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

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠺⣧⢉⣻⡄⠀⠀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠉⠙⢶⡄⠀⣠⡶⠋⠉⠀⠀⠀⠀⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⢠⣼⢇⡄⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⣠⡀⠾⣦⣟⡷⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⢀⣤⢙⡵⠾⣶⣤⡀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⢀⡉⡟⣠⡀⠈⢿⣷⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠈⡄⢧⢿⢻⣐⡠⠋⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣾⡿⠃
⠀⠀⠀⢨⣌⢯⡺⣏⢷⡞⢧⣬⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠌⣡⣾⠟⠁
⠀⠀⠀⢸⣿⠓⢭⣪⢽⣛⡮⢷⣘⣌⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⠘⣃⣴⡾⠛⠁⣀⣀⣤⣴⡞
⠀⠀⠀⢀⣿⡁⠀⠈⣛⣶⡶⣦⣼⣿⠿⠶⢾⣽⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣄⢘⣧⣶⣿⡯⠷⠒⠛⠋⢉⣥⣿⠟
⠀⠀⠀⠘⣿⣷⠊⢉⢔⣿⣾⠟⠉⢀⣐⢶⣛⣛⣛⣟⡯⣛⢷⣄⠀⠀⡀⣣⡾⣛⣽⣯⣷⣶⣶⠶⣶⠾⠿⠟⠁
⠀⠀⠀⠀⠀⢿⣷⡏⢸⢻⣿⠀⠰⠟⢻⣿⡎⢹⣧⣤⣭⣮⡳⡙⢷⣐⡾⢫⣾⣯⣤⣀⡀⠀⠉⠳⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠉⢿⣿⠸⣿⣦⣀⣀⣠⣿⡏⠀⠀⠘⠟⠒⠒⠊⠀⠛⠰⠯⣅⡈⠙⠷⣍⠻⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡾⣆⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⠁⠈⠳⣄⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⡤⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣽⣍⠣⡀⠀⠾⣞⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢊⣻⡴⡶⠀⣴⣿⡿⠛⠉⠀⠀⠀⠀⠈⠉⠳⣄⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⠀⣀⡀⠝⣂⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⣼⢣⡄⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣗⠉⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠾⢴⣟⡷⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣌⢛⡵⠶⣮⣄⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢈⢡⡏⣄⠀⠹⣿⣧⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢉⡘⡇⡟⢧⠠⠬⠏⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⡹⣕⢼⡍⣧⡶⣆⣸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡿⢮⣳⢽⣚⠷⣼⣧⡰⡝⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠙⠶⣭⣤⢄⣩⣿⣶⣤⣭⣻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⠖⠊⡩⣿⣵⡿⠋⠉⠡⣤⠶⠿⠿⡿⢟⡿⣦⣀⠀⠀⠀⠀⣠⡶⢟⣩⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠞⢸⣿⡟⠀⢠⡶⣿⣾⡝⢻⣟⣉⣓⣙⢮⡙⢦⡀⣠⡞⣋⣊⠥⠤⠤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡏⢻⣷⡀⠀⠀⢈⣿⡷⠈⠉⠹⣿⣭⣥⡕⠌⢳⢋⡾⠭⢭⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠻⢿⣷⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⠀⠀⠁⠂⢄⡠⡀⠀⠀⠔⠁⠀⠂⠀⠐⠄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠜⠁⠀⠀⠐⡀⠀⠀⠈⠊⢢⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⠈⠂⢄⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣄⠀⠀⢸⠀⠈⢢⠐⠀⢷⣄⠀⠀⠀⠀⠁⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠃⠈⠉⠻⡄⠀⠘⠀⠀⠈⠀⠀⠀⠀⢧⠄
⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠉⠙⠉⠀⠁⠀⠀⠀⠀⠀⠀⠻⢆⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⢄⣦⠤⠤⠤⠤⠤⢤⣠⣀⣀⡀⠀⠀⠀⠊⠀⠀⣀⠤⠀⠒⠒⠒⠒⠒⠲⠤⣄⣤⣤⣤⣀
⠐⠃⠉⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⢴⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡃

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⣼⣿⣿⡟⠛⠛⠻⣿⣿⣷⣄⣴⣾⣿⣿⠿⠿⣿⣷⡄⠀⠀⣾⣿⣿⠟⠉⠛⠿⣿⣿⣦⣀⣴⣿⣿⡿⠿⠿⣿⣦⡀
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋⠀⠀⠀⠈⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⠃⠀⠀⠀⠸⡞⡇
⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⢉⣿⠀⠈⣿⣿⣟⡄⠀⠀⠀⠀⠈⢻⣿⠏⠀⠀⠀⠀⠀⠀⢡⡇
⠀⠀⠀⠘⣿⣿⣯⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠿⠹⡣⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⢝⡂⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆



⣶⡶⢶⡶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⣭⡍⠉⠈⠉⠓⢦⡀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣵⣿⣿⣿⣷⣶⢬⣠⣴⣶⡝⣄⢠⠞⣡⣤⣤⣀⣀⣠⡀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣮⣿⣾⣿⣿⣿⣿⣿⣿⣯⣤⣀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣴⣿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⢿⣆⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣴⡿⠋⠀⠀⠈⢧⡻⡄⠀⠀⣤⡆⠀⠀⢀⣴⠟⠁⠀⠀⣠⣴⠿⡟⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⡾⠋⠀⠀⠀⠀⠀⠈⢧⢱⡀⠀⠛⠃⠀⣠⡾⠋⠀⢀⣴⡾⠋⠁⠀⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠞⣡⣤⣤⣤⣄⡀⠀⠀⠈⣧⢧⠀⢀⣠⣾⣿⣤⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⡾⠛⠋⠉⠉⠛⠻⣷⣄⠀⠈⢿⣷⢟⣛⣩⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠇
⠀⠀⠀⢀⣀⣤⣤⣶⣶⠿⠟⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢸⠁⠿⠛⢛⣻⡷⠀⠀⣰⣶⣶⣤⣤⣴⣶⣶⣶⣶⣾⠿⣦⣄⣀
⠀⠀⠀⣿⠛⠻⠿⠿⠷⣶⣶⣾⣿⣷⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣾⣿⣿⠟⣿⠛⠁⠀⠿⠿⠿⠛⠻⢿⣿⣍⣩⣭⣭⣥⣤⣶⣿⣿⣿⣿⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⢉⣽⠿⠋⣡⡿⢫⡿⠋⠁⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠛⠛⠛⠋⠉⠉⠉⠉⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣼⠋⢠⡟⠀⠀⠀⠀⠀⣶⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠃⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠘⠻⠿⣿⠿⣿⣿⣿⣶⣶⣶⠟⠁⠀⠀⢸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⣶⠀⢰⡇⠀⠀⠀⠀⠈⣹⠏⢀⣠⡆⠀⠀⠀⣸⢷⣶⣶⣦⣤⣄⣀⡀⢸⡇⢀⣠⣤⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⢠⣇⣴⠿⠋⠁⠀⠀⢀⡏⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⠿⠟⠛⠉⢹⠀⠀⢠⣿⣦⠀⠀⡸⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆⢀⡟⠀⠘⣇⣰⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡⡾⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣦⣤⣄⣀⣀⠀⠀⠀⣸⡇⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣶⣾⣧⣶⣶⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⢀⣐⡍⠑⣠⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢤⠛⠶⢤⠷⢸⡷⠞⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⣀⣾⢦⣷⠠⠂⢉⡳
⠀⠀⠀⠀⠀⠀⠓⡐⢲⡎⠋⣍⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢸⡧⠓⢁⠋⢉⠉⠉⠉⠉
⠀⠀⠀⡀⠀⠀⠠⠀⢨⣇⢠⠈⡑⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⡂⡒⢄⢀⡠⠐⠂⣻⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢰⣁⡀⢸⢀⡀⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⡌⠈⠁⠀⠈⠃⠀⡁⠀⠀⠀⠀⠐⠀⠀⠊⢉⡹⢢⣍⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⠀⠀⠈⠀⠈⣡⣬⠈⠁⠉⠈⢃⣀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢁⡈⢈⠉⡇⠀⡦⠉⢀⡲⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠐⠀⠈⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠈⠀⠀⢡⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣀⠅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠂⠚⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⡁⠀⠀⣀⡸⠀⢸⣀⣀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣄⠀⠦⡀⠀⢀⠀⠟⡀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣀⠀⢀⣀⣀
⠀⠐⢄⠂⠈⠒⠁⠀⠀⣻⡆⠀⠞⠤⠊⠠⠖⠡⠂⠴⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠦⠶⣛⢉⡩⢒⡛⠛⠻⣟⡓⠁⡀⠀⠑
⠴⠂⠀⠒⠒⠰⠷⠾⠲⠿⠿⠿⠒⠒⠢⠔⢲⠄⠈⠰⠐⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⡇⠀⠈⡀⠀⠀⠀⢆⠀⠀⡄⢠⠊⠉⠀⠁⠈⢛⠂⠀⠀⢀⡀
⠐⠀⠂⠉⠀⡀⠀⠀⠀⠰⠀⡎⠐⠂⠀⠀⠀⠂⠲⠘⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⠈⠉⣷⡋⠀⠻⠈⠁⠉⠈⠾⡅⠈⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠀⣠⣄⡀⠀⠀⡀⠄⠀⠘⡆⠀⠀⠣⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡧⠸⣷⠟⠃⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠉⡻⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠩⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⣠⡆⠐⠃⠇⠤⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡶⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⢤⠤⠤⢄
⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣑⣦⡋⣐⡠⠂
⠀⠀⠀⠀⡈⠉⠘⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⢀⠀⡀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠛⡿⠞⡋⠀⠀⠀⢀
⠀⠀⠀⠠⠇⠀⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠲⠶⠶⢧⡀⣅⡞⢁⠴⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠠⠖⠠⠄⣀⠀⢰⠂⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⠂⢤⠀⠀⠉⠁⠀⠀⢠⠄⠙⣀⠄⠀⠀⠈⠉⠈⠁⠓⠀⠤⠆⣀⠀⠇⠀⡨⠦⠛⠉⠁
⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⣀⠀⠈⠀⠐⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰
⠀⠀⢠⠹⣧⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢲⠖⠊⠀⡇⠀⣀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠈⠐⠁⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠸⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢠⢀⠀⠴
