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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢈⠓⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠁⠐⢄⠈⠂⠀⠀⠀⠀⠀⠀⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢐⣾⡇⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⣠⠤⠚⠉⢿⠿⣶⣄⡀⠀⠀⣰⣧⡁⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢈⡆⠀⠀⠀⠀⢀⠃⠘⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢸⣁⡀⠀⠀⡰⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠀⢸⣧⣀⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡏⠛⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⠀⠈⡻⠛⠙⠛⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠊⠀⠀⠀⠀⠀⠈⠉⠻⣷⡄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠔
⠀⠀⠀⠀⠀⠀⡴⠒⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⡎
⠀⠀⠀⠀⠀⠀⢳⡄⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⡄
⠂⠒⠲⠶⠶⠿⠿⠿⠷⠗⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤

⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⣤⠾⣿⣷⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⡄⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⠉⠀⠀⠀⠀⢹⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢸⡟⠃⠀⠀⠀⠀⠀⣤⠙⢻⡇⠀⠀⠀⠀⢸⡏⢈⣦⠀⠀⠀⠀⢀⣼⠁⠺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡾⠉⠀⢸⠃⠀⠀⠀⠀⠾⠇⠀⠀⢠⡶⠂⠘⠛⠉⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢀⣼⡗⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⣴⢾⣿⣿⣴⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⠳⠀⠀⠀⠀⣧⢨⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⠂⠀⠀⠀⢀⠀⢸⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⡙⠻⣦⠀⠀⠀⢀⣠⣾⠟⠿⣦⠀⠀⠘⠛⢧⣽⣿⡆⠀⠀⠘⠛⠳⠼⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠀⠀⡒⠛⠉⠀⠀⢘⠃⠀⠀⠀⠀⠀⠉⠘⠃⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⣴⢻⣿⣤⠖⠀⠀⠀⠰⡿⣿⣆⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⡁⠀⠀⠀⠀⠀⢸⣾⣿⡀⠀⠀⠀⠀⠀⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠛⣿⠄⠀⠀⠀⡴⢿⣿⡇⠀⠀⠀⠀⢀⣼⣟⣩⠿⠆⠀⠀⢀⣴⠟⠋⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⢀⣿⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⣴⠛⠁⠻⠀⠀⠀⢀⡴⠋⠁⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠂⠀⠀⠀⠈⢛⡀⠀⠀⠀⠀⠙⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⢀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢘⡁⡀⠀⠀⠀⠀⠀⠀⣰⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣿⠇⠄⠀⠀⠀⠠⡟⠀⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢰⡆⢹⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣸⣿⡀⠀⠀⠀⠀⠀⢁⣼⣿⠇⠀⠀⠀⠀⠀⣠⣸⣿⡀⠀⠀⠀⠀⠀⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⠀⠀⠀⠀⠀⠛⢿⡟⠀⠀⠀⠀⢀⣠⡿⠋⠛⣷⠀⠀⠀⢀⣴⠟⠋⢉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⠁⠀⠀⠀⠛⠀⠀⠀⠚⠁⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠛⠉⠉⠙⠟⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡀⠀⣠⣶⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⢠⣄⣄⠀⠀⠀⠀⠉⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠘⠛⠿⢶⣤⣤⣤⣴⣿⡿⠋⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠻⢿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣛⡉⠀⠀⠀⠀⠙⠿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⢿⣷⣶⣤⣤⣀⣄⠈⠙⠛⠻⠿⠿⣷⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢤⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡻⠿⠛
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⣿⠻⠟⠁⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⢰⣄⣠⣾
⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣾⠃⠀⠀⢠⣷⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⣾⣿⣿⡟⠘⠻⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠐⣿⣿⣿⣅⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⢦⣄⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣮⡝⠻⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣾⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢃⣾⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠋⠁⠈⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⡀⠡⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠽⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠀⡀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⠠⠊⠉⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠁⠀⠎⠀⠀⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣠⣄⡀⢠⠎⠁⠁⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠊⡏⢂⠀⠀⠀⠀⢸⢾⠀⠀⠀⠀⢉⠟⡿⣷⢤⣄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠘⠾⣭⣬⡑⠂⠠⠀⠆⠼⣇⠀⠀⠀⠀⠀⠀⢹⡤⠃⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠀⢠⠛⡄⠀⠀⠀⠀⠀⠸⢁⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⢱⠀⠀⡀⠀⠊⣰⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⣥⣀⣾⣃⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠁⠈⠉⠁⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠠⠀⠀⠀⠀⠀⣠⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠦⡀⠀⠀⢀⣤⠞⠁⠠⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠉⣷⣦⠿⣧⡠⠔⠉⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢀⠔⠋⠀⠀⠀⠉⢷⡀⠀⠀⠘⣧⠔⠒⠀⠐⠒⠒⠚⡍⠈⢱⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⣀⣀⡀⠤⠤⠀⠓⠀⠘⠒⢤⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠜⠁
⠀⠀⠀⢀⡄⠠⠤⣤⡤⠁⢉⡟
⠀⠀⠀⠀⠓⠤⣀⡀⠐⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚
⣿⣿⣿⣻⣛⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣋⡀⠀⢉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣭⠉⠁⢀⠈⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⢡⣤⡞⠁⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣅⠺⣿⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣉⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡠⢾⡿⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣽⣿⡏⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣷⡾⢿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣿⣿⡄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣍⢙⣦⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠟⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⡠⣤⣴⣶⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⢽⣮⣿⡟⠉⠉⠉⠉⠉⠉⢙
⠀⠀⠀⠀⢀⠞⢁⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣼⣿⣿⣷⡀⢢⡀⠀⠀⠀⠘⡄
⠀⠀⢀⡔⠁⠰⡉⣼⠛⠉⠳⠶⠦⣤⣤⣄⣀⣠⣀⣀⡀⠀⠀⠀⠀⢸⠀⠀⠹⣿⣿⣿⠿⣄⠙⣦⡀⠀⠀⢷
⠀⣠⠏⠀⠐⣘⢀⣡⣠⣤⣄⣀⣤⣀⡀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⡘⠀⠀⠀⢸⣿⡿⠀⠙⣷⡌⠻⣶⣶⣿⡆
⢰⢃⣀⣼⠷⠟⠛⠋⠉⠉⠉⠉⠛⠻⠿⣿⣿⣷⣶⣶⣦⣄⣀⢉⣙⣡⡈⠀⠀⣾⡿⠁⠀⠀⠘⣿⣦⠀⠈⢻⣿⡀
⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⠛⠁⢀⣠⣾⡿⠁⠀⠀⠀⠀⠘⣏⠳⠀⢀⢿⡇
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠛⠒⠒⣻⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢷⣷⠞⠛⢋⠉⠛⠳⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠐⠶⠶⢶⣬⡄⢀⢀⣤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣒⣒⣂⣠⣬⣭⣭⣄⣒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠁⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡠⠄⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠐⢀⠔⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠶⡸⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠊⠤⡀⠀⠀⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠁⠀⠀⠀⠀⣀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠊⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⡠⠂⢀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠠⠤⠀⠐⡖⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇⡀⠀⠀⡠⠚⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠡⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠊⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠱⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠐⠇⠀⡀⠀⠀⠀⠀⠀⢀⠋⠠⡀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠈⠂⠀⠀⠀⠀⠌⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠒⠀⠀⠚⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⠀⠀⠀⠀⠀⡨⠁⠁⠐⠠⢀⡀⠀⠀⠀⠀⠀⡆⠀⠀⠈⡄
⠀⠀⠒⠒⠢⡄⣀⠀⠀⠀⡀⠀⠀⠠⠤⠤⠤⠁⠀⠉⠂⢄⣀⠎⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄⢀⢴⠀⠀⠀⠀⠁
⠀⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢰
⢠⠀⠀⠀⠔⠓⠘⠉⠁⠀⠐⠒⠒⠀⠠⠤⠄⣀⣀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡀
⡄⠃⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠇
⠷⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠞⠂⠓⠐⠒⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⡀⠠⠀⢂⡀⠤⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣤⣤⣤⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⡿⠋⠛⠋⠙⠉⠙⢿⣯⢳⣄⡀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣿⣿⣇⣾⣿⣿⣷⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⡿⠿⣿⠏⠀⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢴⣦⣿⣿⣿⣿⣿⣾⣿⣆⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠦⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⠛⠁⠈⠻⢿⣷⣄⣼⠏
⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⣠⣾⣮⡟⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⡯⣠⣾⣿⣿⠟⠋⠀⠙⣻⣿⣿⣿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣴⣿⣿⣭⣴⣶⣶⣶⣾⣿⣿⣿⣿⣶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⣀⠀⠔⣠⣿⡿⠛⠁⢹⣿⡏⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢥⠶⣟⠋⠉⠀⠀⠀⠀⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡌⠚⠁
⠀⠀⠀⠀⠀⠀⡈⠀⡀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣄⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⢀⠀⠀⠀⠀⠀⠀⠀⣀⣭⢤⣄⣀⠀⠈⢇⠐
⠀⠀⠀⠀⠀⠀⣾⣿⣿⢀⠄⠠⡀⠀⠀⣴⠻⣿⣿⢿⢾⣽⣳⣲⢄⣖⡀
