Funding Emojis & Symbols

⢰⣏⣉⣉⣉⣉⣉⣹⡇⠀⠀⠀⠀⠀⠀⢰⠒⠒⣖⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⣀⡀
⠀⠭⠭⠭⢭⠭⠭⠭⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢈⣿⣽⡅⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠡⡈⢹⡁⠀⡙⢧
⢀⣃⣸⣀⣻⣀⣟⣈⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠓⠐⠂⠙⠀⡇⠀⠀⠀⠀⠀⠀⠳⠤⠤⠚⢓⠒⠋
⠀⠧⢼⠤⢼⠤⠿⠤⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣿⣿⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣿⡀
⠈⠓⠚⠓⠛⠒⠛⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⢠⣴⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⣤⣠⣿⣿⣿⣿⣿⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢦⡀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠺⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠁⠈⠀⣿
⣿⣿⣿⣿⠀⣽⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣿
⠉⠉⠉⠉⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉


⢀⡴⠶⢦⡀
⣏⠀⣫⠀⣿⣚⣛⣛⣾⡆⠀⠀⠀⠀⠀⢰⣶⠒⠒⣲⣖⠒⠒⢶⡆⠀⠀⠀⠀⠀⠀⢀⣀⣈⠉⠉⠑⣂⣀
⠙⠶⣤⣶⡿⠋⠉⠹⣿⡇⠀⠀⠀⠀⠀⢨⡀⠀⠸⣿⣿⠇⠀⢀⡅⠀⠀⠀⠀⠀⠠⡏⢰⠈⡇⠀⣾⠱
⠀⠀⣿⣿⣷⣈⣁⣠⣿⡇⠀⠀⠀⠀⠀⢸⣿⣦⣤⣾⣷⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⠙⠒⣚⠁⠀⠈⡓⠒
⣀⣀⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣉⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠁⣀⣠⡤⠶⠶⠒⠛⠛⠒⠶⠶⢤⣄⣀⠈⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⣠⡴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⣄⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠋⠉⠙⢿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣀⣀⣠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠙⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⡀⠉⠙⠓⠶⠶⠤⣤⣤⠤⠶⠶⠚⠋⠉⢀⣠⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣤⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⡀⠀⠀⣠⢦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠐⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⡇⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⢠⣿⣿⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠁⠀⢠⣯⣯⣿⣯⡮⠜⠧⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡍⠀⠀⠈⠙⠛⠉⠀⠘⠃
⠀⠀⠀⠀⠀⣠⣤⣾⢿⢿⣶⣤⡀⠀⡠⠔⠈⠀⢠⣤⡄⠀⠀⠀⠀⣠⣴⣾⣟⣟⣿⣶⣤⡀⢶⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⢀⣀⡀⣾⣷⣨⢿⡃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⣿⣟⣻⣧
⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣏⢸⣿⡇⣿⣿⣯⣿⣿⠀⠀⠀⠀⣿⣿⣿⣾⢿⣿⣿⣿⡇⠸⠤⢸⣿⣟⣻⣿
⠀⠀⠀⠀⠀⠛⠙⠛⠛⠛⠛⠙⠋⠈⠉⠁⠉⠉⠙⠛⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⢠⡄⠀⢰⣷⠠⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡃⠀⠀⢘⣿⣿⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃⠱⡀⢸⣿⠀⡰⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠘⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢷⠶⠂⢸⣿⠀⠲⢶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡍⠀⠀⠀⠀⠀⢀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡅⠀⠀⠀⢸⣿⠀⣤⠤⣤
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣶⣤⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣟⡟⣿⣶⣤⠀⢸⣿⠸⣿⠯⣿⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣿⣿⣿⠀⣤⣤⣼⣯⡴⣶⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣯⣤⡌⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣷⣿⣿⠚⣿⣿⣿⣿⣿⢿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠈⠉⠉⠀⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠂⠐⠀⠀⠀⠛⠀⠚⠃⠐⠂⠀⠀⠀⠐⠐⠐⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠛⠛⠁⠛⠛⠛⠈⠛⠛⠃⠘⠛⠛⠁⠛⠛⠛⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠈⠉⠀⠉⠀⠉⢑⠈⠁⠉⠉⠁⠈⠉⠀⠉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⠀⠀⢀⠀⡀⠀⠀⣸⢀⡀⠀⠀⡀⢀⠀⠀⢀⠀⠀⣿⡇
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠄⠀⠀⠠⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠂⣺⡄⡸⠁⠁⠀⠀⠀⠀⠈⠁⠁⠈⠖⠉⠉⠉⠉⠲⢎⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⡢⠬⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⡒⠶⢤⡀⣿⡇⠀⡁⠀⠀⢈⠀⡁⠀⠀⡈⢀⠀⠀⠀⡀⢀⠀⠀⢀⠀⠀⣿⡇
⠀⠀⠀⠀⡔⡇⠀⠀⠀⢀⣀⡀⠀⠀⠴⠀⢠⠏⠀⢠⠚⡗⠆⠠⠛⠉⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⡏⡇⠀⠀⠀⡏⢸⡄⠀⠀⠂⠀⣿⠀⠀⠈⠳⡗⠁⠀⠐⠈⢀⠈⠐⠀⠀⠐⡀⠄⠀⠄⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠙⠪⠄⠠⠀⢤⠀⠹⡄⠀⢤⡐⠀⠀⠀⠀⢼⢹⠛⠀⠀⠀⠀⠸⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠿⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠙⠶⣤⡅⠀⠀⠀⠀⠈⢹⠶⡄⠀⠀⠀⠀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠄⢺⡌⢣⢀⣀⣀⣀⣀⣀⣀⣀⢈⠠⣄⡀⠀⡀⠀⠀⠀⠲⢼⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠂⠈⠀⠈⠓⠚⠒⠒⠒⠒⠒⠒⠒⠓⠒⠛⠋⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⡿⠿⠿⠿⡿⣻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⡀
⠀⠀⢿⣜⠐⢆⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠀⠀⠀⠀⠀⠐⠿⠈⠁⠀⠁⢠⠀⠀⠀⠀⣼⠟⢳⣝⠃
⠀⠀⠀⠉⠛⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠈⠛⢦⡇⠀⠀⠀⠀⠰⠖⡠⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⢀⠀⠰⠎⢁⠀⠀⠵⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡈⠳⣦⣶⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢆⣽⣅⣼⠍⠀⢠⣶⠾⠃⠀⠀⠀⠀⣰⣾⣭⣿⣧⡤⠀⠂
⠀⠀⠀⣤⠒⢀⣦⠀⢨⣤⣴⡿⠿⠃⠰⠿⠆⣀⠀⠀⠀⠀⠀⢀⡀⢤⡀⠘⠢⣼⡿⠋⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠉⠀⠙⠙⠻⣷⣄⣑⡆⠀⠰⡇⢳
⠀⢀⠌⢀⡾⠊⠃⠀⢸⣁⠍⠀⠀⠀⢀⢀⠀⠀⠑⠦⣄⠀⠀⠀⢨⣦⣜⡀⣦⣦⡀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣼⣙⡆⠀⠻⣷⠇
⠀⠀⠛⠋⠁⠀⠀⠀⠠⠁⠀⠀⠀⠀⢨⢟⣷⣦⡀⠀⠀⠳⡀⠀⠀⠀⠠⣸⠋⢥⣌⡦⠀⠀⢠⠋⠀⣠⡶⣿⢿⣿⠶⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⣟⣿⣿⡉⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠢⠶⠋⠀⠀⡠⢂⣤⡼⢧⠾⣥⠤⢤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⡿⢦⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢠⠀⠲⢺⡇⣼⠃⠚⢚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠸⣯⣵⣿⡇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⢯⣛⣛⣉⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠹⢾⣿⣿⣯⠟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⣠⠞⠀⠀⣤⣐⠀⡀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠈⠁⠀⠀⠀⠈⠉⠛⠟⠀⠀⠀⠀⠈⠙⠲⠄⣀⣀⣠⠴⠄⠀⠄⠀⠴⠖⠋⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠓⠒⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡾⠹⠇
⠀⠀⠀⠀⣴⠤⣀⠀⠀⠀⢰⣷⠂⠀⠀⠀⠀⠀⢀⠀⠄⣸⠅⠀⡴⠀⠀⠀⠀⠀⠀⢹⠀⠀⠉⠓⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⣿⡠⠉⡁⠄⠀⠀⠙⠛⠀⠀⠀⠀⡔⠁⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠄⠀⣦⠀⠀⠠⠀⢻⣆⠀⠀⢰⡗⠉⠶⡤⡴⠒⣀⡄
⠀⠀⠀⠀⠈⠣⣀⣂⣄⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⡃⠀⠀⠄⠀⢿⡄⠀⠈⢿⢇⠤⠰⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⡀⠀⠀⠀⠀⠀⣠⣀⢲⡄⢸⠀⠠⠀⠀⢸⠁⠀⠀⢀⠀⠘⢷⣄⡀⠀⠐⠳⠶⠀⠀⠠⣦⣆
⠀⠀⠀⠀⢰⠤⠀⠀⠞⠁⠀⠊⣀⠀⠀⢀⡾⠋⡀⠀⠀⣿⠀⢰⠀⠀⣞⣩⡬⠟⠁⢸⠀⢀⠀⠀⢸⠀⠀⠀⠀⠁⠤⡀⠈⠙⠿⣦⣄⠀⠀⠀⠀⠱⠏
⠀⠀⠀⠀⠘⠓⠛⠁⠀⠀⢀⣾⠋⠁⠰⠟⠁⠀⠁⠀⠀⣿⠀⢸⠀⠀⠉⠉⠀⠀⠀⢸⠀⢸⠀⠀⢸⡀⠀⠀⠀⢸⣆⠀⠑⠠⣀⠀⠁⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠈⣶⠀⠀⠀⠀⣾⠀⢸⡀⠀⢀⣀⠀⠀⠀⢸⠀⢰⠀⠀⢸⡃⠀⠀⠀⠘⢩⠀⠀⠀⠀⠀⢄⠀⠀⠛⠛⢿⠇
⠀⠀⠀⠀⠀⠀⠠⣤⠞⠁⠂⢺⡆⠀⠀⠀⣿⠀⠀⠀⢸⠙⠀⢸⡇⠀⢺⣍⠓⠀⠀⣸⠀⠈⠀⠀⢸⡇⠀⠀⠀⠘⢸⠀⠀⠀⠐⡦⠀⠐⠀⠀⠀⠈⣦⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠈⠉⠀⠀⣿⠀⢰⡄⠀⢸⠀⠀⡀⠀⢸⢸⠀⠀⠀⠀⡇⠀⠈⠐⠂⠤⠀⡘⣁
⠀⠀⠀⠀⡴⠋⠄⠰⡄⠀⠀⢸⡇⠀⣤⣀⣿⠀⣀⣀⣺⣇⣀⣸⡇⠀⠀⠀⠀⠀⠀⣿⣤⣬⣤⣤⣾⣧⣤⣧⣤⣯⣬⣀⣰⣄⣰⣇⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠤⠀⠖⠚⠛⠚⠛⠛⠛⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠐⠒⠀⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣠⣴⣦⠀⠀⠀⠀⠀⠀⠞⠹⠗⠶⡄⠀⠀⠐⢀⠁⢢⡀⠀⠰⠛⣈⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠁⠀⣿⣀⠠⡦⢤⡀⠀⣁⡀⢰⡞⠃⠀⠐⠀⠿⠃⣸⠁⠐⠀⣼⢃⠍⠀⠀⠀⣀⣀⡤⠼⠇
⠀⠀⠀⠀⠀⠀⠐⠒⣤⣠⠔⠂⢀⣴⣖⣢⢠⡤⢀⣤⣾⡿⠀⠀⣿⣯⠻⢿⡟⠁⠀⠉⠡⣼⠿⠆⠀⠈⠢⠤⠞⢁⠄⢀⡼⡡⠁⠤⠤⢄⡀⠈⠉
⠀⣠⡀⣀⠀⠀⢸⣶⠋⠁⣠⣴⣬⠶⠐⣘⣳⣴⠿⠟⢻⣗⣀⠰⠟⠓⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢊⠁⢀⣤⣄⠀⣿⠀⠀⠀⠀⠴
⠀⠈⠅⣉⣉⣛⣹⡇⠀⠀⢻⣛⣋⣉⠉⢉⣿⠕⠁⠀⠋⢠⣽⡟⢒⠢⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⣠⡾⠋⠀⢸⠀⠸⠇⠜⢀⣿⠁⠀⠀⠀⢰⣤
⠀⠀⠿⠀⠀⠀⣸⣷⠀⠀⣼⣷⠦⠀⠀⠈⣿⣂⠾⠅⡀⠈⠛⠃⠄⠛⠁⠀⠀⠀⠘⣉⡀⠠⠀⢀⡀⠀⠀⠈⠉⠀⠀⠀⠀⠳⠄⣠⣤⡮⠎⠀⠀⡠⠃⠘⠁⢀⡀
⠀⠀⣠⠿⠏⠀⠈⣙⣻⣷⢹⣤⣒⠘⠓⠂⣄⠁⠠⠞⠁⠀⠀⢀⡀⠀⠀⣀⡴⢳⡞⡍⡍⣠⣤⡯⠖⢢⣴⡍⠀⠀⠐⠀⠀⠤⡀⠀⠀⠀⣀⠤⣮⠙⠆⡀⠐⠎⠁
⠀⠀⠀⡀⠀⡄⠢⠉⠛⠷⠇⠻⠏⠁⢄⣠⢠⣔⢧⣀⡠⠴⠒⠈⠀⠤⠺⣾⣁⣿⠹⠷⣤⣽⣦⠓⣶⠌⠙⠹⠆⠀⣀⣀⠈⠒⠀⠀⠀⠋⠁⠰⢉⡆⢸⣷
⠀⠀⠐⠈⢛⡶⢠⣔⡳⠶⠂⠀⠈⢁⣉⣍⠳⠝⠋⠁⠀⠀⠀⠀⠂⠆⠐⠘⡖⠉⢛⡶⠾⣿⣥⣀⣼⡶⠚⠁⠀⠐⠊⠩⠀⠀⠀⠀⠀⣟⠉⢀⡹⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠢⠴⠒⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠾⢷⣾⣣⣼⠤⠶⠖⠂⢠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠍⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠘⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣉⣬⣹⣅⣠⣉⣩⣭⣁⣿⣟⣽⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣀⣀⣀⣠⣤⣶⠾⡛⠛⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⡿⠿⠿⠛⠋⣡⣤⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇⢰⣟⣁⡀⢹⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⣼⡏⠰⠾⣿⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⢦⢻⣿⣿⣿⣿⣇⠀⠨⢿⠃⣸⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⡙⡄⠀⠀⠀⠀⠹⣿⣟⣁⣾⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠘⣏⢿⣿⣿⣿⣿⣿⣶⡶⠾⠛⣉⣁⣤⣤⣤⡄⠀⠀⠀⢸⣿⣿⣿⣧⠹⡀⠀⠀⠀⠀⠈⠉⢉⣡⣤⡶⠾⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠸⣎⠛⠛⠛⠛⣉⣡⣤⣶⣿⠿⠟⠛⠛⠛⠛⠀⠀⠀⢸⣿⣿⣿⣿⣧⠳⣦⣤⣤⣴⡶⠿⠛⠉⢀⣀⣤⣤⣤⣤⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⡀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⡴⠋⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⢤⡀⠀⠀⡀⠀⠀⠾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠹⣷⣄⢷⢀⣆⡀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣀⣠⣿⣿⣿⣿⠿⢧⣈⣶⣤⣀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠚⣽⢻⢟⠋⠏⠃⡀⠀⠍⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢸⣿⠟⠞⠀⡄⠀⠈⢠⣼⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⣠⣄⣔⣂⡴⠀⠂⠉⠁⠀⠀⠀⠀⠀⠀⣠⣟⠀⠀⠐⣿⠶⠚⠉⢉⠀⠀⠉⠉
⠀⠀⢠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠁⠀⠀⣰⣆⠀⠄⣀⣾⣤⣤⣤⣤⠞
⠀⠀⢸⣿⠛⡿⣿⢿⡏⡇⠀⣤⡤⠀⠀⠀⢀⣠⣾⠏⠀⠀⢠⣾⢿⡟⠿⠛⠉⢤⠀⠀⠀⠀⢱⡀
⠀⠀⠈⢁⣀⠀⣀⣴⣇⣧⠀⢳⣤⣤⣄⣶⣿⣿⠋⠀⠀⠀⠸⣿⣼⡇⣷⣤⣤⣾⣶⣶⣶⣾⣿⡁
⠀⠀⣿⣼⣿⣿⣿⣿⣿⢸⠀⠘⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⢻⣷⣿⣄⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡎⡆⠀⢻⡟⠉⠙⡟⠀⠀⠀⠀⠀⢀⡀⢸⡇⢹⠻⣷⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⢹⠀⠈⠆⠀⢀⣷⡀⠀⠀⠀⠀⠸⣿⣤⣇⣸⣤⣿⠇⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⢿⠀⠀⠀⠀⢹⣿⣶⣄⡀⠀⠀⠈⠛⡿⢻⠛⠉⠀⠀⠀⠀⠀⠀⢀⣬
⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠈⠛⠿⣿⣿⣷⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⠿⠟⠛⠛⠛⠋⠉


⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢹⢠⠄⠀⢤⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠂⠄
⠀⠀⠀⠀⢹⢸⠀⠀⠁⠀⠀⠉⠀⠈⠑⠀⠀⠀⠀⠀⠄⢡⣄⣀⡀⡀⣀⣀⡄⢀⠊⠀⣠⣄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⠢⣤
⠀⠀⠀⠀⢸⢸⠀⠀⠀⢰⠀⠀⠰⠀⠘⠃⠠⠀⠀⠀⠀⢸⣦⣄⠠⠤⣄⣤⡆⢸⠀⠀⣙⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠻⡦⡀
⠀⠀⠀⠀⢸⢸⠀⠀⠀⢼⠀⠀⠀⢰⠒⠀⠲⠄⠀⠀⢀⡴⠤⠄⢀⠀⢀⣤⠄⠀⠁⢄⡈⣁⠠⣂⠀⠀⠀⠀⠀⠀⡠⢊⣴⣶⣦⡀⠀⢀⠜⠁⠈⠄
⠀⠀⠀⠀⢸⢸⠀⠀⠠⢸⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⢄⣁⣉⣉⣉⠨⠤⡀⢀⠠⠀⠀⠀⠀⢈⡀⠀⠀⡠⠐⠁⠀⠸⣿⣿⣿⠇⣠⡆⠀⢀⠊
⠀⠀⠀⠀⢸⢸⠀⠀⢰⢺⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢨⡍⣉⣁⣉⣉⣉⡅⢨⣉⣉⣉⣈⣉⣹⡇⠀⠈⠢⡀⠀⠀⠀⠈⠉⡡⠾⠟⢁⠔⠁
⠀⠀⠀⠀⢸⢸⡀⣀⣘⣸⠀⠀⢸⣸⢀⠀⣨⣀⠀⠀⠀⠸⣉⡀⠀⠀⢀⣈⡅⠨⣁⣐⢀⢀⣀⣩⠇⠀⠀⠀⠈⠢⡀⡠⠔⠈⠀⡀⠔⣡⠞
⠀⠀⠀⠀⠀⠈⠈⠈⠈⠉⠉⠁⠉⠁⠁⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣗⣚⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⠠⠈⠉⠁⠉⠁⠄⠀⠐⠝⠇
⠀⠀⠀⠀⠐⠈⢅⠀⠀⠀⠀⢠⠁⣸⣿⡟⠀⠀⢄⠠⠊⠀⠀⠀⠀⡔⠀⠀⠀⠈⠢⠀⠁⡠⠂⠀⠀⠀⠀⠁⣤⡀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢃⢡⠀⢠⣿⡿⠀⢀⠁⠠⠀⠁⠀⠀⢀⠈⠄⠀⡴⢴⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠰⠁⠀⡔⣄⠀⠑⡀⠀⢸
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡌⢀⣿⣿⠁⠀⠈⢀⠂⠀⠀⠀⠀⠂⠌⠀⠈⢤⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⠐⢾⡂⣵⠗⠀⡇⠀⢸
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⠀⣾⣿⠇⢀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣍⡀⠈⢄⠀⠑⠁⢀⠔⠀⢠⢸
⠀⠀⠀⠀⠀⠀⠆⠀⠀⢅⣸⣿⡏⠀⠄⠀⡐⠀⠀⠀⠀⢈⢄⠢⡁⠀⠈⠓⠚⠁⡀⠌⠐⠀⠀⠀⠀⠀⠀⠄⢉⠁⠀⠀⠈⠉⠉⠀⠸⠜⠈⠘
⠀⠀⠀⠀⠀⠌⠂⠂⠈⠁⠀⠉⠑⠀⠌⠠⠁⠀⠀⠀⠀⠀⠑⢁⣈⣤⣤⣤⣤⣤⣤⣉⡀⠀⠀⠀⠀⠀⠀⢌⠂⣥⡤⠉⠀⠀⠬⠍⠡⣬⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠐⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠺⡿⡀⢀
⠀⠀⠀⠀⠀⠁⡀⢐⣂⣀⣀⣀⣀⡒⠲⣦⡀⠀⠀⠀⠀⢀⠀⡤⠀⠁⠀⠀⠀⠀⠈⠐⢤⢄⡀⠀⠀⠀⠀⠀⢸⣿⡠⠗⢀⣐⣣⡼⢀⠀⢄
⠀⠀⠀⠀⠀⠀⡇⠅⠀⠀⠀⠀⠀⠨⠀⣿⡇⠀⠀⠀⠀⠈⠲⡖⠶⠂⠒⡶⠶⠖⠐⢶⠶⡖⠁⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠉⠀⠑⠐⠄
⠀⠀⠀⠀⠀⠀⡇⢔⡢⢐⡠⣒⠦⣂⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡄
⠀⠀⠀⠀⠀⠀⡇⠂⠰⢈⠰⠈⠶⠀⠀⣿⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠰⣌⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⡇⢊⣑⣊⡑⢩⣓⣉⠀⣿⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄⢀⠀⠀⢀⠀⡀⠧⠀⠀⠀⠀⢠⢸⣦⡀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⢇⠑⠀⠀⠐⠃⠀⠀⢁⡿⢣⠀⠀⠀⢀⠐⠚⠙⠛⠂⠚⠛⠋⠓⠐⠋⠋⠛⠂⡀⠀⠀⠀⠲⠄⠿⠋⡀⠔⠐⠲⡟⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠙⠒⠦⣀
⠀⠀⠀⡴⠋⢀⠴⠚⠉⢉⣩⠙⠢⡄⠈⢣
⠀⠀⡼⠀⣰⠃⠀⠀⢸⡉⠉⠆⠀⠘⡆⠀⣇
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠱⡄⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣚⣋⣉⣉⣉⣛⡒⠲⢤⡀
⠀⠀⢧⠀⠹⣄⠀⢳⠖⠚⠁⠀⣠⠎⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⡞⠭⣔⣀⣠⠖⠛⢳⠾⢙⡆⠘
⠀⠀⠈⢷⣄⠈⠙⠒⠶⠖⠒⠋⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣈⡑⠒⠒⠒⠖⠒⠒⠚⣉⣤⣾
⠀⠀⠀⠀⠙⠿⣷⣶⣶⣶⣶⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣀⠤⠤⠤⠶⠦⠤⠤⢤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢩⠔⢒⡏⠉⣉⠽⠿⣗⣲⢤⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡘⠺⠭⢿⣛⣁⣀⣠⠥⠴⠚⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣦⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⣀⣀⠉⠙⢿⣿⣿⣿
⠀⢀⣠⠤⠔⠒⢒⣒⠒⠒⠦⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡴⠋⠁⠀⠈⠉⠲⣄⠘⢿⣿
⡔⠉⡴⣚⣩⠉⢉⡽⠛⢯⡽⠖⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⣠⠄⠀⢀⣠⣀⠈⣆⠈⣿
⣷⣄⡙⠒⠬⠭⠭⠤⠤⠬⠔⠚⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠈⠹⣄⣠⠏⠀⣸⠦⢸⠀⢸
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢣⡀⠀⠀⠀⠀⠁⢀⡞⠀⡼
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠉⠉⠙⠛⠛⠉⠉⠉⠀⠘⢿⣷⣄⠉⠲⠤⠤⠤⠴⠋⠀⡴⠁
⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣄⣠⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠠⢄⣀⡀⠀⠀⣀⡠⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠦⠄⠰⠳⠑⢤⠠⠈⣉⣁⡀⡹⢢
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠚⠉⠉⠐⠒⠳⠖⠀⠐⠞⠏⠓⠉⠁
⠀⠀⠀⠀⣁⣤⠰⠶⢶⡶⣆⠒⠢⢤⣤⡔⡀⠥⢒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡠⠄⢤⣀⡀
⠀⢀⡔⠹⠶⠾⠷⢄⣊⡄⢓⠾⠢⠚⠁⠀⠀⠀⠀⢩⠐⣄⠀⠀⠀⠀⠀⠀⡤⠐⠉⢡⠌⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣀⠄⣠⠼⠐⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡴⣆⣐⣐⠢⠀⠀⢈⠠⡼⠂⠀⠀⠀⠀⠄⣈⠀⢀⡀⠀⢠⡯⠤⠤⠄⢠⣄⣀⡀⠀⠀⣐⣵⣔⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡟⠤⡌⠁⠈⢁⠄⡀⠕⠀⠀⠀⠀⠠⠎⡐⢁⣀⡛⣲⠀⣰⡏⠀⠀⠀⢀⠯⡏⠛⡑⠉⡉⠉⠀⣧⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠳⠧⠭⠄⠒⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⠁⠌⠉⢌⠁⠀⠅⠒⠂⠉⠉⣡⠖⣒⢿⠶⡄⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠀⠀⠂⠀⢆⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠢⡤⣌⢘⡎⠁⠀⠀⠀⠀⠔⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢄⢉⠢⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠤⣒⣤⠊⠁⠀⠀⠀⡀⢀⣀⣀⣠⡤⣤⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⠣⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠖⣤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢦⠯⢀⡀⠈⢛⡲⣤⠄⣀⡀⠀⣠⢠⣈⡲⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣠⢄⡨⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⢄⡦⢭⠕⠦⢈⠂⠉⢗⢮⡙⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣌⢀⣄⠀⠀⡆⡏⢑⡦⢄⣀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠⠀⠀⡀⠀⠀⠉⠺⠇
⠀⠀⠀⠀⣀⠄⠒⣦⡄⢰⠒⠚⠙⣆⠀⠁⠅⠛⠒⠓⠬⠇⡬⠩⢂⠉⠍⠉⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⢢⠔⠢⠤⠀⠆⡠
⠀⠀⠀⠐⠌⣀⠤⣨⣈⣀⠁⣷⠾⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣉⣻⣛⣉⠁⠁
⠀⠀⠀⠰⣆⢁⡜⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣧⠀⢀⡨⡅⣆⠑
⠀⠀⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠉⡭⢯⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠩⠁⠈⠺⠤⡀
⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠰⠶⠚⠒⠚⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠁⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠊⠉⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⠛⠛⠉⠉⠛⠛⠻⢿⣷⣦⣀⠀⠀⣀⣴⣾⡿⠟⠛⠛⠉⠉⠛⠛⠿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠙⢿⣷⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣤⡀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⢿⡶⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⣠⣾⡿⠛⠉⠉⠉⠻⠇⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⡀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣶⣶⣿⣿⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠈⣿⡇⠀⠀⣿⣧⠀⠀⠀⠿⠿⣿⣿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠻⣿⣦⣤⣾⠿⠁⠀⠀⠹⣿⡆⠀⠀⠀⠀⠙⢿⣷⣤⣀⣀⣠⣴⡆⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠁⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣠⣾⡿⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣤⣤⣀⣀⣤⣤⣴⣾⡿⠟⠉⠀⠀⠉⠻⢿⣷⣦⣤⣤⣀⣀⣤⣤⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠙⢿⣦⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⣠⣀⣀⣠⣤⣀⣨⣿⡿⠛⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢠⡞⠁⠈⠉⣡⣴⡿⠛⠁⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢀⡴⠋⠀⣀⣴⡾⠟⠁⣠⠴⠛⢦⣤⡤⣄⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠏⢀⣤⡾⠟⠉⣀⡴⠛⠁⠀⠀⠀⠀⠀⠈⢷⡀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣀⣤⣾⠟⠋⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢀⣤⣾⠟⠋⢀⠀⠰⣆⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⠟⣡⣶⠿⠋⢁⣤⠞⠉⠀⢠⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⢁⣠⠖⠋⠀⠀⠀⠀⠈⠻⠶⠾⠿⠿⣶⡀⠀⠀⠀⠀⣀⡴⠛⢁⣠⣾⠟⠋
⠀⠀⠀⣠⣿⢿⡟⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣿⠇⠀⢀⡴⠞⢁⣠⣶⠿⠋⠁
⠀⠀⢰⣿⠁⠘⢿⣆⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⡄⠀⠀⣠⣴⡿⠋⠁
⠀⠀⠀⠻⣷⣄⡈⢻⣧⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠈⠙⢿⣦⡹⣷⣄⠈⢧⡀⠀⠀⠀⠀⢀⣠⠖⠋⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⠀⠛⠉⠉⠳⠶⠋⢁⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡀⠀⠀⢀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⡉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⣀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⣾⡿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢁⣠⣴⣶⣾⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣉⣉⣉⡀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣴⣿⣿⣿⣿⣿⡿⠛⠋⠀⠛⠻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣄⣀⡉⠉⠙⠛⠛⠛⠛⠛⠀⣼⣿⣿⣿⣿⣿⡏⠀⢠⣶⠀⣦⠀⠈⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠆⢠⣿⣿⣿⣿⣿⣿⣇⠀⠘⠿⠀⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠉⠛⠻⠿⠿⠿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣤⣤⣤⣤⣀⣀⡀⠘⣿⣿⣿⣿⣿⣿⠛⠙⣿⣿⠀⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⣿⣿⣆⠀⠘⠻⠀⠟⠁⢀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣄⣀⣈⡉⠉⠉⠙⠛⠛⠃⠀⠻⣿⣿⣿⣿⣿⣷⣦⣤⠀⣤⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠷⠆⠀⠉⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢩⠱⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⡃⠀⠐⡘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠐⠆⠠⠔⠀⠤⠓⠒⠈⠈⠀⠉⡉⡙⠒⠶⢤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠤⠄⢺⠀⠀⠀⠀⠀⢀⢀⡐⠛⠛⠉⠀⠀⠖⠒⠤⠤⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠒⠁⠀⢢⠀⠀⠀⠀⠀⠐⠒⢋⠤⣄⠀⢦⣴⠪⢕⣺⣓⠪⢶⡶⢄⡠⣀⠈⣁⣍⣤⡒⠈⠁⠁
⠀⠀⠀⢠⠀⠀⢀⣬⠤⠤⡁⠒⠒⠀⠒⠘⡂⠀⠂⢻⣉⢀⡀⠀⠀⢀⠀⠐⠂⠀⠀⠀⠉⣀⡠⠄⢂⣀⡇⠰⠂
⠀⠀⠀⠀⠈⠉⢍⡀⡈⣉⣅⠀⠀⠀⢒⣸⠷⠤⠬⣟⡁⢭⠅⠉⠉⠛⠛⠻⠷⠆⣀⠩⠭⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠉⠉⠉⠀⠀⠀⠀⣅⠀⠀⠐⠂⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⣀⣸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠁⠀⢼⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣾⠛⠷⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣈⠙⣦⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠉⠛⠉⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⣸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠏
⠀⠀⠀⠀⠀⠀⠷⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠾





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢪⣵⠖⠒⠛⣩⠭⡄⢐⣒⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣪⠋⠁⠀⠀⠀⠀⠐⠻⠈⢠⠚⣡⡾⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⠙⠅⠉⠉⠉⠛⠽⢶⣄⠀⠀⢈⣴⣿⢿⣫⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢻⣿⣿⣿⣿⣿⣯⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠔⠉⠀⢢⠀⠀⠉⠭⡻⢒⠶⣤⣄⡀⠀⠄⡌⣾⣿⣿⣿⡟⢗⢀⣴⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⠕⡾⠅⠀⠀⠀⠀⠀⠙⠢⠤⠤⠴⠔⢊⢟⣥⣿⠈⢞⣼⢿⣿⠿⠗⣠⡾⣿⡿⡿⡧
⠀⠀⠀⠀⠀⠀⣺⣅⠀⠲⠋⢡⠀⠀⠬⠄⠀⠀⠀⠀⠀⠀⠀⢀⠤⣪⣵⣽⣿⣿⠄⣽⣿⡿⠿⠥⠾⣿⢾⣝⣝⡴⠋
⠀⠀⠀⠀⠀⠀⣿⠪⣍⣙⠳⠶⣭⣑⠢⢌⣀⠔⢒⣲⡖⠐⣪⢵⣺⣿⣿⣿⣿⣟⣨⡾⠭⠁⠀⠀⡄⣿⣫⣫⣾⠣⣄⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠩⢝⡳⢶⠬⢿⣶⠤⣒⣫⢴⣾⣽⣾⣿⣿⣿⣿⠿⠿⢎⣁⣀⣀⠀⡄⣶⡃⣾⣿⠿⠧⣭⠁⠘⠿⠓⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠑⠦⢤⣀⡒⠬⢅⡋⠃⠊⢹⠒⠾⢵⣟⣻⣿⢻⠿⠻⠉⠉⢁⣀⠤⣞⡽⡷⣽⢧⠻⣟⡿⠶⠒⢙⡧⢠⡁⠞⢚⣠⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣶⠬⣝⡛⠴⡬⠍⣽⣿⣿⠛⠫⠏⠀⠈⠉⠁⡠⢮⣾⣾⡇⡎⢦⣞⠁⠀⠀⡘⠀⠈⠒⠭⢗⣿⣿⣯⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠿⠓⡡⢍⣉⠛⠴⢼⠿⠭⠅⠈⠉⠀⠀⠀⢀⢔⣔⣟⣽⣿⡇⣦⠏⠀⣀⠤⠊⠀⠀⠀⠀⠀⠒⠀⠉⡏⠉⡓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣶⠥⠤⠪⠁⠀⢃⡀⠤⠬⠷⣒⠂⠀⢀⠤⠄⢲⠖⢒⣵⡷⣳⣫⣿⣿⠛⠍⠁⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⢀⠘⣄⠀⠣⢋⡤⢞⣥
⠀⠀⠀⠀⠀⠀⠀⣯⠍⣉⣀⣒⠒⠂⠠⠬⠍⢉⣙⣒⠊⠥⠤⠬⠥⢒⡽⣫⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣠⠤⡺⠴⣚⡥⢖⣩⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣒⣒⠒⠈⠭⠭⠭⠀⢒⣒⣒⣶⣚⣒⣟⣤⣥⣾⢾⣿⣿⣷⣴⡤⠐⠶⣶⠠⢄⡀⠀⣀⡠⠄⣂⡨⠱⣐⣭⣶⣿⣷⣾⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢱⠶⡰⢬⠭⣭⣩⣀⣐⣒⣒⡒⣲⢒⡒⣒⢒⣚⣺⡿⢕⠅⠀⠉⠺⢗⡦⢭⡙⣂⣨⠭⣔⣲⡯⢷⣿⣿⣿⣟⠿⠒⣉⠤⢖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡓⠉⠢⠕⠦⣑⠂⠊⠴⠩⢚⡕⣨⠌⡁⢩⢌⠄⡹⣷⣵⠀⠀⠀⠈⠛⡗⠪⣻⣿⣯⣼⣾⣿⠿⠟⣋⠡⢔⣚⣭⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠂⠀⠀⠀⠀⠈⠑⠺⢦⡤⠤⠔⠐⣤⣄⣠⢔⣾⣶⣟⣽⣷⣄⡀⠀⠀⠅⠈⡿⠿⣟⡫⣭⣖⣾⣍⡘⠭⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠒⣽⣦⣜⣛⣩⣤⡾⣻⣿⣿⣟⢿⠭⠊⠉⠒⠤⣀⠀⣿⡽⡼⠾⠛⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠢⡢⠄⠑⣯⣿⢿⣯⡷⠗⠋⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⢀⡘⠀⠸⣿⠗⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠠⡠⢔⠀⡀⠐⡨⢀⠁⢊⠔⠁⢔⠀⢂⠔⢁⠄⡀⢂⠔⡀⠀⠠⢀⠔⠠⢂⠄⠀⡠⢒⠔⢀⠄⡠⢂⠔⡠⢈⠔⡡⢂⠔⡀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠐⠡⢊⠤⢂⠤⡢⠐⡡⢊⠥⢊⠔⡈⢐⡥⠊⠔⡠⠀⡡⢂⠔⡠⡪⢊⠔⠁⠐⡡⢊⠔⡡⢊⠔⢡⠎⡔⠡⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠐⠁⠐⠁⠊⠀⠉⠐⠡⠂⠠⠊⠄⠁⠊⠎⠀⠀⠘⠀⠊⠀⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠣⡔⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣧⡰⠃⡼⢨⠆
⡏⢉⠁⢡⡿⢿⣿⡉⠁⠈⡉⢹⠀⠀⠀⠀⠀⠐⣀⢸⣷⠃⠀⠐⠗⡇⠀⠳⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣶⣷
⡇⠘⠁⠹⣿⣬⡿⠁⠘⠉⠃⢸⠀⠀⠀⠀⠀⠀⠉⣸⣟⣠⣴⣶⣶⣷⣶⡴⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠳
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢉⢎⢀
⠀⢿⣿⣯⣤⣿⡷⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⣤⣤⣤⣤⢤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣾⣿⣦⣅⣊⡜
⢰⣾⣶⣿⣿⣷⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⠋⣿⣧⠀⣀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠈⢹⡍⠹⡏⢉⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣾⣿⠋⠀⠉⠉⣁⣸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⡿⢿⣿⣿⡇
⠀⠉⠷⠲⠖⠒⠇⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠙⠛⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠈⠙⠃


⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣶⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣤⣶⣶⣶⣾⣷⣾⣿⣶⣶⣶⣦⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠉⠉⠉⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⡿⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣷⣮⡠⢂⣽⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣹⠏⠭⣽⠩⡍⣿⠩⠀⠀⠀⠀⠀⢸⣿⣿⣄⣭⣴⣾⣽⣿⣯⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣭⣿⣬⣯⣿⣭⣭⣽⣧⣵⡀⠀⠀⠀⢸⣿⣿⣿⣾⣶⣶⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⠿⠽⠋⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠤⠄⣤⣆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠛
⠀⡀⠰⠌⠸⠷⠯⠀⠒⢉⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣷⣶⣾⣶⡖
⣘⠨⢨⡷⡖⠒⠒⢲⣴⡁⠔⡰⠀⠀⠀⠀⠀⠀⢠⠄⠀⣠⣤⡄⠀⠀⠤⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠁⠙⠛⠡⠬⢥⠿⣿⡧⠈⠀⠀⠀⠀⠀⠀⢠⣥⣍⣘⣿⣾⣏⣈⣉⣡⣵⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡄⢠
⠀⠈⠿⣬⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⣄⠀⠀⠀⠀⠠⣀⠀⠀⡀⠘⠓⠀⠀⠀⠀⠈⢁⠀⠀⠀⣆⠀⠀⠰⣋⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⡆⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⣱⠇⣾⠃⢀⣠⡴⠀⠀⠀⠀⣠⣳⣶⣀⠁⠀⠀⠀⠉⠁
⠀⠀⣠⡄⢀⡄⠰⣉⣰⡾⠿⠹⠷⢀⡀⠀⠀⠀⠀⡀⣀⠀⠠⣽⡿⠃⠀⠈⠉⠀⠀⡀⠤⠚⠉⠙⠛⢿⣤⣒⡄⠀⢴
⠀⢰⢀⣴⠊⠀⠰⣭⠋⠁⠀⢀⣀⠀⠈⠢⢄⠀⠀⢈⠦⠅⣠⡾⠀⠀⠀⠀⠀⠔⠁⢀⣀⠀⠀⠀⠀⠀⠱⢾⡹⠀⠘⠿⠇
⠀⠈⠉⠀⠀⠀⡐⠀⠀⠀⠀⣸⣯⣿⠆⠀⠀⠑⡀⠀⠀⠐⠋⢐⡾⠃⠀⠠⠁⣠⢞⠟⠛⠛⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢿⣿⣿⣤⣄⠀⠀⠀⢢⠀⠀⠈⠉⠀⠀⠀⢠⣻⠋⡿⠹⠿⠏⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢀⣀⢻⣿⣿⡆⠀⠀⠈⡅⠀⠀⠀⠀⠀⡀⠀⠈⢿⣓⣒⣚⡦⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡠⠂⠀⠐⢮⣄⣄⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠠⠤⠤⠤⠀⠤⠐⠂⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠐⠂⠉⠉⠈⠉⠉⠀⠀⣴⡶⣦
⠀⠀⠀⣠⠀⡀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⣞⠀⣀⠀⠀⠀⠀⠀⠇⠀⠐⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢿⡰⠋⠆⠀⠀⠙⠃⠀⠀⠠⠊⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⢳⡀⠀⢰⡒⠲⠤⡔⢒⡠
⠀⠀⠀⠀⠑⠮⠴⡄⠀⠀⠀⠀⠀⠀⢠⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠈⣧⠀⠈⢻⣄⣨⠋⠈
⠀⠀⠀⢀⡀⠀⠠⠏⠀⠄⠀⠤⠉⣴⢿⠀⠲⡆⠀⡃⠀⣴⣟⣿⠇⡏⠀⡅⠀⣿⠀⠀⠘⢀⠙⠛⠦⣤⡈⠉⠀⠀⢻⣳
⠀⠀⠀⠘⠦⠚⠀⠀⢢⡾⠃⠤⠞⠃⠸⠀⠀⡇⠀⡄⠈⠛⠛⠀⠀⡇⠀⠆⠀⢿⠀⠀⠐⣦⠈⠲⠀⠈⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠐⠰⡆⠀⠀⠀⡇⠀⡇⠀⣀⣀⠀⠀⣿⠀⠆⠀⣼⠀⠀⠀⢻⠀⠀⠀⠀⠠⡀⠈⠛⠻⠇
⠀⠀⠀⠀⠀⣠⠞⠁⠀⣿⠀⠀⠈⡇⠈⠀⢨⠁⢀⡇⠀⠻⣍⠀⠀⡏⠀⡀⠀⢿⠀⠀⠀⠘⠀⠀⠀⢘⡆⠀⡀⠀⠀⢳⡄
⠀⠀⠀⢀⠔⠁⡀⠀⠀⢿⠀⠀⢰⡇⢀⠀⣾⠀⠸⡇⠀⠀⠀⠀⠀⣇⠀⣇⣀⣿⠀⣆⢀⣀⠀⢠⠀⢸⡆⠀⠀⠀⠀⠀⠷
⠀⠀⠀⣉⠀⠤⠿⠤⠶⠿⠖⠚⠛⠓⠚⠛⠛⠓⠚⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠙⠛⠛⠓⠒⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣄⣤⠀⠀⠀⠀⠀⠚⠷⢲⡄⠀⠠⢀⡀⢤⠀⡠⠛⣗⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣸⣧⠀⣿⣤⣶⣖⡄⠀⣤⠀⣟⠁⠀⠀⠛⢁⡎⠀⢁⡜⠀⠀⠀⠀⡤⠤⠚⠃
⠀⠀⢀⠀⠢⣈⡽⠟⠉⣀⡘⢋⢫⣒⣡⡾⣿⠇⠀⣿⣏⠉⠿⠀⠀⠀⠙⠋⠁⠀⠀⠉⢉⠀⣠⠮⠂⠀⠀⠐⣄
⠀⢛⢀⣒⠒⣾⠁⠀⣾⠟⠛⠂⢹⡿⠍⠀⠞⢁⣤⢥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢰⠀⢾⡉⠀⣸⠀⠀⠀⣠⡀
⠀⢨⠇⠈⢉⣹⠄⠀⣼⣿⠆⠀⠨⣗⡴⠇⡂⠙⠛⠒⠚⠁⠀⢀⢘⢁⠀⠀⠀⢀⠀⢘⠋⠀⠀⠀⠢⢤⣤⢴⠋⠀⢀⠆⠘⠉⣀
⠀⠀⠌⠃⠠⢭⣘⣿⡼⣾⠃⠁⣐⠮⢠⠊⠉⢀⢠⡠⠀⣀⡼⡹⣿⠤⢠⡶⠞⠉⣷⣭⠁⠀⠀⢠⠀⠀⠀⢀⡤⢲⡏⢋⡄⠐
⠀⠀⡐⣀⣈⠁⡬⣙⠀⠉⠥⢶⠶⡲⡿⠖⠂⠉⠀⡠⠈⡛⣗⠶⢌⣙⣧⡿⠈⣿⣀⠌⠁⠐⠶⡆⠀⠀⠀⠥⠤⠁⡮⠈⠃
⠀⠀⠀⠈⠁⠛⠷⢬⣥⢤⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠤⣶⣭⣽⣛⣘⣭⠄⢀⠀⣀⠀⠀⠀⠀⠀⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⢔⠮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢔⣴⡂⠎⠉⠯⠉⠽⢿⠿⡟⠇⡟⡫⢾⠶⠾⠿⠿⠛⠟⠘⠘⣊⡉⠃⣿⣻⡇
⠀⠀⠀⠀⠀⠀⡠⡴⠉⠀⢀⡀⠀⠀⠀⡐⣠⡾⣠⢳⣾⠀⢠⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⡆⢹⣻⡇
⠀⠀⠀⠀⢀⠪⠊⠀⠀⠀⠑⠁⠂⠀⠀⣿⣿⠅⡴⣾⣿⠆⡘⠀⠀⠘⠛⠛⠋⠀⢀⡀⡐⣀⢿⣿⡏
⠀⠀⠀⡠⣰⣧⣤⡤⠰⠠⣰⢐⢢⣤⣤⣾⢏⢜⡇⣿⣭⣮⣤⣀⣁⣀⡀⡁⣼⡿⠍⠷⣴⣏⣾⠟
⠀⠀⠀⣷⣶⣿⢛⣛⣻⣿⣶⣶⣶⣾⣭⣿⣾⣴⣷⣿⣿⡿⠿⢿⣿⡿⠿⠷⣖⣚⣛⢛⣿⣿⡟
⠀⠀⠀⣯⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠭⠽⠷⣶⣶⣶⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠁⢀⣀⠀⠀⠀⢀⣶⡿⠟⠙⢿⣶⡀⠀⠀⠀⣀⡀⠀⠙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢰⣿⣿⣿⡆⠀⣿⣿⣅⡛⠸⢿⣿⣿⠀⢰⣿⣿⣿⡆⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠿⠿⠟⠁⠀⢻⣿⣛⠛⢘⣠⣿⡟⠀⠈⠻⠿⠿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠼⠿⠋⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡤⡤⣤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣴⣿⡿⠿⢿⣿⣿⡿⠉⢀⣼⣧⡄⠉⢿⣿⣙⣛⠻⢻⣿⣶⣤⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡏⠀⠀⠀⢻⣿⠀⠀⠻⢾⣧⡀⠀⠈⣿⡟⠀⠀⠀⢸⣿⡧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣷⣄⣀⣠⣾⣿⡄⠀⠦⣼⡧⠟⠀⢠⣿⣷⣄⣀⣠⣾⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣦⣄⣘⣃⣠⣴⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠀⠀⠀⠚⢒⡆⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠁⠀⠀⢀⣀⡠⠤⠒⠚⠉⡄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡑⡏⠉⠉⠉⠀⠀⠀⠀⠀⠐⠛⠃⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠨⠥⠤⠤⠤⠤⠤⠤⠄⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⠀⠠⠀⠀⠀⠀⢀⣰⣼⡴⠧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⠤⠤⠸⠭⢍⣏⣀⠈⠀⠀⠀⠀⠀⠒⢛⠋⣹
⠀⠀⠀⠀⠀⠀⢠⠘⠦⠤⢀⣀⣀⠀⣀⣀⣀⣀⣀⣸⡹⡆⠀⠀⣀⣁⡴⡼⠶⠋
⠀⠀⠀⠀⠀⠀⠘⣦⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣆⡿⠀⠀⠀⠀⠀⣀⡴⠖⠗⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠰⡀⡁⡌⠀⠀⠀⠀⠀⠀⠀⠠⠘⠚⢙⠀⠀⠀⠀⠀⠒⠛⣅
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠧⠄⠀⠀⠀⠀⠀⠀⣶⣰⠼⢿⠀⠀⠀⠀⠤⢠⡆⡝⣺⣄⡀⠀⠐⠠⢄⡀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⡳⣤⡄⢠⠀⠀⠀⠀⠀⠀⢰⢤⣖⡿⡀⠀⡀⠀⠀⠀⢩⢈⡁⣾⠉⠓⠢⢤⣴⡀⡉⢉⣋⡟
⠀⠀⠀⠀⠀⠀⠀⢇⡀⠈⠉⠁⠀⠀⠀⠀⠉⠈⣩⡧⡖⠀⠀⠃⠀⠀⠀⠚⠚⠚⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⠤⠤⠤⠤⠴⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⠀⠠⠐⠂⠈⠁⢀⠁⠀⠀⠀⠀⢄⠠⠂⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠢⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⡀⠘⢀⠀⠀⠀⠀⢀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠱⡄⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠔⡀
⠀⠀⠀⠀⠀⠀⠄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁⠀⠀⠀⠀⠤⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⢂⣈⣀⠀⠀⠀⠀⠌⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⣀⣘⣤⠄⠀⠀⡀⠀⠈⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢻⣿⠟⡡⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠄⠀⠀⠁⠄⠀⠔⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠂⠀⢀⠠⠄⢈⠂⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡄
⠀⠀⠀⡀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⡀⠀⠀⠈⠁⢀⠀⢀⡐⠈⠀⡀⠀⠀⠀⠂⣀⠀⠀⠀⢀⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠏⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣂⠀⠀⠀⠀⠂⠀⠀⠀⢀⣼⠟⠡⠀⠀⠀⠠⠀⢀⠘⢄
⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⣴⠯⠉⠀⠀⠀⠀⠀⠈⠀⠐⠁⠈⠡⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠸⢿⢿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠐⣨⣤⣬⣄⣀⣀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠥⠭⠛⠗⣐⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⣀⣤⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣦⣄⣀⣠⣾⣿⣿⣿⠿⢿⣿⣿⣷⣶⣿⣿⣿⣿⡿⢿⣿⣿⣷⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠙⠻⠿⠿⠿⠛⠋⠁⠀⠀⠹⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠛⠛⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣦⣀⡀⠀⠀⣀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣴⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠈⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⡿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⡀
⠀⠀⢀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⢠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣿⣤⣤⣼⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄
⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⢿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄
⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠁⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠈⠛⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⣄⣀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣦⣤⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠿⢿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠈⠉⠛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⢈⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣾⣿⣿⣀⣀⣸⣿⣿⣀⣠⣤⣶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏
⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠈⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠙⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠟⠉⠉⡽⢲⠎⢩⡯⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⢋⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠷⢦⣀⡀⣠⣾⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡼⢛⠗⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢈⢻⣿⣿⣿⣿⡽⠧⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠭⠒⠉⠀⠘⢄⡠⠀⠐⢉⡫⢛⣶⡦⣄⢠⢂⢆⣿⣿⣿⡿⠏⣎⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⠖⠙⠓⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡥⣪⣾⣿⠀⢊⣾⣿⡟⠝⢉⣶⢿⣿⣻⢻⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣍⡁⢮⣀⡑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⢁⢿⣿⡿⠟⠚⠻⣿⣝⡕⣵⠏
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡛⠶⣭⣕⣲⠚⣩⡽⢛⡩⢕⣺⣷⣿⣿⣿⣿⣯⡖⠛⠀⠀⠀⣀⡆⣿⣾⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⣑⠪⢝⣒⢻⣾⣽⣓⣖⣿⣶⢿⣿⣿⣿⠿⠿⠏⠓⠒⣶⣤⣄⢮⡟⡄⣿⡿⣉⣑⣦⠀⠩⢃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢭⣍⡲⠬⣅⡂⠭⢙⢻⣿⣶⣿⡧⡟⠓⠈⠤⠤⠖⡩⣫⣯⡟⢾⠑⣹⠟⠋⠉⠀⠺⢕⡳⢜⣨⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⠳⠦⢭⣙⠫⣼⠿⣟⢉⠬⠍⠀⠀⠀⢀⢜⣴⢿⣿⡏⢇⡿⡓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠿⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠤⠉⢰⡁⠒⠂⢭⣙⡋⠨⠁⠀⢀⣀⣀⣀⢔⣕⣞⣷⣿⣿⣦⠏⠄⠐⠒⠁⠀⠀⠀⠀⠤⠄⠀⠀⠜⠂⢔⢨⠉⣳⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣁⣀⠐⠊⠭⠭⠶⢢⡠⠤⠊⢁⠞⣃⢴⡿⡫⣾⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⠒⠀⣸⢄⣠⠴⣊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠅⢃⣒⣒⠒⠂⠭⠭⠤⠀⢀⣀⣈⢉⡭⠭⠔⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⢔⡒⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⡒⣒⣂⡰⠯⠭⠭⠥⠔⢛⣒⡂⢬⠍⡭⡯⣿⣿⠟⡏⠙⠿⣿⠤⣐⠓⢭⠕⠲⣒⣂⠭⣔⣺⣵⣶⣿⢿⣿⠿⠛⣋⢭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠮⣛⢜⡙⠫⢛⢟⡹⣋⢟⠰⡫⢛⠕⡲⢛⢷⣜⡦⠀⠀⠀⠹⣦⢽⡖⠺⠿⠿⠟⢿⣾⣿⣿⢿⣛⣭⣴⣾⣿⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠀⠉⠢⢤⣄⡀⢁⣉⣈⣀⡃⠈⣈⣰⣧⣿⣿⣄⠀⠀⠀⠈⠃⠖⡶⠶⠶⢶⠚⠫⠟⠚⠿⠟⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠨⣟⢥⡠⠶⠚⣋⣭⣾⢿⣿⣿⡿⣟⠝⠒⠄⡀⠀⠘⣟⣛⣿⣿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣿⡿⠟⠊⠀⠀⠀⠀⠈⠁⠒⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠘⡇⢰⣿⣿⠗⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠔⡠⠐⠄⠂⠔⡠⠀⠔⡠⠂⡠⢀⠄⡠⢂⠄⡠⢀⠤⢀⠄⠄⠀⡠⢂⠔⠀⠄⡠⢀⠄⡠⢂⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⠊⠔⠡⢊⠔⢈⢔⡠⢊⠔⣊⠔⡠⢂⠕⡈⠐⠥⢊⠔⡡⢂⠄⡠⡢⢊⠄⡀⢐⡕⡨⢂⡵⢃⠔⡡⢪⠊⠔⠱⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠁⠐⠁⠊⠁⠊⠐⠁⠊⠔⠡⠊⡠⢊⠔⠡⠪⠎⠐⠁⠊⠐⠋⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢞⣛⣛⣯⣭⣭⣭⣭⣭⣭⣭⣛⣛⡳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡵⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢮⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣵⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣮⣛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠞⠋⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⢦⣄⡀⠀⠀⠀⠀⠙⠳⣍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⠟⠁⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠈⠳⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⢫⡾⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠘⢷⡝⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡾⣱⠟⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠻⣎⢷⡀
⠀⠀⠀⠀⠀⢀⡾⣱⠏⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠹⣎⢷⡀
⠀⠀⠀⠀⠀⣾⢱⡏⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢹⡎⣷
⠀⠀⠀⠀⣸⢇⡿⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡞⠋⠉⠉⠙⠻⢦⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢿⡸⣇
⠀⠀⠀⠀⣿⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡇⣿
⠀⠀⠀⢰⡏⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠙⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢹⡆
⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢸⡇
⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠸⣇⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⣿⢸⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⣿
⠀⠀⠀⠀⢹⡎⣷⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⠻⢦⣄⣀⢀⣀⣠⡾⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣾⢱⡏
⠀⠀⠀⠀⠀⢿⡸⣇⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣸⢇⡿
⠀⠀⠀⠀⠀⠈⢷⡹⣆⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣰⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡹⣦⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣴⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣜⢷⡀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⣣⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡙⢦⡀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⢀⣴⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢦⣄⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⣠⡴⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣭⡻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣭⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣝⡳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⢞⣫⡵⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢮⣭⣭⣟⣛⣛⣛⣛⣛⣛⣻⣭⣭⠵⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⢤⡀⠀⠀⠀⢀⡤⠔⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠢⣄⠴⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠱⡄⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣆⣀⡰⢦⣴⣁⣴⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⠀⠀⣀⣀⡀⠀⠈⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠋⠉⢀⠛⡅⠉⠓⠲⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡴⠖⣿⢻⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⢸⡇⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⣻⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⣿⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣦⣼⣡⣯⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠴⠚⠢⢤⣀⣀⣀⣀⠤⠴⠮⠤⠤⠴⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⣴⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠉
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣾⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠉⠉⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠉⠉⠉⢹⡇⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⣿⣿⣿⣿⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣀⣀⣀⣸⣇⣀⣀⣀⣿⣀⣀⣀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣿⡇⢹⣿⣿⡿⢸⣿⣿⣿⠇
⠀⢸⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢠⣶⣾⣶⡄⠀⢠⣿⣿⠡⡆⣶⣼⣿⣧⠀⠀⣴⣾⣶⣦⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⡟⠀⢸⣿⣿⣷⡆⣤⢹⣿⣿⠀⠘⣿⣿⣿⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⡀⠀⠉⠉⠉⠀⠀⠀⠻⣿⣦⡄⣴⣾⡿⠃⠀⠀⠈⠉⠉⠁⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣴⣿⡿⠿⢿⣿⣿⣿⠋⠀⣠⣼⣤⡄⠈⢻⣿⣿⣿⠿⠿⣿⣿⣦⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡏⠀⠀⠀⢹⣿⡇⠀⠀⠷⣼⣄⡀⠀⠀⣿⣿⠁⠀⠀⠈⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣧⣀⠀⣀⣼⣿⣇⠀⠠⣤⣼⣬⠗⠀⢠⣿⣿⣄⡀⢀⣠⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠸⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⣆⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠙⡿⣁⡀⣀⡸⡟⠁⠀⠀⠀⠙⢁⡀⡀⠛⠀⠀⠀⠀⢸⣿⡤⣤⣤⣤⠀⠀⠀⠀⢸⢿⢻⢻⣻⢻⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠠⡄⢠⠀⠀⠀⠀⠀⠐⠠⢀⡀⣀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠀⠀⠀⠀⢸⣼⣽⣸⣿⣸⠀⠀⠀⠀⠀⢀⣀⣿⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠛⠛⠛⠛⠛

