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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠰⣿⡙⠁⠐⢻⣦⣶⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣷⡀⠶⣝⠻⣷⡙⠿⣿⣿⡆⣠⡀⠀⠀⠀⠀⣀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⣉⣿⣼⣿⣤⠰⣅⠊⠛⠒⠛⢷⣤⣿⡿⠀⠀⢀⠀⠸⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡀⠀⠀⠀⠨⢏⣴⣿⡿⠋⠙⢿⣷⡤⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⢉⣤⣾⠛⠃⠀⠰⣦⣦⡞⠉⠁⠀⠀⡀⠀⣿⣿⣦⣤⢤⠀⠀⠀⠀⢀⠐⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠐⣨⣿⣷⣶⣶⣶⡄⠸⣿⣷⣤⡀⠀⣀⣬⠀⣿⣿⣿⣿⣿⣇⣀⣀⣠⣬⡙⠻⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⠶⢾⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣾⣧⣼⣿⣦⣤⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⡿⣿⡿⣿⣿⣿⣿⣮⣅
⠀⠀⠀⢤⣼⣶⣯⣤⣄⣀⠀⠀⠀⠀⠈⠙⠻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⣽⡀⡀⣹⢳⡼⠿⣭⠛
⠀⠀⢰⣋⠹⣯⢙⠛⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠼⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣵⡿⠋⠀⠀⠀⠀⠀⠲
⠀⠀⠈⠛⠗⣌⠀⠀⢀⠿⢩⣟⣺⢷⡀⠀⠀⠀⠀⠀⠈⠁⠈⠃⣙⠁⣿⣿⣿⣷⣿⣿⣷⣄⣤⣴⣿⠖
⠀⠀⠀⠃⠀⢩⣷⣤⡈⠼⣦⣜⣟⡃⠜⠿⣷⠀⠀⠀⠀⠀⠀⡼⠛⠋⠙⠛⠻⣿⣿⣿⣿⣯⣀⠈⠉⠁⠀⡀
⠀⠀⠀⠷⡤⣜⣿⣿⣿⣆⢻⣞⣎⠥⠀⠀⢻⡇⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⢸⢿⠍⠘⢿⣿⣿⣷⣤⣄⡀⠁
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣯⠸⣿⠻⠠⠃⠀⣻⣿⡴⣶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣇⣽⣾⡏⠄⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⡉⢀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣆⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠁⠸⣿⣿⣿⣷⡟⢿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠞
⠀⠀⠀⠀⢸⡆⠀⠀⢸⣿⠀⠀⢻⣿⣿⣿⠷⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂
⠀⠀⠀⠀⠈⣧⠀⠀⠀⣿⡇⠀⠈⣿⣿⣿⡿⣁⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⡇⠀⠀⣷⠈⣿⣅⣥⠻⣧
⠀⠀⠀⠀⠀⣿⣇⣀⣶⠀⣷⠀⠀⠉⠀⣯⠹⣿⢸⣿⣀⣀
⠀⠀⠀⢠⣄⢻⣿⣿⣿⣷⣸⣷⠀⡇⢠⡙⠀⣼⠈⣻⢸⣿⡇
⠀⠀⠀⠀⢻⡾⢿⣿⣿⣿⣿⣿⣶⠇⠘⣇⠘⣧⣰⣿⢰⣿⡇
⠀⠀⠀⠀⠐⡄⠀⠀⠈⠉⢻⣿⣿⣾⡇⠛⠃⠀⠘⢿⣼⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠛⠻⣧⠀⠀⠀⠀⠘⢿⣿⣧⣄⣀⡀⣰⣅⠐
⠀⠀⠀⠀⠀⢾⣧⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣟⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⡀
⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡀⠹⣿⣿⢸⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⢠⠀⡸⣇⢠⠂⠀⠀⠀⠉⠇⠀⠻⣟⣸⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁
⠀⠀⠀⠀⠻⣆⠘⢆⠀⠀⠀⠀⠀⢠⡤⠿⣾⠁⠀⠀⢀⡄⠀⠀⠀⠿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠘⡀
⠀⠀⠀⠀⠀⠈⢷⣼⡀⠀⠀⢀⣠⡄⢻⠓⠛⠂⠀⢠⠛⢧⠀⠀⠀⠀⠘⣏⠙⢷⣦⠀⠀⠀⠀⠀⣠⣿⡧⡀
⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣶⡾⠛⠁⠀⠀⢀⣀⣀⡤⠄⠈⠁⠀⢠⡀⠀⢙⣀⠈⠙⠀⠀⠀⠀⠀⣿⣿⡗⢿⡀⢠⡄
⠀⣀⣀⣴⣦⡐⣦⣠⣿⣿⣷⣋⠀⣀⣀⣀⣻⣿⣥⣤⣤⣤⣤⣠⣼⣧⣤⣦⣿⣤⣤⡶⠞⠃⠀⢴⣿⣿⣧⣌⢑⣜⠦
⠀⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠂⠀⠀⠀⠀⠐⠻⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢦⡀⠀⠨⠵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠀⣌⢀⢀⣱⠐⠄⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⢠⡿⠄⠀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠔⠒⢦⠀⠀⠷⡎⠀⠀⠀⠀⠀⠀⢀⡰⣾⠶⠛⡗⠒⢶⠆⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠢⡴⠁⠀⠀⡿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⣰⠁⠀⠀⣸⠃⠀⠀⢿⣇⡠⠔⠢⡄⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⢀⢡⣶⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢐⠿⣼⣃⠴⠁⠀⠀⠀⠸⠋⠚⢉⡵⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠆⠈⠉⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠢⢀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣠⡠⠖⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⡰⡠⠃⠀⠀⠀⠀⡠⢠⡧⠄⣄
⠀⠀⠀⢀⠠⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢋⠇⠀⠀⠀⣀⠀⠃⢢⡵⢚⡇⢤⡀
⠀⢲⢞⣣⠔⣢⢎⠀⣠⡀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁⢀⠴⠧⣀⣀⡰⠉⡼⠦⢚⡀⢱⣟⠒⠒⠃
⡶⢻⣶⠖⡶⠿⢄⣙⠆⢹⠀⠀⠀⠀⠠⣎⣿⠀⠀⠀⠀⡠⠶⢏⡁⠀⢀⠔⠁⠐⢲⠮⠅⢧⢀⢓⠊⠀⠘⠋⠘⡆
⠑⠓⠂⠁⠀⠀⠸⣉⢤⠮⣉⣐⣲⣖⠉⢴⠈⢦⡠⠔⠋⠀⠀⠀⠉⠒⠁⠀⠀⠀⠘⢤⠀⠀⠉⠉⠀⠀⠀⠀⠀⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⠀⠀⠀⠀⠀⠀⠀⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠉⢓⢲⠒⠀⠀⠀⠀⠀⠒⠒⠂⠉⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⣦⣧⣀⣀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠓⠘⠛⠂⠘⠢⠤⠤⠤⣤⣤⣤⣤⣴⠖⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢰⣿⠀⣨⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠛⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠻⠿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⠀⠉⠀⠀⠀⣼⣿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠛⠛⠿⠿⣿⣿⣿⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⢩⣿⠋⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣞⢶⣶⣶⣶⣦⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⣫⡷⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣡⣶⣾⣿⣿⣦⡀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⠀⣠⣄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡛⠁⠀⠀⠈⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⢶⣄⠀⣠⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣶⣦⣴⣿⣄⣀⣀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⢀⣤⣶⣦⣤⡀⠀⠀⠀⣠⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠛⠋⠁⢹⣷⡀
⠀⠀⠀⠰⠿⠿⠿⠿⣿⣿⣦⠀⠈⢉⠛⠛⠛⢻⣿⣿⡀⠀⠀⠆⠆⠀⠀⠀⠀⠀⠀⠐⠖⠆⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠶⠀⠀⠀⢹⣿⣿⠀⠀⠈⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠈⣿⡃⠀⠀⠑⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⢀⡄⠂⢄⠀⠀⠀⠀⠀⢀⠔⠠⢄
⠀⠀⠀⠀⠀⠠⠒⠠⡀⣿⡇⠀⠀⢀⠆⠉⠁⢫⠉⠀⠀⠀⠀⠀⣾⠁⠀⠀⠁⠀⠀⠀⢀⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢨
⠀⠀⠀⠀⠇⠀⢀⠘⡀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⢄⢀⣁⠤⠀⠀⠀⠀⠘⣀⣀⣡⠔
⠀⠀⠀⠀⠠⢀⠀⡀⠆⠀⠀⠀⠀⠈⠂⠄⠄⠂

⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠎⠱⡀
⠀⠀⢀⡖⠉⠉⠭⣙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠁⠀⠙⣄⡀⠀⠀⠀⢀⠀⣀⢀⢀⡴⣒⣀⡒⢄⡀⣴⣫⡤⠤⢦⣳
⠀⠀⣸⠄⣒⡴⠀⠀⣨⢽⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⣤⠶⠶⣶⡀⢱⠀⠀⠔⡡⠂⠁⠀⡏⡎⠁⠀⠉⢆⢹⠟⠡⠀⠠⠀⢻⡇
⠀⠀⠱⣴⡯⠀⠀⠷⠒⢻⠄⠀⠀⠀⠀⠀⠀⠀⣻⡻⠀⠀⡀⠀⣻⠏⠀⣸⠼⠁⠀⠀⢠⣤⠃⠀⠀⠀⢰⡟⠀⠈⠀⢀⠀⢀⠇
⠀⠀⠀⢀⣶⣄⣀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡄⠰⢂⠞⠛⠀⠀⠓⠄⠀⠀⠀⢸⢫⣑⣀⡀⠀⢼⡿⠇⡀⠀⠁⡀⠎
⠀⠀⡠⠛⣩⠙⠶⠭⣩⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢺⠛⠧⢦⣄⠀⠀⠀⠈⠑⡂⢀⡿⠻⠿⠿⠛⡀⣨⢀⣤⡸⠛⠻
⠀⢸⣝⠃⠀⢠⠂⠚⠀⠸⢇⠀⠀⠀⠀⠀⡠⢞⡏⠠⢸⠠⡁⣾⠄⢙⠢⣄⡤⠚⠍⠉⠙⠲⣄⣤⠚⠛⣦⠞⠛⠠⠌⡤⠊⠑⠢⡀
⣰⠃⠈⢓⠖⡃⠄⠒⠁⠀⠨⢦⠀⠀⣠⢊⠔⠉⣇⠑⠘⠄⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⢀⠮⠀⢀⠉⠒⠇⠐⠢⠤⠄⣤⣀⣄⠀⢇
⢇⠀⣰⢏⠈⠁⠀⣠⡴⠀⠀⠊⢧⣠⢬⣲⣅⠀⣹⠀⠉⠀⠀⢸⣀⡆⠀⠀⠈⠰⡄⠀⠣⡀⠛⣉⣱⡈⠛⢄⣀⡠⢤⣄⣀⣉⣚⣊⠇
⠀⠈⡹⠀⠀⢁⠔⠀⠃⣈⠀⠏⡏⠄⠒⠢⡝⢷⢿⠀⠀⠀⠀⣯⠞⠉⢀⣀⣀⣤⠃⠀⠀⠈⠓⢌⠻⣸⡄⠈⠍⠉⠀⣉⠀⠉⠋⠁
⠀⠀⡗⠀⠀⠁⡢⠀⠁⠀⠀⠀⡇⣠⣴⡄⢸⡸⣾⠀⠀⠀⠐⡆⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠎⡇⠀⠳⠺⠄⠸⠆⠾
⠀⠀⠧⣤⣤⣤⣠⣀⣀⣀⡀⠀⡼⣄⣈⠴⠚⠁⡇⠀⠀⠀⠀⣇⠀⠈⠂⠲⠖⠛⠒⠒⠂⢀⣀⡼⠃⠘⡇⠀⢿⠀⠀⠀⠀⢰⡀
⠀⠀⠀⡇⠀⠈⠉⠉⠉⢹⢉⡭⡚⠅⠃⣄⠀⢠⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡛⠛⠛⠉⡉⠁⡇⠀⢸⠃⠀⢸⠀⠀⠎⠀⠀⡇
⠀⠀⠀⢃⠀⠠⠀⠀⠀⠇⡞⠏⠳⣌⣤⠒⡆⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠸⠁⠀⡇⠀⡾⠀⠀⠈⠀⠀⢡⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⢘⠓⠦⡍⢶⠛⠀⢘⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⡇⢀⡇⠀⠀⡗⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠈⡄⠀⠀⠀⢸⠀⠀⢟⣮⣤⡼⠀⠀⣮⠀⠀⠐⠀⠈⠁⡇⠀⠀⠀⠀⣷⠀⠀⢰⡇⢀⠁⣸⠀⠀⠀⡇⠀⠀⢸⠀⠀⣧
⠀⠀⠀⢠⠁⠀⠀⠀⡌⠀⠀⠈⠀⠀⢠⠀⠀⠹⠤⣤⣤⢤⣤⣤⠃⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⠀⠈⠒⠤⡤⡇⠀⠀⢸⠀⠀⣿
⠀⠀⠀⢰⠖⠒⠲⠖⠳⠤⣀⢀⠀⣀⣸⣀⠀⢀⡠⠇⢼⣸⡤⠾⠤⣀⠀⢀⡠⠘⠤⣤⣴⣷⡊⠀⢂⣄⣠⠀⠃⠀⠀⢸
⠀⠘⠛⠻⠶⠶⠤⠤⠤⠴⠾⠥⠜⠓⠄⠚⠻⠭⠖⠒⠋⠉⠁⠒⠀⠒⠛⠛⠒⠒⠛⠉⠉⠉⠉⠉⢻⣟⣙⠶⠃⠀⠄⣺⠦⠤⠒⠠⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣄⣀⣠⠤⠖⠒⠉⠉⠉⠉⠉⠉⠓⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⠀⠀⠀⠀⡸⢨⣃⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣄⠀⢀⡤⠖⠉⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠙⠢⣀⢀⠇⠈⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠁⠀⠙⡎⠀⠀⠀⠀⠋⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⢘⣦⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⢦⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠓⣤⠀⠀⢀⣠⡅⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠂⢨
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⣏⠀⠤⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣀⡀⠀⠀⠛⠟⠀⠀⠀⠀⠀⠀⣠⠔⠃⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⢶⢻⢻⢲⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠊⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡼⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣅⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⡼⠁⠀⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁⠀⠀⠉⢳⠄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡜⢳⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⠢⣄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡀⠀⠀⠀⠀⢀⠖⠛⠑⢄⡠⠚⠉⠈⢦⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠶⠦⠤⠤⠤⠿⠷⠶⣶⠶⣤⠏⠀⠀⢰⠇⠀⠀⠀⠀⢩⠀⠀⠀⠈⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠉⠉⠙⡏⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠜⠀⢠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢲⠖⠲⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁⢹⠲⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢤⠒⠒⠲⠤⣄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡎⠀⠀⠀⠀⢸⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⠀⡞⣉⡻⠾⠃⠀⠀⠀⠀⡸⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢩⠟⠁⠀⠀⠀⠀⠀⠀⠀⡗⠚⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⣀⣴⣁⣀⣀⣀⣀⣀⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠊⠉⣉⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠐⠢⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⠿⠿⠿⠛⠛⠛⠋⣉⣉⡉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⢙⣛⠶⣦⣀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣴⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⡠⠴⠚⠋⠉⠉⠉⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠆⡐⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⠤⠖⢊⣉⠉⠉⠲⣄⠀⠠⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠁⣰⣿⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠊⠉⠉⠉⠉⠲⣌⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⢀⣿⡏⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣄⣀⣀⣀⡠⠜⠃⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡆
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠀⠀⢀⣠⠞⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠋⠛⣋⣉⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡠⠂
⠀⢀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⡄⠂⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣋⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⠟
⠀⠈⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⠞⠁⢀
⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⣀⣤⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠕⠊⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⣴⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠄⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠷⠶⣦⣀⠀⠀⠀⠸⣆⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢷⣦⣀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠈⠛⢷⣦⣘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢯⡁⠀⡆⠙⠂⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠹⠋⢳⡄⢠⢇⡀⠁⠀⠀⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠲⣄⡀⠘⡡⠖⢉⠁⠀⢳⡀⣸⠀⠀⡿
⠀⠀⠀⢀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡋⠀⠀⢿⠀⠀⠈⢷⠀⠘⠃⠁⠀⠀⢸
⠀⠀⠲⡌⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠓⣤⣴⢚⠛⠓⢆⠀⠀⠀⣸
⠀⠀⠀⠀⡠⣖⣺⠿⣖⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢜⡩⠄⠚⠉⠀⠀⢿⠂⠀⢸⠀⠀⢰⠃
⠀⠀⢀⢎⠞⢱⠂⡀⠀⠈⢪⣳⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠲⢿
⠀⠀⢸⣾⠀⠁⡼⠁⠀⠀⠀⡯⡇⠀⠀⠀⠀⠀⠀⠈⠣⠄⠀⡀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⣤⡀
⠀⠀⠘⡜⣄⠘⠁⠀⠀⠀⡰⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡁⠀⠀⣠⣀⠀⠀⠀⢀⡤⠊⠀⢹⠲⣄
⠀⠀⠀⠈⠪⣗⣶⣤⣖⣮⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢉⠀⢨⡳⣠⠔⠉⠀⠀⣠⠇⠀⢸⠐⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣇⠄⣼⠋⠙⣆⠀⣠⠔⡏⠀⠀⡸⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡴⠃⢘⢹⡶⡞⠘⠞⠁⡼⠀⢤⠊⠀⠀⠀⠀⠀⢀⡆⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣏⠙⠣⣈⠳⡀⠀⠀⠀⠀⠀⠸⡀⠀⡇⠀⡏⡞⠁⡇⠀⠀⢰⠁⠀⠀⣹⠀⠀⠀⠀⢀⡞⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡬⠷⢤⠈⠁⠈⢦⡜⠉⠲⢄⣠⠇⠀⡇⢠⢷⠃⠀⡇⠀⢠⠇⠀⠀⡴⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⡦⣄⠀⡼⠀⠀⠀⠀⢹⠀⠀⡇⣸⡟⠀⠀⡇⢀⡎⠀⢀⡞⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠇⠀⠀⠀⠀⠟⠀⠰⠇⢿⠃⠀⠀⠇⠼⠀⢠⠏⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠈⢷⣦⣤⣀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢈⠟⠛⢛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣶⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣑⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢧⠀⠀⡎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⢤⣄⣀⣀⣠⠷⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠳⢤⡙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⣯⠀⠀⠀⣹⠶⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠉⠲⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠉⠛⠋⠠⠄⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠓⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠆⠉⠉⠑⢦⡀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡿⠀⡴⠋⠙⢦⠀⢱⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⣧⠀⢲⣄⣀⠞⠀⡸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠳⣤⠈⢁⣤⠞⠁⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠙⠦⢄⣀⣀⣀⣀⠠⠤⠿⣁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⣸⠋⠀⠉⠁⠀⠀⠀⠈⢣⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣦⣄⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⡿⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠉⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣠⠚
⣟⠀⠉⠀⠈⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⠎⠁
⠉⠉⠉⠉⠉⠉⠉⢹⠋⡝⢋⡛⢫⡙⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠸⣇⣀⠾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⡰⠋⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣃⠤⠤⢊⡴⠻⣄⠀⠀⠀⠀⠀⠀⣀⠴⠛⠁⠀⠀⠀⠀⠈⠛⠤⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠺⣏⠀⠀⠈⠳⡀⠀⠀⢀⡼⠉⠀⠀⠀⢠⠔⠂⢀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⢦⠀⡌⠁⠀⠀⠀⠀⡅⠀⠀⢀⡀⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⡇⠀⠀⠀⠀⠀⠘⠣⠔⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠱⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⣀⣤⣄⣠⣤⣶⣶⣶⠶⠛⠛⠛⠛⠋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡴⠗⠈⠉⠉⠉⠉⠉⠉⠛⠛⠿⠿⠿⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠠⠶⠶⠶⠀⠀⠈⠉⠀⠲⠶⢆⡀
⠀⠀⠀⢠⣶⠶⠶⠿⠿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⢶⣶⠶⠶⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵
⠀⠀⠀⢘⡁⠀⠀⠀⢀⡇⢸⠀⡶⠶⠿⠿⠿⠶⠶⠶⢶⣶⠄⠀⠀⣀⢀⠰⡄⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⡇⢸⠀⣶⠶⠶⠶⠶⠶⠶⢶⣶⣦⠀⠀⠀⢛⡙⣷⡿⢶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠠⣶⠶⠶⠶⠶⠶⠶⣦⣤⣤⠀⠀⠀⠿⠻⣿⡿⢾⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠠⣴⠶⠶⠶⠶⠶⠶⣤⣤⣤⠀⠀⠀⣠⣟⣿⢛⡈⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣤⣶⡶⠶⠶⣶⣶⣤⣤⣭⡀⠀⠀⠈⡟⠑⠻⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⣤⣤⣶⣶⠶⣶⣶⣦⣤⣭⣤⣀⣀⡀⠰⠆⠸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣉⣉⣛⠛⠻⠿⢶⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣥⣤⣤⣤⣤⣤⣤⣤⣤⣌⣉⣉⠛⠛⠻⠶⣶⣤⣌⣩⣿⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⣭⣤⣤⣤⣤⣤⣤⣤⣌⣉⣉⣉⠛⠛⠷⢶⣶⣤⣍⣉⣿⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⣉⣤⣤⣤⣤⣤⣤⣤⣉⣉⣉⡙⠛⠻⠷⢶⣤⣤⣉⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣉⣤⣤⣤⣤⣤⣤⣬⣉⣉⣙⡛⠛⠷⠶⢦⣤⣤⣉⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠶⢶⣦⣤⣍⣉⣙⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠦⠚⠛⠛⠛⠛⠛⠛⠛⠿⠷⠶⢶⣦⣄⣀⣀⠀⠀⠀⠀⣿⠀⠀⠠⠀⢀⣀⣀⡀⠠⠤⠤⠤⠀⠐⠂⠛⠉⠉⠛⠂⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣉⣉⠿⠿⣶⣴⣿⣷⠀⢺⡷⣌⣩⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠉⠋⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⣄⠀⠀⠀⣠⠾⠋⣠⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡈⠢⣈⣷⣶⣿⠥⠤⠶⣤⣥⣀⡈⢳⡀⢀⣀⣤⠴⠒⠋⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⣀⣀⣸⢃⣠⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢯⡉⠀⡀⠄⠂⠘⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠤⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡀⠀⠐⣟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠂⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣿⢣⡀⠀⢀⡤⠞⠉⠉⠀⣈⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⣧⡈⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠄⠀⢀⠀⢸⣄⠙⢦⠎⢀⣠⠶⠛⠛⠙⠛⠳⢦⣷⠀⠀⠀⠀⠀⠀⣿⠑⠀⢀⣀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⢠⡴⠚⠋⠉⢀⡀⠠⠄⠐⠂⠀⠉⠳⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠐⠤⠀⣠⠟⠃
⠀⠀⠀⠀⠙⢦⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣼⡇⠀⢀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⣼⠁⠰⡀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠆⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠶⠶⠒⠚⣉⣤⠞⠁⠀⠀⠀⠀⠀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢊⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⠘⣷⡀
⠀⠀⠀⠀⠀⠰⠯⣄⡈⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠤⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢁⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⣛⣿⡿⢯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠌⠀⠀⡀⠤⠀⠀⠂⠀⣼⠏⠀⠀⠀⢹⡄⠀⠀⠜⠑⠀⡀⠀⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⠂⢀⣀⣀⣀⡀⠀⠘⠀⣿⣄⠀⠈⢀⣼⡇⠀⠊⠀⣀⠀⠀⠑⠲⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠖⠚⠋⠉⠉⠁⠀⢻⡄⠀⠐⠙⣯⣝⣛⣭⢿⣌⠀⠀⣼⠋⠉⠙⠓⠲⠶⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣴⠟⠉⠀⠉⠀⠀⠉⠻⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡈⢻⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⢹⣿⣿⣿⣿⣿⡟⠀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣿⣿⠁⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣿⣿⣿⣿⣿⡀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⣼⣿⣿⣿⣿⣿⣧⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣧⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⠍⠉⠡⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠊⠀⠀⠀⠀⠀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠠⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢈⠢⢀⣀⠠⢊⢜⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢀⢀⠈⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡒⠒⠒⡆⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠠⠄⠀⠑⠤⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⢄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠓⢀⡘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢄⠰⠀⢠⡄⠀⠀⠈⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⢀⡀⠄⡀⠀⠄⠠⠀⣀⡀⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢌⢂⡔⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠀⠀⠀⠀⠀⠙⠀⠀⠀⠰⠀⢸⠀⠀⠀⠀⠀⠀⠜⢄⠒⢊⠬⢀⡀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠀⠉⠀⠀⠀⠘⠉⡠⡠⠁⢸⠀⠀⠀⠀⢀⠊⠀⠀⡨⠁⠀⠀⠀⠉⠂⢀⠈⠢⡀
⠀⠀⠀⠀⢀⠤⠤⠤⢄⠀⠑⠡⡠⠄⣀⢢⣔⠀⠄⡢⠀⠀⠀⢸⠀⠀⠀⢰⣅⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠐
⠀⠀⠀⠀⠈⠂⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠹⠊
⠀⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢉⡀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠌⠐⠒⠓⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠓⡛⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀
⠀⠀⠀⡈⠎⠀⠀⠃⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠤⡄⠓⠄⡀⠀⡄⠊⠀⠀⠈⠐⢤⠐⠀⡠⠁
⠀⠀⠘⠄⠀⠉⠀⠀⠀⠈⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠢⢄⠈⠉⠀⠀⠀⠀⠄⠀⠀⡄⠊
⠀⡐⠈⢀⠀⠀⠀⠈⠆⠀⠀⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⠠⠀⠄⠀⠀⠁⠀⠀⠈⠐⠁⠀⠀⡇
⡀⡌⠀⠈⠁⠀⠀⠀⠀⠂⠀⠈⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⣷⣾⡿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠐⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠀⠖⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⣀⠔⠀⠠⢀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⠂⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⢀⡀⠀⣀⣀⡀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠉⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠈⠰⠀⠀⠖⠐⠆⠀⠠⠊⠰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠲⡶⡿⢻⣾⠖⠶⢚⣿⡷⠲⠖⢺⣿⠖⢲⣚⣿⡗⣶⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠤⣿⠀⠀⠠⢜⡇⠀⡀⠤⣿⠀⢣⠠⣤⡄⠤⠠⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠒⣿⠀⢰⠒⢺⡇⠀⡃⠒⣿⠀⢸⠐⢻⡇⣀⢀⠀⠀⡄⠀⠀⠀⠀⡠⠀⠀⠀⠄⡀⠀⡠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣉⣿⠀⢸⢈⣹⡇⠀⡇⣉⣿⠀⢸⢈⣹⡇⣾⠀⠒⢀⠀⠐⠀⢠⠂⠀⠀⢀⠀⠀⠈⢊⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠤⢾⠀⢸⠠⠼⡇⠀⡇⠤⢿⠀⢸⠠⠼⡇⣿⠀⠀⠀⠁⠀⠀⡄⠀⠀⠰⠀⡨⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠒⠋⠀⠀⡑⠊⠁⠁⠉⠀⠓⠀⠄⢑⣊⣀⣿⠀⠠⠀⠐⠂⠀⠁⠐⢠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⡇⣒⣚⣛⣒⣒⠀⠠⣴⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠕⠃⠐⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣛⣓⣒⣚⠀⠘⣿⡷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣒⣶⣶⣾⣀⠈⠍⠛⠀⡇⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣒⣒⣒⣒⡿⠭⢿⣗⣲⡇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣒⣒⣒⣺⠿⠭⢿⣒⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠖⠒⠒⠒⠷⠮⠭⣿⣒⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠴⠶⠤⠤⢤⣄⡀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠂


⣿⠿⠿⡿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣻⣻⣿⣟⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡆⢸⣿⣿⡇⠀⣾⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣼⣿⣿⣧⣸⣿⣿⠃⢸⣿⣿⣿⣿⣧⠀⢸⣿⣿⡏⠛⢻⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣸⣿⣿⡆⢸⣿⣿⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⣸⣿⣿⡿⠿⣿⣿⣷⣸⣿⣿⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣁⣈⣛⣛⠛⠁⣉⣉⣭⡅⠀⢉⣉⣉⡈⣉⣉⣀⣀⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿⡄⢸⣿⣿⡇⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣼⣿⣿⠇⣿⣿⣿⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣿⡄⣿⣿⣿⠹⣿⣿⣿⣿⠀⣿⣿⣿⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠹⣿⣿⠷⣿⣿⣿⠀⢹⣿⣿⡿⠀⣿⣿⣿⡿⠟⠋
⠀⠀⢠⣤⣤⡄⢀⣶⣶⡄⢠⣦⣤⣤⠀⢠⣤⣴⣶⡀⢠⣶⣶⣶⣶⣶⣶⡄⢀⣤⣴⣶⣶⡄⣤⣤⣤⠀⣴⣶⣦
