Lifeguard Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⡐⠲⣠⠞⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡴⠊⢉⡞⠁⢦⡄⠠⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⢧⡀⢸⣄⡀⢠⠇⣰⠀⣙⠓⠒⣄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⡂⢸⠀⠘⣦⡈⠓⠈⠹⡆⠹⢤⣀⡀
⠀⠀⠀⡜⠁⠀⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⡼⢠⡋⠀⣠⠚⣹⠛⠲⣄⠈⠀⠀⣀⡇
⠀⠀⠀⢧⣀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠙⠚⠁⠀⣇⡀⠀⠈⢧⠉⠉⠉
⣀⣀⣀⣀⡉⠙⠋⠁⢀⣀⣀⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣰⠛⣆⠀⠘⡇
⠁⠀⠀⠀⠉⠙⠛⠋⠉⢁⣀⠀⠀⠀⠀⠀⠀⡠⠶⠒⠚⠛⠛⡻⠛⠛⠒⠛⢂⡿⠀⢀⣇⡀
⠀⠀⠠⠒⠤⢤⣄⣤⠞⠉⠁⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⢠⢎⠄⠀⢀⠴⠚⣉⣀⣤⢞⠅⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠓⠿⠶⢋⣠⣮⣍⣁⣠⣶⣋⣀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠙⢢⣀⡴⠖⠒⠒⢆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⡄⢳⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⡴⠋⠀⠀⢳⡄⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠘⠿⣦⣄⡴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⠉⢀⡟⠀⠀⣾⠋⠉⠛⠓⠂⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⣤⣄⡈⠻⣄
⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢌⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡣⠞⠁

⠀⠀⠀⠀⣠⢴⣿⣿⠇⠀⠀⠀⠀⢨⣉⣉⡅⠀⠀⠀⠀⠀⠰⣾⡯⢀⣠⡀
⠀⠀⣠⣞⣵⡿⣻⠏⠀⠀⠀⠀⠀⡸⢻⡟⡆⠀⠀⠀⠀⢠⣤⡄⠀⠈⠛⠓⠀⠀⠀⣠⣶⣶⣶⣊⣽⣦
⠀⠀⣏⠿⣯⠞⠁⠀⠀⠀⠀⠀⢠⡋⠤⠄⢻⡀⠀⠀⠀⠈⠛⠁⢰⣾⡦⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠉

⠀⠀⠀⠀⢀⣤⣶⡟⠀⠀⠀⠀⢸⣷⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⢀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠼⠿⢿⠿⠿⠄⠀⠀⠀⠀⣠⣾⣿⣿⠁⠀⠀⠀⠀⠀⢈⡿⠿⠍⠁
⠀⠀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⣘⣉⣻⣋⣛⡀⠀⠀⠀⠀⠙⠿⡿⠟⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉

⠀⠀⠀⡀⠀⠀⡀⢠⠆⠀⠀⠀⡞⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠇⠀⠀⠀⠃⡞⠀⠀⠀⠀⡇⠘⠒⠡⠃⢸⠀⠀⠀⢠⣴⣶⣿⣿⣿⡟⠀⠀⠀⠀⡐⢚⣿⡓⠂
⠀⠀⠀⠰⣤⡤⠞⠁⠀⠀⠀⠀⣇⣀⣉⣁⣁⡽⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠄⠥⠤⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⠠⠀⠀⠀⠢⡀⠀⠀⠀⠀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⣠⡀⣄⢀⣄
⠀⠀⣼⣿⣿⣽⠮⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁⠀⠀⠀⠀⢸⣿⡏⣭⡉⣿⡇⠀⠀⢀⠞⠛⡟⠿⢻⠛⠳
⠀⠀⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⢀⡈⣉⠀⢀⠀⠀⠀⠀⠘⠿⠷⠾⠶⠿⠇⠀⠀⠀⠓⠒⠁⠀⠈⠂⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣒⣯⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⡀⠠⠐⠒⠊⠉⠙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠀⠡⠤⢤⡄
⠀⠀⠀⠀⡠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡣⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡼⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢱⣤⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣢⠄⠀⠰⠈⠛⠒⠒⠛⠁⢀⡀⠀⠀⠀⠀⠀⠉⠰⡄
⠀⠀⠀⠀⠠⠀⠀⠄⠀⠒⠒⠂⠀⠈⢹⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠊⠈⣉⡀⠰⠂⠀⠀⠀⠀⠉⠂⢤⡀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠐⠒⠒⠒⠚⠰⢾⡦⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢒⠒⠓⣚⢙⠁⠐⠒⠒⠒⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣞⣢⣄⣰⠀⢧⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠀⠀⣇⣨⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⠄⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠶⠒⠰⠢⠄⠗⠐⠐⠀⠀⠐⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆⠘⠠⢤⣀⣀⣀⠠⠤⠤⠄⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⠀⠠⣔⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠆⠁⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣇⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡋⢀⡠⠒⢠⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⡿⠁⠴⠁⠀⠀⢣⠈⢹⡄⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⡟⡘⠀⠀⢸⣦⠁
⠀⠀⠀⠀⠈⡇⠀⠁⠀⠀⠀⠈⠀⢦⡃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣇⠃⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⡇⠀⠁⡆⠀⠀⠀⠀⢸⡁⢸⣄⡀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⡤⡄⠀⠃⢉⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢇⠀⢧⡟⣉⣉⡀⠀⢸⠀⢸⠆⠀⢠⣭⣉⠈⠹⠷⢤⠀⡁⠀⠄⡁⢀⠔⠁⠉⠉⠈⠉
⠀⠀⠀⠀⠀⢸⡀⠈⢀⡀⣸⠄⠀⡎⠀⢸⠃⠈⠉⠀⠀⢀⣀⣀⠸⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⡘⠡⣄⠀⠉⠁⢄⣸⠁⠀⢸⠆⠀⠒⠉⠉⠀⠀⢸⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⢇⠀⠉⣷⡂⠒⡙⢹⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⢰⠀⠀⠀⠈⠁⠒⡶⡖⠚⡿⡇⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠋⠀⠀⠀⠀⠀⢰⠀⡇⢀⠃⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠀⠉⠉⠒⠀⢄⠀⢌⠘⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠘⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣆
⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠒⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣾⡄⠀⠀⠀⠀⠀⠀⠀⢾⣧⠀⠒⢿⡷⠂⠀⢀⣼⡿⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⣶⣾⣾⣏⣇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠈⠀⠀⣴⠛⠋
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⣻⣿⡏⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠀⠰⠇
⠀⠀⠀⠀⠀⠀⡟⢹⣷⣶⣶⣿⣼⣧⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣤⣤⣤
⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⢸⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡿⠀⢸⡷⠶⠶⠶⢾⡷⠶⠶⣧⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⣿⡿⢩⣿⣿⠁
⠀⠀⠀⠀⡿⠀⠀⢸⡟⠛⠛⠛⠻⡟⠛⠛⠛⣧⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣿⠀⠀⠀⢀⡀
⠀⠀⠀⢸⠇⠀⠀⠸⠇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣿⣷⠀⣾⣿⡆⠀⣰⣿⣿⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⣿⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⠀⢸⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠃⠙⠃⠀⢸⡿⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢚⡟⠁⠀⠀⠀⠀⢀⣠⡴⠚⠁⠈⢧⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠰⣶⠒⠒⠒⠚⢫⠍⠀⠀⠛⠒⠒⠒⠒⠒⠋⠉⠀⠀⢠⣀⠀⠀⠹⡄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⠉⠢⡀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢈⠐⣙⣷⡡⠀⠀⠀⠀⠠⡤⣄⠀⡇⠁⣼⣷⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠉⣉⠳⠶⠋⠀⠀⠀⠀⠘⠄⣌⠿⠃⠀⠀⠀⠀⠀⣠⠏⠀⠘⠶⠶⠟⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣴⠟⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠠⠈⠉⠠⠘⡹⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣸⠇⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣀⡰⠃⠀⠀⠀⠀⠀⢀⡼⠁⢀⠜⢻⣦⣤⡶⠒⢄⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⣴⡯⢴⡞⠁⠀⠸⣷⡜⣷⡀⢸⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢤⣀⣀⡀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⢋⠿⡏⠀⠸⣇⠀⠀⠀⠹⣟⡮⠟⠉⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡇⠀⠀⠀⠸⣅⡀⠀⠀⠀⠀⠸⡄⠱⡀⠀⠙⢷⣤⠤⣀⣙⣷⣄⡀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⡁⠀⠙⢦⡀⠀⠀⠈⠏⢳⣦⣀⠀⠀⠙⠒⠛⠒⡞⠻⡷⡄⠀⠀⠈⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⢱⠀⠀⠀⠉⠢⠤⠚⠀⠀⡆⠈⠳⣄⠀⠀⠀⣠⢧⢀⡫⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠳⠤⠞⠁⠀⣻⢯⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠳⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⣦⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⢤⠔⠒⠒⠒⣧⠀⠀⠀⢾⣓⠲⠦⠤⢤⣀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡞⠉⠤⣴⡇⠀⢸⠿⠛⠀⠀⠉⠛⠛⠓⠛⠋⠁⠀⠛⠷⠶⠯⠙⠲⢤⡖⢿
⠀⠀⠀⠀⡇⢸⡆⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠇⠈⢧
⠀⠀⠀⠀⡇⢸⠇⠀⣣⠀⢸⡀⠀⠀⠀⠀⢀⣀⣤⠤⣤⣀⠀⡴⠿⣛⣩⠵⠋⠁⠀⠀⠈⢧
⠀⠀⠀⠀⢧⡀⠐⠛⢿⠀⠘⡟⠛⠂⠀⠘⠛⣒⣛⣋⣩⠭⠴⠒⠋⣹⢧⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣁⡽⠗⠦⢤⣀⡀⠀⣰⠋⠀⠻⣄⠀⣀⣀⣀⣤⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠁⠀⠀⠀⠀⣠⠟⠉⠁⠀⠀⠀⠉⠉⢿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢠⡶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠉⠛⠻⠿⣿⣿⣿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⡆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣤⣤⣦⣄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⡟⡟⠻⢿⣿⣿⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣄⣠⡀⠀⠀⠀⠀⣼⣅⣁⣁⠛⠘⢻⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣿⣿⣯⣆⠀⠀⢠⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠛⠛⠿⠿⣿⣿⡿⠿⠛⠛⠿⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠛⠛⠿⠿⣿⠿⠿⠛⠛⠻⠿⠿⠛⠛⠻⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠛⢻⣿⠇⠙⠉⠠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠃⢀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⣴⣿⣏⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢡⣿⠛⠋⠀⠀⠘⠛⠋⠩⣶⣿⡿⠿⢋⣀⣀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⣤⠛⠛⠓⢤⣤⣤⠀⠀⠀⠀⠀⢠⣼⠿⠻⣿⠛⠛⠙⢿
⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⣠⡿⠁⠀⠀⠀⠈⣿⠁⠰⠛⠛⠛⠻⠆⠀⠀⠀⠉⣤⠀⠀⣿
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣰⣿⣷⣶⣤⠀⣤⣤⣶⣦⣤⣤⠀⣿⣀⣴⠟⣄
⠀⠀⣴⠟⠁⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠹⣿⣿⣿⠇⠀⠀⠘⡿⠁⢀⡿
⠀⣾⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣧⣤⠟
⣶⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠿⣄⠀⠀⠀⠀⠀⠀⠛⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⣠⡴⠛⣿
⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣤⡴⠛⠁⠀⠀⣿
⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠂⢤⣤⣤⣤⣤⣾⠛⠿⠁⠀⠀⠀⠀⠀⠛⢃⣤⣤⡀
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠋⠁⢸⠟⠛⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠉⣀⠻⣄⠀⠀⠀⢀⣤⡜⠛⠁⠀⠀⠀⣾⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⢀⣿⠟⣷⠀⣿⠀⠈⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡄⠀⣸⠛⠀⠈⣦⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣿⠀⠀⠀⠉⣤⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⣶⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡆⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠻⣤⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠛⣦⡀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠛⢦⣄⠀⠀⣿⠀⠹⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡈⠛⢤⣿⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣠⣿⠀⠀⠈⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠛⠀⠀⠀⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠘⠋⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣤⡴⠛⠁⠀⠀⢠⡚⠓⠀⢀⣤⣤⣿⡄⠀⠀⠀⠀⣤⠀⠀⠈⠙⠛⠛⠛⠛⠃⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠛⠛⡿⠛⠻⡿⠿⠷⣤⣤⣤⣶⣿⣶⡶⠺⠿⠀⠀⠿⠛⠻⡄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣠⠀⠀⠀⠀⠀⣩⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⢀⣤⣤⣶⣦⢠⣿⡄⠀⠀⠀⠀⣟⠀⠀⡋⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣄⠀⠈⠛⠀⣤⣤⣴⣿⣷⣤⣶⡾⠛⠛⠛⠛⠛⠛⣂⡀⠀⣁⠀⠀⣿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠂⠀⢤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠀⢿⣤⣴⠋⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⠀⠀⠀⠀⠀⢀⣶⠠⣶⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠛⠛⠛⠛⠛⠓⠒⠒⠂⠀⢠⣤⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠁⠀⠙⠁⠀⠀⣿⠃⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⢻⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣾⠀⠀⠀⠀⠸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠁⠀⠀⠈⣄⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠟⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠛⠀⠀⠀⢀⣤⠔⠛⠁⠀⠀⣾⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣤⣀⣄⠛⠁⠀⠀⠀⠀⠀⠀⠀⢛⣀⡘⠛⠛⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⣠⡟⢀⣀⣀⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣄⣀⣀⠘⣧⡀
⢀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⢦⣄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠾⠿⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠺⠿⠶⠶⠶⢶⣤⣤⣄⣀⡀⠀⢸
⣿⡶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⣿⠙⠷⣄⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠖⠛⠛⠛⠛⠛⠿⠿⠿⠛⠓⠶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣀⡴⠟⢹⡇
⠀⢹⡄⠀⠈⠙⣶⠦⠶⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠶⠦⢴⡞⠋⠀⠀⣼⠁
⠀⠘⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡟
⠀⠀⠹⣆⠀⢀⣿⠀⠀⠀⠀⢰⡅⠠⣶⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢰⡆⠀⡷⠀⠀⠀⠀⢸⣇⠀⢀⡾⠁
⠀⠀⠀⢹⡾⢫⣿⡄⠀⠀⠀⠀⠻⠤⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⠼⠃⠀⠀⠀⠀⣼⣯⠻⣾⠁
⠀⠀⠀⢸⣇⠈⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢀⣿
⠀⠀⠀⠀⠙⠳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠛⠁
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⣄⣀⣀⣀⣀⣀⣀⣤⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠙⢍⠉⠉⠉⠉⠉⢉⠝⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣴⠂⢠⣤⠌⠳⣄⢀⢀⡴⠃⢤⣤⠀⢲⣄⡀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⡶⠛⢉⡟⠀⠀⠀⠀⣰⣟⠻⢛⣷⡀⠀⠀⠀⠘⣏⠙⠳⣦⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⢰⡏⠀⠀⢸⡇⠀⠀⠀⠀⠹⢿⣤⠘⢸⡇⠀⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⣧⡀
⠀⠀⠀⠀⢠⡞⠁⢀⡀⠈⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⠦⠾⠃⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⣀⠀⠙⣦
⠀⠀⠀⢠⡟⠀⡴⢻⣿⣀⣈⡷⠀⣿⣛⢛⡛⠛⣛⢛⣛⡛⣛⡛⣛⠛⠛⡛⢛⣻⡇⠰⣏⣀⣼⣿⢳⡄⠘⣧
⠀⠀⠀⠀⠙⠛⠳⠛⠀⠉⠉⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠉⠁⠘⠳⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⠀⢀⣾⠛⢻⣆⢀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠾⠶⠾⡏⠀⠀⡿⠶⠶⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡇⠀⠀⡏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠛⠛⠶⢾⡷⠀⠀⣇⠀⢀⡇⠀⠰⣿⠶⠞⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⣀⠀⣀⣀⣀⣀⡀⣀⡿⠀⠸⣇⠀⣀⣀⣀⣀⡀⢀⣀⣀⡈⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡷⠶⠆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢛⣩⣤⠶⠖⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡟⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠁⣀⠀⠐⠶⣶⡈⣓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠈⢻⣟⣋⠀⠁⠈⢳⣄⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠳⢤⣤⠯⠀⠈⠛⠶⣤⣀⣻⡆⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡏⠀⢻⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⡇⢴⣅⣀⡀⠉⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣄⣸⡇⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠉⣩⣍⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⢀⣟⣉⣻⡠⠾⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠻⠷⠶⠶⠖⠛⠋⣉⣉⣭⣭⣉⣠⣿⣄⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⢀⣀⣤⠶⠛⢙⡏⢸⣤⣼⣇⠈⠋⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠿⠾⠛⠛⠉⠀⠀⠀⠀⡇⠈⠡⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣤⣤⣤⠤⠴⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿
⠠⡛⠛⠛⠛⠛⠛⠛⠛⠓⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣤⣤⣤⣤⣤⣤⣤⡼⠁⠒⢚⠋
⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠁⢠⡇⠀⠀⠀⢸
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣁⠀⠀⠀⣼
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠰⠶⠶⠶⠤⣤⣤⣤⣄⣀⣀⣰⡟⠀⠀⠀⣸⠇⣿⠀⠀⠀⣿
⠀⣀⣀⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⢹⠛⠛⠛⠒⠲⠶⠶⠦⣤⣤⣤⣀⣿⠃⠀⠀⣰⡟⠀⣿⠀⠀⢰⡇
⢤⣯⣤⣤⣈⣉⣉⣉⣉⠙⠛⠛⠛⠛⠻⠿⢷⡆⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⣰⡟⠀⢸⡟⠀⠀⣾⠃
⠀⡇⠀⣸⠉⠉⠉⠉⠉⠉⡙⠛⠛⠛⢳⠖⢺⡇⢠⣄⣀⣀⣀⠀⣾⠏⠀⠀⣴⡟⠀⠀⣾⠇⠀⣸⡟
⠀⡇⠀⣿⠀⣤⠴⠖⠊⠉⠀⠀⠀⠀⠘⡆⠀⣿⠀⠀⠀⠀⠉⣼⡏⠀⠀⣴⡟⠀⠀⢸⡟⠀⢠⣿⠁
⠀⣿⠀⣿⠐⠒⠶⠶⠶⠦⠤⢤⣤⣤⣄⣷⠀⢻⡀⠀⠀⠀⢀⣿⠁⠀⣰⡟⠀⠀⠀⣿⠃⠀⣼⡇
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡇⠀⠀⠀⢸⣿⠀⢠⣿⠁⠀⠀⠀⣿⠀⢠⣿
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇⠀⠀⠀⢸⡿⠀⣼⡇⢀⣠⠖⠁⣿⠀⢸⡷⠶⠶⠦⠤⠤⢤⣤⣤⣤⡄
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⠀⠀⠀⠘⣷⢀⣿⠖⠋⠀⠀⠀⢹⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢃
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⡇⢸⠇⠀⠀⠀⠀⠀⠘⡇⡟⠀⠀⠀⠀⠀⠀⢀⡴⠚⣡⠖⠉
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣧⡴⠚⠀⡇⡼⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣏⣠⣤⣠⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⠞⠁
⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢻⠀⢀⣀⣈⡉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠾⠋⣠⠞⠁
⠀⡯⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠶⠶⠶⠞⠁
⠀⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⠿⠛⢉⣀⣤⣤⣤⣤⣤⣄⣈⠙⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⡿⠿⠿⠿⢿⣿⣿⣷⣦⡀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⣿⣿⣿⠋⢠⣾⣿⠿⣿⣿⡯⠀⣤⠀⢸⣿⣿⠿⢿⣿⣦⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠠
⣿⣿⠇⢠⣿⣿⠋⠀⠀⠉⠃⣴⡟⠓⠘⠋⠁⠀⠈⢿⣿⣧⠈⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠐
⣿⣿⠀⣼⣿⣧⣄⡀⠀⠀⠀⠈⢟⡆⠀⠀⠀⠀⣀⣴⣿⣿⡀⢸⣿⡇⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢈
⣿⣿⠀⢿⣿⣿⡿⠟⠃⠀⠀⢰⡏⠀⠀⠀⠀⠻⢿⣿⣿⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⡄⠸⣿⣯⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣨⣿⡿⠀⣼⣿⡇⠀⠀⠀⠠
⣿⣿⣷⡀⠻⣿⣧⣀⣠⣴⡇⠀⠀⠀⢸⣶⣤⡀⣰⣿⡿⠃⣰⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢧⠀⠠
⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⡇⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⡀⠀⠐
⣿⣿⣿⣿⣿⣷⣄⡈⠛⠿⢿⣿⣿⣿⣿⠿⠟⠋⣀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠢⠈⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⡉⡉⡉⡉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇⠀⣴⡄⠀⢸⣿⣿⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⢸⣿⠿⣿⣿⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣼⣿⠏⠀⠀⠉⠛⠇⣰⢾⠲⠆⠸⠛⠁⠀⠀⠹⣿⣧⠀⠀⠀⠀⢀⣾⠏⢀⣾⣷⣦⣀⢸⠟⢐⡛⣿⢀⣤⣶⣷⡄⠈⢿⣆
⠀⠀⣸⣿⣏⠀⠀⠀⠀⠀⠀⠙⢾⣤⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⣾⡏⢀⣾⣿⣿⣿⣿⣿⣌⠸⢿⣿⣿⣿⣿⣿⣿⡄⠘⣿⡄
⠀⠀⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⣸⠞⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⠀⠀⠀⣿⠃⠀⠈⠙⠿⣿⣿⣿⡿⢛⣸⣿⣿⣿⡿⠛⠉⠀⠀⣿⡇
⠀⠀⣿⣿⣿⠟⠋⠁⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⠀⠀⠀⣿⡆⠀⣠⣴⣾⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣦⣄⡀⠀⣿⡇
⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢻⣧⠈⢻⣿⣿⣿⣿⣿⣿⣱⣿⣿⣿⣿⣿⣿⡿⠃⢰⣿⠁
⠀⠀⠀⠹⣿⣧⡀⢀⣤⣶⡇⠀⠈⠀⠀⢸⣶⣤⡀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢿⣧⡀⠻⠟⠋⠁⢸⣿⣿⣿⣿⠀⠙⠻⠿⠁⣠⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣶⣶⣶⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣤⣀⣠⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣦⣤⣦⣠⣄⢠⣄⣤⣔⢠⡆⠀⠀⠙⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣠⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣾⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠰⡇⣀⡴⠛⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⣼⣧⣤⣬⣤⣬⣭⣭⣭⣭⡍⣉⣉⢨⣤⣤⣤⣤⣭⣭⣬⣭⣭⣬⣥⣭⣭⣤⣭⣭⣭⢬⣭⣥⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⣿⣷⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣿⣮⣻⣧⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⠤⠦⠀⠒⠿⠻⣿⡏⠙⣿⣿⣓⠚⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡶⠦⠴⠤⠤⠄
⠀⠀⠀⠀⠀⠠⠔⠒⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠰⢦⣴⠿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡦⠤⣤⣤⣿⣏⣀⣀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡤⣤⣤⣽⡟⣻⣛⡛⠛⠲⠶⣤⣄⣀⣀⣀⡀⠀⢀⣾⡏⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⡿⠐⠀⡀⣀⣀⠀⢨⣡⣭⠙⣋⣻⣇⣏⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠔⠀⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠁⣀⣬⡤⠴⠒⠒⢛⣧⣘⣋⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠠⠟⠿⠝⠉⠹⠤⠀⠀⠠⢤⣤⠀⠒⣦⣤⡤⠃⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣉⠉⠛⠛⠻⠿⠛⠓⢲⣶⠟⠻⠿⠁⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⡤⡄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢴⠆⠀⠸⢃⠈⡅⠻⣿⡿⣿⣿⣿⣿⣷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣭⣭⣭⣉⣽⣧⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠗⠚⡼⠃⠀⣛⣿⡞⣿⣿⣿⣿⣿⣿⣷⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣿⣿⣿⠻⠿⠿⡾⠟⠛⡐⣾⣶⡆⠀⠀⠀⠃⢀⠞⡟⠛⠁⠀⣤⣿⣿⣤⣍⣉⠉⠉⠉⢉⣩⣭⣭⣤⣤⣄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⢠⣤⣤⣤⣤⡆⣴⣿⣿⡿⠀⠀⠁⣀⠘⠀⠁⠀⠀⠀⠋⠰⡆⠀⣀⣉⠁⢰⣷⣾⠩⢿⣿⣿⣿⠟⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⢿⣿⣿⣯⣽⣿⣇⣈⣈⣩⣥⡤⢴⣿⣻⠏⠁⠀⢰⣾⣿⢿⣰⣷⣾⣶⡶⢦⣤⣤⣼⣿⣿⣷⣘⣛⣛⣂⣉⣭⣽⣿⣦⣥
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⠷⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡿⠋⠙⠛⠓⠚⠛⠛⠿⣏⠪⣿⣿⠿⠿⠿⠿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⣀⠠⠄⠤⠤⢤⣤⣠⣤⣤⡄⠠⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠃⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠗⠛⣾⣷⣘⣈⣿⣞⣭⣤⣤⣶⣶⣶⣶⣶⣴⣶⣤⡬⠉⠁⠀⠈⣴⣦⣴⣦⡔⢻⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⠀⠀⢸⣿⣿⣿⣿⠸⣿⡎⣹⣿⣯⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣦⠀⢸⣿⣿⣿⣿⣴⣾⡷⠒⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣭⣭⣭⣭⣬⣭⣭⣭⣭⣭⣬⣽⣷⣦⣼⣿⣤⣿⣤⣤⣤⣦⣭⣤⢺⣿⣏⠀⣿⣿⣿⣛⣻⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣾⣿⣯⠀⣿⢇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⣷⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⡿⠿⠿⣿⣿⣿⡇⣏⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣉⣉⣹⡋⣉⣩⡏⢾⡿⢻⣿⣿⣶⣾⣿⣿⣿⣤⠿⢿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣾⣤⣤⣤⣄⣉⣀⣨⣭⣍⡉⠉⠉⣩⡟⣩⣮⣿⣿⣿⣵⣿⡇⠰⣿⡟⠉⠉⠛⠃⠀⠉⠙⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠘⣿⣿⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠁

⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




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

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⣀⣠⣾⣇⠀⢀⣠⡎
⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⢠⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣯⣿⣶⣿⣿⣿⣿⣿⣶⣿⣽⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣵⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡇
⠀⠀⠀⠀⠈⠉⠻⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠲⠀⠈⢻⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣷⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⠦⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⣤⣴⣾⣏⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣯⠉⠁⠀⠀⠀⢻⣿⠿⢿⣷⣶⠄⡼⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣼⣯⣽⡗⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⡟⠀⣴⡌⠀⢻⣆⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠙⠚⠆⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢳⣷⣧⡄⣍⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠈⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣾⣬⣏⠓⠛⠂⠀⠀⠀⠀⠀⢀⡜⠁⠿⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⠖⠀⠀⠀⠀⠀⠀⠰⠋⠀⠆⠀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⣀⠀⢨⣯⣤⣄⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⣈⡟⠉⠛⠉⠀⠀⠀⠀⡽⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠁⠈⠀⠀⠀⠌⠀⠀⠘⢶⣶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣷⡄⢭⡉⠈⢀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣤⣶⡄⠀⠀⢻⣿⣇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣻⣿⡄⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⢿⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⡿⣟⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⢾⢿⣿⡄⠀⠀⠈⢿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣿⠻⣧⢫⣿⠟⠀⠀⠀⣿⣿⡝⢇⣿⣿⠋⠏⣾⡋⢘⢟⢀⡄⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⡾⣃⣼⡟⠀⠀⠀⢀⣿⣿⣿⣾⡟⣦⡊⣿⢈⣿⣆⢸⣆⣃⣤⣄⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠸⠿⠿⠀⠀⠀⠀⠹⠿⠿⠥⠿⠽⠿⠷⠞⠛⠉⠻⠿⠛⠛⠛⠛⠋

