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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀⠈⠛⠿⣿⣿⣿⣿⣿⡿⠋⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠋⠀⠀⠀⠀⢀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠈⠻⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⡷⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠁⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠃⠀⠀⣰⡄⠈⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⡇⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠻⠀⠀⣴⣿⣇⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢿⣿⣦⣼⣷⣿⡿⢻⠰⡀⠘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣸⠀⢠⣾⠇⠀⠀⠀⢠⠏⡾⠀⠀⠀⠀⠀⠀⠀⣀⣠⠬⢧⣀⠉⠹⣿⣿⠋⢠⣿⡇⡇⠀⣷⡿⣿⡟⠁
⠀⠀⠀⠀⠀⠿⠋⠀⠈⠁⠀⠀⠀⠀⡾⣼⠏⠉⠓⢠⠀⠀⠐⠚⠁⢀⣀⣀⡈⠀⠀⠹⠇⠀⣿⣿⣿⠁⠀⠘⣏⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃⣐⢞⣻⠶⡆⠀⠀⠀⠐⠉⠺⠟⠋⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⠸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠂⣀⣀⣴⣿⠇
⠀⠀⣠⠴⠞⠉⡀⠀⠀⠀⣿⠓⠄⠀⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⠶⣾⡿⣿⠟
⠀⠀⡏⠀⠆⢀⡇⠀⠀⠀⢹⠛⠂⠀⠀⠀⠘⠃⠀⠈⠂⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠴⠚⠁
⠀⣠⡇⠼⠃⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠋⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⡠⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣉⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠈⠉⠁⠈⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⠞⢶⣄
⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣦⠔⠁⠀⠀⠀⣀⣤
⠀⠀⢀⣤⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠷
⠀⠘⡏⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⣼⠸⠟⠀⠀⠀⠀⢀⣠⠄⠀⠀⢀⣿⣷⡄⠀⣠⣴⣶⣦⡀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⢸⣿⣿⡇⢰⣿⣿⣿⣿⡇⠀⣿⣿⣷⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠇⠀⠀⠈⠻⠿⠃⢸⣿⣿⣿⣿⠃⠀⠿⠟⠃⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⢸⣿⣿⣿⣿⣿⡇⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⢀⣿⣿⡟⠛⢿⣿⡇⠀⠘⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠘⠛⠛⠛
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠆⠐⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⡀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠛⠒⠒⠒⠀⠀⠀⠒⠒⠒⠋⠉⡁⡀⡐
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⢋⠐⠒⠂⠤⠤⣤⠤⠤⠤⠒⠒⠚⠁⠀⠀⠀⡤⢴⠤⠤⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⢨⣈⣉⠐⠒⠂⠦⠤⠤⠒⠒⠒⠛⡉⠀⠀⠀⠹⣉⣉⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠤⠀⢈⡉⠙⠒⠲⠖⠒⠒⠒⠉⢉⡣⠀⡀⠀⠈⠉⠉⠉⠀⢠⡀⡎⠰⡧⠀⠀⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠑⠂⠀⠄⣉⡀⠒⠓⠛⠉⠉⠉⡈⣁⠀⠑⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⢛⡀⠠⠀⠀⠒⠈⠀⣛⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⡉⣉⣉⣉⡉⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢁⡠⠀⠂⡈⣻⠀⠉⢌⠛⠉⡉⠉⠛⠲⠦⣠⢉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⡠⠒⢁⠰⠊⠁⠀⢨⡖⠀⠀⠑⢄⠈⠓⠄⡀⠀⠀⠉⢎⠠⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠁⡰⠊⡠⢀⠠⠐⠀⠁⠀⠀⢈⡌⢄⠀⠀⠑⢄⡀⠈⠢⢀⠀⠀⠁⠄⠢⡈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⣟⣠⠊⡠⠊⡐⠁⠀⠀⠀⠀⢀⡴⠃⠈⢂⠑⢄⠀⠀⠉⠢⢄⡀⠈⠐⠂⠤⢑⣌⣄⡿⠿⢿⣿⣿⣿
⣿⠏⠀⠀⠀⣷⠈⡠⠊⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠑⢄⠑⢄⠀⠀⠀⠈⠁⠒⠂⠤⠄⠀⣿⠀⠀⠀⠙⣿⣿
⡏⠀⠀⠀⠀⡷⠊⠀⠀⠀⣀⠠⠂⠁⠀⠀⠀⠀⠔⠀⠀⠂⠁⠢⢌⡐⠢⠤⢀⣀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠸⣿
⡇⠀⠀⠀⠀⠇⠴⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠷⠀⠀⠀⠀⠁⠐⠀⠀⠀⢠⠀⠀⠰⠇⠀⠀⠀⠀⣿
⡇⠀⠀⠀⢰⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⢀⣿
⣿⣄⠀⢠⡜⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣⣀⠠⡌⢹
⣿⠃⠘⠈⢹⡄⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⠈
⡿⠀⠆⠀⢸⠁⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⡆⠀⣷
⣇⡰⠀⠀⠸⠓⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⡀
⣿⣃⡤⠊⠀⠀⠀⠀⢸⡓⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠈⠑⡀⠈⣷
⣿⡯⠀⠀⠀⠸⠂⠀⠈⠈⠒⠄⡈⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⠀⠠⡇⠀⠈⠲⢼
⣿⡇⠀⠀⠀⠀⠈⣄⡆⠀⠀⠀⠀⠉⠘⡗⠀⠤⠤⠐⠖⠶⠤⠤⠀⠒⣒⠉⠀⠀⠀⠀⠀⡆⢀⠂⠀⠀⠀⠀⢸
⣿⠇⡀⠀⠠⠀⢀⠀⠀⠀⠀⢀⠂⠉⣉⢙⡆⠶⠒⠂⣰⠀⠐⠒⠒⠂⠙⠆⡀⠀⠀⠀⠀⠧⠘⠀⠀⠀⠀⠀⢸
⣿⣷⠇⢠⠁⢀⠀⠀⠀⠀⡐⠁⠀⢠⣉⡤⠈⢺⠁⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠄⠀⠂⠀⢀⣸
⣿⣿⣶⣯⣐⣂⣀⣀⣠⠜⠀⠀⡠⠋⠀⠀⠀⠲⠶⢆⠀⢹⡀⠀⠀⠀⠀⠐⠀⠀⠹⣿⣶⣦⣤⣤⣄⠈⣄⣸⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠚⠋⠲⠈⣧⠀⠀⠀⠀⠀⠆⠀⠀⠹⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⣉⣉⡦⠐⠛⡀⠀⠀⠀⠀⠰⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⠒⠉⠐⠂⠠⣄⣈⡧⠼⠀⠒⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⠠⠽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠈⠀⠀⠀⠈⠛⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠠⣤⣽⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⢹⠁⣠⣄⠠⡄⢹⣦⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠀⠀⠀⠀⠂⠈⠉⠁⠀⠀⠀⠀⠀⢸⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠒⠳⢮⡥⣤⣼⣿⠃⡟⠈⠿⡭⢭⣭⣽⡛⠃⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⡈⠁⠀⠀⠀⠀⠈⠠⡗⠈⠁⠀⠀⠐⠒⠘⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⢧⠀⠀⡀⠄⡉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡂⠂⢀⠀⠀⠀⢀⠐⠃⠀⠈⠀⠀⠀⠀⠀⢣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡴⢋⠄⠈⠑⠀⠐⠠⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⣗⠔⠁⠀⠀⠔⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⢸⡀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠂
⠀⠀⠀⠂⠀⠀⡳⠒⠀⠁⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠈⠀⠄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠠⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠄⠃⠀⠀⠀⠀⠀⠁⠤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠠⢤⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⢇⡀
⠀⠀⠀⢰⣉⠔⡃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠘⣄⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⣴⠀⠰⠀⠉⠀⠠⠀⠀⠀⠀⣀⣀⠀⠀⡀⠐⠈⠀⠀⠸⠀⠐⠃⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⣀⣀⠀⠀⡀
⠀⠀⣲⠾⡳⠲⠒⠟⠒⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠐⡖⠛⠒⠒⠲⠒⠖⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢀
⠀⠈⠉⠀⠁⠈⠈⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣐⡼⠁⠀⠀⠐⠉⠁⠀⠀⠀⠁⠀⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢪⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢤⢤⣀⣀⣤⣄⣠⡤⠤⠶⠶⠦⢤⣀⠈⡀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠤⣤⣀⣠⣤⣤⣠⡤⠶⠶⠶⠶⡤⢄⡀⢠
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣬⣭⣮⣥⣤⣤⣄⣠⣀⣁⣄⣙⣻⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣤⣤⣤⣭⣮⣥⣤⣤⣤⣤⣄⣅⣠⣈⣟⣧
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈


⠀⠀⠁⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠍⠁⠈⠉⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠸⠂⠃⠀⠀⠀⠀⠀⠙⠉⠀⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠨⠀⠮⠀⠀⠀⠀⠄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⡀⡌⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠰⢠⣤⣄⣀⣤⢤⠐⡄⢤⣀⣀⣤⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⡤⠄⠀⢀⢀⡰⣀⣀⠀⠀⢤⣤⠠
⠀⠀⠀⠀⠀⠀⠀⢢⡐⠂⠀⠀⠉⢀⠃⠣⠈⠀⠀⠀⠒⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⠛⠉⠉⠛⠀⠇⠃⠘⠉⠉⠙⠛⢠⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠂⠒⠈⠀⠀⠀⠁⠒⠒⠒⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠈⠀⠐⠒⠒⠀⠀⠀⠈⠀⠐⠒⠒⠀⠀⠀⠀⠀⢄⣀⣀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣛⣛⣛⣛⣛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠩⠍⣉⡉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⠈⣧⠀⠀⠀⠄⡉⠑⠢⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢙⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠈⢀⠀⠀⠀⠀⠜⠂⠀⠈⠀⠈⠀⠀⠀⢀⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡵⢋⠄⠈⠑⠀⠐⠠⠀⠐⠀⠀⠀⠠⡀⠀⠀⢀⡹⠿⣿⣿⣿
⣿⣿⣿⡟⠁⠀⣗⠐⠁⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⣇⠀⠈⢻⣿
⣿⣿⣿⡇⠀⠀⣳⠂⠀⠃⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠄⠘⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⣿
⣿⣿⣿⣧⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠄⢻⣿⣿⣿⣿⣿⣧⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⣾⣿
⣿⣿⣿⡏⠰⢤⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⣿⣿⣿⣷⣉⠔⡃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠘⣄⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡷⠀⠰⠀⠉⠀⠠⠀⠀⠀⠀⣀⣀⠀⠀⡀⠐⠈⠀⠀⠸⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⣀⣀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡳⠲⠲⠟⠒⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠲⡀⠀⠀⢿⣿⣿⣿⡟⣿⣿⣿⣿⡿⠛⠟⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠛⢿⣿⣿⣿⣿
⣿⠉⠉⠉⠁⠈⠈⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⣸⣿⣿⣿⡏⠉⠉⠉⠉⠉⠁⠀⠉⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⢤⣀⣀⣤⣄⣠⡤⠴⠶⠶⠦⢤⣀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⣤⣀⣠⣤⣤⣠⡤⠶⠶⠶⠶⡤⣄⡀⣹⣿⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣤⣬⣽⣮⣥⣤⣤⣅⣀⣀⣁⣄⣹⣻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣥⣤⣤⣭⣮⣥⣤⣤⣤⣄⣄⣅⣠⣠⣟⣿⣿⣿
⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⠀⠀⣿⣿⣿⠈⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⢸⣤⡎⠀⠀⠀⠀⠀⢸⣿⣿⡏⠁⠀⠀⢹
⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠤⢽⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⣿⡧⢠⣤⣀⣀⣤⠤⣽⡄⢤⣄⣀⣤⣄⢸⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⡧⣠⡤⠄⠀⢀⢀⣽⡁⣀⠀⠠⢤⣤⢠⣿⣿⡇⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⣿⣧⠐⠀⠀⠀⠈⢀⣿⡇⠘⠈⠀⠀⠒⢸⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣇⠒⠛⠉⠉⠛⠀⣿⡇⠘⠉⠉⠙⠓⢀⣿⣿⡇⠀⠀⠀⢸
⣿⣿⣧⣤⣤⣴⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣶⣤⣀⣤⣿⣿⣿⣿⣄⣀⣀⣴⣿⣿⣶⣴⣶⣶⣴⣾⣿⣿⣶⣴⣶⣶⣴⣾⣿⣿⣧⣄⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠂⣉⣉⢉⡉⢉⣉⣉⡐⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⣠⠰⠂⠉⠁⠀⠩⠭⠭⠥⠴⠀⣤⠈⠉⠰⠦⣉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⡴⠎⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠤⢁⠒⠠⠑⠂⠀⠀⠑⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⠞⠉⣀⠔⠊⢂⡀⠠⠤⡀⠀⡠⠔⢄⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠈⠢⡑⢄
⠀⠀⠀⠀⠀⠀⣠⠖⠀⠳⡾⢋⡴⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠙⠦⡀⠀⠀⢀⠑⢴⡎⠉⠑⢦⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢾⡇⠀⠀⠀⢱
⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠘⢄⡀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠲⢥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠋⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠈⠑⠢⡤⢀⣀⠀⠀⠀⠄⢀⡀⠀⠀⣀⣀⣠⠄⠒⠉
⠀⠀⢀⣀⣀⣤⣜⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⢴⠃⠠⠄⠀⠀⢘⣋⣉⣉⠀⠀⠤⠄⢧⠤⠤⠶⠤⠀⣀
⠀⠀⢸⣇⡽⠉⠉⠉⠉⠉⡩⠋⠉⢱⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⡆⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄
⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠿⢳⠶⣦⣤⡤⢤⣶⣦⣤⠶⣞⠿⠿⠿⠻⠿⠿⠛⣻⠶⢦⡀⠀⣸⠄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡁⠒⣌⢒⡉⠛⣤⣚⢋⡔⣉⡐⠊⠉⠁⠀⠀⠈⠁⠒⠈⠒⠽⠿⣆
⠀⡷⠰⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠦⠦⠤⠾⠿⠿⠿⠛⠛⠛⠻⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠶⠞⠲⠚⠒⠛⠛⠛⠂⠀⠀⠂
⠀⡇
⠀⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢸
⠀⢸
⠀⢸
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠉⠀⠘⠁⠉⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⢀⠀⡀⣹⠀⢀⡉⡀⣀⡀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢒⡶⢖⣒⣒⣒⣒⣲⡆⢶⠚⢳⡆⢲⣖⣒⣒⣒⣒⡶⣶⠒⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠰⠶⠭⠭⠭⠍⠩⠽⠆⣸⡃⠼⡀⠰⠭⠩⠭⠭⠭⠤⠶⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⡰⣄⡀⠀⠀⠀⢀⡠⢊⡴⠁⠘⢌⠒⢀⡀⠀⠀⠀⣀⡐⢺⠏⠀⠀⠀⡀⢀
⠀⠀⠀⠘⠦⠤⠄⠠⠴⠋⠁⠀⠀⠈⠁⠐⠛⠛⠛⠂⠈⠁⠀⠀⠀⠀⠈⠁⠒⠛⠛⠓⠂⠉⠀⠀⠀⠀⠀⠐⠣⣀⣀⣀⣀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣰⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣆⠀⠀⠀⠙⠛⢻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⣿⣿⣿⣿⣿⣿⣾⡶⠛⣛⣻⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠈⠋⠛⠛⠙⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⢦⡈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠘⣿⣷⡦⢽⠀⢇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣺⣷⠄⠀⠈⢻⢭⣭⡅⠀⠀⠀⠿⢃⣴⢲⠀⢺⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢘⡅⠁⠈⠙⠁⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠔⠂⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⣗⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠤⣀⠀⠀⠀⠀⠀⠀⣸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡟⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠏⠸⠁⠀⠈⠙⠳⢶⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⡾⠉⠀⠠⠃⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⡤⠒⠉⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢰⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢘⡥⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣴⡾⠿⢶⣄⠀⠀⢀⣀⣀⡀
⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠶⠿⣿⣤⣿⡟⠀⠀⠀⢿⣷⣴⠟⠋⠉⠛⢦⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠘⣧⢀⣴⡿⠛⢷⡄
⢸⡇⠀⣀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠈⣷
⠀⣿⠀⣿⣇⠀⠀⠀⠀⢿⣿⣦⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⣿⣿⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⣀⣤⡶⠶⢦
⠀⢿⡄⣿⣿⡀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣄⠀⠀⠀⠘⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⣧
⠀⠸⣷⣿⣿⣧⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠛⠻⣿⡇⠀⠀⠀⣿⣷⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢹
⠀⠀⢻⣿⣏⢻⣆⠀⣰⣿⠃⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡏⠀⢰⣿⠇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢸
⠀⠀⠀⢿⣿⡈⢿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠻⣷⣼⣿⣿⠀⠀⠀⠀⢰⣿⣿⡟⢦⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⢠⣿⣿⠋⠻⣎⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡿⠀⠀⣼⡇⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢀⣿⠃⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣸⣿⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⡟⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⢿⣾⣿⣥⣀⣀⠀⠀⣿⣿⠇⠀⠀⢠⣿⠇⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⣀⠀⠀⣾⡿⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣿⡇⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠀⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠟⠁
⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⣠⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣴⣶⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠐⠆⠢⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠐⠀⠃⠘⠀⡁⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡐⠀⠀⢐⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢰⠢⣄⠀⠈⠀⠠⠀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠈⠁⠈⠉⠁⠛⠈⠁⢡⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠇⠀⠀⠉⠢⣅⢄⢀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⡅⠀⠀⠀⠀⠀⠀⠀⢀⠄⠔⢀⠏⠀⠀⠀⠀⠀⠈⠣⠡⠡⠁⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠐⠂⠀⠀⠀⠀⠁⠂⠀⢉⡈⠂⠀⠀⠀⠀⠀⠂⠀⠂⢪⢃⡤⢀⣀⣀⠀⢠⠠⠄⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢨⣑⠀⠀⠀⠀⠀⠈⡀⠀⡰⢅⠀⠁⡐⠀⠠⣀⡀⣑⠀⠁⠄⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢐⠀⠀⠒⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠂⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢓⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠳⢈⠀⠀⠀⠢⡈⠉⠀⠀⠈⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠘⢢⡀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠘⣿⢆⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⡠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣝⢄⠉⠒⠠⠤⠰⠺⠀⠀⠀⡀⣀⠂⠀⠀⠀⠀⠀⢻⣅⠱⢤⣀⠀⠀⠀⠀⢀⡤⠊⢐⣻⠁⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠐⣮⣕⠪⢀⡄⠀⠀⢈⡷⠶⡟⠉⠁⠀⠀⠀⠀⠀⠀⠈⣿⠤⠖⠧⠣⠴⠶⠿⣿⣧⣴⣾⣶⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡲⢄⣀⢁⢲⣙⢁⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠀⠀⠀⠀⠸⠻⠿⠛⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠑⠐⠘⠃⣿⠀⠀⠀⠀⠀⠈⠢⠀⠀⢀⡀⢠⠹⠛⠛⢋⠀⠀⠀⠀⠊⢛⠟⠿⢍⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⡰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢂⣠⠀⠀⠁⠄⣨⡄⠀⠀⠀⠀⠄⠉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠈⠐⠁⢐⠠⢀⠡⠀⠀⠀⠸⠀⠀⡀⠐⠞⡤⠽⠒⠚⠃⠥⠄⠀⠀⡈⠀⠀⠀⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠈⠄⡀⠄⠀⠀⠀⠐⠫⠞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠀⠀⠂⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣚⡭⠭⠭⠭⣝⡒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⣤⣤⡀⠈⠑⠦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠞⠁⠀⠀⠀⠀⢀⣿⡏⠛⢶⣄⠀⠘⢮⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡎⠀⠀⠀⠀⣀⣀⣸⣿⠃⠀⠀⣿⡟⠀⠈⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇⠀⠀⠀⠸⢿⣿⣿⡿⠀⠀⢰⣿⡇⠀⠀⢸⣸⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠀⠀⠀⠂⠺⢼⡆⠀⠀⠀⣠⣤⡤⠀⢠⣶⣶⣾⣿⠀⢀⡤⠊⠀⠀⣉⣀⡀⠉⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⡙⣷⡀⠀⠈⠙⠛⠛⠋⢠⠂⠀⣠⡶⠿⠛⠛⠻⠿⣦⣄⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠹⠟⠃⠀⠀⠀⠀⠀⠔⠁⢀⡾⠋⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡅⡠⠒⢒⠀⠀⠀⠀⠤⣤⡄⠲⠚⣮⡓⠦⣄⣀⠀⠀⢀⣀⡰⠀⠀⣸⠃⠀⠄⡀⠄⠂⠀⠀⠢⠄⢹⡆⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠠⠄⠀⠀⠀⠀⠤⠀⠀⢣⡀⠈⠑⠲⠤⠭⠭⠥⠔⠺⢠⣶⣿⡖⠙⣉⡀⠀⠀⠀⢈⡑⠐⣾⣿⣦⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⡇⠀⠙⠁⠀⠀⠀⠙⠃⠀⣿⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸⠀⠉⠁⢰⠇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠛⢿⠃⠀⠀⣦⣐⣒⣢⡦⠀⠀⢹⢿⡋⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠒⠂⠁⠀⣠⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠣⡀⠀⠈⠛⠛⠛⠁⠀⡠⠊⠀⢩⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⢶⢶⠶⠖⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠁⠂⡤⠀⢴⠒⠉⡠⣀⡠⠴⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠈⠘⠛⠤⢀⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡤⠆⠉⠣⠀⠘⠉⠉⠀⢸⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⠞⡃⠀⠀⠀⠀⠀⠀⠘⢿⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠌⠀⡅⠀⠀⠀⠀⠀⠀⡅⠀⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⣐⠢⠖⠠⣠⢤⡇
