Hindu Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣤⠀⠀⣠⣄
⠀⠀⠀⣀⠴⠶⠶⠛⠋⠉⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣿⣿⠃⠈⠀⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠀⠀⠀⠛⠀⠦⡄
⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⡄⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠠⣄⠀⠀⢀⡆⢀⠀⣀
⠀⠀⠊⠉⠉⠉⠉⠀⢀⡈⠀⠈⠀⠒⣎⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣿⠀⠀⢸⡇⢸⠈⠀⠂
⠀⠀⠀⠀⠀⢀⡤⣤⠴⠀⢰⠓⠉⠉⠁⠀⢰⡟⠂⠀⠀⠀⠀⠀⠀⠶⠱⠜⠃⢀⠙⠀⠀⠙⣠⣾⣄⠀⢀⢰
⠀⠀⠀⠀⡀⣿⠁⠻⠎⣰⠃⠀⠀⠀⠀⠀⡸⠮⣺⣿⢀⡆⠀⠀⠠⡐⠀⠀⠀⣀⣽⠂⠐⡊⠀⠈⠙⠻⣦⡀⢰⠦
⠀⠀⠠⠿⠰⣧⡛⠀⣽⡇⠀⢲⣤⣤⣤⣤⠁⠀⢸⡟⣧⡀⠀⠀⠉⣀⡀⣀⣀⣤⠤⠴⠤⠄⣠⣀⡀⣀⣹⡏⠉
⠀⠀⠀⠀⠈⢿⠇⢠⣿⣧⠀⣿⣶⣮⠙⣿⣆⠀⠸⣉⣁⡀⠀⠀⡀⠠⡤⠄⠀⠐⠒⠒⠒⠒⠂⠨⢿⣿⣿⡏
⠀⠀⠀⠑⠀⠀⣠⣥⠽⠿⢒⡉⣿⠏⢰⣿⠿⠏⢀⣿⠏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠸⠿⠿⠿⠿⠧

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡋⠀⠀⠀⡝⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠽⣶⡀⠴⠦⢿⣆⠐⠤⠻⣷⡀⠶⠟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠂⣠⡿⠟⠂⣴⡾⠛⠒⠒⠒⣿⣿⣶⡖⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡶⣶⣶⣿⣷⣶⣾⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠒⠚⢻⡿⠛⠟⢻⣿⠛⠛⠛⠛⠻⣿⣿⠟⠟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⡾⠃⠀⢀⣼⠗⠀⠀⠀⠀⢀⣙⣿⣧⠀⠸⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣤⣤⣼⣤⣴⣦⣤⡄⠰⠶⠟⠻⠿⠟⠛⠿⡇⠰⣠⣤⣦⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢄⡉⣿⣧⡦⣌⡉⠀⡀⠀⠼⢷⡆⣰⡾⢶⣷⡀⣯⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⠎⠁⢈⡙⠓⠋⣟⣾⠇⠈⠛⠋⠃⢻⠟⠛⢿⣿⣿⣿⢩⡽⣿⣄
⠀⠀⠀⠀⠠⠊⠀⠀⢀⠀⢹⡄⠀⠀⠀⠀⢻⣤⣀⣠⣿⣿⣤⣾⠈⢩⡅⠀⣠⣶⣤⣾⣦⣴⡟⢙⠛⢿⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢆
⠀⠀⠀⠀⡃⠀⣠⣤⣿⣾⠃⠀⠀⠀⠀⠀⠀⠸⢿⣾⣿⣿⡿⣁⠀⠄⠐⢰⣿⣿⣿⣿⣿⣿⠀⠇⡜⢀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣼⣶⣴⣆
⠀⠀⠀⠀⠓⠶⣿⣿⣿⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢻⠳⣤⠶⣿⣇⡀⠉⣩⣴⣶⡿⢤⣴⣿⢻⣿⡟⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢉⡜⢣⣼⠀⠀⠀⠀⠀⠀⠀⣤⡞⢸⣿⣿⣇⡈⠙⠃⠀⣿⣾⠹⣉⠷⣿⠟⠃⢸⡟⣿⠀⢹⡇⣹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⣤⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠀⠀⠀⣿⣿⡃⣼⣿⣿⣿⠉⡀⢈⣠⣾⢻⣄⡀⠀⣠⡴⠁⣼⢿⣿⢴⣿⣯⢹⣿⣿⣿⣷⡆⠀⠀⠀⠀⠠⣾⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣿⣦⡀⠀⠄⢀⠀⠙⠃⢿⣿⣿⣿⠀⠰⣨⣿⣿⣿⣿⡟⠙⢿⡁⠠⡿⠛⠻⣾⣿⠇⣸⣿⣿⣿⣿⣧⢠⡄⠀⠀⣴⣟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣉⢛⣿⠇⠘⠒⠊⠀⠀⢉⣿⣿⣿⡍⢤⢸⡟⢿⢿⣽⠅⢀⠀⡇⢸⣧⣦⣰⣿⣿⡇⢾⣿⣿⣿⣿⣿⡈⠀⠀⠲⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠄⠀⠀⢀⣠⣴⠿⠓⠚⣿⡛⠿⣷⡀⠨⣟⣀⠔⠉⠀⠀⠁⠀⢸⣧⣿⣾⠿⣿⣤⣾⠿⠿⠉⣍⡙⠁⠀⠄⢠⣿⡏⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⣼⣴⣶⣿⣿⣁⠀⠀⠀⠀⢁⣴⣿⣷⣤⣽⣿⢿⣦⠶⠂⣤⠆⢈⡁⠴⢿⣿⠿⠛⠿⢷⣶⣦⣌⢻⣷⣀⣰⡿⠋⠀⢀⣤⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣼⣿⣿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⡿⣿⡇⢸⡿⢷⡞⢡⣶⠿⣿⣄⣿⣦⣤⣤⣤⣮⣿⣿⣿⣶⣿⣿⣿⣤⣴⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡜⢣⣿⣿⣤⣿⡿⢱⣿⣿⣿⣿⣿⣿⣿⣇⣿⠇⢸⠟⠀⣹⣿⣉⣾⣿⣿⣿⣿⣹⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞⢀⡾⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⡟⢻⣿⣧⡆⠀⣠⣾⣿⣿⢿⣿⣿⣿⣿⠏⢈⢇⠹⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿
⠀⠀⠀⢀⡄⠘⠁⣰⠏⢠⡿⢠⣿⣿⡟⣴⣿⢿⣿⣿⣿⣿⣿⣧⠑⢊⣩⡵⠏⢁⣉⣉⣴⣿⣿⣿⠟⠁⡀⠈⣹⣇⣻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢃⠀⠞⠁⣠⡿⢡⠟⠉⢀⣴⣿⣿⣉⣿⣿⣿⣿⢛⣉⠶⠃⠈⠈⠒⠂⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⢿⣤⠻⣿⣏⠁⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⠀⣴⠟⣠⠃⠀⢀⣼⣿⡿⢣⣿⣿⣿⠋⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠻⣿⠙⠷⠀⠀⠻⣿⣿⣿⣿⣿⠁⠈⡙
⠀⠀⠀⠀⠀⠐⡦⠈⣴⣿⡆⢀⣿⣿⡟⢡⣿⣿⡟⠁⠀⠀⠀⠙⣿⣿⢷⣦⣤⣤⡞⠁⣀⡠⠀⠈⠂⠀⠀⠀⠀⠀⠘⣿⡆⠘⣆⠀⠀⠀⠀⣿⡟⡿⠻⡇⠀⠸⠃
⠀⠀⠀⠀⠀⠂⣤⣾⡟⢁⣤⣿⣿⣿⣵⡿⠟⠉⣆⠀⠀⠀⠀⠀⠉⠻⣷⣶⣿⣿⡃⢰⣿⣵⣾⣏⣀⣀⢀⡀⠀⠀⠀⠘⢻⡌⠿⣷⣀⣴⣾⣿⣟⣀⠾⠀⠀⣾⣰
⣀⣀⣀⣀⣀⣤⣿⡟⠀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠈⠷⣄⠀⠀⠂⠀⠀⣸⣿⣿⣿⣷⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣆⣻⣿⣿⣿⣿⣿⡿⠂⠀⣨⢁⣿
⣿⡿⢿⣿⣿⣿⠟⢡⣾⣿⡿⢻⣿⡁⠀⠀⠀⠐⠁⣴⣿⣷⣦⣀⠀⢠⣿⣿⡿⠿⠋⠀⠤⠤⠀⠂⠀⣀⣤⣾⡿⠋⠈⡉⠉⠻⣿⡔⢿⣯⠉⢛⣿⣧⣤⣾⣱⣼⣿
⠿⣿⠂⣺⠿⠏⣰⣿⣿⡿⠁⠈⢿⣿⠀⡀⠂⢠⣾⢿⣿⡇⣸⠛⠗⠛⠯⠤⣤⣄⣀⣤⡤⢤⣤⣶⠾⠿⠋⠉⠀⠀⠀⠀⠀⣄⣿⣿⡉⣹⣧⣩⣿⣿⣯⣽⡿⢹⣿
⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣤⣤⣄⣻⣷⠀⣴⡋⠀⠿⠿⠇⠉⠀⠀⠀⠀⠀⠀⣠⣠⣥⣶⠾⠋⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⢸⣟⣿⣷⡤⠿⣿⣿⣿⣿⣿⣿⣿⣿
⡛⠉⠋⠛⣹⣟⡛⠛⡛⢻⡅⠀⣹⠆⠈⠷⡿⣿⣶⠖⠀⢀⡀⠀⠶⠶⠾⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⢸⡟⢹⣿⣿⣙⣿⣿⣿⣻⣿⣿⣿⣿
⡀⢠⣇⠺⢿⣿⡿⠿⠿⠿⠿⠿⠷⠀⠀⡀⠀⢀⣀⣀⣴⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢸⣿⣷⣦⠀⠠⡿⠃⢼⣿⠟⠿⠿⠿⠻⣿⣷⣾⣿⣿
⡇⢸⡿⢿⣾⣿⡇⠀⠀⠀⠀⠀⠰⢶⣶⣤⣴⡾⠟⣿⣿⣋⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⢀⡿⣿⣶⣆⣀⣠⣿⡟⢸⡏⠀⠠⣶⣶⣿⡉⠀⠀⠀⠀⠀⣿⣯⣿⢼⣿
⠀⠀⢉⣠⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⣧⡈⣿⣇⠙⠶⠶⠟⣫⡿⠸⢧⣤⣤⣶⠟⢡⡿⠛⢈⣙⠛⠋⢠⡟⠀⠀⢀⣿⡏⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠁⠈⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢯⣿⣷⣶⡾⠿⣏⣀⣴⣦⣉⡉⣠⣴⣾⣥⣠⣾⠿⣿⣿⣿⡿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠙⢻⣿⣿
⠀⠀⠉⠀⠘⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠿⠿⠀⠀⠀⠙⠛⠉⠈⠙⠿⠟⠁⠀⠉⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠟⠉⠛⠛⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠺⢿⣷⡄⠀⢀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⠀⠆⢻⡇⢠⠏⢘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⠀⠀⠀⠀⢀⡴⡆⠀⠀⣠⡟⢃⠀⠀⢀⡉⣧⣘⡆⡼⠃⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠳⡌⣮⠓⢢⠀⠀⢎⠀⡷⡆⢰⣿⢿⠋⠁⠀⠙⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠀⡐⢙⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠁⠀⠀⣸⠀⣠⣣⠋⢸⢈⠣⢄⣘⡠⠟⠀⠀⠙⢢⣀⠀⠀⠀⠀⢀⡧⣿⣯⡽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣟⡶⢤⣀⡻⠶⡾⠁⢠⣈⠦⠨⠋⠉⣁⣨⣤⠴⣄⠀⠘⠣⣄⠤⠚⣝⡤⠬⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠈⡅⢀⠧⢾⣿⣄⠀⢤⠐⠀⣹⠃⢸⠀⠈⠒⠦⡀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⡈⠀⡸⠋⠙⣄⠈⠢⠔⠀⢀⠸⣄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⣀⠔⠋⠉⠁⢳⠒⡦⢤⣼⣽⣍⣩⣍⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⠀⣠⢄⡀⢸⡆⠑⡌⠏⣿⡇⠀⣿⣋⣯⡤⣖⣲⠶⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⢴⠤⠐⠒⠒⠃⢡⣀⡬⠤⠒⠋⠁⣾⠙⣶⣿⢀⠁⠈⣽⠁⠀⢿⣉⣉⣁⡀⠠⠤⣼⣭⡋
⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠔⠢⠜⠋⠁⠀⠀⠀⠀⢰⡖⡺⠧⢿⢘⣧⣿⠛⢦⣀⡀⠉⠙⠋⠉⠛⠛⠷⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⢹⣴⠋⢸⡄⠘⠿⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠋⠁⠀⢺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈⢣⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠐⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠙⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⣠⠀⠀⠀⠀⢼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠁⢀⣴⡿⠃⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣤⣘⠿⣁⠀⣿⡿⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋⠉⠛⠛⢣⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡿⠀⠀⠡⠄⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣿⣷⣦⡷⠷⣴⡄
⣀⠘⠃⠀⢀⡀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣼⠃⠀⠘⠋⠀⠰⠤⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣧⡈⢷⣧
⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠍⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⠃⠉⠛⣿⣿⣿⣷⣼⡏
⠀⠀⠀⠀⠐⢀⠀⠀⠀⢸⣿⣿⣿⡟⠹⠟⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⠂⠐⣗⠀⠹⣿⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⢿⣳⣠⠀⠀⠀⠠⡞⢃⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠾⢀⣤⣠⣴⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣦⠀⣄⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠹⣿⣿⣋⠀⠁⠀⠀⠀⠀⠀⠀⢄⣀⠸⡏⢻⡿⠃⠀⠀⠀⠀⢸⡛⠻⠟⠿⢿⡄
⠀⠀⠀⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡀⢈⡛⢮⠈⠃⠀⠀⠀⢀⡀⠀⢸⣿⣇⠀⢀⣤⣄⣀⢀⣠⠐⠋⠀⠀⠀⠀⡀⢹
⠀⠀⠀⢠⡞⠀⠹⠗⠄⠀⠀⠀⠀⠀⢸⠂⢃⣼⣷⡈⠙⠂⠀⠀⠀⡈⠁⢀⠘⠛⠃⢴⡟⢿⣿⠃⠈⠁⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⡅⠙⠳⠆⠀⠀⢀⡘⠃⡤⠻⣄⠀⠀⠈⣄⠠⠛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠛⣳⠄⠀⠀⠀⣈⠣⠀⠙⠦⣿⣿⡇⠀⠹⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠛⣳⠆⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠻⠁⠀⢻⠃⠀⢀⡈⠿⠀⠠⠀⣀⠀⠱⡦⠴⠐⠂
⠀⠀⠀⠀⢀⡜⠁⠉⠀⠀⠀⠀⠖⢠⡿⠃⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠸⡿⣦⠀⢀⠀⠉
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⠃⠀⢠⠀⠀⠘⠒⠀⢀⡸⠋⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠀⠀⠀⠀⠀⢠⣤⣀⣀⠄⣠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠢⠷⢷⣾⣷⣼⣿⣿⡍⢣⡵⠦⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣄⠠⢼⡟⢻⡉⠉⠹⡛⠧⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣰⣾⣿⠿⣿⣿⣿⡉⢆⠀⠀⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⢀⡀⢠⠄⠔⠀⠀⠀⠸⠿⠿⠋⠅⢸⣆⣼⡿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠈⠉⠁⠁⣀⠀⠄⠀⠀⠀⠀⠀⠀⢢⡀⠋⢁⠄⠀⠀⠙⡃⠀⠰⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠂⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠛⠀⠀⠀⠀⠀⠀⡄⣠⡁⣴⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⠎⠙⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⡆⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠛⠀⠲⠾⠇⠀⠀⠀⠐⣷
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣤⠀⠸⠁⠀⠀⠀⠀⠀⠀⢩⠆
⠀⠀⠀⠶⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⠀⠐⠂⠀⣤⡦⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⣠⡀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠈⠋⠑⠂⠀⠒⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⠂⠀⠀⠀⣴⠆⢠⣤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠊⠙⡿⢠⠀⠼⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡀⠁⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠄⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠠⠈⢁⠀⠻⠆⠀⣄⢀⡀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⢉⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠶⠇⠀⠀⡀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⠉⠠⣾⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣀⠀⠀⠀⠀⣡⠏⡈⠠⠜⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡈⠀⠀⠀⣤⡄⠀⠀⠶⠀⠀⠀⠸⠛⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠂⠉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡄⠀⠀⠀⢦⡈⠀⠀⠀⠀⠀⠀⠁⠀⠖⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⢦⠀⠂⠉⡆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡤⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠃⡈⠙⢋⠀⡸⢉⡉⣯⠹⢈⣿⠟⡻⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄⠀⠰⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣁⠁⠀⠀⠀⠈⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣓⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⠀⠠⢠⣬⣤⡤⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠀⠇⠀⠀⠀⢷⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠰⠹⠃⠿⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⠀⣬⣷⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠂⠀⠀⠀⢀⣠⣤⡘⠰⠇⠀⠀⠀⠀⠎⡷⣄⡀⠀⠀⡼⠀⢷⡀⢄⣰⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⡀⠦⠀⡈⠉⣷⠀⠀⠈⠀⡇⠀⠂⠤⠜⡀⢮⠓⢨⠁⠐⣆⣸⠃⠀⠉⠗⢠⣎⢀⡠
⠀⠀⠀⠀⠀⠀⠁⢀⣬⠄⡻⠤⠁⢰⢠⠔⠉⢀⣄⠀⠀⢻⠀⠀⠂⠳⣾⠄⠈⠁⠀⠀⣽⣇⣀⢀⡀⠠⢃⠿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣴⢀⣠⡴⠆⠢⡘⠛⠡⠀⢸⠏⠀⡐⠉⠀⠀⡄⡠⠀⢨⣩⠃⠻⠆⣹⢦⣍⠀⠈⡃
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠋⠙⢉⡀⠤⣀⠈⠁⠀⠒⢉⣲⣿⡣⡀⡠⢀⡇⠀⠐⢥⣊⠇⠀⡔⠁⠘⠟⠟⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢤⡄⣰⡊⠁⠲⠬⠭⠅⠤⡄⠉⢀⠀⠙⠺⣵⡿⠁⠀⢠⠏⠉⠀⠜⠀⠀⡈⠲⣷⡀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡯⡤⠀⠀⠀⠀⠀⢰⢡⠠⠘⠃⡠⠀⢸⡇⠀⣰⠆⠀⢠⠂⠚⠀⠠⠂⠀⠀⠹⣤⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡜⡶⣡⠃⠀⠀⠀⣠⠚⠈⠀⠻⠃⠀⣀⣾⣪⠷⠯⠄⠀⠻⠀⣿⠃⢤⣧⠀⢀⠀⢈⣹⣿⠧⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠼⣱⡣⠆⠀⡐⢉⡁⠀⠀⠀⠀⣁⠀⡟⡣⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠤⡌⢼⣿⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢀⡾⠻⢼⣢⣋⡁⠀⢀⣞⡢⠏⠁⠀⠀⠀⠀⠀⠀⠑⠒⠶⠢⣴⣶⣷⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡤⡤⡄⢀⠀⠠⢠⣼⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠿⠿⠛⠀⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠀⢠⠈⢠⡌⠐⡖⠁⡆⠂⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⡀⠀⣇⠈⢂⣐⠂⠴⠃⠀⠀⠾⠅⢘⡏⢀⣧⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠐⢦⠀⣃⣤⣶⡿⠉⠀⠈⠒⠒⠒⠶⠒⢋⡄⠀⠱⣦⣄⡋⢀⡤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⢂⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠿⠿⣿⣦⣅⠴⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠡⣀⢈⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⣻⣶⠂⠀⠁⠁⠀⠀⠀⠈⠙⣿⣧⡰⡏⢁⡇
⠀⠀⠀⠀⠀⠀⠠⣀⢀⣾⣿⡛⠀⠧⣠⠀⠀⠀⠀⠀⠀⣿⠁⠸⡿⠠⡀⠀⠀⠀⢀⡀⠖⠀⢹⣿⣿⣌⠊⢀⡄
⠀⠀⠀⠀⠀⠁⣠⢁⣾⣿⣿⠀⠀⠀⠀⠀⠂⠆⠦⠆⠀⠈⠳⠛⠁⢶⡀⡄⣄⠄⠛⠀⠀⠀⢸⣿⣿⣿⡎⠉⢁⡀
⠀⠀⠀⠀⠡⢄⡎⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⣄⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡈⡈⡁⠄
⠀⠀⢀⠐⢂⣠⡴⠤⡤⢭⡛⢷⣤⡀⠀⠀⠀⠀⠉⠁⠀⠁⠀⡈⠁⠀⠀⠀⠀⠲⢦⣤⡴⠟⣋⣉⣉⣙⠻⣇⠉⠁
⠀⣠⢃⠔⠋⠀⢁⡞⠁⠀⠈⠢⠻⣀⡀⠄⠂⠁⠀⠀⠀⠠⠀⠁⠈⠘⠁⠁⠒⠦⣤⣏⣔⠋⠁⠀⠀⠀⠙⢤⠫⣁⡄
⣴⣽⠉⡄⠀⠀⠈⠚⠠⠀⡀⣀⠑⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢹⣯⠼⠀⠀⠀⠀⠀⠀⠀⠃⠳⡄
⠙⠛⢿⡘⠱⠂⠆⠆⠒⠒⠊⠋⠒⢇⣁⣀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⣠⢐⣊⣥⡼⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣄⡹
⠀⠀⠀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠈⡭⡑⠆⠐⠁⠀⠀⠀⠤⠀⠀⡜⠡⠉⠴⠄⢻⢐⡀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡇⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡆⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡜⠁⠀⡔
⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⢠⣿⣷⣀⣠⠎⠀⠀⠀⠀⠀⠀⢀⣆⠹⣦⣴⡎
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⠥⣚⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⣟⢦⣀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡥⠘⡻⡿⠒⢀⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⣭⡓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣭⡄⠀⢠⣾⣇⠀⠀⠀⡆⠀⠀⠀⠀⢠⣾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢖⡦⣼⢻⠀⠀⠀⠗⠀⠀⠀⠀⠘⠁⣠⡶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⢧⠀⠀⠘⢄⠀⠀⠀⠈⠒⠓⢲⠶⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢠⠀⢸⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣥⡿⣦⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⢰⠃⣸⡇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⡴⠖⠊⠿⣛⣛⣻⠗⠛⠲⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢸⣄⣹⣇⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠞⠉⠀⠀⠘⣿⣩⢨⣹⣟⠀⠀⠀⠙⢷⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠙⢹⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⠃⠀⠀⠀⠀⠸⣿⣜⠾⣻⠹⡄⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣸⠃⠀⠀⠀⠀⠀⣾⡿⢣⣷⠺⣿⡆⠀⠀⠀⠀⠀⠸⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣤⡟⠀⠀⠀⠀⢰⠶⣩⣾⣇⠇⣸⣿⡡⢶⠀⠀⠀⠀⠀⡷⢀⣜⢫⣄⠀⠀⠀⠀⠀⠈⣿
