Syrup Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠴⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⢀⡠⠤⠤⠤⠤⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠣⢤⠀⠀⠠⡇⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⢠⢇⣤⠞⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇⠸⠀⡇⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠤⠒⠐⠂⠠⠔⠒⢻⠙⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠐⠵⠀⠀⠐⠀⠀⠀⠳⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⣀⡀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⣀⡌⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠢⠴⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⠤⠀⡄⠀⠀⡠⢄⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⡬⠀⠀⠀⠀⠆⠀⡀⢃⣠⠇⣀⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⡄⠚⢀⣸⠦⠗⠀⠿⠼⠀⠀⠤⠯⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠂⢉⡆⠪⡑⠁⡙⠁⠄⢀⡰⠁⠖⢰⡋⢠⠉⠂⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢈⡂⠄⢀⣀⠜⡰⠀⠀⢰⡈⢙⡄⠀⣸⠀⠒⠐⠃⠀⠀⠀⠀⠀⠸⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠘⠤⠐⠊⠘⠒⠁⠀⠓⠊⠈⠁⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠖⠀⢠⠀⠀⠀⠀⠀⠀⡰⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⠀⠈⠁⠀⠀⠀⠀⡴⠁⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠉⠀⠀⢀⠎⢀⣀⠤⠊⠉⠉⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠶⠶⠤⠤⠤⠤⣤⡤⠄⠀⣀⣀⡸⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠶⠒⠊⠉⠁⠀⠀⠀⠉⠉⠛⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣠⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡃⡀⠀⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠷⡴⠶⠶⡾⠮⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⣿⣿⠿⠿⣿⣿⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣿⣿⣿⠶⢶⣿⣿⣮⡣⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⣿⠀⢸⣿⣿⡏⢻⣿⣶⣮⣵⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⠀⣿⣿⠿⡟⠁⠀⠉⢢⡉⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⠀⣿⣿⢸⡇⠀⠀⠀⠀⣧⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⣿⣿⢸⡇⠀⠀⠀⠀⡘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣿⣿⣇⢿⣿⡜⡇⠀⠀⠀⠀⡇⣸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢸⣿⣧⢿⠀⠀⠀⢰⢁⡏⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡎⣿⣿⡼⡆⠀⠀⡎⠘⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣄⣼⡆⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⣫⣶⣶⣦⣝⢿⣿⣿⣿⡟⢿⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢋⣾⣿⣿⠿⣿⣿⣧⠻⣿⣿⣿⣸⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢣⣿⠛⢿⡍⢘⠨⡿⣿⣧⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢃⣿⣿⣌⣇⠠⡅⣈⠜⢙⣿⣇⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⣼⣿⣿⣿⡥⢰⣆⣉⣰⣿⣿⣿⡄⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢠⡟⠛⣟⠻⢿⠛⡛⠟⡟⢻⠛⡛⣧⠸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣯⣥⡜⢂⣾⣤⣿⣧⣤⣼⠀⣥⣿⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⣿⣿⣧⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢠⣿⣯⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⣿⣿⠿⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⡿⣿⣿⣿⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠙⠻⠿⣷⣶⣮⣭⣭⣭⣭⣭⣭⣭⣽⣶⣶⣿⣿⡿⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠤⠬⠭⠿⠯⠥⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡏⠉⠉⠉⢹⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠙⡧⠤⠤⠤⠼⢦⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⡇⠀⠀⠀⢀⡼⠀⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠱⠤⢄⠀⡏⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⢸⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢸⠀⠀⢸⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⠀⠀⠀⠰⠜⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⡾⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠈⠙⠳⡄
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠹⡆
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠹⡀⢀⣠⡤⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠈⠑⢄⠀⠀⢹⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠤⠶⠶⠶⢼⠇⠀⠀⠀⠀⠗⠉⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠘⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸⠁⠀⠀⡇
⠀⠀⠀⠘⡇⠀⠀⠐⠀⠀⠀⠈⠢⣄⠀⠀⠀⠁⠀⠀⠈⠁⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢠⠏⠀⠀⢰⠇
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠧⣄⣀⡠⠔⠋⠀⠀⢀⡞
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣟⣀⣀⣀⣤⠤⢶⠀⣰⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⢤⣀⣀⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣭⣭⠿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⣿⠁⠀⠈⠹⠏⢹⣧⣤⣄⣤⡉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠡⣿⣶⣶⣤⡴⣣⡜⣿⠋⠀⠈⢡⣾⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⣶⣤⣤⣤⣤⣤⣾⣧⠘⣄⣀⣠⣟⣡⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⣾⣿⣿⣶⠶⠶⠴⢿⣿⣆⠘⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣽⣿⣿⣛⣒⡛⢛⣟⣛⣥⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠜⢷⣿⠿⢿⣿⣿⡿⠿⠿⢞⡓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⣴⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣦⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⣼⣿⣿⣿⢿⣿⠿⠶⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⡿⣿⣤⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⣾⣿⣿⣟⣽⠟⠁⠀⣀⠀⠀⠤⠴⠦⢤⣀⡀⠀⠙⢿⡟⠂⢻⣇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⣿⡿⠁⠀⡀⠊⠁⠀⠀⢠⡀⠀⠀⠀⠙⠲⣄⠀⠻⡇⠀⣿⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⡿⠁⠀⡜⠁⠀⠀⠀⠤⠈⠛⣶⠀⠀⠀⠀⠸⡂⠀⢻⠀⢹⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡇⠀⢠⠁⠀⢰⣦⠒⢾⠀⢰⠺⠾⢂⣶⠀⠀⠛⠀⠈⡇⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡇⠀⢸⠀⠀⠠⣭⡲⣤⡀⣿⢣⠔⢨⡿⠀⠀⢈⠀⠀⡇⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣷⠀⢸⡀⠀⠀⣀⡛⢯⣿⣾⣿⠇⣿⣄⠀⠀⣼⠀⢀⡇⢸⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸⣿⣿⣿⡀⠈⣇⠀⠀⠉⠁⠀⠈⠁⠈⠛⠁⠀⠀⢀⡟⠀⢸