Cute baby Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠛⢿⣿⣿⣷⣶⣦⣤⠋⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⢔⡭⠂⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⡟⠀⠈⠳⣄⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣞⢕⡵⠎⣀⠄⠀⠀⠀⠀⠀⢤⣾⣿⣿⡿⠿⣶⣶⣾⡁⠀⠱⠀⠒⠒⠲⢦⣄
⠀⠀⠀⠀⠀⢀⢠⡟⣠⣿⣴⣯⠷⠭⠶⠤⠤⠦⠬⠿⠟⣻⠏⠀⠀⠀⠙⢿⣿⡖⠓⣖⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⢸⡟⣰⡿⣛⣵⣶⠾⠖⠒⢶⣶⣤⣶⡶⠿⠋⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠷⠰⣦⠃
⠀⠀⠀⠀⠀⢸⣇⢟⣾⣟⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⢀⡞
⠀⠀⠀⠀⠀⠘⡏⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⡟⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠊⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢿⣿⣇⡂
⠀⠀⠀⠀⠀⠰⢿⣿⣿⠀⠀⣠⣶⣶⢶⣄⠀⠀⠀⠀⠀⠀⢠⡶⣿⣷⣦⡀⠀⢸⡿⢟⠣⡀
⠀⠀⠀⠀⢠⣦⡶⢮⣻⠀⠀⠿⠉⢿⣿⡿⠃⠀⠀⠀⠀⠀⠛⢿⡿⠃⠛⠁⠀⢩⣎⢹⣷⣰
⠀⠀⠀⠀⠘⢻⡓⠸⢻⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⢈⡿⠇⠐
⠀⠀⠀⠀⠀⠀⠙⠢⣤⡆⠀⠀⠀⠀⠀⠀⠀⠂⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⢰⡧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⡀⢤⡤⠄⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠤⣄⣠⠤⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⠉⠉⠉⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠊⠉⠉⠑⠒⠠⢀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⡀⠀⠀⠈⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠁⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⠁⢀⠄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⠂⣰⣿⣿⠿⡛⠋⢋⡩⠟⠉⠉⢉⠋⠛⡿⢿⣿⣿⣷⡄⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠸⠀⠁⣼⣿⠟⠁⢀⡠⠀⠁⠀⠀⣠⠖⠁⠀⠼⠂⠀⢠⡉⠛⢿⣦⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡧⠀⣸⣿⠏⣰⠞⠉⠀⠠⠤⠴⠞⠃⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠙⣧⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⢰⣿⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡆⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⣿⡆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿⣿⣯⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⣼⣿⢿⣿⠸⠀⢀⢞⣽⡿⣷⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡝⢆⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢸⠀⣿⣇⢾⣿⠀⠀⠉⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠠⠹⣿⣿⠇⠈⠀⠀⣸⡇⣽⣷⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⣿⣮⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⠟⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠟⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠢⣈⠙⠛⢛⣉⣤⣶⡖⠈⠛⣿⣏⡽⠋⣀⣶⣦⣬⣉⣉⣉⣠⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠉⡏⠉⠉⠉⠛⣷⢺⣧⣌⣉⣠⡌⢿⠿⠿⠛⠛⠻⣿⡿⠟⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣄⠁⠀⠀⠀⡼⠇⡞⠀⢀⡄⠎⠀⢹⠀⠀⠀⠀⠰⢱⠁⠀⠠⠋⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠈⠀⠀⠁⠀⠎⠀⠈⡄⠈⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣺⣿⠇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢦⣀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⡀⠀⣀⡴⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢛⣉⠉⠙⢻⡾⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢷⡟⠋⠉⣉⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠉⠉⠉⠓⣾⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⣷⠚⠉⠉⠉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⡟⠀⣀⣀⡀⠀⠈⣿⣿⣿⣿⠀⠀⢀⣀⣀⠀⢻⣤⣠⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⣅⣀⣸⡆⠀⣿⣿⣿⣿⠀⢰⣇⣀⣨⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠛⠋⠀⢀⣿⣿⣿⣿⠀⠀⠙⠛⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⢳⡄⠀⠀⢀⡾⢁⡴⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣤⣼⣧⣀⣛⣀⣀⣛⣀⣼⣧⡤⠶⢯⠰⡏⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡶⠶⠾⠛⠋⠀⠀⠀⣼⣿⡿⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠈⠉⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢻⣿⣿⣿⣿⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⢸⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢹⠀⠀⠙⠛⠛⢩⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⣀⣀⣀⣠⡇⠀⠀⠀⠀⢸⣄⣀⣀⣀⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡷⠤⣤⣤⠤⢾⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⡤⠤⢳⡐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠂⠀⠀⠈⠀⠀⠁⠀⠈⡦⢄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⢄⣠⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢈⡄⡀⠐⠀⠀⠀⡠⠐⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⣾⣀⣠⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠘⠁⠇⡄⠀⢀⡊⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢣⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣴⠃⠐⠞⠢⣈⣁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢿⣿⣿⣿⣿⣁⠀⣨⡟⠁⠀⠀⠀⠀⠀⢻⣿⠟⠻⣿⣿⣿⣷⣤⡇⠀⢀⠶⠁⡠⠂⠀⡔⠣⠤⡈⠩⠍⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠙⠿⢿⣿⡿⠟⠋⠀⠀⢠⣶⣶⣦⡄⠘⢿⣦⣤⣾⣿⣿⣿⡿⠁⡠⠃⢀⠀⠀⡠⠊⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣛⠩⡅⠀⠀⠙⠛⠿⠟⠛⢋⡀⠊⢀⠔⡁⠀⠞⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣤⣄⣀⡀⠀⠀⡃⠀⠀⢨⠀⠀⠀⣀⣠⣴⣾⣿⣿⣆⣀⡴⠆⠠⠞⣓⠄⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠇⠀⠀⠸⠀⣸⣿⣿⣿⣿⣿⣿⣿⡍⠋⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣘⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠶⢦⣄⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⣿⣿⣿⡿⠿⢋⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠘⣧⣄⣴⣿⣿⣦⡙⠻⢿⣿⣿⣿⣿⡟⠁⠀⠘⣉⠉⠉⠛⠀⣠⣶⣿⣿⣿⣏⠀⢘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣶⣶⠟⠋⠉⠙⣿⣿⣷⠀⠀⠈⢉⡉⠁⢰⠀⠀⠀⡁⣠⠀⣠⣾⣿⠟⠉⠀⠀⠈⢻⡟⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠀⠀⠀⠀⣿⣿⣿⣇⣤⡀⠀⢃⠠⢸⠂⠀⠀⡇⡟⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢷⠖⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⡀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣄⡀⣢⠸⡀⠀⠀⠀⢁⢹⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣦⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣶⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠻⡇⢀⣀⣸⠵⠀⠈⠙⠿⢿⣷⣶⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣠⠋⣠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡦⣡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡰⠋⣠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡱⡡⠚⠁
⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠛⠛⠛⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣷⣀⣀
⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠡⠖⠚⠃⠀⠀⠀⠀⠐⠛⠓⠂⠀⠹⡆
⠀⠀⢸⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠉⣉⠑⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢴⣶⡖⢠⠀⠀⠀⠀⣰⣾⣁⣶⠀⠀⡇
⠀⠀⡜⠀⢸⣿⣿⣁⡄⠀⠀⠀⢰⡿⢇⣼⡇⠀⡿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿⣽⡿⠀⣀⣀⡀⠹⢷⣿⠟⠀⢀⣧⣄
⡼⠻⠛⠀⠀⠻⠶⠿⠁⢠⠤⡄⠈⠳⠿⠟⠀⢀⡉⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⡴⣿⢧⠀⠀⠀⠀⠀⣀⠥⠬⣅⠀⠀⠀⣀⣀⡞⢳⣌
⢃⣀⣐⢲⣀⣀⠀⠀⢠⠴⠒⢦⡀⣀⣠⡔⢢⠃⠔⠲⠞⠀⠀⠀⠀⠀⠀⠀⠘⡟⠓⠀⡗⠒⡄⠀⢰⠃⣴⢢⠈⡇⠀⠀⢇⠘⢁⠀⠠
⢠⠦⠀⡀⠃⣸⣿⣆⡇⣴⣃⡄⢹⣿⣿⣷⠀⠐⠀⢉⠂⠀⠀⠀⠀⠀⠀⠀⢘⡉⠀⠀⠀⣸⣄⣀⠀⠲⣙⣋⡗⢁⣀⣤⣼⣄⠀⠀⠐
⠀⠂⠀⢀⣴⣿⣿⣿⣿⣯⣭⣵⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⣀⣤⣙⢿⣿⣿⣿⣟⣙⣛⣿⣿⣿⡿⣃⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⡹⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⡿⣡⣤⣤⡀
⠀⣸⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠈⠙⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠉⠻⣿⣿⣿⡿⠃
⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⡴⠂⠀⠀⠀⢄⠑⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡇⢀⣿⣾⣿⣷⠴⣀⡤⣶⣴⢃⢀⠳⣾⣷⣄⠘⣦
⠀⠀⠀⠀⠀⠀⢠⡇⣾⣀⠈⣳⣿⣿⣶⣾⠿⠿⣟⣻⡽⢞⡵⢣⢸⡇⠡⡘⣿⣿⣿⣿⣇⠀⡇⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣬⣷⣶⣶⣾⣿⠿⢛⣩⣶⠟⢡⣆⣾⣷⠀⢳⠘⣿⡳⡙⢿⣆⢹⢸
⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⡿⣻⣿⡿⢋⣡⣶⠟⢋⢀⣴⢏⣾⡿⣿⠀⢸⡇⢹⠻⡄⠈⣿⠘⣸
⠀⠀⠀⢀⣀⣠⣴⣾⣿⠟⣡⣾⠟⣡⠶⠛⢉⠠⢂⣵⠟⣵⣿⣿⢱⡿⠀⣸⡇⢸⡇⣷⠀⠋⠰⢃⠀⡀⠀⣦⣄⡄
⠀⠀⠀⠀⠀⢩⣟⡿⢡⠞⠉⠰⠋⠁⠀⣀⣤⡾⢛⣵⣿⣿⡿⢁⣿⠃⠀⣿⡇⣸⠀⣿⡆⠱⣔⢤⣀⣹⣦⣿⣿⠄
⠀⠀⠀⠀⢠⠟⡼⠁⠀⠀⠀⠠⠴⠶⢛⣋⣥⣾⢟⣻⣾⠟⣠⣾⠏⠀⣼⡟⡠⠃⢀⣿⡁⠀⠙⢷⣿⣿⡿⠟⣋
⠀⠀⠀⠀⡞⠀⠀⠀⠀⢰⣔⣲⣶⡿⠿⢛⣩⣴⠿⠋⣁⣴⡿⠁⢀⣼⠟⠀⠀⢀⠞⢈⡄⠀⡆⢀⢹⡛⠉⠉⠁
⠀⠀⠀⠀⠇⣼⠀⣤⠀⢸⠀⠀⠴⠶⠟⢛⣉⣡⣴⡾⠟⠉⢀⣴⡿⠋⠀⠤⠒⠁⣠⠞⠀⠀⣧⢸⡆⣧
⠀⠀⠀⠀⠀⢿⢸⡇⠀⠀⠩⢽⣿⣿⣿⠿⠟⠿⣋⣤⡴⠾⠛⠉⠀⠀⠀⠤⠴⠚⠁⠀⠀⠀⣿⢸⣇⡏
⠀⠀⠀⠀⣀⣘⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⢛⣁
⠀⠀⠀⢰⣿⡟⠛⢧⠀⠀⠀⠀⠠⠶⠖⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠶⠄⠀⠀⠀⠈⡰⠛⠛⣿⣧
⠀⠀⠀⠸⣿⡇⣴⣦⡀⠀⠀⢀⣀⣠⡴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄⣠⡀⠀⠀⠀⣤⣶⣸⣿⡇
⠀⠀⠀⠀⠹⣿⡇⠈⣧⠀⠀⠹⢿⡿⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⡿⠿⠀⠀⢸⠃⢸⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣦⣽⡆⠀⠀⠛⠃⠘⠿⠋⠀⠀⢀⡀⠀⣀⠀⠀⠀⠺⠿⠋⠛⠃⠀⠀⣮⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⣰⣬⣿⠿⠋

⠀⠀⠀⠀⣠⣴⠶⠟⠛⠶⠀⠀⠀⠠⠤⠴⠶⠶⠶⠿⠿⠿⠿⠶⠶⣤⣄⠀⠀⢀⣄
⠀⠀⣠⠞⠋⠀⣀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠈⢷⡀
⠀⠐⠁⠀⠀⠈⣽⠟⠀⠀⢀⡾⣠⣾⡟⢠⡆⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣷⣶⠖⠋⠁
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠼⢱⣿⡟⠀⠙⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⠁
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡄⢤⡀⠀⠀⢸⡇
⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⠀⠀⠈⠢⠙⢦⡀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⡿⠀⠀⠀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⢸⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣴⣿⣿⣷⡀⠀⠀⠀⣼⠁
⠀⠀⢀⣼⣦⡄⠈⠀⣀⠀⠀⠀⠀⠀⣤⠤⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠁⡇⠀⢀⣾⠏
⠀⢀⣿⠋⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠶⠶⠶⠶⠄⠀⠀⠀⠀⠉⠀⢀⣴⣿⣴⣿⠏
⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣾⣿⣿⣿⠟⠁
⠀⠀⢿⣄⠀⠀⠀⠱⢶⡶⠶⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⠿⠛⠁
⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣤⣤⣤⣴⣶⣶⡶⠿⠛⠉⠀⠈⠉⠙⢦⡉⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡿⠉⠀⠉⠙⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⠳⣤⣀⣀⣠⣴⡾⠿⠛⠛⠉⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢻⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇⠀⢸⡆⠀⠀⢀⣠⠄⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⢀⡟⢁⣤⡾⠋⠁⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠇⢈⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣴⣏⣴⠟⠁⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣸⠁⠀⣀⣠⡴⠋⣹⠟⠁⠀⢀⣠⣴⣶⡿⠿⠿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡿⠛⠉⠅⣰⣏⣠⣴⡾⠛⠉⠉⠀⠀⠀⠀⠘⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡏⠛⠛⠻⠿⣷⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⢱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡾⢻⣷⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠁⠀⠻⣿⣦⡀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣖⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠗⢀⣠⣶⠀⠛⠓⠀⠀⢀⡎⠀⠀⠀⠈⠙⠛⠷⠾⠿⠿⣿⣦⣤⣀⣀⣤⣴⣾⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣤⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⢾⡿⠿⠿⠿⠛⠿⠷⠶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⢀⡠⠖⠉⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⣄⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣴⠟⠋⠀⠀⣀⠤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠑⠒⠤⣀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⣴⠿⠛⠛⠉⠉⠉⠉⠉⠛⠒⠶⡾⠋⠚⠓⢄⠀⠀⠀⠀⣀⠤⠂⠛⠙⠓⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠈⢷⣄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⡠⠖⠉⠀⠀⠀⠀⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠻⣧
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⡠⠞⠛⠓⠲⢖⠛⠛⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠹⣷⡀
⠀⠀⠈⢿⡄⠀⠀⠀⣠⠴⠒⠈⠉⠉⠉⣤⠛⠙⢦⡀⠀⠀⠀⠀⠀⠙⢏⠉⠉⠉⠉⠉⠁⠐⠒⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢹⣧
⠀⠀⠀⠈⠻⣦⣀⡎⠁⠀⠀⢀⡠⠔⣉⡿⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⡀⠀⠀⠀⠀⠀⠻⣿⡆
⠀⠀⠀⠀⠀⠈⣿⡷⠶⠶⠚⢉⠴⠋⠁⠃⠀⠀⠀⠀⢸⠙⢦⡀⠀⠀⢀⣀⣀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⠚⠁⠀⠀⢰⡇⠀⠀⠀⠀⡈⠀⠀⠙⢢⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⣸⣿⠀⡴⠁⠀⠀⠀⠀⢸⠀⣠⠶⡀⠀⡇⠀⠀⠀⠀⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢻⣇
⠀⠀⠀⢀⣾⠏⢹⡎⠁⠀⠀⠀⠀⠀⢰⡿⠗⠁⠙⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣦⡀⠀⠀⠀⠀⠀⠀⢱⡎⠻⣷
⠀⠀⠀⢸⡏⠀⠘⠁⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⡇
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠻⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠸⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠷⢶⣦⠤⠤⠤⠀⠀⠀⠀⠤⠤⠤⣴⠶⠟⠛⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠹⡦⢀⡀⠀⠀⠀⠀⢀⣀⢴⠏⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠉⠉⠉⠉⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠠⠤⠄⠒⠒⠂⠉⣿⠀⠀⠀⠀⠀⠀⣿⠉⠐⠒⠒⠠⠤⢄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⣿⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠾⠀⣿⠷⣶⣤⡀
⠀⠀⠀⣀⣤⣴⡟⠛⣷⡄⣠⡾⠛⠁⠀⠀⢸⠀⠘⢇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡼⠃⠀⠇⠀⠀⠈⠛⢷⣄⢠⡾⠛⢻⣦⣤⣀
⠀⣰⡟⠛⠃⠈⠁⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⡆⠀⠈⡆⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⢰⠁⠀⢰⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠈⠁⠘⠛⢻⣆
⣾⡏⠁⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠈⢹
⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢳⡀⢰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⢀⡞⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⣀⣼⠿⣦⣀⠙⢆⣃⠀⣀⠠⠤⣤⣀⡀⢀⣀⡤⠤⠄⡀⠀⣜⡴⠋⣀⣴⠿⣧⣀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢀⣠⡿⠛
⠀⠀⠀⠙⠻⢶⣦⠀⠀⠀⠀⠙⣦⡾⠛⠁⠀⠈⠛⠿⢦⣾⡟⠉⠀⠀⠀⢈⣙⣏⡁⠀⠀⠀⠉⢻⣧⡶⠿⠛⠁⠀⠈⠛⢷⣴⠋⠀⠀⠀⠀⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠹⣷⣀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⠈⣿⡿⢁⡀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⣀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣀⣿⣴⡿⠟⠻⢿⣤⣿⣀⣷⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣼⠆⡀⡟⢷⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡿⡿⣿⣿⢟⡯⠥⠤⣴⣧⢀⢈⣰⠇⠀⢐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡕⠉⠞⠀⠘⠁⠋⠀⠀⠀⠑⠺⠿⠟⠂⠙⢦⡴⠃⡷⠀⠀⣀⣤⣴⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢁⣷⣿⣿⣿⣿⣿⣿⠿⢟⣂⣤⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡯⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢶⣿⣹⡽⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣌⡿⢯⣬⣿⠏⠀⠀⠀⠀⠀⠔⠀⠐⠀⠀⠀⠀⠀⢀⣴⠤⡄⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣇⣿⣦⣶⣦⡄⠀⠀⠀⠀⢠⣿⡿⠍⠂⠀⠀⠀⠀⠉⠉⠁⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠈⣉⣭⣾⣿⣿⣿⣿⣿⣶⣔⣯⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣧
⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠘⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢅⣠⡀⠀⠀⠀⠀⠀⠘⠳⠿⢯⡽⠏⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣋⣥⣴⣾⡿⠋
⠀⣀⣭⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⠽⠶⠦⠤⠄⠀⠄⠀⠠⠔⠚⠓⢾⡉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠌⢢⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠀⠘⡟⠫⠻⣿⣿⠿⠻⠟⠛⠁
⠘⠛⠛⠛⠛⠻⢿⣿⣿⢿⣿⣿⠋⠀⠀⠀⠁⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡄⠀⠱⠭⣁⣀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠠⢎⠀⣜⣟⣏⢸⣻⠇⠀⠀⠀⠀⠈⡄⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⡇⠀⠀⠀⠀⠊⠉⠉⠉⠩⡔⢋⠀⠀⠀⠱⡀
⠀⠀⠀⢠⡌⡢⠘⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠐⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠚⠶⠶⠶⠴⠤⠤⠤⠤⠤⠤⡞⢠⠆⣠⠀⠇
⠀⠀⠀⠐⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠂⡁⠸
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣤⣤⠤⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠛⠷⣄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⣿⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢰⡿⠀⠀⠰⣟⠁⡠⠂⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⡀⠁⠀⠀⠀⢸⡇⠙⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣈⡀⠓⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠈⠁⠀⢠⡀⠀⢸⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠦⣀⠀⠀⠀⠀⠈⠑⠲⣤⣤⣤⣤⣤⡶⠋⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠳⢤⣀⠀⠀⠀⠀⠈⠛⠒⠛⠚⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⡈⠉⠛⠓⠲⠶⠦⣤⠤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠻⢇⡀⠀⠀⠀⡇⠀⠈⢓⣤⠴⠶⣄⠀⠀⢠⠶⣼⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠧⠤⠶⠋⠀⠀⢀⣼⣁⣰⠛⢆⠸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⢀⣜⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡏⠀⠸⣉⠳⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠈⠉⠓⠮⣕⠲⠤⢤⣤⣤⣤⣤⣤⠤⢶⡦⣠⡄⠀⣹⠋⠀⠀⠀⠘⠅⠀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣀⣀⣣⣀⣩⠟⠻⣖⠒⠶⠆⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⣸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣠⠖⠛⠉⠉⠙⠓⢦⡀⠀⣀⡀⠀⠀⠀⠀⠋⠁⠀⠈⣷⡴⠒⠲⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠙⠾⠡⠎⠽⢳⡄⠀⠀⣠⠔⠊⠉⢧⠀⠘⠉⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⣠⣞⣁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠶⠶⠶⠶⠲⠤⠤⠴⠋⠉⠀⠀⠀⠉⠉⠒⠒⠤⠤⠴⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⠲⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠰⠐⠈⠀⠀⠀⠀⠃⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠳⣄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠖⠃⠀⠀⠂⠈⠄
⠀⠀⠀⠠⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠀⡃⠀⠀⠠⢀⠔⠘⠀⠀⠀⠌
⠀⠀⠀⠀⠱⡀⠀⠀⢆⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠔⠉⠀⠀⠀⠈⡸
⠀⠀⠀⠀⠀⠈⢂⠀⠈⠑⠒⠒⠛⢄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠁⠀⠀⠠⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀⠄⠈⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡣⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠖⠈⠀⠁⠀⠀⠀⠀⠐⢄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡄⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢊⠁⠀⠀⠀⠣⡀⠀⠀⠇⠀⠀⠥⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⠤⠊⠒⢄⡀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⣻⣟⣣⣦⡼⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠈⠀⡙⢋⡙⠛⠟⠛⠃⠈⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣿⣆⢀⠀⣾⡇⡀⠀⠙⠓⠦⣴⡌⠟⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⣀⣿⣟⠛⠿⠿⣋⣳⠀⠀⠀⠀⠓⠲⣾⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄⢿⣻⣾⣷⡾⣿⠃⠁⠀⡤⣶⠀⢠⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡙⣿⣷⡿⠉⠁⠀⠀⠀⠁⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⢿⣇⢿⣿⡷⣷⠀⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣻⣿⣾⣟⠣⠈⢷⣦⣤⣄⣀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉⠀⠑⠀⠉⠻⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣏⣿⣿⢿⢻⢟⣭⣼⣿⡿⢫⢞⡯⣚⡽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢟⣫⣭⣿⣾⣿⣿⢻⣸⣼⣸⣸⣸⣿⣿⣿⣿⣷⣷⣪⣵⣾⣿⣷
⠀⠀⠀⠀⠀⠀⣠⡾⣛⠵⣚⣭⣶⠾⣟⡿⢟⣿⣏⣧⣯⣿⣧⢣⢳⢻⣿⢿⣿⡿⢿⣿⣿⣿⠏
⠀⠀⠀⠀⢀⣴⣿⣿⡵⣫⠵⣫⠴⣫⡵⣚⠽⣫⢟⣿⣿⣿⣿⣷⣭⢣⢯⢿⣿⣿⡳⣝⢿⣿⣷⡀
⠀⠀⠀⢠⣾⣿⣿⡿⢞⡵⣋⣵⢟⡽⣚⡵⣫⡴⣫⢟⣽⣿⣿⣿⣿⣷⣼⣿⡿⣗⢝⢦⡱⣝⢿⣿⡄
⠀⠀⢀⣾⣿⣿⣯⣴⣫⣾⣯⣾⣯⣾⣿⣾⣿⣾⣵⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣎⢫⡣⡻⣌⢦⢻⣿⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡱⡵⡹⣎⢧⢻⣧
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢱⢱⣹⣹⡼⡎⣿⡄
⠀⠀⣿⡏⠀⠉⠉⠉⠉⠉⠉⠉⣉⣉⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣸⢻⢿⢻⡞⣏⣇⣇⣿⣷⢹⣧
⠀⠀⣿⡇⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡟⡾⡾⡾⣇⣿⢻⢸⢸⣿⡜⣿
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣇⣇⣇⣿⢸⣿⣿⡟⣿⡇⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⢸⣿⣿⣇⣿⡇⣿
⠀⠀⡀⣿⠀⡛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡇⣿
⠀⢳⣽⣿⠶⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡈⠁⠀⠀⠀⠀⡰⣿⠟⠱⢋⡥⢶⣬⠻⣿⣿⣿⣷⣿
⢠⣀⣻⣿⠀⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠷⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠃⣿⣿⣿⣿⡿
⠀⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠚⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣴⡶⠞⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠁

⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⡀
⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠂⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⢀⣴⣶⣆
⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⠋⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏
⠀⠈⠛⣁⢀⣀⠀⠀⠀⣀⢀⡈⠀⠀⠀⠀⠘⠁⣔⣠⣄⠀⠀⠀⢠⣶⣀⠀⠉
⠀⠀⠰⣷⣿⣿⠇⠀⠺⣿⣷⣼⠀⠀⠀⠀⠀⠈⠻⠿⠿⠀⠀⠀⠹⢿⡽⠃⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡧⠀⠀⠀⢾⣿⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⣤⣤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠛⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⣴⣿
⠀⠀⠀⢠⣾⠧⠀⠀⠀⠻⠷⠀⠀⠀⠀⠸⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣵⣶⡆⠀⠀⣶⣾⡄⠀⠀⠀⠀⢿⣽⣶⣦⡀⠀⠀⠀⣠⣶⣺⣹⡏⠀⠀⠀⠀⢀⡤⣦⣄⠀⠀⢠⣴⢦⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⣾⣽⣿⣿⠇⠀⡀⠀⢿⣿⣿⣷⣄⠀⠀⠀⠀⠘⢿⡿⠟⠀⠀⠻⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠿⠿⠂⠈⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠆⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⢠⠸⣷⣾⡏⡀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣗⠀⠀⠀⠊⠑⠈⠁⢸⣿⡿⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠁⠈⢿⡆⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⢀⡴⣿⣿⡇⠀⠀⣾⣿⣿⢦⣦⠀⠀⠀⠈⠛⣛⢓⠀⠀⠀⠀⣚⣛⠛⠁⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠉⠋⠉⠉⠀⠀⢀⡀⠉⠉⠉⠀⠀⠀⠀⠀⢸⢴⣾⣿⠀⠀⣾⣿⡶⡇⠀⠀⠀⠀⠀⠀⣠⣥⡀⠀⠀⢀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⣶⣶⠘⠛⠃⠁⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠛⠻⠿⠿⠿⠿⠛⠋⠉⠉⠻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠥⠶⢶⣦⣤⣄⣀⠀⠀⠀⢀⣀⣤⣴⣶⠶⠦⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠰⣿⣿⣿⣿⠯⠙⠀⠀⠀⠘⠉⢿⣿⣿⣿⡷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠠⣶⣿⡿⢿⣷⡦⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⡿⠿⠿⠿⠿⠿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢹⢟⣫⣿⣾⠿⠿⠿⠿⠿⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠟⣛⣛⣛⣛⣿⢿⣷⣶⣤⣀⠀⣀⣀
⠀⠀⢀⣤⡄⠀⣴⣿⣿⣿⡷⠶⠦⠀⠀⠀⠀⠀⠉⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣫⣶⣿⣿⣿⣿⣿⠟⠛⠛⠉⠿⠿⣯⣬⡙⠿⡇
⠀⢠⣿⣴⣗⣸⠟⣿⣷⣍⣛⣁⣀⣀⣀⡤⠀⠀⠀⠀⠘⠃⠀⠐⢶⣤⡀⠀⠀⣠⣿⣿⣿⢫⣾⣿⣿⣿⣿⣿⠅⢀⣀⡀⠀⠀⠀⠸⢿⣿⡆⠀⠃
⠀⢺⣿⠉⢨⡴⣤⣽⣿⣦⡬⠭⠭⠥⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⢰⣿⣿⣿⡏⣿⣿⣿⣿⣿⠿⠛⢱⣿⣿⢋⣀⢤⣭⣴⡲⠿⠇⠚
⢀⣀⢟⣴⣿⢣⡸⣿⣿⣿⣆⡀⠀⠀⠀⢀⡀⢀⡞⠀⠀⠀⢦⠀⢰⠀⠘⣿⣾⣿⣿⣿⢱⣿⣿⣿⡿⡫⠖⠉⢸⣿⣿⠿⢈⡌⠈⡗⣤⠀⢰⣦⣤⠄⣰
⢸⡏⡾⠋⠁⢄⠻⣿⣿⣿⣯⣿⣛⣻⣿⣿⠿⠋⠀⠀⠀⢦⢸⣶⣿⢀⠀⠈⣿⣿⣿⣿⢸⣿⢇⡟⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠐⠴⣫⣾⣾⣿⡶⢿⣿
⠘⣿⠃⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⡸⠀⠀⣿⣿⣿⢏⠘⣿⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢡⢃⡾⣸⣿⣟⢿⡳⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⡇
⠀⠈⡄⢀⡤⢤⡀⠀⠈⠉⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠈⠋⠀⣡⡾⣱⣿⢹⣿⣷⣮⡀⠈⢿⠑⠀⠀⠀⠀⠴⠒⠚⠂⠀⠀⠀⠀⠀⠀⠚⠛⠂⢠⠁
⠀⠀⢠⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠀⠀⠀⣠⠄⢀⣵⣾⣿⡏⠈⣟⣭⣭⣑⢄⠈⠇⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⢀⣞⢿⡆⡜
⠀⠀⠈⡆⠿⢡⣶⡄⠀⠀⠀⠀⠀⣠⣿⣿⠷⣦⠀⠀⠀⠁⠴⢛⣙⡛⣿⡇⠀⠟⢻⢁⣌⡁⠀⠀⠀⠀⠀⠸⠋⣿⣿⣿⠄⠀⠀⠀⠀⣿⣿⡏⠃
