Worried Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢆⣀⣤⣤⣖⣤⣀⣤⣶⣶⣶⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⢰⠋⠀⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠚⠁⠈⠳⢤⣿⣿⣿⣿⡟⣠⠌⠁⣀⢻⠉⠀⠀⣻⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⣿⠿⣿⣿⠀⠁⠀⠀⠈⠉⣄⠀⠉⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⠀⠀⣸⠈⢑⡎⠉⠀⠀⠉⠈⢣⠀⢸⡇⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⡽⠛⠛⠛⠻⠶⡤⢀⣀⡀⠦⠀⠞⠀⣸⠁⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⣡⠆⠀⠀⠀⣷⠀⠤⠬⠙⠂⢀⡴⠃⠀⣀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣠⠞⠁⠀⠀⠀⣰⠃⣹⣦⠔⠖⡺⢹⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⡜⠁⠀⠀⠀⠀⡰⠃⢉⣿⡏⡟⢠⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠏⠀⠀⠀⢀⣠⠞⠀⠀⣿⣿⣷⣣⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⡤⠖⠋⠈⠑⠢⠤⢿⡿⣟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢣⡀
⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠰⡀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢠⠏⠒⢤⡀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⣀⣹⢶⡊⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠹⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢰⠞⠁⣏⠀⠀⠀⠀⠀⣠⠴⠋⢀⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⣹⠤⣤⠤⣄
⠀⠀⠀⢻⣀⠀⠘⢄⣀⣤⠖⠋⠁⠀⠀⠀⠉⠲⢤⣀⠀⠀⣧⠀⠀⢠⢎⡵⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⢄⡀⠀⠀⠉⠀⠸⡄⣠⡿⠋⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⡰⠇⠀⠀⠀⠀⠈⠓⠲⠀⠀⠀⢻⡟⠛⠲⢶⡶⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠛⠹⣋⠙⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠔⠃⠀⣑⣀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⡀⠀⠌⠉⠉⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⡾⣧⠀⢠⢚⡳⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠒⠟⠀⢨⡀⠑⠴⠀⠸⣭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⡻⠲⣄⠒⠀⠚⠂⠇⠀⠀⠀⠀⣿⠠⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠔⠁⠀⢠⢀⠀⣀⠀⠐⠀⠀⢹⠅⠊⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⢀⠰⠆⣀⡄⠋⠀⠀⠀⢀⣀⠤⠶⠋⠀⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⢀⢁⠈⠉⢉⠟⢒⠞⠉⣷⡀⣲⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠚⢠⣴⣧⠀⢸⡇⢠⣿⣿⣅⣠⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣾⣿⣿⡀⢸⠃⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢀⣿⣧⣥⣿⡟⢿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠟⢉⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡇⠀⠈⠻⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠻⢿⣿⣿⣧
⠀⠀⢰⣿⣿⣷⣾⣿⡋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣠⣬⣿⣿⣿⡿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣠⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣷⣝⢿⡿⣫⣿⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠒⠀⠀⠈⠉⠁⠩⣿⡿⠛⡄⠙⠥⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠩⠮⠒⠉⠀⠀⠛⠈⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠤⠤⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠦⣄⡀
⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠒⠒⠒⠲⡄⠐⡖⠒⠒⠚⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢀⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢠⠚⠄⠀⢤⡀⠈⠻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⣲⠖⡆⠀⠀⠀⠀⠀⠀⢀⡤⠛⠂⠀⣀⠈⠓⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠦⢊⡞⠁⢈⠒⡁⠀⣀⡄⠀⠀⠎⠀⠀⠀⠀⠉⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠛⢀⢀⣀⠚⣏⢻⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠃⡠⠊⠀⡟⠘⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢃⣴⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠄⠐⠈⡔⣦⠁⣁⡆⢈⣣⠄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⣠⢊⣠⣴⣶⣶⣶⢤⡇⠀⠉⣥⠋⢿⠀⠀⠀⠀⠀⠀⠀⢨⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣰⡿⠛⢩⣿⣿⡟⢠⣾⠀⠀⠀⢰⢀⡈⠀⠀⢀⡀⠀⠀⠀⢘⣷⡄
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⢀⣼⠏⠀⠀⣾⣿⡿⠀⠀⡿⠄⠀⠀⠈⢡⠃⢠⢾⡟⠙⠛⣛⣛⠩⠟⢧⣀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⣼⠏⠀⠀⢀⣿⣿⠃⠀⠀⢠⠀⠀⠀⠀⢰⡴⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡀
⠀⠀⢀⠜⠁⠀⠀⠀⠀⢀⣾⠋⠀⠀⡠⣼⣿⡏⠀⠀⠀⠀⢃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠻⣦
⠀⣠⠎⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠊⢀⣿⣿⠁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⢹⡇
⠀⡏⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣼⠁
⢸⠃⠀⠀⠀⠀⢀⣾⣏⣠⣴⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢠⡿⡀⢀⣠⣴⣾⣿⣿⣆⠀⣰⡿⠁⠀⠀⠀⠀⠀⣠⠏
⢸⣤⣀⣀⣤⣾⣿⣿⣿⣿⣿⠿⠛⠋⢸⣿⠃⠀⠀⠀⠀⠀⠀⡾⠀⢿⣿⣿⣿⣿⠟⢻⣿⣾⣏⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠻⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢸⡇⠀⠘⢿⠟⠋⠁⠰⠟⣹⢻⡏⠓⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠸⠇⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢀⣴⡿⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠾⠟⠁⠀⡏

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠄⡩⠒⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠂⠊⢀⠐⠩⡀
⠀⠀⠀⠠⠀⠀⠀⠘⠈⢀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠂⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠤⠀⠂⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⡠⠔⠚⠋⠉⠅⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠊⢀⠀⠀⠀⠀⠀⠀⡔⠶⠕⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣤⡀⠀⠀⠰⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠢⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢵⠢⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⡆⠀⠀⢦⡂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣏⠐⢄⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠏⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⢌⠳⡄⡑⢄⡀⠀⠈⠐⡤⠔⡦⣀⣀⡉⠂⠀⡀⠀⠀⠀⠀⠠⠄⠴⠺⣏⡀⠀⠀⠀⡀⠀⠠⠀⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⢴⠀⠉⠉⠢⢄⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠓⠒⣂⣨⣟⣛⠶⢶⣶⠶⣒⣫⠭⠗⣲⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⣘⠀⠀⠀⠀⠀⠈⠑⠠⠤⡀⠀⠀⠀⠀⠀⢱⠀⠀⠄⠲⡢⠸⠏⠟⠑⠎⣻⠶⢦⣽⡥⠶⠚⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⢃⠀⠀⢸⠓⣔⢾⡀⠀⠀⠋⣀⠀⠀⠀⠀⠀⠀⠉⠢⡉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠒⠤⠠⡄⠀⢸⠄⢨⠂⠁⡁⠁⠸⣍⠺⡒⠤⠤⠐⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡀⠤⠐⠚⡀⠀⢸⠁⠸⠀⠀⢌⠠⠤⠘⢦⡘⢄⠀⠀⠑⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡀⠀⠛⢀⡀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⣤⠀⠀⠀⠁⠀⢰⠆⠚⠀⠀⢠⠀⠀⠀⠀⠑⠦⣌⡉⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡄⢺⠿⣄⡀⠀⠀⠀⡸⠠⠃⠀⠀⣀⣀⠔⢁⡄⠀⢠⠁⣆⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⡡⠀⡙⠘⡙⠆⠀⠊⠀⠀⠀⠀⢸⠁⣄⠀⠃⢀⠄⠁⠠⠀⠀⢀⡀⠀⠠⠦⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⣻⠉⠈⢻⣀⠂⠀⠀⠀⠀⠀⠀⠘⢄⡍⠁⠀⠁⠀⠰⣗⠢⢤⠄⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠂⠀⠀⠀⡠⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢠⡿⢁⣤⠞⠋⠉⣉⣙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠛⠛⠛⠛⠛⠷⢾⣷⡋⠀⣠⣶⡛⢿⡍⠙⣷
⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠾⣧⣄⠻⡄⢹⣴⡟
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⡤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣼⠟⠀⠀⠀⠀⠀⣠⡤⠤⢤⣄
⠀⠀⠀⣰⡟⠁⠀⠀⢀⣤⠖⠋⠀⢀⡴⠞⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢀⣾⣿⣟⣲⣤⣄
⠀⠀⣰⡏⠀⣠⡴⠞⠉⠀⣀⡤⠞⠋⠀⢀⡴⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢰⢋⡽⣞⣿⣿⣿⠿
⠀⢠⣿⡴⠛⠁⢀⣠⠶⠛⠉⠀⢀⣤⠞⠉⠀⠀⢀⣼⢿⡆⠀⢠⡀⠘⣦⠸⣧⠀⠀⠘⢿⣿⣉⣿⣿⡧⠴
⠀⣾⣇⣤⠶⠛⠉⠀⣀⣠⠴⠚⠉⠀⠀⢀⣤⡾⢋⠁⡈⢿⣄⡈⢳⣄⠈⢷⣿⠀⠀⢀⡾⠶⣌⣻⠋
⠀⣿⣉⣠⣤⣴⠖⠛⠉⣀⣴⣦⣤⡶⠞⢛⠁⣷⣼⠀⡇⢸⠻⣧⣇⠙⢦⣄⣿⡄⠀⠀⢹⣿⠟⠁
⢀⣿⣧⣶⠶⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠘⠀⠇⢸⠙⡟⢻⠛⡏⣟⠻⢶⣬⣿⣇
⣿⠁⣿⡆⠀⠀⢠⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣿⣄⠃⠋⠀⢸⡿⠀⣻⡆
⢿⡄⢸⡇⠀⠀⠈⣽⡿⠿⠿⠗⠀⠀⡤⠀⠀⠺⠿⠿⢿⣯⠅⠀⠀⠀⣸⡇⢀⣿⠁
⠈⠻⣾⣿⣴⣶⣷⣶⡄⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⢀⣿⡶⠟⠁
⠀⠀⠀⠸⣧⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣼⠇
⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⢀⣤⣀⣠⣄⣀⣤⡀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠉⠀⠈⠁⠈⠁⠀⠉⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⡶⠶⠶⠶⢾⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡇⠀⠀⠀⢼⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠙⢧⣄⠀⠀⣀⣤⠞⠉⠉⠻⢶⡄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃

⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣄⡀⠀⠀⠀⠀⢀⡴⢺⡗⠀⠀⢀⠀⠀⠀⢠⠆⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⢱⡄⠀⠈⠛⢦⡀⠀⠀⡞⠀⣼⠁⠀⠀⢸⡀⠀⠀⣼⠀⠀⠀⣼⠁⠀⢀⡞⠁⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣖⣒⠎⢻⡀⣸⠁⠚⣇⣀⣀⣀⣈⣀⡀⠀⠿⠀⠀⣼⠇⠀⢠⠟⠀⢀⡴⠋⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡍⠛⠿⠗⣋⡤⠚⠉⠉⠉⠉⠙⠛⠿⢿⣷⣤⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⠖⣿⠏⠀⠀⠀⠀⠀⠀⢀⣀⣒⠲⣜⠻⡟⠲⢶⡲⣄⠀⠈⠀⣠⡾⠛⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢳⣟⢁⠴⣾⠛⣳⡆⠀⠐⣋⣀⠈⡇⢸⠀⢻⡄⣧⣟⢻⣆⠀⠚⠋⢀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣈⣧⣀⠛⢋⣉⣤⠶⠋⢩⡹⢇⣧⡾⣠⣿⣿⣿⠇⡜⡧⠀⣆⠀⠋⠁⠀⣀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⣿⡑⠲⠝⢛⣫⠟⠀⠀⠀⠘⠿⠟⠹⣿⣿⠟⠋⠡⢞⡴⠃⢠⡟⠀⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⣾⠥⢈⡽⠶⣾⡻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣤⣭⣤⣾⠟⠁⠀⢀⣀⣤⡴⠶⠒
⠀⠀⠀⣀⡴⠞⠉⠠⢂⠔⠉⠳⣿⣶⣿⣿⣿⡵⠂⠀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠈⠉⠁
⠀⢀⡞⠉⢀⠀⠀⠊⠁⠀⠀⢰⡏⢿⡍⠉⠀⠰⠂⠈⣁⣀⣀⣉⣍⠀⠀⠀⠙⢷⣤⡼⠀⠀⠀⠀⠀⠰⠶⠒⠛⠋
⢠⡏⠀⠀⠁⢀⡖⠉⠁⢢⡀⠈⢷⡘⢃⠀⠀⠀⠀⠀⠻⣿⣩⡏⠁⠀⠀⣿⡶⣦⣼⡁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⣾⠀⠀⠀⠈⢸⡑⠢⣰⠾⡄⠀⠻⡷⣟⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡿⠧⠈⣿⠁⠀⢀⣀⠀⠉⠉⠉⠉⠉⠉⠉
⠹⣆⠀⠀⠐⠤⣩⡿⠃⢰⡇⢳⠀⣷⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣱⠀⢀⡟⠀⠀⠀⠉⠙⠓⠶⠦⡄
⠀⠙⢿⣖⠒⠚⢁⣀⣤⠎⠀⣸⣠⣿⠿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⠟⠀⠀⠀⠈⠳⢦⣄⡀
⠀⠀⠀⠻⣯⣉⣭⣖⣡⠤⠞⠋⠽⢋⡼⠉⢳⡀⠀⠀⠀⠈⢻⣛⢻⣿⣆⡰⠋⠀⠀⠈⢿⣄⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠉⣿⣻⣶⣲⡛⣇⠈⠉⠀⠀⠀⠙⢦⣀⠀⠀⠈⠙⠛⢙⠦⣍⡛⠲⣤⡀⠀⠙⠷⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⡘⣆⠀⠀⠀⠀⠀⠀⠉⢳⣦⣀⣀⡴⠋⠀⠀⠙⠶⡌⠙⢦⣄⡀⠈⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡘⢦⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⣠⡴⠛⠛⠛⠛⠓⢫⣭⠙⠶⣤⣀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠹⡄⠀⠀⣠⣾⣿⣿⣿⣿⡷⠞⣁⡴⠛⠉⢋⡿⠛⠉⣠⠤⠀⠀⠉⠛⠲⢤⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠹⣄⣼⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⢠⠏⢠⡶⠋⣀⣤⡀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣆⢻⣿⣿⠿⢿⣿⣿⣿⣿⠀⠀⠀⢀⡞⣰⠟⣰⠋⠁⠀⠙⠶⣤⣀⣀⣀⠀⠀⢻
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠹⣿⣿⠁⠀⠘⣿⣿⣿⣿⠀⠀⠠⡿⠞⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⢻⣅⠀⠀⠈
⠀⠀⠀⠀⠛⠿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠙⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠉⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢹⡄⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠰⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡤⠀⠀⠀⠀⠀⠀⠀⣿⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠟⠀⠀⠀⠀⠀⠀⠀⣸⡿⠙⠷⣦⣌⣻⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⡶⠓⠀⠈⠉⠉⠉⠉⠉⣉⠉⠙⠙⣿⡿⣦⣀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⢀⣰⣿⡟⠁⠀⣀⣴⣶⣿⣦⡀⠀⠀⣠⡿⠛⠳⢦⣜⣷⡄⠙⣦⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢀⣠⡴⢟⣿⠋⠀⢠⡾⠋⠀⢀⣴⣬⡝⣦⢸⡏⢀⣤⣄⠀⠙⣿⣿⣦⠈⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠐⠛⠋⣩⣴⣿⠃⠀⠀⢿⡁⠀⠀⠈⠻⠿⢋⡾⠻⣷⡸⠿⠟⠀⠀⣾⢿⡏⢷⡄⠀⠀⠀⠀⠈⢳⡄
⠈⠉⠉⠁⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢘⣿⠿⣿⠀⠀⠰⣬⣳⣤⣤⣤⣤⡴⠟⠁⠀⠙⢿⣶⣤⣤⣼⣋⡼⣷⠀⠻⡄⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⢰⣆⠀⠀⠀⠀⢹⡏⠛⠛⠋⠀⢻⡟⣷⢻⠀⠀⠀⠀⠀⠘⣷⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣶⠶⠟⠁⠀⠀⠀⠀⢀⣰⡏⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠳⣤⡀⠀⠀⠀⢻⡌⠻⢿⣷⣄⣀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⡶⣶⣶⢤⣤⣤⣤⣄⠈⠻⣇⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣼⣿⣶⣤⣤⣤⣭⣤⣴⣿⡟⠁⢀⣴⣿⠋⠙⣧⠀⠈⣷⠀⠀⣼⡇⠀⣼⡇⠈⣹⣷⡀⢻⡇⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣭⣭⣍⠀⠀⡿⠀⣼⣿⡇⣻⣄⣴⣿⣦⡴⣿⢶⣤⣿⣧⡶⢿⣟⠛⢿⣿⣦⣾⣇⣀⣀⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⢳⣼⣧⠀⠙⣻⡿⢋⣿⠋⢀⡿⢀⣿⠀⣿⡏⠙⢷⡀⠙⣧⡀⢻⡎⢻⡟⠻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠻⢿⡟⠀⣼⠃⠀⣸⠇⠸⠏⠀⢻⣷⡄⠈⠻⠀⠸⡧⠘⣿⠘⣿⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣄⣄⢺⡇⠀⠁⠀⠀⠉⠀⠀⠀⠀⢸⡇⠙⣦⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣹⡎⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢻⣦⡄⠀⠀⠀⠀⠀⠀⣼⠇⠀⣿⡀⠀⠀⠀⠀⠰⣿⣤⡤⠶⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣸⣿⣁⡴⠟⠁⠀⠀⠀⠀⠀⣀⡼⠋⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⠿⣤⣤⣀⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠶⠛⠉⠉⠁⠈⣹⡿⠋⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠉⠻⣍⡛⢶⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣤⡶⠾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠘⠷⠈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⣾⡟⠁⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡆⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣤⡀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠉⠻⢿⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠶⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠟⠉⠙⠛⠛⠛⠻⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⡴⠤⠖⠉⠁⠀⠘⠁⠈⠉⠉⠈⠲⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⡐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡀⡀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠉⣤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⡤⠛⠻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⡟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⢿⣿⣿
⣿⣿⣿⣿⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠤⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢸⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠿⠿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⠀⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⠉⠁⠀⠒⠚⠓⠀⠀⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⠉⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉⠧⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠑⢦⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⠾⢿⣿⣿⣿
