Upside Emojis & Symbols

⠀⠀⠀⠀⠀⣀⡠⢴⣒⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣕⡢⢤⣀
⠀⠀⠀⣶⣫⡷⠞⢋⣽⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣉⠛⢶⣭⣶⠄
⠀⠀⠀⠘⠷⣿⣿⣿⡿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡠⠷⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠴⣅⡀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠊⠁⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠈⠓⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⠶⠖⠚⠛⠛⠛⠓⠲⠶⣤⣐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⣠⠞⠛⠛⢦⡀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠰⠯⠭⠭⠭⠭⠗⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⣄⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣀⣀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⢀⣀⣸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠸⡅⣈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⡇⠀⠈⠛⠲⠶⠤⢤⣤⡤⠤⠶⠒⠶⠦⢤⣤⣤⣤⣤⠴⠞⠁⠀⠀⡟⠁⠈⢣⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⡄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣠⠞⠀⠀⢀⡴⠃⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠙⢦⡀⠀⠀⠳⣄
⠀⠀⠀⢸⠃⠀⠀⣠⠏⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠹⣄⠀⠀⠘⡇
⠀⠀⠀⣿⠀⠀⣸⠁⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⣇⠀⠀⣿
⠀⠀⢠⡇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢸⡄
⠀⠀⣸⠁⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⡇
⠀⢀⡟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢻⡀
⣛⣋⣀⣤⣝⣛⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣟⣛⣫⣤⣀⣈
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣴⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠉⢒⣦⣤⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣤⠴⠦⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡟⠒⠋⠉⠉⠉⠉⠉⠉⡉⠉⠉⢹⠉⠉⠛⢺⡿⠦⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡠⠂⠀⠀⠀⠀⠈⡹⢶⣼⣤⠆⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠚⠒⠉⠉⢉⡭⡉⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠘⡆⢸⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠋⠀⠙⢆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣀⣀⣼⢀⡷⠽⠤⠤⠤⣤⣤⣇⣸⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡴⠋⣀⠀⠀⢀⡀⠱⡄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠗⠉⢀⣠⡼⠿⢧⡄⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⢸⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⡇⠀⠀⣀⠴⠲⣄⠀⠀⠀⣠⣤⡄⠀⣴⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠤⠄⠘⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢁⡤⠚⠁⠀⠀⣈⣙⣂⠴⣾⡉⣱⡾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⠇⢻⢸⣿⠏⢳⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠉⢀⣠⠴⠚⠉⠁⠀⠀⠀⡏⣛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠸⣿⣧⡾⢿⣿⠗⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⠅⡠⢶⡏⠀⣀⣀⣀⣀⣠⠄⢸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠲⣶⡴⠒⠂⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢁⣞⣀⣨⡇⠀⠈⠛⠧⣬⣰⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠴⠚⢉⠁⠠⠀⠀⠉⠉⠒⠦⣄⠀⠀⠀⠀⠀⠀⢸⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⢠⠞⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠸⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠒⠚⠀⠀⠀⠀⠀⣠⠼⠛⢦⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⠶⠖⣣⣴⣿⠀⠀⢀⡾⠁⠀⠀⣼⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⠁⣾⡟⠀⢀⡞⠀⠀⠐⢢⡟⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣸⠁⠀⠀⣠⠋⠑⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡟⡟⠀⣰⠻⣤⡤⠞⠁⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⣽⠀⠁⢸⠇⢰⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢰⠂⢷⡄⠈⠀⢰⡇⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠈⠱⣆⣀⣀⢠⡎⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡟⡎⠑⢤⣀⣿⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡄⢀⡈⠀⠠⢤⡤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⠤⠶⢆⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢻⡑⠢⢤⣈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠷⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣖⡂⠤⣔⣦⡀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡀⠀⠀⠀⠈⠿⢷⡿⣿⢻⣾⡿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⡠⢣⡄⠀⠵⢿⣤⣂⣠⡶⠖⠛⢻⣺⢯⣄⡀⠈⠉⠛⠿⢶⣄⣀⡀⠴⣾⡭
⠀⠀⠀⠀⠀⢸⠃⣿⡇⢸⠛⠏⠛⠾⣧⣄⡀⠀⠀⠙⠻⢾⣻⣶⣶⣦⣄⠀⢸⣿⣶⣦⠀⠘⣇⠠⠄
⠀⠀⠀⠀⠀⠂⠷⡐⣁⣬⣹⠈⣀⠀⠐⠉⡙⠷⡦⠤⠀⠀⠈⠉⢉⣛⣿⣿⣧⣍⣻⡟⠳⠽⢿⠄⡀
⠀⠀⠰⣾⠥⢞⡇⢿⣿⣯⢂⠀⠈⠹⠲⠀⠀⠈⢠⣶⡄⠐⢺⣉⠻⣍⠀⠘⢯⣽⣵⡿⣦⣄⡤⠴⠃⣵⣤⣄⡀⠀⠀⡀
⡀⢀⡀⠻⠶⣤⡅⣂⣬⠅⢄⡀⠀⠂⣤⡄⠀⠀⢸⢿⡡⠀⠃⠉⠛⠾⠿⠾⠉⢉⡈⠭⢍⣀⣨⡥⣾⣿⣿⣯⣝⡓⢓⣶⠂
⠈⠉⠁⠀⠀⠀⠘⠚⠫⡎⣦⣿⣷⠂⡼⠋⠀⢀⡸⢾⠏⢼⣄⠀⠀⠀⢠⣤⣾⡾⢀⠞⠿⠛⠛⢻⡿⠟⠷⠶⢷⠋⣌⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠫⢿⡁⢶⣿⣁⣷⣜⢿⣶⣾⡣⠋⠙⠲⠦⣬⣬⠴⣖⢛⣋⠥⢓⠒⠋⠀⠘⠶⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠓⠋⠐⠉⠉⠿⣥⢤⠶⠾⡛⢓⠿⠙⠚⠉⠾⠀⠀⠉⠚⠃⠴⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠒⠃⡀⠢⣤⠶⠆⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⢀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⠐⢀⡀
⠀⠀⠀⠀⣤⠶⣌⡷⠶⠛⢻⣷⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣿⠙⠳⢦⣍⡱⢶⡄
⠀⠀⠀⠀⠙⠋⣿⣦⠶⢶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣤⣾⠋⡛⠃
⠀⠀⠀⠀⠀⠈⠙⠻⢿⡿⠋⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠻⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣠⡴⠞⠛⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⣠⠴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⠉⠑⠢⣄⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⣠⡖⢀⣀⣀⠰⣤⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣰⡉⣈⣉⢉⢉⣁⣈⣧⠀⠀⠀⠀⠀⠈⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢻⣷⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠖⠁⠀⠀⠀⠀⠀⠟⠙⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⡁⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠄⣽⠀⠙⠒⠲⣤⡀⠀⠀⠀⣠⡤⠒⢲⠆⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠱⣦⠀⠀⠀⠀⠉⠲⣤⠞⠁⠀⠀⢹⣄⣠⠔⠈⠁⠘⢇⠀⢀⡤⠂⠀⠀⠀⠻⣴⣇⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⡠⠂⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢰⠏⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⣧
⠀⠀⠀⠀⣾⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢻
⠀⠀⠀⠀⠏⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⡆
⠀⠀⠀⣼⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢇
⠀⠀⢀⡛⠀⠀⡎⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢸⡆
⠀⠀⣸⡇⠀⢾⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⠀⠀⠀⠀⠀⠀⠀⢁⡀⠈⢁
⣶⠒⢁⡀⡀⣛⣒⣒⣦⠀⢠⣴⣧⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⢒⣤⣀⡠⣀⠀⠘⠁
⣤⣤⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣭⣭⣭⣶⣶⣶⣶⣶⣯⣭⣭⣭⣭⣭⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣬⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣄⢀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⢻⡟⢻⣿⣾⣿⠻⣿⣏⣽⠿⣧⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠟⠁⣿⠀⣿⠻⠷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢦⣤⣤⣤⣤⣿⠶⠿⠶⢦⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡄⠀⠀⠀⠀⢠⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⢠⡿⣟⢷⣟⣷⢿⡟⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣗⣾⠋⠁⠀⢀⣤⠾⠛⠛⠛⠛⠛⠻⢶⣄⡀⠉⠻⢦⡴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠀⢀⣴⠟⠁⣠⡶⠖⠛⠓⠶⢦⣄⠈⠻⣦⠀⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠾⠃⠀⠀⠀⠀⠀⠀⠙⠃⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣤⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠟⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠻⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⠀⣰⢏⣿⣿⣿⣿⡳⠶⣤⡀⣠⡴⠶⣾⣿⣿⣿⣿⡹⣆⠀⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠿⣿⡿⠾⢋⣼⠏⠈⠻⣿⣿⣷⣾⣿⣿⣶⣿⣿⡿⠟⠁⠹⣧⡙⠳⢿⣿⠿⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡎⢀⣠⣤⣶⠶⠟⠋⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠙⠻⠶⣶⣤⣄⡀⢹⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⣿⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⣰⠏⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⢹⡆⠀⠀⢸⡇⠀⠀⣿⠀⢠⡟⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠘⣧⠀⠀⢸⡇⠀⢸⡏⠀⣾⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢿⡀⠀⢸⡇⠀⣸⡇⣼⠇⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⣇⠀⢸⡇⠀⣿⢠⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⠀⢸⡇⢀⡏⣼⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢹⡆⢸⡇⢸⣧⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡇⢸⡇⣾⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢸⡇⣿⡟⢸⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣸⢧⣿⡇⣼⠃⢀⡾⠋⠉⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⣿⡇⣿⢠⡿⠁⣀⣾⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⣿⣿⣤⠾⠛⠉⠀⢀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢿⣿⣿⣿⠛⣿⠃⠹⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠀⣿⠀⠀⠹⣇⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠀⢹⡆⠀⠀⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⡇⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⣹⣿⡇⢸⡇⠈⣿⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⣀⣴⠞⢻⣿⣧⠘⡇⠀⢹⡟⠉⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣉⣤⣤⣾⢻⣿⠀⣿⡶⠾⣿⣄⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⢠⣿⢸⣿⣰⠟⠀⠀⢻⣟⠳⣤⣀⡀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⢀⣾⡇⢸⣿⡿⠀⠀⠀⢸⣿⡄⠈⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣠⠟⣹⡇⢸⣿⣇⠀⠀⢀⣾⠀⣷⣤⡶⠿⠿⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⠶⠟⠁⠀⣿⠀⠘⠛⠿⠀⢀⣾⠋⠀⠘⣿⣧⡀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣾⡟⠀⢠⣅⣴⣀⠋⣤⡀⣠⡾⢻⣿⡻⢷⣦⣤⣤⣤⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢀⣴⣿⠻⣶⡀⣯⡘⠛⢀⣉⣥⡛⠁⠀⣹⣇⣀⣀⠈⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣀⣴⠟⣿⣿⠀⠘⣷⣿⡟⠀⠀⠻⢿⣷⠀⢸⡟⠙⢿⣍⠀⠈⠉⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠛⠋⠀⣸⠏⠸⣧⡀⣸⣿⣿⡀⠀⠀⢈⣿⢶⣿⡀⠀⠘⣿⣿⣷⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⣿⠀⠀⠈⠛⠋⢸⡏⢷⡄⠀⢸⡇⠀⠈⠻⣆⠰⣏⠀⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢠⡟⠀⠀⢀⣤⠶⢻⣧⠀⠙⣶⣬⣿⠀⠀⠀⠘⣧⠻⣆⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⡟⠁⢀⣴⠋⠁⠀⢸⣿⠀⠀⢹⡇⠹⣧⡀⠀⠀⢸⡇⠙⢷⣄⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡼⠋⠀⠀⣾⠃⠀⠀⠀⣾⣻⣧⡀⠀⢻⣤⣌⣻⣦⣀⣸⡇⠀⠀⠉⠛⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⢸⡇⠀⠀⢀⡼⢻⡏⠀⠹⣦⣸⡇⠈⠉⠻⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣠⡶⠛⠀⢸⡇⠀⠀⠈⢿⣿⡀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⣷⠀⠀⠀⠘⣏⠻⣦⣀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣿⠀⠀⠙⠛⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢠⣶⣒⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣵⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣼⠟⠀⠀⠀⠀⠀⠉⢋⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡠⠔⠢⡀⠀⠀⠀⠀⠀⠀⠈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠀⠈⠒⢄⡀⣀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠈⠑⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡤⠤⠤⠤⠤⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠒⡒⢒⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠤⠤⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠚⠚⠛⠃⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠶⠤⠄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠘⠃⠀⠀⠀⠐⠒⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡧⠄⠀⠒⠀⠀⠀⠀⠒⠀⠀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡸⡆⠉⠒⠊⠡⠔⠑⠄⠀⠐⢆⠈⠢⡀⠀⠀⠀⠀⠀⠀⡰⠁⡇⠈⠒⠒⠒⠒⠒⠂⠤⠄⠂⠀⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠙⠀⠀⠀⠀⢠⠊⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠼⡄⠧⠄⠀⠄⡆⠀⠀⣀⣀⣙⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡄⠀⠼⢤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡀⢤⠀⠄⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠉⠉⠐⠂⠀⠇⢡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣁⣀⣠⣤⡼⠏
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁

⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃
⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆
⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃
⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠷
⠀⠹⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡼
⠀⣰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣇
⠀⠁⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡆
⠀⠈⣻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂
⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡋
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠂
⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷
⠀⠀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠆
⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄
⠀⠐⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠘⢿
⠀⢀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡀⠀⢀⣾⡆
⠀⢈⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠀⠹⣿⣇⠀⢹⣇⡀
⠀⠌⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⣤⣾⣿⠃⠙⣿⣿⣾⣿⣶⣶⣤⣤⣤⣾⣿⠃⠀⠀⠀⢸⡟⠀⠈⠉⠁
⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠋⢉⡉⠛⣿⣶⣶⣿⣿⡿⠋⠉⠉⠉⠛⠛⠻⣿⣄⡀⠀⢀⡞⠁⠀⢠⣿⡇
⠀⣘⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣼⡿⠋⠀⠀⢀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣰⣿⣷⣦⣤⣴⣿⣿⣷⣤⣾⣿⣤⣤⣾⣧⣼⣀
⣿⣿⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣿⣿⠁⣠⠄⢠⣾⣿⣿⣋⣹⣿⡟⠀⠀⠀⢸⣿⣿⣿⣟⡛⠛⢛⡛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⠀⠋⠀⠘⢿⣿⣿⠟⢻⣿⡇⠀⠀⠀⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠿⠿⠿⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡏⠀⠀⠀⢀⣴⣿⣿⠀⣼⣿⠃⠀⠀⠀⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠻⣿⣿⣤⣿⡟⠀⠀⠀⠀⢪⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⠀⠈⣩⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢻⣿⣿⠟⠁⠀⠀⠀⠀⠰⠏⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⣘⠁⣤⡀⠀⠀⡀⠀⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣾⣡⣶⣿⣷⡟⢣⣩⣾⣿⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠉⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠶⠶⠟⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠂⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠀⢀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠒⠒⠶⠂⣠⣴⣶⣦⣀⡒⠒⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⣿⡿⠟⠛⠿⢿⣿⡄⠈⠛⠀⠀⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠁⢀⣠⣿⣿⣶⣶⣶⣶⣤⣿⣿⣷⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⢀⣀⣼⡿⠟⠁⣀⣤⣀⣀⠈⠻⣿⣿⣷⡄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢹⣿⠁⣤⣾⠿⠿⠿⢿⣿⣦⠈⢿⣿⣇⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⣿⠀⠈⠁⢠⡶⠀⠀⠈⠉⠀⢸⣿⣯⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣿⣿⠀⢠⣤⣼⡄⠀⠀⣠⣶⡄⠀⣿⣿⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢻⣷⣄⠀⠉⠉⠀⠀⠀⠈⠁⠀⣰⣿⡟⠋⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⣿⣿⣷⣄⣠⣴⣶⣄⣤⣴⣶⣿⣿⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢸⣿⣌⠹⣿⠛⠛⢉⣻⣿⠛⠃⣼⣿⡀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢾⡿⢿⣇⣹⡇⠀⣽⣿⡏⢹⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣷⠀⠹⣿⡿⣿⣿⣿⣿⣾⠛⠁⢹⣇⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠘⢿⣆⠀⠉⠀⠈⠉⠛⠉⠁⠀⠀⣸⡿⠂⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠷⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣴⡾⠆⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣤⡜⣫⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣷⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠐⠂⠀⠀⠀⠀⣠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠀⠀⡠⠞⠉⠙⣃⣀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠃⠀⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡃⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣴⡧⠴⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⢀⣠⠶⠖⠛⠛⠉⠉⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠀⢼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡠⠖⠒⠈⠉⠉⠀⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠⠦⠤⠤⠴⢆⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠛⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⡷⡀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠙⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⠙⠂⢄⡀⠀⠀⣀⠤⢴⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠻⡄⠀⠀⠈⠒⠊⠀⠀⠘⣄⠒⠁⠈⢆⡀⠔⠁⠀⠈⠒⢄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡸⠀⠀⣌⣠⢤⣠⣤⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢀⡀⠀⠀⠀⠀⠂⢠⢷⡀⠀⠳
⠀⢀⡤⠒⠉⠉⠉⠉⠁⠘⡇⠀⡜⠈⠁⠀⠁⠀⠈⠉⠈⠙⠢⢄⣀⠀⠀⠀⠔⠂⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠨
⢰⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⠀⠀⢠⡄⠊⠀⢯⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠡⣀
⣤⣤⣤⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣤⠀⡀⢀⠀⢰⣦⣿⣸⡏⣧⡿⣤⣶⡄⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⡜⠇⠘⠟⠷⠛⠇⢹⠋⠹⢅⣏⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢸⡇⣴⢀⣾⠀⢍⡇⢸⣇⢺⡇⣴⠵⢀⡖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠙⠊⢯⠜⢸⣇⠚⠇⠸⠤⠻⠀⠞⠧⠚⢇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠂⢠⡇⠀⣀⢠⡄⣤⢀⡀⢠⣶⡆⡴⢱⢠⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⢇⣄⠤⡄⢨⡔⠀⡒⠘⣛⣹⡷⠃⢈⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢀⢿⢸⡏⣠⡧⠜⣧⢴⡏⣸⠇⣸⡇⢀⣿⠋⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠘⠸⠇⠀⠀⠮⠾⠀⠉⠉⠉⣈⣓⣊⣙⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⣒⣒⣒⣊⣤⠤⠶⣖⢻⢋⡿⢹⡉⢰⢀⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⡼⣭⣅⣰⣶⣧⠐⠏⣹⡞⢷⡘⣇⢸⡀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠿⠋⢧⡟⢿⠿⠹⠙⣀⣈⣡⠥⢶⢚⣿⡉⣯⣉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠶⠶⠶⠖⢚⣋⣭⡍⠸⣦⣶⡟⣾⠀⡷⢹⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠿⠹⠇⠙⠒⠁⠘⠁⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡄⠀⡄
⠀⠀⠀⢀⣴⢶⣭⠴⠞⢉⣿⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣾⡉⠑⠶⣭⣑⢶⡄
⠀⠀⠀⠀⠙⢙⣷⣶⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣦⣾⣏⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠛⢻⠛⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠙⢻⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠀⠀⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠹⣆⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡠⠖⠋⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠴⠚⠊⠉⠉⠉⠉⠉⠉⠉⠑⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⡠⠒⠀⠀⠒⢄⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢼⣛⣛⣛⣛⣃⣘⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⣀⣀⠀⠀⠀⠙⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠋⠀⠀⠀⠀⠀⠀⣠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢹⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢞⠀⠀⠛⠤⠄⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢸⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠈⠉⠈⠉⠀⠈⠉⠁⠚⠛⠒⠒⠒⠂⠉⠀⠀⠀⠀⢀⡀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠸⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⢣
⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸
⠀⠀⠀⢠⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⡆
⠀⠀⠀⣾⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠂⢤⣤⣤⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠻
⠀⠀⢠⡇⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠀⠀⡆
⠀⠀⡸⠃⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠖⠀⠁⢀⣀⣛⠛⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠁
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡿⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣿⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣾⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢽⣿⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣼⠃⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⣼⠃⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⣾⠃⠀⢰⠃⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⢠⡾⠃⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⣠⡿⠁⣀⣴⣞⣁⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⣰⡿⠶⠛⠉⠁⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⣠⣴⡿⠟⠁⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⢀⣤⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠒⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣄⣀⡀⣄⣤⣾⣉⣀⣀
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠟⠛⠋⠤⠉⠉⠉⠁⠀⠀⠀⠣⠀⠨
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠘
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠄⣁
⠀⠀⠀⠀⠀⠀⠸⡟⣿⣿⣿⡇⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠰⠶⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⢄⠀⠄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣙⣦⢳⡼⣄⣧⢀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡝⠃⠉⠀⠀⠀⠀⠉⠟⣠
⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢀⡗⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣄⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠐⢿⣽⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⡷⠀⠀⠀⠀⠙⡟⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⡀⠀⠀⠀⠀⠀⠀⢠⠂⡌⠀⠀⠀⠀⠀⠀⠰⠈⢆⠀⠀⠀⠀⠀⠀⠀⠌⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠐⠀⡀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⡗⠈⠑⠠⡀⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡤⠤⠄⠤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢉⣤⣤⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⣉⡉⡀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠘⠀⠀⠀⠐⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠽⢄⠐⠂⠀⠀⠀⠂⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠆⠁⠊⠙⠂⠁⠀⠐⠢⡈⢄⠀⠀⠀⠀⢀⠊⢸⠈⠀⠀⠀⠀⠀⠒⠈⠀⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠃⠀⠀⢰⠁⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⡄
⠀⠀⠀⠀⠀⠀⢀⡆⣀⢀⣊⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠁⠈⠀⠀⡍⠀⠉⠉⠉⠉⠐⠂⠀⠀⠀⠐⠂⠀⠀⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢉⣀⣈⡉⠉⠀⠀⠀⠀⠀⠀⠀⠈⣓⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣁⣠⣬⡭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠱⣆⠀⠀⡀⠀⠀⠀⣠⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠤⠤⣼⠀⠀⡄⠀⠀⠀⣧⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⠷⠶⠶⠿⠶⢶⡶⠶⠶⠴⠷⠾⠷⢾⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣀⣀⣼⣧⣀⣀⣀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠉⢉⣏⣉⣉⣉⡉⠉⠉⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠧⠶⠖⠒⠛⣋⣉⡈⠉⠉⠉⠉⠉⠉⠛⠛⠒⠶⠦⢤⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣤⠲⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢨⡿⢆
⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠲⣾⣭⣯⣿⣂⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣮⠵⣹⣇⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⢼⣯⣷⣾⣿⡁⠁⢹⡧⠄⠈⠛⠛⠲⠶⢦⣤⣤⡖⠛⢻⣟⢦⣿⠿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣏⢨⡝⠟⠛⠙⠻⣥⠒⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⣿⣁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⠉⠀⠀⠀⠀⠀⣀⡴⠛⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠔⣠⠖⠉⠉⠉⠉⠉⠐⠒⠲⠤⣤⣌⡓⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⢡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⢦⣄⠈⠻⣄⠲⣄⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡇⠀⡿⠃⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢠⡾⠙⠳⣤⡀⠀⠀⠀⢀⣷⡀⠀⣸⠃⠀⡿⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢴⣤⣠⡞⠀⠀⠀⠀⠙⠦⡴⠞⠋⠈⢿⣴⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠀⢠⡇
⠀⠀⣀⡠⠤⠤⠤⠼⠧⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⢠⡾⠉⠀⠀⠂⠐⠀⠂⠈⠉⠉⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠲⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠾⠿⠗⣀⡿
⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣶⣁⣀⡀⠀⠀⢨⣻⡶⠒⠚⠉
⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣰⠟⠛⠛⠓⣾⡋⠁⠈⢹⡄⠀⠈⢳⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⢠⣴⣤⡿⠃⠀⠀⢨⡿⣄⠀⠀⠹⣭⠧⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠘⠿⠧⣿⣆⣀⣀⡾⠁⢹⡄⠀⠀⠈⠢⣍⠓⠤⣽⠹⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⠃⠈⠉⠉⡀⠀⠘⣧⠀⠀⠀⠀⠻⣇⠀⠀⠐⢇⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣀⣠⡴⠛⠁⠀⠀⠛⠛⠁⠀⠀⣿⠀⠀⠀⠀⠀⢀⡿⠀⠈⠙⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣠⣾⢠⡄⠀⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠞⠋⠁⣻⡜⢻⡄⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸⡇⡸⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠉⣭⣭⣭⡀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣩⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡏⠛⠛⠛⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⡀⠀⠀⠀⠐⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣖⣢⣿⣶⣿⣿⣁⣠⡤⣤⡤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⣈⣭⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠋⠀⠑⠝⡹⣻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⢀⣀⣠⣤⣶⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠓⠵⠫⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠿⠿⡿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⣻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡯⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣠⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡌⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢠⠔⣪⡽⠟⣋⣴⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠓⡉⠁⠚⣈⣥⣴⣿⢿⣫⠿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠀⠒⠊⠁⠀⠀⠐⠀⠀⣠⢀⡜⠿⣻⠭⠓⠉⠁⡈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠶⢶⠶⠶⠶⢶⢚⡛⡉⠉⠩⠁⠀⠀⠀⠀⠀⠀⡐⠰⠰⢀⣀⣡⠴⠖⠉⠀⠀⠀⠀⠠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⣛⠛⡛⢛⠙⠁⠈⠁⡉⡀⣠⢠⠀⠀⠀⠀⠀⠀⣁⣴⠥⠐⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⣲⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣛⣟⣻⡿⣗⠛⠀⠠⠘⠁⠁⠢⠀⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠠⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⠶⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⢀⣠⠴⠚⢋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣉⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⠋⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣩⣥⠤⣤⡀⣀⣀⣠⠤⠴⠶⠖⠚⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣩⡙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠶⠒⠋⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢾⣟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⣠⣤⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣀⣀⣀⡀⠀⠀⠀⠙⣿⣯⣵⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣶⣴⣦⣀⠻⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠀⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠤⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢽⣿⣿⣿⣧⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⢿⣿⣿⣿⡿⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢷⢶⣶⣤⣤⣀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⣟⣻⣯⡽⠿⡗⠻⡿⣦⡠⠐⠁⠀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡟⢻⡏⣹⣿⡄⣅⢹⡄⠘⡜⣷⡴⢶⡄⠀⠄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠄⢴⣇⣽⣿⣷⣄⣱⣿⣾⠷⠾⣧⣾⢷⢚⣦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣸⣧⣶⠼⣿⣛⣿⣯⣿⣿⣶⣿⣿⣿⡋⠌⠌⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣷⡟⢿⣿⢿⣿⣿⣻⣽⣿⡿⢹⢹⣇⢃⢁⢊⡇⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣇⠛⣿⣦⣿⡾⣿⣿⡟⠛⠙⡄⡈⣎⣿⣾⣦⣬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢹⠈⣁⣇⣬⣿⣷⣾⡿⢿⣿⣿⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⢷⣾⣻⣟⣹⣭⠿⠶⢿⠛⠛⠛⠁⢸⣯⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡏⠙⠛⠋⠁⢳⣶⣾⣿⣄⣀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣆⠀⠀⣠⣿⣿⣿⡿⡿⠛⠃⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠉⠉⠛⠿⠟⠁⠀⠀⢀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⣠⣿⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣷⣾⣶⣿⣿⣿⣿⡿⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⡀⡀⢀⢀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡄⢠⡀
⠀⠀⠀⢸⢸⡇⡇⣿⢸⢸⢸⡇⠈⡟⢻⠋⠀⠀⠀⠀⠀⠀⢀⢴⣄⠀⠀⠀⠀⠀⠀⠠⢦⠈⢉⢈⠁⡉⢡⠶⢀
⠀⠀⠀⠈⣦⣤⠁⠁⣈⣈⣰⠁⠀⠇⢸⠀⠀⠀⠀⠛⠛⣛⠛⣿⢛⡛⠛⠛⠀⠀⢘⠓⠤⡐⡬⠂⠪⡔⡡⠄⠚⡂
⠀⠀⠤⢤⣿⣿⣼⣿⣿⣿⡷⠀⢸⡇⢸⡇⠀⠀⠀⠠⣄⠈⠁⣿⡀⠉⢠⡀⠀⠀⠛⠓⠐⡂⢅⠿⢷⡸⣐⠀⠚⠃
⠀⠀⠀⣠⢼⣇⣿⣿⣿⣸⡧⣄⠀⡇⢸⠀⠀⠀⠀⣰⣶⣶⣶⣿⣷⣶⣶⣶⠀⠀⠚⢃⡁⠔⢩⢁⠹⡕⠂⢈⠚⠃
⠀⠀⠀⢇⣉⡹⠉⠉⠈⢏⣉⡸⠀⡇⢸⠀⠀⠀⠀⠉⢉⠉⠉⠉⠉⠉⣌⡉⠀⠀⠀⠙⠃⣰⡇⢠⡀⢱⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠷⡾⠆⠀⠀⠀⠀⡏⠐⠀⠀⡄⠄⠈⡅⠀⠀⠀⠀⠀⠴⠤⠭⠥⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢀⣤⣴⡀⠀⠀⠐⣀⡄⣰⡈⠀⠶⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠈⠉⠉⠉⠀⠀⠰⢶⡟⠃⡇⠀⠒⠧⡶⠴⠆⠃⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣟⡋⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠧⠤⠤⠤⠧⠤⠤⠄⠀⠀⢀⡠⢀⡖⡔⡕⠦⣀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⡿⣿⣿⢿⣿⡿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣘⠉⠉⠁⠁⢠⣶⠏⠀⠀⠺⢻⣷⣤⣤⡀⠙⣿⣿⣶⡄⠀⠀⠀⡇⡇⢸⢸⠀⡇⡇⢸⢸⢸⡇
⠀⠀⠀⠀⠀⠙⢿⣿⣶⣦⣤⣴⣾⠇⠀⠀⠀⠀⣸⣿⣿⢿⡥⠀⠻⣿⠟⠃⠀⠀⠀⡇⡇⢸⢸⣾⣷⣇⢸⢸⢸⡇
⠀⠀⠀⢀⣄⣤⣤⣉⣉⣭⣤⣤⣄⠀⠀⠀⠀⠀⣉⣿⣿⣋⡀⢀⣠⣿⣄⡀⠀⠀⠀⡇⡇⢸⢸⣿⣿⣿⣿⢸⢸⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠁⠁⠉⠈⠉⠉⠉⠉⠈⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢻⢟⣿⠟⠻⠿⠿⢿⣿⣿⠟⠛⠛⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⡟⠁⢸⠟⠀⠀⠀⠀⠀⠀⠙⠲⢷⡀⠀⢿⣄⣙⣿⣄⡿⠉⠙⢿⡟⣿⣿⣿
⣿⣿⣿⢿⣿⣿⣽⢏⣿⠏⠸⣆⠞⠀⡰⠋⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⠁⠀⠀⠀⠹⣮⡟⢻
⠉⠉⣈⠓⠮⣿⣯⣼⣯⠀⠀⢻⢀⡼⠁⠤⠖⠒⠂⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠈⣷⠋
⠁⠀⠀⠀⠀⠘⣿⡏⠈⠙⠲⢼⡟⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠐⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢻⣿⣿⡆⠀⠀⠘⣧
⠀⠀⠀⣀⣀⣀⣸⣿⡄⠠⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣷⠀⠀⠀⢿
⠀⠐⠋⠉⠉⠉⠀⢿⡇⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⢿⣿⣿⣿⣿⡟⠋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡟⠀⠀⠘⣿⣿⣿⣿⠇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⢹⣇⠀⠀⣰⣾⣿⣿⣿⠿⠛⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣸⣦⡀
⠀⠀⠀⢘⣶⣦⣤⡄⠘⣿⡆⠀⠀⠀⠸⣿⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢰⣤⣸⣿⣿⣿⣦
⠀⠀⠀⠈⢿⣿⣧⠀⠀⢻⣷⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣿⣿⣿⣿⣶⣾⣿⣿⣏⢿⣿⣿
⠀⢤⠀⠀⢻⠁⣙⣳⣄⣸⣿⣤⣤⣶⡀⠘⢿⠀⠀⢸⡏⠀⣿⣿⣿⣿⣿⣿⠿⢿⡿⡿⠿⠛⠋⠹⣿⣿⣿⣿⣿
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠀⠀⠀⠀⠘⢩⡿⠯⠁⠀⠀⠀⠚⠋⠁⠀⠀⠐⣶⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⣛⢻⣏⡛⠛⠛⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡿⢛⡽⠂⣩⣿⠟⢿⡳⠯⢷⣦⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⣠⠞⠉⠀⠀⠀⠙⢧⣀⡈⠙⢛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇⠰⢻⣄⠀⢀⣤⣤⣄⠈⢻⣿⣷⠀⣿⣏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⡏⠀⡀⣿⠉⠀⠀⠠⠤⣄⠀⢸⣿⠿⢷⣽⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⠇⢼⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠀⣼⣿⣷⣶⣦⡀⠀⠀⠀⢀⣿⣿⣏⠀⠀⡿⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣧⣼⣿⡇⠛⠟⠛⠁⠀⠀⢀⣾⠟⣸⣿⡄⠀⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢻⣿⣿⣿⣦⣤⣤⣤⠶⢾⡿⢡⣾⣹⣿⠃⡜⣰⢠⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢠⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣵⣥⣾⣿⣿⣿⣿⣄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⣿⣿⢻⣿⣿⠿⠏⠀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠀⠀⠀⣾⣟⣵⣶⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣽⡷⢀⣀⣰⢠⣿⣿⣿⣿⣿⠛⣿⣿⣶⣿⣿⣿⣿⢿⣧⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣠⣉⣽⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⣿⡟⣸⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣖⣾⣿⡿⢡⡯⣹⣧⠿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠀⣼⡴⣥⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡗⣿
⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡟⠀⠀⣿⣷⢛⣿⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⢿⣼⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⠟⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣭⣿⣿⣿⣷⣿⠆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣘⠿⣋⠹⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣻⣵⣿⣿⣷⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣧⡀⠀⠀⢀⣼⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠀⣈⣿⣿⣿⣏⣁⠀⠉⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⣀⣤⢶⣿⠟⠋⠁⠀⠀⠈⠙⢿⣿⠶⣤⣄⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣴⠟⠋⠁⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⠙⠻⣦⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣴⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⣦⡀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣄⣠⡿⠁⠀⠀⠀⠀⠈⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠀⠀⠀⠀⠀⠈⢿⣄⣠⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠛⠛⠿⢷⣄⡀
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣄⣤⣤⣤⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⣯⡉⠙⢿⡍⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠷⣦⠀⠀⠀⠀⠀⠀⠀⠛⢻⡿⠛⢛⣿⠟⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠈⠻⣦⡀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢀⡴⠟⠁⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⠛⢶⣄⡀⠀⠀⠹⣇⠀⠀⠀⢀⣼⠇⠀⠀⢀⣠⡶⠛⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠈⠛⠻⠶⣦⣭⣿⣶⣾⣿⣥⣴⠶⠟⠛⠁⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⡾⠋⠉⠁⠈⠉⠛⢷⣄⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⢠⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠁⠀⠀⢀⣔⡀⣀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢘⠀⠀⠀⠀⠈⠀⠈⡁⠀⠀⠀⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡀⠀⣺⠞⠂⠀⠀⠁⣀⠀⠀⠀⠀⠈⠀⠀⡠⠂⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠄⠀⠘⠓⠀⠘⠛⡀⠀⠀⠀⠈⠰⠄⢀⠔⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⢠⠐⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠀⠤⡀⠀⠀⠀⢀⡄⠀⠎⡄⠀⡴⠀⠀⠘⠠⠀⠀⠀⠀⠀⠈⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠄⠀⠀⠀⠀⠀⠀⠞⡀⠀⠀⠀⠀⡠⣢⠄⠠⠀⠀⠀⠆
⠀⠀⠀⠀⢠⡄⠀⠀⠀⣤⠀⠓⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⣶⡤⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⠀⠀⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠁⠘⠂⠀⠀⢀⣤⠂⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢛⣃⣠⡙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡴⠛⣁⣤⢶⣦⡾⢋⠰⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢋⣴⠾⢛⡋⠙⣡⣾⣏⠻⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡾⢁⡾⠋⠀⠐⠋⠀⠙⢷⡈⢷⡀⠀⠀⠀⢸⣿⣿⣿⡿⢡⡿⢡⣾⣿⣧⣾⣿⣦⡙⣷⡘⣿⣿⣿⣿
⠀⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠁⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⢹⡄⠀⠀⠀⠀⠀⠀⢠⡏⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢻⡆⢿⣿⣿⣿⣿⣿⣿⡟⣸⡇⣼⣿⣿⣿
⠀⠀⠀⠀⠻⣄⠹⠦⢀⣴⣂⣀⡴⠟⣠⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⡌⠭⣬⣹⡿⠋⣿⠿⢋⣴⡟⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠃⢴⡏⠉⣉⣉⣤⠾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣾⡃⢴⣶⡶⠾⠛⣩⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣴⠋⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⡄
⠀⠀⠀⠀⢿⣄⣀⣼⠀⠀⠀⣀⠀⠀⠀⢀⣼⣿⡿⠟⠉⠛⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣹⣧⣀⣠⠇
⠀⠀⠀⠀⠀⠹⣟⠉⠀⣀⠠⠀⠄⣀⠀⢺⣿⡿⠄⠀⠀⠀⠠⠙⢻⣿⣿⡀⠀⠀⠀⠠⣴⠏⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣆⢂⢁⠀⠐⢀⡀⠐⢿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡀⠀⠐⠀⠋⠀⡀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⣿⣿⣿⣦⡼⣿⡆⠀⠳⣶⡾⠀⠀⣼⣿⡿⠋⠁⣠⣴⡦⠀⣀⡰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸⣿⠿⠟⠛⠙⠋⠼⠗⠄⠀⠀⠠⢾⣏⠉⠀⠐⠿⠿⠟⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢀⣠⡤⢄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠴⣿⣾⣷⣤⣤⣤⣄
⣾⡇⠀⠀⣡⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢈⣿⣿⣿⣿⣿⣿
⠹⢿⣶⣾⡷⠀⠀⠀⠀⠿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢸⣿⣿⣷⣶⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠻⣏⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣆⠀⠑⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⢰⡏⠀⢰⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡾⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⠀⢰⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢀⡇
⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠤⠈⠀⠀⠀⠀⣀⣀⣀⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢁⡾⠤⠒⠒⠚⠛⠛⠓⠒⠒⠤⢄⡀⠉
⠀⠀⠠⢤⡀⠀⠀⠀⠐⠢⠄⣸⢀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡐
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠉⠒⠤⠤⠒
⠀⠀⠀⠀⠀⢱⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠠⠐⠈⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠒⠁
⠀⠐⠓⠲⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠢⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢤⣀⣀⡄
⠀⠀⠀⠀⡠⠔⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠜
⠀⠀⢠⠊⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣀⠀⠄⠂⠁
⠀⢠⠇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⡄
⠀⠸⠈⠁⠉⠐⠠⢀⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠔⠂⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡈⠀⠈⠙⠲⠦⢤⣀⣀⣀⣀⣀⣀⣤⡤⠖⠋⠁⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⠤⠚⠩⡀⠀⠀⢀⠞⠉⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡴⠋⠁⠀⠀⠀⠀⠑⢄⡀⠏⠀⠀⠀⠈⠑⠒⠠⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⢛⠙⠛⢉⣉⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢤
⠀⠀⠀⢀⠤⠤⠜⠀⠀⠀⠱⠀⠀⠀⠀⠀⢀⣴⠋⢉⠚⢃⣴⠞⠛⠻⣿⣿⣍⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠛⠀⠀⠀⠦⠄⠠⠤⢀
⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⢠⡟⠁⠀⠀⣠⠟⠁⠀⠀⠀⠈⠹⣿⣷⡄⢻⡄⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡟⠁⠲⠄⣼⠏⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡆⢻⡄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⢨⠆
⠀⠘⠢⠀⢀⢀⡀⠀⠀⠀⠀⠀⠒⠀⣾⠁⠀⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢿⠀⠀⠀⠀⠉⠉⠠⠤⢄⠀⠀⣠⠄⠠⠄⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠑⠂⠐⠊⠉⠁⠀⠉⣿⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⡀⠀⠐⠀⠂⠀⠠⠀⠀⠀⣿⢀⡠⠄⣿⠀⠀⠀⠀⠀⠀⣿⣿⢦⡀⠀⠀⢾⠇⢸⡇⡠⠂⠐⢢⠀⠀⠀⠀⢐⠂⠀⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠁⠀⠀⠀⠀⢿⡀⠐⠀⢸⡇⠀⠀⠀⠀⢀⡟⣿⣇⠙⢦⣄⡛⠀⣼⠃⢻⡀⠀⣸⠁⠀⢀⠀⠈⠂⠦⡇⠀⠀⠂⡁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⢹⣿⡄⠀⢹⡗⠀⣿⠀⠀⠙⡋⠁⠀⠀⠨⠱⡀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡠⠷⠉⠻⣆⠀⠀⢸⠁⣄⢿⣷⡀⢸⠇⠀⢹⡄⠀⠀⠇⡀⠄⣐⠀⠠⠀⠂⠂⠑⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⡠⠤⢄⠀⠐⠀⠀⠀⠀⠀⠘⢷⣴⣷⣄⠙⢶⣄⣾⢸⣿⣯⣿⢷⡿⠀⠀⠈⣇⠀⠀⡆⠩⢲⠠⠤⣀⡀⠀⠀⠀⠀⠀⠁⠋⠲⢄⢄⡀
⠀⠀⠀⠀⠀⠸⡀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣶⣈⣙⣺⣉⣁⣤⣾⠇⠀⠀⠁⢻⡄⠀⡛⠛⠉⠚⣤⠀⠄⣉⣈⠀⠒⠀⠄⠄⣀⠀⠋⠁
⠀⠀⠀⢀⣶⠀⠙⣶⠁⠀⠀⠀⠀⣠⠔⠃⠠⡀⢸⠇⣨⣿⣿⡻⣿⣹⣭⠟⢁⡟⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣿⠈⠋⠈⢉⠿⠉⠒⣶⠠⣤⣄⠁⣙
⢀⠐⠀⡾⠘⣆⠀⣙⠠⡀⠀⢬⠀⣣⠀⠀⠀⣿⣿⣾⣁⣿⠶⣿⣿⢹⣟⣦⡾⠀⠀⠀⠀⠀⠀⠘⣧⠁⠀⠀⠀⣇⠀⠈⠹⠘⠀⠀⠂⠀⠦⠰⠀⠂⠭
⠀⠁⡼⠁⠀⠹⡆⣿⣰⠳⠄⠜⠠⣋⢄⠉⣉⢛⡩⠛⢦⣿⠾⣅⣿⢸⣵⠏⠀⠀⠀⠀⠀⠐⠀⠀⠘⢧⡀⠀⠀⣇⠀⠀⠀⢈⠀⠈⠀⠰⡦⢄⠀⠀⢀
⣀⣼⠃⠀⠀⠀⢻⣯⠘⠚⠢⠤⢀⡆⢸⡳⠤⢰⣷⢂⣸⡏⢳⣜⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⢦⡀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠉⠘⠓⠒⠃
⢳⡏⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⢷⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⣙⠷⣤⠤⠔⠚⠀⠀⠀⠀⠠⠀⠀⡀
⠘⣇⠀⠀⠀⠀⢠⠏⠐⠒⠂⠀⠀⠀⠠⡈⠈⠀⠀⠀⠉⠀⠈⠀⠀⠀⠈⠀⠈⠐⠂⠀⠁⠀⠂⠀⠀⠀⢀⠀⠀⢓⠂⢤⡞⠀⠀⠀⠀⠀⠀⡄⠉⠁
⠀⠈⠓⠶⠖⠚⠁⠀⠀⠁⠀⠀⠀⠀⠀⠙⠆⠠⣾⣶⡶⠀⠀⠀⡀⠀⠀⠐⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⣀⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⢤⡔⠀⢡⣾⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⣀⠀⠀⠀⢈⠛⠟⠛⡡⠚⣀⡀⡀⠈⠀⠄⠀⠀⠀⠈
⠀⠀⠀⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠰⠐⡀⠀⠀⠈⢈⠁⠀⠀⡄⠂⠰⠐⠀⠐⠈⠁
⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡆⠁⠁⠀⠈⠀⠉⣠⣀⡠⠄⠀⣀⣀⣀⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠺⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠾⠿⠿⠿⠿⠿⠿⠿⠶⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣶⣖⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⣘⡿⠿⠟⠿⠿⣿⣿⣻⡏⠁
⠀⠀⢠⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠞⡡⢋⣄⡠⠙⢄⠈⢣⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡜⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡀⠡⠆⣐⡀⠀⠀⠣⡏⠀⢑⡿⠋⢤⣿⠟⠀⠀⡧⢄⠀⠀⠀⠀⠀⠀⠐⡀⠂
⠀⢰⠃⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⣀⠀⠀⠀⠀⠠⡇⢠⡈⠙⠁⣀⣼⠒⠀⢀⡗⠋⠀⠀⠐⠒⢈⠒⠄⢠⢂
⠀⡏⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠄⠀⠀⠀⢇⠀⠙⣉⠉⣵⠃⠀⠀⡼⠁⠀⠀⠀⠀⣀⠼⡤⠔⠃
⢸⠁⠀⣀⣤⠀⡶⠶⠖⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡂⠄⠈⢦⣀⠈⠭⠄⢀⣠⠞⢁⠀⠠⢄⡂⠭⠚⠃
⠿⠒⠋⠁⣿⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠬⠙⡹⠋⠉⠡⠀⠀⠒⠈⠁
⠀⠀⠀⠀⢸⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⡇
⠀⠀⠀⠀⠘⡇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠐⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⠁⢠⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡇⠈⠦⠊⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠐⡉⡬⠁⠀⠠⠤⠴⣄⡉⠒⠤⠤⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⢀⡇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⡇⠀⠀⠀⠀⡠⠔⢻⠤⢄⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⢍⣙⣀⣈⡨⢥⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠹⣄⠠⠐⢉⣠⠖⠋⠀⠀⠑⡄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⢤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣤⣀⣀⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣀⣠⣤⣬⣥⣤⣄⣀⠀⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡦⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠾⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠈⠉⠛⠛⠻⠟⠛⠛⠉⠁⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣸⣿⣷⡆⣤⡀⠀⢠⡄⢀⠀⣀⡄⣤⣤⣠⠄⡀⢀⣠⢤⡄⣤⢤⡄⠀⢠⢄⢠⡄⣤⣄⣄⡄
⠀⠀⠀⠀⠀⠹⣿⠿⠻⠏⣿⣷⠄⣸⣧⢸⣾⢿⡇⣏⢳⣸⠂⢻⣾⣿⠂⡿⣿⢸⡇⠀⢸⠞⣸⣧⣿⣏⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠟⠀⠀⠋⠙⠂⠃⠊⠳⠋⠛⠛⠂⠚⠛⠛⠒⠃⠛⠘⠃⠀⠘⠀⠋⠙⠟⠟⠁⠉


⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇
⠀⠀⠠⢨⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡜⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠘⣼⠀⠀⡇
⠀⠀⠀⢠⠁⠀⠠⠖⠀⠀⠀⣠⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡞⠀⠀⠀⡼⢸⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠠⡚⠀⠁⠂⢂⡀⠀⠀⠀⠀⡰⠁⠀⠀⢠⠎⠀⠀⠀⣰⠁⢰⠀⠀⠀⠀⠀⢠⡍⠵⠀⠀⣇
⠀⢀⠀⠀⠠⠡⠉⠀⠚⠀⠀⠀⠠⢨⠇⠊⠀⠀⠀⠔⡁⠀⠀⠀⢠⠇⠀⡈⠀⠀⠀⠀⠀⢸⣿⠜⡄⠀⡟
⠀⡟⠀⠐⠀⠆⠀⠀⠼⠀⡀⠀⠀⠀⠂⡤⠄⠤⣢⠎⠀⠀⠀⢠⠏⠀⠀⣿⠀⠀⠀⠀⠠⠳⠶⠶⠿⠀⣇
⢠⠃⢀⡑⢒⡚⣇⢂⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⡠⣤⢀⡄⠤⠿⠤⠤⡤⠤⣐⡀⠀⠀⠀⠀⣿⢀⣀
⠈⠀⢸⡏⠈⠑⡀⠀⠀⠠⠄⠀⢀⡀⠠⠄⠔⠠⠂⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠠⣀⡄⠠⠄
⠀⡤⠾⠂⠀⠀⠙⠀⠠⠰⠋⠀⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠺⠦⠺⠶⠮⠲⠴⢶⠓
⠠⠄⠀⠀⡀⠀⢰⠀⠀⢢⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⣀⡖⠀⠀⠈⠃⡁⠀⠐⠂⠈
⢰⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠈⠀⠀⠀⠀⡆⠀⢓⡀
⠀⠀⠀⠀⠐⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠚⠀⠀⠀⠠⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠠⠅⠘⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⢈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⠀⢾⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠑⠾⠀⠀⠀⣇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠘⠆⠠⠄⠀⠈⠀⠀⠀⢠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣮⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢳⡀⠀⠀⡇⠀⢸⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠐⢄⠈⢢⠀⡀⢣⠰⠀⡇⠆⢏⠄⣠⠋⠀⣠⠖⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⡑⢄⠑⠌⠀⠂⠀⡇⠘⠌⡔⡡⡢⠞⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⠁⡂⠀⠀⠀⠀⣀⠀⠁⠀⠪⡈⢀⠠⢚⡡⠀⣀⡤⠴⠂
⠀⠀⠀⠀⠀⠰⠶⠦⠤⢤⣀⣀⡉⠑⠀⠀⡐⢡⠊⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢉⣀⣠⣄⣠⣤⡴⠒⠦⠤⠴⠒⠶⠄
⠀⣀⣀⣀⣀⡤⠴⠤⠤⠤⢄⡀⠀⠄⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣁⠀⣀⠴⠉⠉
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⣀⠨⠤⠤⠤⠤⠔⠊⠉⠀⠀⣀⠜⢁⡤⠂⠀⠐⠀⢀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠐⠄⠄⠀⣀⠐⠄⠀⠀⠵⢦⣒⠂⠀⠀⠐⠒⡴⠊⢉⠁⠂⠉⠀⠶⠀⣒⠍⣁⣀
⠀⠀⠒⠐⠒⠐⠀⠐⠂⠐⠒⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⣰⠚⣡⣀⣄⣖⡰⠀⠀⠂⠁⠚⠀⠘⠲⠗⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⢤⣭⠴⠞⠟⠋⣁⣉⠁⣀⣈⠛⠛⠛⠋⠉⠉⠓⠒⠛⠛⠚⠓
⠀⠀⠀⠠⠶⠒⠒⠛⠒⠛⠓⢾⣍⠉⠁⠀⠩⠤⠬⢒⣐⠈⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠰⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣟⣛⣿⣻⣿⣿⣶⣶⡆⢀⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⡯⠡⠏⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⢛⣵⣿⣿⣿⡟⠿⡧⠴⠶⠿⠛⠻⠟⠟⠋⠁⠀⠀⠘⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠿⠟⠿⣿⠀⢿⣾⣿⣿⠉⠀⠠⠤⠀⠠⠄⠀⠀⢀⣶⣿⣷⣤⣶⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⣟⣿⣶⡘⣦⣌⠛⢃⣀⢀⣠⣀⣄⣀⣀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢹⣿⡿⠟⠁⣨⣿⠁⠰⠿⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡏⠀⠀⠀⣠⣶⣿⢠⡇⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣰⣾⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣿⡗⣾⣿⢟⣀⣈⣠⣾⣿⣛⣟⡃⠿⣿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠆⠈⢻⣿⡟⠋⠉⠀⠍⠻⠉⠙⠻⢿⣷⣦⣄⡀⠀⠀⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡁⢰⢃⣼⡟⠀⠐⠲⠦⠄⠀⠀⠀⢊⠄⣩⡙⠟⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡷⢸⠿⡃⠀⠘⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⣡⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣷⡖⠛⢀⠀⠀⠀⠄⡀⠀⠀⠀⠄⠀⠀⠀⠒⠀⠀⠀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢛⣿⠀⢸⣿⡦⡀⠀⠀⡐⢦⣄⡀⠀⠀⢀⣠⣄⠀⠁⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠀⢸⣿⡅⣄⠀⠀⠀⠀⠉⠙⠛⠉⣈⣻⡏⠧⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⢸⣿⣗⡙⣺⣶⠶⠦⣀⡀⢀⣈⡿⢻⣿⣀⠀⠋⡄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢿⣿⢿⣿⣿⣿⣿⣿⢇⡠⠟⠋⣿⣿⣿⣾⣿⢏⠠⡈⡇⡔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡟⠾⢿⡧⠿⠟⢽⡿⣿⡿⠛⠙⠙⣷⣾⣿⣿⡷⣾⠀⠫⠐⣢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣦⠤⣤⠀⣌⠉⢮⡉⠁⠀⠈⠙⠛⠛⠻⠇⢿⡄⢆⣀⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⠛⠉⡰⠛⠂⠛⠀⠀⠘⠛⢶⠀⠀⠀⠀⠀⢺⡏⠉⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣇⣀⣄⣀⠀⠀⠀⠀⢠⡀⣷⣶⣤⣶⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠛⠻⠏⠙⡟⠣⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣦⠹⣆⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣈⠳⣌⢳⡘⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⠷⣝⠳⣝⠳⣍⡳⢦⣄⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡶⢡⡶⣌⡳⢬⡓⢮⣝⣓⠶⠶⢶⣒⣒⠲⣜⣎⣙⠿⢽⣿⠋⣹⡶⢋⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⡿⣰⢤⣝⡲⢭⣓⠦⠭⠭⠭⣝⣲⠽⣝⣳⢂⡿⣡⠞⣡⠞⣩⡴⢛⣡⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢣⡏⡞⡬⣍⣓⠮⣝⣛⣻⠭⣗⣪⡍⡼⣡⢏⡼⣣⠞⣡⠞⣡⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣼⡹⣻⣱⣨⣝⣒⠶⣒⣭⡇⡜⡼⣱⢃⡞⡴⢣⡞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⢻⡇⡇⣧⢧⢣⢳⢨⡭⡷⣼⢸⢱⢇⡏⡞⣸⢡⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⡟⣼⢱⢇⡏⡾⣸⢸⠑⣛⣛⡘⡜⡜⡎⡇⡇⣿⢼⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣱⢏⡼⣡⢏⡞⡼⢱⢃⣛⠭⢖⡲⠭⣝⣸⡼⣸⣹⢹⢸⢸
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣥⠞⣡⢏⡴⢣⠞⣼⣁⡫⢭⣒⡭⠭⢭⣝⡲⢭⣁⢣⢇⡏⣼⢸
⠀⠀⠀⠀⣤⠶⢛⣡⠶⢋⣵⠞⣡⠟⠴⠯⣌⣛⣶⢭⣑⣒⣛⣛⣓⠲⢭⡓⢮⣙⠺⢸⡵⣿⡀
⠀⠀⠀⠀⠀⠐⠛⠥⣔⡻⡤⢨⣽⣿⣌⠵⣿⡾⣽⣿⣒⣒⣒⠶⢬⣝⠲⣍⡳⣬⡻⠏⡼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠙⠳⢬⡙⢮⡙⢦⡙⢦⡇⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣦⠹⣦⡱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⡌⠻⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦


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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡴⢚⣩⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⡟⢠⡿⠛⠙⠻⣿⣦⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⣠⣄⠀⠹⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣧⠸⣇⠀⢀⡇⣽⣿⠆⠀⠀⠀⠀⠀⡄⠀⠘⢿⣿⠟⠀⢀⣿⠁
⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⢟⡿⠃⠀⠀⠀⠀⠀⠈⢿⣷⣌⣉⣩⣴⣿⠟⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠛⠄⢻⢰⠟⣀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⣉⣉⠛⢿⣦
⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢶⡤⠄⣠⣤⣆⢌⠜⢋⠁⠀⠀⠀⠀⢀⣿⡇⢠⡿⢛⠻⣿⣮⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣿⣿⣿⣞⢿⣷⡇⠀⠀⠀⠀⠀⢠⣴⠊⡀⠙⠉⣁⣚⠙⠃⠀⠀⠀⠀⠀⣿⣇⠘⠧⠼⠃⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣵⣿⠟⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃⣧⠑⣦⠉⠋⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣤⢄⡀
⠀⠀⠀⠀⢀⣶⣿⣟⣿⡹⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣉⡉⠙⠳⣄⠀⠀⠀⠀⠀⢠⢿⣿⣿⡿⢿⣿⣾⣢
⠀⠀⠀⠀⢸⣿⠷⣻⣿⣿⣿⡴⣿⡷⠀⠀⠀⠀⠀⢰⣿⡇⢠⢞⣉⡙⢷⡀⠸⠀⠀⠀⠀⠀⣿⣾⡷⠁⠀⠈⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣏⣾⣿⡿⣿⣙⣿⡟⠀⠀⠀⠀⠀⠸⣿⣇⠈⠪⠭⠃⣼⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣄⢀⣀⣼⣇⢯⠇
⠀⠀⠀⠀⠀⠙⢿⣷⣿⣼⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⡾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣻⣿⢿⣿⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⢠⢄⣞⠕⠛⠀⠀⠳⡝⣆⠀⠀⠀⠀⠀⢠⡞⢁⣾⠿⣛⠿⣝⢷⣄⠀⠀⠀⠀⠀⣾⡿⣫⣽⣶⣯⡹⣿⣿⡄
⠀⠀⠀⠀⣾⢾⢼⠀⠀⠀⠀⢸⣻⣿⠀⠀⠀⠀⠀⢸⡇⢸⣏⣾⣿⣷⢸⠈⣿⠀⠀⠀⠀⢸⣿⢳⣿⣯⡂⢈⡿⣸⣿⣷