⠀⠀⠀⠃⠀⠹⠿⠇⠀⠀⠀⠀⠀⢻⣿⣿⡇⠙⠀⠀⠀⠀⠀⣤⣤⠁⠏⠇⠀⠀⣄⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⢀⠉⠉⠁
⠀⠀⡔⠀⠀⠀⠀⢠⠂⠀⠀⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢿⢏⣤⠀⠀⠀⠀⢿⣿⣮⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡤⠴⠂⠀⠀⠀⢸
⠀⠀⢧⠀⠀⠀⠀⠀⠉⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⡿⠃⠀⠀⠀⠀⠙⠿⠿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣶⡄⠀⠀⠀⢀⡜
⠀⠀⠈⠢⡀⠀⠀⠀⠈⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠋
⠀⠀⠀⠀⠈⠑⠀⠤⡀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣶⣤⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠻⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡅⣿⠞⢛⡛⢳⡄⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡏⠉⠙⠓⠶⣤⣀⠀⠈⡷⢀⠚⠛⣃⡼⠃⠀⣀⣤⠶⠚⠋⠉⢹⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠛⠶⣤⣄⣀⣀⠈⠙⠲⢧⣸⣯⣯⣭⡤⠖⠋⠉⣀⣀⣠⣤⠶⠛⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢻⡏⠉⠉⢻⡟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⣠⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢷⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣾⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠈⠳⠶⠶⠶⡖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢲⠶⠶⠶⠞⠁⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠄⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⣀⣠⣴⠶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠶⣦⣄⣀⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠘⠿⢿⠋⣰⣾⣿⣿⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⢆⠙⠿⠿⠃⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠇⠀⠙⣿⣿⣿⣿⣆⢉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣷⡈⡀⠸⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⡂⢰⣿⡿⠛⠟⠛⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⣷⣿⠛⠛⠛⢿⣿⡇⢘⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠱⡀⢻⣷⣄⢀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⣠⣾⡟⢀⠎⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠢⠭⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠡⠖⠁⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣠⣤⣴⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡿⣡⣴⣶⣦⡀⠀⠀⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣤⡙⠛⠁⠀⠀⠿⠟⠛⠛⠿⣆⠹⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣶⣧⠀⠀⠀⠀⢰⣟⢳⣄⠀⠘⠀⠈⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠛⠿⡿⡀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡜⠀⠀⠀⠀⠉⢉⡜⠁⠀⠀⠀⢄⡈⠛⠋⣸⠆⠀⠀⠛⡉⠈⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⠔⠁⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠂⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⡐⠒⢀⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠰⣿⣷⣿⡷⠟⠉⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠐⠤⠤⠀⠀⠀⠀⠀⠀⠀⣀⡔⠁⠈⠛⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢠⣤⡀⠀⠀⠀⣠⠀⠈⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣾⠀⠀⠀⢠⠃⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠙⠄⠀⠀⠁⠀⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠂⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⣽⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣿⣿⡾⣽⣿⡟⣾⣿⣿⣿⣿⣿⢸⣿⣿⣷⣿⣿⠿⠿⣿⣯⣻⣷⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⢳⣿⣿⢧⣏⣿⡟⣾⣿⣿⣿⡿⠁⠀⠀⠹⣿⣷⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢣⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⢫⣾⣿⡿⢣⣿⣿⢿⣿⣷⣦⣴⣶⣄⡀⠌⠁⠀⠀⢸⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣶⣿⠿⣋⣴⣿⢟⣵⣿⠟⠋⠘⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣭⣶⣿⣿⣯⡷⠟⠋⠁⠀⠀⠀⠀⠛⢻⣿⣷⣶⣾⣿⣿⣿⡀
⠀⠀⢀⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢻⣿⡇
⠀⠀⣸⣿⣿⡿⣿⠿⠧⠀⠀⢀⣰⡷⣒⣓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣚⣲⢧⣀⠀⠀⢸⠿⣽⣿⣼⣿⣷
⠀⢀⣿⣿⢿⣧⣿⡀⠀⠀⠀⠀⠋⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠋⠀⠀⠀⠀⣼⢿⣇⣿⣿⡄
⠀⣸⣿⡿⣿⣿⣻⣷⣦⣤⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⢠⣤⣾⣿⡘⣿⢹⣿⣧
⢀⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣿⡞⣿⣿
⣸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⢸⣿⣿⣿⣧⣿⣿
⣿⣿⡏⣿⣿⡿⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⢿⣿
⣿⣿⡇⣿⣿⠃⠸⠿⠟⠉⠟⠻⠛⠋⠛⢈⣻⣴⣿⣿⣿⣿⣿⣿⣷⣝⡁⠙⠋⠛⠛⠛⠛⠻⠿⠃⢹⣿⡿⣼⢿
⢿⡇⣿⡸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢣⡿⣸
⠈⢿⣜⣷⡈⠻⠦⣄⣀⣀⠔⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣀⣀⣠⠴⠛⣡⡿⣱⡿
⠀⠀⠙⠿⣿⣶⣤⣤⣤⠤⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠤⣤⣤⣤⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠑⠀⠀⠀⠀⠀⣀⣤⠤⠤⠄⠀⠀⠀⠘⠆⠐⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⣀⣴⠞⠋⢀⡴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠉⠛⠛⠛⠉⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠴⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⠇⠀⠀⠀⣈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠂⠀⠀⠀⠀⠀⠀⠳⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢶⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠖⠛⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣷⣶⣦⣄⠹⣶⣆⣰⣶⠏⣠⣴⣶⣾⣛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⣿⣿⣿⣿⣷⣾⣿⣿⣷⣾⣿⣿⣿⣿⣵⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⣙⠉⠳⣿⣿⠈⠁⣿⣿⠞⠋⢛⣿⣿⣿⣿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣟⣻⣭⣷⣮⣺⣿⡇⢸⣿⣗⣴⣦⡬⢽⣻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⠛⠛⢶⣼⣿⣿⣿⣿⣿⣿⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣧⡶⠛⠛⠛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⡀⢸⣿⣿⣯⣿⠟⠛⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠛⠻⣷⣭⣿⣿⡇⢀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢫⠟⣻⣾⣿⣿⠟⣡⣴⣖⣿⣿⠟⠲⢦⡀⠈⠛⠿⠿⠛⠁⢀⡤⠖⠻⣿⣿⣲⢦⣄⠻⣿⣿⣷⣟⠳⡝⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠈⣼⣿⣿⣿⠏⣼⣿⣿⣿⡟⢡⣶⣿⠓⠉⢢⣑⠲⠖⣊⡔⣡⣾⡟⠒⡈⢻⣿⣿⣿⣧⢹⣿⣿⣿⣧⠁⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠹⣿⣿⣿⠀⣿⣿⣿⣿⡄⠸⣿⣿⣶⡿⢀⡟⠛⠛⢻⡀⢿⣿⣷⣾⡇⢠⣿⣿⣿⣿⠀⣿⣿⣿⡟⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠑⠮⠽⣿⡇⢿⡹⣿⣿⣷⣄⡉⠛⠉⣠⣾⣿⣿⣿⣿⣷⣄⠙⠛⢉⣠⣾⣿⣿⢏⡿⣸⣿⠿⠵⠊⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠾⣿⣿⣎⠳⣬⣙⠻⠿⠿⠿⣛⡿⠋⠁⠈⠁⠈⠙⢯⣛⠿⠿⠿⠟⢛⣡⠞⣱⣿⣿⠳⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⣿⣿⣷⣦⣍⣛⣻⣿⣿⣿⡀⣤⣤⡀⢀⣤⣤⢀⣿⣿⣿⣟⣛⣩⣴⣿⣿⣿⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠁⠈⠻⡇⢸⠟⠁⠈⠁⠉⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢻⣇⠢⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠔⣸⡟⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⢿⣧⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣼⡿⠡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⣰⡀⡀⡄⢠⢀⣀⣆⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⡝⢿⣿⣿⣿⣿⡿⢫⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣤⢀⡀⣶⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣶⣾⣿⣿⣿⣿⣶⣾⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣀⠉⠛⣿⣿⣿⡿⠛⠉⣀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠂⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠐⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣼⣿⠏⢀⣤⣤⣤⣐⢄⠀⠀⠀⠀⡠⣀⣤⣤⣤⡀⠹⣿⣧⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡿⠋⠭⣽⣶⣿⣿⡾⠛⠋⠉⠛⠻⢷⣷⣤⣦⣾⡾⠟⠛⠉⠙⠛⢷⣿⣿⣶⣯⠭⠙⣿⡄
⠀⠀⠀⠀⠀⢸⡇⠘⠻⣿⣿⣿⠏⠀⢀⣤⡶⢶⣤⡀⠙⣿⣿⠋⢀⣤⡶⢶⣤⡀⠀⠹⣿⣿⣿⠟⠃⢸⡇
⠀⠀⠀⠀⠀⠘⣇⠀⡞⢛⣿⣿⠀⢠⡟⣱⡀⣻⣦⢻⡀⢸⡇⢀⡟⣅⢘⣷⣎⢻⡄⠀⣿⣿⡛⢳⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠙⣦⣇⠈⢾⣿⣆⠘⣇⠻⣟⣿⡿⢡⠁⡼⢇⠈⡎⢿⢻⣿⡟⣸⠇⣰⣿⡿⠁⣸⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⣿⣿⠧⡌⠳⠬⠥⢖⣡⠾⠗⠺⠷⣌⡲⠭⠥⠞⢁⠼⣿⣿⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣧⣄⣀⣀⡤⠋⠀⣀⠀⠀⣀⢠⠙⢦⣀⣀⣠⣴⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡟⠀⠀⠀⠉⠃⠘⠉⠀⠀⠀⢹⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡉⠓⠒⠒⠒⠒⠚⢉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣤⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠛⠻⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⠤⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠙⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠈⠻⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣴⣿⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡀⢀⢀⣸⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⢠⠒
⠀⠀⠀⢀⣨⣿⣿⣿⣿⠿⠿⠿⠿⠷⠄⠴⢦⠰⣾⡿⠟⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠝⢦⡄⢠⡄⠀⢠⠇
⠀⠀⢀⢩⡶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠱⢤⣀⡷⠉⠙⠛⣢⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠈⠳⣄⢘⡶⠛⠠
⠀⢠⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢷⡆⠛⢩⡀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠝⣥⠤⢤⠒
⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠑⠀⠀⠙⡄⠈⣧⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⢁⡴⠃⠀⠀⢂
⠀⢿⣮⣠⠀⣀⡠⠔⠀⠀⢀⠠⠒⡡⠀⠀⠀⠀⠀⢻⠀⣿⠀⠀⢰⣇⠀⠠⠀⣀⣠⠤⠂⠁⠀⣠⡶⠋⠀⠀⠀⠀⠘
⠀⢸⡟⠛⠛⠒⠒⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡏⠀⠀⢸⡿⠿⠚⠋⠭⠤⠤⠴⠖⠛⠉⠀⠀⠁
⠀⢸⡇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⡇⣰⠃⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⡇
⠀⣤⡑⠀⠀⣴⠿⣧⡀⠀⠀⠀⠀⢀⣼⣿⣦⠀⠀⢔⣧⠀⠀⠀⣤⡓⠀⢀⡴⢿⣧⠀⠀⠀⠀⠀⢀⢼⣿⣆⠀⠀⣇
⠀⠘⠉⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠉⠏⠀⠀⠀⠹⠉⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠘⠛⠃⠁⠀⠐⠩
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⣤⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣤⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠀⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⣀⣠⣤⣤⣤⣀⣀⡀⠀⠀⠉⠉⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⢸⣿⣿⠇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠸⣿⣿⠀⢸⣿⣿⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⡄⠙⢿⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⣀⣀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⢀⣶⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀
⣾⣿⡏⠁⣉⠙⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦
⣿⣿⠀⢾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣭⣻⣿
⠸⣿⡆⠘⢿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠐⠂⠀⢀⣤⣤⣦⣤⣄⡀⠀⠀⢀⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⣿⣿
⠀⠹⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠿⠇⠀⠀⢠⠿⠟⠛⠛⠛⠿⣿⣦⠀⠙⠃⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠡⠿⣻⣿
⠀⠀⠙⠿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⢁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠛⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⣴⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠟⠛⠉
⣤⣦⣴⣤⣤⣤⣤⣴⣤⣤⣤⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡀⠀⢀⠀⠀⢀⣴⣿⣿⣿⣇⡀⠀⣴⠇⠀⠀⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠒⠀⠒⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣾⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠙⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢛⠋⠉
⠀⢀⣴⣶⣶⣦⠀⠆⠀⠄⠀⡸⠋⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⢠⠚⠛⢖⠻⢇⡀
⠀⠸⡿⠋⠉⠙⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⡆
⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣦⣄⠀⠀⡠⠶⠆⠀⢠⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠛⢿⣿⣿⣷⠀⠀⠀⠀⠄⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⢀⠀⠀⠸⣿⣿⣧⠀⠀⢠⣿⣿⣟⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⠿⠿⠇⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢰⣶⣶⣶⣶⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⠛⠋⠉⠹⡇⠀⠀⢀⡀⠀⡀⠀⠀⢸⣷⡄⠀⠀⠸⣿⡆
