Suspicous Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⠿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠐⠒⠂⠀⠀⠠⠀⠀⠀⠘⣬⣧⣼⣿⣦⣽⣶⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣿⣶⣶⣿⣧⣿⣤⣤⣿⠀⠀⣶⣐⡄
⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⢬⣿⡩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣘⣻⣯⣣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣻⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢳⠀⠀⠀⢀⣀⠤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡇⠀⠈⣧⠴⠚⠉⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠙⢶⡦⣄⣀⠀⠀⠀⡼⠁⢰⠇⠀⡼⠁⠀⣠⢴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⢦⡈⠉⠓⢾⠁⠀⢸⠀⣰⠃⢀⠞⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢱⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠳⣄⠈⢧⠀⣿⣰⢃⣴⠧⠖⠊⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠛⠒⠦⢬⣳⣼⣆⣿⡷⡟⣟⠻⣶⣒⠶⠤⠭⣦⣄⣀⠀⠀⠀⢠⠎⠀⣼⠀⠀⣠⠟⠒⠶⠦⠤⠤⣤⣀⣀⣀⣠⣴⠖
⠀⠀⠀⠀⠀⢀⣀⣠⣔⣉⣀⣠⣤⡤⠤⠤⠤⢶⣿⡿⠋⣼⡵⠋⠉⠉⠉⠻⣖⠦⣄⡀⠈⠉⠉⠉⢯⠀⢠⠃⢠⠞⠁⠀⠀⠀⢀⣀⡤⠴⠚⣋⠵⠋
⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠤⢤⣀⣀⣠⠴⠚⣩⣾⣖⠒⠫⡀⠀⠀⠀⠀⢀⣼⣉⣻⣿⣷⣦⣄⡀⠈⣆⣞⡴⠃⣀⡠⠤⠖⠋⠉⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡜⡿⠹⡿⣝⠲⢬⣢⡀⠀⠈⠛⠛⠳⢴⠋⣾⡿⡟⢏⠙⠺⢿⣾⠯⠥⠤⣄⣀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡰⠋⢠⠇⠀⢻⡈⠳⣄⠈⠙⠲⢤⣀⠀⠀⢸⢰⢯⡇⢻⡈⠳⣄⠀⠈⠙⠲⢤⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠞⠁⠀⡼⠀⠀⠀⢳⠀⠈⡟⠉⠉⠙⠙⠋⠀⢸⡟⢸⡇⠈⢧⠀⠈⢦⡀⠀⠀⠀⠈⠙⠲⢤⣀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣁⡤⠖⠋⠙⢦⠀⠀⠈⢧⠀⡇⠀⠀⠀⠀⠀⠀⢸⡷⠋⠹⡄⠸⡆⠀⠀⠱⣄⣀⣀⣀⣀⣀⣀⣈⡉⠓⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⣇⢷⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠘⣆⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠿⠿⠿⠿⠿⣿⣿⣿⣷⣦⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡉⠛⠟⠁⠀⠀⠀⠘⠻⠿⠃⠀⢀⣤⡶⠟⠛⠉⠉⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⣧⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡶⠶⠶⢶⡾⠛⣿⣿⣿
⢿⣯⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠻⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠛⣿
⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠾⠟⠋⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣤⣤⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣄⣀⣀⣀⣀⣀⣤⣶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣦⣶⣤⣤⡄⣴⣤⣤⣴⣤⣤⣤⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣥⣭⣩⣥⣽⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⣠⣴⡟⣛⣛⠛⣷⣤⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢿⣿⣿⣿⡌⢿⣇⠻⠿⢡⣿⢏⣾⣿⣿⡿⢗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡌⣿⣿⣿⣌⠿⢰⣶⠸⢋⣾⣿⣿⡏⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣆⢿⣿⢀⣾⣿⣿⡿⢃⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣆⢡⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⡙⢿⣿⢸⣿⢟⣵⣿⣿⣿⣿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣮⡛⠘⣡⣾⣭⣭⣩⣩⣿⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠂⠛⠛⠛⠛⠃





⣶⣶⣶⣶⣶⡾⢷⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⢷⡶⠶⢶⠶⢶⣶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣶⣿⣿⣿⣿⣿⣿⣮⡙⠃⠰⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢿⣿⣿⣧⢠⣷⣾⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠛⠉⢿⡿⢿⣿⣿⣿⣿⣿⢿⠿⠿⠿⢿⡿⠿⠿⠻⠿⠿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⣿⣿⣦⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠸⣿⣿⡟⠳
⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⡦⠀⢀⣾⡿⣿⣯⣭⡅⠀⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠐⠛⠹⣿⣿⣿⠛⠀⠙⠓⠀⠉⠙⠇⠀⢨⠈⠑⠛⠃⠀⠀⠀⢿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡯⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠈⠁⠀⠀⠀⠦⣤⣤⣠⠔⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡟⢛⠛⠛⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢰⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠁⠠⠤⠄⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⣶⡶⠄⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣾⡋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⢀⠠⢶⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣝⡛⠛⠛⠛⠉⠉⠁⣠
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠐⣿⠇
⠀⠀⠀⠀⡚⢫⣽⣿⣿⣿⣿⣿⣶⣶⣦⣤⣴⡟⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣧⣄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⢢⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣄⣀⡀⠀⠀⠉⠛⠷⣄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠈⠂⣉⡙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢹⣿⡻⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣤⣿⣿⣶⣶⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣸⡇⠋⠿⡇⠙⣿⣿⡿⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣛⠿⠯⢽⣿⣿⣿⣿⣿⣿⠟⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⣧⠰⣀⡀⠀⠿⠋⠀⠀⠀⠀⠈⢿⣿⣿⡿⢿⣿⡿⠉⠙⣛⣿⣶⠦⠶⠿⠿⠿⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢦⣾⣿⣷⠄⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡉⠻⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⣦⣀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⡉⠀⢠⣾⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣽⣿⣦⡀⠀⠀⠀⠀⠀⠉⠀⢦⣬⣍⡽⢳⣿⣿⣿⣿⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⣷⣻⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠩⡙⢿⡿⣶⣄⠀⠀⠀⠀⢀⣰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⠁⣈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣶⣤⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠟⠛⢻⡿⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣶⣀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣶⠄⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⣦⣤⣈⠛⠁⠀⠀⠀⠈⠛⠛⠁⢀⣠⡶⠛⠉⠁⠀⠀⠀⠉⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣾
⣿⡟⠛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠶⢶⡶⢶⣶⣾⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠈⠻⣤⣼⡿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣿⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣤⣤⣤⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⡀⠀⢀⣀⣀⣴⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣝⣏⣭⣉⣏⣹⣟⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠚⠋⠉⠁
⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠈⠙⠛⠶⠶⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢
⣈⠳⣤⣤⣤⣤⣤⣀⣐⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣐⣀⣤⣤⣤⣤⣤⡴⢋
⣰⡿⠛⠋⠉⠉⠙⠛⠛⢻⣿⣶⣍⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣊⣵⣾⣿⠛⠛⠛⠉⠉⠉⠛⠻⣷
⣿⠁⠀⠀⠀⠀⠀⢀⣾⠁⠀⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⣹⣆⠀⠀⠀⠀⠀⠀⢹
⠘⡄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⢻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠋⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⡀⠀⠈⠻⣿⣿⣿⣳⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢴⣻⣿⣿⡿⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠲⢤⣤⣀⣉⣉⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢍⣉⣁⣠⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡓⠀⠀⠀⠀⠀⣀⡤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠞⠀⠀⠀⡴⠋⠀⠀⠀⠈⠀⠀⠀⠈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣽⠀⡰⠀⠀⢀⠴⠚⠃⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠰⠁⠀⢠⡇⠀⠀⠀⠀⠀⢀⡜⣇⢸⡇⠀⣴⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⠒⠀⠀⠀⠳⡀⠲⠒⠒⠊⠁⠀⠘⠾⠹⣄⡟⢆⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢏⠛⠻⢿⣿⣿⠀⠀⠀⠀⢰⠋⡇⠀⢀⡄⠀⠀⠀⠀⠀⠈⠙⠈⢱⠚⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⡄⠀⢈⡴⢺⣿⣿⠀⠀⠀⠀⠘⠢⡇⠀⠈⠀⠀⠀⠀⠀⡗⠀⣀⣀⣬⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣭⢥⡂⠀⠸⣁⢼⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣀⢁⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⡿⠛⠉⠑⢦⠀⠀⠀⠘⣄⠀⠰⠆⢰⡄⢠⠄⠉⠈⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠉⠉⢉⣿⢿⡁⣰⠋⢳⢈⠆⠀⠀⠀⠈⠓⢤⣀⣀⣀⣅⣠⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣤⣬⣷⣾⣿⣿⠺⡑⡏⠀⠀⠀⣠⣴⣖⠣⠤⠤⠙⣻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⢹⣛⠆⢀⣼⣿⣿⣿⣆⠀⠀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠂⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣉⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣋⣉⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠹⣇⣸⠏⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣤⣽⣿⣷⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣾⣿⣯⣠⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠸⣧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⣀⠤⠖⠉⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠤⢽⣀⡀⠒
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⢉⠀⢉⠳⠦⣄
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠐⣤⡀⠀⢀⣠⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢠⣴⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⣆⣄⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠙⠃⠀⠀⠉⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢦⡀⣈⠁⠀⠀⠉⠉⠃⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡐⠀⠀⠀⣀⢀⢸⡀⠇⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠁⠈⠈⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⣲⡂⡉
⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠂⠤⢀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢿⡷⠀⠐⠠⡀
⡆⠅⠑⣄⠀⠀⠀⠀⠀⠀⢸⣷⠰⠄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢄⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡀
⠅⠀⠀⠈⠁⠂⠀⠀⠀⠀⣾⣿⠀⠢⠤⠀⠀⠀⠀⢰⡆⠀⠛⠂⠀⠀⠀⠃⠀⠈⢲⠋⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠂⢩⣿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠐⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠈⠐⠠⠄⡀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠐⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠸⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⡆⢠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠃⠀⠀⠀⠀⠀⠀⠃⣀⡢
⣿⣿⡿⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠠⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠋⠀⠙⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣀
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠚⠛⠒⠚⠓⠚⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠓⠚⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣄
⠀⠀⠀⠀⠀⠐⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⣀⢿⣽⠟⠛⣷⡶⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⡀⢀⣠⣤⣤⣤⣤⣤⡤⢶⣶⣿⣿⠛⠛⠛⠛⠒⣦⡄
⠀⠀⠀⠀⢀⣿⡿⠁⣠⡾⠋⠀⠀⠀⠀⠀⣀⣰⠟⠋⠉⠉⠛⠛⠛⢛⣉⠛⠛⠛⣉⡻⢿⣉⠛⠿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢛⣷⣏⠁
⠀⠀⠀⢠⣾⣿⣤⡞⠉⠀⢀⣠⣤⣤⣴⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠓⠀⠉⠳⢦⣤⣉⠙⠒⠶⠶⠦⣤⣤⣤⣤⣄⣀⢰⣶⠟⢷⡙⣷
⠀⠀⠀⣸⣿⠋⠉⠛⢿⠟⠉⠉⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠘⠟⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠦⢤⣤⣤⣤⠬⠛⠻⣧⠀⠀⠻⣯⣧
⠀⠀⢸⣿⡋⠀⠀⢰⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢹⡆⠀⠀⠻⣿⡆
⠀⢰⡇⢻⡇⠀⢠⡞⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⡄⠀⠀⣿⣷⡄
⠀⣼⠃⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡿⢇
⠀⡟⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠤⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢸⡇
⣠⠃⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇⠈⣷
⡿⠀⡸⠃⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿⠀⢿
⣧⣼⠇⠀⠀⣰⣿⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣻⠁⠀⠀⢻⡀⠘
⣽⡿⠀⣠⡾⠋⢰⣇⠀⠀⠀⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠀⠀⢀⡿⣧⠀⠀⢸⣇
⠿⢶⣿⡇⠀⠀⠀⣿⡀⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡇⠀⠀⠀⢸⡇⠘⠷⣄⡀⢻⡶
⠀⠀⣿⡇⠀⠀⠀⣿⣧⠀⠀⣿⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠋⠁⠀⢸⡇⠀⠀⢠⣿⡇⠀⠀⠈⣿⡿⠃
⠀⢠⣿⡇⠀⠀⢠⡇⠹⣇⠀⣿⠀⠀⠀⠀⠀⠙⠛⠳⣶⣤⣄⣀⣀⣀⣈⣤⣤⣤⣔⣀⣀⣀⣀⣠⣤⣶⠞⠛⠉⠀⠀⠀⠀⠀⠘⡇⠀⢠⡾⢹⡇⠀⠀⢠⣿⡇
⠀⠀⣿⡇⠀⠀⢸⠏⠀⣿⡷⠏⠀⠀⠀⠀⠀⢀⣠⠶⠿⣄⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣼⣦⣄⠀⠀⠀⠀⠀⠀⠀⢿⢀⣾⠁⠸⣇⠀⠀⢸⣿⡇
⠀⡄⢿⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣠⡴⠛⠻⣦⡀⠉⠛⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠾⠟⠉⠀⣹⣷⣤⡀⠀⠀⠀⠀⠻⠿⣧⠀⠀⣽⡄⠀⠈⣿⡇
⠐⣿⣾⠃⠀⣾⠁⢠⡞⠁⠀⠀⠀⢠⡾⠋⠀⠀⢠⡝⠛⢶⣄⡀⢀⡀⠀⣼⢹⡟⢹⡄⠀⠀⠀⢀⣀⣴⣾⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⢻⣄⠀⠘⣧⠀⢠⡿⢧
⠀⢰⣧⣠⣾⠃⢠⡞⠁⠀⠀⢀⣴⠟⠀⠀⠀⠀⣸⠇⠀⠀⠉⠙⠉⢻⠿⠁⢸⡇⠀⠻⣶⠛⠶⡏⠉⠀⢈⣧⠀⠀⠀⠈⢿⣄⠀⠀⠀⠈⢿⡄⠀⠹⣇⢰⣇⢸⡇
⠀⠀⠀⠉⢻⡶⠛⠀⠀⠀⠀⣸⣁⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣸⡇⠀⠀⠀⣀⣴⠿⢧⣴⡜⢿⡀⠀⠀⠀⠀⢹⣷⠤⣤⣤⣴⣿⣦⣀⣻⣾⡿⠛⠃
⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠾⣿⠛⠷⣦⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⢻⡇⠀⠀⠀⢿⠁⠀⠀⠀⣷⢸⡇⠀⢀⣠⡶⢿⡇⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡀⠀⠀⣠⣼⠋⢹⡶⠋⠉⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣭⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢰⡏⠙⠛⠛⠢⣤⣤⣤⣤⣀⣀⠀⣠⣤⣤⣼⣷⣤⣄⡀⢀⣠⣤⣤⣤⡤⠚⠛⠛⠉⠻⣄⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠈⠷⣾⡇⠀⣶⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡇⠀⠀⠀⢈⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡆⠀⣿⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⡄⢘⡿⠶⢦⣤⣤⣄⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠶⠾⣿⢀⣼⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡗⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⢷⣶⣶⣶⠾⠿⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⠤⡤⢤⣀⣸⣧⡠⠶⢶⠶⠶⠶⠶⠶⠶⣤⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣀⣀⠀⠀⠀⠀⢀⣀⣠⠶⣟⣿⣞⣳⠦⣀⡀⠀⠀⠀⠀⠀⣀⣀⡴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⠀⠈⠙⠚⢋⣽⣷⣶⣶⣶⣿⣿⡟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠿⠻⠿⠁⠀⠀⢀⣀⣤⠤⣄⡀
⠀⠀⢠⣾⣿⣾⣽⣷⣶⠄⠀⠉⠀⠀⠀⠀⠀⢰⣾⣿⣿⢿⡿⣿⣷⠄
⠀⠀⠀⠐⠠⠿⠿⠻⠝⠀⠀⠃⠀⠀⠀⠐⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⢀⣼⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣶⣶⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣻⣿⣆⣀⣠⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣷⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣧⣿⣿⣟⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣮⢿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⡼⢻⣧⣿⡏⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣽⣞⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⡇⣿⣾⠡⡇⣿⡟⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢿⣧⡿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡗⡇⣿⢀⢷⡏⢺⢻⣟⣿⣿⣿⣿⡦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢹⣿⣹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⢿⣾⣾⡇⢸⣸⣿⣿⣿⣿⢹⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠓⠿⠋⠀⣸⡿⢿⡿⠞⠃⠈⢿⡿⢿⣿⣿⣿⣿⣯⣿⣽⣷⣿⠛⠿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠆⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⠦⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠋⠁⠀⠀⠀⠀⠀⣤⣴⠿⢦⣤⡄⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⡀
⠀⠀⠀⠀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀
⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⢸⡆⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⡇
⠀⠀⢸⡏⠉⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡇
⠀⠀⣠⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢿⡀
⠀⠰⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽
⠀⢠⠛⣯⡙⠿⠒⠰⠤⠤⠤⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠤⠴⠖⠚⠿⣟⣉⣡⣽⠛⡄
⢀⡽⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠉⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⢷⣯
⣮⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉
⢣⡀
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣄⣀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣟⡑⠶⠶⠤⣤⣤⣤⣤⠤⠶⠶⠂⣻⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠻⣦⡈⠙⠶⠶⢤⣤⣤⣤⣤⣤⣤⠶⠞⠉⢈⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢠⡏⠙⠶⣤⣤⣀⡾⢹⡏⢷⣀⣄⢀⣤⠶⣿⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⡜⠁⠀⠀⠀⠀⠛⡀⢸⡇⠈⠛⠁⣱⣄⣀⣸⡆⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⢠⠿⢦⣤⠀⣸⠃⠀⠀⠀⠀⠀⢸⣯⢾⡇⠀⠀⣿⠁⠀⠙⣤⢣⠀⠀⣠⡴⣏
⠀⠀⠀⠀⢀⡞⠀⠀⠀⢙⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣀⣀⣠⠿⠸⡶⠋⠁⠀⠻⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⡇
⠀⠀⠀⠀⣾⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢰⡶⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡄
⠀⠀⠀⣴⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢷⡀
⠀⠀⡼⠃⠀⠀⠀⢰⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⢳⡄
⠀⢸⣁⡀⠀⡀⢠⡏⠀⠈⠛⠛⠲⠶⠶⠶⢤⣶⠒⠚⠓⢶⡶⠶⠶⠖⠚⠛⠋⠉⠀⠻⡀⠀⠀⢀⣀⣷
⠀⠀⠙⠳⣸⣇⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⢶⡇⢸⣧⡼⠋
⠀⠀⠀⠀⠀⠙⠛⢷⠀⠉⠉⠛⠛⠛⠛⠓⠘⠿⠶⣶⡶⠾⠗⠚⠛⠛⠛⠛⠉⠁⠀⢰⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⣀⣀⣀⣤⣀⣀⣀⣄⣀⣸⣇⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡗⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣶⣶⣤⣤⣤⣶⠶⣮⡷⠛⠳⣽⠶⣦⣤⣤⣤⣤⣶⣾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡿⠟⠛⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣃⠀⠀⠀⣀⣀⡀⠹⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⣽⡃⠀⠉⣭⠁⠀⠉⠛⠛⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣸⠿⢿⣧⣼⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠈⠱⠀⠀⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠤⠴⣤⣐⠀⠀⢀⢔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠋⠉⠑⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⣞⠃⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠘⠂⠀⠀⠉⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡤⠶⠒⠋⠛⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠓⠦⣤⣤⣤⣤⣠⡤⡾⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⢀⡀⠀⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠐⢂⣠⡃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿⠀⢸⣧⡄⣠⣀⡀⣤⣠⣶⠀⠀⢠⣤⣴⠈⣭⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿⡖⠿⠿⠇⣿⣿⣿⠃⠀⠸⠿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⡀

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

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⢸⠀⠀⡇⢸⢸⠷⣿⢸⠉⠁⡿⠉⢸⢭⡇⢹⡏⢰⣿⡄⣿⢸⢧⡇⢉⡏⠈⢳⠋⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠉⠁⠈⠀⠉⠈⠉⠁⠉⠉⠈⠀⠁⠈⠁⠉⠀⠁⠉⠈⠀⠁⠀⠁⠀⠈⠀⠀⠀⡇
⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⠆⠀⠀⣀⠤⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⠀⠀⠀⣰⣫⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⢻⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⣆⠀⠀⣀⣀⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠼⣶⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠙⢋⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⠾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣾⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢛⣻⠇⠛⣦⢫⣿⠟⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠆⠀⠶⠛⠛⠛⠷⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⣺⢖⣶⣸⣞⣿⡀⡏⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⣻⡛⠁⡬⢤⣤⣤⣄⣠⣤⣽⣾⣿⡿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠸⣿⡇⠀⠇⠀⣿⣭⣭⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠈⠓⠒⣀⣀⣿⣒⣲⣿⠟⠋⠙⠺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⢀⠀⢸⣁⢨⣿⣷⡾⠃⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠿⠓⢋⣸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠀⢶⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣾⡄⣶⣽⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⡟⠇⢣⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠀⠘⢸⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠊⠉⣉⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠐⠢⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⠿⠿⠿⠛⠛⠛⠋⣉⣉⡉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⢙⣛⠶⣦⣀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣴⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⡠⠴⠚⠋⠉⠉⠉⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠆⡐⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⠤⠖⢊⣉⠉⠉⠲⣄⠀⠠⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠁⣰⣿⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠊⠉⠉⠉⠉⠲⣌⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⢀⣿⡏⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣄⣀⣀⣀⡠⠜⠃⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡆
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠀⠀⢀⣠⠞⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠋⠛⣋⣉⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡠⠂
⠀⢀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⡄⠂⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣋⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⠟
⠀⠈⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⠞⠁⢀
⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⣀⣤⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠕⠊⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⣴⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠄⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠷⠶⣦⣀⠀⠀⠀⠸⣆⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢷⣦⣀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠈⠛⢷⣦⣘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠈⢷⣦⣤⣀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢈⠟⠛⢛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣶⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣑⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢧⠀⠀⡎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⢤⣄⣀⣀⣠⠷⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢯⡁⠀⡆⠙⠂⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠹⠋⢳⡄⢠⢇⡀⠁⠀⠀⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠲⣄⡀⠘⡡⠖⢉⠁⠀⢳⡀⣸⠀⠀⡿
⠀⠀⠀⢀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡋⠀⠀⢿⠀⠀⠈⢷⠀⠘⠃⠁⠀⠀⢸
⠀⠀⠲⡌⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠓⣤⣴⢚⠛⠓⢆⠀⠀⠀⣸
⠀⠀⠀⠀⡠⣖⣺⠿⣖⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢜⡩⠄⠚⠉⠀⠀⢿⠂⠀⢸⠀⠀⢰⠃
⠀⠀⢀⢎⠞⢱⠂⡀⠀⠈⢪⣳⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠲⢿
⠀⠀⢸⣾⠀⠁⡼⠁⠀⠀⠀⡯⡇⠀⠀⠀⠀⠀⠀⠈⠣⠄⠀⡀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⣤⡀
⠀⠀⠘⡜⣄⠘⠁⠀⠀⠀⡰⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡁⠀⠀⣠⣀⠀⠀⠀⢀⡤⠊⠀⢹⠲⣄
