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⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣦⠀⠀⢠⣾⣿⣿⡿⢿⡇⠀⣿⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⣸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⠿⣿⡆⠀⠀⢸⣿⣷⣾⣿⣿⠀⠀⣿⡿⠁⠀⠀⠈⠀⠀⣿⣿⠀⠈⢻⣿⡆⠀⢸⣿⣦⣶⣤⠀⠀⠀⢿⣿⣤⣤⣤
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡄⠀⢸⣿⣛⣛⣹⣿⠀⠀⢹⣿⣤⣀⣀⣤⡄⠀⣿⣿⣀⣠⣾⡿⠀⠀⢸⣿⠛⠛⠃⠀⠀⠀⢸⣿⠛⠛⠃
⠀⠀⠀⠀⢻⡿⠉⠉⠉⢻⡗⠀⠸⢿⣿⣿⣿⠟⠀⠀⠀⠹⢿⣿⣿⠿⠃⠀⣿⣿⣿⡿⠟⠀⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⢼⣿


⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡄⠀⠀⣶⣦⠀⠀⠀⣴⣦⠀⠀⣶⡆⠀⠀⢰⣶⠀⣠⣶⡆⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⣼⣿⠟⢉⣉⣛⠁⠀⠀⣿⣿⣤⣄⣤⣿⣿⠀⠀⣿⡇⠀⠀⢸⣿⣾⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⣼⣿⣿
⠀⠀⠀⠀⣿⣿⡀⢻⣿⣿⡇⠀⠀⣿⣿⠿⠿⠿⣿⣿⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣿⣿⣿
⠀⠀⠀⠀⠘⢿⣿⣶⣾⣿⠇⠀⠀⣿⡿⠀⠀⠀⣿⡿⠀⠀⣿⡇⠀⠀⢸⡿⠈⢻⣿⡆⠀⠀⢸⣿⣿⣿⣿⡦⠀⠀⣹⡿⠘⢿⣿⠃⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠈⠛⠙⠉⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⣤⣄⠀⠀⢀⣤⡄⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⢠⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⣿⣿⣷⡀⢸⣿⡇⠀⠀⢠⣿⡿⠛⠛⠿⣿⣧⠀⠀⠀⢸⣿⠿⠿⣿⣧⠀⠀⠀⢠⣿⡿⠟⠻⢿⣿⣧⠀⠀⠀⢸⣿⡿⢿⣿⡇
⠀⠀⠀⠀⣿⡟⢿⣿⣾⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣹⣿⠀⠀⠀⢸⣿⣷⣶⣿⡇⠀⠀⠀⢸⣿⠁⠀⢠⣄⣹⣿⠀⠀⠀⢸⣿⣷⣾⣿⠃
⠀⠀⠀⠀⣿⡿⠀⠹⣿⣿⡇⠀⠀⠘⣿⣷⣦⣤⣶⣿⠏⠀⠀⠀⢸⣿⠟⠛⠋⠀⠀⠀⠀⠸⣿⣷⣦⣾⣿⣿⡏⠀⠀⠀⢸⣿⠟⢿⣿⣆
⠀⠀⠀⠀⠛⠃⠀⠀⠈⠋⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠃⠀⠀⠀⠸⠿⠀⠈⠻⠏

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡿⠀⢸⣿⡆⠀⠀⣼⣿⠁⠀⣿⣷⡄⣠⣿⡟⠀⢸⣿⣆⠀⢀⣾⡿⠀⢸⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠸⣿⣷⣶⣿⡅⠀⠀⠀⠀⣿⡏⠀⠀⠀⠈⢿⣿⡄⣰⣿⠇⠀⠀⠘⢿⣿⣿⡟⠀⠀⠈⢻⣿⣧⣾⡿⠃⠀⠀⠉⢉⣽⣿⠟
⠀⠀⠀⠀⣴⣮⣉⣛⣿⡿⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⣰⣿⠿⣿⣷⡄⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠙⢿⣿⡿⠿⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⢿⠏⠀⠀⠀⠸⣿⠏⠀⠘⢿⡟⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⣀⠉⠙⠳⠛⠙⣷⣤⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢷⡀⢰⣏⣉⣉⡿⢀⡶⢾⡿⢁⣾⣉⣀⡀⢀⣀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⡽⠟⣛⣩⡍⠁⣨⣝⡛⠀⣼⠋⣉⠉⡟⢉⠉⢻⠋⡉⠙⡟⢹⠏⢻
⠀⠀⠀⠀⠀⠀⢰⡏⠀⣼⣋⡿⠁⣰⠇⣼⠁⢠⣏⠀⣃⠐⠁⠚⢀⡌⠐⠃⣠⡁⠘⠀⡿
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⠉⠀⣰⠏⠀⣿⠀⢸⣏⠛⢹⡏⢰⡟⣿⠁⣾⠛⣩⢿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠖⠛⠁⠀⠀⠙⠶⠞⠋⠀⠸⣇⣼⠇⢿⣠⡿⠀⢿⣄⣠⠟




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

⠀⠀⠀⠀⣤⠀⠀⡄⢠⠀⣠⠀⠠⠀⡄⢠⡀⢠⠀⠤⠀⡴⢠⠀⡤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠
⠀⠀⠀⡐⠘⣆⠀⣇⣹⠄⢣⡀⡀⢸⣇⢰⠇⢸⠛⣠⠀⡟⠃⠘⠆⢀⡔⠀⠀⣇⡄⢸⠀⠆⢠⣈⠰⠀⡇⠰⣘⠀⡇⢘⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠁
⠀⠀⠀⢰⡂⢰⠀⢰⠂⠀⡔⠀⣄⡠⠂⠀⡄⠀⠀⠴⡄⠰⡖⠀⠦⡀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⠀⢀⣀⢀⠀⠀⢀
⠀⠀⠀⠸⡁⢸⡀⢸⡀⠄⡇⠀⣏⠑⣄⠀⠅⢠⠀⡄⠙⠀⣇⠀⠀⠹⡇⠀⠀⠀⠁⠀⠀⣳⠈⡛⣄⠨⠄⢈⠸⡀⡇⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⡔⠀⢢⡀⠰⢀⠀⡀⠀⢠⠀⢰⢀⡆⠠⣀⠆⠀⢰⠐⠀⡆⠀⠆⠀⠀⠀⡀⢀⣀⡀⢀⢀⠀⢀⣀⠀⡀⠠⡀
⠀⠀⠀⠱⠀⠘⠁⠸⠀⠀⠡⢀⠸⠀⠠⠈⠄⠠⡀⠇⠀⠸⠀⠀⠣⠠⠀⠀⠀⠀⠆⢸⠠⠇⠠⠰⠀⠨⠀⢌⠆⠸⠄⠠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠐⡀⠐⠀⠰⡀⣾⣀⠃⠀⢦⠀⠀⠲⣀⠂⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠀⠀⣠⠀⢀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⠀⠏⠀⠤⠀⠧⠄⠀⠧⠀⠠⠥⠤⠀⠀⠀⠀⠀⠀⠀⠟⠀⠘⠁⠏⠀⠌⠄⠈⡇⠀⠮⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠀⢰⠀⠀⠀⡄⠀⠀⢀⠀⠀⢀⠰⠀⠀⡀⠀⠀⠀⢠⠀⠀⠄⡀⠀⢀⠠⡀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⠸⠀⠀⢨⡀⠀⢠⣎⡀⠀⡀⠓⠀⠀⢼⠆⠀⡀⠹⠀⠘⣅⡇⠀⠀⡆⠀⠀⡈⡳⠀⢈⡴⠃⠀⠇⠸

⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠝⠀⠸⠀⠀⣦⣀⠀⠀⠀⠀⢶⢃⠄⢀⠊⡠⡄⢂⠀⠻⣝⠀⠰⠇⡐⢁⠀⠀⠾⠂⠈⡆
⠀⠀⠀⠀⢹⡀⠰⠀⢰⠀⠐⠂⠀⠐⠀⠀⠃⠀⠈⠉⠀⠳⠞⠿⠂⠘⢌⠓⠗⡢⠀⠶⠟⠀⠀⠎⠀⠳⠃⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⣠⢲⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡤⣶⢦⣄⣼⠟⣴⢲⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⡞⣿⡄
⠀⠀⠀⠀⢸⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⢠⣯⣴⡏⠀⠹⡆⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢠⣮⡆⠀⢀⡟⣾⣿⣿⡇⠀⠀⣿⠀⠈⣿⣿⣿⠀⠀⢰⣽⡆⠀⢸⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⣼⣿⣇⠀⣸⠁⢿⣿⣿⡇⠀⠀⡿⠀⠀⢹⣿⣿⡀⠀⣾⣿⡇⠀⡾⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⡴⠉⢿⣿⣤⠏⠀⠘⢿⣿⣧⣀⡼⠃⠀⠀⠀⢻⣿⣷⡴⠘⣿⣷⣴⠃⠀⠀⣙⡉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢀⣤⣄⣿⠀⠀⠀⢀⣤⣤⠀⠰⠶⣿⡷⠶⠆⠀⣠⣤⣄⣠⡄
⠀⠀⠀⠀⢠⡏⠀⣸⡏⠀⠀⢰⣷⠞⠻⣦⠀⠀⠀⣾⠁⠀⠁⠀⢰⡏⠀⢘⣿⠀⠀⠀⣿⣁⡿⠀⠀⠀⣿⡇⠀⠀⢰⡏⠀⢸⡏
⠀⠀⠀⠀⠈⠷⠶⠻⠇⠀⠀⠈⠻⣦⣤⠟⠀⠀⠀⠻⣦⣀⡄⠀⠈⢷⣤⠞⣿⠀⠀⠀⢿⣭⡴⠀⠀⠀⢹⡇⠀⠀⠈⠿⣶⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡀⠀⠀⣼⡇
⠀⠀⠀⠀⠈⣿⣀⣀⠀⠀⠀⠀⢈⡉⠀⠀⠀⠰⡷⠀⠀⠀⣿⣠⡾⠁⠀⠀⣿⠀⠀⠀⠀⢰⣶⠿⣷⣠⢶⣤⠀⠀⠀⢸⡶⠶⣦
⠀⠀⠀⠀⠀⣿⠋⠙⣧⠀⠀⠀⢸⡇⠀⠀⠀⢰⣦⠀⠀⠀⣿⢯⡀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⢹⡆⠀⠀⢸⡇⠀⣸
⠀⠀⠀⠀⠀⠿⠆⠀⠿⠀⠀⠀⠸⠷⠄⠀⠀⢸⡟⠀⠀⠐⠟⠈⠿⠀⠀⠀⠙⠳⠛⠀⠀⠀⠛⠀⠈⠋⠀⠘⠁⠀⠀⠸⠇⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣴⠞⢷⡄⠀⠀⠀⣶⠶⣤⠀⠀⢀⡴⠛⢶⡄⠀⠀⢰⣧⣤⣤⠀⠀⠀⣾⠛⠶⠀⠀⠠⣼⣷⡤⠀⠀⢰⡆⠀⣶
⠀⠀⠀⠀⢸⣇⠀⢨⡿⠀⠀⢰⣿⠀⣸⠇⠀⢸⣇⠀⣼⡇⠀⠀⢸⡟⠀⠈⠁⠀⠀⠙⠷⣤⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢀⡏
⠀⠀⠀⠀⠀⠙⠷⠟⠁⠀⠀⢸⣿⠛⠁⠀⠀⠀⠙⠛⢻⡇⠀⠀⠘⠏⠀⠀⠀⠀⠀⠳⠶⠟⠀⠀⠀⠈⠻⠦⠀⠀⠙⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⣿⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡀⠀⡀⠀⠀⠀⢀⡀⠀⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⡄⠀⣾⠀⠀⠀⣿⠀⢀⣤⠀⢸⡇⠀⠀⠀⠈⢷⡾⠁⠀⠀⠀⡟⠀⢸⡇⠀⠀⠀⢀⡴⠟
⠀⠀⠀⠀⠀⠘⣧⣠⠟⠀⠀⠀⠸⣦⠞⠹⣦⡾⠁⠀⠀⢀⣴⠟⢷⡄⠀⠀⠀⠳⠶⣾⠇⠀⠀⢴⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡿

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

⢝⢦⡀⠀⢹⣦⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢠⡟⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⠋⠀⣰⠏⠀⠀⡴⠋
⠀⠀⠙⢦⡀⢿⣷⡀⢠⡄⠀⣾⠀⠀⠀⠀⣼⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡼⠁⠀⡠⠊⠀⣠⡾
⠀⠀⢦⣤⣙⠺⣿⣿⣾⣿⣼⣿⣀⣤⣴⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⣠⠞⠁⠠
⠄⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡀⠔
⢤⡶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀
⠶⠾⠷⢶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⠟⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡭⠝
⠠⠁⡵⠟⠋⠁⠸⡿⠹⣿⡏⠈⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣀⡤
⠉⠓⠲⠠⠤⠀⠀⠁⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈
⠑⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⣐⡐⡐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀
⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠴⣴⣴⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠩⢉
⢒⣋⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢼⣿⣾⢀⣰⣶⡄⣠⠀⢸⣿⢰⣾⣿⣇⣿⣿⣶⢀⣰⣶⠀⣠⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⢿⣿⡿⡿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⣶
⠴⠂⣠⠔⠁⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠢⡀⠀⠀⠬
⣤⠞⠁⠀⡀⠀⠀⣠⠋⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠱⣄⠀⠙⢦⡀
⠁⠀⡠⠊⠀⢀⡾⠃⢸⡟⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⢱⡀⠀⠄⠀⠈⢷⡀⠀⡙⢦

⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣀⢀⣀⠀⣀
⠀⢠⢀⠐⠁⠂⠀⣀⡀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡂⢲⢸⡇⢒⣰⣦⣦⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⠉⠉⢡⡾⠇⣴⠟
⠀⢰⢮⡑⡻⠓⠈⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⡈⠀⠀⠀⠀⢫⣹⣹⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶
⠀⠈⠳⣴⠁⣠⡆⠠⢀⠀⠠⠀⢀⡀⠤⡄⠀⠰⠄⠀⠀⠀⠆⢰⠿⣾⠋⠙⠛⠛⠛⠉⢻⡇⠀⠀⠀⠀⠀⠀⡘⠻⢖⡑⠂⠠⠲⢤⣀⠀⠀⠀⠀⠠
⠀⠀⠀⢣⣀⣀⣠⡴⠆⠒⠒⠒⠚⠠⠐⠇⢰⠀⣰⠀⠀⠀⠁⠈⠉⡇⠀⡀⠀⠀⠀⣀⠀⡇⠀⠀⠀⣀⡀⣀⣙⡛⠺⡏⠀⠀⠀⠀⠀⠉⠑⠢⢄⡠
⠀⠀⠀⢘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠟⢀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁⠀⠃⠀⠰⠃⠀⠀⠈⠉⠉⠙⠻⣟⣶⣄⣀⠀⠀⠀⠀⡀⠛
⠀⠀⠀⠈⠉⠉⢛⡷⣤⢠⣄⡀⣀⠀⠀⣴⠂⠾⡀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠾⡟⢳⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠙⠘⠑⠋⣄⠴⠱⠌⠒⣤⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡜⡄⡿⣆⣐⠦
⠀⠀⠀⢀⡀⠀⢀⠠⡤⠀⠀⠀⠈⠁⢰⢶⣶⣤⣄⢀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠒⠲⢦⡀⠀⣀⣀⣤⠂⡾⢠⠃⡇⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠈⡀⠀⢀⡄⢀⣿⣶⣿⡿⠻⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠿⠃⠾⠃⠘
⠀⠀⣆⠀⡀⠀⠀⠀⠀⠉⠉⣾⠁⠘⢻⣿⡟⢹⠂⣘⠁⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠤⠀⣀⠀⢀
⠀⠀⢸⡆⡀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠘⡟⠓⠓⡈⢁⣨⡾⢇⢈⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠈⠏⠙⢷⣂⣠⠴⢥⣾⡁⠀⠈⠉⠃⠀⠀⠉⠛⣉⣀⣠⣿⢧⣴⣼⠀⢀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣳⠀⣆⢀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣽⣷⣾⡿⠧⠄⢀⠀⣀⣀⢸⣇⠐⠂⠰⠀⠀⠀⠀⠰⠀⢠⡄⠠⠉⣁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠦⣀⢀⣉⣤⡀⠀⠀⠀⠀⠀⠀⠒⠚⡶⠂⠀⠀⠈⠑⠁⠉⠁⠈⣄⣉⡴⠷⠋⠑⠛⠃⠈⠃⠙⠒⠚⠭⡇
⠀⠀⠀⠀⠀⢀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠘⢢⠀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⣄⠀⠠⣠⠀⠀⠠⠀⡐⢠⠤⠒⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠁⡀⠁⠀⠀⠀⠀⢀⢀⠠⢠⡄⡀⠀⠀⠀⣠⠤⠐⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠁⠀⠀⠘⠛⠋⠋
⠀⠀⠀⠀⠄⠞⠁⠀⠄⠠⠀⠐⠉⠀⠀⠀⢴⢃⡈⠂⠸⣄⠀⠀⢈⣷⡤⠄⣼⣏⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣦⣀⠀⡠⠀⠁⠀⠀⠀⠀⠀⡈⠀⠀⠘⣿⣶⠎⠀⠀⢠⠋⠀⠀⠘⠟⠛⠁
⠀⠀⠀⠘⠋⠛⠋⡀⠀⡀⠀⣤⣴⣶⣾⣿⣄⠀⢠⣿⣿⣷⣤⡶⠃⠀⠀⠀⠀⠀⡛⠁⠀⠀⠀⠀⠀⡀⢠⣤⠤⠒⠒⠢
⠀⢠⠀⠀⠢⡄⠐⠀⠀⡄⠀⠉⠋⠉⠉⡉⠁⡀⠀⣠⣤⣤⡾⢀⣀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⣠⡼⠃⣀⣀⣶⣶⣿⣿⣿⣷⣦⣤⡈⠃⢄
⠀⠐⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢟⣁⢴⣶⣋⠜⣀⣤⠀⠀⠀⠀⠀⠀⢀⡴⠇⣠⣾⡿⠉⠉⣁⠀⠀⠢⣌⠉⠻⢿⣃⣀
⠀⠀⠈⠠⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣎⣁⣿⠿⣿⣪⣿⣠⣶⣄⠀⠀⠀⠁⡞⢀⣾⠿⣃⠀⠙⡀⠀⠀⠀⠀⠘⠊⠁⠐⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⡤⠼⣿⣿⣷⣾⣿⣿⠿⡁⠋⠛⠀⠀⠄⠸⠀⣾⣿⠀⢻⣷⣾⣯⣄⠀⢀⠀⠀⠀⠀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡆⣶⣿⢿⣾⣛⣿⣟⣻⠏⠘⠲⢄⢸⠀⡇⢸⣿⠿⡄⠀⣿⢹⣿⣇⣰⣿⡇⣾⣀⣤⢸⡿⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⢿⣿⡿⠀⠃⢸⣿⣽⣿⣿⠀⠀⠀⢸⣿⠀⠇⠸⣿⣧⠀⠀⠈⠀⠉⠉⠉⠙⠳⠏⢹⡇⣿⠃⠠⣿⣿
⠀⠀⠀⠰⣀⠈⠀⠀⠀⠀⠠⠤⠈⠑⠀⠀⠾⣟⠓⠀⠃⢐⡻⢬⣿⣿⠀⠁⠀⢸⠀⠀⠀⠈⠿⠿⠂⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢦⡽⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠂⢀⣄⠔⠉⠙⠾⢿⡄⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠙⢦⣀⣀⣤⡆⠀⢀⣠⣦⡴⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⢄⣉⠙⠻⠶⢋⣬⣬⣿⣿⣟⡀
⠀⠀⠀⠀⡀⢀⡤⠄⠔⠈⠘⠠⠀⠤⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⠀⠉⢉⣿⣛⣯⣍⠳⣆
⠀⠀⠀⣯⣀⠲⢾⠂⠠⠀⠀⠀⠀⠀⠁⠀⢂⡀⠀⠈⠃⠀⠀⠀⠀⠀⠁⣄⠀⢀⠀⠀⠀⡈⠂⢠⡴⢶⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⢾⡇
⠀⠀⠁⣾⣬⡤⢾⡠⢤⠁⠀⠀⠀⠀⠀⠂⠤⠳⡆⠠⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠁⠈⠐⢀⡁⠀⠀⢢⡄⠀⠀⠘⠀⠀⢸⠿⣿⣿⣿⢇⡟
⠀⠀⠀⠃⣤⠐⠨⠄⠈⠓⠶⢆⣀⢀⡀⠈⢤⡆⣅⠠⡇⠀⠀⠀⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠼⠃⡀⠀⠀⠀⠀⠀⣴⣿⣿⣧⡾⠁
⠀⠀⠀⠀⢺⣾⠆⠠⢀⠄⠀⠀⠈⠻⠌⡬⠋⠡⠞⣰⣓⠀⢀⣎⠀⠚⣁⣀⣐⣄⣈⣀⡢⠐⡞⠀⠿⠉⡆⠘⠀⠀⠀⠀⠀⠀⠸⣿⠟⠉⡁⠀⣀⣀
⠀⠀⠀⠀⠘⣿⣷⣵⣦⣤⡆⠑⠀⡐⠠⡛⢻⣠⣾⣽⡟⢀⠷⡦⣿⣬⠉⠀⠀⠀⠀⠀⠈⠉⠑⣚⣤⡾⣷⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⣤⣶⠟⠛⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣄⣿⣾⣿⣿⣿⣿⣿⠁⣾⣿⣟⣿⣧⣶⣆⣴⣆⣴⣤⣠⣀⣶⣿⣿⣹⣿⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⣸⡿⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠟⠛⠢⠆⠤⠄⠀⠤⠾⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⡒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣭⣭⡉⠙⢦⣀⠀⠀⠀⠀⠀⡴⠁⣰⠎⢹⣿⡿⠀⠀⣀⣀⠀⡏⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⢰⣿⡿⢋⡹⡄⠀⢻⣦⠀⠀⠀⣰⠁⢰⣿⠀⣸⡿⠧⣄⣰⣇⣈⣋⠁⢀⣉⣳⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣏⠀⢻⣿⢁⠏⣽⡇⠀⢸⣿⣆⡀⢀⠇⠀⣿⡇⠀⢋⡖⠀⣿⡏⢈⣿⡏⠀⣾⣿⡟⠀⢠⣶⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣈⢛⣫⣾⣿⡇⠀⣸⡟⠀⢹⡾⠀⣸⣿⠀⢠⣿⠀⣸⡟⠀⣸⡿⠁⢰⣿⡟⠀⢀⣋⣵⢏⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⢻⠁⠀⡟⣰⡆⠸⠃⢠⣿⣿⠀⣿⣇⠀⢋⣴⡀⢛⣴⡀⠸⢋⣶⡀⠸⠿⣣⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣰⣿⡇⠀⠀⣼⣿⠏⠙⠿⠿⠛⠿⠿⠛⠿⠿⠟⠻⢿⠿⠛⠳⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣼⣿⡟⢣⣀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⣀⣤⣤⡀⣸⣿⣤⣤⡄⠀⣠⣤⡴⠶⠶⠶⢶⣆⣴⠞⠛⠛⠻⢶⣦⡶⠞⠛⠛⠻⣷⠀⠀⣀⣀⣀⡀⠀⣠⡶⠛⠛⠻⣦⠀⢀⣴⠟⠋⣉⣻
⠀⠀⢠⡿⠉⠀⠈⠻⠏⠀⠀⠀⢿⣰⢿⡇⠀⠀⣤⣶⣾⡿⣧⠀⠀⠀⠶⣞⣟⠳⠦⣄⠀⠻⢿⣴⣿⠉⠉⠉⢷⣾⡏⠀⣠⢖⣾⣧⣤⣾⢻⠀⠘⠛⢻
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⡇⠀⠀⣨⣽⣃⣷⣨⣭⡽⠀⠀⢈⣿⡦⠦⠟⠀⠀⢸⣷⠇⠀⠛⠀⢸⣿⡇⠀⠹⠚⠀⠀⢀⣿⢸⠀⠀⣴⣾
⢀⣾⡿⠀⠀⠀⣤⡀⢀⣴⡀⠀⠀⠘⣦⡇⠀⠀⠉⠉⢹⠋⡇⠀⠀⠀⣀⣼⢻⡇⠀⠀⣀⣤⠋⣞⣀⣰⠦⣀⠀⣷⠙⠦⣤⠾⣄⣸⡟⡏⢸
⢸⠟⠷⠶⠶⠾⠃⠹⠏⠀⠙⠛⠛⠛⣿⠛⠛⠛⠓⣲⣿⡀⠙⠛⢛⣻⠿⣏⠀⠉⠛⢛⣹⣿⡀⢉⣹⣯⣤⣩⣭⠿⣦⣤⣦⣀⣹⡟⠐⢿⣌⣛⣛⣻⡿
⠻⢦⣤⣤⣤⣤⠾⢧⣤⣾⠳⠶⠶⠛⠛⠛⠛⠛⠛⠋⠈⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠋⠁⠈⠙⠋⠉⠀⠈⠉⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣀⣠⣄⠀⣠⣿⣤⣤⡄⠀⣠⣤⡤⠴⠶⠶⢶⢆⣴⠖⠛⠛⠶⢶⣤⡶⠶⠒⠒⠲⣦⠀⢀⣀⣀⣀⡀⠀⣠⡶⠛⠛⢻⡆⠀⢀⣴⠟⠛⢛⣻
⠀⠀⢠⡟⠉⠁⠈⢿⠏⠀⠀⠈⣷⣰⢿⠀⠀⠀⣤⣤⣾⡿⣇⠀⠀⠀⢶⣚⡟⠷⠤⣄⠀⠿⢧⣴⡿⠉⠉⠉⣷⣾⡏⠀⣰⢶⣾⣧⣠⣾⢻⠀⠘⠟⢻
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡆⠀⠀⣉⣽⣈⣧⣈⣩⡿⠀⠀⢸⣿⡦⠤⠟⠀⠀⣸⣧⠃⠐⠟⠀⢸⣿⡇⠀⠳⠞⠀⠀⢈⡏⢸⠀⠀⣴⣾
⢀⣿⡟⠀⠀⠀⣤⠀⢀⣤⡀⠀⠀⢹⣼⡇⠀⠀⠛⠋⢻⠉⡇⠀⠀⠀⢀⣾⢿⡀⠀⠀⢀⣴⢋⣞⣀⣰⢦⡀⠈⣧⠳⢤⣤⢶⡀⣸⢿⡇⢸⠀⠀⠀⠈
⣸⠛⠷⠶⠶⠾⠉⠳⠊⠀⠛⠚⠛⠛⣿⠛⠒⠒⠒⣶⣿⠀⠙⠒⠚⣿⢿⡇⠈⠙⠛⢛⣹⣿⠈⠉⣹⣧⣄⣩⣽⠿⣤⣤⣦⣀⣹⡟⠸⣧⣜⣛⣛⣳⡿
⠿⣤⣤⣤⣀⣠⠾⣦⣤⢿⣷⠶⠶⠚⠛⠒⠒⠛⠛⠋⠉⠛⠛⠛⠛⠁⠀⠙⠒⠛⠛⠋⠁⠉⠛⠛⠉⠀⠈⠉⠀⠀⠀⠀⠈⠉⢹⣷⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⡤⢶⣦⣀⠀⠀⠀⠀⠀⣠⣤⣴⠶⠶⠶⠶⠶⠾⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⣈⢻⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⢿⣶⣤⣶⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠛⠻⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⢹⣿⠋⠀⢀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠋⠀⠀⣀⡄⠀⢠⣾⣿⣿⣿⣷⣤⣾⠃⠀⣴⣿⣏⣁⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣦⠈⡟⠁⢀⣾⠟⢋⠙⢿⣿⣿⠃⠀⣼⠿⠛⠛⢻⣿⠿⠋⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣼⠃⣴⣿⡀⠈⢹⡏⠀⠘⢡⣾⣀⣀⣾⠃⢠⠟⠀⣰⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢰⠇⠰⣿⠿⠃⠀⣾⠀⠀⡀⠘⠿⣿⣿⠇⠀⣤⣴⡾⠟⣡⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢸⣄⠀⢀⣴⡀⠀⢿⡀⢀⣧⡀⠀⣀⣼⣆⡀⠀⢁⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢸⣿⠿⠿⠛⠿⣷⣿⠿⠿⠟⠿⠿⠟⠋⠛⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⢉⣉⣀⣀⣀⣉⡉⠙⠛⣿⠛⠋⠉⠉⣉⣉⣉⣉⣉⣉⡙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠡⠖⠊⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⡟⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢀⠤⠄⠒⠒⠉⠉⠉⠉⠉⠒⠀⢰⡇⠀⠤⠔⠒⠊⠉⠉⠉⠉⠉⠉⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⣀⠀⠀⢸⠁⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⣀⡀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⠀⠐⠊⠁⠀⠀⠀⣀⣀⡀⠀⠀⠁⠀⡟⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢀⣿⢧⡇⢀⡤⠔⠒⠉⠉⠉⠀⠀⠈⠉⠓⠀⢠⡇⠀⢀⠤⠔⠒⠊⠉⠉⠉⠉⠒⠒⠦⠄⠘⣿⣿
⠀⠀⠀⠀⣸⡿⣾⠀⠀⢀⣀⡠⠤⠔⠒⠒⠒⠒⠤⠄⠀⢸⠃⠀⠀⠀⣀⣀⠤⠤⠤⠤⠤⠄⣀⡀⠀⠀⣿⣿⡇
⠀⠀⠀⢠⣿⢷⡇⠀⠚⠉⠀⠀⠀⣀⣀⣀⣀⡀⠀⠄⠀⣼⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⣿⣇
⠀⠀⠀⣸⡿⣼⠀⣀⠤⠒⠊⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⡏⠀⠤⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠢⠀⢸⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⡦⢤⣄⡀⢠⡇⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣐⠂⠈⣿⢹⡇
⠀⠀⢰⣿⣿⢁⣤⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣧⣾⣿⣿⠿⠿⠿⠷⠶⠶⠾⠿⠿⠿⣿⣿⣿⣶⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣭⣥⡶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⠇
⠀⠀⠘⠛⠋⠉⠁