⠀⠀⠈⣿⣿⣿⢸⣿⣿⣷⢸⣿⣿⡇⢀⣿⣿⣿⣿⣧⠸⠿⣿⣿⣿⡿⠿⢠⣿⣿⣿⠿⠿⠃⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣾⣿⣿⣿⣾⣿⣿⠁⣸⣿⣿⢻⣿⣿⡄⠀⢸⣿⣿⠇⠀⣾⣿⣿⡃⠀⠀⠀⣿⣿⣿⣀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⡿⢸⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⠀⠀⢹⣿⣿⣷⣶⣾⠆⣿⣿⡟⠛⣿⣿⣿
⠀⠀⠀⠈⠿⠿⠿⠃⠘⠿⠿⠿⠇⠘⠻⠿⠇⠀⠻⠟⠛⠂⠸⠿⠟⠀⠀⠀⠙⠻⠿⠿⠟⠀⠿⠿⠇⠀⠿⠿⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢰⣦⡏⢧⣴⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡜⣿⡶⢀⣼⠟⠉⡿⣯⠉⢻⣆⠀⣶⣿⣱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣛⢿⣿⣅⠀⣿⡄⣠⣺⣇⡀⣸⣟⠀⣨⣿⠿⣃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⣭⣭⡍⣻⣿⡿⠿⠷⠈⠻⣏⢹⡏⣽⠟⠡⠾⠿⢿⣿⣟⣩⣭⣥⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠾⣢⡾⣯⣥⣤⣴⠿⠋⠁⠀⠀⠀⠀⠀⠈⣼⡇⠁⠀⠀⠀⠀⠀⠈⠛⠿⣦⣤⣬⡝⢷⡔⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠋⠔⠋⠁⠀⣤⣴⣶⣶⣶⣶⣶⣄⠀⠀⠈⠁⢸⣿⣿⣷⣶⣶⠀⠀⠀⢠⣤⣌⣙⡦⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣷⡄⠻⣿⣿⡟⠛⢻⣿⣿⡇⠀⠀⠀⢈⣿⣿⣿⣿⣿⡆⠀⠀⠸⣿⣿⣿⣿⠀⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⢘⣿⣿⡇⠀⣿⣿⣇⢀⣸⣿⣿⠃⠀⠀⠀⣸⣿⡟⢸⣿⣿⣷⠀⠀⠀⠘⣿⣿⣷⠀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣾⣿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⢀⣿⣿⣧⣤⣿⣿⣿⡆⠀⠀⠀⠘⣿⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠁⠀⠀⠀⠀⣿⣿⡷⠀⣿⣿⣿⡆⣤⠀⢸⣿⡿⠿⠿⢿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣄⠀⠀⠀⠀⣶⣿⣿⣿⡆⢻⣿⣿⣿⡟⣶⣿⣿⣷⡆⢰⣶⣿⣿⣿⣶⠀⠀⢠⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠙⠛⠛⠁⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠀⠀⠈⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣶⣶⣶⣶⣤⡄⣀⣤⢀⣤⡄⣶⣶⠀⢶⣶⣶⠀⠀⣶⣶⢤⣤⣤⣤⣀⢠⣴⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⡅⢹⣿⢸⣿⣇⣼⡇⠀⣾⡟⣿⡇⠀⢸⣿⠘⠃⣿⡏⠿⢨⣭⣭⣭⣭⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣭⣉⣉⣉⣩⣭⣭⡅⠈⣿⣿⢻⣿⣿⠀⢠⣿⠿⢿⣷⠀⢸⣿⠀⠀⣿⣇⠀⢨⣭⣭⣍⣉⣉⣉⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠙⠛⠀⠻⠟⠀⠿⠿⠆⠿⠿⠷⠿⠿⠀⠘⠛⠋⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⡆⣴⣶⣶⣶⣶⣤⡀⠀⣴⣶⣶⣦⠠⣶⣶⣶⠀⣠⣶⣶⣦⣤⣶⡆⢰⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢸⣿⣿⠉⣿⡇⢹⣿⣿⠉⢻⣿⣷⠀⠙⣿⣿⠋⠈⢻⣿⡏⢰⣿⣿⡟⠙⢻⣿⡇⢸⣿⠉⣿⣿⡇⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⣤⣾⡿⠏⠀⠀⣿⣿⠀⠀⢸⣿⡇⠈⢿⣿⣿⣶⣦⣄⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⢸⣿⣿⠙⣿⣿⡆⠀⠀⣿⣿⡀⠀⣸⣿⡇⢀⣀⣙⠛⠿⣿⣿⣷⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠛⠻⠿⢿⠀⠀⣾⣿⣿⡆⢿⣿⣧⣿⠀⢿⣿⣷⣴⣿⣿⠃⢸⣿⣿⣶⣶⣿⣿⠟⠀⠀⠿⠿⠟⠛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣙⡛⠻⠷⢶⣦⣤⣤⣉⣉⡉⠃⠈⠛⠿⠃⠀⠈⠛⠿⠿⠟⠃⠀⠘⠛⠋⠙⠛⢛⣁⣀⣠⣤⣴⡶⠾⠟⢛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣙⡛⠻⠶⢶⣦⣬⣭⣉⣛⡛⠛⠻⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠟⠛⢛⣛⣉⣩⣥⣴⡶⠶⠟⢛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⢶⣦⣬⣭⣉⣛⡛⠛⠻⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠟⠛⢛⣛⣉⣩⣥⣴⡶⠾⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠟⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⢉⡉⠉⠉⠉⠉⠉⠉⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣾⢼⣠⢃⣳⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣤⣤⣤⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣧⣤⣼⣤⣤⡼⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣯⣏⣩⣉⣯⡅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢋⡉⠉⡙⠃⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣛⡧⠨⠅⢎⣩⠿⠛⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣆⠡⠔⠒⠳⢀⣤⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠈⠁⣿⣿⣿⣿⠁⠀⢀⡀⡀⢀⡀⢠⡉⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⣀⣀⣀⣀⠿⠤⠨⣿⣿⡿⠧⠜⢒⣓⡱⣦⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⣀⣀⠠⠄⠒⠉⢙⡄⠀⠔⠁⠘⡎⠀⢰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣀⠤⠄⣘⠀⠀⡀⠀⡰⠣⠤⢚⠆⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣼⠀⠒⠖⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠄⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⢅⠀⠀⠀⠀⠠⡀⠀⡠⠃⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠉⠦⣶⣤⠀⠀⠁⠀⠒⠒⢺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⡴⣶⡼⢲⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠰⠁⢈⢱⠀⠙⣥⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠤⣶⢶⡸⠃⠀⠀⠈⢆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⡜⠃⢸⠀⠒⠖⠆⠉⡍⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠈⠁⠀⢐⣠⠀⡐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣀⠀⠀⠙⡛⠁⣀⣅⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠺⡿⡦⠀⡈⠀⠐⡏⠡⢴⡶⠓⠤⣄⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠤⠶⠤⠤⠤⠄⠘⠓⠲⠺⠤⠶⠴⠼⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠷⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣤⣦⣤⣉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣤⣄⢀⣴⣮⢷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⡿⠛⢻⡿⠋⢳⡈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣿⠟⣃⢾⡿⠃⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⢏⠀⣠⠾⡁⣠⣾⡇⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠛⠃⠀⠛⠃⠀⢀⣾⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣤⣿⣿⣤⣾⣿⡿⢁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⣤⣄⣀⣤⡴⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⣈⠛⠿⠿⠟⢋⣤⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠁⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⡋⠁⠁⠠⠈⠉⠭⠑⠒⡛⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⣸⡃⠀⠀⠀⣌⢀⡤⣌⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠰⠀⠀⠸⣇⢹⠀⠀⠀⠈⠁⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣧⠼⠀⠀⠀⢜⣩⡤⠤⠔⠊
⠀⠀⠀⣠⠴⣒⠲⢄⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠠⠴⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⣬⡅
⠀⠀⠀⣇⣸⠉⣹⢀⠆⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⢀⣞⡵⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠁⣐⡒⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⢡⡦⠀⠀⠀⢀⣀⠤⠔⠒⠚⡳⡦⠀⣬⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠇⢇⣀⠬⠀⠸⢸⠧⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡄⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠈⢸⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢀⣏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡝⠀⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣇⢠⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢠⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢹⡌⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢸⠀⠀⠀⠻⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⠉⠉⠀⢘⣗⡿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠉⠉⠙⠓⠲⠤⢄⣀⣀⣧⠀⠀⠀⠀⢧⠠⢤⣸⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠘⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢳⡖⠒⠒⠒⠲⠶⠶⠤⠤⠤⠤⢿⡄⠀⠀⠀⢸⡥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⠂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠠⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠠⢀⠠⠠⡤⠠⣀⠄⠀⠈⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠑⠀⠇⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠈⠐⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢶⡀⡇⣱⢆⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⢱⡶⠁⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠬⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⢄⠀⡇⢀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠀⠀⢈