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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣬⣴⣾⣯⣵⣦⣴⣤⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣖⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠿⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠠⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⠄⠀⠀⣼⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡀⠄⠀⠀⠀⠀⠀⡀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠉⠀⠀⠁⠈⠿⠒⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣀⠀⠄⡀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢸⣿⣷⣦⣈⠁⠂⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠿⠛⠿⣿⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠙⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣽⠛⠶⢦⠀⠀⠀⡀⠀⢀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠉⠀⠀⠀⠿⣤⠀⣀⠀⠀⠈⠉⠉⢙⡛⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠉⠐⠒⠒⠒⢉⣉⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⣼⣃⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠈⢻⠿⠿⠿⠿⣷⣦⣤⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣶⡖⠉
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠉⢻⣿⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⡇
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⡇
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⢉⠃⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⣆
⠀⠀⠀⠹⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠀⠀⠈⣂⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⠖⠁⠈⠀⠀⠀⠀⢣⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠃⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠛⠙⠓⡖⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡤⢶⠁⠀⠀⠀⠢⡄⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⣀⣀⡸⣠⣤⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠛⢆⠀⢠⣌⣡⠟⠰⡌⠙⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣀⣀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⣠⡔
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠦⣤⣤⣠⠞⠛⡅
⠀⠀⠀⠠⣄⣎⡾⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠒⠛⠛⠃⠀⠀⠱⡀
⠀⠀⠀⠠⠋⠠⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠌⣤⡤⣄⡀⠀⠀⠀⠑⢄⢀
⠀⠀⠀⢤⡄⢶⢹⡙⣿⡷⠤⠔⠂⠒⠐⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠘⢙⢆⠀⠀⡀⠠⠁⠄
⠀⠀⠀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⢧⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡁⠀⠀⠀⠉⠉⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⣀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⢶⡆⠀⠀⢲⣶⣿⣶⣶⣶⣶⣶⣶⣶⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡀⠀⠀⣲
⠈⠳⣦⣶⣾⣿⣿⣿⣿⡛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣿⡿⠟⠓⢲⡶⠋
⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⣀⣠⣤⣶⣾⣷⣦⣀⠀⠀⠀⠀⢀⣀⣤⣤⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠾⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡈⠑⢄⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⢀⡀⠀⢀⡤⠀⠀⠑⠄⠂⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⡄⠀⢠⣠⠐⠒⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠃⢈⠉⠻⠿⠁⠀⢀⡀⢠⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠛⠀⠀⠀⠀⠦⡕⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠡⡀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⡀⠈⠀⠀⢾⠀⠅⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⠀⠀⠄⢠⡗⠤⣠⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠢⠀⢆⠣⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⠀⢜⠸⣤⣀⠀⠀⠀⠘⠸⠇
⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠑⠀⠀⠀⠀⠁⢢⠀⡀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡀⠀⡇
⠀⢠⡜⠙⠘⢄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⢇⣿⠤⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁
⠀⢸⡆⠀⠀⡇⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⢢⢸⠀⠀⠀⠀⠈⠷⠠⠤⣄⠈⠢⣄⠀⠀⢠
⠀⠈⡇⠂⠀⢹⡸⡆⠀⠀⡇⠰⡀⠀⠀⠀⠈⠆⠀⡀⠀⠀⠀⢸⠦⣄⠀⠀⠀⠈⠋⠑⠦⢷⡀
⠀⠀⣇⢸⠀⠘⡇⡇⠀⠀⢡⠐⢿⣄⠀⠠⠀⠈⢎⠁⡠⠂⢀⠄⢀⡠⠓⢦⣄⡂⠀⠀⠀⠀⢻⡍⠋⠐⠒⠒⡀⠀⢤⢀⠀⠀⢀⠴⠀⠀⠄
⠀⠀⢻⠘⠀⠀⣇⢸⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠈⢧⡀⣴⠡⠒⠁⠀⠀⠀⠀⠉⠉⢁⠖⠛⠚⠛⢍⠀⠐⠒⠂⠀⠔⠋⠓⠴⡒⠒⠦⠤⠦⢤⣤⣤⡀
⠀⠀⢸⡀⠄⠇⢹⠸⡄⠀⢠⡇⠀⠀⠈⢣⡀⠀⠀⣰⠯⠇⢤⣀⣀⣠⣤⠤⠀⡶⠖⠁⠀⠀⠀⠖⠌⣃⠀⠀⠀⠀⠀⢀⠤⠶⠍⡀⠀⣀⡠⠖⠀⠀⠞⠃
⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⣸⠉⡁⠒⠒⠒⠓⣄⠀⠷⠀⠀⠀⢀⠀⠀⢠⡄⠀⡴⠤⠤⠒⠃⡚⠁⠀⠀⠀⢀⡀⢄⡨⡀⠄⣂⡥⠖⠋⠀⠀⠀⠀⢠
⠀⠀⠀⠸⡀⠀⠀⡇⣿⢠⠇⠀⠀⠈⠉⠀⠀⠈⠻⣦⠤⠤⠴⠷⠒⠛⢛⣗⡈⠀⠀⠂⠠⠄⢉⡑⠄⠤⣚⠵⠚⣉⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢧⠀⠀⠇⢸⡏⢀⠔⠁⠀⠀⠀⠀⠀⠀⡟⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠚⡁⠄⠒⡀⢈⣉⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣧⠀⠀⢸⡇⠄⠀⣀⠤⠔⠒⠒⠤⣀⠁⠀⠉⠉⠙⠓⠲⠶⠤⣤⣄⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠄⠁
⠀⠀⠀⠀⠸⡏⣷⠼⡌⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⠑⠒⠲⠦⠄⣄⠠⠤⠶⠖⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⡇⡇⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⡇⣷⠚⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⣅⠠⣿⢸⡀⠀⠤⢤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢠⠃⠁⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⢠⡇⣿⠀⠿⢸⠿⠿⠷⢶⣴⣄⣤⣤⣤⣭⣭⢭⠼⠲⢤⣀⠀⠦⣤⠀⢀⡤⠚⠁⣰⠃⠀⡀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠐⠋⠹⠀⢨⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠨⣗⣀⡤⠞⠉⠀⣠⠞⠁⠀⠸⡇⢀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⡇⠰⠋⠉⠉⠓⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⣶⠒⠂⢈⠁⠀⠀⣀⠞⠁⠀⠀⠀⠀⡇⢸⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⡞⢠⡇⢸⢀⣀⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⣤⣀⣀⣀⣀⣀⣀⣰⠷⠘⡇⠟⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠀⠈⠉⠉⠉⠉⢍⠀⠒⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⡤⣤⠄⢀⡀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠐⠂⣠⠤⢤⣤⠤⠀⢤⠀⠀⢶⣀⠈⢆⠀⢣⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠉⠒⠂⠀⠀⠁⠀⡄⠀⠀⡀⡀⠁⠀⢀⣀⠀⠀⠀⠁⣂⠤⣵⠌⠀⠓⠮⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣵⠀⠀⠀⠀⠀⠀⠀⠰⠣⡄⡏⠃⠀⡔⠁⠀⠙⡄⠀⠈⠁⠀⠈⠂⠀⠀⠀⢀⣰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠀⠱⢿⣆⠔⠀⠀⠘⠿⠆⠔⠁⠀⠀⠀⠘⢡⠀⡔⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢦⠐⠒⠀⠐⠂⠡⡈⠆⢇⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⢠⠀⠀⡂⢁⠜⠀⠀⠀⠀⠀⠀⠁⢐⣄⡀
⠀⠀⠀⢀⣀⠀⠀⠀⠤⠤⠦⡀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠉⠂⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢉⠀⡰⠉⠀⠀⡀⠂⠀⡠⠂⣈⠄⠀⠀⠈⠑⢄
⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠡⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠈⠀⠒⠒⠀⠉⠀⠀⠀⠁⠔⠃⠀⠀⠈⠉⠉⠉⠁⠈⠉⠈⠉⠉⠉⠉⠉⠂⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠁⠒⠷⣵⣀⡀⠀⠀⠀⠀⡀⠄⠔⢊⠳⣄⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠐⠂⠤⢀⣑⠈⡆⠈⠉⠈⠰⠆⠉⠀⠀⣤⡨⠶⠤⠤⠤⠒⠉⠉⠉⠁⠒⠒⠠⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠁⠒⠒⠒⢄⠀⢀⠂⠀⠀⠠⠄⠀⠀⡠⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⢀⠠⠀⠂⢀⣉⠉⠉⠉⠉⠉⣛⡒⠲⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⡖⠁⠐⢼⠀⣀⣀⡀⠀⡤⠔⠐⠒⠒⠀⠉⠉⠀⠀⠁⠒⢤⣠⣤⣤⣤⣤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠠⠆⠀⠀⠀⠀⠀⠈⢁⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡍⣒⠀⣈⣉⡀⣀⣀⣀⠤⠤⠤⠔⠒⠒⠊⠉⠙⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠴⠆⠶⠞⠶⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⢙⢁⡼⢻⢏⡀⣄⣥⣜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣆⡩⡤⣀⣴⠈⠉⠉⢇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣤⢀⠀⠉⠛⠲⢆⢰⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠋⠹⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⢀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠒⡊⠁⠀⢀⡼⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠉⠀⠉⠙⠳⠦⣄⡀⠀⠀⠀⢠⡞⠁⠀⠀⠑⠦⡴⠋⠀⠈⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⢀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣴⢫⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠋⠁⢀⣠⣞⡽⢏⡙⢦⣀⣀⠀⠀⠀⠀⠀⠀⢀⡆⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠳⡀
⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⣠⢔⠊⠀⠡⠛⠀⠣⢀⡼⠐⠉⠉⠉⡉⠀⠂⠠⠞⠁⠇⠀⡀⠀⠀⠀⠀⠀⠀⠄⠐⣦⡀⠂⠀⠀⠀⠹⣄
⠀⠀⣀⡤⠾⠛⠁⠀⠀⠀⠀⠘⠇⠳⣄⠀⠀⠀⠀⠀⠀⠘⢻⡒⢤⣀⠀⠀⠀⠄⠀⡼⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⢠⠁⠙⢦⡘⠀⠀⠀⠘
⣴⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⣌⠳⡄⠀⠀⠀⠀⠀⠀⡿⠀⠈⠙⠢⢁⣴⣾⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⡆⠀⠀⢀
⠀⢹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⢣⡈⢢⡀⢀⡠⠖⣀⠷⠀⢀⣠⡴⠛⢁⡞⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⢀⡇
⠀⢸⡇⠉⠻⣦⣀⠀⠀⢴⢀⠀⠀⠙⣄⠀⠙⠚⣙⠭⠖⠋⢁⣤⣶⠿⠥⠶⠖⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⢀⡞⠀⠀⠠⢠
⠀⢸⡇⠀⠀⠀⠙⢷⣄⠘⠈⣦⠀⠀⠈⠢⠖⠉⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⣠⠄⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠋⠀⠐⠀⣰⠋
⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⣤⡈⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠉⠙⠒⠂⠄⣴⠟⣠⢒⣈⠉⠙⣄⠶⠊⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣤⡶⢻⡇⠀⠀⠀⢀⣐⣤⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⠀⢹⡖⠁⢀⠌
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠁⠀⢸⡇⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣔⣛⠉⠀⣀⣀⣠⣴⡟⠓⠒⠋
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⡁⠀⠀⠀⠤⣱⡶⠚⠉⣿⠇⠉⠉⠉⠉⢹⡏⠀⢻⣇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢳⡀⠀⣿⠀⠀⠀⠀⠀⢸⡿⠀⢸⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡾⢹⡄⠀⠀⠀⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⢸⣧⠀⢸⣿
⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠻⡄⠀⠀⠀⠀⠸⣆⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠹⡄⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢸⣇⣀⣤⡾⠟⠛⠛⢻⡏⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⢸⡏⠁⠀⠀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⢸⡟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣀⣠⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢀⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⣠⡟⠁
⣶⣶⣦⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣼⣿⣿⣿⣿⣥⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠶⠖⠒⠛⠋⠉⢙⣦⠤⠶⠶⠶⢤⣀⣤⠶⠛⣿⡤⠟
⠀⢠⡤⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⢀⢀⣠⠶⠛⣩⣴⡖⠋⠁
⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠀⣀⣤⠶⠛⣉⣤⡶⠋⠁⠀⠹⣆
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣛⣉⣤⢶⢾⣿⡟⠀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣠⣯⣀⡿⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠸⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠉⠉⠓⢦⣤⣀⣀⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣀⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠈⠳⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⣀⣀⡠⠄⠀⠈⠛⢶⡶⢄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡴⠊⢿⡏⠀⠀⠀⠀⢠⣴⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⡤⢾⠀⠌⠀⠀⠀⠀⠀⢤⡀⠐⡶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡞⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠛⢂⡔⠊
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠾⢯⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠐⠋⢀⡀⠀⠀⠸⡆⠀⠀⠹⡀⠀⢈⣦⡀⠀⠀⡾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣧⠀⠀⠀⢳⡔⢯⠈⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡀⠀⣼⠻⢳⣼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢧⡸⣇⣠⠞⠿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⢀⡔⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡎⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡞⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠾⡇⠀⠐⠶⠀⢸⡇⠀⠀⠀⠀⢶⣄⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢘⣧⣀⣤⣿⣎⣀⣀⣀⣤⠶⠾⠷⣤⠤⠒⠒⠛⠽⣿⣦⡀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⢹⠀⠀⠀⠀⠀⠀⠦⣝⠢⣄⠀⠀⠙⠳⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⢀⣼⣥⣀⣀⣀⠀⢬⣽⠞⠉⠛⡦⠠⠀⣀
⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣤⡤⠤⠖⠚⠛⠛⠛⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⡩⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣟⠲⠻⠤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠈⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡤⠺⡅⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣾⠞⣉⡉⢦⢀⡴⠦⣸⠃
⠀⠀⠀⠀⠀⢡⠀⢰⠀⠀⠀⠀⢰⣿⠉⠀⠉⠻⣿⠀⢓⡟
⠀⠀⠀⠀⠀⠈⡀⢸⠀⠀⠀⠀⢸⠘⡄⣸⠷⠒⠙⣀⠀⡇
