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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⢰⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢢⠇⠀⡸⠁⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠣⠀⢀⠀⠀⠀⠀⠐⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⢠⠀⣥⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠸⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠁⠀⠀⡁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢘⠆⠀⢢⡠⠻⠂⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠌⠀⠀⠎⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣉⡐⠙⠊⣀⣀⠔⢂⠸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⡈⠉⠠⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠴⠞⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣀⢠⣮⡳⠉⠀⠦⢼⡷⢤⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠂⠀⣤⠩⠘⠐⢴⡔⡎⠀⠜⠆⣝⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⡈⠁⠀⠀⠁⠀⢙⣀⡀⢀⠌⠀⠓⢄⠀⢀⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠐⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⢄⡐⠁⠐⠉⢉⡐⠀⠉⣨⠁⠛⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠂⠂⠀⠀⢀⡀⠀⠀⡼⠛⠐⠀⡇⠰⠐⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠂⠄⠀⠂⠀⠀⠐⣧⠀⠀⠠⢀⡟⠓⠢⠂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⡽⠂⠰⢦⣾⠀⠀⠰⡟⢿
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠈⠀⠂⢠⣏⠀⢰⡀⠀⡈⠦⢦⡴⠃⣾⡦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡁⠀⠂⠀⠐⠋⡀⠀⣀⡚⠀⢠⠆⣷⠤⣼
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠤⢀⠊⠀⠀⠀⠰⠁⢸⠅⠸⠀⢤⡅
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡁⠀⠀⠀⠸⠀⠐⠀⠀⣬⠒⠐⠙⡷⢊⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢀⡀⠚⠃⠒⠤⡎⢠⠄⣈⣠⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠓⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⢸⠀⠳⠀⡜⠀⠀⠀⣰⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⠢⢄⠀⠀⢸⠈⠂⠀⡀⠃⠃⢠⠀⣴⡏⡠⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡁⠂⠠⢁⡐⠻⡶⣄⡀⠘⠢⣁⠀⢰⡎⢲⠃⠔⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠁⠀⢀⡈⠐⠄⢌⠑⢄⡀⠏⠀⠀⠇⠀⢐⠀⢠⡗⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠁⠐⠈⠳⡄⠀⠁⠀⠈⢢⡀⠀⠈⡄⢸⠀⣆⠃⠁⣠⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠑⢀⣀⠀⠀⠀⠫⡁⠁⠉⠑⢂⡤⢬⢸⡀⠀⡠⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠀⠒⠶⠤⠤⣉⠐⣀⠘⡄⠀⡠⠊⠀⢸⡏⡀⠉⢀⢬⠇⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡓⠂⠀⢁⠂⣌⠑⠠⠅⠀⢀⡸⠷⢀⡴⠁⠓⢀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣞⢡⠄⠐⠷⠐⠋⣀⣁⡀⠠⢃⣷⡔⠪⠍⣁⣉⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣴⠨⠄⠚⢈⡀⡀⠰⡞⠘⣋⡤⠄⣲⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣮⡄⠴⣥⣄⣒⣚⠃⢀⢰⣾⢅⠬⠐⠒⠎⢁⣈⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠂⠀⡄⢇⢀⡽⢭⠤⣈⠀⡗⣩⠆⠀⢈⠆⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠩⡄⠵⠶⠞⠁⢆⡀⠒⠮⣽⠐⠒⠀⣈⣀⡅⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⡄⠠⠒⠒⡄⢀⡤⡀⠀⠒⢃⣠⠁⡸⠁⠠⢀⡌⡄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠋⠘⠣⣠⠷⣀⢄⠈⠀⠞⠁⣀⠀⢁⣴⢁⠎⠀⢠⠦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠑⠢⣌⣠⣀⠃⢀⡤⠤⢤⠚⠀⠝⠁⠀⢈⣤⢡⠁⠠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠃⢀⡁⢀⡨⠄⣁⣀⠀⢤⡴⠤⢤⠊⡐⠁⠀⢈⡀⠹⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣅⡉⢢⡀⠀⢈⣁⠀⠊⡀⠀⠐⣀⣀⠊⡐⠁⠈⣀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠶⠤⣀⠊⠉⠉⡉⠀⠔⠉⠀⠠⠐⠀⠰⠊⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣦⠀⣈⠃⢀⠔⠓⠒⠤⢈⠴⠊⠀⡁⣀⠀⢄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⡁⠓⠒⠄⢀⠰⠶⠨⠄⢂⠔⠋⢁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⡉⠥⢄⣀⠀⠀⠀⠀⣠⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⢢⣀⢹⣷⡀⣼⠇⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠰⠀⠂⠈⡀⠐⠀⠆⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⡹⣷⣿⣿⣿⣷⣿⣾⣟⣤⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠁⠀⢣⠀⠀⣇⠀⠀⠰⠀⠪⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⠀⠀⠀⡸⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠈⠁⠈⠀⠀⠁⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠤⡀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠙⢠⡀⠓⢀⠐⠉⢲⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣾⣿⣿⣿⣿⣿⣿⣿⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠂⠈⢀⠦⠀⠰⡄⠁⢹⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠈⢄⠀⠐⠀⠀⠢⠀⢀⠐⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⡄⠀⠀⡄⠈⠀⠀⠀⠣⢀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⡀⠲⡀⠀⠀⡀⠐⠀⠀⠈⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⠀⠀⠒⠀⠀⠀⠈⠀⡀⠘⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄⠨⠐⢄⠐⠁⠀⡀⠀⠀⠠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡀⠄⠀⠀⠠⠀⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠍⠐⠈⠀⢀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢰⡆⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⢀⣿⣿⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⣤⣀⠀⢸⣿⡎⣿⣿⢠⣿⡇⠀⣀⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀⠈⠻⣷⡜⢿⣿⢸⡇⣾⡿⢣⣾⠟⠁⣀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡉⠻⢿⣦⡘⣿⣎⢿⡇⢰⡿⢱⣿⢏⣴⡿⠟⢉⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢿⣿⣶⣍⠳⠸⣿⡌⣧⢸⢃⣿⡏⠞⣩⣶⣿⡿⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣬⣙⢿⣿⣆⢻⣧⢻⡈⢸⡿⣰⣿⠿⣋⣥⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣍⡳⢌⠻⢸⡟⡘⢇⢸⡇⢟⡵⢞⣩⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣭⣭⣝⡻⢶⣕⠈⡇⣿⡾⢸⠃⣪⡶⢟⣫⣭⣭⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣝⣷⢰⣎⣵⡇⣾⢫⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡹⡎⢩⣵⣮⡅⢱⢏⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⢍⣴⣶⣌⢛⣵⣶⣜⢋⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢋⣭⣜⢿⡟⣩⣬⡻⡿⣫⣬⣝⠧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠿⣿⣿⢇⡸⢿⣿⠟⣐⢿⣿⡿⠠⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⣿⣿⡎⢱⣿⣿⣮⠡⣿⣿⣷⠉⢼⣿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣙⠿⣡⣶⣶⡝⢏⣴⣷⣌⠛⣱⣾⣶⡝⢏⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣂⡻⢿⣿⢟⣘⢿⣿⠟⣀⡻⣿⡿⢃⡺⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠋⣾⣿⣿⡎⢱⣿⣿⣷⠉⣾⣿⣿⡌⢼⣿⣿⡌⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡛⠿⣡⣶⣶⡝⢋⣶⣿⣎⠻⣱⣾⣶⡹⢟⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠆⠻⢿⡿⢛⣜⡻⡿⢛⣤⡛⣿⠿⠃⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⡦⠱⢿⣿⠿⠰⢿⣿⠿⢠⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠺⢿⡦⢱⣿⣿⣦⢱⡾⠷⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠑⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠟⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⠀⢀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⠜⠀⠀⠀⠁
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢶⣄⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣠⠞⠁⠀⠀⡘
⠈⠉⠁⠂⠀⠐⠂⠄⠀⠀⠀⢻⣧⠀⢻⣇⡄⠀⠀⠀⠊⠁⢶⡀⠀⠀⠠⡆⠀⠀⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠈⠒⠄⠉⠐⢤⠀⠻⣷⡀⢿⣐⡄⠀⠀⠃⠀⠈⠑⡆⠀⠀⢡⡀⢸⣿⠛⠀⠀⠀⠀⡈⠀⠀⣠⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⣀⠈⠀⢹⣿⣶⣯⡼⢆⣾⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣯⠁⠀⠀⠀⠀⣀⣴⣿⠟⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⡀⠙⢿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡄⠀⠀⠀⣼⣾⣿⡿⠁⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠈⠛⠁⠐⠀⢀⠀⠀⠀⠈⠱⢆⠀⠉⠻⣿⣿⠀⠀⠂⠀⠀⠀⢀⣼⡏⢩⠄⠀⢀⣼⣿⣿⠟⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⢄⡀⠱⢟⣄⠀⠈⣿⠀⠀⢰⠀⠀⠀⠸⣿⣹⣋⣠⠶⠚⢻⣿⠋⠐⠀⠀⡠⠂
⠀⠀⠀⢀⣴⣶⣶⣶⣤⣤⣤⣤⢀⠈⠀⢺⣄⢳⣄⣹⡇⠀⠈⠀⠀⠀⠀⢻⡿⠋⠀⠀⠀⢸⡇⠀⣀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⢷⣏⣤⡀⠙⢷⣌⢻⣿⡌⠀⣦⠀⠀⣀⠎⠀⠀⠄⠀⠀⠎⢁⣀⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⠉⠶⣄⠉⠛⠷⢦⣹⣶⣻⣧⡒⠟⠀⢰⡏⠀⠀⡘⠀⠀⣰⣤⣟⠴⠦⠶⠀⠦⣀⣴⣾⠿⠛
⠀⠀⠀⠀⠀⠚⠛⠛⢿⣿⣶⣦⣤⣈⢣⣄⠀⠀⢉⣙⣻⣿⣧⠀⣶⡈⠁⠀⣦⠁⢀⣰⡿⠏⠁⠀⢠⣤⣴⠾⠟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⡄⡼⣥⡀⠀⠛⠻⣿⣷⣤⣀⣸⣶⣞⣿⠂⠀⠀⣰⡿⠋⠠⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠿⣿⣶⡭⠿⣿⣿⣿⡄⢨⢿⣖⡀⠀⠀⠙⣿⣿⣿⢷⣾⡇⠀⠀⣸⡋⠴⠋⢲⡦⠀⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠻⢶⣄⡉⠛⠻⢿⣦⣿⣿⣦⣀⠶⠉⠀⠀⠂⠀⣆⣠⠞⣋⣤⡰⠋⢁⣀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⣶⣤⣙⣿⣦⡀⠀⠈⠉⠛⠿⣿⣆⠀⣠⣀⣤⠶⠛⠓⠊⣹⣿⣷⣿⣿⣛⠋⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠛⠿⣿⣷⡄⠀⠀⠀⠀⠈⣿⣦⣟⠋⠀⠀⢠⣈⣾⡿⠟⢛⡥⠟⠛⠓⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣌⣉⡀⠀⠈⠻⠿⣷⣦⣠⡞⠀⠀⠈⠉⠓⣤⣾⠟⣋⡴⣶⣿⣿⣶⣶⣶⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡌⠋⠹⠿⠿⠷⣤⢀⡀⡀⠻⠿⠐⠷⢀⣀⣤⣦⠙⠁⠉⣻⣷⣄⠈⢁⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠀⠉⠷⣿⣇⠀⠀⠀⡀⠿⠃⠀⠰⢆⣈⣉⣥⠴⠓⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣩⣵⣄⠀⠈⢳⣴⡏⠀⠀⠀⣀⠰⢿⣿⣿⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⣉⡉⠳⠖⠿⣄⠼⠛⢀⣸⠦⠴⠾⠿⠧⣌⡙⠿⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⠛⠁⠈⢳⠀⠀⢀⠰⠞⠀⠁⢠⣤⣀⣂⣀⣨⠄⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠤⠠⡄⠀⠀⠀⠛⠰⠃⠀⠘⢦⡀⡀⠠⠀⢀⡼⠋⠀⣌⠓⢄⡀⠀⠂⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢹⣄⡠⢤⡴⠤⠄⠀⠀⠀⠙⣧⠀⠸⢞⡁⠀⠀⣀⠀⢘⡇⠘⢛⠛⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⢀⡴⠋⡀⠀⠱⡄⠀⠐⠀⣠⡞⠃⠀⠀⠀⠙⠆⣀⡀⠀⣤⡑⠀⠈⠀⢹⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠘⠆⣀⡜⠀⡀⠀⠰⠂⠀⡲⠓⠘⠯⣅⠀⠀⠀⠄⠀⡸⠉⠀⢀⠈⠁⢶⣠⣁⣠⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⡀⢠⠉⠙⢦⡁⠀⢠⣀⡎⠁⠀⠘⠀⠈⢳⣤⠄⠴⣤⡗⠀⠀⢀⡀⠀⣨⠋⠉⡙⢣⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⣀⣤⠀⠀⠀⠈⠛⣷⠂⠛⣄⣀⠀⠀⠈⠁⡴⠃⠀⠀⠂⠈⢳⣄⣀⣁⣠⡍⠀⠀⢑⠀⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⢁⠀⢢⠀⠂⢰⡞⠁⠀⠀⠈⠻⣦⣀⣤⣔⠂⠀⠀⠀⠀⠀⠈⢹⡏⠉⠀⠻⣤⠐⠈⣠⠉⢉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠈⠀⠀⠶⠒⠲⡁⠀⡀⠀⠂⠀⣠⠏⠀⠋⠓⢦⣤⠂⠁⡀⠀⡿⠀⠀⠈⠀⠀⣶⠞⢻⡄⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⣀⠡⠞⠁⠀⠀⠙⣆⠁⠀⣠⣼⠃⠀⠀⠀⠀⠀⠈⢱⠖⠛⢶⣄⠀⠀⠀⡄⢸⠇⠀⠄⠹⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⡙⢶⡀⠀⠀⠀⢀⣈⠏⢁⠈⠳⢦⡀⠀⠠⠀⠀⣴⠃⠀⠀⠀⠉⠙⠧⣀⣠⡞⠀⠀⠠⣦⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⡼⠛⢦⠀⠀⣸⠃⠀⠀⠀⠀⠀⢰⣤⠠⠴⢶⡏⠀⠀⠈⠀⠀⠓⣼⠃⠀⠘⢦⣀⣴⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⣦⡼⠀⠀⠈⢳⠄⣉⡀⠀⠀⢀⠀⢀⡴⠉⠀⠰⠀⠈⠳⣄⠀⠂⠀⢸⡏⠀⠀⠀⠠⡛⠣⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢨⠄⠀⠀⠀⣀⠆⠁⢠⠙⠦⡀⠀⣼⠇⠀⠀⠀⠀⢀⠠⠿⠀⣶⠶⢋⠀⠀⢀⠀⢸⡇⠠⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⡀⠀⢸⡇⠀⠀⠀⢀⡀⠙⢲⠦⠀⣀⠀⠀⠀⠄⠀⣠⡞⠁⠀⠈⠹⣆⣋⣀⣾⠇⠠⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⠒⠺⢆⡀⠀⠀⠈⢀⠔⠀⠠⠀⠉⠒⣤⠀⢀⣤⠉⠁⠀⠀⠀⠀⢈⠿⠉⠻⣦⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⣾⠀⢀⠀⠙⢢⠀⠀⠿⠀⠀⠀⠀⠀⢤⠀⢳⣯⣉⡀⠀⠀⠀⠀⣴⠾⠀⠦⠀⢻⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⠀⠈⠓⢊⡓⣀⠀⠀⠀⠠⠀⡴⠊⠁⡉⠙⠓⢦⢀⣤⠏⠰⡄⠀⣠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⢠⣀⠀⠈⠁⣴⠈⠉⠉⠑⠢⣄⣤⣸⠀⠀⡀⠀⠀⠀⠀⠛⣥⣤⣤⣤⣴⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣏⠱⠊⣤⣄⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠠⠀⠀⣤⠞⢁⠀⠀⣀⡽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣦⣤⣈⣦⣀⡀⠀⢀⡴⠚⠋⠉⠉⠙⠶⣴⢋⣀⣀⣥⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠷⠿⣦⣤⣶⡶⠶⠶⠶⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠹⣄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠸⣦⣹⣆⠀⡇⢀⡇⠀⣾⡿⢀⡴⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⠻⣦⡀⢻⣿⣿⣶⣿⣿⣧⠜⢡⣣⡿⢡⡞⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠢⣀⢳⡙⢷⡘⣿⣮⣿⣿⣿⠛⢻⡏⢀⣿⣼⢣⣿⢀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣳⣷⣼⣿⣿⣿⣿⡿⣯⠀⣼⡀⣸⣿⣷⣿⣵⣿⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⠲⢿⣿⣿⣿⣿⣿⣿⢘⣿⣰⣿⣧⣿⣿⣿⣿⣿⣿⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⢿⣿⣿⡍⠻⣾⣿⣯⠙⣿⡿⢋⣴⣿⡿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣻⣦⣝⣿⣿⣇⣨⣿⣿⣦⣿⣧⣿⣿⣭⣴⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢽⣿⣿⣶⣬⠿⣿⣿⡿⣿⡽⣛⡻⠿⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠮⣻⣿⣷⣄⣸⣿⣤⣻⣿⣿⣿⡛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⣿⡿⣿⡿⠿⢿⣿⣟⠿⣍⡉⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠛⠖⠻⡾⢯⣽⣿⣋⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠴⠠⣌⠙⢶⣰⡿⣿⣽⣧⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠙⡀⠀⠹⣶⠎⠽⣦⣿⠿⣿⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠊⡉⠃⠂⢛⢦⡔⠋⣷⣨⣿⢷⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⡀⠂⣄⠈⠛⣇⢠⣷⣬⡿⢻⣷⣨⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⢤⣁⣠⣄⡩⠛⢿⡧⣸⣿⣤⡿⢿⣇⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡜⠓⠋⢹⠀⢨⣧⣂⣾⢿⣿⡋⢹⣷⣸⡿⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣥⣦⠞⢻⡟⢉⣿⠅⣸⣿⣷⣿⣟⢻⣇⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢡⡀⣾⣅⣲⣷⣾⠿⣿⠟⡟⣿⣿⢁⡸⣿⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣯⡍⣿⡛⢸⣷⠂⣿⣿⣶⡿⣿⣿⠑⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⠛⣟⣩⣝⣿⣇⣁⣾⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠛⠋⠉⠁

⣶⣶⢶⢷⠶⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠒⠀⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠓⢶⡆⠀⠀⠀⠀⠈⠠⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠆⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠠⡄
⠀⠀⠠⣄⠒⠛⠀⠤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱
⠀⠀⠠⠀⠁⠀⠀⠐⠈⠀⠀⠀⠀⠊⠉⠐⠀⠀⠀⠀⠀⠀⠀⡆⠂⠀⠀⡇⠀⠃⠀⠀⠄⠀⢰⡀⠀⠒⠁⠌
⠀⠀⠀⢀⠀⠀⠉⠂⠀⠀⠀⠀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⢰⠐⠀⠃⠀⠠⠄⠀⡠⠋
⠀⠀⢈⣀⠁⣀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠈⠀⠀⠀⠠⠋⠠⠀⠁
⠀⠀⠐⠌⠀⠀⠁⡄⠀⠀⣀⣁⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠀⠈
⠀⠀⠸⢒⠂⠘⢂⡉⠤⠤⠀⠀⠐⠈
⠀⠀⠀⠐⠓⠤⠄⠭⠒⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⠀⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠁⠀⠀⠈⠉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⢦
⢠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠃⠀⠀⠀⠀⠄⠀⠀⠒⠂⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⠀⠂⠁⢀⢂⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡇⠀⠀⠙⠢⠀⠀⠀⠀⣸⡇
⠀⡐⠀⠀⠘⠀⠀⡀⠠⣀⣀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡯⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡼⡇
⠀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⠠⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⠀⠀⣀⠀⠀⠀⣼⡧
⠀⠈⠀⠈⠉⠀⠠⠀⠈⠂⠤⡄⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠨⠁
⠀⠀⠉⠀⠠⠈⠃⠀⣀⠠⠀⠈⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⣰⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡈⢃⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⠒⠒⠻⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⡀⠠⠙⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⣀⢀⡊⣌⠠⠖
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠈⢣⣀⡄⠀⠀⠀⠀⠀⠀⢠⠄⠀⠷⢘⡇⠚⠰⠇⠀⠰⠀⠀⠀⠀⠀⠀⠀⢠⣤⠖⠋⠉⠉⠉⠍⠙⠶⣂⠆
⠀⠀⠀⠀⢀⡠⠤⠿⠀⠀⡀⠀⠀⠀⠠⣤⣭⣄⡀⠀⠀⠀⠀⠀⠀⠄⠙⣢⠼⠁⠀⠀⢛⠇⠀⠠⠀⠀⠀⠀⠀⠀⡰⠁⠀⠠⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⡆⠀⠀⠀⠀⢰⠞⠉⠀⠀⠀⠉⠉⠃⡀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠑⡶⠦⠤⠤⠤⢤⡤⠤⠤⠤⠤⡶⠊⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣤⡴⣶⡶⢾⣄⣀⣀⣄⣀⣀⣀⣀⣠⣀⣠⣼
⠀⠀⠀⠀⠀⡸⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⢱⠀⠀⠀⠀⢀⣷⣴⣾⡿⢃⣤⠀⣶⣿⠗⣡⣤⣳⠀⠀⠀⠀⠀⡾⠀⣀⣸⡧⡀⠀⢀⣀⣷⣀⢀⠨⡄
⠀⠀⠀⠀⢀⠁⠀⠀⠘⠛⠀⠀⠀⠘⠛⠀⠀⠘⠆⠀⠀⠀⢨⠀⠙⢫⣴⣿⠟⠉⠛⣥⣾⡟⠁⠘⠂⠀⠀⠀⢈⠁⠀⠈⡁⢴⡍⠀⠉⢉⢴⡿⠁⠀⢣
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⠀⠛⠁⠀⠀⠀⠘⠉⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠉⠀⠀⢰
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠎⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠐⠀⠑⡄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠦⠠⠄⢤⡀⠀⠀⣀⠀⠀⠀⢀⣀⠀⢀⠔⠚⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠮⠄⢹⣦⠴⠏⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠑⠠⠤⢀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⡀⠀⠀⠀⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠐⢆⠤⣀⠌⣀⠔⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠠⡀⢀⠊⣁⢔⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠠⡀⠰⢁⡁⠴⠃
⠀⠀⠀⠀⠀⠀⠤⠀⠘⢆⠸⠀⠏⠴⠀⠠⠒⠀⠀⠀⠀⠀⠀⢠⠄⠀⠳⡀⡇⠘⢃⠖⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⢧⠘⠃⡘⠁⠆⢀⠰⠆
⠀⠀⠀⠀⠀⠀⠠⢊⣑⡤⠷⠸⠦⢴⣉⠠⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢢⡸⠦⠼⣄⡔⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⣢⠸⠄⠶⣀⡘⠁⡄
⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⠀⠀⠀⠀⠙⢦⠃⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠉⠳⡙⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠁⠀⠀⠀⠀⠉⠫⡙
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡄
⠀⠀⠀⠀⡜⠀⠀⠀⣠⡄⠀⠀⠀⣠⡄⠀⠀⣇⠀⠀⠀⠀⣮⣤⣤⡴⠢⣤⣄⣠⣤⠄⠤⣤⣬⡄⠀⠀⠀⠀⢠⣥⣾⡿⠃⡠⣢⢠⣾⡿⠋⠠⣢⣵
⠀⠀⠀⢀⠁⠀⣠⠼⠛⠷⢤⣠⠄⠛⠧⢄⡀⠘⠀⠀⠀⢸⠉⠺⠯⠴⣾⣿⠛⢻⣧⠴⠾⠿⠊⠷⠀⠀⠀⠀⡈⠹⠋⠠⢊⣼⡿⠛⠋⣠⢊⣾⣿⠏⠇
⠀⠀⠀⢘⡠⠊⢀⡀⢀⣀⡰⠇⣀⡀⢀⡀⠉⢶⡆⠀⠀⠘⠀⣀⡰⠋⠀⠉⣷⠉⠀⠈⢦⣀⡀⢰⠀⠀⠀⠀⡇⠀⠈⠁⠉⠁⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠘⡷⡶⠋⠟⠉⠁⠠⠄⠈⠉⠹⠏⢲⠾⡇⠀⠀⢸⠈⠻⠤⣀⡀⠔⣉⡳⢄⣀⡤⠟⠁⠰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠄⢞⡀⠀⠠⠀⠀⢀⣀⡀⠀⠀⠄⠀⠈⡓⠠⡀
⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢎⣀⡠⠈⢙⠒⢤⠤⣀⣀⣀⣀⡤⠤⠄⠒⡉⠈⣄⠈⡄
⠀⠀⠀⠀⠀⠀⠙⠠⢤⣀⠀⣀⣀⡠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⡀⠀⠀⡀⠀⡤⠞⠁⠀⠀⠀⠀⠄⠰⠂⣀⡆⢀⣀⠀⡄⠀⣆⠀⠶⢄⡘⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠈⠉⠚⠉⠒⠋⠙⠁

