Dial Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣶⣄⣀⣀⣀⣠⣴⣠⣶⣶⣶⣶⣾⣷⣶⣶⣶⣦⣶⣠⣀⣀⣀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⢿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⣼⣿⣧⡀⠀⠀⠀⢸⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯⠆⠀⢶⣾⣿⣿⣿⣿⠶⠀⠀⢿⠃⢀⣠⣄⣤⣶⣦⣀⣄⣀⠀⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠹⣿⣿⡁⠀⠀⠀⢸⠀⠀⢸⡇⠉⣿⡋⠉⡏⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠈⠓⠒⠋⠀⠀⣿⡅⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣟⣛⣛⡻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⣩⣶⡋⣻⣉⣹⠋⣳⣮⡙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏⣾⠧⣤⣿⠟⠛⠛⢿⣧⡤⢿⡎⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⣿⡴⠾⣿⣄⣀⣠⣾⡿⠦⣼⡇⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⡹⣦⡖⠉⢻⠟⢿⠋⢳⣶⡾⣡⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣬⣙⠻⠿⠦⠾⠿⢛⣩⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣤⣤⣴⣶⣾⠿⠛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⣧⣽⣿⣄⣤⠤⠤⠤⠤⠤⠤⠶⠦⠤⠤⣤⣴⣿⣧⣾⣇⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠉⢉⣵⣟⠻⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣓⠻⢿⡯⣍⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢵⣀⣦⠴⠚⠋⠉⢀⣰⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣧⡂⠈⠈⠑⠲⢴⣆⡤⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣮⣍⠵⣤⠤⢴⣶⣿⣿⣿⣿⣿⣿⠹⣿⣦⣭⡭⠭⠭⠭⠭⠭⠽⣿⣿⣾⣿⣧⣾⣿⣿⣿⡏⢩⠳⠶⠤⢤⠦⢩⣭⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣆⠀⠀⢈⣿⣿⠿⣻⣿⣿⣿⡄⢿⣿⣿⣿⣀⠀⠀⠀⠀⣀⣿⣿⣿⡟⠱⣿⣿⣼⣿⡳⣦⣕⡁⠀⠀⢠⣾⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡎⠛⡃⠈⠉⠉⠴⢝⢀⣿⣿⠏⡏⠀⠀⠉⢉⣭⠭⠭⠭⠭⣍⡉⠉⠁⠀⠄⣿⣿⣿⡈⣛⠒⠉⠉⠁⠒⠛⢹⣾⠇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⡀⠀⠀⢀⣨⡶⢿⣿⢿⠂⡁⠀⣠⠞⠉⠄⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢸⣿⡏⠻⢷⣧⣀⠀⠀⣀⣀⣼⠏⢠⠖⠖⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⣰⣾⡇⢺⠀⠀⡼⠃⠀⠀⡠⠔⠂⠐⠢⢄⠀⠠⠘⣆⠀⢼⣺⣷⡄⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⣤⣤⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠀⢰⡇⢀⠀⡸⠀⡰⢂⡐⢀⡈⠆⠀⠀⢸⠀⢸⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⣿⠂⠸⡇⠈⠀⢱⠀⠷⣌⡡⠞⢁⡆⠀⠀⢸⠁⢀⣿⣿⣿⣷⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⠁⠈⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣾⠀⠀⢷⡀⠀⠀⠑⠤⣀⣀⠤⠊⠠⠀⢀⠟⢠⠀⡿⣿⣿⣿⠃⣿⡋⣠⡶⣀⠀⠀⣀⡘⠛⠳⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠀⣿⠀⠀⠠⠙⢄⡘⠁⠀⠠⠀⠀⠀⢐⡀⢈⠂⠘⠀⠇⣿⣿⣇⠀⠈⡝⢁⣾⢟⣼⡗⢹⡷⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡏⠐⡏⠀⠀⠀⠠⠂⠉⠓⠲⠶⠶⠖⠚⠉⠐⠋⠀⠀⠐⠇⢻⣿⣿⠀⠀⠀⠸⢧⠼⢿⣤⠜⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣼⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⡄⢿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣙⣛⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣟⣻⣟⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣉⣿⣙⣉⣏⣹⣉⣉⣋⣙⣏⣉⣍⣩⣭⣽⣉⣹⣯⣿⣻⣿⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠙⠲⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢛⡛⣿⣿⣿⣾⣝⠛⣵⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣪⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢔⣿⣿⢛⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣆⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣿⣿⣿⣿⣿⣿⠀⢸⡇⠈⣉⡉⠉⠉⠉⢉⣉⡉⠀⣿⢀⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠞⠛⠛⣿⣷⣶⣶⣶⣾⣿⠟⠛⠋⠘⠿⠿⠿⣿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢿⣁⣀⣀⣀⣀⣀⣴⣿⣷⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠻⣿⠟⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⠛⢧⣀⣸⣀⣠⠟⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠓⣶⣾⠟⠛⠛⢿⣦⡔⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⣿⣿⡷⠴⣿⡇⠀⠀⠀⠀⣿⣷⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡤⢿⣧⣀⡀⣀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣠⠟⠻⡿⠿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣶⣧⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠄⠐⠒⠒⠉⠀⠈⠉⠙⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠏⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣭⣿⣽⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠐⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠀⠐⠒⠒⠂⠤⠄⣀
⠀⠀⠀⠀⢀⣀⠤⠖⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⣄⠀⠀⡀⠀⠀⠀⠈⠑⠢⣄
⠀⢀⠠⣴⣅⣼⠆⠀⠀⠀⠀⠰⢔⣖⡶⡖⣶⣶⣶⣗⠓⠓⠛⠛⠚⠛⠛⠛⠓⢓⣳⣿⣿⡟⢓⣷⣶⣦⠆⠀⠀⠀⠘⢷⡀
⡴⣡⠮⡁⢈⠃⠀⠀⠀⠀⢀⣡⢝⣣⡟⢽⡯⠉⢯⣿⡿⠿⠽⢽⣿⣛⣛⣛⣿⠿⢿⠇⢦⣷⠋⢹⠉⠀⠀⠀⠀⠀⠀⠐⡿⡀
⢧⢼⡧⣷⠞⠠⠀⠀⡠⢠⢪⠢⣑⠜⡨⣻⡇⠀⢨⣿⡳⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠀⠨⣿⠀⢘⣄⡢⡀⠀⠄⠠⢀⢎⢪⢳⣄
⢨⣽⣺⡇⠘⠢⠦⠦⠄⠦⠖⠊⡡⡰⢱⣽⠇⠀⠸⣯⣰⠂⢺⣿⣶⣶⣾⣿⣦⠀⡨⠀⠀⣽⡔⠁⠉⡓⠶⣤⣠⣀⣁⣄⠦⢋⡾
⠈⢖⣛⣷⠀⡇⡆⡆⡔⠘⢰⠘⡱⢔⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣦⡐⠀⠀⠐⠀⠄⠀⢆⣼⠧
⠀⠐⣖⣒⣳⢶⠤⢧⣧⣧⠦⡖⠡⠁⠀⠀⠀⠀⠀⢀⠤⠄⠒⠂⠐⠒⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣗⢶⣥⣠⣠⣤⠼⠎⠁⠄
⠀⠀⠈⢖⣛⣯⣧⠠⠤⠤⣼⠠⠁⠀⠀⠀⠀⠀⠊⠀⠀⣀⣠⡤⢀⣤⡀⢀⣀⠀⠀⠑⠢⡀⠀⠀⠀⢽⠄⠀⠀⠉⠀⠒⠂⣁
⠀⠀⠀⠈⡖⣒⣻⠀⢀⣶⡏⠀⠀⠀⠀⠠⠂⠀⢀⢴⣯⡴⢿⡝⠻⣿⡗⢴⣻⡏⠢⡀⠀⠈⢆⠀⠀⢼⡇⠈⠉⣙⡉⠭⠁
⠀⠀⠀⠀⡧⠸⣶⣷⣿⣿⡇⠀⠀⠀⠰⠁⠀⢠⣥⣬⡛⢗⣬⣦⣬⣱⣎⢝⠫⣸⣹⡻⣆⠀⠈⡆⠀⢸⡷⠶⠭⠉
⠀⠀⠀⠀⠘⣉⢟⣺⣿⣿⡇⠀⠀⠀⠀⠀⠀⠃⠻⠷⢫⣿⠋⣠⠄⢀⣈⠑⡱⡘⡄⠀⢸⡄⠀⢰⠀⠘⣧⠐⠒⠞
⠀⠀⠀⢀⣼⣿⣎⠔⣻⢿⡇⠀⠀⠀⠀⠀⢸⣟⣽⠇⠀⡿⠀⠀⣀⠁⠀⠀⣡⢸⡇⠀⢸⡇⠀⠸⠀⠈⣯⠀⠤⠬⢅
⠀⠀⢠⢻⢾⣿⠛⠻⣔⠿⡇⠀⠀⠀⡆⠀⢸⣴⣷⡾⡰⡹⡄⠀⠀⠀⣀⢔⢍⡼⠀⠀⣼⠀⢰⡇⠀⠀⣻⠀⣀⠀⠄
⠀⣠⢷⣺⣽⠗⠐⢛⡷⠭⡅⠀⠀⠀⢃⠀⠈⢯⠉⣴⡾⡾⢿⣷⣏⣙⢨⠷⠋⠀⣀⡸⠂⢀⡞⠀⠀⠀⢿⡅⠉⠉⠩
⠀⢷⢮⣞⢷⣤⡤⣚⢬⡙⠆⠀⠀⠀⠈⢦⡀⠈⠻⢯⣋⣀⣯⣿⠇⣸⣼⣦⡤⠾⠋⠀⡠⡛⠀⠀⠀⠀⢸⡇⠀⠀⠈
⠀⢛⣯⣼⢸⢸⠾⡾⣷⡿⡇⠀⠀⠀⠀⠈⠻⣦⢄⡀⠈⠉⠁⠀⠋⠓⠘⠈⢀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠺⡅⠀⢀⡀⢁
⠀⠀⠀⠙⠫⠿⠾⠻⠚⠋⣗⠀⠀⠀⠀⠀⠀⠀⠁⠒⠛⠿⠗⠶⠶⠾⠋⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⢰⡇⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⡆⡄⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡄⡆⠈⢸⠇⠉⠉⢙⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣁⠀⠀⢸⢸⠈⠈⠀⠐⠀⡆⠀⠂⢀⡆⠆⠂⠀⠀⠁⠁⠀⠀⡇⠁⢀⡞⠀⠀⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠧⣮⣸⣠⣀⣘⣀⢀⣄⠀⠀⢸⡆⡇⡃⢀⣀⣀⣄⣇⣆⣧⠶⠟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⠍⠉⠉⢉⡉⠙⠛⠛⠛⠛⠋⠩⠭⠩⠉⠀⠄⠀⠠⠀⠠⠠⠄⠤⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠿⣿⡿⠿⠿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⣿⡁⠀⠀⠀⠀⠻⠃⠀⠀⠀⠈⢹⣿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠈⠛⢀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣀⠙⠁⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⡀⠀⠀⣠⣶⡿⠛⠉⠁⠀⣠⡄⠀⠉⠙⠻⣿⣦⡀⠀⠀⢙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠻⠗⢀⣼⡟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠻⣿⣆⠐⠿⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢺⣧⡀⣨⣿⠿⠿⠿⠿⠇⠀⢸⣿⠀⠰⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣾⠗⠀⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠠⣿⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠈⠻⢿⣶⣤⣀⣀⣀⣀⣀⣠⣤⣾⠿⠋⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⢀⣴⠀⠉⠙⠛⠻⠿⠿⠟⠛⠉⠁⣰⣄⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣄⣀⠀⠀⠀⣶⡆⠀⠀⢀⣀⣼⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣷⣺⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣾⣟⢻⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡹⣯⣻⣿⣿⣿⣿⣿⣿⡶⠷⢷⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⢷⣟⣯⣽⣷⣾⣿⣿⣿⣿⣿⣶⣾⣿⣛⢿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⡿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⢟⣫⢝⣷⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⡀⠉⠱⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣤⢻⣞⣯⣶⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣸⣶⣤⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⢀⣴⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣦⣍⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣀⣴⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⡈⠛⠛⠏⠭⠛⠋⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣻⣿⣭⣭⣴⣾⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣟⣿⣿⣟⣿⣏⣿
⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣀⣄⣠⣴⣶⣦⣈⡙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣟⡯⢞⡯⢞⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣷⣭⣻⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⢿⣞⣽⣾⢛⣽⣷⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣷⣶⣯⣵⡿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⣯⣾⣿⣿⣿⣿⣷⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢟⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⣫⢻⣷⣯⣻⡿⠿⢿⣿⣿⣿⣿⡿⢿⣿⣛⣽⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⣵⣳⡿⣹⣿⣿⣿⣿⣿⣷⢶⣾⣷⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⢷⣿⣣⣿⣿⣿⣿⣿⣿⣿⢼⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⠟⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⡿⢽⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⠒⠒⠒⠛⠒⠒⠒⠲⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⣀⣠⣤⠴⠶⠶⠶⠒⠒⠶⠶⠶⠦⣤⣄⣀⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⣀⣤⠶⠛⠉⠁⣠⠤⢤⣀⠀⠀⠀⠀⣀⡤⠤⣄⠈⠉⠛⠶⣤⣀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠞⠋⠀⠀⠀⢠⠋⠀⣠⣶⠈⠳⡀⢀⡞⠁⠶⢶⡄⠹⡄⠀⠀⠀⠙⠳⣤⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣴⠟⠁⣀⣤⣄⡀⠀⢸⠀⠰⠧⣿⠄⢀⡇⢸⡀⠀⠘⢻⡄⠀⡇⠀⢀⡠⣤⣀⠈⠻⣦⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠟⠁⢠⠋⢀⡶⠶⠈⢢⠈⠳⣄⣀⣉⡠⠞⠀⠀⠳⢄⣙⣋⣠⠞⠁⡜⠁⠴⢶⡄⠙⡄⠈⠻⣆⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⣼⠋⠀⠀⢿⠀⠈⠙⣷⠀⢨⠃⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⡅⠀⢀⡼⠃⠀⡿⠀⠀⠙⣧⠀⠱⡄
⠀⠀⠀⠀⠀⢀⡎⠀⣼⠃⠀⠀⠀⠈⠦⣈⡛⣉⡠⠊⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠙⠛⠶⣤⡀⠀⠱⢄⡛⢛⣃⠴⠁⠀⠀⠀⠘⣧⠀⠱⡀
⠀⠀⠀⠀⠀⡼⠀⣸⠇⠀⡠⠒⠒⠒⢦⡀⠈⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠉⢀⡴⠒⠒⠒⢤⠀⠸⣇⠀⢧
⠀⠀⠀⠀⢰⠇⢠⡟⠀⡞⠀⣴⣯⣅⠀⢹⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⡏⠀⠀⢻⠀⠀⢳⠀⢻⡀⠸⡆
⠀⠀⠀⠀⢸⠀⢸⡇⠀⢳⡀⠻⣄⡿⠀⡸⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢇⠀⠀⢸⠀⢀⡞⠀⢸⡇⠀⡇
⠀⠀⠀⠀⢸⠀⢸⠃⠀⠀⠙⠲⠤⠔⠊⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠓⠦⠤⠖⠋⠀⠀⠘⣇⠀⡇
⠀⠀⠀⠀⢻⠀⢸⡄⠀⠀⣠⠴⠒⠲⢤⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡟
⠀⠀⠀⠀⢸⠀⢸⡇⠀⡼⠁⠚⢛⡷⠀⢱⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠸⡆⠘⣷⠀⢧⠀⢀⡾⠁⠀⣸⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⠇
⠀⠀⠀⠀⠀⢳⠀⢹⡆⠀⠑⠦⠤⠤⠞⠁⣀⡀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡞
⠀⠀⠀⠀⠀⠈⢇⠀⢻⡄⠀⠀⠀⢀⠔⢉⣥⣌⠑⢆⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣠⣤⠶⠛⠁⠀⠀⣎⢱⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠘⣆⠀⢻⣄⠀⠀⣾⠀⢘⡷⣿⠀⢘⡄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠱⢄⡀⠀⠀⠀⣠⡟⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠹⣦⡀⠘⣄⠘⠶⠞⢀⡜⢀⡴⠋⣉⣉⠑⢆⠀⠀⡴⠊⣉⣍⠙⢦⡀⠀⠀⢣⡀⠀⠉⠓⢲⣶⠏⢀⡼⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠻⣦⡀⠉⠛⠋⠁⠀⢸⠀⠸⣇⣼⡇⠈⡇⢸⠁⢰⡏⢸⡇⠀⡇⠀⠀⠀⠑⢄⣀⣴⠟⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠛⢦⣄⠀⠀⠀⠘⢄⠀⠶⠛⢀⡼⠁⠈⢧⡈⠳⠾⠃⣰⠃⠀⠀⠀⣠⡴⠟⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠉⠛⠶⣤⣀⡀⠙⠒⠚⠉⠀⠀⠀⠀⠉⠓⠒⠋⢀⣀⣤⠶⠛⠉⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⠉⠙⠛⠲⠶⠶⠶⠤⡤⠶⠶⠶⠖⠛⠋⠉⢀⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⠤⠤⠤⢤⡤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡴⢒⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣬⣐⡒⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⢛⡿⠛⡿⠛⠛⠛⠛⠋⠛⠛⠛⠛⡟⠛⠛⠛⡿⣿⠛⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⡿⣿⣷⣦⠀⠀⢀⡹⡀⡄⠀⠠⣬⣔⣿⢶⣿⣣⣼⠁⠀⠀⡴⠋⠀⠁⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁⠁⠀⠀⠀⠠⠙⣿⣿⣿⡤⠄⠈⠁⠁⠐⣛⣢⡉⠀⢙⣌⢃⠁⠀⠀⠀⠀⠀⠀⣀⡤⣶⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⢻⣉⡥⡿⣫⣞⢩⣽⠛⣳⡇⠀⠀⠀⠀⠀⠉⠓⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆⠀⣀⢻⣿⣶⣽⣧⣤⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣛⣺⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠦⡀⢠⣿⣿⣷⣶⣶⣶⠖⠶⠶⠶⠶⠶⠶⠶⠖⠒⠲⠒⠶
⠀⠀⠀⠀⠀⠀⣶⣮⣭⠳⠶⣄⣀⣀⣀⡀⠀⡀⡐⢯⠱⣸⢷⣼⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠛⢷⣿⣹⣸⣈⣇⣿⣧⣧⡇⣃⣬⣼⠞⠁⠀⠸⣿⣦⣀⡀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⣀⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁




⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠄⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠠⠤⢀⡀⣠⢀⢤⡀⠤⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠚⢡⠀⡄⠀⠉⠀⢸⡇⠻⠖⠼⡁⠈⠁⠀⠀⣴⠉⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠴⠀⠀⢸⠀⡇⠀⠀⠀⡸⠇⠀⠒⠶⠆⠀⠀⠀⠀⣿⠀⠀⠀⠄⠈⡠
⠀⠀⢀⠀⠀⠀⠀⠀⠐⠔⠐⠂⠂⠀⠠⠘⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠟⠀⠀⠀⠁⠑⠐⠄⠄
⠀⠀⢸⠀⠀⠀⠀⠘⠎⢸⡇⡀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡐⢹⠆⢲⠂
⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⣧⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡁⡂⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⢿⠆⠀⠀⠀⠘⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠂⠀⠀⠀⠀⢸⣿
⠀⢀⠿⣅⠀⠀⠀⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠘⠀⠀⠀⣠⡞⠉⣢⣀
⠀⠫⣤⡼⠁⣀⠀⢠⠠⠝⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣩⣅⠀⢨⡀⢸⡁⠨⣿⠀⢹
⠀⠈⢹⠁⠀⠉⠀⠈⡛⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢹⠇⠈⠉⠠⣿⢿⢩⠿⡿
⠀⠀⣾⠀⠀⠀⡠⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⠀⠀⠀⠀⢾⣬⠇
⠀⠀⣿⠀⠀⠀⠀⢠⠀⠀⠀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡄⠀⠀⠀⠀⣼⣿⣄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⢻⡷⣅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⠀⠀⠀⠀⠀⠘⢧⣤⡿
⠀⢰⠛⣦⠀⠀⠀⠠⢆⠘⠮⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⠀⢼⠄
