Croissant Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣄⣀⣠⠴⠒⣒⣒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⣀⣤⠤⣤⡴⠋⠛⠋⠉⠁⡴⠋⠀⠈⠉⠉⠉⠙⠓⠦⣌⠳⣄
⠀⠀⠀⠀⣼⠱⠂⠀⢸⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣈⣧
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠙⠯⢽⡲⣄
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⢠⠪⠀⠀⠀⠀⠀⢀⣠⣤⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠀⠀⠀⡠⠚⠁⠀⠀⠈⠻⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂⠀⠀⡠⠴⠖⠒⠶⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠠⠤⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠙⠛⠋⠓⠒⠛⠲⢤⣤⣀⣠⡾⠵⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⣤⣾⣸⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠰⠟⠙⠾⣷⣶⣮⣾⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠶⠀⠀⠀⠈⢿⣿⡟⡘⠂⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣋⡭⠀⠀⠀⡀⠀⠀⢠⣠⣀⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣠⠚⣽⡁⣴⣿⠏⠁⠀⠀⣠
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠹⣿⣭⣽⠁⣠⠔⢩⡇
⠀⠀⠀⠲⣄⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⣿⠋⠀⢀⡨⠃
⠀⠀⠀⠀⠘⠷⠟⠛⢷⣤⡄⣢⡀⠀⣀⠀⢀⡜⢁⡔⠀⢀⠀⢰⣶⢬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⢇⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⠿⠿⠿⠛⠚⠿⠿⠞⢧⣀⣏⣠⡦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡉⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠁⠀⠀⠈⠓⢄⡀⠀⠀⠙⠀⠒⢦⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠈⠒⢤⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡⡀⠉⡐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠉⠀⠠⢀⠀⠉⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠈⠸⣧⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢠⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣧⠀⡀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠠⠀⠈⣳⠀⢿⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠀⣷⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠸⡄⠀⠀⣹⡆⢀⡀⠀⠠⠀⠀⣸⠀⠀⡎⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⡟⣰⡇⠀⠐⣿⠁⠀⡇⠐⠁⠠⣰⠇⢠⣿⣧⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠷⠷⠶⣿⣶⣾⣷⢂⣀⣴⣏⣠⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠙⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣡⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠛⠛⠿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢾⡏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣠⣿⠀⣰⡟⡇
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠻⣿⣻⣧⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⣀⣼⡿⣷⣤⣴⣾⣿⣷⣾⣿⣠⠇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⣾⣿⢿⣦⣶⣤⣶⣾⣿⠟⠛⠛⠛⠛⠉⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢸⣇⣀⣤⣶⣶⣤⡀⠀⠀⢸⣿⣿⣶⣶⣾⡿⠟⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠈⢻⣿⠋⠉⠙⠻⣷⣤⣾⡿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⣀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠒⠒⠒⠒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⡀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠈⠉⠓⢦⡀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣸⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣀⣀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡿⠀⢀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠉⠉⠙⠳⢦⡉⠳⡄⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡼⠃⢀⡼⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠙⣦⠙⠆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣤⡤⠴⠛⠓⠛⠻⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣸⣧⣄⣠⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡴⠶⠦⢭⡀⠀⠀⢀⡿⠦⠤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⢹⣦⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⡄⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠁⣀⠴⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡆⠘⠀⡀⡂⠀⠀⣠⡳⣸⠞⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀
⠀⠀⠀⢀⠎⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⣰⢣⡟⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⢀⠞⠊⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⡸⢁⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠋⠉⠑⠲⢌⣇
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢢⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡜⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢣⠀⠀⠀⠀⢿⠀⡀⠀⢸⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠄⠄⡀⠒⠈⢯⣆⠀⠀⠀⢸⠘⡡⠀⢸⠀⣠⠀⣰⠃⠀⠀⠀⠀⠀⠈⠠⠠⠄⠀⣀⣀⠀⠀⠀⠉⠣⠀⠀⠘⣆
⢧⠁⠤⣔⡒⡛⠛⣡⣠⠤⠖⠛⠚⠚⠛⠛⠶⠤⣬⣦⣘⣇⣼⠀⠁⢀⠃⠀⠀⠀⠀⠀⢀⣤⠶⠒⠋⡩⠩⠍⠉⠭⠝⠛⠻⣄⡀⠀⠸
⠈⠓⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣸⠇⠂⠀⠀⠀⢠⠖⣩⠤⠒⠚⠉⠉⠉⠐⠓⠒⠢⠤⣀⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⡀⡄⣴⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⢺⣹⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣾⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠴⠖⠒⠒⠲⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠒⠦⣄⠙⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⢘⢧⠀⠀⠈⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡀⡇⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠐⠀⠂⠔⠠⠈⠐⠠⢀⠀⠀⠀⠀⢠⠇⢠⠇⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⣊⣠⠾⠖⠋
⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠙⣼⠀⢀⣼⡞⠀⠀⠀⣀⡤⠤⠶⠶⠦⣤⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⢸⣽⣦⡀⢹⣷⣌⣽⣇⢀⣢⡾⠉⠀⠒⠈⠙⠒⢦⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠶⣌⢻⣿⣧⣠⣿⣿⣿⣿⣿⣿⣷⣶⣞⣭⠄⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠠⢾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣫⣴⡖⠀⠀⠈⣒⢌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣁⠀⠲⣿⣿⡟⠚⠋⠉⠁⠀⠈⠙⠛⢿⣿⣿⡟⠀⠀⠀⢀⣸⣷⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⠤⠀⡸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠻⠀⠀⠒⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠐⣄⠀⠀⢀⣠⡄⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⣦⢠⢟⣳⠿⠛⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠀⠜⠛⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠦⢄⡤⠖⠚⣛⣉⣉⡉⣉⣉⣉⠉⠙⠛⠒⠲⢤⡀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡴⣋⠤⠒⠀⠀⠀⠀⠀⡞⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡠⠖⠊⠉⠁⠐⠊⠭⢝⡒⠤⣀
⠀⠀⠀⠀⠀⡞⠘⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⠙⠢⣄
⠀⠀⠀⣠⠖⡇⢸⠀⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⡞⠁⠀⢻⠈⡆⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⡎⠀⠀⠀⠸⡄⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⢎⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡠⠤⢔⣒⣉⣭⠤⢤⣷
⣼⠀⢀⠀⣀⡀⢧⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⢠⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢔⣪⠵⠒⠋⠉⠀⠀⠀⠀⠀⠀⠙⣄
⡇⠀⠔⢋⣠⠴⠚⢦⣀⣠⡤⣤⣀⣧⠀⠀⠀⠀⢰⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡭⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠳⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠀⠀⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠹⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢕⣦⣔⡀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⢭⣓⣂⣀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠤⠤⠄⣒⣒⡒⠠⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⠔⣡⠤⠖⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠓⢤⡀⠀⠀⠀⠀⢀⡴⢃⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⣉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡠⣠⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠠⣤⣦⣶⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣚⣽⡶⠾⣷⣶⣬⣍⠓⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢚⣉⣀⣀⡀⠀⠈⠉⠻⣿⣦⣝⢦⡉⠲⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠚⠉⠉⠀⠀⠉⠙⢿⣶⣄⠈⠙⢿⣧⣳⡄⠀⢳⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⣃⠀⠐⠶⣤⣀⡀⠀⠀⠈⠻⣷⣦⡀⠙⣿⣿⠀⠀⢷⡄⠀⢿⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣴⠾⠋⠙⠛⠶⣿⣿⣻⣧⡀⡈⠱⡄⠈⢻⣿⣦⠸⠟⠀⢠⢸⡇⠀⣸⣧⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠍⠋⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣷⣿⣿⡿⣿⣶⣌⠀⣈⣀⣿⣼⡾⣿⣷⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡵⠓⠛⣓⠶⢶⣤⣄⠀⢄⣀⠀⠈⣿⣿⣿⡞⢖⣯⣷⣻⣿⣿⣿⣿⣿⡿⠃⣠⣏⣡⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠉⠉⠙⠳⢾⣿⣷⣬⣛⣹⣷⣾⣿⣿⣿⣾⣽⢯⢿⡿⢟⣿⣿⡿⠶⠏⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣠⣲⣽⣿⣿⣾⣿⡻⡟⣿⣿⠙⠖⢛⣸⣿⡿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡣⠖⠛⣉⣉⣙⠺⣷⣿⢻⢿⣿⢟⣟⣀⣿⣷⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣴⣶⡶⣾⣿⢿⣺⣷⣯⣴⡾⠿⠝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣙⠿⠞⣛⣵⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣦⣄⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⠟⠛⠋⠹⠟⢻⣽⣿⣿⣏⠉⠙⢿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⠀⠀⡀⠀⠀⠠⠀⢸⣿⣿⣿⣷⣶⣾⣿⣷⣭⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⣻⣿⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢏⡐⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⠀⠁⣗⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡛⢛⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣹⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠟⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠛⠶⣶⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⢳⡖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⣷⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⠀⠀⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⠟⠛⠛⠛⠛⠿⢿⣶⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⢸⣇⣀⣴⠗⠒⠻⠧⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠘⣷⣀⣠⣴⠟⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣴⣾⠿⠛⠛⠛⠿⢶⣄⠀⣠⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣤⣴⣶⣦⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣟⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠴⠞


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠤⣤⠤⠖⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⢀⠔⡠⢊⠕⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡜⢈⡞⡵⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢍⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢸⠁⢠⠀⠀⠀⠀⢸⠃⡞⡜⠀⠀⠀⠀⠀⠀⢀⡤⠖⣛⣉⣍⣓⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡀⡄⣷⡀⠀⠀⠀⠀⣼⢸⡹⠀⠀⠀⠀⠀⢀⡴⣫⠔⣫⠥⠒⠂⠈⠑⣞
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢣⢳⡸⣷⡡⠀⠀⠀⣿⣾⡃⠀⠀⢀⢄⡴⣫⠞⡵⠋⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⢿⢧⠀⠀⠣⡱⡽⣷⡄⠀⠀⢸⣿⠁⠀⠀⢠⢎⡼⠣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢠⠃⠈⠳⣷⣦⣀⣀⣘⣿⣿⣦⣔⠈⢿⠀⠀⠠⢣⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠸⡆
⠀⠀⠀⠀⠀⢸⣀⣀⡴⠚⠉⠉⠁⠀⠀⠀⠈⠙⠛⠛⢆⡀⢠⠏⠀⠀⠀⠀⠀⢐⣒⡭⠭⠶⠶⠦⢭⣍⡁⣇
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣶⡬⢶⣺⠭⡶⠾⠿⠿⠿⢿⠒⢦⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢀⣀⣀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠛⠯⣟⡦⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣹⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠶⠋⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣍⡳⣶⡆⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢛⣉⣩⣿⣮⡙⢶⡆⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣉⠀⠈⠙⠻⠿⠿⣿⣄⠑⢦⠀⠀⠀⢻⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⢿⣆⠘⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢿⣿⣿⣿⣿⣽⡻⡦⠀⠀⠀⠀⢌⢿⡆⢣⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢚⣋⣀⣰⣿⣿⣿⡿⢿⣿⣦⡀⠀⠀⠀⠈⣮⣷⢨⡀⠀⠀⢈
⠀⠀⠀⠀⠀⣠⣭⣷⣿⣿⣍⡉⠀⠉⠀⠀⠈⠙⢿⣦⠀⠀⠀⢻⣾⣼⣿⠂⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠋⣹⣿⣿⠟⠋⠉⠳⢄⠀⠀⠀⠀⠀⠹⣧⠀⠀⢸⣿⢹⡿⠀⠀⠀⠀⠀⠀⣼⣡⡴⣄⡱
⠀⠀⠀⠀⣴⣿⣿⣟⡛⠧⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢻⡆⠀⢸⣟⣿⠣⠄⠀⢸⣀⣦⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢖⣿⣟⠛⠫⢁⠀⠀⠀⠀⠀⠀⠈⢣⡀⢀⣀⣿⣿⣴⣿⣿⣿⣧⣄⣰⣾⣿⣿⣿⠛⠉
⠀⠀⠠⣿⣿⡿⢷⣄⠀⠑⡄⢀⢰⣄⡀⢀⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁⠀⡀⠄
⠀⠀⠀⢿⣿⣄⣈⣿⣧⣴⣿⣿⣾⣿⡵⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠻⠿⠿⠟⠛⠛⠋⠩⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠒⠒⠶⣦⠴⠤⠶⠦⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠁⠱⡄
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⡵
⠀⠀⠀⠀⣷⣴⣲⣶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣠⠟⡇
⠀⠀⠀⠀⢸⠀⠀⠈⠉⠙⠛⠿⣅⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡟⠁⠀⠀⠀⢀⡴⠁⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠒⠉⠓⠮⣝⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣼⣅⣀⡀⢀⣴⠋⠀⠀⠀⡇
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣝⠳⣄⠀⠀⠀⠀⠀⠀⠀⣤⡼⠋⠀⠀⠀⠀⠉⠉⠉⠑⣆⠀⢰⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠷⡄⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠾⠖⠋
⠀⠀⠀⠀⠻⣤⣒⣲⣤⡤⠤⠬⠿⠭⠥⠤⢤⣄⣀⣒⣤⠟
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⣤⡦⢞⠋⠉⠉⠀⠉⠑⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⣅⡀⠁⠀⠀⠀⠀⢀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠛⠛⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣻⣩⣙⣿⣋⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡤⡤⣦⣤⣴⣦⡤⡶⠿⠷⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⡾⠟⠉⠉⠁⠀⠀⠈⣼⡿⠋⠁⠀⠀⠀⠀⠀⠀⠁⠙⢿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠛⠉⠉⠉⠰⣽⡟⠉⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣶⣿⠍⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⡟⣻⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⣠⣾⣟⡛⠀⣀⢀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣯⣿⠾⠛⠉⠉⠉⠉⠍⠻⣿⣿⣿⣿⣄
⠀⢠⣶⡿⠏⠙⠛⠳⣤⡙⢧⡀⠀⠀⠠⢠⣿⡀⠀⠀⠀⠀⢀⣰⡏⠀⠀⠀⠀⠀⢀⣴⣿⣿⣵⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⢙⣿⣿⣿⣿⣦⡀
⢠⣿⡿⠇⠀⢀⡀⠀⠈⠻⣾⢷⣦⡄⡴⣿⣿⣧⠀⠀⠀⠠⣾⣿⣧⠀⠀⠀⠀⣰⣾⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣷⣄
⢼⠏⠀⠀⠀⢙⢿⣦⣂⡢⢹⣿⣿⣤⣀⣿⣿⣟⡃⢴⡢⣞⣾⣿⣷⡄⣀⣶⣾⣿⣿⡿⠃⠀⠀⠀⠀⣀⠀⣀⣤⡨⠝⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣧
⣼⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣅⡄⠀⢀⣴⣶⣶⣿⣬⣿⣿⣼⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⢻⠀⠀⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⢿⣴⣾⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣿⣿⣿⣿⣿⣧
⠘⣇⠀⠀⠀⣶⣿⣿⣿⡿⠟⠛⠉⠙⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢿⢻⣿⣿⣿⣿⣿⣿
