Goof Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠉⠍⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⣦⡄⠑⡀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠉⠀⢣⢁⡐⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠑⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠔⠂⠀⠀⡑⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣀⣭⣔⣀⡀⠣⠀⠐⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠌⠃⣨⠖⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠜⠁⠀⠀⢠⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢾⡀⠀⠀⡴⠃⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡽⢶⠋⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⡔⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⣠⠞⠀⠀⠀⠸⠀⠀⢀⡀⠠⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⡴⠋⠀⢠⡶⠏⠚⠀⠀⠀⠀⣀⠀⠉⣁⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢺⠁⠀⠀⣸⠀⠀⠀⠀⠀⢠⠞⠁⢠⡾⠁⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠤⠂⠁⠉⠉⠀⠀⠀⠀⠀⣰⠃⠀⢀⡟⠀⠀⠀⢀⠞⠁
⠀⢠⠊⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⡞⠀⠀⠀⠀⣽
⠀⢸⡀⠀⠈⢆⠀⠀⠀⠠⢐⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⠈
⠀⠀⠳⣀⠀⠈⡦⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⣧⠀⠀⠀⠀⢇
⠀⠀⠀⠈⠓⠶⠖⠉⠀⠀⢀⡴⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⢄⣸⣀⣀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⡟⠀⠀⠀⠙⢦⣤⣤⣤⠼
⠴⢶⣶⣶⣶⡶⠦⠤⠤⠤⠤⠬⠽⠤⠶⠶⠤⠤⠤⠤⠤⠤⠶⠶⠶⠾⠟⠓⠒⠛⠛⠓⠒⠒⠲⠶⠦⠤⠤⠤⠄⠀⢀⣀⡀
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠠⣼⡼⠁⠀⠀⠀⢀⠜⠁⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠿⣧⣀⣸⡀⠀⢀⡿⠃⠀⠀⠀⡴⠃⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢠⡆⢴⣶⣶⡆⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣠⠊⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠳⠦⠶⠋⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⢠⡄⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣴⠀⠀⠐⡴⠃
⠀⠀⠀⠳⠄⠀⠀⠈⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠿⣤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣾⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⣠⡼⠃⠀⠀⠀⠀⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠈⠉⠉⠉⠉⠉⠙⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⢀⣀⡀⠀⠉⠩⠭⠉⠉⠐⢶⣤⣀⣈⠁⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⣀⡤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡙⠲⢄⡀⠈⠐⠠
⠀⠀⠀⠀⠀⢀⠞⠁⢀⡴⠋⠁⠀⢀⡀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠶⣍⡲⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣦⢤⡀
⠀⠀⢠⠞⠉⢙⠿⠟⠻⠛⠋⠉⠉⠉⠉⠉⠉⠉⠭⠭⠭⠭⠭⠭⠍⠉⢻⡏⠰⠁⠞⣩⡧⠤⣄⠀⠀⠀⠀⠀⠀⠰⡄
⠀⢀⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢰⡏⡠⢀⢈⡃⠀⠀⠀⠀⠀⠀⠙
⠀⠸⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⡍⠣⡺⠎⠻⢆
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠻⣄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠲⣶⠖⡿⠁⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠈⢦⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣠⠾⣣⠞⠛⡀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⣸⠂⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⣴⠏⡞⣛⡟⠚⠉⣱⣆⠀⠀⡸⠀⠀⡏⠀⣠⠞⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣽⠷⠄⠙⠊⠀⠀⠈⠁⣈⣧⣠⣅⣀⡸⠓⠪⡁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⣶⡆⠀⠒⠢⠔⠚⢁⡾⠁⠀⢸⠇⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠒⠒⠊⡟⣹⠃⠀⠀⣾⠀⠀⠘⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠞⠁⠀⢀⡴⠃⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠤⠤⢤⣄⡀⠀⢰⣷⣦⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⠳⣄⡀⠀⠀⠲⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡔⠒⠶⠖⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠉⠛⠾⠧⣤⣤⠴⠦⠖⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠶⠤⠴⠞⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠉⠉⠈⠉⠉⢹⠂⠀⠀⠂⠐⠒⠂⠀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⡀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠈⠉⠉⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠉⠀⠀⠀⠀⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⣀⡤⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠒⠒⠂⠈⠉⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⡴⢦⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠⠋⠀⠈⡆⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⠃⠀⠀⠀⢸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡇⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⣇⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠠⠴⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣓⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⣠⠶⠚⠉⠁⢀⠨⠔⢻⠟⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠳⠶⠒⠃⠉
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⠿⠏⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣸⡍⠁⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⢧⡃⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠐⠒⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢠⣴⣃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀⠀⠀⠀⣼⣿⡉⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠲⠦⣤⡿⣧⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠙⣾⣅⠀⡀⢀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⢸⣅⡴⢿⣧⣴⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢸⣿⣿⠟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⢸⣿⡟⢠⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⢀⣿⡿⢠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣿⣿⠇⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣡⣿⣿⡟⢸⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠁⣿⣿⣿⠁⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠘⣨⣿⣿⡇⢼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣷⠈⣿⣿⣿⣷⡀
⠀⠀⠀⣀⠀⡀⠀⠀⠤⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢸⢿⣿⣿⡖⢸⣿⣿⣿⣧⠀⠀⠐⠒⠒⠒⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⡇⠀⡇⣹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⢸⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠻⠛⠛⠀⠈⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠦⠈⠇⠠⠶⠦⠀⠀⠀⠀⠀⠀⠀⠤⢠⡄⢈⣉⣀⣠⣤⣼⣶⣿⠯⠠⠟⢀⣦⣶⣤⣤⣤⣤⣄⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠏⡉⠀⠀⠤⠄⠉⠉⠉⠉⠉⠉


⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⡄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⣠⣶⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⢁⣿⠋⠉⠀⠀⠀⠀⣀⣠⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⠈⠓⠀⢸⣤⣴⣾⠟⠀⠀⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣼⡿⠟⠁⠀⠀⣠⣿⣿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡽⠟⠉⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢟⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠙⠿⠛⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⠋⠀⠀⢠⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⢠⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢰⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡿⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡼⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⠁⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣰⠇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢿⠄⠠⠉⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣀⣀⣠⣾⣆⡀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣤⣭⣤⣄⡀⠠⠴⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠀⠀⠀⡞⠈⠐⠒⠃⠀⠈⠀⠀⠈⢻⣍⠉⡁⠁⢀⣈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣶⠆⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⡞⠶⠶⠖⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⠿⣿⡀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣻⡄⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⣷⡀⠀⠀⠀⠀⠀⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⠄⣴⠤⣄⠃⠀⠀⠀⠀⢠⠖⠢⢄⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡏⠀⠈⠙⢦⡀⠀⠀⠈⢀⢀⠈⢉⣀⠀⠀⠀⠙⠮⠑⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠿⠷⠄⠠⠈⠁⠐⠀⠉⠀⠀⠉⠒⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⡀⣴⣾⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⣹⣟
⠀⠀⠀⠀⠘⠛⣿⡿⠿⣽⣿⠿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡆⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣁⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣀⡄⠿⣿⣷⣶⠿⠿⠟⠛⠿⠿⠶⢿⣿⣿⣿⣿⣿⣿⣀⣼⢿⣿⡟⠛⠛⢿⣿⡟⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠸⣿⣅⠆⠀⢸⣿⣷⠾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡁⡴⣆⡼⣱⠃⠀⢠⠖⣰⣦⢠⡦⢦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠾⠉⠁⠈⣹⠋⠀⠀⠈⠁⠉⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⣮⣿⣧⣽⣿⡁
⣿⣟⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣟⣟⣿⣿⣟⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣷⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣹⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⠈⠙⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⣦⡈⠁⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠟⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠁⠒⠂⠀⠤⠤⠤⠀⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⢶⣶⣶⣿⣿⣏⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣙⣿⣿⣿⣷⣶⣶⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠀⠀⠀⠀⠀⠀⠉⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⢀⠠⢆⠀⠀⠀⠀⢠⣀⣆⣼⣥⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢰
⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠚⠁
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠐⠀⠀⢀⠠⠚⠀⠀⠀⢠⠒⠠⢠⣀⣀⠀⠀⠠⡄
⠀⠀⢀⡔⠁⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⡄⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠄⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠧
⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠈⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⢂⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠄⠠⠄⠀⣀⠿⠊⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠲⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⣠⡀⠀⠀⢂⠀⢀⠔⠀⠀⠀⠀⠀⠀⠁⠀⠀⠢⡄⠒⠉⠅⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠄⠀⢰⠒⠤⠀⡐⠀⠀⠀⠀⠀⠓⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣰⠊⠀⠀⠀⠈⠀⠀⠊⠉⠑⣤⠔⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠢⣀⡀⠀⠀⠀⠀⠀⢦⠀⠄⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠢⡀⠀⠠⠚⡄⠀⢀⡠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠤⠂⠀⠈⠒⠀⠀⠘⠂⠁⠀⠀⡆⣀⠔⢲⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢀⡠⠄⠒⠒⠂⠘⢰⠂⠤⠁⢀⡤⠖⠛⠛⠛⠓⠲⢤⡀⠀⠠⢺⠉⠉⢹⡂⢸⠀⢀⢾⠉⠉⠉⠉⠁
⠀⠀⢀⡼⠁⠀⠀⢀⣀⡠⠤⡼⠀⢀⡴⠋⠀⠀⢀⣠⣤⣀⠀⠀⠈⢢⡰⢸⠀⠀⢸⡇⠘⠂⢑⢸⠀⠀⠀⣀⣀⣀⣀
⠀⡐⢹⠁⠀⠀⡴⠛⡃⠈⠈⠀⢀⢺⠁⠀⢀⡴⠏⠂⢄⠈⢳⡀⠀⠀⣷⢸⠀⠀⢸⠃⠀⠀⠨⢸⠀⠀⠘⡏⠀⠂⠐
⢠⠁⡏⠀⠀⢸⠃⠀⠁⣀⣀⣀⡈⢿⠀⠀⢸⠀⠀⠀⠀⢣⠀⡇⠀⠀⣽⢸⠀⠀⢸⠀⠀⠀⠨⢸⠀⠀⠀⠉⠁⠁⠈
⢘⠀⡇⠀⠀⢸⣆⠠⣏⣀⠀⠀⢷⢹⠀⠀⠸⡀⠀⠀⠀⢸⣰⠃⠀⠀⡿⢸⠀⠀⢸⠀⠀⠀⠈⢸⠀⠀⢀⣠⣤⣄⣠
⢠⠀⢣⠀⠀⠀⠣⢴⣤⣹⠀⠀⣸⣾⠷⡂⠀⠓⠂⠤⢤⡞⠃⠀⠀⡼⢱⢸⠀⠀⠸⠤⠤⣀⣜⢸⠀⠀⠰⡧⠖⠂⠂
⠈⢆⠈⢦⡀⠀⠀⠀⠀⠀⢀⣼⢛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢾⠁⠠⣸⠀⠀⠀⠀⠀⠀⣗⡽⠀⠀⠘⠂
⠀⠈⠳⣄⠉⠲⠤⣄⣀⣤⣼⣇⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢿⡖⠲⠲⠲⠶⢶⢇⠷⠶⠖
⠀⠀⠀⠋⠩⠕⡒⠒⠒⠒⣿⡅⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠁⠈⡨⠋⠀⠀⣦⡀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣟⡧⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡏⠀⠉⠀⠀⠀⣠⠞⠁⢙⠦
⠀⠀⠀⠀⠎⠀⣀⠀⠀⠀⢸⣹⣦⠀⣠⠔⠊⠉⠉⠑⠲⢤⠀⠀⠀⠀⠀⠀⠀⣾⡟⡇⠀⠀⠀⢀⡴⠃⢔⠊⠁
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠱⣟⡾⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⡿⠏⣇⢀⣠⠔⠋⠀⠀⣀⠱⠄
⠀⠀⠀⠀⠀⠀⠃⠀⠒⠆⠀⠀⢸⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣰⠿⠾⠛⠋⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢸⠀⠙⢄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⢙⠻⠶⣶⣶⠉⠀⠀⡀⠀⠀⠀⡤⡀⢀⠇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠄⢸⠁⢄⢀⠜⠀⠢⡀⠰⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⣿⠿⣿⣦⣬⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠒⢦⡀⠰⡟⠓⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠻⣦⡀⠀⠙⠆⠠⠬⡃⣰⠟⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⣀⡌⠀⠙⠦⣤⠀⠀⠀⠈⠙⢠⡀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⡁⠀⠁⠀⠐⠀⠛⢤⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢶⣷⠂⠀⠀⠀⢻⣇⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠉⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⣀⣀⡀⠀⣀⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠉⠉⠛⠻⠿⠿⢿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠐⣄⠀⠀⠀⠀⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠈⣷⡀⠀⠀⠀⢻⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣾⡇⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢻⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⣷⡀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⣧⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⢰⡟⠘⣆⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡆⠀⣿⠃⠀⠘⢧⡀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠛⠀⢿⡀⠀⠀⠐⡅⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⣾⣧⠀⠀⠀⠹⣆⠀⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠹⣿⠀⠀⠀⠀⢿⢧⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢻⡀⠀⠀⠀⠘⣦⣼⡿