⠃⣸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣇⠀⢿⣀⡀⣀⣀⠀⠀⠀⠀⣀⣀⢀⣀⣾⠃⢠⡟⠀⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⣿⣿⣾⣿⡄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣾⠁⢰⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣿⣛⡿⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠃⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⣿⣿⣿⣟⣿⣿⠓⠒⠂⠀⠀⠀⠀⠀⢀⠈⠉⠹⠿⢷⣼⣿⡴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣿⣛⣛⣒⣒⣐⠒⠀⠀⠐⢂⣛⡒⣒⣒⣚⣫⣭⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⣿⣿⠿⡍⣿⠿⠿⠿⡋⢹⡿⠿⠿⣏⢉⠿⠻⣿⢰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⠼⣿⣶⣦⣿⣄⠠⠤⡗⢾⠠⠤⠂⣿⣼⢆⣼⠫⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣛⣿⣉⣿⣿⠿⣃⠘⣷⠾⢛⣁⣻⣿⣧⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⢹⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠭⠭⠭⠭⡽⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢫⠥⠴⠤⡉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⡒⠒⠶⠶⠚⢆⠀⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠈⠀⠀⠀⢸⠀⣠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⡠⠔⠒⠒⠢⣄⠀⠈⢌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⠊⠀⠀⡀⣀⠀⠈⠳⡄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡰⠁⠀⠒⠀⠀⠐⣒⠀⠀⠘⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣸⠁⣀⠀⠄⢀⡀⣀⠀⠀⢠⣄⡸⡄⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢡⠇⠀⠒⠀⠁⠈⡍⠉⢦⣀⠎⠀⠀⠹⡐⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠰⠀⡸⢀⠑⠠⣀⠀⠻⡀⠀⡀⢀⠀⠰⡸⠀⢇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡆⠀⡇⠀⠈⠐⠀⠈⢡⡇⢀⠰⡌⠀⢀⠇⠀⢸⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠄⠀⠀⠀⠈⡋⠁⠀⣀⠓⠇⠀⡜⠉⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠈⠢⣀⠈⠁⠒⠂⠤⡁⢦⡜⠤⠆⠀⢸⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠸⣄⠀⠀⠈⠀⠒⠒⠒⣇⡠⠤⠤⣀⣠⠇⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⠒⠶⠶⠤⠤⠎⠴⠖⠒⠊⠁⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠶⠶⠒⠚⠛⠛⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣂⣀⣀⡀⠤⠤⠤⠤⢼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠁⠀⠀⠸⣄⣀⠀⠀⠀⡔⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠂⠀⠁⠀⢸⠆⢀⡠⠛⠉⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡋⠸⠉⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⢀⣠⣀⠀⠓⢁⣀⣡⡤⣤⠤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠉⠁⠀⠀⠨⠉⠉⠀⠘⡀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠑⠊⠀⠀⠀⡇⠀⠀⠀⠛⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠤⣄⡀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢄⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠋⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⡀
⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠪⡉⠉⠀⠀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠹⡄
⠀⢠⠏⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣃⣨⡀⡠⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢃⠀⠀⠀⢻⡀
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠈⣧
⢸⠁⠀⠀⡀⠀⠀⣀⡀⢠⠲⠀⠀⠖⠉⠑⠐⠍⣉⠁⣶⠓⠂⠒⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢹
⡟⠀⠀⠀⠁⠀⠀⠇⠀⠆⢀⠀⠄⠀⠘⡃⠀⡀⢩⠤⠟⠸⢄⡄⢨⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢲⡀⡏⢀⣀⡧⠴⠑⠆⠻⠞⠀⠐⠀⠴⠧⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⣇⠀⠀⠂⠀⠀⠀⠓⠃⠉⠁⠀⠀⠀⡀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⢻⠀⠀⠀⠀⠀⠀⡤⠒⢒⡆⢮⡁⠚⢉⠎⠁⣄⠀⣡⡎⢦⠖⢲⠋⢉⠑⢂⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠺⠃
⠸⡄⠀⠀⠀⠀⠀⠢⢄⡈⠑⠀⢹⠆⡌⠀⠀⢠⠘⠣⡇⠀⠀⣼⠀⡬⠤⠞⠀⠀⠐⠂⠀⠀⢰⠂⠀⠀⡶
⠀⢷⠀⠀⠀⠀⠀⢘⢉⣀⠴⠣⣅⡼⠁⠸⣀⡄⠑⠖⠉⠒⠒⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠃
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢠⠏
⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⢸⠄⠀⠀⠀⠀⠀⢀⡎⠀⢀⡞
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⡤⢤⠀⠀⢀⠀⠀⠀⠀⢠⠎⠀⠀⠘⣧⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠳⣦⠄⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⢀⡴⠁⣀⡤⠔⠊⠀⠀⠀⠀⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠶⠤⠤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣰⡯⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⣀⣀⠤⠴⠒⠚⠛⠉⠉⠙⠓⠲⠶⠦⠤⠴⠶⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠶⠖⠒⠚⠛⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠦⠔⠒⣖⠀⠀⠀⠀⡛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠈⠙⡆⠀⢀⠧⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢐⢡⠚⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢀⠤⣀⠘⢊⣀⣣⠤⠤⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠬⠁⠀⠈⡀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠉⠀⠀⠀⣇⡀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⡀⠀⠀⠉⠲⢄
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠱⡄
⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠲⡙⠁⠈⢩⠙⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢄⠀⠀⠘⣆
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢰⡡⢤⣀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢂⠀⠀⠸⡄
⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠤⢀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢷
⡜⠀⠀⠀⠀⠀⠖⢀⠋⠁⠀⠁⢤⠀⠀⠲⢀⡇⡈⠈⠀⡼⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⡄⠰⢆⠐⣆⡰⢄⠸⣬⠍⢄⢈⣿⣀⣱⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠖⠁⠚⠀⠉⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⢧⠀⠀⠀⠀⠀⠔⠒⣢⠲⡁⠊⠱⠋⢠⠀⣡⠊⠴⢲⡎⢉⠑⡄⠀⠀⠀⠀⠀⠸⠀⠀⢸⠃
⠸⡀⠀⠀⠀⠀⢓⡤⠈⠂⡸⢂⠀⡀⠠⡌⠑⡆⠀⣸⠃⡖⠐⠃⠀⠀⠁⠀⠀⡏⠀⠀⡏
⠀⢳⠀⠀⠀⠀⠠⠐⠒⠁⠓⠊⠀⠓⠒⠈⠉⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣸⠁
⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⢠⠀⠀⠀⠀⠀⠀⡌⠀⢠⠃
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣀⡀⠈⢁⠀⠀⠀⠀⠜⠀⠀⢯⣀⣀