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠿⠿⠀⢀⠉⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⢸⣷⣦⢤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁
⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠛⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⠔⠀⠀⠀⢸⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠢⠀⠒⣈⡁
⠀⠀⠀⠀⢁⠄⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⡌⢀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈
⠀⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⡀⣀⣀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢌⠁
⠀⡀⠤⠤⠤⢠⠐⠁⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠂⡁⠠
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠈⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠈⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣴⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣙⣏⣝⣹⣟⣽⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⣂⣤⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⣿⣿⣿⣿⣿⣿⣷⣦⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠋⠉⠉⠿⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⢀⣴⠀⢀⡤⠚⠁⠀⠀⠚⠛⠛⠻⠿⣿⣿⣿⣿⣿⣶⣖⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢤⡀⢳⠀⠠⣀
⠀⠀⢠⣏⣻⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⢿⡟⠻⢶⡃⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣦⠀⠀⠀⠳⡀
⠀⠀⢸⡉⠛⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠘⢦⡀⢀⣾⠃⠀⠀⠀⢀⡴⠖⠒⠒⠒⠒⠛⠂⠀⢀⡀⠀⢀⣴⠟⣀⡤⠄⠀⢱⡀
⠀⣼⠘⣷⡔⣾⣧⠶⣋⣡⠤⠀⠀⣀⣀⣴⡿⠛⠁⠀⢀⠀⠀⠀⠀⢤⣤⣈⣧⢸⠋⡆⠀⠀⠀⣿⣶⣶⠶⠿⢿⣷⣶⡾⠿⢛⣁⣰⡿⠛⠣⠔⠺⣷⠠⡈⣧
⠀⢿⣶⣼⣿⣿⣿⣿⣋⣤⣶⣾⣿⠿⡟⠁⠀⠀⠀⠀⣾⣴⡀⠀⣄⠈⣿⣿⣿⣾⠀⢷⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⢧⠈⣿
⠀⠈⠻⣿⣿⡿⠿⠿⣿⣿⣿⠯⢑⠈⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣿⣇⣿⣿⣿⣿⡄⠘⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿
⠀⠀⠀⢸⡿⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡏⣇⢀⠘⣆⠀⣇⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣱⠏
⠀⠀⠀⠘⣿⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠈⠻⣿⣿⣿⣿⢿⣿⠀⠹⣆⠢⣹⡄⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣄⠀⠀⠀⠻⢋⠟⠑⢶⠁⠀⢴⠀⡀⢿⡇⠁⠀⠀⣠⠞⢋⢽⣦⠀⠀⠀⠀⠀⠀⢐⣻⣦⠀⠸⠁
⠀⠀⠀⠀⠀⢧⠀⡶⠓⣢⡀⠀⠀⠀⠀⠈⢁⣾⣿⣿⠃⠀⠀⠀⠉⠰⠏⠈⠀⠀⠹⡄⠨⡄⠀⠀⠀⠀⠛⠀⡶⣿⣿⠂⠀⠀⠀⠀⢴⣾⣿⠿⠃⢋⠃
⠀⠀⠀⠀⠀⠈⠀⠃⢼⣿⡷⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠀⢀⣀⣤⡾⠂⠀⠀⠈⢦⣀⡀⠀⠀⠀⠀⣑⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⢁⣂⠀⢊
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠉⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⣀⣀⣠⣤⣴⡶⠋⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠈⠱⣶⣶⣤⣤⡤⠖⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠡⠴⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⣦⡀






⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⠶⠶⠶⢶⣶⡶⠶⡶⠶⣶⡶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⡟⠉⠉⢹⡇⠀⠀⠀⣼⠉⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⣧⠀⠀⠈⢧⠀⠀⠀⣯⠀⠀⠀⠀⠀⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠘⢧⡀⠀⠈⠳⣄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠈⠑⠦⣽⣄⡀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡄⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠈⠛⣿⡀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠉⣙⣿⣿⣛⠉⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⡿⠀⠀⠀⣠⠟⠉⠀⠀⠉⠻⣆⠀⠀⠀⢻⡀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠸⡇⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⢸⡇⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⢿⡀⢰⠇⣿⡀⠀⠀⠀⠀⢀⣾⠙⡆⢀⡾⢁⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣾⠀⣏⠙⠶⠦⠤⠶⠋⣹⠀⣿⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⡙⠲⠶⠶⠖⢛⣡⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣠⣾⠋⠁⠀⠒⠲⢶⣿⣿⣿⣷⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢤⡀⠀⠀⢄
⠀⠀⢠⣾⣿⠁⠀⠀⠀⢈⣉⣙⣻⣿⣿⣿⠟⠙⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣳⣦⠀⠀⠠⡀
⠀⢸⣿⣿⣵⠖⠃⠀⠀⠀⢀⣶⣿⣿⣿⡇⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠊⠀⠀⠀⠀⠀⣀⠬⣀⠀⠀⠈⡀
⠀⣿⣿⣿⣷⣶⣶⢋⠀⢀⡴⠚⠋⡁⠁⠀⠀⣦⣄⣤⣿⣿⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⣒⡄⠀⠀⢀⠔⠛⠁⠀⠀⠀⠁
⠀⣼⠿⣿⣛⣿⡿⣭⡶⠋⠀⠀⠀⠀⠀⢀⣿⡟⣿⣿⣼⣿⠀⠀⠆⠀⠀⡑⠀⠑⠯⢟⣛⡛⠋⢥⡴⠅⠀⠀⠀⠀⠀⢄⠘
⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠟⣰⠋⣿⣼⣿⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠴⠁⣼⣷⣫⣿⠀⠀⢣⠀⢄⠻⠂⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡀
⠀⠀⠀⣤⣬⡀⠀⠀⠀⢀⣥⣤⡀⠀⠀⠀⠀⠀⠘⡉⠹⣿⣇⠀⠀⠙⠆⠀⠀⠀⢀⣤⣬⡀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠁
⠀⠀⠀⠑⠿⠇⠀⠀⠀⠘⢾⡿⠛⠀⠀⠀⠀⠀⣰⡄⡸⠛⠃⠀⠀⠈⡅⠀⠀⠀⠛⠴⡿⠃⠀⠀⠀⠀⠿⠿⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠁⠀⠀⠀⠀⠐⠬⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢀⢹⣲⣤⣀⣤⡶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢦⣄⣀⣠⠴⡺
⠀⠀⠀⠀⢡⣤⠿⢁⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⣦⠈⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣊⣡⠤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⣁⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⢀⡤⣚⡥⠖⠋⠉⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠳⢦⣝⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢺⠁⡀⠀⣄⠀⠀⠀⢀⣵⠞⢉⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡈⠛⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡇⠈⡆⣷⠀⣿⠀⢀⡴⢋⣠⠾⠋⠁⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠙⢦⣄⠙⢦⡙⢆
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⢸⣦⡟⣴⢏⣴⠟⠁⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⡀⠀⠀⠀⠀⠀⠠⣉⢳⣄⠻⣎⢳⡀
⠀⠀⠀⣠⠖⠒⡶⢿⣄⠀⣘⣿⡿⣡⠞⢁⣴⠟⠁⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⣠⣾⡿⠁⣸⠃⣶⢀⠈⠒⢤⡀⠈⠳⡝⢧⡘⢧⣳⡀
⠀⠀⠀⠹⣄⠀⠈⠙⣻⣶⡉⠹⣴⠋⣰⠟⢁⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠚⢉⣠⣴⣿⡿⠋⢀⣼⠃⣸⡟⠘⣇⢦⠳⣝⢶⡀⠘⢎⢳⡈⢷⣷
⠀⠀⠀⠀⠙⢶⣤⣭⣭⠿⣄⠀⣧⡾⠃⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢞⣫⡽⠟⠁⠀⣠⠞⠁⣰⣿⠃⠀⣿⠸⣇⠈⢳⡙⣦⠈⢧⣷⠘⣿⣧
⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣽⢻⣿⠁⢰⡟⠀⠀⠀⠀⠀⠀⠒⠒⠋⠉⣉⣡⣴⡶⠟⠉⠀⣀⣴⠟⠁⢀⣾⣿⠃⠀⠀⣿⠀⣿⠀⠀⠹⡌⢧⠈⣿⡇⢹⣿
⠀⠀⠀⠀⢠⠎⠀⠀⣴⠟⢡⣿⡇⠀⣿⠇⠀⠀⠠⠤⠤⠤⠶⠶⠾⠛⠋⠉⢀⣀⣤⠴⠚⠉⢀⣠⣶⣿⠟⠁⠀⠀⣸⠏⢰⡇⠀⠀⠀⠹⡜⣇⢹⣿⠸⣿
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣾⣿⡇⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣉⣡⣤⣤⣶⡾⢿⡿⠟⠁⠀⠀⠀⣰⠏⢀⡞⠀⠀⠀⠀⠀⠃⢹⢘⣿⠀⣿
⠀⠀⠀⠀⠀⠀⡾⠀⠀⣀⣿⡿⣇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠭⠴⠒⠊⠉⠀⠀⠀⢀⠴⠚⠁⠴⠋⠀⠀⠀⠀⠀⠀⠈⠈⠠⣿⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⢸⡟⣿⣇⢿⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⡏
⠀⠀⠀⠀⠀⠀⢷⣄⣸⠇⣧⢻⡘⣇⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢹⠈⢷⠹⣆⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠒⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⢤⣄⠀⠀⢠⡏⠜⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⢦⣌⢧⡘⢆⠀⠀⠀⠀⠈⠁⠀⠀⣀⣀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠁⠀⡜⠴⠚⠛⢷⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⣤⣤⣈⠂⠳⡀⠀⠀⠀⠀⠀⣄⣻⣦⡿⣿⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡷⣿⣥⣾⠀⠀⠀⣠⠶⠲⣦⢀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠁⢀⣉⣧⠀⠀⠀⠀⠀⠀⣳⣶⢟⠟⠉⠉⢀⣹⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⣉⣿⣦⡹⣿⣵⠂⢰⠷⢶⡄⠸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣇⢻⠀⡟⠁⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⡇⠀⠀⠈⠀⣶⠀⢀⠇⣰⠋
⠀⠀⠀⠀⠀⠀⠀⢻⠹⣆⠀⠣⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠀⠀⠀⠀⢰⡁⠠⣪⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠹⣦⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⣦⣀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⣤⠀⢀⡄⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⠉⠉⢉⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣀⡀⠀⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⠆⠀⠀⠰⢿⣿⣿⣿⣿⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣄⣼⣿⡇⢸⣿⣿⣷⢸⣿⣧⣠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⣗⢸⣿⣿⣿⢸⣿⣿⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣤⣼⣿⣧⡘⢛⡛⢛⣸⣿⣧⣤⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣽⣿⣿⣿⣿⠟⣡⣿⣿⣆⠻⣿⣿⣿⣿⣿⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⢛⣡⣾⣿⠿⠛⠻⣷⣌⠛⠿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣭⣭⣴⣶⣿⣿⣿⡏⣼⢿⡷⢸⣿⢸⣷⣦⣭⣭⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⠶⠶⠟⣡⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⣱⣶⡌⢻⣿⣿⣿⣿⣿⣿⣿⢡⣶⣦⢹⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣛⣡⣿⣿⣿⣿⣿⣿⣿⣿⣌⣛⣋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⡿⠿⢷⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣌⠛⢰⣿⢡⣶⣶⣦⠹⣿⡌⢛⣡⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⡘⢿⣿⠟⣠⣿⢃⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡙⢿⣶⣶⣾⠿⢃⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣦⣤⣶⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣡⠞⠉⢀⣏⣠⠴⠖⠒⢛⣻⠖⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠏⠀⠀⠚⠉⠀⠀⠀⠀⠉⠉⠛⠒⠒⠚⣋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⢤⣀⠠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣏
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣠⡖⢋⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⢀⡆⢀⢼⠇⢀⠜⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠈⠉⢹⠁⠀⠀⠀⢀⠆⠀⢀⡴⠋⡇⠀⣠⠾⠓⠡⠾⠖⠁⠀⠀⠀⠀⠉⠲⣄⠈⣆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣞⠤⠖⢉⣠⠤⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⡀⠀⠳⡸⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠂⠀⢸⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠀⠀⢹⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠧⣄⢸⠀⠀⠀⡴⢋⡥⢶⣅⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⣮⡑⣤⠀⠈⣦⠞⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⡏⠰⢒⠌⢾⠀⠀⢺⠀⣾⣧⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⡧⢸⠀⠘⠃⠸⡉⠃⣽
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⠄⠀⠀⠀⠈⠆⡹⣟⣿⡻⠃⠀⠀⠀⠀⠀⠀⠀⠹⣟⣛⣻⡣⠂⠀⠀⠀⠀⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠂⠤⠤⠔⠊⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣭⡟⠙⠛⠛⠉⢹⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢣⡀⠑⠦⢄⣀⣀⣀⡠⠏⢠⠟⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠙⠲⢤⣀⣀⣀⣀⡠⠖⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣽⣶⣶⣾⣿⣿⣿⣶⣶⣦⣤⣉⠻⣿⣿⣿⣷⣄
⠀⠀⠀⣤⣾⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣾⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣷⣶⣮⣭⣛⠿⣷⣾⣿⣿⡻⣿⣆
⠀⢀⣾⣿⠟⠛⠛⠿⣿⣿⣶⣄⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⣿⣿⠉⠙⠻⢿⣿⣿⣿⣶⣝⡿⣿⣷⡹⣿⡄
⠀⣼⣿⠏⠀⠀⠀⠀⠈⠙⢿⣿⣿⠿⠛⠻⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣀⣤⣤⣼⣿⣷⣭⡛⢿⣮⣻⣷⣿⣷⣶⣶⣶⣦⣤⣀
⢰⣿⡿⠀⠀⠀⠠⠤⣤⣤⣠⣿⠃⠀⠀⠀⠀⢻⣷⠶⠶⠖⠒⠒⠒⠂⠀⠀⠀⢻⣿⣿⠿⠿⠟⠛⠛⠛⠛⢿⣦⠙⢿⣏⣉⣥⣤⣬⣉⡉⠛⢿⣿⣦⡀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣴⣶⣶⣶⣶⣶⣶⣶⣿⣷⡄⠹⣿⣿⣭⣉⣛⠻⣷⣄⠙⢿⣿⡄
⢿⣿⣷⣀⠀⠀⢀⣀⣀⣤⣴⣿⣷⣤⣀⣠⣴⣿⣿⣿⣶⣤⣀⣀⠀⠀⣀⣠⣴⣿⣿⠋⠉⠉⠉⠉⠛⠛⠻⢿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣷⣌⢻⣧⠈⣿⣿
⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⡟⠉⠙⠿⣎⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠹⣿⡄⠸⣿⣿⠿⣿⣿⡆⣿⡆⢸⣿⠇
⠀⠀⠀⠈⠉⠉⢹⡿⠋⠀⠀⠀⠀⠙⠇⠀⠀⠀⠈⠀⠀⠙⠻⠈⠉⠩⣭⣶⡿⠃⠀⢤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⠀⢿⣿⠀⢹⣿⡇⣿⠇⣿⣿
⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⢁⣿⡇⢸⣿⡇⢸⣿⣿⣿⣾⣿⠃
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⣀⣀⣀⣤⣤⣶⣿⡿⢿⡇⢸⣿⡇⣸⣿⣿⣿⣟⢿⣷⣄
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠋⠁⣠⣿⠇⢸⣿⣧⣿⣿⢿⠘⣿⣷⠙⣿
⠀⠀⠀⠀⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⠀⣿⣿⣿⣿⡟⣸⠃⣿⢹⡇⢸
⠀⠀⠀⠀⢿⡏⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢛⣿⣿⡿⠟⣹⡏⢰⣿⣿⡿⣋⣴⢏⣴⢏⣼⠃⣾
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠾⠟⠛⠉⣠⣾⣿⣷⠘⣿⣿⣿⣽⣶⡿⠵⢛⣡⣾⡿
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠟⣫⣿⣿⣇⢸⣿⠻⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠸⣧⣿⣷⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⣿⣋⣥⣾⡿⣿⡟⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢰⡟⢿⣿⠁⠀⢀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣉⠀⠈⠙⣿⡟⢫⣾⣿⢁⣿⠀⣿⣧
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠋⠓⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⠇⠀⠀⣹⣿⣿⣿⡏⣸⡿⢰⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣽⢏⣴⡿⣱⣿⡿
⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣿⣿⣿⣿⣭⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢶⣦⣤⣤⣤⣤⣤⣴⠶⠾⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣉⡉⠙⢦⣀⡴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⡉⠙⢳⡄⢸⠃⠣⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⢴⣷⡿⠤⢤⣤⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠀⢸⡉⠒⠾⡤⢶⠞⠛⢛⡇⠀⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸⣬⣤⠜⠓⠲⣄⣌⣸⡇⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣤⣄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣠⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣈⠑⠀⠀⠘⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠈⢀⣴⢶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣝⢷⡀⠀⠀⠐⠁⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⠀⠀⠀⢠⠏⣴⡿⢿⣯⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⣅⣸⣿⡆⣧⠀⠀⠀⠀⢀⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣧⣴⣿⠎⡆⠀⠀⢀⠤⠤⡀⠀⠀⠸⠀⢿⣿⣿⣿⠃⣿⠀⠀⠀⠠⠋⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢿⠀⠻⣿⣿⠟⢀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⢇⠈⠙⠋⠁⢠⠏⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢸⡀⠈⠳⡄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠢⣄⣀⡴⠋⠀⠀⠀⠀⣀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠐⠊⠀⠀⠀⣠⠞⠉⠀⠀⠉⠙⢦⠀⠀⡀⠂⠐⠠⠀⠀⠀⣠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠴⠀⠀⠀⠀⠀⠊⠀⠀⠀⡄⠀⣇⠀⢠⡀⠀⣠⠀⢨⡇⠀⢄⠀⠀⢀⠃⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠀⠡⠀⠀⠔⠁⠀⠙⠦⣄⣙⣛⣡⡴⠋⠀⠀⠀⠈⢉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠒⠛⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢈⠝⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠐⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⣄⣀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢆⠰⣤⠤⠴⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠁⣉⢁⡼⢧⣄⣀⣀⣀⣠⡤⠴⢒⠍⠰⢄⡤⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠉⠀⠈⢂⢠⡄⢀⡀⠺⢢⠇⠀⠀⠀⠉⠚⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⢀⣀⠀⠀⠙⠤⣙⠁⢶⢼⠀⠀⠀⠀⢰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠉⠉⠉⠉⠙⠛⠓⠛⢦⣀⣀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠒⠒⠒⠒⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠒⠒⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠒⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣠⠴⠞⠛⠛⠓⠶⢤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡤⠶⠚⠛⠛⠓⠶⣤⡈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡞⢁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡌⢳⡄
⠀⠀⠀⠀⠀⢀⡞⣰⠋⠀⣀⡤⣖⣺⣿⣿⣯⣖⣲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣒⣮⣿⣿⣿⣒⡦⣄⡀⠙⢦⢹⡄
⠀⠀⠀⠀⠀⣸⠡⠃⣠⢞⣵⠞⠋⠁⠀⠀⠀⠈⠙⠳⣬⠢⡀⠀⠀⠀⠀⠀⡠⢪⡵⠛⠉⠀⠀⠀⠀⠉⠙⢷⣝⣦⠈⠇⣷
⠀⠀⠀⠀⠀⡿⠀⢰⣿⡿⠁⠀⠀⠀⣠⣤⣶⡤⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣤⣶⡶⠀⠀⠀⠀⠙⣿⣧⠀⢸⡄
⠀⠀⠀⠀⠀⡇⠀⣿⣿⠃⠀⠀⢠⣾⣟⣻⣿⣄⣀⣴⠀⠈⣧⠀⠀⠀⠀⢠⡏⠀⠀⣼⣿⣛⣿⣇⣀⣠⡇⠀⠀⢹⣿⠀⢸⡇
⠀⠀⠀⠀⠀⣷⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠂⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⠀⣸
⠀⠀⠀⠀⠀⢸⣆⠸⣿⣇⠀⠀⠀⠻⣿⣿⣿⣿⣿⠏⠀⢀⡏⠀⠀⠀⠀⠈⣇⠀⠀⠙⢿⣿⣿⣿⣿⠿⠁⠀⠀⣼⡟⢠⡿
⠀⠀⠀⠀⠀⠀⢿⣦⠹⣿⣦⡀⠀⠀⠈⠉⠛⠉⠁⠀⣠⡞⠀⠀⣀⣄⡀⠀⠘⣦⡀⠀⠀⠉⠉⠉⠁⠀⠀⢀⣾⠟⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⡳⣌⠻⢿⣶⣤⣀⣀⣀⣠⣤⠞⠋⠀⠀⠀⠁⠀⠁⠀⠀⠈⠛⢶⣤⣀⣀⣀⣠⣤⣾⠟⣫⠼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣙⠲⠬⣍⣉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣩⡥⠶⣚⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠉⠉⠉⠙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠉⠁⠀⢀⣀⣀⣀⣀⡉⠑⢯⠀⠀⠀⠈⠙⢆⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣴⣒⣛⣛⠛⠒⠲⠤⢤⣉⡉⠉⠒⠦⢤⣀⠀⠘⡆⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠈⠉⠙⠒⠦⢄⡀⠉⠓⠦⣄⠀⠈⠙⠲⢵⡀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡿⣶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠉⠲⢄⡀⠀⠉⠒⢤⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⡇⠙⣆⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣍⡒⢤⣀⠀⠉⠲⢄⠘⡄
⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢠⠇⠀⠈⢧⣄⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠿⢶⣤⣀⣱⣿⡀
⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠎⠀⠀⠀⡼⠉⠳⣦⣀⡀⠀⠈⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠟⠋⣝⠳⡀
⠀⠀⠀⢀⠎⠀⠀⠀⡰⠋⠀⠀⣠⠞⠀⠀⠀⠀⠙⠻⢷⣒⡤⠤⣄⣀⣈⣉⠙⢒⣒⣒⣲⡶⠶⠒⢚⣩⡭⠿⠓⢋⡹⠃⠙⢤⣀⣠⢤
⠀⢀⡴⠃⠀⢀⡴⠊⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠤⠬⠭⠭⠤⠤⠔⠲⢾⣉⡙⠒⠒⠒⠋⢉⣩⠗
⠘⠓⢤⣴⣊⣁⡤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢏⠹⡀⠀⠀⠀⠀⣠⠞
⠀⠀⢀⡏⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢏⠓⠒⠒⠉
⠀⠀⡎⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆
⠀⠀⠣⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠐⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣄⣀⣒⣒⣒⣒⣃⣠⠤⠤⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡯⣍⡳⣄⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⢀⡴⣛⡭⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡤⠤⠿⢸⣦⠞⢁⡟⠙⣶⠒⠒⠒⠒⡾⠉⢧⡙⢦⣼⠰⠧⠤⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣐⣛⣠⣼⡥⠞⠁⠀⠀⠘⣆⠀⠀⡼⠁⠀⠀⠙⠲⣽⣦⣘⣓⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⠤⠤⠶⠚⣇⠀⣳⠀⠀⠀⠀⠀⣸⠀⢸⡃⠀⠀⠀⠀⢠⡏⢀⡿⠳⠦⠤⡤⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢨⠟⠉⠙⠛⠒⠚⠛⠁⠀⠀⠉⠛⠒⠚⠛⠉⠉⢿⡀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠓⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠐⠾⢿⠏⣵⣾⣿⣿⠦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⢎⠹⡿⠶⠂⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⡀⠈⢻⣿⣿⣿⣆⣁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡎⡀⠐⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠃⢸⣿⠋⠛⠙⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⠋⠛⠙⣿⡇⠠⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡄⠀⠀⠀⠈⢄⠻⣷⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡾⢟⡠⠁⠀⠀⠀⢠⠖⠋
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣀
⢠⠞⠉⠈⠙⢦⣠⢤⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣤⣄⡴⠋⠉⠙⠲
⣏⠀⠀⠀⠀⠀⠃⠀⡷⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢿⠀⠘
⠘⢦⠀⠀⠀⠀⠀⢰⣃⠀⠀⠀⠀⠉⠓⠲⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠖⠚⠉⠀⠀⠀⢀⣸⡆⠀⠀⠀⠀⢀⡴
⠀⠈⠑⢦⣄⣀⠀⠀⠈⠉⠓⠲⣤⠤⠤⠤⠶⠚⠉⠙⠦⢤⣍⣉⣉⣉⣉⣡⡤⠴⠋⠉⠓⠦⠤⠤⠤⢤⠖⠚⠉⠁⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⣀⠴⢋⡇⣀⡤⠞⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢩⡷⠛⠁⢀⠾⠛⠉⠀⢀⣼⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡇
⠀⠀⠀⠀⢠⡆⠹⡄⠀⠀⠀⠀⢀⣤⠞⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠳⢶⣤⣀⠀⢀⡼⠁⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠙⢦⡀⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣩⡷⠋⠀⠀⣼⡇
⠀⠀⠀⠀⠀⣿⣆⠀⠀⠙⠻⣿⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⠖⢋⠵⠛⠁⠀⠀⠀⢰⡿
⠀⠀⠀⠀⡀⣸⣿⡃⠀⠀⠀⠀⠉⠓⠉⠛⠾⣷⢤⡀⠀⠀⠀⠀⢀⠔⣸⠟⠉⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⢷⣤⠄
⠀⠀⠀⢈⡛⠿⣿⣿⣦⣀⢀⡀⠀⠀⠀⠀⠀⠈⠑⠈⠳⣄⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠟⠛⢡⡄
⠀⠀⠀⣸⢳⡀⠀⠈⠻⠿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠘⢿⠃⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡟⠋⠀⠀⢀⣮⢹⡄
⠀⠀⢖⠳⠾⠛⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠿⢶⣄⠀⠀⠀⠈⠀⠀⠀⢀⣴⣿⢿⣿⣿⣿⡛⠻⣟⢦⡀⠀⠀⠈⠙⠓⠂⡈
⠀⠀⢸⠦⣄⠀⠀⠀⠀⢠⠟⡹⠋⠀⠛⢿⣿⣿⣮⣳⣄⠀⠀⠀⠀⢀⣾⣿⡅⠀⠈⣿⣿⣿⣷⡈⢧⠑⡀⠀⠲⣦⢶⡏
⠀⠀⢸⡆⢸⡟⠁⠀⠀⠃⢰⢁⣷⣤⣤⢾⣿⣿⣿⣷⢻⡆⠀⠀⠀⢸⡿⢹⣿⣶⣞⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠈⠛⠃⡰
⠀⠀⣀⠷⠋⢀⡀⠀⠸⠀⢸⡈⣿⣿⣿⣿⣿⡿⣿⡿⣸⠓⠀⠀⠀⠁⢿⡸⣿⣿⣿⣿⣿⠻⣿⠇⣼⠀⡀⠀⠀⢀⣠⣞⠁
⠀⠈⠁⢲⡆⣿⠁⠀⠀⢆⠈⢳⣜⠿⣿⣿⣿⡶⠟⣡⡟⠀⠀⢴⡄⠀⠈⠷⣌⡛⠿⠿⠿⠛⣡⠾⠁⡠⠁⠀⠀⢀⣴⠏
⠀⠀⠀⣤⠋⠿⢠⡆⠀⠈⠢⡀⠉⠓⠦⠤⠤⠶⢛⠕⠀⠀⣰⠋⠛⢦⡀⠀⠀⢉⠓⠒⠒⠋⠁⡀⠀⠀⠀⠀⣀⡼⢋
⠀⠀⠀⠈⠻⣶⡟⠆⠀⠀⠀⠈⠒⠤⠄⠀⠤⠐⠁⢀⠠⢚⡁⠀⠁⠀⣿⠀⠒⠤⢈⣀⠀⠀⠁⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠈⠙⠿⣆⡀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠛⡄⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣥⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡇⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢻⡿⠋⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⣼⣟⠉⠙⠛⠛⠳⠶⣤⣦⣀⢠⠀⠉⢠⣄⠤⠞⣠⡴⠞⠋⠁⠀⠀⡞⢀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⢤⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡿⢄
⠀⠀⠀⠀⢀⠞⢷⠀⠀⠀⠀⠀⠁⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⣠⠏⠀⠈⠀⠀⠀⠀⠀⠐⣸⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⠈⠀⠀⠀⠀⠀⠘⡮⢧⡀
⠀⠀⠶⠗⡟⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠸⡀⠀⠀⣤⡀⠀⣀⠀⠀⠀⠀⢠⡆⠀⠄⢠⠃⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⢘⡇⣠⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠙⢄⠀⢱⡍⠢⣏⠑⢄⠀⡶⢩⡆⠂⡰⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢷⡇
⠀⠀⠀⠘⣧⡇⠀⢘⢀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠢⡀⠁⠀⠀⠀⠀⠛⠁⠈⢀⠎⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠂⠀⢸⠁
⠀⠀⠀⠀⢹⡇⠀⠀⢿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⣿⣷⢀⡇⠀⣤⠎
⠀⠀⠀⠀⠈⣧⣀⠀⢸⡆⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⣀⡔⠀⠀⠀⠀⣼⡿⢡⣼⢃⡼⡿
⠀⠀⠀⠀⠀⠘⣿⢦⡀⢿⣦⡿⣷⡀⠀⠀⡀⠈⣧⡀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢉⡬⠴⠆⡴⢀⣼⡟⢁⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣇⠘⢷⡄⢳⡼⠢⡄⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⡔⣡⡿⠋⠀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⣠⠿⢶⡷⠖⠛⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡏⠀⠀⠙⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠁⢀⠀⠀⠀⠘⢿⠿⠷⠶⠶⠿⠟⢿⠁⡴⠀⠀⡀⠀⢆⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣠⣰⠃⠀⢰⣠⡀⣼⠃⠀⠀⠀⠀⠀⢸⣼⡇⢀⣰⡆⠀⢸⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠹⣦⣀⡟⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠻⢿⠉⠻⣼⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠴⠒⠚⣋⣉⠉⠀⠀⠀⣀⡀⠀⠈⠉⠙⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⡀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⢀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⠔⠉⠀⠀⠀⡀⠤⠀⠀⠒⠂⠀⠀⠀⠀⠒⠂⠀⠠⢀⠀⠀⠀⠈⠒⢜⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠔⠁⠀⢀⠠⠂⠁⠀⡀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⡀⠁⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⠤⣄⣀⣀⠀⠀⠈⠲⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠀⡠⠔⠋⢀⠄⣰⢆⠀⠀⠀⠀⡁⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡐⠀⠀⠀⠀⡠⠋⡰⠃⠀⠑⢄⡀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠚⣡⠞⠁⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⡎⠀⠀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣽⢶⣶⣒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢸⢰⠁⠀⠈⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢸⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⡄⡾⠀⢠⡾⠛⠉⠉⢉⠉⡓⡄⠀⠀⠀⠀⠀⠀⢀⠚⠉⣉⠉⠉⠙⢿⣄⣇⠀⡰⡘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⠠⣹⡷⣶⡿⠀⠀⠀⠌⣠⣶⣄⢹⠀⠀⠀⠀⠀⠀⡞⢠⣤⣄⠱⠀⠀⠀⢻⣼⣄⣧⠅⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢺⠁⠈⠁⠀⠀⠀⢢⠙⠛⢃⠜⠀⠀⠀⠀⠀⠀⠣⡘⠛⠋⡠⠀⠀⠀⠘⠋⠈⠓⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠀⣸⠀⠀⠀⠀⠀⠀⠒⠘⠛⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢰⡀⠀⡠⠳⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⣄