⠀⢀⣿⣶⣄⠀⠀⠀⠀⠀⠶⣾⣯⣻⣿⣷⠆⠀⠀⠀⠀⢂⣳⣿⣽⣿⣽⣧⣇⢃⠀⠀⠀⠀⠀⠟⠇⣆⠖⡥⡟⠃⠀⠀⠀⠀⠸⣷⡄
⢠⣿⣿⠛⠀⠀⠀⠀⠀⡀⠀⢜⠛⣿⠋⢻⣠⠀⢸⠀⢸⢏⣿⣿⣿⢾⣿⣿⣏⢻⣀⣠⡤⠀⢠⡇⠈⢚⣉⡆⠀⠀⠀⠀⠀⠀⢸⢿⣿
⠙⣿⣿⠀⠀⠀⠀⠠⠤⠼⢿⣿⣤⠤⠤⠬⠽⣿⣿⡦⢾⣦⣶⣿⣿⣽⣿⣯⣢⣿⣿⣏⠁⢀⡟⠠⠁⠀⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣙⣻⡄⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⢀⠀⣿⣿⢻⣼⣹⣧⠾⣝⠻⣿⢛⣿⣳⣿⢿⣶⣯⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠈⢹⣶⣦⡀⠀⠀⠀⠘⢿⣏⢀⣤⡾⣿⣷⣿⡿⣠⢻⣿⣿⣷⣮⣤⣷⢿⣿⣿⣷⣾⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⡟⢿
⠀⠀⢸⢿⣿⣿⣦⡀⠀⢈⣻⣿⣾⢻⣿⣿⣿⣋⣙⣻⣿⣿⣿⣿⣳⠖⢣⣍⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣴⣿⣿⡶⠀⢰
⠀⠀⢸⠀⢻⣿⣿⣷⢸⣮⢻⣿⣿⣿⣭⡻⣿⡿⣿⢻⣿⡻⣿⣿⣿⣷⣿⣿⣿⡌⣿⠿⠛⠿⣿⣿⣿⣿⡟⣰⣦⣰⣿⣿⣿⠟⠀⠀⠘
⠀⠀⢘⠀⠀⠹⣿⣿⣿⣿⠸⢿⣿⣿⣿⣿⠉⠀⣿⡀⠙⢿⣿⡿⣿⣿⣿⡿⣿⠃⠀⠀⠀⠀⠀⠈⠙⠫⣾⣿⣿⣿⣿⣿⠋
⠀⠀⢸⠄⠀⠀⠈⠉⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⢻⣇⠀⠀⢩⡷⠘⣉⣸⣿⣁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠁
⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡀⠀⠀⢾⠏⣿⣿⣿⠻⢗⡄⢹⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢣⣶⣧⣝⣈⠽⣺⣥⣴⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⡿⢿⣾⣿⣾⣽⣺⣿⣿⣿⣿⣿⣿⡿⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣭⣿⣿⣭⠝⢀⠾⣿⠟⣛⣿⣛⡳⣿⠉⠉⠉⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠋⠀⢀⣴⣿⣿⡾⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢸⡇⠀⠀⣀⡀⣶⣶⣿⣿⠟⠉⠀⠀⠀⠀⣾⣿⣿⣿⣿⣝⣛⣻⣾⠏⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢘⣷⣶⣿⣿⣿⠿⠿⠿⣿⠇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡭⠛⠋⠁⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣟⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⠿⠿⠟⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠘⠛⠃⢀⠤⣿⣿⡷⢄⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠰⠿⠷⢿⠇⢀⠀⠀⠀⢀⡀⢀⣿⣿⡀⢀⠀⠀⠀⠀⠛⣸⣿⡇⠀⢦⠀⠀⠀⠿⣄⣤⣴⣮⣿⣵⣤⣠⡾⠃⠀⠀⢸⡀⣸⣿⡀⡼
⠀⠀⠀⠛⠶⣾⣿⣿⣷⠞⠁⠀⠀⠈⢳⡮⠈⠈⢵⡞⠀⠀⠀⢰⣼⣿⣿⣿⡆⠈⠆⠀⠀⣄⣈⣼⣿⣿⣿⣿⣿⣯⣶⡶⠀⠀⠀⣸⠍⢹⣿⡆
⠀⠀⠀⠀⠀⠘⠛⠛⡇⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠀⠻⣿⣿⡿⠇⠀⠁⠀⠀⠀⠘⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠛⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠿⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣥⡀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠏⢠⢀⣰⡀⢀⠁⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⢘⣿⣇⣀⠀⠀⠀⠀⠀⠱⣼⣿⣯⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⣿⡀⣀⡄⠀⠀⠻⠐⢦⣸⡤⣿⣧⣼⡠⠆⠀⠀⠀⠀⠠⣿⣧
⠀⠀⠀⠃⠙⣿⢿⣿⡇⠀⠀⠀⠀⠠⢠⣿⣿⡿⢧⣀⠂⠀⠀⠀⠈⢓⣷⡠⠖⣿⢃⡀⠀⠀⠀⠘⠓⣫⠇⣀⡸⣏⠓⠋⠀⠀⠀⢀⠇⠀⠰⣆
⠀⠀⠀⠀⡎⠀⠈⠁⠉⠁⠀⠀⠀⠀⠘⠏⠉⠙⠀⠈⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠛⠁⠀⠀⠀⠀⠐⡋⠈⠉⠁⠈⠃⠀⠀⠀⠀⠋⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠟⠀⠐⠉⠀⠿⠀⠀⠀⠀⠈⡛⠃⠘⠂⠙⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⡄⠀⣼⣷⠀⢠⡇⠀⠀⠀⢻⡄⢸⣿⣿⡄⣰⠃⠀⠀⠀⢀⣀⣸⣿⡀⠀⠀⠀⠀⠘⡇⣰⣿⣆⠀⣠⡄⠀⠀⠀⠀⠀⢰⠟⡇
⠀⠀⠀⠀⠙⢻⣿⣿⣟⠋⠀⠀⠀⠀⠀⠙⣳⣶⣷⡟⠋⠀⠀⠀⠀⠀⣭⣽⣿⣡⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠉⠿⠃⠀⠀⠀⠀⣴⣶⣶⠄
⠀⠀⠀⠀⠀⢈⠿⠿⠙⠲⠤⠀⠀⠀⠀⠀⠙⠿⠿⠙⠲⠄⠀⠀⠀⠘⠋⠹⠿⢸⡇⠀⠀⠀⠶⠚⣙⣛⣏⠙⠲⠖⠂⠀⠀⠀⢸⡿⠿⢿⡇
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⢿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠌⢁⡠⣌⢁⠠⠄⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠀⠀⠀⠀⠀⠐⠿⠀⢿⠁⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠠⠤⢶⢄⠤⢤⣀⡀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠉⠉⠉⠁⠂⠰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠈⠁⠀⡤⠀⠀⡀⢀⠈⣌⢨⠉⠓⠤⠀⡠⠎⢅⢠⠂⠈⡄⠈⠀⠐⣀⠁⠀⠀⢉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⡀⢁⡵⠞⠁⠀⠀⠀⠈⠉⠁⠉⠐⠢⡔⣊⡴⠁⡀⠀⢈⣿⠐⢆⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⢢⠈⡱⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠄⣀⠀⠀⠟⠀⠒⣡⠘⡁⠴⠆⢨⣤⡄⢀⠀⠀⠀⠀⠈⠀⠀⠰⠟⢠
⠀⠀⠀⠀⠀⠀⠀⡘⢀⠠⡞⠀⠀⣠⡞⢍⠀⢔⡠⠐⠁⣆⣠⣀⣥⠟⣣⠄⣠⠀⠀⠀⠀⠀⠘⠛⣣⡜⣄⢭⠀⠀⠀⠀⠀⠦⢀⡈⢁
⠀⠀⠀⠀⠀⠀⠀⡇⢀⡈⠁⠀⣸⠇⠠⢛⠈⠁⠀⠀⠀⢈⡩⠀⠸⠏⠉⠀⠴⠤⢀⠠⠀⠀⠄⠘⠧⠘⢏⣿⠁⠀⠀⠀⠀⠀⠀⠀⡨⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠄⣡⡆⠀⣃⠈⠀⡧⠀⠀⠀⠀⠀⠁⠀⢀⣾⡃⠀⠺⠧⢁⠄⢲⡾⡀⠐⢤⡀⢜⣀⠱⣦⡴⠀⠀⠀⠀⢸⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠁⣷⠀⠩⣄⠈⠁⡀⡀⠀⠀⠀⠀⠀⠀⠛⠓⣤⡶⠀⡋⡀⢀⠴⠈⠀⢾⣿⡹⢉⡆⠉⡁⠀⢀⣤⣔⣚⠒⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢁⢇⠀⠈⠻⢾⣇⠳⠄⠀⠀⠀⠀⠉⠀⠄⠀⠓⠈⠀⠀⠀⠀⠘⠓⠂⢈⣧⣸⠷⠄⠀⠀⠜⠑⣿⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢀⡷⠊⠐⢷⣙⡣⢦⡆⠛⠟⢹⠥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣌⣁⠐⠁⠀⠐⡁⠀⠀⠠⣿⠂⣇⠡⣰⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢋⠄⠀⠀⠀⠩⠄⡨⣼⠳⣈⠯⠜⠀⠀⠀⠀⢀⣀⢠⠄⢲⠄⠬⣿⡿⢤⠄⡢⣶⠾⣜⠂⠀⡙⢤⣭⣓⢃⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠆⠀⠀⠀⠀⠰⣶⣳⠀⠄⣠⣤⠀⠂⠲⠄⠁⠀⠀⠀⠰⠂⠀⠉⠳⢛⡌⢻⣴⢀⣷⣾⣦⣈⣾⣙⣞⡌⣧⣾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠀⠀⠀⡶⣯⡉⣀⣀⣀⡀⠤⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⣿⣿⣿⣿⡟⢉⣭⡼⠟⢧⣄⠘⣿⣯⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⢀⠀⠐⠻⣾⣿⡟⠋⠉⠀⠒⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠙⡇⣹⡿⢛⡠⠖⣈⠩⡔⢂⠒⡄⠘⠶⣝⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡅⠹⠀⢸⣿⠀⠀⠐⢲⠄⠀⠀⣤⣤⡀⠹⠇⠀⠂⢠⡴⠖⠓⠠⣴⣫⠔⣋⣴⡾⠋⠀⠈⠐⠡⠀⠀⢀⣈⢻⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⣸⡿⠀⠀⠀⠀⠉⠢⠄⠻⠿⠋⠀⠀⠀⠀⠀⢲⡦⣄⣠⣿⣵⣾⠅⠀⠀⠀⠀⠀⣀⠀⠀⡴⠋⠁⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⡄⡠⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⣟⣻⠟⠉⣀⣤⠴⣦⣤⡀⠉⠀⢈⣵⣦⠈⠂⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣏⣻⣴⠻⠟⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⢼⠀⠞⠁⣀⡀⠀⠈⠁⠀⠀⠈⠙⠃⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⣼⣿⡷⢺⡆⠰⠺⣿⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠏⣄⠠⠀⠣⡀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠄⣀⠀⠀⢀⠸⣿⣿⣿⢘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠟⠿⢯⡸⠟⠄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⣠⣽⣿⣿⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠆⠀⠀⢠⠟
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⢠⣄⠀⣈⢭⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⢋⣹⣿⣿⣄⣼⠻⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⢶⡉⠀⠙⠻⣧⡄⠀⠀⠀⢀⣀⠀⢀⣀⡀⠤⢀⣤⡀⠆⠄⠠⠀⠾⢻⣿⣿⡯⠹⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⡐
⡄⠠⠄⠰⠶⠋⣛⠀⠶⣖⠀⠀⠀⠀⠀⠈⢤⡘⢋⣤⠌⠉⠠⠦⠉⠁⠐⢂⣤⣰⣧⣶⣤⣶⣶⣾⣿⡏⢾⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉
⠁⠈⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠹⣾⣷⣀⠆⣠⣤⠤⣴⣢⣽⠻⢉⣿⣿⣿⣿⣿⠿⠿⣿⣄⣖⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⠒⢲⡖⢤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠂⠈⠌⠛⠙⠟⠛⠉⠀⠒⡠⢾⠁⠈⠉⠉⠉⠀⠀⠀⠈⠻⡧⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠙⢶⠁⠓⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠷⡈⠷⠀⣷⣠⢄⡀⣀⣀⣀⡤⣰⡯⠒⣶⣄⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣝⠓⡦⣀⠀⠀⠀⠀⠀⢢⣍⠀⠀⠀⠞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⣑⡶⢚⠚⣍⠫⡝⢠⡑⠘⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠋⠐⠓⠲⢿⣋⣬⡦⠀⠀⠰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣹⠉⣡⠛⢤⠶⠀⢉⠀⡐⠀⠉⠙⠛⠛⠁⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⣯⢷⣼⣇⠀⠀⡃⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣤⡄⠈⠭⢀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢻⣿⠾⡏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠥⠀⠀⠀⠀⠀⠀⣾⣷⣶⣤⣤⡿⠛⠁⠀⠀⠀⠀⠀⠙⢇⠡⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠙⠃⠀⠸⠂⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⠦⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠀⠠⠀⠀⡴⠂⡶⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣶⣤⣤⣤⣤
⣿⣿⣟⠋⡙⢻⡉⣽⠛⣻⡙⢹⡿⢻⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠚⠿⠯⠽⠟⠛⠛⠒⠒⠲⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⣧⡄⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⡀⠀⣘⢒⠃⠟⢩⣏⠛⠛⠓⣌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢏⠉⠁⠀⢸⢿⠀⠀⠈⠹⣄⠀⠀⠀⠁⠀⠐⠀⠁⠀⠀⡀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠶⠄⠀⠀⠀⡸⠻⡀⠀⠀⠀⢸⣷⣤⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠣⡀
⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⣾⠇⠠⠀⠀⠀⠀⠀⢸⣶⣇⠀⠀⠀⠀⠘⢿⣿⣿⣶⣄⡀⠀⠀⠈⠀⢀⠀⠀⡀⠙⢦
⠀⠀⠀⢠⠞⠀⠀⠈⠀⠄⠁⠀⠀⣠⣾⣿⣿⡟⠀⠠⠀⠀⠀⢠⣠⠟⠛⢹⣤⡀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣦⡀⠀⠀⠁⠀⡀⠁⠀⠈⢳⡄
⠀⠀⢠⠏⠀⠀⠀⠀⠈⠀⠀⣠⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⣼⡏⠀⢀⠀⠹⣇⡄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⣠⡏⠀⠐⠒⠂⠀⠉⠉⡉⠙⠻⢿⣿⣿⠁⠀⠀⣀⣀⣀⣀⣿⡀⢀⣌⡔⠀⣹⡁⢀⠀⠀⠀⠀⣽⣿⠟⠋⠁⠀⠒⠀⠀⠘⠷⢤⣀⠀⠀⢻⡀
⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⡈⠉⣄⣶⣞⡯⠳⠛⠂⠋⠀⠀⢨⠁⠀⠻⢯⡽⣟⣶⢤⡠⠊⢀⠔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠤⠷
⢸⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣡⠀⢘⠚⠁⠀⠀⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠈⠈⠋⣧⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⣾⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⢟⡀⠀⠀⠂⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⡇⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡸⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠀⡀⠸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟
⣧⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡇⠀⠛⢻⣿⣷⠄⠀⠃⠀⠈⠀⠀⡴⢿⡿⡹⠁⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⢻⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠳⠭⠡⠆⠀⠀⠀⠀⠀⠀⠄⠒⠚⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⢸⡄⠀⠀⠘⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢰
⠈⣧⠀⠐⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⡇⢣⡀⣸⡆⠀⠀⠁⠀⠀⠀⠀⠀⠰⠀⠀⣦⣠⡏⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠂⠀⠀⠀⡾
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⣿⠁⢀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢹⣿⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠹⣆⠀⠁⠀⠀⡐⠀⠀⠀⠀⠀⠀⠐⠁⣀⣼⡟⠀⠘⡀⠀⠀⠰⠆⠀⠀⠀⢠⠇⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⣴⠃
⠀⠀⠀⠙⣦⠀⠀⣨⠅⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠁⢀⣼⠃⠀⠀⠒⠀⠀⠀⠀⣾⣧⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠠⠀⠈⠁⠙⠛⣋⣠⣴⣿⠏⠀⠀⠐⠀⠀⠀⢀⣾⣿⣿⣷⣶⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠈⠒⢶⡀⠁⠀⢀⠀⠀⠀⠀⠀⠈⢉⢻⠏⠀⠀⠊⠀⠀⠀⣠⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠰⠀⢀⠀⠀⢡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠠⠀⠀⠀⠀⠀⡉⠀⠐⠃⠈⠀⠐⠚⠋⠙⠢⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⢀⠃⠁⠶⠀⠠⠀⠀⠀⢀⠀⠀⠠⠀⠄⠀⠐⠀⠀⠀⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡘⡈⠀⠒⠀⡘⠄⡀⠀⡘⠤⠰⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⠈⠁⠤⠀⡄⠰⢘⣶⣀⣤⡤⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⢰⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⣿⣿⣿⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣧⠤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⣼⣿⣿⣿⣧⡄
⠀⠀⠀⠀⢹⣹⠿⡏⡏⠀⢸⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇⠀⢹⣿⡿⣯⣿
⠀⠀⠀⢀⣿⣿⣤⣿⣷⡀⢸⣿⣿⣷⣼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣇⢀⣾⣿⣇⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⡿⠀⣿⣿⣿⣿⣿⡅
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢿⣿⢫⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣏⢠⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⡟⣡⣎⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣽⡟⢻⣻⡿⠶⣿⣋⠠⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠋⠀⠀⠀⠀⠀⠈⣿⠃⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡒⢊⣉⣴⡆⠀⠀⠀⠀⠀⠘⠁⣈⣁⡀⠀⢰⣶⠦⠀⠀⠀⡦⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠂⠀⠤⠀⠀⠠⠄⠰⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⠠⣀⠄⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢦⡀⠀⢀⠀⢰⡄⠀⠀⠀⢀
⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣶⣶⣄⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣿⠿⡶⠶⢾⠋
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⣋⡹⠛⠁⠀⡄⠀⡰⠀⠀⢼⡀⠀⠀⠀⠀⠀⠐⢀⡜⠀⠀⠐⠀⠀⠀⡠⠂⠈⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⢦⡄⠀⣠⠐⢂⣩⡿⠀⠀⠀⠀⡠⠊⠀⠀⢀⡀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆⢠⣴⣶⢦⣴⣶⢾⠉⠁⠀⠀⠀⢀⠌⠀⠀⠀⣠⣎⢳⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣻⣇⡀⠀⠀⠋⠀⠀⠠⠀⠁⠘⠍⠛⠞⠛⣹⣾⠁⠀⠀⠀⠴⠋⠀⠀⠀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠂⡽⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡸⢦⠋⢱⣆⠀⠀⣠⠀⠀⠀⢀⣠⣦⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠱⢤⡉⠛⠛⠋⠀⠀⢠⣿⠿⠉⢀⣈⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠠⠈⠀⠀⠀⠀⣀⣦⣄⠙⣀⣀⣾⣿⠿⠛⢂⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⡴⡀⠀⠀⠉⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠉⢨⠟⠀⢈⡿⣿⡁⠀⠀⠀⠀⢂⣀⣠⡶⣻⠇⠲⢀⣼⣟⢄⠀⠠⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⣦⣄⠀⠀⡀⠀⠠⠀⠀⣼⣑⡿⠉⠀⢤⡀⣠⣲⠝⠻⠋⠁⠀⠀⠘⢻⢯⡿⣵⣲⠌⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⢰⡂⠀⠠⠌⠙⠁⠄⠀⠀⠛⠙⠿⢄⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⠏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠷⠀⠀⠀⠰⠀⡤⠦⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠛⠀⠀⢀⡔⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠂⠀⠐⠀⠀⠀⠘⠂⠈⠁⠀⠀⠀⠀⠀⣴⣦⣤⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⣤⠤⠶⠶⠞⠛⠻⠖⠷⡆⠀⠀⠀⠀⠀⠀⠀⢴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡆⠀⢰⡀⠀⠘⠀⢠⠃⠀⠐⢠⠀⠀⠀⠀⠐⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠛⠀⠀⠉⠀⢀⠰⠂⠀⢁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠞⠁⠀⠔⠈⠀⢴⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢬⡀⢠⣾⠈⠀⠀⢣⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⣄⣠⡄⠀⠀⡄
⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠓⠛⠉
