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⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠌⠀⠀⠀⠀⠀⠂⢀⡀⣸⢀⠘⠃⠄
⠀⠀⠀⠀⢀⠀⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠄⠄⢠⠰⠠⢠⠀⠀⠀⠀⠀⢸⠁⠁⠇⠀⠀⡜⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠐⠂⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⡄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠈⠀⠀⠀⠀⢠⢠⠘⠐⠋⠈⠈⠀⢴⣂⠀⠀⢀⣛⢨⠀⠐⢀⡖⡛
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠞⠖⠶⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⡀⠀⠀⠀⣀⢀⠀⡀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡉⠐⠀⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠓⠲⠄⢀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⡰⠋⠀⣠⣄⠀⢀⡤⠶⠶⠶⢶⠿⠖⠚⠁⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢰⠃⠀⠸⡇⠼⢀⢤⡀⡴⢲⢇⡏⡀⠀⠀⣠⡀⡴⡆⠙⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⣠⠀⠘⡆⣟⣫⠾⠷⠛⠚⠛⡁⠀⣼⡽⣼⡟⣁⡤⠞
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⠀⠀⡀⣇⣀⣰⠇⢰⣦⠀⢠⣆⢀⣞⡽⣸⠋⣴⡟⠉⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡃⣸⠀⣸⢁⣄⢉⡄⡰⣨⢻⢏⣞⣼⠛⠧⠞⠉⢻⢧⠇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠻⢴⣿⣏⡿⢾⠼⠷⠃⠘⠋⠈⠋⠀⠀⢠⠇⠘⠋⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢠⠞⠀⠉⠠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠖⠉⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣤⠴⠋⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢁⡄⡀⠀⠀⠀⠀⡀⠄⠀⠄⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠃⠘⠈⠃⠀⠀⠈⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⠂⠉⠙⢷⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠋⢸⡟⠀⠀⠀⢸⣿⢠⡄⠀⠀⠀⠘⠟⠀⠀⠀⢀⠀⠀⣸⡏⠀⢰⡄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣿⠁⠀⠀⢀⣿⠇⢿⣁⡀⠀⢠⡿⠁⣼⠃⠀⢸⡇⢀⣿⢀⣠⠟⠰⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⣾⠋⠀⠀⢠⡟⢀⣿⠃⢰⡿⠀⢀⣿⠁⣸⡇⠘⣧⠀⠀⣿⡦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣠⠟⠁⠀⠀⠀⣿⡠⠊⣿⠠⠉⠛⠀⢸⡏⠀⠙⠋⠀⢻⡄⡠⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠫⠐⠋⠀⠀⣤⣤⡀⣤⠀⣤⣤⡄⠀⢠⣤⡀⢈⣥⣤⠀⣤⣤⡌⢿⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠓⠀⠀⣿⣸⡇⣿⠀⣿⠶⠀⠀⣿⣹⡇⠘⢷⣄⠀⣿⣤⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⣤⣟⣛⣃⢠⡿⠉⣿⢠⣤⣽⠇⣿⣀⡀⠠⡈⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢠⠠⡀⠞⠠⣤⣤⣤⡤⣤⡬⣭⠭⠤⣤⢤⣤⡤⠤⢤⣤⣤⣬⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠁⠀⠁⣀⠂⠙⣿⣿⡇⢛⠃⣽⠀⣛⣿⢸⣿⡇⠘⣀⣿⣿⡋⠀⠠⢀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠋⢠⣾⣿⣿⣇⣿⣇⣿⣀⣋⣿⣈⣛⣇⣸⣿⣿⣿⣿⡄⢀⣴⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢀⡀⣤⠀⠀⢀⡀⠀⠀⣠⠀⣄⠀⣾⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⣶⠀⣠⡆⢰⡄⣼⠇⢀⣾⠁⡯⣀⠀⢸⡇⠀⣼⣇⣴⠇⢸⡇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⢰⡿⠔⣿⠁⣼⢱⡟⠀⠸⣏⠀⢀⡟⢠⢘⣷⠈⣿⠀⠀⡠⣿⠀⢸⡿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⡿⠁⠀⠉⠈⠀⠀⠁⠀⠀⠀⠀⠘⠗⠁⢈⡿⠀⠈⠃⠈⠀⠛⠀⠸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣄⣀⢠⢄⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⢋⢀⠉⠈⠁⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⢀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡷⢮⣀⡇⠸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⣀⡀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⣿⣇⣀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⣿⣿⣿⣷⣶⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣀⣽⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣽⢁⣀⣹⣿⡂⣽⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣏⣸⣏⣹⠟⠟⢻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠤⠤⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠑⢀
⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⡂⢀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⣠⠔⡁⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢀⠠⠐⠒⢢⠀⢀
⠀⠀⠰⠊⣀⠔⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⢀⠀⣠⠄⠁⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠠⠜⠁⠀⠀⠈⠢
⠀⠀⠰⠊⠁⡠⠂⠀⠀⠀⠀⠡⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢠⠄⠂⠀⠀⠠⠈⠀⠀⠉⠁⠀⠨⠀⠀⠀⠂⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⢀
⠀⠀⣠⠔⠈⢀⡤⠀⠀⠀⠀⠀⠁⠀⠀⢀⡗⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⣈⠀⠀⠀⢠⡖⠃⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠁⣠⠔⢁⠔⠀⠀⠀⠀⢀⠄⠀⠀⠈⠀⠀⠀⠀⢀⠠⠰⠋⠀⢀⠀⠀⠀⠀⠀⠠⠈⠀⠀⡀⠀⠀⠀⠂⠀⣀⠀⡄⠀⠐⠂⠨⠁⠀⠀⣀⣀⠂
⠀⠀⢛⡠⠄⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠤⠤⠀⠂⠁⠀⠀⠀⠊⡰⠀⢀⠤⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠚⣉⠀⠲⠒⢂⡀⠡⠀⠀⠀⠀⠀⠰
⠀⠀⠈⣁⠤⠊⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⡐⠉⢀⠴⠦⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⡨⠗⠀⠁
⠀⠀⠀⠁⡀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠠⠄⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⢀
⠀⠀⠀⠋⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠐⢅⠀⣱⠀⠑
⠀⠀⠀⢠⠄⠒⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⢈⠀⠈
⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠐⠖⠒⠒⠂⠀⠪⢉⣀⠀⠈⠁⠑⠀⠀⢂⡰⠀⠐⢤⣀⠀⠐⠂⢐⣀⡁⢤⡄⣂⠈⠰⡀
⠀⠀⠀⠀⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠀⢰⡅⠚⠀⣤⢄⡈⡀⢠⠀⡀
⠀⠀⠀⠀⣄⡠⠐⠁⢀⠠⠒⠁⠀⠀⠀⠈⠄⠀⠀⠀⣘⢥⠆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠀⠀⠀⠐⠨⠁⠤⢀⢀⡀⠐⡀⠠⢸⠰⣆⠥⠂⡒⠸⠆
⠀⠀⠀⠀⠁⠀⡀⠔⠁⠀⢀⠔⠂⠀⠀⠀⢰⠀⠀⠀⠺⠈⡆⠀⠰⠦⠉⡄⠀⠀⢈⣢⠄⠐⠀⠀⠀⠀⡏⠐⣀⠂⢘⠀⢀⠠⠄⡐⠚⠉⠀⠀⣄⠊⡐
⠀⠀⠀⠀⠀⠉⠀⠀⡀⠔⠁⠀⢀⠄⠀⠀⢸⠀⡀⡀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠸⠈⠀⠀⢹⠀⠠⣊⣵⣉⠀⠄⡈⠘⡃⣀⣁⢇⠨⠶⠍⢭⡍⠠⠑⠃
⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⡠⠂⠁⠀⠀⠀⡀⠠⠁⠤⠀⢀⡀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢀⡈⡆⠀⠀⡈⢩⣛⣄⡆⢀⢀⡀⢐⠠⡽⢗⡠⠈⠒⠃
⠀⠀⠉⠀⠀⠀⠀⠀⠄⠊⡀⠔⠀⠀⠀⠠⠁⠀⠀⢰⠉⠇⠀⠀⠀⠇⠄⠀⠀⠀⠀⠂⠀⣠⡄⠧⠀⠈⠀⠀⠀⠀⠀⠘⠐⠊⠉⠀⠁⠉⠑⠂⢄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠤⠂⠀⠠⠃⠀⠀⠀⠈⠀⠒⠈⠀⠀⠀⢀⠀⠀⠀⠆⠠⡜⠂⠶⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠒⠤⣀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⡀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⡤⠀⢙⡀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠺⠇⠀⠘⠘⠀⠀⠉⠈⠀⠒⠀⠄⠉⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢰⡄⡀⣄⡄⠀⣀⣠⡀⡄⢀⠀⣄⣤⡀⢠⠀⣤⡀⡄⠀⡀⢀⣀⡀⡄⢄⣀⢀⡖⠀⡤⢠⡀⠀⡀⣀⡀⣄⠀⡀⡀⣄⣀⠀⠰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠢⡁⠀⠀⠀⢀⡀⠀⠀⢠⠤⠤⠤⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣉⣀⡀⠐⢂⡉⡀⠀⢀⣀⣠⣤⣀⡀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⡶⠚⢻⣯⣤⣤⣤⣄⣙⣏⡋⢉⠋⠓⠒⠚⠛⠛⠛⠒⠒⠒⠉⢠⢐⡰⠃⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⣿⣷⡐⠛⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣦⣄⣐⣂⡀⠄⠒⢺⣀⠠⠆⠠⢄⣿⣟⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠸⢬⣿⣿⡻⣿⣻⣾⣷⣶⣾⣾⣿⣿⣿⣻⣿⣿⡿⡿⡿⠿⠿⢿⣿⣻⡁⠀⠀⠀⠘⢻⣿⠄
⠀⠀⠀⠀⢠⠀⠈⠻⣿⣿⣿⣿⠻⢮⣿⣿⣿⣿⣿⣵⣿⣿⣯⣭⣽⣿⣿⠿⣿⣤⣴⡧⣷⢚⣷⣦⡴⡆⠠⣼⣿
⠀⠀⠊⡒⠘⡇⠀⡠⣽⣿⣿⣿⠷⣾⣿⣿⣿⣿⣿⣿⣯⣿⡛⢛⠛⢷⣶⣦⣽⣿⣶⣶⣿⣿⣿⣟⣷⣧⣰⣿⡿⠁
⠈⠉⠉⠉⠛⢾⣶⣿⣾⣿⣿⣿⣿⡿⢿⣯⣿⣿⡿⣿⣿⣿⣿⣷⣾⣿⡛⣻⣟⣻⣿⣿⣿⣿⣻⣿⣿⣿⣿⡿⠃
⠀⠀⡌⢥⣤⣀⣉⢶⣋⢽⣿⣿⣿⣿⣶⣽⣌⣙⣻⣿⢿⣿⢿⣻⣯⣿⣿⣿⣿⣿⣿⠿⢿⣟⣟⣾⣿⣿⣿⠁
⠀⠀⠐⢤⣔⡩⠁⠀⠉⠓⢶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣴⣻⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⣿⣿⣿⣿⣿⣲⢿⣿⣿⣯⣽⣿⣿⣿⣶⣖⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠿⢿⣭⣭⡵⠰⠭⠽⠛⠛⠋⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⢉⣉⣉⣉⢻⣭⣭⣭⡙⣩⠉⠉⠉⢹⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠔⠛⠀⢁⣠⣄⡈⠉⠙⢶⣤⣟⡂⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠀⢀⢀⠀⠙⠄⡰⢊⠀⠄⠀⡏⠻⡟⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣛⠘⠀⠿⢿⠍⢨⠰⠀⣧⠀⠙⣆⠀⢿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢣⣀⣈⡦⠴⠃⠀⣇⠀⠀⢿⣧⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠤⠶⠀⠳⠎⠱⠃⠀⢀⠀⠁⠀⣧⢸⣿⣾⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⣾⣿⣿⣶⡄⠀⡀⣴⣿⣿⣿⣶⣄⣿⣿⠋⢹⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣿⣿⣿⣿⣿⠇⢀⡀⠻⣿⣿⣿⣿⣿⠟⣧⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠛⠛⠛⠁⢲⣿⣿⣆⠉⠛⠛⠋⢁⠁⠸⡆⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⢣⣂⡢⠓⡄⢾⡿⢿⡇⠀⠀⠐⢂⣁⣀⣼⠏⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢿⡇⣛⡁⠀⣀⠀⠀⠀⠀⣼⣿⣿⡅⡆⢀⣸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⣸⡗⠀⡈⠉⠈⠙⠉⠁⣉⢹⠟⣿⣿⠚⠈⢱⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⢉⠹⡴⠢⣅⣀⢀⣀⡤⠖⠛⠓⣻⠏⣷⣇⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢙⣿⣄⠀⢂⡀⣠⠞⣁⠀⠾⢛⣵⣟⠺⠇⢸⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠈⡹⢿⡷⣄⡙⡏⢿⣿⣷⣶⠋⠀⢠⣄⢀⡘⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣤⣴⣿⣽⣿⣿⣿⣭⣤⣤⣤⣤⣥⣼⣦⣾⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⢿⡿⡿⠿⠿⠿⠿⢿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡦⣴⣴⣴⢤⡄⢀⣄⣄⢸
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡿⣿⣿⣿⢻⡇⢸⣿⣿⢰
⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠗⠻⠛⠻⠛⠃⠘⠛⠛⢸

⠀⠀⢴⣴⡆⡄⠀⠀⠀⠒⠆⠲⠆⠀⠀⠀⠐⣰⣤⠠⣴⠀⠀⠰⠦⠤⠆⢤⠄⠤⠀⠀⠀⡶⠤⣦⠁
⠀⢰⡄⡤⣴⡄⠀⠀⠘⠃⠒⠒⠒⠛⠀⠀⢠⣤⡤⣄⡀⠀⠀⣤⠠⢠⢠⢠⠤⠠⠀⠀⠀⠀⠦⠄
⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉⠃⠋⠋⠃⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣦⣠⣤⡤⡄⠀⠀⠀⢰⠰⠀⡆⢰
⠀⠀⣄⣤⣠⡄⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠦⠰⠀⠀⠀⡀⣘⡉⡁⡉⠀⠀⠀⠀⠲⠂⠂⠂⠀⠀⠀⠀⢄⣶⣤⡄⠀⠀⠀⠀⠀⠛⠺
⠀⠀⠟⠛⠛⠇⠀⠀⠁⠉⠉⠉⠙⠁⠉⠀⠀⡄⢂⢂⠙⠀⠀⢠⣀⣀⣀⣈⣀⣀⠀⠀⠐⠃⠒⠛⠛⡗
⠀⢠⢀⢀⢀⠀⠀⠀⠀⢠⢂⣀⢠⠄⠀⠀⠀⠉⠈⠘⠉⠀⠀⠐⠁⠁⠇⠀⠉⠙⠀⠀⠀⢀⠀⡀⡄
⠀⢈⠈⢉⠉⡹⠀⠀⣠⢀⣁⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⠿⠷⠿⠿⠶⠀⠀⣞⣙⣛⣧⡄
⠀⠘⠛⠓⠛⠃⠀⠀⠈⠉⠉⠁⠀⠉⠘⠀⠀⠹⠋⢙⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠻⠛⠃
⠀⠀⡀⡀⠀⠀⠀⠀⠀⢠⡤⢤⠠⣤⠀⠀⠀⠈⠙⠉⢹⠀⠀⠀⠘⣽⡧⡆⠶⡇⠀⠀⠀⠀⡠⢠⡄
⠀⠀⠆⠛⠘⠋⠀⠀⠀⠀⡶⠲⠰⢀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⣠⢠⢀⡄⢠⢀⠄⠀⠀⢀⢛⢘⡀