⠀⠀⠀⠈⠪⣗⣶⣤⣖⣮⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢉⠀⢨⡳⣠⠔⠉⠀⠀⣠⠇⠀⢸⠐⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣇⠄⣼⠋⠙⣆⠀⣠⠔⡏⠀⠀⡸⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡴⠃⢘⢹⡶⡞⠘⠞⠁⡼⠀⢤⠊⠀⠀⠀⠀⠀⢀⡆⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣏⠙⠣⣈⠳⡀⠀⠀⠀⠀⠀⠸⡀⠀⡇⠀⡏⡞⠁⡇⠀⠀⢰⠁⠀⠀⣹⠀⠀⠀⠀⢀⡞⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡬⠷⢤⠈⠁⠈⢦⡜⠉⠲⢄⣠⠇⠀⡇⢠⢷⠃⠀⡇⠀⢠⠇⠀⠀⡴⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⡦⣄⠀⡼⠀⠀⠀⠀⢹⠀⠀⡇⣸⡟⠀⠀⡇⢀⡎⠀⢀⡞⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠇⠀⠀⠀⠀⠟⠀⠰⠇⢿⠃⠀⠀⠇⠼⠀⢠⠏⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠤⠦⡀
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢦⡴⢮⠝⠻⣵⢦⣴⠶⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣶⠀⠀⠀⢀⡸⣄⠃⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠑⢦⡤⠊⡵⠉⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠶⠉⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠛⠛⠛⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢸⠀⠀⠀⠀⣦⣼⣿⣿⣿⣯⠉⠉⠉⣽⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠦⣀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⠀⡀⠀⠀⠀⠀⠀⢀⣿⣅⡀⠀⠀⢀⠄⡀⢰⣿⣄⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠃⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⢸⠀⢃⠈⠛⠋⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠇⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠎⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠖⠒⠒⠚⠛⢳⡄⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣀⡰⠃⣀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⣉⡶⠒⠒⠒⠒⠲⣏⡁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠛⢦⡀⢀⡴⠋⠋⠀⠀⠀⠀⠀⡏⠳⣄⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠛⠈⠱⠀⠘⢆⡀⠀⢀⠔⠀⠠⠋⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⠟⠀⠀⠛⡄⠀⢀⢷⡀⠀⡀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠐⠶⠈⡇⠘⠾⠁⠀⠀⠀⠀⠘⡇
⠀⠀⢀⠼⠃⠀⠀⠀⢀⡠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣴⡾⠿⣶⡆⢸⠀⠃
⠀⡰⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣄⠀⣼⠇⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠤
⢾⡁⠀⠀⠀⢴⠋⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠻⡾⠛⠀⠸⠀⢸⣤⡀⠀⠒⠒⠲⠒⠸⡇⠻⠒⠒
⠀⠉⠶⣄⠀⠀⠙⢦⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠴⠷⠶⠖⠐
⠀⠀⠀⠈⠓⢤⡀⠀⠙⢦⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡦⠤⠄⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢱⠖⣤⣀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣃⠀⢿⡟⠢⠀⠀⠀⠀⢴⣶⠷⢶⣶⡦⠀⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣵⣤⣤⣤⣤⣤⣼⣷⣄⣸⣿⣧⣤⣤⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⢘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠢⢤⡾⠛⠂⢤⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣴⣶⣶⣶⣦⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⠛⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠈⠁⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠰⣦⡆⠠⠄⠀⠠⡂⠀⠀⠀⠀⠔⠀⠂⣼⣤⣌⠀⣤⡶⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⡀⠉⠠⣿⣿⣶⣾⣿⣿⣿⢰⣌⡀
⠀⠀⠀⠀⠉⢁⠀⠄⠠⢀⠀⠚⢅⣀⣀⣄⡀⠉⠷⠍⢵⣿⣿⣿⣿⡿⠿⠛⠋⠉⢁⠀⢰⣶⡀⠰⠆
⠀⠀⠀⠀⠀⠙⠢⠀⠈⣀⠒⠉⠉⢁⣀⠐⠄⠑⠂⠻⣿⡿⠟⠋⠁⠀⠀⠀⣨⣞⠁⢞⣾⣿⡇
⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⠿⠷⢾⣿⣷⣶⣶⣶⣶⣀⡐⠒⢀⣤⡶⠾⠟⠛⡀⣿⢿⣿⣿
⠀⠀⠀⠀⠽⠟⠋⠉⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣭⣍⡉⠉⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠠⠶⠋⠁⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢈⣿⡟⠉⠛⠻⠿⢿⣿⣿⣿⣿⣶⣶⣶⣦⣤⡎⢿⣯⣷⠶⠶⠆⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠳⢤⡙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⣯⠀⠀⠀⣹⠶⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠉⠲⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠉⠛⠋⠠⠄⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠓⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠆⠉⠉⠑⢦⡀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡿⠀⡴⠋⠙⢦⠀⢱⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⣧⠀⢲⣄⣀⠞⠀⡸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠳⣤⠈⢁⣤⠞⠁⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠙⠦⢄⣀⣀⣀⣀⠠⠤⠿⣁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⣸⠋⠀⠉⠁⠀⠀⠀⠈⢣⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣦⣄⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⡿⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠉⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣠⠚
⣟⠀⠉⠀⠈⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⠎⠁
⠉⠉⠉⠉⠉⠉⠉⢹⠋⡝⢋⡛⢫⡙⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠸⣇⣀⠾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⡰⠋⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣃⠤⠤⢊⡴⠻⣄⠀⠀⠀⠀⠀⠀⣀⠴⠛⠁⠀⠀⠀⠀⠈⠛⠤⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠺⣏⠀⠀⠈⠳⡀⠀⠀⢀⡼⠉⠀⠀⠀⢠⠔⠂⢀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⢦⠀⡌⠁⠀⠀⠀⠀⡅⠀⠀⢀⡀⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⡇⠀⠀⠀⠀⠀⠘⠣⠔⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠱⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⣀⣤⣄⣠⣤⣶⣶⣶⠶⠛⠛⠛⠛⠋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡴⠗⠈⠉⠉⠉⠉⠉⠉⠛⠛⠿⠿⠿⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠠⠶⠶⠶⠀⠀⠈⠉⠀⠲⠶⢆⡀
⠀⠀⠀⢠⣶⠶⠶⠿⠿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⢶⣶⠶⠶⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵
⠀⠀⠀⢘⡁⠀⠀⠀⢀⡇⢸⠀⡶⠶⠿⠿⠿⠶⠶⠶⢶⣶⠄⠀⠀⣀⢀⠰⡄⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⡇⢸⠀⣶⠶⠶⠶⠶⠶⠶⢶⣶⣦⠀⠀⠀⢛⡙⣷⡿⢶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠠⣶⠶⠶⠶⠶⠶⠶⣦⣤⣤⠀⠀⠀⠿⠻⣿⡿⢾⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠠⣴⠶⠶⠶⠶⠶⠶⣤⣤⣤⠀⠀⠀⣠⣟⣿⢛⡈⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣤⣶⡶⠶⠶⣶⣶⣤⣤⣭⡀⠀⠀⠈⡟⠑⠻⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⣤⣤⣶⣶⠶⣶⣶⣦⣤⣭⣤⣀⣀⡀⠰⠆⠸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣉⣉⣛⠛⠻⠿⢶⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣥⣤⣤⣤⣤⣤⣤⣤⣤⣌⣉⣉⠛⠛⠻⠶⣶⣤⣌⣩⣿⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⣭⣤⣤⣤⣤⣤⣤⣤⣌⣉⣉⣉⠛⠛⠷⢶⣶⣤⣍⣉⣿⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⣉⣤⣤⣤⣤⣤⣤⣤⣉⣉⣉⡙⠛⠻⠷⢶⣤⣤⣉⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣉⣤⣤⣤⣤⣤⣤⣬⣉⣉⣙⡛⠛⠷⠶⢦⣤⣤⣉⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠶⢶⣦⣤⣍⣉⣙⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠦⠚⠛⠛⠛⠛⠛⠛⠛⠿⠷⠶⢶⣦⣄⣀⣀⠀⠀⠀⠀⣿⠀⠀⠠⠀⢀⣀⣀⡀⠠⠤⠤⠤⠀⠐⠂⠛⠉⠉⠛⠂⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣉⣉⠿⠿⣶⣴⣿⣷⠀⢺⡷⣌⣩⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠉⠋⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠛⠛⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠶⠶⠾⣧⡲⣶⣤⠀⠀⠀⣼⣿⠃⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⢦⣤⣉⠋⠁⠀⠀⠀⠉⠁⠀⢀⡴⠞⠋⠉⠉⠉⠉⠛⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠃
⠀⠀⣿⡤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠶⠶⠚⢿⠛⣿⣿⣿⠁
⠀⠀⢿⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⢏⣹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣟
⠀⠀⢨⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠁⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡤⣶⣵⢻⣿⣿⣶⣶⣦⣶⣶⡶⠿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣴⣄⠑⣿⣾⣿⣤⣿⣿⣿⣻⣆⠀⣄⡀⠀⠀⡱⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⣠⠀⢠⠀⣘⠆
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢿⣿⣀⠀⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠠⢇⣠⡧⣀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣈⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⠀⠀⢠⣄⡹⠟⠹⠃
⠀⠀⠀⠀⠈⠀⢀⢀⡀⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⣿⠟⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠙⠀⠈⢡⠾⠉⠁⠀⠀⡈
⠀⠀⠀⠀⠄⠀⠨⠤⠄⠀⠀⠀⠀⠀⠈⠛⠻⣶⣷⣯⠀⣀⣴⣿⣿⣿⢿⣻⣿⣿⣿⣿⣷⣠⣤⡼⠋
⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠂⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⢀⠀⠀⠄⠶⠄
⠀⠀⠀⠀⠤⠀⠒⠒⠀⠀⠀⠀⠀⣽⠀⠀⠘⠾⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⡴⠐⠃⢿⠷⠆⠀⡆
⠀⠀⠀⠀⠂⠀⠠⠐⠀⠀⠀⠀⢠⠐⠀⣰⣄⠀⣼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⣡⣞⣷⠃⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣷⣶⣌⡄⢉⣿⣿⣿⣅⡛⠛⢯⣽⣿⣿⣿⣿⣿⣿⣤⠻⣿⣿⣷⡮⠄⠀⡀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠊⠸⠀⠀⠀⠀⠀⠈⠛⠋⠀⢊⣿⣼⣧⠀⢹⣶⡋⢹⣿⣿⣿⣿⣿⣿⣛⣿⣿⢿⣿⣧⣠⡄⢋⠚⠰⣿
⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣽⡶⠇⠐⣻⣀⣹⣠⣿⣿⣿⣿⣿⣿⣿⣶⡩⢿⣿⣿⣧⣤⡏⢀⣤⡿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣿⠋⠁⠠⠊⠝⢩⣿⣿⣟⢏⣿⠿⣾⣿⣿⣧⣌⠈⣻⠋⢿⣿⣿⣏⡿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢠⣦⣷⠹⠋⠀⢠⠖⠀⣠⡾⢻⣿⣯⢿⢃⠀⠻⣿⣿⣿⣿⣧⣉⠛⠴⣟⣿⡟⠀⠐
⠀⠀⠀⠀⣄⠀⢐⣐⠆⠀⠀⠀⠀⠁⠛⣿⡇⠀⣤⠂⠀⢐⣿⢡⣿⣿⠛⠀⡘⡀⡀⠈⠛⢻⣿⣿⣷⣦⠀⢸⣅⢟⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣟⠁⠀⢹⣄⣴⣯⢭⠟⡁⠀⠤⠐⠁⠀⠀⠊⠮⠀⣹⣿⣿⣿⠀⠘⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠀⠀⠀⢠⡄⢻⣋⠀⠀⠁⠀⢇⡺⠆⣠⡶⠛⠛⢶⣄⠀⠀⣴⣿⣿⣿⣿⠿⠀⠀⢈⣉⠀⠂
⠀⠀⠀⠀⠐⠀⢀⣘⠀⠀⠀⠀⠀⣈⢠⢼⣿⡆⠀⠀⠐⠘⢁⠀⣿⠀⠐⠃⠀⢻⠆⣸⣿⣿⣿⡟⠁⠀⣀⣀⢙⣿⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⣿⠄⠄⠀⢀⢀⣤⡀⠹⢦⣄⣀⣴⠿⣴⣿⣿⣿⠛⣶⣴⣿⣶⣽⣿⣿⠋⣽⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣎⡔⣀⡀⠸⢿⣿⣦⡀⣹⣿⣿⣿⣿⣿⣿⣿⣦⡉⠿⣿⣿⣿⠉⢹⣾⣿⣇
⠀⠀⠀⠀⠍⠀⠀⠶⠀⠀⠀⢸⣿⣶⣤⣤⣿⣿⣿⣭⣿⣶⣿⣿⡿⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣯⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣾⠛⠋⠀⠉⢍⣛⣵⢶⣾⣿⣿⣿⣷⣾⣿⡿⢿⣿⡟⠛⠋⣹⣿⣿⣿⡛⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⣻⣷⣷⣀⡀⢀⣾⣻⣿⢘⡿⣿⣿⣿⣿⣿⣌⣛⠛⠦⣄⡂⣶⠟⣿⣿⣿⣟⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣄⠻⣿⡧⠼⢿⣽⣿⣽⣿⣷⣿⣿⣿⣷⣶⣤⣼⣇⣀⡀⠀⠈⢙⣿⣿⡟⣴⣿⣆
⠀⠀⠀⠬⠭⠭⠭⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⢿⡯⠉⠉⠉⠉⠍⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⡀
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⢀⡤⠤⢤⡀
⠀⢸⡈⠉⠶⠤⣄⡀⠀⣀⣸⡄⠀⠀⢸⠀⠀⡴⠁⠀⠀⠀⢹
⠀⠀⢳⡁⠀⠀⠀⠈⠉⠀⠀⡀⢂⠄⠉⠙⠺⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠳⡀⠀⠀⠀⠐⠄⠀⠒⠸⡀⢀⡐⣀⠀⠀⣴⠊⠁
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢀⡴⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⣼⠀⢀⠈⠓⠀⠄⠂⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⢤⣀⣼⡆⠄⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣋⣁⡀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠉⠉⠉⠉⢉⣙⣛⣋⣙⣛⣛⣉⣉⣉⣭⣭⡹⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢲⡀⠰⣦⠀⣰⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣆⠀⣸⣷⠀⡏⢷⠋⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠿⠋⠘⠛⠀⠘⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣶⣳⣾⣟⣿⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⠿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠘⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠈⠹⣿⣷⣄⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⣶⣦⣼⣿⣿⣇⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠈⢿⣿⣤⡄
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠘⣿⡿⠁⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠀⠀⢸⡇⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣀⠀⠘⣷⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡽⠀⠀⠈⠉⠉⠀⠀⣴⡄⣤⠀⣀⠀⠀⢤⠀⠀⢸⡇
⠀⠤⠄⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⣷⠃⠀⢀⠤⢸⡷⣦⡇⢹⢙⡿⠉⡷⣿⣿⡄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⣄⣿⠚⢾⡇⢹⡀⣸⣈⣇⣀⣇⣇⣏⣏⣻⡏
⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⣯⡽⣴⠾⢿⠛⠛⣿⠙⠉⣿⢙⣹⢩⣸⡏⠉⠉⠐⠒⠒
⠀⠀⣆⣠⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣧⢘⡄⢼⣇⠀⣿⣎⣡⣿⡞⢻⠁⣿⠁
⠀⠀⣯⣛⣕⠒⣄⠀⠈⠉⠀⠀⢿⡿⣿⣿⣏⠻⠓⠾⠙⢧⣿⠛⣩⡏⠀⠀⣸⠇
⠀⠀⠉⣿⣿⣿⣿⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣴⣿⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠈⢻⣿⣿⡟⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣄⣀⡐⠋⠁⠀⣰⡟⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣽⣿⣯⡻⣎⣻⣿⣿⢿⣿⣿⡶⠿⠋⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣿⣿⣿⣏⠫⣙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠈⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣴⣿⣿⣿⣿⣸⢶⣿⣿⣿⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠻⣿⣙⣿⣿⣿⣿⣿⣿⡄⠪⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣹⡿⠁
⢀⣠⣤⣤⣀⣤⡶⣿⣿⣿⣿⡿⠻⣷⣟⢮⡻⢿⣿⣿⣿⣿⣿⣿⣶⣆⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣯⣿⠟⠁
⣿⣿⣿⡿⣻⣧⣿⣿⣿⣿⣿⣿⣦⣌⡻⣦⡑⡌⣿⣟⣱⣮⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⣽⣿⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⣿⣿⣤⣾⣫⣾⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠙⠻⠿⠷⣦⣀
⣿⣽⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣶⣤⠀⠉⠀⠁⠀⢠⡾⢛⠿
⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣯⣭⣝⣻⣿⣿⣿⣿⣯⣿⣿⣿⣿⣾⡇⣿⣿⣶⣄⣿⣥⣷⣶⠛⢛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣭⣤⣶⣾⣿⣍⢻⣹⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣦⡷⠖⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⢶⡶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⣁⣀⣸⣿⠿⠿⣿⣇⣀⣉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣡⣾⠟⠛⠛⠙⣿⣄⣠⣿⠛⠛⠛⠿⣷⡌⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣾⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣷⣤⣄⣀⣀⣀⣀⣠⣴⡾⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡆⠉⠉⠛⠛⠛⠛⠉⠁⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⢀⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣽⡻⠿⣶⣶⠾⢟⣯⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⠟⠛⠋⢻⣷⠀⠀⠀⠀⣼⡟⠙⠛⠻⠿⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢠⣤⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⣤⣤⣤⣤⣤⣤⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣰⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣼⡟⠁⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠙⢿⣦⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠶⢶⠾⠿⢿⣶⠶⠾⢿⢷⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡿⣷⠶⠶⢿⣿⠷⢶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⣄⠀⠀⠀⣠⠾⠋⣠⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡈⠢⣈⣷⣶⣿⠥⠤⠶⣤⣥⣀⡈⢳⡀⢀⣀⣤⠴⠒⠋⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⣀⣀⣸⢃⣠⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢯⡉⠀⡀⠄⠂⠘⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠤⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡀⠀⠐⣟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠂⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣿⢣⡀⠀⢀⡤⠞⠉⠉⠀⣈⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⣧⡈⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠄⠀⢀⠀⢸⣄⠙⢦⠎⢀⣠⠶⠛⠛⠙⠛⠳⢦⣷⠀⠀⠀⠀⠀⠀⣿⠑⠀⢀⣀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⢠⡴⠚⠋⠉⢀⡀⠠⠄⠐⠂⠀⠉⠳⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠐⠤⠀⣠⠟⠃
⠀⠀⠀⠀⠙⢦⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣼⡇⠀⢀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⣼⠁⠰⡀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠆⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠶⠶⠒⠚⣉⣤⠞⠁⠀⠀⠀⠀⠀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢊⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⠘⣷⡀
⠀⠀⠀⠀⠀⠰⠯⣄⡈⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠤⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢁⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⣛⣿⡿⢯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠌⠀⠀⡀⠤⠀⠀⠂⠀⣼⠏⠀⠀⠀⢹⡄⠀⠀⠜⠑⠀⡀⠀⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⠂⢀⣀⣀⣀⡀⠀⠘⠀⣿⣄⠀⠈⢀⣼⡇⠀⠊⠀⣀⠀⠀⠑⠲⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠖⠚⠋⠉⠉⠁⠀⢻⡄⠀⠐⠙⣯⣝⣛⣭⢿⣌⠀⠀⣼⠋⠉⠙⠓⠲⠶⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣴⠟⠉⠀⠉⠀⠀⠉⠻⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣒⣒⡶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣭⣾⣿⣿⣿⣿⣿⣿⣾⣭⡲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⢞⣵⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣦⣝⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⡋⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡿⣄
⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⡇⠀⠂⢪⣈⣋⣭⣝⣛⡻⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⢟⣡⣶⣾⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣼⣿⣿⣷⡿⠏⠀⠈⢻⣿⢻⠏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣏⠿⠿⠋⠛⠛⠁⠀⠈⠁⡇⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠛⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢸⣀⣨⡭⠆⠀⠀⠀⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢉⣁⡤⠤⠄⠀⠀⠀⢰⡏⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠘⠋⠤⠒⠀⠀⠀⢀⣠⡟⠀⠀⢻⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⣠⣿⡿⠁⠀⢠⠟⠀⠀⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣿⣿⠿⠋⠀⠀⣴⠋⠀⠀⠀⢞⣈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣿⣬⡀⠀⢀⣤⠞⠁⠀⠀⡠⢿⣏⣿⣿⣶⣬⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠀⠹⣿⣧⠖⠋⠀⠀⢀⡤⠚⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⣇⣾⣿⣷⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠿⣹⣿⡛⠷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠟⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢂⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⠍⠉⠡⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠊⠀⠀⠀⠀⠀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠠⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢈⠢⢀⣀⠠⢊⢜⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢀⢀⠈⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡒⠒⠒⡆⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠠⠄⠀⠑⠤⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⢄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠓⢀⡘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢄⠰⠀⢠⡄⠀⠀⠈⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⢀⡀⠄⡀⠀⠄⠠⠀⣀⡀⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢌⢂⡔⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠀⠀⠀⠀⠀⠙⠀⠀⠀⠰⠀⢸⠀⠀⠀⠀⠀⠀⠜⢄⠒⢊⠬⢀⡀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠀⠉⠀⠀⠀⠘⠉⡠⡠⠁⢸⠀⠀⠀⠀⢀⠊⠀⠀⡨⠁⠀⠀⠀⠉⠂⢀⠈⠢⡀
⠀⠀⠀⠀⢀⠤⠤⠤⢄⠀⠑⠡⡠⠄⣀⢢⣔⠀⠄⡢⠀⠀⠀⢸⠀⠀⠀⢰⣅⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠐
⠀⠀⠀⠀⠈⠂⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠹⠊
⠀⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢉⡀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠌⠐⠒⠓⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠓⡛⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀
⠀⠀⠀⡈⠎⠀⠀⠃⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠤⡄⠓⠄⡀⠀⡄⠊⠀⠀⠈⠐⢤⠐⠀⡠⠁
⠀⠀⠘⠄⠀⠉⠀⠀⠀⠈⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠢⢄⠈⠉⠀⠀⠀⠀⠄⠀⠀⡄⠊
⠀⡐⠈⢀⠀⠀⠀⠈⠆⠀⠀⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⠠⠀⠄⠀⠀⠁⠀⠀⠈⠐⠁⠀⠀⡇
⡀⡌⠀⠈⠁⠀⠀⠀⠀⠂⠀⠈⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⣷⣾⡿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⠒⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⢦⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠒⠒⠲⣄
⠀⠀⠀⠀⠈⡦⠀⢀⡤⠼⣗⠉⠁⠀⠀⠔⠀⠀⣠⠂⢀⡈⣥⣤⠇⠀⠀⣶⡄⠀⠈⡇⢀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠀⣠⠐⠈⠉⠉⠀⠈⠙⠫⡗⠁⠀⠀⠀⣸⠟⠀⠀⢀⣇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⢤⠀⢥⣀⠀⠀⠀⠋⠀⠀⠀⠙⡿⠿⠶⢿⡏⠾⡀⢁⣀
⠀⠀⠀⠀⠀⡄⠰⣄⡤⠛⠁⠀⠀⠀⡤⠒⠀⠀⢆⡀⠀⠀⠀⢱⠀⠾⠀⠀⠀⠀⣠⡧⠄⠼⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⣄⢸⠁⠀⣀⡀⠀⡇⠹⣄⡀⠀⠀⠀⠉⠣⠀⣰⣤⣀⣠⣤⣶⣿⠋⠀⠀⠀⠀⡀⢠⢰⣐⠀⡊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠉⢦⠀⠀⠉⠳⣇⠀⢹⡀⣀⡄⠲⠄⠁⢰⣿⣏⠙⠛⠉⣿⠉⢰⠢⣇⣟⣄⠈⡇⠸⠥⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠆⠀⠉⠁⠀⠀⠀⠀⣼⠀⠀⠑⠒⡖⠘⠀⡘⠀⠘⠁⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⠁⠈⠳⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣤⠴⠃⠀⠀⣠⠜⠂⡀⠀⠣⠀⠀⣤⣄⡀⠀⣀⠴⢳⡀
⠀⠀⠀⠀⡰⠊⠀⡞⠉⡗⠀⠀⠉⣟⠓⠤⠤⡤⠔⠋⠀⠀⠀⠀⢰⠇⠀⠀⠀⡘⠀⡀⠀⢸⠃⠚⠛⠀⣠⠜⠁
⠀⠀⠀⠀⠧⠀⠀⠁⠀⠇⢀⠀⠴⠯⠤⠚⠋⠁⠀⠀⠀⠀⠀⠀⠈⠓⠢⣄⠀⠁⢈⠔⠂⠀⠢⠤⠄⠊
⠀⠀⠀⠀⠀⠹⢄⣀⣀⣀⢉⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⢙⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠐⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠀⠖⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⣀⠔⠀⠠⢀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⠂⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⢀⡀⠀⣀⣀⡀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠉⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠈⠰⠀⠀⠖⠐⠆⠀⠠⠊⠰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠲⡶⡿⢻⣾⠖⠶⢚⣿⡷⠲⠖⢺⣿⠖⢲⣚⣿⡗⣶⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠤⣿⠀⠀⠠⢜⡇⠀⡀⠤⣿⠀⢣⠠⣤⡄⠤⠠⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠒⣿⠀⢰⠒⢺⡇⠀⡃⠒⣿⠀⢸⠐⢻⡇⣀⢀⠀⠀⡄⠀⠀⠀⠀⡠⠀⠀⠀⠄⡀⠀⡠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣉⣿⠀⢸⢈⣹⡇⠀⡇⣉⣿⠀⢸⢈⣹⡇⣾⠀⠒⢀⠀⠐⠀⢠⠂⠀⠀⢀⠀⠀⠈⢊⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠤⢾⠀⢸⠠⠼⡇⠀⡇⠤⢿⠀⢸⠠⠼⡇⣿⠀⠀⠀⠁⠀⠀⡄⠀⠀⠰⠀⡨⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠒⠋⠀⠀⡑⠊⠁⠁⠉⠀⠓⠀⠄⢑⣊⣀⣿⠀⠠⠀⠐⠂⠀⠁⠐⢠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⡇⣒⣚⣛⣒⣒⠀⠠⣴⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠕⠃⠐⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣛⣓⣒⣚⠀⠘⣿⡷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣒⣶⣶⣾⣀⠈⠍⠛⠀⡇⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣒⣒⣒⣒⡿⠭⢿⣗⣲⡇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⣒⣒⣒⣺⠿⠭⢿⣒⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠖⠒⠒⠒⠷⠮⠭⣿⣒⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠴⠶⠤⠤⢤⣄⡀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠂


⣿⠿⠿⡿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣻⣻⣿⣟⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠤⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠨⠄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢁⠨⠤⠤⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢤⡔⠊⠁⠀⠀⠀⠀⠉⠑⠂⢄⠀⠀⠀⠀⠁⢂⠸⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠊⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠⠀⢳⣦⠀⠀⠀⠀⠀⣅⢣⠀⠀⠀⠀⠀⡖⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠀⠀⠀⢰⣫⡿⠛⠛⡆⠀⠀⠀⠈⣾⡇⠀⠀⠀⠀⡏⣺⣀⡀⠀⠠⠄⠆⠀⠀⠉⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⢀⠠⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡯⣗⠀⠀⠀⠀⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢠
⠀⠀⠀⠀⠀⠀⢀⠐⠩⣀⡀⠀⠤⡄⠀⠀⠈⠻⠿⠿⠟⠛⡡⠄⠒⠀⠀⠀⠀⢠⢳⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣤⣶⡥⠞⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⠀⠁⢃⡠⠊
⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⣵⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⠛⡏⠉⠉⠉⠀⠘⠀⡠⠋
⠀⠀⠀⠙⢄⡀⠀⠀⠃⠨⠀⠀⢈⣰⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡀⠅⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠈⠢⣀⠠⠐⠈⠉⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⣿⡦⠀⠀⠀⠀⠀⠀⢀⡀⠘⡂⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⣻⣯⠀⠀⠀⠀⠀⢀⣐⡆⣰⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠂⠀⠀⠀⠀⠀⠀⠙⠻⠟⢓⣉⣀⣀⡀⡠⡤⠴⠄⣄⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣒⣒⣒⣒⣢⡯⠽⠻⢛⢚⠛⠔⠖⠒⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⢃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠂⠁⠀⠀⠀⠐⠒⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠥⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠂⡠⠁
⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀⠀⡀⠄⠠⡀⢀⠀⠀⢄⠠⠀⢀⠠⢀⠀⡀⢈⠱⢶⠁⡀⠀⠀⠀⠀⠀⠀⡀⠔⡀⠀⠠⢀⠀⡀⢀⠄⡀⢄⠠⠀⢄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠈⠀⠁⠈⠀⠈⠀⠁⠊⠐⠀⠀⠁⠂⠁⠈⠀⠈⠐⠈⠐⠁⠈⠀⠁⠈⠀⠈⠀⠁⠈⠀⠁

⠀⠀⢀⣀⡀⠀⠀⣀⣠⡄⠀⠀⠀⠀⢀⠀⠴⠺⠆⠀⠀⠀⠀⡤⠂⠀⠰⢤⠀⠀⠀⠀⠀⠠⠄⠀⠰⠲
⠀⠀⠈⠩⠥⠄⠌⣭⠁⡀⠀⠀⠈⠉⢭⠀⠀⠤⠄⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠐⠛⠻⠄
⠀⠀⠀⢀⡀⠀⠰⠶⠤⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠉⠁⢀⠀⠀⠠⠀⠀⠀⠀⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⢐⠲⠀⠠⢚⡀⠀⠀⠀⠀⠀⠤⠀⠠
⠀⠀⠘⢔⢂⡸⢄⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠋⠀⠀⠉⠈⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⢾⡧⠀⠀⠀⠀⠀⣉⣀⣨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣏
⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⣛⡒⠐⣚⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⣦⡄⠀⣤⣤
⠀⠀⠀⡀⠶⠀⠶⢀⡄⠀⠀⠀⠀⠀⠙⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣿⢰⣶⡆⣻⠁⠀⠀⠀⠀⠀⢆⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠈⠻⠈⠟⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠘⠿⠿⠛

⠀⠀⠀⠀⠤⠀⠋⠁⠀⠀⠀⠀⢀⣤⣄⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡈⠁⠈⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒
⠀⠀⠀⠀⣀⠀⠐⠄⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠞
⠀⠀⠀⠀⠛⠓⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠚⠋⠉⠁
⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣠⣾⠟⢋⡞⠁⠀⢹⣿⣿⣿⣿⣯⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢫⣿⣿⣿⣿⣿⠁⠀⠙⣎⠛⢿⣦⡀
⠈⢛⣿⡿⠃⠀⢸⡁⠀⠀⣠⣤⡍⠙⢿⣿⡇⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⣿⣿⠟⠉⣥⣤⡀⠀⠀⣹⠀⠀⠻⣿⣟⠋
⠀⣸⣿⠁⠀⠀⢸⠇⠀⠀⣿⣿⣿⠀⠀⢻⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠃⠀⢸⣿⣿⡇⠀⠀⢿⠀⠀⠀⢹⣿⡀
⠀⠈⠻⡀⠀⠀⠸⠀⠀⠀⠈⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠋⠀⠀⠀⠸⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⠈⠢⠄⣀⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢀⡀⠤⠊⢀⣀
⣤⣤⣤⣤⣤⣤⣬⣬⣽⣷⣶⣶⣶⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣴⣶⣶⣶⣿⣭⣬⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠷⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣤⣦⣤⣉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣤⣄⢀⣴⣮⢷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⡿⠛⢻⡿⠋⢳⡈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣿⠟⣃⢾⡿⠃⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⢏⠀⣠⠾⡁⣠⣾⡇⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠛⠃⠀⠛⠃⠀⢀⣾⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣤⣿⣿⣤⣾⣿⡿⢁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⣤⣄⣀⣤⡴⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⣈⠛⠿⠿⠟⢋⣤⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠁⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣀⠀⣤⡀
⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡰⣖⣿⣿⣿⣿⣿⡿⠁
⠀⢠⣄⣼⣿⣹⣷⣦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣶⣄⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣶⡾⠃⠘⠿⠟⢹⣿⣿⠁
⠀⠘⢻⣿⣟⠛⠛⠂⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⠀⠀⠀⢾⡟⠋⠉⢿⣀⣤⣶⣷⣟⡛⠛⠛⠻⠿⢿⣿⣿⣷⡤⣺⣷⣿⣇
⠀⠀⠀⣿⡟⠀⢀⣀⣸⣿⡿⠿⠖⣲⣤⣤⡀⢻⠋⠀⣹⠋⠀⠀⠀⠘⠃⠀⣰⡿⠟⢻⠎⣉⢻⣿⠦⠀⠀⠀⠀⠙⠟⠿⣿⣿⣭⠻⢿⠿
⠀⠀⠳⠟⠛⣳⣿⣿⠏⠃⠀⠀⣾⣿⠻⠟⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⢋⢀⣤⠀⠀⠈⢱⣀⡀⠀⠀⠀⢀⡀⣸⠋⢀⣤⡄⠀⠀⣠⣤⠾⠛⠛⠛⢲⣄⣀⠀⠀⠀⠀⢻⣿⣧⡀
⠀⠀⠀⢀⡼⠁⠀⠀⣀⣤⣀⣾⣄⣸⣿⣀⠀⠀⠸⡟⠀⠀⠀⣴⡿⢱⡇⢰⢿⡏⢀⣴⣾⡿⠁⢀⠀⢠⣄⢈⡉⢻⡄⠀⠀⠀⢺⣿⡝⠷
⠀⠀⠀⣼⣧⠀⠀⢸⠿⡿⠏⠙⣁⠈⠀⠈⢢⠀⠀⢻⣆⠀⢀⣿⣷⣿⢀⠂⣾⡁⢨⣿⣿⣥⠀⣼⣧⣾⣿⣴⢏⣤⣷⠀⠀⠀⢸⡟⠁
⠀⠀⠙⠛⡏⠁⠀⠅⢸⣟⠰⠟⠻⣧⠰⢿⠺⡆⠸⡿⣿⣶⣿⢿⣿⣯⢿⡆⡴⡟⠸⣿⢷⣦⡷⣿⡇⡠⣾⡟⢼⣿⣿⠀⠀⢠⣾⣷
⠀⠀⠀⠀⣿⣶⡄⠀⢸⣿⢷⣶⣾⣏⠀⠀⢰⣷⣤⠀⢸⣿⣿⣿⡾⠁⣸⣿⣷⣖⠀⢻⡇⣤⡄⠸⣿⣷⡈⣁⣻⣿⠃⠀⣠⣿⣿⡇
⠀⠀⠀⠈⠙⣧⣀⢀⡀⣡⣿⣿⣿⣿⣷⣠⣿⠃⠀⠀⢈⣿⣯⣷⣹⣿⣿⣿⣿⢯⣷⡄⢷⣏⣠⣴⣾⣿⣍⠉⠁⣩⣧⣠⣿⣿⡟⠇
⠀⠀⠀⠠⣴⣾⢿⡿⣤⣼⣯⣶⣿⢱⣨⣿⣿⣧⡄⣾⣿⣿⣿⣴⣿⣿⣿⣿⣿⡊⠉⠁⠈⣿⣿⣿⣿⡿⣿⣷⣦⣽⠿⢻⡁⣿⣿⣤⡀
⠀⠀⢀⣰⣿⡟⠈⠃⠈⢋⣿⣿⣿⣱⣿⣿⡿⣿⣷⢉⠙⢿⣿⣿⣿⣿⡿⠆⢸⣿⣆⠀⠀⢿⣿⣿⡌⢹⣿⣿⣿⣷⡄⠀⠀⠙⠻⠟⠛⠃
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⣿⡿⠋⣿⣾⣿⣧⣄⡈⠀⣠⣟⣽⣿⣠⣤⡴⣶⣿⣿⣿⣿⣄⡈⠻⣿⣷⣿⣿⣿⣯⣿⣁⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅⠀⠀⠀⠉⠙⠻⡽⠿⠚⠛⢹⠩⣭⣿⣿⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠻⠛⠛⠛⠛⠉⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣟⢛⣉⣹⣿⡟⠛⠻⠿⠿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣞⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠻⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡤⢤⣄⣀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⣀⣉⠙⠳⢾⠟⠁⠀⠀⠰⠯⠏⠀⠀⢀⣤⠶⠶⠛⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠉⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣿⣧⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠴⠶⠶⣶⠚⢿⣿⣿⡇
⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⣿⠙⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠋⢿⠃
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠐⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠦⢀⣀⣀⣀⣀⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠆⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠠⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠠⢀⠠⠠⡤⠠⣀⠄⠀⠈⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠑⠀⠇⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠈⠐⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢶⡀⡇⣱⢆⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⢱⡶⠁⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠬⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⢄⠀⡇⢀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠀⠀⢈⠀⠀⠀⠀⠀⡁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠅⡅⠁⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠙⣄⢃⡜⠑⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠰⠀⠘⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠘⡟⠀⠀⡀⠠⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⠘⠀⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠈⠢⡀⡇⡠⠊⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠠⠀⠰⠆⠀⠀⠈⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⣤⠂⡇⢢⡀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠒⠒⠊⢨⠀⠂⠂⠂⡅⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⢦⣥⠋⠈⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠄⠀⠀⠀⠂⣀⣰⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠀⠂⣀⡀⠈⡇⠀⡀⠄⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⢀⠀⠠⠂⢀⠜⠁⠀⠀⠀⠀⠀⠉⠀⠒⠀⠠⢤⡠⠤⠀⠀⠀⠉⠈⠁⠀⠀⠀⠈⠱⡎⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⠍⠹⠀⣚⣉⣉⣉⣛⣓⠀⠀⠤⣦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠐⣒⣒⣒⠒⠲⠀⠀⢛⣿⡣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠬⠭⠬⠭⢭⣭⠀⠀⢛⠛⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣘⣭⣭⣭⣉⣙⣒⡒⠦⢬⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢐⣒⣒⣒⣒⡒⠤⠭⣭⣑⣒⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠭⠭⠤⠤⠬⠭⢭⣉⣒⡲⠭⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠉⠉⠉⠙⠛⠒⠲⠦⠭⣍⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠤⠤⠶⠶⠦⠤⢤⣄⣀⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠄⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠿⠿⠷⠿⠭⠥⠤⠤⠤⠤⠤⠤⠤⠤⠬⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋⠁⠈⠙⠿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⠾⠟⠋⠉⠀⠀⣿⡿⢿⣿⠀⠀⠉⠙⠻⠷⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠐⣿⣆⣰⣿⠂⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣰⣶⠾⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⠷⣶⣆⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⠶⠿⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⠿⠶⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠿⢶⣶⣦⣤⣤⣤⣤⣴⣶⡶⠿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠀⠀⠀⣤⡀⠀⠉⠉⠀⠀⣤⡀⠀⠀⣿⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣸⣿⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⣿⣇⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⣄⠀⠀⠀⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠈⠻⣶⣤⣤⣴⡿⠃⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠉⠉⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⣶⣤⣤⣤⣤⣶⢾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⠿⠿⣿⡇⠀⠈⠉⠉⠁⠀⢸⣿⠿⠿⠿⠿⠿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠿⠿⠿⠿⠿⠿⣿⣄⡀⠀⠀⢀⣠⣿⠿⠿⠿⠿⠿⠿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣤⣤⣤⣴⣦⣄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⠉⠉⠉⠉⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠉⠉⠀⠀⠀⣀⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⡿⠟⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⠿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢠⣤⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⢿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣶⣶⣶⣶⣶⣶⠶⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⡋⠁⠁⠠⠈⠉⠭⠑⠒⡛⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⣸⡃⠀⠀⠀⣌⢀⡤⣌⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠰⠀⠀⠸⣇⢹⠀⠀⠀⠈⠁⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣧⠼⠀⠀⠀⢜⣩⡤⠤⠔⠊
⠀⠀⠀⣠⠴⣒⠲⢄⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠠⠴⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⣬⡅
⠀⠀⠀⣇⣸⠉⣹⢀⠆⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⢀⣞⡵⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠁⣐⡒⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⢡⡦⠀⠀⠀⢀⣀⠤⠔⠒⠚⡳⡦⠀⣬⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠇⢇⣀⠬⠀⠸⢸⠧⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡄⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠈⢸⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢀⣏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡝⠀⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣇⢠⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢠⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢹⡌⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢸⠀⠀⠀⠻⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⠉⠉⠀⢘⣗⡿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠉⠉⠙⠓⠲⠤⢄⣀⣀⣧⠀⠀⠀⠀⢧⠠⢤⣸⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠘⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢳⡖⠒⠒⠒⠲⠶⠶⠤⠤⠤⠤⢿⡄⠀⠀⠀⢸⡥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⠂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠂⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠛⠉⠀⠀⠀⠀⠀⠀⣀⢰⡞⣩⠗⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠠⣤⡐⠢⡀⠀⠀⢠⣿⠉⠁⠉⠀⠐⣸⡇
