Patrick Star Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠃⠀⠸⣿⡦⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢅⠀⠀⣀⣀⡀⠁⠀⠀⠀⢰
⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡱⠈⠀⠀⠈⢆⠀⠀⠀⢸
⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰⣿⢠⠃⢠⣶⡄⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒
⢰⢀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⢸⠀⠘⠛⠀⠀⡜⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⢏⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⢀⠟⠦⢄⠠⠊⠣⣀⣀⡠⠊⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⠀⠀⠀⢣⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠀⠈⢱⣶⣶⣶⣶⣶⣿⣿⣿⣿⠃⠀⠀⠀⢈⡦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠔⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⠎⠀⠀⣠⣾⡿⠛⠉⠀⠛⠉⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⣠⠃⠀⠀⠀⠈⠙⠂⠀⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢂⡔⠁⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡄
⠀⠀⠀⠀⠀⠀⠉⠢⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣊⠔⡇
⠀⠀⠀⠀⠀⠀⢠⠀⢸⣷⣬⣒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⡨⠔⠊⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣆⠹⠿⠟⣿⣷⣶⣬⣁⣒⠀⠤⠤⠄⠀⠀⠀⠠⠤⠄⢒⣒⠩⠔⠊⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣤⣶⣄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣌⠻⠿⣿⣿⣿⠏⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠈⠉⠓⡦⠤⠄⠄⣀⣰⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠆⠀⠀⣠⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠘⠢⠄⠀⡠⠤⠤⢀⠀⠀⠀⠀⢸
⡎⠉⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⢁⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⡇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠃⠀⢠⣶⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⢱⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠀⠀⠘⠋⠀⢀⠇⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⣴⠣⠄⠀⠠⠞⠁⢀⠀⣀⠠⠊⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣛⣿⠛⠀⠀⢷⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠒⢦⠤⣀⣀⣀⣀⣀⣀⠠⠶⠚⢁⠇⠀⠀⠀⠘⣧⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡎⢀⡤⠄⠶⠖⠒⠉⠉⢩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⣼⡷⠁⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⠞⠀⠀⠀⠀⠀⠾⣽⣀⣀⣀⣀⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⢌⠢⣄⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣧
⠀⠀⠀⠀⠀⢹⠀⠑⢦⣙⠢⢄⣀⠀⠱⢄⣉⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢚⣡⠴⠋⣿
⠀⠀⠀⠀⠀⠘⡇⠀⡆⠈⠙⠒⠤⢉⡑⠒⠒⠤⠤⠤⢤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠒⠒⢉⣉⡠⠴⠒⠉⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢹⡄⠘⠤⠶⣶⠀⠀⠀⠉⠁⠒⠲⠒⠠⠤⡤⠤⠤⠤⠤⠶⠶⠖⠚⠋⠉⠀⣀⣀⡀⠀⠀⡤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢰⡁⠀⠀⠀⠀⣠⢦⡀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠈⠙⠒⠋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡝⠦⡀⠙⠂⠶⠤⠴⠟⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠈⠉⠒⠤⢤⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡜⠉⠉⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⢤⣄⣀⣀⣀⣀⣀⣠⡇⠀⠉⠉⠉⢙⠇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠁⠈⢈⡟⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⣤⣬⠧⠤⠰⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⣼⠃⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠃⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⠀⢠⣼⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠿⣿⡿⠿⠀⠈⠛⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠂⠐⠶⣄⡴⠛⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠃⠀⠀⢀⡀⠘⢧⠀⢀⣤⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⣸⣿⣿⡆⠘⡆⢿⣿⣿⡆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣷⠀⠀⠸⣿⣿⠇⠀⡇⠘⠿⠟⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣆⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠳⣤⣀⣤⠾⠋⠙⠳⠶⠶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠷⠴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠸⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀⡼⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢀⣀⣈⡻⣦⡤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠛⠛⠂⠀⠀⠤⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂⠀⠀⠐⠋⠉⠉⠁⠀⠳⣤⡈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡄⠀⠀⠀⠀⠀⠈⠹⣦⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⣿⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⣹⡇
⠀⠀⠀⠀⣠⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⣙⠳⣦⣀
⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠈⠻⣆⠹⣧⡀
⠀⣼⠏⠻⠃⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠾⣿⠰⠆⣀⠉⠀⠹⣧
⢰⡟⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠀⠀⠀⠙⠻⠶⣦⣤⣤⣤⣤⣶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠟⠛⠋⠁⠀⢻⣆
⢸⡇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣿⡀⠀⠀⠀⠀⠀⠹
⣾⠁⠀⠀⠀⠀⠐⠚⠒⠛⠶⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡇⠸
⠹⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡚⣃⡶⠀⠀⠀⠀⢀⣽⡇⠀⠠⠔⠋⠀⢸
⠀⢿⣄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⡡⠤⠤⠶⢚⣛⣩⣿⠁⠀⠀⠀⢀⣠⠟
⠀⠈⢻⣦⠈⠻⣦⣀⣀⡀⠀⠀⠀⠀⠈⢻⣧⠼⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣫⠴⠶⣭⠽⠕⠛⠛⠋⠉⠉⣽⣧⡴⠖⠛⠛⠛⠁
⠀⠀⠀⠉⠻⢶⣤⣈⣁⣀⣀⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣴⣿⢿⣷⡀⠀⠀⠀⠀⣰⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢉⡟⠉⠀⠀⠀⣤⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⣴⡏⢀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⣀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⢀⣠⣾⠋⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣤⡶⠟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣠⣤⣤⠶⠞⠋⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠙⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢶⣤⣄⣀⣀⠀⢀⣀⣀⣀⣀⣠⡿⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠉⠉⠉⡙⠋⠉⣿⠟⠉⠁⠀⠀⠀⠈⣿⠉⣩⠉⠉⠉⠉⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠀⠀⡼⠃⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⠄⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣿⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣠⡀⠀⣦⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣴⣿⡿⠃⠈⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⠋⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⠀⡠⠼⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡰⠋⠀⠈⠛⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣱⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣿⠀⠀⡇⣀⣤⣴⡶⢻⠈⠛⠋⠱⠦⣤⠤⠤⠤⠴⡄
⡐⢲⠒⠒⠉⠉⠉⠉⠉⠁⢹⠀⠀⠉⣁⣀⣀⣀⡘⠢⠄⠔⠀⣰⠛⢲⡤⠔⢤⣹⡄⣀⣀⡀
⢡⠈⠀⠀⠀⣀⡤⠖⠚⠉⢹⡄⠀⠀⠀⠀⠈⠙⢦⡀⠀⠈⠉⠁⣴⢋⡔⡹⠦⠽⠉⣀⡠⢞⡆
⠀⣆⣰⠶⠛⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⢋⣇⣛⣦⠖⠓⠯⡱⠟⠛⠛⠲⢤⡀
⢠⠟⠁⠀⠀⠀⠀⠀⠀⡀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠾⠶⠞⠉⠉⠹⡆⠂⣾⠁⠀⠐⠓⠀⠀⠉⢦⡀
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢹⠈⠻⣤⣠⢴⠖⠉⠉⠑⠲⢷⡀
⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⣄⣠⠾⣥⣀⡠⠦⠤⠤⣽⣤⡀
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠤⢐⠢⠌⠓⠀⠁⠀⠀⠀⠀⠀⠈⡓⢄
⠀⠀⠀⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠠⢒⠭⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠫⡉⠒⠘⠁⠈⢧
⠀⠀⢸⠁⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡡⢾⠀⠀⢀⡾⠉⢹⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠈⢣
⠀⢠⣧⣰⠀⡇⢹⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣡⠊⠀⢸⡀⠀⣼⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠈⡆
⠀⢸⠁⣭⠀⢾⠀⡇⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡜⠁⠀⠀⠀⠙⠚⠁⠀⢸⡀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠀⠀⠀⠀⡷⠄⣀⣀⡀
⠀⡌⢠⠄⠐⠒⣇⢱⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠉
⢠⠁⡖⠀⠀⠀⠸⠀⠷⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠆⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢠⣄
⠸⠶⠇⠀⠀⠀⠀⡆⢰⡄⠀⠀⢸⡀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡽⠁⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢳⠀⠀⠀⢧⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⣠⠒⢢⠀⠀⢠⠤⣄⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⢰⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠼⡗⠂⠀⠙⣧⠀⠀⠀⢸⠀⠀⠀⠀⠰⠇⠀⠀⡇⠀⡋⠀⠀⢳⡤⠾⣄⠀⢨⠇⠀⠀⠀⠀⠀⡴⠃⣘⠀⠀⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⢎⡳⢦⠀⠈⢷⡄⠀⠘⣴⠀⠀⠀⠀⠀⠀⠀⠙⠶⠃⠀⣠⠞⠀⠀⠘⢦⡟⠀⠀⠀⠀⣠⣾⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠧⠶⠀⠳⣄⠡⣄⠀⠙⠶⣄⢹⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⣧⠀⣀⠴⠊⠁⠉⠳⠤⢴⣚⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⡁⢤⡀⠈⠙⠻⣄⠀⠈⠃⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢈⠳⡀⠀⠀⠙⠶⢼⣄⣈⣦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⣑⠢⡀⠀⠀⠀⠈⠈⠙⠶⠾⠋⠳⣄⠀⠀⠀⠀⡧⠤⠴⠒⠒⠋⠉⠉⠁⢠⡅⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⣑⣀⣀⡀⠤⠤⠔⠒⠒⠊⠉⠳⣤⣤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⣤⣤⡹⣧
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣾⡿⠋⠀⠀⠉⠻⣿⠿⣆
⠀⠀⣠⠾⠋⠁⣤⠀⠀⢈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠛⠁⠀⠀⠀⠀⠀⠈⠀⢿⡄
⠀⣼⠋⠀⠀⠀⣠⡴⠞⠙⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠞⠉⠈⠑⣦⡞⠉⠉⠳⣼⣧
⣸⠇⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠃⠻⠀⢀⣤⡀⠸⡇⢀⣀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠛⠛⠳⢦⣄⡀
⣿⡀⠀⠀⠀⠸⣤⣀⣠⠶⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⠀⢺⣿⡇⠀⡇⢸⣿⡷⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠀⠀⠀⠀⠀⠈⢻
⠘⢷⣄⣠⣤⠾⠋⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠴⢷⡀⠀⠉⠁⣸⠇⠈⠙⢁⡾⠋⠛⠳⣄⠀⠀⠀⠀⠀⠀⣠⡶⠟⠓⠾⣧⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠹⠤⣠⠴⠛⠦⣀⣠⠾⠀⣤⡄⢀⡿⠀⠀⠀⠀⠀⢰⡏⠀⢀⡀⠀⠀⠙⡦
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⣯⡀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣿⠁⣾⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠙⠶⠶⠞⠳⢦⣀⣠⡼
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢀⡿⠀⣾⠀⠙⠲⢦⣤⣤⡴⠶⡟⠁⠀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣸⠇⠀⣿⣷⣄⠀⢀⡇⠀⠀⢀⡇⠀⣸⠇⠀⣿⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢀⣿⠀⠀⣿⣿⣿⣆⠀⠙⠛⠛⠛⠀⢠⡏⠀⠀⣿⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣸⠇⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣿⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⣿⠀⠀⢸⠟⠛⢿⣿⣄⠀⠀⠀⠀⣼⠃⠀⠀⠀⡿⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠸⡄⠀⠈⠃⠙⣧⠀⢀⡼⠁⠀⠀⠀⠀⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠻⣦⣀⣀⣀⣿⡴⠋⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣄⡀⠉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠏
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⠏
⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏
⠀⣰⠏⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢠⣿⠀⠈⢻⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣷⠀⠀⠀⠱⣝⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢿⠀⠀⠀⠀⠈⠑⢯⣓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡙⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠉⠛⠯⣛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠿⠿⢶⣄
⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⠾⢻⣿⠶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠞⠉⠀⠀⠀⠀⣀⣀⢻⡇
⠀⢠⡞⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⢲⡆⠈⠉⠉⠐⠒⠚⠛⠛⠛⠛⣿⠻⠛⠀⠀⠀⣠⠖⠉⠁⠈⠙⢿⣇
⢠⡟⠀⢠⡦⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢰⡛⠓⠚⠃⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢻⡆
⠘⢷⡀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⡜⢀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠈⢻⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣀⣉⣙⣒⣚⣛⣉⣉⣀⣀⡠⣄⡀⠀⠀⠀⣷⠋⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠸⣟⠛⠶⣤⣀⠀⠀⠀⠀⢀⣀⣼⠉⠉⠉⠉⠛⠛⠉⠉⠉⠉⠁⠀⠈⠙⠶⣄⣼⠃⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⢿⡆⠀⠀⠈⠉⠙⠛⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠙⠶⣄⣀⡴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⠀⠀⠲⡄⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡀⠀⡀⢀⠠⠤⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⣠⡆⣀⠀⠘⠀⢇⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠘⠃⢃⠛⠀⢀⠀⠈⡆⠀⠀⢀⠴⠊⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⣀⠁⠀⠐⠁⠀⠀⠀⠁⢠⡀⠘⠂⠊⠁⠀⠀⠘⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠉⠓⠦⠤⠤⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⢀⠎⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠴⡋
⠀⠀⡾⠠⠀⠀⢀⡴⡟⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⡠⠋⠀⢿⠢⢤⣀⣀⣀⠀⠀⠀⠀⠲⠭⠂⠀⠀⠀⠀⣀⣀⢴⡇
⠀⠀⠹⠤⠊⠀⠀⠀⠸⡀⠉⠙⠛⠛⢿⠭⠭⠭⠭⠭⠭⠿⠿⡟⠛⠛⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⠦⠴⡦⠀⠀⣴⠦⡤⠃⠀⠀⠈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠇⠀⠀⠀⠀⠀⠀⠳⠦⠴⠞⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠙⣆⠀⠀⠀⠀⢲⠖⠒⠉⠉⠀⠀⠀⠀⢰⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠛⠒⠒⠒⢶⠞⠀⠀⠀⠈⠖⠒⠒⠒⠋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠂⣠⠃⠀⠀⠀⠀⠀⠘⡄⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⠡⢀⣀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠂⠀⠀⠐⠀⢀⠀⢀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠀⠀⠀⠀⠀⠀⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠛⠿⢧⡀⠀⠘⠿⣦⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠲⠀⠀⠀⠀⠀⠀⣹⠄⠀⠀⠈⠁⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢀⣀⣈⣹⢇⣉⠁⢀⡤⢤⣀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢰⠏⠁⢸⠁⡌⠁⠈⢱⠋⠀⠀⠈⡆⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡿⠀⠀⣿⢰⡇⣾⡆⢸⢾⡷⠀⢀⡼⠶⠶⠀⢻⡀