⠀⠀⠀⠀⠀⠀⠀⡔⢸⠄⠀⢰⣶⣤⣤⣤⣴⣶⠀⠀⢸⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠾⢿⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠰⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡟⠀⢛⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠠⠆⠀⢸⠄⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠾⠃⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠀⠀⢾⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⡀⠠⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⣠⠇⢀⣤⣶⣾⡿⠿⠟⠿⡛⠛⠿⠿⡅⢠⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠅⣲⣿⠟⠉⠁⢀⠀⠁⠔⠹⠀⠈⠈⡁⡀⠀⢬⡀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠎⣼⣿⠕⠀⠂⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠑⠀⢻⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠁⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣼⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠦⢤⠀⣿⠀⢔⢣⠜⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠃⣿⡇⡀⠀⠀⢀⣤⡶⠶⠶⠶⠆⠀⠀⠈⠉⠀⠀⠀⠀⠀⣾⡘⠓⠣⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠀⠃⠘⢷⡁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⢸⣷⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠐⢀⠀⠈⢻⡇⠀⠀⢴⣦⣤⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠀⠀⠀⠀⠠⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠃⠀⠀⠀⢸⡘⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣗⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠘⠆⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡤⠓⠂⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡄⠀⠠⠔⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠈⠀⡤⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠔⠁⠀⢰⡠⠀⠀⠠⠀⠀⠀⠀⡀⠂⠊⠀⠀⠀⠀⠠⢶⠈⠁⠀⠀⠘⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠐⣠⠊⠀⠘⠀⠀⠧⠤⠀⠀⡇⠀⠀⠚⡼⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠁⠀⠀⠄⠀⠀⢀⡒⠐⠆⠁⠀⠠⠀⠉⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢿⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀

⠀⠀⠀⠀⢰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠋⠉⠉⠉⢻
⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡞⠀⣀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡞⠙⢤⣦⣠⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾⠐⡂⠀⢈⠙⢧⣀⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⣼⠀⢿⠀⠁⡀⠀⠀⠀⠈⢹⠇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠊⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠸⣤⣸⡄⠀⠦⠶⠀⣰⠟⢋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡦⠶⠚⣷⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠁⠊⠈⠁⠊⠓⢤⡀⠀⠀⠀⣀⡠⡾⢹⠀⠀⢀⡏⣷⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠡⠀⠀⠀⠀⠀⡇⠀⢠⠋⠁⢀⡇⣏⣉⣉⣉⡁⡏⠀⠀⠙⢲⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡿⠤⠤⠷⢖⡏⠘⡏⠉⠉⠉⠉⠉⠉⢹⠃⠀⠀⠀⠀⡷⠒⠒⠶⠶⠖⠒⠒⠛
⠀⠀⠀⠀⠀⢀⣀⣀⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁⠀⢣⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⡖⠋⠀⠀⢀⡐⢦⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠸⠀⠀⢸⡀⠀⠀⣴⡲⡢⡀⣧⡀⠀⠀⠀⠁⠀⠀⢀⣀
⠀⠀⢰⠟⠁⠀⠀⠀⠀⣧⠀⢷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣇⠀⠀⠛⡀⠈⠈⠙⠹⡄⠀⣠⠖⠋⠉⠉⠈⠑⠶⢤⡀
⣴⠋⢹⠆⠤⠄⠀⠀⠀⡿⠀⠘⡤⠫⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠁⠀⠉⢷⡀⠀⠸⣄⣀⣀⣈⣦⡀⠀⠀⠈⣼⢡⢂⡞⣵⠀⠀⠀⠀⠀⠹⡄
⣏⠀⠘⣇⠂⠀⠀⢸⠉⠁⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⢷⣸⢘⠀⡇⢸⠀⠀⠀⠀⠀⢻
⢿⠀⠀⠈⣷⣤⣀⠈⠛⠃⠀⠀⣀⣀⠆⣀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠀⠀⠸⣄⣀⡀⠀⠀⢸
⠸⡆⠀⠀⢸⠀⢸⣇⡀⠀⠀⢀⡏⠀⠀⠉⠉⠹⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⣀⣀⣤⠤⠄⠀⠤⠀⠀⠀⠀⢈⣠⠇⠀⠀⣸
⠀⢻⠀⠸⢸⡶⠛⠺⠿⠿⠿⠽⠛⣆⠀⠀⣀⠛⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⡇⠀⠀⠀⣧⠀⠀⠀⠀⠸⢤⡄⠀⣀⣀⣨⡇⢠⢂⢄⡏
⠀⢸⣇⣧⠏⠀⠀⠉⠙⠓⠒⠛⠂⠈⠑⠶⣭⣠⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⢿⠀⠀⢰⢿⣤⠀⠀⠀⠀⢠⠟⢛⣛⣋⡛⠀⣗⠛⠋
⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⡛⢸⠀⠀⣸⢸⢸⢀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⡄
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠉⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⢯⠀⠀⠸⡄⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⣆⠀⠀⠀⠀⢀⣷⠏⠀⠀⢸⡇⠀⠀⠹⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙⠚⢦⣀⣸⡤⠋⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠪⠁⠐⠀⠉⠂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⣁⣤⣼⣿⣿⣿⣷⣶⣤⣄⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢋⣴⣶⣿⣿⣿⡿⠛⢛⣏⠭⣝⢿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⣿⠟⠋⠀⠈⠁⠀⠔⠀⠐⠢⠀⢿⡄⠇⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡞⢻⣿⡄⡠⠖⠂⠀⢰⠀⠀⠀⠀⠀⠈⣷⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡅⢸⣿⠀⢀⠉⢘⡇⠈⢦⡀⠀⠠⠃⠆⠘⣧⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡄⠀⠀⢻⡇⠈⠋⠋⠰⠄⠀⢉⣠⡤⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⣁⡤⠣⣀⠀⠀⡀⡐⡄⠘⣀⠠⣡⠤⣉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠹⠀⠀⠙⠢⢄⣥⣭⠅⠃⢩⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⣤⠔⢧⣀⣧⡀⠀⠘⢀⠀⠀⠀⠌⠂⠀⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠐⣔⠊⢁⢀⣇⠀⠈⣷⠄⠀⠒⠒⠶⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣉⠉⢁⡼⠋⣇⢠⢉⢂⡀⠀⠀⢰⠀⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⠉⠀⠀⠈⡏⠈⠀⠙⠋⠒⠂⢸⡇⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣀⠀⠀⠀⠀⠠⢄⠀⢻⣧⣀⡠⠔⠋⠑⡦⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⢿⣿⠁⢲⣶⣶⡌⣖⣿⡀⢝⠀⡀⠘⠆⠠⣝⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⠉⢉⠉⠉⠣⠈⡁⣃⢨⡁⣸⠑⢤⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⡍⠇⠀⠀⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡆⠀⠀⢸⢸⡄⠀⠀⠤⡖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⡏⠀⠀⠈⠸⡄⡇⠀⠀⠀⠈⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⡇⠐⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⠀⠉⠀⠐⠠⠤⠄⣧⠤⡤⠴⠚⣉⣀⣀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠊⠀⠀⠑⠤⣀⠀⠀⠀⠀⢸⢄⣀⡠⠋⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⢀⡉⢩⣽⠟⠋⠈⢷⣤⣀⣀⣠⣤⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠂⠲⠟⠂⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣦⠀⠀⠀⠀⠀⢠⡿⠃⢀⣤⡶⠶⠶⢶⣤⡀⠘⢿⡄⠀⠀⠀⠀⠀⣴⠟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣿⡄⠀⠀⠀⠀⣸⡇⢰⡟⠁⠀⠀⠀⠀⠈⢻⣆⢸⣧⠀⠀⠀⠀⢠⣿⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢸⡇⠀⠀⢰⣶⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣶⡆⠀⠀⢸⡇⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⠀⠀⠈⠛⠿⠗⢻⡇⠀⠀⠀⠀⠀⠀⢸⡟⠺⠿⠛⠁⠀⠀⣾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⡟⠻⣦⡀⠀⠀⠈⠻⣦⣀⣀⣀⣀⣴⠟⠁⠀⠀⢀⣴⠟⢻⣿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠙⢷⣄⠈⠻⣦⡀⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⢀⣴⠟⠁⢠⡾⠋⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⠻⣦⡀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠁⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠃⠀⠀⠈⠻⣦⠀⠀⠀⠀⣴⠟⠁⠀⠀⠘⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸⣷⠀⠀⣾⡇⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⣿⠀⠀⣿⠁⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⢾⠇





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⣠⣤⣀⡴⠋⠉⠉⠶⠞⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠘⡏⠉⠓⠀⠀⠀⠀⠀⠀⠀⢀⣀⡎⠀⠀⣈⣥⠶⠶⠶⠶⠶⢦⣤⣈⠓⠛⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠳
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠁⢁⡴⠟⠉⠀⠀⠀⢀⡀⠀⣀⡀⠉⠻⣦⡀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣴⠏⠀⣀⣀⡼⠓⠒⠋⠉⠉⠉⠓⠤⡖⠈⢻⣄⠉⢢⠀⠀⠀⠀⢠⣶⣶⣿
⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⣰⠋⠁⣸⠏⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡄⢻⡄⢸⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢠⡟⠀⠚⠋⠀⠀⣠⠶⣄⠀⠀⢀⡤⢤⡀⠀⠸⠤⠄⣿⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢸⣿⣽⠀⠀⠀⠀⣷⣿⢾⠀⠀⢸⢸⣷⡇⠀⠀⢀⣿⣿⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⡆⠀⠀⠀⠙⠛⠋⠀⡀⠈⠓⠛⠁⠀⠀⣾⣿⣿⠀⠈⡇⠀⠀⠀⢰⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠜⣿⡿⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⡴⠃⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢻⢹⡀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣄⠀⠀⠀⢸⠁⣨⡟⠀⠀⠀⠀⠠⠶⠗
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠈⠓⢾⠼⢧⠀⠀⠀⠈⠫⣍⡛⠋⣉⡠⠃⠀⠀⠀⡟⠛⡯