⠀⠈⠀⠂⠀⠄⢻⣷⡀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠘⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣾⣿⣿⣿⣿⣷⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠋⠙⠛⠋⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⡿
⠻⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⣿⣿⣿⣿⡹⢿⣿⠟⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣹⣿⣽⣿⣾⣿⣷⣿⣧⣴⡿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡏⢻⡻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣶⣿⣿⣿⣿⠛⣿⣿⡟⠻⠁⡀⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣀⣤⣶⣤⣤⣤⣾⢳⣦⣾⣿⣿⣿⣿⣿⣿⠙⠀⠀⠂⠉⠂⠸⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠉⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠷⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⣴⣾⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣄⠀⢀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡟⣦⡵⠀⠄⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠠⠏⠀⠈⠀⠂⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡁⣰⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣤⣶⠖⠒⣉⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣼⣿⣿⣿⣿⣿⣷⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⠟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣿⣿⣿⣷⡔⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⡿⠋⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛
⠀⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣋⣠⣾⣿⣿⣿⣿⣿⣿
⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠶⠶⠀⠀⠀⠒⠃
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠁⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠿⣷⣾⣿⣿⣿⣿⣿⣿⠏⠻⠛⡿⢻⠁
⠀⠹⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉⠀⠀⠛⢿⢿⣿⡿⠟⠛⠟⠛⠀⠀⠀⠁
⠀⠈⠀⡮⠿⠀⠀⠁⠈⢩⣭⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠈⠻⠿⠓⣈⡁⠂
⠀⠀⠀⡀⢀⠀⠀⢀⠘⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄
⠀⠠⣤⣿⣽⣄⣠⣾⣷⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢄⣀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣾⣿⣿⣷⣶⣶⣶⣶⣶⣾⣷⣦⣤⣄⣀⣀⣀⣤⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⡄⢠⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣄⣤⣿⣷⣼⣿⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠘⠻⣶⣦⣄⡀⠀⢻⣷⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⢻⣷⡶⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⡙⣿⣧⣿⣷⣶⣶⣤⣤⡀⠀⢀⣨⣼⡿⠟⠻⠿⣶⣦⣤⣀⡀
⠀⠀⠀⣀⣀⣠⣤⣴⣶⣾⡿⠿⠟⠛⠛⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡿⠛⠉⠉⠙⠛⢿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⣿⣷⣦⣤⣀
⣶⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣦⣬⣿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣯⣁⡈⠉⠛⠿⣷⣦⣄⡀
⣿⣯⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠉⠉⢛⣿⣷⣄⠀⠀⠀⠀⠀⢀⣿⣯⣀⣤⣤⣄⣀⠀⠀⠈⠉⠛⠿⣿⣦⡀⠀⠀⣉⣹⣿
⣿⣿⠀⠀⠈⠙⢿⣿⣶⣶⣶⠿⠻⣿⣉⡁⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⣹⣿⢿⣷⣶⣶⣾⣿⣿⡿⠛⠋⠉⠛⠻⣿⣦⡀⠀⠀⠀⠈⣿⣿⣾⠿⠛⠋⢹
⢿⣿⠀⠀⠀⠀⠀⢽⡿⣿⣶⣄⢸⣿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⢁⣴⠟⠁⠀⣠⣿⣏⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡷⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⢰⣿⠃⠈⠳⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠏⠀⠀⠀⣿⣿⣴⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣦⣤⣶⣶⣿⠟⢋⣁⣼⣿⠀⠀⠀⢸⣿⡟⠛⠉⠀⢸⣿⠻⣿⣦⣄⡀⠀⠀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡿⠟⠛⠋⠙⢿⣿⡿⠿⣿⣿⠀⠀⠀⠘⣿⣇⠀⠀⠀⢸⣿⠀⠉⠻⠿⣿⣿⢿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢿⣿⡀⠀⠀⠀⠹⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣧⡀⢀⣀⣀⣬⣿⣦⡄⠐⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⢿⣿⡿⠿⠛⠛⠉⠉⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⢿⣧⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢻⣷⡀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠻⣿⣄⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣠⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢸⡇⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⢸⣿⡏⠀⠀⣀⣠⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⣿⣷⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⠋⠁
⣿⣿⣿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣤⣤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠂⠒⠉⠉⠙⠛⠲⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠉⠣⠠⠒⠀⠀⠀⠀⠀⠀⠀⠑⠲⡤⠄⠀⠀⠀⠀⠀⠀⠀⠁⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠈⠀⠀⠀⠀⠀⡀⠒⢢⣤⣤⠀⣿⣑
⠀⠀⠀⠀⠀⠀⠀⠘⢾⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡾⡯⠉⠨⣿⡧⠈⢶⠧⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡞⡠⠒⡾⢭⣉⡙⣿⡒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠞⠁⠀⢀⣾⣿⣗⡀⣘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣞⠁⣳⣷⡛⠉⠓⢟⣉⠲⣲⢤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⡿⡻⠛⢻⣧⠘⠈⣿⡃
⠀⠀⠀⠀⠀⢰⢻⡘⠛⠋⠙⣄⠀⠀⠉⠛⢉⠓⠈⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠂⠀⢘⡮⠌⢠⢄⠖⠋⠀⠀⢠⡏⢀⢠⣿⠃
⠀⠀⠀⠀⠀⠟⢀⠃⠀⠀⠀⠐⠀⠀⠀⢈⡦⠖⠀⠰⠂⢠⠴⠧⠊⠄⠋⠉⠹⠖⢻⣯⢟⡠⢂⡰⠁⠀⠀⠀⢠⠟⢀⡒⣸⡿
⠀⠀⠀⠀⢠⡴⠿⡅⠀⠀⢠⡄⠀⢉⠀⠀⠑⠀⠿⣭⠘⠂⠐⠚⠀⠀⢁⡀⡘⢀⣼⠃⣀⠀⡋⠀⠀⢀⠐⠀⠀⠈⠸⠀⠁⠁
⠀⠀⠀⠀⢸⡄⠀⠀⢀⣉⡛⠳⡤⡏⡍⠙⠛⠛⠻⠧⠄⢀⡀⠀⠀⠀⠊⢀⣴⣫⠑⣦⡾⣿⢁⠹⡏⣤⣧⡏⠀⠈⢠⠂
⠀⠀⠀⢀⠞⡁⠀⢰⡀⡏⠁⠀⠀⢠⠰⠀⠀⠀⢠⠀⠀⠀⠑⢤⣂⡠⠞⢻⣻⠋⢹⠀⠇⢹⠘⡇⣿⠀⣸⠇⠀⢸⠈⡀
⠀⠀⠀⢸⢀⡇⠀⡀⠳⠃⠀⠀⠃⠀⠀⣆⢰⣴⡨⠄⠀⠀⠀⠀⠀⠀⠀⠘⢹⡇⠌⠀⢠⣾⠇⠛⠏⠁⠛⠀⠀⢸⢸⠁
⠀⠀⠀⡌⠊⠀⠀⢳⡀⠀⡰⡄⡦⣴⣠⣽⣆⣙⣻⣾⣶⡟⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣤⣤⡄⠀⠀⢀⠀⠘⠀⠘⡌⡄
⠀⠀⠀⣴⢠⣶⡄⠈⠙⣦⠁⠁⠀⡘⣿⠟⠛⢿⡛⣿⠿⢿⡆⠀⠀⠀⠀⣿⠿⣿⣼⡿⠿⣿⣧⡀⠠⠊⠀⠁⠀⠀⠀⠱⠄
⠀⠠⠟⠃⠿⠉⠻⠀⠀⠈⣇⠐⠶⣵⠃⠀⠀⠘⠿⠏⠀⠈⠓⠀⠀⠀⠐⠛⠐⠿⠋⠷⠀⠟⣫⡶⡂⠀⠀⠃⠀⠀⠀⠲⡖⠺⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⣀⠀⣀⠀⡀⠀⠐⠃⠘⠀⢀⣀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⢀⡀⡉⠢⡀⢀⡀⢰⠁⠄⠀⠁⠈⠀⡔⠁⠢⣀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⢉⡩⠀⠀⡰⠁⠀⠀⢀⡔⠊⠁⠺⠁⢠⣸⠅⠀⣼⠀⠀⢰⠀⢠⢰⠃⣏⠂⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠔⠲⠀⢸⠀⠀⠀⠀⠀⠘⠤⢎⠜⠸⠀⠠⡜⠀⠄⢸⠀⠈⠉⠈⠑⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢠⡀⠑⠀⠀⢄⠀⣀⠔⠀⢸⠀⠇⣈⡉⠉⠀⠀⠈⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠹⡄⠀⠁⠀⠀⠀⠀⠀⣀⣰⠈⠢⠴⠀⠀⠈⠁⠀⠀⠈⠢⠀⠀⠀⠨⠂
⠀⠀⠀⠀⠀⠀⣖⡄⠀⠸⡀⠀⢱⡀⠀⠐⠋⠀⢀⠀⡈⠀⠘⠐⡄⠀⠀⡰⢱⠀⢀⠞⠉⠉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠱⡀⠀⢣⠀⠀⠀⢰⠃⣀⠈⢃⢸⠀⢃⠀⢠⠇⢸⠀⡎⡰⠊⠉⠁⠉⠀⠠⢀⡔⠀⠀⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⡀⠘⢄⠀⠳⠤⢢⡸⡀⡇⢳⠈⣸⡄⢸⠀⡼⢀⠇⡜⠀⠁⢠⡀⠀⠀⠀⠀⠈⠆⠀⠀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣄⠀⠳⠤⠐⠊⠀⣇⢇⢸⡀⣷⡇⠈⣤⠃⡸⢰⢡⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⠄⠀⠀⠀⠀⠸⣸⠀⡇⢸⣧⠀⠉⢀⡇⡆⣨⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠣⣉⠁⡜⢹⠀⠀⡸⠀⠉⠁⠀⠀⠐⢶⠁⠀⠀⠈⠁⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢱⢄⠀⠈⠉⠁⠈⠣⠔⠁⠀⠀⠀⠀⠀⠀⠈⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠤⠤⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠛⢄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡆⠈⢳⡀⠀⢹⡤⠴⠖⠒⠒⠒⠂⠂⢄⡜⠁⠀⠀⣀⣠⣦⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠱⡚⠉⠙⣄⠀⢇⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⣴⠋⠁⠀⠘⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠹⣦⡱⣄⠀⠘⣆⠘⡄⠀⠀⠀⠀⡴⠁⣠⠔⠃⣀⡽⢦⣀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡳⣄⠘⡆⢹⣦⡀⢀⠞⣡⣞⡥⠖⠋⠁⠀⢀⣩⣷⣄⡇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢻⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⡞⠉⢩⡿⠛⠉⢁⣀⡤⠴⠚⣋⣉⣀⣀⣼⠇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣿⡀⠀⢸⠉⠒⠚⠛⠛⠛⠛⣉⣉⣩⡭⠥⠖⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠉⠢⣄⣀⡤⠴⠖⠋⠉⠀⠀⢀⣨⠽⠻⡏⠁⠉⠉⠙⠯⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⢨⡇⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠈⢢⡀⠀⠀⠀⠈⠳⢦⡀⠀⢀⣀⠤⠒⠋⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠢⣀⠀⢸⡇⠀⠀⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⣀⡤⠖⢻⡟⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠼⣇⠀⠀⣿⠈⠑⢦⡀⠀⠀⠀⠀⠀⢀⠠⠄⠊⣯⠁⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠻⣤⡇⠀⠀⠀⠈⠢⢀⠠⠐⠈⠀⠀⠀⠀⢸⠄⠀⢀⣸⡧⠤⠶⠚⠉⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢈⡗⢄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣀⡼⠓⠛⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⠀⠀⣼⠁⠀⠈⠒⢦⡀⣿⠀⣀⣠⠴⠚⠉⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠉⡟⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣸⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⣇⠀⢀⣀⣠⣤⠼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⡤⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢸⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠁
⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⢶⡄⠀⠀⠱⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣀⣀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠤⣤⣙⣦⣄⠀⠙⣆⢣⠀⠀⠀⠀⠀⠀⡠⠃⢀⠔⠉⠀⠀⡘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡈⠙⠻⠷⣤⣈⢙⣠⣤⣤⡀⡠⠜⡡⠔⢁⣀⣀⣀⣔⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠠⠀⡀⠀⠉⠛⠒⠒⠒⠊⢹⣀⠀⢡⡏⣐⠛⣉⣉⣁⣀⣀⣀⠀⠉⠳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣄⣀⠠⢤⣤⡄⠁⠒⠒⠒⢒⠶⠾⣿⣆⣸⡇⠀⠙⣦⠨⢭⣉⣀⠉⠀⠁⣰⣧⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠒⢶⡎⣧⢀⣈⣉⣉⣁⠰⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠤⢄⣀⣀⡤⠖⠇⢸⣦⡀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⢀⠁⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⡸⠀⠸⡇⠉⠛⠲⠒⠋
⠀⠀⠀⠀⠈⠓⠒⠛⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⢸⡃⠀⢠⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠉⠢⣸⠀⠀⢨⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠷⠒⠛⢆⢠⠇⠀⠀⢹⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠀⠀⠀⢸⠀⠀⠀⠈⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠈⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠀⡆⠀⠀⢸⠀⠀⠀⠘⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⡇⠀⠀⢸⠀⠀⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣠⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⡀⠀⡇⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⣿⣏⢹⣟⠻⣿⣟⠟⡻⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠊
⠈⢨⠑⠀⠀⠆⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⢒⢄⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠄⠀⠐⣉⠊
⠀⠠⠀⠄⠀⠒⠀⠀⠕⠂⢂⡙⡀⠠⠄⠀⣌⡼⠢⡆⢟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⠂⠐⠅⠀⠐⠂⠀⠄
⠀⠀⠀⠑⣀⠀⠒⠀⠈⢋⠱⡤⢥⡆⢀⢀⣥⣀⢀⣁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢄⢁⠉⠀⠐⠂⠀⡐⠁
⠀⠀⠀⠀⠠⠀⠀⠔⠀⠑⡄⠐⠆⠙⣀⡸⠀⢀⠝⠊⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣈⣰⠀⡴⠁⠐⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡁⠀⠁⡒⠐⣉⡴⠊⢛⠁⠰⠋⠀⠠⠀⠙⠀⠀⠀⠀⡀⠈⢁⡠⠥⣍⡑⠐⣂⠀⠀⡉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠈⢱⣞⣄⠉⠀⠀⠀⠀⠀⢀⡰⠄⠂⠃⢀⣸⡮⣍⣅⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡐⠃⠐⠠⢫⠀⠠⠐⠆⠈⠀⠀⠀⠀⠀⣌⡀⠟⢚⠽⠁⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠤⠄⠀⠀⠀⠀⠀⢐⠂⢣⠛⢠⠚⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠊⠐⣂⠐⠁

⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢠⡴⠦⣄
⠀⠀⣀⣤⢤⣠⣿⠁⠀⠀⠀⠀⠀⠙⢀⣼⠂
⡰⠋⠀⠀⢠⣿⠉⠙⠲⠶⠤⠤⠴⠶⠛⠁
⠀⠀⠀⢠⣿⠇
⠀⠀⢀⣾⡟⠀⠀⢸⣏⡷⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⣤⣦⣄
⠀⠀⣸⣿⠁⠀⠀⢀⡿⣥⠀⠀⠀⣼⡇⠀⢠⣿⠇⠀⣴⣿⢋⣠⡿⠃
⠀⠀⣿⡏⠀⠀⢠⠟⢰⣿⠃⠀⢰⣿⠃⠀⣾⣿⠀⢰⣿⠇⠉⠉⢀
⠀⠀⠸⡇⠀⣴⠏⢠⣿⡏⠀⣠⢿⡟⢀⡾⣿⡇⣠⠞⢻⣄⣀⣴⠏
⠀⠀⠀⠙⠛⠁⠀⢸⣿⠀⡴⠃⠈⠛⢋⡀⠈⠛⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠀⢠⡾⠋⣩⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⡴⠋⠀⠀⠀⠀⠀⢠⡶⢦⠀⢠⣾⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠛⢀⣤⣤⣀⠀⣴⠀⠈⢳⣿⣤⣿⡯⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠠⣿⠃⠀⣿⣴⣿⠀⠀⣼⠁⢸⣿⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⣰⠟⠶⢶⡟⢸⣿⠀⣼⠃⠀⠀⠙⠶⠚⠁⠀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⣰⠋⣿⠇⠀⢠⣿⢀⣰⡟⠀⠈⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣾⡿⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠠⠃⠀⣿⠀⣠⠏⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠙⢦⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣰⠋⣹⡿⠀⢀⣿⠀⠀⣼⡇⠀⢰⣿⠟⢙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣰⠏⢰⣿⠁⢀⣾⡏⢀⣴⣿⠀⠀⣿⡟⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣰⠋⠀⠸⣇⣠⠟⠙⠳⠋⢹⡇⠀⣰⠛⢿⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⢀⣴⠗⣆⠈⠳⠞⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⣠⣾⢿⣛⡆⢰⡆⢰⡟⣦⠀⣴⡿⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⠀⠀⣿⡇⠀⢹⠷⣾⡇⠀⢹⠗⢾⣿⢥⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⢀⡿⠀⣿⡇⢀⡿⠀⢸⣿⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⠛⠧⠞⠁⠀⠹⣧⡾⠁⠀⠀⠙⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠢⡀⠀⠈⠪⡉⠐⠂⠔⠁⠀⠀⡠⠊⡹
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⡲⠏⢉⡒⠈⢐⣄⣀⡈⣦⠎⣀⣀⡔⠪⠔⣊⠉⠲⢄⣀
⠀⠀⠀⡤⣒⠊⠉⠀⠀⠀⠀⠄⢀⣀⡀⠁⣂⣬⠭⡿⣤⠼⡯⣥⣄⠈⢁⣀⡀⠀⠜⠀⠉⠉⠒⡢⡄
⠀⠀⠀⡆⠀⠉⠑⠢⠤⣠⡄⣒⡩⠥⠒⠉⠁⠂⠉⡠⠋⠢⣀⠁⠀⠁⠐⠢⠍⡒⢤⡤⠔⠊⠁⠀⡆
⠀⠀⠀⡇⠀⠀⠀⠀⢀⠗⠋⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠈⠣⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠢⣄⣀⢀⡴⠁⠀⠀⠀⠀⢀⠔⠉⠉⠒⠢⢄⣀⣀⡤⠐⠊⠉⠢⡀⠀⠀⠀⠈⠂⠀⣀⡤⠂
⠀⠀⠀⠀⡇⠀⢋⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⡗⢄⠀⠀⢠⠛⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠈⢆⠀⠀⢀⠜⠓⡧⢄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⡷⠊⠢⡴⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠈⠣⡤⠃⠀⠀⡇⠀⠈⠉⠒⠤⢄⣀⣃⡠⠔⠊⠁⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠧⢄⡀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠧⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⢸⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁
⠀⠀⠂⢠⡀⢀⠀⠀⠢⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠦⣤⡶⢠⣀⣀⠀⠀⠀⠀⢀⠀⠀⡀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠽
⠀⠄⠐⠉⠳⢎⠀⢀⡆⠀⠀⠀⠀⠀⠠⠹⠌⠀⠔⠀⠀⠀⠀⠀⠈⠐⣗⣰⢩⢿⣉⡈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⡀⠀⠀⠀⠀⠀⠀⠆
⠀⢤⠀⠐⠣⠄⠃⠡⠀⠀⠀⠀⠀⠀⠀⡆⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠈⠀⢠⡤⠊⠀⠀⠀⠀⠀⠠⠀⠀⠤⠁⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠖⠋
⠀⠀⢀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢆⣀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⢄⠀⠂⠠⠀⠀⠀⠀⠀⠀⠤⠷⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢖⡀⢏⠃⠠⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠠⠀⠀⠀⠀⠇⠠⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠐⠣⠖⠀⠀⠀⠆⠀⠀⠀⠀⡨⠁⠁⣀⡄⠈⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠠⢤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡉
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠂
⠀⠀⠀⠳⣄⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠛⠀⢰⠀⠀⠀⡄⠄⠀⢀⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠐
⠐⠎⣠⠔⠙⠀⠏⠁⠀⠀⠀⠀⠀⠀⠂⠐⠒⢀⠀⠂⡄⠀⠀⠀⠀⣧⠀⠀⢣⡼⠋⠢⠀⡀⠀⠀⠀⠀⠀⠀⠙⠣⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠲⠄⠘⠀⠂⢠⠀⠀⠀⠀⠀⠀⠃⠸⠀⠉⠀⠀⠀⠀⠀⢈⠑⠲⢡⣴⠶⠂⠀⡀
⠀⠀⠀⠃⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠈⡇
⠀⠀⢦⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡄⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠐⠀⠀⠐⠐
⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⡀⠤⣐⡖⣲⠤⠚⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡴⠉⢨⣧⣌⣀⡠⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠠⠿⠍⣤⡽⠃⠀⠀⠀⠀⠁⠀⠐⠰⠃⠙⠈⠁⠀⠀⠀⠀⠀⠈⠆⠀⠃⠛⠒⠠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⣀⠜⠁⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠓⢤⠀⠀⠀⠀⢀⠴⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⢲⣄⢳⠀⠀⣰⠃⢀⠴⠒⠦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠙⢎⢧⣠⠇⢰⣋⠴⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠛⠒⠚⠛⠒⢺⠆⠀⠀⢯⠤⠤⠶⠥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠛⠛⢦⠀⠀⠉⠀⠀⡴⠉⠉⠳⣆⠀⢸⠁⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⡀⠀⠀⠀⡼⢁⡔⡂⠀⡧⠀⠄⠀⠀⡇⠠⠙⣦⡘⣆⡼⠀⡴⠋⠀⣀⣀⠙⡆
⠀⠀⠀⠀⠀⡜⢁⣠⣄⡀⠉⢦⡀⠀⡇⣾⠃⠀⡰⠃⣀⣄⣀⣀⢱⢄⡀⠘⢧⢳⣇⡜⢀⡴⢋⡠⠊⢀⡇
⠀⠀⠀⠀⠀⢣⡈⠑⠤⣉⣳⡀⠱⠶⢣⣡⠴⠞⠓⠉⠁⠈⣇⠈⠉⠉⠙⠓⠲⠆⠍⠁⣜⣉⣁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠙⠢⢤⣤⠤⢼⠂⠈⠈⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠉⠉⠙⠓⢢
⠀⠀⠀⠀⠀⢰⠒⠉⠁⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⣇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠆⠂⠈⠀⠀⠀⢩⠀⠀⠀⠙⠀⠀⠀⠗⠀⠠⠄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠤⠀⠀⠈⠈⠄⠀⠀⠰⡀⠀⠀⠀⠈⠙⡆⠀⠀⡄⠀⠀⢀⣀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⠠⠀⠚⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⡈⠀⠀⠁⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣆⣀⣀⠴⠴⠒⠉⠉⠒⠒⠲⠤⠤⠴⣄⣀⣀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠟⠛⠳⣦⡀
⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⣿⡀
⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⢠⣿
⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣽⡿⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⢠⣿⣿⠁⠀⠀⠉⠛⠓⠒⠒⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⣼⣿⡇⠀⣴⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣄⠀⠀⣠⣾⠏⠁⠈⣦
⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣿⣿⠀⠀⢿⣿⡀⣀⣄⠀⠀⣾⣿⠇⠀⣰⣿⡇⠀⢰⣿⡇⠀⠀⣰⠏
⠀⠙⠷⣄⡀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⢸⣿⡇⠀⠀⢸⠉⢩⣿⡏⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⢿⣿⡧⠴⠚⢡⣄
⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⣠⡾⢳⡄⠀⣿⡇⠀⠀⣿⠀⣿⣿⠁⠀⠈⣿⣿⠀⣰⢿⣿⠀⠀⣸⣿⣇⠀⠀⣸⠟
⠀⠀⢀⣤⠶⠶⣶⣄⠀⠀⢀⣿⡇⢸⡇⠀⠹⣇⠀⢠⠇⢰⣿⡏⠀⠀⣰⠏⠙⠋⠁⠀⠙⠓⠚⠁⠈⠙⠛⠛⠁
⠀⠀⣾⠃⠀⠀⣼⡿⠀⠀⢸⣿⣧⣸⠁⠀⠀⠈⠛⠁⠀⠀⢿⣇⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣄
⠀⠀⢻⡄⠀⠀⠉⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠙⢶⡀⠀⠀⠀⠀⠉⢿⣿
⠀⠀⠀⠙⠲⠤⠤⠤⠤⠴⠚⣿⣿⠀⠀⠀⣠⣾⠟⠻⣦⠀⣀⣰⣿⣿⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⡟⠀⠀⢸⡆⠀⣿⣿⡇⢰⡖⠂⢰⣿⣿⣇⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢿⣿⡇⠀⠀⢸⡇⠀⣿⣿⠁⢸⡇⠀⠘⣿⣿⡇⠉⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠘⢿⣧⣀⣠⡟⠀⠀⠹⣿⣆⣸⠇⠀⠀⠙⠿⣷⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⣀⣤⠴⠖⠒⠒⠒⠒⠒⠦⢤⣀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠿⣇
⠀⠀⠀⢠⡴⠚⠉⣡⡾⠟⠁⠀⠀⠈⠉⠙⠒⠢⢤⣀⡀⠀⠀⣠⠞⠁⠀⠀⠳⣤⣀⣀⣀⣀⣠⠤⠖⠚⠉⠀⣠⡟
⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⠯⢄⣀⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈⠉⠉⠒⣿⠒⠒⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⡤⠞⠁
⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠙⢦
⠠⠀⠀⠀⠀⠀⠌⠀⠀⢀⠀⠄⠀⠀⠐⠂⠄⠀⠀⠀⠉⠀⠄⠠⢘⡀⢀⠄⠀⠀⠦⠀⠀⠀⣀⣀⣸⣀⣀⠀⣸⡇
⠀⠀⠀⠁⠀⠀⠀⠈⢉⡉⠀⠀⠀⠀⠈⠀⠈⢄⠀⠶⢦⣘⣴⡔⢁⡠⠖⢒⠂⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠉⠀⠉⢈⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣀⢀⣉⠭⠤⠄⢿⡉⠉⠻⠭⠭⠭⠤⠤⠤⠄⠀⠀⠀⠒⣀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠠⠉⢐⢀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠈⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⡄⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠠⡆⠄⠀⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠑⠶⠄⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠑⣆⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣱⠤⠨⠥⠃⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠆
⠀⠀⠀⠠⠒⠈⠂⠀⢤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠈⠀⠄⠀⠠⢀⠀⠄⠀⠀⢴
⠀⠀⠀⠀⠒⠂⣧⡴⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠂⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠐⠀⠞⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠈⠈⡋⠈⠍⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡂⣲⣤⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠤⡧⠠⡦⡄⠠⣤⠬⢤⣄⠀⠀⠀⠀⠀⠉⠤⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⢤⣄⡀⠀⠀⠀⠀⣸⣿⣿⣯⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠉⠛⠓⢉⡗⣠⣴⣾⡿⠿⠛⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⣾⠇⠀⠀⠀⢰⣾⣿⣿⠛⠁⣠⣶⣶⣶⣤⣽⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣼⡟⠀⠀⠀⢠⣿⣿⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡁⠀⠀⠀⣾⣿⡟⢁⣾⣿⣿⣿⣿⠿⣿⣿⠛⡿⠿⠿⠧⢤⣄⣀⣀
⠀⠀⠀⠀⠀⢠⠖⠦⣀⠀⠀⡼⠁⠀⠀⠀⣸⣿⣿⣇⣾⣿⣿⣿⣿⠉⠀⠀⠘⠀⠉⠀⠀⠀⠀⠀⠀⣉⢻
⠀⠀⠀⠀⠀⠘⢦⣀⣠⣽⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣠⠴⣲⣮⠶⠶⠶⠶⠤⡴⠏
⠀⠀⠀⠀⡔⠊⣩⣿⣿⣿⣿⣿⡿⠿⠿⠛⠿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣤⣟⣵⣿⣷⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠙⢛⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⣹⣿⣿⡿⠻⣦⡀⢀⣾⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣼⣿⣿⡿⠿⠧⠤⢴⣶⣿⣿⣿⣿⣿⣛⣁⣀⡙⣿⣿⣿⣽⣿⣿⣣⣿⣿⣿⣿⣶⣦⣤⣀⣠⠤⣴⠆
⠀⠚⠻⢭⣁⠶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣤⣤⡶⣢⠞⠁
⠀⠀⠀⠀⣿⣷⠮⣟⠄⠀⣤⡄⢠⣤⡀⣤⢠⡀⣤⣤⠀⠀⡄⠀⢀⣤⡀⣄⢠⠄⣤⣤⠀⣠⣾⢟⡥⠞⠁
⠀⠀⢀⡤⢋⣥⣾⡏⠀⠀⠚⠂⠘⠙⠃⠙⠚⠁⠛⠒⠀⠀⠓⠒⠙⠒⠋⠘⠛⠀⠛⠚⠀⢹⣿⣯⡳⢄⡀
⠀⡠⠋⠔⠛⠛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣐⣛⣛⣛⣂⣉⣳⠤
⠾⠓⠛⠉⠉⢹⣿⣿⡄⠄⢹⣦⣈⣾⣽⣿⣿⢿⣿⣿⣯⣽⣿⢿⣿⣿⣿⣿⣯⣿⣭⣽⠿⢋⡅⢹⣇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⣲⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⠏⣼⣿⣿⢟⣿⣿⡿⡟⠍⢒⣰⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⢿⣿⣿⣿⣿⣿⡀⠙⢿⣿⡿⢋⣼⣿⣿⣿⣿⣿⡟⣴⣦⣽⢾⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡌⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡧⠀⣿⣧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡛⠻⢿⡇⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡎⠙⠁⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠂⠀⢉⡽⣫⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣯⠼⢿⣿⢧⡀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⣼⡿⠟⠉⠀⡀⢛⡡⣿⡗⢩⣿⠿⠿⠿⠿⣄⡈⠏⠑⠈⠂⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠬⠡⡞⠹⢀⠀⠸⡀⠓⠻⡆⢸⡋⠟⠻⣷⣦⣀⠀⠀⠙⠃⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠊⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣷⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠲⢄⣠⣤⣶⣤⣤⠔⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣇⣀
⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⢻⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠹⡈⠉⠛⠒⣦⡄
⠀⠀⠀⠀⢸⠳⠦⢤⣀⡀⡞⠀⠀⠀⠀⠀⠀⢀⣨⠧⣤⣤⣠⡼⣀⡀⠀⠀⠀⠀⠀⠀⣧⠴⠚⠉⠁⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⢳⣀⣀⣀⣤⣴⣾⠋⠀⣰⠃⠀⠈⢣⡀⠉⠹⣶⠦⠤⣤⣴⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⣤⣀⣀⠀⠀⠀⡏⠁⠀⢸⡼⠁⠀⢠⠧⢤⣠⠤⠖⢣⠀⠀⠘⢆⠀⠀⣿⣀⣠⠴⠒⢻⠁
⠀⠀⠀⠀⠀⡇⠀⠈⠉⠛⠲⡧⢤⣠⠞⠀⠀⢀⡏⠀⠀⡇⠀⠀⠈⢧⠀⠀⠈⢷⠚⣿⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠯⢴⡆⢀⡾⠦⢤⣀⣇⣠⠤⠖⠛⢦⢠⡖⢤⡧⣻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠷⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢻⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠳⠤⣄⣀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⣠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠣⢤⣀⣸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣨⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⡇⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠲⡄
⠀⠀⠀⠀⠐⠀⣁⠃⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠉⠡⡀⡄⠠⠐⢀⠀⣀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿
⠀⠀⠨⠀⠁⠀⠀⠄⠈⠃⠋⠀⠀⠀⠀⠐⠀⠀⢒⠀⢢⣦⣄⠱⣤⣠⠊⢁⣨⠴⠠⠀⠂⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠋⠁⠉⢉
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠙⡿⠛⠓⢾⣃⣒⣚⣃⣀⣀⣀⠀⠀⠌⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠀⠈⠉⠉⠀⠀⠀⠀⠸⠁⠀⠀⠄⠀⠀⠿⠠⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠈⠀⠀⠐⠢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠐⡄⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠰⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠃⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠀⢹⠂⡀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠜⠐⡇⠂⠀⠀⢰⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠒⠓⠀⠘⠒⠂⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠘⢾⢀⡠⠞⠀⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡶⠀⠀⠀⠀⠀⠀⠀⢀⡁⠂⠀⣳⡤⠨⡭⠖⠐⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠅⠀⠄
⠀⠀⠀⢀⡠⠄⢲⠁⠀⣀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠖⠁⣀⠀⢄⣀⠠⣀⠀⠀⢠⣠
⠀⠀⠀⠀⠤⠠⡰⡀⡠⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠲⡈⠳⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡁⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠙⠀⠘
⠀⠀⠀⢀⣀⠀⢀⣿⢋⡀
⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡀⠈⠠⡀⣜⠄⢸⣀⣄⢬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢨⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠽⡶⢃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⡄⠀⠤⠄⠀⠢⢼⠄⠸⠴⠤⠴⠶⠦⠤⠤⠄⠀⠀⠀⠀⡀⠐⠰⠁⠀⠀⣀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠠⠤⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⢸⠳⣇⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⡟⠀⠀⠣⡜⠀⣀⡀⠀⠀⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⣿⣿⡟⢛⣉⣥⣤⣤⣀⠀⠀⠀⣹⣿⡿⠂⠋⢁⠀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⡶⠲⣄⢿⣿⣇⠀⢹⣿⡇⠀⠈⣷⠀⠀⣿⣿⠃⠀⠀⣼⠀⣿⣿⣿⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⡔⡂⠹⣄⠀⠀⣠⠏⢸⣿⣿⠀⠀⣿⣿⣠⣾⡃⠀⠀⣿⣿⠀⠀⢠⡇⢸⣿⣿⠃⣀⠀⠁⢠⣤
⠀⠀⠀⠀⠀⠀⠑⢤⡄⠀⠈⠉⠋⠁⠀⠘⣿⣿⣆⠀⢿⣿⡏⠙⣿⣦⠀⢻⣿⡀⣀⡞⠀⣿⣿⡟⠙⡏⠀⢀⣿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠐⠛⠋⠉⠀⢸⡿⠷⠆⢹⣿⣆⠀⠈⠉⢁⠀⣼⣿⣿⠁⠀⠀⠀⢸⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⣾⠟⢳⡄⠀⠀⠀⠀⠀⣀⠀⢻⣿⡀⠀⠀⣼⠀⠀⠀⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⣤⠀⠈⢿⣆⣰⣿⡿⠁⣀⡼⠁⠀⠀⠀⢀⣾⣿⡇⠀⠙⠷⠤⠾⠃⠀⠀⠀⣴⠚⠓⣆⠈⣆⢠⡖⠛⢲⡀
