Pup Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠛⠛⠀⠙⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠴⠛⠹⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⢀⡤⠞⠃⠀⠀⠀⣠⠤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠢⢤⣄⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉
⠈⢳⣤⣀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡿
⠀⠀⠉⠉⠡⣴⠶⠊⢹⡇⠀⠀⠀⠀⠀⠀⣰⡯⢶⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡤⣰⠟⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣀⣤⣾⠋⠀⠀⢠⠟⠀⠀⠙⠓⠆⠒⠷⠒⠋⠀⠀⠀⠀⣸⢱⡇
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡴⠋⠉⠀⠾⢦⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢷⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣿⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣤⣭⠆⠀⠀⠀⠀⠀⠈⠓⣦⠀⠀⠘⠦⣤⡴⠞
⠀⠀⠀⠀⠀⠀⠱⣦⣄⣀⣨⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⢰⣬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣭⣞⣿⡶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⠟⠉⠉⢩⣳⣲⢿⣟⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⢦
⠀⠀⠀⠀⠀⠘⢄⠌⠀⢨⢋⣿⣵⣾⣿⣿⣿⣿⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠯⢠⣼⣿⣿⣿⣿⣿⡿⠛⠃⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⣿⡏⠁⠀⠀⠀⠀⠀⠈⢳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⠶⠦⣤⣤⣤⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠇⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⣿⣟⣿⣻⣯⣿⣟⣿⣿⣿⠿⣿⣶⣤⣤⡈⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠁⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡗⣿⠿⣿⢿⣿⣿⣛⠉⠻⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣦⢄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣺⣿⣻⣾⣯⣿⣿⣿⣿⡅⢄⣻⡏⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠘⣆⠀⠀⢰⣟⣿⣿⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡦⠋⠟⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣧⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠟⠟⢛⡉⣉⡁⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢣⠀⠀⢫⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣭⠴⠶⠛⠛⠛⢻⣿⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠇⠀⢸⠇⠈⠉⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢳⠰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡳⣆⣀⠮⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡞⢻⡀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣋⠁⢠⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⠁⢰⠁⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡴⢛⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠊⠁⢀⠠⠣⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⠴⡤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⡍⢻⣿⣿⣿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠁⠙⠁⣼⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠳⡀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡆⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣸⣿⢹⣿⣿⣿⠋⠀⠀⠀⢀⡀⢀⠀⠈⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡏⠁⠀⠈⠁⠀⠀⠀⢀⣼⣿⣿⣷⠀⠸⢿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠆⠀⠀⠀⠀⠀⠀⢈⣿⣿⡿⠟⠀⠀⠀⠀⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⣄⠀⠀⠈⠐⠐⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠠⣍⣀⣠⡴⠇⠀⠀⠀⢠⣽⣿⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⡄⠀⠀⠉⠉⠀⠀⠀⠀⣤⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡠⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⢶⣶⢶⢶⡶⠿⣯⣍⠻⢿⣦⠙⣿⣟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⡴⠾⠻⠉⠁⠠⠀⠈⡜⠉⠈⠁⠀⠠⠙⣷⣄⠻⣦⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣶⠟⠿⢻⣽⣿⠏⠀⠀⠀⠀⠀⣧⡀⠀⢠⣥⢀⣄⠀⢀⠀⠹⣿⣦⠀⢲⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡋⢸⠁⠀⠀⣾⣿⠏⠀⠀⠀⠤⢴⣾⣾⣧⠀⢠⣿⣿⡿⣧⣾⣿⣤⣤⣿⣷⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⣤⠾⠿⢿⣿⣆⢸⣿⠁⠀⠀⣿⡿⢬⠉⠻⣿⠟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢠⣿⣿⠀⠀⠐⣼⣧⡀⠀⠀⣿⣿⠈⢿⣷⣦⣾⣿⣷⢶⣼⣿⣿⡆⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣸⣿⣿⠀⠀⠈⢡⢴⣾⡓⢸⡿⠛⡆⢀⡛⢯⣼⣧⣵⠞⣼⠋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⠏⢸⣷⣠⣶⡞⠻⢌⡙⠷⣾⡁⠠⠐⠀⠉⠘⠙⢷⣶⣾⣿⡄⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢚⡟⠁⠀⢐⣦⣀⣀⣠⠏⠀⢰⣶⣦⠀⠀⠀⠀⠈⠉⠹⠇⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠐⠛⠿⠛⠃⠀⠀⣈⡉⠀⠐⣲⣶⣤⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠁⣴⣾⣿⣿⠗⠀⠀⠀⣼⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡷⢿⣿⣶⣦⡀⠀⠀⠀⠀⠐⣼⣿⢻⠿⣖⣆⠀⠀⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣴⣽⣿⣿⣷⡄⡀⠀⠀⢸⣵⣧⡼⠦⠀⢉⣴⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠼⢿⣿⣿⣿⣿⣿⣉⠐⠛⢀⣠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢈⡋⠈⣩⣾⣿⣿⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣉⠽⣿⣿⣿⣿⢿⡇⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⠋⠀⠘⠲⠿⠞⢻⣏⣾⣸⠃⠹⣿⢳⣿⡿⣏⠻⠿⠛⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠁⠀⡄⢀⣤⠀⠀⠀⠈⠙⠁⠀⠀⠀⠊⠋⠉⠀⠀⠀⢀⣀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠛⢿⣿⣷⣶⣖⣫⣴⣸⡀⠀⠀⠀⠀⠀⢀⣾⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⠄⠚⠛⠻⣿⣿⣿⣿⣿⣿⣯⡄⠀⠀⠀⠘⣿⠏⢙⡿⢿⣿⣿⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠈⢻⣿⣿⡿⠛⠿⠷⠿⠳⠖⠚⢿⡀⠛⠀⠀⠀⠀⠉⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢰⡆⠀⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠀⠀⠆⠀⡌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⠀⠀⠰⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠃⠀⢸⠀⠁⡀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢧⡰⣿⢶⣄⣸⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣄⠀⣿⣄⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁
⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⢰⠈⠀⠀⠀⣮⠉⠦⠠
⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⡟⠀⠀⢀⣮⠡⣀
⠀⠀⠴⡒⠁⠀⠀⠈⠀⠀⠀⢧⠀⣠⠞⣣⠀⠈⠑⡀
⠀⠀⠙⠥⠀⣀⡀⢴⡖⣲⠆⠈⠛⠁⠐⣟⠀⠀⠀⢸⣿⢲⣤⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣍⣠⠊⠉⠁⠂⠠⠀⠀⣀⡷⠀⠀⣰⠟⠁⢸⣿⡇⠘⢿⣿⣿⣟⡛⠛⠛⢋⣙⣉⡉⠛⠛⠒⠒⠠⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠓⠄⠀⠸⡇⠀⠀⠁⠀⠐⢿⣿⡇⠀⢸⡿⠏⠈⢹⣿⡟⢿⠙⠏⠉⠉⠀⠀⢀⠀⠀⢌⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠃⠀⠀⠀⠀⡴⠀⣿⠃⠀⠀⠇⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠈⡆⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢼⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠃⠀⠀⠀⠀⠁⠡⡀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⣦⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠠⡹⣆⠀⠀⠀⣴⢁⠀⠀⠀⠀⠀⠀⣀⠄⠀⢰⡶⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣶⣦⣬⣵⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠓⠷⢌⠀⠀⠒⢩⡇⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⣷⠘⢿⣿⣧⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣼⠃⠀⠀⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠘⣿⣷⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣿⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠹⣿⣿⣧⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠙⣿⡏⠻⡀⠸⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢸⠃⠀⢱⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡆⠀⡼⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡘⠀⠀⠈⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣦⣀⡜⢁⣤⣤⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣙⣲⠇⠀⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠶⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⠇⠲⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠏⠀⡀⠀⠀⠀⠀⢱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠮⠁⠶⣿⠆⢘⢋⡀⠒⠦⠀⠬⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣆⡈⢥⡤⣯⠍⠁⣤⠐⢠⠐⢨⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣯⡿⢿⠷⡒⠗⠀⣤⣴⣷⠶⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯⣿⡷⣞⣼⡏⠀⣆⠀⠉⠁⠀⠀⠀⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣤⢾⣿⣿⣿⣶⡜⣆⠀⠀⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⢼⣄⠂⠀⠀⠠⠸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣍⠘⡟⠓⢻⣿⡿⢿⣓⣻⣏⢸⠛⢛⡱⠞⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢡⠀⢁⠈⡷⠄⠀⠀⠉⠻⢿⠉⠁⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣦⠀⠛⠻⣿⣤⡀⠈⣁⣄⠀⠀⠀⠀⢀⠦⢤⣴⡟⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⡀⠂⠐⠄⠹⣝⣳⣮⠵⠀⠁⠀⠰⠋⠀⠈⢻⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠁⠆⠀⠀⠁⢻⣿⡆⢐⠀⠀⠀⠗⠀⠀⠀⠀⠓⠤⠤⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣀⠄⠀⠀⠰⣿⢿⣤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢶⡌⠀⠀⠈⠉⠓⢮⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⠀⡘⣀⠂⡽⠤⠴⠬⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠁⠒⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡚⠲⡄
⢠⠐⠒⠀⠉⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠠⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠇⠈⠁
⡇⠀⠀⠀⢀⡀⠀⠈⠁⢀⡈⠀⠠⠀⠿⠀⠱⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⠀⠀⢦⠈⢆
⢱⡀⠀⠀⠁⠀⠀⠀⠎⢉⣿⣿⣦⠀⠀⠀⡀⠀⠀⠀⠈⠀⠠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⢀⣀⡀⠀⡼
⠀⢱⡀⠀⠀⠀⠀⠀⠁⢈⣻⡿⠟⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢹⠀⠀⢰⠀⠈⠀⠀⠊⠀⡀⠠⣿⠏
⠀⢀⠃⠀⠀⠀⠠⣀⡠⠴⠢⡀⠀⢀⠀⠀⣸⠇⠀⠀⣀⡤⠖⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢋⠉⠘⠁
⠀⠸⡀⡄⠀⠀⠀⡀⠀⠀⠀⠈⠉⠁⠀⠀⢱⠁⢀⠎⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠢
⠀⠀⠈⠀⣸⠀⠀⠇⠀⠀⠀⣰⠀⠀⠀⠀⢏
⠀⠀⠀⢠⠇⠀⠀⠘⠦⣤⠴⠃⠀⠀⠀⠐⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⠀⠀⢀⠄
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠂⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠠⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⣶⡈⢦⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠈⢧⠀⠀⣀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⣀⠄⠈⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢣⠀⠀⠑⢄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠠⠴⠶⣤⡀⢘⢀⠀⠀⠑⠄⠀⠀⠳⣄
