Iv Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⠻⣿⣷⣦⣠⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠊⢻⢿⣷⣽⣿⣿⣯⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠃⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⣦⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉⠈⢉⣽⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⠄⠀⠀⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠶⣦⣿⣿⡿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣶⣷⣾⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣾⣿⣹⣿⣿⣿⣶⡿⢫⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣝⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣉⡉⠀⠈⠉⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠐⠎⠉⠻⠷⠄⠀⢸⢻⢸⡇⢹⠉⢹⡏⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠘⠃⠙⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⣦
⠀⣀⣀⣀⣀⣀⣀⣀⣿⣄⣀⣀⣿⣄⣀⣀⣀⣀⣀⣀⡀
⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠶⠿⠛⠛⠛⠛⠛⠛⠷⢶⣤⣀⣀⣀⣀⣀⣀⣠⣤⣾⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣀⣴⠞⠛⣉⣉⣉⣉⠛⠳⣦⣄
⠀⠀⠀⠙⠻⠶⣶⣤⣦⣤⣤⣴⣤⣤⣶⠶⠾⠛⠁⠀⠀⠀⠀⢀⣼⠟⣠⡶⠟⠋⠉⠉⠉⠛⢶⣄⠹⣧⡀
⠀⠀⠀⠀⠀⣠⣤⡀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣷
⠀⠀⠀⠀⠀⢿⣼⠟⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠹⣧⣄⠀⠀⠀⠀⠀⣠⡾⠋⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡙⠛⠷⠶⠾⠛⢉⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⣶⣶⢴⣖⢰⡆⣴⢶⠀⢀⢂⠀⠠⠀⠀⠒⢀⠂⠠⠀⣂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣿⣤⣴⣿⣴⣧⣽⣯⣤⣧⣴⣶⣤⣶⣤⣤⣤⣴⣦⣴⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠒⢊⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡑⠒⠦⢤⣄⣀
⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⢯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡽⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⢀⣤⠞⠉⠀⠀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⡀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⢸⡇
⠀⣴⠋⠀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠦⣀⠀⠙⢄⠀⠀⠀⠀⢸⡇
⠸⠃⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⠛⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⢷⠀⠀⠀⢸⡇
⢆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣷⠀⢸⠀⠀⠀⢸⡇
⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⡏⠉⠉⠉⢻⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⣀⣀⣀⣀⣀⡇⠀⠀⠀⢸⣀⣀⣀⣀⣀⡀⠀⠈⣧⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢩⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⡇⠀⠀⢹⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⡟⠀⣼⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠉⠉⠉⠉⠙⡇⠀⠀⠀⢸⠉⠉⠉⠉⠉⠁⠀⢀⡿⠀⠀⣷⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣸⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡅⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢀⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠤⠤⠤⠤⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿⠀⠀⠀⢸⡇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢈⠀⠀⠀⢸⡇
⢣⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⡏⠀⠀⠀⢸⡇
⠈⢷⡀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢠⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠙⠢⢤⣀⠀⠉⠒⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠙⢦⠀⠀⣠⠶⠦⠤⣄⠀⠀⣠⠤⠴⠶⣄⠀⠀⡴⠋⠀⠀⢀⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢸⡆⠀⣇⠀⠀⠀⢸⠇⢰⡇⠀⠀⠀⢸⡀⢀⡇⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣿⢽⣯⣭⡯⣽⣤⡤⣼⡥⢼⡧⢤⣤⣿⢽⣯⣽⣯⣿⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠸⡶⢶⠇⠀⠀⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⣷⠸⡆⠀⠀⣿⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠛⠀⠀⠀⠘⣦⠙⢦⡀⠛⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢉⣤⠞⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⣿⣿⣿⢀⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠛⠛⠛⠛⠛⠛⢛⣶⣦⣤⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⢷⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⢿⣷⣦⣄⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⣿⣿⣿⣆⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠀⠀⠉⠉⢹⣿⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿⣷⣶⣶⠀⠀⣶⣶⣾⡿⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣤⣤⣿⣿⠟⠁⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⡄⢀⣀⣀⠀⣀⣀⠀⣠⣶⡾⠟⠋⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠸⡿⢿⢸⡿⣿⠦⢿⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠞⠛⠛⢷⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⠾⠾⠷⠶⠶⠶⠶⠷⠶⠷⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣽⠀⠴⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣹⡫⠛⠛⠙⠋⠛⠉⠋⠉⠋⠉⠛⠛⠛⡛⢻⡧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡕⢙⠛⢓⠊⡠⠀⠀⠀⠀⠀⡔⢁⠜⢠⢚⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠐⠶⡶⢷⠞⠁⠀⠀⠂⢀⠊⢠⠊⡰⠋⣴⡪⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⢀⠤⣤⠮⠀⠀⠐⠁⠀⠀⡴⠁⠔⢁⠜⣸⣗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠊⠐⣁⣀⠀⠊⠀⠀⢀⠞⠀⠀⠀⠈⠠⢻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠁⡀⠁⠀⠀⢠⠊⢀⠀⠀⠀⠀⠀⢾⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⡰⠁⠀⠁⠠⠀⡀⠀⠀⣸⡃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⣳⠒⠤⠦⠀⠈⠀⠀⠀⠀⠀⠠⠤⠖⠛⣻⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⠖⠲⠲⡖⠲⢿⢻⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⠤⠦⠧⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡉⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠄⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⢶⣾⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠛⢻⣿⠁⠀⠀⠀⠀⠸⠿⣯⠏⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⣠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⣀⣿⣿⣟⣿⣿⣯⣤⣤⡙⡀⠀⠀⢠⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡶⠶⠿⣿⢿⣿⢟⣿⣿⣿⡿⣿⣿⣟⠀⠄⢀⣼⢷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇⠉⣹⢀⡀⣸⢿⣿⣿⣿⣿⣿⣧⠒⢛⣿⣝⣶⣿⠸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣿⠀⠀⢹⡆⣠⣿⣿⢻⡾⣿⣿⣟⠃⠐⢿⣿⣿⣿⡟⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠈⢻⡅⢹⣧⣿⣿⣿⣿⣿⠙⠻⣿⣤⣴⣿⣷⣿⣿⣿⣷⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢺⣿⣦⣿⣿⣿⣿⣿⣿⣾⡷⣉⠛⠿⣿⣿⣿⣿⣿⣿⠀⠆⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣧⣭⣍⠛⢿⣿⣶⣦⠘⣏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠏⢼⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣉⣻⣆⡿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠉⠀⠛⠛⠛⠋⢹⡿⠛⣿⣿⠉⣿⣿⣿⡇⣸⡏⢸⡇⣿⣽⣿⣷⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⠀⠀⠀⠀⠀⠁⣸⣿⣿⣆⢹⣿⣿⢣⣿⠃⢸⣧⡧⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠈⢧⠀⠀⠀⠀⢾⣿⣿⣿⢹⣮⣿⣿⣿⣿⣇⠈⣿⣣⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣷⡤⣄⢨⡛⠂⠀⣼⣿⣿⡇⢸⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⣀⣷⣤⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠷⣦⡙⣏⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠇⠘⠛⠻⠙⠋⠕⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⡼⠁⠹⢷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣿⣿⡆
⠀⠀⠀⠀⢀⡾⠋⠛⠳⠀⠀⠈⠛⠛⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢸⡇⠹⣿⡇
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣾⣇⣼⠟
⠀⠀⠀⠀⠀⠘⠷⣶⢀⣠⠂⠀⠀⠀⠈⣩⣿⣿⣿⣿⡿⢷⠀⠀⠀⠉⠻⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⢨⣰⡏⣄⠸⠿⠃⡴⠋⠉⣿⣿⣿⡁⠈⠀⡀⠀⠀⠀⠙⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⠀⠀⡜⠁⠀⢠⣿⣿⣿⣧⣴⠀⠁⠀⢠⡄⠀⠘⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣠⣿⡿⠻⣿⣿⣿⠀⢠⠀⣸⣿⣇⣀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⢉⠧⢶⣿⢀⡇⠈⠛⠻⢿⣦⣤⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣟⣛⣻⣯⣙⣛⣟⣛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣯⣭⣿⣿⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢠⣀⠀⠀⢠⡀⠀⠀⠀⣀⣶⠤⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣤⠘⢦⡴⠊⠁⠠⠟⠉⢺⡁⠀⣀⡶⠟⠉⢀⡠⠒⠒⠤⡀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠘⠐⢓⠚⣓⠐⣟⠀⠀⠀⠀⢀⡤⠞⢿⣾⠋⠀⠀⢠⠃⠐⢠⠀⠀⠈⢳⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠁⠄⠻⠉⣆⠀⠀⠂⠈⢀⡤⠾⡿⠀⠀⠀⠁⠀⠀⣰⣀⠀⢀⣰⢷⡀⠀⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⠀⠀⢀⣀⠀⠁⠚⣲⠿⢷⣄⠀⠀⠀⠀⣀⣴⣀⡴⠾⠤⠤⢾⣿⣿⣿⢿⣿⡄⠑⡀⡷⠀⡬⡀
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂⢹⣷⣀⣀⣽⢶⠒⣨⣶⣈⣤⢿⣷⣄⠀⠁⠀⣿⣏⠉⢈⣀⣴⣄⣿⡉⢁⠀⠈⠙⣦⣼⡧⠤⠴⠃⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠟⣀⢽⣿⡄⣼⣿⣿⣿⣿⣿⣀⣀⠽⢧⣙⣰⠛⢻⣷⣼⣿⣻⣅⣠⣈⠟⣠⡄⠀⠹⢧⡴⠀⡀⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡑⢰⣧⣤⣈⠻⠛⠙⠛⠉⠉⠉⡀⠠⠛⢿⡟⠛⢿⠛⠛⠻⣏⣭⡉⠃⠈⠁⠀⠀⠀⢠⡌⠁⠰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠡⣀⣠⠀⠐⠛⢙⡆⠀⠀⠀⠀⠘⡀⠀⠀⠉⠓⠲⢾⣷⢄⢲⡄⠀⠀⠘⢻⡄⠀⠀⠖⠀⠀⠀⠀⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⡁⢀⣆⠀⠁⠀⠀⠀⠀⢈⣴⣶⣶⣶⣦⣄⠈⢿⡄⠀⡃⠰⠀⠂⡼⡇⠀⠀⠀⠀⠀⠀⠈⠁⡈⠀⠈
⠀⠀⠀⠀⠀⡐⠂⠀⠰⠤⠀⠀⠁⢠⣾⣿⣿⡆⢆⠈⢿⣿⣿⣿⣿⣿⠂⠀⣿⣤⣙⠘⠃⣷⣾⠇⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠉⠨⢋⠴⠂⠀⠀⢰⡌⠛⣻⠿⢳⣶⡄⠈⠳⢿⠿⠃⡼⠀⠂⢠⢈⡅⠒⢠⣾⠃⠀⡆⠀⠀⠀⢀⡃⢀⠁⠂⠁⣰
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⣼⣇⠰⡁⢀⡘⢻⡿⣦⡴⠏⡠⠙⠑⠦⠀⣼⠉⠀⣷⠉⡏⠀⣼⣶⡦⠀⠀⣌⣀⠀⠁⠀⡰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠉⠀⠘⣿⣇⡊⠀⠋⠙⠛⠁⠂⠄⡄⠇⣸⣿⡹⠀⠘⣿⡎⠀⢠⣿⠟⠀⢀⣼⡿⠋⠀⢠⣾⡀⠁
⠀⠀⠀⠀⢸⠀⠀⣀⢰⣿⣆⠀⠀⠀⠀⠀⢸⣤⣘⣄⠁⢦⠐⠖⠞⢋⠐⡿⠁⠀⠀⠎⠀⢀⣿⠃⠀⢠⡿⠋⠀⠀⣴⣿⠟⠃⠀⣠⠎
⠀⠀⠀⠀⠸⡄⠀⡉⠙⣾⠉⢄⢠⣈⣰⢲⣾⡀⠈⠈⠀⢀⡦⠾⠊⣡⠌⠀⠠⠄⠈⠀⠀⠘⠁⠀⠀⠊⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠱⡔⠻⢃⠙⢴⠃⠀⠀⢮⢀⣼⠿⣞⠋⣀⣉⡃⠤⠂⠁⠀⣰⠃⢀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⣴⠖⠚⠓⠛⠀⠀⠀⢀⣁⠀⠑⡄⠀⠀⠀⠀⠀⠄⠀⣿⠏⣀⠴⠲⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⡀⣇⣀⠀⠄⠀⣠⡀⠀⠈⣡⣴⠀⠣⣓⢲⣶⡆⢠⠀⣷⠋⠔⠁⣤⣴⡶⢶⣶⣾⡷⠃⠀⠀⢀⣼⣅
⠀⠀⠀⠀⠀⠀⠁⢀⢃⣀⠤⠤⡁⠀⠘⠛⠉⠀⠀⢃⣀⣉⡑⠈⠷⠖⢋⣠⡿⠋⠀⢀⣠⡉⠉⠰⠔⠂⣠⠊⠈⢹⣧⠀⠰⠁
⠀⠀⠀⠀⠀⢀⠘⠱⠈⠁⣴⣾⠇⠀⠀⠀⠀⠀⠈⠉⠱⣾⡝⢷⣶⠚⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠿⠧⢤⡴⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⣀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⠀⠀⠀⢀⠐⠘⠀⠀⣿⣯⡌⣤⣤⠾⢿⣆⠀⣠⣤⣤⣀⣀⣀⣠⣤⣴⣶⡟⠉⢻⡷⣦⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣾⣦⡀⠀⠁⠀⠀⢀⣴⣿⣿⣡⣿⣉⠞⢿⡿⡻⢿⣿⠉⢿⡏⠀⠀⢿⣿⠟⠁⢀⡾⠤⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠛⢿⣿⣷⡾⠿⢿⡟⡉⣿⣷⣾⣟⣿⢻⣷⠀⢻⡅⢺⡇⠀⠀⢸⠁⢀⣤⠞⢀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠔⠉⠀⠀⣀⣀⣘⣿⣧⣤⢿⡟⣿⣟⣟⣛⣿⣤⣦⣍⠁⠂⢀⡾⠖⠋⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⢀⡀⠀⡀⠸⣿⣋⡼⢷⣟⡏⣹⣿⠿⡛⠛⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠈⠛⣿⣿⣬⠿⡏⠁⠀⠁⠀⢓⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢷⡀⠈⠳⡖⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡜⠳⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡞⠀⠀⠹⡄⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠎⠀⠀⠀⠀⠘⣆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢠⠏⠀⣸⠉⠉⡝⠀⠘⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣠⠃⠀⠀⡇⠀⣸⠁⠀⠀⠈⢧⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡰⠁⠀⠀⢸⠀⢠⠇⠀⣀⣤⠀⠀⢣⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⡼⠁⠀⠀⠀⡏⠀⣾⠴⠊⠁⡇⠀⠀⠀⠳⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡜⠁⠀⠀⠀⢸⠁⣀⡤⣶⠀⢸⠁⠀⠀⠀⠀⠱⡄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠞⠀⠀⠀⠀⠀⠿⠊⠁⠀⡏⠀⡏⠀⠀⠀⠀⠀⠀⠙⣄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡇⢸⢁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠎⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠘⣆
⠀⠀⠀⠀⢠⠋⠀⠰⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠷⠀⠈⢆
⠀⠀⠀⢰⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢸⡀⢀⡇⢸⢠⠞⠉⠁⢸⡂⢀⡇⠀⢰⡀⢰⢃⡞⠉⢳⡄⡇⠀⠉⡟⠁⢰⢷⠀⣰⠊⠉⠀⣾⠉⠁⡇
⠀⠀⠀⢸⠉⠉⡇⢸⠘⣆⣈⣻⢸⡍⠉⡇⠀⠀⢧⡏⠘⣇⣀⣸⠃⣇⣀⠀⡇⢠⡟⠚⣇⢳⣀⣉⡇⣿⣉⡁⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠆
⠀⠀⠀⠀⠀⠰⡄⢀⣀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⢀⠤⠶⠿⣧⡄⣤⠿⠗⠦⢄⠀⠀⠀⠀⠀⢲⣾⣯⣿⠻⣿⣦⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢨⣿⡄⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣥⣤⣴⠆
⠀⠀⠀⠀⠀⠀⢣⡈⣶⠁⡰⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⢿⠻⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠉⢾⣿⣿⣿⡇⠀⠀⠀⠀⠈⠋⠉⠉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣨⠟⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠈⠀⠀⠀⢸⣷⡾⠁⠀⠀⠀⠀⠈⠻⣿⠏⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣼⣏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠸⠿⣄⣾⣿⣿⣿⣷⡤⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⣘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⡆
⠀⠀⠀⢀⣀⡼⣇⣀⡿⣀⣸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠫⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣟
⠀⠀⠀⠘⠟⠀⠀⢿⣷⡿⠁⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣶⡦⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢣⡆⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠟⣹⠇⠀⠀⠀⠀⢠⡄⢸⣿⣾⣿⢀
⠀⠀⠀⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠾⣶⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣽⠋⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⡏⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠉⠀⢛⣿⠟
⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠿⠿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣶⣤⡙⠄⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⡴⢃⠄⣀
⠀⠀⠀⢀⣄⠀⠘⢲⣿⣗⠃⠀⢀⡄⠀⠀⠀⠀⠀⢠⣾⠋⢀⣿⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠛⣿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⡈⠀⠴⢊⣡
⠀⠀⠀⠙⠛⠶⣤⣀⠶⢆⣠⠴⠛⠛⠀⠀⠀⠀⢠⡿⠃⠀⢸⣿⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣶⣿⠇⠀⠀⠀⠀⠀⠀⢠⣬⣄⠋⡀⠀⢈⡁
⠀⠀⠀⠀⠀⣠⣠⡽⠛⠯⣄⣠⠀⠀⠀⠀⠀⢰⣿⣁⣀⣀⣈⣉⣀⣀⣀⣻⣧⠀⠀⠀⠀⠀⢀⠀⠉⠳⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢻⡷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠋⠉⠉⠁⠀⢐⣤⢴⣤⣀⠀⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢀⣯⣀⡴⣤⡈⠃⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣿⣤⡍⢡⢛⣋⡇⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⡇⢿⡼⢤⣌⣿⡇⡀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢼⡇⢀⣸⣇⣸⣿⣦⣿⣿⢁⠈⡤⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡏⡿⢯⡻⣿⣿⠇⢇⠎⠞⣵⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⢀⡙⠾⣿⣼⣯⣾⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢻⡎⣿⣿⣦⡀⠙⢿⡿⣯⣲⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣷⢸⣿⣞⢿⣆⠀⡀⢈⠉⠻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⣿⣿⠀⣿⡄⣿⣿⣆⠹⣷⣼⣶⣷⣄⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡅⣿⣿⢹⠀⠘⣧⢹⣿⣿⣦⡘⢧⠉⡛⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣇⡟⣿⠈⢠⠀⠹⡆⠻⣿⣿⡳⣄⠱⣌⠃⢸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⣾⣿⡇⢿⠀⣿⢰⠀⢻⣆⠹⣿⣿⣌⢓⢬⡛⢸⡇⠘⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣾⣿⢹⡇⠀⠀⣿⠸⣧⠀⢻⣷⡙⢏⢻⣿⣝⢦⢻⣧⠀⢹⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢰⡟⣿⢸⡇⠀⢸⣿⠀⣿⣆⠀⢻⣟⠦⠀⢿⣿⣮⢻⣿⠀⣼⡇⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣿⠘⠁⠀⢸⣿⡄⠘⣿⡄⢠⠙⣦⠀⠈⣿⠈⠋⠃⣴⡟⢇⢻⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠀⠀⠀⣿⣿⣇⠀⢹⣿⡀⢳⡄⠀⠀⣿⠀⢠⣾⢿⠁⠀⠈⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢱⠀⠀⠀⣿⡏⠹⡆⠀⢹⣧⠀⢻⣆⠀⠁⢀⣿⣷⡈⠀⠀⠀⠸⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⣆⠀⢻⡄⠀⠹⣷⣄⠘⢻⡿⣷⡄⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠘⠂⠀⢧⠙⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣏⢽⣟⠻⣿⢟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⣤⣶⣶⣦⡀⠀⣴⣶⣶⣦⠀⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣅⢈⣿⣿⡄⠀⠿⣿⣿⡿⠁⠀⢿⣿⣿⡿⠀⢀⣿⣿⣏⢙⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡈⠁⠀⠸⠇⠀⠈⢁⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⣿⣷⣀⣠⣄⣠⣄⣀⣼⡟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣤⣿⣿⡟⠁⠀⠀⠉⠛⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢤⣶⣶⠤⠀⠀⠀⠀⠀⠠⢴⣶⣦⠤⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣏⠀⠀⠈⠀⢀⣠⣤⣶⣦⣤⣀⠀⠈⠁⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠤⠄⡤⠤⣶⣶⣶⣶⣄⣀⣤⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣶⣲⡆⠈⠙⣿⣿⣿⣿⠛⣍⣩⣽⣉⣝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡽⠃⠀⠀⠁⠙⣿⠿⢷⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⡜⠃⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣍⠐⠁⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⣿⠀⠉⠛⠿⠀⡀⠀⠀⠀⠀⠰⠿⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⢀⣴⢣⢼⡇⠀⠠⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⣀⣤⠀⣽⣿⠟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⠃⠈⢻⡌⣡⠺⣿⡿⠗⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠐⠉⠀⡰⠧⡄⣦
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣴⣿⠀⣢⣾⣏⡛⠁⠠⠁⠰⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢘⠀⠑⢾⠆⠐⣑⣿
⠀⠀⠀⠀⢠⠃⠀⠀⢠⡞⠁⡟⣰⣿⣏⠛⠁⠶⠤⠄⠠⣴⡂⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢤⣼⣶⡶⠚⡏⠸⡟⠉
⠀⠀⠀⠀⠸⠀⠀⡰⠋⠀⠀⡃⣿⣦⡍⠀⠚⠒⠒⠂⠀⠄⠈⠈⠉⠀⠀⠀⠈⠉⠉⠉⠉⠛⠃⠂⠼⠀⠀⠀⠀⡼⠏⠀⢀⡆
⠀⠀⠀⠀⣠⠤⠚⡁⠀⠀⢸⢠⡿⠶⠤⡀⣛⣗⠀⡇⠀⡑⠦⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⡆⠀⢸⠁
⠀⠀⠀⠸⢆⠄⠠⠀⠀⠀⡸⢸⣿⡷⣡⠭⠈⡉⠀⠀⢆⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣀⡾
⠀⠀⠀⠰⠼⠀⡅⠀⠀⠀⡇⠰⣶⢤⣙⣤⡠⣌⠲⣀⠀⡅⢀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢹⡇
⠀⠀⠀⠀⣧⠄⣀⠈⠀⢰⠇⠰⣿⡙⠷⠈⠋⠈⠑⠚⠁⠋⢁⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⡀⠀⠄
⠀⠀⠀⠀⢹⠀⡇⠀⠀⢸⠀⠐⣿⡁⢜⠠⠀⠀⠀⠤⣐⠺⢿⠛⠉⠀⠚⠴⠦⠀⠀⡌⠀⠀⢀⣠⠄⠋⠀⠰⣿⡈⠁
⠀⠀⠉⠁⢸⡀⡇⠀⠀⡄⢸⣦⣌⠳⠠⠀⠁⣿⣿⡦⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⣁⣴⡖⢀⡰⣄⠀⠗
⠀⠀⠀⠀⠘⡇⢀⠀⠀⠇⠘⣿⣿⣷⡀⠀⠀⣼⣿⣟⠀⠐⠀⠀⠀⠀⠀⠀⣁⣤⡀⠶⠚⠛⠿⠁⠈⠉⣤⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠹⠀⡀⢸⠀⠀⢸⡿⠓⢄⡈⠀⠘⡷⠛⠀⣠⣦⡤⠴⠒⠋⠉⠁⠀⠀⢀⣠⣤⣶⣶⣦⣼⣿⠁⣜⣷⣾⠃⠀⠃
⠀⠀⢀⣤⠶⣷⠂⠙⣿⢠⣾⣿⡷⠀⢸⣧⣠⣆⡐⠆⠘⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⢸⡇⠀⠀⠈⠉
⠀⣄⠀⠀⠀⠈⠆⠐⠋⠈⠛⢿⣧⡀⣠⣈⠉⠉⠉⠲⠲⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⣿⣷⡀⢤⣀⠀⠀⡄⠀⠀⠀⠀⠈⠉⠙⠻⣿⡶⢤⣄⡀⠀⠈⠹⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀
⠀⣿⣻⣧⠻⠿⠘⣿⢆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠈⠉⠙⠒⠦⢤⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⣾⢠⠃⠁
⠀⣿⣟⢿⠉⠁⢸⡇⢸⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⢇⠀⣀⠑⠎⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⢀⡆
⠀⢽⣾⣧⡀⠀⢸⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⡀⠃⡈⠉⠒⠃⠀⠀⠀⢸⠁⡄⠀⠀⠀⠀⠀⢸⣷⣄
⠀⠈⠉⢿⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡠⢖⣩⠔⠉⣤⠶⠶⠀⠀⠀⢠⠏⠀⠁⠀⠀⠀⠀⠀⠒⢹⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣸⠀⢀⠴⣊⡬⠞⡋⠄⠒⠉⠹⣆⠠⠀⡀⠀⡷⡄⠴⡀⠀⢀⣦⡀
⠀⠻⣿⣿⣇⢠⠇⢀⣷⠀⠀⠀⠸⠧⠀⢀⡀⢀⣠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⡠⠺⢿⣿⣆⠈⡄⠉⠛⠻⠷⠶⢦⠤
⠀⠀⠀⠈⠉⢸⠀⣿⣄⣀⡀⠠⠀⢀⡚⢁⣄⠉⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠦⢭⣝⣧⣈⠠⠄
⠀⠐⢤⡀⠀⡇⠀⠸⣿⣿⡏⠘⠛⠿⣇⠀⠈⡏⠀⢈⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠘⠔⠂⠠
⠀⠀⠀⠙⠲⠀⠀⠀⠀⠙⠻⢶⣤⣤⣠⣦⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠷⠶⣧⠤⠦

⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⣤⣀⠀⡀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⣀⣄⡀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢧⢀⠆⠀⢁⣾⣿⣿⣿⣶⣤⣀⣴⠆⠀⢰⣤⣀⣤⣾⣿⣿⣿⣷⡘⢀⢰⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣾⢞⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢀⢤⡘⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⡿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡦⠣⠀⢄⠹⠿⠿⡿⢟⠏⡡⢣⣶⡹⡌⢛⡻⢿⠿⠿⠋⡤⠀⠎⢼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠴⠃⠀⢱⣿⣿⣿⡌⠀⠙⠦⠀⠀⠈⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠶⠶⠂⣀⡀⠤⠀⠀⠀⠭⣿⠍⠀⠀⠀⠄⢠⣀⠐⠶⠶⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⣿⣷⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠹⡿⠲⡖⡶⠒⡞⠉⡟⠉⣷⠒⣖⢲⠒⢿⢋⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣷⢸⡦⣧⠤⣧⣀⣧⣀⣷⠤⣷⢼⠇⡟⠘⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢄⡹⠖⢧⣧⠀⡇⠀⣯⠀⣧⠀⣧⠼⠰⢃⠴⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⡄⠀⠀⠉⠈⠒⠉⠂⠁⠋⠀⠀⣰⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢤⣀⣀⣄⣀⣀⠄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠀⠀⠀⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣔⣶⣒⣒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⢉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣽⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠋⣿⣏⡉⡺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⣿⠀⠈⣹⡏⠀⠀⢀⣀⣤⣴⡶⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣧⣴⣿⣿⡄⠘⠻⠶⠶⠾⣋⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣀⣀⣠⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⢻⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⢿⠈⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣿⣛⠻⠿⢿⣭⣽⣭⣿⠣⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠻⡉⠑⡀⣤⣤⣤⣤⣷⠀⠀⠀⠠⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠈⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣛⡁⠀⢀⣿⣶⣿⣿⣿⣿⣿⣿⢰⣷⣦⡀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣆⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣾⣷⣸⡟⠻⣦⡀⠀⠀⠀⠀⠀⡜⢠⡀⣽⣦⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣆⠀⠀⠀⣼⢁⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡄⠈⣻⠓⠾⠁⣠⣷⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣷⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⡀⠀⢀⢀⠄⣀⣄⢀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⣙⠶⣿⠃⢾⡿⠫⢈⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⡇⠶⣄⠈⢁⠀⠚⢣⣿⣿⣿⣿⠉⠛⠿⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠻⢢⣼⣿⣯⡐⠒⣸⠏⢝⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣴⣂⣱⣾⣾⣿⠀⠀⠤⠀⠙⠃⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣤⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⡇⠹⢿⡇⠹⠈⡏⠏⠿⠿⠙⠿⠿⠿⠙⠏⠁⠈⠁⢠⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⠱⠟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣷⢿⡄⠀⢸⣿⣯⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣿⣿⣿⣿⢹⡆⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣿⣿⣿⣿⡏⣾⠁⠀⣰⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣸⣿⣻⡟⣿⠇⣯⢀⣼⣿⣿⠟⠻⣿⣷⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⡏⣿⣿⣿⡿⠀⢻⡟⠻⣿⠁⠀⠀⠈⢳⣿⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠘⣿⢀⢿⣿⣿⡇⡷⢸⡇⠇⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡇⡂⣿⢸⢸⣿⡿⢸⡇⣾⢿⠀⡆⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠃⢸⡟⢸⣿⣧⣿⠃⣿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣧⠀⠈⡗⠘⣿⢹⡿⢰⡿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢰⠀⡇⡆⣿⣸⠇⣾⡇⡆⢸⡇⠀⡿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⣿⢸⠀⡇⡇⣿⠛⣸⣿⠇⡇⢸⡇⠀⣷⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⢸⠀⡇⡇⣿⡄⣿⣿⢠⡇⢾⣧⠘⡟⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⢸⠆⠃⡇⣿⠀⣿⡟⢸⡇⣼⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠙⠃⠀⠀⠀⠀⠀⢠⣿⠀⣿⢸⡃⠂⡇⣿⢰⣿⡇⢸⡇⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⠆
⠀⠀⠀⠀⠀⠀⠒⠲⠔⠐⠒⠒⠲⠛⢢⡆⠀⠀⢠⣿⠛⠀⣿⢸⡇⠁⡇⣿⢸⣿⠁⣾⣿⣿⣇⡄⣿⣄⠀⠀⠀⠀⢲⠛⠔⠂⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠐⠶⠀⠀⠂⠀⣨⢏⣾⢰⡇⢸⡇⡄⡇⣿⢸⣿⠀⣿⣿⣿⣏⣽⣻⠏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒
⠀⠀⠀⠀⠀⠓⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠈⠁⣸⣧⢸⡇⡇⡇⢻⢸⣿⠀⣿⡿⣻⣿⢸⠷⠀⠀⠀⠀⠀⠀⠀⠛⡤⠀⠀⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠛⠂⠀⣼⠏⠀⠀⠀⠛⣿⢸⣧⡇⡇⣸⠸⣿⠀⣿⡇⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⢲⡆⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠛⠛⠈⢙⣿⢇⠀⠀⠀⠀⠀⡹⣸⣏⡇⡇⢹⠀⣿⠀⣿⡇⣿⣿⠀⠴⠆⠀⢀⠀⢠⡀⠀⠀⠀⠉⠀⢘⡁
⠀⠀⠀⢀⠀⠛⠒⠒⠒⠋⠁⠀⠀⣴⢻⠏⠀⠀⣀⠀⠀⢠⡀⣿⡀⡇⡇⢸⠀⣿⠀⢹⣷⢸⡿⡇⠀⣠⠈⠛⢀⣤⠉⢁⠄⠀⠒⠞⢋⣡⡖⢀
⠀⠀⠀⡄⠀⠊⠠⠴⢶⠶⣤⡤⢾⣷⣾⠀⠐⠚⠛⢁⡀⢸⡇⣿⢰⠇⡇⣼⠀⣿⡇⢸⣿⡘⣧⣿⠀⠀⠒⠒⣿⣀⣀⣀⠐⠲⠶⢶⡞⠉⢛⣉
⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⣠⠀⠸⢿⡿⠀⠀⠀⠀⠀⢀⣾⢹⣿⢸⠀⡇⣿⠀⣿⡇⠈⣿⡇⢻⣾⣷⡀⠴⠊⠉⠙⠳⣿⡿⠷⠶⠚⠛⢱⠿⠛
⠀⠀⠀⠀⠤⠄⠁⠤⣤⠤⠤⠆⠀⣸⣇⠀⠀⠀⠀⠀⣼⡟⣸⡇⢸⢠⡇⣿⠀⣿⣷⠀⢿⣿⠈⣷⣹⣷⡀⠀⠀⠀⠀⠛⠓⠀⠀⢀⣠⣧⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣴⣶⣜⣣⡏⠀⢀⣀⣠⣾⡿⢁⣿⠁⡏⢸⠁⣿⠀⣿⣿⡀⢸⣿⣇⠸⣷⡹⣷⣔⠀⡀⠀⠀⠀⠈⠻⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠉⠀⢺⣿⡏⠈⠁⠀⠸⢿⣿⣿⣟⣡⣾⡟⢸⠃⣾⢀⣿⠀⣿⣿⣇⠀⢿⣿⣆⢹⣷⣽⣿⣷⡤⠆⠀⠉⠐⠦⣾⣶⡤⠐⠊
⠀⠀⠀⠀⠀⠄⠀⠀⠄⢸⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⣠⠇⢰⡇⢸⣿⡄⢿⣿⣿⡄⠘⣿⣿⣾⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢠⣿⠏⠠⣿⣶⣏⣴⡿⢀⣾⣿⡇⣸⣿⣿⣿⣤⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠀⠀⢠⣿⠃⢠⣿⠏⠉⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡿⠁⣰⣿⠃⠀⠀⠰⠟⣿⣿⠛⠛⢻⣿⣿⣿⡁⠈⢿⣦⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣴⢤
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⣯⠟⢀⣼⡿⠃⠀⢀⣶⣤⣤⣿⣿⣦⣾⣿⣿⣿⣿⣷⣄⠀⠹⣿⡄⠀⠈⢻⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡿⠃⢐⣏⡹⠀⠀⠀⢸⣿⣿⣿⢻⡏⠀⣤⣈⡿⣿⣿⣿⣿⠀⠀⠘⢏⣵⠀⠀⠈⢹⣛⣦
⠀⠀⢀⢀⡤⠀⠀⠀⣔⠛⠿⠉⠀⣰⣛⡏⠁⣀⠀⠂⠈⢁⣽⡿⠼⠷⠠⣿⢻⡀⠀⠨⠀⣀⣤⡄⠀⠀⠹⡛⣆⠀⠠⠈⠩⡟⢢⡀⠀⠉⠛⠀⠐
⠀⠀⠈⡜⠀⢀⠴⣾⠟⠋⠀⢀⠶⣿⠉⠀⣾⣿⣷⣦⣰⣾⣿⣄⣠⣤⣴⣾⣿⠟⢲⣾⣿⡿⠋⢀⣶⣦⡀⠈⠿⠛⣆⠀⠀⠀⠉⢻⠟⠀⡀
⠀⠀⢀⣤⣤⡿⠖⠋⠀⠀⣠⣾⠟⠂⠀⣠⠟⠻⠟⠋⣿⣿⣿⣿⣿⡦⣀⣸⣿⣀⣀⣼⣿⣿⣿⣿⡿⠟⠓⠀⣄⠚⢿⣤⡀⠈⠁⠈⠀⠶⢿⣤⡄
⠀⠀⣾⡀⠋⠀⢐⠂⠀⢾⣤⠋⠀⠀⣴⣯⣤⣄⣠⣴⡿⠟⣿⣿⣍⣀⠹⣿⣏⢉⡛⢻⣿⣿⣿⣧⣤⣤⢀⣀⣤⣷⣞⢀⣼⡀⠀⠐⠀⠀⠈⠳⣀⡀
⠀⠀⠈⠀⠀⠉⠉⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠉⠁⠀⠀⠉⠁

⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄
⠀⠻⠿⠿⠁⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⣸⣿⣿⡦⠀⠸⣿⣿⡄⠀⢀⣠⣤⣶⣶⣶⣶⣤⣤⣀⠀⠘⠿⠿⠋
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠛⠉⠀⠀⠀⠀⠉⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣿⣿⣿⠛⠉⠛⣿⣿⣿⡿⠁⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣷
⠀⠀⠀⢻⣿⡇⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⣿⠁⠀⠙⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠻⣿⣦⣤⣶⣿⢻⡿⣿⣷⣤⣤⣴⣿⡿⠁⠀⠀⠀⠀⠀⠘⣧⣀⣰⡿⣿⢿⣷⣄⣀⣀⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣾⣷⣿⣿⣿⡟⠛⢁⣤⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⠿⠛⡀
⠀⣠⣴⣶⣶⠄⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠘⢿⣿⣶⠀⠀⣠⣤⣤⡄⠹⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠘⢿⣶⣦⡄
⠀⠙⠛⣿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠀⠀⠈⠛⢻⣿⠃⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠋⠉
⠀⠀⠀⠀⢠⣤
⠀⠀⠠⣾⣿⣿⠀⠀⠀⣀⣀⣠⣄⣀⡀⠀⠀⠀⠀⢠⣶⣀⠀⠀⠀⠀⣀⣴⣶
⠀⠀⠀⠀⢈⣩⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠸⠿⠛⠀⠀⠀⠀⠛⠿⢃⣠⣤⣴⣶⣶⣦⣤⣤⣄⡀⠀⣼⣿⣤
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠛⠋
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⣿⡿⠿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⡟⠉⠉⠙⣿⣿⣿⡿⠋⠉⠙⢿⣿⡇
⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠸⣿⡇⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⠀⢀⣸⣿⣿⣿⣀⣀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣿⣿⢙⣛⣿⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⣀⣀⡉⠻⣿⣿⣿⣿⣋⣿⣍⣿⣿⣿⢁⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠋⠁⢰⣶⣦
⠀⠀⠀⠀⠺⠿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⣿⠟⠋⠀⠀⠀⢿⣿⣧⠀⠈⠛⠿⠿⠿⠿⠿⠛⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⢠⡾⠟⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣤⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣿⡉⠉⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣠⡄⠛⠑⢶⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣼⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣷⡶⣿⣿⣷⣤⣤⣾⣿⣿⠿⠿⠃⠀⠀⠀⠀⡂⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢸⣿⣿⣿⡇⣤⣤⣀⣠⣴⣦⣶⡆⠀⠀⠀⠀⠨⠿⢁⣻⣿⣿⣿⣿⠿⣿⣿⣿⣶⠄⠀⠀⠀⢸⠇⠀⣶⠀⠀⢸⣿⣿⣿⣏
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠈⠻⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠐⠋⠁⠾⠉⠉⠉⠁⠉⠛⠿⢿⠄⠀⠀⠀⠈⠳⣶⣿⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢐⣿⡀
⠀⠀⠀⠀⣰⣾⣿⣷⣶⣶⣦⣤⠀⠐⠀⢢⠀⠀⠀⠀⠀⠀⠈⠿⢿⡿⠁⠀⠀⢠⣶⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠄⢀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⢿⣿⡟⠰⠶⠶⠶⠶⠀⢿⣿⡟
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡦⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠙⠿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣾⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡤⠤⠤⠤⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠻⣧⡿⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠠⢠⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⢸⠀⠀⢀⣠⡇⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣻⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⢟⣿⠉⠀⠘⠿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠛⢻⢛⣋⣉⣠⡷⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠉⢨⡇⠀⠀⢀⡀⣿
⠀⠀⠀⠀⠀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄⠘⠛⠂⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣤⣄⣀⣠⣇⣸⠃⣸⠁⣿
⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠝⠉⢹⡋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⢒⣊⣉⣩⣭⣭⣥⣤⣤⣤⣤⣤⣭⣭⣍⣉⡁⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⣉⣭⣤⡶⠶⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣊⣩⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠤⠴⠶⠒⠒⠶⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣈⠉⠓⠢⣤⣀
⠀⠀⠀⠀⣤⣤⣴⡞⣩⣴⠾⠛⠉⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠤⠭⢷⣦⡄
⠀⠀⠀⢺⣏⢿⣿⣷⣉⠀⠀⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀
⠀⠀⠀⠀⠻⡾⢿⣿⠉⢷⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⡼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣟⣾⣿⣿⠟⢋⣤⣶⣶⣶⣦⣌⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⡿⢛⣡⣾⠟⠉⠀⠀⠀⠈⠙⢿⣦⣌⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⡆⠀⠀⠀⢀⣿⣿⢏⣴⣿⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⡀⠀⠀⢸⣿⢡⣾⣿⡟⢰⣿⣇⣰⣤⡄⠀⣤⣆⣸⣿⡧⢻⣷⡌⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣷⠀⠀⢸⠉⣾⣿⣿⣧⠈⠻⠿⠿⠟⠀⢀⠛⠿⠿⠿⠁⣼⣿⣿⡈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣿⣧⠀⢸⠀⣿⡟⣿⣿⣷⣄⡀⠀⢠⣿⣿⡆⠀⢀⣀⣼⣿⣿⣿⡇⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⡆⠸⡇⢸⣿⢹⣿⣿⣿⢸⡀⠀⠁⠈⠁⣰⣯⣸⣿⣿⣿⣿⡿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢿⣿⡄⢱⠈⣿⡌⣿⣿⣿⠛⢿⣧⣤⣄⣥⡿⠋⣿⣿⣿⣿⣿⠇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⢿⣷⣈⣧⠘⣧⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣋⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡜⣿⣿⣿⣆⠹⣧⠹⣿⣽⣿⣶⣾⣷⣶⣾⣿⡿⢛⣡⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠹⣿⣿⣿⣦⠙⣇⠹⣿⣮⡻⣿⣿⣿⡟⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣷⡈⢧⡘⢿⣿⣿⣿⡿⣴⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣦⣳⣌⠻⣿⣿⣷⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣩⣭⣭⣽⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⡀⠸⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⢛⣛⣛⣧⡼⣿⣿⠿⣿⣿⣿⣿⣷⣄⡹⣿⣿⣿⣧⠀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢁⣴⣿⣿⣿⣿⣿⣧⢻⣿⣷⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢠⣿⣿⣿⡿⠿⠛⠟⣩⣌⠿⠛⢠⡆⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⢠⣶⡿⠀⢉⣠⡴⣾⠿⢁⢸⣿⣿⣿⣿⣿⣿⣏⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣏⠀⠈⢋⠉⢠⠴⣶⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢿⣿⣿⣤⣄⣀⠀⢒⢸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠸⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⠜⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠸⣿⣿⣿⣿⣿⡿⣱⣷⠹⣿⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣿⣿⣿⣿⡇⣿⣿⣧⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⣿⣿⡿⠂⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⣸⣿⠁⠘⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢇⣿⠃⠀⠀⠁⡼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⡼⠃⠀⠀⠀⠀⠠⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⢿⡿⢿⠿⢿⠿⠋⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⢻⠟⠛⠻⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿
⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⣶⣷⡀⠀⠀⠀⠀⣰⣿⣦⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⣿⣷⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣔⠿⠛⠛⠻⢗⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⣿⣿⡏⣴⣿⣷⡌⢿⣿⣿⣿⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⠶⣾⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠹⠿⠿⠃⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠉⢨⡝⠻⣿⠀⠀⢸⡿⠛⣽⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣾⣶⣾⣦⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣷⣄⢹⣿⣿⣿⢁⣼⡟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢠⣾⣿⣿⣿⣿⣧⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣺⣿⣿⣿⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣭⣙⣛⣛⣩⣥⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠛⢿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣤⠀⠀⠀⠀⠀⠀⢠⡜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣷⣿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⢁⡾⠀⣤⣦⡄⠸⣆⢻⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡘⣧⠈⠻⠿⠏⢀⡟⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠷⠈⠀⢠⣆⠀⠈⠰⠟⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⠁⠿⠋⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣸⣿⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢿⠀⠀⠀⠟⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠃⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁



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

⢀⡆⢠⢠⠏⢾⢦⠪⣃⡜⠁⠖⠒⠋⠀⢠⢠⠇⠠⠃⡷⠼⢂⡛⠀⠞⠠⠆⠴⢀⠀⠄⣈⣀⠠⠤⠠⡷⠈
⢈⡀⡉⣜⠦⣼⠾⠀⠁⡀⠰⠐⠐⠬⣄⢏⠆⠶⢤⠛⠁⠀⠇⠈⡆⠀⢷⠘⠛⠛⠃⢠⡀⠲⠄⠠⠤⠄
⠀⡇⠀⡟⢀⣸⠀⡾⢶⠶⠆⠀⣇⠄⢀⠀⠀⣄⠀⡀⠂⢱⠀⢜⠀⠶⠀⠖⠰⠀⠆⣸⠃⠂⠀⠀⣀⠀⡜
⠘⣀⣘⣁⠀⢰⣀⡇⣸⢀⡄⡴⠀⠀⠒⠒⠒⠀⢰⠉⢀⣁⣀⢈⠉⠀⢰⠀⡉⠉⠈⡁⣀⠀⠃⠈⠉⢠⠅
⡄⡂⢀⢀⠀⠆⠨⠤⠅⢈⠁⠖⠛⠠⠂⠆⠰⠐⠂⠀⣉⣈⡠⠀⠀⠹⣙⡆⢀⡤⡄⠀⠠⠀⠠⠖⠒⠀⠰
⠀⡁⢨⢤⡥⠀⠆⠄⠄⢸⠄⠂⠐⠀⠒⠖⠂⠀⠀⠁⢁⠀⠀⠀⠋⠀⢈⠀⡀⢀⠄⠀⠀⠆⠂⠁⠁⠒⠂
⠐⡇⡀⣸⢀⡄⠀⠈⠓⠀⠀⡘⠀⠠⠠⠆⠄⠀⣀⢀⡘⠀⠈⡁⡀⡉⠡⠀⠀⡄⠰⠃⠀⠆⠀⠱⠀⠤⠤
⣀⡀⠀⢀⠀⠠⢤⡘⠀⠂⠈⠁⠰⠒⠀⢀⡀⠈⠀⠠⠤⠤⠀⠀⠀⠠⠶⠂⠰⠁⠀⣀⣀⠀⣀⠀⠠⠠
⠀⠄⠀⡘⠀⠠⠀⠄⠀⠐⠒⠀⠀⠈⠀⣬⠁⠀⠀⠔⠠⠠⢀⣠⡄⠠⠀⠂⢀⡀⠠⠠⠠⠀⡉⠁⠀⢀⠇
⡄⠀⠈⠁⠠⠖⠂⠀⠀⠀⠈⠀⠸⡀⠀⠀⠀⠉⠓⠀⠀⠀⡈⢀⠀⠀⣀⠀⢀⢉⠀⠀⡀⠀⠀⣬⠄⠉⠰
⢠⠀⠀⠀⠄⠰⠀⠀⠊⡁⠀⠀⠀⠉⠀⢀⡀⠁⠘⠀⠀⡄⠁⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⢀⠀⠄⠀⠰
⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠐⠲⠀⠀⢀⠁⠄⠀⠀⠀⡤⠁⠀⠀⠀⠁⠈⠀⠀⠀⠘⠀⠁⠀⠀⡤⠄⠀⢀
⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢠⠀⠄
⠀⠀⠀⢀⡀⠀⠑⠀⠈⠄⠀⠀⠀⠠⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⠄⠀⠀⠠⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆
⠀⠀⠀⠀⠀⠀⢹⡀⡄⠀⠀⠀⠀⠀⢀⣤⣠⣿⣿⣿⣿⣷⣠⣄⠀⠀⠀⠀⣼⠛⠏⠀⠀⠀⠘⠟⢷⡄⠀⠀⠀⠀⠀⠀⡼⠁⠈⢦
⠀⠀⠀⠀⢀⡆⣼⣇⣃⠀⠀⠀⠀⠀⠈⠛⢛⡏⢙⣿⡉⣻⠛⠋⠀⠀⠀⢰⣿⣶⣆⠿⠂⠺⢆⣶⣿⡇⠀⠀⠀⠀⢀⡞⠀⢀⡴⠉⢣
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣤⣿⡿⠾⢿⣯⣄⣀⠀⠀⠀⠀⢿⡉⣩⡐⠶⢆⣌⡉⣻⠇⠀⠀⠀⢀⠞⠀⢠⠎⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠛⠋⠈⠻⠶⠛⠈⠙⠋⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⢃⣀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡰⠃⢈⣏⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠛⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⣦
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡰⠁⠻⠏⠈⠿⠃⠱⡀⠀⠀⠀⠀⢠⠎⠘⢿⣅⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⠋⠈⣿⡿
⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠛⠒⠀⠀⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠁

⠀⠀⠀⢀⣴⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣾⣿⠿⣿⡿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⡆⢀⡴⣩⠗⢄⠀⠀⠀⠀⠀⡴⠁⢀⣠⣄⡀⠈⢆
⠀⠀⠀⢸⣿⣤⡞⢳⣤⣿⠁⠀⠀⠀⠀⢀⣾⣦⠈⠟⢁⣾⡆⠀⠀⠀⠀⢠⠇⠀⢨⢁⠋⢀⡥⠞⣉⣇⠀⠀⠀⡜⢡⣦⣿⣿⣿⣿⣴⠌⢧
⠀⠀⠀⠀⢠⣹⡿⢿⢃⡆⠀⠀⠀⠀⠀⠸⣿⡿⠃⡀⠙⣿⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡨⠒⠋⠁⡿⠀⠀⠘⡄⠀⣀⣳⣾⣷⣞⣀⠀⣠⠃
⠀⠀⢀⣦⣈⡛⠿⠾⢟⣁⣴⡀⠀⠀⠀⠀⠙⠁⣼⣿⣆⠈⠁⠀⠀⠀⠀⠀⢣⡰⣿⡿⠀⠉⠨⠉⡹⠃⠀⠀⠀⠘⣌⠛⠉⠳⠞⠉⠛⡰⠃
⠀⠀⢀⣀⣈⡽⠗⠺⠯⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠥⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠔⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⢠⠊⠁⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⢀⣧⣴⣶⣶⣶⣶⣶⣆⠀⠘⠀⠀⠀⠀⠀⢀⣶⢸⣿⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠀⠄⠀⡴⠉
⠀⠸⠈⠉⠉⠉⠉⠉⢻⣿⣦⠀⠃⠀⠀⠀⣴⣿⣷⣾⣯⡾⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠙⣿⣿⣿⡦⠀⠀⠀⠀⠚⠓⠀⠈⠁⠀⠀⠠⠄
⠀⠀⢣⠀⠀⠀⠀⠀⠈⠛⠛⡿⠀⠀⠀⠀⣿⣿⠿⠻⣿⣿⣆⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠐⠐⠈⡄⠐
⠀⠀⠀⠑⠠⠀⢀⡀⠀⠤⠊⠀⠀⠀⠀⠀⠈⠻⣦⣤⠿⠿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠏⠀⠈⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⢠⣾⣿⣿⣦⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀
⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⣼⣿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣿⣿⡄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⡟⣡⡖⣤⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⢠⡾⣯⣷⠀⠀⠀⠀⠠⣶⣿⣷⠆⠀⠀⠀⠀⢠⣮⣯⣷⡄
⠀⠀⠐⢶⡯⠸⠟⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠻⠷⠟⠀⠀⠀⠀⠐⠿⢿⡿⠟⠀⠀⠀⠀⠈⠻⠿⠟⠁


⠀⠀⢀⡄⢀⡀⢠⡀⠀⠀⠀⠀⠄⡄⠤⡄⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢀⣤⡄⣀⡀⠀⠀⠀⠀⡀⠂⠀⢀
⠀⠀⢼⡁⣼⢁⣼⠃⠀⠀⠀⢀⠁⢦⠊⠄⠀⠀⠀⠀⢠⣦⡈⢉⡀⠀⠀⠀⠸⣧⠸⣯⣀⠀⠀⠀⠀⢄⠀⠀⣠⡀
⠀⠀⠈⠁⠛⠉⠀⠀⠀⠀⠀⠀⠘⠈⠘⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠉⠃⠀⠙⠂⠀⠀⠀⠀⠈⠁⠙⠛


