Letterbox Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶⣄⠀⣀⠴⠚⠁⠀⠀⡈⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠃⠈⠟⠁⠀⠀⠀⠀⢠⣁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⢰⡀⠀⠀⠂⡀⠀⠀⠀⠌⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⢠⠏⠀⡅⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⡇⢀⠈⠀⡰⠆⠴⣁⠀⡀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⢹⣄⠠⠐⠂⠁⢀⠾⠀⢀⠁⠐⠀⠀⠈⠉⠒⠺⠥⣀⡀
⠀⠀⣀⠤⠐⠊⠉⠀⠀⠀⠀⢀⠎⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⣻
⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠠⠐⠚⠢⡀⠀⠂⠁⠀⠀⠇⠀⠀⠀⠀⢀⣤⣀
⠘⢃⠂⠠⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠒⠀⣠⠄⠀⠀⠀⠀⠀⠀⢠⠀⡸⠀⠀⠀⣠⠞⠁⠁⠈⠓⢄
⠀⠀⢣⡀⠀⠀⠒⠀⠐⠁⠀⢀⣠⠴⠖⠛⠓⠶⠦⠤⣤⣄⢀⣀⣀⣳⠃⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠱⡄⠈⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⠞⠁⠀⠀⠀⠀⠰⠤⠀⠀⠀⢘⡇⠀⣀⣀⡀
⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⠂⠀⠀⢀⡴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠒⠈⠈⠁⠸⠀⠀⠀⠀⡠⠛⠒⠒⠒⠒⠒⠒⡰⠃⣀⠔⠒⠁⠈⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡱⢄⠀⠀⡄⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⢠⠊⢠⠚⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡰⠀⠀⠑⠢⣧⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠃⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⡀⠀⡆⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⢶⡶⢤⣬⣭⣬⣥⣀⣀⣀⣀⣀⣀⣉⣈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⢂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠦⠼⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠤⠤⠤⠤⠤⠤⠄⠀⠤⠤⠤⢤⡤⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⠀⡀⣀⠀⠀⢀⠀⣀⠀⡀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠹⠁⠆⢀⠉⡆⠙⠠⠏⠈⠙⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⠸⢀⣤⣰⢸⢈⠑⢀⠐⡀⠀⠐⣀⡀⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠐⠚⠀⠙⠻⠎⠣⡿⠤⠬⠗⠤⠤⠄⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡏⠀⠉⠉⠉⠉⠙⠛⠛⠓⠒⠲⠶⠶⡄⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⢸⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣘⣠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣼⣶⠋
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡤⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢄⡀
⡔⠒⠋⠀⢈⡁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢙⠆⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠁⠢⢄⡀
⠈⢂⠀⠂⠀⠆⠐⠃⠀⠀⠂⠐⠀⠀⠊⠀⠐⠘⢠⠎⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⠀⠀⠀⠀⠈⢰
⠀⠸⣀⡀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⡜⠀⠀⠀⠀⣀⢀⡀⠈⠑⢄⡀⠀⠀⠀⠂⠀⠀⠌
⠀⠀⠈⡇⠀⠐⠒⠒⠒⠒⠒⠒⠀⠂⠒⠒⠂⢀⡇⠤⠒⠉⠀⠀⠀⠑⠀⠀⠀⠀⠑⠢⢄⡀⢀⡔
⠀⠀⠀⡇⠀⠀⠀⠀⢸⠋⠉⠉⠛⠫⠁⠀⢸⠡⠀⡀⠀⠀⠀⠤⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠲⢄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠒⠒⠒⠀⠉⠈⠀⠈⢃⠀⠀⠈⠀⠀⠀⢀⡀⠜⠁⠀⠀⠀⠀⣀⡠⠔⠚⠉⠀⠀⢀⠀⠈⢆
⠀⠀⠀⢸⠀⠀⠀⠀⡤⠠⠤⠀⠀⠀⠄⠀⠀⠀⡦⡄⠤⠄⠒⠉⠀⠀⠀⠀⠀⠀⢔⣉⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠈
⠀⠀⠀⢸⠀⢀⠀⠀⠀⠈⠀⠀⠀⠈⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠉⠀⠰⠀⠤⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⡐⠀⠀⠀⠀⠀⠔⠒⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠀⠀⡀⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠤⠖⠒⠉
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⢀⠤⠤⡠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢉⠤⡤⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇
⠀⠀⠀⠀⡔⠃⠐⠒⠐⠒⠒⠒⠒⠒⠒⠒⠘⠒⠳⡀
⠀⠀⠀⠀⡇⠠⡀⢄⠠⠄⠠⢄⡠⠄⠠⠄⠠⠦⠄⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠄⠠⠀⠠⠀⠠⠀⠠⠀⠠⠂⠂⠀⡇
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⣶⣶⡶⣶⢾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠈⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⢀⡠⠤⠒⠒⠒⠤⢄⡀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⡜⠁⠀⠀⢀⡴⠊⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⡇⠀⣠⠔⠁⠀⠀⠀⠀⡸⠘⣄⠀⢠⣴⢦⣄⠀⠀⣿⠀⠀⠀⠀⠈⠢⣄⠀⠀⠆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣧⠞⡇⠀⠀⠀⠀⠀⠀⣇⡀⠈⠳⠿⣁⢀⣿⣦⢞⠇⠀⠀⠀⠀⠀⠀⢸⠳⣄⠂⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠁⠈⠢⣀⠀⠈⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠉⠑⣢⣤⣀⣤⣔⠊⠉⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢲
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣒⣉⣁⣀⣀⣀⣈⣉⣒⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⢺
⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠬⠿⠦⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⠛⠛⠛⠛⠛⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠁⠀⠀⢸⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠳⣤⡀⠀⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⣀⣼⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠶⣍⡛⣛⣛⣛⣃⣀⣀⣀⠀⣠⡴⠞⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠉⣩⡵⠟⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⠔⠃⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠁⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⠋⠙⠙⠛⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⣤⣤⣦⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣾⡏⠁⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⢋⠏⢉⢉⡿⠀⢹⣿⡛⠻⠛⡟⠻⠛⠻⠛⠿⠟⢿⢿⠿⠿⢷⢶⡶⣶⢶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⣝⣿⠀⢀⠈⠈⡇⠀⢨⣿⠀⠀⠀⠰⡀⠀⠀⠀⠀⢡⠀⠂⠆⠀⠄⡄⠑⠈⠄⠳⡘⡌⢢⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠻⣯⣿⡇⠀⠀⢰⡇⢀⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠘⠀⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡳⣣⢿⡇⠀⠀⢸⣀⣈⣿⡇⠠⠤⠤⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⣯⢿⡇⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡷⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣾⣸⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣼⣸⣿⣏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡼⣹⣿⡃⠀⠀⠄⠀⡠⢄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠚⢿⣿⣿⣿⣿⣃⠀⣀⢔⡫⢒⡡⠖⣡⠴⣋⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⢸⡇
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠛⠋⣉⣥⣴⠶⠾⣛⠿⣛⣿⣿⣿⣷⣮⣴⣩⣴⣩⣴⣯⣵⣊⣤⣄⣡⣴⣠⣤⣀⣤⣤⣤⣤⣨⣴⣨⣄⣠⣴⣠⣄⣥⣆⣼⡇
⠀⠀⠀⡴⠞⢛⣉⣁⣤⡴⠶⠛⠋⠍⡲⠞⠛⠉⣒⣩⣄⣤⡤⢴⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠹⠋⠉⣻⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⣿⠾⠿⣯⡥⣤⢤⠶⠶⠶⠶⢶⢛⣛⣫⣭⣭⡷⠶⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡐⢹⡇
⠀⠀⠀⠈⠛⠓⠓⠒⠒⠒⠓⠚⠚⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⡎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠠⢆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠀⠀⣻⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⣩⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠈⡷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⣟⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⡫⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠕⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠛⠛⠛⠒⢶⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⡼⠤⠤⠤⠒⠒⠋⠉⠀⠉⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠄⠐⡇⡴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠙⠳⣄⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⢷⡀⠀⡇⢈⣀⣀⣠⡤⠤⠶⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣷⠚⠇⠸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⢀⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⢀⣀⣀⣤⡤⠴⠶⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠉⠉
⠾⣭⣉⡉⠉⠉⠉⠉⠙⠛⠻⠭⢤⣀⡀⠀⠀⣿⠀⠀⠀⣀⣠⣤⣴⡶⣿⡟⠋⠁
⠀⠀⠈⠉⠙⠓⣒⣒⣒⣒⣒⣀⣀⣀⣀⣉⣰⣿⣴⠾⠿⠛⠋⢹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠔⠒⠒⠒⠊⢉⠙⡆⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠓⠉⠉⠉⠉⠀⠀⢀⠌⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠉⠐⢠⠄⠀⢀⡐⠣⢀⠀⠰⠀⠀⠀⠀⠀⠐⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠐⠃⡢⠀⠀⣀⡀⠀⢁⡤⠀⠀⠀⠠⠊⠐⠄⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⢀⠔⠉⠙⠊⡀⠀⡐⠁⠀⠀⠀⠀⠀⢈⣠⡼
⠀⠀⠀⠀⣀⣠⡤⠴⠒⠒⠚⢛⣋⠉⠭⠽⠛⠿⠿⡀⠈⠢⡀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠲⣀⣀⡤⠴⠒⠘⠉⠁
⠀⣠⠖⠋⠉⠀⠀⣠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠘⠐⠤⠔⠈⠉⠒⠀⠀⠀⢀⢀⠀⣀⡀⠀⠀⠀⠈⠳⣄
⡼⠁⠀⠀⠀⠀⡘⠁⠀⢀⣀⠤⠀⠀⠶⠀⠀⠀⠉⢳⣌⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡁⡮⠀⠀⠀⠡⠀⠀⡀⠀⠀⠈⢳⠄
⡇⠀⠀⠀⠀⠀⢰⠀⡛⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡅⠈⡅⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⡅⠀⠀⡰⠋
⡇⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⣠⠞
⡇⠀⠀⠀⠀⠀⠈⠀⢰⠄⠚⠓⠀⠉⠉⠙⠂⠦⢀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠁⠀⣀⡤⠶⠚⠋⠙⠳⡀
⡇⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠵⠾⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠒⠉⠀⠀⠀⠀⢀⠃⠀⠱⡀
⡇⠀⠀⠀⠀⠀⠀⡇⠸⠄⠤⠄⠐⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢁⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠣
⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠤⢄⡔⠊⢉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠉⠀⠐⠦⠄⠤⠂
⡇⠀⠀⠀⠀⠀⠀⡇⠀⡆⠉⠉⢡⡎⠀⠈⢳⠦⣄⠉⡏⡆⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠁
⢹⠀⠀⠀⠀⠀⠀⡇⠀⡇⢠⠤⠚⡅⠀⠀⡀⢏⣁⡠⠃⠑⠂⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣃⣀⠤⠔⠒⠉
⢸⠀⠀⠀⠀⠀⠀⡇⠀⢇⡰⠀⠀⠈⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⡁