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⢀⣤⣿⣿⣤⡀⠀⠀⣶⡶⢲⣶⠂⣀⠀⠀⢸⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠇
⠀⠈⠛⣿⣿⠛⠁⠀⠀⢿⡿⠾⠿⢿⡿⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠈⠐⠊⠀⠐⠆⠀⠀⠀⢠⣀⠎
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⢸⠓⠚⠇⠀⠀⠀⠀⢸⠓⠚⡇⠀⠀⠀⠀⠀⣶⣿⡿⠀⠀⠀⠀⢸⢀⣁⠀⠀⠀⠀⠀⠀⣰⣦⡖
⠀⠀⢘⢢⠴⠂⠀⠀⠀⠀⢘⢢⡴⠃⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⢨⠈⠛⠀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠘⠐⠒⠂⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⡴⢊⡑⢦⡀⠀⠀⠠⠔⢒⡒⠢⠄⠀⠀⢠⣤⣯⣽⣦⡄⠀⠀⠀⠀⣰⠉⠇⠀⠀⠀⠀⠘⡛
⠀⠢⠸⠤⠤⠇⠜⠀⠀⠱⠀⠘⠃⠠⠌⠀⠀⠸⣿⣆⣴⣿⠇⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠸⣿⡿⡻⣿⠇⠀⠀⠀⠀⡅⢨⠀⠀⠀⠀⠀⢰⠈⠁⡆⠀⠀⠀⢸⡏⢉⡉⢸⡆⠀⠀⢸⠢⣤⣤⣤⡆
⠀⠰⣿⣿⣾⣿⠆⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠘⢳⡞⠃⠀⠀⠀⢸⡆⠻⠟⢸⡇⠀⠀⢬⠉⠉⠉⠉⡥
⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠈⠓⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠂⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⢤⣀⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⢤⡀⠘⠀⠀⡰⠀⠀⠸⠁⠐⠒⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣄⣈⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠀⣁⡀⠠⠤⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣄⣀
⠀⠀⠀⠈⠉⣁⡤⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤
⠀⠀⡀⣸⢈⡿⠁⠑⠠⢀⡈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⡆
⠀⠰⠿⠿⣿⠃⠀⠀⠀⠀⠈⠐⠢⣀⠈⠑⠢⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠩⠳⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⡇
⠠⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠀⠉⠐⠄⡀⠈⠀⠀⠀⠀⠉⢐⣲⡭⢅⡂⢽⡀⠀⠀⠀⠀⠁⠀⠀⠀⡇⡇
⢦⣶⣄⢀⠈⠒⠄⡑⠠⠀⠀⠀⢽⡲⠄⠀⣀⠀⠁⠀⠄⠀⢀⣈⣁⠉⣴⣶⣿⢦⢻⡐⠲⢤⣀⡀⠀⠀⠀⡇⡇
⢸⣿⣿⣏⣢⣄⠀⠈⠐⠀⠑⠠⠀⠈⠁⣹⣿⡶⠀⠀⠀⢞⣾⣿⠂⠀⣿⣿⡟⢸⣜⡇⠀⠀⠀⠉⠲⢤⣄⡇⠇
⠀⠉⠉⠲⣭⡺⡝⠢⣄⠀⠀⠀⠐⠤⡀⠀⠛⠋⠀⠀⢸⣿⠿⢟⠁⣤⢻⠓⠉⢀⡿⣷⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣄
⠀⠀⠀⠀⠈⠙⢧⣀⣧⡁⠀⣀⡀⠀⠁⠀⠂⢀⣀⣀⣾⠛⠉⡩⠀⠖⠉⠁⡀⠀⠇⣿⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣦⡀⠀⠉⠘⠋⢹⠀⡀⣴⠤⠖⢚⡡⠔⠊⣁⡏⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣇⠀⠀⠀⠀⢸⣀⡀⠄⢒⡩⣦⣤⣶⡯⣀⡼⠃⠀⠐⠠⡈⠙⠦⣄⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣷⣤⣤⠀⠉⡀⠤⠔⢾⡿⢟⣫⠵⠊⠁⠠⣀⠀⠀⠀⠈⠑⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢀⠀⠀⢀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⠲⣄⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢤⡀⠑⢤⡀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡉⠲⣄⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠘⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⠀⢸
⠀⢷⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡾⠀⠀⠀⡟⠀⠀⠀⠀⡟
⠀⢸⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⡞⠀⠀⠀⠀⢰⠇⠀⠀⢠⡇⠀⠀⠀⢰⡇
⠀⠈⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠹⡆⠀⢠⠞⣡⣄⠀⣀⢠⣄⢸⢧⠘⡆⠀⣠⠏⠀⠀⠀⠀⠀⣾⠀⠀⠀⣸⠀⠀⠀⠀⢸
⠀⠀⢷⠀⠀⠀⠀⠀⠀⣧⠀⢀⡼⠁⠀⠀⠀⠀⢀⣀⠠⣦⣹⠀⠈⠋⠀⡟⢸⢿⣿⢸⣼⠈⠷⠇⣰⠃⡀⠀⠀⠀⠀⠀⠹⣄⠀⠀⡏⠀⠀⠀⠀⡟
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀⠸⠶⠚⠉⢧⠀⣷⣼⡝⡆⣯⠻⠃⠀⠀⢰⣇⡞⢸⣿⠘⠿⠀⠀⠠⣧⢞⡟⣴⡆⣠⠀⡶⠲⢬⣷⢸⠃⠀⠀⠀⢠⡇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡀⡇⡇⣧⢳⡼⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡴⣿⣰⢻⢆⡇⠀⠀⠀⡾⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣀⡈⠳⠃⠻⢏⣠⠤⠖⠒⠚⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠶⠦⢤⣀⡙⠃⠘⠞⠀⠀⠀⢠⡇⠀⠀⠀⠀⡿
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠴⠇⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⢀⠀⠀⠀⠀⢀⠀⢳⠀⠀⠀⠀⡜⠁⠀⢀⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠀⠀⠀⣰⠃⠀⠀⠀⠀⠰⡇
⠀⠀⢰⡇⡿⡆⢰⣿⠀⡟⣇⠈⡇⠀⠀⡼⠀⠀⠀⣾⠤⢼⢁⣶⠂⣼⢰⡴⢲⠀⠀⢰⡰⣆⡴⡀⣴⠖⠠⠁⠀⣼⠀⢠⠞⣁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠸⣿⠃⡇⠘⣿⡇⣧⠹⡄⢻⠀⠀⣧⠀⠀⠀⠃⠀⠘⠈⠓⠃⠛⣸⠛⠁⠀⠀⠘⠁⠉⠀⠃⠓⠚⠐⠀⣰⠃⠰⣧⠞⢸⠀⣤⠀⡀⢀⡀⢻
⠀⠀⠀⠀⠀⠹⡴⠃⢧⣸⠀⠙⠋⠀⠀⠈⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⢸⢸⣿⢰⣿⢸⡇⢸
⠀⠀⠀⠀⢀⣀⠀⠀⠈⢁⣤⡀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⠤⣤⣤⣤⣤⠤⣄⠀⢠⠴⠶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⣸⣿⡾⢷⡼
⠀⠀⠀⠀⣏⡈⠳⡄⠀⠻⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⢦⡾⠀⠀⠀⢠⣤⢠⣶⣠⡄⠀⠀⠀⠀⠉⠁⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠉⠙⠛⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⡀⡞⣧⣿⣷⢿⠄⠀⠀⠀⠘⠃⠀⠀⠀⢸⡘⣾⢸⡇⣧⣾⡆⠀⠀⠀⣠⠖⢦⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⣿⢿⡏⣼⠀⠀⠀⠀⠀⠀⠀⢶⡒⠾⠃⠀⠈⠁⠋⣸⠁⠀⠀⣞⣡⠴⠋⣰⢊⣁⣨⠇
⠀⠀⠀⠀⢠⠶⣄⠈⠓⠲⠶⠷⠀⣠⢤⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠛⣻⠇⠀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⢠⠇⠀⠀⠈⠁⣀⣀⠘⠉⠉⠁
⠀⠀⠀⠀⠘⠳⠾⠷⠀⠀⣀⡀⠀⢧⣀⣈⣳⡀⠀⠘⡇⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡾⠀⠀⠀⢀⠞⢁⣨⠇⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣽⡦⠀⠀⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠛⠋⠉⠀⠠⠾⠧⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣿⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⡖⠒⠛⠓⠲⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣻⣻⣻⣿⣻⣟⣟⡛⣿⢿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⣇⣀⣀⣀⣀⣀⣀⣹⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⠿⢿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣵⡿⠿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣿⣿⢯⣶⡿⢿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⢻⣿⡀⠀⣸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢿⣯⠀⢀⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠚⢠⠔⠁
⠀⠀⠀⠀⠀⡠⠚⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠄⠀⢀
⠀⠀⠀⡠⠺⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⡭⠖⠚⠉⣉⡉⠙⠒⠮⣕⠠⡀
⠀⡠⠾⣀⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠀⠀⣶⣦⡟⣧⣶⡄⠀⠀⠙⢮⡢⡀
⣮⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡱⠁⠒⠀⠐⠀⣴⣿⣯⣿⣷⡄⠐⠀⠀⠂⠱⡑⡀
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⠃⠀⠀⠻⠀⠀⠈⠀⠿⠇⠈⠀⠀⠸⠃⠀⠀⢳⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⠒⠒⠂⠉⠱⠖⠚⢓⠒⢤⡄⠀⠒⠛⠒⢤⣀⣸⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⠋⡢⣞⠑⠀⠀⠀⠀⠀⠀⢸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣆⠠⠄⠄⠠⠄⠀⠨⠀⠈⠀⠤⠀⠠⠤⠀⢲⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠭⠓⣒⣒⣒⣒⠫⠅⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠛⠛⠉⣉⣁⣤⣽⣿⣷⣶⠿⠿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⣷⣶⡾⠿⠿⠛⠛⠋⢹⣿⣧⣤⣤⠶⠶⠞⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⠿⠿⠟⠛⠛⣿⣿⣁⣤⣤⡴⠶⠖⠛⠛⠉⢛⣁⣀⣤⣤⣴⣶⣾⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⢀⣠⣤⠶⠶⠚⠛⠛⠛⣉⣀⣠⣤⣴⣶⣶⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⠟⠁⢀⣠⣴⣶⣾⠿⠿⠛⠛⠋⠉⠉⠀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡏⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⣁⣀⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢸⡟⠉⣿⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣸⡇⠀⢿⠶⠶⢾⡆⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⠇⢸⡟⠉⠉⠁⠀⢀⣀⣤⣼⠇⣸⠃⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡄⠸⠷⠶⢻⡇⠀⣿⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⣇⣠⣿⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣉⣁⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⢀⣀⣀⣤⣤⣶⣶⡾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣶⣶⡾⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠖⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠐⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢈⡭⠊⠙⠶⣀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠙⠐⠒⢬⡀
⠀⠀⠀⠀⠀⢀⢈⢀⡠⠤⢄⢀⡀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡆⠆
⠀⠀⠀⠀⠀⢛⡟⠀⠀⠀⠀⠀⠘⠢⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢀⢃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠐⠉⠳⣀⠄⢠⠃
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⡀⠀⠀⠀⣿⣶⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⣆⠀⣰⠏⠙
⠀⠀⠀⠁⡃⠀⠈⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣴⡋
⠀⠀⠀⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⡞⠜⣠⣞⣿⠟⠁
⠀⠀⠀⠐⢄⡀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠄⠀⠄⡤⠁
⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⣀⡓⠤⠦⣀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠈⠑⠓⡦⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠐⠉⠑⠲⠬⢕⣐⠐⠠⠠⢀⣀⡀⠤⠘⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠊⠁

⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⡶⢶⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⢢⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠈⣧⡠⣂⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣤⣬⣀⣄⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⢹⣄⣈⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠦
⠀⠀⠀⡧⠤⠤⠄⠄⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠒⠂⠒⠂⠒⢂⣇⡗⠒⢷⠛⠓⠒⠒⠒⠋⠂⠀⠛⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⠀⠀⢱
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⡤⠈⠃⠆⠀⠀⠈⠃⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⢀⠠⠐⠁⢀⡀⠀⠈⢡⣀⣀⠠⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣤⣾⣽⠇⠀⠤⣤⠠⢀⡀⠀⠽⣀⠀⣼⡟⠉⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠉⠋⠁⠀⡠⢰⡅⠀⠘⠓⠢⠀⠓⠺⣿⡷⠴⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣤⣀⡶⠁⠀⠀⠀⠃⠰⠀⠀⠀⠈⠥⡠⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠠⡀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀
⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠁⠄⠀⠀⣀⡀⣠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠔⠀⠈⠃⠀⠾⢶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢁⡙⡇⠀⠀⠀⠀⠀⠀⠲⣿⣿⣯⡆⢀⣀⣤⣴⡶⠖⠚⠉⠉⠉⠉⠉⠙⠓⠤⣄⡠⡄⣀⠀⠀⡐
⠀⠀⠀⠄⠀⠤⡀⠐⠊⣿⣽⣧⠿⠃⠀⠀⠀⠀⠀⠀⠙⠿⣉⠛⠋⠉⠟⠉⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⢋⠀⠀⠀⠀⠦⣀⡀⠀⠀⠄⣀⠀⠁⠀⠁⠀⠀⠻⠉⠉⠉⠁⠀⠉⠁⠀⠠⠉⠙⠛⠉⠥
⠀⠀⠀⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠲⠂⠀⠈⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠖⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⣶⣤⣀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣧⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠈⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⣈⣉⣻⣿⣿⣿⣿⣿⣿⣋⣉⣀⣀⣠⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠛⠛⠉⠀⠈⠙⠛⠻⠿⠿⠟⠛⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠁
⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡈⢀⠄⠀⠀⠈⡀⠀⡀⠀⠀⠀⢀⡀⠐⠂
⠀⠀⠀⠈⠂⠑⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡐⠃⠀⢀⡀⡀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠠⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠘⠀⠀⠀⠐⠀⠈⠁⢀⣸⠏⠹⡃⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⡀⠂⡀⢇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠄⠂⠁⠀⠐⠃⡂⢀⠐⢂⠐⠊⠠⠄⠀⠒⠀⠊⠀⠀⠐⠒⠈⠀⠀⠒⠀⠠⠀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡠⠀⠀⠀⠀⠀⠗⢪⠔⢈⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⡄⢀⡠⠒⠢⠐⠂⠈⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠺⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡖⠷⢤⠤⣎⠓⠆⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠀⢱⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢉⡙⠚⠤⠉⠀⠐⠉⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠉⠀⠀⠀⣠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠊⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠈⠀⠠⠀⠐⠂⠈⠀⠀⠁⠈⡉⠉⠀⠀⠠⡄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠂⠀⠀⠀⠀⠚⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⡀⢷⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⡀
⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠁⡮⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠃⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣠⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠹⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⡤⠂⠠⠁⠀⠀⠠⠄⢂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠎⠀⠀⠀⠀⠐⠈⠂⠐⠤⠈⢂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⡀⠄⠀⠄⠠⠘⠀⠀⠀⠀⠐⡌⠀⠀⠀⠀⠀⠀⢀⠁⠀⠂⢀⠦⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢤⣉⡤⢇⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠈⠂⠰⣄⣀⣀⣀⣨⠄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠠⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠬⢩⣤⠝⠰⠇⠀⡀
⠀⠈⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⣀⡀⠙⡑⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠒⢃⠀⢠⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠠⠀⠁⠈⠂⠀⠀⢀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠰⠈⠀⠈⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠂⠊⠠⣤⠀⠀⡈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢆⠀⠒⠀⠀⠃⠘⠀⠛⠀⠐⠃⠀⠀⠤⢀⡀⣀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠰⠀⠰⠠⠤⠴
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠶⠆⠀⠇⢠⠀⠀⢀⠐⠈⠀⠀⠔⠁⠀⠀⠀⢀⡅⡍⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠉⠐⡋⠀⠙⠁⠀⠆
⠀⠀⠀⢀⣠⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⡛⣛⣛⣛⡘⣛⣻⢛⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣽⣯⣭⣿⣿⣿⣿⣿⣿⣉⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⣸⡇⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⣈⣁⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠗⢰⣿⡏⠀⠀⢹⣿⡆⠺⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⣠⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠛⠉⠁⣀⣀⣀⠀⠈⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣠⣴⣿⡿⠁⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⠉⠉⠉⢹⣿⣿⣿⣿⠋⠉⠉⠉⠁⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠈⠛⠛⠛⠉⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣶⣶⣾⠿⠟⠉⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠉⠉⠙⠻⣿⣆⠀⠀⠀⣿
⢻⣿⣄⣀⣀⣀⣀⣠⣿⠋⠀⣴⣿⣿⣷⣄⠀⢻⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⠁⢠⣾⣿⣿⣶⡄⠘⣿⣇⣀⣠⣿
⠀⠙⠻⠿⠿⠿⠿⢿⣿⡀⠸⣿⣿⣿⣿⡿⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠉⠛⠛⠛⠁⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠙⠛⠛⠋⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣴⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣶⣶⡿⠟⠁
⣶⣶⡶⢶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣀⠀⢀⣾⠟⠻⣷⡀⠀⣀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⠋⠀⠀⠙⣿⡿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣧⠀⠀⢀⣴⡦
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠀⠀⠛⠋
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣈⡁⠀⢈⣉⠀⠀⣉⡀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠛⠃⠀⠘⠛⠀⠀⠛⠃⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣶⣤⣤⣶⣾⣿⣁⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠿⠟⠛⠻⣿⣄⣉⣉⣠⣿⠟⠛⠻⠿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢰⣿⣅⡀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠀⠀⠈⣿⡆
⠀⠀⠀⢀⣾⡟⠛⠛⣿⣆⠀⣿⡇⠀⠀⠀⣴⡿⠛⠛⢿⣦⠀⠀⠀⠀⠀⠀⣰⡿⠋⠙⢻⣿⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⡀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠇
⠀⠀⣀⣤⣿⣷⣶⣶⣿⣧⣤⣶⠿⠿⣶⣤⣽⣿⣶⣶⣿⣿⣤⣶⠿⠿⣷⣤⣼⣿⣶⣶⣿⣿⣤⣴⡿⠿⣷⣄
⣴⡿⠛⠉⠁⠀⠀⠀⠀⠉⣿⡇⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⣿⡏⠀⠀⠘⣿⡍⠀⠀⠀⠀⠈⣿⡏⠀⠀⠈⣿⡆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣄⣠⣾⣟⠀⠀⠀⠀⠀⠀⣻⣷⣄⣠⣼⣿⠀⠀⠀⠀⠀⠀⢹⣷⣄⣀⣴⣿⡁
⣿⡇⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠙⠋⠙⠛⠿⣶⣤⣠⣶⠿⠟⠋⠙⠋⠉⠛⠿⣷⣤⣠⣶⡿⠟⠋⠙⠛⠉⠛⠿⣷⣦
⠿⠃⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣶⣶⠶⠶⠶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾

⠀⣀⣀⣀⣀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠰⡶⡶⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⢠⣿⣿⣿⠋⠙⣿⣿⠉⠁⢄⠀⠀⠀⠀⢀⣴⣯⡀⢀⣀⣀⣉⣀⣀⡀⠀⠀⡿⠿⡿⡛⣿⠿⠿⣷⠟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⠚⠀⠀⣼⠃
⢸⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣷⣶⡄⠀⢸⣿⡄⠛⠘⠭⠟⠛⢿⠿⠇⠀⠀⣿⣿⣿⣿⣬⣿⣿⣿⣷⣶⢿⣶⡄⠀⠀⣿⡹⠞⠿⠿⠿⠿⠾⣿⡀
⠛⠛⢿⡽⠛⠛⠛⠛⠛⠛⢻⣭⠟⠃⠠⡞⠀⢿⠀⠠⠴⠞⠉⠑⠦⠄⠀⠘⠛⢻⣽⠟⠛⠛⠛⠛⠛⠛⢯⡽⠛⠀⢰⠏⡧⠀⠀⠀⠀⠀⡼⠉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣀
⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⠀⠀⠀⠀⣾⣿⣷⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢸⠂
⢠⣦⣶⣶⣷⣶⣶⣶⣷⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣵⡏⠀⠉⠀⠀⠀⠀⣿⣿⡿⢀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣠⠞
⢸⣿⣿⣿⢫⣴⢶⣌⢻⣿⣿⣿⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⡌⢀⡠⠊⠁⣴⠒⠚⠒⣦⠀⠀⠀⠀⢀⡤⢄⡤⣌⢳⡞⠁
⢸⣿⣿⣧⢻⣦⣠⣼⢢⣿⣿⣿⠀⠀⠀⠀⠀⢯⣾⢟⡿⠋⠀⠀⠀⠀⠀⠀⢸⢀⠊⠀⠀⠀⣾⣫⠉⣝⣿⠀⠀⠀⠀⡏⠀⠈⠻⠋⢰⠆
⢸⣿⣿⣿⣷⣬⣭⣴⣿⣿⣿⣿⠀⠀⠀⠘⢷⡇⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣦⣤⡤⠤⣿⣽⣻⣫⣿⠀⠀⠀⠀⠳⣄⡀⣀⣠⠞
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣖⣲⣦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆
⢰⡿⠙⢃⣀⡀⢀⡀⣀⠀⠀⠀⠀⠀⢻⡛⡛⣛⢛⡛⡛⣛⢻⡟⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⢀⢀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣟⡶⣾⢶⣺⣿⣿⡷
⢸⡇⠀⢾⣅⣿⢸⡇⣿⠀⠀⠀⠀⠀⢸⠭⡭⠭⠭⠭⡭⠭⢽⡇⠀⠀⠀⠀⠀⠀⢸⡇⠯⣼⢾⣼⢸⡇⠀⠀⠀⠀⠀⠈⠁⠣⣹⠤⠯⠜⠈⠁
⢸⡇⠀⠰⣭⡿⢸⡇⡿⠀⠀⠀⠀⠀⢸⣓⣓⣚⣚⣓⣓⣛⣺⡇⡴⠲⣦⠀⠀⠀⢸⡇⠙⠋⠘⠙⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣡⠶⣚
⠈⣿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣧⡴⡴⣦⢦⣴⣼⡇⠻⡶⠋⠀⠀⠀⢸⣿⣛⡗⢺⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⠅
⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠃⠉⠘⠛⠛⠓⠚⠛⠂⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡖⡒⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆
⠀⠘⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣓⣚⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣟⠃⠀⣠⣦
⢰⠉⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠻⣿⣿⣿⣮⠉
⢸⡇⢀⠸⢿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠈⡿⠿⠿⠿⠿⢾⡫⢝⡇⠀⠀⠀⠀⠀⣈⡛⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣵⣿⣿⣾⡿⠁
⢸⠿⠿⠀⠻⢿⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⣾⣷⢠⣾⠿⠂⠀⠀⠙⠋⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠿⠋
⠸⣀⣀⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠛⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡎⢀⢤⡤⠔⠂⠀⠀⢔⣶⠂
⠀⠀⠀⠀⠀⣶⣾⣖⣀⣀⣀⣀⡀⢠⡤⢾⡷⠤⡀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣶⣾⣧⣴⢶⠟⠋
⠀⠀⠀⠀⢰⠟⠛⠛⠛⠛⠛⠛⠛⣿⠀⠨⣧⣤⣥⣀⣀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣏⡿⢿⣿⠿⢿⡿⣿⡿⠿⠿⠿⢿⣿⣿⢿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠁⢯⣸⠗⢏⡹⠋⠉⠉⠉⠉⠉⢿⣉⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣴⣿⡟⠉⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣤⠀⠀⠀⠀⣯⣿⡇⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⡆⠀⣠⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠿⣶⣶⣶⣾⣿⣿⣦⣤⣤⣤⣴⡇
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⡛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠿⣫⣽⣿⣿⣿⠋⡃⢐⢰⣦⣶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠐⠒⠓⠒⠚⠒
⠀⠀⠀⠀⠀⠀⠳⠜⠈⠉⠁⠀⠀⠀⠀⠘⠦⠞⠳⠜⠁⠀⠐⠀⢀⣀⠀⢠⣤⣄⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣚⣀⣸⣿⣟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣍⡭⣭⣭⣭⣭⣭⣭⣩⢭⣭⡅⠀⣠⣶⣷⣦
⠀⠀⠀⠀⠀⡌⠀⠀⢸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠋⠉⠉⠉⠉⠉⠓⠞⠉⠀⣼⠏⠀⠀⠙⣷
⠀⠀⠀⠠⣶⠿⢿⡄⠈⠀⠀⣼⠿⣶⣆⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣀⣀⣀⣀⣸⡄
⠀⠀⠀⠰⢷⡞⢳⡷⠶⠦⠾⣷⠛⣿⠿⠀⠀⠀⠀⠀⠶⣷⣦⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⡄⠂⠀⠀⣿⣿⡇⠀⠀⠀⣤⣼⡃⠀⠀⠀⠀⠸⠿⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣒⣒⣒⣛⣛⣛⣒⣒⡲⠍⠀⠉⠀⠀⠀⠀⠈⣷⣄⣀⣀⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠹⠏⠉⠿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠓⠋

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⠄⡌⡸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⢀⣤⣤⡄⠀⢀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠃⠀⡼⠛⠿⠓⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⡶⢾⡀⠀⢰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠠⠀⠀⢸⠇⠘⡇⠀⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢀⠄⠀⠎⠀⠀⠃⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠⠤⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠇⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠒⠊⠉⠁⠀⣠⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢐⡂⠤⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠛⠁⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣋⠔⠀⣠⠔⠊⠁⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣠⣶⠗⠋⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⢛⡡⢔⡠⢔⠮⠐⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣟⣉⣔⡩⠦⠥⠬⠤⠤⠥⠤⠤⠤⠤⠤⠤⠀⠀⢡⣀⣀


⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⣀⣿⡟⢻⣿⣀⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⡏⠀⢸⣏⠉⠁⠈⠉⣹⡇⠀⢹⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⡀⠘⠛⣿⣇⣸⣿⠛⠁⢀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⣄⠀⠉⠉⠉⠁⠀⣠⡾⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⡀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠁⠀⠘⠛⠛⠃⠀⠈⠛⠛⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⠄⠀⢠⣤⣤⡄⠀⠠⣤⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⠀⠘⠛⠛⠂⠀⠘⠟⠛⠂⠀⠐⠛⠛⠃⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⡀⠀⢀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠦⠀⠰⠶⠶⠆⠀⣿⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⣿⠀⠰⠶⠶⠆⠀⠴⠶⠶⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠦⠀⠰⠶⠶⠆⠀⣿⠀⢸⣿⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣉⣿⡇⠀⣿⠀⠰⠶⠶⠆⠀⠴⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣿⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⣿⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢶⣶⡦⠀⠰⣶⣶⠆⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠰⣶⣶⠆⠀⢴⣶⡶⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠏⢀⣾⣿⣿⣷⡀⠹⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡏⢀⣿⣿⣿⣿⣿⣿⡀⢹⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠿⠿⢿⣿⣿⡟⠀⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⠀⢿⣿⣿⡿⠿⠿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠃⠀⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⠏⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠻⢿⡿⠟⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢁⣴⣶⣶⣄⣠⣶⣶⣦⡈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠘⢿⣿⣿⣿⣿⡿⠃⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠙⠻⠟⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⡟⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⡥⠤⠖⠒⠚⡇⠀⠀⣸⠛⠓⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠹⡄⠀⠀⠀⠀⠙⠦⠴⠋⠀⠀⠀⠀⠀⠈⠉⣳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⠙⢶⡶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠉⠉⢉⡽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⠖⠒⠛⠛⠛⠛⠒⠶⠤⣄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣼⠉⠳⣄⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣈⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⡇⢠⠇⠀⠀⠈⠑⢤⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣠⠤⠖⠚⠉⠁⠸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⠀⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇
⠀⠀⠀⠀⠀⢠⠚⠉⠉⠉⠳⡄⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢠⠞⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠘⢤⣀⣀⣠⡴⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢦⣄⣀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠀⠀⠀⠀⠀⠀⡏⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡟⠘⡆⠀⠀⠀⠀⠀⣀⡠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠓⠦⢤⣀⠀⢰⠃⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡀⣠⠴⠖⠋⠁⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠈⢙⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⠦⠤⢤⣤⣤⡤⠤⠶⢶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢀⣠⠞⢁⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣄⣀⣀⠀⠀⠙⠲⢤⣀⣰⠃⠀⠀⠀⠀⣠⠖⠲⣄⠀⠀⠀⠀⢧⢀⣀⡤⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠦⢬⣉⠛⠒⠦⠤⢤⡇⠀⠀⢹⣤⣤⠴⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣧⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢢⣴⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠒⠒⠲⢦⣄⡀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠿⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠘⣷⠀⢀⣀⠀⠀⠉⠀⡴⣆⢀⡀⠀⠈⢉⠛⠿⢶⣤⣀⠀⣼⡇⣀⣶⠀⠐⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣴⣆⠀⠀⢀⣼⡟⣴⠋⠈⣇⢸⡆⢸⡇⣿⣼⡇⢠⡄⢸⣧⢠⡖⠺⡍⢹⣿⠋⣉⡅⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⢸⣿⠀⠶⢿⣋⠀⢿⣄⣠⣿⣸⡇⢸⣷⠇⣿⡇⢺⡄⢸⡿⣿⠀⢠⡇⣸⡇⣾⡏⠀⡿⢀⣿⡿⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣼⣿⣆⠄⠀⠈⢻⡎⠉⠁⣈⣭⣽⣾⣿⠀⠈⠛⠋⠙⠛⡁⠙⠖⠻⣧⢿⡇⢿⡅⣀⣤⡏⣿⠇⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣧⠀⢀⣾⠃⠀⡾⠁⠀⠀⢹⡿⢿⣶⣤⣤⣤⣤⠟⠀⠀⠀⠀⠈⠧⠄⠈⠉⢸⣼⠏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠈⠉⠉⠀⠀⠀⠛⠤⠤⠔⠋⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣷⠀⠀⣀⠹⢦⣄⣀⣤⠞⢁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⠀⠒⠒⠒⠒⠒⠒⠒⠀⣿⠀⣾⠋⠁⠀⠈⠉⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠷⠟⠁⠀⣛⣛⣛⣛⣛⣛⣛⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣤⡼⠃⠀⠶⠶⠶⠶⠶⠶⠦⠀⣿⠀⠻⣦⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠶⠶⠶⠶⠶⠶⠖⠀⣿⠀⠀⠻⡶⠶⠶⠶⠶⢶⡞
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡶⢶⡄⠀⣉⣉⣉⣉⣉⡉⠉⠀⣿⠀⠀⠀⡇⠀⢰⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⠾⠃⠀⣭⣭⣭⣭⣍⠁⠀⠀⣿⠀⠀⢰⡇⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠘⣧⣀⣸⣇⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠊⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠲⣾⢹⡘⣿⣾⣿⣿⣶⣿⣴⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠘⣷⣌⠢⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠸⣧⢳⡌⠺⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢻⡌⣇⢧⡙⢿⡿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣎⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠝⠚⠋⠊⠀⠀⠀⠉⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣦⣤⣀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣀⣤⣴⣾⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣃⣀
⠀⠀⠀⠀⡜⠁⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⠈⢣
⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸
⠀⠀⠀⠀⠘⠤⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠤⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠟⠛⠉⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠉⠛⠻⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢍⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣀⡀⠀⠀⠀⠀⢀⣀⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢻⣿⠿⣿⣿⠿⣿⠻⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⣀⠠⠞