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣼⠗⡾⠾⣧⣴⣦⣀⡄⣿⣿⣥⣤⣤⣤⣄⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠐⠚⠁⠀⠈⠉⠙⠉⠉⠀⠈⠁⠀⠈⠀⠀⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⡿⣿⢿⣿⣿⣿
⣿⡯⡋⠝⠕⠉⠊⠉⠑⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠊⠈⢜⢕⡵⠻⡻⣿
⣿⣟⢕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠪⣾⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡎⢹⡟⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣏⣀⠘⡯⢁⠔⠉⢂⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⡉⠁⡸⠋⣠⠆⠀⠀⠀⢃⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⡷⠶⢶⣶⣴⣾⣿⣿⣷⣶⣦⣤⣤⠤⠀⡠⠚⣣⠎⢀⣾⡟⢢⡤⠤⠔⠚⣷⡀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡴⢊⢔⡟⠁⠘⢿⣿⣷⣿⠃⠀⣄⣤⣽⣧⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢸⠸⣄⠀⣈⢯⣿⣿⠃⠀⠀⢻⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣶⠁⠀⠀⠀⠈⢣⡈⢉⣵⣿⣿⠏⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠆⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⠀⠀⠀⠀⠾⡟⠀⠀⢤⠤⢤⣤⣤⣤⣤⣶⣶⣾⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣤⢴⣤⣤⣤⡴⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢼
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢻⣾⠛⠈⠉⢤⡈⠳⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣸
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠙⠿⢷⣦⣄⡀⠀⢸⡟⠀⠀⠀⠀⠉⠓⠛⣤⠂⠉⠛⠿⣶⣦⣤⣠⠎⠉⢻
⢸⡇⠀⠀⠀⡠⠔⢩⡷⠚⠁⠀⠀⠀⠀⠀⠈⡫⠛⠳⢾⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⢻⡿⠁⠀⠀⢸
⢸⣗⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡄⠈⣒⢤⣀⠀⢸⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⢸
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄⣶⣿⣿⣿⣇⠀⠸⢽⣾⣿⣿⣿⣿⣿⡿⢿⡟⣴⣄⠀⣀⠀⢸
⢸⡇⠀⠀⢀⠀⣤⠀⣄⢀⡄⣿⠠⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⢽⣿⣿⣿⣿⡿⠟⠇⣸⣿⠀⣿⡎⢹⣦⣺
⢸⣧⢐⡇⣸⣄⣿⣴⣿⢸⣷⣿⣷⣧⣿⣼⣿⣿⣿⣿⠿⠿⠛⢣⠔⠒⠉⠉⢻⣁⣀⣀⡄⢨⣿⠀⣻⡇⠸⣿⣿
⢸⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⣀⣀⣤⣼⣇⠀⠀⠀⢸⣿⣍⠉⠁⠠⣿⠀⣹⡇⠘⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⠛⣉⣠⣤⣤⡶⠾⠛⠛⠛⠋⠉⠈⢢⡄⠀⠀⢹⣿⣷⠚⡀⣿⡇⢹⣷⠘⢿⣿
⢸⣿⣿⣿⠿⠿⠟⣛⡛⠛⠉⠀⠉⠉⠉⠉⣡⡆⠀⠀⡤⠂⠒⠒⣲⣼⣷⡀⠀⠀⣿⣿⣧⠑⡟⠐⢬⡟⢄⣻⣿
⢸⣯⠤⠖⠊⠉⠁⠀⠀⠀⢀⣀⠤⠚⠀⠀⠉⣢⠀⢰⠀⠀⠀⠀⢉⣽⣿⣥⡀⠀⣿⣿⣿⣷⡛⠢⠴⣆⡀⣫⣸
⢸⣿⣦⠀⠀⢀⠴⠂⠀⠉⠀⠀⠀⠀⢀⣴⠟⠁⠀⣸⠀⠀⠀⠀⠀⠀⠿⣿⣧⠀⡇⠈⠻⣿⣷⣤⣄⡀⠀⠉⣿
⢸⣿⣿⣷⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠐⠛⣿⣧⣵⠀⠀⠈⢻⣿⣿⡿⠀⠀⣿
⢸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠤⠤⠶⠀⠤⠤⠤⠀⠀⠈⣿⡀⠈⣧⡀⠀⠘⣿⣿⠀⠀⠀⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⢿⣦⡁⠉⢳⠀⢰⣿⡟⠀⠀⠀⢿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⢷⡶⠿
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠉
⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠸⠿⠤⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣀⡀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠺⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡅⢰⣿⡇⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⠄⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠿⠛⠉⢙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⢿⣧⣀⣠⣾⡿⢟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⢻⣿⠢⣽⣯⣚⣿⣿⣿⣋⣶⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣅⣿⡍⣿⡟⢵⣾⣶⢻⣿⠃⣬⣍⢶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣝⣯⣙⣿⠙⣿⠟⢠⣿⣦⠻⣿⠇⣻⡟⣤⣭⣲⣯⢽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣏⢻⣏⠻⡟⢸⣿⣷⣨⣿⡁⣾⣿⡆⢿⣿⢙⡛⢣⣯⣰⣟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡹⣟⢿⡿⢾⣿⠇⣾⡄⢿⡿⢇⣬⣍⣼⣿⣷⠻⣿⢏⣛⢣⣌⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣿⢶⣿⣧⣴⣶⣮⣿⣿⡆⢻⣿⡿⣈⡉⢡⣶⣦⢿⣿⠏⡛⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣷⣡⣮⢛⣧⣽⠻⣯⣬⡛⣵⣿⣷⠸⣿⡿⢈⡛⢋⣴⣆⢾⡿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⣼⠟⣘⡿⢟⣈⡿⡿⣃⣬⡛⢩⣶⣦⡰⣿⣿⠎⠿⣫⣤⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⣿⡷⢸⣿⡷⠀⢿⣿⡿⠘⠿⢿⣃⡉⢹⣾⣷⠙⢛⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣴⣶⣄⣴⣮⢠⣶⣶⡍⣶⣿⣦⠸⣿⣿⣨⡉⢽⣿⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣤⣝⢋⣬⣿⣛⣵⣾⣍⣵⣾⣷⣚⠛⣷⣷⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣿⣟⣷⣯⡉⣵⣶⣆⣿⣿⣶⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠛⠻⣿⣿⣿⠟⠛⠛⢿⡿⠃
⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡔⠋⣀⣀⠀⠀⠈⠻⠃⠀⢀⠉⠦⢱
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⣴⣿⣷⡄⠀⣀
⠀⠀⠀⠀⠀⠛⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⠇⠀⠀⠘⣿⣿⣿⡟⠀⣿⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⠇⠀⣿⠀⠀⢀⣀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠛⠻⠧⠀⠈⠻⠟⠁⠀⠀⠀⠀⠈⠛⠃⠤⠐⢛⠶⠂⠈⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⡄⠀⠀⢠⣶⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⢸
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⠀⠙⢄⠀⠀⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠈⣦⠀⠘⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣙⠇
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠰⡄⠀⠘⣿⣙⣻⣿⣿⣿⣿⡛⠒⠛⠛⠛⠛⠿⠓⠿⠈⠀⠀⠁
⠀⠀⠀⠀⠈⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢢⣴⡈⠛⠻⠿⠞⢋⡜⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⡦⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡅⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⢋⡼⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⠒⠈⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡌⠉⠡⠒⠉⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠊
⠀⠀⠀⠰⠾⠿⠟⠛⠛⢿⠿⠛⠉⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⠟⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠙⠉⠉⠉⠉⠙⠋⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠒⠒⠋⠉⠉⠉⠉⠉⠓⠒⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⢨⡤⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠄⠀⠠⠆⠀⠘⠃⡀⠐⢊⡡⠰⣃⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢀⡀⢰⠃⠀⠐⣺⣿⠴⢚⠅⠔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠁⡴⠉⠀⠀⠀⠀⠄⠀⣀⠀⣈⡴⣂⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⠎⠀⢂⠐⣴⠀⠀⠂⣼⡿⡄⠘⢻⡥⠴⠊⠀⢈⠘⣦⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢂⡄⠛⡒⣀⢈⡨⢉⠡⢾⢣⠄⢀⠀⢀⢟⣉⠁⡠⠘⠂⠀⣀⡀⢠⠀⠀⠃⠀⠂⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠂⢠⡄⢠⠁⠸⢃⣴⣿⢃⢂⣴⢋⣤⠞⣡⣔⣈⠶⠶⠊⢀⠖⠁⠀⡖⠀⠈⣠⡘⠀⠇⠰⠀⠀⠆⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢠⣹⠀⣉⠁⠤⠜⣯⢯⡍⠞⢨⣯⣤⠜⢻⡜⠙⣠⣾⡆⠈⣡⣤⣆⠑⠀⠘⠋⡁⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⣄⢃⣿⠻⢢⣄⠻⣿⣿⢧⣜⣿⣣⡊⣿⣷⢾⣿⡿⠁⣸⢿⡍⠁⢠⡞⠳⢀⣀⠀⢀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢆⠚⢻⡜⣤⣦⣼⠿⠿⣉⣕⣾⢉⠗⣧⡿⢻⢿⡿⣷⣳⡟⢲⢏⣡⡶⡼⠤⣄⠈⡿⠀⠸⣄⡀⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢼⡗⢿⢿⢹⣴⣔⡹⣿⡇⡟⣾⣛⡟⡯⣼⣾⣅⡞⢳⡏⠐⣿⡒⢳⣾⣏⣰⠰⠹⠃⡵⣡⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠍⢀⢼⣮⡉⡿⠋⣓⣷⣦⣍⢪⣹⣭⣟⢻⢿⣟⣴⣿⡟⠍⢠⣿⣿⢛⣭⠅⠀⣆⠀⠀⢋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠛⠃⠙⣧⡇⣽⣧⡿⣿⢿⢸⢿⢯⢿⣿⣿⣿⣿⣥⣶⣟⡩⣯⣟⠸⠷⡖⠰⠻⠒⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠨⡿⠿⢈⣿⣿⣿⣿⣿⠢⣿⣿⣿⣿⣯⣷⣿⡯⣽⡾⣿⣿⣴⡟⢉⣀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣈⠿⣟⠿⣿⣿⣿⣧⣻⣯⣷⢿⣾⡿⣿⢷⣿⣶⣾⡿⢞⣳⣿⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢌⣛⠹⠿⣿⢽⢫⣵⣿⡿⣽⠿⣿⢿⠿⣛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⠲⠶⠶⠿⣶⣶⣶⠶⠒⠺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣶⣶⣤⣶⣶⣤⡤⠶⢞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⠲⣄⠀⠉⢳⣾⣷⡄⠈⢽⣶⣚⡋⠀⣰⣿⣇⡴⠁⣠⠖⠀⠀⢀⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢌⢻⡛⢄⠀⢨⣳⣷⣬⣿⣿⣿⡄⢴⢻⣵⠇⢰⣿⣿⣿⣧⠊⢫⠖⠀⡰⠃⣡⣠⡴⠚⠁⢀⠊
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠪⢐⡬⣧⣯⣻⣷⣿⣿⣣⡻⣻⣿⣿⣷⠸⣿⣿⠀⣿⡿⣿⣿⣧⣦⣿⣾⣷⣷⣿⡿⣫⢆⣴⣯⠆⠞⡄
⠀⠀⠀⠐⠠⣄⢀⠀⠻⣶⠲⠦⣄⢉⢢⣿⢈⡛⢿⣿⣏⢹⣶⡽⣿⣿⠐⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⡏⣰⣿⣷⣂⡰⢀⡼⡇
⠤⢤⡤⠈⠦⠈⠛⢷⡈⠩⣿⢙⣦⠱⣬⣹⣟⢿⡦⠏⠹⣽⡞⣧⢻⣿⠀⣿⣿⢸⣿⡿⠟⡿⠃⣿⣿⣼⢿⣴⠿⢏⣴⣿⣯⣾⣿⣷⠏⠞⠁
⠀⠀⠀⠀⠀⠠⢀⣠⡀⠑⠺⠄⠀⠛⠊⠻⠟⠲⣏⠢⢀⣿⡇⢹⡇⠻⡇⠛⡿⠈⡿⡇⠰⠓⢀⡟⡽⠇⠈⠃⠠⠈⠻⠉⠙⠛⠛⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠃⢸⠀⠀⠁⠀⠀⠀⠀⠁⠄⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⡿⠟⠓⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣩⡶⠋⠀⠴⠀⠉⠀⠀⠁⣉⠉⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⢿⡷⠛⢀⡀⠚⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠈⠹⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣿⠚⠀⣠⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣫⠟⠁⣠⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠁⣰⠈⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣫⡆⢉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣇⠛⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣧⢀⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⠀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣱⡌⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⢀⡋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣾⡄⠠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠐⣂⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣬⣄⣤⣤⡤⠴⠖⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠰⠀⠀⠀⠀⠹⡄⠂⠀⠀⠀⡠⣿⡏⢱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⣄⡀⠀⡀⠀⡀⠈⢶⠀⠀⠈⠠⣾⠁⠀⠀⠓⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠷⠶⣿⣧⡀⢆⠀⠈⠣⢀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠉⠿⢓⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠸⠷⣶⣶⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠀⣀⣀⣤⡤⠀⠐⠛⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠧⠀⠀⠀⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢷⣦⣄⡀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡆⠀⠀⠛⠿⠿⠓⡴⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⢀⣤⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡟⠉⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⣆⠀⠀⠀⡇⠈⢧⣠⣴⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢡⢀⠟⠉⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣶⣾⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣤⣶⣿⣿⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠛⠿⠻⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠛⠉⠉⠉⣉⡵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠂⠀⠀⠀⣀⠠⠤⠤⠤⠦⠤⠤⠤⢤⣀⣀⠀⠀⠀⢀⣀⣀⣤⠬⠭⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠔⢩⠃⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⢄⠀⠀⣷⠀⠀⠀⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⡆⢀⡇⠀⠀⣀⣀⣀⣈⣙⡛⠀⠀⠒⠋⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠤⠬⣍⠓⠾⠀⠀⠀⢸⠁⠀⠛⠀⢸⠍⣹⠈⠉⠉⠉⢣⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⠚⠲⢤⡀⣠⠞⠦⣅⣀⠀⢀⣠⠎⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠙⡇⠈⣹⢧⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⣀⠤⠤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠁⢱⢤⡀⠀⠀⠀⠀⠀⠁⠀⢤⣀⣀⣀⡼⠀⣸⠁
⠀⠀⠀⠀⠀⢀⡠⠖⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡴⠁⢀⠏⠓⡦⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢏⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⣠⣋⣀⣘⣁⣰⡭⠭⠟⠒⠂⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⢦⣀⡤⡔⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣹⢃⣀⢿⡄⠀⠀⠀⠉⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡴⡊⠁⢀⡔⠐⠒⠒⠙⢷⠋⠀⠀⠀⢸⣤⠟
⠀⠀⠀⠀⠀⠀⠙⠲⠥⠞⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠸⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⣤⢤⡤⠴⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣸⣸⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡲⣾⣤⣤⣽⡤⢤⣷⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠭⠟⠚⠉⠑⠺⠿⠾⠟⠃

⠀⠐⠂⠀⡄⢳⢄⠀⠀⠀⠀⠀⢼⠂⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⡴⠣⠁⣴⡇⠀⠀⠐⠀⠀⠠⣈⠞⠈⣆⠀⢸⢂⣼⣿⣿⣬⣀⣀
⠀⢀⠀⢰⡅⠘⡇⠈⠀⠀⠀⠀⣸⠀⢠⣾⡿⣿⡇⠀⠀⠀⠀⠀⡼⠃⠀⡰⢁⠃⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠀⣸⠜⣽⠟⣿⠻⣿⣿
⠀⢸⣧⡘⠀⢀⠙⡀⠀⠀⠀⠀⡏⣰⣿⠟⠁⣿⣇⣤⣶⣾⣿⣿⡷⣤⡐⠀⣸⢒⠠⠀⠀⠀⡔⡇⠀⠀⢀⡄⠀⣿⣼⡿⢀⡏⠀⠑⠘
⠀⠸⣿⣧⠀⡀⠀⠸⡀⠀⠀⠀⣳⣿⠏⠀⣠⣟⣻⣿⣿⢫⣬⣄⢩⣷⡟⣆⡧⠈⠀⠀⢀⠎⠀⠀⠀⠀⢸⣷⣴⣿⣿⠃⢸⡏
⠀⠀⠸⣿⡄⠀⠀⠉⢿⡄⠠⠀⣿⡟⠀⠘⣿⠐⣿⣿⣿⡈⣙⡋⠠⣭⠠⣿⡱⠀⠀⢠⠏⠀⠀⠀⠀⠰⠇⢀⡻⣿⠁⠀⣾⠁⠀⢡
⠀⠀⠁⣿⣧⠀⠀⠀⣸⣷⠀⠀⣿⡇⠀⠀⠙⣦⣽⣿⣿⣿⣿⣦⣠⠄⠂⢺⠄⠀⢠⠏⠀⠀⠀⠀⠀⣽⣾⣥⠔⠋⠀⢰⣿⠙⢶⢾⡇
⠀⠀⠀⢹⡏⣷⡀⠀⢸⣿⡅⢠⣿⡇⠀⣐⣶⣿⠿⠿⢿⣿⡟⠻⣿⣿⣿⡿⠀⣠⠋⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⠀⢀⣿⣿⡆⠀⢠
⠀⠀⠀⢸⡇⣿⡇⠀⢰⣿⠃⠸⣿⠀⣸⣿⣿⣿⣶⣶⣿⣿⣿⣦⣤⣤⣿⣀⣰⠡⠴⣶⣾⣾⣿⣿⣿⣿⠏⠰⠀⢠⣾⡿⣷⠈⠀⠀⣻
⠀⠀⠀⣸⣷⣿⠂⠀⢸⣿⠀⢀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣉⣩⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⢏⠅⠀⠀⢀⣾⡿⠁⠈⠀⢀⣼⣿
⠀⠀⢀⡟⣿⣿⠀⢀⣾⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣼⣿⠿⢡⠟⠁⠀⠀⣾⣿⠃⠀⠀⢠⣿⡿⢹
⠀⠀⢸⠁⣿⡏⣠⣾⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠒⠶⣶⣾⡿⠛⢀⡏⠀⠀⠀⣼⣿⠏⠀⠀⢠⣿⡟⡰⠿
⠀⢀⡏⠀⣿⣷⣿⢧⠤⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢀⠀⡇⠀⠀⣴⣿⠏⠀⠀⣴⣿⠏⠀⠎