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟
⠀⠉⠓⢦⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠈⠙⢦⣈⣀⣀⣀⣠⡤⠔⢲⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣇⠀⠸⣗⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⡟⠷⣠⠁⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⠟
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣄⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣄⣀⣀⡀⢀⠀⠀⠀⠀⠀⣿⣿⣟⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣯⡉⠙⢶⣤⠀⠀⣿⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢯⢧⠀⣿⠟⠒⠦⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣯⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠘⡞⡄⠿⡄⠀⠀⠀⠀⠻⡿⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⡀⠀⠉⠉⠉⠉⢉⣻⡟⠃⠀⠀⠇⡇⣠⣁⠀⠀⠀⠀⠀⢱⢱⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠙⢦⣀⠀⠀⠀⠀⠸⡏⠈⠀⠀⠀⠸⣿⠲⢰⠀⠀⠀⠀⠈⡆⡇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠓⢤⡀⠀⠀⣀⣶⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⡇⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠉⠳⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡴⠋⠀⠀⠀⠀⠀⠸⠋⣿⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣈⣉⣁⣀⣀⣀⡠⠴⠒⠋⠉⢉⣧⣬⣍⣉⣉⣉⣉⡉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⣉⣻⠿⢿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠐⠚⢿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢰⠭⠤⠶⠿⠍⠐⠒⠂⠉⠀⠀⢀⣀⠠⠄⠂⠈⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠤⠖⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⡀⣒⠰⠿⡿⢭⠯⣒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠐⠂⠀⠉⠀⢀⣀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠒⠂⠥⢄⠀⠀⠀⠀⠀⠀⠐⡏⠑⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠳⡜⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣟⣵⣛⣛⠷⠶⣦⣄⠉⠢⡀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡏⢸⣟⣁⡄⢀⣥⡶⠙⠏⡀⢡⠀⠀⠀⠀⡇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⢸⡍⠐⢤⣀⠀⠢⣠⣷⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣸⣇⠀⢀⠈⠽⢖⠢⠿⠀⠀⠀⠀⠀⣨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠛⠋⠉⠀⠀⠀⠀⠀⠁⡒⠀⡀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠀⢨⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠉⠑⠒⠦⠤⣀⡀⠀⠀⠀⠀⢽⡄⠃⠄⠀⠀⢸⣧⠘⠒⢠⠀⠀⠀⠀⠀⠀⠐⠘⠀⠀⠠⠀⢉⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠉⠓⡶⠤⢼⡆⠂⠀⠀⠀⠘⣿⠀⠋⡰⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⡈⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇⠀⢀⠀⠀⢀⠫⢻⢾⡅⠀⠀⠀⠈⢀⡀⠀⠀⠀⠂⢈⡁⠀⠼⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⡄⠀⠀⢀⠎⠀⠀⡀⠀⣿⠀⢸⠀⠀⠀⠑⠀⡀⠀⠀⠀⣈⡈⠠⠄⠐⠂⠈
⠀⠀⢀⣀⣤⢤⠶⠾⠛⣋⣉⣤⣽⠽⢄⢀⠎⠀⣀⣠⣤⣶⣿⢻⠟⡰⠙⠤⡦⠆⢒⠊⡋⠡⠺
⡰⡾⠛⢉⣠⠴⠶⠞⠿⠙⠉⠁⠀⠀⠈⠛⠚⠛⣩⣿⡿⠟⢋⣬⠟⠀⠀⠀⣢⠐⠁⢀⠀⢈⢼
⣇⣦⣾⢉⠁⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡶⠞⠋⣁⡤⠗⠋⠀⠀⠀⠀⠀⠘⠂⠐⠑⠡⠀⡽
⠡⡈⠛⠛⠷⣶⢢⣤⣤⢤⡤⠶⠛⠃⠉⣇⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢑⠀⡠⡯
⠀⠀⠉⠒⠐⠰⠤⠤⠤⠤⠴⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⡪⡪⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠙⣆⠀⣀⠀⠇⠀⠀⠀⣠⠔⠊⠀⠰⡀⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠐⠒⡠⠼⠁⠀⠀⠀⠀⠀⠈⢅⠀⡠⠔⠚⠁⠀⠆⡐⠀⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠀⠀⠀⠀⠀⠀⠠⠄⠠⢀⠉⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⠀⠀⠀⡀⡄⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠂⠀⠀⠀⢀⡤⠒⢹⡆
⠀⠀⠀⠀⢠⠊⠀⠆⠡⠇⠈⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢁⠰⠀⠀⢾⠁⣀⣀⠎⢷⠈⠉⠁⣸
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⡶⢤⣆⢰⢣⠀⢈⠀⠀⠀⢸⠀⡄⠀⠈⢄⢀⡠⠴⠊⠁⠀⠀⡇
⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⢸⠀⡅⠀⢰⡜⠘⠄⠃⠒⠊⠀⠸⢀⠇⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⡧⠄⠀⠀⠁⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠠⠤⠤⢐⢶
⠀⠀⠀⠀⠛⠤⠤⠴⠦⢤⡃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠤⠴⢋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢉⠉⡄⠀⠀⠦⠤⠤⠤⠤⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠶⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠤⣀⠠⠊⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠒⠒⠒⠒⠲⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠆⠀⠀⠀⢀⠊⠠⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠣⡀⠀⢀⣄⣘⡁⠠⠌⠤
⠀⢀⡀⠀⠀⡀⠀⠀⠀⡀⠀⢀⡈⠓⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⡀⢤⢄⣀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠰⣿⠉⠐⠒⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠢⣄⠀⠀⠀⠈⠉⠉⠉⠉⠛⡿⢿⡇⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⠙⢦⣄⠀⠀⠀⠀⣠⠎⠀⠀⡇⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⡼⠛⠦⣤⠞⠓⠦⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣠⣾⣷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⠟⠋⣠⠴⢻⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣠⡴⠋⠁⠀⢸⡷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣛⣡⡴⠛⠁⠀⠀⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⣝⡉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣛⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣦⡾⣿⡷⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⣿⢐⣒⠂⠂⢠⡟⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣴⣿⠀⠠⠤⠀⢸⡵⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⠉⠉⠩⢋⡔⡡⣿⡇⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣠⣋⣴⢷⣿⠀⠠⢂⣠⣏⣾⣏⠐⠀⠀⡠⡀⠀⠀⠀⢀⠄⠀⣀⠀⢀⡀⣀⢀⡀⣀⠀⡀⣈⠉⣉⣁⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠉⠙⣡⢎⣿⡄⢠⢻⣿⣿⣿⣿⠆⠀⡜⡑⡵⠑⠠⢀⠎⡜⢠⠋⠠⠉⠠⢥⡬⡤⠅⠴⠴⠂⡴⠂⠖⠲⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠐⢣⢎⣼⡇⢀⢼⣿⣿⣿⡿⠃⢌⠜⡜⠌⠀⢁⢊⡜⠠⠃⢠⠃⠀⢀⠀⡐⢀⠌⠘⢁⠚⢱⣋⣴⢋⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠐⢣⠎⣸⡇⠀⠀⠀⠈⠁⠀⢈⠊⠈⠈⡐⠠⠃⠜⠀⠁⠠⠁⠀⠠⢂⠄⢀⢂⠀⠠⠂⡀⢀⠔⢂⣶⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢃⡜⣹⡆⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠘⠐⢁⠊⠠⠡⠀⡀⠁⠀⠀⠁⡠⣮⣌⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠞⡰⣿⣥⡶⣶⡀⠂⠒⠐⢒⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⡴⣠⠾⡴⡿⢫⢼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⣚⣋⣿⣾⡥⠿⢋⣶⣾⣿⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣤⡠⠤⠤⠠⠀⠀⠈⠉⣉⣙⣋⣋⣋⣹⣉⣼
⠀⣀⣀⣠⣤⣶⣶⣿⠿⢿⣛⣛⣯⣭⡽⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣏⣿⣻⡻⣿⢿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⣿⣛⣛⣭⡭⠵⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣽⢸⡱⣧⢾⣿
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢙⣸⡡⣇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠘⢸⢡⡏⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣽⢸⢃⣏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⢣⡏⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⢠⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⣡⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠘⢡⠞⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢉⢰⢺⡴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠈⢸⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢰⢸⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⢘⢸⢫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⠋⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢸⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠻⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠘⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠐⣀⣀⣄⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣀⣼⠟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡶⠶⢶⠶⠶⡇⠀⣿⣿⠶⢶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⢾⢿⡇⠉⠀⠆⢳⡇⠀⢺⡟⠂⠈⠀⢣⠀⠡⠀⠑⠀⠱⠈⢃⠹⡈⠅⠹⠙⢏⠻⡙⢍⠻⡙⢟⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣻⣾⡇⠀⠁⠀⢸⠀⢀⣾⡇⠈⠀⠀⠀⠆⠀⠁⠀⠄⠀⠇⠈⠀⠁⠘⠀⢃⠈⠆⢱⠈⡆⠰⠈⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢼⡹⣿⣷⠀⠀⠀⣼⠀⠜⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢌⡷⣯⣿⠀⠀⠀⣷⣶⣾⣿⠁⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡮⣵⣻⡿⠀⠀⠀⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⡱⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡾⡤⣿⣏⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢼⣥⣿⣯⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⣸⣡⣿⣏⡠⠊⠠⠊⣀⠴⣂⠄⣠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠖⠛⣻⣿⣿⣿⣿⣏⠀⢀⡤⢚⠕⢊⠴⢊⣡⠖⣋⠔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⠄⢀⣿
⠀⠀⠀⢀⣀⣤⣴⠶⠞⠛⢉⣉⣤⣴⠶⢞⡻⣿⢿⣿⣿⣿⣿⣶⣥⣎⣡⣞⣥⣶⣫⣴⣋⣤⣄⣁⣴⣠⣤⣀⣠⣤⣤⣤⣄⣡⣴⣡⣄⣡⣴⣡⣔⣡⣆⣩⡿
⣠⠶⠛⠋⣉⣠⣠⡴⠶⠛⡛⠭⣴⠶⠛⠫⢙⣊⣉⣰⣬⠤⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⢻⠉⠹⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠁
⣿⣶⣿⣿⣭⣭⣤⣤⣤⣴⢶⠶⠶⠶⢛⣛⣛⣩⣭⣿⡾⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠘⢘⠜⣿
⠘⠳⠶⠦⠦⠴⠴⠴⠶⠶⠞⠒⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⣨⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⣽⢂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢡⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠰⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠼⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢴⡳⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢈⠀⢺⡝⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠐⠀⢹⠎⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡃⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢘
⣿⣿⡿⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠦⣄⡴⠚⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣸⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧⠀⠀⢾⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣈⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣿⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⢯⣄⣀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡧⠶⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠛⠻⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⠉⠉⠉⠉⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢿⡿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢦⣄⠀⠀⠀⠀⠀⠀⣀⣴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠛⢦⣄⣠⡴⠛⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣇⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣟⣿⢻⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢀⠠⡉⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⠟⠛⡄⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠈⠀⠈⠙⣿⡏⠀⠄⠀⠀⠉⢿
⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⠀⠀⣀⣀⡠⠀⠀⣿⠉⠉⡄⠀⠰⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠠⠇⠀⠸⠄
⠀⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⢐⠀⠀⢠⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣴⡇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠠⠀⠸
⠀⠀⠀⡠⠚⣁⣀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄
⠀⠀⡀⠀⣈⠀⠈⠊⡵⠒⠉⠉⠢⡀⠀⠀⠀⠀⠀⡠⠀⠀⠄⠤⠤⡀⠀⠀⠀⠀⠠⠤⠆⢄⠀⠀⠀⠀⠀⠀⢀⠀⠆
⠀⠀⢉⡁⠀⠀⢀⡚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠱⠀⠀⠀⠀⠀⠀⢀⢠⢀⠀⠀⠀⠀⢸⢀⣰⠒⠀⠀⠀⢸⣀⣠⠀⠀⠀⠀⢠⣄
⠀⠀⠈⠍⠛⠢⠀⡇⢀⡀⡀⣀⠀⡇⠀⡀⢀⣠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠈⡀⠉⠀⠀⠃⠀⠈⠀⠈
⠀⠀⠀⠀⠈⠀⢰⠇⠈⠉⠉⣀⠀⡇⠠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣇⡄⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⡄
⠀⠀⠀⠄⠀⠀⠘⡂⠀⠀⣀⡠⠀⡛⠀⠀⠐⡁⠀⠀⠸⠀⠂⠂⠄⠂⠀⠀⢌⠀⠀⢀⠆⠀⠈
⠀⠀⠀⠀⠀⠀⢰⠘⠉⠁⠀⠀⠀⡇⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⢀⠴⢂⣈⣐⠲⡆⠀⠀⠀⠀⢀⡀⠐⠀⠄⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⢸⡠⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢃⠆⠀⠀⠀⠙⠘⡆⠀⠀⠀⠀⠙⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠦⠠⠀⠀⠒⠆⠀⠀⠀⠢⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⣧⠠⡄⠀⠀⠀⠀⠀⡄⢠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠘⠦⣀⢀⠀⠔⠓⠀⠀⠀⠀⠋⠉⠉⠉
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠁⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡏⠀⡇⠀⠀⠀⡔⠂⠀⠀⠴⠪⠒⠁⠀⠠⠀⠁
⠀⠀⠀⡇⢸⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⠀⣀⡀⠀⠛⣧⡄
⠀⠀⠀⠧⠼⠀⠀⠀⠈⠀⠀⢹⠀⠀⠀⠳⠤⠇⠀⠀⠀⡧⠔⠋⠁⠀⠘⣶⣤⠀⠠
⠀⠀⠀⠀⠀⠾⠐⠀⠀⠀⠀⠺⠀⠂⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠐⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⡆
⠀⠀⢘⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠟⠛⠛⢿⣿⣿⣇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣴⠞⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⡍⠛⠖⠀⠁⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣤⡤⠀⠀⠁⠛⠲⢤⣠⣄⡀
⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣴⠋⠀⠇⠀⡌⠈⠀⠈⠲⣄
⢀⠔⢖⠴⠷⠛⠼⠶⠶⠶⠴⣖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣻⣗⢹⠀⢴⠁⢀⠀⠀⠀⠼⣄
⢸⠀⠈⠛⢏⠉⠀⢈⠈⢉⢉⠉⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠁⠙⢻⣿⠊⠀⢠⡠⠀⠀⠀⠀⠀⠹⣦⡀
⢸⠀⠀⠀⠈⣧⣀⣈⣁⣀⣉⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠁⠀⠀⠋⢀⢀⠀⣤⣤⣀⠂⠀⢹⣄
⢸⠀⠀⠀⠀⠁⠀⢀⣀⣀⣀⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣧⢀⠸⠏⠁⠠⠏⠀⣘⡎⠀⠟⣿⣂⠤⣂⣸⣿⡆
⢸⠀⠀⠀⠀⠠⠀⣠⡏⠥⠬⠙⠃⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡼⣆⢤⡖⠀⢀⢘⠀⠀⠁⠈⡿⠃⠈⠹⢝⣿⠇⢄
⢸⠀⠀⠀⠀⢰⠀⠠⠥⠤⠦⠴⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣧⠉⡆⣀⠸⡆⠈⠉⠙⢷⠖⠲⢤⣀⣻⡿⠿⢴⠐⠢⣤⡀
⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣠⡔⠊⠀⢀⣉⡿⠆⣥⡀⢀⣧⠰⠀⢠⠤⣀⠀⣤⣾⠟⠀⠀⠀⠀⢀⣼⣅
⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡠⠼⢅⣀⡠⠠⢴⡉⠀⠙⢆⣨⣭⣿⢇⢠⠏⡉⠈⠉⠁⠀⠄⠀⢸⡧⠟⠁⠀⠀⢀⣤⠶⠏⠁⠈⢣
⠀⠓⢲⣶⣶⣾⣦⣶⣶⡖⠒⢲⣍⠀⠐⠠⠌⠒⠢⣄⢣⢀⣤⣾⠟⠉⠐⠀⣸⣿⣦⣄⣈⠀⠂⠈⣠⡿⠉⠛⠷⠶⠾⠋⠁⠀⠀⠀⠀⠀⣧
⠀⠀⠈⠉⠀⡇⠉⠙⣿⡇⠀⠈⠉⠉⠒⢤⣀⡴⠚⠉⠈⠙⠋⠀⠀⠐⠀⠀⣿⠃⠈⠹⣿⣶⣴⠊⠻⣷⣄⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⡇⠀⢰⠈⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⡿⠀⠀⠀⢈⠿⣿⡄⠀⠙⡟⠷⣶⡀⠈⢆⠀⠀⠀⠀⢀⡠⠋
⠀⠀⢸⡀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⠁⠀⠀⠀⠈⡂⠛⢿⣄⣀⣹⣶⣄⣀⠀⠈⠡⠀⢀⡤⠎
⠀⠀⠐⠂⠀⠇⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠁⠀⠀⠀⠀⣰⠳⡀⠀⢛⠛⠿⣿⣿⡿⣏⣠⠖⠋⠁
⠀⠀⠀⡄⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣾⠏⠀⢱⠀⠈⡀⠀⢿⣿⡇
⠀⠀⠀⡇⠀⡀⠀⠀⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠆⠀⠀⣰⢻⡇⠀⠀⠀⡇⠀⢹⠀⠀⠟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣿⠀⠀⠀⠀⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣤⡀
⠀⠀⠀⠀⠀⢰⣿⣯⣀⢸⡇⠀⠾⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠒⠒⠖⠲⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠀⠀⣿⢀⣨⣿⣷
⠀⠀⠀⠀⠀⢸⡇⠈⠙⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠉⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣬⣁⠈⠉⠉⠉⢉⣉⣉⠉⠉⠉⠉⢀⣩⣼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⡾⠛⠉⠙⠻⣦⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣇⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢿⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠈⣩⣿⡿⠛⠙⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⣿⡇
⠀⠀⠀⠀⠀⢠⣴⣿⣯⣁⣀⣀⣀⡀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡿⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠋⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣴⣿⣟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⣿⠛⠛⠻⠿⠿⠿⢿⣿⣶⣶⣶⣶⣦⣤⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⣴⣿⣷⣄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣿⠟⠋⠀⠙⢿⣷⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣿⣶⣶⣶⣶⣤⣤⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠈⠉⢹⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣤⣿⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣦⣤⣤⣤⣤⣀⣼⣿⠀⠀⣿⣿⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠀⠀⣿⣿⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⢿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠙⠲⢄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⠀⠀⠀⠀⠀⠙⠲⣄⡴⠋⢿⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⡠⠂⠁⠀⠀⠀⠀⠀⢂⡴⠋⠀⠀⠀⠙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡸⠂
⠀⠀⠀⠀⣠⠊⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠞⠣⡀
⠀⠀⠀⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡴⠋⠀⠀⣸⡆
⠀⠀⠀⡏⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⠋⠀⣀⠴⠊⠁⣇
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠞⣡⠔⠋⠁⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⡀⠉⠳⢤⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣴⠗⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣡⠖⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠲⢄⡀⠀⢀⡠⢤⣀⠀⣀⡤⠚⠁⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠽⠛⠉⡀⠄⡈⠙⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠐⠀⠀⠀⠀⠐⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠀⠀⠀⡏
⠀⠀⠀⢸⡄⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⡇
⠀⠀⠀⠈⠷⣮⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣽⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢺⢩⢣⢣⢃⠇⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⢇⢆⠃⡞⠆⠎⡌⡘⡜⡘⡳⣄
⠀⠀⠀⠀⠀⡤⠄⣀⣀⣀⣀⣀⣴⡞⡝⡜⡌⡎⡎⡜⡜⡸⡘⢰⡸⡰⠱⢣⢫⢦⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠓⠓⠳⠧⠷⠧⣧⣧⣧⣧⣇⢛⢦⡀
⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠤⠤⢤⡀
⠀⢀⡠⢶⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢰⠀⡄⠀⡇
⠀⢸⠳⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠇⠀⡇
⠀⡄⠀⠈⠙⢷⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡄
⠀⡇⠀⠀⠀⠀⠉⠻⣥⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣝⣙⣆
⠀⡇⠀⠀⠀⠀⠀⠀⠈⠳⢖⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣿⡼⠟⢻
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⡥⣒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣲⠯⠒⠉⠀⠀⡆⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣚⠭⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠿⠚⠉⠀⠀⠀⠀⠀⠀⡁⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣒⣤⣀⣤⣀⠄⣰⣶⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡇
⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠉⠑⠿⣅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⡿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⢆⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⣒⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡥⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣷⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣂⡀⠀⠀⠀⠀⠀⠀⡆
⠙⠫⠤⣔⣲⣂⠤⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠬⠬⣐⣐⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡠⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀⣀⠀⡀⠀⠉⠉⠐⠒⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠄⠀⢸
