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⠀⠀⢀⠀⠀⠲⢄⠀⠀⢀⢀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠴⠂⠀⠀⠀⢀⠀⢀⠀⠀⠀⠠⢴⣿⡦⠄⠀⠀⢀⡤⣀⠀⠄⠀⠀⠀⢤⡈⡣
⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠁⠼⠧⠐⠀⠀⠀⡠⠊⠦⡀⠀⠀⠀⢬⣉⡤⡄⠀⠀⠀⠨⠀⡍⠀⠀⠀⠀⠸⠭⠁⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠚⢽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠈⡤⠆⠀⠀⠀⢠⡀⣀⣐⡠⠀⠀⠘⠀⡼⠛⠋⠀⠀⠀⡤⡻⠣⠀⠀⠀⠐⢇⢐⢀⡀⠀⠀⠈⠀⠀⠦⡀⠀⠀⠠⠀⣀⣹⡄⠀⠀⠀⡁⠠⠗⢄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠃⠀⠀⠀⠀⠈⠉⠚⠃⠀⠀⠀⠁⠀⠐⠇⠀⠀⠀⠘⠀⠐⠀⠀⠀⠘⠊⠁⠈⠉

⠀⠀⠀⢀⣤⠤⠐⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠓⣶⠖⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⡠⠾⠀⠀⠀⠀⠖⠀⠒⠀⠀⠀⢀⡀⣰⡄
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠄⠐⠠⡄⠀⠀⠀⢀⠐⠉⡁⠀⠀⠀⣀⣯⣄⡀⠀⠀⠀⠀⡐⠈⠂⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⢀⣁⠂⠀⠀⠀⠈⣀⣡⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⡁⠀⢘⠀⠀⠀⠀⢦⠀⡵⠂⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⡴⠷⢶⠄⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠒⡔⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠋⠚⠊⠁⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⢤⡤⣄⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠒⣤⠂⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⣤⠒⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠠⠷⡄
⠀⠀⠀⢀⡀⠋⠄⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⢈⠋⠁⠀⠀⠀⠀⠸⠀⠆⠀⠀⠀⠀⢀⣵⡤⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠚⣒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡦⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣤⡰⠤⡁⠤
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⡀⠂⠀⠀⠀⠐⢀⡂⡀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠠⢺⠠⠀⠀⠀⠐⠹⠯⠌⠃⠀⠀⠀⠸⠄⠔⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠠⣀⠔⠁⠀⠀⠈⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠒⠖⠀⠀⠀⠀⠀⠊⠀⠆⠀⠀⠐⢠⡉⡅⠋⠀⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠰⠈⠈⢁⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠂⠉⠲⠀⠀⠀⠀⠈⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣠⣖⠀⢀⣆⠀⠀⣠⣴⠂⠀⠀⣠⣷⢠⣀⣀⠄⠀⣀⣌⠀⢠⣾⣷⣆⢠⡦⢿⡆⠠⡾⢣⡄⠀⠀⠆⠀⣦⠘⣧⠀⣼⢶
⠉⠁⠉⠀⠀⠋⠑⠀⠿⠇⠀⠀⠀⠛⠃⠀⠋⠙⠀⠈⠹⠅⠀⠀⠻⠟⠋⠘⠛⠛⠃⠙⠒⠛⠀⠀⠀⠀⠀⠛⠲⠋⠀⠙⠟
⣶⣀⣤⠀⢤⣄⠀⠀⢀⡀⡆⢠⣆⣤⡄⠀⠀⡴⠀⠀⢀⡴⠂⠀⣤⠶⣄⢠⡶⢶⡀⢠⣶⣦⠀⣠⡶⢤⡀⣀⣴⣄⠀⣰⡶
⠈⠋⠉⠀⠚⠉⠛⠀⠛⠿⠓⠘⠛⠛⠂⠀⠜⠁⠀⠀⠘⠁⠀⠀⠻⠺⠃⠘⠛⠿⠁⠈⠛⠛⠀⠸⠔⠟⠀⠻⠷⠿⠀⠻⠷
⢴⣶⣦⠀⢶⣤⡶⢠⣤⣀⣤⢠⣄⣀⡄⢠⣄⣠⡄⢤⣀⣤⠆⠀⡀⣂⢀⢠⣴⢶⡀⢠⣤⢶⡀⠀⢦⣴⠀⠐⣦⠆⠀⠲⣤
⠀⠩⠿⠀⠛⠉⠛⠀⠿⠻⠗⠀⠛⠹⠃⠈⠟⠻⠃⠺⠋⠻⠀⠀⠊⠚⠁⠸⠧⡾⠃⠸⠷⠾⠁⠀⠼⠿⠀⠠⠿⡄⠀⠲⠿
⢠⣄⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⢠⣄⠀⠀⢠⡄⠀⠀⣠⣀⠀⠀⢠⣤⡄⠀⢤⣤⠄⠀⣄⡤⠀⠠⣤⡤⠀⢤⣤⡤⠀⡀⣀
⠘⠿⠟⠃⠘⠛⠀⠠⠘⣫⠀⠀⠹⡏⠁⠈⢽⠏⠁⠈⢹⠍⠀⠰⣺⣿⡇⠰⣾⣿⠀⠠⣼⡇⠀⠢⣿⣷⠀⢾⣿⡇⠈⠉⠏
⠀⢀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⢀⡀⠀⢀⢀⣀⡀⠀⣀⣄⠀⢠⣀⣀⠀⡠⠄⢀⠀⣀⠀⠀⠀⣀⣄
⠐⢺⡗⠀⢸⣾⣾⠀⣿⣿⣿⠀⡏⠦⣹⢀⣟⣽⣇⢸⣯⢝⡇⠸⢿⣿⠟⠰⢿⡿⠆⠸⣝⣮⠆⠙⠾⡯⠀⠙⠿⢦⠈⣹⣿
⠀⠈⠀⠀⠀⠈⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠈⠋⠀⠀⠉⠉⠀⢀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢰⡿⣿⡆⢿⣿⣿⠇⢿⣷⡿⠐⣏⠉⡽⠀⠀⢀⡷⢈⠉⠉⡗⠸⣧⣄⡀⠸⣿⡟⠁⠾⢶⡼⠁⢶⡶⠤⠄⢾⣿⣟⢰⣿⡿
⠀⠙⠉⠀⠀⠙⠁⠀⠀⠉⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡀⠀⠀⠀⢀⠀⠈⢁⡈
⢴⣷⣾⡆⢠⣼⣤⠀⠤⠐⠠⠀⣰⣶⡆⠀⡔⠒⡄⢠⣶⣶⡄⢰⠿⠟⠆⠀⠲⡦⡄⠘⢺⠿⡆⢴⠶⡿⠀⡶⣿⠉⠈⠷⢶
⠀⠙⠉⠀⠀⠋⠓⠀⠐⠀⠂⠀⠈⠉⠁⠀⠈⠋⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠂⠄⠀⠦⠰⠄
⣀⣂⣀⣀⣙⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣿⣽⣽⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⡆⢀⠀⠀⠀⠀⠀⢠⡄⠠⣄⠀⠀⠠⠐⡆⠀⠀⢀⠄⠀⢰⠄⠀⠀⠀⠀⠀⠠⡆⠈⢦⡀⠀⠀⠀⣷⠀⠀⠀⠀⡠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡗⠁⠀⠀⠀⠀⠀⠠⡆⢀⡞⠀⠀⡇⠠⠁⠀⢠⡏⠀⠐⠈⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢱⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠂⢸⡇⢀⠀⠀⠀⠀⠀⠐⡗⠊⠉⣷⠀⡟⠀⡀⠀⢸⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⡏⠀⠀⡤⠊⠙⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⡇⣎⠀⣴⡄⠀⠀⠈⡇⠀⣠⠟⠀⣇⠀⡅⠀⠸⣧⡀⠀⡼⢲⡇⢀⠀⠸⣧⢸⡇⠀⣠⡞⢣⠀⠰⣇⠀⠈⢷⡀⠀⡰⠞⣇
⠀⠀⠀⠀⢰⡆⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁⠀⠀⠀⠀⠹⠄⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠉⠀⢘⡇⠀⢀⡿⠀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⣀⠀⠀⠀⡾⠀⠀⡇⠀⢿⡌⠀⠀⠀⠀⣾⠁⠀⠈⣿⠀⠀⠀⠀⢠⠎⠀⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣀⠀⠀⢰⡇⠀⠀⢰⡇⠀⠀⠀⠀⢻⠀⠀⡿⠀⠐⡗⠁⣾⠀⡄⠀⠀⠀⢻⡄⠀⠀⣿⠀⡄⠀⣰⠇⠀⠀⣏⠀⠠⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠩⡆⠀⢸⣆⠀⠀⣾⡃⡀⢀⡀⡰⠃⠀⠀⡇⠀⠀⡇⠀⣿⠀⣿⠀⠀⠀⠀⠛⠦⣰⠇⢰⠂⠀⢿⡀⠔⢠⡟⠀⣶⠀⠠⠋
⠀⠀⠀⠀⢀⣤⠀⠀⣼⠃⠀⢸⢰⡇⠀⠀⠉⠃⣠⡿⠀⠃⣼⠋⠀⠀⠀⠀⠃⠀⠀⡇⠀⠙⠀⣿⠀⠀⠐⠀⠀⠘⠁⠀⢸⠀⠀⠀⢀⠖⢹⠇⡀⣾⠁
⠀⠀⠀⠀⠘⠳⠀⠚⠁⠀⠀⠸⠞⠁⠀⠀⠀⠘⠟⠃⠀⠻⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠳⠛⠀⠿⠘
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢻⡇⣰⠁⠅⢸⠀⠀⠀⠀⠀⠀⢠⠀⠰⠀⠀⠐⠁⠀⠀⠀⢠⠃⡇⠀⡤⣀⠀⠀⠀⠀⠀⠀⣽⠀⠐⣆⠀⠀⠀⠀⠀⡐⠀⠀⠶⣶
⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⣾⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⠀⡇⢸⡀⣿⠀⠀⠀⠀⠀⠀⣽⠀⠀⣿⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡏⠐⡆⠀⢻⠺⡄⠀⠀⠀⠀⣸⡄⠀⠀⢸⡇⠀⠀⠀⠀⣻⠀⢿⡀⠃⡟⠀⡄⢲⡤⠀⠀⣿⠀⠀⣿⠀⡄⠀⡄⢸⡄⠀⠀⠀⢸⡇⠀⢠⡄
⠀⠀⠀⠀⠀⢸⠇⢀⡇⠀⢸⠀⣿⠀⠀⣧⡀⠘⠓⠦⡀⠸⣇⠀⠀⠀⠀⢸⡄⠀⠀⠐⡇⠀⡇⠸⢀⡇⠀⢿⠀⠀⣿⠀⠇⠀⡇⠀⠳⣄⡀⠀⠎⠙⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠄⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⡾⠀⠀⠀⠀⠻⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⠀⠀⠀⡧

⠀⠀⠀⠀⠀⠀⢴⠂⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⢰⠄⠀⠀⢀⡤⠀⠲⠀⠀⠀⠀⠀⢀⠠⠤⠀⣄⠠⠄⢰⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢈⠀⠀⢨⡇⣠⠀⣾⠀⠀⠀⢿⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⡇⠐⠂
⠀⠀⠀⠀⠀⠰⣯⡀⢀⡴⠋⡀⢀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠤⣬⡄⢠⡞⠀⠓⠈⣇⠀⠀⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⢸⡁⢸⡇⠀⢠⡄⠠⢄⣠⠃⠀⡆⢠⡄⠀⠀⢸⠀⠀⢹⠀⠀⠀⣯⠀⢺⡅⠀⠀⠀⣼⠃⠀⠐⠡⡀⠀⠀⠀⠀⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠋⠉⠀⠀⠀⠉⠁⠈⠉⠉⠀⠉⢹⡇⠀⠀⠈⠀⠀⣻⠀⠀⠀⢿⡀⠀⠉⠉⠀⠉⠀⣀⠀⠀⠀⣽⠂⣄⠀⠀⡏⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠉⠂⠀⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢴⠁⡸⠁⠘⢷⡀⠀⠀⠀⠀⠀⣴⠁⣼⠀⠀⠀⠀⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⣷⠀⠀⢀⡄⡦⠀⠀⡿⠀⠀⢸⡇⠀⠈⣷⠀⠀⠀⠀⢠⠴⠄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡇⠀⠀⢀⠀⠀⠸⣇⠀⠀⠀⣿⠀⡆⠘⡇⠀⠀⠀⣇⠀⠀⢸⡇⠀⠀⡿⠀⡖⠀⣄⢸⡆⠀⠀⠀⠀⠀⡼⣇⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠈⠳⠀⠚⢧⠐⠆⠛⠀⠀⠀⠙⠦⠀⠘⠁⠀⠳⠄⠃⠀⠀⠀⠙⠆⠀⠚⠛⠤⠞⠁⠀⠻⠔⠻⠜⠃⠀⣀⠀⠀⣴⠃⠹⣄⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠁⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⠁⣾⠀⢠⡆⠀⠀⠀⠀⢶⡀⠀⠀⠀⣸⠇⠀⠀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡇⠀⡀⠀⠀⢸⠇⠀⠀⣰⡋⠀⠀⠀⢀⡀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⡇⠀⣅⣰⡶⠋⠀⠀⠴⠃⣹⠇⠀⠀⣼⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡎⡽⠁⢠⡎⣹⠀⠀⠀⠀⢠⣞⡟⠀⠀⣞⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⣤⣤⠀⠀⣠⠖⣉⠀⠀⢀⡶⠀⣴⠃⠀⢀⣾⣄⢀⣠
⠀⠀⠀⠀⠀⠀⡀⡰⠀⠀⣿⡇⣉⡃⠀⢀⡏⠀⣻⡁⠀⢸⣇⠀⠘⠀⠀⠈⢳⣿⠘⠁⠀⠀⠄⢀
⠀⠀⠀⠀⢀⡸⠀⠉⠀⠀⣿⡇⡟⠀⠀⠈⣧⠀⠞⠀⠀⢸⢟⠀⣧⣀⠀⠀⢸⣿⢀⠀⠀⠀⠈⠈⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠙⠓⠀⠀⠀⠘⠛⠋⠛⠃⠀⠀⠘⠛⠃
⠀⠀⡄⠀⢠⠢⠀⡠⠠⠀⡄⡀⠀⡆⡄⠀⠢⠄⠠⢠⠀⠀⠆⠀⢰⠀⠀⢀⠂⢰⠠⠀⢰⠀⠀⡄⠂⠀⡆⠆
⠀⠐⠎⠦⠸⠬⠀⠙⠄⠀⠷⠂⠀⠳⠄⠀⠆⠀⠰⠌⠀⠲⠹⠀⠸⠀⠀⠮⠀⠘⠒⠂⠘⠆⠀⠇⠳⠀⠗⠂
⠀⢀⠀⠀⠀⢀⠀⠀⡀⡀⠀⣀⡀⠀⠀⢀⠀⢀⡀⠀⠀⢀⡀⠀⡀⡀⠀⠀⣀⠀⢀⢀⠀⠀⡀⠀⠀⢀⡀
⠀⢨⠸⠀⠀⡆⠀⠀⡄⠁⠀⣇⢤⠀⢈⣂⠀⠀⣇⠀⠈⣜⠃⠀⢣⠁⠀⢣⣉⠀⢸⢣⠀⢀⡇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⢠⠠⠀⢸⣐⠀⣄⠆⢠⣨⡀⢀⠴⠀⣏⠇⢰⠀⠀⢈⡀⢀⡇⢸⠄⢸⢀⠀⣨⠀⠀⣆⣄⣄⠀⠀⠀⢀⠆
⠀⠀⠁⠁⠀⠁⠀⠈⠀⠀⠈⠁⠈⠁⠀⠉⠀⠈⠀⠀⠈⠁⠀⠁⠈⠀⠈⠉⠀⠈⠀⠀⠉⠉⠁⠀⠈⠀⠀⠁
⠀⢠⣄⣄⢠⡰⠀⢸⡠⠀⠐⣆⠀⣃⠂⠀⡦⠀⠐⣆⠂⢆⢂⠀⢡⡂⠀⡆⡀⠀⡎⠀⢘⠂⠀⢰⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⡀
⠀⢀⠇⠀⢳⠀⣖⡀⢸⠂⠸⣤⠀⣟⠁⢄⠆⢸⡃⠠⡃⠀⡵⠀⢠⣀⠀⣄⡀⢦⠀⢸⡄⠀⡇⠠⢄⠀⡙⡁

⠀⠀⡀

⠀⢠⡄⠀⣤⠀⠀⠀⠀⣠⣤⡄⠰⠶⠀⢀⠀⢀⠀⢀⠀⢀⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠹⠹⠀⠈⠀⠀⠄⠁⠰⠁⠰⠖⠁⠀⠀⠀⠀⠸⠿⠀⠞⠀⠿⠀⠐⠀⠐⠂⠀⠂⠀⠠⠀⠂⠀⠞⠀⠲⠖⠐⠶⠂⠀⠂⠀⠀⠐⠀⠀⠐
⠀⠀⠀⢠⣤⠀⢀⣀⠀⡄⠀⠀⠀⡄⠀⣤⠀⣄⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⢀⠀⢠⣤⠀⣀⠀⣤⠀⠀⠀⠀⠀⡤⡄⠠⠀⠀⡀⢀⡀⢀⡄⠀⣠⠀⡠⠀⣤
⠀⠈⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠈⠀⠐⠀⠲⠀⠀⠀⠇⠰⠁⠘⠀⠀⠀⠘⠀⠪⠀⠀⠠⠀⠸⠇⠀⠚⠀⠐⠂⠀⠐⠂⠀⠐⠀⠨⠒⠀⠐⠂⠀⠐⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠒⠒
⠀⢀⠀⢀⣄⢀⣤⠀⡠⡀⢀⡄⠀⣀⠀⡀⠀⡀⢠⡄⢀⡀⠀⠀⢀⣄⠀⣀⡀⢀⢀⢀⣤⡀⣠⡀⢀⣄⠀⠀⠀⠀⠀⠀⡤⡄⠀⡀⠀⢀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠁⠀⠉⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠉⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠈⠀⠀⠁⠀⠈
⠀⠰⠆⠀⠨⠆⠐⠖⠐⠿⠀⠐⠀⠐⠂⠀⠲⠀⠐⠂⠐⠗⠀⠾⠂⠸⠇⠀⠷⠀⠐⠂⠐⠂⠐⠗⠀⠿⠀⠺⠆⠰⠿⠀⠿⠇⠰⠇⠀⠖⠀⠺⠂⠐⠗⠀⠒
⠀⣠⡄⢀⡀⠀⣄⠀⣠⠀⣀⡀⢀⡀⢠⡄⢤⠄⠀⠀⠀⠀⠀⢠⠀⢀⣠⠀⢀⠀⠀⢠⠄⢄⠠⠀⠄⠀⢀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠄
⠀⠈⠁⠈⠁⠈⠉⠀⠉⠀⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⢶⡆⠀⣸⡇⠀⣸⡂⠀⢿⡆⠀⣼⡀⠀⠀⠀⠰⣿⡇⠰⠆⠼⠆⠰⠀⠻⠂⠻⠟⠨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠀⡀⡀⠀⣀⡀⣀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⠀⠀⠀⠀⣠⡀

⠀⠀⠀⢀⣶⡆⠀⠀⠀⢰⡶⠶⣦⠀⠀⠀⣠⠶⠶⣦⠀⠀⠀⣶⠶⣦⡀⠀⠀⢰⡶⠶⠖⠀⠀⠀⢰⡶⠶⠆⠀⠀⠀⣴⠶⢶⡄⠀⠀⢰⡆⠀⣴⡆
⠀⠀⠀⣼⣧⣽⡆⠀⠀⢠⡿⠛⣿⠀⠀⠀⣼⡇⠀⣀⠀⠀⠀⣽⠂⣨⡗⠀⠀⢀⣿⠛⠛⠀⠀⠀⢨⡿⠛⠃⠀⠀⠀⣿⠀⢶⡶⠀⠀⢨⡟⠛⢿⠇
⠀⠀⠘⠋⠀⠈⠃⠀⠀⠘⠛⠛⠋⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠛⠛⠋⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠘⠃⠀⠙⠁
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⣠⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⣤⡀⠀⣠⣄⠀⠀⣤⡀⠀⣤⠀⠀⠀⣠⣤⣄⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⣻⠄⠀⠀⢻⣾⣏⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣻⢳⣄⡿⣿⠀⠀⢻⡿⣦⣻⠄⠀⠀⣿⠀⠘⣧⠀⠀⠘⣧⣤⣿⠄
⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠶⠶⠟⠀⠀⠀⠿⠁⠻⠆⠀⠀⠀⠿⠶⠶⠀⠀⠀⠿⠘⠿⠰⠿⠀⠀⠿⠀⠹⠿⠀⠀⠈⠻⠶⠾⠃⠀⠀⠸⠇
⠀⠀⠀⢀⣠⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⡀⠀⣀⠀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⢀⠀⢀⠀⠀⢀⡀⠀⢀
⠀⠀⠰⣿⠁⠹⣇⠀⠀⠸⣏⠙⣷⠀⠀⠀⠸⣏⣙⠃⠀⠀⠈⠉⢿⡋⠁⠀⠀⠸⣇⠀⢿⡀⠀⠀⠻⣆⢰⡟⠀⠀⢻⡄⣿⣆⢾⡁⠀⠈⠻⣴⡏
⠀⠀⠀⠿⣤⣴⠋⠀⠀⢰⡏⠻⣏⠀⠀⠀⢤⣌⣹⡿⠀⠀⠀⠀⣾⠃⠀⠀⠀⠰⣯⣀⡾⠁⠀⠀⠀⢿⡾⠁⠀⠀⢰⣿⠋⣿⡟⠁⠀⢀⣴⠻⣇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠛⢿⣧⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠛⠻⣿⡟⠛⠛⠀⢀⣾⡿⠛⠿⣷⡄⠀⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢻⣷⡄⠀⣀⣀⣀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⢻⣧⠀⠀⠀⠘⣿⡆⠀⠀⠈⠁⠀⠀⠘⣿⣦⣤⣤⣿⣿⡄
⠀⠀⠀⠀⠀⣾⣿⠀⠀⢹⣿⡏⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⣾⡏⠁⠀⠀⠀⠐⣿⡏⠀⠀⠀⢰⣿⡁⠀⠀⣤⡄⠀⠀⣰⣿⠉⠉⠉⣿⣿⠁
⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠀⠹⠿⠀⠀⠀⠿⠿

⠀⠀⠀⣄⠀⢠⡄⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠙⣧⡿⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣼⣧⣤⠀⠀⠀⢻⡄⢹⡆⠀⠀⠀⠈⢹⡇
⠀⠀⠀⠀⢿⠁⠀⠀⠀⠠⣾⣧⣤⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠻⣦⠿⠁⠀⠀⠀⠀⢸⠇

⠀⠀⠀⠺⠋⣿⡀⠀⠀⠀⠛⢙⣷⠀⠀⠀⠀⣴⢿⡀⠀⠀⠀⢸⣏⣉⠀⠀⠀⠀⣾⣋⣀⠀⠀⠀⠘⠋⢻⡃⠀⠀⠀⢾⣋⣿⠀⠀⠀⠀⣿⠙⣷
⠀⠀⠀⣠⣶⣋⠀⠀⠀⠀⣄⣉⣿⠀⠀⠀⠰⠿⣾⡇⠀⠀⠀⣤⣁⣼⠃⠀⠀⠠⣿⣁⡿⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⣶⣋⣿⠆⠀⠀⠀⢘⣻⡟⠁
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠴⢶⣆⠀⠀⠀⢠⡾⢷⡄⠀⠀⠀⢀⣴⠶⠀⠀⠀⠰⣿⣉⡓⠀⠀⠀⡖⣦⣠⠀⠀⠀⣠⠞⣩⣉⢳⡀⠀⠀⣠⣆⣆
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⢀⠙⢶⡆⠀⠀⠀⢩⡿⠅⠀⠀⠀⠀⣿⣍⣽⠂⠀⠀⠉⣡⣿⣤⠀⠀⣽⢨⣇⣽⢈⠇⠀⢠⣽⣯⣯
⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠈⠛⠛⠀⠀⠀⠰⢦⣬⠿⠀⠀⠀⠈⠁⠙⠚⠀⠀⠙⠮⣭⡍⠉⠀⠀⠀⠋⠘