⠀⢀⣀⢀⣀⢀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠽⣤⣤⣴⡈⡇⠀⠀⠀⠀⡏⣤⣴⣦⠈⠀⠀⠀⠀⢰⣿⣽⢿⣿⣷⠀⠀⠀⠀⠀⣠⣄⣠⢶⣠⡄⠀⠀⠀⠏⠉⠁⠈⠀⠈⠓
⠀⣹⣤⣤⣤⣵⡆⠀⠀⠀⠀⣷⣤⣤⣤⣴⡀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣭⣿⣯⣶⣿⣿⡇⠀⠀⠀⡀⠀⢠⣦
⠐⠛⠛⠓⠛⠛⠛⠀⠀⠀⠘⠛⠛⠒⠛⠛⠓⠀⠀⠀⠀⠈⠛⠓⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠒⠒⠚⠛⠋

⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⣠⠞⣡⠀⠀⠀⠀⠀⢠⡀⠢⣄⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣶⢰⣿⣿⡿⣻⣷⢰⣶⣶⣤⣤⡄
⠀⠀⠈⡀⣷⣿⠀⠀⠀⠀⠀⢸⣷⡆⡈⠛⠂⠀⠀⠀⢸⣿⣟⡶⢶⡦⠀⠀⠀⠀⣿⣿⢟⣵⣿⢨⣽⣵⣾⣿⣷⢘⡿⣻⣥⣿⡇
⠀⠀⢸⣏⣿⣿⠀⠀⠀⠀⠀⣸⣿⣟⣿⢈⠀⠀⠀⠀⠈⣛⢹⡇⣸⣇⠀⠀⠀⠀⠏⠹⠟⠋⠀⠀⠀⠚⠙⠂⠀⠀⠈⠛⠿⠛⠇
⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⠇⣀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢤⡭⠉⠠⠯⠍
⠀⡇⣠⣀⡜⠁⡅⠀⠀⠀⠀⡇⡴⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⢊⠀⣶⠀⠀⠀⠀⢣⣾⢿⣾⢿⣦⠇⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠇⠀⠈⠀⠀⠅⠀⠀⠀⠀⡇⠀⠈⢧⡀⠇⠀⠀⠀⠈⠀⠀⠈⠛⣿⡇⠀⠀⠀⠀⣿⢸⣿⣸⣿⡀⠀⠀⢀⢲⣾⣏⢧⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⢸⣿⢸⣿⠀⠀⠀⠀⢀⣻⠛⠛⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠻⣿⠿⠟⣛⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⣤⡶⠟⠛⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⣴⠿⠋⠁⠀⠀⠀⣀⣙⣛⣉⣛⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣫⣴⠟⠋⢀⣀⣤⣴⠶⠿⠛⠛⠛⠋⠛⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⢐⣋⣭⣤⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠀⢀⣤⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⡃⣍⣹⣿⣿⣿⠀⠀⠀⠀⣴⣿⣷⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠋⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠘⠇⠿⢿⣿⣿⣿⠀⠀⠀⠀⠙⠻⠟⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣶⣦⡀⠀⠀⠀⢸⣿⣿⣿⣷⡆⣶⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠘⢿⡿⠃⠀⠀⠀⠘⣿⣿⣿⣄⣁⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⣀⣠⣤⣴⣶⡶⠶⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣤⣤⣤⣶⡶⠾⠛⠛⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡻⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢲⢃⣞⡆
⠀⠀⠀⠀⠀⠀⠀⠨⠀⢐⣋⣻⠀⠀⡞⢀⣖⡆⠀⠀⢸⠰⠯⡇⠀⠀⠀⠀⢘⡀⣟⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠖⢺⠀⠀⡇⢬⠵⠇⠀⠀⢸⠌⠽⡇⠘⠅⢀⣿⠀⡆⢴⡂⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⡀⠠⠅⢈⣉⣭⠀⠀⡓⠭⢽⠃⢀⠄⠀⡯⢭⡇⢐⡅⢐⣻⡇⡣⠽⣇⣀⠀⢸⡀⢽⡀
⠀⠀⠀⠀⠀⠏⢑⣺⠀⠒⠒⡛⠀⠀⢩⠹⢽⡀⢒⣚⣆⣮⣽⣆⣿⣧⣰⣾⣧⣬⣭⣥⣧⣰⣶⣿⣿⡁
⠀⠀⠀⠀⠀⢱⣐⣺⠄⣉⣩⠵⡀⠀⠈⠉⠙⣇⢀⣐⣷⣯⢈⣁⣀⡁⣉⡛⠋⢁⣩⣕⠾⡟⠛⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⠔⠚⠠⠼⠋⡁⣀⡄⠈⠡⠿⣮⣋⣍⣉⣹⣭⣍⣀⣀⣏⣡⣈⢉⣉⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣤⣴⡀⠐⠒⠒⠁⠀⠀⠠⣀⠭⠍⠉⢉⠽⠯⠜⠋⠉⠁⠀⠈⠡⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠁

⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⡏⢉⡉⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⢻⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⡇⣀⠀⣶⢸⠀⠀⠀⡆⠀⠀⢸⢸⡇⠀⠀⢀⠀⠀⠀⠂⠄⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⡇⢸⡇⣿⣶⣿⢸⣿⡇⠀⡇⣤⠀⢸⢸⡇⠠⠐⠀⠆⡐⠀⠀⠈⠦⠄⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠸⠧⠿⠿⠿⠸⠿⠧⠷⠷⠿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⡃⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣛⢘⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣛
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇⠉⠁⠉⠁⠀⠀⠀⠀⠀⠁⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠁⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠀⣴⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠜⠃
⠀⠀⢀⢀⠀⡀⠀⡀⣀⢀⡀⡀⢀⢀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣛⡜⠣⠤⠤⠤⠤⠤⠤⠦⠄⠀⠀⠀⠈
⠀⠀⠘⠛⠛⠘⠃⠇⠹⠂⠃⠛⠛⠻⠘⠙⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠐⠒⢲⣶⠒⣶⡖⠒⠂⢰⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿⡏⠀⡀⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⢀⢰⣤⣬⣿⣤⠙⣧⣤⣤⣼⣿⣿⣤⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠸⢇⣀⣀⣀⣀⣀⣀⣻⣿⣿⣿⡃⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⣬⣀⣀⣁⣀⣉⣁⡀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠛⣉⠆⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⣶⣷⡦
⠀⠀⠀⠀⠁⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀
⠀⢸⣷⣦⣀⠾⣉⠉⠉⠉⠉⠉⠉⡋⠉⠭⣭⡛⠩⠍⡉⠉⠉⢐⠶⣲⠷⠒⣚⣛⡒⠲⠶⡢⢄
⠀⣿⣿⣿⣿⣿⣦⣅⡒⠀⠤⣀⣀⣀⣁⣒⣬⣭⣧⣤⣼⣀⣠⠺⡦⡵⣶⣤⣘⣛⣃⣠⡤⠚⣈⠉⢉⡐⠢⡀
⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣻⡞⢶⠾⢶⣾⠊⡔⠉⢀⣤⡀⠈⠢⡈⠄
⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢭⣏⣹⣏⣹⡏⢠⠠⠀⠘⢿⡉⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣯⣟⣚⡗⢺⡖⢺⣿⡀⢖⢄⠘⠿⠏⠀⢠⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠘⠯⠽⠭⠽⠭⠑⢄⡑⠂⠤⠄⠊⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠿⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡾⣿⠿⣤⡀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣤⡴⠟⠁⠀⠀⠀⠀⠙⢳⣾⣶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⣿⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣭⣭⣭⣭⣤⣤






