Onomatopoeia Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣷⠀⠀⣼⡇⠀⢀⠀⠀⢀⠎⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⢸⡄⠀⡄⢸⣇⢠⣿⡇⢠⠎⠀⣰⠏⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⢢⡀⢷⡀⢹⣾⣿⣾⣿⣵⠏⢀⡼⠋⠀⠀⣠⠔⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣿⣷⣌⡟⠺⣧⠀⣿⣿⠁⠻⠏⠀⠊⠀⠀⣠⢞⣁⣤⣶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⣦⣬⣿⡿⣿⣟⣀⣤⣤⣾⣧⣶⣶⠀⠀⣾⣿⠿⣿⣿⢁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⠿⣟⣛⣿⡏⠉⠀⣾⡿⣿⡇⠀⣿⣟⣀⣾⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠚⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⠀⣼⣿⣁⣿⣿⣸⣿⡿⠛⠋⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⣲⣿⣿⣿⣿⣴⣿⠿⠟⢻⣿⣿⣿⡇⠀⢹⣿⠟⠶⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣺⣿⣿⠟⠀⠛⣻⣿⠃⠀⠈⠛⠿⠟⠀⠠⠿⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠈⢀⠟⠁⣠⣠⣾⣿⡏⠀⠀⢸⣇⣾⣶⣦⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⡿⢫⢰⣿⠀⠀⠀⢸⡟⣿⣿⢻⣿⢿⡙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠠⠁⣸⠇⠀⠀⠀⢸⠇⢹⣿⡄⠹⠀⠁⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢸⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣦⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢊⣤⢂⣦⢰⡛⠶
⠀⠀⠀⠀⠀⠀⢠⢠⠀⣶⠀⠆⠀⠠⠦⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣤⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠸⠙⠒⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣤⢴⡖⢿⢱⢸⣸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡍⣷⠈⠳⠀⡀⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠐⠀⠀⠢⢡⠀⠀⡥⠀⠁⠀⠀⠀⠀⠀⠁⠱⠬⠬⠶⠗⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡄⠒⠀⠘⠀⡈⠈⢑⡁⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠅⠀⠂⠀⠂⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠁⠀⡀⠐⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠆⠁⠰⣖⣦⣀⢤⡙⣄⣹⢈⠇⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠁⢯⢸⠇⠚⠁⠈⣴⠁⡉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⡀⠀⡀⠀⣠⡀⡁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠔⠡⡀⠈⢁⠜⢣⡀⠛⢦⠉⠀⠀⠀⠀⠐⠆⠀⠀⠀⢀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢀⣠⢠⠅⣸⢠⡍⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠯⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠰⠁
⠀⠀⠀⠀⢠⠦⣠⣤⢀⡀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠘⠂⠋⠃⠞⠇⠼⠶⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⣀⡐⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⣭⠙⡴⢁⠌⢣⡜⢠⡌⣶⠈⢹⢶⠋⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⢐⣠⢀⣀⣤⠀⠀⠀⠀⡃⢘⡋⣼⠃⠆⡈⢸⠀⠃⡆⣿⢰⠘⢃⡆⡘⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢣⣛⣡⡞⢆⠲⢁⠙⡀⠔⢀⠇⠸⢠⡼⣠⠇⠀⠀⠀⠀⢲⠘⠁⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠈⠉⡁⠈⡁⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠁⠀⠈⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⢠⣿⣇⠀⠀⠀⣼⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⠀⢸⣿⣿⣷⡄⣸⣿⣿⡆⠀⣼⣿⣿⣷⢀⣼⣿⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣼⣿⡇⠈⢿⣿⡏⠹⣿⣾⠏⠀⣿⣿⠾⣿⣿⠇⠀⢀⣠⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣀⡀⣿⣿⡇⠈⢻⡇⠀⠈⠻⠀⠀⢻⠏⠀⠀⠹⠋⠀⣿⣿⣤⠾⠛⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡋⠛⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠋⠁⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣦⣤⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⣽⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⣿⣿⣶⣶⣦⣤
⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣷⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⠛⠛⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠛⠛⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⢀⣠⣴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣤⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣶⠟⢻⣿⡇⢀⣼⡆⠀⢀⣾⡄⠀⢰⣦⡀⠀⠀⠀⠀⢿⣿⣧⠀⠉⠛⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠉⠀⠀⢸⣿⣿⡿⣿⡇⢠⣾⠟⣿⣀⣿⡿⣿⣆⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⠏⠀⣿⣿⣿⠏⠀⢹⣿⣿⠇⠈⠻⣷⣄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢿⣿⠏⠀⠀⠀⢿⡿⠀⠀⠀⠘⢿⣷⣄⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⣆⠀⠀⠀⢠⠃⠀⡆⠀⢀⠔⠁⠀⠀⠀⣀⠠⠤⠀⠀⡀⠄⠂⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢧⠀⠸⡄⠀⢠⠃⠀⠰⢀⡔⠁⡠⠤⡀⢀⠁⠀⠀⠀⠀⠁⠀⠀⢀⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠲⢤⣀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠣⡸⠱⣸⠈⠒⠁⠀⠀⠘⢁⢰⠈⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠤⢀⡈⠻⣖⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠂⠀⡀⠀⡀⠀⠀⡆⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⣀⢈⠑⠂⠤⢄⣑⡄⠈⠉⠉⠀⠀⠀⣠⠀⡠⠐⠈⠀⠈⠢⠀⡀⠒⡄⠀⢰⠀⠙⠊⠁⠀⠀⡠⠀⠀⠀⠄⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠭⠔⠂⢀⠠⠀⠠⢀⣀⠔⠁⠀⡎⠀⠀⡀⢀⠀⠀⠺⠀⠄⠀⠀⡀⠀⡨⠀⠐⠈⠤⠀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠒⢢⠀⠠⠊⠀⠀⠀⠀⠀⠑⠀⡠⡲⠑⠀⠌⠀⠘⠀⠀⠄⠀⠀⢀⠴⠔⠁⣀⠀⠤⠤⠄⠒⠒⠒⠐⠒
⢀⣀⣤⣀⠤⠐⠒⠈⠀⠰⠁⠀⠀⠄⠐⠀⠀⠀⠃⠉⠀⢀⠅⠁⠈⠀⠀⡠⠃⠀⢈⠡⠄⠄⠭⠖⠒⠒⠒⠒⠒
⠀⠀⣀⣀⡤⠌⡃⠀⠀⡂⠀⠀⠀⠄⠈⠀⠀⠬⠀⠀⠰⠣⡀⠀⢀⠠⠊⠀⠀⢰⠒⠓⠶⣤⣀
⠀⠀⠀⢀⡠⠄⣋⡀⢀⠈⠀⢄⠀⠀⠀⠀⠨⡀⢁⢀⠔⠁⠀⠀⢀⢀⡀⠀⠠⠂⠵⣄⠀⠀⠈
⠀⠀⠈⠔⠂⠉⠀⡠⠃⠀⠀⠀⠀⠢⡀⠀⡔⠁⠀⠁⠀⠀⡰⠱⠎⠛⠘⡀⠘⡄⠀⠈⠙⠦
⠀⠀⠀⠀⠀⠴⠋⠀⠀⣀⡀⠀⠀⠀⢈⡊⢀⠔⠒⡔⢂⠀⠁⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡠⢔⡩⢊⠔⠁⠀⠀⠁⡵⠁⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠒⠉⠐⡡⠊⠁⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡼⡇⠀⠀⠀⠀⠀⣠⠀⠀⢠⣤⣤⣴⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢸⡷⡀⠀⠀⠀⡸⠀⡇⠀⠀⠀⢀⢔⠏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⢀⠼⡇⢻⣎⠓⡶⠁⠀⣇⠤⠒⡶⠃⣼⡀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠛⢿⣄⡀⠀⠀⡰⠁⠈⡇⠀⢻⡶⠁⠀⠀⣇⣠⠎⠀⡼⠀⠙⡄⢀⣼⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠾⡫⢿⣿⣶⣷⣶⣾⡇⠀⠀⠁⠀⠀⠀⠟⠁⠀⣰⡁⠀⠀⣾⣿⠛⠉⠘⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣸⣦⡀⠈⠉⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣾⡿⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣶⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⡁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠥⢸⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣄⡼⢀
⠀⠀⠀⠀⠀⠀⡰⠉⠀⣀⣴⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣿⠟⠉⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠩⣿⣋⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⡀⠀⠀⠈⠻⣿⣿⣷⡄⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⢠⠾⠿⠿⢿⣿⣿⣿⠃⠀⠀⣠⣶⡀⠀⠀⠀⣿⣿⣿⣿⣷⣦⡤⣀⠈⠢⡴⠠⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠘⣿⠃⢀⣴⣿⣿⣿⣧⠀⠀⢸⣿⡿⠿⢿⣿⣿⣇⣀⣭⡓⠚⠦⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⣴⣡⠞⠋⠉⠉⠉⠙⣿⡀⠀⢸⠁⠀⠀⢀⡜⠉⠛⠀⠈⠻⣿⣶⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣰⣿⠋⠑⠤⠄⠀⠤⠔⠋⠈⢣⠀⢸⣀⡠⠔⠋⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡷⠂⠀⠈⠁⠀⠀⠀⠀⣰⣷⡀⠀⠀⠀⠀⠉⠀⠘⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⠀⣷⢦⣄⣀⣀⣠⣾⠏⡌⠻⣦⣄⣀⣠⣤⢶⡏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢀⣿⠀⠮⠍⣉⢉⡡⠊⠈⠢⣈⣉⡉⠭⠆⢸⣧⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢶⣶⠶⠾⠟⠋⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠛⠻⠷⠶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠘⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠁⣾⡏
⠀⠀⠀⠀⠀⢀⣠⣴⠿⢋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠙⠿⣦⣀⡀
⠀⠀⠀⠀⠀⠉⠙⢷⣦⡉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣩⣴⠞⠋⠁
⠀⠀⠀⠀⢀⠀⠀⠀⣿⡟⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢿⣇
⠀⠀⠀⠀⠙⠃⢠⠾⢯⣤⣭⣌⡁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣉⣭⣭⣤⣽⣦⡀
⠀⠀⠀⠀⣀⣶⣤⠀⢀⡀⠀⢹⣿⢀⡡⠄⢒⠒⠢⡀⠀⡠⠐⠒⠂⠤⣰⢰⣿⠋
⠀⠀⠀⠀⠈⠟⠛⠀⠈⠁⠀⢸⣯⠴⠞⠛⠛⠻⣦⡐⡔⣡⡾⠟⠛⠛⠶⢤⣿⠀⠀⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢷⣴⠏⠀⠀⠀⠀⢠⣤⠉⢁⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⣀⡀⠠⢄⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⠤⣀⣀⡠⢀⠀⢀⡀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠙⠀⠂
⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⠀⠀⢿⠀⠀⠀⢰⠂⠁⠀⠀⠀⠀⠀⠉⠁⠰⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⢈⡔⠁⠀⡜⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⢈⠗⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠰⣄⠀⠀⡎⡐⠀⣀⡆⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠀⠀⠀⠀⠐⠒⠢⡀⠀⡄⠈
⠀⠀⠀⡀⠀⠀⠈⠛⠃⠀⡤⠀⠀⢀⠀⠀⠀⠘⠀⠉⠑⢄⡠⠔⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠴
⠀⠀⠀⠈⠉⠉⠁⠀⠎⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⡄⠀⠀⠀⠀⢀⡀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⡠⠂⠄⠀⠇⠀⠀⠀⠀⠀⠀⢂⣀⠄⠐⠢⡤⠂⠀⠁
⠀⠀⠀⠀⠈⢆⣀⣀⣀⠀⢀⠠⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠑⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠘⣧⠀⠀⠀⠀⠀⠄⠀⠀⢈⡃
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠤⠀⡀⠀⢀⡀⣠⡴⠚⠁⠀⠀⠀⠘⡖⠈⠀⠀⠀⠀⢀⠞⠉
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠤⠚⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠂⠂⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠴

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⠿⢿⣿⣿⣷⡄⢀⣴⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⣀⣾⣿⠟⠉⠀⠀⠀⠀⠀⠈⠙⣿⣿⡿⠟⠛⠋⠙⠻⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠟⠛⠛⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⣀⣤⣤⣬⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆
⠀⣴⣿⣿⣿⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣷
⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠈⠻⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿
⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⠤⠴⠾⠿⠛⠋
⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⣀⣠⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⣿⣦⣀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠷⠶⠟⢛⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣀⣀⣀⣀⣀⠛⠛⠛⠛⠛⢛⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⣿⠏⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠙⠉⠀⠀⠐⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡃⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⠀⠁⡅⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡴⠃⠀⠀⠀⢸⡄⠀⠀⠀⠀⠆⠘⠀⠨⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⡠⢄⣠⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠐⠤⡀⠀⠀⠀⠀⠀⡀⠤⠤⠥
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⢀⠊
⠀⠀⠀⠀⡀⣠⠃⢰⡦⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢀⢠⠈⠙⠀⠈⠀⠋⠉⠀⣸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠐⠛⠃⡀⠀⠀⠀⠀⠀⠀⠈⣩⠆
⠀⠀⠐⣆⡀⠂⠀⡀⠀⠀⠀⠉⠈⠙⢇⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠙⢉⡀⠀⠀⠀⠀⠀⠀⢠⢰⣤⡤⠂⠀⠀⠈⠿⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠉⠉⠛⠀⣠⡄⣠⣶⡀⢸⡶⠇⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠠⠂⠙⠁
⠀⠀⠀⠀⠖⠋⠋⠉⠁⠀⠁⡜⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠦⠤⠜⠁⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡤⠔⠋⠚⠷⠖⠒⠉⠁⠀⢀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢨⡤⣄⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠴⠶⠛⠓⠲⠦⣀⠞⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⠷⣤⣀⣀⡀⠀⠀⣇⡀⠀⣨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠁⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣀⣀⣀⣀⣠⣶⣿⣿⠟⢿⣿⣷⣦⣄⣀⣀⣀⣤⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠠⣿⣿⣿⣄⡀
⠀⠀⠀⠳⣶⣶⣶⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⡙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⢀⣠⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣄⡀
⠐⠿⣿⣿⣿⣟⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣿⡿⠷
⠀⠀⠀⠉⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄
⠀⠀⠠⠾⠿⠿⠿⠿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣁⣤⣶⣾⣿⣿⣿⣶⣄⠀⠀⠀⢀⣴⣾⣦⣄⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠟⠋⠉⠉⠉⠉⠛⢿⣿⣧⠀⣠⣿⣿⠟⠋⠻⣷⣄⡀⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⡟⠁⠀⠀⠀⠀⠙⢿⣦⣀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠚⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠
⠀⠀⢰⡁⠀⠀⢀⠁⠀⠆⡀⢰⠀⠀⡀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠏⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠸⡤⠤⣄⠀⡤⠖⠚⠒⠂⠁⠀⠀⠀⠀⠀⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠆⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠶⠶⣤⡀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠉⢂⡴⠶⡦⠖⠴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⢠⡀
⠀⠀⠀⢀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⡤⣄⠀⠀⠀⠀⡀⢤⣀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠁⠀⠀⠀⠀⠉⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⡴⠁⠈⠉⠛⠛⠒⠂⠁⠈⠑⠒⠘⠁⠀⠐⢄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠸⡀
⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⢴⡊⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠆⠀⠀⠷⠀⠀⡀⠀⠄⠀⠀⠂⢀⡵
⠀⠀⠈⠙⣧⠀⠀⣰⠀⠀⣀⡠⠴⠛⠛⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠘⠂⠚⠀⠀⠀⠀⡇⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⣀⣀⣸⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢀⡠⠄⠠⠤⢄⣀⠀⠀⣠⠤⢷⠀⠀⠀⠀⠀⢃⡠⠄⠤⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠦⠎⠀⠀⠀⠀⠀⠈⠣⡔⠁
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠦⠔⠀⠀⣇⣀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠠⠤⠄⠁⠀⠀⠀⠀⠀⠀⠀⢡⡁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠅⠄⠀⠥⠤⠄⠀⠀⠃⠀⠀⢀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠤⣤⣄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⠀⠀⠄⠒⠀⠐⠀⠀⠚⠀⠠⡊
⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠥⡀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠛⠉⠙⠓⠦⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⡠⠛⠂⠀⠉