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⣀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣤⣤⣀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣤⡀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⣼⠻⣇⠀⠀⢸⣏⣹⠀⠀⠀⣾⠟⠉⠁⠁⠀⠀⣿⡟⢿⣷⠀⠀⣿⣋⠁⠀⢸⣿⠉⠀⠀⢠⡿⠋⠉⠉
⠀⠀⠀⠀⢠⣷⢶⣿⠀⠀⢸⡟⠛⣦⠀⠘⣿⡀⠀⠀⠀⠀⠀⣿⣇⣼⡿⠀⠀⣿⠟⠃⠀⢸⣿⠿⠀⠀⢹⣇⠀⠀⢴⣀
⠀⠀⠀⠀⠼⠟⠘⠿⠇⠀⠸⠷⠴⠋⠀⠀⠘⠿⣶⣶⠦⠀⠀⠿⠿⠛⠁⠀⠀⠿⠿⠂⠀⠸⠿⠀⠀⠀⠀⠙⠷⠶⠞⠁

⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠈⣿⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⣿⡄⠀⣿⣧⣴⠟⠁⠀⠈⣿⠀⠀⠀⠀⣿⣿⡄⢰⣿⣷⠀⠀⣿⣧⠀⣿⡆
⠀⠀⠀⠀⠀⣿⣷⣾⣿⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⣿⣿⣯⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡟⣷⡿⢻⣿⠀⠈⣿⠻⣷⣿⡅
⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⣿⠀⠀⣤⣼⡿⠀⠀⣿⡏⠻⣧⠀⠀⠀⣿⣶⣤⠀⠀⣿⠃⠘⠁⢸⡟⠀⠀⣿⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣰⡾⠿⠿⣷⡄⠀⠀⣿⡏⠛⣦⠀⠀⣰⡾⠟⠻⢿⣆⠀⠀⣿⡏⠙⣷⠀⠀⢠⣾⠛⠃⠀⠐⠻⢿⣿⠟⠂
⠀⠀⠀⠀⢸⣏⠀⠀⠀⢸⣿⠀⠀⣿⡷⠾⠋⠀⠀⣿⠁⠀⠀⢈⣿⠀⠀⣿⣷⡾⠛⠀⠀⠀⠛⢶⣄⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⢿⣦⣤⣤⣾⠇⠀⠀⣿⠀⠀⠀⠀⠀⠻⣷⣤⣾⣾⡋⠀⠀⣻⡟⢿⡆⠀⠀⢠⣤⣴⡿⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⠀⠉⠁⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉

⠀⠀⠀⢸⣷⠀⠀⢸⣦⠀⠀⣶⡆⣰⡦⠀⠀⣶⡄⠀⢀⠀⠀⢠⡖⠀⠀⢰⣦⠀⣼⡏⠀⠀⢺⣦⢠⣶⠂⠀⠰⠶⠿⣿⣷
⠀⠀⠀⢸⣿⠀⠀⣼⡏⠀⠀⢿⣧⣿⠇⠀⠀⢹⣷⢠⣿⣧⢀⣿⠇⠀⠀⠈⣿⣿⣿⠀⠀⠀⠈⢿⣿⠏⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠘⣿⣆⣀⣿⡇⠀⠀⠸⣿⣿⠀⠀⠀⠈⣿⣿⠃⢿⣿⡟⠀⠀⠀⢰⣿⠉⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⢀⣾⣿⣁⡀
⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠚⠁⠀⠘⠓⠀⠀⠀⠘⠛⠀⠀⠀⠘⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠉⠉⠉⠒⢄⣠⠖⠉⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⡜⠀⢀⠀⠀⡀⠀⢣⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠓⠚⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠑⠦⢄⣀⣀⣀⣀⠠⠤⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠤⠔⠊⢁⡄⠀⠉⠑⠒⠠⠤⠄⠒⠉⠀⠀⢀⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⡰⠁⡀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⢠⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡟⠀⠀⠀⢰⠁⠀⡆⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⠎⠀⡇⠀⠀⠀⠈⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠈⠓⠊⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠖⠁⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⢀⠎⠀⡇⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢠⠎⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⠀⠀⠀⠀⠀⠈⠢⠞⠀⠀⠀⡔⠁⢸⠀⠀⠀⠀⠀⠸⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⠞

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣿⣻⣟⣙⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⠀⢀⣀⣀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡀⢷⠀⠀⣿⠠⡇⠠⡏⠀⠈⠀⣾⠀⢹⠄⢸⣴⠈⠀⢸⣥⠙⢰⡏⠀⠈⡀⠀⠀⡀⠀⠀⡆⢀⠀⡄⠤⢀⠀⣲⠀⡀⣄⠀⡏⢠⠀⡤
⠀⠀⠀⠤⠉⠨⠷⠀⠻⠴⠟⠀⠳⠤⠖⠠⠻⠄⠛⠀⠼⠀⠴⠀⠸⠄⠀⠀⠓⠠⠾⠀⠀⠀⠂⠳⠀⠃⠘⠀⠑⠄⠈⠀⠻⠀⠱⠖⠀⠇⠀⠤⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⣶⠀⢰⠀⠀⠆⠀⠠⡆⠀⣔⣠⠖⠀⢠⠆⠀⠀⠶⣆⠀⢶⡆⠀⠶⡀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠆⢠⠀⡀⠀⠆⠀⣀⡀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⣿⠉⢸⡀⢰⡇⢠⢸⡇⠀⣟⠙⢦⡀⢨⡀⢠⠄⣀⠘⠆⢈⡇⠀⠀⠸⡇⠀⠀⠀⠀⢁⠀⠇⠀⡇⢸⠚⡀⠀⡇⢀⡃⢸⠀⡇⠀⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈
⠀⠀⠀⢀⡠⠀⣀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀⠀⢀⠀⣄⠀⠀⡤⢤⠀⡀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠸⡇⠀⢸⠆⢨⠄⠋⠰⡇⠀⢸⠇⠈⠃⢾⠁⠀⠑⢤⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣂⠀⡎⢳⠀⠎⢐⠀⢰⠒⠂⢊⡂⢨⠃⢠⠀⡄
⠀⠀⠀⠀⠁⠀⠁⠀⠘⠂⠀⠀⠀⠠⣎⠀⠀⠃⠈⠃⠘⠒⠊⠀⠀⠓⠀⠀⠘⠒⠃⠀⠀⠀⠀⠃⢀⡗⠚⠀⠑⢺⡀⠘⠂⠈⠒⠁⠘⠃⠈⠓⠓

⠀⠀⠀⠀⠀⠀⠈⡅⠐⠁⠀⢻⣀⡿⣀⠏⠀⠘⡄⠁⠀⠘⣀⠌⠀⠀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡀⢤⠠⠀⢠⡄⠀⢤⢠⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠋⠀⠟⠀⠠⠌⠈⢦⠀⠀⠽⡀⠀⠰⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠙⠈⠏⠀⠌⢣⠀⢸⠇⠀⠮⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠂⠀⠀⠀⢠⠃⠀⠀⠠⡇⠀⠀⠀⣆⠀⠀⠐⢤⡀⠀⡼⢠⡄⠀⠀⢠⠃⠀⠀⠢⡇⠀⠘⡄⣹⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠶⠄⠀⠐⠳⠆⠀⠠⠤⠃⠀⠀⠉⠟⠀⠀⠤⠜⠁⠀⠙⠴⠃⠀⠀⠎⠀⠀⠸⠤⠏⠀⠠⠜⠁⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠁⠟⠀⢰⠃⠠⢤⡧⡄⠀⣀⠀⠀⠘⣞⢡⠄⠀⡎⢠⢠⡄⢡⠀⠹⣧⡁⠀⠳⠇⡔⠁⡀⠀⠸⠿⠁⠀⢷
⠀⠀⠀⠀⠀⢧⠀⠰⠆⠠⡆⠀⠰⠀⠀⠰⠂⠀⠀⠃⠀⠈⠉⠁⠈⠶⠼⠿⠆⠀⢇⠻⠺⠶⡣⠀⠦⡿⠃⠀⠀⠎⠀⠳⠟⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⣀⣀⠀⣠⣤⡀⠀⠀⠀⠀⢀⣤⡀⢠⣦⠀⢰⣶⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⣄⠀⠀⠀⠀⠀⡶⣦
⠀⠀⠀⣸⠁⠈⠿⠁⢀⡇⠀⠀⠀⠀⠈⠷⣿⡼⠿⠤⡿⠃⠀⠀⠀⠀⠀⣿⣏⠿⣹⠂⣞⣯⢳⡄⠀⠀⠀⠀⠈⡽⠃⣸⠀⠀⠀⠀⠀⣧⡏
⠀⠀⠀⠘⢦⣀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⣸⣤⣤⡀⢿⠀⠀⠀⠀⠀⠀⠘⠿⣏⡁⠀⣨⠵⠛⠀⠀⠀⠀⠀⢸⡷⠋⠁⠀⠀⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⠁⠀⠀⠀⠀⠀⠀⠀⢈⣅⠀⠀⠀⠀⠀⠀⠀⣼⣻
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠻⠞⠀⠀⠀⢀⢀⡀⠀⠈⠉
⠀⠀⠀⢸⡉⠉⠉⠉⠀⢹⠀⠀⠀⠀⠀⢀⣴⠚⠾⠹⠖⣦⠀⠀⠀⠀⠀⠀⢠⡿⠿⠿⠿⠿⣌⣷⠀⠀⠀⠀⠀⢀⣀⡷⣤⠟⠚⠧⢴⡆
⠀⠀⠀⢸⣇⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠈⣇⠤⠤⠤⠤⠟⣻⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⠀⠀⢹⠟
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠓⢶⠀⣠⣤⣼⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠛⠛⠋⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡶⠦⣄⡤⣄⡴⠋⠁
⠀⠀⠀⢀⣀⣀⣼⣭⣷⣤⡀⠀⠀⠀⠀⠀⢀⡀⠀⢠⢤⡄⠀⠀⠀⠀⠀⠀⢠⡏⠀⡴⢳⣄⣶⠀⣤⣠⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⣠⡄
⠀⠀⠀⠈⢧⣼⠃⠀⢈⣿⠃⠀⠀⠀⠀⠀⢿⡙⢦⡏⣸⠁⠀⠀⠀⠀⠀⠀⢸⠀⠠⣇⡼⢿⡾⢸⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⢻⠂
⠀⠀⠀⠀⣴⣩⠿⠻⢯⣽⠀⠀⠀⠀⠀⠀⠀⣹⢂⣀⢷⡀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠻⠴⠚⠓⠛⠓⠋⠓⠂⠀⠀⠀⢀⡴⠋⣁⣤⠄⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⡾⠉⠳⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠛⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠗⠒⠒⠛⠉
⠀⠀⠀⢀⣴⡆⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠘⣦⢟⣿⣿⣦⣤⣀⡄⠀⠀⠀⠀⠾⠀⠀⢀⣤⣴⠾⣆
⠀⠀⠀⠈⢠⡷⢻⣷⣿⣶⠀⠀⠀⠀⠀⠀⢸⡛⠒⢲⡿⠋⠀⠀⠀⠀⠀⠀⠠⠞⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⢉⡷
⠀⠀⠀⠀⠈⠁⠈⠁⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣄⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣖⣫⡥⢤⣤⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣄⠀⣰⠋⣳⠀⠀⠀⠀⠀⠀⢀⡞⢀⡤⠀⡀⠀⠀⠀⠀⠀⠀⣠⠖⣫⠗⢋⣡⠴⣶⡴⠃⠀⠀⠀⠚⠉⠉⠳⣤⠴⢻⡄⠀⣀⣤⠄
⠀⠀⠀⠀⣼⠿⢞⡵⠚⠁⠀⠀⠀⠀⠀⢠⠞⣠⠞⣠⠞⣡⢆⡀⠀⠀⠀⢾⣁⣀⣙⣫⡭⠶⠛⠁⠀⠀⠀⠀⠀⠠⠖⠛⢳⣀⣀⠀⠉⠉⢉⡀
⠀⠀⠀⠸⠧⠤⢿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠘⠁⠘⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠘⢋⠈⠳⠖⠛⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠉⠳⠶⠖⠛⠉
⠀⠀⠀⢀⡞⣩⣭⣽⡀⠀⠀⠀⠀⠀⠀⢠⣦⠀⣀⡀⣀⡀⣤⡄⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢸⠰⣟⢋⡿⢻⠀⠀⠀⠀⠀⠀⢸⣟⠀⣇⡿⣯⡟⣷⡏⠀⠀⠀⠀⠀⣿⣿⣟⠁⠀⠀⠀⣹⡿⠀⠀⠀⠀⠀⠀⣴⠞⠳⡄⢠⡞⠉⢳⡀
⠀⠀⠀⠘⢦⣍⣉⣤⠞⠀⠀⠀⠀⠀⠠⠞⠙⠛⠛⠓⠋⠉⠁⠉⠁⠀⠀⠀⠀⠻⣿⣿⣶⣶⣶⡿⠟⠁⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⡞⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⣰⢦⣼⠁⢳⡏⢸⣦⣤⠀⠀⠀⠀⠀⠐⠂⠒⠀⠒⠐⠀⠃⡀⠀⠀⠀⠰⠶⣶⢶⣶⡶⣶⣾⣿⡶⠶
⠀⠀⠀⠀⣠⠞⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⡄⠉⠀⠈⠁⠈⠁⣯⠀⠀⠀⠀⠀⢉⡁⢉⡁⢉⡅⢩⠀⣩⠀⠀⠀⠘⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠈⠛⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⡀⣀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣠⣄⣤⣤⣠⣀⠀⠀⠀⣿⡀⣿⢀⣷⢠⣷⠀⣿⡀
⠀⠀⠀⠀⣿⡿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⢷⠿⣷⢿⠿⠷⠶⠶⠄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⠳⣿⢸⣿⠸⡿⠰⣿
⠀⠀⠀⠀⠉⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠉

⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⢺⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⣆
⠀⠀⠀⠈⢹⣯⠘⣿⡇⣿⠘⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠉
⠀⠀⠀⠀⠈⡿⠀⣿⠋⣿⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⡆⠀⠀⠀⠀⢰⡿⣿⣿⡀⢠⣤⣶⡶⢶⣦⠸⣿⣿⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⣤⠀⢰⣶⠶⠀⠀⠀⠀⡈⢻⣿⠁⠀⠁⠀⠀⠀⠀⣼⣧⣼⣿⣇⠀⠉⣿⣷⣼⣏⡀⢻⣿⣇⣠⡟
⠀⠀⠀⠀⠀⣠⣴⡶⢶⡄⣿⣿⠀⢸⣿⢰⣶⣿⣿⡿⠟⢸⣿⡶⠆⠀⠀⠀⠀⢰⡿⠁⠈⣿⣿⡀⠀⣿⣿⠀⣿⣿⠀⠈⠉⠁
⠀⠀⠀⠀⣼⣿⡟⠀⠘⠃⣿⣿⠀⢸⣿⠀⠀⢻⣿⡇⠀⢸⣿⡇⠀⣄⠀⠀⢀⣾⠇⠀⠀⠸⠿⠿⠂⢿⠿⠴⠟⠃⠀⠀⢀⠀⡀⠤⠀⠂
⠀⠀⠀⠀⣿⣿⡇⠀⠀⣀⠸⣿⣤⣿⠃⠀⠀⢸⣿⡇⠰⠾⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠀⠐⠂⠁⠈
⠀⠀⠀⠀⢻⣿⣇⠀⢠⡟⠀⠈⠉⠀⠀⠀⠀⠸⡿⠅⠀⠀⠀⠀⢀⡀⠠⠀⠄⠐⠀⠁⠈
⠀⠀⠀⠀⠀⠛⠿⠿⠛⠁⠀⠀⡀⠀⡄⠠⠄⠀⠂⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣄⢠⣀⣀⡄⢀⣤⡄
⠀⠀⠀⠀⠀⠠⠄⠂⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢀⡄⠀⢠⠴⣶⣤⣀⣤⢹⣿⠀⢹⣿⣼⠀⣿⢿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⠆⣀⣀⢀⣾⠉⠇⢸⣿⣸⡧⣶⣀⡀⡿⢸⡟⣦⠀⣿⣰⠜⠁⠉⠀⠛⠈⠁⠉
⠀⠀⠀⠀⠀⠀⢲⣿⠂⣀⡤⣄⢠⡶⠲⢠⡶⣦⢸⡗⠈⢿⡥⢸⣿⠠⣤⠸⠏⠸⠇⠿⠛⠞⠃⠘⠀⠙⠁⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡿⢿⡇⠘⡟⣷⢸⣇⢠⠄⣧⠟⠚⠓⠃⠸⠁⠀⠙⠓⠁⠀⠀⠀⠀⠀⣤⡄⣴⡆⣶⢂⡀⢀⣈⢻⣴⡏⡴⣶⠆
⠀⠀⠀⠀⠀⠸⠁⠘⠓⠀⠋⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⣤⡶⠀⠀⠀⢺⣇⢠⡏⠙⣷⡟⣷⡇⠙⣷⣿⠁⢈⣿⠀⣼⡏⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⡤⣄⢀⣤⡜⢿⡏⣿⠀⣖⡀⣿⠐⣿⢸⡏⠈⣿⣼⠀⠀⠉⠁⠉⠀⢸⠏⢿⠄⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⣿⢸⣧⠟⢸⣴⣇⣸⣷⡅⣰⠿⠇⠿⠀⣿⡼⠃⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣿⢠⡟⠘⠁⠀⠀⠀⠀⠘⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⣘⣷⣛⡂⠀⠀⠀⠀⣴⣿⡄⠀⠀⠀⢀⡴⡒⢦⡀⠀⠀⢐⣻⣿⡂⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡿⠃⠀⠀⠀⢻⣮⡻⣆⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣸⣻⣿⣻⡀⠀⠀⢸⡐⠧⠼⠃⠀⢠⣯⡛⢛⣽⡆⠀⢀⣤⣿⣿⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠻⣿⣾⡿⠃⠀⠀⠈⠓⠶⠶⠋⠀⠀⠿⢿⣿⠿⠁⠀⠈⠁⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣴⣾⣷⡄⠀⠀⠀⢀⠤⣤⡞⠁⠀⠀⠈⠳⡄⡀⠀⠀⠀⠀⣰⣷⣴⣧⠀⠀⠀⠀⠀⢀⣿⡀⠀⠠⢾⣟⣟⡷⠄⠀⢠⢴⣤⡾⠷⡄⠀⢰⣖⣟⣟⣗⡆
⠀⠀⠀⠹⠷⠿⠛⠂⠀⠀⢻⣜⣺⠇⠀⠀⠀⠀⠀⢚⣙⡶⠀⠀⠘⠿⢿⣿⠟⠀⠀⠐⢶⣼⣷⡿⠀⠀⠀⢾⣿⣿⡧⠀⠀⠈⣾⡿⠿⣿⠇⠀⠀⠻⠛⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢠⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣀⣴⡆⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⣴⣷⣾⣶⣧⠀⠀⠀⠰⠶⠀⠰⠶⠀⠀⠶⠀⠶⠀⡆⠀⠐⢷⣾⣷⣄⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠙⠛⠿⠿⠄⠀⠀⣸⣟⣻⣇⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠹⠿⠿⠟⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⢠⣤⣠⣤
⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⢠⣄⣀⣠⣤⠀⠀⢀⢀⣀⣀⣀⠀⠀⢘⣶⣇⠀⠀⠀⠀⠠⢴⣶⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠸⣿⣦⣾⡟⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠠⢾⣿⠷⠆⠀⠀⠘⠛⠛⠛⠛⠀⠀⠘⠉⠛⠛⠋⠀⠀⠈⠽⡷⢶⡶⠀⠀⠀⠺⣿⣿⠿⠀⠀⠀⠀⠻⣿⢟⠀⠀⠀⠀⠹⣿⡟⠀⠀⠀⠶⢻⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠟⠙⠇

⠀⠀⠀⠀⢀⣶⣠⣄⠀⠀⢀⡴⣶⣦⡀⠀⠀⠀⣠⡀⡠⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⡤⠀⠀⢠⣤⡄⢠⣤⡀⠀⠀⣿⡆⢰⣿⠀⠀⠀⡀⠀⠄⠀⡀
⠀⠀⠀⠀⢼⣿⠿⠃⠀⠀⣸⣿⣻⣾⣇⠀⠀⢸⡿⣹⣷⡆⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⣀⡠⠊⠀⠀⠀⢈⣟⣿⣟⣿⠀⠀⠀⣻⣷⣾⣟⠀⠀⠀⣄⣠⣦⣀⡄
⠀⠀⠀⠒⡿⠛⠃⠀⠀⠀⠈⠙⠿⣿⠉⠀⠀⠸⠷⠛⠋⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠻⠾⠾⠋⠀⠀⠀⢷⣿⣻⣾⠃⠀⠀⠹⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡀
⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠙⢦⣠⣄⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢠⣦⣠⣦⠀⠀⠀⢠⣖⠀⠀⠀⠀⠀⠀⠀⡠⠺⣄⠀⠀⢀⣼⣥⣠⣤⡀⠀⢀⡤⠿⠿⢤⡄
⠀⠀⠀⠰⠺⢿⣽⠟⠀⠀⠀⢸⡓⢪⡇⠀⠀⢠⡌⢻⠀⠀⠀⠀⠸⣿⣿⢿⠷⠀⠀⠸⣯⣶⣠⡴⠀⠀⢀⣼⣿⡶⠃⠀⠀⠘⣖⣳⣛⣳⡃⠀⢸⣇⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠙⠉⠉⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣶⣶⣶⣶⣶⣦⣤⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⢿⣿⣿⣿⡿⠿⠛⠛⠋⠙⣿⣿⠙⠛⢻⣿⣏⠉⠙⠛⠻⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠉⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⡀
⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⢀⣾⣿⠏⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⣾⣿⠃⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⣠⠞⣻⣿⣷⣄⠀⢠⣿⣿⠀⣠⠖⣻⣿⣷⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⢸⣿⡏⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⢰⡇⣾⣿⡏⠀⢹⡇⣸⣿⡇⠀⡏⣼⣿⣏⠀⢸⡟⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⢸⣿⣇⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⣶⡿⢡⣿⣿⠀⠀⠳⣽⣿⣿⣿⡿⠃⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠻⣿⣷⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣠⣿⡿⠃⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠙⢿⣿⣦⣄⣀⣤⣾⣿⠟
⠀⠀⠀⠈⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠀⠀⠀⠀⠀⣠⣬⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠈⣿⣿⣶⣤⣤⣤⣶⣿⡟⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⠀⣿⣿⠛⠛⠛⠛⢻⣿⡇⠀⢀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⡆⠀⠀⢀⣿⣿⣷⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣶⣴⣾⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉

⠀⠀⢄⣠⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠒⡀⠀⠀⢦⡴⠀⠀⢆⡐⠀⠀⢶⡆⠀⠀⠀⢀⠄⡀⠀⠀⢺⣟
⠀⠀⠀⠰⠆⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠟⠀⠀⠀⠀⠃⠀⠀⠊⠐⠀⠀⠉⠑⠚⠀⠀⠁⠀⠀⠀⠀⠘⠿
⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⢤⠆⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴
⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠈⠀⠀⠀⠀⠓⠂⠀⠀⠨⠁⠀⠀⠀⠆⠀⠀⠂⠀⠁⠀⠀⠁⠐⠀⠀⠀⣾⣻
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀⠀⠀⢠⡄⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠰⣀⠜⠀⠀⠚⠓⠀⠀⠀⡆⠀⠀⢈⡈⠀⠀⠀⠀⠹⠀⠀⢐⡊⠀⠀⠈⡍⠀⠀⠠⢀⡀⠂⠀⠨⠉⠛
⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣉⣹⣸⠀⠀⢈⠁⠀⠀⠀⡁⠀⠀⠀⣓⠀⠀⠀⢦⠴⠀⠀⠤⠤⠀⠀⢀⠀⠁⢀⠒⡆⠉⠒⡆⠀⣖⣱
⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁
⠀⠀⣤⣠⣀⠀⠀⢠⡦⠀⠀⠀⢠⡄⠀⠐⠀⠐⠀⠀⠀⣤⡀⠀⢠⡄⠀⠀⠰⠤⠀⠀⠀⢀⠆⠀⠀⠀⣶⢰
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠁⠁⠀⠀⠛⠀⠀⠀⠊⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠠⠄⠀⠀⠀⡤⢤⠀⠀⢀⡄⠀⠀⠀⣤⠀⠀⠀⡤⠤⠀⠀⠀⢀⠀⠀⣤⣦⡄⠀⠀⢠⣦⡀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠂⠀⠸⠹⠆⠀⠀⠀⠐⠀⠠⠔⠌⠁⠀⠻⠿⠃⠀⠀⠀⠹⠃⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣤⠐⠂⠀⠀⣄⠤⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠈⠟⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠒⠴⠀⠀⠀⣀⠀⣒⣄⠀⠠⣤⣄⣀⠀⠀⠀⣭⣬⡀⠀⠀⣴⠀⢠⡀⠀⠀⣔⣶⠀⠀⠀⠀⣀⡄⠀⠀⠀⢀⣠⠄⠀⠀⠀⣠⣤⠀⠀⠀⢠⣤⣤
⠀⠀⠐⠀⠸⠂⠀⠈⠛⠛⠛⠛⠀⠀⠀⠉⠀⠀⠀⠀⣯⣿⠃⠀⠀⠈⠀⠋⠀⠀⠀⠛⠛⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠋⠛

⠀⠀⠀⣠⡶⠀⠀⠀⣄⣠⣤⡄⠀⠀⠚⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⣶⣶⠀⠀⠀⣤⣦⣤⠀⠀⠀⠀⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠁⠉⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⡶⡆⠀⠀⠀⠀⡔⠀⠀⠀⠀⠠⠞⠀⠀⠀⢤⠤⠂⠀⠀⣠⣤⣤⣤⠀⠀⢠⣶⡄⠀⠀⠀⡀⢀⡄⠀⠀⢀⣦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠛⠃
⠀⠀⠀⠀⡀
⠀⠀⠀⢸⢽⠃⠀⠀⢠⣤⣶⡄⠀⠀⢜⠉⣳⠀⠀⠀⡐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⡇⠀⠀⢠⣿⣷⡀⠀⣀⣃⣐⣠⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠛⠈⠁⠀⠀⠀⣿⣇⠀⠀⠀⠀⣿⠄⠀⠀⠠⣤⣤⠄⠀⠀⠀⢴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⣤⡴⠀⠀⠀⢰⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠛⠂⠀⠀⠀⣀⡀⠀⠀⠀⣠⣰⣶⠀⠀⠀⢸⡿⠀⠀⠀⠘⠶⠣⠀⠀⠀⢨⠹⠃⠀⠀⠀⠺⠇⠀⠀⠀⠺⢷⣶⣤⠀⠛⠃⠰⠅⠀⠀⠸⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⡄⠀⠀⠀⣤⡤⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤
⠀⠀⠀⠿⠿⠗⠀⠀⢰⣿⣿⠀⠀⠀⢹⡴⠇⠀⠀⠀⡿⣟⠀⠀⠀⠐⢿⣆⠀⠀⠀⠾⠿⠆⠀⠐⠟⠟⠛⠻⠀⠠⠖⠴⠷⠀⠀⠼⠿⠷⠶⠂⠀⢸⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡸⣆⠀⠀⢸⠶⠿⠿⠆⠀⡷⠾⠿⠷⠀⢸⠶⠿⠿⠧⠀⣟⣻⣃⣻⡇⠀⠀⣾⣷⠀⢴⣷⡆⣴⣶⠀⣿⠀⠀⠀⠀⢀⣾⣫⣯⣫⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠙⠋⠀⠙⠙⠁⠉⠉⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡄⡀⠀⢠⣤⠀⢤⣤
⠀⠀⠸⠿⠇⠿⠇⠠⠿⠀⠸⠷⠄⠸⠆⠶⠰⠇⠀⠀⠀⠀⠀⠀⠀⠺⠂⠾⠂⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠂⠺⠆⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠉⠙⠉⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢠⣾⣷⡄⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡆⠀⠙⠋⠀⣰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠻⢷⣤⣤⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣠⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣄⣄⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠻⢦⡄⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠘⡏⢸⡁⠀⢀⣣
⠀⠀⠀⠀⠀⠀⢠⣾⡆⢀⡖⠠⠄⠀⡀⠀⠀⠁⠀⠀⣀⠘⢳⠀⠀⣄⠀⠠⠶⡴⠀⠀⢲⠂⠀⠐⡀⠨⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠃⠀⠀⠴⠀⠀⠀⠀⠀⢦⠹⠁⠈⠚⠣⠈⣃⣄⠀⠀⠀⠀⠘⠀⠀⠀⠱⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠁⠀⠀⠀⠀⡀⣏⠈⠁⠀⢀⠀⠂⠐⠍⣉⢀⠀⠀⠀⠀⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⠀⠉⠀⠂⠀⠒⠀⠀⡂⡄⠀⠙⣼⠀⠀⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡄⠠⡆⠀⠀⠼⠀⢀⠀⡀⢁⡄⠀⠘⣡⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠎⠀⠀⠒⡀⠀⡄⠀⠐⠀⠀⠀⢰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠐⠠⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⠂⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠶⠀⠀⠀⠀⠀⠀⡠⠀⠀⢨⠀⠠⠈⢀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⠂⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⠀⠀⠀⠀⠀⡈⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⣀⡀⠀⠀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠁⢠⡤⠦⠆⠨⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⠀⠀⠐⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠄⠈⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⡀⠄⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⢄
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢄⡁⠀⠐⠀⠀⢄⡀⠔⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⢀⠀⠰⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⠈⠘⠳⣄⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⣤⢁⡄⠀⠀⠀⣠⡶⠂⢠⡄⠁⠀⠩⠄⢠⡀⠀⠀⠀⠀⠲⣀⠨⣷⠀⠀⠀⢀⢄⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠐⡅⢴⠄⠀⠀⢸⡃⠄⠹⢤⡀⠢⠀⠀⠀⠀⠀⠀⠀⠂⡜⠉⠁⢸⠆⠀⠀⣒⠋⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡄⠀⠨⣇⠀⠀⢤⠟⣡⠀⠀⠀⠀⠀⠀⠀⠜⠛⠂⠀⠜⢀⣴⠾⠁⠀⠀⠀⢠⠀⡀
⠀⡃⠀⠀⠀⢀⠈⠀⠀⠀⠀⡔⠹⢃⠀⠈⠠⠤⠔⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⠴⠈⢠⡄⠁
⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⠀⢀⠠⠔⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠭⠐⠀⠋⠒⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠉⠈⢉⡀⠀⢀⠁⠀⠀⠄⠠⠀⡄⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠄⢀⠀⠀⠐
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠠⠢⠀⠊⡠⠂⠀⠀⢁⠈⠀⠀⠴⠀⠀⠁
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠰⠉⠀⠀⠀⠄⠀⠐⠀⠀⠈
⠀⣾

⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⢸⡿⢿⣿⠟⣩⣿⠀⠀⣼⡿⢿⡿⢋⣼⣧⠀⠀⠀⣀⠀⣠⡾⠃⠀⠀⠀⢀⣀⠀⣠⠞⠀⠄⠀⣄⡀⢀⣴⠟
⠀⠀⠀⢸⣿⣦⣠⣾⣿⣿⠀⠀⠹⣷⣄⣴⣿⣿⠏⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠁⠀⠙⠿⠟⠁
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠂
⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⢸⣿⣅⠙⢡⣿⣿⠀⠀⣼⣿⣌⠛⣡⣿⣧⠀⠀⠀⠘⢷⣴⠗⠀⠀⠀⢀⠀⠘⣦⣼⠃⠀⡄⠀⠘⢻⣶⡟⠃
⠀⠀⠀⢸⣿⣥⣶⣦⣽⣿⠀⠀⠹⣿⣥⣶⣬⣿⠏⠀⠀⠀⠰⠟⠙⠗⠀⠀⠀⠀⠀⠸⠋⠙⠆⠠⠁⠀⠰⠿⠉⠻⠆
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣅⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠛⣷⣄⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠄⠀⠈⠻⣷⣄⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠈⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠙⢷⣄⠀⠀⠀⠀⠀⣠⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠙⢷⣄⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣀⣀⣤⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣶⠾⠟⠛⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠶⠶⠶⣤⣀⣴⡇⢸⣶⣀⣤⠶⠶⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠶⢭⣷⣆⠰⢦⣍⣸⡃⢸⣇⣩⡴⠆⣰⣾⡭⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⢦⣄⠒⡍⠉⠒⠒⢦⡉⠃⢸⡇⠙⣇⣸⠋⢸⡇⠘⢉⡤⠒⠒⠉⢡⠒⣠⡴⠞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠹⡆⠘⠀⠀⢀⡏⠀⢀⣘⣓⣦⡌⢡⣴⣚⣃⡀⠀⢹⡀⠀⠀⠃⢰⠏⠀⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠘⢧⣸⠇⠠⠀⠀⠸⠗⢋⣩⡭⠭⢭⣭⣭⡭⠭⢭⣍⡙⠺⠇⠀⠀⠄⠸⣇⡼⠃⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣈⠀⢀⡀⠁⠀⠀⠀⢀⣀⣰⣯⠀⠀⢀⡾⠁⠈⢷⡀⠀⠀⣽⣆⣀⡀⠀⠀⠀⠈⢀⡀⠀⣁⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣡⠞⣻⠁⢀⣠⠶⣚⣩⣭⠄⠀⠀⠀⠺⡅⠀⠀⢨⠗⠀⠀⠀⠠⣭⣍⣓⠶⣄⡀⠈⣟⠳⣌⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣡⣆⢸⡴⢋⡴⠚⢩⠟⠁⠀⠀⠀⠀⠀⢹⡀⢀⡏⠀⠀⠀⠀⠀⠈⠻⡍⠛⢦⡙⢦⡏⣰⣎⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣼⠃⢹⡌⢡⡟⠁⠀⡏⠀⢠⣤⣤⣄⡀⠀⢈⡇⢸⡁⠀⢀⣠⣤⣤⡄⠀⢹⠀⠈⢻⡌⢡⡏⠘⣧⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠻⡟⠀⠀⠀⢿⠀⠀⠷⣌⠻⣷⣄⠈⠃⠘⠁⣠⣾⠟⣡⠾⠀⠀⣿⠀⠀⠀⢻⠟⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⠤⠤⣤⠀⠀⠀⡀⠘⢧⠀⠉⠒⠲⡌⣿⠇⠀⠀⠰⣿⢡⠖⠒⠉⠀⡼⠃⢀⠀⠀⠀⣤⠤⠄⣀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣥⢶⠀⡟⢰⡤⠤⠟⠀⢸⡆⠀⠀⠀⢱⠏⠀⠀⠀⠀⠹⡆⠀⠀⠀⢰⡇⠀⠻⠤⢤⡆⢹⠀⡶⣬⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡆⣇⠈⣇⠀⠀⠀⠘⢧⡀⢧⠀⣹⣠⣠⣄⣠⣤⣀⣇⠀⡜⢀⡼⠃⠀⠀⠀⣸⠁⣸⢰⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡌⣧⠸⡀⠈⠢⣄⣀⡀⠈⠳⣤⠇⢣⡁⠹⣄⣰⠏⢈⡜⠸⣤⠞⠁⢀⣀⣠⠔⠁⢀⠇⣼⢡⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣾⣆⠳⣀⠤⢤⠀⠀⠀⠀⣿⠪⣐⠀⠀⢸⡇⠀⠀⣂⠑⣿⠀⠀⠀⠀⡤⢤⣀⠞⣰⣷⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⣠⣥⠀⠘⣆⢠⠀⠀⠘⢶⣶⣶⠶⠋⠙⠶⣶⣶⡶⠋⠀⠀⡀⣰⠃⠀⡬⣄⡉⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⡰⢋⣤⢸⡇⠀⠘⢶⡀⠀⠀⢸⣷⠂⠀⠀⠀⠀⠐⣾⡇⠀⠀⠀⡶⠃⠀⢰⡇⣤⡙⢆⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡏⢸⢸⡇⠀⠀⠀⠀⠠⡄⠈⢿⣶⣄⣠⣀⣠⣶⡿⠁⢠⠄⠀⠀⠀⠀⢸⡇⡇⢹⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡞⣼⠁⣠⡴⢶⡆⢸⣁⣀⠀⠟⠻⣿⣿⠟⠻⠀⣀⣈⡇⢰⡶⢦⣄⠘⣧⢳⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣏⢸⡏⠀⢸⡇⣨⡄⢸⠀⠀⠀⠘⠃⠀⠀⠀⡇⢨⣅⢸⡇⠀⢹⡇⣹⠎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢿⡇⠀⠘⣿⠁⣿⠘⡆⠀⠀⠀⠀⠀⠀⢰⠃⣿⠈⣿⠃⠀⢸⡿⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠹⠀⠸⣆⠙⠇⠀⠀⠀⠀⠸⠋⣰⠇⠀⠟⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻⣆⠀⡴⠋⠙⢦⠀⣰⠟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠀⠀⣼⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣛⢿⣍⠉⣛⢶⢾⡇⢸⡷⠶⣛⠉⣩⡿⣛⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣾⣋⠉⣭⠭⠭⢍⡛⠷⠈⣿⢺⡇⢸⠗⣿⠁⠾⢛⡩⠭⠭⣭⠉⣙⣷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⢷⠈⠆⠀⠀⡞⠀⠀⠿⣤⡙⢋⣤⠿⠀⠀⢳⠀⠀⠰⠁⡾⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠘⣧⣼⠀⠄⠀⠸⠗⣚⣭⣭⣭⣿⣿⣭⣭⣭⣓⠺⠇⠀⠠⠀⣧⣼⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠅⣀⡀⠀⠀⠀⣀⣠⣼⠏⠀⠀⡼⠃⠘⢧⠀⠀⠸⣧⣄⣀⠀⠀⠀⢀⣀⠠⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣟⡴⠞⢹⢀⣤⢖⣫⣽⡶⠒⠀⠀⠈⢧⠀⠀⡾⠁⠀⠀⠒⢶⣯⣝⡲⣤⡀⡏⠓⢦⣻
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡵⢿⡸⢟⡴⠋⢰⠏⠀⠀⠀⠀⠀⠈⡇⢸⠃⠀⠀⠀⠀⠀⠹⡆⠙⢦⠻⢇⡿⢮⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢧⡼⠁⠀⢸⠀⠘⣟⠻⣷⡄⠑⠇⠸⠊⢠⣾⠟⣻⠃⠀⡇⠀⠈⢧⡼⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣇⢀⣀⣀⠁⠀⢀⠘⢧⠀⠛⠥⢍⢻⡆⠀⠀⢰⡟⡡⠬⠛⠀⡼⠃⡀⠀⠈⣀⣀⡀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣉⣥⡄⡼⢠⣤⠼⠆⢸⡇⠀⠀⠀⡿⠁⠀⠀⠈⢿⠁⠀⠀⢸⡇⠰⠥⣤⡄⢧⢠⣬⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠋⡀⡇⢷⠘⣆⠀⠀⠈⢧⡀⢇⢠⣧⣤⡤⢤⣤⣼⡄⡸⢀⡼⠁⠀⠀⣰⠃⡾⢸⢀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢸⢧⣻⡘⡄⠈⠲⠤⠄⠀⠑⣾⣈⠦⠘⢧⡼⠃⠴⣁⣷⠊⠀⠠⠤⠖⠁⢠⢃⣟⡼⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠧⠘⠖⠛⡆⠀⠀⠀⣯⣓⣤⣀⡸⢅⣀⣤⣚⣽⠀⠀⠀⢰⠛⠲⠃⠾⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠴⠊⢳⠀⠘⢤⡀⠀⠈⣿⣛⠉⠀⠀⠉⣙⣿⠁⠀⠀⣤⠃⠀⡞⡑⠢⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣅⡼⢻⢸⠀⠀⠈⠁⢀⡀⠻⣧⡄⢀⡀⢠⣼⡟⢀⡀⠈⠁⠀⠀⡇⡟⢧⣨⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡟⣿⢀⣤⢶⡆⣸⣁⡀⠹⠿⣿⣿⠿⠏⢀⣈⣇⢰⡶⣤⡀⣿⢻⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣇⣾⠁⢸⣇⣬⡆⣏⠀⠀⠈⠁⠀⠀⣸⢰⣧⣸⡇⠈⣷⣸⢏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⠿⠀⠀⣿⠁⣧⠸⣠⠀⠀⠀⠀⣄⠇⣼⠈⣿⠁⠀⠟⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠈⠀⠹⣆⠉⢀⡠⢄⡀⠉⣰⠏⠀⠁⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⠀⠀⣧⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⢀⣘⠆⠀⠀⠤⢔⡄⡀⠀⠀⠀⣀⠀⠀⠀⠘⣆⢐⠀⠀⠀⠠⠄⠤⠀⠀⠀⢺⢿⠒⠀⠀⠐⢢⡆⠂⠀⠀⢐⡮⡽
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠉⠈⠑⠀⠀⠈⠒⠊⠁⠀⠀⠐⠀⠃⠀⠀⠀⠨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠤⠆⠀⠀⠀⣀⣀⣂⠄⠀⠘⢰⠟⠋⠀⠀⠠⢼⠧⠀⠀⠀⢒⣐⢱⠀⠀⠈⠀⡀⠄⠀⠀⠠⢄⢘⡆⠀⠀⢠⠉⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉
⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠰⠟⠁⠀⠀⠀⣁⠀⢈⠀⠀⠀⠀⠐⠃⠀⠀⠀⢸⣃⠀⠀⠀⠀⠘⠲⠀⠀⠀⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⣓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉
⠀⠀⢀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠
⠀⠀⣠⣠⡨⠀⠀⠀⡸⠠⠯⠀⠀⠀⠴⣶⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠁⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⢀⣠⡀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡴⡄
⠀⠀⠀⡅⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⢨⠩⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⢭⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢂⠓

⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡖⡆⡀
⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠻⠟⠀⠀⠈⠉⠑⠏⠉

⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢄⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⣄⣠
⠀⠀⠀⠊⠐⠀⠀⠀⠙⠒⠀⠀⠀⠀⠙⠐⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠑⠁⠀⠀⠐⢐⢃⠐⠀⠀⠀⠘⢑⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠄⠀⠀⠠⠊⠁⡀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠊⠑⠂⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠢⠄⠀⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠐⠷⣤⣅⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢦⡀⠀⠀⠀⠀⡂⠁⠠⠔⠃⠈⠀⠀⠈⠄⠀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠃⠂⠄⢀⠑⠀⠀⠈⢀⠐⠈⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠼⠟⠀⡀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⣼⣇⡄⠀⠀⢠⡴⠖⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⡹⣶⣄⡙⠂⣀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠛⠉
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣤⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆⠀⠀⠀⢀⣾⣿⣿⣛⣟⣻⣿⢿⣿⣿⣦
⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣧⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣶⣝⢿⣿⡆⠀⠀⣐⣹⣣⢟⣫⣿⣟⢮⣄⠿⣿⡿⠆
⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣻⣿⣟⡽⠏⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⡿⣷⣿⣿⣿⣯⢾⣿⡇⠀⠀⣻⣘⣿⣿⡑⠿⣿⣿⣿⠟⣉⡾⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠘⢿⣿⢃⣩⣿⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠸⠇⠸⢻⣿⣿⡟⢋⣭⣴⣿⡿⠀⠀⠀⠹⣿⣾⣧⣭⣭⠽⣿⣵⣿⣿⡞
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣷⣊⣾⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠛⠦⠽⠟⠻⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠰⠀⠒⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢚⣛⣓⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢏⡉⠉⠉⠉⠉⡧⢤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⠤⠀⢸⠀⠀⠀⠀⠀⠀⠐⡚⠐⢧⠀⡸⠉⡀
⠀⠀⠀⢸⣸⠀⠀⠀⠀⡇⢠⠇⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠓⡴⡏⠀⣇⣀⡇
⠀⠀⠀⠈⣇⢀⠀⠀⣠⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠢⠤⣄⡸⠀⠀⠀⠀⠀⠀⠀⢐⠁⡇⠀⢹⡅⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠦⠇⠀⠘⠷⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⠰⠀⠀⠤⣤⠤⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣛⣛⣛⣛⡛⠋
⠀⠀⢠⠦⡤⠀⢸⠀⠀⣤⢾⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡏⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢰⠶⡏⡄⠀⠀⠀⢹⠶⡄
⠀⠀⠞⠀⠣⠐⠋⠂⠠⠏⠈⠃⠀⠀⠀⠀⠀⠀⠀⠞⠚⢳⠐⠂⢸⠀⠀⠀⠀⠀⠀⠀⠈⠚⡇⠁⠀⠀⠀⢸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣉⣸⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣜⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠾⠤⠴⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⡇⠀⡆⡆⣢
⠀⠀⠀⠀⢸⣄⡠⠤⢺⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⠷⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⢸⣥⠤⠔⢺⠀⠀⠀⠀⠀⠀⠀⠈⡦⠤⠤⠤⠤⠒⡺⢅⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠸⣄⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠓⠒⠒⠒⠋⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⣀⣀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⣀
⠀⠀⠀⠘⡯⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⢸⣅⣀⣀⣑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣛⣛⡚⠛⡂⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠈⢇⣸
⠀⠀⠀⢸⡀⠀⠀⢠⣤⡽⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣓⣒⡋⠀⡇⠀⠀⠀⠀⠀⠀⠀⢨⠉⠑⠢⡀⠙⠈⠙⢦
⠀⠀⠀⠀⢧⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡩⠭⠅⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⢀⣴⡎
⠀⠀⠀⠀⠘⠤⠤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⣀⣀⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠤⠤⠊

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡀⡀⠀⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⣼⠟⠻⣷⡀⠀⠀⢠⠤⢤⣤⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⠲⠀⠀⠀⣶⣴⠂⢰⡆⠀⠀⠀⣀⡀⠈
⠀⣼⣿⣣⣦⣤⠀⠀⠀⠐⠟⠟⠀⠀⠀⠀⠻⣷⣴⠿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠐⣿⣽⣿⡇⠀⠀⠀⠤⠀⠠⠶⠀⠀⠀⣿⣻⣇⣹⡇⠀⠀⠸⣿⣿⡰⣙



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠂⠀⠀⠐⠂⠀⠀⠀⣀⣀
⠀⠀⠐⣣⣄⠀⠀⠀⢲⠂⠀⠀⢰⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡾⢿⣿⢶⠀⠀⠀⣾⡟⣭⢹⡄⠀⠀⢰⠒⠒
⠀⠀⠀⠉⠉⠀⠀⠀⠙⠁⠀⠀⠘⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠁⠘⠉⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠈⠀⠀⠐⠁


⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣯⣹⡇⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⢰⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⠀⠀⠀⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣼⣯⣽⠀⠀⠀⣠⣴⣿⢼⡇⠀⠀⣤⣤⣤⣀⣀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⣨⣿⡧⣶⠆⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⡿⠿⢿⡇



