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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⡀⠁⠀⠀⠉⠀⠀⠀⣀⠀⠀⠠⠤⡦⠀⣶⠁⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡴⠁⠀⢠⣴⠀⠀⠀⠈⣠⣤⡀⠀⠀⠀⠀⣿⠀⠁⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠁⠀⠐⠿⠋⠀⠀⠀⠠⠿⠛⠀⠀⠀⠀⢴⡏⠀⠀⠈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⠀⣼⠇⠀⠀⠀⠈⠀⠈⠁⠀⢸⣷⠀⠀⢠⡄⠻
⠀⠀⠀⠀⠀⢀⠀⠀⠸⠋⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣼⠋⠀⠀⠀⠠⣶⣶⣄⠀⠀⠀⢀⠀⠀⡈⠀⢠⣷
⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⣠⠀⠀⠀⠸⣿⡇⠀⠀⠀⠊⠀⢀⠀⣠⣾⣥⣤⠆⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠀⣰⣿⠃
⠀⠀⠀⠀⣿⢠⡷⣤⠄⠘⠁⠀⠀⠀⠘⠉⠀⠀⠀⠀⠊⠀⣠⣿⣯⣇⠉⠡⠄⠀⠀⠐⠃⢀⡤⠀⠀⠀⠀⢈⣦⣾⡿⠃
⠀⠀⠀⠀⢀⣌⡛⠯⡀⠀⠀⠀⠀⡀⡤⠶⠄⢀⣠⣤⣾⣿⣿⡞⠉⠩⡍⠁⢉⠉⠉⠉⠀⠈⠄⢀⣀⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠈⠛⢧⣤⣄⣀⠀⠀⠀⣤⣴⣶⣿⣿⣿⡿⢿⡿⢿⡷⣤⣴⣿⣿⣿⣦⣤⣤⣤⣾⠿⣿⠛⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠐


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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁⠀⠒⢂⣤⣭⣝⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢚⢛⣭⣶⣦⠀⠀⠀⠀⠘⠿⠿⢟⡶⠈⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠘⣘⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠘⠃⠀⠀⠀⣀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣄⣦⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠟⠀⠀⣠⢰⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠰⣿⣿⠃⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠛⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠮⠍⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡴⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣆⠀⠀⠀⠐⠇⢾⣿⣿⡿⠀⠰⢆⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣤⣬⣭⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠛⠳⢦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠶⠟⠋⢀⣶⣟⣻⣦⠀⠀⠀⠀⠀⠉⠛⣷⣶⡦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⡋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢮⣙⢷⡄
⠀⠀⠀⠀⠀⣠⡶⣫⣵⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⢰⢏⣼⠋⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡀
⠀⠀⠀⡰⢡⠞⠁⠀⠀⠀⠀⣿⣿⣷⣌⣳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣧⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⣌⢷⡄
⠀⢀⡾⠕⠁⠀⠀⠀⠀⠀⠀⠸⣿⢿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⢼⣿⣿⣟⣻⡄⠀⠀⢸⢸⡗
⠀⣿⠁⠀⠀⣀⣴⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠁⠀⠀⣾⢸⣇
⠀⡏⠀⠘⣿⣿⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⠀⠀⠀⠀⠀⠀⠈⠃⢿
⢰⠃⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢳⡀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⢷⠀⠀⠀⢀⣶⣆⢆⠸
⣿⣮⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⢸⣿⣿⣿⣦⡇⠀⠀⠀⠀⠀⠀⠘⠦⠿⠿⠋⠉⠀⠀⣴⣻⣿⡿⣸⢸
⢿⣿⡀⠀⠀⠀⠀⣾⣿⣿⣮⣷⠀⠀⠀⠈⠛⠿⢿⣻⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢣⣿⡟
⠈⢧⣛⣦⣐⣶⣤⣜⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⣿
⠀⠈⠻⢻⡝⠿⠋⣙⠿⠛⣷⣶⣄⡀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⣠⣼⡇⣿⠋
⠀⠀⠀⠰⠍⠀⠀⠉⠀⠀⠉⠉⣿⣅⢸⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⡇⠀⠀⣨⣵⠟⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠀⢟⡿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⠁⢀⣾⢟⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⢙⡷⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⢧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣧⣤⠀⠀⠄⣿⣿⣶⣿⡇⣠⣴⣶⠷⣟⣋⣉⣩⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣽⣿⣛⣛⣓⣚⡛⠛⣋⣈⣥⣾⣶⡿⠟⠛⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣬⣽⣿⣿⣿⣯⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠽⠋⠀⠀⣠⡶⠂⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢿⣿⣿⣿⡿⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣤⣠⣴⣶⣦⣄⠀⠀⠀⢻⡄⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠘⠃⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠱⠂⠀⠀⠀⣰⡿⠣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠻⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠛⠋⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢶⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠾⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡴⣶⡍⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣬⠀⠀⠀⣠⣠⡈⠁⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠰⠋⠁⡀⠈⠁⠀⠀⠀⠉⠀⠀⣤⣠⠆⠀⠀⠀⠋⣀⡀⠀⠀⠀⠀⠀⢴⣄
⠀⠀⢠⠁⠀⠀⣴⡦⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⣤⠄⠀⠀⠐⠀⢀⡵⠁⠀⠀⣎⡀⠐⠛⠁⠀⠀⠀⠰⣦
⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠌⠁⠀⡀⣀⠠⠀⠀⢰⡿⠶⠒⠁⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢸⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠞⠓⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⡴⢦⠀⠀⠀⣠⡄⠀⠐
⠀⠀⠸⣦⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉
⠀⠀⠀⠀⠂⠀⠘⠳⠀⠀⠀⠀⣴⠀⢰⡿⠁⠀⠀⢀⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣴⠋⡇⠀⠀⠀⠀⣀⠀⠀⠀⠈⠠⢠⣤⠀⠀⠉⠓⠀⠀⡴⠶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣹⡗⠀⣾⠟⠛⠿⠿⠶⣾⡿⠞⠁⡶⢀⣀⣸⣍⠇⠀⠀⠀⠀⠈⠉⠒⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⣾⡿⠁⣠⡴⡄⢰⡖⢳⢠⣿⢎⣽⢸⡏⣾⠵⢃⠟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠛⠗⣰⣿⡀⣿⣾⠧⣼⢿⡿⣿⡅⠸⢗⣙⡒⠱⣏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣴⠻⣯⡽⠃⠙⠒⢁⣈⣥⣼⠿⣿⣉⣉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠛⠁⠀⠀⠀⠻⠿⠿⠟⠛⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠛⠁⣀⣉⠉⠀⢤
⠀⠀⠀⠀⠶⣀⠀⠠⠆⠀⠀⠀⠈⠁⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⢀⡴⠂⠛⣉⠉⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⢠⣯⡴⠟⠋⠁⠀⡀⠉⠀⠀⠀⠐⠋
⠀⠀⠀⣦⠀⠙⠃⠀⠀⠀⠀⠰⠟⠃⠀⠀⠀⠁⠀⡀⠀⠀⣠⠊⣉⠁⠀⠈⠋⢡⡀⠀⠰⠦⠀⢤⡄⠀⠀⠀⠈⠻⠀⠴⢦⣠⣤⣥⣤⣄⣶⠀⠀⠰⡄
⠀⠀⢀⣿⣦⣀⠀⠶⠀⠺⠷⠂⠀⠰⠟⠶⠀⢈⡞⠁⠀⠸⠇⠀⣩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡷⠞⠁⠀⢠⡄⠀⠀⠈⠑⠠⡦
⠀⠀⢸⣿⣿⠿⠿⠿⠿⣤⣾⣶⣤⣤⡤⢶⡖⠿⡀⠀⠀⢼⡃⠀⠀⢀⢀⠀⠀⠀⠄⡻⠆⠀⢀⡀⠁⠀⠀⠀⠐⡿⠉⢠⣦⠄⠀⠀⠀⠀⠶⠄⠀⠀⡀
⠀⠀⠈⢙⣷⣶⡴⠖⢀⠉⠁⠀⢀⣄⣰⣿⣿⣾⡧⠀⠀⠈⠁⠀⠉⠁⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢻⠛⠻⠟⠋⠛⠛⠛⠛⣻⡋⠉⠉⠀⡀⣠⡄⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠐⠆⠀⠈⠁⠀⠔⠲⠄⠀⠸⣧⠋⠀⡀⠀⠀⠀⠛⠀⠀⠀⢰⡀
⠀⠀⠀⠟⣿⠟⢲⡶⠶⠶⢶⡾⢧⡴⠶⠶⠟⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠻⠂⢀⣀⠀⠀⠀⠃
⠀⠀⢠⣾⡶⢤⣬⣝⣀⣀⣈⣁⣈⣁⣤⡦⠶⠚⢋⢡⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⣠⣀⡀⠀⠀⠠⠗⠀⣀⠀⠀⠀⠱⣄⡀⠀⠀⠀⢠⣠⠀⣀⡀
⠀⠀⠈⢻⣥⣾⣆⠉⠩⣿⠉⠉⠈⠀⠀⠀⠾⣆⡤⠊⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠉⠉⠳⠦⠤⠅⠚⠁
⠀⠀⠀⠀⠁⠉⠉⠉⠓⠀⠒⠖⠚⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠢⠀⠛⠃⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣋⠁⠄⠀⠀⠀⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠀⠘⠄⢀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠶⢲⡷⠃⢴⠶⠀⠀⠀⠀⠀⠋⠁⢀⣴⠞⠹⠆⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⡰⠡⠀⠉⠀⠀⣾⣇⣴⡄⠀⠀⠀⢀⣄⠀⠀⣶⣿⣿⠀⠀⠀⠀⠛⠏⠀⣠⣴⠞
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠶⡶⠂⣿⡿⡌⢁⠀⠀⠈⠛⠛⠁⣼⡿⠘⢃⡄⠀⠀⠂⢀⣶⠻⡿
⠀⠀⠀⠀⢰⣇⠀⠀⠀⢀⣤⣀⢸⣿⣾⣤⣄⠀⠂⠀⢿⣿⣿⣿⠒⠂⠁⠀⢠⣍⣿⠷⠞⠀⠀⣈⠁⠀⣀
⠀⠀⠀⠀⠸⣯⠈⠂⠘⠧⠀⠀⢼⣿⣿⣿⡋⡀⠊⢘⣶⣿⣿⣿⣶⡆⠀⠀⠈⢹⣷⡌⡼⠳⠈⠁⠀⠘⠃
⠀⠀⠀⠀⠀⠹⢿⡀⠀⠀⠀⠀⠼⢟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣍⠀⠀⠀⠸⡧⠇⠃⠈⠀⠖⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠘⣷⡏⠀⠀⣀⠀⢻⣛⠁⠙⣿⡿⠋⠀⠀⠀⠀⠈⠻⠀⠰⣖⠰⣿⠀⠀⣵⣧⡄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠭⠿⠖⠄⠀⠚⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⣀⣹⡆⠀⠉⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⠀⠀⠘⠁⠀⢈⡀⠀⠀⢠⡄⢰⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠃⠀⣰⡆⠀⠀⢀⣤⡄⠀⠀⠀⣼⠄⠠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⠀⠀⠀⠘⠋⠀⠀⠀⠀⡏⠀⠀⠀⢄⠠⠀⢀⣄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⡼⠃⠀⠀⣠⣤⡀⠀⠀⠉⠁⠀⡈⠀⣦
⠀⠀⠀⢀⠘⠀⠀⠀⡀⠀⠀⠰⣿⠇⠀⠀⠊⠀⠀⣠⣾⣥⡄⠀⠀⠈⠙⠋⠀⠀⠀⠀⠈⢀⣼⠏
⠀⠀⠀⠸⢰⣶⡄⠈⠁⠀⠀⠈⠁⠀⠀⠀⢀⣠⡾⣿⣦⡉⠂⣀⠀⡈⠀⠔⠀⠀⢀⣠⣶⡷⠃
⠀⠀⠀⠰⣶⡈⠒⠀⠀⠀⠀⢚⣩⣤⣴⣾⣿⣿⣿⡀⣰⣤⣤⡀⠀⠀⣀⣤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠙⠛⠲⠶⠶⠾⠟⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⢂⡐⠉⡉⡉⠉⠉⣘⡓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡋⠀⠐⠂⠰⠀⠀⠀⠀⠆⠲⠄⠈⠀⠀⠀⠈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠛⠁⠀⠀⠀⠀⠀⣠⡀⠀⡴⡒⠆⠐⠛⠁⠀⠀⠋⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⢡⣒⣦⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠫⣇⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠺⠭⠿⢛⠰⠄⠈⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠴⠂⠀⠈⠉⠑⠤⢀⡀⠀⠀⠀⠀⠈⠁⣀⣤⣀⠠⠴⢲⡏
⠀⠀⠀⠀⣀⠀⠤⠐⠂⠈⠛⢉⣁⠀⠐⢒⠂⢤⠁⠀⡀⠀⠉⠀⠙⠀⠀⢀⡀⠙⠀⠛⢀⡬⣁⠀⠀⠚⠓⠃⡇
⠀⣠⠖⠁⠀⢊⠝⠀⢠⣤⠀⠘⠘⠀⠀⠉⠀⠈⢧⣀⠉⠠⢔⡒⠆⣤⠀⠈⠁⠄⠀⠀⠈⠈⠉⢀⡀⠄⠀⢀⡇
⡰⡁⠀⡓⠀⠀⠀⠀⡈⠀⠀⠤⠀⠀⢀⡀⠀⢠⡀⢉⠶⣄⣀⠀⠀⠀⠉⠀⠒⠀⠐⠐⠋⠉⠉⠁⠀⣀⡴⠍⠉⢂⠀⢀⣀⣠⣤⣄⣀⣀⣀⡀
⡃⠐⢜⠉⠀⠀⠀⠀⠉⠀⠀⠀⠲⠖⠃⠀⠀⣨⡅⠈⠂⠀⠙⠛⠉⢀⡄⠠⠤⠤⠄⢤⡤⡤⠔⠂⠉⣠⠀⠐⡤⠛⠉⠙⠉⠀⠰⠆⠘⠃⠀⠉⠓⠢⣄
⢇⠰⣛⠁⠦⣶⡄⠀⠀⠂⠀⠤⡄⠀⢠⣀⣤⠈⡐⢻⡀⠀⢀⡈⠁⢈⡀⠀⠀⠀⠄⠀⠉⠁⠘⠃⠀⢀⠖⠁⡠⡀⠒⠂⠀⠀⠀⠀⠀⡾⠆⠀⠰⠀⢈⠳⣄
⠈⠲⣌⠀⠀⣠⠀⠒⠃⠀⡈⠉⠀⠀⠀⠈⣀⡘⢲⠀⠑⣦⣘⠊⠀⠒⠀⠈⠉⢩⣏⠁⢀⡤⠀⠀⠴⠞⠀⠀⠁⠀⢀⠀⠰⠲⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⢳⡄
⠀⠀⠈⠉⠓⠄⠀⠀⠀⠀⠀⠉⠁⠀⢠⡾⠁⢙⠀⠑⢦⠀⠀⠽⠲⠤⢤⣀⣀⣀⣁⡀⢀⣁⣠⣴⣶⠀⠀⠁⠠⡄⠈⠀⠀⠀⠀⠈⠙⠀⠈⠁⠀⠀⣤⢀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠤⠀⠀⠀⠐⠒⢻⡀⢀⡀⠈⠒⠤⠀⠀⠀⢈⣅⠀⡈⢁⡀⡀⠀⠠⢾⠀⢃⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠙⠫⢼⡀⠀⠰⠦⠀⠀⠈⠉⠀⠁⠀⠶⠀⠀⠀⠘⢆⠘⠟⠁⠀⠀⠀⠀⣏⡈⠆⠐⠿⠃⠀⢀⣘⠃⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠴⠖⠂⠀⠀⣄⠀⠠⠀⠚⠋⠙⠒⠤⢍⡓⠦⠤⠤⠤⣤⠄⠐⠀⡀⠄⠆⠒⠚⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠧⣼⠃⠀⠠⣏⣹
⠀⠀⠀⣠⠖⠉⠙⠆⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠃⠀⠸⠽⠗⠈⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠉⠲⡛⣆⠤⠀⠀⠀⠀⠰⣯⠆⠰⠗⢀⣤⢞⡽
⠀⠀⢰⠃⠀⠁⠀⠀⠘⠃⠀⠤⠀⠀⠀⠲⠆⠈⠁⠀⠀⠀⡤⡄⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠉⠀⠠⠤⣠⣤⠠⠤⠄⠀⠈⢁⣠⠞
⠀⠀⡄⠀⠐⠠⠘⠃⢰⣑⣴⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠈⠏⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠐⠢⠦⣬⣭⡤⠤⠤⠴⠖⠋⠁
⠀⠀⠃⠀⠈⠂⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠠⠦⠀⠀⠘⠶⠁⢀⣼
⠀⠀⢸⡀⢤⡲⠀⠀⠀⠀⠈⢉⠁⠀⣀⣤⣀⣀⠀⠙⠀⠀⠦⢀⠀⠤⢚⣅⡜
⠀⠀⠀⠉⠳⢄⡀⠀⠿⠀⠀⠀⠁⠈⠓⢲⡤⠟⠀⠠⠤⠔⠂⠈⠴⠀⣈⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠤⢀⣀⡀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣁⣠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠖⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠛⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡴⠋⠀⠀⠀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠘⣆
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠘⠻⠛⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⣠⠏⠀⠀⠀⠀⢀⣴⣶⣤⣀⠀⠀⠀⠘⠛⠓⠦⠤⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⢰⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠲⢦⡀⠀⠀⠀⠀⢻
⡟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⣀
⢿⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷
⠘⣧⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⢰
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠸⣧⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠃⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠀⠀⠀⠈⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣦⣤⣤⣤⣴⣶⣶⣾⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠀⠤⠤⠴⢲⣦⣦⣶⠆⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣊⣍⠁⢠⣶⣿⠏⠐⠀⠀⠩⠍⣁⣀⣀⢠⡑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠿⣿⡟⠈⢉⠀⣀⠀⠀⠀⠄⠀⠀⠸⠛⠿⠟⢤⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣿⢻⠖⢲⣤⠾⡙⠄⠀⠀⣤⣦⣤⣄⡴⠾⣫⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⣉⡛⣉⣸⡇⣴⣤⣀⣉⣭⣥⣿⡯⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠄⢰⣤⣦⢿⡛⠋⢋⣬⣀⠈⠁⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠐⠟⠿⠛⢐⣣⣬⠐⠂⣤⣄