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣰⣾⡿⠝⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠙⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⡄⠀⠀⠊⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠚⠀⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠀⠐
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠉⠁⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠶⠤⠴⡄⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠍⠩⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠀⡐⠀⢸⠂⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠿⠛⠛⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⠀⣿⡇⢸⠀⠀⠩⠀⠀⠀⠀⠀⠀⡀⢠⣤⢤⣤⡀⠰⠀⠀⠀⠀⠀⣴⣦⠄⡀⢐⠀⢠⡇⠀⣤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣹⠀⣿⣏⢰⣀⣀⣰⡀⠀⠀⢀⣯⣒⣾⠀⠇⣀⠛⢸⣆⢢⣧⠀⠀⠀⣰⣦⠖⠛⠛⠒⠚⠛⠓⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠁⠉⠀⠀⠀⠀⠀⠁⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡕⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⡇⡸⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⢛⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣗⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠸⠆⣀⡀⠀⠄⠄⠀⠀⠀⠀⠉⢳⣯⣀⣁⣬⣷⠊⠀⠀⠀⠀⠀⠀⢘⣿⣿⣽⡏⢿⣭⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠰⢉⠀⠀⡀⠆⠀⠀⠰⠀⠀⠀⠀⠁⢹⠤⠀⠁⠤⢿⠀⡅⠀⠀⠀⠀⢠⣿⣽⠿⣼⡄⣯⡴⢿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⠆⠧⠴⠁⢖⢀⠀⠄⠀⠀⠀⠐⠀⢺⣀⡀⠀⣤⣼⠀⠂⠀⠀⠀⠀⣟⣇⣿⣧⡾⠀⢸⣦⣾⣿⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⠀⠀⠈⠈⠉⠀⠀⠀⠀⠈⠀⠘⠟⠛⠛⠛⠛⠀⠛⠀⠀⠀⠑⠟⠘⠗⠰⠃⠀⠀⠳⠰⠿⠓⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣚⣢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠙⠉⡛⠹⡀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠞⠋⠓⣆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠸⠿⠿⠿⠿⠿⠀⠀⣿⠀⠀⠀⠀⠀⠸⠧⠧⠭⠤⠦⠇⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠞⠆⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢨⣿⠀⠀⣶⣶⣶⣶⣶⣶⡆⠀⣿⢠⠄⠀⠀⠀⠀⠟⢻⢉⣹⢿⠃⠀⠀⠀⠀⠀⠀⡀⠠⠇⠀⠁⠃⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢨⠀⢦⡤⢤⠄⠠⣤⢤⡤⠀⣨⠈⠀⠀⠀⠐⢒⣳⣶⣶⣶⣖⡓⠖⠀⠀⠀⠀⢠⣥⡀⣀⡰⠔⣀⠀⣿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⠀⠈⠉⠉⡎⠘⠉⠉⡇⢠⣾⠀⠀⠀⠀⠀⠈⢹⡾⠀⠘⡟⠁⠀⠀⠀⠀⠀⣸⣿⡇⢈⠛⠀⢃⡄⡀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠰⠛⠀⠘⢩⢹⣷⣾⡏⠍⠓⠀⠛⠖⠃⠀⢀⣠⣤⣼⣫⣤⣽⣷⣤⣤⡀⠀⠀⠈⡛⣙⣿⣿⣿⣿⣿⣿⣿⣟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣯⣽⠄⠖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⡏⢻⠿⢷⣰⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣦⣿⣿⣿⣼⣷⣤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⢿⣿⣷⣿⣑⡙⢛⣿⡿⢿⣿⣿⣿⣦⡀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⢶⡄⠠⠀⠀⣻⡛⣿⠛⣿⡧⠿⢿⡿⠿⠦⣸⡿⢿⡟⠻⢿⠀⣀⣾⠱⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡸⠇⠀⠀⣘⠛⢛⠛⠛⣻⣿⣴⡄⣼⣶⢸⣾⡗⠂⠁⡘⠋⣀⠀⠛⣄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠁⠀⠀⠀⢹⡄⠘⠇⢠⡾⢷⡀⢰⡷⠀⣼⣿⣆⠀⢸⠁⠸⠃⠀⠀⠈⠻⢄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠍⠀⠀⠀⠀⠀⣠⣿⣲⣛⣹⢧⡴⣟⡐⢃⣺⣅⣠⣯⡔⣊⠻⢧⡀⠀⠀⠀⠀⠻⣿⡝⠁
⠀⠀⠀⠀⠄⡒⢀⠀⡇⠀⠀⠀⣠⣾⡕⢩⠃⣿⡟⢀⣶⡌⢉⡉⢩⣿⠇⣴⣿⠘⠁⢂⠹⡄⠀⠀⢀⡀⠉⠷⠤
⠀⠀⠀⠀⠀⡸⢋⠀⣻⠄⠀⠀⢧⡿⢦⣜⣻⣿⢢⣿⣿⡟⠦⠶⢛⡟⠀⣸⣿⠤⡄⠀⣏⠘⡆⠀⠿⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠈⢻⠁⠉⣰⣴⠾⡿⢶⣧⢼⣿⡟⠈⢹⡇⠀⠀⢸⡇⢦⣿⣿⣙⣁⣤⢼⣿⣤⣤⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⠾⠷⠛⠮⢧⡴⠐⢊⡝⠉⠁⢸⠇⡇⢰⣿⠠⠷⣲⡿⠋⡉⠁⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠀⠀⠀⣀⢸⢽⠆⠹⡗⢐⡔⠉⠙⣿⠛⠉⠉⠁⠉⠀⠉⠁⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⡦⢣⢨⣷⡆⣿⠀⣶⢠⣴⡟⢻⣶⡖⢤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⡟⣿⢹⣾⣿⠁⣯⠀⣿⣴⡏⠃⣸⣿⡇⢦⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢙⡏⣿⠀⡇⣹⣿⣿⠙⢿⣦⡼⠛⢣⡖⠁⣿⣗⢺⠝⣿⣿⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢌⠑⡿⢀⠧⣾⣿⡏⠢⡀⢛⢶⣖⠊⠀⢀⣿⣿⡚⠆⠹⣿⣇⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠉⢀⢀⠀⡱⠆⡘⢲⣿⡏⢳⣅⠀⠀⢀⡈⠿⠿⣴⠛⣿⣏⢙⠀⢻⡜⠆⠀⠈⢿⢦⣄⣀
⠀⠀⠀⠀⡄⢰⡚⡍⠂⠀⠀⠀⡠⠍⠀⠠⣿⣿⠁⢠⠙⠻⢦⡀⠑⠀⠈⠔⠁⣿⣿⣅⠧⠀⠹⣄⠀⠀⣿⣸⡇⢸⠁
⠀⠀⠀⠀⠑⡀⢳⡘⢆⠀⢀⡤⠀⠠⠦⣼⣿⣁⡦⠀⣁⣀⢾⣿⣷⢤⣤⢶⡂⡟⢿⣿⠴⢃⠀⠈⢦⣬⠃⣸⠇⢸
⠀⠀⠀⠀⠀⠐⣄⠙⢦⣱⡞⠀⠀⠣⣼⣿⠉⣯⡶⡿⠏⣼⣿⡿⣿⢂⣫⠿⣴⡇⠘⣿⣷⠊⡄⠀⠀⠙⣔⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠃⢄⡙⠀⡀⠁⣜⣿⠏⠀⡇⠘⠀⢐⡾⢻⡇⠿⣃⠀⠟⢽⠀⠀⢹⣿⣏⢰⡄⠀⠀⢸⡡⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣉⣾⣿⠃⠀⠀⣧⠄⡰⠏⠀⠘⠁⠀⠻⢂⡀⡾⠀⠀⠀⢻⣿⣦⠘⡄⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⠯⡽⠁⠀⠀⢀⡷⠮⢁⡐⢦⣤⠶⣶⢾⣯⡧⡗⠀⠀⠀⠀⢻⣿⣆⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠠⣈⣞⣿⠿⢄⠖⣠⠛⠡⡝⠲⢤⣈⣉⣉⣉⣛⣹⡝⠒⠞⢣⡜⣁⠖⠀⢿⣿⣗⡊⣢
⠀⠀⠀⠀⠀⢀⠐⠀⣽⡟⢃⣳⡌⠶⢠⡜⠁⢶⡤⠤⠆⣽⠉⠹⣯⠙⠛⢶⡳⠆⣤⣉⠶⠂⣼⠻⣯⣥⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠻⣾⣀⢉⡇⢸⠃⠎⣁⠶⣀⣀⠤⢿⠀⢰⠏⠑⠢⣼⣱⠛⣤⢉⡴⠐⠇⢀⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢤⠇⡸⣉⠖⢈⡰⡥⠄⠰⠟⡦⠛⠶⠤⠤⡿⢡⡛⠰⢏⣐⠀⡗⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠠⠥⠉⠰⠦⠀⠆⠠⢰⠌⠱⠏⠠⠲⠴⠗⠠⢌⡁⠮⠉⠦⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⢦⣤⡠⣤⣤⡤⣤⣤⣴⣦⣀⡤⣦⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣷⣀⣡⣼⣿⣤⣼⣿⣤⣤⣿⣯⣤⣼⣥⣬⣿⣤⣿⣧⣼⣤⣿⣤⣧⣤⣥⣽⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠁⠀⢀⣠⣾⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢶⣶⣶⣦⣤⣀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠛⠛⠉⠉⠉⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠟⠛⠛⠛⠻⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣀⣀⣀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⣠⣶⣿⣿⣿⣶⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⠿⠛⠉⠀⠈⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⢻⣿⣿⣶⣤⣤⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣷⣾⣷⣾⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣌⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣙⢛⣠⣿⣿⣿⣿⣿⣿⠿⣻⣿⣟⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣯⣶⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡏⣶⢽⣿⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣾⣷⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠻⠟⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣬⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⡛⠉⡿⠋⠉⠙⣿⣿⡿⢿⣿⡿⠟⢿⣿⣿⣿⣿⣿⡟⢡⡌⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡅⠶⣸⣿⣿⣿⣿⣾⡇⠀⠀⢠⣿⡏⠀⠀⠉⡇⠀⢘⣿⠾⣿⣿⣿⣿⣦⣴⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⠉⠙⢿⣿⣦⣤⣿⣯⣴⣶⣶⣾⣷⣶⡿⠋⠀⣠⠜⣩⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⣠⠴⠿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣀⣤⠶⠾⠷⠚⣿⣿⣿⣏⠤⢘⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠟⠋⠛⠋⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣾⡟⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠙⠋⠙⠛⠉⠋⠙⠋⠉⠛⠛⠛⠋⠉⠉⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⣀⠈⠠⡀⠀⢤⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠄⠀⠀⠀⠀⠀⠀⠠⠔⢲⡶⢉⠁⠀⡑⡴⠁⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢶⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠂⠌⠲⠶⠦⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⣀⣀⠀⢀⡞⠁⠀⣀⡤⠔⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠿⠻⠛⠉⠁⢀⣌⣉⣀⡀⠀⠀⠐⠉
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠲⠀⠀⠀⠀⠀⠀⠀⢀⠏⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠔⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠢⠤⠀⠀⠀⠐⠻⣷⣤⣀⠀⢀⡀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠤⠀⠀⠊⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠻⢦⡄⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠘⡏⢸⡁⠀⢀⣣
⠀⠀⠀⠀⠀⠀⢠⣾⡆⢀⡖⠠⠄⠀⡀⠀⠀⠁⠀⠀⣀⠘⢳⠀⠀⣄⠀⠠⠶⡴⠀⠀⢲⠂⠀⠐⡀⠨⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠃⠀⠀⠴⠀⠀⠀⠀⠀⢦⠹⠁⠈⠚⠣⠈⣃⣄⠀⠀⠀⠀⠘⠀⠀⠀⠱⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠁⠀⠀⠀⠀⡀⣏⠈⠁⠀⢀⠀⠂⠐⠍⣉⢀⠀⠀⠀⠀⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⠀⠉⠀⠂⠀⠒⠀⠀⡂⡄⠀⠙⣼⠀⠀⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡄⠠⡆⠀⠀⠼⠀⢀⠀⡀⢁⡄⠀⠘⣡⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠎⠀⠀⠒⡀⠀⡄⠀⠐⠀⠀⠀⢰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠐⠠⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⠂⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠶⠀⠀⠀⠀⠀⠀⡠⠀⠀⢨⠀⠠⠈⢀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⠂⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⠀⠀⠀⠀⠀⡈⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⣀⡀⠀⠀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠁⢠⡤⠦⠆⠨⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⠀⠀⠐⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠄⠈⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⡀⠄⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⢄
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢄⡁⠀⠐⠀⠀⢄⡀⠔⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⢀⠀⠰⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⠈⠘⠳⣄⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⣤⢁⡄⠀⠀⠀⣠⡶⠂⢠⡄⠁⠀⠩⠄⢠⡀⠀⠀⠀⠀⠲⣀⠨⣷⠀⠀⠀⢀⢄⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠐⡅⢴⠄⠀⠀⢸⡃⠄⠹⢤⡀⠢⠀⠀⠀⠀⠀⠀⠀⠂⡜⠉⠁⢸⠆⠀⠀⣒⠋⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡄⠀⠨⣇⠀⠀⢤⠟⣡⠀⠀⠀⠀⠀⠀⠀⠜⠛⠂⠀⠜⢀⣴⠾⠁⠀⠀⠀⢠⠀⡀
⠀⡃⠀⠀⠀⢀⠈⠀⠀⠀⠀⡔⠹⢃⠀⠈⠠⠤⠔⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⠴⠈⢠⡄⠁
⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⠀⢀⠠⠔⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠭⠐⠀⠋⠒⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠉⠈⢉⡀⠀⢀⠁⠀⠀⠄⠠⠀⡄⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠄⢀⠀⠀⠐
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠠⠢⠀⠊⡠⠂⠀⠀⢁⠈⠀⠀⠴⠀⠀⠁
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠰⠉⠀⠀⠀⠄⠀⠐⠀⠀⠈
⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠃⠻⡄⠀⡀⠀⠀⠀⣴⢿
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⡘⠀⢷⠊⠀⠀⠀⠈⢇⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠙⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡷⠈⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠈⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡺⡛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠀⠈⠀⡸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⢀⡇⠀⢙⠈⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⢄⡇
⠀⠀⠀⠀⢠⠐⠁⠀⠀⠀⠀⠀⠀⡏⠐⠈⡆⠀⠀⢀⣴⣯⣦⣿⠋⠉⠍⠉⠉⡉⠙⠒⣦⡀⢀⡼⠀⠀⠀⠀⠀⣰⣸⡁⠀⠉⠀⠀⠀⠂⣀
⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢁⠀⠜⠉⠀⣿⠈⠁⠀⠀⠠⠄⠀⠀⠴⢿⠻⣟⢾⡃⠀⠀⠀⠀⢰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠡
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢾⠀⠀⠀⢹⠈⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠘⡆⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢻⣆⡀⠀⠈⠌⠀⠘⠔⠀⠀⠂⠀⠀⠀⠀⠀⠘⠀⢂⡠⠂⠀⠀⠀⠀⠠⠤⠤⠤⢤⣀⣀⣀⡠
⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠈⠉⠋⠓⠋⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⢠⠞⠀⠀⠠⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⣀⣐⣀⣀⡠⣤⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠢⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠐⡀⢀⡀⡀⠀⠒⠒⠁⠉⠉⠑⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⠉⠓⢲⡶⠶⠦⠄⠾⠒⠄⠒⠿⡂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠀⢐⠒⡒⠀⠉⠂⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠒⠒⠒⠚⠉⠋⠉⠀⠀⠀⠙⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠲⢆⠄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠳⢬⣤⣄⢀⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣄⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡆⠰⠀⣤⢜⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣹⡄⢲⡟⣓⢂⣹⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡮⣉⢈⣣⠽⠅⠤⠀⠹⢦⡒⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠳⠦⠤⢤⣝⣋⣈⣉⣋⣛⣺⡁⣈⣤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡟⣫⠶⣒⢢⣽⣥⣧⣿⣤⣧⣻⢭⡤⠤⢚⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣶⣿⡿⡛⠯⠉⢉⠩⠝⡻⣿⣟⣻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⠐⠈⢻⣤⠁⣀⡾⠛⢧⡃⢸⣌⣛⠫⡥⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠿⣿⠆⠀⣾⠝⠗⠛⠃⠀⠚⠛⢺⢟⡧⢀⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⣻⣤⠟⠀⠆⠀⠀⠀⢡⣿⡄⠀⣀⣐⣂⠀⣾⣯⠁⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⡠⠴⠊⡷
⠀⠀⠀⠀⠀⠀⠀⣠⢀⡠⢋⢀⡴⠛⠉⠀⠀⠀⠈⡶⢀⡠⢠⡧⢫⣀⠜⠙⢁⣠⠋⣍⠣⡀⢶⠎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢶⣦
⠀⠀⠀⠀⠀⠀⠀⡟⢇⣠⣚⠏⠀⠀⠀⠀⠀⠀⠀⣡⢀⠰⠁⣠⡎⠙⠦⣠⠞⠙⢦⡈⢦⣀⣴⠀⠀⠀⠀⠀⠀⠀⡠⢫⠇⠀⢸⡚
⠀⠀⠀⠀⠀⠀⠀⣧⡼⠟⠁⡄⠀⠀⠀⠀⠀⣀⠨⢏⣛⡟⠁⢨⡀⠀⠀⠀⠀⠀⢴⠙⠖⢷⣼⠀⠀⠀⠀⠀⠀⢰⠀⠸⡀⠀⢸⡁
⠀⠀⠀⠀⠀⣠⠞⢻⣦⠤⡎⠀⠀⠀⠀⢠⠎⠁⠀⣈⣾⣏⠀⠈⢣⣀⠀⠀⠀⢀⠎⣠⢈⡿⠄⠉⢦⠀⠀⠀⠐⣍⠁⢀⣇⠠⠴⠃
⠀⠀⢀⣴⠟⠁⠀⠈⠀⠀⢀⠀⠀⠀⢠⠃⠀⠀⠀⡤⢹⣫⠄⠀⠀⠈⠻⡿⠛⠁⢀⡄⢸⠗⠆⠀⠘⡇⠀⠀⡜⠀⠀⢻⠁
⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠸⠀⠀⠀⡎⠀⠀⠀⠀⡶⢈⣏⠄⠀⠀⠀⠀⠀⠀⡐⠁⠆⣹⣯⠂⠀⠀⣇⡠⠊⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢃⠀⡞⠀⠀⠀⠀⡰⢛⠠⡧⠕⠒⠒⠒⠊⢀⠼⠄⢴⠠⣼⡏⠀⠀⠀⢹⠁⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢈⡄⠑⠀⠀⠀⢴⠁⢨⡄⢳⡣⠀⠀⠀⡰⠃⠀⠀⠈⣐⣟⢻⠀⠀⠀⣈⡇⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡜⠀⠠⠤⣀⡝⠁⠀⠈⡖⢩⡑⡀⢀⠔⠁⠀⠀⠀⡆⣬⣻⠁⣗⠌⠓⠈⢙⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⣼⠀⠀⠀⠀⠀⢳⡀⠀⢠⠋⢒⣚⣥⠀⠀⠀⠀⠀⠐⠡⡘⣇⠀⢨⡄⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠰⠋⠙⢆⡆⠀⢌⢯⣆⠂⡀⠴⠀⠠⢠⣖⠇⠙⡄⠀⢣⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢥⣦⣀⠈⢏⢵⣄⡤⠘⡃⠞⠰⠧⠿⢊⠋⠀⣠⢇⠀⠈⣧⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠙⣱⢞⣒⡋⠈⠂⠠⢀⠁⠀⢁⠠⡔⠉⠀⠈⢳⡄⠘⢆⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠼⠁⠀⠄⠀⢺⠔⢫⡀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠈⠀⠀⠛⠦⣈⢣⡀⢠⠴⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⢀⣀⣀⣤⣠⣄⣠⢆⣸⡅⣀⣤⠀⠦⠄⠉⠉⠲⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡀⢨⣉⢳⡄
⠀⠀⠀⢀⠔⣊⠔⠋⣀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠈⣥⠀⠀⠀⠂⡆⠀⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⢀⡝⠢⡽⠋⣅
⠀⢀⡔⠁⡜⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢀⡀⠬⠧⠳⠖⠶⢤⣄⣀⡀⠀⠈⢻⣿⡉⠉⠉⠁⢾⠀⠀⠘⡷⡟⠑⢤⡀
⠀⡞⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠒⢉⡀⢀⡠⠤⠤⠤⢼⣂⠀⠀⠈⠑⠒⠒⠀⠓⠀⠀⠀⠈⠳⠶⠒⠁⠘⡄⠀⠑⡄