⠀⠻⣷⣴⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⢤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⣁⣅⣆⡏⣉⣀⣀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣍⡉⠉⠋⠙⠻⣿⡶⣦⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠑⣿⣷⣀⣠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢐⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢀⢀⡤⣆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣤⢀⡀⣀⣤⢂⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠒⠊⠙⠣⡀⠈⠣⡈⠈⠛⠠⣶⠒⢖⠲⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠈⠉⠓⠤⠀⠀⠈⠀⠀⠐⠀⠀⠀⠘⣆⠀⠀⠹⠀⠁⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠭⢭⣉⣓⠢⢄⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⢸⠈⠉⠐⢄
⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠈⠉⠢⢍⠲⣄⠀⠀⠀⠀⠀⠐⡀⠀⠀⢸⡄⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⡮⠤⠄⠀⠀⠀⠀⠀⠀⠀⠑⠀⠑⢄⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⣀⠤⠆
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡆⠀⢠⠞⠁⠀⠀⠡
⠀⠀⠀⠞⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠠⠀⠀⠀⠀⢸⠀⢠⢀⠼⠤⠴⠁⠀⢀⡠⠖⠒⢆
⠀⠀⠸⡶⠋⣁⣀⣀⡀⠉⠉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠱⡀⢃⢠⡄⠀⡘⠈⠁⠀⠀⠀⠀⠑⠲⠁⠀⠀⠀⠀⡀
⠀⠀⠀⢳⣎⣁⠀⠀⠉⠈⠉⠐⠂⠤⠄⣉⠒⢄⡀⠀⠀⠀⢱⣸⣸⡧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠
⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠫⢦⡀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⢸⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢹⠁
⠀⠀⠀⠀⢻⡠⠂⢀⡤⢐⣒⠤⠖⠂⠐⠲⠤⠤⢄⣠⠃
⠀⠀⠀⠀⠀⠳⡀⡏⢴⡫⠀⠀⠀⠀⢀⣀⣀⣀⣀⡙
⠀⠀⠀⠀⠀⠀⠈⠓⠾⡀⠄⢀⢀⠞⠉⠀⠀⠀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣬⣼⡀⠀⠀⠀⡀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠰⠴⢤⣄⣀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣯⣝⣓⡲⢦⣴⣶⣿⣿⡻⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠉⠉⠙⠛⠭⣙⡻⣷⣿⣯⡙⢿⣷⡜⣿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡧⣆⣴⣶⣶⣶⣤⣤⣀⠉⠙⠛⠛⣷⣀⠉⢻⣿⣧⣿⡇⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣫⣽⣯⠭⠭⢭⣝⡻⠯⠙⠷⢦⣤⣤⣬⡿⠀⠈⠙⠋⣿⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠋⠾⠿⠛⠛⠛⠿⣶⡀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣧⣄⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡋⠻⠟⠛⠻⠷⣤⣀⠈⠛⢦⡄⠀⠑⢠⠆⣀⣤⣤⣶⠟⠁⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣼⠃⢀⣴⡞⠛⠷⣿⡻⣧⠀⢀⣀⣀⣠⣴⣾⠟⠋⠉
⠀⠀⠀⠀⠀⢸⡟⢠⡟⠁⡟⠀⠀⣉⠙⣿⣤⡾⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠸⣷⡘⠃⠐⡇⠀⢰⣿⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⡄⢀⣿⣰⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠘⡄⠀⠈⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣼⣤⡀
⠀⠀⠀⠀⠀⡀⠉⠙⡄⠀⠀⠀⠀⠘⠀⠀⣣⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠈⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⣀⠀⠀⠈⠂
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⢀⡇⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠿⠤⠶⠿⠆⢀⣀⣀⣀⣀⠼⠿⠆⠐⠛⠓
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠾⣷⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢀⡀⠀⠢⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠢⠚⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⢉⠟⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃
⠀⠀⠀⠀⠀⠈⠀⢄⡈⠳⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠳⡄⢀⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠈⢻⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠴⡇⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠈⠃
⠀⠀⠀⠀⠀⠈⠀⠀⡠⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠈⠉⠒⢮⣹⠀⠀⠀⠀⠈⠒⠂⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⠔⠂⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠉⠓⠂⠀⣆⠀⠀⠀⠀⠀⢀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠗⠒⠒⠀⠀⠀⠈⡆⠀⠀⠀⠀⡜⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜⠃
⠀⠀⠀⢀⠄⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠴⠒⠂⠐⠒⠠⣆⠀⠀⠀⠀⠣⡀⠀⠀⠀⡇⢀⠀⠀⡠⠛⠁
⠀⠀⢠⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡤⠚⠛⠶⢤⣄⣀⣀⣠⡼⣿⠀⠀⠀⠀⠀⠈⠁⠐⠚⡁⠈
⠀⠀⢸⡆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠂⠀⠐⠀⣀
⠀⠀⡸⠇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠳⣔⣂⡡⠤⠒⠒⠒⠒⠲⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠁⡴⠓⠦⢄⡀⠀⠀⠑⠀⠀⠀⠘⠈⠑⢆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡽⠧⠤⠤⠀⠄⠀⠀⠘⠃⠀⠀⠀⠒⢒⡶⠀⣰⠋⠀⡠⠚⠉⠉⠐⠀⠉⢆⠀⠀⠀⠀⠀⢠⣇⣠⠜
⠀⠀⠀⢀⠀⠀⢀⠄⠀⠈⠂⠀⠀⠀⠀⠱⣦⣤⡤⠤⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠀⠔⠁⠀⡔⠀⠀⠀⠀⠀⠀⠀⠂⡑⡘⡄⣴⡠⠜⠉
⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢟⠀⠀⣀⡠⠴⠒⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠤⠌⠁⣀⠂⢀⠀⠀⡷⠿⠛⠁
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠙⢯⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠂⠤⣀⣀⣹⡦⠄⠋
⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⢀⣀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡈⠐⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠈⠅⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⠤⠠⠤⠠⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠂⠉⠀⠀⣀⠀⣀⢀⣀⠀⠀⠀⠙⠀⠤⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⣀⠠⠆⠉⠁⠁⠀⠀⠀⠀⠈⠉⠐⠆⠀⠀⣠⠞⢁⡾⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠀⠠⠆⠁⢀⣠⠴⠒⠛⠉⠉⠉⠃⠀⠀⢀⡤⢤⡞⠁⡀⢸⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠚⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠚⣏⠙⢲⡉⠀⡴⠃⠀⠐⠦⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠞⠀⢠⠉⠀⣰⠋⠀⠀⠀⠀⠀⣠⢾⡙⢦⡈⠳⣄⠉⢺⡁⠀⢀⠀⠀⠇⠀⠘⣂
⠀⠀⠀⠀⠀⠘⠀⢀⡌⠀⣸⠃⠀⠀⠀⠀⣠⢾⠙⢆⠙⢦⡙⠂⠈⠓⢦⠇⠀⠘⣆⠀⠈⣃⠀⢙⡀
⠀⠀⠀⠀⠀⡋⠀⢰⠀⠀⠇⠀⠀⠀⡠⢎⠹⣄⠳⢄⠙⠂⠀⠀⢀⡴⠋⠀⠀⠀⢹⠀⠀⢙⠀⠈⡅
⠀⠀⠀⠀⠀⡃⠀⢰⠀⠀⠀⣄⡠⢾⠀⠈⠳⣌⠓⠀⠀⠀⢀⠄⠀⠀⢀⡀⠀⠀⢸⠀⠀⢘⠀⠀⡅
⠀⠀⠀⠀⠀⣃⠀⠰⠀⠀⠀⠘⢧⠀⠳⣌⠳⣄⠀⠀⢀⡴⢋⡴⠛⣉⣩⣉⡙⠲⣄⠀⠀⣘⠀⢀⡅
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡠⠚⣳⣄⠈⠳⢬⣧⠴⠋⢠⠏⢠⠋⠁⠀⠀⠙⢦⠘⣆⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⠲⣤⠊⣠⠞⢡⠜⠓⠦⣴⠋⠀⠀⢸⠀⢿⣠⢤⣄⡤⣄⣸⠀⢹⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠳⣄⠜⠁⢠⣀⠀⠈⠁⠀⠀⠸⣆⠸⣭⡛⠵⢙⣫⡗⢀⡏⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣬⡗⠀⠀⠈⠙⠲⠦⠤⠤⠤⠘⠦⣄⣙⠛⢛⣀⡴⠋⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠠⠄⢀⡀⢀⠀⢀⡀⣀⠀⠉⠉⠉⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⢠⣄⢀⠀⠀⠈⠁⠉⠁⠀⠀⢀⡄⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠃⠛⠀⠛⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⠔⠓⢆⡀⢐⣦⣠⠼⠷⡖⠀⠀⢀⠠⡾⠀⣤⣤⣤⣭⣥⣀⣀⡀⠀⠀⠘⠿⠃
⠶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⢶⣮⣷⢶⡚⠙⢯⣀⣈⣶⣤⠀⠀⠀⠀⠀⢱⠀⢒⣒⣒⣿⣿⣿⠿⠇
⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣧⠟⢾⡉⠉⢳⣄⡈⠀⠀⠠⠄⠀⠀⠀⢴⠚⠛⠩⠉
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠺⠗⠀⠀⠀⠀⠀⣿⠶⠖⠛⠛⠩⡈⠀⠐⡀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣶⢆⣶⠁⢀⣀⠄
⠀⠀⠀⠀⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠰⠷⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⢸⣿⡆⢰⣿⢸⠇⣴⣿⣿⣿⣦⢠⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣾⡄⢿⣿⣿⣿⡿⢰⡿
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣾⢿⣁⠀⠀⠙⠿⢦⣽⡿⣿⡁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠀⠀⢸⣧⠀⠀⣀⣠⣤⣤⣤⣤⣤⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠸⠦⠀⠀⠀⠀⠀⣀⠴⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠲⠀⢤⠀⢀⡀⢀⡀⠀⠀⠀⠀⢰⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠢⡀⠀⠄⠀⠀⠀⠀⠀⠃⠀⠆⠀⠤⠀⢠⠀⢈⠁⠘⠁⠘⠃⢐⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⢠⠃⠜⠀⠀⠀⠀⠀⠀⠋⠐⠂⠀⠦⠀⠤⠀⢨⡁⢈⡀⠘⠁⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠐⠊⠹⡖⠀⠀⠀⠀⠀⠠⣂⡀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠈⠁⠠⠀⢀⠶⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠐⠤⠂⣓⣀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠐⠒⢖⠠⠤⠀⣀⠀⠀⠀⣁⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠀⢸⠃⠸⣧⠀⠠⠀⠀⠀⠀⠀⠈⠀⣀⡀⠈⠁⠀⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠚⠀⠄⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠴⠒⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⠞⢺⠉⠁⠤⡀
⠀⠀⠀⠀⠀⠀⣔⣮⠇⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠂⠀⠓⠦⡀⠈
⠀⠀⠀⠀⠀⡐⠉⠃⠀⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠂⠀⠀⠂⠀⠀⠀⠈⠓⠂⠉⠉⢢⠄⠍⠉
⠀⠀⠀⠀⠀⠃⢀⣴⣤⢀⠀⠀⠀⠀⠀⢠⡤⢤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠄⠖⠓⠒⠉
⠀⠀⠀⠀⠀⠃⠘⠩⠞⠂⠀⠀⠀⠀⠀⠈⠙⠊⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⣶⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠄⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⢀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠏⣻⠇⠀⢀⠃⠀⠀⠀⠀⣰⣿⣿⠟⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡠⠄⠒⠙⠠⠤⠐⠁⠀⠀⠀⠀⠀⢿⣿⣯⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠘⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠐⠒⣸⣿⣯⣐⠒⠂⢰⠦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣀⠔⠊⠁⠀⠀⠀⠀⣸⣿⣿⡿⠁⠀⢀⠆⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡟⠀⠀⠀⠈⢛⠿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣿⣿⠏⠀⠀⡠⠃⠀⠀⢠⣶⣄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠾⠋⠀⠀⠀⠀⡠⠁⠀⠈⠙⢿⣿⣿⣷⡀⠀⠀⠙⠿⠄⠄⠊⠀⠀⠀⠀⢺⡟⠁⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠉⠐⠢⢄⣴⣿⣿⣷⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢀⡤⠖⢐⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠑⢄⡀⠀⠀⢀⣠⣿⠟⢡⣿⠁⠀⣾⣿⣿⡟⠀⠈⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⢰⣿⣿⡿⠁⠑⠢⢤⣤⡾⠆⠀⠈⠉⠉⠉⠉⠀⠀⠘⠛⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣈⠁⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⢦⠀⠀⠀⣿⣿⣿⠁⠀⢠⣶⣶⠀⠀⠙⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢀⣴⣾⡿⠷⣆⠀⠀⣾⣇⠀⠀⢠⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⡆⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⢠⣿⡟⠁⠀⠀⣰⣶⣤⡹⣿⣦⣠⠃⠀⠀⠀⣿⣿⠁⠀⠀⡆⠻⣿⣀⡠⠀⠀⠀⢘⣿⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢸⣿⠀⠀⢀⣴⣿⣿⠿⠃⠙⣿⣿⣦⣄⠀⠀⢻⣿⠀⠀⠀⠃⠀⠈⠉⠀⠀⡠⠒⠁⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⠃⠈⠉⠁⣼⣿⠟⠁⠀⢀⠌⠀⠛⢿⣿⣷⣄⠈⢿⣇⠀⠘⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢀⠂⠀⠀⠀⢰⣿⠃⠀⣀⠔⠁⠀⠀⠀⠈⠻⣿⣿⣷⡀⠙⠒⠁⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠐⠁⠀⠀⠀⠀⠈⠃⠄⠊⠀⢀⠤⢤⣄⡀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢹⣿⡄⠀⠀⢹⣿⣿⣧⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠋⠉⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠁⠂⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⣠⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⢀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠢⢤⠀⢠⡠⠀⠀⣠⢀⡲⠶⠔⡉⢅⠉⠀⠀⢭⡉⠐⠦⡀⠀⠀⠀⠀⠨⠤⢉⡀⠄⠠
⠀⠀⠀⠀⠀⠠⠔⠀⠐⠀⠼⡋⠉⢹⠈⠑⠐⣀⠤⣶⣧⠴⠄⠊⢠⡦⢆⡷⠀⠽⠀⠀⡔⣩⣉⢀⡀⠴⣾⠇⣐⠀⠀⠀⠀⠤⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢄⠙⠟⠛⣆⣉⡔⣀⡠⣔⣎⠰⠤⠚⠁⠀⠀⠀⡂⠉⣁⠉⣠⣤⣀⠅⠉⣠⠒⠷⠦⠊⡶⠀⠒⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘⢆⠤⣄⡀⢌⡨⢵⠀⠌⢰⡀⢠⠰⠞⠁⠠⢁⣄⡘⠿⠈⡃⣠⠈⡛⢲⣋⡴⢣⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⠈⠔⠂⡀⠰⣄⣈⠐⠀⡤⠃⠤⣈⣧⡀⠧⠭⠀⣀⣤⠃⠀⠉⠉⢉⠔⠁⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠢⢄⡄⠀⠐⠃⠀⠎⠙⠛⠛⣛⠉⡑⣲⠐⢀⣣⠀⠀⠀⡀⡄⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠐⢠⡠⠖⠀⠴⠆⣤⣤⣠⣤⠅⡥⢨⠟⠀⠀⠀⡆⣆⡗⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠒⠒⠂⡠⢀⣐⡐⠲⠔⠒⠚⢖⠦⠚⠃⠀⠀⠀⠃⠓⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢀⠀⠀⠀⠀⠀⠈⠒⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠍⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡀⠀⠀⠀⠀⠈⣀⠤⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠐⢉⠁⠀⠀⠀⠈⡒⠰⢂⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⠀⠂⠄⢀⠀⠀⠀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⢀⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠚⡆⠀⠀⠄⠓⠁⠑⠀⠐⠀⠠
⠀⠀⠄⠀⠀⠑⠀⠀⠀⠀⠀⡇⠀⠀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠆⠀⠀⠑⠒⠊⠀⠀⠀⢀⣦⠀⢢⢀⠢⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠎⠀⠺⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡤⣄⡀⢀
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⣁⠰⠄
⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠆⠀⠀⠀⠰⠇
⠀⠀⠈⠉⠈⠉⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠊⠔⠁⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⡄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠀⠀⠈⠈⡀⠀⠀⠀⠠⠼
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠘⠀⡃⠈⡀⠄⠀⡀⠁⠐⠀⠀⠠⠀⠀⢃⠀⠀⠀⡄⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠠⠄⠄⠈⠠⠃⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠊⠑⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠄⡰⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠢⣀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢨⠃⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⢐⠆⠀⠀⠀⡀⢂⠀⠀⠀⣮⣁⣠⣤⠆⠀⠀⠀⡆
⠀⠀⠀⠀⡄⢘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⠀⠀⠀⠈⢀⠀⠀⠀⠄⠀⠀⠀⠄⠁⠀⠀⠶⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⡉⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠆⠐⠚⠛⠓⠀⠀⠀⢈⡉⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠄⠠⠂⠈⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⡀⠀⢀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢀⠴⡁
⠀⠀⠀⢦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠐⠀⠀⢠⠀⠀⠀⠤⠴⠂⠀⠀⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡤⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠉⠃⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⡄⡨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣦⡌⠀⠸⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠚⠈⠀⠈⠂⠐⠂⠈⠑⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣐⣻⠿⠟⠓⠁⢻⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⡀⠀⠀⣀⣀⠘⠉⢑⠠⠠⠄⣄⣀⣰⡂⠉⠁⠀⠀⠀⠁⠈⠀⠀⢀⡀⠀⣄⣤⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣷⣶⣷⡎⠸⢦⣶⣾⣷⣶⣷⣶⣷⣤⣿⣯⣭⣄⡨⣉⠀⢂⡰⢦⠒⣊⣥⣾⣾⣿⣿⣿⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣄⡼⠿⠛⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣧⣿⣤⣸⣶⣿⠿⠿⠛⠋⠉⠀⠀⢀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠉⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⣀⠄⠀⠖⠒⢀⡉⠙⠤⠤⠥⣱⠀⠀⣀⠈⠐⠒⣒⠂⠀⣀⣊⠈⢧⡈⢢⣽⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⣀⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⠂⠀⠀⠀⠀⠈⠄⢀⡬⠩⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⢰⠁⢠⡄⣀⡤⠂⠒⠀⠤⢄⠀⠀⠀⠀⠈⠃⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠥⠔⠋⠁⠀⠀⠀⠀⠀⢀⠱⠢⢀⠀⠀⠀⠀⠈⢢⠄⠀⠀⠀⢀⡠⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⡀⠱⠒⠤⠤⠤⠤⠄⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡉⠐⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠦⣙⣤⠀⠀⠀⠀⠀⠀⠀⢮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⣦⣫⡶⠋⠀⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠿⣷⣋⡦⠀⠀⠈⢻⢲⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡲⠆⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⡾⠀⠑⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡔⠁⠀⢀⠄⣢⡆⠀⠀⠀⠀⠀⠀⠘⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⢀⡈⢧⠀⠀⠀⡇
