Dupe Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⢀⢈⣻⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⡆⠀⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣶⣶⣶⣶⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠐⠂⠘⠀⠀⠀⠀⠃⠐⠀⠀⠠⠈⠁⠀⠀⠀⠃⠀⠠⠀⠀⠀⠉

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣿⠉⠈⣉⣉⣉⠉⠉⢉⡉⠉⠉⡉⠉⢉⣉⣉⠉⠉⢉⡉⠉⠉⠉⢉⡉⠉⠉⢉⣉⠉⠉⠉⠉⣉⠉⠉⠉⣉⡉⣉⣉⠉⣉⣉⣉⣉⠉⠉⣿
⠀⠀⣿⠀⠀⣿⠉⠙⣷⠀⢸⡇⠀⢸⡇⠀⣿⣉⣙⣷⠀⢹⡇⠀⠀⠀⢸⡇⠀⣴⠋⠉⠋⠀⠀⣴⠿⣧⠀⠀⠉⢹⡉⠉⠀⣿⣉⣉⠉⠀⠀⣿
⠀⠀⣿⠀⠀⣿⣀⣠⡿⠀⠸⣧⣀⣸⠇⠀⣿⠉⠉⠁⠀⢼⣅⡀⣀⠀⢸⡇⠀⢿⣄⣀⣄⠀⣰⡿⠶⢿⣆⠀⠀⢸⡇⠀⠀⣿⣍⣁⣀⠀⢀⣿
⠀⠀⣿⣀⣀⣉⣉⣉⣀⣀⣀⣈⣉⣁⣀⣀⣉⣀⣀⣀⣀⣈⣉⣀⣈⣀⣈⢁⣀⣀⣉⣉⣡⣀⣉⣀⣀⣀⣭⣤⣤⣨⣁⣀⣄⣩⣩⣭⣩⣀⣼⣿
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠀⠀⠘⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⢠⣶⠶⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⢸
⢸⣇⠀⠀⣤⣤⣤⣄⠀⠀⢠⡄⠀⠀⣤⠀⢠⣤⣤⣤⣄⠀⢠⣤⠀⠀⠀⠀⠀⣤⠀⠀⢀⣤⣤⣄⠀⠀⠀⢠⣤⠀⠀⠀⢠⣤⣠⣤⡄⠀⣤⣤⣤⣤⡄⠀⠀⣿⢸
⢸⣿⠀⠀⢿⡇⠀⢹⣷⠀⢸⡇⠀⠀⣿⠀⢸⣷⣠⣠⣿⠃⢠⣿⠀⠀⠀⠀⠀⣿⠀⢰⡞⠀⠀⠉⠀⠀⢰⡿⢻⣧⠀⠀⠀⠀⣧⠀⠀⠀⣿⣧⣤⠀⠀⠀⠀⣯⢸
⠘⣷⠀⠀⣿⣇⣠⣴⠏⠀⠸⣧⣀⣰⡿⠀⢸⣿⠉⠉⠀⠀⢰⣿⣀⠀⣀⠀⢰⣿⠀⠘⣷⣄⣠⣤⠀⢠⣿⠿⠛⢿⣇⠀⠀⠀⣿⠀⠀⠀⣿⣇⣀⣀⡀⠀⠀⣿⢸
⢸⣿⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠉⠀⠀⠈⠉⠉⠁⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣿⢸
⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸
⣿⣿⡟⠉⠛⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢘⢆
⠤⡔⠁⠀⣱⠄
⠀⠈⢢⡜
⠀⠀⠈⠀⠀⠀⠀⠀⡤⠒⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣇⠀⠀⠀⡠⠊⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡄⠀⠈⠑⢄⠠⠋⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢧⠀⠀⠀⠈⠃⠀⠀⠀⢀⡠⢶⣋⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢰⢄⠀⠀⠀⢳⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⡤⠚⠉⠒⢤⣈⠳⢄⠀⠐⠨⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠴⢅⠀⡙⠁⠀⠘⡆⠀⠀⠀⠀⠀⠓⠤⣤⠴⠯⣍⠲⣄⠀⠀⠈⠳⢄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠹⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⡄⠀⠀⠀⠳⡌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢆⠀⠀⠀⠘⣎⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢧⠀⠀⠀⠘⡎⢧⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠘
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⣆⠀⠀⠀⢹⠘⡆⢸⠁⠀⠀⢹⠀⠀⠀⣀⠴
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄⠀⠀⠈⡇⢷⠈⠢⢤⡤⠞⢀⡤⠞⠁
⠀⠀⠀⠙⡖⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⢧⠀⠀⠀⣧⢸⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⡄⠀⠀⠀⠀⢀⡤⠚⢷⡃⣼⡤⣶⠯⠏⣸⣠⠔⠋
⠀⠀⠀⠀⠀⠀⢏⠙⠋⠁⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⣠⠖⠉⠀⠀⠀⠉⠁⠀⠈⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠲⡶⠒⠋⠀⠀⠀⠀⢀⣠⠔⠋

⠀⠀⣰⠓⠰⠦⢄⡀⠏⠙⡄⡠⠤⠶⠸⢦⠀⠀⢠⠞⠐⠦⢤⡀⡟⠙⡇⣀⠤⠶⠚⢶⡀⠀⢀⡞⠃⠶⠤⣀⠸⠉⢃⢀⠤⠴⠒⠷⡄
⠀⠀⠘⠢⠤⠄⡀⠩⢧⣴⠯⠀⠀⠤⠤⠊⠀⠀⠈⠢⠤⠄⢀⠨⢷⣴⠯⠄⡀⠤⠤⠚⠀⠀⠀⠓⠤⠤⠀⡀⠽⣤⡾⠅⠀⠠⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣠⢫⡏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡏⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣽⠱⡀
⠀⠀⠀⠀⠀⣀⡔⠁⠨⢣⠈⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⢠⢻⠀⠑⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⠝⡀⠙⣤⡀
⠀⠀⠀⠀⠈⠉⣹⠔⠁⠈⠣⣭⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⢩⡥⠃⠀⠓⢬⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⢉⡧⠊⠀⠙⢼⡍⠉⠁

⠀⡠⠒⠉⢣⠀⠀⠀⠀⠀⠜⠉⠲⣄⠀⠀⠀⣠⠔⠋⢳⠀⠀⠀⠀⠀⢰⠉⠓⢄⠀⠀⠀⢀⠴⠊⠙⡄⠀⠀⠀⠀⢠⠋⠑⢦⡀⠀⠀⢀⡤
⡊⠀⣀⣀⣸⣄⠀⠀⠀⣴⢇⣀⡀⠈⢳⠤⡞⠁⢀⣀⣘⣤⠀⠀⠀⣠⢆⣀⣀⠀⢱⠄⢔⠁⢀⣀⣀⣧⡀⠀⠀⢠⡜⣀⣀⠀⠙⡦⢴⠋
⡗⠋⠁⠀⢀⡸⠀⠀⠘⢅⡀⠀⠉⠑⣦⣀⣶⠊⠁⠀⠀⣨⠃⠀⠀⢧⡀⠀⠈⠑⢺⣀⣼⠚⠉⠀⠀⣀⠇⠀⠀⠻⣀⠀⠀⠉⠲⣇⣰⠖⠉
⡌⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⡘⣿⢇⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⢩⢻⢣⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⢿⡟⡈⠉
⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡿⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠸⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⢸⡇⠙⢄
⢆⣐⠛⠶⠀⠀⠀⠀⠀⠀⠠⠖⠛⣆⡜⠀⠣⣰⠛⠲⠀⠀⠀⠀⠀⠀⠀⠶⠛⣂⡰⠃⠳⣀⡚⠳⠆⠀⠀⠀⠀⠀⠀⠰⠚⢳⣠⠃⠘⣄⡞
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⣀⢄⡀⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⢀⢤⡀⠀⠀⠀⠀⠀⢀⡠⣄
⠀⠀⠀⢠⠁⠀⠙⢦⡀⠀⡠⠊⠁⠀⡇⠀⠀⠀⠀⢀⡅⠀⠈⠢⡀⠀⣠⠞⠁⠀⢰⠀⠀⠀⠀⠀⡎⠀⠈⠳⣄⠀⢀⠔⠉⠀⢈⡀
⠀⠀⠴⠛⠒⠢⢤⣀⠋⠙⣄⡠⠔⠒⠚⢦⠀⠀⢰⠛⠒⠂⠤⣀⡟⠉⣇⡠⠤⠒⠚⢳⠀⠀⠀⡞⠓⠒⠤⣄⡸⠉⢃⣀⠤⠐⠒⠻⡄
⠀⠀⠈⠢⠤⠤⠠⠬⢷⣾⠧⠄⠤⠤⠤⠊⠀⠀⠈⠒⠤⠤⠀⠨⢷⣶⠯⠄⠠⠤⠤⠚⠀⠀⠀⠑⠤⠤⠠⠠⠽⣶⡿⠄⠀⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⡠⢩⡇⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣏⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣿⠑⡄
⠀⠀⠀⠀⠀⣠⣜⠁⡸⢣⠀⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠁⢠⠻⡀⢑⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡂⢀⠞⡄⢘⣦⣀
⠀⠀⠀⠀⠈⠁⢸⠔⠁⠀⠓⢧⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⢨⠔⠃⠀⠑⢼⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⡧⠊⠀⠘⠺⡄⠉⠁

⢀⡤⠚⠉⢣⠀⠀⠀⠀⠀⡚⠉⠓⢄⠀⠀⠀⡠⠖⠉⢱⠀⠀⠀⠀⠀⢸⠉⠑⢦⡀⠀⠀⣀⠔⠋⠙⡀⠀⠀⠀⠀⢠⠋⠙⠢⡀⠀⠀⢀⠴
⡋⠀⣀⠠⠼⣄⠀⠀⢀⡴⠥⠄⣀⠀⡱⠲⡎⠀⣀⠠⠬⢦⡀⠀⠀⣤⠧⠤⣀⠀⢹⠆⢞⠁⢀⡀⠤⢧⡀⠀⠀⣠⠾⠤⢀⡀⠈⡖⢶⠁⢀
⣗⠉⠀⢀⣀⠼⠀⠀⠘⠄⣀⡀⠀⠁⣷⣀⣞⠈⠀⠀⣀⡨⠃⠀⠀⠣⣀⡀⠀⠉⢺⣄⣼⡊⠁⠀⣀⡠⠇⠀⠀⠻⢄⣀⠀⠈⢺⣇⣰⣃⠁
⢆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡜⣿⢣⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡠⢻⠣⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢡⢻⡟⡄⠉
⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⣿⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡸⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣸⣇⠘⢄