⠀⠀⠀⢶⣼⣴⠆⠀⠀⠀⠀⠀⡠⡤⣄⠀⠀⠀⠀⠀⣠⣶⠛⢶⡄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢤⣤⣄
⠀⠀⠘⣹⢿⢿⡛⠀⠀⠀⠀⠨⢗⣄⡺⠀⠀⠀⠀⠀⢿⡈⠳⣤⡇⠀⠀⠀⠈⢻⣟⣽⠇⠀⠀⠀⠀⢴⣄⣈⣙⠋
⠀⠀⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⣹⣿⣷⣄⠀⠀⠀⠀⢸⣿⠀⣠⡀⠀⠀⠀⠀⠊⢰⣦⠑⠀⠀⠀⠀⢀⣙⠚⢺⣁⠀⠀⠀⠀⢠⠊⠏⣲⡀
⠀⠀⠘⠿⠿⡟⠃⠀⠀⠀⠀⠈⠛⠀⣿⡇⠀⠀⠀⠀⠑⠄⠡⢾⣦⠀⠀⠀⠈⡴⢢⠶⡌⠀⠀⠀⠀⠈⠺⠡⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⢤⣴⡄⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⣀⢰⢄⡀⠀⠀⠀⠀⢀⣠⢤⡤⣄
⠀⠀⠻⢿⡯⠿⠟⠀⠀⠀⠀⠨⣿⣻⠟⠀⠀⠀⠀⠀⠀⢾⡽⠚⠃⠀⠀⠀⠀⣩⢡⡌⠦⠀⠀⠀⠀⢾⣹⠞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣷⣦⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣠⣴
⣿⣿⣿⣦⡀⠀⢸⡇⠀⢀⣤⣾⣿⣿⣿⣿⣷⣄⠀⠀⢸⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣀⠀⠀⣿⠀⠀⣠⣾⣿⣿
⣿⡿⠋⠁⠙⢦⣸⣇⡴⠋⠈⠙⢿⣿⣿⡿⠋⠈⠑⢄⢸⢀⡴⠋⠈⠻⣿⣿⣿⡿⠋⠈⠳⣄⣿⣠⡞⠃⠉⠻⣿⣿⣿⠟⠁⠙⢷⡄⣿⣠⠞⠁⠙⢿⣿
⡏⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⢀⣽⣏⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⢉⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⠀⣠⣾⣿⣿⣷⣆⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣿⣷⣀⠀⠀⠀⣿⠀⠀⠀⣀⣴⣿⣿⣿⣦⡀
⡇⢀⣤⠞⢿⣿⣿⣿⣿⡿⠟⢦⡀⢸⡇⢀⡴⠻⣿⣿⣿⣿⣿⡿⠳⣄⠀⢸⠀⢀⡔⠻⣿⣿⣿⣿⣿⠟⠳⣄⠀⣿⠀⣠⡞⢿⣿⣿⣿⣿⣿⠿⢦⡀
⣷⠟⠁⠀⠀⠙⢿⡿⠋⠀⠀⠀⠙⢾⡶⠋⠀⠀⠈⠻⣿⡿⠋⠀⠀⠈⠳⣾⠖⠁⠀⠀⠈⠻⣿⠟⠁⠀⠀⠈⠳⣿⠞⠃⠀⠀⠙⢿⣿⠟⠁⠀⠀⠙⢶
⣿⣦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣾⣷⣄⡀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⣿⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣴
⣿⣿⣿⣦⡀⠀⠀⡇⠀⠀⣠⣾⣿⣿⣿⣿⣷⣆⡀⠀⢸⡇⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿
⣿⡿⠋⠀⠙⢦⣀⣇⡴⠋⠀⠙⢿⣿⣿⡿⠋⠀⠙⢦⢸⣇⡴⠋⠈⠙⢿⣿⣿⡿⠋⠀⠙⢆⣿⣠⡾⠋⠈⠻⣿⣿⣿⠟⠉⠈⠳⣄⣿⣠⠞⠁⠙⢿⣿
⡏⠀⠀⠀⠀⢀⣽⣯⡀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢀⣽⣯⡀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⢀⣽⣏⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀
⡇⢀⡴⠛⢿⣿⣿⣿⣿⡿⠛⢦⡀⢸⡇⢀⡴⠛⢿⣿⣿⣿⣿⠿⠛⢦⡀⢸⡇⢀⡴⠻⣿⣿⣿⣿⣿⠟⠳⣄⠀⢸⠀⣠⠔⢿⣿⣿⣿⣿⣿⠟⢦⡀
⣷⠋⠀⠀⠀⠉⢿⡿⠋⠀⠀⠀⠙⢾⡷⠋⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠙⢾⡶⠋⠀⠀⠀⠹⢿⠟⠁⠀⠀⠈⠳⣾⠞⠁⠀⠀⠘⢻⣿⠟⠁⠀⠀⠙⢦
⣿⣷⣄⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣷⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⣾
⣿⣿⣿⡷⣄⠀⢸⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣦⡀⠀⢸⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⡇⠀⠀⣠⣾⣿⣿
⣿⠟⠁⠀⠈⠳⣼⣇⠴⠉⠀⠙⢿⣿⣿⠟⠁⠀⠙⢦⣸⣠⠞⠁⠀⠙⢿⣿⣿⠟⠁⠀⠙⢦⣸⣠⠞⠁⠈⠻⣿⣿⣿⠟⠁⠈⠳⣄⣇⣠⠞⠁⠙⢿⣿
⡇⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⢀⣽⣧⡀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣀⣿⣅⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⣨⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⢀⣴⣿⣿⣿⣿⣧⡄⠀⠀⢸⡇⠀⠀⣠⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⣿⠀⠀⠀⣠⣾⣿⣿⣿⣶⡀
⡇⣠⠖⠙⠻⣿⣿⣿⣿⠟⠉⠣⣄⢸⠇⣠⠞⠙⢿⣿⣿⣿⣿⠟⠙⢦⡀⢸⠀⣠⠞⠛⢿⣿⣿⣿⣿⠟⠳⣄⠀⣿⠀⣠⠞⠻⣿⣿⣿⣿⣿⠟⠢⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠆⠈⣁⡦⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠂⠀⠀⣤⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⡶⠄⢀⠉⡀⠐⢑⢾⣿⠛⠛⣒⣶⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠘⡉⠉⠐⢀⣄⣀⣤⣶⣴⣶⣶⡶⠶⠿⠛⠛⠻⠇
⠀⠐⠒⢀⢤⠲⠄⠀⡈⢁⠢⠘⢀⠁⠔⠮⠒⠀⣀⠤⠚⠋⠉⠀⠀⠀⠀⡏⡶⠶⠿⠇⠀⠀⠀⠉⢁⡈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠁
⠀⢰⠀⠀⠈⠓⠎⠀⢉⠀⠲⢌⠒⠉⢀⡠⠐⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠕⠒⠂⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠘⠀⠀⠀⠀⠰⣛⣻⠟⠀⢀⡰⠜⠈⠀⠀⠀⠀⠀⠎⠁⡉⢳⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⠀⠀⠀⢀⣴⠟⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠞⠁⠀⠀⠀⠀⠀⠀⢀⠆⠠⡀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠈⠀⠀⢀⣀⢀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⢀⠖⢀⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠈⠁⠀⠀⠉⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⡠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠⠠⠦⣤⣴⣶⣶⡖⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠋⡄⡈
⠀⠀⠀⢠⠂⠢⡐⡄⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠁⣠⡔⣿⢿⢟⡋⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠈⠀⠀⠠⠁
⠀⢀⣌⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⣀⠔⠀⠀⠀⠀⢠⣮⣤⣤⣤⣤⡴⠫⢻⣇⠘⡘⣴⢻⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⡠⠃⠀⢀⠀⠀⢠⡄
⠀⠈⠁⠀⠀⠀⠀⠀⠹⡀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⣨⠋⢶⠀⠒⠊⢿⡀⢡⢻⣗⣷⢳⣟⣷⠀⢘⡁⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⣤⢼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⠃⠀⢀⣠⠟⠀⠀⠀⢾⠉⢃⡜⢣⠀⠀⠈⢿⡄⢣⢻⡜⣯⣿⣿⡄⠈⠁⠀⠀⢀⡄⠁⠀⠀⠀⠀⢀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣧⠇⠀⠀⠑⠀⠀⠀⠀⠀⢣⡀⠰⠊⣧⠀⠀⢊⣿⡄⠌⣿⣽⣿⣿⣇⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⢈⣑⠂
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠘⣟⢶⠀⠀⢰⠀⠀⠀⠀⠀⢳⡀⠉⡜⣧⠀⠀⠘⣿⡈⠜⡿⠛⠁⠀⠀⢠⢟⡐⢄⠀⠀⠀⠀⠀⠀⣿⡃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠙⣻⡇⠐⠋⠀⠀⠘⠀⠀⠀⠹⡄⠘⣨⡷⠠⠤⢘⣷⠞⠁⠀⠀⠀⠀⣾⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠖⠀⠀⠀⠀⠙⠤⠊⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠐⠀⠉⠊⠁⠌⠀⠀⠀⠀⢀⡿⡇
⠀⠀⠀⠀⠈⠑⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣫⡕⠚⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡯⡷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡶⠉⢻⣿⣶⣼⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣧⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠀⠀⢿⡇
⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠄⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢶⣖⠀⠀⠘⣷
⠀⠀⢠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⡃⠀⠀⠀⠶⡦⠶⠀⠀⢀⣤⡌⠲⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣩⣁⣀⡀⠀⢹⣇
⠀⠀⠘⠁⠀⠀⠀⢸⣷⣿⢀⡖⠀⢸⠆⠀⠀⠈⡇⠀⠀⠀⠰⢷⣾⣯⣉⣁⣀⣠⠾⣼⠇⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠐⠉⠀⠀⢸⣿
⠀⠀⠀⡄⠀⠀⠀⣠⣧⣿⡜⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠻⠛⠟⠛⠛⠛⢩⡄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠷⠀⠀⠀⠰⠶⠶⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣤⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⣀⣠⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⢸⠂⠙⠀⠀⠀⠀⠀⠄⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⢀⠤⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠐⠀⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢁⣤⡖⠂⠉⠉
⠀⠠⠀⠠⠀⠁⠀⠉⠁⠀⠀⢐⡿⠀⠐⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢃⣼⠶⢋⣌⠀⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠊⣠⡾⠋⠀⠀⠆⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⡤⢠⣆⠀⠈⠀⠀⠀⠁⠀⠀⢀⣼⠃⠀⠀⠀⠐⠖⠛⡟⠛⠏⢁⠀⠀⠊⠁⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣟⠀⠔⠀⠀⠀⠀⠀⠀⠈⠂⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡂⣨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⡈⠥⣀⠀⠀⠀⠀⠀⢈
⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠙⣧⡈⡀⠀⠀⡀⠀⠀⢠⡀⠀⠛⠀⢀⠀⠀⠀⢀⠀⠀⠀⢈⡓⠀⠀⠀⠔⠁
⢀⠀⠀⠀⠀⠐⠂⠘⠀⠉⠀⠄⢀⡷⢄⠘⣦⡀⠨⢽⣦⡀⠱⢄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉
⢁⠤⠴⠥⢤⣀⡠⠄⠀⠀⠀⠀⣨⠁⠁⢨⠁⠑⠀⠀⠹⡝⠿⡸⣇⠀⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⣠⠛⠀⠉⡇⠀⠀⣧⠀⠘⡜⡆⢡
⠀⠀⠀⠀⠀⠀⠐⠃⠈⠀⠀⠀⠀⡴⠂⠀⠀⡰⠂⠀⡀⠊⠃⢀⠂⣴⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢤⠄⢁⡠⠞⠁⢀⠤⡂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠑⢲⠆⠀⠀⠈⠔⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠘⠋⠉⠀⠈⢀⣀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠇⠀⢠⠃⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢤⣾⢁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⠄⣼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠐
⡤⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠚⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⢁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣉⣷⡄⠉⠂
⠀⠀⠀⠀⠺⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⡴⢾⣿⠋⠉⠀⠠⠀⢆
⠀⠀⡀⠀⠀⠨⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠒⠉⢁⠪⠕⠁⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⢀⡀⣀⡄
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⣀⣠⡤⢴⠠⠊⣀⠀⠠⠄⠀⠀⠀⡆⣇⠀⢳⣁⡠⠂⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⡠⢖⣪⡷⠋⢈⡤⠞⠏⠀⠒⠒⠒⠲⠲⠾⠓⠛⠘⠋⠁⠀⠀⠈⢀⣼⣤⠏
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠤⠌⠉⠐⡡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠤⠀⡀⠀⠀⠀⠀⠠⠆⣸⠁⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠠⠀⠈⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢀⠄⠉⠀⠤⠄⠀⠀⠈⠀⢀⣀⣀⣉⠀⠒⠀⠀⡀⢀⡀⠀⢤⠶⠊
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠁⠀⠄⡠⠀⠁⠠⠠⠆⠊⠀⠀⠀⠀⠀⠁⠀⠀⠁⠒⢄⣀⡀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠂⠐⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠉⠀⠘⠠
⠀⠀⠀⣀⣠⣤⣤⡤⡜⠉⠀⠀⠁⠘
⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣧⣿⢹⢳⣿⡿⣿⢸⡟⣿⢸⡇⣿⡈⣿⣸⡏⣇⢹⠻⡇⠀⣷⠀⠀⠀⠀⠀⠀⣰⠇⠀⣠⢞⡥⢖⣊⡩⠭
⣸⡿⡾⡞⣿⣿⡿⣼⠃⣿⢸⡇⢸⡇⣿⡇⣿⢹⢸⡆⠳⢿⠈⠢⣤⣀⠀⠀⢰⡏⢀⣼⢟⣵⣾⣷⠶⢿⣿
⣯⢣⣧⢇⡿⣸⠇⣿⢠⣿⢸⡇⢸⡇⣿⡇⣿⢸⠸⡅⠀⢸⡆⠀⠈⠙⠦⡄⣸⡇⣼⣿⣿⣿⢕⣪⡭⠵⠶
⠃⣼⢝⡾⢱⡟⣸⠇⢸⡟⢸⡇⣼⠇⣿⡇⣿⢸⢠⡇⠀⣼⡅⠀⠀⠀⠀⢌⢻⣽⣿⣿⣿⡷⢋⠤⣒⣨⣭
⣾⢏⡾⣱⠏⢠⡟⢀⡿⢁⡿⣱⠏⢠⣿⢳⣿⣿⡾⠀⠀⡟⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⣱⡷⠛⠉⠁
⡱⢋⡴⠋⣠⠏⣠⠟⣡⠟⣱⠏⣠⣾⣷⢋⣾⠞⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⡟
⡥⢛⡴⢊⡥⢞⣡⢞⣡⠞⢁⣴⣿⣿⣷⡿⠃⠀⠀⠀⢹⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣚⡵⢚⡭⣖⡯⠖⢋⣠⣴⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⣵⣾⠿⣚⣥⣴⣿⣿⣿⣟⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣽⣶⣿⣯⣅⣈⠀⠀⠈⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠐⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⡀⠀⠈⠈⠉⠙⠛⠷⣦⡀⠀⠀⠀⠈⠑⢝⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⡛⠻⠶⣦⣄⡀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⢹⣧⣠⡾⠋⠂⠒⠀⠀⠀⠀⠀⠀⢀⢀⣠⠃
⠙⠷⢤⣀⡉⠛⠷⣦⣄⠀⠀⠈⠳⣄⠀⠀⠀⣿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠠⠀⠐
⠀⠀⠀⠈⠉⠓⠲⢬⡙⢿⣦⡀⠀⠘⢧⠀⣠⣿⠂⠀⠀⠀⠀⠀⠀⠀⢰⣾⢏⠠⠂
⠛⠛⠛⠲⠶⢤⣀⠐⠙⢷⣝⢷⡄⠀⢸⣿⣻⣥⣄⣀⠀⠀⠀⠀⠀⡄⠀⣿⠃⠀⠀⠀⠀⠀⠀⢀
⣭⣭⣭⣕⣢⢄⡈⠓⢆⠀⢹⡎⣷⣠⡟⠡⠄⠀⠀⠉⠙⠲⣄⠀⠀⢹⣾⣏⠄⠀⣀⠠⡰⠄⠊⠀⠉⠉⠉
⣶⠿⠳⠶⢮⡑⢯⣢⡈⢳⠀⣿⣿⣷⣦⠶⠤⣔⡢⠀⠀⠀⠈⠳⡄⠸⣿⢏⣴⠪⠔⠁⢀⡠⣶⠿⣛⣛⣛
⣉⣉⡛⠛⠦⡍⢲⣹⡱⡀⡇⣹⣿⣿⣭⣽⣛⢶⣭⡳⢅⠢⠀⠀⢹⢸⡿⠃⠁⢀⣠⡾⣛⠭⣒⠭⠒⣒⣲
⡤⢭⣉⠓⠦⡈⢆⢹⣧⢱⣧⡿⠉⠀⠀⠀⠈⠙⢪⣻⣮⠳⡱⡀⢸⣼⠃⢀⣴⡿⡫⡪⠔⠉⣀⣴⡿⢗⡯
⠤⢭⣐⠭⡢⡙⡼⡄⣿⢸⣿⣷⣖⡒⠲⢄⠀⠀⠀⢹⣟⢇⢻⣧⣾⠃⣴⣻⡫⡪⠊⢀⣴⣾⢟⡵⣪⠵⣚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣶⣶⣶⣄⠀⠀⣠⣶⣶⣶⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠙⠛⠛⠉⢀⡀⠉⠛⠛⠛⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⣴⣿⣿⡦⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⢢⡀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀⡔⠁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⢳⡀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⢀⡞⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠸⠥⣙⡢⠔⠓⠤⠚⠓⠤⠚⠲⠔⣋⠬⠇⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⠦⢄⣀⠉⠙⠲⠤⢄⣠⠤⠒⠋⠉⣀⡤⠴⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡹⠖⠾⣅⠀⠀⣀⡭⠖⠲⠤⣀⡈⠉⠒⠶⢭⣀⠀⠀⣨⠷⠲⢏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠊⠉⣀⡀⠤⠒⠋⠉⠈⠙⠒⠤⢄⣈⠉⠙⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠈⢳⠀⠀⠀⢰⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡆⠀⠀⠀⡞⠁⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠘⢤⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡤⠃⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⡀⢀⣤⣦⣶⣷⣾⣿⣶⣿⣿⠟⡠⠖⠈⠀⠀⡀⠀⢰⣤⡤⢀⠜⠉⢀
⠀⠀⣠⠔⠒⠀⣀⣽⣿⣏⣴⣿⠿⠛⣋⡶⣊⠤⠒⠐⢚⡛⠉⠉⠉⠩⠍⢛⡀⠂⠠⠤⢀
⠀⠀⠀⠀⠚⣉⣿⡿⠟⣋⣡⡴⠚⠋⢁⡄⣀⣀⣤⣀⣀⣤⣀⡀⠀⣀⡄⠈⠀⠀⠉⠀⠐⢆⢁⣒⢦⡀
⠀⠀⠀⠀⢀⢁⣭⣶⠟⣋⣠⡴⠊⠉⣁⣘⣋⣵⣿⡿⡿⠻⠏⡻⠋⢁⠉⠑⠒⠚⠧⣄⡀⢠⣾⢩⣯
⠀⠀⠀⢀⢰⣿⣿⣣⣶⡿⠗⠀⠠⣴⣟⣿⣿⣶⣶⣾⣿⣿⣵⡃⠀⠈⠈⠑⠂⠀⠀⠀⠁⢈⡿⠓⣿
⠀⠀⠀⣼⣿⣿⣿⢿⣿⠃⠀⣀⠶⣊⣿⣫⣿⣿⣯⠭⠘⠉⣂⡀⠛⢗⡢⣄⡀⠂⠀⠈⠀⠀⡟⢐⡃
⠀⢀⡼⢀⣿⣿⣧⣾⡇⡄⠀⠔⢪⣿⣿⢟⡭⢀⡤⠖⣋⣉⠉⢍⣛⠲⢌⢪⡙⢦⠀⠀⣀⣠⢷⡾⢃⡀
⠀⠋⣠⠞⡿⣿⣿⣿⢃⠁⣠⠾⣿⠿⠣⠏⡐⢁⡴⣛⠣⠩⢉⣹⡇⡌⢲⡄⡌⢈⡆⢐⠚⢹⡞⠋⣸⡇
⠀⠊⠠⠚⣱⣿⣿⣿⠛⠀⠀⠀⣀⣞⠀⡟⣰⢟⡃⢻⢤⣷⡆⡁⢿⠘⡆⢡⡸⣸⡄⠙⠁⢸⠈⢰⠍⠃
⠀⠀⢀⠴⣿⡿⢋⣭⣜⡀⠀⠚⡟⡞⣰⢡⣿⣶⣶⡌⠈⠉⠀⠀⠈⢣⡀⠺⡅⢸⠁⠀⠀⣿⠘⡟⠁
⠀⠀⠀⡖⣾⣿⡿⢿⣿⢰⡀⠞⡐⡇⢸⡟⠿⣿⠿⢃⠈⣤⣶⣤⣰⢎⡗⠂⣯⡆⣤⠐⢁⠏⢸⠃⡄
⠀⢠⠎⢈⣿⣿⣷⣾⣿⣷⡞⡵⢳⣇⣾⠀⡦⣤⣾⣿⢇⠻⣿⣿⣿⣿⠙⣸⣽⠠⠀⣀⣼⢄⡎⠀⠈
⠀⠀⣴⣫⣿⣿⡿⢹⣿⡟⡜⢡⠟⢹⣿⣇⠀⡛⠻⡿⢠⠦⠈⠩⢡⠇⠀⢿⣏⠴⠦⢤⠏⢸⠃
⠀⠀⣠⠴⢾⣿⣷⣿⢏⡞⣡⡏⠐⢺⡿⣏⠛⢦⣤⣤⣶⠓⣢⡝⡝⢠⢃⡼⠘⠖⢠⠏⢠⡿⠁⠲⢦
⠀⠀⣴⢿⣿⣿⡟⣡⠛⣴⡿⠁⠂⠀⡖⠈⣍⣓⣖⣿⣠⡿⢟⣠⣶⠇⡉⢓⠶⢲⡟⢀⡞⠀⠀⡀⠼
⠀⠀⠁⢘⠿⢁⡞⠡⣼⢿⣧⣤⣾⠛⣧⣄⣀⢙⣉⣤⠋⢐⡻⢟⠔⠁⠘⢣⡴⠋⢠⢟⡼⠃⠸⠒⠁
⠀⠀⠰⢇⣤⠏⢠⡾⠟⡿⢉⡶⢿⡔⢛⣛⡻⠥⢄⣐⣋⡭⠶⣧⣦⢀⣴⠾⢁⡔⠁⠸⣶⠠⡄
⠀⠀⢠⠞⠀⢀⡞⠁⡼⠃⣾⣧⣿⠒⣯⡴⠟⡯⠉⣽⣇⠿⣛⣭⣿⡶⠁⣠⡿⣦⣀⠶⠬⣁⠈⠓
⠀⠀⠘⢤⣤⡿⠁⣾⠇⢸⢃⣼⣿⠈⣧⣿⠾⠗⠟⣋⣭⢾⣿⣾⠿⣁⣰⣟⠓⣤⡙⡄⠀⠸⡟⢉⡇
⠀⠀⠀⡰⠛⣀⢄⡟⠀⡿⢸⡏⢻⣶⣿⣽⡶⢛⠿⢉⡱⣻⣏⣱⣟⠳⢮⣿⠀⣿⠇⡼⢀⡀⣈⣁⠠
⠀⠀⢰⠃⠈⢵⣾⠁⢰⡗⢸⢿⡇⢹⣏⣤⣿⣾⡾⢋⣼⣏⡶⠳⣍⣽⡻⣿⡄⠻⣾⡁⠈⠉⠀⣥⠆⣀⡀
⠀⠀⠘⣿⣦⣾⠃⠀⢸⠇⢸⡾⣿⠈⣿⣿⡇⠀⠉⠙⠛⣿⡇⢀⣿⣿⡇⢻⣿⢦⣀⡋⢉⡐⠒⠤⣄⡀
⠀⠀⠰⠋⢰⣇⣀⡀⣿⠆⢸⠃⢻⠀⢻⣧⠉⡻⣶⣶⣴⡿⠀⢸⣿⣷⣱⣿⣿⢣⢹⠿⣿⡙⠯⣤⣾⠟
⠀⠀⡗⠀⢺⢿⣿⠏⡟⠐⠸⣆⣿⠀⠈⢻⣎⠯⠻⠻⣿⠀⠀⣾⣿⣿⠷⠞⢋⡴⡿⢸⠹⣿⣀⠛⠻⡷
⠀⠁⠟⣄⡄⣸⢿⡟⢠⡦⣤⡏⢿⣦⠀⢼⣿⣀⠀⠂⢸⠀⢰⢻⣿⣥⡴⠶⠟⢁⡥⠾⢅⠛⠛⣷⡄
⠀⠀⠀⢨⣿⣿⡜⠁⠉⠠⢿⣯⣲⠟⠀⠈⠻⢥⠔⠆⠈⠓⠊⠚⠻⠯⠤⠰⠖⠀⠀⠀⠀⠛⠾⠞⠋

⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⠀⣀⡂
⠀⠀⠀⠀⢀⡞⠁⠄⠂⡕⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⢹⡟⠁
⠀⠀⠀⠀⠸⡇⠀⡠⠊⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⢼⠂⠀⣄⣀⣀⣠⡤⠶⢻⡥⠀⠀⠀⠀⠀⠀⠀⡴⠁⢸⠓⢰⢆⢀⠁⠀⠀⠀⠀⢀⠄⠀⢐⢸
⠀⠀⠀⠀⠀⠘⠀⠀⠹⠀⠀⠨⣇⣀⡼⠃⠀⠀⠀⠈⠀⠀⡀⠧⡆⡌⣇⣇⠹⡈⠸⠃⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⣷⡿⠋⠡⢃⠄⠀⠀⠀⠁⠀⠠⢠⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⡀⢠⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⣪⢝⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠐⡆⠀⠀⠀⠀⠀⠀⣠⡶⢫⣊⣜⢷⠑⠁⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⢷⣤⠃⠀⠀⠀⠀⢀⠄⠁⣑⣶⠄⢠⡇⠀⠀⠀⠀⠀⡜⠁⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸
⠀⠀⠀⠙⠀⠀⠀⢀⠀⣹⠔⠘⠉⠀⠐⠋⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⢂⠐⡀⠀⠈⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠠⢀⠔
⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣅⡀⠀⠀⠀⡄⠤⠶⠒⠁⢀⡀⠀⠀⢠⠊⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠉⠀⢠⠐⠒⠀⠀⠑⠂⠔⠋⠀⠀⠇⠀⠀⠈⠀⠀⠈⣠⠀⠌
⠀⠀⠀⠀⠀⡀⢡⠀⢀⠀⠀⠀⠀⠀⠛⠣⣀⠀⢀⡠⠀⠀⠀⠀⠀⠀⣠⣤⠤⠖⠁⠀⠀⠀⠀⠀⢀⢽⠄
⠀⠀⠀⠀⣧⣀⣨⡗⠊⡄⠀⠀⠀⠀⠀⠀⠠⠙⣀⣀⣀⠤⠤⣤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠈⠒⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⢰⠀⠀⠀⣠⠄⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠐⠀⢸⣽⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠃⠀⠀⠛⠋⠠⢄⠄⡗⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠺⣿⡿⠃⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢿⣷⡄⠀⠀⠀⠘⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⢻⣿⣧⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⢀⣀⣤⣿⣶⣶⣶⣾⣿⣤⣀⠀⣰⡟⠁⠀⠀⠀⠀⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣤⣾⡿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⢶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡶⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠾⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠷⣦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⡿⠛⠁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠈⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⠀⠀⠀⠀⢀⣼⠏⠀⠉⠛⣿⣿⠿⠿⠿⣿⠛⠉⠀⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⢻⣿⣧⠀⠀⠀⢰⣿⡇⠀⠀⠀⢿⡄⠀⠀⠀⠙⣿⣷⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⣿⠁⠀⠀⢠⣾⣿⡆⠀⠀⠀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠋⠉⠉⠛⠛⢿⣿⣿⡿⠛⠋⠉⠉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⠀⠀⣠⣿⣿⣀⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠛⠉⢻⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⡟⠉⠛⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣦⣼⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠉⠀⠈⠙⠻⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⠄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠊⠀⠀⣠⠀⢀⠴⢀⣴⣤⣀⡉⠓⠀⠀⠀⠘⢷⢶⠖⠐⣊⡉⠴⠒⠉
⠀⠀⠀⠀⠀⠀⣴⠁⠸⣿⣿⣦⡑⠀⢀⡀⠈⢹⡿⠯⣆⠉⠛⠛⠫⠁⠀⠀⠀⠀⠀⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⣿⣿⠇⢰⢿⣽⣶⣾⣿⣷⡇⠠⣁⠤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠾⣿⣿⠅⡀⠈⣸⣿⣿⣿⣿⡿⠁⠀⠀⠓⠀⠐⢢⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣘⠂⠀⠀⡜⠁⠀⠁⠀⣿⡆⠋⠄⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⣤⣄⠀⢠⢖⣹⠀⡀⠀⠄⠀⠀⣄⠀⠁⠀⢀⣐⠓⠄⠘⠂⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠈⢹⡟⡀⠀⠸⠋⣠⠽⡖⢀⢊⡄⣴⣹⣺⠃⣿⢿⣷⠀⠀⡀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⣾⣷⡁⠀⣀⣾⣧⠀⠀⠀⠘⠈⢻⣿⣿⣠⣿⣦⠹⣧⡇⠘⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⠀⢿⠃⠀⠁⠀⠘⠀⠀⢸⣿⣿⣿⠻⣟⠀⣿⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⢄⡀⢰⣄⠀⠀⠀⣿⣿⡇⠙⠟⠙⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⠀⠀⣸⠏⡋⠀⠀⢠⠸⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⠀⠀⣿⠇⠈⠀⠀⢈⠢⡟⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢺⣀⣼⠏⠀⢀⡀⠀⣸⠃⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣏⠀⠀⠈⣠⣞⡰⣇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⠋⠀⠀⠙⠿⠆⢹⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠣⠽⠯⢿⣛⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠉⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣼⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⡀⠑⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⣠⡄⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⣠⣀⣀⣀⣀⣤⣤⡄⠀⢠⠀⠀⠀⠀⣤⡄
⠀⠀⢠⣾⡇⠀⠀⣷⠀⠸⣿⣿⣿⣳⠦⢤⣀⣼⢃⣀⣤⠠⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣦⡄⡄⣼⣿⣿
⠀⠀⣾⣿⡧⠀⠀⣿⡆⠀⠻⢿⠿⣏⠀⠀⠁⠷⠸⢿⠇⢠⡄⠀⠀⠀⠀⠀⠀⠀⢠⡼⠶⠶⢦⣼⠟⠁
⠀⠀⣸⣿⡇⠀⠀⣿⡀⠀⠄⠀⠀⠀⠀⠀⣀⡀⠐⠀⠀⠈⢷⠈⠙⠓⠒⢦⠀⠀⡠⠔⢲⡖⠢⢤
⠀⢻⣿⣿⣧⣀⣤⣿⡟⠂⠖⠂⠐⣂⣥⣴⣿⣿⣏⡁⠈⠁⢸⡄⠀⠀⠀⣼⡀⢠⣀⣀⠀⠀⣀⣀⡄
⠀⠀⠙⢿⣿⣿⣿⠏⠁⠀⣀⣀⣀⣀⡀⠀⠉⠁⠀⠀⠀⠀⢸⣧⠀⠀⠀⣿⡷⠘⠻⣿⣿⣿⣿⠟⠃
⠀⠀⠱⢌⣉⣋⣩⠔⣁⣼⣿⣿⣿⣿⣧⣤⣴⡆⠀⡇⣸⠃⠘⣿⠀⠀⠀⣿⣳⡀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⢸⡏⠀⠀⠈⠛⢿⠿⣿⣿⣿⣿⣿⣇⠀⠀⠘⣦⠠⢻⣆⠀⠀⢿⣸⢫⣷⠆⣼⠏⠀⠀⢱⣆
⠀⠀⢀⣀⣸⡧⠀⠀⠀⠀⢸⠀⠀⡏⠉⢹⡏⠸⠀⠀⣴⠋⠄⢸⡗⠀⠀⢿⣪⢾⡅⠀⡿⠀⠀⠀⢸⡿
⠀⣄⡀⠀⠙⢩⣄⣀⣠⠀⠈⠀⠺⠷⠀⢸⡇⠀⠀⠀⢿⠀⠀⢿⠃⠠⠀⢽⠰⣿⡇⠀⣹⣿⣶⣶⣿⡧
⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠈⠛⠀⠀⠈⠛⠋⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠻⠿⠿⢿⣿⣿⡿⠿⠿⠟⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⣿⣷⣄⣀⣀⣠⣾⠟⠻⣷⣄⣀⣀⣠⣾⣿⡇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣷⣌⠙⠛⠻⣿⣿⣿⣾⣷⣿⣿⣿⠟⠛⠋⣡⣾⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣟⠛⠛⠓⠀⠀⣿⢻⡟⣿⣿⢻⡟⣿⠀⠀⠚⠛⠛⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠷⣦⣀⠀⠀⠀⠈⠁⠉⠉⠈⠁⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⣉⠳⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣀⡀⢰⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⡆⠀⣿⣧⠀⢰⣶⡀⣶⡆⢠⣶⣾⣦⢸⣿⠛⠃⢸⡟⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⣸⣟⣿⡄⢸⡿⣷⣿⡇⣿⡇⣴⣿⢸⣿⠿⠇⢸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠁⠿⠋⠻⠷⠸⠇⠘⠟⠃⠙⠿⠟⠿⠸⠿⠿⠆⠸⠟⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⣁⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⣤⣄⡀⠉⠙⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣡⠞⠋⠁⠀⠀⠀⢀⣴⣾⣿⣿⣷⣤⡀⠀⠀⠀⠈⠉⠛⠦⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡵⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣼⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡟⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠋⠉⠉⠙⠻⢿⣿⣿⣿⣿⣦⠀⢰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇⢸⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢠⣤⣤⣤⣤⠀⠀⣴⣶⣤⣤⡄⠈⣿⣿⡇⣼⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⠇⢸⣿⣿⠁⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⠉⠛⠋⠀⡴⢦⠈⠙⠋⠁⣠⣿⣿⡏⢀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⣠⡄⢠⡄⣰⣄⣠⣿⣿⣿⡟⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠙⠋⠙⠋⣹⣿⣿⣿⠏⠀⠀⢸⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣇⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣯⣡⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⠧⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⢠⠶⡈⠁⠢⡀
⡸⠒⠘⡄⠀⢸⠄⣠⠤⡤
⠉⠲⠠⠧⠴⠋⠈⣩⣿⣜⠚
⡀⠂⢴⡀⠆⢀⠻⣿⣿⣿⣷⣤⡀
⠐⠠⡋⠐⡗⠉⠢⠽⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠉⠊⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢝⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢿⣿⣿⣿⣿⣷⣤⡴⠖⠛⠋⠉⠉⠉⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⢿⡷⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠋⠙⢾⡿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⠻⣿⣳⣝⢿⣻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣟⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣷⣝⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣝⣿⠿⡛⠢⠀⠀⠀⠀⠈⠳⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⡉⠢⡈⠢⡁⢠⣶⡐⠒⢦⡈⠳⣦⣀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⠶⠛⢋⠉⡇⢻⣆⠀⠉⠛⠛⠛⠛⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢦⣀⣠⣄⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⡶⠉⠀⣈⣤⡴⢇⠀⠹⣷⣀⠀⠾⣷⠄⠀⠀⠉⠻⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⢸⠋⠀⠈⠙⠶⣄⠀⠀⠘⡇⠘⠁⠀⠇⣰⠞⠉⢄⠀⢀⠑⢄⠈⠙⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⠶⠶⠤⣄⠘⢧⡀⠀⠹⣆⠀⠀⠘⠁⠀⠑⢄⠀⢄⠁⠤⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠳⣄⠹⡄⠀⠈⢷⡄⠀⠀⠀⠀⠀⠱⣢⡭⠖⠉⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠞⠀⠸⣇⡼⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠙⢦⡘⢦⡀⠀⠀⠀⠀⣠⠞⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠉⠙⠲⣄⠀⢳⠀⠙⣆⠀⣠⠞⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⣷⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢧⡤⠾⢿⠁⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠳⣄⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠛⣧⠀⢸⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠈⠻⢶⣼⣧⣄⣀⣀⣀⣀⣠⣤⡶⠟⠋⠁⠀⣀⡼⢧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣀⣤⡤⠶⠛⠉
⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠁⠉⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢰⣏⣤⣶⣶⣶⣶⣤⡀⠀⠀⣠⣴⣶⣶⣶⣶⣄⢻⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⡜⢿⣿⣿⣿⣿⣿⣿⣿⢻⣮⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠘⢿⣝⢿⣿⣿⣿⣿⠿⠋⣼⣿⣿⡌⠛⠿⣿⣿⣿⣿⠟⣸⡿⠃⠀⠀⢀⣠⠤⠶⠦⣄
⠀⠀⠀⢀⣠⣼⡅⠀⠈⠙⢧⡄⠀⠀⢈⣿⣷⣦⣭⡀⠀⠀⢸⣿⡿⣿⡷⠀⠀⠀⢀⣤⣴⣿⣿⠁⠀⢀⡴⠋⠀⠀⠀⣤⣟
⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠙⠳⢦⣼⣿⣿⣿⣿⡇⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⡶⠞⠋⠁⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠉⠳⠦⠤⠤⠤⣤⣀⠀⠀⢸⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⠃⢀⣀⡤⠶⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡶⢤⣀⡈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢀⣀⣤⣾⡋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠚⠉⠁⠀⠀⣈⣩⠿⠓⠒⠒⠒⠒⠒⠒⠿⢯⣉⠁⠀⠀⠉⠙⠲⢤⣄⡀
⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⠀⠀⣀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣤⣀⠀⠀⠈⠙⠛⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠻⣤⡄⠀⠀⠀⢠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡄⠀⠀⠀⠀⣀⣀⣸⠇
⠀⠀⠀⠀⠸⣇⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣼⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉








⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⣿⣿⣿⣿⣏⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⡀⠀⠀⢀⢀⡷⢠⣼⣿⣿⣿⣿⡟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣾⣶⣦⣄⣒⣒⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⢿⣟⣿⣿⣿⠿⢿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠊⠙⣿⠛⠛⠛⢫⣿⣿⡇⠀⣼⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠣⠄⢀⣈⣓⠈⠳⣠⣴⣾⣿⣿⣯⣿⣷⡀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠚⠒⢀⣈⣭⣛⠷⡹⣿⣯⡙⠿⠭⠿⢿⣿⣦⣄⡙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠊⠀⢀⣈⠄⠂⢀⣡⡶⢟⣫⡥⠀⢀⢀⠻⣯⠈⠄⠀⠀⠀⠀⠈⠙⠻⣦⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢛⠃⠀⣠⠞⠋⣀⣴⢞⣫⡵⠞⣫⡿⣧⣿⣾⣿⡆⢹⡆⠙⢆⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⢶⡶⠔⠁⡠⠞⠋⠊⠈⠁⣠⣾⣫⣾⣿⣿⣿⡟⢿⡄⢳⡀⢆⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠺⢷⡔⠈⠀⠀⢀⣤⣴⣾⣿⡿⢫⣾⠇⣿⣿⣿⠘⣿⡀⣷⠀⠙⢮⡓⢤⡀⠀⠀⠀⠀⠀⠀⠠⠭⠃⡄⠀⠀⠀⢠⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠴⠿⣿⣇⡙⢿⣿⠿⠟⠋⣭⣴⣿⡟⢸⣿⣿⢹⣇⢹⣧⠘⡄⢢⡀⣹⠞⠃⠵⠤⡀⠀⠀⢢⣄⣴⣖⣷⣤⣀⡁⢀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠐⡲⠞⢿⣟⣿⠦⠤⣤⣿⣿⣿⣿⡇⣼⣿⣿⡿⣿⠈⣿⠀⠘⡄⠻⠃⠀⢀⡀⠀⠤⠚⢛⣀⠀⠀⠀⠀⠉⠤⠶⢛⣋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡆⠙⢿⣾⣟⣧⣀⣠⣾⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣠⣾⣷⣾⣿⠉⠁⠘⠂⢡⣠⣤⣴⠾⠛⠉⠓⠒⠠⠟⠃⠬⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⡴⠸⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⡿⣧⣤⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢯⢼⡇⣿⣿⣿⣿⣧⣟⢻⣿⣿⡿⠿⣿⣭⣽⡿⢴⠟⠔⢷⣹⡇⠻⢿⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣽⡿⣶⣿⣿⣿⣿⡟⡌⢸⡟⣽⣿⣿⡟⣿⡿⣿⣴⡦⠄⠶⠋⠑⠈⠁
⠀⠀⠀⢀⠀⠀⡐⠀⠀⡼⠀⣿⣧⣿⣿⣿⣿⣿⡅⠀⢸⢃⣿⣿⣿⢱⡜⢷⠘⢿⣿⣆⠀⠀⠢
⠀⠀⠀⠀⢀⠀⢈⡦⠊⢀⣠⣿⠃⢻⣿⣿⣿⣿⠁⡇⣿⢸⣿⣿⣿⠈⣿⣦⡀⠀⢻⣿⣷⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠀⠠⡀
⠀⠀⠀⠈⠂⣄⠉⣠⣾⣿⡟⠁⠀⣼⣿⣿⣿⡇⢀⣇⠀⢸⣿⣿⣿⣇⢻⣿⢷⡼⣆⠻⣿⣦⡙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⣀
⠀⠀⠀⣐⡤⠜⣹⣩⡟⣫⠀⠀⠀⠹⣿⣿⣿⣧⣼⣯⠀⠰⣿⣿⡞⣿⣦⡙⠧⢻⣜⠆⠹⡻⣿⣶⣄⡙⠃⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠋⠀⠠
⠀⠀⠀⠈⠉⠛⠋⠉⠁⡉⠀⠀⣠⣮⣙⣿⡏⢻⣿⣿⣧⠀⢻⣿⣧⠹⣿⣿⣦⢤⣛⣦⡀⠱⡈⢻⠛⣿⡷⠦⣌⡁⠀⠀⠀⠀⠀⠀⠚⠁⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢉⢻⣿⣟⢷⣸⣿⣿⣿⣷⡌⢿⣿⣶⣿⣿⣷⣶⣮⡙⠻⣦⠰⡸⠂⡿⢻⣆⠀⠠⣀⡀⠀⠀⠀⠀⠠⠖⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣮⡻⣿⣄⢘⣿⣿⣿⣿⣿⡜⣿⣏⣿⣿⠟⣥⣝⢿⣷⡈⢧⠹⣶⣄⠀⣹⡶⡀⠀⠉⠓⠦⠀⠀⠀⠀⢒⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠙⡙⠃⡽⠋⣴⢿⡿⢛⣭⣿⣿⣿⣿⢻⣧⣼⣷⡜⢎⢿⣷⡈⣇⠹⠻⣮⢿⣧⡿⠄⠈⠑⢦⣀⠀⢤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠛⠀⠸⣦⣉⣤⣤⣾⣿⣿⣿⣿⣿⣿⡙⢻⣿⣿⡌⢮⣿⣷⣹⣦⢳⣬⣿⣿⣇⠸⣆⢢⡀⠉⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠡⡄⠘⠛⠿⠿⣿⣻⣿⣿⣿⣯⣾⠃⠀⠈⠙⠛⠦⣜⣈⣛⣛⣳⡛⠛⡧⣻⣆⣿⣦⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⡿⠄⠀⠓⠒⠊⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠄⡑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⠀⠀⠠⠄⢀⠠⠂⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣼⡿⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃
⠀⠀⠀⠀⠀⠘⠿⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡇⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠈⣿⠁⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⡿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⢀⣀⣴⣿⠟⡟⢿⣷⣄⣀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏⠀⣷⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣯⠙⣿⣿⣷⣼⣿⣴⣿⣿⣿⠩⣿⣏⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠀⡼⠻⠇⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⡆⡖⡆⠀⡆⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠐⠀⠀⠀⠀⠁⠁⠀⠁⠁⠁⠀⠁⠁⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣎⠀⢀⠀⢰⠄⠀⠐⠀⠀⠀⠀⢀⠁⠀⠉⣉⠉⣁⡁⣉⠁
⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣴⠀⡌⠀⢸⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠉⠁⠉⠀⠁
⠀⡐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠸⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠐⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⡆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠄⠀⠀⢀⠀⠀⠀⠒⠉⠃⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡉⢁⡀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣡⣤⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⠔⠒⠛⠛⠓⠒⠤⠌⠉⠛⠛⢗⠒⠒⠲⠶⠦⣤⣀⡀⣼⠁⢀⢀⣴⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⡠⠁⣼⠿⠋⠑⢦⣀⠀⠀⠠⡀⡄⠀⠑⣄⠀⠀⠀⠀⠈⠙⠣⠶⣾⣿⠇⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠐⢀⠔⠋⠁⣸⡣⠀⠀⠀⠀⠉⠟⠦⢀⠈⠻⡀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡿⠑⢦⡀⠀⠀⣠⡿⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⡇⠀⠀⢀⡠⢔⣦⣀⠼⢷⡄⠈⠀⠀⠀⢳⡄⠀⠀⠀⠀⣼⠁⠀⠀⠙⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠟⢯⡅⠀⠐⢿⣿⡯⠃⢰⣿⣆⠀⠀⠀⠈⣿⡄⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢻⠟⢱⣦⠈⣉⠁⣠⡯⢻⣿⡄⢠⡆⠀⢹⠏⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⣶⣆⠀⠀⠀⢈⣽⣿⣷⠹⣿⡇⢨⡇⣰⠘⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡸⣿⣿⣧⣶⡞⠀⢠⠟⠉⠀⣿⢣⠟⢁⡿⠀⢸⣿⣾⣷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⣞⢿⣿⣿⠡⠄⠂⠀⣰⡟⠁⠀⠀⠘⠃⢠⡿⣿⡟⠻⡄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣕⣈⠻⣇⣠⣀⣼⠏⡔⠀⡄⠀⠀⢠⣿⡟⢻⡇⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣿⣿⣽⣮⣝⠿⡟⣸⡇⠀⣿⠀⠀⣿⠋⠀⢸⠃⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠀⠀⠇⠁⣿⡇⢹⣶⡇⣿⡇⠀⡔⠀⣼⠇⠀⠀⠘⠀⡆⠀⠁⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⢰⠟⢠⠸⡆⠀⠀⠀⡫⢷⡞⡓⠀⢛⣃⡰⢁⣼⢿⠃⢠⠀⠀⢠⡇⠀⡀⡄⠈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⢗⣠⠎⠀⠀⠊⡞⠁⢀⢠⣾⡑⢿⣿⣿⣤⣿⣄⣺⣾⡇⠘⣸⣧⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡭⠖⠉⠇⠀⢀⠀⢀⡱⡀⣰⡟⢦⣙⡶⡹⣿⡿⢿⣿⠿⣿⣿⣾⡿⢿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⡀⢀⠀⠀⠀⢸⠀⢸⠹⠿⠧⢙⢳⣞⠓⠁⡌⠃⠀⢀⡄⠀⠉⠁⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠚⣧⣾⣗⠄⣠⡿⣦⡟⣦⠀⠐⠀⠙⠀⠀⠰⡇⠀⠀⠈⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠁⠉⠛⠉⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣀⠀⠀⠀⠀⡤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣀⡤⠤⠤⠤⠤⣄⣀⠀⠀⠀⣇⣀⠏⠀⡜⠀⠀⠀⠀⣨⡟⣇⠀⠉⠀⠀⠀⠀⠀⢀⣤⠞⣇⡱
⠀⢎⣸⡇⠀⠀⠀⢰⣾⡠⠃⠀⣮⣖⡠⠤⠴⡀⠀⠀⠀⢸⠒⠒⠒⡇⠈⠀⠄⠀⠀⡔⠁⢑⡾⠋
⠀⠀⠉⠓⠢⠄⠤⠒⠉⠀⠀⠰⣗⣿⡁⠀⠀⣽⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣠⡮⠓⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠁⠀⠀⠀⠸⠤⠤⠤⠇⠀⠀⠀⠈
⠀⠀⣠⣴⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡶⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⢰⣿⣿⣿⡿⢀⠙⣆⠶⠀⡄⠀⠀⠀⣿⣿⣿⡇⠐⢸⢄⢃⠰⠄⠀⠀⠀⣰⣟⣀⣐⣛⣂⡀⢿⡄
⠀⠸⣿⣿⣿⡏⠀⠀⢯⡖⠂⠀⠀⠀⢰⣿⣿⣿⣥⠀⠈⣺⠈⠁⠀⠀⠀⢰⠁⠣⠤⠤⠤⠤⠤⠎⢱
⠀⠀⠘⠿⠿⢿⣁⣀⡞⢠⡖⠒⡆⠀⠘⠻⣿⡟⣻⣤⣼⠁⣾⠉⢹⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠈⠧⠠⠇⠀⠀⠀⠉⠀⠈⠉⠉⠀⠈⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣒⢒⣒⡆
⠀⠀⢀⣀⠤⠔⠒⠒⡄⠀⣀⡠⠶⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠃
⠀⢰⡇⠀⠀⠀⠀⠀⢣⠀⣍⠐⠀⢀⠇⣾⣿⣿⡆⢸⠉⠉⠉⠉⣷⣤⡀⠀⡠⠤⠤⠤⠤⡀⢡⠀⠀⢸
⠀⠘⣿⠀⣀⡀⠀⠀⠈⣆⠉⠑⠒⣉⣄⡈⠛⠛⠁⢸⠀⠀⠀⠀⠁⢸⡇⠀⡷⠶⠶⠶⠶⠆⢸⠀⠀⢸
⠀⠀⢫⣇⣙⣉⣠⣤⡶⠟⠀⠀⠐⡇⠀⣹⠂⠀⠀⢸⠀⠀⠀⠀⢸⠟⠀⠀⡀⠀⠀⠀⠀⡄⢸⣿⣻⡇
⠀⠀⠀⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠰⠿⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⡔⡆⠀⢀⠲⡀⠀⠀⡤⠤⠤⢤⠤⠤⣤⢤⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠉⣂⣤⣀⢠⡄⠈⢁⠀⠀⡞⠀⠳⠤⠞⠀⢣⠀⠀⡇⠀⠀⢸⠀⠐⠡⢻⠀⢀⠤⣤⣴⠞⠋⣈⣯⠆
⠀⣴⣴⡇⠀⠀⠀⢠⣦⣾⠀⠀⡇⠈⡘⠀⡁⠃⢸⠀⣀⡧⢄⡲⣹⠀⠀⠀⢸⠀⡎⠎⠱⣧⡴⢏⡫⢹
⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠁⠁⠀⠀⣸⠀⠀⢢⠵⠊⠼⠤⠤⠤⠜⠘⢯⢉⣉⣉⣉⣉⣉⠽
⠀⠘⠛⠀⠀⠀⠀⠀⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⣗⣿⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣗⢾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⠟⠂⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷⡀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀⣠⣤⣤⣤⣤⣦⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⠿⢻⠟⠉⠉⠉⠙⠛⠿⠛⢛⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣳⣾⣷⣿⠟⡡⠚⠁⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠟⠭⠓⠉⠀⠀⠀⠀⠀⠀⠐⠾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠙⠛⠻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠀⠀⣀⣠⠶⠾⢿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡫⠄⠀⠀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣧⠲⣄⣀⣠⣴⠞⣁⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢋⠁⠀⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣄
⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⢿⣿⣿⣿⣿⠿⣿
⣿⣷⣾⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⣂⣠⣄⡀⠀⠀⠀⠀⠀⠐⠒⠈⠉⠈⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠿⠟⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣤⡘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣤⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⠛⣷⡈⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣿⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢠⣾⣿⠀⣿⣷⡌⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠒⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠛⢐⣟⢻⣿⣭⣿⣿⣿⣆⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣭⣭⣭⣭⣍⣭⣭⣭⣭⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⣠⣤⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠉⢻⣷⡀⠀⣿⡇⢸⣿⠀⢀⣾⡟⠉⠀⠉⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⢀⣀⣠⣤⣴⣶⣶⣿⡇⢀⣿⡇⢸⣿⡀⢸⣿⣶⣶⣦⣤⣄⣀⡀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⣰⣿⣫⣴⣶⡿⠟⠛⠋⠉⠁⠀⠀⣿⣷⡿⠋⣠⡀⠙⢿⣾⣿⠀⠀⠈⠉⠙⠛⠻⢿⣶⣦⣝⣿⣆
⠀⠀⢀⣼⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣠⣾⡿⢿⣦⡀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣧⣤⡀
⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠋⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿
⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣸⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⡇⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⣀⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣄⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⢸
⣿⡇⢠⣶⠀⠀⠀⢀⣀⠀⢰⣦⠀⠉⠁⠀⣀⣿⣷⡾⠿⠿⠿⠿⢿⣷⣤⣀⠀⠈⠉⠀⣴⡆⠀⣀⡀⠀⠀⠀⣶⡄⢸
⣿⡇⠀⠀⠀⠀⠀⠈⠋⠀⢀⣀⠀⢀⣴⡾⠟⠉⠀⠀⣶⣦⣤⡀⠀⠀⠉⠻⢷⣦⡀⠀⣀⡀⠀⠙⠁⠀⠀⠀⠀⠀⢸
⢹⣷⠀⠸⠿⠀⠾⠇⠀⠀⠘⠋⢠⣾⠏⠀⣀⣤⣄⠀⠀⠈⢹⣿⠀⢠⣶⠿⠂⠹⣷⡄⠙⠃⠀⠀⠸⠿⠀⠿⠇⠀⣾
⠈⣿⡄⠀⠀⣤⠀⢀⣴⡿⠀⢠⣿⠃⠀⠈⠛⣹⡿⠃⠸⠿⠟⠃⠀⢻⣧⠀⠀⣀⠘⣿⡄⠀⢿⣶⡄⠀⣤⠀⠀⢰⣿
⠀⠸⣷⡀⠀⠉⠀⣾⣏⣀⣠⣾⡏⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⢠⣴⡿⠋⢠⣾⠟⠀⢹⣧⣄⣀⣹⣿⠀⠉⠀⢀⣾⠏
⠀⠀⠻⣷⣄⣀⣤⣿⠟⠛⠋⣿⡇⠀⠀⠀⠈⠛⠿⠿⠇⠀⣀⠈⠁⠀⢤⣾⠟⠀⠀⢸⣿⠙⠛⠻⣿⣄⣀⣠⣾⠟
⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⢿⣇⠀⢠⣾⠗⠀⠀⠀⠀⠈⢿⣦⣤⣤⣬⡀⠀⠀⠀⣸⡿⠀⠀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⣿⡄⠀⠀⣤⣶⣤⡀⠉⠉⠉⢹⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠈⠁⠀⠸⠿⠀⠙⢿⣦⣤⣴⡿⠋⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠈⠉⠉⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣶⣶⣶⡾⠿⠛⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠈⠉⣉⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠐⠒⠚⠿⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⢰⣿⡅⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⠉⠛⢷⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠐⠒⠂⠉⠉⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠻⣄⠀⠀⠀⠶⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⢠⠀⢳⡀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⢨⠿⠦⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣤⣄⣀⣀⣀⣀⣀⢠⣤⠔⠋⠁
⠀⠀⠀⠀⢸⠀⡀⢢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⢸⠶⣽⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠄⣸⠀⠈⠙⣦⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⡏⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⡜⠁
⡄⠀⠀⡆⢹⠖⠒⠒⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠀⠀⠀⡇⠀⠀⠀⠈⠉⠲⠦⣤⡀⠀⠶
⢧⡀⠈⠉⢸⠆⠀⠀⠀⠈⠳⢦⠀⠀⠀⠀⠀⠀⣀⠴⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢦⣀⣀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠙⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⠀⠀⡘⠃⠀⠀⠈⠉⠛⠛⠢⠤⣄
⠀⠀⠀⠀⠀⣿⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠙⡄⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⣆⠀⠀⡗⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⢧⣄⠐⠒⠈⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠈⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆⠀⠈⣧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠳⣤⡄⠄⠒⠠⠄⠘⢷⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠈⣿⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠙⠠⠤⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠛⣦⣤⣤⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠈⣤⠁⠀⠀⠀⠀⠀⠀⡍⠀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⠁⠈⠁⠈⠀⠈⠉⠀⠈⠉⠁⠈⠈⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣠⣤⣤⣤⣤⣤⣀⡀⠴⠖⠒⠒⠶⢦⣤⠀⠖⠒⠒⠖⠢⣄⡀⠰⠶⠒⠶⠶⣤⣀⡄⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠃⠀⠀⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⡜⢁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣸⡆⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⢀⡆⠀⡄⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠸⡇⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⢀⠀⠀⣦⠀⠀⠀⢠⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⢿⠀⠀⠀⠘⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢰⡀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⢩⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⢸⠁⠀⠈⠀⠀⠀⠀⠇⠀⠀⠸⡇⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⡄⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⢁⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠇⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⠀⠀⠀⠃⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⡄⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⠸⠀⠀⡆⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⡇⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⢄⠀⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⣀⡿⢧⣀⠀⠀⠀⣀⡿⢆⣀⣀⣀⣀⡠⠘⠛⠲⠶⠶⠶⠖⠉⠛⠑⠶⠆⠉⠁⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠀⠀⠀⠈⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠶⡄⠐⠂⠀⠀⢲⡆⠐⠀⠀⠒⠒⠒⢶⠀⠐⠂⠐⠒⠀⠒⠀⠒⣤⠔⠂⠀⠀⠀⠀⢠⢰⣧
⣤⣤⣀⣤⣤⣤⣄⣠⣤⣤⣄⣤⣸⣷⣤⣷⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧⣠⣤⣤
⣿⣿⣿⣿⠿⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡚⣟⣳⠀⠀⠀⠀⢀⡀⢇⣀⡠⠰⣃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣻⠻⢲⠖⣠⢤⣤⡔⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣳⡭⠼⠿⢷⣀⠀⠀⢀⡽⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠃⠈⠉⠁⠰⣄⠀⠅⠘⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⢿⠉⠙⡄⠀⠈⠻⠟⠖⠛⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢿⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠈⠀⢸⡄⠀⠀⠑⠂⠠⡄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⡄⢸⡇⠀⠀⡇
⠀⠀⢀⣤⣀⡴⠖⠀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢸⣰⠞⠙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡀⠀⠀⢿
⠀⠀⢍⢻⡏⠀⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⢸⡿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⢈⣸⣏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣷⠀⢀⣰⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⠀⠀⠀⣿
⠀⠀⠘⢿⡏⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡁⣾⠿⣷⡾⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡀⡏
⠀⠀⠀⢸⣡⠂⢾⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⣿⣿⡟⣷⡾⣿⠿⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⠘⡇⠀⢀⡇
⠀⠀⡇⢸⡏⢀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇
⠀⠀⣧⠀⣇⣼⣽⡇⠀⢀⡀⠀⠀⠀⣼⠀⠀⠀⠀⣸⣿⣿⣾⣿⡅⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⡇
⠀⢠⣿⢠⣿⣼⣿⠣⣀⠔⡴⠀⠀⠀⠆⡀⠀⠀⠀⢻⣿⣟⣿⢻⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⢂
⠀⠀⢹⣻⡇⠻⣿⣷⣏⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠛⢡⠸⠙⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠘⡄
⠀⠀⢸⢸⡇⠈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡎⢦⢃⡻⠀⠀⠤⠤⠀⠀⠀⣤⠀⠀⠀⠀⢠⠀⠈⠀⠀⠃
⠀⠀⠀⠀⣹⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢈⣇⣾⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡛⢺⣿⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⣄⠂⢸
⠀⠀⠀⠀⢹⡇⢰⡀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⡀⠐⠂⠰⠉⠀⠀⠀⠀⠀⠀⠀⣿⣥⡀⢸⣧⠈⣼
⠀⠀⠀⠀⢀⣿⠀⡇⠀⢰⡇⠀⠀⠀⠈⡆⠀⠀⠀⠀⠸⡿⠃⠈⠻⠆⠀⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⡇⢰⡿⠀⡏
⠀⠀⠀⠀⢸⡏⢰⠃⠀⠘⣇⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠐⠀⢀⢻
⠀⠐⠀⠀⠈⡇⠘⡄⠀⠀⢹⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠞⠁⠀⠀⠀⠠⠀⠸⣿⡇⠀⠀⢿⣿⢲⡄
⠀⢀⠀⠀⢰⠇⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢘⣿⣜⣡
⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢋⡵⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠈⠀⠀⢀⠈⣿⣿⣋⠄
⣿⣿⠿⠿⡿⢿⠿⡿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠛⠉⠉⠩⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⡀⠀⠀⣾⣿⣿⣦⠘⡏⣿⡀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⣿⣿⣷⠀⠀⠻⣿⣿⡿⠀⠃⣹⡇⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠻⠿⠃⢰⣤⡄⢨⣭⢀⡎⢐⣿⣷⠀⠀⠈⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⡇⠘⠛⠃⠀⢡⣿⣿⡿⢻⣿⣷⡀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⢰⠘⣿⣿⠇⣼⣿⣿⠟⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⠉⠛⠀⠐⠉⢉⣾⠛⢃⠀⠀⠀⠀⠃⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣇⠀⢀⣀⣀⣴⣿⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣄⠉⠁⠙⠏⠻⢿⣿⣿⣷⣿⠃⡰⠈⠀⠀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠈⠁⠀⠀⠐⣄⠀⠉⠛⣿⣿⠀⠀⠀⠀⢠⠾⠄⠄⠠⠤⠄⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⢿⡄⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⠤⠤⠦⠤⢿⣄⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇⠈⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠄⡠⣠⣤⣤⣘⣦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡓⣷⣿⠋⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠘⠀⠀⠈⠀⢀⣝⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠁⣤⣷⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢹⡎⠉⠉⠉⣠⣾⡛⠉⠂⠀⠀⠀⢄⠠⠀⠃⠁⠀⢠⠆⢰⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⢈⡇⠀⡐⠀⢠⡏⢀⡇⠀⣿⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠉⠀⠀⠀⠀⡟⠿⠇⣶⠀⠀⠈⠀⠸⠀⠀⢿⣄⣀⣈⠹⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⠀⠀⠈⠛⠛⠧⠀⢀⠀⠀⠀⠀⠁⠀⠀⠸⡇⢸⣷⠀⣶⠂⣠⣾⣿⣱⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⡀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠁⠈⠽⠀⡟⢰⠏⠀⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠈⡄⠀⠈⠁⠁⠀⠀⠀⠠⠄⠀⢠⡤⠈⠁⠀⠀⠀⠀⠀⠀⠸⣧⣤⣠⣤⣀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠁⡄⠀⠈⠉⠁
⡳⣿⣸⢀⣴⠞⠗⢶⡄⠂⠀⠖⠀⠀⠄⡀⠀⢐⣦⡆⢀⡀⠤⠂⢀⠀⠈⠉⠻⢧⡄⠀⠀⠀⠀⢠⠴⢿⣳⣇⡔⢠⡶⠞⠲⢦⡀
⠉⠀⠉⢸⡇⠀⠄⠀⣿⠀⢶⠶⡀⠀⠀⠑⢀⡾⠈⠁⠠⡄⠀⠀⢸⣦⠀⠀⠀⠀⠤⠔⢚⠛⠛⠑⠀⠋⠉⠀⠁⢸⡅⢀⠄⢀⡇
⠀⠀⠀⠈⠛⠶⠶⠞⠁⠈⠉⠀⠈⢓⠒⠞⠉⠣⣿⠇⠀⢹⣆⠀⠀⠘⣧⡀⠈⢂⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⡁
⣛⡋⠛⠓⠀⢠⠀⠀⠀⢡⠖⠤⣠⣄⠀⠀⠀⠰⠿⠀⠀⠀⢻⣦⠀⠀⠈⢷⡀⠘⠀⠀⠐⠞⠋⠛⠛⠛⣛⡙⠛⠂⠀⡄
⠀⠈⠓⠦⠀⠃⠀⢀⠀⡀⠓⢶⣎⡈⡓⡤⣄⡀⠀⠀⠀⠀⠸⣯⣧⡀⠀⠈⢿⡄⠀⠀⠀⠀⡀⠀⣄⠀⠀⠉⠳⠄⠀⠃⠀⢀
⠀⠀⠀⠀⠈⠃⠀⠈⢢⠀⠀⠈⠁⠁⠛⠷⠏⠀⠀⠀⠀⠀⠀⠼⢋⡟⠀⢠⠈⢻⣶⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠂⠀⠈⠦
⡄⡈⠉⢻⣟⣋⣙⡀⢨⡰⠶⠆⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠈⢻⣦⡄⠀⠀⠀⢀⡇⠀⡌⡉⣻⣟⡉⣓⠀⣰
⣄⣀⠄⠹⢭⠄⠢⠧⢼⠃⠆⠀⠀⠀⡆⠈⠁⠈⠀⠈⡉⠉⡄⠆⠰⠀⠘⠆⠀⡀⠈⠿⠓⢀⠇⠀⠸⣧⣀⣁⠄⠽⣭⠄⠪⠧⢿
⠀⠋⠀⠀⠀⠐⢰⢨⢸⡀⣷⡀⠀⡀⣱⠀⠀⢠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⠃⠀⠀⠀⠂⢶⠈⢀
⠈⢆⠀⠀⣀⣜⠓⡴⢾⠩⠮⣡⠀⠁⢸⡇⠀⢼⡆⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠃⠘⢄⠀⠀⣀⢺⢢⠴⢯
⡀⠀⠁⢀⠁⠀⠀⠀⢀⠤⠀⠈⠀⡄⠀⣇⠀⠈⡇⠀⠀⠀⠠⠴⠇⣹⣶⠚⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⡀⠀⠁⢈⠁⠀⠀⠀⡠
⠈⠢⡀⠀⠐⠀⠐⠉⠀⠀⠳⠀⠀⠀⠀⢻⡀⠀⣿⠀⢾⠷⠀⠀⠀⠈⠈⠄⢀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠈⠢⡀⠀⠐⠀⠒⠈
⠋⠙⠛⠶⢦⣤⣄⣀⠀⠀⠀⣷⡀⠀⠀⠘⠁⠀⣿⠀⠀⠀⠄⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠙⠋⠛⠻⠶⣦⡤⣀⣀
⠠⠀⠀⠀⠘⠲⣄⠈⠀⢠⠄⢹⢂⡀⠀⠀⠀⡀⢹⡇⠀⠀⠀⠀⠒⠠⠀⣀⡀⠀⢀⡀⢀⡈⠤⠀⠀⠂⠀⠀⠀⠀⠘⠢⣄⠉
⠈⢃⣤⣤⡄⣀⣈⠣⣀⠃⠀⠀⢴⢳⣄⠀⠈⠀⢹⢷⠄⠀⠐⠛⠷⢶⣦⣄⣀⠉⢘⠀⠀⠀⠀⠀⠀⠀⠈⡢⣤⣤⣄⣀⣈⠂⡘
⠀⣎⣝⣃⣁⠡⠋⠀⠈⠀⠀⠀⢘⡾⠷⠒⠀⠀⢈⡯⡀⠀⠀⠀⠀⠀⠈⠉⡛⠛⠉⠀⠀⢄⠀⠀⠀⠀⠀⣇⣟⣀⡐⠰⠃⠀⠈
⢀⣿⣯⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠀⢘⡿⢠⠀⡀⠀⠀⡄⠀⠀⢣⠁⠀⠀⠀⢈⠀⠀⠀⢲⢰⣿⣿⣿⣯⣿⠁
⣼⠯⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⠀⠀⠠⠘⢀⣀⡴⠀⠀⠚⠖⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⢿⢿⣿⣿⡏
⠀⡄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠘⠧⣴⣠⣦⣀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁
⣓⣿⣜⢠⡶⠛⠛⠳⡄⠂⠁⠖⠐⠀⠀⡀⠀⢸⣶⡆⢀⠤⠐⠂⢠⠀⠀⠈⠙⠷⠆⠀⠀⢀⢀⢠⠴⣻⣳⣧⡘⢠⡾⠛⠛⢶⡀
⠁⠀⠀⢘⡇⠀⠄⢀⡿⢀⣶⠗⣄⠀⠀⠐⢂⡞⢀⠀⠰⡆⠀⠀⠸⣧⡀⠀⢀⠄⠴⠒⢚⠛⠋⠈⠁⠁⠀⠀⠀⢸⡄⠠⠄⢀⡇
⠀⠀⠀⠀⠙⠓⠒⠛⢁⠀⠉⠐⠈⠑⢓⠚⠉⠳⣿⠃⠀⠸⣆⠀⠀⠈⢷⡀⠀⢃⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠈⠛⠓⠞⠛⣁
⢯⣉⡉⠃⠀⡠⠀⠀⠀⠡⠒⠢⣄⣄⠀⠈⠀⠒⠞⠀⠀⠀⢹⣧⡀⠀⠈⢷⡄⠘⠀⠀⠘⠛⠉⠉⠋⠛⣏⣙⠉⠃⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⢄⠀⠀⠀⠆⠀⣿⠀⠀⠀⠀⠉⢈⣉⡁⠀⠀⠀⠛⠘⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠐⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠛⠄⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠉⣈⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⢽⡄⠀⠀⠀⢰⢂⠔⠊⢳⡀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠐⣞⣷⠀⠀⠀⠈⢧⣠⡴⡏⠃⠀⠀⠀⠁⠉⠉⠉⠈⠑⠀⢱⠀⠀⠀⠀⠀⢸⣠⠠⠀⢦⡄⠐⠂⠒
⠀⠀⠀⠀⠀⠀⢫⠼⡆⠀⠀⠀⠀⠙⠀⢄⡃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠘⡒⣻⣄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⠖⠃
⠀⠀⠀⠀⠀⠀⠠⢯⠼⣏⠁⠘⠢⡀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⡌⠀⠈⣃⢻⡀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡛⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠂⠀⠀⢡⣀⣧⠀⠀⠸⢠⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠒⠇⠀⠀⠀⠀⠀⠰⢻⢦⠀⠀⠀⠀⠛⠓⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⢰⠀⠈⢾⠇⠀⠀⠀⠀⠀⠀⠀⠘⠑⢺⠀⠀⠀⢀⠐⠁⠀⠰⡌⢣⠀⠀⠀⣉⣁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⠀⠀⠀⠑⠆⠀⠄⠀⠀⠀⠀⠋⠐⠮⠀⠀⠀⠁⠀⠀⠀⠀⠙⡼⡇⠀⠀⣈⠀⠀⠀⠻⣀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠒⠆⠀⠀⡈⠀⠀⠀⠀⠠⠂⠉⠁⠀⠀⠲⠒⠀⠀⢠⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠴⡇⠀⢀⡀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⢆⡀⠸⠃
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠃⠀⠀⠀⠀⡀⢉⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡄⠀⠀⠀⠀⠀⢀⢤⠐⢸⡆⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠔⠁⠀⣈⢸⠅⠈⠢
⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⢰⠂⠈⠉⠉⠉⠛⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢠⠤⠀⠀⠀⠈⠂
⠀⠀⠀⠀⢆⡔⠁⢠⠷⠀⠀⠀⠀⠀⣉⡳⢤⣀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠣⢶⠗⢡⢂⠀⠀⠀⠀⠛⠀⠀⠨⠉⣒⠢⠴⠦⠤⠤⢤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠠⠐⢤⣭⣁⡘⢃⡬⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⢠⡀⢀⣀⣀⠀⣀⣀⡀⢀⣀⡀⠀⠤