⠀⠀⠀⢀⠊⢠⠃⠰⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣾⡇⠀⠀⠀⢠⣿⡟⠀⢳⠀⠀⡇
⠀⠀⠀⠊⠀⠁⠀⠀⠀⢠⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠘⠛⢁⠀⠀⡇⠀⡇
⠀⠀⠶⢀⡵⠀⠀⠀⠀⢨⣼⠁⠀⠀⠀⠀⠀⠀⢀⣠⣖⣮⣽⣦⣤⣴⣦⣤⣇⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⢄⡀⠸⡄⡇
⠀⢘⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠛⠷⠟⠀⢻⡇
⠀⢨⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠻⣅⣀⠤⢊⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠁⠀⠀⠀⠰⠀⣠⡇⣸⡃
⠀⣈⡀⠠⠀⠀⠀⣸⡿⢳⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠒⠈⠙⠲⠤⠤⠤⡤⠤⠤⠔⠒⠚⡉⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⡷⠀⢀⡴⠏⢀⣼⣿⢠⠟⡇
⠀⡍⡄⠰⣴⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠐⠀⠀⠉⠀⠀⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡤⠖⠋⢀⣤⣿⣿⢿⣿⢘⡇
⠀⠳⡙⡄⠈⢿⣷⣤⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠤⣄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣠⣒⠋⠀⢀⣠⣴⣿⣿⣟⣵⣿⡇⢨⡇
⠀⠀⢹⣿⣷⣦⡙⢿⣿⣷⣤⣘⢮⣭⡹⠳⠂⠀⢀⢾⣷⠀⠀⠀⠀⠀⠀⠸⠶⠾⠿⠋⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠚⠀⢀⣴⣦⣴⣿⣿⣿⡅⣴⣿⣿⣿⠇⠰⡇
⠀⠀⠈⡄⠙⠿⣿⣜⠿⠿⠀⠙⢻⢞⡂⢀⡀⠀⠻⠞⣡⣶⡦⠀⠀⠀⠀⠀⠄⠀⣀⠠⠤⠤⠀⠀⠀⠀⠀⣀⣀⣠⣤⢰⣿⣿⣿⣿⡿⠟⠃⣼⣿⣿⣿⠏⠀⢸⡇
⠀⠀⠀⠱⡀⠀⠈⠿⢿⣶⣦⡀⠈⠛⢿⡿⣷⠿⠷⣾⣿⣯⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣯⣤⣴⣾⣿⣿⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠛⠿⣿⣦⣤⣤⣤⣤⣀⣀⡀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠟⠻⠛⢛⣿⣋⣽⣷⣶⣬⣟⣛⣉⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠈⠉⠉⠛⠟⠿⢿⣿⣿⣦⣄⡀⠉⠛⠛⢉⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣴⠀⠀⠀⠀⠉⠈⠛⠿⣿⣷⣦⣤⣶⣿⠿⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⣤⣤⣀⠀⠀⠑⠛⠛⢿⣯⣶⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠾⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⡤⠤⠤⠤⠤⠤⣄⣀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠄⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⠖⠉⡉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⢡⠖⠛⢛⢲⣞⠒⢇⠁⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠋⠉⠉⠓⢦⠤⠤⣤⠶⠤⣤⠤⠖⢉⣨⠽⠛⣿⣆⠀⠀⠇⢴⠊⢱⠛⠉⠁⠀⢓⠩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⢀⣀⣈⣷⢖⣚⡷⢀⡔⠒⠛⠉⠀⢀⣴⡿⠋⠀⠀⠀⠘⠑⠎⡳⠖⢟⡠⠳⡄
⠀⠀⠀⠀⣠⣶⠿⠷⠈⠙⢷⣦⣀⡏⠉⠉⠁⠀⠀⠛⠉⠳⣄⣀⣤⡶⠛⢁⠀⣀⠤⠤⢤⡀⠀⠀⠁⢴⠈⢱⡆⣿⠐
⠀⠀⠀⠀⠛⣿⣄⠀⠀⠁⠀⡀⠉⠛⠷⠶⠶⢤⠶⠶⠶⠛⠋⢉⣀⣤⠖⠃⠰⢑⡄⠆⣤⢹⣄⡤⠦⠉⠙⠈⠡⠃
⠀⠀⠀⠀⠐⠈⠛⢷⣤⡀⣎⠉⠉⠀⠀⠀⠠⠀⠠⣄⢀⣠⣴⠿⠋⠁⠈⠀⠀⣈⠃⣄⠿⣸⠃⣠⠰⢤⡘⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣤⣤⣤⠴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⢿⢦⡀⠗⣤⡺⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣴⣻⡌⡇⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠚⢉⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠏⠐⠂⠒⢄⡀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⢢⡀⢀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠂⢀⠀⠈⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⢀⠀⠀⡫⠵⠀⢤⡔⠌⠀⠀⢱⣦⡈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢼⠡⢠⡶⠋⠀⢀⣀⠀⡀⠀⠁⠀⠀⠀⣸⠃⣝⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠑⠃⠈⠃⠠⠄⣠⡖⢀⡴⣻⠟⠂⠀⣀⣀⣼⣿⠎⣩⠇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠈⠁⡤⠛⠀⢘⣇⣴⣀⣤⣾⣿⣿⣿⠃⠀⡟⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠘⠹⠿⢿⠿⠟⠛⠉⠀⠀⠀⣇⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠀⠀⢀⠰⡀⠀⠀⠀⠀⠀⠀⡏⠀⢢⠈⠉⡇⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⡠⠴⠄⡁⠀⠀⢀⠀⠀⠀⡠⠀⢸⠀⢆⠸⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣂⣤⠖⠁⠀⠀⠀⠁⠀⠀⠶⠀⠊⠀⠈⠢⡁⠈⠂⠀⠀⠸⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣘⠁⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⣀⠀⢀⠀⢀⢱⣀⣀⣀⠀⠁⡀⠁
⠀⠀⠀⠀⠀⠀⣠⠃⢃⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⡀⠀⠀⠀⢔⠛⣻⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⡴⠁⠀⢸⠀⠀⠀⡐⠸⠀⠀⢀⡝⢶⣄⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢒⣒⠒⠒⠒⠀⠀⠀⠀⠒⡒⠞⠋⠉
⠀⠀⠀⠀⡜⠀⠀⠀⠸⠀⠀⡐⠀⡀⡧⠀⠀⠑⣄⠙⠿⣦⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠃⡉⠛⡋⠉⢩⡍⠀⠡
⠀⠀⠀⠰⠁⠀⠀⠀⠀⡆⠰⠁⠀⡇⠱⣄⠐⢄⣀⠁⠀⠈⠛⢿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢒⠃⠠⢾⡇⠀⠀⢣
⠀⠀⠀⡧⠀⠀⠀⠀⠀⠱⠃⠀⠸⠁⠀⣨⠗⠸⠁⢀⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢦⣀⠠⡄⠀⠀⠀⠄
⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⡠⠖⣿⠐⢉⡤⠛⣧⠀⠀⠙⠳⣦⡀⠀⠉⠒⢄⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠘
⠀⠀⣜⣀⡀⠤⠤⢴⠖⠉⠀⠘⠁⡠⠋⠀⠰⠁⠀⠸⡀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⢤⢚⠀⠀⠀⢀⠆
⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠁⠈⠁⠀⠈
⠀⠀⠀⠀⠈⠒⠦⠠⠤⠒⠉

⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⢴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⣷⣮⣽⣛⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢦⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⡿⠿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠿⣿⣶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⢀⡠⠀⠒⠀⠀⠀⢀⠘⠟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⠁
⠀⢠⠃⡀⠀⠀⣀⠀⢀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠁⠀⠀⠀⠀⠀⠄⡀⠀⠠⡀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠔⠀⠒⡺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠒⠒⠀⠊
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⣷⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠋⠀⠀⠀⠈⠙⢿⣿⣿⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠄⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠂⠀⠀⠠⠀⣈⣻⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣀⣠⣤⣶⠂⠀⠀⠀⢀⢀⡀⣀⡀⢀⠀⡀⠀⣀⣀⠀⡀⢀⢀⡀⢀⠀⠉⠀⠀⠀⡀⠐⠁⠀⡠⠂
⠀⠀⡔⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⡼⢺⠠⣟⠢⠿⠇⢧⡤⢸⠇⢸⠓⠾⠀⢹⠇⠀⠀⠀⠀⠁⠀⠀⠀⠊
⠀⠀⠁⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣿⣿⣷⣧⣿⣿⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠁⢠⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠄⠀⠀⠀⠛⠛⠛
⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠙⠛⠛⠛⠁⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⢻⣦⣄⣀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠘⠃
⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⣿⣿⠿⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠉⠛⢳⠆
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⣄
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⣎⢧⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⣀⣼⣾⣇⠀⠀⡾⠁⠀⣠⠞⢳⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣿⣿⣿⣿⣿⡆⠐⣇⢀⡜⠁⠀⣸⠃
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⣿⣿⣿⣿⠇⠀⠈⠻⠤⠤⠖⠃⠀⠀⣀⣀
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠉⢳⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠞⠉⠉⣹⢦⡀⠀⠀⠀⠀⢀⣤⣾⣿⣦⠀⣠⠴⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡞⠀⠀⣏⠀⢠⠞⠁⠈⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣸⠁⠀⠀⠙⢶⣋⣀⣠⠼⠁⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣴⣇⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠙⠒⠶⠒⠋⠀⠀⢰⡇⠀⠀⠙⠒⠶⠒⠋⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠲⣤⣀⣠⡴⠂⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⣄⣀⣀⣀⣀⣀⣀⣤⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣾⣧⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠀⠀⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⠀⠀⣴⣶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣀⣀⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠃⢀⣾⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡀⢀⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠶⠦⣞⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⢠⣦⠀⠀⣀⡤⠴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠱⣿⣶⣶⣶⣿⣿⣿⣿⡿⢫⣿⣿⣿⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣿⡿⠋⢠⣿⣿⣿⣿⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠻⠿⡿⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠳⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⡆⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⡾⠛⣿⡀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⣴⡟⠻⣧
⠀⠀⠀⠘⢷⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣴⠟



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

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡠⠀⢤⣀
⠀⠀⠠⠖⠐⢶⡄⠈⡂⠀⠀⠇⠀⠠⠀⠀⢠⣅⣈⣀⣹⠐⠀⢤⠊⠰⢸⡆⠄⠤⢴⡆⠀⠈⠄⢠⡉⣱⣶⣧⡀⣦
⠀⠀⢀⣄⠀⠤⠀⠔⠁⠀⢸⠀⠆⠂⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠁⠤⢰⠀⢀⠀⠀⠄⠄⠈⠀⠘⢁⣨⠋⢩⡴⠃
⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⢁⡀⠀⠀⠀⠤⠭⠭⠭⠥⠀⠀⠢⠴⠀⠀⠀⠀⢀⢀⡴⠦⠐⠀⠾⠀⢈⣩⠀⣄⡀
⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⢀⠈⢉⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠁⠀⠀⠄⠠⠠⠟⣛⣼⡟⢁
⠀⠀⠀⠀⠑⠁⠒⠀⠈⠀⠠⠀⠃⠄⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⢤⢄⠀⠀⠀⠀⠀⠀⠀⡀⠘⢿⣿⣦⠄
⠀⠀⡀⠀⣀⡀⠀⠢⠀⠀⠘⠒⠒⠃⠀⡀⠀⠀⠀⠐⠀⠀⠀⢀⠄⠀⠐⣃⠏⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠉⠀⢄
⠀⠀⠁⢀⠉⠁⠄⠀⠀⡠⢖⣱⣞⡭⠙⠀⠑⠀⢀⣀⠠⠀⣀⠀⠀⠀⠮⢄⠐⣄⠀⠀⠀⠂⠀⠀⠈⠐⡄⠀⠀⡰
⠀⠀⠀⠀⠀⠈⠂⠀⠨⠑⠒⠀⠐⠀⠀⡍⠁⠀⠈⢹⣿⣿⡏⠀⠠⡀⠀⠀⠁⠀⠀⡠⡀⡀⠀⠸⢀⢠⣄⠉⠉
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⢄⣀⣀⢠⢀⠀⠁⠴⠀⠀⠉⠉⠀⠐⠄⣀⣤⣴⠇⠀⠀⠑⠁⢀⠀⠀⠀⠉⠊⠀⢀
⠀⠠⢄⡠⠀⠀⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⢁⡀⠈⠀⠈⠋⠀⠀⠀⢉⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠁⣀⡅⠀⠀⠀⢀⢡⡾⠀⠀⠂⣀⠀⠀⠀⠠⡰⢆⡀⢐⡌⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠁⠍⠀⠀⠁
⠀⠀⠀⢀⠂⠀⠀⠠⠂⠆⠈⠀⠱⣀⠀⠀⠀⢐⣲⠂⠈⠉⠀⠈⠀⠀⠀⠀⡀⠄⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤
⠀⠀⠐⠀⠀⠂⠀⠀⠁⠀⠀⠀⠠⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⠄⠤⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⢦⠀⠄⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠂⠀⠀⠀⠀⢀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡔⢀⡀⠀⠀⡼⠀⠂⠀⠀⢀⠍⢻⠆
⠀⠀⢸⣿⣶⣤⣤⠀⠀⠟⠀⠀⢀⠔⠊⢶⡴⠋⠝⠀⠀⠀⣷⣦⣤⣤⣴⣾⠀⠀⠀⠪⠙⠆⠃⣰⠃⠀⠀⠀⠀⠀⠁⡞
⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠈⡁⢀⡤⠞⠁⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠈⠁⠀⠀⢀⣣⡀⣠⠟⠉⠀⠀⠀⠘⠀⢃⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣮⣀⡀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢉⡉⠉⠀⠀⠀⠀⢹⣦⡀⠀⠈⠉⠛⠲⠦⣶⡿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡗⠀⣄⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠀⠀⠐
⠀⠀⠀⠠⠀⠀⠀⠀⠰⣿⣦⡀⠀⠀⠀⠀⠛⠃⠀⠀⠁⠄⠀⠀⠜⠀⠀⠀⠩⠵⠀⠀⠀⠻⠀⠰⠀⢢⠀⠀⠀⣴⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⣾⣿⢿⡆⠀⢠⠀⡈
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡂⠀⢰⢦⠴⣶⠙⢰⠳⡄⡏⡔⠱⢰⡆⠐⠀⠀⠀⢠⡄⠀⡠⠀⢐⡇⠀⠀⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⡞⠂⢸⠀⡇⡇⡇⢠⣸⢼⠀⠀⠀⠀⠰⠀⠁⠽⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠒⠚⠋⠐⠓⠘⠊⠋⠈⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣖⣲⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠁⠀⠐⠈⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢄⡴⠊⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⡜⡖⠤⠤⢤
⠀⠀⡀⠀⢀⡋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠁⠀⠀⠀⠀⠀⠈⠑⢒⣂⠛
⠀⠀⠈⠒⢭⡰⢂⡌⠰⣀⠄⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⡤⢈⡥⠊⠁
⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠐⠅⠈⠋⠁⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠰⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⠈⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⡖⣋⠐⠄
⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⣰⣄⠙⢦⠀⠀⠀⠀⠀⠱⣀⠠⠌⣿⠻⠗
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⡀⠀⠀⠶⠆⠀⠀⠀⠈⠙⠓⠀⡐⠀⠀⠀⠀⠀⠘⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⣰⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣲⣾⣽⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⣀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠑⢴⠖⠠⢶⣞⢻⡟⠳⡶⠶⠶⠷⠖⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡆⠳⠔⠂⠉⠹⠷⠶⠶⠆⡭⣍⡒⠢⣄⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠉⠀⠀⠀⠠⡼⠃
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠈⠁⣀
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡀⡀⡃
⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣧⣅⡴⠃
⠀⠀⠀⠀⠀⠀⣸⠀⢰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿
⠀⠀⠀⢐⣤⣺⣿⣴⣦⣾⣿⣿⣷⣦⠀⣀⠄⣠⡀⠀⢶⠿⠁⡸
⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣰⣿⣿⡟⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠯⠭⠏⠋⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠒⠋⠉⠁⠀⠀⠀⠈⠳⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠉⡉⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣄⣀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠱⡄⠀⠀⠀⠣⠀⠀⢣⠀⠀⠁⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⢈⠉⠙⠢⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠘⣆⠀⠀⢳⡀⠸⠄⠀⠀⠀⢃⠀⠀⠇⠀⠀⡠⠂⠀⠀⠀⠈⢆
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠘⣆⠀⠈⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠊⠀⠀⠀⠘⡄
⠀⠀⠀⢨⢧⡀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡆⠀⠘⠀⠀⢠⠒⠀⢸⣄⣀⡤⢦⣀⣀⠠⠐⠀⠀⠀⠀⡇
⠀⠀⢀⡏⠀⠈⠒⠲⢄⡀⠙⠆⠀⠀⠀⠀⠀⢻⠀⠀⠃⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢀⡧⠒⠒⠒⠂⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⡀⠉⠲⢄⣀⣠⡤⠖⠒⠚⠦⠤⠤⠤⠤⠶⠒⠚⠉⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⡠⠊
⠀⠀⠘⣄⠀⠀⠀⠀⣀⠀⢀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉
⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠠⠶⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠠⠂⠉⠉⠑⣄⣠⠤⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠢⣄⣀⡀⠀⠀⠀⢐⣀⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠈⠉⣹⣋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡟⣀⠔⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢐⣀⣩⣤⠤⠄⣤⣀⣀⠀⠠⡄⠤⢤⣀⡙⢅⠢⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣀⡤⠖⠋⢩⠏⠁⠀⠀⠀⠀⠀⠈⠉⠒⢼⡳⡄⠈⠉⠓⠧⠀⢠⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢨⠁⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⡄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠙⡆⠀⠀⠀⠀⠀⢸⠋⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠖⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣄⡀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠑⠲⠦⠴⠖⠋⡁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠄⣤⣠⡤⡤⠼⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠒⡈⠃⠛⢐⠒⠂⠀⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡀⠀⠀⠻⣄⡔⠁⠀⠀⠀⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⣀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⣇⣀⡴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡔⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡉⠂⢀⠠⠒⠈⠁⡴⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠊⢁⡠⠤⠗