⠀⠀⠀⠀⠀⢷⣿⠙⢸⠁⠀⠀⠀⠀⡜⠾⢩⢿⠷⠦⢴⡯⣛⠿⣿⡿⠛⢷⣿⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢠⣷⡴⢩⣵⠛⡷⢾⢦⡏⣹⣏⢫⣶⣾⣿⣿⣷⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠲⠧⠀⠀⠀⠀⠘⡟⠞⠋⢠⠒⡇⢺⠉⢠⣶⡼⢹⣟⣿⣿⣷⡿⣿⢿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⠁⡞⠐⡇⣯⠀⢶⢀⣜⢻⡦⣽⣟⣿⣿⡿⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠬⢀⡿⡗⠀⣤⠈⠁⡻⠿⢷⣬⠛⣿⣿⣿⣿⣇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡂⢸⡇⢰⠄⢍⠁⢨⣉⣟⢠⡜⠓⢻⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣂⣸⣁⣼⣿⣮⣤⣹⡗⢿⠈⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣋⢪⡿⣿⣏⠉⠃⢸⡇⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠁⢢⠀⠀⠀⠀⠀⠺⣿⠋⠙⢻⣿⣿⣾⣾⣷⣤⣼⣧⣼⢿⡿⣿⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⡰⠃⠈⢢⣀⣘⣀⡀⣤⡀⠀⠀⣿⡀⠀⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣆⢷⡹⣧⡘⣿⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠡⠄⠀⠀⠛⢉⣿⡗⠈⠁⠈⠉⠉⠁⠀⠀⠀⡟⠈⠻⣿⣿⣿⣿⣿⣾⣎⢳⣜⢿⣿⣇
⠀⠀⠀⠀⡔⠉⠀⠀⠀⡀⠂⣀⡴⠛⠛⠃⠀⠀⠀⠀⠀⠐⠀⠠⣸⠃⠀⣼⣿⠻⣝⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠡⠄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠠⢤⣄⣀⠀⣀⠘⢦⣴⠁⠀⠀⣘⣇⠀⠀⠈⣿⠋⠻⣿⢿⣿⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⣠⣶⣶⣾⣾⣿⣿⣿⣤⣾⣿⣿⡇⡀⡤⢿⣤⣶⣤⣿⡀⠀⠘⢷⡈⠘⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣙⡛⠉⠙⠛⡛⢛⠿⢿⣿⣿⣷⣷⣧⢸⣿⣿⣿⣿⡇⠀⠀⠈⢹⡆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠏⠳⠦⠤⠀⠀⠴⣿⣿⣿⣿⣿⠘⢿⣿⢁⣿⠁⠀⠀⠀⠀⠗⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠀⢘⣿⢀⡿⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠀⠌⠈⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⢾⣽⣿⣿⣿⣿⣯⡗⠀⠀⠀⠀⢀⣤⠴⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠇⣀⠁⣀⣀
⠀⠀⢿⣿⡿⠁⠎⢻⣿⡿⠀⠀⠀⠀⣾⣿⣷⠰⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠉⠘⠃⠨⠄⠉⠁
⠀⠀⢫⢿⣿⠀⠀⢸⣿⣍⠁⠀⠀⢸⣏⠉⠀⠁⠈⠉⣽⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠘⣿⣤⣤⣼⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡖⣴⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣀⣈⣉⣩⡅⠀⠀⠠⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⣿⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⠀⡀⢀⡀⢀⡀⣀⢀⢀⠀⢀⢀⢀⠀⣀⢀⠀⢀⠀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀
⠀⠀⠀⠀⠀⣔⣆⠆⠀⠀⢶⣌⠉⠀⠀⠀⣱⢸⠀⠺⡇⣿⢾⢸⠇⢿⢸⢹⠀⢿⠈⡇⢸⠀⢻⡇⡟⡇⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠘⢹⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣤⠄
⠀⠀⠀⣀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢀⣾⣶⣶⣿⡎
⠀⠀⠀⠙⠁⠙⠛⠈⠋⠀⠀⢠⣤⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢶⣦⠿⠚⠿⣧⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠱⠶⣿⡇
⠀⠀⠀⢠⣤⡀⢀⣀⡄⠀⠀⠈⣿⡆⠀⠀⠀⣸⡟⠀⠀⠀⣷⣶⠇⠀⠐⠀⠘⣷⣶⠀⠀⠀⠀⠀⢿⣿⡤⡀⠀⠁⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⢐⣂⣀⣉⣭⡅⠀⠀⠀⠘⠇⠀⠀⠀⠿⠁⠀⠀⠀⣶⣾⡀⠀⠀⠀⢀⣿⣶⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⢿⠀⠀⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⢰⣿⢲⣶⣾⢿⡧⠉⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠈⣷⣶⡇
⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣉⡛⠈⠀⠀⠀⠀⣴⣶⡖⠀⣨⣭⣅⠀⠐⣶⣶⠆⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣴⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⢿⣿⣿⠏⠀⠀⠈⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⣾⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢻⣿⣿⣿⣿⣷⣦⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣷⣄
⠀⢨⣿⠃⢠⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣮⣝⠻⢿⣿⣿⣿⣦
⠀⢸⡏⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠛⢿⣦⡙⣿⣿⣿⡄
⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⣷⠸⣿⣿⣇
⠀⢸⠁⠐⠐⠛⠋⠙⠛⠻⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⣿⡇⢻⣿⣿⡀
⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢿⣿⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄⠀⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣠⣿⡟⠀⠀⢸⡧⠉⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠁⡞⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠠⠀⠐⠒⡄⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡿⠃⠀⢄⡀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣂⡔⠁⠐⠤⠸⠁⠀⠀⠀⠈⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡤⠀⠀⠀⢠⠖⡉⠂⠠⠀⠀⢀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠄⡽⠀⡀⠀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠉⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠀⠀⠀⠙⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠒⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢄⡀⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠚⠣⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄⠐⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃
⠀⠀⠀⠀⡄⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⢀⡀
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⢿⣄⠄⠀⢤⡤⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⡶⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠸⣽⣷⣦⣀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢂⣶⣿⣷⣾⣿⣷⣦⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠁⢀⢤⣽⣿⣿⣷⡄⠀⠀⠀⡇⠀⢸⠀⠀⢹⣿⣿⣿⣿⣿⡟⠻⣷⣒⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡿⣏⠋⢹⡇⠀⠀⠀⡇⠀⢸⠀⠀⢸⠋⣿⣪⡟⢸⡇⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⠀⣃⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠉⠁⠉⠉⠐⠟⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡃⠀⢘⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣴⣆⣴⣤⣄⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠄⢀⡀⠴⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⢿⡏⢛⡻⣿⣗⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⣠⣯⣥⣬⠆⠀⠀⠀⠀⡇
⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⣿⣍⣃⡈⢠⣿⣿⣦⠄⠀⡇⠀⢸⠀⠀⢀⣀⢂⣩⣟⣻⣝⡁⠁⠀⠀⠀⡇
⠀⢀⣻⠁⠀⠛⢿⡇⠀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⣶⣾⣿⣿⡟⠀⠀⠀⡇⠀⢸⠀⠀⠀⠈⠁⢿⢹⡯⡟⠐⠓⠁⠀⠀⡇
⣾⡿⠋⠤⠠⠀⢤⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢿⢺⡿⠿⠋⠁⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢀⣷⣶⠄⠀⠀⠀⠀⠀⡇
⠉⠃⠈⣶⣿⣤⡷⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⡇
⣧⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣶⠤⠄⠀⠠⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢀⠇⡇⠀⡠⠴⡄⢀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⡇⠘⠾⠀⠘⠊⠀⠀⠙⠛⢤⠀⠙⢆⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢇⠓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣙⠚⠁⠀⠀⠀⢀⡠⠔⠒⠉⠑⠒⡶⠀⠀⠀⠀⢨⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡗⡒⠉⠁⢀⣠⣢⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⢢⠀⠀⣞⠀⠈⡇⢀⢧⡆⠀⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⡜⠀⠀⢿⢀⡴⠃⣸⠁⠹⠏⣅⢸⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⠊⠐⠉⠀⠀⠉⢀⣴⣿⡄⠀⢀⠡⢞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠢⣤⣀⣠⣤⣾⡟⠋⢹⠇⠀⢀⣤⠞⢛⣣⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⢿⡿⠛⡟⣀⡴⠋⠀⡴⠋⣟⠉⢹⠈⢩⠃⠁⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⠟⠒⡄⠈⠓⠤⣍⣛⣉⣁⣠⠤⠞⠁⢀⣸⠤⠞⠀⢇⣠⠆⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢇⡔⠰⡇⠀⠀⠀⢀⣀⡴⠋⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠑⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠍⠊⠳⣌⡑⠒⣒⣳⠎⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣏⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢫⡉⠉⠉⠁⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣠⠦⠤⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢨⢗⢳⠀⣀⡤⡀⠀⢀⠀⠀⠉⠒⣦⠀⡠⢷⡒⢦⢀⠀⡝⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⠀⠙⠊⠁⠀⠈⠉⠻⣄⠀⢀⡴⣃⡰⢋⡵⠟⠳⢮⣎⡠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠾⢛⣛⡿⢝⡢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠲⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠈⠑⠪⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠤⠴⠃

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⣀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠂⢄⠀⠘⠒⠶⠀⠸⠇⠀⠀⠀⠈⠁⠀⢸⡀⠀⠐⢀⡀⠀⠙⠃⣤⡄⠁⠌⠁⠠⠀⠀⠀⠙⠂⠀⠒⠊⠀⠈⠠⠀⢀⠐⠂⣀⡀⠀⠉
⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠰⠆⠀⠀⠄⢀⣀⠀⠉⠁⠉⠀⣀⡄⠘⠦⡷⢛⣀⠀⠀⠀⠙⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠋⠁⠀⠀⠠
⠀⠀⠀⠀⠀⣶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠯⠀⠘⠗⠀⠺⠿⠀⠀⠀⠀⠀⠀⣠⠈⠈⠀⠀⠛⠀⡐⠀⠀⠀⢀⡀⠀⠠⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠄⠬⡙⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢐⠣⠀⡆⠀⣀⡰⢶⣶⡀⠀⠀⠑⠀⢀⠠⠀⠦⡀⠀⣠⡦⠀⠀⠀⠀⠀⠀⣠⠄⠀⡉
⠀⠀⠀⠀⠉⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢛⣁⠼⠏⠀⠀⠤⠄⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⠄⠀⠈⠉⠀⠈⠉⠁⢀⣂⠒⣤⠀⠀⠀⠠⠀⠀⠀⠤⠾⠀⠸⠀⠀⠀⠀⣀⠀⠐⠂⠴⠆⠀⠠
⠀⠀⠀⠂⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⡺⣿⡿⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠒⠀⠈⠀⠀⠠⠆⠀⠀⠀⠀⠀⠦⠀⠀⠛⣾⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠂
⠀⠀⠀⢀⠄⡀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⠈⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣀⠀⠀⡆
⠀⠀⠀⠀⣀⡀⠀⠀⠤⠤⠦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣠⣿⣿⡿⢿⠿⠿⠿⢿⢿⣿⣧⡀⠀⠀⠀⢰⡲⡀
⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠙⠋
⠀⠀⠀⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠀⠂⠻⠀⣀⠈⠀⠀⢀
⠀⠀⠀⠀⠤⠀⢁⡀⠀⠀⠀⠀⠀⠀⠰⠀⣤⣴⡇⠀⠀⠀⠻⣿⣤⣬⣧⣬⣥⣤⣧⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⣠⡈⡀
⠀⠀⠀⠘⡀⠀⠀⠠⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠆⠀⠘⠓⠓⠀⠀⠀⠀⠁⣈⡁
⠀⠀⠀⠀⠴⡀⠀⠀⢘⣻⣿⠩⣬⠀⠀⠉⠀⠀⠀⠉⣸⠏⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠰⠆⠀⠛⠈⠶⠆⢐⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇⠀⠠⠠⠀⠀⠀⠰⠒⠚⠀⠐⠁⠀⠛⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠈⠀⠄⢠⡄⠀⠀⠶⠐⠀⡴⡆⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠠⠤⠴⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⢸⠀⢨⡀⠛⠀⠀⢰⠛⣆
⠀⠀⠀⡀⠀⠀⠀⡀⠄⠀⠀⠀⠔⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠐⠀⠘⠐⠂⠀⠀⠑⠉⠀⠀⠉⠉⠂⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠲⢷⣤⣶⠟⠀⣀⢀⠀⠈⠠⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⡀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⣀⠈⠉⠀⠈⠀⠸⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠄⠀⠀⢤⢰⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠀⡀⡀⠀⠈⠁