⠀⠀⠀⠀⡐⠄⠀⠀⠉⠹⠿⠻⠍⠂⠉⠻⠿⠗⣢⣀⣻⠿⠷⡀⠀⠀⠀⠀⠀⠀⢨⣫⡇⠐⠶⣾⣶⠄⠘⠛⠛⠁⠐⠈⠿⡧
⠀⠀⠀⢐⡇⠀⠠⠒⢀⣴⣶⡆⠀⢰⣶⣶⣄⠀⣄⡻⣽⣧⢸⣼⠀⠀⠀⠀⠀⠀⠘⢿⣷⣧⠴⣨⣍⠳⣄⠀⠀⣀⡀⠀⣤⡁⢱⡀
⠀⠀⠀⠸⣿⢄⡠⢐⠛⠛⠛⠂⠉⠀⣉⣙⡋⠁⠀⠀⣒⠄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡘⠻⠃⠘⠇⡅⣌⣿⡆⠙⠟⣰⡇
⠀⠀⠀⠀⠻⢿⣗⣤⠀⠀⠀⠤⠐⠿⠯⠛⠓⣂⣰⣭⣷⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣦⡠⢀⢑⡌⠁⣱⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠷⠿⢾⠿⠾⠟⠛⠉⠋⠉⠁⠀⣠⣤⣄⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠚⠿⠿⠼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠰⠾⠶⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⣄⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣠⠴⠊⠙⠋⠉⠉⠓⠶⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⠀⠂⣔⠒⡆⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⢀⠀⠀⠀⠀⠛⠃⢣⠀⠉⠉⠀⠀⠀⠠⣋⡹⠀⡽
⠀⠀⠀⢀⡞⠉⢢⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⢠⠟⠒⠉⠁⠠⠴⢏⣀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠛⠀⣀⣀⡜⠁
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢢⠀⠀⠀⠀⠀⢸⣀⡠⠔⠋⠁⠀⠑⠦⣄
⠀⠀⠀⣳⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠒⠃⠀⠀⠀⡀⠀⡇⠀⠀⢀⣀⠀⠀⢀⠀⠈⡇
⠀⠀⠀⢿⠀⠀⠀⠀⡔⠓⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢁⠜⠀⠀⠀⠧⠬⠂⠀⠀⠀⠀⡏
⠀⠀⠀⠈⢻⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⠁⠀⠀⠀⠀⢀⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⠀⡀⠀⡎⠁⠈⢧⠤⠔⠀⠀⢀⠿⠤⠤⠤⠤⠖⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠙⠺⢄⡀⠀⣉⠈⣉⠳⠤⢖⣁⣀⣠⠴⠚⠁⠀⠀⠀⣠⠖⣦⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠛⠂⠀⠈⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠂⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠉⠉⠀⢠⡄⠀⠀⠉⠉
⠐⠄⠄⠀⠀⠀⠀⠀⠀⡤⠔⢇⠀⠀⠘⠀⠀⠀⠀⠀⠐⠄⠄⠀⠀⠀⠀⠀⠀⡤⠔⣇⠀⠀⠘
⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠎⠀⢀
⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠄⠈⠉⠁⠀⠀⠀⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠄⠈⠉⠁
⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠘⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠄⠀⣀⠆
⠂⠀⢐⠘⠙⢢⠄⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙⢢⠆⠀⢘⠘⠉⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙
⠀⠀⠈⠓⠒⠉⠀⢠⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠒⠉⠀⢠⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠢⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉
⠐⠄⢄⠀⠀⠀⠀⠀⠀⡬⠔⣃⠀⠀⠘⠀⠀⠀⠀⠀⠐⠄⠄⠀⠀⠀⠀⠀⠈⡭⠔⣂⠀⠀⠘
⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠆⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠊⠀⢀⠀⠄
⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠊⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠄⠀⢀⠆
⠀⠀⢈⠘⠙⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙⢀⠄⠀⢐⠘⠉⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙
⠀⠀⠈⠑⠒⠉⠀⢠⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠒⠉⠀⢠⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⡀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠰⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⠀⣰⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠀⢸⣿⠃⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠸⣿⣇⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠈⠉⠉⠁⠀⠀⠘⠟⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢰⣿⡏⠀⢀⣀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠘⠟⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⢴⡆⠀⠀⣠⣶⣶⣄⠀⠙⣿⡗⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢀⣼⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣄⣁⣀⣋⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠯⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠉⠉⠉⠙⠓⠒⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡜⠁⠀⠈⢣⠀⢀⣤⡄⠀⠀⣀⠀⠂⠀⠀⠀⣠⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⠢⠤⠐⠋⡠⠌⠠⢀⠀⠈⢙⠀⠀⠀⠀⠀⢧⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡶⠁⠀⠑⡀⠀⠀⠀⠀⠉⠉⠉⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢀⢇⠈⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⠀⠀⠀⠀⠀⢄⠀⠙⠒⠒⠉⠈⠓⠒⠊⠀⡠⠀⠀⠀⠀⠀⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆⠀⠀⠀⠀⠀⠑⠠⠄⣀⣀⣀⣀⠠⠄⠊⠀⠀⠀⢀⡤⢤⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣷⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠊⢰⠇
⠀⠀⠀⠀⠀⠀⣀⣀⢽⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⡿⢄⣀
⠀⠀⠀⢀⡔⠋⣠⠤⠒⠻⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⡞⠓⠤⣄⠙⢢⡀
⠀⠀⣀⠎⠀⡜⠀⠀⠀⠀⠹⣄⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡴⠋⠀⠀⠀⠀⢣⠀⠱⣀
⣠⠊⢀⣀⠀⢜⠀⠀⠀⠀⠀⠀⠙⠲⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣣⠀⣀⡀⠑
⠧⣴⣫⠏⠢⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⡤⣤⣤⢴⠶⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠔⠹⣝⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣯⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⡇⣿⣿⢸⣀⡤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣈⣁⡻⢟⣈⣁⣀⣀⣀⣀⣀⣈⡆
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣤⠖⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣴⣦⣄⠁⠸⡆⠀⠈⠙⠒⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⡀⠘⠛⠋⠉⠀⠀⠐⡄⠀⠀⠀⠐⠄⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⢿⡿⢻⣶⣏⡀⠀⢀⡀⠀⠁⠀⣀⠀⠘⠦⣄⠈⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠀⢞⠵⠓⠾⣷⣿⣇⢰⣿⡿⠀⠐⢿⣿⡄⠀⠀⠋⠒⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⣆⠀⠹⣾⢻⢶⣤⣴⡆⠀⡀⠁⢠⠀⠀⠐⠂⠉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠻⣗⢄⠀⠈⢉⡉⠀⠀⠈⠳⣶⡿⢋⣿⡀⠐⠄⢈⠓⠘⣇⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠻⡦⢠⣍⡇⣶⣀⣀⠀⠈⢷⡾⢙⣿⣦⡴⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⠀⡄⢿⣮⣭⡀⠀⠀⠉⠀⣀⠤⠈⢣⡸⣹⣇⠰⣖⠀⠀⣗⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⡟⢂⣴⡈⣷⣦⡄⠀⡾⢃⣠⠐⠀⠻⣿⣿⣦⠈⣠⣨⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⣾⡿⠟⢘⠻⠿⢦⡄⠁⠀⢠⠄⠀⠀⠙⠿⠾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣷⣤⣈⣙⡆⠀⢈⡳⣲⠄⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠛⠲⠖⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⢶⣦⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠀⠀⠀⢻⡇⠀⣴⡿⠛⢋⡙⠛⠃⠀⠐⣽⣶⣮⡝⠛⠿⠿⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⠔⣶⡄⠀⠀⡘⣿⣧⣾⠋⠀⣾⣭⣽⡆⢦⣤⠄⠀⠉⢡⣄⣀⢰⣉⣭⣦⠀⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠻⣼⠿⠁⠀⢠⢇⣿⣷⡶⠂⠀⠉⠁⣁⡴⠒⠪⣕⠆⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢠⣾⡿⣩⡦⢀⣲⠆⠀⠀⠀⢦⣾⡿⠀⣀⣀⠀⠀⠀⠙⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⡀⣤⠤⢤⣖⣤⣶⠟
⠀⠀⠀⠀⢠⣿⡿⢸⡿⠃⠈⠁⠀⠀⠀⠀⣼⣿⡿⣷⣶⣶⣾⣦⣄⣀⣠⣄⣀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⣼⣿⡀⠁⢀⠆⡠⠤⠀⠀⠀⠀⣿⣿⡿⠟⠀⣾⣭⣽⡟⢿⠿⠿⠿⠿⣿⠿⠟⠛⠿⠿⠿⠀⢘⣛⡛⢶⣄
⠀⠀⠀⢰⣿⠟⠀⣒⠸⢋⠖⣹⡷⠀⠀⢠⣿⡏⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⣴⣩⣭⣽⡆⠀⠀⠀⠈⠙⠉⠀⣿⡇
⠀⠀⠀⣿⡏⢸⣏⣿⠇⢸⣼⡿⠃⠀⣠⣿⡿⣷⣤⣔⣒⡤⠤⠠⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⢀⣄⣒⣠⣤⣶⡿
⠀⠀⠀⢻⣇⠈⠻⠏⠀⠈⠉⠀⠀⣼⣿⠏⣠⡶⠟⠛⠻⠻⣿⠿⢿⣿⣿⣷⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⡏⠙⠿⢶⣦⡀
⠀⠀⠀⠀⢻⣦⢂⡀⠀⠀⢀⣠⣴⡿⠋⣾⠏⠐⠻⠟⠁⠀⠀⠀⢠⣾⣋⣙⣻⡝⠋⠉⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⢸⡇⢿⠇⠀⡔⣵⡿⠋⠀⠀⢿⣧⣀⠀⠤⠤⢀⠀⠀⠈⠛⠛⠟⠛⠃⠀⠀⠀⠀⠀⠀⡀⣠⣤⠤⢤⣐⣤⣾⠟
⠀⠀⠀⠀⠈⣿⣄⣀⣠⣾⠟⠁⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣶⣤⣤⣤⣤⣀⣀⠀⣀⣀⣴⣶⣶⣿⣷⣶⣾⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠⠀⠐⠐⠈⠉⠉⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢸⣷⠘⠆⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⠀⠠⣧⠀⠀⠀⠀⠀⣠⣤⣤⣤⣹⠦⣀⣠⣥⠽⠶⠶⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠰⢩⡉⢲⠨⣧⠀⣻⠿⠛⠛⠋⠉⠉⠁⠱⣇⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠆⠀⠀⠈⠀⠻⣆⠨⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣽⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠠⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠸⡿⣯⡳⠤⠀⠀⠀⢀⡢⠄⠓⠚⠍⣎⢧⠀⠀⠒⠒⣂⣉⡹⠴⠥⠔⠒⣒⣈⣉⠭⠭⢋⠄⠀⡀
⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⠠⣀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡭⠭⠥⠴⢒⠒⠒⠋⠛⠁⢄⠀⠀⠀⣀⠀⠀⢊⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣮⣙⠢⠀⠀⠀⠐⠀⢀⠄⢳⠀⠀⣞⡳⣄⠀⠀⠀⠀⠀⠑⡄⠐⠃⠀⠀⠀⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣷⣦⡀⡤⠀⢀⠀⠄⠈⠀⠀⠫⣷⣬⡓⠀⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠸⠂⠀⠈⠻⣿⡆⠀⣾⠥⢨⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠁⠀⠈⠓⠚⠃⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⣠⠏⠀⠀⠀⠀⠼⠃⠀⠀⠀⠈⠓⢶⣀⠀⠀⠇
⠀⠀⠀⢀⠐⢄⠀⠠⣤⠠⢥⣀⣀⣀⡼⣀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠀⠀⠑⠶⣶⢁⠂⠀⠀⠀⠀⠀⠓⠨⠁⠀⡠
⠀⠀⠀⠉⠑⠀⡈⠑⠀⠉⠌⠀⠧⣄⡴⠀⠉⠉⢉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢠⠏⠀⠀⠀⢰⣣⠃⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠲⢦⣉⠂⠀⠀⠀⠀⠀⠈⠋⠀⠀⠄⢸⣄⠀⠀⠀⠀⠄⣀⣤⣄⡰⡏⠀⠀⠀⢠⢒⠏⠀⠀⠬⡁⠀⢢⠐⡄⠀⠀⠀⠀⠀⢀⣀⡠⡆
⠀⠀⠀⠀⢾⠂⠀⠉⠃⠐⠠⠀⠀⠀⠉⠀⠐⠀⡀⠀⠀⠀⠠⠂⠀⠀⠈⡙⠁⠛⠶⢶⣤⡋⠎⠀⠀⠀⣄⠈⢦⠀⠇⡇
⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⣨⡋⠀⠀⠀⠀⠉⠓⠚⠂⠃⠀⠀⠀⠀⠙⠓⠒⠂
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠃⠀⠀⢶⣤⣀⣀⣴⠏⠀⠀⠀⠀⢠⠟⠀⠀⣠⣤⢶⣤⣄⠀⠀⠀⠀⠀⠀⢠⡄⢄⢄
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠻⠿⣦⣄⣀⠀⣴⡎⠀⠀⢪⠞⡵⠚⠿⣯⣧⠀⠀⠀⠀⠰⠀⠀⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠀⠀⣴⣄⢘⢀⠀⠈⠁⣠⠦⠀⠀⠀⠸⣤⣠⣀⡀⡰
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠈⠀⠀⠀⠀⣴⠄⠀⠀⠉
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏⠀⠀⠀⠀⠄
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠂⠀⢦⣄⣠⣔⣺⡼⣷⣄⠀⠀⠀⠀⠀⠀⣠⠤⠂⢩⣿⡍⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠇⠀⠀⣄⠀⠀⠄⢀⡿⢟⣯⠾⠛⣫⠼⢆⡀⠀⠀⡀⢺⣿⣦⡀⠈⠙⢯⣌⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢗⣋⣽⡔⢛⣯⣴⣿⣭⣉⣙⡿⠆⠁⠀⠀⠙⢯⡻⡯⠈⠀⠈⠛⠒⠤⣉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠰⠄⠀⠀⠀⠀⠉⠻⠻⠟⠊⠉⠛⠂⠓⠛⠋⠹⠁⠀⠀⠀⠀⠀⠀⠙⠂⠈
⠀⠀⠀⠀⠂⠀⠀⢐⡎⠀⠀⠀⡄⢀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⡟⠒⠂⠀⣸⠎⠀⠀⠈
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⣠⠄⠉⠍⠉⣇⠀⠈⠲⠗⡉⠁⣀⣀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⢲⠁⠀⠀⠀⠠⡀⠀⠀⠠⠄⢨⠇⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠓⠀⠀⠀⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⢄⣀⣤⣤⠤⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠹⢦⡘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠢⡀⠠⠄
⠀⠀⠀⢠⣀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠈⠀⡖⠀⠀⢰⣦⡤⢄⡀⠀⣠⢟⠦⡀⠀⢄⠀⢀⠀⢀⡴⠀⠀⢀⡀⠀⠠⠀⠐⡇⠀⠀⠠⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⢈⠀⠀⠀⢀⣷⠄⠀⠈⠙⠣⠀⠈⢀⡴⠁⠀⠈⢢⣄⣧⣤⠆⠈⠀⠀⠀⢋⠀⠀⠀⠐⠔⣀⡀⠴⠤⠖⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⠀⠀⠀⡠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠘⠈⠂⠀⠀⠂⠚⠰⠰⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠳⠬⢒⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⠻⠶⣂⢀⠀⠀⠀⠀⠀⠀⣇⠈⢙⣿⠃⠉⣱⠀⠀⠀⠠⣦⠀⣔⡀⠒⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣈⣉⣰⣤⢆⠀⠀⠀⢀⡟⠛⠻⣿⠛⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠘⠻⠀⠀⠀⠸⣿⣷⣾⡟⠿⠻⠿⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⠀⠤⣤⣲⣤⣯⣽⣦⣠⣖⣀⡀⠀⠀⠀⠰⠟⠟⠋⠀⠀⠀⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢀⠁⠈⢋⣀