⠀⢀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣀⣀⣀⣀⣀⣀⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡍⠀⠀⠀⠀⠀⢀⣰⡀
⠀⣿⣿⣿⡿⣿⣟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠛⠂⢰
⠀⣿⣳⢿⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠧⣄⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣿⢧⡞⣿⣿⠃⠀⠀⠀⠀⠀⢠⣾⣿⠋⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡂⠀⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢤⡔
⠀⠻⣼⣽⣿⡏⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠴⠋⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⣏⡁⢀⡏
⠀⣾⣿⣿⣿⠇⠁⠀⠀⢀⣠⣾⣿⣦⣤⣶⣿⡛⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠤⣰⣿⣿⣀⠀⢷⡄⠀⠀⠀⠀⣾⡢⠀⡀⢠⣶⣾⣷⣤⣾⣗⣏⡀
⠀⣽⣟⣻⡿⠀⢀⣠⣾⣿⣿⣿⣿⠛⣿⣿⣿⣿⣏⠀⠀⢀⡞⠛⠁⠀⠀⢀⣀⣀⣸⣿⣿⢿⠀⠻⣷⠀⠀⣿⣿⣿⣷⣾⠀⢸⠦⢘⣿⣿⣿⣃⣿⣷
⠀⢾⣿⡟⣤⣶⣿⣿⣿⣿⣿⣿⣿⡀⠨⣿⣿⣻⣿⣆⣤⣼⢃⣻⣿⡯⡀⢼⣿⣟⣻⣿⣿⡿⠀⡀⣿⠀⠀⢸⣿⣿⣿⣯⣇⣸⣇⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠆⣸⣿⡿⢿⣿⣿⣿⢋⢿⣿⡒⢈⣾⣾⣟⠉⠹⣿⣿⡀⡇⢹⠁⠀⢸⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⡿⢿⠿⠁⠀⢹⣿⠕⣫⡾⠿⢿⠘⡿⣿⣿⡇⠇⠸⠀⣀⡸⣿⣿⣿⣿⡿⢈⣿⣛⣨⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠆⠸⣿⣟⠷⡀⠀⢀⢸⣿⣾⣿⠇⠀⢶⡄⠀⢘⣿⡇⢀⡅⢀⠀⠄⣿⣿⣻⢽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠟⣿⣿⣿⣆⠀⠀⠀⣿⣿⣿⣸⣿⣤⠈⠀⠀⠈⠻⡗⠀⢻⠈⢀⣀⣼⠉⢻⣼⣿⡿⢿⠿⠟⠿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠄⢰⣾⢟⣟⢿⢷⣤⣶⣿⣿⣿⣿⣿⣿⣤⠀⡀⣄⡀⠀⠀⠀⠀⣾⣿⣿⠀⢳⣿⠋⣱⡋⢸⡀⠀⢸⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠅⢰⠳⠁⣼⣄⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⢀⠙⣿⣿⣧⡀⠀⠇⠿⣿⣿⡜⢺⣿⠆⡻⣧⡼⠳⠾⠿⣿⣿⣿
⠀⣿⣿⢿⣿⣿⣿⣿⣫⣿⣿⣿⣿⡂⠈⠀⠀⡉⣿⡶⠤⢶⣦⡤⢉⢹⠿⠛⠛⠋⠀⠀⡘⠠⢿⡄⠀⠈⠀⠘⢿⣿⡈⣿⡁⣿⣧⣴⣿⣶⣄⠀⠀⠘
⠀⢹⣿⣷⣿⣍⣿⣷⣾⣿⣿⣿⣿⡿⠄⠀⢸⠇⠉⠙⠉⠉⠀⠀⠀⢀⣤⣶⢿⣶⣦⣄⢠⣄⣘⣷⣀⢀⣧⣀⠘⣧⡴⠬⠟⣿⣿⣿⣿⢿⣟⡀⡀
⠀⢸⣿⣿⣿⣯⣴⣤⣉⣉⠉⡀⠈⠷⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⡋⠀⢶⣿⣿⣿⡸⠿⣟⠛⠻⠁⢀⣬⣭⣿⣶⣄⣾⣿⣿⣿⣿⡿⢻⣿⣧⡄
⠀⠸⠿⠛⠁⢉⡙⠋⠉⠩⠙⠛⠻⠲⣄⢀⣠⡴⠀⠀⠰⣆⠀⠉⠈⠉⠁⠀⠈⠙⠛⠿⠀⠀⠀⠁⠀⠀⠈⢉⣉⣉⣉⣻⣿⣿⢿⠿⠿⢷⣾⢸⡿⠃
⠀⢸⣷⣿⣟⠤⠿⠖⠦⠀⣀⠀⠀⠀⠙⠻⢿⣤⢀⣀⣴⣿⢀⠀⡖⠀⠀⠀⠀⠠⠄⡀⣄⣀⣀⠀⠀⠀⠀⠈⢹⠿⣿⣷⣮⣿⣶⣤⣾⡿⢟⡾⠁
⠀⢸⣿⣿⣿⣷⣶⠆⠰⢳⣟⣿⠛⢂⣀⡀⠀⣹⡟⢻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⢿⣏⠛⠃⡉⠀⢢⡄⠀⠉⠙⠻⠛⠿⠷⠾⢿⣿⣦⣀
⠀⠸⣿⣿⣿⣿⣿⡶⢃⠀⣉⣶⣋⣛⣷⣶⣴⣿⣯⣿⣋⣽⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⡿⣛⡁⠀⠀⠠⠄⠀⢀⠀⠀⠀⠈⠙⠿
⠀⠀⠛⠛⠿⢿⢿⣿⣯⡀⠝⠿⣷⣿⣿⣿⣿⣿⣆⣂⣿⣩⣽⢻⣿⣦⣍⠁⠻⢟⣃⣿⣟⠛⠻⡴⢤⣴⣶⣿⣶⣭⣷⢾⣤⣄⠀⠀⡀
⠀⠀⠀⠉⠁⠶⠦⠌⠟⠛⠷⠶⠈⠹⠿⠁⠿⠿⠿⠍⠛⠛⠻⠿⠟⠛⠛⠿⠿⠿⠯⠿⠻⠿⠿⠯⠴⠅⠩⠍⠛⠟⠛⠋⠛⠛⠛⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠄⠀⠀⠀⠀⠀⠁⠢⢀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢎⡎
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠀⢠⢢⣾⡗
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⣿⣿⡁
⠀⠀⠀⠐⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⡐⣿⢯⣾⣿⡿
⠀⠀⠀⣈⣋⣒⣒⣂⠬⢔⡠⢁⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣼⢿⣿⢏⡾⠁
⠀⠀⠀⠹⠠⠀⣈⠈⠉⠛⠺⠿⣿⣦⣄⠀⠀⠴⠀⢀⣀⣠⣤⡤⡤⠤⢤⡄⣀⡀⠀⠀⠀⢜⡾⣡⣿⢫⡞
⠀⠀⠀⠀⠥⡿⠞⠚⠈⠉⢙⣻⣯⣭⣿⣿⣾⣾⣿⣯⣶⣶⣶⣶⠶⠒⠒⠂⢀⡈⢙⠲⢼⡾⣱⡿⣣⡟⠁
⠀⠀⠀⠀⢷⢶⣏⣹⠟⠛⠋⠹⠯⠝⡶⣓⣒⣹⣽⣿⣷⣿⡭⣟⣻⠕⢒⠶⢄⣈⠉⢊⡹⣿⡟⣰⠟⠀⠀⠀⢀
⠀⠀⠀⠀⡄⠄⢿⢿⠀⢜⢾⠢⠵⠿⠷⠶⠆⠁⠑⠒⠦⠄⢾⣿⣿⡫⠗⠦⠄⡈⠍⠲⣕⣟⣼⠋⠀⠀⠀⠠
⠀⠀⠀⠀⡇⢸⠘⣿⣬⣖⣀⡀⠤⢀⠀⠀⠀⠀⠀⢀⡀⣧⡾⣫⣮⡁⠀⠀⡀⠬⠑⠢⢼⡾⠁⠀⠄⠐
⠀⠀⠀⠀⢸⠀⠀⠙⣇⢉⠉⣉⣉⣽⡽⣿⠶⠶⠭⡿⣐⣫⠾⠍⡝⠁⠀⠀⠀⢀⣀⢤⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢷⢤⣾⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⣧⡄⠀⠀⠀⡡⠤⠚⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠘⢧⢿⡇⠐⠴⠈⠀⣀⢀⣀⢀⡀⠀⣀⡀⢀⣀⢉⣸⠛⠀⠈⠀⡀⠤⠐
⠀⠀⠀⠀⠀⠀⠈⠛⠭⠍⠩⠤⠀⠓⠒⠒⠒⠉⠉⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠶⠶⠚⠛⠛⠉⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠖⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⢸⡁⢛⢻⣆⠐⠆
⠀⠀⠀⠀⠀⢀⣤⡤⠶⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⢸⢸⠈⡅⢸⠸⣿⡆⢠
⠀⠀⠀⢠⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⣄⠀⣾⡿⣲⠌⣧⠀⠀⢀⡿⠀⠀⢸⣾⠀⡆⢸⠀⣿⡇⢸
⠀⠀⠀⢸⡻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⣤⣀⣒⣚⠩⠍⢳⡟⠁⢹⡆⣿⠂⢸⡆⠸⣧⠀⣸⠇⠀⠀⢸⡟⠰⡇⢸⠀⢿⡇⢸
⠀⠀⠀⢸⣿⡏⠀⠀⢀⡀⠀⡴⠄⢰⣷⢀⣾⠟⠛⠉⠃⠛⠈⢉⠉⠀⠀⣼⠁⠀⠈⡇⣿⣁⡾⠃⠀⠹⣧⡟⠀⠀⠀⢸⡇⢰⡇⠈⢀⢸⡇⢸
⠀⠀⠀⢸⠋⡇⠀⠀⢙⡇⠀⠄⡆⠸⣿⠘⠧⠶⢖⣠⣄⠀⠀⠘⢀⠀⠀⣿⠀⠀⢀⡇⣿⠛⢷⡄⠀⠀⢻⡃⠀⠀⠀⢸⡇⡞⢠⠀⢸⢸⡇⢸
⠀⠀⠀⢸⣠⡇⠀⠀⠈⣷⠀⡄⡇⠀⣿⠀⠀⠀⠀⠀⠙⣳⠄⠀⢸⠀⠀⢻⡀⠀⣼⡇⣿⠀⠈⣷⠀⠀⢸⡇⠀⠀⠀⢸⡇⡇⢸⠀⠸⢸⡇⢸
⠀⠀⠀⢾⣿⡇⠀⠀⡤⢛⠭⢤⡃⠀⣿⠀⠀⠀⠀⢀⠠⣠⠄⢀⣸⠀⠀⠘⢷⡾⠋⠀⣻⠀⠀⢻⡆⠀⢸⡇⠀⠀⠀⢸⡇⠁⣼⠀⢰⣾⡇⢸
⠀⠀⠀⢾⣿⠇⠀⠀⡇⡏⠀⢸⡇⠀⣿⠀⣦⣶⠶⠿⠛⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠇⠀⢸⡇⠀⠀⠀⢸⡇⡇⠉⠘⢸⢿⡇⢸
⠀⠀⠀⢸⣯⡀⠀⠀⡇⡏⠀⢸⡇⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⡇⣿⠀⢸⠸⢸⡇⢸
⠀⠀⠀⢸⣾⡀⠀⠀⡇⡃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢇⣿⠸⠀⢸⡇⢸
⠀⠀⠀⢸⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⡰⢶⣦⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠀⡄⢸⡇⡞
⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⡠⠄⠉⠀⠀⠀⡆⢀⠀⠐⠋⣿⡄⠀⣱⣨⣿⡄⠀⠀⣯⣀⣀⢀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠀⡇⣾⡇⡇
⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⣰⠈⢦⠀⣗⣻⡀⠀⠟⢀⡀⢹⡆⠉⠻⠾⣧⠘⠁⡀⠀⠀⠀⠀⠘⣿⠀⢹⠀⡇⣿⡇⠃
⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠁⢠⡀⠀⢀⡐⠂⡿⢙⡆⠰⣶⣷⢬⡁⠀⢠⡼⠟⠻⣆⠀⠓⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⡏⠀⢸⠀⡯⢻⡇⢸
⠀⠀⠀⣸⣼⡇⠀⠀⠀⠀⠀⠀⢨⠗⢤⠁⣤⣾⡄⠉⠀⢀⣟⣀⡀⠙⣷⣬⣽⡶⡆⠈⠳⣶⣂⣤⠞⠀⠀⠀⠀⠀⠀⠀⡟⢰⠀⠀⠀⣸⡇⢸
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⣶⠋⠀⠰⢾⠏⠙⢿⡶⢦⣬⣟⡋⠹⣧⡈⠁⠀⠹⡷⢄⣠⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣀⠀⡀⢿⡇⢸
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠈⠒⢤⡀⢾⣴⢶⣏⢣⠈⠉⠋⠉⠀⠘⢳⣀⣀⣠⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⡇⢸⡇⢸
⠀⠀⠀⣿⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠿⠧⠀⠰⠶⠒⠒⠛⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⡇⢸⡇⢸
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠘⣃⠀⡇
⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⠀⡇⡇⢼⠀⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠁⡇⢸⠀⡇
⠀⠀⠀⢻⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⡀⠃⢸⠀⡇
⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⡇⣆⢸⠀⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⡇⣿⠸⡆⡇
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠇⣿⠀⡇⡇
⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡆⠀⣿⠀⡇⡇
⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡇⡀⡿⢠⠃⠇
⠀⠀⠀⠀⣟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠥⣇⣯⣴⣄⡔
⠀⠀⠀⠀⡟⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣤⠿⠛⠉
⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠴⠖⠒⠒⠀⠀⢰⣶⣶⣶⠿⠿⠿⠿⠿⠿⠛⠛⠛⣛⣛⣁⣀⣤⠞⠋
⠀⠀⠀⠀⠻⣽⣇⣀⣀⣛⣓⣒⣋⣉⣉⣉⣉⣀⣤⣤⣤⡤⠤⠤⠴⠶⠶⠶⠖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠛⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⡀⡀⠀⢀⣾⠛⠛⢤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠙⠃⠀⠈⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠿⣇⠂⠀⠀⠀⢀⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠈⡅⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢱⡀⠀⠀⢀⡐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢀⠀⠀⢐⡏⠉⠲⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠙⢦⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⠀⠀⠀⠘⣃⣀⡀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠛⠋⢹⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣶⡿⣆⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠄⠀⠀⠠⡔⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠂⢀⣭⣭⣭⣵⣾⣿⡿⣛⡛⢛⣉⡭⠿⣛⣫⡄⠀⠀⠀⠀⠀⠀⣗⡀
⠀⠀⠀⠀⠀⠀⡀⢀⣠⣀⠀⠀⠀⢱⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣴⠉⠁
⠀⠀⠀⠀⠔⠫⠒⠕⣫⡵⢛⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠏⣠⣤⣶⡿⡿⠟⠒⠂⠤⢤⣤⡤⢤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠈⠉⠉⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠿⣅⣀⠸⣤⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠛⠛⢛⣿⡿
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠠⡀⠀⠠⡧⢀⡤⢄⠠⣄⢤
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⠄⢌⠛⠆⠀⠀⡇⠸⡄⠸⠀⡏
⠀⠀⠀⠀⠀⠀⣠⣿⣟⣀⣀⣀⣠⣼⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⠸⡏⣿⣿⣿⣷⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡕⠻⠋⠉⠛⣋⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⠀⠄⠈⢢⡈⢻⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣋⠁⠀⠁⠂⢀⡆⢀⠎⡀⠘⠉⠙⠑⡖⠲⠆⠤⢤⠤⢤⣤⡤⢤⠤⢤⠴⣏⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠁⠐⣀⡡⠄⠈⢾⠀⠡⠀⠀⠀⠰⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠒⢀⣀⣸⠖⠁⠀⠈⢇⠃⠀⠀⢠⠃⠀⠀⠺⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⢀⣸⡀⢀⠀⡆⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠄⠀⡞⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣫⠵⠚⡇⢈⣤⣇⣀⣀⣂⣀⣁⣀⣬⣠⡤⠞⢻⣄⠙⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡜⠁⠀⡜⠱⠃⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠰⢐⡇⠈⠦⣈⡼⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⣀⠞⠀⠀⠘⣅⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠀⢠⠇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢡⡶⠋⠀⠀⠀⠰⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⢠⠟⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡶⠋⠀⠀⠀⠀⣴⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣰⠇⠀⣀⠀⢸⡆⡀⠀⡀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⢼⠀⠛⢸⡇⢸⠀⣟⠈⢈⠅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠈⠁⠀⠈⠁⠁⠀⠉⠀⠈⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⠀⣿⣿⢻⣿⡇⠀⢀⣾⣿⡿⢿⣿⠆⠀⣿⣿⣿⣿⣦⠀⠀⣿⣿⠿⠿⠟⠀⢸⣷⡿⠿⠿⠃
⠀⠀⠀⠀⣼⣯⣼⣷⡀⠀⠀⣿⣿⣿⣯⡀⠀⢸⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⡇⠀⣿⣿⣶⣶⠀⠀⢸⣿⣷⣲⡄
⠀⠀⢀⣾⣿⡟⠹⣿⣷⡀⠀⣿⣿⣨⣿⣿⠀⠸⣿⣷⣦⣶⡷⠀⠀⣿⣿⣶⣿⡿⠃⠀⣿⣿⣤⣤⣤⠀⢸⣿⡏
⠀⠀⠈⠉⠉⠀⠀⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠉⠁
⠀⠀⢀⣴⣶⣶⣶⣤⠀⢰⣶⡦⠀⠀⣶⣶⠀⢰⣶⣶⣶⣶⣶⠀⢰⣶⣲⣲⣲⣖⠀⢰⣶⡆⢀⣶⡖⠀⢰⣲⡆
⠀⠀⣾⣿⠟⠉⠙⠟⠁⢸⣿⣷⣤⣤⣿⣿⠀⠈⠉⣿⣿⠉⠁⠀⠈⠉⢹⣿⡏⠉⠀⢸⣿⣷⣿⠿⠁⠀⢸⣯⡇
⠀⠀⣿⣿⣄⣀⣻⣷⡆⢸⣿⣿⠛⠻⣿⣿⠀⢀⣀⣿⣿⣀⡀⠀⢀⣄⣼⣿⡇⠀⠀⢸⣿⡟⢿⣦⡀⠀⢸⣿⣧⣀⣀⡀
⠀⠀⠈⠻⠿⠿⠿⠋⠀⠸⠿⠿⠀⠀⠿⠿⠀⠸⠿⠿⠿⠿⠷⠀⠙⠿⠿⠋⠀⠀⠀⠸⠿⠇⠘⠿⠇⠀⠸⠿⠿⠿⠿⠇
⠀⠀⣀⣀⡀⠀⠀⣀⣀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣀⣀⡀
⠀⠀⣿⣿⣷⣀⣼⣿⣿⡄⢸⣿⣿⡄⢸⣿⣧⠀⣴⣿⡿⢿⣿⡆⠀⢸⣿⡿⣿⣦⠀⣰⣟⣿⣿⣿⣆⠀⢸⣿⡟⣿⣷
⠀⠀⣿⣿⣿⣿⡿⣿⣿⡇⢸⣿⣿⣿⣾⣿⣿⠀⣿⣏⠀⠀⣿⣿⠀⢸⣿⣷⣿⡟⠀⣿⣿⠀⣠⣿⣿⠀⢸⣿⣿⣿⠏
⠀⠀⣿⣿⠀⠉⠀⣿⣿⡇⢸⣿⡇⠹⣿⣿⡿⠀⢻⣿⣶⣾⣿⠇⠀⢸⣿⡏⠁⠀⠀⠹⣾⣷⣿⣿⡏⠀⢸⣿⡿⣿⣷
⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠉⠛⠋⠁⠀⠀⠈⠙⠁⠀⠀⠀⠀⠈⠉⠉⠙⠁⠀⠘⠛⠁⠙⠙
⠀⠀⠀⠀⠀⣠⣶⣶⣤⠀⢰⢶⡶⣶⢶⣶⡆⠀⣶⡆⠀⠀⣶⣶⠀⢰⣶⣦⠀⣴⣶⡆⠀⢰⣶⣆⠀⠀⠀⢀⣶⣶
⠀⠀⠀⠀⠐⣿⣿⣉⠁⠀⠈⠉⢻⣿⡏⠉⠁⠀⣿⣧⠀⢀⣿⡟⠀⠀⣿⣿⣀⣿⣿⠀⠀⠈⢿⣿⡄⢀⠀⣼⣽⠇
⠀⠀⠀⠀⣀⠈⠙⣿⣧⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣄⣼⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠘⣿⣷⣯⣷⣿⡟
⠀⠀⠀⠘⠿⣿⣿⡿⠋⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⠿⣿⠿⠋⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠹⠟⠁⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⣀⣀⠀⣀⢀⡀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣾⡿⠃⠀⢻⣿⣆⣿⡿⠁⠀⠛⠛⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢿⣷⡄⠀⠀⢸⣿⡇⠀⠀⠀⣾⣿⣿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡿⠿⠛⢋⣉⣤⣶⠿⠟⢻⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠟⠛⠉⣁⣠⣤⣶⠾⠟⠋⣉⣤⣴⣾⣿⠿⠿⠟⢻⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠉⠁⣀⣠⣴⡶⠿⠛⠋⣁⣠⣴⣶⠿⠟⠛⠉⣁⣠⣤⣶⣾⣿⣿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⠛⠉⠁⢀⣀⣤⣶⡾⠿⠛⠉⣁⣤⣴⡶⠟⠛⠉⢁⣀⣤⣴⣾⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⣿
⠀⠀⢀⣤⣶⡿⠟⠛⠉⠁⠀⣀⣠⣴⣶⡿⠿⠛⢉⣁⣤⣶⠾⠛⠋⣉⣀⣤⣴⣶⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣿⣀⣀⣀⣤⣶⣾⡿⠿⠛⣉⣡⣤⡶⠿⢛⣋⣡⣤⣶⣾⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⡟⠿⢿⣿⣿⣯⣵⣶⣾⣿⣿⣿⣷⣶⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢠⣿⡇⠀⠀⠈⠉⠛⠛⠛⠋⠉⠉⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠿⣿⣶⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⠇
⣿⣿⠃⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⡿⠿⠛⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⡿⠟⠛⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠿⠟⠋⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⠟⠋⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣤⣶⣿⣿⠿⠟⠋⠉
⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⡿⠟⠛⠉
⠀⠈⠻⣿⣷⣶⣤⣶⣶⣶⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠶⠤⠤⠤⠤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡋
⠀⠀⢠⣴⣎⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠄⠀⢸⡄
⠀⢠⠇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠓⠒⠒⠶⠶⠶⠤⠤⣤⣤⣄⣄⣀⣀⣀⡠⠞⡁⠀⠀⢀⠔⡩⠃
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠀⠈⠀⢀⠔⢁⠎⢀⠜
⠀⠀⡜⠉⠉⠓⠒⠒⠲⠦⠤⠤⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠔⠁⡰⠁⡠⠏⣀
⠀⠀⠘⡆⠤⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠒⠦⠤⠤⠵⣾⣅⣠⠎⠀⡔⠁⢀⠔⡇
⠀⠀⠀⠉⠒⢤⡀⠀⠉⠙⠻⢭⣁⠒⠒⠂⠤⠤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢎⡠⠚⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠉⠙⠲⣤⡀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠀⠤⠤⣷⠗⠉⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠈⠙⠢⢤⣀⡠⠒⠁⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠍⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣠⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⠀⠀⠀⠹⢿⣿⣾⣿⠗⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⡿⠁⠀⠠⣾⣿⠿⠛⠿⠀⠻⠟⠀⠀⠘⠛⠃⠀⣼⣿⣿⣿⣿⡟⠀⣠⡤⣦
⣤⡀⠀⠀⢻⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠛⣿⣿⡿⠁⠀⠀⠀⠘⣿⡷⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠿⠿⠿⠿⠛⠀⢾⣿⣤⣼
⠉⣿⡆⠀⠀⠈⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⣶⡆⠀⢠⣄⡀⠙⠛⠁⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡟⠉⠉⣿⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⣾⠟⠀⠀⠀⠀⠀⢀⣠⡶⠻⣦⡀⠀⠀⠀⢻⣷⣠⣿⠏⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣼⣿⡆⠀⢀⣀⠀⠀⣿⣿⣿⣷⣶⣶⣿⣿⠀⠀⠀⠀⣀⡀⠀⠀⠹⣿
⠁⠀⣠⣄⡀⠀⠀⠹⣿⣷⣶⣿⣷⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⣴⣿⣿⣿⣦⡀⠀⣼⣿⣿⣷⣠⣾⣿⡇⠀⠹⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠘⣿⣷⡄⠀⠀⠙
⠀⣸⣿⣿⣷⠀⠀⠀⠘⢿⣿⡟⠁⠀⠀⠀⢸⣿⠇⠀⢀⣠⡄⠘⣿⡀⠈⠋⠁⣰⣿⡿⣿⣿⡿⣿⣿⠀⠀⠀⣿⡏⠉⠁⠀⠀⠀⢀⣀⠀⠀⠈⢿⣿⣆
⢰⣿⣿⣾⣿⣇⠀⠀⠀⠈⢻⣿⡦⠀⠀⠀⠀⠉⠀⠀⠘⢿⣿⣶⣿⡗⠀⠀⠀⠛⠟⠰⢿⠏⢸⣿⡟⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⠿⠃⠀⢀⣈⣿⣿
⣾⣿⠛⢻⣿⣿⡄⠀⠀⠀⠀⠉⠀⠀⢲⣿⣆⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠿⠋⢻⣿⣆⠀⠀⠘⠿⠟⠃
⠛⠋⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣄⡀⠀⣠⣤⣄⠀⢰⣶⡆⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⡀
⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⠄⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⢸⣿⣿⣧⡈⣿⣧⠀⠀⠀⢀⣿⣿⠀⢀⣴⣄⠀⠀⠻⠟⠂⠀⠀⢠⣾⣿⠄
⢿⣷⣄⠀⠀⠀⠀⣴⣿⣿⡁⣠⣶⣄⠀⠀⠈⠿⠗⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⢸⣿⣯⠻⣷⣿⣿⠀⠀⠀⣼⣿⣃⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠠
⣼⣿⠏⠀⠀⣴⣿⡿⠛⢻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⠀⠀⠀⠀⠀⣿⣿⠀⠘⣿⣿⡄⠀⢰⣿⣿⣿⠟⠁⠀⠀⢠⣴⡾⠛⢷⣄⠀⢻⣿⣷⣤
⠿⠋⠀⠀⠀⠈⠋⢀⣴⣿⠟⠁⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠛⠋⠁⠀⠈⠉⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⣿⣿⣶⣾⣿⠀⠀⠙⠻⠿
⠀⠀⢀⣀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢠⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁
⣄⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠂⠀⠀⢀⣶⠞⠛⢻⣶⡄⠀⠀⢸⣿⡿⠉⠀⠀⣄⡀⠀⠀⢿⣿⠿⣿⣶⣄⡀⠀⠀⢹⣿⣦⠀⠀⠀⠀⠠⣴
⣿⣷⣜⣿⡇⠀⠀⢴⣿⣄⠀⠀⠀⣾⣿⣧⣄⠀⠀⢠⣿⣿⣦⣤⣾⣿⡟⠀⠀⠘⣿⣿⣄⠀⢠⣿⡿⠀⠀⠘⣿⣷⣿⡿⣿⣿⡷⠀⠀⠉⠀⠀⢰⣶⡆⠀⢿
⣧⠙⢿⣿⣿⠀⠀⠈⠻⣿⣦⠀⠀⠀⠉⠛⠋⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠁⠀⠀⠀⠘⣿⣿⡀⠈⠋⠀⠀⠀⠰⣿⣦⡀⣿⡇⠀⢸
⣿⡄⠀⠻⠿⠃⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠹⠟⠁⠀⢠⣶⣶⢦⣄⠈⠻⢿⣿⣧
⠁⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠏⠀⠀⠀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣴⣦⠀⢀⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⣸⣿⣧⣤⣿⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⢿⣿⣿⡖⠀⠀⢠⣿⣟⣴⣿⠟⠀⠀⢰⣿⣿⠋⢀⡀⠹⠿⠃⠀⠀⠀⣿⣿⠿⠿⣧⠀⠀⠀⠀⠈⠀⢀
⠃⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣶⡄⠉⠀⠀⠀⣼⣿⣿⣿⡁⠀⠀⠀⠘⢿⣿⣤⣼⣿⡦⠀⠀⠀⠀⣼⣿⣿⣤⣴⡟⠀⠀⠀⢀⣤⣶⣿
⡀⠀⠀⣾⣿⡄⠀⠈⠻⣿⣆⠀⠀⠈⠙⠿⣿⣿⣭⡙⠁⠀⠀⠀⢀⣿⡿⠘⣿⣧⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⢠⣶⣤⣀⠀⠈⠉⠉⠁⠀⢀⠀⠈⢻⡿⠛⣿
⣿⣄⠀⠈⢿⣿⣦⣀⣀⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⠋⠀⠀⢀⡀⠀⠉⠁⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡋⢹⡇⠀⠀⢀⡀⠸⣿⣷⡀⠀⠀⠀⠈
⠻⠛⠀⠀⠀⠙⠻⠿⠿⠟⠁⠀⣠⣿⣦⣄⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡖⠀⢀⣾⡿⣿⡿⠿⠟⠁⠀⠐⢿⣷⣄⣼⣿⣷⣄
⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡋⠉⠿⠃⠀⠀⠀⣰⣿⣶⣿⣿⣷⣶⣤⣤⠀⠈⠉⠉⣼⣿⠃⠀⠙⠟⠀⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠁⠹⣿⣧
⡿⠁⠀⣀⠀⠀⠀⠀⢀⣴⣿⠟⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣷⡉⠙⠃⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠈⠛⠃⣀⣀⠀⠀⠀⠀⠀⠙⣿⣷⠄⠈⠁⠀⢀
⣁⣴⣿⡿⠃⠀⠀⠀⠀⠉⠋⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⢠⣾⣿⣷⣶⣶⡂⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⢀⡀⠈⠁⠀⠀⠀⢀⣾
⡿⠟⠉⠀⣠⡶⣶⣷⣦⡀⠀⠀⠀⣰⣿⡿⠋⠛⠋⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⡀⠀⠀⠸⣿⣧⠀⠀⠀⣾⣿⠛⢶⡄⠀⠀⣾⣿
⠀⠀⠀⢸⣏⠀⢈⣿⣿⣧⠀⠀⠀⣿⣿⡁⠀⢀⡀⠀⠀⠀⢸⣿⡇⣠⣶⡆⢀⣤⡄⠀⠀⠀⠀⣠⣴⣾⣿⣷⡀⠀⠀⢻⣿⡄⠀⢰⣿⣿⣀⣠⣿⠀⠀⠈⠛