⠀⠀⠛⠺⠐⠖⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠚⠈⠟⠇⠇⠀⠀⠉⠘⠈⠃⠉⠘⠀⠀⠘⠈⠈⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠾⠿⠿⣿⣶⣿⣧⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⢠⣴⡶⣶⣄⠀⣿⣿⠛⠻⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢠⣤⣤⣴⣿⣯⣤⣾⡿⢸⣿⡏⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠛⠛⠛⣿⣿⠛⠛⢋⣁⢸⣿⣧⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣴⣷⠹⣿⣷⣴⣿⡟⠈⠻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠛⠛⠁⠈⢉⣛⠉⠀⣠⠄⢠⡶⠂⢰⡇⠀⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣴⢀⣶⢲⡄⣾⠀⣰⢰⡇⠈⡇⢼⡃⠀⣸⣇⠀⣼⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⡟⣼⡇⢸⢳⡟⠀⡟⣼⢡⡞⠁⠈⣿⡀⣿⠀⢀⡿⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⠇⣿⠀⡜⢸⡇⢰⠃⡿⠀⣿⡤⢀⡿⢰⣇⣀⠸⣧⡄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠟⠀⢿⣴⠃⠸⠧⠋⠘⠃⠀⠃⠙⢉⣀⣀⣀⠀⠀⣠⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⡀⠀⣀⣤⣤⣀⢠⣾⣿⣿⠿⡿⢼⣿⣿⣿⣷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⣉⣿⣿⠇⣸⣿⠟⢻⣿⡜⠿⣿⣷⣄⠀⠀⢰⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⢻⣥⣤⣿⣿⣶⣿⠿⠁⠀⠀⠙⣿⣷⠀⢸⣿⡇⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⣿⣆⡘⠉⠉⣿⣿⣍⣴⣶⡆⣤⣤⣴⣿⣿⠀⢸⣿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠙⢿⣿⣿⣇⠘⢿⣿⣿⠟⢸⣿⣿⣿⡿⠋⠀⠈⠻⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣯⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠈⢻⣻⣿⣧⣦⡀⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣄⠀⡀⣻⣿⣿⡿⣤⣶⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣿⣭⣟⣿⣿⣷⣦⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠝⠛⠛⠛⠛⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣬⠀⠀⣐⡆⠀⠀⠀⠀⢀⠐⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⣳⣦⣾⣿⣝⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠨⠀⢀⣬⣷⡄⢤⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡿⣿⣿⣿⡿⠁⡘⠉⠓⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⢨⣷⣿⣿⣿⣿⡟⢤⣂⣾⡇⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⠠⠜⣇
⠀⠀⠀⠀⢸⣿⣶⣤⣶⣿⡯⠈⣿⣿⣿⠝⠛⢷⡶⠀⠀⢤⣤⣿⠁⠀⠀⠀⠀⠀⣀⣤⡤⣀⣫⣀
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠓⠒⠛⠳⠾⠟⠿⠿⠿⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠁⠀⠈⠀⠀⠀⠀⣿⣿⣿⠟⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠈⢉⡐⢄⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠉⠀⠐⣀⠐⠀⣿⣿⣿⠀⢀⡂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠⡏⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⡂⠂⠀⠀⠤⠀⣿⢿⣿⠀⠒⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⣌⡀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣶⣶⣿⣓⣾⡟⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠀⠈⠉⠘⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⢿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣿⣿⣿⡆⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⠿⠿⠙⣿⣿⠷⣶⣦⣤⣄⣀⡀⣾⣿⣿⣿⣿⠇⠀⢀⣾⢿⣿⣇⠀⢀⣿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢐⠉⣿⣷⡂⠀⣿⣿⣶⠻⣿⣷⡍⠛⢗⣿⣴⣿⣿⣷⣦⣵⣿⣿⣿⣿⠿⠀⣿⣏⢹
⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⢈⠁⠀⣿⣿⣷⣰⣿⣿⣿⡀⠀⠀⠈⠢⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⡀⠹⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⢦⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠁⠀⡄⢨⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠉⡀⠉⠁⠇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠄⢼⠋⣸⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠁⢺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢸⢀⡟⡛⢿⣿⣟⣡⠤⠤⣤⠀⠀⡀⠈⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⠒⠚⠚⠓⠚⠛⠛⠃⠀⠀⠈⠓⠐⠃⠀⠘⠟
⠀⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⠀⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣻⣿⣿⡿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢟⡿⣿⢿⣻⢿⣿⣿⣿⣿⣯⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠋⠀⠈⠛⠛⠛⠿⣮⣿⣿⣿⣟⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠁⠀⠀⠈⠉⠙⠀⠹⣟⣿⣿⣽⡳⣯⡻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠤⠂⡇⠀⢆⡀⣀⡤⠀⠻⠩⢻⣽⣿⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠞⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠒⠒⠂⠀⠀⠛⠃⠀⣾⣿⣿⣽⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣸⡿⠿⢿⣿⣿⣿⣽⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣤⢀⣀⣀⣼⣟⣿⢰⣾⠃⢸⡏⡿⣫⢛⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣴⠿⡻⢿⡏⣾⠇⠞⢸⠁⣾⡟⣘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣵⣻⣿⡆⣷⣿⣾⡧⣾⣼⣜⡇⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⢸⢷⣿⣿⠟⣡⣶⢉⡇⣘⢉⡽⢟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣷⣉⣿⠸⣮⡽⢋⢸⡿⢃⡞⣰⢣⣾⢱⠟⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠈⣷⡿⢻⣯⠷⠶⣶⣿⣷⣾⣦⣾⡃⣿⠗⠠⠾⣫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⡇⢉⡴⣿⢃⣴⣶⠘⣷⠋⣿⣾⡿⢷⡶⠿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⣿⣿⡏⢸⣿⡟⢠⡟⢰⡿⣿⠇⣾⠃⣶⣤⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢾⣿⣿⣷⣈⣋⣠⢿⡇⢸⣿⡟⢰⡟⢀⠈⣤⡾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣟⣻⠣⣶⣽⡏⣙⣻⡿⣿⣦⣤⣴⣿⣇⣾⣇⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⣧⢻⣿⢘⣻⣿⢱⣿⣿⢃⣶⣽⣿⢿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣌⣛⣼⡏⢼⣿⡏⣼⣿⡟⣠⣽⣿⡿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣾⣧⣭⣹⣇⣿⣿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡴⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠉⣴⣋⡥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡇⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠻⠾⠏⠀⠀⠀⠀⠈⠙⠛⠲⠤⣤⡴⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢉⣠⠾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡞⠁⠀⠈⠓⢦⣜⣧⣀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⠇⡄⠀⠀⠀⠆⠀⣷⠉⠛⢻⡆⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⣐⣀⣠⣤⡄⠉⣰⢦⡼⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣬⡟⠉⠙⢦⣴⣾⣿⣯⣭⣷⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣶⢶⡄⠀⠀⠀⠀⠀⠀⢀⣤⡎⠀⠀⠀⣠⣿⢿⣻⡇⠀⣀⣀⣀⣀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢈⡿⠻⣷⠶⣄⡀⠀⢀⡶⠋⠘⣿⡶⢤⣴⡿⠋⠘⣧⣿⠉⠉⠉⣯⢹⡆⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣃⠀⠀⠉⠓⠉⠀⢀⡴⢯⣿⠙⠋⠀⠀⠀⠸⣜⣧⠀⠀⣿⢸⡧⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣤⠶⠋⠀⠘⢿⡀⠀⠀⠀⠀⠀⠈⢻⣀⡴⠿⣯⡀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣀⣤⠞⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠿⠓⠶⠖⠛⠋⠉⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⢀⣠⡴⠶⠒⠚⠛⠛⣷⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⣇⢀⣠⠴⢲⣶⡟⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣠⡏⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠙⠋⠁⠀⣀⡿⢷⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣭⣀⠀⠈⠉⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡤⠶⠛⠉⠀⠈⢷⣀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣶⣾⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣵⣿⣿⣿⠿⠿⠿⠿⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣻⣿⣿⣿⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⢋⣿⣿⣧⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⠀⠈⣿⡏⠀⠀⠠⠅⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠑⠈⠀⠀⠀⠄⠀⢠⢤⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠇⠀⠀⠀⠀⠉⠀⠀⠈⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢋⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠘⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠆⠀⠈⠒⠂⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡐⠁⢀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠎⠑⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢀⠃⠀⠀⠘⠀⢀⡠⠔⢺⠛⠇⠆⣠⠄⠀⠀⠿⣀⢈⡦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠠⠀⢀⡞⠀⠀⠀⠀⠃⠀⢃⠀⣸⡏⡈⣰⠁⠀⠀⠀⠀⢈⣯⡇⠀⢋⠉⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠁⠀⠀⠁⠀⠀⠀⠀⠀⢇⡈⢴⠃⡿⡨⢨⠁⠀⢀⡄⡄⠀⠀⠿⢒⡺⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣌⡑⠢⠄⢤⡀⠀⠀⠀⠀⠀⠀⠑⢌⠞⠐⠁⠀⠘⠦⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⣉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠁⠐⠧⡀⠀⠀⠀⠀⠀⠀⠈⡛⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠒⠴⠃⢀⡀⠀⠀⢠⠤⠉⠀⠂⠄⠀⠀⢀⡴⡇⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠈⠛⠲⠭⣑⠲⠠⢾⣠⣇⠀⠀⠀⡼⠉⠁⠀⠉⠒⢶⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠀⠂⠠⢈⡁⠂⠌⣱⢤⠀⠐⠈⠀⢹⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠁⡼⢳⡆⡰⠚⠉⠀⠀⠀⠀⠀⢇⠀⠀⢀⠄⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡄⣰⠇⡞⠈⠲⡤⠄⠤⠀⠀⠀⠀⠈⠆⣴⠃⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰⠐⡟⠸⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⢂⠊⠼⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⢀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⠀⢐⣨⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠆⠀⠀⠀⠁⠀⠀⠀⠀⠘⠒⠂⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠀⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢲⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⢰⠁⠀⡀⣠⣄⠀⠀⢧
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠠⠀⠀⠀⠷⠾⠿⠤⠤⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠭⠀⠀⡇⠀⣄
⠀⠠⡠⣠⡱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⢄⠀⠀⠀⠀⠀⠀⡠⠄⠊⣀⢴⠏
⠀⠀⡔⠯⢤⠀⠀⠀⠀⠀⡲⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠆⠀⠀⠀⡌⠀⣰⠟⠉⠁
⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⠄⠂⠁⠀⠀⠀⠀⠀⠧⠲⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠀⠀⠀⠀⠀⠀⠀⢀⠼⠜⢿⠉⠀⠀⠲⢀⠈⡘⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢲⠂
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠅⠴⠺