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠤⢤⣀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠉⠀⠀⠀⡏⠛⠋⠀⠈⡳⢒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠇⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢀⠀⠀⠀⣄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⣠⣦⣤⣭⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡼⠀⠘⠂⣸⡟⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠸⠃⠘⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⣿⣿⡿⢩⣿⣿⣿⠈⠻⣿⣿⣷⠀⠀⠀⠀⡀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣸⣿⣿⡇⠀⠀⠹⢿⡿⠁⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠄⠂⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠈⠙⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⢸⣿⣿⡇⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⡁⠄⠒⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠑⠄⠀⠀⠀⠀⠈⢿⣿⣿⣦⣸⣿⣿⣧⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠂⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢧⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠐⢉⡀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠉⠀⠢⠄⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄⠀⠢⣀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣸⠀⠀⠀⠀⣀⡇⠀⠀⢀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡅⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣿⣄⣀⣀⣀⣿⣧⣀⣄⣿⣴⣿⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠠⣾⣗⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⢿⡿⣿⢿⣿⢿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⢉⡄⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⢿⣿⣿⡌⢠⠉⣼⠃⠆⢿⡇⣿⣿⢸⣿⣿⣯⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠀⢀⡄⠀⢉⣤⠀⠀⠀⢸⣿⠀⠀⠀⡂⠀⠀⣆⠀⣤⣄⣁⠀⣄⣿⣿⣿⣷⣿⣶⣷⣾⣿⣾⣷⣦⣾⣶⣦⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢛⣭⠀⠀⠀⣠⡿⠀⠀⣼⡇⠀⠀⡏⣸⣿⣿⣷⣴⠝⣿⡇⠀⠉⢿⣿⠍⠹⣿⣿⣿⣿⠭⣿⣿⣿⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢠⠀⢈⡁⢸⣿⣿⠀⠀⠀⠈⣵⣄⢠⣾⡇⠀⣰⣷⣿⣿⣿⣿⣿⢋⡾⡗⠀⠀⠸⣿⡇⣀⣈⣹⣿⣿⣆⣓⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡆⠟⠇⣈⣽⣿⠀⠀⠀⠀⢸⣿⡟⠛⠃⠀⢸⢏⡿⣿⣿⣿⢣⡟⣱⠟⣤⠀⢑⣿⡧⠵⢶⣾⣿⣿⠉⠉⠙⢹⣿⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢈⡁⣶⡇⣻⣿⣿⡀⠀⠀⢠⣸⣿⣿⣶⢏⡶⣨⡟⣰⢏⡽⢣⡟⣴⢃⡾⠁⠀⢨⣿⡇⠀⢀⠈⣹⡿⢀⣀⣤⣴⣿⠗⠒⠈⠉⣋
⠀⠀⠀⠀⠴⠂⢹⡇⠙⡇⢸⣿⣿⣿⣷⢄⣀⣹⢏⣿⡏⡜⢰⢻⣵⢃⡞⢡⠏⣸⢣⣿⡇⠀⢀⣀⣹⢃⣠⣾⡿⠿⠃⣠⣆⣿⣯⣤⣴⠶⠎⠉
⠀⠀⠀⠀⠀⠀⠀⡅⠨⡇⢸⣿⣿⣿⢇⡞⡽⣿⠸⠸⣙⢡⠃⣼⣷⠏⡴⢃⡼⢣⣿⣶⠇⣠⡿⠿⠃⣀⣿⣿⡾⠟⠛⠛⠿⠛⢿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣸⣿⣏⠟⡌⡜⣰⡇⡆⢰⢃⡌⣸⡟⢋⡾⣡⠞⣴⢟⡯⠡⠀⠲⠿⠋⠁⠁⠈⠉⠁⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⢾⡿⣿⠀⡸⣼⠹⠣⣇⣿⡞⡵⢻⠓⠀⠚⢣⡞⠁⠈⣁⣀⡀⠀⠀⠤⠀⠐⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣷⡠⣡⠃⢀⣀⣉⣉⣷⠥⠤⠤⠒⠒⠊⠉⠉⠉⠁⣀⣀⣀⣀⣀⡤⣤⢴⡶⠶⠶⠶⡾⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠁⠁⠈⢩⣭⣥⣤⣤⣴⣶⡶⢶⣿⣿⣿⢿⣿⡿⠿⠟⣹⣿⠏⠇⠁⣼⣷⠂⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⡿⢿⣿⠙⠻⣿⠃⠀⢻⡏⠀⠘⣿⡇⠀⠀⣿⡏⠀⠀⠀⢺⣿⠀⠀⠀⠀⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠈⠿⠂⠀⠸⡿⠀⠀⠿⠇⠀⠀⠿⠇⠀⠀⠀⠻⠿⠀⠀⠀⠀⠻⠟⠃

⠀⠀⠠⡶⡷⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⢠⠖⠲⡄⠀⠀⠀⠀⠀⠀⣖⡆
⠀⠀⠨⣯⣿⠟⠀⠀⠀⠀⢠⣒⠒⣚⡀⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⠀⠀⠀⠓⢺⡄⠀⠀⠀⠀⠰⣍⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠠⠤⢤⣴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠠⠀⠠⠠⠀⠀⠀⠀⠀⢒⡒⡄⠀⠀⠀⠀⢀⡤⠢⠄
⠀⠀⠀⠀⢸⠤⠀⠀⠀⠀⠀⡀⠩⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠐⠃⡇⠀⠀⠀⠀⠸⡘⢙⣬
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠
⠀⠀⢾⣿⡦⡠⠀⠀⠀⠀⢸⠠⠢⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⣿⠀⠀⠀⠀⢸⠐⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠀⠀⠀⠀⠀⠓⠒⠒⠂⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠘⠒⠒⠒

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣺⣿⡂⠀⠀⠀⠀⠀⠘⠛⡅⠀⠀⠀⠀⠀⢈⠥⢙⠀⠀⠀⠀⠀⢰⠡⠌⡆⠀⠀⠀⠀⢘⢼⣿⠜⠂
⠀⠀⠀⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠈⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⡀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⡀⢸⣿⣿⣿⢀⣀⣀⣀⡀⣿⣿⣿⣇⣸⡿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡇⢸⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣟⠉⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡇⢸⡿⢿⣿⢸⣿⣿⣿⡇⣿⡿⠋⠀⣀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡷⠋⠀⡀⠙⢿⣿⣿⣿⡿⠋⠀⣴⡞⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡿⠋⠀⣠⣾⣿⣦⡀⠙⠿⠋⢀⣴⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋⠀⣠⡞⢻⣿⣿⣿⢻⣦⣀⣤⡟⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠾⠿⠇⠸⠿⠿⠿⠘⠿⠿⠿⠃⠿⠿⠿⠇⠸⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⠀⢠⣤⣤⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣤⠤⡠⠄⠀⣀⣠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠸⣿⣿⠷⣦⡌⠀⢸⣿⣿⠟⠙⣃⣴⣿⣿⣿⣿⣯⣩⣥⣤⣤⣤⣤⣥⣤⣤⣬⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣶⣦⣤⣠⣴⡾⠉⣁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠛⣻⠟⠉⣉⣤⣾⡭⡟⢡⡿⠙⠿⣋⣛⠛⢻⣭⣎⢡⣶⣨⡯⠩⣳⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣴⠿⣇⣠⣶⣷⠟⣿⠛⡬⢏⣤⠘⠿⠏⢨⠀⢤⢹⠜⠈⣁⣤⣷⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⡿⠟⠉⣠⠟⡁⢿⠱⡀⠀⠀⠠⠗⠀⠐⢃⣠⣤⣶⣿⣿⣿⣿⣿⢿⣿⡿⠿⠃
⠀⠀⠀⠀⢸⣿⢿⡇⢀⡴⠃⢼⡇⠀⡅⠀⠀⠀⣀⣠⣴⣾⣿⠿⢛⠋⠙⠏⡩⠙⠁⢉⡠⣶⣾⡇
⠀⠀⠀⠀⠀⢻⢸⣥⡿⢡⡖⠆⠀⠈⢀⣠⣴⣿⣿⠿⠛⠉⠀⠀⠀⠀⠈⢠⣠⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢋⣼⡿⣋⡀⢁⣠⣶⣿⡿⠿⠛⠉⢀⠀⠀⠀⠀⣤⢰⣾⣷⣮⢟⢿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⢰⣿⣿⣽⣷⣿⣿⠿⠋⠁⡀⠄⠀⣈⣀⣶⣶⣿⣿⣾⠟⣿⣿⣿⠁⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⠿⠀⢀⣡⣤⣼⣿⣿⣿⣿⠉⢿⣿⡏⠀⠘⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢰⣾⣛⣩⣴⣶⣿⣿⡿⣿⣿⡇⠈⣿⣿⠀⠘⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⡟⠻⣿⠁⠈⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣶⣾⠁⠀⢽⡇⠀⣿⠀⠀⣿⡇⠀⢻⣿⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⡿⠀⠀⢺⡇⠀⣿⠀⠀⣿⡅⠀⢸⡿⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣧⠀⠀⢸⡇⠀⣿⠀⠀⣿⠆⠀⢸⣷⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⡷⠀⠀⢸⡇⠀⣿⠀⠀⣿⠀⠀⢸⣿⠀⠀⣿⣇⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣇⠀⠀⢸⡇⠀⣿⠀⠀⣿⠂⠀⢸⣿⠀⠀⣿⡏⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⠠
⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⡆⠀⣿⠀⠀⣿⡆⠀⢸⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⠠
⠀⠀⠀⠀⠀⣿⡟⣠⡀⣼⡇⠀⣿⢰⣦⢿⡇⠀⣸⣿⡀⢀⠌⠓⠄⠤⣿⣿⡖⠒⢾⣿⡇
⠀⠀⠀⠀⢀⣿⣷⣿⣇⣾⡇⣚⣿⡻⣟⣾⡇⣨⣿⣿⠀⠊⢠⣟⣀⣴⣿⣿⣄⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⡿⣳⣿⣽⣿⣿⣽⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⢀⡀⢠⡄⠀⠀⠀⢀⣤⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣷⠄⠀⠀⠀⠈⣶⠒⢲⣮
⠀⠀⢰⣺⣷⣾⡇⠀⠀⠀⠺⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⣿⣷⣿⡧
⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⡤⢤⡄⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⢀⣀⣛⠳⡄⠀⠀⠀⠀⣷⣾⢸⠀⠀⠀⠀⠀⣦⣤⣤⣴⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⠿⠅⠽⠿
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠒⠒⠚⠂⠀⠀⠀⠀⠛⠋⠙⠛⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣤⡀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠘⢰⠀⠀⠀⠀⠀⠀⢀⡶⣽⢦⡀⠀⠀⠀⠀⢿⣻⣟⡿⠁⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛

⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⣀⣯⣈⣄⡅⠀⠀⠀⠘⠛⠻⠻⠃⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⣷⣶
⠀⠀⠿⠿⠿⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠰⠠⠼⠿⠇⠀⠀⠀⠘⠛⠷⠛⠛

⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢰⣤⣤⡤⡇⠀⠀⠀⠐⣿⣿⣿⠂⠀⠀⠀⠀⣈⣒⣚⡁⠀⠀⠀⠀⡞⢻⢿⢻⡇⠀⠀⠀⠀⡇⡄⣀
⠀⠀⢸⣟⣟⣇⡇⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠀⠀⠠⢆⡰⠄⠀⠀⠀⠀⢿⣷⣾⡸⡇⠀⠀⠀⠀⡇⡇⡿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠂⠀⢱⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠂⣀⡈⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠁⢉⡷⣫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠕⡰⢿⠅⢦⣼⡯⠀⠠⠀⠀⠀⠈⠀⠁⢀⢊⡴⣫⠖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣠⡊⠀⠀⠛⠛⠊⠙⠁⢀⠀⠀⣠⣶⣄⠐⠆⣰⡟⡫⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⢀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠃⠼⠃⢄⠄⠠⠈⠐⠁⠐⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠊⢀⠴⠊⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠆
⠀⣶⣶⣶⣤⣀⠠⠤⠤⠤⣤⣄⣤⠴⢊⡀⢀⣀⣀⣀⣀⡀⠀⠄⠀⢠⡚⠋⠀⠐⠂⠠⢈⠕⠀⡀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢉⣿⡏⠀⠀⠀⠀⠀⢰⠃⠀⢀⡴⠃⠀⠿⢇⣀⣀⣈⣉⣄⠀⠘⠀⢀⡀⢀⠔⡠⠔⠅⢴⠂⠀⣧⠀⠀⠶⣄⢠⠀⢀⡠⠀⠀⠀⢑
⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠭⠄⠂⢆⣀⠀⠀⠈⠉⠉⢀⣀⣀⡤⠠⢋⠁⠊⠀⠀⠠⠁⠀⠀⢸⣆⠀⠀⢈⢹⡿⠁⠀⠀⠀⠀⠈
⢢⣿⣿⠁⠀⠀⠀⠀⠀⠀⠠⣘⣗⣲⣶⣒⣛⣿⣿⣶⣖⣶⣶⣷⣾⣷⡶⣥⣪⢄⠔⠁⠀⡀⢀⡰⠺⣿⢶⣴⠿⠛⠻⠒⠒⠓⠂⠄⠒
⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⡔⢻⣿⣿⣗⣠⣤⣶⣾⣿⣿⡿⣿⣿⣿⣾⣿⡛⢊⠅⠀⠈⠊⠠⡈⠔⠀⣿⡷⠁⣀⢀⢀⣀⣠⠤⠖
⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⢀⠸⣟⣛⣻⡯⠭⠥⠭⠭⠭⠿⠽⣿⢿⣽⣷⠶⣄⠀⡀⠔⡫⠊⢂⣴⠼⢛⠟⠉⠁⠀⣹⠉
⠀⣾⣿⡇⠀⠈⢉⠉⡇⠀⢸⣷⣿⣷⣂⣈⣉⣉⣉⣩⣍⠭⠩⠽⣯⡽⢞⠁⣿⣂⠔⠀⢀⢴⡿⠁⠀⠀⠀⣠⠴⠚⠁
⢀⣿⣿⣷⠀⠀⠀⡀⠇⣀⡜⢿⣿⣛⣛⣒⣛⣺⣿⣿⣻⠿⠶⠶⠶⢶⣟⣉⣹⡅⠀⠠⢡⣾⠧⠴⠶⠒⠋⠁
⠀⣿⣿⣿⡖⣚⣟⣫⣉⣭⡤⠤⠋⠉⠉⠁⡴⡤⣤⣤⣤⣤⣤⣤⣤⣒⣐⡒⠄⠿⢧⣶⣿⡟
⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⡇
⠀⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣏⡀⡅
⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣽⣶⣶⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣽⣷⣭⣿⣤⣵⣿⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠃⠀⢀⠀⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⠀⢂⠂
⠠⠅⠸⠀⠂⣃⣧⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠁⠠⠂
⠀⠀⠀⠀⠭⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠂⠀⣀⠤⢮⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁⠀⠀⣀⣔⣈⡀⠀⢸⠀⢔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢⠁⠀⠀⠀⠠⠐⠈⡉⡏⠉⡇⠀⣮⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠄⠸⢷⠲⠖⡆⠁⠆⢈⡇⠀⡗⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⡀⠄⠀⠤⡤⠦⠄⢸⠀⢰⠀⢠⡇⢠⠁⠀⡇⠀⢤
⠀⠀⠀⠀⠀⢄⠠⠐⠀⠀⡀⠄⠢⣤⡤⢄⠀⠀⠇⢨⠁⢸⠀⢸⠀⢰⠄⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠘⣂⣄⠠⠐⣠⢥⢦⡄⠀⡇⠀⡂⠀⢨⠁⢰⠀⢸⠀⢸⠀⢸⠀⢸⠀⢸⠁⠀⡇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠠⡄⣴⣤⠀⢸⡀⠐⠅⠀⡇⠀⡂⠀⢸⠀⢸⠀⢸⠀⢸⠀⠸⠀⣺⠀⢰⠀⢀⡇⠀⠀⠀⢁⡀⠙⣃
⠀⠀⠀⠀⡇⠈⢘⠀⢘⡀⢲⠀⠀⡅⠀⡇⠀⢸⠀⡪⠀⠰⠀⠼⠀⡄⠀⠄⠀⢸⠀⠨⠀⠀⠐⢴⣬⣠⣴⠃
⠀⠀⠀⠀⡧⠀⢺⠀⢸⠃⢰⠀⠀⡇⠀⡆⠀⢸⠀⡂⠀⡟⠐⠄⠀⡇⠔⡃⠀⣛⠠⢺⠀⠀⠀⠀⢈⠉
⠀⠀⠀⠀⣇⠀⢼⠀⢸⠠⢺⠀⠀⡇⠐⡇⠀⡃⠀⡇⢀⡇⠀⡇⣴⣇⣊⣇⣴⣧⣤⣼⡄⠀⠀⠀⡂⠀⡀⠐⠂
⠀⠀⠀⢀⠯⠀⢟⢰⣼⣨⣼⡀⢾⣧⣼⣧⠺⠷⠶⠗⠻⠿⠛⠛⠙⠛⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠁
⠀⠠⠀⠈⠛⠛⠛⠀⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣰⣿⡿⢋⡅⣍⠻⣿⣷⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⢸⣿⣿⣇⠸⠇⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⡆⣦⡈⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠘⢿⣿⣇⠘⠇⠟⢁⣿⣿⠟⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⠒⠢⠄⢐⠂⠠⠄
⠀⠀⠀⠀⣸⡟⠒⠒⠤⠤⢉⣀⣂⠀⠅⠂⠤⣀⡀
⠀⠀⠀⠀⣻⡯⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠤⢬⣀⣂⠀⠂⠀⡀
⠀⠀⠀⠠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠲⠤⠨⣀⣂⠀⠁⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⢨⣗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⠤⢬⣁⣀⠀⠈⠄⢀⠀⠠⠀⢀
⠀⠀⠀⠀⢈⣷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠙⠐⠲⠤⣅⣀⡐⠂⠄⢀⠐⢠⣄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡀⠀⠉⠉⠓⢲⠈⡇⡯
⠀⠀⠀⠀⢘⣯⠰⠀⠀⢀⡀⠀⠀⢀⢔⡑⢚⠄⠀⢀⢔⠕⢪⢒⡱⡀⠀⠀⠔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠀⣇⡯
⠀⠀⠀⠀⠠⣯⠀⠀⠔⡡⡜⠤⢚⠕⢁⡈⠈⢺⣒⠕⠁⠀⠀⠁⠀⠕⡤⡪⠊⠨⢆⡠⢦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡯⣏
⠀⠀⠀⢠⠈⡿⡰⢊⠈⠀⡌⠒⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⢣⠔⢣⢂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⡀⣧⣟
⠀⠀⠀⢸⠀⣿⡔⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⣿⣿
⠀⠀⠀⢸⠀⣿⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠁⢸⣿
⠀⠀⠀⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢿⡀
⠀⠀⠀⠀⠢⣝⣉⠒⠲⢤⣁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡂
⠀⠀⠀⠀⠀⠀⠈⠈⠙⠲⠠⢥⣐⣉⠓⠂⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⣹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠢⢥⢌⡉⠀⠤⣁⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠓⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⠤⣀⡂⠤⢀⡀⠄⣀⢀⠀⠀⠀⠀⢃⢣⠀⠀⠀⠀⠀⠴⠀⢸⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣍⣐⠨⠕⠢⣄⠀⡎⠄⢀⠀⠀⠀⠀⠀⠸⢈⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢤⡿⠗⡈⡀⣶⠂⡂⠧⣀⢀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠽⠐⡐⡙⠶⡦⣭⣐⠛⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⡂⠀⠀⠀⡇⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠰⠈⠢⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⠀⠀⡀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⣀⣀
⠀⠀⠘⢰⡦⠬⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠁⠒⠒⠤⠤⣀⡀
⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⣀⣀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⢸⡇⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⡉⠉⠒⠲⠄⡮⡯
⠀⠀⠀⢸⡇⠂⠀⡠⡀⠀⠠⠪⡀⡂⠀⠐⠄⠺⡪⢅⠀⠀⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢩⡏
⠀⠀⠀⠨⡇⡀⠊⠘⢄⠈⠀⠐⠀⠘⠕⠁⠀⠀⠀⠈⡢⠋⠀⠱⢦⡁⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⢰⣷
⠀⠀⠀⠸⣗⠄⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿
⠀⠀⠀⠀⡏⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠂⠀⠸⣿
⠀⠀⠀⠨⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿
⠀⠀⠀⠈⠒⠒⠬⢅⣒⡢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⢠⣒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⠀⠀⠀⢰⣶⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠥⢀⣀⠀⠀⣤⢤⡀⠀⠈⡥⠀⠀⠀⠀⠲⠸⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣅⣒⢢⣵⠀⠀⡀⣀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⢱⠂⢜⠧⣄⣊⠰⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠔⠀⠀⠀⠀⠉⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠂⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⢹⠉⢉⠉⠉⠉⠉⠉⠈⠉⠉⠉⡍⠉⢹⠉⠉⠉⠉⡏⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⢀⣴⡄⠀⠀⠀⠇⠀⢸⠀⠀⠀⢀⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⢀⡀⠘⣱⠞⠀⠳⡎⠀⠀⡇⠀⢸⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡼⠁⠀⠀⠀⠛⢦⣄⡇⠀⢸⠀⠠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⡦⣬⣤⣥⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠻⠻⠫
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡞⢁⣀⡄⣄⣀⣀⣀⣀⣀⡄⣄⣀⢀⣀⣀⣀⡄⣄⣠⣠⣠⡀⢀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠲⢤⣞⠙⡗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠃⠈⠉⠉⠁⠀⠓⠒⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠈⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⣴⡆⢸⡍⠲⣿⠀⡟⠲⣄⡀⠀⣠⠞⠁⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣄⡇⢸⠄⠀⣿⠀⡇⠀⣿⠉⠲⣅⠂⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠳⠇⢸⠄⠀⣿⠀⡇⠀⣿⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠀⢧⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠓⢶⣄⠀⡀⠀⠀⠀⠀⠀⠘⡎⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⢸⡁⠀⣿⠀⡏⣦⠀⠀⠀⠀⠀⢸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⢗⢦⡀⠀⠀⡇⢸⡄⠀⣿⠀⡇⢸⠀⠀⠀⠀⠀⢸⠃⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠳⢵⣄⠀⡇⢸⠆⠀⣿⠀⣧⠏⠀⠀⠀⠀⠀⣸⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡐⠀⠀⠀⠀⠀⠀⠉⠛⠃⠸⠥⠤⠟⠀⠁⠀⠀⠀⠀⠀⢠⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⢈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢶⡄⢠⣤⡤⣄⠀⡶⣏⡁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠄⢙⣯⠀⡗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣷⣤⡏⠻⢽⣦⡿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⡖⠓⠂⡖⠒⠒⢒⠒⠒⢲⠒⠚⠒⠒⠒⠒⡖⠓⠒⡖⠒⠓⢒⠚⠒⢲⠚⠒⣶⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⢀⣀⣀⣓⣀⣀⣓⣀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⣠⠞⠁⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠈⠀⢰⡾⣶⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠘⠀⢠⣼⠏⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⡆⠀⠛⠉⣴⠛⠀⠸⣷⠊⠁⠀⠀⡇⠀⠀⡇⠀⠀⢀⠀⣸⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠙⠂⢰⡞⢩⠀⠀⠀⠈⢷⣀⠀⠀⡇⠀⠀⡇⠀⠀⠸⢰⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⣴⠟⠀⢸⠀⠀⠀⠀⠀⠉⠛⠶⣅⠀⠀⡇⠀⡀⢉⡟⢰⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡧⣶⠾⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢱⣶⣶⣶⣶⣶⣶⣶⡌⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⣠⡟⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠛⡛⣛⠛⣛⢛⠛⣣⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣤⣤⡀⠀⠀⠀⠘⠃⠀⠶⠶⠇⠲⠶⠖⠲⠶⠶⠰⠶⠶⠲⠶⠶⠒⠶⠶⠂⠶⠶⠆⠲⠶⠖⠲⠶⠖⠀⠐⠃⠀⠀⠀⢀⣤⣤⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠷⠾⠿⠶⠶⠶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣾⠛⠛⣿⡏⠉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠋⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢶⣾⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⢻⣟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⢄⡛⠷⣦⣄
⠀⠀⠀⠀⢀⣤⡾⠟⣩⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣍⡀⠀⠙⠿⣦⡀
⠀⠀⢀⣴⠟⠉⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⠉⠻⣶⣄⠀⠈⢿⣆
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠹⣷⡀⠀⢻⣆
⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠻⠿⠶⠾⣿⣦⣾⣶⣦⣤⡉⠛⠷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣼⡿⠀⠀⢿
⢸⡏⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣰⠟⠓⠲⠶⠶⣤⣤⣤⡀⠀⣸⡏⠙⠳⢦⣄⠀⠉⣹⣿⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢉⣀⣀⠀⢸
⣾⠇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣴⠋⣀⣤⠤⠤⣄⡀⠀⢸⡇⢠⡿⠀⠀⢀⡴⠋⢀⣾⠋⠉⠛⠛⠛⠛⠛⢻⡟⠀⠀⠀⠀⠀⣿⣏⣻⡇⠸
⣿⠀⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣼⠏⣰⢇⡼⢋⡀⢀⣤⡀⠹⣆⡾⢀⣾⠇⢀⣴⠟⠀⣴⡟⠉⠉⠛⠛⢻⡆⠀⢠⡿⠁⠀⠀⠀⠀⠀⡈⠙⠋
⣿⠀⠀⠀⠀⠀⢀⣾⠟⠿⣦⣀⣀⣀⣼⣏⣴⣇⣸⣇⣻⣤⣟⣨⣿⣴⣿⣥⣾⣯⣤⣾⣥⣤⣾⣯⣤⣤⣤⣤⣤⣿⣀⣠⣿⣁⣀⣀⣀⣀⣴⡾⠻⣶⡀
⣿⠀⠀⠀⠀⠀⠘⠿⣦⣀⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⡟⠁⣀⣴⠟⠃
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⣿
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⢰⡿⠻⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⠟⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⣾⠃⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⢀⣿⠀⢸⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣿⡷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⠀⠀⢸⡇⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⠀⠀⣼⡇⠀⢿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⣧⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢰
⣿⠀⠀⢿⣦⣤⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢈⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣄⡀⠀⣼
⣿⡀⠀⢀⣴⠟⢁⣴⡿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⡙⢿⣦⣿
⢿⡇⢰⣿⣅⣰⣿⠁⠀⣠⣿⣷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣠⣤⣤⣤⣀⣼⣏⠀⠙⢿⣿⢀⣿⣿
⢸⡇⠀⠈⠉⠛⠻⢷⣾⣿⣉⡉⠛⠶⠶⠟⠉⣹⡟⠉⠉⠉⠉⣿⣏⣛⣫⣭⡽⠿⠋⢉⣹⡿⠉⢉⣉⣉⣉⣭⣭⣭⡉⠙⠻⣯⡉⠙⢯⣻⣷⣴⠟⣡⡾⠋⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣶⣦⣄⣼⠟⢀⣀⣀⣀⣀⣿⠋⣉⣡⣤⣶⣶⣿⣻⣿⡅⢸⣟⣿⣽⣯⣭⠿⢾⣷⣠⣤⣬⣿⣦⡀⠙⠛⢁⣼⠟⠀⢰⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⣴⣾⠙⣿⣾⣿⡿⠟⠛⠻⣿⡿⠿⠿⠛⣿⣿⣿⣿⣿⣶⣾⣿⣬⡿⣴⠿⣶⡾⢛⣩⣷⠀⠈⣿⣿⣦⣴⠟⠁⠀⠀⢸⡇
⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⢶⣿⡶⠶⢿⣿⡏⠀⣿⡀⠀⢹⡷⠶⣴⣶⢿⡾⣷⠿⣶⠿⢶⡾⡿⢷⣟⣿⡿⠟⠛⠛⣿⣷⡾⣻⡿⠈⠁⠀⠀⠀⠀⢸⡇
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠻⣮⣛⣿⡿⣧⡸⠷⣬⠻⣦⠘⠟⣠⡿⠷⠶⠾⢿⣾⢷⣿⣶⡿⣶⣿⣿⣻⣿⡟⠛⠛⠿⣦⣾⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⣾⡇
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣬⣻⣶⣾⣶⡿⠿⠛⠋⠀⠀⠀⠀⣿⣻⣟⣻⣟⣻⣟⣿⣿⣭⣿⣷⣶⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠻⣦⡀⠀⠀⠓⠶⠶⠶⢶⣶⣶⣾⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣟⣙⣉⣉⣉⣠⣠⣤⣬⣭⣭⣭⣤⣤⣤⣦⣶⡶⠶⠶⠒⠀⠀⠀⣸⡏
⠀⠀⠀⠉⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠾⠛

⠛⠋⠀⠀⠨⣟⢻⣿⣿⣿⣿⣶⣄⣈⡙⠛⠉⠀⠀⢸⣿⣿⢹⣷⠀⠀⠀⠀⠀⣰⣿⣿⢇
⡇⠀⠀⠀⢀⠙⠛⠛⠛⠿⣿⣿⣿⣿⣿⣄⡀⠀⢀⡈⠛⠿⠛⠁⠀⠀⠀⠀⣰⠿⠻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠁⠀⣰⠿⢂⣸⡆⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣯⣯⣿⡆⠀⠀⠀⠀⠀⣰⠷⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠑⡄
⠀⠀⠻⢶⡾⠟⠁⠀⣰⣾⡅⣹⡆⠈⠛⠿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠳⣈⣦
⣤⣶⣶⣄⠀⠀⠀⠀⢻⣧⣶⡿⠃⠀⢀⡶⢿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⡿⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⠀⠀⠀⠀⠀⠀⠹⣧⡀⠙⢿⣷⣄
⣿⣷⣿⡿⠀⠀⣠⢤⣤⣄⠀⠀⠀⠀⢸⣿⣷⣾⡿⠉⠻⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠙⢆⣱⣄⠀⠀⠀⢀⠈⠻⣿⣦⠀⠻⣿⣦⡀
⠙⠛⠋⠀⠀⢰⣯⠀⠀⠉⠀⠀⢀⣀⣀⣉⠉⠉⠀⠀⠀⠀⠉⣿⡭⠛⠿⢿⣷⣄⠀⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠈⢻⣷⣄⠈⢻⣿
⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⣿⣏⣜⣿⡆⠀⠀⠀⠀⠀⣰⡿⠚⠁⠀⠀⠈⠉⠛⠆⠀⠀⠀⠈⢻⣤⠀⠰⣤⣧⡀⠘⢿⣷⡄⠈⢢⣀⣄⠀⠹⣿⣷⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⡀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⠙⣷⡀⠈⠻⣿⣄⠀⠻⣿⣦⡀⠙⢿⣷⡄⠈⢿⣿⣦
⠙⠶⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠨⣿⣦⠀⠙⣿⣧⡄⠈⢿⣷⣆⠈⠻⣿⣧⡀⠉⣿
⠀⠀⠀⠀⠘⠛⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠹⣷⣄⠀⢺⣷⣀⠀⠹⣿⣦⡀⠘⢿⣷⡀⠈⢻⣿⣆⠀⠙⣿⣦⡀⠘⢿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠻⢶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠈⢿⣧⡀⠙⣿⣦⠀⠘⢿⣿⡆⠈⠻⣿⣦⡀⠙⢿⣧⣄⠈⢻⣿⣦⡈⠻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢚⠀⠀⠙⠳⢦⣤⢻⣄⠀⢀⡀⠀⠈⢻⣿⣆⠀⠀⠹⣯⡀⠈⢿⣧⣤⡀⠻⣿⣦⡀⠘⢿⣷⣄⠈⠻⣿⣠⠀⠹⣿⣧⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢿⣿⣦⡈⢳⣤⣄⠀⠙⣿⣿⣀⠈⢿⣷⣄⠀⠹⣿⣶⠀⠘⢿⣷⣄⠈⠻⣿⣆⠀⠙⢿⡷⠄⠈
⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠈⢻⣿⣷⡄⠹⣿⣦⣂⠈⢿⣷⣄⠀⠹⣿⣧⡀⠘⢿⣷⡄⠀⠻⣿⣦⠀⠘⢿⠿
⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠘⢿⣿⣄⠀⠸⣿⣧⠀⠘⢿⣷⣆⠀⠹⣿⣀⡀⠀⠀⠂
⠀⠀⠀⡄⢉⠀⠀⠂⠀⠀⠀⠀⠀⠁⠤⠒⠊⠀⠀⠀⠀⠘⣿⣿⣤⠀⠻⣿⣦⡀⠘⢿⣷⣀⠀⠻⣿⣦⠀⠘⠿⠋
⠀⠀⠀⠀⠊⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠒⢀⡠⠖⠀⠀⠀⠈⢻⣿⣆⠀⠘⢿⣿⡄⠀⠻⣿⣆⠀⠙⠟⠃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠈⠻⣿⣄⡀⠙⠋⠁
⣶⣶⡶⢶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣿⣿⣷⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠
⠀⠀⠀⠀⡧⠄⢀⠀⡀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠤⢺
⠀⠀⠀⡾⢧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⣇⣀⣀⠀⠁⠀⠀⠀⠀⠈⠀⠈⠀⡀⣠⡼⠀⠀⠀⠀⠠⠐⠂⠀⠈⠀⠀⠀⠁
⠀⠀⠀⣇⠀⠉⠉⠉⠉⠉⠈⠀⠈⠉⠉⠈⣉⠔⣺⠷⠖⠒⠒⠶⢤⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠠⣎
⠀⠀⠀⠀⣽⠉⠀⠂⠒⠒⠒⠒⠒⠒⠲⢂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⡀⠀⠀⣀⣀⣀⣀⢠⡤⠿⡆
⠀⠀⠀⡆⡈⠁⠐⠂⠐⠄⠠⠀⠤⠠⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠉⠉⠀⠀⠂⠐⠂⢄
⠀⠀⠀⣇⣀⠀⠀⠁⠈⠉⠉⠁⠈⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠈⡳⠤⠀⠀⢀⢀⡀⠀⡀⠀⡀⢀⠀⠀⠤⢰
⠀⠀⠀⣇⠀⠈⠉⠉⠀⠁⠀⠀⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠿⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞
⠀⠀⠀⢸⡛⠉⠀⠐⠐⠦⠂⠤⠂⠲⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢿⣄⣀⡀⢀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣼⠂
⠀⠀⠀⢸⡃⠤⠤⠠⡄⢠⠀⢀⠀⠀⠀⣜⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢷⡀⠈⠉⠉⠉⠉⠈⠁⠀⠁⠈⠉⠁⠀⡸⠃⡄
⠀⠀⠀⠘⠣⠤⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⠈⠹⠶⠤⠤⠤⠤⠄⠒⠉⠀⠀⠀⠀⢸⠉⠁⠈⠐⠂⠒⠒⠒⠒⠒⠛⠋⠉⠈⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠠⡀⠀⠀⠀⠀⠁⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠂⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⠀⣿⣿⣿⠀⣿⣿⣿⠀⣿⣿⠛⠋⠁⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⠀⣿⠿⣿⠀⣿⣿⣿⠀⣿⠟⠁⣀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⠞⠁⣀⠈⠻⣿⣿⣿⠞⠁⣠⣾⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠟⠁⣠⣾⣿⣶⡀⠉⠛⠁⣠⣾⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⣠⣾⠋⣿⣿⣿⠙⣦⣠⣾⠋⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣘⣛⣛⣀⣛⣛⣛⣀⣛⣛⣛⣀⣛⣛⣛⣀⣛⣛⣛