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

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠂⠈⠀⠀⠀⢨⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠨⠈⢀
⠀⠀⠀⠀⠀⠐⠀⠃⠀⠃⠀⠘⠀⠃⠀⠀⠃⠀⠈⠀⠈⠀⠀⠀⠁⠘⠀⠀⠙⠀⠀⠀⠃⠀⠘⠊⠁
⠀⠀⠀⠀⠀⠠⠀⠀⡠⢄⡀⠀⠀⠠⠠⠄⠀⠄⠀⢀⠤⠀⠀⠠⠠⠀⠀⡄⠀⢄⠀⠀⠄⢤⠀⠀⠄⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡒⠀⠁⠀⠀⡀⠀⢀⠀⠀⢐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠁⠀⠀⡀⠀⢀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡁⡀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠉⠀⠀⠀⠈⠀⠀⠁⠘⠀⠀⠁⠀⠈⠀⠀⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠐⠈⠁⠗⠀⠀⠂⠀⠀⠀⠂⠀⠰⠐⠆⠀⢄⠐⠀⠂⠠⠂⠀⠄⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⡄⠀⠐⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠔⡀⠠
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂⡄⡆⠢⠀⡄⡀⢰⠀⡄⡄⠀⢀⠀⠀⠀⠀⠀⠀⠠⠂⢤⠀⠀⡄⠀⠀⠀⠂⠀⠳⡀⡀

⠀⠀⠀⠀⠀⠈⠀⠀⠊⠀⠀⢀⠀⠀⠀⠂⠨⠅⠐⠀⠀⠀⠀⠒⠉⠀⠀⠁⠈⠐⠀⠁⠀⠘⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠐⠁⠀⠒⠀⠀⠐⠀⠖⠀⠁⠀⠀⠀⠰⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⡀⢠⠠⠀⡄⠀⡄⡄⢠⠀⠠⠀⢠⠀⠀⠀⠀⢠⠀⡄⡄⠀⡀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠆⠀⠀⠀⠀⡣⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠁

⠀⠀⢠⣶⣿⣿⣿⠆⠀⠀⠀⠀⠀⣼⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⣿⠿⣾⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⢿⠀⣀⠀⠀⠀⠀⠀⠈⠛⠲⠶⢶⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠉⠹⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠠⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⣏⠇
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⢠⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⢠⣄⣄⣀⣀⡴⠃⠀⠀⡔⠒⠒⠒⠒⣄⣤⡤⠀⠀⠀⠀⣤⣶⣶⣶⣄⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢘⣿⣋⣽⣿⡃⠀⠀⠀⡇⠀⠀⢠⣾⠋⠁⠀⠀⠀⠀⣾⣿⣿⡇⣽⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢈⣿⣯⣿⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣿⣟⠓⢻⣿⡿
⠀⠀⠘⠛⠛⠻⠿⠟⠋⠀⠀⠀⠴⠋⠈⠛⠛⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋

⠀⠀⠠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠐⡍⡏⡄⣌⣟⣝⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛
⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⡇⡇⡇⡿⣿⣿⠀⠀⠀⠀⠸⣧⣀⠀⣀⣼⠇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠹⠿⠿⠏⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⢀⠆⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⣠⣿⣥⣤⣤⠄⠀⠀⠀⠀⣰⣿⠃⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣥⣤⡝
⠀⠀⣘⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢿⠿⢿⣿⣿⠁⠀⠀⠀⠀⡟⠅⢠⣤⣤⠀⡸⠀⠀⠀⠀⠀⠀⡨⠋
⠀⠀⠛⠒⠒⠲⠒⠒⠚⠃⠀⠀⠀⠸⣷⣶⣶⣶⣾⠏⠀⠀⠀⠀⠘⠦⣈⣙⣻⠴⠁⠀⠀⠀⠀⠀⢠⣶⣷⣶

⠀⠀⠐⢄⠀⠀⠀⠀⡠⠐⠀⠀⠉⡀⠀⢤⠐⠤⡀
⠀⠀⠒⠚⠠⡀⡠⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠐⠒⠒⠂⢤⢀
⠀⠀⠀⠀⡠⠃⢓⠀⠀⠀⠀⠛⠻⠛⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⠀⠒⠀⡀⠂⠀⣄⡀⠀⠀⠀⠀⢀⠔⠨⡤⠈⡄⡄⢠⡆⠇⡀⣈⠢⡀
⠀⠀⠀⡎⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⠘⠀⠀⠀⡃⠀⡔⠁⠈⢆⠀⢀⡇⠀⡔⠁⠈⢂⠀⠀⢀⠲⢄⠀⠃⠀⠰⠘⠚⡙⠀⠳⢁⡤⠓⡀
⠀⠀⠀⢳⡀⠀⠑⢄⠀⠀⠀⠀⡠⣿⣿⣿⠏⠀⠀⠀⠀⠃⠌⠀⣶⢴⠈⢆⠀⠀⡔⠀⢸⠀⠀⠂⠀⠂⠀⣤⠛⠢⣀⢀⠧⠰⢇⣠⠔⡅⣠⢀⢠
⠀⠀⠀⠀⠉⠓⠶⠤⠤⠾⢿⠟⠁⠀⠀⠀⠀⠀⠀⡠⠊⠰⠢⠤⢀⡈⠀⠈⡦⠼⠀⠈⣈⡀⠤⠼⠀⣀⠀⠠⠟⠀⢨⠁⠀⠀⠀⢁⣀⣘⣋⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⠤⣀⣀⠀⠠⠊⠀⠀⡆⠀⡖⣀⠈⠉⠚⠀⠘⠉⢁⣀⠀⠀⢸⠀⠀⠀⡟⡫⠉⠙⠀⠀⠀⠀⠀⠀⠐⣤⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠉⠁⠀⠀⠀⡥⠀⣉⣊⣀⡀⡅⠀⡄⠀⠘⠑⠀⠀⡄⠀⢂⠀⣋⣱⠤⠔⠇⠀⠀⠠⠲⠤⠬⢈⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⠁⣈⡀⡐⠀⠀⠀⠀⠀⠀⢰⡒⠀⠀⠁⢠⠉⠉⠉⠉⠉⠉⠁⠀⠘⠁⢰⠉⣳⠀⢠⠀⢀⠁⠀⢦⠂⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠁⠀⠀⠀⢀⠆⠤⣘⣀⠀⠀⠀⢸⠀⠀⠰⢀⣀⠴⠀⠀⠀⡦⠤⠍⠭⠄⢸⠀⠖⠀⠤⠬⠄⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠥⠔⠊⠢⡀⠠⡆⠀⡤⠄⠈⠉⢱⠀⠈⠖⢲⡍⣡⠀⢀⡵⢠⠘⢄⠀⣯⠵⠀⠈⠀⠀⠀⡰⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠉⠀⡴⠒⠀⢠⠁⣀⡘⢄⠫⠁⠀⢀⠂⠀⠀⢁⠘⠙⠃⡣⣊⠀⠸⠀⡠⣕⢧⡀⢀⠂⠀⡀⢀⡁⠕⣚⣄⣀⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠑⠂⠀⡎⠀⣇⠝⠈⢢⠀⡠⠁⠀⠀⠀⠀⠣⣀⠞⠀⠏⢲⠀⡀⠤⠜⠀⡸⠃⠀⠀⠘⣁⠴⢊⠜⡙⡉⢋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠴⠅⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠠⠇⠠⠐⠈⠀⠀⠀⠀⠀⠈⠐⠡⠴⠥⠒

⠀⠀⠀⠀⠀⠀⢠⣷⡄⣠⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡷⢰⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠆⣠⢦
⠀⠀⠀⠀⠀⠀⠟⠉⠗⠻⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠟⠸⠷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠃⠻⠖

⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠀⣯⢈⡷⣼⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⡴⠒⡆⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣴⢠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣄
⠀⠀⠀⠀⠀⠈⠷⠬⠗⢸⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠿⠸⠇⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣏⢰⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠘⠁

⠀⠀⠀⠀⠀⣤⡄⣤⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡄⠀⡀
⠀⠀⠀⠀⠀⣿⣿⢿⢸⡏⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⠰⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣠⠗⢿⣹⠇


⠀⠀⠀⠀⠀⠀⣾⣹⠆⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⡀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠇⣤⠄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠿⠀⠳⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠛

⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢨⣓⣧⢚⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⡇⣇⣸
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢶⠀⡶⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣦⣰⢀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠿⠁⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠹⠏⠘⠟⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⠆⠟⠧

⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⣷⠃⢶⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠘⠛⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡥⢬⣟⣛⠛⢛⡛⣛⠛⣛⢛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠆⢀⠀⠀⠀⠀⣶⠛⠉⢑⡀⠀⠀⠀⣠⠖⠢⠆⠀⠀⠰⡖⠂⠀⠀⠀⠀⠀⢠⠖⠒⠒
⠀⠀⠀⠠⠇⠀⠐⡄⠀⠀⠀⢓⣀⣄⠞⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⢘⣠⡄
⠀⠀⠀⡴⠉⠉⠉⣸⠀⠀⠀⣛⡀⠀⣠⠇⠀⠀⠨⣄⣀⣠⠀⠀⠀⠀⠇⢀⣠⠟⠀⠀⠀⢘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠀⡀⠀⠀⠀⢀⡈⠉⠉
⠀⠀⠀⢠⠖⠋⠁⠀⠀⢠⡞⠉⠐⠃⠀⠀⢰⠇⠀⠀⢁⠀⠀⠀⢸⠁⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⡸⠂
⠀⠀⠀⢈⡴⠄⠀⠀⠀⠞⠀⠠⢠⠀⠀⠀⢨⠖⠒⠒⣾⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠈⡵⠚⠁
⠀⠀⠀⠨⠆⠀⠀⠀⠀⠀⠤⠤⠞⠀⠀⠀⠀⠷⠀⠀⠟⠀⠀⠀⠼⠀⠀⠐⠦⠏⠀⠀⠀⠀⠧⠀⠸⠇
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠤⠤⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼⠀⢠⠞⢰⠀⠀⠀⢸⠋⠂⠀⡆⠀⠀⠸⡇⠀⠀⢰⠀⠀⠀⢼⠀⢀⡆
⠀⠀⠀⠀⠇⠀⢀⡀⠀⠀⢣⠀⠉⠀⢰⠃⠀⠀⠸⠂⠀⢡⡇⠀⠀⠀⠃⠀⢀⡟⠀⠀⠀⠀⠞⠉
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢀⡖⠒⠢⡀⠀⠀⠀⡴⠃⠒⡄⠀⠀⠀⡞⠉⠁⠃⠀⠀⠲⠒⠲⡋⠒⠀⠀⠀⣲⠀⠀⢰
⠀⠀⠀⠾⠀⠀⠀⢱⠀⠀⠀⠇⠀⣀⠗⠀⠀⠀⠘⠤⠄⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢃⠀⠀⢘⡄
⠀⠀⠀⠈⢀⣀⡥⣫⠀⠀⠀⡟⠁⠁⣤⠀⠀⠀⢀⣀⣀⡼⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠘⣀⣀⡾⠁
⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⢸⠁⠀⠀⡟⠀⠀⠀⡿⠀⠀⠀⣾⠀⠀⠀⠱⢠⠖⠋⠀⠀⢱⠀⢀⡼⠀⠀⠈⠋⠁⠉⡼
⠀⠀⠀⠈⢢⠀⣰⠇⠀⠀⠀⢳⡀⣠⣀⡾⠀⠀⠀⣰⠋⠄⠀⠀⠀⠈⢀⡞⠁⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠞⠁⠚⠀⠀⠀⠘⠁⠀⠀⠃⠀⠀⠀⠸⠇⠀⠀⠀⠀⠢⠿⠶⠚⠃
⠀⠀⠀⠀⣠⠤⢄⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⣠⠄⡄⡀⠀⠀⠀⠴⠒⠄⡀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⢰⠃⠀⠈⡄⠀⠀⠀⠚⠁⡇⠀⠀⠀⠈⠁⠀⣠⠇⠀⠀⠀⠀⣀⣀⡟⠀⠀⠀⢸⣁⠰⣀
⠀⠀⠀⠀⠂⣀⣰⠇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣼⣁⡄⠀⠀⠀⣀⣀⣀⡇⠀⠀⠀⠈⢈⡾⠉
⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⢸⠉⠉⠃⠀⠀⠀⠀⡖⠒⠃⠀⠀⠀⠑⠒⣢⠇⠀⠀⢠⡟⠉⢰⠀⠀⠀⠀⡞⠉⠁⣀
⠀⠀⠀⠀⠘⠁⢠⡀⠀⠀⢠⡞⠉⠁⣢⠀⠀⠀⠀⢰⠏⠀⠀⠀⢠⠗⠒⠈⡀⠀⠀⠀⠃⣀⣀⡏
⠀⠀⠀⠀⠦⠶⠞⠀⠀⠀⠀⠙⠖⠶⠃⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⠤⠤⠾⠁⠀⠀⠀⠠⠤⠞

⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⣀⠐⠢⡀⠀⠀⢄⡤⠒⠒⢠⣠⠄⠀⢀⣤⠀⠲⢤⡀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀
⠀⢸⠀⠈⢻⣆⠀⠀⠀⠀⠀⣿⢱⣴⠃⠀⢀⣿⠀⠀⠀⠈⠉⠀⠀⠀⣷⠀⠀⠀⢻⡄⠀⢸⣇⡀⢠⠀⠀⠀⢸⡄
⠀⠘⣽⠋⢸⣿⡆⠀⠀⠀⠀⣿⠞⠉⣧⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠁⠀⢸⡇⠀⠈⠀⠀⠀⢸⡟⠳⠂
⠀⠀⠀⠀⡟⠸⣿⡄⠀⠀⠀⠛⠀⠐⠁⠀⠀⠈⠛⠢⠄⠒⠁⠀⠀⠐⠛⠂⠐⠋⠁⠀⠀⠘⠃⠀⠐⠂⠀⠀⠚⠂
⠀⠀⠀⣸⠁⠀⠻⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤
⠀⠀⢀⡟⠛⠉⠈⢻⣆⠀⠀⢰⡇⠀⠀⣀⠀⠀⣿⠀⢀⢈⡇⠀⠀⠀⡅⠀⠀⠀⠀⢸⡀⠀⢸⡄⢀⠀⠀⠀⠀⠻⠀⢀
⠀⠀⣾⠀⠀⠀⠀⠈⣿⣄⠀⠈⠳⢀⠀⠟⠀⠀⣿⠁⠋⠈⡇⠀⠀⠀⠃⠀⠀⢀⢀⠼⠀⠀⢸⡇⠀⢣⡀⠀⠀⠸⠀⠀⡄
⠀⣼⡧⠀⠀⠀⠀⠀⠧⠿⢦⠀⠀⠋⠀⠀⠀⠀⠛⠂⠀⠀⠉⠀⠀⠀⠛⠀⠀⠘⠀⠀⠀⠀⠘⠃⠀⠀⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⠀⠀⠀⠀⠀⡀⠀⢤⣀⠀⠀⠀⠀⣤⡤⠤⣄⠀⠀⠀⣠⠄⠀⠠⢄⡀
⠀⢀⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⣇⠀⠁⣿⠀⠀⠀⢠⠋⠀⠀⠀⠈⢳⡄⠀⠀⣿⣇⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⡟⢧⡀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡏⠉⠉⠀⠀⢿⡀⠀⠀⠀⠀⣸⡇
⠀⢀⣰⡀⠀⠐⠀⠀⠈⠁⠀⠀⠀⢸⡇⠀⠹⠿⠀⠀⠀⠈⠣⢄⣀⠀⠀⠋⠀⠀⠀⠿⠇⠀⠰⠀⠀⠀⠑⠀⠀⠐⠰⠯

⠀⠘⣿⠋⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⡶⠖⠒⠀⠀⠀⠀⠉⣹⠂⠀⠀⠈⡟⠀⠀⠘⣧⠀⠀⠀⢠⠋
⠀⠀⣿⡀⣀⡼⠞⠀⠀⠀⠀⠀⠀⠹⣢⣄⠀⠀⠀⠀⣴⡏⠀⣰⣦⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⣷⠀⠀⠀⢹⣧⠀⢀⠇
⠀⠀⣿⠁⠙⢦⠀⠀⢀⡀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⢿⡇⠀⠻⠙⠉⡄⠀⠀⣿⡆⠀⠀⠀⠸⡿⠀⠀⠀⠀⢻⣆⠊
⠀⢠⠿⠇⠀⠀⠛⠦⠼⠇⠀⠀⠷⠤⠤⠜⠁⠀⠀⠀⠈⠛⠦⠤⠤⠔⠀⠀⠀⠙⠻⠦⠤⠀⠠⠿⠆⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⡀⠀⢀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠐⠏⢋⡉⠙⠉⠑⡝⠁⠀⢹⣐⠃⠀⠀⠳⣿⣄⠀⠀⠙⠉⠀⠀⠀⣰⠈⣿⣆⠀⠀⠀⣷⡇⠀⠈⠁⣀⠈⠉⠉⣹⡷⠁
⠀⠀⠀⣼⠁⠀⠀⣸⠁⠀⠀⠈⣿⠀⠀⠀⠀⠙⢿⢆⡜⠁⠀⠀⠀⠀⠋⠀⢸⣯⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⢸⣿⠀⠀⢰⣿⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢀⢿⣳⡄⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢀⣿⡇⠀⠀⠀⠀⠀⣜⠎
⠀⠀⠈⢿⣦⡀⠈⢿⣧⡀⠀⣨⣿⡇⠀⠀⣤⢀⠞⠀⠙⣿⣦⠀⠀⠀⣀⡀⠀⠈⠙⠛⠁⣿⡇⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠉⠉⠋⠁⠉⠉⠉⠉⠉⠁⠀⠀⠑⠛⠙⠀⠀⠘⠒⠓⠀⠀⠛⠃⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠾⠛⠚⠒⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈
⠀⠈⠀⢈⠀⠀⠄⠁⠀⠀⠀⠀⠀⠠
⠀⣠⠀⠀⠇
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠰⡀⠀⠀⠀⠀⠀⢸⣃⡁⠈⡄⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠒⠒⠚⠀⠀⠂⡀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⢠⠀⠠⠄⡄⢤⡄⢀⡄⡄⠀⠇⠀⠀⠀⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⢹⢀⢣⢰⡆⢇⠜⢷⠃⠁⡇⠀⠀⠀⠀⠀⠠⠀⠀⠠⢐⠠⢴⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⢠⣠⠀⣤⠄⣤⡄⣄⡄⠠⡆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠸⠐⠙⠠⠛⠢⠏⠇⠚⠧⠀⠇⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠆⢘
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⢂⣀⡀⢀⠐⠂⠀⡀⠀⠀⠿⠀⠄⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠈⠉⠉⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠊⠑⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠔⠉⠐⠛⠂⡀⠀⣨⠛⠘⠋⠒⡪⠬⠭⡕⠚⠛⠚⢢⢄⠀⠀⠀⠀⢠⠀⠈⠐⢀⠀⠁⠀⠀⡄⠤⠤⠌⢳⠖
⠀⠠⡀⠀⠀⠀⠀⠄⠀⢿⠀⠀⡀⠀⡟⠀⠸⡇⠀⠀⠀⢸⠖⠆⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣖⢰⣤⣿⠃
⠀⠀⠀⠑⠠⠖⠂⠀⠀⠈⠒⠲⠀⠒⠁⠀⠀⠐⠛⠒⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠄⠒⠂⠈⠉⠉⠁⠒⠒⠒⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⡠⠤⣖⡒⠂⠉⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠓⠒⠲⠶⠤⢤⣤⣀⣀
⠀⠀⠀⢸⢡⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⣄⣀⣀⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⡟⠁
⠀⠀⠀⠸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⢤⣄⣀⣀⡀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠸⡇
⠀⠀⠀⣀⣑⣲⣤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠷⠓⠤⢤⣄⡀
⠀⠀⠀⠙⡟⠲⠤⣄⣀⠀⠈⠉⠙⠒⠲⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠐⠚⢷⣦⣀⣀⣤⡶⠿⠿⠗⠒⠒⠊⠉⠉⠀⠸⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠈⠉⠉⠛⠓⣶⣦⣤⣄⣀⣈⣇⣠⣤⡶⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡶⠶⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⣈⡿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⢶⣿⣶
⣰⡿⠛⠳⠷⣦⣭⣛⣿⣶⣤⣀⠀⠀⠀⠀⠀⣼⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠔⠒⢒⣋⣩⣭⡥⠶⠶⠛⠛⠋⢹⠃
⣿⠁⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣶⣤⣀⣠⠏⡄⢀⣀⣀⣤⣤⠶⠶⠒⣚⣋⣩⣭⡤⠶⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣯⣭⣴⡶⠶⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⣦
⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣖⣷⣀⣀⣤⣤⡶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢴⡶⠤⢽⣫⢭⡷⢍⣉⡓⠢⠄
⠀⠀⠀⣴⢓⠦⣄⠀⠀⠀⢀⠀⢠⡀⢤⡰⢟⣒⣉⣉⠓⠦⣉⠓⠦⠬⠷⠈⠑⠀⠉
⠀⠀⢰⠏⠀⠀⠈⠙⠶⣄⠈⠓⢶⣞⠓⢬⣓⢤⡒⠶⠿⠂⠀⠁
⠀⠀⣾⣅⠀⠀⠀⠀⠀⠈⠙⠦⢄⡈⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄
⠀⠀⢸⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠏
⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠖⠛⠋⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠉⠻⣧⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠻⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⣠⣤⣶⡶⠟⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠂⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣠⠀⠀⠀⠀⠄⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⡀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠛⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆⢿⡇⢀⣀⡤⢔⠚⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠚⠈

⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡠⠒⢄⠀⠀⠀⢄⡤⠒⠒⢤⣠⠄⠀⠀⣠⠀⠐⢦⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀
⠀⢸⠀⠈⢻⣧⠀⠀⠀⠀⠀⢸⡟⣦⡜⠀⠀⢰⡟⠀⠀⠀⠈⠉⠀⠀⠀⣿⠀⠀⠀⢹⡆⠀⠀⣿⣀⡀⣄⠀⠀⠀⣿⠀⠀⠈
⠀⠘⣿⠗⢠⣿⣧⠀⠀⠀⠀⢸⡧⠋⢹⡆⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⣿⠀⠀⠁⠀⠀⠀⣾⠒⠲⠄
⠀⠀⠈⠀⣼⠘⣿⡆⠀⠀⠀⠘⠇⠀⠘⠁⠀⠀⠛⠳⠤⠤⠔⠀⠂⠀⠀⠿⠆⠤⠞⠋⠀⠀⠀⠟⠀⠠⠴⠀⠀⠀⠿⠄
⠀⠀⠀⢰⠇⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⡾⠶⠒⠒⢻⣧⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⡆⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⣏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⣸⠃⠀⠀⠀⠈⣿⣆⠀⠀⠻⣄⠀⢈⡇⠀⠀⣟⠰⠛⠚⡇⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⠀⠀⣿⠀⠑⡄⠀⠀⠀⣄⠀⠈⡀
⠀⣰⣯⠀⠀⠀⠀⠀⢸⣹⣆⡀⠀⠰⠛⠀⠁⠀⠠⠿⠀⠀⠀⠳⠀⠀⠀⠴⠀⠀⠘⠇⠁⠀⠀⠀⠿⠀⠀⠙⠀⠀⠀⠃⠐⠒⠁
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⡄⠀⠀⠀⠒⢲⡀⠀⠀⠀⠀⠀⣿⡁⠐⢺⣟⠀⠀⠀⢀⠔⠁⠀⠈⠑⢦⡁⠀⠀⢹⡿⠀⠉⣷⠀⠀⣴⠋⠀⠀⠀⠙⢦⡀
⠀⠘⠁⠀⠀⠀⠀⠘⣾⡀⠀⠀⠀⠀⣿⢷⡀⢸⡇⠀⠀⠀⢺⠀⠀⠀⠀⠀⠘⡷⠀⠀⢸⣷⠲⠤⠛⠀⢸⣯⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡄⠀⢀⠀⠀⠘⠧⠀⠀⠀⠀⣿⠀⠻⣼⣷⠀⠀⠀⠹⡄⠀⠀⠀⠀⣸⠃⠀⠀⢸⣇⠀⠀⡠⠀⠀⠻⢄⠀⠀⢀⣀⡾⠁
⠀⠐⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠈⠉⠀⠀⠀⠀⠀⠑⠒⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠉
⠀⢠⣤⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠀⢤⡄⠀⠀⢤⡄⠀⠀⠀⢠⡤
⠀⠀⣿⡇⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢉⣴⠋⠁⡀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢸⡃⠀⠀⠘⣷⡀⠀⠀⡜
⠀⠀⣿⡧⢤⡞⠋⠀⠀⠀⠀⠀⠀⠀⠳⢗⣆⠀⠀⠀⠀⣿⡃⠀⣞⠿⠤⡀⠀⠀⣸⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠹⣧⠀⡰
⠀⠀⣿⡇⠀⠙⣂⡄⠀⣤⠀⠀⢀⡀⠀⠀⣹⠀⠀⠀⠀⢻⣇⠀⠈⠉⢀⡄⠀⠀⢹⣧⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢿⡦⠁
⠀⠐⠓⠃⠀⠀⠈⠙⠒⠛⠀⠀⠘⠙⠒⠒⠋⠀⠀⠀⠀⠀⠉⠛⠒⠒⠋⠀⠀⠀⠀⠉⠛⠛⠋⠀⠛⠋⠀⠀⠀⠀⠀⠁
⠀⠀⣠⣀⣀⣀⣠⣄⢀⣀⠀⢠⡄⢠⠀⠀⢤⡤⡄⠀⠀⠀⣠⣤⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⢤⢀⠀⠀⣄⠀⠠⣤⣀⣀⣀⣤⡄
⠀⠈⠃⢉⡤⠀⠁⠀⡽⠁⠀⠀⣧⣝⠀⠀⠀⠹⣷⣄⠀⢀⡈⠀⠀⠀⠀⢸⠇⢸⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠄⠀⠀⢀⣾⠟
⠀⠀⠀⣾⠃⠀⠀⣴⡇⠀⠀⠀⢸⣯⠀⠀⠀⠀⠙⢿⢢⡏⠀⠀⠀⠀⠀⠁⠀⢸⣻⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣞⠏
⠀⠀⢸⣿⡄⠀⠀⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣠⢫⣵⣄⠀⠀⠀⠀⠀⠀⠈⢿⣇⡀⢀⣼⣿⠀⠀⠀⠀⠀⢀⣭⠂
⠀⠀⠀⢻⣧⣄⠀⢹⣷⣄⣀⣀⣼⣿⠀⠀⢠⡤⢰⣃⠀⠙⣿⣦⡀⠀⠀⣀⡀⠀⠀⠉⠉⠉⢸⢿⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠁⠉⠉⠁⠀⠀⠘⠛⠃⠀⠀⠘⠒⠛⠀⠀⠛⠃⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠐⠖⠓⠛⠛⠲⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠉⠀⠀⠀⠀⠀⠠⠐⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡀⠀⠄⠀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡴⠡⠂⠀⡀⣤⠀⠁⠰⡄⡈⠙⠂⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠶⠁⢀⠊⠄⣷⣢⣄⡀⠘⢿⠀⣖⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⢀⡇⠀⠊⠁⣀⠤⠈⠛⢿⣶⡌⢩⢋⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠨⡄⢿⠁⠠⠘⠁⠀⠀⠠⠒⠚⠻⣆⡌⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⢄⣀⠀⠀⠀⠀⢠⠖⠁⠀⡾⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⡐⠂⠀⡰⠀⢠⡟⠀⢀⠳⡄⠃⠈⠁⠺⠁⠀⠀⢐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣧⠃⠀⠀⢔⠂⠀⠸⠀⠀⠄⠀⠙⢆⡀⢀⠃⠀⠀⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠂⠀⠋⠀⠀⠀⠀⠐⠀⠂⠀⠈⢿⡏⠀⠀⠀⠀⢘⢸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠀⠀⡔⠀⠀⠀⠸⢂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠠⡰⠊⡌⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⢰⠀⠀⠀⠀⡃⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢼⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⡇⠀⡆⠀⠀⠀⡸⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠀⡉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢱⠀⡅⠀⠀⢰⠁⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⢰⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⢄⠡⡀⠆⢁⠀⠀⣇⠎⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⣀⢀⢣⡎⢀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠛⠒⠳⠄⢬⣼⡀⢸⡿⠀⣠⠴⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⠌⠐⡄⣀⣄⡀⠀⠠⠀⠀⠀⠠⠚⠉⠁⠀⣢⣤⣈⣙⣷⣌⣴⢸⠁⠶⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⣀⡤⠖⠉⠀⢀⡀⠠⠼⠏⣿⣬⡂⠀⠀⣀⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⢐⡓⠂⠀⠀⠐⠒⠀⠐⠒⠚⠣⠤⠤⠀⠀⠀⠁⢀⡐⢀⠞⠇⡿⡿⢦⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠁⠀⠀⡏⠀⠀⠍⡉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢘⣶⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⡐⠆⠨⠐⠀⠀⠙⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠂

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





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

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠂⠈⠀⠀⠃⠀⠀⠠⠀⠀⠀⠀⡁⠀⠁⠀⠀⠀⠀⡀⠂⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⡏⢉⣏⣻⢶⣀⢄
⠀⠀⢀⠀⠀⣀⠀⢀⣀⠀⠬⠅⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⢀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠐⠆⠄⠄⣘⠀⠞
⠀⠈⠂⠀⠈⠋⢹⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠦⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠰⡲⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⢠⡤⠁
⠀⠀⠀⠠⠀⠄⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⢀⠀⠑⠂⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⡄⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠁
⠀⠐⠨⣿⡱⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢶⡀⠀⠠
⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠃⠀⠁⠀⠀⠂⢠
⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡆⠀⢰⠀⢠⠖⠓⠀⠀⠒⠒⡖⠒⠃⢀⠖⠛⢢⡀⢰⠛⢢⠀⠰⡄⠀⠀⡴⠀⠀⠠⠀⠰⠀⠟
⠀⠀⠀⢀⠸⠍⠯⠑⠀⠀⣇⣀⣀⡇⠀⢸⠀⠘⢦⣀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⡇⢸⣀⡸⠀⠀⠘⣦⠞⠀⠀⠀⠀⡖⢢⠀⢀⡀
⠀⠀⠀⠘⠐⠀⠐⠃⠀⠀⡏⠉⠈⡇⠀⠸⡆⠀⠀⠈⢳⡄⠀⠀⡇⠀⠀⢿⠀⠀⢀⡇⢸⠉⠳⡄⠀⠀⢸⠀⠀⠀⠀⢀⠃⠹⠁⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠇⠀⠀⠇⠘⠦⡤⠜⠁⠀⠀⠇⠀⠀⠘⠦⣠⠜⠁⠸⠀⠀⠹⠀⠀⠸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢲⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠇⠀⠀⣾⠏⠀⠀⠀⠀⠒⠓⠒⠛⡐⠓⠒⠃⠐⢿⡦⠀⠼⠀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠐⠙⠂⠀⠙⠃⠞⠁⠀⠀⠀⠀⠀⠀⢰⣶⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢁⣀⢀⢤⠄⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⠀⠀⠀⠂⠀⠉⠏⠁⠀⠙⠀⠘⠛⠒⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠠⠀⠠⠀⠀⠀⠀⠒⠠⠖⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠹⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⢀⠀⢈⠀⠠⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠑⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠼⠀⠈⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠘⠸⠀⠀⠂⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⢿⡄⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠉⠉⠉⠓⢦⡞⠛⠛⠛⠛⠒⢶⡄
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⡟⠀⢿⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣸⠁⠀⠘⡇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠻⠤⠤⠴⠇⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠻⡆⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠳⠦⠤⠤⠤⠤⠤⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠴⠤⠤⠤⠤⠴⠟⠀⠀⠀⠀⠀⠈⠛⠶⠤⢤⡤⠤⠖⠋⠙⠶⠤⠴⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠶⠒⠶⠶⠶⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠶⠶⠒⠲⠒⠶⠶⢤⣄⡀
⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠙⡆⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣆⠀⣀⡤⠖⠛⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠙⠛⢋⡉⠉⠈⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠸⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⡞⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠋⠉⠉⠉⠉⠻⡆⠀⢠⠞⠛⠓⠶⣤⣤⣤⣤⠾⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢨⡇⠹⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠘⠲⠶⠶⠶⠶⠶⠶⠖⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠀⠀⠀⠈⠉⠛⠛⠒⠲⠶⠶⠶⠶⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢄⡤⠴⣦⡀
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢶⣀⠻⢍⡙⠚⢿⡺⢿⣒⠦⠌⠑⠂
⢀⡾⠈⠝⡲⣄⡀⠀⢤⣀⠐⣬⣓⢭⣓⠚⡒⠦⣌⠑⠢⠌⠙⠛⠋
⣸⡕⠀⠀⠀⠀⠉⠳⢀⠈⠙⠯⣍⠓⠮⠓⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢺⡗⠀⠀⠀⠀⠀⠀⠀⠘⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡍
⠘⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠛⠛
⠀⠀⠈⠙⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⣀⣀⣤⣶⡶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣿⠟⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⣀⠀⠀⠀⠀⠀⠃⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢀⠆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡘⣿⡇⠀⠀⠀⣀⣤⡤⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣿⣤⣴⡲⠯⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢈⢀⡈⠇⠀⠀⠀⢸⣿⡿⢿⣿⡄⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⢸⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠀⠀⠀⡞⢥⢧⠻⡄
⠀⠀⠀⠀⠀⠀⠃⡊⠃⢰⠀⠀⠀⢸⣿⡇⣠⣿⠇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⡸⣽⠁⠀⠯⠇
⠀⠀⠀⠀⠀⠸⠘⠆⠖⠀⡄⠀⠀⢸⣿⡿⠿⣿⣄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣷⡇⠸⢟⣟⣻
⠀⠀⠀⠀⠀⡔⢰⠀⠀⠆⢁⠀⠀⢸⣿⣧⣠⣿⡟⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣾⣿⠃⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⠀⠀⠀⠀⢣⠺⡄⡜⣼⠁
⠀⠀⠀⠀⠀⠒⠋⠀⠀⠐⠈⠀⠀⠘⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠛⠛⠀⠀⠀⠀⠀⠙⠘⠘⠁
⠀⠀⠀⠀⢸⣽⠀⠀⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⢍⠿⠀⡼⣹⠃⠀⠀⠁⡉⠀⠀⠀⠀⠀⣿⣯⡆⠀⢰⣿⣟⡆
⠀⠀⠀⠀⢸⣺⣀⣀⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⢮⠬⡽⠁⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⣿⣿⢿⠀⢼⣿⡻⡇
⠀⠀⠀⠀⢸⢿⡿⢯⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠤⠁⠀⠀⢭⢎⢿⡅⠀⠀⠀⠀⡆⠰⠀⠀⠀⠀⠀⣿⡕⣗⢤⢽⣿⣪⡂
⠀⠀⠀⠀⢸⣸⠀⠀⣏⡇⠀⠀⠀⠀⠀⠀⠀⠈⡀⠂⡘⢀⠃⠀⠀⣯⠮⢫⡵⡄⠀⠀⠀⡄⠠⠀⠄⠀⠀⠀⣟⡇⢸⡿⣻⢸⣟⡇
⠀⠀⠀⠀⠘⠚⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠛⠃⠀⠻⠹⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠓⠃⠈⠋⠉⠈⠗⠁
⠀⠀⠀⠀⠀⠀⠀⣒⠀⢴⡃⠀⠀⠀⠀⣿⣿⢿⣿⣆⠀⠀⠀⢠⣾⣛⣿⣶⡀⠀⠀⠀⠀⠀⡠⠀⠐⠀⠀⠀⢠⣾⠿⣿⣦⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡎⣲⠀⠀⡆⢰⠀⠀⠐⡷⡇⠀⣿⣿⠀⠀⠀⣾⣿⠁⠘⣿⡇⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠸⣾⣄⠘⠛⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡄⢯⠀⠀⡃⠠⠀⠀⠀⣷⣷⣶⣿⠟⠀⠀⠀⣿⣟⢀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣉⠻⣿⢦⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠁⠭⠀⢠⠅⠌⠀⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠸⣿⣞⣿⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⡸⢻⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠌⠁⠀⠀⠀⠀⠿⠻⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠹⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⠿⠟⠃⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⣿⢷⠀⠀⣶⣶⠀⠀⢴⠈⡂⠠⠠⠠⠀⢒⢐⠀⠀⠸⠛⡧⠀⠘⡗⠀⠀⢲⣶⡆⠀⣾⡿⠁⠀⠀⣇⣈⣀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢻⢹⠀⢠⣯⡇⠀⠀⠸⠀⠅⢐⢀⠀⠂⡅⡸⠀⠀⠀⢩⣏⣴⢨⠇⠀⠀⠀⢻⣿⣼⣿⠁⠀⠀⠀⠀⢀⠌⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⢸⣿⣇⣸⣾⠀⠀⠀⠘⡀⠱⠅⡍⠇⠘⠁⠂⠀⠀⠀⠀⡷⢾⠠⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠎⡀
⠀⠀⠀⠀⠀⢀⠀⠄⡐⠀⠄⠀⠀⠀⠀⢹⢻⢲⡇⠀⠀⠀⠀⠇⠠⡪⠀⠠⠀⢐⡅⠀⠀⠀⣘⣟⠙⣘⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠸⠽⠽⠀⠀⠀⠀⠀⠈⠌⠞⠀⠘⠤⠰⠀⠀⠀⠘⠠⠏⠀⠹⠧⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠇⠆⠄⠆⠀⠰⠀⠰⠣⠀⠠⠡⠘⠀⠈⠰⠀⠀⠀⠐⠒⠉⠰⠂⠂⠲⠀⠐⠈⠉⠾⠀⠰⠀⠇⠀⠐⠒⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠢⠐⢆⠀⢀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⣦⣤⣤⡄⢀⣀⣀⣀⠈⠲⣶⣶⠀⣼⣷⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⢿⣿⣿⣿⠿⢇⢪⠴⠤⠷⢤⡀⣹⠉⠀⣹⠛⠻⡟⠋⡡⠀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠟⠻⢄⢀⣴⣿⢀⣴⠿⠀⠀⠀⠀⠀⠁⠀⠙⠣⠀⠑⠄⠁⠼⠃⠀⠙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣥⣤⣤⣶⡿⡛⢡⣾⠁⠀⠀⣶⣶⣿⣿⡀⠀⠀⠘⣦⢀⡀⠀⠊⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠂⢤⣶⣷⣿⣿⣿⣿⣣⠿⠛⠛⠂⠀⠀⠛⠛⢿⣿⡷⠤⠀⠐⠁⠸⠧⡠⠴⠋⠉⠀⣄⠀⢤⠆
⠀⠀⠀⠀⠀⢰⠌⠒⣼⣿⣿⣛⣍⢳⣰⠁⠀⠐⠟⢻⡄⠀⠀⠀⠈⠋⠀⠀⢀⡖⠀⣀⠀⠦⢆⡈⠐⡿⠉⢀⢀⣀
⠀⠀⠀⠀⠀⠠⠶⠸⣿⣿⣿⣿⡟⢈⣿⡀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⣤⠟⣏⠰⣿⣿⡇⢰⣄⣥⠀⠀⠰⠚⠃⠁
⠀⠀⠀⠀⠀⠠⣤⣤⣘⢯⠟⠋⡀⢼⣟⢿⣟⢿⡋⠀⠀⣰⡄⠀⠀⠀⠈⣷⣿⠆⣈⣇⡴⡎⢶⣴⡄⠠⡦⠘⠓⢶
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⠕⢤⣿⣾⢿⣌⡟⣬⠁⠀⢾⣿⣿⡀⠀⠀⠀⠘⡄⠑⣾⣿⡿⣷⢠⠌⠃⠷⠞⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠈⢹⣦⡘⠶⡆⠻⣯⣿⠟⢻⠆⡀⠀⠀⠟⠛⠃⠀⠀⠀⠀⠐⠀⠘⠏⡄⢻⣶⠀⣤⡸⠃⠘⣫⡄
⠀⠀⠀⠀⠀⠀⠀⣈⠻⡁⠐⣪⡿⡞⣿⣟⠟⠀⠛⠂⠒⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⠁⠝⢶⠀⣀⢿⠒⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠷⠿⢙⡿⢞⠚⡿⠌⢠⣿⣀⣴⡶⠖⢢⣿⣿⡄⠀⠀⠀⠈⠢⣧⠔⠂⠐⡲⣿⣿⣷⣤⣆
⠀⠀⠀⠀⢀⠀⣴⣤⣤⣴⢿⡇⢻⡷⠖⠀⣞⠛⡯⣋⡁⠰⣾⣿⣿⣷⡄⠀⠀⠀⠘⢿⡁⡄⢀⡐⣻⣷⣿⡿⠇
⠀⠀⢀⡴⠋⠛⡆⢈⣿⣥⢶⠾⣟⡙⠀⠸⣿⢶⠴⣿⠂⢀⢘⡉⣿⣿⣿⡄⠀⠀⠀⠈⠁⠦⠍⣁⠹⡿⠋⣤⣶⠒
⠀⠀⣎⠀⠀⣠⣿⣿⣿⣟⣻⠤⠿⠃⠀⠁⡭⠘⢆⠹⠃⣈⣉⠉⢳⣾⣗⣶⡄⠀⠀⠀⠰⣶⠄⡝⠰⢷⠀⠿⣇
⠀⠀⣼⠂⠀⣿⣿⣿⣿⡟⠣⣱⠶⣶⠚⣴⡆⠙⢌⣢⠀⢩⡌⠰⣰⢿⣶⠾⢛⢄⠀⠀⠀⠈⠠⣧⡁⣈⠀⢘⡿⠷⠄
⠀⠀⢹⡆⠀⠈⠿⢿⡋⢀⣠⣤⡀⠘⡇⠬⠶⠋⠀⢸⢡⡸⠇⠼⠷⠭⢿⢎⡐⠋⠁⠀⠀⣀⡀⠀⢀⡁⣤⠞⠁
⠀⠀⠀⠹⠦⠖⣀⣿⡀⠘⣿⣿⠋⢀⡾⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠓⢀⣀⣀⣱⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⡉⢻⡟⣙
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⢻⣄⢹⠿⠿⠿⠿⠿⠿⠿⠿⠷⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠾⠦⠴⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠁


⠀⠀⠀⠎⢥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠄⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⢲⣶⣦⡵⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡿⣴⠟⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣈⣛⣛⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⡧⣜⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⢁⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠆⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡆⠐⠁
⠀⢀⠀⡀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⠀⢀⠀⡀⢀⠀⣠⠀⡀⠀⠀⡀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⢁⡌⣠⣈⣠⢁⣜⣠⣤⣄⢡⣌⣤⣎⣤⣱⣿⣆⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠒⠒⠒⠒⠂⠈⡆⠉⠐⠒⠂⣶⢂⠐⠲⢆⠀⠀⠀⠀⠰⠅⠤⢉⠉⠉⠀⠀⠒⠒⠂⠀⠀⠠⠤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠄⠀⠘⠂⠀⠀⠀⠀⡝⠀⠹⢤⣀⡀⠀⠀⠈⠉⠒⠸⠧⣄⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⢺⢮
⠀⠀⠀⠀⠨⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⠀⠉⠉⠉⠉⠛⠒⠒⠒⠓⠒⠀⠠⠠⠤⠄⠁⢉⣀⣀⠀⠀⠀⠀⠐⠧⠤⡼⠑
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⢀⡶⠤⠀⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡤⠤⠤⠀⠒⠒⠒⠀⠉⠈⠉⠉⠉⠄⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠠⠤⠀⠐⠂⠈⣈⣀⢄⠉⢄⠀⢀⠔⠁⢸⠉⠁
⠀⠀⠀⠀⠘⠀⠀⣠⠀⠄⠰⡌⠢⠀⣀⣀⡠⠤⠤⡤⣤⡤⠤⠴⠶⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠙⠻⡈⠩⡆⠸⡾⠁⠀⠀⡎⠑⠢⢴
⠀⠀⠀⠀⠀⠀⢰⠂⣤⣔⠂⠧⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣼⠿⣶⢕⠀⡇⠀⠀⠀⣇⣠⠔⢘
⠀⠀⠀⠀⠀⠄⠘⣀⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠉⠉⠁⠀⣧⠀⠀⢸⠁⠀⠠⠃
⠀⠀⠀⠀⠀⢠⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣞⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠁⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣤⠰⠔⠶⠒⠉⠉⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡂⠀⣠⣂⢂⣀⠠⠤⠀⣠⡠⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠁⠉⠍⡂⠒⡩⠍⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⠀⠁⠒⢂⣃⠉⠤⠄⠐⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠀⠐⠂⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠄⠐⠒⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠱⠂⠐⠒⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠤⠀⠄⠀⠄⠠⠀⠀⠂⠐⠨⠀⠠⠀⠀⠀⠀⠨⠐⠉⠂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⠢⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⢉⡇⡷⠀⢀⡀⠀⡰⢩⢤⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠇⣠⠟⢿⠀⡇⠰⠬⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡼⠋⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠙⢅⠀⠀⠔⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⠙⠲⣄⠀⢸⠀⠀⠀⠀⠉⢀⣿⠀⠋⠀⠀⢠⠇⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠑⣼⠳⢄⣀⣀⡤⠞⠀⠓⠦⠤⣶⠋⢀⡤⠚⠉⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⣂⠀⢠⡾⠃⢀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢹⡖⠁⠀⠀⠀⠈⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠟⠀⠀⠀⠑⠠⠤⠤⠤⠴⠚⠁⠀⠘⣯⣐⣀⣀⣂⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡋⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠁⠀⠀⠀⠀⠀⢠⡾⡇⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣸⣥⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢳⡀
⠀⠀⠀⠀⠰⠏⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⠁
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⠀⠀⠐⡄⠀⠈⣇⠀⡇
⠀⠀⠀⢠⣞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠔⢠⠉⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⠒⠊⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣶⣶⠾⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀
⠀⠀⠉⠓⠒⠒⠒⠒⠒⢲⣲⢶⡚⠋⠛⠧⠤⠤⠤⠴⠦⠀⠀⣀⡤⠊⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢑⡄⠀⠀⠀⢠⢉⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠤⠤⠀⢨⣉⠁⣳⣾⡀⢀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⣄⠀⠀⠀⠠⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⡌⢠⠁⠀⠀⠸
⠀⠀⠁⠀⠈⠉⠈⠉⠀⠉⡙⠉⡟⠋⠉⠛⠓⠒⠛⠓⠒⢠⠻⠈⠀⣃⠀⠑⢦⡀⡼⠁⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄⢠⠁⡜
⠀⠀⠀⠀⢀⢤⣄⠀⠀⠸⠁⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⢹⡁⢈⣤⠟⠃⠀⠀⠘⠣⠤⠔⠒⠦⢄⣀⣀⡠⠔⠊⠀⡈⢀⠇⠀⠀⠀⡇
⠀⠀⠀⠀⣹⡠⢀⡌⡁⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢑⠢⡀⠀⢈⡽⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⡁⣸
⠀⠀⡴⡾⣋⢵⢏⡸⠃⡤⠋⠙⠃⠠⠲⠂⠀⠀⢀⠄⠫⠠⡢⣑⠞⠉⠁⠀⠀⢀⣀⡀⢈⠻⠎⠀⠀⠀⠀⢄⡤⠙⣉⠞⠁⢀⢃⠇⠀⠀⢠⠀⣀
⠀⠀⠧⠬⠕⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣘⠄⢺⡇⠀⠀⠀⠀⠀⠈⠉⢋⣁⠔⠀⠀⢐⡤⢟⠳⢀⢗⠚⠃⠀⣴⡼⠀⠀⠈⠛⣗⠻
⠀⠀⡠⣤⣤⣀⣀⢎⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⡁⠀⠆⠀⠀⠀⣠⠴⣉⠀⠴⠢⠉⠀⠠⠀⠄⠁⢇⡀⠀⠀⠁⣅
⠀⠀⢱⠛⠛⠛⠚⠐⠢⠭⠭⢶⣒⣒⡀⠀⠀⠒⠒⠒⠒⠒⢒⣦⠴⠆⠀⠉⠁⠀⠀⢀⣤⡾⠗⠋⠁⠀⠀⠀⠀⠀⠀⠚⠂⠠⠂⣬⡹⠀⣖⡾⠇
⠀⠀⠀⠑⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⣀⣤⣠⣤⣄⣄⣀⣀⣀⣀⣉⣄⠀⠀⠀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠃⠀⠹⠀⠀⠀⡀
⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⢀⣤⠄⢠⢀⠀⠀⠀⠀⠀⠀⠸⡅⠠⠆⢀⠆⢠⠆⠀⠀⠀⠀⠁⠈⠙⠲⢬⣆⠀⠀⠀⡀⠀⠇⠀⢀⠆⠀⢠⢀⠀⠀⢀⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠂⠀⠐⠀⠀⠀⠀⠙⣆⢀⠀⠀⠫⣤⣤⣤⠉⠁⡄⠀⡂⠀⠀⢍⠳⣦⡀⠱⡀⢀⠄⠊⠀⠀⠀⠘⠃⠀⠀⠤
⠀⠀⠀⠀⢀⡀⣀⠀⢀⠀⠀⣀⡀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠘⢦⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠈⠂⠙⢦⡙⠁
⠀⣴⠚⠟⠌⠁⠍⠀⡌⠀⢐⠃⠀⠏⠉⢐⠂⢤⣄⡀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⣀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠹⣾⢠⠄⡘⠀⠰⠀⢀⡁⠀⠘⠀⠀⠠⠀⡘⠀⠻⠑⡶⣀⡒⠒⢂⠀⠉⠓⠒⠒⠒⠒⠦⠤⢬⣥⣤⣔⣒⣂⣐⣢⣼⠃⠀⠀⠀⠀⢠⠀⢒⣤
⠀⠀⠈⠻⣄⡂⠀⡒⠀⠠⠀⠀⠉⠀⠀⠂⠀⢤⡄⢧⠤⢤⣌⡛⢦⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠂⠐⠒⠃⠐⡄⣲⠀⣄⣘⡄
⠀⠀⠀⠀⢎⠙⠒⠮⠄⣒⠀⠠⠅⠀⡚⠀⡴⠀⡬⠉⣒⠒⡤⠏⡀⣺⡆⠀⠀⢀⣠⠤⠤⠐⠀⠉⠉⠀⠀⠀⠀⠀⣀⠠⢄⠒⠒⠉⣄⠘⠠⠷⡀⠃
⠀⠀⠀⠀⠀⢈⡁⠐⠒⠋⠀⠉⠐⠖⠂⢚⡤⠚⠃⢲⡓⠻⣛⣛⡻⠁⡄⠙⡥⠉⠔⠀⢁⢀⠌⢢⠐⠄⠒⠂⠈⠀⠀⠂⠁⠀⠀⠀⠀⢘⠀⡧⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠅⠈⠓⣴⠃⠂⢠⠟⡝⢧⠠⠄⡇⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⢣⠐⡄
⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠠⣤⠤⠤⠄⠀⡀⠀⠀⠠⠀⠀⠛⡄⢃⠈⠦⠥⠋⣀⢁⠇⠀⠠⠂⠀⠁⠀⠀⠐⠌⢆⠀⠀⠀⠘⠀⠐⡀
⠀⠀⠉⠂⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠙⢌⠁⠼⠄⠐⣠⠞⠀⠀⠆⠄⠀⠠⠠⡀⠀⠀⢸⠀⠀⡄⠀⡇
⠀⠀⠀⠄⠀⠤⠜⠄⠠⠀⠄⠠⠤⠄⠀⠄⠀⠀⠄⠀⠀⠀⢀⠀⠀⣀⣀⣀⣀⠰⣉⠑⠂⠉⠀⠀⠀⠀⢃⠀⠀⠑⠐⠀⡀⠄⠄⠀⢠⠰
⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠠⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⣀⣈⣈⡀⠀⣀⡀⣀⠀⠓⠀⡀⠀⡀⠤⠊⠀⠀⣚⠀⣇⠀⠀⠀⠁
⠀⠀⠀⣄⠀⠠⠀⣤⠄⠒⠒⠖⢤⡤⠒⠒⠀⢠⣔⠂⠀⠐⠒⣴⡂⠀⠒⠒⠠⣀⡀⣉⣀⡀⠀⠀⠀⠀⠀⠙⠒⠤⡄⡀⠀⢠⠚⠙⠀⠠⡄⠈⠰
⠀⠐⠋⠉⢹⠒⠂⠙⠛⢶⣶⠞⠛⠟⠒⢶⡶⠚⠛⠳⢷⡖⠛⠛⠛⢶⡶⠒⠚⠭⣇⠚⠿⠤⣄⡀⠀⠀⠀⠀⠀⠀⠃⠂⢠⢧⣀⠖⠤⠔⠀⠀⠉⠂
⠀⠠⢞⠑⠈⠉⢻⡏⠉⠉⠙⠳⣶⠖⠞⠛⠛⢲⡆⠚⠛⢳⣶⠖⠂⠉⠓⢲⣤⠒⠺⠥⢤⡔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠚⢧⡄⡈⠀⢀⣠⡀⠤⠄⠘⠄
⣀⣀⣀⣈⣉⣉⣁⣉⣐⣒⣈⣉⣈⣐⣛⣛⣂⣉⣙⣒⣚⣉⣉⣁⣚⣛⣒⣈⣑⣂⣀⣀⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣈⣉⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⢀⣤⣤⣤⣀⠀⠀⢀⣤⣤⣤⣄⠀⠀⢠⣤⣤⣤⡀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⣸⡟⢻⣇⠀⠀⢸⣿⣯⣽⣿⡃⠀⢰⣿⡏⠉⠛⠃⠀⠀⢸⣿⡏⢹⣿⡆⠀⢸⣿⣿⣭⣥⠀⠀⣾⣿⣯⣭⠀⠀⢰⣿⡏⣩⣿⣥
⠀⠀⠀⠰⣿⠟⠻⣿⠆⠀⠘⣿⣿⣾⣿⠇⠀⠈⠿⣷⣶⡿⠇⠀⠀⠸⣿⣷⣾⠿⠁⠀⠸⣿⣷⣶⣶⠀⠀⢹⣿⠃⠀⠀⠀⠈⠿⣷⣾⣿⠟

⠀⠀⠀⠀⢰⣿⣆⣸⣿⡆⠀⢰⣿⣧⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⢸⣿⣤⣾⠟⠀⠀⢰⣿⡆⠀⠀⠀⢰⣿⣷⣠⣿⣿⡀⠀⢰⣿⣦⣰⣿⡆
⠀⠀⠀⠀⢸⣿⡟⢻⣿⡇⠀⢸⣿⣿⠀⠀⣤⣄⣿⣿⠀⠀⠀⠀⠀⢸⣿⠿⣿⣦⠀⠀⢸⣿⣅⣀⡀⠀⢸⣿⢿⣿⣿⣿⠇⠀⢸⣿⠻⣿⣿⡇
⠀⠀⠀⠀⠈⠛⠁⠈⠛⠁⠀⠀⠙⠁⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛⠀⠈⠛⠁⠀⠈⠛⠛⠛⠁⠀⠈⠛⠈⠛⠈⠛⠀⠀⠈⠛⠀⠉⠛⠁
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⢰⣿⠟⠛⣿⣆⠀⢸⣿⣏⣿⣿⠀⠀⣾⡟⠛⢻⣷⠀⠀⠀⢠⣿⣟⣹⣿⠆⠀⢸⣿⣯⣿⡃⠀⠈⠛⣿⣿⠛⠀⠀⠀⢸⣿⡇⠸⣿⡇
⠀⠀⠀⠘⢿⣧⣴⡿⠇⠀⠸⣿⡏⠉⠁⠀⠀⠿⣷⣾⣿⣿⠀⠀⠀⠘⣿⡏⢻⣿⡀⠀⠰⣾⣭⣿⡟⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⢿⣧⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⣠⣄⠀⣠⣄⠀⢀⣤⡀⣠⣄⢀⣤⠀⠀⠀⣤⣤⡀⣤⣤⠀⠀⢠⣤⡄⢀⣤⡄⠀⠀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢹⣿⣠⣿⠏⠀⠈⣿⣷⣿⣿⣾⡿⠀⠀⠀⠈⣿⣿⣿⠃⠀⠀⠀⠻⣿⣿⠟⠀⠀⠀⠈⣩⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠹⣿⠇⠻⣿⠇⠀⠀⠀⣼⣿⠛⢿⣧⠀⠀⠀⠀⢿⣿⠀⠀⠀⠠⣾⣿⣷⣶⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡁⠀⡀⢰⡀⠀⢢⣌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⡄⠀⠚⠛⠒⠴⣯⡜⠂⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⢛⠀⠐⠁⠀⠀⠈⠛⢦⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢠⠄⢠⡀⠰⠋⢀⡌⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠁⠀⠄⠀⠈⠀⠀⠑⣆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠘⠀⠀⠀⡀⠀⠀⠈⣿⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⡸⠀⠀⠀⠀⠠⠀⠀⠀⡅⠰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠁⢀⠀⠀⠇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠆⠀⠀⠠⠀⠀⠀⠰⢌⣄⡀⢈⠀⢸⠎⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠨⡀⣀⡀⠀⠀⠀⠀⠖⠒⠀⢀⣈⢙⣧⣸⡀⠋⠄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀⠀⠀⠀⠀⠀⢂⡀⠀⠐⠈⠀⡠⠀⠘⢙⢷⣅⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⢸⠉⠓⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠈⠀⠀⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢃⢸⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄⡀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠈⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢾⣆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠤⠤⣦
⠀⢻⣆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀
⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣠⠊⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿⣿
⠀⠀⠀⠹⣧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂⠈⠀⠀⠀⠀⢀⣿⣁⣤⠀⢀⣴⣶⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡷⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠘⣷⡄⠀⠁⠀⢀⠠⠐⠉⠀⠀⠀⢀⣠⣤⣶⣾⡿⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠘⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡇⠀⣠⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⠟⠋⠁

⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⡿⠛⠻⣷⠀⠀⠀⢾⣃⣀⡈⡷⠀⠀⢠⣿⠋⠉⢻⡇⠀⠀⡐⣡⠀⢠⣅⠀⠀⢰⡏⠀⠉⣧⠀⠀⠀⣰⠉⠛⢁⣄
⠀⠈⠇⠂⠐⠨⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⢰⣏⢙⡛⢹⣆⠀⠀⠹⣌⣤⣌⡜⠀⠀⢸⣇⠁⢈⣿⠀⠀⠀⠉⠛⠀⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠹⠿⠶⠰⣿⠃⠀⠀⠀⠈⠛⠋⠀⠀⠀⠸⠿⠃⠻⠿⠀⠀⠀⠀⢐⠶⠆
⢀⠀⢀⣤⣤⠀⢀⠀⠀⠀⡤⣶⠤⠐⠀⠀⠀⢠⡖⠀⡆⠀⡀⠀⠀⣿⠀⣶⠀⠀⠀⠠⣦⢰⡆⣶⠀⠀⠀⠀⢠⡁⣉
⠀⠂⠘⠳⢦⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠁⢸⡇⣸⠇⠀⠀⠀⠀⢿⣠⠏⠀⠂⠀⠁⢿⣸⣧⡿⠀⠀⠀⠀⣸⣷⠋
⠀⠀⠈⠳⠞⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠐⠃⠸⠇
⠀⠀⢀⣤⣤⣀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣿⣟⠉⡙⡇⠀⠀⡿⢛⠋⡉⢿⠀⠀⠀⡟⠉⠀⠉⢷⠀⠀⠚⠉⠈⠉⠃⠀⠀⠀⡼⠉⠉⠙⡄⠀⠀⢠⣾⠟⢿⣦
⠀⠀⠁⠉⠀⠁⠀⠀⠀⣿⣄⠠⣀⣾⠀⠀⠀⢃⠀⠀⠀⠄⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠋⠁⠀⠉⠃⠀⠀⣼⠁⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠷⠎⠀⠀⠀⠀⠀⢑⠒⡃⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⣀⣐⡂⠀⠀⠀⠀⠀⣀
⠀⠀⢠⡄⢠⡄⢀⠀⢀⠀⠀⢒⣶⠂⢀⠀⠀⠀⠀⣤⠀⠉⠀⠀⠀⢉⣥⡄⠀⢀⠀⠀⠀⢠⣤⡄⠈⠀⠀⠀⠀⣠⠄
⠀⠀⠀⢻⡾⠁⠀⠀⠀⠀⢠⣞⣁⠀⠀⠀⠈⠀⠀⡟⠀⠀⠀⠀⠂⠈⣠⠏⠀⠀⠀⠐⠀⠀⠺⡅⠀⠀⠀⠀⣼⣱⣇
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠃⠀⠀⠁⠀⠀⠀⠿⠛⢁⣀⡀⣀⣀⡸⠾⠃⠨⡄⣀⣀⡀⠸⠁⠀⡀
⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⡩⣉⡉⠁
⠀⠀⣴⠟⠛⠻⡆⠀⠀⢰⡿⠛⠛⢻⡆⠀⢠⡿⡛⢛⣻⡄⠀⠀⢠⡾⠿⢿⣦⠀⠀⠀⣶⠶⠶⢶⡄⠀⠀⠰⠁⠤⠤⠌⠆
⠀⠀⠥⠀⠀⠀⡅⠀⠀⠈⠟⠁⠘⢺⠃⠀⠸⣧⠀⠀⣼⠇⠀⠀⠙⠘⠀⠀⠹⠀⠀⠀⡏⠒⠐⢊⡃⠀⠀⠐⠈⠀⠈
⠀⠀⠀⢀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⣠⢤⡄⠀⠀⠀⠀⠈⢁⣉⠀⢀⠀⠀⠀⢈⣤⡀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠁⠸⠶⣦⠀⠀⠀⠀⠀⡷⢶⡀⠐⠀⠠⠀⠀⣼⠀⠀⠀⠀⠀⠀⣻⣼⡁⠀⠀⠀⡀⠸⠤⣿⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠐⠶⠛⠀⠀⠀⠀⠀⠛⠞⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠿⠾⠇⠀⠀⠀⠀⠀⠲⠋⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⣀⣀⠀⠒⢡⣄⣀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠉⢄⠠⢠⠤⠐⠤⠎⠀⠀⠝⠛⠃⠤⣉⠒⠄⡀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠐⠀⠀⠀⠀⣈⣤⠆⠪⡵⡀⢀⠃⠃⠈⡆⠀⠀⠀⠉⠦⢌⠒⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⠀⠀⠀⠸⣀⠀⠀⠀⡠⠭⢋⡘⢤⡵⡇⠀⠀⠀⡰⠒⠀⢱⢄⠑⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠁⠈⠀⠀⠀⠀⠀⠀⠁⠢⠜⠀⠀⠚⠋⢿⠀⠳⣄⣢⠞⠀⠀⡀⢄⠌⠓⣄⠢⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠖⠋⠀⠀⠀⠀⠀⠀⢀⠅⠎⠌⢀⡟⢂⠐⡀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⠖⠲⠶⢄⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠂⠈⠀⠸⢤⡾⡄⠰⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⢣⠀⠀⢰⠋⠀⠀⠀⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⠕⠁⡀⠐⡠⡥⠻⠿⡀⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢹⠆⠀⠀⢮⣤⠀⠀⠀⠀⠀⠀⢀⡠⠊⢔⠕⡁⡠⠀⢊⢘⠅⠀⠀⠀⠅⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣄⠂⢸⡆⠀⠈⠃⠀⢳⡀⠀⠀⡠⢔⠡⣊⠔⡡⡈⢈⢄⡑⠋⠁⠂⠂⣀⣀⡀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠱⠦⣀⣀⠀⢀⠠⢒⠥⢂⠥⢊⠔⡀⢊⢔⠅⠈⢇⠀⠀⠀⣸⠉⠌⠆⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⢮⡔⠨⢀⠨⢐⠁⣈⠔⡡⠂⠠⠈⣓⣴⣾⢡⣤⠶⡆⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⡂⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠐⡨⠔⣈⠄⣈⢔⡡⠢⡾⠋⠀⠀⠀⠀⠠⠀⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡐⠀⠤⠄⢂⡨⠅⡞⠀⠀⠀⠀⠀⠀⠀⢰⢒⡊⠅⡐⠅⢒⠩⡠⢑⠠⣺⠁⠀⠀⠀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠌⠩⠀⠢⢕⡃⠀⠀⠀⠀⠀⠀⣠⠭⢀⠀⠁⡀⠈⡐⠅⢊⠤⣪⠌⠀⠀⢀⡀⢀⠀⠠⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠂⠀⠐⡄⠀⠀⠀⠀⡀⠔⠓⠈⠄⠒⠥⠐⠉⠄⡪⢔⠨⢈⢗⢶⡀⡗⠠⠃⡠⠁⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣒⣨⠆⠀⣠⠊⠉⠀⠒⠨⠄⡒⠬⢐⡈⠄⡂⠥⡂⢔⠡⣾⣁⠖⠀⠊⠀⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⣔⣀⠠⠄⠂⠈⠅⢒⠈⠅⡂⠅⠂⠅⣪⡅⠜⢋⠡⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⢉⠘⠃⠬⢍⣉⠒⠒⠀⠶⠒⠀⠒⠛⢂⣉⠠⠔⠂⠁⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣙⣒⣶⠶⡆⠐⠐⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠹⠾⠮⣕⡉⠂⢀⠩⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⢀⣔⡃⠤⠄⠀⢤⣀⣀⣅⣈⣄⣪⢕⣬⣢⣽⢦⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠛⠩⠒⠲⠒⠲⠖⠶⠆⡶⠼⠼⠶⠞⠺⠛⠁⠊⠀⠂⠀⠊⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⣴⣶⡄⠀⣴⣶⣦⡀⠀⣴⣶⡄⠀⢰⣶⣤⡀⠀⠀⣴⣶⣤⠀⠀⣤⣶⣦⠀⠀⣴⣶⣦
⠀⠀⠀⠀⣼⡏⣿⡇⣰⣿⣶⣿⠃⣼⡿⠿⠏⠀⣿⡟⢻⣷⠀⣾⣿⣯⡉⠀⢸⣿⣿⡧⠀⣼⡿⣛⣛
⠀⠀⠀⢰⣿⠿⣿⡇⢿⣯⣼⣿⠆⢿⣧⣾⡆⠰⣿⣷⣿⠟⠀⢿⣿⣿⡁⠀⣿⡟⠁⠀⠀⠻⣷⣿⡿⠃
⠀⠀⠀⠈⢉⡀⢈⣄⢀⣭⣭⣍⠀⣠⣭⣍⠀⣠⡌⢉⡁⠀⣠⡄⠉⠉⠀⠀⣩⣄⣀⣤⡀⠀⣨⣉⢀⣄
⠀⠀⠀⢀⣿⣇⣿⣿⠉⢻⣿⠛⠁⠙⢻⣿⠃⣿⣿⣿⡟⢠⣿⣷⠀⠀⢀⣾⣿⣿⣿⣿⡇⢠⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣀⣼⣿⡀⢠⣤⣼⡟⠰⣿⣿⣿⡄⢸⣿⣧⣤⡄⣾⣿⣿⣿⢿⣿⡇⣼⣿⣿⣿⠇
⠀⠀⠀⠈⠛⠋⠛⠋⠛⠻⠛⠃⠈⠛⠛⠁⠀⠛⠋⠙⠛⠈⠛⠛⠛⠁⠛⠃⠙⠋⠀⠛⠀⠛⠃⠙⠛
⠀⠀⠀⢀⣴⣶⣦⡀⢀⣴⣶⣦⠀⣠⣾⣷⣆⠀⣠⣶⣶⣄⠀⣠⣶⣶⡄⣶⣾⣷⡶⢀⣶⡆⣶⣦
⠀⠀⠀⣾⡿⠉⣿⡇⣼⣿⣤⣿⢻⣿⢃⣸⣿⠀⣿⣧⣼⡿⠀⠙⢿⣿⡁⢰⣿⡿⠀⢸⣿⣇⣿⣿
⠀⠀⠀⢿⣿⣾⡿⠁⣿⣿⠉⠁⠸⣿⣿⣿⣿⠀⣿⡟⢿⣦⠀⢿⣿⣿⠇⠘⣿⡇⠀⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣧⣰⣿⠆⣸⣧⣰⣿⣦⣾⡇⢼⣷⣶⣿⡆⢠⣶⣄⣴⣶⡀⢴⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⡿⠁⢨⣿⣿⣏⠀⠈⢿⣿⣿⡿⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠀⠀⣿⣿⢿⣿⡿⠁⠠⣿⡿⠻⣿⡆⠀⣴⣿⠟⠁⢺⣿⣿⣿⣶