⢒⠆⠀⠀⠈⢠⠒⠄⢳⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⠒⡉⠀⠀⠀⠀⠀⢨⠂
⠈⠀⠀⠀⠈⠉⠔⢁⡼⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣖⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⣽⣳
⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠉⠐⠢⠤⠐⠂⠻⡍⠋⠉⠈⠋⠉⠁⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣨⢯
⠀⠀⠀⠀⠀⢠⢊⡆⠀⠀⠀⠀⠒⡆⠀⠀⠀⠀⠀⠀⠈⠁⠰⠶⡦⣤⡀⣀⡀⠄⡅⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⡱⠀⠀⢠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡓⡾⢿⠄⡀⠀⠈⠀⠀⠀⠊⣠⢄⠙⣆⠀⠀⠀⠀⡁⠀⠀⢿⡺⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡰⡽⠙⣓⣒⢫⠦⡈⠢⡀⠀⠀⠀⠑⠋⡠⢁⡟⠀⠀⠀⠀⠁⠀⠀⠈⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⡇⠀⡎⠀⢀⣀⠀⢈⠴⠇⠀⠀⠀⠀⡎⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⢠⡶⣿⣗⣢⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⠀⠇⠀⡇⠘⢊⡡⡄⡄⠀⠀⠀⠀⡌⠨⡀
⠀⠀⠀⠘⠛⢁⣼⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣁⣡⠀⣾⡫⠊⠀⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣗⣷⠈⠀⠀⠀⣀⠄⠂⠱⠀⠀⠈⠉⠉⠉⠀⠈⠙⠟⠛⠂⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⢀⣥⣥⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠤⠤⠄⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠄⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠹⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠄⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡠⠋⣤⠒⠐⢏⢆⠀⢉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠈⠉⠲⣴⢁⢀⣇⠀⠀⢸⠘⠤⠋⢢⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⢿⣫⣿⠟⢀⡜⠀⠀⠀⠳⡄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢘⣄⣏⡀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡖⠀⢀⡀⠤⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⢳⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⢺⠁⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠉⠉⠒⠢⢄⡀⠀⠀⠀⠀⢈⡆⠀⢸⣠⠤⢤⣀⣀⢀⢀
⠀⠀⠀⠀⠀⢠⠦⢤⡠⢄⣱⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠜⠁⠀⣸⡥⠦⣄⡀⡘⢋⡋⡗
⠀⠀⠀⠀⠀⣸⠂⠀⠀⣼⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠙⡇⠀⠀⢉⠻⣤⠮⠇
⠀⠀⠀⠀⠀⢇⣀⣀⣤⡄⠁⠙⠢⣄⣀⣀⣠⡤⠤⠴⠒⠋⠉⠀⠀⠀⠀⢳⠀⠀⡟⠳⣄
⣠⣄⣀⠀⠀⠀⠀⢀⣤⣙⠢⣄⠀⠈⠑⢦⡄⣀⡠⠔⢛⡩⠔⠂⠀⠠⣦⣨⣧⠀⣳⡀⠈⠷⣄
⠳⡀⠈⠉⠲⢄⡀⣼⣿⣿⢿⣿⡛⠲⢄⡀⠈⣁⡤⠚⢻⣆⣀⣤⣶⣾⣿⣿⣿⢾⣿⣿⣦⡀⠈⠛⠶⢤⣀
⠀⠙⠢⣀⠈⠀⠙⢿⣻⣿⠀⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣧⣼⣿⣿⡗⠙⠢⣄⣀⣀⣀
⠀⠀⠀⠈⠑⠦⣄⡀⠉⢦⠀⠀⠉⠉⠉⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⣆⠉⠉⢹⡏⠉⠉⢰⠖⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡔⠊⠁⠘⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠀⠀⠀⠋⠈⠑⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠈⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢮⡀⠓⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢁⡵⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⠀⠹⣷⣄⠀⠀⠀⠀⠀⠸⢓⣀⠠⠤⢰⡇⣿⡇⠤⠤⣀⡚⠇⠀⠀⠀⠀⠀⣠⣶⠟⠀⠜⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠈⢻⡻⣦⡀⢀⠔⠊⠁⣠⡀⠀⢸⠁⣿⡇⠀⠀⣠⠈⠙⠢⣄⢀⡴⣊⡿⠋⠀⠀⠀⠀⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠀⠀⠀⠀⠙⣎⠻⣷⣄⡀⠀⠉⠁⠀⣼⠀⣿⣿⠀⠀⠉⠀⠀⣠⠞⣩⣾⠏⠀⠀⠀⠀⠀⠀⠉⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠳⡌⠻⣿⣦⣀⠀⠀⡏⠀⣿⣿⠀⠀⣀⡴⠋⣡⣾⡟⠉⢣⡀⠀⠀⠀⠀⠀⠀⢀⡬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⠆⡜⠀⣀⠀⠙⢦⡈⠻⣿⣷⣤⡇⠀⣿⣿⣦⠞⠁⣠⣾⣿⠋⠀⢀⡀⠱⠐⡶⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠧⠄⠀⠀⠀⠀⠀⠀⡘⠀⠘⠛⠀⠀⠈⠳⣄⠈⠻⣿⠁⠀⣿⣿⡇⣠⣾⣿⠟⠁⠀⠀⠙⠃⠀⢣⠀⠀⠀⠀⠀⠀⠠⠴⠆
⠀⡄⠄⢠⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⣤⣤⣤⣤⣶⣶⣾⣶⣾⣿⣷⣄⣿⣿⠟⠛⠛⠛⠶⠶⠦⠤⣤⣤⣄⣸⣀⣀⠀⠀⠀⠀⠀⠀⡆⠀⢠⢀⡄
⠀⢠⡔⣋⠀⠀⠀⢺⠒⠲⠿⠿⠿⢿⣿⣉⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠋⠉⣉⣿⣿⣥⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⡿⠿⠿⠿⠖⠒⡷⠀⡠⢈⡃
⠀⠉⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⠉⠉⠉⠛⠛⠓⣲⣶⣶⢾⣿⣿⠈⠻⣿⠿⠟⢿⠛⠛⠛⠉⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡖⠁⠀⠀⠀⠀⠀⠀⢸⠀⢰⣶⠀⠀⢀⣾⣿⠟⠁⢸⣿⣿⠀⠀⣿⣷⣄⠈⠳⣄⠀⠀⢴⡆⠀⡎⠀⠀⠀⠀⠀⠀⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠞⠇⢡⡀⠀⠀⣰⣿⠟⠁⣠⠖⠋⣿⣿⠀⢸⠏⠻⣿⣷⣄⠙⢦⡀⠀⠀⡜⠰⠷⠀⠀⠀⠀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠀⠀⠀⠀⠀⠑⣤⣾⠟⢁⡴⠚⠁⠀⠀⣿⣿⠀⢸⠀⠀⠀⠙⠿⣷⣄⠳⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠀⠀⠀⠀⠀⠀⣴⠟⣡⣞⠉⠀⠀⣤⡀⠀⢹⣿⠀⡿⠀⠀⣤⠀⠀⠈⣻⢷⣌⢧⡀⠀⠀⠀⠀⠀⢤⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⣠⣾⡷⠋⠁⠀⠑⠢⢄⣉⠀⠀⢸⣿⠀⡇⠀⠀⢉⣠⠴⠊⠁⠀⠉⠷⣵⣄⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⢀⡴⠟⠁⠀⠀⠀⠀⠀⢸⡶⠈⠉⠉⠘⣿⢸⡇⠉⠉⠈⠲⡇⠀⠀⠀⠀⠀⠈⠻⢷⡀⢳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠄⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠄⣰⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⣦⡠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠃⠤⠤⠼⠧⠤⠤⠘⠓⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣡⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠻⣧⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣇⠀⡟⢳⣾⠟⢻
⠀⠀⠀⠀⠀⠀⣠⠀⠀⣿⢀⣇⣸⣿⠀⠘
⠀⠀⠀⠀⠀⠀⣿⠀⠸⠿⠿⠿⠿⠿⢿⡇⠀⢠⣶⣷
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣿⠹⡇⠀⠀⠀⠀⢀⣴⣿⣿⡿⠻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣦⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠉⠉⠉⠙⠻⢿⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠋⠀⠀⣀⣀⠀⠀⠈⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠘⢿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⠀⠀⠀⢠⣶⣿⣿⣆⠀⠀⣠⣄⠉⢻⡇
⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣀⠶⠛⠻⠿⢿⣿⣿⠇⠀⠀⠀⠈⠛⣉⣴⠟⠀⢸⣿⡟⢡⣿⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣤⣴⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⠉⣥⣾⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⣧⣄⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠉⠉⠛⠛⣋⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⣶⣤⣤⣾⠟⠁
⠀⣠⠾⢿⣇⠀⠀⠀⠀⢠⣿⢻⣧⡀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠶⣿⣁⠀⠀⠀⠀⠀⠀⣹⠇
⣴⠋⣠⣾⣿⣦⣄⣀⣴⡿⢿⡄⠹⣿⣆⠀⠀⠀⣠⣄⠀⢸⣿⣿⣆⠀⠀⠀⠀⠈⠙⠿⠷⠶⠶⢀⣼⡏
⣿⡐⠻⠿⣿⣿⣿⠋⠉⢿⡌⢻⣆⠈⢿⣷⣀⣾⠉⠉⢻⣿⣇⠘⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠹⣿⡀⠀⠈⢻⣟⣧⠀⠀⠹⣦⡘⢧⡀⠻⢿⣧⣴⣶⣛⣻⣿⣷⣤⡉⢿⣿⣦⣄⣀⣀⣠⣾⠟⠁
⠀⠘⢿⣦⡀⠀⠻⣯⣷⣄⠀⠈⠻⣮⡷⣄⣸⣿⣥⡶⠛⠋⠀⠈⢻⣿⣾⣿⠛⠛⠿⢿⣿⠃
⠀⠀⠀⠙⢿⣦⣄⠉⢿⣿⣷⣄⠀⠉⠳⠉⠹⣿⡃⠀⣠⣤⣶⣿⣿⡿⢿⣿⣦⣀⠀⠈⣧⣀
⠀⠀⠀⠀⠀⠉⠻⣷⣄⠙⠛⡹⣷⣤⣤⠀⠀⢻⣿⣾⡿⠛⢻⡟⠛⠀⠀⢹⣿⠛⢿⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡏⢻⣿⠁⠀⠀⣾⢻⣿⠀⠀⢸⡟⠀⠀⠀⢸⣿⠀⠀⠈⢻⠁⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣷⣾⣟⠀⠀⠀⣿⠘⡇⠀⠀⠸⣿⠀⠀⠀⣸⣿⡀⠀⠀⣾⠀⣿⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⠈⢻⣿⣄⡀⠀⣿⡄⠀⠀⢠⡀⠙⠀⠀⠀⡿⣿⡇⠀⠀⣿⠀⣿⢠⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⢷⣾⡏⠙⠻⣿⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⣧⢸⡇⠀⠀⢿⠀⠃⢸⣷⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢨⣿⠇⠀⠀⣿⣿⠃⠀⠀⠘⣷⠀⠀⠀⠀⢿⡼⠇⠀⠀⠈⣦⠀⣸⣿⠟⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡟⠀⠀⣠⠻⣷⠀⠀⢠⡄⢸⠀⠀⠀⡆⢸⡇⠀⠀⢸⠀⣿⠀⣿⣇⢀⣀⡾⠿
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢠⣿⠇⠀⠀⣿⡆⠁⠀⠀⠘⠃⠘⠀⠀⠀⡟⠀⠇⠀⠀⠸⢀⠈⠀⣿⣿⠉⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣼⣿⠀⠀⢰⡀⠻⠀⠀⢠⡆⢰⡄⠀⠀⣤⠀⡄⠀⠀⢸⡀⣼⠀⠀⣿⣿⠀⢹⡇
⠀⠀⠀⠀⠀⣰⣿⠃⠀⢠⣿⠇⠀⢀⠘⣷⠀⠀⠀⡘⢃⡘⠁⠀⠀⢋⡀⠃⠀⢀⡘⣧⠈⠀⢠⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠀⣿⣟⠀⢀⣿⡏⠀⠀⠈⣷⡏⠀⠀⢸⡧⠸⡇⠀⠀⢾⠀⠟⠀⠀⠸⠇⠹⠁⠀⢸⣿⡿⠃⢸⣷
⠀⠀⠀⠀⠀⠻⣿⣦⣾⣿⠀⠀⠸⣆⡛⠀⠀⢀⣾⣠⣤⣀⠀⣠⡀⣤⠀⠀⢰⡆⢸⠆⠀⠀⣿⡟⠀⢠⡾⠇
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣤⣴⣷⣿⣧⣤⠴⣾⢻⣿⠉⠀⢀⣹⣷⠁⠀⢀⡜⢳⡄⠀⢀⣼⣿⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣷⣿⣿⡏⠀⢤⣼⣿⡎⠁⣠⡿⠟⠃⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠈⠛⠻⣿⣿⣿⠋⠛⠛⠛⠿⠿⢿⣿⣀⣘⣿⣿⣿⣾⣿⡇⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡟⠉⠀⠀⢹⣧⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣸⡇⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣿⡿⠁⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢸⣧⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠸⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⠟⣿
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢹⡄⠘⢃⠀⠀⠀⠀⠀⠀⢰⡟⠀⠘⣧
⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⡄⠀⠀⠀⠀⠀⡿⠃⠀⠀⢿
⠀⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷
⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢻⣆
⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠐⣿⣯⡙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠨⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⢸⣿⣿⣦⣌⡙⠿⣶⣄⣀⠀⠀⠀⢠⠉⣿⡄⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⡿⠀⠈⠙⠻⠿⣶⣮⣭⣿⡷⠶⣶⡆⠀⠘⣷⠀⢻⣦⣀⣀⣀⡠⠶⠶⠶⠆⠻⣷⡀
⣶⡀⠀⠀⠀⠀⠀⠈⣿⣭⣽⣿⠿⠟⠛⠛⠾⣿⣧⠈⢻⣏⠉⠁⠀⠀⠀⠀⢀⣤⣿⠿⠂⣀⣠⣤⣤⣤⣤⣤⡀
⠿⣿⣦⣤⣤⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⡉⠀⣀⣀⣀⣤⣶⡟⠋⠀⠀⣠⣾⠟⠋⠉⠉⠉⠉⢿
⠀⠈⠉⠠⢿⣿⣷⡶⠄⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡘⠻⣿⠛⠛⠛⠛⠁⠈⠻⠶⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠉⠛⠷⠶⠿⣟⣿⣿⣿⣿⣿⡶⠶⠟⠁⠀⢹⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⣉⣥⣤⡴⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡒⠀⠀⠀⠀⠐⢤⠤⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣦⡀
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⢱⡴⠚⠉⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡀⠀⣠⠀⠀⡀⣀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢫⠀⣿⡉⢹⠁⡇⢰⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢸⢀⡿⡇⢸⢠⡋⠸⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⠉⠇⢇⢸⠈⠉⠷⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⣀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠓⢄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠶⠶⠖⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡴⠃⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⣀⣀⣤⠤⠤⠔⠒⠒⠈⠉⠉⠈⠀⠀⠀⠀⠈⠑⠒⠂⠀⠉⠉⠀⠈⠑⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠑⢦⣠