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠖⠛⠉⣉⠁⣠⣄⢀⣤⡀⣴⡄⣠⣄⢀⣤⡀⢉⠉⠛⠓⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⣠⣄⢰⣷⠊⠿⠇⢙⣋⣈⣉⣀⣉⣁⣈⣉⣀⣛⠁⠻⠏⢰⣷⢀⣤⡀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⣠⡀⠿⠇⢈⣉⠠⠆⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢄⣉⠀⠿⠆⣠⣀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣶⡆⠉⣡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡉⠃⣶⡆⠈⠻⣦⡀
⠀⠀⠀⠀⠀⢀⡼⠋⡀⠿⠗⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠐⠿⠆⡈⠻⣄
⠀⠀⠀⠀⢠⡞⠁⡘⠻⢂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠸⠿⡀⠙⣦
⠀⠀⠀⢀⡟⠀⠹⠿⢠⠎⠀⠠⠤⠤⠤⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠈⢦⠸⠿⠀⠘⣇
⠀⠀⠀⣸⠁⠺⠷⢀⠆⠀⢀⣀⣀⠀⠀⣀⡀⠀⣀⠀⣀⣀⣀⠀⢀⣀⠀⠀⠀⢀⡀⠀⣀⣀⡀⠀⠀⢀⡀⠀⠀⣀⣀⣀⡀⣀⣀⣀⣀⠀⠀⢇⠰⠿⠂⢹⡄
⠀⠀⠀⣿⠀⣾⡆⢸⠀⠀⢸⡏⠙⢷⡄⢸⡇⠀⣿⡅⡿⠉⢙⣷⢸⡿⠀⠀⠀⢸⡇⢀⡍⠉⠛⠀⢀⣾⢿⡄⠀⠉⢿⠋⠁⣿⣏⡉⠉⠀⠀⢸⠀⣾⡆⢸⡇
⠀⠀⠀⣿⠀⣴⡄⢸⠀⠀⢸⡇⢀⣼⠇⢸⡇⠀⣿⠃⣿⠛⠋⠁⢸⣧⠀⠀⠀⢸⡇⢿⣆⠀⣀⠀⣼⣷⣾⡇⠀⠀⣿⡆⠀⣿⡏⠁⠀⠀⠀⢸⠀⣴⡄⢸⡇
⠀⠀⠀⢿⠀⣩⣅⠐⡄⠀⠘⠛⠛⠋⠀⠈⠛⠛⠋⠀⠛⠀⠀⠀⠘⠛⠛⠛⠃⠘⠃⠈⠛⠛⠋⠐⠛⠀⠀⠘⠃⠀⠛⠃⠀⠛⠛⠛⠛⠀⠀⡜⢀⣍⡁⢸⠅
⠀⠀⠀⠘⣇⠘⣉⣀⠁⡀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⠄⠀⡴⣁⣼⡛⢠⡿
⠀⠀⠀⠀⠹⣆⠘⢛⣄⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⣜⠛⢡⡿⠁
⠀⠀⠀⠀⠀⠘⢧⡘⠛⣠⣄⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣤⡝⠛⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡛⠃⣴⡄⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠡⣴⡌⠛⣣⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣉⠁⠾⠆⣠⡈⠑⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠚⢉⣠⡀⠿⠆⢉⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡙⠃⠺⠗⢠⣤⠀⣈⡉⠀⠒⠂⠐⠒⠒⠒⠒⠒⠒⠂⠈⣉⡀⢠⣦⠰⠷⠂⠛⣁⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣌⣉⡈⠛⠃⠻⠗⠰⢿⠂⣾⡆⢺⠗⠘⠿⠃⠛⠋⣈⣉⣤⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⠉⠁

⠰⣤⣤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠
⠀⠹⣿⣷⡄⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣤⣾⠋
⠀⠀⠀⠹⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠠⣶⣶⣦⡀⠀⠂⠀⠀⣠⡰⣷⣾⣿⡏⠁⡆
⠀⠀⠀⠀⢈⣻⣿⣧⣤⣤⣠⣄⣄⣀⣠⣤⣤⡤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⡇⢫⡆⠀⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⠇⠈
⠀⠀⠀⢰⢘⣽⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⠟⠁⠀⠘⠿⣿⣿⣿⣿⣿⡿⣿⣿⡿⢽⢿⣿⣫⣟⢁⡇⠀⠀⢸⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠠⠀⡈⣿⣿⣿⣿⣝⡁⠌⢁⡈⠈⠰⠄⠙⠀⠀⠘⣿⣿⣿⣿⡇⠙⡓⠹⠋⠛⠺⣸⣇⣼⠀⠠⢀⢸⣿⣿⣿⢁⠼⠑⣬⡈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⢈⠁⠠⠐⡉⠐⡂⣤⡰⢀⡴⠀⣼⣟⡋⢻⡇⠰⢀⠀⠀⠁⠀⣾⣿⡟⠀⠀⢸⠁⠟⣸⡏⠀⠀⢸⡿⡇⠙⢻⣿⠇
⠀⠀⠀⠀⠀⠀⣾⡿⡛⠃⠀⠀⠀⠀⢂⠀⠄⣿⣿⣿⠷⠄⠁⢸⣇⠀⡇⠰⠄⠀⠀⢂⠁⣿⣿⡇⠀⠀⢈⠀⠸⠋⠁⠀⠀⣾⡅⣯⠀⠸⣿
⠀⠀⠀⠀⠀⠀⣿⡟⠐⠉⢊⡀⢀⡀⡅⠐⠃⢹⣿⣿⣄⠀⠀⢸⣿⡄⡇⠀⠀⠀⠀⢤⢸⡇⢸⡇⠀⠠⣾⠀⠀⠀⠀⠀⠀⠿⡷⠛⠄⠀⣧
⠀⠀⠀⠀⠀⠀⣻⡷⠂⡇⢀⠀⠏⡁⠗⠀⢈⣽⣿⣿⣟⣓⡥⠼⠿⠖⣣⠀⠀⠄⠀⠀⣾⣧⡾⠀⠀⠀⢹⠁⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠈
⠀⠀⠀⠀⠀⠰⣾⣧⣔⣀⣈⣋⣷⣿⣇⣺⣿⣷⣏⡙⠁⣷⣤⣤⣼⣿⣿⠐⠀⣄⠀⠀⣿⡏⠀⠀⠀⡀⢸⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⠋⢁⣤⠞⠋⠋⣿⣺⣿⣿⣿⣿⣿⣿⠈⠀⠀⠸⢻⣿⣧⣸⣇⣠⣷⣾⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣶⡟⠁⣀⣤⣶⣿⣾⣿⣿⣿⣿⣿⣿⠄⠂⠀⠂⠹⠉⡻⠿⣿⠛⠛⠉⠀⠂⠀⡀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⡿⣿⠿⢿⡿⠿⠿⠿⠛⠛⢻⡁⠆⠐⠀⠀⠀⠸⣀⣠⣤⣤⣤⣤⣌⠀⠀⠀⠀⢸⡇⠀⡀⠀⢸⣧⢀
⠀⠀⠀⠀⠀⢈⣿⡏⠛⠛⠛⠛⢛⡏⠀⢸⠟⡋⠐⢺⡁⠛⠉⠃⠘⠋⠛⡟⠚⠚⠓⣛⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠠⠤⢼⡧⠀⢄⠀⠻⣿⡀⠡
⠀⠀⠀⠀⢘⡞⠁⠀⠀⠀⠀⢀⣼⡇⢠⣾⠆⠓⠀⠸⡇⠀⠀⠀⠀⠀⠠⠷⠀⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣦⡀⡀⠀⠹⣿
⠀⠀⠀⠀⠟⠀⠀⠀⠀⠤⠾⠿⠟⠿⠟⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠹⣄
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣽⠻⢿⠟⣩⣽⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠛⠛⣃⣼⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠘⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠶⠲⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⣡⡾⢻⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠿⠿⠿⠿⠿⠿⠿⠃⢸⡿⠶⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠆⢸⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄⢸⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⣶⠀⣀⣀⣀⡀⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⢸⣿⡀⠉⠉⠉⠁⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠃⢸⣿⣧⠘⠛⠛⠁⢸⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠀⢿⡘⣷⡈⠛⠃⢸⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠈⢷⣌⠛⢶⣄⣘⠁⠀⠀⣶⣄⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⡙⢷⣤⣈⠙⠛⠛⠻⣿⠉⠻⣦⡀⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠈⠙⠛⠻⠶⠶⣿⣴⠞⢋⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⣿⣦⡀⠀⠀⠀⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢿⣦⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣼⣧⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⣿⣶⣶⣿⣿⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⣷⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠙⠛⠛⠋⠛⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣦⣾⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⣸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⣿⣿⣿⣿⡿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⠋⠀⠀⠀⠀⠙⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠀⠀⠀⠈⠻⣿
⣿⡇⠀⠀⢰⣦⣄⡀⠀⠘⢿⣿⣆⠀⠀⢀⣤⣶⣿⣧⠀⢻⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⠏⢀⣿⣷⣦⣄⡀⠀⠀⣴⣿⡟⠁⠀⢀⣠⣤⡄⠀⠀⢹