⠀⠸⠧⠄⣿⣿⡇⠘⣦⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆⢀⠂⣴⣶⣦⠀⠀⠀⠈⣃⠀⢠⡇⠀⣰⣿⠏⠀⣠⣾⡿⠃⠀⠁
⠀⠈⠀⠀⣿⣿⣿⠀⠹⡄⠀⠀⠀⣿⣿⣿⣿⡏⠀⢸⣿⣿⣧⢤⣾⣿⣿⣿⣿⣄⠀⠀⠑⢄⢸⠀⣴⣿⠏⢀⣾⣿⠟⠁
⠀⠰⠤⢀⣿⣿⣿⡄⠀⢳⠀⠀⠀⢿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⢾⣿⢋⣴⣿⠟⠁⠀⠀⠀⠀⠀⡠⠚
⠀⢀⠀⢸⣿⣿⡿⠸⢄⠀⢇⠀⠀⠘⣿⣿⣿⣧⣠⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠻⢷⣆⡀⠀⢱⣿⡿⠃⠀⠀⠀⠀⠀⢀⢞⠱⡀
⠀⠀⢀⣿⣿⣿⡇⠀⠀⢵⢼⡀⡈⠀⠙⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣦⣤⣤⠻⠿⢶⣶⣿⠛⠻⢶⣖⡂⠀⢀⡿⡘⡄⢳
⠀⠀⣿⣿⣿⣿⢇⠀⠀⠀⢧⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠻⢿⣿⣷⡀⠀⠀⠙⢧⡀⠀⠈⠉⠋⣿⠀⣡⠘⢉
⠀⢸⣿⣿⣿⡇⠘⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣟⠿⠿⣿⣿⢿⣷⣦⠀⠈⢻⣿⣿⣶⣄⣀⡀⠙⣷⣄⠀⣸⡿⣍⠀⠀⠈
⠀⣿⣿⣿⣿⠁⠀⠈⢧⡀⠀⠀⠀⢀⠀⠀⠀⡠⠞⠤⣈⡁⠀⠀⣰⣿⡇⠀⠀⣿⣷⡆⠈⣧⠙⢿⣿⣿⣿⣶⣌⣿⣿⣿⠃⠈⠳⢤
⠀⣿⣿⣿⡿⠀⠀⠀⠈⢐⢄⠀⠀⠀⠘⢦⡞⠁⠀⠂⠈⠃⢠⣾⠟⣼⠈⠃⠀⣿⣿⣷⡀⣷⠀⠀⢙⡿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⡝
⠀⢻⣿⣿⠀⠢⠀⠀⠀⠈⢢⠣⡀⢀⡴⠋⠀⠢⠀⠀⠄⣴⣿⡋⢸⠏⠒⠄⠀⠿⣿⣿⣧⢹⠀⠘⠁⠀⡈⣻⣿⣿⣿⣿⣿⣿⣷⣄⣱
⠀⣿⣿⣯⠀⢀⠀⡀⠀⠀⠀⢣⡸⠎⠀⠀⠀⠀⠀⢈⣼⠟⠁⣇⣾⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠂⣀⠞⠠⠃⠎⠳⣌⢉⠉⢻⠿⠿⢟
⠀⣾⣿⣿⠀⠈⠀⠛⠐⢂⠄⡟⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠠⢸⠇⠀⠀⠄⢀⡠⣾⣻⣿⣿⡁⠌⠡⠀⠀⠘⠀⠀⠟⣦⠹⣤⣀⢃⡘
⠀⢻⣿⣿⠀⠀⠀⢠⠜⠇⡄⠀⠀⠀⠀⠐⠀⢰⣿⠃⠀⠀⢀⡏⠀⢀⠤⠚⣡⠞⠁⣸⣿⣿⡗⠈⠀⠀⠀⠀⠀⠀⠀⠁⡷⡀⠈⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⠀⠀⠀⠀⢀⠊⣄⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⣄⠐⣂⠀⠄⡰⠂⠒⠊⠈⠿⠿⠖⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠈⠀⠤⢀⠱⠁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣷⣶⡶⡐⠀⠉⠂⡤⠀⡶⣄⠀⠀⠀⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠛⠿⢛⡴⡁⠀⠀⠀⠧⣀⠀⠈⠉⠀⠘⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠐⠀⠋⠀⠀⠋⠀⡜⢀⠠⠃⠀⠈⢈⠗⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠤⠤⠚⡀⠁⠒⠀⠤⠀⠂⠀⢀⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠀⠀⠀⠀⢆⠉⠐⠀⠀⠔⠁⠀⠀⠠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣠⠀⠀⠚⠒⠄⠈⠢⣀⡠⠀⠀⠀⠀⠀⢠⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠛⠢⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⡠⠘⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠐⠉⠀⠀⠀⢀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢠⢀⠊⠀⠀⠀⠀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⢀⠤⠄⡀⠈⠐⠤⣀⣀⠠⠤⠐⠀⠀⠀⠁⠀⠀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠁⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⢉⠄⠀⠀⠀⠀⠀⠀⢐⠑⠁⢀⠄⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⠀⠈⠙⠄⠀⠀⠀⠒⠒⠒⠁⢀⣔⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠙⠲⠤⠤⠖⠒⠒⠛⠛⠛⠛⠋⠒⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⡴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⢸⣧⠴⠖⠛⠲⣄⠀⣄⣻⠉⠓⠲⠿⡍⠓⠲⠦⠿⠏⠛⠲⢦⣄⣀⠀⢦⣤⣀⠀⠑⠦⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠢⢾⡀⡮⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⡞⠠⠌⠉⠓⠛⠀⠉⠓⠦⣌⣷
⠀⠀⠀⠈⠛⣟⣛⣒⣒⠒⠒⠠⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠶⢦⠀⠉⠁⠀⠀⠀⣀⣀⠀⢀⠴⠒⠓⣄⢠⠞⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⡠⠔⣊⣁⣀⣀⡤⠤⣤⠤⣤⣤⡬⠍⠛⠛⢛⣿⡿⠶⠶⠶⠦⠈⠧⣄⣀⣀⣠⡄⠀⢠⡴⢋⡽⢩⠅⢰⠋⠀⠀⠀⡜⠈⣦⠄⠀⠀⣸
⠚⠋⠉⠉⠉⢻⡅⢠⠖⢾⠉⠹⡇⠀⣵⣔⣚⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢻⡉⣵⢻⠀⢾⠀⠀⠀⡰⠃⠀⠈⢟⠛⣛⡡⠤⣄
⠀⠀⠀⠀⠀⠀⠳⠜⠢⡀⢣⠀⡇⠐⢋⣁⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣⣸⣿⠀⠀⠑⠒⠊⠁⢀⠀⠀⠨⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠉⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠳⣼⡄⠀⠀⠀⠐⢦⠉⠄⠀⢀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣇⢠⣤⣤⣀⣀⡀⣀⡰⢛⣶⡤⢾⣿⡆⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣙⡆⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠒⠛⠋⠉⠉⠹⣄⣈⣉⢙⡄⠀⠀⠉⠀⠁⠸⡅⢀⣼⠉⠹⣆⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⡟⠉⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣏⡃⠀⠀⠀⠀⠀⠀⠘⠲⢋⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢇⠀⠀⠀⠀⠀⠀⣎⠁⠀⠘⠛⠊⠁⠀⠀⠀⠀⠀⡀⠈⢧⣀⠀⠀⠙⠳⢤⣤⣤⣤⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⡾⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⢀⣨⣽⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⢶⢲⠊⡿⠀⠀⠀⠈⠓⠦⣄⣀⣀⡀⠀⣰⠁⣧⠔⣹⢿⣿⣤⡀⠀⣴⢮⣉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣦⠁⠀⠀⠀⠀⠀⢰⠃⡏⠉⢩⠏⠁⠀⠉⠀⠀⠀⠙⠦⣙⢺⢁⣤⡨⢏⡁⣹⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣯⣄⡀⠀⠀⠀⠀⢿⠾⠗⠂⢺⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢰⡶⣄⣹⠁⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡜⠻⠏⡟⡿⡲⠶⡾⢶⣊⠻⠟⠻⠇⠉⢳⠀⣀⡠⠤⠒⠊⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⢀⣃⡤⢖⡽⠿⣧⡀⠈⢑⣲⣭⣛⣓⠶⢲⡏⣟⡓⠒⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⣴⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⠥⣶⣛⡃⢂⣴⡧⠯⢬⢤⠖⠒⢺⠻⡉⠁⠀⠈⠉⡿⠿⣷⣶⣶⣤⣤⣍⣉⣉⠒⠒⠲⠦⠤⢴⠋⣠⣾⠟⣋⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡸⠀⢸⠉⡏⠻⣳⠀⢸⠾⡆⠀⠀⡇⢧⠀⠀⠀⠀⢹⣤⣤⣤⣌⣉⣉⡛⠛⠻⠿⠿⢿⣶⣶⣶⠾⣋⡥⠊⢹⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⠇⠀⢸⠰⡇⠀⠉⠀⠈⠉⠁⠀⠀⢳⠼⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢤⣄⣼⡧⠚⠁⠀⠀⢸⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠟⠀⠀⠀⠀⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠈⠁⠒⠂⢄⠀⡀⠤⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠠⢀⠀⠀⠄⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠓
⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⢉⢀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠃⢀⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡐⠑⠃
⠀⠀⠀⠀⢁⠀⠀⠈⠋⠀⣌⠈⠁⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣆⠈⠁⠀⢀⠂⠀⠀⠀⠀⠀⢀⠀⢡⠀⠀⠀⠺⢫⠀⠀⠀⠀⠄
⠀⠐⠺⡇⠀⠁⠀⠀⠐⠈⠀⠉⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣉⡀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠩⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠈⡦⢀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠉⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⣀⠠⠊⠁
⠀⠀⠀⠀⠰⡄⠈⠀⠃⠀⠐⠃⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠘⢫⡉⠉⠉⠉⠙⠉⠉⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐⠒⠂⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠐⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠁⠈⠉⠉⠉⠉⠙⠃


⠀⠀⠀⠀⠀⠀⣠⠶⠶⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠂⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠄⠀⠀⠀⠀⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠤⡀⠀⣀⠠⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⣀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⡠⠀⠐⠛⠛⠂⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢂⣀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣄⠱⢂⣅⣠⣌⢠
⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡀⠀⡀⠻⣿⣿⡿⢃⠻⢿⣿⠟⢠⠀⠠⢤
⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠔⠉⠢⠄⠤⠒⠁⠀⠀⠀⠀⠀⢱⠀⠀⠐⠀⠈⠉⠠⠊⠑⠤⠀⠔⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⢊⠡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢤⣠⠔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢒⣀⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢃⣠⠒⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⡀⠀⠀⠈⠙⠓
⠀⠀⠀⠉⠍⠉⠛⠿⣷⠶⣴⣦⢶⣿⠛⠉⠁⢸⠀⠀⠀⠀⠀⠀⡞⠉⢠⣤⡄⠀⠇⢤⣤⠀⠁⢱⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠲⠊⠀⠀⠀⠠
⠀⠀⠀⠀⠘⢄⠀⠀⠀⣠⠆⠑⢄⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⣀⠦⢀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⢀⠛⠃⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⢀⡠⠊⠒⠤⠀⠤⠊
⠀⠀⠀⠀⠀⠀⣀⡄⠒⠈⡇⠀⠈⠉⠁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢶⣄⣀
⠀⠀⠀⠀⢀⡞⠐⠒⢲⠐⠃⠀⡎⠁⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢀⠸⡑⡄⠤⠘⠒⠒⠒⠓⠒⠚⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠻⠆
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠘⣇⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠶⢶⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢄⣠⠱⠀⠀⠀⠀⢻⡿⣿⣦⣀⠀⠀⢸⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠈⠻⢿⣷⣦⣬⣄⣥⣴⣿⣷⣄⣀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠉⠙⠛⠛⠛⠉⠈⠙⠛⠟⠛⠋⣿⣿⠁⠀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣧⣀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣆⣠⠞⠉⠀⠀⠀⠀⠉⢻⡿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠀⢀⣿⣇⣀⣀⠀⢠⡶⢶⡀⠀⠀⠀⡏⠀⠘⣿⡅⣴⡾⢛⣻⣷⠶⠿⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⡾⠛⠉⠁⠀⠉⠙⢾⣿⡧⠙⢦⡀⢰⡇⠀⠀⠹⣷⣏⠀⢿⠀⠀⠀⠀⢸⣿⣆⣀⣸⡓⠂
⠀⠀⠲⡶⠺⡄⠀⠀⠀⠀⠀⢹⣿⡠⢀⣤⣴⣦⣄⡀⠀⠙⣯⢂⠈⠳⣼⠀⠀⡀⠀⢻⣟⠦⣼⡷⠟⠁⠀⣼⠹⡏⠀⠈⣷⣤⣤⣀⡀
⠀⠀⠸⠓⢴⠆⠀⠀⠀⠀⠀⣾⣿⣏⣾⠋⠀⠀⠉⢷⠀⠀⢸⡷⣄⠀⠈⠀⢠⣧⠀⠀⢿⣄⠀⠀⣀⣤⡾⣣⠾⠷⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣷⡀⠀⠀⠀⣸⠃⠀⢸⣯⣿⣷⡄⠀⢸⠙⣧⣠⣬⡿⠻⠛⠛⠉⠰⡏⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⡤⠶⠾⠿⠿⠿⢿⣿⣿⣯⠻⠦⠤⠞⠁⠀⢀⣾⣿⣿⡈⠻⡶⠟⠀⠉⠉⠁⠀⢄⣑⣄⣀⣀⣹⣄⠀⠀⠀⠠⢎⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡹⢿⣿⣧⣀⡀⢀⣀⣴⣿⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁⠀⢈⠿⠛⠛⠛⠋⠁⠈⠻⣧⠀⠀⠀⢀⡴⠞⠛⠓⠲⣄⡂⠀⠙⢦⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣰⠋⠀⣠⠴⠶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠔⢃⡀⠀⠀⠀⠀⠀⣿⣰⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣃⡀⢰⠊⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⡦⠐⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠈⢻⣿⢿⡿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⠀⠀⠀⡀⠐⡈⣋⠂⠀⠀⠀⠀⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢻⣿⣿⣶⣄⠀⠀⠀⠀⢨⣴⣿⣿⡿⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠭⠿⡿⠟⠀⠀⠀⠀⠸⢿⡿⠯⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⣀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⣴⣶⠆⠀⠀⠀⠀⠀⠘⢧⣁⣙⣛⣉⣵⠋⠀⠀⠀⠀⠀⠐⣶⣦⣌⢲⣀
⠀⠀⠀⠀⢦⡴⣶⣶⡄⠈⠸⠛⠛⠁⠀⠀⡀⠀⠀⠀⠀⠈⢿⣻⢟⡿⠁⠀⠀⠀⠀⢢⠀⠀⠈⠛⠛⠃⠉⢠⣴⣶⢦⣴
⠀⠀⠀⠀⠈⠾⠿⡿⢷⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⡾⢿⠿⠷⠃
⠀⠀⠀⠀⠀⠈⡖⢄⢸⣧⣴⡶⠂⢀⠀⠀⠸⢷⣦⡀⠀⢀⣀⢀⡀⢀⡀⠀⢀⣴⡾⠏⡀⠀⡀⠐⢶⣦⣼⡇⣀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡬⣷⣟⡀⠀⠘⡦⡄⠉⠁⠀⢸⣿⣿⣿⣿⡇⠀⠘⡏⢀⢴⡃⠀⠀⣙⣶⠥⣼⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠙⢿⣤⣌⣿⣁⡴⠊⠀⢻⠀⠀⢈⠌⠉⠻⡿⠋⠁⡀⠀⠀⡟⠀⠙⢦⣈⣿⣁⣤⡿⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠈⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠏⠀⠀⠀⠻⢿⣿⣿⡆⢰⣿⣿⣿⠟⠀⠀⠀⠘⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣴⢫⡜⠁⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡸⣿⣿⠟⣋⡭⠭⠭⢍⡙⠿⣷⣶⣟⡵⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⢹⣦⢣⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⣱⠊⠁⠀⠀⠀⠀⠈⠳⡝⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣷⢣⣶⡁
⠀⠀⠀⠀⠀⠀⡘⠎⢼⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣏⡜
⠀⠀⠀⠀⠀⠀⢿⢷⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣰⢿⠇
⠀⠀⠀⠀⠀⠀⢈⣳⠉⢻⣷⡀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢲⣽⠏⢷⣋
⠀⠀⠀⠀⠀⠀⠈⢯⡣⠸⣿⣡⠀⠀⠀⠀⠙⣿⣿⡏⠀⠀⠈⠓⠤⣀⣀⣀⣠⠶⠋⠀⠀⠘⢿⣿⡟⠀⠀⠀⠀⢠⣹⡿⠀⢞⡏
⠀⠀⠀⠀⠀⠀⠀⢠⣭⡀⢸⢻⢆⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⢀⣤⣀⣠⣤⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢼⢿⠁⣠⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠮⠄⠈⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠁⢨⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡢⠀⠋⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡼⠇⠡⢶⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣧⡤⠙⠸⢸⣦⡀⣀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⢀⡀⣠⣾⠸⠙⣁⣤⣯⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣻⣧⣬⣽⠟⢿⣿⣴⣦⣤⠀⠀⣼⣿⠀⢀⣤⣶⣴⣿⠿⢿⣫⣤⣿⣛⠛⠁
⠀⠀⠀⢰⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣶⣶⣦⣁⣙⣻⡇⠀⢹⡇⠀⠿⠟⠉⣛⣱⣶⣼⣛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤
⠀⠀⠀⠀⡀⣷⣾⣭⣖⣦⢤⢀⢞⣖⡀⣤⡤⠀⣈⣉⠙⠿⠿⠟⠁⠀⠸⠃⠀⠻⠭⣿⠿⣉⣉⠁⠠⣤⡄⣀⣞⣆⡠⢤⣀⣪⣭⣶⡇⡋
⠀⠀⣀⠔⣡⣿⣿⣿⣿⣿⢿⡶⣸⡧⠈⣿⣿⣶⣶⣶⣷⣬⠿⢧⣭⠭⢭⡤⢭⣥⠤⠿⣿⣶⣶⣶⣿⣿⣏⠀⢿⣴⡿⢿⡿⣿⣿⣿⣧⡳⣄⡀
⠀⠾⢅⣿⢿⣿⣿⣿⣟⠀⣽⣇⡿⣹⣠⣿⣿⣿⣿⣿⣿⡇⢺⣀⠇⢸⠀⡗⠸⣿⠀⢶⣿⣿⣿⣿⣿⣿⣿⣦⡈⣇⢻⠁⣨⣿⣿⣿⡿⠿⣆⡽⠆
⠀⠀⠀⠈⠙⠓⠿⢿⣻⡯⡿⣬⠿⣟⣻⡿⠿⠿⣿⣿⣿⣇⠺⣘⣇⠸⢀⣿⣘⣻⣀⣿⣿⣿⣿⣿⡿⠿⣿⣟⡿⢯⡸⡿⣿⣻⠯⠗⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠈⠉⠉⠓⠒⠓⠛⠿⠿⠿⠿⠯⠭⠿⠿⠿⠿⠛⠒⠒⠋⠉⠉⠀⠀⠀⠀⠋⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠐⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠀⠂

⠀⠀⠀⠀⠀⠀⣠⢴⢖⣒⣾⣒⡒⣤⣄⡀
⠀⠀⠀⢀⡴⣏⠔⠚⡇⠀⡇⠀⠸⠓⠪⣹⢦⡀
⠀⠀⢀⣞⡴⠁⠓⢄⠘⡄⡇⢠⠃⡠⠊⠈⢢⣱⡄
⠀⠀⣼⢸⠉⠐⠢⢬⣳⣽⣷⣯⣎⠤⠔⠊⠉⡇⢷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡟⢛⠒⠒⠒⣒⣾⣿⣀⣿⢗⡒⠒⠒⠒⡛⣻⠀⠀⠀⠀⠀⢠⠿⣆⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⢻⡸⡄⠒⠉⡠⠊⡽⣿⣯⣳⣌⠉⠒⢠⢧⡏⠀⣀⣤⣤⠦⠇⠤⠸⠵⠤⢀⠀⠀⠀⢰⢑⡃⠀⡀
⠀⣠⣌⣷⡙⣤⣎⣤⡼⢸⣿⣏⣆⣈⣳⣤⢋⠞⢀⡏⣁⣸⣹⡏⠹⠉⢻⠘⢳⡁⠤⠰⠲⢸⠼⣱⠀⢺⢽⠂
⠀⠀⠀⠁⣙⣦⣝⣺⠅⣼⣿⣯⡸⡟⣫⠴⢋⣠⡾⣟⡩⢁⡼⢹⣧⣀⣌⠀⢸⢿⢲⣾⣾⡶⣖⡿⠀⠎⠀⠃⠀⠀⡤⢀⣢⡄
⠀⠀⠀⣀⣤⣤⡈⢉⣞⣓⣷⣾⣿⣿⣿⣾⣯⣥⣤⡼⡟⢻⠛⠙⣇⠚⣸⠃⠐⠂⠙⢋⣛⠉⠁⣠⣤⣤⣤⣤⣤⣅⣐⣻⣀⣂



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠻⡉⠉⣙⡛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣠⠤⠤⠤⠤⠤⣝⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢰⢄⠀⠀⠀⠀⠀⣀⣠⢶⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣄⠉⠉⠉⠉⠉⠉⣠⡎⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣿⣿⣶⣶⣶⣾⣿⠟⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⣼⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣃⠀⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣶⣄⣀⡀⠀⠀⢀⣀⣠⡶⠋⠀⠀⠙⠲⣄⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠉⠉⢉⣩⠟⠁⠈⠙⢦⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⠔⠊⠉⣄⠈⠉⣉⠓⠲⠤⢤⣤⠤⠶⠚⡿⠁⠀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⣷⡀
