀⢠⣞⣠⣴⡂⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠙⢧⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⣇⡀⠀⠀⠐⠒⣞⠉⠁⠀⠀⠀⠀⢀⣠⠤⠐⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⢹⠁⠀⠀⠀⠊⢈⢦⡀⠀⠀⣠⠖⢋⠀⠀⠀⣂⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠸⣄⠀⠀⠀⠀⠁⣠⣷⣄⣼⣅⠀⠀⠶⠀⠀⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠈⣷⣶⣦⣴⣶⣿⣿⣿⠟⠿⠂⠀⠀⠀⠄⠚⠀⣤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣌⠓⠀⠀⠸⢆⡀⠨⠄⠺⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣛⣛⣟⣿⣛⣿⣿⣛⣦⣤⣤⡀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠀⣾⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠾⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠟
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠙⠲⠶⠶⠚⣹⣤⠼⠷⠛⢻⠈⠉⠙⠛⠿⣿⣿⣿⣶⣶⣶⡀⠐⠶⠶⢶⡶⠶⠒⠒⢒⡶⠒⠊
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠐⢧⣀⡀⠀⠀⣠⠶⢿⣁⠤⠀⣠⠟⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠠⣴⣾⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠉⠉⢹⠉⠶⣄⣹⠆⠉⣀⠴⠊⠀⠀⠀⢀⠀⠀⠀⠙⢿⣿⣿⡏⠛⠷⣦⣤⣤⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠑⠒⠲⢶⠶⣶⠺⡘⢆⢸⢹⠀⠀⠈⠢⡀⠀⠀⠀⣈⡆⠀⠀⢱⡀⠙⣿⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢰⠙⣆⠰⡑⢷⡀⡸⣺⠀⠀⠀⢀⡜⠀⠀⠀⠘⢷⡆⠀⠀⠇⢠⡏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡄⠀⡜⠠⡿⢤⣀⠈⠛⣥⠇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠸⣿⣿⣿⣿⡿⢁⡴⠀⠀⠀⠈⡿⠚⠧⠰⠧⡀⠈⠉⠒⣾⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⢿⣿⣿⡿⣡⠏⢀⣤⠤⢶⡞⠉⠀⠀⠦⢀⢹⠀⠀⠀⣿⠊⣀⡤⠤⠀⠀⠀⣤⣄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏
⠀⠈⢿⣿⣷⠃⠀⠁⠀⣠⠾⠀⠁⠀⠠⢀⢈⡇⠀⠀⠀⡸⢿⡁⠰⡶⠀⠀⠀⠈⠡⠀⠀⠀⠀⢹⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠈⢿⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠢⢤⠟⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠘⡇⠀⠢⣼⣷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣴⡄⠀⠀⠀⠉⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠱⣄⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣶⡆⠀⠀⠈⠋⠀⠀⠀⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣿⣷⣶⣶⣶⣾⣶⣶⣿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶
⠀⢐⠂⠀⣿⣿⣄⠀⠈⠁⠀⠀⠀⠁⠀⠀⠉⠁⠀⠈⠉⠁⠀⠁⠀⠉⢹⠈⠉⠉⠉⠉⠉⠍⠉⢻⣧⣿⠛
⠀⠀⠀⠈⠱⢾⣿⠀⣀⢠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠓⢦⣸⣿⣿
⠀⠀⠀⠀⠀⣼⠟⠀⠉⠈⠀⠀⠈⠑⣖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⢀⡀⠀⠀⠀⠀⠈⢿⣿⣿⣄
⠀⠠⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠀⠀⣤⡜⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠠⠤⢐⣄⢽⣿⣿⡏
⠀⠃⠀⠀⡤⣀⠀⠠⡀⠀⠀⠀⠀⢠⣜⠀⠸⡀⢾⡇⠀⢀⠐⠶⠄⢳⠀⢿⠀⡀⠀⡄⠀⠌⠀⢹⠛⣿⡀
⠀⢀⣴⡇⠀⠀⠀⢙⣮⡒⢄⠀⣒⣼⣿⠀⠀⡁⢸⣷⣴⣿⡏⠀⠀⠀⢀⣀⣧⡦⢄⠑⢰⣒⡒⣤⣶⣷⡇
⡀⢈⡛⠛⠓⠒⠂⠈⠁⠙⠂⠁⠀⠉⠋⠀⢀⠘⢊⣛⢛⢛⡛⠛⣉⣉⣉⣉⣉⣁⣈⠀⡛⢛⣃⣈⣙⣛⣓⣀⣀
⠈⠙⢠⣠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡴⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⢦⣤⠀⠀⢀⡋⡇
⠀⢠⡼⡄⠀⣻⣦⣤⣶⡌⡏⠁⠀⠠⣼⡷⠟⠁⠀⠛⠢⠀⠀⣂⣀⡶⠄⠀⠀⢸⣟⡀⠺⠻⠿⢿⡆⣤⡜⡇⢰
⠀⢸⣧⠀⠀⢻⣿⣿⣯⡍⠀⠀⠀⠀⣿⣿⠂⠀⠀⠀⡰⣤⣤⣉⡀⠀⠀⢠⠄⢸⣿⣿⣿⠟⠋⠛⡇⢸⡇⠇
⠀⢸⣿⣴⡇⠘⣿⣯⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⡀⢈⣿⣿⣧⣠⣴⣃⢀⣾⣿⣿⡅⠀⠀⠀⠀⡈⣷⡄
⠀⢨⣿⣛⣤⣤⣸⣿⠧⠀⠀⠀⠾⠿⣿⣿⡶⠆⠀⠀⠁⠘⠁⠙⢳⠿⣧⣿⣼⣿⣿⣿⣦⠀⠀⠀⣐⢰⣿⡇
⢄⢸⠿⠯⡍⠙⠿⠿⠷⠆⠀⠒⠀⠀⠙⢟⢻⠻⢿⡿⠴⡴⠶⠷⠾⠒⠾⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⣇⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠓⠚⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀⣠⡰⠊⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⠈
⠀⠀⠀⠀⠀⠀⡀⢀⡠⠠⠄⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠄
⠀⡀⠀⠀⠈⠉⠠⠤⠂⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠
⠀⢁⠀⡀⠀⠀⡀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⠀⠀⠀⠈⠀⢠⡀⠤⠄⠄
⠀⠈⠀⠂⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⡀⠀⠀⠀⠀⣠⡆⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⢀⢀⠀⠀⠀⢠⠠⠀⠀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠈⠁⠉
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡁
⠀⠀⠁⠁⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠈⣠⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⢒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠴
⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⢁⠐⠒⣒⢂⠉⢱⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠔⠀⠠⠠⠁⠐⢀
⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠱⠐⠐⠁⠀⠀⠀⠚⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠒⢂⡐⠐⠊⢠
⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢀⠑⠠⠤⠤⠤⠒⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠉⣁⣀⣀⣤⣀⣀⠉⠙⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠉⠁⢠⣶⣿⠿⠛⠛⠛⠛⠻⢿⣷⣦⡀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠛⠛⠻⢿⣷⣦⡀⠈⢠⣴⣶⣶⣶⣶⣦⣄⡈⠻⣿⣦⡀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⣤⣾⡿⠿⠿⣿⣶⣌⠻⣿⣄⠀⠋⢉⣀⣀⣈⠉⠻⣿⣦⠈⢿⣷⡀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣾⡿⢩⣶⣿⣷⣦⠹⣿⣆⢹⣿⡄⠀⣿⠟⠛⢿⣷⡄⠘⣿⣇⠈⣿⣇⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⣿⣿⠀⠈⠛⠃⠛⠛⠘⠛⠃⠀⠁⠀⠀⠀⠛⠛⠀⠛⠛⠀⠙⠛⠀⠙⠛
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⡇⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣄⣀⡟⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣤⣾⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁



⡶⢶⢶⢶⣷⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⣶⣶⣶⣶⣶⣶⢶⡶⢶⣶⡶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⢉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣄⣀⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⠈⢳⣤⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣶⡟⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣄⠀⠀⣿⣿⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⡆⠀⢸⣺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⢿⡀⢠⠏⢀⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⠻⢦⠈⠇⢠⠟⠉⠉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠄⡀⠙⠀⠀⠋⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⠁⠀⠀⠀⠁⠐⠂⠺⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⢀⣤⠄⠀⠀⠀⠀⢈⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⡋⠀⢀⡄⢀⡠⠀⡀⠀⠲⣦⢀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠸⠁⠀⠰⣦⠈⢷⣌⢳⣶⣶⣾⣿⣿⣿⠛⢿⣿⣿⣿⡏⣰⡟⠁⣼⣧⡄⠀⢉⢸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⣆⠀⢰⡇⣿⡄⠘⣿⣬⣿⣿⣿⣿⣿⣏⠀⢸⣿⣿⣿⣷⣿⠃⣰⣿⡟⣧⡀⢸⡸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣿⣴⣿⡇⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣴⣿⣿⡇⣿⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⢨⡟⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⣿⣿⠿⠛⠉

⣿⠟⠉⠙⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢹⣏⣿⢉⢉
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⡩⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⣀⠀⠀⠀⠐⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠈
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠠⠀⠀⠀⠰⣀⠀⠀⠀⣀⡠⠔⠚⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠐⠀⠀⠀⠀⠂⠉⠉⠉⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠
⠀⠀⠀⠈⠒⠤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠠⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠰⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠀⠀⠀⠀⠀⢰⡖⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⢁⣥⢉⣍⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠺⠻⠿⠟⠻⠸⠿⠀⠀⠀⠀⠀⠳⠤⠄⠒⠁⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠛⠿⠽⠣⠞⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠐⠀⠀⠀⠢⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠓⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⠀⠁⠐⠀⠐⠂⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠆⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠦⠤⠤⡤⠴⠐⢂⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠
⠀⠀⡁⠀⠀⠀⠐⢄⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠠⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠒⠆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀⣀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡏⠙⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠤⠤⠤⠦⠐⠒⠂
⠀⠀⠀⠀⢀⣤⡶⢟⣻⡟⣻⣿⠛⠛⣷⣶⣬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣶⡆⠒⠂⠀⠈⠉⠁⠀⠀⠀⠀⠀⠠⢀⠒⠂⠄
⠀⠀⠀⢰⠿⣿⣿⣟⣻⡿⠿⣿⣿⠟⠙⠻⢟⣽⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣷⠀⠀⠀⠀⠀⠀⢀⠀⣨⢩⠐⠀⢈⡇⠉
⠀⠀⠀⠙⢶⣇⠀⠀⠀⠀⠀⠀⣏⠀⠀⢀⡿⠃⠀⠀⠀⠈⠀⠀⠀⢶⣿⣿⣿⣿⣧⣿⡖⠀⠀⠀⠀⠀⠀⡀⢿⣧⢸⣿⣾⣧⡄⣀⠰⡇
⠀⠀⠀⠀⠀⠈⠙⠒⠶⠴⠶⠯⠝⠒⠒⠉⠀⠀⠀⠀⠀⠀⠂⣶⡄⣤⣿⢿⡿⠿⣼⠿⣜⢢⣷⠀⠄⠀⠀⠁⠾⣿⣿⣭⣿⣿⣬⢿⣒⠁⠂
⠀⠀⠀⠀⠀⠁⠀⣄⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⠿⣽⠃⠀⣐⠎⢇⠀⢿⢿⡄⠀⠀⠀⠀⠀⠀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⢀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⣄⠴⣟⣂⣠⣀⣀⡀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠓⠀⠀⠚⠛⠁
⠀⠀⠀⠀⠀⠰⠆⣾⣾⢱⣾⣵⣿⠏⢷⡔⣽⠦⡀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⢿⡗⢈⣶⡾⠟⢀⠠⣤⠋⢀⠀⠀⠀⢎⣼⣧⠾⡍⢁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠹⢁⠲⠖⠊⠁⠀⠀⠀⢀⡆⣼⣿⡿⢠⣏⢾⡀⠀⠐⠀⠀⠀⠸⠀⢀⣀⣠⣤⣤⣄⠀⠀⠀⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣷⡻⠗⣡⢀⣿⠀⠀⠀⡄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠆⠀⡀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⢼⣧⠞⠁⢈⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⣽⣿⣿⣿⣿⣀⡀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⠒⠁⠀⠁⠃⠀⠀⠀⠀⠐⠠⠿⡇⠤⠖⠀⠀⠀⠀⢠⠃⢀⣠⣾⣿⢿⣏⣿⣿⣷⣟⣿⣿⣷⣦⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⡀⢹⣿⢿⣶⡟⠙⠉⠍⠙⢿⠛⠙⣻⣿⣷⠀⣦⢠
⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⡼⠿⠟⠿⠎⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡏⣳⣍⠀⠀⠀⠠⠂⢨⡇⠈⢉⣿⣏⡜⠉⠃
⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣠⠄⡒⠀⠉⢡⠀⠀⠀⠀⠀⠘⢿⣯⡙⠛⠲⠶⠶⠶⠿⠛⢛⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠷⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⢀⠔⣨⣤⢼⣿⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠛⠛⣿⣷⣶⣦⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⢨⣿⡛⠉⠈⠛⠻⣿⣴⣿⣎⠀⠀⠀⠠⣴⣿⡿⠀⠋⢐⠔⠚⢹⣿⣿⡿⣟⢹⣦⠀⠀⠀⢀⡔⠂⠀⣀
⠀⠀⠀⠂⣿⠃⢶⠾⠃⠓⠻⣇⠀⢨⣿⠀⠀⠀⠀⢻⣿⡀⠀⡴⠋⠀⠄⢈⡛⢯⣶⣿⢻⡿⠁⠀⠁⢸⠃⢀⡴⠁⠀⡴⠂⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠁⣿⣦⡏⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠈⠀⠈⠛⠿⢷⣤⣤⣤⣴⠿⣾⡧⠛⠋⠀⠀⠀⠀⠘⢷⣠⣿⠀⢸⡇⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠸⢿⣷⣀⢀⢀⣀⣰⡿⢿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠷⢤⣄⠌⠄
⠀⠀⠀⠀⠀⠈⠑⠛⠻⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡐⠑⡄
⠀⠀⠀⣉⠑⠀⠀⠀⠀⠀⢀⣾⣴⠀⣠⣀⠀⣀⠤⡄
⠀⠀⠀⢿⡌⠀⠀⠀⢠⠦⡔⡏⠜⢀⣐⣄⣯⣀⣤⠃
⠀⠀⢖⡐⣧⡠⢄⠀⡤⠳⠀⡀⢶⠈⡤⡻⢇⠠⠚
⠀⠀⢀⣀⣩⡗⣸⠓⣌⠐⡀⡥⠊⠐⢠⡉⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠙⢄⣌⠞⣮⠢⡏⡁⠱⡐⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠊⠐⠂⠁⠀⠑⡀
⠀⠀⠀⠀⢙⣻⡢⣉⢹⠂⣄⣈⣺⠴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⢙⠒⠀⢀
⠀⠀⠀⠀⠛⠄⠘⡢⠍⣓⠒⠨⣍⠀⡀⠵⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠤⠼⠄⠺⠀⢴⠉⢊⠀⠀⢀⠗⠄⣀
⠀⠀⠀⠀⠀⠀⠘⠒⠈⠀⠀⠀⢈⠊⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠘⠰⠀⠀⢠⠀⢠⠇⠀⠀⣆⡔⠁⠀⠀⠀⠑⠤⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⣠⠀⠀⠀⠀⠘⡀⠀⠀⠀⡀⢰⠀⠀⢠⠖⠃⠀⢠⠃⢠⠋⠀⠀⢠⠃⠈⠒⢤⠊⢀⡠⠊⠀⢸
⠀⠀⢀⣀⡀⠄⠂⢄⡀⢿⣿⣃⠘⣒⠟⠁⠀⠀⠀⢀⡴⠑⡀⠀⠀⠉⢡⠂⠀⢁⡞⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠁⠀⠀⠀⠾
⠀⠀⡠⢬⣍⢉⣋⠁⠤⡙⡿⢃⠭⠒⢷⡤⠤⡀⠉⠉⠀⠀⠡⡀⡌⠑⠁⠀⠀⠎⠀⠈⢀⣀⡤⠴⠒⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⠊⠀⠣
⠀⠜⠀⢀⠀⠀⠀⠑⠂⠀⠞⠁⠐⠁⠀⠀⠀⠈⢢⡤⠂⠀⠀⢻⠁⠀⠀⠀⣀⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃
⠈⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠴⠊⢆⣤⡶⠟⠋⠁⠀⠒⠒⠒⠉⠀⠒⠂⠠⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠔
⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⡀⠀⠀⠈⡗⠒⠒⣀⡀⠀⠀⠀⢀⡠⠆⡀⠀⠀⢀⣀⡌⠢⡄⠀⢀⠤⢄⣀⠆
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠛⢦⠖⠃⠐⡞⠀⠂⠠⠬⢉⠁⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠈⠂⠂
⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠑⢄⡠⠒⠛⠄⠀⢡⠀⠀⠀⠀⡀⠔⠒⢄⠀⢠⠊⠐⠂
⠀⠀⠈⠐⠂⠄⣀⣀⣀⣀⣀⣀⣀⣀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡇⠀⠐⠈⠀⠀⠀⠀⠈⠁
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢠⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣀⠉⠉⠙⠛⠻⠿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠙⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠋⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⢿⡆⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣛⣛⣛⣛⡛⠛⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠀⡰⠚⠉⠉⠩⢭⣉⡙⢦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠁⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢰⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠑⠆⠀⠀⠰⠚⠉⠉⠁⠀⠀⠀⠀⠀⡎⠀⠠⠴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⢰⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣇⠀⠐⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡤⠶⠒⠛⠛⠉⠙⠛⠛⢻⡇⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠁⢀⡤⠤⠀⠀⠠⠖⠲⠄⠘⡇⠀⠀⠀⠘⢦⠀⠀⠀⢰⠑⢮⠱⢄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⠀⠀⠈⠁⣠⣤⣤⣷⡌⡦⢀⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡄⢹⡀⣠⣴⣾⣿⣶⣿⣿⣿⣶⣾⣧⣴⡗⠀⠀⠀⣴⣋⣀⣄⠈⢡⠃⢰⠟⠀⣱
⠀⠀⠀⠀⠀⢰⠀⠀⠀⣾⠀⠀⠘⢷⣾⣿⣿⡿⠟⢉⣉⣉⠉⠙⠛⣿⠛⠋⠀⠀⠀⢰⠛⡇⠀⢸⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠁⠀⣀⡼⠃⠠⡄⠀⠀⠀⢸⡀⠙⠤⠎⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⡆⠀⠀⠀⠀⡎⠀⠈⠙⠓⠶⠚⠛⠉⣀⠀⢀⡇⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢳⡀⠀⠀⠀⣹⠖⢋⣩⣤⣷⠀⢸⣧⠤⠈⠻⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠱⡄⠀⢰⠇⠀⠈⣈⠅⠛⠒⠋⠑⠢⢄⣀⣈⡷⢰⠚⢲⣀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠈⠢⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡶⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣄⣀⣀⣀⣀⣀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣻⣉⣟⣻⣋⣿⣫⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢷⠀⠀⢿⠀⠀⡿⠀⢀⡎⠀⢠⠆⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠐⣄⠀⠈⢧⠀⠘⣇⠀⠸⠄⠠⠧⠀⣼⡀⣠⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⡆
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠈⢣⡠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢎⠀⠀⠀⠀⠀⣀⣤⣤⡄⠙⣿⡿⢛⣿⠟
⠀⠀⠀⠀⣴⡟⠁⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣤⣀⣴⣶⣦⠀⣴⣟⣽⠟⢀⣼⠟⢀⣾⠏
⠀⠀⠀⢸⣿⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⢀⣤⣤⡀⣴⠾⣿⡆⢀⣿⣿⡟⠀⣿⣿⠃⣿⡇⢰⣿⡿⠋⢠⣾⠋⠀⣾⡏⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠘⢿⣆⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢰⣶⠆⠀⣼⣿⣷⡟⠁⣿⣿⠏⢸⣿⠀⣼⣿⡟⠀⢰⣿⠇⢰⣿⣁⣾⣿⣤⡴⠟⠁⠀⠀⣿⣇⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⢀⣴⣿⣿⠀⢠⣿⣿⠀⣼⣿⣿⡟⠀⢸⣿⡏⠀⣿⣏⣼⣿⣿⠀⠀⠺⠛⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣶⠛⠁⣿⡇⢀⣾⣿⣟⣰⡟⢹⣿⠀⠀⠸⠟⠀⠀⠈⠉⠁⣠⡤⣤⡄⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⣿⣇⠀⠻⠷⠾⠋⠈⠛⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢡⣿⠇⠀⠀⠀⠀⣾⡟⣽⡟
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣟⣰⡿⠋⠀⠀⠀⠀⠀⢘⣿⣿⣅⡀
⠀⠀⠀⢰⣿⠁⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣾⡟⣿⡆⠈⠛⠀⢠⣿⣿⣋⠀⠀⣴⡟⣿⡇⢠⣾⠏⠀⠙⣷⡄
⠀⠀⠀⠈⢿⣦⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠻⣶⣿⠷⣿⡟⠀⣾⣿⢿⣽⣷⣼⣿⠟⠋⣰⣿⣷⡶⠷⢄⣿⠇
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢀⣿⡏⠀⣿⡇⣸⣿⡇⢀⣿⡏⠸⣿⣤⣾⠟⠘⠿⣤⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⣾⡟⠀⠀⣿⣷⡟⠙⠻⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠾⠋








⣿⠛⠛⣿⢿⠿⢿⠿⠿⣿⡿⠿⠿⡿⢿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠿⣿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡀⢀⡀⢀⠀⢀⠀⣀⠠⢠⠀⡀⢀⡀⢀⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠛⠘⠋⠘⠃⠙⠃⠛⠡⠋⠀⠀⠚⠯⡘⠃⠙⢁⠂⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢉⣄⣄⡀⢠⡦⠤⠀⡐⠀⠠⠀⠀⠀⠀⠠⠁⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠘⠛⠍⠿⠆⠃⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠁⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣦⣿⠅⠢⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠸
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡠⠈⠈⢀⠁⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠈⡃⠀⠀⠁⠀⠀⠅⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢦⠰⠀⠀⠀⠀⠀⢀⠀⠴⠤⠦⠆⠀⠀⢀⢄⠀⠀⠠⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠚⣆⡀⠀⠀⠈⠂⠀⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠁⠀⠀⠀⠄⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠌⢠⠂⠀⠀⢐⣦⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⠀⠀⠀⠀⠐⠀⠂⠀⠀⠊⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠘⠄⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠁⠀⢿⢠⣾⣽⡿⠞⣿⠀⠰⡏⠥⠀⠀⢾⣿⣿⣹⠷⠀⡷⠺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡿⣿⣇⣤⣄⠀⠀⢠⣦⣰⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢠⣤⡀⠀⣿⣿⡇⣿⣿⣿⣿⣧⠀⣿⣿⡇⣾⣿⣷⣷⡄⠀⠀⣀⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢻⣿⣆⠸⣿⣷⡀⢻⣿⣧⢹⣿⡇⣿⣿⢸⣿⣿⢰⣿⡟⣼⣿⡇⢠⣾⣿⠟⢡⣿⡇
⠀⠀⠀⠀⠀⢠⣾⡟⠻⣷⣄⠈⢿⣿⣄⢿⣿⣿⡎⠿⠿⠈⠛⠃⠙⢋⠈⠛⠃⠘⠿⠁⣿⡿⢠⣿⣿⣏⣴⣿⠟⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠈⢿⣷⣤⣈⣛⠀⠀⠙⢿⣿⠃⠉⠁⠀⣠⡀⠀⠠⣴⣿⣦⡄⠀⢀⣀⠀⠀⠀⠘⣿⣇⣉⠛⠉⣠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠾⠛⠇⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠁⢠⣾⣿⠟⠉⠀⢾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠛⠀⠀⣠⢖⣿⣿⣿⣿⣿⣶⣦⣤⠀⣤⣴⣶⣿⣿⣿⣿⣿⡲⣄⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣼⣿⣿⣿⣿⣿⣿⣿⠏⣀⡹⣿⣿⣿⣿⣿⣿⣿⣧⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⢡⡾⠉⡇⠈⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠶⠆⠀⠀⠉⠛⠋⠉⠀⠀⣾⠃⢰⠁⠀⠀⠀⠉⠙⠛⠉⠁⠀⠰⠾⠿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠀⠀⠀⠀⢀⡀⠀⠤⢸⣿⢠⢃⣀⡀⠀⠀⣀⠀⣠⣄⠀⠀⠀⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠉⢀⠀⠀⣼⠀⢿⣆⣴⠆⢸⣿⣡⠋⢸⡇⣠⡾⠍⠀⣿⡘⢀⠀⠀⣀⠈⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⢻⡀⡜⠀⣿⠀⡜⣿⠷⢒⣿⠋⢸⡇⣠⠆⣨⢿⡁⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠻⡴⢡⡏⣀⣾⠏⠀⠀⠉⠁⠀⠣⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠤⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆⣣⣤⠾⠓⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⢱
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠁⡑
⠀⠀⠀⡀⠀⠲⠀⠀⠐⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⢠⣤⣆⠟⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠄⠀⠀⠉
⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠈⠀⠀⠔⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠐⠐⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣠⠤⠀⠒⠁⢀⡁⠠⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠰⠐⠒⠐⠖⠒⠒⠒⠆
⠀⠀⠀⠈⠢⢈⠉⣉⢀⣠⡤⠆⠒⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠒⠐⠀⡀⠀⠀⢀
⠀⠓⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠖⠂⠉⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢀⠀⠀⡆⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠃
⠀⠈⠆⠁⠆⠀⠠⠄⡔⠂⠉⠁⡬⣀⡀⠜⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⠁⢠⡄⠃⠀⠀⡇⠀⠀⡇⠀⠀⠒⠂⠛⠋⠈⠃⠁
⠀⠀⠈⠒⠓⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠐⢀⠔⠁⢐⠀⠇⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠅⠈⠀⠀⠢⠀⠠⢭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠤⠤⠈⠁⠀⠀⠀⡄⠀⢀⠀⠸⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠆⠀⠆⠀⠀⠀⠘⠀⠘⠋⠉⠀⠀⠠⠤⠤⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠊⠚⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⢀⡀⡇⡁⢇⡩
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠁⠀⠙⠁⠙⠓
⠀⠀⠀⡄⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠛
⠀⠀⢠⠦⡤⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⡀
⠀⠠⠁⠀⠀⠈⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⢀⠀⡀⠤⠤⠤⠤⠀⠀⠀⡀
⠀⠐⠀⠀⠅⠀⡀⠀⠀⠐⠈⠀⠀⠀⠳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡸⠉⠀⠀⠀⠀⠐⠀⠀⠀⠉
⠀⢠⠀⠤⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⣆⠀⠾⠡⠄⠆⡰⠀⠀⠀⠀⠀⠀⣀⡀⠘⠫⠤⠤⠛⢀⣀⣤⠀⠀⠀⠀⠀⠀⠐⠛⠑⠘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠈⠐⠂⠀⠐⠚⠁⠀⠀⠀⠀⠀⢠⠂⢈⡉⠉⠉⣉⢉⡉⠀⢸⠀⠀⠀⠀⠀⠀⡀⣰⠀⠐⢬⡄⠀⠀⢸⠀⠀⠀⠀⢀⠚⠉⠁⠀⠆⠔⠀⠉⠓⣄
⠀⠀⠀⠀⠥⣭⠈⠀⠀⠀⠀⠀⠀⠀⠀⠧⠑⠚⠞⠃⠸⠀⠄⠃⢼⠀⠀⠀⠀⠀⠀⠁⠋⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠰⠒⠂⠀⠂⠐⠔⠇⣸
⠀⢰⠀⠂⢀⠁⠉⢀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠲⣅⡁⠘⠓⠁⢀⣀⠴⠁
⠀⠈⢄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⡀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⢲⣄⢀⡉⠉⠉⠉⠉⢁⣀⡴
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⣶⣿⣿⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠰⠋⠀⡠⠛⠛⢻⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣅⠀⠔⠀⢀⠊⠁⢀⠼⠋⠙⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣷⣄⣨⠀⠀⠃⠀⣠⠊⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠾⣦⣠⡋⠀⠀⢛⣄⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢈⠻⣦⠀⠀⢰⣯⣴⣿⣿⣿⠛⠀⠀⠉⠷⣦⡤⠾⣷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠻⣿⠛⠀⠀⠘⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⡑⠂⠀⣹⠇⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣾⣷⣴⣶⣄⣤⣀⣿⣆⠀⠙⢿⣟⢉⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣳⣿⡿⣿⠁⠀⢰⣤⣠⣿⣿⣿⣿⣿⣶⣶⣦⠀⣠⠴⠶⣾⣏⠉⠻⡏⠀⢹⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⣿⠻⣿⣴⠛⠃⢀⣀⣹⣿⣿⣋⣿⠿⣿⣿⢟⣿⣽⡇⣄⠀⠈⢋⠀⠀⡇⠀⡌⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢀⣠⠴⢦⣤⡀⠀⠀⠀⠀⣼⠿⣀⠀⠀⢻⣿⣶⡏⠉⠻⡄⠈⢷⣾⠃⠀⠈⠻⡀⠀⢸⣤⣼⠷⢿⣶⣾⠋
⠀⠀⠀⠀⠀⠀⡞⠁⠀⢒⣈⠿⣦⠀⠀⠀⢿⡀⢈⡳⠖⢾⡀⠀⢹⡄⠀⢻⣀⡼⠛⠓⠷⡆⠀⣹⣾⣏⣩⣷⣤⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢿⡀⠈⠉⠁⢀⡿⣦⣤⠴⢿⠉⠈⠹⣆⠀⢹⡀⢀⣿⣶⠾⣻⣧⣠⣄⠀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢹⡖⠒⠒⠉⢀⣼⡇⢀⠀⠱⡄⠀⢸⣀⡸⢿⣿⠋⠀⢰⡏⠀⠀⠈⠳⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⠖⠋⠀⣧⢙⡇⠀⣶⣶⠋⡴⠛⣉⠛⠛⡷⠟⠳⣄⠸⠦⣄⣿⠛⠛⠛⡟⠁⣠⠴⠛⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠠⠄⠊⠀⡿⣤⡾⠋⠹⣾⡀⢰⠁⢀⡼⠀⢀⢴⠃⢀⣄⡈⣿⡆⠂⠤⣼⢾⡁⢀⡤⠖⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠔⠛⠁⣸⣧⡀⠀⠈⠛⠾⣄⣀⣥⠀⠈⢀⠀⠈⣻⣷⡿⠃⠂⠤⣼⡆⣹⣏⠀⣠⠒⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠲⠖⠋⠁⣸⢷⡀⠀⠀⠀⠀⠉⠉⠳⠶⠟⠶⠶⠛⠈⣷⠄⠤⣄⣸⡆⠻⣯⣿⣇⡞⠉⠒⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠴⠚⠁⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⣠⣈⣿⠀⠸⣿⣯⣭⣿⣦⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣶⡊⣰⠟⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠸⣧⣀⣀⡀⢹⡆⠀⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢠⡾⠛⢶⡾⠛⠉⠻⣦⣀⣀⠀⠀⠈⠻⣭⡴⠟⠁⠀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠂⠀⣠⣤⠜⠉⠁⠉⢿⣄⠀⠀⠀⣀⡀⠀⠀⠘⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣰⢿⡇⠀⣴⡶⠘⠋⠛⣿⠛⠻⠍⠙⣷⡀⠀⠀⠉⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣀⣹⡿⠿⣁⢀⣤⣂⠿⠤⣀⠈⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠻⠏⡉⠑⢀⠈⢻⠋⠀⣀⡀⠀⣷⡟
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⢹⡆⠀⠐⠀⢸⣀⣰⡿⣿⠛⠙⠲⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠉⠀⣀⣴⣴⠟⠛⠋⠘⣷⣄⣀⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⢿⡉⠄⢀⡾⠃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠋
⠀⠐
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣻⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣃⣤⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⡿⢿⡿⠋⠛⠿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣁⠀⠀⠀⠀⠈⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠰⠖⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣠⣤⠴⣋⠜⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⣻⡿⠋⢹⣁⣰⣊⣁⣾⡿⠁⠀⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢿⡀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠻⡄⠀⠀⣾⠏⢙⣿⣿⣶⣶⣶⣿⣿⡇⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢠⣇⢀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⠴⣿⠴⠞⣿⠀⢐⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⣦⣤⠖⠚⠛⠛⣾
⣾⠀⠀⠀⠀⠸⣏⠻⣆⠀⠀⠀⠀⣸⠃⠀⠀⣰⠋⠀⠀⠀⠀⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⡟⢳⠀⠀⠀⠀⢸
⣿⡇⠀⢀⠀⡴⠿⠀⠸⣿⠀⠀⢰⠃⠀⢀⡜⠁⠀⠀⠀⠀⠀⣿⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⡼⠓⠉⠀⠀⠀⠀⢸
⢿⣿⡄⢸⣆⢃⠀⠀⢀⡏⢀⡴⠃⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠁⠀⠀⠀⠀⠀⣠⡴⠋
⠸⡄⢻⣞⣸⣦⠑⢒⠋⠀⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠲⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠹⣿⠙⠿⠹⣆⣾⠀⣀⢧⢠⣶⣠⠏⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⣾⣿⡿⠋⠁⠀⠀⠀⢀⣠⣴⣶⣏⠁
⢀⠀⢹⣆⠀⠀⢻⡟⣇⣿⣮⡏⠋⠁⣤⠀⠀⠀⠀⠀⠀⡼⠁⣸⣿⣿⡿⣶⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡀
⢸⣷⣨⣿⡄⠀⠈⠃⠘⠛⠘⠧⣤⡾⠋⢷⡀⠀⠀⢀⡞⣁⡿⠟⣋⣥⣴⣃⣜⣴⣤⣼⣿⣿⣿⣿⣿⣿⣇
⠀⣇⠙⠛⠃⠀⠀⣌⣧⣀⢠⡀⠀⠀⠀⢸⣿⣿⣷⣾⣛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⡄⠀⠀⠀⢸⣾⣿⠟⠿⣿⣆⡀⠀⣸⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢹⣦⡀⠀⠘⢿⣧⣀⣀⣀⡿⢧⣐⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⢿⠿⢻⣶⣬⣭⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠳⢤⡈⠻⢿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⣶⣶⡶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⣤⠀⠀⠀⠀⠀⣀⣄⠀⢀⣀⠀⢀⡀⠀⣀
⠀⠀⠀⠠⣦⠀⠀⣀⣄⡀⠀⠀⠀⣿⠀⣃⢤⡟⢀⣴⠂⢀⣾⡣⣻⡟⢡⡟⢠⣿⡐⣿⠖⢹⡇⣰⡿⠃⠀⣸⡧⠊⠙⠀⠀⣤⢤⣄⠀⠀⡤⢄⡀
⠀⠀⠀⠀⢻⣧⣼⣯⣹⡏⠀⠀⠀⣿⠊⠁⠸⡷⠋⣿⡴⠁⠿⠁⠛⠀⣼⢣⢿⠟⠰⠏⠀⣿⣠⠿⡤⠴⠊⡿⠁⠀⠀⠀⠘⡿⠛⠛⢠⡾⢁⠀⠁
⠀⠀⠀⢰⠛⠛⢛⡻⠻⡁⠀⠀⣮⡾⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣄⠘⠁⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣠⠛⠒⠘
⠀⠀⠀⠈⠃⠈⠹⣧⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⢠⡄⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣠⣤⣄⡀⠀⠀⠀⠀⢀⠀⠐⠃⠀⠁
⠀⠀⠀⠀⠄⠀⠀⠹⣇⠀⢃⠀⠀⠀⠀⠀⣿⡆⠀⢠⣿⠃⣴⠄⢸⣿⣴⣶⣦⡄⢠⡿⠋⢹⣿⢸⣿⣉⡙⠃⠀⠀⠀⠀⠀⠢⢀⠀⠀⡠⠂
⠀⠀⠀⠀⠐⡀⠀⠀⠹⠆⠈⠀⠀⠀⠀⠀⢸⣧⠀⣾⠇⠀⣿⠀⢸⣿⠁⢀⣼⡏⢸⣟⠛⠛⠁⠀⠉⠛⠻⣷⡀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠐⠄⠀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠻⡿⠋⠀⠀⣿⠀⢸⣿⠟⠛⠉⠀⠈⠻⣶⣶⠗⠈⠻⣶⣶⠟⠀⠀⠀⠀⠀⠔⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠿⠛⠛⠛⠻⠿⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⣠⡄⠀⠀⠀⢠⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠉⠀⠀⠀⠠⠀⠈⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠃⠀⠀⠀⠤⠆⠀⠀⡏⠲⠂⠀⠀⣷⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢰⡙⣿⡆⠀⠀⠀⠀⢀⣀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣷⡀⠀⠀⢀⣸⡇⠀⠈⠙⠁⠀⠀⠀⠀⡿⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣈⣧⠀⠀⠀⠀⠙⠢⠤⠤⠤⠖⠁⠀⢸⠇⠀⠀⠀⢸⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢉⡱⠦⣄⣀⡿⣷⣄⠀⠀⠀⠀⡀⢀⢀⠀⠀⣠⠟⠀⠀⠀⡠⣼⣷⣹
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠠⣤⣼⠛⠀⡇⠘⠳⢦⣄⣀⣁⣈⣬⣴⠾⣧⡄⠀⠀⠀⢁⣿⣋⠛⢆⡔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣏⣨⣘⠀⠇⠀⠀⣼⠃⢹⠈⠹⡄⠈⠃⢻⡝⢦⡀⠀⠘⣌⠳⢃⣩⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡝⡏⡉⢹⠛⣧⢀⣼⠋⠀⠃⠀⠀⢃⢠⡠⣄⢻⣈⢓⡄⠀⢸⠀⡎
⠀⠀⠀⠀⠀⠀⢀⣾⢹⠀⡇⡇⢸⠀⡏⢻⡃⠀⠀⢰⡱⡀⡜⠸⢣⣸⠀⠻⣟⠁⠀⣸⠀⣇
⠀⠀⠀⠀⠀⠀⣶⢻⢸⡄⡇⡇⢸⠀⠇⣘⣧⠀⠀⡀⢧⠉⠀⠀⠀⣛⡼⠀⢸⡆⢠⠇⠀⢸
⠀⠀⠀⠀⠀⢠⡇⢸⠈⣷⡇⠀⢸⣤⠿⢛⡁⠀⢰⡇⢸⠀⣠⠖⠋⣡⣀⣀⡼⢣⣼⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣇⢸⠀⡏⣷⣾⣏⠁⠀⠈⣷⠶⢾⣹⠃⡸⡷⡴⢟⡿⠙⢯⡀⠀⢁⡀⢠⢧⣀⡀
⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⡇⢸⠹⣦⠀⠀⠹⣧⠜⠁⢠⠃⡇⡇⣿⠃⠀⠀⢱⠖⣉⣀⣀⠀⢈⠉⢐⢢⣀⣀
⠀⠀⠀⠀⠀⢸⣧⣬⣄⡇⡇⠘⠐⠙⣧⡀⠀⠉⠀⠀⡿⠀⡇⣳⡿⠀⢀⣠⣶⠲⠤⠤⠔⠒⠒⠉⠀⡸⢿⣿⣿⣿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢸⡏⠉⠙⠛⠛⠿⠾⠷⠿⢷⣄⠀⢀⣼⢶⣶⣿⣿⣿⣿⣿⡟⠈⠳⣀⠀⠀⠀⢀⣠⠴⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣉⣉⣉⣁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠒⠛⠛⠛⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⢿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠈⠻⣆⠈⠹⣿⠿⣿⣿⣿⣿⡿⠟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⢦⣤⣤⣤⣤⣼⣿⣤⣀⠀⠀⠀⠘⢧⡀⣿⣀⣬⣿⣿⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠻⣦⠀⠀⠈⣷⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢷⣀⣾⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⡿⢛⡛⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠻⣿⣯⣶⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⡄⠀⢀⣀⠀⠀⣀⡀⠀⢀⡀⢪⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⢠⢶⣶⣇⢀⣿⣿⣤⢠⡏⠿⣶⡼⠗⠁⢸⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠁⠘⠋⠈⠉⠛⠀⠙⠁⠃⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡿⡰⠃⣀⢀⡤⠀⠀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣀⠀⡀⡁⢠⡗⠁⣸⢋⢼⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⠀⠀⠀⣿⡘⠹⡷⢺⡇⢻⠉⢁⠜⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠹⡇⠀⠀⢸⣇⠇⠀⠎⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⢸⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣠⠖⣦⠀⠀⠀⠀⠀⠀⣠⠔⠢⣄⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣾⣤⠴⠶⠒⠛⠋⠉⠉⠙⣁⠙⢳⡞⠁⠀⠀⢿⡀⠀⠀⣠⣾⡁⠀⠀⠀⠙⢗⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡷⠀⠀⠘⠰⡇⠀⡜⠉⠈⠛⠦⣤⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡠⠂⣼⠃⠀⠀⠀⣸⡇⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠐⢁⡾⠁⠈⠀⠀⣀⠟⢀⡾⠉⠀⠙⢦⡀⠀⠀⠀⠀⢀⠔⠫⣻