⢻⣧⠀⠀⠈⣿⡷⡉⢷⣤⣀⠛⢿⣆⠀⠀⢻⣿⣿⣿⣧⠈⢿⣿⣿⡟⣰⡄⢻⣿⣿⡟⠀⣾⣿⣿⣿⡏⠀⠀⣼⡿⠋⣀⣴⠞⠉⠿⣿⡇⠀⠀⣼
⠈⠻⣷⣄⣸⠉⢇⠀⠀⠹⣿⣷⣤⡈⠀⠠⣾⣿⣿⣿⣿⣇⠸⣿⠏⣰⡿⣿⣄⠻⣿⠃⣼⣿⣿⣿⣿⣷⠄⠀⢀⣤⣾⣿⠏⠀⠈⡰⠉⣇⣠⣼⠟
⠀⠀⠀⠀⡇⠀⠀⠣⠈⠂⠹⣿⣿⣿⣷⣤⡀⠙⠻⣿⣿⣿⣆⠋⣴⣟⡞⣯⣻⣆⠙⢰⣿⣿⣿⠟⠋⢀⣤⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸
⣶⡀⠀⠘⡇⠀⠀⠀⠐⠀⠀⢻⣿⣿⣿⣿⡿⠳⠄⡈⠛⡿⠃⡜⣿⣿⣿⣿⣿⣯⢆⠘⣿⠟⢁⣤⠎⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⣿⣷⣤⡀⢇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠈⡷⠀⣼⣷⠹⣿⣿⣿⣿⢇⣾⣦⠀⢦⠀⠘⡆⠘⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⡄⢀⣴⣿
⣿⣿⣿⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⠞⢀⣼⣏⣿⡇⢹⣿⣿⡏⣾⣿⣼⣧⡈⢣⡀⠀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⣰⣴⣿⣿⣿
⢸⣿⡿⢁⣴⣿⠷⡄⠀⠀⠀⠀⠀⠈⢆⠀⠀⣠⠎⠀⣬⡭⢭⣽⣿⡆⢿⡿⣰⢸⣧⣭⣭⣥⠀⠳⣄⠀⢀⡜⠁⠀⠀⡴⠁⠀⢠⠾⣿⣦⡈⢿⣿
⠈⣿⣱⡿⢋⣠⣶⡿⣧⡀⠈⡄⠀⠠⡈⢣⡰⠃⢀⣶⠶⠶⣶⢸⣿⣷⠘⢃⣿⢸⡇⣶⠶⠶⣶⡀⠙⣦⠎⠀⠀⠀⡸⠁⢠⣾⣿⣶⣄⡉⢿⣞⣿
⠀⠘⣿⣵⡿⣿⢟⣾⣿⣿⡄⠰⡀⠀⢈⡶⠁⣠⣿⡁⠰⠾⠟⠸⠟⠿⠃⠸⠟⠸⠃⠿⠶⠆⠈⣿⣄⠈⢶⡀⠀⢰⠃⢠⣿⣿⣿⣿⣏⢿⣦⣻⠃
⠀⠀⢸⡟⢸⡟⣾⣿⣿⣿⣿⠀⢳⣠⠋⢀⣴⡟⢛⣛⣛⣛⣛⣛⣛⣛⠛⢛⣛⣛⣛⣛⣛⣚⣛⡛⢻⣧⡀⠹⣦⡎⠀⣿⣿⣿⣿⣿⣿⡆⢻⡇
⠀⠀⢈⠀⣿⡇⣿⣿⣿⣿⣿⠀⠸⠁⣠⣾⡟⣸⣏⣿⣿⣿⣿⣿⣏⣿⡇⣾⣏⣿⣿⣿⣿⣿⣯⣿⡆⢻⣷⡄⠘⠇⠀⣿⣿⣿⣿⡿⣼⣿⠀⡁
⠀⠀⣸⣧⠸⣿⡹⣿⣿⣿⣿⠀⠀⣴⣿⡿⢁⣙⣛⡛⢛⣛⣛⣛⣛⣛⠁⢙⣛⣛⣛⣛⣛⠛⢛⡉⣉⠀⢻⣿⣆⠀⠢⣿⣿⣿⣿⢣⣿⠇⣼⣇
⠀⢰⣿⡙⢷⣽⣷⡙⣿⡿⠃⢀⣾⣿⡿⢁⣿⢟⣿⡇⢸⡿⣿⣿⣟⡿⠃⠻⣟⣿⣿⡿⣿⠀⣿⠟⠿⣷⡈⢿⣿⣷⡀⠙⢿⡿⢣⣾⣯⣾⢋⣿⡄
⢀⣿⡝⢿⣄⠙⠻⣿⠎⠀⢠⣾⣿⡿⠁⣼⣿⣯⣿⡧⢸⡷⣿⠟⢛⣩⣭⣭⣍⡛⠿⣿⣿⠃⣿⣯⣿⣿⣧⠈⢿⣿⣷⡄⠀⠱⣿⠟⠋⣠⡿⢻⣿
⢸⣿⣷⣄⠙⠿⠖⠀⠀⢠⣿⣿⡿⢁⣀⣀⣀⣀⣀⣀⣀⣠⠀⣰⣿⡿⠛⠛⢿⣿⡆⠀⣀⣀⣀⣀⣀⣀⣀⣀⡈⢿⣿⣿⡄⠀⠀⢲⡿⠋⣠⣿⣿
⢸⣿⣿⡿⢛⠏⠀⠀⠀⠈⠛⠛⢡⣿⣿⣧⣽⣿⣿⣿⣿⣿⡀⢻⣿⡀⠢⠔⣨⣿⡇⠀⣿⣿⣿⣿⣿⣯⣬⣿⣿⡌⠛⠛⠋⠀⠀⠀⠹⡻⢿⣿⣿
⢸⣿⠋⠀⡌⠀⠀⠀⠀⠀⠀⢠⣤⣥⣭⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⡄⠀⠀⠀⠀⠀⠀⢣⠀⠙⣿
⠾⠁⠀⢀⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢸⠀⠀⠈
⠀⠀⢀⣈⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣒⣓⣒⣒⣒⣒⣒⣒⣒⣶⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣀⡀
⢀⣾⠟⠁⢹⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣒⣒⣒⣂⣂⣐⣒⣒⣂⣀⣀⣀⣀⣒⣒⣘⣛⣻⣿⣿⠿⣿⣿⣿⣿⣿⣆⠂⠀⢠⡏⠈⠻⣷
⣼⡏⠀⠀⢸⣷⣶⣐⣁⠴⠛⣩⣴⠆⠀⠀⣼⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣦⠀⠀⢻⣦⣍⠛⠦⡉⢢⣰⣿⡇⠀⠀⢹
⣿⡇⠀⠀⠘⠛⠉⠉⠀⢀⣼⣿⠏⠀⠀⠘⠻⢿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⠿⠛⠃⠀⠀⠻⣿⣦⠀⠀⠉⠙⠛⠃⠀⠀⢸
⣿⣷⣄⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⣷⣶⣶⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣿⣿

⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⣿⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀
⣴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠦⣄⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦
⡇⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣄⠘⠢⡄⠀⠀⠀⠀⠀⠀⢸⠀⢸
⡇⠀⡇⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡄⠀⠀⠀⠀⢸⠀⢸
⡇⠀⡇⠀⡰⡩⠒⠉⠐⢻⢦⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢶⡀⠀⠀⢸⠀⢸
⡇⠀⡇⠀⡇⡇⠀⠀⠀⢈⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢸⠀⢸
⡇⠀⡇⠀⠑⢷⠤⠤⠤⣾⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢸⠀⢸
⡇⠀⡇⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢸⠀⢸
⡇⠀⡇⠀⢯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠂⢸⡇⢸⡇⠀⠀⢸⠀⢸
⡇⠀⡇⠀⣠⠤⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠤⠄⢀⠀⠘⡇⢸⡇⠀⠀⢸⠀⢸⠀⢰⠒⠒⠒⠚⢻⠆
⡇⠀⡇⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⢸⡇⢸⡇⠀⠀⢸⠀⢸⠀⠘⣇⠀⠀⠀⢿⡀
⡇⠀⡇⠀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡀⢸⡇⢸⡇⠀⠀⢸⠀⢸⠀⠀⢹⣀⡴⢦⡀⠙⣆
⡇⠀⡇⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⢸⡇⢸⡇⠀⠀⢸⠀⢸⠀⠀⠀⠉⠀⠀⠹⡄⠘⣆
⡇⠀⡇⠀⣏⣉⣉⣉⣉⣉⣉⣙⡂⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡏⠉⢀⡼⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸
⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⡇⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠹⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠲⢤⣀⡀⠀⠀⠀⠀⣀⣠⠴⠚⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣄⣈⠉⠉⠉⠉⠉⢉⣀⣤⠴⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⡟⠛⠛⠛⠛⠛⢻⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣈⣻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠿⡿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣀⣀⣀⣠⣿⣇⣀⣀⣀⣀⣀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠉⠉⠉⠙⣿⡏⠉⠉⠉⠉⠉⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠿⠷⠶⠶⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⢀⣠⣤⣤⣤⣤⣀⡀⠈⠙⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣀⣴⠾⠛⠉⠀⠀⠀⠈⠉⠛⠿⣦⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢻⣆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡀⠀⠀⠀⠀⢿⡆⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠊⠀⠀⠀⠀⠀⠈⣿⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣷⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠊⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠈⢻⣆⠀⠀⠀⠀⠀⠘⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠹⣷⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠉⠛⠷⣦⣤⣀⣀⡀⣀⣤⡾⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠈⠉⠙⠛⠛⠉⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣦⣄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣿⡇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣼⡇⠀⠀⠀⠀⢸⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣒⣒⣒⣒⣒⣒⣒⣒⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠒⠒⠒⠒⠒⠒⠐⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠲⠶⠶⠖⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠤⡤⣤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡁⣉⣭⣭⣍⠸⣿⡀⠀⢸⠀⠀⠀⠀⠀⢸⠀⡷⢾⠀⠀⠀⠀⠀⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠒⠒⠖⠒⠀⢿⣇⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠤⠤⠤⠤⠤⠤⠤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢿⣏⠉⠀⠀⠀⠀⠀⢸⠈⣭⣭⣭⣭⣭⣭⣭⣭⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⣠⣀⠀⢸⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠷⢼⢐⢐⣒⣒⣒⡂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠠⠠⠤⠤⠤⠄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⣀⣛⣛⣛⣃⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣦⣀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣷⣄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣶⠶⠶⠶⣶⠀⠀⣶⣶⣶⣶⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣧⣤⣤⣤⣿⠀⠀⣤⣤⣤⣤⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⠀⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⣿⠀⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠤⠤⠤⠤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠤⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠹⣿⣿⣿⣻⣾⡯⠋⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⣟⣵⣿⣿⠇⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣾⣿⣿⣿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣤⣿⣿⣿⠏⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣉⣉⣁⡀⢉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠐⢿⣟⠋⠇⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣼⣿⣿⣿⠫⣄⣴⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢟⣵⠋⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢀⡀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⡆⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣾⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⣁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣋⣀⣸⣩⣽⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⣿⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣭⣭⣥⣭⣭⠀⠉⣵⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⣿⣿⣿⣿⢾⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣼⣿⣿⣿⣿⢣⣧⠀⠘⣿⣄⠀⠀⠀⢀⡀⣠⠇⠀⠀⠀⠁⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⠿⠿⠆⠠⠘⠉⠂⠉⠁⠀⠀⠃⠀⠀⠂⠀⢰⡌⠨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⠑⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣲⠀⠀⢤⢤⣤⣤⣴⣶⣠⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠁⢛⣛⠋⠹⠿⠿⠿⠛⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣿⣿⣽⣿⣿⣽⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣴⣾⡀⣿⣦⡀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⠿⣦⡿⠈⣿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⢿⣤⠀⣠⣼⠏⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡀⠀⠿⠂⠿⠁⢰⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴⠆⣤⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⣶⡶⠶⠖⠀⠰⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⣿⣤⣾⡇⠀⠀⠀⠀⠀⠀⢀⣙⣁⣀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⣿⣀⣼⡇⠀⠀⠀⠀⣴⠞⢋⣉⣉⣉⠛⢶⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⠉⣹⡇⠀⠀⠀⣾⠃⣴⠟⠉⠉⠙⢷⡄⢻⡆⠀⠀⠀⠀⣴⣧⠀⠀⠀⠀⣿⠉⣹⡇
⠀⠀⠀⠀⠀⠀⣿⠛⢻⡇⠀⠠⣴⡿⠀⣿⡤⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⣼⡏⠘⣷⡀⠀⠀⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠿⢿⡇⠀⠀⠘⣷⣠⡿⠁⠀⠀⢀⣄⢀⣄⠀⠀⠀⠘⢻⡇⢸⡟⠃⠀⠀⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⣿⣦⣾⡇⠀⠀⠀⠘⠟⠁⠀⢀⠀⠀⢿⣄⠻⢦⣤⣤⣴⠟⢡⡿⠁⠀⠀⠀⣿⣦⣾⡇
⠀⠀⠀⠀⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢷⣤⣬⣥⣤⡶⠛⠁⠀⠀⠀⠀⣿⣤⣼⡇
⠀⠀⠀⠀⠀⠀⣿⣁⣹⣇⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⡀⠀⠀⣀⣀⣴⣀⣀⣀⣀⣀⣀⣀⣀⣿⣁⣹⡇
⠀⠀⠀⠀⠀⠀⠿⠉⠻⠏⠉⠉⠉⠉⠉⠉⠉⠿⠏⠹⠿⠀⠸⠟⠉⠿⠋⠉⠉⠉⠉⠉⠉⠙⠿⠋⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠶⠶⠶⠶⠾⠿⠿⠿⣿⣿⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠚⣛⣉⣭⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣜⠉⠉⠹⢿⣿⣿⣿⣧⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⢛⣩⡥⠶⢒⣛⡭⠭⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣧⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⣼
⠀⠀⠀⠀⢀⣠⡴⠚⣉⠥⢒⣋⣭⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢟⡟⢸⣿⠀⠀⢘⣿⣿⣿⣿⣧⠀⠀⢀⡟
⠀⠀⠀⣰⠟⠁⢐⣩⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⡼⠀⢸⣿⠀⠀⢈⣿⣿⣿⣿⣿⠀⠀⣸⡇
⠀⠀⢰⣏⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢯⠏⣸⠃⠀⣸⣿⠀⠀⢰⣿⣿⣿⣿⣿⡀⠀⣿⠁
⠀⠀⣸⣿⣾⠁⠀⠀⠀⢰⣰⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⣠⣿⢫⠃⢠⡏⠀⢄⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⡇⢰⣿
⠀⠀⣿⣿⡟⠉⢉⣉⠐⠚⠚⠒⠒⠲⠦⠤⣤⣄⣸⣺⠀⠀⠀⠀⠀⣰⠃⡏⣞⠀⡼⠀⠸⢸⣿⡇⠀⠀⣻⣿⣿⣿⣿⣿⡇⣸⡇
⠀⠀⣿⣿⣇⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⢤⣼⠃⠀⠘⢾⣶⣇⠀⣇⣿⣿⡇⠀⠠⣿⣿⣿⣿⣿⣿⡇⣿⠇
⠀⠀⢿⣿⡇⠱⣄⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⢠⠚⢳⠀⡄⠀⠀⠈⢛⠿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⢸⣿⣇⠀⠸⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡷⢋⢎⡀⠀⠀⠀⠀⠉⠲⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠸⣿⠿⡄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡿⠀⡞⡎⠫⠄⣄⠀⠀⠀⣼⡿⠁⣀⣾⣿⠿⠛⠋⠉⠙⠛⣿⡇