⠀⠀⠀⠀⢠⠔⠐⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣶⣾⣻⡇⠀⠲⡄⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠁⢠⡀⠀⠀⠀⠛⡟⠋⠀⠀⡠⠓⠀⠀⠤⠤⠤⠶⠶⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠖⠚⠉⠉⠉⠉⠒⠲⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢸⠀⠀⠀⢀⣶⢦⠀⠀⠀⠀⠀⠀⡴⣶⡀⠀⠀⠸⡄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⣞⣀⡀⠀⠈⠿⠞⣠⠖⠉⠉⠳⢄⡳⠿⠁⢀⣀⡀⢧⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢠⠏⠉⠉⠙⠛⠛⠉⢰⣾⣭⣭⣷⡆⠉⠉⠉⠉⠉⠉⠙⡄⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⢀⡀⢉⣹⢿⣉⣀⡀⠀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠉⢏⡁⢀⡟⠁⠀⠀⠀⠀⠀⠀⣰⡷⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠤⣀⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣠⣤⠚⢹⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠉⠓⠒⠒⠒⠒⠚⠉⠉⠁⠀⢸⢀⡼⠦⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⢸⢯⡀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢳⠀⠀⠘⠦⣄⡀⠀⠀⢀⣠⠔⠁⠀⠀⡼⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣇⠀⠀⠀⠀⠉⢳⠏⠉⠀⠀⠀⠀⢰⠃⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠓⠆⡸⢦⣀⣀⣀⠀⢸⠀⠀⢀⡀⢀⡠⢇⠰⠞⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⡠⢸⣡⡄⠀⡀⠀⠀⢸⡀⠀⠀⢩⠉⠠⡌⡇⣀⠀⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠓⠖⠛⠧⠤⠷⠴⠒⠋⠉⠓⠶⠼⠤⠴⠛⠓⠻⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠶⣦⣄⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠈⠙⢦⣠⡞⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠛⡛⠷⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠀⠀⠀⠉⠛⠷⣄
⠀⠀⠀⠀⠀⡾⠛⠉⠀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣷⡀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠸⣷⠀⠀⣠⡴⢤⡀⠀⠀⠀⠀⠀⢰⠿⣍⠻⣆⢻⣷⡀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣿⠀⢰⡏⢹⡆⢳⠀⠀⠀⠀⠀⢸⣆⣸⠀⣿⠈⢻⡿⠶⠚
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣿⠀⢸⣿⣾⡇⢸⠀⠀⠀⠀⠀⠈⢿⡟⣠⡏⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣰⢿⣄⡀⠻⣿⡤⠃⠀⠀⠀⣀⣀⣀⣀⠉⠁⠘⢾⣠⠴⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠌⢻⣶⡴⠀⠀⠀⢰⣿⣿⠛⠛⢻⡇⠀⠀⢨⠇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⢷⡀⠀⠀⠈⢿⡿⠷⠶⠞⠃⣀⣴⡟⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⡆⠀⠀⠀⠈⠛⠳⠶⠶⠶⠖⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠉⠉⢻⡷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⡶⠶⠶⣦⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⣄⠀⢀⣠⣾⠿⠟⠛⠛⣟⠁⠀⠀⠀⠀⠀⠈⠛⠿⠖⠃
⠀⠀⠀⡜⡀⠀⠀⢸⣆⠀⠀⠀⠀⠈⠻⣿⣿⠟⡟⠀⠀⣿⠀⣿
⠀⠀⠀⢳⠘⢦⣀⠘⣿⣆⠀⠀⠀⢀⣼⣿⣧⣀⡇⠀⠀⣿⣄⠘⠳⣦
⠀⠀⠀⠀⠓⠦⣌⣉⣙⣋⣀⣀⣀⣘⣉⡀⣶⡘⣇⠀⢠⡀⣌⢧⣷⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠙⠛⠋⠛⠉⠛⠋⠛⠋⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠲⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠂⠀⢾⠋⠀⢀⣀⠈⠒⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢳⡄⢀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠄⠀⠀⠀⣶⡎⠀⠀⢠⣴⣌⠀⡀⠀⠀⠀⠀⠀⠀⣤⠦⣄⠀⠀⣿⣦⡀⠀⠀⠈⢧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⡇⢠⠀⠻⠿⠛⠐⠀⠀⠀⠀⠀⠰⡎⠻⠿⠟⠁⠀⢻⣿⡇⠀⡆⠀⠀⠑
⠀⠀⠀⢀⡞⣤⡆⠀⠀⣿⡇⠀⡐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠋⠀⢿⣇⠀⠃⠀⠀⠀⠀⠆
⠀⠀⠀⣰⣿⡁⠀⠀⠀⢻⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠨⡁⣻⣿⡄⣀⠀⠀⠀⡈⠋⠣
⠀⠀⠀⢻⢿⣿⣾⣄⣄⢸⡇⡏⠔⡠⠀⠀⢠⣿⣿⣿⣿⣿⣷⡄⠰⠀⠀⣦⣤⢸⣏⣿⣇⢿⣿⣄⣸⡟⣾⡟
⠀⠀⠀⢸⣿⢿⡏⣽⣽⡇⣏⢧⡀⢿⠢⠀⠈⠛⢿⣿⣿⣿⠛⠀⠀⠁⣴⠟⢁⣾⡷⣿⣿⢸⣿⣯⠁⣴⡏
⠀⠀⠀⠀⠙⢿⣿⣿⣏⢰⣿⠀⠁⠈⣇⠀⠀⠀⢀⣈⠡⠀⠀⠀⠀⣠⠃⠀⢀⣈⠀⣸⣿⡸⣿⣫⡸⢢⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣏⣞⣤⠗⠀⣀⠻⣷⣤⣶⡶⠿⣿⣶⣶⣤⡶⡫⠟⠀⠁⠈⠀⢻⣻⣇⢿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢚⠇⠀⡀⠀⠙⠃⠘⢻⣧⠀⠀⠁⠀⣨⣿⠎⠀⠀⠀⠀⠢⠀⢀⣡⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡀⠀⠀⠀⠀⠀⢻⣷⣦⣤⣶⠟⠁⠀⠀⠀⡀⠀⢆⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢷⡄⢃⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢯⠀⢨⣶⣾⢿⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⠋⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡀⠉⠈⠁⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠳⣶⣄⢰⡆⣀⠀⣄⣠⣅⠀⠿⠂⠁⠀⠀⠀⠙⠂⢈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠙⠛⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠄⠀⠀⠘⠐⠀⠀⠀⡀⠀⠀⠀⢀⠀⢈⠀⠁⠀⠍⠁⠈⠁⡉⠁⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡐
⠀⠀⢠⠁⠀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠈⡆⠀⠀⠀⠈⠀⣠⠀⠀⠀⡄⠀⠁⠀⠀⠀⠈⠉⡅⠀⡄⢀⣖⠀⢠⠀⠈⠉⠀⠀⠀⠛⡛⠀⠈⠃⠀⠀⠉⠀⠘⡛⠃
⠀⠀⠀⠂⠀⡄⠀⠀⢴⠆⠀⠀⣘⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⠀⠀⠀⠛⠀⠀⠀⠀⠃⠀⠀⠀⠀⣁⠀⠀⠀⢙⠃⠀⠀⢀⡁
⠀⠀⠀⠀⠈⠉⠢⢠⠀⠀⡄⠄⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⡀⠀⠀⡘⠀⠀⠀⠀⠠⣀⠀⠀⠜⠙⠡⠀⠀⣄⠀⠀⠀⠀⠀⠹⢄⠀⠀⠁⡍⠀⢀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠠⠀⠄⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠒⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠔⡠⠈⠐⠀⠐⠂⠐⠁⢈⠐⡀⠀⠀⠀⠄⠈⠉⠀⠉⠉⠉⠁⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠑⣀⢀⡐⠀⠈⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠐⠁⠆⠀⠀⠀⠀⠀⠑⡎⠑⠂⠀⠀⠆⠀⠠⠀⣀⠀⠀⠀⣀⠀⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠠⠀⢋⠀⣤⡤⡘⠁⡀⠀⠀⠀⠀⠀⠢⠆⠀⠈⠀⣤⡀⠁⠀⠘⠔⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠐⠂⠀⠈⠀⠀⠀⣀⠀⠀⠀⠠⡦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢡⠆⠀⣩⡀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠐⠐⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⢀⠤⠀⠵⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⣀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠑⢄⠀⠂⠀⠂⢀⠄⠁⠀⠀⠀⣿⢀⠀⢀⣀⢄⣀⠀⠀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠈⠻⢤⢌⠉⠀⠉⣠⡆⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⡎⠀⠀⠐⠖⠀⠀⠀⠶⢐⠀⠀⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠠⠊⠀⠂⡀⠀⠀⢀⠘⠀⠈⢀⠀⠀⠀⠀⠄⠀⠛⠀⠙⠀⠠
⠀⠀⠐⡀⠀⡄⠀⢀⣄⠀⠀⡬⠀⡐⠀⠀⠀⢀⠀⠀⠀⠲⣄⠀⢠⠖⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠁⢀⡀⠈⠁⠀⠀⢨⠀⠀⠀⢶⡀⢠⡀⠀⢀⡄⠀⣦
⠀⠀⠀⠈⠐⠀⢄⡈⠯⣀⡄⠈⠊⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⣶⠀⠈⠦⠄⠀⠀⠀⠀⠀⢄⠀⡦⢀⡘⢏⣀⠰⠀⢀⠀⠀⠀⠀⠘⢦⠀⠀⣤⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠛⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠂⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⢸⡤⠀⣉⡀⢠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⡐⠈⠘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠈⠀⠈⢂⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢖⠀⠀⠀⣲⣀
⠀⠀⡀⠀⠀⠀⠘⠂⣤⠐⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢠⣤⠀⠖⠈⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠀⠉⡀
⠀⠀⠉⠰⠀⢤⡄⠀⠲⠆⢀⣦⠀⠐⠈⠀⠀⠀⠀⢀⠀⠀⢀⠸⢃⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡅⠀⠀⢀⠃⠀⠀⠀⠀⠀⡎⠀⠀⣛⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠠⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠂⠄⠒⠁⠀⠀⠀⠀⠀⠀⠳⠦⣄⡌⣠⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⣶⡴⠚⠉⠉⠉⠉⠉⠛⢶⣤⠤⠶⠶⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢦⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣤⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠄⠈⢐⡂⠀⠀⠀⣒⡒⠄⠀⠀⠈⠙⢿⣆⠀⠀⠀⠀⠀⠑⢢⣄⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⡾⠛⠀⠀⠀⠀⠻⠟⠒⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣀⡴⠚⠉⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠉⠉⠁⠀⠀⠀⠀⡘⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠛⠒⠶⣤⡀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢿⣷⣤⣀⣀⡤⠄⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠛⠉⠀⠀⢀⣠⠖⣫⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣲⠶⠶⢖⣚⣭⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠛⢟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠠⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡟⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠙⣄⡀⠀⠀⢠⠇⠀⠀⠀⠀⠀⣰⠇⠀⣠⣟⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⢶⡞⠀⠀⠀⠀⠀⣰⣿⣶⠟⠁⠈⠓⠦⣤⡴⠃⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡤⠤⣾⡁⠀⠀⠀⠀⣴⣣⡴⠋⠀⠀⠀⠀⠀⠘⢆⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣆⠀⡄⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠶⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠂⠀⠁⠉⠉⠉⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣈⠁⠀⠀⠀⠓⠀⠀⠄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠘⠛⡁⠀⠀⠈⠭⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢸⠀⠀⠐⠄⢀⠀⠀⠀⢀⠀⢠⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠜⣄⢣⢀⡠⠤⠤⡤⢴⠋⡠⡞⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠢⡹⠀⠀⠀⡖⡡⠊⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡇⠀⠀⡇⠀⠀⠀⡋⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⡀⠀⠱⢄⡠⠔⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⡞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡆⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⢰⠀⠀⠀⢛⠀⠀⠀⠀⠀⡜⢇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠄⠑⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⢠⠞⠀⠈⢆⠀⠀⠘⡀
⡔⠋⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢣⠀⠀⠘⡄⠀⡰⢻⠀⠀⠀⠈⢣⠀⠀⢃
⡇⢄⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⡟⠀⠀⠀⠀⠀⠘⡄⠀⠀⢣⠐⠁⣸⠀⠀⠀⠀⠀⢣⠀⠸⡀
⢣⠈⢦⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⡀⠀⠘⡤⢖⣁⠀⠀⠀⠀⠀⠈⢇⠀⢣
⠀⠑⢤⣈⠉⠛⠉⠉⢳⡄⠀⠀⠢⣇⣀⠀⠀⠀⠀⠀⢀⣱⠀⠀⢸⡁⠀⢙⡆⠀⠀⠀⠀⠀⢣⠈⠓⠤⢄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠢⠼⠤⠴⠼⠀⠉⠉⠉⠉⠉⠁⠸⣀⢀⠀⢈⢻⠉⠁⠀⠀⠀⠀⠀⠀⠣⣀⣰⣀
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⡀⠀⣀⣀⣠⣤⣤⣠⣤⣤⣤⣴⢿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣀⠄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣼⠁⠀⢀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠀⠀⠘⠀⠀⣷⣾⣿⠈⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠿⠟⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠠⡀⠀⢸⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⠀⠀⢀⣀⣤⣤⣀⠘⠀⢸⠃⠙⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣥⣶⣶⣾⡇⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠻⣿⡗⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠰⣄⠀⠀⢀⣠⠴⠒⠛⠓⢢⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠸⢻⣿⣧⣤⣶⣶⡿⠛⠉⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠧⢀⡀⠀⠀⠀⠈⠙⠉⠋⠁⢀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠶⣄⣤⣦⣤⣤⢴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠋⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠲⢄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡄
⠀⠀⠀⠀⢰⠀⠀⠀⡴⠁⠀⠀⠀⠈⢃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡀
⠀⠀⠀⠀⢠⠀⢀⡞⠀⠁⣀⡤⠤⠶⠖⠒⠺⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⡇
⠀⠀⠀⠀⠈⠦⠊⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⠞⠛⠛⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡇
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⡠⠴⠒⠒⠒⠒⠒⠤⠄⠀⠘⠀⡟