⠘⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢠⠇⠀⠀⢸⠁⠀⠀⠀⢸
⠀⢷⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡼⠀⠀⠀⡿⠀⠀⠀⠀⡾
⠀⢸⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⠇⠀⠀⢠⡇⠀⠀⠀⢠⡇
⠀⠈⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⢞⣡⣄⠀⣀⣠⣤⢸⢳⡘⡇⠀⣠⠏⠀⠀⠀⠀⠀⣼⠀⠀⠀⣸⠀⠀⠀⠀⢸⠁
⠀⠀⢿⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠁⠀⠀⠀⠀⢀⣄⠀⣦⣹⡀⠈⠉⠀⡟⢸⢿⣿⢸⣾⠀⠳⠇⣰⠃⣀⠀⠀⠀⠀⠀⠙⢄⡀⠀⡟⠀⠀⠀⠀⡿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡄⠘⠒⠚⠉⢳⠀⣿⣾⡌⡇⢿⠛⠃⠀⠀⠰⣇⡞⢸⣿⠘⠟⠀⠀⠀⣧⢎⡟⣼⡇⣠⠀⡾⠓⠦⡵⢸⠃⠀⠀⠀⢠⡇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⡄⡇⡇⣧⠻⡼⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⣿⣰⢻⢇⡇⠀⠀⠀⡾⠀⠀⠀⠀⢸⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⣤⡀⠓⠃⠙⣋⡤⠶⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠲⠦⣤⣙⠁⠈⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⡾
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠃⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⣠⠀⠀⣀⠀⣀⡀⢳⡀⠀⠀⢀⡞⠁⠀⢠⠀⢀⡄⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⡇
⠀⠀⢰⣧⡟⡇⢸⣿⡀⡏⣇⠈⣇⠀⠀⣾⠀⠀⠀⡾⠶⢾⢡⡾⠃⣾⢸⡖⣹⠀⠀⢸⡴⣧⠾⡆⣼⠟⠐⠃⠀⡼⠀⣠⢎⣤⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠘⡿⠁⣇⠈⡟⡇⢷⠘⣆⣹⠀⠀⢷⠀⠀⠀⠁⠀⠈⠈⠙⠁⠉⣼⠉⠁⠀⠀⠈⠀⠈⠀⠉⠙⠉⠈⢀⡴⠁⠐⠧⠋⢸⠇⣶⡀⣄⢠⡄⢸
⠀⠀⠀⠀⠀⠙⠾⠃⢷⡼⠀⠈⠁⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⢸⢸⣿⢡⣿⢸⣿⢸
⠀⠀⠀⠀⣠⢤⡀⠀⠀⣠⠦⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠶⠶⠶⠶⠶⢦⠀⢰⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⢿⣸⢿⡞⠻⠞
⠀⠀⠀⠀⠧⣄⣙⣆⠀⠙⢄⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡀⢀⠀⠀⠀⠀⢳⡼⠀⠀⠀⢠⢶⢰⢿⣤⣆⢀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⢉⣀⣀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢀⣄⡏⣷⣿⣿⢻⠆⠀⠀⠀⠀⠁⠀⠀⠀⢸⡎⠿⢸⡇⣧⢿⡇⠀⠀⢀⡴⠋⣱⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢮⡀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⠻⠇⠿⠛⠏⢸⣀⣀⡀⠀⠀⠀⠀⠻⢍⡛⠁⠀⠀⠀⠀⣼⠀⠀⠀⡾⠴⠚⢁⣴⣩⡤⠴⠃
⠀⠀⠀⠀⢰⡛⠢⡀⠉⠉⠛⠛⠀⡴⠒⢦⡀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣉⡼⠃⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢰⠃⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠉⠛⠛⠀⢀⡤⣄⠀⠓⠦⠤⡽⠆⠀⠈⡇⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡟⠀⠀⠀⣠⣋⣠⠴⠃⢀⡤⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠓⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡭⢉⣳⣤⡀
⣿⠻⠿⣟⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⠛⠛⠛⠛⠛⣋⣾⣿⣿⡿⢻⠛⠛⢳
⣿⢰⠖⠤⠤⣈⣉⡓⢲⡄⠀⠀⢀⡴⢚⠉⠈⠻⡍⠙⠲⢦⣤⣴⠙⠛⠀⠈⠑⣿⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠈⣿⢸⡇⢀⡴⢋⡶⣯⣳⠀⠀⠀⠀⠀⠀⢹⠈⡆⢸⣧⣀⡼⠋⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠀⣿⢸⣷⡋⠀⠘⠓⠛⠁⠀⠀⠀⠀⠀⠀⠈⣧⢿⣿⣧⡈⠀⠀⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠀⣿⡼⡿⠁⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡝⣄⠀⠀⠀⢸
⣿⠘⠛⠿⠶⢶⣴⣿⣽⠃⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡯⢗⣂⠈⣧⠀⠀⢸
⣿⠀⠀⣀⡤⣴⢲⢷⣿⣀⡀⠀⣾⡿⡸⢀⡀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣭⣴⣶⢸⠀⠀⢸
⣿⠀⠀⣒⣿⠏⣼⢸⣿⠖⠂⠀⣿⢣⡇⣸⠀⠀⠀⠀⠀⠀⣼⡿⠟⠉⠉⠛⠛⠾⢦⣀⣸
⣿⠀⠀⠀⠀⠀⠸⡎⣿⠀⠀⠀⣿⢸⠇⡟⢸⠀⠀⠀⢀⡼⠋⠀⠀⠀⣠⠒⠒⠦⡀⠀⠉⠑⠢⢄⡀
⣿⣿⢰⠞⠓⢠⡶⢿⡸⣆⠀⠀⢹⣼⡴⢗⣯⠟⢝⡤⠊⠀⠀⠀⣠⠾⠁⠘⠀⢠⣾⡆⠀⠀⠀⠀⠈⠑⠢⣀
⣿⠿⠘⢦⣀⢸⣟⡛⣧⣻⣷⣴⣾⣵⣾⣷⣿⣠⢎⣠⡖⠚⠓⠲⣽⣄⣀⠀⠐⣟⡿⠁⠀⠀⠀⠀⠀⠀⣠⡞
⡿⠀⠀⠀⠀⢀⣩⣵⣺⣵⣾⡇⠀⣿⣿⣿⣿⠗⣻⠿⢤⣄⡀⠐⢌⣉⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⠏⢁
⡗⠀⣀⢤⡺⠟⠊⠁⢸⣿⣟⡥⠤⢿⣟⣡⠾⠋⠁⠀⠀⠈⠙⢦⡀⠀⠉⠉⠳⣄⠀⠀⠀⠀⠀⣴⡿⠁⠀⡼⢄
⣧⢾⠟⠁⠀⢴⣒⡟⣸⣿⣿⣆⠀⣰⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⣼⠃⠀⠀⣠⣾⠉⠀⢠⠞⣠⣾
⡇⠃⠀⠀⠨⠽⢞⡣⠟⣫⣾⣝⠋⠁⠀⠉⠓⢦⡀⠀⠀⠀⢀⡴⣿⣲⡄⢀⣴⡃⠀⣠⣾⠏⠁⢀⡴⣣⣾⠟⠁
⡇⠀⠀⢀⡠⠖⢉⡴⠛⠉⠀⠉⠙⠶⣄⠀⠀⠀⢳⡤⢔⣾⠗⢮⣁⠈⣾⡏⠀⣹⣾⠿⠉⠀⡰⢋⣴⡟⠁
⣇⡤⠖⢉⡠⢚⣡⠔⠉⠀⠀⠀⠀⠀⢈⣷⢴⣾⣵⣾⣟⣁⣤⣾⣯⢿⣿⣠⣾⢿⡇⠀⡠⢊⣴⡿⠋
⡏⣠⢖⣭⠖⣫⠔⢊⠀⠀⠀⣀⢴⣺⡿⠞⠋⠁⣈⣩⣽⣿⡿⠟⢁⣴⣿⡟⠋⢀⡷⠊⣠⡿⠋
⣿⣿⣫⠶⣋⠴⢊⠡⢀⢔⠽⣪⠕⠉⣠⡤⠶⠟⠻⣏⢉⣠⣴⣿⠻⠻⣿⣀⡴⠋⣠⣾⢿⠁
⣿⣫⡵⣋⡵⠚⠁⢰⡷⠁⣼⣿⣶⠞⠀⠀⢀⣠⣴⢿⠟⠃⠃⠀⣀⣴⠟⠉⢀⣾⠟⠁⢸
⣿⡵⢚⡥⢒⠁⠀⣸⡷⠊⠛⠛⠁⣠⣴⡞⠟⠋⠉⢸⣀⡤⠖⠋⠁⣀⣬⣿⠿⠃⣀⡴⢿
⣿⣚⡵⢚⡡⠔⠋⢹⣇⣀⡴⡶⠟⠙⠘⠀⢀⡠⠔⠋⠁⢀⣤⣶⡿⠛⠉⣠⡴⠛⠉⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣤⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣡⠔⠊⠁⡾⠛⠛⠛⡻⡇⠉⠑⠬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣿⠏⠁⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡾⠃⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣇⣀⣀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⣀⣀⣘⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠧⠼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠦⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠈⢤⠖⠒⠉⠛⠒⠢⢤⣀⡀⣀⡾⠁⠀⠀⢀⡿⠁
⠀⣤⣿⣿⠒⢦⡀⠀⠈⡿⣧⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⢀⡬⠋⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⣠⠀⠤
⠘⠀⠈⠀⠀⠀⡇⠀⠀⢳⡘⢷⡀⠀⠀⠀⠈⣹⣷⣶⣶⣿⠉⠀⠀⠀⠀⣠⡼⠏⠀⢀⡀⠀⠀⢯⠀⠀⠂
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠙⢫⣝⡷⣦⣀⠀⣿⣷⣷⣶⣿⠀⠀⢀⣠⣶⡕⠀⠀⠀⢿⡈⠀⠀⢀
⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⣠⠼⠯⣿⡶⣭⣷⣿⣿⣿⣿⣷⣒⠈⢁⡘⣥⡀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡀⣀⣀⣀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⠃⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠈⣿⣏⠀⠈⠁⠈⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡆⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢀⣀⣇⣀⠀⠀⢸⡇⣠⠤⣷⣠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⢿⣤⠟⠀⠀⠸⡇⠘⠂⢹⣇⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⣄⠨⣤⡰⠀⠀⠀⡇⠀⠠⠀⢻⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⢗⡀⣀⡾⠀⠀⠈⣷⠆⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀⣰⣿⠋⠀⢠⢓⣄⣀⣤⣿⣀⣀⢀⠂⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⠁⢢⣼⣿⣿⣿⡿⣿⡇⡀⢨⠵⣶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢾⣿⣿⠁⠀⢤⣾⣿⡿⠛⠀⣀⣸⡀⠈⠹⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⣿⣿⣅⣀⣰⣿⠟⡿⢧⠀⣸⡏⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠄⠤⠠⣿⣿⡇⠀⢸⣿⠀⠇⠈⠁⢸⡄⠘⣧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣗⡀⢸⣿⠆⠆⠸⢃⣿⡄⣦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⢿⣿⠄⠄⠠⣿⣿⣏⠀⠈⠛⠃⠀⠠⣿⣿⣄⣿⡗⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢿⠀⠀⢀⣿⣿⣿⠀⠀⣀⣀⣤⣴⣿⣿⡿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⣺⣿⣿⡿⡿⠻⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠘⠉⠩⠤⠤⠀⠈⠉⠁⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠾⣿⡿⠷⠶⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⣠⠞⠉⢀⣴⠗⠀⢀⡀⠀⠀⣄⡀⠀⠒⣠⡀⠀⢠⣀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠁⣀⠀⣿⠃⠀⠀⢹⡟⠀⠀⠉⠛⠃⠛⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⠀⡿⠀⠀⣿⠀⠀⠓⣿⡗⠒⠃⢺⣿⠃⠒⠀⣿⠷⠶⠿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢠⡄⡿⠐⠀⣷⠀⠀⣿⡀⠀⠀⣿⡇⠀⠀⠈⣿⠁⠀⠀⢸⠄⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⢻⡿⢿⡦⣤⣧⣀⣀⣿⠀⠀⠀⣿⡁⠀⠀⠈⣿⠁⠀⠀⣸⣄⣤⡾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢛⠁⣦⠀⣿⠀⢨⣟⠛⠓⠛⣶⣿⣷⠶⣶⣾⣿⠓⠛⠛⢿⡁⠀⢱⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⣿⠠⣿⠀⢸⣇⠀⠀⠀⣿⣿⡇⠀⢸⡏⠈⠀⠀⠀⢘⡅⠀⣸⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⠀⢸⣿⠀⠀⠀⣿⣿⡇⠀⠸⡏⢺⠆⠀⠀⠘⡃⠀⣸⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⠀⠸⡟⠀⠀⠀⣿⣿⣇⠀⠘⡇⢺⠄⠀⠀⢰⡃⠀⢹⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢰⡿⠀⠀⠀⣿⢿⣿⠀⢀⡧⢸⡀⠀⠀⠸⠆⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣾⣿⣿⣷⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⣶⠀⢰⣶⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⣉⡀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢈⣿⠀⠀⠘⠉⠋⠉⠉⠉⠉⠉⠙⠁⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⠀⠘⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⢶⠀⠠⡶⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡄⠛⠶⠶⠃⣠⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠹⣿⠀⣿⣿⣿⠏⢡⣤⣄⠹⣿⣿⣿⣿⣿⠏⣠⣤⣄⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢀⣿⣿⠿⣿⣿⡀⢿⣿⡿⠀⣿⣿⠿⣿⣿⡀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣤⣈⠉⢠⣴⣄⠉⣁⣤⣀⣤⣄⠉⣠⣴⣄⠉⣁⣤⣀⣤⣀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠻⣿⡿⠀⣿⣿⣿⣿⣿⡀⢻⣿⡿⢀⣿⣿⣿⣿⣿⡀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⢋⣁⣤⣤⣤⣄⡉⠛⠿⠟⢋⣁⣤⣤⣤⣄⡉⠻⠿⠟⢉⣡⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧




⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣷⣤⠀⢠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠶⠶⢤⣤⣄⣀
⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⣿⣿⠿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⣦
⣿⣷⠶⠶⠶⢾⣿⡇⠀⠀⠀⠀⠀⢀⡼⠻⣄⠀⠀⠈⠻⣦⣀⣀⣤⣤⣤⣄⣀⣠⡾⠋⠀⠀⣠⡾⢳⡄
⣿⣿⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⡾⠁⠀⠈⠳⣄⠀⣠⠞⠋⠁⠀⠀⠀⠀⠉⠻⣦⡀⣠⡾⠋⠀⠀⢻⡄
⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⠀⠀⣷
⣿⣿⡀⠛⠛⢻⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢹
⣿⡿⠋⣉⡉⢻⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠃
⣿⡇⢸⣿⡇⠈⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⣿
⣿⡇⠸⣿⡇⢀⣿⡇⠀⠀⠀⠀⠈⢧⠀⠀⠀⣠⠞⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠻⣦⡀⠀⠀⣸⠇
⣿⣷⣤⣭⣤⣼⣿⡇⠀⠀⠀⠀⠀⠈⢧⣠⠞⠁⠀⠀⣠⡾⠛⠓⠶⠶⠶⠖⠛⠻⣤⡀⠀⠈⠻⣦⣰⠏
⣿⡇⠀⢻⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣠⡾⠃
⣿⡇⢠⡈⢿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟⠋
⣿⡇⢸⣧⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⣤⡤⠤⠶⠖⠛⠉
⣿⣷⣾⣿⣷⣾⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠤⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠄⢂⠀⠀⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⢒⣤⠒⠒⠒⠛⠋⠳⡄⣄⣀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣱⣿⠰⠀⠿⠀⠀⡀⠀⠘⢮⡟⠳⠄⠀⠌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡰⣹⢡⡊⠀⡠⢮⣬⣌⠳⡀⠀⠑⠀⠀⠘⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠅⠀⠙⠟⢀⢇⠹⢿⡿⣈⠆⠀⣰⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠂⠄⠁⠂⡀⠀⠑⠱⠐⠐⠁⠀⠠⢳⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⢒⣣⣦⣀⠉⠂⢀⠀⠸⣿⠇⠀⢰⠇⠀⢀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢛⠻⢿⣿⣿⣿⣶⣦⣬⣉⠀⡐⠊⠋⡐⢠⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⢍⠐⠂⠈⠭⠍⠭⠅⢠⠟⠁⠀⣧⢷⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡬⣐⠂⢰⣶⡖⠈⢀⡠⢆⠀⡈⣼⡆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⡏⠲⠒⠈⢩⡍⣭⠭⢴⡎⠉⠁⠀⠈⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢰⢹⣫⠴⠴⡇⣷⣚⣹⣂⢰⡄⠀⠀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⣿⡄⠀⠀⠀⠀⠀⠈⣘⠤⣏⣉⣧⢹⠭⢶⠟⠁⠘⠒⢩⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⣿⣿⣿⣿⣇⢸⣗⣀⠀⠀⢀⣀⠠⢀⠆⠈⢲⠺⣼⣋⠽⠃⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢹⣿⣿⣿⣿⡿⢿⡇⠘⠛⠋⠉⠐⠂⠀⠀⣀⡘⣗⣷⡔⠒⠢⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⣿⣿⣿⣿⣷⣾⣿⣀⣀⣠⣤⣤⣤⣤⡞⠀⠈⢉⠁⡴⡦⠀⣶⠦⣾⣦⣤⣤⣤⣤⣀⣀⣀⣿⣿⣿⣯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⣥⣴⣶⣶⠶⠶⢯⠀⡷⠾⠷⠷⢷⠀⡷⡶⠎⢴⠤⢤⣤⣄⣈⣉⣿⣿⣿⣿⣿⣆⢳⣄⡀
⠀⠀⠀⠠⠴⠒⢛⣉⡭⡌⢻⣿⣿⣿⣿⠿⢛⣿⣏⣠⣠⠤⠤⠀⠖⡆⡼⠐⠒⠒⠒⠲⠄⠶⣷⣴⡌⠀⠀⠒⠀⠦⣽⣻⢿⣿⣿⣿⣿⠘⣇
⠀⠀⢀⡴⠖⠋⣩⣴⣾⠛⠘⠿⣷⣤⣴⣶⡿⠋⠀⠀⠀⢀⡄⠀⠀⢣⠃⠀⠀⠀⠀⢀⠃⠄⠙⠛⠁⠀⢠⡀⠀⠀⠈⢻⣿⣮⠹⣿⡿⠶⠟⠛⣶⣄
⠀⠀⠻⠄⠀⠈⢿⣿⣧⠀⠀⠀⠀⠉⠻⣿⣄⡀⠀⠀⢀⣀⠙⠒⠀⠁⠀⠀⠀⠀⠈⠁⢈⣀⣀⠀⠀⠠⠚⠁⠀⣀⠤⣚⣼⣋⡴⠟⠁⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣙⡛⠦⠄⠀⠀⠀⢀⡉⠙⠓⠲⠦⠬⠍⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠠⠤⠴⢖⣛⡛⠉⠁⢀⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⣦⣤⣍⣙⣓⠒⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠒⠒⢚⣋⣉⡥⠤⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠰⣿⣿⠆⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣾⠿⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠿⣷⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡇⠀⢿⡶⠶⣶⠶⠶⠶⠶⠶⠶⣶⠶⢶⡿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⡉⢹⡇⠀⢸⡇⠀⣿⠀⠀⢘⡃⠀⠀⣿⠀⢸⡇⠀⢸⡏⢉⡉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠃⢸⡇⠀⢸⡇⠀⣿⠀⢰⠋⠙⡇⠀⣿⠀⢸⡇⠀⢸⡇⢈⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠃⢸⡇⠀⢸⡇⠀⣿⠀⢸⠀⠀⡇⠀⣿⠀⢸⡇⠀⢸⡇⠘⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣀⣤⣤⠶⠶⠶⠶⠶⠶⣤⣤⣈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠀⠈⠻⣦⣀⣤⠴⠶⠶⢦⣤⣄⣴⠟⠁⠀⠀⠀⠀⢙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⠋⠻⣦⡀⠀⠀⣠⡶⠋⠉⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⢀⣴⠟⠙⣿⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣼⠃⠀⠀⠈⠻⣦⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⠟⠁⠀⠀⠘⣧⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢠⡟⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢹⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣧⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣸⠇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⢻⡄⠀⠀⢀⣴⠟⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣦⡀⠀⠀⢠⡟⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣿⣄⣴⠟⠁⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠀⠀⠈⠻⣦⣠⣿⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠀⠀⠀⢀⣴⠟⠙⠛⠲⠶⠶⠞⠛⠋⠻⣦⡀⠀⠀⠀⠀⣸⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⠉⠛⠛⠶⠶⠶⠶⠶⠶⠟⠛⢉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠛⠛⠋⠉





⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠈⠉⠂⠀⠀⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⢤⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⣤⣤⢠⣤⡄⠀⠀⠀⠀⠀⠘⣤⣤⡄⣤⣤⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣉⣉⠈⣉⣁⠀⠀⠀⠀⠀⠰⣙⣛⡁⣙⣛⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⡇⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠛⠛⠘⠛⠛⠀⠀⠀⠀⠀⢸⠻⠿⠇⠻⠿⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠆⡤⢠⢠⠤⠄⡄⠤⢠⠀⠄⠀⠠⠀⠀⠀⠄⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⡀⠀⢸⠀⡆⡇⢠⢰⠄⡄⠄⠠⠠⠀⠀⠄⠠⠠⠀⠄⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀
⠐⠁⠈⠈⠀⠁⠁⠀⠈⠀⠀⠁⠀⠊⠀⠁⠁⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢂
⠠⠤⠀⢤⠀⠀⠠⢤⠀⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠠⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠨⠀⠀⠀⠀⠀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠑⠢⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠈⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⡀⠤
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠠⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠘⠀⠀⠀⠀⡀⠄⠊⠁⢀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⡀
⠀⠀⠀⠀⢀⠤⠒⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣤⣤⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⢉⣩⣴⣿⣿⣿⠉⡇⢹⣿⣿⣦⣄⡀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⣶⣿⣿⣿⣿⣿⣿⠃⡇⢸⣿⣿⣿⣿⣿⣶⣄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⣼⣿⣿⣿⣿⣿⣿⠿⠟⠒⠳⠿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣉⣉⣙⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠿⠟⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣤⣤⣤⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⢿⣿⣿⣿⣿⣿⣷⣦⣤⢤⡤⣤⣴⣾⣿⣿⣿⣿⣿⡿⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠙⢿⣿⣿⣿⣿⣿⡇⢸⢠⣿⣿⣿⣿⣿⣿⡿⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣌⣛⣿⣿⣿⣇⢸⡀⣿⣿⣿⣿⣛⣁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⣀⠀⠒
⠀⠀⠀⠀⠢⠐⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡰⠈⠀⠠⢀⡀⠸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠀⠁⢂⣻⣟⡺⢇⠆⠂
⠀⠁⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⡱⠄⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡀⠀⠀⠁⠸⠥⠊⡀⠰⠄⠀⠔⢊⡁⠀⠶⠄⠐⠀⠀⠀⠶⠤⠊⠀⠀⠤⠂⠀⠀⢀⡀⠠⠄
⠠⠄⠀⠤⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠀⠀⠀⠀⠧⢜⡠⠜⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠐⠀⠀⣠⠤⣤⡀⠀⠀⠤⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠈⠁⠀⠰⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢁⣈⣇⢀⠠⣤⢤⡄⠚⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢠⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠗⠯⢽⣉⡿⠀⠡⠤⠶⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠠⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠰⠀⢈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠘⠀⠀⢠⠇⠐⠋⠐⠄⠑⠀⠐⠂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⣏⡀⢃⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⣤⠀⠀⡄⠄⠀⠀⠁
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡄⢠⠀⡄⠰⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢶⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠃⠀⠀⠈⡇⠀⠆⡿⠀⢷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠓⢃⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠁⠀⠘⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢺⠉⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠄⠁⠈⠐⢈⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠂⠀⠖⢁⡀⠓⠀⠀⠀⠚⡁⠰⢀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⢢⠀⠀⢠⡀⠀⠀⠀⠀⠁⡘⠃⠀⠡⠀⠀⠀⢀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡶⠘⢀⠓⠀⡃⠐⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢡⠄⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢮⠴⠖⢚⡰⠀⡁⠊⠀⠀⠀⠀⠀⠀⠀⠈⢠⡛⡤⡤⣤⠤⣜⠂⠀⠁⠔⠈⠁
⠐⠀⠀⡀⠀⠀⠔⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠆⠉⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠗⠠⠶⠔⠋⠐⠁⣈⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠲⡀⠂⢱⡉⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⣀⠈⠃⣀⣰⠆⠀⠠⠄⠠⠀⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠋⠉⠀⠁⠀⠀⠄⠔⠀⠀⠐⠒⠘⠇⠘⠀⠀⠀⠀⢸⠀⠀⠁⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠰⠀⠀⠠⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠶⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠤⠄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡰⠤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠆⠀⠀⠀⠀⠀⠂⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠃⠐⠒⠒⠀⠉⠑⢤⡄⠀⢰
⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠊
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐⢆⠐⠒⠂⠀⠒⠘⠂⠘⠂⠒⠓⠀⠀⠐⡄⣀⣀⡀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠦⠤⠤⠄
⠀⠠⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐⠈⠲⠶⠦⠆⠀⢰⠀⡆⠀⠀⠀⠒⠁⠀⠀⠌⠁⠀⠀⢀⠈⠉⡋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠉⠀⠳⠖⠀⠀⢑⠛⠐⠋⠁⠀⡄
⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠠⠄
⠀⠀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢠⣀⣠⣤⣾⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠆⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡅⠀⢀⡀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠁⠀⢸⣀⠀⠀⠀⠀⠀⠈⠁⠀⢈⠹⠿⠿⠟⠛⠛⠉⠉⡁⠄
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢸⢿⣿⣷⣶⣶⣶⣾⣉⣁⣠⣤⣤⣶⣶⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⡿⣛⣛⣛⣛⣛⣋⡉⡉⡉⢉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣤⣼⣼⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⣀⡀⣼⣿⢿⢟⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⠛⠛⠚⠒⠒⠒⣒⠒⠚⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣿⣿⣿⣿⢹⣿⣿⣿⣿⠛⠉⣿⣿⣿⡏⣿⣿⡁⢸⡏⠁⣿⡏⠻⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠆⠀⠸⣿⣿⡇⣿⣿⡀⢸⡇⠤⣿⡇⠀⡇⠘⢦⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠸⠿⠿⠿⠿⠀⠀⠀⣿⣿⣷⠭⠭⠤⠼⠇⠤⠤⠭⠶⠇⢀⠀⢹⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⢻⣿⣾⡶⠶⠶⠆⠀⠀⠀⢸⣿⣿⣷⣶⣆⣀⠶⠶⠶⢶⣶⣦⣀⠀⢼⠀⠀⠙⢦⣄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⡟⠦⠄⡀⠀⠀⠀⢸⣿⣿⡆⠋⠋⠀⢀⣠⠴⠂⠈⠻⣿⣧⣼⠀⢄⠀⠀⡇⠑⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣼⡇⠀⣦⡄⠀⠀⠀⢸⣿⣿⠆⢀⡠⣾⣿⣿⠀⢠⣀⣀⣈⣙⣿⣷⣄⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⢳⠀⠀⠀⣾⣿⣿⡀⠀⠉⠛⠿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⣿⣿⣿⣿⡿⠀⠁⠀⡀⠀⢀⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠷⢦⣶⣿⣧⡤⠼⠿⠿⠿⠟⠛⠛⠛⠛⠒⠀⠘⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠉⠙⠳⢦⣄⡀⢀⠀⠀⠛⠒⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⣀⠠⠤⠤⠄⣀⠀⠀⠀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢾⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠘⠢⠄⣀⣀⣀⣀⠤⠦⠀⠀⠀⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡤⠀⣀⠀⠀⡀⠠⠤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣄⡈⠛⠛⠁⢿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣽⣿⠿⠉⠛⢃⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣠⣾⣿⣿⣿⡶⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡆⠀⠻⢿⣿⣿⢻⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣧⣴⡿⠛⠁⠀⠀⣼⣿⠁⠀⣼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣄⠀⠀⠀⠀⠉⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣦⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠈⢛⣿⣿⣿⠩⣿⣿⡟⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠔⡻⢿⡿⢿⣿⣿⣿⣿⣷⠀⣧⠀⢸⣿⣿⣿⡇⣿⣿⡇⠀⢠⡇⢰⣿⣿⣿⣿⣿⠿⣿⠿⡓⠼⠿⠿⠛⠁
⠀⠀⠀⠀⣘⢧⢶⡀⠀⠀⠀⢶⣾⣿⣿⣍⠺⣿⣛⣿⡿⣻⣿⣶⣿⠀⢸⣿⣿⣿⣷⣿⣿⣷⠀⢸⣷⣾⣿⡻⣿⣝⣿⡷⢊⣽⣿⣿⣶⠆⠀⠀⢀⡶⡼⣃
⠀⠀⠀⠀⠹⣗⣪⢷⡄⠀⠀⠀⠈⢉⣥⣶⣿⣷⢌⣽⡒⢻⡼⢫⣟⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄⢘⣯⠻⣽⠓⣺⣍⢼⣿⣷⣦⣍⠉⠀⠀⠀⢠⡾⣑⣲⠏
⠀⠀⠀⠀⠀⠘⢷⣉⠟⣦⡀⠀⠀⠀⠈⠉⠉⣴⣿⡿⢣⣿⡿⢹⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⠹⣿⣧⠻⣿⣷⡌⠉⠉⠀⠀⠀⢀⣴⠻⣉⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠛⣯⣜⡹⢷⣤⣀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠿⠋⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠈⠻⠇⠙⠛⠃⠀⠀⠀⠀⠀⣀⣤⡾⢏⣣⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣧⣎⡏⡛⢳⠶⠀⢀⣀⣀⡀⠀⣀⣀⣀⠀⣀⠀⣀⡀⢀⡀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⡶⡞⢛⢫⣱⣼⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠳⠎⠀⢸⣿⣽⡿⢰⡟⠉⢻⡆⢿⣶⡿⣧⣿⠁⣿⡯⡭⠀⣿⣯⣽⡇⠀⠀⠻⠞⠋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠈⠻⠶⠿⠃⠘⠿⠁⠻⠿⠀⠿⠶⠶⠆⠿⠇⢻⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣦⣤⣤⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣀⢸⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠿⠿⢿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣠⣤
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣷⠀⠘⢿⣿⣿⣿⣿⠟⠀⢰⣷⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠈⠉⠉⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦


⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣤⣄⣀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣀⣸⣿⣇⣀⣸⣿⣀⣀⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⣇⣀⣸⣿⣀⣀⣿⣿⣀⣸⣿⡇
⠀⣿⣿⠿⢿⣿⡿⠿⣿⣿⡿⠿⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡿⠿⣿⣿⡿⠿⣿⣿⠿⢿⣿⡇
⠀⣿⣿⣀⣸⣿⣇⣀⣼⣿⣄⣀⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⣇⣀⣼⣿⣄⣀⣿⣿⣀⣸⣿⡇
⠀⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⡇⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿
⣙⣉⣋⣋⣙⣁⣀⣸⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣀⣀⣙⣉⣋⣉⣉
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⣂⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⠾⠛⠋⠉⠙⠙⠳⢤⡀⠀⢠⠒⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡱⠀⠀⢨⢀⣴⡾⠿⠶⣶⣄⠀⡁⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡠⠊⠀⡨⠆⠐⡿⠁⣠⣴⣦⣀⠙⠦⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠪⢴⣈⡄⠀⠘⢫⠁⢉⠛⠃⠀⠀⣠⠂⠁⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡟⠛⠋⣿⣷⣦⣄⡀⠀⠀⢳⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠉⢻⣿⣷⠀⢀⣿⣿⠟⠉⠻⣦⠀⠀⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣀⢀⠿⠛⠉⠙⠛⢏⠀⣀⣴⣿⣷⠀⢠⠊⠁⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⣷⠟⠿⠿⣿⠋⠀⠀⠀⠀⠀⠀⢹⡿⠿⠿⢻⡆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⠀⠐⣿⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⢸⡿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⠸⣿⣿⣿⡿⠧⡀⠀⠀⠀⠀⣠⠾⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠻⣟⠁⠀⢠⣿⣷⠒⠲⣾⣧⡀⠀⢉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢧⠀⠀⠙⠷⣤⣿⣿⡏⠀⠀⣿⣿⣷⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠻⠷⢶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠜⠀⠀⠀⢀⡔⠀⠐⣀⠀⠀⠀⢢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠠⣆⠙⠻⢷⠿⠟⢡⣦⠀⠀⢞⡡⠊⠁⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠷⣦⣤⣤⣶⡿⡱⠄⠀⠈⠀⡠⠊⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢦⣀⡀⠉⠉⣀⣠⡼⠀⠀⠀⠀⠀⠄⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠙⡋⠯⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠀⠀⠀⠀⠐⡨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢓⠁⠀⠀⠀⠀⠀⠐⠰⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⢶⣒⣒⣛⣛⣿⣿⣿⣾⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣶⣻⠁⠀⣤⡏⠀⠀⡀⣴⢰⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠒⠶⢬⡟⡉⠀⠀⣹⣦⣀⠀⢀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣷⣄⡀⣸⠏⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠱⠈⠀⡤⢿⣟⢦⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⢇⣤⡾⠁⠀⠙⢷⣹⡇⠈⠙⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠳⠈⣟⠁⠀⠀⠀⢠⡈⣿⡇⠀⢠⠈⠉⢂⢺⣿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠁⠀⣏⡔⠀⠀⠀⠀⢻⣿⡇⠀⢸⠀⠀⠈⣸⡟⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢀⣏⠆⠀⠀⣿⠁⠀⠀⠀⠀⠹⣿⠁⠀⡘⠀⠀⣠⡿⣿⠄⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣛⠃⠀⠀⣇⠘⠀⣀⣤⣤⣤⣼⠃⢀⣁⣴⡾⠋⡁⣿⠇⠀⠐⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠠⠄⠐⠁⡁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⠿⠛⠛⡩⠿⡟⢿⠿⣥⠤⠤⣤⣿⡥⠀⢀⣤⢠⣿⣄⣠⠴⠒⠋⠙⢾⣿⣿⣿⡄⠀⠀⠐⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣦⠴⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢻⣽⠛⠋⠁⠀⣴⣾⣿⣿⡆⠈⠀⠀⢨⣿⣿⣿⡇⠀⠈⢁⠀⠁⠀⢀⢐
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⢻⠧⣥⡤⠤⠀⠀⢀⡀⠀⠀⠀⡀⠀⣌⣿⠀⠀⠀⠀⠈⠻⠿⡿⠋⠀⠀⣠⡼⠿⠿⠟⠃⠄⠠⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⢿⠈⠇⠑⠈⠉⠛⠉⠛⠛⠛⠷⠶⣶⠦⣼⣿⣿⣿⣷⡤⠀⣤⡰⠾⡷⢶⡖⠋⠁
⠀⠀⠀⠀⠀⢀⣾⣅⠸⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢸⣧⣾⣿⣿⠿⢹⠏⠛⠶⠖⠒⠛⠉⠉
⠀⠀⠀⠀⣰⠿⣭⣛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡿⠋⠀⢸⣿⢷⡄
⠀⠀⠀⠘⢷⡋⠀⠉⠛⠷⢯⣯⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣧⣶⣚⣯⠷⠛⢝⣦
⠀⠀⠀⠀⠀⠙⣶⢒⢶⣒⣀⣀⣈⡉⠉⢉⣽⣿⣿⠟⠛⢻⣿⡟⡉⣀⡤⠴⣟⢳⡀
⠀⠀⠀⠀⠀⢀⡇⣾⣿⠦⠀⠈⠉⣳⣶⣿⣿⣿⣥⣤⠖⡚⠁⠓⠚⠁⠀⠘⠉⣣⢻⡄
⣀⣀⡀⠀⠀⣼⢡⣯⣷⣴⣶⣾⣿⣿⣿⣿⣣⠋⠙⢦⡻⣼⠮⡂⣀⢀⠀⠀⠀⠛⠚⡷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⡛⠭⡶⢃⡏⠀⠀⠀⠙⠲⣝⡿⣯⣧⣶⡏⠀⠀⠈⣳⡸⡄
⣿⠋⠉⢩⡟⣽⡿⢭⡝⠂⠀⢀⣾⣯⡖⠋⠀⠀⠀⠀⠀⠀⠈⠹⣷⣮⣽⣴⠤⠂⠀⢚⡷⡹⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠚⣩⢞⣋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⢃⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠑⢈⡿⣿⠾⠁
⠀⠀⠀⠀⠀⠀⣰⢃⠎⡍⢆⠳⡀⠀⠀⠀⠀⠀⠀⣰⠣⠀⢸⠁⢸⡉⠮⡳⣄
⠀⠀⠀⠀⠀⡰⢣⠂⡐⠑⡀⢢⠻⡄⠀⠀⠀⠀⢀⡇⠇⡔⠁⠀⠀⠉⢧⠘⠪⠇⠀⠀⣀⣠⡤⠶⢛⡏⠙⠒⠒⠒⠒⠛⠉⡟⠳⠦⣄⣀⡀
⠀⠀⠀⠀⡼⠡⠃⡜⠀⠀⠘⡄⠡⠹⡄⠀⠀⠀⢸⢰⢐⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⠥⠤⠒⠉⢁⠀⠀⠀⠀⠀⠀⠰⠉⠓⠀⠤⢼⠃
⠀⠀⠀⡼⢡⡇⣊⡉⣉⣉⣉⣉⣀⣥⠙⡄⠀⠀⢸⢸⢸⣦⣤⣤⣤⣤⣾⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣼
⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⢸⢸⠘⡖⠂⠀⠀⠂⢺⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡄⠀⠀⠀⠀⢼⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣹⠛⠲⠶⠖⠚⢻⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣠⠤⠤⢤⡤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠓⠦⠤⠤⠶⠊⠀⠀⠀⠀⡷⠒⠒⠒⠒⠲⠢⠒⠒⢒⡗⠒⠒⠒⠢⠂⠲⠒⠒⢲⡇
⠀⠀⢸⠁⣀⣀⣀⣽⣯⠀⣀⡆⠀⠀⠀⠻⠚⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⢠⡛⠛⠛⠛⠋⡀⡝⡇⠸⠀⠸⠘⠓⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠒⠀⠠⠀⠠⠄⠠⠒⠚⡗⠀⠠⠠⠄⠠⠀⠀⠒⢺⠁
⠀⠀⢹⢠⠀⠀⠀⠀⠀⠀⡔⠋⠰⠤⢤⠀⢄⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⠉⠃⡈⠉⠙⠐⠉⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠙⡖⠆⣖⡀⠒⣀⢛⣟⣹⠿⣿⣶⢬⡉⣉⣭⣷⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⡷⠤⡤⠀⣀⣀⡀⢀⡀⢀⣇⡀⡀⠀⣀⣀⡀⠤⡤⢴⡇
⠀⠀⠀⠀⡇⠀⠀⠉⠻⠛⠻⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠧⠼⠀⢠⡇⠀⠀⠀⠀⠀⠓⠤⣤⣄⣀⣈⢉⣈⣁⣨⣇⣈⣉⣀⢁⣀⣀⣄⠤⠾⠃
⠀⠀⠀⠀⠙⠢⢄⡀⠠⠀⠀⠁⠁⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠓⠋⠉⠉⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⣴⣦⣭⣥⣶⣶⣬⣭⣶⣦⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡽⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢷⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣯⡀⠀⠀⠈⠻⣿⠿⠟⠛⠛⠛⠿⣿⠟⠁⠀⠀⢀⣵⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢀⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⢿⣿⣿⠟⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠁⠀⠀⢀⣴⣿⣶⣤⠤⠤⣤⣶⣿⣦⡀⠀⠀⠈⣻⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢳⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⣞⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠙⠻⠟⠛⠿⠿⠿⠿⢛⠿⠟⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢧⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⢀⣤⠶⠛⠛⠙⢛⡓⠶⣤⡜⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣄⠀⢀⣴⠛⢁⠔⠁⠀⠀⠀⠀⠑⠈⠻⣆⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣾⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡯⣀⠃⠀⠀⣰⣾⢿⢧⠀⠀⠀⢠⣒⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⠈⠣⠀⢰⣿⠃⠀⠈⡇⠀⠀⠽⣫⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢭⠀⣿⡟⢀⢄⡀⡄⠀⠀⣧⡱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠿⠀⣿⣿⡇⠀⡱⡇⠀⠀⣷⡐⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢗⡀⠸⣿⣿⢉⣧⠇⠀⠀⢳⡲⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⡀⠙⢒⣍⠹⣿⡾⡟⢁⠁⢠⣯⣷⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⠀⡹⠋⠙⠟⠛⠒⠒⠚⠉⠉⢫⡏⠙⠛⢟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢩⠟⠛⠛⠻⠶⠾⠶⠚⠒⠒⢺⣏⣉⣥⣹⣽⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠧⣘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⠕⣢⣽⣟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣖⠟⠛⠻⠶⠲⠶⠶⠶⠶⠖⠒⠒⢚⡫⢥⣀⠤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣥⣄⣤⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢼⣚⣩⣤⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠷⠿⠾⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣟⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⠢⣀⠠⣀⠀⠀⠀⢸⣾⡠⠄⢰
⠀⠀⠀⠀⣶⣿⢾⠞⠡⡸⠊⠀⠀⠀⠀⠀⠈⠀⠀⠘⣮⣤⠀⠉
⠀⠀⠀⠘⠃⠈⢳⡠⢈⣣⡄⠀⠀⠀⢀⢦⣄⣆⠀⠰⠋
⠀⠀⡁⠁⠀⠀⠰⣏⣿⣿⣷⡀⠀⠀⡀⠀⠉⣿⠃⠘
⠀⠀⠀⠀⠀⠀⠸⢗⣺⣿⣿⣿⣶⠀⠉⠁⠉⠋⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠐⢼⣧⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⣠⣄⠀⠀⢹⣵⠙⠙⠿⡿⣿⡻⠇⠀⢏⡢⠀⢀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠀⠀⠐⠛⠺⣿⣦⠰⠁⠀⠀⠀⠣⠌⢮⡀⠀⠼⠥⠀⠷⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠇
⠀⠀⠀⠒⠟⠛⠛⠒⢶⣶⣄⠀⢀⡀⠀⠁⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⣭⣢⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢟⠂⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠈⢻⡛⠳⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⢞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⠀⠀⢳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢦⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⠀⠀⠀⠸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠿⠃⠀⠸⠃⠀⠀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠛⠀⠀⠀⠀⠀⢰⣯⣹⡆
⠀⠀⢠⣶⣦⠀⠀⠀⢀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠷⣦⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠈⠉⡛⠀⠀⠀⢿⠐⠊⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣷⠟⠀⣰⣦⣦⡀⠀⠘⣿⣿⣧⠀⠀⠀⣀⡀⠀⠸⠟⠃
⠀⠀⠀⠀⠃⠀⠀⠀⣸⡅⣼⡇⠀⢀⡀⠀⠀⠀⢀⣄⣀⢀⣤⣶⣿⣿⠀⣠⣿⣲⣿⣷⠀⢀⡿⣿⣿⣦⠀⣠⣖⣼
⠀⠀⣦⣤⣀⣀⠀⠀⠉⠉⠉⣦⡀⠂⠁⠀⠀⠠⡿⣏⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡀⢸⣇⢻⣿⣦⡀⠹⣽⣏⡀⠀⠁⠀⠀⢀⣀⣠
⠀⡘⣹⣿⣿⣿⣯⡿⠃⣠⣾⣿⠃⠀⠀⠀⢀⣿⡏⣨⣿⢻⣿⣿⣿⣿⣿⣤⣿⣿⣿⣦⡷⣾⣿⡈⣿⣿⡅⢸⣻⣿⣿⣿⠀⣤⣶⣾⣿⠏
⠀⠇⠙⠛⣿⡿⢃⡴⢿⣿⣿⡇⠀⡀⣠⠞⠟⠋⠀⠙⡉⢻⣿⣿⡿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⢀⡀⢰⡆⢈⣤⣶⣶⣦⣠⣦⣦⠀⠱⡇⠀⠀⠀⢠⣾⡇⢸⣿⣿⣶⣤⣾⣄⣾⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠈⠁⠀⣰⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠙⢆⣠⡤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡯⢻⣿⣿⣿⣿⠉⢸⣿⣿⣿⣿⣇⣻⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠉⠈⠁⠈⠉⠉⠉⠀⠋⠉⠁⠀⠈⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠈⠛⠛⠛⠿⢿⡟⠻⠿⠛⠛⠿⠿⠿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣙⡆⢀⣾⡆⠀⢳⣴⠇⢸⣤⣿⣤⡇⠀⣾⣆⠀⠻⣿⠓⣴⠞⠳⠄⣶⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠇⠸⠛⠻⠄⠀⠯⠀⠈⠿⠁⠿⠇⠸⠟⠻⠆⠀⠿⠀⠹⠦⠴⠆⠿⠉⠹
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⡀⠀⠀⢸⣧⠀⠀⠀⣸⡏⠀⠀⢠⢾⠟⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣠⠴⠒⢳⣷⢤⣴⣼⠿⡄⠀⢰⢻⠃⠀⡴⢣⠏⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠷⠚⠁⠈⠉⠛⢿⣽⠿⠟⣟⢻⡸⢠⠞⢠⠃⠀⣠⣴⡿⠋
⠀⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⣠⣴⣴⣀⠀⠘⣿⣀⣀⣀⣸⡷⠁⢠⣣⠔⢋⠝⠉
⠀⠘⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠘⢿⣟⢿⠀⠖⡟⠛⠹⠻⢿⡄⠀⠉⢀⠔⠁
⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣿⣿⡿⠁⠀⠀⠀⠀⣀⣾⣿⢮⣀⣀⣇⡀⢠⣄⣼⠇⠀⠀⠈⣉⣉⣩⠭⠽⠿⠛⠛⠋⠉⠉
⠀⣠⡴⠾⠛⠫⠭⠭⠭⣹⣿⠁⣿⡦⠀⠠⠤⣿⣿⣿⣟⣛⠈⠉⠉⠙⠻⢻⣏⠀⠀⠈⠉⠛⠲⣤⣄⣀⡀
⠀⠀⠀⠀⢀⠀⠐⡶⣶⣾⣿⣦⠊⠁⠈⢁⠞⢻⡿⣿⣿⠿⣧⣤⣠⣴⣄⣤⠟⠀⢀⡈⠻⢍⠉⠉⠉⠉⠉⠉
⠀⢲⣒⣋⣭⣽⣼⣴⣵⣷⣿⣿⡰⣄⠀⠀⠀⠀⠀⠹⣯⢀⡀⠉⠁⠉⠛⢣⠀⣦⠀⢿⠲⢄⡙⢦⡀
⠀⠙⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠷⣭⣓⣲⣤⠤⠤⣤⣈⡾⢭⣤⡶⣶⡣⢾⡀⡇⠳⣌⢧⠀⠉⠲⢌⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠟⠉⠉⠉⣹⡿⠛⠛⠓⡞⡿⠁⠘⣧⡇⠀⠘⢿⣧⠀⠀⠀⠈⠚⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⢀⡾⠋⠀⠀⠀⢰⡿⠁⠀⠀⣿⠇⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣦⠀⠀⣀⣀⡠⠘⠂⠀⠀⣴⣷⣿⣀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⡀⢀⣸⡏⠀⠀⠀⣠⡖⠺⣿⣿⣿⣿⣿⢙⣿⣧⠀⠀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡛⣣⣬⣿⡎⠿⢿⡆⠀⡴⠞⢻⡦⡞⣿⣿⣿⣅⣠⣿⣿⢤⢀⣾⠇⠀⠀⡴⠚⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢇⣾⣿⣿⣯⣭⢠⣼⣷⣤⣇⣿⣿⣿⣶⣸⣿⠛⠉⢉⣿⣿⡲⠾⢟⣃⣄⠀⠁⢀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣦⣿⣿⣿⣿⣿⣿⡿⣿⣏⠿⢿⣿⣿⣿⣿⠄⠀⣸⣿⣿⡃⢠⣾⣿⣿⢧⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡿⣷⣿⠋⡿⣡⣿⣾⣿⣿⣾⣿⣿⣈⣿⠛⠉⠛⡟⢿⡿⣿⣿⣧⣿⣿⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢻⣿⡞⠒⡿⢛⣿⣿⣿⡿⠻⣿⢿⠇⣶⠀⣼⣿⡷⢾⣧⢿⣿⣧⠋⠙⡇⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣾⠈⡄⢸⡇⢿⡏⡆⢸⠛⠛⠻⣧⣀⢈⣿⣾⣿⠠⠿⢿⣧⠀⣿⣿⡘⠛⣾⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣟⣠⣼⡄⢿⣿⣷⢿⡇⠀⡄⠀⢻⣿⣿⣧⣿⡟⠀⣠⣾⣿⣇⣿⣿⡇⢠⣿⡇⣸⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⠀⢷⡀⢸⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⡇⢸⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢛⠻⠿⠿⣻⣿⣿⣿⠀⣿⣷⣿⣿⡿⠛⣿⠀⣾⠿⣿⢿⡯⠙⠿⢷⡘⣿⣽⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠛⣿⣿⢿⣿⣿⣿⣿⡙⢶⣿⣦⡾⠆⠁⠀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠛⠋⠉⠈⠛⠉⠉⠙⠃⠀⠀⠀⠈⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⣀⠀⠀⠀⠀⣀⠀⢠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠁⣄⡁⢈⣢⠈⠀⠀⠀⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠠⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠄⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠐⠒⠒⠂⠰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠂⠀⠈⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡈⠀⠀⠀⠀⠈⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠁⠀⠀⠀⠀⢁⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠘⢀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡋⠢⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⡄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢃⠀⠀⠀⠀⠀⣠⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣄⠀⠀⠀⠀⠀⡘⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣄⠀⠀⠠⠁⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄⠀⠀⠀⠀⠰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠙⠿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠋⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠁⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠈⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠈⠉⢩⡀⢀⠍⠈⠁⠀⠀⠀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠉⠢⠤⠠⠔⠋⠀⠀⠛⠉⠉



