Soya Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⣤⣖⢒⡿⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⡰⠚⢋⡡⠞⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣀⡀⢇⡴⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢀⣴⠏⠉⠈⣹⡏⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠛⠁⢰⠋⠀⠀⢀⡼⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠋⣁⣀⣀⠀⠀⠸⡀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⣀⡤⠖⠋⠉⠉⠁⢀⠔⠉⠀⠀⠀⠙⢆⢀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢠⠞⣫⠖⠂⠀⠒⢄⠀⡎⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡾
⣠⠴⢃⡞⠁⠀⠀⠀⠀⠘⡆⢇⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⣇⠀⢸⠀⠀⠀⠀⣀⡤⠴⠓⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠈⢧⠸⡀⠀⣠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⣀⠤⠖⠒⠒⠲⣄
⠀⠈⠓⢬⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⢀⡴⠋⠁⣤⠴⠚⠋⠀⠈
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠚⠉⠉⠀⠀⣠⠴⠒⣒⡒⠲⠶⠤⠤⣄⠀⠙⠒⠶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠈⠓⠛⠛⠛⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⣀⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠙⠃⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠶⠗⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣀
⠀⠀⠀⠜⠁⠀⠀⠙⢦⡀⠀⠀⢸⡀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⢃
⠀⠀⠀⠈⢤⡀⠀⠀⠘⣷⠀⠀⠀⠉⠐⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠉⠀⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠀⢄⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠈⠳⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠉⢿⣷⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡀⠀⠀⠉⠻⠦⢀
⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⣀⣈⡥⠞⢻⠏⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣄⣀⢀⣠⠎⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠉⣩⠞⢋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠇⡠⠀⠠⡴⠂⠀⠉⢠⠂⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⣦
⠀⠀⠀⢀⡆⣠⡴⠞⠁⠐⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣠⣄⢹⠀⠙
⠀⠀⠀⢀⠀⠛⢀⡀⢀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣿⡏
⠀⠀⠀⠀⠀⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠉⠀⠀⣀⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠀⠁

⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⠶⣫⡉⠉⠁⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⠀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣐⠀⠀⠐⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠀⢀⠀⢈⡉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠸⣇⠀⠀⣀⡀⠠⠖⣋⣉⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠠⢀⠁⠐⠚⠀⠄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠑⠠⡀⠀⠐⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⣄⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠂⢨⠀⠀⠘⠀⠀⠈⢢⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢀⠘⠢⡀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠂⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⡰⠀⢸⡄⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠹⠄⠀⠀⠀⠀⠀⡆⠀⠀⢿⣆⠀⠂⢂⠭⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠢⣄⠁⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠘⣎⢦⡀⡇⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⠂⠂⡼⠁⠀⠀⠀⠀⠀⡇⠐⠁⠀⠘⢢⡳⣔⡠⠀⡀⠀⠀⠀⡂⠉⠲⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠙⣎⢮⡡⠬⠀⠠⠴⠁⢠⠖⠁⠒⠉⠛⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⢧⠙⠦⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢳⡉⠲⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠉⠳⢤⡈⢄⡀⠀⠈⠄⠂⢠⡄⡠⠃⠀⠀⠈⠁⠳⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣀⠀⠀⠀⠀⠈⢆⠈⠀⠀⠀⠀⠀⠊⠳⣄⠀⠀⠀⠙⠲⣌⣐⠠⠤⠔⠋⠀⡅⠀⠀⠀⠀⡠⠀⢸⠠⠀⠉⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡀⠘⣇⠀⠀⠀⠀⠈⢧⡀⠀⠁⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠉⠙⠛⠒⠲⠤⣌⡒⠄⢀⣀⣁⡴⢊⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠄⣱⠇⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⡉⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠠⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣁⠀⠀⠀⠀⠈⢛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠂⠒⠀⠀⣀⠀⠀⡀⠚⢀⠂⠘⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠷⡴⠁⠀⠰⠀⡀⢀⡸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀⠀⠀⠀⠐⢠⠀⠀⠀⢄⣧⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⢸⠃⢀⡆⠈⠠⡠⠁⢖⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠉⠙⢦⡀⢠⡿⠥⠤⢨⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⡋⠁⠀⣉⠦⡀⠀⠀⠎⠀⠦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠈⣤⠁⠀⠘⡄⠀⠀⠀⠀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⡀⠉⡀⠀⠀⢠⡄⠀⠀⡰⡱⠀⣶⠀⡂⠀⠠⠔⠒⠒⠒⠤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣠⠁⠀⠀⠀⠀⠘⠀⠀⢠⡏⠣⢠⡟⠀⠋⠀⠀⠀⠀⠀⠀⣀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠛⠉⠁⣀⠘⠀⠀⠀⠀⠀⠀⠀⣁⠀⠘⠐⢁⣤⠔⠋⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠚⠉⠉⠀⠀⡀⠀⠐⠦⣷⠈⠄⠸⡳⢤⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡁⣺⡄⢠⣝⣞⠻⠇⡆⡀⠇⠀⠁⢒⠠⠤⠄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠀⠀⡟⠁⠀⠸⣿⣄⡀⠀⠻⠀⠀⠸⠧⣤⣆⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⢬⠁⠀⠀⢸⡿⣭⠀⠀⠀⠐⣾⣆⠀⠀⠀⠈⠁⠢⣈
⠀⠀⠀⠀⡴⠂⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢠⠇⠀⠀⠀⣦⠀⢿⡇⡄⠀⠀⠙⠉⡓⠦⣄⣤⢤⡾⠟⠋⠁
⠀⠀⠀⢸⡁⢀⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠠⣿⠀⢸⡘⠓⠲⡄⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠳⡈⠃⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠩⠉⣡⠄⠀⠀⢀⡼⠃⠀⠘⠃⠀⠀⠹⣶⠀⠀⠀⠛⢤⡀
⠀⠀⠀⠀⠀⠉⠲⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣠⣴⡿⠁⠀⡁⠀⠀⡨⡇⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡽⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠹⡀⠀⠀⣸⠏⠀⠀⠀⠀⢠⢻⠶⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠄⠀⣤⣶⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠘⠋⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⣡⠚⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠒⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠌⠢⠀⠀⠀⠀⠠⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⠄⢀⠔⠀⠘⢰⠁⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢌⠓⠦⠀⢠⠈⠀⢀⠴⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠈⣶⠠⠿⠈⣠⢂⠉⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠁⠀⠀⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⣄⠀⠀⢀⡀⣡⠀⠀⢠⠀⠀⠸⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠤⠀⡄⠀⠀⠁⠨⡄⠛⡆⠀⠀⠀⠀⠁⠈⣤⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠑⠲⢄⣀⣠⠀⢀⣄⡀⠀⠀⠐⠤⢌⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⡤⠀⠀⠀⡌⠶⣦⣋⢠⡆⢲⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠔⢃⠀⠚⠀⣤⣿⠀⠈⢣⠀⠙⢆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⣄⣀⢀⡠⠾⠃⠀⢀⣶⢩⡀⠀⠀⡇⠀⠈⠳⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣡⠊⠀⢀⢤⠟⠀⣾⡧⠀⠀⠐⣦⡀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⢰⡟⠁⢀⡸⢏⠹⢦⠀⠀⢺⣉⣀⡀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣴⠏⢟⠈⢆⣀⡁⠀⠀⣂⠉⠛⢳⠂
⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⠀⠀⠀⠈⠁⠀⠸⡄⠀⠀⠙⠱⠀⠈⠱⣄
⠀⠀⠰⡿⠥⢛⠖⣬⢩⢬⠃⣲⣏⡚⡁⣸⣷⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠙⠂⠄
⠀⠀⠀⠈⠙⠛⠻⠷⠿⠶⠿⠶⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢂⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣜⠛⠙⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠘⢄⡌⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠋⡇⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠴⠶⠲⢦⡄⠐⠀⡇⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠀⣤⠃⠀⠀⠀⠀⠀⠈⠡⣀⠀⠀⠀⠀⠀⠁⠙⡄⡀⡁⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣱⠋⠀⠀⠉⣝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⠦⠤⣄⠀⡇⠸⠋⠙⠉⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⢀⣴⡮⠈⠁⠀⠀⢀⢂⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢙⢴⡞⠓⠦⣤⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⢠⣤⢔⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠄⡠⠴⠃⡸⡇⠀⠀⠀⠳⣤⠄⣀
⠀⠀⠀⠀⢠⣃⣀⣤⣤⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠳⣄⡀⠀⠀⣀⣀⠉⡀⢽⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⢀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠹⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠉⢻⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡹⡀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠁⠀⢠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢮⠂⠈⠀⠙⠎⠢⡀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢡⡘⠦⣴⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠁⠀⠀⡇⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⢤⠀⠀⠀⠀⠦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⣶⠿⠸⣄⡀⣀⣠⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⡓⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢀⡀⢺⠀⠀⠀⠀⠀⡏⣩⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠕⡠⠤⡤⠠⠅⠀⠀⠀⠀⠀⠀⠑⠈⣷⣤⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡃⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣳⣄⡀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⠡⣽⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⢻⡄⠀⠀⠀⠀⠀⠀⢻⡐⠊⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⡀⠀⠀⠀⠀⢀⡤⠀⢳⡄⡞⣹⣷⢏⣼⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠘⣄⣷⡀⣠⡇⠀⠀⠀⠀⠈⠓⠒⠛⠉⠛⠁⣙⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠄⡀⠀⠀⠀⠀⠈⡿⠗⠭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⣶⣲⣶⣍⢦⠀⠀⠀⠀⠃⣆⠑⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⠄⡀⠀⠀⠀⡠⠂⣆⠀⢸⣮⡋⠀⠀⠈⣻⡧⠀⠀⠀⠀⣧⠀⠐⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢆⠀⢀⣴⠃⠹⡁⠸⡎⢿⣶⣶⣥⡞⢸⠀⠀⢀⠀⣿⠀⠀⢸⠀⠠⠖⠈⠀⠉⠀⠀⠀⠀⠈⠉⠁⠉⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠄⠁⡈⠈⠢⣈⠁⠀⠉⠳⡙⠛⢁⣀⣀⣦⡀⠈⢷⣿⣆⣠⡌⠀⠁⠀⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⣀⠤⠴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⢄⠀⢀⠀⠀⠂⠠⡄⠀⣼⢀⠀⡀⠀⢰⢴⡿⠷⠶⠶⣼⣧⠀⠑⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⣻⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠘⠮⠆⠀⠀⠀⠋⠀⠀⢀⠀⢀⢿⣄⡀⣀⠀⠀⣿⡇⠀⠀⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡶
⠀⠀⠀⠀⠀⠀⢀⡠⠄⠈⠀⠀⠀⠆⠀⣄⠀⠀⠀⠀⠀⠠⠄⠠⠀⠘⣧⡹⣿⣭⣶⡿⠏⠱⢄⠀⠀⠀⠀⠀⡀⠒⣀⠀⠀⠀⠀⠀⠀⠐⢠⠄⣘⠁
⠀⠀⠀⠀⠐⠈⠀⠂⠈⠢⣀⠀⠀⢰⡀⠀⠂⠀⠠⠀⢀⡄⠀⠀⠀⠀⠈⠣⢄⡉⣤⡶⠟⠛⠶⡱⡄⠀⠀⠀⠛⠐⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠠⡊⠀⠀⡀⠀⠄⠀⠀⠀⠈⠉⠉⠳⣄⡀⣤⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢳⣻⣅⣀⠀⡠⣻⣷⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠓⢛⣡⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠽⣗⠾⡾⠟⢸⠀⠀⠀⢸⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⡞
⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀⠀⠀⠀⠀⡁⣤⣄⣀⡤⠕⠒⠒⢄⡀⠀⡀⠀⠀⠤⠒⠒⠠⠉⠈⠉⠒⠓⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠂⠄⣀⡁⢀⢀⡁⠊⠀⠈⠚⠃⣧⣤⣀⣤⣌⣀⠀⠈⣇⣀⢠⣄⡠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠰⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⢻⡋⠀⠀⠈⠙⡿⠋⠁⠈⣱⢈⠅⠥⢀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢤⣴⠎⠹⢷⡂⠠⠁⠹⢆⡴⠖⠋⣦⡀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⣁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣿⣛⢛⣛⣛⣛⣙⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣠⣴⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣿⡿⠋⢻⣥⢻⣷⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣾⡾⢰⡌⣿⣿⣿⣯⡀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻⣟⣿⡟⠋⠘⣧⣾⣿⠛⡟⢸⣏⠙⠑⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣧⣻⣿⠏⠀⠀⠀⢉⣻⣟⠚⠃⡎⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⡿⣰⠏⠉⠀⠀⠀⠀⢸⣻⡇⢀⡜⠀⢸⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⡋⠡⢀⣤⣿⡿⠀⠀⠀⠀⠀⠀⣰⣿⠁⡞⠀⠀⣸⣇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⡡⠟⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⢀⣾⡿⠃⠐⠀⠀⠀⣹⢻⡄⢌⠲⢤⡀⠀⠀⠀⠀⠀⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢡⠈⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠐⣾⣿⡟⠀⠀⠆⠀⠀⠐⡿⠀⢳⣦⡀⠀⠉⠐⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢖⣟⣿⡄⢺⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠈⠀⠀⠀⢀⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠉⠁⠀⠀⠑⡝⠳⣲⣾⠁⠀⠀⠀⠀⠀⠀⡠⢡⣿⠇⠀⢠⠀⠀⠀⠀⡞⠀⠀⠀⣠⠀⠀⠤⠄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⢤⠶⠉⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠁⠁⠀⠀⠀⠀⠀⠀⠈⣰⡿⠋⠀⠀⠀⠳⢠⠀⠈⠆⠀⠀⠀⠘⡗⣆⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠱⣄⠘⢧⣄⣀⣤⣤⠶⣶⡿⠋⠁⠀⠀⠀⠀⠀⠠⣶⣾⠟⠉⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠻⠿⠿⢿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⣇⠀⠀⣿⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣈⢀⡀⣙⡟⡇⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣦⣤⣤⣤⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠠⢐⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣻⣫⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣭⣿⣯⣿⣭⣭⣯⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⢛⣻⣭⠿⢭⣥⠞⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⠁⢀⡞⠉⢉⣠⠖⠋⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠁⠀⠀⠀⠈⢇⣴⠛⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢀⣴⡞⠋⠉⠉⣳⢞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⡴⠛⠉⠀⠀⠀⣴⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠀⠀⢰⠃⠀⠀⠀⣠⡞⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠾⠛⠉⣀⡤⠤⢤⣄⠀⠀⠈⢧⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⠀⠁⠀⠀⡠⠊⠁⠀⠀⠀⠀⠙⣆⢀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣩⠤⠖⠒⠦⣄⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣰⠏⣠⠞⠁⠀⠀⠀⠀⠈⢣⠀⣷⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⣴⠛⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⣆⣹⣤⣤⠶⠚⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⢿⡀⠀⣿⠀⠀⠀⠀⢀⣤⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠹⣆⢹⡀⠀⢀⡴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠙⠦⣍⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⣠⠞⠋⢀⣀⣠⡶⠟⠃⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠛⠛⠋⠉⠀⠀⠀⠀⣠⠴⠖⠛⣛⠛⠒⠲⠶⠶⠶⢤⡀⠀⠈⠛⠒⠒⠚⠛⠉
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠛⠭⠹⠿⠿⠟⠀⠀⠀⢻
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠤⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣡⠊⢻⡛⡆⣊⠀⡀⠀⣀⠔⠒⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡷⠁⠀⢀⢇⠴⠀⠉⠑⠊⠠⣤⡀⢀⢤⠽⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠛⢰⡇⠀⡠⠃⢸⣀⠀⠀⠈⠑⠢⣄⠙⡓⠦⣄⣈⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠂⢰⠁⢀⠌⢧⠉⠐⠂⠤⠠⣀⢣⡑⣀⣈⣯⠵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠙⢧⠀⠀⠀⠀⠀⣰⣁⠤⠂⠒⠢⡀⠀⡏⢰⠃⠀⠘⠆⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣂⠀⠀⠀⡐⢹⠋⠀⠀⠀⠀⣦⡼⡱⠇⠀⠀⠀⣠⡄⣀⠀⠠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠂⠀⠀⠀⠀⢹⠀⠀⢠⡷⠇⢬⠀⠀⠀⢸⠇⠀⢻⠂⠀⠀⠰⠁⠸⡀⠈⠁⠂⠌⡄
⠀⠀⣀⠀⠀⠀⠀⡴⠿⠿⢳⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⢿⠀⢀⠀⠀⣠⡟⠀⡅⠈⢧⠀⢠⠃⠀⠀⢩⡀⠀⠀⢀⡇
⡰⠛⠋⠻⡄⠀⢸⡆⠀⠀⢸⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠠⣾⠀⠀⢑⠈⠿⢷⣔⡤⠏⠀⢀⠃⠀⠀⠣⣈⠀⠀⠀⢀⢯⠑⠮⢨⠀⠀⠀⠀⢠⢤
⣯⠀⠀⢠⣷⠀⠸⡤⠀⡀⡿⠀⠀⢸⠃⠀⠀⠀⠀⠀⡀⢠⠆⠀⠀⠘⡆⠀⠸⠀⠀⠀⠀⠨⠀⠀⠀⠀⠈⠑⠢⢄⡘⠀⠡⡀⡌⠀⠀⠀⠀⡞⡒
⠘⢶⣤⣵⡼⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠄⣠⠀⠠⠊⢷⡟⠀⠀⠀⠀⡁⢈⡠⠖⠉⠉⠁⢀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠳⠀⠀⠀⠀⠀⡄⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠄⠀⠀⡈⠀⠸⠃⠀⠀⠀⢸⡁⡎⠣⠀⠀⠀⠀⣴⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠁⠄⡀
⠀⠀⠀⠀⠀⠀⡠⠒⠐⠶⣄⠀⢠⡏⠈⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢸⠩⠃⠀⠀⠀⠀⠀⢘⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⠒⠶⠪⣤⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⢠⠇⠀⢰⠴⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠘⡇⣇⡀⢀⠀⠀⠀⣞⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣹⠋⠀⠀⠀⡦⡌
⠀⠀⠀⠀⠀⢻⣦⣠⣿⠎⠀⠀⢸⠃⠀⠀⠀⠀⠀⢐⣿⠁⠀⠀⠀⠀⠱⠹⣾⡤⠃⠀⣲⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⠰⡄⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⣴⣇⠀⠀⢠⢀⠋⣽⠀⠀⠀⠀⠀⠀⠱⠌⠓⠁⠐⠊⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣖⡸⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠴⠄⢀⡠⠤⣸⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢀⣖⠉⠉⠁⡤⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⢙⡶⢘⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⢠⠈⠈⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⣇⢀⣎⠀⠀⠀⠀⠠⡆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢃⡔⠋⠀⠉⠛⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠂⠂⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⢸⠇⠀⠄⠀⠀⢘⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡿⠀⠀⠠⠄⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⣟⣇⠇⠀⠀⠄⢲⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣠⣯⠀⠈⢠⣠⡼⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡅⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢧⠀⣽⠀⡠⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠁⠙⠺⠯⠬⠛⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠠⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠀⠊⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⠞⢠⡼⠟⠳⠲⡄⠀⢐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⡼⠀⠀⠀⠀⠀⠀⠟⣰⠋⠀⠀⠀⠀⡿⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠜⠂⣻⡆⠠⠀⢀⡾⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⠁⣁⣀⠻⢵⠠⡡⠞⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⡽⡿⠉⠈⠉⠉⣷⡀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⠀⢟⡃⠄⠀⠀⣮⡝⠁⢦⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣚⣥⣤⣥⣤⣖⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠂⠁⠀⠀⢀⠀⠜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡅⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠈⠉⠓⠲⢦⣄⠀⠀⡰⠋⠀⠀⠀⡀⠀⢠⠇⠀⠔⠁⠐⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⣀⣠⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠛⣄⠀⠀⠀⠀⠀⠨⠙⢶⢣⠀⠀⠀⢻⡇⢀⡼⠒⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⢠⡿⠋⠁⠀⠀⢹⡟⠿⣮⡛⠷⢦⣄⡀⠀⢠⠀⠠⠀⠈⢳⡄⠀⠀⠀⠰⠠⠎⣿⠂⠀⢃⢈⡶⡫⢚⣳⡲⠀⠀⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠉⠁⠒⠪⠛⠳⢤⣆⣀⣀⣀⠀⠙⢶⣄⡀⠂⠀⠀⣿⢲⣟⣭⠛⠋⡼⠁⠀⠛⠃⠀⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣤⣬⣍⠉⠛⠳⠦⣝⣷⣤⡤⣶⡟⢫⣿⣴⣷⣰⡇⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣴⣶⣶⣶⣶⣶⠶⠤⣤⣤⠤⠤⠶⣒⠒⠒⠯⠷⠒⠛⠉⠀⠀⠁⢸⣿⣟⣀⣀⣀⣤⠶⠋⠑⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠋⠉⠁⠀⠀⠀⠉⠙⠛⠁⠀⠒⠲⠛⠛⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⠟⠛⠛⢋⡄⠀⠰⠀⣀⣀⣠⣄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⣐⠚⣰⡟⢀⡄⣨⠾⠃⠀⠐⠊⠉⢁⡀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⢛⣉⡙⠂⣄⣀⡐⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢳⣖⣺⣥⣴⠾⣫⡶⣫⡞⠉⠁⠀⠤⡀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠎⣀⠀⠺⠿⠟⢃⠔⠛⢠⣦⣬⡉⠲⠴⠦⠦⠲⢉⣡⣴⠿⣅⡠⢭⣭⣉⣉⡤⠚⠁⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣯⠀⠀⠀⠉⠛⠋⠀⠀⡳⠲⠟⠛⠉⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠀⠤⠒⠙⢦⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠤⣖⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉

⠀⠀⠀⢤⡀⢀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⡀
⠀⠀⠀⠀⠱⢤⡀⠀⣠⠞⡀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⣟⣠⣕⠀⠀⠀⠀⡠⠄⠀⠀⡀⠀⠀⠒⠁
⠀⠀⠀⠈⢙⠀⠤⠀⠍⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠆⠀⠀⠈⠈⠈⠁⠀⠀⠂
⠀⠀⠀⢀⢆⡣⠔⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢀⠀⠆
⠀⠀⠀⠈⠀⡀⠄⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠠⠅⠀⡾⠀⠀⠀⠀⠀⠁⣀⠄⡀⠀⠀⠀⠀⠑⠠⠼⠁⣴
⠀⠀⠀⠀⠀⠨⠒⠐⠄⠶⠀⢡⠀⠀⠀⠀⣀⢀⡀⠀⢸⠁⠀⠀⠀⣾⠘⠁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⡀⠌⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣉⠈⠙⠃⠀⢸⠄⠀⠠⠈⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠂⠘⠀⠀⠀⣠⣈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠈⢀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⠀⡐⠲⠖⢂⠏⠀⢀⠀⠐⠺⠀⠀⠐⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠘⡧⠈⠁⠀⠀⠈⢀⢀⡠⠐⠀⠀⠁⠀⡀⠘⡠⠂
⠀⠀⠀⠐⠂⠈⠑⠀⢈⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⣸⠎⠉⠀⠐⠁⠀⢀⢈⡁⠘⠁
⠀⠀⠀⠀⠠⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠟⠛⠂
⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⢄⡉⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢦⠵⠝⣴⣞⣿⣶⣵⣄⣀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠸⡌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠈⣉⠉⠛⠓⠛⠚⠛⡽
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⠤⠤⠀⡀⡄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠁
⠀⠀⣿⣿⠿⡿⣿⢿⠿⡿⣿⣿⡇⠀⠠⣛⣴⣺⡯⡿⢷⣦⣷⣮⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠘⣿⣷⣷⣿⣾⣷⣿⣿⡟⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⢙⡿⠁⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠉⠀⡤
⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠅⠢⠀⠀⠀⠀⠠⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⣀⠤⠀⠑⢰⠁⠀⠀⠀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠠⠐⠢⢄⠀⢀⠘⠀⠀⣠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢁⠰⣤⢈⣧⠐⢀⠔⠛⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠳⠂⠀⠀⡁⠈⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡐⠈⠙⠄⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠁⠀⠈⠁⠐⠄⢎⢈⠯⡀⠀⠏⠀⠀⠂⠰⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠀⠈⠀⠒⠤⣄⠀⠆⣀⠁⠀⠁⠀⠀⠐⢀⢉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⡀⠀⠀⣀⠈⠙⠊⡇⣴⣘⠟⠓⡀⣄⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡠⠀⠈⡖⠋⢩⣿⠛⠼⡇⠘⢧⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠋⠉⠁⢀⣼⠄⠀⠀⣿⠻⠀⠀⢈⡄⠀⠰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠲⠦⢒⡍⠉⠀⢠⡞⠙⣾⡆⠀⠀⢣⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢠⡬⠝⠀⢺⡿⢇⠀⠀⠀⣻⠦⠄⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢂⠞⠁⢀⡴⡎⢀⠈⢹⠀⠀⠘⡚⢾⣤⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⡴⠟⠀⢆⠀⠑⠬⢮⣀⠀⢡⣄⠀⠘⠆
⠀⠀⠀⣠⠶⣾⠟⠿⣿⠿⣿⢟⣿⠿⣷⠖⣤⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠂⠘⢄⠀⠈⠂⢀
⠀⠀⠘⠿⣓⣨⣛⣜⣢⣐⣪⣰⣿⣤⣥⣤⡿⠃⠀⠀⠀⠀⠀⠀⢱⢠⠀⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⠀⠰⡌⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⡀⠀⠀⢀⣀⡠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⢻⡏⠀⠀⠀⣠⡑⠀⡄⠀⠀⢀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡼⠃⠀⠀⢀⠉⡍⠆⠹⣶⣤⠃⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡠⠞⠁⠀⠀⠀⠆⠀⠅⡰⠀⠈⢻⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠔⢉⡴⠀⢠⣥⠀⠀⠀⠀⠀⢪⠀⠤⣀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠆⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠈⠛⣆⡘⢁⣠⠖⠛⠃⠠⠄⠀⠀⠈⠑⠄⠀⠠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠻⡇⠀⠀⢀⡤⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠜⠋⠀⠠⡀⠀⠀⢐⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣤⡴⠞⠁⠀⠀⠙⠄⠀⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠐⠂⠉⠀⠀⠀⠀⠀⠁⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢆⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠖⠶⠖⠀⠁⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡿⢠⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣁⡀
⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⢙⣶⢇⠸⣀⡆⠀⠀⢀⠀⠀⠀⠈⠲⣄⣀⠀⠀⣀⣄⣀⢀⣀⡤⠞⠀⠀⠉⠉⠉⠁⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠿⠿⠁⢀⡑⣝⠓⠳⠖⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢢⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢸⠇⠉⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣿⠇⠀⠀⠀⠳⣄⣈⢉⣡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡤⢯⡀⠀⠀⠀⣹⡄⠀⠀⢠⡞⢹⡭⠂⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢬⠙⠒⠐⠚⣡⣈⠁⠀⠁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠀⢸⣿⢸⡻⡿⠿⢿⡏⠴⠖⢲⠀⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢧⣼⡀⠸⣿⠀⠀⠀⠀⠀⡅⡄⠀⢸⠀⡟⣂⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣹⣧⢠⣿⣾⠀⡄⠀⠀⡇⡅⠀⡆⠀⣿⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣮⣿⢿⠄⢂⣀⣤⣦⠃⠀⠳⢤⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⡿⢿⡆⣿⢸⣿⡿⠥⠤⢬⣥⣶⡤⣜⣹⢴⡆⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡁⣀⢻⡿⠥⠌⠲⣥⣤⡄⣯⣤⣍⣿⣤⣏⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠷⢿⣾⠁⠀⠀⢠⡈⠛⢻⣝⣶⣿⣿⣿⣽⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⣦⣾⣿⣤⡤⢤⡄⡏⢘⣠⣻⣿⡻⢿⠿⣭⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣿⣿⣿⡟⠒⠘⡏⠓⠈⢩⣠⣜⡔⠛⡄⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢸⣿⣿⣿⣷⣶⣾⣷⣶⣇⣸⣿⣿⣿⣿⣿⡀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠷⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⠶⠶⢛⡽⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉⢁⣤⠶⠟⠉⣩⠿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡫⠟⠉⠀⠀⢰⠋⠀⢀⡴⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣀⣀⡀⠘⣄⡾⠋⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⣀⣴⡾⠉⠉⠉⠉⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠽⠋⠀⡼⠟⠋⠀⠀⠀⢀⡾⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⠁⠀⠀⢸⠁⠀⠀⠀⠀⣠⠟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠋⠁⢀⣀⣀⡀⠀⠀⠀⠸⡄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠚⠉⠁⢀⠴⠋⠉⠀⠉⠉⠳⣄⠀⠀⢙⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⢀⣠⡶⠟⠉⣀⣀⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⡴⠟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⣠⠟⢋⡴⠚⠉⠀⠀⠉⠓⣄⠀⢸⡃⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟
⠀⣀⡼⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡇⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⣾⠉⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⣵⠤⢶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⢿⡀⠀⢸⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠹⡆⠸⡄⠀⠀⢠⡞⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⢀⣠⠴⠶⠚⠛⠛⠲⢦⡀
⠀⠀⠙⢦⡉⣳⠞⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⣀⣤⡾⠗⠀⠀⢹
⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣀⣀⣀⣀⣤⡴⠶⠛⠛⠉⠉⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⣉⣭⣭⣉⣉⣉⡙⠛⠛⠳⣄⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠙⠒⠒⠛⠛⠛⠋⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣤⣤⣤⣤⣤⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠀⢲⠈⠀⠀⠀⠀⠀⠀⠚⡆⠀⠀⠀⠀⠉⠹⣁⠀⠀⠈⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠉⠀⡀⠁⠰⠻⡿⡧⠃⠀⠀⠀⠀⢀⠴⡇⠣⠈⠢⠀⠀⠀⠀⠀⠈⠚⢟⢢⡀⠀⠈⠦
⠀⠀⠀⠀⢀⠀⠀⢠⠁⣰⡟⣿⡾⠀⠀⠀⠀⠀⢠⣾⣾⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠷⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠟⠀⠠⠈⠠⠏⠛⡽⠁⠀⠀⠀⠀⠀⠈⢻⣇⣼⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣰⠿⢆⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⡄⢠⡚⣷⡇⠀⠀⠀⠀⠀⠀⠀⣃⣖⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡳⠟⡇⠀⠀⠂
⠀⠀⠀⣠⠀⠀⢰⢰⡊⠛⢩⡇⠀⠀⠀⠀⠀⠀⢰⣿⢛⡗⡀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⣧⠶⢿⠀⠀⡄
⠀⠀⠀⣾⠀⠀⠀⠀⣴⠛⢷⡇⠀⠀⠀⠀⠀⢀⠏⠙⠛⡴⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠷⢟⣷⠎
⠀⠀⠀⠠⠀⠀⠀⠀⢹⡛⠋⢱⠀⠀⠀⠀⢀⣦⡾⣷⡌⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠊⠁
⠀⠀⠀⠀⠁⠠⠀⠀⠀⠉⠒⠾⠄⠀⠀⡤⠲⢯⠶⠋⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⣌⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠄⠀⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠉⠉⠙⡌⠢⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⣠⠄⠠⠂⢀⣀⠀⠀⠈⠃
⠀⠀⠀⢀⡁⠀⠀⠈⢢⠀⠛⢆⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠀⠀⢠⠀⠈⡀
⠀⠀⠀⠎⠑⡀⠀⠀⠀⠂⡀⢸⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠢
⠀⠀⢰⠆⠀⠙⣴⠶⠶⣄⢈⡘⠀⠀⠀⠀⠀⡂⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⠀⠀⠀⠀⠀⠀⠂⠉⠁
⠀⠀⠘⠀⠀⠁⠞⠹⡌⠙⢷⣈⠡⢀⠀⠀⢸⠁⠐⠂⢻⢩⠹⣷⣀⠦⡄⠀⠀⠀⠀⠁⠢⠄⡀
⠀⠀⠀⢠⠀⠆⠀⠀⠀⡶⢀⡙⠌⠠⠄⠀⠀⢀⠀⠀⠉⠈⠣⡈⠙⣧⠄⠢⡀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡃⠀⠀⠀⠀⠀⠸⠊⠉⠀⠀⠀⠻⡠⡆⠁⠀⠒⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠐⠂⠀⠃⠂⠄⠒⠂⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠋⡠⠀⢠⠀⡄⢠⠀⠀⠈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠠
⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠰⠅⠧⡠⠚⠦⡅⢸⠡⠡⡠⠱⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠅
⠀⠀⠀⠀⠆⠀⠀⡰⠚⠁⠀⠁⠁⠙⠀⣄⠀⠀⢀⣼⣯⡀⢠⣤⠀⣂⢂⢄⣀⣠⡄⢠⡠⢼⡖⢀⣹⣮⡀⠀⠀⡠⠆⠉⠀⠀⠀⠀⠙⠲⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⠙⠉⠉⠀⠀⢀⡀⠤⠐⠛⠂⠤⢀⡀⠀⠈⠉⠙⡉⠙⢀⡼⠀⠀⠀⠀⠀⠀⢀⠀⠁⠐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⠐⡁⡀⢾⡶⢠⠁⠀⠁⠀⡠⠤⠤⠤⢄⡀⠀⠀⠀⡀⢾⡷⠂⠈⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠂⠲⠀⠄⠀⠤⠤⠄⠒⠀⠖⠰⠀⠶⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠛⠂⠒⠘⠂⠒⠐⠒⠚⠀⠐⠒⠚⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⠀⠀⡇⠀⠀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠅⠀⠀⠯⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢐⠐⣶⠀⠀⠀⠀⠀⠀⠁⡀⠀⠄⠐⢠⢄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡄⠀⠀⢠⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⠊⠠⠰⠶⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠔⠀⡈⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠃
⠀⠂⠀⠀⠊⠛⠈⠉⠀⣠⣤⣤⡄⠀⢀⣐⣶⣄⠀⠀⠀⠀⠘⠛⠃⠀⢀⠁⠐⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡍
⠀⠀⠀⠀⣠⡀⢀⣀⠀⣉⣉⣹⣙⠀⠘⡓⡶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠚⠓⠛⠓⠀⠀⠀⡁⠀⠀⠀⠦⠄⡄
⠘⠋⠿⣿⠋⠃⠘⠹⠄⠤⣼⡿⠻⢶⣿⡷⠻⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⡇⢺⣽⣿⢻⠛⣿⡷⠀⠀⠀⠐⠃⠀⠘⠀⠒
⠸⡀⡄⣧⠀⠀⠀⠀⠀⠀⣨⢁⠈⢸⣿⠅⢠⡀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠧⠴⠦⠤⠈⠀⠀⣴⠘⣜⢷⢸⠀⠁⡇
⠀⠁⠁⠀⠀⠀⠀⠉⠉⠀⠛⠚⢅⡀⡏⠀⢸⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣻⠼⠖⠛⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠐⠂⢤⡀
⠀⡀⣀⣀⣀⠀⣀⣀⣀⣀⡀⢄⣤⡿⠇⠀⠀⣏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣦
⠰⣶⣶⣶⣶⣾⣿⣛⣯⣭⣭⣭⡧⢺⡃⠀⢠⡟⠚⣄⠀⣐⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣃⠈⢻⣦⣼⡿⠭⡭⣿⣿⣭⣭⣾⣬⣿⣭⣿⡿⡶⠓⣛⣁
⠈⢻⣿⣿⣟⣿⠃⠉⢹⠀⡇⢺⠁⢸⠃⠀⠸⣦⣤⡞⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡏⠉⣿⠛⡟⠙⣿⣿⠛⠛⣿⣿⡿⣿⣿⢿⣿⣛⣻
⠀⣼⡏⣿⡏⢻⠀⣀⣸⠀⠃⢸⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠈⢸⡇⠀⣿⠂⡇⠘⡇⢸⣴⣦⡿⠞⢇⠀⡇⠸⢿⡾⡿
⠘⠮⠴⠉⠀⠈⠂⠙⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡿⠀⢁⣀⡇⠸⢹⣽⣿⡦⣼⠀⡇⢸⢰⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⠔⠁⠀⠉⢻⣧⣾⠸⠃⠺⢸⡿
⣿⠿⠿⠿⢿⠟⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣛⣿⣿⣟⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣀⡀⠙⢦⡀⠀⠀⠀⠀⢀⡾⢋⣁⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣰⠟⣉⠙⢦⡀⢻⡆⠀⠀⠀⣿⠀⠚⠁⠀⣠⡟⠀⠀⠀⣠⡴⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⠰⡏⠀⠘⣇⠀⣿⠀⠀⠀⠙⠶⣤⡤⠾⠋⠀⠀⠀⢸⡇⠀⠶⠶⠌⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢿⡀⠀⠀⣰⠇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠈⠷⣄⡀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⠷⠞⠋⠀⠘⠷⣤⡀⠀⠀⠀⠀⣴⠟⣉⣍⠉⠙⢷⡄⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣴⠛⠛⠳⣦⠈⢻⡄⠀⠀⠀⣿⡈⠋⠁⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⢸⡇⠀⠘⣷⠀⢿⠀⠀⠀⠈⠛⠳⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠻⣄⠁⠀⢀⣿⠀⠈⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠛⠒⠛⣡⠶⠛⠳⢦⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣆⠀⣿⠐⣦⡀⠀⠙⣧⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠘⢷⣌⣁⠀⣠⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡈⠉⠉⠉⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠠⠤⡀⠠⠀⠀⡀⠁⠀⠈⠢⡀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠁⠈⠁⠀⠀⠘⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⡄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠡⢏⡐⡠⢃⣜⡀⢄
⠀⠀⢰⣶⠤⢄⣀⠒⠀⠀⡀⠀⠀⠀⠻⢦⣴⡬⠉⠓⠎⠴⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠈⢳⣝⢢⣄⡀⠓⠂⠀⠀⠀⠀⠀⠀⠉⠛⠓⠷⠾⠶⠴⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠻⣮⣝⣒⣤⢤⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⢀⢀⡈⠱⣆⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⠍⠘⢃⠥⢹⣿⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠆⠀⠀⠀⠀⠈⠋⠳⠂⢤⠁⠤⠹⠯⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠘⢒⣿⢁⠰⣖⢙⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠠⠶⠒⠻⠿⢟⠚⢂⣶⠾⠀⠀⠀⠀⣄⠐⣒⣖⢛⣛⠛⣓⣀⣀⣀⣀⣀⣉
⠀⠀⢤⣠⣌⣀⡠⠀⠠⠐⣋⠭⠳⡇⠋⣸⠆⠈⠀⠀⠀⠀⢔⢀⡐⣛⠭⠖⣋⠁⠬⠀⠀⠉⠠⠔⣺⢿⠄
⠀⠀⠀⠈⠙⠫⡂⢥⣲⡩⠬⢒⣉⣠⡾⠃⠀⠀⠀⠀⠠⠤⣬⣨⣥⣤⣦⣿⠁⢀⠀⠈⠁⣠⡴⣪⡵⠋
⠀⠀⠀⠀⠄⢤⣍⣨⣤⣠⣦⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣐⣾⣭⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣿⣭⠙⠿⠟⠛⠛⠉⠉⠛⠻⠏⡈⠻⣿⣆
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣠⡞⠻⣦⣼⡿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣌⣿⡇
⠀⠀⠀⠀⢀⣿⣿⣄⠰⣷⣬⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆
⠀⠀⣀⡰⣿⣿⣏⣽⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄
⣠⣾⣿⠿⠿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠿⠿⠿⠷⠶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠘⠋⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⡾⠃⠀⠘⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠀⠚⠀⠘⠀⠈⢠⣼⠇⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⢩⣤⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠁⣠⡄⠀⠀⠀⠀⠀⠀⠈⢻⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠘⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠈⠂⠀⠀⠀⠈⢀⡠⠀⠀⠀⠀⠀⠀⡈⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠛⠀⡠⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⢸⣷⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠠⣿⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣀⠀⣄⠀⠀⠹⣄⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⢹⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠙⠀⡀⠀⠘⣶⡄⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⢸⡏⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⣴⡿⣦⣶⠀⢹⡆⠘⠃⣤⣀⣀⣀⣄⣴⣿⣿⡆⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣷⣾⣿⣶⣦⠄⠛⠛⠛⠛⠋⠀⠀⢻⣿⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠐⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠀⠀⠀⠀⠂⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⢿⡅⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠁⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⠀⠀⠀⠀⠀⠀⢠⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⡧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⢛⡿⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⡀⠀⠀⠀⠀⠀⣠⡀⢀⡀⠰⠆⠀⠀⣀⣀⣾⠿⡿⢲⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢆⣤⣤⣤⣶⣍⣠⣄⣠⣄⣼⣦⣸⣿⠉⠻⠷⠒⠿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠇⠀⠀⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠠⣿⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠛⠈⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⠂⠀⠙⠷⠀⣾⣦⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢂⠀⠀⠀⢸⠏⠀⣠⠀⢸⣷⠰⢶⣄⠠⡶⠀⣀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⣦⣀⠀⠀⠀⠈⠁⣀⡀⠁⢀⣌⣹⠿⠀⢀⣿⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⠏⢠⡶⠄⣠⡴⠻⣟⡀⠈⢻⣿⡧⢰⡿⢙⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⡘⢧⣾⣿⠃⣠⣏⡻⢶⣾⣁⣴⣿⣴⣿⡿⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣿⣁⣼⣿⣿⣷⣴⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣿⣶⣿⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠟⠋⠻⠟⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠀⠀⠀⠀⠈⠀⠀⡀⠀⡆⢀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠰⣄⣀⡀⠀⠀⠀⠤⠀⠉⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠰⠔⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠰⠀⡀⠀⠀⢀⡀⡀⢅⠀⠀⠈⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠄⣈⡀⠐⡀⢀⡐⢤
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠀⠀⡀⠀⠀⠢⡀⠁⠄⠀⠀⠀⠄⠁⢀⠀⠀⠀⠀⢀⠀⣀⠀⢠⡀⠉⠀⠄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈⢁⠴⠀⠐⡁⠁⢂⠘⠀⡀