⠀⣧⠀⠀⠙⠤⢤⣀⣀⡠⠴⠒⠒⠒⠋⠻⠥⠴⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡄
⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠤⣤⣄⣀⡀⠀⠀⣀⠼⠁⠀⠀⢸⠃
⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠠⣪⠎
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢤⣀⣀⠂⠀⠠⠤⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣠⣴⣊⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠀⠤⠤⠤⠤⠤⠄⠀⠀⣀⣈⣉⡁⠀⠬⠬⠤⠤⠤⠤⠒⠒⠒⠊⠉⢙⡏⢩⣷⠦⢮⣤⣥

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⣴⣾⣶⡄⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠀⠀⠈⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⣦⣄⣠⣴⣿⣿⣿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣶⣤⣀⣴⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣶⣤⣄⣀⣀⢀⡀⠀⢀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠗⠿⣿⡯⠽⠊⠽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⡛⠛⠳⢦⡴⠒⠒⠒⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣟⣿⣛⡿⠿⠙⣿⡟⠻⡉⠙⠛⠒⠶⠷⠶⢶⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⡈⢹⡟⠛⠳⣦⠀⠀⣻⣾⡛⠛⠶⢶⣤⡤⠶⢛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⣄⠙⠷⣤⣿⡀⠀⢸⣧⣴⠟⠈⠙⡷⠶⠦⣄⠳⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢹⣷⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⢠⡟⠳⢦⡀⠉⡟⠓⢾⡁⠘⣄⡀⠀⡷⠤⣀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⢿⡻⠛⢃⣼⠃⢼⠀⠀⠀⠘⣦⠇⠀⡟⢳⡀⠀⠈⠐⣇⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠂⠀⢣⣠⠏⠹⡆⢸⢦⡀⠀⠀⠘⡳⠮⢵⣄⡳⢤⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡴⠞⠋⠀⠀⢻⡼⠀⠁⠀⠀⠘⠁⠀⣀⣱⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣏⣧⣄⡀⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠛⢻⣿⣿⡟⠛⠿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢀⣴⣿⣿⡿⠋⢸⣿⣿⡇⠙⢿⣿⣿⣦⡀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏
⣴⣦⣄⢇⣰⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣴⣿⣿⡿⠋⠀⠀⢸⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣆⡸⣠⣴
⠙⠻⠿⢿⡿⠛⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡋⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠿⠟
⢠⣤⣤⣀⢣⢠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⢸⣿⣿⡇⣀⣀⣤⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡄⡜⣀⣤⣤
⠘⠻⠿⠿⠿⢿⠛⠉⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠉⠛⡿⠿⠿⠿⠟
⠀⢠⣤⣤⣄⣈⢧⢰⣿⣿⡿⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣿⣿⡆⡼⣁⣤⣤⣤⡄
⠀⠹⠿⣿⣿⣿⡷⠿⡟⠉⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠉⢻⠿⢾⣿⣿⣿⠿⠏
⠀⠀⠀⠀⣠⣤⣄⣀⡘⢦⡸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢇⡴⢃⣀⣠⣤⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡿⢿⡋⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠙⠺⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢺⣷⣀⠔⠋⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⠛⢻⣷⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⣠⣤⣄⠀⣿⣭⡿⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠲⣶⣾⣋⣠⣿⣧⣿⠇⢰⣿⠛⣿⠇⠰⣿⡀⠋⣀⣿⠿⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠉⠉⠉⠻⠷⣦⡶⠿⣷⣾⠿⣧⣤⡿⠛⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣦⣠⣿⣀⣼⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠱⣼⣷⣿⣟⣋⣙⣻⣿⣵⢎⣀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣧⠦⠀⢬⣿⣿⠛⠿⣏⣿⣿⠟⢙⣿⣿⣅⡀⠐⢶⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⠀⠚⠋⠙⠂⠀⠠⠴⢟⣿⣿⣶⣖⣲⣶⣿⣿⡍⠁⠀⠀⠛⠈⠋⠀⢤⣄⡖
⠀⠀⠀⠀⠀⠀⠀⠘⣻⡿⠥⠀⠀⠀⠀⡀⠀⢄⠀⠾⡋⣧⡿⣿⣿⡟⢿⠀⠙⠀⡠⠀⠀⠀⠀⠀⠴⠟⣿⠉
⠀⠀⠀⠀⣄⢀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⡵⢁⣉⣈⠃⠀⠔⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⣦⣠⠆
⠀⠀⠀⠠⠾⣿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⢢⠐⠀⢴⣿⠿⣿⡆⣠⠐⡡⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⡟⠓
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡘⣠⣾⣿⣶⣿⣶⣿⠮⠂⠐⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣠⠄
⠀⠀⠀⠀⢨⡽⣷⠒⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⣠⣿⣯⣿⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠳⠆
⠀⠀⠀⠀⠀⠀⠛⠀⢀⡀⠀⡀⢠⠖⠒⠦⠴⣾⢿⣧⣙⣛⣛⣿⣩⣿⣿⣿⠿⠻⠟⠂⡰⣲⣦⠀⣤⣀⡴⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣽⣿⣁⠀⠛⠶⠆⢧⣤⣤⡿⣋⣩⢹⣿⡍⣼⣡⠰⡄⠤⢲⡭⠗⠛⠋⠐⠾⢿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⢻⣭⣿⠛⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⡴⢣⠀⠀⢀⡤⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠢⡑⢄⢰⠁⠀⢳⣶⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠑⢄⠀⠀⠳⠦⣀⢿⠁⠀⠙⢄⢻⠀⠀⢸⠃⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢈⣧⠀⡄⠀⠈⢻⠀⠀⠀⠈⢾⠀⠀⡎⠀⠀⠀⠀⠸⡀⠀⠀⢀⡠⠖⡩⠊⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠈⠑⠳⠤⣀⢸⠀⠀⠀⠀⠀⢣⠀⡇⠀⠀⠀⠀⠀⡇⢀⠴⠋⢀⠞⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢧⡇⠀⢀⡠⠔⠚⠉⠉⢱⡤⠋⠀⡼⠁
⠀⠀⠀⠀⠀⠰⡤⢤⣤⡤⢄⣀⡀⠘⣇⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠘⣧⠔⠉⠀⠀⠀⠀⠀⡼⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠲⢄⡈⠑⠻⣆⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⣃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠢⢄⡈⢢⡀⠀⠀⠀⠀⠀⢸⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⡀⠈⠑⠳⢄⡀⠀⠀⠀⠀⢧⣠⠇⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠛⢒⡤⠤⣤⡴⡟⡤⠤⢤⡤⠒⠻⠤⣀⣀⣀⣀⣀⣀⣤⢴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠲⠶⢖⣒⣒⣒⣒⣉⣉⠥⠔⣪⠋⠀⡇⡇⠀⠀⠙⢗⡒⠠⠤⠄⠒⠒⠋⢉⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⡇⡇⠀⠀⠀⠀⠉⠓⠒⠂⠐⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢴⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡿⡟⢿⡄⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣄⡨⣊⣙⣤⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡟⣾⣯⢶⣿⣿⣿⣷⣿⣿⣦⡀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣴⣿⣻⣿⣾⡯⣸⣬⣇⣷⡟⢲⢽⣿⣷⣿⣦⡀⠀⠀⣤
⠀⠀⠀⠀⠈⣠⠚⣳⠄⠃⠀⢠⣿⣉⣿⣍⣿⣞⠛⠹⢯⠙⠓⢼⡟⠙⢏⢉⡻⠇⠰⣺⡆⢳⡆
⠀⠀⠀⠀⠀⠟⢠⠏⠁⠀⠠⣥⠈⢄⡈⠉⠿⡷⠾⢂⡶⠶⠠⣿⢃⠉⢀⡦⠁⣸⠄⠈⠲⣈⡃
⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⢹⡀⠘⡀⣰⡞⠻⡄⢸⡗⢀⡾⠿⣾⡀⢘⡁⡈⠃⠀⠀⠀⠘⢡⢤⣄⣠⠄
⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣀⢖⣻⡴⣡⣝⠣⠜⢯⣈⣀⣽⣧⡤⢷⣜⢡⡈⠱⢄⡀⠀⠀⠀⠘⢺⣿
⠒⠮⣀⡄⢨⡀⠀⠀⠀⢴⣷⠋⡜⢲⣿⠇⣀⣼⡆⠠⠄⣠⣿⢳⢰⣾⢂⡁⠈⢌⣣⡀⠀⢀⣤⠀⣫⠓
⠀⠠⠋⣄⣨⢷⠀⠀⠘⣬⠿⠤⣍⢽⣿⣠⠻⣽⡙⢓⠚⢉⡏⢘⡸⣿⡢⠖⠀⢹⣐⡹⠀⠘⠙⠺⡏
⠀⠀⠀⠙⠄⠀⢈⡶⠾⠛⡛⢿⡧⢺⣿⠈⠀⢺⡄⣤⠈⢸⡇⢠⢿⣿⡍⣥⡤⠴⣿⣧⠶⢤⡀⣰
⠀⠀⠀⠀⠀⠀⣿⠂⢀⠴⠛⠚⠛⠋⣷⡆⠐⢋⡍⠀⠀⢸⢁⡎⢰⡿⠖⠷⢺⣿⠞⠉⠁⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠀⣀⠴⡝⠎⠹⡇⢀⠚⠨⠉⢿⠙⠁⠀⠀⠀⠀⠀⠉⢱⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢹⣿⣿⡴⣠⢤⣶⠂⣇⠀⣦⠀⣴⡟⠛⣷⡆⢰⡤⣤⣤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣌⣿⠙⣟⣱⣾⣿⠀⣯⠀⢿⣠⡏⠑⠀⣿⡗⠺⣠⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡎⢢⡉⣿⠈⢅⣹⣿⡏⠻⢾⡤⠼⠛⢃⠔⠁⣿⣿⠒⡅⢻⣿⡏⠿⣄
⠀⠀⠀⠀⠀⠀⢠⠞⢀⠂⡘⡇⠰⠤⣾⣿⡧⠀⠀⡙⡶⣄⡀⠀⠀⢹⣿⡒⠃⠘⣿⣧⣀⠈⠢⡀
⠀⠀⠀⢀⣀⠐⠁⡀⠀⠐⡐⢀⠷⣲⣿⡏⢦⡀⠀⢀⢄⠀⠘⠷⣤⠚⣿⣿⠛⠀⠸⡏⠃⠀⠀⠹⡷⣤⣀
⢀⠀⡄⢮⠑⠀⠀⠀⠀⠐⠀⠠⠤⣿⡟⠁⠀⠙⢶⣄⠀⠀⠀⠈⠀⠑⢼⣿⣏⡹⠀⠘⢆⠀⠀⢨⠓⣿
⠘⡀⠘⠀⢣⡀⠀⠀⠀⠀⠠⢀⣾⣿⠑⡀⠀⠀⣀⡼⣿⣷⣀⣌⠀⠀⣹⢿⣿⡄⠷⠀⠀⠳⣀⡎⠐⣿
⠀⠀⢄⠈⢢⡙⢴⠋⠀⠐⢆⣾⣯⠻⢈⡴⠖⠏⡙⣶⣿⣿⠆⢉⢶⣤⡎⠈⢿⣷⡐⢃⠀⠀⠘⢅⢸⡇⢀
⠀⠀⠀⠳⢄⠘⠏⠀⠐⢠⣚⣿⠃⢸⠅⠐⠈⠠⣽⠻⡿⢹⡎⠀⢒⣩⡇⠀⠘⣿⣿⠉⡆⠀⠀⠈⡿⠂⡞
⠀⠀⠀⠀⠀⠘⢀⠈⢳⣟⣾⠃⠀⠘⡥⠂⣠⠎⡈⡀⡇⠠⠹⠄⠀⢸⠁⠀⠀⢸⣽⣯⡨⢀⠀⠀⢘⠜
⠀⠀⠀⠀⠀⠀⢂⣼⡙⠹⠃⠀⠀⠀⣧⡔⠁⠠⣄⣁⣧⣤⣴⡞⣭⡾⠀⠀⠀⠀⠻⣾⣆⠈⣄
⠀⠀⠀⠀⠀⢠⡼⢾⡎⢠⡄⠀⠶⢈⡣⢤⡉⠐⠚⠃⠲⠚⢻⣟⠥⣥⢆⣀⠒⢠⠀⢽⣿⣧⠠⠃⡀
⠀⠀⠠⢀⢰⣿⡟⠡⣄⢣⡤⢉⠲⢈⣀⣀⣈⡍⣺⠚⢻⡟⠛⠷⣄⠦⣄⣉⠒⢢⡄⢠⠿⣟⣓⡂⣡
⠀⠀⠁⠂⢼⣯⣄⠁⣺⠀⡶⢍⡰⠂⣭⠀⠀⢀⣿⢠⢸⡛⠤⣄⣸⠇⡤⢉⡰⢢⡄⡞⢀⡙⢋⡉⠁
⠀⠀⠀⠀⠀⢼⢃⠀⢽⠀⡑⠂⣄⠓⢴⠀⠈⢁⡿⣌⡸⢁⠀⣀⣹⠓⠢⢎⡰⢫⡄⢐⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠈⠁⣀⡇⣈⡁⠂⢤⠛⢀⡉⠀⢨⡠⢉⡶⢈⠀⠀⣏⠘⠑⠎⣀⠛⢆⠈⣤⠀⠋
⠀⠀⠀⠀⠀⠀⢀⣜⡁⠉⠉⠂⠀⢀⢀⠀⢀⢨⡀⠈⢀⣄⢈⣈⣀⣀⣉⣳⣌⠀⣈⡉⣀
⠀⠀⠀⢀⡀⣐⣉⠐⢁⣎⣰⣊⣿⣋⣈⣛⣳⡞⢹⡆⣴⠉⣿⠉⣿⢉⣿⠝⠯⢙⡏⠛⠙⠓⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⡀⠤⢀⣐⠀⢀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣋⠑⡟⢉⣛⠧⠤⠍⠰⣼⣿⠆⡔⣻⠁⡫⢄⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠈⡰⢷⣹⢆⡊⠀⠀⠀⠘⠁⠀⠚⡴⢷⡎⢟⠲⣷⢀⡄⠲⠫⠄⢄⡀
⠀⠀⠀⠀⠀⣠⣖⢽⡆⢭⣤⡁⢉⡐⠀⠊⠈⠀⠀⡀⠀⠀⠈⠀⠀⠒⠚⠂⡦⡄⠗⠀⢈⡡⣀
⠀⠀⠀⢀⠀⣽⣟⣏⡱⣇⠏⠁⠀⠀⠀⠀⠀⠸⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠍⣀
⠀⠀⠀⠎⠀⣴⡭⡩⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⢂⡰⠀⠀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠
⢀⡯⠦⠱⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⢀⣀⣍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠋⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡀⢣⠄⢠
⠀⠀⠂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠈
⠀⡀⠀⠥⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⡄⠠
⠀⢋⠁⠄⡜⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠰⠁⠀⠀⠳⠂⠀⠠⠘⢃⠀⠄⠀⠀⡄⠀⠀⠀⠠
⠀⠈⢧⣀⠀⠉⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⣾⠃⠀⠀⠀⠟⠠
⠀⠀⠀⠂⠀⢡⣡⢀⣠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠋⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠟⡇⠆⣸
⠀⠀⠀⠀⠄⠀⢻⠟⡚⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢜⢰⡀⠀⠀⠀⠀⠀⠠⠂⠂⠄⠳⢁⣠⡧⠄
⠀⠀⠀⠀⠈⠀⠨⣀⡀⠀⠀⠀⠀⠉⠀⠠⠄⠀⠀⠤⠷⠇⡠⠀⠀⠈⠀⢀⠈⠘⢃⡑⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠐⠍⡀⡀⠘⡂⠂⠁⠀⡴⠆⠀⠀⠂⠀⠐⠀⠈⡀⠹⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢈⣀⠸⠋⢣⠠⠀⡕⠀⠀⠓⠀⣩⠇⠈⠁⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠲⣀⣉⢀⣤⢀⠂⠃⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⣆⠰⠀⠀⢰⠀⢸⠀⠀⠄⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠈⠂⠘⢆⠁⠀⠈⠀⠸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⡀⠐⢄⠀⠀⠀⠈⠀⠀⠀⡀⣀⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢄⠀⠀⠀⣀⣴⠲⡇⢹⣇⣭⣨⡟⣳⣤⡀⠀⠀⡀⠖⠁⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠁⢀⣞⢣⡤⠀⣤⢣⣼⠲⣤⣌⠁⢤⠓⣄⠀⠀⠀⠀⠠⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⣯⣉⢏⣲⣦⡿⠟⠋⠈⠛⢯⣣⡀⡸⠛⣦⠀⠀⠀⠀⠀⢂⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠲⠂⠠⠀⠀⢸⠶⠟⢸⣿⡏⠰⣾⢻⣾⣧⠀⣏⢹⡀⡷⢻⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣢⠰⣾⣇⢠⡽⣿⡟⠟⢀⣽⣻⠗⢮⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⣏⡰⢦⠰⣞⣷⣤⣿⣻⡶⡟⣽⢛⣤⢳⡞⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⢧⠞⠛⢹⠛⡶⣿⡿⣿⢾⢡⡖⣮⠟⠀⠀⠀⠀⠀⠙⠳⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠉⠀⠤⠂⠀⠐⠀⠀⠀⠙⠻⣼⣁⣽⣿⣏⣮⣼⡴⠛⠁⠀⠀⠀⠐⠠⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⠀⠀⠀⠀⠉⣿⣏⡏⠀⠁⠀⠀⠀⠀⠁⠀⠐⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠞⠀⠀⡄⠀⢹⣿⣷⠤⠘⡄⠀⠱⡀⢆⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠆⠺⠿⣷⣄⠂⢡⡄⠀⠡⠀⠁⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⠀⠈⣦⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⣿⣿⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣦⣴⣿⡋⢹⢿⣿⣷⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣇⠹⣿⠀⣈⣻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣥⢻⡄⣽⠎⡛⢣⣻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣄⣹⠊⣱⠿⠀⠇⠈⠋⢻⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢾⠛⠸⣿⡟⠓⠀⣀⠃⠆⣹⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣶⣷⣶⣾⣷⣶⣶⣿⣷⣶⣶⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠶⠂⠉⠈⢉⠍⢳⡈⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⡔⠉⠳⡀⠠⠆⣄⠀⢐⠹⡉⠉⠉⠉⠈⠁⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠚⠄⢀⠐⠁⢀⡄⠸⠂⠈⢠⣅⣀⠀⣀⣀⣀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢸⠄⠀⢨⠃⠄⢠⠔⠈⠐⠺⡅⢇⠀⠀⠀⠀⢀⣉⣀⣀⣈⣣⡀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢿⠀⠀⠃⠀⢆⠙⠀⠀⠀⠀⡟⠊⠀⠈⠈⡉⠀⠉⠉⠋⢉⠜⠑⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠶⠄⠠⣒⣬⠕⠓⠤⠤⠊⠀⠈⠀⠈⠁⠉⠀⠑⢀⢀⠂⠀⠀⢰⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⠖⠉⢦⠀⠀⠀⠑⠀⢀⣳⣀⣠⡀⠈⠠⠄⠸⠏⡄⠀⠀⠀⢇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣤⡀⠓⠢⡀⠠⣄⠀⠀⠈⡍⠁⠀⠒⠒⣼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠢⠤⢴⠄⠄⠙⢆⠀⠀⢠⠁⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡠⠍⠄⡈⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠝⢠⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⣀⠀⠀⠀⠀⠀⠘⡞⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⠌⠓⢜⡆⠀⠀⠀⠀⠀⠇⠀⠀⠆⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⡈⠞⠁⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢌⡯⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⢀⡀⠀⠀⠀⣀⣠⠴⠬⠛⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣦⠀⠀⠀⠀⠀⠸⣶⣤⢁⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣀⠘⡈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣀⣙⣛⠳⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⣴⠟⠉⣉⠙⠻⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠉⠀⢀⣴⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⣶⣶⣶⣶⣶⡶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠐⠷⠶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⣼⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠲⠄⡈⠓⢄⠀⠀⡀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣇⠀⠈⠢⣠⠇⠀⠀⠟⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠀⡤⣑⢦⣸⣷⣦⣀⣀⣦⠎⢻⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠚⠇⠘⠛⢬⣯⠈⢿⣿⠇⠘⢿⡟⠁⠀⢠⢀⣤⠌⠀⠀⠀⠀⠀⠀⢠⣀⠀⠴⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠘⠀⠀⣹⠀⠀⠘⠀⠀⢠⣾⠟⠃⠀⠀⠀⠀⠀⢀⡖⢸⢃⠀⠀⠴⠛
⠀⠀⠀⣀⣀⣀⢡⡀⠈⠀⠁⠀⠠⣧⣃⡟⠀⢀⣄⣠⡶⠏⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⠏⠰⠀⠐⣯