⠀⠈⠉⠁⠀⠀⠀⠀⠠⠡⠠⡄⢀⠠⠴⠶⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠠⠠⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠲⠀⠀⣴⠡⠄⠀⠀⠀⠀⢠⠀⣒⡀⠀⠀⠀⠀⠋⠙⣷⠀⠠⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢌⡛⠖⠀⠀⣀⠀⠀⣿⡠⠍⡗⠀⠀⣀⠀⠉⠐⣋⠄⠢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠅⠂⠉⠀⠒⠀⠈⠛⠂⠀⠀⠒⠀⠉⠂⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⣿⣆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣼⣿⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢈⣉⣛⠛⠛⣛⣉⣉⣉⣉⡁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣈⣉⣉⣉⣉⣛⠛⠛⣛⣉⡁
⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠛⠁⠀⣰⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣄⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡇⠀⡞⠃⣾⢹⠀⢺⠀⣖⡆⢰⢳⠀⡞⡆⣾⠢⢰⠂⢰⢻⠠⠻⠀⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⣇⣷⠒⣇⡗⢺⣼⠘⣿⢷⣧⣷⢺⣼⡚⣇⣷⡻⡜⢻⣸⣻⣼⣛⣼⣏⢧⣿⠉⠉⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠢⢀⠖⠐⡆⢀⠖⠒⠰⡀⠀⠏⠉⠑⠢⡀⠀⢀⠔⠢⠀⢰⠉⠃⢸⠉⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠀⢠⡡⡄⠀⠃⠨⠀⣄⠀⢡⢰⠀⠘⠀⣠⠂⢀⠇⢰⣊⣀⣸⠀⢐⣚⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣀⡼⠀⢇⣀⡇⠃⣠⢄⠀⠈⣲⡆⠀⣤⣀⣑⡜⢦⣀⣠⠞⠡⠤⠼⠸⠤⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⡀⢀⠀⡀⠀⠤⢄⠠⠠⠀⠤⢠⠀⠀⠂⢠⠀⢲⠠⠄⢠⢀⠠⠄⠠⠤⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠠⠀⠅⠠⠤⠄⠠⠠⠜⠒⠐⠀⠄⠔⠘⠒⠃⠘⠄⠐⠸⠤⠜⠠⠤⠄⠠⡀⠄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⢰⠀⠂⠂⢠⠒⠄⠔⠰⡄⠖⢤⠀⠄⢠⠀⡔⢢⢠⠀⢄⠠⠄⢤⠠⠒⠆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠑⠚⠈⠂⠂⠘⠂⠂⠈⠚⠁⠃⠊⠀⠂⠘⠀⠣⠞⠘⠂⠊⠈⠂⠂⠘⠒⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠒⠀⠒⠒⠄⡊⠱⠀⡎⠂⢘⠉⡆⠐⠉⡆⡒⢒⠀⠒⢲⠀⡒⢢⠀⠒⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠚⠁⠉⠉⠁⠉⠈⠁⠁⠁⠈⠉⠁⠈⠉⠀⠉⠁⠀⠉⠁⠀⠉⠁⠀⠁⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠬⠭⠍⢁⣚⣛⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⠒⠒⠒⠀⠀⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣂⣐⣂⣀⣀⡀⠀⠄⠒⠒⣒⣛⣛⣛⣛⣛⣒⣚⣓⣀⡀⠀⠀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠓⠲⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡄⣠⡀⢠⢠⣤⣤⣠⢠⡄⠀⢀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⠿⡿⡿⢿⣿⣿⠿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠓⠛⠓⠚⠊⠛⠛⠛⠚⠃⠀⠀⠛⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣿⣿⣾⣷⣶⣾⣿⣿⣿⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣏⣈⣷⣀⣼⣉⣙⣆⣀⣾⣁⣹⣆⣠⣏⣉⣳⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⠟⠀⠀⠀⣤⠇⠀⠀⢀⡼⠃⠀⠀⢠⡞⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠈⠁⠀⠈⠁⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠆⠀⠠⠄⠀⠠⠄⠀⠴⠀⠀⠤⠀⢀⣀⣤⣄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⢀⡀⠀⢀⡀⠀⢀⠀⠀⣠⠞⠉⠀⠰⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠈⠁⠀⠈⠀⠀⠈⠀⢰⠃⠀⠀⠀⢰⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠂⠀⠐⠂⠀⠐⠂⠀⠒⠀⣿⠐⠂⠀⢀⡼⠀⠀⠀⠒⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣆⠀⠀⠈⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡘⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠿⣿⢿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⠋⠉⠉⠁⠀⠸⠉⠉⠓⠢⢍⡛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠳⡈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠻⣑⢤⡀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⢣⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡳⣄⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠨⡄⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⠝⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡠⠋⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⡀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡵⢊⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⣷⣖⣠⠤⠤⠤⠼⠤⠄⣒⣩⣥⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⣿⡆⠀⠀⣼⣿⠛⣿⡇⠀⠀⢰⣿⠛⣿⣷⠀⠀⢰⣿⠟⢻⣷⠀⠀⢀⣿⡟⢻⣿⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣿⡇⠀⣿⣧⣤⣤⣿⣿⠀⣿⣷⣤⣤⣾⣿⠀⣿⣿⣤⣤⣼⣿⠀⢸⣿⣤⣤⣼⣿⡇⢸⣿⣦⣤⡀
⠀⠀⠀⠀⣴⣿⠟⠉⣿⡇⠀⣿⡏⠉⠉⣿⣿⠀⣿⡟⠉⠉⢹⣿⠀⣿⣿⠉⠉⢹⣿⡀⢸⣿⠉⠉⢹⣿⡇⢸⣿⡋⠛⢿⣷⡀
⠀⠀⠀⢸⣿⠃⠀⠀⠙⠿⠿⠟⠁⠀⠀⠘⠿⠿⠿⠃⠀⠀⠈⠿⠿⠿⠋⠀⠀⠈⠻⠿⠿⠏⠀⠀⠀⠻⠿⠿⠟⠀⠀⠈⣿⡇
⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠃⠀⠀⠸⠿⠿⠟⠀⠀⠀⠿⠿⠿⠃⠀⠀⠺⠿⠿⠟⠀⠀⠐⠿⠿⠿⠂⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣶⣶⣦⣿⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣶⣶⣶⡄⠀⠀⢰⣶⣶⣦⠀⠀⠀⣶⣶⣶⡄⠀⠀⢴⣶⣶⣶⣾⠿⠛⠉⠁⠀⣴⣦⠀⠈⠉⠙⠿⣿⣦⣀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣽⣿⠟⠁⠀⠀⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠘⠛⠛⠋⠀⠀⠀⠛⠛⠛⠁⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⣀⠸⣿⡇
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠸⠿⠿⠷⠀⠀⠀⠿⠿⠿⠆⠸⣿⡇⠻⠿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣤⡀⠀⠀⠀⠀⠻⠿⠟⢀⣿⡇
⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡷⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣦⣤⣬⣥⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⠀⢸⢳⡆⠀⢰⠛⡆⠀⢰⠒⡆⢰⠓⡆⢸⠓⡆⢰⠛⡆⢠⠖⡆⢰⠟⡄⣾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⢼⣈⠤⢼⠰⠧⠄⠸⡁⡷⠐⠾⡀⡳⠰⣂⡗⠺⣀⡗⠺⣀⡷⠲⡀⡗⢺⣀⡓⢻⢘⡛⠋⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠙⠁⠘⠋⠀⠀⠀⠉⠁⠀⠈⠃⠁⠀⠉⠁⠀⠀⠁⠀⠁⠁⠀⠃⠃⠀⠁⠁⠈⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠋⠙⢐⠁⢐⠀⠁⠈⠀⡁⠈⠀⡏⠀⡅⠊⠀⠀⠈⠀⠈⠰⠁⠈⡀⠈⠀⡆⠉⠀⠁⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠉⠉⠀⠀⠀⠁⠈⠉⠁⠈⠁⠀⠈⠁⠀⠀⠈⠈⠀⠙⠒⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠁⡂⠀⠂⠈⠉⠂⡋⠉⠁⠈⠁⠆⢸⠁⠱⠐⠂⠐⠀⠂⠒⠀⢘⠐⠂⠈⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠀⠑⠒⠀⠈⠀⠁⠈⠀⠀⠈⠉⠁⠈⠀⠀⠈⠀⠚⠀⠓⠒⠀⠈⠀⠁⠈⠀⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡖⠀⠆⠘⠀⠐⠰⠉⠉⠀⠏⠙⠀⠉⢹⠀⠋⠉⠃⡊⠉⠀⠘⠉⡀⠐⠂⣂⢰⠀⠒⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠃⠈⠀⠈⠁⠉⠀⠁⠈⠉⠀⠁⠚⠀⠉⠉⠁⠈⠉⠀⠈⠉⠁⠈⠁⠀⠈⠁⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⠄⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⣀⠀⠐⠒⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠴⠴⠦⠤⠭⣭⡭⠭⠥⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣸⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢃⡤⣄⢀⣣⣃⠀⠠⠟⠁⠀⠀⠀⠀⠀⠀⠻⡆⠀⣿⡇⠘⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⣸⠶⠛⠉⠉⠭⣭⡳⣄⠀⢠⡖⠲⡦⣄⣠⠴⠒⠒⠛⢿⣴⣛⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣠⠤⢶⡾⠃⠀⠉⣻⡷⣾⡁⠀⠀⠀⠀⠀⠈⠛⠧⡀⢠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⣾⠀⢀⣠⠖⢻⣽⡶⠞⠓⠒⠛⠛⠻⠿⢯⣉⠉⠳⣄⠀⠀⠀⠀⠀⢢⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠯⣶⠞⠋⢀⣀⣤⡤⠤⠤⢤⣤⣀⠀⠈⠙⠦⣿⣿⣦⣄⡀⠀⢸⠂⡇⠐⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⣠⠞⠁⣠⡾⠟⠉⠀⠀⠀⠀⠚⠀⠈⠙⠲⣄⠀⠈⠳⡄⠀⠙⠳⣦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⠃⣠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⣆⠀⢆⠘⣆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⢣⠏⠀⠀⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠘⣆⠘⡆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡇⡾⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡀⢻⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⠶⠀⠀⠀⠀⠀⠈⡶⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⣇⠀⠀⠀⠀⠀⠀⠀⣱⠶⠃⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⡇⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣇⢸⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣆⢳⡀⠀⠀⠀⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡞⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢦⡙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⡽⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢⣍⡒⠤⣀⡀⠶⠀⠀⠀⢀⣠⠖⠋⠀⢀⣾⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⣀⠉⠛⠒⠲⠶⠒⠒⠛⠉⠀⣀⣤⡖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠈⠙⠓⠶⠶⠶⠶⠶⠖⠚⠋⠙⣦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⢤⡴⡄⣬⣤⢭⣧⢤⢤⣤⣤⡤⣤
⠀⠀⠢⠖⠡⠀⠀⢠⠀⠀⠐⠈⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠉⠩⠉⠉⠨⠍⠛⠛⠋⠙⠉⠛⠙⠋⠙⠙⡟
⠀⡁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⡀⠀⠀⠀⠀⡇⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂⠀⠘⠀⠀⡧
⡟⢀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⡇⢀⣈⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⡏⠀⠀⠀⣿⢓⣶⣦⣐⣖⣀⣶⣆⣶⣶⣾⣆⣶⣶⣳⣶⣶⣷
⠰⡔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠁
⠀⠙⢌⠢⠈⠀⠀⠀⠀⠀⠄⡀⢌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠉⠒⠤⢭⣠⣤⠬⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡍⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢔⡢⠖⠊⢹⡄⠉⠛⠷⣠
⠀⠀⠀⠀⠀⠀⢀⡆⠈⠈⢰⠄⠈⠉⣏⣉⠁⠀⢰⡂⠁⠠⡀⠀⠀⠠⡞⠛⠓⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢺⠋⠀⠀⠀⢾⡗⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠘⢲⡀⠖⣠⠔⠉⠀⠠⠀⠀⢉⠂⡝⢶⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢈⠀⠀⣀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡼⠁⢠⣄⠀⠀⠀⢀⡀⠑⠘⡀⠁⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠁⠀⠀⠀⠘⠁⠉⠋⠁⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠉⠳⣦⡞⠋⠀⠠⠀⠁⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⣐⡅⣀⡀⣀⡠⢦⣔⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠠⠀⠀⠄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠿⡟⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠒⠒⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠙⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⣻⠞⠁⠀⠀⠀⠀⢀⠀⠤⠀⠁⠼⠤⠥⢈⣀⣤⣀⡀
⠸⠧⠥⠴⠦⠤⠄⠄⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⠀⠙⢦⣀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠰⠾⠿⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⡄⠀⠀⠀⠀⠀⢻⡇⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠘⣇⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠏⢈⠁⠁⠀⡀⢀⠀⠀⠀⠙⢏⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣶⠄⠀⠀⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡅⠀⠈⠂⠀⢀⢸
⠀⠀⠈⠐⠄⠂⠂⠈⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡧⠐⠾⢿⡗⢦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡀⠀⠀⢀⡀⠀⣀⠀⠠⢸⠈
⠀⠀⠀⡀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⣰⠀⠠⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢸⠸
⠀⠠⢁⠄⠀⠀⠀⠐⠀⠑⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⣔⠁⠀⠼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠠⠄⢀⠠⡀⠀⠀⡀
⠀⢃⠇⠀⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⡏⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠂⠀⡞⢹⡏⠙⠂⠈⠀⠀⡀⡆
⠀⡀⠃⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠇⠉⣠⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠇⡈⠡⠀⠀⠀⠀⠀⠀⠇
⢠⣗⣐⣒⣆⣓⡒⠚⠒⠒⠚⠆⠀⠀⠀⠀⠈⠠⣈⣀⣠⣤⣼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠓⠄

⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⣭⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡈
⠀⢸⣿⣿⣿⣿⡻⠟⠛⠛⠉⠙⠻⠶⠰⠾⠛⠉⠙⠛⠛⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣶⣦⣄⣹⣏⣠⣴⣶⣶⣤⣤⣤
⠀⢸⣿⣿⣿⣿⣧⣄⣀⣀⡀⠀⠀⢀⣄⠀⠀⠀⣀⣀⣀⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠿⣿⣿⣿⡟⢻⣿⣿⣿⠿⢿⡿⠋
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏⠉
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⢿⣿⠇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠒⠂⠉⠉⠉⠉⠉⠑⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⣠⣤⡶⣲⠶⠻⢟⠟⠛⠒⢲⠶⠦⣤⣀⡀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡤⡾⢛⢹⣴⣿⠶⠿⠻⠿⠛⠿⠻⠳⠻⡿⢾⣵⣬⣝⡛⠲⢤⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡞⡙⣤⣷⠿⠋⠁⡀⣤⢰⡆⢰⣶⡄⢠⣦⠀⣠⡄⢀⡀⠊⠙⠫⢖⣬⣋⡳⢦⣤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⡏⣨⣶⡟⢉⣥⡄⢶⡌⠟⠉⠀⠀⢸⣿⡇⠀⠁⠀⠉⠁⠘⠏⢴⡗⢀⣤⡈⠙⠿⣯⣭⢛⣶⣦⣄⣑⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⣴⠻⣠⣾⠏⠁⣴⡹⣧⢻⣆⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣤⡄⠀⠽⢗⣴⣚⡻⣮⡝⡻⣦⡠
⠀⠀⠀⠀⡠⠃⢀⡞⢀⣾⠟⢁⣀⠻⠎⠀⠸⣧⠹⡧⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠉⠰⠿⢀⡀⠉⠿⢶⣛⡻⣦⡟⢷⣆⠒
⠀⠀⠀⡐⠁⢠⠏⢤⡿⠃⠀⣮⡙⠀⠀⠀⠀⠀⠡⠘⣆⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢡⡶⠀⠙⢿⣯⣃⠙⣎⣿⣆⣐⡀
⠀⠀⡐⠀⣰⠏⣄⡞⠁⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠁⡘⣆⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆⠀⠘⢿⠁⠈⢏⠹⣷⡤
⠀⠠⠀⢰⠿⠐⣾⠅⣀⠉⠻⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠟⠋⢡⣤⠀⠀⢂⠀⠘⣧⢸⣧⠄
⠀⡄⠀⡞⢂⣻⠇⢀⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠈⠄⠀⠸⣧⢻⡇⠀⠅
⢰⠀⢸⣏⣾⡛⢀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠸⠀⠀⢳⢿⡿⠂
⡈⠀⣬⢋⣿⠃⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⢸⣱⣿⣭⡄
⡇⠀⣯⢪⣿⠂⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⠀⠁⠀⢸⢕⣿⡂
⡇⢰⡇⠐⣿⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⡛⠋⠀⠀⠀⢸⢺⣿⣒⡀
⠀⠐⢧⠪⣷⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠆⠀⣞⢽⣿⠒
⢰⠀⠷⢌⣻⣆⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⡸⠀⢀⠗⣹⡿
⠈⡆⠘⣗⠨⢧⡀⠀⡶⠆⢀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣈⠋⠀⢠⠁⠀⣘⡩⣾⣯⡭⠩
⠀⢡⠀⢹⡬⢹⣧⡀⠰⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣶⣄⡈⠛⠀⠀⡌⠀⢠⠇⣺⡟⠉⠁⠄
⠀⠘⡄⠀⢻⣆⠻⣧⡀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⠀⠀⡜⠀⢠⢛⣭⣟⣃⠈
⠀⠀⠈⢆⠸⢳⡌⠹⣯⡄⠀⠺⠓⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⢀⠈⠀⡰⢋⣿⡟⠓⠒
⠀⠀⠀⠈⠣⣞⡻⣶⡞⢷⡢⡀⠈⠋⣴⡆⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠘⠷⠀⠀⠀⡠⠋⢀⡜⢭⣵⡟⠒⠒⠂