⠀⠨⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡴⠟⢻⡎⢀⠀⠐⣠⠄⡼⡏⠐⠀⠀⢀⠀⠀⠀⠠⠀⠀⠉⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⠃⠀⠀⣰⠣⢰⢟⣤⢁⠤⠀⣀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⡴⢿⣦⡿⠄⡿⠾⣊⢾⣊⠴⠃⢀⠄⠁⠀⠀⠰⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿⠀⠀⣠⠺⣇⡰⢻⠁⠀⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠀⠈⠓⠚⠁⠀⠉⠀⠸⠔⠀⣤⠁⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠂⠉⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠐⠂⠀⠀⠠⢂
⠀⠀⠀⣤⠐⣲⣲⣤⡤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢄⣀⣀⣀⡀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠈⠁⠈⠁⠀⠀⠀⠀⠠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⡀⢴⣦⣤⣀
⠀⠀⠀⠀⠠⠤⠺⢥⠴⠛⣤⠌⢳⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠇⠀⠀⢹⡆⠲⠶⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢹⡆⠀⠀⠓⢶⡀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠂⠋⢨⣞⣃⢸⣿⣦⣄
⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⢨⣷⢰⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠘⣚⣿⡏⢻⣿⣆
⠀⠀⠀⠀⠀⢠⠷⠚⠆⠀⠀⠘⢿⣿⢸⣿⡦⠀⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢹⣿⣿⣌⣿⣿⣆
⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⢻⣼⣿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢈⢿⣿⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⠿⠄⠀⠀⠀⠀⠀⡏⠀⠀⠠⣸⣾⣿⣿⣿⣦⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⣿⣿⣏⣿⡟⠻⣷⡀
⠀⠀⠀⠀⠀⢸⣆⣠⡤⠀⠀⠀⢿⣿⣟⣿⣿⡄⠀⠀⠀⠀⠀⠰⠿⢆⠂⠀⠀⢀⠀⠀⠀⠀⠀⠘⢹⣏⢹⣿⣸⣿⣧
⠀⠀⠀⠀⠀⢸⡽⡆⠀⠀⠀⠀⠈⠹⣿⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢸⡄⠀⠀⠀⠀⢀⣿⣿⣺⣿⣷⣽⣿⣧
⠀⠀⠀⠀⠀⠀⡖⢁⠀⠀⠀⠀⠀⠀⢻⣿⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠘⡃⠀⠀⠀⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢈⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⢢⡀⠀⠀⠀⠈⢽⣿⣿⣽⣿⣿⠙⠻⣿⣦⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠈⡍⢷⢄⠀⠀⠀⢀⣽⣿⡇⢋⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡙⠢⡀⠀⢀⣸⣿⣿⡈⢻⣿⣄⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢖⡀⠀⠜⢿⣿⣷⣎⣀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢌⡈⠄⠱⣻⡿⣷⣆⠻⣿⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠻⣿⣯⠃⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢧⣠⡀⠠⠀⠈⠉⣦⣹⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡸⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠐⠘⣿⣏⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡎⠢⡀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠈⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⣿⣆⡘⢆⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⣾⠿⡿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠄⠦⣿⣿⣿⡿⠿⠓⠠⠵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⠤⠴⠖⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⣇⠶⢠⣤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠜⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⢰⡌⠺⢂⡠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠄⠀⡀⢠⡆
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⣋⠘⢃⡸⢇⡸⣧⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⣤⣾⣝⠟⢧⡔⠁⣀⡀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡌⣡⡈⣥⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠭⡴⠞⣛⢠⣿⡋⠀⠀⠀⠈⠷⣶⠶⠆
⠀⠀⠀⠀⠈⠟⡂⢸⡄⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⢧⠄⠀⠀⣠⣴⣿⣷⢧⢿⣧⠈⠛⠓⠐⠐⠛⠉
⠀⠀⠀⠀⠀⠀⢚⡰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠼⠁⠀⢸⡷
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⣄⠀⣶⣀
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠃⠀⠀⠈⠛⠛⠳
⠀⠀⠀⠀⠐⢀⣩⣿⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠻⣿⢿⣇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⢀⢨⣤⣆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠜⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⠈⢛⢻⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⡴⠁
⠀⠀⠀⠀⠀⣀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠙⠦⣄⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠈⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠘⠀⠀⠰⠁⠀⠂⠐⠃
⠀⠀⠀⡐⢛⡹⠀⢳⢛⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠵⠆⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡐⣀⠀⠂⠀⢀
⠀⠀⠀⠃⠀⠂⠂⠈⢫⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢀⠂⠀⡼⡏⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠑⠒⠂⠁
⠀⠀⠀⠧⠀⡢⠟⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠤⠄⠀⠀⢰⡄⠀⠀⠠
⠀⠀⠀⠀⢼⠁⣒⡀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠋⠀⠐⠀⠃
⠀⠀⠀⠀⠑⠲⠿⠷⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢤⠀⢀⠀⠀⠀⠴⠟⠁⠀⠄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠸⠆⠀⠀⠄⠀⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠁⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣤⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⢞⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣿⣹⣟⡿⣻⡗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠤⡤⢩⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⣽⣻⣿⢷⣵⡇⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢂⣤⣀⣶⣁⣀⠤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣿⡀⣀⡠⠟⠿⠈⠉⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣤⡶⣶⣦⡀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠂⢀⢀⡀⠲⢤⣄⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⢀⣤⣞⡿⠷⠝⣻⡧⣸⣷⣷⠦⢤⣄⠀⠀⠀⠀⠀⣴⣿⠿⠿⢳⡌⠛⢏⡨⢧⠈⣿⢿⣷⡄⠀⢻⡟⠿⣿⣶⣄⠀⢾⣷⡌⠉⠉⠋
⠀⠀⠛⠿⣿⣶⣦⣞⣵⠟⣹⢻⡿⡝⣿⣖⡶⢙⢦⡀⠀⣾⣷⣿⡿⢿⢨⠴⠘⠦⣤⡼⠁⣿⣻⠟⠀⠀⠈⠻⣿⣮⣻⣿⣆⠀⠘⠟
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢻⡜⣿⣖⠉⠳⡱⡄⠀⠙⠻⠿⣿⣧⡍⠢⣁⣈⣡⣾⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣹⣿⠀⣀⣀⠀⠀⠀⢸⠃
⠀⠀⠘⢿⣷⠀⢹⣿⡆⠀⣿⢸⣷⠸⣷⠸⣿⣶⡤⠹⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣠⣤⣤⣤⣄⣀⠀⠈⠛⠛⠁⠀⠻⣽⣿⣦⣠⡏
⠀⠀⠀⠀⠀⣶⡆⠉⠁⠀⢿⢸⣿⣇⢿⣧⣿⣇⠚⢤⠇⣷⠀⢠⣶⠀⠀⠀⠛⠳⠶⢞⠍⠭⡭⠭⢙⡛⠿⣿⣷⣤⡀⠀⠀⠀⠀⠴⠛⠉⢹⠁
⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠸⣇⢿⡻⣿⣿⣿⣿⠃⠶⠂⣿⠀⠿⠃⠀⠀⣠⣿⣄⠀⣯⢀⠊⠄⠠⠀⡀⠋⢰⢝⡻⣿⣆⠀⠀⠀⠀⣠⢖⡻⣆
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⣜⣿⣿⠋⠉⠉⠉⠛⠀⠀⠀⢀⣾⣿⢻⣿⣷⡈⠪⡳⣄⠀⠉⣷⠴⣀⠈⡐⠙⣿⠀⠀⢀⡾⣵⣿⣧⢻⡄
⠀⠹⣆⠀⠀⠀⣀⣀⣤⣤⡤⠤⢤⣄⣈⠉⠛⠀⠰⣶⠶⠀⠀⠀⢀⣿⢹⡏⣸⣽⣿⣇⠀⠘⠺⡿⠶⢦⣀⣇⣯⣴⣶⡼⠇⣠⢟⣾⣷⡟⢾⣿⡧
⠀⠀⢻⡟⠛⢻⣿⡟⣩⢲⢒⡖⡶⡦⣌⣙⠶⣄⠀⠀⠀⠀⠀⠀⢸⡇⣼⠇⣿⡇⣿⣿⠀⠀⠀⠀⠈⠒⠳⠿⠒⠛⠉⠀⢠⣟⣾⣝⣧⣽⣾⣿⠇
⠀⠀⠘⠇⠀⠀⢻⡄⣿⠻⢺⠟⢟⢷⣵⣽⣳⣌⠳⣄⠀⠀⠀⠀⣿⡇⣿⢀⣿⡇⣿⡿⠀⠀⣴⢲⣦⠀⣿⣶⣤⣄⠀⠀⢸⣿⣻⠿⣙⣼⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣾⡿⡾⣾⣬⣌⣙⣳⣷⡙⣦⠀⠀⠀⢻⡇⣿⢸⣿⣸⣿⠇⠀⢸⣷⡘⣿⠀⢻⣯⣿⠻⣧⠀⣿⣗⣿⢿⡻⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣽⣳⣷⠿⠷⠿⠿⠾⠿⢈⣧⠀⠀⠈⢿⣿⣸⣿⣿⡏⠀⠀⠈⢿⣧⣙⣧⠀⠛⠿⠷⠿⠀⢻⣙⣚⡯⠞⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠛⠛⠉⠁⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠋

⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⣀⣀⣀⡀⣀⣀⣠⣄⣲⠟⠛⠛⢿⡟⠉⢈⣹⠆⠀⠀⠀⢀⣤⣤⣴⠖⡄
⠀⠀⠀⢸⠿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢾⣿⡋⠉⠉⢹⡟⠉⢀⢻⣧⠴⠶⠶⠛⠉⠈⠉⠀⠀⠀⠀⠀⠈⠓⠶⠖⠂
⠀⠀⠀⣯⠀⢘⡇⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠈⠛⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⢢⠀⠀⠀⠀⠀⠀⡄⠀⠀⠦⡀
⠀⠀⢀⡯⠀⢸⡇⠀⠀⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠋⣠⠎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⣆⡀
⠀⠀⠈⡧⠀⠠⡇⠀⢀⡟⠈⢷⠀⠀⢠⡞⢋⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⣼⠛⢷
⠀⠀⢸⡇⠀⠀⡁⠀⠀⣳⠤⢸⡄⠀⠘⡇⢸⠅⠀⠀⠀⠀⠀⠀⡄⠠⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢹⠀⠀⠀⣏⢀⣠⠃
⠀⠀⢘⡿⠀⠶⡇⠀⠀⢁⡀⢀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⢤⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣆⠀⠀⢻⡄⡌⠀⠀⠀⡏⠀⠉⠃
⠀⠀⢸⡇⠀⢀⠀⠀⠀⢸⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢿⡄⠀⠆⠀⢠⠀⠀⠀⢻⠂⢲⡄
⠀⠀⠘⡧⠀⢺⠀⠀⠀⠈⣿⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⢐⡇⢻⠀⠀⠀⢸⣀⣼⠃
⠀⠀⠘⡇⠀⠘⡀⠀⠀⠀⢹⣄⢸⡇⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⣤⣤⡤⠤⠠⣄⡀⠀⠀⠀⠀⠰⡇⠈⠀⠀⠀⣾⠀⢘⡇
⠀⠀⠀⠐⠂⠀⢃⠀⠀⠀⠈⣗⠀⢷⠀⠀⠀⠉⠒⢿⣿⣉⣉⠉⢹⡟⠋⣉⡛⣿⡀⠤⠤⠴⣟⣃⣤⠶⠞⠃⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣽⠂⣿
⠀⠀⠀⠀⠑⣤⢲⠀⠀⠀⠀⢹⠁⠹⡄⠀⣀⣀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠒⠂⠀⠒⠈⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡇⡰⠀⠀⠀⣿⢀⣾
⠀⠀⠀⠀⠀⠈⢾⠃⠀⠀⠀⠘⡀⠀⡅⠀⠹⣏⠉⠙⠶⣦⣄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⢀⡄⠀⠀⠸⠞⠃⠀⠀⠀⣿⣠⡏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠡⡀⢸⠀⠀⠙⢿⣷⣄⣼⡅⣀⡈⠙⡿⠒⠒⠲⢶⣦⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾
⠀⠀⠀⠀⡤⢤⡤⠀⠀⠀⠀⠀⠀⠙⣮⡆⠀⠀⠀⠈⠛⠿⠿⠿⣿⣾⣷⣤⣦⡤⢴⣧⣀⣀⣈⠙⣷⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣋⠞⠁
⠀⠀⠀⠀⠓⠤⠤⠚⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠛⠛⠛⠛⠛⠉⠁⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣋⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣯⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣠⡤⠞⠁⣀⡀⠙⠻⢿⣶⣶⣦⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠻⠛⠉⡉⠈⠁⠀⠀⢀⠜⠉⣿⣄⡀⠀⠙⠻⠷⡿⣋⠿⣿⣹⢿⠯⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠍⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣿⣿⣯⣦⠀⠀⠀⢈⡀⠓⠢⠉⠛⠶⣾⣝⢮⡻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⠀⠄⠀⢀⠀⠀⠋⠀⠀⠀⠀⡠⠀⣰⣧⡀⠸⠂⣿⡿⢿⣷⡈⠀⠀⠀⠈⠂⠀⠀⠠⡐⡄⠉⠳⡝⣎⢿⣦
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⣾⣙⠻⠟⠞⠀⣿⠀⠀⠘⣸⣧⡀⠤⣀⠀⠀⠀⠀⠘⡀⠀⠰⡴⡜⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⣴⣧⡴⠋⠁⢀⢠⠀⣻⣆⠀⠀⠀⢙⣿⣦⣍⠳⣄⠀⠀⠀⠠⠀⠀⢡⣴⣹⠸⣿⡄
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡘⠀⠀⠀⠀⢀⡄⠀⣠⣿⡝⣏⠀⠀⡠⣸⣿⡀⣿⣿⣤⠀⠀⠘⣿⣿⣿⡷⣌⠳⣄⠀⠀⠀⡄⠊⠞⣿⣇⣯⣷
⠀⠀⠀⠀⠀⡿⠀⠈⠶⠀⠃⠀⠀⠀⠀⠉⡠⠊⣿⣿⣜⠁⠀⣨⣽⣿⣿⠃⣿⢹⣿⣄⠀⣼⣿⡿⠟⠀⠈⠑⢌⠓⢄⠀⠃⠠⢸⢻⣿⢿⢹
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⠯⣿⣧⣼⣿⣿⡿⠋⠀⣿⢸⡿⣿⡿⣯⣽⣦⣄⠀⠀⠀⠀⠙⢦⡻⣄⠀⢸⢸⢾⠘⣼⡀
⠀⠀⠀⠀⠀⡧⠀⠰⠀⠀⠀⠔⢀⠜⠃⠀⠀⠀⠀⠀⠈⢙⣻⣿⠿⣦⠴⠀⣿⡾⡓⠃⠀⠀⠘⠷⢿⣷⡄⠀⠀⠀⠀⠹⣄⢷⣄⢸⠸⣾⣾⡇
⠀⠀⠀⠀⠀⡇⠀⢀⠀⢠⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠛⢺⠅⣿⣇⠁⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠈⠳⣌⢿⡌⣿⡏⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢡⡞⠁⠀⠀⠀⠀⠀⠀⢠⣾⠫⠀⠀⠀⠀⢀⣿⡇⠟⣾⣦⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠙⢦⣿⣧⣷⣿
⠀⠀⠀⠀⣴⠃⠠⢎⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢃⠀⠀⢀⢈⣾⣿⡇⢻⣿⣿⣦⣤⢀⠀⣠⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡄
⠀⠀⢠⣾⡝⢠⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣦⣴⣷⣿⣿⣿⡇⢸⣿⣿⣿⡷⠾⣾⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠃
⠀⠀⠘⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⢰⡅⢸⣿⣿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣿⠀⠈⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡂⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣬⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⣿⠉⠙⡿⠛⡟⠿⠛⢛⢿⣛⡟⠛⠟⠿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⣀⡠⠄⠤⠄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠈⠑⢄⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠉⠁⠀⠈⠄⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠐⠲⣄⣀⣀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡄⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⢆⠀⠀⠀⠈⠑⠢⠤⣀⠀⠀⠀⠱⠀⠠⠀⠄⠀⠁⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⡀⠀⠀⠀⠀⠀⠇⠘⣆⠀⠙⠢⣄⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢹⠀⠀⠀⠀⢸⠁⠀⢻⢦⠀⢀⠔⠛⠒⢤⣄⠀⠀⠀⠀⠀⠀⠉⠙⠢⢄⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⠀⢸⡀⠀⠀⢣⡳⣀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣘⠃
⠀⠀⠀⠀⠀⠀⠳⣄⣀⠴⠃⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⢞⢷⢄⢀⠀⢀⡌⢀⠨⠭⠷⢶⣄⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢣⠙⢤⣁⠀⠀⠆⠀⠀⠀⠀⠈⢪⠓⢤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢣⡀⠈⠑⠢⣀⠄⡀⠀⠄⠀⣠⢠⠎⠁⠀⠀⠙⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠢⡄⠀⠀⠀⠱⡄⠀⠀⠀⠀⠁⠙⢦⡀⠀⠈⠑⠦⢏⣀⣈⡀⠘⢄⠀⠀⡀⠀⢠⠇⢀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠀⡿⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠉⠉⠑⠲⠥⢤⡴⠶⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⠤⠤⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⢻⣿⣷⡀⠀⢀⣶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢎⢿⣿⣿⣷⣰⣿⣿⣷⡀⠀⠀⠀⠀⠤⢀⠀⠀⡏⢂⡀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢸⣿⣿⣿⡟⢿⣿⣿⣷⠀⠀⠀⠀⠐⣏⡳⠸⢏⡟⢞⠐⠐⠃