⠀⠀⠀⠀⠀⠀⡔⣨⣽⢭⣠⣤⣤⣤⣤⡤⠞⠃⠐⠉⠉⠛⠉⡌⠀⠂⠁⣀⡿⢋⣠
⠀⠀⠀⠀⠀⠀⠀⢡⣀⠊⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠈⠉⢄⣼⣟
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣝⠇⠀⠸⠟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠄⠘⣷⣤⣼⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣀⣠⣀⠀⠀⠀⠘⣿⠟⢠⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡉⠀⣀⠀⣀⡄⠀⠀⢸⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡤⣤⣀⣠⣬⣿⡿⠁⠐⠛⠐⠈⠀⠀⠀⠀⣷⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢹⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠀⢠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠟⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⡀⠀⠀⠀⡀⢀⡴⠟⢿⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡠⠄⠀⠺⠀⠀⠀⠀⡡⠊⠀⠀⢀⡷⠋⠀⠾⣿⣶⣢⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣟⣧⠘⣗⠀⣀⣀⣀⣠⡤⠊⠀⠀⠀⠔⠋⠀⡀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠦⠄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠒⠱⠞⢫⣴⣶⠂⠀⠀⡈⠂⡈⠉⠉⠉⢁⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠝⠛
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣿⣿⢟⡅⠀⠀⠈⠀⠀⠀⠀⠀⣀⣀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⢼⢀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠂⠀⠀⠀⠀⠀⠀⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠉⠐⢢⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⢔⠑⢆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠠⢄
⠀⠀⠐⠀⡀⢀⠀⠄⠀⠀⠀⠈⠂⢩⠒⠉⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠂⢄⠀⡇
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠦⡐⣃⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠤⢚⡤⠞⣠⠇
⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⢨⣀⣨⠇⠀⣀⠑⠀⠀⠀⠀⠀⠀⠈⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠠⠄⠂⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠄⠀⠀⠠⠁⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⡐⢢⠈⠄⠀⠀⠀⠀⠐⢂⠀⠀⢀⠔⡀⠌⠂⠀⠀⠀⠈⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⢆⢁⠊⠀⠀⠀⠀⠀⠀⠈⠓⡀⠈⠢⠥⠀⠀⣠⡤⠄⠠⠄⠄⠁⠀⠀⠀⢀⠀⠀⡂⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⠊⠀⠘⠍⠧⡄⠀⠀⠀⠀⢀⡠⢠⡀⡓⠊⠀⠀⠀⠀⠀⠬⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠠⠀⠤⠀⠀⠀⠀⠀⠀⡐⠀⠀⢆⠀⠀⠀⠀⠈⠀⠱⣠⠉⢒⠁⠀⠀⠀⠀⠁⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠂⢀⠄⠀⠀⠀⠀⠊⠐⠲⠚⠃⠀⠀⠀⠀⠀⠀⡨⠃⠉⠀⠀⠀⠀⠀⠐⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⠈⠀⠀⠀⠀⠀⠀⠐⢀⠀⠠⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⡠⠀⠂⠄⠀⠀⠀⠀⠀⠀⢐⢀⡆⠀⠀⠀⠄⠀⠀⠰⠐⡤⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢄⡀
⠀⠀⠀⠀⠀⢀⡐⢑⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢯⣰⠖⡆⢹
⠀⠀⠀⠀⠀⠀⠨⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄⠒⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢉⠕⠋
⠀⠀⠀⠀⠀⠀⠠⡉⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠖⠉⠡⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⢢⢈⠀⠀⠀⠀⠀⠀⣀⡂⠒⠂⢤⡠⠐⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢀⠊⠀⠀⠀⢠⠂⠁⠀⠀⠙⠄⢸⠇⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡆⠀⠀⠀⢧⢀⣀⣀⡀⠀⠈⣩⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠐⠀⠻⠟⠩⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣵⡝⠻⠿⠿⠟⢻⡇⣮⠻⠿⠿⠟⢛⣹⣷⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠿⠀⠀⠀⢾⣜⡃⣣⡿⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⡀⠀⢠⢠⣤⣿⣿⣯⣤⡀⡄⠀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣷⡄⠀⠀⠰⣮⣭⣥⠏⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣦⣠⡀⠉⠉⠁⢰⣦⣾⠟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠻⢿⣶⣶⠶⠟⠑⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣾⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠙⢆⠀⢀⣿⣿⣿⣿⡷⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠓⠉⠹⣿⣿⡟⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠋⠉⠙⠛⠃⠀⠀⠀⢰⣿⣿⡇⠀⠀⠿⠿⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠐⠀⠀⠂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠒⠢⠤⠤⢤⣀⠲⠄⠀⠀⠀⢱⣟⠛⠉⠑⠂⠠⢠⡇
⠀⠀⠀⠀⠀⠀⢠⢄⣲⣂⠥⣠⣤⣤⣀⣀⡈⠵⠊⠕⠒⠺⠿⠫⡀⠀⠀⠂⠀⣼⠗⡀⡀
⠀⠀⠀⠀⠀⠀⠁⠛⡉⠰⠛⠉⠙⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠘⠋⡁⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡳⠆⠀⢰⣿⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠦⠄⠙⡀⠻⣦⣀⡀⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠐⠀⠀⠀⠹⣿⣿⢃⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠑⠀⡀⠀⠀⠹⡅⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⣻⣿⡿⠀⢨⡿⠠⠚⠁⠀⠀⠀⣧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⠉⠗⡖⠺⡿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠉⠀⣠⠂⠀⠀⢀⠄⣠⠎⠉⠙⢇⣸⠠⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⣷⡀⠀⠉⠁⠀⠀⠀⣠⠞⠁⠀⢀⣰⠞⠁⠀⠚⠛⠓⠺⠍⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠶⢟⠟⠵⠂⠘⠇⠰⠤⣤⣶⣶⡏⣀⠀⠀⠀⠊⠁⠀⡐⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠀⢀⣠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠋⢡⣾⣿⣯⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⡝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣵⠂⠀⠀⠀⠀⠀⠠⣀⣀⣤⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠵⢇⡄⠀⠀⠀⠀⠀⠀⠀⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⣿⣏⢹⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠉⢰⠋⠙⡌⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⣶⣶⣶⣷⣾⣧⣤⣤⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢉⣍⠛⠛⣻⡛⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠉⠀⠀⠉⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠳⣖⣲⠖⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⢦⣄⣀⣀⣀⣀⣠⡴⢥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠢⣀⢯⢧⢀⠜⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢀⣀⣠⠈⠸⠀⢁⣀⣀⡀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠔⡆⡇⠀⠀⠀⢸⠀⢸⠀⠀⡇⡴⢄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⢇⡇⠙⠒⠚⠀⢸⠀⠈⠒⠒⠃⡇⡼⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⡸⣇⠀⠀⠀⠀⣸⡀⠀⠀⠀⢀⣿⡁⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⡏⠉⠉⠉⠉⠣⠞⠉⠉⠉⠉⡇⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⣷⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠢⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⠿⢿⣿⠿⣿⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⢙⡁⣤⣶⣾⣾⣷⣶⣬⡄⠀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠈⢉⡽⠟⠛⠉⠁

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠈⠂⣌⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠈⠑⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠁⠂⢄⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠄⠤⢀⠀⠀⠑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠂⠠⠒⠢⠀⢂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢄⠐⠤⠔⢀⠌⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠑⠒⠒⠁⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠈⠂⠄⣀⠀⢀⡀⠄⢎⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠑⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣛⢦⣄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⣈⠉⠁⠁⠉⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⡔
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡓⠀⠂⠀⠀⠀⠀⠀⠀⠂⠂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠣⠃⠀⠁⠍⠌⠩⡙⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⡐⠀⠀⡦⠀⢂⢄⠄⡂⡡⢂⢆⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠐⠁⠀⠐⠋
⠀⠀⠀⠀⠀⡰⡠⠙⠋⠉⠈⠌⠂⠁⠒⢦⡁⠀⠁⠀⠀⠀⡠⠁⠀⠀⡠⠤⢄⠀⠀⠙⠄
⠀⠀⠀⠀⠰⠠⠁⠀⠀⠺⠂⠈⠌⢆⠀⠀⠈⢢⡀⠈⠢⣐⠀⠀⠀⠀⢄⠀⠀⠃⠀⠀⠈⡄
⠀⠀⠀⠠⠡⠁⠀⠴⠀⠀⠀⠀⠘⠌⠄⠀⠀⠀⠈⢦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⢡⠃⠀⠀⠀⠀⠀⡀⠀⠀⠘⡘⡆⠀⠀⠀⠀⠈⢾⠀⠀⣀⣀⢠⢄⣀⠤⠤⠔⠒⠛⠉⠘⠃
⠀⢠⢁⠃⠀⡀⠀⠀⠀⠢⡠⠀⠀⠀⠐⡘⣤⠐⠢⠄⠐⠛⠋⠉⠁⠈
⠀⢃⠃⠀⠀⠊⠀⠀⠀⠀⠀⠀⢈⣈⠀⠐⡐⡀
⣆⣘⣂⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⢿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣧⠀⠀⣸⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⢈⣉⣛⣛⣛⣛⣛⣋⣉⡀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣿⣼⣏⣹⣿⣿⡏⢹⣏⣹⣿⣿⣧⣿⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣿⠹⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⠏⣿⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠲⠶⠶⠖⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣦⡀⢀⣴⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠻⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⢦⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⢀⣾⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣄⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⢷⡀
⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⠉⠉⠁⠈⠉⠉⢩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢦⡴⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠛⣛⣛⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣥⣬⣭⣭⣭⣭⣭⣭⣭⣭⣍⣉⣛⣛⠛⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣏⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⠟⣿⠟⠻⢿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⢰⡟⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⡆
⠀⠀⠀⠀⠀⠘⢷⣀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⢦⣀⡀⠀⠀⢀⣀⡴⠷⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢳⣄⡀
⠀⠀⠀⠀⠀⠀⡀⣠⠚⠉⠀⠉⠉⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠉⠉⠀⠈⠓⣄⣀
⠀⠀⢀⣠⠤⠾⠧⣌⠓⢤⠤⠤⣀⡄⠀⠈⢍⡛⠳⠶⢤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠶⠖⢛⡹⠁⠀⢠⣀⠤⠤⡤⠚⣡⠼⠷⠤⣄⡀
⠀⣼⠁⢀⣠⣄⠀⠀⢱⠎⠀⠀⢀⠇⠀⠀⠀⠉⠲⢤⣀⣀⡴⢿⣿⣿⡿⢤⣀⣀⡤⠖⠉⣀⡀⣀⠸⡄⠀⠀⠱⡎⠀⠀⢠⣄⡀⠈⣷
⠀⠈⠻⣉⠜⢩⡀⠐⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠈⠉⠀⠀⣿⣿⠀⠀⠉⠁⠰⠆⠀⠀⠀⠁⠀⡇⠀⠀⠀⢘⠆⢀⡈⠳⣍⠟⠁
⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢣⠀⠀⠀⠈⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠢⠦⠘⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⢤⣄⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣶⣬⣍⠉⠉⠛⠛⠛⠛⠛⠐⠃⠛⠛⠛⠛⠛⠉⠉⣩⣥⣶⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠉⠙⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⠉⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢹⣿⡇⣠⠞⠛⠛⠛⠛⠛⠛⠳⣄⠰⣿⡟⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠄⠀⠂⠀⠁⣸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⠈⠀⠐⠀⠠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣤⠞⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣴⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠒⠂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠷⠶⢄⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⣶⣿⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠉⠐⠀⠠⠧