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠄⠚⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣖⡉⣉⣀⠙⠐⠲⠖⠀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⡀⠤⠤⠒⠒⠒⠒⠤⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⣿⣿⡟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡈⠁⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠹⣿⠿⠄⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣛⣛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣤⣤⣤⣬⣙⠻⢷⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⣿⣿⣿⣿⡇⠲⣶⣶⣷⣤⠙⠻⠿⠟⠓⠀⠉⠉⠉⠉⠁⠀⠈⠙⠋
⠀⠀⠀⠀⢀⣠⣦⠀⢠⣤⣤⣤⡀⢰⣶⣶⣷⡄⠙⠿⠿⠟⠃⠈⠙⣩⣭⣭⣤⣄
⠀⠀⠀⠺⠿⢿⣿⡆⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣛⣋⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠛⠛⠂⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣻⣛⣟⣛⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠿⠀⠀⢸⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠳⣄⠀⠀⠀⠀⠀⠹⡆⠀⠤⠀⠀⠞⠀⠀⠸⠋⠀⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⢠⡴⠒⢦⣾⣹⣆⣀⡀⢾⣱⣄⣀⠀⠀⣾⣿⠀⠀⡼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠘⣷⣄⣀⣿⠞⢹⣿⠇⠀⡹⢹⣿⠇⠀⣹⣿⡀⡼⠁⢻⣿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⣿⡄⠀⢿⣇⢀⡿⠀⣼⣿⠀⢰⠃⣼⡿⠀⢰⠃⠈⠉⠀⢀⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠘⣿⠀⠈⠉⠉⠀⡀⠹⣧⣠⠏⠀⠻⣧⣠⠏⠀⣰⢊⠞⢣⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣠⡿⢀⠞⠛⠒⠋⠙⠦⠤⠤⠞⠓⢦⣄⣀⠀⢰⡇⠘⠠⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⢀⡀⠀⠀⣀⡀⠘⣇⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⣧⢀⡀⡿⠀⣿⠉⠀⠀⠀⠀⣿⣹⠆⠀⡏⠁⠀⠈⠓⠞⠋⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡿⠶⢂⡀⠂⠀⢹⣼⣷⡇⠀⣿⠉⠀⠀⠀⠀⣿⢳⡄⠀⡏⠁⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠘⠃⠙⠀⠀⠙⠒⠀⠀⠀⠀⠉⠀⠃⠀⠓⠒⠀⠀⠀⠶⠉⠡⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣶⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠹⣷⣄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⢸⡇⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣀⣀⣀⣀⠀⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣸⠇⠀⠀⠀⠀⠙⠒⠤⠤⠴⠶⠒⣾⡟⠉⠉⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢠⡟⠀⣠⣤⣶⡶⢦⣄⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣀⣤⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⠟⠀⣼⣿⡟⠁⠀⣰⣿⣷⠀⠀⠀⣿⡿⠀⠀⢀⣾⣿⠃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠃⠀⠀⣿⣿⠀⢀⡼⣿⣿⣿⠀⠀⠀⣿⡇⠀⠀⣸⣿⡇⣀⡼⠃⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⠿⠶⠋⠀⣿⣿⡇⠀⠀⣸⣿⠁⠀⠀⣿⣿⡏⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⢠⡇⠀⠀⠀⠀⢻⣿⡇⠀⣰⡟⢿⣄⣠⠞⢹⣿⡇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⣸⠃⠀⠀⠀⠀⠸⣿⣇⣴⠏⠀⠈⠉⠉⠀⠀⠙⠿⣦⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣠⡏⠀⠀⠐⢦⣀⠀⠈⠉⢁⣰⠓⠦⠤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⢠⡖⡄⠀⠀⡠⠒⠈⠁⠀⠂
⠀⠉⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣠⠈⢂
⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⢀⡠⠼⠁⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢁⡀⠁⠀⠐⣹⣇⡄
⠀⠀⠀⠀⠀⠀⠀⢫⠸⠀⠀⠀⠉⢈⢣
⠀⠀⠀⢈⡄⣀⠴⠁⢄⡀⠀⠘⣧⠎
⠀⠀⠀⢼⣵⠏⠀⢀⡐⠁⠁⠀⠁
⠀⢰⠁⠃⠃⠀⠀⠈⡇
⠀⠈⠀⠀⢀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢤⠀⠠⢀
⠀⢀⠀⡠⢾⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⡐⠀⡈⢢⠀⠁⠄
⠀⡼⡏⠀⠀⠆⠀⠀⣆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣼⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣖⡆⠀⡟⢠⠀⠀⠀⡆
⠀⢠⠃⠀⠀⠰⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠊⠁⠢⠅⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢂⡉⠀⡠⠄⠀⠀⠀⠀⡁
⠀⠈⠀⠀⠀⠀⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠲⢀⡀⠀⠁⠀⠀⠀⠠⡎⢹⡘⡟⠁⠀⡇⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢃⠀⠀⢸⣇⠀⠀⠱⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣀⠀⠀⠀⠈⡏⣶⡄⠐⠋⣶⣖⠐⠂⠁
⠀⠀⠀⠈⠂⠀⠘⠘⢀⣀⡤⠦⡴⠆⠲⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠒⠠⣀⡇⡆⠈⠂⠒⢛⣏⢀⡇
⠁⠆⢤⠀⣀⣀⡗⠀⠈⣧⣀⣴⢦⠖⠺⣍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠜⠁⠀⠀⠀⠸⡟⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⢿⡟⠀⠐⣽⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⢁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣌⡁⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠑⠒⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠀⠀⠀⠡⡀⠀⠀⠀⢫
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⡀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠀⠀⠀⠈⠐⠀⠀⠀⠣
⠀⠉⠒⠤⢤⠒⠒⠲⠄⠒⢻⠀⠑⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠈⠀⠀⠈⠢⣤⠚⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⠹⡄⠀⠂⢲⠀⠐⠋
⠀⠀⠀⢀⠘⠀⠀⢸⠀⠀⢦⠀⠀⠀⠀⢡⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠰⠀⠂⠈⡇
⠀⠀⠀⠨⢀⠀⠀⢸⠀⠀⠘⡀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⡰⡇⠀⠀⠁
⠀⠀⠀⡙⠶⢌⠀⠸⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠈⠲⠊⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠣⠔⢍⠂⠴⢄⠀⢠⠀⠀⠀⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⡅⠀⠀⠀⠠⡄⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠸⡀⠀⠀⠙⠂⢬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⢦⣀⠈⣂⠀⠀⠀⢡⠀⠈⠂⠀⠤
⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠀⠀⠀⠈⣦⡀⢀⣤⢚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠑⣔⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠆⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⢳⡀⠀⡤⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢧
⠀⠀⠲⡀⠊⢙⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠠⠘⡄⠀⠀⡰⢹⡄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢅⠠⠀⠀⠀⡼⠉⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠗⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⣫⠃⠀⠁⠈⠉⠡⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⡋⢀⡤⠖⠁
⠀⠀⠀⠠⠊⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⡮⠚⠉⠀⠀⢄⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⢻⠃⠀⠀⠀⠀⠀⠀⠀⠁⢉⠎⢀⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⣀⢀⠎⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢠⡄⠊⠙⠛⠼⠗⠃⠉⠒⢀⣤
⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢀⣀⡤⠔⠈⠁⢸⠀⠀⠀⠐⠶⣤⣄⣀⣠⡀
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡃⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⠀⢉⣋⣀⡑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠄⠈⠀⠀⠀⠀⠘⠨⢀⠀⠀⠀⠰⠀⢀⡠⠔⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠼⠚⠃⠀⠀⠀⠀⠀⠑⢠⠀⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣨⢩⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢴⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢡⢊⣀⣀⣉⣿⣿⣿⣿⣿⣿⣿⠷⡀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠻⠿⠿⣿⣿⣿⣿⣧⣌⣀⡀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠰⢉⡙⠠⠀⠘⠃⠀⠀⠆⠈⠉⠉⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠠⡇⠀⣀⠀⠀⠀⠀⠀⠀⠃⠀⢠⣿⠿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠘⠂⠀⠀⠐⣂⠀⠀⠀⠀⠐⠛⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⣀⠀⠀⡀⠀⠀⡸⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣀⣀⣀⡠⠔⠊⣉⡉⠀⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠁⠀⢤⡼⣤⢀⠀⠀⠈⢆⠉⠁⠐⠂⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠉⠈⢹⠉⣃⠊⠘⠳⡠⡀⠈⠆⠀⠀⠀⠄⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠨⠥⠰⢳⢇⡁⠘⠄⠀⡀⡄⠒⠓⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠐⠘⠚⠃⠁⠐⣈⣐⣣⠤⠴⠟⠓⠒⡦⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⠜⠓⢄⠀⡊⠉⠀⠀⠀⠀⠀⠀⠀⣹⠦⡈⠳⣄
⠀⠀⠀⠀⢀⡠⠄⠂⢤⡀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠣⡠⠀⠄⠀⠀⠀⠀⠀⠸⡆⠈⠢⡈⠳⡀
⠀⠀⠠⠊⠣⡀⠀⠀⠀⠘⢄⢀⡤⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢈⠦⠊
⠀⠀⠀⠈⠢⡈⠀⢄⠀⠀⠈⠃⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡀⠀⠄⠀⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠑⢄⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠠⡀⠀⠀⠀⠀⡰⠁⠈⠉⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⢀⠀⠀⠀⢸⠁⠀⠀⢀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠀⠀⠀⠨⢤⠑⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⢢⠊⡃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡗⢄⠀⠀⠀⠰⡄⠀⠀⠀⠈⠠⠌⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣠⢴⠁⢐⠀⠀⠀⠀⠈⡑⠀⠀⠀⠀⠀⣧⢰⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⢠⠀⠀⠀⢠⡎⢨⡴⠚⠁⠈⢦⠰⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⢆⠘⠤⣀⠀⠀⢀⣠⣾⠃⠄⠰⠤⢤⡤⠄⠤⠤⢤⡴⠞
⠞⠀⠀⠀⠉⠰⡁⠀⠀⠀⠀⠀⢳⡐⢄⡀⠀⠀⠀⠀⢀⡴⠀⠈⠑⠤⢄⣌⣭⠅⠖⠁⢀⠛⢒⠒⠒⡒⠒⠒⠚⠉⠀⠀⠀⠀⠀⡀
⠆⢄⠀⠀⠈⠳⢌⣺⣿⠀⠀⠀⠈⡳⣄⡈⠁⠒⠒⣈⡥⠃⠀⠀⠀⢀⠠⡀⠀⢰⠊⠁⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠠
⠠⡀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠉⠉⠀⠉⠁⠀⢀⢦⡀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠤⠤⢄⡠⣤⡄⠀⠀⠆⢀