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠸⡄⠈⢣⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⡼⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⣀⡤⠊⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢰⠃⠉⠙⠒⠒⠚⠋⠹⡄⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣤⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠙⣄⠈⠑⠦⣄⡀⠀⠀⠀⣀⡤⢚⠕⠁⠀⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠉⠉⠉⠉⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢋⠁⠀⠀⠀⠢⡱⡀⠀⠀⠀⠀⣿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢀⠊⣴⠶⠓⣠⡄⣦⣘⡀⠷⡘⡀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠌⣸⡏⢘⡠⣺⠷⢋⠸⢃⠃⡅⣷⡀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⡋⠀⢻⡞⠴⠧⡀⠈⠂⣀⣹⣿⢧⠁⠂⠀⠀⠀⠀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠄⠀⠀⠸⠿⢙⣦⡞⠀⠘⠛⠃⠀⠀⠶⠶⡇⠀⢂⠀⠀⠀⠀⣀⠇⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣸⡄⠀⢠⡄⠀⠑⢆⠀⠀⡀⠀⠀⠀⠀⠠⠳⡹⡿⠀⠀⠀⠈⠋⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠻⠁⠀⠘⣦⣷⢇⢸⡆⠀⠙⠤⠇⢛⡴⣥⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠸⡇⣿⣾⡷⣷⣤⠀⠀⠋⢸⣿⠃⠱⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢹⡾⡟⠰⠏⢉⠀⠀⠀⠄⣹⡄⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⠤⠀⢀⡎⠀⠀⠀⠀⠀⢸⡇⠘⡀⠀⠀⠀⠀⡴⠀⠻⣧⠀⢀⠂⡔⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡟⠀⠉⠂⠀⡠⠜⠀⠀⠀⣿⡇⢨⣶⠂⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣄⣯⣿⣃⠀⠀⠀⡔⠀⠀⠀⠀⠈⠉⠙⠛⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⠿⠛⠓⠦⣴⠀⣀⠀⠀⡄⠂⡀⠀⡸⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠄⠀⡃⠋⠂⠀⢰⡞⢀⠁⡐⢌⠺⡆⠑⣤⠇⢰⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢖⡙⠶⣤⡈⠁⠑⠒⢠⣤⣀⠑⡠⠂⠁⠀⠀⢱⠉⣩⡽⠻⢁⡠⠁⠐⠠⡀⠀⢀⡄⠀⢨⠉⠙⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣪⡼⢦⣑⡄⠀⠀⣾⣏⣿⣷⠤⣰⡄⠀⠀⠀⡄⣺⡅⠓⠈⠀⠀⠀⠀⠈⠑⢌⠀⠠⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣵⡶⣡⢿⣿⣆⠀⠈⠛⢽⣯⣼⣯⣴⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⢠⣴⡏⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣧⣞⣿⡝⡆⠀⠀⣠⣿⣿⣿⣿⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠻⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⢻⣧⣸⡏⠉⢹⣿⣿⣿⢉⠀⣿⡀⠀⠀⠀⠀⠀⡰⢀⣾⠀⡀⢀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⣎⠟⣷⣿⡀⢸⣿⣿⣡⡁⠀⢻⣿⠀⠀⣀⣀⣴⢶⣿⣥⡤⠦⠈⠀⠂⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣯⣴⣿⣿⣯⣿⣿⣿⣿⣴⠁⠘⡟⠓⠚⠉⠁⠀⢶⣦⠛⠻⣆⣄⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⣿⣿⣏⠽⠛⣿⣿⣿⡷⣣⡄⣇⡀⠀⠀⠀⣀⣘⣷⣤⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠙⢾⣿⣾⣿⣿⢿⣱⣿⡿⣿⣿⣟⣿⣿⣿⣿⣶⣤⣀⠀⡀⠀⢀⣴⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢿⡯⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄⣤⢤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣠⣾⣿⣟⡉⠁⠀⠀⠀⠈⠉⠻⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣁⣠⣶⣿⣿⣯⣾⣿⣿⣤⣤⣴⣿⣦⣀⣺⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡏⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⢐⣼⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⢻⣿⡿⡇⠀⠀⠹⣿
⠀⠀⠀⢠⠀⢀⣾⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠃⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣛⣛⠻⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠤⠶⠶⠶⣄⢀⡄
⠀⠀⠀⣰⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣷⣦⠙⣿⣿⣼⣿⡏⠀⠀⠀⢀⣠⣤⣴⡇⠀⣅
⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠋⠀⠀⠉⢿⣿⣷⡈⠙⠿⡙⣇⠀⠀⠀⠉⠉⣹⡿⠀⠀⠘⡄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡁⠀⠀⠀⣨⣿⣿⣧⢠⣤⣄⠈
⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢈⠀⠀⠀⠀⠉⠿⣿⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣧⠸⣯⡷⡄⠀⠀⠀⠈⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣷⣌⠃⠀⠀⣀⣄⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡈⠻⢷⣶⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣻⣷⣶⣤⣄⣀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣸⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠈⠏⢴⠂⣀
⠀⠁⠈⠉⠈⠉⠉⠉⠉⠁⢉⣉⡉⢱⠀⠀⠀⠀⠀⠀⠀⠚⠦⡁⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢠⠀⡴⢀⡤⠄⠀⡄⢀⠀⠀⠀⠀⡀
⠂⠀⢀⡖⢠⠀⡇⠀⠀⡞⣻⣱⠷⣺⠆⠀⠀⠀⠠⡄⠀⠒⠆⡅⠠⠲⡀⠒⢠⢰⡖⠀⠺⠙⠶⢂⠂⡀⣛⢹⠀⠀⠀⠀⠀⠀⣸⠀⢠⠥⠆⢤⡄⠉⡄⡐⡠⣱⠂
⠀⡀⣸⡵⢻⡴⠟⠶⠟⢛⡇⣨⢨⣹⠀⠀⠀⠀⠈⠃⠀⢠⠦⢠⠀⠀⢸⠓⢿⠛⠀⢠⠸⠦⠈⠈⠀⠁⠉⠈⠀⠀⠀⠀⠀⠠⠼⢘⢤⣤⣼⣦⣥⣾⣵⣾⡽⠁
⠀⢉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⢢⢀⠐⢀⡀⠁⠚⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠉⠀⠉⠉⠁
⠀⠀⠂⣠⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠒⠈⠁⠀⠀⠀⠀⣤⡄⠀⠀⠀⠁⠈⠶⣁⡀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠀⠀⠀⠀⠀⢰⣿⠛⣿⡆⣀⡀⣀⠀⢀⣀⠀⣿⣧⣤⡄⠀⢸⡇⠀⣿⢣⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠆⠀⠀⠀⠀⠁⠀⠀⠀⢠⠶⣄⢸⠇⡟⠀⠀⠀⠀⠀⢸⣷⠀⣿⡇⣿⡇⣿⠀⣿⡋⠁⣿⡏⣿⣇⠀⢸⡟⢘⣟⢸
⠀⠀⣸⣿⡄⣿⣀⢤⡄⢀⡴⣦⢴⡎⢶⢰⡇⣿⠻⠶⠃⠾⠐⠃⠀⠀⠀⠀⠀⠀⠛⠿⠛⠀⠙⠿⠟⢀⠻⠿⠀⠿⠇⠻⠇⣀⣜⣻⣍⣻⠞⠀⠀⣀⠄
⠀⢀⣿⠘⣿⡏⢿⣠⠇⠸⠦⠟⠘⠷⠋⠈⢉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣌⣀⡀⠀⠈⠀⠋⠉⠁⠒⠒⠉⠀⢁⣀⣡⠤⠒⠋
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠀⠀⠀⠀⠠⠀⠀⠐⠒⠀⠐⠒⠀⠌⠷⣦⡔⠒⠀⠀⠉⠉⠉⠑⠄⢀⠔⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠐⠂⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠤⢄⠀⠀⠀⡄⠤⣐⡚⢻⢙⣇⠘⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢠⠒⢒⢁⢌⡀⢠⡔⠈⠙⢷⠘⠉⠃⢨⠑⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⢻⠈⠓⠈⣇⠘⠃⣸⢧⢮⢳⠎⡱⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣷⡀⢠⠓⣤⢎⣛⣿⣹⣶⣼⣿⡦⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⣀⡂⠠⡄⣸⠆
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠙⠉⠳⣼⣵⣾⣿⣿⠟⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠲⣶⢍⡤⣵⢲⣞⣶⠊⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡥⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢤⡀⠈⠉⠙⠛⠛⠋⠁⡀⠢⢤⣤⠆
⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣠⠴⠊⠁
⠀⠀⠄⠀⠔⠴⠽⠦⠮⠷⠴⠾⠗⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠤⠄⠀⠀⠒⠀⠀⠀⠠⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠓⣠⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠂⠄⠀⢄⣀⡀⠀⠀⠀⣀⠨⠀⠁⢂⠈⠀⢢⡂⠀⡧⠠⠃⠀⡇
⠀⠀⠀⠀⠀⠠⠐⠊⠀⠀⠀⠀⠀⣀⣠⡄⣿⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⢦⣀⣁⣼⣿⣀⣸⡀⠀⠀⠀⠀⢰⡆⣶⢀⡀⣀⢀⠀⡀
⠀⠀⠀⠠⠊⣤⣆⢠⣄⣰⣦⣾⡟⠿⠟⠃⠉⢡⠂⠀⠀⠀⠀⠀⠀⠀⠈⠞⠿⣿⠿⠿⡿⠛⠛⢉⠀⠀⠀⠀⢸⡿⣿⣻⣷⣿⣼⣷⣿⣿⣾⣷⣿⣾⣇⣤⣤⡄
⠀⠀⠀⠀⠀⣯⠟⠻⠋⠉⠁⠀⠀⢀⡀⠔⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠙⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠁⠠⡀⣀⠀⡀⠐⠐⠃⠂⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠉⠀⠒⠤⣀
⠀⠀⠀⠀⣠⠖⠒⠁⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠈⠁⡀⠀⠀⢀⡤⠤⠤⣄⣀⠤⠒⠀⣀
⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡔⠁⠀⠀⠀⠀⡇⠀⠀⠀⢈
⠀⠀⠀⠸⠀⠉⠗⠤⢠⣀⣀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⡸⠀⣧⠀⠀⠀⠀⠀⡘⠦⣄⢠⡎
⠀⠀⠀⠀⢀⢼⣀⠔⠁⠀⠴⢠⠃⠀⠀⠀⠀⢀⠤⠂⠈⠁⠀⠀⠑⢄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢰⠁⠀⢇⠙⠚⠓⠤⣀⡷⡀⠀⡟⠉
⠀⠀⠀⠘⡐⢼⠁⠢⡀⠀⠉⢸⠀⠀⠀⠀⡰⠁⠀⠠⠤⠤⠤⠤⠤⠄⠝⠶⠠⠤⢠⡀⠠⠤⠒⠢⣀⣀⡣⠀⠸⠈⠀⠀⠀⠈⢱⠆⡰
⠀⠀⠀⠀⠈⣦⡀⠀⠈⠻⣹⠁⠀⠀⠀⠀⡁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡸⠀⠀⢀⢾⠊⠁
⠀⠀⠀⠀⢠⠋⠛⠲⠦⡴⠊⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠖⠋⠁⣸⣀
⠀⠀⢀⠔⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠉⠀⠈⠳⡀
⠀⠔⡕⠒⠢⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠦⠤⠤⠒⠒⠒⠀⠄⠀⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠠⢀⠀⠸⡀
⠘⠘⠀⠀⠀⠘⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡠⠁⠀⠀⠃⠀⢃
⠀⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠙⠞⠀⠀⢟⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢠⠁⠀⠀⠀⠀⠀⢸
⢠⢃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⡠⢲⣗⠥⠂⠀⠀⠀⠀⢱⠴⣗⠠⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸
⢸⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣠⠖⠁⠐⢋⡴⠃⠀⠀⠀⠀⠀⠀⠑⢌⡁⠀⠑⢄⠀⠀⠀⠀⠸⠀⠰⠀⠀⠀⠀⠀⠆⠀⢸
⠸⠀⢇⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⣀⠠⠔⠊⢁⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢄⡉⠀⠐⠠⢠⣄⠆⠀⠀⠀⠀⡌⠀⠀⢸
⠈⠀⠈⡄⠀⠀⠀⠀⢀⠧⠒⠉⠁⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠈⠢⠀⠀⠀⡰
⠀⠀⠀⠘⡄⠀⠀⠀⠎⠀⠀⢀⣠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠢⠄⣄⠀⡐⠁
⡄⠀⠀⠀⠘⢄⣀⣼⠠⠔⠊⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠤⣀⣀⠤⠒⠒⠲⠤⢤
⠸⡆⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡄
⠀⢰⠁⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢡
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⣐⡂⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⠶⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⢦⣄⡀⠑⠱
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠙⢷⡀
⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⢂⠂⣰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⢰⣿⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣾⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡩
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠈
⠀⠌⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⠟⠁
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⣁⡠
⠀⠀⠀⠀⠀⠈⠿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠛⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢍⡉⠛⠛⠻⠷⠶⠶⢶⣶⣶⣦⣤⡤⣄⠀⠀⠀⠸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠹⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠻⢦⣌⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⠠⠤⢀⠀⠀⠀⢀⠤⠒⠀⠈⠉⠉⠁⠐⠢⢄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠐⠀⠀⠐⠆⠁⠉⢂
⠀⠀⠀⠀⠀⠈⡆⠠⡄⠄⠤⠞⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣤⠀⠀⠀⠀⡦⢀⢠⡎
⠀⠀⠀⠀⠀⠐⠇⢰⠀⠈⠄⡆⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠉⠑⠄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠈⡀⠀⠑⠚⠂⢄⡧⠀⡛⠁
⠀⠀⠀⠀⠈⢬⠂⠀⠐⢶⢲⠂⠀⠎⠄⠠⠒⠂⠤⠤⠤⠄⠐⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠀⠀⠉⠠⡀⡄⠀⠀⣟⠴⠁
⠀⠀⠀⠀⠀⠀⡑⠠⠄⢠⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢥⠄⠊⣃
⠀⠀⠀⠀⣠⣊⡉⠂⠴⠊⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠊⠀⠑⡄