⠀⠀⠀⠀⠀⠙⣧⡟⢷⣔⣻⣷⡀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢴⣆⠛⠃⠀⠀⣠⠞⠀⣰⢓⣰⡾⠯⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣎⣹⢶⡀⠛⠦⣀⠀⠈⠁⠾⢂⣶⠀⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⠀⣤⡙⠿⠟⠈⠁⠀⣀⠔⠊⠀⣠⠾⢉⣰⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣅⠋⢦⣄⠈⠑⠢⢤⣀⠀⠁⠘⠛⠸⡷⠘⣿⣷⢰⣆⢸⡆⠙⠃⠉⠁⠀⣀⡤⠒⠊⠁⢀⣠⠾⢍⣽⠟⠓⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⢻⣧⡨⢱⠢⣄⡀⠀⠉⠐⠢⠤⠄⢀⣀⣈⣁⣀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠀⣀⣤⡶⣯⣥⡶⠿⠡⠬⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠽⠷⢧⣡⢉⢉⠱⡰⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢤⡔⠶⢙⣩⣤⡜⠛⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣙⣛⡳⣶⡤⣬⣌⣏⣫⡹⡙⢝⠏⡫⡙⠬⢑⣬⣦⠶⠽⠛⠛⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠖⠚⠛⠛⠛⣛⡛⠛⢛⣛⣛⣛⣛⡡⠬⠌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣾⠀⠀⣷⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⠀⢹⡆
⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⢾⡇⠀⢸⡷⠶⠶⠶⣿⠀⢸⡷⠶⠶⠶⣿⠀⠀⣿⠶⠶⠶⢾⡅⠀⣿⠶⠶⠶⢾⡇⠀⢸⡷⠶⠶⠶⠶⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡇⠀⢸⣇⠀⠀⢀⣿⠀⠸⣧⠀⠀⢀⣿⠀⠀⣿⡀⠀⠀⣸⡃⠀⣿⡀⠀⠀⣸⡇⠀⢸⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⣇⠀⣸⢿⡆⠀⣿⣿⡀⣸⣿⡇⠀⣾⢿⡀⢀⡿⣷⠀⢸⣿⣇⢀⣿⣿⠀⢰⡿⣇⢀⣸⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⢮⣭⣥⠟⠀⠀⠙⢮⣭⣭⠞⠁⠀⠘⠷⣭⣭⠾⠃⠀⠈⠳⣭⣭⡵⠋⠀⠀⠻⣬⣭⡵⠟⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠉⠉⠀⢹⠀⠀⣿⠉⠉⠉⠁⣿⠀⠀⣿⠉⠉⠉⠉⣿⠀⠀⣿⠀⠁⠀⠉⣿⠀⠀⡏⠀⠉⠉⠈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠓⠒⠒⠒⠚⠀⠀⠙⠒⠒⠒⠒⠛⠀⠀⠛⠒⠒⠒⠒⠛⠀⠀⠛⠒⠒⠒⠒⠋⠀⠀⠛⠒⠒⠒⠚⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢰⠛⠛⠛⠛⢻⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⡟⠛⠛⠛⠛⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⣤⣤⣤⣤⣼⠀⠀⢿⣤⣤⣤⣤⣿⠀⠀⣿⣤⣤⣤⣤⣿⠀⠀⣿⣤⣤⣤⣤⡿⠀⠀⣧⣤⣤⣤⣤⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠈⠁⠀⠈⠉⠉⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠀⠈⠀⠐⠒⠤⢄⡀⠀⠀⠀⠀⢀⠔⠁⠀⣠⣴⠶⠞⠛⠛⠓⠶⠦⣤⡀⠈⠙⠦⡀
⠀⠀⠀⠀⡠⠐⠁⢀⣠⣤⡶⠶⠶⠶⠦⣤⣀⠀⠈⠓⢄⡀⢠⠃⠀⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠈⢢⡀
⠀⠀⢀⠎⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⢹⠇⠀⣼⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠹⡄
⠀⢠⠃⠀⣼⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⢧⣸⠀⢰⡏⠀⠀⠀⠀⠀⠀⠸⡁⠀⡁⠀⠀⠀⠀⠀⠀⠘⣆⠀⢻⡀
⠀⡏⠀⣼⠃⠀⠀⠀⠀⠀⢀⣼⣏⠙⠄⠀⠀⠀⠀⠀⠈⢇⠀⢸⡇⠀⠀⠀⢀⡶⠲⢄⠉⠉⠁⡴⠒⠢⠀⠀⠀⠀⢸⡄⠘⣇
⢸⠀⢠⡏⠀⠀⠀⠀⣠⣶⣿⣿⠟⠊⣠⡤⢄⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠈⠳⠤⠌⠀⣀⡀⠳⢤⠴⠀⠀⠀⠀⢸⡇⠀⣿
⢸⠀⢸⡇⠀⠀⠀⠈⢯⣙⠟⠁⠀⣠⣿⢷⣠⠀⠀⠀⠀⢸⣇⠀⢻⡄⠀⠀⠀⠀⠀⠀⣾⠁⠈⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣿
⠸⡀⠀⣷⠀⠀⠀⠀⠀⠀⢠⡞⣿⣿⠟⠉⠀⠀⠀⠀⠀⣸⠋⢆⠀⠹⣄⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⠇
⠀⢳⠀⠘⣧⠀⠀⠀⠀⠀⠀⠛⠞⠃⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠣⡀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣼⠏
⠀⠀⢳⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⢠⡾⠃⢀⡨⠲⣄⡀⠙⠓⠶⢤⣤⣤⣤⣤⠴⠞⠋⠀⣠⡾⠋
⠀⠀⠀⠙⢦⡀⠈⠛⠶⣤⣀⣀⣀⣀⣀⣠⡴⠞⠉⢀⣴⠟⢁⠔⠁⠀⣀⣴⠿⠷⠦⣤⣀⣀⣀⣀⣀⣠⣤⠶⠚⠫⣄
⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠈⠉⠉⠉⠁⣀⣀⣤⠶⠋⠁⡴⠁⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠙⢷⣄⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⡜⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⡞⣩⣽⡆⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⢀⣤⢤⡀⠙⠿⣷⣝⢦⣀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⢿⣴⢾⣳⡀⠀⠈⣮⡷⠟⡡⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠉⠳⣽⣿⣶⡄⠈⠓⠚⠁⠀⠀⠀⠀⢸⠇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠸⣟⣋⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣠⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⠛⠶⢦⣤⣤⣤⣤⣤⡴⠶⠛⠁⢀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⠀⠀⢀⣀⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⠛⠋⠙⠛⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣀⠀⠀⠰⠆⠀⠀⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠈⠁⠀⢸⡇⠀⠀⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⢀⣴⠟⠁⠀⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠈⠁⠀⠀⠀⠀⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⢰⡆⠀⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⡴⠞⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⣿⣿⣿⣿⡁⠰⣨⠃⢠⣄⡟⠀⠛⠀⡏⠀⡄⠘⠃⢠⡄⢹⠀⣾⣿⣿⣿⡦⠀⣶⠇⢸⠀⠘⠏⠀⡟⠀⠲⣾⣦⠀⣶⣿⠃⠀⢻⡇⠀⠂⣽⠀⣾⣿⠀⠐⢶⣿⣿
⣿⣿⣿⣯⣉⣀⣸⣆⣈⣩⣇⣀⣧⣀⣧⣀⣁⣼⣄⣈⣠⣾⣀⣉⣹⣿⣿⣧⣀⣿⣄⣸⣀⣶⣦⣀⣇⣀⣉⣹⣿⣀⣿⣇⣠⣄⣈⣇⣘⣀⣼⣀⣉⣹⣀⣈⣙⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣯⣭⣭⣽⣭⣿⣿⣿⣿⣿⣯⣭⣯⣭⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣿⣿⣿⣭⣯⣭⣭⣯⣽⣿⣿⣿⣿⣭⣯⣭⣽⣿⣿⣿⣿
⣿⣟⣟⡿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿
⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣯⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣏⣿⣿⣧⣀⡀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠉⠀⣀⢀⡀⣀⠀⡉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⠁⢠⡀⠆⠘⠀⠁⠈⡘⣃⠁⠈⠀⠃⠰⢀⡄⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢄⠙⠈⠓⡄⡄⠀⠀⠀⣇⣜⠁⠀⠀⠀⢠⠚⠀⠋⡠⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠟⠀⣄⠑⠀⠀⠀⠀⠇⠇⠀⠀⣤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⣠⠀⠻⣿⣦
⠀⠀⠀⣸⣿⠏⠀⠲⢦⢀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠖⠀⠹⣿⣇
⠀⠀⢠⣿⡟⠀⣈⠁⠀⢸⠻⡸⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠈⣁⠀⢻⣿⡄
⠀⠀⣾⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣧
⠀⠀⣿⣿⠀⢘⣂⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣐⡃⠀⣿⣿
⠀⠀⣿⣿⠀⢨⠍⠀⠛⢀⣀⣠⣤⣤⡶⠶⠟⠛⠛⠻⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠩⡅⠀⣿⣿
⠀⠀⢿⣿⡄⠀⠂⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⡿
⠀⠀⠸⣿⣧⠀⢉⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣰⡆⠀⢀⡉⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣆⠀⠴⠚⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠓⠦⠀⣰⣿⡏
⠀⠀⠀⠀⠻⣿⣦⠀⠋⡠⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢄⠙⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠊⣠⠀⡤⠁⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠐⢥⠀⣄⠑⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠘⠁⠆⢠⠀⡀⢀⢨⡅⡀⢀⠀⡄⠰⠈⠃⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠉⠈⠁⠉⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣚⣛⣭⣭⣭⣛⡒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢛⣭⠶⠒⠚⠛⠒⠲⠮⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡴⢋⡵⠛⠉⠀⠀⠀⠀⠀⠉⠑⠮⡙⢦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣡⠞⠉⠀⠀⢀⣀⣀⣀⠀⠀⠀⠙⠢⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡞⡰⠋⠀⢀⡤⠖⠚⠛⠓⠲⢄⡀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⡰⠃⠀⢀⡴⠚⠉⠁⠀⠈⠙⠲⣄⠀⡀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡿⠀⠁⡀⢠⠏⣴⠶⣦⠀⣴⠲⣆⠹⡄⢣⠀⠀⢷⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠁⡠⢀⡞⠠⣿⣿⡆⠀⣾⣿⣷⠈⣦⠸⡄⠀⠸⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⢰⠁⡏⠀⠙⠺⠟⠀⠙⠺⠿⠀⢻⢸⡇⠀⢸⡄⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⡇⢸⠀⠀⠉⠛⠋⠀⠈⠛⠛⠀⢸⠀⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠘⡇⠀⠘⡆⢧⠀⣴⠛⣦⠀⣴⠛⣦⠀⡟⢸⠃⠀⣸⠁⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⢳⠘⡆⠠⣾⣿⡆⠀⣾⣿⣷⠀⡾⢠⠇⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⢻⡀⠀⠹⣌⠣⣌⠛⠟⠁⠈⠛⣻⠞⢡⠏⠀⢠⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠈⢧⡙⢦⡉⠻⠋⠀⠈⠛⣻⠞⢡⠎⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠳⣄⠀⠈⠀⠈⠙⠒⠒⠒⠋⠁⠀⢁⡔⣠⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠁⠀⠉⠓⠒⠒⠒⠋⠁⠀⢁⡠⢂⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠳⣌⡒⠤⣤⣀⣀⣀⣤⠴⢚⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⡑⠦⣤⣀⣀⣀⣀⣠⡤⠖⣋⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠶⠶⠶⠖⠛⠉⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣭⣭⣥⣴⣶⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢡⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡫⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣒⣒⣒⠶⠤⣤⢫⢎⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣩⣭⣭⣝⣛⢿⡟⡵⢡⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠶⣫⠴⠚⠉⠉⠉⠉⠉⢻⢣⢋⠾⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⡡⠖⠋⠁⠀⠀⠀⠀⠉⡟⠼⣱⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡴⠃⠘⠁⠀⣀⡤⠴⠶⠦⢤⠏⡴⠋⠳⡜⢦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠈⢀⠀⣀⡤⠖⠒⠒⠶⡼⢁⠞⠁⠱⡌⢿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⠁⠀⣰⢁⡞⠁⣴⠒⠒⠒⢯⣼⢳⡀⠀⠈⠈⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⣰⢁⡜⠁⢰⡏⠉⠒⠺⢥⡞⢦⠀⠀⠘⠈⣿⣿⣿⣿
⠀⠀⠀⠀⡏⠀⢀⡇⡼⠀⠀⠙⠷⣶⠦⣄⣸⡆⢷⠀⡀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢀⡇⣼⠀⠀⠈⠻⢷⡶⢤⣀⣽⠈⡇⢰⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⣇⠀⠀⡇⢷⠠⣏⡉⠉⠙⠺⣍⠛⠁⡼⢸⠇⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⡇⢻⠀⣾⠉⠉⠙⠲⢏⠉⠋⢀⡇⢸⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢻⡀⠀⠙⠈⢧⡘⡿⢋⠶⣤⣾⢀⡴⢃⡞⠀⢀⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠱⠈⢧⡘⢻⠟⡲⢤⣤⡇⢀⡼⢡⠏⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢳⡀⢄⡀⠀⡽⢡⠾⠤⠴⠛⢋⠴⠋⠀⢀⡞⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠐⣄⠀⠀⣹⠋⣼⣥⣬⠽⠗⢋⡴⠋⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢦⣙⡞⢡⣏⣀⡀⣀⣀⣠⠄⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣌⠳⣴⢃⡜⠁⠀⠀⠀⠀⣁⡀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠿⠮⢭⣭⡭⠵⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣙⣛⣒⣒⣛⣩⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠀⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⡄⠀⢤⣤⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⣁⣀⣸⣀⡙⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⣩⣷⡾⠟⠛⠛⠛⠛⠛⠿⢶⣿⣙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠀⠈⠙⢢⣦⡄⠀⠤⠤⠄⢤⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠷⣶⣶⣶⣶⡶⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣀⠤⠄⠒⣂⣀⣀⣐⡒⠠⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠔⡫⠔⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⢳⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠇⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⡆⢇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢦⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡡⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⠓⢬⣓⠦⢤⣀⣀⣀⣀⣀⣀⡤⠔⣊⡵⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠈⠉⠒⠲⠮⠭⠭⠴⠖⠚⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⢿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣤⣤⣄⣀⣤⣤⣤⣀⣠⣖⣦⣄⣠⣤⣤⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⣯⢽⣦
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇⢱⢻⡇
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⠀⣉⡀⢀⣀⡀⣀⡀⢀⠀⡀⢀⣠⡀⢀⣀⢀⣉⡉⠉⠉⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢿⡼⠀⣷⡀⠸⣤⠀⢿⡄⢸⠟⢿⢸⣗⡇⢸⣂⢸⠷⡏⠀⠀⠀⡏⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢰⢒⣶⢲⠤⣤⠤⡤⠤⠤⣤⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣷⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠒⢺⠒⠒⢲⠗⠛⠏⡦⠬⠭⠤⡇⢐⣒⣸⣀⣒⣈⡋⠯⢹⠇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢀⣀⣜⣀⡀⢸⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⡆⠀⠀⠸⠀⠈⣹⠀⢸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⡇⠀⠀⡏⠉⠉⢸⠓⠒⠂⡞⠒⠒⢲⠇⠤⠤⡧⠤⢤⡇⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡧⢄⣰⣁⣀⣀⠃⠀⠀⡇⠀⠀⢠⠃⠀⠀⢸⠀⠀⢀⠃⠀⠀⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⠃⠀⣸⠀⠀⡎⠀⠀⡜⠉⠉⢉⠋⠉⠉⡻⠀⠒⠒⠗⠤⠤⢼⠤⠤⢤⠇⢰⣿⠃
⠀⠀⠀⠀⠀⠀⡞⠀⠀⡗⠂⢰⠧⠤⠤⡧⠤⠤⣼⣀⣀⣀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⢸⣿
⠀⠀⠀⠀⠀⣸⠁⠀⢸⠀⠀⠸⠀⠀⢰⠀⠀⠀⡇⠀⠀⢸⠉⠉⠉⡝⠉⠁⢸⠓⠒⠲⡇⠀⣿⡏
⠀⠀⠀⠀⢀⡇⠀⢀⡏⠉⠉⡏⠀⠀⡞⠒⠒⢲⠧⠤⠤⡦⠄⣀⣠⣇⣀⠀⡘⠀⠀⢸⠁⢠⣿⡇
⠀⠀⠀⠀⢸⠀⠀⣸⠀⠀⢸⠀⠀⢰⠁⠀⠀⡜⠀⠀⢠⠃⠀⠀⢸⠀⠀⠈⡏⠉⠉⡟⠀⣸⣿
⠀⠀⠀⠀⢿⣄⣀⡈⠉⠉⠛⠒⠒⠚⠒⠒⠒⠧⠤⠤⠼⠤⣄⣀⣌⣀⣀⣸⣀⣀⡰⢃⡤⣿⡏
⠀⠀⠀⠀⠀⠳⢶⣿⣭⣭⣽⣶⣒⣒⣒⣒⣒⣒⣲⣦⣤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠔⠋⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠭⠭⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⠶⢶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⣀⣤⠴⠶⠒⠒⠒⠒⠚⠶⠶⣤⣄⣀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠹⣦⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣸⠏⠀⠀⠀⠀⠀⣠⡾⠋⠙⢿⣦⠀⠀⠀⢿⡇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡿⠀⠀⠀⠈⠙⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠙⠿⠦⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⢠⡟⠁⠈⢻⣆⠀⠀⠀⠀⢰⡟⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠉⠆⠀⠀⠀⠸⣧⣀⣠⣾⡿⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠻⣧⡀⢀⣠⠇⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⢀⡾⠃⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠉⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠈⠙⠷⢶⣦⣤⣤⣄⣀⣀⣀⣀⣤⠴⠞⠋⠉⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣶⣦⣤⣤⣀⣀⣀⡀⢀⣀⣀⣠⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣟⣟⣻⣫⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣭⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠈⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠐⠤⠀⠊⠁⠀⣸⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠰⠃⠀⠠