⠀⠠⠟⡉⠉⠉⠉⠉⠓⠐⠢⠤⣤⣤⣼⠷⠶⣿⠷⠤⠤⡀⠀⠠⠴⠂⠀⠀⠀⠀⣻⣴⢣⡆⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠂⠄⢂⡤⢦⡴⠎⠁⠀⠀⠐⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢾⣷⣶⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⣀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⢨⠋⠀⠀⠀⡀⠂⠀⠀⠀⠀⠄⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣴⠃⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠚⠀⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣴⠀⠀⠀⠃⠀⠀⠀⢀⠀⢢⠀⡀⠂⢀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠇⠀⠀⠀⠀⠀⠀⠀⠐⠩⡶⢿⣷⣱⣶⣴⠦⠀⠀⠚⠂⠀⠀⠀⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⡄⠂⠀⠀⠀⠘⠁⠶⠺⢧⠩⡭⣝⠀⠠⠀⠐⠖⠶⠤⢄⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠃⠃⠀⠐⢤⠀⠀⢀⣒⠐⠂⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠉⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⠀⠀⠻⣟⠣⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⠄⠀⠀⠀⠈⠙⠲⢴⣶⣆⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢁⠀⠀⠰⢂⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⠉⠉⠋⠚⠛⠲⠤⠬⠦⠤⠵⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠠⡇⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡘⠃⠀⠀⠀⣰⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣁⣀⣀⣀⣀⣠⣆⣀⣀⣀⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡥⢤⡤⣨⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡲⢗⡶⣖⣲⡽⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢰⡔⣷⠄⠐⡁⠐⢶⣿⣿⠃⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠂⠀⢤⠀⠀⠈⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⢀⠈⢄⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠆⢀⡥⠘⠉⠀⡀⠀⢰⡉⠁⠛⠉⣹⠂⠀⠀⠀⠈⠀⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠁⠀⢸⠀⠉⡿⠉⠁⠐⠁⠀⠀⠀⠀⠀⠐⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣤⠴⠆⠀⠘⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠑⠀⠀⡆⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠘⠜⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣰⠀⠀⠒⠀⠀⠀⢸⣈⣹⠄⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠰⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣦⣤⠴⠟⣡⡼⢡⡆⠀⠀⠂⠀⠀⢀⠀⠀⠖⠚⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠶⠚⣡⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠇⠀⡀
⠀⠀⠀⢀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠛⠛⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⡠⠂⠀⠠⠼⠇⠀⠠⠯⠀⠀⠀⠈⠀⠈⠙⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠴⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠦⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣒⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡔⣲⣏⡭⣭⠭⣉⢷⡲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢻⣭⡷⠾⣷⠟⢶⠷⠴⣯⣽⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠟⣋⣤⠶⡿⠁⠀⠈⠻⡖⢦⣍⠛⢧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣹⣿⠋⠁⡼⠁⡠⣔⣄⠀⢳⠀⠈⢹⣏⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣴⡿⠁⠀⠀⡇⠀⢿⣯⣿⠁⠈⡇⠀⠀⠙⣧⠞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢳⡟⠁⠀⠀⠀⢷⣀⠀⠁⠀⢀⣴⠃⠀⠀⠀⢹⡗⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣻⡁⠀⠀⣀⣠⠠⣝⡷⠶⠿⣿⣁⣀⣀⠀⠀⠈⣿⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⣿⡇⡴⢋⣡⣴⣖⣖⣚⠈⣛⣖⣖⣦⣤⣉⠳⠄⣿⣥⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⢿⣇⣼⠟⠋⠉⠉⠉⢲⠋⣷⠋⠉⠉⠙⠛⢷⣼⣿⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠻⣇⠚⢿⠡⢒⡛⠶⢦⣄⡜⠀⠘⣄⣤⠶⢯⡑⠢⢹⠟⣶⡿⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⢧⣠⠀⣽⡓⠒⣆⢩⡧⢤⠴⣿⢣⡖⢙⣮⡀⢀⢰⠏⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠤⠀⠀⠈⢻⡄⠣⠍⠩⠉⣸⠃⣐⣀⢻⡌⠍⠉⠵⠃⡟⠁⠀⠀⠀⠤⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠄⢂⡴⢸⡇⠀⠀⠀⢀⠇⠠⣴⢤⡀⡇⠀⠀⠀⢀⣧⠰⣌⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢀⢴⣯⡤⠞⢷⠀⠀⡀⣸⠀⣀⣘⡀⠀⡗⢀⠀⠀⣼⠛⢦⣼⣧⣀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⣧⣸⡟⠀⠀⠘⣷⡼⣍⡟⠀⠩⠩⠍⢁⡟⣩⠧⣴⠇⠀⠀⠹⣇⣨⣦⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⣿⡏⠀⠀⠀⠀⠯⣤⣼⡇⠈⣁⣀⡁⣼⢧⢧⣀⣈⡆⠀⠀⠀⠸⣿⡟⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⡿⠀⠀⠀⠀⠀⠀⠀⠙⡧⠀⠠⢬⠀⣇⣴⠟⣠⢞⡉⠙⠓⣦⠀⢻⣅⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠔⠲⢒⣟⠁⢠⡏⡐⠀⠴⡀⠼⢦⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠐⠊⡹⢻⢶⣾⠒⠄⢰⠀⡟⢾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⠀⠀⣠⠎⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⡇⢘⡑⢸⡖⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠸⠆⠠⣲⠅⠨⣗⡌⡰⠇⢰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⠀⢄⡘⠀⠉⠷⣀⠴⠋⢠⠊⠑⡄⠙⠿⣄⠺⠋⠀⢡⡠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡱⠧⠀⢛⠁⣈⢋⢀⠈⡐⠁⢀⡄⠀⠑⡚⢀⡠⡀⠈⠣⠀⢀⢘⡁⠀⣛⠀⠴⠏⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣉⡅⡔⡪⠕⠚⠉⠉⠁⠀⠤⠐⡉⠀⠠⠀⠒⠒⠀⠄⠀⢈⠂⠤⡀⠀⠉⠉⠓⠪⢕⢲⠈⣉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠉⠁⢿⠇⡆⠈⡷⠀⠊⢀⠀⠈⡡⢄⣴⣰⣎⣑⣆⣣⡠⢌⠉⠀⡀⠑⠀⢴⠁⢀⠀⣎⠀⠋⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⣼⠀⠈⠉⠑⣐⠂⣀⣜⣲⡧⠴⠋⠀⠉⠉⠀⠙⠷⠼⣗⣁⣀⠐⣀⠪⡉⠁⠀⣿⠀⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⢦⢨⠀⢀⠋⢠⢀⢨⣀⡼⠁⠀⢀⠀⣤⠈⠉⣤⠂⡀⠀⠈⢷⣀⡅⡀⠐⠈⠀⠀⡟⠰⢎⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡈⠈⠀⢈⢠⡀⣈⣶⡏⠀⠠⢁⢜⡿⣟⠿⠿⣛⢿⡏⡈⠆⠀⠘⣶⣁⢀⡄⠰⠀⠁⢇⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣡⠏⠀⠁⠈⠈⢢⣼⠃⠀⡚⣻⣟⠏⢁⣴⢦⡌⠹⣿⣿⠓⠄⠘⣧⡴⠁⠉⠈⠂⠙⣆⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚⠃⣰⠇⠀⠸⠀⠀⠘⣩⣇⠀⢠⠉⢻⣿⡃⢺⣄⣠⡟⢨⣿⡟⠛⡄⠀⣸⣍⠃⠐⠀⠇⠀⠘⣦⡘⠓⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠳⠆⡮⠏⡠⠒⢦⠠⢲⠐⢇⣸⠀⠀⠒⡿⣿⣵⣦⡬⢥⣴⣮⣿⠿⠒⠁⠀⣯⡠⠂⢐⠠⣴⠒⢂⠹⠴⠐⠞⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡦⡘⢄⢦⡁⠒⠊⢰⠀⠠⠘⠰⡷⣤⠀⠐⠛⠉⠙⡿⢾⡻⠉⠙⠂⠀⣤⢾⠇⠓⠤⠀⡄⠙⠒⢊⡠⡡⢃⠴⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠾⠂⠑⠻⢦⣀⠈⠁⡀⠠⡈⠤⢼⢦⣀⠀⠁⠀⠑⠈⠀⠈⠀⣀⡴⡧⠤⢣⠀⢀⠌⠁⣀⡴⠯⠊⠔⢷⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢁⣠⡌⡓⠂⠈⢆⠀⠀⠈⠘⢍⣹⠶⣶⠖⠲⣦⠶⣏⡩⠓⠊⠄⠀⡰⠁⠐⢚⢩⣄⡈⠛⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠈⡀⢇⠀⠑⠤⠀⠖⠆⠀⡉⠁⢑⡊⠈⢉⠀⠠⠲⠀⠤⠂⠀⣰⢠⠀⠀⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡹⠃⠘⣆⠀⠠⠒⢮⠀⢤⠀⠀⠠⢬⠀⠀⡤⠀⡱⠂⠄⠀⣰⠃⠈⢫⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠂⢪⢧⠀⠄⠊⠀⢀⣀⡤⠂⠐⠤⣀⡀⠀⠑⠠⠀⡰⡑⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⢆⠳⠥⠽⠛⢋⡉⠠⣶⠉⠁⣲⠄⢉⡙⠛⠯⠬⠜⡰⢶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⢀⠿⣁⡸⠃⠘⠀⠓⠚⠁⠃⠘⠣⣈⠿⡀⡒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⣀⣤⣤⣌⣤⣤⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣤⣶⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠈⠀⠈⠄⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⡿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⢀⣉⣉⣩⣙⠉⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠁⣴⠟⠋⠉⣙⣿⣷⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡐⠀⢀⣳⢰⡿⠿⣿⣿⣽⣧⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⡄⠙⠋⡀⣹⣄⣿⡟⠻⣄⢹⠷⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣁⠀⠀⠀⠀⠀⠀⠠⣷⡴⠃⠀⢙⣷⣆⣠⡥⠽⣻⣿⣧⡑⢚⣿⣦⣬⡿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠉⢀⣀⣥⢈⣻⣿⠛⠓⠚⠋⣩⣿⡇⢻⣏⡉⠛⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠺⢷⣄⣀⠀⠀⠀⣀⡴⠟⠉⢠⣿⣿⠄⢻⡷⣾⣿⡿⢋⠆⣸⣿⢿⣀⣰⠎⠡⢃⣀⠃⠀⠀⠀⠀⠀⣾⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠸⣄⣠⡟⠻⣿⣆⠀⠀⠘⢣⢀⣾⣯⣹⣷⠘⣷⣼⢯⣿⡿⢠⣟⣻⣿⣿⣿⣿⠓⠈⠻⣆⡀⠀⠀⠀⢠⡿⠁⢹
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠛⢏⡠⠀⣿⡟⣿⣾⢳⣾⡿⠛⢷⡿⠛⣦⢹⡏⠀⠙⡇⢸⡏⠭⣿⣿⠻⣿⣦⢀⣾⢿⡇⠀⠁⣰⣿⠃⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⢸⣦⣀⣼⡇⣼⣿⣧⡍⠀⠀⠘⣧⠀⠰⠈⢃⠀⢼⠃⣿⣿⣶⣿⠇⠀⠙⢁⠺⠃⣨⡀⢀⣰⠿⠁⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⢿⣿⣿⣿⣆⠀⢸⣟⡀⢀⢀⣌⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⢮⠁⢰⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⢰⣧⣄⣴⡆⣿⣿⠟⠀⣰⣆⠀⠀⠀⠀⠉⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⣿⣿⠟⠀⠀⢈⠿⠛⠁⠀⠀⠿⣿⣿⢄⡘⣿⠉⢙⠇⣿⣿⠀⣴⣿⣿⣆⠘⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠂⠀⠀⠘⣿⣷⠀⡠⠄⣲⡿⠋⠀⠀⠆⠀⠀⢠⡀⠠⠀⠀⠙⠻⣷⣅⣈⣑⣡⣾⣿⣿⣾⣿⣿⣿⣿⡇⡀⠀⠀⠀⢡⡆⠀⢺⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢘⠀⣿⣿⡀⠀⠐⠛⠀⠘⠀⠀⠠⡀⠀⠀⠈⠀⠀⠀⣰⡆⡀⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣼⣿⣇⠀⠀⠀⠀⠀⠓⠀⠿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠈⢠⢸⣿⡇⠀⠀⠀⠁⡀⠀⠀⠄⠀⠀⠀⠀⢀⣠⣴⣿⣷⣇⡀⢠⢜⣿⣯⡉⠉⠉⠁⠙⠻⣿⣍⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⢻⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⡇⠀⠀⣄⠀⠐⠄⠁⠂⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣶⣶⣿⣿⡿⠿⠿⠟⠂⠀⠀⠀⠛⣧⡀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠆⠀⠀⢸⣿⣧⠀⠀⠙⣦⡀⠀⠁⠒⠒⠒⠦⠀⣿⣿⣿⣿⡏⡏⡌⠉⢻⣏⢿⣦⠀⠀⠀⠀⠐⠀⠀⠀⣱⡄⠀⠀⠀⠀⠀⠀⠈⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠈⠻⢶⣦⣤⡤⠤⠐⠚⠋⢀⡟⣽⠆⠀⠀⠀⠰⠿⣶⣤⡤⠄⠀⠀⠀⠀⠀⠚⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠿⣿⠀⠀⠀⢀⠞⡹⢀⣴⣤⣠⡀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠏⠔⠁⢸⣿⡿⠀⣿⣿⣧⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠲⠄⡀⡠⠀⡼⠟⣷⣼⡿⠻⠿⠟⢛⡿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠑⠂⠈⠑⠲⣷⢸⣿⡏⠤⠤⠘⠓⢉⡠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⢻⣬⣿⣇⣠⣤⣤⣍⣁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠀⠀⠀⠘⠛⢿⣟⣛⢉⠉⣉⢛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⢿⣿⣄⢠⡄⢠⣾⠟⠉⠀⠀⠀⠀⠺⣶⣿⠿⠉⢿⠧⢠⡬⠼⠂⠂⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢶⣿⠧⠀⠀⠀⠀⡀⢀⡆⠈⡀⠀⠀⠀⢀⣀⣤⣶⣤⣤⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠀⢳⡀⡘⠀⠁⣀⣠⣤⣶⣿⣿⠿⠛⠋⠁⠈⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⣬⣤⣤⣤⣶⣶⣿⠿⠿⠛⠋⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣢⣲⣭⣵⣽⣦⣶⣯⣭⣖⣢⣤⣐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣾⠿⠛⠋⠁⣼⣿⣿⣿⣟⣿⣷⡈⠉⠛⠶⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢯⣀⠀⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢙⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢮⡑⠲⢤⣀⡻⣿⣿⣿⣿⣯⡞⠀⣀⣠⠴⢚⣭⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠉⠒⢤⢌⣉⣙⠛⠛⣉⣉⢩⠡⠖⠜⠢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠈⠸⠀⠈⠈⠂⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠚⠛⠛⠛⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠛⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⠠⠄⢐⣀⠀⠤⢀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣐⣐⣐⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠋⠉⣿⣿⣿⣿⡟⠉⠛⠦⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠴⡛⠉⢿⣿⣿⣯⣿⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠛⠛⠚⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠀⠀⠀⠀⠈⠘⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⣀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠉⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠈⠁⠀⠀⡀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠈⠒⢄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡾⡿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⣠⣾⢟⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣽⠀⠂⠀⠀⢠⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠇⠀⡀⠀⣴⡿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠎⢀⠀⡀⣠⡎⠀⠀⠀⠀⠀⠀⣠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⣾⣧⠀⠀⠀⠀⠀⠀⣤⣠⠿⠐⠸⠿⢦⣆⡀⡄⠀⠀⠀⠀⣠⠦⠂⢀⣬⡞⣿⠏⠀⠀⠀⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⢿⠛⢷⣤⡀⠰⣘⣶⠋⠀⠀⠉⠀⠀⠠⠈⢳⡄⡀⠀⣠⣼⠏⠀⢀⣾⢯⣾⠇⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠐⠝⢿⡖⣼⣇⠁⠄⠈⡀⠀⠀⠁⢠⣼⣽⡥⠞⠋⠀⠀⢀⣾⣷⡿⠃⠀⠀⠀⠀⢀⡾⠋
⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣇⠄⢀⡠⠤⢽⣽⣯⣷⣦⢤⣤⣤⣠⣴⣾⡷⠁⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⣀⡴⠋
⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠩⣢⡆⠀⠄⠀⢿⢿⡿⡟⠛⠙⠷⠻⣷⣿⡄⡀⠀⠀⠀⣠⡾⠟⠁⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠉⢶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⣧⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⢁⠀⠤⠤⠒⠋⠁⠀⠀⠤
⠀⠀⠀⠈⠢⣀⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠙⠲⢽⣻⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⠷⠦⢀⢀⠀⠀⠀⠀⠀⡤⠚⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢶⣶⣶⣶⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁⠈⢉⣿⠀⠀⠀⠉⠫⣅⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠖⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⠶⣾⠿⣿⠟⡫⠽⠯⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⡿⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⡇⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢸⠁⠀⠀⣾⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⢀⡏⠀⠀⢰⠇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡜⠀⠀⢀⡞⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡰⠁⠀⠀⡼⠁⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠐⠀⠀⢀⡜⠁⠀⢀⡞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⡘⠆⢠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠎⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣤⡙⠆⠰⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡐⠁⢀⣾⣿⡏
⠀⠀⠀⠀⠀⢀⡈⠳⢄⠠⠏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⣸⢃⣴⠾⠿⢷⣶⣄⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠟⠀⠒⠂⠀⠈⠻⣷⣄⠠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠈⢿⣦⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠹⣷⡈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⡶⢶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠈⠻⣌⣿⠀⠀⠀⠀⠀⠀⠀⢐⣿⠏⢀⡾⠁⠸⠁⠠⠞⠉⣩⡿⠛⠳⢶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡞⠀⠀⠀⠀⠀⠐⠋⢁⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠏⠤⠔⠊⠀⠀⠀⠀⣀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡆⠀⡔⢀⡆⠀⡆⠀⠠⠂⠠⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠞⠀⠀⠱⡄⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠄⠘⠂⠀⢀⠈⢀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⠀⠀⠀⠐⢀⠀⠠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡰⠃⢀⣘⣠⠀⠀⣀⡆⠀⠀⢆⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢱⠀⠀⡐⠁⠁⠈⠀⠁⠀⠀⠈⠐⠢⣄⡌⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢄⡜⠀⢰⠂⠀⢀⢤⠤⠤⣤⣀⠀⠀⠀⡴⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠜⠢⣤⡇⠀⢠⠶⡟⢩⣝⠳⢯⡁⠀⠀⠃⠀⠱⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠠⣧⣙⣫⣬⡤⠝⠃⠀⠆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⡄⠀⠀⠀⠀⠀⠀⠘⠓⠓⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⢤⡀⢀⠀⠀⠀⡀⠀⠀⠁⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠤⠄⠊⠰⣀⠀⠠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠃⠀⠀⠀⠀⠀⠈⠀⠝⠋⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇⠀⠀⢻⠀⠀⠂⡀⠀⢰⡀⠀⣤⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠂⣦⣿⠀⠀⠀⠈⢳⣾⠇⠀⠈⢷⣦⡞⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⣤⣀⠀⠀⠲⡀⠻⣶⣰⣤⣴⣄⠋⠀⢀⠀⠈⠋⠀⣀⣴⣴⣿⣿⠴⠶⠶⠾⠷⠦⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠓⢄⠹⣆⣿⣟⢀⠹⠋⠀⢷⣿⡶⠆⠀⣾⠟⠛⠻⠉⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠢⡀⠀⠹⣿⡍⠻⣾⣦⣼⢶⣯⣯⣾⠳⣷⣅⠀⠀⠈⠀⠀⠀⢀⠄⣠⠞
⠀⠀⠀⠀⢄⠀⠀⠀⠀⢘⣦⡈⠢⣄⡘⠇⠀⣈⢿⣗⡃⣠⡹⠃⠀⢯⣿⡷⠊⠊⠀⢀⠔⣡⡾⠁
⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⣉⡛⠦⣈⣿⣿⠸⠋⠈⠁⠈⠉⢡⠀⢠⣾⠏⢀⣴⣦⣾⣵⡿⠿⠄⠀⢀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠉⠻⣿⣿⣦⣳⡀⠀⡆⢀⣰⣷⣾⡧⠶⠿⠥⣽⠟⠉⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣐⣦⣀⠀⠀⠐⠈⠛⠿⣿⣷⣀⣿⡏⠉⠀⠀⣀⣀⣤⣾⡿⠤⠄
⠀⠀⠀⢀⠤⠓⠐⠠⠤⠬⣭⣿⣶⣦⣤⠴⠶⠶⠿⠛⠿⠻⢶⣾⣿⣿⣿⣍⡁⠀⠀⠀⠀⠀⢀⣀⣀⠤⠄⠊⠀⣈⠄
⠀⠀⡠⠁⠀⠀⠀⠀⢀⣐⣿⣿⠉⠘⠿⣄⠀⠀⠀⠀⠀⢠⡿⡟⣿⡟⠅⢄⠁⠀⠙⠷⠦⠌⠉⠁⠄⠠⠐⠂⠉
⠀⠀⠣⣀⣀⠀⠀⠈⠙⠻⣿⣿⣷⣤⣄⠘⠓⠀⣀⡤⠞⢣⠀⠱⠄⠀⠀⠀⠑⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⡍⠄⠌⡝⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠁⠤⡠⠔⠠⡄⠤⣉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠴⠚⠛⡁⠎⠖⠙⠛⠲⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠥⠴⠐⠛⠣⠀⠈⠛⠓⠐⠤⠽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⡠⢂⣴⠀⢹⣤⢉⢡⣼⠃⢑⣔⠠⡈⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠻⣼⠛⠊⠉⠉⢹⣿⠀⢸⡿⠉⠉⠉⠚⠻⡼⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⣣⣶⠋⢉⠋⠓⠸⠦⠾⠔⠋⢋⠉⢱⣦⣣⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠒⠒⠒⢾⡿⢻⠯⠐⠁⠀⠀⣄⢠⢀⠄⠀⠀⠁⠨⢿⠻⣿⠖⠒⠒⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣥⡳⠟⢻⣿⡞⢳⠏⢠⠤⣤⣀⠀⢷⣼⣵⠇⢀⣠⡤⢤⠈⢷⠛⣾⣿⠛⠶⣣⣍⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠋⠀⠘⢄⢈⣿⡗⠺⡆⢀⣤⣀⠙⠃⠀⠀⠀⠀⠛⢁⣠⣀⠀⡾⠚⣿⣏⢀⠐⠀⠈⠃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠐⡟⠟⢧⣴⠁⠈⠠⠝⡆⠀⠀⠀⠀⠀⠀⡾⠡⠈⠀⢳⣤⠟⠏⡗⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⠀⢀⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢆⠀⠈⠂⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠇⠙⣦⣄⡄⣰⠀⠀⠲⠀⠀⠀⣷⣄⣤⡞⠀⠇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡐⠄⠀⠀⠀⠀⣿⡿⢃⠇⠀⠴⠤⠀⠀⢠⡇⢻⣿⡇⠀⠀⠀⠀⠔⣊⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣀⠔⢁⡴⡻⠃⣽⠀⠀⡤⡄⠀⠀⣾⣾⡁⠻⡳⣄⠑⢄⡰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⡶⠟⢊⣠⠶⣻⠀⠀⢀⡀⠀⢀⡏⠕⠱⢦⣈⠚⠷⣦⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣽⠛⢯⡢⡀⠘⣇⠀⠉⠁⠀⠸⣇⠂⠀⡡⣪⠟⣻⡅⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢈⡟⠮⣂⠈⣦⡀⠘⠉⢀⠈⠳⣬⠞⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢧⠀⠈⠙⠒⠽⢦⣀⠋⠀⠀⠈⠳⡏⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠈⠳⣄⡀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠘⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⣦⣄⡀⠉⠙⠒⠒⠒⠛⠉⠳⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠟⠻⠶⢶⡖⡖⠖⢚⣻⠇⣸⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⠁⠀⠀⣠⠈⠙⠛⠛⠶⠚⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⡵⠚⠉⠓⠶⠤⠤⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⣴⣦⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢀⠀⠀⠈⢛⠃⠀⠔⣿⣿⡗⠄⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⡀
⠀⠀⠀⡀⠸⢿⠿⡿⠃⢀⠀⠀⠀⣠⠂⣸⣿⣇⠀⣀⠀⠀⠀⢘⣹⣿⣇⠀⢺⠀⠀⠀⠹⣦⣴⣶⣾⣿⣶⣦⣤⠾⠀⠀⠀⢻⣀⣿⣧⣠⠇
⠀⠀⠀⠙⠶⣿⣿⣿⡷⠋⠀⠀⠀⢈⣿⠂⠀⠰⣿⣁⠀⠀⠐⢦⣿⣿⣿⣇⣀⠇⠀⠀⢤⣤⡼⣿⣿⣿⣿⣿⣥⣿⠗⠀⠀⢀⡼⠈⢽⣿⠂
⠀⠀⠀⠀⠀⠈⠉⢉⡇⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⡀⣶⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⣀⡀⡃⢰⠠⣾⡄⢠⠀⠀⠀⠀⠀⠀⢠⣶⠄
⠀⠀⠀⡄⣄⣨⣿⣷⡄⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠠⣄⣀⠻⠟⣢⡤⠖⠀⠀⠉⢁⣑⣾⡞⠿⢳⣾⣚⣁⠀⠀⠀⢀⡚⠿⢃
⠀⠀⠀⠀⡨⠛⠻⠿⣅⠄⠀⠀⠀⠃⣸⠿⣿⡝⠳⠆⠀⠀⠀⠀⣨⡿⠗⠉⠛⣥⡄⠀⠀⠀⠀⢀⡼⠣⣤⡜⢦⡀⠀⠀⠀⢀⡼⠁⠀⠙⢦
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⡄⠀⠀⡤⠀⠀⠀⠀⠀⠀⣤⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄


⠀⠀⠀⠀⡴⠀⡀⠆⠀⣄⠀⠀⠀⠀⢰⣶⠀⣄⠠⣷⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢰⠀⠀⣰⡄⠀⢹⡄⠀⠀⢀⡔⠀⣴⣤⡆⠀⣤⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⢰⡆⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠈⠧⣤⣿⣿⡤⠞⠀⠀⠀⠀⠹⢤⣟⣛⣻⡴⠃⠀⠀⠀⠀⢲⢾⣿⡇⠀⠀⠀⠀⠀⢳⣾⣿⣷⣤⢾⣷⠀⠀⠀⠀⠀⣹⣼⣁
⠀⠀⠀⠀⠀⠸⣿⣿⢧⣄⡀⠀⠀⠀⠀⠀⢿⣿⡿⣤⣀⠀⠀⠀⠀⠾⢿⣿⣿⡇⠀⠀⠀⣀⡼⠿⣿⠿⢧⣠⣥⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⢘⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙
⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⣿⠀⠀⠀⠀⠀⢠⣾⠀⣿⢸⡇⠊⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣻⣿⣿⡟⣿⣿⣿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣷⣽⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠉⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣧⣠⡀⠀⠀⠀⣿⣿⣿⣟⢷⣄⠀⠀⠀⠶⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⡿⠃⠀⠀⠀⠈⣻⣿⣿⠿⢿⠀⠀⠀⠿⠃⢸⣷⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⡄⢸⣿⣿⡀⠀⠀⠀⠀⢀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢴⣰⢺⡿⠏⠀⠀⠀⠀⢀⣼⠇⣸⣿⣿⣷⣤⠀⠀⠀⣸⣿⣿⣇⠀⠳⣄⡀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⣬⣿⠛⠀⠇⠀⠀⠀⢰⣿⣏⣴⡿⢿⣿⣿⣿⠀⠀⠸⣿⣹⡿⣻⠗⠀⠹⣟
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⠞⠉⠘⠙⣷⡜⣿⣿⣿⡄⠀⠀⣿⣿⣾⣿⠀⠀⠀⠈⢧⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⣿⣿⣶⠂⠀⠀⠀⢿⡄⠀⠀⠀⠸⡟⠙⡉⠛⣿⣦⡀⠈⢠⣿⠇⠀⠀⠀⠀⠘⣧⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠘⣯⠁⠀⠀⡇⡂⢀⣏⢰⡧⠀⠀⢻⡀⠙⢄⣼⣿⣿⡦⠘⡇⠀⠀⢠⠆⠀⢀⡿⠄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢠⢀⣧⠛⣿⡿⠘⣿⣤⣤⣾⣧⠄⢼⣿⣿⡧⠀⢠⡁⣶⠀⢾⣦⣄⠚⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣼⣿⣄⣿⡀⣸⣿⣿⡿⠿⣿⡄⠀⣙⣷⡀⢠⣿⠉⠋⠀⠀⠙⢿⣻⣷⡰⠦⡀⣠⡘⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠘⠛⠁⡸⠋⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⡞⣠⠋
⠀⣤⣄⣀⢀⠀⢀⣠⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢼⡀⠀⠀⡀⠀⢰⡀⠀⠀⠀⠀⠀⢰⠿⠧⣤⣴⡶⢛⣁⣠⠄
⢠⣿⣿⣿⣿⣷⣿⠋⠙⠟⠛⠿⠛⢉⣻⣿⣻⣿⣷⣄⣺⣿⣣⣾⣥⣄⣾⠃⠀⠀⠀⠀⢀⣼⡄⠀⠄⠀⠈⠁
⣿⣿⡿⠟⠀⠙⠀⠀⠀⠀⠀⢀⣴⡿⠋⢻⣿⣿⣿⣿⣿⣿⣅⣈⡿⣻⡃⣠⣿⣶⠶⠞⠋
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠴⣿⣟⠀⠀⠀⠙⢻⣿⣿⣿⣿⣟⣿⣿⣿⣏⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⡀⠀⣀⣀⣶⣿⡿⠋⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠿⠛⣩⡞⠃⣸⡏
⣿⣧⠀⠀⡀⠀⠀⣠⣤⣿⣿⣷⠹⣿⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠐⠿⢁⣠⣿⡇
⣿⣿⡇⠀⠠⠀⢰⢿⣿⣿⣿⣼⣦⢿⣷⣼⡆⣄⠀⠀⠀⠀⠶⣬⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠟⠛⣿⡅⠀⠀⠀⠀⢀
⣿⣿⣿⣆⠀⠣⡀⠻⣿⣿⣿⣧⠿⢾⣿⣻⣿⣿⣶⣶⣦⡀⠀⡀⠉⣛⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⣠⣿
⠛⠛⠛⠛⠳⠦⠿⢦⡈⠻⢿⣿⣦⡘⣿⣿⣿⡿⣿⣿⣯⣻⣦⣄⡙⠛⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠺⠿⠟
⣤⣤⣤⣀⠀⠀⠀⠁⠙⢷⣶⣿⣿⣷⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⡠⢀⣀⠴⠶⠿⠅
⣈⣿⣿⣿⣷⣤⣀⠀⠀⠸⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣤⣤⣤⣶⣾⣿⣿⣿⡀⠀⠀⠀⠲⡶⠶⠒⠒
⠀⠈⠙⠛⢿⡿⣿⣿⣿⣿⣿⠃⠸⣿⣿⣿⣿⣮⠿⣿⡿⠿⠿⠿⠿⠷⠿⡿⢿⠿⠋⣿⣯⣽⣛⣏⣉⣤⣴⡶⠋⠁
⠀⠀⠀⢠⡤⠤⢶⡾⠿⣿⣿⣷⣶⣼⣴⣧⣤⣴⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣴⣶⣾⣿⠿⠿⠛⠋⠉
⣼⣿⣶⠀⠈⢠⠀⡧⢠⡿⠋⣉⣳⣄⣉⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣾⣿⣿⠶⣶⣾⣷⣿⣿⣿⣿⣿⣿⣏⡉⠛⠆
⣿⣿⣿⡟⠋⠉⣿⣿⣿⣿⣿⣿⣟⠉⠉⠓⠲

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


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⣀⣽⣭⣭⣭⣭⣯⣯⣭⣽⣍⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⠿⠟⠁⣠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⠈⠳⣦⣤⣤⣤⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠛⠉⠉⠈⠙⣿⣿⡇⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁⠀⠀⢠⣿⣿⡿⠛⠛⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣶⡶⠶⠿⠿⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣧⡀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⢻⣿⣶⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⠟⠀⠀⠀⠙⢿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⣿⣿⠀⠀⠈⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣿⣿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⢀⣴⣿⣿⠟⠁⠀⣿⣿⠀⠈⠻⣿⣿⣦⡀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⣴⣿⣿⠟⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠲⣶⣷⣆⣀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⣿⣿⠀⠀⢀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣷⣄⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⡀⠀⣀⣠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡠⡤⠀⣀⡀⢾⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⢐⣿⡏⡡⣤⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣶⣶⣮⡽⠲⣴⣾⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡟⠋⢤⡀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣥⣀⣠⣤⣀⡀
⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣆⡿⠀⠐⠶⠉⢐⡂⠾⠃⠀⠀⠀⠀⠀⠀⠀⣴⣾⣴⣿⣿⣿⣍⠙⣿⡆
⠀⠀⠀⡀⠀⠐⢀⠀⠀⣼⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⠛⠻⢿⣿⣿⣧⣸⠇
⠀⠀⠀⠁⠂⡀⠀⠂⢸⣿⣿⣿⠟⠽⠂⠀⠀⠀⣡⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⡥⠀⢲⡀⠘⢿⣿⣿⡅⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠻⣿⡣⣄⠀⠀⠀⡿⢇⠙⠛⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣻⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠶⣦⢤⣶⣷⠤⡀⠀⠀⠀⠀⠀⠀⣘⢿⣦⢀⡆⠀⠀⠀⠀⠈⣿⣿⣷⣶⡄
⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠻⣿⢯⢤⡀⠀⠀⠀⠀⠀⢰⣤⣌⠋⠻⡟⠀⠀⠀⠀⣸⡏⠙⠛⠛⢿⡄
⠀⠀⠀⢉⠿⣿⣦⠀⠀⠀⠀⠀⠀⡄⢫⠀⣬⠙⣤⠀⠀⠀⠀⣴⠀⠸⣿⡇⢀⣤⣷⣶⠀⢴⠂⠉⠀⠀⠀⠠⣄⡑
⠀⠀⠀⠾⠀⠀⠉⠁⠀⠀⠀⠀⠀⢉⠀⣾⠻⡧⡈⠀⠀⠀⢴⠀⠠⡆⠀⠀⠹⠉⠛⡯⠀⠀⠀⠀⠀⡀⠀⠐⢾⣷
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⣦⡈⠁⠀⠀⢰⠂⠸⣅⣰⣦⡄⠀⣿⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣤⠘⠁⠀⠀⠐⠆⠀⠀⠀⠹⢿⠇⠐⢶⠃⠀⠀⢴⡆⠀⠀⠀⠀⢚⢁⡄⠆
⠀⠀⠀⠀⣀⠂⠉⠂⠀⠀⠠⠤⢔⡾⠀⠁⠀⠀⢀⠀⡷⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⢿⣦⡀⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠰⡄⠀⠀⠘⠃⠀⡠⠈⠁⠘⠀⠀⣰⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⢀⠀⢀⠀⣤⣤⡀⣤⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⢰⡀⠀⠀⠀⠀⠛⠛⡷⠛⢿⣿⣧⣿⣿⣧⡤⠞⠒⠺⢥
⠀⠀⠀⠀⠀⠀⠹⠀⣤⠠⢒⡆⠸⠁⠀⢦⠀⠀⠇⠘⠙⠃⠀⠀⢰⣦⣤⡷⠃⣾⣯⣿⣀⣨⡿⠁⠛⠛⠀⠀⡐
⠀⠀⠀⠀⠀⠀⡂⠀⠈⠀⠀⠁⠰⡄⠀⣷⠀⠫⠀⠀⢀⡀⠀⠀⢸⣿⣿⣧⠾⣿⡛⣿⣿⣿⣧⠘⠀⠀⠠⠀⢿⡄⢠⠀⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠓⠈⠉⠁⠀⠀⠈⠉⠉⠁⢄⠉⠿⢯⠉⠉⠹⡆⠀⢀⠀⠀⠀⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⠞⠀⠀⠀⠐⣀⠀⠐⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⡌⠿⣿⣾⠿⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⡀⠀⠀⠀⠈⠁⠀⠙⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠿⠃⠀⠶⠶⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⡇
⠀⠀⢀⢆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣆⠀⠋⠀⠀⠀⠀⠀⠄⠙
⠀⠀⠘⠊⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠰⠶⠀⠀⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⣀⣀⠀⠀⠀⢀⡀⣀⡈⠀⠀⠊⠁⠀⠂⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠎⠀⠀⠀⢀⠴⢷⡄⠀⢠⣤⠄⠀⠀⠉⠁⠀⠀⠀⠈⠁⠛⠁⠀⠀⠐⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡀⠂⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠉⠰⠾⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⡀⠀⠃⠐⠷⠀⠀⠀⣀⠀⣤⡄⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠟⠀⠀⢀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠂⠀⠀⠱⣀⠲⠇⠀⠐⠀⠀⠀⠀⠈⠀⠉⠁⢀⣠⠀⠀⠀⠀⡀⠄⠑⠔⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⢀⠠⠊⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠐⠶⠀⠀⠘⠃⠀⣀⠈⠉⠀⠰⠘⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⣤⠬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡀⢀⠠⡀⠠⠀⠱⠀⠀⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠐⠃⠄⠀⠀⠀⠀⠀⣓⠀⠂⢀⠀⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⢡⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠐⠖⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡥⢼⡿⡆⣹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣤⣀⡟⣮⣶⣶⢟⡴⢶⣌⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠀⠀⢤⣝⡷⠂⢾⣷⣬⣟⣵⡾⠿⣴⠋⠀⠀⠀⠀⠀⠀⢫⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡏⢳⠾⠐⡄⠂⠀⠀⡸⣦⡌⠛⣦⡟⢹⡯⠦⠈⣴⢟⣧⣍⡦⠀⠀⠠⡀⢨⡈⠥⠸⡄⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠦⣀⣝⠀⠸⠇⠀⠀⣸⡉⣯⣴⣤⣿⣄⠀⠾⠟⠃⢿⣶⡜⣯⣠⡅⠀⠀⢠⣌⠋⢠⣇⢸⡦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡿⠀⠀⠀⢀⢶⣾⣭⢨⡷⠉⢼⣭⣿⡠⠚⢳⣦⣤⢈⢡⣄⡴⣿⣄⠀⠈⠀⠙⢥⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣽⢻⡇⠀⠀⢀⢮⠟⢻⠦⠻⢃⣴⣶⣿⠉⢣⡞⠀⢰⡆⠀⢸⠇⡠⡟⢛⡆⠀⠀⠐⡀⠛⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⡏⢋⢀⣄⠀⢞⡻⠀⣙⠻⢃⡜⣿⠟⢿⣶⠦⠽⠀⣻⡇⠀⠛⠣⢰⠁⢨⠜⡀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣸⢛⣧⣀⠉⣩⡷⠒⣶⣙⣹⠂⣧⠀⠻⣧⣤⣆⠀⢰⡆⢰⡙⠲⡆⠀⠐⣏⢡⡀⣤⠓⠶⢩⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢶⡟⣳⣤⣶⢟⣷⣬⡝⢿⡌⢢⣴⠉⠛⠙⠀⣺⠁⣸⣛⣋⡱⠄⠘⠛⠀⠀⠀⠳⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡖⣸⠉⣀⢡⠟⠁⡼⣧⣈⡿⠶⣹⡇⣀⣀⠰⣽⡓⢋⣩⣭⡴⠶⠶⣦⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠛⠘⢵⣧⡀⣤⣶⢿⡟⠐⢦⣼⡇⢹⡏⠿⡶⠟⡛⢭⣿⡴⠲⢦⡷⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣱⣿⡿⣤⣼⡧⠀⢈⡇⣀⣄⠰⡇⠚⠃⣠⣽⡀⠀⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢴⢳⣿⣆⣸⣿⡝⠀⢺⡿⠫⠿⢠⡧⠀⢈⣹⣧⣧⣀⣴⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡼⢁⡿⣯⣈⡿⣿⢤⡀⠀⣿⠀⠀⣨⠀⠀⠂⢍⣣⣉⢥⡆⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⡼⢡⡞⢡⣀⣉⡷⣟⠢⡁⠀⠹⠶⠾⠛⠠⢌⠑⠤⣿⠤⢜⣿⣦⠖⢸⠉⠂⠚⠁⠘⣳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⣁⢦⣴⣏⠘⣌⣩⣼⡁⢈⡳⣍⠢⣄⠀⢀⡀⠲⢤⡙⠲⣿⠦⡌⣧⢀⠳⠀⠀⠀⠀⢴⠾⠯⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⡀⣡⢦⡰⢎⡵⣭⣄⢹⣆⠁⠌⠈⢷⣌⠳⣤⡈⠳⣤⡉⢷⡏⡁⡴⢿⠀⠁⠐⡆⠀⠐⠺