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠈⣇⠈⠁⢸⡈⠁⠀⢺⣄⠀⠀⣀⠀⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⡇⠀⠙⢦⣀⣠⢷⣀⠀⣠⠋⢛⠙⣇⠀⢹⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⢦⣠⡇⠀⠀⠀⠉⠁⢀⣌⣉⣁⣠⢏⠀⢻⡄⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢤⣀⢟⡇⠀⠀⠀⠀⠀⣇⠀⠵⠀⠱⡀⠑⢤⡁⠀⠘⠙⢦⡀⠀⣀⣠⠴⠖⠚⠋⠉⠉⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⡏⠸⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠱⣀⡈⣡⠂⡐⠀⢈⡷⠛⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠆⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠩⠔⢋⠔⠁⢀⡜⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣄⠀⡑⢄⣀⣀⡀⡀⠀⠀⠀⣀⡠⠤⠚⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⡠⠀⠀⠈⠓⣤⡾⠁⠀⣴⣿⡛⠋⣉⣉⣠⡤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠳⣆⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⡰⠊⠁⣺⠶⠦⣗⠂⠙⢏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡀⠀⠁⠈⠉⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠻⠆⠀⠀⠛⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠦⣠⣠⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠠⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠐⠠⠀⠀⠀⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⣀⣀⡀⠀⠤⠤⠀⠚⠀⠈⠀⠀⠀⠀⠀⡠⠖⠛⠉⠙⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡾⡇⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣦⠉⠲⠤⣀⡀⢀⣼⠋⠙⠛⠶⢦⣤⣤⣤⣄⣀⣤⣤⣤⡴⠶⠚⠋⠁⠀⠙⣦⣀⣀⣀⣤⠤⠒⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⢀⡽⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⣀⣠⡤⠶⠋⠉⡀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠁⢤⣀⡀
⢠⡄⠛⠉⠀⠀⠀⠀⠀⠁⠀⣀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠈⢁⣀⣐⣂⣤
⠀⠈⠉⠐⠢⣄⣀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⡄⠀⠀⠸⢿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠐⠒⠂⣀⠔⠉⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠆⢀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠈⡄
⠀⠀⠀⠀⠀⣸⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠸⣿⡄⠀⠰⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⡀⠀⠉⠀⠀⢀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠉⠶⣄⠀⠀⠀⠀⢰⠁⠀⣀⣀⡀⠄⠈⠢⠀⠀⠄⠀⠑⠀⠀⠀⠉⡆⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⡏⠀⠀⠉⡿⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡀⠀⠀⠀⠰⡀⠉⠛⠢⠶⠤⠤⠤⠴⠖⠚⡋⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠹⣄⣤⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠙⠁⠙⠢⡄⢣⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠈⢦⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠛⠁⠀⠀⠀⠀⠀⠣⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⢀⠆⠀⠀⢀⡀⠴⣈⠔⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠓⠤⣈⠑⠒⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣒⣩⠤⠔⠒⣉⠴⢛⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠉⠓⠒⠤⠤⣀⣉⣉⡉⠉⠓⠒⠓⠒⠈⢉⣉⣉⡁⠠⠤⠒⠂⢁⣀⡀⣠⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢦⣀⠤⠒⠢⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣹⠉⠀⠀⠀⡀⠀⠀⠀⠀⠘⠇⠉⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣠⠌⠉⣆⣀⢀⣀⠠⠞⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⡤⠊⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⠲⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠈⠁⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠤⠴⠶⠶⠚⡛⠉⠁⠀⠀⠀⠀⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⠀⢀⣰⠃⡀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠙⠲⡖⠂⠀⠒⠒⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠉⠉⠁⠈⢩⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿
⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠃⠀⠛⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠤⠂⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⡤⠀⠐⡀⢀⠀⠊⠁⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿
⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣄⣠⡀⠀⠀⠀⠐⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⣼⢀⡤⢀⣀⣠⠤⣠⠤⠤⣤⠤⢄⣀⢤⣀⡤⢄⣀⣀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⢸⠀⣦⠙⠇⠀⢷⡆⢰⡆⢰⡄⠋⠨⡏⢰⡀⡋⠸⠁⡜⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⢸⠀⢋⣤⠀⠇⢸⡇⢸⡇⢀⠀⣿⢀⡇⢸⠛⡃⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⢸⠄⠛⠿⠐⠃⢸⠇⢸⣧⠘⠂⢭⡤⠷⢤⠤⠧⠼⢧⡼⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢿⢸⡀⠳⣴⣗⠐⣿⢀⠀⣿⠀⡇⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⢸⠋⡦⢸⣿⠸⡏⢈⡁⢹⠀⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣼⠈⠓⠶⠊⠙⠚⠓⠚⠳⠞⠚⠳⠞⠃⠀⠀⠀⠀⠀⠀⠈⡙⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠈⢂⡀⠀⠁⠀⠀⢄⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⡀⡘⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢺⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠿⠁⠁⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠁⠀⠀⠀⠁
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣮
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠃⠀⠀⣀⡠⠖⠁⠀⠀⣤⣄⠀⠀⠀⠆⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣿
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⡍⠀⠀⠀⠀⠈⣄⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣿
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠶⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡰⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠜⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡘⠀⠀⠀⠒⠐⠂⠁⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠖⠀⣰⠁⠀⠀⠀⢀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠈⠀⠂⠀⠉⠍⠐⠒⠒⠒⠒⠒⠒⠒⣂⠀⠌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠠⠤⠤⠀⠤⠤⠄⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⣤⡀⢀⠁⠀⠀⣠⣄⡀⣠⣀⠀⠀⣾⠀⣿
⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⣿⣯⣿⣿⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⠀⣿
⠀⠀⠀⣾⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣄⠀⠀⢹⣏⣀⣉⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣿
⠀⠀⠀⠈⠹⠋⠹⠋⠉⠉⠿⠉⠋⠈⠏⠉⠟⠉⠉⠉⠉⠿⠉⠹⠉⠉⠙⠿⠉⠙⠉⠉⠉⠉⠉⠉⠉⠁⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣩⡁⣸⢰⣿⢯⣓⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢸⡑⣻⣣⡿⠃⠀⠀⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡞⣡⠶⠯⠿⠟⠁⠀⠀⠀⢸⣹⠁⠀⠀⠀⣠⣤⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠳⣿⠋⠁⠀⠀⠀⠀⠀⣀⣀⣀⣈⣥⣤⣤⣤⣞⣽⣿⣿⡿⣿⣟⠿⢯⣍⡓⠂
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⠏⠀⠀⣠⡤⠶⠚⠛⠉⠉⠠⠚⠉⠿⡦⠀⠀⢉⣭⠭⢟⡓⠶⢿⣷⣤⣉⠑⠂
⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠃⣀⡴⠛⠁⠀⠀⠀⠀⣤⠶⠖⠒⠒⠒⠒⠀⠀⠀⠀⠀⠘⢻⠀⠀⠈⠙⠻⢿⣦⣄
⠀⠀⣰⣿⣿⣿⣿⣿⡿⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠙⠻⣷⡀
⠀⠘⢿⢿⣿⣿⣿⡿⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠈⢻⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠶⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣠⠴⠚⣋⣭⣟⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣳⠦⠈⠙⠚⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣽⣂⡀
⠀⠀⠀⠀⠀⠘⠚⠉⡽⠁⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠙⣆⠀⢻⠉⠳⠬⣷⠲⢤⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣸⠀⠀⠀⠀⠠⢔⣀⡉⠙⠒⠤⠤⣘⣯⡉⠙⢲⠒⠦⠬⣷⣦⡀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡟⠀⠀⠀⢀⡞⠉⠈⠉⠓⢄⠀⠀⠀⣀⣰⣤⠈⢧
⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡏⡼⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷⣰⠋⠀⠀⠈⡆⠈⢧⣀⡀
⠀⠀⠀⠀⡞⢉⣀⣉⠑⢾⡼⠁⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣷⠀⠳⠆⠺⡄
⠀⠀⠀⠀⢇⠀⠈⠳⠻⠄⠁⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣠⣀⣸⢠⢴⣄⠀⢀⡇⠀⠘⢯⡍⡣
⠀⠀⠀⠀⠀⠑⠲⠤⠴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣳⣿⠋⠻⠿⠯⠤⠚⣠⠖⠒⡄⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠒⠲⠖⠛⠁⠀⠀⠀⠀⠀⠀⠁⢀⡴⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠄⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⡠⢀⡴⠒⣒⡧⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡸⢀⠆⠀⠀⠀⢸⠉⣉⣹⠭⠼⠧⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⢱⠓⢺⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢖⡓⠄⠛⠀⡞⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠤⠤⠴⠃⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⡖⠒⠒⠒⢻⢹⡉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠼⢾⠷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣠⣤⢾⠒⠲⣍⣹⡆⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣢⠤⠤⠾⠷⢓⣚⡪⠽⠛⠉⠀⠀⠀⢀⣠⣤⣤⣤⠄⠍⠉⠉⠉⠛⠛⠳⠶⠶⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⠿⠯⠭⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠟⠻⠎⠙⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠠⠐⠀⠁⠀⠀⠀⠈⠉⠒⠦⢀⣠⠞⠀⠀⠀⠀⠀⠒⠈⢷⣄⠀⠀⠀⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠳⢤⣀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⣀⣠⡤⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠀⠀⠀⠈⣇⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⢉⡥⠤⠄⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⡀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⡇⠀⠀⠀⠀⠀⢰⠋⠀⣠⣴⡆⠈⢳⠀⠀⠀⠀⡴⢉⣤⣤⠀⠈⢢⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⡶⢸⡇⠀⠀⢰⠀⣿⣿⣷⣶⠀⠀⡇⠀⠀⠀⠀⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⠀⠀⠘⠲⠤⢌⢛⠋⣁⠜⠀⠀⠀⠀⠣⠙⠛⣛⢋⢀⡼⠁⠀⠀⠀⢀⡇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛
⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⣄⡀
⠀⠀⠀⠀⠈⢧⡈⠙⠦⣄⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡤⠖⠋⣠⠟
⠀⣠⠤⠤⠶⠶⠞⠓⠀⠈⠙⣆⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠙⠂⠀⠀⠀⠀⣠⢞⡟⠉⠀⡀⠛⠷⠶⠤⢤⢤⡀
⣾⣃⣈⠭⠅⠀⠀⠈⠉⠙⡆⣜⠀⠀⠉⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⢘⡇⡔⠉⠉⠀⠀⠀⢭⣭⣁⣙
⢯⠁⣀⣀⣀⣀⣀⣠⣤⠴⠃⢹⡀⠀⠀⠀⠀⡽⠓⣶⢤⣤⣄⣀⣀⣀⣤⣤⡴⠶⣏⠁⠀⠀⠀⣸⠃⠳⢤⣤⣄⣀⣀⣀⣀⡀⢨
⠈⠉⠉⠀⠀⠀⠀⠀⠸⡄⠀⠀⢳⣀⣀⣠⠞⠁⠀⣿⢢⡀⠴⣿⣿⡷⢄⡌⣿⠀⠙⢦⣄⣀⣠⠏⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠀⠀⣠⠞⣯⠀⠀⠀⢨⣽⡀⠀⠀⢸⣦⡀⠀⠀⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⠤⠖⠚⠁⠀⡃⠀⠀⠀⠈⠛⣇⠀⠀⠈⡇⠉⠓⠦⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⣿⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣿⡿⠃⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠒⢲⡆⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢰⠀⢸⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠋⠀⠘⢧⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢯⡀⠀⠀⠀⠀⡹⠂⠀⠀⣽
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣧⣦⣤⣤⣷⣄⣀⣠⣾⣥⣤⣤⣴⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⢀⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⣴⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⢾⠏⠉⢻⣷⡄
⠀⠀⠀⠀⣿⡁⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠙⠋⠀⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠑⠦⣄⠀⠀⠈⢯⠳⢄⡀⠀⢀⡀⠀⠀⠀⠀⣀⡛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣴⣶⠀⠀⠀⠠⣤⣤⣤⠼⠀⠀⠀⠉⠒⠒⠋⠀⠀⠙⠦⠼⠏⠲⣄⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣈⣢⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⡤⠤⢤⣀⠀⠀⠀⠉⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡏⠀⠀⠀⠈⢳⡀⢠⠒⠒⢢⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⡇⠀⠀⢸⡀⠀⠀⠀⠀⣸⡇⢻⠉⠀⠀⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣥⣬⡓⠋⠀⠀⠀⠈⢇⠀⠀⠀⠀⣹⠁⠈⠳⠤⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠖⣆⠀⠀⠀⠀⠈⠣⣄⣀⡴⠃⠀⠀⠀⠐⢦⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠒⠐⠋⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠦⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⠀⠈⠙⠒⣦⢤⣄⣀⣁⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠦⡴⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠺⠟⠀⢰⠁⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠭⠀⣀⣯⣀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠁⠀⢨⡭⠉⣹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠉⠙⡟⢒⣶⣶⡾⠖⢺⠟⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⠭⠭⠟⠋⠉⠑⠛⠭⠿⠯⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡖⣶⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⢻⡍⢧⡘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣠⣴⣾⡿⣿⠹⣄⢳⡄⢷⠈⢧⠘⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣴⣿⢿⠻⡍⢷⠘⢧⠘⣆⢻⣨⣷⡾⠟⠋⠉⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⣆⢻⡌⣧⣼⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⢀⣤⣾⣻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣸⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣶⣶⡦
⠀⠀⠀⠀⠀⠈⠳⠺⠿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⡇⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣏⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠳⣌⠳⣄⣻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡈⣳⡾⠋⠙⣿⣆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠰⡄⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠈⣿⣇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢠⣿⡁⠉⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⣧⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠆⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⢤⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⢿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⢸⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣰⣿⠃⠀⠀⢸⣿⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠈⠁⠀⢸⣿
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⣿⣇⣤⡶⠶⠶⢶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢸⣿
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⢸⣿⡏⠥⠀⠋⠀⠒⠀⠯⢉⡻⣦⡀⠠⠏⠀⠀⠀⣠⣿⡏⠀⠰⠆⠀⠀⢸⣿
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⡄⠀⠀⠀⣰⣿⢿⡟⢿⠻⢶⣤⡀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣀⣠⣾⡿⠃⠈⠀⠀⠀⠓⣘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣤⣄⣀⣀⣤⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⣿⣅⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣶⣶⡿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢛⣛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣿⣿⣿⣤⣿⣼⣿⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠙⡟⢛⠛⠛⣛⢻⡛⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠈⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠈⠀⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠀⠀⠀⠀⡃⢰⣁⡉⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢛⣥⣤⣤⣤⣤⣀⠀⠀⢶⡞⠀⣀⣪⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⢿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢹⣿⣿⡇⠀⠀⠀⠈⠻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⠤⣒⣒⠂⠀⠠⠖⠚⣉⣉⣉⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⣡⠖⠉⢁⣀⡀⠠⠖⠋⣉⣉⣀⣉⣉⠙⠒⢤⡈⠑⢦⡰⡄
⠀⠀⠀⠀⠀⠀⡔⡽⢡⠞⢀⡴⠊⠁⣀⣤⠤⠖⠋⠉⠀⠀⠀⠉⠙⠲⣄⠉⢦⡀⠳⡜⣄
⠀⠀⠀⠀⠀⡜⡸⢡⢏⡴⠋⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠳⡄⠹⡜⣆
⠀⠀⠀⠈⣿⣶⣷⣟⣭⢶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡄⢳⡘⡄
⠀⠀⠀⠀⡏⡏⣿⣿⡄⡾⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄⠀⠈⣧⡹⣄⢳⡹⡄