⠀⠀⠀⠀⠀⡁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠅⡅⠁⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠙⣄⢃⡜⠑⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠰⠀⠘⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠘⡟⠀⠀⡀⠠⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⠘⠀⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠈⠢⡀⡇⡠⠊⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠠⠀⠰⠆⠀⠀⠈⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⣤⠂⡇⢢⡀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠒⠒⠊⢨⠀⠂⠂⠂⡅⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⢦⣥⠋⠈⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠄⠀⠀⠀⠂⣀⣰⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠀⠂⣀⡀⠈⡇⠀⡀⠄⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⢀⠀⠠⠂⢀⠜⠁⠀⠀⠀⠀⠀⠉⠀⠒⠀⠠⢤⡠⠤⠀⠀⠀⠉⠈⠁⠀⠀⠀⠈⠱⡎⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⠍⠹⠀⣚⣉⣉⣉⣛⣓⠀⠀⠤⣦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠐⣒⣒⣒⠒⠲⠀⠀⢛⣿⡣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠬⠭⠬⠭⢭⣭⠀⠀⢛⠛⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣘⣭⣭⣭⣉⣙⣒⡒⠦⢬⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢐⣒⣒⣒⣒⡒⠤⠭⣭⣑⣒⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠭⠭⠤⠤⠬⠭⢭⣉⣒⡲⠭⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠉⠉⠉⠙⠛⠒⠲⠦⠭⣍⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠤⠤⠶⠶⠦⠤⢤⣄⣀⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠄⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠿⠿⠷⠿⠭⠥⠤⠤⠤⠤⠤⠤⠤⠤⠬⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡏⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠑⠤⠶⠛⠦⠤⠼⠣⣄⣠⠔⠣⣀⠼⠑⠤⢤⠄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢀⡀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⡈⠀⠀⢠⠀⠀⠈⠁⠀⠀⠘⠀⠀⡟⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠙⠌⡀⠐⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠇⢀⣀⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢰⣖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⡤⠄⠤⠤⠒⠋⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⠀⠀⠀⢀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣸⠀⠀⠀⡼⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡟⠀⠀⣰⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣸⠧⣸⠁⠀⣰⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⡿⡤⠼⠯⢤⠴⠛⠉⡶⢇⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⡇⠙⠚⠋⠉⠉⠉⠉⠀⠀⠀⠉⢉⣒⣒⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣵⡝⠻⠿⠿⠟⢻⡇⣮⠻⠿⠿⠟⢛⣹⣷⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠿⠀⠀⠀⢾⣜⡃⣣⡿⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⡀⠀⢠⢠⣤⣿⣿⣯⣤⡀⡄⠀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣷⡄⠀⠀⠰⣮⣭⣥⠏⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣦⣠⡀⠉⠉⠁⢰⣦⣾⠟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠻⢿⣶⣶⠶⠟⠑⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣾⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠙⢆⠀⢀⣿⣿⣿⣿⡷⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠓⠉⠹⣿⣿⡟⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠋⠉⠙⠛⠃⠀⠀⠀⢰⣿⣿⡇⠀⠀⠿⠿⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣧
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡟
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠋⠀⢀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄
⠀⠀⣴⣿⣿⣿⣿⣿⡟⠁⣠⣴⣿⣿⣷⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠸⣿⣷⣷⣶⣶⣶⣶⣿⣿⣿⣦⣤⣤⣀⡀
⠀⣼⣿⣿⣿⣿⣿⠋⣠⣾⣿⣿⣿⣿⡿⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣹⣿⣿⡟⠛⠛⠛⠻⠿⠷⣤
⢰⣿⣿⣿⣿⣿⣡⣾⣿⣿⠃⣿⣿⣿⠃⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⢀⣿⣿⣿⠁
⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⣿⣿⡏⢀⣼⣿⡟⠁⢀⣠⣤⣄⣴⣦⠀⢠⣤⣌⠉⠀⢸⣿⣿⡇
⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣼⣿⣿⣡⣿⣿⠟⠀⣴⣿⣿⢿⣿⣿⠟⠀⣿⣿⠏⠀⠀⢸⣿⣿⠁
⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠈⢿⣿⡴⢻⣿⡏⠀⠀⠹⣿⣤⣤⠀⢸⣿⣿
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠉⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣤⠈⠛⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⣿⢶⣿⡆⢀⣤⡀⠀⣀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣶⡀⠀⠀⠀⢸⣿⣯⣶⣿⣿⠃⣾⣿⣴⡿⣿⡟⣾⡟⣩⣿⡇
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠛⠁⣿⡏⠀⠈⠿⠋⠀⢿⡇⠈⠿⠟⠙⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⠃⢀⣤⣶⣿⣯⣿⣶⡄⠀⠀⠀⣰⡾⠟⠗⠀⠀⣴⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣾⣿⣿⣿⠟⠉⣿⣿⣿⠀⠀⢰⣿⣷⡿⠆⢀⣠⣿⡟⠁
⠀⠀⠀⠀⠈⠻⠟⠟⠿⠋⠁⠀⢰⣿⣿⣏⣤⣾⡿⠿⣷⡶⠿⠛⠋⠉⠻⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠉⠐⢢⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⢔⠑⢆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠠⢄
⠀⠀⠐⠀⡀⢀⠀⠄⠀⠀⠀⠈⠂⢩⠒⠉⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠂⢄⠀⡇
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠦⡐⣃⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠤⢚⡤⠞⣠⠇
⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⢨⣀⣨⠇⠀⣀⠑⠀⠀⠀⠀⠀⠀⠈⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠠⠄⠂⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠄⠀⠀⠠⠁⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⡐⢢⠈⠄⠀⠀⠀⠀⠐⢂⠀⠀⢀⠔⡀⠌⠂⠀⠀⠀⠈⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⢆⢁⠊⠀⠀⠀⠀⠀⠀⠈⠓⡀⠈⠢⠥⠀⠀⣠⡤⠄⠠⠄⠄⠁⠀⠀⠀⢀⠀⠀⡂⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⠊⠀⠘⠍⠧⡄⠀⠀⠀⠀⢀⡠⢠⡀⡓⠊⠀⠀⠀⠀⠀⠬⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠠⠀⠤⠀⠀⠀⠀⠀⠀⡐⠀⠀⢆⠀⠀⠀⠀⠈⠀⠱⣠⠉⢒⠁⠀⠀⠀⠀⠁⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠂⢀⠄⠀⠀⠀⠀⠊⠐⠲⠚⠃⠀⠀⠀⠀⠀⠀⡨⠃⠉⠀⠀⠀⠀⠀⠐⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⠈⠀⠀⠀⠀⠀⠀⠐⢀⠀⠠⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⡠⠀⠂⠄⠀⠀⠀⠀⠀⠀⢐⢀⡆⠀⠀⠀⠄⠀⠀⠰⠐⡤⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢄⡀
⠀⠀⠀⠀⠀⢀⡐⢑⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢯⣰⠖⡆⢹
⠀⠀⠀⠀⠀⠀⠨⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄⠒⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢉⠕⠋
⠀⠀⠀⠀⠀⠀⠠⡉⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠖⠉⠡⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⢢⢈⠀⠀⠀⠀⠀⠀⣀⡂⠒⠂⢤⡠⠐⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢀⠊⠀⠀⠀⢠⠂⠁⠀⠀⠙⠄⢸⠇⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡆⠀⠀⠀⢧⢀⣀⣀⡀⠀⠈⣩⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠐⠀⠻⠟⠩⠛