⠀⠀⠀⡼⠁⠀⠈⡄⠀⠀⠀⠘⡡⡀⠀⠀⠀⠀⢀⡴⣟⣦⠞⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠑⠘⡄
⠀⠀⡜⠀⠀⠀⠀⠃⠀⠀⠀⠀⠇⠈⢆⠀⡠⠂⠁⠈⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠆⠀⠀⠀⡇⡆
⠀⡘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⢯⠂⢀⠀⠀⠀⠀⢰⠀⠐⠀⠀⠀⠀⠁
⠀⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢀⠑⠀⠀⡀⠘⠀⠁⠀⠀⠀⡐
⢠⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡀⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠐⡄⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠤⢰⡀⡠⠁⠀⠀⠈
⣆⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⢀⠖⠉⠁⠉⠉⠑⠾⣋⠀⠀⠀⠀⠀⠀⢛⡖⠊⠉⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢐⠅
⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣎⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⠤⠠⠒⠁⠀⠀⠈⠢⢄⡠⠤⠒⠉⠉⢰⠊⠉⠀⠀⠈⠉⠱⢄
⠀⠀⠀⠀⠀⠀⠀⡾⣇⡠⠒⠒⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⣠⠇⠉⠀⠀⠀⠀⠀⡇⠀⠀⢀⡤⠖⠚⠙⠒⠦⡀⠑⡄⠀⠀⠀⠀⡀⠀⣠⠃
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⡴⠋⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⠉⠑⢼⣯⡉
⠀⠀⠀⠀⠀⠸⡀⢀⠀⠀⠀⣀⣀⠇⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⣤⣀⣠⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠙⠒⠋⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⣀⠤⠊⠀⢀⡼⠁⠀⠉⠙⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⣇⡜⠁⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠹⢂
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠓⠋⠃⠛⠑⠒⣺⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠚⣆⡀⠀⠀⠀⣘⠄
⠀⠀⠀⠀⠀⠀⠠⣔⠀⠀⠀⠀⠀⠀⠀⢨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⠑⢢⠒⠃
⠀⠀⠀⠀⠀⠀⠈⠁⢳⠤⡤⣠⢤⠴⠲⠉⠁
⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣸⠾⠷⣇⠀⠀⠀⠈⢦⡀⢀⣤⣧⣆
⠀⠀⠀⠀⠀⠀⢈⡆⠙⠒⠒⠤⢄⡀⠀⠲⢤⡤⠞⠈⠀⠀⠈⣟⠀⠀⢀⣨⠎⠉⠀⠀⠙⣆⡀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣼⠀⠀⠀⠀⢠⡇⠀⠀⢤⡗⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠐⢻⢦⣄⣀⣀⡞⠃⠀⠐⢚⡯⣄⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⣷⣄⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⣄⠀⠈⢻⣿⣧⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⢹⣿⣧⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⢿⣿⡆⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⡇⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢸⣿⡇⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⣿⣿⠃⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⣼⣿⡏⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⣾⣿⠏⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣫⣭⣽⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣾⡋⠤⣴⣶⣤⣤⣭⣉⡙⠛⠿⢿⣷⡶⣾⠀⠀⠀⠀⠀⠀⢀⣿⠀⣀⣸⡟⠀⣀⣄⡀⠀⠀⠀⣿⣥⡴⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣟⠋⠋⠉⠉⠉⠉⠉⠉⠙⠻⢷⣦⡀⠈⠛⢿⠷⢶⣦⠄⢰⣿⣿⠇⢾⣿⡿⠃⠀⠁⠈⠛⠁⠀⢀⣿⢀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣆⠀⠈⣧⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⠇⠺⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣃⣀⣄⣠⣄⢀⣄⠀⠀⡀⠀⠀⠀⠀⢸⣿⡀⠀⢸⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣽⡟⡸⠠⠮⣤⠈⠛⠈⣙⣚⡉⠳⣄⠀⠀⠈⡇⡇⠀⢸⠀⢹⣷⠀⠀⠀⠀⠀⠀⠰⣷⢤⡽⠃
⠀⠀⠀⠀⢀⣾⠋⣸⡇⡇⠀⠀⠀⠀⠀⠈⠁⠈⠙⠃⠸⠀⠀⢀⣧⡇⢀⡟⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠐⠋⠙⠲⠚⠃
⠀⠀⠀⢀⣾⡏⢰⠇⣷⡇⠒⠶⡄⠀⠀⣀⣤⣀⡀⠀⣹⢆⡤⠚⣫⢝⡛⠛⣾⣿⣄⠀⠀⠀⠀⠀⠿⠿⠃
⠀⠀⠀⣼⠋⡇⢸⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⠋⣠⡞⠁⡏⢹⣼⠉⣿⣿⡀
⠀⠀⢠⡏⠀⠳⡈⢇⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢳⢸⠀⣸⠇⢰⠁⣼⠇
⠀⠀⢸⠁⠀⠀⠙⣿⣧⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠘⠋⠀⠉⠀⠉⣰⣿⣄
⠀⠀⣿⠀⠀⠀⣸⠏⠈⢻⣧⠀⠉⠓⠒⠚⠀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠏⢩⡿
⠀⢰⡟⠀⠀⠀⣿⠀⠀⠀⠙⠷⣦⣀⣀⡀⠀⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⢀⣴⠏
⠀⢸⡇⠀⠀⢰⣧⢀⣤⡴⠶⠶⠾⠿⠿⠿⣿⡇⠀⣽⣷⣼⡇⠀⠀⠀⠀⣀⣼⠃
⠀⣸⠃⠀⠀⠈⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⢯⣙⣺⠟⠀⣸⠃⠀⠀⢠⣾⡟⠁
⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⡎⠙⢷⡄
⠀⠹⣧⣀⣀⣀⣀⣤⣋⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣸⠀⠀⠈⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⢻⡆⠀⠀⠀⠀⡞⠀⠀⠀⢀⣇⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣸⠁⠀⠀⠀⠈⠈⢧⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢀⣸⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠙⠲⠶⠶⠚⠋⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠓⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠐⣖⠔⠶⢶⣖⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠖⠀⠀⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⠆⠀⠀⢀⡀⠀⣣⡀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⡈⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠲⢄⣠⠖⠤⣰⠃⠀⠀⠀⠀⠀⠀⢀⣀⡠⠃⠈⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠑⣄⣓⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⣀⠀⠀⠈⠛⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⡏⠁⠀⠀⠀⣀⣀⣀⠴⠂⠀⠀⢈⡥⠎⠀⠀⠐⠔⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢳⠀⠀⠀⡜⢀⢉⢪⠃⠀⠀⠀⠸⣀⣀⡀⠀⠀⢠⠟⠋⠁
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⠀⠀⠀⢸⠀⠀⢠⠃⣸⠷⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⡀⠀⠀⢸⠀⢠⠯⣲⠃⠀⠀⠀⠀⠀⢀⣴⣶⢿⡻⣿⡟⣻⣷⡶⣦
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⡇⠀⠀⢸⢀⡏⠀⢸⠇⠀⠀⠀⠀⢀⠞⣗⡀⢉⣼⡿⣿⡤⡨⣼⠋⠲⣄
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡇⠀⠀⢸⡞⠀⢀⡟⠀⠀⠀⠀⠀⡞⠀⡌⠳⠋⢿⠃⠽⠌⠛⠃⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢷⠀⠀⠸⠁⠀⡾⠀⠀⠀⠀⠀⢸⠁⢸⡇⠀⣴⠂⢀⣠⡄⠀⠀⢸⠀⠀⣷
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⡏⠀⠀⢣⠎⠛⠙⣾⣿⣾⠅⠀⡎⠀⠀⣿
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢳⡀⠀⢠⡏⠀⠀⠀⠀⢠⠞⠀⢀⡴⢁⡠⠔⠒⠋⠉⢉⠆⡼⡇⠀⠀⣿
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡷⣤⠞⠀⠀⠀⠀⢠⡏⠀⣠⠊⡴⠉⢢⠘⡃⠀⠈⡟⡼⠁⢇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢈⠇⡖⢁⡜⠁⠀⠈⡇⠋⠀⢲⠂⡇⠀⢹⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠣⠜⠀⠀⠀⡼⢙⠋⠀⠙⡅⢳⠀⢻⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⢀⡜⠁⡈⠁⠀⠛⡇⠈⢧⠸⣃⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠀⠀⠈⠸⡄⠀⠃⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢒⣒⣋⡩⠭⠭⢍⣉⣙⡒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⠥⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡵⠋⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⢤⣀⡀⠀⠀⠀⢳⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠎⠀⢀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄⠀⢧⢻⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⢸⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣸⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠭⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢫⠟⠉⠛⣛⣉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⡟⠀⣰⣿⣿⣿⣷⣄⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣿⠇⢀⣿⣿⣿⣿⣿⣿⡆⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠾⣀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⢻⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⢸⡄⠀⠀⣀⠴⠒⠊⢉⣡⡶⠒⠚⠳⠬⣙⠲⢍⣳⡞
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣾⡇⠀⣿⣿⣿⣿⣿⣿⣿⡀⠈⡇⢀⡼⠁⠀⡠⣪⠗⠋⠁⠀⠀⠀⠀⠈⠙⠢⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢻⣧⠀⠸⣿⣿⣿⣿⣿⣿⠁⢰⢇⡞⠀⢀⢎⡞⠁⠀⢀⡔⣊⣯⢟⣲⢄⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢻⣣⡀⠙⢿⣿⣿⡿⠋⣠⠏⣼⠀⠀⡎⡞⠀⠀⣰⢻⡹⡼⣺⡽⣞⡭⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣁⣉⠒⢤⣤⣤⠴⠚⠁⠀⣿⠀⢸⢸⠀⠀⢰⡛⡭⣖⣫⢵⣻⠵⢺⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠸⡀⢸⢸⠀⠀⠨⢷⣻⠽⡞⣏⡧⢗⡿⠀⠀⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⡜⡆⠀⠸⡹⡔⢆⢫⠵⢞⡼⠁⠀⡰⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠳⣄⠀⠀⠈⠻⠭⠦⠶⠋⠀⢀⠜⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣄⠀⠀⠙⠒⢤⣄⣠⣄⣀⣀⣺⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣭⣍⣉⣉⣩⡥⠤⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⢤⣄⣀⡀⠀⠀⣠⡆⠀⢀⠨⣙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⠁⠀⠀⠀⠀⠊⠉⠿⢁⡴⠘⠉⠀⠖⠀⠟⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣴⠃⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠲⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣴⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⣤⠂⠀⠈⠻⣦⣤⣀⣀⣠⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⠄⠀⠐⠋⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⢠⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡆⠀⠀⠀⠀⠚⢃⣀⡤
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠛⠁⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣠⠆⠀⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠈⠁⠀⠀⣧
⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠿⢿⡿⠿⠿⢿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠀⠀⠀⠀⠀⠀⠀⠿⠏⠛⠛⠋