⠀⠈⠑⠉⠚⠕⠫⠚⠝⠫⠒⠩⠖⠭⠪⠕⢪⠕⡢⢂⢔⡤⣋⢝⡱⢒⢦⡤⣤⣀⣀⡀⠀⠀⠀⠀⠻⣆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠉⠘⠁⠊⠐⠉⠋⠝⠹⠳⠖⠾⠷⡫⠪⠿⠯⠗⡵⡪⢆⢴⠖⠄⡤

⠐⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣰⠖⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠛⠒⠛⠛⠛⠛⠛⢱⡆
⠀⠀⠛⠀⠂⠒⠀⠂⢐⡀⠘⠀⡂⠀⢂⠀⠂⠛⠀⠃⠐⠀⢻⠁⠀⠐⠂⠃⠐⠃⠀⢂⡀⠃⠀⡀⢀⢀⠈⠀⠐⠂⠐⠃⠚⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡉⠀⠘⠀⠈⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀⢀⠀⠈⠀⠓⠀⠃⡀⠈⣁⢒⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣁⣑⣀⣘⣀⣀⣀⣀⣀⣀⣀⠀⢸⠀⠀⠀⠀⢐⣁⣉⣘⣂⣊⣙⣋⣂⣅⣀⣋⣊⣃⣙⣋⣁⠀⠀⡇
⠀⠀⠀⠀⣏⣡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⠀⠀⠀⠀⢸⣿⣵⣧⡆⡨⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇
⠀⠀⡇⠀⣿⢾⠓⠥⠄⠆⠠⠠⠴⠐⠠⠀⢰⠦⠴⠆⠾⠀⢸⠀⠀⠀⠀⢸⡿⡟⠧⠿⠖⢲⠶⠰⡶⠶⡖⠀⡖⠲⡶⠶⠀⠀⡇
⠀⠀⡇⠀⡻⢾⠖⠲⠒⠖⠖⠒⠶⠒⢺⠒⢶⠒⢲⠒⠶⠀⢸⠀⠀⠀⠀⢸⠶⠬⠦⠶⠤⢤⠤⠤⠦⠤⡥⠤⡤⠤⠤⠶⠀⠀⡇
⠀⠀⠁⠀⡇⠺⣿⣾⣷⡆⠁⠀⡽⠀⢸⠀⢸⠀⢸⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⢼⠀⢵⠀⢸⠀⠀⡃⠨⡇⠀⡇⠀⡇⢰⠀⠀⡇
⠀⠀⠀⠀⣅⣸⣯⣩⣼⡷⣷⠀⣼⣿⢽⢢⢸⠀⢸⠀⢠⠀⢸⠀⠀⠀⠀⢸⠀⠡⠀⠨⢰⢸⠞⢒⡁⠀⡇⠀⡇⠀⡇⠘⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣿⣥⣶⠶⡇⠀⢹⡟⢻⠈⢸⠀⠘⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⢘⠀⠀⠘⢿⠋⡸⠀⠀⡇⠀⡇⠀⡇⢠⣀⠀⡇
⠀⠀⠀⠀⣀⣽⣟⣟⣁⠀⠇⢞⡋⠰⣹⢠⢰⡮⢠⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⡰⠒⢹⢄⠒⡅⠀⡇⠀⣧⡀⢹⣧⣌⣀⡇
⠀⠀⡤⠈⠁⠀⠀⠀⠁⠱⣕⠀⣓⠀⢹⠠⢸⡗⢱⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⠈⢐⠆⠀⠃⠀⡇⠙⣆⢩⠈⠛⠀⡀
⠀⢠⠗⠀⠀⠀⠀⠀⢘⠀⠀⠸⡿⣾⡼⢤⡬⠰⠎⠀⣨⠀⢸⠀⠀⠀⠀⢸⠀⠰⠀⢘⠀⢸⡗⠀⠄⠀⡃⠀⡇⠀⠜⠻⠒⣶⣾⠃⠀⠐
⠀⢸⠰⡀⠀⠀⠀⢘⣿⡗⣮⡀⢹⠖⠛⣉⡈⡟⢛⡛⠻⠀⢸⠀⠀⠀⠀⢸⠛⠛⠚⡛⠛⠺⡗⠚⠓⣒⣓⡲⠟⠀⠀⠀⠘⣿⠂⡌
⠀⠀⢻⠛⠖⠂⠋⣠⣾⡇⠀⢉⣥⣶⣴⡴⠶⠿⠓⠀⠈⠀⢸⠀⠀⣀⠀⠘⠊⠁⠀⠁⠈⠉⠙⠛⠐⠬⢈⡀⣠⡞⠛⠀⠀⢿⡱⣧
⠀⠄⣼⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠷⠧⠴⣧⠞⠀⠈⠣⠥⠥⠭⠍⠁⠬⠭⠉⠌⠉⠩⠽⠿⣿⠃⡀⠀⠀⢹⠿⡯
⠀⢰⡇⠀⠀⠀⠀⠀⢿⡧⣤⠤⣬⣤⡤⡤⡤⠤⠤⠤⠤⠤⠏⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠼⢦⡵⠶⠾⣶⣷⠂
⠀⢸⠁⠀⠀⠀⢰⠀⢸⠆⠀⢠⢟⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢸⣻⣧
⠀⡘⢰⠀⠀⡇⠀⠀⢸⡆⠀⡀⣾⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠅⣿⡇
⠀⠇⠈⠀⠀⡇⠀⠀⢸⠃⢰⣧⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⢹⣧
⠀⡴⠀⠀⠅⠇⠀⠀⣰⣀⣭⣿⣿⢟⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃⠘⠸⡆
⠀⡇⠀⢠⡀⠀⠀⢰⣿⣿⢃⣿⢯⣿⣣⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣄⣈⠀⠁
⠀⡷⠶⢰⣤⣴⣶⣿⣿⣿⢸⡷⢱⢻⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⢿⣿⣿⣿⣿⠟⠃⡄
⠠⡇⠀⠀⣻⠘⣿⠹⡟⣿⠃⢿⣷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠃⣽⣿⣿⠶⡋
⠰⠇⠀⠀⠅⢀⢿⡞⣦⣿⠉⢻⠋⣽⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡿⣿⣏⢠⠖⠁
⠀⠀⢤⣀⣀⣀⣞⣣⣿⣾⣿⣿⠿⠛⠛⣛⣛⣛⡛⠛⠒⣛⣿⡟⢛⣛⣛⠛⣛⣓⣛⣛⣛⡛⠛⠓⠛⠛⠛⠛⠛⢛⢽⣿⣿⣿⢿⣽⡿⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣈⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠄⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠔⠒⠒⠒⠈⢉⠛⠀⠀⠀⢀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠉⢀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠛⠋⠉⠉⠉⠉⠀⡠⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠢⠄⡀⢀⠔⠃⢀⠀⢀⠀⠀⠀⠀⢀⠊⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢁⡰⠀⠀⣠⡀⠀⣑⠾⠁⠄⠀⠔⠉⠒⢄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡃⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⡂⠠⠂⠐⠈⠣⡀⢀⠌⠀⠀⠀⠀⠀⣀⣈⠴⠃
⠀⠀⠀⣀⡤⠴⠒⠒⠊⠉⣉⡉⠭⠤⠈⠛⠛⠛⢳⠀⠀⠀⡀⠀⠀⠀⣠⠊⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠓⣀⡤⠔⠒⠉⠉
⢀⡴⠋⠁⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⣸⠀⡠⠀⠉⠒⠐⠁⠀⠁⠂⠀⠀⠠⠄⠤⠠⡄⠀⠀⠀⠀⠑⢤⡀
⡎⠀⠀⠀⠀⠀⡆⠀⣀⠤⠄⠒⠀⠘⠋⠀⠀⠀⠀⠳⠤⡤⠤⡤⠤⠤⠤⠤⠤⠤⣌⠖⠀⠀⠀⠐⠄⢠⠐⠂⠂⢀⡿
⡇⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡇⠀⢡⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⣠⠎
⡇⠀⠀⠀⠀⠀⢸⠀⡀⢀⣀⡀⠠⠤⢆⣀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⣧⠜⠁⠀⠀⠀⣀⣠⣤
⡇⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⣀⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⠤⠖⠊⠉⠀⢀⠂⢳⡀
⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠒⠀⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠆⠀⠀⢣
⡇⠀⠀⠀⠀⠀⠀⡇⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠒⠀⠄⢀⠀⠀⢀⠜⠂⠀⠤
⡇⠀⠀⠀⠀⠀⠀⡇⢀⡠⠤⠤⣀⡤⠒⠤⣰⠊⠩⠿⢄⢠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⣀⠠
⢧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡿⠀⠀⠀⣯⠟⠃⠸⠠⣨⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠂⠀⢀⣀⡤⠴⠚⠉
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠊⠀⠀⠢⠄⠴⠁⠒⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠉
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⠁
⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⢀⣀⣀⠤⣴⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⢸⡀⠀⢻⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠻⠙⠧⠌⠀⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠟⠋⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠛⠁⠀⠀⠀⣸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢀⣿⣿⢻⣿⣿
⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⣿⣿
⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢸⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣦⣤⣄⠀⠀⣿⣿⡛⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⢸⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠛⠛⠻⣿⣿⠀⠀⣿⣿⡄⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⢸⣿⡏
⠀⠀⠀⣿⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣄⠀⠀⣿⣿⣇⣀⣀⣀⣿⣿⠀⠀⣿⣿⠆⠀⠀⠈⠻⣿⣦⣀⠀⠀⢀⣴⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠸⠿⣿⣿⣿⡿⠿⠟⠀⠀⣿⣿⠁⠀⠀⠀⠀⢈⣿⣿⣿⣾⣿⡿⠻⣿⣷⡄⠀⢸⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣾⣿⠏⠉⠉⠀⠀⠀⠈⢿⣿⣦⣸⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠛⠉
⠀⠀⠀⣿⣿⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⡄⣰⣿⡟⠀⠀⢀⣠⣶⣿⡿⠟⠋⠁
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣷⣿⣿⣀⣴⣾⣿⠿⠛⠉
⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠭⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⡳⠀⠀⣠⡀
⠀⠀⠀⠀⣗⢌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⢀⣰⠼⠁⠂⠁⡼⠁
⠀⠀⠰⡂⢝⣆⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⢀⣤⠴⠒⠋⢁⢀⣀⣀⣤⡎⠤⠔⠊⡽
⡄⡀⣤⡹⣤⡈⠙⠶⢤⣄⣄⡈⢉⠉⠐⠢⡄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢢⠋⢀⡴⠚⢁⣠⠤⠒⠚⠉⠉⠉⠀⠀⠀⢀⣰⠞⡡⡠
⠀⠁⠱⣌⠒⠛⠲⢤⣀⣈⣉⢉⠝⠁⢄⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠁⠐⢯⠀⢀⡤⠖⠓⢓⢳⡷⠋⠉⠉⠁⠈⠈
⠀⠀⠀⠈⠳⣶⣀⣀⠀⠈⣉⣉⡀⠸⠿⠤⠀⠀⠠⠤⠀⠠⠤⠠⠤⠄⠠⠤⠤⠄⣀⠀⠉⠢⡀⠀⠃⠈⣠⣶⠺⠭⢓⠂
⠀⠀⠀⠀⠀⢯⣍⠉⠙⠛⠋⠠⠄⠂⠀⠀⠀⠀⠀⡇⠀⠀⡄⢰⣢⣀⠤⢤⣄⣀⠈⡗⢦⡀⠘⡄⠀⢀⣠⣠⡶⡚⠁
⠀⠀⠀⠀⠀⠀⢩⣛⢶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠅⡇⠘⠉⠙⢀⣈⡉⠸⠀⡯⡲⢝⣲⠇⠀⠈⡉⢁⡥⢃⠄
⠀⠀⠀⠀⠀⠊⠊⠳⢶⣤⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠇⢆⣀⡀⠬⠕⠂⠃⠀⠑⢬⡲⠋⠈⠻⣿⣚⠉⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠰⣄⣉⣸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡭⢀⠀⠀⠁⠉⠉⠊⠀⠀⠨⡟⢀⠸⣷⢶⠶⠛⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⠀⠀⠀⠀⠀⠀⠀⢸⠍⣷⡈⣿⠞⣻⡒⢈⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠠⡆⠀⠀⠀⠀⠀⡆⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠝⠚⠻⡿⠁⠟⠽⠻⢁⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡄⣆⠀⠀⠀⠀⠀⠃⢸⠄⡀⠀⣀⣀⠀⣄⠀⠦⣀⠤⣼⠦⣴⢀⣤⡇⠈⡎⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣤⢤⡤⡤⣤⢼⠬⣤⢧⡤⠤⣤⢤⡤⣤⣖⠔⢢⠆⢠⠟⣓⡟⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣵⣙⢟⣳⣿⣾⣶⣟⣟⣻⣤⣟⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠉⠉⠉⠙⠛⠛⠓⠛⠓⠛⠚⠛⠛⠛⠛⠛⣃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠂⠅⠠⠀⠨⠀⠀⠀⠀⠈⠠⠔⠠⠀⠄⠨⠀⠄⠀⠄⠬⠊⠔⠠⠂⠅⡨⠀⠅⡢⢀⠅⡠⡴⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠿⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠻⢿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡍⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠈⠻⣿⣷⣄⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⣠⣾⣿⣿⣷⣦⣴⣿⣿⣿⣦⡀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠇⣠⣾⣿⠟⠁⠈⠻⢿⠿⠋⠀⠙⢿⣿⣦⡀⢿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣴⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣁⣍⣻⣀⣝⣉⣭⣙⣨⣟⣩⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⣾⣿⣿⣿⣷⣶⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡿⢻⡿⠟⠿⢿⣿⣿⣿⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⢠⣌⠁⣤⠀⠀⠹⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡇⠈⢍⣉⣿⠆⢠⣀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣿⣿⢹⠀⠀⠂⠡⠀⠀⣿⣍⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⡏⠀⠀⢀⣀⣐⣠⠟⠀⠃⠀⣄⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠆⠉⢉⡳⢷⠀⠉⠙⠛⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣖⣾⠇⣿⣿⣿⣷⣆⣉⣸⠟⣨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣻⣭⢿⣿⣋⢐⠻⣿⣿⣿⣿⣿⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡟⣿⠻⣷⢉⣹⣟⣿⢿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢞⡻⢿⣷⢿⡄⣸⠟⡿⣟⣿⣿⣿⣷⢿⣮⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠌⠙⠁⠹⡴⡿⠛⠿⢿⡿⠻⠿⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠶⠶⣶⣤⣤⣄⣠⠤⠤⠤⠴⠖⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠀⠀⠀⠀⡴⡶⢤⣀⢀⡟⠁⢀⡀⣀⠀⠠⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣟⣛⣓⣒⣒⣒⣒⣒⣒⣒⡾⠀⢳⠀⠈⢙⠁⠀⠉⠀⠃⠤⠎⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣉⣉⣉⣉⣉⣉⣉⣉⡝⠀⠀⠘⡧⠖⠊⢱⠖⣤⢠⣄⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣥⣤⠤⠤⠤⠤⠤⠤⡜⠀⠀⠀⠀⣹⠀⢠⣏⣠⠀⠉⢧⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠦⠤⠤⠤⢤⣜⣀⡠⠔⢲⠉⠁⢠⠿⠦⠼⠶⠷⠿⢧⡀⠑⠢⠤⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⠶⠶⠶⠾⠿⠤⣀⠀⠈⡇⢠⠿⠤⠤⠤⠤⠤⣤⡾⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣛⣛⣛⣛⣒⣿⣟⡋⢹⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⠭⠤⠤⠤⢤⣤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⢤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠚⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⢷⣄⡀⠀⠀⣠⣴⠟⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠⣴⠟⠻⠶⠾⠛⢷⣄⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⣿⠉⠉⠉⠉⠉⠉⠉⣿⣷⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣷⣿⠋⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣤⣤⣤⡄⢠⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⢹⡏⢻⡇⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⣀⣀⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⣧⣤⣤⣤⣤⣤⣾⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠛⠛⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠾⠷⠶⠀⠶⠆⠰⠶⠶⢾⡟⠛⠛⠛⠛⠛⠛⣿