⠀⠘⢦⡀⠑⠂⠤⠤⠤⠰⠒⣉⠜⢀⠠⠂⠀⠀⠀⠀⠢⡓⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡀⠀⠀⠀⠀⠀⠸⠦⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠑⠲⠤⠤⠤⠖⠊⠁⡄⠁⠀⠀⢀⣤⣄⠀⠀⠈⠀⠑⢄⡈⠒⢄⠀⠀⠀⠀⠠⠁⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠆⢠⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠀⠀⡤⢀⠀⠀⠙⠢⣐⠹⠀⠀⠀⠀⠆⠀⡇⠰⠀⠀⠀⠀⠴⠶⢶⠶⠾⠁⠀⣼⢀⡏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠄⠃⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠒⠒⠒⠒⠲⠀⢀⡗⠚⠛⠲⠦⣄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣴⠒⡀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠰⠀⠁⢸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠝⠋⠉⠚⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠀⠀⠀⠙⠋⠀⠀⠀⢀⡆⠀⠀⠐⠀⡣⡴⠒⡟⠀⣤⠁⠁⠉⠉⠉⠃⠒⠒⠲⠁⢠⢊⠄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡟⠀⠀⠀⠀⠶⠖⠚⠚⠀⠀⠈⠉⠉⠓⠓⠓⠒⠆⠊⠀⠀⢸⠀⢧⡀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⠒⢩⣡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⢀⡍⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⡖⠒⠶⠶⠒⠒⠓⠒⢚⠓⠒⠒⠒⠒⠒⠒⠒⠲⠖⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣬⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⡶⠶⠶⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣲⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣧⣥⣤⣥⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣀⣀⡀⢀⠀⠀⡀⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡿⠛⡿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠀⡭⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠄⣽⠀⣀⣠⠤⠖⡚⠍⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢄⣀⡤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠬⠥⠝⠒⡟⠙⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠸⡋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠇⣿⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠆⢸⠙⣇⠀⢐⡽⠊⣡⣇⣠⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⢬⣢⠸⠙⢢⡇⡈⠉⠙⠟⠂⠄⠁⠘⠰⠊⠿⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣒⣒⣒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠎⠁⠀⠉⠱⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡀⡠⠚⠉⠙⠳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠮⣵⡆⠡⠶⠶⠷⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⢀⣠⢤⠶⠰⠶⠲⠶⠦⡤⣄⣀⠉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠴⣋⡭⠴⠒⣟⠋⠙⠛⠛⠓⠧⢦⣎⣍⡖⠶⣤⣀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⣴⣻⠴⠋⠅⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣶⢤⡉⠲⢥⡀⡑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣰⡿⠋⠀⠂⠀⠀⠀⠀⠀⠐⠉⣳⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠓⢤⡙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢀⣀⠀⠙⢦⠙⣄⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡰⠛⠳⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⢊⡽⠃⠀⠀⠀⠳⡘⣎⢄⠡
⠀⠀⠀⠀⠀⠀⠀⠏⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⣀⠤⠔⠒⠉⢁⢠⠠⣡⡞⠁⢰⠒⢲⠤⡤⠾⣺⠎⠄⠂
⠀⠀⠀⠀⠀⠀⡘⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢾⣩⣄⡆⢐⠊⡄⡰⢈⡾⠁⠀⠀⠀⢻⠏⠀⠀⠄⢹⣇⠆⡀
⠀⠀⠀⠀⠀⠀⡇⡆⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠃⢎⣷⣹⡜⠀⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠌
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⠒⠞⡛⢦⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡏⡔⡁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠧⠀⠀⡠⠿⡀⠀⠀⠀⢠⠊⠀⢀⠠⢐⠀⢀⡠⡔⠋⠁⠀⠀⠀⠀⠀⠉⠐⠠⠀⠀⠀⠀⢐⢀⡗⡰⡃
⠀⠀⠀⠀⠀⠀⠇⠸⡀⠸⡟⠀⠉⠒⠓⠀⠀⡰⠁⠀⠀⢘⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡘⡱⢠
⠀⠀⠀⠀⠀⠀⠘⡄⠱⡀⢫⠀⠀⠀⠀⢀⠎⢐⣡⡴⠞⠉⠀⠀⠀⠀⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢀⣟⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠣⡀⠙⢤⠀⠰⣷⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢀⠀⠀⠀⠀⠀⠀⠨⢫⣿⠎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠙⣄⠀⠙⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⠛⡽⠀⠀⠀⠀⠀⠀⠀⢠⠔⡵⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡐⠱⢦⣀⠈⠑⠲⣤⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⠀⣀⡸⠖⡁⢌⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣆⡎⡕⡢⣤⣀⠈⠙⠒⠶⠤⠤⢠⢀⣄⣤⡤⠤⠽⠖⠖⠒⡉⠕⢐⡤⠖⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣸⢩⢓⢺⢦⠤⣀⣂⣂⡂⠂⠀⣂⣀⣘⣨⠤⠶⢊⣡⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠺⠶⠤⣤⣴⣸⣉⣉⣁⣁⣄⣌⡬⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠄⡠⡂⠔⡠⢒⡠⠒⡠⢐⡠⢀⠤⢀⠀⡠⢀⡉⢉⠉⣉⠄⣀⠄⡠⢄⡠⢂⠤⡀⠄⡠⢀⠄⡀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠊⠐⠁⠔⠡⠊⠡⠊⠔⠉⠖⠡⠊⠡⠊⠡⠊⠐⠁⠔⠡⠊⠰⠊⠞⠰⠋⠔⠑⠁⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⡄⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⡀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⣿⡿⠏⣿⣿⣿
⠀⠁⠀⠀⠙⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢠⡀⠀⠀⠀⢠⠤⠶⠲⣄⣠⠒⠐⢢⠌⢠⠄⠃⠐⢚⠀⢸⠛⠛⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠨⠖⠂⡎⠀⣉⠖⢁⣴⠥⡒⠔⡦⠃⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣶⣶⡆
⠀⠀⠀⡠⠂⠉⠈⠀⠀⠀⡠⢖⠀⠈⠁⠀⠀⠋⢋⡠⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠑⠢⠤⠤⠤⠒⠈⠀⡐⠦⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢰⡀⠀⠀⠀⠀⠐⡔⣮⡇⠀⠀⡀⠟⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠣⡀⠀⠀⠀⠀⡹⢉⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠛⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣞⠀⠀⡠⠞⢀⠋⢂⠀⠁⠀⠀⠀⠀⠀⢊⡴⠈⠁
⢀⣀⡀⢀⣀⠀⠀⠀⠀⢄⠄⠀⣴⠹⢿⣤⡾⠗⠤⠎⠀⠀⠰⡀⠀⠀⠀⠠⢃⠎
⠈⠀⠉⠉⢹⡎⠀⢀⣬⣋⢌⣾⣅⡈⡾⡏⣀⠄⠀⢀⡄⠀⠀⠙⠀⠀⠀⠀⠎
⠀⠂⠤⠔⢃⢠⢼⠃⠀⠸⣔⣊⣛⣱⡵⣿⡦⡀⣠⠊⠀⠀⠀⠀⠈⢀⠀⠘
⠐⠦⣈⠀⠀⠀⠀⠉⠊⠋⠁⢃⡐⣦⠁⠀⠀⠈⠁⣀⡀⢀⣀⠀⠚⠙⠂⠂
⠀⠒⠬⣉⠐⠢⠄⠤⠔⠂⢉⡈⠵⠂⠉⠂⡀⠀⠀⠏⠁⠀⠩⡾
⠀⠀⠀⠀⠉⠒⠠⠠⢔⠈⠀⢀⠠⠐⠈⠁⠀⠀⠀⠂⠤⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⠀⣀⡀⢀⣀⣀⣀⠀⣀⠀⠀⡀⠀⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠟⢸⣿⣛⣿⡇⣙⡁⣿⣿⠛⠛⢹⣿⣷⣸⣿⢸⣿⠿⣿⣆⣾⣿⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠗⠀⣿⣿⣿⣏⠠⣿⡇⣿⣿⣛⣛⢹⣿⠿⣿⣿⢸⣿⣀⣿⡟⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠻⠏⠈⠻⠇⠿⠇⠿⠿⠿⠿⠋⠿⠃⠻⠟⠸⠿⠿⠛⠀⠻⠿⠟⠋

⠀⠀⠀⠀⠀⢰⣿⣦⢸⣷⢰⣿⡿⠿⠇⣾⡇⠀⣾⡆⣾⣿⠿⠿⢀⣾⡿⢿⣷⠀⠀⠀⠀⣶⣿⠿⠆⢠⣾⣷⣴⣷⠀⣼⡷
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⢸⣿⡷⠶⠆⢻⣿⢰⣿⡇⣿⡿⠶⠶⢸⣿⣿⣿⠋⠀⠀⠀⠀⠙⢿⣶⡄⣼⣧⣼⣷⢹⣷⡟⠁
⠀⠀⠀⠀⠀⠸⣿⠀⢿⡿⠸⣿⣿⣿⡷⠈⠻⣿⠟⠀⣿⣿⣿⣿⠾⡿⠉⠻⡧⠀⠀⠀⠀⢶⣾⡿⠏⢿⠏⠙⣿⠆⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⢀⣤⣶⣦⡀⢀⣴⣶⣤⠀⣶⣶⣦⣄⢠⣶⡶⣶⣄⣴⡆⢀⣴⣄⣶⣶⣶⣶⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢀⣀⢼⣿⠉⣿⣧⣾⣟⢹⣿⡇⣿⡏⢹⣿⢸⣿⣷⣿⡋⠘⣿⣼⡟⢸⣿⣿⣤⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⡿⠸⣿⣶⣿⠇⠹⣿⣾⡿⠃⣿⣷⣿⠟⢸⣿⣭⣿⡿⠀⢸⣿⠀⠘⣿⣷⣶⣶⠀⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠀⠀⠉⠉⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠁⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠙⠓⢦⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡼⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⢳⣴⣾⣯⠶⠶⠶⠶⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⣄⠀⠀⠀⠀⢻⠀⠀⠀⠀⡀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢀⣿⡀⢀⣆⠠⣿⣶⢿⣆⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠃⠀⣠⡾⢹⣧⡾⠻⣶⡏⠀⠀⠹⣆⠸⣇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠰⠛⠋⠀⠀⠉⠀⠀⠀⠀⢻⡄⢿⡀
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣆⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣤⣶⣶⠶⠾⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠶⠶⠶⣶⣄⠀⠀⠀⠀⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠈⣷⢀⡴⠋⠁⠀⠀⠀⠀⣠⠶⠚⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠒⠓⠒⠶⣄⠉⠀⠀⠀⠀⠀⣿⣠⡟
⠀⠀⠀⠀⠀⣴⠟⣿⠋⠀⠀⠀⠀⠀⠀⡾⠁⠀⣴⣾⣿⡇⠀⠀⢳⡀⠀⠀⠀⠀⢠⠞⢁⣤⣴⡆⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢸⡿⣦
⠀⠀⠀⠀⢨⣧⡀⣿⣀⡀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣶⡇⠀⡇⠀⠀⠀⠀⣾⠀⣿⣿⣿⣷⣴⡆⠀⠀⣧⠀⠀⠀⠀⠀⢸⡇⢸
⢀⣤⠤⡼⠋⠉⠻⡟⠋⠙⣷⠀⠀⠀⠘⠇⠀⠈⠻⢿⣿⣿⠿⠁⢠⡇⠀⠀⠀⠀⠸⠀⠻⣿⣿⣿⡿⠁⠀⠀⠃⠀⠀⠀⠀⠀⢸⣇⣼
⣿⠁⠀⣿⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠉⠒⠶⠦⢤⣤⣤⣴⠏⠀⠀⠀⠀⠀⠀⠰⣦⣤⣭⠥⠶⠖⠒⠁⠀⠀⠀⠀⠀⠀⣾⠛⠁
⣿⠀⠀⣿⠀⠀⠸⡇⠀⠀⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⣿⠀⠀⢿⡇⠀⠀⣷⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⣿⠀⠀⠸⠇⠀⠀⠋⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⣿⠀⢠⣤⣤⡶⠶⢶⣤⡶⠘⠇⠈⣷⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⢺⡀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢹⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋⠁
⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢨⡟⣷⠶⢦⣤⣤⣀⣀⣀⣀⣤⣤⣤⠶⢶⡟⣿⠁
⠀⠻⠶⠶⣶⠶⠶⠛⠛⢻⣟⠛⠋⠀⠀⠀⠀⣠⠟⠀⣿⠀⠀⠀⠙⠿⣭⣭⣥⠼⠋⠀⠀⢸⡇⠸⡇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠙⢦⣄⣀⣤⠶⠟⠁⠀⢸⡟⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⣷⠀⢿⡀
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⠀⠀⠀⣿⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⣀⢀⣀⠴⠞⠉⢸⠃⠀⠀⠀⠘⡏⠙⠛⠛⡿⠋⠀⠀⠀⢸⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣧⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠋⢻⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡶⣤⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣸⣇⣹⣀⢀⠀⠀⠀⣠⣿⣿⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠉⠉⠁⠀⣹⠈⠁⠀⠀⠈⠈⠉⣻⣷⣴⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⢀⣀⣀⣀⣀⣿⣤⣤⣄⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣉⣹⡏⠀⠀⢸⡇⢈⣟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠙⢿⣄⠀⣼⠟⠋⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣿⣠⣏⣀⣀⣀⣀⣸⣇⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠁⣿⡎⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠳⣦⣀⠀⢠⣿⣿⣿⣇⠀⢀⣠⡶⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣾⣿⡟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣀⢀⡀⣀⠀⢸⠇⠀⢀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡀⠀⠀⣿⠀⠀⣀⣤⡄
⠀⠀⠀⠀⢸⣾⢷⡟⠀⠒⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠖⠂⠀⣸⣒⡂
⠀⠀⠀⠀⠀⠁⠈⠁⠀⢸⡇⠀⠀⠀⠈⠉⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠉⠀⠀⠀⠀⡿⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣴⣿⠿⢻⣿⣿⣿⣿⠻⣿⣷⡀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠼⠋⠁⠀⠘⣿⣿⣿⡏⠀⠀⠙⠳⠄⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣄⢿⢃⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡰⠋⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠡⡐⢌⠳⡀⣷⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢶⣿⡵⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠞⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠂⢌⣿⣠⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠴⠖⠋⠑⢦⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⠘⠛⢻⢇⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⣸⣿⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⣀⣟⡟⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⡬⢧⣿⠾⠓⠁⠀⡠⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⣲⠆⠘⠁⠈⣭⣭⡇