⠀⠀⠀⡀⠀⠂⠀⠀⠀⣠⡄⠔⠂⠉⠀⣀⠀⠀⠀⢀⣀⠀⠀⣀⠀⢀⡇⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠂⠀⠆⠀⢠⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠶⠀⠠⢀⠀⢀⠀⠔⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠠⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠛⠋⠛⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠀⠀⠀⢠⠂⠀⢻⣿⣿⡅⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠠⣀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡄⠀⠀⠈⠄⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣀⠀⠀⣀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠂⠀⠢⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠄⠐⠂⠄⠀⠀⠙⣆⠀⠀⠀⠀⠠⠀⡰⠖⠉⠁⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠐⠀⠠⡀⣠⠂⠐⡀⠀⢸⡄⠀⠀⠀⠀⢰⠁⠀⠢⣠⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠋⠢⡀⢀⠃⠀⢰⠃⠀⠀⠀⠀⢸⠀⠀⠀⣘⠢⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠑⠀⠀⠀⠜⠃⡀⢂⠎⠀⠀⠀⠀⠀⠀⠳⣄⠀⠉⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣐⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣯⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣀⣀⣀⣀⣀⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠉⠉⣿⡏⠉⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣠⣤⣤⣶⣿⣷⣤⣤⣤⣀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⣴⣾⣿⡿⣿⡛⠿⠿⠿⠿⠿⠟⣻⠿⣿⣿⣶⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋⢁⣴⣿⡿⡿⠟⠂⠉⠀⠐⠀⠁⠸⠀⠁⠐⠀⠈⠁⠋⠟⡿⣿⣷⣄⠉⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢀⣴⣿⣟⠟⡈⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠙⢟⣿⣷⣄⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢆⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠐⠦⢄⢼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠠⣪⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣗⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢔⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣦⢜⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢘⢤⣪⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣦⣤⣂⣠⣡⣬⢤⣥⣀⣲⣬⣶⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡤⠴⠒⢚⣉⣉⣉⣉⣉⣉⣉⡙⠒⠲⠤⣄⡈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡠⠖⢋⡡⠔⠚⠉⠉⠀⠀⡄⠀⠤⣄⠀⠈⠉⠙⠒⠤⣉⠓⠦⣀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣀⠔⢉⡴⠊⡅⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢠⡄⠉⠢⣌⠑⢤⡈⠳⣄
⠀⠀⠀⠀⣠⠋⢀⠜⢁⠔⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢄⠙⢆⠈⠣⡀
⠀⠀⠀⡼⠁⡰⠃⡴⠁⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠱⣄⠳⡀⠙⡄
⠀⠀⡜⠀⡼⢁⠞⢠⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⢆⠹⡄⠘⡄
⠀⡼⠀⡼⢁⠎⠀⢸⠀⣯⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆⠹⡄⠸⡄
⢠⠃⢠⠃⡞⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⡆⢳⠀⢳
⡼⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢳⠘⡆⠸
⡇⠀⡇⢸⠀⢀⠄⢦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣀⣀⣤⠤⠤⠶⠒⠒⠻⠖⠉⠀⠐⠦⠀⢸⠀⡇
⡇⠀⡇⢸⠀⠈⢃⠟⠛⠧⠼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢄⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢘⠀⡇
⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇
⢫⠀⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⢰
⠘⡄⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⡼⠀⡜
⠀⢳⠀⢱⡈⢇⠀⠀⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠗⢀⠏⣰⠃⢰⠃
⠀⠀⢣⠀⠳⡈⢦⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠃⢠⠃
⠀⠀⠀⢳⡀⠱⣄⠳⡄⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⡴⢃⡴⠁⣰⠃
⠀⠀⠀⠀⠙⣄⠈⠢⡈⠢⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⢀⡠⠊⡠⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢⣌⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠑⠢⣌⡑⠲⠤⣄⣀⡀⠀⠀⠣⠘⠀⠀⣀⣀⡤⠴⠒⣉⡤⠖⠉⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡉⠑⠒⠤⠤⣍⣉⣉⣉⣉⣉⣉⡥⠤⠔⠒⠉⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇
⣿⠛⠛⠛⠿⡿⠟⠟⠛⠛⠟⢿⡿⠻⠟⠛⠻⠻⣿⠟⠛⠛⠟⠟⠻⡿⠻⠿⠛⠛⠻⢿⡿⠛⠛⠛⣿⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⢀⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⣿⡇
⠛⠀⠀⠀⢸⡏⠀⠉⠀⢹⣿⠈⣿⠁⠉⠉⠈⣿⠀⣿⡏⠈⠉⠁⢹⡇⢸⡏⠀⠀⠈⢸⣿⠀⠀⠀⣿⡇
⣀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠸⠿⠿⠿⠿⠾⠟⠐⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠟⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦⠠⣶⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⣶⡆⢰⣶⣶⡆⢀⣀⣤⣤⣴⣶⣿⣧⣤⣄⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⣇⣤⡶⠟⠋⠉⢀⣀⣀⣀⣀⠈⠉⠛⠷⣦⣀
⣿⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠘⠁⣼⡿⠋⢀⣤⡶⠟⠛⠋⠿⠋⠛⠻⠷⣦⣄⠈⠻⣷⡄
⣿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠃⢀⣾⠋⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠈⢿⣆
⣿⠀⠀⠀⢰⡶⠶⠶⠶⢶⣶⠀⣶⠶⠶⠶⠶⣶⠀⣶⡶⠶⠶⠶⣾⠇⢀⣾⠏⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠈⢿⡆⠀⢿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠈⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⢸⡿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸
⣿⠀⠀⠀⢸⣇⣀⣀⣀⣸⣿⠀⣿⣀⣀⣀⣀⣿⠀⣿⣇⣀⣀⣸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠲⣿⡇
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⢹⣷⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢠⣿⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⣾⠃⢀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⣾⠏
⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⣷⣄⠈⠙⠿⣶⣤⣤⣶⣦⣤⣴⡾⠟⠉⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⡈⠉⠉⠉⠁⣀⣠⣤⡾⠗
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣩⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣒⣦⣰⣋⣳⣤⡮⠭⡦⠾⠭⡷⠿⢿⡷⠿⢿⡿⠚⠓⣺⠗⠓⢲⠞⠛⢒⡾⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠩⠞⠀⠀⠚⠀⠀⠚⠁⠀⠉⣀⡀⢉⠀⡀⣨⣄⠀⣤⡀⡤⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⠀⠀⠀⠀⢰⠋⢠⡿⡄⣇⡀⢿⡄⢹⠓⠇⠳⠜⠐⠛⠃⠛⠙⠀⠀⠀⣀⣀⠈⡇
⠀⠀⠀⠀⠀⠀⡏⠀⢻⣿⡇⠀⢀⣀⣀⣩⢤⡤⠤⠤⠤⢤⠒⠒⠒⢲⡒⢒⣚⣻⣉⠉⢽⡭⠭⢽⠭⠭⡇⢹
⠀⠀⠀⠀⠀⣸⠁⠀⠸⣿⣿⠀⠸⡖⠒⠒⠒⠗⠒⠈⠉⢹⣉⣉⣉⣉⣇⣀⡠⢸⡤⠤⠤⡦⠤⠤⡦⠤⢿⠈⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⣿⡇⠀⢷⠒⠊⠉⢹⠉⠉⠉⢀⣇⣀⣀⡀⢥⠤⠤⠤⡧⠤⠐⢷⠒⠒⢻⠊⠙⡆⢹
⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⣿⣿⠀⠸⡖⠒⠈⠉⡏⠉⢀⣀⣹⣀⠠⠤⠼⡤⠤⠄⢲⠒⠒⢺⡂⣈⣙⣏⣉⣱⠈⡇
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⣿⣿⡆⠀⢷⠒⠒⠈⢹⠉⣁⣀⡬⡦⠤⠔⠒⢗⠒⠒⠚⡏⠉⠉⣏⢉⣀⣣⣀⠬⡇⢸⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢻⣿⣷⠀⢸⡊⠉⠉⢉⣅⣀⣀⣀⢸⠤⠤⠤⢺⠒⠒⠒⢻⠉⣉⣹⣀⡠⠼⠤⠤⠞⠀⣇
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠏⠉⠉⠁⣣⣀⣀⣠⠤⠦⠴⠖⠚⠓⠋⠉⠉⠉⢀⣀⣀⣠⡤⠤⠤⠴⢶⠛
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⣇⠀⠘⠚⠉⠉⠁⠀⢀⣀⣀⣀⡤⠤⠴⠖⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⢰⠃⣀⣀⠤⠤⠖⠒⠒⠙⣿⡼⠤⠶⠖⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠖⠒⠒⠋⠉⠉
⠀⠀⠈⠙⠓⠶⠤⣄⣀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠒⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠾⠦⠖⠒⠚⠉⠉⠁


⣶⣶⠶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣙⣉⣉⣉⣙⣳⠀⠀⠀⠀⢸⣙⣉⣉⣉⣛⡃⠀⠀⠀⠀⢘⣛⣉⣉⣉⣋⡇⠀⠀⠀⠀⣞⣋⣉⣉⣉⣋⡆
⠀⠀⠀⢸⠀⠴⡆⣲⠂⢸⠀⠀⠀⠀⢸⢰⣲⣖⣖⡆⠁⠀⠀⠀⠀⠈⠠⠄⠤⠠⠄⡇⠀⠀⠀⠀⡇⠀⠠⠤⠤⠀⡇
⠀⠀⠀⠸⣀⣀⣁⣉⣀⡼⠀⠀⠀⠀⠸⣘⣚⣓⣓⣃⡄⠀⠀⠀⠀⢠⣘⣁⣃⣘⣀⠇⠀⠀⠀⠀⢇⣀⣈⣉⣉⣀⠇


⠀⠀⠀⢠⠴⠤⠤⠤⠴⢤⠀⠀⠀⠀⢠⠴⠤⠤⠴⢤⠀⠀⠀⠀⠀⢠⠤⠤⠤⠴⢤⠀⠀⠀⠀⠀⡤⠦⠤⠤⠤⠦⡄
⠀⠀⠀⢸⠒⣒⣒⣒⠒⢺⠀⠀⠀⠀⢸⠒⠒⠒⣒⣺⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⣻⠀⠀⠀⠀⠀⡗⢒⠒⢒⠒⠒⡇
⠀⠀⠀⢸⠀⠣⢌⠴⠁⢸⠀⠀⠀⠀⢸⣀⣀⣸⠰⠆⡃⠀⠀⠀⠀⢂⣀⣀⣎⠀⠀⡇⠀⠀⠀⠀⡇⠐⠐⠐⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⢸⣉⣉⣉⣉⣉⣽⠀⠀⠀⠀⢸⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⢨⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⣯⠭⠭⠭⠭⣽
⠀⠀⠀⢸⠀⠀⠆⡖⢲⠞⠀⠀⠀⠀⢸⠀⠀⢠⠊⡙⠆⠀⠀⠀⠀⡀⠀⠀⡰⠊⡙⡄⠀⠀⠀⠀⡇⠀⠀⡴⢊⡛⡄
⠀⠀⠀⠸⠤⠥⠥⠗⠁⠀⠀⠀⠀⠀⠘⠒⠒⠚⠤⠥⠃⠀⠀⠀⠀⠘⠒⠒⠓⠬⠵⠃⠀⠀⠀⠀⠑⠒⠒⠳⠬⠴⠁


⠀⠀⠀⢀⣰⣒⣒⣒⣒⣢⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⡆⠀⠀⠀⠀⡴⠶⠶⠶⠶⠦
⠀⠀⠀⢸⢺⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠘⠀⢀⢀⡀⠀⡇⠀⠀⠀⠀⡏⠉⠉⠉⡡⣶⠄
⠀⠀⠀⢸⠘⠤⠤⠤⢤⠜⠀⠀⠀⠀⢸⠀⠘⠲⠃⠀⡀⠀⠀⠀⠀⢀⠀⠰⠛⠄⠀⡇⠀⠀⠀⠀⢧⡀⢠⣪⠖⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠆⠀⠈⠉⠉⠁⠈⠉⠉⠉⠉⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢺⣶⣦⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢸⣷⣾⡿⠛⣩⡼⣷⣌⡻⠿⠿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠉⠐⢀⡀
⠀⠀⢀⡴⠁⠀⠀⠀⠀⣠⣶⡿⢟⣡⣾⣏⡵⢖⣫⣶⡞⠉⠙⣿⣿⣿⣷⣶⣶⣭⣤⣀⡀⠀⠀⠀⠀⠀⠀⠁⠆⡀
⠀⢠⡾⠁⠀⠀⠀⣴⠟⣩⣵⣾⣿⣿⣿⣷⡞⠋⡙⢿⠛⣀⠀⠈⢻⡛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠀⠈⠑⢄⡀
⢠⡿⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡿⢛⣩⣵⣆⠀⠀⢿⣿⣆⠀⠀⢻⣿⣿⣷⣶⣤⣭⣙⠻⠿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠈⠢⡀
⡼⠃⢀⣠⣾⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿⣦⡀⣰⠿⣉⡙⠒⣤⣤⣄⡈⠙⣉⣛⠿⠿⣷⣶⣬⡛⠿⣿⣿⣿⣿⣿⣶⣄⡈⠢⡀
⣧⣶⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⠟⠑⣨⡷⠞⣩⣭⣴⠶⠶⠶⠾⢟⣛⣤⣶⢄⣉⠙⠻⢿⣶⣍⡛⢿⣿⣿⣿⣿⣷⣼⣆
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⡿⠁⣠⠟⣡⠶⣋⠳⠄⠉⡹⢰⡖⠈⠈⠉⠉⠚⠟⣭⡾⣣⡆⣈⠙⠿⣦⡉⢿⣿⣿⣿⣿⣿⣧
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⡟⢠⡾⣡⠞⣡⡞⠦⡀⠈⠉⢷⣌⡾⣦⠀⠀⠀⠀⠀⠀⠈⣩⡞⢳⢞⣵⣤⡙⠂⣽⣿⣿⣿⣿⣿⣇
⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⡟⢱⡟⢡⠛⠌⠁⠺⣦⠙⠀⠀⠖⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠉⠲⣭⠞⣷⣦⣉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⡏⣼⢱⡌⢇⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⢮⡙⢾⡷⢦⡙⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⠀⢿⠀⣷⠞⠛⣏⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠢⣝⠷⣌⠳⢮⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⢸⠀⣿⠀⠀⠀⠈⠀⠈⣹⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣿⠀⠀⠀⢀⡔⣣⠌⠳⣌⠻⣦⡙⢎⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣷⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠈⣷⠀⠀⣵⠶⠋⠀⠀⠀⠙⠞⠁⠀⠀⠈⠳⡈⢳⣌⡱⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠸⡇⣤⡀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠳⠶⣤⢶⣼⣧⣰⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠙⣦⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢸⢀⡁⣶⠛⠻⣉⠁⠀⠀⠀⠀⠀⠀⠲⠶⢶⣿⣿⣿⣿⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠶⠶⠖⠒⢸⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡿⢠⡇⣨⠁⠉⠉⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠟⣻⣿⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠓⠟⢸⠇⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⣸⠃⣮⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠟⠐⡹
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⠀⠀⠀⣀⠃⡸⡹⣧⣄⣀⡐⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⢊⣵⠞⠁⣠⠞⠁
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣠⠀⠙⡘⢃⢻⣯⠥⣬⡍⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⡄⠀⠀⠐⣡⡾⢋⣡⡴⠊⠁
⠀⠀⠄⠐⠁⠀⠀⠀⠀⢀⣼⡇⢰⡆⠃⠘⣠⡴⠖⠀⠀⠀⠀⠀⢀⣤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠋⠀⠐⣊⡥⠘⠊⠁
⠀⡘⠄⠀⠀⠀⠀⠀⣠⣿⣿⡇⠘⡇⢠⠀⠁⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢂⣴⡟⠉
⠀⠓⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢀⡇⣾⠀⠀⣠⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣤⢲⣼⣿⣿⡇⠀⠀⠀⠀⠀⠄
⢰⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⢰⢿⡞⢫⡵⢫⡜⠋⠀⠀⠀⡰⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⣀⣶⣋⣾⣿⣿⣿⣿⠃⠀⠀⢀⠢⠈
⠘⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠞⡻⣇⠞⠁⡼⠀⠀⠀⣴⢡⡜⡇⠀⠀⠀⠀⠀⠀⠀⠘⢃⣤⠾⠏⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⡮⠁
⢀⡰⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣎⠳⡙⢷⣠⠄⡄⠀⠺⠣⠇⢰⠀⠀⠀⠀⠀⠀⢠⣤⠶⡟⢉⣴⣿⣿⣿⣿⣿⡟⢁⠀⠀⣰⠟
⠘⣧⠂⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡉⠲⣅⡛⠴⠞⣉⣤⡶⠶⠒⠒⠒⠒⠤⡍⠓⢌⣣⣴⣿⣿⣿⣿⣿⠟⠉⠔⠁⣠⡼⠋
⠀⠹⣦⡱⡀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⠶⠦⣭⣅⣀⠀⣀⣈⣭⣥⠒⣢⣽⣾⣿⣿⣿⣿⠟⠋⠁⠀⠀⣠⣼⠟
⠀⠀⢹⣷⣡⢄⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣼⣶⣶⣶⣾⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⣀⣤⣾⠟⠁
⠀⠀⠀⢻⣿⣯⠦⡀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠋⣀⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠹⣿⣷⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣷⣌⠐⠒⢀⣴⣂⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡿⠉⠇⢰⣌⠉⠉⠉⠹⣦⠘⣆⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⣼⡄⢸⣿⣿⡿⠂⠘⣿⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢠⠀⡀⠙⣦⣹⢿⣿⣿⠛⢀⣤⠟⣱⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⡄⢰⠁⠀⠈⠙⠻⠉⠽⠿⢻⠏⣰⠟⠉
⠀⠀⠀⠀⠀⢀⢡⡟⠀⠀⠀⠀⠀⠀⢀⣴⠋⣴⠋
⠀⠀⠀⠀⠀⠘⣼⠃⠀⠀⠀⠀⠀⢀⡾⢁⡼⠃
⠀⠀⠀⠀⢰⠄⡿⠀⠀⠀⠀⠀⣰⠏⢠⠞
⠀⠀⠀⠀⡄⢸⠁⠀⠀⠀⢠⡞⠁⡰⠃
⠀⠀⠀⠀⡇⠘⣄⣀⣀⠔⠃⢠⠞⠁
⠀⠀⠀⠀⠡⡀⠉⠉⢁⣠⡴⠋
⣤⣤⣤⣤⣤⣬⣶⣶⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠻⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠊⠉⠈⠁⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣀⡤⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⣤⣀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠀⢈⠆⠀⢄⢀⠀⠠⡊⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⢊⠡⠀⠀⠀⠃⠀⠀⠄⠑⠠⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠒⡁⠔⢉⣠⡤⠖⠒⠒⡒⠒⠲⢤⣄⠈⠂⠈⠂⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡴⠋⠁⡆⠀⠀⡄⠱⠄⠀⠀⠌⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠏⡀⠀⠀⠇⠸⠀⠁⠈⠀⠀⠇⠀⢀⠀⠹⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠀⢹⢰⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠁⠹⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⣠⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⡀⢠⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠁⠘⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⡈⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⠀⢠⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣆⠁⠀⠀⢨⠁⠀⢀⠀⠀⠰⡂⠀⠀⠈⣠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠳⣄⡐⠂⠀⠀⢛⡃⠀⠀⠰⢂⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢠⡈⠙⠲⠦⠤⠨⠄⠤⠴⠖⠋⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠤⠀⠀⠂⠈⠀⠑⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣵⣶⣦⣤⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠿⠛⠛⠋⠉⢩⣭⣍⠉⠉⠙⠛⠻⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠻⣆⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣰⠏⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠘⢧⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⢴⠏⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣠⡴⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠉⠛⠶⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠞⠋⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠂⠀⠀⠀⡰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣌⣦⣠⠚⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⣿⣿⣄⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠿⠟⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣤⣀⣠⣿⡟⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⢀⡾⠂⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⡀⠀⣰⠏⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠹⣆⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠛⠋⠀⠀⠀⣀⣠⣤⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣶⣶⣶⣶⣶⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁





⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠒⣶⠒⠦⣤⣀⠀⠀⢰⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⣀⡔⠾⣀⠀⠉⠀⠀⠀⠙⢷⣄⣈⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣀⣬⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠹⣿⠂⠀⠀⡱⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⢡⢴⣿⣶⠐⠃⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣠⣠⣶⣿⢏⣿⣿⣿⡆⢀⡀⠀⢀⠠⠤⠾⣞⠁⠀⠀⢿⡅⠆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⡓⠀⠀⠀⠀⠀⢫⠀⠀⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣀⣠⣿⣖⣻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠘⠄⠀⢢⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠸⣶⠀⢸⣦⡾⣿⣤⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠛⠛⠋⠀⠈⠉⠉⠉⣉⣉⣭⣽⣿⣬⣽⠙⠓⠒⠒⠛⠛⠁⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⣾⣍⣻⣿⣿⣿⣿⠿⢿⡽⠖⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣓⣦⣭⣭⣥⣄⣉⣍⡛⢉⣷⣿⡭⡲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣍⡛⠿⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣠⣾⠟⣡⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣷⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡿⠟⣁⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣌⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⣶⣤⣬⣍⣉⣉⣉⣉⣉⣩⣤⣴⣶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠣⣉⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣉⡡⠃⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠖⠶⠲⡆⠀⢠⠖⠶⠖⢦⠀⠀⡄⠲⠖⢦⠀⠀⡴⠲⠒⠲⡄⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠃⠀⠘⠓⠒⠒⠛⠀⠀⠓⠒⠒⠚⠀⠀⠛⠒⠒⠚⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣠⠤⠤⢤⡄⠀⢠⠤⠤⠤⡄⠀⢀⡤⠤⠤⣄⠀⠀⡤⠤⠤⢤⠀⠀⣠⠤⠤⢤⡀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣇⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠓⠖⠶⠒⠃⠀⠘⠶⠶⠶⠃⠀⠘⠒⠶⠖⠞⠀⠀⠳⠶⠶⠞⠀⠀⠓⠲⠖⠖⠃⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠷⠤⠤⠤⠇⠀⠸⠤⠤⠤⠇⠀⠸⠤⠤⠤⠼⠀⠀⠧⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣒⣒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⣡⣶⣶⡝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣵⣼⡟⢿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣓⣛⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣺⠭⠭⣀⣠⠖⣉⠻⣹⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣀⡗⠧⣭⡧⠼⠧⢌⡲⠒⠹⣐⣳⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣴⡟⣊⡠⠚⠁⠀⣶⡶⠀⠈⠑⣤⠞⢼⡻⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠧⣶⡋⢠⠎⠀⣀⠴⠒⢉⢙⢢⡀⠀⠘⡄⡤⢭⠒⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⡰⢣⠃⠀⣴⠁⠀⠀⢻⠿⠀⢳⠀⠀⢹⢓⠜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣽⡞⢘⣛⡇⠀⠀⠸⡽⠀⠀⢸⢺⡃⣸⠛⣏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡜⢥⢼⠀⠀⢹⠀⠀⠰⡇⠀⣠⠏⠀⢀⠇⢸⡸⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣑⣋⡀⢣⡀⠀⠑⠦⠤⢔⠚⠁⠀⢀⣞⢠⢖⠋⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠰⠬⢦⣛⠢⢄⣀⡘⠟⢃⣠⠔⢁⠜⡌⢋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣈⠛⢳⣴⣛⡩⢿⠣⠼⠿⢻⣈⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠢⠤⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⡄⣿⠀⣤⣤⣤⣤⣤⣤⢸⣿⢠⣄⡀⠀⠀⠀⢸⣿⣿⣿⡿⠟⣿⠀⣿⠛⠛⠛⠛⠛⠛⡇⠀⡟⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣧⣭⣼⣿⣿⣿⣿⣿⣿⣦⣭⣼⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠛⠶⠃⠀⢸⣿⣿⣿
⠀⠀⠀⢸⡏⠉⠉⢩⣭⡍⣭⣭⢩⣭⡍⣭⣭⢹⡇⠀⠀⠀⢸⣿⣿⣿⡇⣾⣿⣿⡟⠛⣻⠛⠛⡟⠛⢻⠛⠛⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣀⣀⢘⣛⡃⣛⣛⢘⣛⡃⣛⣛⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠿⢿⡷⠤⣿⠦⠴⡷⠤⢾⠦⠴⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⠿⠿⠸⠿⠇⠿⠿⠸⠿⠇⠿⠿⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⣀⣸⣇⣀⣽⣀⣠⣇⣀⣼⣀⣠⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠀⢸⡇⠀⣻⠀⢀⡇⠀⣸⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⢸⣿⣿⣿⣧⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡂⠀⢸⠀⠂⠂⠂⠀⠀⠀⡆⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⠁⠀⠘⠠⠥⠤⠤⠤⠤⠤⠄⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⣣⣷⢄⠀⠀⠀⠀⠁⣠⠁⠈⠁⡀⡀⡀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⡾⣳⣄⠀⠀⢰⠃⠀⠀⠂⠧⠁⠈⠊⠀⠀⠜⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠻⣦⠀⠙⠧⣀⣀⠘⠃⠂⠀⠀⠀⠈⠈⠉⠉⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣧⠀⠀⢴⡆⢵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣸⡦⠤⣞⡁⠈⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⡿⠶⠛⣬⢻⣿⢋⣍⣷⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⣤⠀⠀⠉⠛⠿⠷⠾⠿⠿⣿⢾⣿⣗⠒⠖⠒⠲⢿⠿⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠸⢿⣿⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠈⠀⠀⢀⡄⡀⢀⣠⠞⠁
⠀⠀⠀⠀⢰⣬⣁⠉⣧⡈⠉⠑⠒⠲⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢹⡅⠒⢘⢉⡳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢿⣧⡌⠎⠇⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢺⣏⠈⠒⠆⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠄
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠸⣿⠏⢘⠿⣷⣿⣦⡀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣴⣂⣀⣀⡀⠐
⠂⠀⠠⠀⠚⢱⣿⠂⠀⠀⠙⢿⣿⢛⠒⠀⠐⢠⡶⠤⡔⣆⠾⠛⠛⠙⠉⠙⠉⠋⠙⠇
⠀⠀⢀⠈⡀⠐⣷⠀⣤⠢⠼⠛⠃⠿⣀⣤⠀⠈⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠉⠹⠷⠀⠥⠀⠀⠀⠀⠷⡀⠀⠸⠎⠄⠀⠀⠰⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠠⠀⠀⠀⠀⠀⠀⢠⣤⢤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⠃⡀⠀⠀⢀⠀⠀⢶
⠀⠀⠀⠀⠀⠳⠄⠀⢁⣸⣺⡇⢐⣀⣀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠓⠀⠂
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠙⣿⠟⢀⡀⠰⢠⡄⠀⠀⠀⠀⠰⠘⠀⠰⠄
⠈⠚⠉⠫⠈⢠⠥⠠⠄⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠓⢦⡀⠘⠗⠖⠄
⣶⣶⡶⠶⠶⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠈⠉⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⡀⠄⠒⠀⠀⠀⠀⠠⣤⠀⠠⠄⢦⡄⠀⠈⠀⠃⠠⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⠄⠈⠀⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⢀⣀⣋⣀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⢤⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠄⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣦
⠀⠀⠀⢠⣾⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⡀
⠀⠀⢠⣿⠏⠀⠦⠀⢠⣤⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠆⠀⢻⣷⡀
⠀⢀⣿⡏⠀⡠⠀⠀⢸⡏⠀⣸⡃⠀⢸⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠀⢻⣷
⠀⣼⡟⠀⢠⠀⠀⠀⢸⣇⠀⢹⡇⠀⢸⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡀⠈⣿⣇
⢰⣿⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠛⠀⠀⠀⢀⠀⢸⣿
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⣠⠂⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃⠀⢿
⣿⡇⠀⠆⠀⠙⠶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣬⡀⠀⠀⠶⠀⢹
⣿⡇⠀⢀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣕⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⡄⠀⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠐⣿⣿⣿⡟⠁⢀⠀⠀⣿
⠸⣿⡄⠀⠀⠀⠀⠀⢀⡀⢀⡐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣾⣿⣿⣿⣦⣈⠀⢸⣿
⠀⢻⣷⠀⠙⡀⠀⠀⠸⣆⣠⠃⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏
⠀⠈⢿⣇⠀⠐⠀⠀⣴⠉⠙⣷⣰⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣦⠀⠖⠀⠈⠓⠚⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠈⢿⣧⡀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠲⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠃⢀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⢠⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠁⢾⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⡇⠈⢻⡆⠀⠀⠀⠀⠀⠘⠂⠈⠁⢠⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⢀⠀⠤⠐⠈⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣧⠁⠉⠀⠘⠂⠀⠉⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⣼⣿⣿⣿⢿⣿⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⣷⣶⣶⣶⣶⣾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣏⣻⣿⣦⣈⣏⣙⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣑⣾⣿⣛⣿⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⣏⡛⠻⠿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠛⠃⠀⠀⠀⠈⠉⠛⠿⢿⣿⣷⣶⣤⣄⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠘⡶⠦⠀⣀⠀⠀⠀⠉⠛⠻⣿⣿⣟⣷⣷⣤⣀
⠀⠀⠀⠀⠀⢀⠏⢀⣤⣀⠀⠀⠀⠐⠒⠃⢠⠾⣽⠀⣰⣆⠀⡀⠀⢹⡦⠈⠉⠛⠻⢿⣿⣆
⠀⠀⠀⠀⠀⡞⠀⡜⠙⢺⠯⣷⣲⠤⣄⡀⠀⠀⠈⠀⠃⠘⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠏⢸
⠀⠀⠀⠀⢰⠃⢰⠉⢑⠗⠤⣠⡏⠙⠳⣼⣉⣖⢢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠈⣇
⠀⠀⠀⢀⡏⢀⣏⡉⡾⠤⢄⣜⠈⠙⢲⠧⢄⣈⠟⠚⠦⣽⣓⡶⢤⣀⡀⠀⠀⢰⣿⠁⠀⠀⢹
⠀⠀⠀⡼⠀⡼⣄⣸⠓⠲⣼⣉⠉⢒⠧⢤⣀⡎⠉⠒⣴⣅⡈⠉⡞⠺⢭⡇⢀⣿⠏⠀⠀⠀⠈⡇
⠀⠀⢰⠃⢰⠧⢄⣏⠙⢲⠧⢄⣉⡟⠢⢤⣜⠉⠑⢲⢧⣀⠈⡝⠓⠤⡼⠀⣾⡟⠀⠀⠀⠀⠀⢹
⠀⠀⣞⠀⠘⠒⠼⣁⣉⠟⠢⢄⣼⠉⠒⡾⢥⣉⢓⠾⢤⣀⣹⠓⠦⣤⠃⢸⣿⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠻⢽⣓⠦⢄⣀⠉⠑⠲⠤⣇⠉⢲⠧⢤⣀⡝⠓⠢⢤⣏⠑⢲⡎⢠⣿⣏⣀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠉⠓⠺⢽⣒⠦⣤⣀⠉⠛⠢⢄⣸⠉⠒⢢⠯⣀⠉⡞⠀⣾⣿⡀⠈⠉⠒⠦⢄⣀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠯⢽⣳⣦⣤⣀⠉⠓⠺⠤⣀⣹⠁⣼⣿⡇⠈⠑⠒⢤⣀⣠⠤⠭⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⢽⣓⠦⣤⣀⠀⢠⣿⡿⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⢽⣛⣻⣧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣀⡤⠖⠛⠉⠉⠀⠀⣄⢀⡀⠀⠉⠉⠛⠲⢤⣀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠚⠁⢀⢀⠀⠀⠀⠀⠀⠟⠰⠧⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠛⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢦⡀⠙⢷⡄
⠀⠀⠀⠀⣰⡿⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢿⡄
⠀⠀⠀⢠⡿⠁⢠⠏⢠⢠⣶⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢻⡄⠈⣿⡄
⠀⠀⠀⣾⠃⢀⡟⠀⠈⠀⠉⠁⠀⠀⠀⠀⠘⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢷⠀⠸⣷
⠀⠀⢰⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡄
⠀⠀⢸⡇⠀⣾⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡇⠀⢹⡇
⠀⠀⢸⡇⠀⢻⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠾⠃⡇⠀⢸⡇
⠀⠀⠸⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠀⢰⡇⠀⣿⠃
⠀⠀⠀⢿⡆⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠘⣿⡀⠘⣧⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡇⠀⣼⠃⢀⣿⠃
⠀⠀⠀⠀⠘⣷⡀⠘⢧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣄⠈⠳⣄⠀⠀⠀⣲⡖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢀⣰⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⢦⣀⠙⠁⠀⠀⠀⠀⢼⣧⡀⠀⠀⠀⠀⠈⠟⠀⣀⡴⠛⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠉⠛⠶⢤⣄⣀⡀⠈⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠟⡿⠀⠈⠉⠉⠉⠛⠛⠓⠲⠶⠶⠶⠦⢤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡏⠀⢠⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⣧⣾⣿⡷⠶⠤⠭⠭⠭⣍⣉⣙⣒⣒⣒⣒⣒⣒⣒⠲⣶⣤⣤⣿⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠋⢷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣼⠿⣷⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⢿⠀⠀⠀⠀⢀⣤⣶⠿⢻⡿⠀⠀⠀⠉⠀⠹⣷⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣇⠀⠀⠀⠀⠁⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣼⣿⠾⠛⠛⢷⣄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢿⡆⠀⠀⠀⠈⠻⡧⠀⠀⠀⠀⠀⠈⠛⠶⠶⡶
⠀⠀⠀⠀⠀⣿⢹⡟⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⡀⠀⠀⠀⣿⣿⣷⢏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⣤⣤⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⣿⡟⣿⣹⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⣿⣧⢻⡜⣿⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⢹⣼⢹⢱⢇⢧⣷⣬⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠘⣇⣎⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠙⢷⣄⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣈⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠙⠻⢷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡘⠀⠀⠀⡰⢻⣆⣌⢛⣦⠀⠘⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣧⣶⣠⣦⣷⣿⣼⣿⣿⣿⠀⢈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣦⠀⢸⡏⠸⡏⠀⢹⣿⣿⡇⠐⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⡇⠀⢸⣿⣿⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣅⣤⣡⣤⣬⣭⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠋⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠷⠒⠒⠛⠛⠛⠛⠿⠦⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠴⠊⠀⠀⠀⠀⣠⠔⠚⡛⠥⠐⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⢐⠵⠁⠀⠀⠀⡀⠐⢉⠤⠂⠁⠐⠂⠀⠀⠀⠀⡶⠀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣀⡀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⣠⡖⠡⠀⠐⣄⢳⡀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠈⢿⣆
⠀⠀⠀⡰⠁⠀⠀⠀⣴⢃⡴⠃⠈⠀⠀⠀⠈⠳⣳⡀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡙⢶⣿⣦⡀⠄⠈⢿⣆
⠀⠀⡴⠀⠀⠀⢀⡾⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠔⠈⠀⠈⢻⣿⣿⣶⣆⠀⢿⣆
⠀⢠⠃⠀⠀⠀⠞⠠⠀⠀⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⣴⠿⠛⠀⠀⢻⣿⣿⣿⣶⠘⣿⡄
⠀⡄⠀⠀⠀⡘⢀⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢻⣧
⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⣿
⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⣿⡇⢿⡇⠐⣿
⠀⠀⠀⠐⣼⡇⠰⠀⠬⠍⠉⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠿⢀⡀⣿⣟⢸⣷⡀⣿
⠀⠀⠀⠘⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⣌⠧⠀⣿
⢠⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⣿⡇⢟⠁⢀⣿
⠈⡤⠀⠀⢹⡿⢣⠀⠀⢀⣄⢖⣫⢱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣄⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣦⡀⠀⢠⣿⡻⠼⠞⠀⣸⡏
⠀⠐⢀⠀⠘⣷⡉⣧⡀⡈⠻⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉⠛⠶⠙⠀⢠⡿⡷⢡⠚⠀⣸⡟
⠀⠀⠀⢂⠀⢻⣷⡹⢿⡇⠀⠋⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⢘⣧⡀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡟⣡⠎⠀⣰⡟⠁
⠀⠀⠀⠀⡣⡈⣿⣿⣌⠢⣾⣦⠀⠀⠀⢀⣤⠏⠗⠀⠀⠘⠃⠈⠛⠉⠀⠀⠀⠈⠀⠻⣄⠀⠀⠠⠄⣠⣾⣯⢏⣴⠁⠀⣰⠟
⠀⠀⠀⠀⠘⢷⠈⢿⣿⣷⣝⠿⣷⣄⠀⠈⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠂⢀⣤⣾⣯⠟⣡⠟⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢂⠉⢿⣿⣿⣮⡝⢷⣦⢄⡀⡀⠀⠀⠀⠘⠂⠰⡆⡀⠀⠠⠀⡀⣠⡴⣞⠿⠟⢋⣴⡟⠉⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⢻⣿⣿⣿⣷⣮⣘⣤⠉⡀⠐⠒⠂⠤⣤⣤⣶⣶⣶⣿⣿⠿⢞⣋⣴⣶⡿⠗⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣍⣉⣁⣀⣠⣤⣤⣶⣶⣿⢿⢛⣉⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠨⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠆⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠈⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠛⠛⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⠀⣀⠀⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⢄⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠈⠁⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣄⣀⣀⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⡀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡆⠀⠀⠀⠀⢀⠌⠁⣀⢀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠳⣋⠀⠀⠀⠀⢸⠀⠀⠀⠀⢨⠀⠀⠈⠻⡀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢳⠀⠀⠠⠀⠀⠀⠰⠄⠠⢀⠇⠀⠀⠀⠀⠀⠢⡀⠀⠓⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢊⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣤⡴⠶⠟⠛⠻⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣡⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡏⠀⠀⠀⠀⣀⣠⣤⢠⣤⣀⠀⠀⠀⠀⠀⣧⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡇⠀⠀⢀⡾⣿⣿⢱⣼⡿⣿⣷⣄⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢣⠀⠀⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⠀⠀⠰⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠣⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⠄⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⠀⢸⣿⣇⠀⠀⠀⣿⡇⠐⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⡝⠀⠀⠀⢻⡷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡿⠿⡿⢿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠶⠶⠾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠂⣀⣉⣠⠌⡻⠿⠟⢻⣥⣄⣉⣙⠒⠲⢤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⡠⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣬⣻⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⣥⣾⣯⣶⠦⠔⠓⢈⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠈⠉⠙⠿⣿⣵⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣾⣿⡿⠿⠋⢀⣤⣶⡟⠛⠋⣉⡗⠲⠞⠳⠟⠊⠁⣈⠩⠿⣿⣶⣤⣀⠀⠀⠀⠈⠹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡞⣣⣾⣯⣻⠟⣀⣴⣞⣻⠾⠛⠂⠉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠈⠩⠿⠿⢶⣄⠀⠀⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⣿⡿⣡⣾⡋⠉⠉⠁⠀⢀⠀⠀⢀⣀⣠⠤⠄⠱⠂⠠⢄⣄⣀⠀⠀⠀⠀⠀⠐⠣⡛⢻⣆⡀⠘⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣼⣧⡟⣡⡿⢫⣾⣿⠛⠋⠀⠠⠀⢈⠰⢄⡁⠈⠁⠀⠀⠀⡏⣻⠀⠀⠀⠉⠁⠒⢤⠂⠀⠀⠀⠀⠀⠉⣿⣄⠀⠘⢿⣿⣿⢿⡄
⠀⠀⠀⢀⣾⣣⡟⠘⠁⣰⡿⠿⠌⠀⠀⠀⡠⠞⢉⠐⢫⣻⡄⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⣴⠃⠈⠳⣄⠀⠀⠀⠀⠈⢻⣷⡀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⣿⢫⣿⣤⢂⣼⡿⠃⠀⠀⡀⢀⡜⠁⠀⠀⠳⣄⢩⣟⡀⠀⠀⣀⣧⣿⡀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠲⡀⠀⠀⠀⠂⠙⣷⡌⢿⣿⣿⣿⣿⡄
⠀⠀⣼⣿⣿⣿⣇⣼⡟⠀⠀⠀⠚⢱⠆⣄⠀⠀⠀⠀⢻⣷⡞⠛⠀⠀⠀⠀⠀⠀⠀⠒⠺⠁⠀⠀⠀⠀⠀⠀⡈⣶⠒⠀⠀⠀⠙⢻⠸⣿⣿⣿⣿⣷
⠀⢰⣿⣿⣿⣿⣻⡏⢀⠀⠀⠀⢀⠃⠈⠉⠛⢦⣤⠀⠠⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⣀⣤⣴⠿⠛⠀⢄⠀⠀⠀⠀⠈⣇⣸⣿⣿⣿⣿⡇
⠀⣼⣿⣿⣿⡟⣿⡁⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠐⠦⣀⣀⡀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⣿⣿⣿⣿⢃⣼⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡽⠆⡀⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣼⡟⠀⠀⠀⠏⠀⢠⣝⣓⣒⣒⣒⠀⠀⠀⠀⠀⠀⠀⠐⣻⡿⢿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⢨⢠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠙⠀⠉⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣾⠀⠇⠀⠀⠘⠂⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⡆⠸⠀⠀⠀⢁⠀⣤⣶⡿⢝⡻⠄⡀⠀⠀⠀⣀⣤⠴⢖⡶⠶⣤⣀⠀⠀⠀⣰⣿⢿⣶⣄⣀⠀⠐⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣧⠀⠄⠀⠀⢬⡆⠸⣯⠾⠋⠁⠀⠘⣦⣤⠞⠁⠀⠀⠈⠁⠀⠈⠙⢦⡤⠊⠀⠉⠻⢦⣴⡟⢠⣃⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⣰⣯⠍⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⡤⠈⠃⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⢀⠀⠀⠀⠱⣄⠀⢀⣴⣟⣯⠶⠀⠀⠀⠠⣾⡇⠀⠀⠀⢀⡀⣶⡀⠀⠀⣠⠚⠀⠀⠀⡠⠁⣠⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡄⠁⠀⠀⠀⠀⠙⢢⡍⠛⠙⣆⠀⠀⠀⠠⠁⠀⠀⠀⠀⢀⡜⠀⣁⡰⠋⠀⠀⠀⢀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⡀⠰⢄⠀⠀⠻⠏⠙⠒⢿⣇⡀⠌⠀⣀⡀⠀⠀⢀⡾⠗⠛⠣⠑⠀⠀⡀⠔⠁⣠⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣦⣄⠑⠀⠀⡀⠀⠀⠀⠈⠻⠶⢦⣽⣦⡶⠟⠋⠀⠀⠀⢀⡠⠀⠊⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣿⣿⣿⣿⣿⣿⣦⣄⡈⠒⠠⠄⢀⣀⣀⣀⡈⣉⣀⣀⠀⠤⠒⠋⢁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠠⣦⠀⢠⡄⠀⣴⠀⠀⣦⠀⢠⡆⠀⢰⡄⠀⣤⠀⠠⣦⠀⠀⠀⠘⢿⣿⣆
⠀⠀⢀⣾⣿⠏⠀⠀⠀⣾⣾⣿⣶⣾⣷⣶⣿⣶⣶⣿⣶⣾⣷⣶⣿⣷⣷⣿⣶⣾⣿⣾⡆⠀⠀⠈⢿⣿⣆
⠀⠀⣼⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⢻⡟⠛⠛⡟⠛⠛⡟⠛⠛⣿⠛⠛⢻⠛⠛⢻⡇⠀⠀⠀⠈⣿⣿⡄
⠀⢰⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣀⣠⣇⣀⣠⣧⣀⣠⣧⣀⣀⣧⣀⣀⣼⣀⣀⣼⡇⠀⠀⠀⠀⢸⣿⣷
⠀⢸⣿⡏⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⡇⠀⠀⣿⠀⠀⢹⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⠖⠒⢺⠖⠒⢺⡗⠒⠲⡗⠒⠒⡷⠒⠒⣿⠒⠒⢾⠒⠒⢾⡇⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣧⠀⠀⠀⠀⠀⣿⣀⣀⣸⣀⣀⣸⣇⣀⣀⣇⣀⣀⣇⣀⣀⣿⣀⣀⣸⣀⣀⣸⡇⠀⠀⠀⠀⢠⣿⣿
⠀⠘⣿⣿⡀⠀⠀⠀⠀⣿⠉⠉⢹⠉⠉⢉⡏⠉⠉⡏⠉⠉⡏⠉⠉⣏⠉⠉⢹⠉⠉⢹⡇⠀⠀⠀⠀⣸⣿⡟
⠀⠀⢻⣿⣧⠀⠀⠀⠀⣿⠤⠤⢼⡦⠤⢼⡧⠤⢤⡧⠤⠤⣧⣤⣤⣿⣤⣤⣾⣤⣤⣼⡇⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⢿⣿⣧⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠈⢻⣿⣷⡀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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




⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠛⠀⠀⠛⠀⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣾⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣷⣾⣿⣿⣾⣭⣍⠛⣦⠰⣧⣤⣼⠇⢾⣁⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⣿⣄⣀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡿⠀⡾⠀⣿⣿⣿⣿⣿⡟⣉⣤⠶⣛⣫⣭⣭⣿⣿⣿⣿⣿⣷⣶⣿⡟⠻⣿⣿⣿⡇⢸⠄⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⡿⢋⡽⠛⣩⣶⣿⠿⠛⠛⢉⣉⠉⠉⠛⠻⠿⣿⣿⣿⣷⣄⠉⠛⠧⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⣤⠞⠁⣰⠋⣠⣾⡿⢋⡅⠀⠀⠀⢸⣿⠀⠀⠀⠀⠖⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠴⠶⠀⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⢸⣿⠁⢠⡏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇⠀⠹⣿⣿⣧⠀⡤⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⢡⣿⠏⢀⡾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣦⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠞⣱⣿⠋⢠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢳⡀⠈⢻⣿⣿⣷⣤⣘⠃⠀⠀⠀⣿⠄⠀⠀⣀⣤⠞⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⣶⣶⠶⣖⣛⣭⣴⡿⢿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠋⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠉⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋

⢀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣿⣿⡿⠻⣿⠛⡿⢿⠿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠻⣟⣹⣿⣝⣫⣽⣋⣻⠟⣶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣟⢹⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⡏⣻⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣻⣾⣿⡿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣟⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣥⣿⣿⣿⣿⣿⣎⣿⣿⣿⢿⣿⣛⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣓⣿⣿⣿⣿⣿⣿⡿⣫⣵⠾⠛⠛⠋⠙⠛⠛⠷⣮⣟⣿⣿⣿⣿⣿⣿⣿⣚⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⣿⣶⣭⣿⡿⣫⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢿⣝⣿⣿⣿⣿⣿⣿⣟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⢻⡟⣿⣿⣿⣿⣿⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣿⣿⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣏⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⣭⣯⡷⣿⢹⡏⠀⠀⠀⠀⢀⣀⡠⢾⡿⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢹⣿⣿⣿⣿⣿⣿⣿⠤⠒⠂⠉⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⡏⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣺⣧⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⣿⣿⣿⣿⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣽⣿⣿⣿⣿⡿⣿⣝⢷⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣤⡾⣻⣿⣟⣻⣿⣿⣿⣯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠭⣿⣿⣿⣾⣿⣿⣷⣝⡻⢶⣤⣤⣤⣤⣤⣴⡶⢟⣯⣾⣿⣿⣿⣿⣿⣿⢭⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⢛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣽⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣷⣿⣿⠿⣯⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣇⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⣽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣴⣯⣭⣟⣫⣝⣻⣭⣽⣦⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢰⣿⡆⠀⠀⠀⢰⣿⡆⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣶⡆⠀⠀⠀⢠⣶⡆⠀⠀⠀⢠⣶⡄⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇
⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⢾⠷⠶⠶⠶⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⠶⠶⠶⠶⡿⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣠⣯⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⢿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠿⠛⠉⠀⠀⠀⣿⢸⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠞⠁⠀⠲⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠖⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⠤⠀⠀⠀⣠⣤⣤⣤⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⣎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠎⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣀⡀⠀⠘⠃⠀⣀⣠⣴⣾⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⡴⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⡤⠶⠶⠶⢤⡞⠋⠉⣉⣙⣯⡤⠴⠒⢻⠏⢧
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⣹⣿⣠⣤⣤⣶⠾⣿⠛⠋⠉⢁⡿⠀⠀⠰⡿⠀⠈⢧
⠀⠀⠀⢀⣠⠼⠷⢒⣚⣛⣭⠭⠞⣟⠉⠁⠀⣀⡟⠀⠀⠀⠘⠃⠀⠀⢀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⣸⢠⠖⢻⠟⢻⡅⠀⠀⣴⣏⠀⠀⠀⠛⠁⠀⢀⣤⡀⠀⣷⢦⣸⣆⠀⠀⠀⠈⢷⡀
⠀⠀⠀⡏⢸⠀⢸⠀⠀⡇⠀⠀⠉⠁⠀⠀⠴⠶⣿⠋⢸⣯⡿⣄⠹⡆⠉⠛⠀⠀⢀⣀⡈⢳⡀
⠀⠀⢰⡇⢸⡇⠸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⣹⠀⠸⠇⠀⠉⢀⣀⣤⡴⠒⣻⡩⠄⢹⡄⠻⡄
⠀⠀⢸⡇⠸⡇⠀⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⡤⢶⠚⣙⣅⠄⢳⣈⠥⠿⣎⠉⠙⣆⠙⣆
⠀⠀⢸⡇⠀⡇⠀⢸⡀⠀⢹⡀⠀⠀⣀⡤⠶⡖⢋⡉⢧⠒⣊⣧⠴⠚⣏⠁⠙⣆⣀⠼⢶⡊⠉⢧⡈⢧⡀
⠀⠀⢸⡇⠀⣇⠀⠀⣇⠀⠀⢷⠀⠈⢇⠰⠒⣱⣬⠴⠚⢏⠁⠈⢧⣀⠼⢶⡊⠉⠣⡀⢀⣱⣤⠒⠛⣄⠙⢦
⠀⠀⢸⡇⠀⢻⠀⠀⢹⡄⠀⠈⢧⠀⠘⣞⠉⠁⠹⡄⣀⡬⢶⠚⠉⠳⡀⠀⣱⣤⠖⠛⢏⠀⠈⢣⣀⡬⠷⡀⠳⣄
⠀⠀⠘⡇⠀⢸⡄⠀⠀⢷⠀⠀⠈⢧⠀⠘⢦⠴⠚⠹⡅⠀⠈⢣⣠⠤⠛⢏⠁⠈⢢⣀⡨⠷⣞⠉⠙⢦⡀⢙⣦⡈⠳⣄
⠀⠀⠀⡇⠀⠀⣇⠀⠀⠈⣧⠀⠀⠈⢧⡀⠈⢧⡀⣀⡼⢶⠚⠉⠙⣄⠀⣈⡷⡔⠋⠙⢦⡀⠈⣳⣤⠔⠛⢯⡀⠙⣦⠈⠑⢄
⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠘⣆⠀⠀⠈⢳⡀⠀⠻⡄⠀⠈⢣⣠⠴⠚⠻⡁⠀⠈⢲⣄⠤⠛⢯⡁⠈⢲⣤⠴⠛⠋⠁⠀⢀⣠⠗
⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠘⣆⠀⠀⠀⠹⣄⠀⠙⢶⠊⠉⠙⢦⡀⣀⡼⠶⡚⠉⠈⠳⣄⡠⠽⠚⠉⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠘⡇⠀⠀⢳⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⢦⡀⠀⠳⣄⡤⠖⠛⢧⡀⠀⣈⡷⠖⠋⠁⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⣷⠀⠀⠘⣇⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⣦⠀⠈⠑⢦⣠⠴⠚⠉⠁⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⢸⡄⠀⠀⢹⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⣧⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣹⢦⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣣⠀⠀⠀⣀⡤⠴⠚⠳⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠳⠒⠛⠉⠉⠳⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠚⣛⣛⣛⣛⣛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣋⣥⡴⠶⠛⠛⠋⠉⠉⠉⠙⠛⠛⠶⢦⣍⡛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣩⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣬⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠤⠤⢄⡀⠀⠀⠀⠀⠱⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠏⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣰⠏⠀⠀⠀⠀⠀⢠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠟⠛⠛⠛⠿⢶⣦⣄⠀⠀⠀⠀⢣⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⡏⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠦⢤⡀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠘⡄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡿⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠈⠳⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠊⠈⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⢻⠀⠀⡀⢙⣷⠀⠀⠀⠀⠀⠀⠀⢸⠿⣦⣤⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠸⡆⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⡸⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⣀⠽⠤⠤⠤⠤⠤⣀⡀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡜⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣸⢃⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣧⣼⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠑⠢⠤⠤⠤⠤⠤⠒⠋⠀⠀⠀⢠⠏⡼⠁
⠀⠀⠀⢠⣶⠞⠛⠉⠉⢀⣀⣠⠄⠀⠀⠀⠀⢸⠋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢠⠏⣼⠁
⠀⠀⠀⠈⠛⢦⣀⠀⠛⢬⣷⠙⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢉⠓⢴⠋⡼⠃
⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣈⢣⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢄⣀⣀⣀⣀⣀⣀⡠⠤⠐⠚⠉⢰⣒⠛⠓⣰⡾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⡜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣄⣀⡉⠃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⡽⠋⠀⠈⠀⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⣛⣥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⣙⡓⠶⠦⠤⠤⠤⠤⠴⠶⢒⣋⣭⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢿⡿⣷⠖⠒⠒⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠤⠾⠡⡏⠰⠯⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠤⠤⠤⠶⠶⠶⠒⠛⠓⠒⠶⠦⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⣤⡴⠶⠔⠿⠃⡀⢠⢠⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⡀⠀⢠⡄⠀⢦⣠⣆⠀⠀⠰⠁⠀⠀⠶⠋⢁⡀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠸⠟⡃⠄⢰⣶⠿⢈⣉⣭⣤⣤⣦⣴⣤⣿⣧⣀⡺⠻⣦⣤⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡈⠁⢀⣠⣴⣾⠛⠛⣻⣿⣿⣿⣿⠛⠛⠛⣻⣿⣿⣦⣄⡀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠶⠀⠈⣁⣿⣿⣿⣿⣧⣄⡨⣽⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⡦⢀⣶⣿⣿⣿⣿⣿⣹⣿⡇⠋⠈⠛⠛⠋⠙⠛⠻⣿⣿⣼⣿⣿⣿⣿⣿⣷⣠⣬⠁⠀⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣿⣿⣿⣿⣿⣿⡄⢘⡁⠀⠀⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠀⠀⠺⢦⣾⣿⣿⣿⣿⣿⣿⣿⢸⣟⠠⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠙⢿⣿⣿⣿⣿⡿⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡻⠇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⡄⠀⠙⢿⠿⠉⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡎⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡹⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣙⣿⣿⠀⠀⠈⠀⠀⢠⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢠⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣷⠀⠀⠀⠠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣟⣈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢋⡈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠛⢶⠄⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣴⣦⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⢋⡄⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠐⢤⡄⠈⢻⣿⣿⣿⣿⣿⢿⣿⣧⣀⣴⣿⣿⡿⢿⣦⣠⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋⢘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠙⠿⣿⣿⣿⣾⠟⠿⢧⣶⣿⣿⣷⠒⠿⠿⢿⣿⣾⣿⣿⣿⡿⠋⣄⡀⢊⡙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⢴⣌⣻⢿⣿⣤⣈⣿⣿⣿⣿⣿⣭⣥⣤⣿⣿⣿⡿⠟⠉⠠⣶⡈⡁⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢨⡿⠻⢂⣤⠉⡙⢻⢻⠿⠿⡿⡿⠟⠛⣛⠉⢡⡄⢀⠛⠀⠂⢀⣄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⢰⣶⡄⠻⠆⠛⢃⣈⡁⠀⠀⣿⣿⠶⣿⠁⠈⠁⣿⠁⠛⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠀⢱⡜⢂⣀⣀⣠⣄⠱⣆⠀⣄⠀⣰⠃⠀⡙⠁⠾⠂⠠⠚⠛⠋⢰⣿⠀⠁⢴⡷⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⢀⣡⣼⣿⣿⣿⡏⣩⣦⣦⣶⣿⣿⡟⠀⠢⠉⠀⠀⠀⠀⠀⡀⠀⠀⠁⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠉⣀⡴⠐⡖⠮⣄⡀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠶⡿⠛⢀⣀⣉⣀⡘⠫⡓⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠊⡁⠀⠀⠀⣩⠶⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠙⢦⣤⠞⠉⠁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡀⠀⠀⠈⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢄⠀⠀⠀⠀⢂⠄⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣦⣌⡁⣀⡉⢄⣶⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠀⠀⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣻⣄⣀⣠⣞⣉⣳⣄⣀⣀⣴⣛⣶⣀⣀⣠⣖⣛⣲⣀⣀⣠⠖⠒⣆
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⣏⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⠙⣯⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⢻⣛⠛⠛⠛⣷⣦
⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣬⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣉⣀⣀⡀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡤⠒⠒⢢⠠⠒⠒⢢⠠⠒⠒⢢⢰⠒⠒⢢⡔⠒⠒⢢⠄⠀⠒⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⢣⣤⣤⡜⢰⢤⣤⡜⢰⣤⣤⡜⢸⣤⣤⣜⢣⣤⣤⡜⣢⣤⣤⡇⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢸⣇⡀⠀⡀⢀⠀⠀⣸⢀⠀⠀⢸⢸⠀⠀⢸⣇⠀⠀⣸⡁⠀⠀⡇⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠒⠋⠱⡔⠒⠒⠂⠐⠒⠚⠂⠐⠓⠒⢢⠐⠛⠒⢢⡔⠒⠒⢢⠒⠒⠚⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⠴⠤⣔⡳⠤⢤⠆⡰⠤⢤⠄⡠⠤⢤⠎⡰⠤⠤⡜⡳⠤⢤⢚⠤⠤⡤⠃⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⣀⣀⡿⣀⣀⢀⠀⡀⣀⣀⠇⣀⣀⢀⠇⡇⠀⣀⢮⣀⣀⣀⢇⡀⠀⡘⠀⣸⣿⣿⡇
⠀⠀⠀⠀⢀⣼⠃⠀⡎⠉⠉⢁⠆⠉⠉⢷⠂⠉⠉⣲⠂⠈⠉⡳⠂⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⠃
⠀⠀⠀⠠⠾⣥⣄⣀⣉⣉⣉⠁⠑⠒⠒⠈⠒⠒⠒⠉⠒⠒⠒⠉⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠴⠞⠁⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣛⣛⣛⣛⣟⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⣀⣄⣈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠚⠶⠿⠓⠦⢍⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣠⣟⣛⡂⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⣐⣾⣿⣿⣿⣷⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡥⢒⡉⠀⢠⡆⠀⢉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣅⣌⣉⣉⠉⣉⣉⣤⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠣⠊⣀⠈⠋⠀⠀⠀⠀⣀⠀⣀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠤⠾⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢠⣸⠟⠈⠁⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⠰⠶⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠾⠂⠀⠀⠠⣤⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠒⢀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠑⠂⠀⠀⠈⠁⠀⣀⠀⠀⠀⠀⠀⢶⣦⡀⠄⡀⠘⠁⠀⠀⠀⠀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡴⠚⠉⢀⠀⠁⠀⠀⠀⠈⠡⠁⠀⠀⠀⠀⠀⠀⡀⠀⠐⠶⣤⡘⢿⣆⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⣠⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡠⠂⠘⠈⠀⠀⠀⠀⠀⠰⡀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠻⠦⣙⣷⡄⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠜⠡⠊⠀⠀⠀⠰⢆⡱⡀⠀⠀⠁⠉⠁⢁⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠢⠀⢿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⠂⡐⠠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠈⢨⣽⣿⡆
⠀⠀⡄⠀⠀⠀⠀⠈⠃⠁⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠰⡀⠉⢹⣿⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⢸⣿⡇
⠀⠰⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⡈⢻⣷
⠀⠀⠀⠀⠅⠀⠐⠀⠀⠐⣉⠀⠀⢠⢠⣠⢔⡧⠖⠒⠢⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠋⢨⢐⢠⣿⣿⠀⠀⠀⠠⠖⠊⠉⠐⠢
⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡇⠀⠀⠀⠁⡀⣴⣶⣦⣀⡰
⠀⠀⡄⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⢦⠀⣀⠄⢀⡮⡀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠉⠛⣻⡟⠋
⠀⠀⠰⡀⠀⢳⠈⢄⠆⡠⠘⠄⢠⠀⠀⠀⠀⡇⠀⠀⢀⡄⠁⠀⠀⢸⠀⠀⠀⠀⠙⠋⠀⣠⢀⠀⡀⣹⣿⠇⠀⠀⠀⣀⣤⠄⠒⠂⠘⠿⢶⣆⡀
⠀⠀⠀⠳⡀⠀⢗⠈⢊⠁⠀⠀⠃⠀⢀⢶⢄⠻⣄⠁⠉⡀⠀⠊⣠⠎⠠⣄⠀⠀⠀⠀⠀⠠⠊⢼⣿⣿⡟⠀⠀⢠⡞⠁⠐⣄⠀⣼⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠱⠀⠀⠙⢦⡑⠄⠀⠀⠀⡙⠃⢸⠀⠈⠓⢦⠤⢴⠚⠁⠀⠀⠀⢅⡀⠀⣀⠞⡀⣶⣿⣿⠏⠀⠀⢠⠏⠁⠢⠀⠀⠀⠀⠀⠀⢀⠤⠚⢻⡆
⠀⠀⠀⠀⠀⠈⠠⡀⠀⠙⢦⣀⠀⠉⠖⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⣆⠥⢊⡠⣢⣾⣿⠟⠁⠀⠀⠀⣿⠦⠤⠄⠀⠀⢠⣤⣀⠀⢀⣄⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠁⠷⠤⣀⠀⠠⠀⠀⠀⢚⠘⣀⡀⠠⠤⢒⣪⣤⣄⣥⣶⡿⠛⠁⠀⠀⠀⠀⠀⣻⡀⣠⣤⠀⢴⠋⠄⢈⠉⣤⡆⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣄⣄⣸⣿⣶⣖⣒⠤⢤⣤⣴⣶⣮⣥⣼⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡈⠆⡰⡇⠀⠂⢀⢆⠈⠙⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣄⣀⢲⡄⢈⣀⣦⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠠⠠⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣄⣀⡐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠟⠋⠉⠉⠉⠉⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠃⠀⠀⡀⠀⠐⢶⣶⡆⠤⣄⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⢄⠀⢀⢸⣹⡇⣶⣬⣇⠀⢨⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⢂⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣿⠛⣿⠛⢻⣿⣿⡏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⠀⣇⠀⢸⣿⣿⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⢀⡋⣀⣐⣛⣟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠻⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠤⠐⠶⠖⠾⠷⠷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄
⠀⠀⠀⠀⠀⢀⢤⡶⠂⠀⠀⠀⢀⠤⠒⡉⠍⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣀⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⢀⠀⠀⠀⠀⠀⠸⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣇⢄⡀⠀⠀⠙⢿⣄
⠀⠀⠀⢠⠟⠀⠀⠀⣠⠀⡠⢊⠐⠀⠈⢦⡳⡀⠀⠀⠀⠀⣆⣸⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⢷⣕⣦⠀⠀⠈⢻⣦
⠀⠀⢀⠋⠀⠀⠀⣴⢃⠞⠁⠀⠀⠀⠀⠀⠑⡷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣄⡀⠈⢻⣧
⠀⠰⠏⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⣤⠀⠘⣿⣿⣿⣷⡈⢿⣆
⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⡴⠟⠋⠀⠀⠀⠘⣿⣿⣿⡷⠸⣿
⠈⠃⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠀⣻
⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⠁⢸
⠀⠀⠀⠄⣿⠀⠄⢠⣒⡒⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⢷⠀⠀⣿⡇⢹⣇⢨
⠀⠀⠀⢻⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣟⡘⠇⢸
⢠⠀⠀⠈⣮⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣸⣿⠷⡏⠀⢸
⠀⡆⠀⠀⢿⠟⡄⠀⠀⣤⡲⡫⢵⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠒⠄⠀⠀⠀⠀⠀⠰⢿⣷⣦⢦⡄⠀⢠⣿⡿⠾⠋⢀⣾
⠀⠈⠀⠀⢹⣧⢹⣆⣀⠙⣫⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠒⠙⠁⣠⡟⡟⢡⠋⢀⣾⠃
⠀⠀⠀⢀⠀⣻⣦⡛⡗⢀⠉⠀⠀⠀⠀⢀⣾⡃⠀⠀⣀⠀⣷⣤⠀⠠⠄⢀⢠⡀⠀⠀⠀⠀⠀⠀⣴⣏⠟⡴⠃⠀⣼⠋
⠀⠀⠀⠀⣦⠘⢿⣷⣌⢻⣷⣄⠂⠀⠠⡾⢉⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⡀⠀⠙⢆⠀⠀⢚⣤⣿⡷⣣⡾⠀⢀⡜⠁
⠀⠀⠀⠀⠈⠃⢈⠻⣿⣷⣍⡻⣷⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⠀⠀⠀⠀⣠⣼⣿⠿⣫⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣧⣝⡻⠳⠲⠤⠀⣀⡀⠀⠀⠅⣀⣀⣀⣤⣴⣖⣭⠮⣋⣡⣾⠛⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢻⣿⣿⣿⣿⣾⣯⣤⣄⣀⡀⠀⠽⠛⠛⠛⣛⣛⣋⣩⣶⣾⠿⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡵⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⢦⡼⢩⣽⣗⣛⡽⠻⡿⣏⡽⣁⣹⣄⠤⡼⠉⡙⡆⣠⡰⣬⣉⠘⡍⣛⣬⣳⢯⢘⣼⢂⡀⠈⣠⡴⣖⣒⡲⢤⡈⠈⠗⣤⠸⠾⠀⣦⡀⠈⣛⢓⠚
⢰⡷⠓⢰⣯⡋⠿⠳⠾⣿⣯⡵⣗⣓⣤⢸⣋⡾⠊⠠⠟⢀⢀⢀⣠⡀⠡⣣⡙⠃⠜⣼⠃⣀⣤⢾⣡⢎⢕⡴⣮⢂⣷⢸⡀⠠⠐⠂⠀⡏⠙⢉⡦⢀⠁
⠈⠟⢮⠿⢍⡉⢶⡆⠙⡳⠿⠟⣴⠥⣎⡭⡀⢟⡛⠲⠼⠀⠎⢋⡰⡷⡦⢩⣿⣿⡛⠉⡏⣍⡼⢞⡵⣻⣋⠀⠻⠼⢏⠼⠀⣑⡀⢀⠈⡄⠰⠀⡀⣭⠅
⠀⠀⣤⠃⢢⠤⣽⠮⠯⠯⣕⠒⠣⠟⢐⠰⠇⢘⠃⢃⠐⢣⡞⢶⣍⢢⢛⡫⡿⣿⠽⢻⠏⡽⡌⠙⢎⠹⡮⠁⣈⡡⠁⣤⣀⠀⠂⢙⠄⢷⠆⠐⡏
⠀⠀⠀⠐⠋⠀⠘⠈⠀⠀⠙⢦⣤⡤⢆⠲⠅⡵⠨⠬⠔⠀⠙⠚⠀⠉⠙⠷⠦⠾⠓⠛⠉⠊⠀⠀⠑⠓⠒⠊⠙⠃⠑⠛⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠺⠛⠟⠛⠺⠊⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠾⠺⠳⠗⠗⠺⠛⠃⠀⠀⠀⠀⠀⠘⠾⠺⠸⠓⠛⠺⠸⠓⠗⠟⠛⠃










⠀⠀⠀⠀⠀⠀⠐⠂⠶⠶⠶⠦⠶⠶⠶⠖⠆⠀⠀⠀⠀⠀⠀⠀⠰⠇⠶⠰⠶⠶⠲⠆⠀⠀⠀⠀⠀⠀⠀⠰⠦⠖⠒⠲⠶⠆⠶⠶⠲⠆




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠋⠛⠛⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠉⠉⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣻⣽⣶⣾⣶⣶⣮⣝⡻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠔⠊⠉⠁⠉⠉⠓⠦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⡄⠀⠀⠀⠀⠀⠀⢠⠋⢀⠞⠀⢀⣀⡀⠀⠀⢀⣀⠀⠈⢢⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⡀⢨⣿⣿⣏⠀⣹⣿⣿⡎⣿⣷⠀⠀⠀⠀⠀⠀⡌⠀⡎⠀⠀⢏⠀⡹⠀⠰⣅⢀⡇⠀⠀⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⣿⣿⡿⢿⣿⣿⣿⠿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢀⠭⡀⠀⠀⡨⢅⡀⠀⠀⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⢿⣿⣿⣀⣠⣿⣿⣧⣀⣼⣿⣿⢳⣿⡟⠀⠀⠀⠀⠀⠀⢡⠀⢣⠀⠀⢧⣀⡼⠀⠘⢅⣀⠇⠀⢀⠇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢦⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣾⣽⣛⣻⣟⣛⣫⣵⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠑⠢⠤⠤⠤⠤⠒⠋⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠤⠤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠂⠒⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡔⢀⡤⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠘⢠⠋⠀⡖⠦⠀⣀⣈⢢⠀⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⣯⠀⢀⡀⡇⠀⣀⣄⠀⠀⠳⠝⠀⡇⠀⠀⠘⡄⠀⠀⠀⣀⠴⠞⠛⠋⠉⠀⢀⡀⠈⠉⠑⠂⢄⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠈⢇⠣⡀⠳⠖⠀⠴⢶⠀⢠⠃⡀⠀⢸⠃⠀⣠⠞⠁⢀⠤⠐⠈⠉⠀⠀⠈⠉⠓⠤⣀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠸⣦⠀⠈⠲⣄⠀⠀⠀⣈⡡⠔⣣⠔⠁⢀⡏⢀⡔⠁⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠈⠑⠒⠈⠁⠐⠛⠁⢀⣴⠏⠀⡞⠀⢀⠎⠀⠀⢠⡖⠛⢶⠀⠀⣀⡤⢄⡀⠀⠀⠘⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣶⣤⡤⢤⣶⠾⠋⠁⠀⢸⠀⠀⣸⠀⠀⠀⠀⠧⠤⠞⠀⠀⢩⡀⢀⡇⠀⠀⠀⢹⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠸⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠚⠋⣉⠁⠀⠉⡉⠳⢦⡀⠀⠀⠇⠀⠘⠀⠀⠀⡐⠉⢱⡀⠀⠀⠖⠲⣄⠀⠀⠀⢀⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠁⠀⠀⠀⠀⠉⠳⣄⠙⣆⠀⠀⠀⠀⠱⡀⠀⠑⠞⠋⠀⠀⠐⠦⠤⠏⠀⠀⢀⠞⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠐⠋⠙⡆⠘⣇⢹⡄⠀⠀⢄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⣺⠀⢸⠀⠀⣀⣀⡀⠘⠦⠜⠁⠀⣿⠘⡇⠀⠀⠀⠀⠀⣀⠀⠈⠉⠒⠒⠒⠒⠒⠉⠁⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢋⢂⠘⡄⠰⡇⠀⡓⠀⠀⠀⠀⢠⡏⢰⡇⠀⠀⠀⠀⠀⠀⠉⠂⠠⠤⠤⠤⠤⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡄⠈⠢⣈⠉⠀⠀⠀⡀⠶⠋⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠉⠉⠙⠉⣀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⠀⢀⣤⣤⣄⠀⣠⠤⠤⣄⣠⠤⠤⣄⢠⠶⠶⢦⡀⢠⠖⠛⠲⣄⣰⠚⣛⣳⣄⣞⣋⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣉⣉⣙⣦⣾⡥⠤⢼⡷⠿⠶⠶⣾⠷⠛⢒⣚⡟⠛⠛⣛⡟⠋⠉⣩⡿⠋⠉⠉⢉⣽⠋⠉⠉⢩⡟⠉⠉⢉⣽⠋⣧