⠀⠀⠀⠀⠀⢠⣇⡸⣆⠰⣄⠀⠈⢇⡼⠁⠀⠀⢆⡿⢸⠗⠒⠒⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠹⢇⣤⡙⠒⠚⠓⠋⠀⢀⣪⠵⣻⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠉⠙⠒⠚⠛⠛⠭⠤⠚⠁⠀⠀⢀⣀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠚⠉⠸⡇⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣰⡀⠀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⡏⢦⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠲⣠⣅⣧⡀⠱⣄⠀⠘⡦⠀⠀⣀⣀⣤⠇⠀⢱⡄⠀⠀⠀⢧
⠀⠀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠛⣦⣤⢷⡱⡯⢭⠤⣤⠤⠶⢾⡷⣶⠆⠀⠸⡦⠤⢴⠎⠉⡧⠒⠒⠒⠶⠶⡶⠶⠂
⠀⠈⠉⠉⠀⠈⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠠⢱⣝⡄⠀⠀⢀⠔⠋⠸⠃⠀⠀⢀⡇⣾⢧⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣤⣀⡀⣀⣸⣀⠀⠀⠶⠿⣿⡄⢠⣃⣀⠀⠀⠀⠀⣠⠞⠀⠙⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠋⠈⠉⠓⠂⠉⠈⣟⠍⠀⠀⠉⠑⢦⠞⠁⠀⡴⠶⢼⡿⠴⠻⡆
⠀⠀⠀⢀⡠⠔⠒⠒⠒⢦⡄⠀⠀⠀⣰⣃⣀⠀⣀⣀⣀⣀⣀⣼⡀⣆⢀⠀⢺⡇⠀⠀⠀⢹⠉⠉⠙⣧⠀⡇
⠀⠐⠶⠥⣤⠤⠄⠀⣠⠞⠤⣀⣀⠀⠀⠀⠁⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠉⠀⠀⠀⠀⠈⣗⠒⠒⠃⢀⡇
⠀⠀⠀⠀⠓⠤⠤⠤⠤⣤⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣦⠉⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡙⠄⠀⠀⠀⠀⠀⢀⠘⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢲⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣗⢦⡀⠀⠀⠙⣷⣤⣀⣀⡈
⠀⠀⠀⠀⠀⠀⡴⠋⠹⣆⠈⠣⡈⠀⠀⢀⡏⠀⠈⠁
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣆⠀⠈⠢⡀⠸⣿⠂
⠀⠀⠀⢠⣾⡥⠂⠀⠀⠀⠀⡀⠠⠶⠈⠲⢻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠻⣀⢀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠟⠁
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠢⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠃
⠀⠀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢹⣯⠓
⠀⠀⠸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠍
⠀⠀⠀⢇⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠿⠃
⠀⠀⠀⢸⡆⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⢀⣀⡤⠂⠀⠀⠀⠐⡄⢀⠐⣩⠉
⠀⠀⠀⠀⢿⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣠⡼⣿⠉⠉⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁
⠀⠀⠀⠀⠘⣇⠀⠘⢿⡄⠀⠀⠀⠀⠚⢩⡇⣻⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⢻⡄⠀⢸⣿⡄⠀⠀⠀⢀⡾⠀⢻⡇⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠳⣝⠀⠀⠀⡸⠁⠀⢸⢃⣠⡶⢿⠉
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢈⡧⣤⣀⡁⠀⣀⣼⠟⠋⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠉⠛⠾⠟⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣾⡦⠖⠚⠛⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⣋⣀⣠⠤⠴⠶⠶⣤⣄⠀⠀⠠⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠤⠀⠀⣸⡿⠻⣿⣷⠀⢠⠎⠻⠿⡷⡆⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⡼⢻⠀⠀⢀⡜⠀⠼⢦⣀⡼⠁⢳⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡖⠀⠀⡼⠁⠘⠲⠖⠋⠀⠀⠀⠀⠀⢱⡄⢸⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⠶⣤⣴⡾⠟⠀⢸⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠘⠳⣆⢸⡇⠀⣠⢶⣀⠀⠀⠀⠀⠀⠀⡼⠛⠘⢏⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣲⣶⣧⠀⢠⡞⠿⠇⠀⠀⠀⠈⠙⠦⠄⠀⢀⠞⠁⣀⡀⢯⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣈⣹⠀⠘⡏⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⢡⡖⠉⣉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢭⢶⣾⡆⠀⠙⠦⠩⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠟⠛⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⡇⢠⠖⠒⠀⠛⠛⠓⢤⡀⠀⠀⠀⠀⣠⣿⠟⢋⠁⢸⠉⠉⠉⠉⠉⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⡇⡟⠀⠀⠀⠀⣠⠆⠈⣿⡇⠀⠀⢸⣧⣿⠾⠋⠁⣼⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⡇⢀⣤⠤⣾⠁⠀⠀⠙⠧⢄⡤⠿⠃⢹⣤⡾⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⠈⣿⣧⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠞⠉⢷⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⢿⡇⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠈⢳⡿⠀⣷⢀⣎⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏⠙⢦⣀⣼⣧⡴⠇⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠈⣧⠀⠀⢸⠇
⠠⠤⠤⢤⣤⣤⡟⢹⠀⢧⠀⠈⣿⣧⣿⣧⣠⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣀⣠⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠻⠶⠾⢦⣨⠷⣴⠟⢷⣾⠷⠤⢤⣍⣉⣉⣛⠛⠛⠛⠓⠒⠒⠶⠤⠤⠤⣤⣤⣼⣿⣯⣭⣥⣤⣄⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠈⠿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠉⠋⡿⠃
⠀⠀⠀⢀⣤⣔⣚⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠶⠤⠤⠤⠤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣠⣤⠤⠤⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣝⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⣅⣌⢧⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣿⡉⠈⠉⠛⠷⣿⣌⣩⡉⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠖⢉⣴⣾⣿⣿⣲⠁⠀⠀⠀⠉⠛⠳⢤⡀⠙⠄
⠀⠀⠀⠤⠴⠶⠖⠒⠒⠒⠉⠉⠀⠀⠀⣩⡇⠀⠀⠈⠙⠀⣶⣤⣀⢀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣶⣿⣷⣤⠀⠀⠀⠀⠀⠙⢿⣾⣵⡗⠐⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣍⠉⠉⠀⠀⡀⠀⠀⠀⠀⠉⠻⣍⡙⢶⣄⡀⠀⢀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠼⣷⣦⣄⠀⠀⠀⠀⠉⠛⠻⣿⣦⣄⡀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡉⡛⠻⠯⢳⣄⢠⡁⠈⢹⡿⢾⣕⣀⡀⠀⠀⡽⠛⢿⡉⠁
⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣴⣿⢰⠿⣿⣧⣀⠀⠀⢛⣿⣿⣧⣾⣀⠤⣿⣿⣿⣿⣿⣶⡶⠾⠻⣶⡀
⠀⠀⢀⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣮⣿⣻⣷⣖⣬⡈⡹⠿⢻⣿⣿⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠘⠛⣛⣛⣛⣹⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠿⠛⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣾⣩⣟⣿⣿⣾⢿⣟⣣⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣾⠿⣻⠟⠟⠻⣻⣟⣿⣷⣿⣯⣍⣟⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣊⣿⣿⣿⡵⢿⠃⠀⠀⠀⠀⠘⠉⠋⣼⢛⢋⣴⣻⠿⡾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣟⣾⠛⣷⡙⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠉⣿⣿⣿⣿⣵⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢿⣝⣀⠉⠃⠀⣋⢉⠃⠀⠀⠀⠈⣉⠁⢠⡚⣩⣷⣿⣗⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⡏⣃⠀⠀⠀⠰⠦⠇⠀⠀⠀⢨⡦⠅⢸⡾⣯⣷⣏⣰⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢣⣧⡙⠆⠀⠀⠀⠀⠀⣀⠒⢀⡀⠀⠀⠈⡋⠉⢿⣯⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⢷⣦⡄⠀⠀⠀⢤⣈⠀⣉⣠⠀⠀⡸⠃⠀⣈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠣⣀⠀⠀⠉⠻⠉⠁⣀⠔⠁⢀⠄⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠈⠁⠒⠒⠒⠏⠽⠀⠀⠘⢒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⡀⠀⢀⠇⠀⠑⡀⡠⠋⠡⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠁⣠⠞⠀⠀⢠⡿⠁⠀⠀⠘⠄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢢⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠳⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠱⢤⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠖⠊⠉⠉⠁⠒⠂⠼⣀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠳⠦⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠘⠫⣵⠀⠀⠠⠤⠤⠤⠄⠠⠀⠀⠀⠘⠛⠢⡀⠀⠀⠀⠔⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠉⠉⠉⠉⠙⡄⠀⠀⠀⠀⠀⢀⣀⡀⣼⠗⢄⠀⠀⢀⡸⠳⢌⡀⠉⠉⠉⠉⠉⠈⠉⠉⢱⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠁⠀⠀⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⢀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠉⠈⠀⣀⠈⠈⠁⠀⠀⡉⢉⣉⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⡀⠀⠀⢀⣀⣀⣈⢀⣁⡀⠐⠁⠀⠘⣁⡁⠋⠘⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⢛⢿⢛⡻⠻⣿⢛⣻⡿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⠫⣾⣿⠟⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⢉⡴⢋⣾⣷⣬⠏⣀⠄⠀⠀⠈⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢧⢀⡴⢋⣴⣿⣿⣿⣿⣴⣧⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⣾⢋⣴⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⡠⠊⣠⠞⠋⡡⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⡄⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⡎⠀⢻⠟⢥⠊⠁⠀⠀⠑⠛⣀⣀⣀⣤⣤⣾⣵⣷⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣿⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⢰⠁⣠⡎⠀⢀⡱⠀⣀⣠⣴⣾⣿⣿⣿⡿⢿⣿⣧⣌⠛⠛⣧⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠎⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠀⣅⡼⣡⡿⣶⡶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣶⣤⣄⣙⣳⣂⣤⢀⣠⣾⣷⡶⠀⢰⣿⣿⣿⠛⡿⣿⢿⣿⡿⡏⡇
⠀⠀⠀⣠⣿⡾⠋⠀⠻⣧⡄⢤⠙⢯⣽⣿⣿⣿⣟⡛⠲⢿⣝⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠯⣼⣿⠃⠀⢸⣿⣿⣿⣇⣁⡇⣿⣿⡇⠇⠃
⠀⠀⠐⠿⠋⠀⠀⠀⠀⠈⠙⢦⣀⣼⠁⠈⠹⣷⠀⠈⠉⣦⠈⠻⠦⣤⣽⣿⣟⣻⣷⢤⣬⣀⣻⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⢸⠟⠒⠚⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠀⠀⢠⠤⣤⣤⣀⠀⣊⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⡇⣿⠄⠀⠀⠀⠀⠀⠈⠳⣶⣭⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⡀⢉⢳⡀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡿⠁⠹⠀⠀⠀⠀⠀⠛⠷⠋⠉⠠⣄⣀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠟⢿⠏⢸⠁⠀⠈⠄⠀⠀⡀⠀⠀⠀⠼⣷⡔⠊⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⢀⠖⠂⢸⡀⠀⠀⢀⠀⠀⣽⡲⢈⠫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣠⠊⢈⣠⣄⢳⠀⢀⣴⡟⠒⠛⠉⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡀⠁⠀⠉⠀⠀⠙⠇⠸⠟⠋⠁⠀⠚⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠃⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠠⠋⠀⠀⠀⠉⢢⣔⣢⡤⠖⠀⠂⠀⠀⠢⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣂⣤⣤⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠛⠛⠀⠂⠀⠀⠀⠘⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠛⠛⠉⠁⠈⠉⠉⠙⠒⠠⠆⠀⢠⠤⠒⠁⠀⠈⠀⠶⣶⣄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣿⡺⠿⠷⠾⣤⣤⣀⣀⣀⣀⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣀⣠⠤⠤⠤⠅⠤⠾⢦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠤⠀⠈⠉⠉⠙⠁⠀⠒⠂⠀⠤⠤⠴⠲⠐⣀⡷⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⣔⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣷⣶⣤⣤⣀⣀⣤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠛⠉


⣿⠿⠿⠿⡿⢿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣮⣟⠿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣯⣿⣿⠇⠉⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢙⠈⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⠀⠀⠀⠀⠀⠢⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠇⠀⠈⢦⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠂⠀⠈⢿⣦⡄⡀
⠀⠀⠀⠀⠀⢠⢞⠃⠀⠀⠀⠀⠀⠀⠑⠲⣧⡐⠢⠄
⠀⠀⠀⠀⣠⡿⠆⢀⢀⠀⠀⠀⠀⠀⠀⠸⢿⣍⢰⡀
⠀⠀⠀⣾⣿⡶⠟⠃⠀⢑⠀⠀⠀⠀⠀⠀⢸⣿⣆⠹⣆⠀⠀⠀⠀⢦⡀
⠀⠀⠀⣷⠉⠀⠀⠀⠀⠀⠁⢸⠀⠇⠀⠀⠀⢿⣿⡆⠘⡄⠀⠀⠀⠈⠃
⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠸⣿⣿⡄⠰⠀⠀⠀⠀⠀⢒⡺⡄⠀⢀⡄⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣾⢸⠀⠀⠀⠀⢹⣿⣷⣀⡦⢄⠀⠀⢀⡤⠉⠈⠀⠉⢀⡗
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠈⣿⠟⠉⠀⠈⠧⠒⠁⣀⠠⢠⡀⢃⣘⠁
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⢀⠌⠀⢰⣏⣸⣤⡥⠚⠁
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠐⠛⢉⠔⠉⠀⠀⠀⠀⢆⣴⠃⠀⠀⢘⡿
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣰⡏⠁⠀⠀⠀⠀⠀⣠⣾⣇⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⢸⢿⢇⠀⠀⣸⢹⠀⠀⠀⠀⠀⢀⠞⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣡⣿⣄⠀⠊⠀⠀⠀⠀⢀⡴⠁⠀⠻⣿⠇
⠀⠀⠀⠀⠀⠀⢠⡉⠛⢻⣧⣀⣀⠀⢀⡖⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠋⠉⠉⠛⠉⠀⣀⣀⣠⣤⡔⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⡿⠿⠿⠿⠇⠿⠿⠛⠛⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⡁⣈⠁⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠠⢌⡣⠥⢠⡌⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡃⡞⠀⠊⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠇⠈⠧⣀⢀⢀⡓⠀⠰⠀⡆⠢⡀⠀⣠⠆⠀⠀⠀⣰⢦⠀⢠⢶
⠀⠀⠀⠀⠀⢁⠀⠁⢖⡛⠛⡓⠊⠁⣠⠟⢡⠀⠃⠙⠀⡨⠬⠥⣼⣆⠀⠀⠙⢾⡤⡾⠋⠀⠀⣰⠉⣇⡐⣲⠀⡀
⠀⠀⠀⠀⠀⠂⡖⣀⠀⢙⡆⠀⡀⠈⣈⠕⠂⡎⡤⠒⠢⡀⠠⠤⣄⢉⡄⠀⠀⢈⠀⠇⠀⠀⢠⠤⠀⡄⠀⠙⢠⡞⢀
⠀⠀⠀⠀⠀⠸⡇⣧⠄⠚⢉⣈⠁⢰⠃⠀⠀⠿⠁⠀⠀⠑⣠⠴⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣦⠀⢡⠚⠁
⠀⠀⠀⠀⠀⠡⠙⢀⡴⠋⠉⠁⠀⠚⠂⠀⢀⡴⠋⠉⡥⠋⠁⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠰⠄⠸⡏⠀⠀⣩⠁⢈⡅⠶
⠀⠀⠀⠀⠀⠠⠀⠈⣠⠀⠘⣇⣠⠴⠒⠒⡿⠁⡔⠉⢧⠀⠀⠀⢈⠐⠲⡅⠩⠥⠀⠬⠅⠌⠀⣄⠀⣀⠴⠏⢀⣘⢢⡄
⠀⠀⠀⠀⠀⠥⠶⠄⣏⠀⡀⠈⠃⠀⠀⠀⡇⠀⠃⣠⠄⠂⠀⠀⠀⢱⡀⠘⡇⠻⣿⠟⠀⠀⢠⠏⠈⠁⠀⣤⢌⣧
⠀⠀⠀⠀⠀⠀⢀⠴⠛⢲⡧⠖⠒⠂⠀⠀⠁⢀⡞⠀⡔⠂⠀⠀⢀⠼⠁⠀⡇⠀⠘⡆⠀⢠⠞⠀⣸⠇⠀⠙
⠀⠀⠀⠀⠀⠀⠏⠀⢠⠏⠀⡔⠀⣠⣤⣄⠀⢸⠀⠀⣇⠀⢄⡀⠀⠀⠀⠀⠀⠀⢀⡇⠐⢁⠤⢰
⠀⠀⠀⠀⠐⣴⠒⠂⢾⠀⠸⡅⠈⢀⡠⠤⣄⣘⣆⠀⠘⣆⠀⠉⠛⠂⠀⠀⢀⡠⠊⠀⣰⠁
⠀⠀⠀⠀⠀⢳⠀⣀⠘⢆⠀⠈⠁⡏⠀⠀⠀⠈⠉⠀⠀⠘⡆⠀⢸⠀⣀⡤⠤⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⢀⣥⠀⠀⠒⠂⣶⠐⠀⠀⠁⢀⣘⣛⣁⣀⣀⣀⣀⣘⠀⠀⠀⢢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠀⢠⡀⠀⣿⠠⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⢠⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠓⠚⠋⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⡀⣀⣀⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣠⠞⠉⠉⠉⠀⠉⢳⡄⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡇⠀⠚⠂⠀⠈⠉⠁⠀⢳⣀⣀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⡇⠀⠐⠃⠀⠀⠘⠀⠀⠈⢩⠄⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣷⠀⠀⠐⠂⠀⠀⠀⠀⣠⣤⡴⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠁⠉⠀⢀⡴⠃⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣶⠶⠶⠚⠹⣇⣤⣼⣤⣤⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠹⢤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠦⣄⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⣠⠞⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡆⠀⠀⠀⠙⢦