⡞⠲⠧⠴⣇⣀⣸⢧⡤⣾⣀⡀⢰⡖⠶⠶⠖⠒⠛⠛⠋⠉⠑⠄⠀⠀⠀⠀⠀⣠⡼⠋⢀⡠⠔⠋⠀⠀⠀⢹⡄⠀⠀⠤⠚⠀⢷
⠓⠶⠦⠤⠭⢭⣥⣄⣀⣀⣀⠉⠙⢷⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠱⡆⠭⣗⡶⢫⡀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣇⠀⠀⠉⠒⣽⡛⡻⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⢸⡇⢸⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⠀⠀⠀⡇⠈⠳⠞⠁⠀⣀⣀⣀⣀⡠⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢻⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢸⠇⡼⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⡿⢀⡇⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠀⠤⠤⣤⣤⣤⡤⠤⠖⠚⠉⠀⠀⠀⢀⡾⣅⣼⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣰⠋⠀⠙⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠙⠓⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢩⠟⠛⠛⠛⠛⠛⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣖⠒⠶⠧⠿⣤⠖⠛⠲⣿⠀⠀⡀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⣼⠶⠋⠁
⠀⠀⠀⠀⠀⠀⡼⠁⣰⠽⣄⠀⠀⡾⠁⣠⢤⡀⢹⡙⠓⠻⠦⠤⠼⠷⠖⠛⠲⠶⠤⣤⣤⣀⣠⠴⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢧⠀⠙⠋⢙⠗⢾⡇⠐⣇⡼⠃⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡔⢢⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠋⠀⠘⣇⠀⠉⠀⢀⡾⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠶⢤⠤⢼⡇⠘⢦⠊⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣷⣄⣀⣀⣴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡆⠀⠁⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢀⣤⡤⢦⢄⡀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠈⠥⣂⡁⢀⡴⠛⠉⡉⠉⣿⣠⣔⠷⠚⠀⣴⣦⡄⢢⣶⡶⠖⢲⣶⣶⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠴⠾⣿⣿⢿⢦⡬⠼⠏⠿⠿⠿⠿⣔⣘⣿⣿⡟⠩⣧⡄⠀⠀⠀⢒⣢⠦⠰⠶⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠀⠀⠀⠀⠀⠀⡀⢀⠠⢄⡀⢠⠶⠞⢛⣋⣹⣿⣏⡠⢠⡜⠁⢐⣤⣶⣯⡵⢶⡿⠤⣤⡖⠀⠀⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⢀⣤⣶⣿⣿⣿⣿⢿⣷⣿⣟⣛⣭⣿⣿⡿⣿⣿⡿⢀⣼⣿⣿⣿⡟⣳⣟⡀⠀⠀⠇
⠀⠀⠀⠀⠀⢠⣄⣀⢀⣈⣙⣾⣿⠿⠛⠿⠟⢻⣿⣿⣿⣭⣽⣽⠿⠛⢫⣥⣤⣿⣧⣞⣿⣿⣿⣋⣿⣿⠛⢿⣏⠀⠐
⠀⠀⠀⠀⠀⣨⣽⣯⣭⣝⣏⣻⣦⣤⣤⣤⣶⣿⣷⣦⡴⣶⣿⣍⡀⣠⣾⢿⣿⣿⣽⣿⣿⣿⠟⣹⣿⢷⣶⠾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠠⠄⣠⣴⠀⡘⣿⢉⣻⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣵⣿⣿⣯⢀⣿⣟⣀⢥⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠈⣹⠿⣿⣿⢸⣟⣿⣿⣿⣻⣾⣿⣿⣾⣽⠃⠀⠀⠀⠘⠉⠂⠲⠴⠠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⣿⣬⣧⣾⢻⣿⣿⣿⣿⣿⣷⣿⡏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠊⠁⠒⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⠟⣯⣿⡛⣿⣿⣿⣿⣿⣯⢽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣄⡏⢰⣿⣿⣿⣿⡿⣱⣿⡾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠧⣼⠁⢼⡿⠻⢾⣿⣾⣿⠈⠀⠙⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⡟⠀⠀⠻⡌⢿⣏⣼⢷⣦⣀⠊⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠀⢿⡿⢀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠏⠁

⠀⠀⢀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⢀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠁⠣⢀⠀⠀⢀⡀⠀⠀⠀⣸⣤⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣶⣿⣶⣼⣿⣷⣧⣾⣼⣿⣷⢂⣤⡀⠀⠀⠀⠀⡠⠔⠋⠀⢠⠖⠋
⠀⠀⠀⠀⡄⢄⠀⠀⠀⠀⠀⠀⢤⡈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣋⡤⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣤⣞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠂⡈⠁
⠀⠀⠀⠠⠤⠤⠴⠒⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠐⠚⠙⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣱⣿⣿⣿⠻⠿⠋⠉⠿⢻⣿⠃⣼⣿⣿⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢈⣩⣽⡒⠀⠘⠋⠀⢘⣩⣤⣼⢿⣿⣿⠛⠁⠀⠀⠀⣀⠀⢈⡀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠈⠁⠛⠋⠀⠀⠀⠀⠉⠀⠛⠉⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠁⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⡤⠄⠀⠀⠠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⡀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣦⣄⡑⠀⠠⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⣈⣤⣾⣷⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣴⡶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠟⠛⠛⠓⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⡀
⠀⠀⠀⡔⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠔⠒⡄
⠀⠀⠀⠳⠤⠤⠴⠄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠴⠃
⠀⡄⠂⢀⠀⠀⠀⠀⠀⠀⠴⠛⠻⠟⠛⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⣿⠷⢦⡀⠀⠀⠀⠀⡀⠀⣀⠀⠂⡄
⠀⠈⠉⣠⣴⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠉
⠀⠀⠞⠉⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠙⠓⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠤⠤⠤⣄⣀⣀⣀⣈⣉⡉⣉⣉⣉⣁⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⠔⠋⠉⠉⠙⠻⠍⠀⠀⢙⡷⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⢤⣀⣀⣀⣀⣤⡤⠶⠚⠋⠀⠘⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂

⣿⠋⠙⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⡿⡏⡌









⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣩⣩⣭⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⡛⢦⣀⣀⣀⣀⣠⣞⡉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠀⠉⠉⠉⠉⠉⠘⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⠿⠽⠆⢠⡄⠘⠉⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠠⣤⠼⠧⡤⠄⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⣴⠀⠀⠀⠀⣧⠖⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⠀⡴⠚⠛⠛⠋⠉⣿⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡼⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢀⣼⠈⢳⡀⠀⠀⠀⠀⠀⠀⣷⢰⠇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠐⣿⣀⡀⠈⠳⡄
⠀⠀⠀⠀⠐⠟⢃⣠⣴⣿⣴⣶⣿⠟⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠓⢿⣤⣶⡴⠿⠖⠲⠶⠶⠶⠖⠋⠁⠈⡿⠀⠀⡷
⠀⠀⠀⠀⠀⢰⣿⡿⣟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣾⢹⣦⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣌⡓⠿⢶⣤⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⣠⠶⠛⠉⠉⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⣀⡀⠀⠀⣀⣀⣀⠀⠘⣷⠞⠁⣀⣀⠀⠀⣀⣀⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣼⠟⢻⣿⣷⣾⣿⣿⣿⣧⠀⠸⣇⣾⢋⣿⣿⣾⣿⣿⣿⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣵⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⠀⠀⢀⣀⣀⡀⠀⠀⠀⢻⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⡇⠘⠻⣿⣿⣿⣿⡿⠋⠀⠀⢀⡿⠀⠀⠀⠴⠞⢻⠷⣦
⠀⠰⠟⣿⡉⠉⠁⠀⠀⠈⢧⡀⠀⠀⠈⠛⠿⣿⣿⠿⠛⠁⠀⢀⡾⣧⡀⠀⠀⠉⠉⠁⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣼⠀⠈
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠈⠙⠶⣤⣄⣀⣀⣠⣤⠞⠋⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡛⠛⠛⠛⢛⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣷⣾⢡⡶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠶⡟⣻⣦⣿⣧⣙⣲⣯⠶⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠐⠦⣼⣸⢿⣵⠊⠁⠀⠈⣙⢯⣔⡃⠄⠀⠀⠀⠀⣷⢀⡟⠀⡀
⠀⠀⠀⢀⡀⠈⢷⣠⡟⠀⠀⢸⡛⣦⢼⡿⢋⡀⠀⠀⠀⢊⡥⢄⢻⡿⠗⠀⠀⠠⣀⣸⣾⡶⠋
⠀⠀⠀⠀⠉⢓⣬⡟⢻⡄⠀⠐⠒⣨⡿⠁⢾⣿⠀⠀⠀⡋⣶⠀⠘⣷⠀⠀⠀⠀⠈⣿⡼⠋⠛⠒
⠀⠀⠀⠀⠛⠛⢉⡿⠛⢷⢦⡀⠀⣿⡇⠀⠠⠤⠂⠒⠀⠘⠒⠁⠈⣿⡆⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠣⡱⣄⠹⡅⡄⠀⠀⣶⡟⠿⣷⡄⠀⣼⠿⠁⢀⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠳⣝⢌⡲⣜⡛⠾⠟⢛⣠⡾⢋⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠘⠋⠹⣿⠮⠵⡶⢾⣁⣴⠏⢉⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⠀⠘⢧⣼⠁⠀⠙⠃⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠉⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠀⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⣩⢿⣖⠒⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⣴⠏⠸⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣰⡟⠀⠁⠀⠀⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠢⣤⣤⣾⠟⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣰⣶⣄
⠀⠀⠀⠀⠀⠀⠸⠿⠶⠋⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣏⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠔⠂⠂⡀⠀⠀⠀⠀⢀⡴⢟⡍⢈⣠⣤⣾⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠁⠐⡀
⠀⠀⡐⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⠏⠀⠀⡼⠋⠀⠀⠙⢿⣷⡌⢧⡀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠲⢠⡾⠁⠀⠀⠀⠀⠀⠹⣿⡞⢷⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠤⡀⠀⣀⠀⠉⢀⡟⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⡇⠀⠀⠀⠀⠀⠒⢄⣀⠔⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠤⣧⠀⠀⠀⠀⢸⣷⣄⠀⠀⢹⠇⣿⢠⠐⠂⡀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡄⠀⠀⠀⣼⢿⡆⠙⢦⡉⢰⡇⠸⣀⡠⠃⠀⠀⠀⠁⠒⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⢷⡄⠀⠀⡏⠘⣿⡄⢈⡇⠸⡇⠀⠈⠀⠀⠀⠀⠁⠀⠐⠂⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠘⢧⣠⣄⠹⣄⣀⡇⣿⣿⣷⣼⠁⠀⢷⠀⠀⠘⠦⢀⠀⠁⠀⠀⠀⠁⠂⠦⣄⡀
⠀⠀⠀⠀⢋⠀⢈⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣮⣉⣓⣏⣩⣴⡏⠀⠀⠸⡄⢠⠓⠋⢒⡄⠀⢉⣀⠐⠀⠀⠀⠀⡉⡠
⠀⠀⢀⢷⠀⢹⠁⠀⠀⠀⡐⠈⠀⢀⣼⡠⢾⡿⣿⣯⣷⠉⡼⠁⠀⠀⠀⢳⡀⠀⠀⠀⡇⠉⢀⠘⠋⠉⠳⠒⠦⠄⠥
⠐⢀⡞⠈⣧⢸⢈⣆⠈⢇⠜⠀⠄⠿⠿⣗⣿⣩⢻⡷⣟⡿⠁⠀⠀⠀⠀⠀⢻⡀⠀⠠⡇⠀⠈⠀⠀⠀⢠⡉⠁⠀⠑
⢀⡼⠀⠀⠘⣿⠹⠘⠤⠀⡁⣴⠯⢨⡧⢈⡟⢪⣻⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠁⠀⠀⠐⠀⠀⠘⠃⠓⠦⠂
⢹⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡆⠈⡷⢚⡛⠉⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⢽⡦⠄⢒⠀⠀⠀⠀⠐⢀⡀
⠘⢦⣀⣀⣠⠎⠀⠉⠀⠀⠀⠐⡄⠀⣀⣠⡀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⢠⡄⠠⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠟⠛⠔⢁⠀⠀⠂⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠄⠀⠠⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠤⣤⣴⣤⣶⣶⡿⢶⣶⣶⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠴⠿⠿⠿⠿⠿⠷⠾⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢼⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢤⣀⡀⠀⠀⠀⢠⣀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⠃⡈⠀⠉⠉⠒⣒⡲⢖⣶⠤⠭⣥⣀⣩⣄⠀⠀⠀⠠⡄⣶⣾⡿⠛⠁⢡⣼⣿⣖⡤⣄⡀
⠀⠀⠀⠀⠀⠀⣼⠸⣷⣮⣉⠁⠀⡴⠋⢉⡡⠖⠉⠀⢀⣤⣔⠛⢉⣉⣻⣿⣛⣓⣙⣿⠝⠒⠉⣠⣿⠋⣿⣷⠀⠈⣿⡅⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣾⡿⠇⠀⣿⣯⡭⣥⣄⠐⠊⠑⠠⣈⣍⢝⣡⣨⡄⠈⣭⣥⣤⣶⣾⣿⡿⣧⣾⠿⢿⠄⠀⠚⢀⡷⢆⠩⠉⠠⠖⠤
⠀⠀⠀⠀⠟⢿⣿⡯⣀⡙⠳⠄⠀⡀⠀⠉⠙⠛⠿⣶⣶⣶⣶⣾⠿⢿⡟⠁⠀⠿⠟⣷⠾⠿⣏⡉⠋⢰⠴⠀⠀⠀⡰⠃⠁⡚⠃⣆⡀⠀⠀⢀⣶
⠀⠐⠢⢤⣐⣾⣿⡇⠈⠉⠛⠇⢠⣉⣤⡀⠀⠀⠀⠀⠀⠉⣁⠠⠤⠊⠀⠀⠀⣠⢔⣁⣴⡀⠘⢁⠀⠀⠀⢀⠤⡞⠁⠀⠴⠈⠚⠉⢠⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠠⣤⣀⣀⣀⣀⣤⡤⢄⣩⢥⢝⣼⣿⣿⡥⠆⠀⠀⡠⡾⠁⢠⡇⡀⠈⡐⢖⠀⣰⣯⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣁⣤⣴⣶⣶⣾⣿⡟⠛⡿⢿⣇⣀⣠⡴⡿⠃⠁⢀⡾⠋⡀⠀⢯⡵⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢿⠿⠿⠯⣙⣻⠿⣿⣿⠋⠉⠉⢸⠇⠈⢻⣆⠀⢈⢁⡶⣴⣏⣡⣤⣈⠀⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠭⣍⡓⠈⢩⣀⡀⠉⠁⠀⠀⠙⢦⡀⠀⢢⡈⢻⣿⣟⠛⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠙⠒⠦⡄⠀⠀⠙⣦⠾⠯⠟⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠃⠀⠀⠄⠀⢼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢠⣤⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⢀⡤⠾⠀⠀⠀⣀⡀
⠀⠋⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠉⠙⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠰⠀⠀⠀⠀⠆⡀⠀⠀⠄⠀⠠⢶⢶⣤⣀⣠⣀⣠⣤⡶⠟⢻⠉⠀⠃⠀⠀⣀⠙⢠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠘
⠰⠀⠸⠇⠀⡇⠛⠀⢀⠂⠀⠀⠀⠀⠀⠉⠉⠩⡁⠈⠁⢀⠌⠀⠀⠇⠀⠀⠿⢠⠙⠁⠀⠌⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠈
⠀⠑⠀⠀⠔⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠒⠀⠀⠀⠀⠀⠀⠉⠓⠀⠙⢒⠀⠀⠀⢠⠴⠒⠀⠀⠐⠢
⠀⢀⣠⠾⠛⢁⡐⠳⣄⡀⠀⠀⢀⣤⣤⣀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠠⠂⠀⠀⠐⠂
⠐⡏⠁⣤⠄⠨⠄⡤⠀⠈⡃⠀⠀⠀⠀⠉⢻⡆⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠠
⠀⢳⠀⠀⠀⢠⡄⠀⠀⢠⠁⠀⠀⠖⠒⠬⣆⠀⠣⠘⠃⠀⠄⠀⠀⠡⡀⠀⣠⡄⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠈⠁
⠀⠀⠑⠄⠀⠃⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⣀
⠀⠀⠀⠐⠶⠐⠁⠀⠀⠀⠀⠀⠀⠀⢘⠉⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠴⠂⡘⠛⠦⣄⡀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠀⠀⠀⠠⢡⣶⢦⡄⠁⣴⣶⠆⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋⠀⠀⠴⠱⠆⠀⠀⡉
⠀⠐⠀⠀⠠⠇⠀⠀⠐⠖⠀⠀⠀⢀⠤⣿⡷⠀⡄⠭⢿⠟⠀⠀⠀⠆⠀⠒⢀⡆⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⣀
⠀⣰⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠤⠄⡀⠀⠙⢛⡢⠀⠀⠀⢘⠩⠅⢀⠀⠈⠍⢃⠀⠀⠀⠀⠨⠀⠠⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠟⠛⠷⠆⠀⠀⢄⣀⣒⡂⠀⠄⠀⠀⠀⠀⠀⢀⠂⠀⢀⣤⢶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣀
⠃⠀⠐⠛⠀⠰⠀⠘⠛⠉⣀⠂⠀⢠⠀⢉⡀⠀⣣⣼⡁⣰⡇⠀⠘⡀⠀⠈⠛⠸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠈⠣⣙⣋⠼⠓⠉⠉⠚⠀⠀⠀⠻⢄⢀⡴⠋⠲⠤⠚⠁⠀⠀⠀⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠲⢄⡀⡴⠉⠀⠀⠀⠀⠈⢢⡀⣠⠴⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠈⠑⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠋⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢀⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⢠⠞⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣀⢀⣠⠾⣄⠀⠀⢀⡠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡴⠋⠀⠀⠁⠀⠀⠀⠉⠉⠉⠀⠈⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⣀