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠄⢀⣴⣿⠀⢀⣠⠶⠘⠷⢤⣼⡷⢦⠀⠉⠀⣟⠀⠀⠀⢀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢀⠤⠊⠀⠄⢁⡴⠋⠁⡷⠈⡀⠁⠀⠀⠀⢀⠀⠰⣾⣀⣠⠞⠁⠀⠈⠛⠫⡉⠉⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠤⠈⠁⠀⢈⣠⠞⠋⠀⡀⠀⠙⣾⠀⠀⠀⠀⠊⠅⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢿⡀⠈⠩⠄⠐⢈⣁⡴⠞⠋⠀⣀⠤⢴⡦⢤⣀⣸⣛⣒⠒⠚⠛⠉⠁⠀⠀⢰⡖⠶⠲⠤⣤⡀⢀⣨⡧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠛⠉⠁⠀⠀⠀⢰⠁⠀⠈⢀⠞⠁⠀⠛⠀⣱⡄⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠙⣯⠈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢘⣄⠀⠀⠀⢸⡏⠉⠉⠛⠲⠦⠶⠛⠀⠀⠀⠀⠀⠐⢸⣇⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⡠⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠶⠖⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠠⢴⡶⠖⠒⠶⣽⡆⠀⠀⠀⠀⣠⢤⠖⠲⠚⠲⠶⣄⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⢀⠐⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠈⠃⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡟⢀⣀⣠⠟⡃⠠⠄⠀⠀⣠⠂⣠⠀⠄⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣏⠠⠀⠀⠀⠀⠀⢰⠏⣠⠏⠐⠋⠴⠁⠀⠀⠀⢰⠋⠛⢯⠀⠀⠀⠀⠀⠀⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣶⡖⠒⢦⠀⠘⡄⠀⢀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠂⠀⠀⠀⠀⠙⣆⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⠇⠀⠈⠳⠤⣌⣁⡤⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢸⣤⣀⠈⠓⠶⠤⣄⣠⣤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⣀⡴⠏⠀⠀⠀⠀⢀⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠶⢆⣀⣀⠀⡀⢠⡤⠞⠓⠒⣆⠀⠀⠀⠀⠀⠀⠀⣿⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠲⣦⠀⠀⠉⠉⠉⢉⣠⡴⠖⣻⠟⠁⠀⠀⠀⠀⢠⣴⠇⢀⡄⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠞⣙⠛⠉⠛⠶⡄⠀⠀⠈⡇⠀⢀⣤⠞⠋⠁⠀⢼⣋⣀⣖⣠⣂⣰⡶⠞⣁⡴⠋⠀⠀⣼⣧⣀
⠀⠀⠀⠀⠀⠀⢀⡞⢉⠀⠈⠁⠀⠀⠀⣾⣥⣔⣀⡴⠃⣴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣩⡥⠶⠛⠉⠀⠀⠄⠁⠄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣸⢁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠁⠀⢸⡅⠀⠛⠶⠶⠦⠴⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉
⠀⠀⠀⠀⠀⢸⡏⠄⠀⠀⠀⠀⠀⢠⣞⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⣀⣀⠀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢻⡀⠀⠦⠀⠀⠀⠀⠁⠈⠳⣄⣀⣀⠀⠀⠀⣠⡽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠤⠀⠀⠀⠀⡀⢄⡜⠀⠀⣼⠁⠀⠀⠈⠉⢁⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢿⡀
⠀⢠⠀⠀⠀⠀⠙⢶⣠⣴⠆⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠈⠳⣬⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡇
⠀⠀⠀⢀⡶⠛⢲⣤⠟⠻⠤⠶⣤⠼⠴⠋⠁⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⢀⠞⠋⠉⠉⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠄⠀⣀⣀⣀⡀⠀⠠⠤⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠐⠀⠀⠀⠀⠀⠀⠈⡷⢟⡙⠲⠦⡴⠖⠛⠶⣄⠀⠀⠀⠀⢹⡇⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⢰⠀⠀⠀⠠⠽⢧⠤⣄⡀⠀⢳⡄⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⢀⣠⣄⠀⠀⠀⣠⠇⠀⠀⠈⢢⡀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢧⠀⠀⠈⠉⠓⠶⠤⠤⣤⡤⠤⠶⠶⠚⠋⠉
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠈
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣋⠉⢹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣃⣥⠍⣽⠇⢸⣿⠟⠉⣤⠄⠀⠀⣰⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡁⢰⣷⣦⠀⠁⠀⠀⠀⢀⣠⡈⠛⣠⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣿⠙⠀⠀⠀⠀⠀⠀⠨⣻⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣈⠘⠻⢿⣿⡷
⠀⠀⠀⠀⠀⢸⠑⠒⠒⠒⠒⡇⣿⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⡻⢿⣿⣷⣞⢟⡄
⠀⠀⠀⠀⠀⠘⠂⠀⡀⠀⠐⡇⡏⠛⢧⣀⡶⢄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣿⣉⡿⠇
⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠐⡇⠇⠀⠠⠋⠃⠀⢳⣄⡀⠀⠀⠀⡉⠉⠙⠛⢿⣿⣿⠜⡄
⠀⠀⠀⣀⠀⠀⡇⠀⡀⠀⢸⣇⡄⠀⠀⠄⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⠫⠖⠁
⠀⠀⠀⠃⠀⠀⠓⠐⠀⠀⠈⠁⠀⠀⠈⢀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠐⠂⡓
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠂⠠⡄
⠀⠀⠀⠀⠀⡠⠒⢠⠖⠒⠒⠒⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⡀⠐⠀⠀⠀⠐⠈⠁⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠁⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⠀⠀⠉⠂⡀
⠀⠀⠑⠄⠀⠀⠀⣀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⠀⢀⡀⠰⠤
⠀⡀⠒⠀⠀⠀⠀⢠⠐⠀⠀⠀⠙⠄⠂⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠤⠀⠀⠉⠁⠈⢡⠀⠀⠀⡘⠀⢈
⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠒⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢠⣤⢁⠀⠄⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⣠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠿⠂
⠀⠀⠁⠈⠈⡀⠀⠀⠀⠀⠀⠀⠰⢤⣤⠆⠀⠐⠂⠀⠒⢢⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⡲
⠀⡀⠀⠀⠀⢀⢀⢤⢠⠀⠀⢀⣀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠉⢪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢶⢄⣀⠐⡒⢋⣂⠊
⠀⢀⡀⠀⣀⠜⠙⠣⡀⠀⠠⠜⢣⣈⠂⠀⠀⠀⠀⠀⠀⠒⢆⣨⡀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡃⠀⠉⠉⠈
⠀⠀⠀⠀⠙⠀⠀⠀⠈⠠⠀⠀⠀⡍⠈⠀⠀⠃⠀⠀⠀⠨⠉⠘⡁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢐⡀⠀⢀⠆⠀⠢⠐⠀⠴⠀⠂⠐⠞⡝⠡⠔⠲⢦⡤⢀⡠⠀⠤⠦⠈⠀⠀⠀⠄⡀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠈⠠⠤⠀⠀⠀⠈⠉⠁⠂⠀⠀⢜⠀⠀⢀⠄⠋⢇⠀⠀⠀⠉⢉⠹⠀⠉⠀⠀⠈⠀⠀⠁⠀⠈⠐⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠂⠚⠥⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁


⡂
⠁

⡿⣿⢿⢿⡿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡠⠤⠶⠒⠒⠒⠲⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠐⠭⠲⢤⣀
⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡖⠉⠀⠀⠀⠀⠀⠀⠑⠪⣍⡲⢤⡀
⠀⠀⡜⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠈⠉⢻⣇
⠀⠀⡇⠀⠀⠂⠀⡀⠘⠥⠄⠠⠬⠟⠁⠀⠀⠀⠀⠀⢸⠀⢠⣔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠻⣄
⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⣸⠀⠘⡇⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠰⡋⠑⠋⡠⠐⢃⡏
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠐⡰⠃⠀⠀⢏⡀⢲⠚⠀⠎⠁⠈⢁⠠⠈⠴⠂⣀⣚⣥⡭⠭⠷⠒⠛⠉
⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠘⢗⣋⣀⣤⡼⠥⠽⠓⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢶⡉⠁⡿⣦⠠⠖⢰⡾⢉⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠚⠋⠉⠉⠉⠙⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⢀⡌⠉⣷⡇⠀⢼⣿⣤⠓⢧⠀⡀⠘⠀⠎
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢦⠀⠀⢀⠴⣧⡈⠲⠈⢣⠂⠀⡿⣿⠀⠀⢣⠀⠀⠰⡄
⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡀⠀⠘⠦⡙⠻⠀⠀⠀⠄⠀⠐⣹⠄⠀⠈⠀⠀⠀⢸⡀
⠀⠀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢁⡟⠀⠀⠀⠈⠲⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⣿⠀⠈⢷⣒⠄⠠⠤⠤⠄⠠⠠⠀⠒⣂⣀⣥⠺⢁⣼⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠘⡦⠄⢀⠘⠧⠉⠙⠁⠀⠀⠉⠀⠀⠄⢁⠀⠄⡋⠉⢦⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⢱
⠀⠀⠀⠸⡀⠔⠣⠄⣀⣀⣀⣈⡙⠦⠄⠀⠀⠔⠧⠐⠚⢀⢤⠊⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠹⡀⠀⡀⣀⠀⠠⠤⠤⢀⠀⢀⡠⠤⠤⠀⠐⣡⠞⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠁
⠀⠀⠀⠀⠀⠈⠒⠤⢤⣀⣀⡀⠀⠀⢈⣁⣀⣤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠔⠒⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠉⠁
⣿⣿⢛⠿⢛⡛⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⡀⠀⠀⢠⣤⡀⠀⠀⣀⠀⠀⣶⠶⠆⠀⢰⣶⣶⣦⡀
⠀⠀⠀⢾⣇⠀⠹⠃⠀⠀⠀⠀⣾⡿⣧⣰⣿⣿⠀⠀⢸⡿⣷⣀⣾⣿⡄⠀⣿⡦⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠈⣉⠻⣧⠀⢠⣶⣶⠀⣿⠃⠹⠏⠀⣿⡇⠀⢸⡇⠘⠟⠁⢹⣇⠀⣿⡀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠛⠛⠃⠀⠈⠻⠃⠀⠉⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠈⠛⠀⠉⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣄⠀⢀⣤⣀⠀⢀⣀⣠⡀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⢸⡏⠀⣿⠀⢸⣿⡟⠀⢸⣏⡉⠀⣿⣁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⠃⠀⣿⠀⢸⣿⣿⡄⢸⡏⠁⠀⢈⡙⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠐⠿⠀⠸⠿⠟⠁⠸⠷⠶⠀⠈⠛⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⢤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⢀⠀⠛⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡆⢻⡄⠀⠀⢔⠁⠀⠀⠀⠀⠀⢀⣾⣵⣛⠶⠀⠀⠀⠀⠀⠀⣀
⠀⠐⠚⠓⠈⢇⡗⡧⠀⠠⣭⡭⣥⠆⠀⠀⠀⣷⠘⢃⠀⡿⠀⠀⣀⠦⠠⡀⠀⠀⠴⣻⠿⣿⠟⠁⠀⠀⠀⠀⢠⡿⣹⠇
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠚⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣄⣠⣀⠀⠀⠀⡠⠄⡐⠉⠄⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠘⠄⠀⠀⠀⠀⠀⢀⠄⢠⡲⠄⠀⠀⠠⣌⠻⠈⣠⠇⠀⠀⠈⠙⠛⠁⠀⠀⠀⣡⠆⠀⠐⢦⠀⠀⠘⠛⢿⡟⠛
⠀⠀⠀⠀⠶⠀⠀⠀⠠⡄⢂⣉⡀⠀⠀⠀⢀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⢀⠶⠀⠸⢋⣀⣉⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠖⠢⡀⠀⠀⠈⠛⠠⣤⠀⠀⠀⠀⠰⠦⠆⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠍⠙⠳⣦
⠀⠀⠀⠀⣠⣀⡀⠀⢠⡆⠀⠀⠀⠀⣶⠀⠨⠃⠀⡠⠈⠁⠀⢀⠀⠀⠀⢀⠀⠀⠐⣲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠁
⠀⠀⠀⠀⠾⣟⠏⠀⠀⠁⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠀⠀⠀⠀⠀⠛⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠴⠆⠀⠶⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠀⠀⠘⣡⠀⠀⢀⠀⠆⣘⡴⠃⠀⠀⠀⠐⠀⠐⠀⠀⢰⠶⠶⠶⠶⢤
⠀⢠⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠤⠠⠤⠦⠜⠀⠐⠒⠒⠋⠀⠀⠀⠀⠂⠂⠀⠤⠀⠀⠀⠈⠂⠄⠀⣊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⠇⠀⠰⠒⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⢸⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠁⠀⠀⠀⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠂⠀⠄⠂⠀⠐⢯⠾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠻⣼⣧⡽
⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠐⠀⠀⠀⣴⡶⣞⢶⣆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠶⠆⠀⠠⠶⠀⠀⢀⡤⢈⢛⠀⡀⠀⠀⢠⣴⣦⡄⠰⠆
⠀⠙⢶⣶⣶⡶⠏⠀⠀⠙⠳⠿⠷⠛⠀⠀⣄⡈⠀⣁⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠠⢺⠏⢺⣅⣱⠀⠛⠳⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠉
⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠰⡆
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢴⠆⠀⠀⠀⠘⢻⣟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠆⠀⠀⠀⠀⢡⡂⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⣄⣀⣀⣸⠟
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠆⠨⠀⠘⡄⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⣬⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤
⠀⠀⠐⠃⡤⠾⡅⠀⠐⠓⠂⠒⠑⠀⠀⠀⣶⣽⣷⠶⡀⠀⠀⠀⢀⡆⢀⡷⠀⠀⢀⣤⠀⠀⠁⠀⠀⠀⠰⠖⠛⠑⠛
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠠⣤⡄⠀⠀⠀⠒⠊⠐⠃⠙⠃⠀⠀⠠⠁⠁⠽⠀⠀⠀⠀⠉⢴⠇⠙⠂⠀⠀⢃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠛⠛
⠀⠀⠦⠀⢠⠀⠀⠀⠀⠀⠉⢡⡀⠀⠀⠀⣠⣤⡶⠶⠀⡀⠀⡀⠀⢀⠰⠆⠀⠀⠀⠐⠲⠖⠠⠄⠀⠀⢘⢃⠀⠠⣞⢳⡀
⠀⠀⠸⠿⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⠉⠁⠀⠀⠀⠀⠀⠀⠐⢺⠗⠀⠀⠀⣤⣄⣠⣧⠖⠈⠀⠀⠀⡠⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⠏⠛⣿⠉⡟⠛⣟⠟⣻⡿⣛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⠀⠒⠈⠉⠐⠲⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡊⢠⠄⠠⣄⠀⢰⡄⡄⢠⠂⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠣⣜⢀⢀⡙⠧⠀⢁⣷⣃⠀⠀⠴⣾⡧⡠⠶⠆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡇⢀⠜⠋⠀⠀⠀⠀⠀⠀⠁⠘⠃⠾⠋⣨⣔⣥⣤⡄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠯⢷⣶⣦⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣬⣤⣻⠶⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⠴⠒⢛⣛⠩⠉⠉⠉⠛⠒⢦⣄⠀⠀⠀⠀⣼⣀⣒⣋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⠛⣡⠶⠶⢦⠀⠀⠀⠀⣿⣿⣦⣀⣰⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⣽⠛⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠶⠄⠀⠀⡀⠠⢒⠤⠀⠐⠂⠀⠀⠀⠀⣷⢈⡇⠹⢻⡄
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣇⠀⠒⡧⠀⠓⠐⠁⠂⠀⠈⠁⠀⠀⠀⠀⠸⡌⠰⣡⡏
⢀⡀⠀⡎⠠⠄⠄⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠸⠞⢻⠃⠀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡶⣿
⢀⠹⡀⡇⠀⣠⠀⠐⣋⡔⠒⠶⢄⡀⠀⠀⠀⡜⠀⢸⣄⡰⣶⣬⣜⠩⠉⠀⡇⠀⠀⠀⠀⠀⣸⠁⣯⡀
⠈⠄⠡⣡⡀⠙⣆⡀⠛⠀⣰⠂⠀⣉⠀⠀⠀⢧⠠⡇⢾⠡⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⢃⠔⣵⠟⠉⢹⡆
⠀⠐⣄⠘⠓⠂⠀⠀⠁⠀⠘⡆⠀⣯⠀⢧⠀⠘⣆⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠑⢱⠞⣩⡤⠀⠈⣿⠲⠖
⠀⠀⠀⠢⡀⠙⠄⡀⠉⠀⠀⠁⠀⣼⠀⡼⠀⠀⠈⢢⠸⡇⠀⠀⠀⡠⠈⠀⠀⣠⠎⢂⠄⠈⠀⠀⠀⠀⠰⠾⠿
⠀⠀⠀⠀⢧⠀⠀⠈⢦⠀⠀⠀⠀⠘⢦⠃⠀⠀⠀⠀⠑⣌⠐⠒⠈⠀⢀⣠⡞⡡⠂⠄⢒⡁
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⡀⠀⣠⡧⢄⡀⠀⠀⠀⠀⢘⣶⣤⣴⠚⣉⢂⣪⠔⠒⠚⠉
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡠⠚⠓⠒⠓⢌⠒⠃⠉⢩⡶⢚⠿⡠⣞⠻⠛⠄
⠀⠀⠀⡔⢿⠁⠀⠀⢀⠀⡌⠀⠀⠀⠀⠀⠀⠑⠦⠤⠗⠋⠁⠀⠀⠈
⠀⠀⢰⠀⠈⠀⠀⠈⠀⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠊⠀⠈⠑⢄⡠⠊⠀⠈⢢⠀⠀⢀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠠⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢥⠀⠀⠀⠋⠀⠀⠀⢀⠃⠀⠀⠀⠔⠂⢀⠀⠊⠀⠀⠀⠀⠀⢄⠀⡄⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠂⠀⠀⠈⢀⠁⣠⢠⠀⡞⢠⠄⡀
⠀⠀⠀⠀⠀⠈⠢⠀⠀⠱⠀⣀⡀⠔⠁⠀⠀⠀⠀⠁⠐⡀⠀⠀⣠⠀⡠⠀⠀⠁⠀⡀⢠⠄⢤⢀⠀⠀⠀⠀⠐⠀⣀⠀⠀⡄⠀⠁⡈⠄⣔⠁⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡄⠠⠀⠀⠀⠠⠆⢁⡀⠐⠀⠀⠉⠀⢤⢠⠆⠁⢀⡐⠂⠀⠀⠀⠀⠀⠋⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⠀⠀⠀⠰⣿⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠠
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠔⠐⠊⠀⠀⠀⠀⠠⠀⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠤
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠊⡀⠀⠈⢀⢀⠄⢄⠀⠀⠄⠈⢢⠀⠀⠀⡀⢀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠘⠄⠀⠔⠲⠁⠀⠀⡀⢈⠆⠀⠄⠊⠀⠀⠀⢁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⣤⡀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠀⠀⡠⡊⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠒⠒⠒⠂⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⠌⠀⠀⠀⠠⠪⠊⠀⠀⠀⠀⠀⠈⢱⡄
⠀⠀⠀⢸⠁⠈⠢⢀⠠⠂⠀⠁⢀⠀⡀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⡉⠛⣦⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⡆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⡞⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣧
⠀⠀⠀⢸⠀⠀⠠⠐⠠⠀⠀⠀⠑⠀⠂⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠂⢠⡴⠋⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢂⠀⠐⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠑⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠖⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠤⡀⢀⠀⠀⠀⡿
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⠁⠀⢠⠇
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠸⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠸⠤⠔⠢⠤⠠⠤⠤⠤⠤⠤⠌⠀⠀⠱⠴⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠤⠤⠤⠄⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠔⠁⠀⠈⠂⠄⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⠄
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⢀⠀⠠⢀⡀
⠀⠀⠀⡔⢉⣤⣀⡈⠳⡚⠉⠀⠀⠀⣀⠰⣞⢳⣄⡀⠀⠀⠀⠀⢀⣀⣠⣴⣞⠁⠀⠀⠀⢀⡔⠁⠖⢆⢒⡈⢄
⠀⠀⠐⡀⢺⢿⣿⣿⠄⡇⠀⠀⠀⠈⢢⠤⢽⠧⠒⠚⠀⠀⠀⠀⢡⠀⡤⠀⡎⠀⠀⠀⠀⠻⡄⢳⠄⢁⣢⡇⡨⠃
⠀⠀⠀⠱⢄⡉⠑⣃⠔⠁⠀⠀⠀⠀⠘⡀⢸⠀⡸⠀⠀⠀⠀⠀⢸⡀⠈⠐⡇⠀⠀⠀⠀⠀⠈⠢⢁⣋⡩⠔⠁
⠀⠀⠀⢄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠉⠒⢤⣤⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠤⠄⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡆
⠀⠀⠀⠀⣤⣤⣾⣯⡄⠀⠀⠀⠀⠈⢹⢹⢹⠉⠉⠉⠁⠀⠀⣴⡟⢩⣤⣬⠻⣦⢰⠀⠀⠀⠀⢸⢫⢮⢻⡇
⠀⠀⠀⣿⣿⣿⣻⣻⣿⡇⠀⠀⠀⠀⠈⠸⢸⠀⠀⠀⠀⠀⠀⢿⡟⢿⣿⡿⢱⣿⢸⠀⠀⠀⠀⢸⡸⡯⢿⡇
⠀⠀⠀⠈⠻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣶⠟⠁⠸⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣄⡀⠀⠀⠀⠀⠀⡀⢀⡀⡀⠀⡀
⠀⠀⠀⣀⠶⣷⣐⣛⣃⡀⠀⠀⠀⣀⠐⠋⠷⠜⠓⢄⠀⠀⠀⠈⡮⠷⣏⡷⠶⢎⠀⠀⠀⣠⢶⣖⣿⢿⣦⣶⠢⡄
⠀⠀⠀⢻⣤⣿⣿⣿⣿⠃⠀⠀⠀⠉⠉⠀⠀⠀⠙⠉⠀⠀⠀⠀⠹⠐⠹⠓⢩⠁⠀⠀⠀⠈⠻⡾⢷⣥⡹⣿⡟
⠀⠀⠀⠀⠉⠻⠿⠛⠁⠀⠀⠀⠀⠀⠐⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠢⢸⠠⠄⠀⠀⠀⠀⠀⠀⡈⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⣄
⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄
⠀⠀⠀⠀⣔⣸⣟⣐⡄⠀⠀⠀⠀⠀⣾⣋⠁⢘⢿⡄⠀⠀⠀⠀⠾⢍⡩⢍⠹⠇⠀⠀⠀⠀⠀⠀⣤⡿⣯⡄
⠀⠀⠀⠀⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⢧⣉⠀⠱⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣯⠇
⠀⠀⠀⠀⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣖⣈⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣁⣲⠄⠀⠀⢀⡀
⠀⠀⠀⠀⣿⠈⠉⠙⠛⠛⠛⠛⠛⡛⠛⠛⣛⣋⣉⠩⣿⠀⢠⠚⣁⠨⠆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⡏⢠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡀⢸⣧⢋⡈⠰⠶⠤⡔⠊⠉⠑⠚⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠶⠀⠀⠀⢀⡤⠀⠀⠀⠀⣻⣾⣏⣀⡀⠠⠤⠰⡀⠀⠀⠀⠠⣀⠀⣀⠄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⣚⠝⢡⠆⠰⢄⡀⠒⢄⡎⠀⠀⠀⠀⠀⢈⡝⠁⠀⠀⠀⢀⡕
⠀⠀⠀⠀⠀⢸⡀⠀⠰⠆⠀⠀⠀⠀⠀⢠⠟⢁⠔⠁⠇⠀⠃⠉⠢⢄⣗⠦⣤⠤⣢⢭⢭⢺⢳⣤⠢⠴⢊⡴
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⣿⠀⠀⠀⠐⠓⢪⠀⠀⠀⢰⠀⡄⠀⠀⠀⠉⠑⠒⢘⠧⣓⠵⠇⠘⡇⠈⠈⠁
⠀⠀⠀⠀⠀⠀⡕⢄⡀⠀⠀⠀⠀⠀⠀⠀⢈⣦⠀⠀⠀⠦⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠑⡄⢀⢳
⠀⠀⠀⠀⠀⠀⠈⠢⢌⣀⣁⣀⣀⣀⣀⣀⡡⠔⠓⠤⣀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⣄⡞⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉





⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠉⠉⠉⠉⠒⠂⠤⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡠⠤⠒⠋⠁⠒⠒⠤⠤⠤⠤⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠓⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠤⠤⠤⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠐⠚⠒⠒⠢⠤⣀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⢅⠀⠀⣠⠯⡂⠀⠈⢘⡇⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⢄⡸⠀⠀⠁⣀⠀⠀⠀⠀⡴⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⠇⠈⢩⠆⠀⠀⠙⠚⠀⠀⠀⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⠀⠀⠁⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢢⠀⡟⠢⢄⠈⢦⠘⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⣇⡚⠣⡀⠀⠱⡄⠑⢤⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠀⣈⡆⢙⠀⡐⠀⠀⡎⢰⣀⠶⠋⢸⠃⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠉⣁⣼⢋⡰⠀⠀⠀⠘⠢⠙⢦⡔⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⢀⣀⣡⠼⡈⢇⠀⠀⠀⠀⠀⠀⣠⡣⢄⠤⢰⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠔⠁⢰⠋⡌⢆⣠⠴⠖⠋⢉⡠⠖⠁⠀⢹⣇⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠴⠛⠈⢀⣠⠴⠚⠁⠀⠀⠀⠀⠈⡟⢆⢀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⡠⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⢰⠋⢷⣧⠞⠁⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣅⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠉⠀⠀⢈⣇⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⣀⣀⣰⡯⣤⣀⠤⠄⠀⡀⢀⡴⠛⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⠀⠀⠀⠀⠀⠙⠓⠛⣛⡲⠯⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠀⢁⣀⣀⣤⣤⣤⣌⡢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⢀⣤⠴⠚⠋⠉⠀⠀⠀⠀⠙⣿⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⢶⣒⣒⣚⣿⣿⣷⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⣀⡤⢒⣩⠵⠞⠛⠉⠉⠉⠁⢀⣀⣈⠉⠉⠉⠙⠛⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢢⠀⠀⠀⢀⠄⠈⡠⠖⠋⠀⠀⠀⠀⢀⣀⣤⠞⠋⠉⠉⠙⢳⡶⢶⣄⣀⣀⠈⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣳⣄⠐⢁⠔⠈⠀⠀⠀⣀⣤⠶⠛⢻⡟⠁⠀⠀⢀⡤⠒⡡⠂⠜⠉⠉⠙⠛⢶⣄⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⡀⠁⠀⠀⢀⣴⠞⠉⠀⠀⠀⠘⡇⠀⠄⠂⠁⢀⠁⣤⡄⠀⠀⠀⡀⠀⠀⠻⢦⡀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠎⣤⢶⠖⠛⠉⠙⠳⣦⠀⠀⠀⢀⡿⠀⠀⠀⠀⣾⣿⣿⣏⠀⣧⠐⠀⠀⢀⠀⣸⣇⠀⠀⠈⢿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⠎⢸⠁⠃⠀⠀⠀⠀⠀⠘⣧⣀⡴⠞⠃⣀⡂⢠⣶⣾⣿⣿⠿⠃⠁⢒⣒⠚⠡⠒⢿⡙⣷⡄⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡾⢀⠎⠀⢸⠀⡄⠀⠀⠀⡀⠀⠀⠈⣿⠀⠈⢀⣀⡀⣀⡛⢿⣿⣿⣿⠟⠀⠀⠀⠀⡀⠀⢸⠷⠾⠿⢲⡄⠀⢿⣷