⠀⠀⠀⣀⠄⠈⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠐⣌⠀⠀⠉⠁⠀⠀⠀⠀⠀⣠⣦⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢰⠁⣠⣤⡀⠀⠀⠀⠀⠀⠀⠧⠞⠀⠐⠀⠀⢀⡠⠴⠒⠒⠓⠲⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢃⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠈⠀⠘⡟⠛⠃⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠥⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⢀⣀⠀⠀⢢⠜⠙⠛⢛⠟⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⡜⢁⡄⠀⢰⠃⠀⠀⡴⢃⣀⣀⡤⠖⠀⠤⢄⣀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠡⡎⠀⠀⠸⣤⡤⠞⠁⠀⠈⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠑⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡤⠤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠳⢄⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠆⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠳⣄⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠄⠐⠒⠂⠠⠄⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⠃⠀⠀⢸⣯⠆⡠⠤⠤⡀⢯⣿⠀⠀⠀⠇⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠸⠒⠒⠒⠐⠒⢊⣤⣒⣠⣌⠑⠒⠒⠒⠐⠺⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠇⠀⠀⠀⠀⠀⣀⣉⡿⣏⣁⡀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠫⣀⡨⠁⠀⠀⠀⠀⠀⡰⡖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠢⠤⢀⣀⣀⣀⣀⠀⠤⠄⠲⡋⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⠃⠉⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢣⠀⠈⠢⣀⠀⠀⠀⢀⡠⠊⠀⢠⠃⢱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆⠀⠀⠈⠉⢲⠊⠁⠀⠀⠀⡜⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠂⡜⠢⠤⠤⠀⢸⠀⠀⢀⡀⠜⣄⠚⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣰⣘⣴⡀⢰⠀⣀⠼⢄⡀⢐⠀⢰⣼⣰⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⢤⣶⣶⡦⡤⢀
⠀⠀⠀⣴⣿⣿⠟⣱⠇⠀⠉⡽⠡⣩⣍⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⣤⣤⣼⣶⣬⡑⢂⢹⣦⡀
⠀⠀⠘⡟⢿⣿⠂⣟⠀⠀⢸⡗⢰⣿⡟⠃⣘⣦⠤⣶⢾⢿⢿⠿⠻⣖⠂⢉⡀⠶⢶⣭⡏⠉⠉⠛⢦⡻⣿⣿⡄
⠀⠀⠀⢣⠘⣿⣀⡡⠂⠀⣾⠃⣸⣿⡷⠾⠋⠁⠴⢁⡴⠀⣸⠇⡐⠈⠓⢦⡌⠓⢄⡙⢇⠀⠀⠀⠀⠹⡝⣿⣿
⠀⠀⠀⠀⠁⢸⣿⠀⠀⠸⠏⢠⣿⠟⠀⠀⠀⠀⠠⠏⢠⡾⠋⢠⡷⡄⠐⣦⡉⠓⢤⡙⢌⢦⠀⠀⠀⠀⠸⣽⣿
⠀⠀⠀⠀⠀⡈⣿⠃⠀⡁⠀⣾⡟⠀⠀⠄⠀⠠⠁⢠⠟⢀⣶⠟⢀⣷⠰⠘⡟⠁⠀⣀⣀⠘⢧⠀⠀⠀⠀⠙⠿⠂
⠀⠀⠀⠀⠀⢀⠉⠑⢸⠁⣸⣟⠀⠀⠀⠤⠀⠀⠀⠁⠀⢁⣠⡤⣄⠙⡆⡆⢷⡄⠘⢩⣿⣫⣍⣦
⠀⠀⠀⠀⠀⠀⢗⣻⣿⢠⣿⣿⣦⣐⠀⠀⠀⠀⣠⠖⣛⣛⣻⣷⣤⣤⡗⠁⢸⣷⣴⣾⣟⡷⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠘⣿⣿⣿⣿⣿⣟⠀⠈⣡⣾⣿⢿⡝⠻⣿⣿⣷⠀⣾⣿⣿⣿⡿⠴⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⢹⣿⡿⠀⠍⠉⣶⣄⠁⠱⢽⡯⠴⢃⣿⣿⣿⡆⠈⠻⣿⣭⡳⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠰⠂⢀⢻⣦⡀⣀⣶⡄⠈⡼⡫⣿⠇⠠⠧⠃⠀⠉⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠨⡄⠻⣿⣿⡿⢁⠎⠜⡴⠁⢀⠄⢀⡀⡀⠫⣤⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣆⠀⠀⣷⠀⢹⡟⢠⠎⢀⡾⣂⡀⠚⠀⢸⡧⢸⢀⠻⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡏⣿⣿⣦⡀⠈⣇⢸⢁⡞⣠⣟⡴⢛⣷⡀⠉⠑⢤⣸⢨⣞⡡⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢱⢹⣿⣿⣷⠰⣹⣸⡸⢠⣿⡯⠶⢋⣹⡇⢰⠖⠁⢸⡂⠀⠛⠂⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠂⠁⠉⢻⡀⣿⢻⠇⣼⣯⠤⠶⠟⢻⡆⠈⠀⠀⡀⠷⢦⡘⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠄⠀⠀⠈⡇⠟⣼⢀⣿⣷⡦⠴⠖⣻⣇⢠⠄⠃⠐⠂⠈⢷⡆⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⠐⠐⠆⠀⠀⢳⣾⣿⢸⣿⣿⣿⣄⣀⣿⡏⠈⡄⠖⠒⠛⢋⠸⣧⣸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⠀⠈⠀⠀⠀⠀⠔⠀⢻⡇⣼⣿⣿⠗⠊⠙⡻⠀⣀⣤⣤⡀⠀⠀⠀⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠛⢧⣸⣶⠀⠇⠀⠌⡖⠀⡤⠀⠳⣽⣟⣧⣴⡶⢏⣵⠃⣿⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢴⣄⣀⠈⠄⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠀⣹⣿⡿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠀⠀⠀⠀⠀⠰⠟⠛⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡠⠔⠒⠉⠉⠷⠿⣶⣦⣭⠭⠉⠉⠉⠙⠳⣆⠠⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣴⠶⠆⠀⠐⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢺⡏⣠⠀⢠⠂⠀⠀⠀⠀⠀⠀⠈⠀⠢⡀⠀⠀⠘⡇⠀⠄⠀⠀⠀⠀⠈⣏⢧⠸⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠁⠀⣨⣴⡇⠀⡇⠀⠀⠀⢀⡀⠀⠀⣀⡆⣠⠇⠀⠀⠀⠁⢦⣠⣶⣀⠤⠀⠀⢸⣾⠀⠃⠉⠻⣄
⠀⠀⠀⠀⠀⠀⣼⢸⣿⡇⠸⣿⣿⡇⢰⡇⠀⠀⣀⡾⣿⣻⣿⣿⣆⢿⠀⠀⠀⠀⠀⠀⠻⣿⣽⣷⠄⡀⠈⣟⡇⠀⠀⠀⠿⢷
⠀⠀⠀⠀⠀⠀⢹⣆⢻⡇⠀⢻⣿⡇⢸⡇⠀⠉⠉⠁⢼⡿⠉⣿⡏⠀⠀⠀⠀⣷⠀⠀⠰⡙⠧⠀⢀⡄⠀⣯⡇⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⠀⠀⢿⣷⠀⡿⡄⠀⢤⣀⠀⠀⢈⣿⠆⠀⠀⣀⣸⡟⠃⠀⠀⢀⣰⠿⠋⠀⠀⣿⣇⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⢸⣿⡀⡇⠀⠀⠀⠙⢻⡇⠀⡏⢀⡴⠛⠉⣠⣤⣄⠐⢶⣦⠙⣦⠀⠀⠀⣿⣏⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⢸⣿⢣⣇⠀⠀⠀⠀⠊⠀⠀⠀⡜⠁⠀⠀⣭⣷⠄⣀⠸⣯⡅⠸⠀⠀⢠⣿⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⣃⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⣿⡷⠞⠀⠀⠠⢇⠀⡿⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⢿⣿⡇⠀⠋⠘⣵⣆⠀⠀⠀⠀⡀⡀⢸⣿⣿⢈⣶⡆⠄⠀⢾⠃⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠈⣿⣷⡀⠀⠀⠻⣿⣆⠀⠀⡄⡄⢤⠴⣿⢻⣾⣿⣷⣍⣀⠌⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⢹⠈⢳⣄⠀⠀⠘⡿⢿⣶⣤⣤⡴⣚⠛⠯⠋⠀⢹⡛⣉⠞⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡼⡆⠀⢇⠀⠉⠑⣦⡀⠈⠙⠛⠻⣏⣴⡙⠦⣀⠀⠀⠘⣿⠃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠈⣆⠀⠀⠈⠛⠂⠰⢤⣀⠙⠻⠿⠶⠶⠖⢚⡶⠋⠀⠀⠀⣼⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠈⠀⠀⢠⢠⠀⠀⠀⠈⠑⢶⣶⣶⠃⠈⠁⠀⠀⠀⠀⡀⢸⡟⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣻⣿⠃⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢀⡾⣿⣿⡀⠀⠀⠀⠀⠀⢠⠀⢸⠃⠀⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⣿⡀⠀⠀⠀⠀⠀⠘⢱⠃⡀⠀⠀⠀⠀⠙⢿⣟⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠁⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢿⣇⣤⠀⠀⠀⠀⠀⢸⣿⡇⢀⡄⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡇⠀⠀⠀⠻⡿⣆⠀⠀⠀⠀⠈⠈⢿⣿⣧⣆⠀⣠⣶⣄⠀⣀⠀⠀⠀⠠⣹⡆⠀⠀⠀⡏⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⡀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠁⠙⠻⢶⡏⠀⠉⠻⠇⠈⠁⠀⠀⢹⠃⠀⠀⡼⠃⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠳⣄⢀⡀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠈⠛⢷⣾⣷⣄⢲⣄⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣷⠖⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⠉⣀⠀⠉⠐⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⢠⡙⣿⠁⣼⠟⣀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠉⠁⠃⠄⠹⠀⠋⢈⣷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢈⣿⣷⠀⠰⢲⢳⠢⠀⢠⣿⢽⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠃⣿⡀⣤⢠⣇⢠⠀⢸⣿⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢧⣥⢟⡹⢦⣀⣸⣏⣷⡍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣽⠃⢲⣿⠿⠊⠑⣫⣾⣿⠇⠘⡆⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⢘⡇⠀⠀⢠⣿⣿⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣾⣷⣆⣴⣿⣿⣿⠁⠀⠀⠀⣾⠁⢠⠀⢸⣄⣤⣤⠀⠀⠀⠻⠦⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⣰⣿⠀⣾⠀⢸⣿⣿⣿⣤⡀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠄⠀⠀⠙⠻⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⣿⣿⣾⡟⢀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⣸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣀⡀⠸⣧⣤⣀⠀⢀⣠⣤⣀⣠⣶⠃⠀⢰⣿⣿⣿⡿⠛⠛⠻⢟⡁⠙⠻⡷⣷⡀⠰⠁⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣠⡝⠁⠀⠀⠈⠛⠿⣿⣿⡿⠿⠟⠋⠁⢀⠀⠀⠛⠉⠀⠀⠀⠀⠀⣼⣿⡤⠔⣃⡼⠃⠀⠁⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡰⠋⠁⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⣤⠞⠋⠁⠀⠀⠀⠀⠀⠈⢌⡉⠟⠋⠀⢸⡋⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿
⠀⠀⠄⠒⠉⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒
⡤⢀⢀⡄⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⠀⠁
⠓⠛⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⠀⡧

⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⢀⡀⠿⠇⣀⡀⠛⠃⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷
⠀⠹⡿⠻⣿⣿⣿⣿⣷⣶⣿⣿⣷⣿⡁⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠙⠻⠷⢦⣄⣀⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠆⢀⡔⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠻⣿⡟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⡿⢹⣿⠀⠀⠙⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠡⣾⠟⠁⠀⠀⠈⣿⠻⣿
⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠿⠄⠀⠀⠀⠀⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠛⠃⠸⠿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠹⠸⡈⠠⠹⢸⢘⠸⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣼⠇⠀⠀⢠⣾⡿⠿⢿⠿⠿⠿⠶⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠤⣤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣽⠀⠀⢠⠄⠀⠘⠇⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡝⠀⠀⠈⠛⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠠⣇⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠓⠀⠀⠄⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡗⠲⠛⠦⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠙⡄⠀⠘⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⠤⠶⢤⡀⠀⢢⠀⠀⠀⡔⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠉⠀⠀⠀⠀⠹⡆⠈⡄⠀⠀⡇⠀⣠⠀⠸⡇
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢧⠀⠀⣧⠞⠹⡄⠀⡇
⠀⠀⠀⠀⢸⡷⡄⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠀⠀⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠰⣧⠙⢦⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡠⠚⡇⠀⡇⠀⠀⢇⠀⠇
⠀⠀⠀⠀⠀⠈⣷⣄⠈⠓⠦⢄⣸⣄⣀⠀⠀⠀⢀⡤⠚⢁⣠⣖⠉⠉⠙⡇⠀⢧⠀⠀⠈⢣⠘⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠄⠠⠤⠾⣅⣀⣀⣀⣈⣉⣉⡀⣠⡠⠏⠉⠀⠙⢄⠀⡁⣠⠀⠀⠁⠠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣶⣶⣾⣿⣷⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⡿⠛⠟⠋⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣶⣶⣦⢹⣿⣿⣿⣿⠿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣆⠀⠘⣿⣿⣿⠏⣸⣿⡿⠋⠀⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣽⣤⣴⣿⠟⠁⣀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢹⣮⣉⣠⣾⠂⠀⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣹⡇⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⡀⠀⠀⠀⢀⣼⣿⠃⠙⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠰⣿⣄⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢹⣷⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠟⠋⠙⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠋⠱⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⢀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠘⠀⠉⠙⠉⢻⡄⠀⠷⠀⠀⠑⠀⠀⠐⠀⠀⠀⠁⠀⠈⠀⠉⠉⠁⣶⣦⣤⣤⣴⣶⣤⣤⣤⣤⣤⣶⣤⣶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠂⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠀⢀⣠⣴⣯⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣭⣨⣸⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣼⣿⠿⠻⠧⢘⣿⡟⣹⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠘⠀⠀⠛⠀⠀⠘⢛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⣤⣀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠘⠿⡟⢿⣿⣿⣿⣿⣿⡿⢿⠿⢿⣾⡿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⢈⣿⣿⣿⣿⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⢏⣿⡿⠤⢄⣠⠶⠊⠉⠀⠘⢿⣿⣿⣿⣷⣬⡛⣾⣿⣿⣿⣷⣄