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡀⡄⠤⠴⠒⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢊⣡⢄
⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⡵⠊
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠤⠤⠤⠤⠶⠶⠒⢻⡉
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣢⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠉⠙⠒⠐⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣠⣤⠤⠼⢰⣤⣤⡤⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⣸⠀⢸⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠁⠀⠀⠘⠢⠚⠡⠀⠀⡘⠀⠙⠋⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⣀⠀⠀⣀⣀⣀⣤⠴⠖⠒⠃⠀⠑⠦⣤⣀⡀⠀⠀⠀⠈⢢

⠀⠀⠀⠉⠀⠀⠀⣤⣷⡄⡀⠁⢀⠀⠀⠂⡉⠸⠀⠀⠀⢹⢀⡀⠀⠉⠀⡀⣌⣠⢁⣸⡟⢅⠀⠀⠀⠀⠀⠉⠒⠁⠀⠀⠠⠀⠀⡁⠂
⠀⠀⠀⠄⠄⠀⠀⠙⠉⢠⣽⠄⠈⡇⠛⠚⠋⠀⠀⠀⠀⠉⠘⠃⠀⠀⠀⡇⣉⠘⠻⠇⡇⡀⠀⠀⠀⠀⠀⠼⠛⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠃⣠⠟⠋⡄⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⡀⠀⠀⣀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⡄⠘⠃⠄⠄⠀⠒⠁⠀⠖⠀⢀⠀⠈⠀⠤⠀⠀⣶⠀⠀⠐⢢⡄⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠠⠀⠀⠈⠀⠞⠀⠉⠓⠀⠀⠛⠀⠀⠀⠀⠀⢀⡀⡄⠀⠀⠁⠃⢀⢰⡾⣷⡈⣀⠠⠐⠀⠀⠿⠁⠀⠀⠀⠁⠀⠀⠄⠀⠀⠆
⠀⠀⠀⠀⡀⠠⠀⠀⣤⣀⠀⠀⢀⠀⡀⠁⣄⠀⠄⠀⢠⠀⡄⠀⠀⠀⠙⠛⠀⠉⠉⣉⠀⠀⢰⡆⠀⢀⡀⠐
⠀⠀⠀⠀⠀⢄⠀⣄⠀⣽⠆⠁⠨⡀⢿⡀⠄⣄⠐⠀⠀⠾⠀⠀⠀⠀⠀⠁⣀⡋⢠⣿⣶⠀⠘⣷⢀⣿⣹⣧⢸⡿⣿⡆⠀⣀⡀⡀
⠀⠀⠀⠀⢀⡄⠀⣿⠀⣤⣀⣤⡰⠌⣆⡄⠀⠰⣆⣀⠠⠀⡀⠀⢤⣤⡀⢸⣿⣿⡆⢹⣷⣿⠀⣿⡆⢿⣿⣿⢸⣿⣿⣧⠀⠉⠉⠉
⠀⠀⠀⠀⠃⠄⡀⢻⣧⣿⣿⣿⣿⡜⣿⣿⣦⢸⣯⣽⣧⠀⢿⡀⣿⡟⣿⣿⣿⣿⣿⡾⠿⠛⢿⣟⣵⣤⣍⡁⠈⠉⠈⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠠⠄⠀⠘⠉⢿⠟⠿⠻⢿⣿⣾⣿⡏⣿⣿⣟⠀⢸⣧⠻⠟⠿⠇⠉⠉⠙⢁⣶⣿⡄⠌⣁⣠⣈⡿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣀⠀⠀⠉⠉⠉⢑⠿⢿⠟⠁⠘⠟⠀⠀⢀⠀⠀⠀⠀⢄⠁⠀⠱⡀⠉⠉⠙⠋⠀⠀⠄⠀⠀⠁⠤⠀⡂
⠀⠀⠀⠀⠈⠀⠀⠀⠰⠘⠓⠛⠀⠀⠀⠀⠆⠀⠒⠀⢀⠀⠐⢀⠀⣬⠸⢛⣀⠀⠃⣀⡀⠀⠀⠀⠀⠀⠀⠶⣐⡀⠤⠈⢲⠀⠀⠀⠃
⠀⠀⠀⠄⠀⠀⠂⡈⠀⣹⡇⠀⠀⠉⠀⠀⠀⢀⠀⠂⠀⠙⢀⣄⢠⠜⠆⠻⠋⠀⠀⣹⠳⠄⣀⣀⡀⡄⠀⠤⢭⠁⠀⠁⠈
⠀⠀⠀⠀⣠⣦⣘⠁⠋⠉⠠⢄⡀⠀⠀⢠⠀⣀⡀⢁⡀⠀⢈⡑⡂⢠⣶⡄⠀⣴⠀⠘⠳⠐⠀⠀⠀⠀⠀⠐⠀⠈⠉⠐⠒⠋
⠀⠀⠀⠀⠁⠄⢨⠙⠠⣿⠠⠁⠀⠀⠀⠀⠲⠟⢡⡄⠐⠖⠀⠈⠑⠻⠈⠀⠀⠘⠀⠀⠀⠀⠰⠒⠃⠀⠀⠀⠀⢠⡒
⠀⠀⠀⡀⠁⠰⠓⠀⠀⠀⢀⡀⠀⠀⠠⠤⠃⠠⢉⣤⠀⠘⠃⠀⣀⡀⠀⢀⣀⡐⠀⠀⠠⠉⠀⢠⣤⠀⠀⠀⠈⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⢠⠀⠀⠀⣀⠀⢀⣸⡛⣿⣶⣧⠤⡤⠠⢬⡁⠐⡃⠀⠀⠀⠀⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀⠀⠉⠁⠀⡿⣿⡉⠉⢀⡀⠀⡀⣶⣤⡥⣤⣶⣶⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡚⠀⠐⢴⡋⠀⢈⣷⠐⣀⢀⠋⡁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠐⢄⠀⠀⠀⢫⣶⣤⡍⠀⠁⠆⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⡀⠰⠀⡂⠈⠀⠶⠶⠃⢀⠈⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠱⠆⠀⠛⠀⠀⠀⠀⠈⢢⠴⡀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠓⠀⠢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣠⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⣀⠀⠈⠀⢀⡄⢲⠎⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠓⢤⠀⠊⠓⠊⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠴⠀⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠐⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣦⣭⣴⣶⣶⣶⣶⠶⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⠀⠀⠀⣠⣶⣞⣲⣶⣄⠀⠀⣠⣴⣤⡀⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⢠⣷⣄⠀⠀⠀⠀⠀⣴⣶⡀
⠀⠀⠀⠀⠀⢠⠤⣼⣿⡶⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣟⣻⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠈⠻⠿⠿⠃
⠀⠀⠀⠀⠀⢨⣿⣿⣿⢿⣦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣀⣴⣿⠀⠀⠀⣠⣶⣄⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⣠⠖⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠀⠀⣾⣿⣿⣿⠀⠀⠸⣿⣅⣠⣿⠇⠀⠀⡇⠀⠆⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠙⠢⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣦
⠀⠀⠀⠀⣀⣴⣶⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⢲⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠉
⠀⠀⠀⠀⠀⣀⡀⠈⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⣶⣿⣿⣿⣿⣿⣷⣶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡞⢿⣿⣿⣿⣿⣿⡟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠉⠙⠛⠋⠉⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠛⠞⠿⠿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⣤⣤⣴⣿⣿⣛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⣀⣀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣰⣿⣶⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⠒⢦⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⢄⠐⠀⡀⠈⣆⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠖⠀⢀⡆⠀⠐⠒⠂⠰⠦⠂⠀⠀⢸⡶⣌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢺⣀⠐⣯⠂⠾⠋⠀⡀⠲⣀⣤⠀⠀⠀⠀⢀⣼⢡⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠈⢄⠉⢰⡞⠀⡾⢛⠛⠁⢀⣤⣤⣾⡿⠃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠰⠉⠀⠀⣷⣾⣶⣶⣿⣿⡿⠟⠁⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠁⠀⠀⠀⠀⡦⠒⡂⢶⢤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠊⠢⡀⠀⠀⠀⠀⠀⠡⠀⢓⢈⠈⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠀⣀⠔⠉⠁⠂⠀⠀⠑⣀⡀⠰⠋⠢⡘⠀⠣⠁⠀⠀⡌⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠖⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠅⠀⠢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠚⡁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⡶⠧⠶⠓⠒⠒⠛⠒⠒⠷⠴⠂⠀⠀⣀
⠀⠀⠀⠀⠀⡠⠃⠀⡇⠀⠀⠀⠜⡄⠀⠀⡲⣤⡀⠀⠺⡄⠀⠈⠀⠀⠀⠀⠁⠒⠻⢧⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⠞⠃
⠀⠀⠀⠀⡰⠁⠀⠀⢡⠀⠀⡄⠀⢱⠀⠀⠙⣈⠻⣶⣄⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⡒⢲⠶⠒⠒⡒⠂⠢
⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⡐⠀⢰⠸⡄⠀⢄⠈⠓⠀⠙⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱⠈⠢⠌⠀⣰⡇⠀⠀⢁
⠀⠀⠀⡇⠀⠀⠀⠀⠘⠴⠁⠀⠞⠀⠈⡲⢈⠏⢀⠀⠀⠀⠈⢃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢢⡀⠈⡁⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⢶⡖⠒⣋⡵⢲⠀⠀⠙⢶⣄⠀⠈⠓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠐
⠀⢀⡜⣀⣀⣀⠤⣤⠒⠉⠀⠞⢁⡔⠁⠀⡎⠀⠸⡄⠀⠈⠻⢦⣀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠠⡄⠀⠀⠈⢠⡐⠂⠀⠀⠀⡀
⠀⠀⠀⠈⡁⠀⠀⠉⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠊⠈⠑⠒
⠀⠀⠀⠀⠑⠲⠄⠤⠤⠚⠁

⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⠠⡆
⠀⠀⠀⠀⣒⡶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠑⢸⡀
⠀⠀⠀⠀⢸⢛⣱⡆⣤⡀⠀⠀⠀⠀⠀⢼⣲⡀⣀⠠⢊⣹⡆⠀⠀⠀⠀⠀⠀⢲⣿⡖⠀⠀⠀⠀⠀⠀⠀⠁⠈⣴⣶⠀⢹⡆
⠀⡀⣾⣿⡿⣻⣻⣛⡀⠀⠀⠀⠀⠀⠀⠈⠉⢣⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⢿⠀⢹⢱
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣀⠀⠀⠀⠀⠀⢀⠀⣘⢇⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⢀⠞⢿⡿⠀⠀⠀⠀⠀⠀⠋⢿⣿⠙⠃⠀⠀⠀⠀⠀⢀⡀⢞⣻⠇⠵⠒⠂⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠄⠀⢀⠀⡸⠁⢠⠞⠂⣵⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠼⠧⠿⡿⡿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⣷⠈⣀⣠
⢠⣀⣙⣁⣛⣛⣋⣀⣀⣟⣗⡀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡄⠀⠀⣈⣋⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡶⠀⠀⢀⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠠⡖⠒⠊⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠙⠃⠐⠒⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⡀
⠀⠀⣯⣤⠀⢸⢿⠉⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⣾⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣂⠀⠀⠀⠀⠀⢀⣷⣿⢦⡆
⠀⢿⣿⡟⣇⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠶⠶⠾⢀⡀⠀⠏⡇⠀⠀⠀⠀⣰⣷⣼⣿⣷⣧⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⠋⡓⠂
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠁⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠠⠤⡀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⢸⠁⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠓⠄⡀⠀⢧⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⢀⡈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⠟⠃⠀⠰⠟⠛⠛⠦⡈⠐⣄⠃⠀⠀⠂⠠⠆⠀⠀⠀⠀⠀⠞⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠽⠀⠀⠻⠥⠀⠀⠀⠀⢹⡆⠀⠀⠀⡨⢄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⢀⠁⠄⠀⢀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⢠⠀⣀⣀⣤⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢊⠄⢁⠀⠀⠸⠀⠀⠐⠿⢿⣛⣛⢻⡷⠖⠀⠀⠀⢸⣷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⠀⣠⠜⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⡠⠚⠀⢨⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⢦⣤⡤⠔⠊⠀⠀⡠⠋⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠐⣾⡇⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠙⠲⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⢊⠁⡘⠃⠀⠉⠿⠦⣤⣀⣀⢐⣐⠧⠀⢀⡴⠊⣴⠁⠀⠀⢉⢑⠖⠦⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⢋⡜⠁⠀⠀⣠⠀⠀⠀⠀⠀⠉⣉⠉⠉⠆⠁⣸⠏⠀⡆⢰⠃⠏⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⣦⡿⠀⠈⢀⡟⣀⣔⢀⡀⠀⢰⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣸⣿⢣⡇⢠⢾⠇⣿⠟⠀⠀⢠⣤⠑⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠰⠙⣧⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢡⠏⠞⢸⣠⠇⡜⠸⢻⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠀⢸⠁⠀⠀⢸⡟⠀⠃⠀⢿⠀⣠⠄⢠⣤⠀⠈⠢⠀⠠⠂⠈⣁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠐⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣸⠃⠀⠀⠀⠈⠣⡄⢰⡿⠋⡁⠀⣤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢰⢸⠇⡀⠳⡄⠘⠧⡀⠀⠀⡈
⠀⢀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠘⡷⠸⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⢀⡄⠀⡇⠀⠀⠘⠀⣰⠀⠀⠀⠀⢠⠂⠀⠁⠷⠄⢁⡀⡀
⠀⡇⠠⠀⠀⠀⠀⢀⠔⠠⠁⡤⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⢐⡅
⠀⢱⠐⠀⠀⠀⡰⠃⠀⡴⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢼⡇⠀⠀⠀⠘
⠀⠀⢣⣄⠀⠈⠀⣠⡌⠔⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠐⠢⠔⠊⠁⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠆⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣤⠀⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠛⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣻⣟
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣏⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣂⣬⣿⣿⣿⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣉⣩⡀⠁⠇⠀⠀⠀⠀⠀⠀⢀⠠⠄⢶⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⣠⣿⢿⣿⣟⠻⣿⣿⣿⣿⡇⡀⢀⣀⠤⠒⠂⡉⣹⣦⣶⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠸⣿⠃⢺⣿⣿⣇⢸⣿⣿⣷⠞⢱⡿⠀⣶⣷⣾⣿⣿⣷⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣭⣭⣿⣿⣝⡿⢧⣄⣰⠏⠀⢸⣿⣿⣿⠿⠟⠁
⠀⠀⣠⠆⠀⠀⣰⣿⣿⣿⡿⠟⠉⠉⠉⠉⠀⠀⠀⢨⣿⣿⠟
⢠⠞⠁⠀⢀⡴⢜⣿⣿⡟⠁⢀⣤⣠⣦⠀⠀⠀⠀⣾⣿⣿⡇
⡞⠀⠀⠠⠞⠁⢸⡿⢿⣇⣴⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⢹⡁
⠙⠶⠤⢤⠴⠚⠀⠀⠈⢻⣿⣿⣿⣿⣿⡀⠀⠀⣤⣿⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡄⠀⠻⣿⡷⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⣸⣿⣇⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⣿⣿⣿⣿⣿⣷⣿⠿⢿⡖⠋⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⠇⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠷⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⣀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⡛⣿⣿⣿⣿⣿⣿⣧⣤⣀⣀⣤⠴⠞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠛⠻⣿⣿⣿⣿⣿⣯⡟⠉⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⣿⠟⠋⠉⣿⣿⡄⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⡿⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⢀⡴⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⡪⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡭⢖⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠋⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣵⠿⣿⡿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⠶⣤⠶⠚⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⣿⣿⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣤⣬⣤⣴⡾⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣯⣹⣹⣏⣥⣩⣹⣯⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣭⣬⣭⣭⣽⣿⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⠿⠿⠿⠿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠈⠻⠷⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀⠀⢾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣀⡀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⣿⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢰⡋⠙⡇⢰⠋⢻⡄⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣆⠀⠙⠛⡟⢻⠛⠋⢀⣴⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣶⠿⠋⠀⠀⠹⣇⠀⠀⡇⣾⠀⢀⣾⠃⠀⠘⠻⣷⡄⠀⠀⠀⠀⠀⣰⣿⠏
⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⡇⢼⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⣿⣟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⠿⠿⠿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠈⠛⠛⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛⣛⡃⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠘⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠙⢿⣷⣶⣶⣾⡿⠿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢰⣫⡄⢷⠃⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣖⢦⠀⠀⠀⠀⠀⠀⢠⣄⡶⢶⣠⡄
⢰⡦⠭⣤⠤⠴⣤⠤⢶⡄⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣵⣿⠂⠀⠀⠀⠀⣸⢷⣭⣽⣭⣭⡾⢇⡀
⢸⡇⠀⠈⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⢸⣿⢘⣠⣤⣤⣄⣘⢸⣿⡀⠀⠀⠀⠀⠀⠀⣸⡿⠮⠥⠞⠋⠀⠀⠀⠀⠀⣻⠿⣶⣾⣶⣶⠾⣟⡁
⢸⣿⣿⣿⡿⢿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣈⠉⠛⠛⠋⢩⢸⣿⠃⠀⠀⠀⠀⣠⡾⠋⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠙⢿⢿⣶⣶⡿⠿⠋
⠚⠛⠛⠛⠃⠘⠛⠛⠛⠃⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡿⠃
⠀⣾⣿⢀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢡⣿⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⡙⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⡀
⠀⠀⢼⣿⣯⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠔⠉⠁⠞⢷⡟⡔⠉⠉⢢⠀⠀⠀⠀⠠⠞⠛⣻⣦⣜⠿⠃
⠀⣀⣀⣙⣿⣟⣀⣹⣇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⠄⠈⠛⠣⣀⣀⠜⠀⠀⠀⠠⢄⣀⣾⡭⠽⠞⠚⠒⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⡆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⡶⠶⡆⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⣥⣬⡲⡀
⢰⣷⣶⣶⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡧⠍⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡇⣴⠇⡇⠀⠀⠀⠀⠀⠀⢠⠃⡈⠿⠿⢃⠄⣄
⢻⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡤⣤⣤⡇⠀⠀⠀⠀⠀⠀⣺⣄⣈⡢⣔⣋⣠⣼
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉

⠀⠀⠰⢶⠝⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣄
⠀⣴⣾⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢀⢄⣤⡰⡶⣀⣄⢤⠀⠀⠀⠀⠀⠀⢿⣿⡷⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭
⠀⣯⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣧⠸⠿⣣⣦⡻⠿⢀⡇⠀⠀⠀⠀⠀⠠⢤⠔⠒⠔⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⢻⣜⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⣿⣶⣦⣤⣶⣾⡿⠁⠀⠀⠀⠀⠸⣷⣶⣶⣿⣾⣷⣾⣾⠂⠀⠀⠀⠀⠀⠀⢹⣖⣖⡿
⠀⠀⠙⠻⠟⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣶⣾⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣰⢄⣠⠤⣶⡯⠴⠖⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠉⠉⠛⢻⣦⡀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠠⣴⡮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣆⠀⠀⠀⠈⠙⢿⣿⢉⠀⠉⠉⠽⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠚⣹⢷⣦⠞⣒⠛⠉⠓⠶⠤⣄⢸⠆⠀⠀⠀⠀⠀⠀⢈⡿⣭⣵⠀⢮⣙⠀⢹⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠩⠋⠀⣖⣢⠀⠀⡤⢌⣼⣟⠀⠀⠀⠀⠀⣀⣀⠐⢧⢸⣤⡴⢠⡖⠀⢀⠭⠃
⠀⠀⠀⠀⠀⠀⢀⠔⠉⢂⠀⣍⣁⡀⠀⣟⣊⠷⠋⠀⢨⢓⣞⡵⠦⡐⡕⣾⢤⣙⢷⠟⢀⡠⡆⠐⠠⡀
⠀⠀⠀⠀⠀⡠⠞⢀⠀⠀⠑⡬⠿⠟⡯⠑⢋⠀⠀⢠⡃⠋⠀⠀⣧⠉⠿⠋⢻⢽⠉⠻⠯⠒⠅⠀⠀⠈⡆
⠀⠀⠀⠀⠰⠀⠀⢼⠀⠀⠀⢱⠀⠀⠀⠰⠉⠁⠀⣤⠀⢀⣄⢀⣀⣀⠀⠖⠋⠘⠤⠤⠐⠚⠋⠑⢢⢠⠃
⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⢀⣈⠄⠀⣜⠀⠦⡄⠠⠤⠤⠨⣽⡿⠏⢏⠉⢓⠦⣤⣤⣤⣤⠤⠔⠒⠛⠁⡄
⠀⠀⠀⢸⡀⠀⠀⠀⢻⡁⠀⢩⡴⠚⠛⠀⡎⢀⡀⣀⣴⡎⢀⣌⣳⠀⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢰⠉⢲⡴⠻⡛⠻⠖⠾⣧⣤⣄⣀⣁⡾⢧⣈⠉⠀⠚⢿⣿⣿⡿⠿⢾⡁⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⢸⣤⠞⢀⠞⠀⠉⠐⠂⠠⠀⣉⣉⠛⣿⢿⠟⠛⠋⠉⠉⠁⠀⠀⠀⣾⠏⠉⠉⣉⡉⠉⠁⠀⠀⢰
⠀⠀⢠⡞⠁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢣⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠹⠀⠀⠀⠀⠸
⢀⡴⠋⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡰⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢠⣤⣄⣀⡄⠀⠀⠀⢀⡠
⢺⣥⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⡡⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⡆⢺⡀⠈⢹⣉⣹⡄
⠀⠑⠊⣽⣲⢦⣄⡀⠀⠀⣠⡾⣣⠎⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠛⠚⣙⣆⣴⣶⣤⠧
⠀⠀⠀⠷⣨⠭⢿⣿⠿⣾⣋⡔⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⢻⡋⠀⠉⢉⣁⣤⠇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠈⠑⠢⢄⡀⠀⠀⠁⠀⠀⠀⣀⠠⠤⠐⠒⠉⠁⠀⠀⠈⠛⠛⠛⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠐⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠠⠀⠀⣈⣡⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢌⡂⠁⠀⠀⠀⠈⠃⢑⡺⢀⠄⠓⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠄⠀⠨⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠃⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠐⠚⠃⠰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⢙⠿⠗⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠘⠤⠒⠉⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡑⣀⡀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⡒⠀⠂⣉⢀⠩⠛⢢⠀⠀⠐⠤⠀⠘⠣⡄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⢶⣦⣌⣍⣀⣀⡀⠠⠒⠁⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉⠙⠉⠉⢉⡍⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⡀⢀⠤⡘⣉⢰⠆⢠⡄⡀⣐⡂⠀⠀⠀⡀⠀⠀⠀⢠⣲⠦⣤⡄⠤⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣤⣈⠙⢳⡞⠀⣀⣿⠁⠈⢸⢙⡿⠈⣿⡀⣀⡼⠀⠀⠀⣦⣀⣤⣅⠀⠀⣀⣤⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠶⠦⢤⣤⣤⣤⣬⣭⡥⡶⠞⠋⠉⠀⠀⠀⠀⠁⠋⠀⣀⠀⠀⠠⠄⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⠌⢶⠱⡶⠑⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣑⣛⣒⠳⠶⢿⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠠⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠀⠀⠠⠠⠤⠤⠄⠤⠤⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠂⠈⠌⡁⠀⠰⠀⠀⠐⠋⠉⠈⠀⠀⡤⣀⣄⡀⢔⡂⠚⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠠⠀⠋⠀⠀⠀⢄⠀⠘⠛⠀⢀⠀⣐⡷⠘⠋⠀⠀⠄⣈⣄⡈⠉⠁⠀⡀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠲⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⠀⡀⠌⣈⢡⢶⠅⠘⠟⠛⠁⠘⢶⠂⠑⠉⠀⠖
⠀⠀⠀⠀⣴⡄⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢰⠈⢋⣁⣀⣰⣚⡉⠛⠁⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠠⠀⠁⠀⠁⠀⠐⠾⢃⡁⠒⠃⠀⢄⠀⠀⠀⠀⠀⠊⠁⠀⠈⠁⠀⠀⠀⢉⠀⢀⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⠠⠁⠐⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠈⠁⠀⠀⠈⠀⠠⢄⠄⠀⠐⠁⢈
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢅⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡏⠉⠉⠛⠛⠲⠶⠶⠶⣶⣤⣤⣤⣤⣤⠄⠀⠰⠀⠦⠀⠐⠶⢶⣶⣶⠒⠒⠲⣶⣶⣶⢶⣶⣶⣶⣶⣴⣤⣦⡴⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠤⡁⣀⠀⣈⣩⣨⡶⠿⠛⠛⠉⠏⠀⠀⠠⠀⠀⠀⢚⠁⢘⣄⣄⣀⢠⣌⡇⢀⣀⢠⣿⣍⡉⠉⠉⠉⠈⠉⠉⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠉⢛⠙⠀⠀⠁⠐⣤⣤⣄⠀⠆⠐⠙⢄⣩⡤⢯⡟⢢⠉⣉⣉⣉⣿⣿⣓⠒⠒⠤⠤⠤⠤⠄⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⣴⡧⡖⠀⠂⠃⠀⡀⠀⠰⠚⢿⣿⣿⡷⠊⠀⡀⠀⣤⣬⣏⣆⣩⣈⣁⡀⠈⠉⠉⠉⠁⠉⠀⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠈⠀⢁⠉⠀⠀⠠⣴⣿⣈⠁⠐⠊⠥⢷⢳⡞⢛⣻⣿⣿⣿⣧⣾⣇⣀⣼⣆⣭⣏⣛⣛⣻⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⡀⠀⠈⠁⠀⠠⢟⣳⣌⣹⡿⠏⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠀⠉⠉⠉⠀⠀⣀⠵⢿⠫⠹⣟⣿⢿⠶⠒⠂⠀⠐⢉⠃
⠀⠐⠿⠏⠰⡀⠀⠢⣈⡇⢫⣫⠀⠀⠁⢀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢺⡄⠀⠃⠀⣞⠿⡉⠀⠀⠀⠀⢀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠻⢣⠍⠏⠐⠀⢀⣰⣶⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠏⠡⡶⠅⠐⠀⠄⠱⢂⡀⡀⠀⠐⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⢁⣀⡅⠀⣯⢹⠻⡟⢦⡀⠀⠀⠀⠀⡠⠎⠁⠀⠉⣒⣀⣀⡀⠲⡆⠀⢨⠶⠤⠟⢀⣘⣚⡳⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠛⠿⠷⠜⠁⠀⠀⠆⠑⢄⠀⠀⠀⢴⠄⠑⠋⠀⠀⠀⠨⣉⠀⠀⠀⠀⠀⠀⠀⠀⢉⠙⠂⠀⠀⠠⠁
⠀⠀⠀⠀⠄⠀⠀⣰⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⢔⣤⣶⣶⣶⣶⠲⠴⠖⠒⢲⢶⣦⣤⣠⡄⠤⠐⠠⡆⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠐⠈
⠀⠀⠀⠀⠀⠈⠒⠼⢤⣀⡀⣀⣀⡠⣄⠀⠀⣀⠉⢚⣛⣓⠛⠒⠒⠒⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⣰⢀⣀⡀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠒⠒⠛⠁⠁⠒⠒⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⣠⣤⣼⣿⣿⣿⠶⠀⠀⠀⠐⠁⠐⡄⠀⠀⠀⡠⠤⠶⢶⣦⣀⡀⠀⠀⠀⠶⡄⠀⠀⠀⠀⠠⠞⠀⠠⠞⠁⠀⠴⠋⠀⠄
⠀⠀⠀⠉⠉⠋⡁⠰⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠄⠀⢾⣼⣿⡇⠀⠀⠀⠀⣿⠀⠀⢰⣦⣤⣴⣦⣀⣰⣤⣀⣠⣤⣄⣒⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⠃⠀⠀⠰⠀⡿⠀⠀⠘⠛⠈⠡⠜⠆⠛⠰⠼⠧⠴⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⠟⠛⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠇⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⠀⠀⢀⡀⠀⣠⡆⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⠀⠀⠀⠈⠁⠀⠀⢀⣴⡿⠛⠛⠻⠛⠢⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣽⣶⣿⣿⠻⠿⠇⠀⠉⠉⠀⠁
⠀⠀⢀⠐⠄⠀⠀⠀⠀⠂⢈⠄⠀⠀⠀⢀⣽⠏⠁⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⡄⠀⡀⠀⠘
⠀⠀⣄⠀⠁⠒⠠⠁⠀⠖⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠷⠀⠀⠀⠀⠇⠀⠀⢰⠘⡆
⠀⠀⠓⠺⢿⣈⡢⠞⠁⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⠀⠀⠔⠀⠀⠀⠀⠂⠀⠀⣠⠁⢀⠘⠞⢣⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⡷⠈⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⡅⠀⢠⡇⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⣻⣿⣿⣟⣩⠁⠀⠀⠈⢂⠀⠀⠸⣿⣏⠰⡇⠀⠀⠀⠀⠀⢁⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⡀⠀⠓⠂⠀⠀⠀⢀⡆⠀⠀⠀⠀⠿⠉⠉⠉⢙⠉⠹⠀⠀⠠⠀⠀⢻⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠇⡁⣹
⠀⠀⠀⠀⠈⢡⡆⢈⠜⢧⠠⠤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣄⣠⣄⠀⢀⠀⠀⣦⠀⠀⢆⠃⣿
⠀⠀⠀⠀⠀⠉⡤⠋⠀⠈⡆⠒⡀⠀⠀⠀⠀⢀⣠⠄⡐⢻⠷⠢⠀⠀⠀⠀⠀⠀⣸⠀⠀⡟⠛⠋⠉⠻⠀⠈⠙⠂⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢰⣮⣿⣷⣽⣭⣦⣴⣿⡤⠀⠆⠀⣠⠋⠀⠀⠷⠂⠀⠀⠀⠀⢀⡀⠀⠀⣀⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣐⠶⢿⣯⣝⡳⣆⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠰⠃⠀⠚⠀⠀⠘⠆⠀⠀⠀⠄⠀⣹⠀⢀⡳⢶⣶⡼⢷
⠀⠀⠀⠀⣀⠀⠀⠛⢷⣦⠋⠟⠁⠛⠁⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⠀⣴⠏⠀⠀⠉⠀⠋⠑⠚
⠀⠀⡆⠀⢹⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⢀⣤⢴⡶⠀⠀⠀⠀⠜⠀⣼⠇⠀⠀⠀⢀⡀
⠀⠀⡇⠀⠈⡱⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠜⠋⠀⠀⠀⠀⠀⠤⣴⠏⠀⠀⠀⠠⢸⢃⠐
⠀⢀⡇⠀⠀⡷⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠄⠀⠀⠀⢀⡴⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⡀⡀⠀⠀⠀⡘⠘⠀⣇⡁⡐
⠀⠸⡇⠀⢀⡘⠂⡁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠁⠁⠈⢹⡷⠁
⠀⠀⡇⠀⢠⠟⠀⠀⠀⠀⣄⡀⣔⡾⢤⠗⣤⢠⣆⠤⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣶⡄⠄⠆⢂⢐⡐⡀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⣾⡾⠂⠀⠀⢃⣀⠐⣛⣂⢛⡚⣘⣋⢳⠞⠋⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸⡟⠒⠐⠂⠂⠃⠐⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⢠⣄⠀⠀⢸⣿⡇⢾⣿⣦⣰⣿⡿⠀⠀⡟⢻
⠀⠀⠀⢀⠀⡘⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⣤⢄⢿⣿⠇⠀⢸⣿⡇⠀⢹⣿⣿⡟⠁⠀⠀⡇⢸
⠀⠀⢀⣾⡞⠀⠀⠀⢀⠔⢲⠀⢀⠠⠶⠂⠀⠀⠸⢿⣿⣶⡄⠀⠀⢸⣿⡇⣰⣿⡿⢿⣿⣦⠀⠀⣧⣼
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠀⠀⠸⢿⠇⠘⠛⠀⠀⠙⠋⠀⠀⢿⡿
⠀⠀⠁⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⣔⣂⠁⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠆⢠⡀⢸⡇
⠀⠀⢠⠤⠤⡄⠀⠁⠀⠀⠀⠀⠀⣴⡾⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠔⠀⢥⡴⢸⠁
⠀⠀⠈⠈⠉⠁⠀⢈⠓⠶⠚⠇⢸⠋⣠⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠐⢃⠠⣴⠌⠁
⠀⠀⠐⠢⠤⠀⡆⠀⠀⢱⣶⠂⠀⠀⠫⠊⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣤⠲⠆⢽⠆⠁
⠀⢀⡅⠀⠀⠀⡇⢈⠀⢘⠦⠀⠀⠀⢰⣶⣤⣤⣤⣴⣾⠀⠠⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⢀⡀⠒⡀
⠀⠈⠢⠶⠒⠚⠁⠀⠒⠺⠕⠀⠀⠀⠈⠻⣿⣿⣿⠿⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳
⠀⡀⠐⢀⠀⡄⠀⠀⠀⠰⢄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠃⠀⠠⠒
⠀⠻⠶⢎⣠⡼⠀⠀⠙⠒⢫⡄⠐⠊⠀⠀⠀⠀⠀⠐⢄⠀⠶⣦⣶⣄⡊⠀⠀⠀⠂⠀⠉⠁⢀⠀⠁⠘⢸
⠀⠰⣤⣤⠁⡀⠀⢀⠀⠘⢒⠀⢤⠄⠀⢀⡘⠛⢀⣿⠈⠀⠐⣫⣿⣿⡿⠀⠀⠀⣀⣤⣴⣦⠀⠀⠀⠅⡇
⠀⠀⠀⠀⠀⠙⠛⠫⣄⠠⠿⡗⠀⠀⠀⠘⠻⠖⠈⠁⢀⡀⠀⠀⠚⠁⠀⠀⠻⣿⣏⣹⣿⡟⠀⠀⢨⣽⠇
⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⡄⠈⠲⣄⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⢰⣶⠆⠲⠿⠟⠙⠙⣷⣶⡄⣼⣿
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⢢⡖⠘⢷⣄⠀⡾⠟⠉⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡀⠀⠀⠈⠀⠇⢠⢐⡴⠮⠀⠋⢸⡏⠉⢹⠀⠀⡠⣿⣄⠀⠀⠸⣼⠃
⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⣏⣿⣹⡇⠀⠀⠰⠬⣕⠻⠦⣵⡚⠣⢾⡇⠀⢸⠀⢰⣵⣿⣿⠀⠀⢠⡏
⠀⠑⠒⠒⠙⠿⠶⠿⠃⠈⠁⠉⠈⠁⠀⠀⠀⠐⠊⠠⠄⠒⠀⠤⠘⠿⠿⠟⠀⠀⠙⠛⠋⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠳⡄⠙⢷⣜⢿⣿⡿⣰⠏⢠⠏⣰⠃⣰⠃⢠⠆⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆⢠⣝⠁⠘⢦⡈⢢⡀⠹⣦⡛⣡⠏⢠⠏⣰⠃⣰⠃⣰⠃⣘⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢈⠉⡐⡌⢇⠳⣄⠙⢦⠙⢦⡈⠻⡟⢠⠋⣰⠃⣰⠃⣰⠃⠸⠿⢣⠴⣛⣉⣙⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠀⠠⠠⡀⠀⠃⠈⠳⡈⠳⡄⠱⣄⠉⣀⡀⠋⢀⣁⠀⣁⡀⠀⠀⠀⠰⣄⠈⠩⠶⠦⣝⢧⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠐⢶⣶⣶⣶⣶⠶⠦⠤⠄⠙⢦⡘⠂⠀⠀⢻⡇⠀⠙⡿⠀⠹⡏⢀⡄⠙⢦⡈⠓⢤⡈⠓⣼⢸⡇⡇⣰⣦
⠀⠀⠀⠀⠀⣶⡆⢀⣶⠀⠀⠉⠻⠟⠷⠶⠶⠶⢂⣤⣴⣧⣤⣴⣤⣶⣤⣤⣤⣴⣦⡀⠀⠈⠓⠒⣦⠙⠢⣄⠙⢦⣀⠉⠼⠓⠋⣁⣀⠀⣀
⠀⠀⠀⠠⠤⠄⠰⠿⠛⠃⢀⠤⢄⠀⠾⠆⢀⣴⢟⣽⢟⣼⣿⣷⡌⡀⢏⡉⢉⢀⣸⣇⠀⠀⠀⠀⠳⠖⢳⠈⠳⣄⡈⠳⣄⠀⠈⢆⠒⠑⢸
⠀⠀⢠⢖⡭⣙⠲⣄⢳⡀⢁⠫⠅⡆⢀⣀⣼⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣷⣴⡾⠻⣶⣤⣄⡘⠶⢲⡄⠙⠦⠈⢁⠈⠒⠪⠆⣈⡀
⠀⠀⡇⢿⣀⡇⢳⢸⡆⠇⠁⠄⠉⠀⣿⠻⣟⠀⣿⢛⣿⣿⡿⣟⡆⢹⡴⡿⢰⠟⣩⢲⡶⣠⣛⣶⣶⣭⣿⠀⠘⢧⠴⠀⣴⣿⡀⠺⠆⠘⢿⣧
⠀⠀⠘⠲⠤⠴⢃⠜⢰⣦⣤⠀⢀⠀⠙⠻⢿⡆⢻⡏⢿⠈⢣⣸⡇⣼⠀⡇⠘⣘⡇⠟⢠⢏⠈⣩⡛⣿⠷⣦⠀⠀⡀⠈⠉⠉⡠⢒⡭⢕⡲⣌
⠀⠸⠟⠐⠒⠚⠁⡀⣤⢠⡄⡄⠸⠟⠋⠀⠈⢷⡈⢣⣌⣾⣼⢿⡇⢻⣧⣿⣼⣿⣿⡇⣻⡿⣧⣾⣁⡾⢃⡿⢰⢫⢎⢰⢡⣸⠰⣱⣫⡕⡝⡜⡆
⠀⢀⡤⢞⣋⡁⠀⡇⡿⢸⠇⡇⢰⡶⠖⠀⠀⠈⠛⠋⢉⠀⠀⠘⣷⠼⢛⣛⢻⣿⣿⣿⠏⠀⠀⢸⣟⣠⡟⣁⠘⢎⠂⠑⠂⠸⣜⢌⣚⣚⠽⡱⠁
⠀⡞⣱⢋⡵⠂⢰⠇⡇⢸⠀⠃⢀⣀⣀⠀⢀⢤⣸⠓⠼⢀⣴⢶⣿⣶⣾⢷⢴⣿⣿⣷⡆⢀⣤⠀⣙⠉⠀⠛⠳⠂⠈⠈⠀⠀⠈⠓⠒⠒⠋⠁
⠸⡇⢯⢸⡘⠇⢸⠰⠇⠈⢀⠀⠘⣋⣉⣀⣈⠀⠀⣄⠀⢸⡇⠰⢈⠁⡙⢰⠘⢷⠁⣿⠁⠘⠈⠛⠉⣛⡁⠀⠀⠈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⠀⠻⣌⠷⣍⡁⢸⣀⣾⠙⠋⠀⢼⣿⠏⣩⡽⠟⠀⠸⠷⠈⢿⣟⠿⠛⡿⠿⠿⠟⢿⡏⠀⢀⣴⢶⡾⢋⣿⠀⣠⣾⢷⣌⠁⣤⣬⡥⠈⠈⢉⣈
⠀⠀⠈⠓⠶⠄⠋⣨⡴⠚⠀⢀⣾⠋⣼⡿⢷⣦⣤⣦⣄⣀⣠⣽⣧⣤⣵⣦⣤⣴⣽⣿⣿⣿⡇⢨⡄⣿⡷⢿⠛⢿⡆⢿⡆⠉⡡⠤⠤⢤⡘⠛
⠀⠀⠀⣀⡤⠖⠋⣁⡤⢖⣿⡾⠇⠐⣛⠁⡴⠈⣵⢀⡿⠟⡍⣿⡿⠋⢹⢓⡿⣿⠟⢻⠟⢹⡇⢸⣇⠘⠀⢸⣧⣸⣧⣸⡇⣾⢰⠉⡍⢢⢹⡄
⠀⠀⠈⢁⡠⠔⣫⡁⢲⠸⠷⠶⠟⠛⠛⠻⠻⠾⣧⣬⣥⣘⣥⣫⣀⠃⣸⣈⣦⡉⣧⠈⣤⣤⡅⠸⢹⡷⠟⠛⣉⡁⠈⡁⢀⡘⠦⢍⠥⢊⡼⠁
⠀⠀⠀⠀⠰⠓⠇⠈⠋⣀⠀⡴⢋⡭⠭⡳⡔⢦⢠⡁⢨⡿⣻⣿⡿⢀⣶⣭⣽⠟⠙⠛⠋⠙⢷⣴⠿⢀⣀⣀⣉⣁⣸⣷⡀⠑⠀⠒⣒⡯
⠀⠀⠀⠀⠀⠘⢿⡃⢧⢸⣸⡁⣇⠺⠤⢃⡆⢸⠀⡇⠘⢷⣿⣿⡷⠟⢡⡖⢦⢰⠛⣆⣰⠋⢳⣠⡆⣈⣉⣉⣉⣉⣿⣿⡷⠐⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠙⢬⣙⣋⠉⠰⠋⠀⠃⠀⠀⠀⢀⠀⣴⡄⠀⠈⢉⡤⠞⣡⠤⠀⢉⠀⠀⠀⠀⠀⣸⠟⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣀⣀⣀⣀⠀⠀⠀⢠⠄⣰⠂⡴⢀⡎⢰⡟⢿⣆⠈⠥⠖⣋⡡⠖⣋⡥⠖⢋⡀⣄⢠⣄⡀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠏⢀⡄⢀⡟⢰⠃⣰⠁⡾⢀⡿⢱⣎⢻⣦⠀⠊⡥⠖⢋⢡⠴⣸⢈⡇⣿⠀⠙⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⢃⡞⢠⠏⢰⠇⣸⠁⣼⢇⣿⣿⣦⠹⣧⡀⣤⠳⣬⣓⣚⣡⠞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠏⢰⠃⣰⣏⣘⣛⣛⣛⣃⣹⣷⡈⠙⠶⠬⠭⠶⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣛⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⣞⠀⠀⠆⠀⠀⢀⠀⠀⠀⡉⠉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⡤⠒⠉⠀⠀⠀⠀⢸⠀⡄⢀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠂⡠⠀⡉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠀⢢⠘⡄⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡄⢸⠀⠂⠤⠀⠀⢀⠀⠀⠆⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢠⠉⠀⠀⡀⠀⠘⡄⠀⠀⠀⠀⠀⡧⠸⠀⢀⠐⠀⢀⠀⠀⠆⠐⠃⠀⠈⡠⠀⠀⠂⡠⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠢⡀⠁⠈⠀⠘⠀⠐⠘⡄⠀⠀⠀⠀⢸⠀⡀⠘⠀⠠⠈⠀⠀⠆⠀⠊⠀⠈⠠⠂⡠⠊⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⠀⠀⠀⠐⠂⠐⠄⠲⠀⡘⡄⠀⠀⠀⠀⠇⠁⠀⡆⢠⠀⡄⠀⠞⠀⠐⣠⠀⡠⠊⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠐⡇⣄⠁⠀⠐⠀⠑⠀⠀⠄⠀⠄⠀⠸⡀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠠⠀⠀⡃⠀⣡⠊⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⠀⠀⠲⠀⠀⠄⠀⠄⠐⠄⠀⠄⠇⠀⠀⠀⠀⠘⡀⠀⠃⠈⠠⠂⠈⢡⠞⠁⠀⠀⠀⠀⣀⠴⠊⡁⠀⠈⠀⠰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⡈⠂⠀⠠⡄⠀⠠⠀⢠⠐⠄⡸⠔⠒⠉⠉⠁⠑⢄⠀⠀⠀⢃⡔⠁⠀⠀⠀⣀⠴⢊⠅⢀⡀⠀⠘⠁⠤⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢦⡀⠠⠀⠀⠠⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠲⣄⡠⢞⠁⢀⠄⠀⠀⠠⠒⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠳⢤⣀⢂⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠤⠀⠀⠐⠃⠀⠰⠂⠀⠀⠐⠚⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠀⠀⠈⠁⠀⠐⠂⢀⣁⠀⢁⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠠⠁⠀⡈⠀⠈⠤⠀⠀⠄⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠙⠀⠒⠒⠒⠒⠒⢛⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⠤⠊⠁






⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣤⡤⣤⣄⣀⣤⢤⡤⢶⣤⣀⣀⣀⣤⠤⠤⡶⢤⣤⣄⣀⣠⣤⣴⠶⠦⣤⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠳⠤⣤⣤⠤⠞⠀⠘⢆⡀⠀⣀⣀⡤⠞⠁⠘⢦⣀⡀⣀⣠⠜⠀⡤⡀⠈⠳⣄
⠀⠀⠀⣼⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠀⠀⠀⠈⠉⠉⢀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⣴⠛⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⢲⠲⠒⠒⠒⢒⡒⠒⠛⣉⣭⣍⡉⢉⣭⣍⣉⣉⣉⣀⣠⣤⣉⡗
⠀⠀⠈⣛⣿⣙⣲⣒⣉⣶⣖⣟⣶⣖⣉⣉⣓⣶⣫⣭⢯⣉⣭⡤⠤⠤⠽⠧⠤⠤⠤⠤⠤⠧⠤⠖⡟⣶
⠀⠀⠀⣏⠓⠒⠊⠓⠤⠴⠒⠲⠒⠊⠓⠒⠒⠛⠒⠒⠛⠉⠉⠈⠙⠋⠙⠋⠉⠙⠋⠉⠑⠒⠊⠉⢀⡏
⠀⠀⠀⠙⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠀⠀⣠⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢰⠏⠀⡈⢀⡀⣀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠸⣇⠀⠀⢠⡟⠀⣸⠇⣸⠇⠇⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣷⣄⡀⠀⠀⠀⠀⢻⠀⠈⠀⠁⢸⠀⠀⡟⠀⡟⠀⠀⠸⠁⠀⠀⠀⠀⠉⠉⠙⢲⣿⣷
⠀⠀⠀⠀⣤⣾⣿⣿⡄⠀⢻⣯⣏⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠦⠶⠶⠿⠿⠿
⠀⠀⠀⠌⠁⢸⣿⣿⠃⠀⠀⠹⣿⣷⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣶⣤⠴⠶⠶⠶⠆
⠀⠀⠀⡀⢠⣿⣿⠟⠀⢀⣶⣶⣾⣿⣿⣠⡀⠀⣴⣀⢀⣀⣼⣿⣦⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⠿⠛⠁⠀⢀⢸⠛⢿⣿⣿⡟⢿⣧⠀⠻⢿⣿⣿⣿⣿⠟⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢀⣀⠈⠈⡄⠈⣿⣿⣧⠈⠳⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⢼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠠⠞⣿⣿⣿⠂⠘⠷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠐⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠤⠀⠀⡀⠀⠀⡄
⠀⠀⠀⢰⣤⣿⣿⡟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣠⣿⠿⢷⣶⣶⣶⣶⣶⣶⣿⣷⡀⠈⠀⠄