⠀⠀⠀⠀⠻⠟⡶⠶⠦⠤⣤⣤⡄⠬⢦⣵⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⢀⣄⠠⢀⠠⠐⠀⢀⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠚⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⡀⠀⣠⡄
⠀⠀⠀⠀⠐⠛⠲⢿⣤⣴⣶⣶⣿⡿⠟⠋⠀⠀⠀⠀⠀⠂⠀⠀⠦⠄⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡄⠉⠉⠉⣔⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠬⠠⠔⢀⡂⢀⡀⠀⠀⢀⡀⠀⠈⣠⠍⠐⠢⣴⡺⠟⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣀⢠⡀⡂⡠⠀⠎⠁⢉⠀⡓⢂⣀⣐⠦⠶⠛⠀⠐⠻⠯⢔⣠⡄⣾⠔⠉⠀⠀⠀⠀⠀⠦⠤⡀
⠀⠘⢇⡀⠀⠑⢌⢡⠄⠢⣓⣠⡵⠶⠚⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢐⡠⠄⠀⠀⠈⢣⡼⣈⠑
⠀⠀⠈⠋⣨⣀⣠⣷⠶⠛⠋⠁⠀⠀⠀⠀⢰⣴⠁⡄⠁⠀⠀⠰⠒⠂⠬⠃⠀⠀⠀⠀⣤⣀⣀⡎⠀⢀⡒⣸⢸⠆⣬⢠⣸⠾⠇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠈⠀⠀⠀⠁⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⠚⠩⢤⣄⣀⡐⠂⠨⠉⠑⢒⣢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣴⡉⠧⠤⣶⣶⠮⠟⠶⠄⠉⢈⣁⡠⠔⢚⣉⡭⠴⡓⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢙⢳⡿⢟⡲⠤⣌⣙⠒⠲⠚⣋⣡⠤⢖⡛⢩⢴⠚⠋⢁⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⣤⣆⣂⠙⠹⡗⠪⣭⣉⣉⢽⠂⠏⠠⠀⣈⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢎⡀⠀⠐⠻⠽⣒⡦⣄⣈⢘⣀⣉⣤⣴⢶⣛⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠹⢓⠺⣝⡲⠤⣄⣀⠩⢑⠺⣭⠭⡝⠛⠋⣉⣩⣤⣾⣡⠵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠙⠛⢶⣅⣈⠉⠛⠓⡻⣿⣖⠤⠬⠤⣶⣞⣫⢷⠿⠩⠝⠛⣀⣸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠫⡧⡀⢪⣍⣛⠶⠮⣄⡉⠁⠉⡿⠉⠁⣘⣀⡤⡤⣖⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡁⣝⣄⠳⡤⠵⠒⠒⠋⠓⢶⣷⣶⡿⣭⣔⣒⣺⣸⣦⠀⠀⠀⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠙⢦⡀⠀⢣⡙⢮⣒⣀⣖⣶⣬⡽⠿⠳⠒⠚⠋⠉⢉⠙⣁⣤⡼⠆⠀⠀⠠⠋⠀⣠⣽
⠀⠀⠀⠀⠀⠀⠀⣡⠘⢓⣿⠀⠀⠑⣬⣉⣁⣁⣬⣬⣤⠶⠶⠖⠻⠿⠛⠛⠋⠋⠉⠀⣀⠀⠀⠙⠻⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠦⣄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⣂⣛⣦⠀⠀⢜⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠐⠀⠐⣶⡈⠉⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠛⣻⡜⡂⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢠⣤⣤⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⢶⠤⣉⣻⡐⢠⣤⣾⡿⠿⠿⠟⠛⢿⡛⠛⠉⠉⠉⠉⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠉⠁⠚⠨⣷⡸⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠩⡒⠦⣤⣘⣷⢹⡄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀⣀⣀⠀⡄
⠀⠀⠀⠀⢺⣿⣳⣤⡀⠀⠀⢤⣄⠺⠿⠟⣣⢷⡀⠠⠤⠶⠒⣒⣦⣼⠤⠤⠔⣒⣒⣛⡿⢃⠀⢀
⠀⠀⠀⠀⠈⠻⢿⣷⣤⠠⣂⠉⠀⠀⠀⠀⠨⠿⠿⡯⠭⠭⠽⠖⠒⠶⠛⠋⢍⠀⠀⠀⣀⠀⠀⣺⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣮⣟⠦⠀⠀⠀⠂⠈⡤⢳⠀⢰⣞⢦⣄⠄⠀⠀⡀⠙⣦⠐⠁⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⢠⠄⢀⠄⠀⠈⠀⠈⠻⣷⣍⠂⠀⠀⡐⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣿⠂⢸⠟⢠⣿⣄⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠲⢏⡔⡠⢋⣝⠀⢄
⠀⠀⢰⣶⠤⢄⣀⠒⠀⠠⡀⠀⠀⠀⠻⢧⣴⣬⠉⠓⠎⠴⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠈⢳⣝⢢⣄⡀⠓⠂⠀⠀⠀⠀⠀⠀⠉⠛⠳⠷⠾⠶⠴⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠻⢮⣝⣒⣤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⢀⢀⡀⠱⣖⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⠘⢃⠵⢹⣿⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠆⠀⠀⠀⠀⠈⠋⠳⠂⢤⠁⠬⠻⠧⢄
⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⠀⢀⠀⠘⢒⣿⢁⠰⣖⢛⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠁⠤⠄⠤⠤⠶⠒⡛⠿⢿⠚⢒⣶⠾⠀⠀⠀⠀⠠⠤⡂⣄⢦⣂⠆⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢄⣤⣌⣀⡠⠀⠠⠐⣊⠭⠳⡇⠋⣸⠆⠈⠀⠀⠀⠀⠔⢀⡐⣓⠭⠖⣋⠑⠤⠀⠀⠉⠠⠔⣲⣿⠆
⠀⠀⠀⠈⠙⠫⡂⢤⣲⡹⠬⢒⣩⣠⡾⠃⠀⠀⠀⠀⠠⠤⣬⣨⣅⣤⣦⣿⠁⢀⠄⠈⠁⣠⡴⣪⡵⠋
⠀⠀⠀⠀⠠⢤⣍⣨⣤⣴⣦⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣤⣤⣖⣾⣭⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠘⠛⠛⠛⣧⠀⢰⡆
⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢠⠄⣀⠀⡀⠀⡄⢠⠀⠲⠈⠁⢀⣿⡇⠀⢠⣤⣄⠈
⠀⠀⠈⣡⡤⢠⠄⠀⣴⡇⠀⠀⠀⠀⠀⢠⡄⠀⡤⡀⢄⠀⠁⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⠀⠹⠿⠟⠀⠀⢀⣄
⠀⠀⠀⠀⠰⠈⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠉⠁⠈⠉⠀⠁⠡⠶⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠐⡅⠀⢠⠃⠀⠀⠃⠈⢀⣴⣷⡀⠛⠛⢁⣀⣀⡀⠀⢀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⢸⠘⠀⠀⠐⠰⠾⣿⡿⠆⠀⠀⠀⠀⠈⠀⠈⠁⠀⠐⢛⡉⣉⢳⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⠀⢠⣈⣠⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠖⢳⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠙⠀⠀⠘⠖⠃⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢀⣠⣶⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠀⠀⠸⠀⠀⠀⠚⠁⠀⠛
⠀⠀⠀⠛⠛⠿⠇⠀⠛⠋⣤⡶⠀⡀⠀⠀⠈⠥⠀⠘⠂⠀⠀⢄⡐⠀⠀⠀⠸⡀⠀⠀⠀⡀⠀⠀⠛⠀⢀⢀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠁⢀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠻⣿⣿⣿⣄⠀⠀⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠸⠇⡌
⠀⠀⠀⠻⠿⢿⣿⡷⠾⠛⠀⠀⢨⣯⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⡎⠀⠀⠀⣌⠀⠤⠀⠀⢠⡄⣬⠀⠀⢠⡆⠅
⠀⠀⠀⠈⠁⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⢐⠒⠫⠇⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠘⠃⠀⢷⠀⠈⠀⠀⡀⠂⢀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠉⡁⠀⢀⠀⠀⠀⠀⠀⠀⠣⠀⣀⠄⠜⠊⠋⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⡨
⠀⠀⢠⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠂⠒⠂⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⠀⢠⠀⠸⠤⠒⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠄⠀⠙⢀
⠀⠀⠀⠐⠤⠤⠔⠀⠀⢀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⢁⣀⡠⠂⠐⠊
⠀⠀⠀⠀⠀⣀⠤⠀⠉⠀⢀⣱⠄⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⡤⠴⠒⠋⠁
⠀⠀⣄⠂⠉⠀⠀⣠⣴⣾⠟⠋⠀⠀⠢⢄⡀⠀⣠⣀⠀⠀⠀⠀⠈⠉⠉⡉
⠀⠀⢿⣧⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠠⠂⠀⠀⠘⠀⠀⠀⠴⠁⠀⡸⠁⠀⠀⠀⠀⠈⣆⠀⠀⠰⠂⠀⠀⠀⢀⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⣟⢶⣤⣄⣀⣤⠶⠿⠀⠀⠀⠀⠀⠈⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣬⣽⣥⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣩⣍⢳⡴⢛⡛⢦⠀⠀⠀⢀⣄⣰⠏⣰⢸⣇⣰⠋⣹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡘⢁⡏⡘⠁⢋⠍⣘⠓⠒⠚⠋⣸⡏⠐⢣⣿⣉⠁⢠⣴⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⣧⣬⡴⠁⠾⠶⢚⡻⡟⡛⠟⢫⠙⠀⡴⢋⠉⣹⢀⡟⣩⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡿⠟⡁⢀⠀⠄⠸⢡⠀⠷⡀⠬⣰⠀⣀⣉⠀⣡⣘⣃⠱⣃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠞⠛⠚⠓⠛⢺⡟⠷⠾⠿⠛⠛⠀⠉⣤⣤⣶⠌⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠴⠖⠛⠉⠳⠦⡭⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡏⠁⠀⠀⢸⡇⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡇⣀⣀⣀⣈⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⡏⠉⠉⠀⢠⡏⠀⠀⠀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠇⠀⠀⠀⢸⠁⠀⠀⠀⢿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡹⣤⠤⠤⢴⡧⣤⣶⠶⢾⡦⠶⠶⠦⣾⢿⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢍⠙⠷⠀⠀⠀⠀⠉⠁⠐⢬⣄⣀⣀⠔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠒⠀⠐⠂⠤⠄⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⣠⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⢀⡀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠢
⠀⠀⢰⠏⠀⠁⠀⠠⣄⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠐⠃⡀⠈⠁⠀⠀⠉⠁⣠⡠⠀⠀⡀⠋⣀⠀⠀⠀⠀⠠⠄
⠀⠀⡆⠀⠰⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠁⠠⠖⠀⣀⠀⣀⠔⠀⠀⣰⠄⠘⠉⠀⠀⠀⢲⠆
⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣤⣐⣈⡠⠤⠛⠉⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⡀
⠀⠀⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠉⣁⠀⠀⠛
⠀⠀⠈⠀⠀⠛⠂⠀⠀⠠⠆⠰⠏⠀⠀⣀⣀⠀⢀⣀⣀⡀⠀⢠⠞⡁⠀⠀⠀⣀⠀⠀⠈⠠⣴⡄⠀⠙⠂⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣡⡿⠀⠛⠉⢉⣛⢛⣿⢫⣡⣇⡤⡜⣧⡀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢿⠇⣼⠋⣆⣿⣸⣾⣿⡞⢹⡿⠯⠔⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡼⢿⡹⠉⠳⢋⣈⣤⡿⣶⣋⣙⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⠀⠀⠻⠿⠛⠋⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡄
⠀⠀⠀⣀⠀⠀⡀⠀⠀⠲⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠰⠦⠀⠀⠀⠀⠀⠀⣠⢣⣬⠤⠀⠠⠄⠐⠀⢠⠄
⠀⠀⢠⠁⠶⠀⠁⠀⠀⡤⠄⠀⠙⠛⠀⠀⠀⢀⠄⠚⠁⠐⠶⢀⠀⢠⡄⢀⡀⠀⠀⠀⢿⠋⣤⢀⣀⣂⣀⢠⠀⠀⠄
⠀⠀⣻⣦⣀⠰⠆⠲⠖⠀⠰⠖⠆⠀⡜⠀⠀⡞⠀⣭⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⣴⡶⠊⠁⢡⡌⠀⠈⠐⢠
⠀⠀⢿⡿⠛⠛⠛⣶⡿⠶⠶⢖⣶⢛⡄⠀⠐⠅⠀⠀⠀⠀⠀⠀⠸⠂⠀⠄⠀⠀⠀⠀⠻⠁⠰⠆⠀⠀⠀⠐⠂⠀⠠
⠀⠀⢰⡿⣿⡯⠤⣤⣤⣤⡶⠾⠟⠛⠃⡀⠀⠀⠀⢀⡁⠀⠀⠀⣀⠀⠠⠀⠀⣀⠀⠀⣇⠤⠀⠀⠀⠠⡄⠀⠀⡀
⠀⠀⠰⣶⠦⢤⠤⡤⢤⣾⣥⡤⠴⠶⠾⠁⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠁⠀⠀⠀⢹⡀⠰⠆⠀⡀⠀⠀⠂⠉
⠀⠀⣼⡷⣤⢭⣄⣀⣨⣤⣥⡤⠖⠒⢊⡁⠀⠀⠀⠈⡀⠈⠀⢠⣄⡀⠀⠀⠟⢀⡄⠀⠀⠳⣄⠀⠀⠀⣤⠀⣀
⠀⠀⠘⠷⠿⣤⣔⠛⣀⢀⣀⣀⣸⡧⠖⠁⠀⠀⠀⢀⠉⣠⡀⠀⠉⠀⢠⡀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡇⠀⠀⠀⠀⠀⡇⠰⠖⠘⡸⢙⡤⢤⢐
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠀⠀⠀⠀⠘⠐⠻⠈⠒⠀⠱⢎⣀⢒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⠀⠀⠀⠀⣰⣀⡀⢠⡴⡴⢤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣽⠀⠀⠀⠀⠩⠸⠿⣿⡖⣲⡀⠿⡗⠳⠗
⠀⠀⠀⠀⠀⠀⣴⣯⣿⠿⣷⠆⠀⠀⠀⢴⣶⢶⡂⣟⣸⣶⣶⡶⠲⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠐⣿⣯⣿⢿⣿⣾⠄⠀⠀⢸⡿⣢⣴⣷⣶⣶⣶⡔⠤⠖⠀⠀⠀⠀⡼⠀⠀⠀⠔⠀⡰
⠀⠀⠀⠀⠀⣼⣹⣹⣿⣿⢻⣿⣦⠀⠀⠐⣪⣼⣿⣿⠟⢻⣿⡇⠄⡀⠀⠤⠀⠈⠑⠀⢀⣴⠃⢀
⠀⠀⠀⠀⢠⣟⣻⣿⣿⢿⣿⢿⡾⢀⠀⠈⠁⠐⣿⠁⢀⡈⢻⣇⠈⡇⢀⡰⢄⣠⣦⣾⢟⠋⢀⠄
⠀⠀⠀⠀⡷⣿⣿⣿⡏⠈⣹⣗⣿⠟⠢⣤⡀⠈⡿⡬⠿⠧⣿⡇⠀⠃⠀⠀⠈⠀⠁⢸⠁⣆⠊⣀⡀
⠀⠀⠀⠀⢛⠻⠁⢘⠟⢱⡏⠛⠀⡟⢰⡄⢀⠠⡴⣦⣾⡧⠾⠭⣦⠀⠀⠀⠀⠀⠀⠈⡇⡃⠀⠈⣤⠰⡃
⠀⠀⠀⠀⢷⡤⠀⣰⠀⠈⠀⢀⣿⡟⣾⠁⣸⡇⠠⠾⠿⠷⠶⠶⠶⠶⠺⠁⠀⠂⡤⠈⠁⠀⣿⠀⢻⠞⣧
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣿⠍⠉⠁⣿⠩⠛⠉⢸⣧⢀⠉⢠⡆⠀⠀⠀⠀⠀⠀⡷⣀⠀⠀⡇⣠⠄⠀⢸⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⡀⣿⠀⢀⣂⢸⣟⠋⡤⢸⡃⠀⠀⠀⠀⠀⢰⡟⡏⠓⠀⡇⡻⠎⠀⢸⠋⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⣿⡛⡛⠛⢹⡏⡉⠋⢹⠀⠀⠀⠀⠀⠀⢸⡏⢛⡛⠛⣿⠛⠛⠛⢱⠙⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⣿⠆⠿⠀⢹⠋⠀⠁⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡿⠐⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠿⠟⠷⠛⠿⠟⠻⠶⠛⠃⠀⠀⠉⠀⠀⠀⠀⢸⡷⠿⠒⠒⡔⠶⠶⠶⢦⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠀⣤⢸⡇⢀⡆⠀⡇⠏⠀⠀⢾⡽⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⡞⠀⠀⠀⠀⠀⢀⠄⢸⣧⡤⠤⠤⡣⢤⠬⠥⡸⠤⠬⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⢀⠠⠀⠁⠀⠀⣿⣿⣇⢸⣇⣷⡦⠀⡇⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠶⣀⣴⣿⣦⣤⣖⣤⣤⠶⢺⠁⠀⠀⠀⠀⠀⣧⣭⣥⣄⣮⣭⣥⣤⣣⣤⣤⣄⡽
⠀⠀⡄⠢⠤⠠⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠿⠿⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⣷⠅⠀⢀⡇⠀⡀⠀⡇⠀⠀⠀⡃⠀⠀⠀⢀⣀⠄⠒
⠀⠀⣻⣦⣤⠴⠖⢻⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣃⣈⣀⣀⢷⣀⣀⣀⡇⠀⠀⠰⠿⣷⣄⢠⣴⡴⢶⢾⠄
⠀⠟⠁⠀⠀⠀⠀⠈⣿⣿⣟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇⠉⠉⠉⣷⠉⠈⠉⡇⠀⠀⠀⠀⠀⠉⠻⠝⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣍⣀⣀⣀⣀⣀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⣇⢀⣀⣈⣁⢀⠀⢀⣹⣀⣀⣀⡇⠀⠀⠀⠀⠀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⣤⣀⣠⣹⣧⡴⡶⠖⠒⠦
⠀⠀⠀⠀⠀⠀⢀⣔⣂⣷⣶⠶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠐⢮⣅⢀⣀⠠⠀⠀⠀⠀⠀⠻⣯⣉⣁⠀⠀⢈⢷⣚⣠⣤⣤⣴⡄
⠀⠀⣠⣴⣮⣤⣸⣿⠿⠻⠿⢭⣿⣿⣿⣶⣤⠆⠀⣠⣄⠀⢰⣞⠫⠀⢀⣙⠦⣔⡲⠚⠁⠛⢯⡀⣀⣤⠤⠀⠀⠀⢻⣳⣿⣿⠿⠭⠵⠞⠉⠛⠒⠀⠁
⠀⠄⠙⠻⣿⣿⣶⣿⣿⣷⣶⣾⣯⣅⡀⢈⡭⠟⣊⣭⣶⣢⠄⠙⢷⣯⡉⢀⣀⣈⡷⢦⣖⠾⠉⠈⠓⢮⣀⣠⣤⣄⠈⠁
⠀⠀⠀⠀⠀⠙⠿⣿⣧⠀⣹⠿⣿⣿⣽⣧⣶⠊⠁⠠⠔⠒⠋⠀⠈⢻⣿⣾⣭⡻⠀⢀⣙⣷⢤⣶⣻⣷⣾⣿⣿⡄
⠀⠂⠀⠀⠀⠀⠀⠈⠛⠓⠂⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⣧⣾⣿⣿⠿⠿⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⣩⡶⠀⠀⡤⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢀⠀⠀⠀⠈⠀⠀⠀⡠⠤⠀⠀⠰⡇⠀⣯⢁⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⠋⠀⢠⣴⡆⠀⠀⠀⣠⣤⡄⠀⠀⠀⢠⣿⠀⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠛⠋⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⢾⠇⠀⠀⠀⢍⠀⣀⠀⠀⣀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⡀⠀⠠⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠿⠇⠀⠀⠇⠈⣆
⠀⠀⠀⠀⠀⣠⠀⠀⠙⠁⠀⠀⠀⠀⣠⣤⠀⠀⠀⢀⠾⠁⠀⠀⣠⡾⠁⠀⠀⠀⠸⢿⣿⡆⠀⠀⠐⠄⠀⡰⠀⢠⣿
⠀⠀⠀⠀⡆⠋⡀⠀⠀⣠⠆⠀⠀⠀⣿⠿⠀⠀⠀⢈⠀⠀⣄⣴⡿⢾⡷⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⢁⣴⡿⠃