⠀⢸⣶⣶⣶⡇⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⢠⡤⠀⢤⡀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⢀⢀
⠀⢸⡇⠤⢘⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣶⠀⠀⣴⠀⠀⠀⠀⣴⣉⠃⠀⠀⠀⢸⠷⠤⠾⣧⠀⠀⠀⣼⢂⡐⡇⠀⠀⠀⠠⣏⣹⡇⠀⠀⠀⠐⣷⡄
⠀⠘⠿⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠘⠄⠤⠄⠏⠀⠀⠀⠹⠦⠴⠇⠀⠀⠀⠠⠿⠿⠦⠀⠀⠀⠀⠿⠉



⠀⢠⠦⠤⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⣰⣶⢶⣆⣀⠀⠀⠀⢀⡄
⠀⢃⠁⢃⠈⠷⠀⠀⠀⠐⣶⣧⡾⠂⠀⠀⢰⣼⣿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠟⣯⣭⣯⡇⠀⠀⠸⣄⠀⡀
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠘⠃⠀⠙⠁⠀⠀⠀⠀⠒⠒⠃⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠘⠛⠚


⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣴⡋⢈⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⣏⡛⢿⣷⠀⠀⠀⢀⣄⣀⡰⠆⠀⠀⢐⣶⣶⡆
⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠈⢱⣮⠁⠀⠀⠀⢀⣐⣿⣷⠀⠀⠀⢸⠳⣌⣆⡿⠀⠀⠰⣿⣿⣿⡥⠀⠀⠀⠐⠿⠿⠃⠀⠀⠀⣿⣿⣿⣷⣶
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠛⠉⠛⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⠁⠀⠉


⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣯⣽⣭⣯⣵⣭⣏⣿⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣿⣿⣿⣿⣯⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⢠⢐⠒⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⡸⠂⢸⢈⢀⢣⠸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡞⠠⡅⢆⢠⡸⢸⢸⠠⢛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠣⣬⠉⡝⣀⠴⠃⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠘⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⢀⣑⣦⠀⠸⡏⠠⣠⣊⣁⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠌⡧⠐⠀⠸⡗⠀⠦⢠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡆⠀⠽⡯⠄⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢶⡶⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⣿⣿⣿⢂⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⠀⢸⡇⠀⡀⢡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣯⣿⣿⠀⣿⣿⣽⣿⡂⠀⠀⠀⠀⠀⢰⡇⠀⠂⠀⢸⡇⠀⠐⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣟⣿⣿⠀⣿⣿⢻⣯⣿⠀⠀⠀⠀⠀⡏⠐⠀⣆⠐⢸⡇⠂⢠⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣻⣿⣾⣙⣷⣿⣟⣽⣏⠀⠀⠀⠀⠀⡷⠀⠂⡀⠈⠠⡅⠉⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⣷⢾⣿⠿⣿⡿⣾⢟⡿⠀⠀⠀⠀⠀⢃⠠⡀⢙⠀⢀⡄⠀⡊⢀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⠛⣿⣿⣿⠎⠀⠀⠀⠀⠀⠀⠈⢳⡁⠀⠀⢠⡇⠀⠀⢈⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠚⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢤⣨⣇⡤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠛⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣇⣧⣿⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡜⢫⡟⠋⠁⠀⠀⠀⠀⢀⣤⡀⢠⣞⣀⣘⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⢦⣤⣽⣆⠀⠀⠀⠀⢀⡞⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⠁⠈⡇⣧⠉⠀⠀⠀⠀⣼⠀⠀⠀⣇⠠⣀⣠⢤⣀
⠀⠀⠀⠀⠀⠀⢈⣙⡺⠶⢚⣛⣛⡲⠼⢛⣋⠀⠀⠀⠀⠐⣿⠥⠴⢯⡀⣠⠗⢸⡀⠀⠀⠀⠀⢹⠒⠒⠒⡏⠐⠦⠖⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠈⠉⠋⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⢀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡗⠛⢾⡀⠀⠀⠀⠀⠀⠀⠠⣏⣃⣈⡧⠀⠀⠀⠀⠀⠀⠈⠙⣓⣾⣶⣿⢶⡋⠉
⠀⠀⠀⠀⠀⠀⠠⣶⣾⡿⠄⠀⠀⠟⠶⠻⠀⠀⠀⠀⠀⢀⣀⣆⣈⠉⢉⣀⣦⣀⠀⠀⠀⠀⠀⣾⡵⢻⣏⠉⠲⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠃⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠬⡿⢤⠀⢀⡬⣿⢥⡀⠀⠀⠀⠀⠘⠁⢾⢻⠀⠀⢈⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠶⠷⢤⣄⣀⣀⣀⡤⠀⠀⠀⢀⠀⠀⣇⠀⢀⡀⠀⣿⠀⢀⠀⠀⠀⠀⠀⠀⢸⡈⢧⢠⡏⡼⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠙⠋⠉⠛⠉⠉⠋⠉⠙⠉⠀⠀⠀⠀⠈⠋⠉⠉⠋⠉⠙⠉⠋⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠤⢤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⠀⣠⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡹⡄⢀⣀⣈⡀⠀⠀⠀⠀⠀⠀⠴⣤⠤⠀⢐⣶⣴⢯⡀⠀⠀⠀⠀⡀⠀⡀⠁⣀⠀⣀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⣤⣤⢤⠀⠀⠈⠀⡿⣆⠀⠀⠀⠀⠀⣉⢋⣉⠋⣉⡋⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠷⡤⠤⠧⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⠃⢸⡆⠀⠀⠀⠀⠈⠉⠈⠉⠁⠉⠀⣠⢾⠗
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⢈⣉⡉⠉⠉⣉⣉⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⢠⢯⠿⠿⠿⢷⡋
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠚⠓⠛⠚⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⢠⣾⣷⣴⣿⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠁⠙⢿⣿⣿⠋⠉⠙⠉⠉⢻⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠈⣿⣿⣆⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠙⠋⠯⠛⠁⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣆⣶⣶⣿⣿⣿⣷⣄⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢴⣿⡆⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⠁⠉⣿⣿⣿⣿⡿⠁⠹⣿⣿⣿⣿
⠀⠀⠀⢰⣾⣾⣿⣷⣷⡆⠀⢾⣿⣿⣿⠆⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠉⠹⣿⣿⠁⠀⠀⣹⣿⣿⣿
⠀⠀⠀⠀⠉⢁⣷⡈⠉⠀⠀⠀⠹⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⠏⢿⣶⣾⣿⣿⣷⡀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡄⠀⠀⠀⠀⢠⣠⠄⠀⠀⠀⠀⠀⢀⡴⠞⣡⡀⠀⠀⠀⠀⢀⠀⠉⠉⢞⠆⡀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⢘⠛⠀⠀⠀⠀⠀⢀⣏⠉⠀⠈⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⣪⣖⠀⠀⢀⣔⠞⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢘⡆⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⡇⠀⠀⠟⢊
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠹⠇⠀⠀⠔⠃⠀⠛⠀⠀⠀⠀⠈⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠯⠋⠀⠀⠀⠘⠾⢦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣀⢠⠦⠀⠀⠀⣴⡏⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢐⡁⠀⠀⠀⠀⢀⢛⠂⠀⠀⠀⣴⠀⠀⣾⠃⠀⠀⠘⠀⠶⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠉⣟⡂⠀⠀⢸⣷⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⢈⣙⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠐⠖
⠀⠀⠀⠀⠠⠆⢠⣍⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢙⡁⠀⠀⠰⠀⠀⢮⡌⠀⠀⠀⠀⠤⠀⣾⡇⠀⠀⢀⠀⠘⠃
⠀⠀⠀⠀⠀⠴⠛⠀⠀⠀⠀⠈⠛⠶⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠉⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠆⠰⢷⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠸⠂⠀⠀⣠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠏
⠀⠀⠀⠀⠀⠀⠀⣘⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠢⠆
⠀⠀⠀⠀⠀⠀⠠⣭⠃⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⡶⠀⠀⠀⠐⠰⠆⠘⣿⡇⠀⠀⠀⠀⠐⢿⠇⠀⠀⠀⠀⠀⠀⠀⡤⠰⡍
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠀⠀⠀⠀⠀⠀⣀⠀⢀⡠⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣾⡁⠀⠀⠀⠀⠀⠀⢸⣷⡘⠂⠀⠀⠀⠀⠀⠘⠆⣼⠏⠀⠀⠀⠀⠃⠀⠀⡔
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡁⠀⠀⠀⠀⠀⠀⡀⠙⢿⣦⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠙⣉⣀⠀⢀⠀⠀⠀⠀⠠⠆⠠⡭⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⡿⠀⠀⠀⠀⠓⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⡀⡔⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⣦⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡤⡶⠂⠀⠀⠀⡀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠘⢃⠏⠀⠀⠀⠀⠀⠀⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠲⢛⠀⠀⠀⠀⠀⠀⠀⠀⠰⢎⠆⠀⠀⠀⠀⠋⠀⢀⢾
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣩⠀⠀⠀⠀⠀⠀⢀⡀⢰⣷⠃⠀⠀⠀⠀⠀⡰⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠑⠤⠂⠀⠀⠀⠈⢱⠆⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⢾⡄⠀⠀⠀⠀⠀⠉⣠⠾⠋⠀⠀⠀⠀⣠⠾⠟⠁

⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⣀⠄⠀⡀⠈⢣⠀⠀⠀⠀⠁⡰⠀⠀⠀⢀⠀⠀⠀⠀⣩⣯⠁⠀⠀⡀⠄⠀⠀⠄⣐⠘⢄
⠀⠀⠀⠀⠀⠂⢀⣍⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠊⠁⠈⢃⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡐⠀⠰⠀⠀⠀⠀⠀⠀⠀⠉⡀⠚⠂⢦⠀⠰⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠹⡉⣅⡁⠀⠀⠀⢛⠂⠀⠈⡄⠘⠀⠀⢦⠴⠒⠚⡆⠀⢄⡆⠀⠀⠀⠀⠀⠀⠁⠘⠚⠁⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠄⠀⠀⠉⠀⠀⠘
⠀⢠⠀⡇⠀⢸⠑⠉⡅⠀⠀⠐⠀⠀⠀⠡⣘⣄⠀⠀⠀⠀⠀⠀⡇⣬⠊⠀⠀⠀⠐⠂⠀⠀⢸⠙⠀⢢⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠐⠀⢀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣄⡀⠈⠸⠆⣖⣰⠶⠵⠞⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠡⣁⠀⠀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢤⠀⠀⠀⠀⢀⠀⠀⠀⣠⠃⠈⠘⢀⠀⠀⠀⠀⠀⠀⠸⠩⢘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠰⠔⡖⠀⠀⠑⠀⠀⠀⣄⠌⠀⠀⠀⠤⡆⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠂⠘⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⢺⡇⠀⠂⠁⢈⠅⠀⡠⠤⠀⠀⠀⠀⠀⠰⠉⣤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠈⠐⢢⠀⠰⣐⡦⣂⠀⠀⠀⠀⠠⠃⠌⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⡆⠀⠀⠠⠄⢉⠙⣁⠆⠀⠀⠀⠀⠀⡰⠀⠠⠂⠀⠀⠁⠋⠃⠀⠀⠀⣆⠃⠀⠰⢀⡀⡀⠀⡠
⠀⢀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠐⠁⠀⠀⡆⡄⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠈⢂⠀⡰⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⡠⠀⠗⡠⢆⢀⠒⠢⠀⠀⠀⠀⠀⡀⠀⠀⠙⢫⠆⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠁⠀⠀⠀⠰⠖⠚⠺⠀⠀⠀⠀⠀⠤⠯⠤⠄⠀⢰⢡⠆⠀⠁⣨⣷⠦⢴⠂⠀⠀⠀⠌⡄⠀⠀⠀⠸⠀⠀⠀⡠⠊⡤⣠⠄⡀⠀⠀⡀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠺⠞⠃⡖⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡀⠸⠋⠁⣫⣆⠀⠢⣀⠠⢂⠞⠀⠀⠀⠀⠊⠀⠀⢠⠀⠐⠀⠀⠀⠸⠀⠀⠀⠀⢙⠁
⠀⠀⠀⠀⢀⡀⡀⢻⠃⡄⠯⠈⠁⠀⠀⠀⡄⠀⠀⠴⠶⢦⠀⠑⢄⠂⠀⠈⡆⠀⠀⠀⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠂⠀⠀⠀⠀⢣⠠⠀⠀⠈
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⢿⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⢀⠄⠀⠈⠀⠀⠀⠀⠀⠢⠤⠀⠀⢸⡄⠀⠠⠁⠀⠀⡀⢀
⠀⠀⠀⠀⢠⡀⠀⠀⠀⣴⠟⠀⠈⠀⠀⠠⠀⠀⠀⠘⡷⢾⠀⠀⠀⠀⠀⡠⠄⠀⠀⢀⢤⡄⠀⠂⠐⠀⠀⠐⠀⠀⠀⢀⣀⠠⡀⠀⠀⠀⠀⠀⠀⣡⡇⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠉⠀⠀⠀⠤⠀⠀⠳⠄⠀⠋⠀⠀⠀⠀⠀⠀⠘⠃⠂
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⡿⠿⠿⢿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠉⠁⣀⣀⣀⠀⠀⣀⣀⣀⠈⠉⠛⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⣀⣤⣶⣾⡿⠿⢿⣿⠀⠀⣿⡿⠿⢿⣷⣶⣤⡀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⣠⣾⡿⠋⢀⣤⣾⡿⠛⠉⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠉⠛⢿⣷⣤⡀⠙⢿⣷⣄
⠀⠀⢀⣼⣿⠋⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠙⣿⣧⡀
⠀⢀⣾⡿⠁⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠈⢿⣷⡀
⠀⣼⣿⠁⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠈⣿⣧
⢰⣿⠇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠸⣿
⣾⣿⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣿
⣿⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹
⣿⣧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼
⢿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣠⣾⠀⠀⣷⣄⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿
⠸⣿⡆⠀⢿⣷⡀⠀⠀⠀⢀⣴⣿⠟⠁⣠⣾⣿⣿⠀⠀⣿⣿⣷⣄⠈⠻⣿⣦⡀⠀⠀⠀⢀⣾⡿⠀⢰⣿
⠀⢻⣿⡀⠈⢿⣧⡀⢀⣴⣿⠟⠁⣠⣾⡿⠋⢸⣿⠀⠀⣿⡇⠙⢿⣷⣄⠈⠻⣿⣦⡀⢀⣼⡿⠁⢠⣿⡟
⠀⠀⢻⣿⡄⠈⢿⣷⣿⠟⠁⣠⣾⡿⠋⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠙⢿⣷⣄⠈⠻⣿⣾⡿⠁⢠⣿⡟
⠀⠀⠀⠻⣿⣆⠀⠙⠁⣠⣾⡿⠋⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠙⢿⣷⣄⠈⠋⠀⣰⣿⠟
⠀⠀⠀⠀⠘⢿⣷⣄⠀⠙⠿⣷⣦⣄⡀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⢀⣠⣴⣾⠿⠋⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠈⠙⠻⠿⣿⣶⣾⣿⠀⠀⣿⣷⣶⣿⠿⠟⠋⠁⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣄⣀⡀⠉⠉⠀⠀⠉⠉⢀⣀⣠⣤⣶⣿⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣄⣿⣠⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣠⠄⠀⠀⠀⠀⠀⠀⢸⠡⠈⡇⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠄
⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠦⣞⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣘⠿⣃⠀⠀⠀⠀⠀⠀⠀⢨⣤⣶⣾⠀⠀⠀⠀⠀⠀⠀⣶⣾⢤⡄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠃⠀⠀⠀⠀⠀⠀⠈⠑⠙⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡶
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠻⡆⣆⠈⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⠛⢡⣿⣿⠏⠘⣷⡈⢧⡑⣼⣟⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⣡⢧⢀⡻⣿⣦⣻⣾⣿⡀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣋⣵⣿⣿⢟⣥⢞⣳⣿⣼⣿⣿⣿⢿⣿⣿⣷⣵⣄⡻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⡄⠀⢉⠛⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⣻⢿⣿⣿⠟⠋⠙⠻⣿⡿⠋⢡⠀⣼⣷⣤⣼⣧⣴⡈⡛⢿⣻⣙⠻⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⢿⣥⣾⣿⣶⣇⢠⡄⣠⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠗⠂

⠀⣀⢀⡀⠀⠀⡐⢉⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠹⠀⠀⠀⠀⠙⢌⡣⠊⠁⠀⠀⠀⡎⠢⠐⠀⠀⢀⡀
⣶⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠐⠀⢃⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠑
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⢀⡀⠀⠀⢀⡴⣚⡂⣄⠀⠀⠀⢤⡀⠀⠀⢰⣄⠀⠀⣠⠦⢤⡀⠀⠀⠀⠀⠠⠄⠀⣠⠷⠀⠀⣀⡀
⠀⠰⠋⠀⠀⠀⠙⠮⠚⠁⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠈⢢⠀⠁⠈⠘⠀⠀⠠⢤⠀⡦⠀⠀⠀⠴⠋
⠐⠪⠅⠀⠀⠆⠀⠠⠆⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠣⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠺⡒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠀⢠⠀⠀⠀⠀⠀⠀⢀⣀⡴⢦⣀
⠀⠀⢠⣤⣤⣤⣄⣄⣤⠀⠀⠀⢠⢳⢀⠚⣀⠞⡄⠉⢁⠉⠀⠀⣮⣠⣤⣤⠥⣵⡀⠀⢀⣀⡀⣀⣀⣀⢀
⠀⠀⠈⠀⠀⠀⢠⣥⣤⠀⠀⠀⠛⠒⠛⠐⠛⠒⠃⠀⠀⠀⠀⠀⠄⠀⠄⠴⠠⠠⠀⠀⠀⠉⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣄⣀⣀⡀⠀⠀⢀⢀⡀⡀
⢄⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⣾⣷⡇⠀⠀⠀⠀⢆⣦⡆⠀⠀⠀⠀⠀⠀⣀⡠
⠒⠋⠀⠀⠐⠩⠀⠬⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠱⠃
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢚⡱⡇⡓⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠊⡇⠀⠀⠀⢀⠠⠀⢀⠀⠀⠀⠈⠑⠊⠃⠀⠀⠌⢱⠆⢡⠀⠀⠉⠙⠙⠋⠉⠀⠀⢠⣏⣧⠀⠀⠀⠛⢣
⠵⠇⠀⠐⠀⠁⠂⠒⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠒⠐⡁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠓⠒⢒⠃⠀⠀⠰⠢
⠀⠀⠀⠀⠀⣈⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠄
⣉⡉⠉⠀⠀⠀⠀⠀⣠⠀⠀⠘⣀⡛⣤⠃⠀⠀⣶⣛⣶⠀⠀⢀⣀⢘⡓⣄⡀⠀⢀⡼⣻⠛⠀⠀⠀⠉⣉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⢀⣤⢀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁⠀⠈⠘⠻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣄⠀⠀⠀⣠⣤⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠁⠀⠀⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠠⠠⠀⠐⠀⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡴⢿⣿⡷⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣻⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢄⡀⠀⠀⠀⠀⢀⡀⣠⣤⣠⣀⡀⠀⠀⠀⢀⣤⠤⢄⠀⠀⠀⠀⠒⠢
⠀⠀⠀⠀⢁⣘⡀⠰⠀⠠⠀⡰⢋⠂⠀⠀⠀⢆⡀⠀⠀⠸⠑⠀⠀⢠⡆⠁⠀⠀⠀⢠⢠⢉⣳⠀⠀⠂⠄⠀⢂⠁⠀⠊⠄⡄⢨⠰⠆⠀⠀⠰
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠈⠓⠦⠤⣤⣤⠤⠤⠤⠊⠀⠀⠈⠑⠀⠐⠀⠀⠀⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠀⣶⡛⠄⠒⠂⠀⠐⠢⡐⠀⣴⡟⠉⢀⡀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢠⣧⡄⠀⠀⠀⣠⠏
⠀⠀⠀⠼⠑⠤⠈⠳⠀⠀⠀⠀⢀⡞⠃⠌⠀⠀⠻⡄⠀⠀⠀⢐⢌⣂⣐⡑⡁⠀⢠⡄⠀⠠⢀⠊⠦⢰⠀⠀⠀⠈⢋⠟⠀⠀⠚⣹⢁⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠭⠉⠍⠃⠀⠘⠀⠀⠐⡶⣷⠷⠟⠀⠀⠀⠠⠮⡤⠀⠀⢠⢿⡏⠀⢀⣾⡷
⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠠⡄⠤⣤⢕⡴⢂⠴⠀⠀⠀⠀⡄⢠⢀⠀⠀⠀⠂⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡼⠀⠀⠀⡾⠁
⠀⠀⠀⠃⠵⠂⠄⠱⠌⠀⠀⠀⠀⡘⠗⢮⣩⠔⠟⢆⠀⠀⠀⡗⠈⠩⢯⠀⠘⣤⠀⢠⠊⠅⠦⠒⠚⠒⠴⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠑⠀⠀⠓⠀⠀⠈⠀⢰⠠⠤⢐⣴⠀⠉⠀⠀⠀⠛⠀⢠⠀⡀⡤⠀⠀⢀⣠⠄⠀⢀⠄⠀⠀⣠⠂
⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⣉⠀⠀⠀⠀⠀⠀⠀⠈⢫⠞⠁⠀⢀⣾⣇⠀⢠⣮⡤⠀⠐⣻⠃
⠀⠀⠘⠿⠁⠀⠘⡶⠁⠀⢩⡏⠀⠀⢈⡥⠒⢢⡇⠀⠀⣄⣴⣠⠀⠀⣤⠀⠈⠈⠀⠘⠀⠀⠰⣤⣦⡄⠀⠀⠀⢰⠇⠀⠀⢘⣿⡇⠀⢀⣾⡅⠀⠰⣻⠟
⠀⠀⠶⠶⠆⠀⠰⠷⠀⠀⠚⠓⠀⠀⢌⡓⢚⣫⠏⠀⠀⠰⠿⠇⠀⢠⡀⡴⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⠇⠀⠀⢉⡟⠁⠀⠈⡟⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢓⣿⠚⠂⠀⠀⠀⠀⠀⠀⡠⡆⣉⣉⠀⡦⡀
⠀⠀⠀⠠⣿⠋⠻⢟⡀⠀⠀⠀⠀⢻⣿⡆⠀⠀⣠⣾⣿⠃⠀⠀⠀⢰⣤⣼⣿⣧⣀⠀⠀⠀⠀⠀⢸⠤⠿⣦⣤⡾⠧⢴
⠀⠀⢸⣿⠂⠾⢆⠹⣷⠀⠀⠀⠀⠀⣿⣆⠀⣿⣿⣿⡏⠀⠀⠀⠀⠘⠉⠙⠛⠋⢹⣇⠀⠀⠀⠀⠰⡀⢣⣸⣿⣠⠃⡰
⠀⠀⠀⠻⠿⠟⠺⠿⠋⠀⠀⠀⠀⠀⠙⠷⠄⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠈⠰⠮⠩⠶⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⣠⣶⣶⣿⣿⣶⣶⡄⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠺⣯⡄⢰⡆
⠀⠀⠙⠿⠿⠻⢿⡿⠋⠁⠀⠀⠀⠀⠻⢿⣏⠀⣴⣶⠀⠀⠀⠀⠀⠀⣿⠫⣄⠈⢻⡇⠀⠀⠀⠀⠀⠀⣀⣐⣿⣷⠟⣱
⠀⠀⠰⠀⠆⠰⠠⠆⠆⠀⠀⠀⠀⠀⠀⠻⢯⣿⠉⠉⠀⠀⠀⠀⠀⢸⣿⣧⣍⣥⣿⣷⠀⠀⠀⠀⢠⣾⡿⢿⣿⣿⡿⠋
⠀⠀⠀⠉⠈⠁⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠉⢻⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⣿⡇⠀⠀⠀⠀⣴⠒⠒⠒⠒⠚⣿⡷⠀⠀⠀⠀⠀⣤⣶⣾⣿⠷⠆
⠀⠀⠀⢀⣠⣾⣦⣀⠀⠀⠀⠀⠀⠀⠀⡟⠛⡛⠛⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡆
⠀⠀⠀⠀⢿⣯⣿⠇⠀⠀⠀⠀⠀⠀⠀⡇⣾⣷⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣶⣶⣆⣤⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣄⢰⠐⠣⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣟⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⢠⡀⠟⣠⡾⠀⠀⠀⠀⠀⠀⠃⠀⠈⡏⠀⠈⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣤⣼⣁⣄⣴⠃
⠀⠀⠀⠀⢙⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⢹⣿⣿⠀⠀⠀⠀⢀⣾⣿⠟⠛⠛⠉
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁

⢼⠦⣀⡇⠀⠀⠀⣀⣀⣀⣀⡀⣀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⠈⣁⠛⠶⠖⠀⠀⠀⠀⠀⠀⢾
⠼⠤⠞⠁⠀⠀⠀⠁⠉⠉⠁⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠁⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣤⡇⢀⣀⡄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡄⠀⠀⠀⠀⠀⠀⣠⡾⡿⣶⣾⣿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠹⡇⠁⠈⣡⣋⡤⠟⠁
⢋⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠋⠉⠀⠀⠀⠀⣠
⠠⢈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠙⡃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⠿⠥
⠀⢁⠀⠀⠀⢰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣶⣶⣤⣄
⠢⠴⠀⠀⢠⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠻⡿⡍⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⡀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠧⠀⡀⠤⣀⠀⢰⡅⠰⠊⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠁⠄⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠂⠀⠺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠇⠰⠶⠇⠀⠀⠀⠀⠀⠀⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠈⠈⠊⠐
⣤⣤⠀⣠⣶⠀⣠⠶⡇⣀⡴⠶⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠤⡦⢀⠀⠀⠀⠀⣯⣭⣭⣭⣥⠉⣶⣷
⠀⠘⠾⠋⠻⠟⠁⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠀⡄⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢠⣀⠠⠃⠀⠀⠀⠀⣰⠀⢠⡆⢀⣠⡄
⢺⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢻⠀⠏⠠⣭⣅⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⠔⠁⠈⠀⣠⠂⠀⠀⠀⠻⠓⠒⠒⠀⠀⠐⠒⠀⠀⠆⠂⠀⠆⠀⠀⠠⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣆⠠⠄
⠀⠀⠀⢀⣀⣀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁
⠀⢰⣿⣿⣿⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠇⠀⠐⠀⠐⢀⠀⠘⠀⠀⠀⠀⠁⠀⣀⣀
⠀⠘⠛⠉⠉⠉⠈⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠂⢀⣴⣤⣴⠄⠀⠀⢸⣿⠿⠄⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⡆⠀⠠⡄⠀⠘⠀⠀⠐⠀⢐⠀⠀⠀⠚⠑⠛⠁⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⢈⡀⠁⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⢀⡤⠂⠠⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⠅⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠁⠠⡄⠀⠸⠀⠀⢀⠆⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⡠⠒⠒⢀⠀⠀⠀⠀⠠⠄
⠀⢻⣼⣠⣾⣤⣄⣿⣿⣸⣶⣄⡆⠀⠀⠀⡀⠁⠈⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠁⠀⠀⡰
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⠀⠀⠄⠀⠀⠀⠀⠀⣠⠶⡦⠀⠀⠀⠀⡔⠋⢢⡀⠐⠉
⠀⠀⠀⠀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⢇⣀⡼⠃
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⡎⠓⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⢰⠂⡘⠀⡀⠀⠿⣿⠀⠀⠰⣶⣷⡶⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠈⠁⠀⠈⠡⠀⠠⠋
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⡀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠈⠀⠀⠐⠛⠿⠟⠃⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠦⠂⠀⠠⡟⠋⠙⡆⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠴⠒⠂⠄⠀⡀⠀⠀⢀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠒⠂⠀⠀⠳⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠐⠀⣀⠜⠀⠉⠀⠈⠈
⠀⠀⠂⠂⠠⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣤⡄⣠⠀⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢀⢤⠀⡀
⠀⠠⠤⠤⠤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠷⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠤⠃⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡇⢰⡤
⠀⠀⢀⣤⣀⡀⠀⠈⣿⣿⡀⢀⠀⠀⠀⣠⡄⢀⣀⠀⠀⢀⣿⠁⡀⠀⠀⢀⣴⠀⣀⠀⠀⢹⣿⣇⠀⡀⠀⣠⡄⢀
⠀⠀⠿⣿⡏⢀⠀⠀⢿⣿⡇⢸⡄⠀⢸⣿⡇⠘⠋⠀⠀⣿⣿⡀⢹⠀⠀⣿⣿⡄⢀⠀⠀⠸⣿⣿⠀⠀⢸⣿⣇⠈⡇
⠀⠀⢼⣿⣧⣄⡀⠀⢸⣿⣷⣄⡀⠀⢸⣿⣿⣄⡀⠀⠀⢿⣿⣧⣄⠀⠀⢻⣿⣷⣄⡀⠀⠀⣿⣿⣇⡀⢸⣿⣿⣄⠁
⠀⠀⠈⠛⠟⠋⠀⠀⠀⠛⠛⠋⠀⠀⠀⠙⠿⠋⠁⠀⠀⠈⠛⠛⠉⠀⠀⠈⠛⠟⠋⠀⠀⠀⠙⠛⠁⠀⠀⢻⣿⣏⠙
⠀⠀⢠⣾⡇⠀⠀⠀⠀⢀⣤⡀⠀⠀⢀⣄⠀⠀⢠⣾⡇⠀⠀⠀⢠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟
⠀⠀⢸⣿⣿⠀⢀⠀⠀⢿⣿⡇⠀⠸⣿⣿⠁⠀⢸⣿⣷⠀⣆⠀⠈⣿⣿⡀⠀⠀⢀⣤⠀⡀⠀⠀⠀⠀⢀⣤⠀⡀
⠀⠀⠀⣿⣿⡄⢸⠀⠀⢹⣿⣇⠀⠀⣿⣿⡄⠀⠈⣿⣿⡄⢀⠀⠀⢿⣿⡇⠀⠀⣿⣿⠀⣧⠈⡇⠀⠠⣿⣿⠀⣧
⠀⠀⠀⢻⣿⣇⣘⣇⡀⠸⣿⣿⣀⠀⢻⣿⡇⠀⠀⢿⣿⣇⣸⣆⡀⢸⣿⣷⣀⠀⣿⣿⣆⣸⣀⣳⣀⠀⣿⣿⣆⣸⣀
⠀⠀⠀⠘⠿⠻⠿⠋⠀⠀⠿⠟⠁⢀⣸⡏⠛⠀⠀⠘⠿⠻⠟⠋⠀⠈⠿⠟⠁⠀⠹⠿⠿⠿⠿⠟⠁⠀⠹⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧
⠀⠀⢀⡄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡀⠀⡀⠀⢈⣿⣿⠀⢠⠀⠀⣀⣴⠀⣦
⠀⢰⣿⣇⠈⡇⠀⢰⣿⣧⠀⡇⠀⠀⣾⣿⡀⢹⠀⠀⢰⣿⣷⠀⠟⠁⢠⣿⣿⣄⡉⠁⠀⢿⣿⡇⠀⠀⠀⣿⣿⡄⢸
⠀⠸⣿⣿⣄⠁⠀⠘⣿⣿⠀⠉⠀⠀⢿⣿⣧⡈⠀⠀⠈⣿⣿⣀⡀⠀⠀⣻⣿⣇⡉⠀⠀⢸⣿⣷⣄⡀⠀⢹⣿⣧⡈
⠀⠀⠻⠿⠟⠉⠀⠀⣿⣿⡇⠈⠀⠀⠘⠿⣿⣿⡇⠀⠀⠿⠿⠋⠀⠀⠀⠙⠿⠿⠋⠀⠀⠈⠻⠿⠟⠁⠀⠈⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠂⠀⠀⠀⠀⢸⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⡄⠀⠀⠀⣠⠀⣤⠀⡄⠀⠀⠀⣠⡄⢠⠀⠀⠀⢀⡄⢠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⡇⠀⠀⣿⣿⡆⢸⠀⢳⠀⠀⢺⣿⣧⠈⠁⠀⢸⣿⣧⠀⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠀⠀⢹⣿⣷⡀⠀⠈⠀⠀⠘⣿⣿⠀⣧⠀⠘⣿⣿⡀⠛⠀⠀⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠛⠀⠀⠈⠻⣿⢿⣿⠟⠁⠀⠀⣿⡿⣿⡿⠋⠀⠻⣿⡿⠷⠄⠀⠘⢿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡄
⠀⠀⣶⠤⡤⠴⠖⣛⡿⠟⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠿⡅⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡶⠶⠶⠶⠶⣶⣤⡄
⠀⠀⢿⡀⣤⢀⡔⢉⣴⣀⣾⠇⣴⣷⣤⣤⡀⠀⠀⠀⠀⠀⢀⡼⣹⢫⠟⢿⣦⠀⠀⠀⠀⠀⠀⠸⣿⠀⣤⣤⣤⣤⣤⣀⠛
⠀⠀⠸⣧⣿⣾⡅⠚⢛⣋⣥⣼⣿⣿⠟⣿⡇⠀⠀⠀⠀⠀⣸⢹⢳⡏⣾⠎⢿⢳⠀⠀⠀⠀⠀⠀⠙⣷⡈⠃⠀⣠⡿⠃
⠀⠀⠀⠘⢿⡏⢀⣾⡿⠙⣿⡧⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠿⣯⡶⢸⡎⡇⠀⠀⠀⠀⠀⠀⠈⢿⣦⡘⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠞⠁⠀⢠⠟⢁⡿⠁⢸⠇⠀⠀⠀⠀⠀⠀⠸⣜⣇⢾⡿⡹⣸⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠦⣼⣱⣫⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡆
⠀⠀⠀⠀⠀⢰⡄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠇⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⡀⢰⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠉⠉⠛⠿⣿⣆⣶⠖⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⠀⠀⣠⣶⡷
⠀⠀⠀⢠⣀⡷⣸⣷⠰⣇⣠⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣄⣸⣷⠆⣠⡼⣿⣧⠀⠀⠀⠀⢀⡾⣿⢁⣾⡀⠀⣸⡆⢹⡛⣧⡀
⠀⠀⠀⠀⣿⢡⣴⣶⣄⢹⡇⠀⠀⠀⠀⠀⠀⣶⣶⠀⠘⢛⣩⠴⢛⣥⠀⠻⠿⠀⠀⠀⠀⢾⡅⢻⡄⠻⣿⣿⡿⠃⣸⠃⣸⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣼⣿⣶⠟⠋⢵⣾⠛⠟⣀⡀⠀⠀⠀⠀⠀⠀⠙⣦⠙⢶⣤⢠⣴⠞⣡⡞⠋
⠀⠀⠀⠀⠸⢿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠉⠁⠘⠿⣿⢶⣤⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣽⠆⣼⠋⠈⢳⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢛⡉⣙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⠁⠈⠀⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⢀⣀⡀⢀⣸⣷⣶⣿⣧⣤⣤⡀⠀⣀⠀⠀⠀⠘⢏⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠻⠷⣄
⠀⠐⣿⡿⠿⣿⣿⣋⠀⣛⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⣿⡄⢀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢨⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣡⣿⡇⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡿⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⣿⣷⣦⣄⣀
⠀⠀⠀⠀⣀⣀⣴⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢿⣿⣿⣉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣿⠿⠁⠻⠋⠸⡿⣿⠀⠀⠀⠀⠀⡀⠀⠈⠉⢀⡌⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⣶⣄⣀⣼⢂⣄⢻⣄⣀⣴⡞⠁
⠀⠀⣴⣿⣷⣤⠐⣶⣶⠀⣤⣿⣷⡄⠀⠀⠀⢻⣄⠀⠀⠀⠀⢿⣿⣏⠁⠀⠀⠀⠀⠈⠉⢻⡏⢸⡿⢡⣾⣿⣦⠹⣏⠈⣿⠋⠉
⠀⠀⠈⠻⣯⣤⠈⢉⠉⢀⣬⣿⠋⠀⠀⠀⠀⠈⢿⣿⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠤⢼⣧⣸⠇⣿⣿⣿⣿⡇⣿⣠⡿⠤⠄
⠀⠀⠀⠀⠿⠿⣦⣿⣷⡾⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡅⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⠟⠉⠀⠹⣏⢉⡿⠁⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣶⣦⣽⣶⣄⠀⣤⣾⣯⣴⡶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⡿⠿⠿⠿⠿⡆⠙⣿⠋⢸⠿⠿⠻⠿⢿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣻⣿⣿⡿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⣟⣋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⣒⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠰⣿⣿⣥⣶⣾⣿⣿⡷⠄⠀⠀⠀⠀⢻⣄⠀⣰⡏⠀⠀⠀⠀⠠⢾⣿⣿⣷⣶⣬⣿⣿⠆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣿⢟⡋⠉⢉⣿⣦⡄⠀⠀⠀⠀⠀⠈⣿⣴⣿⠀⠀⠀⠀⠀⠀⢠⣴⣿⠉⠉⣙⡻⣿⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢀⣾⣿⠟⠀⣄⣀⠀⢸⡄⠀⢸⣿⡇⠀⢰⡇⠀⣀⣠⠈⠻⣿⣧⡀⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⡿⠋⢠⣿⣿⣿⡇⢠⣬⢽⣿⣿⠀⠀⠈⣿⠀⠀⠈⣿⣿⡯⣥⡄⣸⣿⣿⣷⡀⠙⢿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠘⣿⣿⣿⣏⠀⠙⠶⣥⣿⠀⠀⠀⠛⠀⠀⠀⣿⡭⠞⠃⠀⣿⣿⣿⣿⠁⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣠⣴⠀⠀⠘⣿⣿⣿⡄⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣠⣿⣿⡿⠃⠀⠀⣦⣄⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣄⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣰⣿⣿⡟⠁⠀⠀⠀⢹⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣿⣿⢀⣴⡆⠀⠀⠈⢿⣿⣿⡟⠀⠀⡀⣀⡀⠀⢀⣀⢀⠀⠀⢻⣿⣿⡟⠀⠀⠀⢰⣦⡈⣿⣿⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⠀⢄⡀⠈⣿⣿⡁⠀⠀⠙⠻⣿⣿⣿⠟⠋⠀⠀⢸⣿⣿⠁⢀⡤⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠈⢳⣄⢹⣿⡇⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⢸⣿⡇⣰⡟⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⣿⣿⣿⣿⡇⢀⡆⠀⢿⣾⣿⣿⣄⡀⠀⠀⣀⣤⣀⠀⠀⢀⣠⣿⣿⣿⡟⠀⢰⠀⢻⣿⣿⣿⣿⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢁⣼⡇⠀⠘⣿⣿⣿⣿⡿⠋⠉⠁⠀⠈⠉⠙⣿⣿⣿⣿⣿⠁⠀⢸⣧⡘⣿⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⡇⠀⠀⢹⡿⣿⣿⣷⣤⠀⠀⠀⠀⢀⣤⣿⣿⣿⢿⡇⠀⠀⢸⣿⣿⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢁⣼⣇⠈⠁⠙⣿⣿⣿⣷⣧⣤⣶⣾⣿⣿⡿⠃⠈⠁⣸⣧⡘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠻⣿⣿⣿⣿⣿⡄⢲⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⡆⣰⣿⣿⣿⣿⣿⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢻⣿⡟⢹⣿⣿⣌⣿⣶⣤⡙⢿⣿⣿⣿⠟⣃⣤⣾⣿⣱⣿⣿⠏⢻⣿⡟⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⢻⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠏⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⡈⠙⠿⠋⢀⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠉⠻⣿⣦⠀⣴⣿⠟⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⡤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠉⠓⠲⢤⣾⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣴⣿⣿⣛⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣏⣿⣭⣿⣹⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣭⣿⣸⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢿⣿⣾⣭⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣽⣷⣿⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢠⣤⣼⣿⣭⠁⠀⢀⠀⠀⢻⡟⠀⠀⡀⠀⠈⣭⣿⣧⣤⡄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡤⠬⢭⣛⠋⠉⠀⠀⠈⠙⠚⠋⠉⠓⠋⠁⠀⠀⠉⠙⣛⡭⠥⠴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⢾⠀⠀⠀⠀⠀⠀⠀⡿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⠁⠀⠀⠀⠀⠀⠀⢸⠖⠒⠒⠦⢤⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠠⠐⢀⠨⢔⡂⠅⡀⠂⠄⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⢲⡖⠒⢺⠀⠸⠒⡆⠀⠣⠲⠒⠆⠆⠒⠢⠶⠀⠄⠀⠰⢀⠀⠰⠾⠷⠆⢀⡀⠆⠀⠐⠒⠔⠶⠴⠖⠰⠀⠶⠲⠢⠐⠶⠶⠀⡗⠒⢲⡖
⠀⢠⣟⠀⠈⠲⣷⡖⠉⠒⠐⠐⠒⠐⠀⣀⠂⠒⠚⠂⠉⠑⠂⠨⢑⠠⠔⡊⠅⠐⠊⠁⠒⠊⠂⠀⠊⠂⠑⠐⠐⠂⠀⠈⢰⣶⠖⠃⠀⣹⡄
⠀⠈⣩⣭⠉⠉⠉⣠⡴⢶⣶⠶⣤⡀⠰⣿⠆⠰⣦⡴⣚⡲⣄⠀⠀⠈⠁⣠⠞⢀⡴⠚⠲⣄⠰⣦⠀⠉⠈⣰⣦⣤⠶⠶⣦⣍⠉⠉⣩⣍⠁
⠀⠺⡗⠊⢀⡀⣼⠫⠊⠠⡤⠈⢪⢻⡄⣉⠀⢨⡝⠃⢀⠀⢉⠟⠀⠀⠀⣿⠀⢸⠀⠲⡆⣿⠀⢹⡆⠂⠨⠁⡉⢉⢐⠤⢀⢘⣷⠀⠙⠸⡟
⠀⠀⡿⠢⣼⡟⠛⠦⠀⠰⠧⠀⠠⠟⡛⣿⡤⢘⡳⠆⠀⡀⠙⡇⠀⠀⠀⡻⡠⡘⡂⠑⢁⢛⠀⡺⡁⠀⠈⣶⠂⠬⣈⠀⠢⠤⢬⠁⠐⠟
⠀⠾⣗⣠⠀⠀⠁⣈⡂⠂⠀⠒⣂⡉⠀⠊⠀⠸⢧⡄⣤⡅⣹⡆⠀⠀⢈⡇⠠⣤⠄⠈⠈⣤⡄⢀⡅⠀⠀⠘⢷⣄⣀⠉⣁⣶⣦⡴⢀⠃
⠀⠀⣈⠉⢀⢄⡀⠈⡙⠛⠛⠛⡉⠀⠀⠀⠀⠀⠀⠳⠤⠔⠋⠉⠀⠴⠤⣀⠀⠈⠳⠤⠶⠃⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠀⣉⡐⠀⠃
⠀⠐⢁⠀⡀⢂⡀⠈⢨⠀⢠⠊⣡⠞⢻⣛⠻⢦⡌⢳⠀⠀⠠⠆⠄⢠⡄⠡⠡⢄⠀⠀⢠⢪⣶⠿⢛⠿⢷⣮⠢⡀⠀⢎⡸⠉⢃⠋⠩⣆⠂
⠀⢀⢄⠀⣿⢇⡸⠠⢰⠀⣾⢰⡏⣰⡟⡟⣿⡍⣿⠈⡇⠀⡘⠄⡂⣲⣶⢘⡠⢆⠀⠀⡇⣿⡇⠀⣦⡆⡄⢻⣧⠧⠀⢸⠇⢒⡋⡶⡆⠿⡀
⠀⠈⠆⠀⠀⡌⠉⠀⠌⠀⢢⠼⣷⡚⢷⣷⡿⣡⡿⢦⠇⠀⠣⡂⠀⠁⠀⠀⢔⠔⠀⠀⠇⠟⢃⡀⡁⣈⣀⠚⠏⠏⠀⢾⡀⠙⠁⠽⠁⡼⠇
⠀⠀⠀⠐⢄⢀⠔⠈⠀⠰⣾⣦⡌⠛⠶⠾⠟⠛⣤⣼⡷⠀⠀⠘⠢⡂⢐⠜⠊⠀⠀⢶⣧⣥⠅⠶⠶⠶⠆⢨⣥⣾⠆⠈⠛⠢⢖⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢐⡀⢀⣀⠠⡐⠀⠈⠁⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⡀⠉⡀⢀⢀⠀⢀⢀⡀
⠀⢈⠇⠴⠟⡇⠀⣀⣀⡠⣀⡀⢈⣀⠀⠄⠀⠈⢠⡄⠀⠀⣀⣤⣶⣜⣣⣶⣤⣀⠀⠀⢠⡆⠰⡆⠂⠀⢠⠀⠲⠰⠀⠢⠰⠆⠂⠀⠐⢲⠆
⠀⠈⠉⠧⠀⢃⣣⣿⣽⣥⣽⣧⣿⣿⣤⡄⠀⠀⢸⡇⣸⡟⠛⣁⢈⠙⠋⡁⠈⡛⢻⣇⢸⡇⠰⠧⠌⠻⢃⡀⢀⣀⠀⣀⡀⢀⡘⠟⠡⠼⠆
⠀⠀⠀⠖⢛⣛⡛⣛⣛⣛⣛⣟⣛⣥⠒⢂⣀⡀⢸⡇⣿⡃⠀⠈⠀⠀⠀⠀⠀⠁⣘⣿⣸⡇⢠⡄⢀⣴⡌⢁⠈⣉⣀⡙⢁⠈⢡⣤⡀⢠⡄
⠀⠀⠀⠒⠀⠙⠀⠙⠙⠁⠑⠉⠛⢡⣤⠆⢸⡁⠈⠀⠉⠛⠛⠷⣦⣄⣠⣴⠿⠛⢻⠉⣉⡉⣨⣥⡍⢉⡟⣿⣿⣿⣿⣿⣿⣿⣿⠏⣹⢼⡷
⠀⠀⠀⢀⠀⠀⠄⠈⠂⠀⠀⡀⡀⠀⠁⠈⠉⠁⡀⣤⡔⢦⡔⡄⠀⠙⠋⠀⠀⠀⢨⣤⣤⣤⣥⠤⡤⠠⠤⠟⣿⣛⠛⠛⠿⠿⠟⡛⣫⣬⠠
⠀⢈⡶⠆⠀⠈⠀⠀⡀⣠⠀⠠⢶⡍⠀⠀⣠⣌⣳⣾⣿⣿⣾⣏⣠⣀⠀⠀⢀⣴⠋⣡⠀⢠⡍⢳⡄⠀⠀⣞⣿⣾⣶⣶⣶⣶⣶⣿⣿⣿⡆
⠀⠀⢷⣆⣀⡂⠀⠀⠓⠓⣂⣠⡾⠁⠀⢾⣿⣿⣿⣿⣟⣿⣿⣻⣯⣿⡇⠀⢼⡇⢀⢺⣀⢀⢛⠀⣷⠀⠀⢸⣿⣷⣾⣾⣿⣿⣷⣾⣾⣿⡁
⠀⠘⠒⠒⠻⠿⢽⡀⣪⠽⠿⠒⠒⠚⠀⠀⠛⠟⡹⣿⣿⣿⣿⣏⡻⠋⠀⠀⠈⠻⣜⠋⠉⠉⢈⡼⠋⠀⠀⢾⣿⣿⣿⣿⣿⣽⣿⣻⣿⣿⡇
⠀⠀⠀⠠⠤⠀⠀⠭⠄⠀⢀⡀⠀⠀⠀⠀⠀⠈⠘⠛⢣⠜⠻⠉⠀⠀⠀⠀⠀⠀⠘⠛⠿⠟⠋⠀⠀⠀⠀⠈⠛⣉⠀⠠⠤⠀⠀⠨⠝⠋
⠀⠸⡆⠉⠀⠛⠛⠛⠛⠀⢶⢠⣌⣀⡐⠂⠀⠀⠠⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⢂⣀⢩⢤⣶⠆⠠⠛⠛⠛⠀⡉⢰⠇
⠀⠰⠧⠈⠻⠇⢀⣀⣈⣁⡐⠀⠍⠙⣉⡛⠹⠳⠾⠟⠃⡀⠉⡟⠀⣴⣦⠄⢻⠉⣀⠹⠇⠘⠃⠟⢛⡈⠃⠠⠀⠒⣀⣀⣀⣀⠸⠟⠁⠼⠆
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠐⠻⠀⠋⠛⠀⠛⠂⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢦⡀⠀⠀⠀⠀⢰⡆⠀⢇⠀⢄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⣷⠀⠀⠀⠀⢺⢙⣤⠈⠂⠄⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠱⠝⠀⠀⠀⠀⠸⡅⢊⠕⢦⣀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⡄⠀⠀⠀⠀⠀⠀⠻⡤⡪⠔⠩⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡠⢂⣳⠀⣰⠆⠀⠀⠀⠀⠈⢲⡁⡪⢐⠈⢳⡀⠀⠀⠀⠀⢀⡴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠍⡊⠅⣺⠀⡏⣆⠀⢀⠀⠀⠀⠀⠙⢆⠅⡊⠐⢹⡄⠀⠀⠠⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢐⠡⢂⢥⡇⠀⠑⠌⡳⡌⢷⡢⣄⠀⠀⠈⡯⠂⠵⠅⡧⠀⠀⣧⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⠅⢊⡑⠡⣧⠀⠀⠀⠈⠓⢸⡆⠙⣦⠀⢀⡯⢠⡄⠀⣺⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⡆⠄⢈⠱⣄⠘⠷⡀⠀⠀⠀⡼⡐⠈⣽⢀⡾⠀⡌⡇⠀⣊⠀⠀⣟⢯
⠀⠀⠀⠀⠀⠀⡯⢲⢄⡀⠀⠀⠀⢳⡁⠈⡆⠈⠲⡄⠉⢦⢀⡞⠅⣤⠀⣿⠛⢀⠜⠀⠇⢠⠃⠀⢠⠇⠺⡇
⠀⠀⠀⠀⠀⠀⠘⢄⠊⠙⢦⡀⠀⠀⢫⡄⠘⢆⠀⠈⢢⠀⠿⠀⠜⠙⡀⠁⡰⠃⠀⡜⢀⡏⠀⢀⡞⠈⠔⢧⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣉⠒⠨⡇⠀⠀⠀⠙⢦⠈⢆⠀⠀⢣⠀⡜⠀⠀⣧⠊⠀⠀⡜⢀⡞⣀⣴⠋⠁⠀⠨⢺⠀⠀⠀⢀⠴⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡒⢤⣀⠀⠀⠀⠘⡇⠀⠆⠀⠀⣷⠀⠀⠐⠁⠀⠀⠘⠀⡾⠛⠉⢀⡠⡖⠀⢈⡏⠀⠀⡰⢁⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡈⠕⣄⠀⠀⣿⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⡁⠐⡠⠒⠉⢀⠜⠀⢠⡞⠀⠀⠰⠖⢉⡠⣰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣄⠈⣆⢀⡟⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠊⠀⠀⢠⠋⢀⣴⠎⠀⡀⠀⣠⠞⣉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢀⠈⢆⢹⡾⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡏⠀⡼⠁⣠⣾⠁⡼⣋⠅⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠸⠀⠈⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⣇⢞⡭⡇⣸⠇⣤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⢇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⣇⠐⡏⢰⡄⢷⡿⡘⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢆⢁⠁⢣⠘⢠⠁⢼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠑⠌⠀⠸⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⡒⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡀⠔⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠈⠁⠤⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⣀⠐⠩⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠠⠄⠊⠤⢒⡠⠐⡀⠄⢂⠠⢀⡠⢄⡈⠅⡀⠄⡂⠠⢂⠠⢀⠠⢀⠀⣀⠦⣀⠠⢐⡀⢄⠤⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠉⠀⠉⠀⠈⠀⠈⠀⠈⠁⠈⠀⠉⠀⠈⠁⠈⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⣄⠀⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠎⠈⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠻⢄⠉⠑⣦⠀⣄⠜⣀⡀⠀⠀⠦⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠃⠀⠀⠀⠈⢁⠔⠒⠠⠋⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠂⠠⡀⠀⠀⠀⠀⠀⠈⢢⠀⡔⠀⢢⠈⠉⢢⠀⠀⠀⠀⡴⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡄⡦⢀⣄⠷⠀⣄⠀⠀⠀⢰⠁⠀⢑⠒⠊⠀⡖⠁⠀⠀⠀⢰⠀⠰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢇⠀⠀⣣⡤⢃⡀⡜⠀⠀⠀⠀⠑⠐⠈⠓⠄⠔⠃⠀⠀⠀⢀⠘⠤⠃⠀⢠⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠈⣁⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠀⠡⠆⣀⣀⠀⠈⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠠⠊⠁⢀⡀⠁⠀⠀⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⡎⠀⠀⠱⠤⠤⢄⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠤⠊⠀⠈⠙⡄⠀⠀⠉⠁⠀⠀⠸⡀⠀⢠⠀⠀⠀⠉⠁⠀⠀⢀⠵⠀⡔⠐⠄⠀⢈⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⣑⠒⠁⠺⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠁⠉⠉⠦⠀⠔⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠎⠀⠑⠤⠒⠉⢢⠀⢀⠔⢲⠀⠀⠀⠀⠀⠀⡋⠢⡀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢜
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠆⢀⠔⠂⠄⠀⡰⠀⠋⡠⠆⠀⠀⠀⠀⠀⠀⠈⠢⠇⠘⢤⠇⠀⠀⠀⠀⠀⠀⡔⠁⡍⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢸⠀⠀⠠⡀⠀⠨⠀⠉⠆⠉⢤⠂⠠⣀⠀⠀⠀⠀⠤⠀⠤⡀⠀⠐⠁⠉⠢⠀⠄⠈⡠⠚⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠑⠒⠀⠀⠀⠤⣀⡠⠆⠀⠀⠑⠠⠄⠇⠀⠈⠐⠂⠐⠀⠠⠒⠛⡠⠀⢄⠀⡀⢀⡔⠒⠒⢢⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣀⡀⠢⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠣⠀⠜⠀⠈⡀⠀⠁⠈⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠊⢠⠄⢰⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠀⠀⠀⠀⠇⠀⢀⠖⠒⠊⠁⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠔⠁⠀⠨⡀⠁⠀⠀⠀⢰⠓⠄⠀⠀⠀⠀⠀⢘⡠⠂⠀⠀⠀⠀⠈⠉⠁⡘⢣⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠈⢄⠈⠀⢠⠖⠒⠢⣨⠄⠀⢄⠀⠀⠀⠀⠀⠀⠠⣠⠀⠀⡄⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡙⡁⠡⡀⢠⠄⠥⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠴⠃⡔⠁⠐⠄⠰⠀⠫⡀⢏⠑⠢⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⠱⢄⡀⠖⢄⢀⡴⠁⠀⠑⠦⠵⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠢⢤⣀⠀⠀⡄⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢤⣤⠀⢠⠄⠰⠊⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣇⡀⠀⠀⢀⠔⠁⠀⠑⢄⠀⢀⠷⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠈⠑⠲⠋⠀⠀⠀⠀⠀⠑⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢄⣀⣀⣀⣀⣀⣀⣀⠠⠔⠊