⠀⠀⠀⠀⠸⣞⢞⠦⣀⣀⡤⣫⢣⡏⠀⠀⠀⠀⠀⠘⣷⡄⠙⠻⠛⠋⠊⣼⡟⠀⠀⠀⠀⠈⢿⡞⣿⣷⣽⣯⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠳⢭⣟⣛⣯⠷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣶⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⢮⣛⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣛⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠔⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣾⣿⡀⢰⢁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆⠀⣰⣾⣿⣳⣿⣿⣿⣿⣿⣿⣻⣿⣉⣻⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣚⡂⠠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⠉⢹⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⣈⡉⠁⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⢡⡄⠀⢀⣀⣠⣤⣶⣾⣿⡿⢱⠄⢀⠀⣾⠠⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡄⠀⠀⢀⡇⠀⠰⠀⣀⠀⠀⠒⣀⠲⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⡇⠀⢠⣿⡀⠚⡍⠛⠢
⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⣴⣷⠔⠉⢛⡥⡤⣀⠀⣛⠒⠋⠉⡜⠛⣒⠿⠛⠋⠉⠌⡁⠩⠩⣠⠄⡈⠉⡓⠴⠸⢿⡇⠛⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⠧⠘⠉⠁⢀⠔⡅⢀⠈⠁⠀⠁⠀⠸⠓⠒⠉⠀⢀⠄⠘⠆⢀⣿⠀⠀⡀⠉⠈⠑⡷⢤⠀⠀⠈⡖⠉
⠀⠀⠀⠀⠀⠄⠀⠙⠤⠊⠁⠀⡀⠂⠀⣼⢰⡄⠘⠃⠀⠀⠀⢠⡀⠂⢀⠀⠀⣿⢸⣿⡄⣀⣀⠀⢀⠀⠂⠀⡆⠀⠀⠇
⠀⠀⡆⠀⠂⠀⠀⠠⢀⣀⠀⢀⣤⣤⡤⣭⡁⠀⠰⠆⠀⣠⡄⣾⡇⠲⠵⠄⠜⠛⠛⠉⠉⠀⠄⠈⠀⠀⠈⠀⡇
⠀⠀⠁⠨⠁⠠⢈⠀⢀⠀⣀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠀⠒⠠⠅⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⠿⡷⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠃⢂⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠈⠿⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣥⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⢠⠀⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⡆⠀⠀⠀⠀⠀⠀⢠⠠
⠀⠀⠀⠀⣴⡦⠀⠀⢀⠀⢶⠀⠐⠀⠿⠀⠈⠀⣿⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠁⠰⣄⠀⠃⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠧⠀⠀⠀⡀⠀⠔⠀⠤⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠩⡀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠸⢀⣀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⡀⠀⠀⠔⠈⠁⠂⠀⠀⠀⠈⠁⠀⠀⢀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠭⠄⡀⠀⠀⠁⠐⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠷⠀⣕⠦⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡌⢢⣴⣌⠀⢁⣡
⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡞⠳⢄⡀⠰
⠀⠀⠀⠀⣉⠉⡀⣉⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⢀⡆⠀⢂⡆⣄⡒⢆⠀⠀⠈⠠⣭⠓⣒⣃⠔⠂
⠀⠀⣀⡴⠃⣿⠇⢿⣮⠓⢶⠤⠀⠀⠀⠀⠈⠀⠉⠀⠀⡠⢂⣀⣤⣄⣀⠌⠏⠃⠄⠀⠀⠀⣀⣀⡀⣠⡀
⠀⠈⠙⠀⠀⢹⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡆⢀⠊⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤
⠀⠀⢀⠀⡀⢘⡆⠀⢿⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠀⠀⠙⢁⠀⠀⠀⢈⠄⠀⠀⠀⠠⣎⡌⠧
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡲⡻⡿⢆
⠀⠀⠀⠀⢀⢀⣾⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢊⠔⢁⠁⠇⢡⠡
⠀⠀⠀⣰⠉⠀⠉⠈⠉⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠓⠠⡌⠀⣈⠀⣀⡤⡴
⠀⢀⠜⡦⠀⣀⠀⠀⠀⡀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣸⡀
⠀⠀⠈⠐⠒⠀⠒⠂⠀⠘⠒⠀⠀⠀⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠐⠂⣤⣬⠒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠁⠴⢠⠖⣀⠀⠠⠀⢸⡛⢲⣬⣏⡱⡏⢰⣃⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣿⣇⠀⠀⢰⣯⣿⣿⣧⠀⠀⠙⡗⠬⢿⣽⠟⠕⠊⠡⠞⣚⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠿⣿⣆⠘⣖⢿⣽⣿⣇⠀⠀⢀⣿⣞⣃⣈⠃⠾⠤⠀⢀⣝⢚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠎⠀⠉⡴⢿⡷⣄⠁⠈⢉⣉⡀⠴⠋⠉⠉⠑⢍⣻⡇⣦⢠⡄⢱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢔⣋⡩⢄⡟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢮⡆⠊⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠂⣤⠀⠀⣠⠈⠀⠀⣀⣤⠤⣤⣸⠀⠀⠛⠀⠀⡼⢲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠑⠬⣉⠉⠃⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠖⠀⣀⣀⣀⣀⡑
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠀⣴⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠸⠀⡼⠐⣆⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣿⢞⣳⡆⠘⠈⠑⢠⣀⠀⠀⠀⠘⠿⡷⠒⠒⠒⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⡁⣠⠸⣭⡓⠐⢦⡄⠀⠙⡇⠐⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠛⠙⡆⠙⡷⢖⠋⣴⠀⠤⠀⠀⠁⠀⠈⠙⠓⠦⠄⢤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠐⣤⠋⠈⢃⡌⢀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠒⡰⢣⣟⣈⣁⣀⠁⢀⠈⠡⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣀⣉⢘⡏⢉⡁⠓⠄⠉⢐⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣸⠃⣤⠀⠔⢃⠼⣏⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡶⠶⣶⣶⣶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢿⣿⡅⠀⢻⣿⡆⠀⢿⣿⡃⠀⣼⣿⡟⠛⢳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣿⣧⣀⠈⠻⣿⣦⡀⠹⢿⣄⠈⢿⣧⠀⢿⣿⠀⢠⣾⣿⠟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣀⠈⠙⠻⠷⢶⣤⠍⣙⡷⠶⢝⣛⣂⣙⣷⣌⢿⡄⢸⣿⠃⠀⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣏⠀⠉⠛⠛⢛⡛⣻⡷⠋⣴⣿⠋⠀⣾⣿⠋⠁⣠⣤⣬⣤⣿⡄⢿⠀⢸⣿⠋⠁⠀⣹⣦⡀
⠀⠀⠀⠀⢀⡾⠻⠿⠿⠿⠟⢻⠟⣰⣿⠀⠀⠿⣿⣦⣀⡙⠿⣦⡀⢿⣿⠁⢀⣤⣭⣘⠇⢸⡇⢀⣴⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣷⣤⣤⣶⢶⢣⣿⠀⠹⢿⣷⡶⠶⡶⢖⣭⠿⠓⣊⣝⣦⣙⣆⢸⣿⠛⠛⠳⣼⡇⣼⡟⠁⠀⣀⠈⢷⡀
⠀⠀⠀⣼⠏⠉⢉⣡⣴⡇⢸⣿⣦⣤⣤⠶⣲⠏⠰⠿⠷⢤⣈⡻⢯⡀⢨⣍⡂⢿⠀⣴⣶⣌⠣⡿⢀⣴⣿⣿⣿⣿⣧
⠀⠀⢠⣿⣷⡿⠟⢋⢸⡇⠀⠙⠛⣉⣴⢣⣿⣦⠶⣲⠟⢁⠲⢿⡓⠛⠜⣿⠛⢾⢰⡟⠉⠙⢷⡃⣾⠟⠁⠀⠀⠉⢿⡄
⠀⠀⢸⡏⢀⣠⡶⢻⠘⣿⣶⣶⡾⢋⣾⠀⢉⡴⠡⠟⣿⡁⠘⣿⡟⠜⣷⡹⢰⣦⢹⢃⣴⣶⡄⢿⠃⣴⣾⣿⣿⣷⣼⡇
⠀⠀⢸⣿⠿⠋⣡⢺⡄⠈⠉⠉⣰⢏⢻⣷⡟⣸⣤⡎⠿⣤⣄⡸⢁⡄⠃⢡⠟⠻⣏⡾⠋⠙⢿⢈⡾⠋⠁⠀⠈⠙⣿⡇
⠀⠀⢸⣇⣠⣾⠏⡌⢿⣶⣶⣾⠏⡼⣄⠉⢰⣇⠻⠃⣤⣻⣍⠡⠽⣷⢏⣪⣶⢠⢏⣤⣶⣆⢸⢋⣤⣶⣿⣿⣷⡄⢸⡇
⠀⠀⠘⣿⠟⠁⣼⢻⡄⠙⠛⠁⣰⡇⡙⣷⣿⠇⣗⢤⣽⣌⠫⢥⣖⣉⠉⢩⣯⠴⠋⠉⢹⡿⣼⠟⠉⠀⠀⠈⠻⣿⣾⠃
⠀⠀⠀⢻⣤⣾⠏⢠⡻⣶⣤⣶⣿⠃⣿⢦⣉⠀⣸⣎⠪⣭⠤⣴⣛⡫⠶⠷⠶⢿⣿⡆⣸⣣⡴⠾⠿⢿⣿⣦⠀⣿⡟
⠀⠀⠀⠈⢿⡏⢠⣿⢱⣌⠙⠛⠁⢀⣿⠀⣯⢛⠿⠟⠀⠘⣷⣦⠉⠛⣿⣷⠀⣰⢟⣐⣩⣥⣤⣤⡀⠀⠹⣿⢠⡿⠁
⠀⠀⠀⠀⠈⢷⣿⠇⢸⡏⠳⣶⣶⣾⡿⠀⢸⣧⠻⣟⠚⣛⣛⣉⣤⠶⠛⣥⣾⣭⣭⣍⣉⠉⠙⢿⣿⡆⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣾⡇⢻⣦⣉⡉⠀⢀⣼⣿⠀⠘⣷⡌⠛⣶⣌⠙⠳⣶⣤⡀⠉⠻⣿⣷⠀⢸⣿⢃⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣅⣸⡇⢸⡝⠛⠿⠟⠋⠀⣠⣿⣿⠀⠈⣿⣷⠀⠈⣿⣿⠀⠀⣿⡟⢀⣾⣧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⣈⣿⠈⢷⡒⢶⠾⠿⠛⠁⣀⣼⣿⠟⠀⣰⡿⠏⣠⡾⢋⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣼⣷⡌⢻⣏⠙⢿⣉⠩⣴⣖⠛⢭⣤⣉⣻⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⠾⠿⠷⠾⠿⠿⠛⠛⠉⠁

⠋⢀⠞⠀⠀⠺⣿⡷⠿⠿⠃⠀⠀⠀⠾⡁⠀⠀⢀⣀⣇⠀⠀⢿⠿⠇⠀⠀⠀⠶⡁⠀⠀⣠
⣶⡏⣀⠓⠄⡀⢉⠩⠀⠀⠀⠀⢠⡀⠈⢉⠀⢸⣿⣏⣽⣷⡮⠀⠀⠀⠀⢠⡀⠈⢉⠁⠀⠉⠁⠀⣴⡆
⣷⣿⡂⠤⣍⠱⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢁⣿⣿⣿⣿⣿⡷⢀⣄⠰⠆⠉⠀⠐⠢⠄⠀⠐⠀⠀⠀⠀⣀
⠋⠀⠐⠿⢩⣤⣴⣾⣿⣷⣶⣄⠀⠠⠀⠀⠀⢹⣿⣿⣿⣟⠀⠟⠻⠷⢿⣶⣦⣄⠀⠀⠈⠀⠀⠄⠀⠀⠙⠈
⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣸⡞⢡⢃⠈⣿⣿⣿⣿⣷⣦⣤⣀⠀⠈⠉⠙⠛⠦⢠⠄⠈⣤⣤⠂⠀⢠⣶⣶⣄
⠃⡄⢰⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿⠄⢀⣿⠿⣿⣯⣈⣉⡙⠟⠷⢶⣦⣤⣀⡀⣰⠀⠀⢛⣫⣄⠀⠻⢿⣿⣿
⠐⠈⠀⢈⣛⣿⣿⣿⠛⡟⠛⠿⠿⠿⠿⢃⣷⣿⣿⡿⠀⠠⠈⠙⣆⡀⠀⠀⡁⠈⣿⡿⠙⠒⠈⢮⡹⣧⣀⣿⡯⠁⢀
⡀⠓⠠⣿⠟⠉⠻⠿⠀⠙⢀⡵⢶⣾⣿⣿⣿⣿⣿⣿⣦⣄⡀⠈⠉⠉⠀⠀⠉⠛⠛⢠⣿⠈⢐⣾⣿⣾⣯⣷⠆⠀⠠
⠏⠘⠂⠀⠀⢀⠀⣠⠶⠒⢺⢣⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⣶⣿⣡⣾⡿⢟⡝⡿⠹⣿⣎⠁⣠
⣀⢀⡀⠀⢃⣸⣿⣿⣄⣀⢆⠚⣁⠈⠰⠈⠹⢿⣿⠉⠋⠙⠛⠿⣿⡿⢿⡿⣷⣾⣿⣿⡬⡆⢠⠉⣴⣷⣦⠙⢿⡆⢻
⠛⠀⢁⣆⠃⠘⢿⡿⣩⣴⡶⠆⠉⠀⠀⠀⠀⣾⣿⣷⣦⣤⣤⣀⠀⠀⣿⣷⣾⣿⣿⣿⣧⡀⠀⡾⠙⢟⣏⣶⠒⣤⣼
⠀⠀⠈⣿⠀⠈⠐⠹⣿⠛⠁⣀⠸⠁⡠⠓⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢂⠎⢀⡈⠂⠻⠿⠉⠋⠉
⣶⣿⣿⣿⣿⣶⣾⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠚⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⣶⡎⣿⠹⢶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡶⠀⢀⡴⠾⠋⢻⣉⡀⠈⠛⠛⠚⠉⠀⣹⡟⠛⣦⡀⣤⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡘⡇⠠⣏⠀⠀⠀⠀⠈⠉⠓⠒⠒⠛⠛⠛⠁⠀⠀⠀⠙⣦⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢿⠤⢤⣤⣤⡀⠀⠀⠀⣀⡀⠀⣠⣤⠀⢀⣠⣴⠏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠈⠉⠉⠛⠛⠛⠛⠛⠒⠋⠉⢩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⢀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⡴⠶⠛⠁⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠇⠀⠀⢰⡏⢬⡄⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⠀⠀⠀⣉⣭⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⡟⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣀⡴⠞⡻⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⣿⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⠀⠀⠀⢠⡾⢋⡴⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢻⣄⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⣾⠀⡾⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⢹⣄⠻⣄⠉⠛⢶⣤⡀⢸⡇⢸⡇⠸⣇⠀⠀⢻⡄⣷⠈⣷⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠲⣤⡀⠀⠈⠻⢦⣈⠙⠲⣄⠈⢿⣾⠁⢸⡇⠀⢿⡄⢀⣼⠃⣿⠀⢹⡆⠀⠀⣀⡴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⢤⡙⠳⢤⣀⠀⠙⣦⠀⢻⣆⣸⣿⣤⣼⣧⣤⣼⣿⡟⠉⣠⠟⢀⣾⣃⣤⠞⢩⡖⣿⠋
⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠉⠓⢶⣿⡶⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⢿⣤⣾⠟⠋⣠⡴⢋⡼⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣶
⠾⠛⠳⢦⡈⣷⣄⠀⠀⠀⠀⢀⣀⣹⣆⠀⠻⣦⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⢠⡞⠁⠀⢀⣴⠞⢛⣉⣉⣉⠙⠛⣩⡿
⠀⠀⠀⠈⢷⡹⣍⡛⠿⠿⠛⠋⠉⠙⠻⢷⣴⠟⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⣿⣀⣀⣴⠟⣡⠞⢩⡴⠶⠶⠶⠟⠋
⠀⠀⠀⠀⠈⠻⣦⣉⡛⠚⠛⠛⠛⠲⣦⣼⠏⠀⣠⠶⠚⠋⠻⠿⢷⣴⣿⠀⠀⣾⣆⣤⣶⣟⣿⠓⠶⣦⣄⠹⣏⠉⠉⣁⡴⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣦⣼⠇⠀⣀⠁⢠⣶⣿⣿⣿⣿⣿⣿⠂⠀⢻⣿⣿⣿⣿⢿⣿⣦⠈⠙⣆⢻⡟⠛⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⣿⠀⠀⠻⠷⠿⢿⣷⣯⣻⡼⠟⢿⠀⠀⢸⡏⠿⣿⣧⣽⣽⠿⠳⠞⠃⠈⣿⠻⠿⢧⣤⣀⣀
⢀⣀⣠⣤⣶⣒⣚⣛⣉⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⢸⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣈⣉⣙⣛⣿⣶⣦⣤⣤
⠈⠙⠛⠛⠻⠿⠧⣤⣤⣀⣀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣏⠀⠀⠀⣿⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⣀⣀⣤⡤⠴⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠿⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⠟⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡶⠚⠛⠛⣷⡀⠀⠀⠀⠀⠀⣠⣀⣀⣠⣴⣤⣤⢶⣤⣄⣤⣄⠀⠀⠀⠀⠀⢠⣿⣄⠀⠻⢤⣤⣤⣤⣄⣀
⠀⠀⠀⣀⣀⣀⣠⣼⠇⣰⠏⣠⡤⠴⣾⠟⠳⣄⠀⠀⠀⠀⠟⠛⠛⢻⣿⠿⠷⠛⢻⡟⠛⠛⠀⠀⠀⠀⣠⣿⡀⠉⠳⠶⠤⠶⠶⠶⢬⣙⠳⣄
⠀⣴⢟⡭⠥⠤⡤⠴⠞⢁⡾⠃⠀⣰⠏⠀⢀⣿⢦⣄⠀⠀⠀⠀⠀⠀⠙⠷⣶⠶⠟⠁⠀⠀⠀⠀⣠⣾⡃⠈⢿⣦⣤⣠⣤⠴⠶⠾⠶⣌⢣⣹⡆
⢸⣷⡶⠟⠛⠓⠶⠶⠟⠋⠀⠀⣰⠏⠀⣴⠟⠁⣀⡽⠷⣤⣀⡀⣂⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣅⠈⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢦⣽⣟⣳⣶
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠞⢁⣴⢾⡟⠀⢠⡟⠉⣙⣿⡿⠿⣶⡶⠶⢾⠟⢻⡇⢀⡟⠉⠻⣦⣀⠙⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣼⣥⠞⠋⠀⠘⣷⠀⣾⠀⣸⠏⠈⣿⠀⢹⠀⠀⣿⡄⠈⢧⠈⢷⣄⠀⠀⠙⠳⢦⣉⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇⢸⡆⣿⠀⠀⢹⡆⢸⠀⢰⡟⢳⣄⠈⠳⢤⡉⢷⡄⠀⠀⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⡼⢀⡿⠀⠀⠘⣧⢸⠀⣾⠁⠀⠉⠛⠷⣦⠹⡄⢿⠀⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⡜⢁⡾⠃⠀⠀⠀⢻⣼⢰⡇⠀⠀⠀⠀⠀⢸⡆⠇⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⣯⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣰⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣗⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⡀⠀⢸⠃⠀⡄⠀⢐
⠀⠀⠀⠀⣀⡀⠐⠂⢀⣼⣿⡆⠈⠀⢠⠏⣀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⣥⠾⢛⠁⣠⢀⡬⠈⠁⠖
⠀⠀⠀⠈⠀⠀⣠⡿⢷⣶⣿⣝⣋⣉⣀⠀⠚⠁⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⢋⣄
⠀⠀⠸⠀⠀⠀⠀⠀⠸⢻⡿⠷⠀⠉⠁⠈⠉⢷⢄⡆⠙⢉⣭⣭⡭⠽⠿⢓⣒⣠⣤⡄⣶⠶⣦⠉
⠀⠀⠀⠈⠲⣶⠖⠂⠈⠉⠀⠀⠀⠀⠀⠀⢈⠃⢋⣀⢠⡍⣴⡶⠖⢻⡆⣸⡏⣿⣀⡀⣿⣠⡿
⠀⠀⠀⠀⠀⠈⠁⢀⡼⣿⣅⣀⣀⣠⠤⠐⠚⠀⢸⣿⣼⡇⢸⣧⡆⠸⣷⣿⠁⣿⣉⡅⣿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⣀⣀⣰⠀⢾⡏⢿⡇⢸⣧⢶⣀⣻⣯⠀⢻⡏⢱⣾⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠒⠀⠉⠈⠀⠀⢿⠀⢸⡷⣮⣵⢾⡇⢸⠟⣯⣁⠀⢸⡇⢸⣿⣷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⣸⡄⣿⠰⣶⣿⠈⣿⣿⠀⣿⣭⡄⠘⠷⢟⣘⣧⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠒⠒⠒⠚⠋⢹⡇⠻⡶⠟⠻⣂⣛⣫⠤⠅⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠲⠤⠤⠤⠄⠒⠊⢳⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣄⣀⣤⣤⢠⣤⠀⣠⣄⣠⣀⣤⢤⣤⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣇⠄⠀⠀⠀⠀⠟⠟⠛⠳⠸⠚⠀⠳⠐⠛⠛⠆⠘⠘⠟⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⡀⠀⠀⠸⡆⢸⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠨⠀⠀⠀⠛⠿⠘⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⡄⢿
⠀⠀⠀⠀⠇⢀⠰⠀⠀⢀⣸⠋⠀⠀⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠆⠀⢐⠀⣾⠀⢾⡇⠀⡇⢲
⠀⠀⠀⠀⠛⠈⠓⢀⠚⠻⠿⠀⠀⠀⠄⠀⠂⠉⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠘⠓⣾⡿⢹⢲⣼⡇⡆⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠁⠸⣼⢸⣿⢹⣡⡇⡄
⠀⠀⠀⠀⠀⠀⣀⡀⣀⡲⡄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠠⠴⠀⠄⠠⠐⠚⠂⠀⠀⠀⢀⡀⠀⢀⡹⠎⠋⠈⠋⠾⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠁⠈⠉⠉⠠⠀⠖⢊⠁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⣶⣿⣷⡶⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡖⠫⠁⠂⠃⠀⠁⠸⣼⡼⠀⠁⠀⠃⠂⠭⠓⣦⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣋⣙⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⢀⣟⣉⣻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢄⠀⠙⠋⠁⢢⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠊⠈⠛⠁⢀⠜⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠑⠁⠀⠑⢀⠀⠀⠀⢣⠀⠀⠀⡀⠀⢸⠀⡀⡀⠀⠀⢠⠃⠀⠀⢀⠐⠁⠀⠐⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡦⢄⡀⠀⠀⠀⠑⢄⠀⢀⠱⡈⠀⠀⠀⢸⠀⠀⠁⠈⡰⢅⠀⢀⠄⠁⠀⠀⠀⣀⠤⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⠆⠀⠈⠒⠤⡀⠀⢀⠙⢄⠀⠑⡄⠀⠀⢸⠀⠀⠀⡔⠁⢀⠜⢁⠀⠀⡠⠔⠊⠀⠀⢶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠴⠀⠀⠀⠀⠀⠀⠈⡑⠢⢄⠀⠑⠄⠘⡄⠀⢸⠀⠀⡜⠀⠔⠁⢀⠤⠒⡉⠀⠀⠀⠀⠀⠀⠰⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠉⠒⢤⡀⠘⣦⣼⣴⡞⠐⣠⠔⠊⠁⠀⠀⠤⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣈⣻⣿⠟⢿⣿⣋⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⡶⠿⢷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⠿⣿⣷⣿⡿⢧⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⠿⠷⡆