⠀⠀⠀⠀⠙⠛⠉⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⣴⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⣼⠀⠀⠀⠀⡞⠋⠀⠐⠋⠁⠀⠁⠀⠀⢀⢰⡛⡿⡿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⡏⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⡐⢠⡏⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⢻⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣧⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⣼⢁⡿⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢃⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢻⠁⠀⠀⠀⠀⠀⣠⣿⣭⣿⢋⠈⠀⢸⡇⠀⣠⠀⠀⠸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡼⠀⠀⠀⠀⠀⣴⣋⡼⠟⣳⠋⠀⠀⢸⣇⣴⣿⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠟⠋⠀⢀⠄⠀⠀⠀⢸⡏⣿⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢀⠀⣰⣿⠀⠀⠀⠀⠀⠟⡿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⣰⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠁⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⣀⣥⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠒⠂⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡦⢀⣴⣶⣶⣶⣶⣶⣦⣄⡀⢀⣤⠀⢰⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠠⠈⠛⠛⠻⠿⠿⠿⠿⠿⣧⠘⠕⢂⣶⣆⠀⢀⠔⠒
⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠈⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⡁⢀⣤⡄⠀⢠⠂⠀⠀⠦⠀⠀⡀⠀⠀⠀⠙⠠⡆⣸⣤⣤⡄⢏
⠀⠀⠀⠀⠈⣿⢿⡀⠰⠄⠀⠭⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠥⠘⠉⠓⠀⠂⠀⠰⠆⠰⣌⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⡁
⡠⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠌⢃⠁⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿
⠁⡤⠀⠀⠀⠁⠘⠇⠀⠃⠀⠒⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣴⠀⠀⠐⠀⠀⢃⠀⠀⠀⠀⠀⠂⠠⣄⡀⠠⠀⠲⣶⡶⠆
⡀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢠⡀⠉⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠄⠀⠀⠂⠀⠀⠀⠀⢀⡀⢀⣴⢶⣦⡀
⠁⠀⠀⠀⠀⠀⠈⠐⠀⣀⠀⠈⠀⢠⠆⠀⠐⠀⠀⡀⠁⠀⠀⠀⠠⠀⠲⠀⣀⠀⠀⠀⠀⠠⡴⠀⠀⠶⠀⠀⠻⠃⣔⠥⣚⡞⢁⣴⠂
⠀⡀⠀⠀⣂⣥⡤⣤⣐⣒⡂⠀⢀⢀⡀⢰⠄⢳⠞⢳⣤⠄⠐⠀⠀⠀⠀⠀⠀⠐⠠⠀⣤⠀⢀⣶⣶⡄⢀⣴⠀⠀⢹⠚⠞⠀⣾⠫⡂
⠀⠀⠃⠁⠸⢿⡷⠀⢾⣿⠗⠦⠈⠤⠡⠄⠐⠛⠳⡏⠀⠀⠘⠀⠀⣀⠀⢀⡀⠀⢀⠀⠟⠀⠈⠿⠿⠃⠀⠈⠃⠀⠈⠀⠸⠀⠁⠀⠁
⠀⠀⠈⠀⠀⠀⠄⠐⠈⠉⠀⢰⣷⠀⠀⣾⣧⣶⣴⣦⣶⣶⣦⣰⡐⣿⡆⢸⣷⠀⣼⣷⠀⠲⣦⠀⠀⢰⡶⠂⢠⠦⠀⠀⠀⠀⠀⢰⣿
⢀⠀⠀⡠⠀⢰⡶⣗⡢⣀⠀⢈⣿⣄⣤⣿⡏⣩⡾⣦⠁⢭⣬⣍⢀⣿⡗⢿⣿⠃⣿⣿⣠⣄⢠⣤⠀⣴⣦⠀⠓⠁⠀⠀⠀⠀⠀⣈⠉
⠛⠀⠀⠀⠀⠀⠺⠟⢃⠄⠀⢸⣿⠉⠉⣿⠀⢿⣞⣫⢀⣤⣼⣿⠘⣿⡇⢿⡇⠀⣿⡏⠈⣿⡏⣿⣧⣽⣿⠀⠀⠈⠠⠀⠀⠀⠀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⡀⠿⠀⠀⠉⡁⠘⠷⣼⣿⠀⠿⠀⠘⠿⠗⠻⠃⢠⠙⠁⠈⠉⢻⣿⠀⠀⢀⠀⠀⠀⠈⠁⢘
⠀⣰⣦⠀⠐⠠⠀⠀⠀⠀⠀⠘⠀⠀⡁⠀⠀⠐⠀⠉⣠⠐⢂⡄⠒⠆⠢⡀⠠⠤⣠⠀⠆⠀⠖⡀⢰⣼⡿⠀⠀⠀⢰⡄⠀⠀⠀⠠
⠀⣿⣿⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠛⠁⠀⠀⠀⢠⠒⠈⠀⠀⠍⠀⠀⠈⠁⠠⠀⠠⡄⠀⠀⠉⠀⠀⠀⡀⠈⣀⠀⠘⠃
⠀⣿⣿⡇⠀⠀⠉⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣄⠒⠡⠀⠄⠀⠤⠀⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⢾⣶⠆⠀⠄
⠀⣿⣻⡇⠀⠠⠠⠀⠐⡄⠀⠀⢀⡀⠀⠀⠀⠈⠉⠀⠅⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⠀⠀⢀⠀⠰
⠀⢿⣿⡷⠄⠀⠀⣀⠀⠀⠄⠽⡲⢤⠀⠀⠀⠀⠐⠶⠈⠠⠀⠀⠀⠀⠀⢁⠈⠀⡀⠀⠀⠎⠂⣀⠀⠀⢀⣊⠔⠁⠀⠀⠀⠀⠠
⣀⠈⠋⠀⠀⠀⠀⠉⠀⠀⠈⠙⢁⠄⢑⡂⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠐⠈⠀⠀⠀⠀⢀⡄⠀⢠⠂⠀⠁⠀⠤⠀⠁⢠⣴
⠛⠃⠐⢄⠀⠘⠀⠠⢐⣆⡀⠂⠀⠀⠁⠀⠀⠰⠠⡬⣿⠁⣼⠁⢸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡐⠖⠁⠀⠐⠁⠀⠐⠀⣀⡀⠀⠀⠙
⠈⠀⡀⠈⠀⠀⠀⠀⠈⢻⣄⠀⠘⠓⠐⠆⠀⠀⠀⠑⣺⣦⡸⠇⠘⠀⠀⠀⠀⠀⣀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠉⢆
⠆⠠⠀⠠⠀⠀⠀⠀⠀⡈⠻⣦⠈⠀⠀⠀⠀⠳⠀⢠⠀⠀⢀⣀⢀⢻⣄⠴⣶⠻⠃⠀⣠⠈⠀⠈⠁⠀⡀⢀⡀⠐⠈⠀⠀⠀⠈
⠠⠀⠁⠀⠀⠀⠀⣦⣦⠀⠀⠉⠾⠇⠀⠀⠀⠀⠐⠁⠀⠀⠈⠈⠛⠀⠰⡀⣤⣤⣤⡀⠉⠀⠀⠐⠷⠀⠐⠎⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠖⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⢀⣀⣤⣤⣴⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⣾⣿⣟⢰⣿⣷⠜⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⣀⣴⡿⣣⣾⣿⣿⣿⣮⡛⣫⣼⣿⣿⠟⣋⣭⣙⡻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠛⣋⣥⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⢟⣵⣾⣿⣿⣿⣿⡌⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢛⣁⠻⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣷⣌⠛⣩⣴⣶⣥⡸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⡏⣾⣿⡿⢋⣼⣷⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠸⠿⢿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⠇⠻⣏⣴⣿⣿⡟⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣶⣬⡛⠿⣿⣿⠿⢣⣭⣉⣉⣥⣾⣷⣬⣉⣉⣭⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣅⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣙⡛⠿⠿⣃⣬⣭⣭⣭⣭⣭⣭⣥⣶⣶⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡄⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⣀⣀⢨⡤⣀⣀⣖⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⡀⠀⠀⠀⠳⠇⠀⠈⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠘⠋⠋⠈⢀⠲⠘⠚⠁⠈⠙⠃⠀⢠⠈⠀⣤⣤⠀⠓⠦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠤⣤⠄⠤⡤⠤⠰⠦⠄⠀⢀⢀⣤⠀⠙⢀⢷⠀⠰⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠙⢱⠀⠀⠈⠀⠄⠀⢀⠀⠀⠸⠀⠀⠀⠀⢉⢠⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠯⠉⠉⠉⠉⠋⠉⡋⠁⠀⢀⠀⠲⠟⠀⠸⠆⠰⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⣤⣀⣀⡀⣀⣀⣩⡀⠀⠘⠓⠒⠂⠚⠁⠀⠀⠀⠄⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠁⡀⢀⠀⠀⠀⠀⠀⢀⣠⡀⠠⡜⠋⠀⠠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠶⠤⠤⠤⠮⠧⠤⢒⠄⠀⢈⠉⠀⣀⡉⠈⡧⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠞⠀⡀⡀⠀⠠⠖⠀⠀⠀⡐⠬⠀⠈⠉⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠐⠂⠉⠁⠀⠀⠐⠒⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠉⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣀⡀⠀⠀⣀⣀⡀⣧
⠀⠀⣯⣵⢸⠀⠀⡞⡄⡔⠉⠀⣦⠂⠀⠀⠀⠀⠀⠀⣀⣿⢯⣿⠟⠆⠈⢿⣦⠄⠹
⣀⣀⣳⣚⣘⣂⣘⣁⣑⣙⣢⣀⣃⣓⣀⣀⣀⠀⢀⣤⣿⣿⣧⡀⢀⣄⡀⠀⠀⠀⠘⣷⣤
⣿⣿⣻⣿⢿⡽⣿⣻⣻⣿⡇⣿⢹⣿⣿⣿⣿⠀⣼⣿⡟⣿⣿⣿⣦⣦⣤⡀⠀⠀⠘⣿⣿⣧
⠿⠷⠷⠿⠾⠷⠿⠿⠿⠿⠷⠿⠾⠿⠿⠿⠟⠀⣿⣿⣿⣾⣿⣿⣏⣉⣁⣀⣠⣼⣿⣿⣿⠏
⠀⠀⠀⠇⠆⠐⠼⠰⠠⠆⠁⠀⠰⠂⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⠟⠁⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣄⣀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠋⣯⠤⢄⠙⢧⡐⢆⠙⢿⡷⣄⠀⠀⠀⠠⣶⠏⣉⠴⠒⢒⣶⣤⡀⠀⠀⢀⠞⠁⠀⣀⡠⠇⠰⣀⠀⠈⠢⡀
⠀⠀⠀⢀⠔⣊⠁⢸⡒⢢⡀⢻⡅⠀⠀⠈⢷⡈⡆⠈⢧⢸⡆⠀⢠⠁⠯⠀⢡⠐⠿⠛⠛⠛⠣⠀⠀⡞⠀⡴⠉⠀⢀⠀⠀⠀⠙⣦⠀⢳
⠀⢀⠜⠁⢸⡇⠑⠀⢣⡀⢱⡀⢻⣄⠀⠀⠸⡇⢠⢀⣸⡾⠁⠀⠸⣆⠀⠀⡁⣀⡠⠤⠖⠲⢾⡇⠀⣿⡄⠀⠀⠀⢸⣀⠃⠀⠀⠈⠁⣾
⠀⠘⠀⠀⠈⢷⡀⠡⠀⠳⡄⢣⠀⠻⣄⣀⣀⣹⣼⡿⠋⠀⠀⠀⠀⠙⠶⣀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢿⣿⣦⣄⣀⠀⢁⢀⣀⣤⣶⣿⠃
⠀⠘⢦⣀⣀⡀⢈⣀⣿⣤⣽⣿⣶⣶⡿⠿⠛⠋⠁⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠚⠊⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠈⠉⢉⣙⣉⣉⡉⠉⠉⠀⠀⠀⢀⠔⠋⠁⠀⠀⠈⠲⡄⠀⠀⠀⠀⡀⠠⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠚⠋⠓⢤⢤⢄⣀⡀
⠀⠀⢀⡜⢛⠣⣄⠀⠀⠈⣿⠶⣄⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⡔⠁⠀⠀⠘⡿⠀⠙⠆⠀⠀⠀⣠⡾⣍⢄⠀⠀⠀⠀⢸⠀⢠⣀⠼
⠀⠶⠷⠤⢀⡑⣌⡀⠀⢀⡿⠀⡘⢣⠀⡇⠁⠀⠀⠀⠀⠀⠀⣼⠀⢸⣷⢶⣤⡀⡸⠁⠀⠀⢰⡄⢀⣼⡃⠀⠈⢦⡣⠀⠀⠀⣼⡀⠚⠁
⠀⡋⠉⠉⢳⣶⣿⣿⣴⠿⠷⠴⣧⣨⠀⠱⣄⠀⠀⠀⠀⣀⡴⠃⠀⢿⠀⠀⠀⠛⡃⠠⢠⠴⢿⠇⣌⢀⣸⣶⣄⣠⣷⠷⠶⠟⠁
⠠⣄⣤⠴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠸⡀⡄⠀⠀⣧⡀⠀⡠⠋⠀⠀⠀⠀⣀⣭⣴⣖⡉⠉⠉⠒⠲⢤⡀
⠀⠀⢀⡤⠲⠈⠉⠀⠀⠈⠀⢹⢳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢉⠙⠤⠤⠾⠛⠀⠀⠀⠀⢀⠔⠀⠀⠀⠄⠀⠉⠓⢦⣀⠀⠀⠙⣆
⠀⠀⠸⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠻⠿⢿⡟⠒⠤⡀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠀⢀⡼
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠈⡆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⡿⠒⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡞⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⡼⠀⠀⠀⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠓⠦⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣰⠃
⠀⠀⠀⠈⠓⠶⠶⠶⠄⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠙⠓⠦⠤⠤⣤⣀⣀⡀⠂⡿⠚⠉
⣶⠖⢶⡶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⢠⣤⣛⢿⣾⠗⠀⠐⢿⡷⠀⡠⣽⠀⠀⣶⣦⣾⠆⠀⠀⠀⠈⠻⠃⠀⠀⠀⠰⠀⠘⠁⠀⠀⠀⢎⠀⠛⠛⠃
⠀⠀⠙⠛⠉⣿⣿⡇⠀⢀⣴⡺⠝⠁⠀⣠⣼⢇⣵⣿⣿⡆⠀⠀⢠⣶⣶⣦⡀⠀⠀⠖⡀⡀⠀⠘⠀⠣⠀⣠⡄
⠀⢀⣀⡀⠀⠈⠉⠑⠀⠀⠀⠰⡄⠀⠀⠀⠠⣿⣿⣿⣿⣷⠀⠀⠸⣿⣿⡿⠃⠀⠀⠈⠀⠓⠀⣠⣵⣦⠸⠿⠋
⠀⠘⢿⠇⠀⠀⠀⠰⡄⣄⢤⠢⠘⠂⠀⠀⠀⠈⠉⠻⠿⠟⠀⠀⠀⠀⠈⠀⢠⠞⡉⢧⢦⠀⠀⠘⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢤⡽⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⣾⣃⣀⣄⠀⠀⠀⠀⠀⡀
⠀⠀⣵⣾⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣰⣇⠀⣤⠀⠀⠀⠀⣏⠤⢹⠂⠀⠀⠀⠻⢿
⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⡄⢰⣶⠀⠀⠀⠀⣠⣤⡄⣿⡈⣿⣏⡀⣿⣤⠀⡀⣀⠈⠛⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠈⠛⢿⣿⠟⠁⠀⠀⠀⢸⣷⣼⣿⡄⣾⠿⠟⣿⠿⠇⠛⠋⠙⠋⠐⠛⠛⠘⠗⣿⠁⠠⣤⣤⣤⣶⡾⠟⠃
⠀⢀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⠇⠙⠃⢉⡉⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⣴⣴⡟
⠀⣼⢅⣤⢴⠆⠿⠃⠀⠀⢠⣶⠀⢀⡀⢠⡟⠉⣀⣄⠀⠿⣅⢹⣿⠁⣴⣠⡆⢨⡅⢀⣤⡄⠀⠀⠀⠀⠐⠄⠂
⠀⠘⠀⠉⠀⠀⢰⣶⡦⠀⢸⣿⠀⢨⡅⣻⡟⠸⣿⣛⠀⠒⠋⠘⠟⠃⠉⢹⡇⣿⠃⢿⣯⠁⠀⠀⠐⠈⠀⠀⣀
⠀⠀⠠⣄⠀⠀⠀⠉⠁⠀⢸⣿⡶⢸⠇⠛⠁⠀⠀⠀⠀⢀⡠⠄⠀⠸⠷⠟⠀⠛⠛
⠀⢀⣼⣿⠆⠔⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠻⠏⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣿⣿⣿⠀⠈⠢⠀⡥⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⢀⣤⣀⡠⠴⠒⠛⠉⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣜⣷⣦
⠀⠀⠉⠁⠀⠀⠀⠀⢀⡄⠀⢶⣀⣸⡷⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⣀⠠⢄⠀⠀⠀⢈⣿⣹⢯⣆⡀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠆⠀⠀⠀⠀⡳⠿⣤⣀⠀⠈⠉⠉⠁
⠀⠀⠀⠐⡀⠀⡈⠀⠀⣀⣨⠿⠿⠛⠋⢉⡦⣀⡀⠀⠀⠤⠞⠙⠀⠀⠀⢺⣧⠈⠀⠀⠀⠋⣜⢡⠄
⠀⠀⠀⡀⠀⠀⠀⣠⣾⣿⡿⠀⣀⣀⣤⡅⠀⢉⢳⡿⣤⡄⠀⠀⠀⣴⣿⣿⣶⣶⣦⠠⠆⠀⠀⠈⠀⠀⠠⠂
⠀⠐⠿⢿⠀⠀⣴⣿⠟⠁⠀⠸⣿⣿⣿⡯⠀⠈⠉⠀⠈⣠⣤⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⣤⣶⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣻⣾⠿⠏⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣛⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠋⠁⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡛⠿⣦⣌⣉⣉⣉⡉⣻⠉⠉⠉⠉⠉⠁⠀⠀⠀⣉⣈⣀⣀⣠⡤⠤⠖⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣀⡍⠉⠉⠉⠉⠉⠀⠀⠉⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦⣀⡀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣿⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢷⣶⠿⠶⢷⢶⣶⠶⠾⣿⠿⠋⠉⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣅⣀⠀⠈⠉⠀⠀⠁⠠⠴⠀⠈⠀⠰⠒⠀⠀⠳⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣾⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⢻⡛⠭⠽⠋⠿⠿⣷⣦⣀⢀⡀⠀⣀⠀⢀⣀⣠⣤⡶⠖⠛⠛⠛⠉⠉⠁⡀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡶⣦⠄⣀⡀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣤⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣴⣶⣾⡿⠛⠛⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⡿⢿⡟⠅⠁⠀⠀⠀⠀⠀⠀⣠⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣭⣟⠉⠉⠙⠙⠀⠀⠀⠀⠀⠛⠋⠉⠉⠀⠀⠀⣁⣤⣄⣤⣤⠶⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⡳⠦⢙⠻⢿⣿⣿⣿⣿⣾⣾⣶⣶⣶⣶⣦⣶⣶⣶⣾⣷⣾⣿⣿⠿⠛⠋⠡⠀⠐⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣈⡙⠛⠲⠶⠍⢉⡛⠛⠛⠛⠉⠉⠋⠙⠛⠛⠛⠛⠛⠋⠀⠐⠊⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡞⠠⡀⠀⠀⠀⡠⠂⡀
⠀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⠀⠸⣹⣮⠗⡲⠄⠀⡁⢀⠀⠐
⠀⠀⠀⠀⠀⠘⡀⠈⡁⠠⡀⢀⡤⠛⠻⣌⠠⠆⠀⢧⠀⢀⠂⠀⠄⢀⠀⣀⣤⠀⣴⠀⠀⠀⠀⠠⠒⠄
⠀⠀⠀⠀⣀⣀⠡⢤⣑⡄⠀⣾⠀⠀⠀⣼⠀⠂⠀⢰⠄⢨⣀⡴⠤⠚⡋⢰⣿⠀⡿⢄⣀⣀⠚⠁⠀⡀⠔⡆
⠀⠀⣀⠘⣱⠠⢻⡲⠬⢙⡆⣿⠀⠀⣰⣧⣶⢶⡄⠈⡉⠀⠠⠀⢀⠁⡰⢫⡿⠺⠓⠞⢻⡇⠑⢹⡔⢠⠆⡇
⠀⠀⡃⢈⠤⣵⡚⢻⣷⣆⢹⡇⣀⣼⡿⢱⢟⣾⢅⡀⠀⣀⣠⡾⢋⣿⣵⠟⣷⣄⠀⢀⣸⣧⢸⣨⣷⡦⠼⢄
⠀⠀⡰⡘⠄⣞⢯⡓⠉⢻⢸⣿⢩⢿⡇⣹⣿⡟⣷⠶⡾⡛⣿⠁⡾⢸⡏⢼⢯⣭⣻⣯⣾⡵⣽⣟⡅⣒⢊⠅⠤⠃
⠀⠀⠈⠉⠀⢁⠑⢅⠀⢨⣿⣿⣶⣿⣧⠸⣿⠃⣏⣘⣤⣷⣵⣤⣷⣸⣧⣏⣌⠘⣩⡟⠁⠀⠀⠈⢳⠀⢀⢀
⠀⠀⢰⢦⣾⠧⣙⢦⣱⡏⣴⣾⣿⣿⣿⠦⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣤⡛⢧⡀⠀⠀⢠⡿⢀⣴⣾⡿⠏⡄
⠀⠀⠀⢇⢿⣵⣾⣟⣿⣇⠟⠱⣺⠟⡱⣳⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⡦⠖⣻⠋⠉⠉⢻⡈⠉
⠀⠀⠀⠈⣎⢯⡻⣮⢷⠹⣧⣙⠋⣴⣿⣿⣿⠿⣇⡿⢋⣽⣿⣿⡿⣿⣿⣿⣿⡿⣿⣧⣶⡝⣶⠃⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢦⡉⠑⢿⡄⣸⡏⣸⣿⠛⣿⠏⣰⣿⣷⣝⣛⣛⡛⢀⣩⣥⣼⡏⣼⣿⣿⣿⡏⣿⢀⣤⠞⠋
⠀⢀⠤⠄⡠⠄⠵⠦⣄⣹⣿⢀⣿⡿⢹⡟⣰⡿⠋⣹⢋⣿⣿⡁⣼⡟⡟⢹⠀⣹⠟⢷⣨⡾⠟⠋⠀⢀⢀⣀⡤
⠀⠈⠢⢄⠀⠾⢶⣤⣠⣽⣿⣌⣯⣶⣬⠁⣶⢳⣬⣿⣦⡌⢻⣇⣫⣄⡇⢘⡀⢏⠰⡿⣹⠇⠀⠀⠀⢒⣧⠇
⠀⠀⠀⠀⠀⠠⠄⠙⣦⡾⠷⠉⠉⣿⣿⢸⣿⣸⠅⠙⠿⠟⢛⣹⣿⣿⡇⢸⣟⣛⣗⣞⣁⣀⡠⠤⠞⠭⠀⠂⡀
⠀⠀⠀⠀⠀⠀⡀⠺⣏⠃⠀⢀⣴⠟⣿⣘⣷⡿⡆⠀⠀⠹⣿⣿⣿⣿⢇⡾⠉⠣⡉⠣⣤⠌⠑⠆⠤⡀⠀⠀⠀⠂
⠀⠀⠀⣀⠐⠀⠀⣴⠛⠦⠠⢾⡇⠀⠈⠉⡹⣹⣳⡀⠀⠀⢉⣿⣿⡿⠟⡇⡀⣜⢟⡎⢈⢓⡦⠘⠆⠀⠉⠈⠉⠉
⠀⠀⠰⠁⠀⡀⠠⠀⣲⠄⣥⡿⡟⠒⠒⠊⠛⣷⠷⢟⡦⣀⣬⢞⠇⠀⣐⠹⣿⣮⢽⠧⠜⠀⡙⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠋⢸⠐⢀⠀⠀⠀⠀⠓⠈⠁⠀⠈⡓⠻⢤⠎⠩⣀⠰⢹⡄⠕⢫⠘⠛⢄⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⠀⠀⠀⠈⠙⠒⠋⠙⠁⠀⠀⠀⠉⠁

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠚⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⠉⠉⠉⠉⠉⠛⠒⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠈⠓⠦⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢰⢋⡴⠚⠉⠩⠍⠙⠒⠦⣀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠅⠐
⠀⠀⣯⡎⢠⠄⣂⡀⣈⢀⢀⣀⠈⢙⠪⣕⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡗⠒⠂⠀⠈⠙⠒⠒⠲⢤⣀
⠀⠀⣿⣰⡿⠋⠁⠀⠀⠀⠀⠈⠉⠓⠦⣄⡉⠪⣕⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣠⠀⠀⠘⠀⠀⠀⠀⣀⠴⠃
⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⠳⣝⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⠴⠒⠊⠉⠉⠉⠉⠉
⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⢾⠃⠀⠀⠀⠀⠀⠀⢀⡼⠁
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⠈⠀⣀⠤⠒⠉
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⠧⠤⠖⠋⠁
⡏⠀⣀⣀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⣷⢞⣁⢤⠤⠤⠤⠤⠭⣕⡢⣄⠀⠀⡀⠀⢠⠎
⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⣱⡄⢀⡴⠋
⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠉⠉
⢹⠈⠀⠀⠀⠀⠀⠀⢠⠏
⠈⢧⠀⠀⠀⠀⠀⠀⡜
⠀⠈⢆⠀⠀⠀⠀⢰⠃
⣀⣀⣈⣓⣤⣠⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠒⠋⠉⠁⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠉⠀⠀⢀⣀⠀⠀⠀⠈⣧⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⣠⡿⠃⠀⠀⠀⣼⠏⠸⢿⣦⣀⣀⣀⣀⣀⡀⠠⠄⠐⠀⠀⢠⣿⠏⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣰⣿⠁⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢠⣿⠧⠴⠖⢋⣥⠖⡆⣠⡶⣛⠆⣠⡆⠀⣴⡿⢁⣶⠂⣀⢀⣿⠏⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢸⡏⠀⠀⠀⣾⠟⠈⣼⡟⢾⡏⠠⣿⠏⣸⡿⠀⣿⡏⢸⡏⢸⡟⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠈⠁⠀⠁⠀⠀⠀⣿⡃⠀⠉⠁⠈⠉⢸⡇⠀⠀⠈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠒⠒⠛⠒⠒⠒⠦⠤⢌⣁⠀⠀⠀⠀⠀⠀⠣⡀⠀⡸⠃
⠀⠀⠀⠀⠀⢀⣤⣔⣒⣀⣀⡄⠀⠀⠀⠀⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⢒⣿⣿⣉⠉⠍⠙⠛⠳⢮⣴⣆⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠘⠇⢶⣄
⠀⠀⠀⢸⣿⣿⣿⣻⣶⣆⣀⣀⣄⣀⣿⣿⣿⣃⣶⡐⠀⠉⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⣧⣶⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⢹⡆
⠀⠀⠀⠀⠀⠱⠐⣭⣿⣿⣛⡻⣍⠴⠤⠟⠻⠛⠛⢻⣿⠿⡟⠉⠀⠀⠀⠀⠀⠀⢤⡦⠁⠢⠀⢈⣠⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠂⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠐⠀⠙⣿⠇⠀⠀⠁⠀⠀⠀⠀⠀⢀⢯⣿⠇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠓⠈⠀⠀⢀⠀⡀⣷⡟⠃⠀⠀⠀⢃⠀⠀⠀⢠⣎⡾⠟
⠀⠀⠀⠀⠀⢀⠫⣑⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠖⠀⠀⠀⠀⠀⢠⠜⣿⡿⠃⠀⠀⠀⠀⠀⠉⠀⠁⠛⠍⠋
⠀⠀⠀⠀⠀⠘⣄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢀⣤⠆⠁⠀⠐⠀⠀⢀⠀⠐⣤⡟
⠀⠀⠀⠀⠀⠀⠈⠣⣠⡀⠀⠈⠁⠠⠄⠒⠂⠁⠀⠀⠀⣰⠀⠀⠀⠀⠀⡀⠀⠀⠰⢁⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠤⠄⠀⠀⠀⠀⠀⠀⠐⠿⣀⠀⠀⠄⠀⣄⠄⠀⢴⣷⡁⠀⠀⠀⢀⣠⠴⠔⠋⠋⠋⠋⠁⠲⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⡀⠀⣼⠧⡙⠛⠀⠀⠀⡰⡛⠁⠀⠀⠠⠀⠀⣀⠀⠤⠀⠀⣀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢳⠆⠰⠀⠞⢀⠎⡨⣶⡷⠀⠀⠀⢸⠇⠄⠀⠀⠀⠀⠀⠀⠁⠠⢠⡄⣸⡛
⠀⠀⠀⠸⡀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⢐⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂⣾⣻⠅⠁⠁⠀⠀⠀⢿⣷⣃⣀⠀⣶⠀⠀⢰⠂⣄⣭⣾⡿⠃
⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠈⠇⢠⣴⣳⣾⠯⠑⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⡿⢾⣖⣫⠾⡿⠟⠁