⣿⠿⠋⠀⠀⠀⠀⠈⠑⠆⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⣀⠔⠉⠀⠀⠀⠉⠻⣿
⣯⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⠁⣰⠆⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢴⡀⠙⣿
⣿⣿⣿⣏⠀⣼⠃⢠⣿⣷⣿⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣤⠘⣀⣾⣷⡀⠻⣄⠀⢿⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⢴⠄⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣿⣶⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⢠⣤⣤⣤⣤⣤⣤⣤⡄⣸⣤⣄⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⣛⡶⣶⠶⠶⠶⣲⣶⣬⠀⣿⠁⢈⣷⣶⠶⠶⠶⠶⠶⠉⠛⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠉⠀⠀⠀⠉⠉⠉⠉⠉⡠⣜⠣⡈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣒⣀⣀⣀⣀⣐⣒⣨⣴⣿⣷⣌⣐⣒⣀⣀⣀⣀⣀⣒⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⢀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠿⠿⠿⣶⣤⣿⠋⢿⣧⡾⠟⠋⠉⣿⣧⣴⡶⠾⠿⠿⠷⢶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⡀
⠀⠀⠀⠀⠀⢠⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠀⠀⠀⠈⠋⠠⠤⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⣤⣄⡈⠀⠉⠀⠉⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⢀⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⡃⢀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⡴⠛⠉⣉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠟⠁⢀⣤⠈⠙⣦⠀⠆⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠸⠿⠀⣼⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠿⠁⠀⠻⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⠀⠈⢳⣆⣀⣠⡴⠋⠀⣀⡀⠀⠀⠀⠀⠻⣦⣀⣀⣠⡾⠃⣤⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠈⠉⠁⠀⠀⢿⣍⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⣨⡿⠇⠀⠀⠀⠀⠀⠀⠰⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷⠀⠀⠀⠀⠀⠀⣶⠉⢁⣤⣄⠉⣧⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠈⠛⠁⠀⠀⠀⠀⠀⢀⣠⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠙⠿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠾⢿⣿⠻⣷⡄⠀⠀⠀⠀⣾⡟⠛⠁⢀⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡀
⠀⠀⠀⠀⢀⣼⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠻⢧⣀⣀⣀⣠⣿⠛⠻⢿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣾⠿⣷⣀⣀⣾⠟
⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠉⠉⢶⣤⣀⡠⢿⣧⡀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢛⣿⡿⠦⠤⠤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣴⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⣴⣦⣤
⠀⠀⢀⣠⣴⠿⠋⠁⠀⠀⠀⠀⣠⡾⠋⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⣴⣾
⣴⡿⠛⠛⠙⢦⣄⠀⠀⠀⢀⣾⠟⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⢿⣿⣶⣄⣀⠀⣠⠚⠀⠀⠀⢲⣤⣿
⢹⣷⡾⠋⠀⠀⠙⠷⣀⣴⡿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣿⣇⢰⣦⣰⣦⣽⡟
⣿⣯⣴⠟⢠⡄⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⠿⠛⠛⠛⠁
⠀⠙⠿⢷⣿⣧⣴⣷⣼⡟⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣷⠖⠲⠶⠶⠶⠶⠶⠲⠶⠖⠒⠒⠒⠒⠒⠒⠒⠈⢹⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠒⠒⠂⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠙⠻⠷⠶⠶⠦⢤⣤⣤⣴⣦⣄⡀⢀⣀⠴⠞⠉⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣤⣤⣴⣦⣤⣤⣤⣤⣼⠿⠿⣧⣀⣀⣀⣀⣠⣤⣤⣤⣤⣼⡷
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠙⠛⠛⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠟⣛⠛⠉⠉⠩⢉⡉⠙⠛⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣹⠟⠉⠀⠀⠀⠀⠀⠁⠒⠤⡀⠀⠁⠂⠄⡀⠈⠙⠫⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣠⣤⣦⠔⠈⠈⢿⠐⠠⢀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠈⠑⠠⡀⠀⠑⠶⣶⣤⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⡿⠃⠀⠀⡄⠈⢷⡀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠀⠁⠢⠄⡈⠙⢿⡀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡿⠁⠀⠀⠠⠀⠀⠀⠹⢦⠄⡀⠀⠀⠀⠁⠒⠠⠀⡀⠀⠀⠀⠀⠉⠀⠒⢀⡤⠞⢚⡇⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡿⠁⠀⢀⠔⠁⠀⢀⠊⠀⠀⠈⠀⠁⡐⠀⠠⠄⠀⣀⣀⣀⡀⠁⠠⠴⠾⠋⠄⣤⠔⢈⡅⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⢀⡀⠚⠁⠀⠔⠋⠀⠠⠚⠁⠀⠀⠀⠀⠄⠀⠀⠉⠰⠦⠄⠀⠀⠀⠀⠄⠠⠄⠒⠈⠋⠁⠙⠛⡳⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⢻⣦⡴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⢀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠘⣿⣿⣿⣿
⡿⠿⠛⠋⠀⠀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⠈⠻⠿⣿
⣷⣄⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠇⠉⢀⣀⡐⠂⠀⢀⣀⣉⠀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠑⠠⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠄⠊⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠀⠀⠀⠀⢀⠈⣻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⡞⢀⡄⢠⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣀⠀⡄⠰⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣿⣿⣷⣿⠀⠀⠐⠀⠀⠀⠀⠀⣤⠀⠄⠀⠀⠐⠀⠀⠀⢻⣶⣾⣿⣦⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣀⡀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢘⡛⠷⠶⠶⠶⣾⠁⣿⠀⡻⠶⠶⠶⠯⢙⡃⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⠀⠀⠀⠀⡀⢔⣦⡀⢀⠀⠀⠀⠀⠉⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⣿⣾⣿⢿⡿⣇⠰⠴⣿⢾⣧⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⢻⣏⠗⠛⠛⠉⠁⠀⠀⣊⠷⣻⡇⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡿⢬⣷⠘⠦⠭⠁⠀⠀⠀⠠⠋⠀⠻⠋⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡿⠗⣸⢟⡐⣁⡀⠀⢀⠀⠀⠀⢀⠀⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⣿⣛⢾⣨⣱⡾⠁⢼⡗⠀⠀⢀⡤⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣮⠿⢯⡵⠋⠀⠀⠀⠈⠀⠀⠀⠛⡁⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣌⢿⡳⢫⣹⠟⠤⡀⠀⠀⠀⠀⠀⠀⠒⣩⣿⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢘⣺⡶⣝⣿⠗⣘⢛⠀⠀⠀⠀⠰⣾⡶⢪⡿⡉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠳⣶⣶⠍⣿⡀⠀⠀⠀⠀⠀⢀⡰⢇⢠⢃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⣀⡴⠚⠉⠀⠐⡤⠆⢉⡴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⢀⡞⠉⠀⠀⠀⠀⠀⡆⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣏⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⢰⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⣜⣻⣿⣻⣷⡽⣯⣛⠛⠀⠀⠀⠀⢸⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣶⣾⣿⣿⣦⠀⠀⠀⠈⡇⠘⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢭⣲⣿⣿⣿⣧⠀⠀⠀⡸⡂⢷⣈⣠⡶⠆
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠂⠀⠀⠦⠤⢤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣧⣠⠎⠨⠖⠻⠋⠁
⠀⠀⠀⠀⠀⢠⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⡿⠃⠀⡰⢣⠀⢠
⠀⠀⠀⠀⠀⢸⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢻⡿⠁⣠⣞⠀⠸⡄⠈⡆
⠀⠀⠀⠀⠀⠈⣝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣿⣿⣿⣿⣿⡿⠟⠋⠀⢀⡤⠊⣠⣾⣿⣿⣦⠀⣧⣤⣼⣦
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢀⣶⡫⣦⠞⠁⠈⢿⣿⣿⣧⡸⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⡦⠶⠶⠶⠒⣻⣿⣾⣿⣿⣿⡿⠋⠀⠀⠀⢀⡴⠻⢮⣟⠷⠀⠀⠀⠀⠻⣿⣿⣷⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣾⠛⠁⣠⣿⣿⢸⣿⣿⠇⠀⠀⠀⣠⠖⠉⢀⡔⠻⠈⠶⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⢀⣴⣿⣿⣿⣾⣿⣿⠀⠀⢰⣿⣿⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⡟⣿⡿⣠⣾⢼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⠟⠁
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣤⣶⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠛⠿⣿⣿⣿⠏⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⢀⡀⠀⠀⢠⠀⢸⣷⣦⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣠⣿⣧⡀⠀⢸⡀⢸⡙⢿⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠐⠿⡇⠻⡇⠀⠈⡇⠀⠙⠊⠀⠀⠈⢷⡀⠀⠀⠀⢀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠣⠴⠃⠀⠀⠹⡄⠐⣦⣄⡀⠀⠈⢳⣀⡠⠞⣉⣠⣀⠉⠁⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⣆⣀⣀⡰⠃⠀⠉⠢⡀⡸⢋⡴⢫⢞⡥⢿⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠺⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⣀⣿⡴⠛⣷⡷⠋⢠⠏⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢠⡀⠀⠀⠀⢰⠁⠀⠀⠀⣀⣤⠴⠋⠀⠀⠀⠀⢹⢀⡴⠃⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⢻⡀⠀⠀⢸⠀⣠⠴⠞⣡⠴⠀⠀⠀⠀⠀⠀⣸⠏⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢹⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠘⠁⠀⠀⠀⢠⣀⡼⠀⣠⡶⠋⠹⣄⠀⠈⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠶⣄⡘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⣥⠶⠞⠛⠛⣷⠞⢁⡾⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⢻⠻⡆⠀⠀⠀⠀⠐⠋⠘⡆⠀⠈⠢⣄⠸⠅⢠⠟⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⢀⡇⠀⢹⡄⠀⠀⠀⣧⠀⠀⡇⠀⠀⠀⢈⡳⠔⢿⣄⡀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢳⠀⠀⠀⢹⡄⠀⣷⠀⠐⢎⡁⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡡⠒⠀⢧⠀⠀⠀⣷⠀⢸⠀⠀⠀⠉⠐⣦⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠉⠀⠀⠀⠈⣇⠀⠀⢿⡇⢸⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠘⡆⠀⢸⣧⢸⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠸⡄⠘⣿⣸⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠛⠛⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠋⠉⠉⠉⠉⠙⠓⠒⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣙⣲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⢋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⠀⣀⢀⡤⠤⠤⠶⠒⠂⠈⠁⠢⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠓⠀⠀⠈⠲⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢶⠤⠳⡄⠀⠀⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠀⠀⠀⠀⠀⣟⠀⠀⠜⠁⡀⠂⡄⠀⠀⠀⠀⠀⣧⠀⠘⢆⠀⠀⠀⠐⢒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡆⠀⠀⠀⠀⠀⠘⡆⠀⠠⠊⠀⢰⠈⠀⠂⠀⠀⠀⢸⠀⠀⠈⢧⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠸⠖⠒⠉⠁⠈⢦⠀⢱⠀⠀⠀⠀⢀⠜⠲⣤⣤⣠⡤⠃⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⠀⠀⠀⠀⡴⢒⣈⣚⠓⠾⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⡠⠔⠦⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⢸⡁⠉⠑⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢊⠀⣀⡤⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠞⠀⠀⠀⠉⠓⠒⠒⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠠⠄⠰⣚⠟⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢎⠀⣀⡰⠖⣁⣀⣀⡀⠀⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠤⠭⠵⠒⠉⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢠⠀⠀⠀⢰⠀⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣀⣸⠀⠀⠀⡼⢻⡽⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠙⢀⣀⣀⡈⠀⢸⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠈⠁⠀⠐⠆⠢⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠒⠒⠒⠒⣶⢲⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠛⢻⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣀⣀⣀⣠⣼⠤⣤⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⠶⠶⢶⣾⣿⢾⣷⡾⠿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠉⠉⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠐⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⡄⠀⢀⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡆⠀⢸⠀⠀⢆⠀⠀⡇⠀⢸⠀⢠⠇⠀⡇⠀⢸⠁⠀⡌⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠃⠀⠀⢣⠀⠈⠀⠐⠃⢀⡁⠀⠋⢀⠜⠀⢰⣧⡄⠀⠁⠀⠀⠃⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠤⠊⠀⢀⠲⡄⢃⣠⠞⣹⠖⠋⣧⡤⡄⢀⣠⣤⢼⡅⠹⣦⡀⠀⡸⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⢀⠎⠀⠀⠀⡞⢀⠗⠋⣠⠞⠁⢠⡾⠋⢀⣏⠉⢁⣠⣤⣵⠀⠹⡕⣤⠃⠀⠀⠀⢂⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡜⠀⠀⢀⡞⡰⠃⢠⠞⠁⢀⡴⠋⢀⡴⠋⣹⠹⣍⢠⣦⠈⠇⠀⢷⢹⡀⠀⠀⠀⠀⠱⡀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⡰⢠⡏⡜⠁⡴⠁⢀⡴⠋⠀⡰⠋⠀⣠⠓⡄⠈⠳⣤⠴⠃⠀⢺⠀⡇⠸⡀⠀⠀⠀⠱⡀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢰⠁⠈⣇⠁⠈⠀⠀⠈⠀⠀⠈⠀⢀⡴⠳⡀⠘⢦⠀⠈⢲⡀⠀⢸⠀⢻⡄⠱⡄⠀⠀⠀⢳⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠈⡆⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢱⡀⠀⠁⠀⠈⠃⠀⣸⠀⠘⢹⠀⢹⠀⠀⠀⢸⡀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡸⠁⠀⠀⡼⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢻⠙⠲⠖⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⠀⠀⠀⢸⡇⠀⢻⡄
⠀⠀⠀⠀⢀⠖⠉⠀⣠⠞⠁⠀⢀⡴⠃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠒⠙⠒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠙⢦
⠀⠀⠀⠀⡏⠀⠀⢠⡇⠀⠀⠀⡎⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⠀⠘⠀⠀⡇⠀⢳⡀⠈⢢⠈⢳⡄
⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⣠⠃⠀⠁⠀⠀⠀⠀⠀⠀⡾⢙⡓⠛⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⢣⡀⠀⠈⠀⠀⣧⠀⣷
⠀⠀⢠⠞⠁⠀⠀⢸⠇⠀⠀⠀⡗⠀⠰⠁⢀⡔⠀⠀⠀⠀⠀⢀⡼⣅⡀⠉⠁⣸⣆⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⢆⠀⠀⠀⠀⢳⡀⠀⠀⠀⣿⠀⣿
⠀⠀⢣⡀⠀⠀⣠⠏⠀⠀⠀⣰⠃⠀⢀⡴⠋⠀⠀⠀⠀⠀⣰⠋⠀⠀⣩⠿⢏⡁⠈⠳⡄⠀⠀⠀⠀⠀⠘⣆⠀⠈⢳⠀⠀⠀⠀⢳⠀⠀⠀⣧⠀⢸⡀
⠀⠀⠀⠁⠀⢸⡇⠀⠀⢀⡞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⠜⠁⠈⠀⠙⢄⣀⠽⡄⠀⠀⠀⠀⠀⠘⣆⠀⠈⡇⠀⠀⠀⢸⠆⠀⠀⠈⠳⠀⠓
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰⠏⠈⠉⠁⠀⠀⠘⡆⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠘⠆⠀⠓⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠥⣸⣿⡷⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡴⠛⠀⠀⣸⡟⠣⠄⠀⢙⠗⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠋⠀⠀⠀⡀⣿⠈⠂⠀⠈⠢⠐⠄⠹⡅⠀⡀
⠀⠀⠀⠀⠈⠀⠀⡀⠏⠀⠀⠀⠀⠀⢴⡿⢀⠀⠀⠀⠀⠀⠀⡀⡈⠀⢁
⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠀⠀⠤⠐⢺⡇⠠⠄⡀⠂⠀⠀⠈⠐⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠡⢼⡇⢁⠒⠠⠀⠀⠀⠠⠀⣡
⠀⠀⠀⠐⠀⢰⡉⢀⠀⠀⠀⠀⠀⡀⣸⣗⡂⠀⠀⠀⠀⠀⠀⢰⠁⡄
⠀⠀⠀⠀⠐⢨⠃⠘⠀⠀⠀⠀⠀⠐⣻⡇⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠐⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⢀
⠀⠆⠀⠀⡰⠀⠀⠀⢸⠀⠀⠀⠔⠁⣿⠁⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠈⠆⠄
⢰⠐⠀⠊⡁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢂⠀⢄
⠀⠀⠄⢰⠁⠀⠀⠀⠁⠀⠀⢀⣴⣿⡿⢿⡀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡰⠋⡵⢹⠛⢟⢷⡄⠀⢀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⢂