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢳⠊⢻⣿⣿⣇⢮⣿⣿⣿⠂⠀⠀⠀⠀⢺⡉⢞⣄⠖⢙⡇
⠀⠐⠶⢾⣿⣿⣿⣿⣷⣤⡉⢳⡽⣿⣿⣸⣿⣿⣿⠀⠀⣐⣂⡀⠀⠛⢫⠻⠛⠟⠂⠀⠤⠚⠋⠁
⠀⠀⠀⠻⣾⣫⠘⡛⢿⣿⣿⣦⡀⠛⡻⣿⣿⣿⡏⠀⠉⠉⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⠛⠤⣷⣯⣚⡯⣙⣿⣶⠁⠙⠿⢹⠇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠁⠓⠢⢬⡄⠀⠀⢸⡇⠀⠀⣤⣠⠀⠠⣾⣿⣿⣿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠦⡒⢸⣷⠀⢀⠀⠀⠙⠲⠦⢦⣬⣭⣭⡴⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣾⠅⢀⡈⣾⣿⣼⡃⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣾⣹⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⣸⣏⠋⠀⠀⠀⢠⢽⣷⢻⣟⡀⠠⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣤⣿⡇⠀⠀⠀⠀⢸⣿⠛⠂⠹⢿⣦⡐⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠏⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠉⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣭⣭⣭⣽⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡀⢳⠀⠀⠀⠘⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⡇⠀⢘⣿⢸⠀⠀⠀⠀⠀⠰⡄⡔
⠀⢡⠸⣇⠀⠀⠀⢣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⣇⣤⣼⣿⠘⠀⠀⠀⠀⠀⢰⣃⠇
⠀⠀⠀⢻⡀⠀⠀⠘⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠁⠈⡿
⠀⠀⢀⠸⡇⠀⠀⠀⠇⢷⠤⠀⠤⠤⠄⠐⣒⣒⣒⣣⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⢷⠀⠀⠀⠘⠒⠒⠂⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠄⣐⣞⡉
⠀⠀⠀⠀⠈⡇⠀⠀⠀⣀⠠⠀⠄⠀⣀⠒⠒⠀⢈⣍⠳⠀⠀⡇⠸⣦⣀⣈⡓⣜⣯⣧⡄
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠁⢸⠁⠀⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠇⣬⣿⣿⠿⢻⡿⣿⣯⣅
⠤⠤⠄⠲⣆⢘⡆⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣾⣿⣷⡼⣟⢹⣿⡷
⠛⠉⠉⠉⠉⠀⢧⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⠀⣉⣿⣷⣣⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⢸⡐⠤⠀⡐⠂⠀⠩⠍⠉⠚⠀⠀⠀⠒⠟⠿⠙⠀⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⡊⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠃⢀⠤⠀⠒⠀⠈⠉
⠆⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⣀⡀⠤⠤⠔⠒⠀⠈⠁⠀⠀⠓⠂⠀⠐⠢⡀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠀⠀⡰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠑⢤⣀⣀⣀⣤⣤⣴⣶⣾⢻⠀⠟
⠊⠀⣄⣀⣠⡤⠤⢴⡶⢎⡉⠉⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠲⢯⣄⣈⡉⠈⢹⣾⣿⣿⣿⣿⣿⣿⠈
⠒⢁⡠⠜⠁⢀⡴⠋⢀⡴⠋⠂⠒⠠⢑⠂⠤⡶⠒⠂⡴⠋⠀⣠⠚⠋⢂⠀⣤⣀⣽⣿⠂⣿⡿⠿⠛⠛⠉⠉⠉⠀⠀⢀⣀
⠀⠊⠀⣠⠔⠁⢀⠔⡡⠊⣷⠦⣤⣀⢠⡨⡁⠷⣴⠫⠖⣠⣾⣻⠀⡴⡳⠀⣶⣿⠿⡟⡦⠁⢀⣤⣤⣤⡤⠖⠒⠂⠉
⠀⡠⠚⠁⢀⢔⠅⢀⡼⠿⣿⣿⣷⣍⣛⠷⠅⢚⡿⠓⠚⠉⢉⣀⠀⠀⡀⠀⠉⠉⠛⠒⠛⠉⠁⠉⠛⠛⠁
⠊⢀⡤⠞⠁⠀⡠⠊⠔⢡⣾⣏⣛⣷⢮⣍⡁⢈⣠⡴⠾⠟⠛⠋⠉⠉⠡
⡴⠛⠀⢀⡴⠋⠔⢀⡴⠛⠶⣦⣰⡟⠛⠉⠉
⠀⣀⠴⠋⠀⢀⡴⢋⠀⢀⣼⠋⠸⡇
⠜⠁⠀⠀⡴⢋⡴⠋⡰⠛⡳⢤⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣴⢀⡤⣀⠀⠀⡀⠀⢠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲⢶⣦⣤⠞⠁⠀⠀⠀⠀⡀⠀⠄⠀⣡⠔⠉⠁⠀⠀⠈⢀⢠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣊⠅⠄⡀⢀⣿⡇⠀⠀⢀⣤⡖⡁⢀⠀⠄⢸⡍⠀⠀⠀⠀⢀⡠⣣⠎⠀⠀⠀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⢠⢃⠊⠀⠀⠈⢳⢸⡇⠀⢀⢐⠑⠋⠀⠀⠀⣶⡄⠚⠻⢋⣤⠶⠞⣩⠞⠁⠀⣠⠖⠉⠀⠀⠈⠂
⠀⠀⠀⠀⡰⢡⡏⠀⠀⠀⠀⣿⡟⠀⢠⣆⠁⠀⠀⠀⡀⢀⣸⠃⣀⣀⡿⠶⠟⠋⠁⠀⠀⢸⡇⡀⠀⠀⠀⢠⠄
⠀⠀⢀⢜⠁⠘⡂⢀⣄⡦⢻⠟⠀⠀⠃⢺⢦⣄⣈⡶⣖⣫⡥⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠙⣫⠶⠞⠋⠁
⠀⢀⠀⡀⣤⣀⣀⢍⣹⡾⠋⠀⠀⠀⣪⣠⣬⣭⠧⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠐⢢
⠀⣈⡜⠁⠀⠘⠛⠊⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠊⠁⠀⠈⠉⠁⢘⣷⣄⣀⣀⡀⠀⠠⠂⠀⠀⠀⠀⢚
⢰⣿⠁⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⢀⡠⠀⠘⠁⠂⠀⠀⠀⠀⠀⢀⣠⠾⠋⠉⠉⠉⠁⢰⡆⢠⢀⢄⣐⡠⠖
⢘⠹⣢⢀⢀⡰⢿⡿⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠐⠊⠀⠒⢰⡤⠄⠋⠀⠀⠀⠀⠀⠀⠀⠈⠫⠿⠖⠛⠁
⠸⡆⠀⡈⠉⣆⣼⠃⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣣⠖⠈⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄
⡌⠐⠁⠀⠋⣯⣿⡄⠀⠠⠋⠀⠀⠀⣀⠀⢀⠼⠛⠁⠀⠀⠀⣠⡶⠛⠁⠀⠀⢉⢭⢻⡶⠒⠛⠉⠀⡴⠋
⠇⠁⠀⠀⢀⣼⣸⡇⢰⠀⠀⠀⠀⠀⡵⠋⠀⠀⠀⢀⠠⠄⠬⠈⡀⠀⠀⠀⣠⣸⣾⣋⡤⠒⠐⠂⢉⠈⢆
⠀⠣⢀⢀⣜⣿⢽⡇⣼⠄⣀⣀⠴⠊⠀⠀⠀⢴⠾⠛⠉⠈⠁⢠⣝⣯⣿⣿⣿⠿⠯⠁⠒⠀⠁⠣⠉⢄⢈⠣
⠀⣤⠀⠻⢙⠉⡿⠀⠗⠚⠉⠀⠀⠀⢀⣤⡂⣆⠀⠀⠀⠀⣤⡾⣻⠿⠚⠉⢀⠀⠀⢀⠀⠀⢀⡄⠀⠈⡊⠐⡆
⠀⠈⠻⣷⣈⣼⠃⠀⠀⠀⠀⠀⣠⠚⠁⠀⠰⣙⠓⡴⢾⣿⣾⠟⠁⠀⠀⠀⠈⠆⠄⠀⠀⡠⠊⠁⠆⠀⠀⢺⡇
⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⢠⣞⠉⠀⠀⠀⠀⣾⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⢰⡀⠀⡰⠗⡃⠀⠀⠀⠀⠄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣠⢰⣰⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⡜⢁⠀⡠⢔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠙⣻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢾⠁⢈⡐⠑⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣊⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣼⡇⠀⠀⢸⣦⠀⡆⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⣿⢰⡶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⡏⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡎⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⣿⠀⠸⠌⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⠀⠀⢹⡀⣀⠼⠄⠧⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣹⣿⡏⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠶⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⡀⠀⠀⠀⢤⡦⠖⠚⠛⠋⠉⢉⠁⠀⠀⠀⢀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⡇⣁⡰⠂⢶⠸⠘⠐⠛⠃⠘⡘⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⢸⡧⠂⠐⠀⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠛⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣦⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣷⣞⣿⣷⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⠀⣹⣿⣿⡟⠛⠛⠛⠛⠛⠻⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⢁⣀⣤⣄⣀⣠⣄⣤⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⡸⠃⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠃⣐⠒⠚⢻⢛⠟⡃⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⣴⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢗⠀⠀⢸⣿⣿⣿⣿⣧⣰⣾⣷⣶⠤⣀⠼⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠸⠀⠀⣨⣭⡟⠙⠛⠛⣻⠋⡉⠛⢃⡙⠋⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢛⢀⠀⠀⠀⠀⠀⠀⠒⠒⠡⠾⠁⠀⡘⠀⠐⠀⠀⣸⠿⣄⠀⣶⠰⠃⠈⠀⠀⠈⠁⢶⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⡟⠀⡀⠀⠈⠀⠀⠀⠀⠶⠀⠀⠉⠇⠀⠀⠀⢀⡈⠀⠀⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠡⡀⠐⠿⡁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠁⠀⠀⠉

⠙⠛⠿⣿⣷⠀⠈⣷⣷⣶⡶⠶⠶⠶⠲⠟⠋⠁⠋⠙⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⡐⠁⠀⠀⠀⣀⠈
⠀⠀⠀⠈⢻⠒⠂⢿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⣀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⣤⡄⠀⠀⢸⠏⠈⡓⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣾⣿⣿⡟⠉⠀⠀⠀⠀⠀⣅⠠⠄⠀⣐⣀⣨⣥⠤⠶⠖⠚
⠀⠀⢶⣶⣾⡇⠀⣷⣤⣤⣤⠀⠀⢾⠃⢠⠄⠀⢧⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠐⡞⠛⠉⠉⠁
⠈⠄⠀⣿⣽⡇⠀⢫⣭⢹⠉⢀⡦⢾⢧⣮⡀⢀⡠⠿⠟⠃⠇⠀⠀⠀⠀⠀⠀⠀⡇
⡐⡀⠀⢻⢻⡇⠀⢸⣿⢸⠀⠄⣾⣬⠾⣽⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢰⡇
⠲⠀⠀⠀⢸⡇⠀⢸⢼⢸⠀⠀⠸⡽⠀⠛⠀⠀⠀⠀⠀⠀⢀⢸⡄⠀⡇⠀⠀⡠⠧⠤⠤⠤⠤⠶⠖⠒⠒⠒⠒
⠷⠄⠀⠠⢸⡇⠀⢸⠑⢸⠞⠀⠠⣸⣀⣴⣶⡄⠀⣰⣶⡆⠈⠉⠀⣤⣧⣤⣠⡤⠤⠤⠤⢤⣤⣤⣄⣠⣤⣤⣤
⣷⡗⠀⠀⢸⡇⠀⣸⠓⢸⠘⠛⢛⣻⡛⣿⡛⠗⢠⣿⣿⣿⣷⣿⣿⣿⡯⣷⣾⣷⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣳
⣿⡇⠀⠀⢸⡇⠀⣸⠏⢸⢠⣾⣯⣭⣿⣿⡿⠿⢿⣿⡿⣿⢹⣿⣿⡟⠃⠞⠻⠸⠿⣿⣿⡏⠿⢿⢿⣿⣯⣿⣸
⣿⡇⠀⠀⠸⡇⠀⠀⠠⢤⣤⣭⣉⣭⣭⣴⣶⣶⢸⣿⣿⣟⣿⣿⣿⢳⡖⣶⡦⢰⣴⣾⣿⡇⣒⣐⣀⠛⠯⠅⠈
⣿⣧⠤⣴⣶⣟⠀⡀⢾⣿⣿⣮⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⢸⡇⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇
⣿⣋⢁⣛⣻⣏⡁⡅⢸⠿⠿⠿⠿⠛⠛⠛⣛⣛⢸⣟⣿⣿⣿⣿⣿⢸⠿⢿⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇
⣙⣖⣴⣻⣿⡃⠀⡇⢠⡶⠶⢶⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⡿⣿⣷⣶⡆⣷⣿⣿⣿⡇⠿⢿⣿⣿⣿⡇
⣯⣷⣭⣵⡾⡿⠛⡇⢙⣺⣿⣿⣯⣟⣻⣽⡿⠷⡀⡿⠝⠻⣿⠿⡿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣶⣶⣤⣭⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣀⠤⠴⣶⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢠⣴⣦⣔⣨⣥⣶⡿⠟⠛⠋⢛⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣘⣿⣿⣿⣿⠟⠉⠀⠀⠀⣀⠀⠹⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⣿⣿⣷⠀⠀⠀⠀⠀⢱⡆⠀⡌⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠸⣿⡿⢿⠆⢀⡀⠀⠠⠎⢳⣹⣷⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣷⣟⠀⡿⢰⡷⣀⣸⡆⠀⠈⠉⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⣿⢏⡇⠀⠈⠁⢄⠀⠀⠀⢱⣿⣿⡥⠈⢹⣿⣶⣶⡏⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠁⢠⣿⣿⡏⣼⠃⠀⠀⠠⣽⣇⢰⣟⡾⢿⡟⠁⠀⢸⣿⣟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡄⠀⠀⠀⣿⣿⣯⠻⡿⣿⣷⣿⣶⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⡇⠀⣦⣻⣿⣿⣿⣷⡰⣾⡿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢚⠃⢀⣤⣾⣿⣿⡏⣡⣆⠘⠈⣿⣿⣿⣿⣿⣿⠀⠈⠘⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⡿⠟⠋⠁⠀⠀⠃⢰⣾⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣷⣛⣋⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠛⠁







⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢷⣦⣀⠁⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠓⠦⠀⣀⠀⠀⠀⠀⠀⢘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠀⠀⠐⣠⡟⠛⠲⣄⠀⠀⠀⠹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣧⣄⠙⢀⣢⣀⡀⠀⠀⠬⣀⠙⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⢛⠯⣻⣿⢧⣾⣿⡿⣿⣿⣶⣤⣄⠙⠓⠶⣌⡑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⠀⡞⠈⢰⣿⠀⣾⠿⣿⠟⠁⠁⠙⠟⢯⡧⠀⠀⠀⠉⠲⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢈⠀⠀⡿⠀⢏⠀⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⠀⠈⠳⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣀⣀⣀⣠⠏⠈⠀⠀⠃⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠘⠋⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣿⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⢀⢱⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⢀⠣⠐⡄⠠⠀⠀⠀⠀⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠃⠀⠀⡈⠀⠀⠂⠀⡄⠠⠋⡧⠘⣆⠐⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⠀⠀⠀⠁⠀⠀⠀⠸⠀⠠⣉⡃⠐⣜⢧⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⣿⠀⠀⣼⠀⠀⣀⣰⠗⠀⠐⣉⡁⠀⠇⡻⢿⣄⠠⡀⡀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⢠⠇⢠⣶⣟⠙⡃⠀⢈⣥⡄⢰⠀⡇⠈⡟⠷⣬⣮⡳⣶⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣿⣶⡿⡇⠹⣧⠹⡀⠀⢴⡄⠘⠀⠀⢠⠁⠀⠀⠉⠙⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠋⠀⠘⢆⠙⣿⣤⠀⢸⠁⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠿⣧⣼⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠙⠿⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢳⡉⠒⠢⡤⢄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠊⠀⠀⠀⠉⠈⠙⠔⡢⠜⠛⠛⠢⣒⠢⢤⡴⠂⠈⠁⠒⢞⠍⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣠⡀⠀⠠⠀⢠⠇⠐⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⡸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢍⣀⣀⣀⠀⠄⠀⣀⠀⠔⠁⣈⣂⢀⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠒⠚⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠲⡀
⠀⠀⠀⠀⢀⡴⠒⠋⠉⠒⡄⠀⠀⠀⠘⡍⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⡏⠀⠀⠀⠀⢂⠇⠀⠀⠀⠀⠹⡄⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣨⠴⠃
⠀⠀⠀⠀⠑⠦⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠘⣄⠀⠌⠉⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⡄⠀⠀⠈⠳⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠤⠤⣤⣤
⠀⠀⠀⠀⡠⠔⠤⢄⡀⠀⠀⢸⠀⡀⠀⢰⠁⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣠⠞
⠀⠀⠀⢸⠀⠀⠀⢀⠙⡆⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⣀⡀⣀⣀⣀⡀⠀⢀⣀⡠⠴⠊⠁
⠀⠀⠀⠀⠳⣄⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⡀⢀⠀⣀⠀⣤⣀⠀⢀⣀⣤⡀⠀⣀⠀⠀⣄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠋⢻⠸⣇⡟⠀⣿⢸⡇⠸⡏⠀⠀⢰⣿⠀⢸⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⢸⠂⢿⡇⠀⣿⣼⡇⠀⣷⠶⠀⢸⢻⡇⢸⡿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⠀⣸⠀⢸⡇⠀⢹⠈⣧⠀⣿⠀⠀⣿⠀⣇⢸⡇⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢿⣄⡿⠀⢸⡇⠀⢸⣀⡟⠀⣿⣀⢀⡟⠛⣿⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠉⠀⠀⠈⠁⠀⠈⠉⠀⠀⠉⠉⠈⠁⠀⠘⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⣀⣀⡀⠀⠀⢻⡿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⢀⣠⠶⠛⠉⠉⠉⠁⠀⠀⠈⣟⣛⣚⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣍⣉⣉⣷
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠴⠶⠾⡇⠀⣀⣀⣤⣤⠤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢰⠏⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢒⣶⡶⠟⠛⠉⠁⢀⣀⣀⣀⣀⣀⡀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠁⠀⠀⠀⠘⠋⠉⠉⠉⠁⠉⠉⠉⠙⠲⢦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠹⣆
⠀⠀⠀⠀⢠⡇⠀⡾⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⢸⠁⠀⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡴⠋⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡹⡇
⠀⠀⠀⠀⢼⠀⠀⣿⠀⠀⢻⡀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣿⠀⠀⢠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡙⢦⣻⡇
⠀⠀⠀⠀⢸⡀⠀⠘⠲⠶⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣌⠻⣤⡟
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡴⣟⠙⢧⡈⢳⡞
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⠀⠀⠉⢹⡎⢷⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢀⣀⣼⣧⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣨⢷⣦⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠀⠀⢀⡤⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣤⢶⣏⡙⠺⢷⣿⠈⣏⠹⡟⢿⠛⡏⢹⡇⢿⠸⣎⣳⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⢧⣹⣄⢷⡸⡆⢻⡈⣧⣘⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠛⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⡀⡀⢐⡶⢤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠍⠐⡁⣀⣀⡥⣀⣬⣌⢻⣟⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⠴⢿⡿⠿⠿⠷⠿⠟⠿⣿⠿⡟⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠈⢮⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣧⣠⡄⣔⣤⢰⣤⡀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠛⠊⠉⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣿⠃⣀⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⢹⣾⠋⠃⠀⠀⠀⠀⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣯⢀⡤⠋⠉⠀⠀⠀⠀⠀⢠⣿⠀⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠻⡏⠀⠀⣀⠀⠀⠀⠀⢠⠏⣽⠀⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠲⠤⣤⡤⠾⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢹⣷⡊⠁⠀⠀⢀⣼⠀⠀⠀⢀⡿⢠⣿⠀⠸⠙⣧
⠀⠀⠀⠀⠀⢀⠀⢀⡿⠻⢦⣄⣄⢈⣷⡤⠒⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢟⢀⡴⠈⠊⠀⠀⢠⣴⢢⣾⠀⠀⠀⣼⠃⢴⡧⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢷⣿⣠⣴⠞⠻⣿⡋⠁⢀⣠⠿⣄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢼⠻⠉⠀⠀⠀⣠⡾⠛⣿⠟⠀⠀⢰⡟⠀⣿⡏⠀⡇⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠁⣀⡽⣤⡀⠉⠛⢿⠙⢻⡄⠀⡇⠀⠀⠀⠀⠔⠛⢳⣦⡖⠀⠀⠀⠀⣠⠴⠃⠀⣾⠏⠀⠀⢀⡿⠁⠀⣿⠁⢀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣀⢀⡿⠀⠈⠿⠶⠖⠟⠛⡆⠙⠒⠁⠀⠀⠀⡆⠀⣐⡿⠋⠀⠀⠀⢀⡴⠁⠀⠀⣼⠏⠀⠀⠀⣼⠇⠀⣸⡟⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠦⣄⣀⡤⠜⠁⠀⠀⠀⠀⠀⢰⣿⡾⠋⠀⠀⠀⠀⡠⠛⠀⠀⣠⡾⠃⠀⠀⠀⢰⡟⠀⢠⡿⠀⠀⡜⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⠀⢀⠔⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⠃⠀⢸⡇⠀⠐⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⡰⠃⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⢸⡇⠀⢠⡿⠁⠀⡠⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣀⡀⢙⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠁⠀⠀⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠦⠾⠖⠛⠛⠁⠀⠋⠈⡆⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣴⡏⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣼⠃⠀⠀⠀⠀⠀⡰⠃⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣾⡟⠀⠀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⢠⠃⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣠⡟⠀⠀⠀⢸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠒⢦⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⠏⠀⠀⠀⠀⢀⠆⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣴⡄⠀⢳⠀⠀⠀⠀⠀⢀⡎⠁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢠⣿⠇⠀⠀⠀⠀⠀⣜⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠎⠀⠀⠀⠀⠀⢸⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠻⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡤⠤⠐⠒⡄
⠀⠀⠀⡄⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⣠⠇⠄⠀⢠
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠠⣤⣤⡤⢴⡞⠉⠀⠀⢗⣄⠀⠀⠀⠀⠸⢟⣷⣶⣶⡶⠟⠉⠉⠉⠁⠄
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈




⠀⠀⠀⠀⡠⣂⡴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⡀⠀⢀
⠀⠀⠀⣜⣽⢟⣵⠞⣽⠄⠀⠀⠀⠀⠀⠀⢶⣿⣿⣥⣴⠟⡔⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠃⣶⣿⡇
⠀⠀⢸⣿⣣⣿⣫⠞⠋⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⠋⠁⣸⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠻⠟⠁
⠀⠀⠹⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣷




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⣆⠀⢀⡀
⠀⡀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⠿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣬⣴⣦⠀⠀⠀⠈
⠀⠀⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠀⠼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠘⠛⠋
⠀⠀⠀⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣇⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣯⢹⣄⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠘⡘⣿⡄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡃⠀⠁⢹⣿⡀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⣆
⠀⠀⠀⣴⡀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⣇⢸⠐⠘⠺⣇⠀⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⣀⣥⠖⠁⠀⡟
⠀⠀⠀⣻⣥⡀⠉⠛⠿⢶⣶⣶⣐⢒⡒⠤⢄⠀⠀⠀⠇⡸⢾⠸⡇⠆⢃⣿⠀⢰⣿⠀⠀⠀⠀⠀⡠⡔⠋⠁⠀⡨⠝⠂⠀⠀⢀⡰⠁
⠀⠀⠀⠸⣿⣿⣤⠀⠀⠀⠀⠈⠉⠋⠛⢿⣶⣝⡲⢘⣿⣿⡆⣾⣇⠘⢸⣿⠀⣺⣿⠀⠀⢀⠄⠈⣚⣥⠔⠛⠉⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣗⣿⣿⣿⣿⡔⡃⣿⠀⢿⣿⢀⢐⣥⣾⡾⠋⠁⠀⠀⠀⠀⢀⠜⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣤⣢⣆⣤⡀⢀⢀⠀⠀⠙⠻⣿⣿⣿⣷⣧⢻⢀⣹⣧⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠠⢉⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠏⢹⣯⣿⡿⠿⠿⠿⠿⢿⣯⣵⣶⣤⣤⣝⡻⢟⣿⣾⣿⣿⣿⠟⠁⢀⣀⣠⣤⣤⠤⠴⠤⠤⠿⣿⣁⣀
⠀⠀⠀⠀⠀⢀⣠⡤⠚⠉⣡⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡏⠴⣾⡍⣷⠶⡿⠿⠿⠒⠛⠚⠛⠛⠳⠿⠿⠶⠶⣾⣯⣵⣖⡤⣀⣀⣀⣤⠶
⢠⣶⣶⣒⡛⠉⣀⣠⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣶⣿⢟⣷⣦⣬⣶⣿⣿⣿⣷⡤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⢠⡼
⢸⡦⡈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠿⢛⣫⣽⡿⠋⣡⣿⣟⢀⣦⡙⣿⣿⣟⢷⣮⣑⣒⠤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡶⠋
⠀⠻⣟⢶⣤⣀⣀⠤⢠⣤⣤⣤⣴⣴⣾⣿⢟⣡⣾⣿⠟⠁⠀⣰⣿⣿⣿⢾⣿⣷⡹⣿⣿⣷⢮⠙⠿⣿⣶⣅⡲⢶⣶⠴⣶⣶⣟⣿⣞⡷⠟⠉
⠀⠀⠙⠲⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠟⣱⣿⣷⠛⠁⠀⠀⣴⣿⣿⡟⡇⢸⣿⣿⣇⠹⣿⣿⣷⣄⠀⠀⠙⠿⣿⣦⡙⢿⣿⢶⣿⠷⠋
⠀⠀⠀⠀⠀⠙⠛⠷⠾⣿⣿⣿⡿⢋⣾⣿⠟⠀⠀⠀⠀⢰⣿⣿⣿⡅⠀⢸⣷⢿⣿⠀⠹⣿⣿⣿⡄⠀⠀⠀⠈⠻⢿⣦⡑⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣰⣿⡿⠋⠀⠀⠀⠀⣠⣿⣿⣿⣿⠂⠀⠘⣿⢸⣿⡇⠀⢿⣿⣿⣷⣄⠂⠀⠀⠀⠀⠙⢻⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⣾⡿⠟⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⡆⡆⠀⣿⣼⣿⡇⠀⢸⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠀⠉⠳⣍⡢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡾⠟⠋⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟⢻⣿⣿⣇⡃⠀⢿⣾⣿⠇⠀⠈⠻⣿⣿⣿⣷⣿⣿⣿⣶⣦⣄⣄⣀⣠⣉⣓⠚⡦
⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⣶⡶⢶⣶⣾⡷⢿⣵⣾⠟⣿⣿⡜⣿⣿⣿⡶⡀⢸⣿⣿⠀⠀⡸⠀⠈⠻⠿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠉⠉⠀⢸⣿⣿⡆⠸⣿⣿⣷⢣⠈⣿⡿⢠⡾⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠙⢿⣿⣯⢆⢹⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣧⠀⠀⠈⠻⣿⣿⡄⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠀⠀⠀⠲⡀⢟⣼⠂⣸⠀⠀⠼⣆⠄⢸⣧⣼⣟⣉⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣦⡀⣸⣧⡄⢱⣿⣧⠀⣀⣠⠄⠙⢾⡁⢠⣧⣦⣾⣶⣶⣶⣦⣵⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏⡿⠧⣸⢿⣿⣾⠟⢀⣷⣤⡀⠱⣴⣟⠋⢻⣿⣿⢿⣽⣿⣿⣿⣿⣿
⠀⣠⣐⣤⣦⡀⢰⣦⡀⢻⣷⣜⣿⡞⢠⣴⡟⠀⢻⣿⣿⠉⠙⢻⣷⣿⣦⢿⣤⣨⣿⣟⠁⢭⣭⣭⣉⠉⠁
⠀⠀⠉⠽⠛⢛⣈⣻⣷⣌⠛⡻⣿⠓⣿⣫⡇⠀⣻⣿⣖⡀⠠⣮⣹⣥⣄⣸⠟⣛⢙⡁⠣⣾⣿⣿⣿⡷
⠀⢀⣀⣀⣀⣀⠈⠻⣿⣿⠀⣹⣿⡿⢛⠛⣧⣶⡿⠿⠛⠁⢳⣼⢿⣿⠆⢷⡾⣍⡛⢱⡘⠀⠀⠉⠉
⠀⠈⠿⣿⣿⣥⣤⣤⣤⣿⣾⣿⣿⣬⢸⡇⠛⠋⠀⠀⠀⠀⠀⠑⠊⠑⣷⣆⣠⠀⢀
⠀⠀⠀⠀⠉⣉⣉⣉⡛⢻⡿⢻⡟⢍⣾⣦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣉⠿⠾⠧⢦⣤⣤
⠀⠀⣠⣶⣿⣿⣿⣿⣯⣐⠛⠞⡿⢿⣿⣶⣦⣄⣰⡿⣿⣆⠀⢠⣦⣠⣿⣿⡓⠳⢚⠐⠛⠋
⠀⠀⠀⠈⠙⠛⠻⠛⢛⡉⢛⣦⣉⣸⢘⣿⠿⠿⣿⠹⢍⣻⣤⣾⣿⣿⠋⠙⠒⠃⢺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠝⢛⣿⡢⣧⣾⡶⠟⢀⣾⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⢉⣴⠿⠈⠉⠉⠀⢸⢇⣶⢿
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⠀⠀⠀⢸⣿⡞⠁⠀⠀⠀⠀⠀⡿⢸⣿⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣿⢷⣦⡀⠀⣿⣤⣶⡟⠳⠀⢀⢴⣇⠤⠛⠛⠑⡒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣯⣄⣙⡷⠄⣿⣿⣿⡧⢰⣄⡉⣼⠃⠀⠀⠀⠀⠈⠹⠃
⠀⠀⠀⠀⠀⠂⣀⣠⣼⡿⠭⠞⠉⠐⠲⣿⡿⣷⣄⣈⠛⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠈⡟⠁⠀⣴⣾⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢠⡇⠀⣴⢻⠏⠑⢠⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠈⣿⡗⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠘⡇⢀⣤⡄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠅⠌⠙⣙⢿⣯⣷⣻⢟⣥⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠛⠈⣿⣋⣈⣿⣠⣶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡹⠛⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⢿⣧⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠄⣸⣿⡘⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡤⠄⠒⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⣩⣴⣿⠀⣂⣀⠀⠈⠄⢠⠔⢛⠀⢉⠀⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⢡⣅⣀⢄⠟⠉⠉⠈⠓⠈⠰⢂⢒⠃⢰⣏⣾⡆⡄⢠⡥⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⣧⣹⠅⣡⠀⡀⢀⠁⠑⠀⢠⣷⡜⣀⣉⠢⠘⠛⠛⠛⠉⠌⠉⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣟⣶⣤⣰⠿⢻⠃⡹⡆⡿⠞⠃⠁⠇⣼⣿⣛⣋⠉⣀⡀⣄⠁⠀⠀⢀⡴⢠⠇⠈⠄
⠀⠀⠀⠀⠀⠀⡠⠕⢰⣮⣙⡁⡡⢒⣨⣷⢀⠎⠰⣷⣶⢲⡂⢁⣮⣦⣝⢿⣯⣧⠘⣦⣀⣿⣿⡿⠁⠄⢉
⠀⠀⠀⠀⠀⡞⢢⣛⣷⣿⢻⣯⣧⣼⡿⢁⣚⠻⣶⡇⣸⣶⠄⢻⣾⣟⣿⣿⣿⣿⣁⣸⣿⣛⡋⠗⠀⠀⠌
⠀⠀⠀⠀⢸⣦⢨⠿⢯⡁⠸⠟⣋⢤⠥⡈⢿⣿⣾⣿⣿⠟⠀⣀⡉⠻⣿⣿⡿⣿⣶⢿⢳⡸⢹⢤⠀⢀⠁
⠀⠀⠀⠘⠙⣋⡁⢰⣾⡟⡠⢠⡟⠨⣷⣷⠀⠙⠓⠋⣀⣦⡕⢏⡘⣴⠙⠛⢁⣟⣿⢿⣿⣷⣷⣻⠇⠠
⠀⠀⠀⢲⣶⠖⠻⣿⡯⢴⣄⣄⣭⣼⣿⠇⣴⠾⠤⡄⣥⡿⣝⢦⣿⣼⣿⡀⠈⢻⣿⣿⣿⣿⣿⠟⠓
⠀⠀⠀⠘⠻⠀⠀⢨⠇⣘⢿⡹⣯⢩⣐⣶⣿⣄⣿⣷⠈⣦⣼⣿⣿⣿⡿⡅⣴⣦⡘⠻⢿⠟⠛⢀⡔
⠀⠀⠀⢄⠘⠂⠺⠃⡙⢲⡷⣼⡟⣾⣟⣿⣿⣿⡿⢯⣀⣀⠈⠙⠿⠿⡟⣿⡎⣿⣮⡇⠀⠐⢴⠏
⠀⠀⠀⠀⠀⢀⢀⠷⠌⢩⣷⣿⠇⠀⠛⠩⠉⢡⣴⣦⣿⣿⣿⣆⠀⢸⣕⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠘⠈⣡⣾⠿⠿⠏⠀⢀⡀⠀⠀⣿⣯⣿⣿⣿⣿⣿⠇⠈⠻⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠢⠉⢉⠀⠀⡼⣄⠨⠷⣖⠀⢸⣟⣿⣿⣿⣿⡷⡇⠈⠀⠈⠉⠉⠐⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠨⣿⣿⡿⠻⠇⠁⣀⠀⠀⠐⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠚⠑⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠃⠈⠑⠶⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠖⠲⠒⠋⠉⠉⠁⠀⡜⢹⣧⣄⠈⠙⠿⢽⠻⠿⣟⡿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠊⠉⠀⠂⢀⠀⠀⠀⠀⠀⢀⠞⢀⢸⣿⣿⣶⠄⠀⠀⠀⠑⠀⠙⠻⠿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⢀⠎⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⢿⣶⣬⠀⡟⠉⠻⣢⡀⡀⠀⠀⠀⠈⠊⠀⢈⣊⢿⣿⣆
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⣾⡽⠋⠀⢠⠀⣧⡀⠀⠙⣿⣦⡑⢦⡀⠀⠀⠀⠀⢏⣞⡿⣿
⠀⠀⠀⠀⣞⠀⠀⠀⠂⠀⠀⠀⡠⢀⣴⡿⣿⠀⢀⣠⣿⡘⣿⣷⠀⠀⣻⣿⣿⠢⣝⢦⡀⠀⠄⠈⣿⣿⣿⡇
⠀⠀⠀⢀⡏⠀⠉⠸⠀⠀⠀⢀⡴⠁⢿⣿⣄⣠⣾⣿⡿⢀⡿⣿⣶⣴⣿⣟⠁⠀⠈⠓⣝⢦⡀⠀⡜⣿⣻⣧
⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⡰⠊⠀⠀⠀⠙⠻⣿⣿⣏⣠⠀⣧⡿⠏⠉⠻⢿⣷⣄⠀⠀⠈⠳⣝⣆⠁⡿⣼⣿
⠀⠀⠀⠀⡇⠀⠀⠠⢠⠎⠀⠀⠀⠀⠀⣠⣾⠟⠁⠙⢲⠀⣿⡁⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠈⢢⡱⣇⣿⣿
⠀⠀⠀⢀⡇⢀⢀⡰⠃⠀⠀⠀⠀⠀⣰⣟⠃⠀⠀⢀⣿⡄⣿⣷⡀⠀⠀⠀⠠⣿⣿⠀⠀⠀⠀⠀⠙⣾⣷⣿⣆
⠀⠀⣠⡾⢀⣩⠞⠁⠀⠀⠀⠀⠀⠀⢿⣧⣆⣠⣴⣾⣿⡇⣿⣿⣿⣧⣄⣼⣾⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇
⠀⠀⠿⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⢫⠃⣿⣿⣿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣾⠀⢹⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢉⢦⡁⠒⠄⠀⠀⠀⠀⢀⠄⡽⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠥⡀⢲⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⢊⣰⠏⣰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⢠⡄⠀⠞⠀⠀⣼⠥⢈⣵⠁⠀⠀⠀⠀⠀⢀⠴⠋⠀⡀⣀⡠⢀⡀⠠⠚⠁
⠀⠀⠀⠀⠀⠓⠶⠄⠈⠁⢀⡸⠏⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣇⣄⠀⠀⠄⠉⠋⠉⠉⠓⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⢀⡄⣠⠀⣉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⣜⣆⡵⠖⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢠⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠈⢐⠀⢄⡀⡈⠃
⠀⠀⠀⠀⠘⠉⠁⠀⢀⠀⠘⠂⠀⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⠄⠠⣾⡆⠀⠀⠀⠀⠀⠔⠈⠁⠀⡀⠀⡰⠁⠀⠀⠀⠀⠈⠦⢀⣀⡜⠀⢠⡜
⠀⠀⠀⠀⠀⡀⠀⠳⢆⡄⣠⢀⢠⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠰⠀⠀⠀⠀⠀⣰⢀⡴⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠃⠈⠀⡇⠈⠙⠀⠀⠣⠀⠀⠀⣠⣀⠲⣶⣠⡆⠀⠀⢸⡄⠀⡀⠀⠀⡀⡿⠏⠀⢀⣽⠃
⠀⠀⡀⢂⣀⡠⠂⠀⠀⠀⠀⡇⠀⠐⠀⠀⠀⠀⠀⣀⢤⣌⡉⠉⠘⠃⠀⠀⢘⠓⢀⠉⠐⠆⠉⠁⠂⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡇⡆⢸⠀⠀⠀⠀⠰⢀⠚⠉⡁⠀⠀⠁⠀⠀⢃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⠀⠀⠀⠀⣶⠿⠶⠛⠁⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡀⠀⢀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠤⢀⡀⠀⠀⠀⠀⠀⠸⠒⠂⠰⠦⠶⠴⢂⡜⠀⠀⠀⢀⠠⠒⠉⡏⠀⠀⠀⠒⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⡀⠀⠀⠀⢾⠰⣿⡂⠒⠒⠒⠉⠀⡀⠔⠈⠁⠀⡀⢀⡀⠠⠂⡞⠀⠀⠀⢀⡔⠉⡷