⠀⠀⠀⠀⠇⢼⡷⡶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢀⣀⣴⣿⡷⣷⣄⡚⣀⣀⠠⣌⠀⠤⠃⠀⠀⢀⣤⣼⣿⠟⠁
⠀⠀⠀⠀⢢⣶⡈⠐⠀⠀⠀⠀⠒⠒⢋⣡⣤⣴⣶⣿⣿⣿⡟⠀⠐⣦⣤⣤⠀⠀⠀⠀⣀⣠⣴⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠈⠓⠶⠶⣦⣤⣥⣼⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⡿⢿⠿⠶⠿⡛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚⠿⠛⠛⢲⣄⣄⠀⠀⠀⠀⡀⠀⠉⣹⣿⠇⣀⠀⠀⠈⢳⣶⡄
⠀⠀⠀⠀⣿⡀⢀⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡞⠀⠀⣆⠀⣿⣷⢇⠀⠀⠘⢷⣷⠶⠟⠁⠀⠙⠳⠶⠶⠾⠛⢁⣀
⠀⠀⠀⠀⣿⡄⠀⠠⠀⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣾⠃⠀⠀⢹⠀⢸⣾⡘⠀⠀⠀⠀⡀⠀⠀⠐⣿⣿⡆⠀⠀⠀⠁⠈⣸⡇
⠀⠀⠀⠀⠘⢿⣶⡀⠃⢀⠀⠁⠆⢀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠸⠀⢸⣧⡇⡇⠀⠀⠀⠷⣤⣤⡾⠿⠋⠁⠀⢠⡀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣧⣆⠁⠊⣀⠀⢊⡄⠀⠀⠀⠀⠈⠘⠁⠸⠀⠀⠀⠀⢸⡇⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠡⢆⡄⠠⢆⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⡇⢰⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣦⣈⠰⣦⠉⠔⠀⠀⠈⣆⣿⣤⠀⠀⢰⣇⣼⣿⡇⠀⠀⠀⠀⠀⠀⠐⢲⣦⢴⣾⣯⣁⣠⣿⡂⠁⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣬⣁⠸⠄⠀⠀⣿⡸⣿⣧⡀⢸⣿⣿⡿⠁⠀⠀⣀⡀⠐⠳⣶⡏⣀⣼⣿⣍⣤⣝⣿⣆⠉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠑⠀⠀⠘⢷⣿⣿⣇⣾⣿⡟⠀⢀⣀⠈⢉⣿⣶⣞⣫⣤⣴⣿⣿⢙⣿⣿⣻⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠖⣐⠲⠶⠶⠶⣂⠴⠶⠶⠶⣌⠙⠛⠛⠉⠀⠀⠀⠈⠙⠻⢾⣷⣿⣯⣭⣽⣿⣿⠷⠾⠿⠿⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠤⠤⣿⠤⠤⠤⠤⣽⠄⡤⠤⢌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠈⠀⢀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⠀⠀⠀⠂⣿⠇⠀⠐⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡴⣯⢤⣤⡠⢤⣿⢤⣤⡤⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⢰⣿⠀⠀⠐⢻⡯⡀⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⠶⠆⠛⠷⠖⠰⠖⠿⠶⠖⢲⣔⠏⢻⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠀⢠⡴⠁⠀⠀⠀⠀⠀⠤⣴⣶⣤⣴⠒⢶⣖⡖⠚⠃⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠠⣶⣦⣾⣤⣔⣋⣤⣽⣏⠀⡀⠀⠛⠗⢦⣄⠰⣶⣤⠤⣤⣤⣀⣁⡀⠘⡃⣾⣿⣷⠄⠀⠀⣀⣠
⠀⠀⠲⣦⡀⢀⠀⠀⢠⣄⣸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣙⠶⣽⣶⠤⣌⡉⠉⠛⠂⠀⣤⠨⠿⠿⠛⠉⠉⠉⢉⣻⣶⣿⣿⣿⠆
⠀⠀⠜⠋⠻⠮⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠓⢄⠈⠳⢌⣽⠿⠶⠶⣤⣤⣀⡈⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⢀⡴⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠢⠀⠹⢦⣄⣠⣤⣴⣿⣻⣷⣶⣄⣸⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠑⠢⣾⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉

⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡀⠀⣀⠀⠀⠀⣠⢄⡄⢠⣤⣄⠀⠀⢤⣤⠀⢀⡀⠀⠀⣤⡄⢠⠤⢤⣤⣄⠀⣀⣤⡤⡤⢤⣄⡀⠀⠀⠀⠠
⠟⠀⠀⣀⡤⠤⠤⣤⣄⣀⣀⠀⠉⠉⢩⣄⣙⡷⠋⡀⢠⡏⡞⣹⣟⠀⠈⢳⣦⣞⠉⡾⢸⠁⠘⠛⠉⣰⣋⣹⣿⢿⠉⣀⣿⣯⣿⠀⣿⣹⣿⣷⣤⣀
⣠⣾⣏⠀⠀⠀⠀⠀⠀⣙⣿⣟⠛⣇⣄⠉⠉⠀⠀⠅⣰⣹⡿⣇⠈⠲⣶⣾⠋⢉⡿⢡⠞⠀⠀⠀⣰⢥⣼⣿⠛⠀⣦⣿⣿⣾⣿⡹⣿⣿⣻⣿⠉⠻⣷
⣾⠿⠿⢿⣿⣿⣶⣿⣿⣿⣿⠿⣿⠿⣿⣀⣀⣠⠀⠀⣿⡽⢧⣤⡿⣿⠥⠾⣛⣭⣒⠁⢠⣠⡿⢠⠷⣾⢿⠉⢦⣼⣿⣿⡿⠦⣼⣱⣹⢿⣇⠈⠳⡀⠈
⣿⣤⣤⣴⣾⣿⡛⢻⣭⣤⠽⠷⣟⠓⠾⡻⢿⡿⣄⣀⠘⣿⣿⣯⠟⠲⠶⠛⠑⡋⣁⣀⡈⣉⡀⢸⣻⡟⣦⣷⣿⣿⡟⡼⠇⠤⣾⣉⡇⣼⣿⣦⡀⢹⡆
⢷⡿⠫⠤⣤⣈⣿⠿⠛⠓⠢⣄⣈⣻⣶⠿⢾⣻⢏⡏⠀⠈⠉⢁⣠⠶⠟⠛⣡⣴⠾⠋⢉⠅⣹⢨⠟⣷⠿⠛⣿⢻⣇⣷⣄⣠⣿⣈⢷⣧⣿⢿⣧⠀⢳
⡞⢛⢒⣶⣾⣿⠟⠒⠲⣤⡴⠾⢿⡁⠈⢷⣀⡟⣸⣷⠀⠀⡴⠋⣁⣤⠶⠛⢋⣡⣤⣾⣿⣾⠅⠘⠊⠁⠀⠀⣡⣸⣿⠧⠤⣼⣇⡼⣾⡾⣿⣄⢻⣄
⣀⣽⣿⣿⣿⣧⣤⣤⠞⠛⠳⡀⠀⢳⣄⣨⡟⢠⢏⡟⢀⡾⣁⡾⠋⠀⠀⢠⡾⠟⢛⣿⠏⡿⠀⢀⡀⢠⡀⠀⢸⣿⣿⠦⢤⡾⣼⡿⣿⠁⠈⠪⣵⣿⣶
⣿⠿⣯⣿⢣⣄⣴⣛⣳⣤⣤⣿⣤⡾⠟⢋⣴⣧⠞⠀⡿⣿⡿⠀⠀⣠⡴⠋⠀⣠⡿⣡⠔⠁⣤⣌⣽⠟⠋⠀⠘⣿⣿⡶⣾⢶⡟⣽⠏⢀⣤⠤⠀⠉⠓
⣏⣀⣿⣧⣬⢭⣭⣭⣭⣭⣭⣥⠤⢒⡾⣿⡟⠁⠀⣤⠞⢡⣴⡾⣛⢁⣀⡤⠞⢉⠞⠁⠀⠀⠀⠁⠀⠰⠽⠿⠂⠹⣿⣟⣵⡯⠞⠁⠀⠻⠇
⠉⠉⠉⠁⠀⠈⠙⠿⣭⣙⣛⣛⣯⠥⠞⠉⠀⠀⠀⠷⠶⣥⢶⠿⠿⠿⠭⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠐⠄⠃⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠭⢹⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⢰⠄⠀⢀⠀⠀⠠⢠⠀⠀⠀⢀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠐⠀⠀⠀⠀⠠⠉⠀⡀⠀⡄⠀⡀⠘⠀⠀⠘⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⡴⢶⡎⣀⠀⠠⠄⠀⠔⠁⠑⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡙⡚⠣⡀⠀⠀⠀⠀⠢⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠲⠀⣨⠡⡠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣟⣳⠡⠠⠤⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⢀⠀⠀⣠⠐⠫⠁⠃⡄⠀⢠⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠠⢔⡀⠄⠀⠀⠤⠀⠌⢢⣀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠐⢨⢇⠜⡕⠂⠀⠀⣀⠀⠀⠈⠑⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⡱⠒⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⠐⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠒⡒⠀⠀⠈⡉⠩⠙⣿⡄⠀⠀⠈⠀⠄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢻⣆⠀⠀⠀⠁⠌⠐⡐⡡⠁⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢊⢆⠠⠀⠈⠌⠔⠔⡀⠁⠠⠘⢿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡄⢂⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠘⡿⣄⠀⠀⢐⣒⣪⣥⣥⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠛⢿⡌⢆⢤⣤⢬⣭⣿⣶⣶⡶⠿⠽⠕⣎⡵⠛⠛⠉⠉⠉⠉⠉⠀⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢥⣄⡀⠀⠀⢱⡈⢆⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣖⠚⠼⠈⢹⠐⠪⣷⠈⢜⡄⠀⠀⠀⡀⠁⠠⠀⠂⠀⠘⣽⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢉⡗⡦⢤⣀⡘⠃⣦⣧⠈⡿⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣀⡠⠤⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠇⢨⠙⢲⣿⢧⠸⣄⠀⠀⢀⣀⣠⡄⢠⣤⢤⣄⢸⠓⠀⣀⣉⣉⠥⠤⠤⠴⠶⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠈⠉⢀⠀⢣⠹⣄⣀⣀⣀⣤⠭⠄⠖⠒⠒⠚⣉⣉⣉⡤⠤⠤⠖⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢻⢷⠀⢷⣯⠭⠤⠤⠶⠒⠒⠚⢋⡩⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠺⠑⠀⢿⣗⣤⡑⡀

⠀⠀⠀⠀⠀⢀⣤⣔⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡀⡀⠛⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠦
⠀⠀⠀⢠⡶⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⣠⠞⠂⠀⠐⠋⠀⠀⠀⠘⠋⠉⠀⡀⢀⠄⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠁⠀⢀⣤⢀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⡀⠀⠀⢀⠀⠀⠘⡻⠋⠀⠀⡔⠁⠀⣾⠖⠀⠀⠀⠀⣀⡀⠉⠁
⠀⠀⢸⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣀⠴⠂⠀⠀⠀⠀⡛⠁⠀⣤⣄⣀⠤⠚⠁⠀⠀⢰⡗⠃⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⡀⣠⣆⢒⣈⣀⣤⡤⠟⠉⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⠀⠀⠀⠀⠚⠛⠃⠀⠀⠰⡿
⠀⠀⠘⠻⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠂⠀⠉⠁⠀⠀⠀⠀⠞⠀⠘⠟⠀⠀⠀⣠⣤⣄⠀⠀⣠⣤⣤⣤⣀⡀⢀⣾⢃⣼⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠘⠲⣿⠆⠀⠀⠈⡁⠀⠐⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣠⣿⠃⠘⠟⠛⠙⠛⠛⠛⣿⡟⣛⡁⣘⠁⣤⢤⢿⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢸⣿⠇⢀⣶⠲⡄⣼⡏⢹⢠⣿⣏⣼⢣⡿⢸⡧⢊⣔⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⢂⣾⣯⣠⣧⣿⠣⡼⢻⡟⢿⣄⠘⢓⣉⣙⡉⢸⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣠⠞⠻⣧⠟⠁⠙⠚⢁⣀⣥⣼⠿⢿⣍⣁⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠈⠿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠞⠃⠀⠈⠉⠁⢀⡀
⠀⠀⠀⠀⣴⠆⠀⠀⣤⠀⠀⠀⠈⠙⠁⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠒⠓⠚⠃⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⡾⠖⠋⠉⠁⠙⠒⠀⠀⠀⠰⠒
⠀⠀⠀⢠⠀⠘⠷⠀⠀⠀⠀⠀⣴⡶⠂⠀⠀⠉⠉⠀⠄⠀⠀⠀⡠⠂⠛⠁⠀⠐⠟⢃⡀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠘⢿⠁⣠⣤⠀⣀⣘⣀⣀⠠⣤⠀⠀⢤
⠀⠀⠀⣿⣧⠀⠀⣠⣄⢠⣴⣦⠀⠀⢠⣶⣤⠀⠔⢠⠆⠀⠀⣼⠃⠀⢛⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⢦⡤⠒⠉⠉⣉⠉⠉⠉⠓⢄⢀⡀⠁
⠀⠀⢰⣿⣿⣿⣧⣤⣤⣄⡀⢤⣄⣀⣀⣀⣀⣀⣴⡏⠀⠀⢀⡴⠀⠀⠙⠁⠀⠀⠀⠤⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⠀⠀⠋⠀⠀⢀⡀⠀⠈⠡
⠀⠀⠘⢿⣽⣋⡉⣉⡉⠹⠿⠟⠛⠉⠉⢉⣽⡷⣀⣶⠀⠀⠈⠳⠀⠀⠦⢀⠐⠀⠈⠀⠁⠚⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⣏⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⢰⣿⣿⣿⣭⢤⣦⣦⣤⣤⣴⡷⠾⠿⠛⠙⠋⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠉⠀⣀⠀⠀⠤⠀⠀⣀⡀⠀⠀⢸⡇⣴⠄⠀⠀⠀⠀⢴⣆⠀⠀⠀⡀
⠀⠀⠀⢨⣥⣌⣉⠀⠀⣀⣀⣀⢠⠾⠆⠀⣀⣠⣤⣴⡛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⣿⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡄⠛
⠀⠀⠀⢉⠾⠋⠉⢭⡉⠉⠋⢹⠟⢻⡏⠉⠉⠉⢈⣉⣤⠀⠀⠀⠀⠀⣀⣀⠀⠰⠖⠒⠒⠀⠉⠈⠁⠀⢠⡀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠶⠂
⠀⠀⢠⣿⡟⢛⡶⠶⣤⣤⣤⣤⡶⠶⠶⠞⠛⢋⡉⠡⢄⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢴⣦⡄⠀⠀⠀⠙⠁⢠⡆⠀⠀⠀⠈⠳⣄⣀⡀⠀⠀⢶⠆⣀⠤⠔⠁
⠀⠀⠀⠨⣷⣿⣿⣄⣀⠺⠷⠀⠀⠀⣀⡀⠀⣛⣣⠴⠊⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠉⠁⠀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⢀⣶⡿⢿⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣤⣶⣶⣦⣠⣾⡿⠿⠿⣿⣾⡟⠻⣿⡟⠀⠀⠙⠟⠉⣻⣷
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣶⡿⠟⠻⢿⣾⣿⠁⠈⢿⣿⠏⠀⠀⠀⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⢠⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢰⣿⠟⠛⢿⣷⣶⡿⠿⣿⣿⡟⠀⠀⠀⠀⠙⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿
⠀⢀⣴⣶⣿⡿⠀⠀⠀⠙⠟⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟
⠀⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣁⣀⣽⠇⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⡀
⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⢠⣾⡿⠋⠀⠀⠀⠀⠀⢠⣾⠋⠙⣷⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣧
⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⣹⣿⠟⠀⠀⠀⠀⣴⠆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢹⡧⠀⠀⠀⠀⠐⠿⣿⣦⡀
⠀⠀⣿⣏⠀⠀⠀⢀⡾⠋⣴⠟⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠹⣿⣦⣀⠀⠀⠀⠘⠃⠰⠟⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⢶⡶⠟⠁⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣄
⠀⠀⠀⢈⣿⡿⠃⠀⠀⠀⠀⠀⠀⣾⡋⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⣤⣉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣬⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠈⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢾⣿⣧⡀
⠀⠀⠀⠀⠀⠉⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠉⣿⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⢠⡞⠛⢻⣆⠀⠀⠀⠀⠀⠀⠘⠷⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠘⢷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣤⡄⠀⠀⣸⣿⣧⣄⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣀⣀⣼⣿⣷⣶⣾⡿⠃⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣼⣿⣦⣤⣴⣿⠋⠛⠿⠟⠋⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⣀⣠⣼⣷⡀⣀⣼⣿⢿⣷⣶⣶⣿⠟⠉⠙⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣉⣻⣿⣿⣛⣁⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⣀⣤⠶⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠤⠤⢀⢀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠄⡀⠂⠀⠄⠀⢀⠀⢘⡀⠀⠀⠀⡠⠀⠀⠨⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⡋⠁⠀⠈⠂⠀⠀⢠⠃⠀⠀⠉⠀⢤⠈⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢄⠀⠀⠀⢐⢯⠀⠀⠀⠀⠖⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣾⣧⣜⢦⠀⠀⣞⠀⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠈⠙⢦⣠⠻⣧⣾⠇⠀⠀⠀⠀⡀⠀⠠⠂⢀⠀⠀⠀⠺
⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⢰⣧⣼⡦⢿⣦⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⠁
⠀⠹⢄⠀⠀⠀⡄⠀⢳⠀⠀⠀⠀⣠⡆⠀⣨⣍⣭⠖⣩⡿⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠶⠁⠀⠀⠐⠁
⠀⠀⠀⠛⠢⡈⢧⠀⠘⠀⠀⠀⠀⠉⢃⠀⠙⠛⢿⣶⣿⡇⠀⠀⠀⢀⠔⣙⣀⡚⣀⢠⠀⠀⢀⣠⡦⣉⣠⠊
⠠⡄⠀⢀⠻⣤⠿⠗⠀⠘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⡷⠀⣠⡴⠟⠛⠛⠙⡁⠉⣠⣦⣴⣿⢿⠔⣠⠃
⠀⠀⠀⠀⠄⠻⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄⣦⡀⠸⢿⠟⠢⢀⠀⠐⠗⠀⠻⠛⠋⣹⣿⡿⠀⢀⡿⠤⠄
⠀⠀⠀⠄⠈⠂⢒⣇⠀⠀⠀⠠⠀⣼⣷⠄⠘⠿⠟⢹⣶⣿⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡟⢐⠀⠈⠁
⠀⠀⠀⠀⠐⠌⣻⣿⣦⡀⣠⢤⡾⠋⠀⠀⠀⣶⣤⣷⡟⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣍⡻⠦⠈⠁⠐⠒⠀⠀⠈⠉⠉⣶⣤⣿⠀⠀⠀⣀⠀⠀⠠⠐⢀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠛⢠⣤⣀⠀⠀⠙⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠁⠀⢸⡟⡤⠄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠣⣀⣀⡈⢒⠀⠐⠀⠀⠀⡀⠀⠊
⠠⠒⠂⠀⣰⠏⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⡾⠀⢀⣄⣶⣤⣀⣦⢀⡾⠁⡌⠸⠛⠉⠠⠘⠋⠀⠀⠄⠘
⠀⠀⠀⠐⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢈⣻⡿⠍⢻⣿⡊⠀⠀⠀⠊⠁⠠⢄⠀⡠⠒⠀⠉⠛
⠀⠀⠀⢇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⣨⡴⠚⠍⠉⢐⠦⣄⠈⠑⢦⡀⠀⠤⠒⠉⠉⠑⠒
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠍⠀⢾⣦⡷⠻⣾⡿⠀⠀⠀⠁⠀⠢⣤⣞⡁
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠲⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢬⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠀⠀⠀⢸⡔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⡓⠤⣀⠀⠀⠉⠀⠀⢀⠴⠒⠋⠁⠀⠀⠀⠀⠤⠐⣒⣩⠔⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠢⢬⣐⠀⠀⠀⢸⢀⣀⡠⠄⠀⠀⣠⠖⠖⠩⠋⠉⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣀⣈⠀⠐⠀⠁⢠⡄⢈⣉⡥⠴⠚⢋⡟⠁⢠⡤⣀⣀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠀⠁⢸⠁⠀⠀⣰⣎⡠⠖⠛⠛⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡀⢧⠈⠫⡛⠲⣄⣸⡤⠶⠊⠉⠀⠀⠀⠀⠀⠀⠤⠤⠿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡀⠈⢧⢸⠗⢦⠀⠸⠥⠀⣀⣀⡀⣀⣀⣤⢤⠴⠒⠒⢩⠏⠁⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠈⠻⡀⠀⢱⢲⠒⠉⠁⠙⠉⠀⠀⣷⠋⣠⠊⠀⠃⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠎⣿⠊⠉⡗⢀⠑⢄⡈⢸⠀⠀⠀⠀⢀⣀⡴⠥⠾⠁⠐⠒⠒⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠗⡄⠀⠠⠁⠀⠘⣤⠀⢓⠌⠓⠚⠒⠒⠉⠉⢉⡀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠃⢀⡀⠀⠀⢳⡜⠀⠀⠀⠀⠀⠀⠤⣴⣶⢿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⣀⣤⡒⠐⠲⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠶⠖⠒⠢⢄
⠀⠀⠀⠀⢀⡤⠚⠋⣶⠰⠎⠀⠀⢀⡀⠀⠒⠀⢀⣄⠀⠀⠀⢀⠴⠏⠈⠋⠁⠀⠀⠀⣠⠈⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⣩⡜⠷⣄
⠀⠀⠀⡐⠉⣀⠀⠀⠀⠻⡄⠡⠀⠙⠁⠈⠀⠀⠉⠀⠀⠀⡐⠫⠇⠀⠀⠀⠀⠰⠇⠁⠀⠀⠸⠟⠈⠂⠀⠀⠀⡇⢸⠀⣴⡄⠀⠄⠠⡄⠀⠀⡀⠐⢃
⠀⠀⢰⡇⠘⠿⠀⠀⠀⠀⠸⠶⣆⣁⣒⣢⠶⠊⢂⠀⠀⢘⠀⠀⠀⢴⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠐⠃⢰⠀⠋⠀⡀⠀⢤⠛⠀⠈⠛⠀⠘⠆
⠀⠀⣺⠁⡀⠀⠀⠀⣦⠀⠀⣀⠈⠀⠉⠁⠀⠄⢘⠀⠀⢸⠀⠀⢠⠈⠉⠁⠠⠄⠀⠀⠀⢀⠀⠀⡀⢀⡅⠀⠀⠀⢹⠀⠀⠐⠃⠀⠀⠂⠀⠀⠀⠀⠀⡇
⠀⠀⢧⠀⠀⠀⠀⠇⠉⠀⠀⠻⠃⠀⠀⠀⣶⡄⡘⠀⠀⠸⠀⠀⢰⡆⠀⠀⠀⠀⠀⠄⠀⠀⠁⠸⢃⡐⠃⠀⠀⠘⡀⠳⣄⠀⠀⣶⡄⠀⣀⠀⢾⡄⡤⠃
⠀⠀⠸⡅⠘⠿⠀⠀⠀⠀⠀⠀⠀⠒⢾⠃⢩⣰⠁⠀⠀⠀⠳⣄⡀⠀⠀⠀⠈⠲⣶⠖⠀⠀⢠⡤⢔⠎⠀⠀⠀⠀⠘⢦⡙⢧⣄⠀⠀⠀⠻⠀⠀⡘
⠀⠀⠀⠙⠦⣆⡀⠀⠀⠀⢶⠄⡀⡀⢀⣰⣿⠃⠀⠀⣀⡀⠀⠈⠙⠧⠄⣀⡀⠀⠀⢀⣐⠠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⢷⣿⡷⠾⠆⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⢶⣶⠿⠟⠋⠀⠀⢠⠞⠉⠉⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡌⠀⠀⠐⠂⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⢀⡘⠃⠀⢰⠀⠀⠂⠉⣀⡤⢀
⠀⠀⠀⢀⣁⡸⠇⠐⠀⠀⠀⡰⣊⠙⠛⢀⠉⠉⠳⠶⣄⠀⢀⡀⠀⠙⠂⠐⠀⠀⠀⠀⠀⠐⠃⣠⣦⠀⠀⠈⠀⡆⠀⠀⠁⠇⠀⠀⠀⠰⡄⢣
⠀⠀⠀⠀⠁⠈⠀⠀⢀⠖⠁⠀⠛⠀⠀⠛⠁⠀⠀⣀⠘⠄⠘⠓⠀⠈⠛⡄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⠀⡗⠁⠀⠀⠀⠀⠺⠗⠀⢷⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠰⠀⠀⠐⢒⠀⠀⠀⠀⠙⠃⠈⣆⠀⠀⠀⢀⣁⠈⠃⠀⠀⠘⠂⠀⠀⡀⠀⠿⠀⢀⡅⠂⠀⡆⢰⠟⠀⠀⡀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠀⠙⠶⠀⢸⡀⠀⣶⠄⠁⠰⠃⠀⠀⡇⠀⠉⠘⠃⠀⠀⢠⠊⠀⠀⠀⡇⠀⠀⠀⠀⠟⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⠃⠀⠀⠀⠐⠀⠀⠠⢠⠎⡈⠠⠠⡀⠀⠂⠀⠀⠀⢣⣠⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⣀⢄⠀⠀⠀⠱⣄⠲⢀⣀⠀⢐⣀⣠⣠⣤⠖⡠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡠⠋⠀⠀⡤⠶⠉⢂⠀⣆⠲⡶⠀⠀⡠⠁
⠀⠀⠀⠃⠂⠑⠄⠀⠀⠈⠑⠦⣍⣈⣉⣉⡉⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡩⠁⣀⠀⡂⠀⠀⣤⣄⠀⠀⡀⣀⠀⠈⠀⣀
⠀⠀⠀⡾⠷⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠩⡼⠟⠀⣑⠠⠆⢈⡅⠔⠀⠀⠒⡀⠉⠄⢈⠁⠀⢤
⠀⠀⠀⠙⡄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠱⡄⠀⠀⠐⠀⠉⠀⠀⠒⠀⠉⠀⠂⠀⠢⠀⠀⠀⢰⣦⠸⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠁⠙⠒⠊⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣔⠀⠀⢠⣦⠀⢀⠀⠈⣮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡆⠀⠀⣙⡁⠀⣐⠉⠄⠂⠀⠠⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠶⠚⠉⠈⠉⣉⠉⠛⠒⠒⠠⣄
⠀⠀⣠⠄⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠻⠁⢁⠤⠐⠀⠀⠀⠠⠒⠀⠐⠃⠀⠀⠀⠀⠀⢠⠊⠁⠀⠘⠁⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠾⢁
⠀⢸⠀⠀⢠⡈⠉⠁⠒⠲⠄⠀⠀⠀⠀⠲⠀⠠⠄⠒⣈⠁⠀⠰⠂⢀⠀⠀⠀⠠⠇⠤⠦⠄⣠⠀⠀⢠⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡎
⠀⠘⢦⠀⠀⠀⠀⠀⠘⠃⠀⠘⠖⠠⣀⠀⠀⠀⠀⣠⠄⠀⠀⡥⠌⢩⣬⡁⠀⠀⠠⣤⡄⠀⣸⠀⠀⢘⡃⠲⠦⠤⡤⢀⡀⠠⠠⣤⡤⠔⠒⠒⠋⠙⠏⢀
⠀⠀⠀⠙⢄⠂⢀⠀⠀⠀⠀⠰⣄⠀⡈⠁⠀⠀⠸⡀⠀⢠⣶⡇⢠⠆⠁⠀⠐⣤⠀⠀⠀⣰⠏⠀⠀⢨⠷⢟⡳⣶⠤⠀⠀⠀⠀⠀⠀⠉⠉⣀⠤⠔⣚⡽
⠀⠀⠀⠀⠀⠑⢤⡑⠀⠀⣀⠀⠀⠀⠉⠀⢠⠀⠀⠙⢦⣈⠉⠀⠁⠀⠀⠀⠀⠙⠁⣠⠾⠋⠀⠀⠀⢦⠀⠄⠁⠈⠸⣿⠒⠀⢩⣭⡬⠤⠴⠖⠚⢋⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠄⢈⡙⠓⠲⠒⠤⢬⠃⠀⠀⠀⠈⠙⠳⠶⠶⠦⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠑⠢⢤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣨⣥⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠶⠶⣶⡶⠶⠶⠶⠶⢶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠂⠉⠡⡖⠦⠀⠈⠉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⢀⣤⡄⢰⣦⠆⠀⣶⢢⠀⠀⢾⣌⡁⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠈⠛⠉⠀⣠⡴⡆⢀⣤⢄⠫⠄⠀⠀⢠⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠒⠀⡐⠀⠈⠋⠀⠈⠑⠉⠀⠀⠀⠒⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠳⣤⣀⣤⣀⠀⠀⠀⠈⠀⣀⣀⣀⣠⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠀⠠⠈⠉⠛⠛⠒⠲⡶⠚⠛⠋⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣷⣦⣀⠁⠠⡀⠈⡸⠷⠋⠀⣀⣤⣴⠚⣅⠴⠒⠒⠲⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠈⠉⢙⣿⡟⡿⠛⠛⠛⠋⠉⠋⠀⠀⡴⠁⠰⠆⢀⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠘⠃⡭⠅⠀⠀⠀⠀⠀⠀⢤⣠⣼⠁⣄⢂⠀⠸⠿⠇⠀⠀⢰⣿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠿⡲⣷⢿⣶⠶⠶⢲⢗⢾⠉⠽⣿⠀⠙⢉⠀⠀⠀⠺⠈⠰⠆⠀⠄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⢙⠒⠋⠀⠀⠈⠙⠁⠀⢀⣿⣠⠀⠹⡦⠃⠸⣭⠀⠀⢠⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⡊⣿⡿⣧⣤⣀⣀⣀⣀⣀⣴⣿⡿⣧⠁⠀⢀⡤⠄⠀⣆⡀⠀⠁⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢲⣿⣿⣧⡀⠉⠛⡙⠁⠈⢉⠉⠉⠠⣄⣂⣹⣧⣀⠀⠙⠇⠀⠀⠀⠀⣶⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠙⠻⠿⣷⣿⣧⣴⣦⣤⣄⣤⣴⣥⣿⣷⣟⢿⡄⠀⠸⠇⠀⠀⠀⠉⠁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡦⠀⠀⣀⡄⠀⠀⠀⠈⠉⠉⣩⣄⠀⠀⠀⠀⡒⢲⣶⣶⣤⣙⣳⣤⣀⣁⣀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠼⠿⠇⠀⠀⣀⣩⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠈⠁⠀⡀⠀⠀⠆⠀⢀⠀⠂⠀⠀⠌⠁⠀⠀⠀⠈⠀⠸⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠁⠀⠠⠀⠀⠠

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠿⠯⢠⣾⣧⡀⠐⠀⠀⣾⠁⢨⡷⠀⣴⢒⣦⠀⢉⡀⢰⢿⠂⣰⡿⣛⠻⣇⠀⡞⠀⠈⢳⠋⠀⠙⡆
⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠹⢤⣀⡀⠀⢻⣌⣉⢰⠫⣽⠂⠀⠀⠹⣷⣭⣾⡏⠀⠳⡀⠀⢺⠀⠀⡸⠃
⠀⠀⣴⣿⣷⡅⣥⣿⣿⣄⠀⣶⠆⢨⠟⣉⠉⠉⡉⠙⢳⡀⠀⠃⠀⠐⢶⠭⠶⠆⠀⠀⠙⢦⣸⣠⠞⠁
⠀⠈⢉⣽⡿⠇⡿⣿⣏⠉⠀⡀⠀⠘⠦⠿⠀⠀⣉⣀⣰⣧⣄⣀⣀⡘⠻⠖⠿⠃⢀⣤⡀⠀⠙⣡⡶⣤⡀
⠀⠠⠾⢿⣿⡆⣾⣿⡿⠧⠀⡁⢀⠀⠀⣀⣶⣀⠙⣿⣿⣿⣿⣿⡿⠟⠀⠄⠀⠀⠈⠛⠃⠀⠸⣕⡿⣫⡇
⠀⠀⣠⣿⣯⡃⢋⣿⣷⡀⢸⣎⣹⡁⠐⣺⣿⣗⠂⢸⣿⣿⡟⠁⠀⢠⡄⢶⢦⣤⠀⠸⣿⠟⠀⢉⣉⠉
⠀⠐⢛⠛⠛⣷⡟⠛⠛⠛⠈⠑⣋⡁⠀⠀⠛⠀⠀⢸⣿⣿⣧⠀⢸⡗⢠⣶⣦⠐⣷⠀⠀⠀⣰⢿⣿⡷
⠀⠀⣉⠠⡄⢉⠁⠀⣠⡀⠠⣾⣿⣿⣆⠀⠀⠙⠀⢸⡿⡟⡿⣧⠘⢫⡄⣩⣡⣬⠋⠀⢼⣿⣇⣠⡿⠃
⠀⢠⣭⡷⢳⣯⣄⠘⠿⠏⠀⠘⢿⣄⣸⣿⣯⠄⢀⣿⡁⠀⠀⣽⣧⠀⠀⠉⠁⠀⠀⠄⠀⠀⠉⣉⠀⢠
⠀⠀⠴⢋⡏⠳⠀⠀⢀⠀⠀⢀⣀⣉⣉⣉⣤⣶⣿⡿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⢀⣼⣿⣦
⠀⠀⠀⢠⣤⡄⠘⠁⠘⢷⣾⣿⣿⣿⠿⠻⣻⣿⡃⠀⢐⣻⣿⣿⡿⠛⠉⢀⣿⠏⣿⡆⢠⣾⣯⣩⣿⣷⡀
⠀⢰⣿⡟⣙⢻⣷⠄⠀⢸⣿⣿⣿⣧⣀⢀⢸⣿⣷⣦⣷⣿⡿⠋⠀⠘⠃⢸⡿⠀⣿⠇⢀⣼⣿⢙⣿⣧⡀
⠀⠀⢽⣧⣭⣾⡏⠀⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣁⡤⢤⣀⠀⣰⡞⠁⣴⠟⠀⠀⣼⣟⠛⣿⣇
⠀⢰⣶⣭⢀⣭⣶⡆⠀⢰⣿⣿⣿⣿⠋⠉⠀⣠⣀⠀⡼⢃⢮⣷⡂⢧⠈⡀⠀⢡⡀⠀⠈⠉⠉⡿⠉⠉⠁
⠀⠈⠻⠿⠿⠿⠛⠀⢠⣿⢣⣿⡟⠁⠖⠀⠀⠛⠋⠀⠸⣆⡛⢟⣂⠏⠈⠁⣰⣿⣿⡄⢀⡄⠀⠠⡞⢳⠄
⠀⠀⠠⠀⠀⠒⣠⣴⡿⠃⣼⡟⠀⣰⠚⣶⣴⣾⣶⡀⠀⠀⣍⢉⠀⠀⠀⠐⣿⣧⣾⡟⠂⠀⢀⡀⢉⠉
⠀⢠⡶⣲⢶⣰⡿⠋⣠⣾⠟⢁⠀⠹⣾⣿⣿⣿⡿⠀⠐⢶⡿⢻⡷⠂⠀⠰⣿⣧⣾⡿⠂⠐⠶⢟⡻⠶⠂
⠀⠸⣞⣿⣳⠇⠀⠀⡿⠋⠐⣏⣱⠀⠘⢿⡿⠋⠀⠆⠐⠛⡿⢿⠛⠃⠀⠰⠛⢧⡾⠿⠂⠐⢋⡶⢾⠙
⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢸⡄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠠⠤⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠈⠁⠀⠈⠀⠀⠀⠀⠉⠁⠀⠐⠄⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣠⣴⠀⠀⠀⠀⢀⠀⠀⠘⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠈⠓⠚⠛⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠠⠤⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠤⠀⢀
⠀⠀⠀⠀⠀⢄⣀⡀⠈⠁⠀⠀⠀⡀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣁⢀⡀⠀⠀⠁⠒⠁⣄⡤⠄⡀⠈
⠀⠀⠀⠀⠀⠀⢆⢀⠠⣄⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠈⠋⠉⡁⠀⠀⣀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠠⠀⠂⠀⠤⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠯⣅⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠦⢄⡀⠈⠁⠀⢀⠤⠒⠊⠁⠀⠀⠀⠤⢐⣂⡭⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⡒⠢⢅⡀⠀⢀⣄⡤⠄⢂⣀⡠⠒⠛⠙⠙⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣶⣤⡈⠀⠀⢾⠀⡴⠒⠋⢉⠜⢀⣰⣦⡤⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢟⠻⣶⣤⣀⢰⣆⣀⡤⠟⠒⠉⠀⠀⠀⢀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⢲⡸⡧⡈⠂⢜⠉⠉⠁⠀⠀⣀⣀⣠⠬⠭⣽⠓⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠈⠻⡁⠈⠶⡔⠒⠋⠙⠋⠉⢹⠞⣠⠆⠰⠁⠀⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠤⠾⡗⠉⣹⠠⡑⢄⠃⡇⠠⠀⠀⣀⣤⠯⠞⠃⠒⠓⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢉⣟⡶⠀⡐⠀⠈⠱⣄⢻⠆⠉⠉⠉⠉⠀⠠⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠘⠋⠀⠀⠀⠀⠀⠚⠛⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠒⠒⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⣰⠏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢰⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠲⡄⠀⠀⠀⠀⠀⠘⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⡆⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⢸
⢹⣇⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠋⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠤⠀⠀⠀⠀⠹⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⣤⡀⡄⠈⣿⡀⣀⠀⠀⢀⣀
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠈⣦⠀⠀⠿⢻⣿⣷⣿⣿⣿⣷⣾⣾⣥⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⣬⣷⣿⣾⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠩⠳⠤
⠀⠀⠀⠀⠀⠀⠈⠠⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠚⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⡿⢽⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣽⣿⣿⣿⣿⣿⣄⠻⠿⢿⡿⠿⠛⢫⣹⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢻⡿⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠿⣷⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢


⢴⣆⣀⣀⣀⠀⣀⠀⣀⠀⣄⣀
⠉⠛⠛⠉⠉⠁⠉⠋⠉⠁⠉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡾⣿⣷⠴⠶⠶⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣐⡋⢩⠁⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣯⣄⡉⠉⠀⠐⠀⠀⢀⡀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠈⠀⠰⠀⠰⠛⢿⣷⡠⡶⠀⠲⡷⠀⠙⠂⣙⠀⠀⠀⠀⠀⣰⠆⣴⡄⠀⠀⠀⠚⠃⠀⠀⢁
⠀⠀⠀⠀⠠⣄⠀⠰⠗⠀⠻⣻⡶⣧⠀⠀⠆⢀⠀⢀⠀⠀⠀⠀⠀⣿⣶⠄⠀⢀⣠⡄⠀⠂⠤⠔⠻⡄
⠀⠀⠀⠀⢀⠈⢳⡠⠕⠦⠤⠈⠻⡻⣶⡆⡀⠈⡀⡾⠀⠀⠀⠀⠀⢻⣳⠐⠀⠄⢨⣥⠀⠀⠲⠄
⠀⠀⠀⠀⢹⣄⠊⣍⢷⣄⠀⣖⡀⠘⢾⣧⡙⣠⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠐⠢⠤⠂⣁⡀⢠⠃
⠀⠀⠀⠀⠀⠙⠿⣟⡙⡍⠓⢤⡀⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⢰⠃⠴⠄⠀⠀⠀⠋⢀⣴⠺⠌⠁⡙⠋
⠀⠀⠀⠀⢠⠂⠡⠀⠉⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠃⠀⣀⡀⣿⡶⠆⠀⠠⣴⠄⣸⠿⡇⠀⠀⠈⢁⣤⡞⠃
⠀⠀⠀⠀⠁⠁⢠⣤⠄⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⡇⠀⠀⢉⡀⣿⣷⣀⡀⠂⢰⣶⣿⠆⠈⠀⣀⣲⣯⠎⢀⡐⠀⡀
⠀⠀⠀⣇⠀⠀⠀⢤⣀⠀⠀⠿⠀⠀⠘⢀⠁⠀⠀⢧⡀⠈⠃⠀⢾⣿⣿⣥⣈⣽⣿⣿⣿⠆⠀⠈⣿⡦⡏⠊⡀⠀⡃
⠀⠀⠀⢿⡈⠀⠙⠀⠀⠀⠀⠀⠁⠐⠀⠸⠀⠀⠀⠈⢳⡆⠀⢀⠑⡿⠻⣿⡿⠋⠉⠉⠙⠇⠠⡄⢽⠁⢠⣦⡁⠠
⠀⠀⠀⠈⢻⣤⠄⠀⠀⠂⣴⡄⠀⠀⠼⡇⠀⠀⠀⠀⠀⠉⠛⠛⠒⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆⠈⠉
⠀⠀⠀⠀⠈⠻⢦⡤⣶⣀⡀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠻⢀⠘⡀⠃⠀⠶⠧⠊⠃⢡⡴⠀⠶
⠀⠡⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠁⠀⠠⣤⢠⣆⣃⠦⢯⢡⠑⣄⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠉⠓⠁⠈⠀⠀⠘⠑⠀⡄⠉⠉⢠⠀⠊⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠩⠁⠀⠒⠐⠊
⠀⠀⢠⡀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡀⠡⠀⠀⠀⠂⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⢤⡶⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⡨⠂⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡤⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣌⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣯⣍⣉⣛⣛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⢿⣿⡿⠟⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣻⣟⣛⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣶⣦⣥⣤⣀⣀⠀⠉⠁⠀⠐⠒⠖⠀⠀⠀⢨⣍⣉⡉⠉⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣭⣿⣿⠉⠀⠐⠒⠒⠢⠄⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⣀⣀⣀⠀⠀⠈⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄⣀⣀⣈⡩⠛⢻⣿⣿⣿⣿⣥⣬⣭⣽⣟⣛⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿
⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⢿⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠶⣤⣄⣀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⣷⣶⣶⣶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣼⠻⠿⠿⠛⠤⠤⢀⣀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣴⣿⣿⣿⣦⡛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣟⣷⣬⠻⣿⣿⣿⣿⣿⣿⣿⡏⣽⣿⠟⠉⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣵⡿⠿⠗⢶⠾⣋⣿⡿⠟⠛⠉⠀⠀⠀⡆⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣶⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡉⠉⠉⠉⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⢦⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠆⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢀⡆⢸⣏⡝⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠘⡇⡀⠀⠀⠀⠀⣠⠟⠀⣼⠃⡈⣯⢂⠉⠉⠱⠒⢂⢄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠖⠒⠊⠉⠁⠀⠸⠁⠀⡇⢹⡆⡑⠀⠀⠀⠈⠀⠉⠑⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠚⣿⣛⡻⠟⠋⠀⠙⠀⣠⠀⠀⠀⡆⢸⠇⠳⣤⠤⢀⠈⠀⠈⠑⠀⠘⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⣀⠀⠁⠀⢠⡏⣶⠀⠀⡇⢸⠀⠠⡟⢀⠀⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠠⣾⠷⠋⠀⠀⣠⡜⠁⠀⠀⣿⣿⣷⠿⠀⠸⠀⣿⡄⠀⠁⠘⢦⡀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣿⡴⠃⠀⠀⢀⡴⠋⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠘⣿⣇⠀⠀⠀⠈⢳⠄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣎⡁⢠⡟⠁⠀⠀⣠⠊⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⢄⣄⡀⠻⣿⡄⠀⠀⠄⡀⢢⡀⠀⠰
⠀⠀⠀⠀⠈⠛⠶⠾⣿⡾⠿⣷⡟⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⠟⢁⢈⡈⠁⠀⢐⣋⣡⠰⢿⣷⣄⢠⠀⡇⣾⢓⠦⢭⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠉⠀⠀⠀⣶⠃⠀⠀⠀⢤⣼⠋⠠⠀⣨⠄⠀⡀⣺⠤⣼⠀⢿⣿⣿⣦⣇⣰⣄⣹⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡰⠁⠀⠀⢀⢰⣾⠃⠀⠀⠀⠴⠀⠤⢤⡟⠲⠖⠀⢸⢹⣟⢻⣿⣿⣝⡛⠻⠛⠛⢓⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣠⠁⠀⡄⠠⢄⣾⡏⠀⠀⠄⣴⠂⠀⠀⣼⠛⠓⡏⠀⠈⣼⣏⢀⠿⡄⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡉⠐⠤⠀⣠⣾⣿⠃⠀⠂⠠⠈⠀⠐⢤⣌⠉⢻⠃⠀⠀⢿⣿⣇⢰⣷⣦⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠄⣠⣾⣿⣿⣿⠀⠀⠐⠲⠉⠀⠒⢾⡅⠉⡋⠀⠀⠀⡞⠈⠙⢾⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⠀⢉⠐⢊⠈⠀⠲⠅⣀⡸⠁⠀⡴⡼⠁⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟⠋⠁⠀⡄⢨⢨⠄⠀⢀⡾⠦⢤⠃⠀⣰⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠰⣴⠂⠢⣼⣃⡒⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠃⠶⠗⠋⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠶⣦⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡾⠏⠀⣠⣶⠶⣶⣄⠀⠉⠉⠉⠉⠉⠛⠻⢷⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠑⠛⠿⢯⣝⡛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⣫⠆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣹⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⢀⡟⢡⡾⠉⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⢠⠎⣠⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣍⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣬⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⣴⢫⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠻⠿⠿⠃⠀⠀⢀⣾⣿⣿⣿⣦⡀⠀⠀⠀⢹⡘⣿⡆
⠀⢀⣾⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠽⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣼⡿⠀⠀⠀⢸⡇⣿⡇
⠀⣼⡏⠀⠀⠀⣀⡤⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⠀⠀⣾⡇⣿⡇
⠀⣿⠀⠀⠸⣿⣿⣇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⡀⠈⠀⠀⠀⠀⠀⠀⠈⠳⢹⣷
⠀⡏⠀⠀⠀⠙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⣠⣄⢀⠀⢻
⣸⢡⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⣯⠹⣿⣿⣿⡟⡞⠀⠀⠀⣠⣾⣿⣿⢸⡆⢸
⣿⣼⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣶⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠰⢯⣼⣿⣿⢸⣇⣿
⢻⡿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣼⣷⠀⠀⠀⠀⠀⠙⠛⢛⣛⡥⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢃⡾⣿⠃
⠈⢿⡙⢷⣄⡰⠶⣄⣀⡙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣼⡇
⠀⠈⠻⣿⠿⠻⣿⣿⠟⠿⣿⣿⣷⣤⣄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣾⣿⠃
⠀⠀⠀⠀⠘⢧⠈⠁⠠⡶⠉⠀⠸⠿⢿⣷⠦⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡻⣦⠀⠀⠀⢀⡴⣣⡿⢡⣿⠃
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⡄⢿⣿⣿⣿⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣿⠁⠀⢀⣴⣾⠟⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠸⣝⣻⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⣛⡯⠅⠀⢠⣿⢟⣵⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠘⣿⡄⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢇⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢰⣟⢀⠀⠀⠀⠀⢰⣿⣿⣤⣿⣧⠀⢀⣀⣠⣴⣤⡾⠷⠾⠿⠛⣋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣭⣤⣬⡘⣿⣿⣿⣿⠟⣠⠞⣫⢍⣩⣥⣶⣶⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⣿⣭⣴⣶⣮⣤⣭⣉⣥⣤⣶⡾⠿⠿⠿⠛⠉⠁⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣽⣿⣿⣯⣭⣭⣽⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⢀⠀⠐⢠⡄⠀⠀⠀⠠⣸⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⣸⣠⠄⠀⠀⠀⢀⡄⢂⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠱⠘⠀⠀⠀⠠⠄⣻⠄⠑⠆⣲⢒⡐⠂⠀⠐⠛⡝⠒⠀⠈⠒⣀⢀⠰⢀⠄⡠⠎⠀⠐⠊⡍⠓⠀⠀⠀⢂⡚⢖⠀⠎⠡⣞⠠⠄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⢈⡁⠆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠀⠀⠈⠁⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠘⢘⡠⢣⢜⡳⠜⠀⠁⠁⠈⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠄⡹⠃⡭⠀⠀⠀⢀⠐⠑⠠⠍⠂⠑⢦⡀⠀⠀⠨⢐⠟⡠⠀⠀⠛⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄
⠀⢀⠀⢿⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠀⠐⠄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠀⠀⠀⠀⠂⣼⠆
⠀⠀⠂⠠⡄⠀⠀⡀⠀⠀⠠⡇⠀⠆⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠆⢨⠅⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠢⠀⢠⠀⠀⠐⠀⢰⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⡠⠀⠔⠤⠤⠄⠀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠀⣷⠀⠀⠀⠑⣄⠀⣀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⣀⠌⠀⠀⠀⣴⠀⠔⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠒⣀⠋⠰⠀⠀⠙⠛⠀⢣⠆⠀⠀⠀⡄⠀⠀⠐⡞⠂⠛⠊⠀⠀⠒⠘⣀⠐⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⡄
⠀⠀⠀⠀⢤⠘⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣠⠁
⠀⠀⠀⠀⠀⡀⠊⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠤⠑⢂⠄
⠀⠀⠀⠀⠀⢀⠐⠆⢀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠘⠢⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠔⠛
⠀⠀⢠⡄⠀⠀⠀⠉⡈⠂⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠐⢃⡈⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠤⠄
⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠄⠄⠡⠄⠤⠀⠀⠠⠀⠀⠂⠀⠀⠀⡈⢀⣀⠀⠄⢂⠀⠀⠀⠀⣀⡰⠤⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠃⠀⠀⠀⠀⠂⠀⠄⠀⠀⠤⠠⠤⠀⠠
⠀⠀⠀⠐⠈⢀⠂⠀⠀⠀⠀⠘⢣⣽⡀⣁⢒⢡⡤⠭⠒⢠⠊⠉⠀⠀⠀⢉⣇⣀⠟⠀⠀⠀⠘⡙⡄⠐⠭⠤⡌⠐⣀⢀⣼⣔⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠐⠋⣕⠉⠉⠀⡃⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠘⡉⢢⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⢲⢺⠀⡉⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⢒⠢⠔⠂⠀⠀⠀⠠⡄
⠀⠀⠀⢈⣰⠀⠀⠀⠀⠂⠒⢋⠁⡈⠐⠀⠀⠀⠀⠀⠀⠀⠗⠔⠃⠠⠄⠀⡀⠀⠀⠃⠘⠀⠃⠌⠀⠀⠀⠀⠀⠀⠈⠐⢉⠀⡉⠒⠀⠂⠀⠀⠁⣦⠀⠆
⠀⠀⠀⠈⡃⡄⠀⠀⠀⠈⠄⢨⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡁⠠⠁⠀⠀⠀⢁⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀⠄⢰⡈⠣⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠓⡄⠀⠀⠀⠀⠐⠂⢲⠀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠁⠀⠀⢀⠞⠁⠃⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣙⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡙⠿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣤⣴⣿⣷⣆⠈⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻⠟⢻⣿⠿⣿⣿⣿⣿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣠⠴⠛⠃⣤⣾⣿⣿⣿⣬⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣧⠄⢀⣴⣿⣅⢀⣿⣿⣍⡻⡟⣳⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⢆⢀⡷⣼⣿⣿⡿⠟⢁⡰⠎⣩⣿⣿⠏⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⢿⣿⣿⡟⠉⣃⢲⡛⠛⠛⠻⢆⣈⣤⣤⠴⠋⣹⠇⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⢀⢉⡼⠋⠁⢠⣾⣿⣯⠝⢶⡤⡴⠾⠿⠟⢽⢳⣾⣥⣤⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢶⠞⠛⣿⡀⢀⡀⣀⣾⣿⣿⣿⣶⣤⣤⣤⣷⣶⣶⡘⣟⣿⢿⣿⠿⢿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣁⢀⣤⠂⠀⢀⣨⣷⣨⣿⣿⣿⣽⣿⣿⣿⡇⠀⢹⣿⡟⠿⡿⢧⣿⣶⣤⣴⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣒⠉⠙⠓⠲⠧⢬⣁⠛⠉⣫⣼⣄⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠛⢹⢿⣻⣽⠟⠋⢱⣾⠿⢿⣦⡽⠟⠛⠿⣿⡷⣶⡄⢠⡼⢋⢀⡀⠀⣠⣶⢶⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⣀⠠⣦⠰⠟⠉⠀⣀⣠⣾⣿⣄⣼⣿⣖⠶⢤⣤⡬⢿⣶⠊⠁⠙⠿⣿⣿⣶⣟⣁⣾⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⡆⣉⢩⣅⠀⠀⠘⣧⡀⣠⣾⠟⠛⠋⣠⣦⠀⠛⢿⣦⣆⡀⠀⠀⠙⠿⣷⣠⣤⣀⠈⠉⣉⠉⣭⣁⣴⣶⣾⠇
⠀⠀⠀⠀⠀⡸⢸⣿⣿⣦⡆⠀⠙⠣⣾⠟⣿⡷⢀⣷⣷⡟⠾⠁⣀⣴⣿⠧⣿⣷⡀⠀⠙⣿⣿⠂⠀⠀⡖⠉⢻⡟⣿⣿⣶⣿⣶⡏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣷⣾⡿⢣⡟⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠀⠀⢻⣿⣥⣿⠒⢿⣿⡇⠀⠀⠈⢿⡿⣿⣇⠰⢲⠞⠀⠀⠙⢿⣇⡁⠀⠀⠀⠀⠀⣠⡄⢰⣞
⠀⠀⠀⠀⢀⡿⠙⢱⡟⠀⣴⣶⣶⣶⠀⣠⣴⠿⠋⢀⡰⣷⣾⣿⣿⠿⢿⠦⢾⣿⣷⣶⠀⢀⠈⠙⠻⢿⣿⣛⢿⡷⢦⣄⣠⣈⢉⠓⢦⣤⣤⣾⢣⢿⣼⣿⡇
⠀⠀⠀⠀⢸⠃⠀⡾⢸⣿⡘⣿⣿⣿⡟⢉⠀⠀⣰⣿⣷⡸⣿⣷⣿⣤⣴⣄⣴⣿⣟⣻⣾⣿⡆⢰⣶⣾⣯⠉⠳⣄⠸⠟⠛⠛⢾⣿⣤⣦⣤⣿⣿⡜⠛⡿⠤⠴⠂
⠀⠀⠀⠀⢸⡆⢸⡇⠀⣿⣧⠹⣿⣿⣿⡘⢷⣿⠟⢛⣉⢷⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠙⣿⠇⣾⣿⣿⡟⢀⡼⠋⣀⣀⠀⠀⠀⣽⢻⡽⠿⣿⡍⠀⢸⠷⠿⠂
⠀⠀⠀⠀⢸⣇⠈⢷⡖⠛⡿⠃⠉⣀⣀⣰⣞⢳⣼⣿⣿⠿⠿⢿⣿⣿⡇⠀⢀⣤⣤⣄⢸⠟⠀⠿⢿⣿⣧⣾⣵⣿⠟⠻⣿⡆⠀⣷⢰⣇⣴⡟⣠⢆⡇
⠀⠀⢀⡴⠾⣷⡀⠈⢿⡾⣧⣴⣿⣿⠋⠉⣿⣿⣿⣿⡇⠀⠀⠸⡏⢿⡇⣰⠿⠋⠉⢻⡟⠶⠦⠶⣦⣿⣿⣿⣿⠀⠀⠀⢹⣿⢠⡷⠖⢛⣫⣾⣡⣾⣀⡀
⠀⠀⠈⢰⣿⣿⠧⡀⢨⢷⣿⣿⠷⠛⠀⠀⣸⣿⣿⡏⠤⠀⠀⠀⣿⡘⣹⣿⠀⠀⠀⣼⠀⠁⠀⠀⣜⢹⣿⠟⣿⠄⠰⠆⣼⣾⣯⠶⢒⣻⣿⣟⣥⣤⢀⡉⠁⣀
⠀⠀⠀⠀⠉⠃⠀⢻⣿⣼⣟⢣⣀⠐⣠⢤⣿⣿⣿⣿⠔⣄⠶⢰⣿⠇⢻⣿⠣⡜⠃⠻⡇⠶⠀⠀⠉⣻⣟⢦⡀⠶⣤⠴⠂⣠⢿⣾⡟⠋⢠⡅⠈⡿⠏⠛⠃
⠀⠀⠀⠀⠀⣼⢻⡏⠙⢯⠙⠿⢿⣯⣤⡞⠁⣉⣿⡀⠀⠀⠑⠉⠉⠀⣸⡈⣳⣶⣶⡾⠿⢆⠰⡶⠚⢿⣿⣿⣿⣶⣦⣀⠶⠋⣀⠬⠛⢁⡾⠶
⠀⠀⠀⠀⢰⡇⡀⠃⣀⢈⣷⣀⣀⠈⢩⡙⠚⠻⠿⣷⣤⣀⣀⣀⣄⣰⣿⣿⣯⢿⣿⣇⢀⡜⠉⢱⣄⣼⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣸⣧⣄⠀⠈⡄
⠀⠀⠀⠀⣸⢟⣡⠋⣹⣿⣿⣿⡿⢣⢸⡏⡷⣶⠀⠀⠉⠉⠻⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠀⠀⡧
⠀⠀⠀⢀⡾⢹⡇⠀⢿⢻⠝⡏⠀⠀⢸⡇⡇⣿⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠆⢶⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣠⡄⣲⠴⢷⠀⣸⡿⢥⣠⠔⠋
⠀⠀⠀⢘⣷⠟⢳⡀⢀⡟⣰⠀⠀⠀⠘⠃⠗⠿⢸⡏⢸⡇⢸⡇⢰⡆⡆⠀⣿⠀⠀⢼⠐⡆⣷⠁⣷⠛⣤⣿⢸⣿⣿⠷⠟⠃⠈⢠⡏⠀⢠⡗
⠀⠀⠀⢸⡏⠀⢸⣿⣼⠁⠈⠉⠛⠲⠦⠤⣤⣀⣀⡀⠈⠁⠈⠃⠈⠃⡇⠀⠙⠀⠀⠚⠚⠃⠛⠀⠉⠀⠉⠀⠀⠀⣀⣀⠤⠤⠒⠉⢷⣠⠟
⠀⠀⠀⠈⢿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡦⠶⠶⠖⠙⠃⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠾⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠥⠶⠤⡀⡀⠀⠀⠀⠀⠀⢻⣤⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠠⢀⡽⠛⠦⡀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣤⡈⠁⣠⣀⡀⠀⠀⠅⣈⠻⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⡻⢻⢻⣿⢟⣾⣿⣿⣿⠿⣧⣦⣄⠙⠓⠦⣉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠃⠈⡾⠇⢼⡏⣿⠋⠀⠀⠈⠙⢟⠀⠀⠀⠈⠑⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡀⠀⠀⠄⠀⠻⠀⠘⠀⠀⠀⠀⠈⠀⢢⠀⠀⠀⠀⠀⠑⣔⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠓⠲⠶⢶⡶⣶⠇⠀⠀⠀⠀⠀⠐⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢊⠧⠠⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠇⠀⠀⠀⠀⠠⠀⠀⠠⠗⣦⠱⡄⠂⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣌⡃⠀⡹⣆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣿⠀⠀⡞⠀⢠⣠⣞⠀⠀⣣⠁⠐⠈⡿⣷⣄⢄⢀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣰⢃⣴⡿⣏⢩⡀⠀⠦⠄⠰⠀⠇⢨⠙⠳⢷⣝⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⡿⠛⣄⠹⣆⡦⠀⢺⡆⡀⠈⢀⠆⠀⠀⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠘⢆⠙⣿⣆⢸⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠻⣾⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣩⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠒⠚⠓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠲⠦⠤⣴⣯⣓⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢉⣑⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣹⠋⠙⡿⣿⠙⠲⠦⡤⠖⠒⠉⠁⢀⠀⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣨⠞⢽⢷⣄⠀⠁⠤⣔⡢⠭⢴⠞⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠎⠀⡀⣼⣿⠀⢸⡾⠋⠓⡆⠉⠁⠀⢀⣮⡴⢎⠁
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡜⣰⠃⢸⣠⣿⣷⣦⣄⣀⡤⠶⡾⠟⠉⠀⠀⣻⡆
⠀⠀⠀⠀⢀⡾⠳⠤⢤⣀⣀⠀⠀⠀⠀⠀⣠⢎⣾⠉⢣⣿⣁⡀⠀⠀⠈⠉⠑⠆⠀⣠⠔⣡⢰⠁⢃
⠀⠀⠀⠀⣾⣕⣒⠠⠀⠀⠈⠉⠓⣆⢀⡴⢡⡏⢸⣴⡟⠛⠛⠫⠭⢭⣄⣀⡀⠀⢈⡴⢺⠡⣋⡴⠋
⠀⠀⠀⠀⠘⡄⣌⠉⡙⠦⢤⣀⡀⠚⠛⣱⠛⢇⣼⣿⣦⣤⣄⣀⣐⠉⠉⠀⠉⠉⢻⠄⣸⡟⠋
⠀⠀⠀⠀⠀⠈⠉⠒⠛⠓⢄⣙⠻⠉⢻⣧⡀⣼⡏⠀⠀⠀⠀⠀⠉⠉⠛⠒⠷⠤⢼⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠳⠿⠋⠀⡠⠶⣤⡀
⠀⠀⠀⠀⠀⠐⠒⣶⣷⠄⢈⡉⣹⢷⡀⠀⠀⡠⠊⠀⢠⣸⣿⡆
⠀⠀⠀⠀⠑⠶⠥⠾⠇⡈⠁⠀⠉⣂⡼⠁⠘⠅⣀⡀⣋⣠⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠷⠀⠲
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠁⠈⠈⢉⠀⠀⡀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⣒⢀⡄
⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠳⢠⡾⢋⣔⡂⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠿⠀⢰⠇⠀⠀⠀⢠⣦⠄⠀⠁⠀⠀⠀⢀⠈⠐⠆⣉⣤⡤⡯⠝⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⠀⠀⠀⠀⠀⠢⠀⠸⠋⠉⠑⠙
⠀⠀⠀⠀⠀⠀⠀⠐⠁⣤⣾⣵⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⢤⣠⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡻⠿⣶⡫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠤⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠰⣟⡋⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢮⣤⣄⣠⣀⣤⣤⠀⠀⠀⠀⢀⣰⠆⠀⠁⣤⣶⣢⣤⡄⠀⠉⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠼⠁⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠚⡽⠿⣝⢀⠀⠀⢸⡅⠐⠀⠀⡨⠈⠁⠉⡀⠀⣄⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠒⠶⠒⠈⠀⢠⣴⣖⣀⠁⠀⢈⣀⣀⣦⡄⠀⠀⠀⠘⠢⣈⢢⡄⠀⠈⢳⣷⣿⡀⠀⡀⢀⣀⣥⣰⠊⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⡣⠀⠀⠀⠀⠀⠀⠈⡁⢤⠰⠥⣼⣻⠛⡿⠟⠛⠉⠉⢈⡿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠙⠁⠏⠀⠘⠁⠀⠃⡀⢰⣆⣤⣼⠖⠀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⢠⠃⠀⣀⠐⣸⣄⠀⠀⠀⣿⠎⠛⠁⣀⣱⡄⠓⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⡀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣸⠷⠊⠁⢀⡄⢀⠀⠠⣡⡾⣿⡅⠀⠀⠲⢾⣴⣤⡶⠛⢧⡉⢑⡥⠇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠐⠶⠤⠤⢼⣦⣤⣀⣤⣤⣶⣾⡥⠤⠖⠒⠋⠀⠉⠙⠉⣿⣿⣿⠇⠀⠀⠀⠛⠁⠋⡇⢀⣴⢟⠛⠅⠂
⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⢀⡀⠀⠉⠀⠀⠀⠀⠐⠒⠒⠐⠒⠒⠀⠀⠀⠀⠀⠠⡆⠀⢈⠹⡈⠢⠀⠀⠸⠿⡟⣿⡟⠻⠀⠒⠂
⠀⠀⠀⠀⠀⠀⡔⠉⣠⡀⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⢸⡀⢳⣀⢠⡄
⠀⠀⠀⠀⠀⢠⠔⢉⣁⡀⢉⠓⠶⣗⠀⠃⠀⠀⢀⡀⣠⠄⠀⠄⠀⠀⠀⠀⢤⠀⠀⢀⠀⠙⠛⠉⠛⠁
⠀⠀⠀⠀⢠⠡⠾⠀⠿⠃⢠⠷⠀⠘⡆⡄⢀⡖⠈⠁⠀⢀⣤⡶⠀⠠⣼⡷⠆⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠵⠆⢹⠁⠈⠛⠛⢒⣿⡿⠿⠶⠶⠶⣶⣷⣶⡾⠶⠶⡛⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡧⢁⠇⠀⢠⣾⡋⠁⠀⠈⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⢸⠇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠋⠀⠀⠘⠿⢧⣮⣤⣶⣿⣿⢶⣶⣶⡶⠶⢶⠶⠶⠛⠃⠀⠈⢧⣤⢠⠀⡀⠀⠀⠀⢠⡀⣸⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⡋⠀⡀⠀⠈⠉⠉⠀⠀⠻⠛⠁⢈⣢⠀⠀⠀⠉⠛⠚⠧⠼⠦⠒⠛⠉
⠀⠀⠀⠀⠐⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠸⠷⣧⣤⣴⡿⠧⣤⣤⣤⣦⠤⠤⠴⠶⠾⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣧⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⢀⣴⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠚⠉⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣾⣷⢲⣶⣾⣿⣷⡄
⠀⢀⡴⠚⠁⠀⠀⠀⣼⣿⡄⠀⠀⢀⣄⠀⠙⢦⡀⠀⢸⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⡞⠀⢀⡀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣿⡗⠀⠀⠳⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿
⣸⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿
⡇⠀⠀⠀⣴⣦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣿⡷⠀⢸⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠁
⢳⡀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠙⠿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠳⣄⠀⠀⠀⠀⣠⣤⡀⠀⢸⣿⠇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠈⠢⣄⠀⠀⠘⠛⠀⠀⠀⠀⠀⣀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢀⣀⡀⣠⠆⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠄⣿⠁⠀⣤⣤⣶⣳⣎⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⣆⠀⠐⣤⣷⣾⣿⣆⠀⠈⢹⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠲⣀⣄⠘⠾⡁⠀⣀⠤⢉⢻⣷⣠⣾⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⢀⡼⢿⡄⠛⣉⣑⠛⠿⣋⢸⣿⢏⣽⠛⠋⣻⣯⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⢀⡿⠒⠋⢻⠀⣻⠿⢤⢈⡿⢋⣴⡁⣿⣷⣤⣄⢽⣎⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣥⢀⣞⣤⣤⠀⣸⠀⡍⣩⣼⣿⣇⣘⠇⣽⡌⡏⢀⣬⡞⢩⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢠⣆⠳⣤⣍⡳⣶⣾⣇⡩⠝⠿⣯⣷⡾⣿⣣⣏⡘⣛⣴⢫⣾⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠐⠶⣶⡟⠉⣁⡿⢿⡿⡇⠀⢠⡾⠋⢸⣯⢀⣴⣾⠟⣻⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣑⠺⠿⡚⢻⣿⡿⠴⠶⠂⠲⠾⣦⣲⣾⣾⣿⣷⡶⠞⢛⣩⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⠐⠒⡤⣭⣭⣭⣁⣀⣀⣈⣉⣉⣉⣭⣥⣶⣶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠿⡄⠀⣇⠀⠀⠈⠙⠟⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⠁⢸⡄⢐⢰⣷⠆⠾⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠸⡏⢸⣼⣿⣧⢽⣿⣿⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡄⣿⡄⣾⣿⣿⣸⣿⣿⣿⣿⢸⣽⣿⣿⣿⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠧⠧⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠿⠿⠿⠿⠙⡂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠛⠛⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢠⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠀⠸⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠿⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠿⠃⠀⢀⡴⠀⠀⠀⠀⠉⣠⡾⠁⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣶⣿⣦⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠋⠀⠀⠀⠀⠀⣤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣄⡀⠀⠀⢻⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠉⠉⠉⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠘⠇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠀⢸⡿⠿⣿⣿⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣈⠛⠛⠁⠀⠀⠀⠀⠀⠰⠛⠻⠟⠃⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠴⠶⠶⠶⠶⠶⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣶⡀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣤⡻⣇⠀⢀⡤⠴⠦⣤⡀⠀⠀⠀⠀⢀⡠⠴⠶⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⢰⢷⣙⣿⠁⣠⡶⠒⠶⠽⣲⣤⡀⣰⡉⠀⠀⠀⠀⠀⢀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠰⠃⠛⠉⢀⡼⠛⣳⣶⢀⡴⠋⢹⠷⠾⣷⣖⣶⡁⠀⣠⠏⠀⠀⢀⡈⠳⣄
⠀⠀⠀⠀⠀⡴⠃⠀⠤⠶⠓⢼⡀⣾⠏⢀⡄⠊⠁⡼⠧⣾⠁⣀⠖⢁⣸⠿⢿⣷⡉⠠⡈⠒⢯⢤⣀⣀⠙⠂⢤⣄⡠⢤⡀
⠀⠀⠀⠀⡼⠀⠀⠀⢀⡗⠀⠀⠛⠋⢹⡟⢀⡤⠊⢀⠴⠛⢹⣇⣤⠊⠀⡀⠾⣷⣹⡄⠰⡄⠘⠃⠀⠉⡟⠓⠚⣛⠟⠊
⠀⠀⠀⢸⠁⣀⠀⠚⢋⠁⠠⠄⡆⢶⠁⠸⠟⣧⣴⠋⠀⡴⠎⠀⣸⠶⣿⠀⢀⣼⣇⢳⡀⠃⠀⠀⡄⠀⢀⣤⠞⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠐⡆⠀⠀⢻⡜⣷⢀⡔⠉⢹⡶⡶⠁⣠⡼⠃⢀⣵⣾⢁⣈⣿⢨⠃⠻⣧⡘⠛⡛⣯
⠀⠀⠀⢰⠀⠀⠀⡤⠐⢧⠤⠄⠀⢳⣜⢿⣄⡤⠋⢀⠿⠛⣯⣤⠞⠈⢁⣨⣟⢉⡝⣸⠀⠸⠈⠹⣍⠁⢸
⠀⠀⠀⠈⣇⠠⠀⠀⠀⢸⡀⠀⠉⠁⢻⣧⠙⠷⣴⡁⢀⠞⢛⣧⣴⠟⠉⣉⣽⢛⣇⡟⠠⠀⠀⣤⢿⡀⠸⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⢘⡇⠀⢀⡀⢨⠁⠈⠐⠠⢝⡛⠶⣾⣀⠿⣶⠋⢠⣤⣿⣽⠉⠙⠦⣄⠋⠀⠑⢦⡛⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠴⠀⣦⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⠙⠢⠽⣶⢾⣇⣁⣠⣿⡏⠀⠀⠀⠈⠉⠀⠘⠂⠩⣌⡆
⠀⠀⠀⠀⠀⠀⠈⢧⠀⣰⡋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⡙⠻⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠁⣣⡞⠁⠀⠀⢀⠀⠀⠀⣀⣽⠃⠀⠀⠀⠀⠀⠙⢦⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢺⠀⠀⠀⠀⠀⠳⠶⠛⣩⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠦⣤⣄⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⢦⣄⡀⢀⣀⠀⠀⢀⠀⢀⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠒⠀⠒⢤⣥⡀⣈⣙⠋⠉⠀⠻⠦⠆⠚⠏⢶⠀⠈⢉⣀⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤⣽⣾⡯⠶⠷⠚⠛⠙⡾⣟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠒⡐⠚⣡⠀⣠⡦⣥⣼⣯⣁⣀⠀⠀⠀⠀⢰⡆⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡴⠃⠀⠀⠛⣤⡇⠐⠀⠈⡇⢸⡄⢳⠀⠀⠀⡾⠀⢰⡄⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡄⠀⠺⠀⠀⠀⠀⢠⣾⠁⡇⠀⢾⡃⢸⡀⣼⠀⠀⢸⠃⢀⠜⠁⢦⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⣾⣄⡀⠀⢀⡿⠃⠠⠇⠀⠸⡇⢸⡇⠸⡄⠀⠸⡇⠋⠀⠀⢸⣦⡀⠁⠷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⣠⠔⢻⠀⠉⢰⡿⠀⠀⡀⠀⠀⠀⠃⠀⠇⠀⠙⢄⠸⣇⠀⠀⣠⠊⠈⣧⣀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⣧⠀⣻⠀⠀⠘⠀⠀⠀⢠⠇⠀⠐⠀⠀⠈⢂⠙⡆⠰⠁⠀⠀⡈⠡⠉⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣽⠉⣛⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠀⠀⠀⠘⢤⡆⠀⠀⠀⠀⠁⠀⠳⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡴⠹⠀⢹⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢠⡆⢱⡄⠀⠀⠀⠀⠀⣸⡇⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⣦⣀⣧⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⡸⠁⠈⠇⠀⠀⠀⠀⢀⠇⣵⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⠁⢸⡟⢢⡄⠀⠃⠀⠀⡌⠀⣸⠀⠀⢰⠇⠀⠀⠀⠹⣆⠀⣴⠃⣀⣼⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⠴⠋⠀⠀⠙⢦⡀⢦⠀⠀⠰⠁⠀⣰⠃⠀⠀⠀⠀⠀⠈⠓⠇⣀⠋⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠹⣞⡀⠀⢞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⠖⠟⠳⢲⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢰⣶⣶⡆⢱⣶⣶⡆⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠾⢽⢯⠷⢀⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡷⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢰⣿⢾⣿⣄
⠀⠹⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⣸⣿⣧⣤⣄⠀⠀⠐⢿⠿⠀⠿⠿⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠀⠀⠿⠟⠛⠉⠛⠛⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁
⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⣚⣷⡴⠀⠀⢦⣴⣗⠚⠀⠋⣤⠀⠀⠀⠀⠀⢈⡉⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⡻⢟⡀⠀⠀⠈⠁⢖⣿⣶⣶⣾⣳⠋⠁⠀⠈⢃⡄⠀⠀⠀⠀⣈⡠⠄⠀⠸⠿⠆⠀⣀⠂⠀⡀⠀⠺⠿⠂
⠀⢻⣿⣸⣿⣿⣇⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣤⠄⠀⢀⣈⣁⡤⠖⠀⠀⠀⠀⠀⠈⠉⠹⣷
⠀⠈⠿⠋⠀⠀⠙⠻⠋⠀⠀⠈⣿⠛⠛⢻⠏⠀⢠⣦⠀⠀⢡⡆⠀⠈⣡⣀⡠⠵⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣾⣶⣶⣧⣄⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠎⠸⠂⠀⠈⠈⠁⠀⠂⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣾⣿⣿⠟⠃⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠐⠻⠿⠿⠀⠀⣤⣄⣻⣿⣿⣟⣠⣠⠀⠀⠀⢀⣴⣄⡘⠇
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢿⣿⣷⣆⣹⡿⠋⠀⠀⣴⣿⣤⣬⣿⡆
⠀⠰⠿⠿⠿⠿⢿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠋⠈⠿⢿⠷⠀⠀⠿⠿⠀⠸⠿⠇
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⢀⣬⠁⠱⠆⠀⠀⠀⠀⠀⠀⠀⣔⣰⣶⣠⣾⣧⢀⣤⠀⠀⣴⡟⠛⠉⠉⠙⢳⣆⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠘⢆⡛⣡⡿⡿⠳⠇⢀⠀⠠⠀⠀⢀⣹⣷⣽⣿⣿⣿⣿⠃⠀⢘⣿⠀⠀⠀⠀⠀⠀⣯⠀⢠⣿⡿⠟⢻⣿⡆
⠀⠘⠛⣤⠢⣤⠷⢆⠈⣘⠋⠀⢇⠀⠀⠸⢿⣿⣿⠿⠿⠟⠃⠀⠀⠀⢿⣤⣀⡀⢀⡀⣰⡟⠀⢺⣿⣭⢀⠰⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠘⠁⠙⠁⠀⠀⠑⠂⠤⠤⠤⠤⠤⠄⠂⠀⠀⠀⠀⠀⠉⠙⡿⡿⠛⠉⠀⠀⠈⠻⢿⣿⣿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠿⠟⠻⠿⣶⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠟⠋⠉⢠⣶⣿⣷⣄⠀⠀⠉⠉⠉⠉⠉⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠀⠀⠀⠀⠀⠀⠰⣿⣿⣦⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠑⡀
⠀⠀⠀⢀⡾⠃⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠁⠀⠀⣷
⠀⠀⠀⣼⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢻⣧⡀
⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢻⣧
⠀⠀⠀⠀⠀⠀⠘⠿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⡀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⢰⠦⠀⠀⠀⣸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⣀⠀⠀⠀⠀⠻⠤⠂⠀⣴⣟⣻⠿⠟⠁⠀⠘⠦⠄⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉
⠀⠀⠀⠀⠀⣠⠆⢀⠀⠈⠉⠛⠿⣶⣄⣀⣀⣀⣀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⡐⠋
⠀⠀⠀⠀⠀⠹⣄⣸⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⢷⣶⣤⣤⣤⣴⣦⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⡌⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠇⠀⠈⠀⠀⠀⠀⠀⢀⠐⠁⠠⠛⠉⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⠀⠘⠶⣤⡈⠐⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡠⠀⠀⠀⠈⠙⠳⣄⡀⠀⡠⠃⡀⠊⠀⠀⠀⠀⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⢶⡦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⠀⣠⠟⠛⣀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠍⠓⢤⡈⠐⢄⠰⠒⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠁⠀⠘⢿⡄⠁⠈⠳⣄⠙⠢⠄⠀⠀⢀⠞⠁⠖⠁⠀⠀⠀⠉⠳⣄⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⢀⠀⠀⠙⢦⡀⠀⠀⠉⠢⣀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠹⠛⢷⠀⠙⠦⣀⠀⠀⠈⢿⣥⣄⠀⠀⠀⠉⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠔⠠⠀⠀⠸⠖⢳⠀⣨⣷⣄⡀⠂⠀⠈⠳⠤⠀⠀⠀⠀⢂⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠰⠁⠈⠁⠓⢂⡀⠀⠠⠾⠁⣀⣸⡙⠲⣄⡀⠀⠀⠀⢤⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠑⠊⡅⣰⠋⣼⢤⡀⠀⠻⠽⠋⢹⠂⠀⠉⠓⠦⣤⣠⠏⣹⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠉⠡⠞⡁⢖⣮⡤⠂⠀⢀⢎⣀⣠⠶⣤⠴⢤⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡏⠺⣈⠅⠽⡏⠈⠋⠀⢀⡴⠋⠛⠒⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢁⠨⢄⠇⠠⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀⠀⡀⠀⡀⢀⡀⢀⠀⣀⡀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠈⠪⢰⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠒⠂⠤⠤⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠠⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⡀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠄⠀⠀⠀⠀⣕⠨⣐⡒⠀⠀⠀⠀⠁⠐⢂⠀⡪⠅
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢈⠐⠀⠀⠀⠀⠀⠁⠩⢒⠀⠀⠀⠄⠀⠄⠄⠀⡈⠔⠁
⠀⠀⠀⠀⠠⠀⠀⠠⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠠⠂⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠔⠇⠮⠐⠍⠐⠅⠪⠕⠎⠰⠅⠪⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠃⠀⠁⠀⠀⠈⢀⠈⠈⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⡆
⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⠀⢨⡀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⢼⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⢄⠀⠁⠂⠀⠐⠁⠀⠀⠀⡀⠀⠐⢈⠔⠁⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⣀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠁⠐⠀⠀⠂⠐⠁⡀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠀⠂⠈⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⣠⠂⡠⢀⡤⢂⠖⡠⠐⢀
⠀⠀⠀⠀⢠⡀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⡄⢀⠴⠋⣠⠎⡠⠊⡀⢊⠐⠃⠈⡁⠠⡞
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⠀⠘⠁⡊⠄⠤⠒⠁⠈⠠⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠅⠀⠨⠈⠉⠁⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠛⠛⠋⠁⠀⠈⠉⠉⠉⠉⠉⠳⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⢀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠠⠄⠀⠂⠀⠀⠀⢄⠀⠀⠀⠀⢀⡠⠠⢄⠀⠀⠘⠒⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠓⠢⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠤⠀⣇⠀⠀⠀⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⣹⠀⠀⠘⠠⢤⠃⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡀⢀⠀⠀⠀⡀⠀⠁⠁⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠈⠑⠂⠀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠈⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⣯⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡃⠀⠀⠈⠑⢤⠄⠀⠀⠀⠀⢘⡇
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠒⠀⠀⢀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⢺⠃
⠀⠀⠀⠘⢧⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⡀⣾
⠀⠀⠀⠀⠀⢹⡌⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⡏
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⡀⠈⠐⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣀⡀⠀⠀⠈⠑⠀⠀⠂⠁⠀⠀⠈⠁⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠴⢦⣀⠀⠀⣀⡤⢤⣤⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⠀⠉⡁

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠘⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠆⠰⠈
⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠛⡘⠛⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠰⡇⠀⠀⠠⠤⠖⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠠⠄⠀⠀⠠⠠⠄⡄⠀⠀⠀⠀⡂⠆⠅⠀⠀⠀⠰⠰⠀⠀⠀⠀⡀⠀⠀⣠⢭⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠰⠠⠀⣃⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢇⣃⠸⡤⡀⣧⣅⠈⣆⠀⠀⠀⠁⠀⠀⠸⡿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⣩⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⢀⡄⡄⠀⣄⠀⢠⠀⡀⡀⢠⡀⠀⡄⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢀⣿⡇⠀⠈⠫⣶⡽⠃
⠀⠀⡡⢖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣⡀⠐⠇⠧⠀⠷⠄⠸⠀⠬⠄⠀⠧⠀⠧⠉⠆⠮⠐⠦⠰⠥⠀⠀⠠⠻⠛⠃⠀⠀⠀⣿
⠀⠀⠖⠉⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢴⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢄⠐⠀⣠⠆⠀⠀⠀⡆⠀⡆⠀⢀⢀⠀⠀⠀⢀⣀⡀⠀⠀⣄⠀⠀⠀⢰⡿⠁⠀⢈⠀⠄⠀⠀⠁⢈⠄
⠀⠀⠀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀
⠀⠀⠀⣴⡿⠁⠀⠀⠀⣠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡄
⠀⠀⢰⡿⠁⠀⠀⠀⠀⠙⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣾⠃
⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⣶⣤⣄⡀
⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀
⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄
⠀⠀⠀⠀⠀⠉⠛⠿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⢀⣴⡾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠋⠛⠛⠗⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣴⣤⣤⣄⡀
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣶⡾⠟⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣀⣠⣤⣤⣤⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠟⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡾⠿⣶⣤⣤⣤⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣄⢠⡀⣠⣄⢀⣤⡀⣠⣄⢠⡀⠀⣠⡀⣤⣤⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⣿⣼⢿⡇⣿⣾⠈⠃⡇⣿⢸⡇⠀⡿⣇⢸⡇⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⡿⢸⠟⣧⡿⢺⣴⠇⣧⡿⢸⣥⣴⠗⢿⢸⡇⢸⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⠤⠤⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠄⠈⠀⡽⢇⡠⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡂⠀⠀⠀⠈⠉⠉⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠂⠀⠤⠀⠀⠀⠀⠄⠀⠀⡰⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⡞⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠒⢩⠔⠒⠞⡰⠁⢠⣟⠁⠀⠠⠤⠤⣄⢀⣰⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡴⠁⢠⠇⠀⠉⠁⠒⠐⠊⠀⠠⢙⠥⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡖⠀⠀⠓⠂⠤⠼⠁⠀⡿⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠁⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡶⠒⠦⢤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣶⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⢠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠠⠤⠤⣄⣀⣀⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡡⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠶⠷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠂⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⣿⠀⠄⠁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⣿⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠠⠀⠀⠀⠛⠋⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣠⣤⣍⢥⠀⠐⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡄⠀⠀⡄⠀⠈
⠀⠀⠀⠁⣠⣤⣤⣀⡀⠀⠀⠓⠊⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠇
⠀⠀⠀⠀⠋⠉⢙⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣛⡛⣻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠒⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢄⣀⠠⠄⠀⢀⣀⣀⣀⡀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡌⠉⣁⣂⡀⠀⢠⣤⣤⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣯⣭⣛⠿⢷⣮⣿⡿⣥⡀⠈⠉⠀⠤⣒⣄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠙⢶⣄⠀⠀⣠⠀⠈⠙⠲⣤⡀⠀⠀⢀⣠⠀⠀⠀⠀⢻⣷⣹⣶⠙⣶⣆⣀⡽⠿⢷⣽⣷⣦⣤⣀⠀⢉⠀⠀⢀
⠀⠀⠀⣠⣮⣀⠀⠀⠀⠀⡈⠷⣮⡀⠀⠀⠀⠀⠀⢙⣶⡿⠛⠁⠀⠀⠀⠀⠘⣿⣯⠻⣿⣀⡔⠛⣲⢶⣈⡩⠿⣮⣿⣍⠉⠐⠂⠀⠀⠳⢄
⠀⠀⠀⠻⣎⡻⣷⣄⠐⠊⠀⠀⠀⠙⠷⣤⣠⣦⣇⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣝⢿⣧⣼⣁⡀⠛⢣⣔⠉⣻⣿⣿⣶⠶⠖⡠⠄⠊
⠀⠀⠀⠀⠈⠙⢧⣽⣷⣦⡀⠀⠀⣀⣤⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣍⡛⠻⠿⣶⣇⣨⠿⣥⡬⣿⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣗⣮⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣦⣭⣛⣓⣻⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⣤⠤⡀
⠀⠀⠀⠀⠀⠀⡀⢠⣞⡷⣑⢟⡳⢯⢤⢿⣿⣥⣛⡧⢰⣒⡀⠄
⠀⠀⠀⠀⠀⠀⢹⠂⢈⠡⠉⠙⠉⠉⠉⠙⠒⠛⠛⠃⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⣀⡴⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠷⠶⠤⠶⠽⠷⣤⣤⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⣠⣤⣴⡞⡁⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣻⣿⣿⣿⣎⣷⣿⣦⡀⠀⠀⡀⠐⣀⠍⡄
⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡂⠀⠀⠀⠀⠀⠀⠙⣟⣽⣽⣿⣿⣿⣿⣿⣿⣖⣀⣴⣪⣽⡧
⠀⠀⠀⠀⠀⠰⢧⣤⣤⣬⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢿⡿⣿⡿⢻⡁⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⢖⠿⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠒⠉

⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄⣀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠁⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠥
⠀⠀⠀⠤⣄⣀⡤⠤⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈
⠀⠀⠀⠀⣈⠻⢃⢀⡈⣡⡄⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡀⢔⠀⠖
⠀⠀⠀⠀⢈⡷⢉⠦⡞⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⡀⠀⢰⣶⣆⠀⠀⣤⠐⢂⠉⢀⠛⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢴⣶⠄⠀⠀⢰⣶⡄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠉
⠀⠀⠀⠀⣠⣤⡁⢠⣾⡆⠈⠉⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⣼⣿⣶⣷⣶⣆
⠀⠀⠀⠀⠻⠿⠃⠀⠉⠀⢀⣤⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠈⠛⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠨⠀⠀⠠⢀⠀⠀⠽⠀⠀⠀⠀⠀⣤⣤⠶⢶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠁⠀⡄⢠⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⢐⠢⡀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠄⠉⠀⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠁

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