⠀⡄⠀⠄⠀⠀⠄⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠂⠂⠀⠀⠀⠐⠀⡀
⠀⠂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⡀⡀⠀⠀⠂⡀
⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠂⠀⠈⠁⠀⠀⠃⠁⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠤⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢄⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠄⠧⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠠⠀⠀⠀⠐⠂⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣄⣠⡄⠀⠀⢤⠦⡠⠀⠀⠠
⠇⢠⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠄⠄⠠⠀⠀⠀⠀⢀⠁
⠀⠠⠀⠀⠀⠀⠀⠀⠡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠂⠀⠀⠐⠰⠀⠀⠀⠀⠒⠶⠒
⠀⢀⡀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⢀⡀⠀⠀⠀⠀⠀⠀⢐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⣃⢀⠄⠀⠀⢰
⠀⠀⡀⠀⠀⠀⠀⢠⠜⠃⠉⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠈⠄⠀⠀⠀⠠⠁⠀⠀⠔⠁⠀⠀⠀⠀⠸
⡠⠤⠤⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⡀
⢃⢀⢀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠂⠀⡀⡀⢠
⠘⠘⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣁⠀⠀⠀⠐⠠⡄⠀⠀⠀⠀⠆⠃⠘⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡟⣿⣶⣯⣟⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣯⣻⡟⣿⣿⣿⣷⣿⣿⣿⡿⢟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⡿⣿⢻⢿⣿⣿⣃⡿⣰⡿⣡⣾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢩⣤⠇⡵⢡⣿⢑⣎⠻⢹⡇⡿⢱⡟⣰⡟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣦⢸⠸⡸⣇⠸⡇⠃⣾⢁⡿⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢛⡶⢾⣭⣴⣇⢻⡌⢧⢱⠸⡆⣇⢠⡏⣸⠁⡼⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⠴⠶⣶⣬⣭⡙⣆⢻⡌⢿⡇⣧⢸⢸⠁⡏⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠶⠶⠶⣶⣤⠙⣧⠘⣆⢻⡌⢷⡻⡘⠘⠀⠃⢀⡾⠁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠿⠿⢿⣶⣤⡉⢧⠸⣆⢹⣄⣻⣼⣇⠃⠀⠀⢀⡟⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣷⣶⣶⠶⠶⣶⡈⣷⢸⣆⣿⣾⣿⣿⣿⣟⡐⢀⠆⠸⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡞⢠⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⠀⡶⢶⠀⣶⢠⣶⢠⡶⣦⢠⡶⣦⢰⡆⢰⢰⡄⡶⠀⢰⣆⢰⢠⡶⣶⢰⣶⢠⣶⢰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢠⡇⢸⢸⣿⣾⣿⢸⣇⣿⢸⡇⣿⢸⣿⣾⠈⣿⠇⠀⢸⣿⣼⢸⡇⣿⢸⣿⣼⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠘⣇⣸⢸⡟⣿⢿⢸⡏⠁⢸⡷⣿⢸⡏⣿⠀⣿⠀⠀⢸⡏⣿⢸⡗⢿⢸⡟⣿⣿⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣤⣽⣯⣤⣥⣥⣬⣤⣧⣤⣬⣥⣭⣼⣥⣽⣤⣬⣤⣤⣼⣤⣭⣤⣥⣬⣬⣥⣥⣬⣬⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇





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

⠀⠀⠀⣔⠳⡀⠀⠀⠀⠀⡐⠢⠀⡦⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠉⢦⠐⠲⠀⠀⠠⠞⠢⠤⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠀⢀⣤⢀⣄⣀⠀⠈⠀⠀⠀⣴⠔⠂⠙⠋⠉
⠀⠀⠀⠀⠈⠳⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠢⣀⣰⠀⠀⠀⠀⠘⠶⠚⠻⠻⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠞⠛⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡝⢻⣭⡅⠀⠀⠀⠀⠀⣨⠥⣯⣹
⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠈⢇⡤⠔⠁⠀⠀⠠⠛⢿⡆⠀⠀⠀⠀⠠⠂⠘⠁⠙⠋⠋⠃⢀⠂⠀⠀⠀⢣⠀⠛⡏
⠀⠀⠀⠈⠉⠉⠉⠸⠀⠀⠀⢘⡼⠀⠀⠀⠀⠀⡁⣂⠜⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠶⢾
⠀⠀⠀⠈⠂⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣤⣤⡄⠀⠀⠀⠰⠅⢀⡂
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣿⣿⣧⣄⡀⠀⠀⠉⢻⠀⠀⠀⢀⣴⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣰⠿⠛⠛⠋⠉⣙⡆⠀⠀⢀⣄⡀⠀⢰⣿⠷⠓⠛⠛⠻⢿⡀⠀⠀⠈⠄⠀⠀⠸⣿⡿⠛⠙⠋⠙⠻⣿⡆
⠀⠀⠀⠀⠀⣿⠈⣽⠳⠀⠸⡶⣇⠀⠀⠈⣒⡀⠀⢸⡏⠠⠤⠀⢠⠤⠈⡇⠀⠀⠀⠀⠀⠀⠀⢹⢨⣿⡡⠀⣨⣴⠈⡇
⠀⠀⠀⠀⠘⢤⠀⠄⠀⠀⠢⠀⠘⠀⠀⠀⠀⠀⠀⠸⠃⣋⠛⠁⠙⢛⡁⠉⡆⠀⠀⠀⠀⠀⠀⣿⠅⠀⢠⠁⠈⠀⢠⣩
⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠶⠂⡰⠀⠀⠀⠀⡀⠀⢘⣦⠉⠀⢀⡀⠈⣰⡎⠀⠀⢰⠦⢴⠀⠀⠀⢄⠀⢠⡄⠀⢀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢹⠑⢶⠀⠊⠀⠀⠀⠳⢦⡓⠀⠀⠀⠀⡀⠉⠁⠐⠁⠀⠀⠀⠙⢶⠚⠂⠀⠀⠀⢑⡄⢠⡄⠁
⠀⠀⠀⠀⣠⣾⣿⣷⣤⣴⣿⣿⣦⠀⠈⠓⠓⠂⠀⠀⣀⡠⢯⣥⡼⠤⠤⠤⡀⠀⠀⠀⠀⠀⣠⣶⢦⣉⠓⠺⣛⡴⣤
⠀⠀⠀⠀⠛⢻⣏⣹⣿⣿⣿⣿⠛⠆⠀⠀⠀⠀⠀⠞⠀⠀⠀⢺⡀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠷⠠⣶⡍⢡⣝⠋⢴⢮⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠄⠀⢀⡄⠀⣀⡀⢀⡀⢠⢂⢀⠄⢠⠄⢠⣊⡔⡠⠀⢠⡂⠀⠀⡀⠀⠀⠀⠀⠎⢀⡟
⠀⠀⠀⠀⠀⠆⠴⠠⠚⠀⠀⠈⠀⢢⡆⠀⠒⡇⠀⠈⠀⠘⠁⠁⠀⠀⠇⠘⠁⠸⠅⠀⠉⠌⠀⡋⠀⠀⠀⠘⠎⣠⠊
⠀⠀⠀⠀⠘⠖⠳⠎⠁⠀⠀⢸⣀⠎⠀⠄⠉⣃⠀⡙⡁⢀⠀⠀⠀⢰⠀⢰⠀⢀⡀⠀⠂⠀⡇⠃⠀⠀⣠⠾⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠉⠀⠀⠁⠈⠃⠙⠀⠉⠀⠑⠋⠛⠂⠀⠚⠁⠉⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠄⠠⠤⠀⠓⠄⠀⠀⠀⠀⠔⢫⣇⡀⠀⣀⠀⠀⠀⠀⣟⠢⢄⣀⣀⠀⣶⣴⠀⠀⠀⠀⠀⠠⠄⢀⢀⠠⠄⢠
⠀⠀⠀⠀⠤⠦⠴⠓⢧⣤⡼⠀⠀⠀⠀⠙⠞⠁⠈⠉⠉⠀⠀⠀⠀⠈⠓⣤⢀⡤⠚⠁⠀⠀⠀⠀⠀⠈⠒⠖⠃⠀⠑⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⢀⠤⠐⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠂⠂⠀⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢜⠀⠀⠀⠠⠶⢦⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⣄⠀⣀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠀⠀⠇⠀⠀⢷⣄⡀⠀⠀⠀⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠁⠀⠀⠒⠒⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠞⠳⠆⠀⢠⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢀⠿⠇⠀⠀⠀⠀⠀⢀⠐⠒⠲⣉⠀⣠⣿⣤⠄⠀⢀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠻⢓⠀⠀⠀⠀⠀⠀⢈⠈⠤⠘⠂⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⣠⣽⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠰⢠⠟⠀⠀⠰⠞⠁
⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⣀⢠⡆⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⡄⢠⡄⡀⢀⠀⠀⠀⢀⣀
⠀⢠⠊⠉⠋⠀⠀⠿⠀⠟⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣿⠶⢾⠇⠀⠀⠀⡀⠀⡏⠀⡇⠀⣀⡀⠀⠀⢠⠀⢳⣞⠁⠀⢸⠉⠁⢸⠆⠀⠀⢸⣿
⠀⢾⠒⠒⠛⡆⠀⡟⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⣾⠙⢁⡀⠀⠇⠀⡇⠙⢷⡄⢘⢀⡀⢸⡀
⠀⠈⢷⣤⡴⠃⠀⠽⠶⣜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣀⡃⠈⠓⠛⠁⠀⠀⠘⠃⠀⠀⠁⠚⠀⠓⠐⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣄⣄⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⢿⣠⠇⠀⠐⢀⠈⠁⠀⠀⢀⡴⠚⠓⣦⠀⠀⠉⠉⡆⠀⡉⠁⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠁⢹⠁⡇⠀⠸⠈⢰⡆⡁⠀⠀⠀⢀⠘⡇⢠⠃⡶⢲⡄⠀⠤⡇⠀⠐⠖⠀⢾⠀⠀⠀⢸⡆⢀⡟⠉⠀⠀⡙⠒⠉
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⡀⣼⡀⣇⡲⠨⠀⠸⠀⠁⢀⠊⣡⠊⠀⣧⠇⠀⠁⠀⠁⠀⠀⠰⠤⠆⡀⠀⠘⣆⣀⣀⣼⠃⢰⡇⠀⠀⠀⡄⢰⡆
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣈⠠⢬⣄⡉⠀⠈⢀⣤⣴⣧⣤⡉⠁⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠀⠀⡠⠀⢄⠀⠀⠁⢸⡗
⠀⣴⠟⠉⠷⡍⠢⠀⠀⡌⠁⠖⠒⠈⢿⣆⢀⣿⣿⡟⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢛⠀⠐⠀⢤⡀⠀⠀⢀⣀⠀⠀⠀⠀⠡⣕⠃⠀⠀⠀⠘⠁
⢰⡟⠀⠀⠀⡇⠀⠀⢸⠁⠈⠀⠀⠀⣼⣿⢸⣿⣿⠀⢸⣿⣿⡇⢀⠀⠐⠀⠀⠀⠀⠸⠀⡌⠀⠀⠈⠳⠄⠀⠀⠉⠒⠄⢠⣖⠆⡇⠀⠀⠰⣏⣉
⠘⢿⣄⠀⠞⣧⠘⠶⠀⢧⡀⠰⢖⠙⡿⣿⠈⠻⣿⣶⣿⣿⣿⠀⠰⠀⠀⠀⠀⠀⠀⠰⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠕⠁⠀⠀⠰⠦⠟
⠀⣠⣭⣭⣭⣭⣥⡀⡀⠀⠉⠉⠉⠁⠈⠁⠀⠀⠈⠀⠀⠘⠉⠀⠀⢀⠀⠀⣀⠀⢠⠀⢀⡄⠠⡤⢠⡤⠀⣤⠤⢤⣤⠀⢀⣀⣀⣀⡀⠀⠀⡀
⠀⠉⠛⢻⣿⠛⠛⢁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⡴⣿⠟⠀⠀⠀⠀⠉⠀⠀⢁⡽⠃
⠀⠀⠀⢀⣻⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢳⡼⠀⠀⣠⠀⠀⢰⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠹⠆⠀⠐⠤⠆⠀⠀⠀⠀⠀⠛⠷⠾⠟⠖⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠀⡈⠛⠁⣀⠈⠉⠉⢉⡉⠀⠛⠉⣉⣉⣀⣁⣈⣁⣈
⠀⠐⡄⠀⣰⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢾⡿⠈⠿⠏⠻⡟⠃⠀⠀⠀⠁⣴⠟⣿⠄⠁⠈⡆⠀⣿⠀⠀⠀⢸⣿⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢈⡞⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⡀⡀⢰⣀⠮⢷⡼⠀⠀⡄⢀⠀⠃⠀⣰⠋⠀⠀⢠⠁⢰⡇⠀⢀⠀⢸⡇
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⠁⢀⣿⡄⢰⣧⠀⠀⠁⠤⠌⠀⠀⢨⠀⠀⠀⠈⡀⠨⠅⠀⠸⠄⠸⠀⠀⠆⠘⠀⠨⠭⠭⠀⠸⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⠿⠟⢶⡄⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠖⠀⠰⠀⠀⠀⠀⠸⣇⠂⠀⠐⣸⠇⠀⠀⠀⠀⠤⠀⠀⠦⠆⠀⠀⠀⠐⠁⠄⠀⠄⠓⠀⠀⠀⠀⣋⠄⠀⢄⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣤
⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⢧⠀⠀⠀⠀⠀⠋⠀⠙⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉


⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⡡⠀⠀⠦⠉⠀⠀⠀⠀⢡⡆⠀⢶⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠄⠚



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠂⠐⠄⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠸⠑⠀⠀⠊⠀⠀⠀⠀⠀⠖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⢹⣿⣿⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣸⣿⣿⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⣿⠉⠉⢹⣿⣿⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⢛⣿⣿⠛⠛⢻⣿⣿⠀⣿⣤⣤⣼⣿⣿⣤⣤⣼⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣨⣿⣿⣀⣀⣸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠟⠛⢻⣿⡿⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢀⣸⣿⣿⣀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠙⣿⣿⠉⠉⢹⣿⣿⠀⣿⣿⡿⠿⠿⣿⣿⡿⠿⠿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣰⣿⣿⣤⣤⣼⣿⣿⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣾⣿⣿⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢨⣿⣿⠀⠀⢸⣿⣿⠀⣿⣿⡟⠛⠛⣿⣿⡟⠛⠛⣿⣿⡇⠸⢿⣿⣿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣴⣿⣿⣤⣤⣼⣿⣿⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⣿⣿⡟⠛⠛⣿⣿⡟⠛⠛⣿⣿⡇⠈⢹⣿⣿⠉⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⠶⠿⠿⠶⠖⠚⠛⠛⠀⠛⠛⠃⠀⠀⠙⠛⠁⠀⠀⠛⠛⠃⠠⠼⠿⢿⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣁⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣈⣉⡁
⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢋⣄⣠⣷⠤⠞⠒⠛⠋⠥⠤⠬⠤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠁⠀⠀⠰⠆⠀⢀⠛⡀⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣩⠔⠀⠠⣴⠀⡀⢠⡶⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⣠⣾⠝⠀⠀⢀⠀⠐⡴⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢡⡖⠀⠀⠀⠀⣄⠉⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡁⢠⡌⠀⠺⠧⠀⠀⢸⣀⡓⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣀⠀⢧⠀⡀⠠⣾⠦⣬⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢺⢁⡏⠃⠈⡄⠁⠄⢠⠀⢻⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⢸⡇⠈⠁⡀⠀⠹⠀⠀⣀⠀⢸⢠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⢸⠇⠶⠁⠁⣀⠀⢀⠀⠋⠀⢀⢠⡟⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢾⢘⣦⠠⡆⠙⠀⠀⠀⣤⠀⣨⡘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣾⡈⣁⠶⠆⠛⠀⢀⠀⠀⠸⠝⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡉⠣⣴⠰⠶⢌⠡⢀⠀⢪⡸⡄⠈⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠡⠘⠔⢦⡓⠲⢤⢤⣛⣃⣤⣽⢿⣄⢲⣬⡅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⢀⠀⢸⡀⢫⡐⣦⠠⡤⡄⣠⡍⣹⣿⣆⠻⣧⠀⠑⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⢠⣴⡇⠚⣠⣤⣤⡍⢺⠧⠠⣭⣷⣌⠳⠾⠷⠂⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⠙⠦⠁⡄⣸⢋⣥⢬⡛⣏⠡⠐⠦⠄⠀⠰⢦⠉⠛⠷⠤⡹⣆⠠⠤⣀⠑⠦⢄⢰⡿⣡⠴⢚⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢾⢰⡄⣾⣯⡟⠀⢈⣅⢰⣿⠀⠀⡘⢣⡖⣲⢽⠮⣶⣖⠀⣀⣀⣴⢋⣾⣿⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠠⣀⠲⣯⡀⢀⡀⠿⠋⠀⣠⠀⣠⡀⠈⢰⠡⡗⠀⠸⠆⢀⠑⢺⢃⠚⣯⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⢰⠎⣱⡮⢝⠳⣤⢂⣀⣈⠀⠀⠉⡁⢀⠈⠁⠠⡆⢴⠆⠀⣴⠏⠆⢙⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⠽⢦⡦⠽⠓⠭⣌⣣⡼⠃⢢⡜⣦⣤⣲⣐⠙⠻⢛⣫⢉⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⠀⠀⣠⠜⣋⡉⠃⠲⣶⢚⠓⠚⠯⠙⠉⢀⣴⡄⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠙⠨⢁⠃⠀⠙⠈⠀⠀⠀⠀⠀⠴⠛⡿





⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣏⣻⣧⣤⣤⣴⣤⣬⣤⣭⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢸⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢂⠀⠀⠀⠀⠀⠀⠀⡔⡡⠒⠉⡉⠀⠑⠌⠄⠀⠀⠀⠀⠀⢀⣀⡤⣤⠠⣵⠊⠻⢆⡀
⠀⠀⠐⠚⠩⠭⣽⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠉⠑⠂⢄⠀⠀⠀⠀⠀⠐⢰⠄⠀⣀⠁⠀⠀⢨⢰⠀⠀⠀⠀⠀⢯⣀⠼⢏⣲⡟⢆⣲⢤⣽
⠀⠀⠠⠀⠤⢤⡌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⢠⡄⠀⠤⠀⠀⠀⠀⠀⠀⠐⡌⢆⠀⠀⠀⠀⠀⠔⡘⠀⠀⠀⠀⠀⠀⠉⠻⣍⠷⣋⣱⡥⠠⠋
⠀⠀⢰⣚⣁⣆⡈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠐⠚⢬⣭⡽⢂⠴⠉⠀⠀⠀⠀⠀⠀⠈⠢⢑⡂⢓⣒⠡⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂
⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⢠⣤⡀⢀⠄⠀⠠⠀⠀⠠⡀⠀⠀⠀⠀⠀⢀⠶⠾⠿⠿⠿⢏⡁⠀⠀⠀⠀⠀⠀⠀⢀⡀⡐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠸
⠘⡹⠒⠁⠀⠲⡢⣎⠂⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠀⠸⠀⠀⠀⠀⠀⠀⢰⡟⣯⢉⡴⠶⣈⣛⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠨⠀⢘
⠀⣇⢀⠳⣄⣘⣯⠇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠀⡏⠂⠀⢸⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠚⠢⢘
⠀⠞⠈⠓⠬⠥⠅⠂⠁⠙⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⢲⣖⣉⡹⠶⠜⠀⠀⠀⠀⠀⠀⠠⠉⢀⠈⠀⠁⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠆⠠


⠀⠀⠀⠀⠀⡀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀⡠⢄
⠀⠔⠒⠐⠤⣤⡴⠶⢈⠢⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠞⢀⣄⠀⠀⠀⠀⠀⠀⡠⠀⠐⠐⠂⠀⢸⠱⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣐⣂⣨⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠃⠀⠀⠀⠀⠘⣀⠥⠒⢈⠁⢀⡠⠛⠒⠁⠀⠀⠀⠀⠀⠉⠠⣰⡇⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⡀⠋⡗⠀⠀⠀⢉⠀⡇
⠸⢀⡀⠀⠀⠀⠀⠇⠨⠴⠀⠀⠀⠀⠀⢠⣥⠄⢰⠉⠉⠉⠑⠢⡄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠅⠁⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠧⠤⢀⡀⠀⢦⡎⠉⠁
⠀⠀⠀⠁⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠒⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⢀⣐⣐⡂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠀⡈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡘⠺⠁⡱
⡄⠚⠉⠙⠒⠒⠚⠓⠒⢰⠃⠀⠀⠀⠀⠀⠀⢶⡄⠸⠠⡌⠉⠹⠇⠀⠀⠀⠀⠀⠀⢎⣀⣃⠀⢓⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⣤⣄⠀⢆
⡇⠀⠀⠀⢠⠖⢋⡠⠀⢨⠀⠀⠀⠀⠀⠀⢘⣼⡇⠀⠀⠠⢤⡄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣁⢈⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⢨⢀⠆
⠁⣀⣀⣀⣀⣀⣉⣀⣄⠸⠀⠀⠀⠀⠀⠀⡄⢻⣇⣿⣿⠷⢶⣷⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠱⣀⣀⣀⣰⢋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠙⠿⠛⠟⠻⠟⠞⠫⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⣣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠓⠚⠙⠃⠉⠁

⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⡀⣀⢀⣀⣀⠀⢀⢀⢀⡀⡀⡀⡀⣀⢀⣀⡀⢀⡀⠀⡀⣀⢀⢀⣀⢀⡀⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠊⠘⠋⠿⠻⠻⠸⠀⠟⠺⠜⠃⠟⠓⠓⠻⠻⠇⠃⠸⠷⠀⠛⠛⠟⠛⠛⠸⠛⠇⠛⠸⠂

⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⠀⢰⣶⣶⠀⢠⣶⡆⠀⠀⣶⣶⠀⠀⠀⠀⣠⣶⣶⣦⠀⣴⣶⣶⣤⢰⣶⠀⠀⠀⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣾⣿⣿⡇⠸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⣿⣏⠘⠿⢻⣿⡇⢸⣿⣾⣿⠀⠀⠀⣿⣿⣿⡇⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇⣿⣧⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠘⢿⣷⣄⢸⣿⡇⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣧⣿⣿⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⣤⡄⢹⣿⣾⣿⡇⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣼⡿⠉⠹⣿⡦⣿⣷⣶⣶⣿⣿⣶⣶⠄⠀⠻⣿⣾⠿⠉⠿⣷⣿⠟⢸⣿⣷⣶⡶⣿⡿⣿⣷⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⣦⠀⣿⣿⣿⣿⣇⠀⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⠉⠉⠁⣿⣿⣿⠋⠉⢻⣿⣿⡇⣿⣿⣿⣿⣿⡀⢸⣿⣿⡇⠉⠉⠉⣿⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿⣿⣧⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⡀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣼⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⣿⣿⣿⡏⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⠉⠁⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⣿⣿⣿⡇⢹⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣦⣤⣾⣿⣿⠇⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⠋⠀⣿⣿⣿⠇⠀⠀⢿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠀⡄⢠⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣠⣤⣠⡀⠀⣠⣠⣤⣤⣠⡀⠀⣠⣤⣄⣤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⢰⠀⣘⡐⡇⠈⠀⠃⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⡇⠀⣿⣽⣿⣻⣿⡿⠀⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣶⣄⠀⠀⡀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⢻⣿⣿⣿⣿⣿⠀⢠⣧⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣄⠀⠈⠛⠿⠿⠛⠉⠀⣈⣹⠀⠀⠈⠙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠉⠛⠋⠀⠀⠀⣠⢦⠀⢠⣿⠿⣧⠀⢰⣮⣛⠿⣿⣾⣿⢟⣣
⠀⠀⠀⠀⣾⣿⡆⠀⣠⣴⣾⣿⣿⣿⣷⣦⡀⠀⣸⣇⠀⢸⡿⠓⣿⠀⠸⢿⠾⠶⣿⡷⠷⢾⠷
⠀⠀⠀⠀⣿⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⢉⣉⣀⣈⠉⠀⢸⣷⠴⣿⠀⢘⣭⣾⣷⣿⣶⣮⡛⢿⠀⠺⠂
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⢸⣿⠛⣿⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⢸⡿⠿⣿⠀⠀⡠⢄⣸⡇⡠⠤⠀⢰⣿⣿⣿⣿
⠀⠀⣼⠿⣿⣿⡿⣿⡀⠈⠃⠀⠀⠀⠀⢰⠀⠀⣶⠀⠀⢸⣿⣴⣿⠀⠘⢀⣤⢼⡧⣤⡠⠃⢸⣿⠀⠀⣿
⠀⠀⣏⣴⣶⣷⣶⣜⡅⠀⢀⣾⣧⣀⡄⢸⠀⣶⣷⣶⠄⢸⣯⣩⣿⠀⠀⢰⣿⣾⣶⣷⡆⠀⢸⣿⠀⠀⣿
⠀⠀⠁⠛⠛⠛⠛⠁⠁⣤⣾⣿⣿⣯⡀⢸⠀⠚⠛⠛⠀⠚⠛⠛⠛⠀⠀⠀⠀⠀⢀⣤⡀⠀⢸⣿⠀⠀⣿
⠀⠀⢠⡄⢠⣶⡄⣿⠀⠈⠉⣉⡀⠈⠀⢸⠀⢰⣋⣛⣉⣋⣙⣋⣙⣋⣙⣉⣢⠀⢾⣿⣷⠆⢸⣿⣶⣾⣿
⠀⠀⠈⢿⡌⠉⠀⣿⠀⣠⠀⣨⡀⢠⡀⢸⠀⠘⠛⡿⠿⠿⠿⡿⠿⠿⠿⣿⣿⠀⠀⢿⠅⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠘⣷⡀⠀⣿⠀⠀⣀⠙⢁⡀⠀⠻⠀⢸⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⠀⠀⢠⣶⣿⣷⣆
⠀⠀⠀⠀⠈⢇⠀⡇⠀⠀⠉⠀⠈⢁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⢁⣤⣤⣤⡀⠀⠾⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠈⠀⠁⠀⡁⠀⣀⢰⣿⣷⣆⣀⠀⠘⣿⣿⣿⡟⠀⠀⣴⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠠⣶⡗⠀⢸⣼⣿⣿⣇⡜⢀⣾⣿⣿⣿⣿⣆⠀⣿⣿⣯⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⢸⣿⠿⢿⠿⢿⣿⠀⠘⠿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⣿⠀⠀⠀⢸⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⡄⢰⠆⢲⠂⣠⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡷⠇⠳⠎⠀⠃⠀⠃⠻⠤⢎⠜⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠶⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡤⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡆⠃⠁⠀⠀⠛⠛⠛⠲⠶⣤⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠚⠥⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⣀⣠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⢀⣑⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠦⢁⡀⠀⠀⠀⠀⠀⠈⠐⠤⢀⠀⠀⠠⠆⡉⠻⠂⠀⠫⠠⠙⠁⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠈⠑⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠒⠒⠀⠠⠄⢀⡀⠈⠂⠄⡀⠀⢢⣄⠀⣰⡇⠀⣀⠄⠊⠀⠀⣀⠠⠤⠄⠀⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣙⣲⡜⠟⠙⢭⣷⡊⠀⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣃
⠀⠀⠀⠀⠀⠀⠀⠲⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡾⠛⠛⠁⠀⡀⣀⠀⠈⢙⡻⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⣙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣅⠘⠇⢺⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡓
⠀⠀⠀⠀⠀⠀⠀⠬⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣟⣯⡴⣒⣲⣦⣬⣛⠷⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭
⠀⠀⠀⠀⠀⠀⠀⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠾⠋⠟⢈⣡⠤⢤⠈⠙⠻⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠭⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠐⠀⠨⠆⠁⡠⠴⠦⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠄⠀⢇⠈⡠⠖⠒⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠄⠀⠀⠀⠀⠀⠀⡄⠀⠘⠋⠀⠈⢆⡀⠀⠊⢡⠀⠈⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠈⠑⠂⠚⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢼⣥⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣯⠖⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡔⢻⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠟⢱⣣⢎⡴⠦⠀⠀⡀⠀⡀⢀⡀⠀⠀⡄⠂⡞⡇⠀⢳⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠍⠀⠀⢰⢳⣸⢱⠋⣹⠀⠀⠏⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠾⠿⠿⠶⠶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠉⠉⠉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣰⣿⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡀⠀⠀⠀⠀⠀⣸⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢷⣤⣀⣀⣠⣴⡟⠁⠉⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠉⠉⠉⠉⠙⢿⣄⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠹⣷⣀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⠶⣶⣶⡶⠾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢣⠀⠀⠀⠀⠀⠀⠰⠎⠉⠉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣀⣀⠀⣀⠀⠀⠀⠀⠀⠘⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡏⠉⡉⠉⡉⢿⡗⢰⣶⡄⠀⡘⣛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⣼⠃⠸⣿⡿⢃⣸⠋⢀⣽⣿⢀⡀⠀⠀⠉⠢⢄⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⣉⣀⠛⠧⢴⣿⣿⣿⣿⠁⢾⣶⣤⣀⡀⢀⡀⠑⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡄⠀⣴⡗⢘⠛⢿⡇⠀⠈⠙⣷⡄⠨⡀⠀⢹⣿⣿⡏⠀⠐⠀⠀⠹⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢄⠀⠀⠀⠀⢸⣷⣿⢗⡀⢀⣴⣧⡀⠟⠁⠀⠀⠀⢸⡟⢿⣿⣾⠗⠀⢠⣄⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢱⠀⡀⠀⠀⠀⣀⣈⡟⠛⢿⠟⠻⣧⣀⣡⣄⣀⠀⠀⠀⠀⠀⠀⣂⡀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⠀⠀⢸⠏⠀⣴⣶⣤⠀⢰⠀⢨⣿⠟⠃⠀⠀⠁⠚⠱⣿⣿⠟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⠀⠀⠀⡄⠀⣿⡟⠿⠶⠿⡇⠘⣻⣅⠀⠛⢀⣠⠀⠺⣿⣿⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⣀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⠙⠷⠂⠀⠀⣇⠀⠘⢿⣷⠀⠙⢿⡀⠰⠿⠏⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⣼⡀⠀⠀⢚⠁⣀⠀⠀⣿⣷⣄⣀⣀⡴⠋⠉⠀⢘⣨⡿⠀⠀⠠⡏⠂⢀⣐⣚⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢩⡿⠀⠆⣸⠀⠈⠀⢠⣿⡏⠉⠉⠉⠓⠀⢂⠀⣸⠟⠃⠀⠀⠀⣼⠟⠋⢩⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠄⡤⠀⠈⢀⣠⣴⣿⣯⠀⢀⣠⡾⠃⢀⡾⣾⠃⠀⠀⠀⠰⡗⠉⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠸⣼⣇⠀⠀⣾⣿⣿⢏⠟⠀⣿⣯⡀⠐⢟⢠⣷⠀⠀⠀⠀⠁⠲⣤⣄⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣿⡟⠀⠀⠘⣿⣧⠊⡠⠜⠛⠉⠷⠀⠈⣂⠳⠆⣤⡀⠀⢠⣤⣿⡿⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡄⠙⠿⠧⡤⢴⠀⠈⠩⢷⣶⣦⣤⠀⠀⠀⢠⡿⠄⢴⣶⣾⣿⡿⠋⠉⠁⠀⢸⣿⣿⠷⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢿⡶⡀⠀⢀⣸⠀⠀⠔⠀⢻⣿⣿⡀⠀⠀⠂⢀⣤⣄⠈⢿⣏⠀⢿⣿⠆⣸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠈⠢⡀⢱⣄⠀⠀⠀⠈⣿⡟⢁⣀⠀⠀⣹⠟⠻⡇⠘⠛⠀⠈⠁⠶⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠊⠀⠀⠀⠀⢈⠀⠑⠢⠀⠀⠛⠛⠋⠀⠀⠈⣇⣰⡄⠠⡄⠀⢀⣠⣷⠄⠀⠤⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠛⠓⢒⡤⠊⢣⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡄⠀⠀⠈⠉⠀⠀⠀⠠⣾⡿⣿⠆⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢠⡞⠦⣒⡄⠄⡠⣤⠴⠿⠇⠀⠀⠀⠀⢂⡺⠏⠀⠀⠀⢠⣿⠀⠹⠆⢀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠙⢧⠜⠁⠀⠀⣰⡀⠀⣀⡀⠀⠙⢧⠀⢠⣾⠄⠀⠃⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣀⠀⠀⠘⠲⠀⠀⣿⠁⢼⣿⡿⠰⣶⠟⡠⢿⡄⠀⠀⠀⢠⠅⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⢰⠋⠾⡷⠀⣐⠠⠾⢤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠰⠆⠀⠀⠹⡗⢀⡀⠈⠠⡀⠈⢣⣿⠿⣣⣴⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣏⣄⣀⡁⠀⣈⢁⡘⢛⠃⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⣎⠀⢻⣿⠀⠀⣨⠋⠙⠋⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢻⠀⢸⡅⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣧⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠆⢹⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠁⠻⠈⠀⠀⠀⠛⠣⣑⡡⠋⠀⠀⠀⠀⡜⠣⠤⠂
⣭⡉⠉⠉⠀⢀⠖⠉⠁⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠠⠤⠷⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠉
⠨⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⢀⣠⠶⣄⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⣐⣂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡄
⠤⢼⡷⠀⠀⠀⠙⢴⡏⡠⠛⠀⠀⠀⠭⠏⠀⠈⠉⠽⠃⠀⠀⠸⢄⡠⠻⠀⠄⠄⠀⠀⠀⣉⡉⣹⠁⠀⠀⠀⣛⠦
⣀⣨⡤⠀⠀⢀⠀⠀⢉⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⠀⠀⠐⠀⠀⠀⠈⠙⠀⠊⠁⠀⠀⢀⣼⣀⣀
⡀⠘⠓⠀⠀⠈⠀⠁⠉⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠉⡁⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⠐⡆⠀⣶⠀⣤⡀⣠⠀⠀⠀⡴⠤⠋⠣⠴⡄
⠀⠀⢰⢶⢶⣶⡶⣶⣶⢴⠀⠀⠀⢠⠋⢆⡜⢱⣰⠉⡄⠀⠠⠄⠀⠀⠼⠤⠔⠂⠒⠀⠺⠄⠀⠀⢤⡤⣄⠤⣤⢤⣠
⠀⠀⠀⠄⠤⠀⠀⠤⠤⢤⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠐⠆⠀⠂⠒⠐⠰⠀⠀⠀⠄⠤⠀⢠⠄
⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠀⣶⣶⣀⣀⣀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠤⠀⠀⠀⠸⡇⡇⡏⠀⠀⠀⠀⠀⡦⣦⡖⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠺⠿⠀⠀⠀⠊⣃⣀⡩⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠿⠁⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠙⠽
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠂⠐⠂⠀⠠⠀⠐⠀⠀⠀⠞⠟⠦⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣴⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣲⡆⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠰⠷⣭⢾⠀⠀⠀⡐⢊⡉⢦⠀⠀⠰⠤⣶⣶⠶⠶⠀⠀⠀⡰⢧⠀⠀⠀⠀⡶⡒
⢄⡇⠀⠀⡁⠠⢚⣉⣑⠶⠀⠐⠀⠀⢀⠁⠀⠀⠀⠀⠣⣈⣁⠜⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⣴⣛⣛⣣⠀⠀⠀⣇⠜
⠩⠁⠀⠀⠀⠂⠄⠠⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⢩
⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢓⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣎⡱⢄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣲⣀⠀⠀⠀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⢀⣴⣝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠃⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣯⡵⣀⣤⠞⣰⢺⠟⣐⣦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⣤⣌⡄⠀⠞⢁⢘⡟⠀⡘⠛⢛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣿⠿⡟⣀⣀⡀⢸⡜⠠⣾⠃⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣃⡤⠂⠋⠜⣿⣿⢿⡆⣸⣧⣴⠇⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣰⡿⣠⠀⠀⢠⠿⣿⣦⣶⣿⡟⠻⣀⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠒⠛⠛⢿⠙⣃⣤⣥⣤⡈⢙⡳⢤⡁⢤⣹⡼⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣴⢯⡾⣫⣤⣤⡈⢻⡎⣿⣦⠑⢸⣿⠷⢸⣯⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠋⢠⠏⢸⣿⡏⠙⠏⡼⢃⣿⣿⡆⠈⣽⡷⢈⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠁⠀⠀⠀⠸⣿⣧⡀⠲⠶⠿⣿⣿⠇⠚⢋⡃⣶⡹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣘⠻⣿⣶⣶⣿⣿⣿⠿⣷⣄⠙⣿⢁⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⠁⠀⠈⣿⡇⠘⠼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠶⣶⠿⢁⣷⣄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠐⢿⡋⢻⡌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠋⠙⢷⣄⢻⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⡀⠀⠀⠘⣿⣿⣿⡇⠀⢠⡀⠀⢻⣇⠇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣇⠀⠀⠀⣿⣿⣿⣷⣄⣸⣿⣄⠘⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⢿⣦⣤⣼⣿⣿⣿⣿⡿⢿⣿⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠸⣿⡙⠛⠁⢸⣿⡏⠀⠀⠘⣿⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠹⣿⣄⠀⢸⣿⡇⠀⠀⠀⣿⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠘⢿⣷⣄⢿⣧⠀⠀⠀⡟⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠙⢿⣿⣿⣧⠀⠈⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠙⠛⠙⠙⠋⠛⠛⠛⠉⠙⠛⠻⠟⠛⠛⠛⠛⠛⠛⠛⠿⠿⠛⠿⣿⠿⢿⣿⡛⠛⠻⠿⠟⠋⠉⠙⠛⠿⠿⠿⠿⠟⠛⠛⢿⡿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⢻
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾
⣿⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⢿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠂⠀⡀⠀⠀⠀⠀⠀⢀⠀⠁⠄⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤⠀⠀⠀⢠⣾
⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠠⠌⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⠀⠀⠀⠠⠐⠀⠀⢻⣿
⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠤⠄⠀⠀⠀⠙⠉
⡏
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⡧⠀⠀⠀⠀⠀⠒⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣦⠁⠠⠉⣠⢫⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⣘
⣿⠄⠀⠀⠀⠀⢄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠇⠀⠚
⣿⣷⠀⠀⠀⠀⠀⠈
⣿⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠫⢂⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘
⣿⡁⠀⠀⠀⠀⠀⢀⡀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿
⣿⣧⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣶⣶⣿⣿⣧⣠⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠉⠙⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣿⡿⠋⠁⣠⣴⣦⣄⡀⠉⠻⣿⣷⣶⣶⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠛⠛⠉⠉⠉⠋⠀⣰⣿⣿⣿⣿⣿⣿⣦⠀⠙⠉⠉⠉⠛⠛⠛⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⣴⣶⣶⣾⣶⡆⠀⠙⠋⠉⠉⠉⠉⠉⠛⠀⢠⣶⣾⣶⣶⣶⡄⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣸⣿⣿⣿⠟⠁⢀⣠⣴⣶⣶⣿⣿⣶⣶⣦⣤⣀⠈⠛⢿⣿⣿⣿⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⣿⣿⠟⠁⣠⣶⣿⣿⠿⠟⠙⠋⠉⠋⠛⠿⣿⣿⣷⣦⠀⠙⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠃⠀⠈⠉⠀⣼⣿⣿⡟⠉⠀⠀⠠⠀⡇⡀⢀⠀⠈⠙⣿⣿⣷⡀⠈⠁⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃⢀⣴⣿⡗⠀⣼⣿⣿⠇⠀⠀⡐⢌⢦⠇⡧⢣⡣⢂⡀⠀⠈⣿⣿⣷⠀⠰⣿⣷⣄⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⢠⣿⣿⣿⡇⠀⣿⣿⡿⠀⠀⠐⠨⠅⠀⣴⣦⡄⠨⠅⠒⠀⠀⢸⣿⣿⡆⠀⣿⣿⣿⣧⡀⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠙⣿⣿⣿⡇⠀⣿⣿⣷⠀⠀⠠⠄⣓⠀⠻⠟⠁⢒⡁⠄⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⡟⠁⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠈⠻⣿⡧⠀⢻⣿⣿⡆⠀⠀⠡⠊⠎⠆⡏⡜⡑⢌⠁⠀⢀⣿⣿⣿⠀⠰⣿⡿⠋⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⢀⣀⠈⢿⣿⣿⣦⡀⠀⠈⠐⠀⡇⠁⠈⠀⢀⣠⣿⣿⣿⠃⢀⡀⠀⢀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣾⣿⣧⡀⠙⢿⣿⣿⣶⣦⣄⣄⣀⣄⣤⣶⣿⣿⣿⠟⠁⣠⣿⣿⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢹⣿⣿⣿⣦⣀⠉⠛⠻⠿⢿⣿⣿⣿⠿⠿⠛⠉⢀⣠⣾⣿⣿⣿⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠈⠿⠿⠿⣿⡿⠗⠀⣠⣀⣀⣀⣀⣀⣀⣠⠀⠘⠿⣿⣿⠿⠿⠋⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣤⣀⣀⣀⣀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠀⢀⣄⣀⣀⣠⣤⣴⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⣿⣧⣄⠀⠙⠻⠿⠛⠁⣀⣴⣿⡿⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⣀⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠟⠋




⣀⡀⣀⡀⣀⣀⣀⢀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⡀⢀⣀⣀⠀⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡶⢿⣿⣿⣿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁⠀⠀⠀⠉⠛⢮
⠀⠀⠀⠀⠀⠀⠃⠠⣤⠈⠛⠁⢠⡄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⣦⠀⢀⠀⢠⡗⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡄⠀⠀⠀⣤⠀⢰
⠀⠀⠀⠀⠀⢀⠥⢜⠃⠀⠶⠀⠘⡃⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⡃⠀⠈⠁⠀⡛⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢛⠀⠀⠉⠀⠘⠛⠪⡀
⠀⠀⠀⠄⠂⠢⠀⠀⠃⠀⠀⠀⠐⠀⠀⠅⠀⢂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠈⠁⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠉⠈⠀⠁⠙⠄⠀⠋⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠌


⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⢸⠀⡻⠆⠀⠀⠀⠴⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⠀⠒⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⢡⡤⠄⢤⠍⠁
⠀⠀⠀⠀⠀⠀⠀⣬⢃⠀⠦⢠⠻⣤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠸⠀⠀⢰
⠀⠀⠀⠀⠀⠄⠁⠐⠄⠀⠀⠀⢴⠀⠉⡆⠀⡀⠀⠀⠀⠀⢀⠀⠠⠁⠀⡢⠀⠀⠀⠠⠀⠀⠱⠀⢀⠀⠀⠀⠀⠀⡀⠀⠎⠀⠀⠄⠀⠀⠀⢰⠁⠈⠄⠀⡀
⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⢀⠰⠒⠒⠐⠒⠚⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⡐⠓⠀⠀⠀⠚⠆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠄⠤⡤⠄⢤⣤⠂
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣀⠀⡀⡀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠀⠀⠀⠀⠀⠀⠈⢊
⠀⠀⠀⠀⠀⡆⠀⣼⠇⣀⣄⡀⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠏⡀⠴⠈⠘⠶⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣼⠃⣀⣄⡀⢻⡄⠹
⠀⠀⠀⠀⠀⡜⠉⠑⡄⠀⠀⠀⢤⠈⠉⡄⠀⡀⠀⠀⠀⠀⠀⠀⢠⠉⠁⡆⠈⠀⠀⢠⠊⠉⢢⠀⡀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⡄⠀⠀⠀⣤⠈⠑⡄
⠀⠀⠀⠁⠀⠈⠒⠊⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠈⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⣿⣿⣿⣿⣿⣿⣿⡿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠃⠀⠈⢿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠉⠉⠁⠠⡤⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⠟⣵⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⠇⣿⣿⣿⡛⠟⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⠸⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣦⣈⠛⠿⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⡿⠟⠋⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡚⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣉⣹⠃⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣯⣉⡛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢎⡡⠼⣦⡤⠖⠚⡏⠉⠉⠉⢉⡽⠉⠙⠓⠲⢤⣠⠯⢤⡙⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡴⠊⠁⠀⠀⠀⠙⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⠑⢦⡈⡾
⠀⠀⠀⠀⠀⠀⢶⠚⣩⣭⣭⣗⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⣺⣭⣭⣍⠓⡦
⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⣿⠃⠀⠀⠀⢠⡶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠸⣿⣿⠟⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢿⠀⠀⠀⠀⠘⠿⡿⠃⢀⣀⡀⠀⠀⢀⣀⠀⠘⠿⡿⠇⠀⠀⠀⠀⡗⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢴⣶⣶⣶⡄⠠⡏⠰⠍⠋⠙⠡⠆⢹⠀⢰⣶⣶⣶⡦⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠲⣄⠀⠉⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠉⠁⢀⠞⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠓⢦⡀⠀⠘⢦⣀⠈⠛⠛⠁⣀⡴⠃⠀⢀⡴⠚⠉⠉⠓⢤
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⠁⠀⠀⠀⠀⠀⢹⣤⣄⣀⣈⣉⣙⣋⣉⣁⣀⣠⣤⡞⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⡀
⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⠓⣄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠙⠦⢤⣠⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣄⣤⠴⠊⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⡗⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣀⡎⠀⣀⡀⢱⣄⠀⠀⠀⠀⠀⠀⠀⡼⠉⠳⣄⣇⣠⠞⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡠⠃⠀⠀⠈⢀⠜⠙⡄⠀⠀⠀⠀⠀⠀⡖⠒⢒⡾⣿⣗⠒⠒⣛
⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠃⠀⠀⠀⢀⠜⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣴⠋⠀⣇⠈⠲⣤⠋
⠀⠀⠀⠀⠀⠀⣴⠋⠚⠹⠿⠿⠛⢢⠸⣠⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⠓⠀⢐⠋⠀⣨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠛⠒⠋
⠀⠀⠀⠀⠀⡰⠃⠀⠑⠢⠄⠤⠒⠋⠀⠁⣃⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⣰⠁⡀⢀⠀⡀⢀⠀⢀⠀⡀⢀⠈⣗⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣽⣿⡿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠘⠀⠋⠈⠃⠙⠀⠃⠐⠁⠂⠁⠂⠑⠉⠂⠀⠀⠀⠀⠀⠀⣤⢠⡇⢠⡟⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⣿⣷⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⣆⣸⠟⠋⠉⠙⠛⢦⣴⠂⢀⠀⡀⠀⠀⠀⡄⠀⠀⠈⠛⠛⠛⠉⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢖⡿⠡⠶⠌⠀⠘⣷⡄⢹⡎⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⡴⢫⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢺⣧⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⠊⠉⠀⠀⠀⠀⠁⢄⡀⠀⠀⠀⠀⠀⣀⠘⠁
⠀⠀⠀⠀⠀⣼⠁⡾⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠑⢻⡄⠀⠠⣶⠄⠀⣀⡿⢛⠁⠉⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⠀⣿⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⡹⠶⣤⢤⣤⣞⠋⠃⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠘⣆⠙⢷⣄⠃⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠁⠸⠃⠈⠀⠙⠡⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢐⡀⡤⣗⣲⣲⠈⡀⡀
⠀⠀⠀⠀⠀⠀⠘⢧⣄⠙⠳⠦⠤⡤⢤⣴⠞⠁⠀⠀⠀⣀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⡹⠊⣏⣿⢣⠀⡸⣯⡄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠚⠉⠀⠀⠀⠀⠀⢸⠁⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢹⠀⠀⢨⡇⢰⡀⡵⢾⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠢⠤⣀⠀⠀⠀⠂⠠⠚⡘⣶⠁⡎⢻⣦⠀⣴⡿⡲⠆
⠀⠀⠀⠀⠀⣀⣠⡴⠞⢛⣙⣛⡻⢶⡄⠀⠀⠰⡆⠠⠴⣉⠧⠤⢴⠆⠀⠀⢀⢠⣿⠃⠀⠀⠀⠈⣧⡀⠀⠀⠀⠈⢱⠌⡳⢴⣄⣤⡴⢋⠞⢁
⠀⠀⠀⣤⣯⣭⣥⣶⠋⠩⡟⣧⠉⣳⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⡀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠐⠁⠼⠈⠃⠙⠆⠀⠈⠃
⠀⠀⠀⠈⢀⣄⠀⠈⠙⢲⣿⣿⡞⠁⠀⠀⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠰⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠈⠲⠶⠶⠾⠛⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣬⣧⣤⡤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠙⣧⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠦⠄⢈⡀
⠀⠀⠀⠀⠀⠀⣠⡤⠛⣶⠓⠣⣀⡀⠀⠀⠀⠀⠀⠀⢤⠀⡄⢠⡄⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⠁⠂⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢰⠃⠀⠀⣿⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠈⣶⣷⡞⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⣷⣤⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⡂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠻⠙⠂⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠈⠀⢾⠃⣸⠈⢷⣆⠖⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠭⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⢦⠀⣠⠊⠁⠀⢹⠀⢀⡤⠒⠋⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣽⡴⡆⠀⠀⠀⠀⠀⢠⠋⠀⠀⣸⡜⠁⠀⠀⠀⣼⠖⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣉⡘⢓⡿⠂⠀⠀⢀⠏⠀⠀⠀⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠁⠀⠀⠉⠙⠲⢄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠰⡦⢄⣀⡞⠁⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠱⣴⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠙⣦⡼⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡠⠔⠚⠉⠁⠀⠀⢈⡧⠖⠚⠉⠉⠉⠙⡆
⠀⠀⠀⣾⠵⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠋⠉⠉⢢
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⠒⠶⠖⠒⠋⠁⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⣩⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⣁⣀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠀⠀⠀⣀⣀⠤⠒⢫⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠛⠛⠋

⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠐⠁⠀⠀⠈⠄⠀⡐⢡⠀⢠⠐⡀⠀⡌⠀⠀⠀⠑⠀⠀⠌⡀⠀⢀⠑⠀⠀⠊⣀⠀⣀⠱⠀⠀⠊⣀⠀⡄⢡
⠀⠀⠠⡀⠤⠤⢀⠂⠀⠡⠰⠤⠴⠠⠁⠀⢃⠈⢤⠈⡠⠀⠀⢂⠀⠒⠂⡠⠀⠀⢄⠀⠒⠀⡰⠀⠀⢄⠀⠒⠀⡘
⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠁⠀⠉
⠀⠀⢀⠄⠂⠐⢄⠀⠀⢠⠉⠆⠂⢄⠀⠀⢀⠔⠀⠂⢄⠀⠀⠀⠐⠀⠀⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠐⠀⠢⡀
⠀⠀⠀⠀⠀⠐⠀⡃⠀⡌⠐⣀⡐⠀⡆⠀⡆⢐⣀⣀⠀⠀⠀⡁⣀⣀⣀⡈⡀⠀⠀⡲⠀⢆⠈⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠐⠄⡀⠁⠔⠁⠀⠐⠄⠀⢀⠔⠀⠀⠐⠌⠉⠉⠔⠀⠀⠑⠝⠛⠫⠀⠀⠀⠑⠅⠀⠸⠊⠀⠀⠁⢓⠶⠫⠂

⠀⠀⠠⠊⠀⠀⠑⡀⠀⡐⢁⠀⠀⠑⠀⠀⡔⠁⠀⠈⠢⠀⠀⠔⠁⠀⡈⢂⠀⠀⠄⠁⠀⠁⠆⠀⠈⠓⠁⠀⠈⢂
⠀⠀⢀⠐⢠⡐⠀⠆⠀⢃⢨⠥⢭⢀⠁⠀⠄⠐⠤⠂⢀⠀⠀⡀⣍⣀⡅⠠⠀⠘⡀⣅⡀⡁⡘⠀⠘⡁⠩⠤⠁⠸
⠀⠀⠀⠑⠀⠐⠁⠀⠀⠀⠙⠂⠊⠁⠀⠀⠈⠐⠀⠐⠁⠀⠀⠈⠂⠀⠀⠁⠀⠀⠀⠣⠤⠃⠀⠀⠀⠈⠂⠀⠂⠁
⠀⠀⠽⢠⠀⠀⢄⠀⠀⢀⠄⠀⠀⢄⠀⠀⢀⠄⠀⠀⢄⠀⠀⢀⠄⠀⠠⡀⠀⠀⢀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠈⢀⡀⠀⠁⠀⡆⠀⠀⠀⠈⠀⠀⡃⠠⠀⠤⠀⠀⠀⠁⠤⠀⠠⠈⠀⠀⠁⠒⠀⠂⠘⠀⠀⠁⠄⠀⠤⢰
⠀⠀⠀⠤⣘⣃⠔⠁⠀⠐⢄⡀⡀⠔⠀⠀⠑⢌⠀⠩⠔⠀⠀⠡⢙⡛⠻⠌⠀⠀⠠⢑⠐⡋⠈⠀⠀⠣⢀⢛⡠⠊

⠀⠀⢀⠀⠈⠁⠢⠀⠀⢀⠂⠉⠁⠢⠀⠀⠠⠂⠉⠁⠢⠀⠀⣠⠂⠉⠑⣀⠀⠀⡠⢂⠉⡐⢄⠀⠀⡀⠈⠉⠐⠄
⠀⠀⠀⠘⣀⡘⠀⡅⠀⠆⠐⣂⡒⠀⠇⠀⡇⠀⣀⡀⠀⠀⠀⡈⠛⠋⠛⢃⠀⠀⡈⠛⠀⠛⢁⠀⠀⡀⢁⣤⠉⢸
⠀⠀⠈⠒⠀⠠⠊⠀⠀⠈⠢⠀⠠⠊⠀⠀⠈⠢⠀⠀⠊⠀⠀⠈⠂⠀⠁⠊⠀⠀⠐⠨⠉⠁⠂⠀⠀⠑⠈⠛⠄⠁
⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⠀⣰⣤⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠐⢁⠀⢠⠐⡀⠀⡰⢡⠀⢠⠁⡀⠀⠄⠀⠀⠀⠱⠀⠀⠎⠀⠘⠉⠁⠀⠀⠊⡀⠀⢀⠡⠀⠀⠎⢀⠀⡀⢡
⠀⠀⠠⠀⠀⠆⣤⡇⠀⠡⠐⠤⠔⠀⠃⠀⢂⠲⠤⡴⠢⠂⠀⢄⠠⠤⠄⡠⠀⠀⢄⠀⠠⠀⡰⠀⠀⢄⠠⠤⠅⡘
⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠁⠀⠈
⣿⣿⣻⣛⣻⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠚⣭⡔⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠠⠠⠀⢠⣆
⠀⠀⢈⡀⢨⡯⠀⠀⣴⠷⠘⠓⠄⠀⠀⠒⢅⠀⠀⠉⠀⠀⠰⢦⢁⠄⢁⠀⠸⡧⠤⠠⠀⠠⠿⠃⠀⢀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⡆⠉⠀⢠⢟⠀⢱⠀⠂⠀⢀⡂⠀⠈⠙
⠀⢀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⡠⠤⠌⠀⠀⠀⠀⠀⠿⡂
⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠀⠀⠀⠀⠀⡀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡃⢸⠀⠰⣄⣀⠜⠀⢘⡇
⠀⠀⠀⠀⠀⢠⠄⠂⠀⢠⠀⢈⡰⠀⠀⠀⠀⠀⠀⡀⠤⠐⠁⡀⠄⠒⠂⠄⠔⢲⢒⠢⡀⠀⠀⠀⠀⠛
⠀⠀⠀⢀⠄⠀⠃⠐⠄⠂⠁⠢⡄⠠⡀⠐⠀⢰⠐⠀⠄⠁⠀⠁⢀⠀⠈⢂⠘⠀⠀⠌⡀⠀⡀⡀⡂
⠀⠀⠀⠐⠀⠀⠀⠢⡀⠈⠁⠈⠂⠒⠄⠑⠀⠁⠀⠈⠂⠢⠀⠀⠈⡀⠀⠂⡀⡀⠀⠀⠤⠠⠀⠀⠄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠈

⠀⠀⠀⠀⢀⡄⠀⠀⣠⣀⠀⠀⠀⢀⣀⠀⠀⣠⣀⡀⠀⠀⣠⣤⣤⡀⠀⢀⣄⣤⡄⠀⠀⠀⣀⣄⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⡏⡄⠀⠀⡇⠀⠀⠀⢰⡏⠀⠀⠀⣿⠀⡄⠀⠀⣸⠀⣀⠀⠀⢸⡇⢀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡇⢀⡿
⠀⠀⠀⢠⠃⢀⠀⠀⡇⢠⠁⠀⠸⡇⢢⡄⠀⣿⠀⠏⠀⠀⢿⠀⣠⡀⠀⢸⡇⠀⠀⠀⠀⢿⠐⠏⠃⠀⠀⡇⠀⡄
⠀⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠁
⠀⠀⠀⣤⣤⣤⡄⠀⢠⡤⠤⠤⠀⠀⣤⠄⣠⠤⠀⢀⡤⠄⠀⠀⠀⢠⠤⢠⡤⠄⠀⠀⢠⣤⡀⣠⠄⠀⠀⢀⣤⠄
⠀⠀⠀⠉⣆⠀⠀⠀⠈⢰⠀⠀⠀⠀⡇⠘⠁⠀⠀⢸⡇⠀⠀⠀⠀⣼⠀⠀⡀⠀⠀⠀⢸⠀⠀⢿⠀⠀⠀⣿⠁⢰⡆
⠀⠀⠀⣠⠿⠀⠄⠀⣴⠺⠀⠀⠀⠀⣿⠀⡄⠀⠀⢸⡇⠐⠶⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⠀⠘⣆⠀⠀⠀⢿⡀⠘⠃
⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠉
⠀⠀⠀⢠⠤⠄⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⡴⠢⠀⠀⠀⠀⣴⠖⠂⠀⠀⡶⠶⠒⠶⠀⠀⡶⠆⣶⠆⠀⠠⡶⢦⡶⠶
⠀⠀⠀⢸⠀⠐⠂⠀⠀⣾⠀⢰⠂⠀⠀⠀⣿⠘⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣇⠠⡟⠀⠀⠀⣧⠘⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣆⠀⠀⠂⠀⠀⣿⢠⡀⠀⠀⠀⢼⠗⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⢻

⠀⠀⢰⠶⠀⠀⢰⠖⠆⠰⡖⠖⢠⠖⠆⠀⣶⠒⢴⠶⠆⠀⢰⠒⠒⠒⠀⠀⡶⠄⠀⡤⠂⠄
⠀⠀⢸⡆⠰⠋⠺⠀⠀⠀⢹⡄⠈⠀⠀⠀⠘⣦⠈⠀⠀⠀⠀⣠⠃⠀⠀⠀⡇⠀⠀⢠⠟
⠀⠀⠀⣧⠀⢰⡀⠀⠀⠀⡜⠀⠰⡀⠀⠀⠀⡞⠀⠀⠀⠀⢰⠃⠐⠂⠀⠀⡴⠄⠀⢨⠶⠀⢠⡶⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⡀⣠⡀⢀⣤⣤⡀⠀⢀⣀⣀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢰⡏⠐⣦⠀⢰⣉⠀⠀⠀⢿⣤⠀⠀⠈⢁⡄⠀⠀⠀⣿⠀⠇⠀⢸⠀⠀⠀⠀⣟⠀⠀⠀⠀⠁⢰
⠀⠀⠀⠸⣇⠀⠟⡀⢀⣸⠀⠀⢀⡞⡡⠀⠀⢀⡀⠱⠀⠀⠀⠀⠀⡇⠀⢈⣡⠆⠀⠀⢻⠀⣤⠀⠀⠀⡞
⠀⠀⠀⠀⠉⠁⠈⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠂⠀⠀⠈⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⣏⠐⠀⠀⠏⠠⠄⠆⠀⠘⠉⡾⠉⠁⠀⢀⣸⠉⠁⠀⠀⢀⣀⣠⣤⠀⠀⢹⡉⠋⠉⠀⢠⠤⠤⠴⠄
⠀⠀⠀⠰⡏⢀⡆⠀⢀⣀⡆⠀⠀⠀⣸⠁⢀⡀⠀⠙⢠⠄⠀⠀⠀⠙⠀⠀⠀⠀⠀⣰⠃⡀⠀⠀⠀⠖⠒⠛⠃
⠀⠀⠀⠀⠙⠂⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠋⠁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠙

⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠉⠙⠶⣄⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠈⠙⠳⣄
⠀⢀⡾⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⢠⠟⠀⠀⠀⢀⡀⡀⣀⠀⠀⠀⠻⡄⠀⠀⠀⡼⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢸⠃⠀⠀⠀⢋⡔⢰⣿⣷⠀⠀⠀⠸⡇⠀⠀⡟⠀⠀⢰⣌⠲⠥⠿⠈⠣⠀⠀⠀⢻⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⣒⡠⣀⠀⠀⠘⡇
⠀⢸⠀⠀⠀⠀⢸⡄⠈⢛⠅⠀⠀⠀⢀⡇⠀⠀⣇⠀⠀⠀⢻⣷⣶⣶⣿⣶⠀⠀⠀⣸⠀⠀⢸⡄⠀⠀⠀⢳⣤⡳⣦⣴⣿⡄⠀⠀⡏
⠀⠘⣧⠀⠀⠀⠓⠙⠛⠑⠿⠀⠀⠀⣼⠁⠀⠀⢻⡄⠀⠀⠀⠙⠻⢿⡿⠟⠀⠀⢠⡟⠀⠀⠈⣧⠀⠀⠀⠀⠀⠴⠿⠛⠛⠁⠀⣸⠃
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠉⠓⢦⣄
⠀⠀⣴⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠈⢳⡄⠀⠀⠀⣰⠋⠀⠀⣀⣀⣀⣀⣀⠀⠀⠙⣦
⠀⢸⠃⠀⠀⠀⡆⠾⢃⢰⡏⠁⠀⠀⢸⡆⠀⠀⡿⠀⠀⠀⣠⡀⠟⡡⠤⣄⡀⠀⠀⢿⠀⠀⢰⡇⠀⠀⠰⡌⠻⣿⡟⣩⡄⠀⠀⠘⡇
⠀⢸⠀⠀⠀⠰⣝⣛⣡⣿⢣⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⢰⡇⠿⢆⡴⠒⠀⠀⠀⠀⢸⠀⠀⢸⡁⠀⠀⠀⠁⡇⠿⠸⢋⠁⠀⠀⠀⡟
⠀⠸⣇⠀⠀⠀⠈⠉⢭⡴⠋⠀⠀⠀⣸⠃⠀⠀⢻⡀⠀⠈⠣⠰⠎⠀⠚⠂⠀⠀⢀⡿⠀⠀⠘⣇⠀⠀⠀⠼⠇⠀⠀⠹⠆⠀⠀⣰⠇
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⠴⠞⠋

⠀⠀⠀⠀⣠⡴⠶⠚⠛⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠓⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠒⠶⢤⣀
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⣀⡀⠀⠈⠙⣦⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢰⡏⠀⠀⠀⠐⢤⠀⢀⡤⠀⠀⠀⢻⡄⠀⠀⣼⠁⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠈⣧⠀⠀⢠⡟⠀⠀⠀⠀⢀⣀⠀⣬⠀⠀⠀⢹⡆
⠀⢸⠀⠀⠀⠀⣠⢼⠤⣾⣠⠄⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⠤⠄⠈⢥⣶⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⢠⣤⢹⡇⣿⠀⠀⠀⠀⣧
⠀⢸⡄⠀⠀⠀⠀⠈⠠⠌⠂⠀⠀⠀⢸⡇⠀⠀⣷⠀⠀⠀⠀⠀⢺⣷⡄⠀⠀⠀⠀⣼⠀⠀⠸⡇⠀⠀⠀⠀⣔⣫⣼⣿⠀⠀⠀⢠⡇
⠀⠀⢳⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠟⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⠛⠃⠀⠀⢀⡼⠃⠀⠀⠀⠻⣄⠀⠀⠀⠉⠉⠉⠁⠀⠀⢠⡞
⠀⠀⠀⠙⠳⣤⣀⣀⠀⣀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⡀⢀⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠘⠳⣤⣀⣀⠀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