⠀⠀⠀⠀⠀⠙⠲⡤⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⡠⠊⣀⡠⠔⠉⠀⠉⠉⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠶⠦⠤⠤⣤⠤⠤⠤⠤⠤⠼⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠓⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠄⠀⠠⢄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠛⠃⠀⢰⡁⠀⠀⠀⠀⠱⡄⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠋⠹⠲⡆⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢉⠏⠱⡀⠑⠒⠦⢄⡀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠴⠃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣤⡎⠀⠀⡈⡐⠠⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⠵⢮⣤⠈⡀⠈⠀⠀⠁⠀⠨⠱⠄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡴⠃⠀⠀⠀⠉⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣋⠀⠀⠸⠀⠠⠐⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠠⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⢀⡞
⠀⡠⠖⠉⠉⠙⠲⡀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢰⠒⠊⠀⠀⠀⠀⢠⠎
⡞⠀⠀⠀⠰⠉⢀⣇⣀⣠⠴⠚⢅⡠⣆⠐⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠧⠄⠠⠔⢢⠀⢀⠇⠀⠀⠀⠀⠀⣠⠏
⣇⠀⠀⠀⠀⠀⠀⠁⠀⣀⢴⡴⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⢠⠏
⠈⢦⣀⣀⣀⣀⠤⠔⢊⡵⠋⢠⠞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠏
⠀⠀⡠⠋⠁⠀⠀⢠⠟⠀⠀⣏⠘⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⠤⠤⠤⠤⠖⠚⠉⠙⠁
⠀⢰⠇⠀⠀⠐⠀⠙⢆⠀⠀⠈⠉⠉
⠀⠘⡇⠀⠀⠈⠀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣀⣀⣀⣀⣀⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠁⠸⢀⡀⠀⠀⢠⠇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣸⠀⣠⠜⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⡎⠸⠄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⡠⠄⠤⠀⣉⡤⠤⡶⠢⠒⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢔⡂⠀⠀⢡⠀⠀⠀⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠈⠉⠉⠙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠤⡀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠴⡒⠚⠀⠉⡁⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⡀⣀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠡⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢀⡇⠀⠀⡀⠀⠀⡤⠀⡖⠂⠀⢰⢉⠑⠘⠁⡄⢱⡏⠃⠐⠉⣲⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢿
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⣀⠈⡁⠀⡆⠰⠚⠁⠒⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⠷⣨⠀⠤⠛⠚⠀⠈⠙⠉⠀⠁⠒⠛⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⣀⡀⢀⢠⣀⠤⡠⠖⠒⠢⢀⠄⢀⣀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠠⡁⠀⠺⠈⠢⡀⠀⠁⠀⠃⢤⣋⠀⠋⢸⠁⡐⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡤⠛⢀⡤⡔⢁⠆⢰⠀⡟⢄⡼⠆⠀⠾⣀⠉⠉⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠉⠀⢰⡀⠀⠀⠀⠀⠀⡸⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⢤⡀⠀⢀⠀⠀⠀⠀⡼⠁⠀⠘⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡦⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⡠⠊⣀⡤⠔⠊⠀⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠑⠒⠒⠒⠒⠲⠶⠤⠤⣤⠤⠤⠤⢤⣄⠼⠗⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠒⠒⠋⠉⠁⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠖⠒⠛⠛⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⢁⣀⡀⣀⣀⣀⣀⣀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠈⠉⠩⡁⠀⠀⠀⠀⢀⡍⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠄⠀⠐⠈⠙⡆⠀⠀⣸⠤⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⣴⢃⠴⢺⠁⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣦⡸⠀⠠⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣀⠴⠒⠒⠄⣤⡤⠶⠲⡟⠑⠚⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⢢⠀⠀⠀⢱⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠀⠀⠀⠀⠉⠉⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠤⢄⡀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢤⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠫⡉⠉⠀⠀⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢳⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣀⡀⢀⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢷
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⣇
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⡤⠔⠲⠀⠖⠉⠑⢢⡴⢠⠀⡤⠤⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢁⠀⠀⠀⢻
⣼⠀⠀⠀⠀⠀⠀⠸⠉⢢⡇⠀⠡⠀⠀⠰⠆⠀⠀⠘⢃⣰⠇⢸⠀⡀⠐⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠶⡀⠐⣆⢀⠦⣀⣹⡄⡏⠀⡄⢈⣹⡃⠈⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⢠⡠⠙⢤⠆⠑⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⢸⡇
⢷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡠⢄⣀⠦⣄⠖⠚⠓⢦⢀⠤⠀⣀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⡇
⢸⠀⠀⠀⠀⠀⠀⢰⠉⠀⠤⢄⠑⢄⠀⠀⠋⠀⠺⠀⣠⢻⠀⠛⠀⣿⠀⠴⠀⢱⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡜
⠀⣆⠀⠀⠁⠀⠀⠈⣑⡲⠀⠈⡄⡼⠁⡌⠀⠀⢰⢆⠈⢹⡄⠀⢀⣟⠐⡗⠒⠋⠀⠀⠀⠁⠀⠀⠀⡜⠀⠀⢠⡇
⠀⢹⡄⠀⠀⠀⠀⠀⣅⠠⠤⠊⠘⠤⠞⠀⠘⠢⠼⠀⠙⠁⠈⠁⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡾
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣰⠃
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠁⠀⠀⠴⢄⠀⠀⠀⠀⠀⠀⡰⠁⠀⢰⠏
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⢤⠀⠀⢀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢹⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⣠⠎⠀⣀⡠⠔⠉⠀⠀⠀⠀⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⠥⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⣀⣈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠶⣤⣀⠀⠀⢀⣀⣠⡤⠴⠶⠖⠒⠶⠶⢤⣤⣀⣀⣀⣠⣤⣤⠴
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠒⠛⠛⠉⠉⠉⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠀⠀⠀⢀⣀⡤⠤⠒⠒⠲⡄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣳⣹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠜⢯⣷⣷⣿⣿⡗⠛⠛⠳⢦⣀⠀⠀⠀⡇⢀⣀⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠠⠃⢠⢁⠎⠀⠜⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣌⣀⣘
⠀⠀⢰⡯⣉⣩⣉⣉⣍⡭⢿⠂⠀⠀⣀⣤⣤⣤⣤⣤⡄
⠀⠀⢸⡗⠲⢚⠲⢼⣄⣀⣾⠀⠀⠀⣿⠉⠛⠛⠉⠉⡿
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⣄⣠⣤⣀⣴⠇