⠀⠀⠀⠀⠀⠀⠀⢾⡰⢧⠐⢛⣦⠀⠀⠀⡇⢰⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⢹⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣻⠀⠀⠀⣷⢸⡄⠀⠀⠀⠀⠀⠰⠰⠀⠀⠚⠸⠂⠀⠀⠀⠀⠀⠀⠀⡟⠐⡏⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⣧⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢠⠀⡀⠀⢀⡀⡄⠀⠀⠀⠀⠈⠀⢰⠇⢸⠇⠀⠀⠀⣠⢞⡵⠒⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣴⡇⠀⠀⠘⡇⢷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⣽⠀⠀⡴⢛⡵⠋⠀⠀⠀⠱⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢧⢸⡀⠀⠀⠀⠀⠒⠒⠀⠚⠀⠰⠀⠀⠀⠀⠀⠀⢸⡇⢨⡇⠀⠈⠓⣯⠰⡋⢶⠟⣱⠀⣸⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⢄⠀⠀⠀⠀⢸⠈⣇⢀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⣼⠀⠀⠀⠀⠸⡄⠈⠠⠞⠁⢀⡏
⠀⠀⠀⠀⢀⡤⡶⠋⠀⠀⠀⢀⣤⣓⠂⠄⡀⠘⢦⢈⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣤⣤⣤⣤⣤⡤⠶⠃⠀⠀⠀⠀⠀⠙⠚⠉⠉⠉⠉⠁
⠀⠀⠀⠠⡏⠈⠂⠤⢤⣴⡶⢍⠫⠤⠴⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢻⡴⡿⠐⣲⡄
⠀⠀⠀⠀⠙⠶⣤⠤⠤⠴⠶⠾⣐⣶⡾⠟⠁⢀⣀⣀⣀⣀⣀⣀⣈⣷⣶⣖⣒⣒⣒⣿⣁⣀⣀⡠⢤⣤⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢠⠇⣦⠗⣆⠖⢠⢄⡴⡄⣠⠀⠀⠀⠀⠀⠀⠀⠊⠇⠫⠊⢆⠀⢀⡄⡐⡋⠀⠀⠉⠐⠠⣠⣄⡀
⠀⠀⠀⠀⠀⢠⡀⠠⠂⠀⠀⠀⠀⠀⠀⢠⡏⡜⠒⢋⢛⢖⢼⢸⣷⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠘⠂⠫⠘⣉⣡⣿⣣⣤⣴⣴⡿⠀⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢀⣃⣀⣁⣴⣋⠁⠀⠀⢀⡞⠴⡳⠰⣋⠇⠈⢢⠀⠀⠘⢖⣾⣊⣀⡀⠀⠀⠩⠆⠠⠟⢋⣉⣃⠀⠉⠉⠉⠉⠉⠀⠀⠘⢣⡀⠐⠃
⠀⠀⠀⠀⢺⡭⣤⣤⣤⡤⠤⣿⠀⠀⠈⠓⠒⠲⠶⠶⠒⠒⢺⡄⠀⣴⡟⠤⠔⠒⠒⠒⠒⠒⠚⠻⠛⠋⠉⠃⠀⣀⣤⡤⠞⠏⠃⠀⠀⠀⠈⠻⣉
⠀⠀⠀⠀⠀⣣⡀⠀⠀⠐⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⢭⠓⠈⡤⠀⣀⡜⣀⠀⠀⠀⠀⠀⠙⠫⡉
⠀⠀⠀⢰⠏⠁⠭⠖⣛⠻⠭⠆⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⣚⣨⢴⣺⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠈⠀⢀⠆⠃⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⣼⠀⠀⠠⢼
⠀⠀⠀⠀⠀⣠⡴⡶⣲⠖⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠀⠘⠀⢀⠀⢀⠎
⠀⠀⠀⠀⠜⠃⠙⢁⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠓⠬⠤⠥⠤⠚⠁
⠀⠀⠀⠰⠀⠀⠀⠙⠃⠐⠃⢸⠀⠀⠀⠀⠀⠀⠀⣔⣹⠋
⠀⠀⠀⠀⠣⣀⠀⠁⠉⢒⡠⠃⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⢹⣏⠉⠁⠀⢀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⣀⠔⠉⡇⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡠⠊⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠙⢄⠀⠀⠉⠑⠒⠤⠤⠤⠤⢤⡀⠠
⠀⠀⠱⡀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⡌⢈⡁⠀⠉⠒⠤⢄⣀⣀⣀⡀⠀⠬⠛⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠢⣀⢼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠱⡒⠤⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⡷⡶⢀⠀⠀⠤⠤⠤⣤⢰⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣄⡑⠭⠤⠘⠃⠀⠀⠀⠀⠙⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢥⡂⠂⠤⣑⣄⠀⠀⠀⠀⠀⠣⢄⣀⣀⡌
⠀⠀⠀⠀⠀⢰⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠁⠑⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⡄⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠛⢠⠤⢤⡞⡀⠀⠀⠑⣄⠀⠀⠀⠀⠉⠒⠒⢀⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠇⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠀⠁⠀⠀⠀⠈⠑⠢⠤⠄⠀⠠⣔⢋⣀⠤⠒⠉⠣⣵
⠀⠀⠀⠀⠸⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠡⣀⣀
⠀⠀⠰⠭⠭⠄⠀⠀⠀⠀⠳⠞⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⢀⣤⣤⣠⡖⢲⣄⠀⠀⠀⢠⡴⣦
⠀⠀⠀⠀⢠⡿⠀⣀⡀⠀⠀⠉⢛⣶⡴⠋⠀⢸⠇
⠀⠀⠀⢠⡞⣡⠞⠹⡇⠀⢀⣾⡿⠋⣴⡗⠀⣾
⠀⠀⢠⡟⣰⠃⠀⠀⢀⣴⣿⠟⣡⢞⡼⠁⣸⠇⠀⠀⣀⣤⣴⣤⡤⠤⠤⢤⣤⣶⠿⡆
⠀⢠⡿⠁⠏⠀⠀⣠⣿⡟⢁⡴⢁⡾⠁⣰⠏⣠⢶⣻⡿⠛⠉⠀⠀⠀⠀⠀⣾⡏⠘⠷⣦⣀
⠀⢿⣄⠀⠀⠀⣰⣿⠋⣠⠊⢀⡞⠀⡴⠣⢞⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠈⠻⣷⡀
⠀⠀⠙⠳⢦⣴⡿⢁⡞⠁⣠⠏⢀⡞⢁⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⣿⠁⣾⣀⡴⢃⡴⠋⠀⢪⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⣧⣀⣀⣀
⠀⠀⠀⠀⠀⣿⠀⠈⣁⡴⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⡇⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡤⠞⠁⠀⠀⠀⠀⣇⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⠞⣭⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⠀⣼
⠀⠾⣿⡿⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡄⠀⢤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣯⣀⣭⣹⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢀⠛⢦⣄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⣸⠁⠀⢀⡀⠙⢳⡼⣄
⠀⠀⠘⠛⠛⠋⠀⠉⠩⠤⠤⠤⠴⠖⠒⠒⠛⠈⠙⢯⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣯⠀⠀⢸⣟⡖⢋⠇⡼
⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠀⠒⠀⠀⠀⠀⠉⠓⠦⣤⣤⣤⣤⣄⠀⠀⠀⣀⣠⣤⠞⠉⠀⠀⠀⣀⣠⣴⣿⠀⠀⠀⠞⠁⣹⠳⠞⠁
⠀⠀⠘⠳⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣝⣲⢯⡻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣀⣰⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡗⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠲⠶⠖⠒⠒⠒⠒⠒⠒⢒⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⠉⠉⠉⣉⣭⣿⣷⣶⣶⣄⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⣀⡤⢤⣀
⠀⠀⠀⠀⣴⣚⡶⠒⠒⣶⠋⠀⢻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⢁⢔⡯⠛⣻
⠀⠀⠀⣰⠏⠉⢧⡀⣴⠁⠀⠀⠀⠙⠿⠿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡧⡶⠋⠀⢀⡿
⠀⣀⡾⠁⠀⠀⠈⢳⡙⣾⠀⠀⢠⠒⢦⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋⠀⢱⠁⠀⢀⣼⠃
⢺⣹⠀⠀⠀⠀⠀⠀⢳⡝⠻⠦⣨⠃⡾⡅⠀⠀⠀⠀⠀⣀⢤⡄⣀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣿⣿⣿⡿⠁⠀⣶⠃⢀⡴⠞⠁
⠀⠙⣧⡀⠀⠀⠀⠀⠀⠙⣦⠀⠙⢺⡀⠙⣆⡀⠀⠀⣞⡱⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢹⣿⣿⠁⢠⢾⡋⣀⣹⡄
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⡟⡦⢼⠗⢺⠷⠁⢀⡠⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠻⢿⣰⠇⡼⠉⢁⡾⠁
⠀⠀⠀⠘⢷⡀⠀⠀⠀⢠⡞⠉⣡⠋⣰⠋⠀⣰⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣧⣤⠞⠁
⠀⠀⠀⠀⠈⢷⡀⢀⡴⠋⢀⡔⣡⢺⡇⠀⢠⣿⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣯⡽⠷⠚⠋⠁⠈⠉⠀⠀⠸⡟⡀⠀⠘⢛⡟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠉⠉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠈⣎⢧⠠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣠⠤⠴⠖⠀⠀⠸⡘⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⣫⠽⢭⡉⠙⠲⢤⡀⠀⢃⢇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡴⠊⠀⢠⠋⠀⠀⠀⢹⠀⠀⠀⠙⠢⡐⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡝⠔⡴⢋⡤⢄⠀⡇⠀⠀⡀⠀⢸⠀⠀⢀⡆⠀⠙⣿⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⣤⡉⠁⡞⠀⢸⠀⠀⠇⠱⣄⠀⡵⣊⠉⠉⢱⠈⡇⠀⠀⠘⡏⠈⠉⠉⠒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣏⡙⠂⢀⡀⠁⠂⡄⠀⠀⣹⡀⠉⠁⢀⠜⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⣀⣀⢸⠙⢻⠁⠀⠀⠀⠀⠀⠉⠉⠒⠒⠉⠀⠀⠀⠰⣆⠉⠺⡀⠀⠀⠀⣧
⠀⠀⠀⢀⡠⠔⠊⠀⢯⠁⠀⠱⡤⣊⡥⢤⡼⠋⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠤⠤⣇⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⢠⠒⠁⠀⠀⠀⠀⠀⠣⣀⢆⠸⠁⠀⠘⠦⢄⣀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠯⠭⢦⠀⣿⢉⡋⠁⠀⠀⠀⠀⢀⡀⡠⠋⣢⠃
⠀⠀⢣⠀⠀⠀⠀⡀⣀⣀⡴⠞⠙⠀⣀⣤⠖⠮⣄⣀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠉⠉⠙⢭⡉⠓⣦⣤⣤⠴⢧⢀⠈⠉⠓⢦⠀⠀⢸⡶⣁⠜⠁
⠀⠀⢠⠗⠒⠶⣄⠜⠁⠀⠙⢦⡴⢮⣉⠙⠻⢶⣦⣭⣙⠒⠦⢤⣀⣀⣔⣱⣄⡤⠴⠎⣭⣲⣶⡟⠂⠚⠋⡡⠛⡄⠀⠀⠈⡆⠠⣎⡹⠋
⠀⠀⢻⡈⠉⠉⢹⡆⠀⠀⠀⠀⠙⢦⡈⠛⠲⢤⣄⣉⠙⠛⠷⢶⡶⣿⡟⢁⣼⠤⠀⠰⠏⠀⠈⠑⡄⣠⠎⠀⠀⠐⡀⠀⡰⠃
⠀⠀⠘⣆⠀⠀⢸⡗⢀⠎⠙⠢⣤⠋⡨⠶⠤⣀⠀⠉⠙⠛⠒⡾⠁⠙⠃⡾⠉⡞⡓⠃⠒⠦⠤⠴⠟⠁⠀⠀⠀⡠⣷⣞⣀⣀⠤⢤
⠀⠀⠀⠘⣆⣀⣸⣷⠋⠀⠀⠀⠀⠙⠢⡀⠀⠀⠈⠉⠉⠉⠻⡥⣤⣀⠀⠑⢾⢱⣀⡠⢴⣶⠉⡑⢢⡀⠀⠀⠪⢄⡀⢌⡉⠉⠈⠉
⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢱⠝⡾⠻⣆⠈⠲⠥⠶⢅⣈⠉⠉⠉⠁⣀⡤⠔⠋⠀⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣤⣤⡤⠔⠻⡎⠑⠒⢻⢈⠖⠆⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠴⠋⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⡤⠤⣀⢀⡀⠀⢀⣀⡀⢸⢉⠇
⠀⠀⠀⠀⠀⢀⠎⠀⢀⣷⠋⠈⠹⠉⠙⠇⣀⠘⠖⡖⢲⠊⠀⠒⢼⠃⡈⠉⠍⠰⠏⠻⣸⠉⠒⣄⢀⣀
⠀⠀⣀⡴⠋⠉⢄⠘⠉⠉⡆⠁⠠⢈⡁⠦⠀⠔⠀⠀⠀⠀⠀⢁⡠⠀⣃⡀⢀⠈⡠⠠⠆⠀⠀⠈⠁⠀⠉⠒⣄
⢠⠊⠀⠀⠀⠀⠀⠸⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠑⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠒⠶⠖⠒⠒⠉⠁

⠀⠀⠀⠀⠀⢀⠀⠀⢶
⠀⠀⠀⢠⠀⠁⡀⠀⠚⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠦⠴⠴⣦⠖
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⡞⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣤⡶⣦⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠂⣤⡤⠤⠀⢀⣶⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠁⠀⠀⢄⢸⠳⣾⣗⠇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⠐⠒⠒⠖⠒⠖⠒⢛⢛⢓⠒⢺⣟⠀⠀⠀⠀⠁⠒⠚⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡨⠼⠚⠀⠀⠀⠀⠈⡉⡈⡘⡀⡘⠀⢂⡁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣷
⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡄⠐⠐⠈⠈⡀⣐⣀⣀⣀⣠⣿⠀⠀⠀⢠⢷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠑⣄⠢⡈⢻⠀⠀⠀⠠⣸⠀⠀⠀⠠⣤⣀⣨⡿
⠀⠀⠈⠀⠀⠀⠀⢻⣇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠳⣈⣺⠀⠀⠀⠀⢸⣷⠀⠈⠛⠳⠧⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⣀⠤⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠚⡻
⠀⠀⠀⠀⠀⢄⡠⠤⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠀⢸⠀⠀⠀⣖⢦⠀⠀⠀⢈⠀⠀⠀⠀⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠋⠀⠀⠀⡤⡀⠀⠀⠀⡄⠀⠀⠀⢉⡁⠀⠀⠀⢸⠀⠀⠀⠀⠈⠄⠁
⠀⠀⠀⠀⠀⢀⡠⡆⢀⠀⠀⠀⠘⠓⠀⠀⠀⠀⢱⢠⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⡀⠀⠀⢠⠎⡠⠗⠉⠀⠣⡀⠀⠀⠀⠁⠀⠀⠀⡈⢣⡑⠤⢀⣀⡀⢴⠏⢈⠐⠤⠤⠤⠐⢒⣶⠞⠁
⠃⠀⠀⠀⣘⢀⣠⡄⠀⠀⢱⡀⠤⣀⣀⣀⡠⡴⠁⠀⠉⠒⠒⠒⢂⠁⠀⠈⠙⠉⡩⠉⠉⠉⠀⠀⠒⠀⠀⠄⠀⠀⠀⡀
⠀⢠⡀⠀⠈⠑⠛⠁⠀⠀⣰⠉⠒⠤⠤⠴⠚⠀⣀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠣⡈⠂⠤⡀⣀⠀⠤⢒⠟⠀⠠⠀⠀⠀⠀⠰⢌⠳⢄⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠬⢍⣉⡉