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

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⣴⡟⠃
⠀⠀⠀⠀⠀⢠⡶⣦⣴⠶⣶⠟⣷⣶⣤⡆⣴⠀⠀⠀⠀⠀⠀⠀⣤⣸⣾⣧⣘⣿⡄⠀⠀⠀⠀⠀⠀⣀⣠⢶⡟⣿⣷⡞⢩⣍⣿
⠀⠀⠀⠀⠀⠘⣿⣹⡿⠶⠟⠛⠛⠋⢹⣧⡟⠀⠀⠀⠀⣶⡆⣿⣾⡯⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠞⠙⠋⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠠⠤⠸⠀⠀⠀⠀⠈⠸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢠⣤⡈⢛⠆⢠⣄
⠀⠀⠀⠠⡄⣄⡤⠔⣞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠈⠙⡟⡿⡄⠉
⠀⠀⢈⣷⣹⣿⣿⠥⠹⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣄⢤⡞⢁
⠀⠀⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣶⡀⡦⠶⠯⠺⠳⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢰⣴⠄⠦⠺⣴⢉
⠀⠀⠀⠀⠀⠀⠀⣶⢰⡜⣿⡏⢹⣿⢃⡾⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣷⠿⠻⠾⠻⢿⡾⠋⠀⠀⠀⠀⠀⣀⢀⣠⠄⣶⠃⢿⡉⠹⢶⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⣹⡎⢷⣆⣹⡧⠞⠛⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⠀⢠⡆
⠀⠀⠀⣀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠻⡆⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡞⢻⣾⢉⡷⣿⣀⣸⡇
⠀⠀⠰⠿⣾⣾⠻⣧⡾⢿⣰⣆⡶⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡿⢿⠀⠀⠹⣤⠎⠳⠋⠈⠛⡁⠉⠉
⠀⠀⠀⠀⠉⠙⠿⠛⣧⣼⡿⣼⠟⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢶⣷⡞⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣉⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠆⠠⠳⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⢶⡶⠆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⡶⠂⣴⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣼⣏⡙⠛⣻⢾
⠀⢸⣧⣼⡏⣰⢂⣶⣸⡇⣸⠃⣸⢣⠞⠃⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣼⢿⣿⣷⡼⠃⠀⠀⠀⠀⣄⣈⣿⠛⣧⠘⣿⠀⢸⡁⠸
⠀⠸⠟⠿⢷⡟⣻⡷⠏⡇⠛⠀⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉⠈⠉⣿⠀⠀⠀⠀⠀⠀⠹⣿⢿⡆⣿⠄⠛⠀⠘⠂
⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⢀
⠀⠀⢠⡀⠀⠀⠀⣀⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠢⢄
⠀⠀⠈⡁⠔⠁⠉⠀⠀⠈⠒⢂⠐⠁⠀⠀⠀⠀⠀⠀⠠⠔⠔⠃⠀⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⢀⠀⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡠⠃⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⠑⠠⡠⠐⣆⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠄⠔⠁⠀⠀⠀⠑⠦⣀⣀⣀⣀⠀⠧⡄
⠀⠒⠏⠓⠒⢄⡤⣀⠤⠤⠏⠉⠈⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⡶⠖⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀
⠀⠀⢀⠒⠂⠒⠀⠀⠒⠒⠒⢂⠀⠀⠀⠀⠀⠀⠟⠀⠰⠀⡠⠚⠀⠀⠀⡁⠀⢀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠒⠒
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠠⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠖⠀⡜⠁⠀⠀⠀⠀⠣⡀⠀⠀⣨⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠐⢠⠃⠀⠀⠀⠀⠀⢀⡇⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣄⠣⢄⠀⠀⠀⠀⣠⠃⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⠀⡤⢤⡀⠀⣠⣤⣄⠈⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠉⢚⠉⠉⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄⠂⠀⠘⠀⠀⠀⡌⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⣀⠀⡀⠀⢠⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄
⠀⠀⠀⢉⡀⠉⠒⠃⠘⠒⠊⡇⢓⠃⠀⠀⠀⠀⠑⠐⠀⡤⠤⠌⠢⠐⠣⠤⠄⠀⠀⠀⠀⠀⠀⠀⡄⣀⠖⠤⠔⠂⠒⢃⣀⡀⡰⡂
⠀⠀⣤⠨⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⡀⠀⠀⠘⠋⡨⠉⠀⠀⠀⠀⠀⠀⠀⠈⢹⠂⠀⠀⠀⠠⠤⠃⠀⠀⠀⠀⠀⠀⠀⠐⠢⡤⠁
⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠂⠀⠀⢤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠄
⠀⣰⢆⣜⣀⠀⠀⠀⠀⠀⠀⣀⣩⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠀⠀⠼⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯
⠀⠀⠀⠀⠰⠖⡆⡠⣄⡰⠦⣥⠀⠀⠀⠀⠀⠀⠈⠒⣆⣀⡀⠀⠀⢀⣀⢰⠶⡚⠐⠄⠀⠀⠀⠀⢸⠶⡄⣀⠀⢀⡀⣀⣀⠸
⠀⠀⠀⠀⠐⠅⠋⠀⠈⠁⠆⠲⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⠋⠙⠊⠈⠉⠈⠃⠀⠀⠀⠀⠀⠈⠽⠁⠉⠈⠳⠋⠉⠁⠀⠉⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡄⠀⠀⣼⣿⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣿⣧⠀⡀⣾⣷⣿⣼⣿⣿⡀⢀⣴⣿⣇⡾⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣀⢠⣻⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⡿⠧⠶⠦⣠⣤⡞⠒⠒⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣿⠿⠟⠛⠉⠈⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣀⣄⠀⠀⠀⢸⡄⠀⠀⣼⠗
⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⣿⡿⠀⠀⠀⢸⠂⠀⢰⠃
⠀⠀⠀⠀⠀⢰⣯⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⡇⠀⠀⠀⣿⡀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡏
⠀⠀⠀⠀⠀⠀⣿⣆⣀⣠⣴⡄⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠿⠇⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢀⣠⡞⠀⠀⣼⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⢻⣿⡀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣿⣿⣿⣿⣥⣀⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣇⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⢷⠀⠀⠀⣿⣿⣿⠏⠉⠉⢹⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⡄⠀⠀⠀⢹⣿⠀⠀⠀⠀⣾⣿⣧⣀⣠⣤⣾⣀⣠⣴⣿⣿⣯⣀⣀⣀⣿⡛⠿⢿⣿⣦⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠩⠽⢻⣿⣷⠀⠀⠀⣀⣿⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡁⠀⠀⠈⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣦⠈⠉⠙⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡾⠿⠛⠋⠉⢀⣿⣿⢿⣿⣿⣿⡿⡟⠿⣿⣿⣯⠻⣿⠙⠻⣧⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠘⠇⠹⣿⡇⠀⠀⠈⠻⢿⣧⠈⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⣀
⠀⠀⠀⠈⠑⠈⢠⣄⡀⡰⢄⣀⠴⡂⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡀⠸⠀⠀⠀⠀⠀⡸⠈⢩⠀⠀⠀⠀⠐⣠⠴⠒⠒⠓⠒⢦⡀⡀
⠀⠀⠀⠀⡏⠑⠋⠀⠉⠀⠀⠀⠀⠙⠂⡇⠐⠥⠒⠀⠀⠀⠀⠀⣀⠀⠀⣀⢈⡡⠘⠃⠠⢄⡠⠤⢄⠀⠀⠙⠘⠀⠀⠀⠜⡜⠁⠀⣴⣶⡶⡄⠀⠘⢎⢚⡄
⠀⠀⠘⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠃⠀⠀⠀⠀⠀⠀⠉⡑⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⣾⡷⠋⠀⠀⢸⠂
⠀⠠⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⠄⠀⠀⠀⣀⡀⠨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⡀⠀⠘⢳⣤⣴⡛⠀⠀⡸
⠀⠀⢀⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⡔⠁⣁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣂⠲⢄⠀⠀⠀⠙⢄⡀⠈⠁⠈⠁⠀⡔⠁
⠀⠀⠀⠀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠁⠀⠀⠈⠦⡀⠘⠉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⣒⢀⡨⠀⠀⠀⠀⠀⠌⠐⠒⠐⢂⠀⠘⠚
⠀⠀⠀⢠⢤⡤⠸⠔⠲⡄⡠⠒⠢⠊⠒⠤⣡⠀⠀⠀⠀⠀⠈⠉⠀⠀⠲⠤⠤⠄⡀⡠⣄⣠⢄⡤⠞⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠂⡀
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡯⣵⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠹⢠⠃⠀⠀⠀⢠⢤⠀⠀⡋⠀⠉⢹⠀⠀⠘⠄⠁
⠀⠀⠀⠀⡀⣀⡀⣀⣀⠠⣄⡂⢁⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠘⠂⠲⢂⠁⠀⠀⠀⠀⠔⠀⠀⠖⠂⠒⡄
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡄⠀⠀⠻⠇⢀⡶⠤⠞⠑⠒⠉⢦⠀⣤⠀⡴⠀⠀⠀⠀⠀⡄⠪⢀⠀⠀⠈⠠⠖⠒⠒⢄⠀⠀⠀⠀⠄⠀⢂
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢠⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠉⠀⠀⠀⠀⠈⠱⠶⠀⢁⠎⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⢉⡁
⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠰⠉⠀⠀⣀⢀⣀⣤⣴⣠⣶⣼⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠑⠊
⠀⠀⠀⠄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⢜⠀⠸⣧⣿⣿⣿⣄⡿⢿⡏⠿⠀⢰⠋⠀⠀⠀⠀⢄⣁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠠⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠙⠃⠘⠁⠉⠁⠀⠀⠀⢀⠜⠂⠀⠀⠀⠀⠀⠀⠸⡵⠀⠁⠠⣄⡀⢀⡀⠠⠴⠁⠺⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣄⡝⣆⢀⡀⠀⠀⠀⢀⡀⢠⠖⠺⠀⠯⠀⠀⠀⠀⠀⢰⠒⠀⢠⠀⠀⠀⢉⠁⠀⠀⠀⠡⢀⡀⢄
⠀⠀⠀⠠⣁⠠⢀⠠⠄⣀⠤⢀⣀⢤⣸⠆⠀⠀⠀⠉⠀⠀⠀⠙⢊⡉⠲⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡠⠼⠀⠀⠨⠀⠀⠀⠀⡠⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡧⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠃⠐⠀⠀⠀⠑⠔⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢠⠖⠀⠀⢀⣠⢤
⠀⠀⠀⠀⠰⢾⠀⡦⢀⠀⠞⢆⠀⢀⢴⢀⡄⣀⠀⠀⠀⠀⠤⣀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡽⠇⠀⢀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⠄⠇⠀⠀⠀⠀⠀⠁⠈⠈⢨⡀⠀⠀⠀⠈⠆⠒⠀⢤⢀⣠⠘⠤⠴⠚⣄⡠⢤⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠉⠈⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⢤⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢤⡄⠀⠰⢖⠀⡆⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠔⡶⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠀⠀⠀⢤⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢠⣌⠑⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠁⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠚⠁⠾⠐⢒⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠆⠀⠀⠈⠁⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠓⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠘⡉⢁⣠⠂⢢⡸⠈⠲⠇⠀⠀⠀⠀⠀⠐⠓⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠀⢼⡹⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠚⡅⠘⠁⠀⠀⠁⠰⠝⠖⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠙⡤⠊⠲⡔⠉⠁⠃⠀⢞⠆⠀⠀⠀⠀⠀⠀⠈⠠⠖⠲⠆⠀⡀⠠⠖⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⠀⢰⠀⠀⠀⡰⠁⢠⠁⠀⡠⠊⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢆⠸⣆⠈⣇⠀⡰⠁⠀⡀⡠⠊⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠉⠻⢦⣀⠀⠀⠈⢢⡅⠢⠃⠉⠀⠀⠀⡈⠤⠃⠠⡀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠠⢀⡀⠉⠫⢕⠂⠤⠙⠏⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡀⠀⠈⡀⢠⣀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠄⠠⢍⡐⠂⠃⠀⠀⠀⠀⠀⠔⢂⠔⠁⠀⠀⠀⠐⡔⠁⡈⠀⠀⠃⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠠⠀⣀⠉⠁⡠⠀⠉⠀⠈⠐⠎⠀⢀⣜⠀⠠⠂⠰⠀⠀⠉⠈⠀⠀⣔⠄⠂⠀⠀⠀⢉⣀⡀⠀⠤⠄
⠀⠀⠀⠀⣀⠀⠤⠒⠁⠄⠀⠀⡠⢠⠀⠀⢰⠰⠀⠁⢡⠀⠐⠁⠀⠠⠁⠠⠐⠀⠀⠠⢂⠉⠁
⠈⠉⠉⢁⡐⠲⠀⠀⡈⠀⠀⠀⣀⠘⠀⠀⡄⠀⠀⠔⠁⠀⠀⢀⠔⠁⠀⡠⠝⢤⣀⠈⠉
⠀⠀⠉⠉⣀⠬⢂⣀⠈⠢⢀⠀⠀⠀⠀⢎⠐⡀⢀⠔⠀⠀⠈⠀⠀⠀⠠⠌⣦⡀⠀⠉⠂
⠀⠀⠈⠅⠂⠈⢀⠔⠀⠀⠀⠑⠄⠀⠀⠔⠁⠘⠀⠀⢠⠒⠎⠿⠱⡀⠰⠀⠀⠙⠢
⠀⠀⠀⠀⠠⠔⠁⠀⢀⣀⠀⠀⠀⢈⠊⡠⠒⢢⠒⠀⡌⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⡀⠔⡪⢊⠕⠁⠀⠀⢀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠈⠀⠨⠖⠁⠀⠀⠀⢠⠎⠀⠀⠀⠀⢀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⠀⠈⠶⠟⠀⠀⠑⢤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⡀⢀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠂⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢨⠣⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠰⠘⠀⠀⠀⠀⠀⠆⡇⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⡤⢸⠀⠘⠀⣇⡀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣆⢀⣀⡴⠃⠀⠀⠀⠀⢨⡇⠀⠀⠠⡀⠈⠼⠀⠻⠄⠠⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣀⠀⠀⢀⣤⠒⣠⠏⠊⠅⠀⠉⠀⠀⠀⠀⠀⠀⠀⡜⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡠⠒⠢⠀⠀⡔⠈
⠀⠀⠀⠀⠀⠀⠀⣶⠀⣇⡀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠠⠀⢴⣴⠇⠀⣿⠟⣷⣴⡟⠁⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉
⠀⠀⠠⢌⣦⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⣿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡄⠚⠛⠃⠀⠀⠀⢠⠀⢀⠀⠀⠀⠉⣩⡔⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠐⢶⠀⠇⠠⠈⠀⠀⠀⠀⠀⢐⠒⠘⠙⣆⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⡀
⠀⠀⠠⠶⠓⠓⠂⠒⠈⠐⠂⠐⠂⠤⠀⣀⣀⠈⢀⠀⠀⠀⠸⣿⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⣾⠕⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆
⠀⠀⠠⠤⠶⣤⠄⠀⠀⠀⠀⠀⠀⠀⣇⠸⣟⠋⠁⠀⠀⠀⠀⠈⠙⢳⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⡴⠃⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠈⡙⢀⣴⣿⣀⡴⠟⢆⠀⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠒⠉⠈⠀⠁⠀⠀⠈⢢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⡰⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠆⠠⠂⠀⠐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠲⢄⣀⡠⠋⠀⠀⠹⣄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡀⢀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠑⡄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣀⣄⣀⠀⢀⣠⠷⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠶⠦⡴⠂⠀⠈⠲⣆⣀⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣲⠚⠉⠁⠉⠻⠛⠛⠉⠀⠀⠀⠀⣬⠶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⣐⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠀⡀⠀⠀⢠⢠⠀⠄⡀⠀⠀⠀⠀⠈⠉⠲⡀
⠀⠀⠀⠀⠀⢈⣂⡄⠀⠀⠀⠂⠁⠐⠀⠀⠀⠀⠀⠀⠀⠙⠟⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠼⠈⠘⠐⡁⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠚⠙⡁⠀⠀⠀⠀⠀⠁⠁⠀⠰⠀⠀⠀⠀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠄⠀⠀⠀⠁⠀⠀⠀⠀⠒⠂⠐⠀⠀⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠘⠋⠙⠓⣦⠀⠀⣀⣤⣤⣀⣀⡀⠀⢀⣤⠼⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡷⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠗⠙⠛⠋⠀⠈⠉⠛⠳⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⠳⠶⠤⠴⣄⠀⢠⣼⠦⠤⡤⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡌⠢⠤⠖⣇⠀⠀⠀⡀⢠⢄⣁⡟⢠⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠛⢛⡇⠀⠀⡜⠁⢀⣤⠀⠈⠓⠶⣺⢃⣏⠎⡼⢠⠃⡜
⠀⠀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢤⢰⣿⣿⣥⡄⢰⠁⣷⡈⠘⢠⠿⡄⠀⠳⡾⣏⣸⢁⠈⡰⠀⠀⠀⠀⠀⠀⠀⡠⠂⠤
⢠⣄⠀⣈⠒⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⢲⠂⣰⢣⠃⡄⢿⠃⣷⠋⣠⠇⠀⠀⠀⠈⠁⠘⠦⠃⠀⠀⠀⠀⠀⠀⠐⠳⠦⠄
⢷⢃⣄⠀⢈⢧⠤⠲⢀⢤⣀⡎⢉⡑⢦⠀⢀⡇⣧⠂⠘⢻⠇⠰⠇⣿⠀⠁⡜⠁⠀⠀⠀⢀⣀⣤⡤⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⡆
⠘⠱⣣⡀⠧⢞⠀⠀⠃⠘⠋⢀⠤⢤⡼⠀⠈⠛⠋⣄⣤⡀⠀⡴⠆⠛⢀⠀⢳⡀⠀⠀⡔⢉⣔⠁⢀⣠⠎⣱⡗⡲⠀⠀⠀⠀⠀⠉⠚
⠀⠀⠀⠉⠉⡏⡜⠦⠛⢦⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡼⠀⣇⣤⡜⢣⣴⠁⠀⠀⠑⠦⠭⠿⠿⠡⢤⠈⠘⠓⠒⢄⢠⠴⠶⠦⣄
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢒⠒⢦⠀⠀⠀⠀⠉⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⢠⣃⠀⠰⣭⠇⠸⠃⠮⠽⢃⡽
⠀⠀⠀⠀⢀⡤⡆⠀⡴⢦⡄⠀⠀⠀⢀⠎⣰⣹⣣⢯⢤⢖⢆⣀⠀⢀⠄⠀⣀⣀⣀⠤⢄⡀⠀⠀⠐⠋⠈⠢⠤⠤⣆⡴⠓⠚⠋⠉
⢀⠴⠚⡆⢸⠀⢃⠞⠀⠜⠀⠀⠀⠀⢂⠀⣥⢠⠋⠞⢱⣣⢆⡮⠀⣬⠤⠊⣼⠋⠀⣢⠀⣿⠀⡤⠀⠀⠀⠀⠀⠘⠋⠀⠀⣀⡀
⢁⣤⣞⣥⣸⠀⠋⠀⣼⣀⡀⠀⡐⢠⠏⣼⠃⠛⣪⠀⠈⡱⢋⡄⣀⡥⠀⡸⠃⠘⣫⢵⡞⡱⢊⡴⢊⡑⡦⠀⠀⠀⠀⢀⢞⠥⠤⣮⠔
⠸⢃⡔⣦⠈⠀⢠⠤⢀⣸⠇⠀⠈⠉⠚⠉⠓⠚⠑⠒⠛⠒⠉⠉⡔⠁⠄⢠⠶⠀⠀⢉⢚⡙⠋⡠⠭⣯⠖⡢⢄⡀⠀⢧⣘⠖⠚⢂⢤
⠀⡀⠉⣁⣴⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣌⡐⠦⠤⠤⠤⠊⠉⠉⠲⣽⠞⢁⣌⡻⠋⠀⠀⢀⡈⢳⠀⠈⡾
⠴⠋⠉⠁⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣡⠤⣦⠈⣧⡤⢀⣀⡀⠀⠀⠀⠀⠙⢦⣩⠆⠀⠀⠀⠐⣇⠙⠋⠀⣸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠎⠀⠀⠛⢺⠟⢀⢤⡈⡇⠀⣠⣄⡀⢀⡤⣤⠀⠀⡀⠀⠀⠈⠳⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣀⣀⣤⡴⠋⣠⣫⠞⣡⣧⠞⡁⢠⢇⠎⣠⣯⠴⠋⣹⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⣠⠄⠀⢉⡴⢚⠟⠁⠼⠇⣼⠏⢀⠉⢤⠞⠋⠁
⣸⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⡠⢦⡴⢺⠀⠀⠀⠈⠛⢚⡟⢡⡞⠡⡀⠳⠁⣰⠒⣦⠀⠏⢠⠎⢦⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⡕⠋⠉⠉⢹⠆⣶⡠⢦⠔⢡⣞⡇⠸⡾⢸⡄⠀⠀⠀⠀⠘⠶⠏⠀⠀⠱⡜⣶⠇⠀⣇⣰⣴⠋⠀⠈⢧⣀⣹⠄⠀⠀⠀⠀⡰⠋⣠⢖
⢇⢨⠝⠀⣠⠀⡇⢱⠈⡦⢤⡀⠉⢀⣀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠁⠀⢀⡀⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸⡀⠘⢁
⢸⡀⡀⠺⡘⣄⠑⠚⠀⠷⣤⡼⠀⠈⡏⣀⠇⠀⠀⣀⠤⡄⢂⡩⢆⢌⠥⢼⠖⡩⠤⡍⡳⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢸
⡈⡇⢿⠲⡙⣮⠓⠶⠖⠦⣀⣠⠞⠙⠃⠀⠀⢐⠁⠤⠪⠘⡌⠳⡼⡾⡀⢸⠸⣄⠀⢡⠞⡥⠠⣄⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈
⠀⠓⠞⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡛⢹⣠⠇⢸⡅⡇⢹⠈⢷⠈⢳⣰⡀⣇⠀⠄⡶⢠⠤⢤⣀⣀⣀⣀⡀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢠⣀⣀⡴⠈⠋⠀⠻⣠⠼⠁⣨⣇⠳⡌⢿⢠⠃⡎⣀⠈⠉⠉⠀⣀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣖⣂⣀⣀⣀⣀⣀⣺⣴⣃⣈⣸⣼⣰⣛⣛⣀⣀⣉⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡴⠂⠀⣀⣀⣀⣀⠠⠄⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣿⢏⠟⣡⣴⡾⠛⠋⠉⠁⠒⠒⠛⠛⠋⠁⠀⠀⠀⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⢠⣾⡏⢼⣿⣧⢃⣾⣿⡷⠋⣠⣴⣿⣻⣿⣿⣿⣿⣿⡿⢛⣁⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣦⠀⣤⠀⣿⡿⠇⠀⢿⡇⣾⠏⢈⣴⣿⣿⣿⠿⢟⣩⣭⣭⣭⡟⠿⠛⠛⠻⠟⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣧⠀⣾⡀⢸⣇⣿⡄⣿⠃⠀⠀⠘⣿⣿⠄⢿⣿⡿⢫⣶⠟⣫⠴⣛⣭⣷⣿⣿⣿⠿⠋⡀⠀⢼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣍⢻⣿⣞⣼⠀⢤⠀⡀⠹⣿⡀⠸⡿⣐⣞⢁⣾⣵⠞⠋⣱⣿⠟⠋⣀⣤⣾⣇⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠘⢻⣿⣿⡿⠿⠿⠷⠿⢿⣿⣵⣄⠀⣙⣶⣽⢿⣆⣇⣿⣿⣾⣿⣠⣠⣾⢟⣡⠶⠟⣛⠛⠛⠿⢿⣗⠒⠊
⠀⠀⠀⠀⠻⢿⣿⣷⡂⢨⡿⢑⣽⣶⠶⠒⠂⠤⢀⠉⠛⢷⣿⣿⠿⠏⢿⣿⣿⠿⠻⠿⢿⣿⡿⠋⣀⡼⠿⠶⠶⢶⣦⡀⠹⣶⠀⣴⡶⠒⠁
⠀⠀⠀⠀⠀⠀⠈⠉⢁⣰⡇⠸⣿⣛⢿⣒⢠⣤⣤⣽⣾⡷⠈⠁⠀⠀⠀⡁⢉⠀⠀⠀⠈⠙⢿⣟⣡⣤⡤⠌⣛⣛⣩⡇⠀⣏⡀⠙⠁
⠀⠀⠀⠀⠀⠀⡀⠐⠋⠛⣿⣄⠈⡿⠋⠁⣀⣤⣭⣿⠏⠀⠀⠀⠀⠀⢀⣻⣇⡀⠀⠀⠀⠀⠈⢻⣿⣶⣴⣀⠈⠻⣿⣷⣼⣿⡟⠓
⠀⠀⠀⠀⠀⢉⣉⣤⡶⠿⣟⠛⠛⠀⠀⣈⠝⣿⠟⠁⠀⠀⢀⡀⠀⠀⠈⢻⠋⠀⠀⠀⢀⠀⡀⠀⠿⣿⣿⣭⣀⠀⠈⢿⣿⣿⠿⢶⣄⡀
⠀⠀⠀⠀⠈⢛⣩⠿⢿⡟⠉⠀⣰⠿⠛⠁⢸⠋⠀⠠⣶⡶⢾⣧⡀⠀⠲⢺⠃⠗⠀⣰⣿⡷⣶⡶⡦⠀⢸⣿⡍⠛⠦⠀⠬⢙⣛⠲⢦⣉
⠀⠀⠀⠲⠶⠛⠁⠐⠁⠀⠂⠩⠵⠚⡀⢴⣏⠀⠀⠀⠈⠳⣴⣿⠷⠀⡀⢸⠀⠂⠀⠹⠿⣦⣟⠉⠀⢀⣼⣯⣭⣁⠀⠀⢈⡭⠍⠛⠂⢙⢷⣶⠇
⠀⠀⠀⠀⢠⡔⢂⣤⣶⠶⠶⠖⠲⠏⠀⣾⣯⠀⠀⠀⢀⡰⢫⣄⡒⠔⠬⠴⠤⠀⡰⠖⣶⠋⢀⣀⣤⣼⣿⡟⠁⠈⠻⣶⣶⣶⣶⣶⣦⡈⣁
⠀⠀⠀⣆⠉⢴⡟⣿⠾⢛⠫⠉⡀⠀⣠⣿⣿⣶⣿⣿⡿⠀⢰⡟⠷⣶⣦⣤⣤⣶⣴⡿⣿⡆⠸⣿⣿⡿⠷⠾⣷⣄⠠⡀⢈⡉⠛⠿⣿⣿⡙
⠀⠀⠀⠀⠀⢻⡞⠁⠀⢀⣴⠋⡾⠿⣛⣻⣿⣿⣿⣿⠁⠠⢘⣿⣦⣴⣦⣉⣡⣶⣨⣴⠟⠉⣓⢸⣷⡾⠿⠛⠙⠛⠇⠈⠻⣦⡀⠐⡌⢿⡇
⠀⠀⠀⠈⠛⠛⠀⣴⣸⣾⠇⢀⠴⢉⣠⣤⠖⢛⣻⡇⠀⠀⠀⠀⠉⠙⠻⣿⡿⠟⠋⠁⠀⠀⠀⢠⡿⣛⣉⠲⣤⡀⠀⠀⠀⢹⣿⣷⣽⠈⠛⠂
⠀⠀⠀⢀⣴⣶⣾⣿⣿⡏⠀⡎⣠⡿⠋⢀⠶⣻⣽⣿⣦⣀⡀⠀⢀⣀⣤⣿⣧⣄⣀⠀⢉⣀⣴⣿⣿⣯⣟⠀⠈⢻⣆⠸⡆⠀⢿⣿⡿⣀⠲⡆
⠀⠀⠀⠘⢁⣿⣏⣿⣿⡀⠀⠹⡟⠀⠀⣠⣾⣿⣿⣿⣿⣿⠛⢻⡟⣋⣥⣴⣴⣨⠉⣿⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⢿⡠⠁⠀⣸⣿⡇⠙
⠀⠀⠀⠀⠀⠀⡋⢸⣿⣧⡀⣘⠀⠀⣴⡿⣫⠌⣾⣿⣿⣿⡄⢸⣷⠿⣿⣿⣿⣿⢷⣿⠀⢸⣿⣿⣿⣍⠫⠙⢧⠀⠀⢠⡾⣀⣿⣿⡿⣦⣀
⠀⠀⠀⠀⠠⢞⡡⣿⢿⣿⣿⣋⡄⣰⠏⠘⢁⡴⠛⣻⣿⣿⣷⣼⣿⡀⠀⢻⡏⠁⣸⣿⣠⣿⣿⣿⣈⠳⢍⠳⡀⢧⠀⢀⣘⣿⣿⣿⣧⣌⠁
⠀⠀⠀⠀⠀⢀⣼⣿⠀⠸⣿⣟⡽⡿⠀⢀⢪⡴⠛⢉⣽⣿⣿⡏⢻⣿⣤⣤⣥⣤⣿⠏⣿⣿⣿⣯⡉⠉⢢⡑⡁⠀⢷⡝⣿⡿⠋⣹⣿⡟⠁
⠀⠀⠀⠀⠀⠚⠉⣹⢂⡀⢹⣟⡇⡇⠀⢸⡀⢀⢠⠟⡿⢻⡟⠻⠤⡯⠽⠽⡼⠥⢿⡴⠟⣿⣿⡿⣿⡄⠀⣧⢀⠀⡞⣽⠛⠁⠀⠋⠻⡇
⠀⠀⠀⠀⠀⠀⣴⢡⣿⠇⣼⣿⣇⢳⡀⢸⣿⣿⠃⠀⠔⣿⣿⠗⠀⠀⠚⠻⠛⠓⠂⠀⠻⣿⣿⣦⠈⢿⣧⣿⡞⣠⢻⣿⣇⠀⣰⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⣇⠙⣿⣆⠹⣿⣿⡟⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠈⣧⠘⣿⣿⣿⣿⣿⡟⢹⠆⠉⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣦⣠⣻⣇⢴⡄⢰⣿⣼⢷⣶⣠⣤⡀⣤⣦⣴⣶⣿⠛⢇⠀⣿⢀⣹⡿⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠹⣿⣿⡆⣷⠈⣿⢻⡆⠙⡟⠍⢿⠸⣯⠙⠙⣧⠀⢾⣴⣧⣿⣿⣿⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⢿⣷⢹⣷⡈⠣⡙⢦⣽⣆⢈⠇⢻⡆⠁⣼⠀⣿⣿⡿⠛⢨⠃⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢹⣿⣄⢤⡉⠻⢿⣧⣾⣸⣿⣸⠏⡼⠿⠛⠁⣠⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⢷⡅⠀⣠⢹⣿⣫⣿⠏⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠹⡄⡏⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠿⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣿⣉⣭⣭⣥⣭⣩⣯⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣽⣿⣿⣭⣽⣿⣭⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡶⠂⠺⢟⡒⢛⡂⠶⢄⠲⣌⠳⠤⢄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠶⣿⣽⣾⣶⡿⠀⠉⠁⢀⣀⣬⣤⣶⣼⣧⣤⠀⠀⠀⡀⢀⡘⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⢿⣿⣷⣄⠉⠉⣀⡴⠶⣙⡋⠻⠿⣷⣈⣿⡿⠿⠃⠐⢲⣆⣚⣒⠉⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⠹⠒⠙⠻⣿⣧⢻⣭⠗⣤⡍⠁⣠⣤⠀⢄⢀⠀⠀⠀⠐⣿⣧⣧⡭⠍⢓⠄⠆
⠀⠀⠀⠀⠀⠀⣀⡰⢾⢧⣽⢿⣦⡀⠀⠁⢸⠿⣀⠴⠯⢺⡥⣴⣇⡰⠒⠈⣠⠈⠀⣀⣤⣅⣹⣿⠡⢅⣀⡥
⠀⢀⣀⠀⢲⡒⢹⣿⠀⠀⣰⡾⢿⣿⡷⠦⣤⣴⠟⣠⠶⣿⠀⠘⢛⣡⠔⠃⢀⣔⣾⣿⠟⠛⠉⠋⠻⠛⢻
⣏⡉⣹⡇⠈⠛⢈⡥⠶⣈⠿⠶⢛⣉⡁⠈⢻⣴⣶⡍⠛⣿⡿⣁⣍⠁⠚⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠉⢁⣉⣀⡀⢸⡟⣃⣴⡏⢠⣶⠻⣿⡍⠠⡿⢳⢟⣵⣾⡄⠰⢿⢷⡿⡠⣿⣿⣿⣿⣶⣶⣤⡄
⢰⣿⣈⢛⣿⠀⣀⣩⣉⠀⢸⣷⣦⡽⠡⠤⢤⠀⣿⢷⡟⢁⣰⣿⠃⣴⡇⢹⣿⣟⣿⣿⣶⣌⠹
⠀⠈⠉⠈⠁⣾⡉⣉⣙⠓⡠⣤⣤⡀⠛⠛⠀⠐⣄⠰⣾⣿⠟⢉⣤⡟⣱⡆⢻⣿⣿⠙⣿⢿⣄⣠⣠
⠀⣤⠚⠛⣧⠈⠉⠉⠑⣿⣃⣴⣿⠇⢴⣶⣦⠠⢿⣿⣿⣿⣤⣿⣿⣷⣿⣇⣶⡝⢿⣷⣦⣾⣯⣽⣡⣶⡄
⠀⠳⢶⡶⠁⣤⠟⢛⣧⠀⠈⢉⣑⠙⠿⠛⠁⠈⠸⠿⠫⠟⠋⠉⠉⠉⠉⠙⠛⢿⣷⣝⣛⣛⣛⣛⣉⠉⠋
⠀⠀⠀⠀⠀⠿⠦⠾⠋⠀⣼⣋⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⢠⡔⠿⢷⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠷⠾⠟