⠀⠀⢀⡤⠞⠁⠀⠀⢀⣤⠞⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⢦⡀⠀⠀⠈⢷
⠀⡴⠋⠀⠀⠀⣠⠞⠻⢦⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡧⠴⠚⡧⠀⠀⠀⠘⡇
⡼⠁⠀⠀⢀⠀⡏⠀⠀⠀⠑⠫⢷⣦⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢠⡇⠀⠀⠀⠀⠹⡄
⡇⠀⠀⡆⢸⠦⠃⠀⠀⠀⠀⠀⠀⠉⠻⢕⠢⢤⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⢸⣁⣿⠀⠄⠀⠀⢻
⠙⠦⢤⡥⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣽⢰⠀⡇⣀⡼
⠀⠀⠀⠀⠈⠉⠓⠲⢤⣤⣤⣤⣄⣀⣀⣀⣤⠼⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⣿⠙⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⣀⣠⢴⣾⣭⣤⣤⣤⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⢀⣀⡤⠴⠖⠛⠒⠋⠉

⠀⠀⠀⠀⣀⢄⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠚⠆
⠀⠀⠀⠀⠁⢈⡆⠀⠀⠂⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠂⠤⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠆⢉⡀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⢀⡴⠀⡀⡲⠅⠀⠁⠀⠈⡁⠀⠀⠀⠀⠀⠌⠜⠀⠀⠀⠀⠀⠀⣦⠀⠁⣈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢠⣀⣠⣤⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠌
⠀⠀⠀⢔⠒⢂⢀⠀⠀⠤⠤⠌⠀⠀⠀⠀⠄⠀⠀⠠⠄⢀⠈⡀⠁⠀⡇⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠀⠸⠂⠄⢘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠸⣀
⠀⠀⠀⢀⡐⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠚⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠈⠣⡼⢆⠂⠀⠂⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠚⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠁⠂⠈⠄⠶⠀⡀⠘⢄⠃⣀⠂⠀⡀⠄⠀⠀⠀⠀⡀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⡠⠀⢺⠀⠉⠀⠈⠀⠀⠀⠠⠤⢀⠁⣛⡇⠂⠀⠀⢀
⠀⠀⠘⢀⠀⠀⠀⢧⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠒⢈⠤⡀⠀⠡⠐⠀⠀⢂⠀⠘⠋⢸⠆⠀⠀⠆
⠀⠀⠀⠀⠉⠣⠀⠈⠀⠀⠀⡠⡆⠀⠀⠀⠄⠀⠀⠀⠀⠄⠉⠁⠀⠨⠀⠀⢀⠀⡆⠀⠀⣸⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠛⠂⣀⠀⠀⠦⠀⠀⢀⠀⡀⣀⢀⢄⡁⢀⠔⠒⠁⠁⠀⠈⢣⣬⠂⠰⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠉⠀⠀⠀⠀⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠔⠒⠊⠡⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠚⣁⠀⠀⣘⠀⢃⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡃⠀⠀⢸⠀⠀⠀⢀⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠻⡜⣣⡟⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠈⠀⠀⠀⠇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⣤⠀⠀⠀⠀⠶⠀⣬⠀⢷⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠉⠉⢸⠀⢠⠀⠀⠀⢃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⣠⣀⣼⡋⠀⠀⠀⠀⢀⠴⠶⢠⣄⠀⢸⠀⠀⠀⠀⠈⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠁⠚⠉⠑⠶⠖⠋⠀⠀⠀⠀⡈⠑⠺⣄⢀⡀⢀⠀⡸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡤⠔⠊⠀⠀⠀⠀⢰⠚⠉⠉⠲⠄⢈⣉⣿⣉⠁⣡⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢀⡀⠀⠀⠀⠈⠠⠄⠠⣤⠄⣀⡈⠀⠀⠀⢸⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠒⣀⣀⣀⠒⠉⠀⠀⠀⠀⠙⠉⢛⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠡⠉⡍⢩⠌⡗⠶⢲⡟⣼⢧⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡆⠷⠴⠦⠣⡾⢶⠃⠶⠸⠧⠶⠰⠟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢰⡆⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠑⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠎⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡰⠾⠦⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣄⢀⣀⣀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠉⠀⠀⠀⠈⢡⣼⡧⠐⠒⠒⠒⢄⢀⡜⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⡏⠀⠁⠀⠀⠉⠁⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⢀⣀⣀⡄⠀⠘⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⣀⠀⠀⠀⠉⣶⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠙⠀⠀⠀⢠⠛⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠘⠀⢀⡈⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠄⠒⠉⠀⠀⠠⠀⠠⠔⠊


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠁⢀⠀⣿⠻⡉⠛⢹⠟⠛⡉⡟⢫⠉⠙⠋⢙⠋⡿⠛⠋⠙⢉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢀⣀⣀⠀⣸⣿⣿⣿⡄⠀⠀⢠⣶⣶⣾⡂⠀⠀⢀⣤⡀⠀⠀⠀⠰⢾⣿⣿⣿⠆
⠀⢀⣤⡴⠶⣶⣮⡅⠀⠀⠀⣈⡉⣻⡇⠸⠿⠿⠛⠐⢦⡀⠸⣿⣿⣿⡃⠀⠀⠘⠿⠛⠓⠲⠶⠶⢬⠟⣻⣯
⠀⠸⡇⠀⣸⣿⠏⠻⠶⠞⠊⠉⠁⢻⡇⠀⠀⢸⣿⣠⡾⠃⠀⠈⠉⢿⣿⣄⠀⠀⠀⠀⠀⠻⠷⣤⣄⣸⣿⣿⡆
⠀⢰⣧⢀⣿⣟⣀⣀⣀⡀⠀⡴⠀⢸⡇⠀⢀⣼⣿⠋⠀⠀⠀⠀⣀⠀⠻⣿⣿⡛⠛⣿⠂⠀⠀⠀⠈⠉⢸⣿⣷
⠀⠀⠀⢸⣿⠉⠉⠉⢹⡇⠈⠀⠀⢸⣇⣴⠟⣻⣿⠀⠀⢀⣴⣿⣿⣿⡄⠙⣿⡟⠛⠛⠛⠶⣤⣼⠃⠀⣴⡿⢻⡆
⠀⠀⠀⢸⣿⠀⠀⠀⠘⠳⠀⢀⣴⣿⠟⠁⠀⠙⣿⠀⠀⠀⠙⠿⣿⠿⡃⠀⠘⣷⡀⠀⠀⡀⠀⠀⣠⡾⠋⠀⠘⣿
⠀⠀⠀⠸⡿⠀⠀⣤⣶⣶⣴⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⣤⣦⡄⠀⣿⣿⡀⠀⠘⣷⠀⠀⠇⠀⠀⣿⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠘⠿⠦⠀⣿⣿⣿⡟⠙⢷⣀⣀⡀⠀⣴⡷⠀⠈⠿⠿⠛⠲⢾⣿⣇⠀⠀⠐⣷⡴⠆⠀⢀⣿⠀⠀⠀⠀⠈⣿
⠀⣠⣴⣦⡀⠀⠀⠈⠙⣯⡁⠀⠀⠉⠉⢉⡧⠀⠀⢲⣀⣀⣤⣤⣤⣌⣿⣿⡀⢀⣤⣤⣄⠀⠈⠉⠉⠀⣀⣀⠈⠉⠉
⠐⣿⣿⣿⡿⠂⠀⠀⠀⣌⡗⢀⣠⣾⣿⣦⡀⠀⠀⠈⠉⠉⣁⣀⠀⠉⠉⣿⡇⢾⣿⣿⣿⠷⠀⢀⣀⣼⣿⣿⠆
⠀⠈⣛⣯⡀⠀⢀⣀⡀⠈⠀⠈⠹⣿⣿⡿⠃⢐⣧⡀⠀⣾⣿⣿⣷⡄⠀⣿⡇⠈⢻⣟⣡⠶⠟⠋⠁⠀⢉⣡⣤⣄
⠀⠀⣿⣿⣿⡄⣿⣿⡿⠶⠶⠶⠶⠶⣶⣶⠀⠀⠀⠁⠀⠙⠿⠿⠋⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡇
⠀⠀⣿⠈⢿⣿⡆⣉⣤⣶⣄⠀⠀⠀⢹⣿⣧⠀⠀⢸⡟⠛⣿⣿⠀⠀⠀⣽⡅⠀⣿⣿⣠⣴⡶⣷⡀⠀⠀⠙⣿⡏
⠀⠀⣿⠀⠈⢻⣿⠋⠁⠀⠙⠷⣄⣰⠎⢿⣿⡆⠀⢸⡇⠀⣿⣿⠀⠐⠛⠛⠃⠀⢿⡟⠉⠀⠀⠈⢻⣄⣀⡀⣿⡇
⠀⠀⠋⠀⠀⠈⢿⡄⢸⡀⠀⠀⠈⢃⣴⠿⠻⣿⠀⠈⠁⠀⣹⣿⣤⣤⣄⠀⠀⠀⢸⡇⠀⠈⢦⠀⠀⠛⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣠⡾⠛⠁⠀⠀⢻⡇⠀⠀⠀⣿⡏⠉⢩⣿⠀⠀⡶⢟⠁⡀⠀⠀⢀⣀⣤⣤⠶⢹⡇
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣾⡇⢸⣿⣿⣿⡌⢻⣄⠀⠀⣿⡇⠀⠨⣿⣤⠘⠟⠋⣄⠹⠿⠘⠋⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⣠⣿⠀⠘⠿⣿⡿⠃⢀⣹⣷⡀⢿⡇⠀⠀⠀⠀⠀⢀⣾⠋⠀⣀⣤⠀⠀⠀⡇⠀⢸⡇
⠀⠀⡈⢧⣀⣼⡿⣦⣄⡀⠀⠀⢀⣶⣶⠶⣶⠀⠀⠀⠀⠀⠘⢛⣦⣀⠀⠀⢠⡾⢁⣴⠾⠋⠀⠀⠀⠀⠁⢀⣼⡇
⠀⠀⣇⣈⣉⣉⣀⣈⣙⣿⣶⣶⣿⡿⠃⠀⣿⠀⠀⢸⠀⠀⢀⣿⣿⣿⣷⢰⣿⣷⣿⣷⣶⣶⣶⣶⣶⣾⢦⣿⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠀⠀⠀⠀⠀⠈⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣴⢾⣍⣉⣉⣉⢐⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢉⣤⠴⢿⣄⠑⢌⠳⢤⡉⠙⠳⢤⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠎⡠⣚⠥⠒⠊⠙⢻⢦⠀⠙⢤⡉⠳⣄⠀⠉⠢⡈⢢⡀
⠀⠀⠀⠀⠀⢀⠔⣡⢊⠞⠁⠀⠀⠀⡴⠋⢻⡳⣄⠀⠙⠢⣄⠉⠒⠤⢝⣦⢑⡄
⠀⠀⠀⠀⢠⢫⢞⡵⠋⠀⠀⢀⡤⠊⠀⠀⠀⠙⢮⡳⣄⡀⠀⠉⠛⠒⠶⠤⢭⣍⣶
⠀⠀⠀⠀⣹⣧⣊⣀⡠⢤⡖⠋⠀⠀⠀⠀⣃⠀⡐⠉⠲⠭⣓⡒⠠⠤⠤⢤⣤⡶⣏⢦
⠀⠀⠀⣰⡏⠀⡜⠁⠀⢸⠇⠀⠀⠀⠀⠀⠩⠠⠁⠀⠀⠀⠀⠀⠉⠉⡟⠉⠈⢇⠈⣮⢧
⠀⠀⢠⢿⡆⢠⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⡄⢸⡜⡂
⠀⠀⣼⢸⠇⢸⠀⠀⠀⡻⠀⠀⠀⠀⠀⠀⠓⠖⠋⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⢸⡇⢣
⠀⠀⡇⡾⠀⣸⠀⠀⢐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠈⣷⢸
⠀⠀⣷⢃⡴⠹⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣧⡀⢻⣄⡂
⠀⠀⢹⡏⠀⠀⣇⠀⡞⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⡇⠀⢰⠁⠙⣮⡏⠁
⠀⠀⠸⣇⠀⢰⠉⣦⡇⠀⠀⠈⠑⡶⠠⠤⠤⠴⠶⠤⠄⢲⠊⠉⠀⠀⠀⠇⢠⠋⡄⠀⢻
⠀⠀⣼⣸⠀⡌⢠⠁⣠⠖⠉⠉⠉⡟⠒⠒⠠⠤⠤⠐⠒⢺⠉⠈⠉⠳⣄⠘⠻⠈⡇⠀⡛⡀
⠀⠀⠀⢯⡼⢡⠇⡰⠃⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⢆⠀⢇⢹⡀⣷⠇
⠀⠀⠀⠀⠉⠋⡺⠁⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⡇⠙⠢⠟⠋
⠀⠀⠀⠀⠀⢰⠉⠉⠉⠒⠒⠿⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹⡖⠈⠉⠉⠱⡀
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⢩⠟⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⣉⣳⣀⡠⠤⠔⠖⠚⣆⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣽⠓⠢⢄⠀⠀⠠⢤⡽⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠋⣤⣾⡀⠀⠀⠑⢄⣹⠚⠙⣆⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⡠⠊⠁⠹⣿⠀⢾⣿⣻⡄⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⣀⣰⣃⠴⡄⣤⠐⣦⡿⠋⠐⠒⠒⠒⠒⠂⡶
⠀⣠⠊⠀⠀⠀⠀⢿⡄⠰⣿⣿⡟⣄⠀⠀⠀⠀⠀⠀⠈⣳⡉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡘⢩⡏⢩⡏⠉⠉⠙⢮⣦⣉⣤⢧⡬⢆⠀⠀⠀⠀⠀⢀⡠⠿⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⡶⠒⢰⣿⣇
⡇⠠⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⢷⠂⣀⡴⠚⠉⠀⣠⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣿⣿
⢣⠐⡆⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⢀⣠⣖⣯⢿⡿⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣿⡇
⢸⠉⡟⠉⡟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠊⠓⣶⠛⣿⠯⠒⠉⣠⣾⠟⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠛⠒⢺⣿⣏
⢠⠰⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⡶⢶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠸⡀⣃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⢰⠉⡛⠉⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⣿
⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠘⡄⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⠶⠶⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⢉⣠⣤⣤⣶⣶⣤⣤⣬⣙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⣴⣾⡿⠟⣛⣉⣩⣭⣍⣉⣛⠻⢿⣦⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣰⣿⠟⣩⣶⡿⠿⠛⣛⣛⣛⠛⠻⢿⣶⣌⠻⣦⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣸⣿⢣⣾⠟⣥⣶⠾⠛⠛⠛⠛⠿⣿⣦⡙⣿⣆⠸⣇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇⣾⢏⡾⢫⠴⠞⠛⠿⢿⣶⣄⠈⢿⣧⢸⣿⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⢠⡟⠸⢁⣴⣾⣿⣿⣷⣦⡙⣿⡄⢸⣿⢈⣿⠀⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠘⡇⢰⣿⡿⠉⠀⠀⠈⢹⣧⢸⡇⢸⡏⢸⡿⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣛⠛⠲⠄⡠⠋⢉⣉⠙⢆⡰⠚⠛⠻⣆⠀⠀⠀⠀⠸⡄⠸⡇⠇⣿⣿⠀⠀⠀⠀⠀⢸⡟⣸⢃⡿⠀⣼⠃⢰⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣾⠉⠀⡀⠈⠁⢰⠏⠉⡧⢀⠠⢾⡉⠢⠙⠀⠀⠀⠀⠀⠱⡄⢻⡄⢻⣿⠀⠀⠀⠀⠀⡼⢡⠏⡼⠁⣸⠋⠀⠾⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣷⢯⣭⡀⠉⠙⣷⣬⡴⠋⣀⣘⣧⣴⣶⣦⣤⣤⣤⣀⣀⠙⢦⠻⣌⢻⡄⠀⠀⠀⠈⠀⠋⠈⠀⠘⠁⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣧⣀⣨⣿⠀⠀⠀⣠⠶⠟⢋⠉⡉⠁⠀⠈⠉⠉⠉⠛⠻⠿⢾⣷⣽⣦⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣬⡽⠿⠷⠶⠶⡾⠅⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣽⠛⠉⣀⡤⣤⣤⡀⢀⡗⠀⠀⠀⠠⠄⠀⣠⣴⡿⠟⠛⠲⠶⣤⣀⡀⠀⠀⠀⠀⠙⠿⣷⡀
⠀⠀⠀⠀⠀⢀⠔⠁⢨⡇⠰⠙⠿⠿⠆⢁⣾⠁⢈⡉⠐⠲⠄⢰⣿⠏⠀⠔⠚⢶⣄⢠⡉⣉⠛⣛⠲⠶⢤⣀⡉⢿⡄
⠀⠀⠀⠀⠐⠿⠈⠀⠀⣵⣀⣀⣀⣠⣴⡿⠋⠲⢤⣈⠛⠦⠀⣿⣿⠀⠀⠀⢀⠀⣿⡄⢁⠑⣾⡥⢃⡶⣀⣌⡙⠾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⠉⠉⠙⠋⠡⠀⠄⠱⣦⠉⠓⠂⠀⢹⣿⡀⠀⠀⠣⡴⠏⢰⠀⠀⡼⢷⢾⣏⠡⣜⠐⡂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢠⠀⢷⠈⢦⠀⠀⢀⠀⠀⠈⠿⣷⣄⠀⠀⣀⣴⠇⡐⠁⠀⢼⡟⠻⣤⣀⠔⣡⠎⡌
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡻⡻⡟⠋⢁⠄⡰⠒⣂⠸⣤⠂⠀⠙⢧⢐⠚⠷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡞⢩⢝⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡑⢮⡑⠾⣭⣹⣐⢁⠀⠛⠀⠀⠀⠈⣸⠈⡆⢐
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠐⢣⣌⠘⣧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢷⡽⣦⡉⠳⢦⣭⣙⠛⠿⠶⣶⣦⣬⣭⣉⣚⣁
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠤⠖⢃⠈⢣⡘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡛⢷⣤⣈⡙⠛⠿⢶⣶⣤⣬⣭⣉⣛⣛⣋⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⢸⠇⠈⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣬⣙⠻⠿⣶⣶⣤⣬⣍⣉⣛⣛⣛⣋⡁⣇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋⠀⠀⠀⠀⠙⠻⠷⣶⣦⣤⣤⣤⣤⣤⣴⣾⡿⠿⠿⣶⣶⣤⣭⣍⣛⣛⣛⣛⣛⣉⣁⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠉⠩⣉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠉⠒⢤⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠶⠚⠛⠉⣠⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⢀⡈⢩⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠞⡆⠁⠀⠁⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⠐⠇⠀⣀⠇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⠁⠀⡤⠶⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢡⠞⣱⠃⠎⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⡉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠂⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠾⠿⠿⠟⠉⠩⠏⠁⠀⠀⠐⡟⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡶⢶⣶⣶⠶⢲⣶⢢⡖⠒⠒⠒⠒⠒⢶⠶⡶⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣥⣿⣤⣴⣦⣼⣄⢠⡷⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠛⠛⠿⠿⢿⣿⣿⣿⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⠟⠛⠉⠉⠉⠉⡥⠖⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣀⣀⣀⣀⡀⠀⢀⡤⠖⢒⠉⠀⠀⠀⠀⠀⣼⣷⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⡏⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠑⠒⠒⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠢⠤⠄⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣶⣦⣾⣿⣦⣾⣿⣧⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠿⠛⠻⠿⠟⠿⠿⠻⣿⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠐⠲⠶⠂⠀⠀⠀⠀⠀⢠⣤⡀⠀⠘⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⡟⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⡅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡏⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠲⣦⠉⢀⠀⠀⢠⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠁⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠐⢠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠠⠤⠤⠀⠀⠀⠙⠀⢠⣤⠃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣄⡀⣤⠀⠀⠀⠀⠀⠀⠀⠰⠆
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠉⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠈⠉⠁⠈⠁⠉⠁⠉⠁⠈⠀⠉⠉⠉⠉⠀⠀⠀⠀⠛⠀⢀⡉⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠄⢠⡄⠉⢠⣴⣤⣀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣾⣿⣿⣽⠛⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢻⡞⠉⠉⠀⠉⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⢀⡄⠐⠀⠀⠀⠠⠄⠀⠈⠒⠒⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠅
⣀⣀⣠⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⣀⣀⣠⣀⠀⠀⢀⣀⣀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢼⡁⠀⡇⠀⡇⠀⠙⠂⢸⠀⢸⠀⢠⡾⠋⠀⠀⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⣀⠔⡆
⠀⠀⠀⡞⠁⠀⠧⠀⡇⠀⡆⠀⣿⠠⠘⠀⠉⡀⠐⠀⠀⠀⠆⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠄⠀⠐⠒⠉⠁⠀⠀⠋⡖⠒⢲⡏⠂⡁
⠀⠀⠈⠁⠀⠀⠀⠊⠁⠀⠁⠐⠋⠐⠋⠀⠀⠐⠈⠉⠀⠀⠁⠀⠈⠓⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠄⠀⡠⠂⠐⠐⠒⠠⡀⠀⠀⣄⣧⣤⢼⡇⡄
⠀⠀⣀⠂⠀⢐⠂⠋⠘⡒⠀⠀⢰⠈⠉⡆⢀⡄⠀⠐⠖⠀⠠⠲⡂⠀⣀⠀⠀⣠⠄⢠⡤⢠⣄⢠⣶⠠⣴⠌⢰⡀⢠⠄⡀⢰⡀⠀⡄⠉⠀⠀⡀⠈⢁⡄
⠀⢸⡌⠀⣀⢠⠤⠀⠠⡄⣀⠀⠀⠓⠂⠀⠈⠁⠀⠀⠀⢐⡀⠐⠀⠀⠋⠁⠀⠙⠂⠈⠓⠉⠉⠋⠃⠀⢋⠀⠀⠌⠀⠀⠀⠐⠀⢀⡀⠀⠄⢀
⠀⠀⠇⠀⠰⢀⣁⠑⡈⡀⠇⠐⠀⠅⠈⠀⠀⠄⠀⠁⠤⠄⠀⠠⠂⠰⠄⠀⠀⡁⠀⠀⠀⠘⠈⠀⠈⠐⠉⣤⠀⠈⠳⠲⢞⠀⣠⡄⠁⠀⠸⡏⠀⢄⠀⢰
⠀⠀⠄⠀⠀⠅⣙⢒⡋⠰⠀⠀⠀⠁⠠⠀⠠⠀⢀⠀⢰⡆⠀⠿⠀⠀⢠⡠⠆⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠉⠠⠅⠈⠀⢈⡠⡲⠀⠀⠇⠀⠈⠉⠙
⠀⠀⠀⠠⢀⠀⠹⠈⠅⠤⠀⠂⠠⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠉⠀⠉⠀⠀⣀⣀⣈⣀⣂⣀⣁⡀⠀⠀⢄⠀⠠⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⢴⠀⠠⠄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⣿⡁⢀⣀⡀⠀⠀⠐⠉⠉⣉⣉⣉⣉⢉⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠡⠈⠀⠀⠀⣀
⠀⠀⠀⡀⠓⠌⠀⠀⠀⠓⠒⠁⠰⠀⠈⠈⠀⠀⢰⡏⠀⠉⠁⣿⠈⠉⠻⠋⠉⢳⡄⠿⡄⢠⣭⣀⢀⣥⡿⠀⢀⣀⠀⢀⠀⠀⠈⠀⠠⠁⠈⠀⠀⢀⡄⣿
⠀⠀⠠⠄⠀⢸⠀⠈⠀⠙⠀⠀⢀⡀⠈⠀⠀⣀⠘⢳⡀⠀⠀⣿⢸⣷⠀⢠⣿⠀⡇⠀⡇⢸⣿⣿⠸⡇⠀⠐⠛⠛⠒⠛⠛⠀⠀⠀⠀⠀⠂⠀⠁⠀⣤⠄
⠀⢀⡀⠀⠆⠈⢀⠀⢀⠈⠀⠀⠐⢰⠢⠤⠄⡞⠀⠈⢪⣀⠀⣿⢸⡿⠀⠸⠟⢰⡇⣀⡇⢸⣿⣿⠠⡇⠠⢠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⢀
⠀⢨⣧⠀⡀⠃⢚⠀⢐⠒⢂⡀⢀⣼⠀⣼⢀⡇⠀⠀⠀⣽⡀⢿⡤⠴⠻⣤⡴⠎⠘⡯⠤⠤⠿⣷⣴⡇⠀⠁⢈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⢀⣈⡁⣤⡁⠀⢀⡀⡀⠁⠀⠉⠉⠉⠀⠀⠀⣀⣀⣬⣥⣄⣀⣀⣀⣋⣀⣀⣀⡀⢀⣀⢀⣀⣛⣀⣀⡀⢀⣀⣈⣀⡈⠈⠈⢀⠀⠀⠀⡀⠀⠀⠅⣀⠬
⠀⠐⠈⠡⡤⣤⡄⢹⡅⠠⡤⠀⠀⡀⠠⠂⢻⡙⣿⠏⣩⠀⣤⡀⠉⢿⣿⠈⠀⣶⡌⠹⢋⣉⠻⢡⣤⣾⡇⣿⢸⡁⠶⢾⡇⠠⠀⠀⠀⠠⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣶⡀⠁⠄⠀⠈⢷⡘⢀⡏⢸⣿⡇⠀⢸⣿⠀⠀⠛⣡⣴⡸⢿⣶⠘⢛⣿⡇⣿⢸⡇⡶⠺⡇⠀⠄⠀⠀⠄⢸⡄⠲⢖⢾⡄
⠀⠠⠰⠻⢻⠁⠄⠈⠈⠀⠀⠈⠁⠀⠀⠀⠀⢸⡇⢸⣿⠘⠿⠃⡄⢸⣿⠀⠀⣷⡌⣿⢻⡆⠙⢠⣿⣿⠀⣏⣸⡇⣷⠀⡠⢆⡤⡄⠀⠀⠀⢆⢹⣘⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠀⢠⣤⣤⡄⠀⠀⣧⣸⡟⠶⠴⠞⠳⣤⣥⣾⣤⣿⣧⣽⣦⣤⡾⠦⠤⠼⣧⠭⠽⣧⣿⠈⢋⠉⠉⠁⢠⡎⠀⠘⢛⠉⠉
⠀⠀⠈⢶⠃⠀⠀⠀⠄⠠⠀⠀⠸⠿⠿⠇⠀⠀⠀⡉⣁⣀⡤⣠⠀⠉⠀⠈⠉⠙⠀⠀⢸⠇⠀⢴⠒⠠⣀⠀⠀⠁⠀⠀⠘⣆⡇⠀⠉⠀⠀⠀⠀⠀⢢⠶⠂
⠀⠀⢀⡀⠀⠀⠀⠈⠉⠉⠁⢀⣀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠠⡀⠀⠀⠂⠀⢀⠀⢀⠀⡢⣀⡠⠋⢀⣠⣄⣀⣀⠀⠙⠁⠐⣿⢿⣿⡏⠃⠀⢸⡆
⠀⠀⠈⡀⣤⠀⠀⠄⠠⠤⠄⠠⣬⠐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠰⠂⠁⠆⠀⢠⡈⡾⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠏⡁⠀⠀⠘⠇
⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡀⠁⠇⣆⠀⠀⠈⠀⢀⣾⠀⠀⡀⠄⠀⠀⠈⠢⠤⠁⠀⠀⣀⠀⡀⠀⢀⡀⠀⠥⠁⠀⠀⢀⠄⠀⠐⠑⠃⢈⠃⠁⠀⠀⢤⡀
⠀⠀⠀⠀⠁⠀⡶⠂⠀⠐⠻⠇⠀⡇⡴⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢶⡄⣒⡆⠀⠀⠠⠀⠰⡆⠀⠊⠂⠍⠀⠀⡀⣀⡩⠂⠘⠈
⠀⠀⠀⠀⡀⠀⠀⠅⠀⠨⠄⠀⠀⡆⡆⠀⠈⠀⠀⠀⠀⠀⣀⠈⠀⡁⢀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⠀⠂⠀⡀⠀⠨⠅⠀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠸⠀⠇⠀⠀⠀⠨⠐⡀⠀⡄⠃⡧⠀⠀⡡⠁⠀⠀⠂⠈⠐⠀⠄⠂⠀⠀⠀⠂⠀⡀⠂⢄⣐⠄⠀⠀⠀⠀⠀⠈⡅⠀⠉⠁⣠⣤⠀⠀⢠⣤⡀⠀⢹
⠀⠀⢰⠀⢀⡀⡀⠘⠓⠛⢀⣀⡅⠀⠃⠀⠀⡮⠀⠀⠀⠂⠀⠀⠂⠀⠀⠈⢀⢀⠀⠀⢀⠀⠀⡇⠀⣀⢀⡁⠀⣀⠀⡉⢀⣀⡈⡉⠉⠀⠀⠈⣉⣉⣀
⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠩⠐⠂⠀⠀⠃⠀⢠⢀⠈⠀⠘⠁⣀⠀⢀⡘⠀⠀⠠⠀⠀⠀⡇⢸⣏⡸⠇⠀⡷⠀⠿⠌⠉⠁⡇⠹⣶⠁⡏⢉⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⠂⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢠⡟⡠⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠸⢇⡀⠀⣿⡇⠀⠤⠐⠂⠀⠀⠀⠀⠀⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠉⠀⡠⢠⣔⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⢸⣿⠀⠀⣠⡖⢢⣠⠀⣸⡏⠀⣿⡆⠠⠊⠀⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⢸⣿⠁⠘⣿⠀⠀⣿⠀⢸⡟⠀⣿⡇⢀⢿⣷⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⠀⠀⣿⠀⡼⣿⢠⢿⡇⣰⠉⠧⠂⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣠⠟⠀⠀⠙⠖⠁⠉⠁⠈⠳⠃⠀⠀⢠⢹⣿⠀⠀⠀⡠⠂⠁⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⡆⢸⡟⠀⢀⠌⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⢃⣤⠞⠉⠀⠀⠀⢠⣤⠀⠀⢐⡂⠀⣿⠇⠀⠠⠌⠣⠛⠁⢐⠋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠌⠀⠀⡎⠀⠀⢱⠀⠸⣿⡀⢀⣿⠃⢰⣿⠀⠀⣾⠃⢠⡆⢠⢣⡾⠁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡔⢲⡆⢱⣾⠀⠠⠂⠀⢘⣷⠀⣿⠀⣸⣿⠀⢸⣿⠀⣾⠃⡆⣿⠇⠀⠀⠰⣗
⠀⠀⠀⠀⠀⠀⠙⠈⢸⣿⠀⢘⣿⠘⣿⠀⢰⠀⢠⠃⣿⠀⠻⣤⢻⣿⠀⢸⢿⡴⢹⡿⠀⣿⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢠⢊⡟⠀⠉⠂⠁⠀⠈⠀⠉⠀⠀⠀⠘⣿⢀⡇⠀⠀⠀⠁⠀⠈⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠉⠐⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣁⣀⣀⣀⣀⠀⢀⣀⣀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣤⣤⣤⣤⠤⠽⢧⣀⠀⠙⣆⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠑⠒⠤⢍⡑⠢⢄⡉⠒⢿⣄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⡙⢷⠢⣄⡀⠀⠀⠀⠀⠀⠈⠑⠢⠭⣒⢤⡈⠑⠦⣈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡇⠈⢳⣄⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣬⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠏⠀⢀⠞⠀⣠⠞⠈⠛⠶⣤⣄⣈⡑⠒⠢⢤⣤⣤⣤⡤⢔⣶⡯⢴⡟⢦⣀⣀⡀
⠀⠀⠀⠀⣀⡴⢋⡠⠔⢉⣠⠞⠁⠀⠀⠀⠀⣀⠈⠉⡙⠓⠛⠓⠒⠒⠒⠯⠏⣌⣈⣭⠗⠀⠀⠀⡕
⠀⠀⠀⠀⠀⠙⣿⠗⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⡦⠤⠒⠋
⠀⠀⠀⠀⠀⢜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢱
⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠳⣀⠔⣳⡤⠄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠇⠀⠀⠀⠋⣴⠟⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⠋⠀⠀⠀⠀⠀⠀⢚⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠟⠁⠀⠀⠀⠀⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠤⢤⡤⢤⠤⠤⠴⠶⠛⣩⠴⣇⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⢻⠆⢤⣤⣤⣀⣄⣤⠤⠴⠏⠁⠀⠀⠑⢤⡀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢠⠷⠴⠤⢤⣄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⣧
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣃⡴⠚⠉⠉⠙⠢⡄⠀⠀⢀⡴⠋⠁⠀⠈⠑⢾⡄
⠀⠀⠳⢶⢀⡄⢀⠀⣠⣀⡟⠀⠀⠀⠀⠀⠀⠘⣆⢀⡾⠀⠀⠀⠀⠀⠀⠈⣷⠒⠒⠂⠐⠒⠀⠠⠔⠲⠶⠶⠦⠶⡤
⠀⢀⠔⠉⠉⠣⠞⠲⠃⢨⠃⠀⠀⠀⠀⠀⠀⢀⣏⣈⣽⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡅
⠰⠛⡖⠒⡖⠒⠒⠒⠒⡾⠀⠀⠀⠀⠀⠀⠀⠸⠦⢤⠼⡇⠀⠀⠀⠀⠀⠀⠘⡧⠤⠤⠤⠤⠤⢤⡤⢠⡄⠀⣠⣿⡇
⡇⢠⠁⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⢷⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⢰⠀⢰⠁⠀⣼⣿⣷
⢇⠀⡄⠀⡇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⢿⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⢸⠀⠀⢿⣿⡏
⢘⠦⠷⠤⢷⠤⠤⠤⡾⠃⠀⠀⠀⠀⠀⢀⣸⠷⠤⠼⠦⠤⣧⠀⠀⠀⠀⠀⠀⢈⣳⣄⣀⣀⣀⣨⣄⣠⣧⣀⣸⣿⡇
⠸⠀⡇⠀⡇⠀⠀⡞⠉⠁⠒⠒⠒⠤⠤⠤⣭⠀⠀⠀⠀⠀⣻⠄⠒⠒⠊⠉⠉⠁⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⢰⠀⡇⠀⡇⠀⠀⡶⢢⣤⡤⢤⣄⣀⣀⡀⣤⠀⠀⠀⠀⠀⢿⣀⣠⣤⠤⣴⣒⣶⡎⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠈⡦⢵⠄⢼⡄⢼⣇⠰⠿⠟⣛⣳⣿⣿⠈⣇⣀⣀⣀⣀⣀⣨⡅⢹⣿⡻⠯⠝⠛⢋⡼⣃⣀⣀⣀⣀⣀⣀⣀⢘⣿⣇
⢸⠀⡆⠀⡌⠀⠈⠻⣷⣄⣀⣀⣀⣈⣩⡾⠃⠀⠀⠀⠀⠀⠀⠻⣦⢤⣀⣀⣤⣔⠮⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⠀⡇⠀⠇⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠈⢆⣸⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⢏⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠖⣀⡤⠴⠚⠃⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠋⡁⠀⠀⢀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠻⣷⣤⠴⠚⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠐⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢠⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠟⠉⠉⠙⢛⢿⣿⣿⣿⣿⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡟⠛⠛⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⣿⣿⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠍⠉⠉⠉⠡⣿⣿⣿⣿⣿⣿⣿⣿⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠸⣷⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣎⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢻⣿⣿⣿⣿⡁⠀⢀⣿⣿⣿⣿⣉⣿⣿⣶⣄⢠⣤⣹⣿⣿⣿⣿⡿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠙⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠛⠿⢿⣿⣿⣷⣶⣾⡿⠿⠛⠉⢀⡀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⣀⠉⠉⣀⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⣤⣤⣤⣄⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠤⠀⠀⠀⡂⠤⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⢀⣀⡀⠀⠀⣀⡤⣪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢩⠃⣀⣴⣷⣾⣿⣿⠿⠟⠛⠙⠊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢠⣣⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣰⣿⣿⣿⣿⡿⢿⡍⠐⡒⠂⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⣿⣿⡏⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⣱⣿⣿⡿⣛⠠⡲⠋⠀⠀⠀⠀⣰⣇⣠⣴⣿⣿⣽⣆⣡⡔⠀⢢
⠀⠀⠀⠀⠀⢠⠜⠀⢰⣿⣿⣿⣋⠤⠰⠁⠀⠀⠀⢀⣿⠿⣿⠁⣼⣿⣿⣿⣿⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡉⠠⣸⣾⣿⣿⣿⠁⠀⡂⠀⠀⠀⠀⢿⣇⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣶⣴⡾⠁
⠀⠀⠀⠀⠀⠡⣀⣿⣿⣿⣿⡧⠀⢀⡿⠀⠀⠀⠀⠸⠃⠀⠀⠀⠨⢻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣷⡶⠛⠁⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠙⠛⠛⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠛⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⣷⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣸⣇⡠⠤⠤⠀⠀⠈⠥⠴⠾⣟⣛⣛⠲⠦⣤⣤⡥⠴⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠂⠁

⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠄⠀⠐⠒⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠶⠞⠻⠛⠛⠛⡏
⠀⠀⠀⠀⠀⠈⠓⠲⣤⣀⠀⠀⠀⠀⠀⠀⠘⠲⣶⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠙⠲⠦⣄⡀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣶⣶⣒⣺⣿⣯⣽⠇
⠀⠀⠀⠀⠀⠈⠁⠢⠑⠢⢄⡀⠀⠀⠉⠙⠲⠤⣿⣿⣶⣶⣖⣺⣭⡭⠷⠶⠚⠛⠋⠉⠉⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⣑⠢⢄⡀⠀⠀⠀⠈⣿⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡴⣖⣿⠂
⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠈⠑⠢⣌⠑⠢⢀⠀⣿⣄⠀⢀⣀⣀⣤⣴⣶⣺⣿⣽⠷⠾⠛⠛⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠉⠒⢤⠈⣙⣻⣭⡽⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠐⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⢠⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠀⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢸⣷⣆⣀⣠⣤⡤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⢀⠀⠉⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⢈⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡋⠘⣗⠲⠖⣯⡟⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠢⣈⣷⣞⣽⣴⡞⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢈⡋⣿⢰⢹⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⣃⣀⡀⠀⠠⡇⡇⡆⢸⠀⠀⠀⠘⣟⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠤⢄⠀⠉⢻⢸⠃⡃⢸⡆⠀⠀⠀⢻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣇⣀⡜⠓⠂⢾⣎⣠⣆⣸⣧⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣴⣧⡤⠭⠶⢶⣾⣿⠁⠀⠀⠀⠀⠀⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠑⠒⠢⠤⢄⣀⣀⢼⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡴⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠒⠒⠊⠉⠉⠁⢀⡠⠔⠚⠑⠒⠠⠤⠤⠤⢤⠤⠄⠺⢏⣉⣏⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠦⠤⠤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡴⢃⣀⣀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⡛⢹⠀⠀⠀⠀⣠⠤⡤⠐⠒⠒⠛⠛⠒⠒⠒⠚⣿⠉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡇⢸⠀⠀⠀⠀⣟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⠀⡆⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⠃⠀⢇⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢤⢼⡀⠀⠘⣤⠄⢼⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⡠⠔⠒⠋⠉⠀⠀⢀⠇⠀⣀⠷⠀⢾⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠈⠑⠒⠀⠐⠒⢒⡶⠛⠊⠁⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠓⠈⠁⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠢⢭⣙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⠒⠦⣌⡉⠒⠦⣍⡛⠿⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠤⢍⣒⠦⢌⣙⠲⠤⣀⡀⠀⠈⠙⠻⣦⣄⠠⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠒⢬⣝⡲⠬⣉⡒⠤⢌⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⣙⣳⣦⣍⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣛⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠉⠉⠉⠛⠷⢶⣤⣄⣀⠀⠀⢀⣀⣼⠇⢀⣀⣠⣤⣶⠶⠞⠛⠉⠉⠉⠉⠉⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡟⢧⡛⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠸⠀⠈⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⢀⣀⡈⣻⣿⣤⣴⣶⣶⣶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⢹⣿⣴⠾⠟⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣆
⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⣛⣒⣒⣒⣒⣒⡒⠲⠶⠶⠤⠤⠤⠭⠭⠭⠭⠉⠉⠉⠉⢉⣿⡉⠉
⢠⡿⠁⠀⠀⠤⣄⡀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣂⣀⣀⡀⠀⠀⠀⠀⠈⠭⠭⠭⣭⣭⣭⣭⣅⣒⠒⠀⠀⠀⠀⣾⠀⠉⠓⠦⣄⡀
⣿⠁⠀⠀⠀⠀⠀⠈⠉⠂⠀⠘⣷⡀⠀⠀⠀⠐⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⣉⣉⣒⡒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠙⠳⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣐⡒⠲⠶⠶⠶⠶⣤⣤⣤⣤⣄⣀⣀⣀⣒⡒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢷⣀⣀⣤⣤⡴⠶⣶⠟⠛
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠷⠶⠶⠶⠾⠿⠿⠿⠿⣿⣿⣽⣿⣭⣭⣤⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣙⣏⠁⠀⠀⢰⠇