⠀⠶⠆⠰⠶⠶⠶⠶⠿⠷⠶⠶⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠀⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠉⠀⠀⠀⠀⠙⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⣿⣍⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣤⡴⠾⠛⠋⠉⠀⠀⠀⠀⠙⠻⣦⣄⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠉⠻⢶⣄⡀⠀⣀⣤⣴⠶⠟⠋⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠉⣿⡟⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⣦⣀⠙⠻⣦⣄⠀⢀⣀⣤⣴⠶⠟⠛⠉⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣯⣙⢷⡆⠀⠙⣿⠛⠉⠁⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠿⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⣿⠀⠀⣀⣠⣤⡶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠿⠟⠋⠉

⣀⣀⣀⣀⡀
⣿⠿⠟⠿⡇
⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣖⢶⣶⣶⣶⣶⣶⡶⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡄⠀⠀⣠⣶⣶⣶⣄⠀⠀⢀⠤⠞⢸⣿⣷⣭⠻⣿⠟⣭⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠺⠿⠾⠿⠷⠿⠗⢀⠎⠀⠀⢸⡿⣫⣴⣿⣶⣿⣶⣝⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⣠⡔⠋⠀⠀⠀⢀⡙⠛⠛⠛⠛⠛⠛⠙⣫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢲⣿⣇⡈⠉⠀⣠⣿⡖⠲⢄⡀⢸⡇⠀⠀⡤⠤⠴⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠱⣼⡿⠿⣏⠉⠉⠻⣿⠇⠀⠀⠙⢾⣇⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠘⢦⠀⠈⢦⣀⡞⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠳⡀⠀⠹⡄⠈⠉⠉⠁⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠚⠓⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠛⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣋⣀⣀⣀⣀⣀⣈⣙⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢁⣀⣀⢈⡉⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡋⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⠏⠉⠉⢿⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢿⣄⠀⢀⣼⣇⡀⣸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠉⠛⠋⠁⢉⣉⣍⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⠛⠛⠉⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠛⢿⣦⣀⠀⠀⣀⣴⡿⠛⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣾⢛⣁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢈⡛⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⠑⣦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢳⡀⠀⠀⠀⠀⣂⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠋⠁⡛⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡧⠀⣀⡴⠋⠁⠀⠀⠀⢠⠐⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠈⠁⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⣿⠀⣁⠔⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⠈⠁⠀⠀⠀⠐⠨⠷⢄⠀⣠⣤⣠⢤⡞
⠀⠀⠀⡴⠊⠁⠀⠙⠲⣦⡿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠡⡀⡀
⠀⠀⠀⠀⢀⣠⡶⠴⠛⠁⠀⠀⠀⠀⠀⠀⠠⠖⠂⠀⠀⠀⠀⠀⠠⠀⠀⡀⠙⠀⠀⢐⠀⠀⠀⠀⠀⠀⠃⠇⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣔⢀⠂⢠⣾⠞⠀⠀⠀⠀⠀⠀⠉⠀⡀⣠⠏
⠀⠀⠀⠀⠀⡀⠐⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠷⣄⠀⠈⠋
⠀⠀⠀⣰⠀⠀⠀⠰⣄⢠⠂
⠀⠀⠀⢀⠙⠓⠒⠂⠐⠛⡋⠀⠀⠀⠀⠀⠲⡀⠀⢀⣀⠀⠀⠄⠀⠀⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⣤⠀⠀⠀⠀⢉⠀⠀⠘⠟⠁⠀⢸⠀⠀⢁⠀⢀⠄⠀⠀⢀⡄
⠀⠀⠀⠁⠀⣀⠖⠋⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠓⠁⠤⠠⠄⠈⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠒⠀⠀⠀⣠⠔⠊⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⠄⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⢋⣀⠤⠤⠐⠚⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠶⠖⠊⠉⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣤⠤⠖⠒⠊⠉⣉⡤⠚⠅⠰⣄⠀⠀⠠⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠈⠙⣆⠀⠀⢈⣀⣀⣤⡤⠶⢒
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡇⠀⠠⠀⠰⠃⠀⠀⢀⣀⣀⡤⠤⠀⠘⡆⠉⠉⠉⠀⣠⣼⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣠⠖⢱⠖⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⡸⠿⣾⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠤⢶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣈⡩⠽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣬⣄⣠⠤⠤⠐⠂⠀⠀⠀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠈⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢀⣀⢀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⣿⠀⠀⠀⠀⠂⠀⠀⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⠀⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠓⠞⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠇⠤⠤⠥⠾⣤⣤⣤⣤⠼⠤⠤⠤⠴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢹⣿⠃⠸⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣦⡤⢺⣿⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢣⢸⣿⠇⠀⠀⠀⠈⣉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⣴⣷⣼⣿⡇⢸⣇⢠⣾⠉⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠟⣾⣿⣶⣽⣿⡇⢸⣿⣏⠐⣤⣀⣀⠑⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣴⡿⠋⡩⢿⣿⣿⡇⢀⣿⣷⠶⠊⠀⢍⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠈⠀⠓⠠⢹⡟⠀⠀⠨⣿⣕⡄⠀⠠⠀⠌⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠭⠑⣶⣷⠸⡇⠀⠀⠀⠻⣥⣖⡁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠀⠀⠀⠀⠈⠐⠾⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠉⠉⠉⠉⠛⢿⣯⡉⠁⠉⠉⠈⠉⠁⠉⠉⠉⣽⡟⠀⠀⠀⠉⠛⠷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀⣾⣯⣄⣀⣄⣀⣀⣄⣀⣀⣀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⣸⡿⢷⣤⣄⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣍⠉⠉⠉⠉⠉⠉⠉⠉⢉⣭⡿⢿⡟⠀⠘⣿⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠉⠙⠻⢶⣤⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠿⣦⣄⠀⠀⠀⣠⣴⠟⠋⠀⢸⣿⠀⠀⣿⡇⠀⠀⣰⡿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣼⠿⢿⣶⡿⠿⣧⣄⠀⠀⢸⣷⠀⠀⣿⡇⠀⠀⠻⣧⣤⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⡾⠛⠁⠀⠀⠁⠀⠀⠈⠻⢷⣄⣸⣯⠀⠀⣿⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠋⠉⠙⠉⠉⠛⠉⠙⠋⠉⠛⠉⠉⢻⡟⠙⠉⠉⣿⡋⠙⠋⠉⠙⠉⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⡞⠥⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠚⠻⠽⢿⣶⣶⡧⣼⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣳⠶⣤⢤⣉⣿⠛⠒⠯⠭⣽⣒⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃⠈⢁⡼⠛⠛⢹⡏⣿⢹⣶⠲⣤⣤⣌⣯⡽⠲⠖⠒⠂
⠀⠀⠀⠀⠤⠤⠤⠤⠖⠋⠁⠀⣀⠴⠋⠀⠀⠀⢸⣇⣽⢻⡟⠛⡹⠿⠶⠋⠓⠶⠤⠤⠤⠤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⣺⣵⠶⠤⣄⣀⣠⢼⡇⣿⢸⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣉⡴⠶⢯⣍⣛⠓⡿⢾⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠈⠳⠼⠛⣷⠶⣤⣼⡟⠓⠶⠤⣍⣛⠲⠦⢤⣀
⠀⠀⠐⠒⠶⠒⠒⠋⠁⠀⠀⠀⠀⣀⠜⠉⠉⠁⢸⠉⣛⢲⣦⢄⣈⡉⠓⡞⣹⠤⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⢸⢀⣸⢾⠛⠊⢻⣯⠿⠻⢅⣀⣀⣀⣀⣀⣀⠴⠂
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡠⢼⠋⣿⢸⣂⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠖⠒⠋⠁⠀⢸⢀⣿⢼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠤⠤⠒⢻⠉⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡯⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠰⠒⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⡀⠀⠀⠀⠒⠀⠂⠂⠐⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⢈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣅⣤⣀⣠⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣾⣿⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⢹⡟⢿⠧⡽⠋⠛⠁⠀⡀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢳⡶⣤⡄⠀⠀⠀⠀⠚⠶⠚⠉⠁⠀⠀⠀⠀⢧⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣧⠛⠁⠀⠀⠀⠀⠀⠀⢦⡀⠈⠀⠀⠀⠀⠈⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣻⠆⠀⠀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠹⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⣀⣠⡀⠀⠈⢿⠀⠁⠒⠦⠄⢀⣠⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⣧⢴⣶⠈⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡶⠁⠙⣿⣃⡀⠀⠘⡗⠐⠀⠀⠀⢀⠀⠀⠀⠀⢠⣿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠓⠯⠘⣿⣦⣤⣾⣿⣦⣀⣾⣶⣶⣦⣴⣾⣿⣿⣷⢶⠏⣯⡵⠄⠀⠀⢹⡀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠆⠀⠀⣾⣿⣻⣿⢿⠻⠟⡏⣉⣿⣿⣿⣿⣿⣿⡟⣳⡄⠀⠄⠁⢠⠄⠀⠀⠻⠈⠰⠄⠀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠠⢀⡄⠀⢨⣿⡀⠽⠖⣻⡴⢘⣻⣿⣏⡡⢞⠿⢿⢉⠈⠀⠀⠐⠂⠀⣀⣥⡴⠚⠀⠀⠀⠀⠀⠀⠀⡾⠙⠉⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠈⠻⠷⣿⣿⠿⠛⠙⠛⠻⡄⠰⠓⠀⠈⠀⠀⣀⠀⠀⡀⠈⠉⠉⠛⠿⠷⣶⣦⣄⠀⠀⠘⠁⠀⠀⠀⠀⣠⡦⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠤⢤⣤⣤⠀⠀⠀⠩⢁⣠⡮⠘⠂⠈⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⣡⠋
⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡌⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠀⠉⠙⠛⠲⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣰⠛⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠃⠛⠓⠶⠛⠚⠒⠃⠀⠀⠈⠉⠛⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠙⠛⠻⣷⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡏⠠⣤⣄⠀⠀⠀⡀⢀⣀⡀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠛⠂⠘⠛⠃⠀⠌⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣈⢉⡁⣀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠘⠃⠀⠐⠲⠒⠒⠒⠒⠀⠐⢶⠓⠶⣶⠶⠲⠶⠶⠶⠶⠒⠂⠀⠀⠐⠂⠀⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⣿⠿⠋⠀⠀⠀⢀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠠⠀⠀⠄⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠄⠄⠀⠀⠀⠀⠂⠀⠀⠆⠀⠄⠆⠰⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢛⣟⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣷⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⠉⠉⠍⣻⣿⡏⠉⠉⠉⠉⠩⠽⠏⠉⠉⡉⠉⠉⠉⠉⠉⠉⢉⣛⣛⣛⡛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣼⡄⠠⠄⠀⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠉⠙⣩⢊⢿⡇⠀⠀⢰⣇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣄⣼⣵⢟⣾⡇⡐⣡⣼⣼⣿⡀⠀⢀⢄⢀⡀⠀⠠⢀⢀⡄⢠⡄⡠⢠⢠⠄⡤⠤⠤⠤⠤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢡⢎⢾⡇⠐⣹⣿⣿⣿⡇⢠⡣⡳⡓⠀⠠⢡⢃⠎⢀⠀⠐⠲⠲⠂⠘⡵⣁⢏⡙⣩⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⢫⢾⡇⠐⠙⠛⠟⠛⠡⡕⠑⠑⠀⡀⠡⢃⠈⠠⠃⠀⡀⠠⢁⠈⠀⡠⠁⠚⣰⢃⣊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠰⢃⢾⣇⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⠀⠁⠀⠀⠀⠆⠜⠐⠠⠀⠂⠔⠀⢀⠎⣠⣚⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⢃⢾⣇⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⠀⡠⢁⢾⣷⠿⡡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⣖⣚⣩⡷⠧⢾⣋⣶⣿⣇⣁⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣀⣋⣋⣋⣎⣴⡟
⠀⠀⠀⠀⣀⣤⣤⣶⡶⠿⣿⣛⣻⡭⠽⠶⠒⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⢹⣯⣿⢿⢟⠟⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠿⠷⠖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢾⢺⢘⢞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⢺⢺⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⢸⢫⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⢹⢸⢫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⢸⣱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⢨⢼⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢈⣼⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠘⣾⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⢀⠸⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⢘⢼⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠼⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣾⣷⣾⣷⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⣿⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⢸⣿⠈⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠷⣦⣹⠇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠸⣏⣴⠞⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⢶⣄⣤⡾⢷⣤⣠⡶⠛⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡛⠛⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢟⣽⣿⣿⠿⣻⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣽⡿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠛⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠀⣸⠇⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠮⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣻⣿⣃⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⡻⣿⣿⣿⣿⣿⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡿⠿⠿⠁⠀⠈⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠼⠯⠽⠧⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⡰⢦⡀⠀⠀⢀⡴⣆⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⢧⣀⡼⠁⠀⠙⠦⠴⠋⠀⠈⢧⣀⡼⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡤⠿⠿⠿⠿⠿⠿⢦⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⡤⠤⣤⣤⠞⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⡉⢉⡉⢉⠉⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⢿⣯⣽⠿⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡤⠤⠤⠤⠤⠤⠤⢤⡄⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠙⢦⠀⠀⠀⠀⡴⠋⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⣠⠷⠤⠤⠾⣄⠀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣶⣁⣀⣀⣀⣀⣈⣶⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⠏⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⢿⡿⣟⣛⣫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣵⣿⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⣿⣿⣿⣘⠛⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣽⣿⣿⡟⠉⠉⢿⣿⣿⠁⠀⣸⣿⡿⠟⠛⠛⢿⣿⡟⠉⢻⣿⠿⠿⣿⣿⠋⠀⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠈⣿⡏⠀⢠⣿⠏⠀⢀⣤⡀⠀⢻⡇⠀⠘⡟⠀⠀⢹⡟⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠇⠀⢸⡀⠀⢻⠇⠀⢸⡏⠀⠀⠈⠉⠀⢀⣼⣷⠀⠀⠁⠀⠀⠸⠇⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⣿⡇⠀⠘⠀⠀⣿⡇⠀⢠⣶⣶⣾⣿⣿⣿⠀⠀⠀⢠⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢰⣿⣷⠀⠀⠀⢠⣿⣇⠀⠈⠛⠛⠛⠙⢿⣿⣇⠀⣠⣿⣧⠀⢀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⡅⠀⣸⣿⣿⣧⣀⣀⣾⣿⣿⣷⣤⣄⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠴⢂⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠛⠋⠻⣿⠋⠉⠉⠁⠀⠀⢈⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣿⣿⠀⠀⣀⠀⠈⣿⠋⠀⠀⠀⠈⢿⡏⠀⢠⣤⣾⣿⣶⣶⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠋⠀⢀⡇⠀⢰⣿⡆⠀⢸⣧⠀⠈⠛⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⢀⣤⣴⣿⡇⠀⠸⠟⠀⢀⣾⣿⣿⡆⠀⠈⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⡇⠀⢸⣿⣿⣿⣿⣄⣀⣀⣠⣾⣿⠁⠀⠀⣀⣴⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠷⣶⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⢭⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡤⠚⠒⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⣻⠟⢀⡤⠒⠉⠉⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡔⠁⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⡞⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⣧⣶⣶⣒⣊⣉⣁⣉⣉⣉⣉⣉⣉⣉⣉⡇⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠈⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⢲⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢢⡤⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⣸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡤⠴⠦⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠒⠶⢥⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠘⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠙⡎⢃⠀⡎⢱⠙⡆⡜⢸⠉⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⠀⠀⣀⡘⡀⠑⠃⡆⡇⠇⢸⠨⣆⡀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠀⠶⠁⠘⠿⡤⢳⣸⣅⣈⡿⠤⠤⠴⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣆⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠚⠀⠀⢠⠀⠀⠀⢀⣀⣠⡤⠴⠶⢲⡚⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠉⣿⠉⠉⠉⠛⠛⠓⠒⠒⠶⠶⠦⠤⠤⡄⠀⢸⠀⢾⠋⠉⠁⠀⠀⠀⠀⠸⡗⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢨⡆⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣈⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⣀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⢿⣶⣾⠿⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠿⣶⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣾⣿⣿⡜⣿⡞⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡇⣿⡇⣿⣧⣛⣛⣛⣛⣛⣛⣛⣛⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡇⣿⡇⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡇⣿⡇⣿⣿⣿⢰⣒⣒⣒⣒⣲⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡇⣿⡇⣿⣿⣿⡸⠿⠿⢿⠿⠿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡇⣿⡇⣿⣿⣿⠛⢛⣛⣛⣛⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡇⣿⡇⣿⣿⣿⢸⡶⣙⣛⡱⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⡇⣿⡇⣿⣿⣿⣤⣚⣛⣛⣛⣪⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣽⣯⣿⣷⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




⡟⠉⣿⣻⢻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣟⣿⣿⣿⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢴⡇⣿⣷⣯⡷⣿⣹⣿⢺⡆⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣷
⠀⣷⣯⡻⢿⣿⣿⣿⣿⢿⣫⣷⡇⠀⠀⠀⢸⣷⣥⡻⢯⣗⣿⣾⡿⣟⣼⣾⠀⠀⠀⢸⣮⣝⢿⣿⣿⣿⣿⡿⣟⣽⣿⠿⠋
⠀⣿⣿⣿⣷⢝⣻⣋⢷⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⢗⣭⣻⣥⡻⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡮⣛⣟⡹⣾⣿⣿⣿
⠀⣿⢟⣯⣾⣿⣿⣿⣿⣮⣛⢿⡇⠀⠀⠀⢸⡿⣫⣷⣿⣿⣿⣿⣿⣮⣟⢿⠀⠀⠀⢸⡿⣻⣵⣿⣿⣿⣿⣶⣝⡿⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠦⣄⡀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠓⢦⣍⡓⠲⢤⣄⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠳⢭⣙⠒⣺⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠀⠀⠨⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠽⠒⠻⣝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹
⠀⠀⠀⠀⠀⢀⡴⣻⠓⢤⣀⡀⠙⠒⠆⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠘⡎⡆
⠀⠀⠀⠀⢠⣾⣾⠇⣿⠀⠈⢯⠟⠛⢿⡟⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⣷⣷
⠀⠀⠀⠀⡞⣿⣟⡀⢻⡄⠀⠀⢀⡠⢊⠞⠁⠀⠉⠛⠓⠶⠶⢦⡶⠖⠱⣀⡀⠄⠐⢊⠝⠀⢸⣿⠀⠀⠀⠀⠀⣿⡟