⠀⠀⠈⠑⠒⠠⠄⠐⠒⠁⡄⠀⠀⠀⣤⡀⠀⠀⠂⠑⢄⠑⢤⡀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⢀⠂⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠰⠀⠀⠀⠑⠦⠘⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⣤⣤⣶⠶⠀⠀⠆⣘⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠉⠉⠉⠉⠀⠘⢯⣭⠽⣳⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⢀⠀⠀⠀⠈⣀⡠⢤⠀⡀⠒⠒⠒⠤⠤⢄⣀⡀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡎⠀⠀⠀⠴⠷⠒⠃⠀⠙⠒⠒⠐⠢⠤⠔⠀⠀⢰⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠒⣚⡭⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠭⠗⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠴⠶⣶⣶⣶⣶⣶⣶⣒⣖⣒⣒⣒⠒⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⠀⢀⡶⡄⢻⠋⢸⣛⠀⣟⣳⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣠⣬⣭⣭⡭⠭⠭⠽⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠳⠦⠘⠉⠓⠸⠀⠸⠶⠄⠃⠙⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⣛⣛⣛⣛⣛⣻⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠘⢛⣛⣋⣉⣩⣭⣭⡥⠤⠴⠶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⡤⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠈⡥⠴⠶⠶⣒⣚⣛⣛⣉⣉⣭⣭⣭⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⣩⣭⣭⡭⠥⠤⠴⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡛⠛⠛⠲⣄⢠⣦⠙⠋
⠀⠀⠀⠀⠀⢠⡤⠭⠷⠶⠶⠒⢒⣛⣛⣛⣛⣛⣛⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡼⠛⠶⠶⢶⡟
⠀⠀⠀⠀⠀⢀⣀⣈⣙⣿⣭⣭⡭⠭⠭⠭⠭⠭⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⠛⢃⡟⢻⠁
⠀⠀⠀⣀⣤⣬⠭⠥⠶⠖⢒⣒⣛⣛⣛⣛⣛⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢤⡭⢁⡾
⠀⠀⠀⠈⣛⣛⣛⣛⣋⣉⣭⣭⣭⣭⣭⣭⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡏⠻⣾⢻⣥⣀
⠀⠀⠀⠠⣭⡭⠤⠤⠶⠶⠶⢶⣶⣶⣶⣶⣶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠹⣾⡽⡿⠃⠉⠛⠶⣄
⠀⠀⠀⠀⠒⢲⣾⣿⣿⣭⣭⣭⣭⣭⣭⣭⠭⢥⣤⡄⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⠗⣿⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠴⠶⠲⠒⠒⠚⣛⣛⣛⣛⣛⣛⣓⣒⣂⡀⠀⠀⠀⠀⠀⠀⡿⠀⠠⣄⠀⠀⠀⣾⣫⣽⠀⠀⢀⣤⠀⢹⡄
⠀⠀⠀⠀⠀⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⡄⠀⠀⠀⠀⢰⡇⠀⣸⢿⠀⠀⢠⣿⣿⣽⡇⠀⢸⣿⡆⠀⣇
⠀⠀⠀⠀⠀⢒⣒⣲⣒⣛⣛⣛⣿⣿⣿⣭⣭⣭⣭⣽⡅⠀⠀⠀⠀⢸⡇⢠⡏⢸⡀⠀⠸⣿⡵⣾⠃⠀⢸⡏⡇⠀⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣒⣒⣒⣒⣲⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⢸⡇⠸⡇⢸⡇⠀⠀⠈⠛⠁⠀⠀⠘⣷⡇⠀⡟
⠀⠀⠀⠀⠀⠰⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠭⠭⢭⣭⠁⠀⠀⠀⠀⢷⠀⢿⡸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣸⠃
⠀⠀⠀⠀⠀⠤⠿⠭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⡇⠠⠶⠶⠶⠾⢧⡈⠳⣞⣛⡛⠛⣿⣿⣟⣷⠟⢁⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⢲⣖⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣻⡟⠛⣿⢿⣦⣈⣿⡶⠋
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣬⣙⣿⣛⣥⣤⣭⣭⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣀⣀⣀⣀⣀⣀⡀⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⡀⠠⠄⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⣄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⢰⠏⠁
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠠⠀⠀⡄⠀⠀⡀⠀⠀⢀⠘⠄⠀⠠⠂⣠⣀⡰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢂⣀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠈⠁⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⡤⠚⡭⠝⣿⡤⠿⠟⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⢈⠀⣀⡄
⠀⠀⠀⠀⠀⢀⡀⢸⡄⠀⣀⣀⡬⠤⠦⠤⠔⠒⡶⠂⠀⠈⣆⠬⣃⢄⠀⠀⠂⠀⠐⠶⠋⠁
⠀⠀⠀⠈⠀⠀⠣⣰⡠⠀⢈⡷⠀⠀⠀⠀⢠⠞⠁⠀⢠⡀⠀⠀⠈⢢⠑⡄⠀⠉⢠⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠉⡴⠚⠁⣀⡤⡄⢀⠜⠁⡠⠀⠀⢸⠀⠀⠉⠀⠀⠇⠁⠀⠀⠈⠁⠿⠆⠀⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡼⠁⠀⠹⠃⡀⠀⠈⠄⠀⠈⡆⠀⢀⣀⢎⠜⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⡀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣿⠸⠉⠂⠅⠀⠀⠄⠐⠄⣷⣤⣤⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣠⠟⠀⠀⠀⡀⡀⢌⠀⠀⠀⡏⠀⠀⠀⠄⣀⠐⠙⠳⢶⣬⣅⡨⡍⠉⢷⢦
⠀⠀⠀⡁⠀⡀⠀⠀⠈⠚⠁⠀⠀⠀⢀⠄⠈⠀⠀⠂⢼⡄⠀⠀⢠⣀⠀⠄⠀⢰⠀⠈⠁⢛⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠰⠀⢠⡂⠀⢀⢔⠝⠁⠀⠀⠉⠙⢦⣄⣀⣀⡀⠀⢠⠏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠇⠀⠑⠁⠀⠀⠰⠶⠀⠀⠀⠈⠁⠉⢹⠞⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣀⠀⠀⠀⠀⢰⡇⢠⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⣍⠛⠶⣄⡀⣾⠀⢸⡇⠀⠈⣧⠀⠀⣀⣤⠴⢺⡿⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢷⣄⠀⠙⢧⠀⢸⡇⠀⠀⣸⠖⠛⠉⣠⠞⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢧⡀⠈⢇⠸⡇⠀⣰⠃⠀⣠⠞⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠧⣄⡀⠀⠙⣦⡘⣦⡇⢠⠇⣠⠞⠁⠀⢀⣰⣃
⠀⠀⢀⡀⠀⡀⠀⣀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠀⠀⠀⠉⠛⠶⢬⣻⣿⣧⣿⣞⣡⠴⠖⠋⠉⠈⠉⠙⠛⠲⠦⢤⣄⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠈⣿⣼⢷⣼⠋⠀⢠⣴⣶⣯⣭⣤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⣶⣾⣿⣿⣿⣿⣖⠲⠶⠒⠶⠶⠶⠶⠶⠲⠶⠶⠾⢿⣿⣶⠦⠀⠀⣿⠤⠤⠁
⠀⠀⠀⠸⠇⠈⠟⠀⠀⠀⠈⠉⠛⠓⠲⠦⢤⣤⣀⣀⣀⣠⡴⠖⠋⣩⠞⡟⣿⢻⠳⣌⠙⠓⠦⣄⡀⣀⣀⣤⡤⠶⠖⠛⠉⠉⠀⠀⠀⠀⠻⠶⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠁⠀⣠⠞⠁⡸⠁⣿⠈⢧⠈⢳⣄⠀⠀⢹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠞⠁⠀⢠⠇⠀⣿⠀⠈⢧⠀⠙⢷⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠞⠁⠀⠀⣀⡞⠀⠀⣿⠀⠀⢘⣧⣄⠀⠙⢧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣁⣤⡴⠖⠛⠉⣧⠀⠀⣿⠀⠀⣸⠁⠈⠛⠶⣄⡙⢦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⢠⡏⠀⠀⠀⠀⠀⠉⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣠⠴⠒⠋⠉⠉⠉⢉⣁⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡿⠋⠀⠀⠀⠀⠀⠀⠙⣛⣒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣠⡔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡫⠖⠀⠑⠢⢄⡠⠔⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⠆⠀⠀⣠⠔⠋⠀⠰⡴⠒⢤⡜⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣇⢠⡄⠀⠀⠀⢈⡶⠀⠀⢰⡇⠀⠀⠘⣆⡠⠴⡁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣇⣶⠀⠀⠀⠑⢆⠀⠈⣇⠀⠀⠈⡸⠀⠀⠸⠊⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⠀⠀⠀⠈⠉⢦⣹⣦⡀⠈⢧⠀⠈⠲⠤⠔⠋⠀⠀⠀⠒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢑⡦⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠞⠉⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠊⠀⠀⠑⢦⡀⠀⠀⠀⠠⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⡾⣿⠓⠒⠂⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣧⡄⠒⠒⠒⠒⠒⠤⠼⠂⠚⠴⠯⠕⣢⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣋⢀⠀⢀⠾⠧⠒⠒⠀⠉⠉⡵⠂⠀⠀⠀⢠⡿⠄⣀⣀⣀⣀⣀⣄⡜⡈⢛⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡎⠉⠑⠂⠤⠄⣀⣸⡭⡅⠙⡭⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠐⠊⠑⠊⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠒⡆⢀⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢠⣒⡒⠋⡜⠀⠀⢀⠀⠠⡪⠔⠊⡝⠐⠧⠚⠢⠤⣄⣀⣀⡀⢀⣣⠀⠠⠤⠤⣄
⠀⠀⠀⣍⡴⠦⠆⠀⠙⠊⠁⠀⠀⠀⠐⠥⠤⠀⠀⠀⠤⠤⠼⠼⠧⠀⠀⠀⠠⡄⠀⣸⠔⡫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠚⠁

⠀⠀⠀⠀⢀⣿⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀⠈⠙⠿⠟⠀⢼⠘⣄⢸⣧
⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⣤⣀⣀⣼⣿⣴⣿⡆⢀⡀⡇⠀⠀⠀⠀⠀⠀⠒⠂⠀⠂⠀⠀⠂
⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠊⠁⠀⣀⡉⠛⢿⣷⠘⢿⣿⠈⠀⠊⠀⠀⡃⠀⢀⡀⣀⠀⠀⠀⠀⠀⠐
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢋⣴⠋⠀⠀⣼⣿⡯⢡⠀⠀⢹⣦⣸⣿⡄⠀⠀⠀⠰⠛⠛⣿⣿⣿⣁⠀⡀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠖⡒⢀⣴⠞⠁⢰⠘⠀⡀⠻⠿⠛⠁⠀⠀⠘⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣷⡀⠄⠀⠀⠀⢀⣸⢆⡀
⠐⠠⠿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿⠟⠉⠀⠀⠀⢸⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣾⣿⣠⣀⣀⠀⠀⠀⠗⠙⣿⣿⣿⣿⣿⣏⣀⣤⣾⣿⣿⣿⣦⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡯⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡇⠈⠉⠁⠹⣆⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⠄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠣⣂⡀⢸⠳⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⣸⡇⠀⠀⠀⠀⠘⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣷⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠿⠷⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⣦⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠀⢮⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣼⠋⣿⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣀⡀⠙
⠀⠀⠀⠀⠀⠒⠶⣶⣤⠤⢤⣤⣤⣶⣿⠆⣀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⣰⡇⣸⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢯⣭⣭⣭⣤⣨⣛⣳⢦⣄⣤⣴⡍⢀⠤⠞⠋⠰⠁⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⣩⠟⢠⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠙⠛⠛⣋⡈⣩⣌⣁⢀⣀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣀⠀⠀⠀⠀⢱⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣵⡆⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣌⣛⠛⠻⢿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣄⣸⣦⡀⠀⠀⢸⣿⣿⣿⡿⠿⠛⢛⣉⣀⣨⣭⣭⣴⣶⣶⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⠈⠂⠀⠀⠀⠀⠀⠀⠠⠐⠁⣭⠟⠀⢀⠀⠀⠀⠀⠐⢻⣷⠀⣾⡿⣿⣿⣿⣤⣤⣤⣾⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⡻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠈⣷⠾⠄⠀⠀⠀⠀⠇⢀⡀⠐⠀⠀⠀⠀⠰⠂⠉⠀⠚⠓⠸⢫⣤⡛⡉⠁⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⠁⠋⠀⠀⢀⠁⠀⠀⢀⡒⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢡⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⢾⣿⣿⡟⢭⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠂⢀⠀⠀⠀⠟⠲⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠞⠃⠿⢿⢦⠹⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢦⣴⣿⣷⣤⣬⠋⠐⢦⣶⠶⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢽⣷⡄⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⣀⡀⠼⠅⠀⠉⣦⠟⠓⠤⣺⣟⣻⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠐⡀⠸⣿⠄⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠃⠈⠀⠀⠀⠀⠈⠉⣽⠿⠭⢤⣀⠀⠀⠀⠀⠤⣄⡰⠁⠀⠈⠀⣐⣤⣤⣴⣶⠷⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠵⣈⣁⣀⣈⠽⠋⠀⠁⠀⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⠉⣉⣁⣀⣐⡒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⢀⠠⠒⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⢖⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⣹⣆⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢼⡀⠀⠀⠀⠀⠀⠟⠀⠀⣠⠔⠞⢥⠏⠀⠀⠀⠀⠀⠰⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣳⡀⠀⠀⠀⢐⣘⠄⢰⠁⢠⠚⠁⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠂⠉⠀⠀⠀⠸⡆⡇⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠸⡆⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢀⣧⠀⠀⠀⠀⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣻⠀⢀⡀⠈⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡬⠽⠶⣚⡫⠭⠝⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣩⢷⢈⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡪⡸⡄⠀⠀⠀⠀⠀⢸⠄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣈⣉⣉⣷⣦⡇⡀⣇⠀⠀⠀⠀⠀⠀⢻⡿⠿⠦⠾⠶⣀⣀⠉⠀⣸⡇