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠐⠂⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠶⠶⠚⣛⣋⣉⣥⣤⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣌⣛⣬⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡴⠶⠶⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⢠⣀⣰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠰⠈⠁⠀⣐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⢠⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠇⠈⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡠⠋⠔⠀⠀⠀⠀⠀⠸⢀⣀⠀⠀⠀⠀⠀⠀⡲⣆⠀⠀⠀⠄⡄
⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠀⠉⢀⢀
⠀⡎⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣃⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠃⠀⠸⣸⠀⠈⡀⡄⠀⠀⠸⢀⡀⠀⠀⠦⣄⠀⠀⠛⠒⠵⣄⢀
⠀⢸⠀⠀⡏⡀⠀⠀⠹⣆⠀⠀⠹⡀⢀⠔⠠⠀⠑⠲⠠⠄⡠⣹⣬⠀⠈⠀⠐⡢⠄⡀⣂⣀⣂
⠀⠸⡀⠀⠁⠃⠀⢀⠀⠈⠂⠀⠀⡼⠏⢀⣀⠀⡀⠠⠤⠊⠀⠀⠌⠂⠀⠀⠀⠀⠠⠄⠀⢀⠨⠉⠉⠂
⠀⠀⢃⠀⢸⢸⢀⠏⠀⠁⠀⠱⢄⠃⠀⢀⣀⣀⣦⠠⠒⠂⠁⠌⠀⠀⣀⡠⠄⣀⠤⠒⠈⠀⠀⠀⢰
⠀⠀⠸⡄⠈⠸⡞⠠⠂⠀⠀⠀⠀⡏⠉⠉⠀⠀⠈⠐⠂⢀⠠⠀⠀⣑⠯⠒⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣷⣀⡀⡇⠀⠀⠒⠈⠒⠤⠁⠀⠉⠉⠓⠒⠲⠤⣌⣁⠀⠀⠀⠂⠀⢀⣀⣠⣤⠤⠴⠖⢒⡀
⠀⠀⠀⢡⡏⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠏⠉⠉⠀⠀⠀⠀⠀⠀⡌⠡
⠀⠀⠀⢸⠗⢻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢠⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⡸⣇⠹⢸⠦⣬⣄⣒⣒⣒⣒⣀⡠⢄⡀⢀⣀⠀⣀⠔⢡⠂⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡇⠋⠇⣨⣀⣀⠀⠈⠉⠉⠉⠉⠉⠀⠸⡤⠔⠋⢀⠔⠁⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢣⠀⠆⠀⠐⠁⠀⠀⡄⡇⡂⠀⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠤⣤⣤⠤⠴⠆⡇⠃⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢏⣴⢀⣶⣠⠄⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⡕⣶⠸⠹⠻⠸⠣⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣏⢿⡼⠸⣇⡌⠛⠀⠀⠀⠀⠀⠀⠀⠸⡵⠒⠛⠟⠛⢷⠲⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⣉⠀⣒⡶⢚⠀⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠰⠀⠴⠶⠄⠆⠀⣂⠁⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠒⡐⠓⢂⣠⡀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠟⠧⠿⢇⡓⣼⣧⣏⣇⣟⡠
⠀⠀⠀⠀⠀⠀⠀⢘⠁⣷⠀⣷⣷⠀⣷⣿⢹⡇⡍⠀⠀⠀⠀⠀⠀⢏⠙⢛⢃⣛⢚⣈⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈⢒⡃⢠
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠿⠶⠟⠛⠀⠿⠿⠸⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⠀⠘⣛⢛⠃⢸⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣋⠃⡀⠄⠁⠀⡇⠀⠀⠀⠀⠀⢠⡀⢀⣉⠉⢉⡉⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠛⠁⡟⣿⣴⣏⠈⠧⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⣶⣾⡼⠛⢇⢶⠐⠀⣧⠀⠀⠀⠀⠀⠀⠸⠽⠛⠃⠘⠛⠛⠻⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠮⠼⠟⠷⠛⠋⠿⠀⠸⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣠
⠀⠀⠀⠀⠀⠀⣖⡚⣇⣎⠀⠉⢩⣉⠁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢤⣀⡄⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡀⣻⢳⡀⡷⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⣹⢯⡸⠽⠜⢏⠼⣽⡄⠀⠀⠀⠀⠀⠲⡖⠤⣤⢀⣴⠋⢘⣙⣋⠬⢺⡞⠁⣽⠇⠀⠀⠀⣀⣤⠈⠈⡉⠰⠴⠇⠉⢠⣄⣄
⠀⠀⠀⠀⠀⠀⠐⣾⠡⠴⠔⠖⠷⢐⠀⠸⠂⠀⠀⠀⠀⠀⠙⢿⡁⣿⡽⢶⢤⣠⡤⢤⡶⣺⣧⠴⠁⠀⠀⠀⠠⣿⡟⠀⣾⠛⣤⢠⡄⣶⠊⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣒⣛⣻⣰⣓⣙⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠲⠶⠶⠲⠾⠟⢀⡀⡤⠀⡀⠀⠀⠈⠿⣿⠀⢱⣏⠓⠺⢳⠙⠀⢸⡟⠃
⠀⠀⠀⠀⠀⣠⣴⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣦⠄⠀⠀⠀⠀⠠⡶⢸⠍⢰⡆⣾⢰⡄⠸⣼⡆⠙⣦⠉⠀⠀⠀⠀⠉⠀⣸⠚⠀⣴⢋⡀
⠀⠀⠀⠀⠀⠿⠿⠟⠛⠛⠙⠛⠛⠉⠉⠛⠿⠿⠆⠀⠀⠀⠀⠀⠀⠙⠒⠈⠙⠙⠃⠓⠀⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⢀⠘⠤⠿⡯⠿⠷⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡿⠋⢄⡈⠑⠤⣀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⡫⢈⠔⠁⠀⠀⠀⠀⠉⠒⠄⡀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡫⠊⠋⠀⠤⡁⠀⢰⠀⠀⠀⠀⡀⠀⠀⢈⠑⠢⣄⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⣠⡶⠶⣬⡑⠢⢄⠀⠠⠀⠀⠀⠄⠀⠈⡀⠀⠁⠀⢄⡀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠋⠀⡒⣾⠛⠀⠀⠀⠍⠓⢦⣉⠢⢄⠐⠂⠀⠀⠊⠀⢀⠔⠠⠀⠈⠒⠤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⣠⡾⠋⠀⠀⠈⠑⢢⠀⠐⠤⠤⠌⠛⠦⣍⠀⢀⡐⠠⡴⡡⠊⠤⠀⠀⠐⠀⠈⣁⠀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠋⠉⢀⣾⠏⠀⠀⠀⠰⣶⣤⠀⠀⠀⠠⠈⠁⠀⠀⠈⠑⠦⣉⠒⢄⡀⠀⠀⠀⠁⠠⢀⠞⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣪⠆⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠀⠀⠀⠈⠓⠦⡈⠑⠤⡀⢂⠔⢁⣼⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⠞⠁⣰⠟⡅⡆⠀⠠⠂⢀⡆⠐⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢈⣴⡿⣫⢞⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⠾⣱⠐⠿⢀⠸⣄⠒⠺⣵⣮⠀⠾⠂⡨⠀⢄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣴⡿⢋⢜⣵⣿⠿⣷⡿⠂
⠀⠀⠀⠀⠀⢀⠞⣹⠞⠁⣠⠞⠁⠀⠛⠃⢠⣦⠠⡌⠁⡠⢄⢀⠅⣠⠘⡀⡠⠶⢆⣀⡘⠀⢠⠀⣀⢀⣤⠞⢁⣴⡯⢋⢔⣵⣿⣯⣜⣷⣾⠁
⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠈⠻⢦⣀⠠⢶⡂⠷⠖⢘⡏⢁⡤⠠⢤⢠⡶⡎⠀⠀⠘⠁⠈⠁⢻⢔⣵⠟⢁⡴⡯⢋⢔⣵⣿⡻⢝⢷⣽⡿⣻⡀
⠀⠀⠀⠀⠀⠈⣿⢨⡙⠳⣦⣄⠢⢝⠳⣤⡀⠘⠉⢻⡁⢞⣻⠁⠄⢿⡉⣳⠈⠑⠠⡄⢌⡴⣵⠟⢁⣴⠏⢀⢔⣵⣿⠻⢷⣌⣳⣿⠏⢈⣼⣿⠆
⠀⠀⠀⠀⠀⠀⣿⡀⢌⡑⢤⡙⠻⣦⣍⠪⢕⠒⠘⠟⠙⣼⡧⢂⠄⡼⢄⠢⠁⠐⠈⠀⣡⠞⢁⣴⢟⡵⠀⣰⣿⠿⣗⢍⣢⣽⣿⢇⢔⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣌⡓⢌⡒⢄⡙⠳⣦⣉⠲⢄⡐⠋⡣⠉⠋⠃⢌⡗⠀⠀⢠⡞⠁⣰⣿⡿⠋⣠⣾⣿⣭⣿⢾⣿⣿⠟⠕⣱⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣌⡓⢬⡂⢌⠉⠙⠲⣍⡢⠀⠺⢦⣀⠀⠀⠀⡴⠋⣠⡾⠿⠉⣰⡾⠟⡧⣊⢝⣷⣿⡟⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣥⣬⣙⠻⣿⣿⣿⣿⣦⠈⠃⠉⣀⠈⠒⠝⣢⣌⠓⢍⡳⠆⠈⣠⡾⣫⠄⣠⣾⣟⡒⢕⡪⣠⣾⡻⢋⢤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣏⣉⠻⢶⣽⣢⣭⣻⢿⣦⣄⡈⡑⠢⣐⠬⡙⠻⣦⣌⣠⡾⣫⠞⣡⣾⠟⢓⠬⡻⣵⣾⢿⢟⢔⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠸⣏⠠⣉⠢⢌⡙⠿⣯⣻⢮⣝⠻⣿⣗⠤⡉⠢⢕⠀⡽⢋⠜⣡⣾⡟⢁⠪⣑⢉⣾⠟⠕⡱⣫⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣍⠢⢍⡢⢌⠙⠳⣮⣟⡒⠿⢿⣮⣕⠢⠀⡷⣡⣾⠟⡁⢌⡒⠙⣢⣿⠋⠀⣪⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣍⠒⢭⡒⢄⡁⠉⠻⢷⣮⢝⠿⣶⣾⠟⢛⣶⣮⣕⣭⣾⡿⠋⢜⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⡈⠑⠉⢀⠀⠑⠪⢿⣿⣦⢤⡲⣿⣿⣿⣿⣿⠏⡰⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⢉⠒⢤⡒⢬⡙⠿⣯⣄⣠⣿⠟⡡⣪⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⡢⣌⡑⠪⣕⠄⣽⠿⠃⢔⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣭⡓⠤⡁⡇⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠈⢀⡒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⡷⠤⢎⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢰⢀⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠀⠀⢰⢸⠀⠀⠀⣀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⠀⡸⠖⢲⠾⠓⠯⠀⠀⠘⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠦⠤⢄⡠⡌⡹⠊⢁⣀⠤⠔⠒⠂⠉⢃⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣈⣷⠊⠁⠶⣺⠀⠀⠀⢀⣀⣀⣀⡗⣿⠣⣄⠀⠀⠀⠀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡄⠈⠹⣧⣄⣀⣆⠀⠀⠀⠸⣦⠶⠺⢶⣅⠀⠞⠂⡐⠲⡀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢖⡏⠢⣈⠁⠀⠀⠉⣀⡈⠉⠈⣛⠀⠀⣈⣏⣐⣐⣂⣉⣀⣈⣢⡀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡬⠗⠈⠉⠐⠒⡖⠓⠛⠒⠒⠉⠹⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠃⠀⠀⠀⢸⠓⡇⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⡖⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠋⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣌⣳⣔⣦⢀⣠⣼⣄⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠉⠁⠀⠀⠈⠉⠒⠭⣒⠦⡀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠣⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠟⠂⠀⠀⠤⠤⠤⠤⠤⠭⠭⠍⠁⡖⠲⠒⠤⡄⣀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣇⣀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠉⢺⡄⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⢄⣀⢼⡇⠄⣞⠣⢄⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⡊⠁⠀⠉⠀⠉⡷⠊⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠢⡀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢆⡀⠀⠀⠑⡄⠉⠉⠉⠁⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⡀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢆⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⠟⣳⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⣷⣿⣧⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣄⠀⠙⠛⣹⣿⣿⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⠀⠀⠀⠘⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀⠔⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⢀⣀⣀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠁⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⢀⡀⠀⠀⢀⣄⡖⢤⠀⠀⠀⠀⢤⣈⢃⠀⠀⠀⢿⡿⠀⠀⠀⢾⡿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣷⡼⢦⠶⠒⠒⠿⠛⠋⢀⠀⣀⢸⡅⢀⠀⢠⣿⠃⠀⢾⣀⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣒⡒⠐⠂⠀⠲⠤⠌⠁⠀⠀⠀⣨⠀⣯⠁⠉⠉⠡⠀⢀⣼⣿⣿⣷⣭⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣀⣨⣿⣿⣤⣀⣄⠘⣀⣾⣿⡿⣿⣿⣿⣷⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⢶⠰⢶⡾⠟⠋⢽⣭⡄⣱⣾⣿⣿⠟⠁⠀⠉⠉⠉⠉⠉⠉⠐⠙⠈⠻⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠺⢼⢯⣴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠠⠂⠀⣼⣶⣾⣶⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣷⣦⣾⣶⣾⣷⠎⠙⠢⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢾⣿⢻⣿⣿⢉⡁⠀⠀⠀⠀⠈⠐⠃⠰⠤⠒⠀⠀⠀⠀⠀⢾⣷⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢘⣿⣿⣾⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣋⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢈⣧⣾⠁⠙⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⣿⣿⣿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠶⢷⣷⣶⣶⠀⠾⣤⣤⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⣿⣿⢲⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡿⢀⣼⣯⣿⡿⢰⠇⣿⡏⠀⠹⢄⠀⠀⠀⡃⠀⠀⠀⢀⢸⠺⣿⣧⣾⣿⣰⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣿⡿⣿⣿⠗⠋⣸⣿⣶⣤⣶⣦⣡⣦⣸⠁⠀⠀⢰⣿⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⣀⣰⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⣾⣿⣿⣿⣿⣼⣿⣿⠿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠆⠐⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠦⣤⣀⣀⣤⠤⠦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡏⠉⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠈⠉⢹⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣀⣀⣤⣤⣀⢸⣇⣀⣤⣤⣀⣀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠉⠁⠀⠀⠉⠛⠛⠉⠀⠀⠈⠉⠛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠇⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠟⠀⠐⠋⠻⠵⢢⣀⢕⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢑⣀⣠⢻⠀⠀⠀⠀⠘⠈⠈⠠⠤⠠⡐⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣢⠓⡽⣶⠂⡀⠀⠀⠠⡞⢉⣀⠠⠂⠀⢀⠀⣷⠀⡏⣠⡇
⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣷⢄⣄⣢⠇⠀⢀⠁⠀⠀⠀⠀⠉⡀⠀⠀⢹⡄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠢⠽⡿⠀⠀⠀⠈⠀⡄⣶⡶⠈⠁⠀⢸⡀⠻⢃⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⠀⠀⢸⡦⣄⣀⢸⢘⠀⣿⠃⡅⢠⣄⠸⣇⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⠀⠀⠀⠀⣧⡀⠉⠚⠚⢲⠃⠠⡄⠀⢻⣆⣫⣴⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⡀⠀⠐⢀⣿⣿⣦⣤⣠⣏⣀⣴⣷⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣴⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⣀⣠⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠗⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠀⠙⠋⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⠆⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⠿⣿⣿⣿⣿⣿⡷⠀⠀⠠⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡤⡔⠁⠀⠀⠭⣿⡍⣥⣆⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡾⡀⢢⠀⠖⠂⢸⡆⠀⠉⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣾⡇⢻⡌⠀⠀⠀⡈⣰⡃⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠊⣧⣷⠈⠁⠀⠀⠀⠐⠊⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡚⣋⢱⢼⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡶⠔⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⣿⢧⠀⠀⠀⠀⠀⠀⡔⡩⣚⡟⠙⠋⣻⣔⢕⡄⠀⠀⠀⠀⠀⠀⡼⣿⢓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣉⣉⡜⠀⠀⠀⠀⠀⢠⣱⡟⢁⡀⠀⠀⢀⣈⢻⣿⡀⠀⠀⠀⠀⠀⢠⣉⣉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡆⠀⠀⠀⠀⠀⣸⠹⡇⠭⠍⠀⠀⠩⠭⢸⡏⣇⠀⠀⠀⠀⠀⢸⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠀⠀⠀⡏⠒⡀⠀⠀⠀⡄⠀⠀⢀⠒⢹⠀⠀⠀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢃⡀⠀⠀⠀⠇⠀⠱⡀⠻⣉⣉⠟⢀⠎⠀⡘⠀⠀⠀⢀⡸⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠉⠉⠁⠘⠒⠒⠚⠷⡂⢒⠾⠓⠒⠒⠃⠈⠉⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠒⠒⠒⠒⠒⠒⠲⠯⡉⠈⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠤⣀⢀⡀⣸⠀⠀⠀⠀⣄⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠒⢂⠀⠀⠀⢸⠀⠀⠀⢸⠒⠂⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡶⡾⠁⠀⠁⢹⠀⠀⠀⢸⢶⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠃⢸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣀⣀⣀⣀⣀⠏⠉⢏⣀⡰⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⠑⠍⠂⠤⠌⠱⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⢀⠎⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢀⠎⢀⡴⠋⠁⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⢠⠋⣴⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣠⡀⠀⣼⡿⠀⠀⠀⠀⡴⠃⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣾⠟⠁⢀⣿⡇⠀⠀⡠⠊⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠈⣿⠀⠀⢸⣿⠀⡠⠊⢀⡀⢀⣿⠇⠀⢀⣴⢞⠆⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠃⠀⢸⣿⠁⠀⢀⣴⡆⢸⣿⠀⢠⣿⠗⠈⠀⠀⠀⠀⢠⣿⠃⠀⡔⠂⠺⣄⠘⣿⡄