⣿⣿⡿⠿⣿⡿⢿⣿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⡄⠀⠀⠀⣠⠞⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡇⠀⢻⡄⢀⡞⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡓⠦⣄⠀⠀⡇⠀⠀⢳⠏⠀⢠⠏⢀⣠⢴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠙⢶⣇⠀⣠⠏⠀⢀⣾⠖⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣹⡄⠀⠀⠙⢶⡏⢀⡴⠋⠁⠀⣰⠃⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡈⠉⠛⠲⢤⣈⣷⠋⠀⢀⣠⠴⠛⣋⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠉⢻⡤⠞⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣀⣴⠋⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠠⠞⠛⠓⠦⠀⠠⠞⠛⠛⠦⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣤⠤⢤⡀⠀⢀⣠⠤⢤⣀⠀⠀⣠⠤⢤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠀⠀⠀⠉⠀⠈⠁⠀⠀⠉⠀⠈⠁⠀⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠉⠀⠈⠃⠀⠘⠋⠀⠈⠙⠀⠐⠋⠀⠈⠙⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠰⠛⠉⠙⠶⠀⠰⠞⠉⠉⠳⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⠀⠀⠀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⣿⣿⣿⠀⢹⣿⡀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⡉⠛⠛⠿⣷⣦⣼⣿⣿⣿⠀⠈⣿⣧⣴⣾⣿⣿⣿⡿⢋⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⠀⠈⠙⢿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⠟⢁⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠹⣿⣿⠀⠀⣸⣿⣿⠟⠁⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣿⣿⣿⣿⣷⣄⠀⢿⣿⠀⣰⣿⣿⠋⠀⢠⣿⣿⣴⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣯⣄⣀⠀⠀⠉⠙⠻⠿⣿⣿⣷⣾⣿⣴⣿⡿⠃⣠⣶⣿⣿⠿⠟⠛⠛⢛⣉⣛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⣿⣶⣤⣀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣵⣿⣿⠿⠋⠁⣀⣤⣶⠿⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠈⢻⣿⣿⣿⣿⡟⠁⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣶⣄⠀⢻⣿⣿⡟⣠⣴⣾⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣛⣉⠉⠀⠀⠀⠀⠉⠙⠿⣿⣷⣄⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣶⣤⣄⣀⠈⢻⣿⣿⣿⣿⣿⠿⠋⠁⠀⣀⣤⣴⣶⠶⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣥⣤⣤⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠟⠛⠛⣻⣿⣿⣿⡛⠛⠻⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠉⠀⠀⣀⣴⡿⠟⠉⠈⠙⢿⣶⣄⡀⠀⠉⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠛⢿⣶⣴⣾⠟⠋⠀⠀⣠⣄⠀⠀⠈⠻⢿⣦⣴⣾⠟⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢀⣠⣾⠿⠻⢿⣦⣄⠀⠀⠙⠋⠀⠀⢀⣤⣾⠿⠻⣿⣦⡀⠘⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣡⣴⡿⠟⠁⠀⠀⠀⠙⠿⣷⣦⣀⢀⣠⣾⡿⠋⠁⠀⠀⠈⠙⠿⣷⣬⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡋⠀⠀⠀⠺⠗⠀⠀⠀⠈⣹⣿⣿⣿⡉⠀⠀⠀⢰⡶⠀⠀⠀⢈⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠉⠙⠿⣷⣄⡀⠀⠀⠀⣀⣴⣾⠟⠉⠉⠛⢿⣷⣄⡀⠀⠀⠀⣠⣴⡿⠟⠁⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠇⣀⡀⠀⠈⠛⢿⣶⣴⣾⠿⠋⠀⠀⣀⡀⠀⠀⠉⠻⣿⣦⣴⣿⠟⠉⠀⢀⣄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⠀⠛⠁⠀⢀⣤⣾⡿⠻⢿⣦⣄⠀⠀⠙⠃⠀⠀⢀⣴⣿⠟⠻⣿⣦⡀⠀⠈⠛⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⣠⣴⡿⠛⠁⠀⠀⠀⠙⠻⣷⣦⣀⢀⣠⣾⡿⠋⠁⠀⠀⠀⠙⠿⣷⣤⡀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠀⠀⠀⢶⡆⠀⠀⠀⠈⣻⣿⣿⣿⡁⠀⠀⠀⣾⠆⠀⠀⠀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠙⠿⣷⣤⡀⠀⠀⠀⣀⣴⣿⠟⠉⠉⠛⢿⣶⣄⡀⠀⠀⠀⣠⣴⡿⠛⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⡀⠀⠀⠈⠻⢿⣶⣴⣾⠿⠋⠀⠀⢀⡀⠀⠀⠉⠻⣿⣦⣴⣿⠟⠋⠀⢀⣀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠀⠀⢀⣠⣾⡿⠿⣿⣦⣀⠀⠀⠈⠋⠀⠀⢀⣴⣿⠿⠿⣿⣦⡀⠀⠈⠉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⣠⣴⡿⠛⠁⠀⠀⠀⠙⠻⣷⣦⣀⢀⣠⣾⡿⠛⠁⠀⠀⠈⠙⠿⣷⣤⡀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡉⠀⠀⠀⠴⠦⠀⠀⠀⠀⣹⣿⣿⣿⡁⠀⠀⠀⢴⡆⠀⠀⠀⢈⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡙⠿⣷⣄⡀⠀⠀⠀⣀⣴⣿⠟⠉⠈⠛⢿⣶⣄⠀⠀⠀⠀⣠⣶⡿⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠈⠻⢿⣶⣤⣾⠟⠋⠀⠀⣀⡀⠀⠀⠉⠻⢿⣦⣴⣾⠟⠉⢠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣾⠿⠻⢿⣦⣀⠀⠀⠛⠁⠀⠀⢀⣴⣾⠿⠿⣿⣦⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣄⣀⠙⠻⣷⣦⣀⣀⣤⣾⡿⠋⣁⣠⣤⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⡟⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⡿⣿⣿⠿⣿⡿⡿⢿⢿⠿⠿⢷⣿⣿⣿⣿⣏⣿⣹⣻⣿⣿⣿⣿⣯⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠁⠂⠘⠰⠀⠂⠂⠰⠐⠀⠛⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢋⡀⠀⠀⠀⢀⠀⠄⠢⠀⢀
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢁⠂⠠⠠⠀⠀⠀⠀⠀⠢⠔⠔⡒⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠅⠪⠚⠀⠀⠀⠀⠀⢀⠀⠈⠒⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠠⠀⡀⠀⠀⢀⠀⠀⠊⠀⢀⠁⠀⠀⠐⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠚⢀⠀⠀⠐⠒⠀⠀⠀⠂⠠⠀⠴⢁⡡⠀⡄⠘⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠈⢀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⡐⠂⠀⠀⣢
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠉⠀⠀⠀⠀⠀⣆⣈⠠⠘
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠦⠤⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠉⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠠⢶⣟⠀⠀⠀⡠⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⠀⢀⠀⠀⠀⢂⠀⠾⠋⠀⠔⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⠀⠀⠀⠀⣠⠴⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠲⣄⠀⠉⠻⣦⣴⠞⠂⢁⣴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢼⠂⠛⠲⠜⣿⠀⠀⡘⣁⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠲⠶⢷⣄⢀⣤⣬⣣⠊⠁⣠⠞⡁⠌⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣹⣤⣌⣉⣁⣤⡄⢀⣤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢡⢆⣉⣁⣤⣭⣤⣤⣿⣄⠈⡉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣅⡌⣍⣿⠿⣀⡾⠋⢻⡿⢶⣽⣧⠬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢽⡿⢁⣿⣡⠄⠈⣿⠔⠒⠳⡄⣹⠄⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣟⢨⣟⣻⣿⠀⢀⣠⣆⠀⠰⠄⠡⣏⠀⠀⠂⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣇⡍⠁⠈⠞⠉⣄⠈⢳⡀⢀⠐⠙⢏⠘⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⢡⠿⡤⠖⠒⢦⠀⠐⠀⢄⣿⣧⣤⣤⠈⢳⠃⠰⡎⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡇⣄⣘⣿⣅⣺⣂⣈⣧⣠⠖⢉⡀⠹⣆⣀⠤⠸⡆⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣿⠁⠈⢧⠁⢀⡤⠖⢯⡒⠚⠓⠂⡟⠓⢤⣄⢸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⢸⠀⠀⠀⣷⣋⣠⣄⠀⢹⡤⠒⠁⠙⢦⡀⠀⠀⢠⡉⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⢷⡞⢡⠀⠂⠉⠉⠁⢈⡇⠈⣻⠤⠼⣧⡀⡼⢎⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠚⢷⡝⠀⠀⣷⠞⠉⠀⠙⢏⢀⡴⠋⠈⠳⡄⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢃⣴⣿⡶⠶⠿⡢⠰⠿⠀⠘⣿⠀⠰⠾⠄⠣⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⢵⣦⡀⢱⣄⢰⠒⠉⠉⠳⡤⠂⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠙⠵⠎⠼⠁⠀⠲⠄⠘⠒⠲⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠀⠀⠄⠀⠾⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠂⠀⣀⠰⣆⢀⠓⠂⠈⠀⠀⠠⠄⠀⠈⠀⠠⠐⠁⠀⠀⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡞⠁⠀⠀⠀⡀⠀⠀⠀⠘⠀⡀⠀⠸⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢂⣀⣤⣧⢹⢀⡀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠃⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠴⠖⠛⠉⠁⢠⡄⠀⠀⠈⠶⠏⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⣰⣷⡀⠀⠀⠀⠤⠀⠀⠀⢸⠷⠃
⠀⠀⠀⠀⠀⠀⠠⠚⠁⢠⡄⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⢀⠀⢠⡄⠀⠀⠀⠻⠏⠠⠀⠀⠀⢶⠁⠀⠸⠛⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢀⠞⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⠀⠀⠀⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣿⠟⠟⣶⣄⠀⠜⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠐
⠀⠀⠀⠰⢾⣯⣬⣾⡿⡉⢳⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢾⡀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⣴⣦⠀⢀⠀⢠⣾⣿⣿⡄
⠀⠀⠀⠀⢇⠀⣙⢿⣆⣤⣿⣷⠴⣿⡀⠀⠀⠀⠀⢀⣠⣼⣿⠦⣄⢸⡇⠀⠋⠀⠀⠀⠀⣀⣀⡀⠀⠀⢰⣿⡏⠀⠛⡑⠴⣟⠉⢻⡇
⠀⠀⠀⠀⠘⡄⠀⠏⢿⡿⠟⣿⡿⣿⣿⣶⣶⣶⣿⠿⣿⠿⠃⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢁⠶⣶⡄⣿⡄⢸⡆⡭⡄
⠀⠀⠀⠀⠀⠈⢦⡀⠈⢷⣤⣾⠷⡉⠛⠛⠛⠉⠀⠀⠀⠂⠀⣰⡖⣭⠆⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠃⡿⠗⠈⠉⠁⠈⠙⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠻⡿⠛⠛⠦⡤⠀⠉⠙⠁⠀⠀⢠⠃⠇⠸⣄⠀⠠⠜⠂⠀⠀⠀⢀⠀⠉⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠈⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⠁⠀⠀⠛⠹⢶⡆⠀⠀⠀⠘⠚⠀⠘⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠔⠲⡆⠀⠀⠀⠁⠀⠀⠐⠀⢘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⣠⠀⠀⠀⠙⠀⠶⠀⠀⠀⠛⠀⠀⠀⠀⡶⠀⠘⡄⠀⠀⠀⠀⠂⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡄⠀⠘⠋⠀⡀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⠀⠀⠂⠈⠁⡨⠄⠀⠀⠀⢀⠀⡄⠀⠀⠀⢰⡀⢀⣹⣦⣀⠀⠀⠀⠈⠀⠀⠀⠀⠖⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣦⣀⠀⠀⠠⠊⠀⠐⠢⠀⠁⣀⡄⠁⠀⠀⢠⠿⠿⠿⠋⠀⣼⣷⠦⠄⠀⠀⠀⢀⣡⣶⣿⢋⣝⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠻⣿⣷⣦⡇⠀⠀⠘⠟⠀⠀⠀⠈⠠⠀⢀⠂⠀⠀⣀⣤⣾⡿⠋⠀⠀⣹⡿⠛⠋⠉⠉⢈⡥⠍⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢏⠉⣶⠄⠙⠋⠁⠰⠀⠀⢀⣈⣥⣤⣄⠀⠀⢹⣶⣶⣿⡿⠟⠋⠀⣀⣴⡾⠋⠀⠀⠰⡦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣜⣋⡒⢃⠀⠀⣴⣶⠖⠚⠛⠉⢉⠀⠙⠋⠀⠀⡸⠉⠩⢤⣤⣴⣶⣿⠟⠁⠀⠀⠏⢣⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⢦⣤⣀⣎⠀⠀⠀⠀⠈⠀⢀⡶⢘⠒⠀⠀⢀⡀⠰⠊⠀⠀⠀⠀⡉⢤⠚⣥⣤⣤⣤⠀⢠⠀⡀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠉⠁⠀⠀⢪⠀⢀⠀⠁⠀⠀⠂⠀⠔⠀⠈⠀⠀⠀⠀⠈⠀⡇⢠⣤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⢀⣠⠤⠀⠲⠉⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⣠⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠈⠁⠃⠰⠞⢀⡴⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠉⢀⣀⠀⠈⠑⠂⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⠾⠿⠾⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠀⠀⠀⠀⠀⠀⣸⠁⢠⡀⠀⠀⠀⡊⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⢻⡄⣠⡾⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⣿⣿⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠈⢻⣷⡀⠀⠄⢠⡘⠀⠀⠀⢰⢀⣿⣇⣤⡖⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⢦⣼⣾⣧⠀⠀⠀⣸⣾⣿⣿⠋⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠈⠛⡟⠀⠀⢀⣿⣿⣿⡷⠛⠁⣀⣤⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣄⠀⠀⠀⠀⢻⣦⠀⣸⣿⣿⡟⠁⠐⠀⠉⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣍⠀⠉⠻⣦⣀⠀⠂⢿⣧⣿⠟⠋⠀⣀⣴⠊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⠈⢳⣄⠀⠉⢻⣶⡄⢻⠟⠁⠀⢠⣾⣿⣷⡾⠖⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠛⢦⣘⡆⠀⢀⣶⣦⡀⢰⣿⡿⠛⠁⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣀⡈⠌⣻⣶⣾⣿⡿⠋⢀⡀⢸⣿⣗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣉⠛⣷⣼⠋⠉⠀⣶⣾⣷⡿⠛⢂⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣥⣶⣘⣿⣷⣶⡾⠛⠛⣋⣷⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣧⣙⣿⣷⣤⣾⣟⢉⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣟⣿⣟⢋⣿⣧⣼⣿⣌⣷⣶⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠩⠉⢈⡟⠋⠁⢠⡈⠁⢹⣏⠈⡙⢳⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⢨⡀⡀⠀⣟⠀⠀⠀⣨⣷⡴⢟⠟⠻⢀⡈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣹⣅⢾⡉⠀⠀⣰⣆⡶⠋⠉⠉⣇⠀⠀⠀⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⠃⠈⢡⡤⠀⡐⠉⡁⠀⠀⠀⡟⢿⣷⣞⡁⠈⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢹⣀⠄⠈⠁⠀⠀⠀⢰⣤⣠⡞⠀⠀⠁⠙⠅⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡡⢼⠇⠀⠀⢰⣄⣀⣠⠞⠙⠀⡄⠀⠀⠀⣴⣷⣦⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣀⢀⡴⠋⠁⠉⡁⠀⠀⢀⣾⣄⣴⠁⠈⠙⡏⠋⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⢞⠉⠀⠀⠀⢰⣖⣄⣴⠊⢙⠽⣿⠦⠀⠠⣤⡾⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣧⣀⡠⢴⣿⡿⠆⠀⠀⠀⡘⢢⡴⠛⠻⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢠⠂⣻⠋⠀⠀⠉⢹⣦⣤⡤⠿⠿⣯⣄⢀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢶⣽⡧⠀⠀⠀⠈⢿⠏⠀⠀⠀⢻⣇⣠⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⢉⣷⠾⠖⠞⢶⡖⠀⢲⣾⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠛⠓⠛⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⢀⣴⣿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⢸⣷⡄⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⣰⠃⢠⣾⣿⠃⡜⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⡿⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⣠⣿⣰⣿⣿⡏⠚⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⠃⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢻⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⡟⣾⠃⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣶⣬⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣹⡇⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣀⣤⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡏⣰⣿⠟⢁⣠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡈⠙⢿⣿⣿⣯⣿⣽⣿⣭⡉⣽⣿⣿⣿⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣉⣁⠉⠍⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣥⣼⡟⢿⣿⡛⣿⡿⠻⣿⣿⣿⣾⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠂⠀⠾⣿⣍⠀⣀⣽⡟⠲⣤⣤⣬⣿⣯⠙⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣶⠀⠘⢿⣿⣿⣟⠃⠀⠉⢻⣿⡛⠛⣦⣠⣼⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠟⢻⣧⣤⣰⣾⠉⢉⣻⣷⣶⣿⣿⣿⡋⠀⢙⣿⣏⣹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁⠈⣹⠿⡿⣏⠁⠀⢉⣸⣿⣄⢀⢉⢷⡾⠿⠟⣍⣉⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⣿⣿⡧⠀⠄⠸⢷⡞⠛⠉⣿⣄⢀⣠⣼⡷⠶⠖⢿⡟⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣼⢤⡀⣷⠀⣠⠄⠺⡁⠀⠀⠀⢻⡟⠛⠙⢿⣷⢶⣾⣟⠉⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢀⣠⠈⢘⠀⠀⡀⠈⢦⡤⠶⢾⣷⠒⠙⠺⣗⣈⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠀⠀⠈⠀⠀⣀⡀⣺⠀⢀⣀⠙⢷⡶⢛⣿⣍⢁⣿⣷⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠉⠀⠟⠀⠀⠀⢹⣇⣀⣀⣀⣾⢀⢀⣤⢿⡟⣶⣦⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢒⠒⠒⠆⠀⠈⠀⢹⡇⠀⠉⠿⣯⣀⣤⣀⣾⣡⣤⣸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣾⣤⣴⣮⣥⣽⣥⣭⣭⣧⣤⣤⣽⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡐⠁
⠀⠀⠀⣀⠀⠀⢤⢈⠲⡀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢊⠀⠄⠀⢀⡀
⠀⠘⠀⠈⠀⢀⡀⠀⠒⠸⣼⠀⠐⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠑⠀⢸⡼⠐⠂⠀⣀⠀⠈⠀⠘
⠐⠂⠠⠽⣄⡣⠀⠪⠰⠄⣹⡀⡐⠀⠀⡁⣠⠰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢠⡀⡀⠀⠐⡁⣸⡀⠴⠈⠂⠀⢃⡼⠥⠀⠒
⠴⠀⠀⢂⣈⡉⠙⠮⠥⣤⠌⢳⡁⠀⡐⢀⢅⡀⢰⢀⠅⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠅⢰⠀⣀⢅⠐⡀⠀⣱⠋⢤⡤⠭⠞⠉⣉⡀⠂⠀⠰
⠀⠀⠐⠁⠈⠀⠈⢄⢂⠲⠙⢮⠷⣔⠀⠈⢀⡉⠚⣇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡟⠊⣁⠈⠀⢐⡴⢯⠞⠱⠂⢂⠌⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠫⠅⢠⣈⠁⢠⣤⡥⠮⠽⣶⢦⣍⣉⡙⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⡞⣉⣉⣤⢶⡾⠭⠦⣤⣤⠀⢉⣠⠀⠬⠃
⠀⠀⠀⠀⠐⠖⠠⠤⠬⢛⠏⢠⠁⠠⠞⠥⠠⢄⣨⠽⠛⠛⠶⠶⠶⠶⠶⠶⠶⠖⠛⠻⢭⣀⠤⠠⠝⠦⠀⢠⠈⢟⠫⠤⠤⠰⠖
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⣐⠀⠀⡇⡦⠠⡤⠞⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠦⡤⠤⡆⡇⠀⢜⡀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠆⠀⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡇⠀⠦⠀⡀

⠀⠀⡄⠀⠰⠂⠀⠀⠂⠀⢀⣀⠀⠀⠀⠄⠲⠀⠀⠀⠀⡀⠀⣠⠆⠀⠀⠆⠀⢀⠀⠀⢀⠀⠀⠀⠰⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⡀⠁⠀⠀⠁⠓⠀⠀⠁⡄⢀⠆⡰⢡⣷⣊⡄⠀⠔⠀⢀⠀⠀⠀⠄⠈⠀⠐⢤⣄⠀⠀⠀⠐⠈⠁⠀⢀
⠀⠀⠈⣄⡠⠁⠀⠀⠉⠀⠃⠀⡄⠀⡇⢰⠃⠈⢠⣶⡪⠥⠂⠀⠃⠀⠠⠀⠐⠀⠀⠔⠠⠠⠄⠀⠀⠉⠓⠤⡄⠀⠀⠀⠉
⠀⠈⠁⠀⠁⠀⠠⠤⠀⡀⢰⠀⠙⠸⠀⡀⢠⣶⠋⠄⠄⠒⠂⠐⢁⠀⠀⠀⠠⠆⠀⠀⠀⠠⣠⠎⢀⠀⠀⠀⠀⠺⢆⠀⠈
⠈⠀⣄⠀⠀⠀⠀⢰⠀⡇⠈⠀⡆⢸⡰⠷⠇⠀⠀⠀⠒⠒⠀⡀⠀⠀⢀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠘⠀⡤⠀⢠⠂⠘⠃
⠀⠀⠀⢄⠀⡆⠀⢀⡀⠀⢰⠀⣧⣈⣉⠓⠂⠀⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀
⠀⡀⠰⠸⠀⢇⠀⠀⡇⢻⣘⠁⠀⠀⠀⠁⠀⠉⢉⡁⠀⣀⠀⠀⠄⠀⠀⠀⠀⠟⠂⠀⠀⠐⠀⠀⠀⠈⠂⠀⠀⠀⠂⠐⣄
⠀⠓⠀⠀⠀⢸⠀⠀⠣⠤⣬⡉⢀⠁⠒⠦⠄⠀⠞⠙⢦⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⣇⠘⣤⣘⣓⠒⠂⠠⠤⣈⠛⢂⡤⢨⢥⠛⡀⣁⠐⠆⠀⠀⠀⢀⠆⢀⠀⠀⠊⠀⠀⠐⠒⠀⠀⠿⠄⠀⠀⠀⢁
⠀⠀⠀⠀⠐⠓⠒⠀⠠⢄⣈⡑⠂⠀⠀⣉⠝⣇⠈⠆⠁⣨⠀⠘⠃⠈⡡⠄⢀⠀⠀⠈⠀⣀⠄⠀⠀⠀⠛⠀⠑⡄⠀⠀⠈
⠀⡠⠂⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠈⠁⠀⢠⠀⠺⠀⠀⠀⠀⠀⠼⠃⠈⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠐
⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠘⠀⠘⠀⠀⠀⡠⠚⠀⠀⠀⠀⢀⠠⠋⢀
⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⣢⡄⠉⠀⠀⠀⠀⠄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠀⠀⠀⣰⠦⠤⠄⠀⠀⠀⠤⠀⡀⠄⡀⠀⠀⠊⠀⠀⠉⠀⠀⠀⠈⠁⣀⠜⢏
⠀⠀⠀⢀⣀⠄⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⡜⠓⠀⠂⠀⢀⣀⠐⠀⠐⠃⢀⡠⠀⢰⠀⠁⠀⢀⡠⠀⠀⠰⠀⣠⠒⠀⠘
⠠⠤⠒⠊⠉⠀⠀⠈⢀⠀⠈⠀⠐⠄⠀⣴⣋⡁⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠔⠋⠀⠂⠀⠔⠀⠀⠤⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠂⠀⠀⠰⠯⠀⠀⠀⡀⠀⡋⠀⠀⠂⠀⠀⠱⠀⠀⢠⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡲⠄⠀⠰
⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢣⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠦⡀⠀⠊⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⠜⠊⠀⠀⡀⠀⠀⢀⠔⣴⣾⣿⣿⡿⣿⣿⣿⣿⣷⣦
⠀⠀⠁⡀⠀⣐⠆⠂⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠡⠐⠀⠊⠀⠀⢆⠀⠀⠈⡠⠀⠁⢸⣿⣿⡿⢿⡷⣿⣿⡿⣿⢿⣿
⠀⠀⠃⠀⠀⠀⠿⠰⠙⠀⠒⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⢸⣿⢸⡇⣻⡇⢹⠃⡆⣿⡸⣿
⠀⠀⠀⢀⠠⠊⠀⠄⠀⠀⠀⠀⠀⢠⠀⣤⢀⣀⣐⠀⠀⠀⠀⠀⠀⠈⠁⡠⢐⠊⠔⠃⠀⠘⣿⣼⣧⣻⣴⣾⣴⣇⣿⣣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠠⠀⠌⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢔
⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢀⡎⠀⠏⡂⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠤⢆⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣼⠟⠆⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⣀⣤⣀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠶⠾⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠀⠀⠾⢠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠈
⠀⠀⠀⡀⠀⠀⠀⠐⠀⢀⠔⠀⢀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡐⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠠⠄⠀⠀⠠⠀⠀⠀⠆⠀⠀⠆⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠒⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃
⠀⠀⠀⠁⢁⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠈⠀⠄⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠈⠒⠀⠠⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣗⣿⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣄⢌⠀⠰⡘⢃⠀⠀⠋⣰⠟⣱⡿⣻⣿⡟⠀⠀⣰⣿⣿⣿⣄⠀⠘⢀⢀⡄⠐⣶⣌⣿⣿⣿⠿⠟⠋⠁⣿
⡮⣑⣇⠀⠀⠈⠀⠀⠘⠃⠀⠋⣈⣼⣿⣀⣀⣀⣿⣿⣿⡻⡿⢷⣦⠀⠳⣿⣶⣾⣝⣿⣿⡇⠀⠀⠀⠀⠻
⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠋⠿⣿⣿⣿⣿⡿⣿⡝⢿⡄⠈⡯⠳⠀⠀⠉⠛⠿⢿⣿⡏
⠀⠉⠢⠄⣈⢦⡀⠀⠀⠀⠠⠒⠁⠈⣦⣍⠙⠻⣷⠈⢿⡄⠀⠀⠀⠀⠤⠀⠀⣠⣄⠀⠈⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⢦⣄⡀⠀⠘⢯⣝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢙⣧⣤⣄⣠⣼⣿⡿⢷
⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⣐⡽⠃⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⢿⣿⣿⡷⢞⡫⠅
⠂⠀⠀⠀⢐⣤⠀⢀⢨⣦⣿⡆⠀⣠⣶⣬⡄⣀⣀⠀⣀⠠⠔⠒⡫⠉⠁⠀⠀⠀⠀⠈⣉⠭⣭⡶⠋⢴⣿
⣴⠋⠀⡄⢹⣿⣦⣇⢿⣿⠟⠀⠀⠒⠂⢀⡬⢔⣈⠝⠀⢀⠀⠎⠀⠀⠀⡀⠀⠀⠸⣿⣿⡿⠋⠀⠀⣰⠟
⠇⣀⣴⣃⢀⡉⠙⠛⠈⠁⠀⠀⠀⠀⠐⠻⠶⠿⠟⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠦⡄⠀⠀⣼⡇⣿
⣾⣿⣿⡿⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⢠⢀⢠⣀⡄⠀⢧⡀⠀⣿⠀⠀⠘⠛⠛
⠟⠸⠁⠃⠀⣿⣿⠀⠞⠙⠙⢣⣶⣶⣤⣀⠀⠀⠀⠀⠀⢿⢏⡿⣿⢸⡘⣿⠷⣦⢲⣽⣷⣾⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠍⠁⠀⠤⠤⠞⠫⠪⠫⣾⠇⠀⠀⠀⠀⡀⣤⣾⣷⢿⠜⠗⠀⠀⠘⢷⣗⣽⠇⠀⠀⠀⠈⠻
⠀⢀⠀⠀⠀⢀⡀⠀⠀⠤⣤⣾⠛⠉⣟⢿⡗⣄⡀⣠⡹⡌⣇⣿⣾⣦⠀⠀⠀⠀⠀⠀⣈⣠⠤⠀⠐⠂
⠀⢸⡇⡀⠀⠀⠀⠀⠀⠐⣛⣛⣡⣴⣷⣿⣛⠌⠻⢿⣷⠸⠸⢹⡿⠏⠀⠀⠀⢀⣤⣾⣿⠋⠙
⠀⠀⡇⠈⢢⠀⠐⠶⣬⣴⣒⡶⣶⣖⢰⡀⠉⠳⡄⠈⠁⠀⠀⠈⠃⠀⠀⠀⠸⡿⢏⡝⠁⣀⡍⣀⣠
⠀⠀⢡⠀⠈⠂⠀⠈⢺⣿⣿⠝⠛⠉⠈⠳⠦⠀⠀⠀⠀⠠⠀⠀⠀⡔⠁⠀⣘⢤⠩⠂⡴⢶⣞⣉⡀
⠄⠀⠞⡀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⠀⠀⠀⠀⠀⠘⣀⠴⠊⠀⠀⠠⢀⣀⣠⡿⠋⣠⢀⣼
⠄⣶⠀⢷⢹⡿⠀⣀⣴⣿⣦⡀⠀⠀⢠⡿⠭⢉⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⠀⠛⠻⢿⣵⣾⠷⠟⠁
⢸⡟⡄⠀⠻⢁⣾⡟⢸⣿⣿⣧⣀⣠⣤⣿⣿⣿⡦⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠛⠿⣦⡀
⠎⣿⠡⠄⠠⡿⠘⠁⠼⠿⢻⡿⢿⢿⢿⣿⠛⠻⠿⢿⡕⣠⠞⠀⠀⠀⠀⠀⠀⠻⣦⡈⠿⡿⣷⣶⣶
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠀⠀⠉⠉⠛⠛⠛⠛⠃⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣡⠴⡾⠛⠁⢀⡄⠀⠀⢀⠀⠀⠀⠀⢄⡀⠀⠠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⢊⡵⠂⠀⠀⣴⠋⢀⡴⠀⣾⠀⠀⡇⢠⡀⠹⣦⠘⢦⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠋⠀⠀⣠⡞⠁⢠⡾⠁⡸⠇⣰⠀⣿⠀⢷⡀⠈⢷⡀⠹⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢠⡾⢁⡔⢁⣼⠏⠀⠀⠈⡀⠠⠁⢠⡟⠀⠈⢠⡌⢷⠰⡌⢿⡄⠹⣞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⢡⡿⢁⡞⢀⡞⠁⠀⠀⢀⡼⢡⠇⢠⠿⠃⡤⠀⡼⣿⡘⡆⢿⡸⢷⠀⠻⡆⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣿⡇⠃⣾⠁⡞⢀⡼⢀⡴⠋⣰⠋⠀⠁⣨⣞⢁⠘⠀⠙⢧⠃⠈⠃⡞⡄⠀⢱⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠁⠘⣰⣾⠴⠚⠛⠛⠙⠛⠛⠛⠉⠉⠛⠛⠋⠙⠻⡰⠾⣶⢁⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡀⢀⠂⢰⣯⠴⠷⠶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⡹⢦⢹⣯⠀⢰⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢃⢸⡄⣾⠁⢀⣠⣄⡀⠉⠙⠣⠀⠀⠀⠀⠘⠋⠁⢀⣰⣇⡀⠈⣟⠀⡜⠐⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⣿⣇⣿⠀⢻⡟⠋⣿⠷⢦⠀⠀⠀⠀⠀⢀⠴⢻⣿⣿⢻⠟⢠⡇⣰⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣇⠈⠳⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠵⠛⠀⣼⣿⣧⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⡻⣿⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠛⠁⣿⢿⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⠿⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⣲⣄⡀⠀⠀⠀⢠⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣿⣄⠀⠀⠀⠘⢥⣀⣀⣀⣀⡜⠁⠀⠀⠀⣠⠟⢿⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⣻⣦⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡴⠫⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠶⢛⣿⠉⠓⠦⣄⣀⣀⣀⣀⠠⠶⠞⠹⡗⢶⣾⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣛⣿⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⣶⡸⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢣⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠜⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡈⠎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡱⢀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡄⢿⡌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡟⢠⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣧⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡜⠀⣼⢷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢿⠀⠀⠆⢹⡄⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠀⡀⣶⣼⠁⠀⠀⠀⠀⠀⠀⠈⣇⣧⢠⢀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⣷⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⡏⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⡆⢸⣇⠀⣴⣿⠃⠀⠀⠀⠤⢾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠃⠀⢻⡾⠉⠈⠀⢀⡴⣳⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⢷⡄⠀⠀⠼⠻⣿⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⢠⡀⠀⠈⣷⡶⠊⠀⠀⣻⣧⣿⠃⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⢤⣸⢿⣶⣤⢏⠀⠀⠀⢠⣿⣟⣵⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡊⠛⠷⣌⡙⣾⣷⣾⠷⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⢦⣄⣷⣶⣿⣿⡷⣬⣙⡊⢻⡛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⢶⣦⣼⣿⣿⣿⡿⠿⣋⣭⠙⠻⠷⠙⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⢿⡟⠛⣭⣦⡀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⣿⡟⠳⠄⡧⠀⢲⣤⡿⣯⣴⡂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⢒⠾⡧⠔⠀⣹⠠⡅⠸⢵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⡜⣯⡧⠴⠋⣤⢰⡀⣠⠀⠷⠈⣆⣀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠌⣄⠈⢳⡄⠀⣶⡉⢰⡉⢲⡀⡰⠋⡍⢡⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡛⠲⣻⠿⡛⢿⣏⠘⣟⡴⢟⠧⡈⠙⣺⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠳⡄⣿⣴⣧⣤⠷⢤⣿⣦⣿⠤⠼⠄⢰⣼⣏⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣦⠞⢱⠋⠻⣆⢀⡞⠁⢻⡈⣷⠀⡰⢃⡈⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢋⠑⢂⠜⢊⠻⣄⠈⢻⡿⠟⢯⠁⢹⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠲⡈⢳⠷⡄⠞⠣⢽⣄⣿⠠⠚⠦⢧⣷⠄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠛⢤⠚⢧⣙⣆⣼⠛⣿⡛⢶⠈⣿⣫⡝⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣿⡄⠉⡟⠻⣥⣏⣉⠹⢿⢷⣍⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣾⣄⣘⢹⣷⡾⡿⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢹⣦⢸⣦⠀⠀⢰⡀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣌⠻⣦⡀⢿⡆⣿⣧⠀⣿⡇⠀⢀⣴⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠙⣷⣜⢿⣮⠛⣿⣿⣧⣿⣿⣠⣿⣿⣿⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⢪⣍⡘⣿⣷⣿⣿⣿⣟⡿⣷⣿⣿⡿⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣷⢿⣿⣿⢻⣿⣿⡿⣻⣶⢻⣿⣿⠇⣡⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⡼⣿⣿⣿⢿⢏⣾⣿⣿⡏⣿⣿⣼⣧⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢬⣻⣷⡹⣇⣾⣦⡘⣿⣿⣿⣿⢘⣭⣰⡿⣿⣥⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣽⣳⠉⣿⣿⣿⡎⣿⡟⣹⣿⣿⣵⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣷⡽⢿⢿⣿⢈⣾⣿⣿⠏⣻⣽⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣾⣿⠿⢴⣿⣷⣯⠸⠿⢿⣵⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣦⣬⣙⣷⢶⣯⣍⠿⣿⡟⣭⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣤⠻⣷⣍⢹⡶⣿⢿⣾⡷⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣬⣶⣦⡿⣱⣾⢷⣦⣽⣆⡲⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠾⣛⣷⣯⣟⣽⣿⣿⣿⣷⣻⣿⡿⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣾⣿⡿⣫⣾⣿⣿⣿⢿⣿⣿⣿⣿⡏⣽⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣾⣷⣯⣾⣿⣿⣿⡿⣷⣾⣿⡻⣿⡿⢹⣿⣿⣿⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣿⣿⢋⣽⣿⣯⣽⠿⣿⣿⣿⣿⣿⢮⣶⣿⣝⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣋⢿⣿⣿⡿⣫⣿⣿⣿⣻⡿⠛⣿⣿⣿⣿⣧⣿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⡿⣵⣿⣿⣟⠣⣿⣿⣿⣿⣿⣵⣿⣿⣿⢿⡿⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣑⢿⣿⡿⣻⣾⣿⣿⣿⠿⣫⣿⣿⣿⣿⣿⣾⣿⣿⣿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣶⣍⠣⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⡿⢻⣿⣿⣿⣟⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⡚⢿⣿⢣⣾⣿⣿⡿⠿⣾⣿⣿⣿⣿⢗⣼⣿⣿⣿⡿⣻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣷⡙⢽⣿⣿⣿⡿⣢⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡻⣿⢃⣾⣿⣝⠻⢁⣿⣿⣿⣿⣿⢈⣛⠿⢿⣿⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡸⣿⣿⣿⣷⡜⣻⠿⠿⣿⢻⣿⣿⣿⡷⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢥⣟⡻⠿⢫⣿⣿⣿⣿⡆⣿⣿⣿⢫⣶⣶⣶⢩⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⠻⠿⢿⢟⣧⣬⣍⢷⣿⣿⠿⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢠⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⠉
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⣿⣇⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣀⠀⢰⡄⠀⠀⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⡆⠈⣿⣿⡇⠀⠀⠀⢠⡆⠀⠀⠿⠋⠀⢀⣾⡇⠀⢰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⡆⢸⣿⣿⠀⠀⠀⣼⣧⣾⠇⠀⠻⢀⣾⣿⣅⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠈⣿⣿⣇⠀⣿⣿⣇⠀⠀⣿⣿⣿⠀⠀⠀⠻⣿⣿⣿⡆⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢺⣿⣿⣿⠀⢻⣿⣿⡄⠀⣿⣿⣿⡆⠀⠀⣦⣸⣿⣿⠇⣸⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡈⢻⣿⣿⠀⠘⣿⣿⣷⣾⣿⣿⣿⡇⠀⢀⣿⣿⣿⣧⣼⢟⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⣹⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⠛⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡶⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣦⢘⣿⣿⣿⣿⣿⠇⠀⢀⣴⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⢻⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣧⠀⢀⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⠃⠘⣟⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣇⣸⣿⣿⣿⢆⣴⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⣾⣿⣿⣿⣿⠉⢠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⡁⢿⢇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣙⠉⣙⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⡿⠃⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⡇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣾⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣹⠟⢁⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⢃⣴⣿⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢹⣿⣿⣏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟⠙⣻⣶⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⣸⣿⣎⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠷⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⣿⡏⣴⣶⣾⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣄⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠐⠦⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⣠⣴⣿⣿⣿⣿⠿⢃⣿⣿⣿⣿⣿⣿⣟⠙⢿⣿⣿⣿⣧⣤⡈⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⣿⡿⠋⢡⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣧⣄⣤⣿⣿⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣤⢈⡛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣄⣤⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣏⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠏⠀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢹⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣥⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠉⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠙⣦⣀⣀
⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣷⡄⠀⠀⢡⡀⠀⣠
⣿⣿⣦⠄⢀⣸⣿⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⢿⡌⢻⡄⠙
⣿⣿⣿⣤⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠙⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣀⣄⣀⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶
⠀⠀⠀⠀⠀⡀⠛⣍⠈⠁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⢢⡀⠘⣿⠁⠀⠔⠁
⠀⠀⠀⠀⡀⠀⠀⠸⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⣿⠁⠁⠀⠊
⠀⠀⠀⠀⠙⠀⠂⠠⢸⡂⠠⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣿⡤
⠀⠀⠀⠀⠀⠂⣀⠀⠌⣇⢀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄
⠀⠀⠀⠀⠀⠀⠁⢀⠀⢿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠚⠲⠒⠓⠚⠀⠛⠒⠓⠒⠒⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⠠⡀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡄⠀⠀⣸⠅⠀⠀⠀⢤⠠⠀⠙⠀⠀⠁⠉⠃⠀⡀
⠀⠀⠂⠀⠀⠀⢲⡆⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡼⠃⠀⠀⠉⠀⠀⠀⠳⢤⡄⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠠⠤⠤⢴⣾⡿⠃⠈⠉⠂⠀⠁⠀⠀⠀⢠⡼⠄⠀⠀⠀⠀⠀⠈⡓⠀⠈⠈⣙⡦⣄⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣟⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⡀⠐⠉⠹⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡤⠀⠉⠀⠄⢀⠀⠀⠀⠀⠁⠄⠀⣰⡟⠀⠀⠴⠆⣹⠷⢶⣶⣄⡘⠷⠦⠈⠀⠺⣆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⣿⣄⣜⣿⠋⠒⢀⣀⠀⡀⠀⠀⠀⠙⢤⢴⢶⣿⣿⣿⣶⣿⣠⣤⣝⠶⠄⠈⣿⣿⣷⡶⠀⡀⠀⠀⡁⠢⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⡜⣻⠙⣼⣡⣤⣶⡉⠀⠀⠀⠀⠀⠒⡇⠘⢻⣰⣿⣿⢿⣿⣿⣿⣝⡋⠐⠀⠀⠀⠉⣿⣷⡤⠀⠠⣳⡌⡀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠐⠿⣷⣿⡏⢩⠿⢋⣩⠤⢀⣀⢄⡁⢀⣴⣿⡿⣿⠉⣼⣿⣿⣿⠛⠋⠀⠂⠈⠙⠢⠜⢿⣧⡠⠊⢉⠓⠏⠰⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠓⠹⣶⣿⣿⣾⡁⣀⡉⣀⠐⣎⠉⢹⣷⣿⣿⡟⠷⢃⣴⣿⣿⠿⠁⢶⡀⣀⣉⣀⠀⠌⠀⠀⠹⣷⣾⣥⠀⠀⠰⠆⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⠑⢦⣹⣿⡛⣿⡗⠡⠈⠋⠲⡾⠛⢿⣿⣿⣿⡿⠀⣼⣿⢛⣤⣤⣾⣿⣯⣿⣽⣿⣿⣶⣆⡈⢻⣿⣿⣭⡇⠀⠈⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣎⠻⣹⣿⣿⡀⠀⠀⠀⠃⢈⣿⣿⣿⣿⣿⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣾⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢶⣽⣦⣿⣿⣧⠀⠀⠄⠁⢠⣿⣿⣿⣿⣿⣡⣼⣿⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⢹⣻⣷⣬⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⣾⣿⣿⣿⣿⡟⠀⠀⢄⣠⣼⣿⣿⣏⣽⣿⣿⣿⣟⠿⣹⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢃⣶⣿⣿⣿⣷⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣬⡿⢿⣿⢣⡄⠀⠀⣼⣿⣿⡿⠉⣹⣿⣿⣽⣿⣷⣾⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⡜⠿⣀⣹⣷⠐⣻⣾⣿⡟⣛⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣩⣥⣶⢾⠛⠛⡿⣟⠓⢼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⢿⣿⣿⠷⣊⣀⣿⢿⡻⠃⠘⢿⣿⣿⣿⣿⣿⠛⠂⢤⣄⣤⡙⠛⠿⠿⡿⠿⠿⡟⠀⠀⠃⢿⣠⢾⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢼⡿⣿⣿⣿⠟⠿⣿⣞⠀⢐⠀⢸⣿⣿⣯⣿⣿⣥⢤⡀⠙⠋⡄⠤⠀⠤⠀⣀⠀⠀⠀⠰⢄⣿⢀⣼⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⣂⠀⢛⣥⡶⠊⠁⢸⣿⣿⣿⣿⣿⣿⡂⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⣿⣿⣟⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⣿⣿⣿⡿⣇⣰⠞⢃⠀⠠⢳⣾⣿⣿⣿⣿⣿⡛⠿⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⢠⣼⣿⣿⢋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠛⣿⣣⠆⡡⠔⣁⠄⠾⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠋⣠⢾⣿⡡⢷⠂⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⠛⣻⠋⠀⠀⠀⣻⣧⠴⢊⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⣠⣇⣼⣳⣾⠵⢁⠘⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⢘⡷⣟⣡⠀⠀⣄⠉⠈⠶⣼⣷⣿⣿⣿⣯⣷⠀⠀⠀⠀⠀⠀⢠⣀⣴⣶⣟⣽⣿⡿⣫⠐⡁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⢀⡀⠀⠐⣰⣥⡜⢀⣯⠄⠀⣶⡼⣿⣿⣿⣷⣿⣿⡀⠀⠀⠀⠠⠰⣞⠶⠜⠛⢿⣿⢟⡼⢋⠀⠀⠂⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠹⠁⠠⡆⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣞⢶⡀⢀⡀⠀⠲⠀⠀⢷⠈⠿⢣⠈⠀⠀⠆⠀⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⢀⠀⡀⠀⠀⢉⠠⢠⠄⢀⣽⣿⣿⣿⣿⣿⣿⣿⣦⣷⣸⣧⢀⣾⣿⡃⠀⠀⢰⡄⠀⠠⠀⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⢀⡵⠀⠀⠤⠤⠈⠀⡘⢲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠂⢀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⢀⠀⣤⡤⠶⠒⠀⠰⡦⠉⠀⠁⢀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡇⠐⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⠀⠄⠒⠐⡎⠂⠀⢀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⢠⠀⠙⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠾⠂⠉⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⠉⠟⢿⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⢀⣤⡀⠀⠀⣀
⠀⠀⠀⠀⣀⠀⢂⠀⡄⠀⠀⠀⠐⠀⠀⢀⠀⢰⣰⣦⣵⣿⣿⣿⡿⣻⣥⣿⣿⣷⠀⠘⠡⠆⣈⣿⣿⣿⡄⠊⠀⠀⠀⠀⣼⡟⠉⠁⠀⠀⠁
⠀⠀⠀⢀⣒⣿⣶⣤⢁⢀⠀⠀⠀⣀⣀⣤⣴⣿⠿⢋⢛⣉⣿⣿⣿⣿⣿⡟⠋⢭⠤⠚⠁⠀⠈⠠⢿⣿⣀⡄⠀⢀⣦⡾⢋⠞⠀⠀⠄
⠀⠀⠀⠈⣻⣿⣿⣿⣯⣿⣷⣶⣆⣤⣯⣤⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠒⠲⠀⠀⠀⢴⣾⠿⠿⠄⣰⣾⠋⡅⠀⠀⠀⡀
⠀⠀⠀⢺⡿⠛⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⠁⠘⣻⣥⣿⣿⡿⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⣾⣷⢶⣾⣿⠏⠈⠀⠀⠀⡘⠁
⠀⠀⢀⣼⣿⣶⣿⣶⣦⣀⢀⣁⡀⢸⣿⠟⣠⣼⣣⣴⣿⣿⣿⣿⣿⡿⠒⠀⠄⠀⠀⠀⢀⡄⠀⣸⣯⣇⣛⣷⠃⠂⠴⠀⠀⠆