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣦⣤⣭⣭⣙⡛⠿⢿⣶⣄⡀
⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣛⣛⣛⣛⡻⠿⠿⣿⣟⡿⢿⣷⣦⡈⣙⢿⣿⣦⣄
⠀⠀⠀⠀⠙⢷⣤⣀⡐⠲⠤⢤⣬⣍⣉⣙⣛⣻⣿⣿⡿⠿⣟⣛⣛⣛⡛⠛⠛⠛⠲⠮⠝⠳⢮⣝⠻⣮⢷⣿⣿⣿⡳⣄
⠀⠀⠀⠀⠀⢄⠙⠻⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣟⣭⣵⣶⠾⠿⠛⠛⠓⣦⣤⣀⣀⡈⠻⣮⡙⣿⣿⣿⣷⡙⣷⣠⡀
⠀⠀⠀⠀⠀⠀⠻⣦⣄⡉⠛⠻⠿⢿⣿⣿⣿⣿⠿⠿⠿⠛⠋⣉⣥⡴⠶⠚⠛⠛⠛⢶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠠⣝⣿⣿⣷⣶⣶⣶⣶⣶⠶⢖⣢⣴⡾⠟⢉⣁⣤⣴⠶⠟⠛⠛⢛⠙⢙⡛⠛⠳⠦⠐⢮⣍⠛⠿⣿⣷⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣛⣛⣩⣥⣶⣿⣿⠟⢋⣠⣶⡿⣋⣥⣶⣿⣿⣿⣁⠘⢿⣦⡙⠿⣿⣷⣶⣦⣿⣷⣄⠈⠙⢿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⣿⣿⣿⣿⠟⢁⣴⣿⣿⣯⣴⣷⣶⣾⣿⣿⣿⣿⣷⣦⣉⠛⠻⢶⣶⡍⠉⠉⠉⠉⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣻⣿⡿⢋⣴⣿⣿⣿⡿⠛⠉⣩⣴⣿⠿⠛⢛⣻⣿⣿⠿⢷⣶⣿⠟⠀⠀⠀⠀⢤⣤⣤⣷⣶⢤⡽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢡⣾⣿⡿⢡⣿⣿⣿⡿⠉⠀⣴⣿⡿⠋⠀⣠⣾⡿⠛⠁⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⣿⣿⣿⣱⣿⣿⣿⠋⠀⢠⣾⣿⡿⠁⠀⣼⣿⣿⠞⠀⢠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠀⠀⠀⠲⢤⣤⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢻⣷⣿⣿⣿⠃⠀⢠⣿⣿⣿⠃⠀⢰⣿⣿⡟⠀⢠⡿⠁⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠹⣖⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣿⣿⣿⣿⠇⠀⢠⣿⣿⣿⡟⠀⠀⢸⣿⣿⡇⠀⢸⠇⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣻⡿⢰⠀⣼⡏⣿⣿⡇⠀⢀⠸⣿⣿⣷⠀⢸⣾⣿⣿⣿⣿⣿⣿⣴⡾⢡⣶⣿⠋⠉⠉⠩⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣇⡿⢀⣿⠀⣿⣿⡇⠀⣾⡆⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢿⣿⣷⠸⣿⢸⣿⣿⣿⠀⡿⣥⠘⢙⢛⡻⢛⡍⢻⡟⠉⠀⠀⠈⢿⡄⣿⣿⣿⣶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣿⠀⣿⣿⢿⣇⢹⣶⣶⡶⣷⡶⠲⣷⢷⡄⠀⠀⠀⠀⠀⠁⢙⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠹⠀⢿⣿⡈⣿⡄⣿⣾⣴⣤⣦⣄⠁⢽⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠘⣿⣇⠹⣿⣽⣿⡟⣿⣿⣿⣆⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠘⣿⡄⠹⣿⣿⣷⡈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢻⡄⠙⣿⣿⣿⣄⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠈⠻⣿⣿⢦⡀⠙⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡀⠀⠀⠀⠉⠛⠻⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⡀⠀⠀⠀⠀⠀⢩⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣮⡐⠤⡀⠀⠀⠆⠘⡀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⣒⣓⣂⣀⣉⣉⡁⠀⠀⠐⠛⠒⠶⠿⣷⣄⠐⢄⠀⠀⠰⠀⠀⢂⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠈⠁⠠⡉⠀⡀⠆⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠂⠀⠘⣦⡇⠀⣿⣯⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣉⡤⢖⣊⣥⣤⡴⠶⠒⠂⠀⠀⠀⣤⣈⡙⠶⣄⠀⠘⣿⠀⢿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⡿⠟⠋⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡙⠛⢷⣄⠈⠇⠈⠇⠹⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣤⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⣷⣦⠙⠂⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣼⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⢁⠔⠀⣀⣤⣴⡶⠞⠉⠀⠀⠈⢧⣙⠻⠿⣿⣧⠀⠀⠀⢀⡀⠠⣤⡘⣧
⠀⠀⠀⠀⡼⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡴⣡⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣽⣷⡄⣀⠀⠹⣿⣿⣷⣼⣄
⠀⠀⠀⠈⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣴⣿⣿⣿⣿⡟⣁⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⠿⣿⣶⣦⣌⣉⠛⠆⠈⠓⢦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⣿⣿⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣷⣿⣿⣿⣟⠷⣬⣭⣭⡭⠥⠒⠒⠲⢦⣜⠳⣦
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣿⣿⠋⢀⣴⠁⠀⠀⠀⠀⢰⠇⢀⡀⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣶⣾⣥⡀⢤⣻⣦⡈⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⠃⢀⣾⡏⠀⠀⠀⠀⠀⣾⠀⢸⠇⣿⣿⣿⣿⣿⢋⣵⣾⣻⣭⣝⣛⠿⣿⣮⣿⣿⠻⣾⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⡇⠀⠀⡀⠀⢰⣿⠀⣾⣶⣿⣿⣿⣿⣯⣾⡟⠉⠀⠓⠚⠛⠓⢦⣝⡻⢿⡟⠋
⠀⠀⠀⠀⠀⢿⣿⣿⠃⣾⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⡇⠀⢰⡇⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⢀⢁⣾⠟⠋
⠀⠀⠀⠀⠀⢸⣿⡏⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣧⠀⠈⣧⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⠘⣦⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠈⣿⠇⠀⢸⣿⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⡄⠀⣿⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⡇⢸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⣿⣿⣿⠸⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣷⡀⢹⣇⢸⣿⣿⣿⡝⣿⣿⣿⣿⣿⣿⣧⣿⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⢿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⡄⣿⣿⣬⣤⡈⣭⣤⣿⡜⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠈⢿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡌⢿⣼⣿⣿⣯⣦⢼⣷⣯⣦⠀⠹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⣿⣿⣮⣿⣿⣿⣿⣿⣆⠻⠿⣿⣷⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⣿⣮⡛⢿⣿⣿⣿⣿⣿⣿⣮⠻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣙⠻⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⠉⠛⠻⢿⣿⣿⣿⣦⣌⠙⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣦⣀⠉⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢄⣠⣤⡶⢀⣀⣀⣀⣠⣤⣤⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡿⢟⣿⣿⡿⣻⠟⣫⣿⣷⠶⢿⣿⢿⡶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢻⣿⣿⣿⡿⠋⣰⣿⣿⠏⠀⣰⠿⠟⠉⠀⠀⠉⠒⢛⣒⣲⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⢃⣿⣿⣿⣟⠁⢰⣿⠿⠧⠒⠚⠳⠶⠛⣓⣒⣀⡀⠀⠉⢉⡉⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢀⣾⣿⡿⠁⠀⢈⡁⠀⢤⣀⠀⠀⠀⠀⠀⠈⠉⣑⡌⢲⣄⠀⠉⠳⣮⣿⣦⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⡟⠁⠀⠀⢀⠈⡷⠀⠙⠳⢶⣎⣉⡰⠀⠔⠘⢿⣶⣿⣷⡄⢦⣈⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⠗⠀⠐⠵⡄⢀⡞⠛⠁⠀⠀⠀⠈⢹⣿⣃⡀⠀⠀⠠⣽⣿⣿⣷⠰⣽⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⠟⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⣷⠀⠀⠀⢨⣿⣿⡟⠀⠈⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡃⡀⣠⠤⠾⠟⠛⢛⡓⠒⢴⣄⡀⢲⡆⠀⠀⢹⣧⠀⠀⢠⣻⣿⠀⠐⢶⣦⣌⢻⣿⣎⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⡀⠟⡄⢹⣶⣶⣾⣾⣷⠈⣿⣿⣿⡇⠀⠀⠸⣿⣆⢀⢠⣿⣿⡀⠀⠙⢿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣿⣿⣿⣿⣧⣦⣷⣸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⠀⠀⠀⠸⠹⣎⣿⣿⣿⡻⡄⣄⠈⠻⣿⣯⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⢿⣿⣿⡟⠀⠑⠘⣿⣦⣹⣿⣧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⠟⠛⠋⠀⢹⣿⣿⠈⡇⠸⣆⢀⢠⣿⣿⣿⠇⠀⣤⣄⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡌⡟⣿⠀⠀⠀⠀⢸⣿⣿⠈⡇⠀⠙⡄⣿⠟⠛⠁⠀⠀⢹⣿⣷⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⠀⠀⠀⠀⣼⣿⣿⠀⡇⠀⠀⡇⠃⠀⠀⢻⣦⣷⡄⢿⣿⣿⣿⣏⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⢰⡏⠀⠀⢀⣰⡟⠟⢁⠀⠂⠀⠀⠃⡠⡄⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢇⠈⠿⠞⢛⣉⡉⠑⠚⠋⠀⠀⣀⠀⢠⣿⣿⣦⠀⢸⣿⣿⣿⣿⣿⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣧⣿⣿⣿⣧⣾⣿⣿⢿⣿⣿⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣿⣿⣷⣆⠲⣄⠠⡐⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡟⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣿⣿⡍⠛⠉⠙⣿⣿⣿⣿⣿⣿⣿⠏⣿⠏⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⣦⡰⣤⡈⣿⣯⣿⠟⢸⣿⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠚⠛⠉⠀⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⣀⢤⡔⢾⠛⠛⠆⢀⣤⡀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡙⠋⡞⢿⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠑⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡥⢰⡏⠀⠉⠃⠀⠀⠀⠀⠀⠐⠃⣀⣴⣠⣶⠄⡁⠊⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⡿⠠⠟⠁⠘⠧⣄⡐⢀⣠⡖⠀⠀⠋⠀⠀⠈⠉⠁⠈⠍⢹⣶⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠚⠃⠀⠀⠀⠷⠒⠉⠴⠛⠃⠀⠀⣰⠢⠞⣡⠖⣩⠀⠀⢠⣨⠉⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⡀⠀⠤⣶⠶⠾⣤⣦⣦⢰⡏⢠⠀⢐⣫⣤⣞⣳⣆⠀⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠾⠄⡀⠀⠈⠁⠀⠀⢀⣶⠀⠀⠀⠈⠘⣃⠀⠀⠸⢻⣿⣿⣿⠟⣷⣄⢹
⠀⠀⠀⠀⠀⡤⠒⢛⣿⣏⠀⠀⠀⠘⣇⠀⠀⢀⠀⠀⠙⢿⣧⠀⠃⢀⣼⠟⠁⢻⡀⢻⣿⡟⠻⡀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢧⡀⠈⠀⠻⠖⠿⠲⢶⣎⡦⠀⠘⠿⣦⣄⡀⠉⠓⢈⣹⡟⠀⠀⠻⠃⣼⣿⣇⠀⠈⠘⠿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣹⠦⣄⣀⣠⠴⡄⣀⡈⢳⠀⠀⠀⠈⠻⣿⣦⡀⠈⢿⣇⣀⢸⢀⡀⠈⢿⣇⠀⠀⠀⠀⠈⠛⢧
⠀⠀⠀⣀⡴⠊⡾⢸⠁⠀⠀⠀⢘⠟⠉⠶⣄⢤⡀⠀⠀⠀⠀⠀⡀⢈⡇⠈⠉⠉⠉⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢣
⣠⡴⢊⠟⠀⣸⠃⠁⠀⠀⠀⡆⢻⠀⠀⠀⠈⣯⠙⠦⡀⠀⠀⠀⢻⣿⡇⠀⢀⣼⡀⠀⠀⢻⣿⣶⣤⡀⠀⠀⠀⠀⢂⡁
⠏⢠⡟⠀⢠⠋⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠸⣆⠀⠈⢆⠀⠀⠈⠟⠛⠚⢉⣰⣿⠐⠠⠈⢻⣿⣿⣿⣧⡄⠆⡸⢻⢷
⢣⡿⠁⠀⢸⠀⠀⠀⠀⠀⡜⢠⠎⠀⠀⠀⠀⠠⣿⡀⠀⢸⡇⠀⠀⠀⣠⡤⠎⠁⠙⠛⢉⡏⠳⣿⣿⣿⣿⣷⣾⡿⠋⡸
⣾⢃⡆⠀⡏⠀⠀⠀⠀⣸⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⡶⢶⠗⠶⠖⠛⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⡏⢯⣈⣉⣉⣀⠜⠁
⡿⢸⠀⠀⢇⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⣼⡇⠈⢧⡉⠉⠁
⣧⠈⠁⠀⠀⠀⠀⠀⠀⢸⡜⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣷⡀
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢦⡄⠀⠀⠀⠀⠀⠀⢠⣤⣬⣷⡀⢀⡀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢰⡟⢣⡀
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⣰⠃⢸⡟⠀⠀⢠⡿⠁⠀⠙⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠄⠄⠀⢠⡀⡄⠲⢤⡟⠀⠛⠀⣢⠀⣠⠆⠈⠀⢨⢀⠈⠀⠀⠀⠀⠀⠀⡠⠞⠋⣓⣦⣀⠶⣄⠀⠀⡇
⠀⡀⠀⠠⢀⡄⠠⠌⣀⠸⠆⢀⠱⣦⠀⠶⣡⠄⢉⠔⠀⠀⠸⠈⠀⠐⠒⢍⡓⠋⠉⡉⠒⠊⠁⠉⠁⠀⠀⠀⠀⡇
⠀⠀⡀⢰⡆⠁⠀⠀⠉⠲⠆⠚⠑⠳⠀⠄⠀⠀⠀⠀⠀⠀⢂⠔⠀⣤⢴⡶⠼⠟⠁⣙⠲⠖⠐⠢⠤⠬⠥⣤⠞⡇
⠠⠾⡇⠈⠀⠀⠀⣾⣗⠂⠀⠀⠀⠀⢀⢰⡀⠨⠀⠀⠀⠀⠌⠀⡀⡀⠀⠀⠀⡂⠈⠀⠀⠤⣿⡦⠀⠀⠀⠐⠀⡇⠀⠀⠘
⠀⣀⠀⠰⠀⠀⡾⠀⡄⠀⣀⠀⠀⢀⣄⡀⠈⠐⠀⠀⠀⠀⠰⠈⠀⡀⠄⠔⠊⡀⠀⠀⠀⠀⠀⡆⠀⠐⠀⠉⠀⠇⠀⠀⡤
⠐⠓⠀⠀⢠⣼⣷⡟⠐⠛⢻⠀⠀⠙⠟⠁⠀⠀⠀⡀⠂⠀⣀⠀⣊⠔⠉⠀⠀⠁⠀⠀⠀⠀⡸⠁⠀⠀⡀⠀⠠⡄⠀⠀⠒
⠀⢢⠀⢀⢨⣛⣇⢀⠀⡄⣸⠀⠀⠀⠀⠠⠄⠆⢀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠠⣤⣼⡀⠀⠄⠀⢠⠄⠀⡄⠀⠀⠖
⠀⠀⠀⠁⠁⠀⠀⠀⠀⠼⠇⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⡃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⠀⠁⠀⠁
⠈⠈⠀⠀⠀⠐⠀⠀⣾⣿⡰⠇⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠛⣷⠇⠀⠀⡀⠀⠠
⠀⣴⠀⠄⠀⠣⠀⠀⠋⡀⠭⣅⢀⠀⠄⠀⠀⠀⠄⠀⠀⠀⡁⠀⠀⠀⣀⢠⣀⣀⣀⡘⢠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡀
⠉⠛⠀⠀⢤⡀⠀⠀⠀⠘⢷⡇⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⡃⠀⠀⡀⠉⠦⠉⠉⠉⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢇
⠀⣃⠀⠀⠆⠟⠀⠀⠀⡆⠉⠀⠀⡀⠀⡄⢀⠠⠇⠀⠀⡀⣇⢠⡄⣿⣶⣤⣬⣤⣤⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡤
⠀⠤⠀⠄⠀⠀⠠⠶⢆⡀⠀⠉⠉⠀⠀⠀⠁⠢⠀⠀⠀⠀⠗⢦⡀⠠⡄⠠⡀⠀⠠⠄
⠀⠸⠀⠀⠒⠠⠤⣀⡉⠉⠐⠀⢄⠀⠀⠀⠀⢀⠄⡀⠄⠀⡃⠸⣿⠶⠄⠀⠀⣀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⣀
⢐⣆⠈⠁⠠⣀⠀⠠⠤⠤⠀⡀⠂⠠⣀⠀⠀⢀⠎⠁⠀⠀⡏⠀⢳⡄⠀⠓⠙⠋⠁⠀⠀⠐⢚⣃⡀⠀⠀⣀⠀⠀⠀⢀⡄
⠀⠑⠀⠀⠀⠀⠉⠶⢄⠀⠀⡀⠀⠀⢀⠊⠢⠃⠀⠀⠀⠀⡇⠀⠈⢿⣷⣦⡤⠀⠴⠤⠤⠤⠈⠉⠁⠀⠀⠉⠉⣀⡀⢸⢿
⢴⣿⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠉⠉⠐⠓⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠻⡇⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿

⠀⠀⣀⣤⡶⠶⢶⣤⣄⣴⠾⠛⠛⠿⢶⣦⣄⣠⡶⠾⠛⠷⣶⡶⠾⠟⠛⠻⢶⣤⣤⣤⠶⢶⣤⡀
⠀⣼⠟⠁⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠻⣷⣄⣀
⣸⡏⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⣦⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠘⢿⣄⠀⠀⠀⢀⣤⣄⠀⠀⣀⣴⡾⢷⣄⣀⣀⣀⣠⣴⠾⠛⢷⣄⣀⠀⢀⣀⣠⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⣿⠷⠟⠋⠁⠉⠛⠛⠛⠋⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⠤⠀⠀⠀⠀⠀⠀⢺⣏
⠀⠀⠀⢸⣿⠀⢀⡴⠚⢶⡀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⢸⡏⠀⣼⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⢸⡇⠀⠅⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡿⠀⣤⣤⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢨⡿⠃⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⣾⡇⠀⠀⣿⡷⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠿⠿⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⠃⢰⡆⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⠒⠉⠉⠉⠚⢿⣆
⠀⠀⠀⣿⠀⠸⣧⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⣿⠣⣀⣀⣀⡠⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⠀⠀⠹
⠀⠀⠀⣿⠀⠀⠣⣈⣉⡿⠁⡇⠀⠀⠀⠀⠀⢻⡀⠀⠁⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠊⠉
⠀⠀⢸⣿⠀⠀⠀⠞⠋⢀⡜⠀⠀⠀⠀⠀⠀⠘⢿⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸
⠀⠀⢸⣿⠀⢀⣄⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠀⠀⠀⠀⠀⣼
⠀⠀⢸⣿⠀⠀⠙⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈⠧⣄⣀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⠿⠟⠛⠉
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⡄⠠⠤⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⢰⠣⣤⠤⠤⠤⢀⣀⣀⣀⣀⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠳⠧⠤⠤⠤⢤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠳⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠈⠻⠷⣶⣤⣤⣤⣤⣴⣶⠶⠾⠟⠛⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⣀⡀⠀⠀⠀⣀⣠⣾⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠖⠒⠒⠲⠶⠶⢶⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣽⣷⣿⢿⣿⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠑⢪⣷⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠻⣿⣿⣭⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⣿⣿⣿⣟⡭⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣿⣿⣿⣿⣭⣽⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣫⣽⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⡀⠀⢀⠀⠀⡀⢀⣠⣤⣶⣾⣿⣿⣿⣷⡄⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣾⡄⠀⢿⣿⣿⡿⠿⠿⠻⠛⢿⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣯⢗⠃
⠀⠀⠀⠀⠀⠀⠀⢹⢷⣀⣤⣤⣉⢣⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⣿⡟⠠⣿⣿⣿⣿⠿⠛⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠽⠿⢿⡟⠀⠀⠰⣯⣥⣤⣯⣿⣿⢶⠈⠀⠸⣿⣿⡿⠁⣤⠒⢤⠈⢿
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠛⠃⡼⠁⠀⠀⠀⠀⠲⠒⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢼⠃⠀⠘⡄⢸
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⣿⡏⠀⠀⢠⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⡞⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠒⠊⠀⢨⡿⠁⠀⠀⡼⠂⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⣗⠀⠑⢤⣀⣤⠖⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢈⡤⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡐⠀⡤⠤⠖⠒⠢⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣧⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠴⠶⠶⠛⠛⠛⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠰⠶⣤⣤⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠃⠀⠀⠀⠀⢾⣇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⡀⠀⠀⠀⠀⠀⢸⣿⣆⠀⡻⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢹⠀⠀⠀⠀⢀⣾⠟⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠲⠶⣶⣟⡉⠀⠀⠀⠀⢸⠀⠀⠀⢀⣾⠟⠀⠀⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⢿⡙⠶⣤⠀⠀⢸⡀⣠⣶⠟⠁⠀⠀⠀⠀⣼⠋⠉⢛⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠿⢻⡟⠀⠘⢧⡀⠀⢀⣠⣾⠿⠟⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠉⠈⠙⢷⣦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠃⠀⠀⣸⡇⠀⣀⡼⠿⣿⡛⠉⠁⠀⠀⠀⢀⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⢀⣠⡴⠾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠹⢇⣼⠋⣀⡀⠈⠻⣄⠀⠀⣀⡴⠿⠟
⠀⠀⠴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢜⠙⢛⡇⠀⠈⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠙⠋⠀⠀⠉⠛⠉⠀⠈⠉⠀⠀⠉⠋⠉⠉⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⡤⠤⣤⡴⠖⠦⠤⠤⠖⠛⠦⢤⠤⢴⡏⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⣳⠄⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⠋⢹⡄⠀⠀⠀⡞⠀⠀⠸⡆⠀⠀⠀⢿⣀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⡇⠀⠀⢰⠁⣶⠀⠀⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⢠⡇⠀⠀⢸⡆⠀⠀⣠⠃⠀⠀⠀⠀⠘⠖⠈⠉⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣀⠟⡀⠀⠀⠘⣇⣁⡨⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠤⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠑⠢⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣫⠤⠠⠠⠤⠬⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠁⠈⠙⠰⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠖⠒⠛⠋⠉⠛⠳⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣤⡴⠶⠒⠛⠋⠉⠉⠉⠉⠉⠛⠓⠦⢤⣄⣀⣠⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠰⣿⠿⠲⠆⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠀⠀⠀⠀⠀⠀⢰⠾⠛⠛⠻⠷⠀⠀⠠⢤⡦⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠤⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⡐⠦⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠛⠛⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡍⠁
⠀⠀⠀⠀⠀⠀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⣾⣯⡭⠿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣿⢷⢶⡶⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡏⣷⣿⣿⡇⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢣⣿⠈⠛⠀⠀⠘⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠀⢁⣉⠁⠀⠀⣀⡙⠱⢄⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡤⢀⡴⡊⠁⠀⠀⠀⠀⠀⢈⠉⠢⡈⠱⢄⡀⠍⠐⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢰⠣⢠⠊⠀⠀⠀⢀⣀⣀⣀⡀⠀⠂⠀⠘⡄⠠⡁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠴⣊⠀⡎⠀⠀⢠⠚⣛⠸⡵⠾⢩⠄⠀⠀⠀⠸⠀⠱⡀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡠⠂⠀⡇⠀⠀⠀⢡⠀⠀⠀⢀⠃⠀⠀⠀⠂⢸⠀⠌⠀⠀⠀⠀⠀⠀⢀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⢌⠀⢃⠀⠀⢂⠘⡀⠀⠀⡎⠀⠀⢀⠠⢀⠃⠀⠸⡀⠀⠀⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡊⡀⠀⢣⡀⠀⠈⡇⠀⠰⠁⠀⡆⠂⡠⠃⠀⢀⠔⠛⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢄⡀⠀⡆⠀⠀⠀⠀⠀⠉⣄⣄⡙⠢⣀⣄⠀⣺⣈⡠⠔⠊⠀⠂⠀⡌⠀⠀⠀⠀⠀⠀⠀⠠⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢘⠀⢣⠀⠀⠀⠀⠀⠄⠁⠀⠀⠰⣯⣀⣩⠟⣳⠀⠉⠢⡜⠁⠉⠃⠀⠀⠀⠀⠀⠀⠀⡅⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠁⠀⠔⠈⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠪⣥⠠⠶⠞⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠁⢀⠂
⠀⠀⠀⠀⠀⡠⠒⠂⠐⢢⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠾⣤⣴⣴⣿⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠄⠆⢠⠊
⠀⠀⠀⠀⠀⠡⣀⡀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠄⠀⠀⠀⠀⠘⠺⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠔⠁
⠀⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠔⠠⡠⡀⢀⢀⠀⡀⠀⡀⡀⡀⢀⠀⠇⠃⣁⠔⠁
⠀⠀⠀⠀⠀⠨⣄⣀⠤⠀⠀⠀⠀⡏⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠥⣌⣈⣘⣐⣁⣁⣁⡌⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠔⠐⢄⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣼⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠄⠡⣾
⠀⠀⠀⠀⠀⠀⠀⢠⢊⡖⠃⢔⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠘⢆⡷⢃⠅⡪⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠥⡊⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠾⡾⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠰⡼⢳⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⢪⠇⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⡠⢀⠠⡠⢀⠠⡀⢔⡠⢀⠤⡠⢄⡠⢀⠄⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣰⣟⢢⡾⣄⣹⡦⡀⢄⡠⣀⡤⢤⠄⠀⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠀⠁⠈⠀⠁⠈⠁⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠓⠒⠢⢄⡀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠙⠶⠋⠁⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠛⠷⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⢀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⣀⣀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⢀⡼⢳⣄⠀⠀⠀⠀⠀⢀⡼⠃⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⡤⠴⣿⡤⢤⡈⠓⠲⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠿⢷⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⣀⣿⣄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⠛⠛⠛⠻⣿⣿⣯⠀⠀⠀⠀⠀⠻⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢺⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣀⣤⠶⠋⠁⠀⠀⢀⣠⠖⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠉⠉⠀⠂
⢀⡾⠋⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢀⣾⣥⣄⡀
⢹⡄⠀⠀⠀⠀⠈⠙⢲⣦⢤⣄⣀⠀⠀⠉⢳⠤⣤⣤⡤⠖⠉⠀⠀⣉⣙⣿⣶⠶⠀⠀⠀⠀⠀⡴
⠀⠙⠶⣤⣄⠀⠀⣠⠟⠁⠀⠈⠉⠛⠲⣦⠞⠠⣀⡠⠀⢴⡶⠞⠛⠉⠁⠀⠹⣧⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠉⠙⣾⠏⠀⠀⠀⠀⠀⠀⠀⢻⡆⣀⡜⢣⣀⣾⠇⠀⠀⠀⠀⠀⠀⠙⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⠀⠀⠀⠀⠀⢸⣿⢏⣤⡌⢻⣿⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠈⣿⠘⣷⠂⢀⡍⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣿⠸⣯⠅⢸⠇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⢰⣟⡂⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠉⠀⡏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣦⣿⣶⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⡿⡿⢿⠿⠿⣿⣿⣿⡿⠛⠩⠈⠹⣿⣿⣿⣿⣿⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠀⢀⠀⠀⠀⠀⠀⠀⠘⠛⣷⡄⠦⠀⣠⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⠀⠸⣦⣄⣤⣤⡠⠄⠀⠀⣸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⡀⠀⠉⠉⠁⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⢦⣤⣴⠴⠇⠀⠄⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⢨⣄⣹⣄⣀⠀⡀⢀⣤⣤⡄⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠊⠉⠉⢿⣿⣿⣿⣷⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠟⣩⡿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢦⣾⣣⠾⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣯⣤⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠋⠀⠉⠉⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣄⠀⠀⠀⢾⣿⣷⣹⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⠁⠀⠀⠀⠀⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⠇⠀⠀⣠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⣹⣿⣿⣹⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡀⠀⠀⠐⠀⠀⢠⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠲⠤⣀⠀⡞⢳⣀⡼⢹⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡀⠀⣠⡆
⠀⠀⠀⠀⣀⣀⣀⣈⠙⠃⠀⠋⠀⠾⠚⣽⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣄⣸⠀⠱⠞⠁⣇⢀⣠⣴⠆
⠀⠀⠀⠀⠈⠑⣢⡀⠀⠀⠀⠀⠀⠀⠈⠙⣒⣲⠤⠄⠀⠀⠀⠀⠀⠀⣤⡤⠤⠄⠀⠀⠀⠀⠀⠀⠉⠁⣰⣃
⠀⠀⠀⢠⠔⠋⠁⣄⠀⠀⠀⠀⠀⠀⠘⢯⡁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠉⠛
⠀⠀⠀⠀⠀⠀⢠⣇⠔⠛⠀⡔⠀⢸⡀⠀⠹⢄⠀⢄⠁⠀⠀⠀⠀⠀⠐⠂⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡼⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⠔⠂⠀⡴⢀⠀⢧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⡰⠁⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠐⡟⣆⠀⣠⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣰⣇⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠙⠓⠾⠇⠈⠶⠁⢻⠴⢚⡟⠁⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀⡀⠀⠸⡏⢧⣴⠃⢹⣀⡤⢶⡟⠁⠀⠐
⠀⠀⠀⠰⣖⠒⠃⠀⠀⠀⠀⠀⠀⠠⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠻⡝⠲⠇⠀⠁⠀⠈⠉⣠⣞⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⠤⠒⠋⠁⠀⠀⠀⠐⠀⠀⠀⣀⣠⡤⠾⠆⠀⠀⠀⠀⠀⠀⠀⣠⡤⠂⠁
⠀⠀⠐⠂⠒⠒⠒⠀⠀⠀⠀⠀⠠⡄⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⢀⡀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡸⠁⠀⢡⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠮⠖⠒⠃⢀⣤⠀⢠⠄⡈⠣⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠊⢡⠄⠀⢰⠃⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠂⢸⠀⠈⠑⠲
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡾⠛⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⢀⣤⠿⠶⠶⢶⠏⠀⠀⠀⠀⠀⠀⠹⡷⠶⠶⠶⣤⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠋⠉⠉⠉⠙⢷⡄⠀⠀⠀⠀⠹⣆
⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠻⠞⠛⠛⠷⣦⣿⡀
⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠉⠛⢛⡛⠛⠛⠛⠛⠛⠛⠛⠻⣟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣠⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡴⠋
⢀⡾⠋⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣍⠉⠁
⠘⣷⣀⣀⣼⠃⠀⠀⠀⠀⠀⣴⠟⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⣠⡴⠟⢻⡄
⠀⠈⠉⠉⠁⠀⠀⠀⣀⠀⠸⣏⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⣿⠀⠀⣾⠋⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢀⣤⡾⢻⡇⠀⠙⠛⠛⠋⠀⠀⠀⢀⣴⣦⠀⢸⡏⠀⠀⣸⡏⠀⠀⠻⣤⣤⣼⠏
⠀⠀⠀⠀⣰⡟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⣿⠀⠈⠻⠶⠾⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠹⣦⣀⣠⡿⠁⠀⠀⠀⠀⠀⠰⣟⠀⠀⢠⡿

⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣇⢀⣰⠦⠤⣤⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣵⣿⢟⢇⣾⡏⢸⡿⢿⣗⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢺⣤⣬⢠⢄⠀⠑⠎⠓⠀⠀⠀⠄⣰⣿⣷⡟⡞⠁⣰⣟⡷⢾⣟⠀⠀⠠⠀⠂⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠱⣄⣿⡿⢿⡧⣤⠀⢀⠄⠀⢀⠛⠀⡍⢹⠟⣾⠠⢾⡿⠻⣟⠁⡇⠀⠀⠀⠋⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠐⢡⠖⣰⣿⡷⣿⠁⢨⠀⠜⠀⠀⡿⠀⠀⠀⠀⢀⣿⡔⠘⣧⣼⠟⠀⡷⠀⠀⠐⠞⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⣦⠀⠉⢴⠏⠛⠛⣶⡶⣾⡄⠀⠀⠀⠀⠀⠀⢀⠀⢨⠻⣿⠛⠁⠴⠶⡌⠀⠀⠀⠠⡦⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢡⡥⠒⢠⠀⠀⠀⠀⠘⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⠟⠁⠀⠀⠀⠀⡄⠀⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣦⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣯⣩⡹⣿⣿⡶⣼⢣⣴⣯⣱⣅⣉⠳⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⢁⡚⠻⣧⣲⡖⢟⣛⠺⣮⣽⣿⠿⣻⣏⣲⣿⡅⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢻⣥⡴⠾⠷⣍⠀⣠⣝⣦⣌⣿⣭⢭⣳⡀⠘⠙⡿⣿⣯⢞⠟⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣦⣛⣿⣿⣷⣟⡺⢿⣼⣩⣽⡿⠿⠭⠲⠃⠙⢾⣛⣿⡿⣯⣉⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣐⣠⡏⠡⢾⡿⠛⠙⠛⢿⣙⢛⣛⡴⠂⠀⠀⠀⠀⠀⠀⠀⠻⣯⡾⢿⣉⡴⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢣⡞⣻⠿⠋⢹⣬⣀⡚⢿⣿⣭⣭⡏⠀⠀⠀⣀⣐⡂⠀⠀⠐⠖⢺⢤⣄⠙⢧⣀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⡴⣷⣶⣶⠾⠿⣭⣤⣺⣿⣿⢷⠀⠀⠀⠀⠀⠈⠀⠁⠀⠒⠂⠉⠀⢰⢷⣬⡽⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⢚⠞⢫⡿⢥⣤⣼⣿⣿⣏⣫⡷⣏⡀⠀⠀⠀⠐⠀⠀⣀⣤⣤⣶⡗⢶⣭⣿⣻⡒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠏⠀⢃⡴⣫⠎⣻⢷⣾⢿⣯⣷⢲⠲⢀⠀⠀⠀⢀⣾⣫⠀⠋⢸⡗⢸⡌⣿⣎⠻⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⠔⠛⣰⠟⠋⠠⢞⣹⣣⣞⣿⣯⣴⣓⣀⠀⠀⢸⠻⣿⣤⠀⣿⣹⡀⣷⣽⣤⣥⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⢀⠂⠐⣡⣿⠬⣥⡾⣿⢿⣿⡿⠟⢛⣿⡷⠀⠀⠸⣧⠘⢿⢷⣯⣼⡻⣿⣿⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣶⣾⡃⣰⣾⣧⣴⠖⣋⣿⡤⡶⢯⣿⣧⡠⠀⠀⣿⣆⢂⣀⡿⠊⡻⣿⣿⣥⣭⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣽⣁⣠⣽⡵⠟⠤⣀⡟⢃⡼⣿⢻⢙⢿⣧⣹⣶⣄⡙⠺⠿⢛⡿⢧⣋⠻⡍⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⣽⠟⣥⣾⡟⢉⣿⣽⣦⣾⣿⣿⣼⡟⡋⣿⠀⠀⢻⡗⢲⡿⢷⢑⠘⠇⢹⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⣈⣡⣟⣠⢾⣄⣸⡿⢻⣿⡿⣿⡜⡀⣿⣷⣤⣠⢿⡌⠃⠀⢌⣿⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⡝⡋⢩⣾⢷⡾⡿⠷⠨⣾⣿⠁⠈⣿⣜⠞⠀⠉⣿⡇⣹⣀⡇⠈⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⢣⣤⣶⠋⣸⢃⡀⣫⣥⣿⠌⣸⠉⢿⣣⠀⢠⡏⣠⠻⣿⣳⣦⢏⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⣅⢾⠯⣰⡖⠿⢃⣿⡿⣏⠀⠘⢯⣷⣿⡾⣿⠀⠳⠙⢹⠈⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠛⠿⠃⣰⠃⠑⣨⣿⢀⣠⡾⣽⠃⡇⠛⣧⠀⠂⠾⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠑⠋⠑⠈⠁⡼⠊⠉⠀⠾⡄⠐⣈⢟⢈⠜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢆⠩⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠘⠁⠀⠒⠀⠰⠄⠠⡦⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠲⠀⠠⡆⠀⣤⠀⢠⡄⠀⣀⠀⠀⠁⠀⠘⠀⠀⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠓⠀⠰⠆⠠⣤⠀⣤⡄⢀⣄⠀⣈⡁⠀⡉⠀⠘⠃⠀⠗⠀⠰⠆⠀⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠲⠀⠠⠆⠀⣿⠢⣠⣄⢀⣈⠀⢈⡉⠀⠛⣃⠘⠟⠀⠾⠆⢰⡦⠀⣤⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⠀⠰⠂⠀⡤⠀⣤⡄⢀⣼⡆⠈⠳⣌⠛⠁⠛⠃⠸⡟⠈⢿⡷⢠⣶⠀⣤⡄⢀⣄⡠⣼⠃⠀⠉⠀⠘⠂
⠐⠀⠠⠄⠀⣤⠀⢠⡄⠀⣈⠀⢈⡙⣷⠀⠀⠈⠙⢴⣿⢆⠎⠀⠀⠘⡆⣠⣄⢀⣩⠵⠚⠁⣰⠛⠁⠐⠃⠀⠶⠀⠠⠄
⠤⠀⢠⠀⢀⣀⠀⣈⡁⠈⡋⠀⠛⠃⣿⡄⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠐⡬⠛⠉⠀⠀⠀⣰⠛⠿⠀⢴⠆⠠⡤⠀⢀
⡄⠀⣀⠀⢈⠁⠀⠛⠁⠘⣟⣐⣿⣆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢰⡶⠀⣤⠀⢠⡄⠀⡀
⠀⠀⠉⠀⠙⠃⠐⠿⠀⢾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⣴⡄⢀⣄⠀⣈⠁⠀⠉⠀⠘
⠀⠘⠃⠀⠾⠂⢰⡦⢠⣶⡌⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣳⢈⡋⠀⠛⠁⠘⠃⠀⠶
⠀⠰⠆⠠⣦⠀⣴⡄⢠⣤⣠⠽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠻⠃⠐⠿⠀⢶⠆⠠⡄
⠀⣤⠀⣠⡄⢀⣬⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠐⠿⠆⢴⡦⠀⣤⠀⢀⡀
⢀⡀⠀⣸⠿⢭⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⠀⣤⡄⢀⣀⠀⠈⠁
⠀⠁⠀⠙⠁⠘⠟⠐⠿⠏⣿⣿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠙⢦⣉⠀⢈⠉⠀⠙
⠘⠃⠐⠟⠀⢾⠆⢰⣶⠀⣶⡄⣸⠇⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣻⣿⠛⣛⡛⠒⠛⠂⠛⠃⠐⠗
⠲⠀⠰⠆⠀⣶⠀⣤⣄⢀⣭⣡⣿⡾⠿⡿⠛⠿⢿⣆⠀⠀⠀⠀⣼⣦⣀⠀⠀⠀⠘⡇⣙⡋⠈⠛⠃⠺⠟⠀⠶⠀⢠⠄
⠄⠀⣤⡀⢀⣄⠀⣈⠁⢈⡛⠁⠻⠋⠸⠿⠲⣿⡿⢿⡄⠀⠀⡼⣡⣯⢻⣷⣄⠀⠀⢣⠻⠋⠸⠿⠀⢶⠆⢠⡦⠀⣠
⡀⠀⡀⠀⢈⠁⠀⠛⠁⠛⠟⠰⠿⠆⢾⡶⢰⣶⡄⣼⣷⡀⣼⡁⣙⡋⠘⡛⠋⠳⣦⡘⣿⠆⢶⡦⢠⣦⠀⢠⡀⠀⡈
⠀⠀⠁⠀⠙⠃⠘⠟⠀⢾⠆⢰⣶⠀⣴⡄⣠⣭⢀⣩⡉⢿⡛⠁⠻⠋⠸⠿⠀⢿⠆⢹⣷⠀⣤⡄⢀⡀⠀⢈⠁⠀⠉
⠀⠈⠃⠀⠒⠀⠰⠆⠠⣶⠀⣠⡄⢀⣍⠁⣙⡋⠈⠛⠃⠻⠟⠸⠿⠇⢶⡦⢠⣶⠄⣠⡄⢀⣈⠀⢈⠁⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⣤⠀⢀⡀⠀⣈⠁⠈⠛⠀⠛⠃⠸⠿⠀⢿⠇⢰⣶⠀⣤⡄⢠⣄⠀⣉⠁⠈⠋⠀⠛⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠘⠃⠐⠿⠀⢴⡦⢠⣶⠀⣤⡄⢀⣉⠀⢈⠁⠀⠛⠁⠘⠃⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⢰⡦⠀⣤⡄⢠⣄⠀⣈⠁⠈⠉⠀⠙⠃⠐⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠢⡀⠀⡜⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠂⠌⠁⠉⠈⠑⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠋⠐⠂⠰⠀⠀⠒⠓⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠃⠘⠀⡄
⠀⠀⠀⠀⠀⠠⠀⠁⠃⠀⠀⠀⠀⠀⡀⠐⡒⠀⠐⠒⢲⠂⠠⠄⠄⠒⠒⣖⡲⢊⠍⠉⠀⠈⢱⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡥⠆⠒⠂⠒⠀⠐⠓⠊⠀⠘⠒⠚⠓⠒⠒⠀⠉⠘⢒⠞⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠠⡇⠀⠀⣰⢠⠄⠀⢀⠀⡄⠀⠰⡇⢰⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⢰⡀⠀⠀⠀⡇⠁⠀⠀⡍⠰⠁⠐⠐⠈⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢀⣠⠃⠀⠀⠀⡫⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⢀⠀⠃⠀⠀⠀⠚⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠀⠂⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠂⠇⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠁⠠
⠀⠠⡴⣄⠀⠁⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠤⠀⠉⠬⠄⠇⠀⠀⠂⣀⣀⣀⣈⡈⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢐⣀⣃⣤
⠀⠀⠀⠈⠀⠁⠉⠁⠈⠉⠈⠀⠀⠄⠠⠀⠈⠉⠀⠀⠃⠁⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠀⠀⠐⠈⠉⠀⠀⠈⠀⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠙⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡵⠀⠀⠀⠀⠀⣀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢧⢀⣀⠀⠀⠸⣄⣀⣀⡴⢲⣚⣉⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢺⡁
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⠉⠈⣇⠀⠀⠀⣠⠝⣇⢸⠀⠀⠀⠈⠳⣤⠖⠚⠉⠓⢆⠀⠀⢀⣞⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡴⠋⠳⣼⡀⢀⡜⠁⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣘⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠀⠀⠀⠸⣾⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡜⠀⠀⢴⠋⠀⠙⠚⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢈⣉⣱⠤⣼⠁⠀⠀⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠷⠯⠭⠖⠒⡖⠛⢦⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠎⠀⠀⠀⢀⡜⠁⠀⠈⠒⠚⠁
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣇⣀⣤⡶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣄⠀⠀⠃⠀⠀⠀⠺⡁
⠀⢀⡞⠉⠉⠉⠲⠖⠲⡤⢒⡭⡶⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⠳⣄⠀⠀⠀⠀⠀⡇
⠀⠈⢧⡀⠀⠀⠀⠀⢀⡗⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⣌⣦⣠⠖⠲⣤⠇
⠀⠀⡜⠁⠀⠀⠀⠀⠈⠉⣧⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠰⡁⠀⠀⠀⠀⢈⡇
⠀⠀⢣⣀⢀⣀⣤⠀⠀⣠⠃⠀⠀⠀⠉⣓⡖⠒⠒⠉⣇⠀⠀⠀⠀⠀⠀⠀⡰⠓⠦⢼⡁⣀⣀⡖⠋
⠀⠀⠀⠈⠉⠀⠈⠲⠒⢁⡤⠴⠒⠋⣡⠏⠀⠀⠀⠀⠈⠳⠤⣄⣀⡤⠔⠋⠀⠀⠀⢦⣍⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠲⠖⠋⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠺⡙⠉⠉⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡤⣄⠀⠀⢈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⢦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠦⣤⠴⠚⠉⠉⠉⠉⠉⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⢀⣤⠴⠒⠒⠛⠓⠶⢤⣀⠀⠀⠸⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠴⠋⢀⡤⠖⠚⠉⢓⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⣀⣠⠼⣥⣀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣷⣶⣶⡲⠦⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢰⠃⠀⣠⠖⠋⠁⠀⣀⣀⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⠿⢿⣿⡷⠀⠈⠻⣆⠀⠻⠤⣄
⠀⠀⠀⢀⡤⠊⠀⣞⡀⠀⠀⢰⡿⠿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠙⢦⠀⠀⠙⠁⢀⡤⠒⠛⠒⠢⣄⠀⠹⣄
⠀⢀⡴⠁⢀⡴⠋⠁⠀⠀⠀⠀⠁⣼⠁⠀⢸⣟⣿⣦⣀⣴⢷⣄⣠⣾⡿⢻⣿⠄⠀⠈⡇⠀⠀⣰⠁⢠⡀⠀⠀⠀⠈⢧⠀⢹⡄
⠀⡞⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡀⠈⠛⢛⣻⡏⠁⠈⠉⢻⣍⠛⠛⠋⠀⢀⣼⡗⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠈⡇⠈⡇
⠰⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠻⠛⠉⠀⠀⠀⠀⠀⠙⢝⠓⠲⠞⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡇
⠀⠳⣄⠈⢧⣀⡀⠀⣀⡤⠞⠋⠭⠇⠀⢀⣤⠶⠿⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡼⠁⡼⠁
⠀⠀⠈⠳⢶⡌⠉⢻⡁⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠙⠿⣷⣤⣠⣽⣦⠀⠀⢀⡆⠀⠀⠙⠉⣽⠲⠶⠶⠒⠋⣠⠞⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠐⠋⠲⣄⠀⠀⠀⠀⠈⠉⠉⠁⢹⣀⠴⢎⣀⣀⣠⠤⠞⠁⠀⠀⣹⠞⠛⠁
⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠒⠛⠉⠙⠲⣤⠤⠤⠤⠶⣞⠉⢀⣠⠴⠲⠶⠶⠶⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⡀⠀⠀⢀⣀⡴⠞⠛⠶⢤⣤⡏⠀⠀⠀⢰⣏⣭⠭⠶⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣀⣠⡤⠶⡆⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠀⠀⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⣱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠎⢀⣀⣽⣏⣈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣶⠋⡿⣭⠥⠿⠛⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⡿⠀⣏⠛⣙⠰⡆⣿⡟⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢻⠀⣿⣿⢠⣿⣲⣶⣮⠷⣾⡿⡇⠿⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⡿⠇⢻⠀⣿⡏⣿⣷⣿⣿⣿⣭⣿⠃⡇⢠⠻⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⢿⡀⠸⢧⣈⠀⠆⣿⡇⢻⣿⣿⣿⣿⣿⣿⢴⡇⠀⣠⡿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠰⣾⣇⣠⣄⠙⠦⠀⢿⣇⢼⣿⡟⠟⠿⣿⣿⢸⡇⠀⠉⣤⣠⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠛⠋⠿⠿⠿⣦⡀⢸⣿⡌⣿⠋⢠⡄⠙⣿⢸⡇⣰⣾⡿⠿⠓⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⠀⠀⠀⠀⠶⠶⢿⡌⣿⣿⣁⡄⠀⠉⠀⠸⣿⣾⣿⡄⠀⠀⠀⣤⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣠⣤⣾⣦⢤⣤⣳⣿⣿⣿⣀⣤⣀⣤⣀⣿⣃⣤⣀⡀⣤⣶⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⣿⣧⣤⣤⣆⠈⠛⢿⣿⠉⠉⠿⠏⠘⠋⠉⠁⠀⢿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⠀⠙⣿⣿⡛⢿⣄⢿⣦⣀⢀⣀⡀⠀⣀⣀⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡇⠀⢻⡆⠀⠘⢿⣷⡀⠙⢷⣍⡻⣿⣿⣿⣿⠋⠉⢸⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⡇⠀⠈⢷⡀⠀⠈⠻⣿⣄⠀⠉⠙⠋⠉⠙⠏⠀⠀⣾⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⠘⣧⠀⠀⢬⣿⡄⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢻⠀⠀⠀⣿⠀⠀⠀⠹⣧⠀⠀⠀⠻⣿⣿⣦⠀⠀⠀⢀⣾⡿⠃⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⢸⡇⠀⠀⢹⣦⣤⡀⠀⠈⢷⣄⠀⠀⠀⠙⠻⣧⠀⠐⠟⠋⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣿⠀⠀⠀⢿⡀⠀⠀⢿⣿⣷⠀⠀⠈⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⢻⡄⠀⠀⠈⢧⠀⠀⠘⣯⢻⣧⠀⠀⠀⠀⠈⠻⡧⠀⠀⠀⠀⠀⢠⡾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⣿⣆⠈⠓⠀⠀⠀⠈⠧⠀⠀⠙⠂⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡯⣿⡀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⢀⣤⠒⠛⠉⢸⡿⡇⢿⣷⡀⠹⣿⣆⠀⠀⠀⣾⣷⡆⠀⠀⠀⠀⠀⠠⢿⣿⡿⠆⠀⠀⠀⠀⠀⣠⣶⠿⢻
⠀⠀⣠⠾⣿⣟⠢⣄⡘⣷⡙⡄⠻⣷⡄⠙⢿⡆⢀⡀⢹⣿⡇⠀⠀⠀⠀⢶⡀⠀⠈⣷⡀⠀⠀⠀⢀⡾⠋⠁⠀⢸⠆
⠀⡴⠁⠀⠸⣿⣥⡀⢡⡘⣿⡘⣆⠘⣿⣦⠈⢻⡌⣿⡄⢻⣿⠀⠀⠀⠀⠈⠻⣶⣴⣿⣿⣶⣾⡇⠘⣀⣀⣀⡀⣼
⢰⣿⣶⣶⣦⣌⠻⣿⣓⡙⢻⣷⣿⠂⠘⣿⣧⠀⠻⣼⣷⡈⢿⣦⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣾⡿⠋⠉⠙⡿
⢸⠉⠉⠉⠛⠻⣷⣬⣻⣧⣌⣿⣿⡀⣶⡘⣿⡇⢠⡈⠻⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⠷⠋⠀⠀⡇
⢸⣿⡿⠿⠶⣶⣾⣿⡿⢿⣿⣿⣿⣇⢹⣧⠹⣷⠈⢿⣷⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⠀⠀⠀⠀⢠⡇
⣸⣇⣤⣴⣶⣾⣿⣿⣿⣀⠀⠙⢿⣿⡄⢻⡆⢻⣧⡀⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⠋⠀⠀⣿⣴⠄⠀⠀⢸⠁
⣿⡿⠋⢉⡙⠛⢿⣿⣿⣿⣿⣿⣶⣿⣿⡌⢿⣄⠻⣿⡄⢙⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠃⠀⠀⠀⣾
⣿⠀⣰⣿⣿⣷⣿⣿⣿⣿⣧⣤⣬⣽⣿⣟⠀⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠀⠀⠀⣿
⡏⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⢹⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⡀⠀⠀⠀⠀⠈⢳⡄⠀⢸⠀⠀⠀⠀⠹⣆⡀
⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⢞⡹⠟⠋⠉⠉⠀⠀⠀⠀⠀⣀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠙⢦⣼⡷⠖⠀⣀⠀⠈⠻⣆
⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠾⠷⢤⣤⣤⣄⣀⣀⣀⣼⣶⣿⣿⣿⣧⠀⠀⢻⣧⣤⣾⣿⣤⣿⣷⣶⣾⣿⣷⣤⡴⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣥⣤⣤⣭⣿⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠋⠉⠉⠙⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠙⢦⡀⢀⣠⡤⠶⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢠⣤⠀⠀⢠⣤⠀⠀⠀⠀⠈⠷⠋⠁⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⢀⡤⠶⠶⢦⣀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠰⠆⠀⠀⣴⣦⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⠙⣦⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡿⠀⠀⠰⠶⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⣀⠶⣤⠀⣼⢀⡇⠀⢀⣄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢸⡇⠀⠻⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡴⢷⢦⠀⣰⠋⡼⠀⠘⡇⣿⣸⠃⠀⠈⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⠀⠀⢠⣄⠀⠀⠀⣸⣣⠏⠘⡇⣯⡾⠁⠀⠀⡇⢿⡟⠀⠀⠀⣴⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠛⠀⠘⣦⠀⠈⠁⠀⠀⢀⣿⠋⠀⢰⡇⢻⡄⠀⠀⣸⠃⠸⣇⠀⣠⠆⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠳⢤⣤⣤⠶⠋⠙⠦⠴⠋⠀⠀⠉⠓⠚⠁⠀⠀⠈⠉⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣤⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠘⠟⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠘⠛⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠚⠁⠀⠀⠀⠀⣰⠋⠁⠰⡶⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠶⠒⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠠⣶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣀⣼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠶⠶⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢀⣠⠴⠖⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⡟
⢿⡀
⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠙⠶⣤⣀⣀⣀⣀⣤⢤⣀⠀⠀⠀⠀⢀⣀⠴⠢⣄⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⢀⣀⡴⠞⠁
⠀⠀⠀⣀⠀⠈⠉⠉⠁⠀⠀⠉⠙⠛⠛⢃⡉⠁⠀⢠⠃⠈⠁⠐⠂⠉⠀⣼⠀⠉⠒⠤⠤⠴⠶⠚⠉⠀⠉⠉⠙⠛⠉⠁
⠀⠀⢠⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣷⠀⢀⡟⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆
⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⡿⡆⠀⠀⠀⠙⠋⠀⣸⠃⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⡀⠀⠀⣼⢻⡆⠀⠀⠀⠀⠀⠀⠈⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠀⠀⠀⠀⠀⠠⣯⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢠⣟⣀⣀⠀⠀⣼⢷⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠿⠀⠀⠀⠀⠀⠀⢿⣤⣤⡄⠀⠀⠈⠉⣹⠋⠀⠀⠻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠴⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢡⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⣤⡀⠀⠙⠁⠀⠿⠗⠀⣷⡶⠀⠀⢀⡤⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠄⠀⠀⠁⠀⠘⠧⠀⠀⢀⣴⡖
⠀⠀⠀⠀⠀⠀⠐⠖⠒⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢏⣀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡂⣀⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠋⠁⠀⠀⠀⠰⢂⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⢈⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣤⣦⠀⠀⢸⡟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣴⠀⢀⣄⠀⠀⣶⡄⣹⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠟⠋⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠠⠆⠛⠉⠂⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠀⡀⠀⠀⠀⡠⠉⢄⠀⠀⠀⢀⡀⢄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢄⣀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠐⠒⠒⠩⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡵⠖⠒⠂
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⠜⢀⡀⠤⠤⣄⡀⠀⢀⣠⣤⣤⣀⣀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠈⠀⠀⠀⠀⠀⠙⣶⠟⠉⠉⠉⠉⠙⠛⠾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠠⡀⠀⠀⠀⠀⠀⣠⠶⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡰⠚⠛⢷⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢇⠀⠀⡾⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠃⢄⠀⠈⢇⡀⢸⡆⠄⡹⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠢⢈⡀⠙⠀⠘⠁⠘⠃⢰⡇⠔⡰⠋
⠀⠀⠀⠀⠀⠀⠈⠃⠢⠄⣀⠄⠈⢳⡄⢀⠄⠊⢡⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄⠠⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠄⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣁
⠀⠀⠀⠀⠠⡄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣇⡀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠒⠛⠁
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣶⠄⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⠑⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⣁⠴⠆⢀⠀⠀⠀⢀⡀⢰⡤⡘⢦⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠀⣠⣴⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠈⠀⡐⡰⠛⠀⡼⠂⢸⡇⠄⢇⠀⠁⠳
⠀⠀⢀⡤⠠⠀⠀⠈⠁⠀⢡⣠⣾⡷⡀⠀⣾⡟⠉⠐⠢⠬⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠐⡰⠁⠀⢌⠀⠀⠻⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙
⣶⣾⣶⣶⣶⣷⣿⣶⣾⡿⣿⣶⣶⣿⣶⣾⣿⣾⣷⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣾⣷⣿⣿⣿⣶⣶⡶⡶⠶⠶⢶⣷⣶⠷⠶⠷⢶⡿⠷⠾⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠦⣤⣀⡴⠞⠋⠙⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣀⠤⠤⠒⠒⠒⠒⠒⠒⠲⠦⠤⣤⣀⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⠛⠭⠥⣤⣄⣈⣹⣦⣀⣠⡤⠤⠤⢤⣤⠤⠴⠟⠛⠉⢹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣀⣠⡤⠶⠖⠒⠛⠛⠛⠒⢶⣤⡤⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠗⠋⠉⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⣠⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⡴⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡄⠀⠀⠀⠀⠀⠀⢀⣎⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢰⠀⠀⠀⠀⠀⠀⢠⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡏⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡇⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⡜⠀⡠⠄⢰⠇⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠈⠳⢦⣠⠞⣀⡼⢄⣡⣞⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠠⠄⠙⡄⢠⠜⠁⣀⡟⠛⣛⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠐⠋⠀⢠⣼⣾⠀⣴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣰⠏⣐⠁⠀⠃⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣠⡴⠊⠀⠀⡟⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠏⠀⠀⠠⠚⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⡀⠀⢀⣠⠶⠾⡇⠀⠀⠀⠀⠀⢀⠇⠀⠀⠤⠀⠠⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠠⡏⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⣄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠲⢦⡀
⠀⠀⣀⣤⠤⠄⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢆
⠀⣀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚
⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠉⠑⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠁
⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠙⠓⠒⠒⠚⠛⠓⠒⠒⠒⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠀⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣤⣤⣀⣤⠞⠛⠉⠉⠉⠉⠉⠛⢷⣤⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣠⣾⠋⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠁⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠶⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠛⢷⡀⠀⠀⠾⠷⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⣴⠚⠋⠉⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠐⠋⠙⢳⡄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⡄⠀⠈⠃⠀⠀⠀⠀⣚⠉⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠈⠙⣷⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠘⣿⣿⣿⣧⣤⣴⠆⠀⠀⠀⣦⣤⣴⣿⣿⣿⡋⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⣯⣿⡿⣿⠟⠀⠀⠀⠀⠙⣿⡿⣿⣭⣿⠇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⣀⣀⣉⣥⡾⠁⠀⠀⠀⠀⠀⠀⠈⢳⣬⣉⣁⣀⣀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠀⠀⠀⠉⠛⠛⢶⡶⠶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠛⠾⠿⠻⣿⠁
⠀⠀⠀⠀⠀⣠⣴⡋⠹⢧⣄⣀⠀⠀⠀⢠⡟⠀⠀⠈⠳⣦⣤⣤⣀⣀⣀⣀⣠⣼⣷⡶⣦⣤⣤⣤⣤⣤⣤⣴⠶⠟⠋⢳⡄⠀⠀⠀⢿⣶⠶⠶⣶
⠀⠀⠀⠀⠈⣉⡿⠋⢀⣠⠔⠉⠀⠀⣰⠟⠀⠀⠀⠀⢼⣇⣀⣈⠉⠉⠉⠉⠉⠁⢸⡆⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣠⢞⣩⠶⠚⠉⠀⠀⠀⠀⣼⠃⠀⣀⡀⠀⠀⠀⠀⠈⣉⡿⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠶⠞⢛⡟⠀⠀⠀⠀⠈⢷⣤⡶⢿⠀⠀⣸⡇
⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠉⡹⠃⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣄⠀⠀⠈⠉⠀⢸⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣀⣤⣴⠀⠀⢠⡏⠀⠀⠀⡿⢦⡴⢦⠄⠀⠀⣷⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⡴⠟⠁⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠻⠛⠉⠀⢸⠀⠀⣼⠀⠀⠀⢀⣇⡀⢠⡏⠀⠀⠀⢸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⡴⠋⠀⠀⠀⠀⠀⠀⣠⣏⡈⠛⡿⠂⠀⠀⠀⠀⠀⢸⠀⢰⠇⠀⠀⠀⠈⠁⣇⡞⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⣹⢣⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⣾⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠞⣩⣴⣶⡿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⣼⣿⣾⡿⠋⡡⠚⠉⢉⣉⣩⣭⣽⣿⣿⣿⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣿⠃⣾⢫⣿⡟⠁⠀⢀⡴⠾⣿⣿⣿⠿⠿⠿⢿⣿⣿⣭⣛⣛⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⡏⠐⠁⣿⣿⡇⣤⠀⠈⣴⡿⣋⠁⢀⣀⣀⠀⠀⠐⠲⠶⣿⣿⣿⣿⣛⣒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣬⣿⣿⡀⠀⣿⣿⡇⣿⠀⢸⣿⣿⣣⣾⣿⣯⣶⣾⣿⣷⣶⣦⣤⣉⣛⠻⢿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢈⣩⣽⣿⣿⣿⣿⣿⣿⣧⡁⠀⢿⣿⣿⣿⣧⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡁
⠀⠀⠀⠀⠀⠀⡰⠛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣻⣏⠻⣿⢿⣾⣿⣿⣿⣿⢿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡛⢦
⠀⠀⠀⠀⠀⠀⠀⠈⣠⣾⣛⠉⠉⣉⣉⣛⣿⣿⠷⡌⠉⠉⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠉⠉⠉⠴⢿⣿⣟⣋⣉⡉⠉⢝⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠟⣉⣠⣤⣤⣤⣭⣽⣻⣿⡀⠀⢀⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢀⡀⠀⠀⣾⣿⣫⣭⣥⣤⣤⣤⣈⡛⢷⣄
⠀⠀⠀⠀⣠⢶⣿⣶⣿⣿⠿⢟⣛⣭⣿⣿⣿⣿⡧⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠰⣿⣿⣿⣿⣭⣟⡛⠿⣿⣿⣷⣾⡷⢤
⠀⠀⠀⠀⣠⣾⣿⡿⠋⣠⣶⣿⣿⣿⣿⢿⣋⠀⠀⠠⣤⡄⣄⣴⠀⠀⠀⠀⠀⠀⢰⣠⠠⣤⠄⠀⠀⢉⣿⢿⣿⣿⣿⣷⣄⠉⠻⣿⣿⣦
⠀⠀⠀⣰⣿⣿⠋⢀⣾⣿⠿⣿⣿⣿⣿⣆⠈⠁⠀⠘⣎⠓⠴⠷⠄⠀⠳⡔⠁⠀⠼⠧⠐⣡⡎⠀⠀⠉⢠⣿⣿⣿⣿⡿⣿⣷⡄⠘⢿⣿⣧
⠀⠀⠀⣿⣿⠇⢀⡿⠋⣰⣿⣿⣿⣿⣿⣿⠦⠀⠀⠘⢉⣡⡖⣒⡒⠀⠀⠀⠀⠐⣒⣲⣤⣉⠓⠀⠀⠠⣿⣿⣿⣿⣿⣿⣦⠙⢿⡄⠘⣿⣿⡆
⠀⠀⢸⣿⡿⠀⠈⠀⣼⠟⣩⣴⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣦⣍⠛⢧⠀⠁⠀⢹⣿⡇
⠀⠀⢸⣿⡇⠀⢀⠀⢀⣾⣿⣿⢿⣿⣿⣟⣿⣿⣷⡄⢸⣌⠦⠀⡀⠀⠀⠀⠀⢀⠀⠠⢋⣦⢀⣾⣿⣿⣿⣧⠻⣿⣿⣿⣷⡄⠁⣄⠀⠸⣿⣧
⠀⠀⢸⣿⢁⠀⣿⠀⣾⣿⡿⢣⣾⣿⠏⣼⣿⣿⣿⣷⠘⣎⠙⠀⠈⢷⣶⣶⣶⠏⠀⠘⢡⡇⣸⣿⣿⣿⡙⣿⣇⢹⣿⡟⣿⣿⠀⣿⡀⡄⣿⣿
⠀⠀⢸⣯⡾⢸⣿⠀⣿⣿⢃⣾⣿⠏⢰⣿⣿⡿⣿⣿⡀⠘⢶⡤⣤⣀⣹⣯⣠⣠⢤⣴⠏⢀⣻⣿⡸⣿⣧⠘⣿⠈⣿⣿⠸⣿⡇⣿⡇⢹⣜⣿
⠀⢀⡿⠋⡇⢸⣿⡆⣿⡇⢸⣿⡿⠀⢸⣿⣿⢹⣿⣿⣿⣦⠘⣧⣸⡶⠿⡷⠷⡇⣸⡇⢠⣿⣿⣿⠃⣿⣿⠀⠛⠀⣿⣿⠀⣿⣇⣿⣿⢸⡟⠿⡄
⠀⠀⠀⠀⢷⣿⣿⣿⣿⡇⢾⣿⡇⠀⢹⣿⡏⢸⣿⡿⣿⣿⡆⣿⡟⢿⣶⣶⡾⠟⣿⠁⣿⣿⡇⠻⢠⣿⣿⠀⠀⢀⣿⣿⢰⣿⣿⣿⣿⣼
⠀⠀⠀⠀⠘⣿⢿⣿⣿⣇⢸⣿⡇⠀⢸⣿⠇⢸⡟⢸⣿⣿⠁⠘⣷⣴⣮⣥⣦⣴⠏⠀⣿⣿⡇⠀⣼⣿⠇⠀⠀⣸⣿⣧⣿⣿⣿⣿⢿⠏
⠀⠀⠀⠀⠀⠘⠸⣿⣿⣿⣞⣿⡇⢸⡀⣿⠀⠀⠀⣿⣿⣿⣧⡀⠈⠙⠛⠛⠋⠁⠀⣠⣿⣿⠃⣰⣿⠏⢀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠹⣿⣿⣧⠘⣷⡘⠆⠰⣄⠸⣿⣿⣿⣷⣄⣀⠀⠀⣀⣠⣾⣿⣿⢃⣼⣿⠏⢠⡞⢀⣾⡿⣿⣿⡟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠹⣿⣿⣆⢿⣧⠀⠀⠹⣷⣽⣿⡿⣿⣿⣿⣷⣾⣿⣿⢿⣿⣷⣿⡿⢃⣴⡟⠠⠞⠁⣰⣿⡟⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣞⣿⣧⣹⣄⠹⣿⣿⣷⡌⠻⣏⢿⣿⡿⢁⣿⠟⣽⡟⣠⣾⠟⣀⣤⡎⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣈⢿⣝⠿⣦⠉⠈⣿⠁⠸⢁⣾⠏⣴⣿⣿⣾⣿⣟⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡛⠿⣿⣦⣿⣷⡌⠳⡀⠀⡄⠀⣼⣿⣾⣿⡿⢛⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠈⠙⢿⣿⣿⣦⠀⢸⡇⢸⣿⣿⠟⠉⠠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠹⣧⣸⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢻⣿⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠀⠰⠀⠀⠰⠀⠀⠰⠀⠀⠰⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠺⣿⣦⡓⢀⡀⠛⢀⡀⠛⢀⡀⠛⢀⡀⠓⢀⡀⠃⢀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠁⣤⢿⣿⣷⣌⠁⣤⠈⠁⣤⠈⠁⣤⠈⠁⣤⣿⠁⣤⠈⠀⡄⠈⠀⡄⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠰⠆⠀⠰⠆⠀⢸⣿⠈⢿⣷⡄⠲⠆⠀⠶⠆⡀⢶⣾⡿⣿⡇⠀⠲⠆⠀⠶⠆⠀⠰⠀⠀⠰⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⢀⡀⠛⢀⡀⠛⢀⡀⠛⢸⣿⡆⠀⠙⢿⣷⣜⠛⣠⣸⣿⡿⠋⠀⢻⣿⠛⣠⡘⠛⣀⡈⠋⣀⠀⠃⢀⠀⢃⣠⣴⠞⠉
⠀⠀⠀⠀⠀⠀⣤⠈⠁⣤⠈⠁⣤⠈⠁⣤⠈⣿⡇⠀⠀⠀⠉⠻⠿⠿⠟⠋⠀⠀⠀⠈⣿⣦⠉⢡⣦⠉⢡⣤⢉⣠⣤⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠐⠆⠀⠰⠆⠀⠰⠆⣀⠲⠆⣀⢾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⣾⠿⠟⠛⣩⣿⠟⠁⠰
⠀⠀⡀⠉⢀⡀⠛⢤⣤⣿⣤⣤⣿⣤⣼⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣼⡿⣃⡘⠃⣀⠀⠃⢀
⠀⠀⠀⠤⠈⠁⣤⠈⠹⣿⣿⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣦⠉⢠⡤⠈⠀⡄
⠀⠀⠂⠀⠰⠆⠀⠰⠆⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠆⡀⠲⠂⠀⠲⠀⠀⠐
⠈⢀⡀⠙⢀⡄⠛⢠⡄⠛⣠⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡜⠛⣤⡘⠋⣠⠈⠃⢀
⠠⠀⠀⢤⠈⠁⣶⠈⠡⣶⠌⠡⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣉⠡⡦⠈⠠⡄
⠀⠐⠂⠀⠐⠆⣀⠸⠇⣀⣺⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣶⣾⣀⠀⠒
⠉⢀⡀⠙⢠⡄⠛⢠⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡟⠋⣩⠈⠃⣀⠈
⠠⠀⠀⠤⣀⣥⣾⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⢱⡦⠉⠠⡦⠈⠠⠆
⢀⣰⣶⣾⠿⠿⣿⠿⠿⣿⠿⠿⣿⡿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⠿⢃⡀⠻⠂⡀⠺⠀⡀⠐
⠀⠀⡀⠉⢀⡄⠙⢠⡄⠛⣠⡌⠛⣡⣌⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣤⡌⠋⣤⡈⠋⣠⠈⠁⢀
⠀⠀⠀⠠⠀⠀⠶⠀⠀⢶⠈⠡⣶⠌⠡⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢸⣷⡈⠰⡦⠈⠠⠆⠀⠀⠄
⠀⠀⠀⠀⠐⠂⣀⠘⠃⣀⠸⠇⣀⠺⠇⣿⡟⠀⣀⣤⣶⣶⡿⣶⣤⡀⠀⠀⠀⢀⣴⡿⢟⣛⠿⢿⣶⣤⡀⢻⣿⠃⡀⠻⠂⡀⠐
⠀⠀⠀⠈⢀⡀⠉⢠⡄⠉⢠⡄⠋⢠⣬⣿⣷⡿⠟⣻⡍⠛⣤⡌⠻⣿⣄⠀⢀⣾⡟⣤⡌⠛⣤⡌⠋⣿⡿⣿⣿⣏⠉⣠⠈⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠶⠀⠀⠶⠀⠹⠿⠉⠡⡶⠈⠱⡶⠈⠱⡶⠘⢿⣦⣼⣿⡷⠈⠱⡶⠈⠰⠶⠈⠠⠮⠙⢿⣆⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘⠃⣀⠘⠃⣀⠘⠃⣀⠺⠃⣀⠺⠃⣀⠻⠋⣿⣿⠃⣀⠻⠃⣀⠚⠃⡀⠚⠃⡀⠘⠀⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠉⢠⡄⠉⢠⡄⠉⢠⡄⠉⢠⡄⠉⢠⡌⠉⢿⡌⠉⢠⡌⠉⢠⡀⠉⢠⠀⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠦⠀⠀⠆⠀⠀⠆⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠂⡀⠘⠁⡀⠘⠃⡀⠘⠀⡀⠘⠀⡀⠘
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣿⣶⣶⣿⣶⣶⣿⣶⣶⣿⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠒⠂⠠⠤⠤⠤⠤⣀⣤⣤⡎⠁⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠤⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠒⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⣉⣉⣐⣒⣒⣀⣒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠦⠠⠤⠤⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠦⠤⠤⠤⠚
⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡷⠒⠚⠲⠖⠘⠻⡌⠙⠢⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⣀⣤⠤⠤⠴⣺⣳⣏⠉⠉⢉⠖⠈⠉⢲⠉⠉⠉⡶⠓⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡹⣇⣉⣁⡶⠂⢉⣱⠒⠈⣑⡆⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠁⠉⠉⠁⠈⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠉⠉⠑⡄⠀⢠⠖⠉⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⡄⢁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣠⠜⡇⢸⠀⠀⠣⡀⢰⣄⡀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⣀⡤⠋⢁⡀⢀⠘⠀⠀⡀⢣⢸⠀⠉⠉⠣⡀⠀⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣀⣀⠀⠀⣸⡠⠦⠤⠥⣶⢸⠀⠀⠀⠀⠈⠁⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠈⠉⠺⣷⣤⠖⠒⠒⠺⢍⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⢀⣹⣿⣇⠓⠀⠀⠀⢀⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣿⡿⠿⠋⠝⠋⠉⠛⠇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣇⠀⢫⠙⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡄
⠀⠀⠀⢀⣀⣀⣀⡀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢻⠀⠀⠀⠀⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡄⣀⠀⠀⡠⠋⡠⠁
⡠⠂⢉⣀⡀⠀⠀⠉⠑⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢀⢸⡀⠀⠀⠀⣇⣰⠙⣢⠓⠂⢄⣠⠖⠋⠀⠀⠀⡇⠈⠙⣏⡠⠞
⡇⠸⡁⠀⢉⡆⡠⢄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⢠⣿⡄⠀⠀⠀⣇⢀⠉⢩⡜⠭⢊⠈⢦⠀⠀⠀⣠⣇⢄⠀⢹
⠘⢄⠈⠉⠉⠀⠑⠊⠀⠀⣸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣁⡠⣿⣿⣿⡇⠀⠀⠀⡇⢻⣌⣹⢃⢤⢸⠀⠀⠑⠒⠚⠁⠳⠴⠁⢸⠃
⠀⠀⠙⠒⠤⢄⣀⣀⠠⠾⢅⣀⣐⠀⠋⠀⠀⠀⣀⣀⣀⠤⠔⠊⠁⠀⠀⣵⢸⡟⢙⡁⠀⠀⢠⠀⠀⠓⠑⠮⢥⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⠀⢸⡇⠀⠀⡸⠀⠀⠀⠀⠀⢸⣷⠦⠀⢠⠤⢠⣤⡴⠚⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⠁⠀⠀⠇⠀⠀⠀⠀⠀⡾⢿⠸⡆⢸⠀⠀⣿⣧⠀⢱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣼⣀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠈⠀⡇⠨⠀⠀⠸⡜⢆⠀⢣⠘⠎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠋⠀⠀⢀⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⢠⡀⢣⠀⣄⡀⢀⡴⠈⠉⣦⠗⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢰⡥⠞⠀⢠⠋⠀⠀⠀⣠⠴⠦⠰⠃⠀⠙⣼⠄⠀⠈⢾⠀⠀⢰⠃⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⢀⠃⠀⠀⠀⠀⠀⢘⡀⣠⠴⠃⠀⠀⠀⠀⡇⡠⠀⡠⠄⣠⠞⠁⠀⠀⠀⠈⠀⠀⢇⠀⡄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⢙⠀⣸⠁⢰⡃⢀⠀⠀⠀⠀⢸⡆⡇⠈⣲⠳⣄⣈⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠈⠓⠞⠂⠀⠓⢾⠀⠀⠢⢤⠞⠉⠉⠉⡁⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡛⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡴⠁⠀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⠑⠂⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⢺⠓⠀⠀⢀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⡟⢦⡀⠀⠀⠀⠈⠐⡄⢀⠎⢷⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠰⠀⡄⠀⢹⡀⠙⢦⡀⡰⣆⠀⢆⠌⠀⠈⢻⡄⠀⠀⠀⠀⠀⣴⡾⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⢧⠀⠀⠙⠇⠈⢳⡜⠀⠀⠀⠀⠻⣆⠀⠀⠈⣰⡿⡁⠀⣺⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠘⢧⠉⠓⠲⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣰⡿⣻⠄⢀⣁⣈⡀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢯⣁
⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣬⣽⣶⣤⡄
⠀⠀⠀⠀⡈⡉⢉⣻⠿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣟⠻⠉⠋⠉
⠀⠀⠀⠐⡿⠛⠉⠀⠀⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⡈⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣠⣶⠿⣇⠀⠀⢀⡀⠀⠀⠀⢠⣿⣦⡀⠀⣿⢿⠞⣟⢙⡛⢾⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠀⠐⣻⢀⣴⣿⢧⠀⠀⢀⣾⣿⣯⣷⣄⣿⠚⡃⢉⡀⠅⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢹⡿⠏⠰⠞⡆⠀⣾⣿⡄⣷⠸⠟⢿⡈⣁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠖⠀⠀⠀⠐⠀⠃⢻⣼⣿⠀⠆⠐⠂⢋⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡿⠀⠀⠐⠀⠁⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠂⢹⣷⣶⣦⣤⣤⣄⡶⡾⣳⢸⣷⣞⣿⣿⣿⡆
⠀⠀⠀⡀⠀⡄⠀⠀⠐⠈⠀⢀⡠⣀⡄⠀⠀⠀⢿⣿⡿⣿⢻⠇⡇⡿⢻⡄⢿⣿⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⡇⡋⠙⠀⠠⠀⠀⠀⠟⠀⣣⣼⠇⡁⢄⣼⢁⠘⣿⡱⣂⢺⡏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⢣⡃⠀⠀⠀⠀⠀⠂⣿⠁⠀⡄⣻⢇⣼⡧⢸⣿⠷⢸⡇⣄⠘⡓⠂⠀⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠘⠀⠁⡇⣋⠨⣿⡗⠘⡙⠒⠿⢿⣥⡂⠧⡧⠃⠐⣿⢷⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⣼⣯⣇⢠⣯⣿⢰⣶⡆⢠⣞⠀⠄⡀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⡏⢟⢠⡿⣿⡽⣿⡧⡛⠳⡄⠀⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠈⣿⡻⡃⠆⠈⠸⠛⠁⣿⠃⠀⠁⣿⠖
⠀⠀⡷⣤⡤⠀⠄⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢮⠴⠃⠀⢰⠎⠀⠈⠰⠁⠀⠀⠉
⠀⠀⠀⢿⠁⠀⠄⠈⣿⣶⣶⣶⣾⣶⣤⡄⣀⣠⠀⠂⠀⠂⢈⡴⠊⠘⠂⠀⠀⣈⠄⠀⢀
⠀⠀⠀⢹⠀⠀⠀⠀⣿⣿⣿⠫⣾⣿⢃⠣⠙⡁⠀⣠⠎⡡⠈⠀⠀⠐⠄⡎⣠⡙⢶⣗⢹⣀⣤⠀⠀⠆
⠀⠀⠀⢺⠀⠀⠀⠀⠩⢿⢿⡄⣿⢿⢀⠔⡋⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣹⣶⡉⢯⣇⣽⣶
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⢸⡗⠛⠘⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⣿⠘⠉⢻⡄⢻⣿⣼⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢠⠀⠀⠃⠀⠀⡀⢠⠀⠀⠀⠀⠀⠁⠀⠘⠀⡀⠸⢿⡀⣿⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢨⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠘⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠘⠀⠀⢀⠴⠂⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⢰⠀⡵⠖⠃⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⢰⣿⠻⣦⡀⠀⠀⢀⣾⠏⢿⣆⠀⠀⢀⣴⠟⢹⣧⠀⠀⠀⠀⠀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⢸⣷⣤⣀⠀⠀⠀⠀⣿⠙⢿⣦⡀⣾⡇⠀⠘⢿⣶⣠⣾⠏⠀⠈⢿⣷⣴⡿⠋⠀⠈⣿⣧⣄⣀⣴⡾⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡈⠛⠿⣶⣤⣰⣿⠀⠀⠙⢿⣿⠁⠀⠀⠀⠙⠿⠋⠀⠀⠀⠈⠿⠏⠀⠀⠀⠀⢹⣿⡿⠛⠁⢀⣿⣇⣤
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⣿⣿⣯⣤⣤⣤⣤⠄
⠘⢿⣿⣶⣶⣶⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⣩⣿⠋
⠀⠀⠙⠻⣦⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡟⠁
⠀⠀⠀⠀⢈⣻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣧⣀⣀
⢠⣴⣶⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢛⣻⣿⣷⠆
⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠯⠁
⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⠿⠟⠛⠉
⠀⠀⠀⢀⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣶⣤⣤⣀⡀
⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⡦
⠘⠛⠛⠛⠛⠻⢟⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡀
⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⣶⣦⣤⣄⣀⡙⢿⣦⣀
⠀⢀⣤⣾⣟⣡⣤⣶⣶⣿⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠈⠉⠉⠛⠛⠿⠿⣿⣷
⠐⠿⠟⠛⠋⠉⠉⣱⣿⠋⠀⣠⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣴⡟⣡⣴⡿⢿⣿⣿⠀⠀⣠⣶⠆⠀⠀⠀⣀⣤⡀⠀⠀⣠⣤⠀⠀⠀⢠⣿⣿⣿⣷⣄⠈⣿⣿⡄
⠀⠀⠀⠀⢀⣾⣿⡿⠛⠁⠐⠁⣿⣇⣴⣾⣿⣟⣠⣤⣶⣿⡿⠋⠀⣠⣾⣿⣿⣷⡀⢀⣿⣿⣿⠃⠈⠻⣷⣼⣿⡁
⠀⠀⠀⢠⡿⠟⠁⠀⠀⠀⠀⢸⣿⠟⠋⢸⣿⠟⠛⣩⣿⠟⢀⣤⣾⣿⣿⠟⠁⠹⣷⣾⡟⠟⠃⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣾⠟⢁⣴⡿⣟⡿⠛⠁⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣁⣴⡿⢫⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⣾⠟⠁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⢀⣴⡿⠛⠁
⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠠⠤⠤⢀⢀⠀⠀⠀⠀⠀⡀⢠⠔⢈⠡⠉⠐⠠⠐⠈⠉⠁⡁⠂⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⠂⠐⢀⠑⠀⠂⠁⠀⠀⣸⢁⠄⠙⠁⢀⣶⡗⠒⠀⢤⣾⡏⠉⠀⠲
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢀⣉⡀⠢⣤⡤⠄⢤⣴⠞⠉⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣴⠿⠂⠀⠸⠿⣷⣶⠀⢀⣨⡿⠂⡘⠀⠀⠘⠇⠀⠀⢁⣐⣠⣀⡀⢿⡇
⠀⠀⠀⠀⢸⣏⠀⠀⠒⠀⠀⣼⡟⠀⠀⢀⠒⢿⠃⢸⣇⠀⠀⠀⢠⣾⠋⠉⠉⠉⢻⡇⠀⢠⠀⠌⠀⠘
⠀⠀⠀⠀⢸⣿⠃⠀⠀⣠⣀⣿⡇⠀⠈⣸⣷⠀⠀⠾⠿⠀⠀⡀⠀⠀⠀⠀⠃⠒⠸⠇⠀⠄⠀⠅⠀⠌
⠀⠀⠀⠀⣾⡏⢠⠉⢀⣿⠁⠈⠀⠀⠓⣉⡕⠁⠠⠄⢴⣄⠀⢡⠀⡀⠀⠊⠀⠀⣴⡖⠀⠤⠀⠀⢩
⠀⠀⠀⠀⢹⣷⡈⠢⠀⠉⠀⠀⡈⡉⠉⠛⢃⠂⠂⠀⠠⡉⠄⠡⠀⠀⠀⠈⠁⠀⠹⣇⢀⡄⠀⠀⡼
⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⣀⣀⣄⠈⠀⠉⠁⣠⣶⣤⣀⣀⣠⣾⡆⠀⠀⠀⠀⣼⣶⣤⣤⡤⠴⠚
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⣿⣷⣶⣶⣿⣿⠟⠛⠛⠿⢿⣿⣧⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠐⣀⠒⠒⠒⢲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠋⣡⣤⣁⠉⣿⠏⠀⠶⡌⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠉⠉⠁⠘⢥⢄⡿⠀⢿⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠐⠊⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠆⠛⢧⣘⣆⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢰⡄⢠⡶⠛⡑⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠆⢛⣧⡼⣧⣋⡽⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⡀⠀⠀⢀⡞⠛⢶⡀⠀⠀⠀⠀⠠⡌⢿⡀⣿⠁⠀⠀⠉⢸⢫⠄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠚⢷⡀⠀⠈⠓⠒⣼⡇⠀⠀⠀⠀⠚⢣⣹⠧⢿⣄⣀⣄⡴⠻⣌⠛⢋⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠓⠒⣻⠁⠉⠠⠄⡀⢀⣀⣼⡇⠀⠀⡍⠉⠉⢸⠀⡈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡴⢻⢀⣇⠀⠀⠀⠀⠀⠀⡅⠀⣻⠶⠀⠂⢠⠄⠈⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠻⣤⠸⠀⠀⠀⠀⠀⠀⠀⡬⠁⠀⠀⠀⢀⠀⠸⠆⠀⡄⠀⠈⠀⠀⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢱⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⠆⠀⠀⠀⣴⠔⠀⠘⠃⣠⠄⠀⢷⡀⠀⢯⡉⣷⣀⠀⠀⣰⡆⠀⠘⠷⠾⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠛⠲⠈⠳⢦⠀⣸⠋⠀⣀⠀⠀⠏⠀⡀⠈⠳⠧⣦⣾⠿⣛⠻⠟⢛⣿⠶⣦⠰⠾⠛⠦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠁⣰⠟⠁⠀⠀⠀⠐⠁⠀⠀⠀⠘⠁⠀⣠⠞⠩⠐⠩⠐⠀⠉⡾⠁⢞⣀⡠⢄⡀⠉⠐
⠀⠀⠀⠀⠀⣠⢊⡴⠊⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⣀⠀⠁⡂⢀⡤⣪⣵⠐⠿⢿⠆⢠⣿⣿⣿⣦⢀⣒
⠀⠀⠀⠀⡐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠀⢈⠄⣠⡾⠋⠀⠀⠐⡫⠆⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⡼⠡⢣⠣⢋⣾⡟⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠡⠃⠣⢁⣿⣿⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠠⠤⠆⠀⠠⠄⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⢀⣠⣄⣀⣀⣤⡄⠈⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠑⠉⠁⠙⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢤⡤⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣉⣩⣤⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣶⣟⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣮⡻⣿⡿⣿⣿⣿⡇
⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⡻⣿⣿⣮⣹⣿⣿⣿⣇
⠀⢰⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⡏
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠉⢻⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠉⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠒⢄⡜⠁⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠤⡀⠀⢀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠈⡄⠀⢣⡀⠀⠀⠁⠀⠀⠀⠀⡤⠊⠉⠀⠈⠉⠓⠦⣀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⠼⢒⡉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡤⠊⢡⣾⡿⠃⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢧⡀⠀⠠⠀⢀⡴⠋⠀⠀⢙⠋⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠉⠓⠒⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠘⡔⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠹⣄⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠀⠀⠀⠀⠀⠀⢈⣷⠤⠤⡴⠋
⠀⠀⠀⠀⠀⠀⢣⠀⢹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠡⢐⣀⣂⣀⣀⣠⡤⠤⢒⠭⠊⠀⢀⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠳⣄⡉⠀⠄⠁⠀⠀⢀⡠⠖⢋⠀⠀⠈⠀⠒⠒⠒⠒⠒⠀⠉⠀⠀⠀⢠⠞⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡳⠒⢒⠒⠉⠁⠀⠀⣠⠲⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⢸⠀⠀⢀⠖⠒⢇⠀⠀⠈⠉⡙⠛⠙⠛⣋⡉⠉⠁⠀⠀⠀⣀⣀⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸⠀⠀⠸⡄⠀⠀⠁⢀⣀⣀⣁⣁⡰⠋⠀⠙⡆⠀⢀⡴⠋⠉⠉⠉⠳⣄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡼⠀⢀⠖⠛⠂⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠐⠅⢀⡞⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠒⠒⠚⠛⠛⠹⡀⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢄⠆⠀⠑⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠆⠀⠀⠀⠀⠀⠀⠢⢤⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⠔⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⣄⣀⡤⢦⡤⢄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠴⠚⠁⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠁⣿⠀⠀⠀⠀⡎⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⢤⣦⡤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⣀⡏⠀⠀⠀⠀⢇⡄⡀⡜⠑⠢⠤⣤⣄⣀⣀⣀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠳⠦⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⣰⣾⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣴⣶⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⠙⠓⠀⠀⠒⠛⠛⠛⠉⢉⣉⡙⠿⣿⢿⡻⠍⢉⣿⣿⠿⠟⠛⠋⡙⠛⠛⠛⠛⠩⠁⠀⠺⠋⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠄⠀⠀⠀⠀⠻⡟⠋⠉⠭⢤⡲⢕⠈⠀⡴⢋⡴⠃⣠⠭⠉⠙⣛⡗⠂⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⢀⣀⣤⣴⣿⡿⠋⢠⠀⡆⡐⠓⠤⠠⠤⠦⠄⠶⢦⡘⠣⠈⠰⠋⢁⣤⠶⢂⠤⠤⠄⡠⠔⢂⢰⠀⡄⠑⢿⣿⣧⣄⣀
⠀⠀⠀⠀⠈⠻⢿⣿⣿⠊⣴⡟⡀⠈⠳⣾⣦⠶⠦⣠⡀⣄⠀⠘⣷⡀⢀⣶⠋⠀⢀⢀⣄⡤⣴⣤⣦⠞⠁⢀⢻⣦⠲⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⣰⣿⢟⠁⠀⡾⠉⡇⢧⡀⠳⢝⣢⣄⣀⣽⣿⣴⣞⣟⠀⠀⢿⣳⣦⣿⣟⣁⣀⣴⠋⠟⢀⡼⢸⠐⣯⡀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⣿⣿⠃⢸⡇⢠⣠⡂⠸⠁⠀⠲⣭⣤⣶⠺⠋⠛⠁⠈⡇⢸⠁⠘⠛⠻⠿⣾⣍⣉⡴⠀⠘⡇⠐⣄⡌⢸⡂⠘⣿⣿
⠀⠀⠀⠀⣼⣿⡥⠂⣼⠀⡜⢱⠃⠀⣧⡘⣶⣼⠋⠁⢝⠂⠀⣀⣶⠄⢠⣶⣀⡀⠀⣚⠜⠙⣧⣶⠃⣼⠀⠸⡎⢣⠀⣇⠰⢬⣿⣇
⠀⠀⠀⠀⣿⣟⢀⢀⡿⣠⠇⠀⠠⢀⠈⠳⣸⡌⠢⠀⠘⠗⢒⣁⣉⠁⠀⢋⡉⡲⠺⠅⠀⠴⢣⣇⠞⠁⠀⡄⠀⢸⡁⢿⡁⡈⣿⣿
⠀⠀⠀⠀⣿⣷⠟⠸⡀⡅⢸⣄⠠⡄⠳⣄⢳⡙⢢⡙⠂⠀⠀⠈⠙⢄⡠⠋⠉⠀⠀⠐⠋⡐⠋⡝⢠⡜⢀⡄⣀⡇⢸⢈⠇⠻⣽⣿
⠀⠀⠀⠀⢹⣿⣦⠸⢻⣧⢠⠙⢇⢐⡤⠹⡜⢿⣂⣈⣁⣀⣠⣤⣴⢞⣳⣤⣤⣀⠀⠀⢉⢐⡿⢃⠏⢤⠊⣸⠋⡄⣼⡿⠆⣤⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣤⠀⢸⣷⢰⣀⡈⢷⡶⡁⢀⡽⡌⠁⣿⣦⣷⣶⣾⣷⣾⣼⣵⣿⠉⢩⢏⡀⢈⢴⣾⢁⣀⣄⣽⡇⠀⣤⣿⣿⡇
⠀⠀⠀⠀⠘⠛⣿⣿⡄⠀⣏⠀⣿⣿⣦⣝⠸⣦⣿⣿⣜⡏⣿⣿⣿⣿⣿⣿⣿⣿⣻⢣⣿⣿⣵⠇⣮⣵⣿⣿⠀⢻⠀⢤⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⣿⣢⠇⠛⣿⣿⢠⣿⣿⣿⠅⠁⣸⡟⠁⡝⢻⠉⢿⣆⠈⠩⣿⣿⣿⡄⣿⣿⠟⠸⡤⣿⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠻⢻⣿⣿⣿⣿⣶⣷⣥⡻⠀⢿⣿⡏⣼⣆⣟⠀⠀⠀⠀⠀⠈⣻⣰⣦⢹⣿⡿⠀⢟⣭⣶⣦⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣿⣿⣿⠛⠻⣿⣼⡈⢩⠁⡏⢿⣿⣷⣟⢿⣿⣳⢶⣿⡿⢹⠈⡙⢁⣧⣿⡿⠻⣿⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠀⠀⣿⣿⡇⠀⢠⡇⢈⡷⠋⠠⡤⠤⠈⠛⣼⡁⢸⡄⠁⢸⣿⣿⠀⠀⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣤⠸⡟⡋⠔⡾⠛⠉⠙⠻⢷⠠⠙⢛⠇⣠⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⡆⡀⠀⠃⡄⢠⠐⠈⠀⢨⠀⣀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣷⣇⢀⢄⡟⢿⠀⡀⢸⣾⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣌⣯⠁⠘⣵⣡⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣷⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣠⣴⣶⣶⣦⣄⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠿⠿⣿⣿⣿⣭⣤⣤⣌⡙⠻⣧⣭⣛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢉⣠⣥⣤⡴⠶⡶⠛⢉⠉⣉⣤⣤⣶⣶⡾⠿⠿⠷⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⣡⡾⠫⠕⣂⣠⠖⣫⣶⣒⡟⠐⣿⡏⠀⠀⠲⢶⣶⣒⡒⠀⡀⠀⠐⠒⠮⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⢛⣛⣋⠋⣽⠿⢖⣛⡭⠉⠀⢨⣾⣿⡟⠁⠀⠈⠻⣦⡀⡀⠀⠀⠀⢀⣾⣄⡀⠀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠛⣩⡿⠋⣡⡵⠞⣋⣭⠄⡰⢠⢸⡎⠛⠿⠷⠚⠀⢲⡘⣿⣧⠀⠀⢠⢋⠔⣠⣾⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⣾⡟⢠⠞⢁⣴⣿⡟⢁⡴⢣⡟⠈⣿⣶⣰⡀⠀⣧⣸⣿⣿⣿⡆⠀⠃⣧⣾⣿⠟⠉⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣡⣾⠟⣽⡟⢐⣡⡶⠟⣠⠾⠛⣉⠻⠿⢠⡿⠋⢸⢻⣿⠀⠀⢸⡍⢿⠃
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣶⣿⠟⣡⠀⣿⣷⠟⡋⢀⣴⣾⠟⢋⡥⠂⡀⢋⡴⠀⢀⣼⠇⢀⠀⢸⣟⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⡏⢰⡟⠀⣿⠏⡾⢱⣿⡿⠁⣴⡟⠁⣼⠃⣾⢁⣾⡿⣿⢠⡇⠀⠈⣿⣦⣨⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⢠⣿⢸⡇⢸⣿⡇⢸⡟⠇⡤⣿⠀⣿⠈⣿⢁⡷⣿⣡⡆⢀⠈⢯⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡾⠁⣼⡏⣸⢃⣿⣿⣧⢸⠁⢀⢷⡿⠀⣿⣼⡟⣀⠰⠟⣿⣵⡿⣠⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃⢀⣠⣾⡟⢠⢃⣾⡟⢸⣿⠀⢠⢮⡾⠃⢠⣿⠟⠁⡹⣄⣤⠀⣾⣿⡵⠞⠋
⠀⠀⠀⠀⠀⠐⠲⠿⣿⣿⣿⣿⡿⣿⣿⡇⠀⢸⣿⢰⢸⣿⠀⣴⣟⠀⢰⠟⢁⢰⣆⠙⠮⣻⢷⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠸⣿⡸⡼⣿⢰⡟⢸⢷⣄⠠⢼⡌⣿⣿⠶⢶⣦⣽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⣷⣅⠻⣿⣷⡀⠀⠉⠻⠶⠶⣾⡿⠷⠦⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠙⠓


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣽⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠛⠶⢦⣄⡀⠀⠀⣀⣠⣤⠴⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⣀⣤⡴⠶⠶⠶⠦⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⢦⣀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠈⠻⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⣀⣤⣤⢤⣤⣄⡀
⠀⠀⠀⠉⠙⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣶⣦⣤⠞⠉⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⣀⣠⣤⣤⣴⡟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⡾⠋⠁⠀⠀⠈⠀⠀⠀⣀⣤⠶⣷⡶⠂⠀⠀⠀⠀⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⢀⣠⡴⢞⣩⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⠾⠋⠁⠀⠛⠷⠶⠦⣤⣄⠀⠀⠀⢀⣠⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣻⡶⠀⠀⣀⣤⠾⠿⠷⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⢀⣴⣟⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠧⣤⣤⣀⣀⡈⠉⠉⠙⠛⠛⠳⠶⢶⣦⠄⠀⠀⠀⣀⣴⣾⠂⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⡷⠂⠀⠀⣀⡴⠟⠁⠀⠀⣠⠾⠋⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⠏⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⠞⠋⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠘⠷⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠈⣿⠛⠳⠾⠋⠁⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠘⠛⣳⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣠⣤⡴⠶⠛⠛⠉⠁

⠀⠀⣡⡀⠀⠀⢀⣿⣄⠀⢀⣠⠛⠁⣄⡀⠀⣠⣾⠀⠀⠀⠀⣻⠁⠈⣿⠀⠀⠀⢱⡀⠀⠀⢀⣿⣄⡀⠀⣠⠚⠁⣄⡀⢀⣀⣾⠁⠀⠀⠀⣻⠁⠈⣻
⣤⡶⠋⠉⢲⡖⠀⠀⠈⢹⡿⠟⠲⠶⣿⠋⠁⠈⠙⣧⣄⣠⣴⠟⠓⠲⣿⣷⣤⣴⠋⠉⢲⡞⠀⠀⠀⢹⡿⠟⠲⠶⣿⠋⠉⠉⠙⣧⣄⣠⣴⠟⠓⠲⣿
⠀⢹⣤⠄⠾⢧⡀⠀⠀⣸⠀⠀⠀⠀⢹⡀⠀⠀⢀⡟⠉⠙⢇⠀⠀⠀⣸⠁⠀⢹⣤⡄⠾⢧⡀⠀⠀⣸⠁⠀⠀⠀⢹⡀⠀⠀⢀⡟⠉⠙⢏⠀⠀⠀⢸
⠒⢿⣏⠀⠀⠀⠟⠃⠉⠙⢦⡀⢀⣠⣾⠛⣶⣶⡿⣧⣄⣠⠿⠶⢤⣶⡿⠓⠒⢿⣏⠀⠀⠀⠟⠃⠉⠙⢦⣀⢀⣠⣾⡛⣲⣶⡿⢧⣄⣠⠾⠶⢤⣶⡿
⠀⠀⡿⠗⠶⡇⠀⠀⠀⠀⠀⡿⠉⠀⠀⠉⢻⠃⠀⠀⢙⠁⠀⠀⠀⢹⠀⠀⠀⠀⡿⠗⠶⡎⠀⠀⠀⠀⠀⡿⠋⠀⠀⠉⢻⠃⠀⠀⢹⠁⠀⠀⠀⢹⠁
⣤⣴⡀⠀⠀⢰⣄⠀⠀⢀⣴⣇⠀⠀⠀⠀⢸⠧⣤⣤⣾⣄⡀⠀⣠⠾⢷⣤⣤⣴⡁⠀⠀⢰⣄⠀⠀⢀⣴⣇⠀⠀⠀⠀⢸⡧⣤⣤⣾⣄⡀⠀⣀⠾⢷
⠀⠀⠙⣶⠖⠉⠉⠙⢿⠋⠀⠈⢲⣤⣤⣴⠁⠀⠀⢹⠋⠀⠈⢻⣇⠀⣨⠏⠁⠀⠙⣶⠖⠉⠉⠙⢿⠋⠀⠈⢲⣤⣤⣴⠀⠀⠀⢹⠋⠀⠈⢻⣇⠀⣠
⠀⠀⢀⣿⠀⠀⠀⠀⠀⣤⣀⣠⡎⠀⠀⢹⣦⣀⣠⡾⠆⣀⣠⠾⠛⠛⠿⣄⠀⠀⢀⣾⠀⠀⠀⠀⠀⣦⣀⣠⡞⠁⠀⠹⣦⣀⣠⡾⠦⣀⣠⠾⠛⠛⠿
⡷⠂⠉⠙⢦⣀⣀⣠⣾⠁⠀⠙⣷⣤⣤⡾⠁⠀⠙⣦⣤⣿⡇⠀⠀⠀⠀⢹⡷⠂⠉⠙⢦⣀⣀⣠⣾⠁⠀⠙⣷⣤⣤⡾⠁⠀⠙⣦⣠⣿⡏
⡅⠀⠀⠀⠘⠋⠀⠀⠈⢦⣤⡞⠉⠀⠈⠻⣄⣀⣴⡏⠀⠈⣀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠘⠋⠀⠀⠈⢦⣤⡜⠉⠀⠈⠻⣤⣀⣴⡏⠀⠀⣡
⢹⣶⠤⠶⢧⠀⠀⠀⠀⣬⣬⡅⠀⠀⠀⠀⠋⠀⠀⠉⢶⣶⠟⠛⠲⣶⡏⠉⢹⣶⠤⠶⢧⠀⠀⠀⠀⣨⣤⡄⠀⠀⠀⠀⠋⠁⠀⠉⢶⣶⠟⠛⠲⣶⡏
⠛⠇⠀⠀⢈⡷⠖⠲⡞⠀⠀⢹⠆⠠⣴⣾⡀⠀⠀⠀⣼⣇⠀⠀⠀⣸⠟⠓⠛⠏⠀⠀⢈⣷⠖⠰⣞⠁⠀⢹⠆⠠⣴⣾⡀⠀⠀⠀⣼⣯⠀⠀⠀⣸⠟
⠀⢸⣶⣶⣾⣇⠀⢀⡼⠦⠰⢶⣀⡀⠛⠉⠉⠐⢶⡏⠀⠀⢱⠖⠚⣇⠀⠀⠀⢈⣷⣶⣾⣇⠀⠀⡼⠦⠰⢾⣀⡀⠛⠉⠉⠐⢶⡏⠀⠀⢱⠖⠚⢯
⠴⢿⡟⠁⠀⠀⠘⡟⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢈⡿⠆⠐⢿⣀⣀⣼⣷⡤⠴⢿⡿⠁⠀⠀⠈⡟⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢈⡿⠆⠐⢿⣀⣀⣼⣷
⠀⠀⣡⡀⠀⠀⢀⣿⣄⠀⠀⣠⠛⠃⣀⡀⠀⣀⣾⠀⠀⠀⠀⣻⠉⠈⣿⠀⠀⠀⢱⡀⠀⠀⢀⣶⣄⠀⠀⣠⠚⠁⣄⡀⠀⣀⣼⠁⠀⠀⠀⣻⠉⠈⣻
⣤⣴⠋⠉⢲⡖⠁⠀⠈⢻⡿⠟⠢⢶⣿⠏⠉⠉⠙⣦⣄⣀⣴⠟⠓⠲⣿⣶⣤⣴⠋⠉⢲⡖⠀⠀⠈⢹⣿⠿⠲⠶⣿⠏⠉⠉⠙⣦⣄⣀⣴⠟⠓⠲⣿
⠀⢹⣤⠄⠾⢇⠀⠀⠀⣸⠁⠀⠀⠀⢹⡀⠀⠀⢀⡟⠉⠙⢇⠀⠀⠀⢸⠁⠀⢹⣤⡤⠾⢧⡀⠀⠀⣰⠁⠀⠀⠀⢹⡀⠀⠀⢀⡟⠉⠙⢏⠀⠀⠀⢸
⠒⢿⣏⠀⠀⢈⠟⠃⠉⠛⢆⡀⠀⣠⣾⡛⣶⣶⡿⣧⣀⣠⠾⠶⢤⣶⡿⠗⠒⢿⣏⠀⠀⠀⠟⠃⠉⠛⢆⡀⠀⣠⣾⡛⣲⣶⣿⣧⣀⣠⠾⠶⢤⣴⡿
⠀⠀⡿⠖⠲⡆⠀⠀⠀⠀⠀⡿⠋⠀⠀⠉⢿⠃⠀⠀⠙⠁⠀⠀⠀⢹⠁⠀⠀⠀⡿⠗⠶⡎⠀⠀⠀⠀⠀⡿⠋⠀⠀⠉⢻⠃⠀⠀⢹⠁⠀⠀⠀⢹⠁
⣤⣴⡁⠀⠀⢰⣄⠀⠀⢀⣴⣇⠀⠀⠀⠀⢸⡧⣤⣤⣾⣄⡀⠀⣀⠾⣷⣤⣤⣴⡁⠀⠀⢰⣄⠀⠀⢀⣰⣇⠀⠀⠀⠀⢸⣧⣤⣤⣾⣄⡀⠀⣀⠾⣷
⠁⠀⠙⣶⠖⠉⠉⠙⢿⠋⠀⠈⢢⣤⣤⣴⠁⠀⠀⢹⠋⠀⠈⢻⣇⠀⣨⠏⠁⠀⠙⣶⠖⠋⠉⠙⢿⠋⠀⠈⢢⣤⣤⣴⠁⠀⠀⢹⠋⠀⠈⢻⣇⠀⢨
⠀⠀⢀⣟⠀⠀⠀⠀⠀⣤⣀⣠⡞⠀⠀⢹⣦⣀⣠⡾⠆⣀⣠⠾⠛⠛⢿⡄⠀⠀⢀⣿⠀⠀⠀⠀⠀⣤⣀⣠⡞⠁⠀⠙⣦⣀⣠⡾⠆⣀⣀⡼⠛⠛⠿
⡷⠂⠉⠙⢦⣀⣀⣠⣾⠁⠀⠙⣷⣤⣠⡼⠁⠀⠙⣦⣠⣿⠇⠀⠀⠀⠀⢹⡷⠂⠉⠙⢦⣀⡀⣠⣼⠁⠀⠙⣷⣤⣠⡼⠁⠀⠙⣦⣠⣿⡏
⡅⠀⠀⠀⠘⠋⠀⠀⠈⢦⣤⡜⠉⠀⠉⠳⣄⣀⣴⡏⠀⠈⣁⠀⠀⠀⠀⣸⡇⠀⠀⠀⠘⠋⠀⠀⠈⢦⣤⡜⠉⠀⠉⠻⣄⣀⣴⡏⠀⠈⢁
⢹⣶⠤⠶⣧⠀⠀⠀⠀⣨⣬⡅⠀⠀⠀⠀⠋⠁⠀⠉⢶⣶⠟⠛⠲⣶⡏⠉⢹⣶⡤⠶⣧⠀⠀⠀⠀⣨⣬⡄⠀⠀⠀⠀⠋⠁⠀⠉⢶⣶⠟⠛⠲⣶⡏
⠛⠏⠀⠀⢈⣷⠆⠶⡞⠁⠀⢹⠆⠠⣴⣾⡀⠀⠀⠀⣼⣧⠀⠀⠀⣸⠟⠛⠛⠏⠀⠀⢈⣷⠆⠰⡞⠁⠀⢹⠦⠠⣤⣾⡀⠀⠀⠀⣸⣯⠀⠀⠀⣸⠟
⠀⢸⣶⣶⣾⣇⠀⠀⡼⠦⠰⢶⣀⡀⠛⠉⠉⠒⢶⡏⠀⠀⢳⠖⠺⣏⠀⠀⠀⢈⣶⣶⣾⣇⠀⠀⡸⠦⠰⢾⣀⡀⠛⠉⠉⠐⢶⡎⠀⠀⢱⠖⠚⢏
⠴⢿⡿⠁⠀⠀⠙⡟⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢈⡷⠆⠰⢿⣀⢀⣼⣷⣤⠴⢿⡿⠁⠀⠀⠙⡿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢈⡿⠆⠐⢿⣀⢀⣼⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⢨⣶⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠳⠆⠀⠀⠀⠘⠛⠁⣸⣿⠀⠀⡴⠂⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠈⠁⠉⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀
⠀⠀⠀⠤⠄⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣚⡴⠶⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠶⠶⠶⠦
⠀⠀⠀⠀⠀⢀⡀⣈⣁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⢻⣤⡀⠀⠀⠀⠀⠀⠀⠋⠛⠛⠛⠀⢀⣠⡀⠀⢰⡶⠶⠶⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⡄⠀⣀⠀⢰⣤⡘⣿⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠟⠋⠀⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠖⠘⠉⠄⠚⠋⠄⢸⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡐⠄⠀⠀⢀⠠⠊⠂⢀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⡀⠈⠂⠄⠊⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⢁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉
⠀⠀⠀⠐⠒⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⣠⠴⠐⠈⠉⠈⠉⠙⢆⡼⠛⠛⠛⠛⠲⠮⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⢬⣦⡀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠚⠛⢦⡸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠹⡀⠀⡼⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠑⢄⠀⠱⡔⢰⣇⠄⡇⠀⢀⡄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠤⠀⠁⠀⠁⠈⠁⠸⠇⣠⠎
⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠙⠦⠀⠊⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣂⠄⠋⠁
⠀⠀⠀⠀⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣤⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⡀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠠⡤
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⣄⡀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠢⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠚⠁⠈⠡⠈⣠⠀⠀⠀⠀⠀⠀⢠⡈⠣⡁⠀⠙⠂
⠀⠀⠀⠀⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⡴⠃⢠⠆⢰⡇⡘⡇⠁⠱⠄
⢀⣠⠄⠈⠐⠊⠉⠈⣀⣴⡷⡀⢀⣾⠟⠒⠢⠄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⢠⠎⠈⢄⠁⠀⠳
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠴⠚⠛⠛⠶⠚⠋⠙⠓⠤⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡀
⠀⣠⠔⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡀
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠀⣀⣀⢀⣀⣀
⠀⠙⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠙⠁⠀⠉⠋⠁⠈⠉⠀⠈⠉⠉⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠘⠦⣄⣀⡀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣀⡴⠃⣠⣤⣄⠀⠀⢀⣀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠘⢧⣄⣾⠀⠀⠳⠼⠀⠀⢳⡀⠀⣀⡀⢀⣤⡀⠀⣠⢤⡀⠀⣀⡴⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠉⠉⠀⠀⢀⣤⡀⠀⠉⠀⢧⡀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠉⢇⠀⠀⢰⠁⠀⠸⠀⠀⠀⠻⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠶⢸⠀⠀⢰⠶⠀⠀⠁⠀⠀⠀⢇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢠⡎⠀⠀⢸⠠⠀⠚⠀⠀⠀⠀⠈⢁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠉⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠏⠀⠄⣀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠖⠒⠚⠛⠲⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⢀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣹⣶⣶⣶⣯⣤⣤⣤⣤
⣿⣿⣏⣉⣩⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⢶⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠙⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⠾⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⢀⣸⣧⣤⣤⣠⡟⠁⠀⠀⠀⠀⠀⠈⢻⣄⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠻⣦⣴⠶⠛⠛⠛⠶⣦⡀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠈⢻⣄⣤⣤⣤⣀⡀⢿⡄
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠻⣾⡇
⠀⠀⢀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣶⡀
⠀⠀⠈⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⢻⡿⠛⠛⠛⣻⡿⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⣴⣿⣤⣬⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣤⣤⣤⣤⠀⠀⢠⡿⠁⠈⠙⠛⠛⠛⠛⠛⠛⢛⣿⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠐⠛⠛⠛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⣿⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟

⣴⣶⣶⠀⠀⢠⣦⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣽⢾⣲⣽⠏
⣧⣿⣿⡀⠀⠘⣏⣻⣿⠀⠀⠻⣟⣛⡷⠶⣶⢶⣦⣄⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠟
⣿⣿⣿⠁⠀⠸⣿⣿⣿⠀⠀⠀⠈⠙⠳⠶⠧⠶⢿⣻⣿⣿⣿⣾⣿⣷⡦⣀
⢻⣿⡏⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠛⠞⢦⣼⣟⠶⢷⣶⣶⡤⢤⣤⢤⡤⠀⠀⠀⠀⠀⡞⠳
⠈⣿⠧⠆⠀⠀⠸⣿⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠖⠀⠀⠀⠈⠛⠾⠧⡼⠟⠷⠖⠋⠁⠀⠀⠀⠀⠘⠹⠶

⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⠄⠄⡤⡤⠤⣤⠤⠤⠄⠀⠠⠆⠦⠤⠶⠴⠠⠤⠤⠤⠦⠠⠤⠀⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣖⣶⣆⣲⣶⣶⣒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢸⣧⠬⠃⢠⡟⠀⠀⠀⣤⠈⣀⣴⠶⠋⠁⠐⠉⠀⣀⠴⠶⠈⠉⠉⠁⣀⢀⣀⡀⠀⠀⠀⣈⣉⠉⠀⠦⣄⠀⠁
⢸⡧⣤⣈⠳⡄⠈⡀⣠⡴⠟⢋⠠⠀⠀⢀⣤⡶⠋⠀⠀⠐⠲⠶⠦⠀⠙⢿⡿⠿⠶⠀⣶⣿⣿⠇⠐⠀⠈⠓⢤⡀
⢸⡇⣠⣝⣓⣬⡾⠛⠉⠴⠊⠁⠀⠖⣠⣿⣏⠀⢀⣠⣄⣀⣀⠀⠀⠈⠻⢿⣷⣦⡀⠀⠀⠙⢿⣆⠀⠀⠀⢰⣾⣿⣠
⢸⣇⣷⠖⠋⣡⠖⠋⠀⠀⠀⢀⣤⣼⠿⣿⣯⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⠀⠉⠻⢿⣦⡀⠀⠀⠙⢷⡀⠀⠀⢿⣿⣿
⢸⣏⣤⠐⠋⠁⠀⢀⣾⠄⠀⠙⣻⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠈⠻⣿
⢸⡇⠀⠀⠀⠀⢠⠈⠁⣀⣤⣤⣝⣿⣿⣿⣿⡿⠿⠛⠙⠋⠙⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠐⢦⠘
⢸⡇⠀⢀⣀⠀⠇⢀⣴⣿⠻⣿⣿⣿⠿⠋⠁⠀⠀⣀⣤⣴⣾⠿⣿⣿⣿⣿⣿⣇⠀⢠⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⢹
⢸⡇⠠⢼⠙⠀⢃⣾⣿⣫⣾⠿⠋⠁⠀⠀⠀⣴⣿⡿⠋⠉⠀⠀⠀⢉⡛⣻⣿⣿⡆⠈⢧⡀⠀⠹⣷⣦⡀⠀⠀⠀⢻
⢸⣧⣄⠘⠇⠀⢹⣿⣿⣿⠁⢀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣷⡀⠈⢿⣦⡄⢹⣿⣿⣆
⢸⣧⠘⠃⠀⠀⠸⣿⣿⣿⡀⠋⠁⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣧⠀⠘⢻⣿⡄⠹⣿⣿⣷⡄
⢘⡃⠀⠀⠀⠀⡀⢸⣿⣿⠀⠀⠀⠀⣤⠀⠀⣾⣿⣿⣿⣿⡟⣿⣿⠷⠚⠉⠀⠈⢻⣿⡆⠀⠀⠈⢿⣆⠹⣿⣿⠿⣶
⠈⠁⠀⠀⠀⠀⠁⠀⣿⣿⣄⣠⣴⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠘⣿⡆⠘⢿⣷⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠘⣿⣿⣿⣿⣿⣟⣷⠀⠸⠋⠉⠻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠨⢻⣆⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠀⠈⠀⠀⠀⠛⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⢢⣿⣧⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠻⣿⣷⠼⣿⡿⠀⠈⠃⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣆⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠙⣿⡇⠀⠳⣀⣀⣴⣾⣿⣷⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢿⣿⣦⣀⠰⣎⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠹⣿⡄⠀⢈⣿⣿⣿⣿⠛⢤⣀⠀⢠⣄⣀⠀⣄⠀⠀⠀⠀⠀⠈⠙⣷⣦⡈⢢⡙⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠘⣿⣄⠀⠿⠛⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⡎⣻⣿⣤⡙⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠘⣿⡆⠀⣿⣿⠟⠋⢉⣉⣥⣬⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠹⣿⡞⠻⣿⣶⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⠃⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠹⣿⡄⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢻⣿⣆⠀⠀⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢛⣿⣿⡿⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢸⣿⣿⣆⠀⠻⢿⣿⣿⠃⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃⠘⣿⠟⢁⡰⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡃⠀⢻⣿⣿⣄⢀⣴⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⠟⠀⣠⠞⢁⣴⣟⣣⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⠞⢁⣴⠋⣭⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⡞⠁⢀⠬⣠⣿⡟⠁⡘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢄⡹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣶⣿⣟⣰⠊⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣷⡻⣿⣿⣧⣌⠙⣛⠛⢿⣿⣿⣿⡇⠐⣯⣽⣿⣿⠀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⡀⣡⢿⣿⣌⠟⠇⠀⠻⣿⣿⡇⠀⣿⣿⣿⣿⡶⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢡⣾⣿⣿⣷⣾⣷⣄⠹⣿⣿⣈⣿⣿⠟⣫⣾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⣸⣿⣿⣯⣻⣿⡿⢿⡿⠿⠻⢻⣿⣴⣿⠁⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⣼⡟⣸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣩⣾⣿⡿⠛⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠰⣿⣿⢿⣿⣿⣿⣿⡿⠛⠁⠰⠿⠋⢛⣁⣴⣿⣿⣿⣿⣿⣿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⣄⠀⠀⡀⠙⣿⡄⢿⣿⡿⣯⣶⣈⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡋⣧⠀⢠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⡜⢻⡂⠈⢻⣦⣿⣿⡞⣋⣿⣮⣽⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⡄⢡⣼
⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠄⠀⣬⡵⠀⠀⢹⠿⠋⣀⣿⣿⣿⡿⠿⢿⡛⠋⢀⡤⠂⠀⠀⠛⠉⠳⢿⣿⣾⣿
⡇⠀⠀⠀⠀⠀⢀⠔⠁⣀⡀⠀⠀⠀⣠⣿⡇⣠⠔⢉⠄⠀⠀⡈⠙⠋⢀⡼⠉⣰⣶⣇⢠⠖⠖⠀⠀⢠⣦⣼⣽⣿⣿
⡇⠀⠀⠀⢀⡴⠃⠀⠀⠈⢻⣶⢄⣤⣹⣿⡟⣡⣾⠟⠀⠀⣴⠀⠀⣴⠟⣡⡾⠛⠀⠁⠀⠀⢀⣤⣬⣠⣿⣿⣿⣿⣿
⡇⠀⠀⣴⠋⢰⠆⢻⠠⠄⠈⠁⢨⣭⣽⣥⣞⣿⡅⢀⣀⢰⡟⠀⣼⣧⡾⢫⡐⣀⠀⠀⣀⠰⣭⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⣸⠇⠀⠀⣀⠀⠀⠀⠀⢀⡼⠛⣿⣧⣈⡉⠵⠈⠛⣸⡏⣰⣿⠋⣐⣰⠟⠀⢀⣠⣭⣷⣿⣿⣿⣿⣿⣿⠿⣦⢸
⡇⣠⠏⠀⠂⣸⣿⡀⠀⢸⣦⠞⣿⠎⣷⢻⡏⠱⠒⣿⠀⣿⣴⡿⠡⡀⠛⠁⣀⣶⣾⣿⣿⣿⣿⣿⣿⡿⢯⡅⢴⣿⠻
⣷⠟⠤⠀⢠⣿⣯⣵⣴⣿⢋⣾⣿⣼⣧⠀⠀⢷⠀⣿⡀⣿⠟⣀⣶⠁⢀⣴⣿⣿⣿⣿⣿⣿⠿⠟⠧⣀⠌⠑⣿⠇⢰
⡏⢀⠀⠘⣾⣿⣿⣳⡿⠁⣾⡏⠹⠿⠛⠀⠀⠸⣧⠘⣿⡿⢢⡟⢁⣠⣾⣿⣿⣿⣿⣿⣯⡥⠖⠤⠄⠈⠷⣌⡿⠤⣸
⡷⣀⣀⠀⣿⣿⣿⠿⡷⢞⡿⢃⣤⠠⡆⣁⠰⣄⢻⣶⣿⠇⣾⠙⣤⡟⣳⣿⣿⣛⣩⣀⣀⡀⠀⢠⡀⣠⠄⢩⠇⣴⢿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠉⠀⠀⠀⠉⠁⠉⠙⢿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠉⠛⢿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⡖⠀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠛⠿⠿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣤⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣶⣆⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠤⢀⠔⠉⠐⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⡠⠁⠈⠔⡠⢂⠂⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⠂⠄⠀⡠⠠⠂⠀⠀⠠⠊⠀⠀⠀⢀⠔⡡⡀⠁⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⣀⠀⠀⠀⠀⢀⢀⠁⠈⠈⠈⠉⡠⠁
⠀⠀⠀⠀⠀⠀⠃⢠⠀⡠⠀⠴⠀⠀⠖⠀⠰⠂⢀⠄⠀⠀⡐⠁⢀⠟⠉⠐⠁⠀⠤⠂⠀⠀⠀⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠠⠂⠠⠀⠰⠀⠀⠃⡔⢀⠊⠀⠀⠠⠮⠄⠀⢲⣂⠀⡰⠂⠀⢈⡀⠀⠚
⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⢀⠀⢠⠀⠀⡰⠂⠀⠰⠇⠒⠒⠒⠒⠀⠀⠀⠄⠀⠊⠠⠆⠀⠈⡠⠀⠐
⠀⠀⠀⠀⠀⠀⡌⠀⠀⡄⠀⠉⠀⠃⡤⠀⢀⠀⠀⠀⠰⠂⠠⠊⠀⠀⢀⠔⠁⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⠀⠰⠁⡄⠀⠀⠊⠀⠸⠁⢀⠔⠁⠀⢀⠔⠁⡰⠀⡀⠀⠊⠀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠘⠀⠈⠀⠠⠂⠀⠔⠁⠀⠀⠀⠀⢀⠀⠀⠈⠀⣀⠀⠋⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢠⠇⠀⠀⠄⡠⠂⠀⡀⠈⠀⠀⠀⠋⢀⠀⠀⠈⠁⠀⠜
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⢠⣊⣀⡀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡨⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⡔⠀⠀⠀⠀⡠⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠈

⠀⣠⠤⣤⣄⠀⠀⠀⢰⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣺⣷⣶⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣶⣶⣄⠀⠀⠰⠖⠗
⠀⣷⣿⣿⣿⢷⣲⣶⣌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣾⣿⣿⢿⡆⠀⠀⣀⣀
⠀⠙⠿⠋⣣⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢧⣼⣿⡆⢰⣾⠿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣟⠗⣲⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⣻⠟⣿⣿⣿⣦
⠀⠀⣪⣾⠗⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡟⣷⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣿⣿⣷⡀⣿⣼⣿⣿⣿⣿
⠀⠈⠿⣧⢷⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠙⠿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠘⢿⣿⣿⣿⠟





⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠸⠿⠟⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠀⢀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣵⣿⣷⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⢠⣾⢷⣏⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣊⣿⣿⣷⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠝⠀⠀⣬⣾⡷
⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣹⣿⣇⢛⣋⣭⣭⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⣈⣭⣄⡀
⡠⣪⣿⡻⣿⡿⠎⠛⠿⠛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⣟⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣷⡜⣵⣾⣿⣿⡆
⢿⣿⣿⣿⡷⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡝⠿⠟⠊⢻⣿⣿⣿⢟⡇
⠈⠙⠛⠋⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉





⠀⣼⣿⣧⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣠⢖⣶⣦⡄⠀⠀⣮⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢠⣪⣿⣿⣦
⠀⠙⠛⠁⠀⢀⣀⢧⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢇⣼⣿⡆⠸⣿⣿⣿⡿
⠀⠀⠀⠀⣄⣽⣿⣿⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣛⡷⠋⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣯⣽⣁⡔⣚⣿⣷⣄
⢀⡔⣺⣿⣷⣟⠿⠋⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣶⣿⡷⣫⣽⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⢿⣾⣿⣿⣿⣿
⢸⣴⣿⣿⣿⣿⠀⠀⣷⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠿⠿⣷⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠙⢿⣿⣿⣿⠟
⠈⠻⣿⣿⠿⠋⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠿⣿⣿⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡷⠀⠀⠀⠀⠈⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⣲⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠡⣶⣿⢋⠉⠁⡬⠁⠴⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣾⢛⡍⣇⠀⠀⠀⡄⣡⣤⣴⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠺⣿⣬⣵⣇⠀⠀⢈⣿⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡻⣏⣘⢿⠇⠀⠸⣯⣝⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡰⣍⠻⠟⠿⠿⣶⣦⣤⣀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣷⣶⢶⣾⣄⢍⢿⣿⠻⢶⡀
⠀⠀⠀⠀⠀⠀⣠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡄⠀⠈⢛⢻⣷⣾⣿⣦⣿⣿⣧⣼⣇
⢠⣾⠛⠛⠻⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠙⠂⠀⠈⠑⢦⡀⠙⢹⣿⣿⣿⢻⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠠⠄⠀⢠⠀⠀⠀⠀⠉⠀⡠⣩⣿⡿⢸⡟⢄
⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁⢠⠀⠀⠀⠀⠀⠀⠢⠑⠀⠀⠀⠐⠀⢡⡿⡟⣡⡿⠃⠀⠑⠄
⠀⠀⠈⠛⠷⣦⣤⣤⣴⣶⣾⡋⠀⢀⡶⠀⠘⢆⣠⣀⣀⣀⣀⡀⠈⢢⡀⠘⠐⠂⠻⢿⡿⠿⠭⣄⣑⣶⡄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠜⠀⠀⠀⢀⣤⣤⣀⣉⠈⠉⠉⢉⣀⣰⣶⣷⠐⣄⠙⠀⠀⢣⠀⠀⠀⠀⢉⠁
⠀⠀⠀⠀⠀⣼⣧⠀⠣⠇⠀⠀⢊⡶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠀⠈⠛⣌⡙⠒⠄⠀⠣⡀⠀⠀⢸
⠀⠀⠀⠀⣰⠿⣿⠿⠷⣤⡴⠖⠋⠀⠫⡙⠙⢿⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣅⠀⠀⠈⢄⠀⠸⢶⣀⢠
⠀⠀⠀⠐⠆⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠉⠛⢖⢲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠂⠈⠢⠀⠀⠉⠁
⠀⠀⠀⠀⠢⠼⣤⠧⠽⠂⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⠀⠀⠐⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠴⠶⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠫⢻⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣽⣯⣿⣿⣶⣄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠛⠛⠻⠿⢾⣿⣿⡟⢷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣝⠛⠶⢦⣶⣖⣂⡀⠄⠈⣿⠥⠶⠛⡀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣀⣀⡀⡉⠀⠀⠀⠀⠤⠂⠀⠉⠉⢰⡿⠀⠐⢤⡑⠺⣡⣻⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⡵⠠⠄⡄⠠⡀⢤⠈⣀⢀⠀⠢⣿⢁⠈⠒⢄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣤⣀⡈⠑⠂⠠⠦⠤⣄⡤⣀⠈⠃⣼⠛⡄⢈⡀⢄⠈⠢⣙⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣄⡀⢰⣏⣀⢹⣈⡥⠖⡗
⠀⠀⠀⠀⠀⠀⠘⣦⣄⣀⣀⣉⣛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣷⡮⢕⠄⡀⢠⢘⠱⠦⡀
⠀⠀⠀⠀⠀⠀⠀⣽⢿⡍⡆⠜⠹⢛⠿⣶⣤⡹⣟⠿⣿⣿⣷⣤⣤⣉⠲⢍⡪⣛⠡⣀⣭⣚⣷⣦⠤
⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⠺⣥⢀⡄⡐⠁⡋⡟⣦⡀⣽⠟⠛⢿⢾⣟⣿⣦⡙⠲⣝⠢⢄⣀⣠
⠀⠀⠀⠀⣄⠀⠀⠈⢿⣿⣿⣿⣄⣬⡕⠒⣶⡡⠥⢻⡟⡖⠤⢌⢑⡈⠙⢶⣟⣷⣬⣙⠦⣄⣀⡀
⠀⠀⠀⠀⣿⣦⡀⠝⣀⠙⠻⠷⣿⣿⣧⣾⣿⡙⣃⡿⡿⣯⣭⣟⡲⢯⣓⠤⣈⠙⠻⠿⠿⠛⠉⠁
⠀⠀⠀⠀⢹⣟⢝⡶⢶⣶⢦⣼⣶⣴⣉⠿⣿⣿⣿⣓⣐⣶⠶⠤⣯⣓⠶⣝⡦⣕⡲⢤⢤⢤⠄
⠀⠀⠀⠀⠈⢷⣾⣽⣿⣿⣻⣽⣿⡿⣿⣳⣜⣿⡗⠛⠋⠘⠭⢀⠒⠪⢝⡒⢭⡛⠢⢅⣳⠃
⠀⠀⠀⠀⠘⣦⡻⣿⣻⣶⣯⣾⣿⣿⣯⡽⣻⣿⣶⢶⣿⣿⣶⣦⡉⠓⠒⠭⢥⣬⠷⠚⠁
⠀⠀⠀⠀⠀⢻⣼⡗⣿⣽⣛⠚⠧⣾⣾⣻⣿⠟⡃⣶⠷⢤⣿⣋⣿⠷⣦⣄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⢿⣽⡿⣿⣿⣿⣾⡋⠳⣾⢟⣀⣴⣞⣏⣰⣯⠍⣽⡛⢻⠾⠿⣟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣽⣿⢿⣥⣭⣛⠋⠉⠛⠻⠶⠵⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢬⠛⣿⣿⣿⣉⠦⣁⢚⡥⢏⡝⣒⠖⣲⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢅⣬⣿⣿⠾⠶⠩⠧⠽⠿⠶⠼⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢜⣿⣿⠇⠀⠐⠒⣃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⡜⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠉⠶⠆⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠑⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡠⠄⠀⠐⣓⡐⢆⣴⣶⡙⢂⣚⠂⠀⠤⢄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠶⠖⠦⣤⣀⠀⠹⣮⣿⣿⣵⠏⠀⣀⣤⠴⠲⠿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡡⠤⠀⠀⠒⠢⢀⠀⠈⠲⣽⣿⣿⣯⠖⠁⠀⠠⠔⠒⠂⠀⠠⠌⡀⠖
⠀⠀⠀⠀⠀⠀⠐⡰⡊⠥⣴⣖⣦⠴⠶⠚⠻⢦⣴⡀⢼⡟⣿⡯⢀⣤⡴⠞⠓⠲⠤⣤⣶⣦⠬⣐⢖⠂
⠀⠀⠀⠀⠀⢠⡞⢡⣤⣤⡀⠛⠁⠤⣴⣿⣷⣶⣦⣳⣼⣧⣸⣧⣞⣴⣶⣾⣿⣦⠤⠉⠋⢀⣤⣤⡈⢷⡄
⠀⠀⠀⠀⡐⣸⠀⠸⣿⣿⣟⡥⣴⡿⢋⣁⡀⠨⡙⠛⢿⢿⣿⡟⠛⡩⠀⢀⣈⡛⣷⣦⢌⡻⣿⣿⢆⠈⡇⠆
⠀⠀⠰⠚⣷⣿⣮⣧⡿⢛⡁⢚⣿⣿⣿⣿⣿⣄⡀⠀⣴⣈⣸⣄⠀⢀⣴⣿⣿⣿⣿⣿⣓⣈⠺⣿⣼⣴⣿⣾⠒⠂
⠀⠀⠠⢪⢾⣿⣿⠟⠊⠉⣠⣿⡿⠁⠈⠿⣷⣿⡿⢝⠛⠋⠙⢛⠿⢿⣧⣾⠿⠀⠘⣿⣿⣄⠉⠒⢿⣿⣿⠷⡑⡄
⠐⠌⡥⢀⣾⣿⠕⠀⢀⠔⡩⣿⣿⣦⣴⡞⠋⠋⠉⠄⠀⠀⠀⠀⠔⠉⠉⠙⣶⣤⣶⣿⡿⠝⢢⡀⠀⢻⣿⣷⠀⢬⠱
⠀⡼⢁⣴⣿⣯⡞⡰⠋⠀⠈⡵⣿⣿⡁⠙⢂⢸⣶⣄⡠⠤⠤⢀⣤⣾⠁⠜⠉⢈⣿⣿⢮⠂⠀⠙⢄⢳⣿⣿⣧⡈⢆
⠀⣷⠟⣽⣿⣿⠁⢀⠀⠀⠀⣸⣿⣿⣷⣄⠈⠦⠘⢿⣿⣆⣶⣿⠿⠁⠞⢀⣴⣿⣿⣿⣇⠀⠀⠀⡈⠘⣿⣿⣿⠿⣾
⠀⡉⢸⢫⣿⣿⣧⠏⠀⠀⠘⠿⣛⣿⣿⣮⣽⣶⣶⣤⣽⣿⣿⣯⣤⣶⣶⣯⣷⣿⣿⣛⠿⢂⠀⠀⢻⣼⣿⣿⡽⡆⠉
⢠⢃⠇⢸⣿⣿⣯⡎⢀⠎⣴⠋⢟⣿⣿⣾⡏⣹⡟⢻⢿⢿⢿⠿⠟⣿⡏⢹⣾⣿⣿⡻⠱⣄⢳⡀⢳⣿⣿⣿⠇⠹⡘
⠀⢸⠀⠘⢹⣿⣿⣷⣿⠀⡇⠀⡞⠩⠟⣿⣧⢸⣿⣿⣾⣾⣾⣿⣷⣿⠀⣿⣿⠿⠝⢧⠀⢹⠀⣷⣼⣿⣿⣇⠃⠀⡇
⠀⠰⡆⠘⢸⣿⣿⣿⠟⣼⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠛⢻⡛⠛⢻⣿⣿⣿⣿⡆⠀⠀⡀⢸⣆⢻⣿⣿⡿⡇⡆⣷⠇
⠀⠀⢁⢰⠀⢻⣿⣿⣿⣿⣧⡾⣠⡇⢋⣿⣿⣷⣿⠀⠀⠨⠄⠀⢈⣿⣿⣿⣏⠑⢸⡄⣷⣸⣿⣷⣿⣿⡇⠀⡇⡆
⠀⠀⠘⡎⣆⡜⢿⡏⣿⣿⣿⣿⣿⣧⠎⣸⣿⠙⢿⣿⣷⣶⣶⡾⣿⠿⠹⣿⣆⢳⣼⣿⣿⣿⣿⣯⢻⠿⢳⡼⣸⠃
⠀⠀⠀⠀⠈⠳⡈⠇⢻⡟⣿⢳⣿⣿⣿⣽⠃⠴⣘⣫⠤⠦⠶⢬⣝⡠⠆⢹⣯⣿⣿⣿⡼⣿⢹⠏⡘⢠⠞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡈⣧⠹⡸⡿⣿⣿⣿⣯⣭⣽⣾⣿⣿⣿⣿⣶⣯⣭⣿⣷⣿⣿⢿⢧⠋⡾⢠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠘⢆⢧⠀⢻⣿⣷⣟⡿⢿⡿⣿⣿⣿⣿⢻⡿⢟⣵⣿⣿⡟⠈⡜⡜⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠹⣿⣿⣿⣼⡿⡘⣿⡟⡡⣿⣴⣿⣿⣿⡏⣰⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⣝⠙⣿⣿⣿⣿⣶⣿⣿⣿⡏⠼⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣤⠑⢹⣿⣿⣿⣿⠟⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢱⢸⠿⠋⢛⡡⠚⠁

⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢠⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠈⠙⣡⠖⠋⠉⠉⠉⠓⠦⡄⠸⠿⠁⡴⠚⠉⠉⠉⠉⠉⠉⠉⠓⠲⡄⠻⠛⢁⡴⠋⠉⠉⠛⣦⠀⠀⠰⣶⠤⠤⣤⣄⣀⣒⡲⢾⠀⠛⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠘⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠙⠲⢤⣀⣀⣀⡤⠔⠃⠀⢀⠀⠀⢠⣥⠖⠒⠒⠒⠤⠤⠤⠒⢁⣀⢀⠀⠳⣄⣀⠀⣀⠴⠁⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠈⠟⠏⠀⠘⠉⠀⠀⠀⠀⠀⠿⣁⠇⠀⠀⢀⣀⣄⣀⠀⣠⣄⣀⠀⠰⣌⣠⠇⠀⠀⠈⠉⠀⠀⠰⢿⠄⠈⠀⠈⠉⠉⠁⠉⠉⢡⣤⣆
⠀⠀⠀⠀⢴⠲⠶⠶⠤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⡞⠁⠀⠈⠛⠁⠀⠈⢳⣀⣀⠀⠀⠀⣤⠤⠄⠀⠠⢤⡄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣋⠛
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⡦⠀⣠⡽⠦⢤⠀⠀⠀⠀⠀⠈⣁⣠⠃⠀⠀⠃⢀⣤⣤⣄⠀⠃⠀⠀⣏⠀⠈⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠰⡏⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢱⠀⠀⠀⢸⠁⠀⣸⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠙⢳⠀⣠⠤⠤⠤⠤⠞⠀⠀⠀⢀⣉⣒⠒⠈⠉⠳⠤⠤⠓⠦⠤⠴⠋⠀⠀⣆⡈⠛⠚⠛⠁⢀⠀⠀⠯⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠚⠓⠚⠁⠀⠀⠀⠀⣠⡀⠀⠀⢸⣂⣅⡣⠀⠀⠀⣀⡀⠀⠀⠀⠀⣄⡀⠀⠉⠉⠁⠀⠈⠉⠉⠀⠀⠛⠛⠋⠉⠁⠈⠉⠀⠉
⠀⠀⠀⠀⠀⠀⣀⡀⣄⣤⠀⡀⠈⠻⠛⠁⠀⠀⠀⣀⣀⡀⢠⠏⠉⠉⠳⣄⠀⠘⠛⠃⠀⠀⠠⠤⣄⠀⡠⠀⠢⡀⠀⠀⢀⡤⠖⠒⠒⠶⣤⡀
⠀⠀⠀⠀⣠⡤⣷⠙⠛⠋⠷⣛⣠⡄⠀⠀⢀⡔⠋⠉⠈⠉⠻⠀⠀⠀⠀⠋⠀⠉⠢⡀⠀⠀⢄⠀⠀⣿⠀⠀⠀⣱⠀⠠⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⢠⣝⣂⠀⠀⠀⠀⠀⣃⣠⡄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠁⠀⠀⢠⠇⠀⠈⠒⠤⠤⠤⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠤⢲⠞⣾⠢⡼⠛⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠂⠂⠀⠀⠀⣀
⠀⠀⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣤⠀⠀⢀⡼⠖⠋⠀⢀⣦⡀⠀⠀⠀⠀⠀⠸⡿⠇
⠀⠀⠀⢠⡒⠋⠉⠉⠉⠉⠙⠓⢲⡄⠀⠁⠁⠤⠥⠅⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡈⠀⠀⠀⣄⣠⡀⠀⠀⠘⠋⠃
⠀⠀⠀⠀⢉⣷⠤⠖⠒⠒⠒⠒⠋⠀⠦⡄⠀⢀⣰⣄⠀⢠⠒⠉⠉⠉⠀⠀⠀⠀⠈⠉⠑⡀⠈⠢⡡⠃⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠈⠁⠀⠀⠀⠀⠘⠛⠒⠚⠛⠛⠛⠛⠛⠛⠛⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠄
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⡤⠁⡡⠊⠀⠀⠀⠀⡠⠄⢠⠆⠀⠀⠀⠐⡠⠀⢤⠀⠀⠀⢀⡀⠀⠄⡀
⠀⠀⠀⠀⠀⢀⠖⠂⠀⠀⢀⢀⣀⢀⡀⠀⠀⠀⠀⡀⠀⠔⠂⠀⠀⠀⡰⡀⠐⡀⠀⠀⠀⠀⢲⡋⢀⡉
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠎⠤⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢀⠁⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⣀⠀⠰⠐
⠀⠀⠀⠐⠔⠢⠠⠂⠀⠀⠀⢔⡖⠂⠊⠀⠀⠀⠈⠁⠀⣀⠀⠀⠀⠀⣀⡒⢀⡐⡀⠀⠀⢈⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⠀⠪⠀⠀⠀⠀⠀⠈⠪⠇⠀⠀⠀⠀⠀⢡⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⢋⡤⠂⠢⠀⠀⠀⠐⠋⠁⢀⠤⠀⠀⠀⢐⡂⢁⠀⠀⠀⠀⢂⠀⢀⡤⠀⠀⠀⢠⠖⠀⠂
⠀⠀⠀⠸⠁⠄⢀⠠⠀⠀⠰⠁⠀⡰⠋⡀⠀⠀⠀⠩⠀⠁⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠈⠀⠢⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠅⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠻⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠋⣽⣿⣀⣤⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⢋⣼⣿⣿⡿⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠛⠁⢀⣾⣿⣿⡿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠁⠀⠀⢰⣿⣿⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⣾⣿⣿⡀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣾⣧⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⣿⣿⣯⢿⡋⠻⣿⣿⣿⣿⡟⢛⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⢝⡻⣓⣝⢠⣿⣽⣿⣿⡴⣫⣾⢟⣫⠿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣶⣾⣿⣿⣿⣷⣿⡿⠿⣿⣾⣿⣿⣿⣷⣶⣾⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⡏⢸⣿⣟⠿⣿⣿⡿⠟⠉⢙⠰⡆⡋⠙⠻⢿⣿⣿⡿⣻⣿⡏⢻⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣸⣿⣋⣰⣿⣿⠀⠄⠘⠂⢂⡔⠀⠃⠀⠀⣿⣿⣧⣙⣿⣷⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⣁⣴⣿⣯⣿⣥⣴⣶⣦⣀⣄⣡⣀⣴⣶⣦⣬⣿⣿⣿⣦⣉⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠚⢿⣿⣿⣿⠋⠨⣻⣿⣟⣡⣶⣶⣌⣻⣿⣟⡅⠉⣿⣿⣿⡿⠓⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣾⣿⣿⣿⣿⣷⣰⡛⠿⣟⠋⠃⠻⠙⣻⠿⢛⣦⣾⣿⣿⣿⣿⣿⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣻⣿⣿⣿⡟⠁⠉⠀⠙⢿⣷⣾⡿⠋⠀⠉⠈⢻⣿⣿⣿⣟⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣅⣿⣿⣿⣿⣿⣿⣿⣧⣈⠀⢀⣀⣽⣯⣀⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⢿⡘⠛⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⠛⢣⡿⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠀⢸⡿⢺⣿⣿⣿⡇⣿⣿⡿⠿⡿⢿⣿⣿⢸⣿⣿⣿⡗⢿⡇⠀⡟⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⣬⠀⡜⢠⢿⢻⠋⣿⣿⢸⣿⣻⣤⣤⣟⣿⡇⣿⣿⢙⡿⡿⡌⢷⠀⣧⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⠀⠀⠈⡞⠈⠆⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⠰⡍⢿⠃⢈⢀⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⢸⣇⢂⣼⣟⣿⣦⡀⠈⠉⠉⠁⢀⣴⣿⣿⣷⡔⣸⡏⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡄⢿⢿⣿⠏⣟⣹⣿⣷⡦⣶⣾⣿⣿⣿⠹⣿⣿⣿⢠⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡄⡘⢀⣰⡿⣿⣿⣿⡿⢿⣿⣿⣿⢿⣆⡁⢃⢩⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⣼⡟⠀⣻⣿⡉⡇⣸⣉⣿⣿⡀⢻⣧⣸⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⡗⣿⡟⡟⠙⣏⢻⢹⣿⢻⣎⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢱⡿⣾⣿⣿⣿⣿⡿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⠘⠋⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡀⠀⠀⠒⠒⠚⠃⠀⡠⢄⠀⠐⠒⠒⠒
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣌⢧⡹⡼⡆⡇⡟⣰⢀⡀⠀⢀⠜⠀⠀⠣⡀
⠀⠀⠸⣿⣀⠤⠒⠈⠀⠀⠀⠠⢌⠓⠪⠓⠁⠁⠁⠙⠱⣣⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⣠⡀⢀⢀
⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡑⠀⣐⠁⠀⠀⠀⠀⠀⠀⢐⡯⢒⣉⠀⠀⠀⠀⢀⢔⣵⣿⡿⠿⠿⢿⣿⣷⣤⡀
⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢊⡠⠨⢆⠀⠀⠀⠀⠀⢀⢬⠢⣙⠂⠀⠀⠀⠀⣽⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠴⢊⡵⣫⢆⣆⣆⡞⡎⣇⠳⠌⠁⠀⠀⠀⠀⣹⣿⣧⡀⠀⠀⠀⠀⢀⣼⣿⡃
⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠞⢸⠁⠇⠻⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣴⣴⣾⣿⡿⡋
⠀⠀⢄⠈⠂⠌⠀⠈⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠽⡟⣿⢿⠿⠟⠚
⠀⠤⠄⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄
⠀⠤⠤⡄⠀⠀⠀⠀⠨⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠔⠈⡠⠄⠤⡘⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠁⠸⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠑⠲⠶⠾⠶⠶⠾⠛⣅

⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣤⡀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠋⠁⢸⣇⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢾⣿⣷⣤⣈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣤⡤⣿⡈⠉⡉⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣈⣈⣻⡀⢺⡇⠀⠡
⠀⠀⠀⠀⠀⠀⢠⣀⣈⣷⠉⠉⠉⠉⠙⠿⠛⠚⠓⠦⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠈⠋⠉⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⣠⣤⣶⣶⣶⣿⣿⣷⣲⡀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⢿⠿⢿⣭⣽⣿⣟⣿⣿⣛⣿⣀⣀⡀⠈⠁⠀⠀⢀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣱⡗⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠓⠲⠶⢶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣼⣁⣀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣀⣠⣤⣤⣤⠤⢀⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⡤⣒⣹⣁⣽⣿⣤⡤⠴⠶⣾⡿⠓⠛⠒⠂⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠉⢈⣿⣇⣀⣤⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣿⣿⣾⣿⣿⣏⣧⡙⢆⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣿⣿⣿⣿⣿⣿⣧⠻⣿⠆⢉⢶⡿⢟⣟⣥⣄⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢃⣚⣭⣽⣶⣶⣶⣿⣿⠳⣦⣰⣿⣧⣿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⡛⠿⣿⣿⣉⠻⢿⣿⣿⣿⡇⠀⣼⣿⡇⣿⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣗⣴⠂⠸⠿⠯⡉⠁⠀⠹⢿⣤⣴⣿⢿⣴⣿⣧⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣸⣏⡀⠙⠉⠛⣿⠿⠿⠷⣄⡀⠘⠛⠐⠆⢀⠀⠀⠉⠉⠀⠀⢻⣿⣷⣝⡳⠶⠤⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠉⠀⠘⠙⠂⠀⠠⣍⠛⠛⢷⣦⣀⠐⠾⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠛⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣬⣥⣄⣀⡉⣛⡛⠂⠀⠀⠠⣤⣶⣦⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⡿⠿⠿⠛⠋⠉⠒⠀⠈⠉⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣀⠀⢀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢀⡀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣦⣼⣷⣿⣟⣿⣯⠿⣿⣃⣍⠉⣭⣿⣞⢿⣿⣿⢿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⣄⣹⢷⣿⣿⣿⣧⡤⠃⠉⠻⣤⠟⠉⢡⣾⣭⣿⣿⣮⣍⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢨⡿⢛⣾⣳⣾⣿⣇⠀⢀⣤⣶⢿⣦⡄⠀⢚⣿⣿⣻⣿⣟⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣾⣾⣿⣿⣫⣿⣾⣿⣤⣯⢼⡋⢉⡟⣼⣦⣿⣿⣯⡉⢙⣿⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢻⣟⣶⣿⣯⠴⠛⡁⣿⢻⣾⣿⣾⣿⣿⣿⢰⡇⠹⢿⣦⣿⣿⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⠃⣿⣻⣼⢀⣿⡇⣿⢸⣿⣿⣿⣿⣿⣿⠈⡇⠀⠀⣹⣿⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣸⣱⣐⣯⣿⣾⡏⣄⢹⠘⣿⠿⠻⢻⢳⡯⢸⣧⢀⣴⣿⡟⢀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⠟⢁⣴⡟⢿⠐⣗⢸⣧⡟⠀⠀⢸⣿⣡⣷⣼⣋⣿⣿⡇⡜⡆⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠘⡟⣧⣾⣿⠀⠀⠀⣿⣾⣿⣷⡄⠀⡐⣿⣽⣿⡿⣿⣿⢻⡷⢠⢃⡞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢿⡟⣿⣿⢠⠆⣸⣿⠏⣿⣿⣿⣿⣷⣿⣻⣿⢷⢹⢯⢆⠛⢀⡼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠾⢿⣿⡎⣾⣿⣿⡾⠟⣿⣿⣿⠛⢻⣿⠟⠸⣘⠈⠈⣾⣟⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⣿⡿⢿⣾⣯⠿⣷⣶⡍⣼⣿⣴⣿⢏⠂⠀⢹⣴⣀⣫⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣨⢀⣰⡴⣪⡏⣀⠀⠹⣿⣿⡿⠛⡅⣠⡇⢠⡞⡿⠛⠃⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠀⢾⠿⡿⠟⣷⣿⣷⣿⣷⣧⣷⣱⣂⣆⠜⣮⣦⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠁⠈⠁⠀⠀⠈⠀⠹⠻⠛⠛⢿⠟⠿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⡁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⣙⡅⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢄⡀
⠀⠀⠀⣄⣷⣿⣷⣂⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⠀⢾⣿⣷⠛⢁⡀⠀⠀⠀⠀⠀⢹⡄⢀⠀⠀⢀⣀⣘⡀
⠀⠀⠀⠘⠿⣿⣿⣿⠓⠢⣐⣶⣾⣿⣿⣥⡀⠀⠀⢠⣀⣂⣴⣾⣿⣷⣿⡾⢸⣿⣿⣆⣨⣄⠀⠀⠀⠀⢸⣾⣻⣿⣻⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⡝⠛⠛⠋⣡⡳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣷⣿⣿⣿⣿⣿⣶⣶⣾⣿⡿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⣿⣮⣛⠿⣿⣿⣿⣿⡿⠛⠁⠘⡟⠿⠿⠿⠛⠁⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠙⣆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠛⠛⠛⠋⣁⣦⣑⡄⠀⠘⢧⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢰⣦⡀⠀⠀⢀⡾⣦⠀⠀⠀⣰⢿⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⣿⠻⣦⡀⣿⠈⠻⣦⣀⡾⠁⠘⣿⣤⡾⠁⠘⣷⣇⣀⣴⢾⡟⠀⡀
⠀⠀⠀⠀⠀⣿⠉⠛⠶⣤⡿⠀⠈⠻⠇⠀⠀⠈⠛⠁⠀⠀⠘⠋⠀⠀⠀⠻⠟⠋⠀⣾⣷⡟
⢠⣤⣤⣀⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠗⢻⡿⠃
⠀⠉⠳⣮⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁
⠀⣀⣠⣤⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠷⢤⣤⣄⡀
⠻⢯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⠥⠄
⠀⠀⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⠿⠛⠉
⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣛⣳⣶⣤⣤⡀
⠘⠛⠓⣻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⣠⡴⠛⣁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡷⠶⣶⣤⣽⣷⣄
⠠⠾⠿⠛⣻⣿⡟⠁⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠙⣷⡄⠀⠀⠈⠉⠙⠓
⠀⠀⠀⠀⣰⣏⣴⢾⣿⣿⠃⢀⣴⡆⠀⠀⣠⣤⡀⠀⢤⣤⣀⠀⠀⢸⣿⡟⠶⣜⣿⣄
⠀⠀⢀⣾⠟⠋⡰⠟⠁⣿⡴⢻⣿⢿⣀⣾⣿⡿⠻⢦⡀⠙⢿⡙⠳⢾⣿⡇⠀⠈⠙⠿⠦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⠀⠘⠁⠈⠋⠈⠟⠁⠀⠀⠙⢶⣌⠻⣦⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠾⠿⠿⠿⠶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣴⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣴⣖⣒⣐⣠⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⠀⠀⠀⠀⣸⠟⢿⣶⠞⠛⠛⠻⣧⡤⠍⠉⠁⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⣿⡆⠀⠀⠀⢀⡠⢞⣅⣀⣀⣠⠤⠀⠀⠙⢻⠇⠀⠀⠀⠀⢠⣿⣷⣦⣀⣀
⢀⣤⣶⠿⠿⠛⠛⢿⣿⣿⣿⣷⡀⠀⠀⠀⢸⠋⢀⣀⣀⣼⡤⠤⠖⠚⠋⠀⠀⠀⠀⣠⣿⣿⣿⣿⣻⢿⣷⣽⣦⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠨⣇⠉⣉⣁⣀⡠⠐⠉⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⢋⣩⣭⣝⡛⣧⣤⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⣫⣶⣿⣿⣿⣿⣿⣦⠹⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣷⣽⣿⡿⠋⠁⠀⠀⠉⠙⠛⠿⣿⡛⣱⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⡿⠿⠿⠟⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⢬⣻⣿⣿⣿⣻⣿⠿⠛⠛⠉⠉⠁⠀⠉⠀⠀⠉⠻⣿⣿⣿⣿⣧⣘⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢐⣑⢢
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠑⢊⣁⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣠⠒⠂⠒⠢⠀⢅⠀⠀⢀
⠀⠀⠀⠀⠴⠂⢀⠤⠐⠒⠋⡀⢀⠈⠋⠉⠲⡀⠈⠀⠀⠀⠀⣔⣲⢰⠁⠀⠀⠂⠉⠡⠤⡀⠉⣆⠁
⠀⠀⢀⠀⢠⠊⡑⠀⠀⢀⠀⠀⠀⠋⠉⡆⠀⢈⠀⠀⠀⠀⠀⠀⢀⠊⢇⠀⠀⡆⠀⠀⡄⢀⠀⢠⠃⣠
⠀⠀⠀⠀⡜⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠨⠨⠀⠀⢀⡀⠀⡆⠀⢠⠀⠑⠂⠀⠀⠔⠀⠀⡇⡀
⠀⠀⠀⠀⠳⢄⡈⠀⠀⠑⠀⠊⠀⠁⢀⠠⠈⠁⠈⠂⠀⠀⠀⠐⠀⠘⢄⠀⠑⠒⠤⠀⠀⠀⢀⠏⠘⠜
⠀⠀⠀⠰⠑⡄⠉⠒⠚⠂⠤⢐⢒⣒⠁⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⠐⠦⠤⠤⠓⢋⠅
⠀⠀⠀⠀⠉⠀⠀⠀⠃⠂⠀⠈⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣀⣁⡱⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠈⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢰⠚⠒⢢
⠀⠀⠀⠀⠀⢎⣀⠤⢞⠔⠒⠍⠏⠀⠙⡆⢔⢲⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡤⠒⠒⢯⡁⠈⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⢀⠼⡙⠀⡠⠠⢄⠀⠀⠈⠱⡀⠀⠀⠀⠀⣆⡆⣘⠀⠈⡡⠤⡄⠒⠄⠋⠉⢱⡀⠈⠐⠂
⠀⠀⠀⠀⠀⢰⠁⠀⡀⠀⠀⠀⠐⠓⡄⠠⡴⠁⠤⠂⠀⠀⢠⠊⠉⠂⠀⢃⣀⠀⠀⠀⠀⢤⡀⠙⡄⡠⢦
⠀⠠⠤⠀⠀⡜⠓⡎⠀⢰⠒⠢⠀⠀⠁⠀⡸⢀⠀⠀⠀⠀⢸⠤⡎⠁⠀⢪⣀⠀⠀⠉⠀⣀⡇⠐⢧⢟⢘
⠀⠀⠀⠂⢰⠉⡤⠈⠀⠈⠀⠀⠀⡸⠀⢉⠀⠓⠃⠀⠀⠀⢇⠀⠈⠀⠀⠀⠣⠤⠂⠀⠀⠤⠇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⡏⠑⠀⠀⢤⣀⠗⠚⠁⡤⢊⢄⠀⠀⠀⠀⠀⠈⢻⣁⡀⢎⠀⢀⡀⢀⠀⠀⠀⠀⡖⠁
⠀⠀⠀⠀⢀⠽⡶⡀⠀⢠⣀⡤⠤⠞⠰⣡⠎⠀⠀⠀⠀⠀⣠⠄⠘⣄⣀⡉⠁⠉⠁⡆⠀⡱⠒⠁⠈⠒⠄
⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠀⠀⡠⢝⣒⠋⠒⠋⠉⠸⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⠀⢀⣤⡶⠶⠶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠉⠉⠉⠙⠻⠟⠁⠀⠀⠀⠀⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⡀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢀⣀⣀⡀⠠⠤⠤⡤⠄⠀⣀⣀⣀⡀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⡤⠚⠥⠤⠴⠧⠤⠠⠴⠣⠤⠤⠚⠤⠤⠼⠓⠦⡀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠴⠬⠤⠤⠤⠤⣤⠀⠀⠢⡤⠤⠤⠤⠤⠤⠤⠀⠀⠒⠚⠋⠀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⣀⠀⠀⠀⠀⠀⣹⢤⠀⣠⢻⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠘⠁⢸⠀⡇⠸⠀⠀⠀⠀⠀⠁⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⢶⠄⡷⠛⠛⠲⠶⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⣶⣤⡀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠛⠻⠿⠿⠿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠋⣀⣠⣴⢾⡆⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠋⠉⠀⠈⣿⡀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠙⠿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠈⠛⢿⣿⣟⠛⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿⣏⠙⢀⣤⠈
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣷⡌⠉⢴⡆⠁⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠱⣶⠌⠁⠿⠆⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣁⠺⠷⣀⡘⠟⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⢇⣄⠻⢃⣤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡌⠛⣴⣎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡅⣶⠎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣀⡘⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⢀⣠⣶⣿⡟⠩⣿⡿⣿⣶⣤⣀⠀⠀⠀⠀⠀⢻⣟⠁⣤⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣠⣴⣿⢿⡿⢉⠺⡷⣀⡸⠿⣨⣝⠻⣿⣶⣤⣀⠀⠈⣿⣦⠉⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢀⣴⣾⠟⡋⢿⢇⣀⠻⢇⣤⡛⠋⠀⠈⠉⠶⠌⠉⣿⠿⣿⣦⣼⣿⡿⢃⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣠⣾⡿⠿⣁⡘⠟⣠⣌⠛⢠⠌⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠟⣩⡟⠿⣷⣄⠉⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣟⠛⠃⣤⠙⠃⣴⡌⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢶⠌⠁⠶⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠉⢠⡦⠉⠰⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣠⣠⣠⣀⣀⣀⣀⣤⣀⣀⣀⣀⣤⣶⣀⣨⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣄⣀⣄
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠒⠉⠐⠛⠿⣷⣶⣆⠀⠀⠀⢸⡿⠟⠛⠛⣻⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠦⣀⣨⣭⠗⣠⣤⣭⣭⣿⡇⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⢿⣿⡋⠀⠀⠀⠈⠉⠀⠺⠂⠐⠚⠿⠿⠿⠃⠀⠀⠀⠉⣿⡈⠉⠙⠋⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢿⣄⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢃⠀⠀⠀⢀⣀⣱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣼⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠙⡿⠛⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⣦⣾⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠀⠀⣀⣈⣍⣩⡏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣤⡀⠀⠀⣀⣀⡀⠀⢀⣀⣤⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢇⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢿⣿⣿⣿⣿⣿⣿⣧⠫⡈⣿⣿⠏⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠟⠛⠇⢙⣄⣸⣿⣯⣴⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⣦⡙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡝⠾⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⣿⣿⣿⠳⠶⢶⣶⣤⣤⣶⣶⡶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⠀⠈⠉⠓⠒⠚⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠋⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢺⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢻⠀⡀⡼⠀⠀⠀⠠⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⡀⢸⡀⣿⣤⣤⣯⣦⣥⣧⣄⣢⣽⣶⡶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⣏⣠⣼⣿⣿⡏⣠⣤⡶⠿⣿⣿⣭⣿⣿⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⠟⠉⠀⢰⣿⣿⣿⣿⡏⢻⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣤⡀⠀⠀⢉⢻⣿⣿⣿⣩⠇⢀⣠⡾⠟⠉⣼⣿⣿⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣹⣿⣿⣿⡿⠀⢀⣰⡷⢛⢁⠈⣡⠶⠋⠁⣀⣼⣾⣿⣿⠏⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣧⣤⣼⣿⡷⣾⢋⡍⢸⣿⣿⠛⢁⣴⣾⣿⣿⣿⠟⠛⣿⣿⣿⣿⠛
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⢉⣽⣿⣆⣾⣿⣿⣾⣿⣭⣅⣠⣼⡟⠛⠁⣀⣅⣴⣿⣿⣿⣿⣿⣦⣀⠀⠖
⠀⠀⠀⠈⢿⣦⣵⣦⣀⣤⣀⣶⣶⣦⡌⠙⠿⣿⣿⣤⣿⠏⠉⠻⣿⠿⠟⠛⠉⠀⠀⠀⠈⠉⠃⠀⠙⠻⠋⠻⣿⣯⣅⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠿⣿⠿⣿⢿⣿⣿⣿⡿⠿⢿⣿⡿⠟⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣿⣋⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠱⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠛⡀⢛⣿⠟⢉⡿⠉⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣤⡈⣿⠟⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⣸⠃⠀⠀⠀⠀⠁⠀⣀⣀⠀⢀⠀⠄⠀⠀⠀⠀⠀⢀⠂
⡷⠟⢁⡤⠊⠀⡆⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⢼⣷⣶⡿⣿⠿⢿⣶⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⣧⡴⠋⠀⠀⠘⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢐⠄⡀⠀⠁⡀⠀⢀⠀⣿⢋⢀⣁⣜⣴⣸⣧⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⢀⣴⣃⡀
⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⡄⠀⠈⢀⠄⡠⠂⠀⠀⠈⠙⢻⡟⢿⣟⠛⠛⠃⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡥
⠀⠀⢀⠀⠀⠈⠀⠂⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⣠⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠈⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠈⠁⡐⠀⠀⠀⠀⠆⠀⠀⠀⢀⠀⠀⠀⡀⠀⠘⠂⠀⢠⡀⠤⠤⠤⠄⢀⡀⠀⠀⠤⡖⠀⠀⠀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠺⠗⠀⠀⠀⠘⡀⠈⡀⠀⠀⠗⠂⢐⡦⠀⠀⠁⠀⠀⠀⠀⠈⠁⠂⢀⠀⠀⢀⢠⠀⠘⠀⠐⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⢂⠀⡀⢅⠶⠆⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠌⠃⠀⠀⢀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠲⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⢰⡀⠒⢰⣔⠂⣦⡀⡁⢠⠄⠀⠀⠀⠀⠀⠀⠀⠚⠁⠁⠀⢠⠀⠀⣄⡀⠀⠀⠀⢩⠂
⠀⠀⡴⠁⢀⠀⠀⠀⠀⠠⠆⡆⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⢅⠸⠸⠼⠁⠀⠘⠙⠀⠊⠙⢃⣀⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠈⠀⠀⠀⠷⠰⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢴⣶⠤⡆⢰⢰⡆⣶⢸⣦⣿⢸⠻⡆⣿⡉⠉⣯⣼⠇⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⣦⠀⠠⠴⢂⣗⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢻⡟⣇⣸⢸⡟⣿⢸⣶⠇⢿⠤⠄⠿⠙⠆⠀⠀⠀⠀⠀⠀⡀⠀⠈⠆⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠈⠁⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣀⠀⠀⠀⠞
⠀⠀⠠⠀⠀⡀⣀⣀⣤⣈⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⡄
⠀⠀⠀⣦⣤⣿⠿⢿⠉⣿⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⢠⣴⡄⠀⠀⢰⠆⡴⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠠⠈⣿⠹⢦⢳⣰⣷⣾⡟⠀⢢⣶⣿⡇⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⠛⢺⡆⠀⠀⠀⣿⣿⣿⣿⣧⣄
⠀⠀⠄⠞⣷⣷⠿⡿⡍⠀⠀⠀⠘⢿⣇⠀⠀⠀⠟⠻⡄⠠⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠂⠘⠀⠉⠈⠓⠀⠀⠈⠡⠊⠀⠀⠀⠀⠿⠟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠈⠁⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠃⠒⠀⠀⣀⠈⢠⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠐⡄⠀⠀⣠⠇⠈⠈⡄⠀⠀⠈⠉⢀⡟⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠀⠀⢀⠀⣾⣻⠿⣿⣿⣷⣄⡄⠠
⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠁⠀⠀⠀⠉⠀⠀⠀⠀⠎⠀⠀⢠⠀⠂⠀⠀⣴⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠩⣯⣯⣼⣱⡝⡙⣽⡷
⠀⠀⠀⠀⠀⠈⢈⠀⠀⠀⠀⠀⠀⠀⠲⠆⢀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠋⠹⢿⠿⠿⣿⠷
⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⠀⠀⠀⠀⣀⠐⠍⠀⠀⠀⠈⢀⠠⢀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⢻⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢶⣄⢻⣧⠈⠉⠛⠛⠳⠶⠶⠶⠶⠦⣤⣄⣀⠀⢠⣿⡿⠋⢉⣿⡿⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣆⠀⠐⠦⣤⣤⣤⣤⣤⣄⡀⠈⠉⠛⢿⣿⣤⣀⠈⠁⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢻⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠈⠻⣿⡻⢦⣄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⢷⡄⠈⢷⣿⣿⣿⣿⣿⣿⡏⠀⠀⣰⠞⠉⠀⠀⠀⠤⣤⣤⣤⣿⣿⣦⣽⣷⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠘⠺⠿⠶⠄⠉⠻⠿⢿⣿⣿⠁⠀⣼⡏⠀⣶⠀⠀⡀⠀⠀⠈⠉⠉⠙⠛⢿⡇⠘⢿⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠿⠋⠁⠀⣸⣿⣧⠀⠻⣷⣤⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⠆⠀⠀⠀⠀⠀⢀⣴⣾⠟⡋⠁⠀⢀⡄⠀⠴⠛⠻⣿⣧⣀⠈⠉⠛⠛⠿⣧⠀⣼⡇⠀⠀⠀⣀⠀⠀⢌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⢠⡿⣿⣿⠟⠉⠀⣴⡟⠀⠀⠀⠀⢀⣿⡿⠛⠓⣶⣶⢦⣤⣀⠁⠙⢧⠀⠀⣠⣿⠀⠀⠈⣷⡀⠙⣦
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢸⡄⠀⢰⣿⠃⠀⠀⣼⣿⡇⠀⠀⣇⠀⣼⠏⠀⠀⠀⠘⣿⡀⠼⠿⠻⣦⣤⣴⣾⣿⣿⣷⣤⣴⣿⣧⣀⣽⠂
⠀⠀⠀⠀⠀⣼⠁⠀⢀⣠⠆⠀⠀⢸⡇⠀⢸⠇⠀⠀⠀⣿⣿⣇⠀⠀⢿⣦⣿⡀⠀⡀⠀⠀⠈⠉⠉⠁⠀⠀⢀⠼⠛⠉⠁⢀⣀⣀⣈⣈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⣿⢠⣾⣿⠇⡤⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠈⢿⣿⡇⠀⣷⣦⣤⣤⣤⠤⠀⠀⠀⠁⠀⠀⠐⠋⠉⠁⠀⠉⠉⠻⢶⣾⣇
⠀⠀⠀⠀⠀⣿⡿⢿⣿⣾⡇⠀⠀⠸⣿⡆⠀⠀⠀⠀⢠⣄⡻⣿⣝⡳⠤⠀⠉⠻⠄⠻⠿⣭⣭⠄⠠⠤⠤⠤⠤⣤⣄⣀⡀⠀⠀⢠⡶⠛⢦⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠛⠀⠈⣿⣿⠃⠀⠀⢦⣽⣿⣦⡀⢷⣄⡀⠈⠛⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠤⠤⢤⣈⣉⡃⠀⢸⡇⠀⠀⠉⠛⠶⠞⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣠⠀⠀⠻⣿⣿⣿⣶⣿⣿⣷⣶⣤⡀⠉⠛⢷⡄⠀⠀⠀⠀⣰⣿⣿⡟⠛⠻⢿⣶⡶⠶⠤⠬⡅⠀⠀⠙⢿⣦⢀⡄⢀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣥⠾⣿⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣧⣭⣉⠓⠀⠀⠙⠀⠀⠀⢀⣿⣿⣿⣷⣾⣷⣶⣷⣷⣶⡒⠒⠒⠂⠀⠀⢀⣽⡿⠛⠛⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣰⠃⠀⠀⠀⠀⠀⠉⠉⠙⣛⣻⣿⣦⡀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠉⠻⣿⣿⡿⠈⢙⡟⠶⠶⠶⠾⣿⢯⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠀⠀⣀⠴⠋⠁⠀⠀⠉⠉⠉⠛⣻⣿⣦⠀⠀⠀⠘⣿⣷⣄⠀⠢⡀⢈⡟⣁⠴⠋⠙⠓⠲⢲⠞⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⣴⣯⣶⠂⠀⢀⣠⣴⣾⣿⣟⣛⡛⠛⠃⠀⠀⠀⢻⡄⠿⣄⢀⠹⣎⣹⣧⠀⠀⠀⣀⣠⣋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃⢿⣀⣴⠟⠉⠁⠀⠀⠈⠉⠙⠳⣦⡀⠀⠀⠈⠻⣦⣽⣦⡦⠟⣱⣈⣧⡀⡰⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⣿⣤⣴⣧⣍⣉⣹⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠐⣦⡀⠈⠙⠛⠛⠛⠻⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣶⡞⠀⠀⠈⠙⠓⠒⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⣋⣀⣀⣤⣶⠇⠀⠀⢀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠶⠿⠶⠖⠛⠋⠉