⠀⠀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢸⣿⠀⣿⡟⠀⣸⣿⠀⠀⠀⠀⠀⠀⢸⡿⠀⡌⢀⠊⠀⣿⠀⢻⣷
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⣼⣿⢠⣿⡇⡰⣿⡇⠀⠀⢀⡆⠀⠀⠈⠧⠠⡗⠁⠀⢸⠟⠀⢸⡟⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⠰
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⠃⠻⠼⣿⡟⠀⠹⣷⠀⠀⡼⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣴⠟⠁
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠉⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠉⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⢠
⡇⠀⠀⠀⠀⠀⣴⠊⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠁
⢹⠀⠀⠀⢀⣾⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⡠⠊⠀⠀⠸⣿⡑⣄⠀⠀⠀⠀⠀⠀⣴⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠂
⠀⢣⠀⠀⣼⡏⠀⠀⠀⡆⠀⢀⣴⠋⢱⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠘⡄⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠑⢤⣿⠃⠀⠀⠰⠀⠀⣼⡇⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⢸⣿⠂⠄⢠⠁⠀⠀⢿⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠂⣽⡿⠀⢀⠤⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⢸⣿⠀⢠⠃⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡧⠀⠀⠀⠀⢠⣿⡇⣰⠃⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⣴⠀⢸⣿⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⡤⠀⢸⣿⠁⠀⠀⣠⠀⠀⣠⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡀
⠀⠙⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⢀⡄⠀⣴⡆⠀⣿⡟⠀⠀⣴⡟⠀⢸⣿⠀⠀⣼⡟⠀⣸⡿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⠿⣄⠘⡖⠁
⠀⠀⠀⠸⣿⡄⣀⢤⡀⠀⣰⡟⢹⢠⣿⠁⢱⣿⠁⢠⣿⠁⢸⣿⠀⠀⣿⡇⠀⢠⣿⠃⠀⣿⣿⠀⢸⣿⠁⢀⣿⡇⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣷⠀⠘⣿⡤⣿⡗⠁⣾⡏⠀⣾⣿⠀⣼⣿⠀⣿⡏⠀⣸⣿⡇⠀⢸⣿⠀⡀⣿⡿⢀⠘⣿⠀⠌⢿⠀⢀⠆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢹⣿⢀⢜⣿⠀⢿⡇⡰⠻⣇⡼⠘⢿⡜⠀⠛⠊⠘⣧⣰⠂⣿⡇⠀⡎⠻⠔⠁⢿⣿⠀⠀⠈⠉⠀⠈⠳⠊⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠉⠠⠼⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣇⡸⠀⠀⠀⢀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠔⠃⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢻⣇⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁
⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠀⠠⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠦⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠐⠒⠦⠤⢍⣁⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠬⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠉⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠲⠤⢤⣤⣠⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀
⠀⠀⠀⠀⢀⡀⠨⠶⠒⠶⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠠⠤⢀
⠀⠀⠀⠀⠀⢻⠓⠒⠤⢤⣀⡀⠀⠀⠈⠉⠙⠓⠲⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⠀⠈⠉⠻⣿⣆⣠⣤⣶⡾⠿⠿⠓⠒⠊⠉⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠻⠶⢤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⣤⣤⣀⣀⡀⠹⡆⣀⣤⣴⢾⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣤⣤⣤⠴⠶⠒⠚⠋⠉⠉⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⣮⣄⣀
⠀⢠⣶⣖⣋⣭⣉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠐⠒⠊⢉⣁⣠⣬⡵⣿⡞⠉
⢀⣿⠏⠀⠉⠉⠛⠻⠶⣤⣍⣛⡳⢦⣤⣀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠖⠒⠚⣉⣉⣡⣤⠤⠴⠖⠒⠋⠉⠉⠉⠁⠀⠀⣾⠁
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣶⣤⣀⡜⢁⣆⣠⣤⣤⣶⠶⠾⠛⢛⣋⣉⣭⣥⣶⡶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣯⣥⣶⣶⠾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣐⣈⣧⣤
⠀⠀⠙⠛⠿⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⢷⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡾⠶⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⢶⣤⣀⣀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣮⣤⡶⠶⠟⠛⠋⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠀⠄⠀⠀⠀⠠⠀⠄⠄⠀⠀⠀⠀⠀⠐⠠⠂⠀⠆⠄⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠙⠁⢸⣿⠀⠀⣀⠀⠀⠈⠋⢸⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡆⢸⣿⠁⣾⡧⠃⠀⠀⣿⢠⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠀⠘⠒⠙⠛⠀⠙⠃⠀⠀⠀⠛⠁⠠⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⡀⠀⠀⠰⠦⣤⡴⢶⣀⣀⠀⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⡃⢠⣶⢢⠀⣿⡇⠉⣿⠉⢠⡶⡲⢸⣿⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠸⣟⢁⠜⣿⡇⠀⣿⠀⠻⣏⡡⢻⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣁⣀⣈⣁⣀⣉⣁⣀⣄⣀⣀⣀⣀⣈⣉⣀⣙
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠿⢻⣿⣿⢻⣿⣿⢟⡷⣾⣟⠀⣿⠇⣿⢿⣿⠿⠿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣄⣾⣿⣿⠘⣣⢉⣾⡇⠿⣣⠠⣿⡀⡵⠀⣿⣧⣼⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣉⣉⡉⠉⠛⠛⢛⡛⠛⠛⠛⠛⠛⠛⠓⠐⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⣶⢦⣄⠈⠋⠀⢀⣀⠀⣾⡆⢰⠀⣿⠉⣧⢠⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣾⠟⢰⣷⠀⣿⠷⢡⣿⣿⣼⠀⣿⠀⣿⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠿⠹⠆⠈⠿⠞⠙⠓⠁⠻⠃⠛⠀⠿⠾⠃⠠⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡰⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠤⠖⠉⠀⠠⠜⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣅⠈⠁⢠⠀⠈⠁⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠠⠄⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠀⢗⠂⠐⡿⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠄⣡⣌⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡴⣹⡠⠊⠉⠉⠙⠉⠉⠉⠑⠄⠏⢆⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠂⢄⠀⢀⡼⠥⠛⠂⠄⡀⠀⠀⠀⠀⢀⠠⠒⠚⠬⢧⡀⠀⠠⠐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⢢⡠⠒⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠒⢄⡔⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡑⡐⣠⢠⠁⠈⠒⠒⠒⠒⠒⠒⠒⠂⢤⠄⠒⠒⠂⠒⠒⠒⠒⠒⠁⠈⡄⣄⢂⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠖⠂⠁⠢⢄⠀⠀⠀⠀⠘⠁⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣈⣐⣤⣀⣀⠤⠔⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣰⡄
⠀⠀⠀⠀⠀⢠⡞⣲⡀⠀⠀⠀⢀⠀⠄⠀⠊⠈⠃⠁⠎⢱⠀⠀⢱⠁⠀⠁⠀⠁⠀⠀⠐⠀⠂⠀⠀⠰⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠁⠀⠋⣠
⠀⠀⠀⠀⠀⠁⠀⠀⣣⠀⡆⠐⠐⠆⠠⡆⡀⠠⠀⠄⠀⠈⠀⠠⠈⠀⠀⠠⡀⠀⠀⠀⠀⠃⠸⠄⡰⠶⠀⠘⠅⠴⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠇⢃⠈⠁⠀⠀⡄⠀⡀⢃⠀⠠⠁⠁⠌⠙⠕⠀⠠⠀⢈⠉⠀⠄⠀⠀⠀⠀⠀⠁⠰⠈⡄⠒⠂⡀⠀⠀⠀⠀⢀⠂⠀⡀⠀⠀⢌⠀⠁⠘⢁⡀
⠀⠀⠀⠀⠀⠀⣀⡂⢀⡀⠐⡄⠠⠁⡠⢄⠀⠀⢠⠴⠉⠃⠀⠀⠀⠀⠤⠀⠼⢠⠈⠀⠀⠀⠀⠀⠀⠀⡋⡶⠁⢤⠰⠆⠀⠀⠰⠀⠀⠀⠀⠀⠂⡀⠀⠠⡠
⠀⠀⠀⠀⠈⠀⢨⣞⠡⠠⠀⠭⠗⠜⠀⠀⠂⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠳⠤⠊⠀⠐⠀⠀⠀⠒⠀⠀⣤⡇⠀⠀⡠⡠⠁⠀⠠⠀⠄⠈⠂⡀⠀⠈⢔⠆
⠀⠀⠀⠀⠀⢀⠈⠁⠀⠰⠈⠤⠀⠤⠀⢠⠤⢠⢠⡤⣄⠀⠀⡠⢀⠀⠀⠀⡔⠀⠀⢄⠀⠀⠀⠀⠠⠀⢁⡀⠀⠐⠠⠁⠀⢠⠀⠀⠀⠀⠀⡀⠐⠀⠈
⠀⠀⠀⠀⡠⠀⣍⠏⠀⠀⠀⠀⡀⠀⠀⢸⠋⠉⠉⠉⠹⠀⠐⠁⠁⠀⢀⢀⠁⠀⠀⡄⠀⠆⠀⢰⠈⡀⠈⠀⠄⢈⠄⠐⠀⠸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣀⡆⠀⠐⢆⡦⠀⠀⠙⠉⠉⢉⠀⠀⢠⠄⠈⠁⡁⢠⡀⠂⠸⠁⠀⠐⡀⢢⠀⢀⠀⡘⠀⠀⠀⠀⢀⠂⠄⠀⠀⠈⠀⠐⠀⠠⠆⣤⠖⠁
⠀⠀⠀⢀⠕⢫⠐⡄⠀⡁⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠐⠀⠛⠓⠂⠈⠚⠀⠉⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠁⠤⠀⣻⣀⡄⠠⠄
⠀⠀⠀⠀⠀⠈⠆⠼⠂⠠⠀⠠⠀⠂⢼⠂⡄⠀⠘⠀⢀⠀⠄⠀⠠⠄⢜⠠⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢉⠄⡜⠀⡌⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢐⣛⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠠⢏⠄⡀⠀⠰⠅⠻⣘⠕⢮⢀⡐⠀⢐⡅⠀⠀⠀⠀⠀⠀⠀⡟⢀⠁⠀⠀⢃⡀⢐⠀⢀⢶⣷⠊⠀⠆
⠀⠀⠀⠀⠐⠚⠃⠐⠀⠂⠐⠀⠀⠀⡀⠁⣀⣀⡀⠀⠀⣴⠀⠀⠅⠀⠈⢶⡁⢣⡈⡈⠀⠦⠀⠀⠀⠀⠀⡀⠀⣋⡊⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠠⠀⢀⠄
⠀⠀⠀⠂⡀⠠⠀⠡⠁⠞⡆⠀⠀⢈⠀⠀⠁⠁⠀⠊⠀⠘⠶⠰⡆⠀⣀⠄⠱⣂⠱⡀⠐⠀⠀⢀⡠⠂⠀⠀⠀⠄⠀⠀⡀⠀⠙⠂⡁⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠂⠀⠈⠃⠐⠀⠀⠀⠀⠀⠒⣀⠄⢴⠀⠈⣹⡄⠂⡃⠀⠀⠀⠀⣀⢀⠀⠐⠃⣾⢿⡀⠅⠀⠄⠀⠠⠀⠀⠈
⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠀⢠⠀⡃⠐⠀⠠⠀⠀⠀⠈⠛⡀⣀⡼⠁⠀⠀⠀⢤⠑⡀⠀⠈⠀⡈⠉⠀⣦⡶⢹⡎⠀⠘⡀⢀⠂⠒⡆
⠀⠀⠀⠀⢀⣄⡀⢀⢀⠨⡶⠀⢀⠀⠀⠀⢲⠀⠃⡀⠀⢐⠂⡀⠀⠀⠀⠈⠀⠠⡖⢲⠀⠄⠐⡂⠁⠀⠀⡈⢠⠀⠀⠁⠀⣸⡗⠀⠀⠀⠉⠅⢀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⡸⠣⡀⠀⠉⠀⠀⠠⠐⢀⢴⠀⠀⠄⠀⠠⡀⠁⠀⡄⠀⠀⢀⡀⠑⠉⠀⠀⠀⡥⢘⠀⠀⣂⣀⡇⠀⠀⠃⡞⡏⠀⠒⠂⠀⠁⠸
⠀⠀⠀⠓⠀⠀⠀⠲⡇⠀⠀⠀⠀⠐⠀⠹⠿⡦⠤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⡀⠚⠈⠀⢀⣠⡄⠧⠄⠒⠒⠻⢇⢤⡐⣦⣀⣀⠄⡠⠆
⠀⠀⠀⠀⠈⠀⠀⠅⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠁⠀⠁⠀⢌⠀⠆⠐⠒⠐⠂⠒⠈⠒⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠽⠤⢄⠈⠉⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣠⠀⣀⠀⠄⣈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠀⣉⠀⠀⠀⣉⡁⢸⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⢨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠂⠀⠀⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠬⡓⠒⠒⡫⠄⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢄⡈⢆⠎⣀⠄⠀⠘⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡄⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⢠⠟⠒⠢⡀⠀⢠⡾⠴⠓⠂⠄⣀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠳⠼⣦⠀⠀⡠⠒⠚⢧
⠀⠀⠀⠀⠀⡰⠋⠐⠢⡀⡱⠐⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠑⠲⡅⡠⠒⠈⠳⡀
⠀⠀⠀⠀⣸⡱⢠⢀⡄⡹⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠸⠁⣄⢠⠰⣹⡀
⠀⠀⠀⠀⠹⡄⠇⡘⠥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠤⣄⡠⠤⠤⡀⠀⠀⠀⠀⠀⠡⠝⡀⠇⡼⠁
⠀⠀⠀⠀⠀⠹⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠇⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠙⣹⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⠦⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⡼⠁⠀⠀⠀⠀⠀⠀⣠⣴⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⣸⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⢠⠇⠀⣠⣿⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢀⡟⠀⢠⣿⠇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⢀⣀⣤⣶⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⣿⠀⢀⣤⣶⣶⠀⠀⠀⠀⠀⢸⣿⣿⡞⠀⢀⣿⡿⢠⠇⠀⠀⠀⢀⣠⣄⡀⣠⡶⠋⠀⠀⣀⡴⠒⠋⠉⢀⡾⠛⢸⡇
⠀⢀⣀⡀⢀⣀⠀⠀⠀⠜⠁⠀⣿⣿⣠⠏⣠⣿⡟⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢸⣿⣷⠏⠀⠀⢠⡆⠘⠧⣿⣿⠁⠀⠀⢀⣾⠁⠀⠀⢀⡴⠋⠀⠀⣼
⠀⢯⠳⡝⠾⡈⣇⡠⡀⠀⠀⠀⠻⡿⠃⢰⣿⣿⠁⣴⣿⠛⣆⠀⢀⣿⣿⠀⠀⣼⣿⠇⠀⠀⠀⣼⡅⠀⢰⣿⡟⠀⠀⠀⢸⡁⠄⣠⠶⠋⠀⠀⠀⢸⡇
⢠⡎⠙⠌⠦⣇⠁⠄⣿⣤⠤⡄⠀⠀⠀⣿⣿⡇⢰⣿⡇⣰⠃⢀⡞⠁⠛⠷⠛⣿⣿⠀⠀⠀⣸⠃⠳⠤⠾⠟⠀⠀⠀⠀⣠⠽⢻⡇⠀⢀⠀⢀⣠⠏
⠀⠉⡟⠋⠂⠈⠳⣎⡙⠀⠀⣼⠀⠀⠀⣿⣿⠁⢸⣿⡟⠁⣠⠏⠀⠀⠀⠀⠀⢿⣿⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠙⠒⠒⠛⠋⠁
⠀⠀⠓⠂⢐⠶⠀⠬⠛⢦⣄⡩⠇⠀⠀⠘⠿⠖⠚⠻⠿⠚⠁⠀⢀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣴⠟⠋⠀⢲⡄
⠀⠀⠀⠐⢧⣀⢀⡼⢧⡽⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⣤⣄⠀⠀⢯⡀⠀⠀⢄⣱⢛⣧
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⣀⣀⣀⣀⠀⣰⣿⣿⠁⠀⠀⠀⢀⣀⣀⡤⠶⠂⠁⠀⢀⣴⣶⠋⠁⢻⣿⠁⢀⣿⣿⠀⠀⠀⠉⠉⠓⠉⠙⠺⠇
⠀⠀⢀⣤⣶⠶⣄⠀⠀⠀⠀⠀⢴⠛⠉⠉⠉⠉⠉⠛⠛⠻⣿⣿⡿⠿⠿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⣸⣿⡇
⠀⣠⣿⡟⠁⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠘⣿⠇⠀⣿⣿⠃⠀⠀⣼⣷⣄⣤⠤⣤⣄
⢠⣿⡟⠀⣠⣾⣿⠃⠀⠀⣰⣿⡇⠀⠀⢠⣾⡷⠀⠀⠀⢠⣿⣿⠀⠀⢠⣿⣿⠀⠀⢠⣤⡄⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⢀⣿⣿⠋⠀⠀⣨⣿⡆
⠘⣿⡷⠚⢹⣿⡿⠀⠀⣰⣿⣿⠁⠀⢠⣾⣿⡗⠀⠀⢠⡟⣿⣿⠀⣠⣾⣿⡇⠀⠀⣿⣿⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⡇⠀⣴⢻⣿⡏⢀⣴⢟⣷⣿⡅⠀⣠⠟⠀⠘⠿⠟⠉⢸⣿⡇⠀⣠⣿⣿⠁⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⠀⠀⣾⣿⡇⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠘⢿⣧⠞⠁⠀⠻⠿⠛⠁⢸⣿⣿⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠻⠷⠞⠁⢿⣿⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠦⠴⠋⠋⠁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣟⣯⡙⠛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠻⠿⣿⣿⣧⢹⠏⢫⡹⡇⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢰⣿⣿⣿⣿⣿⣿⣎⣧⢀⡘⢙⣯⠀⠀⢸⡃⡄⠀⢀⠀⠘⠺⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⣿⣟⣿⣿⣿⠿⡿⣿⣿⣿⣿⣴⣿⣤⣦⣾⣱⣤⣀⣮⣴⠂⠀⣻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⣡⡦⢘⠻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡻⣿⣫⣩⡷⠎⠀⠀⠀⢹⣿⣟⠻⠭⠉⠙⠛⠛⠿⢿⣿⣿⣌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠈⠙⠻⠻⣿⣽⣷⢦⣄⠁⠀⠻⣟⠻⢶⣦⣀⠀⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠘⠻⠷⢶⣦⣌⠐⠈⢢⡌⢠⣿⣷⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⡂⠀⠀⣼⣿⠻⣦⣤⡝⣦⡀⠐⣻⣾⡟⠉⠦⣀⠀⠀⠀⠀⠐⠲⣤⣀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡛⠉⡘⣧⠀⠀⠙⢷⣄⣸⣿⣟⣿⡻⠊⣝⡟⠃⠀⠀⠈⠂⠠⣁⠀⠀⠀⠀⠈⠉⠠⡘⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠃⢁⠀⢀⡘⢳⢠⡄⠀⠈⠉⠉⠉⠘⠁⠀⣹⣆⠀⠐⢄⠐⠀⠀⢙⢷⣄⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠸⡻⢷⡈⠶⣤⢤⡛⢶⠀⠀⣾⣿⣿⣦⣀⣾⠿⠻⣿⠀⠀⠈⢳⡀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⢀⡼⡿⠊⠙⠀⠈⠂⢳⣟⢠⣠⢸⣿⣿⣿⠗⢾⣧⡀⠘⣆⠁⠀⠀⠘⠦⠀⠀⢿⡿⣿⡿⠆⠀⠀⠞
⠀⠀⠀⠀⠀⢻⣋⡟⠀⠀⠀⣢⣤⣤⣜⠀⡁⠈⠀⢉⢀⡄⡉⠛⠿⠆⢈⡳⣀⠀⠀⠀⠀⠀⢸⣷⢹⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣸⣿⣿⣿⣿⣷⠀⠀⢀⣠⣾⣿⣿⣶⣶⣿⣿⣿⣏⢷⡀⢀⢀⡌⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⢯⡻⣿⣿⣿⢿⢿⣧⣾⣶⣾⣿⣿⣿⣿⣿⣛⡛⣿⢿⡿⣧⣹⣦⢣⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠸⠶⠊⠁⠀⢻⡆⠀⠀⠨⡍⠙⣻⣿⣯⣿⣿⣯⣿⣿⢸⡇⠸⠇⢻⢾⣿⠃⡸⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡄⣿⣷⣘⠄⣐⣠⣲⡿⠟⠉⢁⠿⣹⣿⡧⢸⠇⠀⠀⠀⣸⠏⠀⠃⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣵⣿⣿⣿⣷⢫⢟⡟⡷⠂⡀⠃⠀⠻⠛⢀⡿⣠⠀⠀⢸⡏⠀⠀⠀⣠⡾⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠈⡟⠛⠑⣿⢻⡿⣷⣯⢟⡞⣽⠃⠆⠀⠀⣰⣶⡿⠁⢣⣀⣴⣿⠁⠀⢀⣾⠋