⠀⠀⠀⠀⡀⠀⢀⢰⣀⠀⠦⡀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠀⠹⠃⠀⠀⠀⠀⢀⠀⣀⢦⣙⠦⠛⠀⣀⠀⠈⠀⠀⢀⡆⠁⢹⠊⠁
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠛⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣯⢛⣽⠆⠁⡀⠀⠊⠁⠀⠀⠀⠊⢀⡼⠀⠉
⠀⠀⠀⠀⠀⠀⣠⣠⠆⠀⠚⠉⠀⠀⠀⠁⢀⡀⠀⢀⠀⢤⢀⠤⡒⠀⠀⠀⠀⠀⠀⠀⠘⣿⡽⠶⠮⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⡄⣤⣶⣄⣶⣧⣦⡀⠤
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢢⣠⠄⠀⠀⠀⠀⠀⠀⠓⠢⠄⣀⣀⠀⠀⠀⢄⠙⢕⣾⠖⣟⣿⠎⡾⣿⣿⣼⣻⢵⣄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠯⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣈⡉⠉⠙⠛⠽⠽⠷⠲⠛⢋⡵
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠂⡀⠀⠐⠀⠀⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣐⡋⣉⢉⣤⣴⣂⣧⢃⢒⡌⢱⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣿⣿⡿⠻⡿⢻⡿⢿⡿⢿⡟⢿⣿⣿⡇⠀⠀⠮⣋⡶⣮⣹⣺⠶⡟⠾⣾⣸⣷⣻⣾⡷⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠸⣿⣿⣶⣷⣾⣷⣾⣷⣾⣷⣾⣿⡟⠀⠀⠀⠀⠈⠉⠑⠋⠑⠛⠓⠛⠛⠚⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠟⠁⠀⢀⡀
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡯⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠄⠀⠒⠒⠂⠀⠤⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣚⡉⠹⢯⡍⠈⣠⣤⡶⢶⣶⣾⣿⡆⠀⠀⠰⠾⠶⠒⠂⠁⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠛⠟⢫⣥⣶⡶⣶⣤⣀⣿⡏⠀⢠⣿⣿⣿⡿⠀⣴⣾⠟⢛⣷⣶⣄⡀⠀⠙⠻⠿⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⣀⣀⣤⣀⣀⠛⢏⣀⣹⡿⢟⣡⡗⠈⠛⠛⢉⣘⡉⠁⠀⠛⠆⠀⣹⡟⠿⣋⡼⠀⣀⣀⣀⣈⠋⠊⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠂⠀⣿⠟⠉⠹⣿⣿⣷⣿⡶⠀⠈⠉⠀⠀⣴⡿⠛⢿⣹⣿⣷⣥⠀⢀⣀⣀⡚⠛⠉⢰⣿⡟⠛⢻⣿⣿⡆⢀⠀⠬⠑⣄
⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⣀⣀⣈⡒⠐⠶⠿⠛⣻⣥⠶⠿⣿⣶⣄⠀⠈⠑⢒⣚⠻⠿⠟⠃⢼⠟⠉⣻⣿⣿⣧⣘⠛⠦⢤⠼⠿⠋⠅⠛⠀⠠⢤⡈⣧
⠀⠀⠀⠀⣰⠋⠁⢀⣶⣾⣿⠿⣿⣿⣶⣄⠀⠀⠻⠁⠀⠀⠩⠞⠋⣴⡿⠟⠛⣙⣿⡆⢶⠀⢈⡑⠐⢦⠿⠛⠛⠋⢠⡾⠃⠀⢠⣶⣿⡿⢿⣷⣶⠁⠸
⠀⠀⠀⠀⢹⠀⠀⠸⢿⣁⠀⠰⣿⣿⣿⡟⠀⠀⢀⣤⣤⣤⣤⣄⣠⡙⠒⠒⠒⠛⣛⣁⣚⡁⠈⠒⢠⡶⠛⢻⣿⣷⡆⠀⠀⠀⢿⡏⠁⣠⡶⣻⠟⠀⠀⡇
⠀⠀⠀⠀⢸⡜⢦⣄⠀⠀⠉⠉⢋⣙⣉⣀⡀⠀⢿⠋⠙⠻⣿⢿⣿⢿⡆⢠⣿⠟⠛⣻⣟⣿⡇⠁⠘⠂⠤⠄⣉⠉⢀⣠⣤⣤⣄⡉⠙⠛⡛⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠈⢿⣌⢩⡀⠀⠀⣾⡿⠛⢛⣿⣿⡇⠀⠁⠐⠒⠛⠿⠟⠋⠁⠈⠙⠂⠤⠠⠼⠏⠂⠀⣀⣠⣤⣄⡀⠰⣿⡟⠃⢸⣯⣿⠄⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠈⢿⠳⠦⡀⠀⠈⣁⣶⣶⣿⡿⢋⣵⣾⣿⢿⣷⣶⣄⠀⢀⣴⣶⡾⣶⣶⣦⡀⢀⣾⡿⠟⣿⣹⡿⢀⡙⠆⢀⣀⠽⠋⠔⠃⣠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢳⣶⣈⠳⢄⣀⠀⠉⠁⠤⠼⣿⡏⠀⠚⢻⣿⢿⡀⢿⠏⠁⠀⣾⣿⣿⣳⣆⡻⠄⡀⣤⣫⣴⠿⡉⠀⠠⠀⠀⣠⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣆⣈⣯⢻⣶⣤⣀⠀⠈⠙⠛⠋⠛⠋⠁⣀⠀⠀⠐⠲⠿⠿⡛⠋⠉⠋⠉⠉⢁⣠⣤⢴⣶⣶⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣗⠲⡶⠤⣤⣤⣴⣤⠀⠖⠰⢖⣾⣿⣋⣉⠉⣸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣷⣽⣶⣼⣿⣿⣿⡔⢠⣄⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠖⢊⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⡜⠉⣧⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠸⡀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠑⠬⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣿⣷⡸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠋⠀⠀⢹⣿⣧⠙⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣷⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⡿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⣾⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⣸⡿⠁⠀⠀⣠⣤⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢡⣿⠃⠀⠀⢰⣿⣿⣷⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⣡⣿⣿⠀⠀⠀⠸⢿⣯⡿⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⣡⣾⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⡿
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢋⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠞⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⡄⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⡿⠁⠀⠀⠀⢻⣧⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠄⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠿⠆⠀⠀⠸⣿⣿⣹
⠀⠂⠀⠛⠻⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⢠⠟⣉⠉⠉⣛⠿⣥⣙⢿⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⢶⣶⣽⡶⣿⣄⠈⠑⠐⠁⠀⠀⢉⠻⣤⡠⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠩⡻⠁⠀⡈⠀⠃⠙⠂⠰⢤⣀⠀⠈⠳⣤⣠⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⡋⢿⠀⠀⢀⠃⣠⡴⠿⢿⡞⠛⠛⠛⣷⣿⣿⣦⡀⠀⠹⣤⠂⠈⢷⣆
⠀⠀⠀⠀⠀⠀⢀⡿⠄⠀⢸⠄⢦⡈⣾⡏⠀⠀⣽⣦⠖⠉⠉⠀⠀⢀⣠⣿⣦⡀⠈⠻⣠⠚⠻⢷⡀
⠀⠀⠀⠀⠀⠀⢸⠗⢦⠀⠸⡆⢸⣿⣿⡡⠋⠁⠀⠈⢳⡄⢠⠔⠊⠉⠁⠀⢈⢻⣄⠀⠈⠳⣌⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢾⠷⢀⡀⠀⡇⠈⣿⡟⠀⠀⠀⢀⡠⠘⠿⣇⡀⡀⠀⠀⠀⢀⣡⣽⣦⡀⠀⠈⠲⣠⡌⠳⣄
⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠘⡄⢹⣤⠀⣀⠔⠉⠀⠀⠀⠀⠙⣟⣆⣠⠎⠁⠀⠀⠘⢻⣄⠀⠐⠙⣆⠀⠻⣆
⠀⠀⠀⠀⠀⠀⢿⡶⠀⢀⡀⠃⠉⡄⠹⣾⠁⠀⠀⠀⠀⣀⣀⠴⢿⣿⠃⠀⠀⣠⠄⠚⠛⠻⣆⠀⠀⠈⠳⡀⢻⣆
⠀⠀⠀⠀⠀⠀⠸⣆⢆⠀⠈⠀⡀⠘⡄⠹⣦⠆⣠⠖⠉⠁⠀⠀⠀⠈⠛⣶⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢳⡀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠉⠐⠘⣦⠘⣿⣃⠀⠀⠀⠀⠀⢀⣀⠼⢧⣤⣄⡀⠀⣠⠴⠖⢿⣆⠀⠀⠊⡗⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡄⠀⠀⢀⠀⠀⠣⠈⢧⡀⢀⣠⡴⠚⠉⠀⠐⠀⠀⢈⡿⠋⠀⠀⠀⠀⣹⣄⠀⠀⢹⡄⡠⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⠆⠀⢀⠀⠈⠀⠠⡈⢳⣟⠁⠀⠀⠀⠀⠀⣀⣀⣼⣁⡀⠀⣠⠖⠉⠉⠻⣆⠀⠀⢻⡀⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⢠⡀⢠⡀⠀⢐⠆⠙⣧⠂⣀⡤⠒⠁⠀⠀⠀⠀⠙⢷⠁⠀⠀⠀⢀⣽⣆⠀⠀⣷⡀⠂⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⢤⠐⠀⠀⠀⠈⢀⣦⠈⢿⡋⡀⠀⠀⠀⠀⢀⣀⠠⠼⣦⣀⣠⠊⠁⠀⠘⢧⡀⠘⣎⠠⠈⡉⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡀⠀⠀⠀⠀⠀⠁⠀⠻⡇⠀⣠⠖⠊⠁⠀⠀⠀⠀⢽⠃⠀⠀⠀⠀⣼⢧⡀⠘⢶⡆⠈⡐⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡍⢀⠀⠀⠐⠀⡀⠑⡀⠙⣯⠀⠀⠀⠀⠀⠀⣀⠤⠺⢶⣆⢀⠴⠊⠐⠟⣷⠀⠆⠹⡔⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠁⠈⠃⠀⠀⠄⠘⡄⠈⢶⣄⣠⡤⠖⠉⠀⠀⠀⠀⡿⠁⠀⠀⢀⣠⣿⣇⠀⠀⢻⡄⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠄⠀⢠⡀⠀⠀⠙⠄⠀⢻⡁⠀⠀⠀⠀⠀⣀⡠⣧⣤⣄⡔⠋⠀⠈⠘⡄⠈⠀⢣⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡌⠈⠠⠀⠀⠐⠀⢂⠀⠻⣄⢀⡤⠒⠊⠀⠀⠀⠀⢸⡀⠀⠀⠀⣰⢻⡄⠀⠀⡧⢬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⠀⠀⠀⠀⠀⠀⠂⡀⠙⣿⡣⠀⠀⠀⠀⣀⠤⠞⢂⣀⠤⠊⠁⠰⡇⠀⠀⢻⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠘⠀⠀⠀⠀⠐⡄⠈⢷⡀⠤⠐⠉⠀⠀⠀⠘⠁⠀⠀⠀⢀⡇⠀⣈⢸⢡⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠈⠙⣄⠀⠙⣧⡄⠀⠀⠀⠀⡴⠀⠘⢲⠖⢊⡇⠀⠀⢸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠈⠢⡀⡀⠙⢶⡞⠉⠈⠀⠀⢠⡆⠀⢰⡇⠀⠄⢸⠄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠄⠀⠒⠙⢦⡀⠀⠙⠲⢤⣤⣠⣄⣀⣠⣾⣿⡀⢀⡀⢧⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⢀⠀⠀⠀⠈⠰⢄⡀⢠⠈⠙⠋⣿⣥⡽⠿⢧⠈⡄⠘⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣠⠀⠀⠀⠉⠂⢄⡀⠄⠈⠙⠧⣔⡼⠇⠈⣇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣄⣀⠁⢆⡄⠀⠀⠙⢿⣆⢀⡍⠀⣿
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣽⣆⡀⠄⠀⢿⣌⠻⢠⡟
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣈⣙⡷⠟
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠴⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠀⢀⣤⠀⣀⠈⠀⠀⠲⠀⠀⡆⠀⠀⠀⠀⠘⠰⠀⠀⠀⠀⠈⠁⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣷⡠⢹⡿⠀⢸⡏⠀⠀⡖⠠⣴⠠⠄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠀⠼⡇⠀⠸⣇⠀⢀⡇⠀⠙⣧⡀⢸⡏⢀⡾⠁⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠌⠉⢁⠀⠘⠐⠛⠁⠼⠇⠀⠷⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠉⠀⠀⠀⠂⠠⠄⢀⣀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠃⠀⠰⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠐⠀⠠⠈⠁⠀⠀⠀⣅⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣂⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠉⠒⠚⣷⣄⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢠⠀⠀⠀⠀⠀⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣶⣶⠆⠐⣶⠀⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠸⠏⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⠀⠈⣀⢆⣿⡿⠻⠭⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠀⠀⠀⢀⣄⣠⣄⣀
⠀⠀⠀⠐⢀⡀⡀⢉⠩⣄⠶⠴⠄⠁⠀⠀⠄⢾⠻⣿⣿⣻⠇⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠼⠀⠀⠂⠀⠀⢾⣿⣿⣍⣝⣿
⠀⠀⠀⠀⠈⠁⠀⠙⠋⠙⡁⠀⠢⠢⡀⡤⠤⠬⠦⠤⠠⠍⠈⠏⠀⠀⠀⡇⠀⢰⠀⠁⠀⠀⠄⠤⠠⡄⠀⠀⠀⠀⠈⢸⠇⠀⠛⠚⠋
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⢀⣀
⠀⠀⠀⡼⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⠋⠁
⠀⠂⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠡⡑⡒⣊⡙⣆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⡀
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣴⡯⠀⠀⠀⢺⠀⠀⢟⢁⡘⣿⣮⠛⢦⣄⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠦⢈⣁⠀⣁
⠀⠀⢸⣷⣄⠀⠀⡀⣴⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣬⠋⠀⢢⡙⡆⠢⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢄⠃⣉
⠀⠀⡈⠙⢟⠁⠀⡿⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠰⠶⢃⠷⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢯
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠙⠀⠀⠠⡀⠀⠀⠀⣀⣴⡁⠤⠤⢤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢁⣀⣀
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠀⠀⠸⠋⠆⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠶⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣤⣤⣤⣤⡤⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣤⣤⡴⣄⠐⠒⠒⠀⠦⢄⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠰⠟⢻⣿⣾⣿⣦⣼⣿⣿⠀⠀⠀⢸⣿⣿⣿⠟⠋⢉⣶⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠐⢶⣄⡀⠀⠀⠀⠀⠈⠉⠡⠾⢿⣿⡿⣿⠀⠀⣀⡼⠛⠛⠻⠚⠻⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⣯⠙⣿⣷⣿⣧⠀⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣟⣁⣀⣠⣤⣴⣶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠙⣫⣿⣿⣦⣿⣄⣿⡇⠀⠀⠀⠀⠀⠀⠸⣻⣿⣷⣰⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⢷⡀⠻⣯⠻⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣭⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣌⠙⠿⠋⢠⣶⣶⣶⠖⠀⣼⠃⠀⠀⠀⠀⠀⣻⣦⣈⣷⣬⡿⠷⠾⠿⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⠁⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣷⣀⡄⣿⣙⡿⠃⢀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠋⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⠈⠉⠻⠿⠛⠉⢽⠀⢈⡀⠀⠞⠁
⠀⠀⠀⠀⢀⡴⢿⣿⣿⡃⢉⠀⠀⠀⢠⣠⠟⢻⣿⣿⣿⡟⠃⣤⣔⠀⠐⠀⠀⠘⢦⣽⣇⣤
⠀⠀⢰⣾⠋⣠⣿⣿⢛⣷⠋⠀⠀⠀⢹⣄⣀⡾⠋⣿⣿⣿⠶⣼⣿⣌⣿⣆⠀⠀⣘⠛⠉⠁
⠀⠀⢸⣿⣿⠟⣿⣿⣿⠃⠀⠀⠀⠀⠀⠻⣿⣇⣰⣿⣿⣿⣷⡌⣿⡆⢹⣿⠀⠀⣿⠃⠀⠀⠀⠀⣰⣿⣷⣶⣦⣤⣴⣶⡚⠛⠂
⠀⠀⠘⣿⡏⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⣻⣿⣯⣻⣿⡀⢠⣥⠚⠛⢿⣷⣶⣶⡆⢾⣿⠟⠀⣿⣿⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⢠⣶⢀⡀
⠀⠀⠀⠹⣿⣿⣿⣾⡇⠠⠶⠦⠀⠀⠤⠠⡄⠀⣦⡄⣻⣿⣿⣿⠟⠉⠿⣿⣿⣿⣟⣉⣤⣶⡤⠚⣟⠀⢠⣿⠃⠀⣾⣿⡿⣿⣿⡁⠉⠙⢇⠀⣟⠻⣿⣶⡄
⠀⠀⠀⠀⢿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣿⣿⣿⡉⠲⣾⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⣾⡇⠰⠏⠀⢸⣿⠋⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⢹⣿⣿⣿⣿⠃⢀⡿⢃⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠺⠋⠀⢠⡿⠛⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢻⣿⣿⠛⣶⠟⣠⣟⠀⢹⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛⣿⣿⠓⣾⠋⣸⣷⣿⠀⢸⣿⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣸⣿⢠⣅⣾⣿⡿⠋⠀⠘⣿⣇⠀⠠⢀⣶⣿⠖⠒⠢⢤⣀⠀⠀⠀⠀⠀⠐⠁⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃⢼⣿⠏⠀⠀⠀⠀⢸⣿⣆⣴⣿⣿⣧⡀⠀⠄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣾⣿⣿⢾⣿⣿⣿⣿⣥⣄⣤⣤⣾⡟⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣶⣶⣶⣿⣿⣿⣶⣤⠀⠀⠀⠀⢿⠀⠀⡠⠶⠷⢟⣿⣿⣾⣿⣻⣏⣹⡉⠛⣏⡙⢻⣟⡛⠻⠿⡿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⢉⣩⣽⠟⠋⠁⠀⢀⠀⠐⠓⠦⢼⣧⠞⠀⠀⠀⣾⣿⣿⣿⣿⣟⡿⢿⣿⠂⠉⠀⠀⠈⠙⠶⠄⠶⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⣴⠀⠀⣰⣿⣤⣶⠀⠀⠀⢿⡀⠀⠀⢉⣿⠘⣿⣿⠙⢿⣷⣀⠻⡄
⠀⠀⠀⠀⠀⠀⠈⣁⣤⡿⠋⠀⠀⣰⡇⠀⢸⣿⣿⣿⡟⠀⠀⠀⢸⣇⢰⣿⣿⠇⠀⢸⡿⠱⣿⣿⢹⡇⢻⡀
⠀⠀⠀⠀⠀⠰⣿⣿⡟⠀⠀⢀⣶⣿⡷⠴⠿⠟⠛⠁⠀⠀⠀⠀⠈⣃⣸⣿⡟⠀⣷⣿⡇⠀⠀⣍⡿⣧⠈⣧
⠀⠀⠀⠀⠀⢠⢻⡟⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠏⠀⠀⠁⢸⡁⠀⠹⢿⠷⢾⣿⣿⡀
⠀⠀⠀⠀⠀⠸⡟⢀⡀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⠿⣿⠀⠀⠀⣷⢼⡇⠀⠀⠠⣰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⡇⠈⣿⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣾⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⣿⣿⣿⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣴⡏⠀⠈⢻⣿⣿⠀⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠟⠀⢰⣴⡿⣿⣿⡇⠈⠳⡆⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡿⠋⠀⠀⠈⠹⡇⠹⢿⣧⠀⠀⠉⢄⣬⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⠾⠋⠀⠀⠀⢰⣦⣾⠃⠀⠈⣿⡄⠀⠀⠀⠙⢤⣀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⢻⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡴⠂⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⡆⠀⠀⠀⠀⠀⢠⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠸⡄⠀⠀⠀⢠⠋⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢱⠀⢀⡴⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠀⠀⠀⣼⣤⣊⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠗⠿⣿⣶⣒⣋⣉⣩⣝⣋⣙⣛⣿⣟⠓⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⡿⠗⢦⡀⠀⠀⠀⠀⠐⠒⠂⠬⠭⢗⣲⣤⣔⡚⠓⠶⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⣀⡤⠖⠋⠁⠀⠀⠀⠉⠷⣦⣄⣀⢀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠳⢶⣬⣧⡈⠑⠢⣄⡀