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠠⡤⠔⠋⠀⣰⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⡼⠉⠀⣠⣙⢆⠀⠀⠀⠀⠀⠀⠀⡴⠚⢉⠛⢷⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⢇⠀⠸⢿⠟⣸⡠⠒⠉⠉⠑⢦⣸⠀⢸⣿⡷⠀⡟⠂⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⠓⠤⠤⠚⢹⠁⠀⠀⠀⠀⠀⡏⠣⣄⣀⣠⠜⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠶⡏⠀⠀⠀⢀⡴⠒⠒⠒⠺⢆⠀⠀⠀⠀⣠⣃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⣀⣀⣀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢉⠓⢒⠋⠁⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠘⣆⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠈⠉⠒⠲⠤⠤⠤⠤⠤⠖⠚⠉⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢄⣀⣀⣀⣀⣀⣀⠤⠤⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠤⠶⠶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠈⠑⠦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⢱⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠜⠁⠀⠀⠀⠀⠠⠤⠂⣡⠶⣄⠀⠀⠀⠀⠀⠀⡖⠲⡄⢀⢀⡀⠈⠧⢤⡆
⠀⠀⠀⠀⠀⠀⠈⠦⣄⣀⠀⠀⠀⡤⠠⠀⠔⠁⠀⠇⠀⠘⠀⠀⡏⠉⡏⠈⠀⠀⠓⢸⡡⠎⠀⢠⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢎⣁⣀⡀⠀⢦⣀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⣵⠀⠀⠀⠀⠀⢱⠀⠀⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢒⡶⢀⣀⣀⡱⢄⡀⠀⠀⠀⠀⠀⢰⠈⣿⠀⠀⠀⠀⠀⢰⠗⠒⠒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠀⠀⣩⠗⠦⣄⡀⠀⠀⠑⠂⠀⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠉⠀⠈⠉⠉⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⡼⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣳⠀⠀⠀⠀⠀⢰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⣏⢘⠄⠀⠀⠀⠀⢌⢙⡞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⠀⠀⠀⠈⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣀⣤⣄⣀⣠⣄⣠⣄⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠇⠀⠀⠀⠀⠀⠀⠧⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠉⠙⠿⣿⣶⣤⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠙⠻⢿⣷⣶⣶⣶⣶⣾⡿⠿⣿⣇
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⣼⣿⠟⠉⠁⠀⠀⠀⢻⣿
⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⣠⡆⠸⣿⡇
⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣷⡙⠻⣶⣤⣀⠀⠀⣀⣀⣠⣤⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⣾⣿⡇⠀⢿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣷⡀⠈⡙⠿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠊⠁⠀⠀⠀⣀⣴⡿⠟⠁⠈⣿⡀⠘⣿⣇
⠀⠀⣿⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠉⠛⠿⣦⠹⣧⠀⠀⠀⠀⣀⣴⡾⢿⣉⡀⠀⢠⠀⠸⣧⠀⢹⣿⡄
⠀⠀⠸⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢷⡀⠀⣿⡇⠈⠀⢠⣴⣾⠟⠋⢤⣬⢿⣿⣷⡏⠀⣀⣹⣧⠀⢻⣿
⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⣸⣿⡆⣿⣇⠀⠀⣾⣿⠀⠀⠀⠀⠉⠂⠋⠉⠓⠀⠈⢿⣧⣀⣾⡟
⠀⠀⠀⢸⣿⡆⠀⠀⠈⣿⣿⠀⠀⠀⠀⢸⣷⡀⠀⠀⣙⣛⠁⢸⣿⠀⢀⣿⡏⠐⢦⡀⠀⢀⣰⣶⠦⠄⠀⠀⠀⠻⣿⡟
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠹⣿⣄⠐⣿⠙⣧⢠⣿⠀⣸⣿⠇⠀⠈⢻⣶⡿⠻⣇⢠⣶⣄⠀⠀⠀⢻⣿⡀
⠀⣤⣶⠿⠟⠃⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⠻⣷⣿⠀⣿⣿⠏⢰⣿⡟⠀⠀⠀⠸⠟⠁⠀⠈⣃⣈⡉⣷⣄⠀⢸⣿⡇
⢸⡿⢀⣤⣤⣤⣤⣤⣴⣿⣿⣧⣄⡀⠀⠀⠀⣿⡄⠘⠛⠀⣿⠇⣠⣿⡟⠀⠀⠀⠀⢀⣠⣴⣞⣛⣉⣍⣉⣻⣿⣷⣼⡿
⢸⢇⣾⠟⠛⠛⠉⠉⠉⢙⣻⣿⣿⣿⣷⣦⣄⠈⠿⣶⣶⡾⠋⡴⠟⣿⣷⠀⠀⠀⠀⠉⠉⠙⠓⠿⢬⣍⣋⣙⣹⢿⣧
⣼⡿⠃⠀⢀⣠⣴⣾⡿⠿⠿⠛⠛⠻⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠐⠶⢤⣤⣤⣈⣉⣁⣼⣿
⠁⢀⣴⣾⣿⣿⣭⣄⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠀⠀⠀⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣧⡀
⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠉⠛⢿⣧⡀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣤⣀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⣄⡀⠀⠘⣦⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠙⢿⣿⣷⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣈⣻⣷⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣆
⣿⣷⠿⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣿⣿⣷

⣀
⣿⣀⣀
⣿⣿⣿⣷⣀
⢻⣿⣿⣿⣿⣀⠀⠀⠀⣠⣄
⣶⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣦⣄⣄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⣄⡀⣀
⡿⣿⡄⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⢴⣤⣄⣀⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣄
⠋⠀⠀⠈⢸⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣾⢻⣿⡿⣿⡟⠏⡟⣿⣿⣿⠛⣿⣿⣯⢽⣿⣯⣿⣟⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⢾⡟
⠀⠐⠀⠀⠀⢸⠟⠹⣿⡇⠱⠆⠸⠻⠏⢩⡝⠛⠏⢁⠃⠀⠈⠓⠠⠿⠶⠤⢬⣽⢓⠞⡌⣻⢻⠥⣽⣿⣿⣟⡋⠀⠩⢷⡻⠽⠟
⠀⠀⠀⠀⠀⠀⠃⠐⢪⡇⠠⣤⡃⣷⠅⠀⠁⢢⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁⠐⠃⡁⠂⣀⠋⣀⢒⡐⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠀⠈⠀⢠⠂⢀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⠓
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⡀⠀⢀⣀⣠⣤⣶⣶⣿⣟⣉⣉⣀⣤⣀⣀
⣿⣿⢛⠿⡛⠛⢿⠟⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠿⠟⠃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠴⠄⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠉⠁⠉⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⡀⠀⠀⠀⠀⠀⢀⡸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠒⠐⢶⠺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠘⢂⣴⠀⠀⠀⠀⠀⣀⣠⢄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠈⠀⠘⠣⠀⠀⠀⠀⡶⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠙⠃⠀⠀⠀⠀⠀⢴⣆⠠⠦⠄⣀⡀⠈⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣼⠆⠀⠀⠀⠀⢸⠟⢿⣶⡄⢠⠤⣶⠄⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠇⠘⣄⠀⠀⠀⠀⠳⡄⢸⡀⠀⣠⠉⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠰⡶⠀⠀⠈⠳⣄⡀⠀⠀⠀⠉⠉⠉⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⡛⠛⠛⠻⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⣀⣀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡗⠲⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣟⠁
⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠠⢤⠄⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠚⠁
⠀⠀⠀⢀⠜⠀⠙⢦⡀⣀⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠁
⠀⠀⡰⠋⠀⢀⠴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡜⠁⠀⣴⠎⠀⠀⠀⢀⣈⡀⠀⠀⠀⠀⢀⣠⣼⣇⢀⣠⡤⠀⠀⠀⠀⡆
⡆⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠠⠈⠀⠉⠙⣿⣿⣷⠆⠀⠀⠀⠁
⠷⠠⠤⠠⠤⠤⠤⠀⠀⠰⡎⠉⠉⠉⠀⠀⠈⠁⠀⠼⠿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠠⠤⢤⡤⠤⠁⠸⠤⣤⠤⠤⠤⢤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⡀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣾⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⣿⠿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡌⠓⣤⡀⠀⠀⠀⠀⠀⢰⣦⣀⣠⣾⣿⣿⠃⠀⢸⠓⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡔⠉⠙⢠⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⢶⣶⣎⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⡀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠙⢲⡀⠀⣰⠞⠛⠛⠋⠙⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣔⣩⠤⣤⠀⠀⠀⠹⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠛⢉⡴⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⢀⢈⡗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⠛⠁⠀⠀⠉⠓⠦⣄⣀⠐⢶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣄⡉⠳⣄
⠀⠀⠀⠀⢀⡀⠀⠀⢹⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣓⠮⣷⣄
⠀⣠⠶⠛⢉⢉⠙⠲⣼⠃⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠠⡀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠙⠂
⢰⡇⠀⠀⠉⠘⢳⠀⠹⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠻⣦⢹⡀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⢰⣤⣀⠀⠙⢷⡀
⠈⠛⠲⠤⠤⠤⢤⡄⠀⠀⠀⣀⣤⣴⠲⡖⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧⠁⠀⠀⠀⠀⠀⣀⣀⠘⣆⠙⠳⢦⣽⣏⠓⢦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣾⣿⣿⣇⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⢀⣀⣉⣉⣉⠀⠸⡆⠀⠀⠀⠉⠀⠀⠈⠛⢾
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⠀⠻⣇⣹⣾⠀⢹⣆⠀⠀⠀⢷⠀⠀⠀⠀⠀⣿⡴⠚⠉⠉⠀⠈⠉⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢧⣈⡷⡁⢸⢦⡀⠀⠘⣆⡀⠀⢠⢀⣿⠘⠀⠀⠀⠀⢀⣠⠞⠀⠀⢧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡄⠀⠀⠙⢦⣼⠧⣄⣹⢦⡀⠀⠉⠓⢶⡛⠙⠲⠴⠶⠖⠛⠉⠀⠀⢀⣠⠼⠛⠉⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⠻⣦⡀⠀⠀⠻⣦⣼⠳⣦⣽⡷⣤⣜⡂⢃⠀⠤⠆⣀⠀⢠⠢⠀⢀⢤⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠀⠐⣿⠻⢦⡐⠈⠀⠙⠓⢾⣟⠛⢮⣿⠛⠶⣦⣌⣀⣁⠀⠈⠈⠀⢈⠉⠉⠙⡏⠁
⠀⠀⠀⠀⣠⠞⠀⠀⢰⠃⠀⠀⠸⣆⠀⠙⠦⣀⠀⠀⠀⠈⠙⠺⢤⣉⣳⠟⠤⢤⣼⣉⣉⡟⠉⠈⠀⠀⠀⣿
⠀⠀⠀⡴⠃⠀⠠⠽⡟⠀⠀⠀⠀⠙⢆⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠉⠙⠒⠾⠧⢤⣯⡀⠀⠀⠀⠀⣰⠇
⠀⠀⡼⠁⠀⠀⠀⣸⠃⢀⣠⣤⣶⣚⡿⠃⠀⠀⠀⠀⡼⠻⡝⠛⠲⠦⢤⣤⣀⣀⣀⣀⣈⣀⣀⣠⡤⠞⠋
⠀⡼⠁⠀⠀⠀⠀⣯⠶⢋⣿⡿⠟⠛⢦⣀⣠⠖⣲⠴⠗⠒⠛⠒⠶⡞⣽⣶⠀⠀⠈⠈⠉⠉⠁
⡼⠁⠀⠀⠀⠀⠀⠛⠋⠉⠀⠙⠦⣄⡀⠉⠉⠈⡉⠓⠶⣄⠀⠀⠀⡗⢙⠿⠶⣦
⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⠀⠠⡈⠓⢦⡼⠀⠀⠀⠈⠋⠙⢺⠃
⠀⠈⢳⡈⠙⠓⠶⠦⣤⣄⣀⣀⣀⠀⣼⢈⠳⣄⣸⠦⡼⠃⠀⠀⠀⠀⡴⠒⠻⡆
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢉⣉⠉⣿⢸⡀⠀⠁⠀⠀⠀⠀⠀⢰⡋⠙⢶⡞⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠐⠀⠀⠀⠿⢿⡈⢧⠀⠀⠀⠀⠀⠀⠐⣺⡛⠛⠻⣄
⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠳⢄⠀⠀⠀⠀⠀⠹⡟⠲⠶⠋
⠀⠀⠀⠀⣼⢷⣶⢤⣤⣀⣀⣀⢀⣀⣠⣏⣀⣙⣦⣤⣀⣀⢀⣀⡴⢃
⠀⠀⠀⠀⠙⠶⢍⣛⠳⠤⣬⣭⣏⣁⣀⣀⣭⢿⣩⣃⣈⣉⡿⠁⢀⡾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⢸⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣣⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⡲⠞⣁⡴⠖⠛⠉⠉⠉⠉⠉⠛⠓⠦⣄⡀⠀⠀⠉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⣠⠖⠉⠀⠀⢀⣾⡵⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⡶⠒⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⣻⠁⣴⠷⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⢿⡄⠀⢀⡞⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡰⠋⠀⡇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠃⠀⠀⠀⢀⡿⢄⡀⠀⠀⠀⠀⢀⡠⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⢀⡀⠙⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠙⣦⠀⠀⠀⣾⠁⠈⢧⠀⠷
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣉⠭⢍⡀⠀⠀⢿⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠓⠦⡀⢸⠀⠀⠀⠀⠈⠳⣄⠀⣼⠋⠉⢢⣏⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠙⢦⠀⢸⠀⡴⠋⠀⠀⠙⣆⢀⣤⢤⣄⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⡇⠘⣇⠀⠀⣠⣀⣤⣈⡄⣏⠀⢀⡏⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣷⡀⠀⠸⣇⠤⣇⢧⣠⣤⣤⣄⠀⢾⠀⠀⠘⠃⠀⠠⠴⠖⠛⠛⠛⠿⢿⡷⠀⠙⢦⡜⢡⡏⠀⠈⠁⢉⡒⠋⠀⠀⢀⡜
⠀⢰⡆⢀⡴⠞⠓⢶⠋⠀⢀⠙⠀⠀⣇⠀⢸⠛⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⢸⠁⠈⢳⡖⠓⡆⠘⠃⠀⢠⡴⠋
⡄⠀⠀⢸⠀⠀⢀⣼⡀⠀⢹⡄⢠⠀⠉⠀⡇⠀⠀⢀⠔⠚⠉⠉⠉⣱⠶⠤⢄⣀⡀⢀⣀⡡⠏⠀⠉⠁⠀⠀⠀⣤⠤⠚⢀⡜⠀⠀⠀⠀⢸⠁
⢱⡀⠲⡀⠳⠖⠁⠀⠙⣶⠚⠀⠘⢦⠀⠀⣇⠀⠀⠻⡀⠀⠀⠰⠋⠀⠀⠀⠀⢀⡽⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠎⠀⠀⠀⠀⠀⡸
⠀⠹⢄⠙⠆⠀⠀⠀⠀⠈⠛⠒⡞⠀⠀⠀⠘⠦⣀⠀⠙⠲⠤⣀⣀⣀⠤⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣀⠜⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠈⠉⠒⠶⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⣰⠖⠒⠙⠋⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⢀⣇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣐⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣌⠙⠲⠦⢤⣀⣠⠤⠤⠖⠛⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠲⣶⠶⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⢤⣀⡀⠀⠀⠀⣘⢻⡀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡏⠉⠛⠶⣄⡟⠘⡇⣠⡴⠶⠖⠲⣞⣸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠳⣄⠀⠀⡟⠃⠀⠛⠒⣷⠀⠀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣤⣤⣬⠦⢤⣄⢸⡇⣸⡳⡄⠀⠀⠀⣷⠶⠚⠛⢛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⣭⡟⠁⠀⣎⠩⠷⠤⣒⡶⠾⠷⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠢⣤⡀⡀⠀⠀⠀⠀⣨⠇⣀⣠⠔⣫⠃⠀⠀⢸⣇⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢌⣉⣑⣒⣒⣚⣉⣉⠭⠔⠊⠁⠀⠀⠀⠈⢯⡛⠧⣄⣀⠀⠀⠀⢀⣠⠼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⠉⠉⠉⠁⠀⣤⠖⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡛⣃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⡠⠔⣤⠞⢡⡇⠸⢦⠙⠦⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢠⣾⢼⠧⡧⣷⢰⠀⠀⠀⠀⠀⠀⣠⢸⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣿⣽⣯⣴⣴⣍⣴⣴⣥⣶⣴⣥⣦⣴⣿⣯⣽⣶⣿⠁
⠀⠀⠀⠀⠀⠀⡜⢸⡇⢊⣆⢿⢸⡄⡄⠀⠀⠀⡗⣾⠇⢟⢸⡟⣗⠚⠀⠀⠀⠀⠀⠀⢿⢿⡿⡿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣻⡿⣿⢿⣿⡿⡿⢿
⠀⠀⠀⠀⠀⢠⢃⡿⢀⣼⣿⣌⠁⢧⠠⠀⠀⡰⢡⡟⢀⣾⣧⡑⠹⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⣿⣿⣿⢿⣿⣿⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠈⠀⠚⠛⠛⠛⠂⠈⠓⠁⠘⠁⠁⠀⠛⠛⠛⠛⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷⣿⡿⠟⣿⣿⣿⣸⣿⣿⣿⠀⠀⠀⢀⣴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⣿⣿⣿⣧⡀⣿⣿⣿⡿⠋⠉⣛⣴⡦⣤⣾⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⡀⢀⣀⣤⣾⣤⣀⣀⠀⢀⡶⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠯⠀⠘⠏⠀⠋⡙⠛⠁⠀⢰⣿⠸⢃⣿⠛⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡾⢿⠉⠁⠀⢹⠁⠀⠉⢛⡾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡟⢠⣶⡆⣦⣿⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠠⠤⣄⣀⡼⠋⠀⢈⣷⡒⠋⢻⠋⠓⢲⣞⠀⠈⠳⣄⣀⡤⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣿⣷⣾⣿⣷⣿⣿⣀⣀⣄⣀⣀⣀⣠⡀
⠀⠀⠀⠀⣤⣤⣬⡟⠳⢤⣴⠁⠈⢳⡀⣸⠀⣠⠏⠀⢑⣤⠶⠛⣧⣤⣤⠀⠀⠀⠀⠀⢲⣼⣦⣾⣦⣿⣴⣷⣼⣧⣾⣦⣿⣴⣷⣼⣧⣾⣦⣿⣴⠆
⠀⠀⠀⠀⠈⠉⣼⠁⠀⡸⠉⠙⠦⢄⣱⠼⢔⢉⡤⠞⠉⠸⡄⠀⢹⡀
⠀⠀⠀⠶⠶⠶⣿⠶⠒⡷⠒⠒⢒⠒⣇⠀⢀⢗⢒⠒⠒⠒⡗⠒⢺⡗⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠸⣶⣶⠆⢹⡄⠀⠸⣀⡤⡮⣬⡖⢩⠑⣮⠾⡶⢤⣰⠁⠀⣼⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⡧
⠀⠀⠀⠀⠀⠀⠀⣳⡴⠚⠙⢤⣍⡟⠀⢺⠀⠘⣯⢥⡴⠉⠓⣶⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⢻⠁
⠀⠀⠀⠀⠀⠒⠋⠁⠹⣦⡀⣜⡺⠓⠦⣾⠤⠖⠻⣫⡄⣠⠞⠁⠈⠉⠁⠀⠀⠀⢀⡠⣠⢎⡏⣧⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⢸⡆