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠉⢉⠙⠛⠛⠲⢶⣦⣤⣄⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⠟⣿⣿⣿⣿⣿⣿⡟⢉⠤⢦⠘⣿⣿⣿⡟⠀⠀⠀⠀⠠⠞⠛⢿⣶⣤⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⡔⢠⡄⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢤⣠⡄⠈⠙⡿⠿⠿⠿⠿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢿⡿⠂⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠷⣶⣄⣿⣿⣿⣿⣿⣿⡆⠈⠃⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠟⠲⣶⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣤⣤⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡛⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠋⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡋⣉⣀⡉⠛⠛⢉⣀⣈⢙⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⣉⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣤⣾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣦⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡄⠻⣿⣿⣿⡇⢠⣶⣶⣶⣶⡄⢰⣿⣿⣿⠏⢠⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⡀⣿⣿⣿⣿⢀⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⣀⣤⣶⣿⣦⣄⡙⠻⢿⣿⣿⣿⣿⣿⣆⠘⣿⣿⡇⠈⣉⣉⡁⢸⣿⣿⠃⣰⣿⣿⣿⣿⣿⡿⠟⢉⣠⣴⣶⣦⣄⡀
⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⡄⣹⣿⣿⣿⣿⣿⣆⠘⣿⡇⢸⣿⣿⡇⢸⣿⠃⣰⣿⣿⣿⣿⣿⣏⠠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣧⠘⠃⣾⣿⣿⣿⠘⠃⣼⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⣿⣿⣿⣿⣿⣿⣿⣧⡀⢿⣿⣿⡿⢀⣼⣿⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣷⡈⢻⡟⢁⣾⣿⣿⣿⣿⣿⣿⠟⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣷⡀⢀⣾⣿⣿⣿⣿⣿⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠙⢿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⣿⣿⣿⡇⢸⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠻⣿⡇⢸⣿⠟⢁⣾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠃⠘⢁⣴⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢀⣀⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠁⠉⠛⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣻⣿⡂⠀⠀⠀⡰⠋⢁⣠⣤⣄⠈⠻⡄
⠀⠀⠀⠀⡜⠁⠀⡤⠶⠶⣄⠀⠀⢸⠀⠀⠀⠀⠀⢸⢿⡟⠁⠀⢹⣩⣹⠃⠀⠀⠀⠁⠠⡃⠀⠀⠈⠇⠀⠘
⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⣸⠀⠀⡸⠀⠀⠀⠀⠀⠸⣾⣷⢀⣴⣿⡿⠋⠀⠀⠀⠀⢦⣤⠵⠄⢀⣀⠞⠀⢠
⠀⠀⠀⠀⠙⠚⠁⣤⠖⠚⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢡⡻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢉⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢰⠚⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⠇⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⢸⠉⠁⠀⣀⣤⣄⡀
⠀⠰⠊⠉⠑⠄⠀⡇⢸⡀⠀⠀⠀⠀⠘⠉⠙⢧⠀⠀⠀⢹⣿⠦⠀⠀⠀⠀⠯⠈⠙⠄⠀⠸⢬⠆⠀⠸⠅⠀⠀⢙
⠀⠀⠀⠀⣠⠆⠀⠘⣒⣃⠀⠀⠀⠀⢀⡤⢤⡾⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⣀⠄⠀⠰⣋⠀⠀⠀⠀⢀⡤⠎
⠀⠀⠐⠉⠁⠀⠀⠀⣁⣨⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢈⣛⣁⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠘⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠰⠀⠀⠀⣠⠶⠛⠉⠉⠉⠛⠗⢦⡀⠀⠁⠀⣀⣤⣶⣶⣶⣶⣦⣄⠉
⠀⠀⠀⠀⢠⠞⠋⠉⠉⠉⠛⢷⠀⠀⠀⠀⠀⢸⠁⢠⣾⠛⠲⣤⡀⠀⠀⠁⠀⠀⣼⡿⡟⠛⠛⠛⢿⣏⠙⡆
⠀⠀⠀⠀⢫⣀⡴⠚⠛⢆⠀⢈⡁⠀⠀⠀⠀⠘⢤⣠⡜⠀⠀⠀⡇⠀⢀⡖⠀⠀⢼⣟⣿⠀⠀⠀⢸⣿⣴⠃
⠀⠀⣀⣄⡀⠀⠀⠀⢀⠎⣰⠟⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠉⠁⠀⠀⣴⣿⠟⠁⢀⣀⡀
⠀⢨⣀⠀⠙⠀⠀⠀⡞⢸⠁⠀⠀⠀⠉⠉⣻⠀⠀⠀⢀⠴⢋⠔⠊⠁⠀⠀⠊⠙⠢⠀⠀⠀⢸⡟⠁⠀⠀⡁⠀⠈⡆
⠀⠀⢀⡤⠖⠀⠀⠀⢻⡞⠀⠀⠀⠀⢸⠉⠉⠀⠀⠀⠘⢶⠃⠀⠀⠀⠀⠈⢁⠤⠎⠀⠀⠀⣘⡁⠀⠀⠀⠀⠊⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠸⣌⠇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⢶⣤⣤⣀⡀⠀⠀⢀⣠⣤⣴⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠀⣀⣀⠀⠈⠉⠛⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⢀⣴⠾⠛⠃⠈⠉⠙⠛⠷⠆⣰⠟⠁⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⣴⠟⠁⢀⣤⣴⠶⠶⢶⣤⡀⢰⡟⢀⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣾⠃⠀⣴⠟⠉⠀⣀⣀⡀⠈⠁⣾⡇⠈⢿⡄⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡏⠀⢰⡟⠋⠉⠻⣦⠀⢹⡇⠀⠈⣿⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣸⡇⠀⠀⠀⠙⠀⠈⢿⡄⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠃⠀⢸⡇⠀⢸⡇⢰⡆⠀⣿⠀⠀⠈⠻⣦⡀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡿⠁⠀⣼⡇⢸⡇⠀⠛⠀⠀⣿⠀⠈⣿⡷⣦⣤⣤⣤⣤⣴⠾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠾⠋⠀⠀⢠⡿⠀⠸⠇⠀⠀⠀⣼⠏⠀⠀⣿⡇⠀⠀⠉⠉⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢀⣴⠟⠁⠀⢀⣀⣠⣴⠾⠋⠀⣠⡆⠘⠛⣿⠟⠛⠛⠻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⢀⣴⠟⠁⠀⢀⣴⠟⠉⠉⠀⣀⣠⡾⠋⠀⣠⡄⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠟⠁⠐⠶⠾⠛⠋⠁⠀⣠⡾⠛⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣴⠟⠁⣀⣀⣀⣀⣠⣤⣤⠶⠟⠉⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢁⣀⡀⠉⠉⠉⠉⠉⠀⠀⣀⣠⣴⠾⠃⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠻⣦⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⠤⠤⠠⠄⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠠⠤⠤⠄⠤⠄⠤⠤⠤⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠀⠓⠀⠀⠰⠘⠃⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠃⠙⠀⠑⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠻⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣄⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣀⣀⣀⣈⠙⠻⢿⠟⠁⠀⠀⠾⣿⡿⠁⠀⣀⣤⠶⠶⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠉⠉⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣧⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⠶⠶⣾⠿⣿⣿⣿
⢿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⣿⠙⣿⣿⣿⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣿
⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣤⣤⣤⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⠀⠀⢀⣀⣤⣴⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢀⠀⠀⢀⡀⠤⠔⣢⡄⠀⠀⠀⣀⠤⠡⠡⢀⡄⠀⢀⢀⣀⣀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢄⡀⠊⣁⣀⡙⠀⠀⠁⣀⣤⣤⣄⠀⠀⠀⠀⠁⣀⣀⣀⣀⡀⠀⡈⢁⣀⣈⣁⠀⢠⠀⠈⠑⢢
⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠈⠁⠊⣙⣻⠍⠀⠀⠈⢀⠘⠞⠀⠁⠀⠀⠀⠈⠀⠐⠃⠁⠁⠀⡁⠃⠐⠛⠉⠀⢘⠀⠀⠒⢸
⠀⠀⠀⠀⠀⠄⠀⡇⠀⠂⠀⠆⠀⠀⠉⢀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠑⠄⠀⠁⠁⠀⠨⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠥⠄⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠄⠑⠆⠀⡀⠀⠀⠀⠀⢁⠀⠀⢀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⡠⢀⠄⠀⠀⢰⠀⠀⠀⠀⠀⡰⢀⠔
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈⠳⡄⠀⢺⣄⢤⡀⠀⢀⡶⠒⠀⠀⠀⠀⠐⠀⡐⠁⠀⢠⣄⡷⠀⠀⡀⠋⠀⠀⡰⢁⠊
⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡐⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡜⢀⠄⡂⢀⠀⠐⠀⠀⠄⡀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡙⠿⠕⡳⠋⢀⣠⣿⣟⠎⣠⡾⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠐⡧⡤⠤⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠀⠀⠈⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⣀⣀⢠⡿⣦⢀⡀⡀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠉⠁⠀⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠞⢄⠀⢀⠘⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣙⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠃⢀⣉⣉⣉⣉⠉⠉⠉⢉⣉⣉⣉⣁⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠈⠉⣭⠉⠁⠀⠀⠀⠈⢩⡍⠉⠁⠀⡀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢰⣸⠀⠀⠀⠻⠟⠀⢠⠋⢢⠀⠘⠿⠃⠀⠀⢣⣰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡇⠀⠀⠀⠀⠀⠀⡌⠀⠀⡄⠀⠀⠀⠀⠀⠀⡦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠑⠦⠖⠁⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⠀⠀⠀⠀⣀⠠⠒⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠖⠦⣾⠀⠉⠁⢹⠦⠖⠶⢠⣼⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠊⠉⡈⠀⠀⠸⡁⠸⡀⠀⠀⡸⠀⡽⠀⠀⣿⡉⢽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢉⡄⠀⠀⠀⠀⢀⣅⠀⠀⠀⠙⢆⢈⡷⡎⢀⠞⠀⠀⠀⣌⡉⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢰⠒⠂⠉⠁⠀⢀⠀⠀⠀⠀⠈⠋⠀⠀⠋⢀⣤⡀⢸⡘⠿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡆⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⢻⡏⡻⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠛⠛⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⢸⠤⠤⠠⢤⣤⠴⣤⡤⠤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣴⣦⣶⣷⣶⣿⣷⣦⣤⣦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠿⢿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠰⠆

⠀⠤⠂⠉⠉⠉⠐⠢⢄
⡌⠀⠀⠄⠒⠠⢀⠀⠀⠑⡀
⢇⠀⠸⠀⠀⠀⠀⢡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠌⠀⢀⠃
⠀⠀⠀⠀⡀⠊⠀⠀⡠⠊
⠀⠀⠀⠘⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡙⢄⠀⠀⡔⠄⠀⢀
⠀⠀⠀⢠⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠄⠁⠀⠀⠁⠀⠀⠀⠃⠈⠰⠁⢆⠰⠄
⠀⠀⠀⠈⢄⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠨⠀⠢⠀⠀⠀⠀⠀⠀⠀⡠⡶⢠⠄⠀⠤⢀⠀⠀⠀⢀⡀⠠⢤⡀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢰⡄⠂⠀⠂⠀⠀⠐⠂⠐⠠⠀⠀⠀⠀⡠⢤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣸⢠⣖⠒⠢⡀⠀⢠⣄⠠⢄⠀⠀⠐⠎⡠⠂⡆⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⢄⡀⡁⢻⠘⠥⠀⠔⠇⠀⠫⣉⣀⠜⠀⠀⠀⠀⠑⠀⠇⣠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⢀⠈⠙⢸⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠸⠀⠀⡀⠀⠀⠁⠄⠤⠀⠀⠀⠀⠀⠀⠀⢘⡁⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠁⠀⠀⢃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⢠⠁⠐⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠠⡀⣀⠀⠤⠀⠈⠀⠀⠀⠀⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⠐⠀⠤⠄⠤⢴⠀⠠⡀⠀⠀⠀⠀⠀⡠⠂⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⡰⠤⡒⠉⠀⠀⠀⢀⠄⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠐⡠⠚⢠⢀⡘⢆⠀⠠⠂⠁⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠂⠀⠀⠒⠀⠀⢀⠂⠑⢄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠊⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠚⠀⠢⡀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⡀⠀⠀⠇⠀⠀⠀⢈⠀⠈⢢⣤⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡘⠀⠀⠀⠀⠀⠂⠄⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠄⢤⠀⠀⢠⠀⠀⠀⠀⠀⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠀⠉⠉⠁⠒⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢁⠂⠀⠀⢀⣀⣀⡀⠀⠀⠀⢈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠁⣾⢉⡤⣬⡹⠀⠀⠀⠀⢀⡷⠤
⠀⠀⠀⡠⠐⠈⠨⠁⠉⠉⠀⠉⠈⠐⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠓⠉⠉⠉⠀⠀⠀⠀⠀⣠⣾⡗⠛⠂
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⣠⡤⣤⡤⠤⠒⣡⡞⠁
⡼⠀⠀⠀⠀⠀⠀⢠⠖⠒⢀⠒⠤⡀⠀⠀⠀⠀⠈⢦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣐⡓⠂⡤⣂⡴⢡⣴⡴⠶⠶⠦⠟⠉⠈⠀⢀⡀⠄⠐⠒⠒⠒⠠⠤⣀
⡇⠀⠀⠀⠀⠀⠀⠹⡖⠚⠶⣥⣄⠈⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡠⠂⠁⠀⠈⢁⢳⣴⣴⠯⠁⠀⠀⠀⠀⠀⢀⡔⢪⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢈⡿⠉⠀⠀⠙⡆⠀⠀⠀⠀⢠⣿⠄⠀⠀⠀⠈⠓⠤⢀⡀⠀⣠⢿⡯⡅⠁⠀⠀⠀⠀⠀⠀⠀⡈⠐⠤⣀⠠⠤⠆⢤⠀⠀⠀⠀⠨⠈⡇
⢀⠀⠢⡢⡤⠐⡀⠖⠉⡟⠀⠀⠀⠐⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠐⠄⣀⠀⢈⠙⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠑⢢⣖⣤⣾⣺⠶⠁⠀⠀⠀⠀⢠⡯⠉
⠈⠢⠀⠙⢢⡊⠁⣈⡼⠃⠈⠁⠀⠙⢄⠀⠀⠀⠀⠨⠻⣤⠀⠀⠀⠀⠀⠀⠑⠒⠼⠛⠂⠀⠀⠀⠀⠀⠀⣠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠇
⠀⠀⠑⢄⣽⡴⠞⠋⠐⠀⠀⠀⠀⢠⠀⠑⢄⠀⠀⠀⣡⢺⣇⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢌⠲⠀⠀⠀⢀⡤⠴⠖⠢⠤⠤⠤⠤⠰⠚⢉⣴⠏