⠀⠀⠀⠀⠀⠻⢿⡆⠀⠀⢹⣿⣿⠛⠛⠉⡴⠦⢤⡀⣠⢿⣿⡿⢱⢶⠀⠀⣠⣾⡟⢳⡄⠀⣇⠀⠒⠃⢀⡏⠸⢤⠆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⣿⣿⠀⠀⢀⣓⠤⠶⢻⠁⣾⣿⠇⠘⣿⠁⣸⣿⡟⠀⣸⠃⠀⠈⠲⠶⠖⠋⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⢺⡻⠆⡇⠹⣿⣿⣤⠴⣿⣿⠃⠀⣼⠀⢻⣿⠀⢠⡟⠳⣿⣿⡷⠞⠁⢀⡆⠀⠀⣀⣀⡤⠤⠖⠋⢰⡚⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠇⠀⠀⠀⠀⠀⠻⢿⡤⠞⠁⠀⠸⣿⣤⠞⠀⠀⠻⣿⣄⢀⣠⠞⠀⠀⠎⠵⠒⠲⠤⣤⣀⣨⡭⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠀⢈⡥⠶⠒⠒⠒⢦⠉⢉⣀⣀⢀⡀⠀⣀⢀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣤⠀⣾⡆⣰⡦⠄⢉⠛⠲⠦⢤⣬⣀⣀⣀⣠⣤⠞⠀⢸⡏⠁⢸⣇⢸⣿⢸⣿⢻⣆⣿⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢳⣿⡇⣻⡷⠄⢸⣧⢠⠆⣤⡤⠄⢠⣤⢤⣄⠀⠀⣼⣇⡀⢸⣿⢸⡏⢸⡇⢸⡇⢻⣎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠉⠁⠙⠓⠂⠸⣿⠏⢠⣿⠧⠀⢸⣿⡴⠋⡀⠀⣿⡏⠁⢸⣿⣾⡇⢸⡇⣼⡇⣀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠉⠉⠉⠉⠛⠒⠶⢤⣀⡘⠻⠶⠀⠸⠟⠙⠞⣡⠀⣿⡇⠀⢸⡇⣿⡇⣾⡇⣿⠰⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⣤⣤⣤⠴⠞⠁⢀⣀⠉⠙⠓⠒⠲⠶⠚⠛⠁⠀⠙⠛⠃⠘⠃⠙⠃⠛⠛⠃⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⠀⠠⠞⠉⠉⠛⠲⠤⠤⠤⠖⠛⠉⠙⠲⠤⣄⣀⣠⠶⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣍⣬⣹⣤⣏⣩⣏⣼⣟⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⠔
⢀⡀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁
⢃⡴⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⢃⡠⠂⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⣀⡀⢀
⠉⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠹⢟⡦⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠹⣋⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠰⣦⡄⡢⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠰⣶⡄⣀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠠⠃
⠀⠈⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠑⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢁⡰⠃
⠀⢀⢀⠤⠀⠀⠀⠀⠂⠀⠀⡐⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⢀⢀⠤⠀⠀⠀⠀⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠘⢠⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠞⠁
⡔⢂⠀⠀⠀⠀⠀⠀⠠⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⢂⠀⠀⠀⠀⠀⠀⠀⠁⡠
⠔⠋⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⢶⡦⡰⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢶⡤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⡀⢀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠈⠻⢡⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠈⠻⢡⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠌⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋
⠀⣐⣲⣉⡄⠀⠀⠀⠀⢀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣲⢉⡆⠀⠀⠀⠀⢀⣀⡂
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠏⠉⠑⠀⠀⠉⠢⡀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠤⠼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣃⣀⣀⣀⣀⡀⠀⠀⢤⡈⢦⡀⠀⠀⡞⢀⡠⠂⠀⠀⠀⠀⢈⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠀⠀⠀⠉⠲⣤⣙⢦⣳⡒⡞⣠⠛⣁⣤⣤⣶⠶⠶⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣈⡽⢶⣦⠤⣤⣤⣤⣿⣿⣿⣥⣍⣹⣷⣟⣋⡁⠀⣀⣀⣀⡠⢼⣃
⠀⠀⠀⠀⠀⠀⠀⡏⠓⡶⠋⠀⠀⢀⡤⠖⠋⡩⠋⠙⠛⠻⣟⠛⠻⠭⣉⡉⠀⠀⠀⠀⢢⠝⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠞⠁⠀⢀⣼⣿⣷⣶⠮⢄⣀⠀⠀⠀⠀⠑⠢⣀⣀⣤⣵⣤⡀⠻⠃⠇⠠
⠀⠀⠀⠀⢀⡴⠚⠕⠀⠀⢠⣾⣿⣿⣿⡿⠀⠀⠈⠛⠲⡦⠔⠒⠉⢹⠉⢉⠙⣿⡇⠀⠀⠸⣴
⠀⠀⠀⠀⠉⠉⢉⡿⢀⣴⠿⢿⣿⢻⣿⡇⠀⠀⠀⢈⣾⡇⠀⠀⠀⢸⠀⠈⠀⠘⡇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⣾⠵⡿⠃⠀⠀⡟⢈⢿⣿⠶⣤⣀⡀⣪⡇⠀⠀⣀⣸⡇⠀⠀⠀⣿⣄⠀⣶⡄⣣⡀
⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⠈⡞⡟⠀⠀⠉⠛⢿⡷⠖⠛⠉⠘⡇⠀⠀⢀⠈⠻⣦⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⠀⠁⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⡀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⣿⡀⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⢀⣴⣾⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠢⢄⡀⠀⠨⡇⠀⢀⣀⡤⠧⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠷⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣀
⠀⠀⠀⡐⠒⣲⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣧⣧⢜⡆
⠀⠀⠀⣘⣬⡼⢦⠾⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⣀⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠉⡝⢉⠟⡇⠀⠀⠀⣀⣴⢶⠚⢒⣊⣳
⠀⠀⢀⠀⠀⡄⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠚⡀⡎⠀⡇⠀⠀⠈⡷⢧⡬⢶⣶⣮⣅⣄
⠀⠀⢸⡄⣠⠇⠸⠤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠉⠉⠉⠋⠉⠉⣉⠉⠉⠍⡇⠀⠀⠀⡄⠀⣠⠃⡇⡴⡇⠀⠀⠀⣠⠊⢠⡎⢇⡱⣘⣾
⠀⠀⢸⠤⠤⡀⠀⡤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠀⠁⠀⠈⢀⠀⠀⠀⠄⠀⢁⠀⠀⠀⡗⠊⠁⡠⢻⠀⡇⠀⠀⢸⣁⠜⠁⡳⠚⢀⡴⢻
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠐⠂⠀⡄⠀⠐⠀⠀⠆⣸⠀⠀⠀⡗⠒⠉⠀⡼⢀⡇⠀⠀⠀⡁⡠⠊⠀⡴⠋⡐⡻
⠀⠀⠈⠉⠀⠓⠚⠓⠒⠒⠒⠈⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⢀⣀
⠀⠀⣀⣫⣄⣦⣠⣯⣖⡀⠀⠀⠀⠀⠠⡀⠐⡆⠀⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠤⢤⣤⣿⣧⣤⣤⠜⠀⠀⠀⠰⣒⢞⣛⣛⣯⡭⢀⣁⣱⠀⠀⠀⠀⠀⠀⠠⣈⣇⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢄
⠀⠀⠀⠘⡀⢸⡇⢀⠇⠀⠀⠀⠀⢠⣡⠁⠀⠀⡀⠀⠗⠾⠂⠀⠀⠸⡁⠩⠍⠤⠤⠭⠯⠯⠭⠭⠭⠭⢽⠀⠀⠀⡏⠉⠉⠉⠉⠉⠷⡀⠀⠡⡀
⠀⠀⠀⠀⣇⢸⡇⣸⠀⠀⠀⢠⣔⠟⢹⢠⠀⠀⠇⠀⡄⠀⡅⠀⠀⠀⡄⢒⢐⡒⡂⢒⢖⠒⡒⣒⣒⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⣤⠊
⠀⠀⠀⠀⠘⠾⠷⠃⠀⠀⠀⠀⠈⠀⠸⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠃⠠⠥⠬⠬⠤⠥⠤⠤⠤⠥⠤⠞⠀⠀⠀⠣⠠⠤⠤⠤⠤⠤⠦⠼

⠀⠀⠀⠀⢀⢆⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡠⠉⢢⡀
⠀⠀⠀⢠⢸⠴⣟⢰⣦⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣋⣄⣊⣁⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠄⠰⠛⠛⠂⠀⠤⠂⠀⠀⠀⡆⠀⠙⠛⠛⠛⠛⠋⠀⡂⠀⠀⠀⠀⠈⠀⠠⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠄⠀⡃
⠀⠀⠀⠸⣀⢀⢘⠀⡸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣃⠀⠀⠀⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡀⡀⣈⣈⣀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⢀⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⣀⡀
⠀⣿⠃⠖⢰⠆⣴⡆⣼⡆⢀⣾⢩⠇⢠⠂⠀⠀⢸⣿⠀⠀⢰⡿⠙⣷⠈⣿⡼⠀⣿⡦
⠀⠉⠉⠀⢿⡴⠿⠧⠿⠡⠚⠿⠤⠔⠁⠀⠀⠀⠸⠿⠶⠦⠘⠿⠞⠃⠀⠙⠃⠀⠻⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠢⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢠
⠀⢀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⢀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠣⣀⡟
⢀⡾⠁⣼⠇⣴⡏⠰⠃⠠⢊⡇⠀⡄⠀⢠⣶⢋⡝⠸⣞⣀⠀⠀⠀⢰⣟⡉⠀⣼⣇⣸⡇⢘⣿⢻⡄⣿⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠛⢁⣾⢋⠐⠻⠀⠄⠀⠀⢼⣇⡔⠁⠊⠹⣇⡡⠀⠀⣴⠋⠀⠀⠀⢸⣏⣡⠄⣿⠟⢿⠇⠸⣿⠟⠀⢉⣿
⣶⣶⡿⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠒⠒⠛⢲⡴⠚⠙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢄⣸⠇⠀⠰⣶⡀⠀⣿⣤⣠⠴⠋⠉⠉⢳⢀⡴⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢤⠏⠀⠀⠀⠀⠉⠲⣤⢄⠈⠉⠀⣿⡞⢢⠀⠀⠀⢀⣾⠇⣀⣤⣀⠀⡇
⠀⠀⠀⠀⠀⠀⣾⣦⡄⢀⣀⣸⣴⣠⢀⠀⢠⣶⣶⣮⣯⡭⠄⣼⢟⠛⡽⢀⡀⣐⣼⣿⣿⢷⡶⣿⣿⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣟⡶⠚⠛⠿⣿⣟⡀⢸⣿⣧⣷⣻⣯⢹⡯⣢⢼⣷⠟⠋⠉⠀⠉⠙⢷⣷⣿⢚⠍⠀⠉⣆
⠀⠀⠀⠀⠀⠀⢀⣞⡁⢀⡠⣠⢾⡿⡿⡿⣮⣿⣹⣿⣏⣿⣿⣞⣵⣿⣿⣽⡆⠀⠀⢀⠀⠀⣟⣥⢥⡀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠸⣯⣻⢮⣍⡏⠈⠛⠺⠼⠼⠚⢛⣛⣿⣾⣿⣿⣿⣿⣟⣵⢁⣴⣧⣒⣠⣴⣷⢿⣽⣧⡚⠁
⠀⠀⢀⣠⠤⠖⣪⣿⣟⣛⣻⠀⡸⣟⡿⠛⠛⠉⠉⠉⢛⣿⣿⣿⣿⣿⣧⠉⠉⠉⠙⢿⣿⣛⣛⠋⠉⠉⢉⣉⣷⢶
⢰⠺⢥⣀⡀⠈⠁⠈⠻⢿⣿⣄⣙⣈⣇⢄⠀⠀⠀⠀⣾⠿⠛⣹⢿⣾⣾⢀⢄⠄⡄⡼⠳⠤⣄⣠⡴⠚⣫⡿⡡⣿
⣿⠀⠀⠀⠉⠙⠒⢤⠞⠉⠉⠉⠏⢛⢷⣾⣄⣇⣆⠀⣿⣿⣼⣛⠿⠾⠓⠿⠿⣾⠛⠏⠉⠀⠀⠙⣗⢕⠍⢜⢍⣿
⡧⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⣴⣯⣉⡙⠉⠙⠛⠛⠟⠉⠀⠀⣀⡤⣴⡞⡩⣧⠀⠀⢔⠅⠀⢻⡣⡪⡢⣲⡟
⢧⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠒⠦⢤⡠⡴⠚⡋⡯⡫⣫⢮⣺⣿⠀⠀⠣⠊⠄⢸⣗⣵⣟⡿
⠀⢸⠿⢷⣦⣤⣀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⢶⢎⢮⢎⢖⢕⢕⢕⣿⠀⠀⣠⡠⠆⣼⡫⠪⡶⡇
⠀⢸⠀⠀⠈⠉⠻⢷⠀⠐⣤⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣵⠕⠑⢁⡂⣰⣽⣿⣿⠠⠚⠁⠀⠀⣿⢕⣴⢾⡇
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⢶⢦⣤⣀⡀⠀⣿⣫⣪⣮⣾⢾⢟⢕⢽⡇⠀⠀⠀⠀⠀⣿⡎⡱⣹
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠙⢻⡿⣫⣝⣕⠕⡱⡣⡺⣻⡇⠀⠀⠨⡂⠀⢻⢟⡴⢿⣄⡀
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⢸⣯⡪⡺⣣⡪⡪⢊⢞⢽⡇⠀⠂⠙⠄⠀⢸⡾⣱⣿⣿⡁⢁⠀⠠
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣟⢔⢔⢝⢝⠕⡵⡱⣻⡇⠀⢈⢂⠄⠀⢸⣝⣯⣾⠒⠒⠀⠀⣀⡀⠁
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡧⡣⡻⡣⡫⡪⢊⢞⣽⡇⠀⠀⠫⠊⠀⣸⡿⠟⢈⣉⠉⠋⠀⠐⢒⡒
⠀⠙⠛⠶⣤⣄⡀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢎⢊⢜⢕⡵⡕⣻⣿⡇⠀⢀⣠⣴⡿⢛⠋⠁⠉⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠈⠙⠻⣄⣦⡄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣟⡵⡱⡫⡪⢪⣾⣿⣿⣧⠾⠟⠋⠀⠀⠀⠀⠠⠤⠂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠶⣆⣷⣄⡀⠀⠀⠀⠀⣯⣮⢺⣞⣵⣟⡛⠓⠒⠂⠀⣙⠋⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠗⠶⣤⣄⣿⡵⡟⠉⠀⠤⠤⠀⠈⠉⠉⠁

⠀⠀⡀
⣀
⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣤
⠀⠀⠈⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⡀⠀⠠⠀⠂⢀⣀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢼⣿⣿⣿⣿⣿⣽⣷⣶⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣷⣄⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⡙⠀⠀⠀⣿⠏⠃⠈⣿⣿⣌⣿⣻⠧⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣄⣠⣠⣤⣿⣤⣤⣤⣤⣴⣶⣶⣴⣮⡛⣻⡆⠉⠓⢲⣿⠂⠙⠀⠀⣿⣿⣿⣿⡯⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠿⠟⠻⠋⠛⠛⠉⢉⣩⣵⣿⣿⡄⠂⠀⢘⡁⠠⠀⠀⣠⣿⣿⣿⣿⢁⣼⠿⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠺⣿⠉⠁⣤⣾⣿⣿⣿⡿⠟⠛⢿⣷⢱⣼⣿⠃⠀⠈⠙⠛⠛⠋⠉⠉⠉⢹⣿
⣉⣉⣉⣙⠿⠿⠿⠿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⡄⠀⠹⡿⢿⢿⡟⠁⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿
⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⣽⡿⡟⠁⠀⢀⢠⣿⣿⣿⣿⣿⠂⠀⠀⠀⡀⠀⢠⣾⣿⣿⣿⣿