⠀⠀⠀⠀⠲⠶⠾⠀⠀⠈⣹⣢⣧⣄⣀⣸⣀⣀⣤⣗⣻⠁⠀⠀⠘⠿⠟⠀⠀⠘⠛⠛⠛⠛⣿⡟⠛⡟⠛⠋⠀⠀⠀⠀⡠⠊⠀⠀⠀⡀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡯⡝⠀⠀⠀⢹⠉⠀⠀⠸⣖⡷⠄⠀⠀⠀⠀⠀⣠⡄⣴⠀⣤⠀⣿⣇⠀⢳⡀⢀⣄⠀⢤⣾⣁⡀⠀⠀⠀⡇⠈⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢽⠃⠀⠀⠀⠉⠁⠀⠀⠀⢯⢵⡖⠀⠀⠀⠀⠀⠙⠃⣿⡀⡷⢀⣿⡟⡶⠈⢿⣄⠛⠁⠈⠻⣿⣮⣭⣍⣒⠃⠀⣄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡟⠀⠀⠀⠘⠿⠟⠀⠀⠀⠘⣏⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢚⣿⣇⠙⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢺⡀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠿⠿⠿⠿⠿⠿⠦⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⡻⢿⠛⡻⠿⡿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣁⡖⠀⠀⠀⠀⠀⡠⠀⠰⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠠⢈⣁⢆⠾⠀⠀⠀⠀⠀⠀⠀⠀⣘⠂⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣦⠀⢋⡌⠀⠉⠑⠂⠀⠀⠀⠉⢉⠀⣨⠍⣀⡌⣀⡠⠤⠴⠶⠌⠙⠛⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠁⠁⢀⡀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐⠀⠀⠠⠀⡸⠀⠠⠤⠀⠀⠀⠠⠀⠤⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠠
⢀⠀⢀⠀⢀⣀⠀⠈⠀⠀⡀⠀⠀⢀⠄⣄⡀⢧⠄⠀⠀⢀⠀⡀⡀⠠⠀⡀⠄⠄⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⡀
⠀⠉⠛⠠⠂⠀⠀⠡⠰⠀⠔⠐⠐⠶⠁⠈⠉⠁⠑⢪⠤⢢⠔⠉⠩⠢⠴⠂⠀⠀⠈⠂⠀⠀⠐⠸⠊⠐⠁⠲⠉⠄⠆⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣶⣤⣄⠹⣿⣿⣿⡿⢋⣠⣤⣦⣤⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠛⠛⠻⣦⠀⠀⠀⢀⣴⠟⠛⠛⠛⢿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣬⣿⣿⣵⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⠓⠀⠀⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠈⠻⣿⣿⣿⠿⠋⠀⣸⡇⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⠀⣀⣴⣿⡇⢸⣿⣦⣀⠀⠀⣀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣽⣯⣯⣿⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⠦⢤⢄⡀
⠀⠀⠀⠀⢀⡴⣫⠽⡖⠋⢹⠉⠓⡾⢭⡛⣤⡀
⠀⠀⠀⣴⢫⠞⠣⡀⠘⡄⢸⠀⡸⠀⣠⠞⢣⠱⡄
⠀⠀⢰⢓⠗⠤⣀⠘⢦⣹⣸⣰⣣⠞⢁⡠⠔⢳⢹⡄
⠀⠀⣿⣼⠀⠀⠀⠩⢵⣿⠋⢻⣷⠯⠁⠀⠀⢸⡤⡇⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢹⢸⡀⣀⠤⠒⣪⢿⣿⣿⢯⡓⠢⢄⣀⢰⢒⠇⠀⠀⠀⠀⡠⡞⠈⢗⢀⠀⠀⠀⠀⠀⠀⣔⡕⠂
⠀⠀⠘⣎⢫⠀⢠⠞⠁⡎⣿⣿⢗⠛⢦⡀⢠⢏⡟⢀⡴⠚⠿⢯⡬⣥⠭⡬⢭⢥⠕⠀⢀⣀⢸⢴⢂⠀⣰⣤⡀
⠀⠲⠦⠿⢧⡟⢿⡞⡟⢀⢿⣿⠾⡶⣼⠟⣣⠞⠀⣸⣤⢶⠗⣾⢦⠅⠀⠳⠀⢸⣦⣈⢨⣨⢘⣋⣿⠀⢰⠋⢏
⠀⠀⠀⠀⠐⠘⠳⢼⣑⣾⣺⣛⣶⣽⣴⣏⣥⠶⠟⠙⣷⣇⣶⣣⣾⡖⣖⢲⡀⢸⡛⢿⣯⣿⠿⠿⠋⠀⠁⠀⠘⠀⢠⠙⣣⡜⢚
⠤⠤⠤⠴⠶⠶⠦⠤⠾⠤⠽⠽⠿⠷⠿⠿⠷⠶⠾⠿⠿⠧⠬⠤⠼⠧⠥⠾⠥⠤⠥⠤⠤⠴⠤⠤⠴⠷⠶⠶⠶⠾⠾⠶⠷⠷⠾⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⣠⠶⠛⠉⠉⠉⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⠀⠀⠀⢹⢀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢀⣀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢗⣺⣿⠀⠀⠀⢸⠻⣴⡿⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠠⠶⢶⢦⡄
⠀⡴⠛⣏⠉⠃⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠀⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠳⣤⣀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣤⣤⡤⠶⠋⠁⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢳⠀⠀⠀⠀⠀⠀⡈⠒⠀⠀⠀⠀⠀⠀⠀⣿⠐⠂⠀⠀⠀⠀⠀⠀⠐⠏⠆
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠆⠐⠐⠀⠐⠠⠀⠀⠁⠀⠁⠁⠀⠀⠡⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠈⠁⠀⠆⠀⠄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⠀⠀⠀⠀⠀⣀⡔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠆⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠀⠀⠠⠘⠆⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⢄⠘⠀⠀⠀⠀⠀⠀⢢⠄⠤⠂⠲⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⡀⠨⠃
⠀⠀⠓⠀⠀⠀⠋⠀⠀⠀⠀⠃⡀⠀⣀⡀⠀⠀⠀⢠⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⢀⡀⡀⠅⠀⠀⠀⠀⠁⠀⠀⠝⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠲⡀⠀⠀⠀⠀⠀⠀⡒⢲⡄⠀⠀⡀⣠⡔⠦⡄⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⠀⠠⢤⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠑⠀⣰⠀⠀⠀⠀⡀⢦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠒⢀⠀⡦
⠀⠀⠀⠙⠀⠈⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⢠⠄⠀⠀⠀⠀⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡀
⠀⠀⠀⡴⠢⠰⠀⠀⠀⠀⠀⠀⠐⠚⠲⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠂⠀⠀⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠒⠂⠐⠒
⠀⢀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢤⣤⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠠⠀⠄⠀⠌⠁⠀⠀⠐⠀⠐⠀⠀⠂⠀⠚⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⠑⠖⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠢⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠐⠂⠀⠀⠀⠂⠀⠀⠀⠐⠀⠂⠀⠂⠀⠀⠐⠀⠐⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⠀⠀⠀⠀⣀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢃⡿⠀⠀⠀⢸⡟⡉⠁⠀⡙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣸⠃⠀⠀⠀⢸⡇⠲⣄⠲⡝⣆⠠⣍⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣄⣤⣄⡀⢸⡇⠀⠸⣦⣑⠘⢧⡸⣕⠂⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠉⢀⣈⡙⠎⣰⠀⠀⠘⣿⣧⠈⣷⠈⠓⢤⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⡈⠻⡀⡇⢰⠀⠀⢸⡏⢀⢹⡄⠀⣼⠇⠠⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠀⠈⠻⣷⠀⠑⢸⠀⠀⠀⢻⣌⠪⡷⠈⠀⠀⣾⠹⣸⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⢀⣾⠟⠀⠱⣄⠀⠀⢻⡀⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣿⠀⢠⠞⠁⠀⣶⠈⢿⠀⣠⠿⠃⠘⣺⡷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⢰⡄⢻⡀⡅⠀⠀⢠⡟⠀⠻⠋⢁⡠⠖⢉⢹⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣃⠜⣸⠇⠘⣷⠳⡙⣆⠈⢧⣄⡀⠀⡨⢠⣆⢨⡸⣧⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠴⣫⡵⠚⠁⠀⠀⠈⠳⢑⠈⣷⠀⣿⢃⠞⣵⠏⢹⡼⣧⠀⠈⣹⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣁⣤⠾⠋⠀⠀⠀⠀⠀⣠⠞⠋⣰⠃⢠⡇⠊⣾⠃⠀⠀⠛⣎⠁⠸⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡉⠉⠀⠀⣀⣀⣀⣠⣴⡾⡯⠀⣸⠇⠀⢸⡇⠀⡿⠀⢀⣠⠄⠈⣀⠀⣹⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⡴⣾⠛⠉⣉⡁⠠⢔⣚⣥⠾⠋⠀⠀⠸⡇⠀⠃⣴⠏⠀⣠⣞⠥⠚⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡠⠞⣉⡥⠞⠋⠉⠐⠛⠉⠉⠀⠀⠀⠀⠀⣠⡤⠶⠟⣉⡤⢊⡵⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣶⠀⠀⢀⣿⡗⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠶⠀⠿⣿⡀⠀⠀⣶⣶⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⢻⣿⠀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⢰⠚⣩⠥⡀⣿⣧⡀⠀⠀⠀⣶⣶⡤⠀⠀⢾⣿⡆⠀⠀⠀⠓⠶⢬⣍⢲⡄
⠀⠀⠀⠀⠀⠀⢸⡀⣷⣶⠁⠸⣿⣿⣦⠀⠐⢮⡻⣿⣖⠀⣾⣿⣧⠀⠀⠀⣶⣶⡎⢹⢸⠁
⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠀⠀⣿⣿⣿⡀⠀⠀⢙⣿⡏⠀⣿⣿⣿⡄⠀⠀⢿⣿⡇⡾⢸
⠀⠀⠀⠀⠀⠀⠀⢇⢸⠀⣿⡑⢿⣿⣿⡟⠂⠀⣼⡿⠀⠀⢿⣿⣿⠗⠀⠀⠀⠀⢸⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠸⡈⣧⠛⢛⣼⣯⣀⠀⠀⣾⣿⣿⡦⠀⠈⢉⡀⠀⠀⠂⠀⣤⣼⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⡀⢸⣿⣿⣿⣛⣀⣼⣯⣁⡀⠀⠐⠻⡟⠁⠀⠀⠀⣻⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⣶⣶⡆⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⣀⣶⣶⠈⢉⡟⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢿⣿⡇⢠⠖⠖⡇⣶⢰⠒⢰⠒⢰⡓⣸⣿⢀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢣⣼⢸⡸⠧⡏⣿⢸⠛⠾⢳⡌⠻⡄⢀⡟⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠹⣯⠗⢅⡃⠛⠘⠺⠙⠺⢝⣚⣡⠏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⢦⣸⣷⠀⡇⡇⢸⡇⢸⡿⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢷⡐⠁⣤⣬⡴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡙⢦⠿⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠉⢉⡓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣎⠉⠙⢻⠀⠙⣇⠀⠀⠑⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠳⣄⠈⢻⠙⠋⠀⠀⢸⡄⠀⢀⡀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠘⠰⣄⣸⠖⠛⠀⠀⢀⡔⠋⠀⣤⣈⠫⠑⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠓⢦⡀⢀⣠⡤⠶⠶⠯⣅⠀⢠⠀⠀⡜⢀⠞⠙⠿⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⡀⠀⢸⡉⠑⠒⠢⣄⣿⣋⡁⠀⠀⠀⠀⢸⠀⠀⢀⡾⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠶⠺⡍⠉⢉⣙⣓⣦⡴⠋⠀⠀⣸⣓⠏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠙⠚⠁⠀⢀⡈⠉⠀⠀⠀⡿⠛⢉⣁⡉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠠⢀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⢀⡼⢭⡟⠂⠀⠀⠀⠞⠋⠚⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⣠⡎⠑⣾⡇⠀⠀⠀⣶⠠⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁⢈⣿⣿⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠫⣄⣸⠋⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⠀⠀⠀⠀⠀⠠⠚⠉⢳⡠⠞⠹⣀⠟⡿⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠏⡼⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡞⠁⠀⠀⠀⠀⠀⠈⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⡤⠴⠺⡇⠀⠀⠀⠀⠀⠀⠀⠰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡰⣎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠈⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⣖⡛⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠖⠚⠋⠁⢀⡠⠴⠒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⠜⠻⡄⣠⠞⡉⠉⠉⠉⠁⠀⠀⢀⣀⡤⠖⠊⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠞⢡⡖⠒⠺⢥⡀⢰⡀⢀⣀⡤⠔⠚⠉⣁⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠖⠛⠫⠽⠆⢹⠆⠉⠉⠀⠀⠰⡞⠉⠀⠀⠀⠀⠀⠀⠀⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢈⡿⠥⢛⣀⠀⠘⠠⠤⠤⠔⠒⠚⣿⣦⡀⠀⠀⠀⢠⠆⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠲⠤⣀⡘⠦⠖⠋⠈⣇⣀⠤⠤⠤⠤⡤⣤⡸⠿⠿⠦⣄⣠⠋⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⢆⠀⡞⢩⠟⠒⠲⠤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣁⣀⡤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠉⠀⠀⠈⢻⡁⡎⣀⣀⣠⣤⣤⠤⠤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡤⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠉⠉⢻⢳⠒⠤⣤⣤⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠘⠋⠉⠈⠀⠀⢀⠠⢽⡦⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠴⠴⠄⠤⠄⠛⠛⠉⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⠆⠀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠾⠿⠟⠛⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠃⠚⠃⠀⠂⠀⠏⠙⠀⠀⠋⠉⢘⠗⠃⠐⣻⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣄⣸⣿⣧⠀⠀⠀⠀⣀⣈⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡁
⠀⠀⠀⠀⢀⣾⠟⢉⡉⠙⢻⣿⠀⣤⣞⣋⠠⠤⠤⣤⣹⣿⣧⡀⠀⠀⠀⠀⣴⣿⠟⣉⣁⣉⣛⢶⣄⠀⠀⠀⢠⣾⠛⢉⣉⡛⣦
⠀⠀⠀⠀⣼⠃⠀⢀⣹⡆⠸⠟⢸⡏⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⣾⡿⠟⠛⠉⠉⠀⠀⠀⢛⡇⠀⠀⠀⠁⠀⣾⡟⠀⣼
⠀⠀⠀⠀⣿⡀⠀⢾⣯⣀⣠⡀⠈⢛⠲⠴⠤⠤⠤⠤⣼⣿⣿⠇⠀⠀⠀⣿⣦⣤⠀⢀⣀⣀⡠⠴⡟⠁⠀⢀⣀⣀⣀⣿⣷⢀⡿
⠀⠀⠀⠀⠸⠇⢀⣼⡿⠟⠋⢁⠀⠀⠘⠒⠂⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⣹⡿⠟⠛⠀⠩⠥⠄⠚⠀⠀⠀⠀⠉⠛⠿⣿⣧⡼⠃
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⣀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⣾⠋⠀⠐⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠺⠶⠿⣦⣤⣠⣾⡿⠉⠛⠋⠀⠀⠀⠀⠀⣠⡶⠛⠉⠈⠉⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠘⠛⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠉⢹⡇⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣀⣼⣷⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠛⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠼⠛⠉⠙⠛⠻⠷⢶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠋⠉⠀⢠⡿⠁⠀⠈⠙⠻⣦⠀⠀⢻
⠀⠀⠀⣠⣴⠞⠋⠉⠉⠉⣀⣤⡶⠾⠶⣦⡀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣶⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢸⡅⠀⢀⣀⣤⣄⣼⡇⠀⠘⣷⣤⣄⡀
⠀⢀⡾⠋⠀⠀⠀⠀⠀⢾⣏⣁⡀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠞⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⣦
⢰⣿⠗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣷⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡄⢹
⣿⠃⢀⣴⠇⠀⣠⣤⣄⠀⠀⢀⣠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⣤⠖⢻⡇⠀⠀⠰⣆⠀⠈⣿⡿
⣿⠀⢸⡏⠀⠀⡏⠀⠙⠷⠾⠛⠋⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⠸⠀⠀⠀⠀⣿⠀⣰⡟
⠸⣧⣼⡇⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣾⣿⡆⠀⢺⡆⠀⠀⣠⡿⠞⠋
⠀⠀⠈⠙⠶⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣤⣿⡧⠀⠈⠻⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⠄⠐⠒⠒⠶⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠉⠉⢹⡿⠉⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠉⠙⠙⠛⠛⠛⠋⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣀⣀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠉⠙⠛⠿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣤⣤⣤⣶⣶⡶⠟⠋
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡔⠚⠩⣕⠦⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⣦⠉⠉⠛⢯⡑⢠⡀⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⡌⡇⢹⡄⡠⣄⡀⠀⠀⠀⢀⡿⢴⢋⠈⠳⣤⣀⠀⠉⠓⠉⠻⣆⢻⡀
⠀⠀⠀⠀⠀⠀⢀⡞⣤⡧⠃⣾⡇⠹⣄⢙⢦⣤⠴⡯⠞⠰⠃⢀⣄⠀⢭⡳⡄⠀⠀⠀⠹⡄⠙⣆
⠀⠀⠀⠀⠀⠀⢸⡀⠋⣇⠀⠻⡅⢲⠘⢧⡱⡄⠎⠀⠀⠀⣠⠶⢬⡻⠳⡅⠙⠀⠀⠀⠀⢓⢆⡟
⠀⠀⠀⠀⠀⠀⢸⠃⣾⠘⢧⠘⢁⡞⠀⠀⢻⣎⣆⠀⠀⡠⣁⡤⠤⣉⠒⠌⢳⠀⠀⢠⢞⣟⡞
⠀⠀⠀⠀⠀⠀⠸⠀⢿⣄⠪⣛⠋⠀⠀⠀⠀⠙⠮⠁⠞⠀⣡⠤⠒⢈⡷⢠⡟⠀⡴⠋⣨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⣧⠀⠀⠀⢀⣴⣾⢏⠉⠉⠁⠀⢠⡞⣀⡟⠀⣸⠁⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠌⢦⢀⣤⣮⢚⡕⠁⠀⠀⠀⠀⠘⡴⣹⢁⡴⠋⠜⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⠓⠋⠀⠀⠀⠀⠀⠀⠸⡜⠃⣨⡴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠛⠿⣷⣄⠀⣀⣴⣾⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣏⣽⣷⣤⣹⡍⣼⠋⢉⣉⠷⣆⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠒⣀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠟⠀⣸⠃⣿⣺⣿⡆⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣾⠟⠛⣿⡟⠁⠀⠀⣀⠀⠀⠀⠀⠈⠷⠶⠶⠾⢷⣦⣿⣏⣉⣀⡤⠟⠀⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡇⠀⠀⠀⠘⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀⢘⣿⠀⢻⡆
⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⠁⠀⠀⠀⣠⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⢰⡿⢸⡇⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⢀⣿⡇⡠⢿⡄⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⢸⡇⣼⠁⠈⢻⣦⡀⠀⠈⠛⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣼⡇⠀⠀⢸⡏⠈⢦⡔⠁⠈⠻⣦⣀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⢿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⢠⡄⠈⢿⣤⠊⠑⢦⣄⡀⠈⢙⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣧⡀⢷⡀⠈⠛⠷⠶⠶⠶⠿⠛⠋⠁⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡏⠻⣤⡙⠶⣤⣀⡀⠀⠀⠀⠀⢀⡴⠛⠋⠛⢦⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠠⣿⣧⣤⣌⣛⠶⣤⣀⠀⠀⠀⢀⣴⠋⠀⠀⡀⠀⡈⣧⣤⣤⣤⣶⠶⠿⠛⠃⢠⠇