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠀⢀⠀⠄⠀⠀⠈⠀⠀⠀⠂⠀⢀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⢤⠞⡶⠨⢡⠃⡞⢁⢀⠂⠀⠲⠀⠒⢀⠐⢀⠠⣄⠀⠀⠀⢀⠁⠌⡀⠀⠂⡌⠀⠀⢀⠈⢀⠐⠂⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠠⠾⢄⣅⠇⢸⡀⡆⠒⡎⢠⡄⠊⠐⠀⢲⠀⡏⠐⢸⠇⠀⠀⠘⠀⠂⠐⡆⡀⠀⠀⠐⣁⠀⠁⠀⠀⠄⠁⠋⢀⠈⡀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣇⣅⢰⠅⠰⣄⠀⠀⠀⣿⡌⠁⠀⠀⢡⠐⡎⠀⠀⠀⠀⣅⠃⠀⠀⡀⢨⠊⠀⠀⠀⠀⢀⠅⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⢮⠐⣯⠀⠀⠀⡿⠀⠀⠀⠀⠀⣧⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡆⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠳⡆⠀⣸⣿⡂⠀⢀⠀⣸⠘⣄⠀⠠⠀⡀⠀⠀⠢⣀⡴⠀⠀⠀⠀⠀⢼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡧⢿⡅⠀⢾⡀⣯⠠⠿⠈⢣⣌⡀⠀⠀⢀⣸⡎⣲⡀⣀⣀⡀⣣⣴⠀⣀⣀⣘⣀⢀⣤⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⣼⡤⡔⠀⠰⠀⢸⠆⡃⠀⠀⠀⣿⢟⢁⣀⡉⣉⢁⣙⣿⢸⣽⣦⣾⣿⣽⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠜⠛⠛⠀⢠⣨⡉⣲⣾⠀⡀⣅⢿⣤⡅⣠⣤⣅⣭⣭⡅⢨⣿⣿⣷⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡴⠤⠽⠉⠁⠂⢀⠰⠴⠶⠀⠀⢹⠈⠈⡗⠳⠖⠩⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠂⠐⠈⠓⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠐⠠⢉⡋⣍⡑⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠩⠶⠶⠤⠀⠀⠈⠀⠁⠀⠈⠲⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠀⠐⠒⠂⠀⠀⠀⠀⠀⡀⡀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠛⠋⠠⡄⢆⠀⠀⠀⡀⢀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠒⠀⣠⣤⣐⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠈⠱⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠂⠈⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⢀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⢀⢑⠀⠄⠀⠂⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⢇⠠⠃⠀⠀⡀⣶⡽⢿⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢠⠤⣦⠀⠈⠀⠀⢰⡾⠷⠃⠊⠁⠸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠇⠀⠀⡸⢍⠿⢏⡸⢿⣙⡾⠠⠀⠀⠈⠂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⠀⣔⠀⠀⠱⡜⢣⣼⠛⠀⠀⢀⡤⠀⠄⢢⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠄⠤⡠⠂⠘⡛⠀⠀⠈⢧⠌⠀⠀⢺⠧⠀⠀⠀⣤⡁⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡨⠦⡀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠄⠡⠆⠇⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢰⣈⣐⡠⠤⠀⠀⠀⠀⠉⣋⣠⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠂⠀⠀⠀⢈⡉⠉⠉⠉⣉⣉⣉⣉⡉⠁⠀⠀⠀⠐⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠢⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⠿⢿⣿⠿⣿⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⢙⡁⣤⣶⣾⣾⣷⣶⣬⡄⠀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠈⢉⡽⠟⠛⠉⠁

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣦⣾⣿⣿⣯⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠉⠛⣻⣿⣿⣿⠿⠿⠿⠿⠿⠿⠷⠶⠤
⠀⠀⠀⣠⣾⣿⣿⠋⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠟⠀⠀⣰⣿⣿⣿⠏
⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⢀⣤⣄⠀⠀⠀⣰⣿⣿⣿⠇
⠀⣸⣿⣿⡿⠀⠀⣠⣴⣿⣶⠀⠀⣼⣿⣿⡇⠀⣠⣶⣶⣶⣶⣾⣿⣿⠃⢠⣿⣿⡿⠁⠀⢠⣿⣿⣿⠏
⢠⣿⣿⣿⠃⣠⣾⣿⣿⣿⣿⠀⢰⣿⣿⣿⢁⣾⣿⡿⠋⢁⣾⣿⣿⠇⢀⣿⣿⣿⠃⠀⠀⢸⣿⣿⡟
⣼⣿⣿⣯⣾⣿⣿⢿⣿⣿⡏⢀⣿⣿⣿⠇⢸⣿⣿⣁⣴⣿⣿⣿⡟⠀⢸⣿⣿⡟⠀⠀⠀⢿⣿⣿⠇
⣿⣿⣿⣿⣿⡟⠁⢸⣿⣿⡇⣼⣿⣿⡏⠀⠈⠻⠿⠿⠋⢸⣿⣿⠃⠀⠸⣿⣿⠃⠀⠀⠀⢸⣿⣿⡀
⣿⣿⣿⣿⠋⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣿⣷⣀
⠙⠿⠿⠃⠀⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣀⣤⣶⣶⣷⣶⣤⡀⠀⣀⣀⠀⠀⣀⠀⠀⠉⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⢰⣿⣿⠛⣉⣥⣶⣦⢀⣾⣿⣿⣷⣼⣿⠀⣼⣿⠛⠛⠛⢲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣷⣤⡀⠀⠀⠈⠻⠿⠟⠋⣽⣿⡏⠸⣿⠟⠸⣿⣿⠇⢰⣿⣿⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠉⠀⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠷⠖⢢⣿⣿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⣿⣿⣷⣶⣤⠄⠀⣾⣿⣿⡿⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣛⣛⣿⣿⣿⣿⣿⣦⡀⠹⣿⣇⣁⣀⠀⠀⠹⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡒⠉⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⠙⠀⠈⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠤⠀⡀⠀⠀⠀⢀⢫⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠐⣋⣾⡭⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⣸⢿⣧⠞⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⠀⣴⠟⠶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⠉⢉⠠⠸⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠃⠀⠀⠀⠀⢀⠀⠤⠀⢀⠄⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠄⠀⠄⡀⠀⠀⡌⠀⠀⠀⠀⠀⡄⠀⠂⠀⡈⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡶⠡⣄⣉⡰⢴⠁⠀⠀⠤⢀⣀⠁⢠⠀⠠⠀⠀⠀⠠⠰
⠀⠀⠀⠀⡀⠀⠄⠒⠀⠀⠀⠀⠀⢀⠀⠀⠁⠉⠙⠓⠐⠚⠤⢼⡠⣪⠀⡠⢀⡈⢳⠊⠀⠀⠀⠀⠀⢀⠃
⢀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⣚⡡⠎⠄⠉⠉⠉⠀⢄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⢎⠋⣋⠆⠀⠀⠀⠐⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠂⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠵⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⢝
⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⠤⠤⠚⠀⠀⡀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠇⠀⢀⠀⠀⠀⠀⡀⣃
⠀⠀⠀⢰⠀⠣⡀⠀⠀⠀⠀⢀⡠⠤⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⠁
⠀⠀⠀⠄⢠⠀⠈⠂⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠂⠄⠤⢴⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠈⠂⣌⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠈⠑⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠁⠂⢄⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠄⠤⢀⠀⠀⠑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠂⠠⠒⠢⠀⢂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢄⠐⠤⠔⢀⠌⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠑⠒⠒⠁⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠈⠂⠄⣀⠀⢀⡀⠄⢎⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠑⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣛⢦⣄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⣈⠉⠁⠁⠉⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⡔
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡓⠀⠂⠀⠀⠀⠀⠀⠀⠂⠂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠣⠃⠀⠁⠍⠌⠩⡙⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⡐⠀⠀⡦⠀⢂⢄⠄⡂⡡⢂⢆⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠐⠁⠀⠐⠋
⠀⠀⠀⠀⠀⡰⡠⠙⠋⠉⠈⠌⠂⠁⠒⢦⡁⠀⠁⠀⠀⠀⡠⠁⠀⠀⡠⠤⢄⠀⠀⠙⠄
⠀⠀⠀⠀⠰⠠⠁⠀⠀⠺⠂⠈⠌⢆⠀⠀⠈⢢⡀⠈⠢⣐⠀⠀⠀⠀⢄⠀⠀⠃⠀⠀⠈⡄
⠀⠀⠀⠠⠡⠁⠀⠴⠀⠀⠀⠀⠘⠌⠄⠀⠀⠀⠈⢦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⢡⠃⠀⠀⠀⠀⠀⡀⠀⠀⠘⡘⡆⠀⠀⠀⠀⠈⢾⠀⠀⣀⣀⢠⢄⣀⠤⠤⠔⠒⠛⠉⠘⠃
⠀⢠⢁⠃⠀⡀⠀⠀⠀⠢⡠⠀⠀⠀⠐⡘⣤⠐⠢⠄⠐⠛⠋⠉⠁⠈
⠀⢃⠃⠀⠀⠊⠀⠀⠀⠀⠀⠀⢈⣈⠀⠐⡐⡀
⣆⣘⣂⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣴⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠒⠂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠷⠶⢄⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⣶⣿⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠉⠐⠀⠠⠧
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠉⢉⠙⠛⠛⠲⢶⣦⣤⣄⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⠟⣿⣿⣿⣿⣿⣿⡟⢉⠤⢦⠘⣿⣿⣿⡟⠀⠀⠀⠀⠠⠞⠛⢿⣶⣤⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⡔⢠⡄⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢤⣠⡄⠈⠙⡿⠿⠿⠿⠿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢿⡿⠂⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠷⣶⣄⣿⣿⣿⣿⣿⣿⡆⠈⠃⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠟⠲⣶⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣤⣤⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡛⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠁⠉⠛⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣻⣿⡂⠀⠀⠀⡰⠋⢁⣠⣤⣄⠈⠻⡄
⠀⠀⠀⠀⡜⠁⠀⡤⠶⠶⣄⠀⠀⢸⠀⠀⠀⠀⠀⢸⢿⡟⠁⠀⢹⣩⣹⠃⠀⠀⠀⠁⠠⡃⠀⠀⠈⠇⠀⠘
⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⣸⠀⠀⡸⠀⠀⠀⠀⠀⠸⣾⣷⢀⣴⣿⡿⠋⠀⠀⠀⠀⢦⣤⠵⠄⢀⣀⠞⠀⢠
⠀⠀⠀⠀⠙⠚⠁⣤⠖⠚⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢡⡻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢉⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢰⠚⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⠇⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⢸⠉⠁⠀⣀⣤⣄⡀
⠀⠰⠊⠉⠑⠄⠀⡇⢸⡀⠀⠀⠀⠀⠘⠉⠙⢧⠀⠀⠀⢹⣿⠦⠀⠀⠀⠀⠯⠈⠙⠄⠀⠸⢬⠆⠀⠸⠅⠀⠀⢙
⠀⠀⠀⠀⣠⠆⠀⠘⣒⣃⠀⠀⠀⠀⢀⡤⢤⡾⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⣀⠄⠀⠰⣋⠀⠀⠀⠀⢀⡤⠎