⠀⠀⠀⠹⣦⡉⢻⣤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠃⡸⠀⠙⢤⣠⠟⠀⢀⣼⣿⣣⣴⡟⠉⠀⠠⣤⣤⣄⣀⠀⣿⠃
⠀⠀⠀⠀⠈⠙⠳⢯⣟⡿⢿⣶⣶⣤⣤⣀⣇⣀⡀⠀⢰⣿⣤⣇⡀⠀⠀⠁⣀⣴⣿⡿⣵⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢮⣭⣙⣛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣫⣾⣿⡿⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⠶⣭⣽⣿⣶⣶⣶⣶⣿⣿⣿⡿⠿⠷⢤⣤⣄⣀⣙⠻⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠀⠠⠠⠀⠠⠄⠀⠤⠄⠀⠠⠤⠤⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠬⣿⣯⢩⠭⣥⠀⠀⠀⠀⠀⠀⠄⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠘⠛⠚⠓⠓

⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⢠⠠⢤⡤⢤⡀⠀⠀⣤⣤⡤⢠⣤⡤⣤⣄⠠⠤⡄⠀⠠⢤⠤⡤⣤⡤⣄⢀⡤⠤⣠⡄
⠀⠀⠀⠀⠐⠀⡐⣀⣀⡀⠀⠰⠛⠶⣬⠇⠀⠘⠡⠛⠷⠀⠀⠀⢹⠘⣆⣈⡇⠀⢘⢬⡆⡠⠀⠠⠙⣇⠀⢂⠘⡇
⠀⠀⠀⠀⠈⠀⡁⠀⠖⠀⠀⢁⠀⠀⠂⠀⠀⢠⠞⠀⠂⣄⠀⠀⢸⡇⡯⠤⡇⠀⠠⢤⡜⢠⡀⠀⠘⡧⠈⠌⠠⡇
⠀⠀⠀⠀⠐⢀⠃⢐⢠⠔⡪⢕⠀⠠⠐⠂⠀⠀⠆⢿⡆⣿⡀⠀⢸⢣⣃⣒⡇⠀⢨⢿⡟⢘⣞⡠⢠⡗⠀⠔⢠⡇
⠀⠀⠀⠀⠰⢭⠀⠼⠂⠺⠀⠀⡇⢐⣧⠀⠀⠀⠘⠛⣛⠁⠄⣠⢏⠎⣠⣼⡇⠀⠐⡱⠋⠿⣿⡵⣺⢀⡄⡔⢸⡇
⠀⠀⠀⠀⠘⠐⠀⠀⠈⠀⠀⠘⠀⠙⠃⠀⠀⠀⠀⠂⠀⠀⠰⠃⠸⠿⠛⠁⠀⠀⠸⠃⠐⠃⠑⠀⠋⠀⠀⠀⠸⠇






⠀⠀⠀⠀⢎⠘⠋⠑⡙⠙⠍⡗⠂⠰⠖⠀⠀⠀⠙⡩⢁⢛⣈⠈⣹⠿⣿⣷⡄⠀⢐⡚⠛⠲⠀⣻⠈⡏⣠⣿⣹⡇
⠀⠀⠀⠀⢈⣠⠀⢃⠈⠄⣀⣺⠖⣿⣌⠁⠀⠘⠷⠒⡋⠩⠤⠀⢸⠃⣈⣻⡇⠀⢸⣧⠀⢠⠄⣂⡾⢃⢉⠋⠛
⠀⠀⠀⠀⢯⠔⡧⠀⣀⠔⢭⠌⣤⡴⠛⠃⠀⠀⠸⢀⠴⠿⡉⠃⢴⣿⢿⣷⠀⠀⢰⠍⠐⣊⣴⣽⠁⠇⢴⡅⠀⡀
⠀⠀⠀⠀⣬⠋⠁⣠⠇⠀⢀⡼⡏⣾⠁⠀⠀⠀⠀⠀⢀⠈⠁⠘⣻⢈⣳⠟⠀⠀⢸⡏⠀⡽⢡⠹⢀⣠⡌⠚⡂⡇
⠀⠀⠀⠀⠐⣲⠀⡋⠤⢠⣿⢹⡧⣤⢤⠆⠀⢀⡤⢐⠈⠘⢀⢁⣿⣏⠀⣾⠁⠀⢸⣧⢰⣇⣴⠞⠂⢀⡠⢀⣠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠋⠛⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⣿⣿⣇⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣼⣿⣿⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠻⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡟⠛⠿⠻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⡄
⠀⠀⠀⠀⠀⠀⠎⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢙⠀⠨⣐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⡐⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠠⡀
⠀⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣀⠀⠪⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⠀⢷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠐⠒⠐⠐⠒⠂⠀⠀⡀⠀⠀⠀⠀⠁⢀⠓⠀⢀⠀⠀⠀⠀⠒⠒⠐⠒⠒⠒⠂⠀⠀⠀⠸⣴⡁⣗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡉⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣌⡳⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⡐⠁⠀⣄⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠀⢅⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢖⠀⠀⠘⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠆⡰⠢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡂


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠛⠉⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⣠⣴⠾⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠳⣦⣄⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠻⣦⡀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⣠⠞⠻⣦⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠆⠀⢀⡾⠃⠀⠀⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠘⣷⡀⠀⠈⡄
⠀⠀⠀⠀⠀⡄⠀⠀⣼⠃⠀⠀⠛⣡⠞⢁⡤⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⣠⠞⠙⣷⠀⠀⢁⡀
⠀⠀⠀⠀⢠⠀⠀⢸⡏⠀⠀⠀⠀⠁⠰⠛⠁⠀⠀⠀⢀⣴⠟⠀⣠⠟⠁⠀⠀⢀⣤⠞⠁⠀⠀⢻⡆⠀⠘⠇
⠀⠀⠀⠀⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⣤⠞⠁⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⢸⡇⠀⠀⢰
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀
⠀⠀⠀⢰⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⣠⠞⠁⣀⡾⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⣠⠞⠁⢀⡴⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡿⠁⢀⡴⠋⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⣃⡴⠋⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣽⡏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠸⣿⠀⠀⣿⣦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠃
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣿⣅
⠀⠀⢀⣴⣿⠀⠀⢿⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⡟⠁⠹⣷
⠀⢠⡿⠉⣿⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⢠⡞⢷⢻⡇
⠀⢸⡇⠀⣿⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠘⢧⠟⢸⡇
⠀⠸⣧⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣆⠀⢀⣾⠁
⠀⠀⠙⢷⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣷⡟⢿⡄
⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠸⣧
⠀⠀⠀⢀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡙⠧⣄⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⢿⡄
⠀⠀⠀⣸⣿⠀⠀⣿⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠙⠦⠈⠃⠀⠀⠀⢸⡇⠀⠀⣿⣧⠀⠘⣿
⠀⠀⢀⣿⣿⠀⠀⢿⡇⠀⠀⡴⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡄⠀⢹⡇
⠀⠀⢸⡟⣿⠀⠀⢸⣧⠀⠈⣡⠞⢡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠘⣷⠀⠘⣷⡀
⠀⠀⣾⠃⢻⣇⠀⠘⣿⡄⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡏⠀⢹⡇⠀⢹⡇
⠀⠀⣿⠀⢸⣿⡄⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣾⠁⠀⠘⣷⠀⠈⣿
⠀⢸⡟⠀⣿⠋⢷⡄⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⢀⣾⡇⠀⠀⠀⢿⡄⠀⢻⡆
⠀⣼⡇⠀⣿⠀⠈⢿⣆⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⡞⣿⠁⠀⠀⠀⠸⣷⠀⢸⣷
⠀⣿⠃⢠⡿⠀⠀⠈⣿⣷⣄⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⣠⣾⣿⠀⣿⠁⠀⠀⠀⠀⢿⡀⠀⢿