⠀⠀⠀⢸⡇⡇⡟⣿⣿⡇⠀⠾⠋⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠁⠀⢸⣟⠮⣀⣿⣷⣀⡀
⠀⠀⠀⢸⢣⡇⡇⣿⣿⡇⠀⠀⡴⡖⡛⢳⣆⠀⠘⡆⠀⠀⠞⡏⠈⢹⢳⠀⠀⠀⡿⣿⡄⡀⣏⠙⣏⠁
⠀⠀⠀⢸⢸⡇⡇⣿⡏⣧⠀⠀⠀⠉⠀⠈⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢯⠁⢇⢹⠀⢸⡄
⠀⠀⠀⢸⢸⡇⡇⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡀⠸⡈⣇⠀⣧
⠀⠀⠀⢸⢸⡇⡇⡿⣧⣏⠀⢀⡀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣿⠀⣇⠀⢧⠸⡄⢹⡀
⠀⠀⠀⣸⢸⡇⡇⡇⣿⣇⠳⣿⠙⣄⡀⠀⠀⠈⠉⠂⠉⠀⠀⠀⠀⣀⡞⢸⡷⢁⣿⣇⠸⡄⠈⢆⠹⡄⢷
⠀⠀⢀⡟⢸⢣⣇⡇⣿⡸⡄⢹⡄⢿⠹⡄⢀⡴⠒⠉⠉⠢⡀⠀⡾⢹⠀⣼⠀⡼⡸⠹⡄⢱⡀⠈⢧⠙⣌
⠀⠀⣼⠁⡼⣸⢸⠁⡏⢣⣧⠘⡇⢸⡄⢻⣎⣴⣭⣭⣭⣶⡹⡼⠁⣾⠀⡏⢰⣧⠃⠀⢳⠀⠳⡀⠀⠳⡌
⠀⣰⠃⣰⢣⡏⡟⠀⡇⢈⡿⡄⣧⠈⣇⠀⠻⣿⣿⣿⣿⣿⡿⠁⢀⡇⢰⠇⡼⡏⠀⠀⠈⢧⠀⠁
⡼⢁⡴⢁⡞⣸⠁⢰⠃⡸⢻⡹⡜⣆⢹⡀⠀⢹⡄⠀⢀⡼⠁⠀⣼⢁⡟⡼⣼⠁⠀⠀⠀⠈
⠰⠋⣠⠏⡴⠃⢀⠏⣰⠃⣸⣧⠀⠀⠀⠁⠀⠀⢻⡉⣹⠃⠀⠀⠁⠈⠀⢰⡇
⠀⠀⠀⠚⠁⣠⠋⡴⠁⡴⠃⣿⠀⠀⠀⠀⠀⠀⠘⣧⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⡡⠊⣠⠞⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⣠⠄⠀⣀⠤⣦⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡠⠀⢼⣤⣾⢟⡦⠄⠤⠜⠻⢷⠛⣷⣶⣄⠀⢲⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⡯⠐⡚⠛⣻⢖⠈⠁⠂⠒⠦⠐⣎⡁⠾⡖⢩⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⠶⢗⠀⠠⡎⠉⠀⢀⣀⣤⣄⣀⠀⠁⠠⠓⣷⣀⡹⠶⢒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠏⠀⣑⠀⢀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠋⡰⠂⠀⣀⣁⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢄⡶⣿⣼⠃⠀⠀⠸⠿⠿⠟⠀⠀⢹⣿⣿⠀⠸⣀⠄⠸⡄⠈⠉⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣞⡿⣃⡀⠀⢤⠀⠀⠤⢀⡔⠀⣠⣿⣿⡟⠀⠐⠋⢓⠸⣀⣀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⡅⠀⣦⠀⡰⠋⠠⡟⠍⢀⣾⣿⣿⠟⠀⠀⡀⢠⡎⠈⢉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⢁⣀⡭⣦⣳⠸⣏⠀⠀⣾⣿⣿⠋⠀⠀⠰⠛⢻⠷⡄⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⠂⠀⠻⣼⣋⣀⠀⠉⠉⢉⠤⣠⡒⠒⢢⠁⠹⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⡄⠊⣱⣿⣿⡁⠀⣿⣿⣿⠀⠈⣕⣦⠬⠀⠀⠈⠉⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⢀⡧⢰⡟⣷⡿⢷⠀⠉⠉⢉⣠⢼⡥⢤⠵⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⠞⠛⠀⠤⠉⠌⢉⡀⠀⠀⠰⠄⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠉⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⠄⠀⡤⠤⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠿⠂⠀⠀⠷⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⣴⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠁⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡤
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠉⠲⠶⢤⠄⠉
⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠒⠲⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠃⡀⠀⠀⠀⠀⢄⠀⠀⣠⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠐⠂⠒⠶⢤⣄
⠀⠀⠀⠀⠃⠀⠀⠀⡿⠳⠞⣿⠀⠘⡴⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡶⠖⠻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⢀⠀⠀⠀⠀⠀⡸⢷⣤⣤⠀⠀⡄⠉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⣶⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠰⣀⠀⠀⢰⣀⣀⢀⣼⣆⠀⠑⢄⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⠟⠀⠈⡟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡀⠀⠀⠀⠈⠀⠀⠈⡟⠉⠹⣇⡁⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠒⠂⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠑⠀⠀⠀⠀⠀⢀⡀⠷⢠⣾⣿⡇⠀⡄⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠱⣆⠙⢦⠀⠀⣇⠀⠀⠒⠂⠀⠀⠀⠀⠀⠎
⡀⠀⠀⠀⠠⠘⢷⡀⠀⣼⣿⡇⢠⡄⠀⠀⠀⠘⠀⠈⢆⠀⠀⠀⠀⠀⠀⠙⢦⣄⠉⠁⠈⠳⡄⠀⠀⠀⠀⠀⡠⠊
⠁⠀⠀⠀⠷⣦⣈⡻⠞⠁⢿⠃⣠⡼⢆⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⣤⣈⠷⣄⠀⠒⠉
⠀⢀⢰⣿⣄⣀⡙⠁⠀⠀⠈⠻⣋⢀⣬⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢀⣤⡀⠀⠀⠀⠀⠀⠀⡈⠳⣄
⠆⠈⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠙⡋⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠂⢤⣤⣇⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠙⠲⢤⣌⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠒⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠠⠤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣶⣶⣶
⣿⣿⣿⡿⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣴⣦⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠐⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡐⠒⢤⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠻⠿⠋⠀⠹⠋⠀⠉⠛⠙⠿⠋⠈⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⢠⠔⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⡐⠁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠁⠉⠻⡏
⠀⠀⠇⠀⣀⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠀⠐⠀⠀⡁
⠀⢰⠀⠘⠄⠈⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⣿⡀⠀⠠⠈⠠⠀⣶⠆⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠁
⠀⠈⠢⡀⠀⡀⠃⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢀⣼⣿⣧⡠⠃⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣤⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⠟⠀⠀⠁⠂⢀⠀⢀⠎⠀⠁⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠸⠥⠄⠀⠤⠄⠠⠀⠘⠛⠿⣿⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠄⠀⠠⠀⠀⠀⠀⠀⠈⠣
⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠁⠈⠅⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂
⡇⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠈⡂⠎⠱⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠂⠂⠀⠀⠀⠀⠀⠈⠁⠒⠀⠈⠁⠉⠐⠄⠀⠀⠀⠀⠀⠑⠠⠀⢀⠠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠣⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡎⠞⠛⠛⠳⠆⠀⠀⠀⠘⠛⠋⠉⠙⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠸⡄⢀⠴⣶⡦⠀⠀⠀⠀⠀⠔⣶⡲⡄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢣⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣠⠔⠚⠁⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⣶⠆⣰⡶⡏⠀⠀⠀⠀⠑⠀⣀⡀⢿⠆⠀⢠⣶⡌⢱⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⢸⠀⢰⠉⠀⡏⠁⣿⢦⠀⠀⣠⡎⢰⠹⡏⠀⠀⠀⡌⠉⡇⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢸⠀⢰⡇⠀⡏⢸⠟⢻⣏⠀⡎⠀⡇⠀⠀⠀⠇⢠⠇⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⢸⠀⢸⠀⠠⠁⢸⣴⣮⢹⠀⡇⠀⠇⠀⠘⠸⠀⣸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⡀⠀⠀⠀⡇⠀⠈⠀⠀⠀⢸⠃⢸⠀⡇⠃⠀⠀⠀⢠⠇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠘⣆⠀⢠⠇⠀⠀⠀⠀⠀⠞⠀⡿⠀⠘⠄⠀⠀⠀⠈⠀⠀⡇⠀⢀⠔⠋⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠢⠋⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⢠⠏⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠒⠒⢒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣓⣼⡃⣽⣟⠀⠀⠀⠀⠀⡀⢀⣠⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠙⠛⠉⠀⠀⠀⠂⠁⠀⠘⣶⡐⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⡲⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢈⣰⡎⠁⠠⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠴⠄⡀⠂⠀⠀⠀⠀⠀⣿⣿⣿⣦⣥⡆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⡤⢤⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⢄⡀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢲⢰⠃⡖⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢵⣏⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⡀⣤⣶⣦⡄⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⡟⠋⠉⠻⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣾⡿⠋⢿⣿⣿⠏⠻⣿⣦⣄⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⢀⣴⣄⠀⠀⢀⣦⡀⠈⠻⣿⣿⣿
⠀⠀⠀⠻⠿⣿⣷⣶⣤⣭⣥⣴⣶⣿⡿⠿⠃⠀⠀⠀⣿⣿⣿⣧⣀⡀⠈⠉⠛⠓⠚⠛⠋⠁⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⡀⢉⣿⣿⣿⣿⣿⣏⠁⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡼⠉⠉⠀⠀⢸⠀⠀⠈⠉⠹⡄⠀⠀⠀⠀⣿⣿⣿⣿⡏⣴⣶⣿⣿⣿⣾⣿⣿⣿⣷⡹⣿⣿⣿
⠀⠀⠀⣼⣧⣤⣒⣒⣒⣼⣔⣒⣒⣢⣤⣿⡄⠀⠀⠀⣿⣿⣿⡟⠸⠿⠿⠭⠭⠟⠹⠭⠭⠿⠿⡇⢹⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⣤⣤⣶⣶⣷⣶⣶⣶⣤⣴⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠻⢿⣿⣿⡿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢠⠴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠈⣇⠀⠀⠀⠙⠶⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⢹⠆⠀⠀⠀⢀⣤⠶⠖⢶⠖⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠛⠋⠀⠀⠀⠀⢸⡁⣀⣤⣀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⢶⡟⠁⠀⠈⢧⣰⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢰⠋⠙⡦⢤⠀⠀⠀⠀⣠⠤⣤⠞⢛⣓⠒⠶⡶⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⣀⠈⡷⢠⠖⠋⠀⠀⠀⠀⠈⢻⠁⠈⠋⠉⣀⡴⠃⠀⣠⡤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣠⠔⠊⠙⣏⠈⠻⡇⡼⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⣀⣀⣧⣀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠓⠒⠛⠒⠒⠒⠛⠒⠚⠓⠛⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⢳⡜⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡶⠖⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠛⠓⣶⠚⠹⣄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⣀⠀⠀⣠⢄⠀⠀⢀⡤⢤⡀⠀⢀⣀⡀⠀⢀⣀⠀⢠⡇⠀⠀⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠘⡀⠰⡇⠈⡆⠀⡞⠀⠀⣧⠀⠏⠀⠃⢀⠋⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⡄⢃⠀⡇⠀⢡⠀⠁⠀⠀⢸⢸⠀⠀⡀⠘⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⠸⠀⡁⠀⢸⠀⠀⠀⠀⠈⣸⠀⠀⡇⡄⠀⡼⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⡇⠃⠀⠈⡇⡀⠀⠀⠐⡿⠀⠀⡇⡇⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡆⢰⢸⠀⠀⠃⠀⠀⠀⢀⠳⠀⠀⠃⡇⢰⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢡⠘⠀⡇⠀⠀⢰⡀⠀⣸⢸⠀⢰⠀⡇⠸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⠶⠃⠱⠤⠃⠀⠑⠶⠃⠈⠳⠋⠀⠛⠃⢸⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠴⠃⠀⠀⠿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⢻⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣾⣾⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠒⡄⠀⠀⠯⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣄⣀⡷⠀⠀⢀⡴⠒⠶⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢼⡦⠒⠢⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠇⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠚⠁⠀⢀⢄⠀⠀⠀⠀⠀⠀⢠⣶⣀⣀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢰⢀⠭⠃⠀⠀⠘⡟⠁⠖⢂⡨⠁⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠐⣅⣀⡔⡊⢻⡉⣻⣸⣀⣀⣀⣀⣀⣇⣤⣤⣭⢄⡤⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣴⠒⠒⠒⠒⠶⠦⠤⠤⠤⠤⠄⠒⠒⠒⢲⠞⠳⣜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢔⠄⢠⠒⡄⢀⠖⠲⠀⡔⠢⠀⠔⡄⣸⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡭⠠⢀⠀⢀⠸⠀⠀⡆⠁⢀⠰⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⡀⡀⢸⠀⠀⠀⢣⠀⢀⡀⢸⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠁⠀⠀⠀⠀⠀⣾⠀⠈⠁⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣀⠰⠀⡆⢆⢀⠙⣀⠆⣄⠇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣉⣀⠉⠀⠀⠁⣀⣀⣀⣀⣀⠟⠛⢲⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣭⣭⡭⠭⠭⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡬⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠉⠓⠢⠤⠏⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡖⠒⠒⠋⠀⣀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢳⠀⠀⢀⡎⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠠⠒⠋⠀⠀⢈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠤⡄⢠⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⡄⠀⢰⣀⡀⠀⠀⢏⠴⠛⠐⠒⠒⠂⠀⠀⠀⠀⠢⠀⠀⠀⠀⣠⠄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⣆⠀⢣⡀⠀⠀⡛⠐⠶⠄⠀⠀⢤⠠⠐⠀⠁⠀⠀⠀⢀⡞⠀⠀⠀⠀⠨⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣞⡄⠀⠉⠒⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠐⠲⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡃⠀⠀⢧⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣉⡀⠀⢣⠀⣀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⢭⣳⡀⢳⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡀⠙⣆⠀⢳⠃⢹⠉⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⢸⠦⣼⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢳⠼⣦⡇⠀⠀⠌⠀⠀⠀⠤⣆⠀⠀⢐⡲⠶⠾⠛⠛⢾⣋⣼⢣⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⠓⠨⠄⢀⠈⢁⡐⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⡟⡀⠱⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠈⠃⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡈⠠⠤⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠰⠖⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣄⡤⠇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣇⣠⠁⢀⡀⠀⢀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢒⣛⠀⠀⣏⣸⡏⠀⠀⢰⡛⠭⢀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⣉⡁⠁⠈⠁⠀⡐⠂⠢⠴⠀⠀⠐⠭⢓⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠼⠭⠯⣿⣑⣐⠇⠀⢀⡰⠶⡞⠉⡇⠀⠉⠐⠮⢑⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣚⣹⡛⠤⠤⠀⠠⠴⢦⣚⣻⣉⣲⠄⠀⠀⠈⠒⢬⡷⠦⣄⠀⣀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠂⠈⠋⢀⣀⣤⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⢉⡠⠔⡒⠟⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠀⠠⠠⠀⠠⠰⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠂⠠⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠙⠛⠓⠶⢦⣴⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⣀⠀⠀⠀⣀⣀⡀⠈⠳⢆⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⢀⡴⠋⠁⣠⣶⣿⣿⣿⣿⣿⣦⡄⢀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣠⠏⠀⠀⣼⠿⠛⠉⠁⠀⠀⠀⠀⠁⠀⠉⠓⢦⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠀⠀⢸⡇⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡞⠀⠀⠘⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⠁⠀⠀⠀⢻⡄⣸⠇⢀⣴⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⢀⣴⣿⠟⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣠⡞⠁⠀⠀⠀⠀⠘⣷⠋⠀⠋⠀⠀⠀⠉⠉⠉⠙⠁⠀⠀⠀⠈⢉⣠⣤⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠾⠋⠀⠀⢀⡎⠀⠀⠀⢹⡆⠀⠀⢀⡶⠋⠉⣯⢦⠀⠀⠀⠀⢰⠀⠉⠀⢿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⠈⣷⠀⠈⠉⠓⠰⠖⠚⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣄⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡇⠀⠀⢀⣿⡋
⠀⠀⠀⠀⢸⡏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⢸⡇⠈⠳⡄
⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⣶⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⠀⠀⢸⠁⠀⠀⡇
⠀⠀⠀⠀⠘⣷⡀⠀⠸⣧⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⣼⠿⠛⠉⠹⣿⠀⢀⡿⢲⡦⠘⢧⡀
⠀⠀⠀⠀⠀⠈⠻⣄⡀⠹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣶⡖⢄⠀⠀⠀⢸⠇⢠⠀⢾⣁⡀⠀⣀⣴⡏⠀⠈⣠⠞⣣⣄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⣿⡀⠘⢿⣿⣿⣿⣿⣿⣿⣧⠘⡇⠀⠇⠀⠀⡇⠀⢸⣰⡿⡿⢿⣿⣿⡿⢃⣠⣞⠁⢰⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⡇⠀⢸⡀⢸⠁⢀⡾⠋⢠⠇⠀⠈⢁⠔⠁⣸⠏⠠⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣣⠀⠀⣃⠃⣠⠎⠀⣠⠏⠀⠀⢠⠎⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠏⡴⠁⠀⡴⢃⣤⡶⣠⠋⠀⣰⠃⠠
⣴⣞⣉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠜⡜⠀⠀⣾⠵⠋⢀⡜⠁⠀⡰⠁⡴⠁⠀⠀⠀⠀⠀⠀⣠⡴