⠀⠀⢀⣰⠛⠉⠉⠉⠉⢷⣤⢖⣫⢶⡏⢶⠂⠀⠤⠤⢀⣀⡀⠈⠓⠒⠒⠂⠤⠤⠤⠊⠀⠀⢨⠟⠶⠾⠿⣭⣭⣿⣧⠤⢤⣤⡤⠤⢤⠴⠶⠶⠶⠶⠶⠶⠶
⠀⢀⣾⡇⠀⠀⠀⢀⢄⡾⠿⣯⣤⣊⣀⡈⠣⠴⠒⠈⢁⠼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⣿⡇⠀⢸
⠀⢸⠉⢷⣄⡄⣀⣨⠏⠀⢰⡇⠉⣖⠒⠤⠥⢍⣀⣒⢊⣜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣄⠀⠀⠀⣠⣾⠟⠀⠀⠀⣿⡇⠀⣸
⠀⠘⣆⠘⠿⠟⠉⠀⠀⠀⡾⠀⢠⠏⠀⠉⠀⠀⠐⠂⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣤⣶⠿⠛⠁⠀⠀⠀⠀⣿⡇⠀⡿
⠀⠀⠈⠓⠦⢤⣤⠤⠶⠾⢥⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣇⣄⣧⣶⠏
⠀⠀⠀⣠⣞⣛⣙⣳⡆⢀⣾⣛⣋⣛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⡟⠋⠛⢱⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠄⠀⠈⣙⠦⣄⡀
⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠂⠀⠀⠀⠀⠂⠁⢀⠤⠒⠉⠀⠉⢂⡝⢲⣄
⠀⠀⣾⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀⠐⠈⠀⠀⠀⠀⠈⠐⠤⡀⠄⣪⠵⠚⠁⣿
⠀⠀⣿⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⣀⡤⠒⠉⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⡇⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⣇⣠⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣤⣀⣸⣿⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢛⣛⣛⠛⡛⠛⠻⣿⡆
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣸⣿⣛⣛⠛⢿⣿⢀⣿⡇
⠀⠀⢠⣾⡿⠿⠛⠛⠉⣿⡇⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣥⣽⣿⣉⣉⠀⢸⣿⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⣾⣿⣭⣭⠀⢸⡿⢸⣿
⠀⠀⠀⢻⣧⠀⠀⠘⠉⣿⡇⠀⠀⠀⠛⠛⠛⠒⠚⠒⠒⠒⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠾⣼⣿⣿⣿⣿⠏⢸⣿
⠀⠀⠀⢸⣿⡀⠀⠀⠆⣿⡇⠀⠀⠀⠒⠛⠓⠒⠒⠒⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⣠⣤⣤⣤⣀⡀⠀⢀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠁⣿⡇⠀⠀⠀⠛⠓⠒⠶⠶⠶⠶⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢹⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠴⠦⠤⠤⠤⠤⠤⣤⣤⣤⣄⣀⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠸⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⣤⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠋⠘⠁⠘⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢀⣀⣠⣤⣤⣶⣶⡾⠿⠿⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠿⠛⠛⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁

⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⢸⣿⠿⣿⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣽⣿⣿⣷
⠀⠀⠀⠀⢸⣿⠃⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣷⣄⠀⠀⠀⢀⣴⣾⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠙⢿⣿⣦⣾⡿⠟⠉⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⢃⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣼⣿
⠀⠀⠀⠀⠸⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣻⣿⣿
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⡠⢘⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⡆⠀⣀⣀⠄⠀⠀⠀⣀⠔⠊⠁⠱⡀⠀⢀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠐⣀⠼⠁⠀⠀⠀⠀⠀⠈⢅⠀⣠⠔⠒⠁⠀⠄⡐⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠠⡄⠤⣈⠉⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⠀⠀⠀⣀⡠⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠆⠀⠀⠀⢀⡠⠒⢹⠆
⠀⠀⠀⠀⢠⠊⠀⠅⠁⠇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⠀⠀⢾⠁⠀⣀⠆⢷⠈⠉⠁⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡶⢀⡆⢠⢣⠀⠀⠀⠀⠀⢸⠀⡄⠀⠈⡄⢀⡠⠴⠚⠁⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡜⠘⠄⠀⠖⠚⠀⠸⢀⠇⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⡁⠀⠀⠀⠉⠈⠒⢸⠀⠁⠀⠀⠀⠀⠀⠀⡀⠤⠀⠘⠉⣀⣀⠤⠤⠄⣶
⠀⠀⠀⠀⠳⠤⠶⠤⢤⣤⠇⣘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⢠⡀⠤⡊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⢀⠔⠊⠙⡀⠀⠀⣧⠠⠤⠤⠤⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢄⠀⡠⠒⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠤⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠄⠀⠀⢀⢄⡊⠀⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⢄⠀⠀⠀⡀⠐⣁⣀⣔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁
⣿⣿⠻⠿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠂⠉⠁⠀⠀⠉⠑⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠮⣍⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣠⠤⠜⠛⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⠀⠀⠀⠀⠀⢈⡳⠦⢤⣀⢠⣖⣩⡴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⡶⠶⡺⠛⢿⡧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠻⡀
⠀⠀⠀⣰⡗⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠐⠋⠀⠀⠀⠈⠁⠈⠓⠦⢄⠀⢀⣠⠤⠖⠊⠉⠀⠀⠀⠀⡇
⠀⠀⠀⡇⢱⠀⠀⠀⠈⠙⠒⢤⣴⣊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⢸⠀⠈⠉⠒⠤⣄⡀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⢒⢶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⢣⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡼⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⡠⢻⠀⠀⠀⢸⠀⠀⠀⠀⣠⠞⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⢀⠔⠁⠀⠳⢄⣀⢸⠀⠀⢀⡼⢯⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠈⢹⠓⠒⠛⠁⠈⢧⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⡤⠊
⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢦⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣸⡀⠀⠀⠀⠀⠀⠀⢣⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⠀⠳⡀⡇⠀⠀⠀⣀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠽⠧⠄⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⠒⠒⠒⠒⣲⢋⡜⠁⠳⡘⢖⠒⠒⠒⠒⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢀⡔⣡⠊⠀⠀⠀⠙⢆⠳⡀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⠶⠶⠶⠶⠶⣎⡴⢡⣤⣤⣤⣤⣤⣌⢣⣙⡶⠶⠶⠶⠶⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠒⠒⠒⢮⠀⢸⣷⢝⠿⡋⣵⣿⠀⢩⡖⠒⠒⠒⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠤⠤⠤⣽⠀⠸⠵⠿⠿⠿⠦⠿⠀⠘⡧⠤⠤⠤⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⠀⢀⡤⠒⢲⠒⠦⡀⠀⢸⡇⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⢠⠏⠀⠀⣨⡀⠀⠈⡆⢸⡇⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⢸⠀⠀⡴⠧⠷⡄⠀⣿⢸⡇⡿⠿⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠤⠤⠤⠞⢸⠀⠀⠙⢛⠛⠁⠀⣿⠘⠇⡇⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠼⠤⠤⠤⠿⠤⠄⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣾⣧




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠈⢉⣽⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣯⡉⠁
⠀⠀⠀⣿⢹⠀⠀⠀⡶⢲⡄⡴⢶⡀⡶⠂⠐⣶⠂⠀⣰⠲⡄⣴⠒⢰⡖⠂⣦⢠⠶⠂⡶⠒⠀⠀⠀⡏⣿
⠀⠀⠀⣿⢸⠀⠀⠀⡗⠚⠁⡇⢸⡇⠙⢦⠀⣿⠀⠀⣿⠀⡇⣿⠒⢸⡗⠂⣿⢸⠀⠀⡟⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠀⠀⠃⠀⠀⠓⠞⠀⠲⠚⠀⠛⠀⠀⠙⠞⠃⠛⠀⠘⠃⠀⠛⠘⠳⠂⠳⠖⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⣦⠀⠀⣆⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⢰⡄⠀⢰⡀⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⢹⡀⠀⢻⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⠈⣇⠀⠘⡇⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⠘⡇⠀⠘⡇⢸⡇⠀⡇⠀⣤⡄⢸⡇⢠⣤⠀⢸⠀⢸⡇⠀⢻⠀⠀⢻⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠿⠶⠶⠶⠶⠶⠶⠾⠇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠸⠷⠶⠶⠶⠶⠶⠶⠿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⣠⣿⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣼⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣿⣄
⠀⠀⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⠄⢀⢀⣀⣀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢆⠀⠀⠀⠚⠰⠭⣙⣄⠀⠈⠁⠁⠀⠐⠀⣀⠀⠀⠀⠀⠉⠀⠀⡤⢶⣔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⡓⠀⠀⠂⠒⠂⠨⠥⠴⠤⠀⠀⢀⣀⡀⠀⠀⡀⠀⠀⠀⠠⣇⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⢀⢀⡤⠄⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠄⠐⠂⠀⢀⣉⡤⣤⡈⢆⢀⠔⠁⢸⡉⠀⠆
⠀⠀⠀⠀⠀⢀⠒⠐⠈⡇⠁⠀⠐⠒⠒⠒⠒⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠍⣵⣀⠇⢸⡇⠀⠀⡆⠈⠑⡆
⠀⠀⠀⠀⠀⠈⠸⠙⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠯⠽⠁⠀⡇⠀⢀⠗⠈⠠
⠀⠀⠀⠀⢀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠶⠒⠘⠋⠁⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠧⠐⡂⠲⠤⠐⠂⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠠⠨⠅⢀⡀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⠉⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⣀⣀⡀⠀⠈⣉⣉⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡡⠊⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠝⢁⠔⠋⠁⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡜⠁⠀⠀⠸⠀⠀⠀⠀⠀⢀⡀⠀⠀⡰⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣇⣀⣀⡠⠤⠔⠒⠢⠴⠶⠶⠤⠤⠤⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠃⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠠⠒⠒⠒⠒⠒⠒⠂⠀⠀⠐⠒⠒⠒⢆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⢥⡬⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⣀⣠⠀⠀⣀⠀⡠⠄⡀⣀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠘⠃⠃⠰⢀⠆⠁⢸⠁⡆⡇⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠶⢤⣼⢠⠘⡼⠸⢄⠇⠣⠄⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠉⠀⠈⠙⠃⠓⠓⠒⠚⠉⠒⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠦⠤⠤⠤⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠖⣶⠶⠶⠶⠤⠤⠤⣤⣄⣀⣀⣀⣀⡀⠀⢀⠀⡤⠶⠒⠛⠉⠉⠀⢸⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠿⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣘⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠤⠤⠤⠤⠴⠚⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠋⠁⠀⠀⠀⠀⡴⣚⣫⣷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣀⣠⣤⢴⣶⡿⠽⣛⡻⠽⠓⠒⠲⠤⣤⡤
⠀⠀⠀⠀⠀⠀⠴⠿⢤⣴⠾⠿⢛⣺⣽⠭⢖⣊⣉⡤⠤⣤⠶⢶⢻⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⣏⣽⣿⢿⡞⠀⠀⠀⠀⠀⠀⠹⣀⡾⠀⠀⠀⠈⠉⠉⠓⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⢿⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠤⠶⠖⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡈⠘⠦⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠢⠔⠸⡆
⠀⠀⢀⣤⣄⣤⣏⣁⠥⢀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣠⣿⡉⢹⠉⢹⡥⠀⣎⣨⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣺⣉⣉⠉⠉⠉⠉⠉⠉⠉⢹⣻
⠀⢑⡼⢧⣸⣄⣼⣁⣸⡟⠉⡉⠑⠌⡏⠉⠛⠛⠲⢖⠒⠒⠲⣶⡛⣿⡀⠀⠀⣀⡼⢡⠴⠚⠁⠤⠤⠄⠄⢀⠂
⠀⠈⡏⠁⠚⠓⠒⢲⠏⠀⡼⠀⢀⣆⠇⠀⠀⠀⠡⠀⣙⠲⣀⢸⣼⢸⣏⠑⠚⣯⠱⣼⠀⠀⠀⠈⠓⠓⠛⣒⣂⣀⣀
⠀⠀⡇⠀⠀⠀⠀⣼⠀⠀⣇⣠⡾⢊⣦⣄⣀⠀⠀⠉⠘⠤⠬⠹⡿⠃⢹⡦⠤⠴⠟⢙⡒⠛⠋⠭⠭⠭⠉⠉⠉⠉
⠀⠀⢻⠀⠀⠀⠀⢹⡀⠀⠹⣏⣀⣞⣙⠃⠀⢩⣧⠀⠀⠀⠀⠀⢰⠆⠈⡇⠀⠀⠀⠀⠈
⠀⠀⠈⠳⣄⠀⠀⠀⠳⣤⣪⠞⠀⡶⢾⡁⣰⣿⠏⠀⠀⠀⠀⠀⠘⣀⠀⣷
⠀⠀⠀⠀⠈⠙⠒⠒⢲⠏⠉⡵⠢⠽⠏⣱⡿⠃⠀⠀⠀⠀⠀⠀⠀⡏⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢾⣁⡀⠀⣰⣿⣁⣀⣀⣀⣀⣤⠤⠤⣶⠗⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠃⣦⡀⠀⠈⠻⢽⣛⠁⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠢⠤⠤⠖⠋⢀⡖⣆⠀⢶⡚⠉⠀⠀⢀⡤⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡶⠦⠤⠤⠴⠋⠀⢈⠳⢦⡈⠓⠢⠴⠋⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠳⢤⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠿⢷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⣿⣯⣅⣀⠀⠀⠀⠈⠉⠛⠿⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠛⠻⢿⣶⣤⣀⡉⠙⠛⠿⣶⣦⣄⡀⠀⠀⠀⠈⠉⠛⠿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣋⠁⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣤⣀⠈⠙⠻⠿⣶⣤⣄⡀⠀⠀⠀⢈⣉⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠛⠻⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣤⣀⠉⠙⣻⣿⣶⡾⠟⠛⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣤⣀⠀⠀⠀⠀⠀⠀⣈⣭⣿⣿⡿⠛⠉⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣤⣄⣀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣴⡾⠿⠛⠉⠀⢹⡇⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡟⠛⠿⣶⣦⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠿⣷⣦⣄⣀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢀⣀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠙⠛⠿⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢼⡿⠟⠛⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀⡀⠀⢸⡇⠀⢀⣀⣤⣶⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣾⣷⡾⠟⠛⠉

⣶⣶⣶⣶⣶⡆
⣿⣛⢛⣙⣻⡇
⣿⣷⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡠⠠⠠⠒⠒⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠘⠛⢤⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠠⠒⠂⢈⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠡⠤⠀⠀⠀⠀⠀⠀⢀⡠⠄⢻⡇⢀⡀⠄⠒⠂⠁⢉⠠⠄⠐⠀⠀⠁⠀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⢸⠇⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠃⠀⠀⣀⣀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠀⣿⡇⠈⠉⠁⠈⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠣⡁⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠐⠖⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠐⠂⠃⠁⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠒⠈⢉⠁⠀⠄⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣘⠉⠁⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⠒⠒⠛⠛⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⣀⡤⠖⠒⠉⠉⠀⠈⠙⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡬⠟⠲⠦⣤⣠⣎⣥⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣒⡶⢖⡺⠛⢿⣓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠚⡇
⠀⠀⠀⠀⣼⠧⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠁⠔⠋⠀⠀⠀⠈⠙⠀⠉⠒⠤⣀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⡿
⠀⠀⠀⢐⡇⣇⠀⠀⠈⠉⠒⠦⢤⣀⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⢶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⡇⢸⡄⠀⠀⠀⠀⠀⠀⢸⠉⠑⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠉⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⡇⠀⢣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⣀⣀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠈⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⡇⠀⠀⢹⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⢇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⢀⠼⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⡠⠋⠀⠳⣄⠀⠀⢸⡀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢱⠜⠁⠀⠀⠀⠈⠙⠒⢾⣧⣀⣠⠾⠃⠘⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⡀⠀⠀⠀⠈⢦⢸⠀⠀⠀⠀⢀⣠⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⣌⣻⣄⠠⠐⠊⠉
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣺⣧⣤⣦⡤⠤⣀⠀⠀⠀⣿⡿⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⡀⠘⢿⡆⠀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣻⣿⣿⣿⣿⣿⣿⣧⠀⠘⡞⡀⡟⠀⠀⠀⠈⢲⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠫⣍⠉⠉⠛⠛⠛⢛⣛⠿⠀⠀⠁⠇⣀⡀⠀⠀⠀⠀⢫⢇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠳⢄⡀⠀⠀⠀⠹⠅⠀⠀⠀⠸⣿⠂⡄⠀⠀⠀⠸⠸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⠢⣄⠀⢀⣴⡇⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡀⠀⠀⠈⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉⠀⠀⠀⠀⢀⣈⠼⠧⡖⠶⠶⠶⠶⠶⠶⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⣀⣠⠤⢴⣶⣾⣿⣿⣶⣶⠶⠖⠒⠉⠁⠀⠠⢴⣾⣿⡷⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⣔⣉⣥⢴⠒⠭⠅⠑⠋⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠄⠐⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣭⣭⣭⣭⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⢦⣄⣀⣀⣀⣀⣤⣴⣶⠟⣫⣿⣿⠋⢀⣴⣿⠇⠀⣴⣿⠋⢩⣿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣉⣴⣿⣿⠟⠁⣠⣾⣿⡿⠀⣸⣿⡟⢠⣿⣿⢸⣿⡟⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⡿⠿⠟⠋⢁⣠⣾⣿⣿⡿⠁⣰⣿⣿⢃⣿⣿⡟⠸⣿⣷⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣶⣾⣿⣿⠿⠛⢁⣠⣾⣿⣿⠃⣼⣿⣿⠃⠀⣿⡏⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠾⠿⠿⠟⢋⣠⣾⡿⠟⠁⠀⠀⠟⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠈⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⠻⣦⣄⠀⠀⠀⠀⣠⣴⠞⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠉⠻⢦⡴⠟⠋⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉







⣿⣿⣶⣾⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⢷⠷⠾⠿⠶⣶⣾⠷⠶⠿⠶⢶⡶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡒⠒⢲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⢾⡿⠀⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣷⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⣀⣀⡤⠛⠁⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣒⣒⣒⣶⣶⣶⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠂⠤⣝⣦⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣭⣭⠭⣭⣭⣤⠄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢬⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠂⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⠞⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢀⣄⠀⠂⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣀⣴⡆⠀⠸⢤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧
⠀⠀⠀⠀⠀⣾⠙⠆⠀⠄⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠘⠙⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⣹⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⢀⡀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣧⢠⣀⣀⢀⣀⣀⣀⢀⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠌⠙⠛⠘⠛⠛⠋⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⠉⣉⣉⢉⣉⡉⣉⡉⣉⣉⠉⢉⣉⠉⣉⡉⣉⡉⡉⢉⣉⢉⣉⠙⠉⢹⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⣿⠞⢹⣀⡇⠙⣦⠀⡇⠀⢾⣈⡇⡷⠂⡷⠂⡇⢯⡀⣽⡂⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠻⠿⠿⠿⡿⠧⠤⠤⠤⠤⠤⠭⠤⠬⠥⠤⠤⠤⠤⠭⠤⠤⠤⠤⠤⠤⠤⠭⠬⠥⠤⠤⠼⢿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢲⠀⠀⣶⠒⠒⢲⠀⠀⡖⠒⠒⣶⠀⠀⡖⠒⠒⡆⠀⢰⡖⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣸⠀⠀⣿⣀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⣇⡀⠀⡇⠀⢸⡆⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹⠉⠉⠙⡏⠉⠉⣷⢸⣿⣿⣿⣿⡇⣿⠉⠉⢹⡏⠉⠉⠏⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡟⠀⠀⢸⠁⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢠⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡆⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⣏⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⢀⡀⠀⢀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣙⣉⣁⣸⣿⣿⣿⣿⣇⣉⣉⣉⣉⣉⣙⣉⣉⣉⣉⣉⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢤⣤⣤⣤⣼⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣧⠠⠤⠄⠒⠒⠈⠉⠁⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠀⠤⠤⢼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⢀⡤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠷⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠁
⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⢤⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⡉⠉⠉⠉⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠠⣖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢠⠴⠋
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
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⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