⠀⠀⠀⠘⠇⠀⠀⠀⡰⣧⠀⠀⣿⣄⠁⠹⣷⣏⣼⣱⣿⣤⠀⢠⣿⡟⡱⢃⣾⠉⣸⠇⠀⠀⣼⠃
⠀⠀⠀⠀⠐⣤⣴⣿⣿⡿⡄⠀⣿⡈⡄⠀⠙⠛⣻⣟⣿⣿⡄⠷⢟⠔⣵⣿⡏⢀⡟⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠁⣴⣿⣸⡇⠀⠀⢸⠿⣯⣿⣿⣷⠀⡊⠀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡣⠁⠀⠀⢸⣿⣿⡙⣿⣿⡄⠁⠸⠏⠿⢷⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢋⠁⢄⡤⠠⠌⠶⣦⡹⣾⣿⣿⡀⠀⠀⠀⠸⣿⠃⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠠⣶⠠⠌⠀⠰⠀⠐⡦⢷⢿⣿⣿⡦⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⡡⠈⠴⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠄⢒⢒⠎⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⡿⠓⢫⡾⠀⠀⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣺⣾⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣙⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⡀⠈⠲⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠠⢈⠄⡡⠎⡸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⠀⠀⠌⡠⢀⡔⢡⠎
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠤⠤⠒⠒⠈⠉⠁⠉⠀⢀⠄⠀⠀⠀⡀⠀⠀⣠⢋⡰⠁
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠤⠄⠐⠂⠁⠉⠀⠀⠀⣀⠠⠠⠀⡠⣡⠎⣠⠎⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠤⠄⠐⠀⠉⠉⠀⠀⠀⠀⢀⢮⡼⣇⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡤⠤⠤⠐⠒⠀⢀⠀⠀⢠⠏⣠⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠀⠀⠀⠣⣎⡴⣣⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠄⠂⠀⢂⡀⠠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢆⠀⠀⠒⠐⠛⠁⠀⠀⠀⣀⣀⣄⣄⠄⠀⠠⢒⣋⣋⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⣀⢀⡀⠤⠄⢒⣊⣉⣥⠤⠤⠶⠒⠶⠾⠛⢛⣚⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⣮⡶⠶⢻⠩⢍⠍⣁⡬⡥⠴⢶⠒⠾⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣏⣤⡴⠖⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢀⠀⢉⣀⣀⣀⠀⠠⠀⠀⠄⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀

⠠⣀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡶⢀⠀⠀⢀⡄⠀⠀⠀⠐⠀⠀⡆⠀⠀⠀⡄⠀⣀⣀⠀⠀⠂
⠀⠀⠀⠀⢠⠄⡄⠀⠁⠀⠀⠄⠀⡂⠈⠰⠃⠂⠠⠀⡌⠁⠀⠀⠀⢺⡠⠀⠁⠀⡀⠐⠂⠈⠀⠡⠄⠁
⢠⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⢃⠀⢀⡀⣠⡀⠀⠀⢲⣖⣾⠀⠈⣻⠖⠀⠐⠀⠀⢁⠈⢀⠁⠁⠐
⠈⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⠀⢸⠘⠐⠀⠀⠉⠀⠀⣠⠟⠛⠛⠀⣄⠉⠀⠘⠃⠀⠀⣤⢀⡄⣤
⠀⠰⠆⠀⠘⠀⠀⠀⠀⠀⠀⢀⠸⠐⠀⢦⠐⠄⠀⡀⠀⠰⠐⡀⠀⢄⠀⡄⠂⠀⠀⠀⠁⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⡐⠀⢴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⣌⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠰⠂⠀⠀⡦⡆⠀⠀⠀⡰⠀⠀⠅⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠆⣀⢈⠳⡷
⠀⠒⠀⣀⠉⠁⠀⠀⠀⠁⠀⠶⠀⠀⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠃⠠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⢆⠀⠀⢈⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂
⠀⠀⡀⠀⠀⠀⢀⠀⠐⠀⠠⠄⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠀⡀⠄⠀⠃⢘⠀⠀⠀⠀⠐⠒⠙⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠐⠟⠐⠀⢘⠃⠘⠁⠬⠁⠀⢠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠐⠀⠃⠘⠐⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠘⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠖⠒⠞⠉⠓⠊⠓⠶⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣰⠴⠞⠓⠒⠋⠙⠒⠛⢦⣄⣠⣄⠀⠀⠀⠀⠈⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⢠⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⢤⡀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢾⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠓⣄⠀⡴⠶⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠆⠈⠛⠀⠾⠁⣰⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠋⠀⠀⠀⠀⠀⠘⢁⡴⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⡽⠋⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡶⠤⣤⣀⣀⣀⣀⣀⡠⠤⢶⠞⣉⡴⠻⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠊⠑⠒⣧⠤⣄⣀⣀⣀⣀⣀⣀⡤⢼⡗⠁⠀⠀⠈⠳⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠙⠛⠒⠶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡂
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠇⣠⠴⠒⠒⠒⠢⣄⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠑⢦⣟
⠀⠀⢾⣁⣠⠀⠀⠀⠀⠀⠀⠘⡶⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣨⢧⣼⡇⢰⡆⣰⠓⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡷⠢⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿
⠀⣠⡞⠁⠀⠀⠙⠉⠉⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠒⠺⢇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⡸⠋⣿⠋⢙⠟⠉⠛⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢳⡖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠲⠶⠶⠶⠶⠶⠶⡶⠶⢶⠶⠶⢶⣿⣧
⡇⠀⡏⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⠁⠀⡟⠀⠀⣿⣿⣿
⡇⠀⣇⠀⢸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣿⣿⣿
⠸⣄⣹⣄⣀⣇⣀⣀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡀⢘⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠘⡄⠀⢻⠀⠀⢻⣿⣿
⢰⠃⢸⠁⠀⡏⠀⠀⢀⠿⠤⢤⡤⣀⣀⣀⡀⠀⠘⠿⠉⠉⠉⠉⠉⠙⣟⣀⣀⣠⣤⠤⠴⠖⠒⠚⢻⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿
⢸⠀⣿⠀⠀⡇⠀⠀⣟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣇⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⠄⢹⡀⠀⡇⠀⠀⣽⠉⣿⣷⣖⣒⠶⢤⣤⡤⣤⡇⠀⠀⠀⠀⠀⠀⢳⡤⣲⣶⣚⣻⣭⣽⣿⣏⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⡷⢸⡧⠶⣷⠄⢼⡿⡐⠛⠛⠫⣽⣿⣿⣯⡃⢸⣦⣤⣤⣤⣤⣤⣤⣼⡄⢹⣾⡭⠿⠖⠚⠉⠉⡰⣻⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿
⢸⠂⢸⠁⠀⡇⠀⠀⠻⣝⡦⣀⣀⠀⠀⠈⣉⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⢤⣀⣀⣀⣠⣴⡻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠘⣇⢸⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⣀⣉⣈⣛⣚⣛⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠂⠐⠒⠒⠂⠤⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢔⣾⠯⠷⠦⢰⠋⣤⣔⣒⣀⠀⠀⡀⠄⢀⣀⣀⣀⣀⣠⣤⣤⣔⣲⣬⣷⡿⠟⠃
⠀⠀⠀⠀⠀⢸⣼⠂⠀⠀⠀⠀⠀⠀⠤⠄⣉⡉⠉⠙⠓⠛⢛⡞⠛⢙⠩⠩⡫⡀⠀⠐⡨⡇
⠀⠀⠀⠀⠀⠀⡾⠶⢦⢤⣤⣄⣀⣀⣀⠉⠉⠉⠉⠀⠀⠀⢸⡈⢈⠀⣨⣪⣠⡦⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠁⠉⠉⠙⠙⠒⠒⠛⠲⠲⠶⣿⣿⣿⣛⣛⣛⣭⣉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠴⠁⢑⣦⣁⣀⣀⣤⣤⣤⣤⣶⡶⡶⠲⡻⡻⣻⠞⠝⠝⡝⠝⠝⢝⢍⠻⣧
⠀⠀⠀⠀⠀⠀⠈⢔⠄⢫⠙⡯⡉⠁⠑⠑⠁⢔⠕⢔⢌⢔⠩⠪⠊⠂⠪⠂⠢⠢⡢⡀⣩⣽⡇⢀⡀
⠀⠀⠀⠀⢀⠠⠔⠚⢮⡄⠀⡏⡨⣠⣢⣢⣢⣠⣠⣡⣵⣰⣵⣵⣵⣵⣵⣴⢷⣶⣾⣾⣯⣿⣴⡿⠃
⠀⠀⣠⣞⣑⣧⣤⣤⢀⡙⠛⡻⠿⠿⠟⠛⠛⢙⣛⠹⠭⣍⡉⠈⠉⠅⠀⠉⠠⠀⢂⣴⡾⠛⢻
⠀⢰⣿⠉⠁⢨⢉⡉⠉⠒⠒⠓⠾⠭⠶⣶⣒⣒⡠⠤⠤⢀⡀⠀⠀⠀⠀⠀⣠⣴⠻⠏⠂⡄⣬⣶⣄⡀
⠀⠘⢿⣄⣀⡳⣾⠀⠠⠤⠀⣈⠀⠰⠂⡀⠀⠀⠉⠉⠛⠒⠒⠿⠽⣦⣶⢿⡏⠀⡁⣀⣶⣾⡛⠁⠀⠀⢉⣴
⠀⠀⠀⠀⢉⣩⣿⣿⠷⠶⢶⣦⣤⣀⣉⢀⡒⠀⠉⠁⠀⠀⠀⠀⠀⡇⠁⠨⠁⣤⣷⠟⢉⡀⢉⣠⣴⠞⡿⠁
⠀⠀⠀⢠⣻⠛⠛⠛⠯⣔⡂⠠⢀⡈⠉⠛⠛⠛⠲⠲⠶⣤⣄⣀⣀⣷⣠⣶⠟⠛⢂⣠⣶⢾⡿⠫⠁⠁⣧
⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠌⡛⠓⠦⢅⣒⠠⠀⡀⠀⠀⠀⠈⠉⠉⠛⢋⡹⣖⡿⠟⢙⠍⡠⠁⠐⢐⣠⣿⠦⠍
⠀⠀⠀⢀⠻⣦⡀⠀⠀⠄⢀⠀⠑⠢⢀⠈⠉⠓⠤⢭⣐⡀⢐⣤⣴⢾⠻⠫⡁⠎⠀⠀⣂⣱⣴⣾⠟⠉⠁⠀⠭⠁
⠀⠀⠀⠀⠀⠐⠙⠶⣦⣀⠈⠀⠒⢤⣀⠀⠁⠀⠀⠀⠀⡝⠉⠩⡢⡁⡁⠀⢌⣂⣶⣻⡽⠚⠓⠂⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠒⠶⢢⣄⡉⠙⠀⠀⠀⠀⡟⡑⠐⠔⣔⣴⣾⠿⠟⠉⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠪⢤⣒⠤⢹⣤⣾⣾⠿⠋⠉⠁⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠁⢀⣀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠉⠉⠁⠀⣸⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠼⣿⣿⡿⠿⠋⠙⠻⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠰⣤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⢀⣠⣶⣿⡆⠀⣀⣤⣶⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⣴⣿⣿⣿⣿⣿⣿⣅
⠀⣀⣤⣶⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠙⠻⠿⠿⠿⢿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀⣀⡀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢄⠀⡀⠀⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⠿⣉
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠄⠀⠀⠀⠀⠀⢀⣴⣶⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⡠⢟⠟⠡⣄⠜
⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⡀⠀⠀⢀⣀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣀⣀⠈⠿⠿⠿⠟⠁⠀⠀⠀⠀⣀⣤⡔⠁⠀⠂⢀⡷⠁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠰⠀⠀⠀⠀⠉⠉⠙⠛⢿⣿⣁⣠⣄⡀⠀⠀⠀⡿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠇⠀⢀⣴⠟⣡⡄
⠀⠀⠀⠀⠙⠿⠿⡿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠖⠛⢻⠛⠉⠁⠀⠀⠀⠃⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⡦⢴⣯⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠰⠀⠀⠀⠀⠀⠻⢾⠶⣆⣐⡦⠀⢘⠛⠳⠆⠀⠀⠠⠀⠈⠐⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠉⣿⡆⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⣿⡇⢠⣟⡿⠀⠀⠀⠀⣴⠞⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⣿⣧⡟⣼⠃⠀⠀⢀⣾⠋⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣰⡏⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣿⡟⢠⡏⠀⠀⠀⣾⠇⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠃⠘⡧⠞⣷⠀⣿⡴⠋⠀⣀⡤⠀⠀⠀⢀⡴⠆⣾⡟⠁⣠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⠼⡏⠀⢀⣾⠋⣴⡗⠀⢰⡏⢀⡴⣿⣇⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⡄⠀⠀⠈⣿⠀⢀⡴⢺⡇⠀⠻⠀⠸⠻⠞⠹⣇⡴⠋⠓⠋⠀⠉⠉⠈⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡿⠀⠀⣠⡄⢠⡄⡀⢠⡶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠘⠧⠞⠁⠀⣼⣿⢁⡿⣿⠁⣿⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⣀⠀⠃⠈⠀⠀⠀⠀⠉⠀⠀⠀⣠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⣧⠀⣸⡟⠀⠀⠀⠀⠀⠀⡾⣀⣀⣠⣾⢁⡼⠁⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣰⠿⣿⡆⠀⣿⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⢏⡿⠁⠀⢠⡾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣼⠋⢸⣿⠃⢀⡟⢸⣿⠀⢀⣴⣶⠀⠀⣰⡇⠀⢰⣿⡟⠀⠀⢀⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⡿⠀⣼⠃⢸⡇⢠⡾⢿⡏⠀⢠⣿⡃⢀⡼⢿⣄⣀⣠⡴⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⢹⣇⣰⠏⠀⠈⢿⡿⠁⠹⣇⡴⠋⠈⠛⠋⠁⠀⠈⠉⠁⠀⠀⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣶⣶⣆⡀⣠⣄⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡿⣷⣍⠙⠶⣿⣿⣿⣙⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣮⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠓⠶⠖⢶⣯⣝⣻⣟⢿⣿⣿⣿⣿⣿⣿⣿⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⡿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠦⣙⠛⢿⡿⣿⣇⣻⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣮⢉⠲⣿⡜⣿⣿⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠂⠘⢿⡜⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣻⣿⣿⣆⡙⣿⡜⣿⣿⢸⣿⣿⠟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⡻⢿⡜⣿⣿⡟⠋⣼⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⣿⣿⣿⣿⣶⣿⡝⣿⣵⡶⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⢿⣷⡼⣿⡇⠸⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡻⣷⣤⣤⣬⣤⣈⣷⡹⣿⡇⢸⡟⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣛⣛⠛⢻⣿⣷⣹⣿⡿⠁⠀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡿⣿⣿⣿⣿⣿⠿⣿⣧⢹⣿⡗⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣶⣾⠟⠛⢶⣿⣧⢻⣷⣿⠏⠼⢿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣯⣅⣀⠻⢿⣧⢿⣯⡤⣶⣿⣿⡿⠃
⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⣠⡎⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣯⣙⢿⣯⣿⣷⠺⣿⡿⠛
⠀⠀⠀⣰⣿⣿⣷⣆⣄⢀⣠⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⣙⣿⣿⣾⡻⠁
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣟⣴⣦⣤⡀⠀⣸⡿⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠉⠙⢻⣏⣻⣟⣿⡅
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⡟⠁⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡙⣿⠃
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⢰⣿⣿⣿⣿⠟⡛⢻⣿⣿⣿⣿⣿⣿⣾⣿⢇⢀⡟⠀⠀⠴⣄⡀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀
⠘⠿⢿⣿⣏⠀⢒⣠⡟⠡⣀⠙⣿⣿⠿⠉⢞⣼⠷⠀⠀⠀⠈⠙⠷⠆⠀⠀⠀⠪⡵⠐⠋⠀⠀⢀⡙⠦⢄⡀⠀⠀⠀⠁⡀
⠀⠀⠀⠠⠬⣙⣛⡻⠷⣤⣤⣼⣿⣿⣶⠞⠉⠀⠹⠓⢤⢀⠀⠀⠀⠀⠀⠀⠀⡴⠤⠄⠀⣀⠀⠀⠘⠣⣶⡀⠀⠀⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⡝⣀⠃⠀⠀⠀⠀⠀⠀⠈⠑⠐⠢⠄⠀⠀⠈⠇⠀⠀⢀⡙⠁⠀⠀⠀⠈⠉⠶⣄⡀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠰⠀⠀⢀⣴⠀⠀⡸⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⢰⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣬⣧⣴⣧⣴⣦⣴⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢰⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠁⠀⠀⠀⠰⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⡀⣀⣴⣄⠀⠀⠀⠀⡀⠀⢀⣀⣀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⣠⣤⡄⠀⠐⠀⠀⠀⣡⡾⣫⡟⠉⢉⠀⠈⠁⢁⠒⠀⡉⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠴⠆⠈⠀⠀⡇⠀⠀⠠⣤⡼⠛⠛⠓⠂⠀⠀⡴⠪⠁⠨⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⣴⠁
⠀⠀⠀⠀⠀⢈⠀⠶⠀⢠⣶⣤⠀⠤⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⣥⡄⠀⡛⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⡠⠃⠀⠀⠦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠘⠀⠐⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠁⠁⠉⠀⠀⠀⠀⡄⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⠸⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠃⠀⠀⠇⠀⡸⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠄⢠⠖⠀⣤⠆⠀⣀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠂⢀⣁⢀⣁⠀⠸⠁⠁⠤⡬