⠀⠀⡀⠀⠀⠀⢀⠤⠄⠈⢙⣒⣛⣟⣛⣛⣛⣋⣭⣍⣻⣿⣿⠯⣾⠀⠀⠀⢀⠀⢀⡈⠲⠾⣿⡿⠟⠛⠛⠻⠿⠟⡠⢄⡀
⣐⠓⠹⠯⢉⡉⢉⢙⣿⡙⠛⣒⠷⠿⠶⠶⠾⠟⠛⣛⣻⣭⣿⣿⡟⠀⣸⡇⠈⢿⣦⡙⠳⣶⣦⣬⣉⣐⣒⡒⠒⠚⠁⠀⠀⠁⣲
⠀⠀⠀⠐⠚⠓⠒⠀⠉⠻⣷⣤⣀⡿⠿⠿⠿⠿⠿⠿⠟⣋⠕⢁⡠⡾⢋⡟⠀⣸⠽⢿⣆⠘⢯⡉⠛⠛⠛⠛⠻⡇⣀⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⢀⡠⠔⠊⣠⣴⠿⠋⠀⣸⠃⢠⡟⠀⠀⠹⢷⣦⣈⠲⢄⡀⠀⠘⠛⠻⠑⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠤⠂⢁⣠⣴⠿⠚⠁⠀⠀⡰⠃⣰⡿⠁⠀⠀⠀⠀⠉⠛⢷⣦⣌⡑⠢⠤⣤⠎⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠈⠑⠊⢀⣤⣶⠽⠊⠁⠀⠀⠀⢀⡜⠁⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡶⡀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⠀⠀⢀⣉⣲⣄⠀⠀⠀⠀⢠⠞⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣤⣴⡶⠶⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠛⠿⠟⠛⠁⣰⠑⠂⠤⠋⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢀⠀⠀⠀⠉⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣢⣥⣴⡶⣟⠟⠉

⠀⠀⠀⠀⡧⡷⢗⢛⠡⡠⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠐⠂⠈⠁
⠀⠀⠀⠀⢱⠏⣛⣁⠬⠔⠒⠸⠀⢂⠠⠄⠀⠒⠈⠈
⠀⠀⠀⠀⢘⠉⠁⠤⠐⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣻⡿⣿⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠠⣤⠠⠠⠀⠀⣿⣛⣷⡿⠫⣿⣿⣼⠀⠀⠀⠀⠐⠒⠓⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⣰⡂⠀⠀⠀⣯⣭⣭⣤⣬⡍⢙⡙⠀⠀⠀⠀⢰⡂⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡤⡄⠀⡷⢶⡟⢻⣿⠖⣾⡿⠀⠀⠉⠁⠉⠉⠉⠉⠉⠁⠀⠀⢰⠆⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠉⠁⠀⠀⠀⠐⡆⠀⡀⠀⣿⣿⡟⠉⢹⣸⣿⡆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣃⠀⠁⠀⣿⣿⡇⠀⠸⣿⣿⣷⠚⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⢅⡀⠀⠀⠋⠙⠛⠀⠀⠻⠿⠿⠲⠶⡤⠁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⡀⠀⠀⠀⠀⠀⠀⠏⠀⠙⠀⣐⣶⣸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠘⠒⠀⠸⠍⠛⠈⠘⠂⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠀⠀⢰⠀⠄⢀⠀⠀⡆⠀⠀⠒⢑⢂⡷⡁⣞⠀⣎⢆⣾⠀⠀⠄⠀⠀⠠
⠀⠀⠰⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⡁⠘⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠐⠀⠐⠐⠀⠀⠁⠀⢀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠦⠢⠖⠀⠆⢐⠆⠔⠞⠀⠆⠶⠀⠀⠀⠀⠀⠄
⠈⠀⢘⡚⠀⠀⠀⠀⠀⢌⡀⠤⣠⣀⢠⢤⣠⢤⡠⢤⢠⠄⢤⣸⠀⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠐⠻⠀⠀⠀⠀⠀⢤⠗⠮⠇⢿⠸⡠⠿⠸⠷⠽⠿⠰⠭⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⡬⡁⠉⢀⡀⠀⠀⠀⡄⢀⠀⠀⡃⠀⠀⣴⠶⠶⣦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠈⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠂
⠀⠀⠈⠧⠿⠀⠃⠠⠿⠇⠈⠨⠵⠟⠌⠄⣿⠀⠀⠛⠀⣤⠶⢦⡄⢀⣴⠦⣤⠀⣠⡴⢤⣿⠀⣿⡴⢶⡄⢴⡄⢀⡤⢀⡴⠶⣄⢠⠄
⠀⠀⠠⠄⠤⠆⠤⠠⠄⠆⠀⠄⠀⠀⠀⠀⣿⠀⠶⣶⢰⣿⠀⢸⣷⢸⡇⠀⢸⡇⣿⠁⠀⣿⠀⣿⠀⢸⡇⠸⣇⢸⡇⢸⣧⣤⣿⢸⠂⡈⢵⡏⢸⠐⢠⢳⠀⠅⠂
⠀⠀⣶⢀⣤⣇⢀⣄⢀⡄⠀⠀⠀⠀⠀⠀⣿⡀⢀⣿⠸⣿⠀⢸⡿⢸⡇⠀⣸⡇⣿⡀⢀⣿⠀⣿⠀⢸⡇⠀⢿⣾⠀⢸⡇⠀⣠⠈⡠⢬⠡⠁⠈⠁⠈⡄⠀⠉
⠀⢠⠾⡼⣠⢿⢻⣸⢏⡢⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠙⠓⠛⠁⠈⠛⠛⠋⠀⠘⠛⠋⠛⠀⠋⠓⠛⠁⠀⢸⡇⠀⠈⠛⠛⠋⠀⡇⢀⣤⡩⣗⡏⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠰⠠⠐⠀⠔⠀⡆⠆⠔⠰⠈⠀⢘⢔⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠁⠁
⠀⠂⠀⠀⠐⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠈⡀⢀⠀⠀⠀⢀⠀⡀⢠⣥⡆⡂⠸⠱⠑⢲⠰⢉⣶⡏⠀⡀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠀⠁⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠈⠈⠀⠀⠀⠈⠀⠀⠠⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠀⢈⠊⠈⠀⠃⠛⠘⠋⠀⢀⠐⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⢀⢀⡀⡀⠀⠐⠐⠀⠐⠐⠀⠒⠀⠀⠼⠴⠀⠼⣆⡀⠐⢐⠲⠆⡰⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⢟⣻⣛⢳⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠻⠿⣿⣿⣿⢿⠿⠟⣳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠠⢤⣄⡀⠀⠀⣀⣀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡿⠀⠶⠀⢷⠀⢨⣍⡉⢃⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣷⠀⠀⢠⡆⢀⡀⠀⠀⣸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠊⠤⢄⣀⠐⢄⡴⢣⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣆⠀⠀⠓⠀⢀⠎⠔⡱⢈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⢻⠀⠟⢷⣤⣤⡤⣼⣾⢸⣧⣿⠇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣶⣿⣿⡿⠿⠟⢛⡝⢻⡟⢸⠀⠈⢢⡀⠀⣠⣿⠈⠁⡌⠁⢀⣾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢩⡀⠀⠈⠀⠀⠀⡼⠀⠸⠁⠸⡄⠀⣠⣿⣦⠀⢿⣷⠈⠃⠀⠘⣉⠉⠛⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠈⡇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣇⠰⢉⣿⣏⠃⢰⣿⣧⡀⠀⣀⢘⢦⠀⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣛⠃⠀⢱⠀⠀⠠⠤⠥⠶⣤⡄⠀⠀⢻⢠⣿⣿⣿⡞⣼⣿⣿⡥⠈⢠⠏⠀⠳⡀⠀⠜⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠃⠈⠙⢦⣸⡀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⣷⠂⠐⠻⣀⠀⠀⠙⢌⣠⡾⣿
⠀⠀⠀⠀⠀⠀⣾⡅⠉⠐⢦⡄⣠⣤⠿⣷⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢻⣿⣿⣆⠀⠀⢱⠁⠀⠀⠀⢻⡴⠛⠆
⠀⠀⠀⠀⠀⢰⣿⠐⢄⡀⠈⠙⠳⡜⣻⡿⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠹⣿⣿⣆⠀⠀⠁⠈⠀⠀⠀⢷⡀⢨⡀
⠀⠀⠀⠀⣠⠿⣋⣠⣤⣭⣝⡛⣧⣴⣿⡇⠀⠀⣿⠙⡆⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠹⣿⣿⣶⣄⠀⠀⠀⠀⠀⢨⡙⢦⢱⡀
⠀⠀⢀⣌⣵⣪⠽⢿⡧⣄⡈⠉⣸⣿⢿⡇⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠹⣿⡿⣟⢧⡀⠀⠀⠀⠀⢇⠀⢠⡇⡀
⠀⠀⢸⣿⢟⠝⠀⠀⠀⠉⠻⣼⣿⣿⣷⡡⣀⣾⠁⠀⠀⠹⡄⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢀⣿⣇⡿⣦⡁⠀⠀⠀⠀⠈⡆⣼⢠⠇
⣀⣀⣰⡀⠎⠀⣠⠔⠋⠀⢀⡠⣹⡿⠃⠈⢞⢛⣲⣀⣀⣀⣙⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣠⣿⣿⣯⣄⡹⣷⣦⣀⠀⠀⠀⠀⣱⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠙⢷⣤⣴⠃⠀⢠⠊⢡⢾⣿⢀⡙⢣⠈⠙⠧⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠻⠿⣟⠉⠙⠛⠿⢮⣽⣿⣄⣤⠾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠈⠙⠛⠿⢿⣶⣷⡸⣨⣧⢠⣙⠇⠄⣁⠒⠸⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⣀⠤⠤⠟⠛⠀⠉⠳⠭⠤⠀⠐⠉⠁⠉⠂⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⡀⢤⣄⣒⣦⡠⠤⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⡠⠼⠷⠞⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⣀⣀⡠⠤⠖⠚⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠐⢢⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠹⠋⠉⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣤⣥⣶⣷⣧⣦⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⢨⡙⠛⠋⢀⣀⣰⣆⡀⠀⠀⠀⣀⠰⣀⣤⣴⣶⣄⣀⣰⣦⣂⣖⣀⣤⣶⡀⠀⠀⢐⣂⠙⡶⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡄⡯⠟⠇⠊⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⡏⠳⣄⡐⠻⠖⢊⠐
⠀⠀⠀⠀⠀⠀⠀⡆⠐⠀⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢷⡾⣷⣁⡀⠀⠀⠅⠀⣑⡭⣶⢾⠡⠺
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠸⠀⠀⣤⣆⠀⠀⢶⣷⣀⣶⣷⡀⠀⠀⠀⠀⣾⡟⢩⢟⠃⠀⠀⠓⠙⠁⠀⡟⠑⢪⡌
⠀⠀⠀⠀⠀⠀⠀⡆⠀⣰⠂⡄⠠⠼⠏⣱⡌⠉⠉⠈⢉⣉⠁⠀⠀⠀⠀⠀⠀⠀⡑⢣⡀⣄⣀⣀⡀⠀⡧⢔⢿⢉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣁⡇⠀⠀⢠⠃⠛⣀⡮⠘⠈⠚⠰⣓⡄⠀⠀⠀⢀⠴⣚⢌⠬⠚⣟⠓⠚⠠⡇⠉⠁⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣀⢐⠲⡇⠀⠀⠞⠷⠀⠖⢠⣷⣀⠀⠀⠀⢒⣄⠀⠀⠁⠈⠀⠈⢈⡇⠀⠀⠀⠠⡇⣀⠛⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡉⠉⠁⢰⠀⢰⣶⠀⠐⣉⠀⢀⣳⣇⠀⠀⣌⢥⢶⡿⠞⠀⠀⠠⢢⢠⠤⢦⣦⠀⡇⢍⠢⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠛⠛⠁⠀⠰⡢⠘⠛⠋⠀⡰⠫⢦⠉⢜⡧⢰⡲⠈⠁⠈⡖⣷⡏⠀⠅⠠⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠞⠘⣽⠀⠀⠀⠀⠀⠀⣼⣟⡍⠄⠀⡦⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⣃⣸⣧⡀⡧⡄⠀⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣕⣄⣀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⢞⡟⢠⣰⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠚⠛⠉⠩⠬⠠⠭⠉⠭⡭⢭⣭⣭⣭⠭⠭⠭⠥⠤⠤⠴⠤⠖⠲⠯⠯⠤⠤⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢧⡃⠀⠀⠀⡇⣰⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣸⣧⣷⣾⡦⣏⢾⣿⣿⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡻⣿⠇⣿⢳⡟⣿⢻⢿⡗⣀⣀⢀⡀⢠⢀⠀⡀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠳⠳⠷⠈⠈⠾⢰⠹⢱⠟⡶⠼⠞⠇⠿⠘⠆⠃⠍⠐⠞⠘⠠⠨⢢⢿⠙⠹⠬⠽⠻⠷⠝⠎⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⢩⣇⠖⠛⠲⣤⠖⠲⣼⠉⠉⠙⣦⣀⣀⣀⡤⢿⠀⣀⠉⣷⠋⠳⠴⠒⡛⢛⡻⠦⡇⢸⠶⢤⡀
⠀⠀⠀⠀⠀⣀⣤⣼⠁⠀⡄⠼⣿⡀⢠⡄⠘⠠⡄⠘⠀⠶⠀⡸⠈⠢⠤⠃⢸⠀⣄⠘⠇⠑⡆⠰⠊⠃⣬⠇⠀⠓⠒⠀⠀⠘⣦⠤⠤⠤⣄⡀
⠀⠀⣀⡤⠾⠁⠀⠈⢆⡀⠁⠀⠀⣇⣀⡠⠜⠤⠤⠜⠂⠐⠊⠁⠀⠀⠀⠀⠠⢀⣀⡠⠂⠀⢥⡬⠀⢅⣀⣸⠀⠪⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⢀⡞⠁⠀⠀⠀⠀⠀⠀⠨⣁⡸⠉
⡞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠈⠓⠦⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠓⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣤⡤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣀⡤⠤⠤⣄⠀⠀⠈⠫⣔⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⣉⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠋⠈⠳⢤⣦⣀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡏⠄⠀⠀⠰⠀⡌⠑⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣤⢤⡖⠂⣠⣶⣾⣁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⢈⡹⠁⠀⠈⣿⣿⣿⠉⠙⠛⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⣞⢸⠶⣧⠤⣀⠀⢀⠴⠋⢷⣤⣤⣾⠏⢸⣼⡒⠛⢻⡙⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣇⡀⠀⠉⠁⢀⡴⢻⣿⠉⠁⠀⠀⠣⣳⠀⢀⣇⠷⠼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋⠀⠀⢻⠀⠀⠀⠀⠀⠘⣶⠊⠹⢦⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢀⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠗⠶⠶⠒⠋⠉⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣠⠴⠒⠚⠋⠉⠹⡄⠀⠈⡇⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠹⡤⠞⠉⢿⣇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡋⠉⠓⠓⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⡤⠖⠋⠘⢆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠙⠃