Balls Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠖⠒⠒⠒⣤⣤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁⠀⢀⡤⠴⠒⠉⠉⠉⠀⠈⠙⠲⠤⣉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣬⣻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⣀⣤⣴⡾⠟⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢳⣴⣶⠿⠟⠋⠁⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⠤⠤⠤⠖⠒⠒⠢⠤⠤⢤⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢀⡏⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⣸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣆⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⡧⠖⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⢠⠃⠀⠀⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠦⠤⠤⠤⠤⠔⠻⡛⡄
⠀⠀⢸⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠘⡄⠀⠀⠀⠘⣾⣄⣠⡤⠤⠖⠒⠉⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢹⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠈⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢦⣠⣾⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡏⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠉⠓⠒⠦⣴⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠈⠻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠛⠷⠤⠤⠴⠒⠒⠒⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀⠒⠂⠀⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⣂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠿⠿⠿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⠞⠋⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀⠄⠊⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⢵⡀
⠀⠀⠀⠀⠀⠀⡸⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠰⠁⣠⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢣
⠀⠀⠀⠀⢀⠃⣴⣿⣿⣿⣿⣦⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡆
⠀⠀⠀⠀⢘⣰⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠅⢻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⣀⣤⣶⣿⣿⣿⡆⠀⠀⢸
⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸
⠀⠀⠀⠀⢸⠀⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣘
⠀⠀⠀⠀⠈⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡅
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡸
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⠋⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡿⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⡀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⡀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⠤⠀⠀⠀⠤⠶⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⡟⠛⠒⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠛⠓⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢾⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠹⣆⠀⠀⠀⠀⠀⠀⣼⠁⠀⠁⡀⠄⠀⠒⠀⠀⠀⠀⠒⠂⠀⠄⡈⢳⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⡀⠘⣆⠀⠀⠀⠀⡾⢡⠀⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⣾⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⢻⠀⠀⠀⢸⡇⠊⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⡏⠀⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠉⠻⣿⣿⠟⠁⠀⢼⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢻⠀⠸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣾⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠈⢇⠀⠹⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢹⡆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣀⡴⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠐⠀⠄⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠚⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⣿⡉⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣰⠷⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢳⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⣤⣍⠛⣿⣆
⠀⠀⠀⠀⠇⠀⠀⠀⠈⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢣⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢿⣏⣹⡇⠈⣿⡆
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠀⠀⠀⠀⠀⠉⠑⢿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⠛⠻⣧⠀⣿⣷
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠘⢿⣦⣴⠏⣰⣿⡿
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣴⣾⣿⣿⠇
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⢀⢀⢀⣟⣀⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠈⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠶⠞⠊⠑⠒⠒⢤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⣴⣶⡄⠀⢀⣤⣄⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣦
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠋⠁⠀⠘⠿⠟⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡏⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠈⠀⠀⢢⠈⣇
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠃⠀⠈⠀⠀⠀⠀⠃⢸⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠠⠤⠤⠀⠤⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠒⠊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⢿⠿⡹⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣛⡛⣻⣿⣿⣿⣿⠟⠻⠟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠈⠉⠛⠿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣭⣤⣤⣶⣶⣶⣤⣤⣤⣤⣀⣀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠛⢿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠛⠿⣿⣿⣶⣤⣴⣾⣿⠟⠁⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠟⠉⠀⠀⠀⠀⠉⣻⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⣧⡀
⠀⠀⠀⠀⢠⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠉⠻⣿⣷⣄⠀⠀⢀⣴⣿⣿⠿⣿⣿⣷⡄
⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠈⠻⣿⣷⣦⣾⣿⠏⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⣾⣿⠃⠀⠀⢀⣴⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⢸⣿⡇⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⣿⣿⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⣿⣷⡀⠀⠀⠀⠀⢿⣿
⠀⢰⣿⣧⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠹⣿⣷⡀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠹⣿⣷⡀⠀⠀⠘⣿⡇
⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢹⣿⣧⠀⠀⢸⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⢿⣿⡆⠀⢸⣿⡇
⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠸⣿⣿⠀⣼⣿
⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⢠⣿⡇
⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⢁⣠⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⣀⣤⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣀
⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠉⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉
⣿⢿⠿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠚⠉⠁⠈⠉⠲⣄⠀⠀⢀⠔⠊⠁⠀⠉⠑⠢⡀⠀⠀⣠⠒⠋⠉⠀⠉⠲⣄⠀⠀⢀⡴⠈⠁⠒⠤⡀
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠆⢠⠃⠀⠃⠄⠀⠀⠀⠀⠑⡀⠰⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡏⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⡅⠀⣀⡀⠀⠀⠀⠀⠀⡐⢸⠰⠂⠀⠀⠀⠀⠀⠀⠀⡇⢅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⣄⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠱⡐⠿⠃⠀⠀⠀⠀⡠⠃⠈⢆⠀⠀⠀⠀⠀⠀⠀⡔⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠢⣄⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠈⠢⠄⢠⠠⠤⠚⠁⠀⠀⠀⠑⠢⠤⠤⠠⠔⠊⠀⠀⠀⠈⠓⠤⠤⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠉⠒⠀⠚⠁
⠀⠀⡠⠆⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⠒⠤⡀⠀⠀⠀⢀⣠⠔⠒⠒⠒⢤⡀⠀⠀⠀⢀⡤⠔⠒⠒⠦⣄
⠀⠀⡇⠀⠀⠖⢄⠈⠑⢄⠀⠀⠀⣼⠇⠀⣾⣿⣶⠀⠘⣦⠀⢠⠊⠀⠀⠀⠀⠀⠀⠘⣆⠀⣰⠃⠀⠀⡔⢀⢀⠀⢳⡀
⠀⠀⢣⠀⠀⠀⠀⠑⣄⠀⢳⠀⢸⠃⠀⠀⠙⠛⠋⠀⠀⠈⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃⠀⠐⡡⢈⠠⠂⠀⠀⡇
⠀⠀⠀⠣⡀⠀⠀⠀⠘⠃⠀⡇⠘⡄⣶⣶⡄⠀⠀⣼⣿⣦⠁⢱⡀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢧⠀⠁⡀⠊⠂⠠⠐⢀⠇
⠀⠀⠀⠀⠈⠒⠤⣀⣀⠀⣀⠆⠀⠈⠻⣛⡋⠀⠀⣙⠟⠁⠀⠀⠑⢤⣀⣀⠀⢀⡠⠞⠁⠀⠈⠳⢌⣀⠀⠀⣁⠴⠋
⠀⠀⠀⠀⢀⡠⠄⠤⣀⠀⠀⠀⠀⠀⠀⣀⠩⠍⣅⣀⠀⠀⠀⠀⠀⢀⣀⣭⠩⣅⡀⠀⠀⠀⠀⠀⠀⠈⣩⡉
⠀⠀⢀⢖⣡⣤⡀⠀⠀⠉⢢⠀⠀⡠⠊⠁⠒⠀⠀⠈⠓⢄⠀⠀⠔⠋⠀⠈⠀⠀⠉⠲⡀⠀⠀⠀⠀⡌⠄⠙⠢⡀
⠀⠀⠇⡾⣁⢊⣿⡄⠀⠀⠀⠣⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⡌⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣰⣇⠀⠀⠀⠈⢢⡀
⠀⠀⡆⠻⣦⣾⠟⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠐⠀⠀⠀⢠⠇⢰⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⣴⣿⣿⣷⣄⠀⢀⣀⠹
⠀⠀⠘⢄⠀⠀⠀⠀⠀⢀⡔⠁⠀⠣⣀⠀⠀⠀⠀⠀⣠⠎⠀⠈⢢⡀⠀⠀⠀⠀⢀⡠⠃⠀⠊⠙⢿⡿⠟⠋⠉⠉
⠀⠀⠀⠀⠉⠒⠀⠐⠒⠁⠀⠀⠀⠀⠈⠑⠒⠒⠚⠈⠁⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠑⠀⠊
⠀⠀⠀⣠⠖⠉⠉⠉⠑⠢⡀⠀⠀⢀⠔⠊⠉⣏⠉⠑⢦⡀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢄⠀⠀⠀⡠⠒⠉⠉⠉⠑⠢⡀
⠀⠀⡜⢡⡆⠀⠀⠀⢰⣦⠘⡆⢠⠋⠀⠀⠀⠈⠢⣀⠀⠱⡀⠰⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⡜⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡅⠈⠀⠀⠀⠀⠀⠁⠀⡇⢸⠴⠤⡀⠀⠀⠀⠀⠉⠘⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠳⡀⠀⠀⣀⡀⠀⠀⢠⠃⠈⢆⠀⠈⠳⡄⠀⠀⠀⡰⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠎⠀⠳⡀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠉⠲⠤⢉⠥⠤⠚⠁⠀⠀⠀⠑⠤⠄⣨⠤⠔⠊⠀⠀⠀⠈⠲⠤⢄⡀⠤⠔⠁⠀⠀⠀⠙⠢⠤⢀⠠⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢒⠚⢿⣏⣢⡀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣆⠀⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⢀⣴⠞⠃⠀⠙⢿⡇⠀⠀⠀⠀⠀⡰⠁⠀⠀⣿⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⢠⠃⢀⣼⠟⠁⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢠⠁⠀⠀⢠⡿⠀⠀⠀⠀⣻⢸
⠀⠀⠀⠀⠀⠀⣾⣦⡛⠁⠀⠀⠀⠀⣠⢊⠎⠀⠀⠀⠀⠀⠸⡀⠀⣠⡿⠁⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⡏⠻⣿⣦⡀⠀⡠⢚⡡⠊⠀⠀⠀⠀⠀⠀⠀⠱⣾⠟⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠧⣀⣈⣿⡿⠯⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣀⡠⠤⠊⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡲⠖⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⣠⣀⣀⠈⠻⢦⠀⠀⠀⠀⠀⠀⣠⠋⢳⠾⠤⠴⠒⠒⠒⠳⡄
⠀⠀⠀⠀⠀⠀⣷⣶⡀⠀⢴⣿⣿⣿⡀⠀⠈⡇⠀⠀⠀⠀⢠⠋⣉⡟⠒⠢⠤⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠙⠋⠉⠀⢠⣶⡇⠀⠀⠀⠀⠸⡏⢸⠉⠲⡀⠀⠀⠉⠓⠦⣸
⠀⠀⠀⠀⠀⠀⠹⡀⠀⢀⣀⣄⡀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠳⣸⡀⠀⠘⢆⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢼⣿⣿⣟⣁⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⣀⣱⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠤⠤⠤⢤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠉⠀⢀⣀⡤⠔⠒⠋⠉⠉⠉⠑⠢⢭⣉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠀⠀⢀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣈⣙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⣀⡤⣖⡿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣇⢀⣠⢴⣺⠽⠛⠉⠁⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡼⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⢠⠏⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⠔⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⡞⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⠁⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⡿⠚⠙⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⡇⠀⠀⠸⡽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠷⠤⠤⠤⠤⠴⠚⢷⣻
⠀⠀⠀⡇⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⠀⠀⠀⢳⣣⣀⣀⣠⠤⠴⠒⠋⠉⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣧⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠘⣆⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠯⠿⠿⠟⠛⠛⠿⠿⠟⠛⠛⠛⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠘⢦⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⠚⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⠤⢶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠒⠦⣄⣀⡀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠉⠉⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠈⠻⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⠀⠙⢾⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠛⠛⠒⠒⠒⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠋⢉⡁⠉⠻⣷⣦⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⢿⣿⡤⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣴⣾⠿⠂⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⣹⣿⡂⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⣨⡿⠏⠉⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣸⣟⡂⠀⠀⠀⠀⠀⢿⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣶⣾⠛⠁⠀⠀⠀⠀⠀⠀⢀⠟⡟⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣾⠷⠀⠀⠀⠀⠀⠀⢽⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⣲⠯⠁⠀⠀⠀⠀⠀⠀⢘⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⠞⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢠⣦⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢿⣿⣷⡄⠀⢀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠙⣿⣿⣖⣫⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠛⠉⠈⠉⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣫⠿⠋⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠹⢦⣼⣁⣀⣠⣤⠤⠶⠶⠦⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⠤⣤⣼⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⡄⠀⠀⠺⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡿⢀⣀⣸⠃⠉⠉⠛⠲⠦⣄⣀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠃⠀⠀⠀⠈⠻⠟⠛⠉⠀⠀⢀⣴⣾⡇⠀⠀⠀⠀⠀⠀⠀⣷⠋⠁⡟⠙⢦⡄⠀⠀⠀⠀⠉⠛⠶⣄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡇⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣤⣤⣤⣦⣀⠀⠀⠀⠀⠛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢷⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣹⣿⣿⣿⣿⠟⠀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠙⢦⣄⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠚⠓⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠉⠉⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⠳⡄

⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠿⠛⠉⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⢰⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⣸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠐⠈⠻⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⢹⣠⡄⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⡆
⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣟⣛⣃⣀⣀⡀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣐⣂⣉⠁⠈⠻⢅⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠉⠓⠦⣄⡽⠾⣍⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣀⡤⠤⠶⠶⠦⢤⣠⠞⠉⠙⢦⡀⠙⠢⢌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠰⠆⠤⢤⣄⣠⢃⡴⠊⠁⠀⠀⠀⠀⣠⠞⠉⠳⡄⠀⠀⠙⢆⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣾⣿⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⢀⡾⠃⠀⠀⠀⢹⡄⠀⠀⠈⢳⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⡠⣺⣿⣿⣿⣿⣿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣠⠟⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢳⡀⠀⡇
⠀⠀⠀⠀⠀⢀⡼⠘⠻⣿⡿⠟⠉⠀⠀⠀⠀⠢⢄⣀⣠⣤⣤⣦⡀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢷⠀⡇
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡄⠀⠀⢡⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠸⡰
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠸⡆⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⠋⠀⠀⠘⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠸⠀⠀⢀⣴⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠁⠀⠀⠀⢠⡇⠀⠀⠀⢠⠇⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠈⡄⠀⢺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠘⡄⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠙⠟⠛⠛⠛⠓⠄⡀⠀⠀⠀⢀⣠⣶⣿⣶⡶⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢀⣀⣀⣀⣜⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⢰⣶⣶⣄
⠀⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⢿⣿⣿⠆⣀
⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⢠⣾⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠿⣿⠏⠀⠀⠀⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠉⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣴⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠘⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⣀⣀⣀⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠻⣿⣷⣦
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣼⣿⣿⣿⣆⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⢀⣧⡀⠀⠀⠙⠿⣿⠟⠋⠀⠀⣀⣧⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⣀⣀⣤⡀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⣀⣤⡀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⢱⣄⠀⠘⠿⢿⣿⠏⠀⠀⠀⢘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡀⠀⠀⠀⢀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠻⢟⠉⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣤⠚⠁⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢇⠀⠀⠀⠀⠀⠀⠀⡹⠛⠋⠉⠀⠀⠙⢦⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣦⣤⣤⣤⣴⣇⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣧⣄⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢀⡤⠖⠒⠀⠐⠒⢤⡀⠀⠀⠀⠀⠀⡠⠔⠒⠂⠐⠒⠤⣀⠀⠀⠀⠀⠀⣀⠴⠖⠒⠀⠒⠦⣄⠀⠀⠀⠀⠀⣠⠔⠒⠤⣀⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣠⠊⢠⡄⠀⠀⠀⠀⠀⠈⠳⡀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⢀⠞⠁⠀⠀⠀⠀⠉⠲⣄
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠇⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⢱⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⢸⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠱⡄⠀⠈⠠⡀⠀⠀⠀⠀⠹⡀
⠀⠀⠈⢆⠰⣿⠷⠀⠀⠀⠀⠀⠀⡰⠃⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⠢⣀⠀⠀⠁⠀⠄⠀⣠⠇
⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠑⢤⣀⠀⠀⠀⢀⣀⠴⠃⠀⠀⠀⠙⠦⣀⡀⠀⠀⠀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⢰⠋⠉⠉⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⣉⡉⠉⠓⢦⡀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠉⠙⠲⣄⠀⠀⠀⠀⡤⠞⠋⠉⠉⠉⠉⠲⣄
⠀⠀⢸⠀⠀⠀⠘⠥⣠⠀⠈⠳⡄⠀⠀⠀⣴⡟⠀⠀⣾⣿⣿⡷⠀⠘⢿⡄⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⡞⠁⠈⠀⢠⠀⢀⠰⠀⠌⢳
⠀⠀⠘⡆⠀⠀⠀⠀⠑⡵⡀⠀⠘⣆⠀⢰⠟⠀⠀⠀⠙⠛⠛⠃⠀⠀⠈⢻⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠀⠂⢑⠈⡆⠘⢀⠥⡨⠠⠀⡇
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠑⡷⠀⠸⡄⠈⡆⢠⣤⣤⠀⠀⠀⠀⣰⣶⡆⡸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡀⠀⠃⡨⢠⠃⠔⢠⠁⡌⢀⡇
⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠁⠀⡇⠀⠱⣼⣿⣿⡷⠀⠀⠰⣿⣿⡷⠁⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢣⡀⠂⡒⠜⠰⠁⠆⠀⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠄⠠⠴⠃⠀⠀⠈⠓⠭⣄⣀⣀⡠⠼⠋⠀⠀⠀⠀⠀⠙⠢⢤⣤⣀⣠⠤⠚⠁⠀⠀⠀⠀⠙⠳⢤⣄⣀⣄⠤⠞⠁
⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⢠⠖⣉⣀⣀⠀⠀⠀⠈⠲⣄⠀⠀⢀⡴⠋⠑⠒⠂⠀⠀⡈⠳⢆⠀⠀⠀⡠⠚⠁⠀⠈⠀⠀⠈⠙⢢⡀⠀⠀⠀⠀⠀⢠⢁⠈⡙⢦⡀
⠀⠀⢠⢃⣾⠏⡉⢻⣷⠀⠀⠀⠀⠘⡆⠀⡞⠀⠀⠀⠀⠠⠄⠀⠀⠀⠨⢇⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠘⠢⡀
⠀⠀⢘⠸⣇⠲⢀⣾⣿⠀⠀⠀⠀⠀⡇⢰⠆⠂⠀⠀⠀⠀⡀⠀⠀⠂⠄⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣾⣿⣦⣀⠠⠀⠀⠀⠙⣤
⠀⠀⠸⡀⠙⠿⠿⠟⠁⠀⠀⠀⠀⢀⡇⠈⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡞⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣴⣿⣿⣿⣿⣿⣦⣀⣀⡤⠤⠊
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠈⢦⡀⠂⠀⠀⠀⢀⠁⣠⠞⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠈⠀⠙⢿⣿⠿⠛⠉
⠀⠀⠀⠀⠈⠙⠒⠦⠤⠄⠖⠋⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠒⠚⠁⠀⠀⠀⠈⠢⠤⠐⠁
⠀⠀⠀⠀⢀⡤⠔⠒⠒⠒⠲⢄⡀⠀⠀⠀⠀⠀⣠⠴⠒⡖⠒⠲⠤⡀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠲⠤⣄⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⠢⢄⡀
⠀⠀⠀⡰⠛⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢠⠎⠀⠀⠀⠑⢆⠀⠀⠈⢳⡀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⢸⡃⣾⠗⠀⠀⠀⠀⠘⣿⠆⢈⡇⢀⠇⠀⠀⠀⠀⠀⠀⠳⢦⣀⠀⢣⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡖⠢⠤⡀⠀⠀⠀⠀⠀⠀⠉⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⢆⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢧⠀⠀⠈⠳⡄⠀⠀⠀⠀⢀⠎⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠈⠳⢄⣀⠘⠟⠃⠀⣠⠖⠁⠀⠀⠀⠱⢄⡀⠀⠘⣆⠀⣀⠴⠋⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠂⠀⣤⣀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠿⢿⣿⣿⣦⡀⠀⠀⡐⠑⠀⠀⠀⠀⠀⢀⠜⠁⡀⠀⠀⡠⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠀⠀⡠⠾⠛⠁⠀⠀⠀⠙⠻⠷
⠀⠀⠀⠀⣸⣿⣿⠏⠀⡴⢦⡀⠙⣿⣿⣷⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠠⠀⠠⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⣠⣶⣄
⠀⠀⠀⠀⢿⣿⣿⡀⠀⣿⢻⠄⠀⣿⣿⣿⠀⠒⠀⠒⠐⡃⠐⠒⠐⠇⠀⠒⢒⠀⢀⣀⠀⠀⡀⠀⠃⠀⠀⠀⠀⢀⠀⢰⣄⠀⠀⠸⣿⣿⣿⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠘⣿⣿⣷⣄⣉⣉⣠⣼⣿⣿⠏⠀⠐⡀⠀⡐⠀⠀⠀⠀⠀⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠈⢿⠄⠀⠀⠉⠉⠁⠀⠀⠰⡟
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⠈⠒⠀⠂⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⠀⣤⣤⣤⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠁⠐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⡂⠀⠐⠂⢀⡀
⠀⠀⠀⠀⢀⠜⢦⠀⠀⠀⠀⠀⠀⡠⠓⡄⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠠⠄⢀⠀⠀⠀⠐⠝⠀⠀⠑⠀⠃⠀⠐⢉
⠀⠀⠀⠀⡈⠀⠀⢧⠀⠀⠀⠀⢰⠁⠀⠘⠀⡀⠀⣿⢠⣤⣤⣤⣤⣤⡄⢸⡍⠢⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⡰⠀⠒⠀⠀⠀⠀⠀⠒⠠⠆
⠀⠀⠀⠀⠃⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⢠⠀⠣⡀⣿⠰⠾⠺⠾⠾⠶⠆⢸⠇⡈⠁⢸⠀⠀⠀⠉⠉⠉⠁⠀⠀⢐⡇⠀⢡⠄⠀⣀⡀⠀⣀⣀⠀⠠⡄⠁
⠀⠀⠀⠀⠘⢄⣠⠃⠀⠀⠀⠀⠘⢆⣀⠎⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠁⠒⠠⠤⠤⠤⠄⠄⠒⠉⠀⠀⢀⢀⠀⠀⡀⠀⡀⠀⢀⠐
⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⠀⣀⠠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢀⡃⠀⢁⡀⠔⠃

⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⣠⠒⠉⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⣀⣴⣖⣿⡪⣿⠰⡦⡀
⠀⠀⠀⠀⠀⠊⠀⠀⣤⡄⢀⣤⡀⠀⠀⠄⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠎⠙⣆⠀⠀⠀⠀⢀⡞⠉⢆⠀⠀⡤⡗⢽⢎⣭⡱⠿⢘⣗⣽⣀
⠀⠀⠀⠀⡄⠀⠀⠀⠙⠁⠈⠛⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠘⠀⠘⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⠈⡄⢰⣏⣏⢾⢇⣭⡸⠿⢪⣏⠾⣺⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰⠀⠀⣸⠁⠀⠀⠀⢹⠀⠀⢀⠃⠸⡯⣯⢿⢎⣭⡱⡿⢨⣭⢛⣺
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠋⠀⠀⠀⠠⠂⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠢⣰⠋⠀⠀⠀⠀⠀⢷⣠⠂⠀⠀⠙⣗⣻⢋⣽⡱⡿⢨⡧⣽⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠉⠂⠤⠀⠀⠠⠤⠊⠁⠀⠀⠀⠀⠀⠛⠇⠿⠘⠿⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⣴⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦
⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠻⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠟
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁
⣶⣶⢶⢶⢷⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣾⣶⢶⣶⢶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡿⠗⠀⠁⠀⠉⠀⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⢆
⠀⠀⠀⠀⠐⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠠⢡⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⣀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠚⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡀⠀⢸
⠀⠀⠀⡆⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠸
⠀⠀⠀⠁⢸⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⢀⠂
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⠟⠀⠄
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠃⢀⠌
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠠⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⠖⠒⠒⠒⣒⣶⡦⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠉⠁⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⡁⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⢻⡄⠀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢠⣿⡿⠟⠳⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⣀⣰⣿⠋⠀⠀⢠⠏⡼⠒⠒⠒⠒⠒⠒⠒⠶⠦⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⡤⠶⠚⠉⠁⢹⠻⣤⣤⠖⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣾⠴⠋⠁⠀⠀⠀⠀⠀⢈⠗⠒⠒⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⢻⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⣧
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⠿⣿⠋⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠉⠀⠀⠀⣤⣤⣕
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢺⣤⣦⠀⠀⠀⠀⠀⢠⣦⣤⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣀⠀⠀⠀⠈⠙⠿⠟⠀⠓
⠀⠀⠀⠀⠀⢸⣿⣿⡦⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⠿⠁⠀⠀⠀⠈⢻⣿⡿⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠿⠀⠀⠀⠀⢀⣠⣄⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠄⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⣶⣶⣶⣦⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠙⠛⠛⠓
⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⢿⣿⣿⣿⣿⠆⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣾⣿⠆⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣶⣶⣿⣦⡀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣤⣶⣦⣄⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠈⠙⠛⠿⠋⠀⠀⠀⠀⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⠀⠀⢀⣀⣤⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠋⠉⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠰⡄⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠓⠢⣄⡵⠶⡉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡤⠴⠒⠒⠲⣤⣔⠋⠙⢦⡈⠑⠠⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣐⡠⠐⠂⠒⠲⠤⣧⡖⠁⠀⠀⠀⢀⡴⠃⠈⢧⡀⠀⠙⢆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠲⢄⠀⣠⠏⠀⠀⠀⠈⣇⠀⠀⠈⢣⠀⠀⡄
⠀⠀⠀⠀⠀⡐⠱⣿⣿⡿⠟⠉⠁⠀⠄⠀⠀⣀⣀⣀⠀⠀⠰⡃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠇
⠀⠀⠀⠀⠘⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠰⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡜
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠁
⠀⠀⠀⢨⠀⠀⠀⣠⣦⣄⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⠟⠁⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠄⠀⣴⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠧⠄⠀⠠⠧⠄⠒⠉
⠀⠀⠀⠀⠠⡀⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠑⠄⠈⠿⠿⠟⠛⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠤⠄⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠚⠛⠉⠉⣉⡭⠿⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢳⡀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢳⣤⣶⣶⡺⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⣴⡿⠋⠀⢀⡇⡿⠤⠤⠤⠤⠤⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣠⠏⢀⣀⠤⠖⠚⠙⡿⣇⣀⡤⢞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢰⡧⠖⠉⠀⠀⠀⠀⠀⡹⠶⠶⠒⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⢹⡄
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⣧
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠱⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠱⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠦⠤⢤⣤⣀⣀⣀⣠⣤⠤⠴⠖⠚⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡏⠉⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠛⠉⠉⠉⠉⠑⠢⣄
⠀⠀⠀⠀⢀⡔⠉⠻⠟⠋⠉⠀⠀⠀⠀⣀⠀⠙⢦⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⠈⢧⠀⠀⠀⢠⠋⠀⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⢸⠀⠀⣠⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠸⡄⠀⠀⣷⠈⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠀⣾⣿⣿⣿⣶⠀⠀⠀⠀⠈⠻⠿⠋⠀⢸⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⡄⠸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠙⣄⡈⠉⠉⠁⠀⠀⠀⣀⣠⣤⣄⢀⡼⠁⠀⠀⠀⠀⠱⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⢠⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⠀⢀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠉⠐⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠲⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢠⠦⢤⣤⣀⡀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡆⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⣨⣭⡙⢿⣆
⠀⠀⠀⠇⠀⠀⠀⠈⠙⠳⢄⠀⠀⠀⠀⠀⠙⠦⣀⢸⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣬⡟⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣏⢀⣿⢀⣿⡇
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡸⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣙⣋⣥⣾⣿⠃
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⣸⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠤⠟⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠄⠀⠈⠉⠲⣄
⠀⠀⠀⠀⡰⠁⠀⠀⠀⣶⣶⠀⢠⣤⡄⠀⠀⠈⢳⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢦
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢀⣀⠈⠋⠁⠀⠀⠀⠀⢳⠀⠀⡼⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⢇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢅⣀⣨⣁⡠⠤⠚⠁

⣿⠿⠿⠿⢿⢟⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣿⣿⣿⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⢤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠶⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⡿⠟⠋⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⣠⣤⣤⣴⣶⣶⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⢠⠇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡼⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣦⣄⠀⠀⠀⠀⠀⣼⣿
⠀⠀⢸⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠙⢿
⠀⠀⢸⡆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸
⠀⠀⠀⣧⡆⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠆
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠉⠻⡅⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣽⣶⣶⣤⣄⡀⣠⠔⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠦⠤⠤⢤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠟⠛⠋⠁⠀⠉⠳⣄⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠘⣆⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣶⣶⣶⣶⣶⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠟⠓⠋⠉

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

⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⣶⣄⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠔⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⢰⣦⡀
⠀⠀⠀⢀⣾⣿⡿⠋⣉⠛⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠉⠁⠀⢀⡀⠀⠈⠉
⠀⠀⠀⢸⣿⣿⠀⢸⢾⡃⠀⣿⣿⡇⠠⠀⠀⠤⠀⠤⠠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⣿⣿⡗⠀⠀⣠
⠀⠀⠀⠘⣿⣿⣦⣈⣋⣁⣼⣿⡿⠁⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡇⠀⠀⠉⠉⠁⠀⠰⡟
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠂⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄

⠀⠀⠀⠀⢠⡂⠁⠀⠀⠀⠈⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠄⠠⠀⠀⠆
⠀⠀⠀⠀⠁⠙⡄⠀⠀⠀⢰⠁⠈⠀⠀⠐⡏⣀⣀⣀⣀⣀⠘⡇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢨⠀⠀⠀⠃⠘⡇⠾⠷⠿⠿⠾⠀⡇⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢂⡰⠁⠀⠀⠀⠘⢆⠔⠀⠀⠈⠓⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠑⠂⠐⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⡀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⢀⠀⠀⠀⠀⠀⢀⣀⣤⢠⡄⡀
⠀⠀⠀⠀⠐⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢧⠀⠀⠀⠀⢠⠓⠄⠀⢀⣼⣹⣕⣟⢼⡕⡿⣶⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠛⠁⠀⠀⠀⢘⠀⠀⠐⠂⠐⠒⠒⠒⠒⠀⠀⠀⠈⡇⠀⠀⠀⡏⠀⠈⠀⣸⣺⡶⢎⣯⠾⢎⣟⢿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⢠⠇⠀⠀⠀⢇⠀⠠⠀⢸⣿⢾⣫⣭⢺⡧⣭⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠏⠀⠀⠀⠀⢘⠖⠁⠀⠀⠹⢽⡵⡷⣹⡗⣿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣤⠤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⣦⣄⣀⠀⠀⣼⣃⣀⣀⣤⡤⠤⠶⠖⠒⠚⠓⠂⠐⠠⢤⣌⣣⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠉⠉⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⣀⣀⣤⣤⣴⣿⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡴⠞⠛⠉⠉⠁⠀⣾⠁⠀⠀⠀⠈⠉⠉⠙⠛⠳⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⣀⣰⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⣤⣀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡇⠀⢀⣤⠞⠛⠉⠉⣿⠉⠉⠙⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⡇
⠀⠀⠀⠀⠀⣇⢠⡟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠐⡇
⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠁
⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⡀⠀⠀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣿⣷⣤⣄⠀⠻⣦⠀⡀⠀⠀⣀⢀⠀⣀⣤⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⡿⢿⠿⢿⣿⣿⣿⡿⠿⠿⢿⠿⠿⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠋⠉⢉⣭⠟⠋⠉⠉⠉⠉⠛⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣄⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡤⠶⠶⠦⠤⠒⠤⠀⣀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠛⠻⠷⣶⣤⣤⣼⣿⠶⠶⠖⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣮⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⣀⣀⣠⣤⣤⣿⣥⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣤⡶⠿⠟⠛⠛⠉⠉⣿⡏⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⢶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣸⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣴⠾⠟⠛⠋⢹⣿⠉⠉⠙⠛⠻⠷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣾⠟⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠙⠷⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣷⣠⡿⠁⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀⠀⠀⣀⣤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣴⣄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣾⣿⣿⣦⣤⣌⣿⣧⠀⣦⣄⠄⢄⣲⣲⣴⣂⣶⣾⣿⣿⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠹⠿⢟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠈⠃⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠌⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣠⠰⠹⢿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⢀⡴⠲⣶⣶⢶⣄⡀⢀⣿⡶⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⣠⢤⡀⠂⠈⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⢏⡈⡔⢡⣁⠊⠑⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠄⣸⣥⣤⣤⣤⣭⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣷⣾⣿⣿⡿⢃⠀⠁⢹⠛⡁⢁⣤⡄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⡾⠁⢈⡼⢣⡈⠀⠁⠈⣌⠁⠀⠀⢿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢢⢄⡄⠀⠀⢸⠁⠉⡀⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⡠⠈⠹⠧⠖⠋⠀⢀⠑⠀⠀⡔⠙⠀⠀⣤⠈⣿
⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⡏⢡⢇⡃⠃⠀⠀⠈⠀⠘⣧⠀⣀⣠⠤⠚⠁⠀⠀⠀⠠⣎⡀⢀⣀⠠⠤⠔⡚⢿⢃⡤⠊⠀⢀⠃⠀⠰⠀⡉
⠀⠀⠀⢄⡶⡾⣆⠀⠀⠹⣿⡟⡅⣵⡎⠙⢓⢠⠄⠀⠀⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡸⠀⠀⠀⠀⡇
⠀⠀⠀⠾⣿⢾⣧⠀⠀⠀⠙⢷⣴⡹⡇⠂⠈⠁⠀⠀⠀⣿⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠁
⠀⠀⠀⠰⣿⣷⠁⠀⠀⠀⠀⠀⣿⣇⠈⣺⠀⠀⠀⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠁⠀⢸⡆⠇⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⡌
⠀⠀⢀⣄⠘⣻⡅⠀⠀⠀⢀⣉⠁⠹⢆⣿⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⢌⣣⠸⢸⠀⠀⢰⠀⠐⠃⠀⠀⢀⡠⠖⣹⣷⡄⠀⠀⠐⣉
⠀⠀⠀⠏⠳⠹⣗⠀⠀⠈⠙⠛⠋⠀⡜⠛⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⡄⣏⠦⡘⠀⠀⠸⣄⢸⣤⡠⠖⠁⢀⣼⣿⣿⣷⠀⠀⠀⠈⠐
⠀⢠⣶⡆⠀⠀⢋⣄⠀⠀⠀⠀⢰⣾⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣠⠀⠠⢸⠄⠃⠀⠀⢠⣿⣿⣯⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⢠⠂⠠
⠀⠀⠻⠇⠀⠶⡀⠉⠀⢀⡠⠴⢏⡈⠙⠳⢷⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⡧⠔⢻⡄⠀⠘⢃⣿⠻⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⢸⠁⠀⣿⠙⠓⣿⠌⠉⣲⣤⣀⣀⣀⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡧⠶⠀⠸⣶⣾⣿⣷⡀⠀⣸⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⣸⡀⢼⡷⠟⠀⣈⢻⢫⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢻⣿⣿⣿⣷⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠟⣡⣤⠀⢰⣯⢓⠾⡀⠡⠊⢀⣀⣀⠀⠀⠀⢀⡴⠴⠦⠼⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⡄⢀⡎⠀⠀⠀⡎⢸⠯⠜⡲⠆⠞⠁⠀⣀⠤⣊⠖⠲⣭⠻⢆⠀⠈⣞⣙⣋⠱⠂⠀⡝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣠⢿⠀⡜⠀⠀⠀⣸⢁⣤⣦⠞⠀⣴⣶⣶⣿⣶⣶⣶⣾⣷⣿⣶⣾⣶⣶⣿⣷⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣰⣷⡾⣷⠁⠀⠀⠀⠧⣛⣃⣍⠀⠀⠈⠉⠹⠋⠉⠉⣉⣉⣭⣭⠭⢭⣿⡉⠉⠉⠉⠉⢩⣿⣿⠉⠉⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⢉⣿⣸⣄⣀⣀⣀⣀⣀⢠⡭⣍⡁⠚⠻⢞⣁⢾⡏⢩⣟⠷⠀⢠⣞⠛⣿⡀⠀⠄⠀⢸⡇⡀⠀⠀⠀⣠⠊⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠛⠛⠛
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡠⢀⢽⢏⡀⣤⠀⢀⣐⡄⠀⠀⢹⣯⠆⢤⣴⣾⣿⡇⠀⠀⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⢾⡎⣧⡻⠃⠀⠺⠙⠁⠀⣾⠎⢹⣯⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣦⣸⡆⢹⢁⣶⡿⠂⢠⣶⡇⠀⣴⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣾⣿⣷⣇⡈⢉⣤⠤⠽⡿⠒⠋⠙⣇⠁⠀⠉⢻⣿⣿⣿⣿⣿⣏⠙⣷⢀⠤⠀⠰⠄⠂⢀⣩⣽⡮⠁⠈⠈⠁⠚
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⡰⠉⢦⠀⠀⠹⣆⠀⠀⣾⠉⢿⣿⣿⣿⣿⣦⢹⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⠛⠉⠀⡆⠀⢠⡅⠀⠀⠳⡀⠀⠹⣦⣀⣿⠀⠀⠙⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⣇⣀⣀⡈⢻⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⣁⠀⠘⣆⠀⠀⠀⠙⣿⣿⢿⣿⣿⣶⣤⣤⣀⢀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⠀⢸⣿⠻⢿⣿⣶⣄⣀⠀⠈⣧⠀⣼⠀⠉⠛⣻⣿⣿⡿⠋⠈⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠚⠁⠀⠀⠀⠀⠃⠀⠀⠈⠙⠛⠿⠿⠗⠛⠚⠋⠠⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⠚⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣶⣿⣶⣯⣭⣿⣧⣿⣿⡿⣿⣿⡧⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⠟⠹⠿⠟⢻⡟⢻⣿⡶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡷⡌⣿⣿⡆⠀⠀⠀⢈⠉⠉⠈⢳⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⠀⠀⢺⣿⣿⣿⠀⣸⣇⡇⠀⠀⠀⠀⠀⣶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⣿⣿⣿⣿⡟⠀⠀⠀⠉⠉⠛⢀⣿⠿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠀⢤⣾⣿⣻⢋⣿⣿⣶⣤⣀⡀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢰⣿⡽⣱⣿⣼⣿⡇⠙⠋⠉⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠙⢶⣤⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⠀⢀⣠⣤⣴⣶⠿⣿⡿⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠘⡟⢜⢿⣿⣿⣷⡻⣧⠄⠀⠀⣾⣧⣭⣿⣿⣿⣿⡟⣼⣿⢟⣿⣿⣿⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⣰⡿⢸⠘⠿⣿⣿⢿⣟⠛⢦⣠⠏⠀⢺⣿⣿⣿⠟⣱⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣍⠀⠀⠀⢠⣿⢣⣯⣴⡇⠨⣿⣿⡿⠃⠀⠘⠂⢀⠀⢹⡿⢻⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⠚⠁⠀⠀⠀⠀⠁⠘⡀⠀⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣇⣀⣩⣿⠟⠉⢻⣿⣿⡏⢇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢿⣿⣇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢣⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣭⡤⠀⠀⠀⠰⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠶⠿⡥⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⠳⣄⠀⠀⠀⠀⠀⣠⠞⠑⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠖⠉⠙⠦⣀⠀⠀⠀⢸⠁⠀⠈⠓⠦⣄⡀⢰⠃⠀⠀⠈⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠙⠲⠤⣬⡦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠓⢶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠶⠚⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠘⠦⣀⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⢢⠀⣀⠀⠀⠈⠁⠀⡜⠉⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢹⡹⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡿
⠀⣴⣤⣶⢀⠀⠈⢇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⡀⣶⣤⣦
⣻⣽⠞⠻⢯⠇⠀⠘⢎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠃⠀⠸⡽⠟⠳⣿
⠙⢯⡀⠀⠸⢤⡀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⠃⠀⢀⡤⠃⠀⢀⡽
⠀⠀⠈⠳⡀⠀⠙⠦⣀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠉⠉⢉⠏⠀⠀⠀⠀⠀⢀⣠⣯⠞⠀⣀⠴⠋⠀⢀⠞⠁
⠀⠀⠀⠀⠙⢦⠀⠀⠈⠳⢄⡀⠙⠻⣦⣄⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⣠⣴⠿⠋⢀⡠⠞⠁⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠳⢤⡀⠉⠛⠷⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠾⠛⠋⢁⡤⠞⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠳⡶⣄⠀⠀⢘⡇⠀⠀⢸⡃⠀⠀⣠⢶⠞⠉⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⡰⠃⠀⠙⠒⠋⠱⢤⡤⠞⠙⠒⠋⠀⠈⢆⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡠⠐⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⢄
⡘
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣼⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢀⣿⣿⣏⠀⣸⣿⣿⣿⣿⡆
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣹⣿⣿⣿⣿⣿⣿⣧
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠀⢈⣁⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠟⠁
⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⡇
⡇
⢡
⠀⠑⠠⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠊

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠋⠉⣇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣍⠀⠀⠀⣿⡈⠀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠔⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡏⠐⢤⣴⣿⣿⡆⠀⠀⠀⣀⠀⠀⠀⠀⠘⣦⠀⠀⢀⠃⠠⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉⠁⠀⠈⠙⢻⣿⣿⠆⠀⠰⠂⠈⠀⠄⡀⠀⠁⠐⠠⠊⠀⠔⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⡀⠀⠀⠀⢹⠀⠘⣿⡷⠀⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠙⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠠⠁⠀⠀⠀⡆⠀⠀⠡⠀⠀⠀⠁⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢜⠀⠀⠀⣰⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣼⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠖⠀⠈⠒⠇⠀⠀⠀⢠⣤⣤⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢉⠙⠉⠀⠈⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⠆⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

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

⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢗⢀⠃⠁⢐⠸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⡰⣲⡄⠀⣤⡟⣥⡏⠐⢀⠀⠄⠂⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠐⣿⠁⣼⡟⠫⣴⠀⢃⠄⠐⠈⡌⠎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡜⢳⡆⠀⡖⣿⠀⣞⣀⠢⠉⢐⠒⠠⠁⢀⡂⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢳⡄⢰⡀⡇⢸⠀⣿⠂⡎⠃⠐⠂⠠⠀⠠⢂⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠻⡄⢳⣼⢸⡄⣸⣆⢣⠁⠀⠀⠄⠀⠄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⡄⠹⣶⠇⠘⣷⡾⡊⠲⠆⢀⠰⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢿⣷⠀⠘⡿⣵⣬⢁⣂⡲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡿⣎⠌⢦⠀⠈⢮⠉⢉⠽⠊
⠀⠀⠀⠀⠀⠀⡇⠘⣇⠈⢳⡀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠙⣯⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠒⠖⠒⠤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡰⠋⠑⠀⠀⠀⡀⠂⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁⡄⣷⡀⠀⠀⠈⢦⠀⠀⠀⢀⡞⠀⠀⠀⠈⡶⠊⠀⠀⠀⠀⢀⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⣿⣧⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠠⠐⠈⠀⠀⠀⠀⡠⠂⠁⠀⠀⣀⡤⠞⠓⠒⠒⠒⡆⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣧⣿⣿⢧⡀⠀⠀⠀⢳⡖⠉⠚⣅⠀⠀⠀⠀⠀⢀⠚⢄⠀⠀⣠⠞⠁⢣⡀⠀⠀⠀⠈⠙⠢⡈⢢⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣏⠁⠘⢷⡄⠀⠀⠀⠱⡀⠀⠘⣦⠄⠀⠀⠀⠂⠀⠀⠑⠴⠁⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡔⠛⠃⠷⣄⠀⠻⣄⠀⠀⠀⠰⡄⠀⠘⡄⠀⠀⡆⠀⠀⠀⢀⣧⡠⠔⠈⠻⣿⠿⠗⢀⠀⠀⡀⠄⠛⠿
⠀⠀⣠⠒⠉⠁⠙⣇⠬⠷⢀⠀⠀⠈⠳⠤⠈⢂⠀⠀⠀⠘⢄⡰⠃⡏⠈⠁⠂⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠉⠒⠤⠄⠀⠀⠘⢧⡀⠀⠀⠈⢣⣀⠘⢆⠀⠀⠐⠀⠀⠷⢄⠀⢀⣀⣄⠀⠀⠀⠀⡀
⠀⠀⢣⡀⠀⠀⠀⠎⠙⡇⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠹⡑⡄⠉⠈⠹⠤⠤⠒⡀⠙⣿⣿⣿⣷⡄⠀⠈⠁⠀⢄⣀⡰
⠀⠀⠀⠉⠀⠀⠒⠛⢄⣘⣦⡽⠂⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⢀⠀⣧⣸⠀⠔⠃⠀⠀⠀⠑⢤⠘⠻⠿⢏⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠊⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠈⠧⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣛⠋⠉⠉⠉⠉⠙⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠈⠙⠻⢶⣤⡀⠀⠀⠀⢸⡟⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣠⡿⠁⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣴⠿⢿⣄⠀⢀⣠⡶⠟⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⡶⠶⠟⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⢀⣻⣶⠟⠉⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠙⣷⡀⣀⣀⣀⣀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⢀⣠⡾⣿⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⢠⡾⠋⠀⢹⣇⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⣿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣿⡆⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣴⣾⣏⣠⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⡔⠶⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠐⠈⠱⡤⢈⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡆⢠⣻⣓⢲⣹⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⠙⢻⠶⠊⡁⠀⠙⠃⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⣿⠡⣤⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠴⠆
⠀⠀⠀⠀⠀⡿⠿⣶⣷⣾⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠜⣿⣧
⠀⠀⠀⠀⠀⠌⠂⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠚⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠞⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⣭⣧⣀⠀⣀
⠀⠀⠀⢰⣤⣀⡈⣯⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣖⣐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡽⢉
⠀⠀⠀⠀⠉⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣾⣇⣹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢤⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠙⠋⠉⢉⣿⣷⣴⣿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢴⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⠁⠉⠉⠢⠀⢀⡐⢀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠛⣿
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠁⠀⠀⠀⠀⠘⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠆
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢤⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣥⠞⠀⠀⡠⢴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠚⠶⡀⠀⠀⠀⠀⠠⢿⣀⠸⣄⣁
⠀⠀⠀⣀⠀⡷⠟⠁⠀⠀⠀⠹⠯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠈⠳⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡬⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠈⠹⠾⠇
⠀⠀⠀⠀⠀⢻⣿⣶⣤⣀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣇⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣮⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡟⠛⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⠟⢋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠈⡹⣿⣴⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣤⣧⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢰⣥⣤⠆⠈⠙⠂⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠜⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣠⣴⣶⡆
⠀⠀⠀⠀⠀⠀⢀⠠⠠⣀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢡⠀⠀⡈⠀⠀⢠⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⢸⣿⣿⡇⣀⡀⠀⠉⠀⠈⣁⣈⠉⢻⣿⠀⠀⠀⠀⢿⣿⠟⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠡⠀⠘⣿⡿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢄⠀⠀⠀⢸⡏⠀⠤⠄⠀⠀⠀⠀⣤⡀⢈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⣜⠻⡅⠀⠔⠒⠀⠀⠀⠀⠐⠂⠀⢰⠋⡈⠀⠀⠲⢿⠀⣶⡶⠂⠀⠀⠀⣠⡄⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠦⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⡀⠀⠀⠂⢸⠀⠻⠟⠀⠀⠀⠸⢿⠇⠀⢸⠗⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡨⠤⣄⠀⠀⠰⣶⣶⣾⠇⠀⢀⡾⠛⣃⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠘⠆⢄⡀⠀⠈⢀⡀⠐⠁⠠⠠⠈⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠉⠁⠈⠀⠓⠀⠄⡀⠄⠑⠀⠀⢀⣀⣀⣀⡁⢂⣀⠀⠄⠐⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡂⠄⠀⣽⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠶⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠿⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⠀⡤⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠂⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠑⠌⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠚⠹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠤⠄⠀⠀⠀⠀⠤⠄⠐⡎⠀⠀⠀⠀⠀⢸⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⡀⠌⠀⠀⠂⣀⠀⠀⡄⠀⠀⠀⠀⠀⠃⡄⠀⣀⠔⠀⠈⠄⡀⠀⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⡆⠀⠀⠀⠀⠀⠀⢰⠠⠃⠀⠀⠀⠀⠀⠠⠦⢴⠀⠀⠀⠀⠀⠀⡄⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠲⠀⠀⠀⠀⠀⡀⠒⢂⠀⠀⠀⠀⠀⠀⠐⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⣢⠀⢀⡌⠀⢀⡤⠀⠀⠀⠀⠀⠀⢦⣀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠂⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠐⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠖⠒⠚⠛⠛⠛⠛⠛⠓⠲⠶⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠀⠈⠉⠉⠉⡉⣉⣉⣉⣉⡉⠉⠉⠙⠁⠀⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾⠳⣶⠶⠞⢛⣉⣩⢭⣭⣍⣉⡛⠲⠦⣴⠞⢻⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠿⢋⣡⡖⠋⠁⠀⠀⡄⣡⡀⠀⠈⠙⠲⣌⡙⠾⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⠔⠉⠀⠱⣄⠀⠀⢘⡇⢁⠟⠀⠀⣠⠈⠀⠙⢢⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣰⠃⠀⠀⠀⠀⠈⠀⠀⠚⠓⠛⠒⠀⠀⠉⠀⠀⡀⠀⠙⢆⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡼⠁⠁⠶⠄⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢀⡴⠈⠈⢧⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠃⠀⡀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⢀⢴⡾⠋⠀⠀⠀⠀⠀⠀⠘⡆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⢺⣽⡇⠀⠀⠀⠀⠈⠻⣟⣶⣊⡴⠯⠀⠀⠀⠀⠀⢫⣇⠀⠀⡇⠸⣿⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠐⠖⠁⠀⠀⠀⠀⠀⠉⠉⠛⠻⠂⠀⠀⠀⠀⠀⠺⣤⡼⠃⠀⡇⢠⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⡄⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣈⠂⠘⠲⠀⢠⠎⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⢆⠀⠀⠀⢀⠶⠀⠀⠀⣮⣍⠀⠀⠰⡄⠀⠀⠀⢀⡰⠋⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⠳⣤⡀⠄⠀⠀⠀⠀⠛⠞⠁⠀⠀⠁⠄⣀⡴⠋⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢳⢤⡙⠲⢦⣄⣀⣀⣘⣃⣀⣀⡤⠴⠚⣁⡤⢾⡏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⡤⠾⠗⠲⠤⣬⣍⣉⣍⣥⣤⠴⠖⠛⠓⠢⣼⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡄⠀⢸⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠤⢤⣤⣄⣤⣤⣤⣤⠤⠴⠶⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠁⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⢚⡿⠀⠀⠀⢀⣶⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⢳⡄⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠺⡏⠙⠹⠓⠗⠛⠉⡷⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠇⠀⠀⢀⣠⡴
⠀⣀⡀⠀⠀⠀⠀⠈⢳⡝⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡄⠀⣧⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣾⠤⠶⣾⢕⡠⠶⠚
⢀⣬⡿⣷⣶⡖⠶⠒⠉⠀⠀⠙⡆⠀⠀⠀⠀⢀⡾⠀⡾⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠠⡿⠟⡒⠒
⠀⢬⣭⣻⣯⠶⣠⡤⠦⢤⡀⠀⣷⠀⠀⠀⠀⣼⠃⢸⠇⠀⠀⠈⡷⠶⠶⢶⡞⠀⠀⠀⢹⠀⠀⢻⡄⠀⠀⠀⠀⠀⣸⠃⠀⡴⠛⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠾⠖⠋⠁⠀⠀⠀⣧⠀⢸⡄⠀⠀⢰⡏⠀⢾⠀⠀⠀⠀⢷⠀⠀⣸⠁⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⢠⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⣿⢄⡀⠘⣆⠀⠀⠀⢸⡄⣰⠃⠀⠀⠀⢀⣿⠀⠀⠀⠸⡆⠀⠀⠀⡞⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠀⢰⠇⠀⠉⣱⠘⢧⣤⠤⢬⣧⣏⣀⣀⡀⢀⡾⠙⣧⣤⠴⠞⣧⠀⠀⣸⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⡏⠀⠀⣰⠏⠀⣎⣵⣤⡴⠷⠾⢤⣀⡿⡾⠁⠀⠘⣇⠀⠀⠸⡆⢠⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡿⠀⠀⢠⡏⠀⠀⠀⢈⣆⣀⣀⣀⠀⠈⣹⠇⠀⠀⠀⠘⣆⠀⠀⠙⠞⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠀⠀⣠⡴⠋⠁⠀⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡞⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢦⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡾⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣤⣴⣦⣴⣆⣀
⠀⠀⠀⠀⠀⢀⡟⣵⣾⣿⣿⣿⣿⣆⠀⠀⣶⣿⣿⣿⣿⣿⣿⣷⠂⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣀⠀⣀⣼⣿⡿⢿⠿⠟⠻⠟⢻⠀⢰⣿⢋⠉⠛⠻⠛⢿⠋⠀⠀⣻⠃⠄⠉⠛⡛⢻⣿⠂
⠀⠘⠉⠋⣿⣿⡟⡟⠀⠀⠀⠀⠀⢘⠁⢸⡇⢸⡄⠀⣰⡆⢸⠆⠀⠈⡃⢾⠃⠀⢴⡆⢸⠏
⠀⠀⠀⣀⣙⢿⣿⣇⠀⠠⣤⡤⠀⠀⠀⠀⠃⠈⢁⣀⡈⠁⠸⠛⠄
⠀⠀⣼⣿⣿⣦⠈⠙⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⢀⣤⣀⣠⣄⢀⣄⡀
⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⣸⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠛⠛⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠤⠤⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣎⡠⠂⠀⠀⠐⢤⠐
⠀⠀⠀⠀⠀⠀⠘⠐⠊⠀⠀⠀⠉⠒⠁⠀⠐⠋


⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠉⠻⠿⠿⢿⣿⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠻⣿⣿⣶⣤⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠘⠛⠻⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣤⣀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⠀⠀⠀⢀⣠⣤⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣻⣯⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣇⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠟⠿⣿⠟⢻⠿⢿⢿⠻⣿⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣷⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠻⡿⠰⢀⠐⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⣤⣤⣀⣾⠋⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢷⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⣀⣀⣀⣀⣀⣠⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣰⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣎⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠉⠉⠛⠻⢿⣿⣿⡿⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣉⣻⣿⣷⣬⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⠛⠛⢿⣿⡁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⡀⠈⠻⣿⣄⠀⣘⣿⣿⣦
⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⣤⣿⣿⣿⣿⣿⡿⣿⣷⡀
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⡿⠿⣿⣏⠁⠀⠹⣿⣷⡀
⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⠿⣿⡇⠀⠀⢻⣿⣆⣀⣀⠹⣿⣷
⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⡿⠟⠛⠉⠀⢀⢈⣿⡇⣠⣴⣿⢿⣿⡟⠿⣿⣿⣿
⠀⢸⡏⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⡜⢸⣿⣿⡿⠋⠀⠀⣿⣿⡄⠀⠹⣿
⠀⣾⡇⣀⣤⣶⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣾⣿⡟⠁⠀⠀⠀⢿⣿⣷⠀⠀⢻
⠀⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⣾⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⣸
⠀⢸⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿
⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣿⡿
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢀⣴⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⢀⠔⢀⡠⢤⠖⡔⢢⡄⡀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣀⢠⡖⢢⠶⡤⢄⡀⠢⡀
⠀⠀⠀⡀⢁⣴⠫⡜⠣⢛⠌⡡⢎⡰⣉⠦⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠴⣉⢆⡱⢌⠠⣛⠜⢣⠝⣦⡈⢀
⠀⠀⠀⠀⣜⠶⣙⢎⡱⣉⠶⡑⢪⡑⣥⠓⣜⢢⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡔⣣⠚⣌⢪⡵⢊⠶⣉⢎⡱⣉⠶⣣
⠀⠀⠀⠀⣏⠒⣥⢢⡜⢢⢛⡬⡣⢜⠴⣉⠎⡱⢧⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡼⢎⠳⣙⠞⡣⢜⠤⡛⡔⢣⡔⣬⠲⣹
⠀⠀⠐⠒⢧⢙⠤⣑⢎⡱⢉⠶⡑⢎⡑⢩⠺⡔⢣⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡜⢢⠖⡭⢊⡱⢊⠶⡡⢎⡱⣊⠤⡃⡼⢐⠂
⠀⠀⠀⠐⡀⢊⠶⣍⢊⡕⢫⠛⢤⢣⢜⡢⣛⠼⡱⢞⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⡳⢌⠣⣛⢄⡣⡜⣤⠛⡝⢪⡕⣉⠶⡑⠀⠂
⠀⠀⠀⠀⠈⠢⡘⠤⡓⢜⡱⢍⠶⣑⢎⡱⢭⠒⡕⠪⠀⠞⠀⠀⢀⣠⡶⠛⠁⠀⠓⠀⠕⢪⠖⡩⢎⡱⣊⠶⡹⢎⡡⢚⡤⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⡚⠱⣮⠒⣥⢣⡜⣣⡛⡤⠃⣰⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⢆⠘⢤⢛⣔⢣⡜⣬⠒⣵⠎⠓⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠌⠉⢈⣁⡡⠐⠊⢰⠀⠀⢿⡀⠀⠀⢀⣠⡶⠶⡇⠑⠂⢌⣈⡁⠉⠡⠅⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠈⠻⠶⠾⠛⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⠤⠤⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠄⠀⠀⠀⠈⠠⡀⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢷⣄⠜⠁⠀⠀⠀⠀⠀⠀⠈⠣⣠⡾⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡺⣦⡹⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢏⣴⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠉⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⠀⠀⠀⢀⣀⣀⠀⠀⠀⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣰⡿⢾⣿⣿⣿⣆⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣄⣠⣬⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠭⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡋⣩⣤⡶⠶⠲⠾⠒⠲⠶⠴⣶⣥⣂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣺⣿⠿⡋⠉⠀⠀⠀⠀⠙⠃⠃⠀⠀⠀⠈⢙⠷⣄⡙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢠⡿⠋⠙⠴⠚⢀⠈⠀⠀⢲⡀⡇⡆⠀⠀⠀⢀⠋⠀⠀⠙⢄⢜⣷⡀
⠀⠀⠀⠀⠀⠀⢨⣾⠞⠀⠀⠀⠡⢮⡈⢦⠀⠀⣨⢧⠇⡇⠀⠀⢠⠎⠀⠀⠀⠀⠀⢑⣻⣷⡄
⠀⠀⠀⠀⠀⠀⣸⠃⡐⠀⢀⠀⠀⠀⣩⠒⠁⠀⠀⠀⠀⠠⠼⡀⠋⠀⠀⠀⢀⣌⠘⣭⡤⢹⣽⡄
⠀⠀⠀⠀⡀⣿⠏⠉⠃⠠⠶⠦⢤⠄⠀⠀⠀⠀⠀⠀⠤⣬⡁⠀⠀⠀⠰⣪⠕⠊⠀⢀⠀⠀⢻⣷
⠀⠀⠀⠀⠱⣾⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⢰⠐⣿⠂⡂⠀⡥⠶⠞⠃⠀⠀⠀⠀⠀⠀⢴⣧⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡂⠰⠐⠆⣿⡇
⠀⠀⠀⠈⢀⣿⠈⠁⠀⠓⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠙⡆⠀⠀⠀⠋⠉⠉⠁⠈⠉⢁⣧⡇
⠀⠀⠀⠀⢩⣾⡆⠀⠀⢠⣤⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢱⣶⢄⡀⠀⠀⠀⠀⢸⣽
⠀⠀⠀⠀⠐⡛⢿⣤⠶⡀⠙⢋⡴⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣷⡞⢁⠄⠀⢠⢯⠏
⠀⠀⠀⠀⠀⠘⡴⡷⡜⠃⠀⠈⠀⢀⢜⢝⡕⠀⠀⡄⡄⠀⠀⢦⡀⠀⠀⠀⠉⠀⠲⠊⡠⢮⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⡀⠀⣀⠈⠛⠠⠇⠀⠀⡇⢻⠀⠀⠀⠙⠂⠀⠀⠀⠀⢀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⠅⣐⠂⠀⠀⠀⠈⠁⠈⠀⠀⠀⢠⠺⡀⠀⣀⢔⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣝⡐⢤⣠⣄⣀⠈⠈⠧⠀⢀⢠⣀⡡⣖⣾⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠛⠶⠬⠭⠷⠭⠵⠖⠒⠛⠛⠛

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡾⠛⢁⣠⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⣿⣿⠟⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⠀⢀⡀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠁⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⣴⣶⣤⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠋⡉⠁⠸⢿⣿⣿⣿⣿⣷⣦⣄⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⣠⣾⣿⡏⢠⣶⣄⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⣼⣿⣿⣧⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡏⢠⣿⣿⣿⣿⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠠⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠂⠀⠑⢀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⢺⣿⣿⣷⣶⣶⣶⣶⡆⠀⠀⠀⠐
⠀⠀⢀⠀⠈⠟⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢄
⠀⠀⠀⠢⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠈⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠉⠍⠁⠐⠘⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠂⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠊⠉⠀⠉⠑⢦⠰⡁⠀⠀⠀⠀⠠⠛⡄⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⢀⠀⠀⠀⠀⢱⢑⠮⠀⠀⠀⡀⢠⠓⡄⠀⠀⠀⠀⠀⠀⢘⣿⣿⡖⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠉⠁⠐⠒⠊⠇⠸⡀⠈⣢⡔⠁⠁⡠⠃⠀⠀⠀⠀⠀⢀⠎⢨⠛⣇⣀⣀⣀⡙⣷
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡸⠀⠒⠀⠀⢛⠀⠀⠀⢹⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⢰⠊⠂⣠⣀⠸⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣲⣧⣀⠀⠒⣠⠎⠀⠀⠀⠊⣿⣿⡂⠀⠀⣾⣶⠀⠀⠀⢻⠀⠀⣀⠭⡇⠻⣿⣿⡿⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠿⣿⡿⠟⠿⠿⢷⠀⠀⣠⠃⣿⡏⠀⠀⢀⣿⡇⠀⠀⠀⠈⠢⣄⠀⠀⣇⣀⣙⣯⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠎⠈⢣⠀⠀⠈⢸⠀⠈⡍⠚⠀⠀⡿⠀⠀⢀⣾⣿⣿⡀⠀⠀⠀⠀⣨⣶⡿⠿
⠀⠀⠀⠀⠀⠀⡌⠀⠀⡘⠀⠀⠀⢸⠀⠀⢡⠀⢀⠌⠀⠀⠀⢸⠉⠛⠋⡅⠀⠀⢀⠼⡏⠁⠀⠘⡀
⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⠀⠀⠀⢸⠀⠀⠸⡋⠁⠀⠀⠀⠀⠘⣆⣀⣠⠃⢤⡔⠁⠀⠈⡆⠀⠀⢃
⠀⠀⠀⠀⢠⠁⠀⢀⠎⠀⠀⠀⠀⢸⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠙⠛⡇⠀⠈⠀⠀⠀⢠⠃⠀⠀⠸⡀
⠀⠀⠀⢀⠜⠀⢤⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⣀⠤⠊⠀⠀⠀⠀⢣
⠀⠀⠀⠺⢶⡤⠢⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣦⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣀⣀⠠⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣷⣶⣶⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⡏⠸⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠇⢰⣔⣚⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠎⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣼⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠛⠿⣿⣯⣿⣿⠀⠀⠀⠣⠀⠀⢉⠉⠉⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠠⡄⠤⠤⡄⣠⠖
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡔⢡⠲⢠⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠃⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠄⠘⠘⠈⣂⠀⠀⠀⠀⠀⠀⠠⠔⠺⠋⠀⠀⠀⠀⢸⡇⠀⠈⠀⡖⠀⡆⠸⠓⢠⠀⠀⠀⠀⢀⢀⡤⣄⠀⢠
⣮⠁⠃⠀⠈⠁⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢢⣾⠃⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠓⠋⠀⠈⠀⠉⢹
⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣀⠀⠀⠀⠌⠁⠀⠀⠀⠐⢰⡿⠁⠀⠀⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢒⠀⠀⠀⠀⠀⢸
⠇⠀⠴⠶⠀⢀⣤⣦⡀⢐⠂⡌⢦⠀⠀⢠⣄⡀⠀⠘⠤⠄⠔⠀⠀⠀⠀⠀⠀⢀⣴⣆⣀⣀⣀⡀⠀⣠⠀⠢⠔⠀⡀⠀⠀⠀⢿
⠀⠀⠀⡶⣠⠈⠛⠉⣁⠉⠐⠀⠈⣧⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⠛⠛⠛⠁⠀⠉⠀⠀⠀⠂
⠀⠀⠈⠙⠃⠀⡈⠤⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢀⡇⠀⠀⢀⣀⠄⠀⠀⠀⠀⠃⠀⠱⠀⠀⡄⠀⠻⢿⠿
⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⢸⡇⠀⠀⢍⡉⡄⠀⠀⠠⠀⠄⠀⠀⣨⠔⠋
⣀⡀⠀⠀⠀⠀⠊⠀⠀⠐⡄⠀⠀⠀⢩⡿⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇⠀⠀⠈⠁⢰⠀⠀⠀⠓⠊⠡⢈⣸⠄⠀⠀⠀⠀⢀
⠠⢴⠆⡱⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠋⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠊⠓⠄⠀⠀⠀⠀⠁⠀⠀⠀⡴⠾⠉⢁
⠀⢸⠖⠀⠀⠀⢀⣠⣤⣤⣀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠂⠰⠆⣠⠂⠀⠀⠀⠀⠠⡀⠫⠋⠉⠁
⡠⠏⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠈⠛⠛⠛⠓⠒⠚⠋⠀⠀⠀⠀⠀⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⠣⢄⢀
⠀⠀⠀⠀⠀⠈⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠶⠶⢶⣶⣤⣤⠀⠀⠀⠀⠈
⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣿⣾⡿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⣴⠇⠀⠀⠀⠀⠀⡴⡒⠄⠀⣿⡇⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢿⣿⣟⣿⣿⣗⣿⣽⣿⠧⠀⠀⠀⠀⠀⠀⠣⠥⠂⢰⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡀⢠⠃⠀⠀⠀⢠⡲⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠇⢠⣀⢀⠀⣼⡿⠀⠀⠀⠀⢀
⠀⠆⠻⠀⠀⠒⠐⠀⢸⠀⠀⠀⠀⠨⠤⠤⠴⠀⠖⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠝⠞⢀⣿⡇⠀⠀⠀⠀⠀⡀
⣠⠔⠒⠒⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠠⠤⠄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣅⣠⣄⢄⣀⣈⣷⠀⣾⡷⠀⠀⠀⠀⠀⠰⠂⢲⠄⠀⠀⠀⠀⠀⠰
⠀⠂⠀⠀⠀⠐⠃⣠⣶⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⢸⡟⠛⢯⣛⣼⠛⢻⡿⠀⠿⠉⠁⠂⠀⠀⠀⠀⠀⢸
⢁⠀⣀⠀⠀⠀⢰⣿⣿⣽⣿⣿⣿⣿⣿⣿⣧⡀⣀⠀⣘⠻⣦⣼⣿⣠⣴⠟⣁⣀⣀⡀⢀⡴⠀⠀⠀⡀⣀⠛⠀⠀⢀⡀⠀⠀⢀
⣿⣿⢛⢻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡄⣿⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⠟⠋⠁⡱⠒⡑⠊⣁⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡠⠚⣡⣾⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠰⢯⣟⡣⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⣠⡅⣰⠟⠋⠀⢀⣤⠀⠈⣹⠶⡀⠀⠀⠀⠀⢈⡎⠱⡄⠉⡿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢡⡴⠀⠀⠋⢀⠀⠀⢄⢰⠜⢆⠀⠀⠀⢀⣷⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡘⠇⠀⠀⠀⠙⠁⠀⣤⣉⠆⠈⣇⠀⢀⣾⣷⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⡄⡰⢊⡁⠀⣀⡠⠤⠀⣿⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠁⠁⠸⣶⣶⣶⣾⠏⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡐⢄⠈⠛⠛⢉⢄⣾⣿⣿⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⣥⡒⢂⣵⣿⣿⣿⣿⣿⢵⣾⢫⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⡇⠈⠑⠼⡏⠙⠉⠙⢟⠛⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠁⢃⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠿⠿⣿⣿⣿⣶⠶⢤⣤⣀⠀⠀⠀⠒⠒⠰⠀⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⠇⠀⠈⠸⣩⢻⢦⡀⠀⣠⡼⢤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣼⡀⢀⢃⢇⠀⠄⢋⢞⠃⢀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⢩⠇⡸⠌⠈⡄⣴⠟⡌⣄⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⠈⠊⡤⢇⣄⣠⠟⢁⣤⡾⡿⡱⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⢰⠀⢡⣠⣟⡥⢶⣏⠁⠁⠁⠃⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⡆⠆⢾⣠⡾⠛⠁⠀⠀⠙⣶⡰⡆⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⡞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⣥⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣖⣒⠒⡒⠒⠠⠄⣀⠀⠀⢸⣤⡿⢽⣏⣾⢹⢛⣦⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠉⢁⠀⠀⠀⠉⠉⠁⠒⠠⠉⠒⢤⠛⣳⣼⣈⣽⢿⣷⡟⠁
⠀⠀⠀⠀⠀⠀⢀⠞⣽⠟⠁⠀⠰⢤⡐⢦⣀⠀⠀⠃⠀⠀⠠⠄⠀⠀⠀⣵⣿⡿⠿⠷⠿⣾⣶⠤⠊
⠀⠀⠀⠀⠀⣰⠃⣰⡇⠀⠀⠐⠒⠀⠀⠀⠹⣿⣦⣤⠶⠻⠋⠠⠀⠀⠀⢹⡿⣿⣦
⠀⠀⠀⠀⠀⡏⠀⣇⠃⠀⢠⠀⠀⠀⠀⠠⠒⠩⠯⠛⠁⠀⠀⠀⠠⠄⠀⢸⢧⡿⢿⡀
⠀⠀⠀⠀⠀⡔⡀⠌⠦⣄⠀⠀⠀⠰⠂⠀⢀⠀⠀⡀⡀⠀⠐⠦⠀⠀⢔⣣⠟⠇⢱⡇
⠀⠀⠀⠀⠀⠳⡐⢄⠀⠈⠛⠶⣤⣁⡀⠂⠹⠤⠠⠵⠔⠂⣐⣨⡤⠖⢋⠡⣢⢊⣾
⠀⠀⠀⠀⠀⠈⢎⡒⢕⡢⣀⠀⠀⠈⠉⠙⠛⢛⡛⠛⠛⣉⡉⣀⠀⣊⣤⣺⠕⣹⠏
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠛⠿⢷⣖⣲⣦⣤⣬⣤⣭⣤⣴⣶⣶⠿⠟⢋⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⣈⣥⠉⠉⠉⠉⢩⢭⢶⣒⣯⣷⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢛⠖⠯⡿⠶⣤⡙⣦⠀⠀⠀⠀⣰⡿⢫⣉⡉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⡉⠚⠛⡒⠶⣝⣾⠂⠀⠀⢸⡿⣉⡟⡾⣽⡃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⣾⠀⠁⠀⠀⠁⠀⠩⣻⠀⠀⠀⣿⡇⡾⣽⣟⡟⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⡠⠖⠒⠒⠛⣄⡀⠐⠃⣀⡴⠋⠁⠀⠀⠀⢸⡱⣌⣛⣚⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠓⠊⠀⠀⣠⣴⠁⠀⠙⢿⣻⡅⠀⠀⠀⠀⠀⠀⠀⣿⣋⣿⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⡞⠙⢄⠀⠀⢠⠾⣧⠀⠀⠀⠀⠀⢀⣀⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠞⠀⠀⠀⠙⠒⢧⡀⠈⠓⠒⠒⠒⠚⠁⠈⠁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⣇⠀⠀⠀⠀⠀⠀⠀⡹⠢⢤⣀⡤⠴⠒⢒⡶⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣀⠀⠈⠑⠦⢤⣀⣀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠁⢠⠆⠀⠀⠀⠀⠀⠁⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠙⠒⣧⣀⣀⡟⠀⠀⢧⠀⠘⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⡛⠒⠢⣤⣠⡇⠀⠈⠉⡷⢲⡚⠛⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡯⣿⡿⠢⣴⠇⠙⠀⠀⠀⡼⠁⠀⠳⣄⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⡹⠁⠀⠈⠑⢦⣠⠴⠚⠀⠀⠀⠀⠈⠳⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠞⠉⢳⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⢙⣶⣿⠶⠖⡻⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣀⣀⢴⡯⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠉⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⠘⠓⠒⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⡄⢶⣦⣄⡀
⠀⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣷⣦⡀
⠀⠀⢀⣴⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣦
⠀⢠⣾⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷
⢀⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⢰⡀
⣸⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⢡⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⣏⢿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⢻⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⡟⣡⣿⣿⣿⡟⠈⢳⡀⠀⠀⠀⠀⠀⡾⠁⢹⡆⠀⠈⢙⣿⣿⣿⣿⣷⣤⡀
⠘⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⢸⣿⣿⣿⠏⣴⣿⣿⣿⡿⠁⠀⠈⢷⡀⠀⠀⠀⣼⠁⠀⠀⣷⠀⠀⣾⠁⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠘⣿⣿⣿⣷⣦⣍⡛⠿⢿⣿⡇⣿⣿⣿⡇⣼⣿⣿⠏⣼⣿⡿⠟⣫⡄⠀⠀⠀⠈⣷⠀⠀⢰⠇⠀⠀⠀⢹⡄⢰⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣶⢠⣬⣭⣍⢁⣍⣭⡉⣨⣭⣴⣶⡟⢹⠃⠀⠀⠀⠀⠸⡇⢀⡟⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⣸⣿⡻⣿⡿⠿⠿⠿⢟⣡⣿⣿⣿⠇⣾⣿⣿⡇⣿⣿⠟⢯⡀⢸⠀⠀⠀⠀⠀⠀⢻⣼⠃⠀⣀⣤⣶⣿⣿⡿⣿⠋
⠀⠀⠸⣿⣿⣌⠻⢿⡛⠿⣿⣿⣿⣿⠏⣼⣿⣿⠿⠟⠈⠁⠀⠈⢳⣼⠀⠀⠀⢀⣀⣤⣼⣿⣾⣿⢿⠟⣿⡇⣸⣷⠏
⠀⠀⠀⠈⠛⢿⣷⣶⣿⣦⣽⣆⣀⣀⣀⣀⣙⣿⣆⣀⣠⣤⣤⣤⣴⣿⣶⣾⠿⢿⡿⣿⡉⣽⠋⣸⢿⣤⡏⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢿⡟⣿⠿⣿⣻⠿⣟⢻⡿⣟⠛⠛⣻⠿⣍⠉⣩⠏⠙⢦⣠⠟⠀⢀⡿⢧⣴⠏⢠⡟⢧⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⣈⡿⢦⣈⡿⣦⡈⠻⣶⡃⠀⠈⣿⢯⡀⠀⣰⠛⠷⣤⡞⠁⣰⠛⢳⣾⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣧⠈⣿⢧⡀⠙⣾⣇⠙⢳⣾⠃⠀⢙⣾⣇⠀⣠⠟⠳⣶⡃⠀⡾⠉⠳⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣻⣇⠀⠙⣾⡃⠙⢳⣞⠉⠛⢦⣞⠁⠈⣻⢯⣀⣼⠃⠙⣾⣅⠀⣸⠏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢧⡼⠃⠙⣶⡏⠈⠳⣦⡟⠙⢳⣼⡃⠀⡽⠷⣄⣸⠃⠈⢳⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠛⣆⣴⠋⠙⢦⣴⠋⠙⢦⣾⠉⢛⣾⡁⠀⣸⠿⢦⡀⣼⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡼⠳⣄⢀⡼⠛⢦⣠⡟⠉⢳⣾⠁⠙⢳⣏⡀⠀⢹⡿⣄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⢙⣿⡁⠀⣰⠟⢦⣀⡞⠉⠳⢦⡟⠈⠙⢦⣾⠀⠈⠳⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⠏⠀⠻⣾⠃⠀⢀⡟⢧⡀⢀⡞⠙⢦⡀⢰⠏⠳⣄⢠⠏⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⢇⠀⢀⡜⠉⠣⣀⠞⠀⠀⠙⣾⡀⠀⠀⠈⡿⣄⠀⠀⢹⠦⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢳⡞⠀⠀⢠⠟⢦⡀⠀⡼⠁⠙⢦⡀⣸⠁⠈⠑⢤⡟⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠏⠹⣄⢠⠏⠀⠀⠳⣼⠃⠀⠀⠀⠙⣯⠀⠀⠀⢈⣷⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠙⣏⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⡟⢦⡀⠀⣸⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⡾⠻⣦⠀⠀⣰⠇⠻⣆⠀⠀⣸⠇⠀⠙⠛⠁

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠃⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⢀⡔⣛⣳⣦⠀⠀⠀⠀⠀⠀⠀⠨⢏⡀⡄⠀⠀⠀⠀⢠⡞⣭⣭⢳
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⣾⢰⣿⠿⡾⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⡿⡄⠀⠀⠀⣼⡸⢿⣟⡾
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢠⡯⠷⠚⠋⠁⠀⠀⠀⠀⠀⠀⢸⣧⠀⢁⠀⢄⠀⠀⣠⠷⠛⠋⠉
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠦⣄⡁⠐⢂⣺⠟
⠀⠀⠀⢠⣿⣿⣿⣿⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠂⠀⠠
⠀⠀⠀⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⠀⠐
⠀⠀⠀⠘⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡁⠈⢻
⠀⠀⠀⠀⠘⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘
⠀⠀⠀⠀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⢀⠂
⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠌
⠀⠀⣤⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⢿⢀
⠀⠀⠙⠻⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣍⣓⣀

⡐⠂
⠘⢄⡀
⠀⠈⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⡄
⠀⠀⢘⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⣠⣶⣶⣶⣦⠀⠀⠀⠀⠼⣿⣿⣿⡃
⠀⠀⠀⡀⠀⠆⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⠀⠘⣿⣿⣿⠏⠁⠀⠈⠿⣿⣿⣿⠄⠀⠀⠀⢲⣬⣍⡉⣀
⠀⠀⠀⠘⢄⠄⠀⠀⠨⣿⣿⣿⣿⠇⠀⠀⠀⢀⡀⢈⡉⠁⠐⠀⠀⠀⠀⠈⠛⠟⠀⠀⠘⣦⣠⣿⣿⣟⡻⣷⣄
⠀⠀⠀⠀⠀⠣⡁⠀⠣⠉⠛⠉⠉⠀⠀⠀⠀⡀⡘⠘⠂⠁⠛⠛⡛⠛⠲⠦⣴⣤⣤⣤⣤⢼⣿⣿⣿⡿⠼⣿⢿⡄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠳⠶⠶⠶⠶⠚⢷⣴⠁⠁⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠈⡠⠋⠀⠀⠙⣿⣿⣿⠿⣻⡏
⠀⠀⠀⠀⠀⠀⢸⡍⠀⠀⠀⠀⠀⠀⡰⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠳⣦⣄⠀⠀⠀⠀⢘⠀⠸⣿⣿⣴⣿⣈
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢀⣷⠀⠀⠄⠀⠀⠀⡀⠀⢠⠊⠁⢰⣄⠉⠓⠂⠣⠀⢸⡀⠄⠈⠻⣿⡁⡇
⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⢊⣽⡿⣠⠀⠈⠒⠛⢛⣡⡤⠎⠀⢠⡞⡡⠆⠀⠀⠀⡴⢼⣷⣦⣄⣂⣘⠐⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠩⢼⠀⠘⠷⠶⠾⠛⠉⣁⣨⣧⠜⢋⣈⠒⠂⠀⠀⠤⠄⢸⣿⣿⢿⣿⡿⣿⣷⠄⠋
⠀⠀⠀⠀⠀⠸⠔⣠⣤⣴⣶⣦⣤⣈⠆⢀⠀⢀⡤⠚⡛⣋⣁⣀⢃⡉⣩⣋⣀⣀⣀⣀⣀⣸⣏⡍⠛⡉⢽⣿⡏⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠨⣶⠿⠿⠛⠙⠛⢻⢿⠀⡙⣻⣿⣧⣿⣿⡛⢟⠛⣧⡶⢚⣨⣿⡏⠁
⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⠏⠀⠸⣿⠆⢀⠈⠈⠀⠀⠒⠀⣡⠜⠀⣲⣽⣿⣭⣍⢩⣉⣾⡁⣿⣵⢿⠟⢁⡀⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣄⢰⡇⠀⢘⠀⠀⢀⢸⣧⠈⢡⡆⠀⣍⣿⣹⣿⠛⣿⣿⣽⠧⣿⣽⡿⣟⠋⢸
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⢻⡏⠁⡃⠀⠈⠂⠀⠈⣾⣿⡆⠁⠀⠀⠿⣵⣷⣥⡜⢿⣣⡅⢰⣿⡟⠡⢶⣦⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠘⣷⡀⣒⠀⠀⢿⡞⠳⠀⡭⣧⣠⠀⠀⠉⢨⠼⢷⠃⠚⠫⣤⣬⠛⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠀⢾⣿⠨⡄⠀⠄⠂⠐⠀⠰⠈⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠸⣿⢋⡇⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⢸⣿⠈⠄⠀⠘⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃⡀⠀⠰⡏⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⣀⡁
⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠹⠂⠀⠀⠠⣈⡷⢰⣶⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⣠⣾⣯⣿⣽⡀
⠀⠀⠀⠀⠀⠀⠸⣷⡄⢲⠀⠀⢰⣶⡎⠛⢧⣾⡇⣸⣷⣣⠤⠀⣶⣶⣦⣀⡄⠀⠠⣤⣿⣶⣟⣻⣿⣿⣹⡧
⠀⠀⠀⠀⠀⠀⠰⠿⠿⠉⠀⠀⠘⢿⣷⣼⡾⠉⠀⠯⠴⠶⠆⠀⠙⠛⢛⡿⠃⠀⠀⠘⠉⠀⠈⠁⠉⠑⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣷⣶⣿⣛⣟⡻⣿⣷⣾⣭⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⣿⣿⣿⢿⣿⣿⣧⡘⢿⣿⡏⠹⣷⣾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠻⠿⠿⣿⣯⣴⣿⣿⣿⣿⣿⣶⣮⣍⣀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⡿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⠿⠿⠧⠀⠀⠀⠀⠤⠬⠭⡙⢿⣿⣇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⡿⢁⣤⣤⡀⢶⡀⠀⠀⢀⣤⣀⡀⠀⢿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢁⣹⣿⣿⡇⠘⠋⠀⠀⠀⠁⠀⠀⠉⠀⠈⠉⠀⢸⣿⣿⣄⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⢿⠀⠀⠀⠀⠀⢾⣇⠰⠄⢀⠀⠀⠀⠀⣾⣿⣿⣿⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⡇⠀⠀⢀⣤⠤⠀⠶⠦⣌⢳⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠈⠻⢿⠿⣿⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⡾⠋⠀⠀⣸⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣴⣞⠀⠀⣠⣿⣿⣿⣆⠀⠀⠈⣉⣉⡋⠍⠀⠀⠀⢠⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⠾⠟⠈⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣤⣴⣤⣤⣤⣿⣿⣿⡟⠻⣿⠋⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⠃⠃⠀⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢈⢷⣹⣿⣿⡏⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠙⢿⣿⣿⣿⠟⠛⠁⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣦⡸⡞⣇⢻⣿⣦⡀⠑⠺⢻⣆⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⢸⣿⢿⠟⠀⠀⣀⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣹⡜⣿⣿⣿⣷⣦⣄⡙⠳⡄⠀⠀⢀⣼⠟⠁⢀⣠⣶⣿⡿⠀⠀⠀⣼⣿⣵⠏⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⢻⡏⢻⣿⣿⢿⣿⣿⣷⣌⠢⣠⣞⣡⣶⣾⣿⠛⡉⡁⠀⠀⣇⣼⣿⣿⡏⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠃⠈⠋⠛⠁⠂⢻⠛⣿⣷⣽⣏⠋⠉⠁⠀⠈⠀⡐⠀⣆⣿⣿⣿⡾⠀⣤⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠀⠀⠐⠀⠈⠃⠈⠃⠘⠛⠀⠀⠀⠀⠀⠘⠁⠀⠛⠛⠛⠛⠁⠘⠙⠋⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡲⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢎⣴⣾⠟⠿⢦⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠙⠋⠀⠢⠀⣈⠚⠁⠈⡙⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠒⡂⠖⠲⠁⠀⢿⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠠⠀⠠⢀⠀⠂⠀⣼⠃⠹⠁⢲⡏⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡰⠒⠒⠀⠄⡃⠀⢢⠐⣅⠀⣧⡸⣅⡀⣠⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠍⢀⠄⠀⠀⡴⢹⠋⠀⢈⠉⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡕⠀⠀⠀⠱⠂⠁⠈⠀⠄⠤⢠⠋⠶⡌⠀⢀⢴⠆⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡈⡀⠰⠀⠀⠀⠨⠀⡐⠆⠰⢀⠀⡺⠀⠀⠀⠄⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⠀⠀⢀⠀⠀⠀⠀⠀⣷⣤⣽⡀⠀⠀⢀⠀⠀⢨⣤⡤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⡃⢦⡸⠄⠀⣂⣀⠀⢉⠈⡻⡇⠀⠀⠀⠀⠀⠐⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢁⠀⠀⠁⢧⠀⠀⠀⠀⢸⡴⣿⢵⡖⣒⡲⠒⢪⣇⠼⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠂⠀⠔⣆⠀⠀⢀⠈⠳⠀⠀⠀⠈⠀⠀⠀⢈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⣂⠀⠀⠈⣄⠀⠀⠀⠀⠐⡆⠀⠀⣄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⣍⠂⣈⢆⠈⢀⠀⠀⢂⠀⢀⡉⠆⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠠⠁⠑⠀⠐⢤⡎⠀⠸⡇⡘⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⢱⠇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⠆⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡥⠀⠹⠇⠨⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡤⠴⠆⠰⠤⢤⡤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⣢⣤⣤⣅⡼⠂⠿⣿⣿⣿⣷⣿⣯⣙⠫⣉⡟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⡿⣿⣿⠛⣿⣻⣀⡀⣸⡷⣼⡏⠻⢿⣿⣷⣌⢻⡿⣗⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣻⣽⣿⣿⣿⣿⣷⣌⣻⣷⣽⡿⢯⣼⣿⣿⣟⢿⣿⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⠟⠀⣀⣤⣺⣿⣿⣿⣿⢿⣿⣿⣿⢿⣝⠿⢯⣹⣷⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣉⣹⣟⣿⣿⣿⣿⣿⣷⣿⡯⠼⡿⠇⢿⠀⢹⢙⣿⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠚⠉⢉⠛⢿⣿⡿⢦⢞⣿⢿⣯⣿⢿⣟⡿⣾⠃⠈⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠰⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣤⣤⠴⣫⣬⣷⣿⣷⣼⣿⣯⣿⣿⣿⣷⣹⣷⣿⣄⠠⠞⣿⡿⠋⡜
⠀⠀⠀⠀⣈⣲⣿⣿⣟⠧⣷⣦⣤⣴⣄⣤⣴⣿⠿⢻⣟⣭⣿⡿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣟⣿⣍⣤⣠⣿⡿⠁⠀⠁
⠀⠀⢦⣈⢠⢼⣿⣿⣿⣗⠓⠿⣿⣿⣿⣤⣤⣶⣾⣿⣿⠟⣫⢴⣾⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⠟
⠀⠀⢠⡴⢋⠠⠛⣿⣿⣿⣷⣟⣿⣿⣿⣴⣾⡟⠭⠑⠒⠊⠁⠀⣿⣿⡏⣸⣿⣿⣿⣽⣿⣟⣿⣿⠿⠃
⠀⠀⠀⠀⠀⢠⠊⡼⠉⡝⡿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⡙⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠙⠊⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣽⣿⣿⣿⣿⠻⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣠⣼⡏⠉⠩⢹⡋⢉⣀⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠨⠋⠛⠛⠛⡒⠲⢄⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⡦⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⠀⡀⠀⠄⢹⠀⠈⠀⠀⠀⠀⡨⠰⣀⡄⢀⠀⠀⡇⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠈⢹⠀⠁⠀⣀⠀⠀⢀⣠⠾⠃⠁⢀⠿⡞⠀⠀⠇⠉⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠳⣄⣀⡀⠀⠀⠐⠀⢤⣀⡄⠀⠁⢰⠤⡀⠟⠀⠘⢿⠟⠘⠀⠀⠐⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢒⣩⡩⠭⠉⠉⣋⠴⠂⠀⢌⣛⣃⣞⡀⠐⠒⠉⠀⠀⠀⠠⠄⠀⡠⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠋⠀⢀⣀⠀⢼⠀⠀⢀⠅⠀⠀⠈⠑⠀⠀⣀⠀⠀⢳⣤⣄⠊⠀⢦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⢤⡄⠈⠋⠀⢂⠉⠀⠸⣇⠀⠁⠄⠀⣀⠀⣀⣀⣀⠀⠀⠉⢲⠉⠂⠠⠠⠀⠑⣬⣦⡀⡀⠀⢄⣠⡎
⠀⠀⠀⠀⠀⠀⠈⠀⠈⡁⢁⠀⠠⣤⣴⢀⠁⡄⠀⠀⠀⠈⠄⠄⢌⠀⠤⠴⣦⣦⠄⣈⡀⠠⣠⠀⠀⠀⠀⠈⠙⠁⣴⡿⠋
⠀⠀⠀⠀⠀⡀⢁⠀⠉⣥⢠⠑⠦⣤⠡⠎⡄⢰⠀⠀⠐⢄⣶⣾⡇⡇⠀⠀⠈⠈⠓⢀⡙⠰⠉⠂⠶⠞⣠⠜⠁⢪⠁⣀⡀
⠀⠀⠀⠀⠀⡄⠘⢄⡉⠀⠁⣀⠀⠀⢀⣀⣁⡤⠖⢶⠖⡎⠪⠍⠴⠃⠘⠓⠈⠛⠂⢈⠃⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⣸⠀⠀⠀⢀⠀⠀⠈⠗⠉⠀⠀⠀⠀⠀⢸⠀⠘⠦⣀⡀⠀⢄⢀⣀⣡⠀⢸⢄⠀⠀⣰⡄⠀⠀⠀⠀⢀⠆⢀⠀⠀⠳
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠉⠓⠒⠀⠠⠤⢤⡤⠤⠴⡆⠀⠀⡀⠀⠀⠐⠈⠻⡿⠟⠀⠊⠈⠀⣺⣛⡀⢢⠀⠀⠀⠎⠀⠘⡄
⣴⣴⣦⣶⣤⣴⣴⣦⣶⣴⣴⣦⣤⣤⣤⣤⣤⣤⣴⣷⣤⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣷⣷⣶⣴⣶⣶⣾⣴⣶⣴⣧⣴⣴
⣿⣿⣴⣭⣦⣤⣽⣵⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⢸⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡏⠀⠀⠀⠀⠀⡿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣀⠀⢰⠛⢦⠀⠀⣀⡤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠓⠶⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⢀⡴⠛⠁
⠀⠀⠀⠀⠀⡏⠉⠳⣾⠀⢸⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡴⠶⠦⣄⡀⠀⠀⠸⠶⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣀⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠈⣇
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡴⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡴⠋⠀⠀⢹
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⢰⣄⣀⡸⠁⠈⠒⠀⠚⠁⠈⠑⠦⠴⠈⠳⢄⣀⣀⣀⣠⡶⠋⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢘
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢀⡴⠒⠒⠦⡀⠀⠀⠀⠀⠘⡆⢀⡤⠚⠛⢤⡘⡇⠀⠀⠀⠀⠸⠧⠶⠶⠒⠛⠋⠉
⡏⠀⠀⠦⠼⠂⠀⠀⠀⠢⡄⠀⠐⣦⣤⠜⠀⠀⡎⠀⣠⣤⡀⠹⠀⠀⠀⠀⠀⢹⡏⠀⣶⣶⠀⢷⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢁⠀⠀⠀⠀⢧⠀⠙⠿⠃⢠⠇⠀⠀⠀⠀⠈⣧⡀⠙⠋⢀⡿⡇
⢻⡀⠀⢀⠀⠀⢠⠞⠉⠀⠈⠦⠤⠎⠀⠀⠀⠀⠈⠳⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⣧⠉⠒⠒⠉⠀⢹⠀⠀⠀⠀⠠⣶⠒⠒⠲⠶⠤⢤
⠈⣧⠀⠀⠁⠀⡇⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣸
⠀⠈⢧⡀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⢰⠇
⠀⠀⠀⠙⢦⣀⠀⠙⠓⢲⡆⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏
⠀⠀⠀⠀⠀⠉⠛⢦⡄⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⣯⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣄⣀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢀⣤⠤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢑⡞⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣴⠋⠀⢰⠋⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡾⢄⠀⠀⠀⢠⠞⠁
⠀⠀⢰⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠜⠀⠀⢠⠏
⠀⠀⠀⢷⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠘⡇⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢠⡞⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠑⢄⡀⠀⢀⡴⣏⠀⠀⠈⠙⠳⢤⣄⣀⣠⡴⠟
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣢⠞⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣾⣁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⢀⣠⠞⠁⠀⠈⠙⠶⣄⣸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠉⠙⠫⢉⠉⠁⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠩⡀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⢀⠄⣀⠁⢠⡠⠦⠄⣀⣀⠀⡀⠀⠀⠀⠀⠀⠠⢊⠰⠄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠙⠂
⠀⠀⢀⡀⢣⡀⠳⡀⣡⠄⠛⣀⠰⡜⠂⣕⠀⠀⠀⡔⢁⠎⠀⢠⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠
⢀⢀⠎⠋⡀⠱⠀⠙⡀⠰⠂⠉⣄⠘⢎⠈⡷⢡⠎⢀⠀⠉⠁⠈⠁⠐⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⢈⠉⠁
⡈⡌⠰⠀⢁⡤⠚⠀⠀⡄⠘⢅⠠⠖⠈⣁⠀⢾⡁⠈⢁⠀⠀⠀⠠⠄⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠠⠄⠀⠀⠀⡨⠁
⠇⣳⡄⠚⣀⠰⠄⠉⣄⠰⠆⠉⣤⠔⢊⣠⢖⣁⣀⡀⣩⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠙⠂⣠⠶
⠇⣄⠰⠆⠉⣄⠘⢆⠀⣦⠘⢃⡀⢶⠈⢡⡀⢖⣧⠔⠁⢀⣈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⠁⠀⠛⠉⡛⡅
⠠⢹⡄⠘⢃⡈⠆⠈⢃⠀⠖⠈⣠⠀⠓⡀⠰⢟⢱⠂⠁⠹⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠱⠀⠀⢈⢀⠇
⠀⠡⠠⠆⠉⣤⠐⢋⠀⠤⠀⠃⣀⠴⠊⢁⡀⠸⠋⢆⠀⡇⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⢠⢠⠏⡏⠀⠀⠀⠀⠀⣀
⠀⠀⠁⠚⢃⡀⠶⠊⢡⡀⠶⠈⢡⡤⠚⣁⡠⢼⠀⠈⢄⣅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠀⡇⠀⠀⢀⠀⠉⠀⠀⠁
⠀⠀⠀⠐⠌⠣⡀⠓⠀⠠⠄⠛⣀⠴⠆⠉⣄⡼⠀⠀⠈⡟⠑⠢⠄⡀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠀⠀⣀⠠⠔⠛⡈⠀⠀⠰⡀⠀⠸⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠠⣐⠡⣄⡰⡂⢉⣤⢘⠵⢯⣀⢀⣠⡜⠀⠀⢀⠤⠄⠛⠖⢖⠒⠂⠀⠒⠒⠲⠚⠓⠀⠄⠀⠀⠸⡦⣀⢀⣨⡦⠀⠠⡀⢀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠩⠭⠀⠐⠀⠁⠀⠰⢊⠀⠀⠀⠁⠀⡇⠀⠀⠠⠁⡀⠀⠀⠄⠐⠁⠀⠀⠀⠀⢄⠀⣈⠖⠀⠁⠀⠀⡠⠀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⡉⠋⠉⠀⠀⠃⠀⠀⠀⠐⠄⠤⢈⠊⠀⠀⠀⠀⠀⠀⠀⠉⠤⠂⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⣠⠔⠋⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠘⠂⠠⠄⢀⣀⠀⠀⢀⣀⠀⠠⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢒⠀⠤⠀⠀⢀⣀⣀⣀⣀⠀⠀⠤⠄⡐⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠐⠒⡒⠒⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢃⣀⡠⠄⡏⠢⢀⢀⠌⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣠⠿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⣀⡴⠃⠀⠙⢦⡀⠀⣀⡼⠃
⣶⣶⣤⣶⣦⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣤⣶⣿⣷⣴⣶⣶⣶⣦⣾⣿⣶⣤⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣤⣶⣶
⣿⣿⣇⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠
⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠈⢷⠀⠀⠀⡏
⠀⠀⠀⣴⡡⠤⠒⠒⠒⠒⠲⡾⠤⣄⣸⠃⠀⢀⡴⠦⡀
⠀⢀⠈⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡟⠙⠲⣾⡅⠀⠸⡀
⠀⠘⠀⠀⠀⠀⠀⣠⠶⠉⠀⠀⠀⣸⠃⠀⠀⡧⠈⠑⢄⢠
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⢷⠀⠀⠀⢻⡄
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⡆⠀⠀⢘
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⡇⠀⠀⡘
⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡧⠀⡰⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡧⠞⠁⠀⢀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠒⠄⣰⣋⡀⠀⡀⢀⡀⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠓⠲⠤⣤⣀⣀⣀⣀⣀⣀⣠⡤⠖⠊⢁⠞
⠀⠀⠀⠀⠀⠐⠒⠒⠲⠤⠤⠤⣴⣦⠤⠒⠒⠋⠀⠠⡄⠀⠈⠢⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⣀⣀⣀⣀⠀⠐⠄⠀⠀⠀⠙⠒⠠⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠂⠀⣀⡤⠖⠒⠊⠉⠉⠈⢹⡏⠁⠀⠀⠈⠉⠓⠲⡄
⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⣀⡤⢴⣷⠦⢤⣄⡀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡧⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⠄⡀
⠀⠀⠀⢆⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡃⠁
⠀⠀⠀⠀⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣀⣀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣤⣶⣿⣿⡿⢿⣯⡿⠛⠻⠟⠛⠛⠛⠛⠿⠟⠻⣿⣿⡿⣿⣿⣷⣦⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⡿⠊⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠺⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣰⣿⣿⣻⡾⠋⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠛⢾⣻⣿⣷⣄
⠀⠀⠀⢀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣆
⠀⠀⢠⣿⣿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧
⠀⢠⣾⣿⣻⠏⠀⠺⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠂⠀⢿⣿⣿⣧
⠀⣼⣿⣟⠿⠀⠀⠀⠀⠉⠛⢷⡦⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠟⠋⠀⠀⠀⠀⠀⢿⣿⣿⣆
⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿
⣾⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀
⣿⣿⣿⠆⠈⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡀⢀⣏⠘⠿⠿⠗⠀⢸⣿⣿⣿⠛
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⣿⣿⡿⠾
⢹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣿⣿⠃
⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⡿
⠀⢹⣿⣿⣿⡀⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⡀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⢻⣿⣿⢧⡀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⣠⢿⣾⣿⠏
⠀⠀⠀⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣦⣀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣠⣾⣻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣽⣷⣤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣴⢾⣽⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣛⣶⣦⣀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣀⡠⣴⣾⣻⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠙⠿⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣯⣿⣿⣿⣿⡿⠟⠋⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⣿⡿⢿⣿⡿⠛⠛⠋⠉⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠻⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠛⠛⠋⠉⠉⠻⣿⣟⠛⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣫⣥⣤⣤⣤⣤⣄⣀⣀⠀⢀⣾⡿⠀⠀⠀⠈⠙⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣥⡀⠀⠀⣠⣾⡿⠋⠙⢿⣷⣄
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⠟⠋⠉⠛⢿⣷⣾⡟⠉⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⢀⣾⡿⠛⢿⣦⣄⠀⢀⣴⣾⣿⣿⣷⡄
⠀⠀⢠⣿⠏⠀⠀⢀⣠⣶⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠙⢿⣷⣾⡿⠁⠀⠀⠹⣿⡄
⠀⠀⣾⡏⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⢹⣿
⠀⢰⣿⢁⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢿⣷⡀⠀⠀⠀⣿⡇
⠀⣾⣿⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣷⡀⠀⠀⢸⣿
⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣷⡀⠀⢸⣿
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⢿⣧⠀⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡄⣿⡇
⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠈⣿⣷⡿
⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣁⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢷⣶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠐⠒⠊⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⢠⠶⡶⢦⡄⢠⠆⣴⣦⢰⡴⣄⢠⢤⠄⢠⣶⣤⠦⣤⠄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠁⠈⠈⠉⠉⠉⠉⠁⠈⠈⠁⠈⠉⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠁⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠠⡔⠀⠀⠀⣀⠀⠀⠀⠀⣠⣄⣀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠊⠀⠈⠁⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸
⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠁⠒⠒⠂⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⠀⠀⠚⠛⠀⠀⠈
⠀⠀⠀⡠⠊⠀⠀⠀⠑⢄⢀⠠⠀⠒⠒⠒⠒⠀⠄⣀⠊⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢐⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⠙⢦⡀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠒⠠⠤⠀⠄⠀⠄⠒⠀⠈
⠀⠀⠀⠀⠈⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠁⠈⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠠⠐⠈⠰⠶⠀⠐⠶⠀⠁⠀⠄⢀⣀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⡁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡁⠀⠀⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣶⣾⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠑⣀
⠀⠀⠀⠀⠀⠈⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠊⠁⠁⠁⢈⠒⠠⠄⠂⠁⠚⠀⢄⠀⠀⣠⣾⣿⠿⠛⠛⠛⠛⠛⠛⠻⠿⣿⣦⡀⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⢀⣀⡄⠂⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠈⠻⢿⡄⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠙⠉⠉⠉⠀⠀⠈⠂⠀⠀⠀⠀⠴⢀⠀⠀⠀⢀⠐⠒⠠⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠱⣄⡀⠀⠀⠀⠀⠀⠀⡄⠔⠉⠀⡀⢀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣠⠂⠠⠁⠀⠀⠀⢀⡀⠠⠤⠀⠀⠀⠤⠤⠀⡀⠀⠀⠀⠡
⠀⠀⠀⠠⠤⠊⠁⠉⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠤⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⡶⠂⠁⠀⠀⠀⠶⠆⠀⠶⠆⠀⠀⠀⠈⠀⢦⡴⠀⠀⠀⠨
⠀⡀⠂⠀⠀⠢⠀⠂⠀⢀⡀⠰⠀⠀⠀⠀⠀⠀⣶⡀⡄⠀⡀⠀⢠⠀⢠⠀⢠⠀⠲⠆⠀⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠀⠀⠀⣠
⠘⠋⠈⠋⠉⠉⠉⠉⠀⠉⠉⢹⠛⠛⠛⠛⠛⠛⡏⠁⠉⠀⠉⠀⠙⠀⠸⠁⠘⠁⠈⠃⠈⠃⠀⡍⠑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠋⠉⠙⠉⠙
⠘⠃⠀⠃⠀⠃⠀⠛⠀⠘⠀⠸⠓⠚⠓⠒⠓⠓⡏⠛⡗⠀⠛⠀⠘⠀⢘⠀⠘⠃⠘⠃⠀⠃⠘⠤⠀⠛⠈⠳⢠⣀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠛⠀⠘⠀⠘⠀⠘
⠒⠂⠀⠂⠀⠒⠀⠒⠀⠒⠒⢺⠀⢰⠚⠒⡆⠐⡗⠂⠓⠒⡖⠀⠚⠀⢘⠂⠘⠂⠐⠃⠀⡇⠀⠓⠒⠀⠀⠶⠀⠀⠈⠙⠀⠀⠋⠁⠀⠑⠖⠀⠒⠀⠒⠀⠘⠀⠐
⠠⠆⠀⠆⠀⠆⠀⠶⠀⠰⠀⠰⠀⠰⠄⠠⠆⠀⡆⠀⠦⠀⠶⠦⢴⠀⢸⠀⠰⠄⠠⠆⠀⡆⠀⠆⠀⠼⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠶⠀⠰⠀⠰⠀⠰
⠤⠤⠤⡤⠄⠤⠄⠤⠠⢤⠠⢤⠤⢤⠤⠤⡄⠤⡅⠤⡤⠠⠤⠤⢼⠤⢤⠤⢠⠄⠤⡤⠤⡥⠤⡤⠤⠤⠠⢬⢼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡤⠤⢤⠤⢤⠤⠤
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠂⠀⠂⠀⠀⠀⠐⠀⠘⠀⠀⠀⠀⠂⠀⠂⠀⠂⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⡤⠓⢄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢈⠟⠛⠟⣦⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⡠⡠⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⡀
⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠙⣿⡀
⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⣸⣧
⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠘⣿⣇
⠀⠀⠘⣿⣿⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠹⣿⣿⡆⠀⠀⢾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⢻⣿⣿⣦⣤⣼⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠈⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣼⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⣿⠇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠀⣼⠏⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣷⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⡿⠀⠀⠀⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⢿⣿⣿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⠀⢴⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠈⠻⠇
⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠏⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⢸⣿⡋⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⢀⠤⠂⠀⠙⢏⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢄⡀
⠀⠴⠋⠉⠈⠁⠀⠀⠀⠀⢀⡷⡀⠀⠀⠀⢀⣾⣽⣿⣿⣽⣇⣻⠿⣿⣆⠀⠀⠀⠀⣠⡞⠥⠀⠤⠤⢵⡀⡉⠑⣄
⡌⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⢟⣽⣿⣹⣦⣝⣧⠀⠀⡸⠁⠀⠀⠀⠀⣠⠎⢈⠓⠮⡈⢆
⡇⠀⣀⡴⠊⠁⠀⠀⢰⠀⢰⠀⠀⠀⠀⣿⣿⣿⢟⣯⣾⣿⣿⡏⣿⣟⣿⣯⠀⠀⡇⠀⠀⣀⠴⠊⠁⠀⡸⠀⠀⠈⢾
⢣⡴⠁⠀⠀⠀⠀⢠⠏⠀⠀⡆⡰⠀⠀⢹⣟⣵⣿⣿⣿⣿⡿⣹⣿⣿⣿⡟⠀⠀⢧⢀⠞⠁⠀⠀⠀⢠⠇⠀⢠⠀⡼
⠀⠣⡀⠀⠀⠀⣠⠏⠀⠀⣼⠔⠁⠀⠀⠀⠹⣿⣿⣿⣿⡿⣳⣿⣿⣿⠟⠁⠀⠀⠈⢧⡀⠀⠀⠀⢀⠎⠀⠀⣈⡴⠁
⠀⠀⠈⠑⠢⠶⠥⠤⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣔⣁⣀⡠⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⣾⡶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⠶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⣶⣛⣿⣿⣭⣭⣭⣵⣶⣬⣥⣴⣶⣤⣭⣭⣭⣍⣙⣛⣛⡶⣾⣧⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠛⠋⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⢿⡿⣿⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣤⣶⣿⣿⣛⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣶⣦⣤⣾⣧⣀⣀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⢻⣿⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠈⠉⠉⠉⠉⠙⠛⠛⢿⡿⠿⣿⣿⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡶⠿⠿⣶⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡿⠿⠿⠷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⠶⠾⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣤⣴⡶⠿⠛⣋⣉⣥⣤⣤⣴⣶⣶⣶⣶⣤⣤⣭⣍⣛⠛⠿⢶⣦⣄⣿⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣾⣿⣿⣿⣋⣥⣴⠾⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⣶⣬⣙⣿⣿⣿⣶⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢋⣡⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣌⠙⢿⣿⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣷⣶⠿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⢷⣶⣤⣀⠀⠀⠀⠈⠛⢿⣾⣿⠻⣦⡀
⠀⠀⠀⠀⣠⣾⢋⣽⠟⣵⢶⣆⠀⠀⢀⣤⣶⣿⠛⠉⠁⠀⠀⠀⢠⣶⠠⡾⢿⣶⠀⠀⠀⠀⠀⠉⠛⣿⣦⡀⠀⢀⣤⣽⣿⣦⠙⣿⡄
⠀⠀⠀⣼⡟⢡⣾⠋⠀⠻⠾⠟⣠⣶⠟⠋⠀⠻⣧⠀⠀⠀⠀⠀⠈⣿⠀⣴⣿⣋⠀⠀⠀⠀⠀⢀⣾⠏⠉⢻⣦⡸⣷⣿⠀⠻⣷⡌⢿⣆
⠀⠀⣼⡟⣰⣿⠃⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⣿⣄⠀⠀⠀⠘⣿⡌⢻⣆
⠀⢰⡿⣰⡿⠁⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠸⣷⡀⢻⣷⣶⡄
⠀⣿⢇⣿⠇⠀⠀⠀⢠⣿⠿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠛⣿⡄⠀⠀⠀⢹⣷⠘⣿⡈⣿⡄
⢸⡿⢸⣿⠀⠀⠀⠀⣾⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⣿⡆⢻⣷⣿⡿
⣾⡇⢸⡏⠀⠀⠀⠀⣿⠁⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⣿⡆⠀⠀⠀⣿⡇⢸⣿⡟⣿
⣿⡇⢸⡇⠀⠀⠀⠀⣿⢼⣿⣹⣿⡄⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣾⣿⣀⣽⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣟⠀⢻⡇⠀⠀⠀⣻⡇⢸⣿⣿⣿
⣿⡇⢸⣷⠀⠀⠀⠀⣿⡀⣩⣽⡿⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⡍⠛⠶⢤⣄⡀⠀⠀⠀⠀⠀⣤⣼⡿⠀⣸⡇⠀⠀⠀⣿⡇⢸⣿⣿⣿
⢻⡇⢸⣿⠀⠀⠀⠀⢿⡇⠈⠉⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠟⠋⠀⠀⠀⢿⣿⣷⡀⠀⠀⠈⠙⠳⠶⣤⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⠁⣼⣿⣿⣯
⠸⣷⠀⣿⡆⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣾⡏⢠⣿⢁⣾⠇
⠀⢿⡇⢹⣷⠀⠀⠀⠀⢿⣧⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣿⠃⠀⠀⣸⡿⢀⣾⣷⡾⠃
⠀⠘⣿⡀⢿⣇⠀⠀⠀⠈⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⣴⡿⢁⣾⠇
⠀⠀⠘⣿⡈⢿⣆⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣁⠀⠉⠉⠀⠀⣠⡀⠀⠀⠀⣠⣾⣟⡁⠀⢀⣾⠟⢠⣿⠋
⠀⠀⠀⠘⣿⣌⢻⣦⡀⠀⠀⢀⣈⡛⢷⣦⣄⡀⢠⡶⠀⠀⠀⠀⠀⠈⢿⣭⣿⠇⠀⠀⠀⠀⠸⣷⣠⣴⡾⠟⠹⣿⣿⣤⡿⠋⣴⡿⠃
⠀⠀⠀⠀⠈⠻⣦⣽⣿⣦⡀⢸⣿⣿⠀⠈⠙⠻⠿⢷⣦⣤⣤⣀⣀⣀⣀⣉⣡⣤⣤⣤⣴⣶⠿⠟⠛⠁⠀⠀⢀⣤⣿⣿⣧⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣏⠙⢿⣦⣉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣶⠿⢋⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣈⠛⢿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠿⠛⢙⣿⣶⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠉⠛⠿⣿⣿⣿⣦⣍⣙⠛⠛⠿⠿⠷⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉⢁⣠⣤⡶⢿⣿⠉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣹⡿⠾⠿⠿⠿⠿⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾⠿⠿⠿⠿⠶⢾⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣿⡷⠶⣶⡶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⠶⢿⣿⣄⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠛⠛⠁⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⢙⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⣶⣦⣾⣧⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣧⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⠛⠶⣿⣯⣭⣭⣭⣽⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⣀⣀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⢀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢿⣿⣿⣻⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣬⣿⣯⣭⣍⣩⣿⣷⣶⣶⣶⣶⣒⣒⣚⣛⣛⣛⣛⣙⣿⣋⣀⣀⣀⣸⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣯⡅⣢⡆⠐⡇⢐⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠃⠁⠀⠘⠛⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣉⣡⡞⠙⣦⣿⣦⣿⢸⣏⣉⠁⢸⣿⢉⣿⣿⡏⣹⣇⡟⠉⣷⢾⡏⢩⡷⣾⣙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠿⠏⠉⠿⠿⠈⠹⠸⠧⠤⠄⠸⠿⠉⠁⠿⠟⠿⠄⠧⠤⠟⠻⠏⠉⠀⠶⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠑⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠂⠀⠐⠂⠂⠐⠀⠀⠀⠐⠀⠀⠐⠂⠀⠈⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⢰⣶⠰⣷⡗⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠠⣤⠀⠄⠀⠀⠀⡀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠼⠿⠿⠘⠿⠏⠀⠀⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⣤⣤⣤⣤⢠⣤⡤⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠐⠀⠂⠂⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠀⠉⠁⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠆⠀⠀⠀⠀⠀⢐⢶⠂⠀⠰⠆⠀⠠⠄
⠀⠀⠀⠀⠀⠀⣫⣿⠀⠀⢀⡀⠒⣀⠐⢀⠐⠿⣂⠀⠀⣀⠀⠐⠀⠢⠀⠀⢀⡀⢀⡀⠀⠠⠀⠀⠀⡀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⡄⠀⡄⢀⣀⡁⠀⡈⠀⣀⡁⢀⠈⠁⠀⠈⠁⠀⠀⠀⠀⢐⡀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⣠⠈
⠀⠀⠀⠀⠀⠀⡗⢹⡆⠘⠿⠃⠀⠁⠘⠛⠋⠘⠁⠠⠤⠄⠀⠀⠀⠀⠀⣿⣿⡇⢿⡿⠀⠀⠀⠚⡿⠃⠰⠾⠁⡰⠞⠁⠀⠛
⠀⠀⠀⠀⠀⠀⣏⣿⡇⢠⣶⡄⢰⣿⣦⢀⣀⡀⠘⠉⠀⠀⠰⠂⠀⣀⡀⠈⠉⠀⠀⠀⢤⡠⠀⠀⠀⠀⠄⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠉⠁⠈⠛⠋⠈⠛⠁⠘⠋⠀⠈⠀⠀⠀⣩⣡⣤⠀⠀⠀⠀⠒⠁⠂⠀⠠⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡗⢲⡄⠀⠀⠀⠈⠈⠋⠀⢚⡽⠀⠀⣭⣭⣭⣤⣄⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠘⠛⠂⠀⠁
⠀⠀⠀⠀⠀⠠⣴⡤⠀⣴⡇⠄⣴⡧⠄⣴⡆⠀⠀⡀⢀⠀⠠⡑⠀⣀⢀⠀⣿⠀⢀⠂⠀⠀⠀⠀⠀⢉⠀⠐⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠈⠀⠀⠈⢁⡀⠈⠀⠉⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡜⠀⡌⢠⠀⠸⠃⢐⡯⡯⠀⢿⠇⠀⠉⠈⠆⠁⠀⠀⣸⠟⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⠋⠉⠉⠉⠙⠛⠛⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣀⡿⠟⢛⣉⣩⣩⣍⣙⠓⠶⣧⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢊⡡⠖⠉⠀⠀⢀⡅⠀⠀⠈⠑⡦⣉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠁⠀⠐⠆⠀⠀⠀⠇⠀⠀⠀⠖⠀⠈⠱⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠎⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢀⠈⢦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡎⠀⠀⠈⠈⠳⣦⣀⠀⠀⠀⠀⣠⡶⠋⠀⠈⠁⠀⠈⡇⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⢁⢀⣀⡀⠀⠀⠀⠙⢿⣢⣔⣠⡯⠀⠀⠀⠀⣀⣀⠀⢸⠘⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡀⠀⠉⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠈⠉⠀⢸⢠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢃⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠖⠀⢀⡀⠀⢀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢦⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⣠⠎⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡳⣄⡀⠀⠋⠀⠀⢠⡇⠀⠀⠙⢀⠀⣠⠔⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠪⣛⠲⢤⣀⣀⠀⠆⢀⣀⡤⠔⢊⣡⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⡷⠶⠦⢭⣭⣭⡥⠤⠶⠺⡏⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠦⢤⣄⣀⣀⣀⡤⠶⠶⠇