⠀⠀⠐⠉⠁⠀⠀⠀⣁⣨⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢈⣛⣁⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠘⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠰⠀⠀⠀⣠⠶⠛⠉⠉⠉⠛⠗⢦⡀⠀⠁⠀⣀⣤⣶⣶⣶⣶⣦⣄⠉
⠀⠀⠀⠀⢠⠞⠋⠉⠉⠉⠛⢷⠀⠀⠀⠀⠀⢸⠁⢠⣾⠛⠲⣤⡀⠀⠀⠁⠀⠀⣼⡿⡟⠛⠛⠛⢿⣏⠙⡆
⠀⠀⠀⠀⢫⣀⡴⠚⠛⢆⠀⢈⡁⠀⠀⠀⠀⠘⢤⣠⡜⠀⠀⠀⡇⠀⢀⡖⠀⠀⢼⣟⣿⠀⠀⠀⢸⣿⣴⠃
⠀⠀⣀⣄⡀⠀⠀⠀⢀⠎⣰⠟⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠉⠁⠀⠀⣴⣿⠟⠁⢀⣀⡀
⠀⢨⣀⠀⠙⠀⠀⠀⡞⢸⠁⠀⠀⠀⠉⠉⣻⠀⠀⠀⢀⠴⢋⠔⠊⠁⠀⠀⠊⠙⠢⠀⠀⠀⢸⡟⠁⠀⠀⡁⠀⠈⡆
⠀⠀⢀⡤⠖⠀⠀⠀⢻⡞⠀⠀⠀⠀⢸⠉⠉⠀⠀⠀⠘⢶⠃⠀⠀⠀⠀⠈⢁⠤⠎⠀⠀⠀⣘⡁⠀⠀⠀⠀⠊⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠸⣌⠇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡠⠖⢀⣀⣀⠒⠒⣩⠟⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⠁⠠⠤⢄⡀⠐⢚⣋⣁⣐⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠄⠀⠀⠀⠀⠐⠲⢄⠀⠩⢭⣭⣭⣧⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠇⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠲⣖⠚⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣤⠴⠆⡀⠀⠐⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠠⡄⢪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⣀⠀⢻⠀⠀⠀⠀⠀⠀⠲⣄⠈⢦⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⣿⡟⠁⠀⢸⡀⠀⢠⡀⠈⢷⡀⠙⣷⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠸⣷⡀⠀⠀⢣⠀⠀⠻⣄⠀⢻⣆⠘⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠙⣷⣄⠀⠀⠳⣄⠀⠹⣧⡀⢻⣧⠘⣿⡌⡇
⠀⠀⠀⠀⠀⠐⠲⠿⠿⠿⠶⣶⣶⣶⣶⠶⢤⣤⣀⡀⠀⠀⠹⣆⠱⡌⠻⣷⣄⡀⠈⠑⠄⢹⣷⡈⣿⡆⢸⡇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣯⣽⣛⣶⡶⠤⣤⣹⣆⣈⣛⠿⣶⣦⣤⣀⣻⡇⢸⡇⠈⠁⢀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⣿⣿⣿⣿⣾⣿⣷⣿⢾⢸⣱⣧⡇⣿⣇⣴⣴⣤⣷⣼⠷⠼⠿⠶⠦⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠙⠉⠛⠛⢿⡿⣿⣿⣿⣿⣷⣶⣶⣤⣀⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣛⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⢇⠈⠻⡉⠛⢻⡿⣿⣿⣿⣿⡶⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⣋⡽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠀⠘⠀⠈⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⡀⠤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢰⠃




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠤⠂⠉⠉⠉⠐⠢⢄
⡌⠀⠀⠄⠒⠠⢀⠀⠀⠑⡀
⢇⠀⠸⠀⠀⠀⠀⢡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠌⠀⢀⠃
⠀⠀⠀⠀⡀⠊⠀⠀⡠⠊
⠀⠀⠀⠘⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡙⢄⠀⠀⡔⠄⠀⢀
⠀⠀⠀⢠⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠄⠁⠀⠀⠁⠀⠀⠀⠃⠈⠰⠁⢆⠰⠄
⠀⠀⠀⠈⢄⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠨⠀⠢⠀⠀⠀⠀⠀⠀⠀⡠⡶⢠⠄⠀⠤⢀⠀⠀⠀⢀⡀⠠⢤⡀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢰⡄⠂⠀⠂⠀⠀⠐⠂⠐⠠⠀⠀⠀⠀⡠⢤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣸⢠⣖⠒⠢⡀⠀⢠⣄⠠⢄⠀⠀⠐⠎⡠⠂⡆⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⢄⡀⡁⢻⠘⠥⠀⠔⠇⠀⠫⣉⣀⠜⠀⠀⠀⠀⠑⠀⠇⣠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⢀⠈⠙⢸⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠸⠀⠀⡀⠀⠀⠁⠄⠤⠀⠀⠀⠀⠀⠀⠀⢘⡁⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠁⠀⠀⢃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⢠⠁⠐⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠠⡀⣀⠀⠤⠀⠈⠀⠀⠀⠀⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⠐⠀⠤⠄⠤⢴⠀⠠⡀⠀⠀⠀⠀⠀⡠⠂⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⡰⠤⡒⠉⠀⠀⠀⢀⠄⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠐⡠⠚⢠⢀⡘⢆⠀⠠⠂⠁⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠂⠀⠀⠒⠀⠀⢀⠂⠑⢄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠊⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠚⠀⠢⡀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⡀⠀⠀⠇⠀⠀⠀⢈⠀⠈⢢⣤⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡘⠀⠀⠀⠀⠀⠂⠄⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠄⢤⠀⠀⢠⠀⠀⠀⠀⠀⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⢶⣤⣤⣀⡀⠀⠀⢀⣠⣤⣴⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠀⣀⣀⠀⠈⠉⠛⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⢀⣴⠾⠛⠃⠈⠉⠙⠛⠷⠆⣰⠟⠁⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⣴⠟⠁⢀⣤⣴⠶⠶⢶⣤⡀⢰⡟⢀⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣾⠃⠀⣴⠟⠉⠀⣀⣀⡀⠈⠁⣾⡇⠈⢿⡄⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡏⠀⢰⡟⠋⠉⠻⣦⠀⢹⡇⠀⠈⣿⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣸⡇⠀⠀⠀⠙⠀⠈⢿⡄⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠃⠀⢸⡇⠀⢸⡇⢰⡆⠀⣿⠀⠀⠈⠻⣦⡀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡿⠁⠀⣼⡇⢸⡇⠀⠛⠀⠀⣿⠀⠈⣿⡷⣦⣤⣤⣤⣤⣴⠾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠾⠋⠀⠀⢠⡿⠀⠸⠇⠀⠀⠀⣼⠏⠀⠀⣿⡇⠀⠀⠉⠉⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢀⣴⠟⠁⠀⢀⣀⣠⣴⠾⠋⠀⣠⡆⠘⠛⣿⠟⠛⠛⠻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⢀⣴⠟⠁⠀⢀⣴⠟⠉⠉⠀⣀⣠⡾⠋⠀⣠⡄⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠟⠁⠐⠶⠾⠛⠋⠁⠀⣠⡾⠛⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣴⠟⠁⣀⣀⣀⣀⣠⣤⣤⠶⠟⠉⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢁⣀⡀⠉⠉⠉⠉⠉⠀⠀⣀⣠⣴⠾⠃⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠻⣦⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⠤⠤⠠⠄⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠠⠤⠤⠄⠤⠄⠤⠤⠤⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠀⠓⠀⠀⠰⠘⠃⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠃⠙⠀⠑⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⢀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢠⡏⠀⠀⣾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠸⣧⠂⠀⣿⢠⡏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⠐⣇⢀⣤⠰⠉⣶⢸⡇⢸⡇⠀⣸⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠘⢸⡇⠀⢰⡟⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠶⠙⠄⠀⠘⢦⠄⠈⠧⠘⠇⢼⡇⠰⣏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡏⠀⠀⠀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⣿⠀⢸⠇⢀⡀⠀⣰⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣿⠀⢹⡅⢰⡇⠀⣿⠀⠀⢠⡏⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢻⡆⢸⡇⠸⡇⠀⠻⠄⠀⣾⠃⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠧⠀⠓⠀⠛⠀⠀⡀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣀⣼⠃⢀⡤⣀⠀⣴⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⠹⡄⣾⢀⡟⢀⣿⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⠜⢿⠐⠀⠙⠈⠳⠈⠻⠌⢸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠠⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠀⠉⠉⠁⠒⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢁⠂⠀⠀⢀⣀⣀⡀⠀⠀⠀⢈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠁⣾⢉⡤⣬⡹⠀⠀⠀⠀⢀⡷⠤
⠀⠀⠀⡠⠐⠈⠨⠁⠉⠉⠀⠉⠈⠐⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠓⠉⠉⠉⠀⠀⠀⠀⠀⣠⣾⡗⠛⠂
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⣠⡤⣤⡤⠤⠒⣡⡞⠁
⡼⠀⠀⠀⠀⠀⠀⢠⠖⠒⢀⠒⠤⡀⠀⠀⠀⠀⠈⢦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣐⡓⠂⡤⣂⡴⢡⣴⡴⠶⠶⠦⠟⠉⠈⠀⢀⡀⠄⠐⠒⠒⠒⠠⠤⣀
⡇⠀⠀⠀⠀⠀⠀⠹⡖⠚⠶⣥⣄⠈⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡠⠂⠁⠀⠈⢁⢳⣴⣴⠯⠁⠀⠀⠀⠀⠀⢀⡔⢪⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢈⡿⠉⠀⠀⠙⡆⠀⠀⠀⠀⢠⣿⠄⠀⠀⠀⠈⠓⠤⢀⡀⠀⣠⢿⡯⡅⠁⠀⠀⠀⠀⠀⠀⠀⡈⠐⠤⣀⠠⠤⠆⢤⠀⠀⠀⠀⠨⠈⡇
⢀⠀⠢⡢⡤⠐⡀⠖⠉⡟⠀⠀⠀⠐⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠐⠄⣀⠀⢈⠙⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠑⢢⣖⣤⣾⣺⠶⠁⠀⠀⠀⠀⢠⡯⠉
⠈⠢⠀⠙⢢⡊⠁⣈⡼⠃⠈⠁⠀⠙⢄⠀⠀⠀⠀⠨⠻⣤⠀⠀⠀⠀⠀⠀⠑⠒⠼⠛⠂⠀⠀⠀⠀⠀⠀⣠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠇
⠀⠀⠑⢄⣽⡴⠞⠋⠐⠀⠀⠀⠀⢠⠀⠑⢄⠀⠀⠀⣡⢺⣇⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢌⠲⠀⠀⠀⢀⡤⠴⠖⠢⠤⠤⠤⠤⠰⠚⢉⣴⠏