⠀⠀⠀⠀⠀⠀⣸⠃⡌⠀⠀⢸⣧⣐⠊⡁⠉⠁⠀⡠⣰⠏⠀⠀⠀⠻⣷⣿⣿⣿⣿⡿⠻⣄⠆⢢⡀⡀⠆⣃⠄⠀⠀⢀⠸⣧⠀⠀⢻⡇
⠀⠀⠀⠀⠀⢠⡿⢠⠁⠀⠀⣼⠀⠹⢦⣄⣀⣐⣨⡼⢷⠀⡄⠀⠀⠘⠛⠿⢿⢿⣿⣴⣶⣟⠻⢷⣌⣁⠈⠁⠀⠀⠀⠈⠐⠘⣧⠀⡜⣿
⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠁⢀⡾⠃⢌⣂⣼⣿⣷⣆⣁⣼⠃⠀⠱⠙⣆⠀⠈⣇⠀⠀⠀⠀⠐⠠⠀⢸⡏⠀⢸⣿
⠀⠀⠀⠀⠀⣾⠁⡆⠀⠀⢀⡷⠞⠛⠛⠛⠛⠛⢶⡼⠁⢀⣿⣿⣿⣿⣿⡏⠀⠀⢠⠃⢀⠀⡿⠶⠞⢉⡂⠀⠀⠀⠐⠁⣠⣿⠁⠀⠈⢸
⠀⠀⠀⠀⠀⣿⠀⣷⠀⢠⡾⠁⠀⠀⠀⠀⢸⠆⠆⢡⠰⠻⡿⠿⣿⠃⠛⠃⡤⠔⠁⠘⣧⣬⣴⣖⡂⠀⠃⠐⠏⡒⠐⣟⠉⣿⠀⠀⡄⢠
⠀⠀⠀⠀⠀⢹⣷⣿⡀⣿⠈⠀⠀⠀⠒⠀⠺⠀⢀⡜⠄⠳⠤⣂⠀⠁⠤⣤⣵⠖⠚⠛⢹⣿⣿⣿⣿⣆⠀⠈⠢⡙⢄⣿⢰⡏⠀⠀⠃
⠀⠀⠀⠀⠀⠈⣿⡘⣇⢿⡘⢄⠀⠀⠀⠀⣀⠗⡙⠁⡀⠌⠓⠂⡇⢰⡞⠁⢀⡤⠖⣋⡹⢿⣏⠻⠿⠗⠀⠀⠀⠙⠸⣏⡿⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣎⠳⣤⣈⣀⣀⣩⣴⠊⠀⠀⠀⠀⠀⠈⣠⢙⣧⣤⠞⡡⠊⠀⠀⠀⠀⠑⡄⢤⣀⣀⣤⣄⣠⡿⠁⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡝⣆⠀⠉⠉⢻⣤⡿⠀⠀⠀⣀⠄⢂⡄⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠁⢸⠏⠉⠀⣨⡟⠀⠀⠀⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣞⢷⡀⠀⠀⠙⢧⣤⣀⡉⠄⠂⢁⠈⠂⠀⠀⠈⢂⠣⣐⢄⣴⠋⠀⡀⢡⡞⠀⢀⡼⠋⠀⠀⡔⠡⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣀⡻⣄⠀⠀⠀⠉⠻⣽⣓⠒⢻⡄⠀⠀⠀⠀⡀⠑⠌⠦⣌⣉⣁⡴⠋⢠⠶⠋⠀⠀⡠⢊⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡳⢄⡀⠀⠀⠀⠉⠛⠾⢷⣄⡀⠀⠠⠈⣀⠀⢸⣧⣭⠵⠚⠀⠀⠀⢀⡠⣊⣴⠿⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡙⠂⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠚⠋⠙⠛⠁⠀⠀⠀⢀⡤⠖⢋⠿⠋⠁⠀⠀⠹⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢄⡉⠒⠢⠤⠀⣀⣀⣀⣀⣀⣀⠠⢤⣴⣶⣿⡥⠖⠋⠁⠀⠀⠀⠀⠀⡸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠠⠀⠠⠤⣤⣤⡴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣎⠀⠀⠀⠀⠀⠀⠀⠊⣁⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⣦⣠⣤⡤⠶⠖⠛⠁

⠀⠀⠀⠀⠀⢀⣴⠞⠛⢳⠀⠀⠀⠀⠀⣀⠀⣤⡄⢀⣄⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣿⠃⠀⡠⠋⣠⡄⢰⣿⠀⣿⡾⢸⣿⢫⣿⠀⣾⣇⠎⣿⣧⢶⣦⠀⣀⡀⠐⣧⡇
⠀⠀⠀⠀⠀⢿⣦⣉⣀⠀⢿⣇⣸⣿⢀⢿⡇⠘⠃⣾⡏⢠⠿⠏⠸⠿⢡⣿⠃⣼⡿⠝⠀⡸⣳⣶
⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣆⠉⠀⠙⠋⠀⢀⣀⡀⠙⠛⣡⣴⠶⡄⠀⣿⣃⡸⣿⣄⡠⠈⣾⠟⢡⡀
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡿⠀⠀⠀⠀⣰⠃⠠⣸⠆⢰⡿⠁⡰⠃⠀⠈⠁⠀⠀⠉⠀⢸⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠑⠶⠒⠋⣰⠋⠻⣷⡀⡇⠒⢚⡋⠀⣿⡗⣪⢤⣄⠀⠀⣤⣦⠀⣴⠟⢦⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢹⣷⣧⠀⣿⡇⠀⣿⡷⠥⣼⣿⠤⣾⡟⠊⢀⢿⣆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⣿⠀⢿⣧⠜⠘⢥⡴⠟⠁⠀⠻⠧⢖⡁⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⡴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠚⠹⣆⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⢀⠀⠀⠀⠀⢠⣶⣿⡃⠀⠀⠀⠀⠀⠀⠀⡀⢿⣦⣶⣿⣿⠗⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣰⣾⠃⠀⢸⣿⡇⠀⡯⠀⣿⠈⢹⠀⠀⣤⣤⣤⡀⢴⣿⣿⠆⢿⡎⠉⢀⠀⢆⠀⠀⠀⠳⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⡶⣹⣏⠀⢸⡏⡇⠀⡇⢠⡏⢀⡟⢠⣼⣿⣿⣿⣿⣦⣀⠀⣰⣇⣡⣶⠀⠀⡇⠀⠃⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡘⠉⠀⢸⣇⣷⣴⣧⡾⢁⡾⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⡏⣾⠏⣠⡶⠀⠀⠀⠀⢀⣿⣌⠋⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡟⢠⠀⣸⣿⡿⣻⣾⣷⡿⠁⠀⢹⣿⣿⣿⣿⣿⣿⠃⠀⡟⢀⠱⠞⢋⣴⣶⣤⣤⣀⡀⠛⠁⢀⡹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠘⠳⣿⣿⠉⠁⠀⠀⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣶⡄⠟⠿⢄⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢰⣿⣶⡄⠀⠈⠻⠿⣿⣿⠟⠀⢠⠀⠀⠀⠀⠻⠟⠀⡔⠀⡴⢰⠂⠀⠀⠿⢿⣿⢩⣿⣿⣿⣿⣧⠀⣴⣿⢾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⢷⡀⢰⠀⠀⢰⡄⠀⢸⢐⢀⡎⢀⡆⢰⢐⠃⢰⠃⡆⠰⢰⢲⡆⠀⣿⣿⣡⠿⢿⣿⣿⠳⠟⣣⣴⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢿⣿⣶⡘⣧⢸⡄⢷⡈⣷⠀⠈⠈⠀⠠⠀⠁⠈⠀⠀⠀⠀⠁⢉⠈⢸⠁⣸⣿⡿⠃⠀⠀⠉⠁⢀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠐⢻⠟⠀⠈⠉⠉⠁⠹⣎⣷⠘⣷⠘⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⠲⠁⢠⠋⠰⣿⣿⠅⠀⠀⠐⠀⠁⠀⢀⣤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠳⣄⣻⣿⣧⢹⣧⢻⡄⠀⢸⡄⡜⠀⡞⠀⣴⢐⡇⠀⠀⣴⣿⣷⣦⣍⠀⠁⠈⠀⠀⠀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡞⣿⠀⠀⠉⣿⣻⣇⢻⣿⣿⡄⠘⠄⠉⣀⠈⠀⢈⣀⠁⠀⢸⣿⣿⣿⣿⣿⡇⣠⣶⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡿⠀⢠⣦⣼⣇⢿⣾⣿⡜⣷⡀⠀⢸⣿⣦⣴⣿⣿⡀⡀⢸⣿⣿⣿⣿⣿⠃⠘⢋⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠈⠛⠉⣿⣼⣿⣿⣧⠙⠗⣤⣼⣿⣿⣿⣿⣿⢿⡇⢸⣿⣿⡿⢋⣤⣤⡴⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⢸⣷⡄⠙⠋⠉⠀⠀⠀⣤⣿⣿⣿⡟⢿⣿⣿⣾⠇⠘⠿⠛⣱⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠳⠀⠀⠀⠀⠀⠀⠈⠹⠿⢿⣿⣿⣿⡟⠉⢸⣿⣷⠈⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣤⣤⣀⠀⠻⢟⠉⢀⣠⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣷⣄⠋⠠⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢄⣤⣀⡀⡀⢀⣀⣀⢀⣀⠀⣄⡀⣀⢀
⠀⠀⠀⠀⠉⠉⠁⠁⠉⠉⠈⠈⠈⠀⠉⠈⠁⠉⡠⠔⠐⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣄⡀⠀⢀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠉⠓⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠒⠤⠀⢀⡀⠀⠼⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠒⠒⠀⠒⠚⡓⠉⢠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⢀⡄⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⠂⠀⠘⢆⠀⢤⠀⠀⠀⠀⠀⢉⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢢⡀⠀⠀⠉⣲⣤⣤⣤⠶⠖⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡆⠀⠄⠬⡌⠁⠀⠛⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠊⠀⠀⠀⢡⠖⠒⢲⠄⢠⠐⠀⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢈⠦⠄⠂⠁⣠⢆⣀⡄⠠⠔
⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠐⠄⣀⢠⣅⡀⠔⠈⠀⢸⠁⠀⠀⠀⢠⣂⠀⢀⡠
⠰⡁⠀⠀⠀⠀⠀⣈⠄⠀⠀⠀⠀⠀⠂⡤⠠⠀⡀⣀⠀⠀⠛⠀⠀⠀⢸⠀⠈⠀⠀⢀⠀⠀⠀⠂⠀⣀
⠀⠉⠉⠛⠒⠛⠉⢉⠂⠀⠀⠀⠀⠀⡐⠀⠀⠀⠃⠀⠀⠀⠀⣃⠀⠀⢸⠀⠀⠀⠈⢣⠄⠔⠒⠂⠴⠤⠂
⠀⠁⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⢾⠉⠒⠀⢼⠀⠐⠅⠀⠁⠀⠀⠀⠀⠀⠐⠄⢀⠎⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠢⠤⠤⠎⠑⠒⢄⠀⠀⠀⠀⠀⠀⠠⠐⡀⢸⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠐⠀⠀⠀⠀⠀⠊⠑⠂⠠⠄⠒⠁⠀⠀⠀⠈⠒⠒⠒⠓⠂⠀⠤⠀⠂⠀⠀⠀⠀⠒⠒⠠⠤⠤⠄⠒⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢻⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡾⠀⠈⠳⣄⠀⢀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣠⠞⠁⠀⠈⣧⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢦⣾⡾⠁⢀⣤⠴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢤⣄⠘⣿⣷⣶⢤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠁⣀⣰⡋⠙⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡀⣼⣦⣬⣙⣲⣄
⠀⠀⠀⠀⢀⣾⡶⣚⣫⡽⠟⢳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠙⠲⢬⣭⣿⣆
⠀⠀⠀⠀⣼⠉⢯⡉⠀⠀⢠⡟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣴⠋⢻
⠀⠀⠀⠀⣿⠀⠈⠳⣄⠀⣾⠀⠀⠀⠀⠀⠀⣻⣆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠞⠁⠀⣸
⠀⠀⠀⠀⢸⡄⠀⠀⠈⣻⠃⠀⠀⠀⠀⠀⠀⡿⠈⠳⢦⣄⣀⣉⣳⣦⣄⡀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠹⣤⣄⠀⣿
⠀⠀⠀⠀⠈⣧⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣉⣭⠉⠀⠀⣩⣍⠛⠋⠉⠁⠈⠛⢦⣀⡀⠀⠀⠀⢀⣠⠟⢰⡇
⠀⠀⠀⠀⠀⢹⡌⠻⣦⣤⣤⣤⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠉⣠⠀⠀⠀⣠⡈⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⡏⢻⡀⣾
⠀⠀⠀⠀⠀⠈⣧⣾⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⣀⣀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⣼⠁⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢻⡀
⠀⠀⢰⣤⠤⠾⠃⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠳⠤⢤⡶
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣶⣤⣴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠳⣤⣀⣀⣀⣙⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣞⣻⣿⣿⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣃⣀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠳⢦⣀⠀⠀⠀⢀⡿⣷⣶⣶⣚⣻⡇⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠶⢼⡇⠉⠉⠉⠉⠃⣧⣠⣤⡤⠶⢺⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢺⡿⠶⢤⣽⣦⡀⠀⣠⣞⣁⣀⣀⣤⡴⠾⣷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣸⠃⠀⠀⠀⠈⢻⢰⠋⠉⠉⠉⠁⠀⠀⠀⢻⡄⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⡿⠀⠀⠀⢀⣀⣼⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠴⢾⣧⣤⠖⠛⠉⠉⣿⣘⣇⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠈⢉⣩⣧⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣉⣠⠟⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⡀⠀⠸⣇⠀⠀⣀⠀⠀⣸⠟⠛⠳⣆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢹⠏⠛⣛⣋⣿⠋⠉⠻⡟⢻⣄⣀⣠⠟⠓⢲⣿⣿⣗⠶⣤⣹⣤⠞⢁⡿⢧
⠀⠀⠀⠀⠀⢀⡴⠦⠶⠤⢶⣾⣍⣹⡿⢿⣫⡽⠋⠹⣦⣤⣴⠇⠀⠈⠉⠁⠀⠀⣏⠀⠀⢹⡆⠈⠻⣿⣷⣾⢧⣾⣷⠶⠶⠶⠶⢦⡄
⠀⠀⠀⠀⢀⡾⠁⠀⣠⠶⠋⠁⣙⣿⣹⣿⡇⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠖⠋⠀⣠⣤⠏⢀⡟⣷⣀⡉⠳⢦⡀⠀⠀⢻⡀
⠀⠀⠀⠀⣼⠁⠀⣾⠃⠀⠀⢸⡿⣿⡷⣴⣿⣛⣋⣥⣼⣇⠀⣀⣴⠟⣿⣷⣄⣀⣠⠾⠟⠛⠶⠋⣠⡴⢻⣟⠉⣨⠏⠀⠀⢻⡄⠀⠈⢷
⠀⠀⠀⣸⠃⠀⠀⢻⣄⠀⠀⠈⢷⣄⠙⠳⠦⣭⣍⣉⣛⡛⠛⠛⠉⠉⠉⠉⣁⣉⣀⣠⣤⡴⠖⠛⢉⣴⣿⡙⡏⠁⠀⠀⢀⣼⠃⠀⠀⠘⣧
⠀⠀⣰⠇⠀⠀⠀⠀⠙⠳⣤⣀⠀⠈⠛⠶⢦⣤⣈⣉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⣉⣁⣠⣤⠶⠚⠉⠀⠀⠙⠃⢀⣠⡴⠛⠁⠀⠀⠀⠀⠘⣇
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣄⣉⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢀⣀⣀⣀⣤⣤⠴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⣄⠀⡀⠀⣦⠀⣿⢹⡿⢹⡇⣼⠏⣽⠎⣿⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠻⠀⣿⡀⢿⡄⣿⢸⡇⠸⡧⠟⢰⠏⢸⣇⣾⢏⣠⡾⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⣽⠧⠜⠛⠁⠀⠀⠀⠀⠀⠀⠀⣈⣁⠙⠦⢤⡶⠟⠉⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⢀⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⣄⠄⠀⠀⠀⢀⠀⠀⣠⣴⣶⣶⣤⣘⣿⣷⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⡀⠀⢸⢀⡾⢿⠛⠋⢉⣩⣿⣿⣿⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢄⠀⢱⡀⢸⡋⢠⠇⢀⣶⣿⡟⠁⢸⠹⣧⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠈⠣⠀⢃⣈⣁⠚⠠⠞⣿⣯⠆⠀⣿⠀⣿⡇⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⠓⣠⠞⠋⠉⠉⠙⠲⡄⢋⣻⠤⣼⡟⢸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢰⠃⠀⠀⠀⠀⠀⠀⠈⠀⢤⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠘⠀⠀⠀⢀⣀⣀⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠥⣤⡀⢀⢾⡟⠛⢛⡉⠉⠙⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢠⡿⠀⢀⡾⠁⠆⣼⠇⠀⠀⠀⣀⡀⠈⣿⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⣿⠁⢠⡾⢁⡿⢠⣿⠋⢹⣦⡿⢁⡿⢠⡝⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⣴⠏⠀⢸⣇⠼⣇⢀⡿⠙⢯⣉⡤⠛⠧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠠⢠⠀⠀⠀⠀⠀⠈⣷⣦⣦⣤⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⢈⠃⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡐⠀⠀⠀⢸⡄⣠⠞⠠⠀⡴⠂⠀⡀⠀⡀⠀⣠⠄⣢