⠀⠀⠀⠀⠟⠛⢉⣿⣿⣿⣶⣉⣋⣠⣷⣌⣩⣾⣿⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣸⣟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠿⠛⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⠧⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⢀⢀⣽⠛⠛⢠⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠿⠏⠀⠀⠀⠀⠈⠀⠀⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⣿⣿⡏⠁⠀⠀⠀⠀⠲⣤⡀⠀⠀⠀⠀⠀⡠⠀⠈⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣆⠀⠀⠀⠠⢀⠀⠀⠀⠃⠐⠒⠆⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠁⠄⠀⠀⠀⠀⠘⠏⠘⠀⠁⠀⠀⠀⠀⠀⠈⠒⠘⣦⡀⠀⠂⠀⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⡵⣼⢠⠒⡤⢠⠆⣠⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡑⠶⠶⡷⢈⢀⠄⡐⢻⠸⢻⠟⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢁⠨⡀⢃⠄⠸⠨⠀⢠⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣄⡣⠁⠀⠃⠀⡗⠀⡀⢈⠉⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⡿⢿⣂⡂⠈⠐⠠⠁⠀⣧⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⣤⡌⠆⢁⣒⠐⢀⢨⣭⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣫⣿⠁⡀⡈⠐⢲⣞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠋⡝⠛⣱⠡⢵⣆⡉⠨⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⠁⠡⠐⠠⠘⠢⢰⠁⠨⡑⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠂⠁⠐⢁⡐⠀⡗⠀⠱⠆⢃⠄⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⠒⠠⠐⠉⠐⢀⢀⠡⢸⠀⠀⡴⢇⠀⠈⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠤⡐⡀⠂⠀⠁⠀⠠⠘⣄⠎⠀⠀⠀⠀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠶⠀⠄⠄⠁⠈⠠⠀⠁⡇⠀⣀⠀⠀⢀⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠇⠄⡀⠄⠀⢁⠀⡀⠂⡀⠝⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠈⠂⠐⣀⡁⠀⠐⠂⡏⠀⠀⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⣤⠄⠀⠀⠂⢨⠃⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣤⣀⣀⣀⣤⣴⡖⠚⠉⠉⠉⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣏⠀⠀⠉⠉⢢⡉⠉⠓⣶⢭⣀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣹⡛⠋⠉⠩⠽⢿⣄⠀⠙⠀⠈⢻⣏⠁⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠉⠀⢲⣿⣷⡶⠉⠀⠀⠀⠘⢧⡀⠀⠀⠈⠀⠱⡦⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠐⠁⠂⠔⠋⠉⠉⠙⠀⠀⠈⠀⠀⠈⠻⣶⣄⠀⠀⠀⠙⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠰⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠲⣤⣄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⢸⠸⡇⢀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⡞⠫⣝⠛⡿⠧⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⡆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⣂⡤⠞⠛⢏⡀⠘⢒⣲⡶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠎⡰⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠈⠑⣛⣽⢟⠀⢶⢹⢡⡑⠠⣹⣭⡓⢦⣄⡀⠀⠀⠀⠔⢁⠔⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢈⡶⣻⢇⡣⠛⡱⢸⡜⢎⢧⠳⢄⢢⣩⢟⡾⣿⠀⠀⠀⠔⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⣴⡛⡱⢺⠷⠇⠃⢘⣨⠒⠼⢛⣵⠟⢁⣿⣤⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠉⠛⠓⠛⠃⠛⠚⠚⣁⡤⠖⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠈⠉⠉⠁⢀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣷⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠀⠀⠀⠀⠀⠸⢿⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⢺⣿⡇⠀⠀⠀⠀⢀⠀⣒⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠶⢶⢦⠄⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⢯⣿⡇⠀⢀⡀⠀⢰⡄⣿⡷⠖⣘
⠀⠀⢀⣤⠀⠀⠀⠀⠈⡢⣀⡀⢊⠄⠒⠰⠾⠿⢿⡿⠿⠶⠀⠐⠀⡀⠀⠀⢾⣿⡇⠀⢄⠠⡄⠄⣿⡥⠶⠄
⠀⣰⠏⠉⡇⠀⢴⡞⠟⠟⡇⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⠿⢧⠀⣀⠁⢸⡞⠁⡔⠁⢀
⠸⠯⠀⠤⡀⠀⠀⣑⣦⢜⠀⢰⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⣈⠐⠌⢳⣏⡡⣍⢉⡃⣴⠂⣠⠄
⠀⣠⠾⣆⠑⣀⡾⣋⡩⠈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠸⢏⠑⢶⠄⣶⣠⡾⢿⣿⣀
⢸⠏⠁⢸⣤⡿⠳⣄⠐⢄⠘⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠣⡄⠈⠛⠀⠊
⠀⠀⠀⠈⠛⠖⡚⠂⠉⢲⡷⠟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠐
⠀⠀⠀⠀⠀⡴⠿⢷⣤⣤⡶⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⢠⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣵⠤⠠⠀⠚⠐⠃⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠀⠀⠀⠀⡄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠱⣀⠀⠂⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠹⣀⢷⡘⢠⠞⠷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠙⣄⠀⠀⠛⠀⢷⠋⠀⠔⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠈⢿⣷⠀⠀⠸⠠⠀⣠⣀⠆⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⢣⠙⡉⠢⢄⡀⢿⠀⠀⢀⡿⠋⢀⡰⠆⠚⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠑⡀⠀⠑⢰⡄⡸⠋⠀⠰⠋⠀⠀⠄⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠠⡄⣷⠀⠀⠘⣿⡉⣀⢀⣤⢖⣄⣤⠦⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡛⠒⡒⠛⠷⣞⠉⠻⠭⠀⡹⠟⠃⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠤⠤⠒⠚⠢⢄⡈⡑⢄⣰⣮⡴⠾⠚⠳⠬⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠮⠉⠁⠈⢦⡀⠀⣀⡿⠾⣚⢛⠡⣶⣶⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣡⠚⣃⡄⠛⢡⣄⢈⣡⣴⠕⠀⣤⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡠⢾⡃⣴⠖⠒⠫⣤⡤⠐⣊⠻⡄⠂⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⡴⠛⢇⠈⠁⡠⢚⠳⣄⠀⢀⢞⡙⡌⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⢶⡀⠉⢀⠷⡚⢄⠀⠀⡜⣋⠳⡀⠁⠈⢲⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡋⠁⢱⠤⡎⠀⠁⢈⡦⠮⡀⠀⠁⡰⠲⡀⠁⣹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢵⣀⠃⠒⠈⣢⣤⠋⠈⠂⢀⠤⣎⠀⠁⢘⣴⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡘⠹⢦⣀⡔⠑⠂⠑⣀⡐⠁⠚⠈⢂⣀⠊⠒⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡳⠀⠜⣍⠣⡀⢀⠞⠱⠈⠆⠀⠰⠁⠄⠂⠀⣘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡿⡄⠀⢠⢞⡻⡄⠀⢀⠞⣉⠣⡀⠀⡴⠙⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣰⡶⣅⠀⠀⡰⡖⢧⠀⠀⢀⡷⡺⡀⢁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡁⠉⠈⣡⢶⡀⠈⠀⡱⠤⡍⠈⠁⡰⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢙⢢⢼⡀⠁⢈⡶⢮⠀⠁⣈⢤⠘⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⣰⠒⢮⠈⢀⠵⠶⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠁


⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣵⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⠸⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢎⠂⠉⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⢄⠈⠄⠈⠀⢀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠀⡀⣴⡖⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠂⠠⠀⠈⠀⠀⡇⠀⠀⢉⠀⠔⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠤⠄⣤⠤⠤⡀⠀⢀⣠⡤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⡄⠀⠀⠀⠀⢱⠖⠁⠀⢀⠕⠢⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠥⠴⠈⢀⡉⠐⠓⠢⡄⠀⠊⠀⠐⠪⣤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠦⠒⢄⠈⢂⣠⣠⢈⡀⣠⠄⠈⠀⠀⠑⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣸⠄⠁⡢⠴⠪⡤⠾⢦⣄⠀⡖⠶⠠⢈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⡄⠆⢠⠞⢻⡷⠻⠗⠀⢦⣷⠛⠧⡀⡒⠠⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠄⠊⡸⠃⠀⠀⠀⢠⡀⠁⠄⣡⠀⠘⠛⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠸⠀⠀⠀⠀⠘⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⡄⠀⠁⠄⠈⢱⠆⠐⠀⠁⠐⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⡀⠀⠣⣀⠀⠀⡠⠈⠁⠀⢀⠀⠀⢚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠠⠁⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢀⠄⠀⠉⠠⠑⠂⢠⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠠⡄⠀⠐⠈⠂⢀⡀⠀⠀⠀⠠⠄⠀⠄⠌⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠐⣄⠀⠀⠀⠈⢃⠀⠀⢀⡀⠀⢀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠤⠷⠄⣀⣀⣀⠀⠤⠁⠠⠐⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡘⢄⡆⣿⢀⠀⢀
⠀⠀⠀⠀⠀⢀⡤⠤⠶⠂⠘⠳⢿⣶⡄⠀⠀⠀⠀⠀⠠⣤⣀⠘⣆⣸⣿⣾⢾⢛⡘⡤⡾
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⢀⡈⠻⣷⣾⣿⣿⣯⣾⣿⣫⣴⠇⢀⡀
⠀⠀⣠⠖⠁⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣦⣹⣿⡼⣿⡿⠋⣿⠟⠁⣶⠋⡔⠀⠀⠀⠀⠀⠀⣀⣴⠷⠞⠉⠓⠒⠲⢤⣤
⠀⠀⣿⠀⠀⠀⠀⠚⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⣿⣿⡇⢌⠙⣦⠀⢸⡷⣴⣯⡞⢁⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠘⡄⠀⠀⠀⢰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢾⣿⣿⣿⣿⡇⣿⣸⣾⠻⣿⣿⡬⢥⠆⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢷⡄⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣷⡿⡞⢻⡿⣡⣆⣻⣿⢋⣠⣤⠂⠀⢀⡟⠀⠀⠀⠀⣀⠛⠉⠠⠀⠀⠂⠀⠰⡇
⠀⠀⠀⠈⠻⣤⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠳⢶⣾⣟⠉⠻⣤⢤⡻⠟⢻⣯⣴⣾⣉⡁⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠈⠳⣄⠀⠀⠈⠳⢾⣿⡟⢛⡻⣿⡿⢲⣄⣾⣿⠗⠋⠭⠀⠀⠀⠐⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢻⣿⣿⣿⠻⠢⣠⠿⢫⣯⣤⠆⠂⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⠈⠀⠀⢀⣠⡤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⢈⠃⠀⠀⠀⣤⡿⣿⣿⡻⡖⠺⢛⡄⢠⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⡶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⢶⣿⣷⣭⣭⣱⣿⣟⣿⠿⠿⣿⣭⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣠⡃⠀⠀⢰⣿⡟⣿⢾⣧⣷⣴⣬⣿⣽⣿⡟⠟⢿⠿⡇
⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⢀⡀⣀⠞⠁⠀⠀⠀⢘⣿⣷⣿⣷⣾⡺⠿⢿⣷⣶⣤⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⢻⡟⢿⣄⣈⣿⣿⡿⡟⠚⠛⢻⣽⣯⡥⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠻⢻⣦⡤⣤⣟⣽⣷⡟⠛⠿⣷⣿⡧⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⢹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⡗⢹⣋⣉⣹⣿⣿⡷⠿⣿⡅⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⣹⣿⣷⣷⡎⣽⣿⣿⣡⣌⣽⣿⡟⢀⣴⠚⠃⠀⠀⠎⠀⠀⠀⣠⠄⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⡀⠛⠛⠋⢉⠀⠀⣠⠄⠀⠀⠀⠓⠶⠶⢬⣽⣿⣯⣯⠿⢿⡿⢿⣿⠿⠿⠟⠴⠟⠁⠀⠀⠀⠰⠃⠀⠀⣄⠁⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠁⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡞⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠐⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⣴⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠆⠊⠖⠋⠦⢆⠆⢲⠱⠋⠰⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠤⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠥⠤⡀⠀⠈⢢⠀⠀⠀⠀⠀⠂⢀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⢀⣀⣀⡀
⢀⠂⠀⢀⠀⡀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⡆⠀⠈⡇⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋⠁⢀⣍⠉⠒⣄
⠘⢄⣀⡠⠄⢄⣀⡠⠃⠀⠀⠠⠀⠀⠀⢀⠇⠀⢀⠇⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠸⣄⣀⡾
⠀⠄⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠲⠖⡁⠠⠔⠀⠀⠀⠀⠘⠀⠀⠁⠐⠁⠀⠚⠀⠀⠀⠀⠠⠀⠀⢀⣠
⠀⠑⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠐
⠀⠀⠀⢢⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁⠂⠀⠀⠀⠀⠀⠀⣀⠤⠒⠲⢄⠀⠀⠀⠀⠀⠂⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⢄⢐⣦⡈⡠⡆⢠
⠀⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⡄⠀⠑⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣠⠇⠀⠀⠐⢖⠞⠁⠀⠈⠱⣆⠆
⠀⠸⣿⣿⣿⡟⠹⣿⣿⣿⡟⠀⠀⠀⠀⠑⣄⠀⢠⠁⠀⠀⠀⠀⠀⢹⠉⡉⠈⡽⠀⠀⠀⠀⢃⡼⡀⠀⠀⠀⢀⠧⠐
⠀⠀⠈⠛⠋⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠢⠎⠀⠀⠀⠀⠀⠀⠀⡇⢀⢠⠃⠀⠀⠀⠀⠈⣨⢩⣢⠤⣔⠎⢁⠁
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠀⠀⠀⠀⠘⣄⡏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠈⠁
⠀⠀⠠⣼⡄⠀⢡⠉⠂⠀⠀⢀⣴⣶⣶⣶⣶⣶⠄⠀⠐⠋⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⢁⠀⠀⣠⠁⠀⠀⠀⠈⢦⠂⠀⠀⠐⠐⠃⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦⣀⣹⣦⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠌⠀⠀⢻⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠰⠀⠀⠀⠀⢀⠀⣼⡿⠟⠋⠉⢁⣦⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⡀⠀⣀⡼⡀⠀⠀⠀⠀⠸⠀⠁⠈⠀⠂⠀⠀⢐⠨⠊⠀⠉⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⣸⠀⠀⠀⡀
⠀⠀⠀⠈⠘⠓⠈⠀⠀⠀⠀⠀⠀⠁⠐⠐⡈⠅⠀⠔⠁⠀⠀⠀⠀⠀⡀⢀⣾⣿⣿⡷⠀⠀⢠⣤⣶⣿⣤⣤⣴⣿⡄
⠀⠀⠀⠠⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⠭⠙⠋⠛⠚⠳⠶⢦⣤⣄⣀⣀⠀⠀⢙⣿⣶⡴⣚⠩⠐⠂⠈⠉⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡡⠊⠁⠀⠀⢀⠤⠀⠠⢄⠀⠀⠉⠉⠙⠛⠛⠃⢚⡙⢷⣿⣷⣄⠀⠀⠀⠀⢀⣀⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡰⠋⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⢿⡿⣷⣀⡴⢛⣩⣭⣿⣿⣦⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢃⠎⠀⢀⠔⠂⠒⠒⢄⠑⢤⣀⣠⡴⠏⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢮⢻⣽⣿⢁⣾⣿⣿⣿⣿⣿⣿⠂⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢰⢁⠃⠀⠀⢊⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⠘⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠆⡈⠀⠀⠀⠈⠳⠤⣤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣦⢠⡘⣜⢧⡀⢀⡜⠉⠻⠟⣛⣛⡉⠕⠊⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⢀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣭⣽⣿⣿⣿⡖⠛⡏⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠃⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠠⣏⢰⣿⣿⣿⣿⣿⣿⡟⡀⠘⡀⠀⠘⢄⠀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠱⢄⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⡠⠁⠀⠱⠀⠀⠈⣮⡴⠖⠒⠀⠈⠳⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠛⠭⠚⠁⠀⠀⠀⢳⣴⣾⣿⣆⠀⠀⠀⠀⠀⢹⡆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⠄⣒⣲⡒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⢠⢎⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠰⢡⡟⠉⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣔⠪⠕⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢃⠘⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠀⢠⡞⢉⣴⣶⣿⣿⣷⣄⠀⠀⠀⠈⢿⣿⣿⣿⣷⡄⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠓⣄⣀⠀⣀⢀⠴⡡⠃⠀⠀⠀⣿⣠⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠈⣿⣿⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⢡⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠭⠅⠒⠁⠀⠀⠀⢀⡠⠖⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡔⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢚⡀⢀⠸⣧⣀⣼⣿⣿⠃⠀⠀⠀⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⡇⠀⠀⣿⣿⣿⣿⠇⠀⠀⢀⡞⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⣄⠀⠀⢠⣿⣿⣿⠃⠀⠀⣿⠿⠋⠁⡠⢀⠴⠋⠀⢀⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠸⣷⡄⠀⢻⣿⡏⠀⢀⠀⣡⣾⣶⣥⠞⠁⠀⢠⣶⣶⠟⢁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀⢻⡇⠀⠀⣾⣿⣿⡿⠁⣤⠚⢛⡿⠟⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣎⣧⠀⠀⢸⣿⣏⣶⣿⠷⠚⠣⠄⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡹⠿⠀⣄⣼⡿⠋⠁⠀⢀⣠⣤⠶⣛⣽⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⢂⣽⣿⣿⣦⣾⣾⠋⣤⣄⣠⣴⣿⠿⢷⣿⡿⠛⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠶⠒⣂⣈⣿⣿⣿⣟⠛⢿⢃⣾⡿⢛⣩⣄⣀⣤⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠋⠀⠚⠛⠉⠓⣛⣿⣻⣿⡆⢀⣾⡿⠻⡿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠛⠉⣽⠟⢉⣾⣿⣦⠀⠀⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠃⢀⣏⠛⢯⢻⠀⢠⣿⣗⣷⣦⡞⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄⠀⢳⣄⣾⠷⢿⣧⣀⣸⢼⣀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⢘⣿⣃⣼⣿⡿⠟⢋⡙⣿⣿⣷⣦⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡵⠖⣀⣼⣿⠿⠋⠙⢿⣾⣿⣿⣷⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠻⣿⣿⣿⠟⠁⣀⣤⣳⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⠟⠉⠻⣷⣜⣉⣱⣿⣿⣿⡿⠿⣿⡛⣭⣅⣹⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣾⠃⣠⠦⣄⡀⢻⣿⣿⣿⠟⣩⢤⣤⣜⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⢿⣧⣶⣼⣥⣾⠟⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⢀⡀⢘⣿⣿⠟⠁⣠⣦⣾⣿⣿⣿⣿⣿⠿⢿⣿⣦⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣼⣷⣶⠋⠛⣦⣾⣯⣿⣿⣿⣿⣿⣉⣠⣶⡀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢙⣿⣿⠁⣰⣄⢸⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿⠿⣶⣿⣎⢿⣿⣿⣡⣴⣿⣧⣿⣿⣿⣿⡟⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢠⣸⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⡿⣿⣿⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢾⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⢀⣠⣶⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⢀⣴⡟⣿⠏⠀⢀⣴⡿⠋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣰⣿⡟⠀⠀⠀⢠⣾⠋⣼⡟⢀⣴⡿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠹⣿⠀⠀⢰⣿⣿⡇⠀⠀⣰⡿⠁⢠⣿⣱⡿⠋⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⣸⡟⠀⣿⡄⢀⣿⠃⣿⠁⠀⣼⡟⠁⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⣿⠇⠀⢿⣇⣼⡏⠀⣿⠀⣼⡟⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣼⠟⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠹⣿⣿⠀⠀⠸⣿⣿⠀⠀⣿⣾⠟⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠋⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣿⡀⠀⠀⣿⡏⠀⠀⣿⡟⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣿⣄⠀⣿⠇⠀⣸⡟⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣰⡿⠁⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⠻⣷⣿⠀⢰⡿⠁⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⣼⡟⠀⢀⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠈⠛⢿⣿⣧⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⢀⣾⣏⣤⣶⠿⣫⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣠⣤⣤⣤⣬⣿⣆⣸⣟⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⣠⣿⠿⠟⠋⢁⣴⡟⠁⠀⢀⣠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠛⣿⠉⠉⠉⠉⠉⠉⣿⡟⠻⢷⣦⣄⠀⢀⣴⠟⠁⠀⢀⣴⡟⠁⠀⢀⣴⣿⣫⣤⡴⠾⠛⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠉⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢻⡇⠀⠀⠈⠛⣿⣿⡁⣀⣀⣤⡿⠋⠀⣠⣴⠿⠟⠛⠉⠁⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⣿⡆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⣿⣀⣤⣶⡾⠟⠛⣿⡟⠻⢿⣦⣴⣾⠛⠁⠀⠀⠀⣠⣴⡿⠋⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣤⣀⣀⣿⣇⣀⣀⣀⣀⣀⣸⣷⣤⣤⣶⠶⠾⠿⢿⡟⠉⠀⠀⠀⠀⠈⢿⣆⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣰⣿⡏⠉⠉⠉⠉⠙⢻⣿⠛⠛⠛⠛⠛⠙⣿⡏⠁⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⢀⣾⠏⢹⣇⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢀⣠⣤⡶⠿⣿⣤⣤⣤⣴⣶⠾⠟⠛⠉
⠀⠀⠀⣰⣿⣇⣀⠘⣿⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢀⣀⣬⣿⣶⠿⠛⠋⠁⠀⠀⢿⡇
⠀⠀⣰⡿⠉⠙⠛⠿⢿⣷⣶⣶⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣾⣿⣶⠿⠿⠟⠛⠉⠙⣿⡄⠀⠀⠀⠀⠀⢸⣧
⠀⣰⡿⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⠉⠉⢹⣯⠉⠉⠉⠉⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢀⣸⣿
⢠⣿⣧⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠈⣿⣦⣴⡶⠟⢻⣿
⣸⡟⣿⡿⢶⣦⣤⣄⣀⡈⣿⡄⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠹⣷⣀⣠⣤⣴⡶⠿⠛⢻⣧⡀⠀⣸⡇
⣿⡇⢸⣷⠀⠀⠉⠉⠛⠛⠻⣿⡶⢶⣶⣶⣶⣶⣾⣿⣶⣶⣶⡶⠾⠿⠿⠻⣿⡏⠉⠀⠀⠀⠀⠀⠙⣷⣄⣿⠇
⣿⡇⠀⢻⣧⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠈⢻⡿
⢻⣷⣤⣈⣿⣆⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⣀⣠⣴⣾⣿⠃
⠘⣿⡉⠙⠛⢿⣷⣦⣤⣤⣀⣀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⢀⣀⣀⣤⣤⣾⢿⣿⡛⠉⠁⣼⡏
⠀⠹⣷⡀⠀⠈⢻⣧⠈⠉⠙⠛⠛⠛⠿⢿⣷⡷⠶⠶⠶⠿⠿⠿⠿⣿⡟⠛⠉⠉⠀⠀⠀⠙⢿⣦⣼⡟
⠀⠀⠹⣷⡄⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⣽⡟