⢀⢠⠀⢠⠀⠀⠀⠀⠀⢰⣱⠞⠞⠃⠁⠈⡜⣣⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠈⠠⠀⠀⠀⠀⠀⢠⠀⣿⠃⠀⠘⠀⠀⠀⠸⣷⡀⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠁⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡂
⠀⠀⠆⠀⠀⠀⠀⠑⠈⣧⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠐⢃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄
⠀⠀⡆⢀⠀⠀⠀⢠⠀⣿⠀⠀⠀⠀⣦⠀⠀⠀⢿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⡀⠁⠀⠒⠀⠠⠀⠀⢸⡘⠀⠀⠀⠘⠀⠀⠀⢸⡀⠀⠀⠀⠃
⠀⡇⡾⠀⠀⠀⠀⠇⢰⠸⢇⠀⠀⠀⠀⠈⠀⠀⢈⡇⠀⠑⠀⠠
⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠘⡀⢀⠀⠀⠀⠀⠀⠀⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⢇⠀⠀⠀⠀⠀⠂⠀⠙⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡄⠀⠀⠀⠀⠀⠰⠀⣿⢱⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⠀⠀⠀⠀⠀⠀⢂⡉⡀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⢰⠀⠀⠀⠀⠀⡸⡅⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡇⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠂⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⢰⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠈⠀⠀⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣶⠶⣦⣄⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣶⣥⣀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⡀⢻⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢟⡿⠉⢿⡀⢹⡇⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⢋⡞⠀⠀⢸⣧⠀⠇⠸⣿⣿⣿⣿⣿⣿⣿⣶⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣹⡏⢰⡞⠀⣰⡇⠈⠿⠀⠂⢀⣿⡿⣿⡏⢾⣿⣦⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⠙⢻⣿⣿⣿⣯⣿⣿⣿⣾⣿⣿⠃⢸⠀⢰⣿⠁⠀⠀⠀⢀⠞⣿⣿⣟⡹⣿⡄⢀⡙⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠳⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⠿⠀⠈⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠟⠉⠛⠟⠙⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢉⢐⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢰⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠓⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣦⡊⠀⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿⡧⠀⠀⠀⠀⢻⣿⣿⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⠙⠛⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠘⢿⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣹⣿⣿⣿⠃⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣓⣦⣼⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣶⣃⡄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠿⠛⠛⠉⠛⠛⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠔⠒⢄⡤⠂⠀⠀⠀⣰⣿⣿⣿⠟⠉⠀⠀⠐⠀⠂⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠿⠿⠀⠀⠀⠀⠰⠀⠀⠴⢄⠀⠀⢠⠄⠁⢸⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠄⠀⠊⠁⠙⢄
⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⡀⠀⢀⡠⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠁⠀⠀⡀⣀⢉⠀⠠⠀⠁⠊⠁⠀⠀⠀⠀⢀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⢠⠀⠀⠤⠀⢁⠀⠈⠀⢠⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠄⠀⠀⠀⢱⠀⠀⠀⠪⡀⡀⠀⢠⣀⠜⠀⠒⠒⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⠤⠤⠤⠤⢄⣀⣀⡠⠤⠒⠁⡠⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⡆⠢⢀⠐⠀⠀⢰⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠈⢸⠀⡇⠇⢨⠀⠀⠀⠀⠀⠀⠀⠘⠄⠂⠀⠀⠀⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠐⠀⠣⠀⡊⠠⡀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣋⣡⣤⠤⠤⠴⠒⠒⠒⠦⠄
⣀⣤⣴⣶⣾⣿⣿⣿⣗⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢟⡻⠟⠉
⠋⠉⠉⠈⠀⠀⠀⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠨⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡀⠀⢠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⣰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠔⠛⠳⣶⣿⣿⣷⠀⠀⠀⠀⠀⣤⢄⡐⢶⣤⣾⣿⣿⣇⣼⡇⠀⠀⠀⠀⣠⢺⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢻⡟⠻⣿⡿⢿⡿⢿⡿⠛⠛⠛⣻⣿⣿⠆⠻⢿⣿⣦⡀⠀⡜⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⢸⣿⢆⠀⣀⣾⠀⠀⢣⣀⣴⡾⠛⠻⣿⣇⠀⠀⠉⣿⣿⡈⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⣠⣿⠃⠀⣹⠟⠉⣳⠶⠉⠨⣿⡁⠀⡀⠹⣿⣷⣤⣴⣿⣿⠉⠙⢻⣦⡀⠀⠀⠀⠀⠀⠀⡠⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠋⠀⢠⠇⠀⣼⡷⠀⠉⠘⣿⡅⠀⠘⠀⠈⠛⣯⣍⣽⢧⠀⣀⣼⡏⢳⡀⠀⠀⠐⢤⣜⣣⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢯⡿⣤⣄⡆⠀⢸⣧⣿⠀⠀⠀⢸⣿⣤⠁⠀⢠⠞⠋⢻⣻⣧⣴⣿⠟⠁⠈⠀⠀⠀⠀⣐⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢛⠋⠀⠘⠛⣷⠀⠈⡏⣹⡷⠚⠷⣟⠟⢻⡀⡰⠂⠀⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⠀⠀⡠⠸⣿⡟⠿⢻⣿⣆⠀⠀⢰⠦⣾⣷⡖⠀⠀⠀⠀⣼⠿⣧⣀⣀⣀⣠⣤⠟⣿⣿⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣿⡟⠻⣆⠀⠀⠀⢀⡄⠰⣿⣀⠀⢉⣿⣯⡤⠀⣠⡟⣠⠟⠱⢀⣤⡤⢾⡟⠀⠄⠉⠋⠩⡝⣣⣾⣿⣿⣿⠁⠀⣾⣿
⠀⠀⠀⠀⢀⣿⠏⠀⠀⠈⢻⣦⣴⡟⠀⠀⠹⡧⣼⣿⡟⢿⣿⡿⠋⠁⠘⣦⠶⠫⠁⠀⢸⡇⠘⠀⣴⣷⣷⣾⠟⠉⣰⣿⣿⠄⢀⣿⣿⠂
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⡏⠀⠀⠟⣧⡀⢀⣼⠇⠀⠀⠀⠀⢸⡇⠀⠘⢋⣿⠋⠑⠖⠚⣹⡿⠙⠀⣸⣿⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⣿⠀⡀⠀⠀⢸⣿⢌⡁⢈⡤⠀⣀⡴⠻⣧⠀⣼⠋⠀⠀⠀⣠⡾⠋⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⢸⣷⣤⠶⠛⠉⠉⢻⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡇⠀⠀⡞⠻⣧⡿⠀⠀⢿⣌⠁⠀⠀⢰⡿⠋⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⢻⠀⠀⠀⠀⢻⣅⠀⠀⣧⡴⠪⡟⠲⡄⠈⣿⠀⠀⢠⡟⠀⣠⡾⠛⠉⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⢀⣿⠿⢿⣆⠀⠀⢸⠀⠀⠀⠀⠀⠙⠛⣬⣿⣆⣀⣾⡀⢹⡄⠘⣆⠀⢸⠇⣼⣿⣶⣶⣶⣦⣤⣞⣛⣻⣿⣽⣿
⠀⠀⣼⣿⡿⠋⢻⡆⠀⢀⣤⠞⠁⠀⠀⠹⣧⡀⠀⡆⠀⠀⠀⠀⢀⣴⠟⠁⣿⠛⠉⢉⣿⡇⠀⠙⡄⠈⠀⣿⣿⡿⣶⣤⣾⡉⠙⠛⠛⠛⠛⠉
⢀⡼⢿⣿⠁⠀⡘⢧⣶⣿⠃⠀⠀⠀⠀⠀⠈⢿⣄⣿⠀⠀⠀⠀⣼⠛⠳⣄⡿⠀⠀⢸⣿⡇⠀⠀⠘⣆⣀⠘⣿⣀⡼⠋⠙⢯⣷⣄⠀⠀⠀⢀
⣼⠁⠀⡇⠀⠀⢀⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢧⠀⠀⢰⣿⠀⠀⠀⠉⠳⣤⣿⣿⣿⡀⠀⠀⢿⣿⢀⣿⣯⣴⣶⣶⣦⠉⠛⠿⠿⠿⠋
⡇⠀⠀⠃⠀⠀⢿⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠀⠈⣿⡀⠀⠀⠀⢀⣿⠟⢙⢻⣷⣤⣄⣤⡿⠿⠿⡿⣟⠀⠈⣿
⠀⠀⠀⢸⠀⠀⠸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢿⡇⠀⠀⢀⣾⠏⠀⠀⠑⠿⠷⢿⣧⡄⠀⠀⢈⣿⠀⠀⠏
⠀⠀⠀⠀⠁⠀⠀⣽⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣇⢀⣠⣿⡿⠀⠀⠀⠀⠀⣠⣾⣿⣁⠀⣀⣼⡇
⠀⠀⠀⠀⠘⣤⡾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⡇⠀⠀⠀⠀⡼⠏⢀⣼⢏⣸⣿⠋
⠀⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡇⠻⣆⣀⣀⣤⣼⠷⠞⠋⠑⣦⡿⠁
⠺⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⣶⠾⠛⠛⠛⠋⠉⠀⣷⠄⡀⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠋⠑⠀⠀⠀⠀⠀⠀⠀⢠⠤⠐⠋⠀⠙⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠿⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠠⠅⠰⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⠃⠠⠄⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⡇⠀⠀⠀⠰⠂⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠐⢂⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡸⠁⠤⠄⠙⠋⠀⠀⠀⢢⡈⠁⡸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢣⡈⠁⡸⠀⠀⡿⢏⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠉⠁⠀⠀⠘⣧⡼⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡄⢤⡀⣭⠱⣄⠀⣀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠊⣭⠀⡤
⠑⢂⡈⠻⠆⠈⢶⠉⣨⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠘⡅⠈⡶⠃⡰⠟⢁⡀
⠷⢄⣉⠒⠈⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠬⠽⠿⠿⠿⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠁⠒⣋⡠
⠀⠀⠉⠙⠢⠄⣀⣀⡸⠛⠦⢀⡀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡀⠴⠚⢇⣀⣀⠠⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢆⡀⠀⠈⠃⠦⣀⠀⢠⡤⣤⣈⣙⣛⠲⠶⣦⡤⠤⠤⢤⣤⠶⠖⣛⣉⣁⣤⢤⡄⠀⣀⠴⠚⠉⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠉⠋⠛⠀⣀⣀⠀⠛⠉⠉⠀⠀⠀⠀⠀⣟⠉⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣤⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⡴⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢳⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢸⢿⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠈⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣣⢀⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢹⢸⡇⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣾⢸⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣹⠋⠀⠀⣍⣹⣼⡇⡅⠀⠈⢙⣏⠻⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣶⣶⣾⣭⣷⣶⣶⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠃⣀⣤⡴⠶⠶⠶⠶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀⠀⠀⠀⢀⣤⡶⠞⠛⠛⠛⠛⠳⠶⣤⡀
⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢦⣄⠀⢻⣆⡄⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣂⣿⡿⠀⢸⡟⢇⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏

⠙⡦⠄
⡞⠁⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠁⠀⠀⠀⠀⢀⠊⡠⠖⠚⠉⠉⣉⠂⠀⡥⢤⡤⠤⠤⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭
⡂⣷⠤⠀⠀⣤⠘⡥⠀⠂⣥⠖⠁⡠⢾⣤⠂⢂⠄⢀⠈⣄⣶⢰⡟⢸⠁⠀⠄⠀⠀⠀⣠⢠⡀⠀⠂⠀⠀⠂⡈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰
⠛⠀⠀⠀⢀⣿⢢⠑⢀⡴⠋⣠⠊⠠⣄⠈⠦⢓⠀⣀⣿⠸⣿⢸⡇⣼⢾⣅⣀⠠⠊⠀⠀⠀⠑⠀⠀⢈⡉⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣼⠁⣼⣷⠎⠠⡾⢏⠂⠀⡍⠀⣞⠶⠎⠀⠈⢀⠛⠘⢫⣹⣿⣲⢗⠾⠿⣿⡦⠀⠘⠀⣇⠈⠀⠀⠀⡘⢠⠀⠀⠀⠂⠘⠙⠎⠑⣦
⢀⣦⡀⢰⣿⢸⣿⡇⢠⠁⢱⡦⢉⠂⡍⠉⠁⡀⠀⠀⠀⡀⢥⠰⠙⠓⠘⠗⢩⣿⠿⠿⡇⢤⡠⡄⢸⣕⡋⢤⣡⠄⠈⡌⣀⠀⠀⠲⢀⠀⢸⣟⣷
⡉⠀⠀⢸⣿⡜⣿⠀⡆⠀⣄⣹⡾⠍⠀⠀⢀⣴⠝⠈⡐⢏⡌⠄⠸⠇⣴⢳⣤⠯⡄⣄⠚⡆⠳⣀⠋⡳⣴⠆⠑⠓⠀⠀⡭⠐⠂⠀⡈⠀⢀⠹⣾⡇
⠧⠄⠀⠈⢿⣷⡘⡆⠘⢄⠘⠁⠀⠀⢀⣴⢩⣱⣦⠀⠀⢄⠁⢄⠀⠀⡙⢮⣏⢑⠻⡃⢚⢡⡆⠻⠆⢹⣝⣿⣷⠟⡀⠠⠀⠀⠀⠀⠀⣠⣾⣷⣿⢻⡀
⡀⠀⠀⠀⠀⠙⢿⣬⣦⣀⠉⠁⢀⠀⢮⣶⣿⣿⣿⣦⠐⡈⢲⣦⡑⡀⡘⠉⠻⡬⠷⣀⣿⠇⣿⣿⣣⢘⡮⢈⡟⢎⢀⠀⠀⠀⢀⠀⡐⢿⣏⠂⣽⣿⠇
⠷⠀⠀⠀⠀⠀⣸⣿⡏⣭⠀⠀⡪⠶⢀⡉⠁⠋⠟⠟⢡⢨⠮⣧⣷⠋⠀⣾⠛⡆⣿⢮⣵⠀⢿⣌⣉⠀⢐⠟⢘⢄⡼⢀⣾⣿⡏⠀⠌⣠⣿⣤⣿⠃⣆
⠀⠀⠀⠀⠀⣰⣿⠛⠋⢓⡆⠀⠀⡷⢆⠉⢉⠛⡔⠒⠴⠟⠋⢠⠃⠀⣾⣫⣶⡧⣝⢶⣎⣄⢽⣟⣿⠀⢘⠓⣰⠃⠀⠘⠋⠉⠀⣡⡆⠈⡷⣸⡏⠀⠿
⠀⠀⠀⠀⢰⣿⣿⣆⠘⠿⢷⠀⠀⣏⣀⡈⠙⡳⠼⢷⣶⣯⠀⠎⠀⠀⣿⣽⣞⣎⠁⣐⢹⡄⣍⠁⢿⡆⢩⡜⢪⡀⢘⠆⠀⠀⣰⠿⠇⣌⢳⣟⠁
⠀⠀⣠⠀⠀⣿⣿⣿⡆⠀⢸⣇⢠⣬⣩⣵⣒⡀⠙⠛⠍⣿⡿⠀⠀⠘⢤⣿⣿⡇⢣⡉⠸⠇⢹⡗⢿⡷⠈⠖⠉⣠⣇⠴⢄⣴⣯⠴⢦⠤⠘⠈⠁
⣁⣴⠏⠀⠀⠹⢿⣿⣿⡄⠈⠋⣿⣇⣱⣥⡛⣥⠀⡗⣀⣩⣦⣼⣷⡦⢌⡻⣿⣵⢾⠉⠛⠶⢤⣤⣤⣤⡴⢶⠛⠁⠉⠳⣤⣈⣁⣀⣤⡔⠁
⣿⣃⣠⡾⢁⣼⠃⠈⣻⣷⢀⠀⠉⠁⠀⠀⣿⠦⣍⠛⠻⠯⠀⠚⠋⢰⣟⣛⡻⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠒⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀
⣎⡿⣿⢛⣾⣃⣃⢠⡟⣽⣧⣁⠩⠔⢚⣭⣶⡿⠛⢿⣶⣤⣴⡶⠗⠁⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⣿⣿⣿⣿⠃⢉⢱⡿⠳⠿⣿⣷⣾⣿⠟⠋⣁⠠⠐⠁
⣿⣿⣿⣷⣵⢾⠬⣧⣀⣀⣹⣿⡭⠴⠚⢋⡡⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⣯⠛⣻⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣠⣶⠟⠉⢐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵⣿⣿⣧⣀⠆⠠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠂⡠⢂⢏⡠⠈⣀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠋⠛⢤⢾⣷⠀⠬⠄⢲⡛⠒⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⠿⠀⠓⠤⠊⠀⠉⠋⠀⠀⠀⠀⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠉⡀⠀⠄⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠆⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡅⠀⠀⠠⠀⠐⠤⡀⠀⠀⠀⠠⠀⠀⠀⢠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠀⢀⠀⠀⠈⠑⠢⠄⣀⣀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⢠⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢁⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠑⠢⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⡤⣀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠰⢀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠷⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⣀⡄⢀⡞⣧⠀⠀⠀⠀⠀⠀⠀⣤⠊⠥⠈⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⡶⡊⠑⠃⡀⠘⡀⠀⠙⠉⠀⡀⢊⠈⠑⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠧⣤⢦⡀⡀⠀⢰⠠⠂⠀⢰⠀⠈⠫⠬⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⡼⠀⠈⠙⢼⢃⣠⢼⡖⢶⣲⣺⠿⠲⡆⠐⢢⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠃⠀⠚⠿⠿⠿⣯⢴⣯⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠛⠻⠟⠛⠛⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⣿⢡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⠀⠀⠈⠙⢿⣿⣿⣿⡄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⡇⣴⣧⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⣿⡿⡷⣿⢯⣶⣄⠀⠀⠙⢿⣿⣿⡄⠀⣿⣿⣟⣻⣿⡿⢿⣿⣿⠀⠀⠀⠀⣿⣸⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡎⡆⢈⢆⠻⠂⠉⠀⠠⢌⠀⠻⠹⡆⠀⠀⠈⢿⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⢰⢾⢻⠘⠜⠦⡱⢠⠀⠘⠀⠀⡙⡏⢲⠀⠀⠀⠀⠘⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⢀⣎⠾⡫⠞⠔⠊⠉⠛⠀⠀⠀⠂⠀⠐⠑⠊⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠞⢈⡁⠀⠀⠀⠀⢀⡀⠀⠠⡀⠐⡒⠀⠈⢡⠊⣁⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠗⡆⢱⠀⠀⠀⠀⠘⢥⣠⠇⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⡄⣏⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⢇⡽⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠸⢘⠤⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⣿⣿⣿⣧⣄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠐⢪⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣉⣉⢉⡛⡟⢃
⠀⠀⠀⠀⠀⠀⠐⢸⡀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⡻⣶⡇⢳⢺⣯⣷⣻⢷⡐⡴⡇⢄
⠀⠀⠀⠀⠀⠀⠠⢈⣯⠁⢬⣢⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⢓⠿⣽⣷⣬⡟⢻⣿⡿⣇⢷⠀⢹⡄⢂
⠀⠀⠀⠀⠀⠀⠀⢄⠳⢸⡒⠌⡛⠷⣦⣄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢸⣿⣯⣿⡮⣧⢡⠊⢲⡑⠭⠮⢜⢘⢸⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣼⡆⠀⠀⠀⠉⡛⠶⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⠸⣿⠃⢚⠃⣽⢸⡆⠀⢳⣌⣻⣭⠈⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠰⢫⡆⠞⠁⠀⠀⢀⣶⣿⣿⣶⣬⣙⠓⠶⢤⣄⣀⠀⠀⠀⣤⢻⡜⣿⠄⡉⢜⡟⡆⠇⠐⠀⠠⣺⣿⣆⠳⡞⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣤⡙⠻⣥⡀⠀⣁⣤⠹⣌⠳⠤⡞⠱⠀⢠⢈⢠⡴⠙⠋⠉⣵⠰⢣
⠀⢠⣀⠀⣦⡀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠉⠠⢈⣿⣷⠞⠉⠓⠁⠈⠀⠈⠁⠀⠀⠀⠀⢸⡇⢸⣏⠁
⠀⠬⣽⣷⣮⣿⣦⣄⣀⣠⣾⣿⠟⢁⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢘⡒⢤⡽⠯⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠉
⠀⠛⠿⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⠇⠐⣟⢥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣋⣿⡟⠁⠀⠀⠀⠀⠀⢄⠘⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⢾⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠧⣈⣳⡀⢠⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠿⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠐⢏⡩⠽⠃⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢛⠻⢿⣿⡏⠀⠀⠀⠀⣶⠖⡤⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⣿⡿⠁⠶⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢠⣿⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡴⠁⢘⢷⡀⠈⢦⣉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡸⠃⠀⠀⠀⠀⠀⣤⡼⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⡴⠾⣿⣿⠏⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠚⢷⣄⠀⠀⠀⠀⠀⠉⠀⠒⠙⠀⠀⠀⠀⠀⢠⡾⠳⣶⣦⣀