⠉⠙⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⡇⠀⠀⢻⣿⣶⣤⣴⣶⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣧⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷⠀⠀⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠐⢠⠂⢀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠒⠬⢘⢭⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉
⠀⠄⠐⠂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠄⠀⠀⠀⠀⠁⢀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘
⠠⠀⠌⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡤⠓⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠰⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠄⣖⡶⢔⡠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠌⢘⣦⡀⠄⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠔⠚⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠈⠀⠈⠄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⡒⠒⠒⠶⠦⣤⣀⠀⠀⣠⠔⢚⢉⡉⢩⡟⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡀⠲⡝⠊⡔⠈⠉⠒⢤⡉⠳⣴⠁⢰⣥⠀⠈⠆⠠⠌⠙⠟⢷⡄
⠀⠀⠀⠀⠀⢠⢞⣤⠀⣖⠀⠈⠁⠀⠓⠄⠓⠀⠀⠱⡐⣿⠀⠖⠂⠀⠀⢁⡠⠒⢻⡘⠄⢻⡄
⠀⠀⠀⠀⠀⡞⠉⣀⣁⠀⠉⡓⠚⠒⠒⣀⣒⠒⠄⠀⡇⣌⠆⠀⠒⠁⠊⠉⠀⠀⠈⢙⢦⠀⣷
⠀⠀⠀⠀⠀⡇⡜⠈⠘⠁⠓⠀⣠⠊⠅⠀⠀⡉⢦⠀⠁⡀⠀⡀⠈⡆⠐⡲⠀⡴⢀⡀⢢⠃⢸
⠀⠀⠀⠀⠀⢷⠘⢄⣀⡀⠀⠊⠸⣠⡄⣀⡀⠐⠊⡆⡌⠀⠀⠑⠈⢀⠥⠖⠂⠉⠒⢊⣱⠎⣼
⠀⠀⠀⠀⠀⠘⣧⢠⠦⠄⡔⠘⠓⢦⡰⠅⠃⡈⠰⠀⠀⠀⡀⠀⠲⡌⢓⡤⠄⢠⠀⢧⠄⣣⡏
⠀⠀⠀⠀⠀⠀⠘⢯⡋⠀⡄⠀⢀⠉⠁⡀⠀⠂⡆⠀⠠⠀⠁⣈⡠⢃⡀⠈⠳⢀⡑⠀⠂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⢆⡀⠀⠃⢸⠀⠁⡄⠄⡃⢀⠤⠥⡀⠘⢄⡀⠀⠀⢈⠕⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡈⠒⠠⣸⡀⠀⠱⣥⢠⡓⡀⠀⠈⠎⡠⠒⠐⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠙⠀⠠⠄⡄⠉⠁⠀⢀⠏⠀⠲⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡑⠢⠇⠀⠀⠀⠁⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠐⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣶⣾⣯⣄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣻⣿⣍⣡⡤⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⣾⣷⣾⣿⣷⠴⠀⢀⣶⣤⣤⣾⣿⣿⣿⣷⣬⣥⣮⣷⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠉⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢏⠈⠹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣿⡏⠙⠈⠉⠉⠀⠈⠈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠿⢆⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⡤⠴⠚⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠖⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡁⢀⣉⡠⠂⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋⠉⠉⠉⠋
⠀⠀⠀⠀⠀⣼⣿⡶⠛⠁⢠⡄⠀⠀⠀⠀⢀⠄⠀⠀⠢⢄⠀⠀⠈⠉⠁⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢴⣿⣿⣾⣧⣤⣷⣿⣶⡆⠀⠀⠀⠊⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠊⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠈⠛⠋⠁⠀⠈⠿⠿⠿⠿⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠽⠿⠋⠸⣿⣿⠏⠻⠏⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡲⢤⣉⠳⣄⠉⠛⠄⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⣱⣾⢡⡀⠁⣎⠉⠉⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⣷⠀⠙⢂⣘⢧⣶⡶⣂⡿⣠⢦⣄⠀⠀⣠⣶⠘⣧⣀⣠⠇⠻⣤⣑⣂⣨⡽⣛⠁⣟⠉⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠞⡵⠿⢋⠈⠒⠚⣡⡶⠶⢶⣯⡻⣍⠉⠉⠠⠟⠘⣿⣧⣼⢱⡏⠹⠈⠉⡁⠀⡐⡎⠉⠉⣵⠞⠉⢻⡟⠳⢶⣮⣅
⠀⠀⠀⠀⠀⠀⠀⣤⡶⣦⣄⢸⡄⢿⣸⡏⠀⠲⣦⡈⣿⡌⠃⣴⠀⣄⣠⢟⡿⢻⡜⠧⠀⣰⢸⣄⣖⣳⠇⣾⠜⢁⣠⣴⣯⣄⠀⣟⠁⡈⠳⡀
⠀⠀⠀⠀⠀⠀⢸⠁⢰⡆⢻⡏⢿⣎⢳⡑⠂⠀⣸⡿⢸⡇⠰⣌⠳⠦⠶⣟⡁⣧⡙⠳⣾⣥⣤⡉⠉⢱⠞⣿⢠⣿⢋⡵⠊⠙⣷⠈⠉⣠⠶⣄
⠀⠀⠀⠀⠀⠀⣴⡄⣸⠇⣼⠇⣸⡟⣀⣉⠂⢀⣿⠃⣾⠁⢀⣿⠘⣦⣾⠁⠈⠈⡱⠸⣏⠀⡀⠙⢦⢩⡆⢿⢸⣿⢸⠀⢄⣠⠟⣞⠓⠻⣄⢨⡀
⠀⠀⠀⠀⠀⠀⣿⠰⢫⡾⠋⢘⡉⡀⠉⠙⠛⠆⠇⢸⠃⣠⡾⢋⣼⣃⡈⠛⠛⣯⣵⢇⣭⣭⣥⣄⠈⠸⣧⡈⠣⢿⣎⠀⣠⠶⠶⣍⢓⡤⣬⡻
⠀⠀⠀⠀⠀⠀⠇⡴⣫⣤⣤⡈⢳⡝⠻⣿⣿⡟⠛⠾⠼⢡⣾⠟⣩⣭⣍⠳⠄⠏⠘⠩⠞⠛⢮⡝⢿⣆⠙⠻⢷⣦⣿⣆⣿⠀⠀⠟⠸⠰⡆⣷
⠀⠀⠀⠀⠀⠀⠀⣼⡏⡴⠦⢹⠯⢿⡾⠉⣴⠿⠿⣦⡀⣿⡇⣼⠃⠀⠸⠀⠀⣠⡶⠶⣦⡄⠀⢻⡌⣿⡄⠲⠶⣬⢿⣿⣿⡄⣠⢬⣻⢄⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣧⣳⣄⠈⠀⢿⡀⢀⠿⣄⡄⠘⣇⢿⣧⠹⡄⠀⢠⠀⠀⣿⡀⣰⢸⣿⠀⣸⢣⣿⢱⠏⠙⣿⠘⡹⠁⠀⠀⢀⣽⢂⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⡛⠒⠒⡉⠁⠀⠐⢶⣴⡎⠈⠻⣷⣝⠂⠸⣆⠀⠈⠉⢁⣼⡟⠐⣣⣾⠋⠀⡙⣡⠟⠠⢭⣿⣛⡛⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣽⢟⣿⠿⠷⣦⣸⣿⠇⠸⣄⠈⠙⠷⣄⠈⠛⠿⠿⢛⣩⣴⠾⢋⣡⡾⠾⢟⣵⠟⠛⢷⡿⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⢰⣽⡷⣋⣜⡷⣝⠻⠶⣶⣬⣵⡄⠀⣺⡿⢿⣿⠛⠛⣵⠶⢦⣸⣧⡶⣤⡼⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢹⢰⣏⠀⠁⣰⡶⢛⣳⠦⠈⠓⠸⠃⣴⠟⠛⢻⡎⠧⠀⠈⢟⡏⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣦⡀⣿⡰⠏⠁⢠⠞⠛⢶⡜⠇⢸⡆⠼⣣⢆⣴⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣟⠫⣄⡘⢦⠀⢀⡇⣠⠟⣠⣾⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣯⣿⣆⢇⣾⢟⣥⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⡻⡜⣱⢛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠣⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⡟⠋⠉⢿⣴⠶⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢣⠀⠀⡼⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣳⡦⣤⡷⢶⣷⠲⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣊⡴⠚⢉⣴⠞⢹⡇⢸⠈⠳⣄⠈⠻⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣁⠀⣰⠟⠁⠀⢸⡇⢸⠀⠀⠘⢧⡀⣽⡇⠃
⠀⠀⠀⠀⠀⢀⣀⣐⠒⣠⠤⠤⠶⠶⣮⡭⠿⣞⣹⣄⠀⠀⢸⡇⢸⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⣠⠟⠡⡍⢻⡇⠀⠀⠀⠀⡿⣸⠃⠈⠛⣷⡖⠒⢾⠀⢸⠲⠶⠶⢶⡟⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠐⣇⠀⠀⠙⣾⣇⠀⠀⠂⢸⣿⠏⠀⠀⣰⣿⠀⠀⢸⠀⢸⠀⢀⠀⢸⠇⠀⣤⠶⠶⠛⠛⠛⢷
⠀⠀⠀⠀⠈⢧⣤⠖⠚⠿⢤⣴⠤⠟⠛⢶⣡⣾⣋⣿⠀⠂⢸⡇⣿⠠⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⠀⢸⣷⠞⡛⠳⣦⡀
⠀⠀⣠⡤⠶⣾⠃⠀⠀⠀⣴⠛⣆⠀⠀⠈⢿⡉⠛⠛⢻⡆⢸⠃⢻⣠⡴⠾⠋⢰⡂⣇⠀⠀⠀⠛⠀⢸⡇⢠⠇⠀⠈⠙⣷
⠀⠀⡇⠀⢰⣏⣴⣦⠀⣼⠃⠀⢸⡆⠀⠂⠈⣧⠀⠀⢸⡗⢺⠀⢺⣹⡇⠀⠀⠀⢳⡸⡄⠀⠀⠀⠀⢸⣧⠏⠀⠀⠀⢀⡿⢰
⠀⡆⢿⣠⣼⣯⣤⣽⣴⡇⠀⠀⠀⢿⠀⠀⠀⢸⡶⢶⡾⠛⣽⠀⢸⠀⢳⡀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⡋⠀⠀⠀⣠⡾⠁⠁
⠀⠘⡄⣿⠀⠀⠁⠀⠉⣧⠀⠀⠀⢸⡇⢰⠷⣬⡇⢸⣇⡀⢸⠀⣸⣀⣀⣿⣦⠔⠒⠒⠚⣳⡤⡖⠋⠉⠉⠳⣶⠾⠿⠶⠶⠶⠶⣦⡈
⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣷⡏⠀⠘⠁⢸⡇⠁⢸⣼⡏⠉⠁⢠⠏⠀⠀⠀⡼⠁⠀⠈⢦⡀⠀⠀⠘⢷⣀⣀⣀⣀⣀⣼⠃
⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠁⢸⡏⠀⠀⠀⠀⣸⠄⠀⢸⡝⣷⡤⢤⣿⡴⢶⡄⢸⠗⠖⠛⠛⠻⢷⡀⢠⣄⡈⢿⠉⠉⢉⡏⠁
⠀⠀⠀⢸⡇⠀⠈⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠠⠀⣾⠠⠀⢸⡇⣿⢹⡄⠁⠀⠀⠙⠟⠀⠀⠃⠀⠀⠈⣧⠟⠈⠙⠻⠆⠀⣾⠁
⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⡋⠀⢀⣀⣿⣤⣤⣼⣇⣿⣄⣱⣤⣤⣤⣤⣤⡤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠋
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠶⠶⠶⠾⠋⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⣿⡿⠿⡿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⠿⠿⣻⡿⠿⣿⣿⣶⣤⣀
⠀⠀⠀⣠⣾⡿⣿⠏⠁⣀⣀⠀⠈⠁⣷⠈⠻⣍⠛⢿⣷⣄⠀⠀⠀⢀⣠⣾⡿⠛⣩⠋⠀⣿⠈⠀⠀⣀⡀⠈⠻⣟⢿⣷⣄
⠀⠀⣴⣿⠋⢠⡏⠀⡞⠉⠉⢳⡄⠀⣿⡀⠀⠘⣇⠀⠉⠻⣷⣄⣰⣿⠟⠁⠀⣼⠃⠀⢠⡿⠀⢠⡞⠉⠉⢷⠀⣿⠀⠙⣿⣦
⠀⣸⣿⠃⠀⠀⢿⣄⠙⠒⢂⣼⠃⠀⢹⣇⠀⠀⢹⠀⢀⣀⠈⢻⡟⠁⣀⡀⢰⡇⠀⠀⣼⡇⠀⠸⣧⡐⠒⠋⣠⡟⠀⠀⠈⣿⣧
⢀⣿⡇⣠⣴⠾⠿⠻⠿⣿⡛⠳⠶⠦⣼⣿⣄⠀⢸⣼⢫⠈⣳⡄⢰⡏⠁⡝⣾⠇⠀⣠⣿⣥⠴⠶⠞⢛⣿⠿⠟⠿⠷⣦⣄⢸⣿
⢸⣿⣿⡟⠁⣠⣤⣄⡀⠈⠹⣆⠀⠀⠀⢹⣿⡛⠷⣿⣬⣙⣁⡼⢧⣈⣋⣴⣿⠾⢛⣿⠏⠀⠀⠀⣰⠏⠀⢀⣤⣤⣄⠈⢻⣾⣿
⠘⣿⣿⡀⢸⡁⠀⠈⣿⠀⠀⢹⠀⠀⠀⠀⠹⣷⡀⠀⠙⢿⡅⠀⠀⣨⠿⠋⠀⢠⣾⠏⠀⠀⠀⠀⡇⠀⠀⣿⠁⠀⢈⡇⢨⣿⣿
⠀⢿⣿⣷⣄⣉⣉⣴⣿⡴⠋⢘⣠⣤⣤⣄⠀⠙⢿⣄⠀⠀⠙⣆⡰⠃⠀⠀⣰⡿⠃⠀⣠⣤⣤⣄⡂⠙⢦⣿⣦⣉⣉⣠⣾⣿⡿
⠀⠈⢿⣧⠉⠛⠋⢡⡏⠀⣰⠟⢁⣤⢤⡙⢷⡀⠀⠙⠳⢤⣀⠘⠃⣀⡴⠞⠋⠀⢀⡾⢋⡤⣄⡈⠻⣆⠀⢹⡌⠙⠛⠉⣼⡿⠁
⠀⠀⠈⢿⣷⡀⠀⣾⠀⠀⣟⠀⢸⡀⠀⢃⣼⠃⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠘⣧⡘⠀⢀⡇⠀⣻⠀⠀⣧⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠙⢿⣦⣹⠀⠀⠹⡄⠀⠙⠛⠛⠁⠀⣀⣀⡤⢖⣿⠇⠹⣿⡲⢤⣀⣀⠀⠈⠛⠛⠋⠀⢠⠏⠀⠀⣧⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠛⣿⣶⡚⠒⠿⢶⣶⣖⡒⠛⠋⠉⣠⣴⣿⡃⠀⠀⢙⣿⣦⣌⠉⠙⠛⢒⣲⣶⡶⠷⠒⢃⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠈⣙⣿⣷⣶⡿⢻⢟⠭⡙⢦⡼⢫⠭⢻⡟⢿⣶⣾⣟⣋⠁⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡛⠋⠉⣉⠽⢿⣾⡧⠴⠃⡸⣇⠹⠤⢾⣷⡿⢯⣉⠉⠉⢛⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣸⢣⡖⢦⠙⣿⣆⠈⠁⠈⠂⣴⡿⢋⡖⢲⢜⣆⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⡷⠛⠀⠈⢿⣆⠀⠀⣼⡟⠁⠀⠳⢿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⢻⡆⢰⡟⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠈⣷⡿⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣸⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢧⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣓⣤⣀⢳⣀⡤⠚⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⡤⠠⡄⠀⠀⣿⠀⠀⠈⠉⠙⡟⠺⡷⢮⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡀⣈⠳⣄⣠⣼⣖⣙⣤⣯⣤⠀⠀⠀⢰⠀⠀⠇⠀⠀⠀⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣭⡿⡥⢞⠻⢍⡀⠀⠘⣧⣤⣄⡜⠀⢸⠀⠀⠀⠀⢀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⣏⠉⠰⢯⡴⠇⠀⢳⡄⣾⠷⠶⣏⠀⠀⡇⠀⡼⢍⠙⢯⠁⣹⡇⠀⡱⠂
⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣸⡥⠾⡀⠀⢏⠃⠀⠀⠹⣆⠠⠇⣸⣀⣀⣙⠦⣧⣼⣴⣋⣠⣤⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠙⠓⠦⣧⣀⠈⡆⠀⠀⠀⠹⣼⠀⢸⣁⣨⡭⠿⠿⣿⢿⠍⠀⠈⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣄⠀⢠⡶⠛⠛⠉⠁⠀⠀⣀⣀⣽⡎⣶⠚⠋⢻⡀
⠀⠀⠀⢰⣧⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⢧⣠⣤⠴⠖⢛⡏⠉⠀⠼⠛⢇⠐⣂⣈⣷⣠⣄
⠀⠀⠀⣰⠃⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⢹⡄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⡾⠀⠀⠀⡐⡗⣀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠃⠀⠀⠿⡪⡻⣗⠓⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠋⣧
⠀⣾⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠛⠛⡛⠛⢻⠛⠛⠉⠉⠁⠀⢠⡄⠀⠀⢹
⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⣀⠀⠀⠀⠤⣶⣶⠄⠀⠀⠀⢀⣇⠀⠈⠀⠀⠀⠀⠠⡮⣶⢕⠵⠆⢸
⠀⣇⠀⠀⠀⠀⠀⢀⢄⣹⣳⣿⣼⣏⡠⡀⠀⠀⠘⠀⠀⠀⠀⢰⠁⠀⣱⠔⠢⢄⠀⠀⠀⠈⠡⡇⠁⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠈⠉⣽⠟⣿⡻⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠴⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⣸⠤⠤⠤⢀⣀⠀⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠑⠤⠤⠚⠁⠀⠀⢀⡀⠤⠒⠉⢧
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠉⠙⠛⠓⠒⠒⠶⠶⠦⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⠀⠀⢀⡀⣾⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡆⡈⠁⣈⠱⠵⣬⣱⠀⠀⠀⠀⠀⠀⣶⠭⠒⡴⢊⠉⠉⠑⠻⡟⠿⡾⠾⢃⣤⣄
⠀⠀⠀⠀⠀⠀⢀⢀⢅⡸⣇⣻⣧⠣⣥⠞⠀⠁⠉⣀⣽⢦⡀⠀⠀⢀⣴⢹⠑⢲⢩⡍⢳⡄⣬⣆⠔⣁⢤⡄⠱⡄
⠀⠀⠀⠀⠀⠀⢞⣏⠉⠀⠛⢫⢡⡵⠤⣡⡾⢟⣶⣤⣼⡻⢟⡄⠀⢛⣫⢎⣧⣀⠉⣁⣼⢇⡚⢽⣀⢧⡉⢉⡼⠇
⠀⠀⠀⠀⠀⠀⣮⣻⣶⡂⠘⠃⢈⠀⣼⢻⢣⡟⠁⣠⡄⢹⠈⢸⣄⠁⠡⡌⢃⣨⣭⣉⡀⠸⡿⢸⡟⠃⢍⠩⣾⡀
⠀⠀⠀⠀⠀⠀⢘⢬⣛⡡⢻⣇⣌⡀⠻⣸⡼⡄⣼⠧⠥⠞⠀⡼⣫⣱⣾⡁⠞⢉⣍⣻⡝⢷⠀⣾⢠⢳⠚⠘⠈
⠀⠀⠀⠀⠀⠀⠀⢕⢱⡆⢸⡷⠿⠃⣰⡎⠙⠛⠦⣴⣢⡥⠞⠁⣉⠐⢈⡠⣀⣀⢡⡟⢿⡮⣰⠳⠶⠄⢅⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷⢈⣁⠟⢍⡓⠨⣶⠀⠈⣁⠐⠂⣼⡋⠀⠔⠓⣂⣀⠊⢻⡠⠁⢳⡷⠦⣄⣄⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡇⠐⢩⡝⢿⡖⠛⣘⠫⠀⣶⡿⠇⢹⣴⣾⠱⢦⢇⣠⡗⣆⣤⡄⡠⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠀⢁⡸⡁⢀⡙⠀⠼⠿⢷⡷⠎⢵⣞⢶⣤⡼⢛⣠⣄⠉⠉⠁⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⣢⢠⣌⡫⡐⠀⠂⢜⠀⠤⠈⠃⣻⠏⢿⡜⡟⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠩⡁⢸⡭⢱⡄⡆⣻⣍⠇⠬⠀⡍⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢴⡞⠀⢾⠱⢩⠑⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠻⠏⠂⠄

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡶⢯⣰⡮⣽⠦⣮⣾⡈⣿⣉⡃⠀⢀⡷⢸⡅⣶⠐⣶⠰⣦⣷⡀⣿⢾⡇⣾⢽⡦⢾⣸⡂⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠞⠈⠳⠟⠘⠟⠞⠁⠛⠘⠃⠀⠈⠳⠾⠉⠛⠐⠛⠚⠇⠟⠑⠛⠸⠇⠛⠟⠓⠟⠸⠋⠲⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡤⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀⡉⠉⢹⠐⡁⠀⠀⢨⠃⠀⠑⢈⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡀⠀⠈⢳⣀⣈⣼⣿⣦⡴⠁⠀⢀⠔⠋⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠬⠵⢦⡎⠁⠀⠙⡟⢻⣡⠔⠊⠉⠐⠢⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⡽⠀⣀⠧⢄⣠⠔⠣⡞⠛⠛⠻⡖⠢⣄⠀⠉⣢
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⢰⣕⡋⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⡇⠳⠚⢹⣀
⠀⠀⠀⠀⠀⢰⠣⢜⣄⣤⣷⣾⠴⢳⣸⠀⠀⠀⠀⠀⡇⠀⠀⡠⢓⡢⣇⣠⢔⣵⡒⠲⡵
⠀⠀⠀⠀⡟⢻⢦⣼⠿⠇⠻⣦⡤⢮⠭⠷⢦⢀⣠⣤⠧⠒⠒⡷⣇⣭⠷⡟⢲⣶⣛⣫⠜⠓⠒⠤⢤⣀⡀
⠀⠀⠀⠀⡇⢸⢸⠁⢸⡏⠉⠀⡇⢸⡀⠀⢸⡼⢋⣿⠍⢱⠖⢿⠮⣷⢮⠟⠉⢉⣢⡍⢻⠗⠖⠋⠉
⠀⠀⠀⠀⡇⢸⢸⠀⢸⣇⠤⠤⠧⣮⣬⣝⠉⣱⣋⣻⡥⠼⠦⠼⣀⢸⢸⠉⡏⠁⠀⢸⠸⡆
⠀⠀⠀⢀⡧⢼⢸⠀⢸⢑⠶⣶⠲⠯⣧⠾⠭⢶⠼⢦⠕⠒⠒⠒⢻⣾⡸⠀⡇⠀⠀⠘⢦⣱⡷⠂
⠀⣀⣤⣾⠉⠒⣿⡖⣼⠓⠊⣸⠠⠄⠀⠤⠤⢼⠀⢸⠦⠴⠒⢲⡎⠘⠁⠀⡇⠀⠀⠀⢸⠀⡏
⠀⡇⠈⠉⠑⢻⢻⠿⡟⡟⢻⠃⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⣣⠤⠒⠁
⠀⣅⠀⠀⠀⢸⢸⠀⡇⡇⢸⢀⠀⠀⠀⠀⠀⢐⡇⢸⡀⢀⠀⠼⠣⢀⠀⠀⡇⠀⢀⠠⠚⠈
⠀⠀⠈⠉⠒⠚⠼⠀⡇⠗⠚⠁⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠒⠗⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠊⣉⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠸⠀⠈⢪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠐⢄⠀⢣⠀⠀⢿⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡀⠀⠐⢤⡑⠄⠱⣄⠈⣧⡠⠤⡒⠁⠀⠀⣀⡀⠤⠊⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣌⣉⣁⡀⣉⣉⣐⡊⣑⣦⣬⡗⢻⢡⠀⠘⣍⠉⠁⠀⠀⠀⠀⣀⣨
⠀⠀⠀⠀⠀⢸⠉⠉⠁⠐⠒⢲⠐⠂⢉⠭⠽⠟⠉⣉⢁⡉⣲⠰⠀⡴⠮⢍⠉⠉⠛⠛⠂⠈⠉⠉⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠂⠀⢸⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣤⠄⢀⡀⢀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⢸⠀⠀⢇⡀⢀⡀⠠⡤⠆
⠀⠀⠀⠀⢀⠜⢸⠀⠀⠀⠀⢸⠀⠀⢸⠉⠉⠉⠁⠐⠚⠒⡇⠀⠀⢸⠉⢹⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠓⠂⢹⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢣⠔⠂⠤⠄⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠂⠐⠧⣀⡠⠄⢀⡀⢀⣀⣀⣀⣀⡀⢀⡀⠠⠴⠀⠠⠔⠐⠒⠂⠀⠁

⠀⠀⠀⠀⠀⣠⠤⠶⠶⠒⠒⠒⠚⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠶⢤⡀
⠀⠀⠀⠀⢠⡇⠀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⣄⠀⢠⣤⣤⣤⡤⠀⠘⣆
⠀⠀⠀⠀⢸⡇⠀⠉⠉⣿⠉⠉⠁⠀⡇⠀⠈⠉⢳⡄⢸⡇⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹⠀⠀⠀⢸⡇⠀⢀⣤⠟⠀⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡄⠀⠀⢸⡷⠞⢻⡅⠀⠀⠈⡇⠀⠀⠀⡿⠀⢸⡟⠉⠉⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠹⣄⠀⠀⣧⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⠀⠙⠆⠀⠻⣤⣤⠟⠀⠀⠈⣧⣤⣤⠤⠶⠀⡾
⠀⠀⠀⠀⠀⠀⠙⠢⠤⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⣠⠶⠲⢶⣄⠀⣶⠀⠀⠀⠀⢀⡶⠀⡶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠙⣧⢸⡆⠀⠀⠀⢸⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡇⣷⠀⠀⠀⣾⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⠇⠸⣇⠀⢰⡏⠀⠀⣿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⡿⠀⠀⢻⡄⣾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⣤⣤⡄⠙⢦⣤⣤⠞⠁⠀⠀⠈⢿⠇⠀⠀⠀⣿⣤⣤⣤⣤

⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢾⡉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠳⣞⣀
⠀⠀⠀⠢⠊⣠⣶⠖⠛⠒⠦⡀⢀⠔⠒⠒⠲⣦⡀⠀⠀⠀⠀⢀⣴⠞⠋⠙⠢⠄⡠⠒⠋⠛⢿⣄⢇⢦⣧⣄⡄⣤
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⡏⠺⣴⠎⡿⢹
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣩⣯⣼⣿⣯⣱⠖
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠻⢤⡀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠺⣤⣵⡦⠙⣦⣸
⠀⠀⠀⠀⠀⣦⠬⠳⣦⣀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⢠⣴⡯⢥⣖⠋⠁⢰⣟⠇⢐⡿⠁
⠀⠀⠀⠀⠀⠈⢛⣲⠖⢛⣷⣄⣰⣟⠩⠷⠂⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠀⣖⣀⣿⠄⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⠉⠁⠈⠁⠀⠀⠀⢨⠮⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⣠⢤⢀⠀⣠⡴⠶⠦⢄⡀⠀⡠⠤⠶⠦⣮⡀⠘⢃⢤⡀⠀⢀⣤⠤⠤⢄⡀⠀⣠⠤⠤⣤⡀⠀⢀⣤
⠀⠀⠀⢈⣸⣅⣾⠋⠀⠀⠀⠀⠉⠎⠀⠀⠀⠀⠈⢿⡆⠀⠁⠀⢰⡟⠁⠀⠀⠀⠑⠊⠀⠀⠀⠀⠹⣦⣬⡈
⠀⠀⢬⠖⢯⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠗⣧
⠀⠀⠹⣟⣌⣋⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣅⣮⣏
⠀⠀⢾⠊⢝⣻⢯⣝⣦⣄⡀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠸⣶⡢⠽⠾⣶⡄⠀⠀⠀⣠⣴⣯⣵⣺⡽⠃⣿
⠀⠀⠸⣆⠀⣿⠟⣉⡿⣻⠿⡶⣄⢀⡴⠿⢷⡤⠄⠀⠀⠀⠀⠀⠈⣉⣽⣛⣙⣿⣦⡰⠫⣽⠷⣷⠦⢹⣇⣼⠏
⠀⠀⠀⠙⠷⠻⣄⣩⡿⠉⠛⠓⠈⠋⠊⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠃⠀⠁⠀⠀⠁⠀⠙⠶⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠤⢤
⠀⠢⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠢⡈⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡀⠶⠀⢀⣠⣴⡶⠶⠶⠶⢤⣄⠀⠀⠀⢀⣠⠴⠶⠶⠶⣶⣯⣜⠀⠀⠛⢕⡒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⡗⠛⣰⣿⠟⠁⠀⠀⠀⠀⠀⠈⠳⣄⡴⠋⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠄⢊⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣹⣧⢱⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⣀⢙⡷⠶⣯⡁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⡌⣿⣤⡀⠹⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠐⠹⣯⣱⢤⣉⡈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⢠⣷⢝⡿⣺⢯⣹⣏⢟⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⣿⡂⠈⠓⣯⡬⣈⣿⣷⣮⣽⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⢿⣷⠋⡁⣿⣿⠟⢁⣀⣲⣟⣉⣿⡻⣟⢶⣄⠀⠀⣤⠞⠯⠶⠲⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣤⢻⣯⠀⠚⠙⣿⡆⠻⢶⣶⣾⣑⠙⢦⠞⣵⡒⢦⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠻⣷⣶⣶⠿⠃⠀⠀⠉⠉⠉⠀⠈⠈⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠲⡖⠲⢄⠀⣀⣀⠀⠀⠀⢰⠁⠀⠀⠊⡥⠮⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⢶⣤⡀⠀⠀⠈⠫⡔⠢⡉⠑⢤⠇⠀⠀⠀⡼⢽⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣞⢍⠢⢝⢦⠀⠀⠀⡘⢦⡀⠁⡜⡄⠀⠀⣰⣫⢍⡜⡌
⠀⠀⠀⠀⠀⢀⡠⠤⠚⠛⢞⡻⣷⣧⣭⣳⣄⡒⢭⡖⠋⠉⠺⣦⢀⣴⣯⢕⣫⢞⣉⢶⣄
⠀⠀⠀⢀⣴⠏⠀⠀⣀⣠⣤⠤⠥⡭⢩⢧⣟⣶⣺⣴⣀⠠⣾⢴⡿⠿⢿⣿⡛⠉⠀⠀⠀⠉⢐⠢⢄⡀
⠀⠀⠀⢸⠸⣡⢶⣟⡕⠑⠁⠊⠈⣀⠥⠛⠋⠙⠑⠝⢿⣿⡶⡥⣀⠀⠀⠀⢨⡓⢄⠀⠀⠀⠀⠈⠖⣀⣽⠂
⠀⠀⠀⠀⡆⠘⠚⠊⠉⠒⡤⠒⠝⠀⠀⠀⠀⣀⠤⠒⠋⠁⠉⠲⢅⡋⠔⡢⡀⠕⣈⣆⠀⣀⡤⣖⢏⢜⢾
⠀⠀⠀⠀⠘⢦⡀⠀⠀⢸⠀⠀⠀⡀⢀⣴⠎⠉⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠩⠛⠥⠁⠙⣮⡮⠫⡪⣡⡏
⠀⠀⠀⠀⠀⡇⠉⠳⠦⣸⠀⠀⠀⠀⣼⠉⠑⠤⣄⠀⢀⠀⡀⠄⠀⢀⣠⢴⢻⡆⠀⢸⡃⣿⡕⣕⡴⡿⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡀⣗⠀⠀⠀⠀⠉⠒⢍⣁⡴⡚⡕⡱⣱⢿⡇⠀⢔⠂⣾⢏⢼⣫⡇
⠀⠀⠀⠀⠀⢩⠀⠀⠀⢸⠀⠀⠀⡀⡟⠓⣤⡀⠀⠀⢀⠀⣏⢮⢊⢎⢚⣯⣷⡇⠀⢕⡂⣭⠳⡟⣱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄⠀⠀⠂⣽⠀⠀⠈⠑⠦⣜⣂⣷⣡⢷⢗⢟⢝⢼⡇⠸⣓⠄⣿⢏⢜⣼
⠀⠀⠀⠀⠀⠀⣧⢀⠀⠀⡇⠀⢨⠀⢾⠀⠀⠀⠀⠀⠀⢹⡯⡡⡣⣫⢪⡮⣻⠀⠮⣐⠠⢋⢎⡾⡏
⠀⠀⠀⠀⠀⠀⠑⢤⡑⠀⡇⠀⠀⠂⣿⠀⠀⠀⠀⠀⠀⢸⢕⠝⡰⡕⣕⢝⣽⠀⡩⢔⢸⡣⣟⢵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢇⠀⠀⠅⢸⠀⠀⠀⠀⠀⠀⣸⢫⢎⢎⢎⢮⣮⣿⠐⡪⠕⢸⣯⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢔⠅⢸⠀⠀⠀⠀⠀⠂⢸⡱⡡⡣⣣⡳⣕⡏⠠⣲⣥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠓⠤⡀⢀⣴⡂⢸⡔⡕⣕⣝⡼⠾⡣⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⡀⠀⡈⠑⠪⢔⣸⣎⠾⠊⠁
⠀⠀⠀⠀⠠⠲⢔⠩⣂⠕⡪⠔⡨⠔⡈⠔⢉⠔⡨⠔⡠⠆⡠⢚⠔⡢⠀⡠⢂⡠⢂⡠⡠⢀⠔⣀⢀⠄⡠⢀⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠁⠊⠐⠊⠐⠈⠴⠊⡠⠊⠠⢊⡄⢊⠔⡨⢊⡤⢊⠔⢈⠔⡡⠔⡡⣪⢖⡡⢊⠔⠡⠊⠐⠑⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠐⠁⠀⠁⠘⠁⠊⠀⠊⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⠛⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡯⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⣒⡒⢷⡀
⠀⢀⡞⠀⡴⠶⠦⠤⣌⣙⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣳⣏⣀⣀⠀⠀⠀⠀⠀⣀⡤⠖⢋⣡⠴⠚⠉⠁⢙⡆⢹