⠀⠀⠀⠠⠀⠐⠂⠀⠀⠀⠀⠈⠈⠉⠀⣠⠈⠉⠀⢤⣰⠔⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡠⠀⠀⠀⠀⠉⠁⠀⠉⠈⣀⠈⠉⠁⠀⠈
⠀⠀⠀⠀⠀⢀⣀⣀⠆⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠥⠠⠀⠀⠀⠄⢈⠀⠀⠀⠐⠀⠀⢀⠌⡉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠢⠀⠁⠉⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠄⠲⣄⠀⣨⠀⠀⡠⠀⠀⠀⠈⠂⠁⠂⠀⠀⠀⠀⠁⠀⢀⠀⠀⠠⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠐⠀⠀⠈⠃⠃⠀⠴⠆⠀⠶⠀⠀⠀⠂⠉⠀⠰⠆⠈⠛⠀⠀⠀⠘⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⡷⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠒⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠢⡺⣯⡆⠀⠀⠀⢰⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢠⠉⠈⣀⠙⣷⡀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠐⢹⠀⣿⠀⢸⣿⡑⢾⡇⠀⠀⠀⣿⠃⠀⠐⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣠⠄⠐⢸⣧⣄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠨⠀⠀⢸⡟⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠒⠚⠀⠀⠘⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⣛⠶⠥⠾⠶⢦⡴⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣼⣿⣿⡿⣿⣿⣷
⠀⢸⢹⢳⡒⣿⣥⣄⡴⠛⣁⠀⠀⠄⠤⠊⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁⣸⣿⡇
⠀⢸⣏⠀⠀⢸⡟⣡⠂⢩⠃⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⣿⡟⠁⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠘⠋⠀⠀⢸⢸⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⠀⠀⠀⠀⠀⠛⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠿⢯⣑⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⢒⡲⠀⠀⠀⠀⠀⠀⣽⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢦⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠛⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⡤⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇⠀⠀⠀⢀⣶⣄⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⣾⣿⣿⣿⡿⠙⢿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡄⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣧⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣤⣀⡀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠛⠿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⣸⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣼⣿⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⢀⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠀⠀⠀⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⢠⠃⡇⠀⢀⣠⣄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⢧⡼⠀⠳⡴⠁⠀⠹⠴⢣⢸⠈⠳⡄⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⢦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⡾⠯⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⣀⣹⡄⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣠⣀⣀⣀⣀⠴⠒⠉⠀⠀⠀⠀⣨⠋⠀⠀⠀⠀⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠁⠀⠀⢀⠤⢬⡂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠛⡆⠀⠀⣏⠀⠀⢻⠀⡀⣴⡇⠀⣀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⢠⠃⠀⠀⣿⠀⣠⠏⢀⡏⠀⢱⠏⣅⡈⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⠶⠋⠖⠉⠀⠈⠉⠁⣠⣾⣧⠀⠀⠀⣋⣴⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠢⢄⣀⠀⠀⢀⣀⣤⣾⠿⠛⣻⠀⠀⠀⢀⡨⠧⢮⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⢿⣿⣿⡿⣿⠃⠀⣰⠏⠀⣠⠞⣿⠴⠲⡒⠒⢪⡕⠊⠑⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⣀⠈⠳⢤⡀⠻⢧⣤⡥⠴⠛⠁⠀⡰⠃⠀⠱⡀⢀⠇⢠⠏⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣛⡖⢁⠀⡇⠀⠀⠉⠓⠲⠶⠶⣶⠶⠚⠉⠀⢀⡤⠖⠋⠉⠀⠈⠲⡞⠁⢰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣉⡷⢊⡈⠣⣀⠀⠀⠀⡤⢤⠞⠁⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠘⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠉⠙⢦⣈⣉⣓⣋⡷⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⣀⣀⣀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢯⡉⠉⠛⠋⠁⠀⠀⠀⠀⠈⣙⡿⠀⠀⠀⠀⢠⣴⣦⣶⣢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⡭⢷⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⣯⠴⢤⡀⠀⠉⡳⡕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠙⠺⠦⠚⠉⠈⠒⠶⢾⡀⠀⠀⠀⣠⠏⢀⠿⢢⣳⠴⣃⠆⠔⠁⣸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠤⢾⡙⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠴⢾⡉⠒⣣⠔⠋⠁⠀⠈⠉⠒⠉
⠀⠀⠀⠀⠀⠀⠰⡋⠙⠒⠒⠮⢽⣒⠬⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠈⠙⠮⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣒⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⢟⣭⣭⡉⢙⣛⣉⠉⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣟⠻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠛⠛⣿⣿⣷⣿⣾⡄⠈⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⡀⣀⢼⣿⣿⣭⣴⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⣄⢈⣿⣿⣿⣿⣷⠈⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⠍⠡⣨⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣇⢸⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⠃⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣯⣶⣾⣶⣾⡀⠙⣿⣼⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣾⣧⡀⢪⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⠦⢹⣏⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⢾⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⢹⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠏⠀⠉⠠⠈
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠠⡐⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢨⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣧⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢨⠁⠩⡤⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⡼⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢚⡰⠾
⠀⠀⠀⠀⠀⠀⠀⠰⢻⠋⠹⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢡⠋⣉⣉⢉⡉⠁⡀
⣦⡖⠊⠂⠀⠀⢠⠂⠐⠀⠐⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠈⡏⠁⡏⠀⡇
⢿⢷⡇⡞⠁⠒⠁⠀⣇⠀⠀⠀⠀⢠⣇⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⠔⡖⠒⠊⠈⣮⡄⠀⠀⠘⣯⠉⠙⠃
⠈⠉⠉⠓⢤⣀⣀⠾⡏⡄⠀⠀⡼⣁⠀⡹⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠒⠓⠤⣀⠔⠁⣿⠀⠀⠀⠻⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣆⠀⠈⠛⠙⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡄⠄⠐⠊⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠓⠒⠀⠒⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⠔⠈⠀⠑⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠞⠫⡀⠀⠀⠀⠉⠉⠉⠑⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠋⠈⠢⣀⠀⠀⠀⠀⠀⠈⠑⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡾⠁⠀⠀⠈⠲⢄⣀⡠⠤⠶⣄⣏⠠⢛⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⠀⠀⠀⠀⠈⠉⠒⠒⠲⠦⠤⠴⣧⡇⢠⢿
⠀⠀⠀⠀⢀⡼⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇⠀⠀⢀⠀⠀⠀⠀⡼⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣮⠏
⠀⣀⣀⣾⣉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⠀⠀⠀⠸⣏⠳⠆⢰⣙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠏
⠑⠪⢿⣍⣉⠙⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡢⢄⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⣼⣿⣿⣿⣿⠟⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⢂⢸⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣿⣿⠻⠟⠃⠀⠄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠈⠁⠀⠀⠙⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣿⠃⠀⠀⠀⠘⠟⠒⠀⠀⠀⣄⠔⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠛⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⢀⣀⠂⠀⠀⡠⠀⠘⠀⠀⠀⠀⠀⠀⠴⠕⠈⠈⠟⠦⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠁⠀⠀⣿⡗⠠⢀⡁⠀⠀⠀⠀⠠⠂⠁⠀⠀⢀⠀⢀⠇⠈⠀⠈⠙⢒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⢀⡀⠀⣐⠏⠸⣀⠀⢸⠛⠐⡀⠀⠆⠀⠔⠲⠁⠀⠒⠁⠀⠀⠈⠐⠂⠀⡀⠂⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢒⠀⠀⠀⢰⠾⠋⠀⠀⠑⠤⠜⠀⠂⠀⠀⠐⠀⠈⠀⢑⠤⢄⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⡄⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠡⠀⠀⠇⠀⢀⠅⠀⠀⠀⠀⠀⡠⠊⠀⡄⠀⢀⠔⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⢰⡸⠀⠀⠀⠀⠀⠀⣀⠔⠁⢠⠀⠀⣱⠖⢉⠶⠥⣀⠀⠀⠀⠀⡔⠀⡄⠀⠠⠐⠁⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⡀⠀⣼⡅⠀⠀⠀⠀⠀⠠⠁⠄⠀⠀⠄⠔⠁⡀⢳⠀⠀⠀⠀⠉⢁⠊⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⢄⠀⠀⠉⠀⠒⠀⠴⠁⠀⠈⢄⠀⠀⡠⠊⣡⠟⠓⠂⠄⣤⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⢦⣤⢀⡀⠀⠀⡘⠀⠀⠀⠀⠀⠑⠊⠀⠀⠁⠀⠀⠀⠀⠀⠒⠃⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠈⠉⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠃⡀⠀⡀⠀⠀⠀⠈⠰⠀⠄⢀⠀⠀⢀⠂⠀⠀⠈⠰⠂⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢐⠦⠈⠐⠂⠌⠀⠂⠀⡄⠀⠀⠀⠈⠑⠢⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠀⢀⠌⠉⠁⠀⠒⠒⠚⠒⠀⡤⠀⠀⢠⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⢅⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠆
⠀⠀⠀⠀⠀⠠⠂⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠼⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠷⠾
⠀⠀⠀⠀⠠⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠱⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣀⠀⠁⢄
⠀⠀⠀⡠⠁⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠐⣠⢦⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠒⠤⠵
⠀⢠⣒⣅⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡅⠀⠀⠐
⠔⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠢⢄⣀⡐⡄
⠀⠁⢱⣀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣍⣳⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠙⢆⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠁⣀⡬⠗⢲⣶⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠟⠉⠁⠀⠀⠀⢿⣿⠃⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠙⠦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⣏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣶⣴⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠉⠉⠉⠉⠙⠛⠿⠿⠷⠶⠶⣦⣤⡀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘
⠀⠀⠀⠀⠀⣤⣤⠀⠀⣀⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⢻⢧⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣠⡀
⠀⠀⠀⠀⢸⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⣶⡿⣶⣦
⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠉⠁⠒⢤⡤⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⣤⣶⠶⠶⠶⠶⣶⣤⡈⢿⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⡁⠤⠤⢀⡀⠀⠀⠈⢻⣌⠻⠦⡀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣁⡀⠀⠀⠤⣄⠀⠈⢆⠂⠀⡆⠆⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⣦⠀⠀⠶⠀⠀⠀⠘⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡤⠄⠊⠅⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡀⠉⠀⠀⣀⠔⠋⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣩⠀⠀⠀⣒⠖⠂⠁⠀⠸⠉⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠀⠀⠊⢠⠀⠀⠀⠀⡧⠀⠀⠠⢄⠀⠈⠢⡀
⡔⠀⠚⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠁⠂⠀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠑⡀
⠁⠀⢠⣿⠉⠐⠀⠠⢀⡀⠀⠀⢀⠄⠈⠑⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠈⠉⠀⠀⡠⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠂⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⡰⠉⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠒⢄⠀⠀⣀⠤⠊⠀⠀⠈⢆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢏⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠎⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣿⠃⠀⣱⠆⠀⠀⠀⠀⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠁⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢏⡴⡚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢖⠻⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠿⠒⣴⠞⣴⠛⣱⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⣿⡏⣀⠁⡐⢃⡴⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠔⠚⠘⣿⠿⢆⣿⣴⢳⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣈⣴⣿⣦⠀⣘⠙⠿⣟⠻⠈⣾⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣤⣤⢾⣴⣫⣍⡻⡎⣶⡕⢸⡿⢼⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠋⣾⣏⠙⢓⣣⣿⣷⢈⣽⠀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⢿⣦⣭⣭⣿⣧⣤⡰⣾⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⣮⣽⣿⡟⠁⠘⣷⠘⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⠛⣋⡾⣮⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠉⠙⣿⣿⣿⣿⠟⠳⣬⢳⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡀⠀⠸⣿⣿⡇⠀⡄⠘⣷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣇⠀⣠⣿⣿⣿⣶⣿⣆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⢻⡟⠛⠉⣿⠏⠉⠙⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢻⣦⡀⣿⡆⠀⠀⡿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⢿⣾⣷⡀⠐⠁⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣟⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡩⢸⠋⡁⠖⣻⣤⠴⣶⣶⣶⠀⣀⠄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢏⡸⣽⣀⣼⡇⢈⡭⠆⣚⣉⢈⢛⢯⡥⢂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡮⢀⣿⠷⣻⣿⣿⣿⣾⣶⣿⣾⣿⣿⣿⣯⣴⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢋⢼⣷⣾⣿⣿⡿⠟⠛⠛⠛⠻⠛⣙⣛⢿⣭⣿⣗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⣿⢗⣷⡿⠛⢁⣴⡿⠿⠆⠀⠀⠘⠛⠻⡇⢻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣿⣾⣾⡇⠀⠚⡡⠒⠢⠀⠀⠀⠠⢒⠢⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠛⢿⡟⠀⠀⠀⢸⡿⠀⠀⠀⠀⢿⠇⠀⣯⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠙⠀⠁⠀⠀⠀⡀⠀⠀⢀⣀⣠⠂⠀⠀⠘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣠⣄⠀⠀⠋⠹⣶⣀⣀⢀⣀⡶⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡬⢽⣿⣷⣄⣀⠀⠉⠉⠭⠍⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠛⡆⠀⠀⠀⠹⠋⠙⣿⠓⠶⠤⢤⠴⠞⠭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠈⠓⡆⠀⢠⠄⡤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠜⠁⠀⠘⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡄⣰⠛⠦⠼⢆⣷⠒⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠘⠃⢄⠀⠀⠀⠀⢀⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠊⠑⠤⣀⡀⢀⣨⣿⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢐⣇⡎⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⢸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠁⣿⣤⣀⡀⢀⣈⣭⣶⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⠷⠤⣏⣸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢀⣄⣇⡀⠀⠙⢿⣿⣿⠿⠋⠈⢻⣿⠶⢼⠁⠀⢀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⠦⣼⠋⠉⠉⠦⠤⠞⠀⠀⠀⠀⠀⠁⠀⣏⣤⡶⠛⡡⠊⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠊⠀⡽⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣾⠃⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠩⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁

⠀⠀⠀⠀⠀⠀⣴⠀⠝⠓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣯⡤⠀⠴⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⣠⠀⠀⠈⠃⣴⠃⣴⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⡿⣿⠏⠀⠀⠀⣄
⠀⠀⠀⠀⠈⣷⠀⠀⣤⠃⣰⡇⠀⠀⢠⠀⠀⠀⣀⣀⠀⠀⠀⠈⠁⣿⣄⡀⢀⡴⠊⠀⠀⠀⠀⠀⠀⣀⠀⣠⠆⠃
⠀⠀⠀⠀⠀⠸⣆⠀⢁⣠⢿⠁⠀⠀⠀⠀⠀⠀⠙⠙⠗⠀⠀⠠⠆⠹⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢻
⠀⠀⠀⠀⡄⣰⠹⣶⡈⣧⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠂
⠀⠀⠀⠀⢱⡏⠐⠋⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠈⠁⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣆⣠⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠄⠀⠀⠀⢀⠤⢴⠲⠶⠶⠛⠉⠃⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⢿⡟⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⡿⢦⠄⠤⠼⠛⢻⠀⠇⣠⠀⠀⠀⡈⣠⢆⠀⠈⠇⠀⠤⠆⢠⡖⠀⠀⠀⢸⡇⣠⠛⠢⠀⠐⠁⢀
⠀⠀⢠⠃⣰⡇⠀⠀⠀⠀⠀⠀⣇⣼⠇⠀⢀⣴⠿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢰⠃⠀⠀⠀⠀⠀⣸
⠀⠀⢸⣾⡁⠧⠶⠄⠀⠀⠀⠀⠸⠿⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠉⠐⠢⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠛⡖⠂⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠐⠀⠃⠐⠂⣀⡀⢠⣴⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡠⠊⢀⠄⠀⠤⡈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⡕⠀⠀⢀⡔⠋⠠⢀⠛⠏⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠔⠁⠀⠀⠀⠈⠢⠀⠀⠁⠀⠒⠠⠤⠄⠒⠉⠀⠀⠀⠈⢣⡀⠀⠀⢱⠐⠆⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣃⣀⣈⣱⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⡘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠒⡖⠉⠄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠰⠁⠀⢀⡼⠙⢌⠉⠂⢄⡀⠀⠀⡃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⠀⠀⡰⠁⠀⠠⣾⡅⠀⠀⢃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⠀⠀⠀⠘⠀⠠⣔⠅⠀⠈⠓⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡄⠀⠀⠀⠀⢀⠃⠀⠀⢀⣀⣀⡀⡠⠂⠁⠀⠠⢲⠐⠆⠱⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠙⡄⠘⠒⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣐⣰⢖⣰⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⣀⠀⢀⣀⣀⠤⠇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠈⠈⠁⠈⠐⠰⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⣀⣉⣉⣁⠘⣁⢀⢀⢀⡀⠠⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⠀⠀⣠⡆⠀⠀⢀⠉⠉⠉⠀⠀⠀⠀⠀⣜⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠰⠀⠀⠐⡂⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡔⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠒⠈⢡⠀⠀⠤⡀⣀⠀⠀⠀⠈⠉⠉⢀⠞⠀⠀⠀⢀⠀⠿
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠀⠐⠢⠐⠢⠤⠄⠓⣦⢄⡀⠀⡠⠂⠀⠀⠀⠀⠀⠢⡀⠇
⠀⠀⠀⢠⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠤⠊⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⢀⠔⠆
⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠁⠈⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣉⣁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠒⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⠐⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣿⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠉⢻⠟⠛⠻⡄⠀⠀⢠⣤⣀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣷⣶⣧⣤⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⢿⣿⣿⡿⠿⠏⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣈⣉⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠘⣿⣿⠏⠋⠁⠀⠀⠀⢀⣠⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⠀⠀⠈⠉⠛⠿⢿⣷⣦⣤⣀⣠⣾⣿⣿⣷⣶⣶⣶⣶⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⣿⣿⣿⠟⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠏⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⢨⣿⣿⣿⣿⡿⠛⠛⠋⠉⠁⠀⣸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⠀⠠
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢴⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢭⠀⠄⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢆⢀⠒
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠴⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⢈⠀⠀⠀⠈⠀⠈⠠⡤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⢠⠝⠐⠀⠁⠀⢀⠀⠒⢠⡀⠀⠀⢀⠀⡲⠀⠀⢀⠄⠀⠀⢐⡀⠈⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠀⠀⠀⠀⠀⢠⡡⠎⠀⠚⠁⠀⠀⠀⠀⣰⠁⠀⠠⡀⠁⠆⠀⠀⠀⠀⠀⠀⠙⢠⠖⠀⡘⣠⠀⠀⠀⠀⠀⠀⡜⠫⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢀⡤⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠖⠠⠤⠀⠀⢀⡀⠀⢳⠀⠠⡄⠈⠉⠁⠀⠐⠀⠀⠀⢀⣸⠁⢠⡄⠀⡁⠀⠀⠀⢀⡀⣀⡏⠀⢀⠘
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠄⡸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠐⠠⡀⠀⠀⠀⠀⢀⡤⠛⣦⢒⡱⣆⠀⠀⢀⣴⡋⠀⠈⠻⠆⠛⡄
⠀⠀⠀⠀⡀⠖⡆⠀⠀⠀⡄⠲⢀⠁⠀⠀⠀⠀⠖⠀⠀⡄⠃⠀⠀⠀⠀⡄⠀⠀⠑⠢⠀⠉⠒⠀⠀⡴⠛⣤⠆⠈⠉⡟⣹⠀⢠⠎⢠⡟⠁⠀⠀⢀⠂⡇
⠀⠀⠀⠀⠀⠀⢰⠂⠈⠱⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠣⠀⡀⣠⠆⠘⣄⠀⠀⠀⠀⠀⠐⠤⢄⣵⠁⠞⣿⠀⠀⡘⡎⠁⠀⠈⡀⢸⣇⠀⠀⠀⠿⠉⡇
⠀⠀⢠⡄⠂⠀⢀⠀⠀⠀⡇⠀⠀⢠⠦⡔⠀⢀⠀⢀⣀⠠⠜⠹⣴⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⡞⠉⠉⠒⢻⡁⠄⠀⠀⣗⢘⡌⠉⡀⠏⠈⠀⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠡⣀⠹⠂⠀⠌⢈⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠊⠀⠀⠀⠀⠀⠈⢂⠐⠋⠙⣎⠀⠀⠸⡄⠐⠤⠆
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠜⢀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡖⠳⡤⠰⡖⠒⠟⠀⠀⢠⠛⢦⠀⠀⢱⡀
⠀⠀⠀⣠⡆⠉⢀⠀⠂⠀⠀⠓⠄⠀⠒⢢⠁⠀⠀⠀⠀⠀⠀⠠⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡴⠀⠀⠸⠀⠺⠀⣠⠏⠀⢈⢦⠀⠀⡇
⠀⠀⣷⡁⡚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡜⠀⠀⠀⠀⢆⢀⣴⠃⠑⠲⠁⠘⣆⠀⠴⡄