⠀⠀⠀⠀⠀⠠⢶⡄⠀⠀⠀⠀⠀⠀⠀⠈⢞⠁⠀⠀⠀⡇⠀⠀⡇⠀⣹⠀⠻⠈⠁⠀⠀⠼⠈⠃⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⡦⠈⢆⠀⠀⠀⡇⠀⠀⠛⢠⠏⢀⠀⠀⢆⠀⢀⣤⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⡀⠀⠐⠃⠀⠐⠃⠠⠤⠵⠀⠤⠖⠉⠀⠀⢁⣠⠞⠀⠀⣿⣅⣠⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣄⠀⢰⡄⠀⡤⠴⠆⢢⠀⠀⡖⢠⡆⠀⡆⠀⠀⢿⣦⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⣴⠀⡇⠀⣡⠆⠀⢨⢦⠀⡇⢸⠀⠀⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⣸⡇⠀⢀⡇⣀⣃⡤⢀⡸⠀⣀⡇⠈⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢐⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡈⠀⠀⠀⠀⢰⠁⠀⠛⠟⠀⢀⢀⣄⠀⠀⠀⠀⠉⠑⠂⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⡄⠁⠀⠾⠿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡜⠀⠀⠀⡼
⠘⠣⣀⠀⠀⠀⠠⠋⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⡞⠀⠀⠀⣰⣧⠤⠴⠦⠤⠤⣄⣀⡀⠀⠀⠀⢀⡀
⡏⠀⠈⠙⠢⢤⠃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⡋⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡹⠁
⡇⠀⠀⢀⠀⢸⢀⠀⢰⢀⡀⠸⠀⠀⡀⢀⣀⠀⠀⣀⣀⣀⠘⣷⡽⣠⡞⠁⠀⠀⠀⠀⡀⢀⣀⣠⠤⣐⢂⠁⣀⣀⡜⠁⠀⣀
⣇⠀⠀⠀⣢⣸⠹⠀⠀⣟⠇⠀⣄⠇⠀⠀⠀⠀⢾⠀⠀⠀⢾⣭⠀⠀⢠⡞⡉⠉⠉⠉⢹⡏⠀⠀⡎⡇⠀⠋⠁⠀⠑⣴⠏
⠸⡄⠀⠀⠀⢭⢐⠀⠀⣷⡀⠀⣗⡀⠀⡴⠞⢐⡼⠀⢰⠀⠛⡯⠄⠀⠨⢿⣣⠤⠀⢠⠾⢧⠀⠀⡇⡇⠀⠶⠀⢄⠀⠈⠀⢠
⠀⠹⡄⠀⠀⢸⢨⠀⠀⠛⠁⠀⣇⠄⠀⠀⠠⢸⠚⠀⠸⠀⢸⣗⠅⠀⢠⣄⣰⠩⠀⢠⣠⣩⠀⠀⣇⡇⠀⠀⢱⡈⣆⠀⢀⠃
⠀⠀⠙⢦⠀⢸⣺⠀⢀⣀⠀⠀⣇⠂⠀⢀⣀⢼⡆⠀⣠⠀⠀⡄⡀⠀⠀⠈⢙⡢⠀⠈⠀⣻⠀⠀⠀⠀⠀⠀⠀⢳⠁⠀⢸
⠀⠀⠀⠈⠓⢾⢘⠀⢰⡟⠀⠀⣷⠆⠀⠋⠁⣸⠃⠀⡏⡆⢀⣗⣇⣀⣐⠀⢼⢈⠀⠀⠀⢚⠀⠀⣖⠆⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢴⠀⢸⡇⠀⠀⣇⡄⠀⠀⠀⣿⠀⣀⠷⠛⠉⠁⠀⠀⠀⠉⠙⠲⢄⢠⣼⢹⠀⠀⣿⠄⠀⠸⠀⢸⡅⠀⢸
⠀⠀⠀⠀⠀⠸⡴⠲⠞⣧⣗⡶⢿⡓⣐⣖⣶⠟⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢻⣸⣝⣒⡖⢸⡽⠶⢾⡀⠸⠼⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠈⠉⠀⠁⠐⠂⠀⠐⠒⢺⠍⠭⣔⠢⢤⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⣀⡠⠤⠭⠗⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣏⠀⠀⠀⠀⠉⠓⠄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠋⠛⢻⠉⠉⠉⠘⡆⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠘⠋⠉⠉⠓⠒⠦⢤⣀⣀⠀⠀⢀⣀⣠⠤⠖⠻⡄⠀⠀⠈⢣⠀⠀⠀⣷⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⢠⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣄⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣠⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⣷⡀⠰⠟⠛⠉⢹⣿⡆⠀⣿⡇⢀⡀⢀⣀⠀⣠⣤⣄⠀⢀⣀⡀⢠⡶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠿⠁⣾⠀⣰⡆⢸⡿⣿⣸⣿⣇⢸⣧⡿⢻⣿⠋⢰⡟⣰⣟⣽⠗⢸⣇⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣿⣀⣿⠁⢸⡧⠸⡿⠸⣿⠸⣿⠇⣿⠏⢀⣿⠁⣿⣉⣡⡶⢸⣿⠙⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠻⣦⠈⠛⢁⣀⣼⠇⠀⠀⠀⠿⠀⠀⣴⣦⡀⢸⡏⠀⠈⠛⠉⠀⠈⠉⠀⢿⣄⠙⣿
⠀⠀⠀⠀⠀⠀⢠⣿⠙⢻⡆⢹⣇⠀⣈⡉⠁⢠⣤⠀⠀⠰⠄⢸⡏⣼⠇⠘⠛⠀⢀⣶⠿⣶⠀⣿⣦⡀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⣸⡏⢀⣿⣧⠀⢸⡿⣧⣤⣶⠀⢸⣿⢏⣤⣦⣤⣤⣾⣧⡶⠟⢠⣿⠈⣻⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⣠⡾⠃⢻⡆⣿⠇⠀⠀⣿⠀⣾⣿⡿⢡⡾⢻⡇⠹⣧⡀⢀⣾⠃⣼⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠈⣿⣿⠀⠀⠀⠙⠿⠃⠙⠃⠸⣧⡿⠁⠀⠉⠛⠛⠁⠀⠘⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠳⢦⠶⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⡆⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣷⡀⠀⠀⠀⠀⠀⠈⢷⣀⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣀⣴⠟⡉⠉⠙⣏⠉⠙⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠛⠉⢹⡜⠟⠀⢀⡿⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣹⡶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⡈⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠳⠄⠐⣿⣀⣀⣴⣿⡿⠿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣀⡀⠀⠈⠙⠛⢩⡏⠷⣄⣈⣉⠉⠉⣿⣿⠛⣛⠒⠛⠛⠛⠉⣠⠊⠁⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠘⣷⣄⣀⣹⠀⠀⠀⡟⠻⣤⡾⠀⠀⠀⠀⠀⢷⢫⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠷⠖⠚⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⢶⡇⡇
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⣶⡶⠶⠶⠶⠤⣤⣶⠶⠶⢒⣛⡿⠛⣿⠶⣤⣄⣀⣀⣀⣼⣷⣿⣷⠆
⠶⣤⣬⣷⣶⣄⣤⣤⡤⠶⠟⠋⣿⠀⠀⠉⠛⠒⣶⢾⡉⣉⣷⡛⠉⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⢻⢻⠿⠇
⠲⠶⣚⡿⠛⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣁⣰⠏⢻⣄⣻⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣸
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠀⠤⢸⠈⠉⠀⠀⠀⠀⠀⢻⠀⠀⣏⢄⠤⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣤⣤⣤⣤⣄⣤⣤⣼⣤⣤⣤⣤⣤⡤⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠀⢸⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣥⣉⡉⢙⡦⠀⢰⣋⣥⣄⣀⣀⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢂⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣰⠁⠀⠳⡈⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀⢠⡆⠃
⠀⠀⠀⠀⠀⠒⠒⠀⠀⣀⡔⢸⠀⡏⠆⣀⠀⠑⣦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠁
⠀⠀⠀⠘⠐⠉⠀⠈⠁⠄⡁⠀⢆⠘⣂⡩⠴⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠄⠣⣀⠀⠀⠀⠋⢁⠈⢢⣌⡃⠄⠂⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠛⠛⠳⠶⣧⠄
⠀⠀⠀⠀⠈⠢⢄⡉⠀⠂⣈⠥⠚⠁⠀⠉⠑⠒⠂⠀⠐⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⢀⣀⣠⠥⣤⡤⣤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠐⠒⠒⠒⠂⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠞⠉⠸⠇⠀⡏⠀⠟⠀⠸⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⣀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⠒⢦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠑⠲⢄⣀⣠⣧⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⢠⡞⢺⣤⡖⣠⢖⠀⡴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣯⣤⣇⣼⢙⠃⠛⠘⠚⠂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⢆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠈⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢳⡀
⠀⠀⠀⠀⠠⠈⢳⣠⠊⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠳⢌⣏⠴⢃
⠀⠀⠀⠀⠄⢄⠐⡆⢀⠄⠀⠀⠀⠀⢀⠤⠠⠀⠁⠀⠈⠉⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠢⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡀⢉⣋⠁⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠴⠮⠤⠒⠁
⠀⠀⠀⠀⠁⠙⠢⠧⠞⠁⠁⠀⢅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠂⠦⣈⣇⡠⠂⠂⠀⠀⠁⠀⠁⠀⠈⠁⠀⠀⣀⡀⣩⡤⠠⢡⡶⠒⢲⡞⠉⢹⡏⠀⢸⡏⠀⠀⣿⠁⠀⣼⠋⠀⠈⠈⠑
⠀⠀⠀⠀⠈⠄⠠⡄⠀⠈⠀⠀⠀⣀⣠⣤⢀⣲⣏⠉⢽⡧⠤⢾⡤⠤⠼⣧⠄⠸⣧⠤⢼⣤⠤⠬⣥⠤⠄⢀⣀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠃⠁⠀⢀⡴⠛⠉⢸⠃⠀⢸⠃⠀⢸⠁⠀⠸⠁⠀⠸⠇⠀⠸⠇⠀⢸⠃⠀⠰⠏⠀⠀⣾⠁⠀⢰⠟⠁⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢦⣌⣿⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⢿⣯⡌⠈⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠋⠉⢠⣾⠟⢸⣿⠀⠈⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠉⡉⠙⠢⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢘⣿⣀⣀⠀⢀⣤⣤⠀⣀⣤⣀⠀⢀⣤⣄⢰⣿⢀⡀⠘⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢸⣿⠟⢻⡇⣾⣯⡿⠃⣋⣭⣿⣆⣿⠋⠿⢸⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠘⣿⣷⣿⠃⠻⣷⣾⠟⣿⣷⣿⡏⢿⣤⡶⢸⡿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣠⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢸⣷⠀⣴⡷⣚⠀⣿⡆⣀⠀⠀⣠⣄⠀⢀⣤⡔⠫⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄⠀⠘⣿⣷⣿⠃⣿⡇⣿⡿⢿⣧⣾⣯⣿⠇⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡑⡀⠀⢻⣿⡏⠀⣿⠇⣿⣷⣾⡏⢻⣿⣵⣶⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣄⠀⠉⡀⠀⠉⠀⠉⠈⠉⠀⠀⠈⠁⠀⠉⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⢒⣥⡔⠒⠒⢲⡐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣯⡿⠛⠫⣝⡆⢤⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢠⣷⣾⣿⣶⣇⣰⡘⣯⡈⣿⢆⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣤⣶⣿⣿⣿⣿⣯⣿⣿⣿⣿⣶⣿⣏⡏⢠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⣣⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣾⣿⢙⠏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⣿⣯⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⢩⢃⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⣿⣿⣴⣿⢿⢿⡿⣿⣿⣿⣿⣿⡿⣿⠿⠛⠃⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⠀⠁⡐⣤⣖⠛⠿⢿⣛⣿⠏⢈⣁⠴⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠉⢧⢑⣶⠚⢝⠛⠟⠉⠁⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠒⠖⠏⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠘⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⡀⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠐⠤⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠢⢀⠄⢀⣀⡆⣀⠀⡀⠀⣀⡌⠛⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣀
⠀⠀⠀⡀⠀⠚⠱⣏⣹⣫⡇⢀⢋⣸⣿⣎⣞⡡⢸⡢⡛⠀⣿⡷⣋⡶⣏⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⠀⠀⠀⠀⠈⠳⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣷⢤⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠈⠙⠓⠒⢛⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⣀⣀⣠⠤⠞⠁⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠗⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣒⠒⠊⠀⣀⡠⠂⠀⢀⣀⣤⣀⡤⠞⠉⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠠⠘⠉⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢉⠉⠓⠒⠋⠉⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠩⣅⣀⣀⣀⣀⣀⣀⣀⣀⡽⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢇⠀⢀⠞⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⢀⣼⠢⠬⠥⠤⠤⠤⠤⠤⠤⠤⢴⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠶⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣄⡾⠚⣍⢧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⠤⠖⠚⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠈⠋⠁⠀⠀⠀⠀⠀⢀⣠⠖⣡⠟