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠉⢶⢋⡵⠛⠁⠀⠈⠑⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂⠁⠉⠢⣈⡒⢤⠞⢅⣴⣥⡶⢶⣶⣦⣤⣤⡶⠾⠯⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠟⣻⣼⠢⡀⣀⡀⠀⣀⠴⣿⡁⠀⠀⠀⢰⠯⡀⠀⡀⠀⠀⢠⣿⣼⣾⡫⠉⠉⠉⠀⠀⠀⠐⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡀⠈⠢⡤⢞⠝⣸⠇⠀⠀⠀⠀⠰⡀⠈⠑⠢⣡⡞⣵⣿⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣋⣼⣞⣓⠒⠀⠀⠀⠀⠀⠈⠑⠀⣀⣮⠼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠢⠀⠀⣠⣀⠀⠠⠖⠄⠀⠀⠀⠀⠀⠀⠠⠲⠄⠀⣀⣄⠀⠀⠔⠋⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣘⣓⣈⠀⠀⣄⡤⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⣁⣚⣃⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⠉⠳⢄⣀⠀⠀⠈⠫⣛⠦⠬⠌⠳⣄⠀⠀⠀⠀⣠⠖⠡⠥⠴⣻⠝⠁⠀⠀⣀⡠⠞⢉⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠣⠀⠀⠈⠉⠑⠒⠲⠮⠽⠿⣿⣒⡢⣷⡀⢀⣼⢕⣒⣻⠿⠯⠥⠖⠒⠚⠉⠁⠀⠀⠜⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣛⢧⡾⢛⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⠑⠢⠤⣤⣔⠉⠙⠀⠀⠀⡠⠚⠉⡉⠉⠉⠉⠉⠉⡉⠓⢄⠀⠀⠀⠋⠉⣢⣤⠤⠔⠊⠉⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠈⠀⠀⠺⣉⣀⠀⠀⠀⣈⡹⠆⠀⠁⠀⠀⠀⢨⠄⠀⠀⠀⠀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣉⡁⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠈⣉⣽⠀⠀⠸⣍⡉⠀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⢈⣉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠀⢀⣤⠊⠀⢻⡅⡄⠈⠓⠂⠀⠀⠀⠒⠛⢠⣪⡟⠀⠑⣤⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠊⠙⠶⡃⠀⠀⠰⡶⠾⣦⣄⣀⣠⣄⣀⣠⡴⡷⢶⠆⠀⠀⢘⠾⠋⠑⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠱⣄⠀⠈⠀⠀⠀⢶⡀⢰⣶⠀⠀⠀⠀⣶⡆⢀⡶⠀⠀⠀⠂⠀⣠⠞⠀⡶⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠊⠀⠀⠀⠑⠀⠀⠀⠀⢤⣀⡴⢋⡅⠀⠀⢸⣹⢧⡀⡤⠀⠀⠀⠀⠊⠀⠀⠀⠑⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⢀⠀⠀⢀⡔⠀⠀⠀⠚⠂⠠⢤⣜⠀⠀⠀⠀⢣⡤⠀⠐⠓⠀⠀⠀⠢⡀⠀⠀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⢯⢀⣀⠀⢠⣤⠀⠀⠸⣟⠋⡁⢈⠙⣻⠇⠀⠀⣤⡤⠀⢀⡀⡽⣄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡳⠊⠀⠲⠃⠈⠁⠀⠀⠀⠀⠐⣾⣯⠁⠈⢹⣷⠂⠀⠀⠀⠀⠀⠁⠘⠖⠀⠑⢞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡤⠄⠀⠀⣠⠀⢀⡠⠀⠀⠠⠟⠉⠀⠀⠉⠛⠆⠀⠐⢄⡀⠀⢄⠀⠀⠠⢤⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⣀⠆⠀⠀⠁⠀⡠⠄⠀⠀⠀⢫⣻⣤⣤⣟⡕⠀⠀⠀⢠⢄⠀⠈⠀⠀⠰⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⡠⠴⠀⣠⡙⠃⢀⡠⠚⠁⠀⠀⠀⠀⠈⠓⢤⡀⠘⢋⣄⠀⠤⢤⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠄⠀⢄⠘⠛⠿⣥⣤⠴⠒⣛⡒⢒⣛⠒⠦⣤⣬⠿⠛⠇⡠⢄⡠⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⢩⢠⠞⠁⠀⠱⢤⣀⠚⠻⣯⠉⠉⠉⠉⠉⠉⣭⠟⠓⣄⡤⠊⠀⠈⠳⡄⡍⠱⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠦⣄⣈⠒⠉⠳⠦⠈⠳⣄⠀⠀⣠⠞⠁⠴⠞⠉⠒⣀⣠⠴⠦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠈⢈⠂⠀⠘⠂⠀⠀⠈⠉⠉⠁⠀⠀⡐⠃⠀⠐⡁⠁⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠀⠀⠛⠒⠚⠁⠀⠄⠀⠀⠀⠀⠀⡀⠈⠓⠒⠚⠀⠀⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠒⠊⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣯⠥⠤⠤⠤⠬⣭⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⠀⠀⢀⠀⠀⠑⢮⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠢⣀⣀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢲⡜⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣭⣽⠆⢀⠞⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣇⡀⠉⠒⠤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠋⡏⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠒⡄⢠⠃⠀⠈⠀⠀⢀⠀⠀⠀⣀⡤⠚⠀⠀⢧⠀⢀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⢠⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠤⣀⣀⣀⣀⣨⡏⠁⠀⠁⠀⡰⣪⠻⠿⡒⠉
⠀⠀⠀⠀⠀⠀⣀⠷⠤⣀⠈⢱⢒⠆⠀⠀⠀⠀⠐⢤⢳⠐⢇⢀⠵⡴⢁⠝⢦⠀⣠⠊⢳⢍⡒⢰⠇
⠀⠀⠀⠀⠪⣉⣁⠤⠤⡀⠹⡁⡜⠀⠀⠀⠀⠀⠀⠰⠀⢣⡌⠈⡖⢰⠁⠀⠀⠋⠀⠣⣈⣦⢬⠃
⠀⠀⠀⠀⠀⠑⠊⠀⠀⠉⢒⠟⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⢣⣜⡠⠚⠀⡄⠀⠀⠀⠀⡨⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠀⠀⣼⣀⠀⢀⡤⠊
⡦⠤⣀⣀⣀⠤⠔⠒⢉⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡜⠀⠀⠉⠁
⠘⢦⠀⠀⢀⣠⡠⠔⠁⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇
⠀⠛⠒⠋⠁⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢄⣀⣀⣀⡀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠁⠙⢢⠀⠀⠈⢢

⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠛⠻⣦
⠀⠀⣿⠀⠈⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢀⡴⠋⠁⠀⠀⣀⠠⡀⠀⠀⠀⠀⣿
⠀⠀⢹⡆⠀⠀⡎⠙⠲⢄⠈⢷⣤⠶⠟⠛⠛⠛⠛⠛⠛⠻⠿⢧⣄⣀⡠⠔⠉⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠸⣧⠀⠀⠇⠀⠀⢀⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠄⠀⠀⠘⠀⠀⠀⠀⡟
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢀⡴⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⢤⠤⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⣑⣢⡤⠤⠔⠒⠒⠒⠀⠀⠀⠀⢠⠏
⢿⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠤⡴⠯⠴⠒⠀⠀⠀⠻⡽⠃⠀⠀⢀⣉⡀⠤⡧⠤⠤⠤⠤⠤⠤⠤⢄⣴⠏
⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡗⠒⢉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠼⣀⣀⠀⠀⠀⠀⢀⣤⡞⠁
⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⡸⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⠶⠛⠸⣷
⠀⠀⠀⣸⠍⠛⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⢻⡆
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⡷⢶⣿⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣏⡁⢠⣻⣿⣿⣮⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣹⣿⣿⣿⣿⣿⣿⠏⢡⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣽⣿⣿⡾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢿⠿⢿⠉⢻⣿⣿⠃⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⠀⣠⣷⣶⣾⣦⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣼⠟⠛⠍⠙⠻⢷⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄⣀⣀⣀⣴⠇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣶⣾⣿⡿⢿⠈⢿⣿⣿⠟⠛⠋⠉⠀⡘⠀⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠙⠋⠁⠀⠀⠀⣧⠈⣻⡋⠀⠀⠀⢴⠁⢀⠈⠀⠀⠙⠻⠿⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⢻⣴⠏⠣⠙⠊⡠⠛⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⣤⣴⣶⣶⣶⡄⠀⠀⢠⣤⣤⣤⣤⣤⣄⣾⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⢿⡽⠿⠟⠛⠃⠀⠀⠘⠛⠛⠿⣯⠿⠿⢿⣇⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡤⠒⠋⠉⠉⢱⠆⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠠⠃⠤⢀⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⠀⠀⠀⠀⠀⣸⣤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠑⠚⠀⠰⠁⣄⠀⠀⠀⠀⠈⢲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠊⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣄⡀⠀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡤⠀⠂⠀⠀⣀⣠⣴⣾⠿⢿⣿⣶⣤⣑⢚⣦⣄⡀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠉⠛⠷⣮⡿⢿⠿⠿⠟⠉⣸⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣟⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣴⣮⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢄⣤⣀⣀⣤⡄⣤⣤⣤⣤⣴⣶⣶⣾⣷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣽⣿⠇⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣴⣾⣿⣿⣿⣿⣿⣿⣳⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣽⣿⣿⠻⠇
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣟⡿⢭⣛⡋⠉⠉⠉⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⡰⡀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⢸⠁⠈⣀⡈⠀⢸⠀⠀⠀⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠈⠉⠀⠀⠀⣀⣀⣠⣤⡆⢸⣿⣿⠂⣬⣤⣀⣀⡀⠀⠀⠈⠉⠐⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⣁⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠛⠛⠻⠿⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⡿⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⠐⠒⠂⠻⠁⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⠒⡄⠀⠁⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⣠⣦⡀⢸⠀⢸⠒⠒⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠇⠸⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⡄⡄⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠆⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢱⠇⠀⠀⠀⠀⠇
⠀⠀⡠⡄⡔⡆⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢤⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡘
⠰⢩⠃⡟⢠⠁⡤⡄⠀⠀⠀⠀⠘⠁⠒⠊⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢈⠓⠒⠈⠇
⡇⢸⠀⠀⢈⠀⡇⠃⠀⠀⠀⠀⠀⡆⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⡌⠀⠀⠸
⡇⠘⠀⠆⠘⠙⠀⠰⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⡄⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡆
⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠛⠒⠖⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⡇
⠐⣀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠐⢠⠄⣀⣀⣀⡀⢤⠒⠉⠀⠀⠀⠀⠀⠀⠡
⠀⡇⢀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣀⢀⣀⣀⣸⠀⠀⠀⠀⠀⢸⡀⣀⣀⣀⠀⠀⠀⠀⠀⠁⣢
⠀⠃⢿⡿⠀⠀⡇⠀⠀⠀⠀⠀⢨⣀⣀⣀⣀⣀⣉⡄⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⢀⣏⣀⣀⣀⣀⣀⣈⡇
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠘⡀⠀⠀⠡⡀⠀⠀⢠⠂⠀⠀⡌⠀⠀⠀⠀⠀⠙
⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢈⠖⢲⠁⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⢸⠀⠀⠀⠀⠀⢸⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡠⠊⠀⠀⠑⡄⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⢸⠀⠀⠀⠀⠀⠈⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⢆⡉⢀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⣠⣤⡈⠁⠇⠀⠀⢴⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠄⢎⠀⠡⠘⢿⣿⠇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠘⠀⠀⠑⠄⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠂⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⡰⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢡⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠈⣆⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⡠⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣇⣀⣀⣬⣤⣤⣤⣤⣤⣤⣤⣊⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣺⣭⠿⠷⠶⠶⠾⠯⣿⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡫⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⠔⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠘⣯⢧
⠀⠀⠀⠀⠀⠀⢀⡴⣞⣯⣾⣷⠶⠶⠶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣶⣄⢸⡾⡆
⠀⠀⠀⠀⠀⢀⣏⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⡁⠀⠀⢻⡿⢻⣿⢣⠇
⠀⠀⠀⠀⠀⠘⣎⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠛⠀⠀⠈⣿⡾⣿⣘⣿⡲⣄
⠀⠀⠀⠀⠀⠀⠈⠑⠲⡯⣿⠀⠀⠀⠀⠀⠀⠀⠒⠛⢡⣿⡅⠀⠀⠀⠀⢸⣿⡿⡡⢀⢹⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⡇⣴⣿⣿⡶⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⣿⡿⠈⣼⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⡇⠈⠁⢸⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⠃⠀⣼⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⣆⡇⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡼⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡇⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣷⠀⠀⠀⡤⠘⠃⠂⠐⠛⠉⢙⠢⡀⠀⠀⠀⢀⡀⣷⣬⣭⣭⣟⢦
⠀⠀⠀⠀⣠⢖⣻⣭⣽⣗⢿⢻⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢆⠙⣤⠄⢲⡀⠠⠤⠤⠄⠀⣿⣸
⠀⠀⠀⡼⡷⢩⠋⠀⠂⡍⢷⡤⡽⣦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢦⠈⠓⠸⡄⠀⠀⠀⠀⢠⡟⣿
⠀⠀⠐⣷⡇⢠⠀⠀⠀⠐⢸⡇⡿⢮⡳⣤⡀⠀⠀⠀⠘⢧⠀⠀⢘⠳⣄⡀⠙⠀⠀⠀⢀⠞⢳⡞⡆
⠀⠀⠀⢳⡳⣄⠑⢦⠶⣣⡾⣳⠃⣰⢣⣿⣿⣶⣤⣀⣀⣀⣳⡀⠈⣆⠀⠉⠐⠛⠓⠛⠁⡴⠻⣞⢧
⠀⠀⠀⠀⣹⢞⣹⡧⣾⢻⣜⡇⠀⠈⠓⠮⣽⢛⣿⣿⣿⣿⡏⡁⠀⣏⠁⠀⠀⠀⠀⠀⢀⡆⠀⠘⣯⢣
⠀⠀⠀⢸⡷⣏⣡⡕⢸⠈⢧⡳⣤⢤⣖⣫⣶⣿⣿⣿⣿⣿⣟⣇⠀⠹⡦⠀⠀⠀⠀⠀⢨⢳⡀⠀⠘⣯⢧
⠀⠀⠀⠐⡗⣯⡤⢮⠸⠀⠀⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣏⠠⠀⠙⠦⣄⡀⠀⠀⠀⠀⡅⠀⠀⢸⡎⡆
⠀⠀⠀⠀⢯⢻⡄⢛⣶⠀⠀⠈⠉⠛⠟⠀⠈⠛⠻⠿⠛⠉⢰⠘⢄⣀⡠⠀⡀⠉⠉⠒⠒⠶⡆⠀⠀⣼⣿⣸⡀
⠀⠀⠀⠀⠀⠳⣽⣟⢻⠁⠙⠺⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⡼⠃⠀⡰⢻⣿⣧⢧
⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⡜⠀⠀⠀⠀⢠⢞⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠁⠒⠛⠀⠖⠋⠀⣿⣿⣿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢯⡳⠾⠶⣶⣶⣶⠒⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⢟⣾⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠴⣺⣵⣾⣿⣿⡟⣠⠁⠀⠀⠋⠀⠀⠀⠀⠀⠠⡀⠀⠀⠄⠀⠀⠤⠆⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣎⢿⣿⣿⡿⢟⣉⣏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⣿⣿⣿⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⠭⠭⠖⠉⠳⣝⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣿⣿⣿⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠯⣽⣟⣲⡶⠦⣤⣤⣬⣥⣤⣴⣦⣭⣤⠤⢶⡖⣿⣿⡿⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠒⠒⠒⠛⠛⠋⠉⠙⢾⣫⠞⠁