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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠂⠧⠤⠞⠁⠀⢀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀
⠀⠀⡠⢀⠀⠃⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⢴⠾⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠘⢈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⣀⠄⠐⠀⠀⠉⢁⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠀⠠⠴⠒⠈⢘⡉⠀⠀⠀⠐⠂⠈⠉⠉⠀⠀⢀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠄⠀⠀⠈⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⡠⠄⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡠⠀⠀⡄⡀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠰⡀⢢⡀⠀⣀⢀⠤⠐⠂⢘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣀⠤⠀⡂⠀⠀⠀⡆⠀⠠⠤⠐⠴⠂⠆⠐
⠀⠀⠀⠐⠄⠀⢀⠘⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⡡⠀⠀⡇⠆⠀⠀⠃⠀⢀⣴⠄⠀⠀⠀⢰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠋⠡⠀⠀⠠⠀⠀⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠂⠀⠀⠀⠚⠉⠁⠀⠀⠤⠄⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠀⠀⠀⢘⢈⢸⡬
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⠀⠀⠁⠈⠉
⠀⠀⠀⠂⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠜⠐⠂⡈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠈⡀⠆⠀⠀⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠁⠀⠀⠀⢐⠀⠀⠀⠀⡂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⡀⠁⡄⢀⢈⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡞⡉⢳⡄⠀⠀⠒⠀⠀⠀⠀⠀⠠⠤⡄⠴⠦⠀⠄
⠀⠈⠑⠤⠁⠀⠠⠔⠀⠀⠇⠀⠀⢀⠈⠒⠀⠂⠶⠒⠒⠚⠛⡆⠀⠀⠀⠀⠐⠀⠀⠀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠈⢁⢀⠁⠒⠤⡀
⠀⠀⠀⠀⣀⡂⠂⠀⠀⠀⠀⠀⠀⠘⣄⠋⠄⠴⠌⡃⠉⠀⢠⡇⠀⠀⠀⠀⠐⠁⢇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠌⠀⡤⠤⠀⢠⠀⠄⠄⢸
⠀⢠⠀⠀⣘⠑⠀⠀⠀⠔⡃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠒⠒⠀⠀⡠⠊
⠀⠀⡀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠄⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠑⠂⠂⠀⠀⠀⠀⠀⠀⠀⢤⡀⠉⠑⠒⠚⠊⠉⢀⡠
⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⡲⣄⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⣀⢀⡀⠐⠂⠀⠀⠔⢁⡄⢀⠀⠀⠿
⠀⠀⠂⠀⡤⢈⠀⡀⠀⠤⠀⠀⠀⠠⠀⠈⠐⠀⣠⣄⠄⠀⠀⠀⢈⠀⠀⠀⠀⠰⣇⡠⠙⠛⠨⠚⠁⠆⠀⠈⠄⠀⠈⠂
⠀⠃⠐⠀⡀⠘⢀⠁⠀⠀⠉⠀⠀⠠⠀⠀⠀⠈⢀⡀⠱⠤⡅⠀⢀⠀⠀⠀⠀⢸⠛⠿⣷⣦⣱⠈⠀⠀⠁⠀⡀⠑⠀⠞
⠀⠰⡆⠀⠀⠣⠀⠀⢂⠀⢀⠤⠍⠐⣀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠁⠀⠀⠀⠀⠻⣷⡢⢄⠉⠛⠈⠀⠀⠀⠈⠋⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠁⢀⡁⢤⠀⠀⠀⠄⠀⠀⠈⠀⠘⠻⠿⠀⠀⠀⠀⠀⠀⣀⠈⠛⢷⡽⢀⠁⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠂⢀⠃⠀⠒⠀⠂⠀⠀⠂⠀⠀⠀⠀⢠⠀⠄⠖⡀⠈⠐⠂⢁⠐⣄⠘⠛⠀⠀⠈⠀⠀⠀⠀⠄⠀⠠⠀⠀⠠
⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⡀⠀⠀⠀⠁⠁⠒⠂⠀⠠⠀⠀⠀⠀⠀⢡⠀⠀⠀⠂⣠⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠
⠀⠀⠁⠀⠴⠆⠠⢠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠢⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠠⠀⢰⢹⡿⡏
⠀⠀⠄⠀⠀⠀⠀⠀⠙⢖⡂⢁⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⣤⣤⣤⣤⡀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⣸⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⡠⠀⠀⠀⣰⠻⢿⣿⣿⣷⣭⣭⣏⣨⣿⣿⣿⣿⠛⡆⠀⠀⠆⠀⠀⠀⠀⡔⠀⢿⠀⠀⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠙⠀⠀⠀⢿⢘⣻⠻⠛⣿⢸⣿⡇⠾⢛⠛⡍⠿⡃⡇⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣀⠀⠘⢼⣧⣄⣷⣾⣬⠿⠧⠿⠶⠿⠾⠷⢖⣁⠀⠀⠠⣀⠀⠀⡀⠴⠀⠀⠠⠢⡀
⠀⡐⠁⠄⠂⠀⢄⠀⠁⠀⠠⠀⢠⡾⠻⠎⠁⠠⠀⠀⠀⠤⢄⠀⢤⠀⠀⠀⠀⠊⠀⠌⠐⠀⠀⠤⠀⠘⠂⣀⣠⣤⣤⣤⠁
⢰⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠁⢳⡤⢈⢤⠀⠂⡀⠀⠀⠀⡁⠁⠀⠀⠀⠀⠀⢀⠀⠂⠔⠀⠀⠀⠀⠀⢰⡷⠲⠴⠞⣩⡷
⠈⡀⠀⠂⠤⠀⠁⠀⢠⠃⠀⠀⠐⢀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠑⠀⠉⣠⠂⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠰⠀⠀⡔⠡⠘⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⢠⠓⠄⠔⠲⠒⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡰⠀⠄⠁⠀⠀⢪⠒⢹⠂⡄⠀⠑⠲⠷⢦⣤⡤⠠⠜
⠀⠀⡀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢨⠉⠁⠀⠀⡀⠘⢐⠀⠀⠅⠀⠀⠰
⠀⠀⠙⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠙⠃⡀⠀⠀⠒⠀⠈⠀⠒⠀⠀⡀⠀⠀⢂⠀⠀⠒⠀⢀⠈⠀⠢⠤⠂⠀⠂
⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠃⠀⠀⠁⠀⠰⠀⠀⠀⠀⢂⠀⠀⠀⠈
⠀⠀⠀⠀⣠⣄⡈⠘⠲⣄⣄⠞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠿⠻

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠄⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠
⠀⠰⠀⠀⠀⠀⠀⠀⢑⣄⠉⢠⠁⠔⠀⠈⢢⠀⠀⢀⠇⠠⠔⠀⠠⠝⠀⠁⠀⠀⠀⠈⠀⠄⠀⠐⠀⠐⢈⠀⠩⣎⡀⢀⠄⠒⣢⢠
⠀⠀⠀⠀⠠⠄⠒⠀⠒⣆⠀⠈⢆⠀⠰⠠⡀⢣⠀⠢⣀⣤⠠⠄⠀⠀⠀⠄⠀⠀⡀⢀⠀⠀⠁⠀⡀⢁⠎⠀⠤⡌⠀⠙⠒⠒⠊⠉
⠀⠀⠀⡦⠄⠀⠀⠀⠨⠐⠀⠀⠀⠑⠠⡐⢈⠐⠀⣀⢸⢞⠵⠀⠀⠀⠠⡡⠤⠀⠇⡀⠀⠁⠀⠀⡰⠁⢎⡀⠀⠀⠀⢀⠀⠀⣀
⠀⢸⡀⠈⢈⠀⠀⣀⠴⠲⢙⠌⠜⡐⠢⡀⠀⠀⣾⠆⠀⣧⣟⠀⢀⣢⡌⠀⠀⠘⠈⠀⠇⠐⡠⢊⠀⠀⢘⠈⠉⠉⠀⠀⠀⠀⠀⢤
⠀⠀⣀⡀⠈⠀⡘⢑⣠⡀⣄⡄⠰⣔⡢⠽⠀⣀⣁⡀⡀⠘⡇⠀⠀⠈⠒⠠⠤⠻⠑⠀⠀⠁⠀⠈⠜⢀⠈⠁⠉⠀⠰⠀⠁⠆
⠀⠀⡇⠘⣆⠈⠐⠂⠀⠊⠐⢒⣉⡠⠔⠁⠸⠇⠀⠈⣡⠂⢀⠀⠀⠀⢀⢈⠁⠀⠀⣀⣠⣤⣤⣌⠀⠙⠀⠀⠑⠀⠀⠀⠀⠄⣀
⠀⠀⡇⡔⠈⡆⠀⠉⠈⢉⠉⠁⠀⠀⠀⠀⢺⡀⠀⠀⠟⠀⠘⢐⡀⠐⠁⠈⠀⠀⠺⠿⣯⠿⠷⠽⢧⠀⠆⠀⠀⠆⠀⢰⣅⡠⠂
⠀⠀⠣⡈⣠⠇⢀⠀⠈⠧⣈⠁⠐⠈⡩⠃⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠤⠔⠁⢀⣀⠀⠤⠂⠁⠠⠒⠁⠀⢀⣀⠀⡀
⠀⢀⠀⡀⠀⡐⠑⠄⠈⠆⠀⠀⠈⠀⡀⢀⡄⢴⣯⠒⡄⢀⢔⡀⠄⠀⠠⢏⠝⠀⠀⣀⠀⠀⣀⢀⠀⠀⠔⠈⠄⠀⡴⠁⠈⣆⣄⡅
⠀⠓⠒⠀⡐⠀⠀⠀⠀⠀⣀⠀⠨⠌⠰⠖⠘⢁⡠⠊⠔⠗⠐⠠⢤⡶⠂⠀⠀⢄⠀⢠⡄⠀⢀⠀⠀⢁⠀⠀⠀⠐⠠⢀⠔⠁⠀⠁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠚⠀⠐⠂⠀⠒⠈⠀⣀⣩⣄⠦⠔⠛⣡⡀⠀⠀⠀⢀⠇⠀⢀⠀⠮⠩⠠⠍⢡⠘⠂⡈⢒⣈⠦⠀⡀
⠀⢀⠀⠆⠊⠣⡀⠀⠀⠀⡰⠀⠤⠀⠀⠘⠃⠀⠉⢠⡄⠀⠀⣚⠭⠁⣄⡀⠐⢁⡀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠁
⠀⠘⠦⠀⠀⠀⡤⠀⠀⠀⠚⡟⠠⠈⢀⡴⠖⠋⠃⠚⣳⣠⡄⠈⡀⠄⠈⠀⠀⠈⠁⠀⣰⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠖⡄⡠⡄
⠀⠠⢄⡈⠡⠊⠀⠃⠀⠀⠀⠈⠂⠀⠈⠀⡀⠀⠀⠀⠀⡀⠀⠈⠐⠄⠂⡄⠀⠂⡬⠂⠀⠁⠐⠒⠒⠒⠒⠊⠉⠃⡀⠀⠀⠀⠀⠃
⠀⠐⠀⡀⠀⢁⠀⢊⠔⡠⢄⠐⠀⠠⠄⠀⠀⡠⡴⠄⡈⠀⠀⢠⠀⠀⠀⢀⠋⡀⢄⠀⢠⠆⠀⠀⠀⠀⢀⢀⠀⠸⣰⠀⢨⠙⠈
⠀⠀⠂⠒⠈⠀⠀⠘⡌⠃⠈⠀⢑⠀⡄⠄⠠⡀⠀⠀⠀⠀⡀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠪⠩⠉⢬⡇⠈⠀⠠⡀⠀⠀⠂
⠀⢀⠀⡒⠒⡋⢆⠀⠣⠠⠊⠀⠁⡐⠁⣀⠄⠀⠀⠀⡐⠀⠀⠻⠀⠈⠐⠂⠀⠀⢀⠤⠊⠁⠀⢀⠤⠀⠘⢲⠔⠁⠀⠰⣶⢂⠡⠄
⠀⠈⠀⠀⡄⠀⢈⠀⠀⠊⣉⠢⠀⠈⠁⠃⠃⠀⠢⠀⠀⢀⣀⠄⠀⠀⠀⠃⢀⠄⢃⠀⠀⠠⢀⠀⠀⢀⡠⠃⡆⠀⠀⠄⠋⠉
⠀⠀⠈⠀⠉⢀⠀⠀⠀⡈⠀⢠⠀⢀⠀⠀⠀⠀⢀⣀⣀⠉⢀⠀⠀⠀⠕⠀⢘⠠⠂⠉⠁⠉⠡⠊⠛⠩⠐⠊⠀⣐⠀⠀⠀⠀⢠
⠀⠀⠀⠄⠀⠀⠀⠀⢀⠰⠀⣀⠉⠂⢠⠔⢠⠸⡁⠀⡩⠆⠀⡜⠀⠈⠀⠀⠀⠀⠀⠀⡴⣲⠀⠔⠢⢠⠐⡆⠀⠀⠀⢀⣀⡀⠀⠁
⠀⠠⠴⠀⢁⠈⠀⠀⠀⠑⡃⠀⢀⣀⡀⠙⠉⠀⠥⠤⠧⠀⠠⠀⣀⠈⠂⠒⠃⠀⢀⠀⠈⠁⠀⠂⠁⠀⠘⠀⠀⡚⠃⠀⠀⠀⠑⢲
⠀⠀⡴⡺⠀⡼⠃⢀⣄⡀⢀⠄⠁⠀⢘⠀⠀⡠⠐⠀⣂⣄⠀⠀⠈⠀⢠⠄⠀⠀⠈⠠⠄⠂⣤⣴⣴⠀⠀⠀⠀⠱⡀⠀⠀⠀⡠⠋
⠀⠐⣇⠤⠑⠁⢠⠀⠀⠀⡁⠀⠀⡠⢊⡀⠈⢠⠊⠁⡀⠀⠂⠤⠤⠄⠊⠀⠠⠀⠁⢀⣀⡠⠂⠀⠠⠀⠀⠀⠀⢀⠁⠀⠀⠀⡀⢴
⠀⠀⠈⠐⠒⠂⠚⠀⠀⠀⠈⠀⠀⠀⠈⠀⠘⠁⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠉⠉⠁

⠀⠧⠀⠀⠀⡠⠒⠉⠀⠀⠁⠐⢀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡈⢡⠀⠰⠁⠀⠀⠀⡀⠀⠀⠀⠡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢧
⠀⠀⠀⠀⢈⠀⠀⠀⠈⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠓⠦⣤⣤⠴⠋⠀⠀⠄⠀⢀
⠀⢀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⡠⢃⣀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢈⠦⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⢀⠀⠀⠀⢸
⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠂⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢀⡀⠀⢀⠠
⠀⠀⠀⠀⢀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠄⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢂⡶⠛⠲⡄⣴⠚⠳⣆⢰⡗⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⢧⣀⣰⠇⢷⣄⣠⠟⢸⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢠⡀⢀⡤⣄⢀⣀⠀⣠⢤⠀⣤⣄⢀⡤⡀⢠⢀⠀⣄⢠⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂⠞⠳⠸⢤⡤⠸⠣⠀⠧⡶⠆⠿⠇⠻⣤⠟⠸⡼⠀⠏⠾⠸⡦⠇