⠉⠉⠁⠀⠀⠀⠀⠀⠐⠛⣻⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡸⠀⣠⠞⠁⢀⡴⠋⠀⢀⡞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡰⠁⠐⠁⠀⣰⠋⠀⠀⠀⠘⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠉⠻⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢰⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡴⣷⣬⣥⣾⣿⣿⠄⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣠⣯⣤⣤⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣿⣤
⠀⠀⠀⡿⠻⠆⠀⠀⠀⠀⠀⠘⣿⠟⠉⡉⢻⣿⡟⢠⣄⠀⠀⠀⠀⠼⠿⠆⠀⠀⠀⠀⠀⠀⣿⠿⠏⠉⢿⣿⡟⢠⣤⡀⠀⠀⠀⠸⠿⠷⠀⠀⠀⠀⠀⠀⢻⡿⠿
⠀⠀⢠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠘⠓⢴⣶⣾⣿⠆⠘⠛⠁⠀⠀⢀⡄⢀⣀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣶⣶⣾⠗⠘⠛⠃⠀⠀⠀⣤⢀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠰
⠀⠀⢸⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣷⡀
⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⣾⣯⣩⣭⣽⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣬⣭⣽⣯⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣭⣭⣭⣿⣽⡦
⠰⣿⠟⠋⠙⣿⣿⠏⢠⣄⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠙⢿⣿⡿⢁⣤⡀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠙⠻⣿⣿⠅⣤⡀
⠀⠉⠠⢴⣶⣼⣷⠄⠛⠛⠃⠀⠀⠀⡾⠿⠇⠀⠀⠀⠀⠀⠀⠙⠁⢤⣷⣶⣿⡇⠘⠛⠃⠀⠀⠀⠼⠛⠇⠀⠀⠀⠀⠀⠀⠈⠓⢶⣶⣶⣿⡦⠘⠛⠓⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣿⣯⣽⣿⣷⡄⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣭⣿⣽⣿⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣏
⠀⠀⠀⡿⠻⠄⠀⠀⠀⠀⠀⠰⣿⡟⠋⠉⣿⣿⡟⢠⣄⠀⠀⠀⠀⡾⠿⡆⠀⠀⠀⠀⠀⠀⣿⠿⠏⠉⠻⣿⡿⢁⣤⡀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⢺⣿⢿
⠀⠀⢠⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣶⣿⡆⠘⠛⠃⠀⠀⢠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⣷⣶⣾⡆⠘⠉⠋⠀⠀⠀⣦⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⣼⣿⣉⣍⣽⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣝⣋⣿⣽⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣉⣹⣽⣿⣵⡆
⠰⣿⡟⠋⠙⣿⣿⡿⢡⣄⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⢙⣿⣿⡿⢀⣀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢶⡿⠛⠋⢻⣿⣿⢁⣤⡀
⠀⠉⠀⣤⣶⣼⣿⠄⠛⠛⠃⠀⠀⠀⡾⠿⠆⠀⠀⠀⠀⠀⠀⠙⠃⢴⣶⣤⣿⡇⠘⠛⠃⠀⠀⠀⣼⠿⠇⠀⠀⠀⠀⠀⠀⠈⠳⠾⣶⣶⣼⣇⠐⠿⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣩⣭⣭⣿⣷⡄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡟⣉⣭⣽⣭⣿⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣾⣥
⠀⠀⠀⡿⠁⠄⠀⠀⠀⠀⠀⠰⣿⠿⢟⠛⢿⣿⡯⠠⣄⠀⠀⠀⠀⠺⠹⠇⠀⠀⠀⠀⠀⠀⢿⠿⠋⠙⢻⣿⣿⢡⣄⠀⠀⠀⠀⠰⠿⠇⠀⠀⠀⠀⠀⠀⢻⣿⠟
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢼⣿⣿⣿⣿⣾⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠴
⠀⠀⣸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣾⣿⣷⡀
⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡈⠻⠇⢸⡇⢀⠃⢠⡄⠘⠁⢠⡶⠉⠠⠈⡟⢀⡄⢀⡄⠘⠏⠰⠄⢹⣿⠀⢸⣿⣿⣿⣿⣇
⠀⣼⣏⣉⣹⣿⣭⣷⡄⠀⠀⠀⠀⠀⠀⣿⣏⣉⣁⣼⣄⣈⣁⣾⠀⣉⣁⣴⣀⣾⣧⣀⣋⣸⣁⣸⣇⣸⣇⣰⣆⣘⣃⣿⣿⡄⢮⣏⣹⣿⣭⣽⣦
⠀⣿⠟⠛⠛⣿⣿⡏⢡⣄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢻⣿⣿⠁⣄⡀
⠀⠛⠤⢽⣤⣼⣿⠆⠘⠛⠃⠀⠀⠀⡼⠿⠋⠉⠉⠉⠉⠉⠉⠛⠛⢿⣿⣿⣿⡟⠛⠛⠛⠉⠉⠉⠻⠻⠟⠉⠉⠉⠉⠉⠉⠉⠡⢬⣵⣴⣿⣷⠘⠛⠋⠀⠀⠀⠸
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣉⣽⣯⣽⣶⡄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⢾⣋⣩⣉⣯⣿⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣍
⠀⠀⠀⡽⢿⠄⠀⠀⠀⠀⠀⠘⣿⡿⠋⠙⢿⣿⡟⢠⣀⠀⠀⠀⠀⢸⢿⡇⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⢻⣿⣿⢃⣀⡀⠀⠀⠀⢰⠿⠧⠀⠀⠀⠀⠀⠀⢾⡿⢿
⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢰⣶⣶⣿⠄⠙⠛⠁⠀⠀⢠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠈⠖⢲⣶⣾⣿⡆⠈⠛⠃⠀⠀⠀⠆⢀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠸
⠀⠀⢸⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⣴⣿⣋⣉⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣍⣉⣹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣉⣹⣿⣹⣷⣆
⠰⣿⡿⠟⢻⣿⣿⡿⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠻⣿⣿⣿⢁⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⢿⣿⣿⡁⣄⡀
⠀⠛⠀⣤⣴⣾⣿⠄⠸⠟⠀⠀⠀⠀⠸⠿⠆⠀⠀⠀⠀⠀⠀⠙⠇⣈⣶⣤⣿⡇⠐⠿⠃⠀⠀⠀⢰⠻⡆⠀⠀⠀⠀⠀⠀⠈⠉⢠⣤⣾⣿⣽⠓⠿⠛
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣉⣽⣿⣭⣷⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣙⣋⣩⣯⣿⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣉
⠀⠀⠀⡿⣿⡀⠀⠀⠀⠀⠀⠠⣿⡿⠟⠛⢿⣿⡿⢡⣀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⠀⣿⠿⡟⠛⢿⣿⣿⢣⣄⡀⠀⠀⠀⣠⢻⡇⠀⠀⠀⠀⠀⠀⢾⡿⠿
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣶⣦⣽⡅⠘⠛⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⢤⣶⣦⣽⡏⠙⠛⠃⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴
⠀⠀⢸⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⣸⣿⡋⣹⣯⣭⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣽⣙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣉⣻⣿⣯⣽⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠞⠉⠉⠉⠉⠛⠛⠒⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠂⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⣿⣦⠀⠀⠀⠀⠀⠹⠿⠿⠛⠻⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣠⡤⠴⠶⠶⠦⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⢉⣠⠾⠁⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠁⠙⣦⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠒⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠴⣞⣽⣃⣶⠀⠀⠀⠀⠰⣟⣛⣻⣿⣿⡑⡄⠀⠀⠀⣿⣉⠀⢊⠁⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠊⠉⠉⠉⠈⠀⠙⡄⠀⠀⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⣿⣀⠄⠉⣰⣇⡀⠀⠀⠀⢰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣀⠠⣤⣤⣄⠀⢸⠀⠀⠀⡀⠀⣤⣤⡀⠀⠀⠀⠀⠈⢷⡀⣸⠋⠈⢳⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⠀⠀⠈⠉⠉⠀⡌⠀⠀⠀⠑⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⣧⠃⡴⠀⢘⠀⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠐⠀⡟⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠃⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠋⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠠⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⠶⠀⠀⠀⠒⠒⠻⢧⡀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠸⠤⠔⠒⠒⠒⠒⠂⢐⠂⠘⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢰⠲⣄⠀⠀⠀⠀⠀⠈⢶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣠⣤⣤⣤⡬⣷⡌⠳⣄⠀⠀⠀⠀⠀⠈⠓⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡼⠁⢀⠀⡀
⠀⠀⣾⠿⠶⠾⠀⠀⠀⠴⢤⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⣾⠛⢛⠛⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⡙⠁⠉⠿⠽⣟⣷⡀
⠀⠀⠳⠤⠤⢤⣤⣀⣀⣀⣠⠷⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠏⣠⣎⠁⠀⢀⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣘⠁⠀⠀⠀⢀⣁⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡅⠀⢀⡽⣄⠀⠀⠀⠀⢀⡴⢿⠀⠃⠘⢻⣿⡿⠆⠴⠋⠀⠀⠉⡿⢦⠀⠀⠀⠀⠀⠀⣠⠋⠀⠉⠳⠚⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡁⠀⠈⠳⣄⠀⣠⠞⠁⢹⠀⡀⢠⣾⣿⡇⠔⠀⠀⠀⠀⣸⠃⠈⢳⡄⠀⠀⠀⠴⠉⠑⢢⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢹⠋⠀⠀⢰⢀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣰⢦⡀⠀⠀⠙⢦⡞⠁⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣾⠀⣀⡴⢻⡆⠄⢸⣿⠃⠀⠀⠀⠀⢰⠏⠀⠙⢦⡀⠀⢸⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠚⠉⠀⠘⡇⠀⢠⠁⡠⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠦⣌⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⠀⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠⡅⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣾⠻⣄⣀⣀⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠲⡍⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⠄⠘⣷⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢤⣀⠀⣷⣶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠚⠛⠁⠀⣀⡈⣻⠋⠠⠀⢹⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠀⠺⢿⣦⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠤⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇
⢸⠁⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⡆⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣄
⠘⣆⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⢸
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⢰⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⢹⡄⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣠⠇⠀⢄⣀⣀⣀⣀⣀⣀⠤⠄⠒⠉⢰⠈⠀⠀⠈⣇⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡈⠀⠀⣀⣀⣀⣀⡀⢀⡠⠃⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣼⠃⠀⠐⣤⡊⢠⠞⠉⠀⠀⠁⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⠛⠒⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣄⠀⠀⠀⠀⠈⠢⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠒⡉⠂⠀⠀⠁⠀⠀⠁⠀⠀⠒⡒⠒⠒⠒⠂⠀⠀⠈⠐⠂⠈⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠐⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠶⣄⠐⠠⠤⠠⠴⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⠀⠀⠀⠀⠉⠙⠓⢶⠤⢤⣀⣀⡀⣠⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓⠦⠤⢄⣀⣀⣤⠞⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠠⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⠙⢲⠶⠶⠤⠤⠲⠶⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡤⠖⣋⣩⠤⠤⠄⠀⠐⢋⣁⣠⠤⠤⠤⠤⢤⣀⡀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⢋⡴⠚⠁⢀⣀⣤⣤⠀⠞⠉⢁⣀⣠⣤⣤⣤⣀⡀⠉⠓⠦⣄⠈⠑⣄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣱⠏⣠⠏⠀⣠⠞⠉⠀⢀⣀⣤⠤⠞⠋⠁⠀⠀⠀⠀⠈⠉⠛⢦⣄⠈⠳⡄⠈⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏⣰⠃⣠⠞⠁⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠙⣆⠀⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣡⣏⣼⡧⠚⣁⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⣆⠀⢷⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⡿⢿⢷⠞⠏⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣆⠈⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡇⣾⢸⣆⢡⡇⠀⠀⣠⣶⣶⣶⠶⣤⣄⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠿⢿⣦⡀⠀⠘⣷⡜⢧⡈⢧⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⢸⣿⣾⡇⠀⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣏⠲⣅⡘⣷⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿⢸⢹⣿⡇⠀⠀⢀⢤⠶⡒⢲⣄⠀⠀⠸⡄⠀⠀⠀⡴⡟⠩⠉⣳⢦⠀⠀⠀⠀⣿⣿⡄⣄⢹⡟⠛⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⣿⢸⢸⣿⡇⠀⠀⠉⠘⠦⠁⠘⠉⠃⠀⠀⣷⠀⠀⠀⠀⠉⠀⠀⠁⠈⠀⠀⠀⢸⡇⡏⠧⢸⠀⣇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⠀⠘⡇⢻⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⢳⠈⣧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⣿⣇⠀⢀⣀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣸⣿⠀⢷⠀⠈⣇⠸⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡟⣼⢸⣿⡌⢳⣾⡉⢷⠀⠀⠀⠀⠀⠘⠘⠂⠘⠉⠀⠀⠀⠀⠀⢀⡞⢩⣧⠞⢹⣿⡇⠘⣇⠀⠘⣆⠹⡄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣸⠃⣸⢠⡇⣿⢸⡇⢷⠀⢹⡇⠈⣿⢶⡄⠀⠀⣀⡤⠤⠦⢤⡀⠀⠀⢠⡾⣿⠀⣸⠏⠀⡞⣼⢻⡀⠹⡄⠀⠘⣆⠙⣆⠹⠆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⡿⢸⢇⡏⢸⡿⡜⡇⠀⣷⠀⣿⠈⢷⣀⠞⢁⡠⠤⠤⢄⠙⣆⢀⡿⠀⡿⠀⣿⠀⣸⢡⠇⠈⢧⠀⢹⡄⠀⠈⢦⡈⢧⡀
⠀⠀⠀⠀⠀⠀⣼⠁⢰⠇⡾⢸⠃⢸⡇⢳⣻⠀⣿⠀⢸⡄⠈⢯⣴⣿⣶⣶⣾⣿⣷⣜⡿⠁⢰⡇⢀⡇⢀⣏⡞⠀⠀⠘⣆⠀⠹⡄⠀⠀⠳⠄⠙
⠀⠀⠀⠀⠀⣼⠃⢠⠏⣸⠃⡟⠀⢸⠀⢸⡟⣇⢸⡆⠘⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⠀⣸⠇⡼⣹⠁⠀⠀⠀⠹⡄⠀⠉
⠀⠀⠀⢀⡾⠁⣠⠏⣰⠏⣸⠃⠀⡾⠀⡼⢻⡜⣆⢻⡀⢻⡄⠀⠈⢷⡈⠉⠉⢀⡿⠁⠀⢠⡏⢠⡏⣸⢣⡏⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣠⠞⠁⣴⠋⣰⠃⠀⣸⠃⣰⠃⢠⣷⠈⠈⠃⠈⠳⠀⠀⠘⣷⠒⠒⣿⠁⠀⠀⠞⠀⠘⠀⠁⡟
⠀⠀⠀⠀⠀⠀⠚⢁⡼⠁⠀⣴⠃⣰⠃⢠⡟⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⢁⡼⠁⣰⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⣠⠞⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢲⠀⠀⠸⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣄⣈⠇⠀⠀⢠⠖⠒⠖⠒⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⣽⠖⠒⢢⢀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢼⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠦⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠙⠁⠀⢠⢄⡀⠀⠀⠀⠀⠀⣰⣧⣀⣀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⡀⠣⢨⠝⠀⠀⠀⠻⡏⠐⠒⣀⠜⠀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⣖⣉⣻⣈⣳⣎⣀⣀⣀⣀⡠⠤⠤⢤⡧⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠙⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⠊⠣⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠄⢰⠓⡄⢠⠊⢣⠀⠖⠆⠠⢲⠀⡏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠸⠀⠃⠘⠀⠈⢰⠀⠀⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡈⠀⠀⢰⠀⠀⠀⡄⠀⡆⡅⡘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠆⠆⠀⢀⠀⠀⡇⠀⠃⠇⠇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⣠⢠⢀⠈⢄⡠⠰⣠⠀⣤⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⡼⠉⠉⢷⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠶⠦⠴⠶⠲⣶⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠉⠻⠟⣿⡋⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⡴⠦⠀⠈⠛⠁⠛⠿⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⡆⠀⠀⢠⠞⠀⠀⠚⠂⠀⠀⠀⠒⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⠀⣤⣶⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣄⡀⣠⡀⠀⠀⠀⣿⠟⠀⠀⣾⣿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣆⣾⣶⣄⠘⠃⠼⠧⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⣠⣎⣁⣀⠀⠈⡃⠀⠠⠤⠛⠀⣀
⠀⠀⠀⢸⡆⠀⣴⠀⠀⠀⢻⣏⠉⠉⠐⣾⠀⢀⣀⠠⠶⣄⠉⣀⠀⠀⠀⠀⠀⢀⣀⣀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡆⠀⠀⠈⠓⠀⠈⠛⠛⠶⠿⣿⡷⣤⣴⣿⡖⠃⣀⠀⠀⠋⠀⡈⠘⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡛⠉⠋⠀⢋⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⣯⣄⠀⠀⡹⠇⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣹⣶⡄⢀⣀⠀⠀⢀⣉⡹⠷⢦⣤⣄⡀⠀⠀⠀⠀⣀⣤⡄⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡏⠀⡆⢸⣆⠀⢠⡌⠃⠀⠀⠀⠈⠈⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠿⣆⠘⢿⡀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠲⠟⠁⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣿⡟⠛⠘⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⢻⣶⡞⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣽
⠀⠀⠀⢠⡄⠀⢾⣀⣉⣁⣤⣤⣿⠀⢸⡇⢠⡆⠀⠀⠀⠀⠀⠀⠠⣆⠹⠀⢸
⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⡘⢻⠟⣦⣼⡇⠘⡇⠀⠀⠀⠀⠀⠠⠆⢸⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡴⠃⠀⠙⢹⣿⠆⢣⡀⠀⠀⠀⢀⠇⢀⡌⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡏⠀⢸⡇⠀⠀⢀⠟⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣀⠁⠀⠸⡇⠀⠀⠿⠀⠀⡈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢸⠀⠀⢰⡇⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠶⠶⠶⠖⢀⣼⡇⠀⠀⠀⠀⠀⠈⣧⣤⣤⠄⠒⠒⠒⠂⣠⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣶⣶⣶⣶⣶⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣶⣶⣶⣶⣶⣿⣤
⣿⣿⣿⣿⣿⠿⠛⢿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢺⣤⣤⠐⣦⣰⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿
⠀⠺⣟⠻⠀⠛⠙⠛⠛⢿⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠉⢅⡀⠉⣄⠈⠁⣷⣄
⠀⠸⣯⣶⣶⣶⣦⣶⣶⣿⣶⣶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣴⠬⠿⣙⣎⠯⢅⣿⣿⠇
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⡞⢡⣽⡟⣻⣿⠟⠉