⠀⠀⠀⣀⡴⠖⠲⠞⠉⠀⢀⣀⣀⡾⠉⠛⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡿⠠⣄⡼⠃⢙⢧⡀
⠀⠀⢸⡋⠀⠀⠀⠀⠠⡶⠋⠁⠉⣹⠖⠂⠀⠙⠿⢭⣿⣷⣶⠖⠒⠻⣏⠁⢀⡿⠀⠈⠓⢶⡁⣰⠇
⠀⠀⠀⢿⣀⡀⠀⠀⢀⡟⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⡜⠷⠋⠀⠀⣴⠦⠴⠛⠁
⠀⠀⠀⠀⠙⢧⡴⠷⠋⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠶⠞⠁
⠀⠀⠀⠀⢀⣤⢤⡀⠀⠀⢸⡦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⣠⠞⡱⠋⢹⣆⡀⠘⣇⠀⠈⠉⠙⠛⠶⠦⣤⣄⣀⠀⢀⡟
⠀⣠⠞⡡⢊⡤⣴⠏⠈⠙⢦⣾⡳⣤⣀⠀⠀⠀⠀⠀⢠⠟⠻⣿
⣼⣣⠎⢱⣭⡵⠛⢦⡀⠀⠀⠈⠙⢲⡽⠛⠓⠶⠦⢤⡏⠀⠀⢈⡇
⠈⠛⠛⠋⠁⠀⠀⠀⠙⠶⣤⣤⠴⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢧⣄⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠠⣀⠸⢭⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣅⠢⢍⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣍⡾
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣴⣾⣿⢭⣓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣭⣽⣿⣿⣷⣶⣶⣿⣯⣭⣍⣙⠺⣷⣝⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⠛⠉⠉⠀⢠⣾⠋⠀⠀⠀⠙⢿⡄⠉⠉⠛⢷⣮⡻⢟⡺⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠿⢸⡇⠀⠀⠀⠀⠙⢿⣆⠈⢮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠈⢻⣆⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣛⠛⠛⠋⠀⢀⣦⡀⠀⠀⠀⠀⠈⣿⡘⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⢀⡞⠀⢿⠀⠀⠀⠀⠀⢻⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠿⠀⠀⠀⠀⢀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⣶⣶⣶⡶⠶⠶⠾⢿⣿⠛⠛⠉⠉⠁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⡤⠶⠛⣋⡉⠉⠛⠳⣦⡀⠀⠀⠀⠸⣷⣶⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠞⠉⠙⢦⠀⠀⢸⣷⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢹⡄⠈⠉⣽⢻⠙⠛⢛⡿⡟⠛⠛⢻⡟⠉⠉⠉⠉⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠈⢷⣀⡼⠉⣆⣧⣠⣼⡴⢳⡀⢠⡞⠁⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠈⠛⠁⠘⢧⡀⣠⠴⠃⠈⢷⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢇⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⢆⡽⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⡀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⠛⠛⡛⠛⠛⢻⣿⠷⢦⡤⠾⠿⠾⠾⠿⠿⠿⢿⣿⠿⣿⡿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⣠⣤⠶⠒⠚⠛⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠸⣧⣀⣰⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⡇⠀⢴⣷⣶
⠀⠀⠀⠀⠘⠿⢧⣀⣀⠀⠀⠀⢀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠈⣉⣉⣉⣀⣀⣸⣿⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣬⣿⡀⠈⠛⠛⠛⠛⠁
⠀⠀⣀⣤⠤⠤⠤⠤⠄⣉⡩⠉⠁⠈⠉⠁⠐⠒⣲⡿⠂⣴⡋⠉⠁⠀⠉⠉⣩⡙⢶⡀⠀⢠⡞⠉⠉⠀⠀⢀⣀⡈⢳⡄
⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁⠀⠀⠈⠻⠶⠶⠶⠞⠛⠙⠷⠾⠃⠀⠈⠛⠶⠶⠷⠛⠋⠉⠷⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⡇⢸⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣤⣠⣶⣼⣧⣤⣄⣄
⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠻⠿⢿⣆⡀⠀⠀⠀⠀⠀⠀⣼⡟⡁⠁⠀⠈⠉⠉⣙⠻⣷⣯⡀⠀⠀⠀⠀⠀⠀⣤⠀⣄⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢸⣿⠙⠋⠀⠀⠀⠀⠀⠙⠓⠘⢻⣇⠀⠀⠀⠀⠀⠀⠻⣿⣷⡏⠀⠀⠀⠀⠠⣷⣿⣾⠃
⠀⠀⠀⠀⢸⣧⣤⣤⣤⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇⠰⠆⢠⣾⠀⠀⠀⣴⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢼⣿⡆⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠷⣦⣤⣠⣿⠀⠀⠀⣶⣾⡇⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣤⣤⣬⣿⣿⣆⠀⠀⠻⣿⡇⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⢐⡿⣷⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡄⠀⢹⡧⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢘⣣⣿⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠃⠀⠈⣿⠀⠀⣤⣶⣶⣤⣄⠀⠀⢠⡿⠋⠁⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⣾⣷⣆⣶⣄⠀⠀⠀⠀⠀⠀⠙⢧⣀⠻⠟⠛⠿⠿⠃⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣦⣿⡾⠟⠛⠛⠛⠙⠛⢿⣷⡄⠀⠀⠀⠈⠙⠛⢶⣶⡶⢶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⣀⠀⠀⣰⣿⡅
⠀⠀⠀⠀⠀⠀⣸⣟⣡⣄⠀⠀⠀⢀⣀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡁⠀⣀⣀⠀⠀⣠⣴⣶⠿⢿⡟⢋⣤⠚⠻⣧⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡟⢁⡈⢡⡄⠀⠀⠉⠃⠀⢸⣿⣄⠀⠀⠀⣾⣷⣴⡟⠀⠘⣷⣾⣿⣿⣧⣾⠿⠟⣁⣰⠶⠀⠿⢿⣤⣀⠀⣬⣽⣿⠁
⠀⠀⠀⠀⠀⢀⣿⠀⠈⠁⣸⠇⠀⠛⠁⠀⠀⠸⣿⢿⣧⠀⣸⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⡶⠛⠋⠁⠀⠀⠀⠀⠀⣉⠓⠦⢭⣧⢀⡀
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⢿⣶⡦⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣟⢻⠄⣷⠀⠀⢰⢞⣉⠃⠀⠀⠹⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⢀⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⢻⢟⣿⣿⣿⣿⣿⣿⢹⣿⢻⣏⠀⠀⠉⠀⢠⣿⠃⠀⠀⠈⠉⠀⠀⠀⠀⣹⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠛⠃⠀⢀⣴⣿⣿⣿⣿⠇⢎⡎⣽⣿⣿⣿⢿⣿⡇⣿⡄⢿⣆⠀⠀⠀⣸⣷⣾⡟⠀⠀⠀⠀⠀⠀⣠⣿⡿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣦⣤⣀⣤⣿⠟⠛⠻⢿⣿⣆⠘⣰⣿⣿⣿⣿⣿⢿⠇⠈⠀⣿⡏⡟⢹⢃⡟⠉⢸⠟⢻⣟⣛⣿⣷⠞⢻⡟⠁⠀⠈⣳⣄
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠈⠉⢿⡏⠁⢨⡿⠀⠀⠀⠀⠻⣿⣤⠟⣡⣿⣿⣿⣿⡄⠀⠀⢠⣿⣇⣀⡉⢸⡇⠀⠉⠀⠀⢨⣿⠋⠀⢠⡟⠁⠀⠀⠀⠟⠹⣆
⠀⠀⠀⠀⠀⢰⡟⠀⡄⠀⠀⠘⢿⣶⠟⠁⠀⣤⢀⠀⠀⠙⣧⣰⣿⣿⣿⣿⣿⣿⣤⣤⣴⣿⣿⣟⠿⣿⣿⣿⣿⠖⢷⣾⡟⠀⣰⡿⠇⠀⠀⠀⢀⡄⠀⢻
⠀⠀⠀⠀⠀⣿⣧⣠⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣷⡶⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⡏⠻⣧⠈⢿⣿⢀⣿⣇⣀⡀⠀⠀⢿⡇⠀⠈
⠀⠀⠀⠀⣴⠏⢠⣿⠀⠀⠀⠀⠀⠀⠠⢀⠰⠆⠸⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⠿⣇⠀⣹⣧⠈⢻⣿⣿⣿⡿⢿⡆⣴⣾⠃
⠀⠀⠀⣰⡟⢀⡾⢻⠀⠀⢠⠀⠀⠀⡄⠘⡄⢀⠀⣿⣿⣇⠀⢿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⠁⠀⠚⠁⣠⣿⣶⣿⣿⣧⠀⢻⣿⠏⠀⢸⣧⣾⣿⣾⣷⣶
⠀⠀⢰⡏⢀⣾⠁⣼⠀⠀⠚⠀⠰⠚⠇⠀⠛⠘⢂⣸⣿⣿⠀⠘⣿⡟⣽⣿⣿⣿⣽⣿⣿⡇⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠁⣾⠃⢠⣿⣿⠿⣿⡿⠿⣿⣿⣷⣶⣾⣿⠻⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣟⣛⣿⣿⣿⣷⡀⢀⣴⣿⣿⢿⣿⣿⣿⣿⣿
⠀⢰⣿⢀⡇⠀⢸⣿⣫⣾⣿⣿⢶⣿⣿⣿⡝⣿⣿⠀⣿⣿⣿⣆⠈⣿⣿⣿⡿⣣⡿⣫⣿⣿⣿⣿⣿⠻⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣿⠷⣿⣿⣿⣿⣿
⠀⢸⡇⢸⡁⠀⣾⣿⣿⣿⣿⣿⡺⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⡾⠻⣟⣿⣿⣿⣿⠿⠛⠻⣦⢹⣿⢟⠉⢻⣷⣦⡝⢿⣿⣿⣿⣿⣿⡿
⢠⡟⠀⢈⣻⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⠋⡁⠀⠘⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⠀⠀⣰⣿⡆⣿⣿⣇⠀⠸⣿⣿⡤⣿⣿⣿⣿⠏
⢠⣴⣦⣾⠏⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣷⣿⡇⣀⢴⣼⣿⣿⣷⢟⣿⣿⣿⣿⡏⠀⣴⣿⣿⠿⣿⣿⣿⣆⠀⢻⣿⣷⣿⠿⠛⠁
⠋⡿⠻⡿⠀⠀⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⣿⣿⣷⢸⣿⣷⢿⣷⣿⣼⣿⣿⣿⢲⣿⣿⣿⣿⣿⡇⢠⡏⠀⠀⠀⠀⠉⠙⢿⡆⠸⣿⡋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⣿⢃⣩⡾⣿⣿⣿⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⢿⡀⢿⡇
⠀⠀⠀⠀⠀⢼⣿⣏⣭⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣟⡳⣿⣿⣿⣿⣿⡿⣋⣴⣿⣿⣿⡿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⡇
⠀⠀⠀⠀⠀⠈⠉⢹⣿⡆⢻⡏⠀⠀⠀⢹⡧⠀⢻⡏⢸⣿⣿⣿⢿⣿⣿⣿⣋⡾⢋⣿⣿⣿⣿⠃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⠀⠀⠀⢸⣧⠀⣻⡇⢸⣿⢟⠅⣸⣿⣿⣿⣿⡴⢛⣵⣿⣿⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡏⠀⠀⠀⢸⣿⠀⣿⣇⣸⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⢟⣛⡙⣧⣼⣿⡆⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⣧⣼⠟⠳⣦
⠀⠀⠀⠀⠀⢀⣀⣸⣿⣷⣾⣷⣤⣴⣶⣿⣿⣶⣿⡟⠛⠋⠁⠀⢹⣿⣿⡿⠿⢛⣿⡎⢿⣷⣿⣿⣿⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⣄⣩⣤⣾⣿
⠀⠀⠀⣠⡾⠛⠛⣻⡟⢹⣿⡿⠋⠀⢉⣙⠃⢹⣿⣇⢀⣀⣀⢀⣾⣿⣿⣄⠛⠻⠿⠷⠾⠛⠛⣿⣿⠀⠀⠀⠀⠀⢸⣿⣒⠚⠛⠟⠛⠛⣛⣉⣿
⣀⣀⣀⣙⣷⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣉⣉⣀⣉⣀⣀⣈⣉⣉⣁⣀⣀⣀
⣿⣿⣿⢿⠻⢿⡿⠻⡿⢿⡿⢿⠻⢿⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣼⣷⣴⣧⣦⣾⣤⣼⣿⣤⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⠠⠶⠍⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⢴⣤⣄⠀⢙⣇⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆⠀⠀⠀⠀⠀⣨⣟⡍⢀⠇⣠⢆⣴⣶⣖⣠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠈⠉⠉⠁⠀⠀⢰⡏⢿⠀⠀⠀⢠⠀⠀⢉⡀⣼⣴⣱⣷⣾⣿⣿⣿⣿⣿⣿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠽⣦⣤⣴⡯⣤⠴⠋⣸⣿⠟⢉⣥⡀⠀⠉⠉⣿⣶⣾⣿⣯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠋⠀⠀⠀⣼⠏⠁⣠⣤⣤⠀⠀⠀⠀⠀⣽⣿⣿⣟⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘⠳⣄⡀⠀⠀⠀⠀⡋⢀⢿⣿⠇⠀⡆⠀⠀⠠⡄⠙⠛⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠐⣠⡿⠁⠀⢠⡟⠙⢢⣠⠂⠸⠂⣿⡀⠀⢀⡰⠀⠀⣿⣷⢰⡀⠀⡷⣶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡶⠞⠛⠁⠀⢀⣴⠟⠁⠀⠀⢹⡀⡀⠀⠈⠙⠛⠁⠀⠀⠘⣍⠁⢨⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⢻⣷⣦⣤⡴⠞⠛⠁⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⢷⣄⠀⠀⡙⠻⠃⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢸⡇⠀⠀⠀⢻⣷⡈⠛⠒⠀⠀⠀⢀⣤⣾⣿⠁⣠⣶⡶⠠⡄⢈⠛⠿⠛⠁⠀⠀⠀⠀⢄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠘⣿⡀⠀⠀⠀⠹⣮⡁⠀⠀⠀⠀⠻⢿⣿⣷⠐⢿⣽⣳⣶⣀⡜⠑⠦⠀⠀⠀⢀⣒⡐⠂⠀⠸⣦⣌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⡆⠀⠀⠘⣷⡀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠈⣹⣷⣤⠄⠀⠀⠉⠻⣟⣷⡇⢠⡴⠋⢁⣿⠶⠠⠠⠧⣹⡧⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣹⣄⠀⠀⠈⠻⣄⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠘⠻⣿⣿⣄⡀⠀⠀⠀⣉⣉⡠⠊⠀⢀⣾⠟⢶⣶⣶⣦⡀⠀⣰⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡛⠿⣳⣼⣦⣄⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠉⠉⠁⠙⢿⠟⠉⠁⣀⣠⣴⡿⠃⣎⣀⣀⣀⠀⠀⣰⣿⡇⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠣⠜⠂⠉⠙⢳⣬⣟⣲⢤⣀⣀⠉⠛⠳⢶⣤⣀⣀⠀⠀⠈⠉⠛⠛⢶⠶⠖⠚⠛⠛⠛⠛⢥⣄⠀⢈⡟⠛⠉⠁⣰⣯⣽⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠔⢋⡠⢀⠠⠄⡒⠬⠳⠙⡫⢝⡳⣿⢽⣏⣿⣒⣶⣤⢭⣙⣛⣻⣶⣶⣶⣮⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢏⣹⡿⣿⡿⠞⠋⠀⠀⠀⣠⠞
⠀⠀⠀⠀⢀⣴⡻⠝⡂⢥⡐⠬⢓⠨⢐⢈⡁⣄⢢⣥⣬⣦⣍⣭⣿⣿⣿⣿⣿⣿⠿⣍⠷⢉⣣⣤⡾⠿⢓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠦⠤⠤⠒⠋⠁
⠀⠀⠀⣰⡿⢅⡲⠍⢂⠡⣐⣨⡴⢞⠯⢒⠉⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠿⡿⠶⣳⣶⣾⣉⠫⠕⢢⡑⢮⡳⠬⢛⣄
⠀⠀⡴⠏⠐⢡⠀⢌⡠⠕⠋⠡⢚⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣟⡋⢀⣡⣴⠶⣛⢃⣍⠦⣔⠶⠟⠛⣛⣦⡀
⠀⠀⣷⠐⢈⡀⠈⣠⠴⣲⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣦⡷⠛⠋⠈⡑⠬⡑⢊⣉⣤⡽⢿⡀
⠀⢀⣿⢂⣡⡲⣝⣶⣿⠻⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠺⣷⣼⣷⠿⢟⡻⢍⡳⢸⡇
⠀⠸⣯⢿⣿⠿⢛⢩⠲⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣽⣧⣙⣦⣽⣶⢿⣿⠃
⠀⢸⡿⣻⡷⠞⣺⣿⢾⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⡹⢌⣣⣴⣟⣿⠋
⠀⣷⠞⣿⣤⠞⢽⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⣷⢾⣻⣿⣿⣼⡇
⠘⣷⢺⣯⢔⡪⢼⡟⣧⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⠿⣿⢟⣿⠁
⠀⢹⣯⢲⠟⡩⣴⠟⠺⢑⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡻⢝⢢⣽⣟⣧⡿
⠀⢨⣿⡾⢓⠭⢂⣵⠞⢪⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢜⢣⣝⣾⣿⡿⢛⣸⡇
⢀⣼⣷⣻⣷⣾⠿⢶⣶⣿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⡿⣟⣣⣼⣿⣿
⠸⡀⢉⣿⣿⣧⣤⣤⣸⣿⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠻⣧⣿⡾⣿⣵⣾⣿⡄
⠀⠈⣿⣾⣟⣛⡛⠿⣟⠻⢢⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠙⢿⡟⣉⣀⣼⣁
⠀⣼⣿⣿⣭⣭⣿⣷⡚⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣦⣤⣤⡶⣻⡶⣋⣉⣽⡇⢀
⢰⡿⠟⠉⠀⠀⠀⠀⠈⠙⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣯⣄⣡⣿⣷⣏⣠⡴⠶⣧⣄
⡏⠑⠶⢤⣤⡀⢀⠀⠀⣀⡸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠞⠛⠉⠑⢦⡀
⠘⠢⠤⠤⢀⣀⣀⣀⠀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣶⣶⣤⣤⣬⣍⣙⠛⠻⠿⣡⣀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣤⢭⡭⠽⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠂⠉⠉⠉⠉⠉⠉⠐⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠎⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠉⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⡘⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣰⣶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⣇
⡇⠀⠀⠀⠀⡰⠃⠀⢀⠊⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠱⠈⠙⠢⣀⠀⠀⠀⠀⠉
⡇⠀⠀⠔⠊⠀⠀⠀⠈⠄⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠈⠀⡰⠀⠀⠀⠈⠑⢢⠀⠀⢳⠄⠄⠤⠤⠤
⢓⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠁⠀⡿⠀⠀⠀⠈
⠘⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠐⠀⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⡇
⠀⢇⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡼
⠀⠘⡄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⠃
⠀⠀⠱⡄⠀⠀⠀⠙⢄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⡰⠃⠀⠀⠀⢠⠃
⠀⠀⠀⠹⡄⠀⠀⠀⠈⠳⣄⠀⠡⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠌⠀⡠⠞⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠙⡉⠐⠒⣲⠀⢠⡖⠒⠂⢉⠉⠀⢀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⣹⠻⠯⡍⠉⠀⢀⠤⠚⠀⠈⠗⢤⡀⠈⠉⡩⠭⠛⣍⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⡘⢢⡓⠂⠴⠃⣄⠀⢊⠃⠀⣆⠙⠤⠐⢳⡐⢣⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠑⠒⠊⠀⠇⠀⠀⢸⠀⠑⠒⠈⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠒⠒⠚⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⢀⡞⠀⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⠤⠤⠤⠤⠤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣴⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠙⠳⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢻⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⡀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡞⠁⠀⠀⠀⠀⠀⠀⢠⣿⠻⣦⣀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⣠⣤⠉⠛⠛⠿⠿⠿⠿⠷⠶⠶⠶⢶⣿⣟⡋⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠉⢀⣤⣤⣄⠀⠀⠀⠀⣸⣶⣦⣤⡈⢿⣍⠛⢷⡄⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠀⣠⡾⠋⠉⢛⣛⠳⣄⠀⣼⠋⠀⠀⠈⠙⢾⣿⣦⡀⠳⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⢠⣤⠶⠟⣫⣾⡟⠀⠀⣼⡏⠀⠀⠀⣿⣿⡷⢸⣦⣿⡀⣾⣿⡆⠀⠈⣿⣿⢳⣄⠀⠀⠀⠀⠈⠻⣦
⠠⠤⠤⠤⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠀⠀⢀⠹⣇⠀⠀⠀⠀⢉⣠⡟⠁⠻⣷⣌⡉⠀⠀⣰⡟⢹⡆⠙⣦⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠛⠀⠀⠈⠻⠾⣿⣿⡿⠿⠛⡁⠀⠀⠀⠈⠻⣿⢿⣿⣿⡾⠛⣷⣤⡜⣧⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⢀⣠⣿⠂⠀⠀⠀⠀⠹⢩⡿⠘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣦⡀⠀⠀⠀⠀⢸⣇⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢶⣾⠃⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣈⠻⣦⣄⠀⠀⠀⠻⣤⠙⠻⣿⣦⡶⠀⠀⣠⣶⠾⣿⠟⠻⣿⠟⠛⠻⣿⠛⠛⣿⠿⢷⣦⡀⠙⣧⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠷⠶⠶⠶⠶⠿⣿⠋⢀⣤⡾⣿⠁⠀⢿⡀⠀⣹⡆⠀⢰⣿⠀⢀⣿⣀⣀⣾⣿⠀⣿⡇⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢤⡀⢰⡏⠀⣿⣿⣧⣼⣷⣴⡿⣿⡟⢿⠟⢷⣾⣿⣾⠛⠿⣇⠉⠻⣿⢷⣿⣧⣤⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠻⣾⣧⣀⣈⣽⠟⢁⡾⠃⠀⣿⠁⣾⠆⢸⣿⠀⠙⣷⡀⠙⣷⠀⢻⡆⢻⡏⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠀⣾⠃⠀⢰⡇⠀⠛⠀⠈⣿⢧⡀⠈⠓⠀⠸⠇⠘⠿⠘⣿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡤⣄⢻⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⢻⣆⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣸⡇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣻⣶⠆⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣯⠀⠀⠀⠀⠀⠸⣿⣤⠶⠶⠶⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣤⣤⣼⣿⣡⡾⠛⠁⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠻⣤⣤⣄⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠞⠋⠉⠀⠀⠀⣹⡿⠋⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠈⠻⣭⡙⢷⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣀⣤⡶⠾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠈⠿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣾⠏⠁⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠙⢿⣷⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⣤⠶⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣶⠭⠿⣃⣀⣸⠟⠉⣩⣉⣠⣤⠖⠋⢿⡛⢭⠵⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢿⡷⠋⢿⣀⣀⡈⠀⠙⠿⠃⣀⡁⢠⠀⢀⣠⡴⠛⠒⢤⣀⣴⠇⢀⠉⠑⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡞⣩⡤⠇⠈⣿⠳⡟⠉⠉⠉⣻⠖⠲⢿⡏⣠⠞⢰⡏⠁⢰⣁⠤⠤⣽⣇⣀⣸⡷⠄⣈⠟⠹⠦⣀