⢠⣿⠀⢸⡇⠀⠀⠀⢸⣇⠙⢿⣦⣀⠀⠈⠙⠛⠻⠷⠶⠶⠶⠶⠶⠟⠛⠋⠁⠀⣀⣴⠾⠋⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸
⢸⡇⠀⢸⡇⠀⠀⠀⠀⢿⡄⠈⢿⡍⠛⠶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠛⠋⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠈⣿⠀⢸
⢸⡇⠀⣿⠁⠀⠀⠀⠀⠈⣿⡀⠈⢻⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⠀⠀⠀⠀⠀⠀⣻⡆
⣸⡇⢸⡟⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⠀⠀⠀⠀⠀⠀⣽⠇⢀
⣿⠃⣼⠇⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡇⢸⡿⠶⠶⠶⠿⠿⠛⠋⠀⣸
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠁⢸⣷⣀⣀⣀⣀⣀⣠⣴⠾⠋
⣿⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡏⠀⢸⣿⠉⠉⠉⠉⠉⠉
⢹⣇⠀⠙⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠋⠀⢀⣸⠇
⠀⠙⠷⢶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠒⠒⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⠆
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⡀
⠀⠀⠀⠀⠀⠀⢠⣇⣄⣀⣠⣀⣀⣀⣀⣀⣠⣄⣀⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣄⣷
⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢸⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢸⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣸⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣶⣶⣶⣶⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣶⣶⣶⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠼

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⠛⣡⡶⣾⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠃⣴⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠂⣤⠐⠿⠿⠿⠀⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡶⢸⣿⠀⢶⣶⣶⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⢸⣿⣧⠰⣶⣶⠀⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡄⢿⡝⢷⣀⠠⠀⣿⣿⡷⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣬⣥⣌⠻⣆⠙⠷⣦⣍⣉⣁⣾⣦⣀⠩⣤⣤⣤⣤⣤⣤⣥⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠦⠌⠻⢦⣤⣈⣉⣉⣿⠀⣹⡷⠀⣭⣭⣭⣭⣭⣭⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣉⡉⢿⠞⢋⣀⣈⣉⣉⣉⣉⣉⣉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⣛⢋⡉⣉⡉⣉⢙⡓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢊⡉⣤⠰⠖⢙⡃⠉⠈⠡⠬⠥⠉⠈⠁⣛⠃⠶⢠⡌⡉⠲⢤⣀
⠀⠀⠀⠀⠀⢀⡴⢋⣄⠶⢈⠡⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢉⠰⢆⣌⠳⣄
⠀⠀⠀⢀⡴⢃⡰⠎⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠉⠶⢀⠳⡄
⠀⠀⢀⡞⢠⡌⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⣥⠘⣆
⠀⠀⣸⠰⠦⠠⠁⢀⣀⠀⢀⡀⢀⡀⣀⣀⠀⣀⠀⠀⢀⠀⢀⣀⠀⠀⣀⠀⢀⣀⣀⢀⣀⣀⠀⠠⠠⠦⢸⡀
⠀⠀⡇⠸⠃⡄⠀⢸⠉⢳⢸⡇⢸⡇⣏⣩⠇⣿⠀⠀⢸⢀⡌⠉⠁⢠⡿⡆⠈⢹⠉⢸⣯⠍⠀⠀⡄⠿⠀⡇
⠀⠀⢧⢘⣃⢀⠀⠸⠶⠟⠈⠷⠼⠃⠇⠀⠀⠿⠶⠦⠸⠀⠳⠶⠂⠟⠉⠳⠄⠸⠀⠸⠷⠶⠀⢀⠀⣛⢠⠇
⠀⠀⠸⡌⢩⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣬⠉⡼
⠀⠀⠀⠹⣄⠰⢆⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠶⢀⡜⠁
⠀⠀⠀⠀⠈⠣⣘⢣⡦⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢐⠡⡦⢛⡤⠋
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣛⠰⠆⣬⠀⡂⠠⠄⠀⠀⢀⣀⣀⠀⠀⠀⠤⠀⡂⢩⡀⠶⢘⣣⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠭⣌⣋⠚⠂⠶⠰⠆⢴⠄⠖⠰⠆⠚⢊⣋⣬⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣶⣿⣿⣿⣿⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⣤⣿⣿⡿⣿⣿⣷⣶⡀⢀⡖⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡩⠽⣶⣼⡿⠷⠿⣿⣿⣿⣾⣌⡡⠤⢄⡐⠉⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⢺⡿⣡⡿⠋⠁⠀⠀⠀⠀⠈⠙⢿⣷⣧⡴⣾⣃⠐⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⠈⢸⣿⠋⢀⡄⠀⠀⢀⣾⡆⠀⠀⠀⠹⣏⢻⣾⣥⡀⣸⡄
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣤⣶⠇⠼⠃⣠⣧⣗⡀⠀⣼⣿⣿⡄⠀⠀⠀⠹⡏⢧⡌⢉⣿⡃⢃
⠀⠀⠀⠀⠀⠀⢸⣿⣻⣦⣽⣿⠀⡀⢢⣟⣟⣥⠀⠀⣣⣿⣧⣅⡀⠀⠀⠀⢣⡀⢘⣝⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⢉⡄⣠⡀⠉⠀⠀⣼⣿⣷⣟⠃⣴⣿⣿⣿⣿⣿⠀⣐⡆⠀⣼⣿⡝⠾⠿⠆
⠀⠀⠀⠀⠀⠀⢠⡀⠛⠿⠭⠀⠁⠀⣿⡿⣿⡉⠀⡙⠻⣿⡿⠛⠁⠀⢾⣗⠀⢟⡁⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣄⠀⣭⠀⢿⣿⣯⣤⠱⠀⠀⢿⡇⢀⠀⢀⣸⡟⠀⣿⠷⠀⡆⢤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠐⢃⡆⠸⣿⣿⣠⡆⠠⠀⢸⠇⠀⢠⣿⣿⠇⣸⢇⠤⠈⣿⠶⡁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣟⢀⣭⣿⡄⠻⣿⣿⣷⣶⣴⣾⣶⣶⣿⣿⠏⢠⣿⡋⠀⠀⣸⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⢿⣿⣀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣫⣷⣴⣤⢴⡟⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⡿⣿⣿⣿⠟⣧⣄⠉⠛⠻⠟⠛⠉⣀⡖⢓⠉⣙⣏⣉⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠉⠁⠀⠴⣽⡿⠖⣶⣴⠶⢿⣯⡷⠛⠻⠁⠉⠻⢟⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⠐⠀⠐⣄⣸⠠⠘⠸⣿⣿⠏⠂⢰⣿⡖⠃⢠⠴⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⡁⠀⡤⠄⠠⣿⣷⠤⠤⠈⠳⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⡃⢀⣐⣿⣿⣥⡀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣠⣾⢿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⠀⣴⣿⣅⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣿⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⡦⠀⣿⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⠀⣰⡆⢀⣀⣀⣀⣀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠀⣿⣇⠈⠉⠉⠉⠁⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⠗⠀⣿⣿⡄⠻⠿⠿⠗⠀⣿⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⠀⢿⡞⣷⡄⠠⣤⡄⠀⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⣀⠘⢿⡌⠻⣦⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠃⠈⠻⣦⡈⠛⠷⣦⣤⣀⣀⣀⣿⣷⣤⡉