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠋⠑⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⠄⠁⠀⣄⠀⠀⠈⠄⠀⠀⡀⠀⠀⠀⠀⠠⠘⡀⡀⢀⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠑⠀⠀⠀⠀⠰⣿⣦⣄⠀⠀⠀⠂⢡⠠⢄⣾⠃⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⠙⠿⣷⣦⣐⣈⣄⣶⣾⠃⠀⢸⣧⡀⠀⠀⠀⢀⣤⣶⣿⡞⠀⠀⡰⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⢿⣿⣿⠿⠃⠀⠀⠀⢻⣿⣷⣶⣿⠿⠛⢁⡞⠀⠀⠈⠀⠀⢀⡄⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣦⣤⣤⣤⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡞⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠷⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣈⣿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡄⠀⠀⠀⠈⠛⢿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠾⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣋⣤⣤⠴⢤⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠈⣧⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⣠⠴⠚⠲⢦⣄⡀⠘⣿⡄⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠀⢀⠄⠀⠀⠀⠀⠀⣿⠀⣠⡾⢫⢁⡃⡛⣳⣄⣰⠏⠀⠀⠀⠀⠀⠉⠛⠾⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⡀⠀⠀⠀⢻⡾⠋⡀⠰⢨⠡⠁⠄⠜⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡴⠁⠀⠀⠀⠉⠀⠀⠀⡁⠦⠐⠀⠀⠀⠀⠀⠀⠀⡆⠀⢣⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⡁⠈⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠈⠆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡈⠄⠀⠀⠀⠀⠑




⣿⠿⠿⡿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣛⣻⣿⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⠾⢿⠛⠟⡟⡿⠟⢿⠿⢷⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢻⡛⣏⡁⡵⠉⣈⣰⣧⣄⣤⣶⣦⣄⣤⣦⣈⡈⢘⡅⣩⡟⣻⢳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠻⠛⢀⢸⣷⣬⣿⣿⣿⣻⡿⠿⠿⠯⠿⠯⠿⠿⠿⣿⣻⣿⣿⣧⣴⣷⢠⠘⠙⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⡟⠑⣤⣬⡾⣛⠯⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠫⣟⡻⣮⣦⡖⠙⡿⣶⡀
⠀⠀⠀⠀⠀⢠⣾⡛⣡⣄⣼⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣄⣤⡙⣻⣦
⠀⠀⠀⠀⣰⡟⠒⢴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⠔⠚⣧⡀
⠀⠀⠀⢰⣟⠃⣸⣿⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣿⠀⢛⣧
⠀⠀⢠⣟⠠⣲⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡂⢘⣧
⠀⠀⣼⡟⢳⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠟⣿⡄
⠀⠀⣹⣿⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⡁
⠀⠀⡿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⡇
⠀⢰⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰
⠀⢸⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣹
⢀⡼⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣄
⡎⠀⢰⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢷⠀⢈
⠘⢲⣿⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣾⣷⠚
⠀⢸⠟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠿
⠀⠀⣦⣼⣷⠀⠀⠀⠀⠀⠀⢀⣀⣄⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⢰⣿⣤⡆
⠀⠀⢿⣿⣿⡄⠀⣠⣶⣶⣾⣿⡭⠍⣽⡿⢋⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⠻⣿⡭⠬⣽⣿⣶⣶⣦⡀⠀⣼⣿⣿⠃
⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣛⣒⣿⣿⡇⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣿⣗⣊⣻⣿⣿⡿⢿⣷⣿⣿⣿
⠀⠀⠀⣿⠿⣇⡇⢰⠻⢻⣿⣞⢛⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣟⢛⣾⣿⠻⢳⠀⣇⡿⢿⡇
⠀⠀⠀⢡⠀⢻⡅⢸⠀⢸⣿⣌⣰⢿⣿⡇⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠀⣿⣿⢗⣈⣼⣿⠀⢸⠀⣿⠁⢠⠁
⠀⠀⠀⠈⣆⣈⣷⣸⣤⣼⣯⣭⡁⣼⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣸⣿⡄⢩⣽⣿⣤⣼⣰⣏⣀⠆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣒⣢⣼⣿⡀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣸⣿⣤⣒⣺⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣛⡯⢿⣿⡅⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣿⣿⠯⣟⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⡇⡏⢹⠉⢹⣿⣶⠖⣺⣿⡅⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⣽⣿⡒⢶⣾⣿⠋⢹⠉⡇⡇
⠀⠀⠀⠀⠀⢧⣧⣸⠀⠘⣿⣶⠟⠛⣿⡇⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⣿⣟⠛⢳⣾⡏⠀⢠⣰⣥⠇
⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣶⣞⣿⣿⠊⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢺⣿⣗⣶⣿⣿⣿⣿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⣀⠀⠀⠘⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⣀⣀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢃⠀⠀⠀⠀⡀⠄⠒⠂⠒⠖⠉⠀⠀⠀⠀⠀⣀⢀⠴⠃⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠚⠛⠀⠀⡐⠋⠀⠀⠀⠀⠀⠛⡄⢀⣀⣠⢴⠬⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡠⠰⢌⡤⡀⡠⠀⠜⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠐⠀⠈⠐⠡⠀⠀⠀⠀⣀⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⢆⠀⠀⠀⠀⠈⠀⠂⠨⡌⠒⠢⣀⠀⠈⢂⣈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠡⠀⢿⡉⠈⠉⠁⠘⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠔⡲⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠢⠄⠀⢤⠤⠤⠔⠰⡰⠃⠒⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣒⠂⠀⠁⠀⠀⠀⠀⣶⣄⠧⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣩⢁⡕⢂⢤⢵⠀⡠⢀⠏⠠⠐⠈⠒⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢨⣀⠁⠴⠁⠀⠀⠀⢠⡄⢀⠀⢀⡔⠀⢀⣤⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢴⡁⠈⠄⠒⡡⠋⠐⢜⠄⢣⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⠠⠆⠈⠊⠀⠀⠀⠀⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠐⡉⠄⠀⡄⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠹⣀⠈⠀⠂⠠⠷⠖⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠔⠁⢈⠀⢄⠁⠔⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠈⢄⠀⠁⡀⠀⡘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡐⠄⠠⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡊⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣤⠶⠷⠚⠛⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⢩⠁⢀⡀⢀⡤⠀⠾⢀⠎⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢿⠄⠈⠀⢠⡄⠀⠾⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠻⠀⢀⠀⠸⣷⡀⠀⢠⣤⡄⠐⠦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠴⢀⣀⠀⣠⡆⠀⠀⠰⠒⠃⠀⠴⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣾⣧⣬⣯⡄⢰⣿⠩⠵⣄⣀⠀⠆⢸⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⢦⣍⣛⠴⠁⢰⣷⣀⣼
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢻⣿⣿⣿⣷⣤⣀⠛⢆⠀⢀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⡿⡍⠳⠀⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣷⣸⡞⠃⠴⠃⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣼⣿⣿⣿⣿⣿⣅⣚⣚⡃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣏⠈⢰⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⡽⣿⣯⣿⣿⣿⣿⡟⢠⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⢾⣽⣿⣿⡿⠛⡟⠋⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⠀⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡋⢹⣿⡋⣻⣿⠉⣿⣿⣿⣿⠄⠀⠀⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠈⠛⠶⢤⣤⣤⣤⣤⣤⣄⣺⡏⠁⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⢸⣿⣿⣿⣿⡟⠿⢿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⡿⠿⢻⣿⣿⣿⣿⡇⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠙⠿⠿⠷⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠾⠿⠿⠋⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣿⣿⣿⣧⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀⣼⣿⣿⣿⣿⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣿⣿⣿⡏⠉⠙⢿⣿⣷⡀⠀⠀⢸⡇⠀⠀⢀⣾⣿⡿⠋⠉⢹⣿⣿⣿⣿⣿⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⣿⣷⠀⠀⢸⡇⠀⠀⣾⣿⣿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣿⣧⠀⢸⡇⠀⣼⣿⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⣿⡇⢸⡇⢸⣿⣿⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠆⠀⢠⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠰⠆⠀⣤⠀⢠⡄⠀⣀⠀⢀⡁⠀⠈⠁⠀⠋⠀⠐⠂⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠠⠆⠀⣤⠀⢀⡀⠀⣀⠀⠀⠉⠀⠘⠋⠀⠛⠃⠰⠖⠀⢶⠀⢠⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⡤⠀⣠⡀⢀⣀⠀⢈⠁⠀⢛⠁⠘⠛⠀⠿⠇⠰⡶⠀⣴⡄⢀⣤⠀⢀⡀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣄⠀⢀⡀⠀⣈⠀⠈⣉⠴⠛⠉⠉⠉⠉⠉⠓⠶⣶⠄⣴⣦⢀⣤⡀⢀⡉⠀⠉⠁⠀⠛⠀⠐⠂⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⢀⡀⠀⠈⠁⠈⠛⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣬⡥⠚⠉⠁⠉⠙⠰⡿⠂⠰⠆⠀⣴⠀⢠⡄⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠸⣷⠾⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠘⣦⣴⡄⢀⣤⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠻⣢⠔⠉⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠀⢀⠉⠀⠙⠁⠀⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠛⠀⠰⠂⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⠀⠰⠆⠀⠤
⠀⠀⠀⠀⠀⠰⠆⠀⣤⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⠀⢀⡀
⠀⠀⠀⠀⠀⢠⡄⢀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠉⠀⠘⠁⠀⠂
⠀⠀⠀⠀⠀⠉⠀⠘⠃⠐⠿⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠻⠇⠀⠶⠀⠠⠄
⠀⠀⠀⠀⠘⠃⠀⠶⠀⢰⡦⠈⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠟⠀⠶⠆⢠⡤⠀⢀
⠀⠀⠀⠀⠰⠀⢠⣤⠀⣠⣤⡤⣉⠉⢛⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣶⡆⢠⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⣼⠋⠀⠀⠈⠀⠻⠇⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⢀⡁⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⢿⡄⠀⠀⠀⡰⢿⠆⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣉⠀⠘⠁⠀⠛⠀⠀⠆
⠀⠀⠀⠀⠀⢀⡻⠀⠸⠟⠷⢶⠞⢡⣦⠀⣤⡄⢹⣿⣦⣤⣀⣀⣀⣀⣀⣤⡶⣿⠏⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠙⠛⠀⠻⠂⠰⠆
⠀⠀⠀⠀⠀⢸⣄⣀⣦⠀⢀⣤⠀⣀⡀⠀⣉⠁⠘⠛⠉⠻⠟⠻⡿⠛⣿⡿⢴⣶⠄⣴⣽⢻⣿⣷⣶⣶⣶⣾⣿⠿⠟⠀⠾⠇⠠⡶⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠉⠁⣀⠀⠀⠁⠀⠉⠁⠘⠛⠀⠻⠗⠰⣿⠆⣶⣶⢠⣶⡄⣠⣭⢀⣉⡁⢈⡛⠉⠻⠛⠙⠿⠃⢴⠆⠠⣶⠀⢠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠸⠗⠀⣶⠆⣠⣦⢀⣤⡄⢈⣍⠀⣙⡋⠈⠛⠃⠻⠟⠀⠿⠆⢰⣶⠀⣤⡄⢀⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢴⠄⢠⣄⠀⣨⡁⢀⣙⠁⠘⠛⠈⠻⠃⠸⠿⠀⣶⡆⢠⣦⠀⣤⡄⠀⣀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠘⠛⠀⠻⠇⠰⠿⠀⢴⡦⢀⣤⠀⢀⡀⠀⣈⠁⠈⠉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠺⠇⠀⣶⠀⢠⡄⠀⣤⡀⠀⠉⠀⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣾⣿⡿⠿⣿⣶⣿⣿⣽⣟⡲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⡿⣻⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⢟⣵⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠛⠻⣿⣦⡀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡾⣵⣿⣿⣿⣿⢟⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⢟⣾⣿⣿⣿⡿⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⢏⣾⣿⣿⣿⢟⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣫⣼⣯⣆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣯⡟⣾⣿⣿⣿⣏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣹⣿⣿⣿⡎⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣹⣿⣿⣿⡿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⢰⣿⡏⣿⣿⣿⣿⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⠀⣸⣿⢳⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⡿⣿⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⣷⣿⢸⣿⣿⣿⣗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠛⠋⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢳⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠘⣇⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢛⠉⣉⣙⣲⣼⣿⣿⣿⣿⣿⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢀⣠⣤⣴⢟⣿⣷⠿⠿⠿⢲⣤⡀⠀⠀⢀⡴⠋⢠⣀⣼⠟⠉⣁⣤⠜⣿⣾⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣿⣻⣿⡟⢀⣴⣶⣶⣄⡐⢹⢆⣰⠏⠡⣄⣶⠏⠁⣠⣾⣿⠏⢀⣾⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡟⢸⡏⣿⡿⠀⢸⣿⣿⣿⣿⣿⣧⣾⠧⣶⣶⡿⠁⢠⣾⣿⡟⠁⣴⣿⣿⣿⣿⣿⣿⣿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⡇⢸⡇⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⡯⣶⣿⣿⠁⣠⣿⣿⠏⢀⣾⣿⣿⣾⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠀⣿⡇⣾⣿⢻⣿⣧⠈⢿⣿⣿⣿⣿⣿⢧⣾⣿⣿⣶⣿⣿⠃⢠⣿⣿⡷⣡⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣿⣏⢿⣿⣷⣌⠿⣿⣿⣿⣿⣈⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣞⣿⣿⣿⣿⣿⣞⠻⣿⣿⣷⣼⣭⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣟⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⣷⣌⡻⠿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣇⣴⢟⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣙⠻⢷⣤⣙⠻⠷⣶⣿⣿⠶⠋⠙⠿⣷⣈⢻⣿⣦⡹⣿⣿⣿⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠓⠹⠿⠿⠾⠿⠿⠿⠿⠛⠉

⣤⣤⣤⣠⣄⣠⣤⣤⣄⣤⣀⣠⣠⣠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣤⣠⣄⣤⣤⣤⣄⣀⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠀⠀⠀⠀⠈⠑⢦⣠⣴⣶⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡞⣠⠖⠒⠖⠒⠊⠁⠈⠻⢿⣿⣿⣿⣿⣿⢿⣿⣦⣀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠘⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⡿⠃⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠔⠋
⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⠉⠉⠉⠉⢹⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢿⠀⠀⡄⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠈⠧⢼⡃⠀⢸⠀⠀⢹⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣿⠀⠀⡇⠀⠀⡷⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡅⠀⢸⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡇⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⢰⡀⠀⠀⠠⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠐⣀⠂⠁⠸⠀⠄⢀⠀⢺⠠⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠈⠋⠹⢀⠀⠀⢿⠂⡖⠁⢄⠅⢰⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⣄⠀⠀⠀⣸⣤⣤⣶⣶⣾⣿⣾⣿⣿⣿⣷⣶⣬⣬⣀⠀⠀⢁⠔⠃⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⠈⣦⣴⣿⡿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣿⣶⣄⡈⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠘⠃⠀⠀⣤⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⡈⠂⠀⢠⠁
⠀⠀⠀⢑⢢⣤⡀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡂⠁⠂⠀⠀⠠⠄
⢠⠀⢀⠀⠠⠀⠈⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠉
⠀⠉⠁⠀⠪⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠠
⠀⠀⠀⠠⠆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠤⠄
⠀⠀⠀⠀⢀⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠄⠋⠔⠣⠂
⠠⠤⠄⡀⠁⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠐⠁
⠠⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⢀⠀⣀⣤⢬⡜⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⡄
⠀⠐⠊⠑⠀⠐⠈⠀⣼⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠋⠂⠤⡐
⠀⠀⠀⠀⠀⠐⠫⠈⠀⠀⠻⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢔⠠
⠀⠀⠀⠀⠁⠀⢀⠔⠠⡀⠂⢀⠘⢙⠿⣿⣶⣤⣤⣀⣀⣀⠀⠀⣀⣀⣀⣠⣤⣴⣦⣤⣄⡀⠀⠀⠀⠀⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠌⠀⠤⢠⠎⠈⢹⠙⡿⠟⠿⠟⠿⠿⠛⢻⠛⠉⠋⠉⠛⡛⠻⠿⣶⣤⣀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣀⠢⠀⠀⠀⡣⠇⠇⠀⠲⠀⠠⢀⠐⠈⠆⠀⠈⡆⠀⠈⠄⠀⠀⠈⠝⠳⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠀⠀⢠⠖⢀⠘⠆⢘⠀⠀⢲⠀⠀⢡⠀⠀⠌⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠄⠀⠐⠐⠀⠀⠂⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣩⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⣶⡖⠒⣶⣾
⠀⠀⢸⡟⠣⠀⠝⣿
⠀⠀⢸⣷⠒⠶⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣼⣦⣤⣶⣤⣠⣤⣤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⢸⣷⠴⠶⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⢠⡼⣿⣿⣿⣿⣷⠄
⠀⠀⢸⡿⠤⠴⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⢰⣤⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢋⣼⣿⣿⣿⠟
⠀⠀⢸⣧⣦⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣷⣉⣘⢤⣶⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠁⠀⠈⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘⡩⠕⠊⢰⡋⠍⠋⣞⣻⣽⣿⣿⣿⡿⠟⠛⠟⣿⣿⣥⣽⣀⠀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⣴⣾⣿⣿⣿⣿⡟⠉⢀⣠⣤⢞⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠿⢛⣿⣍⣿⢿⣿⡿⠀⢰⣿⣿⣯⣾⣿⣿⠿⢻⣧⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⣤⣒⣢⢤⣤⣴⣋⣼⣿⣿⣷⣿⣿⡄⢺⣿⣧⣼⣿⣿⡟⠀⡼⣿⣾⣿⣿⠷⠂
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠒⠢⠤⣀⣤⣽⣿⣧⣤⣾⣟⣙⣿⣿⣿⣿⣿⣿⡀⠈⠻⠿⠿⠿⠋⢀⡼⣡⣴⣿⣿⣅⡀
⠀⠀⠀⠀⠀⡾⠁⠀⠰⠶⡀⠀⠀⢠⣽⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣤⡀⣀⣴⣿⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣰⠇⠀⡤⡀⠀⠀⠉⢶⣄⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣶⣿⣧⣀⠀⠈⠑⢤⡚⢻⣿⣿⣿⣿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡖
⠀⠀⠈⣿⣿⣿⣿⣧⣄⣀⢠⣹⣮⣿⣷⠙⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠋⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣢⣤⣴⣶⣶⣦⣤⣔⣅⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠹⣿⣿⣿⣷⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡢⢳⢻⣿⡟⠛⠛⠛⠋⠉⠀⠀⠀⠀⠙⣿⣿⡟⢻⠠⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢰⠀⢸⢸⣿⠀⠀⠞⠛⠦⠀⠀⠰⠚⠛⠆⠈⠻⡇⢸⠀⢡⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡠⠋⢄⢸⢸⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠎⢂⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠔⠀⠀⠀⢹⣾⡀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⣸⣾⠁⠀⠀⠡⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢠⠊⠀⠀⠀⠀⣾⣿⣷⣄⡀⠀⠀⠑⠠⠄⠊⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠑⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡰⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⣄⢀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⢄⠀⠈⢢
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠘⠠⢀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⡈⠆⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠏⠀⠈⠉⠙⡟⠛⡏⠉⠁⠑⠄⣀⡀⠔⠁⠈⠉⢻⠻⠟⠛⠉⠉⠀⢙⠉⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⡌⠀⠀⠀⠀⢀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢠⠀⠀⠀⠀⠀⢆⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢄⡀⠀⠘⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢆⠀⠀⡴⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠳⡞⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠋⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⡀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢢⠤⡄⠀⠀⠀⠀⢀⠙⢅⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠂⠀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠣⠀⠂⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠻⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠣⡀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠃⠂⠀⠀⠀⠈⠀⠀⠀⠀⠁⠐⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⢀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠔⠁⢁⠨⢂⡀⠀⠀⠠⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⡡⠂⠀⠁⠀⠀⠀⠀⠠⠤⠤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠠⠂⠊⠀⠀⠠⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠉⣧⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⢰⠛⠀⠀⠀⠀⢀⠀⠀⡃⠄⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠉⠁⠀⠀⠘⠁⠀⠀⢨⠀⠀⣱⠰⠭⢄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠂⠉⡧⠀⠠⢣⠐⠪⠀⠂⡀⠀⠈⠀⢰⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠹⣄⣀⠈⠀⠀⠀⢆⡰⢀⠀⠀⠀⠂⠂⠀⠀⠀⠀⢀⠀⠀⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠂⠈⠀⠀⠈⠉⠀⠉⠑⣄⠀⠸⠀⡀⠢⠀⠀⠀⢀⠀⠀⠊⠀⠀⠃⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⢠⡒⠀⠄⠐⠀⡴⠊⠉⣦⠐⠀⠃⠀⡀⡩⠀⠑⠀⢀⠠⠀⡀⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢠⠂⠈⠂⠀⠐⠁⠀⡀⠀⠠⠀⠀⠩⠄⠀⠀⠤⠌⠀⠀⠀⠠⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢈⠀⠀⡄⠀⠰⡂⠀⢀⡄⠀⢔⠃⠀⠀⠀⢠⠦⣀⡠⢄⠐⠀⡇⣀⠀⠀⠀⠀⠀⣠⠄⠀⠂⠒⠋⠉⠉⢸⡗
⠀⠀⠀⢢⠀⠠⠉⠀⠀⠓⠀⣞⡀⢀⠁⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠙⢀⡆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠠⢄⡀⠃⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⠤⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠰⣶⠛⠉⠁⠀⠉⠉⠲⣄⠀⠀⠀⠀⠀⠀⢠⣬⠇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢰⠃
⠀⠀⢊⡉⢉⣣⠶⡄⡴⣄⡞⢋⠛⠓⠀⠀⠀⠀⠀⠀⠸⠞⠳⢲⢠⡞⠛⠲⢷⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠋⠀⠀⢿⠀⠈⠓⠀⠲⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠉⠀⠀⠀⠃
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠒⠒⠖⠓⢦⠀⠀⣰⢀⠀⠀⠀⠀⠀⣠⠄⠦⣄⡀
⠀⢸⡅⠈⠁⠀⠀⠒⠂⠐⠒⠒⠶⠦⣤⠀⣀⠀⢳⡞⠁⠀⠀⠀⠀⠀⠱⢞⠁⣀⡀⢀⡤⠴⠋⠃⠀⠀⠀⠈⠉⠢
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⡇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠤⠸⣆⣀⡀⠀⠀⠀⠀⣀⠀⣠⠀⡀⠀⠙⢲⡀⢀⠀⠀⠀⠀⣀⠀⡠
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠖⠀⠀⠉⡙⠶⠖⢻⠒⠉⢍⠀⠀⢌⠀⠀⠀⠉⠉⠳⠖⠊⠒⠉⠁⡄
⠀⠀⠀⠙⠁⠁⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠐⢱⠁⠀⠈⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠒⠦⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⠴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠠⡄⠀⠀⠀⠀⠁⠀⠀⠀⢹⡇
⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠑⠦⣄⣀⡀⣀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠈⠓⠶⢤⣤⣤⡤⠖⠋⠀⠀⠀⠀⢽⡇⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⢉⣭⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠀⠈⠉⠉⠉⠉⠀⠀⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⢀⠀⠀⠤⠄⠀⠀⠤⠄⠀⢀⠀⠀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⢄
⠀⠀⠀⠀⠀⠄⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠄
⠀⠀⠀⠀⠈⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈
⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠆
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠘⠀⠀⢀⠀⠀⠐⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡐⠀⠠⢀⠀⠀⠘
⠀⠀⠀⠀⠁⢰⠢⡀⠀⠀⠈⠄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠌⠀⠀⠀⡠⢲⠀⠃
⠀⠀⠀⠀⠰⠘⠀⠐⡀⠀⠀⠈⢂⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠈⠀⠀⠀⡐⠀⠘⠰
⠀⠀⠀⠀⠀⠃⢂⠀⠘⠄⠀⠀⠀⠠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⠂⠀⠀⠀⠌⠀⢀⠂⠂
⠀⠀⠀⠀⠀⠠⠀⢂⠀⠈⢂⠀⠀⠀⠡⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡠⠁⠀⠀⢀⠊⠀⢀⠊⢀
⠀⠀⠀⠀⠀⠀⢁⠀⢂⠀⠀⠢⠀⠀⠀⠐⡈⢄⠀⠀⠀⠀⠀⠀⠀⢀⠌⡐⠀⠀⠀⢀⠂⠀⢀⠂⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠡⠀⠀⠡⠀⠀⠀⠈⠄⢂⠀⠀⠀⠀⠀⢠⠂⠌⠀⠀⠀⠠⠁⠀⠠⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠐⠀⠀⠐⡀⠀⠀⠈⢂⠡⠀⠀⠀⠀⢁⠊⠀⠀⠀⡐⠁⠀⠐⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⢆⠀⠀⠄⠀⠀⢀⠓⠀⠀⠀⠐⢃⠀⠀⠀⠌⠀⢀⡎⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠈⠂⠂⠁⠀⠀⠀⠀⠀⠀⠁⠂⠊⠔⠈⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣤⣤⣀⡀⠀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠙⣿⡏⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⣘⣿⠀⢧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢛⣭⣭⣍⡛⢿⣿⣿⣿⣿⣿⣷⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣦⣴⣾⣿⣿⣿⣷⣦⡙⠻⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢡⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⡏⣾
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢹⣿⡘⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣜⢿⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣮⣭⣭⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠋⠴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠦⠉⡤⠄⠀⢀⣼⠦⡅⢀⣄⣠⠀⣀⡀⠀⠀⣠⣿⢄⣀⠀⠀⢀⣠⣀⡄⡀⣀⣠⠄⣉⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠀⠈⣳⠄⠀⠀⠸⠀⠇⠿⠻⠹⠺⠯⠖⠀⠀⠀⠇⠺⠞⠀⠀⠸⠿⠏⠏⠳⠏⢳⡆⠿⠺⠥⠖⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢤⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠆⠀⠀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢡⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠠⠈⡀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠬⠁⡼⠀⠀⠀⠠⢤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠠⠈⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠠⠀⠀⠤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠄⠀⠈⠀⠀⢠⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⣀⡠⠀⠠⠀⠀⠀⠤⠤⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⣦⠀⠄⠀⠀⠀⠀⠀⠈⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠⠤⣀⠀⣀⠀⠀⠀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠀⠀⡀⣄⠀⡇⠀⠀⠔⡇⠠⠙⢀⠈⠇⢀⠎⠀⣤⢠⡀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⢂
⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⣠⡀⣆⡿⠟⢀⡅⠀⢈⠀⣃⠁⠂⠄⢈⠀⢸⡀⠰⠟⠿⢸⣟⣠⣟⡀⣰⠃⢀⡴⢶⢀⡆⠁
⠀⠀⠀⠈⠁⠀⢀⡀⣄⡇⢧⠞⠈⢀⠔⠉⠀⠀⠀⠠⣠⣀⢤⠔⠃⠀⠘⣝⠦⡀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠛⠛⠰⠀⠀⠈⠢
⠀⠀⠀⢂⠘⡀⠈⠷⠏⠁⠀⠀⡠⠃⡠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠑⠬⣁⣒⠂⠤⠄⢀⣀⣀⣀⠀⠤⠀⡂⢁⠢⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠀⠀⠀⡠⠄⠠⠤⠂⠉⠁⠂⠀⠀⢄⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠒⠂⠈⠁⠀⠀⠑⠙⡄⡆
⠀⠀⠀⠀⠠⠀⠄⠀⠀⢊⠔⠁⠀⢀⠠⠌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⡂⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠒⠁⠀⠀⠀⡰⠀⠀⣀⢤⠀⢄⠰⣆⢰⢖⡄⢹⠀⠀⢈⠀⠀⠃⠀⠠⡀⣆⢲⡂⢦⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⢌⠀⠀⠀⢽⢸⠇⣼⢱⠽⠀⠯⠁⠰⠄⠀⢠⠀⠀⠀⠃⠀⠆⠛⠘⠒
⠀⠀⠀⠀⠀⠀⠡⠀⡄⠀⠀⠀⠀⢁⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠁⠀⠀⠀⠄⢐⠠⡀
⠀⠀⢀⠂⡀⠀⠀⠀⠉⠀⢀⠂⡠⠀⠂⠠⠐⢄⡀⡠⢀⠀⡠⠠⠀⠔⠁⠀⠀⠢⡪⠀⠀⠀⣠⠔⠀⠀⡘⠀⠸⢴⡆⡀⡀⠀⡐⠰⠁
⠀⠀⠈⠂⠌⠂⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠜⠐⠱⠃⠐
⠀⠀⠍⢐⣠⠄⠉⠀⠀⠀⠀⠀⠀⠁⢀⠀⣀⣀⣀⠀⢀⢀⡀⡀⠀⡀⣀⢀⠀⠀⠀⠈⡆⠀⠐⡪⢀⠈⠀⠀⠀⠀⠀⣀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠋⠚⠋⠛⠃⠉⠀⠙⠙⠚⠃⠈⠓⠃⠈⠀⠀⣀⡠⠃⠀⠀⠐⠁⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⡀⢀⠤⠀⠒⠒⠒⠂⠤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠒⠂⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠤⢒⡤⠉⠢⠤⠤⢠
⠀⠀⠀⠀⢋⠞⠁⠀⠀⢀⡀⢀⠀⠀⢘⠀⠀⠀⠀⠘⢀⠄⢀⡀⣀⡠⡄⠀⠐⠀⠀⠀⠀⠀⢀⠁⠉⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⡅⠀⠀⠎⠹⠁⢙⠇⠀⠀⠢⡀⠀⠀⠀⠀⠐⠺⠑⠊⡓⠹⢎⠀⠁⠀⠀⠀⠀⠆⠁⠀⢂⠀⡀⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠀⠀⠰⠁⠀⢠⡤⡄⣠⣠⣤⠄⠀⠀⠘⠀⠀⠀⡆⢈⣰⣄⢻⢰⣆⠸⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠢⠈⠠⠀⠄⠀⠐
⠀⠀⠀⢠⠀⠀⠈⠁⠁⠉⠉⠛⠀⠀⢀⡜⠀⠀⢠⠀⠈⠀⠁⠀⠈⠀⢀⠤⡈⢆⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠦
⠀⠀⠀⠀⠑⠤⣀⡀⠀⠀⣀⡠⠤⠐⠉⠀⠀⠀⠐⠂⠀⠀⠉⠉⠁⠀⠀⠀⠈⠈⠀⠀⠀⠈⠈⠄⠀⢀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⢀⡠⠄⠬⠅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠠⠤⠄⠦

⣀⣀⠀⠀⠀⢀⣾⠇⠀⠀⠀⣠⠴⠶⢤
⠙⠚⠗⢠⠴⠦⡅⠀⠀⠀⢰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠹⢤⣰⡇⢤⣄⠀⠈⠣⠤⣄⠘⢧⠀⠀⠀⣀⡠⠤⠖⠒⠒⠒⠒⠶⢤⣄⠀⠀⠀⠀⠀⣸⠀⠀⠈⡆
⠀⢠⣾⠇⠀⢿⣇⠀⠉⠁⠀⠀⠀⣈⣧⠈⣷⠖⢋⣡⠤⠲⠶⠾⢯⣍⣓⠒⠦⣌⠙⠦⣠⠔⢉⣠⣀⢀⣠⠇
⠀⠀⠀⠀⠀⠈⣿⡆⠀⢀⣀⡔⠋⠉⣠⠟⣡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣌⣹⢦⠙⣖⠋⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠸⡽⣴⠋⠉⠀⠀⡴⢃⡞⢧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡼⢳⠘⣆
⠀⠀⠀⠀⠀⠀⣠⢷⢣⠀⠀⠀⡼⢡⠏⠀⠼⠦⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠋⠛⢧⢹⡀
⠀⠀⠀⠀⠀⣰⠃⠘⡏⢇⠀⢰⠇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⢸⡌⣧
⠀⠀⠀⠀⢰⠇⠀⢠⡞⠉⢙⣾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢿⠀⣰⠓⠛⣧⡀
⠀⠀⠀⠀⣼⠀⠀⣈⠷⠖⠺⡉⢹⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢀⡴⠁⠀⣠⠞⠁⠀⢸⡀⡇⠀⠀⠙⣍⠓⢦⡀
⠀⠀⠀⠀⢹⠀⡼⠁⠀⠀⠀⢱⠘⠳⡄⢀⡾⣀⠴⠣⣀⣠⠔⠉⠓⠖⠊⠁⠀⠀⠀⠀⠳⢷⠀⠀⠀⡌⠀⠀⠙
⠀⠀⠀⠀⢸⣼⠁⠀⠀⠀⠀⠸⡇⠀⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡏⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⡇⠘⣍⠀⠀⠀⠀⠐⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠐⢇⠀⢀
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⠇⠀⣿⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠶⢺⡁⠀⠀⢀⣾⣴⠋
⠀⠀⠀⡴⠋⠁⠙⢦⣀⠀⠀⡸⠀⢾⠁⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⣳⠀⠀⢀⡟⢻
⠀⠀⠀⠉⠛⠛⡿⠀⠈⢽⠛⠁⢀⡼⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⣤⠷⡀⠀⠀⠀⠀⠀⡿⠦⠴⡾⠒⠋
⠀⠀⠀⠀⠀⠀⠳⠤⠖⠻⣷⣋⣩⢴⣶⠠⣵⠦⡀⠀⠀⠀⠀⠀⠀⢇⠘⣂⠇⠀⠀⠀⣠⠞⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠳⡈⠳⣄⠉⠒⠠⠤⣤⣀⣤⣥⣥⡤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠙⠲⣌⡢⣄⣠⠞⣷⠋⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⡇⠙⠛⠓⠋⢠⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠘⢦⠀⠀⠀⢿⠂⠀⠀⢰⠋⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⢧⡀⠀⠸⡆⠀⠀⣸⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣍⣙⡿⠓⠚⢷⠀⠀⠛⣟⢿⠉⠓⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣽⠁⠀⠀⠈⠳⣤⡴⣿⠋⠓⢺⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢹⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣻⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