⠀⠀⠀⢹⣤⣀⣄⣀⣀⣴⡇⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡍⠁⠈⠉⠁⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡑⡀⠀⠀⠰⠖⠐⢛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣠⠀⠀⠀⠀⢀⣸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠿⠄⠁⠬⠄⠤⠜⣇⣠⡤⠄⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢇⠊⠀⣀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣤⢶⠖⠒⠲⠶⠶⣶⣬⡟⠋⠀⠙⠂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠚⠂⠒⠀⠐⠀⠹⣿⡅⠀⠀⠀⡼⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠠⠂⢁⣤⠴⠒⠒⠲⠢⢤⡄⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠈⢀⡔⢹⣶⠞⠉⠀⠉⠁⢀⡈⠳⡄⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠄⠀⢰⠏⢀⡕⠁⠀⡰⠛⠆⠀⠀⠠⠀⠹⣆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⡋⡀⠂⠀⣤⡴⠁⠀⠀⢺⡄⠄⢀⡂⢹⡄⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⢸⣷⡏⠓⠒⢿⡄⠈⠖⠀⠸⠖⠛⢉⣿⣻⣷⠆⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠀⠉⢸⠋⢿⠀⢠⡄⠀⠀⡄⠀⢀⡀⠀⢨⡇⣁⡶⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠀⠀⢸⡆⣾⠃⠀⠀⠀⠀⡇⠀⠊⠁⢐⣿⠋⢹⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⣧⠋⢉⡿⠀⠀⢨⣏⠀⠀⠈⣻⡏⠠⢿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢹⠀⠘⠷⠓⠛⠻⣿⠛⠛⠙⠁⢀⡦⣿⠋⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢷⣀⠀⣁⣈⣀⣉⣀⣨⣤⠄⡸⣿⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⡀⠘⣇⡀⢉⡁⣀⠀⢀⡀⢀⠄⣁⡎⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠑⠀⠘⠯⢴⣤⣼⣤⣦⣴⣦⠦⠟⠀⠀⢀⡜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠳⣤⡀⡀⠀⠀⠀⡀⠀⠀⢀⡄⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⣠⣤⣤⣄⡰⠶⠀⠀⠀⠀⢤⣤⣤⣤⣉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠍⣉⣙⠛⠂⠒⠶⠤⠶⠷⠟⢛⣋⠥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⠁


⣶⣶⣶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⣿⡄⠀⠀⠀⠀⢸⡟⠻⠿⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⢻⡷⠤⠶⠴⠶⠾⠿⡇⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠁⠀⢸⠇⠀⠀⠀⠀⠀⣠⡟⠀⠈⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠘⣆⠀⢀⠀⠀⣼⠋⠀⢰⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠛⡆⠀⣿⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠀⢸⡄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡇⢸⡇⠀⠀⠘⠷⠶⠶⠶⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠳⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⣧⣀⣤⡤⠶⠶⣤⣤⣀
⠀⠀⣼⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⢰⡏⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠶⠶⠦⣄⠀⠀⠀⠀⠸⡆
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢳⠀⢀⣤⠴⠶⠖⠚⢻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢿
⣿⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⢤⣼⡇⠀⠀⠀⠀⠀⠸⠷⠋⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠰⡄⠀⠈⠃⠀⣠⠄⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢸
⢸⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣸
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠔⠀⠀⠈⠀⠀⠀⠀⠀⠈⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⣀⡴⠛⠀⠀⠀⠀⣰⠏
⠀⢸⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣤⣤⣀⣀⣀⣀⣠⡴⠟⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠶⠶⠶⠒⠚⠋⢹⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⢻⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠶⣤⣀⣀⣀⣀⣀⣠⡾⠋
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣾⠃⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣽⠆
⣦⣶⣤⣶⣤⣴⣦⣴⣤⣤⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣴⣴⣤⣤⣤⣴⣴
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣻⡏⡍⣝⠛⠛⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠙⣿⣿⣿⣿⣿⣿⣏⣿⣇⣿⣧⣿⣹⣹⣼⣵⢹⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣷⣿⣧⠻⢟⣛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⠋⠁⠀⠀⠀⠙⣿⣿⣿⣿⠿⢿⣿⣿⣿⡿⠿⠿⣟⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠦⣤⣀
⠀⠀⠀⠀⣀⣤⡶⠶⢶⣶⣿⠁⠀⠀⠀⠀⢈⣿⣶⠶⠶⣶⣶⣶⣤⡀
⠀⠀⢰⡿⣿⣧⠒⡒⠂⠉⠹⠓⠶⢶⣤⠖⠉⠁⠀⠤⣀⣶⣶⣿⣻⣿
⠀⠀⠘⢷⣜⠻⢿⣿⣷⣶⣶⣶⣤⣤⣤⣒⣒⣲⣾⠿⠿⠟⠛⣁⣹⡏
⠀⠀⢠⣾⣟⠻⢷⣴⣦⣦⣶⣤⣍⣥⣀⣀⣠⣤⣴⣶⠶⠾⠟⠛⣻⡟
⢀⣤⣤⣽⡿⠷⣶⣷⣂⣀⣨⣭⣉⣉⣉⣉⣉⣥⣤⣤⣤⡤⠶⠿⠟⣷⣀
⣼⡟⠛⠝⠻⠿⣶⣶⣬⣩⣭⣋⣉⣩⣭⣭⣩⣭⣤⣴⣤⣤⣴⣾⠿⠟⠻⠟⠛⣿⣦
⠙⢷⣦⣄⣀⣀⣀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⠟⠃
⠀⠀⠈⠙⠿⠿⠿⠿⣿⣿⣶⣶⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡴⠶⠶⠶⠒⠛⠛⠛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⣇⣀⣀⣠⣄⣠⣥⣤⣿⣟⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠈⠀⢠⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⢠⠖⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⠤⡾⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣄⡀⠀⠀⠀⠀⢀⡴⠶⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠙⣦⠀⠀⠀⣸⠁⠀⠀⠷⡴⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠲⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠖⠒⠊⠛⠛⠛⠛⠛⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⠠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⢸
⢰⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣼⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⡤⠤⣀⣀⡀⠀⠀⠉⠀⠀⠀⠙⣆⢀⣠⣤⢶⠶⢶⣤⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⠙⠳⠶⠇⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠸⠏⠀⠀⠀⠀⢸⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢠⡀⠀⠀⢰⡇⠀⠀⣠⠄⠀⢀⡴⠃⠀⠀⠀⠀⠀⠸⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⣯⡀⠀⠀⠀⠀⠀⠀⢀⡇