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⡉⠻⣶⣤⣀⠉⠉⠉⠛⣿⠀⠙⣻⣦⠀⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠙⠛⠛⠿⠿⣿⣤⡾⠋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⠒⠒⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⢤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⣄⠀⣠⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡞⠁⢀⣟⣹⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣻⠟⣿⣿⡋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠛⣿⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⣾⣿⣼⣿⣿⣿⠿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⣿⣿⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠃⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣟⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⠀⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠈⢟⠀⠀⠀⠀⣀⡀⠀⣠⣄⣴⢷⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⡀⠀⠀⠈⠁⡠⠤⠄⠓⠒⠀⠉⠉⠀⠀⣰⢧⣿⢇⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠦⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⣾⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣥⣾⣿⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⣿⡿⠟⠛⠿⠿⣿⢿⣽⣿⣿⣷⣻⣟⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠃⠠⠬⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⢿⠿⢿⠿⠻⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣻⣿⣿⡟⣿⣿⣿⠿⠿⠿⣿⠿⠿⢿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠽⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⢹⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⠀⢸⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠶⠶⣿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠸⠤⠤⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣸⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠛⠛⢻⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣤⣤⣾⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠿⠷⢿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣸⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣤⣤⣾⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣽⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣵⣯⣽⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⢿⣏⣻⡽⣿⣿⣿⣿⣿⣿⢯⣟⣹⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⢡⣀⣌⣡⡴⠛⠉⠉⠉⠙⠛⢦⣌⣡⣀⡌⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠿⣿⣿⣿⣿⠟⠁⠀⠀⣤⡦⠂⠀⠀⠀⠉⠻⣿⣿⣿⣿⠷⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠞⠀⣿⡟⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠐⠲⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣤⣶⠏⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣹⣿⠃⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡏⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⡠⢾⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢸⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠞⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⣿⡏⠳⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣱⠎⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡛⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡝⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠿⣟⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⢻⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢫⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣿⣷⡟⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣀⣈⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⡜⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⣿⣁⣤⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡼⢣⠞⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣆⠀⠀⠀⣠⠞⣴⠃⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⣸⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡾⣿⡀⢀⡼⠋⡜⠁⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⢠⣿⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⢡⠞⠀⠀⠀⠀⠀⢀⣾⠛⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣠⠟⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⣿⣿⣷⣄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣧⣼⣿⣿⣿⣿⣶⣤⣀⣠⣤⣶⣿⣿⣿⣿⣥⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣏⣿⣿⣿⣿⡿⢿⣿⣿⣿⣯⣻⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠀⠸⠛⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢿⣿⣿⣷⣄⣀⣾⣿⣿⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠋⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠁⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡗⠀⠀⢾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣶⣶⣥⣤⣤⣼⣶⣶⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣯⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⡘⣃⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣟⣻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠘⢆⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⣴⠚⠉⠀⠀⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠈⠐⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠆⠄⡀⠀⠀⠀⠀⡠⠄⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⠀⠀⠈⠲⣄⠀⠀⠀⢠⠊⠀⣀⣰⡄⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠀⣀⠤⠃⠀⠀⠀⠀⠈⠁⠀⠉⡁⠄⠸⠭⠻⠿⠦⠶⠶⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡼⠀⠀⠀⠀⠈⠓⠦⣤⣄⡤⠖⠋