⠀⠘⣇⠀⠙⢤⣀⣀⠀⠈⠉⠛⠲⠿⢷⣤⣀⠀⢰⠋⠁⠀⠀⠀⠈⢳⠀⣠⣴⡯⠵⠒⠋⠉⠀⠀⣀⣠⠴⠋⣠⠞
⡖⠒⠛⠓⠦⠤⣤⣍⣉⣉⣙⣻⣿⡿⠿⠟⠛⠛⢛⣦⡤⣤⣤⠤⠴⠿⣟⠛⠛⠿⠿⠿⠿⠿⠿⠿⢽⠖⠚⠛⡟⠛⡟⠛⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠞⠉⠈⣇⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠋⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣸⣄⣀⣤⣧⣤⡧⠤⠤
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠃⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⣀⡤⠚⠉⠁⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠘⡄⠀⢀⡠⠞⠁⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢦⡀⢀⡞⡇⠀⠀⠀⠀⠀⠀⠹⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⣠⠧⠶⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢀⠂⠀⠀⠄⠀⠀⣀⣀⡀⠀⠀⠀⠀⠰⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠠⠄⠀⠐⡀
⠀⠀⠀⠀⠀⠐⠦⠀⠀⠦⠠⣘⢧⠀⡄⠀⣈⣣⠆⣄⢱⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣃⠄⠴⠀⠀⠴⠂
⠀⠀⠀⠀⠒⡀⠠⠂⠀⠙⣀⢡⠬⣤⢀⡀⣬⣉⠀⣉⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⡜⣐⠏⠀⠐⠀⢀⠒
⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⢧⠄⠢⠐⢃⣀⠄⠀⡸⠚⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡲⠠⠜⠀⠢⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠈⡀⠀⢀⠂⢘⣡⠖⠙⡋⠀⠌⠀⠀⠈⠈⠃⠀⠀⠀⠀⠈⢁⡠⠥⡉⠃⠐⡀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠀⠉⣶⣳⠀⠀⠀⠀⠀⡀⠠⡐⠆⠀⠁⠀⣸⡮⣏⣅⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⠀⠂⠞⡍⠁⠄⠶⠀⠁⠀⠀⠀⠀⢆⡀⠟⢞⠼⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠒⠰⠒⠀⠀⠀⠀⠀⣠⠂⢳⠊⢲⠊⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⡒⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠒

⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡞⣿⢦⢤⠶⠟⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⡌⢳⢿⠦⠰⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢜⣲⢭⠋⠹⢚⡇⢢⣾⣿⣿⣶⣦⠂⠀⢰⣖⣿⡗⠶⠛⣣⡂⢅⢛⠸⠡⢲⠦
⠀⠀⠀⠀⠀⠀⠀⡴⣺⠐⣀⠉⠓⢬⡇⢸⢿⠈⡍⠚⢿⣠⣴⠋⠉⠙⣙⠤⢉⠛⣉⣿⣭⡀⣁⣶⡢
⠀⠀⠀⠀⠀⠀⠀⢜⠿⠓⡵⠄⣶⡄⠳⣘⣟⠴⣀⡸⣿⣿⣿⣋⢩⣥⢸⣦⠀⢈⣱⣼⣷⢶⢜⣧⣽⠷⠂
⠀⠀⠀⠀⠀⠀⠀⣀⡰⡒⡀⢣⡴⠶⠶⢽⣭⠰⡎⣥⣿⣱⡀⢵⣜⡿⢿⣫⠆⢲⡵⠃⣸⡟⢩⢦⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠹⣵⠏⠀⣮⡠⡊⣤⣴⣅⠠⠌⣻⣇⠛⣟⣿⣴⡾⡫⢐⣁⡤⢶⢿⢋⣬⢳⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⣿⣿⣿⠐⠨⠗⢸⣓⠊⢻⣿⣿⣸⠋⣴⣺⣿⣋⠉⣙⡳⡤⠲⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣻⢷⡬⣠⣿⠟⣋⣭⣍⡈⢩⣿⢿⢘⣭⢙⣶⣹⢷⣝⡴⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣟⢻⡥⢈⣿⣻⠟⣿⣲⣝⢧⠼⢩⣎⢙⢡⣿⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢲⣤⠊⠯⡥⣾⣸⡉⣙⣿⠃⢿⡡⢮⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣘⡿⢿⢿⣯⡝⠀⣺⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣸⠿⠻⡿⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠲⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣦⠀⠀⣴⠶⠶⣶⣤⣄⡀⠀⠀⢀⡆⠀⠀⠀⠀⠀⣶⠀⠀⢰⡶⠶⠶⠶⠶⠗⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢻⣦⠀⢸⡇⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣸⡿⠀⢸⣧⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣾⠟⠁⠀⠘⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣤⣶⢿⡏⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⠛⠛⠛⠛⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠠⣿⠁⠀⠀⢻⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⣆⠀⠀⠀⢻⠀⠀⠀⠀⣸⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢦⠀⠀⠸⣇⣀⣀⣰⠏⠀⠀⠀⠀⣿⣄⣀⣠⣤⣤⣤⡄⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⢦⣤⣤⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⣀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠉⠉⠻⣷⣄⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠈⢿⣆⢸⣧⠀⠀⠀⠀⠀⢰⣿⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⡀⣿⡄⠀⠀⠀⠀⣼⡏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣷⠀⠀⠀⢀⣿⠃⠀⠀⣿⣇⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢿⣇⠀⠀⣼⡟⠀⠀⠀⣿⡟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠘⣿⡄⢠⣿⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⢹⣷⣾⡏⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⣶⣶⣶⣶⣶⣶⠄⠈⠙⠿⣶⣶⠾⠟⠁⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⢿⣷⣶⣶⣶⣶⣶⠆


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠛⠤⡀⠀⠀⠈⠲⣖⠛⠒⠦⠤⣠⠞⠁⠀⠀⠀⢀⠼⠛⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡼⠦⣄⠈⠢⢄⠀⠀⠈⠳⡀⢀⡴⠁⠀⠀⠀⡠⠔⢁⡠⠴⢥⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠒⠉⠁⡔⠛⠒⠢⠤⣉⡒⠢⠭⣶⣤⣤⡬⠿⣤⣤⣤⢶⠭⠐⣊⣡⠤⠐⠒⠛⠳⠦⢤⣄⣀
⠀⠀⠀⠀⡴⣖⡋⠉⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣀⣀⣀⠈⣑⣢⣤⠴⣿⢤⡤⢞⡳⠤⣴⣒⠉⣁⣀⣀⡀⢀⣀⠔⠁⠀⠀⠈⠉⢑⣲⣦
⠀⠀⠀⠀⡇⠀⠉⠙⠒⠤⣄⣀⠀⢀⣀⠤⠒⢋⣉⠤⠔⠚⠉⠱⠒⠋⠁⡼⠳⣀⠉⠉⠀⠈⠑⠒⠤⢍⣙⠒⠤⣀⡀⣀⡠⠴⠚⠉⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢹⢛⡶⠒⠋⠉⠀⠀⠀⠀⠀⢀⣠⠴⠊⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠈⠙⠢⡖⢹⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣤⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⡀⠀⠀⠀⠀⠀⠈⠺⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠳⢤⣀⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⠙⠒⠦⣄⣀⠀⢀⣠⠴⠒⠉⠁⠈⢦⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣀⡤⠞
⠀⠀⠀⠀⠀⢸⠀⠉⢱⠋⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⢠⡷⠒⠋⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠈⢇⠀⠀⠀⠀⠀⢀⣼⡁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣇⢈⣦⠀⠀⣠⠏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢦⠀⠀⠀⣠⠋⠀⠉⡗⠲⢤⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡠⠔⡟⠉⠀⠱⠔⢹⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠳⣄⡜⠁⠀⠀⠀⡇⠀⠀⠀⠉⠓⠲⢤⣀⡀⢸⠀⣀⡤⠖⠋⠁⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠲⠤⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣀⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢼⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⡼⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣯⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢄⡀⣿⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⣿⣿⡇⠀⠀⠙⣧⡙⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣠⣤⠶⠶⠶⠶⠶⢦⣄⡀
⣿⣿⣿⠀⢀⣿⣾⣷⡈⠣⡀⠀⠀⠓⠴⠿⠇⠀⢸⠃⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⢿⣿⣿⣀⠘⣿⡿⠈⢷⣤⠈⠢⢄⠀⠀⠀⠀⠀⠁⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠳⣌⠛⡏⠐⠋⠀⠀⢄⣻⡵⡄⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⣿⣮⠳⠀⢀⣴⣿⣿⣷⡝⡿⣟⢮⣿⣿⣿⣦⡴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⡌⡆
⣿⣿⣧⠀⠸⠿⣿⣿⣿⡇⠀⠈⢯⡳⣍⠉⠁⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⣳⠁
⣿⣿⣿⠀⣠⣴⣼⣿⣿⣇⠆⠀⠀⠙⣌⠢⡀⠀⠀⠀⠀⠀⠱⢤⣀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⢀⡏⡇
⣿⣿⢃⢇⣿⣿⣿⣿⣟⡞⢀⣤⠴⠶⢬⡳⣌⠲⣄⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⡼⣱⠁
⢋⣥⢏⣼⣿⣿⣿⣿⣟⡴⠋⠀⠀⠀⠀⠈⢆⠙⢮⣑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⠃
⣛⣵⣿⣿⣿⣿⣿⣫⠏⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⠓⠮⣽⣲⣤⣄⣀⣀⣀⣀⣀⠤⠔⣊⡵⠋
⣿⣿⣿⣿⣿⢟⡵⠃⠀⠀⠀⢂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠶⠒⠚⠉⠁
⣿⢿⡫⢛⠕⠋⠀⠀⠀⠀⠀⠈⠣⣄⣀⣂⠜⡁⠀⡄⢠⣄⠀⠂
⡂⠅⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠃⠈⠉
⠀⠀⠀⢰⠋⠀⢙⣶⡋⠀⢹⣶⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠈⠁⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠁⢤⠂⠀⠁⠀⠀⠀⠀⠠⠹⣿⠃
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⢀⠀⢀⣀⢠⣄
⠀⠀⠀⠃⠀⠈⠁⠀⠙⠁⠀⠀⠀⠀⢀⠄⠀⡄⠶⣆
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡄⢰⡶⢦⣄⡀⠀⠀⣶⠀⠀⠀⢠⣄⠀⣴⡶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⡏⠉⠀⠀⠀⢸⡇⠀⠈⢿⣆⠀⣿⠀⠀⠀⠸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⢸⣿⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⣼⡟⢸⣿⠀⠀⠀⢠⣿⠀⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣷⣶⡿⠋⠀⠘⣿⠀⠀⠀⢸⣿⠀⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠈⣷⡀⠀⠀⣿⡆⠀⠀⣼⡇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⡇⠀⠘⣷⡀⠀⠸⣷⡀⢠⣿⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠘⠃⠀⠀⠈⠁⠀⠀⠙⠛⠛⠁⠀⠈⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⢰⡟⠻⣆⠀⣰⠟⢹⣷⠀⣿⡀⠀⠀⢀⣿⠀⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⢹⡇⠀⢹⣶⡏⠀⢸⡟⠀⢸⡇⠀⠀⢸⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠻⠁⠀⣾⡇⠀⢸⣷⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⢠⣿⠀⠀⠈⣿⠀⠀⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄⠀⠀⠀⣼⡏⠀⠀⠀⣿⡇⢀⣿⠀⠀⣿⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣧⠀⠀⢠⣿⠁⠀⠀⠀⢸⣧⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡄⠀⣾⠇⠀⠀⠀⠀⠸⣿⣾⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣷⣸⡿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⠀⠹⠿⠁⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠿⠶⠶⠾⠗