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⢞⣷⠦⡢⠉⣴⠳⡄⢿⠀⣿⣠⢢⠕⢊⣉⡴⢦⡙⠢⣌⠻⡾⠃⢑⠴⠎⣄⠴⠋⠘⠲⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠠⢿⣴⠞⣿⣾⣭⠐⠛⠃⠀⠘⢃⣰⠖⣯⡴⠖⣤⠙⠂⣤⡙⠇⠀⠈⠳⠞⡣⠀⠴⡀⠀⠠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡴⢿⡿⣤⣹⣄⠛⡇⠀⠀⠐⣏⠷⢞⣁⡀⢠⣟⢠⣴⡆⢻⡆⠀⠀⡾⠲⢁⣄⠀⠀⠉⣭⡻⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⡈⢿⠇⣘⡩⠀⠀⠀⢺⡅⠸⣏⡽⢃⣀⣿⡘⠻⠃⢈⡇⠀⠀⠈⡶⠎⣰⡄⠐⣀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠞⢀⣩⠇⠀⠀⠀⠈⣷⣀⣤⣄⢻⡿⢿⣛⠦⢀⣾⠇⠀⠀⠀⠀⠛⢈⢆⡁⠙⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣄⣈⠕⠀⠀⠀⠀⠀⣴⣯⡹⠾⠋⣀⣈⠙⠶⢤⣷⡀⠀⠀⠀⢸⡙⠟⣸⠋⣰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⣁⢈⡘⠂⠀⠀⠀⠀⢀⡲⠇⣠⣶⡆⣹⠏⣠⠶⢀⡇⡅⠀⠀⠀⠀⠉⠻⢉⠠⡈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢌⠉⢠⡇⠀⠀⠀⠀⠀⠈⠉⢃⠹⠟⠋⣡⡄⠛⠺⠋⡏⠀⠀⠀⠀⠀⠈⠁⢉⠶⣿⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠏⠼⠛⠃⠀⠀⠀⠀⠀⠀⠀⢨⢂⣤⠶⣋⡟⠀⠤⣰⠇⠀⠀⠀⠀⠀⠀⠀⢙⣛⡛⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣩⡵⠶⣻⡟⠯⠒⠦⢀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠋⡐⢳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣁⣴⣾⡏⠟⢠⣤⣶⠈⠀⠀⠀⠀⠀⠀⠀⢦⡀⢈⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣻⠁⣽⡬⢧⣥⠴⢄⣀⣀⠀⠀⠀⠀⠀⠈⣉⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡿⠋⣻⠁⢀⡀⢛⠀⠀⠀⠀⠀⠉⠛⠿⠦⣄⠀⠀⢴⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡲⡞⠛⠛⠛⠛⣋⠋⣹⠛⠹⢉⠓⠈⠋⣙⠛⡘⢃⡟⠐⠏⢚⢇⡀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⣁⣄⣤⣆⣐⣀⣌⣁⣨⣅⣌⣀⣀⣁⣨⣁⣎⣀⣏⣰⣍⣠⣅⢈⣝⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠁⠉⠉⠉⠁⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡲⣟⣿⠲⠶⡄⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⢲⢞⠠⣮⣯⣵⡏⠻⠏⠳⢙⠳⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣕⠱⣢⢽⣿⠸⣿⣏⡋⠀⣀⣰⣠⣆⣽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣔⢙⢾⡜⣯⡗⣦⣼⡹⣖⣽⣿⣿⣿⣿⣿⣷⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡱⡬⢥⣬⣮⢿⣿⣶⣼⣿⡻⢻⡻⠹⣿⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⢶⣿⡿⠿⠿⢶⣿⢋⠖⢿⣠⢾⣽⣿⠟⣡⢶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⣷⢚⢊⣾⡿⡹⡟⣿⣙⡿⣿⠁⠀⠻⣛⢰⠝⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣜⣥⠷⣯⡭⢾⢛⣷⡟⠸⣿⣿⣪⣦⢄⡀⠉⠻⣿⢻⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣉⠐⠾⣥⢀⡊⢿⣟⢿⣇⣽⣜⣩⣿⢿⣽⣯⡻⠆⠀⣿⣽⡛⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠯⡽⣸⠶⣿⣏⢉⣀⣿⡍⣚⢿⣴⠽⣛⣇⣭⣻⣿⣿⡾⣧⡷⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠐⢴⠗⣀⣿⣿⣨⣉⣿⡇⠊⢯⠈⢻⣷⣏⣻⠉⢯⠟⣥⡶⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢲⣛⣯⣄⣠⢰⢯⡠⢀⠙⠇⠘⡛⠱⢶⡳⣭⣿⠛⠿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡒⠍⢽⢤⡦⠟⣓⢂⡨⠈⣦⣦⡇⣚⣸⣎⣟⡿⣿⣝⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠷⠟⣇⡅⠜⣷⣄⠼⡣⡐⣹⣥⠟⠳⡟⠐⡷⡫⢻⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣟⡡⣥⡦⣉⣿⢷⣣⢜⠿⢥⠐⠦⣙⡽⠷⡟⠥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠾⣅⣀⠀⠀⣭⢥⠈⣹⣩⣮⣥⠖⢲⠒⠋⠋⠋⡿⠀⠠⠜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠢⠀⠈⠓⠲⣯⣀⣂⡿⣽⢃⠀⡅⠄⢃⠄⠀⡼⠉⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠢⢄⠂⠀⢀⢂⠘⡉⢫⢌⡽⡀⠀⠐⠀⠀⢀⣀⣀⡤⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠑⢄⠀⠀⡱⡘⢨⢯⢯⣿⠄⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢀⠀⠑⠀⡮⢄⡬⡆⠳⣝⢷⣷⣀⡤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠠⠀⠈⠀⠚⢈⠙⣮⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣒⣢⣀⠀⠀⠠⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢼⠋⣷⡟⣧⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠶⣮⣁⣀⡰⢚⠵⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠤⢤⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠎⣀⠈⠘⠟⠱⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠒⠄⠬⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⣾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⣀⠀⢹⣄⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠈⠀⠀⠀⢰⣶⣬⡟⠻⠞⠛⣤⠞⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⢀⡤⢃⠨⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡠⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠒⠘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⢈⡁⢀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⣀⡄⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢫⠋⠝⢂⠜⠤⣄⠀⢨⠁⣀⠀⠀⠀⡀⢀⡀⡻⣝⣿⣟⠋⢇⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣾⣿⣿⠀⠂⠀⠀⠀⠸⠢⠤⠊⠥⠘⡅⠈⠻⠊⢩⠛⠀⠁⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢽⡟⠋⠀⠀⠀⣄⡀⠁⠀⠀⠀⠀⠠⠀⣠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠴⠀⠀⠃⠀⠀⠑⠄⠘⣃⠂⢠⡄⢱⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⢷⣀⡤⠴⠃⡷⠉⡀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠭⠟⣂⡀⠉⠒⡔⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢆⠠⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠤⠘⠛⢁⡀⠀⠞⠆⠀⣀⠀⠐⠀⠀⢀⡀⡀⠀⠀⣼⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣠⠀⠀⠀⠀⠈⠒⠄⠀⠂⠠⠀⣤⡄⠀⡈⢉⠷⢶⠋⠈⢣⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠣⠁⠐⠂⠄⠠⢀⣠⠈⠻⡼⣇⠀⠀⠓⠀⠀⠁⠀⠀⠤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠄⣓⡐⠐⠁⠀⠉⠁⣀⡀⠀⠈⠀⡠⠐⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠠⠈⠉⠁⠀⠉
⣿⣿⠿⢿⢻⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⢻⣯⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣋⣧
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡧⣀⠻⣿⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠛⠀⣯⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⠆⠾⢹⢧⣆⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⣶⡿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢐⣄⠜⠘⣹⠆⠁⠀⠢⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠸⡄⠀⡀⢸⡀⠸⡤⠖⢷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠸⠋⠳⣇⣀⠿⡁⠡⠘⡆⢀⢣⠀⢸
⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡞⢸⡀⠀⠀⠄⠀⢀⢀⣐⢈⠀⠈⠀⢃⠈⡄⠁⠀⡁
⡇⠀⠀⡀⠀⠀⠀⠀⠀⠨⡵⣇⢸⣦⡠⠊⠀⣠⠭⠩⣐⡵⠤⣤⡖⢺⠆⠀⢀⡴⠦⠌⢤⣀⠀⣤⠶⢦⡄
⡇⠠⠄⠀⠀⠀⠐⠀⠀⡀⠑⢙⣾⠯⣀⠀⠈⠀⠙⠓⠻⠀⢴⡧⠔⢃⠤⠂⠁⠀⠀⠀⠀⠙⢶⡿⠶⠶⡽
⡇⢀⠄⠀⠐⠀⢀⣀⣀⠀⠀⣁⣀⢀⠊⠀⠀⠤⠤⠀⠀⣠⠈⠀⠐⠿⠀⠠⠀⠠⠀⠀⠠⠀⡐⠃⠀⠈⢣
⣷⡅⠀⣀⣠⠥⠘⠉⠁⠀⢤⠀⠙⠀⠀⠀⠄⠠⣀⡀⠀⠈⠒⠀⠀⢠⠀⠀⠀⠲⢀⡀⠓⠺⢇⠀⠀⢀⡾
⣿⣩⠀⢸⣻⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠒⠂⠒⠋⠁⠀⠀⣠⣵⡈⠀⠀⠀⠚⠐⠒⠚⠓
⡇⠀⠀⠀⢀⠆⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠐⠉⠋⠉⠉
⡇⠀⠀⠀⠈⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠶⠂⠑⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠆⠰⠖⠀⠀⠶⠂⠲⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠒⠒⠀⠐⠒⠒⠒⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣳⠓⣿⣿⣿⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡛⣩⠆⣩⡽⣼⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⢻⣿⣯⣥⣽⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⣀⣀⣀⣀⣀⣈⠉⣉⣋⣉⢉⣉⠋⣉⣉⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈⠃⢀⠈⡈⠁⠀⠉⠀⡁⠈⠉⠀⠁⠉⠀⠈⠀⠉⠈⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠈⠀⠉⠈⠉⠁⠀⠉⠉⠁⠘⠁⠀⠉⠙⠈⠃⠀⠈⠉⠁⠈⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠛⠐⠂⠂⠐⠋⠘⠃⠐⠃⠐⠒⠀⠓⠒⠀⠂⠘⠂⠂⠀⠐⠒⠂⠐⠂⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠒⠐⠒⠐⠒⠂⠐⠀⠀⠰⠖⠲⠆⠆⠀⡆⠦⠰⠖⠦⠖⠶⠀⠶⠆⠦⠀⠆⠀⠶⠂⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠄⠀⠀⠤⠄⠠⠤⠤⠤⠤⠤⠄⠀⠤⢤⠄⣤⡤⠀⡄⠀⠄⠀⣤⠀⢤⠄⠠⣤⠄⠠⠤⡄⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⢤⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀⣀⣀⣀⡀⠀⣀⢀⡀⣀⠀⢀⡀⢀⠀⠀⣀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠈⢀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠈⠀⢉⠁⠈⠀⠈⠉⠀⠉⠉⠁⠀⠁⠈⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠉⠁⠈⠁⠁⠀⠈⠉⠁⠉⠀⠉⠉⠀⠉⠉⠁⠘⠋⠀⠀⠘⠉⠋⠛⠂⠘⠛⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠃⠀⠋⠀⠀⠀⠈⠂⠘⠀⠒⠀⠀⠐⠛⠙⠀⠒⠒⠀⠘⠃⠐⠐⠓⠒⠀⠆⠀⠒⠖⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠃⠐⠒⠒⠀⠐⠒⠂⠀⠒⠲⠲⠀⠶⠀⠴⠰⠆⠀⠲⠀⢰⠤⠄⠀⠀⠶⠠⠆⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠄⠰⠠⠄⠠⠆⠰⠂⠀⠤⠔⠤⠴⠦⠀⠤⠤⠤⠄⠠⠤⠤⠠⠄⣤⠠⠤⣤⣄⣤⠀⡀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠤⢤⡤⡄⢠⢠⢤⠀⢠⠄⣀⠀⣀⣀⠀⡄⡄⠀⣀⡀⢠⠀⢠⣀⡀⠀⢀⡀⢀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⠀⣀⠀⣀⣀⢀⡀⠀⡀⣀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⢁⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⣈⠀⠀⠀⠀⠀⠀⠁⠉⠀⠁⠈⠀⠀⠈⠉⠉⠁⠈⠉⠁⠈⠁⠈⠁⠉⠁⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⠀⠉⠈⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠘⠓⠚⠂⠀⠃⠐⠂⠐⠒⠂⠰⠖⠲⠄⠂⠀⠆⠖⠖⠀⠂⠀⠀⠒⠀⠲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠀⠐⠒⠀⠐⠂⠐⠖⠶⠀⠲⠤⠒⠴⠄⠲⠤⠤⠀⠠⠤⠤⠤⠀⠠⡤⠀⠀⠠⠤⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠰⠀⠰⠆⠀⠤⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢀⣀⣀⣠⠀⣄⢀⠀⣀⡀⣀⡀⣀⡀⢀⣀⣀⡀⠀⠀⡀⢀⣀⡀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠉⠉⠉⠁⠘⠋⠁⠙⠓⠚⠋⠛⠛⠀⠛⠑⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛⠓⠂⠛⠀⠛⠀⠒⠺⠒⠂⠂⠔⠒⠂⠐⠒⠒⠆⠐⠦⠒⠒⠲⠀⠒⠰⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠒⠒⠂⠒⠒⠀⠐⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣆⠀⠀⠀⠀⡀⠀⠀⠀⢀⠋⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠤⠤⣄⣀⠀⠀⠀⠀⠈⠢⢌⠙⠃⠂⠸⡏⠐⠘⠛⣉⡴⠊⠀⠀⠀⠀⣀⣠⡤⠄⠄⠀⠤⠄⢀
⠀⠀⠀⢰⣦⡅⠀⠀⠀⠀⠀⠄⠒⠂⠉⠉⡙⠂⣀⡀⠀⠈⠣⠔⢷⣴⣷⡴⠣⠔⠁⠀⠀⣀⠀⠘⡌⠉⠀⠒⠂⠀⠈⠀⠀⠀⠀⣵⣶
⠀⠀⠀⠁⠰⣊⠄⣠⠄⠡⠀⠀⠀⠀⢀⡀⠄⠀⠉⣉⢉⠖⠶⣾⡿⠞⠙⢾⣿⡶⠖⣋⢉⡉⠠⠀⠄⣀⠀⠀⠀⠀⠀⠁⣄⠀⣌⠲
⠀⠀⠀⠀⠀⢱⡾⠖⠀⠀⠴⠖⠂⣈⡵⠚⠁⠐⠊⠀⣉⣉⣻⡿⠂⠔⠓⠀⠻⣿⣋⠉⠁⠐⠂⠈⠙⠲⣭⡀⠲⠦⠄⠀⠲⠿⣖⠈
⠀⠀⠀⠀⠀⠈⠐⢺⣷⡂⢀⠀⠄⠋⣴⠞⠛⠂⠀⡐⡯⡛⠿⢢⣞⠀⠀⢸⢔⠾⢃⡙⣖⠀⠀⠚⠳⢶⡈⠃⠀⢀⠀⣶⡟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠫⠊⡆⠀⠀⣀⠀⠉⣁⡄⠾⠁⠤⠠⣬⡦⢠⣯⣤⠤⠆⠹⠆⣄⡉⠀⢀⠀⢱⡳⠌⠚⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠐⡾⠋⠀⠐⠦⢰⣾⠇⠥⠠⠈⢳⣦⠠⠖⠀⠉⠳⡆⠠⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠐⠊⡠⠉⠜⠀⠆⠀⡄⠸⡍⠁⠘⡒⠂⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢀⢀⠀⡶⢐⠃⣐⠳⣀⢆⡼⢃⠀⣗⠠⠻⠦⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠸⠀⠀⠋⢼⠻⠿⢻⠿⠀⠀⠀⠀⠙⠀⠀⢈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠦⠀⠙⠘⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡐⠀⠐⠀⠀⠀⢀⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⢂⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠙⠁⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣿⡿⠛⣻⡿⣿⣿⣿⠿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣥⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣄⠀⠱⣤⣀⠀⠀⠙⠛⠋⣀⣤⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⠻⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣧⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣤⣀⣀⢀⣀⣠⣴⣾⣿⣿⠟⠋⠉⠉⠙⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠻⢿⣿⣿⠻⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⣰⡿⠿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣄⡀⠀⠀⠀⣀⣴⣿⣿⠇⠀⢹⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡄⠀⠀⠀⠀⠠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠀⠀⣀⡀⠤⠄⠀⡀⠀⠀⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠁⠀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠘⠢⣄⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠑⠀⠐⢈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⢀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⡾⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⠳⡀⠈⢻⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⣴⣴⠟⠁⢠⠜⠠⠖⠀⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠈⠻⠭⣄⣀⣀⣀⣤⣴⠿⠟⠁⠀⠄⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠠⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠘⠤⢀⠀⡀⠉⠈⠉⠁⠀⠀⡀⠤⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠
⠀⠀⠩⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠌
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⠀⠀⠒⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡨⠀⠀⠀⠀⠒⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⢀⡶⢺⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢀⠎⠀⠀⠀⣤⣿⣿⣿⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣴⣿⠀⠀⠀⠀⠥⠹⡟⣻⠆⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡓⡏⢩⠀⠀⠀⢘⡀⢰⣾⣇⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠈⠜⢀⣠⠄⢸⣏⣼⡯⠆⢱⠀⠰⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠇⠐⠩⢯⡠⢺⠿⢟⠰⣌⣬⢦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠐⢠⡞⢗⣿⢀⡾⠀⠧⢒⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠉⠉⢀⣄⠂⠘⣸⣤⣯⡖⣩⢵⡈⢻⡂⠑⠂⠀⣤⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠺⠇⠀⠛⠲⡤⠟⢣⢿⣤⠇⢸⠇⢀⠙⠢⢄⠀⠸⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠂⠀⠀⠀⠀⠐⠀⢰⡇⢠⠋⠠⢀⡸⠋⠀⠀⢸⠊⣔⣦⠙⠢⠁
⠀⠀⠀⠀⠀⡀⠀⠀⣀⠠⠀⠀⠀⣀⠶⣤⡘⣿⠉⠒⡖⠁⠀⠀⠀⠈⢸⠈⣻⢵⢈
⠀⠀⠀⠀⠀⠿⠖⠒⠃⠈⠀⢩⣯⡵⠟⢻⢿⣏⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡍⢸⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⣼⣿⡛⢻⠿⢥⠬⠥⠤⠄⠀⡞⡈⣿⣷⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠱⣖⣫⡿⠟⠠⣿⠿⠿⣭⠼⢿⣿⠤⣾⡿⠿⢇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠉⠈⠁⠈⠋⢛⣒⡒⣒⣲⠀⠀⠐⡂⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⢨⠀⠈⠉⠉⠉⢉⠁⠀⠀⠂⠀⠀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠐⠀⠐⠣⠀⢠⠆⢁⠀⠀⢸⠀⠀⠀⠀⠀⠀⡐⠀⠀⡘⠀⣄⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢟⣰⠀⠀⠃⠀⠀⠁⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠊⠐⠒⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢾⠁⠀⠀⢈⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⢄⡀⠀⠙⢦⠔⠋⠀⠀⣠⠞⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⢄⡀⠙⢄⠉⠓⠒⠤⠴⠒⠒⠉⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠉⢢⠈⠓⢤⣀⠀⠀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣀⠤⠒⠒⠒⠢⡀⠀⡇⠀⠀⠈⠉⠉⠉⢉⡭⠕⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠋⠀⠀⠀⠀⠀⠀⢹⠀⡞⠀⠀⠀⠀⠀⡰⠋⠀⢀⠴⠒⠒⢤⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠴⠃⣴⠁⠀⠀⠀⢀⡜⠁⠀⡰⠋⠀⠀⠀⠀⠱⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⠉⠀⢀⡤⠊⢿⢄⡀⠀⣠⠎⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢣⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⠒⢌⡲⢍⣉⢀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢦⠀⠱⡄⠀⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⢀⠇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⠀⢇⠈⠦⣀⠀⢀⡠⠋⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⠤⣀⡀⠀⠀⠀⣀⠴⠋⠀⠀⣸⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⣀⠀⠀⣀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢰⢷⡀⣸⠧⠞
⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠉⠉⠙⠳⣦⡀⠘⠾⣳⢏⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠶⣶⣦⣄⡀⠀⠀⠀⠀⣼⠋⠀⠀⠀⢠⣦⠀⠈⠻⣦⣤⡏⠻⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⢰⡟⠀⠀⠀⠀⠈⠉⠀⠀⣰⠏⢸⢣⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣴⡶⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠉⠛⠛⠻⠶⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⣴⠾⠋⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⡞⡉⠁⠈⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡶⢶⡄⢠⡟⠉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠶⠶⠛⣡⡶⠶⣦⠀⢠⡾⠋⠉⣼⡆⠀⢰⡏⠀⠀⠀⢿⣤⠾⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣟⣀⣴⠏⠀⣿⡇⠀⣠⣿⣦⣠⡼⢷⣤⣤⠶⠛⠛⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠈⠻⣭⣤⡴⠞⠋⠙⠛⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⢶⣤⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡉⢿⣦⠀⠀⢠⣿⣇⠀⢰⡖⣀⠀⠀⣠⣾⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣷⡆⢰⠈⣿⡇⢰⣿⠋⠻⣶⣿⣿⡃⠀⣰⣿⣿⣿⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⣸⢻⣿⣧⣸⠀⢹⡿⣾⣃⣤⡀⡘⢿⡜⣿⣦⣿⣿⣻⣿⡞⣧⢰⡇