⠀⢀⣀⣀
⠀⠘⡔⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠄⠀⠀⠀⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠠
⠀⡡⠂⠀⠑⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠑⠂⠤⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡠⠀⠀⠀⡠⠀⡀⠀⠀⣀⠀⠀⠀⠲⢁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠔⠁⠀⠀⠀⠀⠉⠐⡄
⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⢀⣠⠴⠶⠦⣄⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣄⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢳⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⠀⠀⠀⠀⠁⠀⠀⠀⡘⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢐⡀⠑⠄⢀⠀⠀⢀⠠⠞⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⣀⡼⠉⡆
⠀⠀⠀⣠⢿⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣀⡠⢼⣃⣰⣇⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⣀⡠⠖⠒⡄
⠀⡰⢦⣉⣹⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⣼⡸⢀⣖⣛⣦⣝⡞⣇⠀⠀⠀⠀⠀⣎⠀⠀⢀⣡⠤⣄⠀⢳⡀
⠚⠓⡏⠀⠈⠳⢤⣀⠀⠀⠉⢑⣞⠀⠀⠀⡿⠷⠽⣭⣤⣵⠎⠀⠸⡄⠀⠀⠀⠀⠈⣳⠀⡏⠀⠀⠈⡇⠀⠙⢦
⠀⠀⠙⢄⠀⠀⠀⠉⠛⠛⠻⣅⣬⠀⠀⠀⠸⡄⠀⠂⣠⠋⠐⠀⠀⢳⠀⠀⠀⢠⠞⠁⠀⠳⢤⣠⠴⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠴⢧⡞⠀⠀⠀⠀⠙⣄⣰⠃⠀⠀⠃⠀⠈⣇⠀⠀⢸⡀⠀⢀⣠⠤⠤⣤⣀⡀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡄⠀⡀⢸⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠂⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠰⢺⠻⠛⠀⠀⠀⠀⠀⠀⠸⣙⡆⠀⠙⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⣀⠖⢤
⠀⠀⠀⠀⠀⠀⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠳⡀⠀⠀⢠⢂⡀⠀⠀⠀⣠⡀⠀⠀⠘⢧⡼⠿⣼⢷⡛⣶⢦
⠀⣠⠶⠒⣒⣚⣛⣒⣒⣒⣲⡄⠀⠀⠀⠀⣞⣓⣒⣛⡇⠀⠸⡘⢄⡤⣄⣴⠟⠁⠀⢰⣟⣚⠣⣜⣹⢸⢷⠈⢧
⠀⠉⣏⠉⠉⠉⠉⠉⠉⢉⡇⠀⠀⠀⠀⡔⠉⠀⠀⠀⠙⢦⠐⣁⣼⣆⡀⣻⠀⠀⠀⠀⢠⣼⣧⣈⣳⣜⣸⡧⠴⡗
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⣸⣅⠀⣹⠉⠁⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠈⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⡏⠉⢻⡝⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⡴⠋⠙⢻⡗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠈⠓⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠳⠶⢦⣾⠦⣦⠴⠁
⠀⠀⠀⣀⣤⡤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠓⠒⠦⣄⡀
⠀⠀⡰⠛⠩⠉⠓⠦⡛⢺⠷⡶⢤⡀⠀⠀⠀⠀⣠⠖⣋⠉⡉⠏⣓⠦⡀⠀⠀⠀⠀⠀⢸⡚⠦⢤⣞⠀⠀⠉⠳⣄
⠀⢸⠀⠀⠀⠀⠃⢀⠈⢮⡑⡿⣢⢹⡀⠀⠀⢰⡁⠆⠈⠀⢁⡄⠁⠴⠙⡆⠀⠀⠀⠀⢸⡙⣦⠀⠬⠙⠻⣄⡠⠈⢳
⠀⠈⢦⡁⠐⠂⠀⡀⠀⠆⢳⠉⣿⣾⠇⠀⠀⠀⣟⣓⣒⡒⢚⣒⣶⣒⣲⣟⠀⠀⠀⠀⣼⠕⠃⣔⣭⣽⣿⣢⡵⠖⠚
⠀⠀⠀⠈⠓⠦⣀⠁⠀⠄⠈⣷⠷⡟⠀⠀⠀⠈⢯⠥⠤⠾⠿⣶⡶⠿⠷⡾⠀⠀⠀⠀⡷⢤⣈⣯⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣷⠞⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⣀⡽⠦⠞⠁⠀⠀⠀⠀⠷⠚⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⠤⠀⠀⢀⠤⡤⣠⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⢀⣀⠶
⠀⠀⠀⠀⠀⠈⠀⠀⠀⡗⠒⠒⠒⠺⠀⠀⠀⠀⠀⠀⠀⢠⣪⣀⡊⠂⠀⠀⠀⠀⠀⠀⢀⡤⠒⠲⠃⠤⠉⢁⡸⠵⠖⠂⡩⣠⣞
⠀⡰⢧⣤⡤⠄⢤⡤⢤⣿⠐⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢛⠀⠀⠀⠀⠀⢠⠓⣧⠐⡤⠕⡲⠋⠁⡀⣀⣠⡴⢊⡡⡏
⠒⠁⣉⠯⠥⠤⠬⠭⠽⠣⠤⠦⢄⣈⠀⠂⠀⠀⠀⠀⠀⠈⠀⣉⠀⠈⠀⠀⠀⠀⠀⠘⣄⠙⠿⢷⣩⡇⣰⣪⡭⢔⣿⣉⠭⠚⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠒⠂⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣙⣉⣙⣒⣉⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⣀⣄⡀⢀⠤⢪⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠢
⠀⠀⠀⢻⣯⡟⠉⠂⠄⠨⠗⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠔⠒⢤⡰⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠤⠯⠬⠦⡀
⠀⠀⠀⣠⢧⣀⣈⣀⣃⣠⣀⣸⣄⠀⠀⠀⢴⠒⢒⢢⣬⡿⠛⠐⠒⣎⡤⣸⣄⢔⡞⡆⠀⠀⠀⠀⠀⡏⠀⠂⠀⠀⣼⡀
⠀⠀⠀⠘⡗⠒⠒⠒⡖⠒⠒⠚⡏⠀⠀⠀⠈⣶⢞⢫⡶⣧⣀⡀⠔⣹⣏⢥⡸⢻⣶⠀⠀⠀⠀⠀⠀⢹⠿⠯⠭⠽⠏
⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⢀⡇⠀⠀⠀⢸⣼⡳⠮⢮⣥⠊⠟⠋⡷⢤⣤⠞⢋⣼⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⡀⡇
⠀⠀⠀⠀⢳⣀⡀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠈⢦⡙⠓⠶⠦⢽⡀⠀⣰⠯⠶⠚⡩⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢲
⠀⠀⠀⠀⠀⠙⠭⠭⠩⠭⠝⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⢤⣽⣿⡥⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡀⣸
⠀⠀⠀⠀⣀⣠⣤⣄⣀⣠⡄⢠⣄⣠⡀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠋
⠀⠀⡴⠋⢒⠀⢀⣬⡿⠟⣠⡮⢚⡴⠗⠀⠀⠀⠀⠀⠀⣈⡷⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⣸⠀⣰⣨⣾⠟⠁⠤⠴⡦⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⠉⠉⠑⢦
⠀⣿⣞⡭⣻⢾⡀⠘⠥⡤⠖⢀⣴⣿⠀⠀⠀⠀⠀⠀⡎⡟⠙⠛⠉⠙⡏⡇⠀⠀⠀⠀⠀⠀⡀⠀⢠⣅⣀⣀⣀⣀⣨⡀
⠀⣿⡳⠿⣿⣶⣿⣶⣶⣶⣿⠿⠛⡟⠀⠀⠀⠀⠀⠀⡇⢳⡀⠀⠁⣠⠃⡇⠀⠀⠀⢠⢤⠏⢀⡀⠈⠂⠀⠀⠀⠀⠘⡁⠀⠈⡤
⠀⠘⢿⡦⢤⣀⣀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠙⠦⢭⡿⣯⠥⠞⠁⠀⠀⠀⠈⠉⡇⢾⣿⣶⡶⠀⠀⠀⠀⠀⡠⠀⢀⠁
⠀⠀⠀⠙⠲⠦⢤⣤⡤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣶⣾⡦⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡀⠠⠀⠜⠠⠆⢒⡀⠀⠀⠀⠀⠀⠀⠘⠀⢘⡆
⠀⠀⠀⠠⠀⠁⢲⠢⡠⢨⠀⡦⠀⠀⠀⠀⠄⣤⣤⠞⢷⣴⣦⡤⣄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡔⠃
⠀⠀⠀⠆⣹⠀⣠⡀⣁⢏⢐⢃⠀⠀⠆⠀⠀⠈⠀⠀⠀⠁⠀⢿⣶⣤⠀⠀⠉⠂⠒⠄⠄⠄⠤⠄⠊⠉⠀⠀⠀⠑⠠⡤⢤⡄⠐
⠀⠀⠀⠀⠀⢀⡘⣥⣄⡐⠋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠙⢤⡍⣹⠉⡓⠐⠒⠐⣆⣹⠏
⠀⠀⠀⠈⠐⠌⡃⠉⠈⠠⠰⠀⠀⠀⣄⠀⠀⡀⠀⠀⠀⠀⣰⣷⣿⡏⠀⠀⠀⠉⠙⠓⠻⠒⠛⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢄⡀⣷⡀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣷⣷⣻⣿⣿⣿⠏⠀⠀⠀⠆⠀⠙⠉⣃⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⡤⠒⢲⣦⡀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢤⣄⣄⡼⠀⠀⡜⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢠⠃⠀⠀⢹⡇⠀⠀⠀⠀⢠⣨⠀⠀⠀⠀⠀⣄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠤⣾⠁⠀⠀⠈⠣⣤⣄⣠⣤⣼⡿⠃
⠀⠀⠀⠀⠘⡀⠀⠀⣾⠃⠀⠀⠀⠀⠘⠽⠀⠀⠀⠀⠀⢿⣿⣶⣿⡿⢃⡾⠀⠀⠀⠀⠀⢀⣼⠂⠀⠀⠀⠀⠈⠉⠋⠉⠁
⠀⠀⠀⠀⠀⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠥⠭⠖⠋⠁⠀⠀⠰⠂⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡀
⠀⠀⠀⠀⠀⢸⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠈⠀⢀⡿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠐⣨⡷
⠀⠀⠀⠀⠀⠘⠆⢠⡇⠀⠀⠠⣦⡀⢸⡀⣠⣾⡾⠀⡠⠀⢳⣟⣦⣿⣶⣶⣄⠀⠀⡆⢰⡇⠀⠀⠙⢷⣄⡀⠀⠀⢀⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠁⢸⠁⠀⠀⠀⠈⠉⠙⠙⠋⠁⠀⠰⠉⠁⠙⢋⠎⠀⠀⠘⢻⣇⠀⠁⠘⣧⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠃⠀⠀⠀⣀⣤⠀⠀⠣⣄⠀⠀⠀⣠⣤⣴⡿⠃⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠄⣀⣤⡴⠋⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡀⠀⠀⢀⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣠⣀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⡀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣿⣷⠺⡏⢱⠄⠀⢈⠁
⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣻⣿⠀⠀⠘⠀⠀⠀⠀⠀⠘⠂⠀⠀⠁⠘⠁⠈⠁
⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⠿⠿⠯⠁⠀⠀⠈⠛⠶⠦⠤⠤⠤⠤⠴⠿⠶⠾⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀⠀⠀⠀⠎⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠀⠀⠀⠀⠀⠀⢀⣼⣀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠉⠀⠒⠂⠀⠀⠀⠀⠒⠂⠈⠉⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⢺⠤⣄⣀⣀⣀⣀⣀⠀⠀⠠⠐⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⠚⠒⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⣸⣀⡠⢄
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠤⠒⠋⠍⢀⡠⠚⣳⢦⡀
⢀⣀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡀⠀⠀⠁⠀⠉⠀⠉⠙⠳⢄⡀
⠀⢆⣿⡑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠤⢀⣀
⠀⢌⠛⡙⢷⣦⣤⣠⡤⠄⠀⠀⠀⠀⢀⣀⣀⣬⢴⣯⢠⣼⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾⡷⠤⠒⣟
⠀⢀⣷⡊⠐⠶⠒⠿⣏⣹⣛⣻⠋⢭⣍⣡⣤⣀⣉⢩⣾⠁⠀⠈⠐⢄⠀⣠⠶⣾⡿⠋⣹⣏⣉⣉⡤⣤⡴⣞⣋
⠀⢸⠩⡓⠶⣄⣀⠀⠀⠀⠈⣑⠀⢀⡰⠀⠈⠭⠤⠎⣿⡀⡀⡄⠀⠀⣼⠉⠲⡌⠀⢮⡌⢿⠑⢺⢥⡂⠌⣓⡽
⠀⠈⠳⠄⠀⣄⡈⠛⠀⠤⡄⠀⣄⠉⣀⠀⠀⡀⢠⠞⢱⠙⢤⢣⠆⢸⣿⡇⢸⣌⣵⠪⠷⣘⡆⢘⠶⠼⠷⣧⡂
⠀⠐⡀⠀⠈⠉⠉⠉⠁⠒⠚⠃⠈⠛⠁⠀⠁⠈⠁⡀⠎⠀⠀⠙⢄⣋⠹⣧⠀⢿⠮⣉⡛⢒⠯⡴⣷⠒⠒⠓⠁
⠀⠀⠈⠁⠐⠒⠒⠂⠤⠤⠤⠤⠤⠤⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠙⡄⠹⡦⡞⢡⣈⢙⠆⠑⢌⣻⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⡐⡴⢳⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂
⠀⠀⠤⠔⢀⠀⠁⢀⡎⠈⠓⠀⠀⠀⠀⠀⠀⢀⠄⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠠⠀⠁⠒⠄
⠀⢡⡀⠦⠀⢈⣒⣀⠀⢤⡆⠀⠠⠀⠀⠀⠰⡅⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⢇⣤⡍⠉⠀⠀⠀⡀
⠀⠈⢜⠧⡀⠀⠀⠀⠐⠀⠠⠀⡭⡐⢢⠀⠀⠀⢀⡀⠀⠠⠀⠀⡠⢔⠐⠀⠀⠀⠀⠀⠈⠉⡉⠀⠘⡄⠐⠆
⠀⠀⠐⠷⣌⣀⠓⡀⢀⢣⣤⣼⡭⠝⠋⠀⠀⠐⠏⢀⢄⠎⠥⣀⡈⣛⡛⠉⠉⠀⠀⠀⠀⡐⢀⡔⢀⠅⠀⡁
⠀⠀⠀⠄⠀⠉⠉⠀⠠⠈⠀⠀⠠⠄⠀⠀⠀⠈⠀⡇⢼⠁⠀⠀⢀⠉⠀⠀⢀⠀⢀⣠⠚⡀⠁⢠⠞
⠀⠀⠀⠀⠤⠂⢴⣂⣔⡒⡠⠀⠀⠀⠀⡀⠀⠀⠑⠃⣏⠀⠀⠀⠀⠀⣀⣀⣤⡶⢊⠡⢻⡥⠖⢁⢀⠴
⠀⠀⠀⠐⢦⠀⠀⠀⠙⢾⡁⠩⠢⡀⠀⠀⠀⢠⣄⡇⠁⠀⠘⠋⢉⡥⠡⠛⢈⠒⠀⠐⠂⠐⠂⢁⠔⠀⠀⠀⢀
⠀⠀⠀⢄⠀⠀⠀⠈⠐⠀⢱⠀⢟⡃⠀⠀⠀⠀⠁⠹⣤⣈⣙⣋⢁⣀⡤⠚⠷⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠘⢇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠤⠳⠄⠠⠄⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠈⠉⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⣀⡀⠔⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠂⠀⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⢀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡇⠹⣆⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⢀⡾⣫⠖⢦⠀⠀⠀⠀⠀⠉
⠀⠠⠤⡶⡄⢰⢀⡆⡆⠀⠈⣐⣒⣒⣈⡄⠀⠀⠀⣧⠀⠿⠀⢑⡆⠀⠀⠈⣀⢀⠀⠁⠀⠀⡠⢻⢟⣼⡻⠂⠀⠀⠀⠀⠀⢠⣖⠛⣦
⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢹⣄⣀⣠⠜⠁⠀⠀⠁⠉⠁⣈⠀⠀⠀⠑⠛⠾⠟⠁⠀⠀⠀⠀⠀⠀⠘⠣⠚⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡰⠆⢀⠀⠀⠰⢶⣶⣾⠶⠆⠀⠀⠠⠀⠑⠠⡀⣀⠀⠀⠀⣰⣦⣴⣶⣦
⠀⠀⠈⠘⠀⡀⠀⠀⠀⠀⠀⡒⢐⡮⠙⠀⠀⠀⠐⠷⠈⠁⡠⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣼⠃⣀⡀⠙⠂⠀⠀⠈⢉⡙⠏
⠀⠀⠀⢀⠘⠇⠀⠀⠀⠐⠆⠁⣈⣠⠀⠂⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⢋⣁⣤⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⡖⠶⢀⠀⠀⠀⠈⠙⠠⢤⡀⠀⠀⠀⠀⠰⠦⠦⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡔⠡⠐⠀⠀⠀⡄⠀⠘⠧⠀⠈⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⣀⡴⠜⠃
⠀⠀⠀⠀⢈⣿⣻⡃⠀⠀⠁⠀⢤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠷⠶⠿⠇⠀⠀⠀⠀⠀⢖⡆
⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠛⡶⡀⠀⠀⠀⠀⡄⠰⠀⣡⠄⠀⠀⠀⢀⠀⢠⠀⠀⠀⣀⣀⣀⣄⣀⡀
⠀⠀⠈⠁⠀⢀⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⣀⣐⢀⣡⠔⠋⠁⠀⢘⠀⡀⠀⠀⠈⠀⠀⠀⢏⠉⠉⠉⣡⡽⠃
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠐⡇⠀⠀⠀⠀⠛⠀⠀⠉⠚⠁⠀⠀⠉⠈⠁⠀⠀⠀⠀⠈⠀⠁⠀⠙⠁⠀⠀⠀⠀⠉⢀⡈⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢰⡆⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⡀⢬⣿⡟⠀⠀⠘⠛⠀⠀⠠⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⡆⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⡠⠈⠀⠀⠀⠀⢀⠁⠀⡀⡀⠀⠀⠀⠀⠀⠀⠐⠠⣄
⠀⠀⠀⠀⠀⠀⢀⡄⠆⠀⠀⢺⣶⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠀⠀⠀⠀⠀⡀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣸⡧⢸⠁
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠄⠀⠉⠐⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣠⡀⠀⠀⠀⡀⠐⡿⠀⠀⠀⠀⠀⢀⣀⡀⡀⢀⡄
⠀⢣⣤⣀⢀⣀⣤⡔⠀⠀⠻⣇⣰⣦⣽⡇⠀⠀⠀⠰⠉⠂⠀⠀⠀⠈⠉⠁⠀⠀⠁⢡⠀⠀⡶⢁⢴⡶⢦⡖⠢⡀⠰⣿⣝⣹⠇
⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠳⢦⡤⠴⠞⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠈⣼⣁⡴⠷⠖⠁⠀⠀⠈⠉

⠀⠀⠀⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⡀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠓⠲⡖⠀⠀⠀⠀⠈⢻⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠚⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢈⣰⣏⡄⠀⠀⠀⠀⠀⢰⣀⣀⣀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠠⠀⡂⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣈⠛⢥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀
⠀⠀⠀⠶⠀⣠⣶⠀⠀⠠⠤⠤⠌⠙⠇⠀⠀⠀⣄⣻⣿⣤⡙⠀⠀⠀⠀⠀⡄⠐⢳⠄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣤⡤⠤⠤⠶
⠀⠀⠀⠄⠘⠆⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡹⠉⣯⠩⡷⠄⠀⠀⠐⠘⠃⢸⠉⠀⠀⠀⠘⠛⣀⡀⢴⡄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠒⢤⣤⣤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢀⣀
⠀⠀⠘⣁⣤⣄⠀⠀⠀⠀⡀⠀⠀⡧⠀⠀⠀⢠⣴⡶⠟⠋⠁⠠⠀⠀⠂⠀⢢⣈⠁⠀⠀⠀⠀⠉⠉⡉⢀⠂⠀⠀⠀⠀⠠⠀⠘⡟⢠⠇
⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⣹⠟⠀⠀⠀⣲⣶⠤⠤⡷⠎⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠻⠛⢻⡟⠙⡿⠻⣟⠻⢛⣻⠟⡋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣾⣿⣾⣷⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠠⠀⠀⠠⣶⠄⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠄⢤⣄⣀⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢠⠀⠀⢀⡀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠆⡈⢁⠀⠀⢀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠜⠢⠀⡂⠀⠀⢆⠀⠒⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⡀⠡⠴⡂⠀⠀⢩⡸
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠒⠐⠁⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⢀⠀⠖⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠶⠶⠚⡠⠁⠀⠀⠀⠀⠁⠀⡐⢁⠐⠁⠀⠀⡠⠖⢦⠀⠀⠑⠦⠤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⣀⠀⠀⠀⠂⠰⠁⠁⠀⠀⣀⠀⠧⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠇⠀⠀⠀⠀⠀⠀⢑⣰⠉⠀⠀⠀⠀⢔⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠁⠈⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠎⠀⠐⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⡆⠀⡀⠀⢷⣀⡀⢀⡠⠂⠀⠀⠐⠋⠀⠀⠈⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠤⠒⠀⠀⠀⠄⡂⠇⠀⠀⠉⠉⠉⠀⢄⢠⠷⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⣀⣴⣾⣥⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠁⠀⢀⡰⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠋⠀⠀⠶⠶⠖⠀⠈⠀⠀⠀⠙⠋⠉⠀⠀⠀⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⣴⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣲⠠⡄⠀⠀⠠⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⠾⣿⣿⡦⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣥⣲⣤⠁⠀⠀⠀⠀⠀⠀⢀
⠀⢠⠖⠀⠀⠀⠁⠀⠀⣠⣘⡣⢛⠃⠀⠛⠮⠽⠇⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠟⠀⠀⠀⠀⠀⠀⠀⢨⢀⠀⠈⠀⢓⠀⠠⠀⠞⠛⢻⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡁⢘
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠁⣿⣾⡇⠀⠀⠀⢀⡵⣣⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⡁⠀⠀⠀⠀⠢⠥⠙⠃⠀⠄⠀⠀⠀⠈⠈⢁⡨⠿⠏
⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠐⠀⠁⠀⠰⠖⠀⠂⠀⠀⠀⠀⣿⣿⡷⢀⢀⣤⣶⡄⠀⠀⠀⠀⠺⠿⠀⠀⠀⢀⠉⠃⠀⠐⠿
⠀⠀⠘⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣤⠄⠰⠀⠄⠂⡀⢻⣿⠇⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠀⠠⠄⢃⠂⠴
⠀⠀⠀⠀⢉⣉⡁⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠘⠛⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠈
⢶⣶⡆⠀⠀⠛⢀⣤⣴⣆⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⡿⣽⢿⠀⠀⠀⢀
⠈⣀⠀⠀⠀⠀⠸⣄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠙⠛⠁⠀⠀⢄⠀⠀⣀⡀
⠀⠀⠀⠀⡀⠀⣤⠈⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⣶⢿⢿⣿⣶⣄⠘⠂⠀⠀⠠⠇⠘⢴⡿⠁
⠀⠀⢠⣾⣿⠘⠛⠃⠀⠂⠀⠀⠀⡲⣦⣄⠠⣠⠀⠀⠀⠀⠀⠁⠀⠀⠸⣿⣯⣴⣾⣷⣿⣿
⠀⠄⠸⣿⠋⢀⣠⡀⠀⠀⠀⠀⠀⠙⠃⢂⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡖⠀⠈⠀⣸⣿⣿⡀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣦⠀⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠁⢰⡃⠀⢿⣿⣿⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣷⡯⠀⢀⣤⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠈⠋⠘⠙⠀⠈⠻⠋⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠘⡿⠀⠀⠀⠀⠂⢀⣤⡀⠀⢀⡐⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠂⣠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠘⠋⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠸⠿⠠⠛⠀⠳⡄
⢰⣶⡀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⣀⠀⠀⠠⢲⣶⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⠂
⠈⠋⠁⠀⠀⠀⢠⣤⡄⠀⢣⠉⡗⠀⢠⠀⠈⠀⠝⠀⠀⠀⠀⠀⠠
⠀⠈⠉⠀⠀⠁⠀⠹⠃⠀⠀⠀⡀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡄⢄⠀⠰⣞⠀⠀⠀⠀⠀⠀⣤
⠀⢰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡘⠀⠀⢉⣁