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠉⢶⢋⡵⠛⠁⠀⠈⠑⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂⠁⠉⠢⣈⡒⢤⠞⢅⣴⣥⡶⢶⣶⣦⣤⣤⡶⠾⠯⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠟⣻⣼⠢⡀⣀⡀⠀⣀⠴⣿⡁⠀⠀⠀⢰⠯⡀⠀⡀⠀⠀⢠⣿⣼⣾⡫⠉⠉⠉⠀⠀⠀⠐⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡀⠈⠢⡤⢞⠝⣸⠇⠀⠀⠀⠀⠰⡀⠈⠑⠢⣡⡞⣵⣿⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣋⣼⣞⣓⠒⠀⠀⠀⠀⠀⠈⠑⠀⣀⣮⠼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢰⠇⣠⠁⠀⠃⢈⣀⣀⡀⠀⠛⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢰⠄⣿⡴⠁⢀⣤⡄⠀⠀⢠⠎⣩⣟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⢠⡏⢀⡤⢺⡋⡄⠀⠀⡟⢠⠻⡶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠦⠤⠷⠋⣿⠻⡄⠈⠉⠀⠀⠀⠓⠊⠸⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠠⣴⠠⡞⣦⢾⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⠃⣴⢛⡆⠀⠀⠂⠘⠀⠈⠁⢀⣤⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣇⣜⡀⠀⢈⡴⡄⢠⣄⡀⢸⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠈⣿⡿⢻⡇⠀⣿⢳⡗⢻⠏⠧⢼⡇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡼⠀⣉⠁⢸⡇⢰⢉⣩⣤⣬⣶⣶⠘⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠐⠁⠘⠧⠏⠈⠻⢿⣿⡿⠏⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠏⣳⠀⠀⣶⠀⠀⡶⢠⠶⣄⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠻⠤⠟⠀⠀⣿⣰⣆⡇⣾⠉⢻⡌⠧⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠒⠊⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣯⠥⠤⠤⠤⠬⣭⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⠀⠀⢀⠀⠀⠑⢮⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠢⣀⣀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢲⡜⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣭⣽⠆⢀⠞⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣇⡀⠉⠒⠤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠋⡏⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠒⡄⢠⠃⠀⠈⠀⠀⢀⠀⠀⠀⣀⡤⠚⠀⠀⢧⠀⢀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⢠⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠤⣀⣀⣀⣀⣨⡏⠁⠀⠁⠀⡰⣪⠻⠿⡒⠉
⠀⠀⠀⠀⠀⠀⣀⠷⠤⣀⠈⢱⢒⠆⠀⠀⠀⠀⠐⢤⢳⠐⢇⢀⠵⡴⢁⠝⢦⠀⣠⠊⢳⢍⡒⢰⠇
⠀⠀⠀⠀⠪⣉⣁⠤⠤⡀⠹⡁⡜⠀⠀⠀⠀⠀⠀⠰⠀⢣⡌⠈⡖⢰⠁⠀⠀⠋⠀⠣⣈⣦⢬⠃
⠀⠀⠀⠀⠀⠑⠊⠀⠀⠉⢒⠟⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⢣⣜⡠⠚⠀⡄⠀⠀⠀⠀⡨⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠀⠀⣼⣀⠀⢀⡤⠊
⡦⠤⣀⣀⣀⠤⠔⠒⢉⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡜⠀⠀⠉⠁
⠘⢦⠀⠀⢀⣠⡠⠔⠁⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇
⠀⠛⠒⠋⠁⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢄⣀⣀⣀⡀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠁⠙⢢⠀⠀⠈⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣺⣭⠿⠷⠶⠶⠾⠯⣿⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡫⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⠔⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠘⣯⢧
⠀⠀⠀⠀⠀⠀⢀⡴⣞⣯⣾⣷⠶⠶⠶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣶⣄⢸⡾⡆
⠀⠀⠀⠀⠀⢀⣏⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⡁⠀⠀⢻⡿⢻⣿⢣⠇
⠀⠀⠀⠀⠀⠘⣎⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠛⠀⠀⠈⣿⡾⣿⣘⣿⡲⣄
⠀⠀⠀⠀⠀⠀⠈⠑⠲⡯⣿⠀⠀⠀⠀⠀⠀⠀⠒⠛⢡⣿⡅⠀⠀⠀⠀⢸⣿⡿⡡⢀⢹⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⡇⣴⣿⣿⡶⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⣿⡿⠈⣼⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⡇⠈⠁⢸⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⠃⠀⣼⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⣆⡇⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡼⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡇⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣷⠀⠀⠀⡤⠘⠃⠂⠐⠛⠉⢙⠢⡀⠀⠀⠀⢀⡀⣷⣬⣭⣭⣟⢦
⠀⠀⠀⠀⣠⢖⣻⣭⣽⣗⢿⢻⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢆⠙⣤⠄⢲⡀⠠⠤⠤⠄⠀⣿⣸
⠀⠀⠀⡼⡷⢩⠋⠀⠂⡍⢷⡤⡽⣦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢦⠈⠓⠸⡄⠀⠀⠀⠀⢠⡟⣿
⠀⠀⠐⣷⡇⢠⠀⠀⠀⠐⢸⡇⡿⢮⡳⣤⡀⠀⠀⠀⠘⢧⠀⠀⢘⠳⣄⡀⠙⠀⠀⠀⢀⠞⢳⡞⡆
⠀⠀⠀⢳⡳⣄⠑⢦⠶⣣⡾⣳⠃⣰⢣⣿⣿⣶⣤⣀⣀⣀⣳⡀⠈⣆⠀⠉⠐⠛⠓⠛⠁⡴⠻⣞⢧
⠀⠀⠀⠀⣹⢞⣹⡧⣾⢻⣜⡇⠀⠈⠓⠮⣽⢛⣿⣿⣿⣿⡏⡁⠀⣏⠁⠀⠀⠀⠀⠀⢀⡆⠀⠘⣯⢣
⠀⠀⠀⢸⡷⣏⣡⡕⢸⠈⢧⡳⣤⢤⣖⣫⣶⣿⣿⣿⣿⣿⣟⣇⠀⠹⡦⠀⠀⠀⠀⠀⢨⢳⡀⠀⠘⣯⢧
⠀⠀⠀⠐⡗⣯⡤⢮⠸⠀⠀⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣏⠠⠀⠙⠦⣄⡀⠀⠀⠀⠀⡅⠀⠀⢸⡎⡆
⠀⠀⠀⠀⢯⢻⡄⢛⣶⠀⠀⠈⠉⠛⠟⠀⠈⠛⠻⠿⠛⠉⢰⠘⢄⣀⡠⠀⡀⠉⠉⠒⠒⠶⡆⠀⠀⣼⣿⣸⡀
⠀⠀⠀⠀⠀⠳⣽⣟⢻⠁⠙⠺⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⡼⠃⠀⡰⢻⣿⣧⢧
⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⡜⠀⠀⠀⠀⢠⢞⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠁⠒⠛⠀⠖⠋⠀⣿⣿⣿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢯⡳⠾⠶⣶⣶⣶⠒⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⢟⣾⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠴⣺⣵⣾⣿⣿⡟⣠⠁⠀⠀⠋⠀⠀⠀⠀⠀⠠⡀⠀⠀⠄⠀⠀⠤⠆⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣎⢿⣿⣿⡿⢟⣉⣏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⣿⣿⣿⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⠭⠭⠖⠉⠳⣝⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣿⣿⣿⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠯⣽⣟⣲⡶⠦⣤⣤⣬⣥⣤⣴⣦⣭⣤⠤⢶⡖⣿⣿⡿⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠒⠒⠒⠛⠛⠋⠉⠙⢾⣫⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣶⣶⠀⢠⣿⣿⡏⠀⢀⣾⣿⣿⠀⠀⠀⣼⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣼⣿⣿⠀⣾⣿⡟⠀⢀⣾⣿⣿⣿⠀⠀⢀⣿⣿⡏⠀⠉⠉⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣼⣿⣿⣿⣸⣿⡿⠁⠀⣼⣿⡟⣿⣿⠀⠀⢸⣿⣿⠃⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⠃⠀⣸⣿⡿⢠⣿⣿⠀⠀⣾⣿⡿⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣿⣿⣿⠇⠀⣰⣿⣿⣿⣾⣿⣿⠀⢠⣿⣿⡇⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⣿⣿⡟⠀⢠⣿⣿⡟⠉⢹⣿⣿⠀⢸⣿⣿⠁⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣉⣉⣁⠀⠉⠉⢉⣀⣀⠈⠉⢉⣀⣉⠙⠋⢀⣀⣀⣙⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⣼⣿⣿⡇⠀⣼⣿⣿⠀⢀⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣷⠀⣿⣿⡇⠀⢸⣿⣿⠁⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⣿⣿⠀⠀⢸⣿⡿⣿⣿⢰⣿⣿⠁⠀⣾⣿⡏⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⢰⣿⣿⠀⠀⣿⣿⡇⢿⣿⣾⣿⡟⠀⢠⣿⣿⡇⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⠃⢸⣿⣿⣿⠇⠀⢸⣿⣿⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⢹⣿⣿⠀⣼⣿⣿⠀⠸⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣬⡉⠁⠀⣠⣬⣭⣤⣤⡄⠈⣭⣭⣭⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠻⣿⣿⠀⠀⣿⣿⡿⠿⠿⠃⢠⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠘⠛⠛⠀⢰⣿⣿⠁⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄⠀⠀⣼⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⢹⣿⣿⡄⢀⣿⣿⡏⠉⠁⠀⢸⣿⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣸⣿⣿⠇⢸⣿⣿⣤⣤⡄⠀⣼⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⠟⠀⢾⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢣
⠀⠀⠔⠁⣀⠒⠌⠄⡀⠀⠀⠀⠀⠀⠀⠠⠀⢂⣀⠀⠢⡄⠀⠀⡠⢐⠄⢠⡔⡩⠥⢄⠢⣴⡯⠖⠒⠣⣧
⠀⠀⡋⣴⠏⢠⣔⣮⡇⠀⠀⠀⠀⠀⠀⢥⡼⠋⠉⠙⣦⡕⠀⣌⡜⠁⠀⢸⢸⠀⠀⠀⣣⡟⠀⠀⠀⠀⢻
⠀⠀⠈⢫⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠾⢂⠀⡄⡰⠾⠀⠰⢅⠀⠀⠀⣼⠃⠀⠀⠀⣿⣦⠀⠀⠀⢀⠎
⠀⢀⡴⢛⠛⠯⡅⣻⡀⠀⠀⠀⠀⠀⠀⢀⡤⠵⠺⢥⣀⠀⠀⠀⠐⠄⢀⠷⢿⡿⠿⡀⡩⢠⣀⠒⢲⠁
⠀⣪⣋⠀⢠⠂⠋⠠⠳⡀⠀⠀⠀⡠⢔⡏⠀⡧⠀⢸⠄⠑⢢⡤⠚⠉⠉⠙⢢⠤⠚⠛⡴⠛⠓⠎⠴⠋⠐⢄
⡞⠁⢈⡟⠅⠀⠉⠀⠀⠳⡀⠀⣎⠔⠁⠀⠁⡁⠀⢸⠀⠀⠚⠀⠀⠀⠀⡜⢃⣀⡄⢩⡃⠀⠒⠂⠢⠴⡄⡘⡄
⠁⠠⡟⠂⠀⢀⠲⠃⠄⡠⡕⢚⡒⠳⣄⣤⠀⠀⠀⢸⣰⠧⠀⠀⠀⠇⠀⠐⠄⡐⢾⢂⠉⠶⠾⠛⠶⠒⠶⠶⠃
⠀⢠⠄⠀⠀⢄⠀⠐⠂⠀⡇⢀⣀⠹⢘⣽⠀⠀⠀⡽⠀⠰⠒⠉⠉⠀⠀⠀⠀⠈⡦⡘⡆⢠⡤⡀⢠⠄⡄
⠀⠸⢤⣄⣠⣁⣀⡀⡀⠀⡰⣁⡉⠔⠋⡇⠀⠀⠀⢱⠀⠐⠀⠶⠶⠃⠀⠀⢀⣠⠇⢸⠇⠸⠀⠀⠀⠀⢠
⠀⠀⢸⠀⠈⠉⠉⢉⠏⣽⡘⠈⠣⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠈⡟⠛⠋⢉⠍⢱⠀⢸⠀⠀⠀⠀⠆⠀⢸
⠀⠀⠨⡀⠈⠀⠀⠈⢸⣃⠑⣦⢎⡇⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⢇⠀⠀⢈⠀⡄⠀⡏⠀⢠⠀⠀⢠⠀⢸
⠀⠀⠀⡇⠀⠀⠀⡄⠈⣬⡙⠂⡏⠀⢸⠀⠀⠀⠀⠄⡆⠀⠀⠀⢸⠀⠀⢸⠀⡇⢠⠃⠀⢸⠀⠀⢸⠀⢸
⠀⠀⠀⡄⠀⠀⠀⡇⠀⠘⠛⠋⡅⠀⠸⣀⣀⣀⣀⣀⠇⠀⠀⠀⢸⠀⠀⢸⠀⡇⠘⠄⣀⣸⠀⠀⢸⠀⢸⡇
⠀⠀⠀⡥⠴⠦⠴⠧⣀⢀⠀⡀⣇⠀⠀⣠⡋⣅⣏⡽⢄⡀⠀⢀⠐⠤⣤⣼⡔⠃⠀⣀⣨⠙⠀⠀⢸
⠀⠘⠛⠓⠲⠶⠤⠔⠚⠭⠘⠓⠐⠛⠯⠖⠒⠉⠉⠀⠀⠒⠛⠓⠒⠚⠉⠉⠉⠉⢻⣟⡹⠚⠀⠀⣺⠦⠐⠂⠠