⣿⣿⣿⣿⣿⠿⠛⠉⠉⣿⠿⠿⢻⣿⣿⣏⣽⡿⠃⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠉⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣾⠛⢿⣿⠟⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⢠⣾⣿⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣹⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣤⡄⠈⢿⣿⡿⠀⣀⣀⠀⠸⣿⣿⣿⠁⠀⣿⡏⠉⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿
⣄⢀⡞⣢⣴⡛⢄⣀⣀⠜⢳⣤⡚⡄⢿⣿⡿⣌⠀⣤⠄⢾⣿⣿⣷⠀⣉⠋⠁⣠⣬⣿⡅⠀⠀⠀⠀⣀⠤⠺⡶⠊⠉⠁
⣿⠘⡇⠉⠉⠙⠳⠌⠲⠛⠉⠉⠁⡜⢸⣿⣿⡎⠱⠦⣀⣈⠛⠛⠁⠀⣙⡠⠞⢨⣿⣿⠄⠒⠂⣉⡀⠀⠀⠀⠙⠂
⣭⠀⢹⣾⣯⣿⣿⣿⣿⣿⣯⣿⣾⠁⢸⣿⣽⡹⣆⠀⠀⠀⠙⠋⠉⠉⠁⣁⡼⣿⣿⡏⠀⢀⣴⠿⠃⠀⣀
⢉⣧⣼⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⠄⡉⣿⡟⠣⡉⠻⢷⡤⣾⣶⣤⣶⠶⠋⣠⣿⠋⠀⠀⠈⢡⣴⢮⠀⠉⠀⠀⠀⠀⢠⣴
⠊⣿⣉⣽⣿⡏⠘⠶⠟⠈⣿⡟⠁⠀⠀⠸⠃⠀⠑⢤⣀⣀⣀⠀⠀⣀⣴⢿⣿⡇⠠⡀⠀⠀⠘⠻⠚⠀⠀⢀⡤⠖⠛⠁
⠋⣭⣿⣿⣿⣷⣶⣶⣶⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⣸⠟⠁⠀⠛⠀⢠⣤⣦⣄⡀