⠀⠀⠀⠀⠀⠀⣠⣴⡟⣿⡉⢡⡴⠃⠀⠀⠰⠟⠀⠀⠀⠛⠃⠀⠀⠘⠁⠀⠀⠀⠉⠀⠀⠀⣈⣀⡀⠀⣤⡀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢀⡏⠸⣧⣿⣇⠀⠀⠀⠀⠀⠀⣠⠴⠆⢀⣶⡄⢰⡆⠀⢸⣯⠅⠀⣿⢦⣹⠀⣯⢈⡷⢰⡧⢿⠀⣷⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠇⠀⣿⣾⣿⠀⠀⠀⠀⠀⠀⢿⣀⡄⢸⠛⠷⠈⠧⠤⠘⠓⠒⠀⠛⠀⠙⠀⠙⠋⠁⠈⠀⠈⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡀⢸⡆
⠀⠀⠀⠀⠀⡿⠀⠀⢹⣿⡟⡇⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠴⠶⡶⠖⠒⠒⠛⠛⣿⠛⠛⢉⣉⣹⣉⣉⣉⣭⣯⠤⠤⠤⢽⠤⠤⠼⡇⠀⣷
⠀⠀⠀⠀⢰⠇⠀⠀⠘⣿⣷⢻⠀⠀⢸⣍⣡⣤⡤⠤⠼⡦⠤⠴⠶⠒⠒⢻⠒⠒⠒⠒⠛⢻⠉⠉⠉⠉⠉⡇⠀⠀⠀⣸⣀⡀⣀⣘⣆⣀⣀⣻⠀⢸⡄
⠀⠀⠀⠀⡿⠀⠀⠀⠀⢿⣿⡿⡇⠀⠀⣧⠀⣀⣀⣀⣀⣷⣀⣤⣤⠤⠤⠼⡧⠤⠶⠒⠒⠚⡗⠒⠛⠋⠋⢻⠉⠉⠉⠉⡏⠉⠉⠉⣿⣉⣁⣘⡆⠀⣧
⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⣿⣿⢿⠀⠀⢸⡍⠉⠁⠀⠀⠸⡆⠀⠀⠀⣀⣀⣷⣀⣀⣠⡤⠤⢿⠤⠴⠶⠖⠺⡖⠒⠒⠛⢻⠉⠉⠉⠹⡍⠁⠀⢻⠀⢸⡄
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⣿⡿⡇⠀⠀⣷⠴⠒⠒⠒⠚⢿⠉⠉⠉⠉⠉⠸⡀⠀⠀⠀⢀⣸⣄⣀⣠⣤⠤⣧⠤⠤⠴⠾⡖⠒⠒⠚⢿⠉⠉⠙⣇⠀⣷
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢹⣿⣧⣷⠀⠀⢹⣀⣀⣀⣀⣠⢼⡤⠴⠒⠒⠒⠛⣟⠉⠉⠉⠉⠀⣇⣀⣀⣀⣀⣸⣤⠤⠤⠤⢷⠒⠒⠒⢻⡖⠛⠋⢻⡀⠸⡆
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠸⣿⡿⣾⡆⠀⠈⣏⠉⠀⠀⢀⣀⣧⣀⣠⠤⠤⠶⠺⡖⠚⠉⠉⠉⢹⡉⠀⠀⠀⠈⣇⠀⠀⣀⣸⣀⣠⣤⠤⣧⠴⠶⠞⣇⠀⢷
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣇⣧⠀⠀⢹⡛⠉⠉⠉⠁⢹⠀⠀⠀⠀⢀⣀⣧⣀⣤⠤⠤⠼⠷⠲⠒⠒⠚⢻⡋⠉⠁⢸⣇⣀⣀⣠⣽⣤⣤⣤⠿⠄⢸
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⡄⠀⠈⡷⠶⠶⠖⠒⠚⡏⠉⠉⠉⠉⠀⢹⡀⠀⣀⣀⣀⠶⠤⠴⠶⠶⠾⠓⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⣤
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡎⣇⠀⠀⢻⠀⢀⣀⣀⣤⡽⠤⠴⠶⠒⠒⠛⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠶⠶⠶⠚⠛⠛⠋⠉⠉⠉⣧
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢻⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠶⠖⠒⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠁⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠖⢿⣿⡼⣇⣀⣀⣤⣤⡤⠤⠶⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠶⠖⠚⠛⠉
⣿⣤⣶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠸⡟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠶⠖⠒⠛⠋⠉⠉
⠀⠀⠉⠙⠓⠲⠦⣤⣀⣀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠶⠖⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠤⣼⣧⡤⠤⠶⠖⠚⠛⠉⠉⠉

⠀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠾⢶⡤⠴⠿⢦⠤⠴⠷⣦⠤⠶⠶⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠲⠚⠀⠀⠲⠊⠀⠐⠖⠃⢀⣲⣞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣭⡭⠇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⢠⠤⡄⣤⢤⠀⡖⢦⢰⠒⡄⣖⢲⠀⡖⣆⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⣒⡃⢘⣚⠀⣉⢉⢈⡭⡅⢨⠤⡀⡬⢤⠀⣇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠧⠼⠘⠤⠇⠙⢚⠂⢓⣛⢘⣒⡃⢙⣚⡀⢸
⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣇⣀⣳⠀⢻⣹⠄⣏⣳⠸⠭⠇⠯⠼⠀⠧⠗⠘⠤⠇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⠀⢰⡒⡆⢴⢲⡀⣖⣲⢰⣚⡆⢯⣹⠀⣏⡧⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣩⡥⠬⠭⠤⠬⠤⠤⠤⠤⠤⠶⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣶⣪⠃⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡉⢁⣀⣀⣼⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠭⣅⢀⣴⠾⣧⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡜⢡⣴⡄⠛⢸⡟⢀⡾⠀⠀⢩⢿⣿⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡷⢾⡟⠀⠀⣠⠝⠏⠓⠲⣾⡗⠚⠀⢹⣦⡀⣠⠖⢉⣠⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⠛⣿⣟⣷⣌⠁⣠⡴⡆⠀⠀⠙⣦⣶⣻⠟⣷⡏⠀⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⢾⣿⢱⣿⣏⣹⠋⠶⠽⠞⠁⠀⠀⠀⠉⢧⠀⣀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡈⣋⣛⣣⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⢿⡏⠃⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⣟⢋⡟⢠⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡿⡀⠈⠳⢦⣤⣀⣀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠃⢸⣸⠇⠀⠁⠀⠀⠀⠀⠀⠀⢠⡶⢀⣻⢯⣤⠿⢇⠁⠀⠀⢀⠀⡠⢻⣦⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⠦⡀⠀⠀⠀⠀⠀⠀⠈⢙⠟⣫⣻⣶⡖⣠⣶⡿⠿⠀⠐⠣⢾⡛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣦⡈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠛⢛⣁⡉⠟⣇⠤⠤⠶⠖⠚⠛⢉⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠋⠑⠶⠉⠙⠒⠿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠙⠻⢯⣹⠿⠃⠛⠛⠀⠀⠀⠀⠀⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢀⡄⠀⠇⠀⠀⠀⠀⠘⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠿⠞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠘⠞⠀⠀⣶⣆⡴⠾⠗⠉⠈⠀⡅⠀⠀⠀⠈⠣⡀⠀⠀⠀⣠⣤⠀⢱⣤⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⢴⠟⢿⠀⠀⠀⢠⡄⢠⠃⢀⠃⠀⢴⠶⢷⠛⠀⠈⠉⠉⢿⡟⣃⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣡⠃⠀⠀⠀⠀⠀⠂⢈⢣⠀⠀⠸⢻⣧⣿⣙⣛⡤⠀⠀⠀⠀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣧⡀⠻⠀⢠⠀⠀⡀⠈⠀⣣⠞⠁⢀⣼⡀⠠⡀⠀⡷⢸⠠⢀⣠⠞⠋⢀⡟⢉⣽⣋⠳⡆⠀⠘⠿⢇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠳⣄⣀⠀⠀⢀⣣⣴⡾⡁⢀⣴⣿⣿⣿⣶⡄⣷⣾⡏⠸⣇⣶⠄⢤⣶⣿⣟⣻⣿⣿⣥⠤⡴⠚⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣀⠈⢩⣛⡋⠯⢿⣽⣡⣾⣿⣿⣿⣿⣿⣬⠧⣿⡿⡱⠁⠘⠿⣟⠉⡺⢿⣤⣀⣄⣄⡀⣾⢺⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⣰⠿⡷⣹⠿⣧⣼⠟⠋⠉⠉⠀⠉⣈⠁⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠓⠊⠁⠀⢸⣏⠀⠀⠀⠀⣴⢋⣀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣷⣌⣋⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣈⠓⣠⠏⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣝⣧⡶⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⠆⣿⣿⣿⣿⡏⠉⠁⠉⠁⠈⠀⠉⠉⠉⠁⠉⠁⠉⢹⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⢿⢧⣿⣟⣛⡿⣧⣤⡤⠤⠤⠤⠤⠤⠴⠤⣤⣄⣠⣤⣼⣿⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⢸⠷⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠿⠙⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡈⠋⠀⠀⠀⠀⠀⢶⣾⠡⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢄⠀⠀⠀⠀⢀⡾⠁⠁⠀⢀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⣿⠀⠀⠀⠀⠀⢄⠀⢀⡮⠊⠀⠀⠀⠈⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢿⣷⠀⠀⠀⠀⡀⠈⢸⠀⠑⠀⠀⠀⠀⠀⡾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠈⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠑⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡇⣸⣦⢤⣤⡤⠀⠀⠀⠀⠀⠠⠧⣴⣃⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣾⣾⡞⣿⣯⣽⣷⡟⠛⠳⠒⠒⠒⠒⠲⠖⠒⠛⠋⠑⠛⢻⣷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⣿⢆⣿⣿⣿⣿⣇⡀⠀⠀⡀⢀⠀⠀⢀⡀⣀⣀⣀⣀⣸⣿⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠒⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣶⢲⣟⡿⣲⣦⣄
⠀⠀⠀⠀⠀⠀⠠⣠⡾⠋⠉⢀⡂⠀⠍⠓⢎⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣰⣯⣽⠟⠛⠙⠓⠺⣟⢻⣶⡄
⠀⠀⠀⠀⠀⢀⢰⡏⢰⣴⢢⣤⣩⣤⠓⢦⡌⣅⢠⠀⢠⠇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠸⡆⠀⢰⣿⢫⠉⠀⠀⠀⠀⠀⢈⣷⡼⣻⡄
⠀⠀⠀⠀⠀⢸⢾⠁⢸⠁⢨⣍⢨⣭⠀⣸⠁⠘⢸⠀⢸⠀⠰⠛⢋⢀⠛⠙⣉⣙⡛⠛⣀⣾⠀⠘⡽⣿⠀⠀⠀⠀⠀⢠⣶⣾⣇⣽⡇
⠀⠀⠀⠀⠀⠈⣆⠄⠘⢧⡈⠋⣜⢋⣠⠿⠀⠀⡌⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣿⠃⠀⠰⠺⡻⣷⣷⣶⣶⣶⣿⣿⣿⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠛⠛⢿⠛⣁⠀⢀⠞⠀⠀⠀⠙⢄⠀⠀⠀⠀⢀⣠⣴⣾⡿⠋⠀⠀⠀⠸⣩⣦⣻⣿⣿⣿⣿⢟⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠁⠩⠅⠔⠊⠀⠀⠀⠀⠀⠀⠀⠁⠂⠾⠻⠽⠻⠕⠊⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣽⠿⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⡒⢶⣤⣀⠀⠀⠀⠀⠀⢀⡴⢨⡴⣧⣾⣧⠿⣇⣼⠢⡀⠀⠀⠀⠀⠀⣀⡴⠒⠲⠶⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⢾⣭⣦⣧⣯⣿⣷⡀⠀⠀⢠⠎⠀⢀⡷⠛⠛⠉⠛⢾⣁⠀⠙⡄⠀⠀⠀⡴⠉⢂⣬⣷⣧⣦⣿⠫⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⡿⣿⣿⣿⣷⣽⡧⠀⠀⢸⣠⣤⣾⡁⠀⠀⠀⠀⠀⢻⣥⣤⣱⠀⠀⢰⡋⣦⣿⠟⠉⠩⠉⢻⣶⣉
⠀⠀⠀⠀⠀⠀⠈⣿⣿⢽⡹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢨⠘⠉⢷⣌⠀⠀⠀⠀⢀⣿⠉⠋⡟⠀⠀⠘⠣⣾⡏⠠⠆⠠⠂⣸⡟⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢠⠈⣠⣝⡲⣶⣤⣴⣾⡅⠀⣤⠃⠀⠀⠀⠻⡂⠻⣷⡤⠶⠞⠃⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠑⢽⡹⢿⢛⡿⢹⣿⡽⠚⠁⠀⠀⠀⠀⠀⠈⠑⠓⠶⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⠴⢂
⠀⠀⠀⠀⠀⠀⠀⣔⣴⣿⡛⣿⣻⣦⠀⠀⢀⠞⢫⡲⣄⠀⠈⠀⠀⠀⠀⠔⢠⠖⠋⠑⢄⠀⠀⠀⠀⡔⠀⠈⠀⠀⠈⠛⠛⠋⠑⠄
⠀⠀⠀⠀⠀⠀⠨⢸⣏⡕⣭⠎⡁⣁⠀⠀⢠⠀⠨⢽⣿⣭⣆⢀⠀⠀⠨⡀⢏⠀⠀⠒⢀⠃⡨⠀⠀⢠⣀⣀⡀⣠⢤⣄⢀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠛⣷⣽⣿⣿⣿⠌⠀⠀⠈⠢⡀⠀⣿⣷⣼⠋⠀⠀⠀⠈⠠⡀⠶⠔⡠⠊⠀⠀⠀⠈⠉⠛⠛⠿⠶⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠒⠒⠒⠀⠀⠀⠑⠀⠐⠂⠒⠒⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓⢹⣟⡲⠀⠀⠀⠀⠀⢀⣀⣭⡷⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⣻⢻⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠷⠾⠶⡶⠶⠶⠶⠦⠶⣶⠶⣶⠿⠿⠿⠟⠉⠉⠉⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣀⣀⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠁⠀⠀⢈⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⠤⣖⣲⡭⢿⣛⣋⣁⣀⣠⡶⠦⣤⠀⠈⢙⣁⣀⣀⣀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⣶⡭⠷⢚⣻⠿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠓⠿⣄⡀⠸⣏⢆⠀⠉⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡴⠞⠉⢀⡠⠤⣒⣺⣭⡭⢭⣭⣙⣒⠤⣄⠀⠙⠳⣜⢯⣣⡀⠀⠈⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⠀⡠⣖⡽⠞⡋⠁⠀⠀⡆⠀⢀⡆⢭⡛⢦⣑⢄⠀⠈⢿⣷⡵⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⣪⠞⡉⠀⠀⠛⠀⠀⠀⠀⠀⣨⡕⠾⠃⠀⠙⢦⡳⡀⠀⢻⡽⣞⢆⠀⣼⠰⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⡜⡴⢃⣺⣿⠂⠀⠀⠀⠀⠀⠀⣼⢫⠇⠀⠀⠀⢠⠌⢳⡙⡄⠀⢷⠘⢮⡾⠃
⠀⠀⠀⠀⠀⢀⡟⠀⡜⡼⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⣰⣷⠏⠀⠀⠀⠀⠀⠀⠀⢷⢱⠀⠸⡇
⠀⠀⠀⠀⠀⣸⠃⢰⣿⡇⠀⠀⠀⠀⠀⠀⣠⠦⢤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⡇
⠀⠀⠀⠀⠀⢹⠀⠘⣿⡇⠚⠁⠀⠀⠀⠘⢧⡴⠞⠛⠻⡀⠀⠀⠀⠀⠀⠀⠘⠃⣸⢣⠃⢀⡇⢰
⠀⠀⠀⠀⠀⠘⣇⠀⢳⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⡏⡜⠀⣼⠁⠏
⠀⠀⠀⠀⠘⠄⢹⡄⠀⢿⣧⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣀⡾⢣⣞⡜⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⠻⣷⣄⠀⠀⣴⣶⠆⠀⠀⠀⠀⢀⠀⠀⠀⢉⡵⣫⠎⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣠⠾⣝⣦⡀⠈⠻⣷⣤⣛⡛⠀⠀⠋⠀⠀⢈⣁⡤⢖⡫⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⠴⠞⠉⠙⢦⣄⠈⠙⠻⠭⠭⠽⣛⣛⣛⠭⠕⠚⠉⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣀⣀⣀⣀⣀⣤⣤⣶⡟⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣿⣿⣿⣿⣿⣿⣿⣻⣦⣀⣄⣀⣀⡀⣀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠉⠁⠀⠙⠛⠷⠿⠿⠿⠷⠿⣿⣶⣶⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠣⣤⣤⣤⣤⣀⣐⡭⠋⠉⠀⠈⠉⠛⢏⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠷⣠⣬⠵⢏⣿⣿⣦⠀⣠⣴⡶⠶⢲⡄⠣⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⣷⡌⢁⡾⠉⠼⠁⢝⢹⡿⢠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠈⣴⢣⡖⣂⣷⠿⣵⣾⠏⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣙⡛⣻⠃⠀⠀⠚⣦⠤⠶⠙⠃⠈⣉⡉⠁⠉⠒⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠘⠉⠃⣀⣴⠾⢉⣠⡴⠒⡀⠈⠉⠉⠁⠀⠒⠲⢶⣄⢙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠯⣻⠷⢋⢄⣨⣔⠀⣈⠀⠀⢁⠐⢦⠄⡠⢉⠋⢄⠚⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠐⠣⢊⣔⡻⣻⠁⠀⠀⠇⠀⠀⠰⠌⠒⢮⡂⡔⢦⠵⡈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠸⠎⠖⢰⡿⠋⠘⠂⠋⠀⠀⠀⣈⣧⠉⠀⢀⠠⡙⢼⡆⢠⡿⡁⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⡎⢠⢀⣿⠁⠀⠀⠀⢔⡵⠂⠁⢩⠓⢢⡀⡈⠋⠀⠈⣿⠆⣍⠢⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢈⡟⠘⢺⣇⠣⡍⠀⠀⡂⠀⠀⠀⣯⠀⠠⣣⠐⠀⣤⣤⢸⣷⠘⢼⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠈⠁⡀⠻⣇⠀⠀⠀⣤⣧⢷⣷⠛⠛⠢⣤⣘⠀⠀⠉⡉⢸⡽⢐⠾⠀⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⢛⠓⠘⡑⡐⠞⠂⠈⢄⠺⢄⠐⣰⣤⠖⡠⠉⠊⠓⢀⣾⡇⣚⡴⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠊⢄⠀⠁⠐⡄⠶⠻⠀⠈⠀⠀⠀⠀⠀⠀⠂⡗⢂⣬⡜⡀⡤⠁⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣧⠀⢂⠁⣁⢀⡘⠀⡀⢰⣸⠔⠄⠈⠯⠑⡀⠴⡿⢪⢐⡝⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣀⠑⢌⠀⠨⡐⠾⢇⠐⠖⠒⣠⠐⠻⢛⡥⣒⡽⠛⢠⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣠⡬⡐⢲⣍⣀⣐⠒⢂⠒⣲⡋⠕⠈⢁⡀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠳⢾⣼⣄⣛⣙⢛⡓⢏⣀⠴⠶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⣶⠒⠲⣶⠶⠶⣶⠶⠶⣶⠶⣶⠶⠶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⠶⠶⠶⢶⡶⠶⠶⢶⣶

⠀⣀⠉⢻⡅⠁⠀⢂⠀⣁⣾⠋⠀⠉⠑⡍⠈⢹⣦⡾⢛⣿⠛⠛⠋⢿⡀⡸⠏⠛⠛⢻⣟⠿⣖⣾⠉⠉⡕⠉⠁⠈⢹⣆⡁⠀⠐⠀⠁⣽⠋⢁⡀
⠏⠁⠐⠚⠓⠒⠁⠈⢫⠈⠙⠃⠘⠃⠀⢹⠀⠒⠓⠛⠈⠛⢀⡀⠀⠒⠛⠙⠒⠀⢠⠀⠈⠚⠓⠀⠀⣹⠁⠑⠋⠒⠛⠋⢨⠋⠀⠑⠒⠛⠂⠀⠉
⠠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠦
⠲⡆⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡖
⠑⠃⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠓
⢐⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢒
⣿⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢼
⣾⠃⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⢹
⡶⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠴
⠲⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⢝⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡴
⠳⠄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠴
⢸⡅⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿
⠒⣳⣒⣶⢲⡶⣒⣶⣶⢆⠐⢲⠐⠒⠀⢛⣷⠂⠒⢂⠀⠀⢺⣅⠒⠒⠀⠐⠒⢀⣼⠂⠀⢀⠒⠂⢲⣞⠃⠐⠒⢰⠒⢀⢶⢶⣖⡲⡶⣲⣖⣲⡛
⠈⠉⠉⠉⠁⠉⠉⠉⠈⠉⠀⠁⠉⠀⠀⠉⠁⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠉⠁⠀⠈⠉⠁⠈⠉⠈⠉⠉⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