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

⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠉⡷⠒⠒⠒⠒⠀⠀⠀⠠⠤⠤⢄⡀⠀⠀⠀⠀⣀⣀⡀⠠⠤⠄⡒⢢
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠥⣴⠶⣄⠈⠉⠢⢅⠸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⠪⣧⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣀⡗⠐⠒⠒⠙⠓⠦⢄⡀⠀⠀⠈⠲⡐⠈⠘⢯⣢⣄⠀⠈⠢⣠⠋⠄⣇
⠀⠀⠀⠀⠀⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⢦⣄⣀⠈⢲⣂⣈⢩⡭⢽⠦⣀⡸⢛⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣬⣜⠉⣛⣚⣒⡛⠺⠿⠯⠤⠠⠞⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠬⢕⣒⠸⠧⣭⡥⠟⠉⠉⢁⠖⡒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠭⣓⠒⠊⡍⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣒⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢄⠀⠘⢆⣀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⣟⠛⠷⠀⠀⠰⠖⢒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠱⣀⣀⣀⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⠚⠉⠉⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠁⢠⠞⢉⠍⠙⢄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢹⠀⠈⣲⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢇⠀⠈⠓⠬⠥⠔⠃⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣆⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡶⠿⣿⣿⣿⣶⣶⣦⣤⣤⣄⡀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆⡱⣿⣯⠳⣯⡉⠉⠀⠀⠀⢀⠍⢉⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣿⣧⠈⢳⡀⠀⠀⣴⡧⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⢹⣧⠀⢰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⣻⣾⣿⣿⣿⣾⣿⣄⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⢛⣿⣿⣿⣏⣏⣉⣉⣤⡖⢐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣅⣀⣴⣿⣿⣿⣿⣿⢏⡏⠀⠻⣿⣆⠑⠝⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⡿⠃⠸⠃⣀⠀⡙⠈⠂⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠃⠀⢀⣀⣿⣼⣧⣀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⢻⣿⣿⣷⡄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣿⣿⣟⢀⠀⢿⣿⣿⣽⢸⣿⠉⠁⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣿⣿⣿⣟⡎⠀⠈⢿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⣿⣿⣿⣯⡀⠀⠈⠀⣿⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⢠⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠐⢷⣿⡇⣿⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠠⠽⣿⣿⣽⣿⣿⡃⠛⣿⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⡠⣼⣿⣷⣿⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢀⡀⠙⢛⣿⣿⣷⣿⣶⣿⣿⣷⣾⣿⣿⣿⣤⡴⠶⠚⢹⣿⣷⠈⣿⡟⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣏⣟⣿⣿⣿⣿⣿⣶⣽⠿⢛⣉⣠⣤⣤⣼⣿⣿⣗⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠉⠉⢿⣿⣯⣭⣭⣩⣉⣉⡉⠉⠛⠻⠧⠂⠉⠉⠀⠀⣉⠙⢻⡿⣿⡄⢀⠱⣦⣀⣀⣀⠀⠀⠀⠀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⢿⣯⣿⣿⣿⣶⣶⣤⣤⣤⡤⠚⠉⢀⣀⣺⣿⣿⣿⣴⢞⣓⠻⢿⡇⠐⣀⣶⠶⠛⠉
⠀⠀⠀⠀⢘⣷⣶⣤⣄⣠⣀⡒⠈⠀⠉⠛⠛⠻⠿⣿⣯⡤⠖⣛⣉⣽⣿⠻⣿⣿⣿⣿⣯⣿⣿⣏⢩⠿⠋
⠀⠀⠀⠀⠘⠻⠷⠶⣮⣿⣿⣿⣷⣶⣶⣤⣤⣀⣤⠼⠗⠛⠉⣉⡠⠤⠃⠀⠀⢀⡀⠀⠀⠀⣸⠧⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⢿⣿⣇⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⢸⠯⣿⡅⠠⠾⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⣿⠂⢐⠾⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣽⣥⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⡄⣄⠀⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠥⠤⠌⠁
⣦⣶⣤⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣤⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣴⣏⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡽⢃⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠃⢤⣶⣾⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣷⣿⠿⣿⠿⢿⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢋⡅⠀⢀⣤⠀⠉⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢀⣄⠘⢃⡄⢀⣼⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⣄⣄⠀⠀⠀⢠⣿⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢸⣮⡻⣍⡀⢠⡄⣰⡟⣹⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡴⣦⢻⣿⣶⣽⣶⣾⣿⣾⣿⣷⢰⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠱⣬⣷⣿⣿⣿⣩⡉⠙⠿⢻⣟⣼⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠻⠿⠟⠟⠀⠸⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢣⣿⠾⣿⣥⣤⠄⠇⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣘⡁⠀⠀⡄⡷⣾⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣄⡠⠛⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣛⣣⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢻⣿⡟⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⢿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢀⣀⠁⠌⣁⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣯⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠸⠆⠴⠳⠶⠫⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡄⠀⢀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠉⡈⡍⡈⠁⢀
⠀⠀⠀⠀⠀⠀⠁⠒⠊⠐⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠁⠀⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢻⣿⣿⢿⣿⣿⣻⣿⢿⢿⡟⣿⣟⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣧⣿⣼⣾⣿⣿⣿⣾⣾⣿⣿⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠠⣠⣤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠁
⠀⠀⠀⠀⠄⠒⠈⠀⠂⠄⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣆
⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⠀⠀⠁⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠷⡴⢇⠳⡸⠦⠥⠞⡸⠶⠞⠘⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⠀⠉⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣠⠎⣏⢡⡌⡏⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡀⡈⠉⣠⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠃⠁⠐⠘

⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶
⠀⠀⠀⣿⠀⠀⠀⠀⢰⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⡆⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⢀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣫⠵⠖⠒⠒⠶⢭⣝⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⠋⠁⠀⣠⡶⣦⣀⠀⠈⠳⣜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⣼⠃⠀⠀⣾⠁⣾⣆⣹⡆⠀⠀⠹⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠘⢦⣈⠙⢷⡀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⠀⠀⠀⢿⠉⣿⡇⢠⡇⠀⠀⢠⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢷⡀⠀⠈⠳⣦⡶⠛⠀⠀⣠⢞⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡶⢤⣄⣀⣀⣤⡴⣞⡵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠒⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢉⠁⠐⠂⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠲⢮⡁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⣠⠴⠯⢤⣀⡀
⠀⠀⠀⠀⠤⣤⣤⠤⠤⠀⠀⠐⠒⠀⠀⠈⠙⢲⣤⣀⠀⠀⠀⠀⠉⠒⠪⠽⢆⣀⠀⢀⣠⣴⣿⣶⣶⣿⣋⣭⡄
⠀⠀⠀⠀⢰⡦⢍⣙⠲⠦⣄⡀⠀⠀⠀⠀⠀⠙⠻⠷⠎⠓⢢⣄⣁⡲⠤⠔⠒⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠈⡷⢤⣈⠙⠒⠤⣍⡓⠶⠖⣒⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠈⣿⣿⡿⠿⠿⠭⠭⠍⠴⠄
⠀⠀⠀⠀⠀⢿⡀⠀⠁⠀⠀⠀⠉⢙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⠦⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢾⣑⠢⢄⡀⠀⠀⠈⣟⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠫⢍⡒⠤⢄⣒⡀⠈⠭⢖⡿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢄⠀⣀⠁⠈⠀⠀⠉⠑⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢭⣉⣉⣁⡼⠁⠀⠀⠀⠀⠀⢀⡤⠄⠒⠒⠂⠈⠁⠉⣶⣤⢀⠀⠈⠉⢛⡀⢀⣤⣤⣴⣶⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠬⡑⠢⢄⣀⢀⣀⣀⣤⣤⣴⣶⢸⠍⠁⠀⢸⣿⣿⠟⣛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣙⣲⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⢼⣦⡴⠞⠛⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠤⢀⣀⣀⡦⣁⣒⣻⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⠐⠀⠠⠽⠿⠛⣩⣤⣶⣿⣯⡠⢴⡾⠿⠛⠉
⠀⠀⠀⠀⠀⢁⣒⣺⣿⢕⠢⠬⣽⣿⣿⡇⢀⣀⣠⣶⠀⠀⠀⠀⠘⡗⠢⠬⣉⡂⠀⠀⠀⢰⡿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⡐⠢⠬⣿⡧⠭⣙⣺⣿⣿⡇⣒⣺⣿⣿⣧⠀⠀⠀⠀⠳⠀⠀⠀⠈⠉⠑⢒⣲⣽⡿⠟⠉
⠀⠀⠀⠀⠀⠠⢉⣛⣿⣟⡐⠒⠾⣯⣽⡗⠰⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠐⠦⠗⠉
⠀⠀⠀⠀⠀⠋⠒⠶⣿⠷⠤⢈⣻⣿⣻⡧⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠾⠿⠟⠃





⡶⣶⣶⢷⣿⠾⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠢
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⣀⣋⡒⣒⣤⠴⠆⠂⠀⠐⠀⠈⠢⢄⢳
⠀⠀⠀⠀⠐⠋⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⢢⠁⠀⠀⠀⠀⠀⠀⠀⠀⡋⠁⠴⠒⠲⠈⠀⡠⠀⡀⠀⢄⠀⠀⠤⢄⠀⠘⡏
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠄⡡⡤⡌⠐⠄⠀⠀⠀⠈⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠠⠀⠀⠀⠀⡈⢀⢉⣁⡀⠀⠁⠈⠢⠘⠀⢠⢹⠂
⠀⠀⠀⠀⠀⠀⠀⡂⢐⠀⠀⠀⠢⠤⢄⠁⠀⠀⢐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡥⠃⠀⠄⠀⠀⢁⠀⠌⣁⣥⠁⡀⠀⠀⠀⠠⠁⣀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⢄⠲⠤⠖⢁⠄⠀⠀⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄⠀⠁⠊⠀⠀⠑⠀⠀⠄⠈⣀⡂⠴⢖⣂⣀⡠
⠀⠀⠀⠀⢀⣀⢘⠤⠤⠤⠤⠤⠤⠀⠀⠤⠠⠤⠤⠤⢊⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠁⠐⠄⠠⠐⠀⠄⠒⠛⠛⠲⠭⠭⠭⢠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠌⠐⠂⠀⡀
⠀⠀⠀⠀⠀⢀⣀⢀⠆⢀⠈⠐⠂⠁⠁⠂⠡⢂⠀⣀⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣸⠈⠉⠀⢀⠐⢀⠀⢀⠠⠀⠀⠀⠈⠐⡢⢤⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠡⣀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠄⡝⠘⠀⠀⠀⠀⢀⠁⠐⣉⢉⢃⠐⡀⠀⠁⠀⠜⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠁⠔⡋⠛⠀⠑⠀⢀⠤⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⠈⠣⠄⠀⠀⠀⠀⠈⡀⢢⢂⠃⠀⢀⠃⢀⢀⠀⠐⡄⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠄⡏⣋⠂⢀⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠙⠲⠠⠁⣐⣁⡲⢥⣀⠈⠠⠈⠡⠐⠁⠀⠣⠠⠀⠐⠈⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠑⠠⢉⠉⠔⠁⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠑⠒⠊⠒⠮⠔⡀⢀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠅⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠢⠥⣀⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡁⡄⠂⠁
⠀⠠⠀⠄⠠⠐⠀⠀⠀⠠⠀⠀⠀⠐⠀⠂⠀⠂⠐⠂⠐⠂⠐⠀⠂⠀⠀⢀⣠⡐⠉⠈⠀⠉⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⡸⠛⠁⠀⠀⣀⡤⠄⢀⢄
⠀⠀⠀⠀⠐⠀⠂⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠅⠀⠀⣀⡲⠞⠃⠉⠁⠁⠀⠀⠀⠀⣯⡀⣒⣻⣿⠀⠂
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢔⣀⡼⠊⠁⠀⠀⣔⡒⢂⣰⣢⠀⠀⠀⡯⠡⣯⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⣰⠽⠚⠋⠀⠀⠀⠀⠀⠀⠀⢰⠷⠤⢯⠿⢿⠀⠀⠀⢸⡒⢺⣺⣻
⠀⠀⠀⠀⠀⢀⣠⡰⠌⠒⠋⠀⠀⠀⠐⠮⠭⡅⠤⢵⡀⠀⠀⣍⡛⣻⣿⣿⡇⠀⠀⢨⢉⣭⣭⣽⠆
⠀⠀⠠⢐⠬⠚⠉⢀⣀⣀⠀⢀⡀⠀⠀⣑⡒⣟⣛⣛⡏⠀⠀⠼⠦⠼⠿⢿⡅⠀⠀⢸⠶⢶⡶⣿⠁
⠀⠀⠁⠋⠀⠀⠀⢋⡀⣀⣴⣵⡶⠀⠀⡵⣤⣬⣭⢽⡇⠀⠀⢸⡒⢺⣿⣿⠇⠀⠀⢸⡓⢻⣛⣻
⢀⣤⠤⠀⠠⢀⠀⠸⠉⠩⡭⣭⡎⠀⠀⢱⡀⣒⣒⣛⡇⠀⠀⢫⣉⣭⣭⣿⡇⠀⠀⢱⠤⢭⣭⠽⠁
⠀⠦⢨⡿⠭⣿⣄⣸⠲⠲⠶⢿⣧⣠⣄⡼⠉⣭⣭⣯⣃⣀⢀⣔⠂⣾⣶⣾⢁⢀⡀⣌⣓⣚⣟⣿⣀⢀⠀⣤⠤⠐⠒⣀⠢⢄⠀⢠⣀⡀