⠀⠀⠀⠀⠀⡰⠋⠀⢻⣠⡴⠞⠃⠀⡝⠀⣰⠋⠁⢘⡆⠐⣬⣭⣦⣸⡷⠆⠚⠁⠀⣤⡀⣀⠀⠉⠀⢦⣀⠀⠚⠓⣌⠳⣄
⠀⠀⠀⠀⢰⡷⠲⠖⠲⢿⣄⠀⠀⠼⣄⠀⢻⡀⠀⠈⢻⣄⡡⠄⣨⣿⠀⠀⢻⡏⠀⡿⣿⠁⡤⠤⣴⠀⢸⠁⠀⣀⡌⢻⠋⢷⡀
⠀⠀⠀⢠⢾⡤⢤⠴⠶⣄⠘⣷⠤⣄⠀⠉⠁⠀⠙⣶⠚⣁⣴⠛⡁⠹⡄⠃⢠⡇⠀⠀⣿⠀⠀⠀⠹⣆⢈⡤⠛⢧⠀⢸⡆⣸⢹⡆
⠀⠀⠀⠉⣼⢀⡾⠶⠆⠀⠀⠃⠀⠀⠀⠠⢀⣠⡴⡟⠄⠈⣧⠀⣿⣀⣀⣤⠞⠻⠄⣠⣿⣦⣀⠀⠀⢉⣩⡖⠀⠠⠀⢸⣇⣀⡼⠹⡀
⠀⠀⠀⡼⡃⠏⢰⡀⠀⠘⣶⣄⣀⡠⠤⣀⡴⠟⣰⡇⠈⠀⢻⣿⠟⠉⠉⣡⠴⠚⠉⢉⠀⠀⠙⡿⠛⢉⠉⢻⣆⡀⠀⡿⠉⠀⣿⣤⣧
⠀⠀⠀⢳⠘⠦⠶⠿⢶⣄⣀⣨⡟⠀⢠⡀⠀⣸⣿⣿⣄⣴⠾⠿⠶⣾⣍⣁⠀⠀⣠⣾⠀⠀⢰⡀⣠⠟⠃⢘⡟⠶⡿⠷⣤⣀⠩⣟⣽
⠀⠀⠀⠀⠙⢶⣦⣔⠀⠈⠻⣀⠙⠂⠈⣿⡟⠉⢉⠉⢹⠁⠀⠀⢀⣯⣀⠁⣀⠞⠉⠙⠒⢀⡼⢻⡇⠀⠄⠘⠀⠀⡧⠀⠀⢙⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠉⠙⠦⢀⠀⠙⠁⠂⣠⣿⠀⢠⡏⠀⠀⠀⣀⠀⠸⠁⠀⠀⠈⠙⠶⠤⠄⠀⠀⣼⣅⣀⣀⣰⣁⡤⠀⠘⣉⡽⠟⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⣿⠈⠛⢧⡀⠀⠀⠀⢩⠞⠳⢆⣀⣀⡤⢤⣤⣆⠔⠂⠈⡏⠉⢉⠉⠉⠀⢀⡾⠉⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⠄⠉⠀⠀⢀⣀⣤⠴⠶⢤⣤⣠⣴⣿⣿⣷⡶⠾⡿⣦⣦⣴⣶⣤⣤⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠴⢾⣛⢁⠀⠀⠀⠀⠛⣿⣿⣽⣿⣿⣿⣿⠿⢿⣫⣽⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢇⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⢛⠻⢿⣿⣽⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠈⠉

⠀⠀⡀⠀⠀⠀⢀⠀⡀⢀⡀⠀⠀⢀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⠒⢦⣤⣀⠀⢀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠲⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀⠀⠀⠠⣤⡤⠤⠤⠶⠒⠒⢒⠋⠉⠉⢰⠤⣝⣒⠦⠘⢦⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠁⠀⠀⠀⠳⡄⠀⠀⠀⢠⡞⠁⣀⢠⠏⠀⠀⠈⠙⣖⠆⠱⣄⠀⠀⠀⠉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣠⡄⠀⠀⠀⠀⠠⡟⠀⠀⠰⠋⢀⠔⠣⡼⠀⠀⠀⠀⠀⠸⡆⠀⠘⣆⠀⠀⠀⠀⢲⡒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⡼⠀⠀⠀⠀⠀⠀⠸⡆⠀⢀⣴⠋⠀⣲⢹⡀⠆⠀⠀⠀⠀⡇⠀⠀⠈⢧⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠤⠔⠒⠒⠲⡄⠀⢹⠀⠀⢸⠀⠀⢉⡼⢦⣀⡀⠀⠀⣰⠃⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡽⠃⠀⠀⠀⠀⠀⣀⡤⠤⠼⣄⣸⠀⠀⠘⠦⢶⠟⠁⠀⠙⠭⠉⠉⠁⠀⠀⣀⣀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠡⠶⣓⣲⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠈⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢾⠀⠀⠀⠷⣄⣀⣀⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠩⠦⠤⠴⠚⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣀⠄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣡⡔⢲⠋⢻⡥⠟⠉⠀⠉⠉⠁⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢯⠀⣧⣈⠷⢋⣠⣀⣀⣀⣠⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡭⠭⠵⠚⠉⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⡀⠀⠀⠀⢸⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇⠀⠀⠀⣸⣶⣾⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡶⢶⣒⣃⡀⠀⠜⠁⢸⠋⢃⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠤⠤⠴⠔⠒⢲⠂⡀⢸⡧⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⠉⡙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠒⠒⠒⢲⡛⡏⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠚⠻⡟⠂⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠁⠀⠀⠀⢀⣀⣇⣀⣠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣷⣿⣯⣭⣽⣿⣧⣷⣮⣿⣷⣼⠆⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢈⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢁⠀⣄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠐⠤⠀⠉⠻⢕⣀⣀⠀⠀⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⠞⣡⠆⠀⣁⠠⢖⠀⠠⠄⡛⢿⣷⣇⠀⠦⠀⠀⢀⠞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣧⠋⢤⢖⡠⢄⠦⠠⣤⡶⠦⡉⠚⢻⣬⣳⣧⠔⠒⠌⢀⠀⠀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⢀⡾⢃⣠⠖⠀⠉⠛⠂⠁⠀⠀⠈⠀⡩⢴⣿⣮⣿⣦⣐⣀⡘⡀⠀⢀⡀⢀⡠⠰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⢸⢋⠀⠁⠀⢈⣤⣶⣶⣶⣃⠀⠀⡘⡴⠃⠙⣿⣷⣝⣿⠖⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠈⢦⠀⠀⢀⢴⡿⣊⣤⣡⣾⣿⡟⢹⠉⠉⠉⠛⠀⠴⠁⢄⠀⡞⣿⣜⣧⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⣸⡟⢛⣋⣈⢸⠟⡉⠀⠀⡀⠀⠀⢀⠂⢰⠆⠀⣠⢹⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠈⠁⠀⣠⣿⠛⠟⠛⠙⠉⠀⠃⠀⠀⠀⠀⠠⠁⠀⠋⠰⣰⢧⣿⣿⣳⣤⡄⡀⠀⠀⠀⠀⡊⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣡⣿⢿⢳⠀⠀⠀⠉⠀⠠⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠀⠀⠀⠁⠀⠢⢀⠀⠀⣤⣷⡆⠀⠀⠀⠀⠀⣠⢾⠟⠀⣈⡸⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡿⣦⣧⠀⠀⠀⠀⢾⡟⢠⡇⣿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠠⠄⠀⢀⠀⠀⠀⠙⢿⡟⣧⠀⠀⠀⠘⠆⣾⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⣎⠆⠀⠀⠄⢲⣽⡿⠘⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⣠⣜⠞⠀⠀⠀⠈⢿⠇⠀⠀⠁⢀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠘⢿⠝⠻⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠙⢄⡃⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠚⠶⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠊⠉⠉⠉⠉⠉⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⠤⠂
⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢀⠠⠊⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠎⠀⠀⣀⠴⡞⣁⢀⣀⠀⠀⠀⣀⠭⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⣀⡀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠒⢏⠀⠀⢻⠃⠀⠀⠑⢄⡼⣁⡀⠀⠀⢳⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⡆⠀⠀⠀⢨⣷⠾⠟⠀⠀⣸⡞⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠐⡀⠀⠀⠈⠀⡄⡀⣀⠴⠉⠃⠀⠴⢅
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠦⣄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠐⠂⠀⠀⢀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⣨⡬⠁⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⢀⡤⠚
⠀⠀⠀⠀⠈⠀⡔⠀⠰⠁⠀⠀⠀⠀⢇⠈⠁⣀⢴⡀⠀⠀⣼⡩⠝⢂⡀⠠⠚⠉
⠀⠀⠀⠀⠀⠀⠑⢄⡆⢡⠀⠀⠀⠀⡠⠜⠉⣹⠀⠀⠀⠉⠙⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠣⠐⠊⠀⠀⠀⠀⠻⣀⣠⠖⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⡄⠀⠀⠀⢈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣍⡀⢸⠖⠒⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣘⣿⠤⡤⢴⣖⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠓⠚⠓⠒⠛⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠉⠀⠀⠀⠈⠉⠙⠓⢦⣴⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⠟⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡾⠀⠀⠘⡇⠀⠀⠀⠀⡟⠀⠀⠸⡆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢷⡀⠀⣰⠇⠀⠀⠀⠀⢷⡀⠀⣰⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣀⡴⣦⢀⡴⣄⠀⣠⣤⡀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠺⠋⠀⠈⠛⠁⠈⠛⠃⠀⠙⠆⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⣀⣀⣀⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢀⠦⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠋⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡄⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡴⡶⢤⠞⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠐⣄⣀⣼⡕⠁⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠸⠂⢀⠄⠁⠀⠀⠉⠓⢄⡀⢰⠲⠤⣀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠉⠁⠈⠀⠀⠉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢆⠘⢄⣆⠀⣦⠸⣆⡳⠔⠈⠠⡀⠀⠐⠐⠺⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠘⠉⢄⡀⠣⠆⠉⡤⠀⠀⠨⡄⠀⠁⠀⡄⠙⠁⠄⠀⠀⠀⠀⠀⡐⠈⠑⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡁⠀⠀⠈⠄⠐⠙⠀⢠⡀⠈⠀⠠⠰⠀⠘⡀⠀⠀⠀⠀⣰⠄⣄⠀⠀⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠙⢀⡈⢆⡀⠀⠈⠒⠄⠀⠃⠀⡀⠰⠀⡇⠀⠀⡇⠀⠀⠀⡎⠁⢄⠀⡗⢄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⣠⠋⢀⠀⠙⠀⠀⠀⠀⠀⢀⡀⠑⡀⠈⠁⠀⠀⠃⠀⠀⠀⠉⠒⡄⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⡌⠁⠁⢠⡃⠀⠀⠀⠀⠀⠀⡸⡉⢱⠀⠀⠀⢘⠀⠀⠀⠀⠀⢠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠁⠁⠈⠢⡀⠀⠀⣀⣚⢅⠈⠲⣄⠀⠀⢁⣬⣠⡄⠀⠀⡈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠋⣁⣁⠀⠀⠑⣀⠎⡙⢤⠃⠁⠻⠿⣃⡘⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠉⢠⠿⣦⣶⣦⣠⣄⣉⣮⣈⣦⣼⡄⠀⠈⠉⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⢸⡉⠀⠈⠁⠙⠋⠹⠟⠛⠿⠏⠛⠉⠐⣂⠀⡠⠒⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠟⠓⠐⠀⠸⠈⠂⠂⠔⠖⠚⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀⡀⠀⠸⡀⠀⠀⠀⠀⣠⠤⠤⠀⠀⠀⠀⠀⡤⠤⠤⠄⠰⠦⢄⡀⠀⠀⠀⠀⢈⡵⠋⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣨⣓⠺⣧⡀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⣤⠞⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣥⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢀⡀⣀⣀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠨⠥⠚⠛⠛⠉⠙⠓⠀⠀⠋⠉⠀⠙⠀⠉⠛⠀⠀⠈⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣠⡄⠀⠀⠀⠀⢠⡄⢀⣠⣤⣤⣤⣀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣀⣤⣤⣀⣀⠀⠀⠤⠬⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠂⠀⠬⠉⡀⠒⠂⠤⢀⠈⠙⠛⢿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⣈⠑⠂⢄⡀⠈⠙⠠⢀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⡎⢻⡐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢁⠢⠄⡀⠀⠐⠦⠀⠙⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠇⠀⠱⣄⠀⠈⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⠖⣤⣄⣈⠀⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡰⠀⠀⠀⠘⠻⢦⣀⠀⠀⠀⠀⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠛⠛⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⠀⢀⠎⠀⠀⢀⠎⠀⠀⠀⠉⠛⠓⢤⠤⠀⣀⣀⠈⠉⠀⠠⢤⣤⠴⠶⠆⣈⣙⠿⠛⠊⡹⠀⠙⢿⣿⡿
⣿⣿⣿⠟⠁⠀⡀⠔⠁⠀⡀⠞⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⢡⡤⠤⠒⠂⠠⠤⠄⠐⠞⢉⡉⠁⠀⠉⢁⡀⠓
⣿⣿⣷⣶⡦⣁⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠈⣄⠀⠀⠀⣠⣴
⣿⣿⣿⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠹⣿⣿⣿⣿⣿
⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠛⠛⠛⠛⠛⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠤⠀⠀⣀⣠⣤⣀⠀⠠⠤⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠓⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠾⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⠟⠋⠀⠈⠢⢄⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡠⠔⠁⠙⠿⣿⣿⣿⣿
⡟⠁⡀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⣿⡟⢁⣤⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⡀⢳⣾⣿⣿
⣿⣿⣾⡃⣰⡏⢠⣿⣷⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣤⠆⣠⣶⡆⠸⣆⠈⣷⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢶⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠶⣌⣋⣩⣉⣽⡛⠻⠃⣾⠉⢛⣛⣛⣛⣛⣛⣛⡶⠆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⠛⠀⠀⠐⠂⠐⠛⢀⡟⠀⠚⠛⠂⠂⠐⠒⠛⠒⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⣂⣀⣀⣀⣀⣐⣂⣥⣶⣬⣐⣂⣀⣀⣀⣀⡐⣢⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢄⠀⠀⢀⡈⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠒⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⢀⣀⣀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢨⠀⡠⠄⡸⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠁⠁⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠈⠉⠉⠉⠉⠀⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠠⠄⠐⠀⠐⠀⠒⠀⠀⠠⢄⡀⡠⠀⠒⠒⠀⠤⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⣀⣀⠀⠤⠤⠤⠀⢀⣀⡈⢆⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣒⣛⣉⡉⠀⠒⠒⠒⠀⠤⣀⡀⠀⠉⠐⠒⠤⢄⡀⠀⠀⠇⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⢄⡀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠐⠤⣘⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢺⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠈⠑⠦⡀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠸⠸⡎⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠈⠑⠢⣀⠀⠀⠀⠉⠢⢀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡄⠀⠙⣆⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣑⠢⠄⡀⠀⠀⠑⠦⡈⠆
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⠁⠀⠀⠈⠳⣄⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠱⢦⣤⣤⢘⡘⡄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⡰⠉⠛⢦⣄⡀⠀⠀⠀⠈⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠉⡙⠘⡄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⠜⠀⠀⠀⠀⠉⠊⠕⡒⠠⢄⣀⠀⠀⠀⠈⠁⠒⠀⠀⠠⠤⠤⠤⢠⣤⡤⠤⠐⠒⣛⣱⡚⠤⠚⢁⠇⠘⢄
⠀⠀⠀⡠⠊⠀⠀⠀⣀⠔⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠬⢉⣀⠒⠒⠒⠒⠐⠒⠒⠂⢈⣁⡠⠤⠴⠖⢉⣉⡀⠠⠤⠐⠃⢤⠀⠀⠉⠀⠈⠁
⢀⡴⠊⠀⠀⣀⠄⠊⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠒⠀⠤⠤⠀⠀⣠⠤⡄⠒⠁
⠈⠉⠐⢶⡭⢤⠤⠔⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣄⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⡸⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢆⠉⠐⠒⠈⠁
⠀⠀⠰⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢂
⠀⠀⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜
⠀⠀⠀⠑⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠠⠤⠀⣀⣁⣒⣀⣀⣒⣉⠀⠠⠤⠐⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⢶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣰⡟⠉⠀⢙⣷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣴⠾⠃⣀⣤⣤⣀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠈⢻⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣧⣀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⣋⣭⣿⣿⣿⠉⠉⠛⠿⢷⣶⣶⣶⡾⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⢁⣠⣾⠿⠋⢹⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⣠⣾⠟⠁⠀⠀⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⢀⣾⡿⠁⠀⠀⠀⠀⣿⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢠⣿⠏⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢿⡟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀
⠀⠀⠀⣰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣤⣶⣿⣿⣿⣿⣶⣤⣄⢀⣀⣠⣤⣤⣤⣶⣾⣿⣿⣿⣯⣭⣭⣥⣦⣴⣶⡶⠶⠒⠒
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⣛⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣯⣥⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠠⠻⠟⠛⠛⠛⢋⣽⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⠄

⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣟⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣏⣿⣿⣿⣿⣿⣿⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡟⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣾⣿⣿⣿⣷⣷⣽⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣶⣿⣿⣿⣿⠟⠁⠀⠉⠛⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠋⠀⠁⠀⠀⠈⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠄⠀⢀⡀⠀⠠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠂⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⣸⣧⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠛⠛⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡄⢠⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⢹⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠉⠉⠁⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠄⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣠⠤⢀⡀⠀⠀⠀⠄⠡⣀⠀⠀⠀⠐⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⠁⠀⠒⠀⠀⠄⢀⠈⠐⠀⠄⡀⠈⠐⠂⢀⢁⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⠑⠠⡀⠀⠈⠀⠄⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⢇⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⡁⠠⢀⠀⠈⠀⢄⠈⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⠈⠳⣄⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⢤⣀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠉⠓⢤⣀⠀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⠋⣩⠀⡀
⠀⠀⠀⠄⠀⠀⢀⠔⠁⠀⢀⠔⠀⠀⠀⠀⠈⠀⠁⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⢂⡁⠠⠖⠋⠈⣠⠁⠈⠢⠀⠄
⢀⠄⠀⠀⡠⠐⠁⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢅⠀⠒⠀⠀⠒⠀⠈⠁⠠⠀⠀⠀⣀⢀⠄⠊
⠀⠀⢳⠀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠢⠀⠀⠀⠠⠂
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠄
⠀⠀⡀⡄
⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠀⠉⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⣸⠞⢼⣿⣮⡓⢀⠀⠀⢰⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⡞⣿⠻⡏⠙⢦⣉⣲⣿⣍⢻⣿⣿⣗⣹⣿⣿⣿⣿⣿⡟⠉⠙⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣇⠀⣀⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡄⡀
⠀⠤⡤⢤⣄⠀⡀⠀⠀⠈⠁⣘⣽⣿⣟⣯⣿⣿⣿⣿⡿⢿⣷⣿⣯⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠃⠤⣶⡿⣿⣿⣿⣿⣧⢽⣿⣿⣯⣿⣿⣿⣟⣿⣿⡀
⠀⠀⠀⠈⢳⣠⢠⣤⡈⠀⣤⡀⠀⢀⡐⢀⣈⡽⠛⠛⣿⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠈⠔⠁⠀⢊⠽⠿⢦⣴⣏⣙⣿⣧⡇⠴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠈⠀⠀⠀⡠⠉⠀⠠⡀⢼⡷⠿⣩⣑⣠⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⣠⠤⠇⢠⡄⠀⠟⠋⠠⣠⣿⣿⣧⣿⣟⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⣿⠆
⠀⠀⠀⠀⠀⠘⠁⢸⣴⡥⠊⢁⣠⣤⣄⣤⣤⡵⣿⣿⣿⣿⢻⣿⡿⣿⡿⣿⣿⣿⣿⡏
⠀⠀⠀⠐⠀⠀⠴⡾⠋⠤⠌⠠⣶⣿⣭⡷⣬⢻⣿⠿⣿⣿⣿⣿⣲⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠣⡤⠀⠀⢶⣶⡾⡉⠐⠀⣌⣈⣰⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣈⣉⣙⣩⡷⠶⢭⡋⢰⠐⣻⣿⡿⠻⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠳⡄⠀⣠⡿⠚⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣴⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣰⠀⠀⠀⠀⠀⡤⠖⠃⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄
⠀⠀⠀⡼⠛⠛⠛⠛⠙⢦⠀⠀⣀⣠⣿⡿⢿⣆⣿⠀⠀⢀⢤⣤⣷⣤⣄⠀⠀⠀⢀⣽⣿⣿⣿⣾⣿
⠀⠀⠀⡇⠀⠔⡀⠢⠀⡸⠀⠀⣿⠀⠀⡾⢻⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⢲⠀⠀⠀⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠘⠦⣀⣉⡠⠔⠁⠀⠀⣏⠀⠸⠃⣠⠚⢂⠀⠀⠈⠻⣿⣿⠿⠋⠁⠀⠀⠀⠈⠉⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⢦⡀⠀⠀⠀⠈⣧⣿⣧⡀
⠀⠀⢠⠀⠀⣰⣷⡀⠀⠀⠀⠀⡯⠐⡖⢀⢠⡄⢹⡀⠀⠀⢰⢿⣿⡏⣧⠀⠀⠀⠀⢰⠉⠉⠉⣿⡂
⠀⠀⢸⣠⣼⣹⣿⣽⣦⣄⡄⠘⢧⣈⠁⠚⠈⠁⡴⠁⠀⠀⠀⢸⡏⣷⠀⠀⠀⠀⡄⣈⠩⢉⡉⢉⢀⡄
⠀⠀⠸⠉⠉⠹⠩⠏⠉⠉⠇⠀⠜⠋⠙⠋⠉⠉⠀⠀⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠁⠉⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠈⠓⠑⠀⠀⠀⠀⠀⠀⢀⡾⣿⣷⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣋⣻⣷
⠀⠀⠀⣿⣿⢻⢻⣹⡟⢦⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢰⠀⠠⠏⣰⠪⠇⠀⠀⠀⠐⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⢧⣽⣿⡿⠉⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣷⠀⠀⠻⠿⣿⣿⡏
⠀⠀⠀⠐⠒⠚⠓⠒⠂⠀⠀⠀⠀⠀⠈⠋⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢙⠛⠉
⠀⠀⠀⠀⠀⣠⣦⠀⠀⣤⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣿⣗⡀
⠀⠀⠀⢀⣨⣿⡿⣷⣄⣿⠀⠀⠀⠀⢾⡿⣿⠆⠀⠀⢀⣼⣿⣿⠿⠋⠻⢷⡄⠀⠀⢠⡟⣿⣿⢧
⠀⠀⠀⣿⠀⢀⡾⣻⣿⣿⠀⠀⠀⣠⣤⣿⣷⣤⠀⠀⢸⣏⠀⠤⢀⠠⠄⢸⣿⠀⠀⠈⢠⡟⣿⠈
⠀⠀⠀⣇⠀⠼⠁⡠⠓⣄⠀⠀⢰⡟⣿⣿⣿⣿⠇⠀⠸⠿⠦⣄⣙⣀⡤⠿⠿⠀⠀⠀⠸⡇⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⣀⣀⠈⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣈⣿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣟⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣷⣬⣿⣟⣽⣿⣿⣽⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠀⢀⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⣼⣿⣿⣟⡿⣽⣿⣿⣿⣿⣿⣯⠀⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⣸⣿⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⡈⠃
⠀⠀⠀⠀⠀⠀⠀⢹⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠻⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⠻⣿⣿⡇⠀⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣆⣠⣿⡇⠀⠀⠀⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣋⣉⣿⣻⡁⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠻⣿⣷⣄⡀⢀⣀⣠⣤⢿⣿⣿⡦⣤⣄⣀⡀⢀⣠⣾⣿⠟⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢠⣀⣈⣙⠻⢿⣿⣇⠀⠀⢸⣿⣿⠁⠀⠀⣼⣿⡿⠟⣋⣁⣀⡤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣈⣉⡉⠀⠀⠈⠿⠀⠀⠀⣿⡟⠀⠀⠀⠟⠁⠀⠀⢉⣉⣁⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠛⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⠀⠀⠀⠙⠛⠛⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⡀⠀⠀⣀⣤⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣀⠀⠀⢀⣀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠃⠀⠉⢹⣃⣀⣵⡄⠀⠀⠀⠀⠀⠀⢰⣯⣀⣹⡍⠁⠀⠘⠛⠛⠛⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣠⠇⠀⢀⣀⣀⡀⠀⠹⣶⣆⡼⠁⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⠿⣿⣿⣷⡶⠶⠖⠀⠀⠀⠀⠉⣡⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠀⠀⠐⠲⠶⢶⣿⣿⡿⠿⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣀⡤⠖⠒⠛⠒⠦⢤⡀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣀⡤⠖⠒⠛⠒⠲⢤⣀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡄⢠⠞⠁⠀⠀⠀⠀⠀⠀⡀⠉⠻⣶⣤⣄⣀⣀⣀⣀⣤⣴⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⣤⣤⣾⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠂⠀⠉⠛⢻⡟⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⢳⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⢦⡀⠀⠀⠀⠈⠓⢤⡤⣤⡤⠤⠞⠁⠈⠳⠤⢤⣤⢤⡴⠚⠁⠀⠀⠀⢀⡴⣋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⣀⣀⣀⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⣄⠈⠉⠉⠀⣠⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠛⠁⠀⠀⠀⠀⠉⠛⠿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣀⣀⣀⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣉⠤⡤⠤⣄⣈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣷⠀⠀⠀⠈⡦⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡞⠀⣎⣀⡜⠁⠀⠀⠈⢣⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠒⠒⠒⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢆⠈⣿⠉⡉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠦⠆⠀⢣⠀⠀⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠈⠀⠀⠤⠶⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠐⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠐⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠲⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢰⠲⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⣛⡛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣛⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣉⣵⣯⣿⣿⣧⠌⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⢤⡘⢩⡷⠠⣤⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣻⡾⣿⢿⣿⡟⠺⠿⠆⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢺⠀⢸⠀⢰⠋⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣹⠞⠛⠛⠑⠀⠀⡀⡀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠉⠲⠶⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⠖⠈⠑⣤⠀⠀⠀⠀⠈⠓⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡀⠀⠀⢀⡘⠈⠙⢦⣀⠀⠀⠀⠀⠈⠃⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣩⡷⣦⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠂⠀⠀⠀⣠⡟⠀⠀⣠⡶⠿⠄⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⠀⠰⠿⢶⡄⠀⢀⣠⣾⠋⠀⠀⣿
⡄⠒⢤⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⠷⠶⠋⠁⢸⡇⠀⠀⣿⠀⢀⠐⠒⡀
⠁⠰⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⠘⠃⠐⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠶⠶⣦⣸⡟⠀⠀⠀⢰⠏⢀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠿⠄⠀⠀⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⠁⣤⠀⠀⠛⠁⠀⠀⠀⠘⣄⠈⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⢀⡶⠀⠀⢸⣿⡄⠀⠀⡟
⡏⢀⠜⠀⠀⠀⢸⡄⢸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⢿⣀⡿⠀⢸⡇⠀⠘⢃⠈⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⠇⢠⡝⠃
⠀⠀⠀⢀⣀⣰⠳⢆⠀⣠⢄⠙⢶⡠⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⡾⠁⠘⣾⠁⠀⠀⡠⣄⠀⣠⠞⣦⣀
⠀⠀⠀⠻⣍⡙⠳⠈⠛⠁⢸⠁⠈⠻⣆⠀⠀⠙⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋⠀⢂⡿⠁⠀⠀⠈⠣⠈⠟⠁⠾⢛⣉
⠀⠀⠀⠰⣯⣁⡀⠀⠀⠀⠘⠧⣤⣄⠉⢳⣄⠀⠀⠀⠈⠉⠙⠓⠒⠶⠒⢶⡤⠀⠠⡤⠶⠶⠶⠞⠋⠉⠀⠀⠰⣹⠞⠁⠀⣀⣤⡸⠇⠀⠀⠀⠀⣉⣽
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠚⠲⠦⣄⡀⠙⠋⠻⠷⠌⣡⣄⣀⣠⣶⡶⠶⠶⠾⣄⠀⢀⡴⢶⠶⠶⢶⣆⣀⣀⣀⣴⠿⠔⠛⠋⢀⣀⠴⠖⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢤⡄⣀⡀⠀⠁⣹⡇⠀⠀⠀⠂⠀⠤⠌⠀⠀⠀⠀⢸⡏⠈⠀⢀⣀⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠻⢷⣴⡆⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⡿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠛⠛⠛⠛⠛⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡀⠀⡀⠀⣀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠋⠻⠟⠻⠟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⣿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠹⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣾⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⣿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⠀⢸⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠾⣟⠛⣿⠀⣸⡛⣿⠿⠶⣦⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣷⣶⣶⣦⣤⣟⣀⣻⣤⣤⣶⣶⣾⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠐⠂⡐⠖⠒⠈⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠋⠀⠀⠀⠒⠤⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⣶⠀⠙⢃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠠⠭⠀⠀⠀⠀⠠⠄⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠈⠑⠤⡝⠓⠦⠤⢀⡀⠤⠔⠚⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢫⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢆⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠘⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠈⢢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠄⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠠⠤⣀⠀⠀⢸⢀⠔⠡⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠑⠚⠁⠂⠁⠀⠤⣄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠈⢢⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢡
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠔⠀⠀⠀⠀⠀⠀⠁⢇
⠀⠀⢦⣴⠂⠠⠀⠀⠀⢄⠂⡀⠴⠘⠦⠰⠂⠄⠂⠠⠁⠠⠀⠠⠀⠀⢔⠀⠀⡠⠄⠀⠠⠘⡄⠈⣀⡀
⣄⢀⠀⢄⣀⣀⠀⠃⡀⠀⢀⠈⠀⠀⡀⢀⠀⠀⠐⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠤⠗⠀⢀
⠘⠚⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣠⠖⠛⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠐⠧⠖⠋⠁
⠀⠀⠀⠀⠀⢀⣾⣿⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣧⠀⢀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⣼⣿⢇⣾⣿⣿⣿⣿⠟⠁⠈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠘⠛⠂⠈⢿⣿⣿⣿⠀⠈⠙⠲⠦⠞
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⡇⠀⠐⠙⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢿⡃⠀⠀⠀⠀⣴⠶⣦⣀⣀
⡴⠶⠶⢤⡀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢿⣤⣧⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠘⣷⣶⠶⢶⣾⠃⠀⡼⠋⠉⣷
⠑⠒⠚⠉⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⣀⣨⠇⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠻⣤⣜⠁⢀⠜⠋⢳⡀
⠀⠀⣠⠴⢺⠇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣄⠀⠀⠀⠀⢸⣿⣿⡀⠀⠛⠈⠻⠁⢀⣴⠞
⠀⠀⠧⡤⠏⠀⠀⠀⠀⠈⢻⡟⣡⠤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣆⠀⠀⠀⣸⣿⣿⠟⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⣿⣿⠆⠀⢠⡿⠟⠁⠀⢀⣴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠔⠋⡁⠀⠀⣠⣿⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⡞⠁⠈⠻⣦⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠿⢧⠀⠘⢿⣿⡟⠲⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠶⣛⣽⠟⠀⠈⠓⢾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠤⠤⢄⡼⢧⠀⠈⠿⠷⠶⠒⠛⢻⡍⠉⠉⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⣀⣠⣤⣤⠾⠛⠛⠛⠛⠛⠛⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣀⣤⡴⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⢶⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⠻⢶⣤⣤⣤⠶⣶⠛⠋⠁⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠛⠻⣶⠶⠶⠶⠷⠀⠀⠀⣿⠀⠀⠀⠀⢈⡿⠒⠛⠻⣡⣏⣼⠟⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡧⠤⠤⠤⠀⠀⠒⠒⠂⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠰⣌⡻⠶⠶⠤⠤⠤⠶⠋⠀⠀⠀⠀⠀⠈⠛⠒⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡗⠺⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⡴⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣠⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠿⠛⠛⠛⠛⠛⠿⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⢸⣿⣸
⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠛⠛⠛⠛⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⢸⣿⣿
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢸⣿⣿
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣄⠀⠀⢹⣇⢀⡠⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢸⣿⣿
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢀⣼⣿⡆⠀⠸⣿⢸⡆⢰⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣿⠟⠁⠀⢠⣿⠈⠓⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿
⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢸⣿⣿
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⣿⣿
⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠙⠻⣷⣦⣤⣀⣀⣀⢀⣀⣀⣠⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⢿⣶⣄⡀⠀⠀⠀⣀⣴⡶⠟⠛⠛⠓⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠻⢿⣶⣄⣀⣀⣀⣴⣿⠟⠁⠀⠀⠀⠈⠛⠻⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠙⠻⠿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠒⠒⠒⠒⠒⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠖⣀⣤⣴⣶⣖⣀⣤⢦⡀⠀⠀⠀⠀⠀⢰⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠚⠁⣼⠷⠋⠁⠈⠉⠉⠉⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⢾⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⣿⡈⠿⢁⡤⢤⡀⠀⠀⠀⠀⢀⡴⠒⠲⣄⠀⠀⠀⠀⣿⠞⢻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡇⢠⡏⠀⠀⢳⠀⠀⠀⠀⢸⣁⣀⠀⠈⢧⠀⠀⠀⠀⠀⣼⡤⢤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠸⣯⣷⠀⢸⠀⠀⠀⠀⢸⣷⣼⡆⠀⢸⠀⠀⠀⠀⠀⢩⣤⡀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛⠋⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⣠⠇⣼⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⡼⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢤⣖⠒⠒⠓⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⣼⢉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸⣤⠤⠤⣤⣤⣈⣷⠀⠀⠀⠀⠀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⢮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⡾⠀⠀⠀⠉⠉⠉⠉⠀⠀⣰⠃⠀⠈⡿⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢉⡟⢹⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⡞⠁⢀⣠⣤⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡞⠀⢸⡄⠀⠸⣄⠀⠀⠀⠀⣀⡴⠋⠀⢀⡰⢋⡴⠚⠉⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠁⠀⠀⠹⣄⡀⠈⠙⠓⠚⠋⠁⠀⣀⡴⠋⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣼⣀⣀⣀⠀⠀⠉⠓⠲⠶⠶⠖⠚⠋⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠒⠒⠲⠶⠶⠤⠤⢤⣽⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢫⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢋⢿⡾⢾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢫⣿⣭⣼⣿⣿⣶⣿⣭⣭⣭⣧⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠻⣿⣿⣿⣭⣭⣽⣿⣿⡟⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣦⡄⣤⣤⣤⣤⣤⣤⢠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣮⣟⣻⣟⣻⣵⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠈⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠓⢦⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣨⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⣾⠂
⠀⠀⠀⠀⠀⠀⢀⡤⠾⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⣹⣀⡴⢾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣉⣿⢶
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡉⠹⠄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠖⢶⠖⠛⠁⠴⠀⠀⠈⢁⡷⠶⣄⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡇⠀⢠⡞⠉⡳⣄⢸⠇⠀⠈⢣⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢠⡇⠀⠀⢸⡄⠳⢄⡠⠞⢄⠀⠙⠲⢤⣀⣀⠀⠀⣀⣀⠤⣤
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢱⠀⣼⢧⡀⢀⡸⠃⠀⠒⠒⠒⠢⣷⡄⠀⠀⠀⠈⠉⠁⣀⣠⣔⣃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠁⠀⠀⠢⣄⣀⣴⠃⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⣹⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠻⠇⠀⠀⠀⠀⠀⠀⡤⠤⢤⣉⣀⡀⠀⢹⡄⠀⠀⠐⠶⣖⡚⠋⠁
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠸⢤⡼⢦⠀⠀⠀⢸⡶⣦⠤⠤⠋⠀⠀⢸⡇⠀⠀⢄⣀⠀⢉⡷⠂
⠀⠀⠀⣠⠞⠁⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢳⡀⠀⢸⠀⣆⢀⣤⠶⠶⡄⣸⠁⢀⣤⠞⠉⠙⠉
⢤⡔⠊⠁⣀⣤⡞⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⣴⠋⢀⣀⣀⣠⢿⡿⠞⠉
⠀⠉⢉⣩⠟⠁⠀⢀⡤⣾⠃⠀⢰⠀⠀⣶⡀⠀⠀⢳⠀⣇⢀⡟⢀⡀⠀⢸⠃
⠀⠀⠻⠦⣤⠴⠞⠋⡼⠃⠀⠀⡞⡄⠀⢹⡟⢦⣠⡬⣇⣫⡼⠷⡟⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠾⢅⣀⡠⠞⠀⠘⠶⠶⠗⠀⠀⠀⠘⡿⢆⡼⠁⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⢤⣰⠏⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢦⠙⠀⠶⠊⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⢸⣄⡀⠀⢀⡴⠚⢦⡔⠛⣀⡀⢶⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣧⣀⣀⣀⡸⠏⠋⠙⠋⠀⠀⠀⢿⠈⠉⠙⣦⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡖⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡤⠤⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⡀⠀⣠⠶⢻⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⣇⠘⣦⠞⣡⠴⢟⣇⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⣉⠉⠀⠁⠼⠓⢲⠈⠋⣸⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠀⠀⠀⣠⠖⣩⠏⣀⣤⡶⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⢉⣠⠟⠛⠳⣤⠜⠀⠀⣯⡤⠤⠖⠚⠉⠁⠀⠀⠀⠀⢀⣴⡏⡠⠻⣥⢪⠖⠉⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⣠⢾⡡⠖⠻⣆⠀⣞⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣠⠤⢴⠖⠋⣉⡤⠤⣤⡄⠹⡗⠛⢢⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠴⠚⣷⠚⢡⡴⠋⢠⠇⠀⠙⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⢠⣧⠜⠁⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡱⠿⢤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣁⠴⠋⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⠦⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⢅⠢⡀⠀⢆⠑⢄⢹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡑⠢⣄⠳⡀⠑⢄⢣⠈⢢⡆⠱⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡢⢄⠀⠈⢢⠀⠑⠬⣆⠀⠳⣇⠀⢻⠀⢣⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⠢⡉⠒⢄⡱⡄⠀⠈⠳⠀⠈⠂⠈⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⢄⡲⣬⣢⣄⠉⠺⢆⠀⠀⠀⢀⣀⠀⢤⠂⢀⣴⠆⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠊⠻⢍⡓⠄⠀⠀⣴⠉⠀⠀⠀⠠⠃⣈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠆⣐⠒⠒⠷⢄⠀⠁⠀⠀⠀⠁⠀⠀⠀⢀⠗⠀⠀⠈⠀⣀⠔⠒⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣛⣁⡀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢠⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠠⠀⠈⠉⠂⠀⠀⠀⠀⠀⠈⡀⠀⡆⠀⠀⠀⠀⠀⠘⢣⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠄⣀⠃⠀⣠⠔⠒⠒⠢⢄⡇⠀⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠁⠠⠀⢠⡄⠀⠁⠀⠙⠢⠤⠤⠒⠊⠙⠢⠤⢤⠔⠊⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋⠁⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⣀⡀⠁⠺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣟⣓⣒⣂⠒⠋⠑⠢⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡎⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡜⠉⣠⣤⡈⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣄⡈⢟⡄⢇⣀⠀⢀⣀⠀⢠⡀⠀⢀⡀⠀⢀⠔⠉⣀⠀⠀⠀⡜⢀⡊⠁⢀⣈⠢⠄⠠⠜⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠑⠻⠆⠀⠀⠈⣉⠁⢤⡈⢑⠀⠉⠁⠸⡀⠁⡰⠎⠀⢂⠀⠀⣏⠉⣉⠝⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠂⠜⠁⠀⢀⣯⠋⠀⠀⠀⠀⠈⠉⠁⠘⠢⣈⣦⡀⠈⣉⡤⠖⡡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠆⡀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢙⡛⢄⠀⡰⢁⠞⠐⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠜⠀⠀⠈⣂⠉⠁⢓⡀⠤⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠑⠒⠢⢤⢤⠄⣆⠀⠀⠀⠀⣀⡧⢠⠀⠀⠉⢉⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠢⠄⠑⠒⢶⠀⠀⠀⠀⠠⠦⣼⣘⠀⠈⠈⢑⠀⠘⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣖⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣢⣔⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠀⠉⠉⡠⡶⠖⢶⠘⢻⠉⠀⣈⢽⡈⠸⢒⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠐⠊⢥⣍⣬⢷⣒⣒⡒⠈⠅⠰⠖⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣳⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠠⡀⠀⠀⢀⣀⠀⣀⣀⠀⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠄⠈⠁⠐⠲⠌⠑⠒⠊⠉⠀⠁⠉⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠶⠶⠶⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠉⠉⠓⣆
⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠲⠤⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡈⠀⣠⠊⢁⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢑⠶⠤⣄⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⡇⢠⠇⠀⢸⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠉⢙⠓⠦⢤⡖⣿⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣇⡼⠀⠀⠸⡆⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠠⠁⠀⠀⣀⠴⠚⠉⠑⠢⣄⠀⠸⣝⠒⠺⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⢳⡀⠀⣀⠀⠀⠀⠑⣄⠀⡸⠓
⠀⠀⠀⠀⠇⠀⠔⠋⠁⠀⠀⠀⠀⠀⠈⢣⠞⠈⠳⣄⠀⠀⠀⠀⠀⠉⣽⡃⠀⠀⠀⠀⠀⢹⡀⠀⠑⠦⠏⠣⣀⠀⠀⠀⠉⠀⣠
⠀⠀⠀⠘⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠺⣿⠀⠀⢀⣬⣳⠄⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠈⠁⠐⠂⠂⠈
⠀⠀⠀⡇⠀⠆⠈⠀⠙⠓⠦⠤⣤⣤⠤⢿⠀⠀⠈⠛⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡄⠄⠊⠀⠀⠀⠀⠀⠷⠐⠒⠒⠒⢞⠉
⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⡜
⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣟⠲⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠒⠒⠲⠶⠶⠒⠒⠚⠋⠉⢁⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣁⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⢶⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠦⠤⣤⠀⣤⣤⣤⣤⠤⠤⠴⠒⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⢄⣀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤⠽⠿⠭⠝⠛⠓⠦⠭⢉⣛⣿⣃⡠⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠟⠛⢿⡟⠋⠁⠀⠀⠀⠀⠈⠸⣟⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⣀⣀⣀⣌⣥⣤⣿⠆⣠⣾⣿⠴⠤⠿⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠈⠻⣦⣤⣶⣴⣤⣤⣶⣶⣶⣤⡄⠈⠒⣦⣬⣷⣶⠶⠶⣤⠤⣤⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠈⢦⣾⣿⠁⠈⠉⢧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢸⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⣟⡄⠀⠀⠈⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣾⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⡀⠀⠀⠀⠀⡟⡔⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⣡⠀⠀⢠⡀⡷⢄⣠⠄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡅⠠⣤⠀⣿⠁⡾⠴⠀⠰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠓⠀⠛⠛⠓⠊⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠴⢄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⣾⣡⡴⠎
⠀⠀⠀⠀⠀⠀⠀⣿⠏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢿⣿⣿⣿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣴⡿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⠾⣍⣉⣉⡆
⠀⠀⠀⠀⠀⢠⣿⣷⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠻⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⣀⣀⠈⢻⣿⣇⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢻⣿⣤⣀⣤⣤⡀
⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣄⠀⢿⡿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⡆⠀⠙⠉⣿⣿⣿⣿⣇⡀
⡴⠚⠒⢶⡄⠀⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠶⠶⣄⣀⣀
⠙⠒⠚⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢿⣄⣿⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠿⣿⡟⠀⢠⠟⠉⠙⡧
⠀⠀⢀⡴⠂⡷⠀⠘⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣀⡽⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠸⣤⣤⡏⠀⣠⠞⠙⢦⡀
⠀⠀⢯⣤⠞⠁⢠⣶⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣤⣤⡀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠘⠂⠙⠞⠁⣀⣴⠟
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢋⣉⡉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⡟⣻⡿⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣾⡿⠋⠀⠀⢠⡾⠛⠉
⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣷⣦⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢉⡴⠚⠉⠀⠀⢀⣾⠋⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⡴⠋⠉⢷⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⣿⣿⡁⠀⢻⡟⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⣻⡿⠁⠐⢦⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⡀⠈⣷⠀⠀⢹⣆⣀⣤⣭⡿⢿⡶⠶⠶⠶⠿⠿⠿⠷⠶⠛⠉⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⢻⡍⠛⠀⠀⠉⠉⠀⠀⠀⠘⣧⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠘⢳⡶⠒⠒⠛⠃⠀⢀⣠⣤⣤⣤⣤⣤⡤⠿⠶⢶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⡤⠶⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠞⠛⠉⠉⠉⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣀⣤⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠻⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠈⠙⢻⡶⠶⠶⠛⣿⠉⠁⠀⠀⠀⣿⠀⢀⣀⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡏⠉⢻⠟⠛⠛⠛⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠛⠉⠉⠣⠼⠠⡟⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠁⢀⣸⠃⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⢒⣒⣓⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⠀⠧⣭⠛⠳⠶⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠚⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣇⣠⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣈⢈⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣄⠀⢲⢀⣠⡶⢶⣿⠣⣽⣆⣮⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢰⠿⣄⡰⢣⢞⣽⣿⣿⣴⣿⣿⣡⡴⣣⠶⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢔⠫⣹⣮⡿⣷⣿⣿⣿⣤⣷⣿⣿⣿⣿⢟⣱⣦⡴⢤⣴⢟⣾⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣨⣿⣿⣻⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢨⡙⢮⣻⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢱⣼⡼⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⣶⡟⢁⡔⣠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠋⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣭⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠶⣾⣿⣿⣿⣿⡿⠻⢡⠛⠙⠟⢿⣿⣿⣿⣽⠿⠟⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡴⠛⢻⣿⡽⣯⠁⠀⠀⠜⡀⠈⠈⣿⡟⣿⢿⣻⣦⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢒⡀⢹⣟⡇⠉⠀⠀⠀⢠⡀⠀⠀⠘⠃⡛⠒⠈⠙⠓⠵⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠭⠄⠱⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢤⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠂⠄⠀⡴⠂⠉⠀⠀⢣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡆⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠊⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⠤⠽⠓⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⢀⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠢⠁⠀⠀⠁⠀⢀⣀⡀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠒⠐⠁⠀⠀⠀⢀⠎⠀⠈⠀⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⡠⠤⡖⠁⠀⠀⠀⠀⠀⠀⠀⢀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁⠀⠀⠀⠌⠀⠀⠀⠀⠁⠀⢣⠣⡀⠀⠀⠀⠀⡘⠀⢀⠆