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠞⠁⠀⠀⠀⠳⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢀⣀⣤⣆⠀⢠⣄⣀⣀⣀⣀⣠⣽⠄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠛⠒⠛⠛⠋⠉⠀⣿⠀⢸⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⡤⠈⠂⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢺⣍⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣧⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣣⣿⣿⡿⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣥⣿⣿⣿⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣯⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⣿⡷⣻⣿⣿⣿⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣻⣿⣿⣿⣿⣏⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣿⣿⣭⣿⡿⣷⣦⣤⣤⣤⣤⣤⡤⠐⠒⠲⠶⠾⠿⠛⠋⠿⠿⢿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠂⣤⣤⣬⣿⣧⣌⢻⣿⣭⣥⣤⣤⣾⣿⣿⣿⢷⣶⣶⣶⠶⠾⣷⣿⠋⣈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠈⣡⣴⣟⣿⣿⣟⣿⣿⣿⣧⠹⣿⣿⣋⣽⣭⣮⣩⣶⣏⣡⣾⣯⣤⣾⣿⣿⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⢾⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠻⣧⠹⣿⠛⠛⠛⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⣾⢏⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠹⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣼⠏⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢹⣇⠀⠀⠀⠀⠀⠘⣿⣽⣿⣿⣿⡆⢿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢰⡟⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢻⡆⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⡘⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡿⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣼⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠸⣧⠀⠀⠀⠀⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢻⡇⠀⠀⠀⣼⣿⣿⣿⡿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⣿⡀⠀⠀⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⡇⠀⠘⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⣧⠀⠀⢹⣿⣿⣿⡀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡎⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣏⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⣠⣤⣴⣶⣶⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⣶⣾⣿⠘⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣼⣿
⣴⡿⠛⠋⣉⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⢈⣿⣇⢻⣿⠌⠙⠛⠛⠿⣿⣿⡟⣿⣿
⠻⢿⣷⣶⣯⣭⣭⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣛⣉⣉⣙⣛⣿⣿⢻⡄⣿⣶⣴⣶⣶⣶⣿⣿⣷⣿⣿
⠀⠈⣿⡁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⣉⠉⠉⠛⢿⣿⣿⣄⣀⣿⣟⣛⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣿⣿⣶⣶⣿⣿⣿⣿⣟⣋⠉⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠶⢾⣿⣿⣿⣿⣿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠉⠺⠟⣿⣿⣿⣿⣿⡿⠶⢿⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡋⢐⣿⣿⣿⣿⣿⣿⣟⣛⡛⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣻⣿⣿⣿⣯⣥⣬⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⣠⣭⣽⣿⣿⣿⠿⠉⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣬⣥⣿⣿⣿⣿⣶⠒⠂⣸⣿⣿⠹⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠲⢴⣶⣶⣶⣾⣿⣿⣿⣷⣬⣭⡍⢀⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣹⣿⣿⣿⣿⣿⣿⠿⠇⣼⣿⣿⣿⣿⣿⢿⣿⣿⣿⢹⣿
⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣭⣽⡟⢱⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⣽
⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣯⣿⣿⣿⣿⠿⠿⠁⣼⣿⠀⣽⣿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⣴⣶⣾⣿⣿⢿⣿⣿⡛⢠⣿⠇⠀⢿⣿⠻⣶⣹⣿⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣒⣒⣿⣿⣿⣿⣿⡭⠁⣾⡟⠀⠀⢸⣿⣦⡈⣿⣽⣿⣧⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠅⣀⣙⣛⣿⣿⣿⣿⠿⠃⢰⣿⠃⠀⢰⣿⣿⣿⣧⣼⣿⣿⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣿⣿⡿⣿⣉⡔⢀⣾⡿⠀⠀⠸⣿⣿⣿⣸⣿⣿⣿⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠾⠿⣿⣿⣿⣿⣿⣟⠉⣾⣿⠃⠀⠀⠀⣸⣿⣟⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣯⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣉⣭⣭⣥⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣶⣶⣶⣶⣾⠿⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠛⠀⠋⠈⠋⠀⠁⠉⠀⠀⠁⠈⠀⠋⠀⠁⠈⠀⠁⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠈⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⢀⣧⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠟⠛⠿⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢀⢤⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠃⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠛⠀⠀⠘⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡼⣿⣿⣷⠀⠀⠀⠀⢻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢀⡼⡇⠀⠛⠁⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠯⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢠⣤⣶⠾⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠉⣁⣀⣤⡞⠓⢤⣀⠀⠀⢀⠴⠖⠘⣿
⠀⠀⠀⠀⢠⡀⡀⡠⠤⢈⡀⢉⠉⠻⠘⠿⠋⠁⠀⠀⠀⢹⣟⠷⠋⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⢶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⣁⣘⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠉⠉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⣸⣟⠉⣤⠀⠀⣰⠊⡱⠆⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠐⢠⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣇⠀⢿⣀⠶⠙⠦⣽⠀⠀⠀⠘⢫⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣰⡿⣻⣿⠋⠐⠀⠀⠀⠀⠀⠀⣎⣻⢷⣾⣇⣠⣤⣿⠁⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⠀⠀⠀⣼⣿⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠺⠝⠛⠋⠉⠀⠀⠀⠳⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⢻⣿⢿⡟⠀⠀⠀⣸⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠠⣄⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⢨⣿⣿⡀⠀⠀⢀⣹⡟⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠀⠀⢀⠀⠘⣇
⠀⠂⠀⠀⠀⠀⣾⣿⡟⢡⡀⣀⣿⣉⢰⣿⢤⠀⡤⠀⠀⢘⣃⠀⣰⠄⢐⢿⠙⣧⡤⡄⠀⢀⣆⠀⢃⠀⢱⡀⠀⢀⡸⢆⠀⢹⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⣿⣿⠁⠀⣸⡿⠁⠀⣿⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⢳⠀⠀⠁⠀⠃⠀⢹⡆
⠀⠀⠀⠀⠀⢸⣿⡿⠂⢠⡿⠁⠀⢰⠏⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⣼⣿⠇⢀⣾⡟⡾⢠⡿⢠⠇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠈⣿⣿⠀⣼⡿⢹⠃⣿⡁⠊⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢠⣿⣿⣶⣿⠃⡎⣰⡟⠳⡇⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⠀⠀⢸⣇⠀⠀⠀⢠⡀⠀⠤
⠀⠀⠀⠀⠀⣿⣿⣿⡿⣼⠃⣿⠀⠀⠇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠻⢛⣻⣛⠋⢸⡇⠀⠀⠀⣼⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⡇⠀⠀⢰⣦⣀⠀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢢⣾⡿⣿⡿⠀⣾⣁⠀⠀⢠⡟⠁⠀⢀⣮⣄⠀⠀⠀⠀⠀⠀⣼⡇⠉⡶⣦⣀⡀⠀⠀⣤⠹⡀⠁⠒⠀⣀⡄⢀⡠⣄⠀⠙
⠀⠀⠀⣼⠨⣿⣏⠻⣿⣿⣟⣿⣆⡀⠈⢗⣤⠴⣿⠁⢸⠗⢤⣄⣀⠀⢀⣿⣿⠀⢣⠈⣏⠻⣿⡂⢻⡄⢳⡀⠀⠀⠈⡿⠇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⣼⣧⡙⣿⣷⣿⣿⣿⣿⡟⢡⣴⣿⡏⢰⡿⠀⢸⠆⠀⣿⠉⠙⠛⢻⣿⠀⢸⡄⢹⠀⢏⠹⣌⣷⠈⣧⠀⠀⠀⢐⠂
⠀⠀⠀⢿⣿⣷⣦⣉⣙⣛⣋⣥⢀⣿⡇⣿⠃⢸⣇⠀⢸⠀⠀⣾⠀⠀⠀⢹⢹⡇⠈⣇⠀⠀⠘⠆⠉⢻⣇⠸⡄⢀⠀⢈
⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⢻⡟⢸⣿⢰⣿⠀⢸⡟⠂⢸⠄⠀⢸⡀⠀⡀⢸⠀⣇⠀⢻⡄⢰⡀⠑⠀⢻⣿⠀⢳⠀⡆⠸⡆⠀⠇
⠀⠀⠀⣿⣿⣿⠿⢱⡟⠀⣿⠁⢸⡟⢸⡿⠀⣿⡃⠀⢀⡇⠀⢸⣇⠀⠀⠀⠀⣿⠄⢿⣿⠈⡇⠀⠀⠀⢿⡁⠈⠀⠙
⠀⠀⠸⣿⣿⣤⠀⠀⠀⢸⠃⢀⡞⠀⡾⠃⠀⣿⣶⠀⠸⠿⠀⠀⠉⠀⢸⠀⠠⣧⠄⠸⠅⠀⢃⠀⠀⣀⣸⣧⣀⣤⣄⣠⣤⣤⣠⣤⣶⣶⡶⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣴⣦⣤⣤⣀⣠⣾⣁⣀⣀⣦⣀⣀⣴⣶⣶⣆⣠⣴⣧⣤⣴⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡉⠁
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠻⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⡆⠀⣿⣿⣿⠿⠿⠧⠤⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙⠛⠛⠛⠛⠛⠃⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⡴⠋⠉⠁⠠⠄⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⣴⠊⠀⠘⠀⠈
⠀⠀⢀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠠⠀⠀⠀⠀⠀⢠⡂
⡀⣰⡟⣩⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⡇⠃⢀⣀⣀⣀⡀⡀⢯⣀⠠⠄
⢢⣿⢸⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⠀⠀⠘⠈⡖⠉
⠀⠀⠘⣎⠁⡀⣠⣶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠟⢉⣉⠙⢷⣤⣀⣀⣀⣴⡄⠀⠀⠀⠀⠀⢀⣿⣿⣷⡆
⠀⠀⠀⠨⢉⣴⠿⠃⠀⠀⠀⠀⠀⠒⠂⠤⢤⣐⣂⠉⢕⣡⣾⣿⣿⣿⣿⠃⣴⠛⠩⠙⢆⢹⣿⣿⣿⣿⣿⡆⠀⠀⢈⣷⣦⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⡇⢄⣴⡌⠌⡌⣿⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣇⢘⠛⠃⠀⠇⣿⣿⡿⢿⠿⠿⠿⠿⢻⣟⣛⠩⢍⣀⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⣦⣅⣡⠞⣸⣿⣿⠃⠐⠯⠉⠉⠁⠀⠓⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣉⣠⣴⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠁⠀⠉⠩⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠤⠤⠤⠤⢤⡟⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠘⣏⣉⠤⢄⣈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣀⣀⣀⣿⡀⠀⠀⠈⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠰⠄⢠⠀⠀⠀⠀⢳⠀⠀⠀⡏⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠂⠀⠀⠀⠀⠀⠀⠀⢾⡀⣠⠞⢡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡀⠈⠀⠀⣠⠤⠤⠤⢤⣀⠀⠀⠙⢆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡴⠋⠀⠀⠀⢀⡀⠈⠳⣄⠀⠀⠳⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢠⠎⠀⡆⠁⠰⠄⠦⠄⢁⡆⠈⢣⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢠⠋⠀⠀⠈⣉⠀⡀⠉⣉⠥⠀⠀⠀⢳⡀⠄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠏⢀⣀⡀⠤⠀⢀⠀⣠⠀⠀⠀⢠⣄⣀⢷⡘⡄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠃⡞⠀⠠⠥⠒⠚⠀⢹⠛⠉⢦⠀⢀⡜⠀⠀⠀⢧⢡⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡐⠌⢸⠁⠢⣄⠀⠀⠀⠠⡴⠀⠀⠀⠙⠋⠀⢠⠀⡆⠘⡆⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠃⠀⡎⠈⠢⣀⠑⠲⠤⠤⠸⡄⠀⢣⠀⣴⠀⠈⢳⠁⠀⢣⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⢠⡇⠀⠀⠀⠉⠐⠂⠀⣰⠇⠀⡀⢷⠁⠀⢀⡎⡠⠀⢸⠀⠈⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠄⠀⠀⠀⠀⠈⢻⠉⠀⠀⠀⠙⠺⡀⠀⡼⠋⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠈⡆⠐⠤⡀⠀⠐⠋⠠⠤⢄⣉⠑⢦⡇⡰⢁⣰⡀⠀⢸⠀⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢣⠀⠀⠈⠓⠢⠤⣀⣀⣀⣈⣳⠀⠛⠉⠉⠀⠀⠀⡞⠀⠀⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢎⡴⠚⠉⠛⠲⣤⠞⠁⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠒⠚⠒⠒⠚⠉⠉⠀⢀⡀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠉⠉⠉⠙⢃⣀⣀⢀⠀⠀⠀⠀⠀⠤⠤⠤⠒⠒⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣄⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⠔⠓⢆⡀⢐⣦⣠⠼⠷⡖⠀⠀⢀⠠⡾⠀⣤⣤⣤⣭⣥⣀⣀⡀⠀⠀⠘⠿⠃
⠶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⢶⣮⣷⢶⡚⠙⢯⣀⣈⣶⣤⠀⠀⠀⠀⠀⢱⠀⢒⣒⣒⣿⣿⣿⠿⠇
⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣧⠟⢾⡉⠉⢳⣄⡈⠀⠀⠠⠄⠀⠀⠀⢴⠚⠛⠩⠉
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠺⠗⠀⠀⠀⠀⠀⣿⠶⠖⠛⠛⠩⡈⠀⠐⡀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣶⢆⣶⠁⢀⣀⠄
⠀⠀⠀⠀⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠰⠷⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⢸⣿⡆⢰⣿⢸⠇⣴⣿⣿⣿⣦⢠⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣾⡄⢿⣿⣿⣿⡿⢰⡿
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣾⢿⣁⠀⠀⠙⠿⢦⣽⡿⣿⡁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠀⠀⢸⣧⠀⠀⣀⣠⣤⣤⣤⣤⣤⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⠀⣀⣀⣠⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡀⠀⠀⠀⠀⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⣇⣁⣐⡀⠬⢿⣁⣠⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠸⡝⢁⣀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠶⠟⠒⠲⠶⢶⣦⡿⠋⠀⠙⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠉⠉⠉⠉⠁⠀⠻⡇⠀⠀⢀⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢲⣴⠋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⣀⠐⠈⣁⣀⣀⣀⣀⠀⠀⠀⠉⠳⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠖⢁⠔⣿⣥⢖⠠⠓⢌⡙⠶⡀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡀⠀⣰⠏⣸⠟⠀⣹⠲⣄⡴⠉⢆⠹⣆⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢠⠃⡐⠡⣠⡼⠏⠀⠘⢿⠀⢈⣂⣹⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠁⣀⢸⣿⡏⠙⠛⠧⠀⠎⠀⠟⠋⠉⣿⣿⡿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡙⣧⠀⠶⠀⠀⠖⠀⡠⠀⣠⣿⣧⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⣿⣯⢤⡀⠀⠀⡦⠀⠀⢤⣿⠏⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠀⢾⣥⣤⣾⣿⣶⠦⠼⣋⣤⣽⡷⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠰⣄⠋⠐⠐⠋⡛⣐⣀⣀⢈⣷⡏⠀⠀⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢀⠀⢻⡁⣉⣉⣉⡉⣉⠉⡤⢌⡼⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠁⢀⠛⠾⠧⢽⣶⠤⠾⠶⠟⠁⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⠳⠤⠤⠀⠠⠤⠄⠤⠤⠤⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢶⣶⢾⣴⡋⣀⡀⠁⣘⣿⣷⣶⡦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠈⠌⠉⠉⠉⠩⠑⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡄⢠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠂⠀⠘⠿⠿⢿⣟⡙⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⡿⡿⠿⠿⠏⣿⣷⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠄
⠀⠀⠀⠀⢰⣷⣄⠈⠉⠉⠒⠒⠒⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢸⠁⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣠⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣾⣿⠇⠋
⠀⠀⠀⠀⢸⡆⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⡀⠀⠀⠀⠀⢠⡖⠀⠈⣿⣿⡟⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⢀⣤⣄