⠀⠀⠠⣤⠤⠤⠤⠖⠚⠛⠋⠉⠀⠀⠀⠀⣀⣀⣀⣀⣴⠶⠟⠓⠪⢭⣙⡷⠒⠒⠒⠲⢴⡤⡀⠀⠀⠈⠉⠛⠷⣤⣠⣍⣢⢄
⠀⠀⣀⡬⠗⠦⣄⡀⠀⠀⠀⠀⠀⣀⣴⣴⡶⠗⣻⠋⠀⠀⠀⣀⡤⠤⠽⣄⠲⠿⣶⠄⠀⢹⡶⠒⠉⠉⠒⠦⡀⠀⠈⠑⠮⣷⡕⣄
⠺⠭⣥⣀⡀⠀⠀⠋⣻⠶⠒⠚⠛⠾⡋⠁⠀⠀⠹⣴⣤⠔⢉⡤⢤⣤⠀⠈⢳⡶⠖⠒⠒⠻⢧⡀⠀⠀⠀⠀⠘⣆⢀⠀⠀⠀⠙⢮⣧
⠀⠀⠀⠈⠉⠉⠋⠉⣇⣀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⢹⡠⠬⠭⢍⣀⢀⣴⡏⠀⠀⠀⠀⠀⠀⢹⣦⣤⠀⠀⣠⠟⠳⠮⢭⣄⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⣶⣦⠤⠞⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠈⢫⠀⠱⢤⣉⡯⠽⠤⣴⡊⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰⠯⠽⠃⣀⢄⡼⠀⡴⠉⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠾⠍⠒⠉⠀⠀⢧⡀⠀⣀⣤⣤⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣙⠀⡖⠉⠉⢲⠈⠻⣀⠟⠁⠘⣏⣉⣷⠀⢹⣉⡍⠃⠀⢨⠻⡄⠀⠈⡟⢦⡈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠽⠃⠳⠤⠤⠞⠀⠀⠿⠄⠀⠠⠧⠤⠼⠂⠼⠤⠥⠖⠠⠏⠉⠹⠤⠠⠧⠀⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠐⠐⠀⠀⠀⠀⠂⠐⠀⠐⠀⠂⠀⠀⠀⠀⠀⠈⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡿⢉⡿⢹⡟⣹⡟⢩⡟⢹⡟⢉⣿⠙⣿⠙⣿⠉⣿⠉⣿⠹⡏
⠀⠀⢠⠞⢠⡞⢡⡟⢠⡿⠀⣼⠁⣾⠁⢸⡇⠀⣿⠀⣿⠀⣷⠀⣿⠀⢻⡀⣿⡆
⠀⠀⣾⣦⣼⣦⠸⠇⠸⢇⣤⣿⣠⣿⣄⣼⣇⣰⣯⣠⠿⠀⣿⠀⣿⣀⣼⣀⣼⠇
⠀⠀⣿⣿⣿⠃⠀⣀⣴⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣈⣩⣿⣿⡇⠀⠀⣀⠀⠀⣀
⠀⠀⣿⣿⣿⠶⠟⠃⠀⠀⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⠁⠈⠁⠘⠋⢻⣿⣿⣧⡼⠐⠉⠉
⠀⠀⣿⣿⣿⣼⠇⠷⠸⠿⣿⣿⡏⡠⠁⠂⢸⣿⣿⣯⣜⣋⠀⠀⢀⢹⣿⣿⡇⢀⣠⢄
⠀⠀⣿⣿⣿⡆⠠⠂⠤⠆⣿⣿⡇⠐⠂⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠿⠿⠃⠀⠀⠀⢠⠀⠀⠀⠈⠓⠡⠤⠄
⠀⠀⣿⣿⣿⠀⠀⠂⡃⠀⣿⣿⣧⣤⣀⣤⣼⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠰⠀⠀⠀⢀⡶⠘⠀⠀⢀⢀⡖⠠⠐⠁
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠤
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠁⠀⠤⡄⡠⠀⢐⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⢁⡀⠋⠈⠂⡀⢠⠏⠀⠴⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠋⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠻⠿⠿⠟⠛⠛⠻⠟⠻⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⣀⠄⠀⠦⢀⣀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠐⠒⢢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣙⢂⠀⡰⢂⣠⢌⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣉⠁⣀⠸⠉⠁
⠀⠀⠀⠀⠀⠀⡀⣐⡏⢽⠤⠀⠁⠀⠡⠀⠀⠈⣹⣿⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⡍⠳
⠀⠀⠀⠀⠀⣰⠿⢯⠁⠈⠊⠀⠲⠰⠂⠀⠀⠀⠈⣇⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⢠⡤⡀⢰⡒
⠀⠀⠀⠀⢠⣟⠻⡻⠚⠀⠀⠀⠐⠐⠂⠐⠐⢠⡘⣧⠸⣋⡃⠀⠀⠀⠀⠠⣤⣤⣴⣬⣥⣬⣭⣴⣤
⠀⠀⠀⠀⠀⠙⠃⠻⣲⠊⠀⠀⢀⣀⣀⠀⠀⠀⣰⣟⡅⠉⠁⠀⠀⠀⣠⡤⠀⡿⡍⢀⡀
⠀⠀⠀⠀⠀⠀⠠⣧⠛⢿⡟⠛⠉⠉⠈⠛⠛⠛⠛⠻⡇⠀⠀⠀⠰⣟⣥⡤⠶⠄⠀⠳⠟
⠀⠀⠀⠀⠀⠀⢠⠎⠀⣀⣅⡀⠀⠀⢀⣀⣀⣀⣀⠰⡧⠀⠀⠀⢼⣭⣤⣤⣬⣖⣥⣸⡄
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⣉⣉⣐⣧⣀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⢀⣀⣽⣿⣿⣿⣿⣿⣧⣴⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢿⣿⠿⡿⡿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡄⣘⢾⣥⣥⣤⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠿⢏⣎⠾⢟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣄⠈⠳⢤⣀⣀⡀⠀⠀⠀⠀⣀⡤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠉⠉⠳⠦⣠⠞⠁⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⣠⠶⠛⠉⠉⠁⢀⡠⠞⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣀⣀⣠⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣶⣄
⠀⢀⡂⣀⡈⣉⣰⠦⣼⣧⠶⣷⠔⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢿⠿⣿⣿⠿⣿⣿⠟⣿⡄
⠠⠠⠀⠀⠀⠀⠃⢩⠟⠛⠘⠿⣏⠉⠀⠀⠂⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⣶⣾⣿⣶⣿⣷⣾⣾⣿
⠀⠑⠢⣴⡆⠀⠀⣀⣤⠀⠀⢴⣄⠙⢆⣈⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⣄⣤⣼⣀⣐⣀⣀⣸⣿
⠀⠀⢀⣸⣿⡿⣿⣿⣿⣶⢲⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠉⠉⢙⡏⠉⠉⠉⢹⣿⡿
⠀⠀⢻⣷⣽⣷⣟⡿⠿⣟⣼⣿⠿⢷⣼⣿⣭⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣷⣿⣿⠟⠁
⠀⠀⠾⣿⡟⢿⡿⣿⣿⣿⠈⢻⣿⣿⡇⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⡂⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠐⠚⠉
⠀⠈⢠⣿⢂⣤⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠐⠁⢀⡀⠈⠙⣫⡵⠖⢰⣭⣻⢶⣶⡦
⠀⠀⢘⣿⠀⠋⠘⣿⡿⣿⠀⢸⣿⣿⣿⡧⣾⡗⣹⣿⠀⠀⠀⠀⠀⠈⠉⢀⠀⣤⠞⣥⠀⠀⠃⠀⠁⠀⠀⠉⠈⠛
⠀⢳⢻⣿⠀⠀⠀⢨⣧⣠⡄⣾⣿⣿⣿⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣾⣯⣀⠙⠀⢸⣷⡈⠓
⠀⠘⢏⠹⡀⠰⠄⢺⡟⢻⣇⠺⣾⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢁⣿⣌⣿⣧⠿⣿⢿⣿⡟⣿⣿⡏
⠀⠀⠈⠢⣽⣤⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣹⣿⡟⢁⣿⣶⣾⣷⣤⣿⣯⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠛⢿⣦⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠃⢨⣿⠏⠀⢸⣿⣿⠟⢿⣿⡏⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠯⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⢻⠃⠀⠀⢸⣿⡿⠀⠀⠸⡇⢹⡌⣿⡆⠀⠀⠀⠀⠀⠀⢚⢣⣤⠲⢂⠀⣀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠀⠀⢒⢿⡇⠀⠀⢸⡃⠐⡆⢹⣇⠀⠀⠀⠀⠿⠃⠺⠁⠉⠁⢿⠃⠛⣠⣉⠰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⣠⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣠⣾⠀⠀⠀⣼⠛⠀⣇⣤⣿⠀⠀⠀⠀⠀⢀⡀⠀⠺⡶⢀⣀⠀⠈⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⣄⣿⠀⡀⠀⣾⣿⠟⠀⠀⢰⣿⣿⠄⣿⣾⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣠⠍⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣷⣰⣿⣬⣄⣴⣿⣿⣿⣿⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠻⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣶⣤⣀⣤⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⠫⣿⡌⠷⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣖⣬⢷⠀⠀⠘⠿⣯⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠒⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⠀⠀⠈⠅
⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠠⣿⣷
⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠆⠀⠀⠀⣀⢠⢦⢦⡖⣔⢲⢰⠐⡆⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠊⠁⠀⠘⠘⠀⠁⠉⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⣁⣁⡀⠀⠀⠀⠀⠀⠘⠀⠃⠛⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠂⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠴
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠈⠉⠁⠀⠄⠁
⠀⠀⢀⣩⣉⠉⠀⠀⠀⠀⠛⠛⠋⢹⠀⠀⠀⢠⡄⠐⠷⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠶⠶⠰⠶
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠀⠀⠈⠁⠰⠦⠀⠈⠉⠰⠆⠀⠀⠀⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⢶⣦⣤⣄⡀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠙⢿⡖⠦⣄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠀⠀⣀⡤⠖⠛⠙⢻⣆⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⢳⣿
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⡴⣚⣡⡤⠴⠶⠦⢤⣿⡆⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⡿
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢠⠿⠛⠉⠀⠀⠀⠀⠀⢀⣿⠁⠀⠈⠻⣶⣤⣀⡀⠀⠀⣀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣀⣀⣀⣀⣤⡶⠟⠉⠀⣠⡶⢾⡿⠻⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁⠀⠀⢠⡾⠋⢀⡞⢀⡴⠋⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢸⢁⡞⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡾⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⡶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣉⣀⢠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠐⠒⠶⢂⡤⠘⠁⠠⣤⠀⠈⠀⠀⢠⡄⡤⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣤⣤⠈⠙⢁⠶⣁⢹⣄⣐⣶⢿⡍⠓⢢⣤⠄⠁⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢠⠈⢡⡍⣍⣀⠀⠹⣿⡿⢧⣀⠙⠟⠛⠛⠳⠓⠀⠐⣞⣶⣶⠿⠛⠃⠀⢤
⠀⠀⠀⠀⠘⠏⣡⣴⠄⢈⡤⣤⠵⠈⠉⠉⡼⠏⠁⠀⠠⣌⡀⣀⣶⣤⠰⠆⣀⣷⢂⣤⡌⠻⣷⣆⠀⡐⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⢀⡃⠀⡘⣫⠀⠀⡎⠘⢿⣾⣏⡤⢭⣼⠿⢦⣴⣯⣽⢿⠟⣱⣤⡉⠁⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⣻⠎⣥⣀⠬⡁⣲⠊⢱⢆⠘⢿⣿⣿⠶⢶⣦⣈⢛⠿⡟⢻⣿⣿⠃⢈⠳⠶⡤
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⣈⠉⠀⠁⣉⣩⡀⣀⣀⣘⣤⣀⠀⡀⠠⣾⠯⠅⠋⠛⢙⣿⡶⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠙⠛⠇⠈⠉⠙⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉


⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⡀⠀⠀⠀⠀⠀⡠⠐⠂
⠀⠀⣀⣠⡤⠤⠤⠤⠶⠶⠶⠄⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⢹
⢠⣿⠿⢯⡀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢠⠀⠒⢤⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠖⠈
⣿⠈⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠈⠡⡀⠀⠉⠀⠀⣤⣄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⡈⠰⠈⠀⠀⠸⠋
⣿⣦⠀⢀⣀⠆⣤⣀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠉⠻⢧⠀⠀⠀⠀⠀⠀⢰⠀⢸⡟⠀⠀⠀⢰⠀⣴⠃⠀⠀⠀⠀⠀⣀
⠛⢻⡆⣸⡟⠀⠀⠙⠳⠦⠀⠈⠁⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣤⠏⠀⠀⠀⠐⠃⢠⣅⠀⠀⠈⢉⣁⡈⠉⠁⠐⢀⡀
⠀⠸⣷⠟⠀⠀⠀⠀⢀⠠⠄⠛⠚⠛⠛⠛⠿⣦⡀⠀⠀⠀⠀⢀⣤⣶⣿⣶⣄⠀⠀⣃⠀⠀⠀⣀⡾⠅⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⢠⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠞⢉⠁⠀⠀⠀⠲⠁⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⣦⣄⣀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠉⠉⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⡀⠀⠐⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⡆⠁⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣿⡛⠋⢉⠀⢀⣾⣟⣿⣿⡿⠃⠀⠀⢀⣀⣀⣀⠈⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠰⠎⠀⠀⠀⢀⣀⣀⣀⠀⠈⢿⣿⣿⣿⣯⣽⠀⢀⣼⣶⣾⣿⣥⡿⠋⠀⣤⣴⣶⣄⣨⣤⣼⣇⠘⠛⠁
⠀⠀⠀⠀⠀⢰⡆⠈⣷⣴⣶⠾⠛⠛⢛⠿⠟⠁⠀⠈⠛⠿⢿⣿⣿⣤⣿⣿⣿⠿⠛⠁⠀⠀⠀⠈⠉⠁⠀⠀⠈⢿⡹⣆⡀
⠀⠀⠀⠀⢠⣯⠐⠀⠉⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⣀⣤⣴⣍⠉⠉⠉⠉⠀⢰⣶⣤⣀⠀⠀⠀⠀⠙⡌⢠⠀⠀⢃⠙⢧
⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⢀⣄⣠⡟⠀⠀⢀⣴⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⣼⡿⢛⡛⢻⣶⣄⡀⠀⠸⣇⠀⠁⠄⠀⠀⡇
⠀⠀⠀⣰⠀⠀⣤⡄⠸⠿⠛⢿⣿⣿⡁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢠⠟⠁⠀⠙⠇⠈⢻⣿⢆⡀⢹⡆⠀⣀⠀⠻⡄⢀⡀
⠀⠀⠠⣿⠞⠉⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡷⡀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⡇⠀⢡⡀⠀⢻⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠹⠎⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⣇⠈⠀⠀⡈⢠⠀⡀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠻⣿⣿⣿⣿⣆⠀⠀⠀⠐⡇⠀⠀⣠⣾⠇⠀⢻⣆⠀⠀⢀⠈⣤⠁⠈
⠀⠀⠀⠀⠀⠀⠀⣠⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣰⣿⣿⣤⣿⠛⢿⣦⠀⠀⠀⠁⢀⣾⠛⠀⣷⠀⠀⠉⢀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠿⣤⣾⡿⣿⣿⣿⣿⣿⢹⣿⣿⠀⠀⣶⡾⣿⣿⠀⠈⣿⣆⡀⠀⡰⠊⠀⡇⠀⠻⣤⠀⠀⠉⠀⠆⠀⠀⠇⠘⠀⠐⢠⡄
⠀⠀⠀⠀⠀⠀⢸⡃⢨⣿⣿⣿⣿⣿⣿⡏⣾⣿⣏⠀⠀⣽⡀⣿⣿⠀⠀⠈⣿⣦⢾⡇⠀⠀⣷⠀⠀⢿⠀⠀⠀⡇⠀⠀⠀⠘⠇⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠘⣧⡼⣿⣸⣿⡏⠈⣿⠃⣿⣿⣿⠀⠀⣙⠓⣌⡿⠀⠀⠠⠉⠉⠈⠃⠀⠀⢹⡆⠀⢸⡇⠀⠀⢰⠇⠀⠀⠀⢰⡄⠀⠀⡠
⠀⠀⠀⠀⠀⠸⣇⠉⢸⣿⣿⣿⣽⣾⣿⡷⣿⣿⡟⠀⡀⣻⠀⠋⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⢸⡇⠀⠈⢳⡀⠰⠀⢰⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣹⣦⣴⣿⣿⣿⡁⣿⣿⡄⣿⣯⣿⠀⠁⡽⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢠⡄⠀⠀⠀⠀⠹⠀⡀
⠀⠀⠀⠀⠀⢀⣈⣏⣹⣿⣟⣿⡟⢿⣯⠀⣹⣟⡁⠀⠀⠀⠀⢼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡀⠀⠀⣌⠁⠀⢠⠄⡀
⠀⠀⠀⠀⠀⢸⣿⣷⠛⣿⣾⣿⠀⢠⣏⠀⣽⣿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠙⠀⠂⠈⠀⣿⠀⠆
⠀⠀⠀⠀⠀⢸⣿⣷⣼⣿⣿⣿⠁⣸⠛⠀⣸⣿⢷⡀⠀⠀⠀⢸⣿⠀⠀⡄⢠⡀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⣿⡄⢰⡆⠁⠀⠀⢀⠈⠆
⠀⠀⠀⠀⠀⠸⣿⡟⣿⣿⣿⣿⠀⣿⣦⠀⢨⣿⠋⠇⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠁⢠⠀⠀⣄⠘⡇⠀⠀⠛⠀⠃⠈⠀⠇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⣿⣆⠘⣋⠀⠀⣽⡄⠀⢀⡀⠀⢸⣿⠀⠀⠀⠘⠁⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠸⣿⡆⠀⠀⣧⠀⢠⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣇⣾⣿⠃⣼⣿⣷⣄⣎⠀⠀⢸⣿⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠠⠀⢸⡇⠀⠰⡿⠀⠠⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⢸⣿⡶⣿⣿⣿⣏⢸⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠺⠀⠀⠀⢿⠃⠀⢀⣶⠆⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠄⣿⣷⢹⣿⣿⣿⣾⠀⠀⠈⣿⡇⠀⠀⢸⡀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣿⣿⣌⠻⣿⣟⣿⡆⠀⢠⣿⠀⠆⠀⢸⡀⠀⠀⠀⡟⠀⠀⠀⣠⡀⠀⣰⣿⣦⡜⣿⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⠀⣠⣿⡆⠀⠀⠈⠋⠀⠀⠀⠻⠀⠀⠀⢸⣇⢀⣿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣧⣿⣿⣿⣿⣧⣿⣿⣿⠄⠄⠀⣿⡄⠀⠀⣤⠀⣠⡀⣿⣿⣿⣿⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣿⣇⣼⣆⣭⣄⣿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠉⠉⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠟⠛⠻⠿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⡀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