⠀⠀⠀⠘⢿⣦⡀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠙⢿⣿⣶⣾⣿⣷⣄⣀⣀⣀⠀⠀⠀⠘⢿⣦⡀⠀⣀⣀⣀⣀⣤⣼⣿⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣉⠻⣿⡛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣟⠛⠋⠉⠉⣁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⢈⣻⣷⣤⣶⠿⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠈
⡈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⢠⡡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣧⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⣛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣌⠳⣻⡵⡃⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠐
⠀⠦⡆⠘⢄⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠈⠯⢳⡵⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⡄⠙⠠⠂⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢰⠀⠀⠁⠘⢃⠉⣀⡀⠛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⣨⠃⠀⢲⣁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⡴⠂⠈⠈⣌⠁⢋⣠⠆⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⢐⠀⠱⠀⠈⢑⠀⠈⠉⠀⠀⠠⠖⡄⠀⠞⠀⠀⠀⠀⠈⣄⠩⠖⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾
⠠⠀⡀⠀⠀⣨⠆⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣷⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⣽⣿⣿⡿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣤⣞⡋⠁⢸⠏⢸⡟⢿⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣩⣿⡟⣮⠻⡄⠈⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⠹⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡴⢒⣖⣒⡂⢥⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⣀⣤⣽⡝⢻⠁⢩⣟⣿⣿⣷⣬⣄⡀
⠀⠀⠀⠀⠀⢀⣩⣽⣿⠿⠛⠋⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠟⠻⠿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠈⠛⠟⠥⣤⠀⠒⢲⣾⣿⣿⣒⣒⡒⠲⠶⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⢿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠂

⣹⣦
⠈⢳⡀⠉⠲⣄⡀
⠀⢸⡇⣄⡀⠼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⢤⣄⣀
⢀⡏⠸⠀⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⢸⠁⠀⠀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⢀⣀⣀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠉⡧⠀⠀⠀⠀⢀⡤⠤⢤⡄⠀⠀⠙
⠈⣆⠀⠀⠀⡼⠁⠀⠹⣄⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⡴⠛⠉⠉⠉⢻⡆⠀⠀⠀⠱⢤⣤⣤⠤⠞⠁⠀⠀⠀⡔⠁⠀⠀⠀⡿
⠀⠈⠳⢤⣴⠃⠀⢀⡴⠟⣦⡶⠛⠁⠀⠀⠀⠀⠀⠸⢅⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⡤⠞⠁⠀⠀⢠
⠀⠀⠀⠀⠀⢇⢠⠏⠀⠀⠈⢧⣠⠴⠶⠶⣦⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠈⢷⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⢠⡿⠳⡀⠀⠀⠀⠀⢳⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⡆⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠈⣇⠀⠙⣆⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢀⡴⠃⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠹⢦⣀⠈⢳⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠶⠋⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡙⣆⠀⠀⠀⠈⢧⣠⠤⠔⠒⠒⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠳⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡟⢦⡀⠀⠘⣧⡀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⢦⣀⣉⣹⣦⣼⡷⠞⠋⠁⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣄⣀⣀⣀⣠⣤⣤⣤⡴⠶⣾⠛⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣿

⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠤⠀⢀⣄⠸⠀⠀⠀⣤⣶⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠻⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢰⡖⢰⣶⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡽⠟⠛⠘⡅⢆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤
⢀⡌⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠦⣄⠀⠄
⣿⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠄
⠋⠀⠘⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⢀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢄⠀⠠⠀⡀⠀⠑⠩⣢⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀
⠈⠀⠀⣦⡈⡢⠄⡀⠀⠀⠀⠀⠀⢉⣉⣙⡛⠙⠋⠀⠀⠀⠀⠐⠠⢀⠀⡠⡀⠰⡃⠀⠠⠎⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠂⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢶⢶⣦⣃⡀⠀⣤⣀⣉⣩⣭⣍⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠲⣆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹
⠀⠀⠀⡃⠸⣿⢿⣿⣿⣿⣿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠁⠘⣄⠀⠄⠀⢀⡀⠀⠀⠀⢀⠀⠂⠁⠠⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⢀⠀⠀⠘⠛⢿⣏⣛⠙⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⡁⠀⢀⡀⠈⠧⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⢤⢀⠀⠀⠀⠀⠀⠈⠙⠳⠀⠀⠀⠀⠐⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠺⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠰⠂
⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⣦⡀⣿⣄⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠘⣣⠈⠓⠘⡦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠢⠀⠐⢄⠐⠀⠀⠀⠘⠋⠁⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠄⠘⡀⣄⢀⢃⠀⠀⠀⠀⠁⡞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢠⠼⠏⡐⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠈⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠰⣟⣿⣿⣿⡿⠟⡻⠶⠖⠲⢆⣀⣀⠀⠀⠀⢃⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃
⠀⠀⠀⠀⠀⠇⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣄⠀⠀⠀⠀⠀⠃⢠
⠀⠀⠂⢄⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉
⠀⠀⠀⠀⠀⠙⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⢀⠔⣩⠖⠈⠉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢠⣶⠠⠀⠀⠀⠀⠀⠁⠠⢁⠎⠀⢀⠔⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⢋⠞⡴⠁⢀
⠉⠀⠀⠀⠀⣘⠿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠐⠀⢠⠃⡠⢀
⠀⠁⠀⠀⠀⠉⠲⠠⠄⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠂⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⢁⠂
⠀⠀⠀⠀⠀⢀⡄⣧⠀⠀⠁⠄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⢸⠇⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠠⣤⢄⣀⢠⡄⠠⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣢⣄⠉⢧⠀⠆⢀⢞⡃⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠱⣾⡄⠈⡇⢸⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠮⣁⠠⡈⠳⡐⡇⠸⠀⢸⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠳⠬⣤⣱⣿⡀⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠕⠲⠀⠀⡉⠻⣇⠘⢮⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠦⣄⠈⠑⡮⢓⠋⠐⠩⡉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠓⣞⠁⠀⠙⠀⠀⠀⠀⠀⠲⠂⠌⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⡀⠀⠀⠀⠁⠀⢀⡀⠀⠂⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⢠⠄⠀⠀⠀⠀⠀⠐⠒⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡀⠀⠀⠀⠒⠀⠀⢀⡀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡀⠚⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠖⠀⠀⢀⡄⠀⠀⡀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⡁⠐⡀⡠⠀⠀⠀⠈⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠪⠥⠤⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⣹⡇⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠐⠀⠉⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢤⡉⠙⠻⣥⣠⡤⠔⠋⣰⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢙⠀⠲⠌⢿⡄⠀⠘⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠶⢷⡀⢠⣬⣡⠀⠁⡴⠋⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢦⣤⡵⢤⡤⠠⣤⠆⢠⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⢢⣤⣴⡖⢲⡶⣿⣆⡠⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⣷⠶⣾⠋⠓⢿⢀⠙⣏⠛⡟⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠇⣿⢶⡟⠉⠀⣸⠀⢠⠈⢲⠋⠁⠤⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣆⡹⠁⠹⠔⢋⡉⢢⡀⠀⠘⢿⠒⠂⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡉⠹⣥⠔⠒⢆⠈⠁⢀⡿⣃⡤⡄⠱⠆⠰⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣠⣀⣴⣇⡻⢆⠸⢦⠔⢁⡀⠹⣆⠐⠀⢇⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⣃⡂⠑⣄⡤⠒⠙⢮⠉⢉⢁⡍⠁⠶⠜⠦⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⣇⣩⠄⠚⠰⠾⢇⠀⡟⠡⣄⠈⣷⠀⢀⡰⡄⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⢿⡰⠀⠀⣄⠤⠂⠣⡔⠊⡩⠘⠳⡊⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠑⣀⣿⣤⣤⡧⣀⣴⡀⠘⣾⠁⣠⠄⠡⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢧⣦⡀⣧⣀⠖⠂⠉⢢⠄⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠘⠃⠏⠀⠲⠂⠘⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡤⣤⣤⠄⢠⢤⣤⠀⣤⡤⣤⡤⣤⠀⢤⡤⢠⣤⡤⠤⢤⠠⣤⠠⣤⠄⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣌⡂⢠⡄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⣉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠀⠒⠐⠐⠂⢠⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠂⠐⠀⠀⠀⠀⠐⠺⣀⣀⡀⡀⢀⣠⠔⠃⠀⠀⠀⠀⢣⠠⠐⡆⠐⠒⠒⢂⢀⠐⡒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠆⢠⣼⣧⡼⠇⠉⠁⠀⠀⠀⠆⠀⠀⢋⠦⠀⠇⠀⠀⠀⠀⠀⠠⡄⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠄⠎⢙⣉⡀⠀⠀⠁⠀⣄⣠⠈⠀⠀⠀⠀⠀⠃⠀⠀⠴⡄⠀⢤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡙⠉⠁⠀⠀⠀⠀⠠⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⡀⠀⠁⠀⠦⢀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠄⠀⢠⠀⣂⡀⠦⠀⢀⠠⠤⡄⠀⠀⠀⠀⠀⡀⠦⠘⠛⠂⠁⣀⠀⠀⡀⢇⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⡀
⠀⠀⠀⠈⡀⠀⠂⠉⠀⠉⠀⠀⠀⠑⡀⠀⠐⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠰⠀⡃⠸⠀⠀⠀⠀⠂⡬⢁⡤⠂⠘⠓
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢱⡾⠃⠀⠀⡇⠀⠉⠀⠀⠰⠀⠀⠀⠀⠊⠀⠀⠀⡀⠌⢀⣉⡤⠂⠀⠀⠈⢚⠁⢸⡄⣀⠀⠀⠀⡤⢠
⠀⠀⠀⠀⠲⠄⠈⠀⠀⠀⠀⠀⠀⢀⠀⢠⠜⠂⠀⠀⠀⠀⠐⠀⠐⠂⠠⠊⠁⠀⠊⠁⠀⠀⠀⠀⠀⠛⠀⠈⠀⠀⢀⢀⡴⠇⠺⠢
⠀⠀⠀⠀⠀⠤⣄⠀⠁⠒⠐⠒⠀⠀⡠⠐⠠⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠈⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠘⠒⠈⠁⠂⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠰⠶⠲⠆⠐⠗⠶⠂⠲⠖⠲⠖⠶⠀⠶⠶⠰⠶⠲⠚⠲⠀⠖⠒⠆⠳⠆
⠀⠀⠀⢀⠔⠛⠃⠀⠧⠔⢠⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⢢⠀⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡁⠀⠀⠀⠀⠀⠀⠈⠤⢤⡄⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠉⣥⠀⠀⡄⠄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠱⣀⡀⠀⠀⢠⠄⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡞
⠀⠀⠀⠙⡇⠲⢲⠒⠆⠀⠒⣠⠛⠄⠀⠀⠀⠀⠀⠀⠉⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁
⠀⠀⠀⠀⠀⠈⠋⠠⠛⠃⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠆⢆⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⠀⡀⠀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣈⠀⠀⢀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠂⢈⠉⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠐⢢⠘⠓
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠈⣈⠀⡜⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠈⠘⠁
⠀⠀⠀⠀⠀⠠⠄⠀⢀⠂⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⠀⠉⢈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣌⠁⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠂⠀⠀⠀⠀⠀⠀⡀⠚⠢⡄⠀⠀⠴⠜⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢰⠔⠀⣘⣧⣠⣄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⣿⣿⢙⣏⣿⣯⣵⠋⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠥⡀⢿⡟⢺⣿⣿⣿⣷⣾⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣐⣦⡈⣧⣄⠙⣿⣿⡿⢟⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢸⣯⣛⣿⠛⠛⣿⣯⣴⣿⣴⡒⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣀⣴⣯⣍⣿⣿⣷⣾⣿⣿⣛⣻⡇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⢿⣿⣷⣿⣿⣟⣯⢹⡛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣟⣿⠉⣷⣭⡎⢻⣿⣿⣿⣿⣧⡈⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣽⢟⡫⢿⡿⢿⠯⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⡜⠁⢀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⡌⣿⣾⣓⣛⣿⠴⣶⣭⣽⣿⣿⣿⣿⣿⣿⡀⠀⠒⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣩⣿⣿⢛⡓⣿⣿⣬⣽⣿⣿⣿⣿⣿⡅⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣥⣻⣿⠟⢷⣿⡿⠿⠟⠉⠉⢹⡛⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡅⠀⠀⠈⢳⣾⡗⡀⠀⠈⠹⣾⡶⢬⣽⣿⢷⣦⡉⠉⠁⠰⡄⠀⠈⣷⠐⢆⢀⡄⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠰⣌⠑⣾⣟⢠⠀⠈⢻⣷⣷⡄⠀⠀⠹⣇⣪⣿⠾⢻⣿⣿⣶⠶⣴⣾⣷⠇⣠⣶⣾⣿⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⢹⠓⠈⢻⣿⣷⠀⠈⣿⣿⣃⠀⠀⣴⡿⢋⣿⣾⡟⢼⢿⣿⣧⣿⣵⣿⣶⠟⣉⣿⣤⢖⣾⡥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠃⠈⢿⣿⣦⠀⣹⣿⣿⡄⠀⠙⣿⢧⣽⣿⣟⣵⣾⣿⣿⣿⣁⢸⣏⣸⣿⣽⣿⣿⠿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣿⣿⠀⠀⢼⣿⣿⣧⢹⣿⣿⣷⠀⠐⣟⣧⣿⣽⣿⣿⣿⣿⣿⣟⠟⣛⠛⠻⣿⣿⣯⡋⠙⠾⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠉⠀⠀⠀⢻⣿⣿⢸⣿⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⡇⣿⣿⡶⢮⣿⡿⠧⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⢸⣿⡟⣸⣿⡿⠁⠀⣾⣿⣿⣿⣿⣿⡿⠛⠡⠔⢸⡟⠋⣤⠙⣄⠛⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠁⠀⠂⠀⠀⠀⣿⡏⣸⣿⠟⠀⢀⣴⣿⣿⡿⠟⠛⠋⠀⠀⠀⠐⠚⠁⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⡀⣀⠈⠁⠉⢀⡂⠉⠀⠈⠉⣁⢀⢠⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠟⠨⡄⡆⢹⢹⠀⢌⠆⢇⢡⠈⢿⠄⠇⡇⢃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠇⠓⠈⠘⠂⠀⠁⠁⠸⠀⠘⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠐⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⡠⠀⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢣⡄⠀⠀⠀⠄⠀⠀⠀⢠⠄⠀⠠⠀⠀⠌⠀⣸⠃⠀⠀⣠⠜⠁⠀⡐
⢠⣤⡄⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⢿⣀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠂⠀⠀⠈⣿⠀⢠⠞⠁⠀⠀⠄
⠀⠀⠀⠀⠂⠈⠐⠢⠄⠀⠈⢿⣆⠘⢯⡄⠀⠀⠀⡁⠀⠻⣄⠀⠀⠸⠇⠀⢨⣿⡆⠁⠀⠀⠀⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠢⠀⢻⣧⣜⣷⢷⡀⢠⠀⠀⠀⠈⠇⠀⠀⣶⠀⣾⡿⠁⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠐⢿⣿⣿⣎⣡⣾⠀⠀⠀⠀⠀⠀⠀⢸⣶⣟⠳⠀⠀⠀⣠⣤⣾⡿⠋⠀⠀⠀⡀⡰
⠀⠀⠀⢠⣤⠀⢀⠀⠀⠀⠀⠈⠰⡀⠙⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂⣸⠿⣈⡁⠀⠀⣼⣿⣿⡟⠁⡀⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠀⠀⠹⣢⡀⠈⠻⣿⠀⠀⠀⠀⠀⠀⢰⡿⢦⡌⠀⣀⣼⣿⣿⠋⡀⠊⠀⠀⠄⠂
⠀⠀⠀⠀⣀⣀⣀⣀⠀⠂⠀⠀⠀⠄⣱⣟⣦⠀⢻⡆⠀⠘⠀⠀⠀⠘⣿⣶⠞⠋⠁⠀⣿⠣⠈⢀⡠⠆
⠀⠀⠀⠘⠋⠉⠛⠻⢿⣿⣿⣿⡦⣄⠈⠻⣞⡛⢾⣷⠀⠀⡄⠀⠀⢀⠛⠁⠀⡀⠀⢠⠟⠠⣔⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢯⡙⠛⠷⣦⣈⢻⣦⢿⣇⣠⡿⠀⢠⡟⠀⡀⠢⠀⠀⣀⣰⢞⣀⣀⡀⣀⠀⢀⣠⣴⣶
⠀⠀⠀⠀⠀⠶⠶⢶⣶⣦⣤⣀⡘⢷⡀⠀⠉⠛⠻⢿⣿⣇⠀⣄⠸⠄⠀⡘⠀⠀⣰⣿⠟⠉⠀⣀⣁⣤⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣷⣿⣧⡀⢀⣋⠈⠉⠿⢷⣿⣧⡈⠀⣿⣤⣾⡛⠀⠀⢀⣼⠟⠋⠁⠁⠚⠶
⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⣶⣶⣟⢻⣿⣿⣿⡇⢘⣷⣤⠀⠀⠉⠻⣿⣿⣿⣵⣿⡁⠀⠀⠞⣡⣤⠂⣤⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⢶⣌⡙⠛⠿⣾⣬⣿⣿⣦⣀⣶⠉⠀⠀⠉⠠⣁⣤⠞⢉⣠⠄⠋⠁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣶⣦⣙⢿⣦⣀⠀⠈⠛⠛⢿⣿⡄⠀⣀⣀⣤⠾⠿⠂⢛⣿⣿⣶⣿⠿⠛⠛⠉⠉⠗
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠉⠙⠛⠿⣿⣿⡄⠀⠀⠀⠀⠈⣿⣦⡟⠁⠀⠀⣄⣰⣿⠿⠛⠫⡼⠛⠓⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣮⣉⣀⠀⠀⠛⠿⣷⣦⣠⡎⠀⠀⠉⠉⢢⣴⡿⢛⣩⢴⣿⣿⣷⣶⣶⠶⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠍⠉⠛⠛⠿⢦⣀⡀⣀⡹⠿⠘⠓⠀⣀⡴⠦⠉⠈⢉⣿⡦⠄⠉⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡈⠿⠿⣧⡀⠀⣀⠄⠛⠃⠂⠒⣆⣈⣥⠤⠚⠂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣭⣵⣦⡀⠀⣳⡾⢁⡀⠀⠀⠤⠚⣻⣿⣿⣶⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣨⠍⣙⠛⠛⠢⠞⢋⠠⠿⠒⠚⠛⠛⠲⣤⠙⠓⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⢀⠙⣀⠀⢠⠔⠋⠀⠀⠱⠶⣤⠤⠤⣖⠀⠀⠀⡙⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡔⠂⢢⡀⠀⠀⠉⠘⠁⠀⠈⠳⣤⠄⠀⣀⣰⠛⠀⠐⠈⠑⣤⢀⣀⣾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠠⠀⣷⠖⠐⢟⠒⠁⢀⠀⠀⣨⡟⠀⠜⠣⣀⠀⠠⠆⠀⠩⠀⠁⠌⢫⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣄⠈⢠⡶⠁⠀⣀⣈⢳⣀⣀⣠⣼⠏⢀⡀⠀⡀⠈⣡⠄⠂⠐⢶⣀⠀⢠⠀⣆⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠃⠛⠻⠿⠁⠲⢄⡭⢉⣠⡿⠉⣭⠛⢶⣬⡉⠛⣅⣸⣏⠀⢰⠟⢦⠉⣿⠷⢶⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣼⠄⠀⣤⢦⠤⣠⣶⣤⣼⣿⡿⠀⢈⠑⢶⠄⣹⡟⠛⣿⠟⣧⣄⠈⢷⠆⢸⡟⠀⣶⡙⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣟⣠⠀⣤⡀⣤⢈⣠⡌⣀⢀⠀⣤⣄⠀⣀⣸⡏⠀⠰⠌⠷⠌⠿⣿⡶⠞⢿⡄⢠⡼⣧⣿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⡙⠁⠃⠈⢃⣀⣈⡋⠀⠀⠈⠀⠈⢋⡿⠛⡷⢤⡀⠁⠀⠐⠀⢰⡟⠀⠀⠇⠙⢣⡤⢾⡁⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣀⠆⠀⠀⠳⣄⠂⠀⠀⢠⡞⠀⠀⠀⠀⠈⠳⢄⡰⣦⣈⠀⠀⠀⠀⢀⡾⠁⠀⢻⣶⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢦⡄⠀⠀⠀⠀⠘⡦⠂⠘⢫⣀⠀⠀⣀⠀⠀⡰⠊⠀⠀⠘⠳⢦⠀⠈⣰⠇⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢀⡷⢦⠀⠀⢀⡟⠀⠈⠀⠀⠈⣠⣀⢀⣠⣤⡇⠀⠀⠀⠀⠰⢄⡟⠉⠑⢦⡀⢠⡽⢶⡟
⠀⠀⠀⠀⠀⠀⠀⢈⢠⣀⠎⠀⠀⠳⣄⡛⠁⠀⠀⠀⠀⠀⣨⠋⠀⠀⠈⠱⢦⡀⠀⠀⠀⣾⠃⠀⠁⠀⡻⠏⣀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⣹⠈⣋⠀⠀⠀⠀⡠⠉⢉⠒⢦⡀⠀⣸⡇⠀⠀⠀⠀⠀⠠⠧⢠⣤⠖⠋⠀⠀⠀⠀⡟⠀⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡏⠀⠀⣮⠀⠀⠈⢀⡀⠛⢶⡯⢀⡀⠀⠀⠠⡄⠀⣤⠞⠀⠀⠙⢦⡘⣁⣰⠟⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠶⠒⢧⣀⠀⠀⠈⢁⡴⠂⠀⠀⠉⠲⣄⠀⢀⡜⠉⠀⠀⠀⠀⠌⡻⠋⠙⣦⡀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣴⠇⠀⠀⠈⢳⡄⠀⡹⠀⠀⠀⠀⠀⡤⠈⢲⣏⣁⠀⠀⠀⠀⢠⡶⠃⠰⠄⠸⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠂⠀⠀⠂⡀⠑⢊⣗⣀⡀⠀⠀⠠⢀⡴⠋⢨⠉⠉⠲⡄⣠⠜⠀⢦⠀⢠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢄⣀⠌⠁⢴⠈⠁⠉⠑⠢⣤⣤⡼⠀⠀⠀⠀⠒⠀⠈⣩⡦⣤⣤⣤⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⡉⠛⠠⢴⠀⠀⠀⠀⠀⠈⢄⣀⣀⣀⡀⠂⠀⣠⠞⢁⠀⠤⣀⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣷⣤⣄⡀⢠⡞⠋⠁⠩⠉⠙⢓⣎⣡⣤⣴⡶⠟⠋⠁

⡀⠀⢹⡄⠀⠀⠀⢀⡀⡂⠀⠀⠀⣠⡄⠊⠀⠀⠀⡀⠈⢿⣇⠀⠃⢀⣀⠀⠛⠻⣐⠿⢤⠔⠒⠤⠄⣿⡾⠟⠀⢊⣿⣿⣿⠏⠈⣯⡐⢀⠀⠀⠀⠈⠒⠤⣤
⠁⠀⠀⠙⡄⠀⠀⣤⡀⠡⠶⢶⣼⣯⠕⠀⠀⢂⠀⠀⡀⣸⡿⠀⠘⠀⢩⣄⠀⠀⠎⣤⢤⡴⠶⠟⠘⠈⠀⠀⢀⣌⢶⡟⡻⣄⠀⠋⠐⢦⣣⣄⡀⠀⠀⢀⣀
⡀⠀⢀⠀⠙⠂⠒⢿⡈⠳⢤⡈⠛⠻⠀⡀⠠⡤⠀⠀⠀⡿⠃⠐⠀⠀⢾⠋⡥⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠋⠑⠫⡀⠀⠀⠈⠉⠻⠻⠆⠉⠉
⡇⠀⠸⣆⠀⠀⡀⢈⣷⡀⠀⠙⠣⢀⠈⠈⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⣇⢀⠀⠘⡆⠀⣇⠀⢿⠛⠢⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠟⡌⠀⠀⠋⠂⠄⠀⢠⡀⠀⠈⠓⢆⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠠
⠀⡀⠀⠀⠠⠀⠀⠀⠀⠈⠒⠄⡀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣘⠀⠀⠀⢀⠀⢂⡐
⠟⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⡄⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣿⣦⣤⣀⠉⢷
⠀⠀⠀⠀⠃⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠈⣉⣙⠛⠶⢬
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢠⡤
⢆⠀⠀⠀⣶⣠⣥⡀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡀
⡛⣷⠀⠀⣮⣽⣷⠞⠁⡀⠀⣠⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠤⠄⠀⠀⠀⢀
⡻⣾⡇⢸⣿⣯⣿⠀⣸⠗⠊⠖⠉⠉⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⠓⠲
⣿⠿⣿⣼⡇⣸⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢹⢶⣽⠙⣷⣿⣥⣄⣤⡴⣆⠀⢰⠴⣦⠠⢄⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠠⡀⢀⣦⠀⠀⠀⠠⠀⣴⣟⠉
⠀⣿⣽⣷⡌⠻⠟⠛⠉⠁⠀⠀⡀⢅⣨⣤⡀⠀⠀⠀⠀⢀⠃⠀⠀⠂⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣃⠀⠀⠀⠀⢀⠀⠈⠻
⠂⢿⡏⠙⠟⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⡜⠀⡠⠀⣼⠏⠀⠀⢀⡤⠤⠎⠀⠀⠀⠀⠀⠁⠀⢀⡅⠀⠀⠀⠀⠀⠀⢰⡖⠒⠁⠊⢄⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⡏⣼⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⣿⡿⠿⠋⢩⣩⠆⢀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⡏⣋⠄⢠⣿⣿⣷⣦⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡰⣿⠺⠟⢃⣀⣿⣌⣡⣿⣻⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠇⡀⠃⠀⢰⣾⣿⠟⡻⣿⣿⠿⠟⠻⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣎⠀⠟⡋⣻⣿⡁⣼⣷⣝⣿⣶⣿⠆⠀⠒⣶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠒⠲⣄⡀⠀⠀⠀⠈⢙⠿⠉⢠⣄⣻⣾⠛⠾⠿⣟⣿⣿⣿⡧⠒⠀⠀⠀⠀⠉⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢹⣿⣪⣟⣵⡾⠗⠀⠒⣼⣿⣿⣿⠶⣶⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢠⣾⣿⡏⠉⠀⠀⠀⠀⠰⣿⣿⣿⣿⣾⣿⣿⣶⣶⡀⣀⡀
⠀⠀⠀⠀⡠⠀⠀⠿⣿⣿⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⠛⠛⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣶⠞⠂⠲⣄⡠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⣾⡁⠀⣹⠏⡰⢻⣿⠃⠀⢀⡴⠋⢀⣀⣀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⣄⠙⣶⠁⣸⠁⡃⣿⡼⢀⣿⣶⣿⣿⡿⣿⣿⣶⣤⣽⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠿⠛⠂⠀⢨⣧⣿⣆⣿⣿⡇⣸⣿⣿⣏⣀⣠⠤⠾⠿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⣀⠀⠀⠀⣈⣿⣿⢿⣿⣿⣿⡿⠿⠛⠛⢿⣟⠒⠐⠀⠈⠺⣿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠋⢈⣩⣿⣿⣿⣿⠋⠀⠀⠈⠻⠟⠀⠀⠀⠀⠈⠛⢿⣗⡂⠀⠀⠙⢿⣇⣈⣢
⠀⠀⠀⠀⠀⠀⢠⠞⢁⣤⠖⠋⣁⣠⣾⣿⣿⣿⠇⠀⠀⠀⢠⡄⢀⡀⠀⠀⠀⠀⢻⡙⢿⣿⡟⠛⠛⠢⡁⠀⢣
⠀⠀⠀⠀⠀⢠⠃⡤⠋⠀⢠⣾⣿⣿⣷⣿⠟⠉⣀⡀⠀⠀⠀⠙⡟⠀⠀⣀⣠⣄⡀⣧⠘⣿⠙⠷⣦⣄⠈⠢⡈⡃
⠀⠀⠀⠀⠀⡌⠀⠀⠈⣉⣵⣾⣿⢃⣼⢻⠑⢿⣿⠿⣞⡧⠀⠀⠀⢀⠐⣿⣿⠟⠁⠹⣆⣿⠀⠀⠀⠙⠦⡀⠈⣇
⠀⠀⠀⠀⠀⠁⠀⠔⠀⠀⣩⣿⣿⣿⡏⠀⠀⠀⠀⣦⢹⡷⠂⠀⠀⠊⠻⣯⣷⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⢉⠆⢼⡆
⠀⠀⠀⠀⠀⢰⠁⠀⢠⡞⠉⠁⢻⣿⣿⡄⠀⠀⠀⢫⣟⣵⠃⠀⠀⠀⢲⡌⢿⠂⠀⣰⢻⣿⣆⠀⠀⠀⠀⠈⢂⢸⣷
⠀⠀⠀⠀⠀⣿⠊⣰⣾⠁⠀⠆⠘⡟⣧⣑⠀⠀⡶⠀⠻⣿⡟⢛⠛⣹⣿⠗⠈⢱⡄⠁⢀⠟⠙⢆⠀⠀⢦⡀⠈⢧⣻⡄
⠀⠀⠀⠀⠠⣿⡸⢱⡇⠀⠘⠀⠀⢃⠘⢿⣶⣾⣇⡀⣠⡬⠿⢿⡟⢡⣤⣤⣤⡬⣷⣾⡏⢳⡀⠀⠣⠀⠈⢷⡀⠈⢿⠛⠦⠄
⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⣸⡤⠖⣿⣿⡥⠖⠚⠫⠀⣸⣧⣸⣿⣿⣚⣳⣾⡿⠤⢌⣿⠀⠐⠂⠀⢸⣧⠀⢸⠂
⠀⠀⠀⠀⠀⠹⠀⠘⣆⢀⡀⠐⢿⠁⣀⠼⠟⣹⣦⢂⣡⠴⢛⣿⣿⡿⠳⢦⣭⣿⣿⡧⠀⠀⢸⡆⠀⢸⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⠰⡼⡏⠀⣠⣿⣿⣿⡏⡏⢀⡈⠛⠈⠀⠀⣄⢨⡏⠁⠻⣦⡀⠘⠇⢀⣞⠀⠀⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢩⣿⠞⢽⣿⡿⠈⡇⣇⡎⠻⣄⠀⢀⡜⢹⠏⠁⠀⠀⠈⢷⣼⣷⡿⠃⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⡾⣿⡇⠀⠀⢸⣄⠀⠈⠀⢀⠀⠀⡆⢸⣆⠀⠀⢸⣿⡿⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⢙⠁⠀⣦⡟⠹⡆⠀⠀⡿⡇⠀⠀⠸⠻⣇⠀⢠⣿⠇⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⣿⡇⠀⠹⡄⢠⠇⢿⠀⠀⠇⠀⢸⣄⡾⣿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⡀⢸⠀⠃⠀⠀⠹⣾⠀⢸⠀⡜⠀⠀⢸⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣸⡀⠀⠀⠀⠀⠙⡀⢸⡟⠀⠀⠀⠎⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠁⠘

⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠊⠁⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠃⠀⠀⠀⠀⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣠⠤⠤⠤⠤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡖⠒⠛⠦⠤⣀⠀⠀⠀⢸⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣷⠀⠀⢄⡀⠈⠳⢄⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢳⡀⠀⠀⠙⢆⠀⠘⢦⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠉⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠐⠒⠢⠤⢭⠦⣌⣧⠀⠀⠀⠈⡆⠀⢸⠄⠀⣠⣴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⣿⣀⠼⠶⢟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⡬⣶⠖⠉⠙⠢⣄⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠁⠀⢀⠂⠀⡞⣧⡀⠀⠀⠘⣆⠀⠘⡆⠀⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⡂⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⣄⠀⠀⠘⠆⠀⠉⠑⠒⠒⠀⠀⠉
⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢸⣈⠒⠤⡀⠀⡸
⠀⠀⠀⠁⠁⠒⠢⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠤⠊⠑⡼⠇⠇⠀⠀⠈⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠓⠂⠐⠉⠉⠀⠀⠀⠀⠀⠀⠘⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣸⠠⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⢱⢸⢸⠁⠳⡄⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣧
⠀⠀⠀⣤⡶⢶⣿⣿⣶⣶⣤⣿⣿⣿⣷⣤⣴⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⣾⣟⡓⠶⢦⣼⡃⠀⢹⣤⡤⠶⢶⣶⣦⣤⡄
⠀⠀⢀⣀⣤⣤⣤⣈⣻⣿⣿⣿⣿⣿⣿⣿⡿⣛⣉⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣬⣙⣦⡀⠈⢻⡶⠋⠀⣠⢞⣋⣥⣤⣤⣤⣄
⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣄⠈⠉⠻⢦⡟⠀⢀⡼⠟⠋⣡⠶⠛⠉⠁
⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠾⠷⣦⣀⠀⢹⡶⢋⣠⡴⠾⠛⠚⢒⣶⣤⣄
⠀⠀⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⢦⡀⢀⣈⣻⣾⣵⣟⣁⡀⢀⡴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣟⣫⢍⣿⢟⠿⣯⣹⡿⡿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⡋⠉⣡⠟⠙⢧⣍⡽⠛⠶⣄
⠀⠀⠀⠀⣠⡾⣯⣮⣝⣿⣿⣅⠒⣿⣼⡿⢷⣿⣫⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠈⢙⣾⣏⠀⠀⢠⡟⠳⣤⡀⣼⠳⣄
⠀⠀⠀⣼⢟⠿⣾⣭⡾⣿⣾⡝⣿⣾⣋⢿⡷⣽⣿⣧⡪⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠙⠶⣄⣴⠏⠀⠈⠛⣶⡏⠀⠀⢈⡿⢧⡀⠘⣧
⠀⠀⣸⣏⡻⣯⣾⢟⣿⣮⣟⣽⢿⣽⡻⣿⣾⢿⣶⡽⣿⣿⢻⡄⠀⠀⠀⠀⠀⠀⢠⣏⡀⢀⣴⠟⠙⢦⣄⣠⠞⠁⠙⢷⣤⠟⠁⠀⠻⣾⡛⣇
⠀⠀⣿⣹⡿⣿⣝⠿⢛⣽⠿⣷⣽⡿⣵⣿⡻⣿⣟⣽⣿⣽⣿⣇⠀⠀⠀⠀⠀⠀⣼⢉⡽⠿⣅⠀⠀⣠⡾⠻⣤⡀⢀⡼⠛⢷⣄⢀⡾⠉⠳⣿
⠀⢸⣿⡻⣾⣾⢟⣿⣿⡫⣾⡿⣹⣿⣯⣺⡿⣫⣿⣯⡺⠿⣼⣿⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠈⣻⣾⡋⠀⠀⢈⣿⣿⡀⠀⠀⣹⢿⡀⠀⢀⣼⡇
⠀⢘⣟⠻⣮⣾⢿⣵⣮⡻⣯⣾⢟⣷⣟⢿⣽⣟⣽⣝⢿⣼⢏⣿⠀⠀⠀⠀⠀⠀⣿⠙⢦⣤⠞⠉⠀⠙⢶⣴⠟⠁⠈⠻⣦⡞⠁⠀⠹⣦⡞⢹⡇
⠀⢸⣷⡾⣫⣿⣷⣅⣿⡿⣏⣻⣶⣟⣵⡿⡻⣿⣝⣿⣿⡻⣧⣿⠀⠀⠀⠀⠀⠀⣿⡴⠋⠉⠳⣄⣀⡴⠟⠙⢷⣄⣠⡾⠋⠳⣄⢀⡴⠋⢷⣼⡇
⠀⠘⣿⣮⡻⣻⣴⠿⢿⣮⠿⢟⣽⠿⣿⡻⡿⣪⣿⣯⣺⡿⣹⡟⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⣠⠾⠻⣦⡀⠀⣠⡿⢯⡀⠀⠀⣽⢿⡀⠀⠀⣻
⠀⠀⢿⣾⡿⣿⣵⢿⡿⣻⣷⣟⢿⣷⡝⣿⣾⢟⣷⡹⣷⣴⣿⠇⠀⠀⠀⠀⠀⠀⠸⣦⡽⢿⣅⠀⠀⢈⣻⣾⠉⠀⠀⢻⣦⠟⠁⠈⢳⣤⡾⡿
⠀⠀⠘⣷⣺⡿⣻⣶⡿⣫⣵⠿⣷⣭⣾⣛⣿⣧⣟⣵⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⣷⡴⠛⠁⠈⢳⣄⣴⠟⠉⢷⡀⢀⡼⠻⣾⠃
⠀⠀⠀⠘⢷⣿⢻⣭⣿⣮⣃⣴⢟⢻⣾⣿⢗⣽⣿⡟⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠛⠁⠙⢦⡀⣠⠶⠛⣧⠀⠀⣠⣿⡏⢀⡼⠃
⠀⠀⠀⠀⠀⠙⢿⣽⣵⡿⢿⣝⣿⡷⣻⣷⡟⣋⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣠⡼⠻⣅⠀⠀⢈⣷⡞⠋⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠿⠷⠶⠿⠟⠛⠋⠉

⣿⠿⢿⠿⣿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢢⠀⠀⠀⠀⠀⠀⢸⢆⠀⠀⠀⠀⢀⣴⡡⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⢂⠀⢠⠀⠀⠀⠀⡆⠂⠀⠀⢠⣾⠋⣀⣴⡀⡆⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠺⣦⡀⢆⠀⠀⠀⢎⠀⠄⣰⡟⠁⡼⠟⠛⣷⣿⣿⠿⠿⠛⠉⠙⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣻⣧⢀⠃⠀⠀⢜⢀⠘⡟⣠⣷⡔⠀⢠⠉⢁⠤⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠭⣴⣰⠎⢁⣴⠶⣿⣷⣄⡞⠈⠰⣰⣿⣿⣄⡠⢃⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡾⠣⠄⠐⠊⠈⡴⠛⣷⡄⠸⢿⣿⣡⠀⢰⣿⣯⢿⡿⠵⠚⣘⣃⡤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠾⠗⠛⠛⠉⠁⢀⣀⣿⣿⠀⢸⣿⡿⢃⣤⡢⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⣀⣤⣴⣾⣿⣧⠀⢹⡀⣼⣿⢶⡟⡋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠉⢚⡿⢓⡺⣿⠿⠧⣸⠃⢬⣴⣇⣽⣿⣶⡞⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢘⠛⠿⣿⣶⣗⣞⡆⠙⠋⢋⡽⣿⣿⠇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡼⢿⡿⠛⠈⡉⠀⢁⡔⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣡⠸⣸⡷⠊⠟⢃⠌⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣁⡿⣗⢿⣾⠈⢁⡿⢦⣼⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣼⣟⡁⢀⣷⡴⡈⢿⣁⣬⠿⠁⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠋⠙⠃⢸⡟⡿⢿⡉⠁⢀⣿⡀⣠⠟⢉⣵⡆⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣯⢢⡶⠷⣮⠀⠘⢦⡽⠾⠩⣿⢧⣄⣶⢤⢘⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡶⠄⠻⠿⣦⡶⠟⢻⣷⣄⠀⢦⢽⡅⢠⣌⡏⢉⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⣼⡗⠾⡓⣟⠀⠚⠁⠈⣿⠀⠊⢸⣿⣄⣸⠷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⡤⣿⠗⢈⢉⠻⣆⣾⣿⣭⢭⡄⠤⣉⣥⣿⠁⣴⡞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣀⠼⣦⣀⣙⢓⢸⡟⣉⣀⣀⠙⣦⠟⠩⠉⡐⢡⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣐⡄⢬⣽⠧⢴⣿⠙⢶⣿⣤⣿⡀⠀⠿⠎⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⢻⠚⡃⠠⣀⠸⣶⣜⠿⡛⠛⠛⢷⣶⡤⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⣤⣍⠁⠸⣤⣾⢧⣤⠴⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⣠⡶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠈⠉⣼⣷⠀⠀⡀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠄⠀⠀⠀⡰⠀⠐⠋⠀⠐⢊⡤⠖⡋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⣧⡴⠀⠀⠀⠀⢀⣤⠞⠋⠐⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠛⢦⡀⠀⠉⠛⠿⣦⣴⡾⠗⠚⠁⣴⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⣸⠆⠰⠦⣤⡹⣿⡄⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⣀⣈⣦⡀⠀⠀⠉⠹⣷⠀⡠⠦⠚⣡⣤⠌⠹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠘⠛⠛⢿⡤⠰⢾⣶⠶⠆⠀⠀⡼⠋⠀⠒⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣲⣤⣤⣵⣦⣤⣤⠣⣴⣤⠆⠀⣴⡊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠏⣰⡄⠙⠉⣀⣤⣬⣤⣤⣴⣷⣀⠀⡉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡘⠛⣿⣿⣗⠀⣼⡿⠻⣿⣾⣦⣘⣷⣄⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⠇⣿⡿⢉⠉⠻⣿⠂⣀⣹⡯⠉⠻⡟⠋⠉⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡏⣽⣇⣤⣿⡧⠚⠀⠀⢻⠋⠀⠀⠈⢧⢰⠟⠐⠊⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⢿⣁⣼⣣⣻⢿⡀⢀⣀⡤⢿⡀⠀⠛⠂⠈⣾⣀⡀⠀⠛⠀⠰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣸⣿⣠⡏⠀⠈⠳⠋⢁⣄⠈⠻⣄⠀⢀⠔⠉⢿⡉⠘⠃⠼⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠋⣿⡏⣀⡤⠦⢼⡀⠈⡭⠃⠀⣸⣶⡃⢀⣠⡀⠹⣤⠞⠀⣴⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⠁⡀⠐⣧⣼⠁⣤⡄⠀⢳⡀⠀⢀⡤⠚⠉⢮⡉⠉⠁⣀⣸⣀⠀⠹⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣃⣧⡴⠾⣿⠤⠜⠺⠤⠾⢳⡶⣁⢀⣶⡀⠈⣷⡤⠚⠁⠈⢧⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⠀⠀⠘⢧⡀⢀⡴⠞⠉⠳⣍⠉⠉⠉⠁⣨⠉⠓⡶⣦⠸⢧⡤⣐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡅⢺⠀⠀⠀⠈⣿⣋⣀⣤⡄⠀⠘⣦⠔⠊⠉⠙⢦⣤⠐⠀⠀⠀⣄⠉⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⠟⢁⠀⠣⠙⠉⠙⠂⠐⣯⠀⠐⢷⣀⣀⡿⣄⠀⢰⣦⢻⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢀⡌⣿⣔⢿⠀⠀⣸⣀⡴⠖⠂⠻⣤⠒⠋⢁⡤⠘⠻⣗⠁⠀⠁⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⣥⠈⢹⡆⠀⠈⣸⡏⠀⢀⣤⠀⠈⣧⣰⠏⠀⠀⡄⠘⣇⠀⠠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⢂⣴⣿⣿⡿⠻⠟⢫⡤⠞⠻⠆⠀⢈⣿⡀⠀⠛⠓⢂⡙⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⢴⣷⣀⠀⣵⡀⢰⠞⠋⠁⠈⠙⣤⠔⠀⢈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠌⠑⢇⣶⡏⣨⠋⠀⠠⣤⡀⠠⠧⠤⣴⠃

⠰⢶⣶⣦⡀⣶⡄⣤⡀⠀⠀⢀⡤⣠⡶⣠⣶⣦⠶⢦⣆⢠⡀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⣄⣠
⠀⠀⣛⠋⢽⠟⢫⡿⢓⠶⢶⠛⢿⡝⠿⡯⠛⣓⠀⠠⣿⡟⣽⡾⣱⣯⢣⢰⣄⠀⢠⠆⡴⣹⣞⡟⣯⢯⣷
⣤⣾⣌⢃⡀⣀⠀⣤⡀⠀⠀⢠⡄⢀⡀⢀⢘⣼⣧⣤⣠⣾⢿⢷⡿⡻⣿⢽⣼⠀⣧⣯⣷⢷⣻⡿⡿⣗⣤
⠀⢙⣿⣽⣧⣹⣷⣼⣗⡶⢶⢿⣧⣿⣇⣾⣹⣷⡋⠀⠉⠀⠟⢫⡮⠞⡵⡺⠛⣶⠛⢿⣪⡳⢽⣍⠺⠂⠈
⠀⣈⠁⠸⠃⠘⠋⠉⠘⠀⠀⠋⠉⠙⠀⠹⠆⠈⣁⠀⠀⠀⠀⠀⠀⠈⠉⠀⡜⠀⢳⡀⠉⠃
⠀⣻⣴⡆⣤⢀⡀⠀⠀⠀⠀⢀⠀⢀⠀⣦⢰⣦⣟⢰⢲⣀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⢠⣄⡔
⠀⠉⣩⣷⣿⣼⣇⣸⡇⠀⠀⣼⣇⣼⣧⣿⣮⣉⠁⠈⠙⣻⣿⢾⣶⣷⣴⡄⢄⠀⣠⢤⢶⣞⣷⣿⣿⣟⠛
⠀⠀⠀⠀⠰⠟⢡⠿⠋⡶⢖⠙⢿⡍⠻⠄⠀⠀⠀⠀⠺⠛⣡⣿⢗⣿⢾⣿⠚⠋⠓⣿⡷⣿⣞⢿⣬⠓⠓
⢰⠓⡄⡠⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢠⣄⡰⢻⠄⠀⠀⠉⠀⠛⠁⠛⠉⠀⠀⠀⠈⠛⠈⠛⠀⠉
⠀⠛⢯⣧⡷⡟⡆⡴⢄⠀⠀⡴⣆⣰⢹⢷⣾⡿⠋⠀⠀⠐⢖⡄⠀⣶⡄⡴⡄⠀⢀⣦⢀⣶⠀⢠⣶⠂
⠀⢮⠽⠋⣓⣿⡧⣵⣾⠀⠀⣷⡯⢿⣿⡊⠻⢯⡧⠰⣿⢿⣾⡿⣲⣿⢧⣳⣷⠀⣼⣜⣸⣿⣶⠻⡷⡿⢿
⠀⠀⠀⢰⠥⠛⣴⣻⠏⡽⢯⠹⣟⣎⠻⠼⡀⠀⠀⠀⠀⠺⠊⠾⠟⡵⣊⠝⢛⡶⡛⠻⣕⢦⡳⠷⠙⠿
⠺⣷⣦⣿⣆⣤⠉⠀⠈⠀⠈⠀⠀⢁⣤⣸⣷⣴⣾⠇⠀⠀⠀⠀⠀⠈⠁⢠⠋⠀⠘⣆⠀⠁
⢠⣺⣿⣿⣻⣽⣷⣦⡀⠀⠀⢀⣶⣾⣯⣿⣿⣿⣳⡄⡶⡀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡶
⠈⣉⣭⢤⣽⣿⣉⣿⡇⠀⠀⢾⣿⣸⣿⣧⡤⣍⡉⠀⠳⣿⣿⡔⡶⣜⠿⣢⣤⠛⣧⣴⡿⣣⢴⢆⣾⣿⠞
⠀⠙⠛⠛⠉⣤⣾⡿⡹⠓⠺⣝⢿⣶⣄⠉⠛⠛⠋⠀⠀⠈⢛⣿⡬⠾⣷⡚⢏⣷⣹⠓⣶⠷⢥⣾⡟⠁
⢀⣀⣐⣷⣈⣯⢁⡀⠀⠀⠀⠀⢀⣈⣽⣠⣿⣀⣀⡀⠈⠓⠿⠚⠁⣴⡷⠃⠘⠁⠓⠀⢯⣧⠀⠑⠾⠞⠃
⢉⣿⠟⠛⢻⡿⣿⣧⡄⠀⠀⣠⣼⣿⢿⠟⠛⠻⣿⢁⢤⣿⣷⣼⣷⣾⣦⢶⡄⠀⢠⣶⣰⣶⣾⣿⣼⣿⡤
⠀⠉⠀⠀⠀⠙⠿⢿⣿⡷⢿⣿⡿⠿⠉⠀⠀⠀⠉⠈⠉⠾⠋⠾⠿⡿⢫⡿⢳⠶⢞⢿⡝⢿⡟⠵⠙⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠓⠺⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠌⢁⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⣿⠱⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠓⠂⠨⡉⠓⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢢⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢧⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠷⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⢀⠚⠃⠄⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠆⠀⠀⠀⠀⠀⠀⠢⠀⠇⠀⠀⠀⠀⠀⣸⠀⢠⠐⠿⠤⠛⢢⡔⠤⣔⣓⠴⣷⡼⠲⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡄⠀⠀⠀⠀⠀⢀⡇⠸⣄⠀⠀⢠⡀⠛⠀⣨⠐⢁⠈⢄⢙⣋⣉⣉⠙⡘⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠧⠀⠀⠀⠀⠀⠈⠉⠁⠙⠓⢶⡞⠀⣕⡤⢤⢆⣁⣒⣁⡶⠴⣷⢚⣖⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠧⣄⣠⣄⣀⠀⠀⠀⠀⠀⠀⣀⣰⡾⠓⠒⣏⣀⣼⡟⠶⠾⠶⠮⠭⠽⠿⠿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠲⠛⠛

⠀⠀⠀⠀⠀⠀⠀⢰⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠉⠙⢶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣴⣀⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢓⠳⣤⠀⡀⠀⠈⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣆⣿⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠷⣧⣴⡂⣀⠻⡇⠀⠀⠀⠀⠘⠠⡀⠀⠘⠶⢦⣴⣿⡧⠔⠁⠀⠀⢳⣸⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠻⣥⣈⣻⣿⣿⡾⠀⢠⠤⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣿⣗⠀⠀⠀⠀⢹⣿⣃⡠⠉⠙⠋
⠀⠀⠀⠀⠀⢠⢁⡤⠴⠆⠶⠶⠿⣿⠿⣿⣧⡠⣷⠀⢀⠀⠀⠁⠀⣠⣤⠶⠋⠀⠀⠛⢆⡀⠀⣤⣾⣿⢻⣍⢛⠠⢄
⠀⠀⠀⠀⠀⠈⠈⠒⠤⠀⢀⣤⠤⠾⣄⠀⠉⠧⢿⣄⡽⠒⠀⠂⠀⠑⢆⠀⠀⠀⠀⠀⠈⠁⠀⠘⠿⠿⠏⠉⠙⠷⠬⢵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣟⢠⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠄⡄⡄
⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢣⠈⣄⠀⠀⠀⡞⠀⡄
⠀⠀⠀⠀⠁⢀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⣼⡀
⠀⠀⣀⠀⠀⠈⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢄⠀⠀⠀⢨⣀⣴⣿⠷⣠⠀⣠⠔
⠀⠈⠀⠀⣀⠀⠀⢧⠀⠀⠀⠀⠐⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⢀⣄⠤⣙⣿⣿⣿⣷⣶⡈⠀⠀⠂
⠀⢀⡠⠜⠁⠀⠀⠼⡖⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡇⣆⡏⠀⡀⠀⠀⠀⠐⠂⠀⠉⠿⡿⠟⠿⠿⣿⡂
⠀⠸⠀⠀⣄⠍⠀⠀⠉⠀⡀⠀⠀⠲⠄⠀⠀⠀⢀⡃⢀⠹⣿⣠⣨⣾⡶⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠁⠀⠀⢀⠀⢁⠉⠚⠉⠀⠀⠀⠀⠀⠰⢄⡪⣤⣰⢳⣿⣿⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠠⣄⠈⣑⢿⡻⡘⣿⣗⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⡇⠀⢸⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⡰⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡀⠘⣿⠇⢇⠘⢻⣃⠌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠃⡀
⠀⠀⠀⠀⣤⡀⠆⠸⠀⢐⡤⠂⢐⠀⠀⠀⠀⠀⠠⠄⠉⠙⢳⠄⢧⡘⡵⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠠⣸⣿⣰⠁⡠
⠀⠀⠂⠀⠨⣍⠳⡟⢤⣠⣦⠄⠚⠚⠀⠘⠁⠀⠀⠀⠄⠚⣉⡥⠄⢣⣘⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡟⣿⣇⠎
⠀⠀⠀⠀⢢⣝⣦⣶⠈⣿⣡⣴⣷⡶⠂⢀⠀⡀⠀⠀⠀⠈⠐⠚⠿⠶⣛⠅⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⢈⣻⣷⣿⠃
⠠⠀⠀⠀⠀⢉⡽⠿⠿⣿⡿⠑⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠀⠬⢝⣿⡏
⠀⠀⠀⠀⠀⠨⣀⠀⠠⡿⠛⣦⣤⣴⣶⣍⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠴⠾⢿⡦⠂
⠀⠀⠀⠀⠀⠀⣦⠀⠀⣏⠉⣡⠋⠯⠙⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣤⣤⣤⣤⣤⣤⣬⣤⣤⣥⣤⣤⣤⣤⣤⣦⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤
⣿⣿⣁⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⣀⢀⣩⣦⠀⣀⠀⣊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⡀⠂⠀⡰⠀⠀⠈⠀⢀⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣄⡈⠙⠻⢿⣀⢀⣠⡠⠞⢁⣤⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢉⠆⠠⢤⡙⣿⡁⠀⠠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣤⣵⣀⠀⠀⠙⠇⠀⠂⢊⡤⠂⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⡉⠙⣆⠙⠛⠋⠀⡀⠈⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⣈⠛⠛⢉⣙⣉⣈⣽⡁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⢁⠛⢻⣿⡉⣹⠏⢿⣿⣦⣱⣆⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢻⠟⢸⡟⡀⠈⢿⠠⠀⠷⡀⢹⠉⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠃⣹⡖⢿⡏⠀⢀⣼⠀⢰⠄⠡⡏⠀⠀⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⣶⡉⠃⠀⠳⠊⣉⠈⠳⡀⢀⠄⠙⣏⠒⠀⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⢉⠘⢇⡤⠒⠲⡄⠘⠁⠀⣿⣇⣠⣤⠈⢶⠃⢰⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⣀⣀⣮⢐⣣⣀⢳⣄⠔⠋⠀⠘⣆⢀⠄⠸⡆⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣹⡏⡄⠙⣄⠀⣠⠔⠺⣔⠚⠛⠂⠉⠓⢠⣄⠸⣄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡅⡷⠛⢀⣈⡏⣁⣤⡀⠈⡦⠒⢁⠙⢢⡄⠀⠀⢠⠈⡱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⢶⣋⡄⠀⠀⠀⢉⡀⣅⢀⡹⠖⠚⣷⡄⠜⠊⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢞⠈⢷⡅⠀⢸⡞⠁⢀⠈⢳⣀⡔⠁⠀⠹⡆⠀⣬⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⣾⣿⠞⠛⢡⠀⠛⠀⠀⣹⡄⠐⠛⢀⡓⠐⠂⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢾⣦⣀⡦⣴⠋⠀⠀⣈⡶⠠⠄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠐⣶⣤⡄⠀⠀⣀⢰⡶⣾⣶⠀⠀⠀⢠⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⡀⢿⣼⢟⣿⣛⡆⠀⣤⢹⣼⣿⣿⡩⠄⣠⣴⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⣿⡿⡃⠀⠀⠐⢿⣷⡄
⠀⠀⠀⠀⠀⠲⣿⠂⠀⠀⠀⠀⠻⣾⣫⣾⠿⢦⠀⢰⣸⣿⣿⣽⣿⡗⠀⣿⣿⣿⣿⣿⣷⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣯⣴⡶⠂⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⢯⠷⠀⠀⠀⠉⠙⠿⠟⠃⠀⠀⠻⢿⣿⣧⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠙⣿⡓⠀⠀⠀⠠⣾⣿⡇
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡡⠄⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠹⢿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⢠⣿⠂⠀⠀⠀⠀⢀⡴⡛⠈⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⡀⣄⣀⠀⠀⢠⣄⠀⠀⠀⠀⣀
⠀⠀⠸⢿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣦⣿⡟⢁⣴⡿⠀⠀⢀⠎⡀⠁⠀⡀⠄⠀⢸⣿⡆⠀⠉⠀⠠⣉⣻⡥⢆⡀⠘⣿⣷⠀⣠⣾⣿⠀⠀⠀⢀