⠀⠀⠀⠀⠀⠈⠓⠦⠄⠀⠀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠋⠼⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⠤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠂⠀⠀⠂⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⣀⣀⣐⣒⣒⣂⣀⣀⣠⣠⣤⣤⣤⣀⡄⡀⢠⣤⠤⠤⠤⠤⠐⠒⠉
⠀⠀⠀⠀⠠⠔⢶⠀⠀⠀⠈⠙⢲⣤⢀⣀⣀⣤⣤⠶⠛⠉⠉⠁⠀⣀⣴⡀⠀⠀⠀⠀⠐⠉⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⢍⡉⠁⠀⠉⠉⠀⠀⢀⣀⣤⣤⣖⣋⣁⡀⡁⠂⢐⠀⣋⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡎⢭⠀⠀⠒⠬⠍⠉⠓⠒⠒⠊⢉⡀⠤⠛⠉⠉⠩⠉⠉⠻⠄⠀⡠⠂⠀⠹⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠀⢀⣠⣴⠶⡋⠉⠉⠉⢀⠀⢀⠀⠀⠀⢀⣀⣀⠰⠀⢠⠀⠀⢀⡀⠄⠁⢀⣀⣀
⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⠴⢋⠉⠙⠂⠈⠛⠛⠒⠀⠀⠘⠁⠤⠄⠈⠀⠀⠂⠀⠐⠶⠚⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠔⠛⠇⠀⠀⢠⠶⠶⣿⡆⠀⠀⠀⢀⠴⠖⠶⠄⠀⢀⠞⠳⠄⠀⠟⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢾⠀⠀⠀⠐⠤⠐⠁⠀⠀⠈⠃⠐⠂⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⢀⣀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣋⡥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠛⠁⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠾⣇⢀⣀⣤⣤⡿⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⢀⡶⠛⠛⠳⢦⡀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⢶
⢸⠁⠀⠀⠀⠀⣙⣃⠀⢀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠹⡀
⢸⠀⠀⢀⡴⠋⠁⠀⠉⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢣⡤⠒⠠⣀⠀⠀⠀⠀⡠⠞⠙⡆⠀⠀⢀
⠈⣇⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠆⠀⢠⠞⠀⠀⠀⠇⠀⣠⡟
⣠⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠞⢠⣧
⢇⠀⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠈⢻⠆⠈⠉⠐⠒⠤⠤⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠖⠊⠉⠀⢴⠞
⠀⢸⡆⢠⠒⠂⠤⠄⣀⣀⣿⠀⠀⣼⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⣷⠀⢰⣏⡠⠤⠔⠲⡄⢸
⠀⠈⡇⠘⡀⠀⠀⠀⠀⠀⠉⠲⠖⠋⠀⠒⠲⠶⠖⠒⠖⠒⠒⠒⠖⠒⠒⠛⠛⠉⣿⠀⣼⠁⠀⠀⠀⢰⠁⢸
⠀⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣸⠀⡏
⠀⠀⠸⡆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡏⢰⠃
⠀⠀⠀⢻⡀⢷⠀⠀⣠⠔⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠹⠆⠀⠀⠀⣼⢁⡟
⠀⠀⠀⠈⢷⡈⢧⠀⠘⣄⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢀⢰⠇⠀⠀⣰⢃⡾⠁
⠀⠀⠀⠀⠈⢷⡈⢳⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣰⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠻⣆⠙⣄⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⠉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄⠀⠀⠀⠀⠀⠈⠳⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣌⡑⠢⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⢖⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⠀⠀⠉⠉⠉⠉⠉⠁⠠⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣍⣒⣠⡤⠦⢤⣤⠤⠤⢤⡤⠤⠤⠤⠶⢶⣒⣫⡽⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⢻⡉⠀⠀⠀⠀⡏⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⡔⣿⣿⣿⣿⣿⢸⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢧⠀⠀⠀⠀⠱⡄⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣦⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⡽⠋⠉⠳⣤⠇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⡿⠂⠀⠀⠈⠛⠃
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⠟⠁
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣹⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡻⢿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⡏⠀⠀⠉⠓⠲⠤⢤⣤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣭⣛⡻⠿⠿⠟⠁
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⡏⠉⠉⠉⢉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣴⡆
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⡄



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⣄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠖⠙⢮⡳⢿⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠭⠥⠬⠆⢀⢔⣫⣑⢄⠀⡛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠚⠁⢾⣿⠆⠈⠢⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⠟⠉⠙⢷⣝⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡽⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣵⠟⠁⠀⡏⢵⠀⠙⢷⣝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⢠⣻⣳⠀⡖⠛⢳⢰⣻⣳⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⢙⠁⠀⠀⠀⠈⠁⠀⠀⠀⢹⣵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⠸⣼⣬⠀⡇⢀⣸⢸⣼⡼⢸⢺⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⢸⠀⠀⠉⠀⢸⢠⠁⢸⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣺⣹⡀⠀⠀⢀⡇⠀⢹⠀⠀⠀⢸⣸⣒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣾⠸⠀⢸⠀⠀⠀⠀⠈⠛⠁⢸⣾⣡⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠟⠀⠀⠀⠀⠀⠀⠀⠀⠒⠙⠞⠛⠂⠚⠐⠂⠛⠐⠒⠒⠒⠘⠛⠘⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲⡆⣠⠖⢋⣌⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣌⠳⣄⢯⡭⣭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⢋⣡⠞⠉⠈⠙⢦⣉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⡡⠚⠁⠀⠑⢌⠻⢄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣡⠶⠋⠀⠀⠤⠤⠄⠀⠈⠳⢦⡙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡴⠋⠀⠭⢾⣹⠭⠄⠙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⠋⠁⠀⠀⠀⠀⠒⠚⠂⠀⠀⠀⠀⠉⢳⣬⡇⠀⠀⠀⠀⠀⠀⠀⣴⢋⡔⠋⣀⣐⣒⣒⣀⠀⠐⠒⠒⠀⠙⢦⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠉⠉⠉⡇⠀⣏⢉⡏⢙⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡇⢸⢱⠒⢶⠒⡎⣷⢈⡥⠬⡉⠀⢽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣀⡇⠀⡏⢉⡏⢙⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⣀⣿⣀⡇⣿⢸⠀⠀⡇⠐⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣉⣉⣉⣉⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⠷⢬⡓⠒⠒⣒⡚⠁⢸⡀⠉⣧⣞⣵⣙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠶⠶⠷⠶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠾⠶⠆⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠿⠿⠿⠧⠾⠿⠿⠿⠶⠿⠿⠷⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣋⣉⣩⣤⣤⣾⣿⠿⠟⠛⣻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡶⢾⣿⠛⠛⣉⣉⣥⡤⠼⠿⠒⠛⢋⣉⣡⣽⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢉⣩⣤⠤⠶⠾⠟⢛⣉⣉⣤⣤⣴⡶⠶⠿⠛⠛⠋⠉⠁⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡴⠋⢀⣤⣶⠶⠾⠟⠛⠛⠉⠉⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡇⢠⣿⠋⠀⠀⠀⠀⠀⢀⡴⠛⣩⣤⣤⠀⠈⠳⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⢠⡞⠀⠀⣿⠀⣿⣀⣤⡄⢻⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⣾⢠⡖⠛⠋⠀⢈⣀⣸⡇⣸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⣿⠘⠗⠚⣿⠀⣿⠁⠀⣠⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⠘⣦⡀⠀⠻⠶⠛⣀⡴⠋⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠛⠉⠀⢀⣀⣀⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠶⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠾⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣰⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣶⣦⡀⢠⡾⠋⠁⠀⠀⠀⣿⠀⠀⠀⠈⠛⢷⡄⣸⡟⢃⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣤⣤⣶⡿⠷⠿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠁⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠗⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣶⣶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣴⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄⢀⣤⣤⣤⣤⣤⣤⡀⢰⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃⠈⠉⠉⠉⠉⠉⠉⠁⠘⢿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠓⢄⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣀⠄⠀⠀⠍⠄⠠⣝⠅⠀⠀⣦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠷⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠠⠀⢀⡐⡁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⡏⣼⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢂⣄⠀⠀⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠆⠀⠀⠋⡋⠉⣠⣤⣴⣿⠃⠸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⢿⣷⣾⣿⠿⠟⠃⠀⢸⢇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⡔⠁⠀⠢⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠋⠀⠀⢠⠄⠀⠐⡂⡘⠀⠘⠧⠤⠤⣀⣰⡀⢀⠄⣠⠀⠀⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠴⠃⠀⠀⣠⢆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡀⠀⠜⠁⠀⠐⠀⠀⠀⠀⠰⠀⠁⣰⠇⠘⢰⠁⠀⠀⠀⠀⠙⠂⠠⢶⣎⡁⠀⣤⡶⠿⠿⠉
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⣠⡏⡄⠀⠀⠀⢰⣿⣦⣡⣾⢿⡇⠀⣸⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣿⠀⠀⠀⠤⠀⠀⠀⠀⢠⠀⠰⠃⠉⣷⠁⠙⠁⠀⠀⠀⣿⢏⣿⣿⡿⠀⠙⠔⢁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡇⠀⠀⠸⢇⠀⠈⣿⣿⣿⣿⣧⢀⡄
⠀⠀⠀⠀⠀⢀⠔⠟⠁⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠠⠁⠀⠹⣿⡷⠿⠿⠞
⠀⠑⠄⣀⣤⣶⣊⡤⠖⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠄⠀⠀⠀⠀⡀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠄⠀⠁⢀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⢳⠀⣿⣿⡦⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡀⢺⡗⠈⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠉⣴⣷⣶⣄⠀⠀⠀⢀⠀⠀⠀⠈⠛⠁⣴⠏⡀⠈⠋⠀⠀⢀⠙⣧
⠀⠀⠀⠀⠀⠀⠸⣿⣛⢷⣤⡄⡰⢀⣯⣽⡈⠀⠀⠀⡤⠀⣿⣿⣿⣿⠆⠀⠀⠈⠁⢀⠀⢢⡄⠐⠃⠸⠝⠶⡗⢰⢶⡏⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⠖⢻⡿⠟⠃⢀⠓⣿⣿⣿⣿⠀⠊⠀⣀⠀⠙⠛⢻⡁⡄⠀⠠⠃⢆⣉⠓⠀⠀⣠⢸⡇⢴⠀⣠⠀⢨⣀⡲⣼⠀⠀⢳
⠀⠀⠀⠀⡴⢂⣶⣿⠀⠐⣶⠿⣤⠌⠛⠋⣁⠘⢧⡀⠁⠄⠲⠀⠀⠷⢀⣠⡄⠀⠀⠈⠁⣀⡀⠀⠛⠠⣾⣽⣷⣾⣷⣍⠁⣠⣦⡀⢘⡣
⠀⠀⠀⠸⠇⣸⣿⠟⡀⢿⣿⣿⣷⠀⠀⠸⠏⠀⢈⠉⠀⣴⣤⢀⣀⡉⠉⠁⠀⡀⠀⠀⠀⠈⠻⠲⠂⣼⣿⣿⣿⣿⣿⣿⡄⠿⠿⠃⠀⠘
⠀⠀⠀⠀⠀⣿⠏⠘⠇⠸⢿⠟⠁⠰⡄⠄⢀⢤⠘⣇⠀⣴⣷⣿⣿⣿⡄⠀⡍⠀⠀⠀⠒⠀⠀⠀⢠⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⠄⢄
⠀⠀⠀⢴⠀⣥⣄⠰⣧⣀⠀⠀⠀⣎⡷⠘⠘⢂⣀⣀⣄⠹⣿⣿⣿⡿⠃⠀⠀⠀⠐⣆⠀⠀⠀⠀⢈⣄⡬⠻⣿⣿⣿⠟⠁⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠸⡆⢸⣿⣦⣝⣿⡾⠷⢀⣠⡦⢦⣰⣌⠉⢉⠠⠀⠈⠉⢤⣀⠸⠾⠺⠂⠐⢦⣀⡀⠀⠀⠺⣿⡿⠀⢀⠁⢰⠞⠉⠈⠂⠀⠀⢀⡾⠂
⠀⠀⠀⠀⠀⠀⣤⠀⣩⣿⣷⡆⢸⠁⠆⢈⣧⠏⠀⢐⡄⠀⣶⣿⡇⠀⢰⣦⣴⣾⡆⢠⡍⠁⣰⣦⡀⠀⢁⣰⣿⡇⠀⢰⣾⣿⣾⣿⡆⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠙⠿⣿⠃⠈⠛⠒⠛⠛⢲⡞⣛⡹⣧⠈⠿⠁⣠⣴⣶⣽⣏⡀⡌⢢⠘⠧⠿⠃⠀⣿⣿⣿⠇⠀⡈⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⡈⠀⢶⣿⣿⠁⠀⠘⢧⣌⣡⠏⣸⢠⣾⣿⣿⣿⣿⣿⠁⢅⠀⠑⠀⣰⣶⡄⠈⠉⡁⠠⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⢦⣀⠉⠉⠐⠀⠀⠀⠀⡀⠰⠃⠈⠻⣿⣿⣿⠿⠋⢀⣨⣴⣦⠸⢿⠿⠃⠚⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠼⣯⡦⠶⠀⠀⠀⠀⣧⠸⠀⠀⠀⠀⣠⣶⣾⠀⠘⢾⣿⡟⠠⣶⣿⡞⢠⣶⣧⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢰⡎⠓⠀⠀⢶⣶⡧⠀⢈⣙⠋⠀⠀⠀⡠⠀⠤⠈⠉⠃⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠄⠀⢀⠀⠘⠛⠀⠀⡀⡀⠑⠠⠓⠀⡇⠀⠀⠘⠃⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠲⢻⣾⣶⣄⠀⠰⠄⠃⢀⣤⣤⣄⡈⠓⢀⣰⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡆⠀⠠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠟⠁⢀⠀⠻⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣖⡛⠛⠉⠙⢆⠀⠑⠄⠀⠙⠐⣿⠙⠟⠲⡖⢤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⠀⠉⠑⠀⢀⠀⠁⠀⠈⠀⠀⠀⠈⣧⠀⠀⠁⠀⠀⠱⡄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠿⠒⠲⠭⢝⡲⢤⡀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡇⠈⠈⠑⠄
⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠈⠒⢍⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⢹⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠣
⠀⠀⠀⢀⡴⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⡑⠈⠱⣄⠀⠀⠀⠀⠀⠆⠈⡇⠀⠀⠀⢀⠇⠀⢀⡤⠞⠋⢄
⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⣼⠀⣠⠊⠀⠀⢀⣐⡆
⠀⠀⢰⣁⡤⠶⠶⠶⠶⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣧⠐⡄⢀⠀⠀⣇⠠⠒⠚⠉⠙⢧⣀⡠⠞⠉⠀⠙
⠀⠀⠘⣏⡴⠒⠒⠒⠦⠤⠤⣀⣉⠓⠦⣄⠀⠀⠀⠀⠈⢧⢱⢸⠀⡴⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀⠀⠀⠀⡇
⠀⠀⠀⢸⠖⠂⠀⠀⠀⠀⠀⠀⠐⠉⠐⠒⠭⣦⣄⠀⢂⣸⠾⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠁
⠀⠀⠀⣿⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠞⠁
⠀⠀⠀⠹⣞⢀⠄⠀⣀⡤⢤⣤⣤⣤⣤⣀⣀⡀⢈⡎
⠀⠀⠀⠀⠙⣇⢠⠎⡵⡊⠉⠀⠀⠀⠀⠀⠈⠉⠛⡀
⠀⠀⠀⠀⠀⠈⠺⢧⣏⢀⠂⠀⣰⠖⠋⠉⠉⠉⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣐⣧⡇⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠤⠤⡿⢅⡀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣌⠻⣿⣍⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠛⢿⣿⣯⡃⠘⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠿⠿⠛⠻⠶⣄⠀⠀⠈⠻⣿⣷⡀⠈⢣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⢸
⠀⠀⠀⡖⢒⡆⣶⠀⠀⢀⡶⡄⢀⡖⠛⠂⣶⣠⠖⠀⠀⠀⢸⣿⣿⣇⣴⣾⣷⣶⡄⠀⠀⠉⠁⠀⢀⣤⣶⡾⠻⣷⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⢸
⠀⠀⠀⡏⢙⡆⣿⠀⠀⡼⠒⢧⠸⣄⠀⠀⣿⠹⣄⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠶⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⢷⣧⠸
⣤⣤⣤⣭⣭⣤⣬⣭⣥⣥⣤⣬⣤⣬⣭⣥⣬⣤⣬⣤⣤⣤⣼⣿⣿⣿⣧⣤⡞⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆
⣿⣿⢗⢣⣿⡼⣇⣸⢧⡇⣻⡇⣿⣿⡇⣾⡇⣻⣿⣿⣿⣿⣿⣿⣿⡇⢿⡏⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇
⣿⣿⣼⣧⣼⣧⣼⣧⣾⣧⣿⣧⣿⣿⣧⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣷⠀⠀⠀⢦⣀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿
⠀⠀⠀⢠⢠⢀⠀⠀⡀⣄⢀⠀⠀⢀⡀⠀⣀⠀⣀⠀⠀⠀⢸⣿⣿⣿⡄⢿⡆⠀⠀⠘⠛⠁⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠘⠸⠸⠤⠈⠆⠣⠸⠰⠸⠘⠄⠙⠤⠟⠇⠇⠀⠀⢸⣿⣿⣿⡇⢸⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⡃⢀
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠘⣿⣿⣆⠀⠻⣿⣿⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⢿⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⣿⣿⣿⣦⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⡼⢿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠁⠀⢀⠞⠀⢸⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣾⣿⡿⠋⠀⠀⠀⠀⠊⠀⠀⠀⠹⣆

⠀⠀⠀⠀⠀⢀⣤⣔⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡀⡀⠛⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠦
⠀⠀⠀⢠⡶⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⣠⠞⠂⠀⠐⠋⠀⠀⠀⠘⠋⠉⠀⡀⢀⠄⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠁⠀⢀⣤⢀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⡀⠀⠀⢀⠀⠀⠘⡻⠋⠀⠀⡔⠁⠀⣾⠖⠀⠀⠀⠀⣀⡀⠉⠁
⠀⠀⢸⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣀⠴⠂⠀⠀⠀⠀⡛⠁⠀⣤⣄⣀⠤⠚⠁⠀⠀⢰⡗⠃⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⡀⣠⣆⢒⣈⣀⣤⡤⠟⠉⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⠀⠀⠀⠀⠚⠛⠃⠀⠀⠰⡿
⠀⠀⠘⠻⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠂⠀⠉⠁⠀⠀⠀⠀⠞⠀⠘⠟⠀⠀⠀⣠⣤⣄⠀⠀⣠⣤⣤⣤⣀⡀⢀⣾⢃⣼⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠘⠲⣿⠆⠀⠀⠈⡁⠀⠐⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣠⣿⠃⠘⠟⠛⠙⠛⠛⠛⣿⡟⣛⡁⣘⠁⣤⢤⢿⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢸⣿⠇⢀⣶⠲⡄⣼⡏⢹⢠⣿⣏⣼⢣⡿⢸⡧⢊⣔⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⢂⣾⣯⣠⣧⣿⠣⡼⢻⡟⢿⣄⠘⢓⣉⣙⡉⢸⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣠⠞⠻⣧⠟⠁⠙⠚⢁⣀⣥⣼⠿⢿⣍⣁⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠈⠿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠞⠃⠀⠈⠉⠁⢀⡀
⠀⠀⠀⠀⣴⠆⠀⠀⣤⠀⠀⠀⠈⠙⠁⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠒⠓⠚⠃⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⡾⠖⠋⠉⠁⠙⠒⠀⠀⠀⠰⠒
⠀⠀⠀⢠⠀⠘⠷⠀⠀⠀⠀⠀⣴⡶⠂⠀⠀⠉⠉⠀⠄⠀⠀⠀⡠⠂⠛⠁⠀⠐⠟⢃⡀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠘⢿⠁⣠⣤⠀⣀⣘⣀⣀⠠⣤⠀⠀⢤
⠀⠀⠀⣿⣧⠀⠀⣠⣄⢠⣴⣦⠀⠀⢠⣶⣤⠀⠔⢠⠆⠀⠀⣼⠃⠀⢛⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⢦⡤⠒⠉⠉⣉⠉⠉⠉⠓⢄⢀⡀⠁
⠀⠀⢰⣿⣿⣿⣧⣤⣤⣄⡀⢤⣄⣀⣀⣀⣀⣀⣴⡏⠀⠀⢀⡴⠀⠀⠙⠁⠀⠀⠀⠤⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⠀⠀⠋⠀⠀⢀⡀⠀⠈⠡
⠀⠀⠘⢿⣽⣋⡉⣉⡉⠹⠿⠟⠛⠉⠉⢉⣽⡷⣀⣶⠀⠀⠈⠳⠀⠀⠦⢀⠐⠀⠈⠀⠁⠚⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⣏⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⢰⣿⣿⣿⣭⢤⣦⣦⣤⣤⣴⡷⠾⠿⠛⠙⠋⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠉⠀⣀⠀⠀⠤⠀⠀⣀⡀⠀⠀⢸⡇⣴⠄⠀⠀⠀⠀⢴⣆⠀⠀⠀⡀
⠀⠀⠀⢨⣥⣌⣉⠀⠀⣀⣀⣀⢠⠾⠆⠀⣀⣠⣤⣴⡛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⣿⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡄⠛
⠀⠀⠀⢉⠾⠋⠉⢭⡉⠉⠋⢹⠟⢻⡏⠉⠉⠉⢈⣉⣤⠀⠀⠀⠀⠀⣀⣀⠀⠰⠖⠒⠒⠀⠉⠈⠁⠀⢠⡀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠶⠂
⠀⠀⢠⣿⡟⢛⡶⠶⣤⣤⣤⣤⡶⠶⠶⠞⠛⢋⡉⠡⢄⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢴⣦⡄⠀⠀⠀⠙⠁⢠⡆⠀⠀⠀⠈⠳⣄⣀⡀⠀⠀⢶⠆⣀⠤⠔⠁
⠀⠀⠀⠨⣷⣿⣿⣄⣀⠺⠷⠀⠀⠀⣀⡀⠀⣛⣣⠴⠊⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠉⠁⠀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠖⠒⠲⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⠯⠥⠤⢤⣀⡀⠀⠀⢀⣀⣤⣤⣈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠋⡼⠁⠀⠀⡀⠀⠀⠉⠙⠉⠉⣀⣤⠀⠈⢦⠀⠀⠙⢦⣀⠤⠤⣤⢄⣀⡀
⠀⠀⠀⠀⠀⣴⣃⣀⣰⠃⠀⠀⠻⠟⠀⠀⠢⠔⠀⠀⠈⠉⠀⠀⠀⠳⣤⣔⠷⠐⠒⣈⠁⣠⡖⢉⣦⢄
⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠠⠤⠤⠀⣀⣀⣀⣀⣤⣄⣀⠀⠀⢺⠀⡔⠉⣥⠘⣧⠀⣬⠀⣩⠱⠮⢡⡌⢂
⠀⠀⠀⣼⠇⠀⠀⠀⣀⣀⠀⠀⡠⢊⡥⠞⠋⠉⠉⠓⠺⣵⡄⠈⣰⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢠⠏⡇⠀⠀⠀⠀⠀⠀⡼⢡⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⠀⢳⠀⠀⠀⠰⣆⢰⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢸⠀⢸⠂⠀⣀⡤⠉⠈⣇⣇⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠘⡆⢸⠀⠀⠁⠀⠠⣤⡜⢿⣆⠀⠀⠀⠀⠀⠀⠀⢉⡼⠃⢡⣀⠔⢄⡀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡄⠒⠈
⠀⠀⠀⢳⢸⠀⠀⠀⠀⢠⡀⠀⠀⠙⠳⠦⣤⣀⣠⠤⠖⠋⢀⣘⠀⠀⠀⢀⣀⡈⢇⡀⡠⠃⣀⡉⠂⠈⠀⡇
⠀⠀⠀⠈⢯⢧⡀⠀⠀⠀⠙⠀⠀⣷⠀⠀⡀⠀⠀⢦⡀⠈⢹⠻⡆⠀⠀⠘⠻⠃⠀⠴⠂⠀⠛⠃⠀⡆⢰
⠀⠀⠀⠀⠀⠳⡝⠢⢤⣀⡀⠀⠀⠁⠐⠛⠁⠰⣄⠀⠁⠀⡜⠀⢳⠀⠃⠀⠀⡀⠀⡄⠀⢀⠀⠀⢰⠀⡂
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⢢⡀⠀⠀⠀⠀⠀⡸⠋⠙⠾⠥⡖⠺⠋⠉⢢⠀⠁⠀⠃⠀⠈⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡙⢦⣀⣀⣀⡤⢧⠀⠀⠀⢀⡇⠀⠀⠀⣸⣀⠀⠀⠀⠀⢀⣀⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠳⠶⠶⠶⠟⠒⠒⠋⠉⠓⠒⠋



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣦⠠⣶⡄⠀⢀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠿⠟⠂⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠚⠛⢰⣖⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠋⠀⠀⠀⠀⣠⡴⠒⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠇⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠈⠁⠀⠀⠠⠤⡀⣴⠣⢒⡀⣴⡄⠀⠆⠸⠆⠆⠋⠀⠀⠀⠀⠀⠀⠺⠏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠈⠁⠀⠀⠀⢀⠀⠀⠀⣼⡄⠿⠿⣶⣾⣷⣆⠀⠆⠀⠢⠀⠀⡀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠃⠀⠀⠈⠋⠁⠀⠚⠀⠀⣿⡧⠆⣾⣿⣿⣿⣿⣷⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠴⣿⠃⠀⠀⠀⠀⠨⣺⠀⠀⠀⡀⣿⡇⠀⣿⣿⣿⣿⣧⣾⠀⠀⠡⠈⠀⣤⣤⣤⠐⣿⠆⠀⠀⠐⣾⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⢀⣀⠀⠒⣦⣼⠀⠘⠁⢀⠻⣿⣿⣿⣿⠟⠀⠘⠀⠀⠀⣿⣿⣿⠁⡄⠀⠂⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠾⠾⠀⠀⠀⠀⠛⠈⠿⠁⠀⢀⣾⣿⡇⠀⠧⠓⠹⣿⣿⠏⣴⣿⣦⠀⣶⣶⠉⠉⠉⠀⢰⡄⣤⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠙⠀⠗⠀⡀⠘⠀⠀⠸⣿⣿⣿⠇⠉⠉⣤⣶⠀⠀⠀⠀⠠⠄⠀⠀⢸⠛
⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠀⣠⣤⣀⣤⡀⠀⠀⠀⡀⠀⣶⠀⠃⠀⠀⡈⠀⠙⠛⠉⠐⠀⣄⣉⣩⣤⠀⠀⠈⠁⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠷⠇⠀⠀⠀⡌⠀⡁⠈⡇⠀⠀⡀⠐⠀⠹⢠⡄⡀⠀⠘⣄⠀⠀⠀⠀⠀⠛⠋⠻⠟⢁⣄⡀⢴⡿⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠑⠄⠄⠠⠂⠀⠀⢀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠠⢸⣿⣇⣦⡄⠀⠀⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⢰⣿⣿⣿⣿⡧⠀⠀⢠⠇⠀⣴⣦⣶⣦⢠⣤⣄⠶⠀⠤⠌⠈⢛⡃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠿⠿⠿⣿⠇⠀⠀⠃⠀⢀⣘⠻⠟⠃⠙⣶⣯⣤⣬⣀⠀⠐⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠿⢻⡗⠀⢺⡇⠰⡷⠘⣛⣃⠺⠷⠰⣯⣿⣿⣿⣿⡆⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠛⠛⠀⣀⠠⠀⠤⡐⠈⠛⠛⠃⣀⢀⣙⣻⠻⠛⠋⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡻⠃⠀⡠⡀⠀⠀⠀⠀⢈⣏⠀⠁⠀⠀⠀⠀⢻⠁⣋⠹⠿⠶⠿⠃⠀⠀⢠⠀⠀⠸⢿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠜⡔⣡⠒⢀⡄⠀⠀⠁⠀⠀⠀⠂⠀⠀⠚⠂⠛⠀⠀⠀⢀⡀⣠⠲⡀⠱⡄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢔⠁⣠⡋⢠⠃⢰⢂⡄⢄⣀⠀⠀⡀⢀⡄⢠⠠⠆⢂⠀⢣⠈⢣⠘⢤⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠡⡃⠀⡆⢸⠺⠈⠇⠀⠀⠃⠀⡇⢸⠀⢟⠀⢱⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠰⠀⠀⠀⠥⠀⠳⠈⠂⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆
⠀⢸⠉⢳⢸⡇⠀⢀⢷⠀⣾⠉⢳⢸⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡀
⠀⢸⠓⢣⢸⡇⠀⣸⢸⡆⣿⠀⢀⢸⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠠⠶⣦⣤⣀⠀⠀⠀⣀⣤⣶⠶⠦⡈⢿⣿⣿⣿⣧
⠀⠸⠤⠜⠘⠧⠤⠏⠀⠇⠹⠤⠞⠸⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢀⣤⣶⡤⡉⠀⠀⠀⠉⢡⣤⢀⠀⠀⠘⣿⣿⣿⣿⡆
⣿⣿⢡⠙⣿⢹⠋⡏⢹⠙⡏⢹⠉⣯⠉⣹⠉⣭⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠈⠓⠂⠁⠀⢠⠀⠀⠀⠀⠛⠊⠀⠀⠀⢿⣿⣿⣿⣿⡄
⣿⣿⠀⠞⢻⠀⢠⠁⣿⠀⡅⢸⠀⣿⠀⣿⠀⣭⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆
⠿⠷⠈⠤⠹⠆⠾⠠⠿⠠⠇⠸⠄⠿⠄⠿⠄⠉⠿⠿⠇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⡀⢀⠀⠤⢀⠀⠀⡄⡀⠀⣀⢀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⢰⡀⠹⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠃⠀⠃⠀⠑⠀⠀⠐⠁⠃⠃⠀⠓⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠛⠒⠲⠶⠂⠀⠀⠀⣠⡈⠃⠀⠙⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⣀⢀⣴⣾⣿⣿⣦⡀⠀⠀⠙⢿⣿⣿⣥⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠿⠛⠋⠁⠀⣹⣿⣿⣿⣿⣿⣶⡄⠀⠀⠙⠿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠙⢿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⢠⠀⠀⣦
⠀⠀⠀⡀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉
⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠈⠉⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁⠶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠂
⠀⠀⠀⢶⡄⠀⣀⣼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠈⠉⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄
⠀⠀⠀⠀⢠⠘⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠔⠀⠀⠀⠀⠸⠀⠂
⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠠⠒⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠁⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠂⠀⠐⠉⠁⠀⠀⠀⠀⠀⢀⠤⠆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠲⠆⠀⠈⠑⠒⠒⠲⠀⠀⠀⠀⠀⠠⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣺⠄⠲⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀
⠀⠀⠐⠻⠁⠀⠀⠀⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣀
⠀⠀⠀⠾⠖⠀⠀⠀⢲⣤⡆⠀⠀⠸⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⢹⡦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣟⠉⠁⠀⠀⣰⡆⠀⠋⠀⠈⠁⠀⠐⣶⡄⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠙⢿⣶⠿⣷⡀⢀⠀⠀⠀⠀⠀⣹⣿⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠉⠁⢹⣇⣄⣀⠀⣴⠿⠃⢀⣿⣏⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣾⡏⠙⠉⠀⠀⠀⠀⢸⣿⡉⠀⢀⣻⣷⣄⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⣶⡀⠀⢠⣶⡿⢷⣾⠿⠋⣿⣿⢟⣿⣿⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣯⣿⣆⢸⣿⣷⣤⣾⣿⢿⣿⣿⣿⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⡏⠙⢿⣿⣿⣿⣿⡿⠟⣿⣷⣼⣿⣫⣽⣿⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢿⣿⣦⣿⣿⣿⢿⣿⣿⣿⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⣿⣷⡌⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾
⢳⡀⠉⠛⠶⣤⣤⣀⣀⣀⣀⣤⣤⣤⡶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢦⣤⣤⣄⣀⣀⣀⣠⣤⡴⠞⠋⠁⣰
⠀⣹⢦⣀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⣠⢾⡁
⠀⠻⣄⠈⠉⠛⠒⠒⠶⠶⠶⠒⠒⠉⠁⠀⠀⠘⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠚⠋⠉⢀⡼⠃
⠀⠀⠈⢻⣶⣤⣤⣤⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⣌⠳⣀⠀⢀⣤⣄⠀⢀⣴⢋⡜⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣠⣤⣤⣴⣾⠋
⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡟⠁⠀⠀⠙⣯⠴⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛
⠀⠀⠀⠀⠀⠈⠛⢷⡶⠤⠤⠤⠒⠊⠉⠀⢀⡠⠀⠀⠀⠈⣇⠀⠀⠀⢀⡏⠀⠀⠀⠀⡀⠀⠈⠁⠒⠒⠤⠤⠴⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣄⣀⣤⡴⠞⠋⠀⣀⡤⠶⠶⠾⢷⡶⣶⠿⠶⠶⠦⣤⣀⠀⠑⠦⣤⣀⣀⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣤⣤⡤⠖⢋⣤⣴⣦⣤⡀⠀⡇⡇⠀⣠⣤⣶⣦⣍⡓⠦⣤⣤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⢿⣾⣿⣆⡇⢇⣾⣿⣾⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣤⡀⠈⠙⠉⡇⡏⠙⠉⠀⣠⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣤⡇⣧⣴⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⠉⡏⡏⠙⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⡇⡇⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⡇⡇⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⡏⡛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁⠀⡇⡇⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⡇⡇⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣇⣥⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⡆⡏⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⡇⡇⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⣇⣧⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⡇⡇⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⡇⡇⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡇⣣⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⡏⡿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⡇⡇⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⡇⡇⠀⠠⠞
⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣿⣭⣭⣭⣭⣬⣭⣽⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣥⣽⣿⣧⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⠒⠂⠘⡷⠊⠉⠉⡩⠋⠄⠈⠉⠭⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⣴⣶⣶⢖⣶⣲⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣠⠊⠉⠈⠀⠀⠘⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⣤⣄⣨⣧⠀⠀⠀⠀⢀⣤⡞⠛⢿⡿⢻⣟⣿⡟⣵⣿⣿⣿⡿⠿⣟⣟⣛⣿⣦
⠀⠀⠀⡿⠁⠀⠀⡀⡀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⣿⣿⣿⢷⡀⣆⣿⢯⡇⣼⠷⣿⠟⠉⢐⣫⣭⣿⣿⣿⣿⡀
⠀⠀⠀⣇⠀⠠⡀⡱⠳⠃⢀⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⣿⡬⣍⡀⢻⣿⣿⢸⢰⡇⣼⣧⡢⢪⣾⣿⠿⣿⣿⣿⣟⠳⡄
⠀⠀⠀⠘⡆⠀⣹⠀⠀⠀⠀⠉⢼⣠⡇⠀⢀⠔⠀⠀⠀⠀⠀⠒⢦⠰⠀⠀⠘⣿⠄⣿⠎⠀⠀⠈⠙⢿⡷⣿⣠⣾⣿⡦⢶⣶⣶⣶⡯⣝⢻⡀
⠀⠀⠀⠀⠈⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⡏⠀⠀⠀⠀⠀⣐⢤⣸⡏⠀⠀⠀⠘⠢⠟⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⡵⠿⠿⣿⣿⣿⣿⣾⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠡⡁⠁⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣞⡿⢷⡒⢭⡙⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣦⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠚⠉⣹⣿⣿⣿⣦⠙⣦⠟
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠐⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠷⠿⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⠉⢉⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⠟⠋⠉⠄⠈⠁⠀⠠⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠐⠈⢀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣰⠄⠀⠀⠤⠈⠀⠀⠉⠀⣀⣤⣴⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⡈⠷⠆⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠊⠁⠀⠒⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⣀⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠘⠁⠀⢁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣭⣴⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠒⠂⢌⣭⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⡁⠞⠁⠄⠀⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠂⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡦⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣥⣶⣿⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣯⠙⣭⣽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⣀⣄⡀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣫⢉⠥⠚⠢⣌⢽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⢸⣇⢀⡿⢸⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣏⠀⠀⠀⢸⡌⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣟⡘⢆⢠⡠⣞⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢰⣶⠄⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡸⣌⣷⣽⣷⣹⣮⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣵⣶⡜⢤⣽⣿⢿⡇⠙⣾⠽⢛⡵⡞⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⣠⢿⣯⡾⠃⡇⠀⣻⡿⣧⢣⣤⣿⡴⠈⠉⣻⠂⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢄⣀⡀⡀⠀⠀⠀⠀⠀⡰⢡⡿⢋⡴⠻⡘⣴⣿⣛⡁⢿⣿⣿⡿⡰⠳⣌⢿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠻⠴⠛⠧⠇⠀⠀⠀⠀⢰⡣⣦⣌⣯⡀⠀⠹⡈⠛⣿⣷⣌⡟⠋⣰⠁⠀⣨⣟⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣠⠀⣤⣤⡄⠀⠀⠀⠀⠙⠺⠿⣿⡿⠓⢦⣇⡠⣿⣿⣿⡅⢀⣧⠖⣫⣿⡽⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⣰⡯⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣿⡿⣾⠿⢿⣿⣾⣿⣿⡧⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⡁⢿⣿⣯⠻⣊⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠿⠿⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⣀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⣲⣶⠢⠀⠀⠀⢀⠋⣴⣿⠏⢣⠀⠀⠀⢠⣶⣶⣖⢄⠀⠀⠀⠀⢻⡾⠁⣏⠈⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢠⢃⠈⡽⠿⠔⠀⠀⠀⠈⣴⣿⣶⣷⡜⠀⠀⠀⠘⠬⠽⡏⣠⣧⠀⠀⠀⠀⠃⠀⠙⠛⠉⠈⠛⠋⠁
⠀⠀⠀⠀⠀⡐⠻⣿⣷⣇⡀⠀⠀⠀⠀⠀⣏⡛⢋⣉⡅⠀⠀⠀⠀⠀⣀⡿⠟⠻⢷⡀⠀⠀⠀⣤⠀⣄⡄⣤
⠀⠀⠀⠀⠀⡟⠶⣿⡿⠿⠟⣉⣿⢽⣎⠒⣩⡟⠆⣧⡉⢲⢮⣙⣭⡉⣠⣾⢯⣿⣾⡇⠀⠀⠘⠉⠛⠋⠃⠛⠂
⠀⠀⠀⠀⠀⠈⠒⢽⣺⣺⠽⢹⣿⣮⣽⣾⠈⣴⠟⠻⢿⡿⠟⠛⠛⢹⣿⠏⣈⠝⠋⠁⠀⠀⠀⠀⠶⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⡳⠿⠿⣿⡼⣇⣴⡂⢸⢅⣴⢻⡿⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⣿⠏⢐⢿⣿⠿⡀⢸⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠹⠤⠗⠧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢤⣷⣦⢩⣶⣿⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢽⣿⣿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠆⢠⣟⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠁⠉

⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠚⠛⠛⠛⠳⠶⣖⡒⠒⠛⣛⣓⣲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣦⠀⠀⠉⠉⠉⠉⠙⠓⠊⠃⠀⠀⠉⢻⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⠒⠒⠾⣿⣿⣿⢿⡆⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⣈⠉⠿⢾⣿⡿⠟⠛⠛⠻⡷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⡇⠠⢤⣴⣺⣿⣿⡸⣷⣀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠉⠙⠃⢸⣿⡷⣦⣤⡤⣤⡈⠀⢹⣧⡀
⠀⠀⠀⠀⢀⡴⠶⣴⣾⣿⡞⠇⠀⠀⠉⠉⢛⣿⣇⠙⣷⡇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡗⠁⠀⠁⠙⣿⢀⣽⣟⣉⣛⡛⠲⢦⡀
⠀⠀⠀⣠⡟⣰⠋⠈⠃⢸⣿⣿⡄⠀⠀⠀⠀⢸⣿⣮⣿⠁⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⠏⠀⠀⠀⢀⠟⢨⣿⡿⠋⠉⢡⣾⣿⣷⡀
⠀⠀⠀⢿⣽⣿⠇⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⣺⣿⡏⣿⡆⠀⢸⣿⠀⠀⠀⠀⡀⠀⢈⣿⣿⣿⡀⠀⠀⠀⠈⣼⣿⡿⠒⠀⠀⣮⣽⣿⠿⠿⡷⢶⣶⣄
⠀⣠⠶⣾⣿⣿⣆⠀⢢⡠⣿⣿⣿⣿⣄⢀⢸⣾⣿⣧⢿⡷⠀⡀⠟⠀⠀⠀⢀⡇⡀⣾⡿⣿⣿⣷⡄⣶⢀⣼⣯⣿⣗⠀⠀⣶⣿⣿⣟⠓⢦⣌⠻⣯⣿
⣸⣷⣚⣚⣙⣿⣾⣿⣦⣿⣿⣿⣿⣿⣿⣼⣼⣷⣿⣿⣿⣿⣾⣡⣦⣄⣄⣀⣸⣥⣿⣿⢷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣀⣤⣿⣿⣿⣾⣀⡀⣉⣙⣈⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⢿⡿⠿⠿⣿⣿⣿⣿⠿⣞⡻⢿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⠟⠛⠛⠻⠿⠷⠾⠛⠁