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠉⠙⠛⠿⣯⣝⠓⣶⠞⠁⠀⢀⠀⢻⡀⡂⢹⡀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠘⠿⠶⣶⣶⠶⠶⠶⢶⣤⣀⠙⣿⠋⠀⠀⢀⣼⡇⠸⡇⢷⣼⠛⠛⠛⠛⠛⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠳⣤⣀⣤⡶⠋⡘⢣⠀⣷⡞⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣘⢉⡽⠀⢠⠃⠈⡷⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣠⡏⠀⣼⣁⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⢿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⠈⠉⠉⠉⣹⠋⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣧⠀⠀⢰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣷⡤⠹⣧⢀⡿⠀⠠⣾⣧⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⡤⠶
⣤⣤⣀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢙⣿⠶⠟⠋⠁⢀⣼
⢳⡀⠉⠛⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣠⡿⠃
⠀⠻⢦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⠾⠛⠻⠿⠶⣶⡶⠶⠶⠶⠶⠶⢶⣦⣤⣤⣤⣤⣴⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⣘⡅⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠁⠀⠉⠀⠳⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠓⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⣀⡴⠀⠀⠀⣠⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡠⠾⢥⣀⠀⠀⠀⡹⠦⣄⡀⠹⡆⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢳⡄⠤⠄⢀⡧⠚⠀⠀⠀⠙⡆⠀⡾⠁⠀⠀⠙⣦⢹⡖⠒⠲⠚⠁⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡸⡀⠀⠀⠀⠀⠀⢻⡸⠃⠀⠀⠀⠀⢿⠈⣇⠀⠀⠀⢸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡄⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢾⣿⠀⠀⠠⢐⣀⣺⡇⣿⠀⠀⠀⣾⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⠀⠀⠀⠀⣯⡀⠀⠀⣰⣿⡛⣿⣿⠀⠀⣴⣿⣏⣿⡇⣿⡀⠀⠀⣿⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⡇⡇⠀⣰⣿⣿⣿⡿⢸⡆⢰⣿⣿⣿⣿⠃⡿⢣⠀⠀⠏⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃⠀⠀⢀⡜⢣⢱⡄⢿⣿⣿⣿⣃⠀⢻⣿⣿⣿⣿⡟⢰⠇⠈⢦⡀⠂⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⢀⠔⠊⠀⠘⣄⠻⣿⡟⠟⢁⠈⠀⠀⠙⠛⠿⠟⢡⡟⠀⠀⠎⠩⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠀⠱⡀⠀⠘⣆⠀⠀⠀⣄⣠⣄⡀⠀⠀⠀⣰⠏⠀⠀⢸⠀⠀⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⠀⠀⠀⡇⠀⠀⠈⠳⣄⠀⠘⠀⠈⠉⢧⠀⠚⣧⠀⠀⡤⠚⢄⠀⡄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⡆⠀⠀⠗⠈⠑⠆⠀⠈⠳⡇⠀⢀⠀⠀⡇⠀⣿⡤⠶⡀⠀⠐⠀⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣷⠀⠀⢠⠀⠀⠐⠂⠀⡊⠳⢤⣶⣤⣿⡇⠀⣿⠀⠀⠄⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⢹⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠿⣿⣿⡇⢠⡏⠀⠀⠀⠀⠀⠀⣿⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠋⢻⠀⡾⠁⠀⠀⠀⠀⠀⠀⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢸⡀⠀⠀⠀⠀⠀⠀⠀⢈⡄⢀⡴⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⡇⠀⠀⠀⠀⠀⠀⢰⡁⠿⠊⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡰⠀⠀⠈⡄⢧⠀⠀⠀⠀⠀⠀⠀⠑⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣠⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⣀⠤⠦⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢀⡈⡠⢀⣈⠀⠳⣄⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⢠⡿⠠⡇⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠈⠘⢁⠈⠙⢧⣄⠈⢧⡜⠀⡸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⡘⠃⠈⠁⠀⠀⠀⢢⡀⠠
⠀⠀⠀⠀⠀⠀⠀⡖⠉⠳⢶⣄⣉⠳⠄⠁⢀⠃⣇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢠⠇⢀⠀⠦⠄⠀⠠⠖⠐⣢⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠒⠐⠒⠋⠀⢸⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡞⠀⢸⣀⣀⡀⠀⠀⠙⠉⠁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢸⠁⠀⠀⠀⠀⠈⠛⠒⠲⠶⠖⠁
⠀⠀⠀⠀⠀⢀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⠀⡿⠙⠒⢰⡖⡶⠒⠒⠒⠲⠶⠦⣾⠁⠀⠇
⠀⠀⣰⠚⠉⢻⡆⠰⡈⢳⡀⠀⠀⠀⢀⠞⣡⠬⡇⠀⠀⡛⠤⠠⠜⡰⠁⠀⠀⠀⠀⠀⢰⡏⠀⢸
⠀⠰⠃⠀⠀⢠⡇⠀⠩⡄⢳⣀⢀⡰⣡⠎⠀⠀⡶⠀⡼⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢾
⢀⠃⡄⢠⠀⠸⡀⠀⠀⠙⢄⣉⣩⠾⠁⠀⢠⣞⠁⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠀⠻⡀
⠘⠁⠁⣎⠘⣄⡙⠦⣀⡀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠒⠒⠲⣬
⠀⠀⠀⠉⠀⠈⠙⠓⠉⠁⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⡆⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⡇⠀⠀⠈⢳⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣘⣷⣤⣤⣴⣒⣚⣁⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣳⣶⣶⣒⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⠷⣄⠀⣠⠶⣄⠀⠀⣀
⠀⢀⡤⣾⢻⡏⣷⣀⠀⠀⢀⣀⠀⢸⠉⠳⢤⡏⠀⠈⠛⠁⠀⠹⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀⣀⣴⠳⣄⣤⡀
⠀⠸⣄⡈⠀⠀⣉⣹⠆⠀⢸⡍⠙⠛⠀⣀⣀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠘⠛⢹⠇⠀⠀⠀⢸⣍⠛⠀⠋⠼⠷⡄
⠀⠀⠈⢧⣀⠀⠻⣅⠀⢶⠤⠇⠀⢠⢎⠏⣯⢱⠀⠀⠀⠀⡏⣼⢹⢹⠀⠀⠛⢲⠆⠀⠀⣀⡾⠁⢀⣠⠤⠞⠁
⠀⠀⠀⠀⠉⢧⡀⠀⠙⠾⠇⠀⠀⠘⢾⣦⡯⠞⠀⠀⠀⠀⠑⠿⠾⠋⠀⠀⠴⠯⠶⠶⠛⠁⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⢠⠴⣞⡿⠉⠉⠉⠹⠽⠋⠳⡀⠀⠀⠀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⡇⠀⠀⣀⣀⡀⣀⣤⣀⠀⠀⡇⠀⠀⢙⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠀⠀⠀⠳⣄⣞⡁⠀⠉⠁⢀⣈⡦⠞⠁⠀⢘⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⢶⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⣀⡀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⡇⠀⢤⣀⡤⣄⣀⡴⢶⠀⠀⣿⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⠀⠈⠉⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡇⠀⢸⠀⠀⠀⠀⠀⢸⠀⢀⠟⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⠤⠭⠥⠤⠾⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠉⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⡆
⠀⠀⠀⠈⠓⠒⠒⠒⠓⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠚⠁
⠀⢠⠒⠀⣀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠫⠿⠛⠛⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⣟⠻⢦⡄⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠒⡄
⠀⠈⠉⠉⣀⣤⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠉⠈⠁
⠀⠀⢠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠈⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠉⠛⠲⠦⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠤⣄⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣠⠴⠞⠳⢦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢀⡞⠉⠀⠀⠀⠀⠈⠳⠀⠀⠀⣠⡿⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣤⣄⣀⣀⣠⣤⡤⠶⠞⠛⠉⠀⠀⠉⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠄⠀⢀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠑⢠⠈⢊⠀⣀⠂⢀⡀⠀⠀⡁⡄⢀⢀⡉⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠺⠇⠀⠀⠐⠒⠒⢶⡀⠂⠀⠀⣀⠗⠀⠀⢹⡍⠀⢀⠇⢀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠐⠉⠙⣦⠰⠀⠀⢀⠀⠹⡆⠀⠀⣿⢠⠀⠀⠀⣧⠀⢸⠀⠀⢀⡤⢀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠄⠀⢰⡄⠘⣇⠺⠀⠸⡄⠀⢿⠀⠀⣿⠀⠀⠀⡆⠈⠀⡸⠀⠀⣿⠀⢀⣉⡁
⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⡠⠀⠀⢰⡁⠤⢶⡄⠀⠀⢁⡀⠘⣆⠀⠀⠁⠀⣼⠔⠀⠛⣆⠀⢀⡏⢀⠀⣧⢀⠀⠈⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢹⡄⠀⠀⠀⡠⠐⠁⠀⠀⢸⡍⠀⢘⠆⢀⠀⠈⠁⠉⠍⠉⠁⠐⠠⠀⠀⠜⠁⠂⠑⠠⠖⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⢧⡐⠀⠀⠛⠂⠉⡇⠀⠘⠁⠀⠀⢉⡀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⣀⠐⠀⠀⠀⠂⠀⠀⡐
⠀⠀⠀⠀⣠⢀⢐⠀⠀⠀⣄⠀⠙⣦⠀⠀⠀⠄⠀⠇⢀⠀⠰⠂⠀⠀⠀⣧⢼⠀⠀⢻⠀⠀⠀⣠⠀⠈⣷⠀⠠⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⠄⢌⠁⠉⠈⠀⣀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣿⠈⠀⠀⢸⠀⠀⠀⠓⠀⠀⡿⠀⠀⠃⠈⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠢⠀⡈⠊⠉⠉⠻⣆⠀⠀⠀⢿⠀⢀⢁⠀⠀⢿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⢀⠐⠊⠀⠀⠀⠸⠀⠀⣀⠀⢀⡼⠁⠀⠀⢸⡀⠈⢁⠀⠀⠸⡄⠀⠀⢸⠀⠀⠀⢠⠖⠀⠀⢹⡄⢑⠀⡘⠁⠈
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠄⠀⠀⢠⡇⠐⡌⠀⠐⠂⠀⠈⡇⠀⠀⠀⠀⠀⠳⠀⠀⣸⠐⡄⠀⠈⠃⠀⠀⡼⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣧⠐⠀⠒⢶⡀⠀⠀⠓⠊⠹⡇⠀⠀⠀⠀⣠⠏⠴⠇⠀⠀⠀⠀⠊⠁⠀⠤⠈⠊⢆
⠀⠀⠀⠀⠀⠀⠐⠠⢈⠐⢒⠀⠀⠀⠈⠙⠁⠀⣸⠇⠀⠀⠀⠀⠐⠃⠀⡀⢈⠁⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠄⣀⣠⠐⠁⠀⠀⠀⢠⠀⠀⠀⢀⠀⠌⠒⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠐⡊⠁⠀⠀⠀⠀⠐⠂⠂⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢄⣈⣐⠂⠌⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠐⠀⠀⠀⠬⠬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⢀⠀⠀⠀⠀⠉⢀⣀⣉⣀⣯⡤⠤⠔⠂⠀⠒⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣥⡤⠤⠖⠒⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡈⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⡂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡃⠀⠀⢀⠇
⠀⠀⠠⠀⠂⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢸⠅⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠄⠀⠀⠀⠀⣸⡆⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⠀⠂⠄⠀⠀⠀⠀⠀⠀⠨⡇⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢈⣼⠔⠀⠀⠀⠀⠀⢀⠀⠀⡼⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠠⠀⡀⠐⠂⡥⢘⡩⠀⠀⠀⠀⠀⠀⠊⠀⢰⢧⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢀⠄⣀⠐⢢⠇⡾⠘⠔⢌⣀⠀⠀⠀⠀⠶⠲⠂⠀⠶⠖⠐⠱⠁⠂⢀⣀⣀⡠⠤⣀⣒⠠⠬
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠓⠂⠚⠁⠀⠀⠀⠀⠁⠀⠀⠀⠂⠐⠒⠒⠒⠐⠒⠀⠐⠒⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠡⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠐⠊⠀⠀⡀⠀⢠⠇⠀⠒⠒⠀⠐⠀⠐⠂⠀⠂⠀⠘⠀⠂⠂⠀⠒⠀⠀⠀⠀⠂⠀⠐⠒⠒
⠠⣀⢄⡁⠀⢄⣄⠤⠀⢀⠀⣀⠀⢀⠒⢈⠂⣀⣠⢸⣔⢀⠀⠀⠠⡀⢡⢠⠄⠄⠀⡄⢆⠄⢀⢐⠀⠀⠀⠀⢐⠆⢀⠀⠀⠀⠄⠀⡌⢀
⠀⠈⠛⢃⠔⠂⠀⠀⠡⠀⠆⠦⠂⠠⠐⠶⠁⠁⠉⠉⠈⠳⣪⠄⢢⡜⠊⠈⠕⡤⡰⠀⠀⠈⠀⠱⠐⡀⠀⠐⠰⠟⠐⠈⠜⠦⠁⠆⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢠⠇⠀⠉⠱⠀⠀⠀⢠⠞⠓⠄⠀⠀⠀⠳⢤⡠⠂⢄⡀⠀⠀⠀⠀⠀⡴⠀⡑⠀⠀⢰⣾⠀⠀⠐⠛⠼⠥⣤⣀⣠⠤⣄⣀
⠀⠀⠀⠀⢀⡝⠀⠀⢀⠀⠀⠀⢠⡿⠀⠀⠀⡐⠀⠀⠀⠀⠈⠑⠶⡏⠁⠄⠀⠀⡼⢈⠞⠀⠀⠀⠸⢻⠀⠀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⣜⠀⠀⠀⡄⠀⠀⠀⠀⠙⠦⣄⠂⠀⠀⠄⠃⠀⠀⠀⠀⠀⠁⠀⠀⠘⠡⠊⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⠀⠀⠖⠀⠀⠀⠠⠀⠀⠀⠈⠓⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣈⣀⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷⢰⣳⢴⡲⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠀⠀⠀⠀⠀⡠⠊⠉⠁⠀⠂⠠⠄⢀⠀⠀⠀⡠⠄⢀⣄⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡀
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠐⠊⣀⠀⠘⠦⠄⠸⠰⠞⠸⠶⠀⠀⠀⠀⠀⠀⠠⡄⠁⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡔⠃⠒⠂⠀⠀⠀⠀⠒⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⣷⠀⠀⠀⣀⠁⠀⡀⠀⠀⠀⠀⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⢁⠀⢢⠀⠀⠐⠄⠈⠂⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⡴⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠐⠀⠹⠀⢀⠄⠊
⠀⠀⠀⠀⠀⢠⠀⢙⠋⠈⠀⠀⠀⠀⠘⡂⠀⠀⠂⠀⠀⢠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠀⠈⠐⠤⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⢀⠐
⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⢀⠄⠈⠀⢸⠄⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠄⣁⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀
⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⢄⠀⠠⡀⠀⠀⠀⠔⡉⠖⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠹⣦⠞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠠⠄⠀⠠⠀⠐⠈
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⡸⢺⠄
⠀⠀⠀⠀⠀⢀⣀⠂⡀⠀⠀⠀⠀⠠⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠐⠎⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢸⡂⠀⠀⠀⠀⠅⠀⠀⡴⠤⡄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠈⢄⠀⠀⠀⠺⠂⠒⠀⠈⠀⠀⠀⠀⢀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⡈⠆⠰⡀⠁⠀⡀⠠⠀⠀⠀⠀⠐⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⢉⠁⠀⢸⡆⠀⠀⠀⠀⠀⢄⠀⢄⠀⠀⠀⠀⠡⠀⠀⠠⡠⠤⣰⠀⠀⢀⠤⠀⠀⢀⣀
⠀⠀⠀⠀⢀⠐⠠⠠⢹⠀⠠⣜⠉⠛⠂⠀⠢⡀⢀⠀⠀⠀⠀⠦⠀⠀⠁⠀⠀⠀⠴⠀⠀⠀⠀⠠⠔⠉⠃⡀⠀⠀⠄⠀⠉⠉⠉⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠉⠁⠐⠚⠀⠀⠈⠂⠊⠀⠀⠀⠈⢀⡏⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⠀⣀⣀⣀⢀⡀⣀⢀⣀⡀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣄⡸⢔⡧⢇⣠⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⣋⡡⣹⢶⢼⣷⡶⣏⢄⣙⠓⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣶⣶⣿⣷⣾⣿⣟⣷⣾⣿⣶⣶⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣯⣍⣉⣉⣉⣭⣿⣿⣭⣉⣉⣉⣩⡽⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⢿⣿⢿⢛⡟⡟⣿⠿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠾⠷⠿⠿⠶⠷⠿⠿⠶⠿⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠘⣇⠀⠀⠁⠀⠈⠈⠀⠀⡀⡄⠀⠁⣸⠃⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡇⢻⡀⠀⠀⠀⠠⣻⣧⣼⡿⠁⠀⢀⡟⢸⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣗⠟⠈⣧⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⣼⠁⠻⣺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⣂⣤⣼⣧⡀⠀⠈⢻⢻⠙⠆⢀⣼⣧⣄⡐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⢿⣿⣷⡄⢐⠃⠘⠂⢠⣾⣿⡿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠸⣿⡟⣿⣿⣿⣿⣿⣻⣾⣿⠇⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠉⠉⠛⠋⠁

⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣼⡇⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣿⠀⣠⡀⠀⠀⢠⡄⡆⢀⡄
⠸⡇⢿⡄⢸⣿⡄⠀⠀⠀⠀⠀⠸⡟⣿⢸⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣾⡏⢠⣿⡇⠀⠀⢸⢷⡇⢸⣿
⠰⣿⢾⣇⢸⡟⣿⡀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡿⠀⡾⢻⡇⠀⠀⠈⠘⠃⠟⠘