⠀⠀⠰⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠀⠠⢒⠀⠀⠀⠀⠀⠀⢀⣞⠁
⠀⠖⠶⠖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡜⠁⠀⠐⢤⡹⠀⠈⡀⠀⠀⠼⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡀⠀⠀⢠⡄⠀⡠⠀⠀⠀⠀⠀⠉⠡⠀⢁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⡀⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⠀⠀⠀⠀⠀⡀⢤⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣭⡢⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠈⡀
⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⡀⠠⠀⠤⢠⠀⠀⠀⣀⡀⢠⣠⠄⠀⠀⠀⠀⠀⡀
⠈⢀⠰⢒⣶⠀⠀⠀⠀⢘⠓⢀⠿⡅⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠐⠻⠋
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⣠⡄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠀⢀⣀⣀⠀⡀⠀⠀⠀⢀⠁⠈⠀⠀⠀⣼⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠂⠀⢀⠁⡀
⠀⠢⠀⠀⠀⠀⠘⠁⠀⠰⡄⡀⠁⠀⠀⠀⠀⢉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠔⠈⠀⠈⠐⠠

⠀⠀⢀⣀
⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠆
⠀⢰⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠉⠀⠀⠀⠂⠀⠀⢠⣄
⠀⠸⠀⢸⠀⠁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚
⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠐⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠒⠋⠉⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠡⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠁⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠠
⠀⠀⠀⢤⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠙⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⠀⠒⡀⠀⠀⠤⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠐⠒⠐⠒⠐
⠀⡀⠀⠀⠀⠈⠄⠠⠴⠒⠈⠀⠀⠀⠀⠀⠀⡄⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁⠀⠀⠀⣤
⠀⢁⠠⠀⠀⠀⠀⢀⡀⠄⠠⠀⠀⢉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⢀⡂⠀⠀⠀⠀⡁⠀⢀⡀⣄⡤⢠⠠⠀⠁
⠀⠈⠠⠬⠀⠈⢀⠃⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠁⠀⠘⡁⠀⠁⠀⠀⠀⠇⠀⢀⣤⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠃⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⢛⠉⠁⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⡉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⠂⢀⡀⠸⣁⠘⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢨⢀⢀⡀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡁
⠀⠀⠁⠈⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠆⠀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠲⠠⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠠⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠘⠇⠘⠓⠂⢂⠀⠀⠀⠀⠀⠀⠤⠄⢙⣒⣂⣉⡠⠤⡆⠀⠀⠀⠀⠀⠉⠉⠈⠀⠁⠀⠀⡀⠀⠀⠐⠀⣀⠤⠀⠠⠠⠤⣀
⠀⠀⠀⠁⠠⠀⠁⠀⠀⠀⠀⠀⠸⡀⠖⣀⣄⠲⠐⢂⠀⡇⠀⠀⠀⠀⠰⢠⠂⠈⠺⠂⠀⠀⠀⠀⠀⢀⡈⠀⣀⡀⢃⡁⣀⣀⢱
⠀⠀⠀⠈⠛⠄⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠈⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠠⠤⠄⠀⢀⠜
⠀⠀⠀⠀⠁⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠒⠒⠒⠒⠒⠁⣠
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣯⣭⣽⣧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠶⠉⠉⠴⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢀⡀⢀⡀⠀⡀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⣤⣼⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠄⡀⠠
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇⣿⣷⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⣀⠸⠀⠀⠀⠀⠀⠀⠇⡭⢶⠠
⠀⠀⠀⠀⠀⠀⢹⣿⡿⠟⠀⢸⣿⣿⣿⣿⣿⣷⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⠟⠙⠉⠋⠻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠂⠀⠁⠐⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠁⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣤⣄⢀⣿⣿⣿⡀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⠸⠃⠘⠃⢀⠠⠄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠑⠶⠕⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠁⠓⠁⠐⠁⠀⠀⠀⠰⠈⠀⠄⠀⠀⠀⢠⠊⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚

⣰⣖⠀⣺⡂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠰⢀⡁⠀⠀⠀⠀⠀⢀⠄
⢨⠇⠀⢹⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⢡⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡤⣀⠀⡄⠀⠀⠀⠄⢘⠀⠢⠀⠀⠀⠀⡁⠀⠀⡀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠉⠉⠃⠀⠀⠀⠀⠦⢀⡁⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠈⠀⠀⠀⠆⠀⠀⠀⠘⠙⢐⠂⡀⠀⠀⠀⠀⡬⢁⡃⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⢀⣤⣦⢄⡀⠀⠀⠀⢈⠀⠀⢀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⢵⡅⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠋⠃⠃⠀⠀⠀⠀⠀⢐⡀
⣿⣿⠿⠿⠻⠿⡿⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢠⣶⣶⡄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡆⣈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣌⣻⣟⣁⣼⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣃⣹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⠉⠀⠀⠠⠀⠀⠀⠀⠠⢶⣦⡀⣰⣮⣭⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣯⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣴⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⠋⠏⠹⠟⠀⣿⡇⠀⠸⠏⠿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⣿⠉⡏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠻⡟⢿⡛⠛⠛⣻⣿⡛⠛⠛⠐⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⢸⣿⣆⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⣸⡇⣴⡆⡇⠀⠹⣆⠀⢻⣿⠁⣀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣿⣀⣬⣭⣄⠀⠀⠀⢀⣠⣴⣿⣷⣾⣷⣧⣀⣀⣘⣿⣿⣿⣿⣷⣿⣀⣀⣀⣈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⡇⣾⠉⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠉⠀⣿⣿⡇⠀⠀⠀⠘⠿⣯⣇⣹⣿⡿⠿⠿⣇⣇⣿⣿⡿⠿⠿⠿⠿⠿⡿⢿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠘⠃⠀⠀⠛⠙⠁⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡞⠉⢹⡄⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⢙⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣭⣤⣄⢠⣤⣤⡄⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣄⣀⣀⣀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⠀⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⣿⢸⣿⠀⡇⠀⠀⠀⠀⠀⠛⢻⣿⣀⢀⡂⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠉⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠉⠂⠀⡛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠒⢤⡀⠘⣜⣷⣦⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡿⢶⣄⠀⠀⠀⠉⢦⡈⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⣀⣼⣿⡻⢇⢤⡀⠀⠀⠀⠹⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⣛⠇⠫⠻⠏⠁⠀⠹⢿⣶⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⡀⠀⠀⠠⡤⣤⣸⡟⠋⠉⠛⢯⠈⠳⣄⡄⡀⣼⣿⣿⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠈⣡⣶⣺⡟⢶⣜⠦⣴⣿⡝⢯⣾⣺⣿⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣴⡾⠟⣿⣿⠷⢋⣙⣷⠉⠿⠻⡾⠉⠉⠀⠀⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⢀⣨⡍⠀⠜⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⡀⠀⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡳⣄⠀⠈⢛⡞⠁⠀⠀⠀⠀⢠⠾⠋⡀⠄⣠⣾⣿⣬⣾⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢠⡛⢦⣄⡀⠀⠏⠀⠀⠉⠀⣘⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡀⢷⣄⠈⠙⠛⠓⠒⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀
⠀⠀⠀⢀⡴⠒⢋⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠀⠀⠠⢹⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠄
⠀⠀⠀⡎⠤⠠⠂⠲⠅⠀⣧⠀⠀⠀⠀⠐⣿⠽⠯⠿⠽⢿⡇⠀⠀⠀⠀⠀⠐⣏⠴⣠⣄⣰⡿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸
⠀⠀⠀⢿⠀⠘⠢⢴⡁⢀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⠀⠸⡆⠡⠈⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣰⠁
⠀⠀⠀⠈⠓⠧⣤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⠃
⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢠⠖⡛⠉⠉⢉⡉⡙⢦⠀⠀⠀⠀⠀⠀⢀⣖⢒⣆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠓⠲⣄⠀⠀⠀⠀⠀⠀⣴⣞⣿⣿⣿⡶⡄
⠀⠀⠘⣎⡔⠜⠓⢼⠵⢈⠟⠀⠀⠀⠀⠀⢰⡉⠤⡠⠭⣿⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣘⡇⠀⠀⠀⠀⠀⢿⣯⢿⢹⢻⣽⠇
⠀⠀⠀⠘⣄⣸⡃⣭⣰⠏⠀⠀⠀⠀⠀⠀⢸⠄⣀⡼⡀⣼⠂⠀⠀⠀⠀⠀⣷⣦⣤⣼⣧⣤⣴⡇⠀⠀⠀⠀⠀⢻⠛⠛⠛⠛⡗
⠀⠀⠀⠀⠘⣏⡵⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡤⠜⠁⠀⠀⠀⠀⠀⠈⠲⣀⣓⠉⢀⢛⡸⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠂⠀⠀⠀⠀⢠⠋⣿⢁⣾⡇⢳⠀⠀⠀⠀⠀⡶⢷⣾⡶⣄
⠀⠀⢠⢴⠞⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⡟⠻⣴⡇⠘⡇⠀⠀⠀⠀⡏⢦⢻⣿⣿⣆
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢴⡖⠚⣛⣛⡒⢲⡆⠀⠀⠀⠀⠀⣧⠀⣫⡴⢛⡆⠀⡇⠀⠀⠀⠀⡇⠀⠱⡻⣿⣿⣄
⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⢸⣤⣀⣀⡀⣀⣸⠃⠀⠀⠀⠀⠀⢣⠀⢿⠙⢺⡇⢠⠃⠀⠀⠀⠀⡇⠀⠀⠙⢝⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡏⠛⠛⠻⠟⢻⠀⠀⠀⠀⠀⠀⠘⣄⣸⣞⢋⣧⡼⠀⠀⠀⠀⠀⠧⠤⣤⣄⣈⣮⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡏⠉⢳⠏⠉⢷⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢱
⠀⠀⠀⢠⣇⡀⣀⢆⣀⣾⠀⠀⠀⠀⢀⡤⠚⠉⠁⠆⠉⠑⢦⡀⠀⠀⠀⠀⣀⢔⠚⠿⠟⢷⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡾⡛⠁⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡞⠀⠀⠀⠸⣽⡀⠀⠀⠀⠀⠀⠀⢀⣞⡶⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠸⡀⠀⡸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠻⣖⡒⠒⣲⠾⠛⠁⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠓⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣯⣓⢦⡀
⠀⠀⠀⢀⡴⢋⠡⠈⠄⡀⠍⠙⣦⠀⠀⠀⠀⠀⣰⣒⣚⣛⣛⣶⡀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⣠⢞⣩⠴⣿⣼⣚⣍⣳⣽
⠀⠀⠀⢸⡧⠬⠤⠤⠤⠤⠤⠤⣼⠆⠀⠀⠀⠘⣧⡤⠤⠤⠤⣤⠇⠀⠀⠀⠀⠀⣞⡉⠉⣧⠀⠀⠀⠀⣼⢡⢏⠤⡅⡏⠿⢟⣫⡵⣆
⠀⠀⠀⢸⡗⠛⢒⣒⣛⣒⠒⠓⢺⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⡇⢹⡇⣸⠀⠀⠀⠀⣇⢸⠘⢒⠋⠷⢚⣭⡀⡇⣿
⠀⠀⠀⠀⢿⣙⣯⡉⠉⢉⣽⣋⡟⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡟⠉⠀⣿⠀⠀⠀⠀⠹⡜⢧⡻⠗⣡⣌⣋⡼⣡⠏
⠀⠀⠀⠀⠀⠉⠉⠙⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣸⠃⠀⠀⠀⠀⠀⢠⣇⣀⣀⣻⠀⠀⠀⠀⠀⠙⠦⣍⣛⣓⣛⣩⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣞⢻⡏⣿⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⡳⣄
⠀⠀⠀⢸⠷⠾⠮⠷⠧⠾⠶⣿⠀⠀⠀⠀⠀⠀⣤⣤⣯⣯⣤⣄⠀⠀⠀⠀⠀⣠⠖⠋⡀⢀⠐⡈⠓⢦⡀⠀⠀⠀⠀⣰⢋⠀⡁⢈⠙⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⢛⡧⠀⠀⠀⠙⣶⠓⡶⠚⠲⡖⢳⡞⠁⠀⠀⠀⠀⣏⠠⠀⠄⣠⡼⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣿⣀⣀⣀⣀⣟⣋⣀⠀⠀⠀⠀⢸⡄⢹⠀⢸⠃⣼⠀⠀⠀⢀⣤⠞⣩⠷⠖⠛⠁
⠀⠀⠀⠀⠸⢤⡤⠤⠤⢤⠼⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣀⣀⣀⣤⠟⠀⠀⠀⠀⠀⠳⣼⣧⡾⡴⠃⠀⠀⠀⠘⢦⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣳⣆
⠀⠀⠀⠀⠀⢀⣀⡤⠞⣧⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣄⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⡄⠀⠀⠀⣸⣥⣤⣽⣧⣬⣧
⠀⠀⠀⠀⣰⠯⠤⠤⠤⠬⢷⡀⠀⠀⠀⣰⠋⠀⠀⠏⠀⠀⠟⠁⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣠⡏⠙⠓⢿⠀⠀⠀⠹⡞⠛⢚⣷⢶⡏
⠀⠀⠀⠀⡷⠶⣤⠴⢤⡴⢶⡇⠀⠀⠀⣟⣶⣒⣲⣒⣲⣒⣒⣖⣒⣶⣻⠀⠀⠀⠀⠀⢀⡞⠉⠙⠦⣄⣸⠁⠀⠀⠀⢻⣚⣩⡴⡾
⠀⠀⠀⠀⢹⡀⢸⠀⢸⠀⣸⠁⠀⠀⠀⣿⠦⠭⡽⠯⢥⡬⠵⠮⡭⠴⣻⠀⠀⢀⣠⢾⡉⠙⢦⡀⠀⢈⡗⠀⠀⠀⠀⠈⣏⣡⢴⠃
⠀⠀⠀⠀⠈⣧⣸⣄⣼⣠⡇⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠚⠛⠛⠉⠁⠀⠀⠘⠲⢤⣷⣀⣠⠿⠖⠋⠀⠀⠀⠀⠀⠀⠹⣄⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⢶⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⢀⣤⡀⠀⠀⠀⠀⢠⡤⠤⣤⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣧⡾⡶⣴⢧⡀⠀⠀⣴⠋⣄⠙⣶⠋⣄⠙⣆⠀⠀⠀⠈⡏⠉⡇⠸⡷⢾⡇⠀⠀⠀⠀⠀⠀⣠⢞⡼⣳⠋⠈⢹⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠶⢿⣹⠋⠀⠀⠰⡇⢀⡼⠀⢸⢀⡞⠀⢸⠀⠀⠀⢀⡇⠸⣧⡞⠁⠈⢳⡆⠀⠀⢀⡴⢞⡵⣫⠟⠁⠀⠀⣸⠃
⠀⠀⠀⣴⠋⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠀⢳⡈⠃⣠⢾⡈⠇⣠⠟⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢿⢚⡿⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠘⢦⣀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠙⠶⠃⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠈⠻⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⠚⠓⠒⠶⠚⠃⠀⠀⠀⠀⠈⠉⠉

⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣛⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⢛⡯⠉⢉⣉⠉⢩⡍⠉⠛⠲⢦⣄⠀⠀⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠒⠂⣀⣠⣤⠤⠤⣤⡀⠀⠀⠀⠀⠒⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⠃⠀⡴⠛⣉⡤⠴⠶⠶⣶⠟⠘⠂⠀⢠⠄⠀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⣴⠃⠐⠃⠤⠀⠀⠀⠙⠻⠷⠶⠖⠚⠋⠁⢠⠀⠀⠀⠀⠀⡞⠉⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⢶⠀⠀⠀⠀⠴⠆⠀⠀⠀⢀⣠⣀⣀⣀⣼⡃⠀⠀⢸⡇⠀⠀⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⣦⠀⢀⡀⠀⢀⣠⣤⣄⡀⠀⠀⣰⠋⠐⠁⠒⠒⠉⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠳⣬⣠⣴⠋⠀⠀⠒⠭⠿⠯⠕⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠳⠦⣤⣀⣀⣀⣀⣀⣀⣀⣁⣤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⡿⣠⡶⠛⣦⢰⡾⠛⢷⡰⡇⠀⢸⡆⢺⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠶⠛⠁⢿⣄⣀⡼⠻⡇⠀⢸⡇⢿⣀⣾⣇⠘⣧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠀⠉⠀⠉⠀⠈⠁