⠀⢰⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠘⣇⠀⠀⠀⠀⢀⣠⣤⣴⣆⢿⣾⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣤⣤⣄
⠀⠰⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣥⣶⡆⣼⣿⣿⣿⡟⠻⠿⠟⠿⢿⣾⣿⣿⠿⢣⣿⠿⠿⠟⠛⠿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠪⠿⠛⠋⠁⠈⠀⠀⠀⠰⠿⠟⠛⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⣠⣤⡄⣠⣄⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣶⣿⣿⣷⣤⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡟⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠤⠻⠿⠿⠿⢻⣿⣍⣿⣿⣿⢿⠿⠟⠋⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⠉⢹⣿⣻⣟⣿⣿⣿⣿⣿⣟⣟⣿⢟⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣷⣴⣶⣢⡄⠀⠀⠀⠀⢠⣔⣾⣦⣼⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣽⣧⣺⣿⣿⣿⣿⣿⣿⣇⣀⡀⢀⣀⣸⣿⣿⣿⣿⣽⣽⣗⣼⣯⣿⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⠛⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⠟⠿⠋⠉⠻⡿⡏⠃⢀⠐⢶⣶⡂⡀⠈⠹⣿⠟⠉⠙⠿⠻⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣟⡛⠀⠀⠀⠀⠁⠀⠀⣈⣿⣿⣿⣿⣁⠀⠀⠈⠀⠀⠀⠀⢛⣻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠭⣾⡛⠁⠀⠀⣐⠄⠀⠀⠀⠉⢻⣿⣿⡟⠉⠀⠀⠀⠠⣆⠀⠀⠈⢻⣿⠭⠅
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣲⠖⠀⠀⠀⠈⣽⡀⠀⠀⠀⠀⢸⡂⢐⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠲⣞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢀⡀⢀⠄⣨⡞⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢻⡅⠠⡀⠀⡀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠐⣻⢻⠟⠣⠀⠘⠿⢀⣄⣶⣴⣶⣤⠀⠀⠈⠀⣤⣶⣧⣶⣠⡄⠿⠃⠀⠜⠻⡿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⢾⡃⠀⠁⡄⢠⣿⣿⣿⣽⢿⠃⠀⠀⠀⡿⣯⣿⣿⣿⡄⢠⠈⠀⢘⡧⡌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⣻⣷⠀⠀⠐⠀⠹⡿⠂⠀⠊⠀⠃⠘⠀⠙⠀⠐⢿⠟⢠⠂⠀⠐⣾⣟⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⢿⠇⠀⠀⠆⠀⠋⠀⠀⠀⠇⠀⠀⠸⠀⠀⠀⠙⠀⠸⠀⠀⠸⡿⣷⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠣⠔⠀⣿⠀⠀⠠⣊⠑⠄⠀⠀⣿⠀⠢⡜⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣰⡆⠀⠀⠠⣼⢘⠁⠀⠀⢰⣆⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣢⡤⣿⣻⢄⣔⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣧⣿⣿⣷⣿⣾⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠙⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡏⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⠞⠋⠁⠸⠇⠀⠀⠀⠀⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⢀⣤⠶⠦⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⢀⣀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄
⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠙⣳⣤⣄⠀⠀⠀⠀⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣾⠃
⠀⠀⠈⠉⠉⠡⣤⣶⠞⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢀⠄⣼⠟⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠀⢀⣾⠃⠀⠈⠻⠶⡤⠠⠴⣶⠤⠞⠁⠀⠀⠀⠀⢠⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⠶⠛⠋⢹⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⢀⣴⡟⠁⠀⠀⠀⠈⠙⢷⡀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢹⡇⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣤⣦⡼⠇⠀⠀⠀⠀⠀⠀⠀⠉⠹⣄⠀⠀⠀⠛⢦⣤⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⠀⢀⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣦⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣻⡢⢀⢠⣿⣿⣿⣿⣿⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢉⣛⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⣀⣙⣿⠏⠙⢿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⢠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⡶⣤⡀⠀⠀⠹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢠⣿⣿⣿⣿⡄⢔⣯⣥⣀⠀⠀⠀⠀⣟⠔⠒⠒⠮⡀⠀⠀⠹⣿⣿⣿⣿⣿⣷
⠀⠀⢀⣾⣿⣿⡏⠉⠁⣸⢋⡭⠭⢵⡀⠀⠀⡎⠀⠀⠀⠀⠹⡄⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⣹⣿⣿⣿⣇⠀⠀⢱⠇⠀⠀⠀⠈⠢⡀⢇⢀⣶⣷⣢⠀⣧⠒⠉⠀⠀⠨⡽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢠⣾⣿⣿⣿⣿⣿⡀⠀⠈⡆⠀⠠⣶⣶⡄⢱⠘⣦⣙⣟⡳⠔⠓⠄⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣈⠀⡲⢊
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣘⣦⡐⢜⠿⣣⢟⣿⣿⣿⡆⠉⠀⠀⢰⠒⠂⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢰⠱⠉⠀⠀⢀⠌
⢻⣿⣿⣿⣿⣿⣿⠟⠊⠁⠀⠠⠞⠉⠉⠃⠈⠛⠛⠋⠀⢀⣀⡔⠁⠀⠀⣠⣮⣤⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣤⠀⠈⢈⠃
⠈⠛⠻⠿⠿⠿⢿⡶⡀⠀⠀⢠⢆⣀⣀⠀⣀⣀⣴⣿⣿⣿⣿⢀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠀⠀⠀⣉⠛⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠒⠈⢻⣷⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁⢀⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠒⠠⢴⣿⣿⣿⣿⣿⡿⠟⠓⠠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣤⡆⠹⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣎⠤⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠏⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠈⠛⠿⣇⠀⠙⢿⣿⣿⣿⣿⣿⢿⠈
⠀⠀⠀⠀⠀⠀⠀⠡⣀⣠⠂⠀⠀⠀⠀⠀⠀⡨⠛⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⡠⠓⠈⠀⠹⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⡄⠀⡠⣮⠔⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠏⠈⠢⠄⢠⠄⠠⠤⢶⠔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠊⠀⠀⠀⠀⠀⠀⠀⠉⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡎⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⢀⣀⣀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠒⠒⠊⠁
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⠀⠀⠀⠀⠀⣰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠈⡇⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⣾⠀⢱⣤⣤⣿⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⡇⠀⣿⣿⣿⣿⡻⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡇⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣆⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠉⣭⣽⣆⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⢹⡿⠃⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢻⣿⣧⡀⠠⣄⡀⠀⠀⠀⠀⢹⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠙⢿⣿⣶⣤⣭⣷⣶⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣯⣍⠉⠛⠉⠉⠁⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⡄⠀⠀⠀⢠⣤⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⠀⠀⠀⠉⠛⠿⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⣤⣴⣶⣿⣿⣧⣼⣿⣿⣿⣿⣿⣻⣇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢟⣽⣿⣿⡄
⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡼⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⡟⣿⣿⢟⣫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢋⣴⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⣿⡟⠹⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢊⣿⣇⠀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⠾⠿⠟⠸⣫⣯⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠃⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠟⠛⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡤⣤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣾⣿⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠛⠛⢋⠁⠀⠉⢡⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠃⢠⣐⣋⣽⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢄⠀⠘⢶⡛⣀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠂⠀⢦⣄⠀⠀⢿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠋⠐⠚⠓⠳⢾⠋⠍⢾⣆⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⠀⠀⠀⠀⠻⡄⠀⠈⠻⣀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⢀⠠⠀⠠⠀⠀⠀⠀⢤⣾⣷⣴⣿⣯⠀⠀⠀⠀⠠⢤⣿⣿⣿⡇⠀⠀⠀⠀⢀⡼⠆⠀⠀⠉⠀⢀⣤⣤⠀⠀⠀⠠
⠀⠀⢀⣎⠀⠀⢠⠀⢠⠄⠀⣠⣿⠐⢟⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣶⣄⣀⣠⡿⠋
⠀⠀⣸⡦⡙⡃⠀⠀⠀⢀⣠⡾⠿⣿⢿⡿⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐
⠀⢸⣯⣀⢸⡇⢸⠀⠄⢸⢿⡆⠶⠙⢻⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠌⢀⣶⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠸⢟⣛⢸⠃⣀⣄⡄⠀⢢⣻⣷⣿⡆⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣶⣿⣯⣿⣄⣀⡠⠄⠠⡶⠦⠀⢀⣴⣀⡄⣤⡀⣠⣀
⠀⠀⢀⣻⣾⠁⠋⠉⠀⡆⠀⠛⠇⠟⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⡶⢊⣴⡿⠿⠛⢸⠟⠁⠉⠉⠃⢤
⠀⠀⠨⢝⣘⣇⠀⠀⠀⠠⠤⣧⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢡⣒⣩⣥⡶⠊⢀⣤⠀⣠⣤⢀
⠀⠠⢈⢰⣿⣿⠟⠂⠀⠀⠀⠚⠛⠫⠍⠉⠉⠏⠉⠉⠉⠉⠉⠉⠉⢁⣈⠛⠩⢽⣿⣽⣿⣿⣷⣶⣶⣶⣦⣤⣴⣿⣿⡾⠉⠀⠉
⠀⠀⠀⠀⠟⢹⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣷⡀⠉⢹⡄
⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⡟⠋⠉⡉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠉⠉⢉⠴⠷⠎⢿⣿⣿⠀⠀⣿⣿⡃⠀⢸⣯⡖
⠀⠀⠀⠀⠀⠀⠄⠘⢿⡿⠀⠀⠀⠁⠀⠒⠁⠀⠀⢨⡐⡇⢸⠉⠨⠶⡅⠀⣷⠙⠅⠈⠀⡇⠈⡅⡉⠀⠀⠀⢸⢻⣿⠀⠀⠙⣿⣿⡀⢸⣿⣿
⠀⠀⠀⠀⠁⢀⠀⢰⡦⢠⡆⠀⠀⠀⠈⠀⠁⠀⠀⠘⠀⠁⠈⠘⠐⠀⠃⠀⠁⠘⠙⠀⠚⠀⠒⠂⠲⠀⠀⣇⣼⠘⡇⠀⠀⠀⠈⣿⡄⢸⣿⠟⡓
⠀⠀⠀⠀⠀⠀⠁⠘⡇⢺⢧⡤⠄⠠⣤⠤⣤⣤⠀⠀⣴⣦⠀⠠⣤⣤⣤⡤⠦⠤⠤⠦⠤⢤⣴⣤⣶⣤⢴⣦⣿⠀⠁⠀⠀⠀⠀⣿⡇⠈⠃⠀⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⣭⡭⢄⣭⣿⣤⠉⠉⠀⠙⠛⠞⠁⠀⠀⠉⠁⠀⠋⠉⠐⠒⠀⠘⠻⣾⣿⣶⣮⣷⣾⣤⠀⠀⠀⠀⢰⡟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣶⣾⡿⠀⠀⠀⠀⠈⠻⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠿⠿⠛⠻⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠛⠛⠛⠛⠋⠙⣲⠀⠀⠀⠀⠀⠀⠀⠻⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠃⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠑⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣌⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢟⣥⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⣷⠄⢦⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢋⡔⠠⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣗⠀⠻⣦⡙⠷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣴⠏⠀⣺⣿⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠦⠀⠙⣿⣦⡈⠻⣦⡶⣴⣴⢶⣴⠟⢁⣴⣿⠋⠀⠴⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⡍⠀⠀⠘⣿⣷⣴⠂⠀⠀⠀⠀⠐⣦⣾⣿⠁⠀⠀⢩⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣗⡀⠀⢀⣼⠟⠁⠀⡐⠀⠀⢄⠀⠘⠻⣇⡀⠀⢀⣻⣿⣇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⣿⠛⠁⠀⠉⠀⠀⠀⣰⠇⠀⠀⠸⣄⠀⠀⠀⠉⠀⠈⠛⣿⣿⡄⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣀⣀⣀⣈⠳⣄⡀⢰⠀⠀⡄⣀⣠⢞⣁⣀⣀⣀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣥⠀⠀⠀⠀⠀⠀⠉⠉⢹⡿⣿⣌⢱⡏⠀⠀⢻⡌⣡⣿⢿⡏⠉⠉⠀⠀⠀⠀⠀⢀⣬⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⠄⠀⠀⢽⡆⠀⠀⠀⠀⠓⠒⠻⠀⠀⠀⠀⠀⠐⠟⠒⠋⠀⠀⠀⠀⣸⠇⠀⠀⠠⣬⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣁⠀⣠⣄⡀⢻⣄⠀⠘⠶⣄⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⣠⠶⠃⠀⣰⡟⢀⣠⡀⠀⣈⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣥⠖⠀⠈⠛⢦⠻⠦⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠴⠧⡴⠋⠁⠀⢲⣬⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⣤⣤⣤⣀⠀⠐⣶⣤⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢻⡄⠀⢀⣤⡶⠂⠀⣀⣤⣤⣤⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡟⠓⠀⠈⢿⣷⡀⢺⠁⠀⠀⠀⠀⠀⠀⠘⡇⢠⣿⡿⠁⠀⠛⢻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣯⣤⡀⠈⣿⣧⡀⠀⣠⣤⣶⣦⣤⣄⠀⢀⣾⡿⠁⢠⣤⣽⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣶⣦⣼⣏⢳⣄⠹⣭⡍⢩⣭⠏⣠⡞⣹⣧⣶⣾⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠽⣿⣿⣆⠻⣷⣤⣭⣭⣴⣾⠏⣰⣿⣿⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⣜⠛⠛⣡⠋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣉⣩⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡤⠄⠀⣀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠁⠀⠀⠐⠉⠉⠀⠀⠈⠁⠀⠀⠐⠀⠒⠒⠂⠶⡶⢄⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢀⣿⡏⠠⠂⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢀
⠀⠀⠀⠀⡀⢀⠀⠀⠀⢰⡆⠀⠀⣤⠏⡤⠀⠀⠀⠐⠂⡰⠦⠀⠀⠀⠄⠂⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⠀⢹⣷⠈⠛⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⢀⣴⣏⣜⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢳⢦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣡⡄⠀⠀⣀⣤⣔⢄⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡆⢿⠘⢠⠁⠀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⣼⣟⣵⠶⠓⢠⣿⣿⣧⣿⡌⠹⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⢷⣶⡀⠀⠈⠛⢾⡇⠀⡇⠀⠀⠀⢉⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠉⣿⣿⡿⣀⠀⠀⠑⠘⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠺⠻⣟⡽⠳⠀⣸⠸⣿⠁⠀⠀⠀⠀⠀⠈⢸⠃
⠀⠀⠀⠀⠰⡟⡀⠀⠀⠀⡟⢰⣿⣿⠀⠈⠹⣿⣶⠀⠀⠀⠀⢀⣤⡄⠠⠀⡀⣀⠀⠀⠀⠣⡀⠀⢀⣤⠀⠘⡓⢿⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠈⠻⣆⠀⢀⠁⣾⠏⡿⠀⠀⣀⢹⣿⡀⠀⠀⣰⡿⣋⡀⠀⠌⠁⠈⠻⣄⠀⠀⠐⣶⣿⡏⠀⠀⣿⣿⢸⣇⠀⠀⠀⢰⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢇⡜⢴⣿⢸⡇⠀⠀⠛⡿⡭⠁⠀⢠⣿⣶⣿⣿⠁⡀⢺⣿⣷⣼⡄⠀⠀⠻⣿⠀⠀⠀⣽⣿⡘⢿⡀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣼⡇⣸⠗⠋⠁⠈⠀⠃⠀⠀⠀⠑⣧⡈⢡⣼⣗⣠⠉⣻⠿⠇⠀⠀⠀⠉⠲⣀⢀⡿⣹⣇⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⢃⣿⢀⡴⣶⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢠⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⡿⣿⠐⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣻⡯⠸⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠈⣧⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣩⣿⠚⣧⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣤⣿⣦⣡⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⡼⢏⣼⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠸⡇⢻⣿⣿⣤⣄⣀⣠⣴⡿⣿⣟⣙⣧⣀⠀⠀⠀⠀⢠⠎⠁⠔⠃⠛⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣇⣿⣿⣿⣿⠗⠒⠂⢺⣿⡋⠀⠿⣷⣶⣶⣶⡿⠐⠀⠀⠛⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡷⠞⣨⠛⠁⠀⠀⠀⠙⠋⠁⠀⠀⠸⣯⣽⣟⠓⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠹⠿⣿⡿⠿⠛⣷⡶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠃



⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠁⠈⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣤⡖⠒⠢⠄⢀⣀⣀⠀⠀⢠⡶⢛⣿⣷⣄
⠀⠀⠀⠀⣰⣿⣿⠃⢀⠀⠀⠀⠀⠀⠈⠀⠉⠀⢺⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⡇⢀⡜⠰⣫⣦⠀⡠⣂⠀⠀⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣀⡀⠀⠹⠿⠕⠋⠀⠃⠻⠃⢸⢸⣿⠀⠀⠳⢌⣛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⣾⣿⡿⠒⠶⠒⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⡀⢐⠛⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⠀⠀⠀⢠⣿⣿
⠈⣉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⣀⡉⠠⠀⡀⠠⠤⠤⠶⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡴⠛⠋⢹
⠀⠈⠑⠒⠤⠤⠤⠤⢄⠒⠒⠂⠁⠀⠀⠀⠀⣠⣿⣿⡿⠁⣐⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⢿⣿⣿⣿⠋⠉⠉⠀⢀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠢⡄⠀⠰⣿⣿⠟⢀⣼⣿⠀⠀⠀⠀⠀⣀⣤⣤⣀⣠⣄⡉⠉⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠟⢁⣴⣿⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠔⠊⠉⠉⠀⢠⣶⣷⣦⡀⠀⠘⠿⢿⣿⣿⠿⢿⠿⠃⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠈⠀⢦⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠉⠻⠿⠟⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣤⣀⠀⢀⠠⣀⣀⣤⠴⠞⠁⠑⣄⠀⠀⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡄⠀⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⣰⠀⢸⣄⠰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇⢀⣿⣿⡇⠀⡄⠀⠀⠀⣠⠚⠁⣠⣾⣿⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢠⣾⡿⠃⠀⠀⠡⠀⠀⠀⠑⠄⢸⡉⠉⠁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠐⠤⢄⠀⢀⠤⠴⠁⠀⠀⠀⠀⠀⠈⠀⠐⠒⠉⠉⠁
⣶⣶⠶⠶⠶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡺⠑⠤⢲⡞⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⠒⢤⣐⡊⠁⠠⠀⠀⠀⠉⢽⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣤⣽⠁⠞⠀⠀⠀⠀⠀⢠⢜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⠝⠛⠛⠛⠋⠀⠀⠀⢠⠄⠐⠂⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠸⡅⡜⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠼⢶⠤⠔⠋⠙⠁⠀⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⡘⠀⠀⠈⠙⢦⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣏⣉⠉⠒⠒⣚⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠐⠣⠋⠉⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠀⠀⠘⠳⢤⡀⠀⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠸⠞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⠀⠀⣦⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠸⡄⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⢸⠀⠀⡇⣠⠖⠚⠛⠓⠲⢄⡠⠁⠹⢄⠀⠀⠀⢳⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣇⠀⢻⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢳⡀⢠⠏⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠸⡄⢸⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡄⣀⡬⠽⢧⣸⡆⢀⡵⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡰⠂⠀⠀⢀⣾⠀⢀⠀⠀⣠⣀⡀⠀⠀⢀⣀⣀⣸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠉⠈⠙⠛⠒⠚⠁⠀⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣖⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⣿⣿⣿⠏⢰⣾⠂⢀⣀⠈⣙⠻⠆⠀⠀⠀⠀⠀⣀⣀⠀⣿⣦⣄⣈⠛⡆
⠀⠀⠈⠁⠀⠀⠀⠀⠠⠏⠀⣼⣿⣿⡿⠀⠚⠁⣀⡻⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⣽⣿⣀⠈⠙⢿⣿⣗⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠹⣿⣿⣃⠀⢀⣾⣿⣿⣿⣾⡿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⠸⣿⣿⠀⠹⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⡏⠙⠛⠛⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⢉⣿⣿⡇⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⣽⣿⡟⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⣼⡏⠸⣿⣦⡀
⠀⠀⠀⠀⢀⣾⡀⠀⢠⠿⠀⠀⢹⣿⣷⣤⡙⢻⡟⠀⡞⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢠⡘⠿⣿⣬⣿⠁⢀⣿⣧⠀⢻⣿⢳
⠀⠀⢠⣿⡿⠟⠀⠠⠀⠛⠰⣀⠈⢿⣿⡿⠃⠀⢠⣤⠃⠀⡄⠀⠀⠀⣀⣀⣙⣶⡀⢻⡇⠀⢸⣿⣿⢀⣿⣿⣿⣦⣙⣿⡟
⠀⠀⠈⠉⠀⠀⠀⠀⠐⢀⠘⣿⠀⠈⣿⡁⣠⡀⠐⠋⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⠇⠈⢷⣤⣈⣿⣇⣿⡏⠋⠻⢿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⢀⡄⢻⡿⢡⣿⠀⠀⣿⣷⣿⢿⡀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⠿⠇⠀⠀⠈⣿⡤
⠀⠀⠀⠀⠀⢰⣏⡀⠀⠀⣼⣷⡆⣰⣿⣿⣷⡌⢷⡀⠀⠀⢐⣻⣿⣿⣿⣿⣤⠃⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⠁⣴⣿⣿⣿⣿⣿⣤⣷⡀⢀⣀⣠⣽⣿⣿⣿⣿⣄⣢⣾⣿⣿⣿⣿⣿⣿⣿⣆⣀
⠀⠀⠀⠀⢀⡄⠀⠄⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⢿⣿⣿⡿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠺⠋⠀⠀⠬⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠙⠉⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⢻⣧⡄⣿⢹⣧⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠚⠉⣿⠟⠁⠈⢻⣿⣿⣌⣹⣷⣤⣀⣀⣦⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⠿⡿⠛⠻⣿⢿⠛⠛⢿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠉⠋⠀⢀⠁⠀⠀⠀⠀⠐⠚⠀⠙⢿⡿⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡦⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠖⠛⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠙⠒⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⡀⠀⠀⠈⢧⠀⠰⡄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⡠⠀⠀⢀⣐⣾⠛⠀⠀⠀⠘⠀⠀⠃⠀⠀⢠⡌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠐⠀⣰⣾⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣦⡘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣻⣿⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⡟⣳⡘⣿⡀
⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠉⠉⠉⠁⠘⠁⠀⠀⠀⠀⠀⠄⠀⠀⠤⠍⠻⢿⣇⣿⣧
⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⢙⠶⢤⣄⡀
⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣧⡀⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣷⣤⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⠟⠋⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣆⠡⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡜⠄⠀⠀⠀⠀⢠⣿⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⡀⠀⠀⢀⣾⡯⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣿⣿⣿⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣿⡏⠁⠐⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣪⣴⣿⣿⣿⣿⣿⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠠⡪⠀⠀⠀⢀⣴⣶⣶⣶⣦⣤⣴⣶⣤⣶⡶⢿⣿⣿⣿⣿⣿⣿⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⠃⠀⠸⣇⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠰⣬⣻⣿⣿⣿⣻⡿⠿⢛⣯⣷⣿⣿⣿⣿⣿⣿⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠄⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣛⡟⠉⠁⡠⡾⡿⣿⡟⠁⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⢹⡄⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⠊⠀⠊⡜⡑⢑⣿⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡆⢳⡀⠀⠀⠀⠀⠑⡀⠀⠠⡐⢄⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⠁⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢳⡀⠀⠀⠀⠀⠈⠀⠀⠘⢮⠣⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠜⢀⠏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⢀⠎⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠒⠉⠳⠄⠀⢳⡀⠀⠀⠀⠀⠀⠀⢰⣄⢰⣄⠀⠀⠀⢰⠃⠀⠀⠀⣠⣀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⣤⣾⣿⣦⠀⠀⠀⢳⠀⠀⠀⢀⣀⣀⡼⢹⣼⠹⣆⡀⠀⠘⠀⠀⡀⣼⠿⠿⢷⣦⣀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣴⠿⠋⠁⠀⠈⡇⠀⠀⠈⠶⠒⠋⠉⠝⠋⠀⠸⠃⢰⠏⠉⠉⠓⠦⣄⡇⠇⠀⠀⠀⠈⠙⠳⢦⡈⠂
⠀⠀⠀⠀⠀⠀⠰⠁⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠋⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⢸⠗⠄