⠀⠀⢻⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⢬⣷⡿⣋⣤⡤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠒⠓⠒⠀⠀⢐⠲⠶⣽⠋⠩⠀⠀⠈⠛⠃⣿⣿⠟
⠀⠀⠻⣿⣿⣿⡗⠀⣤⣤⠀⣼⣷⠀⣠⣤⠸⠿⡶⠦⣿⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠀⠀⠠⠭⢭⣯⠴⠞⣋⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠁
⠀⠀⠈⢿⣿⠆⠀⠀⡻⣿⣷⣿⣿⣾⠿⠋⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠘⠀⠀⡀⠀⠀⢠⠀⠐⣻⡭⠀⠠⣤⣀⡒⣶⣚⣯⠗
⠀⠀⢠⣀⠀⠀⠀⠘⠿⠿⣿⣿⣿⡷⠿⠷⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠐⢢⡤⠙⠀⠄⠀⠀⠀⢻⡯⠀⠲⠦⣤⣄⣋⣡⠴⠚
⠀⠀⠸⣿⣿⣦⠀⠀⠀⠰⠿⠛⠻⠿⠀⣴⣄⢠⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠂⠐⠠⠀⠀⠀⠁⠀⠑⠶⢦⣄⣉⣴⠾⠋⠀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⢰⠀⠀⠀⣬⣽⠿⡇⠀⠀⠀⠀⠀⠀⣾⡆⠾⠡⡤⠆⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠁⠀⠀⠀⠛⠏
⠀⠀⠀⠀⠈⠉⠁⠀⢠⡀⣴⠀⢁⣀⣠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⣦⡘⠳⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⡀⠀⢰⣴⡀⠀⣄⠀⠀⣶⡧
⠀⠀⠀⠀⠀⣀⠸⣆⢸⠇⣯⣴⣿⡉⠁⠀⠀⢀⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿⠛⠀⢼⣯⠄⠈⠉⠀⠀⠿⢀⡀⠈⠁⠀⢠⡐⣴⢃
⠀⠀⠀⠀⢦⡙⢦⠻⣼⣿⠯⠄⠀⢀⢠⣶⠶⠘⣿⣖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢀⣀⡙⠶⣋⣤⠄⠀⠀⠲⣝⣗⣫⣔
⠀⠀⠀⠳⣮⡻⣦⣍⣛⡛⠁⠀⠠⢜⠛⠁⣀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⡇⠀⢠⣛⠯⢈⠥⠶⡂⠀⣈⡻⢮⡭⢟⡋
⠀⠀⠐⠲⣞⣿⡓⢦⡄⠀⠀⠀⠀⠀⣶⡿⣿⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⡿⠇⠀⠀⢭⣍⡀⢚⡋⠁⠀⠠⣭⣹⣛⡭
⠀⠀⠀⠉⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠱⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠑⠋⠁⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣭⣥⣦⣭⣷⣥⣿⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠂⠁⠀⠀⠀⠀⠉⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢰⠶⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠠⠄⡐⠂⠠⠈⠀⠈⡉⠞⠁⢐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠒⠀⠀⠀⠀⢀⠀⣠⣤⡤⠴⠊⠉⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡔⠀⢸⡀⠰⠂⠺⣟⠁⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠁⠈⠀⣨⠀⠀⠤⡀⢤⠀⢤⣇⣤⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠀⠀⠀⠰⠁⠁
⠀⠀⠀⠀⠀⢰⠀⠃⠀⢹⠀⠀⠁⠀⠙⠀⠀⠘⠁⠀⠀⠀⠀⠈⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠁⠠⠀⠀⠀⠀⠠⠂
⠀⠀⠀⢀⣠⢈⠀⠰⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⢐⣦⣀⠀⠐⠂⠀⠤⠀⠠
⠀⠀⠀⠸⡕⠪⢄⡀⡆⠀⠠⠀⠀⠐⢄⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠠⠃⠀⠀⠀⠄⢃⡀⠸⡆⠈⢠⡈⢂⠐
⠀⠀⠀⠀⢀⠀⠀⠘⠐⠢⠀⡁⠀⠀⠀⠑⠂⠀⠀⣀⠠⠀⠀⠠⠄⡤⠀⢀⢠⢀⠂⠀⠑⢰⡇⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠘⣦⠄⢀⠀⠀⠐⠉⠉⠐⠀⠀⠁⠀⠀⠰⠀⠀⢐⡂⠀⠀⢰⠓⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣆⠨⡇⠀⠀⠠⣤⠀⠀⠀⠄⢦⠀⠀⠀⠲⡇⠀⠈⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠐⠄⠀⠀⠙⡅⠀⠀⠀⠐⡀⠀⠀⠀⢿⠀⠀⠀⠸⡄⠀⠀⢨⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡉⡀⠀⠀⢣⠀⠀⠀⠐⢁⠀⠀⠀⢸⡄⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠀⠀⠀⠇⡀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⠀⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠚⣆⠀⠀⠀⠃⡀⠀⠀⢯⠀⠀⠀⠈⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣯⣦⣧⣾⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢄⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⣷⣠⣿⡆⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠹⣿⣄⣿⣿⣿⣇⣠⡾⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⣿⣿⣿⣿⣿⣿⣿⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠿⠷⢦⣌⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⣿⣿⣿⣿⣿⣏⣤⡴⠶⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⣿⣿⣿⣿⣿⣿⡛⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⢿⣿⣿⠟⣿⣿⣿⣿⣷⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿⣿⡤⠆⠠⠹⣿⡏⢈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⣿⡟⠀⠈⢿⣿⣶⠃⠙⢷⣾⣾⢫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠘⣧⢀⡀⣼⢿⣇⠀⠀⠀⣻⡿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⢈⣯⣾⠃⠈⠛⣷⣷⡶⠋⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠉⠻⣦⣀⣀⡜⠻⣇⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣧⠀⠀⠀⢹⣿⡏⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠻⣷⣤⢆⠜⠛⢷⣦⣀⣀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⡼⢿⣇⠀⠀⠀⢠⣻⣿⣿⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡌⠙⢷⣦⣴⠟⣫⣧⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠀⠀⠀⠀⣀⣤⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣀⡀⠀⠀⠀⠀⠈⠌⢀⠀⡠⣨⣞⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⠀⠀⣶⣿⡿⡽⠁⠄⠂⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣖⣒⣂⠠⢄⣁⡀⠀⣤⡆⢀⡄⠀⢸⣿⣽⠋⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣷⣮⣧⡀⢿⡇⣸⣇⠀⢸⡟⠁⠀⢠⠊⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⢿⡿⠋⠹⣾⠀⠰⡿⠂⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⢶⣦⣿⣿⣿⣿⠃⠀⠀⠘⣼⣿⣰⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠬⠿⠿⣿⣿⠀⠀⠀⢀⡹⠛⢁⣟⡠⠒⢂⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣯⡒⣤⣉⣲⣴⡠⠋⠀⠀⡞⣵⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⢿⣿⣯⠈⠛⢿⡃⢠⠄⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣿⡄⠀⠄⢳⣾⣷⠟⠉⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡸⢳⡀⢨⡸⠏⠐⠉⣪⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢼⡿⢸⡥⡶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣠⡶⠒⠖⢐⡀⣠⠀⡛⠹⡷⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⡃⣹⣿⡛⢿⣿⣩⡾⡏⠂⣘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡶⣺⡿⠿⣿⣏⣈⣽⣿⣛⣻⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡯⣿⣿⣿⡟⢛⣿⣿⠟⢿⣿⣯⣽⣧⡷⢾⣿⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⢿⣷⣿⡯⢿⣿⣿⣿⣟⣉⣿⡿⣟⣻⣿⣯⣿⣿⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣟⣻⣷⣿⣷⡷⢺⣿⠿⣿⣿⣿⣽⣿⣭⡶⢿⡿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣼⣿⣿⣿⣯⣙⣿⣿⣿⣷⣶⣺⣿⡿⡛⢿⣿⣷⣿⣷⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣿⠿⣿⣤⣿⣿⠬⣽⣿⣿⣿⣿⣿⣽⣿⣿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣾⣿⢿⣿⣶⣿⣯⣦⢼⣿⠿⠻⣿⣿⠿⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⡴⣶⡿⡿⣿⣿⣿⣿⣾⣿⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢿⣿⣿⣾⣿⣻⣿⣻⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⡿⣿⣿⠿⠛


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠠⠀⠤⣄⠀⠠⡄⠠⡖⡄⣄⠀⠀⠀⠀⠀⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⡄
⢰⣤⣄⡈⠙⠢⠈⢆⠁⠁⡯⢢⠀⠀⠀⠀⣑⣀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⠒⠀⠉⠠⠋⠀⠀⠀⠀⡇
⢨⣀⣀⣈⣙⡆⣀⡝⠀⠊⠁⢀⣴⡀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⣀⣤⡶⠶⠶⠄⠃⠀⠀⠀⠀⡇
⢘⠉⠁⣠⡤⠶⠀⠀⠀⠀⠀⠘⠠⣤⡤⠀⠀⠀⠀⢀⡀⠀⣀⠀⢠⣮⣤⣤⡄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠟⠛⢉⡲⠆⠀⠀⠀⠀⠀⠀⠚⣄⡀⠠⠀⠀⠀⠀⠡⠖⠃⠀⠉⠉⠉⠉⠁⠄⠀⠀⠀⠀⠐⠓⠫⣷⠄⡐⡇
⢀⢀⣴⠿⠛⡷⣀⡀⠀⠀⠀⠀⢎⠉⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⡇
⠸⠿⠁⢀⣼⠟⠉⣷⠋⢳⣿⠓⢬⣣⡀⠈⠀⠤⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠘⢠⠀⢀⠩⠥⠀⠛⠂⠀⠿⠄⠀⢹⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⢘⡀⠐⠀⠰⠈⢠⠈⢀⠰⠆⠠⠆⠐⠄⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣀⣀⣀⣀⡀⠀⠖⠁⠀⠂⡇
⢀⠀⠦⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡈⠁⠈⠈⠐⠐⠂⠄⠌⡀⣈⠙⠋⠉⢉⠉⠙⠢⠤⠄⠀⠠⡇
⢈⠁⠄⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⠠⠒⠅⠀⠀⢀⠀⠈⠁⠘⢂⣀⠤⠭⠉⠑⢒⠺⠇
⠘⢀⠐⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠷⠀⠡⠤⠀⠀⠐⠒⠺⢋⠉⠃
⠘⠀⠐⠈⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠋⠛⠋⠃
⠐⠀⠄⠈⣠⠐⢀⠠⠐⠤⠀⢀⠀⢀⡤⠀⠤⠤⠎⠀⠀⡀⠠⠀⢦⡀⠀⠒⠒⣶⣶⣖⣶⠆⠀⢿⢽⣛⣻⠷⡇
⠘⠁⡀⠓⡀⠂⠀⠤⠈⣀⡈⠥⠤⠒⠒⠘⠈⣉⣉⢭⢓⢄⡀⠀⠀⠁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⠇
⢠⠀⢀⡀⣥⣐⠦⠻⣽⠒⢙⡆⠀⠀⠀⠀⠀⡱⡀⢠⡗⠁⡊⢆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠰⠦⢽⡍⠀⣿⡄⠀⢿⡀⠈⡇⠀⠀⠤⢀⡀⠙⠙⠓⢀⣄⡀⠅⣺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢘⠒⠒⡇⠀⡯⡇⠀⣿⡇⠀⡇⠀⠹⣤⣼⡅⠀⠀⠰⠟⡣⠈⠰⡒
⢨⡭⠭⡇⠀⡭⣇⣀⠿⢧⡴⠷⠀⠀⠁⠀⠄⠄⠀⠀⠀⡀⠀⠀⠀⠀⣂⡀⢀
⠰⠖⢒⡓⠠⣭⠉⠐⠾⣁⡨⠆⠀⡀⠀⠀⠶⠞⠀⠀⠀⠀⠂⠀⢠⡐⠠⠯⢭⣆⠆
⠘⠛⠙⠂⠐⠛⠑⠛⠐⠒⠒⠒⠂⠠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠙⠒⠈⠛⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣐⣦⣄⢸⣧⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⢠⡀⠈⣿⣿⣿⡟⠀⠀⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣷⡀⢻⣿⣿⣟⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣈⢻⣿⣄⢻⣿⣿⣷⣿⡟⠃⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣽⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢿⣿⣿⣿⣿⠇⠈⢻⣿⡄⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣿⡙⠻⣿⣿⣦⠀⠀⢻⣇⠀⠀⡼⠏⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⢿⣿⣆⠈⠻⣿⡆⢀⣸⣿⡆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠲⣤⡻⣿⣷⣤⣿⠁⠘⢿⣿⡅⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⢿⣾⣿⣿⣿⣦⠀⠀⢿⠃⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣙⣿⣿⡿⢿⣦⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣿⣿⣿⣿⣦⠀⣉⣤⠀⢨⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣟⢛⣿⣿⣷⣤⣴⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣀⣤⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⠉⠉⠻⣿⣷⣿⣿⣯⣀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣀⣤⣶⠿⣿⡛⠁⠈⠙⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠇⣸⣿⣿⣿⡏⡈⠀⠸⠿⢶⣀⣾⡏⠉⠢⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣧⡀⠀⣠⣶⢾⠋⠻⣄⠈⠀⠉⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠾⡿⣷⣼⣿⠟⠛⠃⠀⠿⣦⣾⠛⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣷⣶⣶⣾⠟⠋⠙⣷⣦⠀⣠⣴⣿⡍⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⠀⠀⠛⢻⣿⠃⠈⠈⢻⣆⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⠉⣉⢿⣿⣆⣠⠶⣿⠻⢧⡀⠀⠀⢀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃⢻⣤⣤⣿⡏⠠⠦⠀⠀⢿⣶⡆⠀⠐⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⢿⠟⢹⣿⣤⠀⣀⣤⣼⣏⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠐⠛⢻⣿⠋⠋⠀⢻⣷⣶⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣻⣿⣿⣧⣀⣤⣤⣾⡟⠐⠇⠀⠈⣿⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⠛⠟⠁⣽⣦⣀⣤⣞⡛⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠜⢋⡽⢟⠛⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣴⣿⣿⣋⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠉⠙⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣷⢿⣿⣶⣯⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠈⣷⣄⠈⣿⣷⣄⠑⠀⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣯⣾⣿⠵⠤⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠰⣿⢹⣷⣾⡁⢹⣷⡀
⠀⠀⠀⣠⣖⢻⡯⢍⡿⠯⡞⢉⠻⣆⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡌⢮⢿⢋⢿⡌⠿⠁
⠀⠀⣾⣯⣵⠎⠙⢿⡠⢦⣹⠿⢾⣁⡇⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠆⠀⠀⠀⠙⡄⠁
⠀⣸⠏⡉⢻⣜⣃⡼⠛⢿⡇⠰⢀⡿⢿⡆⠀⠘⣶⠀⠀⠨⠀⠀⡀⠀⠀⠀⠂⢁⡀⠀⠀⠀⠀⠈⠀⠀⠈
⢰⣿⣧⣽⡏⣉⠹⡇⠛⢠⡿⣿⣿⣷⣷⡇⠀⠀⠈⠱⣆⡀⢀⠀⠀⢀⠈⣀⠴⠋⠀⠀⠀⠀⠐⠀⢀
⣼⣿⡤⢈⣷⣭⡴⠛⠛⣯⡠⣿⠻⢿⣿⣿⠀⠀⠀⠀⠀⠉⠉⠐⠚⠛⠛⠉⠀⠀⠀⣄⠀⢀⣠⡶⠿
⠛⠿⢿⣿⣧⣼⣧⠚⢀⣿⠿⣿⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢠⣼⣷⡻⠍⠈⠀⣀⠄⠐⠒⠒⠒
⠀⠀⠀⠀⠈⠙⢻⣟⠻⣇⣰⣿⠀⠀⠐⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⢀⣀⣀⣀⣀⡾⣿⣷⡟⢻⣿⠀⠀⠀⠈⠢⠀⠙⡏⠀⠀⠀⠀⠠⠆⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠇⣰⡂⣤⣀⡀
⡁⠤⣹⣯⣵⣬⣷⣿⣿⢧⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣭⣻⣏⣠⢄
⠉⠻⣿⣿⡿⠃⠈⠉⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⢀⡰⣲⣾⣉⣁⣼⣁⣚⣲⣤⡀⠀⠀⢀⠃⠀⠀⠀⠈⠉⠋⠉
⡶⠞⠛⠉⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⠙⡏⠻⢙⣏⠐⢋⣟⠩⣵⢷⢄⠀⠘⣆⢀⣶⣒⣶⡗⢲⡈
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣻⣦⡞⠟⢿⡏⣭⠻⡟⣉⡳⣤⣾⣈⣇⠀⠈⠉⠉⠑⠈⠉⠉⠉
⠒⠶⠢⠀⠀⠀⠠⢄⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠸⣷⣸⡉⢃⣼⣧⣄⣴⠷⣄⣠⣟⠐⢏⡝⡆⠀⠀⠀⠀⠀⢀⡠⠖
⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⡴⠃⠀⠀⠀⣿⣯⡍⢻⡇⠖⢸⡇⠒⢈⡏⣭⢱⡞⢉⡁⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣴⡯⣾⠗⡅⡄⠀⠀⠀⢻⢷⣞⡻⢶⣛⡓⢶⣟⡳⣤⡾⣧⣶⡇⠀⠀⢀⣎⠀⠀⠀⠆
⠀⠀⠀⠀⡀⠀⠀⠀⣼⣆⣿⣫⠀⡟⡸⣶⡇⠐⠀⠀⠀⠃⢷⣥⡼⠧⣤⠿⣍⣡⣧⣩⣼⣹⣀⣀⠀⠈⠙⢷⣄⠁⠈
⠀⠠⠀⠀⠠⠀⢀⢈⠻⣿⣿⠃⡼⢿⣱⢃⣧⡆⢀⠀⠀⠀⠈⠓⠫⣣⡼⢜⡴⢇⣡⣿⣭⣻⣿⠋⠢⠄⠀⠀⠙⢷⣄
⣦⠀⠀⠀⠀⢧⣨⣯⠀⢹⣿⣆⡆⣸⣿⢾⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠂⣹⡟⣿⣄⣿⣿⣍⡙⣶⣦⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⢿⢿⡄⣘⣿⢿⣠⣿⠁⣠⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠋⣿⣏⢸⣯⣍⡉⠻⢷⣍⠒⢤⡀
⠀⠀⠀⠀⠀⠀⢨⠀⢷⣿⡃⣸⣿⡷⣻⡟⠋⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⣏⠈⢻⡟⠟⢻⡿⠷⣞⠻⣄
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣰⡦⢴⢴⢄⠴⣴⡦⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⡠⡤⢀⠀⣤⢄⡀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⣽⣿⣗⠀⢸⠀⢄⡮⣛⡇⠀⠀⠀⠀⠀⢸⣾⢿⣿⢧⣳⣽⢠⣟⡞⣷⡏⡿⡇⠀⠀⠀⠀⣼⡻⠤⠽⠷⢾⣿⠴⠿⠥⠴⣟⡀
⠀⠀⠀⠀⠀⢹⣖⠍⠀⢸⠁⢐⣭⣶⡂⠀⠀⠀⠀⠀⢸⡜⡜⣟⢯⣻⣿⣿⣯⡞⡽⣱⣱⠆⠀⠀⠀⠀⣿⠅⠀⠉⠍⢹⣿⠉⠩⠥⠬⢽⡇
⠀⠀⠀⠀⠀⢿⣛⡫⠁⢸⡄⠉⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡝⣌⢳⡙⣿⣿⣫⢞⡵⢣⢿⠃⠀⠀⠀⠀⣿⡛⣛⣛⣒⣻⣿⣛⣛⣛⡓⣻⡇
⠀⠀⠀⠀⠀⣿⣿⣓⠂⢸⡇⣐⣿⣿⡃⠀⠀⠀⠀⠀⢨⢳⡹⣌⢳⣽⣿⣿⣣⢞⣵⢋⢎⠂⠀⠀⠀⠀⣿⠄⠀⠠⠤⢾⣿⠤⠤⠤⠄⢽⡇
⠀⠀⠀⠀⠀⣲⣿⠿⠗⢸⡇⠾⣿⣿⠇⠀⠀⠀⠀⠀⢸⣧⣳⣌⢷⣽⣿⣿⣫⠟⣡⢟⣾⠇⠀⠀⠀⠀⣿⡍⠉⠩⠭⢽⣿⢭⣭⠍⢩⣽⡇
⠀⠀⠀⠀⠀⣸⡷⢏⠁⢸⡇⠨⣿⣿⡇⠀⠀⠀⠀⠀⢸⣧⠳⣽⣦⡙⢿⡿⣡⣾⣿⢋⢾⠇⠀⠀⠀⠀⣿⣄⣒⣒⣒⣚⣿⣒⣒⣒⣒⣺⡇
⠀⠀⠀⠀⠀⣻⠽⡣⠂⢸⠀⢔⣽⣿⠃⠀⠀⠀⠀⠀⠀⢧⠳⣬⡳⣽⣿⣿⡵⣛⣵⢏⡞⡀⠀⠀⠀⠀⣿⣒⠲⠶⠶⢺⣿⠒⠒⠒⢒⣺⡇
⠀⠀⠀⠀⠀⣼⣭⡶⠀⢸⠂⢨⣒⣿⡇⠀⠀⠀⠀⠀⢸⡘⠷⢮⣽⣿⣿⣿⣞⣯⠵⠟⣡⡇⠀⠀⠀⠀⣿⠧⠤⠭⠭⢭⣿⠭⠭⠭⠴⢿⡃
⠀⠀⠀⠀⠀⣾⣿⣡⣀⡸⣠⣤⣝⣿⡅⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠻⢿⡿⠖⠒⠚⠻⠟⠁⠀⠀⠀⠀⠛⠁⠀⠀⠈⠸⠿⠘⠛⠁⠈⠙⠃
⠀⠀⠀⠀⠀⠈⠁⠈⠈⠀⠈⠈⠁⠉

⠀⠀⢀⣀⣀⣀⣀⠀⠀⢠⠀⠀⢄⠀⠀⠐⡄⠀⠀⠀⢀⡠⠤⠄⠠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣄
⠊⠉⡀⠀⠀⠀⠀⠈⠱⠒⠦⡀⠀⠆⢀⠀⠚⠀⠀⣀⠨⠤⠀⢀⠠⠀⠛⠁⢄⡀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠂⠀⠈⠤⠶⢤⣖⡀⠀⠉⠈⠀⠀⠲⡀⠔⠓⢤⠚⠃
⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠈⠢⠚⠀⠀⠀⠄⠀⠀⠐⡖⠂⣠⣤⠀⠀⠀⠀⠙⣤⠤⠀⠒⠂⠀⠉⠁⠀⠀⠀⡀⠐⠀⠀⠀⠉⠢⡀⠀⠀⠴⠦⡁⠀⠀⠀⠡⠶
⠀⠀⠀⠀⢌⠀⠀⠀⢁⢀⣀⠀⠀⡡⠀⣛⡀⠀⠀⠐⠃⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⢄⠀⠀⠀⠀⠀⠂⠁⠀⡠⠠⠀⠀⠄⠀⢈⠳⢄⠀⣀⡕⠄⡠⠔⢥⡀
⠀⠐⠒⠂⣀⠔⠂⠉⠈⠀⢉⠈⡙⣤⢤⠇⠙⢤⠠⠄⢀⡀⠋⠀⠐⢄⠒⠒⠀⢀⠠⠼⠀⠀⠧⠄⣀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡀⠀⠀⠡⡀⠑⢌⠀⠐⠗
⠀⠀⡠⠀⠓⠀⠀⡤⠀⠒⠉⠀⢀⡈⣎⠀⡠⠊⠉⠉⠀⡇⠀⠒⢀⠀⠑⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠡⠁⠀⠀⠀⠧⠀⠀⠀⠐⠄⠀⠑⢄⠀⠀⣀⠄
⢀⠎⠀⡰⠉⠈⠣⣀⠀⠀⠀⣩⣄⡄⠈⣱⣷⡄⠀⢴⠏⢀⣄⣀⠀⠈⠀⠈⠆⠈⠂⢀⣤⠀⠀⠀⠀⠀⠀⠀⠝⠳⠄⠀⢤⣀⠀⠈⡀⠀⠀⠀⠄⠀⠠⡑⢎
⢃⠔⢨⠅⠠⠀⠊⠀⢁⡦⠁⡸⠂⠘⡏⠀⠀⢰⠀⣈⣶⠂⠈⡄⠀⠉⠁⠒⠬⡄⠒⠀⠀⠀⠤⠤⠤⠤⢤⠎⠰⠇⠀⠀⠀⠀⠓⠢⣄⠠⠤⠤⢈⣂⠀⠀⠄⠳⡄
⠠⠄⠚⠀⠀⠀⠀⠈⢣⡀⠀⢠⣴⡈⠉⠑⠲⡗⠻⣄⢈⣶⣍⠈⢑⠶⠀⠀⠀⠀⠒⠢⠀⠐⠒⠒⠒⠒⠚⠒⠓⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠠⠀⠈⢩⠉⠉⠁⠚
⠀⠀⡆⠀⠀⠀⠀⠀⠀⢹⠂⠉⠛⠉⢲⠖⠋⠀⠀⠈⠁⣉⡇⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠆
⠀⢠⠄⠀⠀⠐⠠⠄⠀⠨⠤⢤⠚⠤⣈⡆⠀⣴⣦⠉⠉⣶⡎⠉⠁⣰⠲⡀⠀⢰⠋⠉⠉⠙⢲⣼⠋⠉⠉⠑⢦⢰⡖⣄⠀⢠⢤⡄⠀⠀⠀⠀⡁⠪⠥⠤⠤⠤⠄
⠀⢸⡠⠀⠀⠢⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⣏⣿⢀⣠⡏⡇⠀⢠⠏⣀⢳⠀⢸⡇⠀⠛⠀⣸⢻⠀⠈⠛⠀⣸⠃⠹⣼⢶⣫⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠖⠆
⠀⢻⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣉⢹⠛⠒⡗⡇⢠⣏⣀⣉⡀⢧⢸⢰⡖⠒⠋⠁⢸⢰⠖⠒⠋⠁⠀⠀⠘⣲⡇⠀⠀⠀⠀⠀⠀⣄
⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⢨⣿⠀⠀⣷⠇⣾⠼⠁⠀⠱⠼⠦⠶⠇⠀⠀⠀⠸⠾⠀⠀⠀⠀⠀⠀⠘⣯⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣀⢠⣤⡀⣠⢤⠀⠀⡴⢦⣴⢲⡀⠀⠰⠒⠆⠖⣦⠀⢀⡖⢶⢠⣤⣤⣤⣤⡀⣤⣤⣤⣀⡀⠀⠀⠀⡗
⠀⠀⡆⠀⢀⠠⠀⡀⠀⠀⠾⠀⡇⢠⠏⡴⠒⢦⣿⣜⢺⠀⠀⡏⣿⡿⣀⣇⢠⢷⡀⣆⡆⣸⡄⡸⣧⢸⢾⣰⠒⠚⠚⠃⡆⠀⣤⠀⠙⡆⠀⠀⡇
⠀⠈⣵⠚⠀⠀⠀⠈⠁⠀⢸⠀⡇⠈⠳⢯⣷⠲⣄⢸⣼⠀⠀⡇⣼⣯⣿⣸⡼⣸⠞⢃⣤⣧⣷⡇⣿⢺⢸⢈⣯⣉⣹⠀⡇⡀⢀⣠⠜⠁⠀⠀⡃
⢀⠞⢰⡀⠆⠁⠀⠀⠀⠀⢺⠀⡇⠸⡟⢧⣬⡷⣹⢳⡜⠦⠴⢃⡽⡇⣺⣏⢣⡟⣤⣾⠀⡿⡈⣼⢋⣸⢸⣹⣀⣀⣀⡀⠇⡟⢦⡑⢦⡀⠀⠀⡅⠀⠈
⠊⠀⠀⣇⠰⠀⠀⠄⠀⠀⢸⠀⡇⠀⠉⠓⠒⠛⠁⠀⠙⠓⠒⠚⠁⠻⠏⠙⠚⠀⠛⠛⠿⠇⠓⠋⠘⠛⠸⠾⠶⠚⠺⠋⠛⠃⠀⠉⠉⠁⠀⢠⠄⣀⠤⢄
⠠⡀⠀⢹⠀⢃⠘⠀⠀⠀⠘⠤⠇⡀⣀⠀⢀⢀⡀⠀⠀⡀⡄⠀⠀⣠⢠⡄⠀⠀⠀⠄⠀⠄⠒⠀⠀⠀⠤⠤⠤⠄⠤⠰⠀⠠⠖⠀⠐⠐⠒⠘⠁⠁⠂⠀⠀⠐
⠀⠈⠚⠀⢃⠈⢄⠠⠂⠀⠀⠀⠄⠠⠀⢀⠸⣀⣠⣶⡄⡅⡇⠐⠴⢁⠎⠄⡀⠀⠀⡀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠠
⢀⠖⠀⠀⠘⣄⠚⠄⠀⠀⠀⠀⠤⠠⢰⡁⡀⠈⣧⡉⠀⢱⣇⣠⢁⡆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⢠⣔⡂⠁⠀⠀⡤⠓⢌⠑⢤⡀⠀⠀⠀⠈
⠁⣀⠀⠀⠀⠸⡀⠚⠀⠀⠀⠀⠀⠠⡉⠀⠀⠠⠿⢶⠀⠉⠐⡄⠉⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡗⠉⠀⠀⠀⠀⠀⠦⢤⡼⠁⠐⠙⢆
⠙⠛⠛⠁⠀⠀⢱⠀⠀⠀⠀⣶⡀⠐⠉⢢⣰⠁⠀⠀⣣⣠⡾⢔⡊⣡⣷⡆⠀⠀⠀⠀⠀⢀⡠⠒⠒⠆⠀⠀⠀⡎⠀⠀⠀⢀⠂⡀⠀⠀⠀⠀⠀⠀⠱⡈⠆
⠀⠀⠀⠠⠄⠤⠀⢧⠀⠀⠀⠽⡓⠉⠞⠙⠴⢇⢲⡖⣏⠀⠀⠀⠵⠈⠁⠐⢄⠀⠀⠀⢠⠋⢀⠎⣁⠛⡄⠀⢸⡇⠀⠀⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠈⠐
⠀⠀⠀⠀⢀⠠⠀⠀⢣⡀⠈⠓⠿⣶⣀⠀⠀⠘⠷⠎⠁⠀⠈⢳⣤⡆⠀⠰⠦⡀⠀⠀⠘⢆⠈⠎⠾⣰⠃⠀⠘⡇⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠃⠒⠊⠁⠀⢣⡀⠀⡰⠋⠛⠄⠀⠀⢴⡰⣀⠀⡠⠊⠙⢃⠐⠀⠀⡘⠴⠆⠱⠀⠙⠒⠊⠁⠀⠀⢀⡐⣆⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡰⠁
⠉⠀⠀⠀⠀⠀⠄⠂⠀⡐⠁⢲⡀⡀⠀⠘⠒⠖⡁⢠⡌⠉⢀⡀⠀⠀⣦⠴⠚⠁⠀⠀⠀⠀⠀⠐⠒⠂⠁⠂⠁⠀⠈⠢⣀⠀⠀⠀⠀⢀⠌⠀⠀⡠⠞⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠤⠘⠉⠑⠠⠤⠤⡼⢀⠄⠈⠉⠣⢈⠁⢀⠁⠀⠀⠀⠀⢀⣠⠄⠀⡤⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠤⠾⠐⠒⠉⢀⡀⠤⠂
⠀⠀⠀⠁⠉⠐⠈⠀⠀⠁⠀⠀⠀⠐⠒⠀⠐⠒⠂⠈⠁⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠘⠓⠒⠒⠒⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣁⠀⠀⡀⣀
⣿⣿⡟⠻⠟⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⣀⡀⠤⠒⠈⢈⠇⠀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⠀⠐⠀⠐⡄
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠷⡤⠸⢗⡈⠁⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⡇⠝⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠐⠈⠁⠀⠀⠈⠀⢀⠄⠀⠀⡇
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆⠀⠀⡴
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣼⠗⡯⣄
⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠁⠀⠀⠀⣤⣷⣖⠅⣅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢨⣿⣿⣇⠀⠀⠀⠈⠀⠀⠀⠿⡿⡔⠈⠀⠀⠀⠀⠀⠀⠠⠄⡄
⠀⠀⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣰⢧⠀⠀⠀⠀⣀
⠀⠀⠀⡀⠀⠀⠁⠇⠀⠀⠀⠀⠀⢸⣟⣿⣷⡃⠀⠀⠀⠀⠐⠈⠉⠊⠀⠀⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡅⠀⢘⡀⠀⠀⢰⡌⣽⣾⣿⣿⢫⢡⠀⠀⠀⠆⠀⠀⠀⠀⠐⠈⢠⡔⠠⢄
⠀⠀⠀⠀⠀⠀⣀⠀⠃⠀⠀⠀⠄⢸⣻⣿⢿⠏⠀⠀⠀⠰⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠸⣯⡿⡟⢠⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⡀⠹⡷⠃⢸⠀⠀⠀⠈⠀⢠⠁⠀⠀⠁⠼⣄⠀⣀⡀⣀⠖
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠁⠀⠀⠅⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠟⠁⠀⠀⠀⠏⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢀⠀⠀⠀⠀⢀⠈⠀⣠⡘⠁⠀⠀⠀⠀⣠⠎⠁⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠈⠀⢀⠹⠇⠀⠀⠀⣠⠊⠁⠀⠀⢠⠄
⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠙⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣠⠚⠁⠀⠀⠀⢠⠄
⠈⠉⠉⠉⠁⢀⠀⠐⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠻⣷⡆⠀⠀⠀⠀⠀⢢⠘⢢⣤⡃⠀⠀⠀⠀⣶⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠈⠀⠄⠀⠀⠹⣿⣧⠀⢿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠙⠻⣄⠀⠀⠀⢻⠀⠀⢸⣿⣶⠟⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠘⣿⣷⣄⣿⡏⢱⡄⠀⢀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠘⣦⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠂⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠁⢸⣿⣿⣿⣿⡀⢰⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣿⡏⠁⠀⠀⠀⠀⠀⣄⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⡤⠀⠙⠿⣿⣿⣿⣅⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⠈⠁⠀⠀⠀⢠⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠴⢶⣦⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⢶⣀⠀⠈⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠈⡀⠀⠀⠀⢠⣾⣿⣿⣿⠏⠀⠀⠠⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⠀⣀⠀⠀⠈⢉⡦⠀⠀⠈⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⢡⡞⠁⠀⢀⣠⣿⣿⣿⡿⠃⡠⠎⠁⠀⠀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠂⠀⠀⠻⠟⣦⡀⠀⠸⣿⡀⠀⠀⣰⠀⠀⠀⠀⠀⣿⣿⣈⣤⣤⠶⠎⠉⢹⣿⡿⠁⠄⠁⠀⡀⠰⠂
⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣤⣤⣤⣠⣤⡀⠀⠀⠀⠰⣿⡋⣡⠆⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠋⠁⠀⠀⠀⢸⣿⡄⠀⠀⠀⣤⠙
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠙⠿⣤⡜⠛⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⡸⠟⠀⢠⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠛⠛⢿⣶⣦⣀⠉⢻⣦⡘⣿⣷⠀⠀⢴⠀⠀⠀⣸⠃⠀⠀⠀⠂⠀⠀⠀⠁⣠⡞⢉⡉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠈⠛⣦⡀⠀⠘⠛⠷⣶⣿⣷⣿⣿⣷⠀⠘⠀⠀⢸⠇⠀⠀⠀⠛⠀⠀⠀⣶⣿⣷⡾⠛⠛⠛⠷⠞⠃⣠⣴⣿⣿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠘⠿⠛⠿⢿⣿⣿⣷⣶⣤⣤⡈⠻⣤⠀⠀⠀⠀⠀⠙⢻⣿⣿⣥⠀⢠⣶⡆⠀⠀⠀⠛⠂⠀⢀⣴⣿⠟⠁⠀⠀⢀⣄⣴⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣷⣿⣿⣆⠀⠐⢞⡉⠈⠁⠹⢿⣷⣾⣿⣃⣠⡄⠀⢤⣀⣴⡿⠋⠋⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⠐⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣈⣹⣿⣿⣿⣿⣿⣷⣁⡈⢻⣶⣀⠀⠀⠈⠙⠻⣿⣷⣤⣾⣿⡟⢻⣶⠆⠀⠀⢀⣰⠿⠋⣀⣄⡈⠀⠤⢤⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠉⠛⢿⣿⣿⣧⠘⠿⣿⣿⣿⣿⣀⠦⣹⣿⣥⣄⠀⠀⠀⢈⣿⠿⠿⠿⣿⠿⠿⣄⠀⠀⣈⣉⠴⠾⠋⡀⣻⠷⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠉⠙⠻⠿⣿⣧⣼⣿⣿⣷⣄⣰⠏⠁⠀⠀⠀⠁⠀⠀⢁⣠⠾⠋⠁⢠⡤⠚⠋⠀⢀⣀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⣿⣿⣶⣦⣄⠙⢿⣷⣄⠀⠀⡀⠙⠻⠿⠿⣿⣿⡛⠂⠀⠀⢠⠀⢀⣤⣼⡿⠟⠶⠿⢿⣿⣶⣶⣶⣿⣿⠿⠿⠛⠛⠙⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⢀⡈⠹⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⠹⢿⣧⡀⢠⣼⠿⠋⠉⠁⠀⡀⢠⣶⣿⣿⣿⣿⣿⣩⣷⣶⡶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⣀⡀⠀⠉⠛⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡅⠀⠀⠀⣀⣀⣼⣿⠿⠛⠉⣡⣤⠛⠋⠉⠁⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣞⣡⣄⡀⠀⠀⠀⠈⠻⢿⣶⣶⣄⣠⡾⠁⠀⠈⠉⠈⠙⠣⣠⣶⡿⠛⢋⣄⢰⣾⣿⣿⣷⣶⣶⣶⣾⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⡻⠿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠙⣿⣿⠁⣶⣆⠀⠀⠀⣠⣤⠟⠋⠀⠛⢿⣿⡄⠉⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠠⠈⡛⠋⠰⢿⣦⣴⣆⠉⠉⠀⠀⠀⠀⣠⡿⠋⠉⠁⡀⠀⠠⣾⣿⠉⣣⣄⢉⣾⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣦⣀⠀⠿⠿⠿⣇⡀⠀⢀⣴⠞⠛⠁⠈⠉⠉⢿⣃⣠⣾⣿⠟⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⣉⣉⣭⣤⣤⠀⠀⠘⣿⣶⣾⡁⠀⠀⠀⠀⣀⠴⣿⣿⣿⣿⣿⣶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⡛⠛⠿⢷⣄⣴⣿⡉⢩⣿⡶⠀⢠⣤⣀⣀⣼⣿⣿⢟⣛⡛⢻⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⣮⣉⡱⠖⠶⣌⠉⠈⠉⣁⠘⠉⣀⠐⠚⠉⣉⠉⠉⢀⡀⠀⠉⡅⠀⠉⣉⣽⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠈⠃⢀⠀⠘⠇⠀⡀⣠⡄⠛⠃⠀⠀⠀⡈⠷⠂⣤⣀⣀⣰⡇⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠚⢧⡀⠀⠀⠀⠈⠃⠀⠛⠀⠀⠈⠱⣤⣀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠘⠳⡄⠈⠃⠘⢛⣹⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢛⠀⠀⠀⣿⠀⠀⠈⠁⣤⣤⠀⠀⠀⠀⣤⡀⠻⣤⡄⢠⣴⣤⠉⠀⠀⠀⠀⠀⠊⠹⡟⠛⠒⠻⢿⠟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠈⠀⠐⠀⠙⡿⠛⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⡀⠉⠳⣄⠀⠀⠀⠀⠀⠐⠇⠀⠀⠠⠀⢴⠟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡀⠀⠈⠀⣴⠟⠀⠷⠀⠀⠘⠓⡀⠀⠀⣀⣀⡿⠏⠀⠀⠀⠀⠀⠄⠀⢶⣤⡤⠀⠤⣶⣆⠀⠀⢀⡄⠈⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠹⠀⣀⣤⠉⠀⣀⠀⠀⠒⠀⠈⠙⡟⠛⠛⠻⣦⣀⠀⠀⠀⣤⠀⠀⠀⡶⠀⠀⠀⣠⡄⠈⠻⣦⣠⣁⣀⣼⣶⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠙⣀⠰⢀⠙⠙⠶⣄⠉⠙⠃⠀⢀⣤⠊⠀⠀⡸⠃⠀⠙⣷⡄⣀⡀⢀⣀⣴⠄⠀⠀⠀⠁⠀⠀⠀⣙⠋⠉⠛⠻⠀⠈⢛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠙⢀⠈⠁⠀⠀⠹⣷⣄⣀⣱⣾⠇⠀⠀⠀⠀⠀⠶⠀⢀⣿⠋⠁⠉⡙⠛⢦⣀⠀⠀⠰⠆⠀⢰⡿⠀⠀⣴⠀⣸⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢦⡔⠶⠎⠀⠀⠀⠀⠀⢈⣿⠋⠀⠐⢠⣴⣦⡀⠀⠞⠂⠀⠚⠁⠀⠀⠀⠁⠀⠀⠙⣶⣤⣤⣀⠴⣾⠁⠀⠀⣀⡀⣿⡀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠈⢡⡀⠘⠀⠀⠛⠀⢶⡾⠁⠀⠀⠀⠀⠉⢹⣧⣀⣤⣤⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉⠁⠀⠘⠷⣄⠘⠉⢀⣿⠃⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⣧⡀⠈⣀⠐⠠⣾⠀⠀⠀⠀⠀⡀⠂⠈⢁⡜⠉⠹⠿⠶⣤⡀⣰⠒⠚⠋⠀⠀⣿⠟⠀⠀⡼⠇⠀⠙⣧⡾⠿⣷⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⢻⠀⢰⣤⠀⠀⠈⢦⡀⠀⠀⠛⠃⠀⣰⠟⠁⠀⠀⠀⠀⠀⠙⢿⡄⣠⣴⣶⠚⠋⠀⠀⠀⠀⠀⠀⢰⡟⠁⣠⠟⣧⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⢦⡀⠹⠀⠈⠃⠀⠀⠀⢿⣤⣤⠠⣴⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⢘⠁⠀⠏⠀⠻⠋⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⠀⠿⣆⠀⠀⠀⠀⠀⢀⣀⡀⠉⣄⠀⠉⠳⣤⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠘⠀⠀⠙⢷⠀⢠⣞⠋⠀⠀⣠⠄⣦⡀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠸⠃⢀⡿⠧⣀⠀⠀⠀⠸⣿⠀⠀⠙⠀⠀⠀⠀⣀⣀⣀⠀⠀⣤⣤⠟⠀⠠⠀⠂⠀⠀⠐⢦⣸⠟⠀⠈⠳⣄⠀⠈⣤⡿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢠⠀⢠⡟⠀⠀⠉⠶⢀⣠⠶⠃⠀⠀⠀⠀⠁⠀⠀⠉⡻⠋⠁⢿⡍⠙⢶⣄⠀⠀⣠⠀⠀⠀⣴⡿⠀⠀⠀⠀⠘⣷⣾⠋⠰⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠘⣷⡞⠁⠘⠀⠀⠀⠛⢁⠤⠤⣀⠀⠀⠀⠀⠀⢠⡞⠃⠀⠀⠈⠷⠀⠀⠈⢷⠀⠉⠀⠀⣀⡿⠁⠀⠀⠀⠀⢀⡘⠁⠘⢧⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⣀⡀⠀⢀⡴⠃⠀⢀⡏⠙⢦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠠⠞⠗⢰⣶⠾⢋⣁⠀⠀⠀⢀⠀⢸⡇⠀⢶⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⣸⡏⠉⠙⠀⣿⠀⠀⠀⠸⠇⠀⠈⠙⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠂⠀⠀⠉⢲⡄⠘⠁⠀⣿⡗⠀⢀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠈⠳⣄⡀⣴⠟⠀⠀⠀⠀⠀⠿⠀⠀⠀⠉⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⢾⡟⠁⠀⠀⠀⠀⠀⠻⣿⡾⠿⣯⡀⠀⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⠀⠀⠛⠢⣄⠀⠀⢠⣤⡤⠂⠀⠀⠘⠇⠀⠀⠈⢿⠆⣀⣶⡦⠈⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠈⣿⡆⢰⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠂⣾⠇⠀⢾⠀⠀⠈⠷⡀⠀⢀⡅⠀⠀⠀⠀⠈⠀⠠⡄⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠅⠸⢧⣀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⠀⠀⠙⠿⠀⠀⠛⠀⠻⣥⡄⠀⠀⠀⠀⠀⣤⠀⢠⡴⠋⠁⠈⠻⡿⠃⣤⠀⠁⢠⠇⢠⡄⠀⠀⢠⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⢀⠀⠀⠀⠀⠟⠀⠀⣤⠈⠁⠰⠷⡤⠀⠠⠀⠀⢸⡇⠀⠀⢸⠷⠀⡀⠀⠁⣶⣿⣁⠀⠀⠈⠁⢠⣿⣣⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡄⠰⣦⣀⣠⡆⠀⣼⠃⠀⠀⠀⠀⠀⠈⢲⡶⠶⠌⠀⠀⢴⡈⠀⠀⠁⠀⠀⢀⣼⠿⠟⠛⠷⣶⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⠉⠁⠶⠶⣿⠀⠀⠀⠀⠘⠃⠀⠈⣁⠀⠀⣀⣀⡀⠉⠲⠀⠀⣀⡴⠞⠁⡀⠤⡄⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣤⠀⢦⣀⠀⠀⠀⠀⠀⣀⠹⠿⠋⠉⠉⠉⠳⢦⣠⠖⠋⠀⠀⢘⣥⣤⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⢿⣷⣦⣤⣼⡋⠀⢀⣤⣀⣉⣦⣄⣠⣿⡷⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁
⠀⠀⣀⣀⠀⡀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠉⠈⠉⠀⠉⠁⠀⠀⠉⠁⠀⠈⠁⠀⠀⠉⠁⠀⠉⠀⠀⠉⠁⠀⠈⠁⠀⠈⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡶⣠⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⡄⣀⡔⠃
⠀⠀⠀⠀⠀⢀⠰⣿⠃⢿⣮⣶⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣸⠀⠀⢰⠀⠀⠀⣰⡏⣴⡿⡛⣾⠟⡉⣠
⠀⠀⠀⠀⢳⠈⡀⣿⣶⠛⣿⢃⡇⠾⠂⠀⠀⠆⠀⠀⠀⠙⣦⡀⠀⠀⠀⢂⡀⠀⠀⠀⠠⠀⠀⠀⢀⡾⠃⠀⠀⠈⡀⢀⡈⢻⣿⣿⠏⣰⣯⢜⣶⡏⠠
⠀⠀⠀⢣⠘⣗⢷⣾⡿⡄⢹⡿⠃⢠⡄⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀⠀⠀⠙⠲⣦⡤⣤⣬⡼⠂⠉⠀⠀⠀⠀⠀⣧⣾⠁⢸⣿⣥⡊⢹⣵⢋⣾⡔
⠀⠀⠀⠀⠣⣿⣆⠹⣿⣷⣾⠁⢀⡾⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠀⡛⢻⣀⣼⡿⡟⣠⡿⠛⠿⣿⠗⠛
⠀⠀⠀⠀⠀⠹⣿⠢⣨⡟⢩⣤⣾⠗⠁⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡼⠁⠀⠀⠀⠰⢦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣼⢿⠁⣀⣰⣏⣠⣴⣾⡁
⠀⠀⠀⠀⠀⣆⣘⣦⣻⣿⢻⣯⣤⣤⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢐⠀⠳⣦⣾⡆⠀⠀⠀⣀⣸⣿⣏⣼⡿⣿⣯⣴⠟⠉⠉⠐⠂
⠀⠀⠀⠀⠀⠙⢿⣧⣹⣷⣾⣿⡾⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠂⠠⠀⠉⠐⠚⠉⣀⣤⡖⢻⣟⡛⣿⣟⢻⣾⣿⣯⣉⠁
⠀⠀⠀⠀⣀⣤⣷⣛⠛⣶⣤⢓⣻⣰⡄⠀⠀⠀⠀⠀⠠⢀⣄⠀⠀⢠⡀⠀⣀⣤⠾⠇⠀⠀⠀⣼⢿⣧⡶⠛⢲⡎⣉⢻⣋⡝⣷
⠀⠀⠀⣀⣛⣠⡽⠺⢏⠰⣌⣯⣽⠩⣽⣄⠀⠀⠀⠀⠀⢸⣿⠄⠀⢨⣿⣞⠫⠁⠀⠀⠀⢠⣾⠇⢄⣸⣔⣃⡼⠧⣭⠞⠛⢿⡩⠆
⠀⠀⢸⣯⣋⠙⣇⣃⠼⠓⢿⡔⢮⡟⠛⣾⠀⠀⠀⠤⠀⣿⠟⠃⠀⠀⣀⣀⣀⠀⠀⠀⣠⣿⠟⠻⣿⢡⠈⡿⠠⠀⣿⣀⣁⣼⠿⡇
⠀⠀⣾⢹⣯⡴⠛⠻⣄⢓⡼⢛⡙⣷⣷⣏⡆⠀⠀⠀⠀⠀⠀⢀⡄⠉⠀⠀⠀⠈⠁⠐⠿⣿⣮⣴⠿⣦⡶⠛⢲⡞⢉⣿⣿⣷⣷⠇
⠀⠀⢸⡯⣄⢻⣙⣰⠟⠉⢳⣬⣵⣏⢠⣿⡃⠀⠀⠀⠀⠀⠀⢸⣱⡆⣀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣿⡐⠃⣸⣯⣿⠃⠻⣿⣿⠂
⠀⠀⠘⠾⢶⣞⣉⠛⣦⣩⣾⠁⢶⣸⠖⢿⠁⠀⠀⠀⠀⢀⠴⢶⣿⣙⣿⠖⠂⢀⣀⡀⡀⠀⠀⠀⠈⣿⡟⠻⠃⣠⡟⠀⠀⠹⣮
⠀⠀⠀⠐⢒⣹⠶⣶⣏⡤⢹⠿⠺⣷⣬⠋⠀⠀⠀⠀⢠⡇⠀⠀⠀⠉⠻⠷⢯⣼⣇⣤⣹⣗⢲⢲⡞⣻⣿⡾⢳⣾⠇⠀⠀⢀⠹⣷⣄
⠀⠀⠀⠀⠀⠙⠲⣤⣿⣟⣻⣬⠽⠛⠁⠀⠀⠀⠀⠀⠈⢧⣽⣧⣴⣤⠤⡀⢀⡀⠉⢻⣿⣿⣿⠛⠿⡿⠿⣇⣿⡟⢀⠀⠀⠈⠄⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠛⠚⠓⠺⠛⠛⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠄⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡀⠈⠀⠀⠀⡠⠐⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠄⠂⠀⠈⢁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠔⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡒⠂⠀⠅⠀⠁⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠑⠄⠀⠀⠁⠀⠀⠀⡠⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠰⠀⠀⠀⢄⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⢠⠊⠀⠈⣢⠂⠀⢢⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠁⠐⢀⠔⠀⠑⡄⠀⠐⢄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠊⠢⢀⠔⠈⢀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⠘⠣⠀⡠⠛⠄⠀⡀⠃⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⢆⠀⠀⠜⠄⠀⢈⠆⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢄⠀⠀⠱⡀⠀⠈⡢⠀⠀⠰⠆⠀⢀⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠢⡀⠀⠀⢀⠀⠀⠀⡔⠀⠀⡠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⣀⠈⠀⡀⠀⢀⠀⠤⠐⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠰⠀⠀⠾⠁⠀⠀⠀⠀⣠⠆⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠀⠀⠐⠀⠀⢾⡆⠀⠀⠀⢀⠞⠁⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠘⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠁⠀⣠⠜⠁⠀⠀⢀⡈
⠐⠛⠛⠛⠉⡀⠐⠀⠀⠀⠀⠀⠈⢿⣦⠀⠻⣧⡄⠀⠀⠀⠀⠀⡐⢂⣤⠃⠀⠀⠀⢠⡄⠀⢀⣿⡇⠀⠈⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠀⡀⠀⠈⢿⣧⡀⢿⣇⣀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠘⡇⠀⢸⣿⣴⠟⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠈⢻⣷⣄⣿⡋⣴⡄⠀⡄⠀⠀⠀⠀⠙⠀⠀⠀⢳⡄⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠰⣿⣿⣿⣿⣀⠶⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣏⡀⠀⠀⠀⠀⣠⣠⣶⣿⡿⠁⠀⠀⠀⠀⣀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡴⢀⠈⠛⢿⣿⣿⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠈⠁⠀⠀⠀⣾⣿⣿⣿⠏⠀⠀⠀⢀⠀⠂⠈
⠀⠀⠀⠀⠛⠛⠛⠀⠀⠄⠀⠀⠀⠀⠀⠙⠦⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣼⠿⣀⠄⠀⠀⢀⣼⣿⣿⡿⠋⢀⠄⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠆⡀⠀⠐⢯⣤⡀⠀⠙⣿⡇⠀⠀⢀⠀⠀⠀⠀⢸⣿⡴⠋⠀⣀⡰⠾⣿⣿⡟⠁⠖⠁⠀⢀⡀⠊⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠁⠀⢀⠀⠀⠀⠀⢈⣤⠻⢆⠀⠹⣷⠀⠀⠊⠀⠀⠀⠀⠸⣿⣷⡿⠛⠉⠀⠀⣼⡟⠀⠀⠀⡀⠶⠉
⠀⠀⠀⠀⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣤⠀⠀⠈⠻⣶⣉⠶⢤⣿⡀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠀⠀⠀⠀⠀⣿⠏⠀⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡿⠷⢶⣤⣄⡈⠻⣶⡌⢿⣿⠀⠀⣠⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⠐⢀⣠⠖⠙⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⠉⠉⠓⢦⡀⠉⠙⠿⢦⣬⣿⣾⣿⣷⠀⠘⠀⠀⡿⠀⠀⠀⠲⠀⠀⢀⣶⣿⣤⠿⠛⠻⠷⠶⠂⣠⣴⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠿⠛⠻⢿⣿⣿⣶⣦⣤⣄⠙⣦⠀⠀⠀⠀⠉⠛⣿⣿⣧⠀⢰⣶⠀⠀⠀⠘⠃⠀⢀⣼⡿⠋⠁⠀⠀⣤⣤⣶⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣦⢀⠘⣉⠁⠀⠘⠿⢿⣾⣿⣁⠤⠀⢰⣤⣶⠟⠉⠁⠀⠀⢠⣾⠟⠋⢀⠀⠀⠚⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣤⣬⡿⢿⣿⣿⣿⣿⡤⠉⢿⣦⡀⠀⠀⠉⠻⣿⣷⣾⣿⣯⣹⣿⠁⠀⠀⣠⠟⢁⣤⠦⣀⡀⠒⠦⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠻⠿⣿⣈⠛⠻⢿⣿⣧⣜⣼⢻⣷⣷⣀⠀⢀⡼⠛⠛⠛⠿⠛⠛⠆⠀⣰⡦⠔⠛⢁⣤⠼⠋⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⡈⠛⢿⣦⣄⠈⠉⠛⠿⣿⣿⣿⣿⣶⣟⠀⠀⠀⠀⠀⠀⢠⣤⣎⣁⣤⣤⣟⣀⣀⣤⣤⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣿⣷⣦⣽⣿⣦⣄⠐⠀⠀⠉⠉⠻⢿⣤⡀⠀⢘⣦⡴⠿⠛⠉⠈⢉⣭⣿⣿⣿⡿⠿⣏⣉⣁⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠐⠒⠀⠉⠉⠙⠻⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢻⣿⣶⡟⠁⠀⠀⠀⠀⣰⣿⡿⠟⠛⢛⠿⠿⠛⠋⠁⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢊⣁⠀⠀⠀⠈⠻⣿⣧⣤⡀⢀⡴⠀⠀⠛⠋⠛⢦⣀⣤⣿⠿⢋⡁⣤⣶⣾⣿⣤⣤⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⠿⣿⣿⣿⣶⣄⣀⠀⠀⠈⠙⣿⣿⢃⣤⡄⠀⠀⢀⣀⡿⠏⠁⠶⢿⣧⠉⠻⠟⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁⠀⢈⡛⠁⠶⢦⣤⣤⠈⠉⠀⠀⠀⢀⣼⠟⠉⠁⣀⠀⠠⣿⡟⢙⣄⠈⣲⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣦⣀⠸⠿⠿⢷⡀⠀⢠⡶⠋⠉⠀⠉⠉⢻⣅⣴⣾⡿⠋⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣤⣴⣷⣶⡄⠀⠈⣻⣿⣯⡄⠀⠀⠀⠠⠔⢿⣿⣿⣿⣷⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⢉⡉⠛⣷⠾⣿⠄⣸⠿⠃⢀⣶⣦⣤⡾⠿⠟⠩⣬⠉⠿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠷⢠⡌⠉⠻⣄⠀⠀⠲⢀⣤⠆⠉⠀⠰⣄⡀⠠⠆⠀⢰⠄⠀⠰⠞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡄⠀⠀⠀⠀⠦⠀⣭⠀⠂⠺⢇⠀⠀⠀⠀⠐⠈⢠⣷⠒⠒⠛⣤⡀⠀⡤⢠⡿⢷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠀⠀⠈⣦⠀⠀⠰⠀⣀⠀⠀⠀⠀⠈⠛⣆⣀⠀⣀⣀⠿⠃⠀⠀⠀⠠⠾⣦⠤⠄⣴⣯⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠐⠃⠠⠀⠻⡶⠖⠶⢾⠉⠁⠀⠀⠈⠉⠀⣨⡟⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⡧⠀⠀⢀⠈⣴⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠉⠀⠐⠀⣠⠞⠁⠆⠀⠈⠳⠀⠀⠀⠀⢠⣾⠋⠀⠀⠈⠀⡀⠉⣤⣤⡄⠀⣤⣄⠀⠀⢀⡀⠉⠀⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠘⠇⢀⣠⠉⠀⡀⠀⠀⠂⠀⠙⡻⠛⠛⢻⣤⡀⠀⠀⢠⠀⠀⠀⡦⠀⠀⢀⣤⠈⠛⣆⣨⣁⣀⣶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⢁⡀⢂⡈⠉⠶⣄⠉⠙⠀⠀⣠⠊⠁⠀⠼⠁⠀⠹⣦⣀⣄⣀⣠⣴⠀⠀⠀⠈⠀⠀⠀⣙⠋⠉⠙⠃⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠁⡄⠁⠀⠀⠙⣷⣤⣤⠿⠏⠀⠀⠀⠀⢐⠓⠀⣸⠏⠀⠀⡉⠙⠲⣄⠀⠀⠳⠀⠀⡟⠀⠠⠆⢠⡿⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⠘⠛⡀⠀⣀⠀⢀⣠⡏⠁⠀⠁⢶⣷⡄⠀⠋⠀⣈⠁⠀⠀⠀⠀⠀⠀⠸⣷⣶⡦⠚⢻⣄⠀⢀⡤⢸⣇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣶⠀⠀⠀⠈⠀⡻⠃⠀⠀⠀⠀⠀⠸⠷⢴⠾⣿⣉⡀⠀⢀⣀⣀⡄⠀⢀⣼⡅⠀⢀⣄⠙⢶⡈⣠⣼⡏⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡖⠀⡐⠀⠀⠳⡀⠀⠀⢠⣴⠀⢀⣴⠇⠀⠈⠉⠙⠣⣼⡁⠀⢀⣀⣼⠏⠀⠀⠘⠁⠀⢀⡿⠋⢹⣷⡀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠱⡌⠛⠦⠀⠀⢹⣆⠀⠀⢀⣠⡾⠁⠀⠀⠀⠀⠠⠀⠘⢃⠞⠛⠛⣤⣄⠀⠀⠀⠀⠀⢸⠃⢀⠟⢸⡿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠹⣦⠀⠀⠀⠀⠀⠀⢉⠙⢃⠉⠛⢧⣀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠄⠉⠑⣶⠀⢀⡴⠞⠀⠈⣀⢠⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠾⠀⣸⠷⣀⠀⠀⠀⢾⡏⠀⠘⠇⠀⠀⠈⢀⣀⠀⠀⢀⣄⡸⠀⢀⠀⠄⠀⠀⠠⣀⡷⠊⠙⢦⡀⠀⢋⣼⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⢠⠀⣰⠋⠀⠈⠣⠀⢠⠾⠁⠀⠀⠀⠀⠀⠀⠙⡿⠋⠱⣏⠙⠷⣄⠀⠀⣀⠀⠀⢠⣿⠇⠀⠀⠀⠹⣶⡿⠉⠄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠸⣶⠋⠀⠁⠀⠀⠛⡁⠤⠄⣀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⠄⠀⠈⢷⠀⠁⠀⢀⣸⠃⠀⠀⠀⠀⢀⠋⠀⠳⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢀⣀⣤⠀⢠⡔⠁⠀⣠⡍⠹⣆⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠐⠚⠛⢰⠿⠋⢡⡀⠀⢀⣀⠀⣽⡇⠐⠇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢿⠀⠀⠀⣻⠀⠀⠀⠙⢀⡀⠈⠛⣶⠦⠀⣀⠀⠀⠀⠀⠀⠀⢠⣶⠋⠀⠀⠀⠘⣶⣈⣁⣠⡿⠃⠀⠄⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠱⡆⠺⢏⡀⠀⠀⠀⠘⠁⢀⠀⠀⢠⡄⠉⠳⢦⣀⠀⢀⡀⠹⠋⠀⠀⠀⠀⠀⠘⢻⠟⠙⠷⣄⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣦⢀⣤⠀⢀⡈⠉⠲⣄⠀⠘⠳⠁⠀⠀⠈⠡⠀⠀⠈⠛⣴⡞⠛⠀⠀⠀⠀⠀⠀⢀⣤⡟⢠⡄⠀⣿⣦⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⡏⠀⠘⠣⣤⠀⠹⡆⢠⣞⠀⠀⠀⠀⠀⠀⡈⠁⢀⡼⠟⠀⢤⣤⡄⠀⠠⠀⣿⠉⡁⠀⠙⢃⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠌⠁⠀⠀⠀⠀⡤⠀⠀⡀⠞⢉⣤⣀⠀⢀⠀⠉⣴⠋⠀⠰⣦⠈⠁⠘⢋⣤⡖⠉⠀⠛⠄⢀⣾⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠁⣤⡀⢠⡄⠀⣼⠁⠀⠀⠀⠀⠐⢤⣴⣤⠉⠀⢠⡜⠀⠀⠃⠀⠈⢉⡿⠷⠶⢶⣤⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣤⡙⠁⠦⢴⡇⠀⠀⠀⠀⠃⠀⠈⣁⠀⠀⢀⣀⠉⠲⠄⠀⣀⡴⠎⠀⠠⡄⢀⣨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣄⠐⣆⣀⠀⠀⠀⠀⣈⠻⠟⠉⢉⠉⠱⠦⣤⠖⠉⠀⠀⣫⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠻⢿⣶⣤⣼⣃⣀⣰⣤⣀⣱⣤⣤⣿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠉⠉
⠀⠀⣤⠄⢤⠀⢠⡄⠀⠀⠤⠄⠀⠠⠄⠀⢤⠄⠀⡤⠀⠠⡄⠄⠠⡆⠀⠀⠀⢤⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⡀

⢱⡄⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠑⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠁⠀⠀⠹⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠸⣄⠀⠀⠱⡀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠉⠢⣄
⠀⠸⣇⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠙⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠡⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠘⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠈⢳⡀⠀⢧⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠂⠀⠀⢒⠆⠙⣦
⡅⠀⠀⠀⠀⠀⠻⣄⠈⣆⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠉⠀⠈⠀⠀⠀⠀⠀⠂
⠘⢦⠀⠀⠀⠀⠀⠹⣆⠘⡄⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠊⠁⠀⠀⠔
⠀⠀⠳⡀⠀⠀⠀⠀⠘⢆⠹⠀⠀⡂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠿⣷⣤⡇⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠠⠖
⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀
⠐⠀⠀⢀⠀⠀⠈⢧⡀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠁⠠⠌⢷⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⢀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠘⢿⡌⣿⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠪⣷⡟⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⢀⠞⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠠⡀⠀⠀⠀⠀⠈⠠⡀⠀⢹⡇⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⢸⡇⠐⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣟⠉⣿⣻⣛⣛⣛⣟⣛⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣻⣟⣛⣛⣻