⠀⠀⠀⠀⠠⠀⠀⠠⢄⠄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠻⣷⡶⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣀⢆⣂⡠⠧⠄⢀⡀⠀⠀⠀⠀⠈⠐⠨⠻⣭⣑⢲⣤⡀⠀⠀⠀⢰⣿⡒⢂⡀⠀⢀⣴⡿⡀⠀⠀⠀⠀⡄⠐⠀⡀
⠀⠀⢹⡀⢙⢴⣤⣾⣾⣿⣽⡁⠀⠀⠀⠫⡇⣰⠶⡧⢢⣜⣰⠂⠀⠀⠐⣭⣛⡻⢟⠶⠗⠚⣡⠀⠀⠀⢀⡀⡀⠁⠠⠁⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠉⠍⠋⠀⠠⠀⠀⠀⠀⠀⠚⢦⣄⡔⢌⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠒⠆⠀⠐⠂⡀⠀⠀⠀⠀⢠⣶⣦⣄⡔⠦⢤
⠀⠀⢀⣄⡟⠛⠂⣸⢻⣵⠦⡀⠀⠀⡀⢂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⢻⢿⢿⣷⣞⠨⣧⡄⠀⠀⠈
⠀⠀⠿⣀⠀⠀⠰⣿⢲⢵⣡⡶⠀⠀⠀⠀⡈⠐⠀⠀⠚⠈⢠⠀⠀⠀⠀⠀⠀⠐⠯⠷⠏⠉⠛⢳
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠲⠤⠠⠠⠤⠶⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⡐⠀⢄⠀⠂⠈⠁⣀⠀⠀⠀⠠⡚⠢⠈⠙⠴⠿⠆⠀⠀⠀⠀⢀⠀⢔⠂⠀⠂⠀⠀⠀⢰⡎⣻⣷⢿⣯⣗⣦⣄⡀
⠀⠀⠀⠣⠩⠄⢄⢠⣤⠔⠀⠛⠀⠀⠀⢷⡖⢂⣂⠀⠀⡀⠢⠂⠀⠀⠆⡀⢀⠀⠐⠀⡀⠀⠀⡀⠀⠉⠉⠦⢤⠿⠒⠣⣴⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠀⠁⠀⠀⠀⠈⠑⠦⢄⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⢠⡌⠀⠀⠀⠀⠀⣾⠶⡀⠴⢄⠠⠐⡆
⠀⢀⡈⣿⠀⠀⠀⢨⡙⡐⢈⠰⠚⣶⠖⢲⠁
⠀⠀⠐⠘⡾⢱⢄⠃⠅⢐⠨⠀⢁⠊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢋⢉⠤⢝⢤⡛⠀⠈⡀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠑⠄
⠀⠀⠀⠈⣻⡣⣋⠓⢠⣈⣲⠴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠂⠀⢀
⠀⠀⠀⠘⠂⢈⠖⠨⠌⠀⠾⢀⠄⠓⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠐⠒⠂⠒⠀⠇⠘⡀⠀⡐⠃⠤⢀
⠀⠀⠀⠀⠀⠀⠁⠀⢀⣤⡀⠁⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⢀⣁⠀⠀⡐⠀⡜⠀⠀⡷⠎⡀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⡀⠀⢠⣾⡟⢨⢤⣴⠎⠀⠀⠀⢠⠆⠀⠀⡔⠔⡀⢀⠊⡄⢀⡜⠀⠈⠀⠀⠀⠁⠀⠈⢃⠠⠊⠀⠀⡀
⠀⠀⣘⣖⠒⠦⠙⡒⢿⡟⣑⠭⣄⣀⡀⠀⠒⠁⠈⠄⠀⠢⠐⠀⢀⠎⠀⠊⠀⣀⡠⠄⠀⠀⠀⠀⢠⠂⠀⠀⠀⠔⠑
⠀⠌⠀⠈⠀⠈⠀⠬⠰⠎⠄⠀⠀⠀⠈⠢⡠⠄⠀⠘⡞⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠄
⠈⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠖⠁⣤⠶⠛⠉⠀⠐⠒⠂⠉⠀⠒⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠠
⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡀⠀⢀⠙⢉⠉⠥⠀⠀⠀⠀⠔⠂⠄⠀⠀⡤⠆⠐⠄⢀⠔⠀⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢟⠉⠉⣚⡁⠀⠃⠀⠀⠁⠐⠀⣀⠀⠀⢀⠀⣀⠄⠀⠀⠀⠁
⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠁⠀⠀⠀⠀⢈⡄⠀⠠⠐⠀⠀⠑⠀⠂
⣶⣶⠶⠶⠶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡤⣤
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠲⡟⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⣤⣋⡷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⠤⠞⣻⡥⣤⡷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⠦⣤⠴⠚⠉⠉⡇
⠀⠀⢀⡼⢣⡤⣼⣆⠀⠀⠙⠢⣄⣀⠀⠀⣀⠀⠀⠀⢸⣡⢇⣴⣏⣭⣷⢯⡽⠛⡆⠀⠀⠀⠀⠀⠸⣅⠀⠀⢀⡤⠶⠦⣄⠀⢳⣀
⠀⠐⠛⠲⡏⠀⠀⠉⠳⣄⡀⠀⠀⠈⠉⣻⣹⠀⠀⠀⢾⠻⠿⢾⣁⣄⣽⡞⠁⠀⢻⠀⠀⠀⠀⠀⠀⠈⡷⠀⡏⠀⠀⠀⢸⡆⠀⠈⢳⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠙⠛⠻⠿⣏⣁⡍⠀⠀⠀⠘⣆⠀⠐⠂⢠⠏⠐⠀⠀⠈⣇⠀⠀⠀⠀⣠⠞⠁⠀⠳⣄⣀⣀⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠙⠦⢤⣀⣀⣀⣀⣀⣼⣁⡇⠀⠀⠀⠀⠘⢦⠀⢠⠏⠀⠀⠰⠀⠀⠸⡄⠀⠀⠀⡇⠀⠀⠀⠀⣀⣉⣁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⢳⠇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⠲⠶⠒⠋⠉⠁⠉⠉⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠘⠀⠀⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣄⡀⠀⣄⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡽⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠧⣽⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⢠⡤⣼⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⡀⠀⠀⢠⡞⣠⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⢳⡼⠛⢦⡼⢾⠙⢶⣳
⠀⠀⠀⣠⠶⠶⠶⠖⠛⠛⠒⠒⠒⠒⣦⠀⠀⠀⠀⠀⣾⣛⣚⣛⣛⡇⠀⠘⣇⢳⢀⣤⣄⣴⡿⠋⠀⠀⢸⣛⡛⠳⣴⡋⡇⣾⢧⠈⢻⡀
⠀⠀⠀⠙⣟⠉⠉⠉⠉⠉⠉⠉⠉⡿⠉⠀⠀⠀⠀⣠⠟⠉⠉⠉⠉⠓⢦⠀⠟⣀⣻⡀⠈⢿⠄⠀⠀⠀⠈⣠⣸⣳⡀⠙⣧⡏⣸⣇⣠⣷⠆
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡞⠉⠙⢿⠳⠛⠀⠀⠀⠀⠀⠘⣏⠉⠉⠛⠋⠉⠉⠁⠀⡟
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠶⢿⡶⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⡤⣤⣷⣠⣁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣹⠀⠀⠰⣇⠐⠀⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠚⠛⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⢛⣿⡿⠲⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⣤⠤⣀⣠⣀⣠⣀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⣠⠟⠛⠉⠛⠲⢤⡿⠶⣿⣶⠦⣄⠀⠀⠀⠀⠀⠀⣠⠶⡛⠉⠉⠯⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⢸⠾⠧⣤⣠⠆⠀⠀⠉⠳⣄
⠀⠀⠀⣼⠁⠀⠀⠀⠰⠀⠀⠙⢾⠓⣼⡧⣌⢧⠀⠀⠀⠀⡾⢡⠆⠙⠀⠛⠀⠘⢁⣤⠹⡄⠀⠀⠀⠀⠀⣼⠹⢦⠀⢈⡙⠲⢦⡀⢀⡀⠳⣄
⠀⠀⠀⠹⣄⠂⠀⣤⠀⠀⠈⠀⣌⢳⡘⢾⣻⣸⠄⠀⠀⠀⢹⡦⠤⠤⢤⣼⣭⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⣿⣃⡾⢀⡴⡒⣒⡶⣝⢻⣄⡤⠿
⠀⠀⠀⠀⠈⠳⢤⣀⠀⠠⠆⠀⠀⠀⢧⣸⣹⣾⠀⠀⠀⠀⢼⡳⠒⠛⣲⣶⣖⣛⣓⣶⣞⣹⠆⠀⠀⠀⠀⣟⠡⠀⣟⢾⣹⣽⠽⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠃⠀⢸⡏⡽⠃⠀⠀⠀⠀⠈⢿⡛⠉⠉⠉⠛⠯⣍⠉⢉⡟⠀⠀⠀⠀⠀⡏⣙⡦⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠛⠋⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢡⡠⠀⠀⠂⠑⠀⢀⠀⠀⠀⠹⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠅⣀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠈⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠧⡃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠁⠀⠈⠂⠀⠀⠉⠁⡀⠆⢰⠁⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠓⠈⠀⢀⠀⠀⠀⠀⢸⡽⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⢠⣀⠘⠒⠀⠀⠐⢄⢀⡁
⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⢀⠆⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠄⠀⠀⠀⠉⣉⠈⠁⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⠤⠤⠄⠓⠐⠒
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡴⠆⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢹⡖⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠻⢇⡀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⢰⠀⠈⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠀⣀⡤⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢘⡁⠀⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠀⢹⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⢠
⠀⠀⠀⠀⠱⣄⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⡄⠀⠀⠀⠀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠔⠈⠀⠢⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠠⡄⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⢠⢀⠆⠀⠀⠀⠀⠀⢀⠄⠁⡌⠈⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡀⠀⠀⢀⢀⡼⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⡌⡀⣣⠀⢁⠀⠊⢘⣀⢠⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠘⣆⠠⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠸⢧⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⣜⠀⢀⠆⠀⠀⠀⠀⠀⡐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠁⠐⠉⣦⠠⠊
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠚⠙⠛⠚⠉⠀⠀⢀⠀⡀⠀⠀⠈⠓⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠦⢀⠀⠀⢀⠄
⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⢠⡀⠀⠀⠀⠐⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠆⠀⠀⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡐⠁⠉⠚⢐⠈⠁⡂⠙⢄
⠀⠀⠀⠀⠉⠐⠦⠄⠀⠀⠀⠠⡇⠀⠀⠀⠀⡇⢀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⢁⠀⢠⠀⠀⠄⠆⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠁⠈⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⢾⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠆⠀⠈⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠢⣤⡀⠀⢀⡀⠀⢀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠠⠀⠤⠀⠀⠈⠂⠀⠀⠀⠀⠀⢰⣿⠅⡚⠁⡈⠁⠉⠉⠉
⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢰⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠃⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠉⢉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠱⣥⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠸⣟⠀⠇⠸⠀⡀⡘
⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⠀⢀⠀⠄⠉⠒⠒⠒⠒⠃⠁
⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠐⠂⠁⠀⣀⢀⡀⠀⠈⠛⠂⠂⠀⢤⠀⠀⠀⠀⢠⠂⠀⠓⠌⢀⠉⢀⣀⠀⠄⠀⠀⢀⠄⠠
⠀⠀⠀⠂⢙⠓⣦⠤⣀⣀⡀⠀⢀⣠⠀⠀⠀⠈⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠤⡄⠐⠀⠀⡉⠁⠀⠀⠈⠶⠂⠉⠁⠀⠉⠛⠣⠤⠀⠄⣰⡀⠠⠤⠤⠤⡀
⠀⠀⠀⢰⠑⢖⡙⢆⠀⠐⢙⠀⠠⠀⠄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠀⠇⢠⠀⠁⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐
⠀⠀⠀⠈⢇⠀⠈⠁⠀⣐⠁⠋⠈⠒⠀⠀⠀⠀⠧⣄⣀⠠⡇⠀⡀⢠⠀⠁⠠⠈⠀⡄⠘⠆⠴⠃⡇⠀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠑⠂⠤⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠢⠤⣨⡀⠈⠀⣁⡀⠷⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢀⣀⠀⠀⠀⣀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢘⡲⠀⠙⠉⠛⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠘⣂⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠓⢂⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜⠀⡄⠀⡷⢶⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⠀⢐⡖⠀⠋⢉⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠀⠁⣈⠐⠀⢀⣠
⠀⠀⠀⠀⠀⠛⠋⠁⠖⢠⡻⡄⠹⠁⠀⣠⣤⣤⣬⣠⣬⢇⣾⡤⣏⣀⣤⣀⢠⣆⠁⠀⠀⡼⠃⡤⠴⠛⢉
⠀⠀⠀⠀⠀⠘⠂⠀⠈⠀⠐⠳⠠⠀⡀⣈⢳⣿⡾⢫⡇⠉⣻⣿⡀⠘⠀⢇⣸⣿⣿⡿⢰⣣⣄⠄
⠀⠀⠀⠒⠉⠁⠉⠡⠦⠀⢠⡀⠀⣾⠑⠟⢈⣣⡆⣼⡟⠛⠻⣤⡠⠾⠟⠃⠙⡟⠒⢴⢿⣏⣥⣖⣴⢂⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠙⣴⣯⣰⢷⡴⡄⠀⠈⢢⣼⠾⠿⣷⣼⠦⣤⣴⣾⣶⣶⣾⣿⠛⠛⣿⣿⣋⣀⣤⢀⡖⠁
⠀⠀⠀⠀⣀⡀⢈⠀⠀⠠⢀⢀⠁⢼⣿⣿⣿⠽⣦⡰⣮⣄⢀⣄⠩⣿⠐⢋⣴⣿⣿⡿⢿⡿⣿⢿⣿⣷⣿⣿⠷⠲⠄
⠀⠀⣴⣾⣿⣿⠂⠀⣤⡀⠀⣈⣳⣂⣸⣿⡇⠀⣙⠀⠀⠰⣿⣿⣿⣃⡴⠿⠛⠉⣿⣧⣿⠁⢻⣦⢿⣿⣿⣶⣶⣙⣆⠂
⠀⠈⣟⣤⣌⠉⣶⡀⠋⢙⡿⠉⠉⢩⣿⣿⣇⠀⢩⠀⠐⠘⢌⠉⠷⢛⣷⣦⣴⣾⠃⠀⠀⠲⣾⠏⣉⣉⣨⢉⠛⡁⢫
⠀⠀⠙⠛⠁⠀⢲⡄⣴⣾⣿⠆⠀⠀⢁⡀⠈⠳⣦⠀⠀⠀⠈⢰⡶⠋⣹⣥⣿⣿⣄⠀⠀⣽⢧⣼⣏⣩⣿⠇⠈⣁⣠⡄
⠀⠀⠀⠄⠀⡀⢼⠗⠰⢾⣁⠊⠀⠀⠀⣋⡙⠆⢠⡀⠠⢤⡈⠁⠀⣠⠋⠉⠀⠉⢻⠿⢿⣿⣤⠍⠙⠉⠉⢈⣠⣛⠛⠀⡀
⠀⠀⠈⠀⠘⣋⣴⣷⠀⣾⡟⣿⢥⣠⠎⣹⣿⠐⢀⢳⣤⣴⡏⢡⠋⠈⠖⢒⣿⣿⠛⠲⠊⣹⣿⣿⣶⣗⣲⣿⣟⣛⡉
⠀⠀⠀⠀⠀⠁⠘⢋⢰⣿⡣⠙⢛⠃⠾⡿⠃⣴⢿⠟⠛⢿⣧⡀⠠⠞⠃⠈⠉⠈⠁⠛⠉⠺⣽⣿⣯⣼⣿⣿⣿⣿⡅⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠋⣨⡛⠳⢊⡡⠾⢿⡯⣤⡄⠀⠋⠀⠀⠀⠀⣀⢠⣤⣀⣀⣤⣤⣤⣙⣿⣿⣿⣿⠋⡿⠑
⠀⠀⠀⠀⠀⠀⠀⣘⣤⣜⠀⠀⣽⢻⡏⣰⣧⣮⢀⣀⣤⡆⠀⣀⣤⣶⣯⠉⠀⢀⡁⠈⠘⣿⣿⢏⣽⣿⠿⠿⣦⣄⠀⢨
⠀⠀⠀⠀⠀⢀⣸⠿⣿⣧⠀⣰⣿⡓⠻⡿⠋⠀⠀⠈⢉⣵⣶⢿⡯⠙⡟⠓⣶⣿⣿⣦⣤⣿⣷⣾⣷⡦⠂⠈⡟⠿⠷⠾⢅⠠
⠀⠀⠀⠀⠒⣻⠟⡦⠸⢷⣠⠛⣿⡨⠂⠀⠀⠀⢀⡴⠟⣿⡟⠸⣧⣼⣿⣦⣹⣿⣿⣿⣿⣿⣿⠏⠙⣿⢾⣿⣷⣖⢃⣠⡀⡼
⠀⠀⠀⠀⠀⡰⣦⡀⠤⠀⣩⠆⠉⠀⠀⠀⣠⣾⣿⡠⠾⠛⢿⣿⡿⠛⠙⣛⣿⣿⡟⠉⠁⠀⣽⡄⣤⡿⢀⣿⣿⣿⡛⣿⣆⢰
⠀⠀⠀⢀⣴⢂⣅⣰⠤⠛⠁⠀⠀⢀⣤⣾⢿⠯⣭⣄⡈⣈⡀⠓⡇⠀⠉⢉⣿⠡⡤⣴⠆⢼⠋⠀⢨⡷⢿⡃⣼⣿⣿⣤⣿⢈⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠶⢶⡤⠰⢾⠟⠉⣅⡀⠘⠷⠦⣀⠉⢭⣙⣹⠿⠿⢿⣿⣾⣀⣤⣀⣼⡗⠚⢻⣅⡼⢿⣿⣿⣿⣿⣿⣬
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠲⠀⡀⠀⠀⢾⡏⠟⠀⠀⠀⣸⣷⣿⠋⠀⣠⠀⠀⠀⠀⢹⣧⡀⣠⣀⣠⣼⠏⠀⣠⣿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠶⠒⠓⠈⠵⠆⠈⠆⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⢰⣿⡇⠀⠀⠀⢀⣾⣿⢻⣇⣀⣀⡀⠀⢸⠏⢻⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⡀⠀⠀⠀⣀⠀⢀⣾⣿⣷⣧⣀⣻⡆⠰⢢⣤⣴⣿⣿⣏⡁⣾⣿⣷⣶⠿⠲⣿⠁⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣶⣿⣿⣥⣾⣿⡿⠿⠿⠋⢿⠉⠑⠲⠶⠿⣻⢋⣽⣿⣿⡟⡙⠦⣹⣷⠎⠉⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⡉⠂⡄⣤⣀⣈⡀⣠⣵⢾⠾⠃⠀⠛⠛⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠋⠐⠻⠚⠫⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢦⠳⣄⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠈⠳⠜⢠⡀⢰⡖⠈⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞⢁⡄⢀⠀⠀⣴⠚⣹⠂⢿⡇⠀⣰⠀⠀⠀⡀⠹⣞⣿⣄⣩⠒⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣆⣹⣦⠉⣰⠋⠀⠀⣉⠳⣿⣤⣄⣉⡈⠁⢀⡴⠂⠹⣷⠾⣧⠉⠉⠀⠰⢯⡀⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠀⠈⢷⡈⠓⣠⣞⠁⣦⠙⣦⡙⠆⠀⣄⠠⣄⠉⠴⠆⣰⠞⠀⠀⣀⡀⣿⠙⡎⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢷⠀⠀⠁⠘⣇⠈⢷⡌⠓⠋⠙⠀⣀⠙⢧⡙⣦⠀⠚⠁⣠⡶⠈⣧⠙⡞⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡘⢧⡘⢷⡀⠙⢂⣀⣴⢋⠙⣿⠾⢷⠌⠁⠀⠰⠻⣤⠆⠙⠦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⣾⠁⠀⠙⠚⣅⢀⡈⣷⠹⣏⣠⡌⠃⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⣠⣜⣧⠙⢾⣧⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡶⣌⢿⡛⠷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠷⡼⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⢿⣿⣿⣿⣿⣿⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢛⣻⣏⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠟⠛⠛⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⡀⠀⠀⠀⣴⠾⠛⢁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⡀⠀⠀⠀⣴⡾⠿⣿⣷⣄⠀⠀⠀⠈⣿⣿⣿⠿⠛⠻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢲⡿⠀⠀⠀⠘⠏⠀⠘⠛⠛⠁⠀⠀⠀⠀⠿⡿⠋⣴⡻⠆⢹⡿⣣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠃⢀⠮⠟⠁⣠⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⣠⣶⣶⣿⣿⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢈⠁⢀⡉⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⣟⠛⠛⠛⠳⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠀⣀⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠈⠉⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢐⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢍⣉⡭⠔⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡑⠀⠀⠀⠀⠘⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⢻⣤⢦⣄⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡻⠁⠘⠁⠀⠀⠀⠀⠀⠈⢿⡁
⠀⠀⠀⠀⠈⡾⢿⣿⣊⣿⣞⣾⣇⠀⠀⠀⠀⠠⣤⣀⣀⣤⣤⣶⣶⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠉⠀⠹⠟⠋⠙⠋⣁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⢒⣟⢡⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⣠⣞⠋⠁⣩⢏⠛⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀
⠀⠀⠀⠀⢠⠞⠋⠈⢻⣾⣓⡾⢀⡟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠁⠀⠀⣠⣾⡿⣍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⡀⠚⠏⢥⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡾⠟⠃
⠀⠀⠀⠁⠀⠙⠀⡤⠟⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠿⣿⣿⣿⣿⠋⢉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁
⠀⠀⠀⠀⠀⡀⠖⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠹⡆

⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⡆⢀⠀⠲⠒⢲⡶⠶⠶⣶⠶⠖⠠⢀⣀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠓⠀⠈⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠤⠀⠀⠰⠀⠈⢩⣟⠻⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠞⠃⠉⡀⠀⠀⠊⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠀⠐⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠤⠀⠀⠀⠙⠁⣽⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢤⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠐⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣟⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣷⣤⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢛⠿⢶⣦⣤⣀⣀⣀⣀⣄⣀⣀⣠⣤⣀⡀⠀⢀⣀⣀⣀⣀⣀⣤⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⠿⣟⣛⡭⣉⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣷⣦⣈⠉⢉⣛⣛⠛⢛⣻⣟⠛⠟⠻⢿⢿⣿⣯⣿⣯⣍⣭⣭⠽⠈⢦⡉⠁⢀⣰⡟⢉⠾⠋⢗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠰⠚⣉⡙⠿⠿⠍⣉⣿⣿⣿⡟⢻⣿⣿⣦⣤⣤⣴⠿⣷⠾⡯⠟⠛⠿⢿⡴⠋⠳⢍⣻⡿⠻⠷⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣭⡥⣄⡀⠀⠋⠙⠿⠻⠿⠛⢻⠿⢻⡿⡯⠋⠀⠋⠉⠀⠐⠂⠒⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠧⣿⣿⣷⣶⣏⡋⠉⠀⠀⠒⠓⠀⠊⢀⣠⣤⣴⣶⣶⠶⣶⣶⣶⣾⣿⣿⣿⣿⡿⢛⣽⣿⣇⣼⢧⣼⡏⣨⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣌⢙⣧⣄⡙⢟⣷⣄⠀⠀⢀⣠⣾⣿⣯⣤⣶⣿⣭⣼⡿⢃⣾⠏⢀⣾⠿⣿⢹⣟⠉⣸⣿⣏⣾⡯⣴⡏⢀⡀⠘
⠀⠀⠀⠀⠀⠀⠂⠈⠉⡃⠛⢮⠛⠿⢙⣛⢻⣿⣧⣴⣿⣿⣛⠉⢻⢿⣭⣼⣿⣋⣾⡿⠿⠿⠿⢛⣴⣟⣨⡿⣿⣿⠿⡿⡗⠓⢄⣁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⡥⢄⡠⢀⣀⠀⠀⣈⡉⣉⣉⣤⣤⣞⡀⠒⡢⠈⠟⠀⢐⠊⣀⣀⣉⣡⣽⣄⣤⣤⣏⣀⣿⡿⠶⠾⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠙⠛⠻⠙⠿⠟⠋⠛⠻⠗⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠁⠉⠹⡟⠀⡀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣄⠀⠀⠀⣰⣀⣼⣦⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠤⠤⠤⠤⣤⣤⣤⣄⣤⢀⣀⣠⣤⣄⣴⣶⣴⣤⠤⠴⠾⠿⠿⠒⠶⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢆⣼⠀⠀⠀⢸⡟⣗⣿⣿⣺⣄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡆⢚⠀⡄⣿⣉⣹⣿⣿⣯⣿⣧⣤⠶⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⡘⣿⣿⡧⢾⡶⢧⣶⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣦⠐
⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⣿⣿⣟⢻⣿⠟⠛⠃⠀⠀⠈⠙⠋⡭⠿⣟⣗⣺⣿⡎⣉⠘⠿⣿⣗⢀
⠀⠀⠀⠀⠀⡀⠀⠚⠛⠿⠥⡈⢈⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣿⣸⠟⠟⠛⠏⠙⠻⢿⣯
⠀⠀⠀⠀⡄⠀⠀⣀⡄⠀⠀⢸⣘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⣛⣵⡃⡄⠠⣰⠀⠀⣰⢻⣿⣿⡆
⠀⠀⠀⡄⣾⣄⡀⣀⣀⣀⣠⣿⣩⣭⣽⣿⣷⠄⠀⠀⢠⣿⣿⣿⣿⣿⣙⠓⠴⠿⠷⠄⣚⣿⣿⣿⡇
⠀⠀⠀⠉⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠷⠾⠶⢾⡶⢶⠾⠿⠿⠾⠷⠿⠿⠿⠾⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢹⣆⣀⣀⣀⢀⣀⣀⠀⠀⡴⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣤⠞⠳⣤⡤⠜⠉⠀⠀⠀⠀⠀⠀⠙⠢⣷⣀⠤⠾⣴⠶⡟⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⡠⠊⠉⠛⢄⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡇⠀⠿⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠙⢦⡀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⠆⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀⢀⡴⠢⠤⠖⠱⢆⡤⠖⠲⠶⣄⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⢦⠀⠀⢸⠀⠀⠀⠀⠀⠀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⣄⠀⡀⠀⠀⠈⡇⠀⠀⠀⣯⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠹⣄⠀⢸⠀⠀⠀⠀⢰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣄⣀⡄⢳⣠⠖⠛⠁⠀⠀⠚⠉⠉⠃⠀⠀⠀⠀⠰⠋⠁⠙⠂⠀⠉⢙⣿⠀⢀⣠⣤⠊⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠻⢬⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠖⠉⠀⠀⠀⢀⣾⠀⣾⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⡟⣁⣀⣀⣀⣀⠀⠀⡼⢸⡞⢸⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢳⠀⠀⢸⡗⠒⠋⠉⠙⠛⣧⠀⠀⠀⠀⠈⠳⣄⣀⣀⣸⠞⠁⠀⠀⠀⡸⢿⡅⠈⠉⠉⡏⠀⠠⡇⠈⠀⠘⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢀⡼⠀⠀⠀⢧⠀⠀⠀⢠⠞⠁⠳⢤⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⠞⠁⠀⣧⡄⠀⢸⠃⠀⠀⠷⡄⣀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡘⣆⠀⠀⠀⠈⢧⡀⠀⣿⠀⠀⠀⠀⠈⠙⠒⠦⣄⡀⠀⣀⡤⠖⠋⠀⠀⠀⢠⠏⠙⣶⠃⠀⠀⠀⢰⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠼⠒⢆⠀⠀⠀⠉⢻⠟⠓⢤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⡼⠀⠀⢰⡒⠛⠺⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡤⠞⢙⣆⠀⠀⠀⠸⡀⠀⠀⠈⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⡋⠙⠛⠲⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠶⠚⠋⣸⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣞⠉⠓⠶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡶⠚⣹⡇⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣟⠳⠶⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣇⠹⡄⠀⠀⠀⠀⠀⠈⠑⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⡟⢹⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠸⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀
⠀⠀⠀⣀⣴⣬⡽⠿⠲⣷⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢂⠀⠀⡠⣄⠀⠀⣀⢀⢀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⢨⣏⠙⠒⠂⠉⢁⢹⡆⠀⢀⠞⠁⠀⠀⠀⠀⠀⠢⡑⠄⠀⠀⠀⡠⢤⠤⡀⠀⠀⢨⠀⣯⠀⡎⠀⠹⡇⠀⢟⣏⡮⠁⠀⠀⢸⠁⢩
⠀⠀⠀⣾⠐⢒⣈⡉⠤⠄⣿⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠇⡘⠀⢀⠎⠀⠀⠱⣑⡀⠀⠈⠄⢸⠀⠘⠢⡔⠁⠀⢣⢠⠚⠀⠀⠀⢸⠀⢸
⠀⠀⠀⢹⡁⠤⠤⠐⢒⡀⢸⡀⠀⠘⠢⠤⠤⢤⠤⡤⠤⠊⠀⢠⠃⠀⠱⠓⠀⠈⢷⡀⠀⠇⣨⡇⠀⠆⡇⠀⠀⡆⡈⠀⠀⠀⡠⠃⠀⠀⣷⡄
⠀⠀⠀⠀⡷⠖⠬⠵⢂⣒⠭⠇⠀⠀⠀⣇⣂⣈⣀⣁⣀⠀⠀⠧⢤⣦⣤⣤⣤⡤⠬⠇⠀⠀⢐⣠⠀⢁⡇⠀⠀⢇⠇⠀⠀⠀⠧⠤⠤⠤⠬⠬
⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⠀⢸⠀⠠⡤⡂⠀⠈
⠀⠀⠀⠀⠀⣀⠤⠄⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⡉⣿⣻⣉⢉⣉⢉⣏⣟⣉⡉⠉⠉⠀⠀⢸⠄⢈⣀⣀⡀⢤
⠀⠀⠀⡠⢊⡤⠊⠉⠉⠁⠢⡑⢄⠀⠀⠀⠀⣠⣖⠉⡄⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠉⢁⣴⠞⠛⢶⡌⠉⢳⡇⠀⠀⠀⠀⠈⣄⣀⠀⠀⣀⡄
⠀⠀⢰⢁⢋⠔⢲⡄⣀⢀⢀⠘⡘⡄⠀⠀⡐⠕⠁⣼⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⣵⣶⢠⣤⣯⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⠘⣇⠀⢸⠅⣋⣿⣸⢀⢃⠃⢀⠤⠥⠤⠤⡤⠤⠶⠶⠶⠶⠶⠼⠤⡀⠀⠀⠀⠀⢻⡧⠤⣿⠃⠀⢸⡇⠀⡆⡶⡆⣖⢆⡶⡠⡆⢒⡞⡄
⠀⠀⠀⠣⣩⢲⠉⠀⠳⡄⡵⢋⠜⠀⠈⠋⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠁⠀⠂⠓⠒⠀⠁⠀⠈⠀⠒⠃⠇⠀⠥⠥⠥⠭⠭⠤⠥⠥⠭⠗⠃
⠀⠀⠀⠀⡄⡸⠭⠉⠘⠃⡆⠁⣠⡎⣙⢢⠀⠀⠀⢀⣀⣀⣀⠠⠤⢒⠒⡄⠀⠀⢠⠒⠒⠂⢐⣒⡦⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠳⠁⠀⠀⠀⣄⠃⠀⢟⢲⠲⢽⠂⠀⠐⠁⡜⡐⢠⢰⠀⠓⢁⡿⠀⠀⢨⣠⡤⠤⣤⣤⡄⠀⠀⣠⣴⣡⣤⣤⣤⣤⣤⣤⣤⣼⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠈⠀⡌⠀⠀⢻⣄⣀⠃⠘⣀⡡⢴⡩⠎⠀⠀⠘⡇⢀⣰⣿⢷⡇⠀⠀⠇⠊⣠⠤⣄⡤⣤⡤⢠⣄⡌⢱⠈
⠀⠀⠀⣀⠤⠤⠤⠠⣄⠀⠀⠈⠙⠋⠙⠁⠀⠀⠈⠑⠣⠷⠾⠐⠋⠉⠀⠀⠀⠀⠀⣷⠻⠿⠀⣸⡃⠀⠀⡏⢇⢙⠖⢋⢋⡘⠧⠘⠘⢃⡼⢋
⠀⠀⠀⡗⠒⢀⡀⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠋⠉⠙⠋⠀⠀⠀⠁⠂⠐⠀⠂⠀⠂⠀⠀⠒⠀⠒⠁
⠀⠀⢠⢸⠟⠂⢒⠘⡇⡀⠀⠀⡏⡇⠤⠤⠤⠤⠴⠥⠭⠽⠥⠵⠤⠤⠤⠤⢄⢈⢸⠀⠀⠠⠒⠊⣑⣒⣉⣓⠒⠤⢦⡠⡀⠀⠀⠀⣀⣀⡀
⠀⠀⠈⢌⢆⡀⢀⠰⡱⠁⠀⠀⡇⡇⠉⠉⠉⠉⠉⠉⠛⠙⡏⠉⠉⠉⢉⠉⠁⡀⢸⠀⠀⡸⢊⠉⡇⢂⢠⢠⠈⡑⠤⣰⠗⡂⠀⠮⠄⠄⠽⡀
⠀⠀⠀⠀⠑⠲⡖⠈⠀⠀⠀⠀⡇⡤⠤⠤⠈⡆⠀⠀⠀⠀⡇⠀⠀⢸⢠⠀⠀⠄⢸⠀⠀⡣⣀⠀⡇⠸⠸⠸⠀⣃⡴⠳⡀⠅⠀⡟⢻⠙⠋⡅
⠀⠀⠀⠀⠀⣨⣇⡀⠀⠀⠀⠀⡇⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸⢸⠀⠀⠀⢸⠀⠀⠉⠢⠽⢓⡒⣒⡲⠩⠔⠁⠀⠈⠁⠀⣆⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠈⠈⠁⠀⡴⢌⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠊
⠀⠀⢠⢰⠰⣠⠤⠤⢤⡴⠖⠚⠛⠉⠀⠀⢀⠤⣒⠒⠀⡰⠖⠉⢳⠁⠀⠀⠀⠀⠀⠀⠀⢴⣶⡖⢲⠮⠅⠒⠀⠀⠀⠀⠀⠀⣀⢴⣂
⠀⠀⡬⠤⠤⠤⠥⠬⠭⠤⢤⠀⠀⠀⠀⡴⢁⠞⡝⠀⠀⡁⠀⣁⠎⢡⠀⠀⠀⠀⠀⠀⠀⠨⣿⠗⠃⠀⠀⠀⠀⠀⠀⠀⢀⠏⠙⠒⢋⣉⢶
⠀⠀⢇⢆⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⡇⠘⡄⠙⢦⠀⠈⢉⠡⢂⠜⠀⠀⠀⠀⡠⣺⣟⣻⣻⣿⣛⢷⢦⠀⠀⠀⠀⠀⡼⠔⢚⣉⡡⠤⠚
⠀⠀⢠⡓⠕⠀⠀⠠⠤⢺⡄⠀⠀⠐⢄⣉⣀⣈⣉⣉⣀⣀⣁⣈⣉⡰⠂⠠⡆⠐⠓⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⡤⠀⠀⣇⣐⣁⠬⠤⠒⠋
⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠐⠀⢀⠀⠀⠀⠀⡄⡀⠀⠀⢀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠀⠃
⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢓⠀⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠱⡀⠀⢀⣀⠀⣴⠂⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⢡⠶⠂⢠⠆⠀⠀⠀⠀⠜⡁⣈⡈⠀⠀⣀⢠⡀⡀⣄⣀⡠⠀⠀⠀⢰⡆⠀⠄⠈⠀⠉⠁⠈⠀⠀⢻⣄⡀⠀⢀
⢀⠀⠀⠀⠀⠂⠀⣄⠀⠀⠀⠡⠆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠁⠈⠀⠀⢠⡄⡒⠚⠀⠀⠀⠉⠁⠄
⣄⠀⡆⠀⠀⢀⡀⠀⠶⠀⡠⠐⠁⣀⣄⣰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⣾⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠸⠛⠓⠲⡀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠛⠀⠄⠀⠀⠛⣠⠄⠠⠆⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⢰⡀⢠⣰⡀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡘
⠀⠀⠠⠤⣭⣿⣿⢿⣤⡤⠀⠀⠻⡏⠀⠀⠀⣷⡞⠀⠠⠰⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⡾⢠⣉⠅⣿⡀⠀⠙
⠀⠀⠀⠰⠤⠞⠸⠦⠤⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠐⠀⠠⠁⠰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠉⠐⠤⠗⠹⠇⠀⠈⠡

⠀⠀⠀⡀⡄⢀⠆⡀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡀⠀⠀⠀⠀⡀⠤⢂⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⢀⢠⠈⡀⡏⡀⣇⡆⠀⠀⠀⠃⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⢰⡶⠂⠀⠀⡠⠊⠠⡆⠉⠀⡷⠀⠀⠀⠀⢀⣤⣤⢤⣤⣿⡿⢦⡀
⠀⠸⠘⠋⠛⠋⠉⠁⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⢰⡎⠁⠀⠀⠀⠁⠾⣦⠛⠀⠐⠀⡆⠀⠀⠀⠚⡇⠀⠀⠀⡀⡀⠈⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠒⠠⠀⣠⠟⠀⠀⠀⠀⠀⠀⣆⠃⠀⢀⡄⠄⠀⠀⠀⠀⠀⠀⠈⠋⣁⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⡈⢡⠄⠀⠀⠀⠚⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠼⡀⠀⠀⠀⠁⠀⠀⠀⣀⣀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣠⡔⠤⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠡⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣐⠰⣆⡁⢀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⢄⠀⠀⠀⠀⡀⠺⠃
⠀⠀⢀⠄⠀⠀⠐⠄⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠀⠙⠦⡼⢚⣅⠀⠀⠀⠘⢄⠒⡀⠃⡀⠘⠀⢨⠀⠲⠀⠄⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢰⠓⢶⣁⠀⠀⠀⠀⠑⠨⠄⣒⣠⠥⠀⢊⠀⠀⠀⠀⠐⠀⠙⠀⢀⠖
⠀⠀⠑⠀⠤⠒⠠⠤⠜⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠐


⠀⢀⠄⠒⠖⠀⠒⠒⠤⠀⠀⠀⠀⢄⣠⡠⠐⢦⣸⠇⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠉⠀⢠⡴⠶⣶⡤⣀⠀⡀⠀⠀⠸⠿⠷⠄⣼⣯⣛⠀⠀⠀⠀⠀⠀⠠⠴⠉⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠰⠀⡄⡀⠀⢀
⠀⠀⠊⠀⢠⠞⣻⠆⠀⠁⠀⠀⠀⠀⢃⠀⠀⢲⠀⠉⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠧⢙⣡⣤⣤⡜
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠭⣀⠀⠀⠀⠀⠀⠀⠁⠰⠄⠒⠀⠔⢦⠍⠀⠀⠀⠀⠀⢈⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⡁
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⡄⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⢛⠀⡄⠀⠀⠀⠀⠀⣠⠞⠃⠐⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⢄
⠀⠀⠃⠠⠀⠀⣀⣀⣀⠐⠀⠀⠀⠀⠀⡐⡐⠠⠁⠀⢀⠀⠀⠀⠀⣸⠇⠁⠀⠀⠀⠀⠙⡆⠀⠀⠀⡠⠂⠉⠉⠀⠀⢄⡠⠐⠀⠀⠀⠔⠉⡄⠀⠀⠈⠑⡀
⠐⣤⠀⠀⠂⠋⠉⠉⠉⢒⠛⠀⠀⠀⡄⡀⠠⡁⠃⡠⠀⠀⠀⠀⠀⢿⠂⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠢⠀⡀⠀⠀⢀⡠⠀⢀⡀⠀⠈⡱⠔⠓⠀⠙⠢⢄⡁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠀⠔⠁⠀⠀⠀⠀⠀⠸⠦⠄⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈


⢀⠉⠃⢰⣴⣶⡊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣤⣀⣀
⠈⠀⠀⠸⢟⠇⠀⠤⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢀⠀⠀⣸⣿⣿⠏
⠀⠀⢰⠀⠸⡅⠀⠀⡄⠀⠀⠀⠀⠠⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣻⠛⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⣇⠀⢠⡇⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣾⠅⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢐⣋⣶
⠀⠀⠸⠀⠀⡏⠀⠈⠇⠀⢀⡤⢤⢤⣀⣀⣠⣀⣀⣀⣀⣠⣦⡄⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡜⠈⠀⠠⡀⠀⠀⠀⠀⠀⠥⠤⠀⢠⡉⠉⠁
⠀⠀⠀⠆⠀⠃⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡀⣂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠃⢠⠀⠀⠀⢈⡐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠑⠀⠀⢈⠤⠀⢠⠈⢦
⠀⠀⠀⠀⡠⠔⠁⣀⣀⣀⢀⡀⠀⠀⠀⢀⠁⠀⠈⣠⠀⣀⣀⠀⠀⠀⠀⠐⠂⠀⠀⢱⠁⠘⠀⡈⠳⠦⠤⠀⢀
⠀⠀⠀⢆⣁⡨⢉⠀⠠⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠉⢤⢌⡉⣀⠀⡁⣦⠘⠀⠀⠀⠀⠙⠬⡤⠤⡀⠐⠒⠊⠁⡀
⠀⠀⠀⠀⠀⠁⠘⠑⠞⠀⠮⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠰⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠤⣀⠀⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⢉⠀⠀⠘⠄⢧⠀⠘⢷⣄⠀⠀⠀⠀⠄⠀⠀⠠⠀⠈⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠐⠁⠀⠀⠀⠀⠂⠀⠣⠀⠈⣉⡉⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⢄⣀⣀⠀⣀⣀⠀⢀⣙⣎⠙⠦⣠⠊⠀⠀⠀⠀⠁⠀⡀⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⡠⣄⢐⣀⠆⡀⢸⡄⠁⠈⣳⡞⢁⣠⠔⠒⠒⠛⡉⠁⠀⠠⢄⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠾⠄⠟⠀⠀⠀⠸⣄⣀⡤⠄⢄⡁⢈⡀⢁⡀⣠⠦⠊⠀⠀⠈⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠛⠀⠙⠋⠀⠀⠀⠀⠀⣠⠘⠀⠀⠐⠐⠄
⠀⠀⠀⠀⠀⠄⠁⠀⠀⢈⠑⠢⡀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠔⠁⠀⡀⠀⠀⢀⠁
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠰⠀⡆⠀⠀⠒⠦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠆⢠⠤⠔⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠆⣄⠠⠤⠀⢤⣀⡀⠄⡀⡀⠀⠾⣁⡘⠄⢰⡀⠈⠛⠶⠶⠶⠂⣠⠞⠀⣈⠀⠄⠀⠀⡈⡱⠋
⠀⠀⠀⠀⠀⠈⠃⣰⡄⣤⠕⣠⢤⡈⠙⠁⠀⠀⠸⢌⡓⠄⠙⠀⠑⠀⠀⠀⠉⠁⢀⢠⡵⠂⢴⡦⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠦⠥⢼⡉⠙⢂⣍⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠂⠉⠉⠁⠀⠀⠀⠐⠒⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⣀⣀⣀⡠⠶⠾⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣥⣤⣤⣤⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁⠁⠍⠻⡇⠀⠀⠀⠀⠀⢀⠤⢤⣀⡀⠀⠉⠹⢦⡀⠀⠀⠈⠂⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡇⠀⡀⠐⠉⠛⠛⠳⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡸⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⠶⠛⠁⠀⠀⠹⢦⣀⣀⡀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⢀⣤⠄⠀⠠⠴⣤
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠸⠿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠋⢀⡀⠀⠀⠀⢩⡿
⠀⠀⢠⣀⠀⠀⠀⠀⠉⠛⠲⢿⣆⣻⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠰⠤⠀⠀⠈⠁⠀⣀⠼⠀⠀⠀⡞⠀⠀⢀⣿⠀⠀⠀⠀⣸⡅
⠀⠀⠼⢿⣗⡦⣄⣀⣀⡀⠀⣤⣀⣠⣭⣥⡿⠋⠁⠀⠀⠀⠀⠀⠤⢄⡀⠀⠉⠀⠀⠀⣀⣠⠴⠒⠉⠁⠀⢠⠾⠇⢣⣤⢀⠔⠀⠀⠀⠀⢀⡿
⠀⠰⣤⣄⣀⠀⠉⠈⠛⠃⢀⣿⠀⠀⠘⠁⠀⠀⠀⠀⠐⠒⠲⠤⠀⠀⠀⠀⣀⡀⠐⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠁⠿⠀⠀⠀⢀⣠⡏
⠀⠀⠈⠙⠛⠛⠤⠶⣶⡄⠸⠃⠀⠀⠀⢀⣀⡀⠉⠉⠀⠀⠀⢀⡤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣷⢀⢀⡴⠚⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢞⡘⠇⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠤⠀⠐⠒⢤⣀⡀⠀⠀⠀⠀⠀⠈⢂⣾⡿⠉⠁⠀⠀⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠈⠹⢿⣤⣤⡶⠶⠋
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⣶⣦⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣆⠀⠀⠀⠙⠻⢿⣾⣽⣛⠳⢦⣄⣀⠠⠔⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠦⣄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠉⠛⠻⡷⢶⣏⠉⢠⣄⠴⠒⠛⠁⠀⠀⠀⡠⠂⠂⢬⣀⠀⠀⠀⠀⢈⡶⣄
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠴⠛⠋⣿⠿⣶⡉⠀⠘⣇
⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⡄⠀⠀⠀⠙⣾⠟⠀⠀⢠⡏
⠀⠀⠀⠀⠉⠻⠿⣶⣶⣀⣐⠲⣶⣶⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠁⠠⢄⣀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠘⢤⣠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠷⠶⠿⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⡿⢶
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠛⠛⠛⠛⠋⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠰⠓⠀⠀⠀⠓⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠤⠄⠀⢰⠠⠀⠀⣧⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢯⡰⡙⠷⠀⠀⠀⠠⢄⣸⠀⠀⠀⣼⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡉⢻⡀⠀⠐⠳⣄⡀⢀⣠⠄⠀⡏⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠐⢛⢤⣌⣩⢷⣄⠀⠀⠀⠀⡀⠀⠀⡼⠁⠀⠀⣸⣏⡆
⠀⠀⠀⠀⠀⠐⠛⢠⠾⠉⢸⠈⠓⣦⢤⠤⡭⢶⢻⣦⡀⠀⢀⢾⣍⠣⢞⠂
⠀⠀⠀⠀⠀⠀⣴⣿⠚⢲⣤⠀⣰⠃⠘⠀⠱⢀⢁⢳⠱⢄⠀⢱⢉⠖⠁
⠀⠀⠀⠀⠀⣼⠋⠸⠀⢸⢸⢻⠃⠀⢰⢆⢀⠺⢇⡇⠻⡞⠀⠸⠘⡀
⠀⠀⠀⠀⢰⢹⢰⡀⠀⢸⢈⣨⠆⠀⡄⡇⠁⢀⣨⠕⠁⢸⢠⠃⠀⡇
⠀⠀⠀⠀⣾⢸⢘⢿⣠⡼⠋⠹⣄⣠⢧⠇⣴⣉⣴⡶⡖⠋⠛⠀⢠⡃
⠀⠀⠀⠀⣿⢸⢸⢸⢹⢳⡄⠀⢳⡤⠋⣸⢡⣭⣾⠀⠈⠢⢂⣁⠉⠉⡂⠠⣀⣀
⠀⠀⠀⠀⣿⢶⣼⣼⣈⣈⢹⣄⠀⠀⢀⢏⣸⣼⣇⣠⣤⢷⠒⠀⠀⠈⠉⢀⠛⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⢻⠀⠀⠉⠉⠉⠉⠙⢦⣠⣾⣿⣿⣿⣿⣿⣧⣀⠑⠢⠤⠔⠒⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⠉⠛⠓⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢰⡿
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠻⣍⠛⢿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠤⣤⣤⣤⣴⣿⣀⡀⠀⠈⢧⠸⣇⣩⣿⡟⠁⠀⡿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠙⢦⠀⢈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣷⣿⡿⠿⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⢿⣿⣵⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡠⠚⢁⣴⠮⠭⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡠⠔⠊⠁⢀⢀⠀⣸⢽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⠘⠚⠁⠀⠉⠉⣡⣛⡛
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠋⠀⠃⠀⠀⠐⠀⢉⠤
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠠⣀⠀⢀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢⣀⢀⡠⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⠈⠂⢀⠂⣖⠀⢀⡀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣅⠀⠀⠀⠀⠀⡗⠀⠏⠀⠀⣀⡀⠀⠰⠃⣤⣏⠀⢀⣠⡂⠀⠀⠀⣶⡃
⠀⠀⠀⠀⠀⢀⠘⠋⠀⠑⠄⢀⣀⣂⣀⡢⠐⣻⡟⠳⣀⠌⠀⠱⠀⠀⠀⠀⠀⠀⠀⠸⢿
⠀⠀⠀⢀⡄⠀⠀⠀⡠⠊⣝⡓⣒⣒⣒⡙⠾⠿⠷⠐⠁⠀⠀⠀⠂⠀⠀⠀⠠⠌⠉⠀⠄⠐⢠⡀
⠀⠀⡀⣅⣱⡀⠀⠀⠐⣠⢺⣷⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡐⠦⠄⠈⠀⠀⡘⣸⣂
⠀⠀⠀⠀⠀⠀⣀⣉⣀⡤⠀⠙⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣐⣤⣽⠒⠛⣤⠀⠄⣤⠖⠒⠋⠉⠉⠉⠉⠉⠙⠒⠷⣔⢄
⠀⠀⠀⠀⠀⣿⣿⡟⠀⠈⠻⢄⡀⠙⠖⠒⠆⠦⠦⠤⠦⠶⠖⠒⠚⢁⣀⠴⠁
⠀⠀⠀⠀⠀⠈⠉⠧⠤⠤⠠⣀⣃⣑⣀⠖⠲⠶⠶⠶⠶⠒⠒⠚⠋⠉

⠀⠀⠀⡀⠄⢀⠆⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⠀⠀⢀⡠⠔⡠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠯
⠀⢀⢢⢨⣰⠁⢐⣤⡆⠀⠀⠘⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⢰⠶⠂⠀⠀⡰⠉⠴⡀⢁⠸⠆⠀⠀⠀⠀⣤⠤⠤⠤⠾⠿⢦⡄
⠀⠸⠈⠉⠉⠉⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠀⢰⡆⠀⠀⠀⠀⠐⠿⣾⢉⡀⠒⢰⠀⠀⠀⠀⠣⠀⠀⢀⡀⠤⠀⡁
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣤⠁⠀⢀⡄⠄⠀⠀⠀⠀⠀⠀⠈⠡⠄⠀⠉⡄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⢡⡖⠀⠀⠀⠋⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⢣⠀⠀⠀⠁⠀⠀⣀⢀⣀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⢠⠖⠑⠊⢤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢨⠋⠁⠀⠈⠀⠄⠀⠀⠀⠀⠀⣰⠈⢷⠀⢠⣠⠀⠀⠀⠀⡄⠲⡀⢀⡀⠀⡀⠀⠄⢀⠸⠙⠀⠀⠀⠀⠠⠈⡃
⠀⠀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⣘⢦⠀⠀⠀⠈⠢⡑⠄⢠⠀⠀⠀⢂⠀⠁⠂⠀⠀⡀⢀⠀⠀⢁
⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⢄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠈⡿⠔⠀⠀⠀⠀⠀⠐⠀⠄⠂⠀⠠⠔⠁⠀⠀⠸⡀⢀⠀⠀⠏
⠀⠀⠀⠀⠈⠉⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⣀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⢀⠶⠁⠈⣀⣀⣀⠀⠉⠀⠀⠀⠈⣲⣷⡀⠘⣶⢧⠖⠀⠀⠀⠀⠀⠀⢈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⡀
⠠⠀⠤⠘⠉⡽⢏⠉⠐⠀⠀⠀⠀⢹⠀⠁⢸⡙⠳⠀⠀⠀⠀⠀⡠⠉⠉⠀⠁⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⡠⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠉⢰⠉⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠑⠚⢻⠟⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠈⡘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠖⡄⠀⠀⠀⠀⠀⢠⡴⡆⠴⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⢌⠄⡝⠁⠉⠀⠀⠀⠀⢠⡇⠂⠀⠀⠀⠀⠹⡀⠀⠀⠀⡠⠂⠉⠀⠀⠀⢀⠀⠆⠀⠀⡠⠚⠀⠀⠀⡌⠃
⠀⣀⠀⠐⠒⠛⠛⠛⠓⢲⠃⠀⠀⢀⢈⠂⢜⠖⢀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠄⠀⠀⠀⠀⠀⠨⠀⡁⠀⠀⢁⠄⠒⠀⠘⠢⣄⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢃⡠⠌⡠⠂⠀⠀⠀⠀⠀⠘⡆⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈


⢀⠉⠃⣶⣴⣶⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⣄⣀⡀
⠈⠀⠀⠹⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⢰⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡟⠛⠁⠀⠀⠀⠠⠂⠀⠀⠀⠀⣀
⠀⠀⢾⠀⠀⡇⠀⢸⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣉⣷
⠀⠀⠘⠀⠀⡇⠀⠸⠀⠀⣠⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠄⠀⠀⠀⢁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢀⠀⠀⠀⠀⢀⠐⠤⠀⠠⠌⠁
⠀⠀⠀⠂⠀⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠁⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠐⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡄⢥

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠰⠏⢀⠀⠀⠀⠈⠉⠑⡄⠈⠁⡜⠀⠤⠀⠘⢂⠀⠀⠀⢰⠉⣀⠴⢄⠀⡼⠮⠑⠀⠀⠀⠀⠀⠁⠰⠀⢀⢄⠈⠀⠈⡆⠸⣡⠑⠀⢀⡠⠤⣤⣤⡀
⠀⠀⠀⡀⠀⢀⣀⠠⠄⠊⡍⠀⠀⢱⠀⠈⢠⢄⠀⠡⠀⠠⡁⠀⣁⠀⡈⠀⠀⠀⠀⡄⠁⠀⠐⠀⠀⠀⠐⠅⠀⠁⠂⡜⠀⢀⡈⠙⠀⠷⠦⠄⡸⢺⠁
⠀⠀⠀⢥⣁⡐⠠⠀⠀⠐⢈⠇⠀⠀⠑⢄⠀⠢⡑⢄⠇⠀⠈⠍⡹⢪⠅⠀⠀⡀⡰⠀⠀⠀⠇⠀⠂⠨⠀⠀⠀⢢⠎⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢰
⠀⢠⡄⠸⠄⠤⠀⠈⠑⠈⣀⣤⡀⠄⡤⢀⠉⠁⠐⠀⢠⣞⡃⢸⡞⣵⠀⠀⢀⠀⡕⠉⠈⣄⠣⠂⠰⢀⡈⢈⠔⠁⠀⢱⠶⠤⢤⡄⠀⠐⠀⠀⠒
⠀⠈⠃⠀⠠⡆⠀⢀⠖⠍⡀⠀⠌⡐⠠⠄⣘⢦⡀⠀⠸⠋⠀⠀⢯⡖⠀⠀⠀⡟⠀⠀⠀⣦⠀⠄⠐⠄⠐⠁⡐⠀⠀⠸⣄⣀⠀⠀⡀⠈⡅⠀⠀⠋⠂
⠀⠀⣠⠦⡄⠀⡠⢁⡨⠴⢤⡸⣒⣒⣫⠭⠒⡏⠀⡤⠴⠦⡐⡀⠸⠇⠀⠀⠀⠈⠉⢁⠊⠙⠊⠁⠀⠀⠀⠐⡠⠁⣤⠀⠀⠀⠀⠀⠁⠂⠂⠁
⠀⠀⢷⠀⠸⡄⠈⠣⣉⢈⢉⣉⣍⣍⠤⠴⠚⠀⠀⡇⠀⠀⢀⡜⠀⠠⠈⠐⠀⢀⣀⡀⠀⠀⠀⣀⣴⣶⣶⢾⣄⠀⠈⠀⠈⠀⠁⠀⠀⠀⡠⠆⢤
⠀⠀⡯⢀⢢⠘⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⡀⠻⣀⠀⡀⡠⠗⠀⠘⢜⡥⠀⠃⠀⠈⠀⢀⠚⠿⣶⠽⠿⠿⠿⢧⠀⡂⠀⠀⣐⠀⠀⠶⠤⠠⠂
⠀⠀⠹⣄⣉⡰⠃⢀⠀⠈⠣⢀⡉⠀⠒⢉⡹⠋⠀⠀⠈⠉⠁⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠰⡤⠤⠙⠄⣀⡀⠠⡤⠀⠁⢠⠐⠂⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠄⠑⡤⠀⢆⠀⠀⠈⠉⠀⠀⣀⣠⢴⢮⡫⢆⠀⢀⡔⠠⠀⣀⡈⣩⢋⠕⠀⢀⠀⠀⠀⠀⠁⢰⠀⠀⠤⠘⠠⠀⢀⡜⠀⠘⡄⡅⢰
⠀⠪⡤⠔⠀⡌⠀⠀⠀⠀⡔⠀⠀⢠⠉⡈⣷⡼⠽⠞⠉⡠⢊⠴⣎⠨⢅⡠⣥⡤⠀⢀⢀⡀⠀⠀⣉⠉⠀⠉⠁⠀⠃⠀⠀⠀⠀⠬⡀⢀⡴⠃⠈⠓⠁
⠀⠀⠀⠀⡘⠀⠠⠠⠁⠈⠀⠪⠇⠀⠅⣀⡀⠤⠄⠂⠉⡠⡦⢂⣀⣠⡮⠞⠁⢀⠢⠂⠀⠈⡆⠈⠉⢀⣀⠖⡆⠐⠚⠠⠠⡆⠀⢦⠤⠸⡀
⠀⠀⠀⢀⠠⢤⡐⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⡦⠀⠾⠒⠋⠉⠉⠀⢠⡺⡷⠀⠀⠀⢄⠜⠀⠀⢠⠛⠘⠁⠭⠬⠁⠨⠤⠌⠶⠀⠀⠈⢉⠔⠀⢄
⠀⢰⠀⠈⠀⠀⠱⠂⠀⠀⠠⡐⣥⣬⠁⠠⠀⢀⣀⣀⣠⠀⣿⣆⠀⠀⠹⠏⠁⡐⠦⠂⠁⢰⠦⠀⢘⢂⠀⠀⠀⠤⠄⠤⠐⢠⡄⠀⠀⠀⣌⠀⠀⢸
⠀⠀⠉⢀⣀⠀⣰⠚⡀⠀⠀⠀⠼⠁⠡⠀⠠⠟⠋⠁⠀⠈⠱⠿⠿⠃⠀⡄⠊⠀⠀⠀⠀⠀⢠⠀⡸⢄⣂⡈⠀⠀⠀⠀⣀⣠⠦⠀⠀⢠⠁⠀⠔⠺
⠀⠰⠢⠤⠤⠊⡀⠀⢁⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠃⢀⠄⠠⠀⠀⠑⠀⢠⠀⣉⡠⠖⠂⠀⠀⠀⠀⠩⠉⠭⠭⠀⠐⠈⠠⠄⠠⡀⠀⢀⠇
⠀⠘⡆⠠⠀⣀⡠⠈⢁⠊⡥⠀⡁⢱⠀⠢⢄⣀⡀⠤⠶⠗⢠⠃⠀⠀⢀⡀⢠⠀⠀⡘⠀⠀⢠⠀⢠⡟⠩⠀⠀⠀⠀⢀⠠⠀⠀⠳⡟⠀⠈⢸⠁
⠀⠀⠑⠐⠀⠁⠀⠀⠘⡄⠃⠐⠁⠀⡃⢀⠄⠄⠠⡀⠀⠀⠁⠂⠀⠀⢨⠇⠀⡁⠀⠀⠀⠠⠀⠁⠀⢀⡄⠭⠈⠡⢬⡇⠀⠀⠐⢄⠀⠀⠐⠂
⠀⠠⠂⡐⡢⢲⠁⠄⠀⠡⣀⠔⠂⠈⠀⡘⠁⢀⠠⠀⠈⠀⢀⠶⠀⠀⢹⠀⠐⡀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠈⢀⠤⠤⠜⠒⡥⠂⠉⠀⢤⡔⠀⡈⠄
⠀⠐⠃⠈⢰⠈⠃⠚⠁⠀⡔⠀⠒⡄⠀⠒⠉⡯⠆⠀⠠⡀⠀⠉⠐⢀⡘⠀⠀⠀⠸⠆⠀⡠⠏⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⢰⠀⠀⢀⠘⠛⠉
⠀⠀⠠⠤⠘⠒⠐⠉⠃⠀⠈⠃⠁⠀⠀⠀⠀⠇⡈⠀⠀⠈⠐⠒⠒⠉⠀⠀⠀⢒⣄⠀⠪⣁⠌⠒⠠⠄⠶⡉⠐⠶⢚⣁⠤⠊⢀⡀⠐⡀⠀⠠⠂⣀
⠀⠀⠀⠀⠀⠉⠑⠀⠀⢀⢀⡀⠀⠣⢄⡊⠀⣀⡀⢀⡖⠒⠚⢳⠀⠀⢠⠀⠀⡄⠀⠀⠑⡄⠀⠉⠁⡠⡀⠀⣠⠄⠀⡠⡀⠀⠈⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⡀⠠⠀⠁⠐⠀⠘⢄⣢⠐⠂⠀⠀⠈⠥⡬⠂⠹⠀⠀⡏⠀⠀⡁⠀⠀⢄⣀⣀⡔⠀⠀⠀⠸⠬⠋⠋⠰⠋⢸⢠⡘⠀⢁⣀⠔⠐⠒⠂⠄⠀⡀
⠀⠀⠒⢀⡠⠤⢄⡀⠀⡀⡄⠄⠀⢀⠄⠂⢄⠈⠀⠀⠙⠒⠚⠓⠒⠒⠁⠠⠄⠀⠀⠈⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡉⠈⠉⠉⢩⠃
⠀⠀⡴⢯⣗⢸⠎⢁⠀⡷⠄⡠⠐⠁⠈⠀⢸⠀⠀⡴⠊⣀⡀⠱⠆⠀⠀⠀⠠⠐⡖⠀⠀⠀⡠⠔⠈⠉⠑⡾⠿⠷⣂⠉⠁⠀⠀⢱⠤⠀⠀⠀⡔⠃
⠀⠀⢿⠷⠑⠁⠀⢸⠀⠀⠀⢀⡀⠀⣀⠔⢡⡀⢸⢠⠎⠁⠀⢤⠈⠐⠂⠀⠐⠊⠁⠀⠐⠀⡀⣀⡄⠤⠔⠀⠀⠑⢄⠀⠀⠀⠠⠄⠀⠀⠀⢈⣁⣰⠆
⠀⠀⠀⠁⠀⠀⠂⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⢂⠀⢀⡐⢤⡀⠀⠀⢸⠁⠙⡏⢀⣄⡠⠊⠉⠀⣀⡤⠤⠤⡞⢁⠀⠑⢄⠀⠠⠤⡀⠀⠀⢰⠛⣆⠀⠀⣼⠛⠲⢤⡒⠒⠲⢤⣀⡀
⠀⠀⢠⠡⢄⠙⣄⢸⡇⢀⠽⠀⠀⠈⠓⠊⠁⢇⠀⣉⠆⠀⢸⡅⠀⠀⠂⠈⠉⠁⠉⠁⠀⠤⠔⠻⣄⠀⢣⣀⡟⠀⣸⠃⠀⠀⠀⠉⠓⠲⠤⣌⠉⠉⠉⢳⡀
⠀⠀⠀⠂⢂⣹⣾⣟⠟⢁⡠⠔⠒⠙⠒⠦⣄⡀⠉⠀⠀⠀⣸⡬⢀⣀⠀⠀⠀⠀⠀⠀⠀⠐⢈⣰⡟⠀⢨⠋⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢀⡞⡄
⠀⠀⠀⢐⠉⡠⠈⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⣾⡣⢀⣀⠉⠉⠉⠉⠉⠉⠉⠉⠁⣠⡟⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⢞⡽⠁⠀⡾⡰
⠀⠀⠀⠀⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢹⡄⠈⠳⠤⠄⣀⣉⣉⣉⡉⠉⠉⣉⣀⡠⠇⠀⠀⢾⡀⠀⠀⠀⠤⠀⠀⠀⡸⢡⠏⠀⠀⢰⢇⠇
⠀⠀⠀⠀⠀⠐⣇⠐⡀⠀⠀⠀⢠⠟⠉⠉⠙⢲⡀⠀⠀⡇⢀⣤⣴⣴⡄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⢯⣓⡶⣤⣀⠀⠀⣰⠃⣟⠀⠀⢀⡼⢸
⠀⠀⠀⣄⡀⠀⠙⢦⡁⠀⠀⠀⣿⠀⠀⠀⠀⠄⡷⠀⢸⠇⣼⠁⠂⢠⡇⠀⢀⠤⠤⢄⠀⠈⠉⠙⣷⠀⠀⠀⠀⠀⠉⠛⠮⣝⠶⣋⡼⠥⣭⣍⣭⡤⠟
⠀⠀⣴⠋⠙⢷⡄⠀⠙⢦⡀⠀⠘⢷⣝⣒⣩⠾⠁⢠⠏⠀⠳⣤⠴⠋⠀⠀⠸⣔⠀⠀⣷⠀⢠⣴⠚⣟⠛⠋⢉⣛⣛⣶⠀⠈⠉⠁⠀⠀⢀⡤⠤⣄
⠀⠀⣿⡀⠀⠀⠉⠢⣄⠀⠱⣄⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣤⠤⠤⠤⠤⣤⣀⠈⠙⠛⠃⠀⢸⠉⠉⢻⡏⠉⠉⠉⠁⢰⠀⠀⠀⠀⠀⢠⠏⠀⠀⢈⢧
⠀⠀⠹⣷⠀⠀⠀⠀⠈⠙⠦⣌⠑⠲⠶⠶⠚⠋⣠⡞⠉⠀⠀⠀⠐⠀⠀⠀⢈⡗⢄⡀⠀⠀⠸⡇⠀⠀⣻⣀⣀⣀⣀⢸⠀⠀⠀⠀⢀⡾⠀⠀⠀⠸⠘⣀
⠀⠀⠀⡿⣇⠀⢰⠋⠉⡳⣄⠀⠱⣄⠀⠀⢀⡞⠁⢳⡀⠆⠀⠀⠀⠀⠀⠀⡼⠀⠀⠙⠢⡀⠀⣿⣭⣭⣬⣯⣬⠭⠽⢿⠀⠀⠀⠀⣿⠳⠤⠤⠤⠤⠴⢞⠂
⠀⠀⠀⣿⠹⡆⠈⢧⡀⠘⢸⠇⠀⣿⠀⠀⣸⡀⡄⠀⢷⠀⠀⣤⡤⠀⠀⢰⠇⠀⢠⠞⣰⢇⠀⢹⠀⠀⠀⠸⣇⠀⠀⣾⠀⠀⠀⠀⠘⢦⣧⣄⠀⢀⣤⠞
⠀⠀⠀⠸⣆⠹⣦⡀⠉⠙⠉⠀⣠⠏⠀⢠⡇⡙⢦⣀⣸⠧⠖⠒⠒⠒⠒⠊⠙⠶⠤⣞⠁⠈⡆⠘⣇⠀⠀⠀⠻⠆⢀⡇⢠⡴⢲⡀⠀⠀⠉⢛⠛⡋⠁
⠀⠀⠀⠀⠈⠳⢤⣉⣓⣲⣶⠿⠋⠀⠀⠀⠳⠴⠛⠉⠀⠀⠀⣤⣀⡀⠀⡄⠀⠀⠀⠘⠲⠞⠁⠀⡘⠲⠶⠶⠖⠒⠋⠀⠈⢧⣠⡇⠀⢴⣛⡃⠀⢙⣷⡆
⠀⠀⠀⠀⣀⡀⠀⣀⣠⠤⠤⠴⠒⠒⠲⣄⣠⠖⠒⡄⠀⠀⠀⠘⢎⡻⣷⡇⠀⠀⠀⠀⠀⠀⢠⡾⣿⣗⡶⠤⠤⠤⠤⠤⣤⣄⡀⠙⠒⠶⢬⣉⡉⠉⠁
⠀⠀⠀⠘⡏⠉⢿⠦⣀⠐⢒⣒⠂⠀⠀⠈⠳⡄⠘⢧⠀⠀⣠⠞⡩⠭⠛⠳⠞⠛⠙⠓⢦⠀⠘⣷⣮⣝⣛⡻⠶⠶⠶⢶⣶⣦⣍⡳⢦⡀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⣽⡆⠈⠛⣶⣿⣶⣄⡀⠒⠀⠀⠀⠱⡄⢞⠀⠀⡏⠘⠀⠀⠀⠀⠀⠀⠀⠘⡌⡇⠀⠈⡙⠻⠿⠿⠿⠿⠿⠶⣶⣮⣭⡛⢦⡽⠂⠀⠀⠈⣧
⠀⠀⠀⠘⣏⠫⣒⢆⠻⢷⣪⢟⡿⢷⣦⣄⡀⠀⠘⢌⠃⠀⣧⠀⠀⠀⠀⣠⢀⡀⠀⠀⠁⡇⠀⢸⣯⣙⡒⠒⠲⠶⠤⠤⢤⣈⡙⠻⡟⠀⠀⢀⣤⡤⠿⣀
⠀⠀⠀⠤⡈⠓⠮⣕⠦⣈⠀⠸⢄⣀⠉⠛⢽⡒⠤⣬⠇⡀⠸⣆⠀⠀⠀⠁⠈⠁⠀⠀⣸⠃⠀⢸⣗⣚⣛⣛⣛⣛⠛⠛⠶⢦⡉⡢⠀⠀⠀⣞⠀⣀⡀⣾
⠀⠀⠀⠀⣝⣆⠀⠈⠙⠪⣝⠦⣄⠀⢣⣀⠤⡈⢙⠋⠀⠀⠀⠙⢮⡒⠄⣀⡀⠐⣂⡴⠋⠀⠀⠀⠙⠛⠿⠭⠽⠽⠯⣍⡑⠢⢈⡇⠀⠀⡼⠋⡋⢩⣽⣿
⠀⠀⠀⠀⣹⣿⣾⠭⢋⡷⠈⠑⢮⡙⠦⢤⡤⠋⠁⢀⡠⢤⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⣠⡶⠚⠛⠉⠙⠛⠲⢤⣀⠀⠙⠲⠬⠇⠀⠀⠙⠛⠻⠾⢿⡼
⠀⠀⣴⣻⠿⠃⠈⠉⠁⠀⠀⠀⠀⠈⠑⠋⢀⡠⠞⠉⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠌⢧⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀
⠀⠀⢀⣠⣤⠒⣒⣒⣒⣢⣴⣖⣒⢠⡀⠀⢸⡓⠒⠒⠒⠶⢶⡶⢶⠤⣤⣉⣓⢦⡀⢸⡿⡀⠀⠀⠀⠀⠀⠀⠀⠠⣣⠎⠀⣠⠞⢋⡟⠒⠁⠀⠈⠙⢦⡀
⠀⠀⣏⢸⡍⢻⠉⠛⠛⠛⠛⠉⠉⢹⠀⠀⢈⣭⠇⠀⣀⣀⠀⠉⠁⠀⠀⠀⠈⣹⠁⠀⠳⣝⠲⣄⠀⠀⠀⠀⠀⠀⡇⠀⣸⠃⠸⣅⢀⡴⠒⢶⣄⠀⠀⢇
⠀⠀⠘⢮⣷⠈⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠸⣇⠀⠀⠳⠟⠀⠀⡤⢤⣄⠀⠀⢳⠀⠀⠀⠈⣳⢹⡀⠀⠀⠀⠀⠀⢹⠀⢻⠀⡞⢁⠸⣆⣀⣼⡿⠀⠀⡼
⠀⠀⠀⢰⣟⣳⣀⣀⣀⣀⣀⣠⣞⣢⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠓⠞⠋⠀⠀⢸⠀⠀⠀⠀⢹⡈⡇⠀⠀⠀⠀⠀⠈⡇⠘⣇⠣⣠⠤⣌⣉⣭⣠⠞⣳⠃
⠀⠀⠀⠀⠈⠙⠻⢯⣿⠿⠿⠛⠉⠀⠀⠀⠀⠙⠛⠛⠓⠒⠒⠋⠙⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠳⢿⣤⣤⣤⣤⡤⠴⠃⠀⠈⠓⠶⣤⣌⣁⣤⡵⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢉⣘⠉⠀⠛⢉⣁⠘⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⠈⣀⠀⢀⡀⢀⣀⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠚⢂⠋⢣⠚⠂⣸⣿⠏⢹
⠀⠀⠀⠀⠀⠀⢸⣿⠏⡈⢿⣿⣿⣿⣿⣿⣿⣿⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠆⠀⠀⠀⠈⢩⣤⠋
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠉⠉⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠤⠄⠒⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠶⠀⠀⠀⠶⠀⠀⠈⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⣤⠤⠤⠴⠖⣺⡁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡀⡀⠀⠀⢀⣶⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⡿⡿⣏⢳⣶⠋⠉⣀⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⣄⣈⣀⣠⡴⠋⢹⣿⣿⣿⠇⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣿⣿⣅⠠⠚⠈⡟⠹⠶⢊⣽⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⡍⠉⠉⠛⢳⠸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣠⣶⣶⣶⣶⣿⣾⣼⣿⣿⣧⠀⠀⣧⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⣿⣷⡀⣤⠴⠞⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢸⡹⣿⣿⡟⢷⠁⢸⡇⠀⠀⢠⣿⢿⡿⢋⡿⠋⣿⠛⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠙⣿⡟⠀⠸⣆⡼⣇⠀⢠⡿⣿⠀⠀⠁⠀⠀⠀⠈⠀⠈⣿⣿⠋⠑⠁⠀⠉⠉⠻⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⡟⣇⠀⠀⣿⠀⢹⣤⡟⢸⡙⡆⠀⠖⠀⠀⠀⠶⠀⢀⣿⢹⠀⠤⠄⠀⠀⠠⠤⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⡇⢹⡆⠀⢸⡄⠀⢿⠀⢠⣿⢇⠀⢴⣐⣒⣐⡆⠀⢈⣿⣿⡀⠀⢠⣤⣤⣄⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢷⠀⠀⣰⠏⣿⣴⣿⣾⣿⣯⣵⠒⠚⢯⡀⢉⣿⣶⣾⣿⣭⣥⠤⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠗⢻⡇⠀⠀⠈⠳⠴⠀⣠⢿⣿⣿⡟⠹⡏⠀⠀⠀⠀⣱⣿⣿⣿⡟⢿⠋⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠻⣟⣻⢿⠣⣤⠷⠤⠴⠂⢀⣷⠾⣿⡿⣄⣸⢦⣀⡤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠴⠤⠤⠦⠶⠒⠛⢶⠋⠀⠙⠲⠤⠤⠔⠋⡏⠯⣵⠃⠈⠣⢤⣀⣠⠴⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⢤⣤⣤⣤⣤⠤⠴⠇⠀⠻⠤⣤⣤⣤⣤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠘⠀⠀⠀⣰⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⡄⠠⠀⠀⠀⢀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠉⢉⣡⢿⡅⠁
⠀⠀⠀⠀⠀⡆⢐⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠁⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠐⠁⠎
⠀⠀⠀⠀⠀⠔⠉⢈⠂⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⢀⢢⠀⠅⠇⠉⠆⣑⢄⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⢀
⠀⠀⡠⠥⠰⣉⠂⣌⠛⠂⣈⠾⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⡀
⠀⢠⠁⠆⠂⡀⢠⢤⡀⠭⠐⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⡀⡀⠀⠀⠀⠈⠉⢣⠔⠂⠘
⠀⠀⠉⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠔⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢉⠐⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡤⠐⢂⡈⠒⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠼⠉⢆⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠤⠄⠀⠀⠀⠀⢂⠉⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠒⠂⢁⡀⠆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠺⡀⠀⢠⠬⢴⠴⠁⠒⡱⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠈⠈⠢⠐⡘⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠠⠤⢤⠄⠠⣈⠡⠉⣠⠛
⠀⠀⠀⢠⠤⠀⡐⢂⡠⣄⠀⠀⠀⠀⠀⠀⠌⢂⢠⠐⡈⡀⡵⠀⠀⠀⠀⢀⠔⡁⠀⠤⠄⢀⠄⠀⠀⠀⠀⠀⠈⠦⣄⣀⣉⢊⣁⡀⠔⠁
⠀⠀⣞⠢⣀⡿⠁⣸⠁⠘⠳⠄⠀⠀⠀⠀⠀⠄⡂⠣⠘⠉⠀⠀⠀⠀⠀⠁⢤⡈⢠⠄⠀⡄⠀⠴⠂⢀
⠀⣀⡐⢄⠘⢳⡸⡛⣺⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⣤⠌⢀⠀⠂⠐⠦⠀⣸
⠀⠛⠬⢐⠦⢀⢳⠹⣜⡽⠊⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠀⢀⡛⠁⣈⠃⠄⢊⠈
⠀⠀⠀⠱⠒⠛⠚⠉⠉⠑⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢖⣀⠄⠀⠀⠂⠁
⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠈⠂⠠⠀⠐⠈⠁⠀⠀⠀⠀⣀⡀⠤⠐⢉⠠⢋⢉⡰⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡫⠀⠈⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⡶⠒⠒⡠⢉⣔⠩⠋⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠑⠌⠄⠀⢹⠗⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠖⠃⠈⠀⠁⠀⠀⠀⠀⠀⡨⠊⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⡀⡬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⠀⠀⢀⡤⠖⠂⠀⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠄⠀⠐⠚⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠁⢀⡀⠓⠒⠒⠒⠒⠒⠒⢀⡀⠀⠀⠀⣶⡷⣿⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡠⢺⢀⣁⠆⠁⠀⡂⡾⡀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡟⣰⠋⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠰⣀⠀⠇⢐⠈⠼⠀⠀⢻⡥⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠤⠲⠶⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢨⣉⢿⡇⠄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠙⠚⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠰⠒⠀⠀⢸⠀⠆
⠀⠀⠀⡠⠀⠰⢂⠛⢺⣳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠡⠄⠄
⠀⠀⠀⢉⣀⠀⠀⠌⠀⠨⠌⢻⠍⠀⠀⠀⠠⠄⠐⠀⠀⠁⠀⠀⠀⠈⠀⠀⢣⠀⠀⠀⢀⢀⠀⠠⠄⠤⣀⠀⠀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠁⠰⢤⢀⠀⠈⠓⠂⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠰⡁⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠭⠙⢲⡄⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒⠊⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂
⠀⠀⠀⠀⢀⣀⣄⣀⣀⡀⡀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠄⠀⠀⠠⠶⠀⠀⢂⠀⠀⠠⠄⠀⠀⠈⢀⠠⢀⣀⠀⠒⣤⡀
⠀⠀⠀⠀⠨⡉⠁⠙⠀⠛⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣰⠆⠀⢀⡀⠀⠀⠀⢀⠀⡡⠀⣁⢀⡴⠎⠁⡱
⠀⠀⠀⠀⠀⠱⡒⠲⠲⠒⠒⢚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣀⢠⠴⠙⠘⠀⠀⠀⡀⠋⢸⠅⠀⡀⡤⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⠀⠀⢀⠉⠃⠀⠀⢒⠈⢡⠀⠠⠇⠀⠘⠀⠂⠁⠑⠀⠀⠀⢀⠀⠈⠀⠀⡀⡁⠀⠄⠊
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠐⠈⠀⠈⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⢠⠀⢀⣄⠀⠀⠀⠉⠁⠀⢀⠐⢉⡀⠀⠀⠀⠀⢀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⡘⠀⠀⠠⠀⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⢀⣠⠰⠃⣀⢶⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡘⡷⠀⢠⣄⠀⠀⠀⠀⢀⠨⠁⠀⠄⠄⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠾⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢂⢀⣔⢴⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠈⠿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠁⠀⠛⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠒⢀⠀⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠈⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠀⠉⠀⠀⠀⠈⠀⠀⠄⠀⠠⠠⠔⠲⢖⠒⠀⠥⠀⠀⠀⢢⢢⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠠⡏⠀⠀⠀⠀⠈⠉⠑⢆⠀⠋⢰⠋⡠⠄⠀⠓⣄⠀⠀⠀⢘⠁⣀⠥⠀⠀⣌⢮⠑⠁⢀⠀⠀⠀⠊⢄⡊⠀⠠⠀⠁⠄⢱⠀⢛⣌⠂⠀⢀⡠⠤⣤⣄⡁
⠀⠀⠀⠀⠀⠀⡀⣀⠠⢤⣋⠃⠀⠘⡄⠈⠂⡄⡀⠈⠆⠀⢠⠁⠀⣁⠀⡸⠀⠀⠀⠐⡂⠀⠐⠀⠠⠀⠀⠀⠆⠀⠀⠐⢀⠆⠀⣀⠑⠃⠰⡯⡤⠄⡹⢺⠁
⠀⠀⠀⠌⡄⠡⠀⠀⠔⡠⠌⡳⠀⠀⠘⢆⡀⠠⣈⠢⣘⠀⠀⠙⡝⡣⢬⠁⠀⠀⠀⠠⠅⠀⠀⠸⠀⠀⠠⠁⠀⠀⠑⡠⠋⠀⠀⠉⠓⢄⠀⡄⠀⠀⠀⢸
⠀⢀⡀⠰⣉⣉⡀⠤⠂⠂⠉⣀⣀⢀⣀⡀⠈⠒⠈⠔⠀⢠⢶⠀⢛⠽⣳⠀⠀⠀⠡⡡⠊⠘⢄⢀⠰⢀⠅⡀⠀⢂⠜⠁⠈⠦⢇⣀⣀⡘⠀⠃⠠⠤⠔⠈
⠀⠈⠇⠀⠀⣤⠀⠀⡠⠔⠋⠃⢠⠃⠌⠠⠉⢲⡀⠀⠀⣿⠙⠀⠈⣞⡜⠀⠠⠬⡶⠁⠀⠀⣸⡀⢀⠐⢄⡩⠔⢁⠄⡀⠀⣀⠀⣀⠌⠀⠄⢠⡀⠀⠀⢒⠄
⠀⠀⢀⣄⡀⠁⠀⡈⢁⣀⣎⡀⠦⠬⠐⡦⣮⠵⡎⠀⣀⣤⣄⢀⠀⢻⡃⠀⠀⠀⠙⡂⠐⡦⠛⠬⠂⠀⠀⠀⠀⠌⡔⢀⣀⠀⠉⠉⠀⠐⠂⠁⠀⠆
⠀⠀⢿⠀⢹⡄⠘⠰⠒⠠⠐⠚⠒⠒⣉⣀⡤⠞⠀⢘⠇⠀⠀⢉⡗⠀⢠⠄⢠⠀⠀⡀⠼⠀⠀⠀⠀⢀⣤⣄⣤⣎⠀⠈⠋⠀⠊⠑⠀⠀⠈⠀⣀⡀⣀⠂
⠀⠀⢸⢀⠤⠑⡄⠀⠉⠓⠃⠉⠉⠉⠉⠀⠀⠀⠐⡼⠀⠀⠀⢸⡇⠀⢠⠰⠀⠀⠎⠁⠃⠀⠀⠠⣿⣿⣝⣿⣵⣿⣷⠀⢠⠄⠀⠀⡆⠀⠤⣼⡁⢠⠍
⠀⠀⢱⡈⠒⢀⡇⠀⠀⠀⢳⠚⠒⠀⠤⠄⠒⢶⠂⠈⠃⠒⠶⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡀⠀⡼⡅⠀⠀⠀⠀⢃⠀⢄⣀⡠⠃⠀⠀⠉⠉
⠀⠀⠀⠙⠙⠋⢀⠴⡀⡀⢄⠉⠀⠐⠒⠒⠊⠁⠀⢀⢠⡠⢂⡀⠀⠀⣀⠁⠀⠀⠒⠒⡶⢦⠀⠉⠉⠀⠀⢘⠻⠀⠉⠁⠀⢀⠇⠀⠀⠀⡀⠀⢣⠈⠂⢢
⠀⣰⠂⢐⠀⢠⠃⠀⠉⠁⢈⠀⠀⠀⠠⠄⠤⡄⡖⣆⡿⠿⢉⡕⢀⡔⢱⠊⠉⠌⠉⠉⠐⠁⠀⠈⠒⠀⠤⠀⠀⠐⠀⢠⠌⠁⠀⠁⠀⠞⠁⠀⢀⠗⠲⣎⠄
⠀⠈⠉⠀⢀⠆⠀⠀⠀⠀⠄⢠⢤⠀⢲⠒⠀⠛⠁⣀⠠⠔⢋⡔⠹⠃⠈⣰⣹⠟⠉⠀⠒⠀⠒⡄⠀⠶⠀⠀⣠⣀⡀⣠⡄⠀⣀⠀⢰⠁⠂⣰⠁
⠀⠀⠀⢀⠎⠀⠀⠐⠀⠈⠀⠀⠁⠀⠈⠁⠀⠈⢉⡀⠀⣠⣈⡥⠴⠖⠚⠋⣡⣄⢠⠀⠀⠀⢀⠇⠀⠀⣤⢒⠇⠨⠄⠤⠀⠅⠘⠠⡀⠙⢁⠼⠂⡀⣀
⠀⢀⠀⠰⢈⠋⠣⣀⠄⠀⠀⢀⡂⠀⠤⠄⠀⠀⠙⠁⠀⠁⠀⣤⠀⠀⠀⢞⣭⠍⢸⡀⣀⠔⠁⣀⠀⠀⡀⠀⠠⡈⠉⠉⠁⠉⠉⠁⡁⠀⠀⠀⡘⠀⠀⢁
⠀⠘⢤⡀⠀⠈⠁⣁⡀⠀⠀⠰⠺⣾⠪⠀⠁⢠⣶⠶⠒⠒⠠⢟⡦⣄⡀⠘⠁⡠⠀⠉⠀⠀⠈⠊⠀⠀⡆⠁⠀⠀⠀⠉⠀⠁⠀⢹⠁⠀⠀⠀⠇⠀⢀⡘
⠀⠀⡀⠄⠲⢈⠜⠁⢀⠀⠀⠀⠀⠑⠄⠀⠠⠋⠀⠀⠀⠀⠂⡈⠉⠉⠀⠐⠏⠄⢀⠀⠄⠠⠔⢈⣆⠈⠑⠂⠠⠤⠤⠤⠤⠤⠔⢚⠳⠀⠀⠰⠈⠀⠁⡇
⠀⢨⠁⠙⠀⡁⠆⠀⡠⢀⡠⠄⠤⢄⠀⠀⠁⠀⠀⠀⠀⢈⡄⠀⠔⠈⠉⢂⠀⠈⠁⠐⠀⠄⠂⠉⠀⠀⢀⡄⡄⠀⠀⠀⠀⠀⠈⠀⢀⣎⡆⠐⠤⢤⠰
⠀⠀⢅⢐⡂⠠⠔⠁⠠⡃⣰⠉⢱⠀⢧⠀⠛⠀⠀⠐⠚⠛⠁⠜⠀⠀⠐⠸⡄⠐⡄⠠⠜⠀⢂⠉⠆⠀⠏⠐⡀⠄⠀⠀⣀⠐⠠⠀⠀⠓⠇⠀⡇⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢱⠈⠈⢁⢀⡠⠃⣾⠑⠒⠒⠄⠠⠊⠈⠀⠀⠀⣸⠃⢀⠀⠀⠀⠀⠘⠁⠀⠀⣀⠼⠒⠀⠀⠒⠺⡀⠀⠀⣈⡆⠀⠀⠀⠀⡀
⠀⠀⡁⢨⠽⢹⣀⢓⡀⠈⠢⠚⠁⠀⠀⢰⠁⣀⡤⠊⠀⡇⡀⠰⠿⠀⠃⢸⠀⠈⠄⠉⡁⠀⠀⢀⠴⠊⠁⠀⠀⢀⠔⠂⠀⠉⡹⠊⠀⠀⠀⢾⣟⣀⠤⠂
⠀⠈⠁⠀⠠⡀⠸⢘⡀⠀⠰⣉⣈⡳⠀⠀⠁⠘⠘⡄⠀⠡⡐⠀⠀⢈⣠⠎⠀⠀⠀⠈⠁⢀⠄⢃⠀⠀⠀⠤⢀⠀⠀⠀⢀⡼⠡⡮⠀⠀⠒⠊⠉
⠀⠀⠀⠓⠀⠃⣉⠉⠁⠀⠀⠁⠀⢠⠄⠀⡄⠀⠤⠂⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠨⠦⠀⣑⠬⠤⣉⣒⣒⠚⣥⢊⡛⠋⠥⠐⠊⠁⡄⠄⠀⠄⠀⠀⢀⡀
⠀⠀⠀⢀⠀⠁⢈⠀⠀⠀⡉⡕⢀⠈⠒⠬⠀⢀⣤⡀⢰⡋⠉⠉⣷⠀⠉⢱⠀⠀⠃⠀⠀⠀⡀⠀⠀⢀⡠⣤⠀⡠⢦⡀⣀⠤⡀⢀⠉⠀⠀⠀⠀⠀⠈⠖
⠀⢠⠀⡀⠢⠀⠠⠐⠀⠀⠱⢡⠞⠁⠀⠀⠀⠱⡤⠜⠀⠇⠀⢸⠇⠀⠀⡆⠀⠈⠢⠀⣠⡔⠁⠀⠀⠈⠠⠋⠈⠄⠎⠀⠅⠠⠁⠀⣴⣤⢂⡘⠃⠁⢆⣀⡄
⠀⠀⠉⣀⣤⠐⠤⡀⠀⠀⢀⢰⠀⠀⡠⠒⠒⡄⠀⠀⠀⠉⠉⠉⠉⠀⠈⠀⠰⠆⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢇⠠⠤⠀⠀⠀⠀⣠⠃
⠀⠀⣼⢳⣽⣐⡟⠉⠀⠼⠄⢈⠔⠉⠀⠀⠂⡄⠀⢀⠔⢁⡀⠤⠵⠆⠀⠄⠠⠄⠂⡚⠀⠀⠀⡀⠔⠈⠈⠉⡶⠻⡿⠥⠀⠀⠀⠀⠈⠒⠂⠤⠤⠤⠔⠁
⠀⠂⢻⡶⠒⠉⠀⠀⡄⠀⠀⠸⢀⠀⣀⡠⠊⣤⠀⡇⡰⠋⠀⢠⢤⠀⠑⠒⠒⠒⠊⠀⠀⠐⠀⣀⣀⠤⠤⠔⠁⠀⠈⠢⣀⠀⡀⠀⠠⠄⠁⠀⠀⣀⣀⢺⠃
⠀⠀⠀⠈⠐⠐⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠿⠿⠿⢿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣽⣦⠀⠀⠀⠀⠀⠀⠀⠿⣣⣤⡀⠀⠀⠀⠀⣀⣤⣟
⠀⠀⣾⡿⣾⣷⡾⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢽⣿⢿⢧⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣶⣾⣿⣿⡟
⠀⠀⣸⣟⣿⣽⣟⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠉
⠀⠀⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣰⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄
⠀⠀⠀⣿⢁⡻⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠹⠯⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣈⣀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣀⣀⠑⠤⡀
⠀⠀⠀⣿⡜⢣⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢀⣟⣛⣻⣿⣿⣿⣛⣛⣣⠀⠀⠀⠀⠀⠰⠋⠀⣿⣿⣶⡆⠀⠈⡄
⠀⠀⠀⣿⢸⡆⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠀⠃⠀⢰⡇⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⢄⠀⠙⠛⠛⢁⡀⠠⠁
⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠉



⠀⠀⠀⢀⡀⠂⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡓
⠀⢐⣛⣋⣀⣀⣉⣀⣈⡂⠀⠀⠀⠀⠀⠠⠷⣌⣉⣳⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣽⣦⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣽⣥⣤⣤⡄
⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣙⣛⣛⣛⣛⣛⣛⡃
⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁


⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧
⡀⣴⣿⢹⣿⣿⣿⣿⣦⣠⣴⠀⠀⠀⠀⠀⠀⣶⣾⣿⣙⣿⣿⣿⠀⠀⠀⠀⠀⠀⡏⢠⣴⣄⠀⠀⢊⣴⣦⠀⣷⠀⠀⠀⠀⠀⠀⢠⡞⠐⠊⢣⡀
⠹⢿⣿⣦⣛⣛⣼⣿⠟⠻⣿⠀⠀⠀⠀⠀⠀⣿⣥⣿⣿⡋⢈⣿⠀⠀⠀⠀⠀⠀⢷⡘⠿⠟⠀⠇⠘⠿⠟⣠⠏⠀⠀⠀⠀⠀⢠⠿⠄⢁⡘⢀⠷⡀
⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠁⠀⠁⠒⠋⠁⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⣶⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⢀⣴⢞⣿⡧⢻⣿⣷⡀⠀⠀⠀⠀⠀⡀⢢⡐⣶⠆⢠⣤⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷
⢀⡾⢿⣟⣹⢃⣿⣿⣿⣿⡇⠀⠀⠀⠀⢇⠻⢞⡣⢾⣷⢋⡴⣎⢻⣷⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣛⣟⣟⠃
⠸⣷⣿⢟⣱⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠓⠈⡥⠿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟
⣿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠌⠉⠐⠒⠂⠁⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠂⠀⡄⠀⠀⠀⠂⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠄⠂⠀⠀⠀⠀⠀⠀⢠⢄⡄⠀⠠⠤⠄
⠀⠀⠉⠋⠀⠀⣀⣤⠊⠀⠀⠀⠀⠀⠀⠂⠀⣀⣀⡤⠀⠠⣴⡒⠒⢂⣏⠀⠇⢷⡀⡀⠀⠀⠀⠀⠀⠀⠀⡱⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡞
⠀⠀⠀⠀⠀⠈⠣⡠⠤⠤⠐⣲⡖⠀⣨⣥⣶⣀⠀⠠⠈⠉⠙⠚⠉⠁⠠⠀⠙⠻⣛⢕⠂⢀⣁⡠⠤⠖⠉⠀⠀⡎⠀⠀⠀⠀⠀⠈⠰⢶⡿⢖⣠⣤⡤⠤⠤⠀⢸
⠁⠀⠀⠀⢀⠀⠀⠑⠲⠶⠛⠉⠀⣈⣁⠤⠄⠀⠀⠀⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠧⠤⢄⣒⠉⠉⠥⣤⣏⡀⠄⠒⠁⠀⠀⡀⠀⢸
⠀⠀⠀⢄⠀⠀⠂⠀⠁⠀⢉⡩⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠶⠨⠤⠄⠀⠂⠀⠈⠁⠀⡌
⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⣠⠒⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠉⠂⠀⠀⠇⠀⠀⢀⠀⠀⠀⢀⠀⣙⡦⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢲⣦⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡃
⠢⠀⠀⠐⠁⢀⡞⠊⠀⠀⠒⠀⠈⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣿⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠔⠒⠈⠓⠒⡂⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⣦⠀⠀⠀⠀⢀⠠⠐⠀⢀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⡠⠘⠁⡠
⠀⠀⢹⢿⠃⠀⠀⠀⠀⠀⠁⠀⠈⠄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠠⠃
⠀⠀⡈⠓⠲⣦⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠈⢇⣦⠤
⠀⠚⠁⠀⢷⣏⡄⢀⣀⣤⠤⠄⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠡⠐⢋⣤⠄⠈⠀⠖⠀⢀⣀⠀⠁
⠀⢄⠀⠀⠘⣦⢿⡧⠄⠀⠀⠀⠀⠀⣠⠴⠞⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠠⠇⢀⠀⠀⡨⠋⡠⠐⠃⣀⡤⠊⢩⡇⠸⡅
⡀⢀⡠⠀⠀⠘⣮⢤⠀⣈⡄⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠓⠒⢄⠀⢀⢎⠀⠀⣀⠄⠈⣹⣥⠀⡼
⡀⠀⡸⠀⠄⠀⠘⡭⣦⣬⠴⠒⠀⠐⠀⠀⠀⠀⡄⠀⢀⠴⠀⠚⠃⠈⠈⠁⠐⠠⡀⠀⠀⡀⢀⡰⠀⠁⠈⣰⠶⠒⠂⠲⣄⢀⣠⠔⠊⣧⡆⠀⣿⠟⠋
⠁⠈⠀⠀⠀⠠⠀⠘⡎⠀⠀⠀⠀⠀⡃⠀⢀⡠⠃⡴⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠻⣿⡉⠐⠶⠒⠧⢤⣀⣀⣦⠚⠉⣇⡄⠀⡿⠗⠈⠀⠀⠀⠄⡠⠤
⠀⢠⡀⠀⠐⠀⠃⡀⠙⠤⣀⣀⣀⢠⡅⠂⠁⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⢧⡀⣲⣦⡀⢘⣿⡀⢼⣣⠀⣯⠟⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⠇⠀⡄⠀⣀⠔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⠿⣵⢆⡤⠀⣀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⠀⠀⠀⠈⠁⠒⠛⠷⠼⠿⠊⠁⢀⠔⠁⠀⠃⢡⠀⠀⡠⠖⠉

⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⢘⣛⣛⣛⣛⢛⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⠀⠀⠀⠀⠀⠛⣷⣦⣄⣀⣀⣠⣴⣾
⠀⠘⣿⣿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣿⣆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋
⠀⠐⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢰⣞⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠑⢄⡀
⠀⠀⢸⣿⢯⣽⡇⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠈⢳⠀⠀⠀⠀⠀⣰⠶⣶⣿⣿⣷⣶⣆⠀⠀⠀⠀⡠⠊⢰⣿⣧⡄⠈⢆
⠀⠀⢸⣿⣿⣹⠇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠃⠃⠀⣸⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠱⣄⠈⠛⢛⡠⠤⠎
⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⣐⣛⣁⣀⣈⣉⣁⠀⠀⠀⠀⠀⠶⣍⣙⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⣤⣤⣽⣧⣤⣤
⠀⠹⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣿⢿⣿⡿⡿⢿⣿⣿⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭
⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿


⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣀⠀⠀⠀⠀⢀⡤⠀⢀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡼⢧
⣤⣿⣗⢿⣿⢿⣿⣦⣾⠀⠀⠀⠀⠀⣿⢿⣯⡿⢿⡿⠀⠀⠀⠀⢸⠀⣶⣶⢡⠈⣶⣷⠀⡇⠀⠀⠀⠀⢀⡞⠀⢁⢳⡀
⠈⠙⠿⢷⠾⠿⠋⠀⠙⠀⠀⠀⠀⠀⣿⣿⣿⣧⣴⡇⠀⠀⠀⠀⠀⠳⣍⡉⠌⠠⢉⡥⠞⠀⠀⠀⠀⢀⣮⣁⣐⣂⣐⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⣀⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣂⡀
⠀⣴⣯⣿⡟⣿⣿⣧⡀⠀⠀⠀⢰⣥⣯⢟⣴⡶⢆⢶⣤⠀⠀⠀⠀⠀⠀⢀⣼⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷
⢸⣯⣿⢞⣽⣿⣿⡟⠃⠀⠀⠀⠈⠊⠽⠮⢻⣧⡻⣧⣿⠃⠀⠀⠀⠀⠸⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡗
⠈⠉⠑⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