⠀⠀⠀⠀⠀⠀⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠶⢦⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣥⠤⠤⣅⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣴⣾⣿⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⡀⢀⣮⢳⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡏⣿⣿⣿⣦⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⢻⣿⣿⣿⣿⣿⢇⠃⠀⠀⠀⠀⣀⡀⠈⢯⡻⠿⠿⠟⢁⡞⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠘⢦⣉⠛⠛⢛⣡⠊⠀⠀⣶⣿⣭⣭⣭⣿⣷⠉⠒⠒⠚⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣍⠀⠀⢀⡀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠖⠚⠁⠜⠛⠛⠉⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢀⡴⢋⡀⠀⠀⠀⠈⠉⠛⠓⠶⠶⠶⠶⠶⠶⠖⠚⠛⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡘⢧⡀⠀⢠⠏⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠓⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠤⢾⡄⠀⠀⠒⠂⢄⠀⢸⠀⠀⠀⠀⠘⠢⣀⠀⣰⠃⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡀⢈⣇⣸⠀⠀⠀⠀⠀⠀⠘⣾⣇⠀⠀⠀⡀⠀⡀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠈⠉⠳⠶⠶⠶⠶⠾⠚⠁⠈⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⣠⣴⣶⣶⣶⣶⣿⣿⣿⠛⠉⠁⠉⠀⠀⠈⠹⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⡴⢋⣵⣿⣿⣿⢏⣽⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⠳⣄⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣴⡏⠀⣼⣿⣿⣿⡟⣸⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠒⠲⢿⣿⣿⣿⣿⡇⢻⣷⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⣿⡇⠀⣼⣿⣿⣿⡇⣿⣿⣿⣫⣭⣭⣄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡗⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢿⣿⣿⡄⢹⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠃⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠻⣿⣿⣆⣿⣿⣿⡇⣿⣿⢿⣿⣿⣽⡿⠃⠀⠀⠀⠘⢿⣿⣿⣻⠛⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇⣿⣿⣷⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⢧⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣤⣤⣄⠀⠸⠃⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠙⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣤⣾⠷⠤⠤⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢸⣿⣿⡟⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠈⣿⣿⣿⣌⠻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡙⠻⢿⣦⣌⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣉⣙⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢠⠒⠀⠒⠠⠂⠀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠀⠀⠰⠐⠀⠀⠀⠀⠀⠁⠀⠢⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠊⠐⣀⣤⡀⢄⠂⠀⠀⠀⠑⠢⠀⠂⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡈⢈⠕⠒⠢⡀⠀⠑⢄⠀⢀⠀⠞⠡⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠇⠈⠀⠀⠀⠈⡄⠀⠀⠡⠂⠀⠀⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠰⡀⠠⣤⣶⡿⠀⠀⢀⠀⠀⠰⢤⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⡧⠐⠉⡉⠁⠈⡏⢀⣠⣽⡦⠿⣂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠯⢅⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠌⠀⠀⠀⢀⠀⠀⠻⣿⣿⣿⠇⠐⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠑⡤⠒⢠⠃⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠘⠦⠃⠀⡘⠀⠀⠀⠀⢁⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢸⢢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⡇⠈⠒⣅⣂⠠⢤⠀⠊⢰⠄⠐⡺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⡍⠑⠀⠂⠰⠉⠫⣓⡴⠓⠄⠊⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣤⡀⠀⠈⢆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠆⠀⢀⡁⠑⢤⣀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡘⠀⠈⠒⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠁⠀⠀⢉⣁⠆⠬⡑⠒⡆⠈⠐⠤⠄⠖⢤⠤⠒⠁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠒⠁⠀⢀⡤⠃⢀⣸⠀⠀⠀⡜⡎⠀⠀⠀⠺⣀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠒⠒⠒⠉⠀⠀⡇⠈⠀⠀⠼⡅⣸⡀⠀⠀⠀⠄⠀⠂⠠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⠋⠐⠂⠉⠓⡤⠤⠄⠀⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⡀⠀⠀⢀⣀⡴⠃
⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣾⣷⣦⣽⣿⣵⣶⣶⣶⣶⣦⣶⣴⣶⣦⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠂⢀⢀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⠃⠀⠻⠃⠐⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠋⠃⠀⠁⠀⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢰⣤⠸⠟⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢠⠀⠰⠀⠁⣤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡷⠀⢠⡖⠀⣾⠈⠀⣀⠄⠐⠏⠀⠀⠠⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠰⢆⠉⠒⠀⣀⠀⢀⠈⠖⠀⠙⣃⠀⠀⣠⢴⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠿⠁⠀⠀⠈⠷⠿⠃⠀⢰⡄⠉⠀⢠⣬⣶⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀⠀⠒⠒⠀⠀⢻⣾⣦⢀⡶⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠱⠒⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢀⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⢀⣀⣀⣼⠿⣤⣾⡿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠂⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⡀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⢝⣲⡤⣤⣤⠼⠉⢹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⡆⠀⠀⡆⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣅⣠⠀⠀⢸⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣴⣧⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣦⢻⣿⣿⣿⣿⣿⣿⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢿⣟⢛⣱⠋⢈⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡆⣫⡁⠀⠀⢈⣧⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⣦⣈⢿⣦⢶⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢾⣿⠻⣿⣾⡆⠀⢻⣿⣿⣿⣿⠋⡟⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⣿⣆⠘⢻⣷⡀⠀⠹⡝⢿⡇⢰⠁⠈⢿⣿⣿⣿⣿⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⣿⡄⠘⢿⣷⡀⠀⠀⠈⢳⠃⠀⠀⠈⠉⠛⠻⢷⣤⠄⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣦⡀⢸⡇⠀⡀⢀⢸⠀⠀⠀⠉⠀⣴⡾⠶⠧⣤⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡺⠛⣿⡇⠙⢿⠀⢡⣘⣮⡀⠀⠀⢀⡴⠁⠀⢠⣴⣶⣾⣿⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣃⠹⣷⠀⠀⡇⠠⢹⣻⡁⠈⡆⣼⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣇⣿⡄⠀⡇⠹⣼⣿⣿⣷⣼⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⣿⡇⠀⢈⠘⣿⡯⠙⣿⣿⣿⣇⠀⠸⣿⣿⣿⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡛⢿⠇⠀⠘⣤⣿⣇⣄⣈⣻⣿⣿⣦⠀⠈⠻⠿⣿⡞⠁⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⢶⡯⢆⣤⣿⠀⠀⡾⢠⢻⣿⣿⣿⣿⣿⣿⠾⣷⣄⠀⠰⢽⣿⣶⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠿⠤⠟⠓⠁⢠⣞⡴⣡⠋⣿⠀⢰⣣⢋⡵⢞⣻⣶⣶⠾⠿⠟⠛⠛⠛⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠘⠛⠛⠷⠾⠛⠧⠀⠀⠒⠒⠒⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠓⠀⠀⠰⣶⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⡀⠀⢀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠐⠗⠀⢀⣀⠀⢹⠟⠀⠀⠀⠀⠀⢀⢸⡿⠃⠄⢀⣿⣄⠲⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠄⠿⠀⢺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠀⠀⠉⠀⣼⠀⠀⠀⠀⠀⣴⣿⠛⠳⣆⠀⠀⠉⠃⠀⣴⣆⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⣶⣦⠉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠰⣆⠀⠀⣀⣄⡝⠀⠀⠀⠀⠀⠸⣿⢄⠣⢄⡀⠀⢶⣀⣀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢽⢥⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠀⢩⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⢻⣿⣿⣿⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⢒⡤⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠲⢀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣂⠘⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⣴⢰⡆⠛⠀⠈⠓⣿⢺
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⢀⡙⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣀⡀⣀⡰⠿⠁⠀⠀⠀⠀⠀⢻⠀⣷⣄⣀⢀⢸⣏⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡏⠐⢿⣏⢈⣿⣿⡇⠀⠀⠀⠀⠀⠐⢄⠀⠈⣿⡇⢉⣛⣛⡛⠟⠀⠀⠀⠀⠀⠀⢀⡀⢋⠀⢼⣧⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⠐⠛⠙⠉⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠑⠊⢛⡀⢸⠏⢹⠁⠀⠀⠀⠀⠀⠀⠀⢘⠳⣄⠲⠸⠷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠰⠄⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡼⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠂⢄⠐⡆⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡄⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠀⠀⢠⣄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠶⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢸⡇⠛⠀⠀⠛⢸⠁⠡⠀⠀⠀⠀⠀⠀⣠⡶⣿⣅⠈⠁⠚⠻⠀⠐⣄⠀⠀⠀⠀⠀⠀⠈⠠⡆⠀⠀⠂⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣅⠈⢻⠟⠀⣨⠷⠂⠀⠀⠀⠀⠀⠘⣿⣇⠳⠈⠂⠀⢂⡀⠀⢸⣿⠂⠀⠀⠀⠀⢠⡇⠀⢿⣄⠲⠶⡦⣭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⢿⣾⣷⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠁⠈⠳⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠲⠺⣇⠘⡟⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣲⣥⠃

⢰⣿⣿⣿⣷⠖⣶⣶⣶⡆⠀⢀⡲⢶⣶⣦⣰⡶⢶⣦⡆⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⢾⣿⣿⣺⣿⣷⣼⣶⣾
⠸⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣟⠚⣿⣯⣼⣧⣿⢛⡇⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡶⠿⣟⡟
⠰⢿⣾⣻⣿⣿⣿⣿⣿⡇⠀⠀⣐⣶⣤⠼⣿⣿⣗⣻⠇⠀⠐⣿⣝⣿⣿⣿⣿⡅⢸⡇⠀⠀⣿⢯⣾⣿⡟⣳⣶⣠⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⡿⢟⣵⡿⢿⣿⠀⡇⠀⢰⣶⠄⠙⢧⣿⣿⡟⢻⡇⠀⠀⣾⣿⣿⣏⣹⠟⢿⣿⣷
⢈⣻⣟⢛⣿⣷⣾⣿⣾⡇⠀⢘⣷⣶⣾⣏⣿⡟⣹⣾⡇⠀⠠⣷⣦⣰⡿⡿⡟⢿⣟⡇⠀⠀⣽⣿⣿⣿⢿⣾⣿⣿⣿
⢸⣿⣿⣿⣿⣶⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣽⣿⣿⡇⠀⠘⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢭⣩⣿⣿⣯⣿⣿⣿⡿
⠘⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠐⠛⠓⠒⠛⠛⠛⠛⠛⠃⠀⠀⠛⠙⠛⠛⠛⠊⠛⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⢀⡼⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣆⠈⣦⠲⠾⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣧⡾⠛⠅⠀⠀⣆⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⢻⣿⡿⠵⠀⠀⠀⡀⠀⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⡀⠁⠀⠀⠀⠻⠆⠙⠊⠟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⢯⠵⠍⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠋⠹⢏⣰⣦⣶⠄⠀⠙⠿⣿⡏⠓⠤⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡶⠲⠶⠖⠚⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢴⣾⣷⣶⣴⠀⢀⢀⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣻⡗⣮⣿⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠍⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣴⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣶⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣿⣿⣠⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠠⣷⣾⡯⠀⢈⣧⠀⠀⠀⠀⠀⠀⣼⠆⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⣽⣷⣾⣿⡿⢿⡿⣷⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡰⠀⠈⣿⡷⠀⠀⠀⠀⠐⣿⠇⣠⣴⣤⣄⠻⠔⢀⣤⣦⣾⣿⣿⣿⣿⡇⢼⠐⣻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠰⣻⣧⠀⠀⠀⠀⣾⠟⣼⠟⠛⠛⠛⠿⣾⡿⠿⠛⠛⠛⢿⡟⣿⡃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣣⠆⣉⢻⣿⣿⡀⣠⡜⠃⢠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣿⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⣸⣿⣿⡟⠃⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⢃⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣍⡤⣾⣿⣿⠁⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠀⠄
⠀⠀⠀⠀⢀⡠⠀⣿⣿⣿⠿⣿⣿⡿⣡⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⢠⡏⡀⣼⣿⣿⢧⣺⣿⣿⠀⠃⡔⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣸⠓⣿⣿⣿⣿⣿⣽⣿⣿⡷⠆⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠠⠀⠀⠀⢀⡀⣿⠿⣿⠉⠉⣻⣿⡿⠿
⠀⠀⢠⡟⢰⣿⣿⣿⣿⣿⣿⣿⠿⣇⠐⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣤⢄⣶⣶⣶⣶⣾⡾⠍⠉⠙⠐⠂⠐⠂⡠⠴⢲⡾⠇⠐⠁⢔⣿⣿⡿⠄⠒
⠀⢠⣿⣃⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⠀⢃⠤⣀⣀⣤⣤⡔⠺⠿⠈⠋⠁⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⠁⠀⢀⣤⣿⡿⠛⠁
⠀⢸⣿⣛⣻⣿⣿⣿⣿⡟⢛⣧⠀⢷⣄⠠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⡀⠀⠀⠀⢠⣿⠟⠋⠁
⠀⠈⠉⠉⠀⠻⣿⢿⡿⢁⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⠀⠀⠀⠀⠈⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡃⠀⠈⠀⠀⠀⠀⢀⠀⣶⣠⣄⢀⡀⢒⣐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⠤⠒⠦⠔⠒⠴⠲⠒⠒⠤⣄⣠⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⡿⠀⠠⠀⠀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠙⣿⢿⣿⣷⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠹⣯⡿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠈⠻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⢀⢄⣤⣶⣴⣶⣿⣷⡄⠀⡇⠀⡀⣿⣿⣷⣦⣤⡀⠂⠀⢸⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⣿⡿⣿⣭⣿⣿⣿⡾⠿⠿⢿⣿⣿⡿⣿⣭⣿⣿⠀⠀⢸⡈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⡇⡀⠘⡻⣿⣿⣿⣿⣿⣿⣶⣶⣿⣽⣿⣷⣾⣿⣿⣁⡀⠐⠌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠻⣷⣮⡻⢿⡿⠿⣛⣵⣮⣛⢿⣿⠿⣟⣵⣿⣿⠃⢀⡴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠳⡄⠙⢿⣿⣿⣿⡏⠉⠙⠋⠉⠹⣿⣛⣿⣿⠟⢁⡠⠚⣰⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢷⡀⠀⠀⠀⠉⠛⠿⣷⣤⣀⣀⣠⣼⡿⠟⠋⠁⠀⠀⢀⡴⠃⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢳⡀⠀⠀⠠⣀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⠔⠉⠀⢀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⢦⡀⠀⠀⠑⠀⠀⠉⠉⠉⠁⣀⡄⠀⠰⠃⠀⠀⠀⠊⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠦⠀⠀⠙⣄⠀⠀⠀⠀⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⡷⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢠⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣸⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢀⡞⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣿⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢚⡉⠀⠀⠀⠀⣼⡇⣴⠀⣀⠞⠙⠢⠤⣀⣀⣀⣀⣀⡤⠴⠚⣿⠀⠀⠀⠀⣿⡒⠚⠉⡙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡀⣰⣇⣀⡼⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠀⠠⡀⢰⢙⣖⣄⣈⡦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⠴⠷⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠖⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣀⣀⣀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣿⣿⣿⣿⣷⡄⠀⢱⡜⠒⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡞⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠜⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡀⠀⢀⠜
⠀⠀⠀⣇⠀⠀⠙⠒⠦⠤⣤⣀⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣷⠀⠙⠿⣿⣿⣿⠿⠓⠒⠒⠔⠚⠚⠁
⠀⠀⠀⠈⠓⠦⣀⣀⠀⢰⣿⣿⣿⣿⣿⣿⣯⠙⠻⠿⠿⠿⠋⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠛⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠔⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⡠⠊⡨⠋⠒⠒⠂⠤⠒⠒⡆⠀⠀⡔⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢠⠊⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣷⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠘⢤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢘⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡇⠀⠀⣘⢠⠀⣴⣶⣾⣿⣆⠤⣤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⢰⢿⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣿⣇⣼⣿⡿⣷⡙⣻⣿⣿⡟⣧⣾⣧⢠⣯⣤⣴⡄⠀⠀⠀⠀⣾⣿⣭⣹⡘⠶⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠋⣭⠉⠙⠏⢁⣤⡄⠈⡉⠽⠃⢀⠋⢹⣿⢷⣾⣿⠿⣄⣀⣀⠠⣿⣿⣿⣿⠃⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⢁⣨⡇⢀⣾⡐⡀⠀⠈⠹⠐⠃⢀⣠⣶⠋⢠⣟⠟⠛⣧⣀⣶⣧⠼⣿⣀⣈⠉⠛⣋⣩⡆⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣨⣾⣿⣿⣿⣿⣿⣷⡶⣦⣶⣿⡿⠻⠃⣰⠓⣿⠂⢀⣼⣿⣛⣷⠆⣿⣿⣿⣿⣿⡿⠟⣓⡠⡼⣣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⠇⢰⡿⠀⣿⣗⣿⡷⣫⣾⣟⣄⣿⣿⢳⣿⢿⣏⣁⣿⡉⣻⡿⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠀⣶⣿⣿⣿⣿⣿⣯⢹⣿⣿⣿⣻⣿⣾⡤⠾⠃⣰⣿⣿⡱⣿⣭⣾⣫⣿⣿⣿⣟⣿⣿⡿⢛⡧⠘⠧⢶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡞⢛⣛⡿⣿⣿⣿⣷⣾⣿⠿⠞⣿⣿⣿⣷⣶⣾⣿⣿⣿⣧⣯⣤⣾⣿⣿⣿⣿⡦⡛⢳⠾⠋⢀⣶⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣻⠟⣿⣿⣿⣿⡟⢩⡏⠛⠛⢿⣿⠃⠀⠘⠙⠉⢿⣿⡿⡉⠙⡻⡿⣿⣿⣿⡇⣒⠒⠲⡊⢩⣿⣿⣯⣛⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡒⠋⣹⣿⡀⣤⣶⣶⣶⢋⡿⣧⢠⣾⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣾⣏⠂⠌⡙⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣵⢻⣯⠈⠙⢿⢻⢋⣿⣧⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢉⢉⡛⠿⠻⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠘⢿⣀⣶⣿⢀⣼⣿⣷⣂⣿⣿⣿⣿⣿⢟⣹⢷⣿⣿⣿⣿⣿⣿⣟⡛⡂
⠀⠀⠀⠀⠀⡀⠀⣀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡿⠛⠟⠛⡟⠓⢿⡟⠉⠙⣿⠿⠻⢿⣿⣿⣿⣿⣹⣿⣾⣿⣭⡿⠿⣾⣿⣿⣿⣧⡄
⠀⠀⠀⠀⢸⣿⣿⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⡸⠁⠀⠐⣼⠀⠀⢸⠇⠀⢀⣿⣦⡀⠈⠀⢸⣿⣿⣿⣦⠀⠘⣧⣀⠀⠘⣯⣿⣭⣿⣯
⠀⠀⠀⠀⢸⡿⠋⢈⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠁⠀⢀⣴⣿⠀⠀⢸⠀⢀⣼⣿⣿⣷⣀⣀⣾⣿⣿⣿⣿⣷⡀⠘⢿⡆⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣄⡅⠧⠀⢸⣿⣿⡟⠀⢀⡀⠀⠈⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣷⣶⣠⣌⣿⣖⣤⣽⣿⣿⣿⣏
⠀⠀⠀⠀⠀⢈⣿⣏⣵⣿⣿⣿⣿⣿⣿⣿⣿⡟⠩⠑⠤⣿⣟⣿⣷⣤⣿⣧⣤⣽⣿⣿⣿⣿⣷⣿⣿⣿⣟⢹⣿⣿⣿⣿⣿⣿⣽⣿⣙⣏⠍⠙⠉
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⣦⡘⠿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣧⠒⡏⠸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠿⠭⠟⢻⣿⣿⢿⡯⠿⠿⠿⠿⠡⠏⠈⠿⠻⠟⠛⢹⡿⠀⠈⠉⠹⠛⠛⠛⠟⠿⠀⠶⠶⠼⠿⣿⣿⠿⣿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠈⢹⡄⣤⠙⢓⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠁⠈⣁⣀⣭⣽⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣷⣶⣶⣶⣿⡿⠿⠟⠋⠉⢻⣿⣾⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⢏⠉⠁⢻⡿⢛⣛⠂⠀⠀⠀⢿⣿⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠉⠻⣿⣿⣿⡷⣿⡟⣷⡿⠀⠀⠀⣿⠛⣿⡾⣷⠦⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠻⣥⣍⢉⡿⠀⠀⠀⠠⠀⠀⢠⡉⡉⠁⢠⡞⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⡘⣿⣷⣤⠀⢀⢀⣀⡀⠀⠀⠁⠘⢶⡟⢀⢴⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁⢰⡿⢋⢠⢀⡙⡆⠀⠀⡌⡿⣋⡸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠹⣿⣿⣿⣿⡇⠀⠸⣿⣻⣿⣎⣻⠇⠀⠀⢹⣿⠟⡇⠈⠛⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠈⠀⠀⢹⣿⣿⣿⠃⠀⠀⠈⢻⣿⣿⡯⠀⠀⠀⢸⠇⣼⣇⠀⠀⠀⢻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡻⡇⢾⡷⠀⢼⣿⣿⣽⠀⠀⠀⣠⣾⠿⢿⣶⣤⡀⠀⢨⠞⣽⣿⠀⢠⣤⠸⡿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⣤⢴⣻⣿⣿⡏⠳⡴⢾⠯⠀⠠⠀⢀⡭⠛⡦⠂⢠⣿⠉⠀⢈⣉⣴⣷⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⠀⣀⠀⠙⣿⣿⡄⠨⡻⣷⣄⠀⣠⢎⡴⠋⠀⢀⢺⡿⠀⠄⡀⠙⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣿⢌⠻⠃⠀⠹⣿⣿⡄⠀⠈⠙⢤⡟⠉⠀⠀⠀⢀⣾⡇⠀⠀⢰⡆⢰⣽⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠙⣿⣷⡀⢀⣶⣿⣿⣦⡀⠀⢀⣾⣿⠁⠀⠀⠀⣠⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣷⣄⠀⠀⠘⣿⣆⡘⡏⢻⣿⠁⣷⣠⢎⣾⣿⠆⠀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣄⠀⠸⣿⠉⡆⣼⣿⠀⣽⠁⣼⣿⢋⢔⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣄⣿⣷⣷⣿⣿⡇⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢠⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠡⣤⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇⠀⠘⠆⠈⠀⣀⢤⡀⠀⠠⡎⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⢀⣄⣀⣚⣀⠄⠀⠳⢤⡞⢉⡼⠐⠈⠁⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠲⠀⣤⣛⠈⠁⡙⠁⢀⡀⠀⠶⠖⣾⣏⠶⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠸⡁⠢⣀⠠⣿⠈⠛⢫⡤⠀⠁⠀⢰⡾⠃⠉⣦⣀⡖⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡀⠛⢀⢸⣤⣿⠠⣥⠀⠉⠁⠒⢠⡾⠓⣂⡶⣿⠋⣳⠤⠆⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢦⠀⠉⠂⠀⠀⠁⠀⣀⠰⣤⡿⢀⡄⠁⢦⢀⣴⣿⠷⠚⠡⠈⠛⠋⠙⠋⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠀⠤⠈⢲⡟⠡⡤⢀⣀⡾⠃⠀⢼⡇⠄⠀⠀⣽⣽⠏⠁⠠⠠⠴⢦⠀⠀⠀⡴⢵⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⣁⡶⠂⠀⢘⣶⡇⠀⠀⠀⠛⠖⠐⠒⣸⠃⠙⠦⣠⣿⣋⠀⠉⢀⡰⠀⠈⢤⣤⠤⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢸⢀⠉⠀⡄⠀⠀⢨⡇⠀⢰⠂⢠⠀⢠⣄⣿⡄⠀⣴⡿⢿⣧⢀⠴⠛⠋⣳⡴⢿⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢌⡁⠸⠖⠀⠻⣄⠀⢸⣇⠀⢀⠁⠈⣀⣠⣼⣹⢠⣾⡿⠳⠎⠁⢚⣳⣴⠿⣟⢈⡾⠇⢁⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠟⠾⡆⣤⣦⠄⠀⣸⣟⣸⠈⠉⣴⡈⠀⠀⣸⣿⡿⠇⣹⡴⠾⣛⢿⡖⢲⣷⣏⢦⣤⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⡌⠛⣧⡨⡇⠀⢱⢿⣟⠁⣠⠂⠀⣀⠀⣸⣿⣿⠖⠛⡉⠠⠈⢉⡛⣻⠏⡀⠦⠙⢫⡙⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡥⠀⠐⠀⣿⠀⣀⣰⡟⠸⣿⣀⡨⠗⠲⢋⣾⣿⡏⠀⠀⠠⣆⣠⣴⠀⢉⠉⠄⠁⠐⢁⡿⠳⣇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢲⣿⠐⢧⣽⢀⣰⣯⠈⠀⠀⢲⣿⡿⠉⢣⡀⠰⠖⠪⠁⠈⣿⠉⠁⣀⣴⢾⣟⢀⣤⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢸⣿⡗⠛⡤⠈⢙⣿⣠⡤⣴⣿⠿⡗⠀⠘⣀⠀⠂⠐⢳⣾⣅⡴⢿⡏⢻⡟⢏⣉⣱⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠔⠂⢸⡟⠉⠋⠉⠞⣉⣿⢢⣼⣿⠛⡶⠻⠖⠁⢉⣗⣿⣽⡿⢿⠭⣷⡘⣁⡹⢀⡈⠛⠉⠄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⠋⠃⢸⣧⡐⠙⠲⠀⠀⣿⣾⠿⠿⣄⢲⣦⣤⠾⣿⣿⠃⠉⣀⠸⠀⠈⠄⠿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠀⢂⠈⠀⡀⢸⡇⠁⠀⠃⠀⢠⣿⣯⡤⠾⢟⡿⠉⠞⢿⠛⠡⣴⡋⠻⠒⠶⠃⢰⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⢸⡇⠀⠀⠀⣰⣿⠿⡏⣀⣰⣼⠢⠐⣦⡮⢈⠄⢤⡯⠁⣉⣰⣖⣾⣷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣰⣿⡇⠀⠙⣄⣅⠈⣄⠄⠛⣶⠀⣠⣬⣷⠟⠉⠹⡏⠠⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⣿⠀⣰⡿⠁⠀⠡⣄⠀⠈⡠⢠⣤⣴⡟⠛⠍⠉⢠⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢸⠿⣽⣥⣿⡧⠤⠆⠀⠰⢀⣤⣶⣋⡟⠲⠠⣄⠐⠟⡉⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠁⣠⢙⣿⡏⢀⠀⢀⣴⡾⣿⠇⢘⢉⡀⢱⠞⣫⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⣿⠏⢀⣿⣿⠋⠑⠶⠁⠐⠚⠉⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣾⣿⢾⠛⠁⠀⠈⣠⡀⠀⡘⠰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣡⠞⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣯⡶⢒⣯⣭⣉⣭⣭⣭⣉⣥⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠐⠠⣀⠀⣀⢀⣀⠀⠀⠀⠐⠁⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⣠⠟⠈⠁⠀⠉⠉⠁⠚⠷⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠏⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠀⢀⣂⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠁⠀⠘⠛⣛⣷⣤⡟⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠀⡀⠘⣋⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⣠⣿⠥⠚⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⠀⢸⣾⡟⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠤⠂⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⣶⠀⡄⣾⣿⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣧⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿
⠀⠀⠀⠀⠀⠀⢀⣔⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡤⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⡩⠁⠀⠀⠀⢀⣿⣦⣷⢷⠋⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠉⠠⣔⠉⠀⠈⡀⠀⠀⢸⡟⠉⠀⠰⠀⠀⠈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠞⠛⠉⠁⠐⡀⠀⠈⡓⣂⠀⠠⠀⠀⠈⢆⠀⠀⠀⠣⣀⣀⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⢢⣀⣀⣀⣠⣵⠀⠀⠀⠈⠙⠚⠉⠉⠀⠀⢆⠀⠀⠀⣈⡆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡀⣠⡀⣤⠂⠄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⢧⡇⠳⠇⠏⢀⠠⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠖⠐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣿⣿⡿⠀⠁⠀⣀⣀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠈⢿⠟⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣿⣿⣿⠇⠀⣀⣀⣤⣄⠀⠀⢀⠂⠙⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠿⠿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⠀⣿⡿⣲⠀⠀⢀⣀⠀⠀⠀⢀⢴⣥⣃⠁⢀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣹⣿⡿⠶⡞⢻⣟⠩⢰⣿⠸⢿⣠⣷⡂⣼⢿⣯⣄⣀⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣍⣅⡄⠔⡶⠙⢄⡘⠋⣞⠺⣿⣿⡉⢻⠚⡏⠰⢾⣿⡩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠙⠆⠁⠀⠺⠀⡄⢠⢸⣿⣿⢦⡏⡁⢠⣯⠛⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⢌⡒⠾⠘⣷⣿⡟⢾⠳⡇⢸⢋⡐⢛⠛⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠛⣫⣀⣷⣘⠙⣷⠀⠀⢸⢣⡞⣉⡥⡶⢲⡶⣶⣿⣌⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠀⠀⠀⠛⠿⣿⣏⣹⡆⠀⢈⣾⣧⣿⢶⣴⣯⣶⠷⠎⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠑⠀⠀⠀⠙⢻⣿⢿⡇⠀⠀⣾⣿⣿⡿⢿⣿⣿⡗⠾⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠦⠀⠀⠀⠋⠁⠉⠋⠴⣶⡿⠿⠿⣿⣆⡙⠹⠶⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⡀⠀⠀⠀⠀⠀⣈⡋⠀⠀⠀⠛⠋⣹⣷⣿⣿⣟⣶⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠢⣄⣀⠀⠀⠀⠉⠉⠓⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣟⠻⣶⣤⣤⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢷⣾⣿⣿⣿⣿⣾⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣱⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢠⠎⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠿⣿⢟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⠁⢈⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣸⣿⡏⣿⣿⡿⢿⣿⣿⣭⣿⣿⢿⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠛⠻⢻⣥⡾⣿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⡀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣦⠑⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⣾⣿⣿⡄
⠀⣰⣤⣠⣴⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣄⡀
⣿⠿⠿⡛⠛⠛⠛⠛⠿⣿⣿⣿⡀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⢦⣴⣶⡟⠀⣾⠻⢎⢿⠇⢀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⡟⡍⠙⢋⡻⣿⠀⠀⠻⢥⡟⠋⢠⣾
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠤⠜⠃⠀⠀⢀⡈⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⡞⠉⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡏⠀⠀⠀⢠⣿
⠸⠵⠾⠀⢸⡞⠯⠗⠧⠇⠀⣷⣄⠀⠈⢿⣿⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⣿⣿⡿
⡀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣿⣿⡟⠁
⣿⣶⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⠁⣼⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢠⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣾⣅⠀⣾⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⣿⣿⣿⣿⣿⣿⣿⣻⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⢻⡛⠛⠛⢟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⣤⣦⣿⣶⣦⣄⠙⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠈⡻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠁⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⣇⠘⢦⡙⢦⡀⠀⠀⠀⠀⣀⡴⢋⡴⠃⣺⣿⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⠁⠈⢿⣦⣙⠛⠛⠛⠛⣉⣴⡟⠁⠘⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡂⠀⢈⡿⠋⢀⠀⠀⡀⠙⢿⡀⠀⢐⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣾⠟⠉⠀⠁⠀⢀⠞⠀⠀⠣⠀⠀⠈⠀⠉⠿⣷⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⢤⡀⠀⡀⢀⠀⢀⡤⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠀⠀⠀⠀⠈⠉⢻⠷⣍⡽⠁⠈⢮⣩⠾⡟⠉⠁⠀⠀⠀⠀⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠂⠀⠘⣇⠀⢀⡀⠉⠉⠀⠀⠀⠀⠉⠉⢀⡀⠀⣸⠃⠀⠐⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣗⡄⠙⠳⣜⢦⠀⠉⠀⢠⠀⠀⠀⠀⡄⠀⠉⠀⡼⣡⠞⠋⢠⣺⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢤⣄⡀⠰⣦⡀⢠⡏⠀⠀⠀⠀⢹⡀⢀⣴⠆⢀⣤⡤⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⡅⠀⠹⣷⡘⠀⠀⣀⣀⠀⠈⢃⣿⠏⠀⢨⣽⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣿⠳⡀⢿⡉⢙⡟⢠⢞⣯⣤⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣧⡹⡷⣶⣶⢾⢋⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠂⠐⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠒⠒⠒⠒⢤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠙⢦⣩⣼⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠚⠂⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢸⠇⠠⠀⠀⠘⢿⣽⣿⣿⣿⣾⣦
⠀⠀⠀⠀⠠⣺⣻⣿⣏⡔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠎⢀⣀⡀⠀⠀⣌⣿⠋⠛⠛⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⣠⡾⣿⣷⣦⡀⠀⠀⠀⠀⠀⣬⣾⣿⣿⣿⣦⠀⠺⣿⠐⢧⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⡟⠉⣽⣇⣦⣵⣶⣾⣦⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⢻⣆⡴⣾⣿⣿⣽⡗
⠀⠀⠀⢸⣿⠁⠈⢿⡟⣿⣿⣿⣿⡿⣻⣿⠃⠀⠀⠀⠀⠸⣿⣽⣿⣯⡿⠇⠀⣾⡏⠀⠀⠀⠀⠉
⠀⠀⠀⢸⣯⣿⣤⣼⡖⠈⠻⣿⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⣿⡿⢿⠀⠀⠀⢘⡇
⠀⠀⠀⠘⣟⢿⢷⣼⣿⡇⠀⢈⣼⡟⢁⣴⣶⣚⢛⣿⣷⣄⠀⠻⣇⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠈⠓⠉⠀⢻⣟⠀⡏⠈⠁⢸⣿⣿⡿⣿⢿⡿⣿⡀⠀⠀⡀⢰⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣷⢠⠀⢲⣿⣿⣿⣿⣿⣿⡿⡏⠀⠀⡇⣸⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡀⡾⠼⣿⣿⣿⣿⣿⡷⠁⠀⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣭⣽⣿⣿⣷⡦⠤⢴⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠤⠤⠄⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠖⠒⠒⠒⠒⠛⠒⠒⠒⠒⠛⠛⠛⠛⠛⠻⠿⠟⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣶⣤⣴⣤⣴⣶⡀⠀⠀⠀⠀⠀⠠⠟⡍⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⢀⣄⡀⣸⣿⣿⣿⣿⣿⣿⡇⣀⣀⣴⣦⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣭⣉⣌⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⣿⡏⠛⠻⠁⠀⠿⠑⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣁⠀⠉⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣥⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠉⠉⠁⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⢀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣰⣆⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡶⣶⡷⢶⠞⢻⡿⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢀⠀⠀⣤⡐⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣮⡴⠛⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠛⠛⠻⢶⣄⡀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢸⣿⡵⠂⠀⠀⢀⣠⡴⢶⣶⣦⣄⠀⠀⠀⠀⣠⣶⠿⣿⣶⣦⣄⠀⠀⠀⢻⣷⡄⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠚⣿⣁⡀⠀⣠⢿⣿⣷⣾⣽⣿⣿⢀⣀⢠⣠⣿⣿⣶⣿⣿⣿⡿⢷⠀⣸⣿⣿⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠿⣿⡿⠷⠁⠘⣿⡿⠿⠛⠛⠃⠁⠀⠀⠀⠉⠉⠉⠻⠛⢿⣁⠘⠛⠛⢃⡈⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣏⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠣⡄⠀⢀⡾⠁⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⣟⠀⠀⠀⠁⠀⠀⠀⠀⣴⣿⣿⠛⠛⣻⣿⣶⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠻⣿⠖⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣾⣿⠿⣷⡆⠀⠀⠀⠀⠀⠀⢀⢀⡤⠋⠙⣆⣀⠀⠀⠀⠀⠀⢀⠀⠀⣼⣦⣀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠻⣄⡀⢳⣤⡤⠖⠛⠋⠀⠀⠀⠈⠋⠙⠾⣦⣀⣠⣞⣀⡴⠃⠈⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⢤⣤⣠⣀⠀⢀⢀⣠⡤⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⠀⠓⠀⠞⠀⠘⠃⠛⠀⡠⠚⢛⠋⠀⢀⡀⠀⠀⢰⣿⠿⡿⠂⡀⠒⠁⠂⠂
⠀⠀⡠⣤⠖⠂⠐⠲⣮⡻⢿⣷⠊⣴⣷⣦⣧⣴⠸⡾⠠⡴⠛⠃⢠⠘⢺⣿⣶⣿⢿⣫⡶⢀⡏⣅⢴⣤⡀
⠀⠀⣸⠃⠄⣠⣄⠐⠘⣿⠖⠙⠃⠈⠻⣿⣿⣿⢰⢿⡆⠁⠄⢀⣀⣴⣿⡿⠛⠱⠚⣻⡰⢀⣥⡉⠤⢻⠂
⠀⢀⠘⣦⠀⠘⣁⠂⠸⠳⣄⣠⢤⣴⣶⠘⠛⠏⣀⣿⡃⠀⠈⡝⢿⡛⠛⠁⢀⡀⠠⢻⠉⢐⡙⠀⢠⡯⢢
⠀⠈⣴⠏⣶⠗⠀⠀⣲⠁⢿⡉⠐⣀⡐⠘⣿⣿⣿⡿⢷⢠⡤⠛⡚⢹⣿⣶⡟⢋⢰⡏⢲⠀⠐⢷⡖⢷⡄
⠀⢰⣏⣰⠻⡁⠐⢀⠏⣱⢈⡇⣀⢹⠈⠀⢹⠻⠟⠡⣄⢱⡗⠀⢳⣼⠀⠽⡏⠀⢿⣷⣌⠐⠈⡙⢳⣁⣧
⠀⣀⣽⣿⢦⣿⠁⢈⣶⣿⠉⡀⠈⢛⠁⠀⠁⠤⠤⠀⠐⢿⣿⢤⣼⠿⣶⠀⠹⠀⠋⠸⣏⠀⢹⣧⢾⣿⡁
⠀⣁⠈⠉⢸⡟⢳⡞⢻⣿⣤⣮⠀⣀⠀⣤⡛⠘⠄⠀⠀⠘⣏⣀⣓⣤⡿⢣⠛⡶⠛⠀⡟⣶⠛⣿⠈⢉⡀
⠀⠃⠀⠘⢻⣶⣆⢰⣾⣿⣃⣈⠄⠈⠃⡉⠡⠀⠀⠀⣀⡀⠋⠙⠋⢁⠀⢘⠀⡛⠀⠨⣶⢀⣶⡾⠛⢐⣇
⠀⠀⠀⠀⢘⠃⠈⢀⢀⠈⠻⢩⣀⣀⣠⠀⠀⠀⢀⡄⢹⣠⡌⠄⠊⠋⢸⣾⣄⠀⠀⠀⡃⢸⣀⣇⡔⠊⢁
⠀⠀⡁⡀⠀⢀⣼⣾⠉⠁⠣⢤⡏⠴⠀⣦⠁⠒⠚⠷⠀⠈⠁⠀⡀⠂⠈⣿⣿⡇⠠⠀⠀⣾⣿⣦⣤⠀⣿
⠀⠀⣿⡇⠀⣿⢿⡧⠀⠀⢀⣾⠃⣾⣧⣭⢲⣷⣀⠀⠀⠀⢠⡄⠘⠿⠃⠸⠿⠃⠀⠀⠹⣿⠿⡷⠉⢀⡟
⠀⡴⠁⠀⠀⠿⣾⢡⠀⣤⣾⠟⠀⠀⠘⠻⠃⡙⢿⣿⣾⡶⣿⠀⢸⢿⣷⣦⡀⠀⠀⠀⠀⠄⠘⠏⠀⠈⣳
⠀⠀⠀⠀⠀⡢⢻⣄⠠⣝⣰⠒⢢⣤⠀⠀⡞⢹⣦⣿⣿⡟⠋⢺⠊⠛⣿⣿⣧⠀⠀⠀⣀⠀⡄⣀⢹⣿⣿
⠀⣿⠆⠀⠐⠗⠾⠃⠀⠂⠀⠡⣼⣏⠈⠙⣧⣊⢸⣿⣿⣷⣶⠿⡆⣰⣿⣿⣿⣻⢗⣎⡛⠁⢁⠉⢻⠉⢻
⠀⣿⣿⠐⠿⢆⢰⡖⢈⠁⠆⣰⣿⡿⣤⡼⣿⡯⠉⠻⡿⢿⠋⠼⡉⢾⠻⠿⣧⡅⠈⠏⠀⠀⣯⣥⣿
⠀⣿⣿⠈⠀⢻⡃⠐⢸⣏⣀⣽⣯⣀⣚⣂⣨⡿⠃⠀⠣⣠⠤⠄⠘⠿⣳⣤⡽⢷⣄⠈⠀⠀⠃⢸⣿⣦
⠀⢿⠹⠆⠀⠀⢀⠠⠛⠻⣿⣿⣟⣿⠠⡅⣀⡄⠠⢞⠳⠁⢰⡀⢀⣊⠓⠿⣣⣬⠿⡘⢳⡞⠃⢸⣿⣿⠆
⠀⠀⠀⢦⣠⠦⠐⢠⠀⠀⡁⢸⠛⠁⠀⢠⡟⢿⣄⠀⣀⢀⣻⢋⡌⠉⡀⠞⢉⠁⡀⠐⢸⠃⠀⢠⣻⠋
⠀⠀⠀⢀⢏⠠⠄⡘⠁⠀⠀⡄⠀⠀⠀⠘⠠⣜⠛⠙⠉⠀⠿⢠⣧⡈⠧⡔⠚⠉⣾⣧⡀⠀⠀⠸⢏⡐
⠀⠀⠀⢀⣾⣿⡇⠰⠊⠀⠾⣧⣈⡟⢀⠈⠀⠀⢀⣠⡷⠆⣤⣿⣌⠃⠠⣶⡄⠀⢸⣿⣿⠀⠀⠀⢿⢟⣦
⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠉⠀⠾⣦⣑⢤⣴⣿⣽⢇⣄⢯⣿⣿⣷⣐⠛⠻⠿⠀⢿⡿⠀⠀⠘⢏⣿⡟
⠀⠀⠘⣿⣿⠃⠻⣿⣿⣶⣶⣾⢇⣤⢿⣽⣿⣿⣿⡀⣀⢇⠀⣹⣿⣿⡏⠀⣿⣶⣶⣄⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⣈⠃⠀⣀⣠⣿⣿⣿⡀⢀⢆⠀⣸⣿⣿⡿⠀⠀⠸⣿⣿⣿⣿⠗⡖⠛⢻⣿⣿⡄⠀⣠⠄⠀⠻⠟
⠀⠀⠡⠄⠶⠊⢉⠋⣿⣿⣿⠁⠀⠀⡀⢹⣿⣿⡇⠀⠀⠀⣹⣿⣿⠃⠐⠙⠀⢺⣿⣿⡇⢸⣿⣤⣴⠆
⠀⠀⠁⠒⠀⠀⠛⠀⠹⠿⠿⠄⠁⠀⠄⠸⠿⠟⠃⠄⠀⠀⠘⢿⠿⠁⠘⠊⠀⠘⠿⠿⠡⠥⠾⠿⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠉⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⡴⠒⣢⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⠚⠀⠀⣴⣾⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢳⣦⣤⣤⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⢘⣷⣦⣄⡀⠀⣼⣿⣿⣿⣿⠛⣿⣿⣿⣿⣷⣤⡤⠀⠀⠀⣀⣀⡀⣀
⠘⠛⠛⠿⡉⢹⣿⣿⣿⣿⣏⢰⣁⣿⣿⣿⣿⣿⣿⣿⣼⣃⢿⣿⠿⡿⠁⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⠁
⢐⣤⠰⢲⣶⣸⣿⣤⣼⣿⣅⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠠⠿⠒⠀⢼⡟⠛⣻⣥⠀⠨⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠐⠀⠀⠀⠈⠀⢀⢄⡉⠀⠠⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⣀⠀⠀⠀⠀⠘⠁⡘⠄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠀⠙⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢰⣿⣤⣤⣤⣤⣤⣤⣤⠤⠀⠀⠀⠀⠀⠀⠈⠑⠈⢻⣿⢻⣿⣿⣵⣤⣴⣼⣿⡿⠛⠁⠀⠉⠿⣿⡿⣿⣿⣿⣷⣦⣍⢸⣿⣿⣿⣿⣿⣿⣦⡀
⣸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠖⢻⣿⠀⠉⠉⠉⠀⠀⠀⣠⣀⣴⣦⡸⣿⣼⣿⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣦⠄
⢸⣿⣿⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⠀⠀⡈⢹⣄⣴⣤⣤⣄⣴⣿⣿⣿⣿⣿⡧⣾⣿⣿⣿⣿⠉⢻⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠁⡀
⢸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣸⣧⠶⢿⣟⣼⣟⣿⣿⠿⡏⠉⠛⠋⠉⢉⣉⣿⣟⣻⣿⠏⣄⠴⠿⠋⠈⠀⠀⠀⠀⠈⠉⠛⠻⠿⠟⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠀⠠⠈⠹⠁⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