⠀⢠⡄⠀⠀⠄⠠⠀⣴⠆⣾⠀⠀⣠⠀⢰⡄⣄⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⣿⣿⣻⣻⣶⠏⣹⡟⣹⣇⣠⣤⣴⡶⣦⣤⡀
⠀⢸⣷⠀⢄⡴⣠⠀⣿⢀⡟⠰⢰⡏⠒⣸⡇⣿⢸⡄⣼⢃⡷⣤⢠⡴⢠⡆⣤⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⣻⠏⡭⢻⡟⠀⣿⢀⡿⣻⠃⣴⣻⠇⣸⠈⣿⢷
⠀⠸⣿⠀⢸⡇⢹⡀⣿⢸⡷⣆⢸⡇⠀⣿⢰⡏⣾⢀⡿⢸⢇⡟⣾⠇⣼⢃⡿⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠀⡇⢸⡇⢸⡇⠘⠻⡟⠀⣿⣿⠀⡿⢠⡇⢸
⠀⠀⠻⠀⠘⠷⠸⠇⠻⠘⠇⠿⠸⠇⠠⠏⠸⠁⠛⠼⠃⠟⠘⠃⠿⠀⠋⠘⠃⠀⠀⠀⠀⠀⠀⠘⣧⣸⡟⢻⣤⡷⢬⣧⣸⣇⢸⣀⣷⣘⣿⣏⣼⣇⣼⣧⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠦⠶⠶⡓⣢⠀⠀⠀⠠⣔⣛⠶⡶⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇⢸⡇⣹⠉⡟⢉⡿⠀⡿⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠛⠙⠃⠸⢿⡿⠂⠉⠉⠛⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣼⠀⡟⢰⠁⣰⡇⢠⡇⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣴⣯⡾⣧⣼⣤⣽⠷⠾⠿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢶⣦⣤⣄⡤⣤⢶⣞⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠧⠘⠳⠌⠛⠂⠉

⠀⠀⠀⢠⣶⡆⠀⠀⠀⠄⢤⠀⢀⣴⠆⢠⡶⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⣿⣿⠁⠠⢄⣴⠞⣦⠀⣸⡿⠀⣾⠃⡠⢄⣾⠏⡤⢰⣿⠁⣿⠄⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡦⠀⣼⢠⠇⢀⡤⢠⡆⣄⡀⢠⢠⡀⠀⣀⢀⢀
⠀⠀⠀⣿⣿⠀⠀⢸⣿⠀⣿⠀⣿⠇⢰⣿⣬⡀⢸⣿⠀⠀⣾⡟⢰⡿⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢸⡇⡏⣼⣄⢸⠃⢸⢧⣇⡇⣾⢸⢹⢻⡇⣿⢸
⠀⠀⠀⠸⣯⠀⠀⠘⢿⠔⠿⠄⢿⡄⢸⡏⢹⡇⠸⣏⠀⢀⡿⠃⡼⠃⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠁⠉⠈⠈⠀⠉⠈⠀⠁⠁⠁⠈⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠐⣶⠀⣤⠁⣤⡤⢠⣶⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⢄⡆⢀⡄⢠⣄⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢰⡿⢰⡿⣸⣿⠀⢸⡿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣾⣷⢺⢸⣧⢸⠃⣼⢹⢹⢣⣇⣯⣦⡞⣸⢲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⣼⠃⢾⡇⢿⡇⠀⠻⠃⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠙⠘⠈⠈⠈⠈⠁⠁⠀⠈⠉⠈⠈⠈⠁⠁⠚


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠐⠂⠈⠉⠉⠁⠐⠒⠒⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⢀⡤⠴⣒⣀⡉⠁⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠓⠒⠲⠦⠤⣤⣀⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠈⠉⠑⠒⠒⠦⠤⢤⣀⣐⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⢙⡏
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⣤⣀⣀⣀⣀⠤⠴⠚⠋⠉⠀⠀⠀⠀⠸⡄
⠀⠀⠀⣄⣭⠷⠶⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⡀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⢉⣒⡢⠤⡄
⠀⠀⠀⢸⡏⠙⠲⠤⣄⣀⠀⠀⠉⠉⠓⠲⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠉⠙⣷⣦⣤⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⢹
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠉⠙⠒⠦⣤⣀⡀⠀⠀⣀⣀⣉⣭⠽⠷⠶⠶⢤⣼⣤⡶⠟⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⢀⣴⣶⣏⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⢒⣒⣫⣭⣽⣿⠟
⣾⠇⠀⠉⠙⠛⠷⢮⣿⣿⣶⣤⣀⡀⠀⠀⡾⠐⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠶⠒⣚⣋⣩⡥⠤⠶⠖⠛⠛⠉⠉⠁⠀⠀⡏
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣥⣧⡴⠶⢒⣛⣛⣭⣭⣴⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠘⠿⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠛
⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⣀⣧⠀⠀⣀⣀⣠⣤⣴⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠟⠉⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⢻⡟⠉⠁⣤⡀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣀⠀⠀⢀⡀⠀⠀⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠋⠀⠀⠘⠃⠀⠀⠀⠈⢻⣿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣃⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠠⠉⠉⠁⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣀⣀⣀⣀⣀⡀⠠⠶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡏⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠓⠂⠒⠒⠂⠉⠀⠉⠁⠒⠂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠑⠆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢱
⣤⠤⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⣠⠂⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠇⠀⠈⠒⠂⠤⢤⠜⢱⡆⠀⠀⠀⠀⢰⠒⠢⣄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⢣⡀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠘⣤⠀⠈⠉⠀⠒⠚⠁⠾⠋⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠈⠁⠰⠶⠄⠀⢀⡀⠀⠀⠉⠁⠶⠸⣦⣄⠀⠀⠀⠀⠀⠀⠱⠾⢦⣀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠰⡆⠀⠀⠀⠀⠀⠀⣠⠊⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⢄⣀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠤⢀⡀⠀⠀⠙⠃⠤⠄⣀⡀⢀⡴⠋⠀⠀⠀⠘⣀⠀⠀⠀⣠⠞⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⠀⠀⠀⠀⠀⣠⠜⠃⠒⠤⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠔⠉⠲⠄⠚⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠀⠒⠁⠀⠀⠀⠀⠸⣿⠒⢠⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠉⠑⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠐⠒⠓⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠶⠀⠀⠀⠀⠀⠈⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠄⢛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⢀⣀⠀⠀⠀⠀⠐⠶⠦⠤⠤⣤⣤⠤⢤⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠀⠀⠀⠀⠀⠀⠰⡦⢤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⢤⠤⠤⢤⠤⠤⠤⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢠⡧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣾⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠆⣀⣀⣀⡀⠶⠞⠁⠀⠘⣦⣀⣀⡀⠀⠀⢀⣠⠛⢀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⢀⠆⠀⣠⠞⠁⠀⠀⠀⠉⠁⠈⠁⠀⢠⣄⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠧⠀⠃
⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⣀⠔⠛⠁⡔⠋⠀⠀⠀⠀⠀⠀⣠⠄⠘⠃
⠀⠀⠀⠀⠀⠓⠤⠤⠤⠤⠄⠒⠁⠀⠀⠀⠧⠀⠀⠀⠀⠀⠤⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠈⠋⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⣤⡶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠟⠛⠛⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢀⣾⠋⠀⠀⠀⠈⣹⣿⣆⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⢠⡿⠉⠻⣷⣄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⠇⠀⠀⠀⠀⣼⡇⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣿⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠘⢿⣦⡀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⠁⣴⠏⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡉⠙⠛⠛⠛⠛⠉⠁⣀⣤⠾⠃⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣶⣶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣤⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠃⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣶⣤⣤⣤⣶⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⡛⡛⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣴⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠿⠿⠻⣿⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠻⠻⠿⠡⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⠀⠀⡴⠿⠟⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠇⡀⠀⠘⡏⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠⠀⠀⠀⠠⠃⠃⠀⢰⠇⢀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⡀⣰⣦⠀⠀⢀⡏⠀⣼⠀⠀⢀⠤⠨⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⢿⠇⣽⠋⠀⠀⠈⠀⠀⡗⢀⡴⠁⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⢘⣼⣿⠀⠠⠀⢠⠖⠚⢱⡟⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⣿⡇⠈⠀⠀⡈⠃⠀⣸⣇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣠⡄⣼⣿⠃⠀⡤⠊⠀⡀⠸⢿⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣙⣟⣿⣹⡇⢠⣞⡁⠀⢷⣸⣄⣰⣷⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⢀⠀⣴⣿⣿⣿⣿⠁⠈⠀⠈⢻⣶⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠂⠀⠈⠻⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠁⠈⠙⠛⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠀⠀⠀⠀⠀⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⠳⠦⠀⠀⠀⢰⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣰⠄⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣄⣀⡀⠀⠀⢆⠳⠤⠀⠀⠀⡼⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣬⣑⠂⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢀⣤⣤⣄⣀⣀⣦
⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣾⠇⠀⢀⣀⠈⣁⢈⣍⠉⢻⡿⠛⠁
⠀⢀⣴⡿⠋⠁⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⣤⢸⣟⡴⣲⣤⣿⠀⣴⡏⣸⣿⠃⢸⣏⢀⣾⠃⢲⡄
⠀⣼⡿⢁⡀⣾⢃⣿⠇⠀⣶⡴⣶⢀⣾⢉⠆⣼⠇⣼⣿⠃⣼⣿⣴⠛⢸⣿⢧⢿⣇⡼⣿⡴⢁⣿⡏⣿⣀⡾⠁
⠀⠹⣇⣰⡟⣿⠘⠻⣷⣄⣿⡇⣿⣸⣿⠋⡰⢿⣠⠞⠿⠴⢿⡝⣿⡄⠀⣿⡼⢲⡆⠀⠀⢰⢃⣿⠃
⠀⠀⢠⣾⠛⣿⢻⡆⢹⣿⢿⡇⠀⠀⠙⠛⢁⣠⣤⣶⡶⠾⢶⣆⠈⠻⠦⣤⣤⠾⠃⠀⠀⠈⠛⠁
⠀⠀⢸⣇⠠⠟⠚⢁⣿⡟⢰⣭⣠⣤⡶⠿⠛⠉⠁⠀⠀⠀⠚⠃
⠀⠀⠀⠛⠶⠤⠶⠟⠋⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠈⠉⠉⠀⠀⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠠⠲⠠⠒⡂⠀⠄⡀⠀⠀⠉⠢⣄⡄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠠⠄⠐⠂⠠⠀⡀⠀⠀⠀⠑⠀⠀⡐⠋⠀⢀⣠⢤⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣀⠀⠀⠂⠈⠴⣂⠥⣂⠍⡢⠄⠀⠀⠜⠀⠀⠘⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡼⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠁⠀⠂⠤⢐⡂⠄⡀⠀⠁⠀⠀⢀⠎⠀⠀⠈⠀⠈⠉⠈⠁⠠⠄⠀⠈⠁⡐⠂⠀⠀⠀⠀⡜⣡⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⡀⢀⣀⠀⠤⢈⡑⠂⠅⠂⠤⡑⠪⡑⠀⠀⢀⠊⠀⣀⠨⠀⠀⠀⠒⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡜⣰⢿⣿⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠄⢐⡂⠤⢁⠁⠂⠍⡒⠤⡀⠈⠑⠀⠀⠠⠁⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⢀⢞⣼⣯⣿⡏
⠀⠀⠀⠀⠠⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⡂⠀⠈⠀⠈⠂⠀⠀⡐⠁⠀⠀⢁⡀⠠⠤⠄⠉⠉⠁⠰⠆⢀⡂⠀⠀⠀⠀⢠⢮⢮⢯⣾⣿⢫⡇
⠀⠀⠀⠀⠠⣮⣌⣉⣉⠒⠢⠤⣀⠀⠤⠀⠀⠀⠈⠑⠄⠀⠀⠀⢀⠜⠀⢀⡄⠰⠀⠤⠤⠤⠤⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⡰⢣⢣⣣⣿⡿⣣⡻⠁
⠀⠀⠀⠀⢴⢥⡤⠤⢭⢭⣍⣒⡢⠝⡒⣤⡀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡱⢳⣿⡿⣱⡗⠁
⠀⠀⠀⠀⢹⠥⡆⢀⡀⠀⠉⠁⠛⠿⠷⢿⣾⣷⣤⡀⢀⠀⡠⠗⠀⠀⣀⣀⣤⣤⡤⡤⡤⠤⣤⣤⣀⣀⠀⠀⠀⠀⠀⡰⡱⡽⣱⣿⡟⣴⠟
⠀⠀⠀⠀⠀⢱⢘⣕⠯⠕⡖⠚⠓⡒⠷⣶⣟⢛⣻⣿⡷⣴⣶⢶⡿⢟⣳⣾⣭⣭⣥⣤⡴⠍⡠⡔⠂⠌⣉⠓⠶⣬⡴⣼⡝⣵⢿⠏⣾⠿
⠀⠀⠀⠀⠀⢯⣙⣨⣭⢤⡴⣶⣶⣮⣷⣾⡿⣿⣛⣻⣛⣛⡛⡻⣿⣿⣟⢛⣿⡿⡭⣭⡴⣒⣠⠀⡉⠛⠓⢦⠴⣠⢜⣟⣾⣿⢋⣾⡟⠄
⠀⠀⠀⠀⠀⡎⠟⢹⣧⣼⣏⠁⣑⡁⠀⢉⠛⠒⠒⠲⠲⠭⢭⠭⠿⠿⠻⠿⢿⠟⣫⣭⣍⣑⣨⣍⣛⣲⣤⣄⣈⣦⣭⣿⣼⣡⣿⣓⣀⣅⣂⣬⣐⡠⢀⡀
⠀⠀⠀⠀⠀⣤⠱⠂⢻⡗⡇⢒⣚⡿⠯⠭⠽⠿⠿⠟⠛⠋⠉⠉⠙⠓⠒⢲⣖⠿⣿⣿⢟⡵⢿⣃⣠⣌⣁⠨⣍⢻⣭⣿⣷⣿⣿⣿⡽⣛⣯⣽⣷⣾⣿⠇
⠀⠀⠀⠀⠀⢿⠀⡟⠸⣏⢏⣥⡤⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣸⣾⡟⣡⣟⠀⢽⡤⣤⡥⣼⣵⠤⣅⣬⣤⣭⣭⣤⣼⠿⠿⣿⢿⠿⠏
⠀⠀⠀⠀⠀⠹⡄⠋⠀⢺⣿⠻⠛⠿⠿⠶⡶⠾⢶⢴⣤⣀⣀⡤⢤⣒⡃⣽⡿⣣⠾⠷⡾⢀⠡⠂⢁⠩⠁⡀⠌⠀⣊⠍⠑⡀⠅⡀⠠⢁⠐
⠀⠀⠀⠀⠀⠀⣣⡀⠀⢀⠽⡬⢴⣖⣂⡤⠾⠾⠛⠻⠿⠋⠉⠉⠛⠛⡱⠶⠚⠉⠉⣼⢁⠀⠀⢄⠂⢉⡤⠶⠖⢃⡠⢤⣂⣤⣂⠜⠊
⠀⠀⠀⠀⠀⠀⠘⣯⢲⢪⢿⠉⠁⠤⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⢹⣧⠆⠁⠐⢊⠰⣒⠧⠖⠋⢩⢁⡠⠍
⠀⠀⠀⠀⠀⠀⠀⠙⣭⢿⣽⠀⢒⣒⡊⠉⠀⠀⠀⠀⢀⣀⠀⠀⢀⢀⣀⡀⢒⠐⠉⠡⣟⠷⠀⢊⡉⠠⡐⣪⡤⠆⡅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣛⡼⠶⠶⣒⢒⢂⣛⣙⣉⠭⠬⠽⠫⠖⠔⠃⢘⠗⡫⢑⠩⠑⡂⠌⢀⠌⢁⠭⢀⠠⢁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⢸⡆⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⠠⠊⢡
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠯⣕⣦⡀⢀⡄⣊⣀⣀⡀⡠⠂⠁⠀⢀⡾⠃
⠀⠀⢸⡷⠦⠤⢤⣀⠀⠀⠃⢀⣠⣤⣴⣒⡈⣿⠀⠀⠀⠀⠀⡩⠛⠿⠿⣟⡒⠤⣁⠀⠀⢀⠔⠉
⠀⠀⠺⠿⠿⠷⠶⠶⠿⠷⡾⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠓⠾⡦⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⡓⢤⡀⠀⠀⠀⠀⠀⢑⠥⠆
⠀⠀⠀⣶⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⣞⡂⠀⠙⠲⣌⡒⣤⡦⠐⣈⠤⠒⡀⠒⠂⠀⠠⠤
⠀⠀⢀⠱⡈⠑⢄⠀⠀⠙⠢⣄⠀⠀⠀⣴⣿⠽⣮⡄⠀⠀⠈⠙⠷⠛⡍⠀⣠⠞⠁⠀⠀⠀⠀⢀⡰⡯
⠀⠀⠙⢇⠈⠢⡀⠀⠤⠄⠀⢈⣻⣆⣼⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⡠⢊⠞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⠰⣾⣻⠿⠟⠛⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠦⢉⣁⠒⠂⠤⢎⠔⠁
⠀⠀⠀⠀⠀⠈⠳⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠒⠲⣄⠉⠐⠂⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣄⠈⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡆⠀⠠⠐⠂⠀⠀⠀⠀⠐⠀⠀⠐⠁⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⡙⣶⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣴⠏⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡾⠃⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠢⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢲⣤⣤⡄⠀⠐⠀⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⢴⠞⠁⣿⡅⠀⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⡶⠶⣿⡀⠀⠸⣿⣶⣤⣤⣤⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢹⣧⠀⠀⠘⣻⠿⠟⠛⠛⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢳⡀⠀⢿⣿⣶⣾⣥⣤⣴⡶⠾⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠴⠟⠻⠛⠋⠉⠁⠀⠀⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿⣦⡀⠀⣾⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣏⣈⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠳⠶⠶⠶⠶⠞⠛⠋⠉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣤⣤⠾⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⣉⡉⠉⠉⠁⢀⣠⣾⠿⣆
⠀⠀⠀⣴⣶⣶⣶⣤⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⡉⠙⠋⠉⢉⣁⣤⣤⣤⣾⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⢀⣤⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