⠀⠀⠈⢣⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣌⠀⠀⠀⠀⠀⠈⢃⠉⢵⠲⡀⠀⠀⠙⣆⠐⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠩⠄⠀⠀⠀⠀⠀⠈⠔⠋⠙⠁⠀⠀⠀⠈⠣⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⡈⠙⠲⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣴⣶⣶⣶⣒⣴⣧⡀⠀⠙⣿⣷⡄⢹⡎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⣮⣿⣿⣿⢷⣄⠀⠈⣿⣧⡀⠁⠂⠹⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣋⣾⠁⣿⣷⣶⣦⣾⣿⠍⠀⠠⠄⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣤⡛⠿⣿⣿⡟⠛⠀⠀⠀⢀⣴⠀⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡧⢀⠀⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⢻⣿⠛⠻⣤⠉⠛⢠⠈⠁⠈⠉⠁⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢠⣿⣿⣿⣷⣾⣦⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⣿⡿⠋⢉⣿⡿⣷⣀⣀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄⣾⡿⠃⠸⠛⠛⠋⢻⣿⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠛⠁⠉⠛⠛⠛⠉⠉⠀⡀⠀⠀⠀⠘⢿⠀⠀⠘⣦⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⡤⠄⠀⠀⢀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣿⡔⠃⠀⠀⠈⠉⣉⣱⡦⠶⠆⠀⢀⣴⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢀⡀⢁⡠⢔⣰⣿⣿⣏⣴⣶⣶⡾⠟⠛⠁⠀⠀⠀⠀⠃⠀⠀⠀⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⠿⣬⣶⣼⣿⣿⣿⣿⠟⠛⠁⠀⠰⣤⣄⠂⠀⠀⠀⠄⠀⠀⠀⠀⢀⣷⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣶⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠘⠋⠀⢠⣰⠂⠀⠀⠀⢂⣤⣿⣿⣿⣧⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣶⣾⣯⣧⣿⣿⢻⠛⣿⠁⡉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢰⠹⡀⠀⣀⢤⡄⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠣⡟⠀⠳⠞⠀⠀⠳⠞⣄⡇⠙⢦⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⡟⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠤⠼⠆⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⢀⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢤⣀⠤⠔⠚⠁⠀⡀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠀⠀⠀⡴⠛⢲⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠈⡇⠀⢰⣧⠀⢀⡇⢀⡸⢏⣀⠤⠶⡄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠟⡰⢁⡄⠘⢧⡤⠞⠀⣼⠀⠀⠋⢴⠄⡿⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠉⠀⠁⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠐⠚⣅⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠑⠲⣤⣤⣴⣶⣿⠟⠁⢨⠇⠀⢀⣤⠖⣉⣓⣒⡢⢄⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠹⣿⠟⠫⡇⣠⡴⠋⠀⣴⠋⢸⠉⠉⡇⠀⡸⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣯⠝⠓⡆⠀⠙⠲⢬⣉⣉⣉⣁⣀⡤⠞⠁⠀⣀⡧⠴⠃⠘⣅⣰⠆⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢎⡠⠀⠇⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠣⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣐⠪⠔⠲⣄⠑⠦⠤⠼⣯⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⢹⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⠒⠒⠶⠖⠋⠉⠉⠉⠉⠑⢶⡇⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⣒⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⢸⡩⣉⠉⠉⠓⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠙⢬⣧⠴⠊⠙⠤⠤⣖⠀⠀⠀⢀⡽⠀⣰⢭⣳⣤⣇⠄⡠⠃⣱⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣴⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠴⣋⠳⢞⡡⠚⠉⠀⠉⠛⠒⠊⠁
⠀⠀⠀⠀⠀⠀⢾⠛⠒⠒⠭⢵⣒⠭⣚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠈⠙⠪⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠒⠀⠉⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠀⠤⡀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠡⡀⠲⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠋⢮⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠂⠀⠀⢀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠐⣚⣁⣀⡇
⠀⠀⠠⠈⠑⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢢⠀⠀⠀⠈⢃⠉⠐⠒⠠⠄⣀
⠀⠀⠀⠦⣀⣀⠗⣤⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠈⠓⢇⠀⠢⢄⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠉⠒⠤⠊⠀⢠⢞⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⡰⠉⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⢊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⠁⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠑⠦⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠒⠒⠂⠤⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠑⢄⡀⠀⡀⠔⠊⠉⠢⢀⠀⠀⠀⢠⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⠔⠊⠉⠈⠀⠀⠀⠀⠀⠀⠉⠂⠔⠥⠀⠤⠄⠀⣀⣀⠀⠀⠀⠉⠁⠂⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡌⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⢀⠄⠄⢀⣀⠀⡰⠀⡐⠁
⠙⠓⠦⣬⣉⠉⠣⠜
⣀⣀⣀⣀⣈⣙⣽⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣠⠖⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠈⠛⠿⠿⠛⢉⠻⣿⡿⠟⠁⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢘⠀⠴⠛⠂⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⢁⠈⢻⣿⣿⣿⣿⣿⠁⠀⠀⠙⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠘⠀⠀⣶⡄⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠇⢈⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠇⠀⠛⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠁⠀⠀⠀⠀⣀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠘⠂⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⢡⠦⠤⢤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠙⠀⠤⠤⠤⠤⠤⠈⠛⣿⠉⠀⠠⠃⠀⠀⢰⠀⠀⠀⠙⢋⠙⠐⠦⠄⣀
⠀⠀⠀⠀⠀⠀⢀⣴⢋⠉⠄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢧⡔⠁⠀⢠⠀⡈⠀⠀⠀⠀⠀⠑⡄⠀⢀⠉⠉⠑⠦⢄
⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠠⠊⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠸⣶⣌⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⣠⠊⡠⠊⠀⢀⡄⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⢦⡀⠀⣠⠒⠃⡀⡠⠊⡡⠊⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠘⠋⠀⠀⠀⠘⠤⠀⣠⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡰⠿⠛⢿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⢤⡄⠰⠊⠁⠀⠀⠀⠘⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠈⠉⠛⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣏⣠⣤⣤⣦⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠈⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣷⡈⠛⢛⣋⣭⣤⣶⣶⣶⣶⣶⣦⣤⣬⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠛⠻⢿⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⡈⡁⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⡟

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⣀⠀⢀⠀⠳⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣼⣿⣿⢿⣿⣿⣍⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⠷⠶⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⠁⠀⠀⢁⣭⣫⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠷⢶⣾⠋⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣍⡉⢡⣼⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣤⣀⣀⠀⣀⣸⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⢻⣿⢿⣿⠿⠇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⠋⠀⢠⡀⢱⠿⠟⠛⠻⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⢀⣾⣿⠟⢿⡎⣠⣤⣤⣤⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣤⣿⣧⠀⢀⡠⠽⠮⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠿⠀⠸⣿⡿⣿⣮⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠙⠛⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣄⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣶⣶⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠉⢻⡇⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠂⠸⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⡀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⡀⠈⠁⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠂⠐⠢⡑⠄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠄⠠⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠂⠄⢀⠀⠀⠀⢀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠀⢀⠀⠈⠁⠈⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢈⠦⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⡀⠀⣀⣠⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣿⣿⣿⣿⠛⠻⠋⠐⢄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠌⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⢀⣀⠄⢆⠁⠀⣀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠘⡀
⢻⡀⢲⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢡
⠀⢳⡈⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠆
⠀⠀⠻⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢼⣄⣀⡤⠒
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠛⣋⣤⠖⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣭⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣠⡄⠂⠀⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣰⣿⢀⡆⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣼⠋⠀⠘⠄⠙⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⠀⠀⠀⡼⠛⠿⠟⠋⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⠀⠀⣀⣠⣤⣾⣿⣯⣷⣞⡿⣿⠀⠀⠐⣡⠀⠀⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠀⣴⣏⠀⣴⣾⣷⣾⡿⣿⠛⠻⣿⡿⣁⣿⣆⠀⣎⠀⠀⠀⠀⠘⠀⠀⣼⣻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣷⣿⢿⣼⣿⣿⣿⣿⠋⠋⠒⠶⠿⠟⠛⠋⢹⠀⠹⡄⠐⣄⢳⢁⣼⠏⠙⢿⣜⡇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢿⠟⣾⣿⣿⣿⡟⣿⠘⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠸⡆⢸⡎⢈⠃⠀⠀⠀⣿⠁
⠀⠀⠀⠀⣴⠟⠻⣿⣉⣿⣿⣿⣿⣷⣿⡿⢻⡟⢻⣿⣭⠿⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠗⠉⠀⡀⢀⡀⣿
⠀⠀⠀⢸⣿⠀⠀⢹⣿⣿⡿⠿⠟⡛⣋⣀⠟⢁⣸⣿⣿⠀⠘⣦⢻⡿⠓⠿⢳⡶⠆⠀⠀⡆⠀⠘⢆⡀⢠⠁⠀⠀⣿
⠀⠀⠀⠸⣿⣦⣶⣿⣿⣿⣷⣤⣤⡿⠯⡉⣀⣾⠈⠻⣿⣔⣛⡁⠈⠐⢦⣆⣈⣧⠀⠀⢤⣷⠐⠂⢨⣿⣿⠂⠆⢰⣿