⠀⠀⠀⠀⠀⠀⠲⠶⡞⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡠⠔⠉⠀⣼⠛⢹⠙⢻⡄⠈⠑⠠⣀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠘⡶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⣀⠴⢖⠁⠀⠄⢀⡾⠁⠀⢸⠀⠀⠹⣆⠐⠀⠀⢑⠶⢄⡀⠀⠀⠀⠀⠀⠘⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠂⠀⢀⡠⠔⠊⠀⠀⠈⠰⣀⢀⡾⠁⠠⠀⢸⠀⠠⠀⠹⣆⢀⡰⠌⠀⠀⠈⠓⠤⣀⠀⢀⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⡟⠠⠀⡀⠀⢸⠀⠀⣀⠠⠘⣧⡀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⠷⡐⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠐⡰⠿⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠙⢦⡃⢀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⣣⠞⠁⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣅⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡒⠬⣐⣠⠆⠀⠀⢀⣸⣀⠀⠀⠀⢦⣐⡨⠴⣺⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣭⣭⡅⠀⠀⠈⠉⠙⠓⠒⣾⡖⠒⠛⠋⠉⠀⠀⠀⣭⣭⡝⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢳⡄⠀⢠⣲⣲⣲⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢹⣄⠀⣿⣩⣹⡇
⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠥⠤⠤⠤⠴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧⠤⠤⠤⠤⠴⠽⠦⠿⠿⠿⠧⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⣀⣀⠀⠉⠉⠁⠀⠀⠀⠀⡉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠒⢤⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣠⠊⠀⠀⡠⠐⠀⠀⠀⠀⠀⠢⢀⠀⠈⠉⠒⠢⢄⡈⠢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡰⠁⠀⡴⠉⠀⠀⠀⠀⠀⣀⠀⠂⢄⡈⠐⠢⠄⣀⣀⠈⠢⠈⠈⠆
⠀⠀⠀⠀⣀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠰⠁⠀⠊⢀⠄⢀⠔⠁⡠⠊⠀⠑⠢⣀⠈⠐⠢⠄⣀⠀⠈⠁⠈⠂⠸
⠀⠀⢠⠤⣣⠹⡄⢂⡐⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠐⢊⡡⠖⢁⡠⠊⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⣀⡉⠉⠉⠁⠀⢀
⠀⠀⢀⡵⡌⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣊⡩⠄⠒⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠉⡋⡙⠏⡀
⠀⠀⠀⠑⠤⣀⠀⣀⠔⠉⠑⢄⠀⠀⠀⠀⠰⠹⡨⢁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠴⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡧⠊⢀⡇
⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⣄⠈⠚⢆⠀⠀⡔⠒⡄⠀⠀⠐⠤⣠⠆⠀⠀⠀⡆⢐⠀⠀⠀⡌⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠑⠢⡜⠀⠐⣠⡨⠢⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠞⠂⡅⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⠲⢅⣀⠀⠈⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠁⠀⠀⠘⢮⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠈⠁⠒⠒⠀⠀⠀⠒⠂⠉⠀⢠⠑⢄⠀⠀⠀⠸⡀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣬⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⢀⣀⠠⢤⠤⣀⣀⣀⠠⠔⠁⠀⠀⠑⢄⠀⠀⢈⡖⣈⣽⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⠏⡿⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⡟⡏⢇⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠑⠤⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠣⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⡣⠀⠀⡀⠔⠊⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢠⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡝⣠⡏⡀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⢠⣄
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢠⣤⣶⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⠀⡀⠀⠸⡿⠀⠀⠀⠀⠀⠀⣤⣼⣁⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠴⣀⣀
⠀⠀⢋⣾⠏⠹⣆⠀⢀⡄⠀⠀⠀⠀⠀⢸⢿⣿⡏⠋⠁⢠⠄⠀⠀⠀⠀⠀⠀⠀⣿⣷⣆⣧⣤⣼⣧⡀⠀⠀⠀⠀⠴⢶⣦⣿⢿⡞⣿⣻⡞⠟
⠀⠀⢸⠟⣧⠀⠈⢩⡿⠦⠀⠀⠀⠀⠀⠈⢸⣿⡇⠀⠀⢸⠗⠂⠀⠀⠀⠀⠀⠀⠿⠿⠛⣛⠿⠟⠽⠅⠀⠀⠀⠀⠈⣹⡿⠛⠛⢿⡟⠉
⠀⢠⡏⠀⢸⡄⠚⠋⡽⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣿⣿⣿⣆

⠀⠀⢀⠠⠆⠐⠂⠲⢄⡀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⣠⠉⡐⠈⠀⠀⠁⢊⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠙⢷⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣤⣀⡀⠀⠀⠀⠸⠋⠉⠹⡄⢀⣀
⠀⡧⠠⠀⠀⣿⡷⠀⠀⠀⡧⠀⠀⠀⢠⠀⡻⠐⢺⣿⣷⠤⣄⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⡿⠿⠿⠗⠀⠀⠀⠃⠀⠀⠠⠘⡅⠀⠱⣄⠞⠉⢱
⠀⠸⡀⠡⣴⠁⠘⡆⠌⠰⠇⠀⠀⠀⠀⠈⠁⠚⢺⣿⣿⠖⠁⠚⠀⠀⠀⠀⠀⠀⠀⣀⢠⣿⣇⣠⠀⡀⠀⠀⠀⠈⠐⠊⠀⠀⠘⢄⠀⠀⠀⢀⠜
⠀⠀⠘⣶⡅⢠⡀⢼⣞⠃⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⣚⣉⣈⣿⣧⣈⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠈⠁
⠀⠀⠘⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠚⠚⠓⠒⠒⠒⠒⠒⠒⠒⠚⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⡄
⠀⠀⢻⣽⠀⠀⠀⠌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣴⣦⠻⣿⣀⡀
⠀⠀⢸⠿⢿⡄⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⡅⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⣨⣷⠯⢻⣻⠟⠃
⠀⠀⣿⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣄⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⣴⣰⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢟⠝⡪⠕⠊⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⢿⡀⠈⠁⠀⢠⡄⣿⡅⠀⠀⠀⠀⠠⡿⣿⠟⠛⠀⠛⠓⠀⠀⠀⠀⠀⠀⣻⠟⡵⠊⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠠⠞⠁⠸⡇⠀⠀⢠⣼⣷⣿⣧⠀⠀⠀⠰⢿⣿⣿⡷⣚⣛⣛⣿⣿⡷⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⡟⠛⠛⠟
⠀⠀⣠⡀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⢀⡀⠀⠀⠘⣿⣿⠖⠂⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠹⠦⢿⡶⠶⠖⠀⠀⠀⠠⠀⠀⠀⠰⠞⣶⣮⠁⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢸⣿⡾⢛⠛⢿⣿⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠛⣿⡘⠚⠛⠛⠛⠓⠃⠀⠀⠀⣰⣿⡉⠳⠤⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠈⠿⣧⣈⣈⣾⠿⠁⠀⠀⠀⠀⠀⠀⣿⣧⣴⣿⣷⣿⡇
⠀⠀⣼⠏⢳⡀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢸⡇⢰⣶⣶⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⠀⣉⣏⡀⢀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠈⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠦⠄⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢋⣠⣤⣤⣀⠀⠀⠰⠶⠂⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠶⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠠⡀⣤
⠀⠀⣀⠀⠤⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢀⣤⠀⠤⠆⠀⡀
⠒⠁⠀⠀⠑⠈⠩⠖⣨⡥⠔⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠁
⠀⠐⠊⠁⠈⠔⣊⠁⢀⡴⠖⢉⡤⠄
⠆⡠⠖⣂⡴⠏⣡⡶⠋⣴⠞⢉⣴⠞⢋⣡⣤⠀⢀
⣉⡤⠞⠡⠤⠖⠋⢀⡼⠁⠔⢉⡴⠚⣉⠥⠴⠚⢋⣠⠴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠟⠀⣁⠖⢚⣩⠶⠋⣠⠶⠒⣩⡤⠶⠒⣠⡤⠞⠛⢁⣴⡶⠗⠋⣉⣤⠔⢂⠄⢀⣀
⣤⡿⢃⡴⢋⣤⠶⠛⣉⡤⠞⣋⠴⣠⡿⢋⣤⠾⠁⣸⠟⠀⣠⡾⠋⣠⡾⠋⣠⡾⠋⢠⣼⠃⣠⡤⢀⣠
⣩⣔⣯⣖⣋⣴⣞⣉⣡⣶⣛⣡⣞⣩⣴⣛⣈⣴⣚⣁⣤⣞⣉⣤⣾⣋⣠⣾⣛⣡⣴⣛⣀⣜⣡⣾⣋⣡⣤⣟⣃⣤⣶⣂⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⣹⠃⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠷⢤⣀
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠶⠋⠁⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠳⡄
⠀⠀⠀⠀⢰⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⣿⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠲⣄
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⡅⠠⣄⡀⠀⠀⢀⣴⠋⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢢⣄⠈⢳⡄⢀⣴⢿⠀⠀⠈⢟⢆⢠⢻⢃⢎⣏⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢸⡀⠛⢶⣤⡑⢯⣹⠑⡄⢻⣊⣃⣸⣀⣀⡀⠈⢪⠃⣎⠎⣸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢸⡇⠲⢦⣝⢝⢺⣿⡷⠋⠉⠉⠉⢀⠌⠙⢿⡗⡞⢸⡏⢠⠏⣿⡿⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢧⣀⡤⠽⠮⣧⣻⠁⠀⠀⠀⠀⠉⠀⠀⠀⢻⡆⢸⡇⣬⣼⣽⣧⣧⠤⣄⠀⠀⡇⠀⠀⠀⠀⢀⡤⠟
⠀⠀⠀⠀⠀⠀⢯⣀⣂⣀⡹⠻⢤⣀⣀⣀⣀⣤⣤⡤⠴⠿⢧⡼⢿⣅⠀⢀⣠⣇⣀⣼⣄⣠⣧⣀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣲⡂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡇⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⡞⠉⢹⠉⠙⢲⢯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣬⡄⠀⢀⡇⣀⡤⡟⠙⢯⣉⡉⢑⡶⢴⡖⠒⠀⢰⠒⠢⣄⡀⠀⠀⠀⠀⠀⠀⠐⢦⣴⡁⣠⠿⡌⢹⠉⣱⠧⡀⣙⣤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡇⠁⠀⡇⠀⠀⠈⡷⢇⡀⠀⠓⠀⠀⢸⠀⠀⠨⠑⡀⠀⠀⠀⠀⠀⠛⢻⠀⢹⠓⢤⣽⣾⣶⣧⠴⠛⡁⠸⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡧⡄⠀⡇⠀⠀⠀⡇⠀⠈⢙⡶⠦⠤⣼⣀⠀⢰⢀⠀⠀⠀⠀⠀⣐⡒⢻⠒⢻⠒⢺⣿⣿⣿⣿⡒⠛⡗⢚⣟⣓⡀
⠀⠀⠀⣀⡠⢤⠄⠒⢺⠓⠒⠘⡇⠙⠣⣏⠉⠉⠉⠛⠛⠛⠉⣃⣁⡲⢾⠀⠀⣹⠏⠘⣯⢹⠒⠒⠛⠛⡞⣦⠼⢮⣿⣟⢹⠉⢿⣏⠟⠦⣼⠈⠉⠁
⠀⠀⠸⣅⣀⣰⠀⣄⣸⡇⣀⣠⣧⠤⠤⡿⠷⠶⢶⡗⠋⠉⠉⠇⠀⠀⢸⣤⠞⢸⠄⠶⡤⢸⠀⠀⠀⣒⡏⠙⠦⣴⡿⠒⢻⡖⠚⣿⡠⠞⢀⣉⡀
⠀⠀⠀⠀⠀⢉⣼⡋⠉⠁⠀⣰⡇⠀⠀⣇⡀⠀⠀⠉⣽⣖⣲⣦⣴⣮⣿⣿⣷⣶⣤⣀⣀⣸⣀⣄⢀⣉⡏⠠⢤⣾⣿⠚⢻⡗⠚⢻⣧⠀⠀⠉⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣶⣧⣾⣿⣿⣦⣴⡿⣷⠒⢚⠟⢹⡏⠀⠋⠛⣛⠛⣿⣿⠛⠛⠛⢉⣽⣿⣿⣧⣄⡇⠰⢾⡿⣿⢿⣿⣧⢠⡼⣿⣆⣴⣶⣀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⠿⠲⢦⣄⣸⣧⠠⠴⠶⠿⢠⣿⣿⢴⡾⡯⢠⣿⣿⣿⣿⣿⡧⠦⣿⡇⣿⡄⢯⣿⢛⠶⢿⣿⣿⣿⣿⣇
⠀⠤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠶⢶⣿⣿⣷⣶⢷⣶⣶⣾⣿⣿⣾⣶⣶⣾⣿⣿⣿⣿⣿⣷⢾⣿⣿⣿⣿⣿⡷⠾⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⠉⠙⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠛⠉⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣈⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠋⠀⠀⠀⣠⣴⣾⣿⣿⡿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠷⣦⣌⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⣿⠟⠋⠁⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣤⣉⠛⢦⣙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⣠⡶⠟⠋⢉⣁⣀⣀⡀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣶⣌⡳⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⣰⣿⣿⣿⠟⠀⠀⠀⡠⢚⣡⣴⡾⠿⠿⠿⠿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠉⠻⣿⣿⣿⣮⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⣼⣿⣿⡿⠃⠀⠀⢀⣪⡴⠋⣁⣀⣤⣤⣀⣀⡀⠀⠀⠉⠻⢿⣿⣿⣶⣄⠀⠀⠈⢻⣿⣿⣷⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⣸⣿⣿⣿⠁⠀⠀⢀⠜⣡⠾⠟⠛⠛⠛⠿⢿⣿⣿⣷⣤⡀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⠈⠮⡙⢶⣤⣤⣤⣤⣴⣾⣿⣿⡿⠛⠁⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⣰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⢻⣿⣿⣷⡄⠀⠀⠈⢝⠲⣄⡉⠙⠛⠛⠋⠉⠁⠀⠀⣀⣀⣛⣛⣻⠛⠋⠀⠀⢀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠱⢬⡙⠻⢷⣶⣶⣶⣶⣿⣿⣿⣿⠛⠋⠉⠀⠀⣀⣰⣿⣿⣿⣛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⠙⠷⣤⣄⣈⡉⠉⠉⠁⠀⠀⢀⣀⣤⣴⣿⣿⣿⠿⢋⡵⡫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠈⠻⣿⣿⣿⣦⣄⡀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⠛⢉⣤⠞⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣄⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⢋⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣤⣀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⢉⣡⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠰⣿⣷⣶⣴⣶⣤⣀⡀
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣤⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠛⠛⠿⠿⠟⣠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠖⠉⣭⠿⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣎⣀⣼⡠⠃⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣛⠯⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢆⠀⠀⠄⠐⢄⠹⡂⢠⠈⢖⣠⠂⢠⠀⡀⣀⡀⠤⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⠦⣸⡄⢄⠻⣇⡜⢣⣸⢁⢂⣾⠟⠋⠁⣀⠔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⡀⠰⠈⢻⣬⣶⣿⠞⠳⣿⣿⡏⠀⠰⢀⣼⡉⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⠛⠻⣦⣴⣯⠝⠉⠀⠀⠀⠀⠉⢋⣷⡶⣋⠥⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢈⣑⠶⠔⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⠶⠿⠒⠉⠉⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠠⠐⠉⠀⠔⠚⢅⣻⣿⠀⠙⠒⠂⠀⠀⠑⠚⠋⢈⣿⢌⡵⠒⢀⣤⠄⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣀⣠⠔⢲⡿⢋⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣒⠻⢍⠑⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠖⠁⢂⣡⡞⠟⢭⣷⣤⣀⣀⣀⣠⣴⣯⠟⠉⠻⣦⡅⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢋⠘⠂⣡⣿⣿⡟⠷⢻⡟⢯⠻⣅⡘⠀⠘⣷⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠄⠉⣀⣠⣴⢾⠏⠌⣸⡏⢶⠋⣿⣄⠣⠹⡏⢦⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠁⠀⠀⠐⠀⡔⠁⢧⠀⠀⠸⣄⠑⠄⢩⠀⠌⠢⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⡶⢯⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠙⠃⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢾⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠙⠶⠄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⠀⠀⠀⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠈⣀⠀⠀⠀⠀⡆⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠀⠃⠀⠁⠀⠀⠈⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⢀⠀⡏⢱⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡜⣙⣎⣸⢇⡇⡀⠀⡀⠆⠀⡔⠀⠀⠀⠈⠁⠀⠀⠀⣀⡀⠀⢠⠖⢦⠀⠀⣀
⠀⠀⠀⡋⣇⡇⢸⣠⢏⡇⠀⠀⠀⢠⠶⡄⠀⡇⠉⢩⠦⢊⡀⠘⢆⣡⡠⠜⣀⡔⠉⢀⠀⠠⠖⠀⠀⠁⢀⠀⢸⠁⠸⠀⡰⠁⠱⡀⠀⠠⡄⢀⡀
⠀⠀⠀⠳⠤⡄⢰⠓⠊⠉⠉⠉⠓⢣⠀⡧⢬⡇⠦⣿⢰⢳⠈⢲⠊⠀⠀⠀⠀⠙⣏⠥⠒⠐⠒⡀⠀⠉⠁⠀⠸⡄⠆⢧⣇⠀⠀⡇⠀⡦⡇⢨⠀⢶
⠀⠀⠀⠀⠀⠁⢸⡄⠀⠀⠀⠀⠀⠈⣆⠉⠒⠁⠀⠻⠞⣸⢅⣣⠀⢀⣀⠤⠔⠒⠛⠂⠉⠉⠀⠀⣫⡄⡿⠋⠉⠙⠤⢠⡀⠀⠀⠳⢖⣡⢇⣈⡄⡇
⠀⠀⠀⠀⠀⠘⠈⠀⠀⠰⠷⠆⠀⠀⠈⠑⠒⡄⠈⡴⠉⠁⠀⠉⠙⠢⠤⢀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠈⠁⠠⠄⠀⠀⠀⢣⠀⠀⡏⠉⠀⠀⠈⠇⠀⢤⣀