⠀⠀⠀⠀⠀⠀⢄⠀⠀⢻⢹⡯⠚⣿⠘⢿⣷⣼⠁⢸⡟⣄⢹⣿⡌⣧⠜⣿⣿⡿⢻⣿⣿⠇⠘⣇⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⢸⠁⢀⡟⣽⠛⢸⠇⠠⠀⠻⡻⣄⣿⡴⢉⢿⣿⠿⢻⣆⣿⡟⠁⣿⣟⣿⡇⠀⢹⣤⠞⣻⣿
⠀⠀⠀⠀⠀⢀⡇⢠⢸⣿⡧⢠⡿⠀⠀⡄⠀⢳⣸⠟⠀⡘⠀⠹⣇⠀⢻⡏⡇⠀⢹⣿⣿⣷⠀⠀⣏⢀⣿⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⡇⢸⢸⣿⠆⣿⡇⠀⢀⠀⠀⣼⠃⠀⢠⠇⠀⠀⣿⠀⠈⣿⡿⠀⢸⣿⣷⣿⠸⢰⣧⣼⣧⣴⡶⠶⣾⡏
⠀⠀⠀⠀⠀⢸⣇⢻⣼⣿⣂⣿⡇⡀⢸⠃⣼⠇⠀⢠⡏⢀⡄⡀⣿⢸⢠⡿⢇⠆⢸⣿⣧⡿⠀⣼⠿⣋⡥⠄⠒⣠⡿⠁
⠀⠀⠀⠀⠖⠈⠻⣾⣿⣿⣏⣿⣷⣱⣸⡠⣿⠀⠀⣾⠀⡜⡔⣰⣿⢢⣾⢍⡞⢠⣿⣿⡾⢡⣾⣷⣾⡷⣶⣦⣾⠋
⠀⠀⠀⠀⠘⠄⢐⣨⣿⣿⣿⣮⣿⣿⣿⣿⣿⣇⠀⢻⣾⣿⣴⣿⡿⢛⣥⣞⣴⣿⡿⣫⣴⣟⣛⡒⠒⠲⠦⢬⣝⣻⣦⣄⡀
⠀⠀⠀⠀⣠⠖⠋⠉⠀⡀⢉⣙⣻⣯⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠽⠿⢭⣍⡛⢳⣦⣀⣀⡉⢉⣉⣙⠋
⠀⠀⠠⢾⣁⡤⠀⠂⣻⣭⣿⠟⠛⡩⣽⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⣿⣿⣿⣿⣯⡩⡓⢮⡁⠢⠉⠻⣄⠜⢿⣿⠩⠽⠛⠉⠁
⠀⠀⠀⠀⠙⠿⣶⢶⣾⠏⠀⠠⢊⡼⠛⢡⡿⣵⣿⢏⡾⠟⡝⠀⢹⡄⢹⣿⣿⣿⣿⡌⠌⢻⣦⡀⠀⠈⠣⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠐⢡⡟⢁⣤⡟⣸⣿⡟⢸⡧⠀⠀⠀⢸⡿⠀⣿⣿⣿⣿⣷⠀⠀⢿⣷⣄⠀⠀⠙⢎⣷
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣰⠏⡴⣻⣿⡇⢹⣿⣇⠀⢇⠀⠀⢠⡿⠁⣰⡿⣿⣿⣗⣿⠂⠀⢸⣷⡙⠿⠶⡦⠤⠼⠇
⠀⠀⠀⠀⠀⣠⢞⣀⡴⣛⠵⢋⣵⠟⣿⡇⠘⣧⢻⣆⠐⠀⣴⠋⣠⣴⢿⠁⣿⣿⡯⣿⠀⢀⡾⠁⠉⠓⠶⠆
⠀⠀⠀⠀⠾⠛⠛⠛⠚⠗⠒⠋⠁⠀⢹⣷⠀⣿⠦⢹⠄⠀⣏⣼⡏⠀⠸⡄⡸⣿⣷⣿⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣾⠀⢀⣽⡿⣿⠀⠀⠙⠃⠈⠛⠻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣷⠀⡼⠋⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠇⢰⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⢰⡄⠀⣀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢠⡮⠓⠚⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢂⠀⠀⠀⠈⠁⠀⠀⠀⡔⠉⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠢⡈⠢⠤⠤⢰⡆⠠⠤⠔⣀⠌
⠀⠀⣤⡆⠀⠀⠒⠐⠀⠄⠩⠑⠂⠄⠀⠀⠈⠑⡠⢤⣸⣄⡤⣀⠎⠁⠀⠀⠠⢐⠚⠭⠤⠐⠀⠀⠀⠀⣂⣄
⠀⠀⠨⢂⡀⣠⠀⠀⠀⠀⠀⡀⠀⠀⠘⡃⠒⠴⣶⣾⠟⠻⣷⡶⠆⣒⡚⠃⠀⠀⡀⠀⠀⠀⠀⣀⡀⢀⡸⠉
⠀⠀⠀⠠⠾⢆⠀⠨⠓⠂⣡⠞⠉⠀⠊⠈⣉⣙⡿⠂⠘⠂⠸⢿⣋⣉⠉⠂⠀⠙⠲⣅⠐⠢⠀⠀⠸⢾⠉
⠀⠀⠀⠀⠀⠹⡯⢠⠀⠀⢁⠞⠉⣁⡀⠼⡎⠙⠻⢮⠀⠀⡷⠛⠋⡗⠄⢀⡙⠓⠷⠈⠀⢀⡠⣷⠏⠉
⠀⠀⠀⠀⠀⠀⠰⠐⠁⠂⠀⠀⠂⢀⡶⠊⡁⠐⣒⡾⠇⠺⣷⡒⠃⡘⠰⣦⣀⠐⠀⠀⠲⠐⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀⠜⡱⢿⠀⡁⠈⢰⠿⡈⢣⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⡄⠁⣶⢠⠈⡦⠁⡬⢀⠢⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠁⠈⢾⢿⡿⡏⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣴⣧⠀⠀⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⢣⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠙⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⣧⠋⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢣⠀⠁⢀⡸⣧⡄⠀⣸⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⣴⠋⠀⢹⣿⣦⠙⠁⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣠⣾⣏⣀⣀⣈⣿⣿⣷⡄⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢹⣿⡟⠛⠛⠛⣿⣿⣿⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⣾⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣶⡶⠟⠛⠻⠷⣾⡟⠉⠰⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣶⣋⣤⣥⣍⣠⣭⣤⣥⣄⣶⣶⣶⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣿⣏⣤⣿⣷⣿⣷⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢳⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⣤⠏⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⣤⢿⣹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣣⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⡿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣼⣿⣿⢿⡿⣿⣿⣿⡟⣤⣿⣿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠃⢀⣨⣽⣛⠛⠿⢿⣿⣿⣷⣬⣙⣭⣷⣻⣦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠛⠋⠉⣿⣛⣻⣶⣟⣛⣿⡏⠛⠟⠛⠉⡟⢿⣮⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⢄⠘⣿⣟⠉⢻⣿⡿⠁⢠⠇⣰⠀⡇⣸⡞⠉⣿⣦⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠾⢿⡿⠟⡆⠀⠀⠀⠠⠶⠦⡀⠘⡀⢠⣸⣷⠀⠀⢀⠈⢠⠈⠻⣶⣿⠟⠁⢠⠋⢰⣿⣿⣇⣿⠏⡁⠬⢻⣿⣶⡄⢀⡀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⠀⠀⠀⠀⡼⠆⠰⡄⠱⣾⣿⣧⠀⠆⠀⠂⠀⣉⣀⣼⣥⣀⣐⠁⢠⣿⣿⣿⣿⣿⣁⠤⢈⡁⠿⣿⣾⡿⠁⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⡀⢠⡤⠾⠆⠀⣤⣿⣾⡿⠋⢹⣦⣴⠶⢈⡁⠀⢈⣧⠾⣯⣌⣻⣿⣿⣿⣿⣿⠏⠁⠆⠀⠀⣀⣽⣧⣴⣄⣀⡀⠀⠀⣷⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣏⣠⣤⣤⣼⣿⢿⣿⣄⠀⠈⢿⣿⣿⣿⣿⠀⢸⣿⣷⣿⠻⣿⣿⣿⣿⡿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⡏⢩⣽⣶⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠾⣻⣿⣿⣿⣿⣿⣿⣯⣠⠀⣠⣾⣿⣿⣿⡄⢸⣿⣿⣿⠀⢸⣿⣿⣁⣴⣿⣤⣀⣼⣿⣿⣿⣿⣿⣿⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⠀⣼⣿⣿⡟⠀⠉⠛⠻⠿⣿⣿⣿⣿⠿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⡿⣉⣿⣿⣿⣿⣧⡈⠛⠛⠉⣠⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢰⠆⣰⣭⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠻⣿⢟⠻⣟⡿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣶⣶⣶⣤⣀
⠀⠀⣀⠐⣂⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣤⠀⢤⣿⣏⣳⡘⢧⡉⠮⡛⠿⢿⣿⣿⣿⢿⣿⣿⡿⠿⠿⠟⠛⣛⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠧⣀⠀⠀⠁⠠⡉⠻⠿⣿⡤⢽⣦⣬⣶⣤⠙⣛⣛⣓⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠑⠒⢤⣤⣀⣉⠱⢦⣄⡀⠉⢛⣻⣿⠿⣷⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣻⡌⡤⠊⠉⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣉⣁⣹⣶⣿⡏⠻⡙⠣⡈⠛⡉⠳⣿⣿⣿⡟⠛⢶⣭⣿⣟⣻⣿⣿⣿⣷⣶⣶⣴⣶⣿⠟⠛⠉⡩⠛⣀⡞⢁⡴⠛⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⡟⠛⢭⡾⢿⡟⢧⡈⢢⡈⠠⠈⢢⠈⠢⠀⠀⠁⢈⣳⡶⠲⢾⣿⣿⣿⣿⣿⡯⠉⣻⣶⡾⠁⠀⠀⡔⠁⢠⠏⢠⠎⣠⠞⣡⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⠏⠰⣦⠈⢷⡋⠳⣄⠱⡄⠱⡀⠱⡀⣣⣀⣇⣀⡀⠘⣇⣀⣦⠀⣜⠓⢻⣿⣿⣃⣼⣽⣿⠃⠀⠀⣼⠀⢀⠇⢀⡇⣰⢋⡞⣡⣏⣬⣥⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠘⢷⣶⣷⣾⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣧⣿⣶⣤⣿⣿⣿⣿⣿⡋⠀⠁⢠⣧⣴⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⠖⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⠻⢮⣙⠻⢿⣿⣯⣽⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢛⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⠋⠉⠙⠻⣛⢶⣍⠓⣾⣿⣿⣿⣿⣭⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠟⢛⣡⡶⢛⡽⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠉⢾⣿⣿⣿⢷⣤⣤⣬⣭⣍⣉⣉⣉⣭⣭⣭⣭⣷⣴⠶⠿⢛⣫⣶⣫⣶⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣶⣤⣀⠀⠁⢄⣰⣿⣿⣿⣟⠲⠶⣶⣤⣬⣍⣉⣉⣉⣉⣩⣤⣤⣤⣶⠾⠿⣛⣫⣥⣿⣟⢀⡠⠖⠉⢈⣹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡐⢠⣼⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣭⣭⣭⣭⣭⣭⣤⣶⣶⣾⣿⣿⢿⠿⠋⠙⣏⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⠿⣄⣐⠀⠉⠉⠉⠉⠛⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⢈⣠⡴⠞⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠉⠛⠻⠷⠶⠶⠶⣤⣤⣠⣤⣤⢤⣤⡤⠶⠶⠟⠛⠋⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣈⣉⣁⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣉
⣿⣿⣿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠶⢶⣤⣤⣀⡀⠀⠀⠈⠙⢿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠉⠉⠉⠙⠛⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣶⣄⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠃⠀⠀⠀⠀⠀⠈⠛⢶⣤⣤⣤⣤⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⣶⣶⣶⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠤⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⣀⡀⠀⠀⠀⠀⠀⠆⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢰⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠐⠃⠀⢠⣀⠤⣄⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠃⢀⠌⣥⡔⣈⡭⣤⡛⢂⠀⢀⣀⡂⠀⢀⠞⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⣂⠠⠄⢋⣠⠤⣌⠡⣆⠘⠁⢀⡀⠂⠀⠔⢠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠐⠀⡄⡹⡆⠉⢠⡟⢸⠁⠘⠇⠈⠂⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠇⠁⠀⠈⠛⠇⠀⡤⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠁⠀⡐⠛⡀⠛⡠⡠⠞⠀⢀⡈⠂⠀⡠⠐⠁⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⢖⠀⣤⡛⠐⠒⠉⠉⡸⠟⢛⣥⠀⠀⠈⣁⡱⢶⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢰⠀⠀⠈⠛⠙⠶⡄⢻⠄⢠⡄⢹⠀⡴⠋⠀⠁⠈⠀⠀⠸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⠐⠀⠀⠆⠀⠀⠀⠠⠀⢴⠀⠀⠀⠀⠈⢀⡅⠀⡀⠁⠀⠀⠀⠀⠆⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠡⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠋⠁⣴⣶⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⣠⠄⠀⠀⠀⢄⠐⠗⠌⢁⡀⢡⡀⠀⠀⠀⠰⣦⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢠⡜⠧⠀⠀⠀⢀⠃⠀⠀⠁⠀⠢⠈⣧⠀⠀⠀⠀⠈⠣⠂⠀⢧⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠉⠂⠀⠀⠀⣈⠀⠄⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⢀⣤⠀⣀⢠⣄⠀⢿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠏⠀⡿⠁⠙⠛⣿⠙⠟⢸⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣤⢰⣷⠀⠀⠀⣻⡄⠃⢸⡆⠀⠄⠀⠀⠀⠈⠛⠓⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⢹⢴⠉⢳⡀⠀⠘⣷⡿⢸⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⢀⡼⡀⢀⠳⠀⠀⡻⢷⣸⡄⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⢸⣰⠟⢰⠟⠀⣰⠇⢨⣼⠃⠀⢠⣤⣇⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⡀⠀⠼⠀⢀⣠⢄⡴⢊⡙⢠⡟⠀⡰⠛⢀⣾⡿⡂⡀⠫⡀⠀⣠⡀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠊⢅⠀⠰⠦⡄⠉⠠⠈⠰⠋⢀⣼⠀⠸⡇⠀⢸⡇⡷⣇⠀⠀⠈⠀⢻⡀⠠⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⢤⣀⣱⠤⠘⠐⠢⡅⠀⣤⠴⠋⠛⡀⢠⣿⡀⠈⠳⠽⣏⠤⠄⢀⡖⠤⠹⢆⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠃⠀⠐⠋⣠⠔⢣⠀⠀⠛⡀⠠⡐⠀⠀⡇⢸⠷⣴⡂⢠⣀⣤⣤⣶⠚⠀⠀⠰⣄⠲⠟⠀⠡⠤⠴⠋⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠙⠂⡞⠀⠤⠂⠠⣤⣈⠉⣀⣀⠰⠄⢹⠃⠌⡉⠁⠀⠀⠙⠂⠀⠀⠈⣀⠀⠋⠔⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠥⠀⠁⠠⢀⡀⢰⠋⠡⠸⡀⣶⠖⠉⠉⠋⣙⠃⠊⠁⠘⢦⣟⡡⠀⠀⠀⠀⢸⣀⡇⠦⠠⣬⡁⠈⠙⠋⠶⠂
⠀⠀⠀⠀⠀⠀⢆⣰⣆⡀⠀⠈⡠⠂⠘⡒⠲⠞⢡⡞⠁⠁⠀⠆⠑⠒⠒⠀⠀⠈⢄⡘⠧⡀⢈⡀⠁⠤⣦⡔⠉⠤⠄⠀⢂⡰⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⠃⠀⠢⠺⠉⢀⠀⠀⡌⠀⠀⢠⡀⡙⠀⠀⠛⡀⢸⡎⠁⡶⠍⠋⠀⡀⠆⠈⡁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤
⣿⣿⠿⢿⣿⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣥⣬⣽⣿⣯⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⡿⢿⣿⣦⣀⣀⡠⠐⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⢘⠀⠀⠀⠀⠀⠂⢹⣿⣿⠋⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢷⣾⣆⠀⠀⠀⠀⠀⠘⠟⣧⣤⣤⣀⣀⡀⠀⠀⠼⠛⠁⠈⠀⠉⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⢸⣿⠛⠋⠁⠀⠀⠀⠀⠀⢠⡖⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⣠⣾⣀⣤⣴⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠖⠶⠶⠤⢄⣀⣀⣀⣀⣠⣄⠙⣶⣿⣿⣷⣤⣤⣀⣤⣤⣶⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⣤⣄⠁⠀⠀⠀⠀⣀⣀⣠⡬⠀⠈⠉⠙⠿⠿⠗⠘⠋⣡⠈⠩⣤⣾⣿⣿⡿⠓
⠀⠀⠀⠀⠀⢠⡆⣺⣠⣟⣼⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠙⠛⢿⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠉⠉⢻⣽⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣇⣤⣤⣴⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⢻⣯
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠔⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠙⢿⡿⠁⣠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠘⢷⣶⣤⣤⣶⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠿⠛⠛⠛⠛⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡗⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠋⠉⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣭⣄⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⣿⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣏⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣿⠿⡆⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⡷⠷⣶⠀⠀⠀⣿⣿⢶⣤⣤⡴⣾⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⣶⣶⡿⠀⠀⠀⠰⣿⣶⣶⣿⣿⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⣀⣀⣀⣤⣴⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠙⢿⣿⠟⠁⠀⢀⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⠀⢻⣿⣷⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡄⠀⠀⠈⠉⠉⢉⣉⣭⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟⠉⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣤⣀⣤⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠿⣿⣿⣿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠸⣿⣦⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⠇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠛⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⠀⠀⠀⠀⣠⣦⣄⠀⠀⠀⣠⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠈⠁⢀⣴⣿⣿⣿⣿⣷⣄⠀⠉⠀⠈⢻⣿⡆⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠷⢦⡀⠀⣸⣿⠃⠀⠀⠀⣴⣿⣿⠟⠁⠀⠙⣿⣿⣷⡄⠀⠀⠀⢻⣿⠀⠀⢀⠴⠶⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠈⣿⡆⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⢿⣇⠀⠀⠀⠀⣿⡇⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⢠⣿⠃⠀⠀⠀⢀⣿⠏⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡈⠻⣧⡀⠀⠀⠸⣧⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣼⡏⠀⠀⠀⣠⡾⠋⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠈⠻⠿⣿⣦⣈⠻⢦⣄⠀⠙⣧⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀⣼⠏⠀⢀⡴⠞⣉⣤⣾⣿⠿⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣤⣤⣤⣤⣄⣈⣁⡀⠈⠳⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢡⠞⠁⢀⣈⣁⣠⣤⣤⣤⣤⣴⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡌⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠹⡄⠀⠀⠀⠀⠀⠀⢠⠟⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢩⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠿⣦⡙⠂⠀⠀⠀⠀⠀⢉⡴⠿⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⣿⡿⠃⢀
⠀⠀⠀⠀⠀⢰⣶⣟⠁⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠟⠀⠀⠀⢙⣷⡄
⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⡀⠀⠉⠛⠿⠶⣦⣤⣤⠄⠀⠀⠀⣀⣠⠴⠂⠀⡀⢠⠀⠙⠶⣤⣀⠀⠀⠀⠀⣀⣤⣤⡶⠿⠛⠉⠀⠀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠀⠀⠀⣶⣶⣶⡶⠾⠛⠉⠀⠀⣠⣾⠃⢸⣷⣄⠀⠈⠙⠻⠷⣶⣶⣶⡶⠀⠀⠀⠚⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣤⣤⣴⣾⠟⠁⠀⠀⠙⢿⣷⣶⣶⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢻⡀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⡇
⣿⣷⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⢷⡀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠘⢿⣿⢸⣿⡄⢠⣄⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢣⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⣠⣤⣬⣾⣿⡟⢸⣿⣷⡀⠀⠀⣀⠼⠤⣄⡀⠀⠀⢠⠞⣡⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠈⠛⠿⣿⠟⢧⡘⣿⣿⡇⢀⠌⠁⠀⠀⠀⠹⣦⣰⠋⡼⠁⠀⠀⠀⠀⠠⠴⢶⡍⠁
⠀⠀⢀⣤⣶⣶⣾⣿⡿⣤⣸⢤⠸⠗⠀⠀⠀⠘⠃⠸⠀⠀⠀⠀⣀⣠⡤⠴⢋
⠀⠀⠙⠿⣿⣿⡿⠋⠀⠀⠗⠚⠗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⡈⢳⡀⠀⠀⢀⣀⠴⠊⠁⠘⠀⡄⣧⣹⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡏⠉⠉⠀⠀⠀⠀⠀⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