⠀⢀⠈⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠸⠉⠁⣹⣿
⠀⠻⢂⣀⣠⣠⣴⣶⡿⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⠋⢹⣿⡿⠛
⠀⢰⣿⣿⡛⠛⠋⢠⣢⠄⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠏⠀⢘⣟⡁
⠀⠈⢿⣿⣧⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡠⡏⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣀⣀⠀⠀⣀⣀⣠⠤⠴⠼⠮⣍⠒⠃⠋⠁⠚⠐⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠒⠼⠦⠬⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣛⠂
⠀⠀⠀⠀⠀⠀⠶⣶⠦⠴⠞⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣀⣤⣤⣤⣤⣀⠈⠙⣆
⠀⠀⠀⠀⣠⠤⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡘⢾⡜⣡⠖⠋⠁⠀⠀⠀⠀⠈⠳⡄⠀⠑⢄
⠀⠀⠀⠙⠓⠒⣦⠖⠁⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠙⢾⡟⠁⢀⣄⣀⡀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢳⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣙⠳⣤⠀⠀⠀⠀⡇⠀⠀⢻⠁
⠀⠀⠀⢠⣣⢴⠆⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠉⠳⣎⠁⠀⠀⠀⣿⠀⠀⢸⡄
⠀⠀⠀⠋⠁⢸⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⣄⣀⡀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞⡋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢍⡁⢀⡇
⠀⠀⠀⠀⠀⢸⠈⠩⠭⣷⠀⠒⢶⠶⠶⠾⠟⠛⠋⠉⠀⠘⠿⠂⠀⠘⠗⠀⠀⠀⠀⠀⢠⡿⠀⢠⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢹⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣾⠀⠀⠀⡾
⠀⠀⠀⠀⢀⣀⣸⡄⠀⠈⡆⠀⠀⠈⠛⢦⣤⣀⣀⣤⣤⡶⠂⠀⠀⣀⠀⠀⠀⠴⢾⡯⠀⠀⠀⣇⠀⣀⠴⠗⢒⢶⡄
⠀⠀⠀⣴⠋⠉⢀⠈⠓⠊⠁⠀⠀⠀⠀⠀⠀⠠⠴⠿⠋⠀⠀⠀⠘⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣴⢾⡉⠀⢹⡄
⠀⠀⠀⣇⠀⠞⢩⡿⣆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢻⠇⠀⢸⡇
⠀⠀⠀⠹⣄⠀⠘⠀⠹⠀⣧⠀⠀⠀⠀⢀⣤⣖⡋⠉⠉⠉⠉⠀⠀⠈⠉⠉⢙⣶⣄⠀⠀⠀⠀⢀⣷⣄⠀⠉⠀⣀⡾⠁
⠀⠀⠀⠀⠈⠳⠦⣤⡤⠞⠹⡆⠀⠀⠀⣾⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣶⣶⣿⣿⣿⣇⠀⠀⠀⣼⠃⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠙⢟⣿⠿⠿⠭⢌⣙⣉⣩⠿⠿⠟⠻⣿⣿⠏⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠉⠒⠦⠤⠤⣄⣀⣠⣤⣤⠤⠖⠋⠁⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁
⣿⣿⣿⣿⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣾⣫⡤⠶⠖⢒⣲⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠐⠋⠉⠀⠀⠒⠛⠛⠙⠛⣛⣛⣒⣶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣓⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⠤⠴⠓⠒⠦⠤⢤⣠⠔⠒⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡷⡒⠲⠖⠒⠒⠓⢒⣿⡛⠋⠉⠙⢛⣻⠇
⠀⠀⠀⢀⡴⠊⠉⠉⠑⢦⡇⠀⢀⡀⣉⣀⠈⠉⠑⠒⠂⠉⠁⠀⠈⠉⠉⠙⣯⠖⠋⠉⣳⠄
⠀⠀⠀⡞⠀⠚⠛⠛⠂⠀⣀⠀⠈⢳⡀⣹⠲⢤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠒⠚⢥⡀
⠀⠀⠀⢧⡀⠤⠔⠂⠁⠀⠈⠑⢄⠀⠙⢇⠀⡸⠀⢠⠏⠀⢩⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠙⠧⣤⣼⣠⡤⠞⢳⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡇⠀⠀⢀⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⡚⠢⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⢾⣁⣠⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡀⠙⢦⡼⠂⠀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠉⠗⠀⠐⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⣹⠶⣾⡀⠀⠀⢑⠄⢑⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⡄⠀⠀⢧⡀⠀⠀⣸⣉⣉⣓⠒⠒⠒⢒⣒⣒⣒⣈⣉⡀⠁⢀⡿⠑⢤⠀⢺⡀⣏⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⡉⠀⠀⢸⡉⠳⢤⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠲⠛⡇⠀⠀⡟⠀⡝⠈⡆⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣀⣀⣨⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡠⠀⠀⡇⠀⠀⡇⣼⠀⠐⠁⠀⠀⠐⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⢿⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢣⠀⠀⡇⠀⣸⠀⢹⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣸⠘⣦⡤⢀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢈⡠⣴⠇⠀⡇⠀⠁⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⠭⢿⣤⡭⠉⢹⠶⡦⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⣸⢰⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠇⡞⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢱⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣀⣀⣮⣴⠅⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣀⡆⠀⠀⠀⠀⠀⠀⠧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠈⠉⢩⠏⠉⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢀⢀⣳⠀⠀⠀⠀⠀⠀⢀⣈⣃
⠀⠀⠀⣀⡤⠤⠤⣯⣤⣤⣀⣀⣀⣀⣼⣀⣀⣤⣷⡴⣶⠶⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣼⠼⠋⠀⠀⠀⠀⠀⠀⠛⠛⠟
⠀⠀⣾⣀⣀⣀⣀⣸⢶⣇⡩⠴⠞⠋⠙⠻⠷⣶⡮⠴⠧⡤⣤⣤⡾
⠀⠀⠈⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠾⠆⠀⠀⠀⠀⠀⠀⠰⠖⠚⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡃⠀⠀⠐⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⣦⠚⠉⢢⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⠉⠓⣏⡉⠑⠲⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⣀⢏⠁⠀⡰⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠼⢄⣉⡀⠀⠙⠋⠀⣀⡄⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠒⠂⠉⠁⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢏⠢⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠘⡕⠮⣑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢵⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣱⡀⣏⠉⠒⠪⠭⣐⣒⣒⣒⣒⣒⣒⣒⣊⠩⣥⣖⠁⠵⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡑⠈⠉⡽⠀⠀⠀⡖⠤⠬⠋⠀⠀⠀⢀⡜⠁⠘⠞⢀⡼⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠲⠤⠴⠃⠀⠀⠀⠀⠀⠀⠸⠤⢤⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢄⣀⠀⠀⠀⠒⡤⠤⠤⡀⠀⠀⠀⠀⢰⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠈⠉⢓⡶⠲⠃⠀⠀⠱⢤⣀⣀⣤⠬⠗⢚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⡤⠊⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣶⣤⣤⣀⣤⣤⣶⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡘⣿⣿⣿⣷⣿⣿⣿⣯⣭⣭⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣭⣿⣿⣿⣿⣷⣶⣼⣿⢿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣷⣮⠻⢿⣻⣭⣽⡃⣿⣿⣿⣿⣿⣿⡟⠻⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡉⣿⣿⣿⡿⢿⣿⣷⣿⣿⢿⣛⣷⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣧⣸⣿⣯⣭⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡿⠟⢿⣵⣿⣹⣿⢟⣻⣿⣿⣿⣿⣿⠿⢿⣿⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣟⣓⣒⣒⣶⣾⣿⣿⣿⣿⣿⣿⣯⣭⣶⣶⣯⣿⣻⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣽⣿⣿⣶⣶⣶⣶⣟⢿⣿⣟⣿⣟⣛⢿⣿⣿⣿⣿⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢟⣽⣾⣿⣶⣮⡻⢿⡏⣽⣿⣿⣿⣿⣷⣌⢵⣿⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣷⢻⣷⢿⣿⣿⣿⣿⣿⣿⣧⡻⢿⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⡾⢟⡸⣿⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢿⣿⣿⣿⣿⣿⣿⣷⣻⣿⢻⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⣼⢠⡟⣽⣿⣡⣾⣵⣶⣾⣶⣶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⡻⢿⣿⣾⠳⡇⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣫⠵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠋⣩⣽⣶⣶⣶⣾⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣮⣍⡳⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣭⡓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣌⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⡶⠟⠛⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠉⠀⣀⣀⡀⠙⢿⡂⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢸⣷⢸⡿⣮⡛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣩⢿⡿⠿⣷⣄⢹⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠸⣿⡏⢿⣆⡙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠸⠧⠂⠟⠙⡿⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⡀⠀⠀⠙⠀⠀⠀⠀⠀⣿⢿⡌⠻⣟⡂
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠫⣴⣏⠄⢸⠿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠻⣄⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⡟⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣌⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣙⣦
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢧⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡻⣍⣠⣤⣄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢉⣠⠄⠘⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠿⠷⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⣽⠟⠙⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣸⠇⢉⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢶⢉⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡯⣷⡀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣄⣀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⡇⡏⠛⠦⠤⠤⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹⣷⣟⡒⠒⠶⢶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⢿⣿⣿⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠛⠋⠉

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣶⣾⣿⣶⣶⣄⡀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠉⠀⡀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡈⠀⠊⠀⠀⠀⠐⠀⠀⠨⠀⠀⠀⠄⢼⣯⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠿⠿⣿⡿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄⠀⠀⢀⠀⠀⠠⠙⠿⠟⠿⠻⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡀⠙⣇⠀⢹⡄⡐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠋
⠀⠀⠰⠶⠀⣠⠀⠀⠀⠀⠀⠀⠀⠂⣠⣏⣨⢀⡼⢀⣠⡇⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠐⠉
⠀⠀⠀⠀⢀⣤⣤⣀⣤⡏⠀⠀⢀⣴⡄⠀⠘⠃⠀⣀⠀⠘⠁⠀⠀⠀⠀⠘⠻⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⡁⠀⠀⢼⣿⠀⠀⠀⠀⠘⠛⠀⠀⢀⣤⢄⣠⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⠠⣤⣾⣷⣿⣿⣿⣷⣶⣤⠀⠀⠀⢻⡿⢿⡿⡶⠀⠀⠀⠀⢀⠤⠔⢮⠒⠄⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⣀⣿⣟⣿⣿⣿⣿⣿⣿⣿⣟⣧⣄⠀⠈⠃⠘⠀⢧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠸⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⢀⣠⣄⠀⠀⠉⠁⠉⠉⠈⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢤⣀⡀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⠁⠁⠐⠒⢀⡀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⢀⠠⠀⠄⢈⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⢠⠆⠀⠀⣤⡀⠈⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠛⠛⠛
⠀⠀⠀⠀⠘⠀⠃⠀⠐⠀⠀⠀⢠⠛⠃⠀⣴⣿⣿⣿⣶⡀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣙⠚⠀⠸⣿⣿⣿⣿⣿⡇⠀⠘⠄⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠊⠉⠀⠀⠀⠙⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠉⠉⠳⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⣀⠜⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡬⠂⠀⠀⠀⠀⠀⠀⠀⡃⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠉⠁⠒⠒⡆⠀⠀⠀⠀⠀⡝⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡨⠀⠀⠀⠀⢀⣤⣄⡅⠀⠀⠀⠀⠀⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠹⠀⠀⠀⢀⠨⣀⠨⡄⠀⠀⠀⠀⠀⡀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠢⡀⠀⠀⠀⢀⣀⢀⡞⠉⡗⠀⠀⡁⠀⠀⠀⠀⠸⡶⠅⢪
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠠⠉⠐⠐⠶⣚⠁⠀⠀⠀⠈⠉⠉⠀⠈⠛⢻⣻⠃⠀⠀⠀⠀⠀⣫⣁⡯⣀
⠀⠀⠀⠀⠀⠀⠀⠅⡠⡍⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠣⠄⠀⠠⠔⣊⢙⣯⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠒⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠉⠘⣼⣻⠀⠀⣷⠘
⠀⠀⠀⠀⠀⠀⠀⡘⠶⣤⡞⢉⡱⠈⠉⣽⡄⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠜⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠘⢄⢠⠄⠀⠈⠓⠒⠦⠏⠢⡜⠗⠊⠙⡀⠀⡔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣀⣀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠚⠈⠉⠀⠀⢁⠀⢠⠐⠉⠀⠀⠀⠀⠈⢱⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠉⠉⣤⣹
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠖⠛⠀⠈⢄⡤⠶⡄⢀⣀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⠀⡟⠋
⠀⠀⠀⠎⠉⢒⣁⠀⠀⣠⣬⡓⠈⢁⠀⠈⢡⠐⠁⢺⠀⣥⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⡅
⠀⠀⠀⠣⠤⠏⠉⠀⠘⣇⡀⣹⣤⠼⠀⠀⠈⢄⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣘⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠒⠒⠒⠂⠐⠚⠛⠛⠓⠂⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠀⠀⠀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠉⠻⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠠⠀⠠⡀⠀⠀⠀⢸⡇
⡐⠛⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠁⠀⠀⠀⠈⠂⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠇⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⡿⠀⠀⣾⡷⠀⠀⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠛⠉⠉⢱
⠸⡄⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⡜
⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⡌⠑⠤⠄⠂⠉⠢⠀⠀⠀⠀⢀⡀⠀⠀⠸⡀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⡸⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⡏⠀⠀⠀⠃⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢈⠉⠁⠀⠈⠉⠉⠀⠀⠀⡰⠀⠀⠀⠀⡸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⢀⡼⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀⡞⠀⠀⠦⡜⢀⠄⠈⠁⠘⠉⡀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢠⠏⠀⠀⠀⠀⠈⠉⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢸⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⢸⠁⠌⠢⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠚⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠑⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠩⠒⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠐⠲⠀⠤⠶⢀⣤⣤⣤⡤⠠⢤⣤⣄⠐⠢⠌⠀⠂⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠉⠛⠲⢦⠄⣤⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠢⠤⢤⣀⣀⣠⠜⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠈⠙⢷⠒⠤⠤⠴⠦⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⡀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣥⣤⡤⣤⣤⢤⡄⣤⣤⣠⡤⢄⡄⢠⢠⢤⡤⢤

⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡛⡟⣟⠿⣛⢻⢻⢻⢿⡛⠟⠛⡟⡓⠿⣛⢿⣸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣧⣽⣿⣿⣿⣾⣾⣿⣷⣷⣽⣷⣷⣷⣽⣿⣮⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢟⣾⣿⣿⣿⣷⣶⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⡗⠚⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣼⠀⡛⢀⠴⠓⠦⡀⣀⣀⣀⠐⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣟⢿⡆⣀⠿⠷⠆⠀⣿⣤⠤⢼⠄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣷⣷⡁⠡⡀⠐⠈⢹⣆⣠⡼⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢏⢶⢠⠌⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣿⡶⠉⠒⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⠿⠇⢈⣐⣒⣀⡤⠾⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⢸⣿⣤⣤⣤⣤⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣭⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣱⣶⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⡘⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢀⣾⣿⣿⣆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠘⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⣀⡤⠤⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠋⠉⠀⠐⠚⠛⠓⣒⣒⠶⠶⠤