⠀⠀⠰⠉⠀⠀⠀⠀⠀⠉⢁⡶⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠐⢰⠦⣄⡀⠀⠀⠀⠀⣩⢷⠶⢶⣴⢶⡶⠛⠛⠳⡍⠉⠁
⡖⠲⣶⠀⠀⠀⠀⢀⡀⠀⠰⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠒⠒⠒⠚⣏⢷⣄⠙⠶⣒⠒⣞⣡⠛⠂⠤⠤⣼⠃⠀⠀⠀⠸⠒⠒⠒⠒⠒⠒⠒
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⡗⠠⠤⠄⠀⠀⠠⠤⠄⠐⠒⠒⠉⠠⠀⠖⠀⠈⠲⢽⠀⠤⢀⠉⢁⡛⠀⠀⠀⠀⢼⠀⠀⠀⠀⢸
⡇⠀⢿⡀⠀⠀⠀⠀⠀⠀⢰⠇⠄⠀⢄⡀⠁⠠⠀⡠⠀⢠⣄⠀⠀⡄⠀⠠⠀⠀⠀⡴⠒⠛⣋⠈⠛⠀⠈⠉⠉⠉⠙⠓⠂⢦⣬⠋⠓⢦⠴⠋
⡇⠀⠀⠳⠖⢤⣤⣤⠴⠲⠛⠴⡀⠀⠀⣇⠀⠀⠀⢄⠀⠸⠈⠉⠰⠡⢤⣄⠀⠀⡀⢣⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠴⠚⠉⠀⢀⣿⣴⠟⢷
⡇⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠄⠠⠀⢞⠋⠉⠀⠀⠰⡄⠤⢌⡡⠄⠷⠀⠀⠀⠀⠀⠀⢘⣭⠍⠁⠀⠀⢀⣠⠔⠋⠈⣧⡴⠾⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⡌⠀⠀⠈⠁⠀⠈⠀⠲⠄⣀⣀⣀⢳⣖⣠⣄⠤⠚⠉⠀⠀⠀⠀⣧⠹⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡇
⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢉⡏⠈⠃⠒⢲⠖⠂⠠⠤⠤⠜⠸⡄
⡇⠀⠀⠀⠀⠀⠈⠒⠤⢄⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷
⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⣀⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠃⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡄⠂⠉⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⡀⠘⢦⣀⣀⣀⣀⣼⠉⠓⠢⠤⣤⣀⣀⣀⣀⡹
⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠦⠀⠀⠠⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠀⠀⠀⠀⢀⣉⠽⠃
⠀⠀⣀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣿⡟⡉⣻⠟⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣾⠀⠁⠠⠀⠀⠊⠀⢀⡴⢋⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠂⠀⠀⡟⠀⠀⠁⠀⢀⣀⣀⡾⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢀⡴⢛⢡⡙⣦⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠉⠙⢿⡋⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠰⣏⠰⣌⢧⡹⣜⣧⠀⢂⠀⠀⠀⡇⠀⠀⣴⠏⠀⠀⢀⡼⠃⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢷⡙⢦⠻⠍⠘⡇⠀⢂⠀⠀⣿⢠⡾⠁⠀⠀⢢⡞⠁⢀⠔⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠈⠀⠄⡀⠀⠙⠶⣤⣴⣾⡃⠀⠀⠀⠀⣿⠏⠀⠀⠀⣴⠏⠀⠀⠁⠀⠀⠀⠁⠀⠀⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⡾⠃⠀⠀⢈⡾⠁⠀⢀⣀⣠⣤⣤⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠶⠶⠶⠶⠶⠶⠖⠛⠛⣛⣻⣿⡄⠀⠀⠹⣦⠟⢀⣀⣠⣴⡿⠶⠞⡛⢉⣭⠤⡄⠀⢻⠀⣀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣠⠶⠟⢛⣉⣁⣸⣧⡤⠶⠞⣛⠛⢉⠉⣤⠶⣆⢸⣋⠁⠈⠓⠶⣤⠀⢸⡟⠉⠉⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⣼⠷⠞⢛⣋⣉⠉⢠⡴⠂⢠⡞⠙⠃⣸⠀⣿⣴⠏⢸⣯⡤⠀⠓⠞⠋⠀⢸⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣿⠀⠀⣿⠁⣹⠇⢸⣧⠀⢸⣇⣠⡆⠻⠀⠋⠀⠀⢄⣐⣠⣤⠴⠶⠖⠒⣾⠇⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣁⣀⣤⣤⣿⡀⠀⣿⠚⢯⡀⠸⡧⠶⠀⢉⠉⣀⣠⣤⣶⡶⠛⠋⢹⡇⣀⣤⡶⠶⠛⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⢸⡇⠀⠉⠀⠀⠀⣀⣠⣤⣷⣾⠛⠉⠁⣿⠀⢻⡄⠀⢸⡟⠉⠀⠀⢀⣀⣤⡤⠶⠶⠚⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠈⠷⠴⠶⠖⣿⠛⠉⠀⢰⡾⠁⠀⠀⠀⣿⠀⠀⢻⣆⠈⠓⠖⠛⠛⠉⠉⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠄⠀⢠⡞⠁⠀⠀⣰⠟⠀⠀⠀⠀⠀⣿⠀⠀⡀⢻⣷⡄⠀⠀⠠⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢿⣝⡳⣄⣼⠋⢀⣤⠶⠚⠛⢓⣿⠀⠀⠰⠈⢷⡙⣦⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣽⠷⠟⠁⠀⣿⣤⡴⠶⠟⢋⣿⠀⠀⠀⠁⠈⠻⠾⠇⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⠃⠀⠀⠠⠀⠸⣆⠀⠀⣀⣾⠇⠀⠀⠀⠀⠄⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠋⠀⠀⠠⠁⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠠⠀⢤⡤⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⢀⠠⡄⢰⠈⢢⠀⠘⠦⠜⠴⠒⠁⠀⣴⠂⠉⠈⠴⠀⠜⠀⢉⢲⡀⠰⡁⡇⢀⡏⠈⠙⠣⢌⣑⠒⠤⠤⡄
⠀⠀⠂⢈⣾⡾⠊⡡⠐⠒⠒⠤⣈⠀⠁⠀⣿⠄⣀⡀⠀⠀⠀⠀⣀⢠⡟⠀⠏⢀⡞⠀⠀⠀⠀⠀⠀⣹⠀⢠⠏
⠀⠀⠨⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠙⢢⡀⢯⡠⠄⠀⠀⠉⠉⠀⠀⠠⡇⠀⢠⠎⠀⠀⠀⠀⠀⡰⡫⠂⠀⡎⠂
⠀⠀⠀⠀⣞⠀⠀⠀⠀⡴⠒⠒⢤⡀⠀⡇⢀⣨⣅⠀⠒⠂⠀⢀⠐⠈⠀⠀⠸⣶⣤⣀⠁⠀⢰⢱⠁⠀⢠⢹
⠀⠀⣄⣀⠘⢦⡀⠀⠐⡇⠀⠀⠀⡇⢠⠇⡎⠀⣸⠇⢠⠔⠢⡀⠉⠉⢧⣀⣀⣀⣉⠙⠯⣶⡣⠻⠭⠭⠥⠂
⠀⢸⡁⠈⠳⣀⠙⢦⠀⠈⠒⠓⠋⢠⠎⠀⢙⣋⣁⡀⠈⠢⠤⡞⠀⠾⠽⣧⠤⠤⠔⢛⠀⠀⠀⠀⡰⠊⠓⡄
⠀⠘⣧⠀⠀⠈⠑⠦⣑⠢⠤⠴⠚⣡⠒⠉⠉⠁⠀⠉⢙⡆⡀⠀⠀⡇⠀⢸⣄⣀⣀⣸⠀⠀⠀⣰⠁⠀⠀⠘⡀
⠀⠀⢹⣆⠰⡋⠉⢢⠈⢣⠀⠀⡞⠀⢧⠀⠀⠀⠀⠀⡸⠀⠈⠱⡄⢸⠺⠶⢿⡖⠚⢻⠀⠀⠀⡟⠒⠒⠒⠒⡃
⠀⠀⢸⡘⣄⠑⠢⠞⠁⣸⠀⢰⠳⣇⢈⣆⡬⠥⠤⠤⠧⣄⣨⠊⢣⠘⡆⠀⠈⢧⠀⡏⢀⣤⠀⠈⠻⠶⠶⠚
⠀⠀⠀⠑⠮⣽⣶⡶⠞⠁⠀⠘⠶⠚⠉⠀⠀⣄⣀⢀⠀⠀⠈⠲⠚⢀⣙⠲⠶⠒⠚⠁⠹⠤⡇⠰⣖⣃⣘⡶⠄
⠀⠀⢠⠤⢤⡔⠒⣒⡈⠉⠙⢢⠊⢹⠀⠀⢀⡱⠽⣿⣀⡤⣄⡀⠀⣾⡻⢿⣖⣒⣒⣒⣒⠦⣌⠉⠒⠦⣄
⠀⠀⠀⣧⠘⠻⣶⣤⣀⠀⠀⠀⠡⡀⡇⢀⡏⠊⠀⠀⠀⠀⠀⡙⡄⠈⠻⠿⠶⠿⢿⣭⣭⣟⡾⣵⠄⠀⠈⡆
⠀⠀⠘⢏⠁⣂⠻⠮⣿⠳⣦⣄⡀⠐⣍⠀⣇⠀⠀⢠⠠⡀⠀⢀⠇⠀⣯⣕⣒⣒⣒⣂⠤⢍⠻⠃⠀⡠⠤⠧⡄
⠀⠀⠙⢦⠉⠺⢵⡤⣈⠒⠄⢉⠋⠵⠏⠀⠘⣆⢄⠀⠀⠀⢀⡜⠀⠘⠷⣿⣓⣲⣈⡩⠉⠲⡆⠀⢠⢧⠄⢴⡇
⠀⠀⢈⣿⣦⣖⡲⠈⠪⢕⠦⣥⠜⠀⣀⠤⣀⠈⠙⠛⠛⠛⠉⢀⣤⠖⠒⠒⠒⠦⣄⠉⠳⠬⠇⠀⠙⠺⠶⢯⠇
⠀⠰⠟⠋⠀⠀⠀⠀⠀⠀⠉⠀⣔⠋⠀⠀⢀⣉⠒⠤⢀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⡳⠀⣀⣤⡴⠶⠶⢤⡀
⠀⢰⢺⠳⡶⠶⠿⠿⠟⠚⡃⠀⣳⡄⠀⡀⠙⠛⠉⠙⠒⢳⠀⢯⡦⣄⠀⠀⠀⠀⢹⠁⡼⠡⡚⣀⠤⣄⠀⠙⡄
⠀⠈⠻⣇⠁⠀⠀⠀⠀⡸⠀⠀⣯⠀⠘⠛⠀⢠⢦⡄⠀⢳⠀⠀⠙⡾⡄⠀⠀⠀⠘⡄⣇⡔⠁⢧⣀⣿⢃⡀⡇
⠀⠀⠘⠯⣷⣤⣤⣤⡼⠕⠀⠀⠧⢤⣄⣀⣠⠬⣉⣀⣀⡼⠀⠀⠀⢧⣇⠀⠀⠀⢀⡇⠘⢮⣒⠲⠔⠲⣪⠟
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣷⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢇⢸
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠈⢿⣆⠀⣀⠴⣿⠀⠀⣴⡀⠀⣴⣤⠏⢸⣧⠔⢻⡄⢀⣷⣰⠃⣿⣠⠞⣿⡆⠀⣼⠃⡼⠀⠀⡽⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡠⠻⣿⡄⠀⣿⡆⣰⣿⡇⠀⣸⡏⠀⢸⡟⠀⣾⡇⢸⣿⠃⠀⣿⠏⠀⣿⡇⢰⣿⠁⠀⢀⡼⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢹⣿⠀⠙⠛⠁⠹⣧⡴⠛⠃⠀⠘⠃⠀⠹⣧⠜⠋⠀⠀⠀⠀⠀⠻⣧⠞⠿⣦⡴⠞⠀⠀⠘⢿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢠⡏⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⣀⠀⠀⣼⡇⠜⠁⠀⠀⢠⡾⢹⠆⠀⢿⣧⡟⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢹⣷⠀⣸⠋⢩⣿⠀⠀⣿⡧⣆⠐⣆⡀⣿⡟⠁⠀⠀⢈⠟⢿⣦⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡿⠀⢸⣿⠀⢀⣿⡇⠀⢸⡏⠀⢿⡇⠀⢀⡴⠋⣠⠊⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠛⠒⠋⠘⠿⠶⠋⠀⠀⠈⠙⠚⠉⠀⠘⢧⣤⣾⠟





⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣠⣤⣤⣤⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣏⢽⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠿⠿⡿⠭⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣬⣤⣤⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢨⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠦⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠘⣿⣏⣏⣿⣷⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠘⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠇
⠀⠀⠀⡀⠀⠀⠘⠿⠿⠟⠛⠋⣉⠉⠄⠐⢀⣈⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠭⠑⡤⠤⠀⢤⡀⠀⠀⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠇⠂⠀⢐⠂⠛⢢⣻⢤⣖⡘⠃
⠀⠀⠠⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠢⢰⠄⠀⠀⠃⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣽⡇
⠀⠀⠰⠂⢸⣷⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⡥⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠐⢉⠉⢁⠁⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢙⡇
⠀⠀⢀⡄⠀⠿⣷⣧⣄⣀⠀⣀⣨⣭⡶⠶⠾⠛⠛⠋⠁⠀⠀⠀⠀⡀⢱⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡼⠃⠀⠈⡄⠀⣼⠁⠁
⠀⠀⠀⠇⠀⠀⠀⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠱⣄⡀⠀⡀⠀⢀⡄⠀⠀⠀⠀⢰⡶⠶⠀⡰⠀⢰⣿⠶⣶
⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠌⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⡆⠀⢀⡜⢁⠀⣾⡿
⠀⠀⠀⡂⠀⠀⣤⠤⠄⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⢿⠀⠀⠰⠀⠀⠀⠀⠀⠰⠀⠀⢀⠠⡀⠂⠀⠀⣰⣟⠛⣁⠌⢠⡿⡟⠁
⠀⠀⠀⠂⠀⡿⣿⠀⠀⠄⠀⠀⠠⠄⠠⠌⢰⠶⠤⠀⢿⠀⡀⢈⠀⠀⠀⠈⠀⢠⠀⠀⠰⠀⠀⠀⠀⣰⣿⠏⠻⠋⠐⠉⠁⢴
⠀⢀⢤⡆⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠑⠿⠿⠷⠷⠿⠶⠦⢴⡀⠀⠘⠦⠦⢀⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠰⢿⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣼⣷⣠⠄
⠀⠸⣾⡇⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠁⢰⠚⠀⢠⣿⡗⠂⠀⠀⠀⠀⠀⠁⢀⠸⣽⣿⣧
⠀⠘⠻⠇⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⠀⠀⠀⠠⠼⠸⠸⠀⠈⠀⠀⠐⠃⠀⠀⠀⠀⠴⠀⠀⠠⠿⠴⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⢀⡟⠀⢻⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡆⢸⡇⠀⢸⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠤⠄⢀⠸⠿⠋⠲⠦⣈⠃⡄⠀⠀⠀⠀⢰⠀⢰⠀⢴⠀⢲⠀⢴⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⠂⣠⡔⠘⣀⠂⠀⠘⢆⢱⠄⠘⠆⠀⡁⠀⠸⠆⠸⠆⠸⠄⠾⠀⠖⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣼⡟⠃⠁⠀⠀⠀⠢⠀⠈⢻⣇⠀⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢰⠏⠘⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠓⠓
⠀⠀⠀⢠⡟⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣨⠉⢙⠠⡄
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⡀⠀⠀⠀⠀⡼⢁⢚⡥⠄⢸⠀⠛
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡆⡞⡴⠉⢤⠀⠀⠀⠀⢄⠸⡀⠠⠞⣠⠚⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⠀⠰⢰⡀⠇⡴⠁⠸⠆⠀⠀⠀⠈⠓⠐⠷⠶⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠣⠰⣄⠳⠘⣁⠜⣀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣰⡆⠀⠀⠀⠀⠒⠲⠮⢙⣀⡥⠌⣁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡇⠠⠀⠀⠀⠀⠙⠛⠲⠄⠀⠘⠁⠀⠀⠀⠀⠀⠀⠐⢀⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠀⠀⢄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡐⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨

⣿⠋⠉⠛⠉⠙⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢏⣍⠛⠛⢿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⡇⡷⣸⠈⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠁⢙⠀⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⠐⠐⠀⠃⠲⠑⠈⠻⠁⠜⠀⡰


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠀⠀⠐⠀⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠠⠂⠠⠤⠒⠄⠄⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