⠀⠀⠀⠀⠰⠒⢤⠀⡄⠀⠀⠀⠀⠀⠀⡤⠖⠶⡄⠀⠀⠀⠀⠀⢀⠠⣉⣴⠠⠀⠀⠀⠀⠀⠀⣴⣶⣆
⠀⠀⠀⠀⠰⡃⢨⠴⠁⠀⠀⠀⠀⠀⠨⣇⠁⢀⡿⠀⠀⠀⠀⠀⠈⠰⢿⣿⡇⠥⠀⠀⠀⠀⠀⢻⣿⡉⢳⡆
⠀⠀⠀⠀⠀⠙⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠋⠘⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁


⠀⠀⠀⠀⢀⠤⣄⠰⡶⠀⠀⠀⠀⠀⠀⣄⣴⣄⡄⠀⠀⠀⠀⠀⢀⠖⡛⠲⡄⠀⠀⠀⠀⠀⠀⠰⠆⣠⡄
⠀⠀⠀⠀⠸⡄⠘⠲⡄⠀⠀⠀⠀⠀⠐⣯⡌⣽⠗⠀⠀⠀⠀⠀⠸⣌⠄⢄⡟⠀⠀⠀⠀⠀⠐⢿⠇⣨⣅⠂
⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠉⠹⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠀⠀⠀⠀⠀⠐⠂⠙⠋


⠀⠀⠀⠀⠀⢠⡤⣤⡀⠀⠀⠀⠀⠀⢀⡴⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⣴⠀⢤⠀⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠀⠀⠀⡘⢾⣉⡇⠀⠀⠀⠀⠀⢸⠰⣯⣿⢀⠇⠀⠀⠀⠀⠈⡷⢙⡃⠞⠀⠀⠀⠀⠀⠀⢧⠀⠙⠲⡄
⠀⠀⠀⠀⠘⠛⠀⠻⠃⠀⠀⠀⠀⠀⠈⠓⠬⠤⠊⠀⠀⠀⠀⠀⠀⠁⠏⠱⠞⠁⠀⠀⠀⠀⠀⠈⠙⠲⠶⠃


⠀⠀⠀⠀⢀⡠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀
⠀⠀⠀⠀⡣⠂⡀⣴⠆⡇⠀⠀⠀⠀⠀⣞⠉⠳⠀⠀⠀⠀⠀⠀⠀⠘⢾⡦⠃⠀⠀⠀⠀⠀⠘⢠⠖⣲⣦⢱
⠀⠀⠀⠀⠙⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⢸⣿⣏⠁⠀⠀⠀⠀⠀⠀⠨⠖⠺⠅⠀⠀⠀⠀⠀⠘⠢⠭⠉⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠻⢿⣿⠿⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣾⣎⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣴⣿⣤⣤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⡿⢿⣯⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣤⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⣿⡟⢀⣾⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⠿⢿⠿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⣿⡟⠀⠾⢻⣿⣿⣷⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣀⠛⠛⠞⠛⠋⣠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣽⣿⣿⣷⣶⢀⣾⣿⣿⣿⣿⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠻⣿⣶⣤⣾⡿⠛⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣾⣿⣿⣿⣿⢁⣽⣿⣿⣿⣿⣿⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⡿⠛⠛⠻⣿⣶⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠁⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⣿⡁⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣀⣠⣼⣿⣷⣄⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⣴⣾⣿⣿⣿⠟⠳⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣹⣿⣿⠟⣛⡛⢿⣿⣿⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⡹⣿⡇⣾⣿⣿⡆⣿⡿⣰⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠹⢷⣬⣙⣩⣼⠿⣱⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣙⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠃⠀⠉⠛⠋⠁⠀⠻⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣷⣶⣄⠀⠀⠀⠀⢀⡾⠋⠁⣤⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⡻⣿⣿⣿⣿⣿⡆⠀⣠⠴⠋⠀⠀⠀⠓⢤⡀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠑⢻⣿⣿⣿⣿⢿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⣿⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⣹⣻⣿⣷⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠃⠉⠙⢻⣿⣿⣿⣟⢿⣿⡿⣿⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⢦⣄⡀⠀⣠⡤⠂⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢺⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⠇⠁⠑⠺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⢿⣿⣿⣿⣻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⣿⢿⣿⣿⣹⣿⣿⣿⣿⣿⣿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⠟⠀⠉⠨⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣌⣵⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠻⠏⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣷⣶⣶⣶⣶⣶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⢿⣿⣿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠀⠀⣀⡀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠶⠶⠀⠀⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⡖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣀⣀⣀⣀⣀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠉⠉⠉⠉⠉⠉⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡿⠛⠛⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⡿⣿⠿⠿⠿⣿

⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣀⡄⠊⠁⠀⠀⠀⠀⢹⡄
⠀⠀⠑⠢⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠘⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⠶⠒⠂⠸⠋⠈⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠀⠀⠀⠙⠛⠐⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢠⠀⠀⠠⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠛⠓⣄⠀⠀⠀⠀⠀⣀⡤
⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡀⠀⠀⠀⠀⣀⠀⠀⠉⠑⠄
⠘⡀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠘⢄⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠧⠀⠀⠀⠀⠈⠳⣤
⠀⠃⢑⠀⠀⠀⠀⠈⠩⡓⡀⠀⠈⠀⠀⠀⠘⣆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢨⠠⠄⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠄⠀⠈⢧⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⣾⡆
⠀⠀⡆⠈⡄⠀⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠁⠵⡀
⠀⠀⠁⠁⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠆⠄⠀⠀⠀⡄⠀⠀⠀⠀⠘⠶⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠨⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠡⢳⡀
⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⡀⢉⣄⠀⠀⠀⠀⣡⢻
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢁⣈⣀⣁⡴⠚⠁⢠⠀⠀⠀⠀⢂
⠧⢄⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈⠋⠉⠀⣀⡠⠾⠋⠇
⠤⠆⠀⠀⠁⠀⠀⠀⠀⢰⠘⠁⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠸⣉⣉⡁⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠉⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠑⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠒⠛⠛⠓⠲⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢛⣩⣤⣶⣶⣾⠛⠋⠉⠀⠀⠉⠙⢛⣿⣶⣶⣤⣍⣙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣴⣾⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⣿⣿⣿⣦⣍⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⣾⣿⣿⣿⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⣿⣿⣿⣿⣿⣷⣦⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⣾⣿⣿⣿⣿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⢰⡸⣿⣿⣿⣿⡯⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢽⣿⣿⣿⣿⡟⡄⣠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⡆⣧⢿⣿⣗⠪⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⣺⣿⣿⢱⠃⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣇⢸⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟⡟⢸⣿⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⣿⠀⣿⣿⣿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⢿⣿⢿⡇⢸⡏⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⠀⣿⣞⠻⠿⠶⠯⣥⣤⣀⠀⢰⡏⣆⠄⠀⠀⡀⠀⠀⡄⣾⣆⠀⢀⣠⣤⡭⠷⠾⠿⣫⣿⠀⣿⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣿⠀⣿⡿⠟⠒⠀⠀⠀⠀⠠⣄⣼⡁⠽⢰⢸⣠⡇⣿⣦⠐⠇⣿⣠⡔⠀⠀⠀⠐⠒⠚⠿⣿⡀⣿⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⣿⠠⢋⣶⠀⠀⣀⣀⠀⠉⠉⠙⠻⠷⣦⣾⣿⣿⣿⣿⣿⣶⠾⠟⠛⠉⠉⠁⣀⣀⠀⠀⣰⣌⠇⢻⣇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⢰⣿⣧⣾⣿⣿⣿⣿⣿⣶⣄⡈⡑⢄⡹⣿⣿⣿⣿⡟⣡⢄⡁⣠⣴⣾⣿⣿⣿⣿⣿⣮⣻⣦⠘⣿⠃
⠀⢀⠀⡀⠀⠀⢀⠀⣀⠀⢠⣏⣀⢨⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣝⣷⣽⣿⣽⣿⣽⣯⣵⣫⣽⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣏⣀⡙⣦⣀⢀⠀⠀⠀⡀⢀⡀
⠀⠈⠈⠁⠈⠁⠈⠉⠉⠀⢹⣙⣿⡏⡛⢙⣿⣿⣿⣀⣘⣻⣿⣿⣿⣿⣿⣿⣋⠈⠉⢛⣿⣿⣿⣿⣿⣿⡛⠄⣘⣿⣿⣿⡟⠙⠋⣿⢇⣿⠁⠈⠈⠀⠀⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⠇⣤⣰⡌⣿⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢰⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡄⠛⠈⠲⠶⠿⢿⣛⣛⣿⣿⣟⣛⣣⣿⠏⣸⣿⣿⣷⢘⣿⣮⣛⣛⡿⣿⣟⣛⡻⠿⠷⠖⡃⠘⢁⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣿⣿⠀⠀⢽⣿⣭⡁⠀⠉⠉⢻⣿⣿⣿⣿⢃⣿⣿⣿⣿⡎⢿⣿⣿⣿⣿⠋⠉⠁⠀⣭⣿⣿⠇⠀⢸⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠉⣿⢇⣾⣿⣿⣿⣿⣿⡌⣿⡍⠉⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⣹⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡷⣦⣦⣤⣶⣲⠦⢆⣀⡀⠀⠀⢸⡜⣿⣿⣿⢿⣿⣿⢇⡟⠀⠀⠀⣀⡐⡤⣶⡶⣤⣤⣶⢾⣻⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣻⡿⣿⣿⣛⣥⣾⣿⣿⣿⣷⣆⠈⠙⠘⠿⢻⡟⠿⠏⠚⠁⢠⣾⣿⣿⣿⣿⣮⣛⣻⣿⡿⣿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⢿⠉⠉⠁⣿⢿⣿⢽⡇⠀⠀⢀⠘⠇⡀⠀⠀⢠⣿⣿⡿⣿⡇⠛⢻⣿⢿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⠀⠁⢹⡿⣷⣼⣷⣆⣼⣬⡀⠀⣧⣹⣤⣼⣿⣴⣿⣿⠇⠀⠈⠈⠀⠀⠀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⣿⡏⢿⢻⠙⢿⢳⡶⢻⠋⡟⢹⠙⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣼⣆⡘⠀⡇⢸⣀⣧⣼⡾⠻⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣾⣿⣿⠟⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠉⠋⠘⠉⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⡂⡀⠀⠀⠀⣠⠤⢤⣄⠀⠀⢰⢐⠰⠠⠄⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠂⠀⠬⢠⡔⠲⡅⠀⠸⠈⠀⠀⠐
⠀⠀⠀⡄⠤⠀⠤⠤⢠⠈⠐⠀⠀⢠⢠⠄⠀⠀⠀⠀⠦⠠⠀⡆⢀⠓⠚⡂⠀⠀⠀⠈⠀⠀⠀⠠⠁⠀⢱⠀⠀⠀⠀⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⢠⠘⠀⠀⢠⠰⡈⢂⠆⠀⢀⠀⠀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⠀⢀⠃⠀⠀⠀⠁⠀⠀⠀⠈⣠⢤⠤⠆⠀⠤⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⢘⠀⠂⠀⠀⠀⠀⠂⠈⠀⣿⠀⡜⠃⠀⠀⠀⠀⠀⠀⠰⠋⠡⠁⠀⠀⠀⠀⠈⡆⢀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠐⠀⢀⡢⠀⣠⠀⠠⠀⠀⠀⠀⡄⠀⡑
⠀⠈⠐⠅⠀⠄⢀⢤⠪⠀⠀⠀⠀⠀⠑⠠⢄⢀⠀⠤⠂⠁⠀⣿⠸⠀⡀⠊⢉⣰⡆⠁⣀⠀⠀⠀⠑⠀⢀⣀⠠⠊⠔⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠚⠀⠀⠀⠀⡿⠐⠉⠒⢎⠉⠤⠆⡁⠀⠀⠀⠄⠀⠀⠀⠱⠤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠠⠈⠁⡠⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠈⠀⣀⠀⠊
⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠢⠀⠀⠈
⠀⠀⠳⢬⣀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠉⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⡆
⠀⠀⠀⢀⠀⠀⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⡄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠁
⠀⠀⠀⡒⣀⣈⣉⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⣀⠉⠂⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠁⢷⡰⠁⠀⣅⡇⠀⠀⠀⠤⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠸⠊⠉⠈⠉⠙⠆⠀⠐⠶⠔⣇⢬⠐⠳
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢲⠖⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠑⠒⠊⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢘⠉⠉⠉⠉⠙⠃⠀⢂⣠⣤⠁⠁
⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⢀⣃⠀⠀⢱⣀⣤⣀⠂⠠⡸⡼⣚⡘
⠀⠀⠀⠀⠉⠛⠒⠶⠒⠚⠉⡐⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⣿⡉⠁⠀⠈⢡⠅⠁⠀⠠⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣄⣠⡄⠀⠀⠀⡉⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁

⠶⠚⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢌⡑⢌⣿⣿⢷⣝⡡⡢⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡤⠶⢛⣉
⠞⣋⣩⡄⠀⠀⠆⣠⠆⣠⠆⣠⠦⣹⠦⣹⠦⣽⣿⣻⢮⣼⠔⣡⡦⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⣋⣥
⣛⣩⣤⠦⠀⠀⠀⠁⠞⣡⡾⣍⡷⣍⡶⣍⡶⣿⣿⣝⡷⣡⣾⢋⡴⣿⣿⣳⣀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣏⣭⣴
⣏⣥⠶⠞⡀⠀⠀⠠⡘⢮⡽⢯⡵⢯⡵⢏⡵⢿⣿⣯⣿⢟⡵⢫⣾⢿⣿⣿⡿⣿⣽⣹⣱⣱⣻⣹⣻⣿⣿⣿⣿
⣽⣟⣾⣿⣅⡠⣌⢳⣌⣲⡌⣫⣽⣯⣽⣯⣽⣯⣹⣮⣱⠊⣠⠊⣠⢟⣛⣻⠑⠳⠳⢣⢣⢣⢣⢧⢯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⣿⢿⣷⠿⣫⠞⣡⠞⣡⣟⣻⣿⠁⠀⠀⠀⠀⠀⠉⠋⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠞⢁⠜⢁⠜⢁⠔⢁⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⢋⡴⢋⡴⢋⣹⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠋⡠⠋⣹⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠢⠢⠘⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠞⣩⣭⣽⣄⣀⣀⣀⣀⣄⣄⣤⣄⣤⣌⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣩⣽⣿⣿⣿⣿⢿⠝⠿⠿⠟⠽⠽⠿⠝⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢰⡄⢻⣄⢿⣿⣿⡿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⣶⠈⣷⡈⣷⡈⢷⡈⢷⣬⣛⢿⣿⣿⡿⣫⠿
⠀⠀⠀⣀⠠⣄⠰⣦⠸⣇⢸⣧⢸⣧⡸⣧⠘⣧⠘⣧⠘⣧⣸⣽⣻⢿⣿⣿⣟⣷⠇⠀⠀⠀⠀⠀⠀⢀⡤
⡄⢻⡄⢻⡄⢻⡆⢻⡆⢹⡆⢻⡇⢹⣇⢹⣆⠻⣶⠹⣇⣽⣾⣝⡻⣷⣾⣿⣯⡞⠀⠀⠀⠀⠀⠀⢰⣿⠶⢿⣏
⢿⡈⢿⡀⢿⡀⢿⡀⢿⡄⢿⡄⢻⡄⢻⡄⢻⣾⣿⣶⣿⣿⣿⣿⡿⣿⣯⣿⡻⡝⣆⢧⢳⡸⡎⣷⢮⠶⢺⣋⣿
⡘⣧⠘⣷⠘⣷⠘⣷⠈⣷⠈⣷⡈⣷⡈⢿⡈⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⣿⢧⢹⡸⡌⣇⢧⠹⠘⠶⠞⣋⣻⣿
⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢻⡹⡜⣇⢧⠹⠸⠜⠃⠃⠉⠀⠀⠀⠀⠐⢛⣩⡤⠶
⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡟⠆⠃⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣥⠶⠞
⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⠢⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⠴⠞⣛
⣶⣚⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣌⣿⣯⣷⣽⣣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣴⣖⣛⣩
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡆⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠄⠀⠀⠀⠀⠎⠉⠉⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⡄⠀⠀⠀⠀⠊⠉⠉⠉⣩⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣱⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀
⡇⠀⠀⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠄
⡇⠀⠀⠀⡾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠹⡿⠋⠸⣿⣦⣄
⡇⠀⠀⠀⠱⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣺⠙⠻⣷⡄
⡇⠀⠀⠀⠀⠠⣿⢷⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣱⡿⠛⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡯⣀⣀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠈⠚⠛⠓⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠣⠄⠀⠀⠀⠠⠍⠋⠤⠤⠤⠄⠠⠠⠤⠁⠀⠀⠀⠀⠣⠄⠀⠀⠄⠄⠀⠬⠍⠠⠄⠄⠀⠀⠤


⡆⠀⠁⠁⠉⠉⢈⣄⠉⠉⠉⠉⠁⠉⠐⡄⠀⠀⠀⠀⠆⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⡄⠀⠀⠀⠀⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑
⡇⠀⠀⠀⠀⠀⢈⡍⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⡇⠀⠀⠀⠀⢀⣸⡃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣤⣦⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⡇⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠋
⡇⠀⠀⠀⠈⣿⣿⡿⠁⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⠦⠄⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⠛⣿⡇
⡇⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡾⠋⠁
⡇⠀⠀⢿⣿⣧⣾⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠇
⠣⠀⠀⠀⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄
⡇⢠⡿⣿⣶⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣠⣾⣿⡿⣷⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⡆
⡇⠀⠀⠀⠀⢉⣿⣷⣧⣏⡔⡠⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣬⣾⡿⢛⡩⢼⡆
⡇⠀⢠⡶⢤⣾⣴⢯⣟⡿⣿⣷⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣽⣿⣷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⢿⣿⣷⣵⡾⠗⠃
⡇⠀⠀⠈⠙⠾⣿⣿⣾⣵⡃⣨⡟⠲⠆⡆⠀⠀⠀⠀⠀⠀⣴⠛⡋⠸⢛⣽⠿⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⣾⠋⠀⢀⣼⡿⠟⠋
⡇⠀⠀⠀⠀⠀⠀⠉⠑⠋⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⢤⡼⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠳⠤⠞⠁
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠃⠀⠀⠀⠀⠆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⡶⠶⠶⠿⠿⠟⠻⠛⠛⠻⠿⠿⠿⠷⠶⠶⣶⣶⣤⣤⣀⣀⡀
⠀⠀⢀⣠⣴⡶⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀⣀⣴⡾⠿⣶⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢷⣦⣄⡀
⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠛⠀⠀⠈⠛⠛⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⣀⣤⣶⠶⠾⠟⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠿⠶⢶⣦⣤⡀⠀⠀⠀⠀⢸
⢸⣷⠀⠀⣠⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⡀⠀⣾
⢸⣿⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿
⠘⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⣿⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⣿⡇⢸⣿⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿
⠀⢻⡇⢸⣿⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⣿
⠀⢸⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟
⠀⢸⣷⠀⣿⡇⠀⠚⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠸⣿⠀⣿⡇⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⣿⡀⢸⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⠃
⠀⠀⣿⡇⢸⡇⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿
⠀⠀⢿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿
⠀⠀⢸⣧⠸⣿⠀⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿
⠀⠀⢸⣿⠀⣿⠀⢤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⣿⡀⠘⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣼⡇
⠀⠀⠀⣿⡀⣿⡇⢤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⢿⡇⢿⡇⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣿⠃
⠀⠀⠀⢸⡇⢸⣇⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⢸⡇⢸⣿⢠⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿
⠀⠀⠀⢸⣿⢸⣿⠀⣀⣀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠸⣿⠈⢿⣎⡉⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢸⡏
⠀⠀⠀⠀⠹⢷⣤⣉⡛⠻⢷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡶⠿⠛⣋⣡⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣶⣶⣦⣭⣭⣭⣭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣩⣭⣭⣭⣵⣶⢶⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠈⠉⠉⠉⠙⠙⠛⠛⠛⠛⠙⠛⠉⠉⠉⠉⠀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠛⠛⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣶⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣍⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡴⢦⡀
⠀⠀⠀⠀⠀⠀⣤⡿⠛⠻⣤⠀⠀⢠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠼⣇⣀⣠⡿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢤⡾⠋⠛⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⣄⠀⢸⡇⠀⠀⠾⢧⣀⣠⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣴⠖⠚⠛⠛⠋⠉⠉⠁⠀⠀⠙⣦⣸⠃⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠁⣀⣠⣤⠤⠀⠀⠠⣤⣤⣀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣀⠀⠁⣤⣤⡄⠀⠀⢠⣤⣤⠈⠀⣈⢻⡆
⠀⠀⠀⠀⠀⠀⢐⣦⣤⣴⡂⠀⠘⣧⣿⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⣿⡾⠀⠀⠀⢰⣦⣤⣴⡂
⠀⠀⠀⠀⠀⢲⡟⠁⠀⠈⢻⡞⠀⠈⢹⡆⠀⠀⠀⠀⠶⠶⠀⠀⠀⠀⣸⠏⠀⠀⠀⢲⡟⠁⠀⠈⢻⡖
⠀⠀⠀⠀⠀⠼⢧⡀⠀⢀⣼⠧⠀⠀⠀⠙⣦⡴⠶⠶⠖⠲⠶⠶⢦⣴⠋⠀⠀⠀⠀⠼⣧⡀⠀⢀⣼⠧
⠀⠀⠀⠀⠀⠀⠸⠟⠛⠻⠇⠀⠀⠀⠀⢠⡏⠀⢠⡞⠛⠛⠲⣤⡀⢹⡀⠀⠀⠀⠀⠀⠸⠟⠛⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⢿⠀⠀⠀⠀⠈⠻⣼⣇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠚⠛⠋⢻⣽⠀⠀⠘⣧⡀⠀⠀⠀⠀⠹⣯⡟⠙⠛⠒⠶⠶⢤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠈⠳⣄⡀⠀⢀⣠⣿⡁⠀⠀⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⣟⢲⣏⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣄⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣿⠉⠹⣆⠀⠀⠈⢷⡀⣠⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠘⢷⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠻⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣏⣀⣸⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠀⢀⡴⢞⣋⣭⣭⣙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠛⡟⠛⢻⡟⠛⠋⠁⠀⠀⠀⢠⠟⣴⠋⠁⠀⠀⠈⠻⣎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠷⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡶⣾⠃⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁⠀⠀⠀⠀⠀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⣤⣀⣀⣀⣤⠞⣱⠏⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣭⣭⡭⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣄⣤⢠⢠⣄⠀⠠⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠋⠈⠈⠃⠈⠀⠀⠀⠀⠀⠐
⣿⣿⣻⣻⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣟⣻⣿⣛⣛⣻