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠭⠭⠷⠶⠄
⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣀⡠⠄⠀⠀⠀⢰⡒⠒⠒⠒⠺
⠀⠀⠀⠠⠀⠀⣀⠀⠺⠀⢀⡤⢠⡄⠀⠉⠊⠉⠀⠀⠀⠉⠉⠣⠂⠉⠀⠂
⠀⠀⠀⠃⠀⠀⠀⠀⠐⢤⡀⠙⣤⠋⢑⠒⢲⠦⠤⡤⠤⠄⠀⠀⠀⠀⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠲⣜⠀⡜⢀⣰⢥⠀⠀⠀⠀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⠉⠀⠀⣈⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣸⠀⠀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢖⠓⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠳⠤⠎⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⡀⢱⠞⠀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠄⠉⠈⠉⠇⠀⠀⠀⠀⣀⡀⠀⠀⠐⠀⠋⢻⢆⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠘⠀⠀⣨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢥⣀⣀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢻⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠌⠂⠀⣃⠀⡆⠠⠆⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢤⢸⠸⡟⠰⣄⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡺⠋⠀⠃⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠈⠉⠁⠀⡇⠋⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣸⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⢠⣤⣤⣴⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⢼⠀⠀⠀⠀⠀⠐⢤⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⡜⠀⠀⣄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠠⢠⠇⠀⠀⠀⠀⢠⣤⣦
⠀⠀⣴⠊⠉⠉⢻⣿⢿⣓⣲⠿⠷⣶⣟⣛⣩⡋⠉⢉⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠉⠁⠁⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⠀⠀⠒⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠐

⠀⠀⣠⠴⠚⠋⣉⣉⡙⠶⠞⢋⣡⣤⣤⣤⣄⠙⠛⢉⣤⡿⢿⣍⣴⠟⠛⢷⣤⣽⣯⣤⣶⣌⣩⣤⣤⣤⣤⣤⠉⣧
⢠⡞⢁⣠⡾⠛⠉⠉⠛⢷⣼⠟⠁⢀⣄⠀⠉⢷⣤⣾⣇⠀⠀⠙⣿⡄⠀⠘⣯⠀⠀⠈⠀⣿⡁⠀⠀⣀⣤⣿⠃⠻⡄
⡟⢀⡾⠃⠀⠀⠀⣠⠄⣸⣿⠀⠀⠈⢉⣤⣀⣼⠃⠈⢻⡆⠀⠀⠌⠁⠀⠦⣿⡀⠟⠀⢰⣼⡇⠀⠀⠛⠋⠛⣷⠠⣇
⠀⣾⠁⠀⠀⠀⠈⣡⣾⡿⠿⣦⡀⠀⣾⠿⢋⣡⣴⠒⢾⠟⠲⠿⠖⠒⠶⠖⠛⢓⡆⠀⠀⠻⣿⡄⠀⠀⢼⣾⢿⣤⣌⠙⣦
⡀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢿⡄⢿⣤⠞⣷⣿⢤⣟⣲⣄⣤⣤⣶⣿⣷⣴⣾⣄⣠⡄⠀⠈⢻⡄⠀⠀⠀⣀⡀⢹⡇⢹
⢧⡙⢿⣶⣶⠶⠗⠀⠀⠀⠀⠀⢸⣇⣨⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣯⣭⣿⡶⢳⡀⢻⡶⠶⠿⠿⠿⢛⣡⠏
⢸⠃⣿⠉⠀⠀⠀⠀⠀⠀⣀⡶⣼⣿⣿⣿⣿⢿⣿⡇⣿⣄⠸⢿⠟⣁⠹⣷⠇⡟⢁⡼⠫⢂⡴⣣⡾⣡⠞⠛⠛⠉⠉⠁
⠸⣆⠙⢷⣄⡀⠀⠀⢚⣛⣭⡾⠋⠙⣿⡿⢻⣎⣿⣿⠀⠠⠄⠤⢄⣓⣺⣅⢀⣿⡏⠈⡖⣿⣏⣉⡴⠏
⠀⠙⠳⠦⢭⣝⣛⣛⡛⠉⢁⣠⣤⣤⣿⣁⣸⣿⣿⣿⠀⣀⣤⠀⢈⣉⢈⡙⢿⡿⢿⣦⡹⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠉⠉⢷⣰⡏⠉⠀⠀⠉⣿⣇⢀⡿⣽⠠⠉⣡⠀⣬⡍⠒⢰⣂⣽⣶⢯⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⡀⠀⠘⣿⠱⣏⣿⣿⣿⣶⣶⣿⣶⣿⣿⣿⣿⡏⠉⠛⠋⢾⣇⠹⡆⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⣇⠀⠀⢨⣷⡌⠙⣻⢿⣿⣿⣿⡿⣿⣿⣿⡟⠁⠀⠶⠀⠀⢹⡇⣷⢰⡞⠳⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⣿⠀⠀⢈⡻⠷⠖⢻⡆⣿⣽⡏⢹⡄⢿⣾⣧⣀⣀⣀⣾⣧⡿⣱⠇⠈⣷⠞⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⢿⣀⠀⠀⢠⣶⣦⡾⣿⣿⣽⣿⢿⢿⡏⢉⠱⣿⣿⣿⣿⡿⣟⣁⣄⣀⣡⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⢾⢿⣛⣻⣿⢛⣿⣧⣬⣻⣿⠉⠉⣹⣧⠈⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣷⣼⣿⣶⣾⣿⣿⣿⣮⣽⣿⣛⣛⢛⣛⠁⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠉⠛⠋⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡀⢠⡀⠋⠘⠀⡄⠀⠉⠀⠈⠐⠓
⠀⣀⠀⠴⠇⠀⢀⣤⣴⣶⣿⣦⡀⠀⠀⠰⠀⢀⣮⣀⠀⠀⠀⠀⣸⣖⣀⣠⣴⣶⣿⣶⣿⣶⣭⡀⠀⠀⠀⢰⠆⠀⠀⢀
⠸⡇⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣇⠀⢰⡆⢀⣾⣿⣿⣷⡀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠣⠀⢠⣿⣿⡿⣿⡿⢹⣿⡿⣿⡄⣸⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢻⡏⠉⠉⣿⣿⣿⡟⣿⣦⡀⠠⠀⣀
⠀⢀⠀⠹⣿⣿⠀⣿⡇⣨⣿⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢳⡉⣘⢁⣀⣴⣿⣯⣩⣀⣿⣿⣷⠀⠀⠈
⠀⠀⠀⢸⣿⠏⢸⣿⢳⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣌⡁⢈⣿⣿⣿⣿⣿⣿⠿⣿⣿⡆⠀⠀⠀⠘
⠀⠀⠀⢸⣿⡀⠈⠋⠘⠛⢁⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⡉⠉⠈⣠⣼⣿⣿⡀
⠀⠀⠀⣸⣿⣿⣷⢲⣶⣶⣿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣇⠀⠀⠀⢠
⠀⠀⣰⣿⣿⣿⡟⣸⣿⣿⠉⣿⣿⣿⠆⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠂
⠀⢠⣿⣿⣿⣿⠁⣿⣿⡟⣿⣿⣿⡿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣾⣿⣿⣿⠻⣿⡿⠓⠀⠚⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢸⣿⣿⡏⠀⢘⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠂⠀⢹⣿⣿⣿⡏⣿⣿⣿⣿⣷⣾⠀⠀⠀⠶⠧⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠠
⠀⠈⣿⣿⣧⠁⠘⣿⣿⣿⣿⣿⣧⠀⠀⠀⣀⡄⠀⠘⣿⣿⣿⠇⣿⡟⠋⠉⠙⠉⠉⠂⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣟⠀⠀⡀⠸
⠀⠀⣿⣿⣿⠁⠀⠘⣿⣿⣿⣿⠇⢈⢁⣀⡀⠀⠀⠀⠹⣿⣿⢰⣿⣷⠀⠀⠀⠀⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⢻⠁
⠀⡄⢻⣿⡏⡀⠀⠀⠈⠙⠛⠋⠀⠟⠛⠛⠛⠿⠆⠀⠀⠈⠻⢸⣿⡇⠀⠀⠀⠀⢀⠃⠀⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠐⠈⠙⠁
⠀⠀⣿⣿⠁⣿⣄⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠷
⢠⢸⣿⡿⢸⡿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣘⠀⠀⢠⠶⠠⢄⡶⣄⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋
⠀⣸⣿⠇⣾⣷⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠋⠠⠄⠀⠀⠀⠀⠀⢀⣍⠀⠀⡀⠀⢠⣿⣿⣿⣿⡆
⠀⢸⣿⢀⣿⣿⣧⣴⡔⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⢀⠀⣶⣦⠀⡀⠀⣀⠀⠀⣤⠈⠉⡤⠀⠁⠀⢤⢻⣿⣿⣿⠃⠀⢀⣠⡔
⠀⣸⣿⣿⣿⣿⣿⣿⢁⠛⠋⠁⠐⠦⠠⣄⠀⠀⠐⢲⡞⠻⡀⢸⣿⣶⣷⠈⠙⠛⠛⠻⠛⠿⠧⣸⢟⣻⣿⠃⠙⢻⡿⠀⠀⠉⠁
⠀⠘⢿⣿⠛⠻⣿⣿⣿⡀⠀⠀⠀⢀⣀⣵⡄⠀⠀⠈⢩⣶⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡟⢰⣷⡌⢿⣿⣿⣤⣼⡇⠀⠀⠀⠀⠈
⠀⠀⢈⣁⣀⡀⠙⠿⠿⠿⠀⠀⠀⢘⠛⠛⠋⠀⠀⠀⣏⠉⠙⠙⠛⠛⠛⠛⠆⢛⠛⠛⠛⢧⠈⢻⣿⣞⣻⣿⣿⣿⠀⠰⠗⣤⣀⡀⠆
⠀⠀⠈⠉⠈⠛⠁⠀⠀⠀⢀⣀⣴⠈⠠⣤⣶⣦⣄⣊⠁⢺⣶⣶⣶⣶⡶⠂⢈⣈⣶⣶⡶⠶⠁⢀⣤⣉⠛⠿⣿⠃⠀⠀⢀⢃⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠻⠿⠶⠶⠾⠿⠿⢿⣯⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠿⠋⠀⠀⠀⠀⠋⠈⠁⠈⢀⠀⠘
⠀⢀⠀⠁⠈⠐⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠚⣀⠠⠀⠀⠀⡁⠋
⢀⡌⠄⠀⠀⢠⠤⠄⠀⠀⠐⠢⡀⢄⠀⢤⢶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⢠⣤⠶⠷⢀⠀⠀⠠⠆⠔⠆⠂⠁⢀⠁
⣼⡯⠤⠭⢁⡀⢀⠀⠀⠀⠀⢀⡀⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠘⠷⣆
⠐⠂⠠⡔⠆⠀⠐⡄⢀⡤⠀⠠⠤⠴⠁⣀⠀⢤⡀⣀⣀⠀⠐⠀⠀⠀⠐⠂⠀⠤⠄⠀⠀⣄⣿⠆⣠⣄⠀⠀⢘⠁⠀⠀⠀⠀⠀⢸⡧
⠠⠀⠀⠀⠀⠀⡤⢂⡈⠀⠀⠀⠀⢀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠄⠻⠟⠃⠀⡀⠀⢠⣤⠀⠀⣠⣿⠥
⠀⢫⠀⠻⠁⠺⣧⡘⢂⡴⠠⡄⣴⠏⠀⠈⠱⠀⠀⠀⠀⢠⣤⡀⠀⠀⣤⣤⣌⡀⠀⣨⣴⡀⣠⣴⡦⣴⣶⣿⣗⣤⣄⣥⣴⣶⡽⣧⡄
⣶⡆⠦⠀⠀⠀⠈⣹⡆⠠⣤⣴⣾⣿⣿⣷⣿⣶⣶⣶⣿⣿⣿⣷⣤⣄⣈⣭⣿⣿⣿⣶⣭⣙⡻⠿⠷⠿⠻⣿⣿⣿⣿⡷⢿⣿⣷⣿⣿
⣿⣃⡠⠀⠀⠠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣷⣄⣈⠛⠉⠉⣻
⣿⣿⣇⢉⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣠⡄⠀⢠⣤⣀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣋⠀⠀⠀⠈⡁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⠀⠀⠙⣤⠊⠀⠈⠱⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⢰⡆⡃⢼⡆⠀⠀⡿⣧⠀⡄⠀⠒⠒⠒⠒⠦⠤⠖⠒⠒⠒⠶⠶⠒⠒⠲⠴⠖⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⣄⠀⠈⣱⠻⣌⠁⠀⢠⠇⠸⡆⢳⠀⢀⠒⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣤⠈⠑⠚⠁⠀⠈⠉⠉⢁⡰⠄⢻⣸⠀⠘⠤⠊⢀⠶⠖⠓⠶⣆⠀⢀⡴⠖⠓⠶⣇⠀⠸⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⠀⠉⠓⠒⠒⠶⠖⠒⠛⠉⠀⠀⠀⢻⡀⢠⠄⢰⠋⠀⠀⣠⣤⡈⢷⠋⣠⣤⠀⠀⠈⢧⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠁⠸⡄⠀⠀⠻⠿⠇⡾⣄⠿⠿⠃⠀⠀⡞⠀⠀⣿
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⡙⠦⢤⡤⠴⢾⠀⢸⠦⢤⡤⠴⣚⣠⡀⠀⣾
⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢨⡉⠉⠛⠲⢤⣤⣤⣀⣀⣤⡤⠖⠋⠁⠈⡇⢠⡇
⠀⢠⠞⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠳⣄⠀⠀⠰⣇⣸⠇⣇⣠⠇⠀⠀⣠⠞⠀⠈⡇
⢠⡏⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠃⢈⠓⠦⣴⣛⣑⣒⣒⣒⣦⡴⠞⠁⣎⣹⢠⡇
⣾⠀⠀⠀⠀⠀⣠⠎⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠲⠴⠁⠀⠀⠠⠭⠭⠭⠅⠀⠀⠀⠀⠟⠁⠘⢧
⣿⠀⠀⠀⣀⠔⠁⠀⡿⢒⡤⠤⣀⣀⣀⠀⠀⠀⠀⠘⠮⠀⠀⠀⢀⣀⣠⠤⣂⣹⡖⠒⠛⠒⠒⠲⣒⣲⣒⡶⣎⠽⠛⠛⠛⠉⢻⣥⡼
⠈⠳⠶⠚⠁⠀⠀⠀⡇⠀⠈⠉⠓⠒⢻⠯⠿⠿⠿⠿⠿⠿⠿⣿⠿⠖⠚⠋⠁⢸⡏⢹⣿⣟⠛⢻⣭⣽⠋⢷⣽⡏⠋⣿⣿⣭⢹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢲⡄⠀⠀⠀⠘⢧⡤⢤⠀⠀⢀⣄⣀⡸⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣈⣓⣋⣀⣀⣀⣀⣀⣀⣸⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⣧⣄⡀⠀⠀⠀⠀⣞⣀⠀⢘⡆⠈⠀⠀⠀⢀⡖⠛⠦⡿⠀⠫⢤⢤⠼⠃⠀⠀⠀⠙⢤⢤⠴⠋⠀⢧⣤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⡷⠀⠀⠀⠀⠠⡭⣭⡥⢴⡀⠀⠀⠀⠸⣇⣀⣠⠇⠀⠀⢰⣼⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠻⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠿⠥⠤⠤⠤⢶⠞⠁⠀⠀⠈⢛⠓⠒⠒⠒⠛⢹⠃⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢠⠏⠀⠀⡤⠂⠘⠓⢳⠀⠀⠀⠀⠠⠉⠚⠴⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠓⠶⠖⠃⠀⠀⠀⠣⣀⡠⠶⠚⠀⠀⠀⠀⠈⠑⠚⢤⣤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠙⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⢠⠠⢠⠀⣀⡄⡀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣄⠀⢀⣄⠁⠀⠰⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⢁⠀⡌⠛⢀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⢀⢘⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠤⠀⠄⠚⠆⠀⠐⠃⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⣊⣁⣐⢂⣲⡖⠒⠒⠒⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⣀⢼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠁⠀⠈⢛⠃⠀⠀⠊⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠃⠀⠀⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⢀⡠⠐⠈⠉⠁⠀⠉⠉⠛⠢⣀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⢻⣁⣠⣔⣉⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠶⠛⠉⠉⠀⠀⠀⠈⠉⠉⠓⠦⣀⠀⠀⠀⢸⠄⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣼⡄⢠⡀⣾⣦⣿⡘⢧⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣃⡀⠴⢋⣧⠞⢳⡟⠀⠈⢳⠘⡆⠀⠀⠀⠀⢀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⠀⠀⠀⠠⣦⣤⣀⡈⠁⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⠘⠛⣛⣛⣉⡉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠃⠀⠀⠀⣷⢸⡆⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡴⠋⠀⠀⠀⠀⡴⠋⠁⣀⣀⡉⠳⡄⠀⠀⠀⠀⢀⠔⠊⡉⠉⠓⢦⡀⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⠀⠀⠀⠀⢸⠁⠀⣼⣿⣿⣤⡄⢹⠀⠀⠀⠀⡏⣴⣿⣿⣀⡀⠀⢳⠀⠀⠀⠀⢹⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣄⣿⠀⠀⠀⠀⠀⠘⣀⠀⠘⠿⣿⠿⠁⣘⠀⠀⠀⠀⣇⠻⣿⣿⡿⠃⠀⡸⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠞⠁⠀⠀⠀⠀⠘⠦⠤⠶⠶⠒⠚⠁⠀⠀⠀⢀⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢠⠴⢄⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⢦⣀⠙⠲⣄⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⠒⠲⠦⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠖⠋⣙⡆
⢀⣤⣀⣒⣒⠂⠒⠉⢁⣀⠀⠘⣆⠈⠱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⡔⠉⠁⠀⠴⣮⣉⣀
⣾⣩⡭⠭⠤⠀⠀⠀⠈⠉⢹⡄⡟⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠁⠀⠀⠀⢸⢄⡶⠛⠛⠀⠀⠀⠩⠭⣽⣶⡄
⢻⣀⣀⣀⣀⣀⣀⣀⣠⡤⠾⠁⢿⠀⠀⠀⠀⠀⣠⠏⡿⠙⠛⢶⣖⠒⠒⣒⡿⠉⠹⡎⣧⠀⠀⠀⠀⢀⡟⠘⣧⣀⠀⠀⠀⠀⠘⠓⠶⢬⣿
⠈⠉⠉⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⣆⠀⠀⣀⡴⠃⢀⡇⠀⠀⠀⠈⣉⣉⡉⠀⠀⠀⢷⠈⠳⣄⣀⣠⠞⠀⠀⢀⡟⠉⠛⠓⠒⠶⢤⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠙⠋⠁⠀⠀⣼⠃⠀⠀⠀⠈⠛⢻⣿⡆⠀⠀⢸⡆⠀⠀⠉⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣤⠴⠋⢹⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⣿⠲⠤⣤⣤⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠒⠒⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠰⣶⠂⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠛⠐⠒⠂⠂⠀⣻⠒⠲⠖⠦⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⡀⠀⣿⣀⣀⣀⣀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠼⣯⡉⠉⠉⠉⠙⣧⡽⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢻⠀⠀⠀⣾⠁⠀⠀⠙⡆

⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠠⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠸⣿⣿⣶⣭⡻⣿⢿⠿⢿⣿⡋⡀⠀⠰⣇⡀⠀⣿⢀⠀⣹⡟⠏⠁⣀⣴⡤⠲⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢶⣾⠿⠟⢿⣶⢦⡹⢷⠀⠁⠖⠼⣉⣴⣶⣋⣯⣀⣿⣶⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠋⠀⠰⢶⣾⣿⣿⣶⣬⣤⣄⣠⣽⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁⠀⢀⣀⡓⢲⣿⣿⣿⣯⣹⣿⣿⣿⣻⣿⣿⣿⣿⡏⢿⣿⡟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⣠⣿⣿⣿⣿⡏⠿⠿⢿⡿⣿⡿⢿⡿⠿⢻⣧⢸⣷⠸⣿⠁⣿⢏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢻⣿⣟⠟⠀⠀⣠⣿⡄⠿⠂⠛⠘⠏⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣈⡀⢀⣰⣿⣿⣧⠀⢀⣀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣇⠛⠀⠈⠍⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢛⠋⠉⠀⠀⠈⠀⠈⠁⠀⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠬⠛⠿⢿⣿⡄
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠹⠋⠓⠚⠻⢷⣤⣀⠀⠀⣀⣠⣴⡶⢶⠤⢤⡀⠐⠲⣅⠀⠀⠀⢠⣤⣤⣤⣤
⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⣰⣿⡇⠀⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⣿⣇⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⣶⣤⡤⠀⣠⣄⣀⣀⣀⣀⢀⣠⣴⣶⣾⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⣍⡉⢋⣴⣿⣿⣿⣿⡿⣯⣿⣭⣿⣿⣿⣮⠿⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡟⣰⢿⣿⣷⣿⢯⣭⣭⠝⠟⢿⣿⠟⡿⢿⣿⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠙⡞⠻⣟⣿⠠⡉⠛⣿⣄⠈⢹⡶⠃⠘⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣀⠀⠈⡙⠈⣃⣛⣁⠟⠀⡀⠃⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠄⠀⠉⠉⠋⠀⢠⠄⠀⣶⡆⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣧⡄⢀⣀⣀⣀⠈⢠⣴⡿⠁⠀⠀⣄⣀⣀⡤⠤⣄⢶⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⢛⣿⠁⢀⣰⠟⠀⠀⠀⠀⠈⠛⠿⣦⢸⡿⣙⡾⠶⠀⠁⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢘⡇⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⣀⠄⠒⠒⠒⠒⠚⡗⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⣄⣀⣀⡀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡀⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣀⠤⠒⠉⠉⠹⡀⠀⠈⠑⣄⣎⡠⠤⢄⣀⣀⣠⡤⣖⠉⠉⡇⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠊⠁⠀⠀⠀⠀⠀⠙⠢⢤⡶⡿⠛⠈⠉⠀⡠⠊⠁⠉⠉⠳⢤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⢀⠜⡇⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠱⡀
⠀⠀⠀⠀⠀⢀⡎⠀⢰⠀⠀⠀⠀⠀⢀⣾⠋⠀⢠⠎⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⠼⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⠀⠀⠀⡘⠀⠀⠀⢣⠀⠀⠀⢀⡿⠁⠀⣠⠧⠴⠒⠒⠒⠒⠢⣄⠉⠒⠊⢁⣸⠴⠆⠦⣖⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠣⡀⠀⢸⢃⣴⠟⠁⠀⠀⠀⢀⠔⠉⠁⢸⠉⢢⡔⢩⠃⠀⠀⠀⡼⢢⢀⠤⠒⠉⠉⠑⢾⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠢⣾⠉⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡸⢀⠏⣳⠃⣀⠤⠒⠢⡆⡼⠁⠀⠀⠀⠀⠀⡇⠙⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠏⠑⠢⢄⣀⠀⠀⢸⠀⠀⠀⢠⢃⣸⢖⡷⠊⠀⠀⠀⣸⣿⡿⠀⠤⢄⠀⠀⢠⠃⠀⢰
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡌⡄⠀⠀⠀⠀⠉⠁⠚⡖⠒⢲⠋⣡⠼⣡⠃⠀⠀⠀⡰⠋⡿⢆⠀⠀⠀⡇⢀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠃⢡⠀⠀⠀⠀⠀⠀⠀⠈⡲⣁⠞⠁⢠⢇⠀⠀⡠⡾⢁⢼⠃⠈⠳⢄⠀⠇⡜⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠀⠀⢢⠀⠀⠀⠀⠀⣠⠎⡰⠋⠁⡲⠃⠜⢖⠈⠀⡧⠋⠸⠀⠀⠀⠀⢹⠟⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⠀⠀⠀⠱⣄⠀⡠⠚⠁⡐⠁⢀⠔⠁⠀⠀⢀⡱⢞⡇⠀⡇⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⡆⢀⣀⠤⠚⠛⢄⡀⣰⡡⠔⠁⠀⢀⡠⠖⠉⠀⠀⠟⠲⠂⠒⠒⠉⡸⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣭⠤⠄⠒⠒⠃⠉⠉⡛⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠢⡀⢀⠆⠀⠀⠀⢀⡠⠊
⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢈⠟⠒⠒⠒⠊⠁
⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠎
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣈⣢⣀⡠⠤⠒⠁⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⣀⣀⣀⡠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢀⠀⣄⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠠⡾⠶⣶⡷⢬⣻⡿⢛⡂⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠤⣤⣈⠉⣽⣿⣣⣾⣿⠿⢿⡇⠀⠀⠁⠞⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⠉⠙⠛⠛⠛⣷⣾⣿⣿⣤⡀⠀⠸⡏⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠙⠷⠼⣿⣿⣾⢿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢀⣶⣻⣿⣿⣿⡿⠿⣿⡟⢧⣦⣤⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⣀
⡀⢀⣠⣼⡿⢟⣿⠿⡛⢻⣿⣒⣒⠓⠂⠸⣿⣆⠀⠁⠨⣭⣹⣿⣿⣧⣤⣼⣿⣿⣿⣿⠿⣧⡀⠀⠀⠀⢀⡀
⣾⣏⣨⣠⣾⣿⣿⣿⡷⠴⠶⢤⣬⣅⢀⠀⠉⣿⣿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⡿⢄⣖⡀⠾⣿⡟⣂⡀
⠈⠹⣯⣽⡟⣿⣿⣿⣧⡀⢀⣼⡿⣿⡿⠀⠐⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣛⣛⣿⣟⣯⣸⣿⣷⣶⣦⠀⠤⣀
⠀⠀⢹⣿⠀⠀⠹⠛⠛⠏⠉⠉⠈⠁⠀⠀⢰⡟⣾⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣶⣿⣭⣽⣻⣿⠻⠷⠾⠇⠉
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠁⠀⢈⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣷⡝⣷⡍⠩⣜⣻⣿⠟⣀⡀⠤
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⢸⣿⣿⣿⣿⣯⣶⣿⣿⣿⣯⣛⣻⣽⣶⣿⡀⡍⣤⠀⣽⣌⣛⣛⢛⣂⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠈⣿⣿⣏⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡻⣷⢠⣼⣶⣾⣿⣿⣿⣷⠿⣻⠋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠈⢹⣿⣿⣿⣿⣿⣿⣿⣽⣯⠙⣻⣿⣿⣿⡟⠛⠛⣇⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠳⣔⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣉⠛⠒⠬⢷⣄⠈⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠒⠚⠋⠉⠳⢤⡀⠀⠀⠀⠈⢧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣚⠭⠕⠒⠒⢤⠀⠀⠀⠀⠙⢟⠛⠦⠀⠀⠀⠀⢳⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⡠⠊⠀⠀⠀⠀⠀⢸⠁⠀⢀⣀⣀⣀⠑⠦⡀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⡴⢋⢀⡀⠰⡇⠀⠀⠀⢠⣦⡴⢃⣀⠤⠤⠤⠀⠀⠁⠀⠈⠓⣄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⡐
⠀⠀⠀⠀⠀⠀⢰⠃⢫⠀⠘⠆⠙⠒⠤⠤⠚⠉⠐⠫⠤⢤⣀⠀⠀⠀⠀⠀⣴⡋⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⢀⡀⠀⠠⠃⠀⡠
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⢰⠀⠀⠀⢠⠐⡁⣑⠀⠀⠀⠈⢑
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⡴⢳⢣⠀⠀⠀⠀⢱⠋⡩⣱⣀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⡴⠚⠛⠒⠲⣖⠒⠒⠲⣄⣀⢤⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢶⡋⠀⢸⠋⠀⠀⠀⠀⠁⠸⠴⢺⠠⣜⠁
⡀⠘⣅⢣⡀⠄⢀⣁⣁⠀⠀⠀⠀⡈⠑⠢⢄⡀⠀⠀⠃⠀⠀⠀⠀⣀⡀⠉⠋⠓⠆⠘⣆⢈⡹⠀⠀⠀⠀⠈⢢⢔⣯⡵⠧⡀
⢑⡦⠤⢀⣀⡚⠑⣤⠈⢆⣠⠤⢰⠃⠀⠀⠀⠈⠉⠑⠒⠒⠒⠈⠉⠀⠈⢷⠒⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠈⢮⣇⠴⠮⣷
⠈⠑⠊⠁⠀⠈⠉⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠋⢪⠲⡊⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠒⠒⠒⠒⢠⠠⠄⠀⠀⠀⠀⠀⠒⢲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⣸⣀⣣⣀⡀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢲⡔⣟⣤⣀⠀⠀⢁⠀⠀⡏⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⣒⡨⠭⠥⠤⠬⠉⣏⠀⠀⢳⠲⠖⠒⠋⠉⠉⠒⢂⡢

⠀⠀⢀⣤⣾⣿⡿⠛⠻⠟⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠋⠙⠋⠉⠉⠙⠛⠉⠉⠑⠒⠤⠴⢌
⠀⣰⣿⣿⣿⣿⠀⠸⣷⣆⠈⢷⣦⠹⣦⠀⡀⠀⠄⠀⢀⡴⠒⣦⠇⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢢
⢀⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⣄⣡⠞⠉⠋⠛⠀⣰⡟⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠠⠀⠀⢰⣿⠇⠀⠀⠀⣼⣧⠀⠀⠀⣾⡄⠀⠀⠀⠀⠠⢠⡄
⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢿⣿⣿⣿⣷⡀⠀⣀⣀⣼⣿⠀⠀⠀⠄⠻⠇⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠛⢁
⣿⣿⣿⣿⣿⣿⣿⣥⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⡘⣻
⣿⣿⣿⣿⣿⣿⠿⠛⣉⣩⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣷⠀⠀⢤⠄⠀⢸⡿⠀⠀⠂⠿⢳⣿
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠛⠀⠀⠘⣿⣶⣿⣿⣿⡿⢁⣀⢀⢀⣤⣸⣿
⣿⣿⣿⣟⣭⣶⣿⢶⣍⢫⠀⠉⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠝⣠⣀⠀⠉⠛⢛⣉⡀⠑⠃⠀⠈⣿⣿⣿
⣿⣿⣫⠝⡉⠉⠁⠒⠀⢋⢱⣆⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⣹⣿⣿⣿⣿⣿⣿⡃⠀⠀⠐⢀⣸⣿⣿
⣿⣿⣷⣾⣷⣶⣦⣾⣿⣿⣿⣿⣿⢻⣿⡿⢷⣦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠂⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⠀⠐⠀⢀⠐⢀⠀⠒⠈⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⢇⠒⠠⠓⠀⠀⠐⠁⠉⠀⠀⠀⢂⡀⠈⠀⠐⠦⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⢶
⠀⠀⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⡀⠀⡌⢀⠀⠀⠀⠀⠂⢀⠠⠴⠀⠀⠀⠀⠀⠐⢠⠄⠀⠖⠀⠒⠀⠀⡀⠀⠱⣂
⠀⠀⠀⠀⠀⠀⠅⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠁⠀⠀⠀⠄⠑⠁⠀⠀⠀⠀⠀⠀⢐⠀⠤⠀⡀⠠⠐⠉⠂⠈⠑⠀⠄⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠀⠀⠂⠸⠄⠐⠀⠀⠀⠀⠂⡀⠄⠐⡀⠀⠀⠚⢁⠄⡘⠀⠀⠀⠂⠀⠈⠁⣄⠈⠐⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⣀⠐⢀⠂⠌⠐⠀⠉⢡⠀⡄⠀⠠⠉⢩⠀⠉⠉⠁⠀⠉⠀⠈⠁⠀⠈⠀⠀⠀⠀⢀⠀⡄⠀⠀⢸⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀⠠⠲⠐⠈⠁⠄⢀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠀⠈⠀⠀⠀⠠⠠⢠⡶⡦⣦⢴⣤⡔⠁⠐⠂⠁⠠⠀⠀⠀⠐⠀⠀⢁
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠤⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⠆⢳⣗⣮⠀⣙⣣⡀⠀⢈⠐⢀⡠⡡
⠀⠀⠀⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠉⠉⠁⠀⠐⠰⢦⢯⠈⢐⢚⠉⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢥⡄⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠊⣽⠂⠀⠀⠀⠀⠀⠀⠀⠘⢘⠀⠀⠀⠀⢀⡠⠐⠠
⠀⠀⠀⠀⠀⠀⠁⠂⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⠀⠀⢀⡄⠀⠀⠈⠉⠀⠁⠈⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⠈⠁⠈⠁⠠⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⢧⡀⠀⠀⠀⢰⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣧⢿⣶⣤⣤⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣱⣿⣻⣿⣷⠩⣽⡟⢩⣿⡇⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡻⣿⣽⠯⠶⣟⣿⣿⣿⡇⠀⠀⢸⣿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦
⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠞⠙⡆
⠀⠀⠀⠀⠀⠸⣥⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡼⠁⠀⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⡽⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠁⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡼⠁⠀⣠⠷⠖⢦⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣠⠞⠛⠒⢒⡶⠶⠶⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⠀⠘⠁⢀⡤⠞⢲⣤⡀⣠⣴⣻⣤⡄⠀⠀⠀⠀⠠⠶⠋⠀⣀⡼⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠷⠤⠴⠶⠶⠤⠤⠬⠤⠀⠻⠟⣿⠋⣻⣤⠿⠷⠀⠀⠀⠀⠀⠀⠐⣛⣉⡴⠞⠁
⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠒⢿⠋⢉⣷⣄⠀⠀⢀⡴⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣀⡀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠠⠼⢷⡋⠁⠈⠀⠀⣼⠁
⠀⠀⠀⢀⡟⠀⠀⠘⢿⣷⣿⡿⠁⠀⠀⠈⢻⣶⡀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⢸⠁⠀⠀⠀⣾⣿⠿⢿⠇⠀⠀⠀⠀⠹⣇⠀⢻⡆⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠒⠶⠶⠶⠒⢾⡅
⠀⠀⠀⢻⡄⠀⢀⣠⡤⠶⢤⡄⠀⠀⠀⠀⠀⢰⡿⠀⠁⠀⠀⠀⠐⠶⠤⠤⠶⡇
⠀⠀⠀⠈⢿⣾⣋⣀⣀⡀⠀⢹⡄⠀⠀⠀⣠⡿⠁⠀⠀⠀⠙⣳⣄⡀⠶⠦⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣧⣸⠇⠀⠀⠀⠀⠀⢀⣀⣠⢴⡞⠁⠈⠙⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠤⠶⠖⢶⠻⣏⠁⢠⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣘⣶⡤⠽⠞⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⣊⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣀⠀⠀⠐⣠⣤⣤⣀
⠀⠀⠀⢈⣿⣭⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⠂⢀⢴⠖⠸⠛⠡⣴⣿⣿⣿⡄⠀⠀⠀⠀⢀⡤⡤⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢨⣽⠴⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢰⠋⢀⡄⣠⣾⣿⣿⢉⡀⠀⠀⠀⠀⠀⠀⢰⢫⣿⣧⠀⠀⣼⣿⣿⡦
⠀⠀⠀⠰⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠘⠂⣠⣾⣿⣿⣿⣿⣷⣿⡾⠀⠰⡶⠀⠀⢠⢥⡜⠋⠀⣸⣿⣿⠋⡀
⠀⠀⠀⠚⣿⣛⠀⠀⠀⠄⠀⠀⠀⢠⣭⣬⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢰⣳⠀⠀⠀⠀⠀⠉⣰⣿⠟⣡⣾⠟
⠀⠀⠀⣙⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⢀⣾⣄⣰⠟⢁⡼⠏⠁
⠀⠀⠀⣭⣯⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⣿⣿⣿⡏⠸⣿⡿⠀⠀⠈⡑⠀⢀⣾⣿⣿⣧⣴⢋⣀⣤⡄
⠀⠀⠀⢴⣷⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣻⠇⠁⡈⠉⠁⠀⢻⡇⠀⠀⠀⠃⢀⣾⣿⣿⣿⣿⣿⣯⡉⠉
⠀⠀⠀⠶⡷⠶⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣴⣿⣤⡔⠋⠀⠀⠀⠀⢱⡅⢀⡜⠁⣼⣿⣿⣿⠏⠹⠿⠿⠷
⠀⠀⠀⠻⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄
⠀⠀⠀⣙⣛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠿⠿⢿⡏⠁⠀⠀⠀⣤⠂⣿⣿⡿⡿⠿⠿⠿
⠀⠀⠀⣉⣋⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠄⢴⣾⣿⣷⣿⣶⡆
⠀⠀⢠⣬⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠛⠋
⠀⠀⢠⣤⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⠶⠟⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢀⣀⣤⠾⢿⣄⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠖⠒⠛⠉⠉⠀⠀⣀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣦⣀⣤⣤⡶⠾⠛⢋⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢀⣤⣶⣿⠿⠟⠛⠛⠛⣻⣿⠟⠛⠛⢛⣿⣿⣿⠿⠿⢿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⢸⣿⣿⡎⢻⣇⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠉⠉⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⡀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣷⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⣹⡇⠀⠀⠀⠀⢰⣿⡇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠙⠛⠛⠀⠀⠀⠀⠀⢸⣿⠀⠺⣿⣽⣻⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⠦⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⢾⣯⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⢰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡆⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿⢿⣿⣷⣄⡀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⢀⣠⣴⡾⠟⢉⣿⡇⠀⠀⠀⠀⣰⠏⠀⢸⡟⡀⠀⣼⢿⡄⠀⠀⠀⡿⣧⠀⠀⣼⡇⠘⣿⣇⠙⢿⣦⣄⣀⠀⠀⢹⣷
⢀⣴⡾⠟⠋⠁⠀⠀⣸⣿⠀⠀⠀⠀⢠⡏⠀⠀⢸⡇⠳⣴⡇⠈⢿⣄⠀⣼⠇⢸⣂⡴⢉⡇⠀⢹⣿⡄⠀⠈⠙⠻⢿⣶⣾⠟
⠛⠿⣶⣤⣀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣾⡀⠀⠀⢸⡇⠀⠙⠁⠀⠀⣹⣿⣇⠀⠈⠋⠀⢸⡇⠠⢴⣿⣧
⠀⠀⠀⠉⠻⣷⡄⢸⣿⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠏⠀⠙⣦⠀⠀⠀⢸⡇⠀⢰⢿⣿⡀
⠀⢀⡀⡀⠀⣼⡏⣼⡟⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠘⣷⡀⠀⢸⡇⢠⠟⠈⣿⡇
⠀⣼⣿⣿⣿⠏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⣿⠀⠀⣸⡟⠀⠀⠀⠀⠀⠘⣷⠀⢸⣷⠏⠀⠀⢻⣿
⠀⠀⠉⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⢿⣦⣤⣀⣀⣀⣠⣴⠿⠃⢸⠃⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠸⠿⣿⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣸⣆⣀⣀⣀⣉⠙⠛⠛⢉⣀⣀⣀⣿⣀⣀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⢻⣿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⢤⣤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⡶⠶⢶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⡆⠀⠀⢀⣀⠙⣷⡀⠀⠀⠀⠀⠀⠀⣴⠟⢉⣁⡀⠀⠀⠆⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠷⠾⠛⠋⣿⣄⣼⡷⠀⠀⠀⠀⠀⠘⢷⣤⣾⠏⠛⠛⠛⠛⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠉⠉⠙⣍⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⢬⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡑⢢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢠⣘⢁⡒⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⡎⠄⣈⣄⠠⠂⢹⣊⠔⣁⠠⠄⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠚⢫⡡⠤⠄⠀⠉⠒⠴⠉⣀⠔⠉⢀⠜⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠦⣀⠀⠀⠀⠀⣀⠔⠁⠀⠉⠀⠀⠀⢈⡩⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⡀⠀⠈⢆⠀⠀⠤⢤⣄⠀⠀⠀⠀⠀⠯⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⠈⠑⠁⠠⠨⠦⠠⠀⣈⣁⣀⡀⣀⠄⢊⡰⢠⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠐⠂⠙⠊⠂⠄⣀⠀⠀⠀⡘⠀⠀⠉⠉⠐⠉⠀⢸⠠⠀⠑⢲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠖⠁⠀⠀⠀⠈⠆⢠⡉⠉⠀⠐⠁⠤⣀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠘⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⠀⠀⠀⠀⠀⡸⠉⠙⠳⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⡒⠲⣏⠀⠀⠀⡹⠓⠀⠀⠉⠉⠉⠉⢉⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢻⣄⡠⠂⠣⡀⠀⠀⠀⠀⠀⠘⢁⢓⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠒⠀⠀⠀⣇⡘

⠀⠀⠀⢘⡧⡮⠷⣏⡧⠷⡶⣏⡴⡏⢾⡷⢎⡷⣿⡇⠰⣿⢾⡀⡾⡴⣏⠰⠆⠀⡾⡆⠷⡷⣏⠆⠸⡾⡞⡗⠶⠆⡏⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠠⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⢿⡿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⠰⠆⡰⢇⣼⣿⠀⢠⣀⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠆
⠀⠀⠀⠀⢤⣴⠀⠈⣿⣿⣿⣷⣶⣿⢿⣿⣿⢀⡴⠿⠯⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⡿⠃
⠀⠀⠀⠈⠛⠛⠙⠲⠜⢿⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⢟⣛⣻⣿⣯⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⡟⠛⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⠿⠿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣦⠄⠐⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠖⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠄⠐⠁⢀⠀⠂⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠢⡀⢀⣀⣀⠀⣀⡃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡂⣬⡂⠙⢿⠿⠿⠿⠀⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠂⠒⠀⠂⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⠈⣠⠱⠃⢀⣲⢥⣶⣯⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠠⣤⣤⣽⣿⠿⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠳⡈⡻⠛⠀⣼⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠂⠰⡆⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⠦⠀⠀⠀⠉⠀⠠⡄⠀⠀⠐⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⠂⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⣤⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠘⠃⠀⠀⠀⠀⢾⠆⠀⠙⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⢠⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠈⠁⠦⠀⠂⠀⣼⣿⣿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣄⡀⠀⠀⠀⣺⣿⣿⣿⣷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣶⡄⠀⠀⢸⣿⣿⣿⣿⡟⢉⡻⣿⣿⣿⠡⠝⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⣷⣬⣴⣿⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⠛⠛⠉⠉⠁⣿⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⣿⡿⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠐⠒⠒⢒⡀⠒⠒⢂⡀⠒⣤⣄⡤⠆⠉⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡘⠗⠛⠟⢠⣿⣿⠀⢠⣿⣿⠀⠉⡇⠈⠲⡀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠭⠄⢆⠀⠘⣿⣷⠀⠘⣿⣧⢀⠀⠘⡦⣀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠡⠲⠓⢁⣸⠀⣧⡉⠉⠀⢐⣉⡩⢔⡁⡗⠓⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠜⢶⣀⡘⠋⠀⠸⣿⠶⠶⣤⡶⠂⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡸⠓⠂⠐⠧⠤⢤⣷⣿⣯⢠⠤⠠⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣶⡶⣿⣿⣿⣿⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⡿⠃⠀⠀⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⣇⠀⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡔⠂⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⣀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠀⠘⠓⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⣾⠆⠀⠃⡄⢸⡀
⠀⠀⠀⠀⡀⠀⣀⡤⠀⠤⠀⠀⠴⠀⠐⠒⠀⠐⠲⡄⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠐⠱⡄⠀⠀⠀⠆⠃⠶⣀⠀⠈⢐⢄⠀⠀⢄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣎⠀⠀⡄⠠⠀⠀⠀⠀⠺⢪⠀⠀⢙⡋⠇⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠐⠇⠠⠄⣀