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡈⠳⣄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠂⣠⡞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠉
⠀⠀⠀⠀⠀⠈⢡⠊⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢿⣦⠀⠀⠀⠀⠀⢸⠃⡀⠀⢀⡀⢿⡄⠀⠀⠀⠀⣠⡾⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣷⡀⠀⠀⠠⢾⠀⠁⠂⠈⠁⣸⢧⠀⠀⣴⡾⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⠻⢦⡀⠀⢻⡄⠺⣯⡗⢀⡿⠃⢀⣾⠟⠀⠀⣠⠞⣠⣴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢦⡀⢻⠦⣤⡴⣿⣠⣴⡿⠋⠀⣠⣾⢫⡞⡀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠈⢷⡈⢰⣦⣴⠛⠛⠁⠀⢀⣾⠛⠁⢸⠀⣉⣁⡀⠀⢙⠆⠀⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⡾⠁⠀⠀⠸⣆⠻⠾⢁⢠⡋⠀⠀⡂⢿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠈⢓⡖⠉⠸⣀⠀⠀⢹⡄⠀⡼⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣆⣤⣶⣋⢳⣤⡾⠉⠳⣶⠟⠀⡸⠁
⠀⠘⡕⢦⣄⣏⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣻⣿⢋⠀⠀⠀⠙⢿⣿⠀⠤⠤⠄⠒⠒⣲⡾⠁⠀⣼⠃
⠀⠀⠙⣦⡈⠁⠘⣇⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣼⣿⡇⢠⡄⡄⢠⡦⠸⣿⣿⡆⢒⣛⣛⣻⠏⠀⢀⡾⠃
⠀⠀⠀⠈⠙⠶⣤⡈⠛⠷⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⢻⣿⡏⣉⣁⣼⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠷⣦⣸⣿⠲⢤⣀⣙⢿⣿⡀⢿⣥⣽⠃⣰⣿⠟⢀⣠⡿⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠙⠿⣧⣤⣀⣀⠀⠉⠳⣄⣉⣡⢴⡏⠀⣴⣿⡋⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡿⠙⢻⣿⣷⣦⣼⠉⠁⢘⣧⣾⣿⣿⣿⣦⡾⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢧⣠⣿⣿⣿⣿⡏⠛⣶⠛⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠻⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⣀⡀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⢿⡿⠿⠿⠿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠛⠛⠋⠻⣿⣿⡁⠀⣠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⡟⠀⠀⠐⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⡈⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⣀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢀⣿⡏⠀⢈⣉⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠿⣿⣿⣿⠃⠀⣼⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⢦⣤⣤⠀⣠⣿⣤⣀⣀⣠⣤⣶⣶⡄⠀⠀⠀⠀⢼⣿⡿⠛⣶⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣽⣇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⡿⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠻⠁⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⡆⢲⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠈⠛⠛⣿⡿⣿⢿⣿⠿⠿⠟⢿⠀⠀⠘⠁⢻⠻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠉⠙⠟⠀⠀⠀⠫⠀⠀⠀⠏⠀⠀⠀⠘⠀⠀⠀⠀⡞⠀⠈⠙⠿⠟⠹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠸⣧⡀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⢠⡆⠀⢠⣾⡟⠀⠀⢰⣿⣟⣄⠀⠈⣁⣠⣤⣿⣾⠉⣴⢃⣾⣉⣛⡀⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡀⠀⣀⣿⡀⠾⢛⡇⣀⣀⡻⣃⠀⡄⣽⣿⠻⠘⡟⠆⠙⠛⢻⡾⠉⠿⠟⠋⠛⠉⠉⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠢⠤⠴⠏⡉⠋⠐⢩⠂⠾⠛⠟⠛⠻⠟⠛⠋⠁⠀⠃

⠀⠀⣠⣶⠾⠛⢟⠛⠳⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⢛⠟⠻⢶⣤⡀
⢀⣾⠛⢄⠀⠀⠀⠑⢄⢀⠔⠉⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⣶⣷⣶⡆⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠑⢄⢀⠔⠁⠀⠀⢀⠜⢻⣆
⣾⠃⠀⠀⠑⢄⠀⢀⠜⠛⢄⠀⢀⠔⠙⢦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⠞⠑⢄⠀⢀⠜⠓⢄⠀⢀⠔⠁⠀⠀⢻
⣿⠑⢄⠀⠀⢀⠵⢇⠀⠀⢀⠵⢅⠀⠀⢈⢷⡄⠀⠀⢀⣿⣿⣟⣿⣿⣇⠀⠀⠀⣴⢏⠀⠀⢀⠵⢇⠀⠀⢀⠷⢅⠀⠀⢀⠔⢹
⢹⡆⠀⢑⢶⠁⠀⠀⢑⢶⠁⠀⠀⢱⢖⠁⠈⣷⠀⠀⠈⠉⠙⠋⠛⠉⠉⠀⠀⢰⡏⠀⢑⢶⠁⠀⠀⢱⢖⠁⠀⠀⢱⢖⠁⠀⣾
⠀⢿⡔⠁⠀⠑⢄⠔⠁⠀⠑⢤⡔⠁⠀⠑⢄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠔⠁⠀⠑⢤⠔⠁⠀⠑⢄⠔⠁⠀⠑⣼⠇
⠀⠀⠻⣆⣀⠔⠁⠐⢄⣀⠔⠁⠑⢄⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⠔⠁⠑⢄⣀⠔⠀⠐⢄⣀⡾⠃
⠀⠀⠀⠈⠳⣤⡀⢀⠜⠙⢄⠀⠀⠀⠑⢄⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠔⠁⠀⠀⢀⠜⠙⢄⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣄⣱⣤⣤⣶⠿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⢷⣦⣤⣴⣡⣤⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠋⠀⠀⠀⠈⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠉⠉⠁⠀⡄⠀⠈⠉⠉⠛⠻⢿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⡅⠀⠀⠀⠀⠀⣠⠀⡴⠲⣄⠀⠀⠀⠀⠀⡌⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠠⣴⠀⢴⡆⠀⠀⢸⠀⣧⠴⠋⠀⠀⠀⢀⣤⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⣟⠁⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠘⠘⡟⠓⠂⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⣸⣿⠯⣥⢀⡀⣀⢤⡀⠈⠀⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠀⠀⣠⠤⣄⠠⠄⢻⣿⡄
⠀⠀⠀⢰⣿⡏⠂⠀⠉⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⠞⠀⠀⠀⢻⣿⡀
⠀⠀⠀⣿⣟⠀⠀⠀⠀⠇⠳⠤⠋⢤⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠀⠀⠀⠈⣿⣧
⠀⠀⢸⣿⡇⠀⠀⣀⣤⡀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⡇⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⢹⣿
⠀⠀⢸⣿⠃⡤⠀⢧⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⣏⠀⠄⢸⣿
⠀⠀⢸⣿⡇⠀⠀⠲⠤⠚⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠚⠀⠀⢸⣿
⠀⠀⠈⣿⣧⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⢹⣿⡄⠀⠀⠀⣫⠤⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⡇⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⢻⣿⡄⠤⠀⠧⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠏⠠⠄⣰⣿⠏
⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠒⢒⠖⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣏⣉⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣏⠤⣉⠀⠀⠀⢠⣀⣨⠇⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠖⠈⠀⠀⠀⠀⠀⠧⣠⠾⠀⠀⠀⠀⠈⠁⠆⢀⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⣀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣀⣤⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢶⣶⣶⣶⡾⠿⠿⠛⠋⠉

⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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