⠀⠀⠀⠀⣀⠘⢆⠀⠰⠀⠀⠁⠒⢠⠐⠂⠓⢄⡈⠀⠀⠔⠁⠀⠀⠀⠘⡀⠱⡀⠀⠀⠰⠁⠀⡌
⠀⠀⠀⠄⠉⠁⠈⠉⣠⣴⣴⣿⠖⠘⠌⡈⠁⠒⠭⡈⠀⠀⠀⠀⠀⠀⠀⠃⢀⢸⡄⢠⠃⠀⡘
⠀⠀⠀⠀⠀⢏⡐⠤⣈⠁⣭⣤⢄⠓⢛⡀⠅⡒⠄⡀⠌⣐⠂⠦⢀⡀⠉⢩⠀⠀⠀⠎⠀⢠⠁
⠀⠀⠀⠀⠂⠁⠈⠐⠤⣉⠊⠅⡀⠤⣌⡙⣷⠾⣫⠴⠀⢐⡠⠄⡀⠈⠉⠒⠤⢄⡘⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡁⠂⠍⡃⠦⣔⣁⠉⠁⠂⠀⠀⠍⠂⠀⠀⠀⠈⠉⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠑⠢⠄⠁⠠⠐⡈⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⡛⢻⣿⣿
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⢀⣄⣀⣀⡤⠶⢴⣄⠀⠀⠀⠀⣿⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⢷⣤⣠⡷⢋⣹⡿⣷⣿⣶⣤⣶⣶⣴⣦⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢱⣿⣖⢿⣿⣿⣿⣟⠽⣿⣿⣿⣿⣏⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣝⣴⣿⣿⣿⣿⣧⣤⣴⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⣵⣷⡊⣬⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣯⠙⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⣿⣿⣿⣶⣄⣀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡤⣤⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⣻⣿⣿⣿
⡋⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣶⣤⣴⣾⣿⣿⣿⣤⣿
⣿⣿⣿⣶⣶⣶⣶⣾⣿⣳⡦⠝⡛⠛⠛⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡉⠉⠉⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣛⣛⣻⣿⣟⣏⣉⣉⣉⣈⣁⣚⣊⣁⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣛⣻⣿⣿⣛⣉⣉⣉⣉⣋⣁⣀⣰⣾
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡑⢦⡀⠀⠈⢣⡈⢣⠀⠀⠀⠀⠀⢀⣀⡔⡪⢗⣩⣔⣮⢭⡷⠤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠦⣄⠀⠙⠀⠁⢀⣀⡠⠞⠉⠀⠀⠈⠑⠺⢽⠖⠫⣤⠰⠈⠆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⣷⠀⠀⣠⣴⣤⠀⠀⠈⠉⠉⡆⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢦⡀⠀⠀⠀⠀⢀⠎⠔⢄⡄⠀⠀⣈⠀⠈⠁⠀⠀⠀⠸⣄⣀⣀⠼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⡇⠀⠀⠀⢠⢺⣵⣦⡞⠀⣠⣁⢸⠃⠀⠀⠀⢹⠷⠦⠠⣎⣤⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣦⠶⠶⠶⠶⢸⣇⣀⣤⠞⠁⠀⠀⡰⠁⢀⠔⠃⡀⠀⠛⠋⠀⠀⠀⠀⠀⠞⡿⠇⡀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡷⠋⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠃⠀⠳⠤⠤⠋⠀⢄⠀⠀⠀⠀⠀⠀⢠⣆⠁⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣠⠴⠒⢢⣔⠒⡒⠢⡀⠀⣼⠀⢀⡴⠆⠦⢠⡄⠀⣣⠀⠀⠀⠀⢠⡀⠀⢃⢼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠀⠀⠙⢂⡌⠙⣧⡤⠘⢠⠇⠀⢹⡄⠀⠙⠒⠣⠤⠜⢀⢏⠆⠀⠀⠀⠀⡹⠋⣡⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⢺⡔⡀⠽⡙⢲⣎⣀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣴⣫⣴⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠘⣤⣀⣀⠜⣿⡍⠛⢦⢄⠀⠉⠛⠒⣲⢶⢖⠊⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠑⢦⣾⣿⣿⣦⣀⠀⡱⡀⣰⣾⡏⢸⠞⢣⣾⣿⣿⣿⣿⣿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⠤⠬⣁⡶⠀⢸⣿⣿⣿⣿⠿⡿⢿⡜⣿⣿⡷⠉⢧⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⠼⣤⠴⢦⣤⠀⢸⣿⣿⣿⢹⣿⣿⣷⣌⢿⣿⣧⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⣾⠟⣃⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠐⠬⠝⠀⣄⣠⣈⠙⠛⠂⠽⠿⠛⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢰⡋⡁⢂⣉⣒⣂⠒⠒⠚⠓⠚⣢⠙⠦⡀⠀⠀⠀⠈⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠁⠀⠘⢮⣦⣀⠀⢀⡧⠁⠀⢈⣉⣉⡿⠿⠿⠿⢿⡟⡛⠛⣿⠟⠛⠉⠉⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠉⠉⠁⢳⠀⢸⠀⠀⠀⠀⠈⠉⣉⡥⠖⠚⠉⠉⠀⠀⠀⠀⠀⣀⡬⠥⠜⠛⠑⠒⠂⠚⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⣿⠁⠀⠀⠀⠀⣄⡀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⣸⠀⢸⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⢠⠀⠀⠀⠉⠢⢄⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢠⠔⠒⢤
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⢠⣶⣭⣷⣦⣍⣉⣀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⢄⠀⠀⠁⠀⢀⡴⢆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⠘⢦⡀⠉⠉⠉⠋⠉⠉⠁⠙⠛⠉⠓⠦⢄⣀⠠⣀⣂⠄⠉⣀⣴⣗⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠧⠤⠤⠤⠀⠠⠇⠀⡄⠀⠈⠳⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣽⣿⣿⠟⠋⠁⠀⠉⠀⢀⠀⠀⠒⡞
⠀⠀⠀⠀⠀⠀⣰⠷⠶⠶⠶⢶⣆⣀⣀⣀⣀⣧⡀⠀⠀⠈⠙⠓⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠀⠀⠀⠀⠀⠘⠉⠣⠄⠒⠁
⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠚⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠒⠶⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡟⠈⠀⢽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣾⣿⠉⠉⠈⣹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡝⣿⣿⠐⠚⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⡟⠀⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠄⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡀⣸⣿⡇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡯⠁⠀⢀⡠⠐⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣼⣋⣉⣿⣿⢱⣶⣿⣟⣿⠿⠟⠿⢿⠛⢿⣿⣼⣃⠉⠒⠉⠀⠀⢀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣤⣻⣿⣿⡋⠀⡀⢀⠀⠀⠄⠜⠋⠁⠀⠀⣀⣤⡾⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠓⠴⠐⠴⠆⠲⣚⠉⠀⠪⠥⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠛⣈⣉⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠭⠇⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠗⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆⠀⠾⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠲⢾⠂⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠻⣿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣿⣿⣿⣿⣿⠿⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣫⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿
⣿⣿⠀⠀⠀⠀⠀⠳⠢⠄⠤⣀⠠⠤⠤⠤⠴⠾⠿⠿⠟⠁⠀⠛⠋⠙⠃⠖⠋⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠠⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠁⠈⢱⠉⠀⠹⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠀⠠⠀⠀⢀⠀⠔⠤⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠁⠀⢀⠈⠀⠀⠆⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢠⠟⠶⣿⢿⠶⣶⡄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠲⠖⠻⠃⢺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⡁⢘⣀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠺⠄⢔⡆⢘⠒⠄⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣊⠁⠖⠂⠉⢀⡥⠸⠌⠉⠓⠨⢉⡍⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡊⠅⠀⠰⠖⠘⣢⠶⢋⡇⢘⣀⣛⣒⢀⣘⠃⠀⠠⢄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠚⠠⠔⠂⡡⠞⠉⠭⣍⢐⡇⢨⠔⡢⠬⢙⢓⠬⡁⠒⢤⡁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡴⠁⠀⠈⢁⠀⢀⣀⡵⢸⢉⠀⢀⡀⠀⠆⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠀⣬⠅⠀⠒⣃⢸⠒⠄⠭⢤⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑⢊⠀⢂⠐⢉⣵⢸⠬⣀⣀⡒⢄⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⢃⡀⡀⢄⢸⠅⠀⠨⠍⠀⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠡⠀⠌⠀⡀⠼⢸⠅⡀⠤⡅⠤⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠁⢉⡁⠐⣻⢨⡑⠌⠐⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣉⠔⣋⢔⢸⢌⡖⠠⡑⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⡎⢀⣴⡇⣮⢸⢅⠯⢤⡈⠶⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡑⠀⡎⡿⠐⣹⠸⡀⠹⡀⢄⠒⠢⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠩⢰⢋⠸⠗⠊⢴⠘⠬⡀⢭⠀⠑⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠠⢀⠀⠔⠀⣤⠊⣴⠅⠀⡈⠃⠠⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠔⡃⢀⠜⠀⢂⢀⠈⠫⠀⢰⠈⡌⢄⠈⠓⢄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⠆⠎⠀⠈⠈⠀⢰⠈⠁⠀⠁⢸⠙⢈⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠖⠘⡰⠣⡀⠀⠀⢀⠆⣆⠀⠀⢀⣼⠀⠀⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣆⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⣠⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠻⣿⣿⡄⢰⡣⠔⠃
⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⢠⣄⠘⢿⣧⠀⢀⡤⠤⣄
⠀⠀⠀⠀⣾⣧⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⡄⠈⠓⠲⠊
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣆⠘⠫⢳⡄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠐⣾⠁⠀⠀⠀⡤⠦⣄⣀
⢎⣉⡭⠖⠈⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢹⣦⡧⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠸⣦⡴⠶⣼⠁⢠⠏⠉⣿⡀
⠀⠀⢀⠠⡄⠈⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⣿⢧⠀⠙⡖⢧⣠⠜⠉⣹
⠀⠰⣅⡼⠁⠀⠀⢹⡿⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣄⠀⠀⠀⣿⠘⣧⠀⠀⠀⢁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⢠⣷⠞⠁⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣩⠕⠋⠉⠀⣠⣟⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢿⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⡞⢁⠈⢳⡴⠋
⠀⠀⠀⠀⠀⠀⣞⠉⠛⠻⡄⠈⢷⡈⠙⢲⣦⣤⣤⣤⣤⣤⣴⣖⣿⠽⠛⠀⠀⠙⢯⡀
⠀⠀⠀⠀⠀⣰⠛⠑⠒⢲⠓⠀⠈⠉⠉⠉⠁⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠘⣶⠒⠒⠛⠀⣀⣤⠤⠤⠤⠤⠾⠷⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠙⠶⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⡟⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣀⣤⠴⠚⠛⠉⠁⠀⠀⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣅⠀⠀⠀⠀⣀⣠⡴⠖⢻⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⢤⣄⣀⣈⣹⣟⣛⣉⣿⠀⠀⠀⠀⣧⣤⣤⡿⡠⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡄⠀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣇⢀⠀⠉⠉⠭⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠇⣦⣻⠶⠤⢤⣤⠴⠞⠀⠀⠀⠀⠀⠳⠖⠒⠒⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠙⢷⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣼⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠉⠉⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⠈⠀⠀⠀⠀⡤⠀⠠⠔⠒⠐⠙⠄
⠀⠀⠀⠠⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠊⠀⣈⢵⣂⠢⠀⠐⠈
⠀⠀⠀⠄⢈⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠇⠀⢠⣈⢡⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠁⠀⠈⠐⠃⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠢⠤⡀⠀⠀⠀⠐⠂⠑⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⠐⡀⠀⠠⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⣀⡀⠀⠀⠀⣀⣀⠀⡈
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠠⠠⠄⡐⠁⡢⠥⠈⡅⢰⢠⡄⢐⠁⠊⠁⠉⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠂⠁⠘⠒⠀⠅⠃⠀⠀⠀⠀⢼⠐⠁⠀⠀⠀⠀⠀⠀⠀⠂⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠡⠀⢘⠚⠀⠠⢀⠔⠉⠉⠑⠄⠀⠸⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢈⠄⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠉⠚⠒⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠪⢀⣠⠀⣀⣀⣀⡀⠀⠠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠢⢄⣀⠆⠈⠀⢠⢾⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠂⠄⠀⠀⠀⠈⠲⢇⣀⣄⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠶⣄⠀⠀⢹⡿⡇⠀⡗⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡒⠈⠀⠀⠀⠀⠀⠐⠓⠙⠂⢀⣣⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⠖⠉⣠⠔⠁⣼⡡⠖⠚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠘⠋⠀⢀⡞⠁⢠⠞⠉⣀⡤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⣠⣯⠴⠖⠛⠚⠊⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⡠⠠⢄⡀⠀⠀⢀⣀⣼⠁⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠣⠤⠤⠼⠆⠀⣴⠟⠙⣦⠟⠁⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠾⠤⠤⠴⠃
⠀⠀⠀⠀⠀⢠⠤⠤⣤⠀⠀⢰⡇⠀⡜⠁⠀⠀⠀⠸⠟⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢻⡀⠀⠀⢠⡤⠤⢤
⠀⠀⠀⠀⠀⠘⠶⠚⠁⠀⠀⢺⠏⣸⡇⠀⠀⠀⢠⠖⠋⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⢀⣀⣉⠀⠈⣇⠀⠀⠀⠙⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣇⠀⠀⢠⡏⠀⠀⠀⠀⣀⣹⡆⠀⠀⠆⠀⠀⣿⣁⣈⣳⠀⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠖⢻⡀⠀⠈⣷⠖⠚⠋⠛⠋⣼⠀⠀⠀⣦⠀⠀⠣⡌⡩⠋⠀⢸⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⠀⢧⠀⠀⠘⠦⣤⣀⡤⠞⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣁⣠⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⡴⠋⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⡼⣿⣿⣷⣼⠆⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠰⠦⣌⣻⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⢀⣸⡁⢸⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣄⡀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠙⠁⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⢈⡇⢀⠎⢙⡆
⠀⠀⠀⢀⣠⠤⠤⠴⠊⣉⠴⠚⠉⠉⠉⣿⣉⠉⢻⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢳⠀⠙⠉⠙⡶⠊
⠀⠀⡾⠋⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠈⠉⣻⠇⢛⡗⠶⠦⠤⠴⠚⣏⠙⣦⠀⠀⠀⠀⢀⡰⠋⣠⠖⠀⠁
⠀⠀⠳⡞⠀⠀⠀⡀⢀⠇⠀⠀⠀⠀⠀⠀⢠⡟⠁⠘⠀⠀⠀⠀⠀⢨⢻⡷⠛⢶⣦⣤⣶⡯⠴⠋⠁
⠀⠀⠀⠙⢾⣁⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡀⠀⣸⠇⢰⣦⢀⠘⡇
⠀⠀⠀⠀⠀⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣀⡈⠃⠐⠾⠃⠐⠀⡇
⠀⠀⠀⠀⠀⠀⣰⠟⡹⣧⠀⠀⠀⠀⢰⡿⠶⣤⣤⣉⣉⠛⠛⠚⠚⠃⢠⡇
⠀⠀⠀⠀⢠⡾⢣⠞⢀⡼⡆⠀⠀⠀⣸⣯⠀⠀⠀⠈⠉⠙⠛⠳⠶⠦⣼⠃
⠀⠀⠀⣴⠟⡴⢁⣶⣿⡔⢻⡄⠀⣾⠉⠙⢦⣀⡀⠀⠀⠀⠰⢤⣤⣴⡋
⠀⣠⡾⣡⢮⠀⢠⡿⠁⠹⡄⠙⣛⡿⠳⡤⠞⠉⠛⠷⢦⣤⣀⣤⣶⣿⣧⣤⣀
⣴⢏⡔⢁⣠⡷⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⢳⡦⠉⡷
⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠯⣴⣿⣶⡶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠢⠤⢤⡬⣉⣀⣀⣰⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠺⠶⠶⠦⣤⣭⣭⣉⣗⣻⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