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⠀⠀⣀⣠⣀⡀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⢀⠠⠤⢄⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⢀⠠⠀⡀⠀⠀⠀⢀⣠⣄⣀
⢠⣿⣿⡿⢻⣿⣆⢀⣾⠟⠿⠟⢿⣷⠀⡔⢁⡄⠀⢤⠘⡄⢠⠚⡅⠀⢨⠍⢂⠀⢎⡀⠀⠀⢈⠣⠀⡔⠁⠀⠀⠈⠳
⢸⣿⠈⡁⢀⣿⣿⢸⣿⣆⠀⢀⣼⣿⠃⡇⢶⠒⠒⣶⠆⡄⢀⠀⠀⡶⡆⠀⡸⠸⠀⠶⠀⠀⠃⢀⠀⡁⢶⠆⠐⡶⠂
⠀⠛⠿⣿⡿⠟⠃⠀⠙⢿⣿⣿⠿⠋⠀⠐⠌⣫⣽⡫⠔⠀⠈⠢⢄⡳⠣⠔⠁⠀⠑⠄⠉⠁⢼⠂⠀⠑⠤⣉⣛⡠⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡶⠿⠿⠿⠿⠿⠿⠿⠿⢶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙⠻⣦⡀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠉⠉⠙⠻⢶⡀⠀⠀⣤⣾⠿⠛⠛⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⣀⣠⣶⡿⠁⠀⠙⢦⣄⡀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⢸⡿⠀⠀⠀⡄⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣶⣶⠾⠃⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁
⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⢀⣀⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠄⠠⠠⠀⠀⠠⠄⠄⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆
⠀⠀⢰⡉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⣱
⠀⢀⠀⠉⠉⠉⠉⠀⠀⢀⣤⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠈⠉⠉⠉⠁⢀
⠀⠣⠄⠒⣀⣤⡄⠀⠀⠉⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠙⠂⠀⠀⠀⢰⠀⠐⠂⠤⠃
⠀⠀⡴⠛⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠉⠓⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠒⠒⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠀⠀⠀⠀⣀⠤⠤⣄⣠⠤⠚⠶⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡎⠀⠀⠀⠀⠈⢂⣀⣤⠞⠉⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠞⠛⠛⠛⠛⠛⠛⠳⠶⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⣼⠋⠀⠀⠀⠀⠀⢠⡾⠟⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠻⣦⡀⠀⠀⠀⠀⠙⣧
⠀⣸⠏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠹⣇
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠿⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠉⠙⠂
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⠉⠙⠛⠛⠳⠶⠶⠶⠶⠶⠶⠞⠛⠛⠋⠉⠁⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⢋⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⣠⡶⠶⣦⣄⣴⠶⠶⣦⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣼⡏⠀⠀⠀⠉⠀⠀⠀⢸⣧⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢶⣶⣦⣤⣤⣴⣶⣶⠶⠛⠛⠉⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢴⣶⣶⠀⣦⠀⢰⡆⢶⠦⣶⢲⡆⠰⡖⢰⠆⠀⣾⠲⡄⣶⠺⡄⢶⠐⡖⣶⠴⡶⣶⠤⣴⣦⢠⣦⢠⡆
⠀⠀⢸⡟⢻⠀⣿⠀⢸⡇⣼⠀⣿⢸⡇⠀⢿⡾⠀⠀⣿⢀⣷⣿⣶⠀⣿⠀⡇⣿⣇⡇⣿⢰⡇⢻⠀⣿⠘⡇
⠀⠀⢸⡇⠈⢰⢿⠀⢸⣧⣿⡇⣿⢸⡇⠀⢸⡇⠀⠀⣿⢾⡃⣿⠈⠀⣿⠀⡇⡇⣿⡇⣿⢸⠃⢸⡇⣿⡆⡇
⠀⠀⠘⣷⡇⢸⢹⡄⢸⢿⢿⡇⣿⢸⣧⡼⠸⠇⠀⠀⠿⠘⠷⠿⠶⡎⢿⣀⡇⡇⠸⡇⣿⢸⡃⢸⡇⡟⣧⡇
⠀⠀⢨⡟⠇⢸⣾⡇⡿⢸⢨⡧⠿⠈⠉⠁⣀⣤⡤⠤⠶⠶⠶⢤⣤⣀⡀⠉⠐⠷⠀⡇⣿⢸⡇⢸⡇⡇⣿⡇
⠀⠀⢸⡇⠀⡾⠹⣇⣷⠈⠈⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⡀⠀⠁⠿⠸⣇⣸⡇⡇⢹⡇
⠀⠀⢸⡇⠀⡇⠸⠟⠁⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠻⣿⠀⡇⢸⡇
⠀⠀⢸⡇⠰⠏⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠸⡇⠘⡇
⠀⠀⢸⡇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣰⣿⣿⣿⠄⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣤⢤⡄⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢹⠀⠀⢠⣀
⠀⠀⠀⠀⣣⡄⠀⠀⡇⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⢸⠃⠀⢨⢤⠁
⠀⠀⠀⠀⠰⢮⠀⠀⣿⠀⠀⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠁⠀⣿⠀⠀⡜⡍⠁
⠀⠀⠀⠀⠐⠛⠆⠀⠸⣇⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣰⠇⠀⢠⣭⠉
⠀⠀⠀⠀⠀⠉⠀⣠⡄⠹⣆⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⣰⠏⢀⣻⠶⠍
⠀⠀⠀⠀⠀⠀⠹⢅⡀⠀⠘⢷⣄⠀⠀⠈⠙⠳⠦⣤⣤⣤⣤⠴⠞⠋⠁⠀⠀⣠⡾⠃⠀⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⣱⣀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⢠⡶⣼⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠓⠀⠀⢀⡉⠛⠲⠶⠤⣤⣤⠤⠶⠖⠛⠉⠀⢀⢦⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢟⣀⡠⡄⡀⡀⡀⡤⣄⠀⠀⣤⡐⠟⠎⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⢶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢠⡿⠋⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠈⠹⣧⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⣾⡇⠀⣀⡀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⢀⣀⠀⣿⡆⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢻⡇⠰⣿⡗⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢾⣿⠆⣿⠁⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢻⣦⣬⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣥⣾⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠃⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣤⣶⠾⠟⠋⠀⠀⠀⠀⠀⠀⣠⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣄⡀⠀⠀⠀⣠⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⠿⠃⠀⠀⠚⠛⢋⣉⠀⠉⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠧⣄⠀⡠⠚⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⢠⣾⡆⠀⠀⣸⢠⣦⡤⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⡀⠀⠀⠈⠋⠀⠀⢀⡿⠸⠁⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢀⡼⣄⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠒⠉⠀⠈⠓⠶⠶⠞⠋⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⠀⠀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠛⠋
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢀⣤⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⡾⠀⠀⢀⣴⣿⡿⠛⠛⠻⢿⣷⡄⠀⠀⠀⠀⠰⠟⠛⠀⠀⢀⣤⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠉⠉⠻⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸
⠀⠀⠀⠀⠉⠛⢓⣛⡀⣼⡿⠿⠿⠿⠿⣷⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠟⢩⣀⣤⠶⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢀⣈⣉⣉⣛⠛⠿⠀⠀⠒⣒⡚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠉⠷⠶⠤⠳⣦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⡀⠀⠀⠀⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠈⠛⠀⠀⠀⢸⡷⢤⣄
⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⠤⠴⠶⠶⠾⠿⠿⢀⡀⠀⠀⠀⠀⠈⠁⠀⢹⣦
⠀⠀⣠⣶⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⢀⣠⡿⣷⣄⠀⠀⠀⠀⢀⣄
⣼⢡⣿⠁⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⠀⢠⡆⠀⠀⠀⠉⠛⠛⠛⠛⠋⢿⡄
⣿⡈⢿⡀⠉⠉⠉⠛⢿⣝⢻⡶⢦⣄⡀⠀⠀⠀⠀⠹⣿⣷⣶⣶⠛⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢸⠀⣤
⠘⢷⣄⠀⠀⠀⠀⠀⠀⠉⠻⣅⢠⣼⠉⠛⡶⠦⣤⣀⣀⣈⣉⣀⣀⣀⣤⠴⢶⣋⡿⠋⠉⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡟⠁⢀⡁⠀⠿⠀⠉⡟⠉⠹⣁⣀⣼⠀⣼⠟⠁⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⠉⠁⣤⠛⠛⣀⠀⢨⠉⢀⣼⡿⠋⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣿⣀⣀⣃⣠⣿⠶⠛⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣬⣉⣉⣡⣤⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⠀⠀⠀⠀⠀⢀⡤⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠴⡄
⠀⠀⠈⠻⠷⢂⠀⡐⠚⠋⠉⠀⠀⠀⠰⠶⣒⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠄
⠀⠀⠀⠁⠉⠈⠀⠁⠉⠀⠐⠀⠀⠀⠀⠀⠉⠁⠠⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛
⠀⠀⠀⠀⣀⠀⠀⠘⠛⠶⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡄
⠀⠀⠀⠈⠁⠀⢤⠀⠀⠀⡀⠀⠀⠀⠀⠁⠉⠀⠀⠉⠁⠅⠀⠀⠀⠀⠀⠀⣙⠢⠀⠰⢛⡅⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄
⠀⠀⠀⢂⠒⠀⡸⣔⠂⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠝⠋⠀⠀⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣴⣤⣶⣶⣶⣦⠀⠀⠀⠀⠀⠈⠚⠀⠀⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠃
⠀⠀⠀⠀⠰⢶⢠⠶⠀⠀⠀⠀⠀⠀⠀⠴⣦⡀⠀⢀⣴⠦⠀⠀⠀⠀⠀⠀⠐⠋⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⣦⡍⠀⢩⣴⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠒⠀⠒⠛⠦
⠀⠀⠀⣠⠘⠃⠀⠛⢀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂
⠀⠀⠀⢸⣇⣷⣶⡎⣹⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠙⠏⠨⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠁

⠀⠀⠀⠀⢀⡀⠠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠆⠀⠞⠋⢉⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠄⠀⠀⢾⢍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣄⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⣿⣶⡞
⠀⠀⠀⠀⠈⠉⠉⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣀⡤⠶⠛⠋⠉⠉⣿⠀⠀⠀⠀⣿⠉⠉⢙⡛⠲⢦⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⢠⣾⡿⠖⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠈⠛⠛⠉⠉⠛⠓⢿⣷⣄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠈⠛⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⢠⡿⢁⠎⠀⠀⣸⣿⠟⠋⠉⠙⠻⣿⡆⠀⠀⠀⠀⠀⢀⣿⡟⠋⠉⠉⠉⠻⣿⡆⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⣸⡇⠈⢀⡇⠀⠛⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣿⡇⠀⠘⠁⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢀⣴⣿⣉⠉⠙⠛⠛⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⡴⠶⠾⠛⠛⠉⠉⣨⣿⣿⣷⡄⠀⢸⣿
⠀⠀⠀⠀⢻⡇⠀⠘⠛⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⠋⠀⠀⣸⡏
⠀⠀⠀⠀⠘⣿⡀⠀⠀⢻⡿⠋⠁⠈⠉⠛⠛⢶⣶⣶⣶⣶⣶⣶⣶⣟⠛⠉⠉⠀⠀⠉⠛⣿⡟⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⢹⣧⠀⠀⠈⢿⡄⠀⠀⠀⢀⣶⣿⡿⠿⢿⣿⣿⠿⢿⣿⣷⣄⠀⠀⠀⠀⢠⡿⠁⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⢿⣄⠀⠀⣾⣿⠏⠀⠀⠀⠙⠀⠀⠀⠈⢿⣿⡆⠀⠀⣠⡟⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠹⣧⡐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣾⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠻⢿⣿⡶⠦⠤⣤⣤⣤⣤⣤⠤⣾⣿⡿⠛⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠉⠻⢶⣤⣀⣀⣀⣠⣤⠾⠟⠉⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣌⣉⣉⣉⣉⣤⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⢀⡴⣿⡿⣦⠀⠀⠀⢀⡴⣿⢷⣦⠀⠀⠀⢀⠔⢿⠗⠦⠀⠀⠀⢀⡶⢿⠷⣦⠀⠀⠀⠀⣴⢿⢗⢦
⠀⠀⠘⣷⣞⣻⡾⠀⠀⠀⠘⣧⡾⢶⡿⠀⠀⠀⠘⣾⣿⣾⡾⠀⠀⠀⠘⣷⣾⣧⡿⠀⠀⠀⠀⢷⣾⣷⡾⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠔⣿⠓⢠⠀⠀⠀⢀⡴⢿⠷⣦⠀⠀⠀⢠⡞⢻⡛⣢⠀⠀⠀⢀⡔⢻⠷⣄⠀⠀⠀⠀⣴⠚⡟⢲⡀
⠀⠀⠘⣷⣾⣾⡿⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⠈⢿⣯⣹⠟⠀⠀⠀⠸⣷⣞⣲⡿⠀⠀⠀⠀⢿⡿⠻⣿⠃
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⢀⡔⢻⠓⣆⠀⠀⠀⢀⡴⢻⠓⢆⠀⠀⠀⢀⡴⠿⠷⣦⠀⠀⠀⢀⠴⢿⠷⢆⠀⠀⠀⠀⡔⠺⠟⢢⡀
⠀⠀⠘⣿⣛⣹⡿⠀⠀⠀⠘⣿⣿⣷⡿⠀⠀⠀⠘⣷⣾⣶⡿⠀⠀⠀⠘⣷⣞⣷⡿⠀⠀⠀⠀⢿⣞⣳⣾⠃
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⢀⡔⢻⡗⢂⠀⠀⠀⢠⡞⣻⡛⣦⠀⠀⠀⠀⠒⣻⡟⠂⠀⠀⠀⢀⠐⢻⠓⢄⠀⠀⠀⠀⡴⠚⡟⢦⡀
⠀⠀⠘⣷⣿⣷⡿⠀⠀⠀⠘⣿⡛⣻⠟⠀⠀⠀⠸⣿⣟⣻⡿⠀⠀⠀⠸⣷⣿⣶⡿⠀⠀⠀⠀⢿⣿⢿⣿⠃
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⢀⡔⢻⠓⢢⠀⠀⠀⠀⡔⢻⡓⣦⠀⠀⠀⢀⡔⢺⠓⢂⠀⠀⠀⢀⡴⠻⠗⢦⠀⠀⠀⠀⡴⠺⡟⢢⡀
⠀⠀⠘⣿⣟⣻⡿⠀⠀⠀⠘⣿⣛⣻⡿⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⠘⣿⣺⣷⡿⠀⠀⠀⠀⢿⡾⢷⣾⠃
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢲⡄⠀⢀⠀⠀⡠⡤⢰⠤⡀⢀⡀⢀⣠⠄⠀⡀
⠀⠀⠀⠸⡇⠐⠄⠀⠀⠀⠀⠀⠆⠈⠀⠀⠰⠀⢸⡇⠈⠃⣚⠁⠸⠑⠌⠋⠃⠦⢀⢸⠉⢢
⠀⠀⠀⠀⡇⢀⠠⢄⠀⢀⠄⣀⣀⠀⢀⣄⡨⠀⣾⠀⠀⢀⡀⠁⠀⠀⢰⣄⣠⠄
⠀⠀⠀⢰⡇⠀⣲⠆⣇⠀⣴⡶⢸⠀⠸⢿⡁⠴⠃⠀⣳⢻⠖⠀⠀⠐⠚⣿⠿⠦⠀⠉
⠀⡀⠐⠾⠁⠀⠡⠒⠋⠰⠃⠠⠞⠀⣤⠞⠀⢱⠆⠀⠀⠁⠀⠀⠻⠀⠀⠃
⠀⠁⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⡀⠀⠀⠒⠀⠁
⠀⠀⢀⠀⠀⠇⡧⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⡀⠀⠀⠀⠀⢀⣀⡄⠀⢀⢠⣤⠄⣤⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣶⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠉⠀⠀⠈⣷⠠⠠⠴⠘⠉⠲⠃⠙
⠀⠀⠀⠄⠒⠚⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⣀⠠⠀⢠⡀⠆⣄⡄⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢐⠀⠀⠀⢀⣉⠁
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⣀⣁
⠠⡄⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠇⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⢠⠀⢠⠀⠠⠄⠀⠀⠀⢠⠄⠀⠀⠐⠂⠁
⠠⡔⢠⡤⢀⡆⢰⢴⠆⠀⠀⢲⢄⢠⠀⣉⠘⠀⠏⠆⠀⠀⠀⢀⠀⢸⡄⠀⠀⠀⡀⠀⠁⢀⣄⡀⡆⢰
⠀⠃⠘⠧⠘⠉⠸⠉⠢⠀⣀⠈⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠃⠈⠂⠈⠁⣀⠀⠀⠱⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠙⠁⣠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⠷⠂⠀⠀⡀
⠀⠀⠀⠀⠑⠀⠐⠈⠀⠄⡀⠀⢀⠎⠀⠀⠈⣎⠟⠂⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠀⠠⠄⠀⠠⠀⠧⠄
⠀⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⢠⡀⢀⠀⠀⡄
⠀⢀⢠⠀⠀⠂⢰⠂⠀⠀⡀⠀⠀⠀⠀⠀⡶⠇⠀⡄⠗⢇⠆⠸⡂⢸⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠐⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠿⠿⠿⠿⠿⠿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⡴⠋⢉⡉⠻⣦⠀⠀⠀⠀⣰⠛⢉⡉⠙⢦⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⢰⣿⠃⠀⠀⠀⠀⣤⣤⣼⣧⣴⣿⣿⣤⣼⠄⠀⠀⠀⣧⣤⣿⣿⣦⣼⣧⣤⣤⠀⠀⠀⠀⠘⣿⡆
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⡤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢸⣿
⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⣿
⠀⠀⢿⣷⠀⠈⣿⣶⣶⣿⣷⣶⣾⣿⣶⣶⣿⣶⣶⣿⣿⣶⣾⣿⣶⣶⣿⣷⣶⣾⣿⣶⣶⣿⠁⠀⣼⣿
⠀⠀⠸⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇
⠀⠀⠀⢻⣿⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣾⡟
⠀⠀⠀⠀⢻⣿⡄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠙⢿⣿⡟⠛⠛⣿⠛⠛⢻⡟⠛⠛⣿⠛⠛⢻⣿⡿⠋⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠈⠛⠶⢦⣾⣀⣀⣸⣇⣀⣀⣿⡴⠶⠛⠉⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⣀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣷⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⣻⠃⠀⠀⠀⠀⠀⠀⡼⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠁⠀⠀⣿⠀⠀⣠⡴⢤⠀⣸⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃⠠⠾⢠⡏⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣄⠀⠀⣼⢁⣤⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⢺⣧⣄⠀⠈⠛⠚⡟⠉⠀⠀⢀⣤⡀⠀⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣸⠁⠈⠗⠀⠀⢰⡇⠀⠀⣰⢿⢦⣽⡦⢾⡇⢀⡿⣇⣼⢋⡿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⢸⡀⣸⠃⠈⣧⡞⠀⢸⡟⠉⣠⡴⠂
⠀⠀⠀⠀⠀⠘⢧⡀⢀⣠⠞⠀⡄⠀⢀⠀⠀⠀⣧⢠⡏⠀⠀⠉⠁⠀⠀⠛⠁⠀⠀⢧⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢷⣀⡾⢀⣀⠀⡘⠛⡀⢸⣏⣹⠆⢀⡴⢳⡀⠀⠀⣠⡤⠶⠦⡄
⠀⠀⠀⠀⢠⠞⠉⠙⢷⡀⠀⠀⠀⢹⠀⠸⡽⠂⠹⠶⠃⠸⠋⠹⠀⣼⠁⣸⣃⡴⠋⠁⠀⡴⠀⢻
⠀⠀⠀⠀⣿⠀⠀⠀⢨⡇⠀⠀⠀⠘⠀⠀⢀⡴⠞⠛⡆⠀⠘⠀⢰⡇⢠⠟⠁⠀⣠⢤⡄⠻⠶⠋
⠀⠀⠀⠀⠹⣆⠀⠰⠟⢀⣠⠾⣿⢀⡞⢳⡾⠁⠀⣼⠃⣰⡗⠀⣼⠃⠀⠀⠀⣾⠧⠞⠁⢀⣠⡤
⠀⠀⠀⠀⠀⠈⠳⣄⡴⠛⠁⠀⢻⡾⠀⣼⠃⠀⣼⠃⢠⡟⡇⢰⣿⡀⠀⠀⣼⣿⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⡴⠋⠳⣄⠀⠀⠸⠇⠀⠀⠀⢠⡇⢀⡟⠀⠙⠋⢸⡇⠀⣼⠃⠘⠛⠉
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠈⢧⠞⠀⠀⠀⠀⠈⢧⣰⠃
⠀⠀⠀⠀⠀⠻⣄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣤⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣁⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠋⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢠⡾⠛⠛⠛⢷⡆⠀⠀⠀⠀⠀⠀⣰⡾⠛⠛⠛⢷⡄⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠰⣿⡁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢈⣿⠆⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠻⣿⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣿⠟⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⡀⣀⣀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢶⣦⣤⣤⣤⣤⣴⡶⠾⠟⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠿⠿⠿⠿⣶⣶⣶⡶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠈⢉⣽⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠛⠁⣀⣠⣤⣶⡶⠿⠟⠛⠛⠹⠿⠷⠶⠶⠶⠶⠤⠤⠤⢤⠴⠚⠛⠋⠉⠉⠉⣉⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⢀⣶⡿⠛⢁⣤⣶⣿⡿⠟⠋⣡⣴⡶⠿⠿⠿⠿⠿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣠⣶⠟⠋⠉⠉⠉⣩⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⣰⣿⠏⠠⠶⠿⠿⠟⠉⠀⣠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠘⠿⠿⠟⠀⣿⣿
⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣤⣤⣤⣄⡀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣄⠸⣿⣿⡿⠟⠁⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⣀⣀⣀⣀⣤⣤⣤⣶⣿⠁⣿⡇
⠀⠀⢠⣿⡇⠀⠀⠀⠀⠈⠉⠀⣠⡾⠛⠛⠷⢦⣤⣤⣀⣀⣀⣤⣴⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠉⠁⠀⠈⠉⠙⠻⢿⣿⣄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⣾⣿⠋⠀⣠⣾⠟⠛⠛⠻⣶⣄⠀⠙⣿⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⠶⠶⠂⢀⣼⡿⠁⢀⣾⠿⢿⡶⢲⡶⢶⣼⣿⡆⠀⢸⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠳⠶⠶⠶⠶⠒⠁⢾⣿⠃⠀⣼⡿⠶⠿⠶⠾⠷⠾⠿⢼⣿⠀⢸⣿
⠀⠀⠘⣿⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠒⠒⠒⠒⠒⠂⠸⣿⡄⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣸⡿⢀⣿⠇
⠀⠀⠀⢻⣧⠀⢹⣆⠀⢰⣆⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠃⣾⡟
⠀⠀⠀⠈⣿⡆⠀⠹⣧⡈⣿⡄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃⢰⣿
⠀⠀⠀⠀⠘⣿⡄⠀⠘⢷⣾⣷⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠐⠶⠶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀⣿⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢻⣿⣷⡀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⣉⣙⣛⠿⢷⣦⡀⠀⠀⠀⢠⣿⠃⢸⣿⠤⣤⣤⣀⡀⠀⠈⣿⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢻⣿⣷⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢶⣬⡙⠂⠀⢠⣿⠃⠀⣿⡇⠀⠀⠀⠈⠙⢶⣄⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠻⣌⢻⣄⠀⠀⠙⠦⣄⠀⠀⠀⠉⠙⠛⠲⠦⣤⡈⠛⠶⠄⣸⡟⠀⠀⣿⡶⢶⣦⢠⣶⢶⡶⠛⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠙⠻⣿⡷⢦⣄⡀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣿⡇⠀⠀⣿⣷⣼⠿⢼⣿⣰⣿⣴⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠈⠛⠦⣤⣉⡛⢦⣤⣈⠙⠦⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⠀⠈⠿⣦⣤⣄⣠⣭⣽⠟⠁⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠀⠉⠙⠛⠿⢿⣿⣶⣦⣤⣤⣤⣤⣄⣀⣀⣈⣿⣷⣄⡀⠀⠀⠈⠉⠉⠉⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣦⣤⣄⣀⠀⠈⠉⠙⠛⠛⠷⠶⠭⢭⣭⣭⠉⠙⠛⠿⠶⢶⣶⣶⣶⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣰⣶⣶⣶⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠛⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⣀⣠⣤⡤⠤⣤⣄⣀⠀⠀⠀⠀⣤⣤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⢤⡄
⠀⠀⠀⡇⠀⠀⢸⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⣿⠃⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⢸⢠⠞⠀⠀⣠⠶⠛⣯⠉⠙⢳⣄⠀⠀⠙⣦⡏⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⡇⠀⠀⢸⡞⠀⠀⢰⠏⠀⠀⠸⡆⠀⠈⣿⣧⠀⠀⣼⠁⠀⢰⢿⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⢻⡀⠀⠘⣿⠀⢰⠇⠀⢀⡏⠈⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢻⠀⠀⠀⠀⠈⣧⠀⠀⢻⠀⡾⠀⠀⡼⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠇
⠀⠀⠀⡇⠀⠀⢸⢷⠀⠀⠘⣆⠀⠀⠀⠀⠸⡆⠀⠈⣿⠃⠀⢰⠃⠀⢸
⠀⠀⠀⡇⠀⠀⢸⣬⣧⣤⣤⣬⣷⣤⣤⣤⡤⢿⡀⠀⠉⠀⢀⡏⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⣧⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣉⣙⣛⣛⣛⣁⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠉⠀⠃⣴⠲⢠⡆⣦⠀⠀⠀⠀⠑⠀⣠⠔⡏⢰⡢⠀⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⡏⠀⠀⠀⠀⠀⠰⣁⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢈⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⣿⣿⠁⠀⣀⣀⡀⠀⣠⣤⡀⠀⠀⡠⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⢰⣿⡟⢠⣾⣿⠁⢸⡆⣿⣿⡇⠀⡜⢠⣾⣿⡩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠸⣿⣇⢀⡼⠁⢹⣿⡇⡸⠁⢸⣿⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠻⠿⠤⠤⠌⠉⠉⠀⠀⠈⠛⠋⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡄⢀⡤⣄⢠⠄⡀⠀⠀⠀⢀⡾⠁⢀⡤⣤⠀⣠⠴⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⣸⠇⠘⠗⠋⠛⠘⠣⠊⠀⠀⣸⢣⠀⠻⠂⠙⠐⠿⠐⠘⠧⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁




⣿⣿⣿⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣿⣻⣿⣟⣻⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠙⣧⣀⣤⠆⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣼⠛⠓⠒⠛⠛⣯⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠳⣤⣤⡴⠋⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣤⣤⡤⠀⠀⠀⠀⠀⠀⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠄



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢠⡄⣤⡄⠀⡀⢐⡒⠒⢒⡒⢒⠄
⠀⠀⠈⠃⣘⠃⠀⠀⠀⠀⠀⢠⡶⠀⠀⢨⡿⡇⠰⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡡⠤⢌⡴⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⣠⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⢀⡀⡁
⠀⠀⠀⠀⠠⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠤⢀⠄⠀⠀⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⡀⠀⠀⠟⠐⠛⠀⠀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡶⠄
⠀⠀⠀⠀⠀⡄⢀⠀⡀⠀⠀⡎⠀⡀⠀⢀⡀⠈⠀⠀⠀⠀⠁⠀⢈⡍⠁⠀⠀⠀⠀⠀⡀⡀⠀⡀⠠⠤
⠀⠀⠈⠀⠀⡇⠈⡀⠃⠀⠀⢱⠀⠋⠀⠈⠀⠀⢰⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠈⠀⠆
⠀⠐⠄⡀⠀⠓⠀⠀⠀⢀⠀⠀⠑⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂⠘⢀⡠⠤⠤⢀⠀⠄⣀⠙⠁⡠⠀⠁
⠀⠐⠡⠀⠀⠀⠀⠀⠀⠄⠀⠐⡀⠀⠉⠀⠉⢉⡉⠈⠉⣉⠁⡆⢠⠋⣼⡇⢰⣧⠱⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠠⠀⠄⠀⠀⠀⠜⠀⠀⠀⢡⠀⠀⠀⠀⠈⠉⠠⡌⠉⠀⠃⡆⠀⠙⡁⣘⠋⢀⠇⠀⠀⠀⠉⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⡉⠙⢷⡄⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⠀⠀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⠈⠙⢿⣷⣼⣿⣀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃⠀⣀⣠⣤⣤⣶⣿⣿⣿⣿⡇⠀⠀⠀⣀⣼⠟⠙⠛⣻⣿⣿⣿⣿⣧⣤⣴⣶⣶⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣾⣿⠿⠛⠉⠁⠀⠀⠈⠛⠿⠿⠶⠾⠟⠛⠁⠀⠀⠀⠙⢿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⡀⣀⢀⢀⣠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢟⡰⠭⠓⠷⠃⠯⠄⠀⠀⠀⢀⡞⢉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠚⣔⣢⢄⣀⡀
⠀⠀⠀⠀⠀⠀⢲⠀⡴⣦⠷⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠈⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡤⢤⡀
⠀⠀⠀⢰⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡯⢤
⠀⠀⢠⡲⡝⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡀⠀⠀⢀⣴⣶⣿⣿⣶⣦⣀⠀⠀⠀⢈⡯⣅⡀
⠀⠀⡶⢝⠁⠀⠀⠀⢠⣾⣿⣿⣿⠟⠉⣿⣧⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠁⠈⠁
⠙⢤⣙⡏⠀⠀⠀⠀⣾⣿⣿⣿⢋⡾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿⣿⠇⢘⣼⣿⡟⢉⡤⠈⣿⡿⠉⣸⠟⢩⣿⠟⣡⠆⣸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠴⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⣿⠟⣫⠠⣿⠇⢐⣫⠀⡾⠋⣀⣛⡅⠠⣷⡿⢛⣽⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢾⣳⡄
⠀⠲⣄⡆⠀⠀⠀⠀⠀⠀⠙⢦⣤⣾⣿⣦⣤⣾⣿⣿⣧⣤⣾⣿⣿⣷⣤⣤⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢞⡢
⠀⠀⣖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⡍
⠀⠀⢴⣗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠠⠀⣨⡵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠙⢦
⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⠳⠇⠈⠃
⠀⠀⠀⠀⠀⠀⠁⠩⣗⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡞⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠼⡭⠇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠚⠁⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⢲⢺⡽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠟⡽⢳⣫⠆⣽⠀⠀⠀⢠⠟⠘⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⡾⠉⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⠮⢕⠶⣠⠞⠀⠀⣀⣀⠀⡞⠀⠀⣼⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠟⠭⢔⡪⠕⠊⣀⣤⠶⠛⣿⠉⢿⡁⠀⠀⠁⡾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠴⠒⠋⣁⣤⠶⠛⠉⠀⠀⢀⡇⠀⠈⣧⠀⠀⠀⢰⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠘⣧⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⡉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⡏⠀⠀⠀⠀⠘⣧⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠉⠙⠓⠺⣧⠈⠉⠓⠒⠦⠤⡤⠒⠛⠀⢠⣇⣀⠀⠀⠀⠀⠹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣤⡄⠀⠘⣷⠀⠀⠀⠀⠀⠳⣤⣀⣠⡼⠁⠉⠉⠉⢉⣩⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣤⣤⠀⠀⠸⣇⠀⡀⠀⠀⡸⠁⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣶⠄⠀⠘⣦⠀⠀⣰⠃⠀⢆⣤⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠙⢧⣰⣇⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⢀⣴⣾⡿⠿⠛⠛⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠻⢿⣷⣦⡀⠀⠀⠀⠘⢿⣷⡄
⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠀⠀⠀⠀⠈⢿⣿⡄
⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣿⡿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⣰⡿⢿⣿⡿⠉⠙⣷⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢠⣿⡇⠀⠈⠁⠀⠀⢸⡇⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⠄⢿⣧⡀⠀⠀⠀⢀⡾⠁⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⡾⠿⠟⠀⠀⠙⠷⣶⣴⠶⠋⠀⠀⠸⠿⣄⣀⣀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢉⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣇⠀⠀⠀⠀⠀⠀⠰⣤⠀⣰⡃⠀⠀⠀⠀⠀⢀⣿⣾⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⢧⡀⠀⠀⠀⣶⠀⠘⠁⠀⠀⠁⢀⡆⠀⠀⠀⢨⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠘⠷⣤⣀⣀⣀⡴⠟⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⢀⣠⠤⠠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣴⠋⠀⠀⢀⣼⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⢀⣴⡿⠃⠀⠀⠀⠈⠉⠀⠁⠀⠀⡠⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⣰⣿⡿⠁⣠⡴⣤⠀⣤⣄⠀⠀⣠⢊⣠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⢁⣼⣿⡟⣱⣿⣿⠀⣿⡇⢻⣿⡀⡼⢡⣿⠋⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⢠⣾⣿⡟⠀⢿⣿⠈⢳⣿⠇⠘⣿⣿⠁⣿⣿⠤⠚⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⣤⣤⣾⣿⣁⣰⣿⣿⠟⠀⠀⠈⠻⠦⠞⠋⠀⠀⠻⠇⠀⠈⠻⠦⠄⠠⠒⠁⢀⡀
⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⢀⣴⡿⠋⠉⣻⣿⣿⣿⣶⣶⣤⣄⣠⣾⣶⠂⣰⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⣼⣿⣿⠋⠀⠉⠙⠛⢿⣿⣿⣷⣾⣿⣿⣀⣀⣀⣀⣀⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠈⠒⠒⠚⠉⠁⠀⠀⢀⣾⣿⡿⠁⠀⣀⠀⠀⢀⣿⣿⠃⣼⣿⠏⠉⠉⢉⡉⠉⢐⣂⠀⠀⠀⠀⠀⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢠⣶⠟⠉⣹⠄⢸⣿⡇⢰⣿⡏⢀⣴⡟⠉⢹⡆⠈⠛⢻⣿⣿⠀⠀⢹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣿⡏⣀⡴⠏⠀⣿⣿⠀⣸⣿⠀⣾⣿⢀⣠⠟⠁⡠⠁⣼⣿⠇⠀⢀⠎⠘⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⣾⣿⠟⠀⠀⠀⢻⣿⣁⠀⢀⠀⢻⣿⠀⢹⣿⠀⢻⣿⣍⠀⣀⠔⠁⠀⣿⣿⣀⡴⠃⠀⠀⠀⢻⠇
⠀⠴⠊⠁⠀⠀⠀⢀⣽⡿⠟⠿⢿⣶⣦⣤⣀⡈⠉⠉⠀⠀⠀⠻⢦⡈⢿⣤⠆⠈⠉⠉⠁⠀⠀⡀⠈⠉⠁⠀⠀⠀⠀⣠⠟
⢈⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠈⠙⠛⠿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⢀⡼⠃⠀⣠⡶
⠀⠁⠂⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣶⣶⣤⣤⣀⣈⣁⣀⣀⣁⡀⠤⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠙⠛⣿⣿⡟⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠉⠙⣿⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣿⠇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⡠⢰⣄⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⢏⣭⡍⢣⠙⠛⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠛⠛⠛⠛⠛⠛⠷⢶⣤⣀⠸⣿⣿⣿⣤⣽⣿⣻⣄⢠⣿⠃⢹⣿⣿⣿⣿⡇⠘⠁⠀⠀⠙⣿
⠀⠀⣿⣿⣿⢫⣶⡌⣻⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⢻⣁⣲⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⣿⣿⣧⣛⢿⡿⢾⠇⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸
⠀⠀⣿⣿⣿⣶⢽⣟⣿⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⣫⣤⣁⠘⠀⠀⢸⣿⣿⣿⣿⡇⠀⢀⣀
⠀⠀⣿⣿⣿⣿⣮⢹⡿⠀⠀⠀⠀⠀⣤⡾⣯⣴⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣪⣍⣹⠁⡨⢳⣾⣿⣿⣿⣿⡷⠶⣾⣿
⠀⠀⢿⣿⣿⣩⣶⣤⡀⠀⠀⠀⠀⣠⣿⡷⣾⣭⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠀⠉⠁⠀⠀⠈⢸⣿⢟⣛⠻⡇⠀⠸⡿
⣤⣤⣾⣿⣿⣯⢶⣶⣿⣤⣤⣴⣾⣿⣿⡟⢛⣿⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⣻⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣬⣽⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠻⠺⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡤⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⠒⠇⠀⠀⠀⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⡷⣻⡟⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣷⣫⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⣾⠛⢳⡄⠀⠀⠸⢁⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠵⣿⣿⣶⣤⣤⣀⣀⣠⣤⣴⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⠿⠿⠛⠿⠿⠶⠶⠾⠿⠿⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣽⣤⣯⣽⣭⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣾⣿⣿⣿⣶⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣶⣾⣾⣿⣯⡟⢻⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣗⣛⣛⣛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠟⢿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⢺⡿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠇⢿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣥⣼⣿⣷⣬⣿⣿⣿⣿⡖⣿⣿⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠀⢨⡀⠀⠀⠀⠀⠐⡆⡘⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠴⠃⠀⠀⠰⡥⠤⠤⠄⠀⠐⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢱⢃⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⢞⡃⠀⠀⠀⢠⠆⠐⢂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⢠⡄⠀⠀⠀⠀⠐⡸⠶⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⠀⢀⡒⠖⠀⠀⠀⠀⢦⠆⠃⢊⣙⠀⠀⠀⠀⠀⠀⠄⡅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠐⠀⠊⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⡠⠁⠡⠳⠀⠀⠑⠄⠇
⠀⠀⠀⠀⠀⡀⠎⠈⢀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣦⣴⠄⠀⠀⠀⠀⢀⠰⠆⢳⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⡇⡄⠦⠴⠀⠀⠀⠀⢆⣤⡇⠂⠀⠀⠀⠀⠀⢿⡿⠀⠀⠐⠀⡀⠊⠁⠀⠀⠀⠀⠀⠈⠀⠧
⠀⠀⠀⠀⠀⣀⢠⡤⡤⡆⠀⠀⠀⢸⠀⡀⠀⢀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⠚⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠐⡂⠀⠌⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠀⡀⠀⠀⠀⠀⠈⠉⠡⠂
⠀⠀⠀⠀⠀⠀⡴⣧⣹⣀⠀⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⠇⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⡀⢀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣖⠂⠲⢤⡀
⠀⠀⠀⠀⢀⠀⠀⠢⠐⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⣄⠄⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⡄⠀⠀⠸⢿⣿⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠉⠈⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠂⠀⢀⣀⣠⣶⠀⢰⣿⣿⣿⣿⣿⣿⣿⡄⠀⠇⠈⠻⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣶⣤⣿⣿⠀⠀⠀⠀⡖⠠⠆⠀⠀⠀⢀⡤⠾⠿⢿⡿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠘⣿⣻⣿⠇⠀⠀⠀⠤⢴⡠⠤⠀⢀⡼⠉⠀⠀⠀⠀⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠠⠑⣆⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠈⠀⢠⠰⠈⠀⠂⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠠⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠴⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⢢⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⡈⠌⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠆⠂⠀⠀⠀⢠⠀⠀⠈⠠⣄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⡤⠶⠂⢀⠄
⠀⠀⠀⠀⠀⠀⢰⠐⠎⡃⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢺⡏⣰⠞⠉⣹⡄⠀⠀⠀⠼⡀⠄⠀⢠⡀⠊⠀⠀⠹⣿⣿⣿⣿⣿⣿⠶⣛⢁⠎
⠀⠀⠀⠀⠀⡌⠀⢀⣀⠀⠀⠀⠀⠀⠀⢨⣭⡍⠁⠀⠀⢠⣄⡀⠀⠀⢰⡇⠀⠘⠂⠖⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⡿⣿⣿⣿⠈⡆⢹⠎
⠀⠀⠀⠀⠀⠁⡁⣈⣉⡠⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠠⣶⣶⣶⣤⣀⠀⠀⠈⠳⢦⣀⠀⠀⢰⠙⢿⡟⠀⠙⠋
⠀⠀⠀⠀⠀⠸⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⢈⣽⣿⣿⣿⠗⠀⠀⣀⣀⣙⠳⡄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠄⠀⠀⠈⠉⠩⠤⠼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⣶⣿⣿⠿⠿⠷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠛⠛⣿⣿⣿⠿⠶⠤⠴⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⢀⣿⣿⡇⠀⠀⣀⣀⡀⠀⠀⠀⣠⣶⣿⠿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠉⠉⠉⠂⠀⣼⣿⣿⠁⢠⡾⠟⠛⠛⢷⣤⡾⠋⠁⠀⠀⠀⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⢸⡇⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⢰⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⣾⣆⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⢀⣼⣿⠏⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⢀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠙⢿⣦⣈⠻⣦⣄⣀⣀⣠⣴⠿⠋⣁⣤⡾⠟⢋⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠈⠷⣤⣤⣤⣾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣷
⢀⣠⣴⣶⣶⣶⣤⣄⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠂⠀⡴⢦⡀⠀⢠⣾⢳⠀⠀⢠⢴⣄⠀⠀⣀⣴⣶⣤⠀⠀⠘⠉
⣿⡏⠁⠀⠀⠉⠙⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⣿⢋⠀⠀⠀⠙⠬⣿⣆⣼⣿⠋⠀⢀⡟⣼⣿⠀⣴⣿⡿⠋⢸⡇⠀⠀⠀⣤
⠙⣧⣀⣴⠖⠻⣶⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⡏⢨⡆⠀⠀⠀⢀⣿⡯⢹⣿⡆⠀⠈⣷⣿⡿⢰⣿⣿⠇⢀⡾⠁⠀⠀⠀⣿
⠀⠀⠙⣟⠛⠛⠛⢻⣿⣎⠻⣿⣿⣦⣀⠀⠀⠀⠻⣿⣾⡀⠀⠀⣀⣾⣿⠃⠘⢿⣧⠀⢠⣿⣿⡃⢸⣿⣿⡶⠛⠁⠀⢠⡇⠀⣿
⠀⠀⠀⠈⠓⠀⠀⠀⣿⣿⡆⠈⠛⠻⠿⠿⠿⠿⠟⠉⠙⠻⠿⠿⠿⠋⠁⠀⠀⠀⠻⣷⡿⠟⠁⠉⠉⢻⣿⣦⣀⢀⣠⡟⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⣰⣶⣦⣠⣶⣶⡄⠀⠀⠀⣀⡀⠀⠀⠈⠛⠛⠛⠁⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢀⣾⣿⠁⢸⠀⠀⢻⡁⠀⠛⠀⢸⠇⢀⣴⡿⠿⠟⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⢸⣿⡇⢀⡞⠀⠀⠈⠳⣄⠀⠀⠋⠀⠚⢁⣤⣤⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⣾⣿⣷⠏⠀⠰⠛⠋⠁⠈⠓⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⠾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣠⣾⠏⠁⢴⠛⣦⡀⠀⣾⡧⠃⠀⣾⣿⣷⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⣆⠀⠀⠀⢰⣿⡿⢸⣿⡇⠀⠀⠈⠓⢺⣿⢻⣿⡇⠀⠀⣿⣿⡟⠀⠹⠿⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡇⢠⠏⠀⠀⢀⣾⡿⠃⢸⣿⡆⠀⠀⠀⠀⣼⣿⠃⢿⣧⡀⠀⣻⣿⡅⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⣀⢀⣀⣤⡿⠟⠁⠀⠀⠙⠿⣦⣤⣤⡾⠟⠁⠀⠀⠛⠿⠟⠋⠙⠿⣦⣤⡤⠞⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣰⣾⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦
⠀⠀⠀⠀⠀⢿⣷⣤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣅⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠉⢩⣤⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠿⠿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⡀⠀⢀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⣈⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣋⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠟⠛⠛⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣤⣀⠀⠀⣀⣤⣤⣶⣶
⢸⣿⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡿⠿⢿⣿⠆⠀⠀⠉⠉⠉⠻⣷⣤⣿⠋⠉⢉⣿
⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⢿⣷⣶⣄⡀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣼⣿
⣾⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠉⠻⣿⡄⠀⣸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⠃
⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⣿⠏⠀⠀⣠⣶⣿⣶⣤⠀⠀⠸⣿⡆⣿⣿⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣾⡿
⣿⣿⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡃⠀⢈⣿⡇⠀⠀⣿⡇⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢰⣿⠇
⣿⡟⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⡾⣿⡄⠀⠀⠻⣿⣶⣾⠟⠁⠀⢠⣿⠇⣿⣿⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⡄⠀⠀⣿⣷⣴⣾⡟
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠹⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⠇⠀⣀⣭⣽⣿⡉
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠈⠛⠿⣷⣶⣶⣶⣾⡿⠟⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣾⡿⠛⠉⠛⣿⣦
⣿⣷⣶⣶⣶⣿⣿⣿⡿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣿⣧⣤⣴⣶⣶⣶⣶⣶⣾⣿⠿⠸⣿⡅⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠈⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢸⣿⣷⣦⣤⣤⣤⣶⣿⠿⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⣀⣀⣀⣼⣿⡆⠀⠀⠀⠀⠀⠀⢿⣷⣦⣤⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠙⢿⣦⣄⣀⣀⣠⣴⡿⠋⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣭⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⣠⣤⣄⠈⢻⡄⠀⠀⠀⠀⣼⠏⢀⣤⣤⡀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⢴⣦⢦⣾⣷⣴⡿⣿⣿⣦⣾⠇⠀⠀⠀⠀⢿⣦⣾⣿⣿⣷⣤⣿⣦⣤⣤⠄⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⢸⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⣿⠉⠉⠉⢹⡏⠉⠉⠉⡏⠉⠉⠉⣿⠉⠉⠉⢹⠉⠉⠉⢹⡏⠉⠉⠉⡏⠉⠉⠉⣿⠉⠉⠉⢹⡇⠀⠀⣿⣿⠂
⠀⠀⢸⣿⡇⠀⠀⣿⣄⠀⢀⣸⣇⡀⠀⣠⣷⡀⠀⣀⣿⣀⠀⢀⣼⣄⠀⢀⣸⣇⡀⠀⣠⣧⡀⠀⣀⣿⣀⠀⢀⣼⡇⠀⠀⣿⣿
⠀⠀⠸⣿⣧⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢠⣿⡿
⠀⠀⠀⢿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣾⣿⠇
⠀⠀⠀⠘⣿⣿⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠘⣿⣷⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠙⠿⣿⣿⡉⠉⠉⣿⠉⠉⠉⢻⠋⠉⠈⢹⡏⠁⠈⢙⣿⡿⠟⠁⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠈⠙⠻⠷⣦⣿⣄⣀⣀⣸⣀⣀⣀⣼⣧⡶⠿⠛⠉⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣤⣀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣠⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠛⠛⠛⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