⠀⠶⣄⡀⣲⣤⣶⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿
⠦⣄⡍⠋⣷⣼⡟⠘⢻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⣧⣀⠆⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿
⡄⠙⠿⡷⣆⣸⡏⢛⣴⣿⣿⣇⣴⣿⣀⡀⠀⠀⢰⣀⠂⢻⣾⡄⢛⢛⡃⢀⣽⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠙⠈⠻⠀⣯⣹⡇⢸⣿⠉⣿⡧⠿⠟⠛⠃⠀⠀⢘⣿⡃⢰⣾⡶⣿⡿⠃⢈⣻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠋⢽⣿⣿⣿⣿
⣦⠀⠰⠀⠩⠹⠗⠘⠉⠀⠀⠀⢀⣀⣠⡤⠴⠂⢸⣟⣁⣩⣧⣴⣾⠿⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣾⣿⣿⡿⢿
⣿⣷⡀⠀⠀⣀⢀⣤⣀⣶⠚⠋⠉⠁⠀⠀⠀⠶⢛⣛⠋⣩⣥⠀⣶⠷⠆⣼⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠛⢻⣿⣿⣷⣿
⣿⠿⠟⠋⣩⣽⢿⠏⣿⣿⢻⣶⣒⠤⠄⡀⣀⡀⡿⠉⠐⡏⢀⠀⣭⠠⠀⢐⢸⠈⣿⣿⣿⣿⣿⣷⡄⠀⠰⡄⠀⠀⢸⡇⠀⠀⠘⣛⣟⣿⡏
⠛⢁⣴⠪⠿⠣⠋⢀⣈⣁⣬⣼⣷⣶⣿⠿⢿⡇⣿⠾⠀⠇⠂⠀⠋⠈⠀⢀⣘⣤⣿⣿⣿⣿⣿⣿⣀⣠⣤⡇⠀⠀⠸⡃⠀⠘⠛⠟⣿⣿⡇
⠀⠶⠶⠶⠛⠛⠋⠉⠉⣀⣸⣿⣿⣤⣴⣤⣼⡇⡤⠴⠀⣶⣶⠀⣿⣿⠇⠠⢽⡟⣿⣿⣿⣿⣿⣿⠀⠀⠀⡇⠀⠀⢸⠇⠀⠠⠶⠖⠿⠀⠁
⣶⣤⢴⣶⣶⣾⣿⢿⣿⢻⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣶⣯⣤⣤⣶⣶⡆⢰⣶⡇⣿⣿⣿⣿⣿⣿⣠⣠⠀⡇⠀⢠⣧⠀⠀⠐⡛⣛⡁
⣿⡁⢸⣿⣿⣿⣿⣾⣿⣼⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣧⣟⣙⣀⢈⣙⡷⣿⣿⣿⣿⣿⣿⠀⠀⡅⡇⠀⢸⣷⠀⠀⠈⡉⠉
⠙⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⢻⣿⡇⣿⣿⣿⣿⣿⣿⢴⠴⠦⡇⠀⣼⣿⡏⠱⡀⢄⢀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣿⡿⣿⣿⣿⣿⣿⣿⣬⣠⣅⡇⠀⣿⡏⠁⣸⡇⠘⠛⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣉⣻⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡗⡄⣿⣿⢉⣉
⠀⣿⣿⣿⣿⣿⣿⢻⡟⢻⡏⢹⡏⢹⣿⠉⣹⣿⡿⡿⣿⣿⡿⠇⡿⡿⢿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠸⠇⢿⢿⣆⡀
⠀⠙⣿⡿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣷⠀⢹⡿⠷⠶⠾⠿⠧⠞⠧⠤⠼⢼⣿⡇⢿⠿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠹⠸⠠⣼⣼
⠀⢿⣿⣿⡿⣾⣷⣾⢷⠿⠿⡿⠟⠻⠿⠄⠸⡏⠿⠿⠷⠶⠶⠶⠶⠶⡶⠜⠛⠃⠚⠉⠁⠀⠀⠀⠀⠀⣀⡖⠂⠀⠒⠘⠋⠃⠀⠀⠀⠐
⠀⠹⠿⣿⡿⠿⣿⣿⠿⢽⡏⣿⠉⣹⡇⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⣴⡶⠀⠢⡑
⠉⠉⣃⣠⢀⡀⠘⠛⠁⠀⠀⠁⣿⠸⠃⠀⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢢⣬
⣤⣴⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⣥⣄⣠⣀⣀⣀⣀⣤⣄⣀⣈⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣼⣯
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⢐⣒⣓⣲⠇⠀⠀⠀⠀⡄⠒⠂⣀⣀⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠀⣇⠈⠉⠉⠉⣉⣠⢤⠤⠀⠀⠀⠳⣄⠀⢈⣭⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣛⣧⣀⠀⡉⡉⣉⣽⡗⠒⠐⠒⠚⢊⣿⠀⢨⢤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⢨⣩⣿⣈⠉⡉⣋⣉⢥⣯⠁⠀⢙⣙⣩⡟⠀⠤⠾⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⣀⣰⣧⠀⠉⠋⠀⢂⣹⣿⠀⠉⠉⣋⣹⣓⠂⠲⠶⢾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⢠⣤⣼⣤⣨⣽⣷⣀⠀⣉⣉⢉⣱⣿⡀⠀⣉⣉⣿⢟⠂⠒⢚⢻⡶⠒⠶⢼⠂
⠀⠀⠀⠀⠀⢠⣤⣐⣊⢿⠉⢔⠦⠤⠆⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⢉⠭⠷⣄⣀⣭⣤⣿⢛⠀⠒⠚⢩⡟⠐⣟⣿⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠈⠉⠙⢛⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣚⣛⣉⠁⠀⠠⠄⠀⢀⣡⣶⠭⠿⡶⠲⠻⠿⢿⢚⣯⣿⣼⣖⣀⣶⣦⣶
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⢹⣟⡿⢶⣤⣤⣹⠦⠴⣊⣻⣥⠴⣷⡖⣒⡖⠒⢒⣶⣿⣉⣿⠿⢿⣧⣨⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠈⠿⠯⠽⢿⣿⣷⠐⠀⠀⠀⠀⠀⠀⢸⠒⠄⠀⠀⠁⠀⠉⡷⠚⠝⠚⠝⠛⠛⣻⡟⠛⠛⠯⣝⠛⠀⠤⠤⣽⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⣀⡀⠀⢸⣦⣍⣀⠉⠀⠀⠀⡇⠀⢀⠁⢀⣀⡤⢼⣷⣂⣀⣀⣉⠲⠶⠶⠞⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠛⠛⠿⠻⠶⣦⣤⣷⣶⣶⣾⣿⣞⣛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⣀⣀⠀⣀⣀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣶⠀⡇⠸⣇⣸⠿⣇⣀⢸⢿⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣩⡙⠉⠉⡉⠙⠉⡉⢻⠊⣉⣉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢁⣄⣠⡤⠎⠛⡅⢸⠀⡓⡂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡇⡰⢫⢠⡍⡆⡇⡇⡇⢸⠀⣛⣃⢸⡧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣇⣓⣚⣚⣓⣓⣓⣓⣃⣸⣐⣒⣒⣸⠇⠈⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠧⣀⣀⣠⣀⣀⣀⣀⣠⣀⣀⣀⣀⡼⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⣖⣶⣒⠤⡀⠀⠀⠠⠾⠤⠤⠽⠄⠀⠀⢀⠔⣒⠢⡠⠊⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⠋⣀⣠⣄⠉⢈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡈⣳⠄⡟⠀⠀⠀⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢟⣓⣒⠷⡀⢨⣗⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢙⠒⣏⢰⣢⢾⠇⢷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡕⣂⠴⠦⣄⠙⣼⡯⠛⠭⣟⢄⠀⠀⠀⢰⣿⢸⡳⢷⡇⢯⠞⡼⠈⠥⠖⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⡀⠹⠤⠴⢁⡼⡹⡷⠲⣄⡈⢯⢇⠀⠀⢸⣿⠀⣡⣠⠴⠒⠒⠒⠋⠉⡍⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣲⣲⣾⠽⣪⠞⣇⣀⡞⠁⡼⡝⢦⡀⠀⢩⠎⠉⠀⡖⠀⡗⠀⠀⠀⣇⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢥⣈⣙⢯⣲⠀⠠⠤⣺⠜⠁⠀⠑⠖⠙⠚⢉⡲⣃⠞⠓⠒⠒⣩⣫⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠁⠿⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⣴⠀⣴⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣌⠀⣈⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠛⠀⠛⢀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠉⣿⡏⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⣿⡷⠟⠋⠙⠛⠻⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠙⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⣿⣿⣿⡏⠁⠈⢻⣿⣿⡇⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠘⢶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⣿⣿⣿⣷⣄⣤⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣉⣉⣉⣉⣉⣉⣉⡉⠛⠻⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣶⣶⣶⣦⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡿⠇⠀⣠⣶⣿⣿⣦⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠶⠶⠶⠶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠉⢸⣿⠈⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣿⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⣸⣿⣀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠶⠶⠶⠶⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠉⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⣠⣾⡿⠋⢀⣄⣠⣤⣤⣤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣠⣾⡿⠋⠀⠀⣿⣿⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢀⣴⣶⣴⣦⣶⣶⣤⣦⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡿⠿⠿⠿⠿⢿⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠉⣿⣿⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣏⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣿⣿⠀⢘⣿⣷⣶⣶⣶⣶⣾⣿⠀⠀⣿⣷⣶⣶⣶⣶⣾⣿⡇⠀⣿⣿⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⣿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⢀⠀⢀⡀⢙⣿⣇
⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠁⠀⠀⠀⢰⡖⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢰⡄⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠲⣆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⡄⢀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠂⢺⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣷⣾⣿⡄⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣅⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠘⠃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⢀⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣿⣷⣶⣶⣿⣶⣶⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢩⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣠⢹⣓⣹⡿⢿⠸⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⢀⡇⠀⣠⠀⢰⢆⣾⣦⣄⣾⡟⠛⣿⡟⢻⣿⣿⡟⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⡇⠀⠈⢷⣤⣿⠀⢸⣿⣿⣿⡿⡽⡇⠀⠸⣇⠀⢹⣿⡧⡽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢨⣴⣿⣇⠀⠀⣸⣿⣤⢤⢸⢎⣟⡟⡜⡽⡵⠃⢰⡿⠉⠙⢿⠇⢀⣠⣿⠠⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠆⠛⢸⠀⣿⣿⡷⣤⣼⢻⣯⠣⢻⣎⠎⠜⡼⣽⡇⢠⣤⠇⣴⣿⣛⣠⣼⣯⣴⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⡿⠓⢍⡏⣀⡆⣁⡿⢣⢪⢎⣾⡟⠁⣽⠍⠐⠛⠛⠁⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣿⡇⠰⠟⠳⡼⣿⠟⠃⠐⣱⡋⠈⠤⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠑⠊⠐⣒⣂⣉⣭⣤⣤⣤⡶⣶⡶⠶⣶⠶⠘⣹⡍⠉⠉⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢿⠋⢻⡃⠸⣇⠀⣿⠀⠈⣿⠀⠀⣸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠉⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢻⣀
⠀⠀⠀⠀⠀⠀⢀⠤⠆⡤⢌⠑⠀⠀⠀⠀⠒⠒⠒⢲⠒⢲⠒⢲⠒⠒⠒⡖⠒⡖⠲⡖⢲⠒⣶⠄
⠀⠀⠀⠀⠀⢰⠑⢦⣸⡃⠀⠱⠀⠀⠀⠀⠰⣶⣶⣶⣶⢸⢀⣼⡀⠀⠀⡇⠀⡇⠀⡇⢸⠏⠀⡇
⠀⠀⠀⠀⠀⠸⡀⠀⠽⣗⠀⣠⠀⠀⠀⠀⠀⠀⠀⠰⠀⣠⠏⠀⠻⡄⠀⡇⠀⡇⠀⣇⡏⠀⠀⡇
⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠚⠁⠀⠀⠀⠀⠀⠀⢁⣀⡼⠃⠀⠀⠀⠉⠓⢧⡀⣃⣈⣟⣀⠀⠀⡇
⠀⠀⠀⢀⠎⢰⠊⡁⠀⠙⡄⠀⠀⠀⠀⠀⢀⡜⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⡇
⠀⠀⠀⠘⡄⠘⢷⣉⣠⠀⠃⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠇⠀⢀⣀⡀
⠀⠀⠀⠀⠈⠢⠤⠈⡤⢪⣍⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢷⣄⠀⢸⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠠⠿⢂⠜⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁




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

⠀⠀⠀⢠⠀⠀⠀⠠
⠀⠀⠀⢸⠨⠀⢘⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀
⠀⠀⠀⢸⠀⠀⠨⣰⠀⠐⠄⠘⠀⠀⠀⠀⠀⣧⣀⢀⢈⣡⡇⠀⠀⢙⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⢸⠈⠀⢸⣿⠀⠀⢸⠉⠈⠂⠀⠀⠠⠂⠄⠤⠤⠴⠂⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⣴⣿⣷⡀⠀⠀⠀⡐
⠀⠀⠀⢸⠀⠀⢈⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠖⠲⠒⠲⠈⠆⠖⠄⠀⠀⠤⡆⠀⠰⠈⠀⠀⠙⠿⢟⣡⡟⠀⠄
⠀⠀⠀⢸⢀⠀⢸⣿⠀⠀⣸⠀⠀⡂⠀⠀⠀⡓⠃⠀⠘⠙⡃⡛⠓⠒⠒⠛⡇⠀⠀⠀⠀⠀⣀⠀⠉⠉⢀⡴
⠀⠀⠀⠀⠉⠈⠉⠉⠉⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠄⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢁⣾⡟⠀⡀⠀⠀⠀⠀⠀⠈⠀⣠⠄⠀⠐⠀⠀⠀⠀⠀⠀⢩⡁⠀⠀⢠⡄
⠀⠀⠀⠀⠄⠀⠀⠀⠆⣼⣿⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠘⢧⡴⠓
⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠂⠀⠀⠀⣠
⠀⠀⠀⠀⠠⠀⠀⠐⠛⠋⠀⠀⡐⠀⠀⠀⠀⠐⠀⣂⣀⣀⣀⣀⣁⠀⠀⠀⠀⠀⢀⠀⣂⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢄⣊⣀⠀⣀⣀⠀⣀⣙⡰⠀⠀⠀⠀⠀⡿⠀⠁⠒⠓⠍⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⠀⠀⣿⠀⠀⠀⠀⠀⠁⠁⠀⠈⠈⠀⠈⠈⠀⠀⠀⠀⠠⠔⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡑⢈⠂⡑⢈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠈⠊⠑⢉⠊⠀⠀⣿⠀⠀⠀⠀⣁⣄⣄⢀⣀⣠⡀⣠⣠⣄⠂⠀⠀⠀⡀⣷⡄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠈⠀⠂⠀⠈⠁

⠀⠀⠀⢀⢤⠶⡶⣤⡀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢤
⠀⠀⢠⡋⣛⢙⡏⣻⢙⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣨⡽⢭⣅⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠑⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⣹⣤⢼⡁
⠀⠀⠘⣅⣭⣩⣏⣽⣩⠃⠀⠀⠀⠀⢠⠗⡖⡒⢒⢲⠺⡀⠀⠀⠀⠀⠀⢸⣿⢄⡒⣿⡇⠀⠀⠀⠀⠀⠀⠈⠳⠶⢶⣿⠀⠀⠀⠀⠀⠀⠠⣿⠉⢙⡧⢯⡽
⠀⠀⠀⠈⠛⠶⠶⠓⠁⠀⠀⠀⠀⠀⠼⠿⠷⠤⠤⠿⠿⠧⠀⠀⠀⠀⠀⠺⠿⠶⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠙⠳⠛⠃


⠀⠀⠀⣤⣤⣤⣤⡀⣶⠀⠀⠀⠀⠀⢀⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⢀⡀⠠⣀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⢀⡖⢓⡒⢲⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣄
⠀⠀⢸⡇⠤⠤⠀⡇⢻⠀⠀⠀⠀⠀⢸⠀⡤⢤⠕⠒⢄⠀⠀⠀⠀⢸⠀⠀⢸⢠⡄⡇⠀⠀⡇⠀⠀⠀⠀⢸⡇⠻⣅⢸⠀⠀⠀⠀⠀⠀⢠⡏⢰⢄⣤⡸⢲⠂
⠀⠀⢸⡇⣀⣀⠀⡇⡿⠀⠀⠀⠀⠀⢸⠀⣈⡶⠀⠀⢀⠇⠀⠀⠀⢸⠀⡀⢸⠚⠒⡇⢀⠀⡆⠀⠀⠀⠀⢸⡇⠙⠋⢸⠀⠀⠀⠀⠀⠀⠘⣇⠛⠈⠉⣥⡏
⠀⠀⠈⠷⠶⠶⠶⠃⠙⠀⠀⠀⠀⠀⠈⠓⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠈⠧⠬⠥⠼⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠋⠁


⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⡄
⠀⠀⢠⣴⣶⣶⣄⠀⠈⢣⠀⠀⠀⠀⢸⡇⠀⣀⡀⣠⠶⠀⠀⠀⠀⠀⢸⡇⢠⢦⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡇⠀⠀⠀⠀⢸⡇⢦⣀⣠⡀
⠀⠀⠻⣷⣿⣷⡿⠠⢴⠁⠀⠀⠀⠀⢸⡇⠞⠁⠙⠁⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⢇⡜⠀⠀⠀⠀⠀⠀⠀⣾⣦⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠉⠁⠙⠶
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶


⠀⠀⠀⠀⠀⣠⢀⣤⠀⠀⠀⠀⠀⠀⠀⡠⢀⣐⣐⠂⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠰⢶⡿⢾⡷⠆⠀⠀⠀⠀⠀⠀⡇⣙⡛⡛⡋⢸⠀⠀⠀⠀⠀⠀⢰⠋⣠⣄⠙⡆⠀⠀⠀⠀⠀⠀⢰⢋⣀⣀⠙⡆⠀⠀⠀⠀⠀⠸⢋⢸⡟⣿⡇⡙⡇
⠀⠀⠀⠰⢾⡷⣾⠷⠆⠀⠀⠀⠀⠀⠀⡇⣒⡒⡂⣯⢸⠀⠀⠀⠀⠀⠀⠸⣄⠙⠋⣠⠇⠀⠀⠀⠀⠀⠀⠸⣄⠉⠉⣠⠇⠀⠀⠀⠀⠀⢰⣌⣸⣯⣼⣇⣡⡇
⠀⠀⠀⠀⠙⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⣿⠀⠀⠀⠀⠀⠀⣾⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠀⠀⠀⠀⠀⠀⠘⠷⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠲⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⣄⠀⠀⠀⠀⠀⢠⠶⠞⠻⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