⠀⠀⠈⡀⠀⠘⠀⠀⠘⠒⢢⡀⠀⠠⠀⠁⠟⠀⠀⢠⠀⠀⢀⠃⠀⠀⠀⠀⠀⠂⠐⠢⠄⣀⠀⠀⠀⠉⠐⡄
⠀⠀⠀⢘⠀⠀⠦⠀⠀⠀⠉⠀⠀⡤⠀⠀⠀⠀⠀⠀⡄⡰⠃⠀⠀⠀⠀⠀⡜⠀⠂⠬⢀⢀⡠⠤⠄⠤⢀⣡
⠀⠀⢰⠈⠁⣀⠐⠖⠂⢀⠀⠁⢒⠀⠠⠀⠀⠀⠀⠀⠴⢀⠀⠀⠀⠈⠢⣸⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠂⠳
⠀⠀⠈⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⡆⢀⠃⠀⠀⠀⠀⠉⠢⡜⠀⠀⠀⠸⠀⠀⠀⠇⣀⠀⠂
⠀⢀⣫⣾⣧⢀⡀⠀⠀⠀⢰⠄⠠⠄⠔⠲⠄⠀⠧⢀⡘⢀⠥⠂⢄⠀⠀⠀⠀⠈⠢⣄⠀⡄⠀⠀⢀⠀⡌
⠀⠀⠙⠷⠟⠸⢏⣉⠙⠓⠾⠦⢤⣭⡤⢤⡄⡤⠦⠘⠳⡄⠄⠐⠪⠝⣶⠀⠤⠤⠤⠞⢀⠇⠀⠀⡌⣰⠁
⠀⠰⢀⠰⠄⠀⠀⠀⠉⢁⡖⠤⣤⠀⡀⠀⠀⠀⠦⠆⢀⠃⠀⠈⠀⠀⡏⠀⠀⠀⠀⠀⡸⠀⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠠⠤⢴⠂⠃⠀⠀⠀⠱⠦⣀⢀⠂⠀⠀⢄⡀⠒⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⡄⠀⠀⠀⠀⠀⠀⠈⣦⠀⢘⠀⠉⠒⠤⠤⠛⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠛⠂⠁⠀⠀⠀⠉⠒⠄⠤⠤⠤⠤⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧
⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠄
⠀⠀⠀⠐⠁⠀⠐⠀⢀
⠀⠀⠀⠠⠂⠀⠀⢀⠈⣠
⠀⠀⢴⠁⢄⠥⠖⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣴⣆⢀⠠⠀⠑⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡉⠋⣹⢭⣿⣤⣦⠴⣦⡤⠬⢧⣄⣠⠄⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⠊⠀⠼⣦⣶⣿⣋⣿⣾⠭⠶⣿⣹⠲⢊⠛⠩⣹⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠻⠌⠄⢰⠿⠟⢢⣶⣿⣿⣷⣽⣿⢷⣿⣿⡿⢷⣿⠛⣤⣤⡘⣿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣰⣾⣿⣉⢉⣽⣿⣿⣾⣚⣾⡿⣷⣶⣍⣯⠉⣼⣿⣯⡰⣤⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣸⡥⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠹⠭⣛⡛⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢃⠛⠀⣿⣥⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⢿⣶⠛⡟⣍⢻⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⢴⣿⣿⣿⠟⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣿⣤⠇⣙⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠶⠀⠳⡌⢧⡹⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣁⡨⡝⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⢁⠀⠃⢈⣷⡘⠳⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⠤⢧⡿⣍⡴⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⢐⡊⠙⣿⣇⣲⣮⡿⣿⣿⣿⣿⣻⣯⣿⠟⠉⣱⣸⠿⠘⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⢷⣄⡈⣹⠟⠓⠂⠻⠿⠟⠛⠻⠛⣧⢚⣥⡿⠉⠐⡀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠛⢭⡋⠑⠒⢻⣥⢤⡤⠉⠀⠀⣨⠿⡨⠧⠤⣤⣥⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠾⠀⡾⢯⠋⠛⠯⠀⠬⠁⠆⠀⡛⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⠀⢀⠓⠀⠀⠀⠄⠛⠳⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠤⠤⠤⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠖⠊⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠧⣀⡀⣀⡀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠀⠘⠞⣻⡟⢭⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠸⣤⣄⢀⡛⡁⠀⢸⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⡃⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⠀⠠⠖⠀⠀⠘⠦⣀⣀⣀⣀⠀⠀⠠⠤⠤⠄⠠⠀⠀⠀⢀⣀⡀⠀⠀⠠⠤⠤


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠣⡄⣶⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠿⣆⠳⣶⣼⣾⡗⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠐⠻⣦⡎⠛⣚⣧⣴⣾⣷⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢑⣿⣿⣿⠿⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠋⠀⠀⠀⠀⠀⣠⢀⡂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⢤⣀⠀⠀⠀⢀⠈⠒⣨⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⢫⣀⠀⣕⣾⡏⣧⠀⠀⣿⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⣿⣿⣿⣿⣿⡿⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠀⣠⠄⠀⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠋⠁⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢀⣄⠀⢠⠋⠀⣀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢦⡀⠀⠀⠀⠀⠀⠀⣾⣿⣋⣀⣸⣿⣧⠈⢆⠘⢿⡇⠛⠛⠋⡞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣶⣶⣿⡄⠀⠀⠀⠀⠸⠻⣟⠿⣿⡿⡷⠚⠇⠀⠑⢸⡇⠀⡠⠴⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡦⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⢀⣴⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⣼⠟⣡⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢘⠀⠀⠀⢀⣀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⣧⣾⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⠈⡙⠑⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠐⡟⢿⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⣈⡐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⢀⠞⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡀⣿⣿⣿⣿⠿⠿⠿⣿⣿⢿⣿⠿⠿⠂⠀⠸⠿⠇⠋⢠⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⣿⣷⣦⣀⣴⡷⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⢀⠔⠊⠀⣠⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣀⣀⡤⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠿⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣀⣀⣀⡠⠶⠦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠉⠉⠰⠶⠆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠠⣾⡟⠉⠛⠙⠛⢧⣄⠀⠀⠀⠉⠙⠛⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠤⣤⣤⡤⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡾⠛⠃⠀⠉⠁⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⣀⣀⠽⣷⣤⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠟⢠⣄
⠠⣾⣿⠛⠶⠶⠀⠒⠀⠀⣀⣉⣀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠰⠆⠈⠙⢿⡀⠀⠀⠀⠀⠀⠈⠛⣏⠀⠀⠀⠺⠷⣀⠀⠀⣀⡀⠰⠂⠀⠀⠀⠀⠁⠰⠆
⠀⠈⠉⠀⠀⠀⠠⠄⠀⠈⠙⠛⠉⠁⢰⡿⢷⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠙⣀⡀⠀⠀⠀⠿⣿⡆⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⢦⣄⠀⠀⠀⠀⠀⠀⠈⣶⣄⣠⣤⣤⠀⠘⢿⠟⠙⠁⠀⠀⠀⣼⡆
⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⠀⠀⠀⠈⠃⠀⠀⠉⢷⣤⣀⠐⠛⠓⠂⠈⠙⢧⡀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⢣⣄⠀⠀⠀⠀⠀⠀⢾⡏⠀⠀⠀⣰⡶⠶⠖⠀⣶⠟⠻⣿⣷⠶⢄⣀⠀⠀⠈⠹⢶⡀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠈⣳⡤⠈⠉⠛⠁⠀⠀⣤⠀⠀⠀⠘⢃⣀⠈⠻⣷⣤⠠⣤⡴⠿⣶⣤⣀⣹⡿⠂⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣧⠀⠸⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡿⠃⠀⠉⠁⠀⠀⠀⠀⣀⠛⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣽⡇⠀⠀⠀⠀⠀⢿⣄⠀⠐⠀⠈⠁⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢉⣹⠶⠶⠖⠂⠀⠀⣠⡄⠐⠒⠖⠊⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠉⢿⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⡀⢀⣶⡆⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠁⠀⢶⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⡄
⠀⢀⣴⠀⠀⠚⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠿⠿⠲⠂⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⢀⣀⣀⡠⠶⠶⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢷⣦⠄⠠⠾⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣦⣤⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡄⠀⠀⠀⠀⢀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠂⣀⣤⣀⡀⠀⠀⢀⣀⣀⡀⠘⠛⠃⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢻⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢄⣀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⠦⢀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⡄⠀⠀⠈⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣀⠈⠋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠾⣿⣄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⢀⠀⠀⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣀⣠⣤⣄⣠⣤⣀⣤⣤⣄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠟⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⣧⠀⠀⠀⠀⠀⢿⣅⣠⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠃⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣀⡀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠿⠶⣶⠾⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠾⠋⠻⠶⡶⠾⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⢿⣤⣤⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠀⠈⠉⠛⢶⡟⠁⠀⠀⠀⠈⢷⣀⡴⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠉⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠛⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⢀⣴⠟⠋⠁⠀⠈⠉⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠈⣷⠶⢦⠀⠀⠀⠀⠀⠀⠀⣤⠶⣤⡴⠞⠋⠉⠙⠛⠲⠶⢶⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⣤⠾⠁⠀⠀⠀⠀⠀⠀⠳⣶⡟⠁⠀⠶⠟⠻⠏⠀⢀⡼⠃⠀⠀⠀⠀⢀⣀⣴⠾⠃
⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⣀⡀⢀⡴⢤⡀⣼⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⢰⡾⠛⠛⠉
⠀⢸⣇⠀⠀⠀⢀⡴⠾⠋⠉⠙⢿⠀⠀⠙⢷⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⢠⠀⣀⣿⣤⡀⠀⠀⣠⡟⠁
⢀⣠⣿⣦⡤⣤⣿⣤⣤⣤⣤⣤⣿⣤⣤⣤⣿⣤⣿⣤⣤⠤⠴⠶⠶⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠛⠿⠿⣄⠙⣿⡛⠉
⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣇
⢹⡄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢿⡄
⠈⣷⣀⣀⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡾⢿⡄⠈⢿⡄
⠀⠘⠻⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠻⣦⣄⣻⣦⡀
⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠉⠛⠛⠃
⠀⠀⠀⢻⡀⣯⠹⡆⠀⠀⣾⠁⠈⣧⠀⠀⢀⡟⠁⠀⠈⣇⠀⠀⡾⠁⠈⡧⠀⠀⡾⠁⣿⠀⠀⣾⠇
⠀⠀⠀⢸⡇⢹⡀⢽⠀⠀⣿⠀⠀⢸⡄⠀⢸⠁⠀⠀⠀⢹⠀⢸⡇⠀⠀⣷⠀⢸⠁⠀⣿⠀⠀⣿
⠀⠀⠀⠀⣇⢸⡇⠈⡆⠀⣿⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⣿⠀⡜⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⣿⠀⣧⠀⢧⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠈⡇⣼⠀⠀⠀⣿⠀⡇⠀⢠⡇⠀⣸⠃
⠀⠀⠀⠀⢹⡄⢻⠀⢸⠀⢸⡄⠀⠀⢹⡀⠈⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⡏⢸⡇⠀⢸⡇⠀⣿
⠀⠀⠀⠀⢸⡇⢸⡇⠈⡇⠘⡇⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⡇⣿⠀⠀⢠⡇⢸⠇⠀⣼⠁⢠⡏
⠀⠀⠀⠀⠀⣿⠀⢳⠀⢷⠀⣿⠀⠀⠈⣇⠀⡇⠀⠀⠀⠀⡇⣿⠀⠀⢸⡇⢸⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡄⠸⡀⢸⠀⢸⡄⠀⠀⣿⠀⡇⠀⠀⠀⢠⡇⣿⠀⠀⢸⠀⣼⠀⢰⡇⠀⣾
⠀⠀⠀⠀⠀⢸⣇⠈⡇⠈⡇⠈⡇⠀⠀⣿⠀⢱⠀⠀⠀⢸⠀⣿⠀⠀⡞⠀⣿⠀⢸⠁⢀⡿
⠀⠀⠀⠀⠀⠀⣿⡀⢳⡀⣿⠀⢹⡀⢀⡟⠀⠈⢷⣄⣠⠞⠀⠸⣆⣴⠇⠀⢸⣦⡟⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠁⠀⠀⠙⠛⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠸⣷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣠⣤⣤⣤⣤⣤⣤⡶⠃⠀⠀⠀⢴⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣤⣴⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⢻⣿⡃⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣷⣦⡄⠀⠀⠀⠀⠀⠘⢡⣿⠆⠀⢀⣀⣀⡤⠄⠀⠀⠀⠀⢀⣻⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⢀⣿⣿⣀⣠⠤⠆⠀⣀⡤⠶⠚⠋⠉⣀⣤⠤⠶⠒⠛⠉⠉⠉⠙⠋⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠋⢁⣀⠤⠒⠉⣀⣀⣤⣶⣚⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣩⣭⡠⠖⢋⣠⣴⣾⣿⣿⣿⣿⣿⣿⡟⣰⣾⣶⣦⡀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠋⢀⡠⠟⠛⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠸⣿⣿⣿⣿⣿⣦⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⡴⠋⢀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⣀⣀⠀⠹⣏⡙⢿⣿⣿⡄⢋⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠉⢸⣿⡿⠟⣿⣿⡟⠁⠀⣀⣉⡙⠛⣩⡤⠂⠀⠉⠁⠀⣿⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡠⠚⣁⣀⠀⠘⣿⡅⢠⣿⠏⠀⠀⠉⢁⣈⠉⠛⠁⢀⣴⣶⡄⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠞⠀⠘⢛⣿⡷⠀⢻⡇⠘⠿⠀⠀⣴⣾⠿⠻⣿⠀⢰⠟⠋⠹⠇⠀⢸⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣀⠀⠀⢨⣥⠀⠀⠘⣷⠀⠀⠀⠸⠋⠁⠀⠀⠁⢠⡀⠀⢀⡀⠀⠀⢸⣿⣭⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠀⠀⠀⣠⠶⠦⣿⡄⠀⠀⠀⠀⢠⣶⡆⠀⠀⢷⠀⠻⡿⠀⠀⣾⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡀⠀⢰⣿⠀⠛⢻⣷⠀⠀⠀⠀⠀⠙⢡⠆⠀⠸⣇⠐⢦⡀⠀⣿⣿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⣿⣇⠀⠀⠀⠀⠴⠋⠀⣀⠀⢻⡀⠀⠁⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⣹⡟⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⣁⣤⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠺⢿⡟⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠠⠿⠶⣦⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣼⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣴⡧⠾⠋⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣦⣤⣍⣩⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠻⠿⠟⠛⠻⠿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⢠⡀⢀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡠⠀⢴⠗⠃⢌⢁⠛⠉⠉⠕⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⠉⠳⠛⡖⠃⡤⠸⡤⠀⠈⠞⠒⡆⠀⠑⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠝⣳⡋⢈⠤⠔⠏⠈⢠⠒⡇⠀⠰⢄⠈⠃⢱⡠⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⡀⠣⡘⠓⠀⠀⣤⡈⠀⢸⣀⠒⠐⢄⡀⠀⠩⠇⡈⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣺⠖⠋⠈⣚⠩⠉⣄⣰⠓⣤⠇⡰⡰⠂⠭⣂⢃⡰⣉⠩⣉⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⢿⡴⣆⣉⣤⡀⢢⡜⢅⡘⡟⣙⡡⡁⠳⣐⠎⡢⢑⢈⠂⠌⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⠏⣻⢧⡋⠳⠘⠛⢧⣹⢏⠗⡺⢵⡑⠠⡅⢿⣶⠄⣤⢇⣜⢹⡀⢔⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣻⣃⠛⡆⠥⣠⡃⠨⡼⣻⠞⣠⣅⡼⠈⠟⠲⢏⠸⡆⠢⡖⠐⠴⠀⣼⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠯⢉⣋⣯⠇⠁⣐⠉⡓⡄⠳⡏⠲⠩⠋⠜⡻⢤⣀⡠⢋⠈⢲⡉⣗⣷⣄⠁⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠖⢩⢯⡜⢷⣚⡧⡘⠶⠃⢀⠈⢐⡃⡎⡠⠻⢁⣸⠶⢪⠏⠰⣄⡠⡉⣠⠧⢊⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⢲⠇⠘⠀⢾⣽⣾⠲⣆⡁⢂⡼⣣⣖⡌⠑⢣⣱⡞⣵⠴⡁⡠⢄⡠⠄⢰⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠴⢺⡈⢠⠚⢢⠀⠈⠯⢍⢁⡔⢝⢶⢻⢯⡽⢋⡹⠓⠤⠶⠤⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⠦⡸⠂⠀⠀⢨⡞⠹⠨⢺⣝⣯⣿⡎⢿⣛⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⣤⢵⣽⠙⣦⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣤⠀⠀⠀⠀⠀⠀⢀⡥⠈⠲⢝⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣠⠔⢋⠀⠀⠀⢠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡴⠚⠳⣄⠈⢳⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣤⡴⠉⠛⣦⣞⣋⣻⣦⠤⢄⡠⢤⡄⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠋⠁⠈⠁⠁⠠⠊⠃⠁⠛⠋⠹⠛⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣤⡰⠊⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠻⡉⠙⡅⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡖⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣾⡀⠀⠀⢀⣰⣤⡀⠀⠀⢀⣸⣇⠀⠀⠀⠀⢠⣿⡏⢀⣀⣤⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⡿⣶⣶⣿⣿⡿⢿⣶⢿⠿⡏⠻⣿⣶⣶⣾⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠺⡄⢰⠇⠀⣹⠃⣼⡿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠇⠀⠀⠀⠉⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡾⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⠿⣀⢀⣠⣀⡴⠞⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠳⠦⣤⣅⡀⢀⣤⢶⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢾⠁⢻⣾⡿⢾⢄⣻⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠀⠀⠸⢤⠋⢹⣿⡞⣠⡣⠀⢀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣷⣶⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⡟⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢊⣙⠿⣿⣿⣿⢿⡟⠋⠉⢻⣯⡩⣄⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⣀⠀⢸⡏⣁⡉⠛⣞⡉⢉⣤⣏⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠓⠛⠻⠟⠛⠻⠛⠻⠾⠿⠿⠿⠾⠟⠛⠛⠿⠟⠛⠛⠙⠒⠘⠂