⠀⠀⠀⢶⠚⠉⢀⡀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⠀⠉⠒⡶
⠀⠀⠀⠈⠳⣄⠀⡀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠓⢬⣁⠀⠀⠀⠀⠀⠀⠈⠂⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠁⠀⡀⠀⠀⠀⠀⣈⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠚⠉⠉⠁⠀⠁⠂⠤⠀⣀⠀⠀⠀⢀⡀⠀⠄⠒⠉⠀⠈⠉⠉⠓⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢀⢹⣧⠀⡀⢸⣇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣾⠀⠀⣸⡟⡀⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙⣿⣼⣿⣿⣦⣷⣼⣿⣠⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⣿⣼⣶⣿⣿⣧⣿⢁⡄
⠀⠀⠀⠀⠀⡀⢀⠳⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡶⢀⢀⡀
⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠛⠻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⡇⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢻⣿⣿⣿⡇⠀⠀⠈⠻⣿⣿⣿⣿⣥⠖
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣷⠟⢋⣿⡿⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⢋⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡻⠿⠶⠟⠋⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡛⠿⠶⠞⠋⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⣦⣤⣤⣤⣤⣤⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⣀⣴⡤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣄⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠲⣿⣏⠛⠛⠻⠟⢿⣿⡷⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠷⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠶⠂⠀⠀⠂⢄⠰⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢆⣸⠀⢾⣿⣿⣿⣶⣂⡤⡪⣶⢠⢠⠳⣾⣷⣄⢹⡄
⠀⠀⠀⠀⠀⠀⣼⣄⣧⡀⢨⣿⣿⠿⠿⠿⣟⣻⣭⠷⣫⡿⠡⢸⡆⠡⡘⣿⣿⢿⣿⡄⢸⢠
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿⠟⢉⣤⡾⠋⣠⢧⣿⣧⠀⣧⠸⣿⡣⠙⣿⡈⡇⡇
⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⠟⣫⣿⠟⣋⣴⠾⠛⡁⢀⡾⣣⣿⣟⣿⠀⣿⠀⣿⢳⡀⢸⠇⣰⠃
⠀⠀⠀⠠⠤⣶⣿⣿⠟⣡⡾⢋⡵⠞⠉⠀⠐⣨⡾⣫⣾⣿⡟⣸⡇⠀⣿⠀⣿⠘⡇⠀⠀⠋⠄⢠⠀⣷⣦⠆
⠀⠀⠀⠀⠀⡼⣱⠃⠚⠁⠀⠁⢀⣀⣤⡶⢟⣩⣾⣿⣿⠟⣰⡿⠀⢰⣿⢀⠇⢠⣿⠀⢱⣕⡶⣤⣷⣿⣿⠇
⠀⠀⠀⠀⡼⠡⠃⠀⠀⡀⢀⣀⣉⣭⣵⣾⠟⣫⣾⠟⢁⣴⡟⠁⢠⡿⠃⠊⠀⡼⠗⠀⠀⠉⠻⣿⣿⣯⠅
⠀⠀⠀⠀⠇⡀⠀⠀⠀⡗⠚⠛⢛⣋⣭⣶⠟⠋⣁⣴⡿⠋⠀⣴⡟⠁⠀⣠⠞⢁⡜⠀⢸⠀⡌⣧
⠀⠀⠀⠀⠀⡇⢸⡆⠀⠣⢤⣤⣭⣭⣤⣴⣶⡿⠛⠁⣀⣴⠿⠋⠀⠀⠉⣀⡤⠊⠀⠀⢸⡆⣿⣹
⠀⠀⠀⠀⠀⢧⣾⠀⠀⠀⠈⠛⠛⠛⠉⠁⠈⠐⠚⠋⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⢱⣿⠟
⠀⠀⠀⢠⣶⣶⣽⡀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠏⡼⣷⣶⣦
⠀⠀⠀⢸⣿⠀⣀⡁⠀⠀⠀⠀⠉⢀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⠉⠀⠀⠀⠀⠀⣀⡀⣾⣿
⠀⠀⠀⠘⣿⣾⠋⢻⡄⠀⠠⣾⣿⣯⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⣽⣿⣷⣿⡄⠀⠀⡾⠉⣷⣿⠇
⠀⠀⠀⠀⠈⢿⣦⣈⣣⠀⠀⠥⠿⠨⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠿⠧⠁⠀⢠⣃⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠻⠿⢿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀
⠀⠤⠤⠤⠀⠀⠠⠼⠛⣛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠃⠈⠙⠛⠟⠛⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⢛⠋⠉⠉⠉⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⡁⠀⠀⠀⠉⠓⣦⡀⢀⡠⡢⢄⠀⠀⠀⠉⠙⠒⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠾⢫⡄⡀⠀⢈⣶⠀⣿⣤⠀⡾⠘⠦⡱⣝⣎⢣⠑⠜⠂⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⠞⠡⡴⣼⣇⣿⡞⡆⢻⣷⡇⢿⠀⣧⠀⠹⣌⠚⣏⠇⠃⠀⠀⠀⠀⠀⠀⠱⡝⢆
⠀⠀⠀⡰⠋⠀⠀⠀⠏⢻⡿⣿⣜⡎⣯⢻⣼⣧⣿⣆⢳⡜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢧
⠀⠀⡼⠁⡔⠀⠀⠀⠘⠀⠣⡹⡜⢍⠈⠃⠙⠻⢏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⢰⠃⡞⠁⠀⠀⠀⠀⠀⠀⠐⠘⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣿⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢸⠂
⠀⠘⡇
⠀⠀⡇
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠦
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠁⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣼⣿⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠘⠚⠋
⠀⠀⠀⠀⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⣤⠔⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⣾⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠂⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣴⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣤⣶⣶⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣤⣤⣴⣶⣴⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠿⠟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠿⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢨⣿⣿⠟⠋⠁⣀⣀⠀⠈⠑⠺⣿⣿⣿⣿⣿⣿⠗⠊⠁⠀⣀⣀⠀⠙⢿⣿⣿⡅⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⢏⣴⠾⠛⣉⣉⣉⡙⠢⣄⠈⠻⣿⣿⠟⠁⣠⠔⢚⣉⣉⣉⠛⠷⣦⡙⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣵⣾⣿⣿⣿⣿⣿⡿⠁⣴⡾⠛⠛⠛⠻⢷⣎⠢⣦⣿⣏⣴⠜⣵⡾⠟⠛⠛⠛⢿⣦⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⣿⣿⣿⣿⣿⠃⣼⡏⠀⡴⢿⣿⣷⡄⢻⣧⢻⣿⣿⡟⣼⡏⣠⠺⣿⣿⣦⠀⢹⣧⠘⣿⣿⣿⣿⣿⡟⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠹⣿⣿⣿⣿⠀⣿⡄⠀⣷⣾⣿⣿⡟⢨⣿⣾⣿⣿⣧⣿⡄⢿⣿⣿⣿⣿⠀⢠⣿⠀⣿⣿⣿⣿⠏⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⢛⣿⣿⡄⠹⣷⣄⠈⠛⠛⠋⣠⣾⢣⣿⡟⢻⣿⡜⣷⣀⠙⠓⠛⠁⣠⣾⠏⢰⣿⣿⡙⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣿⣿⡀⠈⠛⠿⠶⢶⣿⢟⣵⢟⠍⠀⠀⠉⠻⣮⡻⣷⡶⠶⠿⠛⠁⢀⣿⣿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣦⣤⣭⣽⣾⠟⢁⣀⡀⠀⠀⢀⣀⡈⠻⣶⣯⣭⣥⣤⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠉⠀⠈⠻⢷⠀⠀⡿⠟⠁⠀⠉⠻⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣤⣍⠉⠉⠙⠛⠛⠋⠉⠉⣩⣴⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧⣄⣉⠛⠛⣁⣠⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⠀⣀⣤⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠟⠋⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠺⣿⣿⡿⠷⠀⠀⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠿⣇⠀⠀⠀⣠⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⡞⢻⣿⣿⠳⠄⠀⢸⣧⠶⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⢳⣿⠀⠀⠘⠉⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢸⡉⣀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠻⢥⡤⠞⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠙⠛⠛⠛⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠙⣄⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⢀⡞⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⢦⡀⠀⣰⠏⠀⢀⡞⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢹⣾⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⣰⠛⡏⣧⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⢸⡇⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⠦⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠶⠾⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⣀⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠛⠛⠛⠻⢄
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠁⠀⠀⠀⠀⠀⢂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂
⠀⠀⠃⠀⠀⠊⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢡⠴⠀⠀⠀⠆⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡄⠀⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠷⠀⠀⠀⠀⠀⠀⠀⠐⠹⡄⠀⠀⠀⠀⠀⡀⠂⠀⢀⠞
⠀⠀⠀⠀⠀⠐⠤⣀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⣀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠈⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣭⣿⣶⣿⠶⠮⣭⡢⡄
⠀⠀⠀⠀⡠⠐⠉⠳⡀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⣀⣀⠀⠠⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢚⡾⢫⠒⠉⠉⠉⠉⠙⠲⡝⣦
⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⠀⠀⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠄⠀⠀⠀⠤⠀⠘⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠠⠄⠀⠀⠠⠄⠀⠀⡬⠕
⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⡀⠀⠑⡢⠤⠄⠠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠒⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠓⠀⣀⢘⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢀⠄⠂⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡀⠀⠀⠑⠢⣄⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠁⠐⠀⠁⠀⠀⣸⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠌⠠⠄⠀⠠⠀⠄⠙⡄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠤⠂⠀⡰⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠛⠄⠀⠀⠈⠣⢄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠴⠊⠀⠀⠀⠀⠈⠃⢄⣀⠀⠀⠀⠀⣈⠦⣀⣀⣀⣠⠴⠃⠄⣀⣀⡀⠤⠐⠁
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⢷⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠶⠶⠶⣷⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣶⡶⠖
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠟⠉⠀⠀⠀⠀⠀⠉⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡿⠿⠿⠂
⠀⠀⠀⠀⠀⣛⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠙⢿⣿⡄
⠀⠀⠀⠀⠈⠙⢛⣿⡿⠁⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣷⣦⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⢈⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣸⣿⠁⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡟
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢀⣾⣿⠃
⠀⠀⠀⠀⢻⣿⡀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠁⢠⣾⣿⠏
⠀⠀⠀⠀⠸⣿⣧⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣿⣿⣿⣿⣿⣿⣟⣡⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠙⢿⣷⣤⣤⣽⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡇
⠀⠀⠀⠀⣾⣿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠟⠋⠉⠛⠷
⠀⠀⠀⠀⢹⣿⡀⠉⠻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠉⠻⢿⣿⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠶⠶⠾⠿⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⢉⣁⣀⠀⠀⠀⢄⡀⢄⡀⢀⣀⠈⠙⢛⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣫⠴⠒⠊⠉⠀⠀⠀⡀⠀⠁⠙⠂⠑⠸⣿⠸⠎⠀⠐⠒⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠏⡶⢓⣄⡀⠀⠠⠄⠒⠀⠀⠀⠀⠀⢀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⢰⣿⣿⢏⡵⢱⣾⣿⣟⡿⢟⣴⡆⢀⢮⡄⠀⠀⠀⠀⠀⠀⠀⠀⢌⣷⡀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⣿⣿⣷⡿⢁⣿⢟⢻⣿⣭⣿⣿⣿⣿⣿⡧⠀⢀⣄⣠⡤⢆⡀⠻⣿⠻⠇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⣿⡟⣿⣡⡿⠋⠀⣾⢿⣿⣿⣿⣿⣿⣿⣿⣶⢟⡿⠩⠞⢉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠠⢹⡼⣿⡟⠁⠀⣸⠇⠸⣿⢹⡟⣿⣿⣿⡟⠀⠉⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿
⠀⠀⠀⠀⠀⢸⣷⡟⢡⣦⡄⠟⠀⠀⠀⡿⠃⠈⠀⣹⢿⣤⡤⠀⠀⠀⠀⠈⣿⡤⠀⠀⠀⠀⠀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠉⢇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣼⡗⠀⠀⠀⠐⠟⣽⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣷⣶⣦⡀⢄⠀⠀⠀⠀⣠⣶⣿⣶⡂⠀⠀⠀⠀⣻⣿⡟⢀⡀⠀⢀⠄⠼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠿⣿⣻⠳⠀⠀⠀⠀⠀⡟⢿⣛⣻⠟⠂⠀⠀⠀⠈⠙⠳⢿⢗⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⢉⠁⣸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠛⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⢉⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢰⠶⠤⣤⣤⣤⣤⣤⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠠⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠺⣿⣿⣿⠟⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠿⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⣼⡿⢡⣾⣿⣷⣦⣝⠻⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣩⣴⣾⣿⣷⡌⢿⣧
⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠿⣿⣦⡙⢀⣶⣿⣿⣶⡄⢫⣴⣿⠿⣿⣿⣿⣿⣧⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⡿⠶⠦⠁⢸⣿⣿⣿⣿⡇⠈⠴⠶⢿⣿⣿⣿⣿⡿⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣧⢸⣿⣿⣿⣿⣿⠿⣛⡄⠸⣿⣿⣿⣿⠇⢠⣛⠿⣿⣿⣿⣿⣿⡇⣼⡏⡀
⠀⠀⠀⠀⢀⣾⡌⣿⣄⠛⣛⣫⣥⡶⢟⣩⣾⣷⣮⣭⣭⣥⣾⣷⣌⡻⢶⣮⣝⣛⢛⣰⣿⢡⣷⡄
⠀⠀⠀⠀⣾⣿⢇⣌⣛⠛⣛⣫⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣙⣛⠛⣛⣡⡜⣿⣷⡀
⠀⠀⠀⢸⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⡇
⠀⠀⠀⢸⣿⢱⣿⣿⣿⣿⠿⣛⣛⣭⣭⣭⣿⣿⣿⣿⣿⡿⠿⠿⠿⢟⣛⠕⠈⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⢸⣿⣾⣿⣿⣯⣵⡿⢟⣋⣭⣭⡶⠶⠶⠶⠶⠒⠚⠛⠛⠉⠉⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣟⡵⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇
⠀⠀⣠⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧⣄
⢠⣿⣿⡏⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⢹⡿⣷
⣿⡇⣿⣧⠀⠀⢸⡇⠀⠀⢀⣴⣦⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣶⣄⠀⠀⢰⡏⠀⠀⣼⣿⢸
⢿⣇⢿⣿⣷⣤⣠⣿⠀⠀⣽⠿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⡃⠀⣼⣇⣤⣴⣿⡟⣼
⠈⠻⣷⣽⣿⣿⣿⣿⣧⡀⠀⠀⠙⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁⠀⠀⣰⣿⣿⣿⣿⣯⣾⠟
⠀⠀⠈⢻⣯⢻⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡟⣽⡟⠁
⠀⠀⠀⣸⣿⣼⣿⣿⡿⠀⠙⠻⢷⣦⣤⣄⣀⣀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡾⠿⠋⠁⢻⣿⣿⣧⣿⣄
⠀⣴⣿⡿⢹⣿⣿⣿⣿⣦⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⢀⣴⣿⣿⣿⣿⡝⣿⣿⣦
⢸⣿⣿⡇⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣸⡿⣿
⠘⣿⣾⣿⣮⣿⣿⡿⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⢿⣿⣿⣶⣿⣷⣿
⠀⠈⠛⠿⣿⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠤⠤⠤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⢤⣄⡀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠉⠓⢦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣰⢻⡀⠀⠀⠀⠀⢰⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢀⡼⠁⠈⣧⠀⠀⠀⠀⠸⡷⣄⠀⠘⢷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠈⢧⡀⠀⠀⠀⠹⣮⣑⠒⠲⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠉⠛⠒⢒⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠸⡇
⠀⠀⠀⠀⠀⠀⡜⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠲⠶⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢁⣤⣶⣷⣶⠦⠀⠀⠀⠀⠀⠀⠠⢶⣶⣶⣦⡌⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⡏
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠙⢦⠀⠀⠀⠀⠀⠀⠈⠀⣰⡇
⠀⠀⠀⠀⢰⣫⣭⡱⡄⠀⠀⠀⠀⠀⠘⠋⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠘⠓⠀⠀⠀⠀⠀⢀⡞⣩⢭⣷
⠀⠀⠀⠀⢸⠀⠃⡏⢿⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠉⠉⠀⢀⠀⠀⣀⠀⠈⠉⠛⠛⠓⠋⠁⠀⠀⠀⠀⠀⠀⢸⠹⡅⠁⣿
⠀⠀⠀⠀⠘⣆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⠏
⠀⠀⠀⠀⠀⠈⠢⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣦⣤⣤⣴⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠻⣿⡛⠛⠛⠛⠛⠛⣻⠟⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤⠤⠤⠖⠋⠁
⠀⠀⢀
⠀⡁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠄⠀⣀⣾⠛⠙⢻⣦⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢹⣇⠀⠈⠛⠋⠉⠉⠛⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⣤⣶⣯⣶⣶⣶⣶⣶⣶⣾⡟⠀⠀⢢⡀⠀⣿⣦⣤⡴⠶⠿⠿⢷⡆⠠⣤⣝⡻⣷⣄
⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⢀⣼⡿⢟⣋⣭⠭⠤⠶⠶⠶⠶⢿⣷⣄⡘⠲⣤⠖⠻⡟⠁⠀⠀⠀⠀⢸⣧⣀⠀⣙⣻⣿⣿⣧
⠀⠀⢠⣾⡿⢟⣽⢿⡷⣴⠛⠁⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢿⣄⣾⠡⣄⡀⠀⠀⢀⣾⣿⣿⣿⣿⣟⠛⠛⢿⣧⣄⡀
⠀⠀⠘⣿⣇⣀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢀⠀⠀⠈⠙⠻⣷⣌⡁⠀⢀⣾⣿⣿⣿⣿⣿⣛⠓⠂⠀⠈⠻⣿⣄
⢠⣏⣤⢾⣿⠟⣡⡔⢂⡄⠀⠀⠀⢀⣠⠔⠂⣀⣤⡴⢞⣩⡴⠚⠁⣠⠔⢀⢀⡀⠈⣻⢿⣿⣿⡿⠻⣿⣿⣿⣿⣯⣉⠙⠲⢤⡀⠘⣿
⢸⣿⠃⣾⡏⣰⣿⣾⡏⣠⣖⣭⡾⣫⣴⡾⠟⠋⠑⠚⠉⢁⣠⣴⠞⢁⡴⢃⣾⠇⡀⢻⣧⡙⣷⠀⠀⠘⣿⣿⣿⣯⡙⠳⡄⠀⠹⣆⢹
⠘⣏⠈⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⡶⠶⠲⠶⣌⠛⠛⣉⡤⠞⢋⣴⡿⠋⢠⠇⠈⣿⣷⣽⢀⠀⡆⣿⣿⣿⡿⣿⣦⠀⠀⠀⢻⣾
⠀⢀⢄⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢠⡾⢷⣶⣤⣀⣀⣠⣴⣾⠟⠋⠀⢀⡞⠀⠀⢻⣿⡟⠀⢠⢇⡟⣸⣿⣿⣦⣌⠑⠀⠀⣼⣿
⠀⢸⣌⠻⣿⣿⣿⣿⠿⠿⠿⢿⣾⡾⣿⣦⣥⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠐⠋⠀⠀⢀⣾⡟⠀⠐⠱⢋⣴⣿⣿⣿⣿⡟⣠⣆⣾⣿⣿
⠀⠈⠻⣷⣦⣽⣿⣿⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⠴⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⢻⡆⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣐⣩⡿⢣⣿⣻⣿⠟⠋⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠠⣀⠀⢀⣿⠘⠉⢉⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣵⢎⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠙⠛⠁⣰⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣄⣌⡑⠀⠀⠀⠀⠀⠀⢙⣛⠛⣛⣡⣾⠿⠿⢿⣷⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⣾⣿⣥⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⡿⣿⣶⣢⠄⠀⠀⠀⠀⠈⠉⢻⡿⠗⠻⣶⡈⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃⠙⠿⠟⠀⠀⠀⠀⠀⠀⠘⡇⢿⣿⣷⡏⠻⠓⠀⠀⠀⠀⠀⠀⠀⢈⣤⠄⠀⢸⠇⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠈⠙⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢎⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⠤⣀⣄⣠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠈⠀⠉⠿⠷⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣤⡴⠖⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣴⣿⣿⡟⡻⠋⠉⠩⠽⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢀⣾⣿⡟⠉⠈⠀⠈⠀⠺⠏⠉⢶⠟⠻⠿⠿⠟⠛⠛⠉⠉⠉⠁⠙⠻⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡆
⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⢸⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⣿⣿⣿⣿⣿⡇
⡠⠤⣼⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⣿⣿⣿⣿⣿⣇⣤⣤
⢰⣶⣌⠁⠀⠀⠀⠀⠀⠀⠟⣿⣭⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣿⣿⢉⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢿⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠛⠁⢀⣿⣿⣿⣿⣿⣿⡿
⠀⠈⢧⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣘⠛⡧⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠤⠴⠿⠿⢿⣻⣾⣿⡆⠀⠀⠀⠀⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠲⠶⠾⠷⠿⠟⠀⠀⠀⠀⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠐⠤⣤⣤⣴⣿⣿⠿⠛⠉

⡿⠉⢻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿
⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢨⣦⠆⣰⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⣠⡉⠀⠠⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠑⡀⠀⢾⠁⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡘⠂⠁⠀⠋⠉⠘⠅⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⠇⠀⠐⠟⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠛⠛⠛⠛⠎⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊

⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣖⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⣀⣀⣀⣀⣲⣶⣶⣶⣶
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠶⠶⠶⠶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣋⣭⣤⣶⣶⣶⣒⡓⢶⣶⣶⣦⣬⣍⣙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣫⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣯⣟⡻⢿⣿⣿⣶⣬⡙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣭⣟⣛⣛⠿⠿⢷⣮⣿⣻⣿⣷⣾⣽⣿⣿⣿⣷⣌⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⣿⣿⣿⣿⣿⣾⣭⣝⡛⠻⠯⢟⣓⠲⠮⢭⣝⣛⠻⠿⣿⣿⣿⣿⣿⣷⣌⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⣶⠭⢭⣤⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣍⡓⢮⡈⠙⢿⣧⡙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡿⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣭⣽⣓⣒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⠀⠀⠘⣷⡘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣕⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣟⣓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠂⠀⠀⠀⠀⠀⠀⡉⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⣍⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⣾⣽⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣿⢿⣿⣿⣿⣿⣿⣿⣿⡻⢦⣍⣉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣷⣿⠄⠀⠀⠀⠻⡿⠻⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠸⣎⢿⣿⣿⠟⢡⠴⢤⡉⠳⣄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠈⢙⡆⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠹⣎⢿⣿⠀⠎⢠⣶⣦⣤⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠒⠉⠉⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠙⣮⠻⣧⡈⠀⠻⣿⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠶⠀⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠈⠳⣝⣿⣦⣀⠈⠛⠛⢃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠈⠻⢿⣿⣿⡛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠉⠙⠛⣻⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣰⣿⣿⣿⠹⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣌⡙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠷⠶⠶⠾⠿⠛⠛⠉⠁

⣟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⢀⣀⣤⣤⣴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣀⡀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⣿⣿⠛⠋⠙⠋⠉⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣲⡄⠀⠀⠀⠀⠀⠆⠀⠐⠉⠜⠁⠀⠉⢂
⠀⠀⠹⠃⠀⠀⠀⠀⡀⠙⠆⡠⠀⠀⠀⠀⠀⠀⠀⡋⠀⠛⠏⠁⠶⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠦⠄⢀⡄⠀⠊⠀⠙⠀⠄
⠀⠀⠀⠡⡘⠸⠗⠀⠘⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠀⠀⠄⠊

⠀⠀⠀⠀⠠⠂⠀⠂⢀⠀⠀⠄⡀
⠀⢠⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠂⡀
⠀⠐⠄⠀⠀⠀⣀⡀⠑⠢⡄⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠑⢦⠀⠀⠈⠄
⠀⠄⠀⠀⢀⠼⠛⠃⠈⠛⠇⠀⠀⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡖⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⢃⠀⠀⢈
⠀⠰⠀⠀⠰⢄⡀⠀⠀⣀⠘⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⠰⣶⠶⢚⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠠⠀⠀⠀⣸⠀⠀⠸
⠐⠀⠀⠀⠀⠀⠈⠈⠀⠀⠂⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠂⠄⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠⠌⠒⠐⠒⢏⡁⠀⠀⢀⡣
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠄⣢⠴⠋⠉⠀⠉⠢⣀
⠀⠀⠀⠐⠂⡀⠀⡁⠈⠂⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠖⠹⠋⠀⢀⡀⠀⢠⠀⢡
⠀⠀⠆⠀⠸⠃⠺⠃⠀⠀⢐⠁⠀⠀⠀⠀⠀⢠⣥⠀⠀⠀⠀⠀⠂⠁⢀⠋⠈⠀⠀⠀⠈⠀⠀⠀⠀⢸⣐⠄⡀⠷⠀⡈⠀⢸
⠀⠀⠐⡀⠀⠀⡀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠠⠤⡊⠀⠀⠌
⠀⠀⠀⠈⠂⠠⠤⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⣀⣀⠤⠊
⠀⠀⢀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣾⣋⣴⣿⣿⣿⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣥⣀⢙⣿⣿⣿⣿⣿⣿⣋⣴⣷⣿⣤⣴⣶⡶⣦⣷⣇⣷⠺⠽⢶⡋
⠀⠀⠀⠀⠀⠀⢠⣾⠛⠛⢻⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣏⣿⡟⢿⣮⠧⣤⣀
⠀⠀⠀⠀⠀⡴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣮⢹⣇
⠀⠀⠀⡴⣶⣿⣎⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣤⠀⢀⡀⠲⣄
⠀⠀⠀⠷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣌⣿⡷⢤⣥⡴⠿⡆
⢀⣴⠛⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⠁⠀⣠⡇
⣹⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣾⠋⠷⠶
⣮⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⡇⠉⠁⠈⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣷⣤⠄
⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⡇
⣼⣿⡁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡾⠃
⣯⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣭⡀
⢟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡇
⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢉⣵⣾⠿⠿⠿⠿⠻⠆⠀⠀⠀⠀⠘⠋⣛⣉⣄⣈⠻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠷⢿
⠸⢻⣧⣼⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⠇⠸⠋⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢿⣷⠙⠘⣿⣿⣿⣿⣿⣷⣼⣿⣄
⠀⠘⠿⣍⣹⠿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢠⣾⡿⣿⣿⣿⢉⠂⠀⠀⠀⠀⠀⠘⠿⣿⠷⠋⠀⠀⠀⣿⣿⣿⡿⣏⠛⠟⠛⣷⠤
⠀⠀⠈⠁⠃⢀⣿⣿⢏⢻⣿⣿⠛⣿⡀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⢠⠛
⠀⠀⠀⠀⢰⣞⡟⠻⣿⡿⡿⣿⣧⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⡀⢤⡄⠁⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⡀
⠀⠀⠀⠀⠈⠛⠉⠉⠉⣴⠰⢯⡿⢆⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠈⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠷⡔⠒⠦⠤⠀⠀⠀⡠⠂⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⢀⣀⠈⠑⠢⠤⠤⠒⠀⢀⣴⣶⣾⣶⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⠛⠿⣷⣶⣤⣄⣀⣠⣤⠴⠋⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⣦⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣛⡻⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡾⢁⡀⠀⠸⣯⣤⣤⣤⣤⣤⣭⣉⣙⠛⠿⣿⣿⣿⣿⣿⡄⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⠁⠀⠤⠽⣿⡀⣸⣿⣭⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⢿⣿⣿⣿⡄⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠛⣷⠀⠀⠀⢚⡿⣿⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣷⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⢸⣿⣧⠀⠀⠈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢻⣿⡇⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠙⠿⣿⣿⣿⡇⢸⡇⣼⣿⣿⣿
⣴⣿⠿⣷⡀⣸⣿⣿⣿⣿⡆⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠈⢿⣿⡇⢸⡇⣿⣿⣿⣿⡆⢀⣶⡿⣿
⢿⣿⣦⣙⣿⣿⣿⣿⣿⣿⣧⠸⣿⡇⠀⠙⢿⣿⣿⣟⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣋⣼⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⡆⢿⡅⠀⠀⠀⠉⠛⠛⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⣼⠿⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠘⢿⣿⣿⣿⢿⣿⡟⠀⠸⣿⡘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣿⣿⡿⣿⣿⣿⡿⠃
⠀⠀⣾⣿⡋⣷⣿⣿⣷⡀⠀⠈⢳⠀⠀⠀⣀⣤⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣶⣦⣆⡀⠀⠀⠀⠞⠀⠀⢠⣿⣿⣿⣶⢉⢻⣷
⢦⣀⣸⣿⠇⠹⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠸⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠅⠀⠀⠀⣤⣤⣶⣿⣿⣿⣿⠇⠸⣿⣏⣠
⠀⠉⠉⠁⠀⠀⡟⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠿⠋⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⢫⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣦⣬⣿⠿⣿⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣿⠻⣿⣥⣴⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠉⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢀⠔⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡦⡈⠑⢤⡀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣷⡀
⠀⢀⢣⠞⣡⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠙⡄⠀⠀⠀⠀⢰⣿⡿⣻⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣿⣿⣦
⠀⡼⣡⢊⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⡀⠀⠀⠀⡿⣫⣾⣿⠁⠄⢹⢀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⡴⢡⢧⠃⢠⡀⠀⢣⡀⠐⣆⠀⠀⠀⠀⠀⠀⣈⢆⠀⠀⠀⠀⢡⠀⠀⢸⣷⢏⡞⡟⡄⣄⠀⢲⡄⠁⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⠏⡸⠀⠀⠻⢆⡈⠣⠘⢎⢷⢤⣀⠀⠀⠀⠈⠝⡆⢀⠀⠀⢸⠀⠀⠸⣧⡟⢸⠃⢹⠙⢷⣄⠛⠄⠻⣟⣷⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⠀⢻⡰⠁⠀⠀⢀⣀⡀⠀⠀⠀⠁⠁⠀⠈⠐⠢⠀⠀⡆⢸⠀⠀⢸⠀⠀⠀⣟⡴⠄⠀⠈⢀⣀⡀⠀⠀⠀⠈⠛⠶⠿⢿⣿⣿⡁⣿⢻⣿⣿
⢀⠌⢇⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠌⠔⠀⣄⠀⠀⣠⠿⣧⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢠⢾⣽⣿
⢰⠐⠪⠀⠀⠚⠙⣾⡧⠀⠀⠀⠀⠀⠸⣾⠗⠹⠀⠀⠀⢀⣴⡲⣌⠆⠀⢧⠚⢻⠁⠀⠺⠙⢾⡿⠀⠀⠀⠀⠀⠸⣷⡿⠹⠃⠀⠀⣨⣗⣦
⠀⠱⣄⢠⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣸⠏⠀⠀⠈⠳⣄⣰⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⡿
⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⡼⠒⠋⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⢀⡟⠚⠉
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⡀⠀⠀⣀⠤⠤⠤⠒⠋⠀⢀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⠠⣄⡀⠀⢀⣠⠤⣤⣴⣾⣷⣶⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣤⢶⡻⠟⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠗⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠷⣦⡽⢾⡷⠾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⣀⣽⣧⣼⣧⣀⣀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠛⠉⠀⠀⠀⠀⠉⠛⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠋⠀⠀⠰⠛⠉⠋⠀⠀⠀⠀⠙⠉⠛⠆⠀⠀⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠺⣦⣄⣀⣀⣠⣴⠗⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⣀⣀⣀⣀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠸⣄⠀⠒⠂⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣺⠶⡿⠷⠦⠒⠑⠒⠒⠂⠄⡀
⠀⠀⠀⠀⢀⠔⠊⠀⣄⡀⠉⠁⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠈⢖⠐⠄
⠀⢀⣀⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢣⠀⠀⠀⠀⡸⠀⠀⠀⢠⡀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠀⠎
⠀⡁⢸⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠑⠈⣆⠀⠀⠀⢀⠞⠉⡀⢀⣈⠑⠄⠀⠀⠀⠀⠀⡜⠐⠀⠠⠂⢃⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠄⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠘⢆⠀⠀⠈⠀⡃⣁⠈⠈⠀⠰⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⢀⠜⠀⣠⠎
⠀⠀⠀⠀⠀⠑⠦⣀⠑⠬⠒⠒⠊⠔⠁⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠉⠒⠒⠊⢁⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠔⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣷⡤⠀⠀⢼⡏⠴⠀⠀⠒⠀⠀⠀⣲⣜⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣄⠀⢸⠷⠄⠤⠀⣢⠅⠤⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠶⣆⡓⠀⣰⢁⣀⡔⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⠿⠁⡀⠀⠀⣂⠀⣄⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠎⠀⢠⣾⣟⠀⠉⡁⢸⠏⠢⠄⢠⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⣀⣴⣿⢠⠶⢵⣮⡍⠍⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠻⣿⣿⠄⠀⢻⣿⠁⠈⢷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⡀⠘⠿⢇⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠀⠐⠒⠒⠀⠀⠀⠠⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠠⠤⠤⠤⠤⠤⠄⠐⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⣿⣶⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡿⠿⠿⠟⠛⠉⠀⠀⠙⣿⣧⠀⠀⠀⠮⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⡟⢀⠄⠐⠂
⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⡇⠀⠀⢸⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⢀⣀⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢈⣧⠀⠻⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠎⣰⠀⠀⢀⠔⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣷⣦⣤⢀⣴⠂⠈⠓⠺⢁⠀⠀⠀⠀⠂⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠛⠁⠀⠓⢠⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠤⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⢠⡠⠄⠂
⠀⠀⠀⠀⠐⠦⠀⠀⠀⠈⠉⠁⠀⠂⢤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣠⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢤⣀⣀⡀⠀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣤⣏⣭⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⡀
⠀⠀⠀⠀⠀⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⢄⢃⠀⠀⠀⠀⠀⠈⣈⢢⠆
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠀⣠⢢⡀⠀⠀⠃⠀⠀⠈⠂⠀⡆⠀⠀⠈⠃⠀⠉
⠀⠀⠘⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡴⢸⠿⠛⡱⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠄⡀⠀⢀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⢄⠠⠐⠂⠀⠀⡄⢲⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠇⠀⠀⠀⠀⠁⠦⣠⢀⡀⠀⠠⠞⠀⠀⢀⠐⣄⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢁⣽⠁⠀⠀⠀⠈
⠀⣤⣀⠀⠐⠀⠀⢀⠀⠀⠐⠚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⢈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⡰⠀⠀⡈⠀⡠⣤⠃⠀⠙⠀⡦⣄⠀⠀⠀⠀⠀⠃⠉⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠂⠀⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠠
⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠑⡌⠀⠐
⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠠⠀⠀⢂⠀⠀⠀⠀⠀⠀⢉⢲⣿⡾⢢⡄⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸⣤⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠁⠀⠋⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠐⣄⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠉⠉⠉⠁
⠀⠀⠀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⢠⡖
⠀⠀⠐⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠢⠖⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠲⠞⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣝⣛⠿⢿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡛⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⡄⠀⠀⠹⣿⠿⠟⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠿⠻⠿⣿⡿⠟⢷⣏⡤⠄⠀⠀⢻⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄⡉⠛⢿⣿⣿⣿⣿⣧⡈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠘⠛⢺⣷⣄⣼⣿⡶⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⠻⣿⣿⣿⣧⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⣿⡄⠀⠀⠘⣹⢻⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⡆⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢰⣿⣷⡄⠀⠀⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⣷⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⢸⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠙⠻⣿⣿⣿⣿⣧⠈⣿⢀⣿⣿⣿⣇
⢠⣶⣶⣦⡄⠀⣾⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⣿⢸⣿⣿⣿⣿⡀⠀⣠⣶⣶⣤
⣿⣿⣆⠹⣿⣾⣿⣿⣿⣿⣿⣇⠸⣿⡟⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣿⣾⣿⣿⣿⣿⣷⣾⡿⢁⣿⣿
⠙⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡄⢿⡇⠀⠀⠀⠙⠿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟
⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠉⢨⣷⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣏⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢙⣿⣿⡿⢟⣿⣿⡇⠀⠈⠿⣇⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠿⠏⠀⢀⣿⣿⣏⠿⢿⣿⣟⠁
⠀⠀⢸⣿⣡⠸⣿⣿⣿⣿⣄⠀⠀⠘⠄⠀⠀⠀⢦⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣧⠄⠀⠀⠀⠊⠀⠀⢀⣾⣿⣿⣿⡟⢠⣹⣿
⠳⣤⣤⣿⠟⠀⢻⣿⣿⣿⣿⣷⣶⣶⠀⠀⠀⠀⠘⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠀⠀⠀⠀⣰⣶⣾⣿⣿⣿⣿⣿⠃⠘⣿⣧⣤⡴
⠀⠀⠀⠀⠀⠀⢠⡛⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⣣⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣷⣤⣬⣿⠟⢿⣻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣻⠟⢿⣯⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠉⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠉⠁⠀⠀⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣤⣤⣴⠶⠛⠶⣦⣤⣤⠶⠟⠶⢦⣤⣤⣤⣴⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠲⠶⠶⣦⣤⣤⣤⣀⣀⣀⣀⢀⣠⡤⠤⢤⡤⣄⠤⢒⣀⣀⣠⣤⣤⣤⣤⣤⠴⠶⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣤⡴⢶⣒⣺⣟⣛⠛⠻⠿⠿⠿⠿⠟⠶⡶⠶⠿⣾⠿⠿⠿⠿⠿⠛⣛⣛⣓⣒⠂⠄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⣲⣿⣿⣿⡟⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢿⣿⣿⠯⣻⡿⣿⣿⣿⡿⢿⣿⣿⡿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣖⠂
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⠺⢾⣖⡒⠊⣷⡖⣂⡐⢂⣀⢲⠷⠆⣿⣿⣧⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠠⠔⠊⡩⢿⡉⠙⣷⣠⡄⢴⣿⣿⣿⣿⣻⣿⠧⠛⠻⠿⠶⠷⠾⠾⠿⠿⠿⠾⠿⠟⠁⠹⣿⣿⣿⡒⣾⣿⡟⢉⣴⡿⠋⠉⡽⢏⠑⠢⠄
⠀⠀⠀⠀⠀⠊⠀⠀⡷⡀⠈⠳⠸⠋⣽⠟⢿⠿⠛⠁⠀⠈⠉⢉⣙⣛⣛⣛⣛⣛⣋⣉⣉⠉⠀⠀⠈⠛⠉⠹⢿⡿⠟⠘⠃⠀⣠⠾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⣀⣠⡤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