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

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣄⡀⠀⠀⢀⣤⣤⣄⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⢸⣇⣈⡷⠀⠀⣾⠁⠙⠃⠀⠀⣿⠀⢹⡆⠀⢸⣇⣀⠀⠀⠀⢸⣁⡀
⠀⠀⠀⠀⠀⣼⠿⣧⠀⠀⢸⡏⠙⣧⠀⠀⢿⠀⢀⡀⠀⠀⣿⠀⢸⠇⠀⢸⡏⠉⠀⠀⠀⢸⠋⠉
⠀⠀⠀⠀⠀⠛⠀⠛⠀⠀⠘⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠛⠛⠋⠀⠀⠘⠛⠛⠛⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⣴⠀⢰⡆⠀⠀⢶⡶⠄⠀⠀⠀⠰⣶⠶⠀⠀⢴⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⣅⠀⠀⣿⣤⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡟⠀⠀⣿⠀⢸⡇⠀⠀⣸⣇⡀⠀⠀⣦⣀⡿⠀⠀⠀⣾⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠈⠁⠀⠀⠉⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⡄⢀⣿⠀⠀⢸⣧⠀⣿⠀⠀⣴⠟⢳⡄⠀⠀⣿⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣧⣸⣿⠀⠀⢸⡟⣧⣿⠀⠀⣿⠀⢸⡇⠀⠀⣿⠶⠛
⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⡄⠀⠀⡿⠹⡏⢿⠀⠀⢸⠇⠸⡿⠀⠀⠹⢦⡼⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣧⠀⠀⢸⣇⢈⣿⠀⠀⢸⣏⠙⠃⠀⠀⠀⣽⠁⠀⠀⠸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⢸⡏⠙⣷⠀⠀⢀⡉⢹⡆⠀⠀⠀⣿⠀⠀⠀⠀⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠘⠃⠀⠛⠀⠀⠈⠛⠛⠁⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⢰⡆⢠⡆⠀⠀⢶⠀⢰⡆⢀⡆⠀⠀⢰⡄⣰⠆⠀⠀⢶⡀⣴⠀⠀⠀⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠈⣧⣼⠃⠀⠀⢸⡇⡿⣷⣸⠇⠀⠀⠀⢿⡏⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢠⡟⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣫⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣭⣭⣉⠹⢤⠤⠴⠶⠖⢶⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡉⠙⣿⣿⣦⣴⣺⣿⢟⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢁⣴⡿⠿⢿⣿⣏⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⣠⠟⠁⠀⢠⣼⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢀⡼⠃⠀⡜⠓⠦⣄⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⡞⢀⡾⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⣴⣶⣿⣿⣿⣷⣶⣬⣉⠒⠦⠼⠃⣼⠁⢠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣷⣦⡀⢠⡏⠀⡏
⠀⠀⠀⠀⣠⠤⣤⠀⢰⡇⠀⣿⣿⣿⣿⡿⠂⢰⣿⣿⣿⣿⣿⣿⣿⣾⡇⢸⠃
⠀⠀⠀⠘⠷⠀⢸⠇⢸⡇⠀⢿⣿⣿⣿⣶⣦⣌⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠻⣦⠘⣇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣦⣀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⢿⣿⣿⣶⣤⣄⣀⡀⠀⠈⠉⠛⠻⢿⣿⣿⣅⡛⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠈⠙⢿⣿⣦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠙⣿⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⣿⣿⣟⣚⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣴⠟⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣼⡿⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⠿⠶⠶⠶⠶⠶⠿⢿⣿⣿⣿⣿⣿⣯⣵⠾⣟⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠶⠖⠛⠋⠉⠛⠶⠶⠶⠦⣤⣤⣭⣭⣭⠴⠶⠛⠉

⠁⠀⠀⡤⠀⠀⠀⠀⠀⠐⠘⠀⠀⠀⠀⠀⠃
⠀⠰⢸⡀⡄⠰⠐⢲⠶⠀⢰⠂⠀⢰⡆⢰⣶⠀⠠⣖⠐⠂⠁⠀⠀⠀⠀⠀⢠⠶⠎⣅⡀⢠⣄⣠⣤⡤⠄
⠀⠶⠃⠀⠦⠿⠆⡰⠄⠀⣬⠰⡦⠸⠇⠘⠤⠈⠁⠽⠄⠻⠆⠀⠀⠀⠀⠀⠘⢤⠆⡇⠦⠸⠇⠸⠅⠷
⠀⠀⠀⠀⡀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⢠⡄⠀⠀⠀⠀⠀⡄⠄⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡄⢀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠂⠈⠀⠒⠃⠀⠀⠃⠃⠐⠂⠀⠀⠂⠀⠀⠂⠀⠀⠃⠊⠀⠐⠂⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠄⠀⠀⠀⡆⠀⠀⠸⠀⠃⠘⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⡀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠀⠈⡆⠀⠀⠀⠓⠁⠆⠀⠀⠀⠀⠀⠂⠐⠄⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠠⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⣀⢆⡐⠀⣀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠃⠀⠄
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠁⠀⠁⠀⠉⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠃⠀⠀⡀⠀⠁⠀⠈

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⡏⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠈⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠓⡶⠛⠛⠓⠒⠒⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠒⠒⠛⠛⠓⡖⠋
⠀⠀⠀⠷⠤⡶⠶⠶⠶⠤⠤⢤⣤⣉⣛⠲⢤⣀⠀⠀⢸⠀⠀⢀⣠⠴⢚⣋⣡⣤⠤⠤⠴⠶⠶⠶⡶⠴⠇
⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠓⠒⠚⠛⠿⠷⢿⣷⣦⣼⣤⣶⣿⠶⠿⠿⠛⠒⠚⠛⠛⠛⠛⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉




⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠈⠑⢦⡠⠚⢶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠑⢍⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠑⢬⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡑⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠢⡀
⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⢘⡆
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣠⠚⢄
⠀⠀⠸⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⠞⠁⣀⠔⡇
⠀⠀⡆⠀⠀⠈⠒⢄⡀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⡔⣡⠔⠋⠀⠀⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠈⠒⢤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⠞⠉⠀⠀⠀⠀⠀⢰
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡠⠔⠢⢄⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄
⠀⠀⢸⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⡇
⠀⠀⢸⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⡇
⠀⠀⠀⠙⠒⠒⠒⠒⠒⠀⠀⠀⠠⠤⠴⠶⠶⠦⠤⠤⠤⠤⠴⠶⠶⠤⠤⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠛⠁

⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⢠⠀⠀⡀⠀⡀
⠀⡌⠀⣄⠀⠀⠀⠀⢰⠀⠘⠃⡄⠀⠀⢰⠁⠠⠈⠖⠀⠀⠠⡀⠀⠀⢠⠀⠀⠀⠼⠀⠐⠒⠊⠀⠀⠀⠀⠀⠙⠛
⢠⠃⠀⡀⡀⠀⠀⠀⠸⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠏⠀⠀⠀⠆⠀⠠⠰⠃⠀⠀⠐⠀⠀⠓⠁
⡴⢤⠄⠐⡁⠤⠀⠀⢠⠀⠀⡠⠔⠀⠀⠀⠠⢀⠀⠤⠂⠀⠈⠀⠀⢀⠠⠊⠀⠀⢠⠢⠤⠤⠤⠀⠀⠀⣀⡤

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡔⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⢰⠁⠀⠊⠀⠀⠀⠀⡇⠀⠀⠓⠀⠀⠀⠸⠥⡄⠀⡤⠀⠀⠀⠗⠔⢰⠀⠀⠀⠀⠁⠀⠊
⡟⠀⠰⣂⡤⠀⠀⠀⡇⠀⠘⡼⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠠⡆
⠀⠄⠀⠀⢀⠔⠀⠀⡧⢀⠀⡧⢀⠀⠀⢠⣎⡁⣀⢀⠀⠀⠀⣴⣀⣀⠤⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠘⠧⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠉⠁⠉
⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⡉⠀⠀⠈⠁⠀⠀⠀⠰⠀⠀⠀⢻⠀⠀⠀⡘⠀⠐⠩⡄⠀⠀⠀⢀⠀⠀⠙⠁⠀⠀⡰⠁⠀⠒⡂⠀⠀⠀⠈⠁⠀⠐⠃
⠉⠀⠠⠇⠴⠁⠀⠀⢠⡄⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠒⠋⠀⠀⡀⠀⠀⠠⠉⠀⠀⠀⠐⡆⠀⡢⠀⠁
⢰⣄⠀⠀⠀⠆⠀⠀⢸⡇⠀⠀⣱⠄⠀⠀⠈⠅⠀⠀⣀⠄⠀⠀⢸⢥⡄⠀⠀⠀⠀⠀⠄⢀⢀⣤⡄⢠⠄⠀⣧⠄⠈⣇⢤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⢀⡂⠀⠀⡠⠀⠀⠀⢠
⠊⡀⠈⠓⠀⠀⠀⠒⡆⠀⠀⠂⠀⠀⡄⠀⠀⠇⠀⠀⠈⠀⠀⢀⠇⠀⠀⠀⣇⠀⢠⠃⠀⠀⠀⡇
⢀⠈⠠⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⠇⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢋⠀⠀⠀⠰⠀⠀⠁
⠡⢀⠀⡠⠀⠀⠀⠀⣧⠠⠀⠀⠀⠀⠁⠀⠀⣠⠄⠀⠀⠈⣔⣤⠄⠀⠀⠀⠈⡇⠄⠀⠀⢑⣤⠤

⠀⠀⠀⣠⠀⠀⠀⢀⠄⠀⠀⡀⠀⠀⠀⡄
⠐⠐⠀⠀⠀⠀⠀⠀⠡⡀⠀⠁⠀⠀⠀⠉⠀⡱⠁
⠀⡸⠀⡠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡐⠁
⠴⠢⠂⠐⣀⣀⠂⠀⠀⠀⡗⠀⠀⠀⠀⡜⠀⠀⠀⣀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⠤⠴⠶⠶⠶⠦⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⢤⠀⣤⠀⡤⠤⠤⣀⠀⣀⡀⢀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠈⠤⠌⡆⠃⢂⢰⠚⢸⡁⠠⢾⡘⠦⡄⢀⠤⣷⢊⣉⠑⣖⣩⣳⡞⢦⠜⡆⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⠀⠶⡄⡇⠀⢸⢸⣀⠀⠉⣙⠆⢸⠀⠀⢸⠀⡇⢾⢀⡇⣭⣬⠳⡘⣦⡞⠁⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⠀⠀⠀⠈⣤⡄⣇⡠⠶⠭⣤⡤⠗⢯⣥⠤⠋⠀⢀⣘⠇⠱⢤⣭⠴⠻⠇⠱⠟⣟⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠙⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠍⠉⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢉⣛⣓⣒⣒⣒⣒⣒⣒⣒⡀⣀⡀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣭⣉⣉⣉⣙⣓⡒⠒⠲⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⢀⣀⣀⣀⣠⣄⣠⣤⣤⣤⣭⡭⠭⠭⠥⠬⠉⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⣿⣿⣷⠀⠀⠀⢎⠀⣧⢈⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢹⣿⣿⣇⠀⠘⠀⢩⡾⢃⠴⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢿⡿⠿⢄⠄⡐⠋⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢸⣿⣷⣦⡈⠓⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠄⡈⠛⢛⣿⣷⡌⣒⣀⠥⡒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢾⣿⠀⠀⠂⢀⠔⢨⠋⣻⣀⡤⠖⠁⠈⢣⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡛⣽⣏⣀⣘⣇⢘⣲⠑⢂⣀⣸⡇⣰⠊⣁⣘⢻⣵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡢⠀⠑⣿⣿⣻⡟⣟⢻⠛⠷⡏⣟⢹⣛⣿⣿⣿⡿⡟⣚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠛⠉⢉⣿⣉⡆⠁⠈⠀⠀⠀⠀⠀⠛⢤⣿⣿⣿⡇⠈⠩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⠰⠉⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠔⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨⣐⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⣸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⠢⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⢉⡇⡷⠀⢀⡀⠀⡰⢩⢤⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠇⣠⠟⢿⠀⡇⠰⠬⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡼⠋⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠙⢅⠀⠀⠔⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⠙⠲⣄⠀⢸⠀⠀⠀⠀⠉⢀⣿⠀⠋⠀⠀⢠⠇⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠑⣼⠳⢄⣀⣀⡤⠞⠀⠓⠦⠤⣶⠋⢀⡤⠚⠉⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⣂⠀⢠⡾⠃⢀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢹⡖⠁⠀⠀⠀⠈⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠟⠀⠀⠀⠑⠠⠤⠤⠤⠴⠚⠁⠀⠘⣯⣐⣀⣀⣂⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡋⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠁⠀⠀⠀⠀⠀⢠⡾⡇⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣸⣥⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢳⡀
⠀⠀⠀⠀⠰⠏⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⠁
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⠀⠀⠐⡄⠀⠈⣇⠀⡇
⠀⠀⠀⢠⣞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠔⢠⠉⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⠒⠊⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣶⣶⠾⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