⠀⠀⠀⠀⣸⣹⣹⣿⠿⠥⠾⠷⠦⠶⣾⣿⣌⡙⠛⠚⠚⠛⠛⠓⠒⠂⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠋⠉⠉⠱⠚⠋⠉⠉⠙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⠑⠲⢄⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⢋⡤⢤⠀⠀⡤⠄⠀⠀⠙⢦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡐⢁⠢⣀⢠⢎⡈⠋⠘⠀⠀⠀⠀⠀⠐⠒⡄⠀⠀⢳⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⢲⣜⣠⠈⠃⢸⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⢀⡜⠁⢧⠀⠀⠀⠈⣆⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠳⣄⠀⠀⠈⠢⢄⣀⡠⠞⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⠀⠀⠀⣀⣀⡤⠔⠊⠀⠀⠀⠀⠀⠠⠋⠙⠈⠱⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠬⠏⠉⡯⣅⣀⣀⣀⠀⠀⠀⠀⠀⢠⠊⢀⠀⢁⠄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠀⡎⠉⠉⠀⠀⠀⠘⢦⠳⡀⠀⠀⢀⠍⠈⠁⠉⠒⢄⡀⠀⠀⠣⡀⠫⣩⠤⠖⠁
⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣀⣠⠃⠀⢀⣀⡀⠀⠀⠀⠑⢌⢂⠀⠌⠀⠀⠀⠀⡆⠀⢈⣢⡀⠀⡈⠀⡎
⠀⠀⠀⠀⠀⠀⡌⠀⠀⡜⠀⠸⠒⠋⠉⠈⡆⠀⠀⠀⠀⠈⠢⠊⠀⠀⠀⠀⠀⠘⡎⠉⠳⣄⠜⠁⠸
⠀⠀⠀⠀⠀⠀⠇⠀⡰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⠀⢠⠇
⠀⠀⠀⡠⠄⠐⠁⠘⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣖⠢⠞
⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠒⠒⠂⠀⠉⠁⡠⠦
⠀⠀⠀⠓⢄⡄⣘⣠⠋⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢠⠎⠀⡄⠀⠰
⠀⠀⠀⠀⡀⠬⢌⠁⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠃⢀⠃
⠀⠀⠠⠪⠊⡔⠀⢱⠀⠀⠀⠀⠀⠀⠀⡝⠓⠒⠦⠤⠤⠤⠤⠤⡖⠒⠚⠉⠀⠀⠈⠙⠒⠤⣄⣀⣠⠋⢸⠀⠀⠀⡔
⠀⡰⡱⠁⠀⠈⢡⠎⠒⠤⣀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⡐⠀⠀⠀⢀⣀
⠐⢰⠁⠀⠀⠀⡼⠤⡄⠀⠀⠈⠐⡇⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⢠⣃⠔⠊⡍⠀⢀⠈⠲
⣆⠃⢣⠀⠀⡼⠁⠀⠈⠓⣄⠀⠀⠸⡄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠞⠁⠀⠀⠀⠁
⢿⡀⠀⣩⠟⠀⠀⠀⠀⠀⠈⠓⢄⠀⠱⡄⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠔⠈⠀⠀⠀⠀⣀⠤⠊
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠉⠢⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠁⠀⣀⡠⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⢛⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡠⢤⡾⠉⠁⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠠⣧⠄⠀⠀⠀⠁⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠇⠀⠐⠄⠀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠶⣤⣤⡿⠀⢠⠀⣼⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⠉⠀⠈⢳⣄⠀⠈⠳⣤⢸⣾⣿⣧⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣠⠄⣸⣇⠀⠀⠀⠈⠺⣷⣿⡏⠀⠈⠑⢀⡀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠋⠉⠀⠀⠀⠀⢹⡿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡏⠀⠀⢰⠋⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⡎⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⢀⡇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⣆⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠹⣿⣿⣶⣿⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⣇⠀⠀⠀⠀⢀⣴⣿⠛⠻⠷⣤⣤⣤⣤⣠⣆⡀⢠⡇⠀⠀⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠹⣆⠀⠀⣴⠋⣾⠇⠀⠀⠀⠀⠀⠙⠻⠿⣿⡷⠟⠁⠀⠀⠘⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠆⢰⠃⠀⣿⠀⠀⠀⠈⢶⡌⢶⡄⢶⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢠⡇⢠⡇⠀⠀⢹⣆⣠⠶⠶⠦⣿⡜⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⢧⠀⢀⣼⠟⠁⠀⠀⠀⠀⠉⠻⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠶⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠈⢙⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠻⣶⡶⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⣠⠾⠋⠀⠀⠀⣀⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣀⣴⣿⠏⠀⠀⢠⡞⠁⠀⠀⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⢀⡾⠁⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⡾⠁⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⠀⢠⣶⣿⡿⠇⠀⠀⠀⠀⢀⡼⠁⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣴⢃⣴⡿⠋⠁⠀⠀⠀⠀⢀⣤⡾⠁⢀⡼⠋
⠀⠀⠀⠀⠀⢀⡴⢿⣷⣿⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⡾⢫⣤⣠⡟⠁
⠀⢀⣠⣄⣴⡟⠁⠈⣹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⠙⠇
⠀⡏⠁⠈⣿⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢰
⠀⣧⠀⠀⣿⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠠⣾
⢀⡟⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠞
⣸⠁⠀⢠⠟⠁
⢻⠀⠀⡏
⠸⣧⡜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠰⣄⠀⠀⢃⠀⠀⠀⠈⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠲⣶⣾⣿⣿⣿⣿⣧⠀⠀⠀⠉⠀⠀⠐⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣴⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠐⠋⠀⢸⡖⡄⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⣀⣴⣦⡀⠀⠀⠀⠘⣿⣷⣦⡄
⠴⠖⠀⠀⠀⠀⠘⠛⢠⠀⢀⣠⣤⣄⡀⣱⣿⣯⣶⣾⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⣶⣔⡄⠀⠈⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣁⡉⠉⠙⠛⠛⠻⢿⣿⠿⠿⠿⠷⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣿⣿⡛⠂⠀⠀⠀⠀⠀⢀⣶⡀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣷⣶⣦⣤⣶⣾⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠲⢶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠏⠉⠉⠉⠉⠁⠈⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠛⢿⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠉⠻⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠽⠞⠋⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠲⠶⠶⣄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡀⣀⡀⠀⠀⠀⠀⠤⣤⣭⣵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣟⠻⡟⠷⣤⠄⠀⠀⠀⢀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡂⠉⣄⡟⠀⠸⣖⠶⠤⠠⠤⣼⡭
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣀⡀⠀⠀⠘⢹⡿⠿⠿⢋⣶⣶⣀⡀
⠀⠻⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⢻⣿⣿⣿⣷⣶⡶⠋
⣠⣴⣿⠿⣟⢦⣤⣤⣤⣤⣄⣴⣤⡤⠤⣤⣰⣟⣁⣴⣟⣋⡏⠉⠉
⢠⣿⣟⣛⠻⢧⠙⠸⠦⠤⠤⠬⠿⢿⣿⡟⠛⠋⠉⠉⠁⠀⠙⢦⡀
⠈⠉⢻⣽⣿⣧⣀⣀⣀⣀⣀⣀⣀⣤⣷⣽⣆⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠈⠉⢰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⢷⡲⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⢦⣄⣀⣠⡤⠶⠚⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⢰⢤⡀⠀⠀⢠⡴⠛⠁⠀⠀⣠⠴⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠄⠀⠀⡇⠀⠀⠀⠀⠀⢀⣠⣤⠐⠛⢻⡇
⠀⠀⠀⠀⠈⢯⣿⢤⣀⡼⡇⢀⣠⡴⠛⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⢣⠀⠀⠀⢳⡀⠀⢀⡠⠖⠋⠀⢹⠛⠳⣾⣿
⠀⠀⠀⠀⠀⠈⢿⣷⣄⣀⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢰⡿⠚⠉⠀⢀⣠⠴⠚⠃⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⢁⣠⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣷⣧⣀⡀⠀⠀⠀⠀⠀⢀⣶⣶⣆⢇⠀⠈⢆⠀⠀⠀⠀⣾⣿⡿⣿⣷⣾⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣄⣀⣀⣀⣠⣿⣿⡄⠀⠀⠀⠀⠀⣀⣈⡿⢿⡉⠉⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠘⣶⣾⣿⡄⠀⠀⠀⠻⡿⣧⣿⢿⣼⣿⢿⡿⠏
⠀⠀⠀⠀⠈⢿⣿⠉⠉⠉⠉⠉⣿⡿⠁⠀⠀⠀⠀⠈⠉⠉⢁⣼⠃⠀⠀⠀⠀⠀⠀⡜⠀⠈⡝⠀⠀⠹⣿⣿⠇⠀⠀⠀⠀⠙⢿⡇⣾⣿⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣬⣥⣤⣤⡄⠀⠀⠀⠀⢧⣀⡼⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢈⣿⡏⠹⠉⠉⠉⠍⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⠿⠿⠿⠀⠀⠀⠀⠀⣿⣧⣼⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢈⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⡀⣀⣾⣽⣷⣜⣿⣃⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣶⡶⠛⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠘⣻⣿⣾⣿⣷⣿⣷⡇⠀⠀⠀⠀⢀⣀⠖⠀⠀⢰⡀⠀⠀⠀⠀⠀⣶⣾⢿⣧⣤⣤⣾⢿⣷⡄
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⠁⠀⣸⠇⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠎⢠⡍⢾⣄⠘⠁⣥⠀⠄⠀⠀⠀⣈⣁⣘⣛⣛⣛⣋⣀⣙⣋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠉⠀⠄⠁⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣷⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣴⠀⠀⠀⠀⠀⠀⣀⣠⢻⢀⡼⠇⢀⣀⣀⡀⠀⠀⠀⣧⡇⢠⣾⣯⣿⣝⣦⢸⣿
⠀⠀⠀⠀⠀⠻⠟⣶⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠃⢸⡼⠀⠀⠀⠀⠀⠀⢸⠁⢺⣺⣿⣿⣿⣿⢂⡏
⠀⠀⠀⠀⠠⠤⠤⠬⠭⠭⠭⠤⠤⠤⠄⠀⠀⠀⠀⠀⠶⣴⣤⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⠀⠈⠳⠯⠯⠷⠋⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠉⠓⠂⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠟⠒⠦⣄⣀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠲⡄⠙⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⣠⠊⣨⡆⠀⢣⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠘⠒⠛⠢⠿⠃⠀⠈⠙⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⣀⣠⠖⠀⠀⠀⠀⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠚⢻⠉⠀⡠⠄⠀⢲⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣺⣿⣿⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⡐⣚⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⢤⠤⠭⣤⠤⠺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠸⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠐⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠛⡉⠀⡧⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣃⣠⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⠁⠃⠀⠈⠲⡇⠠⠤⠄⣀⡀⠉⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⠤⢘⣡⠞⠀⠀⠀⠀⠀⠀⢀⠌⡔⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⢄⠈⠀⠀⠄⣀⠃⠘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡘⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣘⠂⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣄⡸⡀⠀⠘⢦⡀⠉⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠁⠒⠒⠒⠒⠁⠀⠉⠒⠦⣤⣀⣀⣀⣀⣀⣀⣳⠀⠀⠀⠀⠀⠱⠤⠤⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⠀⠒⠒⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠀⠉⠉⠉⠉⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠤⢤⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢦⣀⠀⣀⠠⠄⠒⠀⢦⠼⡄⠀⠀⢡⣀
⣀⠠⠀⠂⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣀⠀⠀⠀⠀⠀⠀⠈⡦⠲⣄⠀⠀⢣
⠉⠙⠓⠲⠶⠦⠤⠤⠶⠶⠚⠛⠒⠢⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠛⠁⠒⠶⠷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠉⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣢⠤⢀⣠⡤⠟⢉⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢠⠆⠹⡅⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⣦⡀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⡄⢉⢲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠤⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠠⠴⠤⢤⣄⠒⡉⠄⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠁⠐⢌⠢⢠⣄⢼⠣⡉⠄⠀⠀⠀⠀⡠⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠐⠈⠀⢱⡈⠁⢠⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⣅⡠⠄⠀⣃⠈⢻⡀⠀⠀⠙⠓⠢⢄⡀⠀⠀⠀⡇⠠⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠃⠀⠣⠀⠙⠂⡴⠁⠀⡠⠠⠌⡁⠒⡼⠧⣅⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣀⠀⠀⠀⠀⡠⠊⠀⠀⠘⢧⡀⢀⠴⠋⠀⠀⠀⠈⠑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠰⢋⠀⠀⢦⡀⡀⠄⠋
⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣿⠛⠐⠠⡀⠀⠈⠀⠐⠉
⠀⠀⠀⠀⢀⣮⠊⠀⠵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠐⠃⠀⠀⠁⠀⢀⡔⠁
⠀⠀⣀⡶⠋⣀⠸⣁⠀⠈⠠⠄⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠡⠀⠀⡀⠀⠀⠑⣤⡀
⢀⠀⠛⠀⡀⡥⠙⢍⡠⢔⠀⠀⠀⠉⠂⢄⠀⠀⠀⣠⠂⢂⠀⠠⡑⢵⡌⠢⠔⠉⠀⠙⢦⡀
⠙⠒⠂⠉⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠱⣄⠎⠀⠀⠈⠑⢄⠈⢢⠝⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢱⢇⡀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⠺⠤⠴⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⣈⡀⡉⠀⡰⠀⠀⠈⢦
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⡈⡆⠈⢆
⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⣄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⢷⡀⠘⡄
⠀⠀⠀⠀⠐⢧⣀⣝⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⠊⠀⠀⠀⢰⣿⣿⡿⠛⠉⠀⠀⠘⡧⠒⠓⢄
⠀⠀⠀⠀⠀⣀⣵⣿⣿⣦⡈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆
⠀⠀⠀⠀⣼⣿⣿⡟⠋⠀⠈⠢⡀⠈⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⡆⠀⠀⠀⠈⠻⣿⣶⣤⡤⠤⠺⣧⢀⣀⠀⢸
⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⠀⠘⢷⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣀⣀⣤⡤⠄⠽⠿⣿⡃⠀⣀⡬⠃⠈⠳⡜
⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⣀⡠⠾⢤⡔⠒⠢⠽⣄⡀⠀⠀⠀⠀⠀⠈⢇⢰⠉⠀⠀⠀⠀⠀⣰⠃⠉⠉⠁⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢣⠀⠀⡰⠁⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠈⠳⢆⣀⣀⣀⡤⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣠⠔⠚⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣅⣀⣀⣀⣀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣠⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠈⠑⢤⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⡠⠦⡀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⠀⠀⢀⠴⠒⢦⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢰⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣠⠊⠀⠀⠙⢆⠀⠀⠀⢁⠀⠀⢠⠃⠀⠀⣠⠞⠁⠀⠀⠀⠈⠲⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠼⣄⠀⠀⠀⠀⢠⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⢣⡀⠀⢸⠀⠀⠸⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣸⠁⠀⠀⢀⡏⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⡇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⠀⠐⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢂⣚⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡾⣒⣚⡻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⣒⣒⢤⡀



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠘⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣟⠳⣦⣀⠀⠀⠀⡠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⢶⠀⠀⠀⠀⠐⣼⠀⠀⠀⡠⠔⠙⢛⣿⣄⠀⠀⢈⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠁⠀⢰⠉⢃⠀⠰⠎⣸⠏⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣀⠄⢹⣿⣷⣖⢾⢺⣿⣛⡆⠀⠀⠀⠀⡀⠀⢀⠈⡆⠈⢤⠄⡀⣇⡃⠾⠹⠂
⠀⠀⠀⠀⠀⠁⣠⣤⣿⣿⣯⣮⣿⠿⢟⡿⢠⡿⡄⠀⢸⣧⠀⣌⣀⠱⣀⣘⡀⣻⢻⣇⠀⣠⡤⠖
⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⠜⠟⠉⠠⠏⠁⣼⠁⠣⠀⠎⢿⠀⠙⠛⣧⣤⣭⣴⢏⡼⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠁⠀⢹⣤⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢦⣾⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⠃⠘⠇⠄⠀⠀⠀⠀⠀⠀⢲⠈⡀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠾⡁⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠼⠃⠀⠀⠀⠀⠂⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡸⢇⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠉⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠑⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠘⠒⠒⠣⠤⠤⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠑⡀⡂⠀⠒
⠀⠀⠀⠀⢠⠀⠀⢈⣬⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠻⣇⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠉⣧
⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠁⠀⠐⠃⠒⠀⠀⠀⠀⠀⠀⠒⠚⠒⠃⠠⠂⠀⠀⠀⠀⠀⠀⠂⠀⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠖⠋⠉⠉⠙⢷⠖⢀⣤⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣏⠁⠀⡀⠀⠀⢀⣤⠾⠛⡁⢹⡆⠉⠉⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢏⣟⡵⠛⣽⣿⣷⣾⣏⠁⠀⠈⠁⠀⠁⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣼⢃⢊⡾⠋⠀⠙⠋⠃⠀⠀⢦⣀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢁⣼⡇⢠⡁⠀⠈⠀⠀⠀⠀⠈⠑⠢⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⣿⣟⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠠⠤⠤⣀⡀⠈⠉⠻⢿⡏⠀⠀⠀⠀⢀⣀⡤⠴⠋⠀⢠⠔⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⣼⠇⠀⠀⠀⠀⣿⠁⠀⠀⠀⢠⢧⣴⣡⠜⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠰⣇⠀⠀⠀⠀⢣⡀⣸⢧⡖⠊⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣠⠶⠶⠦⣤⣀⠀⠀⠀⠈⢿⣄⠀⠘⡏⢷⣦⡀⠀⠀⠹⣇⠀⡟⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⢹⣷⡀⠀⠀⢸⣿⡆⠀⠀⠘⣧⠙⢦⡀⠀⢹⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣾⠃⠀⠀⠀⠀⢀⣾⡏⠙⠄⠀⢸⣿⣿⣄⠀⠀⣿⡄⠀⠱⡄⢸⡇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⣿⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⢸⣿⣯⣽⡶⣾⣿⡇⠀⠀⠘⡿⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠹⣆⠀⢀⣾⣻⣿⡟⣇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⣸⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⡏⢳⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣞⣠⠖⣰⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠻⣄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠁⠀⢈⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠻⠶⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡟⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⠀⢸⡇⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣃⣤⣤⠶⠶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⡇⠀⢸⡇⠀⠀⠀⠘⠻⢿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡾⠇⠀⠘⢷⣤⣤⣄⠀⢀⣀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⡄⡆⣀⠀⠀⠉⢁⠒⠲⡇⠊⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⠁⠀⠀⠀⠀⠀⠈⢣⠀⠀⣀⣤⣴⢿⣏⠀⣀⣀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡶⠛⠋⠉⢀⡿⠛⠛⠛⠋⠉
⣤⡀⠂⢰⣶⣄⡀⣀⣀⣀⣀⣀⣄⠀⣀⣀⣀⠘⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣧⡶⠶⠶⣶⠟
⢹⣇⠀⡫⢛⠀⠈⠉⠉⠉⠁⢸⡏⠀⣿⠈⠉⠛⠺⣄⠀⠀⠀⠀⠀⣼⠇⠀⠀⢸⡇⠀⠀⠀⣴⠏
⠈⢻⣆⠀⠀⢀⣀⣀⣀⣀⡀⢸⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠀⠀⠀⢰⡏
⠀⠈⣿⡆⠀⢸⢩⡏⠉⠉⠉⠉⠛⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⠀⠀⠀⣼⠳⠤⠤⡲⠎⣿⡇
⠀⠀⢿⣷⠀⠘⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠲⠤⣤⠞⠁⠀⠀⣠⣾⠃⠀⡠⠔⠉⠁⣼⣷⣂⣴⡿⠃
⠀⠀⠈⣿⣶⢧⡄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡴⠚⠁⠀⠀⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠈⣿⣿⡁⠞⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣟⣀⣤⣴⠾⠟⠉
⠀⠀⠀⠀⠈⠹⣝⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠹⠂⠀⠠⠖⠉⠀⠀⠀⠐⠂⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠊⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡰⠁⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠸⠁⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠣⡄⠠⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠰⠒⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⡀⠤⠈⠁⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡠⠀⠀⠀⠑⠢⣤⣀⡀⠀⠀⠀⢀⣀⠠⠔⠊⠉⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⢳⠀⢀⡀⢀⠔⠹⢸⠀⠀⠀⠩⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢤⡈⠃⠀⠈⠉⡜⠀⠘⡀⠀⠐⠁⠀⢀⠀⠀⠀⣠⠚⠓⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⡀⠀⠀⡠⠃⠀⠈⠉⠁⠀⠀⠀⠉⠄⣀⠈⢘⠀⠈⠢⠦⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠁⡀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠈⡀⠈⠢⢀⢀⠀⢀⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⠄⡀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⢱⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣠⡂⠀⠀⠈⠢⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠑⠢⡾⠁⠈⠆⠀⠀⠀⢠⢾⡼⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠑⢄⠈⠢⢤⠀⠁⠁⠄⠀⠀⠀⠀⠔⡶⡥⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠂⣠⠄⠀⠀⠀⢣⠀⢀⡠⠴⠊⠀⠂⠈⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠀⠀⠀⠀⠀⠙⠣⢄⠀⡀⠂⢀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣧⠤⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡂⠈

⡿⠿⠿⠿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢍⣀⣀⠆
⠀⠀⢀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⡄⠐⠄⠀⠀⠀⠀⠙⠇⢠⠏⠁⠀⢠
⠀⠀⠀⠠⠀⠀⠈⠢⠄⠀⢀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣡⠀⠴⠴⣏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⢡⠃⠀⠀⠈⠁⠀⠳⠒⠲⠀⢵⠞⠉⠂⠚⠃⠀⠀⢐⠀⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠼⠀⢰⡀⠠⠗⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⡀⠠⠄⠀⠀⠀⠉⠘⠈⠱⠤⡴⠶⠶⠚⠿⢹⠗⠄⠀⠀⠀⠀⠀⠐⡄⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠻⠴⣄⠀⠀⠀⠀⠀⢡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⣰⡀⠘⡄⣸⡄⠀⠘⠛⠏⢃⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⣀⣀⢠⡀⠀⠄⠀⠀⠀⢠⡀⢀⠀⠇
⠀⠀⠀⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠈⠑⠀⠂⠀⠀⠀⠀⢙⠘⠻
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠺⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠖⠒⠉⠉⠉⠲⠚⠉⠉⠉⠓⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠉⠙⠲⠤⣄⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠋⢀⡀⠀⠀⣀⣀⠀⠈⠑⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠠⠦⢄⠀⠀⣠⠋⠲⠃⠞⠁⠈⠀⠀⠁⠀⠤⣀⠀⠀⠳⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣧⠸⠀⠙⠶⠁⡗⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⡀⠀⣠⠜⣗⠒⠀⠀⠀⣇⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⡴⠁⠀⠸⣄⠀⠀⠀⠘⢆⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠙⠒⠒⠚⠁⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⡤⢄⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣀⣠⣠⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⡘⠀⢈⣄⠈⠄⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⢤⠶⢦⠐⠀⠀⠗⠦⣤⣄⣠⣀⠀⠀⠀⠀⠀⢸⠀⢠⣀⣠⠀⢰⢸⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠆⠐⠂⡏⠀⠀⠀⠀⠀⠀⠙⢌⠣⡀⠀⠀⢀⠃⠀⠀⠀⠉⠢⢄⠀⠀⠈⠣⣀⠱⢤⠦⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⣀⣠⠂⠀⢀⣀⣀⠀⠀⠀⠀⠳⡌⢤⢀⠎⠀⠀⠀⠀⢱⠀⢀⣁⡦⠀⠀⡄⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡎⠀⠼⠖⠋⠁⠀⡇⠀⠀⠀⠀⠀⠈⠦⠂⠀⠀⠀⠀⠀⠀⢷⠉⠙⠦⣠⠎⠀⡸