⠀⠙⠘⠟⠀⠃⠀⠀⣀⢠⡄⢀⠀⠀⠀⠀⠀⠐⣀⠤⠔⠗⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⠁⣼⡇⠀⠀⠀⠀⠊⠁⠄⠀⠀⣀⠠⠄⠒⠠⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⣧⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠇⡿⢠⡟⡧⠀⠀⠀⢀⡀⠀⣠⠐⣉⠀⠀⠉⠀⠂⠀⣀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠀⠈⡏⢿⠀⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢅⠚⠰⠈⠀⠲⡄⠀⠀⠈⠒⠀⡀⠀⠀⠀⢀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠈⠛⣆⠟⠀⠀⠀⢀⣀⡀⠙⠠⡀⠈⠁⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⠄⠁⢀⠀⠀⠀⢉⣭⣍⡉⠀⠈⠢⠀⠐⡄⠈⢁⠛⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠑⠄⣀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠱⡀⠀⡀⠀⢣⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢠⠯⡁⠈⣄⡈⠨⠀⡇⠀⠀⠀⢠⣿⣷⣦⣤⣉⣉⣩⡽⠀⠀⠀⢀⢠⠔⡠⠸⠀⠇⠀⠀⡇⠀⢸⠄
⠀⠀⠀⠀⠀⠋⠐⣈⠔⠈⠀⣠⠀⢻⡀⠀⠀⣿⣿⠿⢿⡿⣿⣿⣿⡇⠀⠀⠀⠈⠚⠠⠁⠨⡇⡆⠀⠀⡇⠀⠋⠂
⠀⠀⠀⠀⠀⠑⢅⠁⠀⠀⢴⡃⠀⠀⠳⡀⠀⠻⣿⣤⣀⣀⣼⣿⠟⠀⠀⠀⠀⠀⠀⡠⠒⠢⠈⡇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠣⠐⠈⠙⢄⠀⠀⠈⠢⡀⠈⠉⠙⠉⠉⠀⠀⠀⠀⣀⣀⡴⢄⣀⣀⣀⡤⠓⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⢳⠒⠒⠈⠉⠒⠲⠤⠀⠀⣀⠖⠚⠛⠩⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠂⠈⠉⡋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠤⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⢀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠰⠉⠉⠀⠉⢴⠂⠉⠀⠈⠀⠅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠆⠀⠀⠀⣀⠀⠀⢀⣀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢤⣀⣀⣀⣀⣀⣀⣀⣉⣀⡸

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

⠀⠀⠀⠀⠀⠀⢀⠄⠠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣶⣀⠀⡤
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣱⢤⡀⠤⠄⣀⡀⠤⠤⡄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠟⢫⡄⠶⠀⣜⠻⣅⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⡅⠉⠁⠀⠉⡧⢎⠉⠓⠀⠀⠁⠀⠓⠀⠀⠀⠀⠐⢺⠋⡒⠤⣱⣶⣾⠤⢒⠙⡖⠂
⠀⠀⠀⠀⠀⠀⣿⠀⢀⣷⣀⣂⠀⠀⣇⣀⢀⠒⠀⠀⠀⠀⢠⢀⡀⠀⢀⣬⣹⡉⢋⣩⣿⣿⣿⣍⡙⠉⢯⣥
⠀⠀⡔⠀⡋⡉⢹⠀⢀⣀⣈⣳⣄⣀⡤⠤⠀⡄⠀⠈⣀⡀⡀⣤⡂⠀⠀⠀⣭⠻⣉⣿⢧⣦⠸⣷⢉⠟⠤
⠀⠀⠉⠈⣣⡟⢛⠈⢨⡅⠀⢀⡀⠀⠁⣶⣄⣤⣤⣼⣿⣦⣤⣀⣁⣆⣀⣀⣻⢀⣬⣿⠒⢿⠖⢻⡅⠀⠙⠁
⠀⠀⠀⣴⣿⣿⣿⣾⣿⣿⣶⣟⣟⠉⠏⢹⠁⠉⢉⡍⣿⡏⠙⣉⢩⣿⣿⣷⡌⠘⣿⢿⠻⣿⣤⣾⣿⣴⣷⣄
⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣙⣷⣿⣇⣍⣉⣡⣿⣯⣿⣁⣼⣿⣿⣿⣯⣽⣿⣼⣯⣿⣷⣯⣿⣿⣿⣿⡆
⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠙⠛⠛⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠂⢠⡀⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢛⠋⠉⠀⠀⠀⠤⠧⠀⠀⠀⠒⠀⢸
⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠂⢸
⠠⣴⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢀⣄⠠⣬⠁⢸
⠀⠀⠀⠀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠸⠲⠻⣴⠆⠉⢉⣿⠁⠐⠂⢠⣶⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⣠⣴⣀⡄⠀⢀⢠⣀⡀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠿⠿⠟⠛⠛⠻⠿⠿⠶⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠉⠉⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠉⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⣶⣶⣷⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⢀⣠⡾⣿⡄⠀⡼⠟⠋⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠙⠻⢧⠀⠀⣤⣄⠀⠀⠀⠀⠀⠸
⣿⠀⠀⢀⣴⠟⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠙⢷⣦⡀
⣿⡆⢠⣿⠁⠀⠀⠀⠸⣷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡏⠀⠀⠈⠻⣦⡀⢰
⢹⡇⠘⣿⡀⠀⠀⠀⣰⡿⠀⢸⣟⠻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⣿⠇⠀⣾⡇⠀⠀⠀⠀⢹⡇⢸
⠸⣷⠀⠈⠻⠶⣶⠾⠛⠁⠀⠈⣿⡀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣶⣶⣤⣤⣤⣴⣶⣶⠶⠶⠾⠛⠛⠉⠁⠀⠀⢰⡿⠀⠀⠘⣷⣄⣀⣀⣠⡿⠃⣾
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠉⠛⠛⠉⠀⢰⡟
⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣴⠟⠛⠻⢶⣦⡶⠟⠛⠻⣦⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⢶⣶⣤⣶⡶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡻⢷⣶⣶⣶⣶⣶⣶⣶⣶⣾⢛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⣠⠤⠶⠖
⠀⠀⠀⠀⣿⣿⡿⢿⣙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⢞⣡⣴⣶⣦⣤⣀
⠀⠀⠀⠀⠉⣵⣾⣿⠟⠁⠀⠀⠀⠀⢀⠀⠀⠀⣿⣾⡿⢋⠔⠒⠢⢍⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠘⠻⣶⣄⡘⣿⣷⡧⢊⠭⣑⢄⠑⡜⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣿⣦⣶⠶⠿⠟⠛⠛⢛⣿⠿⠛⠿⢿⣿⠷⣶⣤⣦⡣⠈⠢⡙⢿⣆
⠀⠀⠀⣀⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣶⠦⢹⡆⠀⠈⠙⠻⣿⣢⡙⢦⢹⣷
⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣀⣈⣡⡾⠁⠀⠀⠀⠀⠈⢿⣞⢸⢸⣿
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⢸⣟⠣⠜
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⢸⣿⣧⣿⣦⡀
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣏⠁⠀⠀⣠⠾⠿⣿⣿⠿⠞
⠀⠀⠀⠉⠛⠿⠶⠶⣶⣦⣤⣤⣶⣶⡶⠶⠶⠀⠀⠀⢀⣴⠋⠀⢿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⣀⣠⣤⠶⠛⠁⠀⣠⣾⠀⠀⠀⠺⣶⣶⡿⠋⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⡄⠀⣀⣠⣾⠿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣼⠏⣠⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣿⠟⠋⢀⡼⠃⡀⠀⠀⠀⢸⡏⠀⠀⠀⢴⣿⣶⣾⣿⣾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠛⠉⣀⣴⠏⠀⠀⠀⣿⠃⠀⠀⠀⠀⢀⣼⡿⣿⠟⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⣤⣄⣀⣀⣀⡀⠀⠀⠈⠉⠀⢀⣀⣀⣰⡟⠀⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠒⠲⠶⠦⠬⠭⣿⣤⣾⣉⣉⣉⣉⣉⣉⣿⣧⡶⠿⠛⠉
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇
⠀⣰⠋⠙⡇⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⢸⠇⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣠⣿⢀⡞⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠷⠴⠚⠛⠻⠶⣦⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⣶⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⢦⣤⣤⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡾
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⡠⠐⢢⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢼⠤⣀⣠
⠀⠀⠀⠀⠀⠀⡇⠀⢸⢀⣨⠓⢆⡒⢀⣤⡤⠀⠀⡄⢀⡀⠀⠀⠀⠀⠠⢤⡜⢉⢦⢺⠂⡴⡉⣣⡤
⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⢹⠂⠀⣈⠀⠀⠀⠀⠉⠀⠀⠀⠀⣉⣟⣈⣑⣠⣿⣾⣔⣉⣀⣏⡁
⠀⠀⠀⢀⡀⠀⡧⠠⢼⠡⢴⠀⠀⠸⠤⠤⠯⠁⣘⠃⠀⡶⠠⡤⠄⠰⠦⣭⣈⣤⣾⢿⢿⣷⢤⣀⠗⠶
⠀⠀⠇⢀⡀⣄⣇⡀⢼⠤⠬⠷⠦⢶⠒⠀⠋⠀⢀⣤⠊⠇⠶⠅⠀⠀⢐⡏⠳⢼⡟⢺⡆⢻⡤⠊⣁⡀
⠀⠀⠀⢀⣾⣅⡁⣠⣾⡀⢀⣤⣀⣠⢼⡧⠷⣶⣾⣿⣿⣷⡶⠤⣤⣦⠈⡇⣒⣿⣯⣹⡏⢉⣷⡀⣀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠤⣄⣸⡇⠠⠤⠆⣿⡧⣤⡆⢸⣿⣿⣿⠠⣽⡏⣏⢿⣷⣯⣿⣾⣿⣷⡀
⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⢾⣿⣷⠷⣶⣾⣿⣷⣷⣶⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠀⢲⡀⠀⠠⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⠶⢺⠋⠀⠀⣤⡤⠄⠀⣀⠄⠆⣐⣠⢂⠠⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡘⠀⠀⠀⢳⣦⠤⠄⢀⡠⠔⢫⠸⠒⠁⠀⠀⠈⣗⢈⡆
⠀⠀⠀⠀⠀⠀⠈⠉⢁⣤⠾⠤⢤⢀⠞⠉⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠈⡟⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⠿⠶⠷⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡤⠴⠖⠒⠒⠒⠒⠒⠶⠦⢼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠔⠚⠛⡿⠻⠿⠿⢿⣿⣿⣶⣦⣤⣤⣀⣰⠀⢀⣠⣤⣶⣿⠿⠟⠛⠉⠁⣧⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠡⠿⠟⠉⠉⠉⠉⣿⠀⠈⠟⠿⠿⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠢⢀⡀⠀⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⡤⠀⠀⠀⠉⠒⠤⠤⠀⠠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢱⡀⢨⡗⠒⠶⠤⣤⣤⣤⠤⡤⣤⠤⠐⠒⠚⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠱⢦⡀⠀⣏⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣧⠔⣆⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣄⣀⡀⠀⠀⣷⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠃⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠋⠉⠙⠛⠛⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠍⠡⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⢤⡀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠉⠀⣀⠀⠀⡀⢀⡽⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢠⠏⠀⡔⠈⠀⠀⢀⣰⠏⢠⠏⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣿⠀⠎⠀⠀⠀⢠⠁⣿⢠⡏⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⣀⣀⣁⣢⣀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⢿⡘⠀⠀⠀⠀⠸⠀⠻⣾⠀⡠⠔⠁⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢼⡆
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡄⠀⣿⠊⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⣀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠖⣒⡛⠛⠛⠓⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡇⠀⢻⡀⡠⠐⠉⠀⢸⠇⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠈⣿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡟
⠀⣀⡀⠀⠀⠀⢸⠇⠀⠀⢠⠁⠀⠀⡿⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣟⡼⠁
⡼⢻⡃⠀⠀⣠⠎⠀⠀⢀⠆⠀⠀⢸⠇⠘⡶⠖⠒⠶⢧⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠋⠁⠀⠀⢻⡁
⣧⠀⠉⠛⠋⠁⠀⡀⠔⠁⠀⠀⢠⡟⠀⢸⠁⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠘⢷⡀⠀⠀⠉⠁⠀⠀⠀⢀⡴⠋⠀⠀⠸⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⡇
⠀⠀⠙⠓⠶⠶⠶⠶⠖⠚⠉⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣶⣾⣿⣿⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⢿⡟⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠥⠤⠤⠒⠉⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⣾⠛⠖⣶⠤⠤⠤⣤⣤⡤⠔⠒⠓⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠉⠛⠒⠒⠚⠛⠛⠛⠉⢉⡀⠀⠀⢹⡄⠀⣇⠀⠀⣀⣀⣀⣀⠀⠸⢈⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡂⠤⢀⡶⠀⠀⠀⠀⠊⠉⠒⠀⢈⠀⠀⠀⢿⠒⠋⠉⠉⠉⠉⠉⠈⠙⠢⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⣶⡃⢀⡴⢳⣀⣥⠄⡀⢀⠎⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠈⠉⠋⠀⠈⠡⢼⡁⠈⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⠛⠓⠦⣄⠲⡖⠃⠀⢀⣀⣀⣀⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠠⣤⡤⠄⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⠤⢼⡧⠤⠤⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⠉⠉⢹⡏⠉⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⢸⡇⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢂⠀⠀⠐⠀⠁⠀⠀⣈⣀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠈⠀⠀⠀⠚⠃⠀⠀⢈⡀⢸
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⢸
⠈⠛⠋⠁⢀⣀⠀⠀⠀⠀⠀⠀⢠⣐⣆⣀⠀⠄⢶⡆⢀⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠚⠂⠙⠀⢸
⠀⠀⠀⠈⠾⠯⠁⠀⠀⠀⠀⠀⠀⠰⠾⠏⠀⠀⠛⠂⠐⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠮⠳⠦⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⢠⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢑⠔⢰⣃⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠘⠀⢼⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⠀⠲⠂⠀⢰⣷⡀
⠀⠀⠀⠀⢀⣀⣄⠀⣡⣘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄⢤⡄⠀⠀⡀⠀⣦⢁⡂⠂⢠⣤⣾⣿⡘⡷
⠀⠀⠀⢠⣿⣿⣿⣿⣦⣿⣿⡃⠀⡀⢀⠀⣀⡀⣀⢀⠼⡿⠿⡻⣦⡏⠀⠀⢧⣤⣽⣴⡟⠻⠿⠛⠉⠑⢀⣤
⠀⠀⠀⠀⠻⣿⣿⣻⡙⢻⣉⠉⢀⡀⠀⠄⠀⠀⠀⠀⠀⠁⠈⠀⢈⣀⡲⠤⠤⠩⠽⠭⠠⡆⠤⢴⠶⢠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣿⠓⢀⣀⠀⠀⠀⠒⠉⣉⣡⣤⠤⠤⠤⠤⠤⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠋⠉⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠦
⠀⠀⠀⠀⠀⡀⠤⠐⠀⠀⣀⣠⡤⠤⠤⠶⠶⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡎⡞⡕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣡⡞⢰⣇⠹⣆⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠁⣾⡟⠀⢸⣿⠀⢹⣷⡄⠙⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃⠐⡿⠓⠠⣾⣿⡆⠘⠛⠓⠀⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡂⢀⠀⣿⣺⡇⣠⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠿⢿⠚⠂⣿⣺⡇⠁⡿⠿⡇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣀⣆⣀⣴⣿⣾⣷⣄⣀⣦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠁⠈⢻⣿⡆⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⡿⡄⠀⠀⠹⠿⠿⠿⠿⠿⠤⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠒⠋⠁⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠠⣤⡀⠈⠙⠻⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⠛⠁⢀⠴⠦⣀⣀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁⠘⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡟⠁⠀⠀⠁⠀⢢⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠴⢿⣿⡁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠗⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⣈⠁⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⠀⣀⣀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠢⣌⣳⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣡⠟⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣠⣦⡀⠀⠀⠀⠀⠀⠀⠘⡿⠛⢿⡿⠋⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠀⠹⣄⠀⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢈⡓⠚⢉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⣄⣀⠉⠉⠁⢀⣠⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠠⠤⠀⠀⠴⠆⠐⠶⠒⠲⠖⠒⠒⠒⠋⠉⠙⡷⠄⢀⡼⢱⠄⡀
⠀⠀⠀⠰⣋⣉⣡⣤⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠋⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠯⣙⠚⠀⣸⠀⠒⠠⠄⣀⠀⠀⠀⡄⠀⡇⠀⢠⠂⠀⠀⠠⠤⢤⣄⣀⠀⠀⠐⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠒⢧⡠⢧⣴⠡⠔⠒⠉⠉⠉⠉⠉⠉⠉⠂⠀⠈⢄⡠⠂⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⣄⣤⣄⡀⠘⡄⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠀⠀⠀⠀⠀⣀⠤⠞⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣵⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⢀⡤⠒⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡞⠣⣀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢀⠤⠎⠁⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⢿⣿⡀⠀⢣⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠁⠀⠈⣆⣀⣀⣀⣀⢠⣆⡠⠤⠒⠂⠉⠉⠉⠉⠉⠉⠛⠒⠢⠤⣸⡇⠀⠈⢿⣧⠀⢸⣁⣀⡀⢄⣤⡭⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢈⣿⡴⠋⠀⠀⠉⠑⠪⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