⠀⠀⢀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⣶⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠁⠒⠀⠒⠀⢠⠀⢠⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⠈⠁
⠘⠚⠚⠓⠚⠒⠒⠛⠛⠛⠚⠒⠒⠚⠒⠚⠒⠓⠂⠛⠓⠚⠚⠚⠓⠓⠚⠓⠒⠚⠒⠒⠓⠒⠒⠒⠒⠒⠚⠒⠛⠚⠒⠓⠛⠒⠒⠓⠛⠒⠒⠒⠒⠚⠛⠚⠒

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

⠀⠀⠀⢠⣄⣰⣆⣠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡠⠀⠀⠀⠀⠀⡄⣐⣰⣂⡀
⠀⠀⠀⣤⣿⣿⣿⣿⣤⠀⠀⠀⠠⣿⣿⣿⡧⠀⠀⠀⠀⠬⣿⣿⣿⡎⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⣽⣿⣿⣿⡧⠀⠀⠀⠐⣾⣿⣿⣿⡒
⠀⠀⠀⢠⠿⢿⡿⢯⠀⠀⠀⠀⠀⠹⠿⠟⠃⠀⠀⠀⠀⠈⠽⠿⠿⠓⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠐⢚⠿⡿⢟⠓⠀⠀⠀⠈⠝⢿⠿⠛⠌
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⣄⣂⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⡄⣂⡐⢀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⢀⢹⣿⣿⣿⡧⠄⠀⠀⠀⠠⣿⣿⡯⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢒⢾⣿⣿⢀⡒
⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠚⡻⢿⠿⡓⠂⠀⠀⠀⠀⠋⠛⠑⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠁⠆⡉⢁⠲⡌⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢤⣰⣦⡆⡀⠀⠀⠀⡀⢦⣴⣤⠆⡀⠀⠀⠀⡀⢆⣶⣠⢄⠀⠀⠀⢀⡄⣴⣇⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣅⡂⠄
⠀⠀⠀⣽⣿⣿⣿⣿⡥⠀⠀⠠⣽⣿⣿⣿⣿⠥⠀⠀⢠⢽⣿⣿⣿⡯⡄⠀⠀⣀⣻⠁⠀⠙⡟⠆⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠅⣾⣿⣿⣿⡎⠄
⠀⠀⠀⠚⣻⢿⣿⡟⠂⠀⠀⠀⠊⣻⢿⢿⡛⠂⠀⠀⠀⠊⡻⠿⢿⡓⠂⠀⠀⠈⣙⣷⢤⡞⠻⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠂⠝⠿⠿⠟⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠠⡰⠘⠰⢢⠀⠀⠀⠀⢀⣠⣤⣆⡀⠀⠀⠀⠀⣠⣢⣖⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣷⣶⣄⠀⠀⠀⢀⠰⣀⣆⡐⢀
⠀⠀⠠⠅⢰⣿⣷⡄⠩⠀⠀⠀⢺⡏⠀⠈⣿⡂⠀⠀⢸⣹⠁⠀⠈⣳⡅⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠌⣿⣿⣿⣿⡅
⠀⠀⠀⢫⢌⠙⢋⠠⠑⠀⠀⠀⠙⠷⡦⣶⠟⠀⠀⠀⠘⢭⡷⢤⡴⡿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠀⠀⠀⠐⢙⠿⠿⢟⠐
⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣵⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣿⣿⣾⣽⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⢗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠈⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠰⠦⣀⢨⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠈⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡄⠛⠒⠣⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⣠⡀⡾⠀⠀⠀⠀⠀⠀⠊⠁⠀⣒⠀⠀⠠⠖⠀⣠⠯⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⡗⢒⡦⠭⠛⠛⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⢋⡀⠤⠢⡀⠀⠅⠠⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠥⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠡⠿⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡠⠐⠂⠉⠀⠀⠁⡇⠀⠠⠀⠀⠰⠂⢤⠧⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣲⣀⣿⣴⣍⣹⠂⢄⠀⠀⠀⠀⠀⢀⣠⠔⢃⠜⠁⠀⠈⡐⠰⢀⣀⡀⠤⠀⠛⠉⠉⠀⠒⠲⠦⠤⢤⡤⣤⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠋⠁⡇⠉⢿⠳⠤⢀⡁⣀⠤⠒⠈⠀⠁⠀⠀⠆⢄⠀⠀⠀⠱⠀⠀⡠⠐⠀⠉⠁⠉⠀⠀⠀⠠⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⠀⡇⠀⣼⣷⠀⠤⠀⣀⡀⡀⡉⠀⠀⡠⡆⠀⠀⢑⡄⠀⡘⠠⠊⠀⢀⠠⠐⠈⠉⠀⠈⡀⠂⠀⠈⠰⠇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⣠⠄⠒⠹⠶⠦⠤⠶⠟⠁⣀⠤⠐⢙⠀⠀⠁⠀⠀⠃⠃⠀⠀⡦⡆⠀⢀⠃⠀⣰⠡⠤⠠⠤⢀⣀⠀⠀⢀⡤⠠⠔⢁⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠂⠤⠌⢁⡐⠀⠠⠔⠂⡁⠄⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⡐⠀⠈⠉⠀⠒⠂⠒⠈⠭⠅⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢷⠐⡈⠄⠛⡞⠛⠁⠒⠀⠤⢼⣀⡄⣀⡰⠄⠨⠇⢠⠀⠀⡄⠈⠄⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠠⠐⣈⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠠⠊⡱⠊⠀⡸⠣⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⣀⠔⠈
⠀⣋⠠⠐⠉⠀⠈⠉⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠂⠠⠤⠀⠀⢀⣄⠔⠊
⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡢⠒⠒⠒⠤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⢆⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠐⠄⠀⠀⠀⠠⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠈⠀⢳⠂⠀⠘⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⢂⡀⠀⠀⠀⠠⠈⠄⠐⠄⠀⠀⢸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⡢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⢸⡃⠀⠣⡤⡤⠄⠄⢄⢠⢀⡤⢴⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠒⠀⠠⠀⠀⠀⠀⠀⠢⠈⢀⠀⡀⠀⢸⡇⠀⠀⠘⠋⠋⠉⠹⠩⠽⠿⠿⠟
⠀⠀⠀⠀⡀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⠐⠀⠀⠄⢰⢱⡄⠀⠐⢤⣤⣤⠤⠤⠤⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣦⠀⠀⠀⢀⠰⠦⣀⠀⡂⠈⣿⡇⠀⠀⠀⠀⠀⠀⠨⡩⢙⣤⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠂⢁⠠⠐⠀⠀⣸⣷⠀⠀⠐⣤⣤⡤⣤⣤⢤⣤⢽⣯⠀⠰⠁⠀⠀⠀⠒⠶⠶⢿⠰⠠⠬
⠀⠀⠀⠀⠐⢦⣤⣤⣤⣤⡽⡇⠀⠀⠀⠈⠛⠓⠓⡛⠋⠟⠛⠙⡂⠀⠀⠀⠀⠀⠀⠐⠂⠘⢉
⠀⠀⠀⠀⠀⠀⠉⠉⠚⡚⡟⠃⠁⢡⠐⠀⠀⠀⠀⡌⠈⢚⠩⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⡠⠠⠄⠀⠀⠀⡀⠤⠄⠀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣦⣤⣤⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⢟⡋⣉⡙⠛⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠉⠛⠛⠿⠿⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⣁⠀⠤⠾⣿⡭⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⣴⣿⣯⣿⣻⣿⠛⢃⣀⣀⣠⣤⣤⣤⣤⣴⣶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠙⠿⣶⣀
⠀⠀⠀⠀⠀⠀⣠⢊⡱⢎⣉⣿⣿⣿⣿⣿⡟⠛⠉⠉⢉⣁⣀⣠⣀⣀⣀⣀⣠⣤⣤⡤⣤⣄⣀⣀⣀⣀⣉⣉⠉⠛⠛⠳⠶⣦⣄⣀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⣼⠃⠶⢾⣿⣿⣭⣿⣷⣾⣧⠥⢤⣗⣒⡲⡿⠩⠭⠁⠤⠴⠶⠶⠆⠸⠥⠿⠿⠿⣿⣿⣿⣿⡟⠳⢶⣧⣤⣆⣀⣈⠉⠛⠷⣦⣄⢻⣄
⠀⠀⠀⠀⡼⠁⢀⣼⣿⠿⣟⣛⣻⣿⡿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⢿⣿⣿⣷⡶⠤⡉⠉⠻⡆
⠀⠀⠀⣸⠁⢀⣿⡿⠿⢛⣋⣭⡥⠶⠒⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣄⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⣷⣮⣧⡀⢻⡄
⠀⠀⢰⡇⠀⠨⠛⣚⣉⠭⠔⠒⢂⣉⣀⣤⣴⣶⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠈⠙⠯⠸⣇
⠀⠀⣼⠀⠀⠚⢋⣀⣤⣴⣶⠿⠿⠛⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣶⣦⣤⣄⠀⠀⠀⣿
⠀⠀⣿⠀⣴⣶⣿⣿⣯⠭⣖⣒⠿⠭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣶⡄⣿
⠀⢸⡟⠀⠋⠉⣿⣱⢾⣿⠤⠘⠛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿
⠀⠐⣷⠀⠀⠙⠒⣉⡥⠤⠒⣊⡩⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⣧⠀⠀⠀⠋⣡⠴⠚⠉⣁⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠸⣇⠀⠀⠁⠀⠄⠚⢉⣠⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⢸⡄⣤⣬⠀⠀⠞⠃⠀⣠⣤⣤⣤⡤⠶⠶⠒⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠻⣟⠿⢿⠄⣠⣴⣾⣿⡿⠁⣒⠂⢀⣀⣀⡠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠤⣤⣀⣀⡀⠀⠈⠉⠙⠛⠿⣿⣷⣦⣄⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠘⠙⠟⣉⣻⣷⡿⣿⣿⣏⠉⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠋⠻⠙⢻⣶⣶⣤⣄⡀⠀⠉⠛⠻⢿⣶⠾⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣄⠐⠋⠉⣻⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣩⣖⣶⣦⣤⣄⣈⡉⠛⠻⢿⣶⣦⣄⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣀⡛⠿⣿⣻⣭⣭⣵⡿⠶⠖⠚⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠉⠉⠛⠛⠿⢷⣦⣤⡈⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡴⠾⢿⣭⣤⣶⣶⣾⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠒⠒⠊⠩⢭⡿⢶⣶⣤⣄⣀⠀⠉⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⣹⣿⣿⣿⡿⠟⠛⠛⢉⡁⠀⠀⠀⠀⠀⠀⠀⠬⠉⢉⣙⣛⣶⣶⣤⣄⣈⡉⠙⠛⠿⣿⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣥⡔⠒⢈⣈⣉⣤⣤⠤⠤⠄⠀⠀⠒⠶⣆⡀⢤⣤⣤⣤⣄⣉⣉⠙⠛⠛⠿⣶⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠞⢻⠿⠟⠛⠛⢂⣀⣀⣀⣀⣀⠀⢀⣀⠀⠐⠚⠛⠛⠩⠿⠿⣷⣦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡶⠶⢛⡛⠉⡭⠄⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠻⢷⣶⣦⣄⡰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣦⣈⠤⠶⢒⣛⡚⠛⠒⣒⣒⣒⣲⣶⡶⠶⢤⣄⡉⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⠞⠛⠛⣷⣶⣶⣶⣤⣤⣤⣤⣄⣈⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡖⠋⣉⣉⣀⣀⣈⣉⣉⣉⣉⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣏⣉⣉⣩⣍⣉⣙⣛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠃⣀⣀⣈⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣹⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢳⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⢠⡖⠚⠋⠉⠉⠉⠉⠙⠓⠲⠤⣄⡀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠦⣌⡉⠓⠲⠦⣄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠑⡆
⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⢁⡦⣀
⠀⠀⠀⠀⠀⣰⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠀⠈⠓⢤⡀
⠀⠀⠀⠀⢰⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣉⡷⠄
⠀⠀⠀⢀⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠶⡄⠀⢹⡀
⠀⠀⠀⢸⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⡇
⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧
⠀⠀⠀⢻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⢸⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡇
⠀⠀⠀⠈⣇⠀⠘⠶⠒⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣸⠁
⠀⠐⢾⣉⠁⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠈⠳⢤⡀⠀⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⠺⢁⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠲⠦⣤⣈⡛⠲⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠈⠙⠲⠦⢤⣄⣀⣀⣀⣀⣠⡤⠼⠃⠀⠀⠀⢈⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⢧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⢸⣿⡀⢸⣦⡀⣿⡏⠀⣾⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠹⣶⣦⣿⣷⣤⣿⣿⣿⣷⣼⣿⣧⣾⡟⠀⣤⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⣿⡿⠟⠛⠋⠉⠉⠙⠛⠿⣿⣿⣧⣾⣷⣀⣴⣾⠟⠁
⠀⠀⠀⠒⠶⢶⣤⣤⣀⠀⠙⠾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⡿⠀⢀⣀⣀⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣩⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⠤
⠀⠀⠒⠛⠛⠛⠛⠛⠛⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡛⠛⠛⠻⠿⠷⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠓⠂
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⢸⣿⠟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠻⢿⣦⣄
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠋⢁⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⢦⠀⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⢻⡟⠀⣿⡿⣿⣿⠻⣿⡿⣿⣧⠈⢻⣷⡀⠀⠀⠀⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠀⣸⣿⠁⠉⡏⠀⣿⣇⠀⠘⠂⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⣿⠁⢀⣀⣄⣀⡀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣷⡄
⣿⣿⣧⠀⠀⢀⣴⣿⣿⣿⡆⠀⠀⣀⣤⣄⡀⠀⠈⠛⠛⠁⠀⣿⣿⣿⣿⣷⠀⠀⠸⣿⣿⣿⠿⠛⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⡁⠠⣾⣿⣿⣿⠇⠀⢀⣀⣀⠀⠀⠻⣿⣿⠏⢁⣀⡀⠀⠈⠉⠁⣠⣤⣄⡀⠀⠈⠻⠿⠿⠛⠛⠛⠁

⣿⣿⣿⡇⠀⠈⠻⣿⣿⣿⠇⠀⠹⣿⣿⣿⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣷⠀⠀⣼⣿⣿⣿⣿⠄⠀⠀⢀⣠⣤⣄
⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠿⠿⠇⢰⣿⣿⣷⠀⠈⠿⠿⠋⣁⠀⠀⠹⠿⠟⠛⠋⠀⠀⢠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣠⣶⣶⡄⠀⣰⣿⣿⡇⠀⣴⣾⣷⠀⢀⣠⡀⠈⠿⠟⢠⣴⣦⡀⠀⣾⣿⣿⡆⠀⢀⣴⣷⣦⡀⠀⠸⣿⣿⢿⣿⣿⠃
⣦⡀⠀⢰⣿⣿⣿⣧⠀⠻⣿⣿⡷⠀⠻⣿⡿⠀⣿⣿⡇⣤⣤⡄⠘⠿⠟⣁⡀⠙⠛⣉⡀⠀⢺⣿⣿⣿⠇⠀⠀⢀⣀⡀
⣿⣧⣀⠈⠻⢿⣿⠿⠁⠀⠀⠉⠀⣠⣄⠀⣠⣤⠈⣉⡀⠻⠟⢰⣶⡄⠸⡿⠟⠀⢼⣿⣿⠆⠀⣠⣤⡀⠀⢀⣾⣿⣿⣿⡆
⣿⣿⣿⠀⠀⠀⠀⠀⢀⣶⣷⡆⢸⣿⣿⠈⠿⡿⠘⢿⠇⣶⡆⣈⠉⣶⠆⠰⣿⠷⠀⣴⣦⡀⢸⣿⣿⣿⠀⠈⠻⠿⢿⣿⠟
⠛⠋⠁⠀⢠⣶⣷⡄⠸⢿⣿⡿⠀⠈⠉⢠⣤⢠⣶⠠⣶⢨⣄⡛⢴⠀⠾⡂⢴⡶⠈⣛⡛⠁⣠⣤⡄⠀⢀⣤⣶⣦⠀⠀⠀⠀⣀
⠀⢀⡀⠀⢸⣿⣿⣿⡇⠀⢀⡀⢰⣿⣧⠙⠛⢁⡍⣴⠰⠦⠬⣝⢢⠩⠌⣏⠰⠶⢘⠻⠇⠈⠛⠿⠟⠀⠸⢿⣿⣿⡇⠀⢠⣿⣿
⣼⣿⣿⣦⣄⠉⠉⠉⠀⠀⣿⣿⣦⠉⣉⠰⣿⡎⣉⠴⢜⡨⠍⠂⠁⢁⢣⠃⡝⢃⡛⠗⠐⢿⣿⠀⢠⣶⣶⡆⠈⠉⠀⠀⠈⠻⠿
⢿⣿⣿⣿⣿⠂⠀⣴⣶⣄⡉⠉⠁⠀⢿⠿⠀⣶⡍⣡⢋⡖⣦⡀⠀⠀⠭⢚⡰⠎⣃⡸⣿⠆⢀⣀⣈⠻⣿⠏⠀⢠⣶⣶⣶⣦
⠈⠙⠛⠉⠁⠀⠀⢻⣿⣿⠟⠀⢾⣿⡶⠀⣶⡦⠰⠶⢘⡊⣇⢆⠪⢭⣫⢩⡵⡶⠙⠃⣠⣄⠛⣿⡿⠀⢀⣀⡀⠘⢿⣿⣿⡿
⣿⣷⠀⠀⢠⣶⣶⣤⣄⠀⢀⣴⣤⣅⠀⣴⣶⡄⠺⠟⠨⡦⠹⢆⡛⠶⢰⡆⣶⡄⣿⡦⠹⠿⠁⣀⡀⠀⣿⣿⣿⡇⠀⠙⠛⠁
⣿⠟⠀⠀⠘⣿⣿⣿⡿⠀⠀⠻⡿⠟⠀⣈⣩⡀⠺⠿⠀⠺⢇⣈⠡⣾⢠⣤⡈⣤⡄⢠⣦⣄⠸⣿⣿⡆⠈⠿⠿⠃⠀⢀⣀⡀
⠁⠀⠀⣀⣀⠈⠉⠁⠀⠀⣶⣦⣶⣄⠀⠻⠿⠃⠐⣿⡿⠀⠻⠟⣠⣤⠈⠋⠐⠿⠟⠘⠿⠟⠀⠛⠛⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠻⣿⡿⠃⠀⣴⣶⣿⡆⠀⣴⣿⡇⠈⠛⠋⣸⣿⣆⢠⣶⣄⠀⣠⣶⣄⠀⢰⣿⣿⣷⣦⠈⢻⣿⣿⣿
⠀⠀⠈⢻⣿⣿⡿⠋⠀⠀⣠⣄⣤⣤⡀⠈⠛⠛⣀⣀⠉⠉⣡⣾⣦⠀⠉⠋⠁⢾⣿⡿⠃⣹⣿⣿⡗⠀⣿⣿⣿⠏⠀⠈⠻⠿⠋
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⠀⠀⠻⠿⠟⢠⣾⣷⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠈⠛⠁
⠀⢠⣾⣿⣷⣿⣿⣷⠀⠀⠈⠙⠛⠁⣀⠀⠈⠛⠛⠋⣠⣤⣄⠀⠀⣿⣿⣿⠇⢰⣿⣿⣧⠀⠀⣴⣶⣦⡀⠀⠀⣴⣾⣿⣶⣄
⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣶⣿⣿⣷⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠈⣿⣿⣿⣿⡆⠀⢿⣿⣿⣿⣿⣷
⠀⠀⠈⠛⢿⠿⠋⠀⠀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠛⠿⠿⠛⠀⣶⣿⣶⡀⠀⠈⠉⠉⠀⠀⠸⣿⣿⣿⠟⠁⠀⢼⣿⣿⣿⣿⠏
⠀⠀⠀⢀⡀⢀⣤⣤⣄⠀⠀⠀⠉⠙⠋⠀⣠⣤⣄⠀⠀⠀⢰⣾⣿⣿⣿⡇⠀⢀⣠⣄⡀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠄⠄⡀⠀⠀⡠⠀⠀⢀
⠀⠀⠀⠀⠄⠀⢆⠊⠼⠛⢩⠇
⠀⣤⠀⠀⠠⠀⢨⠴⠀⠀⣀⡄⠒
⠀⠀⠐⠃⠀⠂⣴⠀⠀⣄⣀⠀⢤⡀⠀⢀⢠⡄
⠀⠀⠀⠀⠄⠋⢿⡡⠀⠽⠉⠉⠂⠐⣔⠡⠛⣀⠆
⠈⠉⠀⠀⠀⠀⠀⠘⠂⢠⢤⡤⠂⣠⡏⠐⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠛⠱⠀⠘⠁⠁⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⢐⠀⠀⠀⠁⠀⠀⠀⠀⢠⡼⢏⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠦⠀⠀⠀⠀⠀⣠⣴⣗
⠀⠘⠧⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠟⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠏⠀⠉⠐⠃⡒⠰⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⢛⡁⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠖⠒⠀⠤⠂⠀⠀⠀⠐⠂
⠀⠁⠀⠀⠀⠒⠃⠀⠀⠀⠨⠥⠄⠀⠀⠀⠠⠀⠁⠒
⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠤⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢀⠀⠀⣀⠀⡉⢀⠁⠀⠀⠀⢀⠀⠁⠀⠐⣾⣣⣠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⡀⠈⠉⢀⡀⢠⣠⣀⢀⣀⣹⣄
⠋⠡⠀⣀⠀⢀⣠⣷⣿⡾⠵⠞⠛⠓⠒⠛⠉⠉⠉⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠶⠶⠤⠶⠾⠻⠿⠛⠛⠛⠟
⠛⠛⠛⠛⠋⠉⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠉⣀⣀⣀⣀⡀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠉⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⡶⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⣶⣶⣶⣶⡆⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠃⠀⠀⠿⠿⠿⠿⠃⠀⠘⠿⠿⠿⠿⠀⠀⠘⠿⠿⠿⠟⠀⠀⠸⠿⠿⠿⠟⠀⠀⠻⠿⠿⠿⠇⠀⠀⠻⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡌⢳⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠹⣦⠻⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠶⣟⠢⣌⡑⢒⣴⠶⢋⣠⡴⠾⠛⢋⣁⣤⣤⣴⣾⠷⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡴⣌⠳⣌⠳⣌⡓⠶⢤⡤⠴⣒⣮⢗⠯⡣⣽⢊⡼⢋⡴⢋⣵⠞⣋⣥⡶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣱⢏⢛⢮⣙⠲⢭⣛⣛⣻⢭⣻⡺⡖⣵⢠⢏⡞⡴⢋⡴⣋⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⣫⢞⠓⠮⢭⣒⡲⣶⣒⣭⡮⢺⣥⡇⠏⡞⡼⣱⢏⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣳⠬⣛⡓⣒⡾⢿⢖⡿⣺⡆⢎⢹⢸⠑⡇⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⣽⢿⣏⣭⢤⢶⡒⣪⣄⢇⢎⢧⢧⣣⣧⢻⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢷⢳⢱⠹⣱⣵⣹⢘⡭⣦⣷⣑⣕⣝⢕⣋⢇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼⢸⢃⠇⣿⣿⡘⡯⡫⢖⣚⣛⡺⠤⢧⡵⡣⣪⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣴⢃⡟⡼⡸⢸⢙⣿⣻⡪⢕⣒⣒⠮⢍⣓⠦⣵⣱⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢛⣡⠞⣡⢏⡼⣱⠃⢟⣿⡻⢟⡻⠭⠭⢭⣝⡳⢬⡓⢦⣱⢣⢟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⠶⠛⣋⣥⠶⢋⣵⠞⣥⣟⡠⣝⢮⣻⣕⡯⠝⠒⠛⠛⠓⢦⣝⢶⣝⢦⡙⠏⡼⠁
⠀⠀⠐⠛⠿⠿⠿⢛⡛⠉⢩⣤⣴⡶⠛⠉⡰⠾⠿⠤⣌⣓⠮⠝⠚⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⢦⡹⣦⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢷⡘⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⢷⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀
⣿⡟⡉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⠒⣛⣹⣋⣛⠓⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣞⣡⠔⠻⡉⠁⢸⠀⠉⣹⠓⠦⣙⣦⡄
⠀⠀⠀⢀⡜⢡⠞⠳⡀⠀⢳⠀⢸⠀⢠⠃⠀⡠⠛⢣⡘⢆
⠀⠀⠀⡾⢲⠧⣄⡀⠈⠢⣀⢣⣸⢠⠏⡠⠊⠀⣀⡠⢷⠚⣇
⠀⠀⢰⡇⡟⠀⠀⠈⠑⠲⣬⣿⠿⢿⣿⡴⠒⠉⠁⠀⠈⡇⣹⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠸⡏⣏⠉⠉⠉⢉⣩⠿⣿⣤⣼⣿⢯⣉⠉⠉⠉⠉⡏⣹⠀⠀⠀⠀⠀⠀⢀⡾⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡀
⠀⠀⠘⣇⣹⡤⠒⠋⠁⡠⠊⡹⣿⡟⣌⣣⡀⠉⠓⠢⢴⣁⡏⠀⢀⣀⣀⣠⠔⠽⠁⠇⠱⠕⠤⢀⠀⢀⣀⣀⣀⡞⡺⠁⠑⠄
⠀⠀⠀⠙⣆⠳⡀⣠⠊⡀⣰⢡⣿⣿⠻⡉⠙⢦⡀⣰⠇⡼⠁⣴⠛⠁⡱⠻⡷⠲⡖⠖⣶⠒⡖⡖⠉⣀⢠⢤⣸⣗⣏⣢⠀⠼⣷⣦⡀
⠀⠘⠛⠽⠛⢦⡟⢻⣍⢹⠃⢸⢹⣿⠖⢷⣶⣾⠟⣣⠞⠀⣀⣧⣴⢾⠻⢻⡷⣶⡇⠀⠼⠀⢰⣿⣭⣁⣰⣴⣂⣉⣭⣾⠀⣰⠃⢳⡀⠀⠀⡀⠁⠈⡀⠐
⠀⠀⠀⠀⠀⠒⠘⠷⢤⣏⣒⡞⣾⠛⣷⣏⣯⣼⣋⣥⣴⠾⠋⠙⣶⣇⣴⣟⣴⣷⠓⡖⢳⡄⠈⣟⠿⣿⣿⣿⣿⡿⠗⠋⡀⠌⠀⠀⠃⡀⣀⠟⢑⡤⠗⡗
⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣭⣽⣭⣬⣿⣬⣯⣼⣭⣤⣭⣤⣬⣥⣤⣤⣤⣤⣴⣶⣦⣶⣶⣶⣶⣮⣤⣭⣿⣤⣥⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠐⠀⠀⠀⠀⠀⣀⡀⠘⢃⣀⣉⠀⠰⠇⠀⠀⠀⠀⠀⠀⣄⠹⣇⠹⠀⠀⡜⢀⡀
⠀⠀⠀⠀⠀⡀⣄⣴⠟⢯⡓⣄⠀⠀⠀⠀⠀⠀⠙⠁⡄⠀⡀⠀⡀⠀⠀⠶⠂⠀⠀⠀⠲⠀⠁⢈⣤⣤⣄⠐⣡⡴⡆
⠀⠀⠀⠀⠀⢀⠸⡧⣸⣅⣼⣇⠀⠀⠀⠀⠀⠛⠃⠰⠀⠢⣀⣀⡡⠄⠁⠠⡤⠀⠀⠨⠭⠄⠀⣿⣿⣿⣿⡇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠛⠏⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠶⠒⣡⠌⠛⠛⠋⠀⢌⡙⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⣤⠀⠲⠄⠀⠀⠀⠀⠀⠀⠺⠃⠀⡆⢰⠀⠘⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⠄⠸⡄⠀⢠⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢃⣀⠃⢠⠆
⠀⠀⠀⠐⠶⣄⠢⠄⠐⠈⡔⢀⣤⡄⠀⠀⠀⠀⠀⢠⠿⣟⣛⡛⠿⣷⡀⠀⠀⠀⠀⠀⠐⠄⢠⣾⣿⣿⣿⣦⡀⠤
⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠠⠄⠀⢤⠄⠀⠀⠀⠀⢿⣜⢾⣜⣟⡶⣿⠁⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣿⣿⣿⡇⠤⠄
⠀⠀⠀⠙⠋⠁⠔⡀⠄⣀⠢⠘⠲⠄⠀⠀⠀⠀⠀⠈⠮⣯⣿⣿⡵⠃⠀⠀⠀⠁⠀⠀⠐⠂⡘⠿⣿⣿⣯⠟⠡⢄
⠀⠀⠀⠀⠚⠁⡸⠁⠀⠘⡄⠐⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⢀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣠⠆
⠀⠀⠀⠀⢠⣀⢻⠢⡸⠣⠞⣏⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣤⣀⣹⠟⠀⠀⠀⠀⠑⠧⢤⣤⣤⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠍⠫⣝⢦
⠀⠀⠀⠀⠈⣲⠄⠀⠀⠀⠀⠀⠺⣍⣀⠀⠀⠀⠀⠀⢰⠡⠄⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢆⢞⣻⣾⣿
⠀⠀⠀⠒⠛⣡⣟⣀⠀⠀⠀⠀⣏⠈⠁⠀⠀⠀⠀⠀⠈⠢⣈⣋⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠧⣂⠥⣕⣵⠏
⠀⠀⠀⠀⠘⠉⠀⡶⠉⣦⠲⡏⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠰⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⢹⠈⠉⠈⡇⡞⠀⠈⠁⢻⠀⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⠈⠁⠂
⠀⠀⢰⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢹⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠘⣷⠛⠓⠒⠒⠒⠒⠒⠲⣿⢸⠛⠛⠉⠉⠛⠀⠀⠐⢰⠄⡎
⠀⠀⢸⠀⢠⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⡆
⠀⠀⢸⠀⠸⡆⠀⠀⠀⠄⢰⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠄⠀⢀⠀⠀⠀⢸⡇⢸⠈⠰⠀⠀⠀⠀⠰⠁⡎⢰⠇
⠀⠀⢸⠀⠀⡆⠀⠀⠀⡤⢄⠀⠀⠀⠈⠁⠁⠀⠀⣾⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠠⠄⢰⡇⣈⠀⢀⠀⠀⠀⠀⢀⠀⠁⠸
⠀⠀⢸⠀⠀⠛⠀⠀⠀⡇⡄⠀⠉⠶⢂⡁⡃⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠈⠀⠐⠤⠤⠄⠊⢰⠁⣶
⠀⠀⢸⠀⠀⢸⡄⠀⠠⠃⠉⠀⠉⠉⠈⠀⠀⠐⢄⣹⠀⠀⠀⡆⠀⠀⡇⢸⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣁⡽⠤⢣⠀⠀⠀⠀⠂⠀⠀⡘⠀⡍
⠀⠀⢸⠀⠀⠀⣧⠌⢤⣠⣤⡤⢔⠒⠒⠂⠉⣋⠉⡥⠄⠀⠰⠇⠤⠄⣇⣸⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠈⠡⣴⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⢀⠁
⠀⠀⢸⠀⠀⠀⠀⠀⡞⠁⠠⠒⠋⢓⣤⠔⠋⢥⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠀⣼⡾⠛⡛⠲⡄⠀⠀⠀⠀⣼⠀⠇⠀⠀⠀⠀⠀⠀⢠⠁⢸⠃
⠀⠀⢸⠀⠀⠀⠀⢸⠀⢀⠔⠊⠉⡝⠁⡄⠀⠐⠉⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⢠⣀⠀⠀⠸⠀⠀⠀⠀⣿⠀⡆⠐⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⢸⠀⠀⠀⠀⠸⠀⡎⠀⠀⡜⠀⡸⠀⠀⣶⡀⠂⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠙⡀⣿⠱⡀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠄⡆⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠷⠀⠀⢰⠀⠀⠃⠀⢸⠀⠙⢤⡀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⣿⣿⠀⢃⡸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⠀⢠⠃⠀⠀⠀⠈⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠃⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠑⠀⢸⠀⢰
⠀⠀⢸⠀⠀⠀⠀⡴⠊⠉⠈⡇⠀⢀⡴⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⢢⡀⢸⠉⠉⠓⣤⣠⣀⣤⣖⡇⢸⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⢸⠀⠀⠀⠈⠃⠀⡶⠰⣇⢠⠋⠀⠀⠔⣴⡀⠀⠀⠀⠀⠀⠀⢀⢠⣶⠀⠀⠀⢁⣸⡀⠀⠀⣏⠀⢀⠀⠀⠁⠈⢈⣉⣉⣁⠙⠚⠃⠀⡄
⠀⠀⢸⠀⠀⠀⠀⢡⠀⢱⡞⠉⢿⠀⠀⠀⠘⠋⠁⠀⠀⠐⠂⠀⠀⠀⠙⠋⠀⠀⠀⠙⠉⢓⠀⠀⡇⠀⠀⢀⡤⠬⠭⠭⣍⠉⠉⠉⠉⠉⠉⠁
⠀⠀⢸⠀⠀⠀⠀⠘⠀⠈⠳⠄⢄⠁⠀⠀⠀⠀⠀⠀⡒⠀⠒⡂⠀⠀⠀⠀⠀⠀⠀⡠⢤⠊⠀⠀⠇⠀⠀⠏⠑⠒⠒⠊⠙
⠀⠀⢸⠀⠀⠀⠀⠀⡆⠁⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⢁⡀⣀⠂⠀⠀⠀⠀⠀⣀⠞⠁⠀⡇⠀⢠⠀⠀⡸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⣷⠛⢢⣀⣀⠀⠈⠑⠦⢄⣀⠀⠀⠈⠀⠀⠀⣀⡠⠴⠚⠉⢀⣀⣠⠟⠲⣼⠀⡤⢏⠐⠂⠠⠤⠀⠲⡧⣄
⠀⠀⠸⠀⠀⠀⠀⠀⢹⢶⠶⡏⠈⠉⠙⠐⠒⠖⠾⢿⣏⣉⢉⣻⠟⠏⠒⠊⠉⠉⠙⠁⢹⠶⡖⢹⠀⣇⠀⠹⣶⣴⣴⡖⠉⠀⠈⢦
⠀⠀⢈⠀⠀⠀⠀⠀⢀⡼⠀⠷⢂⣄⣀⣠⣄⢰⡆⠀⠀⠉⠁⠀⠀⢰⣤⣤⣤⣤⣤⣤⠎⠀⣣⡘⢸⢸⠂⠠⠘⠛⠋⡠⠄⠤⠤⠤⠷
⠀⠀⢠⣄⣀⣀⣀⣀⣠⣿⠊⠉⠉⠉⠉⠉⠉⣿⣀⠤⠶⠶⠖⣀⡀⡀⡇⠀⠀⠀⠀⠀⠀⠉⢡⡕⢺⡈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⢠⠀⠀⠀⠀⠀⢀⣇⣀⠠⠤⠤⠤⠤⠔⢉⡀⠤⠤⠤⠤⢀⣈⠉⠛⠤⠤⠤⠤⠆⠀⠄⠌⠿⡟⠛⡌⠉⠉⢸⣿⣷⠀⠒⢳⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⠀⣘⣀⣀⣤⢤⡤⠤⠤⠔⠁⠀⢠⠀⢤⠒⢀⣀⡉⠐⠲⠤⠤⠤⡦⠤⠤⠤⠀⣿⠆⠃⠀⠀⠀⠚⢀⣀⣀⠸⡟⠉⠉⠉⢿⡀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⢠⠉⡀⡈⠁⠀⠃⣠⠶⡂⠠⠀⠤⣋⣙⣌⡱⣄⠊⠀⠀⡀⢀⡔⠿⢶⡷⢤⠀⠉⢶⡦⣬⠭⠤⠭⠭⠽⠽⠿⠥⠤⠤⠤⠼⡧⠀⠀⠀⠀⠿
⠀⠀⢸⠀⠀⠀⢀⡏⡟⢀⠜⠒⢆⢁⡌⡶⢶⠀⠀⠀⠀⡀⠀⡉⠘⠀⣴⠶⠶⣆⢘⡂⢸⡅⠾⡐⣂⣈⢇⠀⠀⠀⢀⡀⣤⠤⠤⣄⡀⠀⠀⠀⠰⠄
⠀⠀⢸⠀⠀⠀⡾⠁⠼⣀⣘⣏⠱⡎⠀⠀⡞⠲⠶⠦⣠⣥⣤⣥⠤⠔⠛⣄⠀⠈⠷⠤⢤⣉⠤⠋⡙⠳⠛⣆⠀⠐⢯⡀⢀⣄⣀⠴⠃⠀⠀⠀⠀⢳
⠀⠀⢸⠀⠀⣘⠓⠛⠀⢉⣁⣀⣀⡱⡴⠞⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠆⠤⠠⠀⢀⣈⣉⣉⣉⡉⣦⠀⢀⡔⠋⠙⠀⠈⠐⢢⠀⠀⠀⠈⢇
⠀⠀⠸⢰⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢊⠁⠉⠉⠆⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠑⠒⠒⠓⠒⠊⠁⠀⠀⠀⠀⠸⡄
⠀⠀⢠⠀⣄⠀⠠⠒⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠂⠀⠀⡾⠠⠄⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⢶⠀⠁
⠀⠀⢸⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⢸⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢈⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏
⠀⠀⢈⡀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⡏⠑⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⡇⡆⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠇⠀⢸
⢶⡶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠂⠀⠀⢀⠀⠁⠀⡇⡇⠀⠀⠀⠀⡏⡅⠀⠀⠀⠀⠠⠆⠀⢸
⠈⢷⠀⠵⡄⠐⠒⢰⡆⠀⠐⡆⠀⠀⢙⠔⠁⠀⣀⡴⠆⠀⠀⠀⡇⡅⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠈⠳⢤⡉⠳⢄⣸⡇⠀⢀⣗⠤⠚⣢⡤⠒⠊⠀⡇⡀⠀⠀⠀⡇⡅⠀⠀⠀⠀⣇⡇⠀⠀⠀⢀⢸⡂⠀⢸
⠀⠀⠂⠀⠉⠻⠯⠿⠷⠾⠟⠛⠛⠫⢛⠓⠀⢀⡒⠓⠂⠀⠄⠡⠗⠣⠤⠄⠈⠀⠷⠇⠀⠀⠀⠲⠼⠤⠤⠼⠶⠦⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠀⠀⠀⠀⠀⠀⠀⡆⠉⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡿⡖⣄⠀⠀⠀⠀⠀⣇⢀⣀⠀⠀⡄⡾⠀⠀⠀⠀⠀⠀⢀⣀⣠⠒⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⣿⣆⠀⠀⡀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⣄⢠⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡄⡀⠆⠘⠀⣳⠀⠀⠀⠃⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠈⠁⠀⣄⣃⣌⠏⣄⡈⠀⠱⠄
⠀⠀⠀⣰⡀⢸⡽⠁⠈⠦⣽⡉⣿⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠐⠶⠇⠐⡆⠸⣄⣸⠖⠃⣦⠇
⠀⠀⠀⠙⣍⡩⢀⡀⢤⠀⣼⠀⡆⠀⢀⡀⠀⠀⠀⢀⢤⡀⠀⠀⡤⣤⡄⠀⠀⠀⠀⣟⣉⣀⣇⠖⠈⠩⣤⣴⠁
⠀⠀⠀⠀⠀⣇⠀⠈⡿⢺⣼⣿⠗⠀⠈⠁⠀⠀⠁⠋⠉⢳⠄⠁⣄⠀⡇⡰⠙⠂⠀⠈
⠀⠀⠀⠀⠐⠚⠛⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠼⠤⠤⠞⠚⠁⠓⠀⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠙⠛⠦⣄⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢱⡀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣿
⠀⠀⠀⠻⠿⠛⠛⠀⠀⠀⠀⣇⠀⠀⠀⢺⣄⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⢦⣉⠓⠒⠶⠶⢒⡿⠋⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠈⠙⠒⠒⠋⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠚⠉⠁⠀⠀⠀⠀⠀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣤⣤⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢿⣿⣿⣿⢛⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠁⢺⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠈⠉⠙⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣺⣶⣾⣿⣿⣶⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣀⣤⣤⣀⣈⣙⣯⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⠻⠛⠿⠿⠟⠟⠿⠿⠿⠟⠻⠿⠿⠿⠛⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢿⡷⠀⠰⣿⠇⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠙⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠋⠀⢀⣴⡄
⠀⠀⠀⠀⣠⣀⠀⠉⠁⠀⠀⣴⣦⠀⠀⠿⠿⠿⣿⣿⣷⣶⣤⣀⠀⠉⠁⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠛⠋⠀⢀⣴⣦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣷⣄⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⡧⠰⣿⠟
⠀⠀⣶⡆⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠠⣶⡆⠀⠀⠀⠀⠀⢀⣤⣄⢠⣤⣄
⠀⠀⠈⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠈⠻⠟
⠸⡿⠆⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠾⡿
⢀⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣤⣤⣀⣀⣀
⠿⠿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⢿⣿⡿⢿⣿⣿⣿⣷⣦⣄
⠀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⣫⣿⡿⠋⠀⢸⣿⡇⠀⠈⢻⣿⣯⠛⢿⣿⣦⡀
⠘⠿⠃⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠀⢰⣿⡟⠁⠀⠀⢸⣿⡇⠀⠀⠀⠹⣿⣧⠀⠉⣻⣿⣦⡀
⠀⠀⣤⡄⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⢿⣿⣶⣿⣿⣀⣀⣀⣀⢸⣿⣇⢀⣀⣀⣀⣽⣿⣷⣿⡿⠿⣿⣷⡀
⠀⠀⠛⠃⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⣽⣿⠟⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠛⣿⣿⠁⠀⠀⠘⣿⣷
⠀⠀⠀⠀⣾⡷⠀⠀⠙⠻⣿⣷⣦⣤⣀⣠⣿⡟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠉⠛⠻⠿⢿⣿⣇⣀⣀⣀⣠⣿⣏⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⣶⠀⠀⣀⣼⣿⡿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⢿⣿⡇⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⣿⣿⡀⠀⢀⣀⣀⣸⣿⣇⣀⣀⣀⠀⠀⣼⣿⠃⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⣀⣠⣼⣿⣿⣿⣿⠿⠿⢿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣤⣀⠀⣼⣿⠏
⠀⠀⠀⠀⢀⣤⣄⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠛⢿⣿⡄⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣾⣿⠛⠻⢿⣿⣿⠏
⠀⠀⠀⠀⠈⠻⠿⠀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⡈⢿⣿⣄⠀⠀⢸⣿⡇⠀⠀⢀⣾⣿⠃⣠⣶⣿⠟⠁
⠀⠀⠀⠀⢀⣴⣶⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣿⣿⣷⣄⣸⣿⣇⣠⣴⣿⣿⣷⣿⡿⠛⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣭⢩⣭⣭⣭⣩⡭⣭⢭⣭⡭⡭⢭⣩⢭⠍⢩⣍⣍⢩⡍⣭⡍⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣀⣉⣈⣉⣉⣉⣉⣁⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣦⠀⠀⠀⠀⠀⠠⣾⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠈⢻⠀⠀⠀⢰⢺⢸⡷⡄
⠀⠀⠀⠀⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⢴⡤⠀⠀⠈⢹⢸⠷⠁
⠀⠀⠀⠀⠛⠋⠛⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⢀⡆⢦⠀⠀⠀⠀⠀⡞⠁⠀⡀⠀⠰⠿⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⣾⢦⡀⠀⠀⣸⠀⢸⡄⠀⠀⣀⢾⡆⠀⠘⠉⢦
⠀⠀⠀⣀⣄⡀⢀⡠⢺⢻⡢⣄⠀⠀⣠⡀⠀⠀⣀⣤⡉⠐⢤⣧⡀⣀⣧⠴⠈⠁⠀⠀⢠⠁⠀⠀⠑⢄⣠⠄⠒⠂⠠⡄
⠀⠀⠀⡏⠋⡏⢸⡷⣛⣚⡛⣾⣧⠉⢹⠉⢁⣩⠉⣯⠍⡍⠸⠀⠀⠀⠸⡀⠀⠀⠀⢀⠎⠀⡀⠀⠀⠀⠉⠛⢇⣉⣭⠷⠒⠒⢢
⠈⠋⠀⣷⣶⡧⣾⣿⡿⣿⢻⣿⡇⠈⣼⡄⠁⣿⣾⣿⣴⣷⣥⣿⣾⣿⣧⣥⣴⡿⠶⠿⠼⠟⠟⠉⠉⠉⠓⠈⠁⠀⠀⠉⠉⠉⠉⠘⠓
⠀⠀⠀⠀⠀⠀⠘⠐⠂⠀⠒⠙⠀⠀⠀⠀⠀⠈⠀⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