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠁⠀⢠⠃
⠀⠀⠀⢀⡀⠄⠐⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠦⢤⠞
⠀⠀⢰⡁⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠴⠖⠀⠀⠀⠀⣤⡀
⠀⠀⠈⠣⡄⢀⠀⠆⣠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡠⠊⡀⠈
⠀⠀⠀⠀⠈⠙⠚⠳⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⢀⡜⠀⠀⢳⠀⢀
⠀⠀⠀⡠⢊⠔⡉⠐⡄⠀⠀⠀⠀⠀⠀⠀⢺⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠚⠉⠑⠦⣄⡀⠀⠀⠀⢀⡞⢢⠆⠀⠀⢀⠆
⠀⠀⡔⡱⠁⠘⠤⢀⣼⡀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠈⠉⠉⠉⠉⠉⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠎⠀⡼⠀⠀⠀⡌
⠀⠌⡘⠀⠀⠀⠀⡎⠀⡈⠑⠲⢄⡀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⢀⣠⠤⠀⠦⢄⡀
⡘⡰⡇⠀⠀⠀⣰⠉⠚⢅⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⢒⡧⠊⠁⠐⡂⠀⠀⠂⠙
⣷⠁⠘⢄⣀⡱⠁⠀⠀⠀⠑⢦⡀⠀⠈⢆⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⢊⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠻⣄⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈⣦⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⢀⠔⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠁⠀⣀⠠⠤⠒⠊⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣧⣷⣤⣶⣵⣾⣽⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⢶⠶⠶⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡬⣀⠞⡣⠂⡡⡠⣣⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⣡⣮⣴⢞⣿⣞⣉⣙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣽⢿⣷⣿⠈⠛⠨⢤⣥⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠙⢿⣿⣦⠌⠉⠠⠹⠤⣸⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠚⢉⣥⡈⠛⣦⢿⡋⠀⠀⠀⣀⣀⣉⡷⠀⠀⠀⣤⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⣐⣰⠛⠛⣋⡴⡿⢉⡱⠀⣈⠥⠶⠿⠿⠭⠵⠖⠒⠒⠶⠞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢙⣫⢶⡛⠥⢃⠜⠃⠠⠋⠠⢤⣶⠶⠶⠶⠶⠒⠒⠚⠚⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣁⠀⠈⠉⠁⠀⠀⢘⣤⡶⠛⠁
⠀⠀⠀⠀⡀⣀⣠⡄⠀⠀⢠⣿⣿⣿⣟⣿⣷⣶⣶⡷⠾⠯⠥⡤⣤⣀
⠀⠀⢴⣿⣿⣧⣤⣿⡀⠀⠈⣿⣿⣿⣿⣶⣭⣿⣿⠃⠀⢀⢀⣤⣤⣈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢹⠘⣷⡀⣴⣿⣿⣿⣿⡿⣿⣿⡿⠿⠶⠟⠛⠉⠈⠛⢶⣤⣌⣙⠢⣄⣤⣶⡽⠇
⠀⠀⠀⠀⠀⠀⠸⡇⠘⣿⠃⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠐⠃⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡄⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠘⠀⠀⠀⠀⠐⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣶⣿⣿⣿⣦⣤⣄⣀⣄⡠⢆⠀⠀⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⠛⠉⠥⠀⠀⠉⠁⠉⠁⢄⠘⢻⣶⣾⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡤⣦⡀⠀⠀⠀⣠⣄⠁⠈⣿⠏⠀⠀⠀⠀⢀⠀⡐⠁⠀⠀⠀⠀⡀⢀⣀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⢸⡄⠀⠀⠛⠁⠀⠀⠈⠱⠿⠃⢰⡏⠀⠀⠀⢀⠊⢀⡼⠀⠀⢀⠼⠿⠿⠉⠉⠛⠚⠿⠞⠛⣷⣦⣀
⠀⠀⠀⠀⢀⠀⠄⠀⠀⠁⠈⠀⠁⠀⠀⠀⣄⣀⣀⣤⠀⠀⠀⠈⠠⠁⠀⠀⢸⣠⣿⣧⣦⣴⠏⠀⡠⣤⡄⠀⠀⠀⠈⠃⠀⠻⠿⣷⠂
⠀⠀⠀⠀⢹⠀⠀⠀⠈⠀⠀⠀⠈⠢⠀⠀⠙⠋⠙⠃⠀⠀⠔⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠻⠀⠀⠸⠟⠀⠀⠀⢠⣿⠂⢀⣆
⠀⠀⠀⠀⠸⠀⠀⠀⠰⠀⠀⣀⠀⠀⠚⠉⠗⠀⠀⠛⠉⢀⠂⠀⠀⠀⠀⠀⠀⡠⠘⣷⣶⣆⠀⠀⠀⠀⣤⡀⠀⡀⠁⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠐⠎⠀⠈⠈⠁⠀⢀⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠠⠀⠀⠐⡀⠁⣸⡿⠿⠷⣀⡀⠀⠉⠻⠟⠁⠀⣀⣾⣆⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠈⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠢⠀⠀⢀⡅⠀⠏⠀⠀⠈⠉⠉⠀⠐⠒⠶⠞⠛⠿⠟⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠄⠀⠹⢻⡀⠀⢃⣰⣿⠷⠄⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠁⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢡⠚⠙⢣⠀⠈⡘⢿⠀⠀⠀⠐⡴⢠⠂⠀⠀⠀⠀⠀⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣇⣰⡋⠀⠀⢡⠀⠀⠀⠀⠀⠁⡰⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⠖⠤⢄⡀⠀⠀⠀⢀⣠⡄⠱⣄⣀⠌⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠢⢀
⠀⠀⠀⠀⠀⠐⠉⠀⠁⠀⠀⠀⠀⣸⠴⠒⠲⢾⣿⣿⡿⡏⢘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡿⢿⣿⣿⣄⡼⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠔⢒⣶⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠐⡖⠊⢭⠋⠁⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⡀⠱⢿⠀⠀⠀⠀⠀⠐⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⢠⡀⠀⠑⠦⢠⠄⠀⡇⠀⠀⠀⠀⠀⡠⠄⠀⠀⡄⠀⠀⠙⢄⡊⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢰⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠳⢄⠁⢳⣀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡼⢇⣀⡀⠔⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡔⠀⢀
⠀⠀⠀⠀⠳⠀⠘⣤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣈⢯⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⠇⠀⢠⠷⡄
⠀⠀⢀⠙⠻⠳⠟⠿⠙⠃⠐⠀⠀⠀⠀⠶⠀⠠⠀⢀⠋⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⣀⠀⠀⠀⠀⠀⠀⠘⠄⠠⠔⠃⢀⡀⠀⡀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⢑⡀⠀⠀⠀⠀⠄⠡⡀
⠀⠀⠀⠚⠀⠀⠄⠤⠤⠀⠀⠁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠁⠀⠀⠚⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠤⠬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠾⠿⠷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣽⠀⡇⠀⡇⣿⣦⣿⠀⠀⢸⠈⢻⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡤⠂⠈⡒⢃⡀⢈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠸⠃⠹⠀⠛⠞⠃⠛⠈⠛⠀⠀⠸⠀⠸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠉⠂⠁⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠾⢠⠔⠁⡄⠀⠀⠀⢰⡶⡻⣄⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡆⢠⡅⠀⣀⣀⣀⢀⣀⡜⠀⠘⢯⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢏⡄⢠⠙⠋⡱⣈⣉⣉⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⡇⠚⢋⣡⣉⣡⣬⡑⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⡟⠋⠁⠈⠁⠀⠉⢸⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣀⣀⣤⣶⣾⣿⣿⠀⠀⠀⠀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⣤⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠈⢹⣿⢹⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡏⡞⢻⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⡇⡁⢸⡿⢰⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣛⠁⡇⢸⡇⡘⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⢁⢸⡇⠇⠀⠀⡠⠀⣠⣾⠏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠈⢸⠀⠀⢀⣴⣣⣾⣿⣡⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣸⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠰⢫⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⡀⠀⠀⠀⢸⠟⠀⠀⢳⣿⣿⣿⣦⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣾⣿⣦⠊⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡀⠀⢶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠠⣤⣹⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⠖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠁⢀⠀⠠⢈⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣇⠀⠐⣿
⣸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠞⠃⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧⠀⠀⠀⢀⣼⣿⣿⣷⣄⡈⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⠄⢻⣿
⣿⡆⠀⢠⣧⠀⠀⡀⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢠⣿⠿⠛⣻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠉⠙⣿⣿⠉⠉⠉⣽⣿⣿⡤⣾⣿
⣿⡗⠀⣸⣿⡀⢰⡇⠀⠀⢀⣸⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢾⣿⣿⠶⣿⣿
⣿⣿⡀⣻⣿⡃⣼⣧⠀⠀⠠⣽⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠘⢿⣿⣿⡿⣿⣿
⣿⣿⣽⣿⣿⣿⣿⣿⠄⠀⠴⣾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣭⣿⣿⣿⣿⣿⡃⢀⣤⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢠⠀⢀⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢱⣾⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⢄⡄
⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⢠⡴⠶⠾⠋⢹⣷⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢯
⠀⠀⠀⠀⠀⠈⢣⡏⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄⠀⢰⠿⡆⠀⠀⠀⠀⠀⠀⡏⢻⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⢿⠸⡄
⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠘⠆⠀⠀⠀⠀⠀⠀⠐⠁⠘⠀⠀⠀⠀⠀⠀⠀⠲⠒⠲⠾⠷⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⠃⠀⠀⠀⠀⠀⠀⢰⢿⣿⡶⠓⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣆
⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡿⣏⡏⠀⠀⠀⠀⠀⠀⠀⠸⢶⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡁⠸⣿⡄
⠀⠀⠀⠀⠀⣸⠉⢹⠁⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⡼⢻⠀⠀⠀⠀⠀⠀⢠⠋⠀⡼⠃⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢘⣇⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⢨⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣤⡦⠀⠀⠀⠀⠀⠀⣹⡟⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠠⣶⣿⣿⡧⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠿⡇⠀⠀⠀⠀⠀⠀⠰⢿⡧⠄⠀⠀⠀⠀⠀⢰⣿⣾⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠹⢿⠿⠃⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧

⠀⠀⠀⢀⡄⡀⢀⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠳⣧⣼⣥⡷⠉⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠐⠒⠚⢻⣿⠛⠒⠒
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠶⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠃
⠀⠀⠀⠀⠀⠀⠇⠧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠛⠲⠂⠀⠀⠀⠀⠀⢸⠁⠈⡆⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠛⠉⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠂⠀⣰⢷⠀⢹⣯⡹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣢⣶⠜⠋⠁⠀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡏⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠉⠀⠒⢦⣴⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⣠⣆⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣇⣧⣴⣿⣷⣽⡟⢦⡐⠓⠀⠚⠘⡛
⣂⣀⠤⠆⠀⠀⠀⠀⣀⣀⡀⠀⣀⡀⢀⣤⣶⡶⠖⠈⠉⠉⠀⠸⠛⠛⠛⠛⢛⣿⣿⣿⣿⡿⣿⣧⣸⣿⠷⢿⣿⣷⣾⡷⢄⠰⣬⡼⣽
⣿⣿⣿⣿⣶⣿⣿⣿⣾⣷⣥⣭⣤⣸⣿⣿⣿⣶⣂⣤⣄⣀⣀⢀⣠⣤⣴⣴⣿⣿⣿⣿⣞⣏⡀⣹⣿⣿⣷⣿⣿⣿⣿⣿⣷⣽⣛⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