⠀⠀⠀⠀⠀⠀⢠⡆⢲⣄⠤⡀⠀⠀⢰⠉⠀⠀⠉⠁⠈⢹⠈⢺⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠁⠀⠀⡇
⠀⠀⠀⠀⠀⢬⠌⠓⠐⢏⣾⠁⢄⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠘⢿⡀⢀⣀⡀⠀⢀⡀⠀⢺⠇⠀⢘⠀⠇⠀⠀⠀⠀⢠⠔⠒⠒⠒⠶⠖⠒⢶
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢙⠀⢑⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠋⠀⠉⠀⢠⡀⠀⢸⠀⠰⠒⠒⠒⢠⠈⢸
⠀⠀⠐⠀⠉⣣⡀⠀⠀⢀⡞⠀⠊⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠀⢸⠀⠀⠀⠀⠀⢸⠀⢸⡆
⠀⠀⠀⠠⣌⢀⣿⢻⣿⢉⢯⠔⠀⠀⠸⠤⠤⠤⠄⠀⠀⠐⠀⠲⠋⠉⠀⠈⠉⠙⠶⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⡸⠀⢸⠀⠘⠀⠀⠀⠘⠀⢸⡇
⠀⠀⠀⠀⠈⠃⠐⠢⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠈⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣶⣿⣿⣷⣷⣶⣶⣶⣿⣷⣄⢸⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣟⠛⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡆⠋⠀⠀⠀⠀⠀⠀⠒⠒
⡇⠄⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠓⠒⠒⠲⠶⠶⠶⠶⠤⠤⠤⢤⣤⣤⣀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠛⠒⠛⠒⢤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣡⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠌⠀⠀⠀⠀⠀⠀⢿⣄
⡇⠀⠀⠀⠀⠀⠀⢰⢼⢏⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⡇⠀⠀⠀⠀⠀⠀⠘⢀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠿⣧⣀⢙⣦⣁⣼⣿⣿⠏
⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢋⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡿⣿⣿⣿⡛⠁
⡇⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⢿⣿⣿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠈⣿⡙⢙⠢⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣷⣾⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣶⠶⢿⣿⣿⣿⠋⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⢸⣿⡄⣿⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢿⡇⢿⡇
⡇⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⡿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢾⡇⢸⣇
⡇⠀⠀⠀⠀⠀⢀⡜⣾⡏⣿⣿⣿⡟⣡⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⡿⠸⣿
⡇⠀⠀⠀⠀⣠⠞⣹⣿⢸⣿⣿⡟⣴⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠟⠠⠄⠀⠀⠀⢁⣀⣹⣦⡀
⡇⠀⠀⠀⣿⣯⣼⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⢿⠿⠛⠖⠒⠊⠉⠉⠁⠀⠠⠟⠀⠀⢠⡄⠀⠀⣠⣧⠙⡜⣧
⡇⠀⠀⠀⡟⢿⣿⣿⠿⠿⠟⠋⠓⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⣸⣷⠆⠘⢨⠣⡀
⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠃⠀⠢
⠣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢢⠄⠀⢠⣤⣤⣰⣤⡄⣤⣾⣶⣆⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠛⠛⠛⠒⠒⠲⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣭⣥⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣾⣿⡿⠿⠿⠿⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⣋⣥⠶⠛⠉⠀⠀⠀⠉⠛⠷⣦⡀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⢁⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠃⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣼⠋⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⣼⠃⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣰⠏⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⢠⡟⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⡿
⠀⢀⡿⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⣼⠃⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣸⡏
⢠⡟⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⣠⡤⢤⣄⣰⡟
⣸⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣴⠋⠀⠀⠀⠈⢿⡆
⣿⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠻⢦⣄⠀⠀⢰⡟
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡾⣿⠀⠀⢸⣿
⢻⡆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣠⠟⠀⣿⠀⠀⠘⣿
⠸⣇⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢀⡼⠃⠀⠀⣿⠀⠀⠀⣿
⠀⢿⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠈⢻⣦⡀⠘⣧⠀⠀⠻⡔⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠙⢿⣦⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠉⠻⢷⣿⣦⡀⠀⣀⣀⣤⡴⠞⠋⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣤⣤⡴⠶⠶⣶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⣷⠀⠀⢸⣧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣠⣴⠶⠞⠛⠛⠉⠉⠀⠀⢿⡀⠀⠸⡇⠈⠉⠉⠛⠛⠓⠶⠶⠦⢤⣤⣀⣀⡀⣸⡆⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢰⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⠟⠃⠀⠀⢀⡟⠛⠛⠛⠿⠶⣦
⠀⠀⠀⠀⠘⣿⠛⠓⠶⢦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠖⠚⠋⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠹⣧⣤⣀⡀⠀⠈⠉⠙⠛⠻⠶⢶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠳⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠶⠚⠋⠀⠀⠀⣀⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠷⠶⠶⣶⣤⣤⣤⣤⣴⠾⠋