⣾⠓⠛⣿⠶⠶⢾⠿⠶⣿⡟⢿⠷⠿⢶⠾⠶⠶⢿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⡷⣷⡶⠾⠿⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢸⣿⢤⣤⣤⣤⡤⢤⣤⣤⣤⡤⢤⣤⣤⣤⣰⣿⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⣿⣋⣉⢹⣷⣾⡋⢉⠻⣷⣾⡟⠙⠻⣿⣿⣴⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣧⣭⣿⠟⢻⣿⣭⣼⠿⢻⣧⣬⣽⡿⣿⠀⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠁⠀⠀⠀⠂⠈⠉⠁⠀⠀⠈⠉⠁⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣥⣥⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣬⣤⣤⣿⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠙⢷⣄⠀⣀⡴⠋⠁⠀⠀⠀⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⡟⣿⣿⠀⠀⠀⠀⠀⠀⣹⡿⣯⡀⠀⠀⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⠀⣿⣿⠀⠀⠀⠀⣠⠾⠋⠀⠈⠻⢦⣄⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣿⣿⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⠀⠀⣿⣿⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⣿⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢿⡿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⠇⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠘⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⠛⡟⠛⠛⠛⣿⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⢠⣿⣿⢷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠾⣿⡋⢸⣿⣄⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡟⠛⠛⠛⠛⢻⠲⢦⡀
⠀⠀⠀⠀⠈⠃⢸⣿⠟⢦⠀⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡟⠒⠒⠒⠒⢻⠀⠀⠉⢳⣤⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠳⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢿⣿⡿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⣶⣤⠀⢸⡇⠀⠀⠀⡟⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠾⣿⡇⠀⠀⠀⠈⠙⠿⣾⣇⠀⠀⠀⡇⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡇⠀⠀⠀⣹⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣾⣿⡇⠉⠛⠿⣶⣤⣤⣴⠿⠛⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⣿⠀⠀⠀⢸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢀⣠⣴⠾⠛⠛⠿⣶⣤⣀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣛⣉⣀⣀⣀⣀⣀⣀⣉⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⠤⢤⣀⣀⣀⣠⡤⡤⠤⣤⡴⠒⠉⠒⠒⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣡⡴⠷⠶⠾⠴⣦⡯⠶⠾⠚⠒⡟⠓⠒⠲⠶⢴⣥⣦⢀⡗⣦⠤⢤⡤⠔⢲⠖⠂⢠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢇⣴⠋⠀⢀⠤⠒⠉⠁⠀⠀⢀⢀⢰⠁⠀⠀⠀⠀⠀⠀⠉⡽⢉⠛⠲⢾⣶⠶⠷⠶⠖⠳⠒⠶⣍⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣴⠞⠁⡠⠊⠁⠀⠀⢤⠀⡀⡆⢸⠘⡄⢘⠰⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠈⠑⠦⣄⠀⠀⠀⠀⠈⢫⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡾⠁⢀⡜⠀⠀⠠⡈⢢⡘⠆⢃⢸⠘⡀⡇⢀⢠⠀⢀⠂⢄⠀⡸⠠⢀⠆⢀⠀⠀⠀⠁⢊⡓⢤⡀⠀⠀⠈⣇⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⢀⠎⠀⠀⣀⠳⡌⢦⢳⡘⡌⣆⢣⣣⡿⠤⠤⠌⣀⣈⡈⠈⢱⠀⡘⢀⢊⡔⢠⠄⠀⠀⠐⣀⠙⢦⡀⠀⢷⣀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⡇⠂⠀⠀⡐⢮⡳⡌⢦⢳⢣⡵⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢧⣌⡌⡐⢡⠞⡀⠀⠀⢒⠀⠨⢷⡄⠈⣇⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⡵⠔⠒⠒⠚⠒⠓⠺⢤⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⢃⠞⡡⠀⠀⢐⠂⢈⣁⢻⠀⢸⡉⢻⡇
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢈⠃⠁⠀⠀⠀⢀⡐⠊⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢊⠔⠀⠀⠀⠀⠀⠖⠬⡇⣎⢙⣾
⠀⠀⠀⠀⠀⠀⡇⠀⡀⠘⠀⠀⠀⠀⠀⠀⣀⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠬⠖⠒⠒⠚⠚⠛⠒⢟⣯⣱⡇
⠀⠀⠀⠀⠀⠸⡀⠠⣂⣇⣀⣀⣤⣀⣀⣐⡂⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠹⡀
⠀⠀⠀⠀⠀⠀⢹⢿⡇⡇⠀⠀⠀⠈⣓⠢⣍⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡈⢸⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡸⢜⡇⢋⠀⠀⠀⠀⠀⠉⣙⣛⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣀⣐⠀⢨⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣷⢬⡇⠸⣡⠆⠀⠐⢄⠀⠀⠐⠊⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠉⠩⢉⣑⡒⠭⣛⢠⡓⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢰⠘⣇⠀⢣⡌⡰⢀⠀⠢⢀⠀⠀⠀⠨⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠰⣜⠒⢍⠳⣤⢋⢿⡷⠋
⠀⠀⠀⠀⠀⠀⠈⢯⡉⣳⣄⠳⡀⠇⠎⠰⢀⠂⠀⠀⠀⠀⢠⠛⠒⠤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠖⠉⠀⠀⠀⡐⢮⡙⢦⡙⢆⣱⢃⠞⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣘⣧⠙⢾⣀⠃⡅⡜⢠⠀⡄⡠⠆⠀⠀⠀⠀⠀⠀⠀⢨⠁⠂⠀⠀⠀⠀⠀⢀⠃⣈⠳⡙⢂⢓⡜⣡⠿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼⡄⠀⠙⢶⣅⡂⡆⡘⠰⠱⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⢠⠠⡐⡄⢦⠹⡌⢡⡈⡄⢹⡼⣫⠞⣯⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣌⡳⠦⣤⣈⣓⠧⣄⡇⢀⢘⢰⠀⡄⠀⡄⣀⠀⢠⢸⢸⠀⡃⢱⠈⡇⢡⣀⡧⠞⣡⠞⣇⡾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠴⢩⣭⣭⣟⣛⣿⡦⢬⣄⣀⡀⠃⠁⠀⢸⡸⡄⣆⣹⣀⡧⠜⠊⠉⠀⣰⠋⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⣀⠁⠊⠊⠩⠉⠏⣟⢛⡿⠛⠷⣤⠤⠤⣤⡤⠚⢇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠋⠉⠙⠛⠚⠲⠶⠴⠵⠶⠾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡠⠔⡠⠔⣀⠔⣀⠔⣀⠄⡠⢀⡠⢂⡠⠀⡠⢄⡠⠀⡠⠄⡀⠄⡠⠄⣀⠀⡠⢀⡠⠂⡠⠀⡀⠔⡠⢀⠤⢀⠔⣠⠔⡠⢂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠋⠴⠊⠤⠊⡤⢊⠔⣡⠔⢡⠔⢁⠔⢉⠔⢉⠔⢋⠔⢊⠴⢊⡤⢊⠕⣪⠔⢉⠔⢉⠄⢊⠴⣋⠔⣡⠖⡡⠊⡴⠊⠔⠡⠊⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠁⠀⠁⠀⠁⠀⠉⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡠⠒⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⠆⢄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣷⣤⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣷⣆⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⣠⣼⣶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠈⠙⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠉⠻⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⠤⠚⠛⠛⠛⠛⠻⠇⡀⣼⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣀⢿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢿⣿⡿⠋⠀⠈⣱⣤⣄⣀⡀⠀⣀⣀⡠⠛⠛⠻⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⢀⣴⣿⣿⡾⠛⠻⠿⢿⣿⣿⠿⠿⣧⣶⣄⡀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠻⡅⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠑⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠤⠊⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣥⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⠀⠀⣴⠄⠀⣴⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠆⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣼⣿⣧⣼⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⠵⣤⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠭⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⡇⠀⠈⢷⡀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣰⠏⠀⠀⣇⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢯⡗⠒⠢⡀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡠⠒⠒⣮⠇⠈⠢⣀
⠀⠀⠀⠀⠀⢀⡠⠔⢺⠖⠊⠁⠀⠀⠀⠘⡞⡄⢀⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣇⠀⡼⡜⠀⠀⠀⠀⠉⠒⢾⠒⠤⣀
⠀⠀⠀⢠⢾⠁⠀⠀⠸⣼⡀⠀⠀⠀⠀⠀⠹⡝⡍⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡝⡽⠁⠀⠀⠀⠀⠀⡸⡜⠀⠀⠀⢹⢦
⠀⠀⠀⠸⡈⡄⠀⠀⠀⠙⣷⣀⠤⠤⠒⠒⠒⠚⠮⣓⢒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⢒⡪⠞⠒⠒⠒⠢⠤⢄⣰⡗⠁⠀⠀⠀⡞⡸
⠀⠀⠀⠀⠱⣘⣄⣀⡠⢤⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⠤⣀⣀⣜⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠈⢆⠈⢣⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠑⣌⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡔⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⠳⣌⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡴⠋⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡜⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠛⠛⢛⣻⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣟⡛⠛⠛⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⢀⣠⣴⣾⣿⡟⠋⢉⣩⣤⣤⣤⣤⣍⡉⠙⢻⣿⣷⣦⣄⡀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⢀⣴⡿⠛⠁⠀⢻⣷⠀⠘⠛⠉⠉⠉⠙⠛⠃⠀⣾⡏⠀⠉⠛⢿⣦⡀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢀⣼⡿⠋⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠙⢿⣦⡀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣾⠿⠿⠿⠿⣷⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢰⣿⠿⢷⣶⣦⣤⣀⣀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⢀⣀⣠⣤⣴⣶⡾⠿⣿⡆⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡏⢀⣄⠀⠉⠉⠛⠻⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠛⠛⠉⠁⠀⣠⡀⢹⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⣼⣿⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⣿⣇⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⢻⣿⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⣿⡇⢠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣇⠈⠉⠀⣀⣀⣤⣤⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣤⣄⣀⡀⠀⠉⠁⣸⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣟⠸⣿⣶⡾⠿⠟⠛⠉⠉⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠈⠉⠙⠛⠿⠿⣶⣶⣿⠇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢿⣶⣶⣶⣶⡿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠈⠻⣷⣄⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠈⠻⣷⣤⣀⠀⣼⡿⠀⢀⣄⣀⣀⣀⣀⣀⡀⠀⣿⡇⠀⣀⣤⣾⠟⠁⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠈⠙⠻⢿⣿⣧⣄⣈⣛⠛⠛⠛⠛⣛⣁⣠⣼⣿⡾⠟⠋⠁⠀⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⣶⣤⣤⣬⣽⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣯⣥⣤⣤⣶⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣴⣶⣦⣤⣤⣤⣿⣿⣿⣷⣤⣴⡶⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢈⣉⣭⣭⣁⣉⣁⣌⣤⣀⡀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠋⠉⢉⣿⠛⢻⣿⡉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⡛⠿⠿⠿⠟⠛⠛⠻⢶⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣴⠾⠛⠛⢷⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡉⢻⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⢻⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⣼⣶⣿⣡⡟⠀⠀⠀⠀⠀⠀⢀⣴⡾⠷⠾⠿⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⣿⣍⡙⠛⠛⠛⠛⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⡉⢿⣦⣀⢀⣀⣀⣀⣿⠟⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠙⠶⢭⣙⣛⣛⣛⣩⡥⠞⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⣤⣬⣭⣭⣥⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣦⠈⠉⠉⠉⠉⠉⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⠀⠀⡄⠀⠀⠀⣿⡇⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡧⠀⠀⠀⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⢰⡇⠀⠀⠀⢸⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⠀⢸⡇⠀⢠⡄⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠃⠀⢸⠇⢸⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠀⠀⢠⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡟⠀⣀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⡏⢸⠀⠀⠀⢸⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⠀⡇⠘⠀⠀⠀⢸⣇⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⡟⢠⠇⠀⢀⡀⠀⢸⣿⠀⠀⢰⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣀⡀⠀⢸⡇⠀⠀⠀⢸⠃⠀⢸⡇⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣴⡟⠛⠻⠇⠸⡿⠀⠀⠀⠘⠀⠀⠘⣧⠀⠘⠋⠉⣠⣄⠀⠀⠘⠋⠉⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠇⠀⢀⣤⡄⠀⠀⠀⠀⢀⡀⠀⠞⠁⠀⠲⣦⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠸⣤⠀⠀⢰⡟⠁⠈⠙⠃⠠⣇⠀⠀⠀⠀⠐⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣠⡄⠀⠈⠀⣀⠀⠀⣀⠀⠈⢉⣤⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠿⠛⢿⣤⣀⣠⡿⠃⠀⠻⣶⣶⠾⠛⠿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣠⣴⡚⠉⠑⠒⠚⠛⠉⠉⠛⠓⠒⠊⠉⢓⣦⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣱⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣴⣮⡻⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⢟⣿⣿⣿⣿⣿⣿⣧⠀⣀⣀⣀⣀⣀⣀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠱⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⡞⠈⠓⢾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠛⠁⢳⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠀⠀⠀⠉⠛⡿⠁⠀⠀⠀⠀⠀⠰⣏⣹⠆⠀⠀⠀⠀⠀⠈⢿⠛⠉⠀⠀⠀⡎⣇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢶⣾⣷⡶⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣶⡄⠀⣸⣇⠀⢠⡶⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⡸⠀⠀⠀⣀⣤⣷⡀⠀⠀⠀⠀⠙⠻⢿⡿⠟⠋⠀⠀⠀⠀⢀⣾⣤⣀⠀⠀⠀⢇⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢧⢀⣤⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⢤⡀⡼⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⢇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠉⠉⠉⠉⠉⠀⢻⣿⣿⣿⣿⣿⣿⣵⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⣿⣮⡻⠟⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠻⢥⣀⡠⠤⢤⣤⣀⣀⣤⣤⠤⢄⣀⡬⠟⠋⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠋⠉⠁⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠛⣣⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣜⠛⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣀⡺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢗⣀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠰⣿⣿⣶⣝⠻⣿⠿⠛⠉⠁⠀⠀⠈⠉⠛⠿⣿⢟⣫⣶⣿⣿⠆⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣼⣷⣮⡛⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣛⣵⣾⣧⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢻⡿⢟⣩⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣬⡛⢿⡟⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠰⣿⣿⠿⣋⣵⣿⣶⣤⣀⡀⠀⠀⢀⣀⣤⣶⣿⣦⣝⠻⣿⣿⠇⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠉⢵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡮⠉⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣤⡝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢫⣤⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⠿⣒⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⢛⣙⢻⣿⠟⣉⡻⣿⡿⣋⡙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣌⣋⣼⣿⣮⣛⣵⣿⣧⣙⣣⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⣿⣿⣿⣿⣿⣶⡍⠛⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿

⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣷
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿
⣿⣿⣿⢋⣥⣦⣬⣭⣭⣭⣭⣭⣭⣭⣥⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿
⢻⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣮⣭⣍⣛⣛⣛⡛⠿⣿⣿⣿⣿⠈⣿
⠈⢿⣷⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡝⢿⣿⢰⣿
⠀⠀⠹⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡎⠟⣼⣿
⠀⠀⠀⠀⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢙⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⠿⢟⢱⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⠿⠸⢿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣮⣍⣛⠿⠏⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡘⠿⢛⣫⣭⣶⣿⣿⣷⣮⠉⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⡧
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣾⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠄⠀⠛⠁⠀⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⢶⣤⡀⠀⠀⣀⣤⣄⡀⠀⠀⣠⣴⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠈⠀⠀⣾⣿⣿⣿⣿⡆⠀⠈⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠇⢻⣴⠶⠶⠖⠀⠀⠛⠛⠛⠛⠛⠃⠀⠰⠶⠶⢶⣼⠋⠷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣶⣶⠈⣿⣿⣿⣿⣿⣿⣿⣿⢰⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣿⣿⣿⣿⠿⠿⠿⠇⣿⣿⣿⡀⠿⠿⠿⣿⣿⣿⣿⣿⢸⣷⣼⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⢸⣿⡿⠿⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠠⣶⣶⣾⣿⣿⣿⣇⠸⠿⠿⠿⣿⣿⣿⡿⠿⠿⠟⣸⣿⣿⣿⣿⢸⣿⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣛⣠⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣋⣭⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⣋⣭⣙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⢸⣿⣿⣿⣧⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠋⣼⣿⣿⣿⡧⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⢣⣤⡄⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⡎⠉⠁⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠶⠶⣉⡱⠶⠆
⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠾⠿⣉⡹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣀⣀⣤⣴⡿⠋
⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⢀⣀⣀⣤⣴⡶⠿⢿⣿⠉⠙⠛⠉⣿⡿⠿⢶⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠛⠋⠉⠀⠀⠀⠈⠻⣷⣦⣴⣾⠟⠁⠀⠀⠀⠉⠙⠛⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⢠⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡄
⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠿⠟⠛⠛⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⢼⣿⡶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣿⣟⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣻⣿⣶⡶⠶⠶⠶⠶⠶⠶⠶⢶⣿
⠀⣿⡀⢠⣤⠀⣤⡄⢠⣤⠀⢹⡟⠛⣿⠛⢻⣿⠛⣿⡟⢻⣿⠛⣿⡟⠛⣿⠛⢻⡏⠀⣤⡄⢠⣤⠀⣤⡄⢸⣿
⠀⣿⡇⢀⣁⠀⣈⡀⢀⣀⠀⢸⡇⠀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⠀⢸⡇⠀⣀⡀⢀⣁⠀⣈⡀⢸⣿
⠀⣿⡇⠘⠋⠀⠙⠁⠈⠋⠀⣸⡇⠀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⠀⢸⣇⠀⠙⠁⠈⠋⠀⠙⠃⢸⣿
⠀⣿⡇⠀⠀⢸⣿⠿⠿⠿⢿⣿⣷⣶⣿⣶⣾⣿⣶⣿⣷⣾⣿⣶⣿⣷⣶⣿⣶⣾⣿⡿⠿⠿⠿⣿⡇⠀⠀⢸⣿
⣾⡿⠿⠿⠿⢿⣿⠀⠀⢠⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⡄⠀⠀⣿⡿⠿⠿⠿⢿
⣿⣧⣤⣤⣤⣼⣿⠀⠀⣸⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⣇⠀⠀⣿⣧⣤⣤⣤⣼
⣬⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭
⣿⣯⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢰⣾⣶⠆⠀⠀⠀⠀⢰⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⣿⣶⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠄
⠀⠀⠀⠀⢀⣤⣝⣥⣄⠀⠀⠀⢀⣬⣝⣥⣄⠀⠀⠀⠀⣠⣤⠀⠻⡿⢇⣬⣝⣥⣄⠿⡿⠃⢠⣤⡀⠀⠀⠀⠀⠀⢠⣶⣾⣛⣧⣄⢀⣤⣄⠁⠂
⠀⠀⠀⠀⣾⣿⣿⣿⢿⡆⠀⠀⣿⣿⣿⣿⢿⡆⠀⠀⢀⣝⣫⣾⣿⣾⣿⣿⣿⣿⣿⣿⣾⣷⣌⣻⣅⠀⠀⠀⠀⠀⠸⢻⣿⣿⡿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇⠀⢠⣧⣿⣿⣿⢸⡇⠀⠀⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⣸⣿⣿⡟⢿⡄
⠀⠀⠀⠀⠁⣿⣿⣿⠎⠁⢀⡾⠛⣿⣿⣿⠈⠁⠀⠀⠟⣿⣿⡃⣿⣿⡏⣿⣿⣿⠇⣿⣿⡇⣻⣿⡟⠇⠀⠀⠀⠀⠀⣼⡟⢸⣷⠹⣿⣿⣷⣜⠗
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠛⠚⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⣿⣿⣿⡅⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⡀⣸⡿⣸⡿
⠀⠀⠀⠀⠀⠻⠏⠿⠀⠀⠀⠀⠀⠿⠏⠿⠀⠀⠀⠀⠀⠿⠻⠁⠿⠻⠇⠹⠏⠿⠀⠿⠻⠇⠻⠻⠇⠀⠀⠀⠀⠀⠀⠿⠁⠀⠿⠇⠿⠃⠿⠃

⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠛⠛⠛⠛⠓⠒⠲⡖⠲⠶⠶⠶⠶⠶⠶⠶⠤⠤⠄⠀⢰⣿⣷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠒
⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢠⣶⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡍⠁
⠀⠀⠀⠀⣸⣿⠓⠾⠿⣿⣿⣿⣿⣟⠩⢹⣿⣿⣿⡇⢠⡆⣿⣿⡀⠠⠙⢦⣄⠀⠀⠀⣿⣿⠃⢀⣤⣤⣶⣴⣶⣶⣶⣶⣶⣶⣦⣴⣷⡆⣾⣿⠃
⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠉⠉⠙⠓⠛⠿⠿⣿⣧⣬⣥⣿⣿⣿⣷⣶⣾⠿⠷⠀⠠⢿⣿⡧⢾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⡷⢿⣿⣦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⣀⣀⣈⣷⣀⣀⡠⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠛⠀⠀⠀⠋⠀⠀⠛⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣴⡿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡆⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣭⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣈⣿⣷⣴⣶⠶⠶
⠀⠀⠀⠀⠀⠀⢠⣤⣄⣼⣁⣀⣹⣁⣹⣧⡤⣤⣤⠀⠀⠀⠀⠀⠀⠛⢹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠭
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⠀⠺⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⣼⡇⠸⣿⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢂⠀⠀⠀⠈⠀⣴⣷⡦⢾⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠻⠃⠀⠿⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣥⡴⠏
⠀⠀⣶⣶⣶⣶⣶⣶⣶⣞⣲⣒⣞⣲⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⣴⣿⣷⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿
⠀⠀⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣴⠿⠷⣄⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠉⣵⣤⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢸⡇⣿⣿⣿
⠀⠀⣿⡀⠛⢀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣶⣴⣾⣷⣤⡀⠀⠀⢩⣷⡆⣠⣤⠀⠀⣤⢶⣾⣻⣿⣷⠀⠀⠀⠀⣼⣿⣉⢿⣍⠻⣿⠀⠁⣿⣿⣿
⠀⠀⣿⣿⣿⢟⣿⡻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣻⣟⣿⣇⠀⠀⢸⣿⣿⡿⢻⣧⠀⣤⠬⣿⢻⣿⡿⠀⠀⠀⣸⠃⣿⡙⢷⣝⢷⣿⠀⠀⣿⣿⣿
⠀⠀⠉⠉⠉⠻⣤⠟⠉⠉⠉⠉⠉⠉⠈⠉⠸⢤⡼⠈⠉⠀⠀⠘⠿⠟⠁⠈⠻⠃⠿⠤⠿⠿⠿⠿⠿⠃⠰⠏⠀⠿⠿⠶⠿⠷⠿⠀⠀⠿⠻⠿