⠀⠀⠀⠀⠀⢳⣾⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⢿⡿⠁⠀⣹⣦⠀⠀⠀⡌⠓⠀⠀⠛⠉⠀⠔⠛⠛⠛⠛⢛⣋⣉⣉⣉⣩⣤⡤⠶⠶⠾⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠻⢿⣶⣤⣼⡇⠀⠃⠈⣹⣷⣀⠈⠙⠶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣇⠀⠂⣀⢹⣿⣿⣷⣦⠀⠀⠤⠆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣶⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠆⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠘⠓⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠌⣀⠀⠀⠀⡀⣀⡼⣯⣄⠀⠴⠀⠀⠀⠀⣤⡄⢤⣤⡵⠶⢻⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠤⠀⠉⠠⠀⠈⠀⠸⣧⠀⠀⠀⠀⠀⠀⡌⠀⠙⠀⠘⠿⢷⡴⠞⣡⣠⠀⠤⡄⠀⠀⠰⢞⡿⢦⠙⠿⣿⣿⠆
⠀⠀⠀⠀⠀⢀⡀⠺⠀⠂⠂⠀⣀⣀⣀⣾⡆⠋⠀⠀⠀⠀⠀⠀⠃⠀⠀⠦⠔⠳⢶⣦⣀⡏⠁⠴⡆⠀⠀⠀⠀⢦⠶⠶⢼⣋⣟⠻⣙⠃⠀⣰
⠀⠀⠀⠀⠂⠈⠙⠧⣦⠀⠐⢢⣀⡎⢍⠀⠈⠃⡰⠀⠀⠀⠀⠀⠨⡀⠀⢠⣤⠓⠂⠉⠀⠀⣠⠶⣴⠇⠀⠀⢠⠘⣴⡆⢯⠘⢣⡋⠼⣧⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠑⠫⠶⠜⣀⠤⠖⠁⠀⠀⠀⠀⠀⡀⢰⣐⣤⠀⣀⣀⣠⣠⡞⠉⠀⠀⠀⠀⠀⠀⠁⠈⠳⠞⠖⠈⠅⠓⠈⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠔⣊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠁⠘⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣀⡔⠚⣄⠀⠆⠀⠀⠀⣀⡈⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢀⣿⠃⠀⢯⣀⠀⠁⠀⠀⠈⢁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣽⡇⠀⡏⠘⢠⡀⢠⡄⠀⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⣿⡉⡀⢳⠈⠁⠀⠀⠀⢀⠀⢠⡿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠈⢻⠁⠘⠀⠈⠧⠀⠀⠙⠁⠞⢁⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠷⢄⠀⢀⡀⠀⠀⠀⢀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⣿⡿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣼⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠋⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢁⠋
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣐⠂⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⢀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠢⢤⠀⢠⡠⠀⠀⣠⢀⡲⠶⠔⡉⢅⠉⠀⠀⢭⡉⠐⠦⡀⠀⠀⠀⠀⠨⠤⢉⡀⠄⠠
⠀⠀⠀⠀⠀⠠⠔⠀⠐⠀⠼⡋⠉⢹⠈⠑⠐⣀⠤⣶⣧⠴⠄⠊⢠⡦⢆⡷⠀⠽⠀⠀⡔⣩⣉⢀⡀⠴⣾⠇⣐⠀⠀⠀⠀⠤⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢄⠙⠟⠛⣆⣉⡔⣀⡠⣔⣎⠰⠤⠚⠁⠀⠀⠀⡂⠉⣁⠉⣠⣤⣀⠅⠉⣠⠒⠷⠦⠊⡶⠀⠒⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘⢆⠤⣄⡀⢌⡨⢵⠀⠌⢰⡀⢠⠰⠞⠁⠠⢁⣄⡘⠿⠈⡃⣠⠈⡛⢲⣋⡴⢣⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⠈⠔⠂⡀⠰⣄⣈⠐⠀⡤⠃⠤⣈⣧⡀⠧⠭⠀⣀⣤⠃⠀⠉⠉⢉⠔⠁⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠢⢄⡄⠀⠐⠃⠀⠎⠙⠛⠛⣛⠉⡑⣲⠐⢀⣣⠀⠀⠀⡀⡄⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠐⢠⡠⠖⠀⠴⠆⣤⣤⣠⣤⠅⡥⢨⠟⠀⠀⠀⡆⣆⡗⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠒⠒⠂⡠⢀⣐⡐⠲⠔⠒⠚⢖⠦⠚⠃⠀⠀⠀⠃⠓⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢀⠀⠀⠀⠀⠀⠈⠒⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠍⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡀⠀⠀⠀⠀⠈⣀⠤⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠐⢉⠁⠀⠀⠀⠈⡒⠰⢂⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⠀⠂⠄⢀⠀⠀⠀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⠀⠀⢀⠀⢶⣄⣀⣀⣤⣄⣀⡀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍
⣿⣿⣿⣿⣿⣯⣴⣶⣶⣾⣿⣿⣿⠿⠋⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠛⢻⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟
⣿⣿⣿⣿⡿⣛⢰⣿⠋⢓⣾⣧⣤⣴⣷⡖⢦⣄⠀⣀⡀⠀⠀⠀⠀⣀⣀⢻⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣷⡟
⡿⠟⠛⠙⠇⠊⣿⠧⠙⠺⣷⣾⣏⣽⣟⣻⣿⣿⣿⣿⣶⣦⣤⣀⡀⠈⠉⠋⠻⣿⣿⣿⣿⣿⣿⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿
⡿⠀⠀⠀⠀⠀⣿⠂⠀⣀⣙⣛⣿⣿⣿⣿⡿⣫⣿⣿⣧⣌⠉⠻⢿⣷⣦⣄⠀⠉⠙⣿⣿⣿⣿⣷⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠙
⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠉⠛⠛⠛⢿⣿⡘⣿⣿⣿⣦⣿⡆⠀⠈⢿⣿⣿⣷⣄⡀⢘⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣮⡙⢿⣿⣿⠃⠀⠀⡀⣿⡿⢿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⡇⠀⣄⣴⣤⣤⣶⣿⣿⣿⣿⣷⣶⣿⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⢿⣶⣬⣀⡀⠐⠀⠁⢈⠇⠈⠛⢿⠛⠃⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠙⠁⠁⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠛⢛⡛⠛⠻⠻⠟⠁⠀⡈⢹⣿⠇
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⡛⠻⢿⣿⣷⣦⠀⠘⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣶⣶⣶⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠰⣟⣿⣿⣿⣽⣷⡄⠈⢿⣿⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣄⠛⢿⣿⣿⣿⠇⠀⣸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠂⠠⣉⠛⠡⠴⠞⣻⠋
⠁⠀⠀⠀⠀⠀⠀⠈⣿⡎⡆⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣧⠀⠀⠙⢿⣶⣶⠞⠁
⣶⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⣿⣦⡀⠀⣽⣿
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠋⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⠟⠛⠛⡷
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣀⣠⣤⣼⡃⠀⠀⠀⠀⠀⠀⠀⣻⠃⠀⠀⠀⠀⠀⣠⡿
⣿⣿⣿⠃⠀⠓⠦⣀⣀⡀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⣀⣀⣀⣀⣹⡏⠀⠀⠀⣤⣾⠟
⣿⣿⡿⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⡈⠻⣷⣄⣀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣀⣶⣿⣿⠃
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣸⣻⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⢀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣷⣤⣤⣶⣁⡻⠼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⠿⣽⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⠏⢸⣧⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⣿⣿⣷⣄⣼⣿⣶⣾⠟⠋⠁⠀⠀⠀⢀⡉⠛⢿⣿⣿⣿⣿⣿⣿⣍⡙⠻⠿⢿⣿⣿⣿⣿⣿⣦⣼⣿⡿⠁⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠟⠉
⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠸⡻⣦⡀⠈⠉⣿⠿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣠⣤⣤⣤⣶⡶⠟⠛⠉⠉
⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣀⡄⠻⣿⣧⡔⠀⠀⠀⠛⡉⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣻⣿⣦⡀⠀⠀⠀⠐⠿⢾⣷⣿⣿⣿⣷⣤⣔⠒⠀⠂⠀⠒⠆⠻⢹⣿⡿⣿⣿⠇
⣿⣿⣿⣿⣿⣇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣿⣿⣿⣶⣶⣶⣶⣶⡟⢀⣠⣾⠏
⣿⣿⣿⣿⣿⡋⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣻⣿⣿⣿⣿⣿⣷⣾⣿⠟
⣿⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⣴⣬⣶⣄⣄⡴⣀⠔⢀⣼⣿⠿⠋⠀⠀⠀⠀⠀⠀⢀⡞⠑⠀⠀⠀⢠⣴⡄⣨⡟⠀⡀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠙⠋⠛⠃⠻⠞⢧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠴⠿⠦⠦⠤⠤⣤⣼⣤⠀⢀⣀⣤⠴⠶⠞⠓
⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁
⣿⣿⣿⡿⠛⠁
⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡄⠢⠀⠂⠐⠀⠃⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆⣰⡰⡄⠀⠀⡸⠠⠄⠀⠐⠀⠀⠀⠀⠖⠄⠀⡄⠀⢠⠀⠀⠀⠴⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⢀⠇
⠀⠀⢀⠀⠀⠀⠁⠆⠀⠀⠀⢠⠈⠁⠙⡀⠠⡇⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⢀⡀⠈⠁⠠⠄⡀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠈⢀⠀⠀⠘⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⠈⠁⠀⠁⠀⠀⠧⠀⠀⠢⡀⠀⠀⠂
⠀⠀⢀⣀⡤⠀⠈⠀⡠⠄⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠉⠉⠉⠙⠒⠶⣄⠀⠠
⠀⠀⠈⠉⠀⠠⠀⠒⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠⡄⠀⠀⠀⠀⠀⢠⡇⠀⠐⠀⡀⠁⠀⠀⢰⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠁⢀⡀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠀⢢⡔⠢⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⠀⠀⡀⠐⠀⠢⠄⠀⠖⠚⠃⠁⠀⠀⡈⠂
⠀⠀⠀⠀⠀⠀⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠀⢀⠠⠀⠀⠤⢀⣀⠠⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠒⠚⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠒⠒⠉⠉⡏⠁⠀⠀⠀⠀⣨⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⢀⠀⠐⠈⠛⢧⠀⠀⠀⣣⡤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⡜⢀⠴⡞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⣀⡝⠀⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣀⣠⠴⠦⢄⣀⣉⣠⠤⢴⠖⠶⠖⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠴⠅⠀⠀⠈⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢇⠀⠀⠀⠘⠓⠲⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠤⠤⣄⣀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠙⣆
⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢤⠲⠶⠃⠈⠓⢒⠶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⢧
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠈⣧
⠀⣸⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠧⠒⠲⠤⠏⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠸⡆
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⠤⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⡄⠀⠀⠀⣿
⢸⠀⠀⠀⢀⠀⠀⠀⡤⠄⢰⠋⢂⠀⢘⠁⣀⠑⠈⠀⣤⠀⢳⠁⡇⠘⠁⢉⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢸⡆
⡟⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⢀⠀⠠⣸⠀⢋⠀⠰⡄⢠⠶⢛⠀⠇⣤⠀⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡏⠡⣀⡽⠄⠎⠑⠒⠳⠚⠀⠘⠂⠠⠿⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇
⣇⠀⠀⠐⠀⠀⠀⠘⠒⠃⠙⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⡇
⢹⠀⠀⠀⠀⠀⠀⠀⡠⠆⠶⡄⢤⠒⠠⠊⢱⠜⠁⡀⠈⢁⡖⢂⡰⠤⡴⠒⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠇
⢸⡄⠀⠀⡀⠀⠀⠸⣄⠀⠘⠳⡄⠑⡄⢀⠃⠀⠀⠉⠰⢏⡇⠀⠁⢰⡇⢀⣉⣀⡼⠀⠀⠀⠄⠀⠀⠀⡸⠀⠀⠀⣾
⠀⢧⠀⠀⠀⠀⠀⠀⡤⠜⠃⢀⡄⡘⠁⡜⠀⡇⠀⡛⢆⣀⠷⣄⣀⠼⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇
⠀⠘⣇⠀⠀⠀⠀⠀⠒⠒⠒⠉⠀⠙⠚⠀⠀⠉⠒⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡜
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡔⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣼⠁
⠀⠀⠀⠘⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⢀⡞⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠖⢆⠀⢀⡰⠁⠀⠀⠀⢀⠎⠀⠀⠀⣸⠦⠤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⠆⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠶⠴⠤⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡯⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠲⢦⣀⠀⠀⣀⣀⡤⠤⠶⠒⠒⠛⠓⠒⠦⠤⣤⣤⣤⣤⣤⠤⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣿⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⢀⣀⡙⣁⣤⠤⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⢀⠀⠀⡆⠀⠀⠃⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠁⣀⠀⡢⠴⠇⠀⠀⠃⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣤⠀⠦⠀⠀⠀⠀⠀⠂⠉⠀⠀⣀⣠⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⡀⠂⠀⠓⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠐⠋⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠛⠀⢀⣦⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠦⠀⠀⠀⣸⣿⣿⢻⣿⣟⣿⣿⣿⡟⣿⣿⣿⣿⡇⠀⢀⢨⠾⠡⠁⠀⠀⣠⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⠃⠀⠸⡉⠁⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠖
⠀⠀⠀⠀⣀⠀⠻⠃⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠃⠀⠀⠀⠼⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠸⠆⠀⢠⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠟⠀⠈⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠰⡆⠀⠀⡀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠰⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⣀⡀⠀⠰⠀⠀⠀⠀⡀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

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

⠀⠀⠀⠠⣶⣄⠀⠀⠀⣤⣶⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⢿⡟⣧⣤⡾⢻⠇⠀⠀⠀⢻⡏⠑⢆⣄⡔⠋⢻⠇⠀⠀⣧⣏⣢⣀⣀⣠⣊⣷⡇⠀⠀⣿⣿⣷⣄⢀⣴⣿⣿⡇
⠀⠀⠀⠀⣴⢃⡜⠟⢥⢸⡆⠀⠀⠀⠀⣾⣽⣿⢿⡿⢽⡦⠀⠀⠀⢀⣭⣯⣛⣟⣫⣿⣍⠀⠀⠀⢨⣿⣏⢿⢿⢏⣿⣏
⠀⠀⠀⠀⠽⠷⠀⠀⠀⠾⢇⠀⠀⠀⠀⠓⠞⠁⠀⠙⠲⠋⠀⠀⠀⠀⠻⠿⠁⠀⠙⠿⠃⠀⠀⠀⠀⠻⠿⠃⠀⠻⠿⠃

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢸⣿⣟⣄⣀⣴⡻⣽⠃⠀⠀⠰⠀⡀⠀⡀⠀⡀⠀⠀⠀⠀⣿⠁⠁⢄⣀⠄⠁⢸⡇⠀⠀⠀⠠⢤⣄⣄⣤⡤
⠀⠀⠀⠀⣽⣿⡽⡿⢿⣿⣇⠀⠀⠀⠀⢀⠄⠐⡿⠀⢀⠀⠀⠀⠀⢨⡴⢂⣿⣿⣇⠲⣮⠀⠀⠀⠳⣤⣤⣾⣿⣦⣤⡤⠇
⠀⠀⠀⠀⠻⠟⠃⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⢿⠾⠋⠈⠻⢶⠏⠀⠀⠀⠀⠈⠒⠊⠑⠒⠊


⠀⠀⠀⢰⠖⢤⣄⣀⣠⡄⢶⡄⠀⠠⣖⠖⠲⣤⣤⡴⠒⠆⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⣦⣀⣀⣠⣶⡀⠆
⠀⠀⠀⠈⢣⣀⢹⣿⣅⣠⡟⠀⠀⠀⠈⠑⣆⣠⢳⣀⡴⠈⠀⠀⠀⠀⠀⢀⡈⠋⣀⠀⠀⠀⠀⠀⠀⢻⡟⠿⣿⠟⣿⠃
⠀⠀⠀⠀⠘⠚⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠅⠀⠀⠀⠀⠀⠀⠙⠃⠀⠛⠁⠀⠀⠀⠀⠀⠘⠻⠟⠙⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡉⠐⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠦⣙⣤⠀⠀⠀⠀⠀⠀⠀⢮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⣦⣫⡶⠋⠀⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠿⣷⣋⡦⠀⠀⠈⢻⢲⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡲⠆⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⡾⠀⠑⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡔⠁⠀⢀⠄⣢⡆⠀⠀⠀⠀⠀⠀⠘⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⢀⡈⢧⠀⠀⠀⡇
⠀⠀⠀⢀⠊⢠⠃⠰⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣾⡇⠀⠀⠀⢠⣿⡟⠀⢳⠀⠀⡇
⠀⠀⠀⠊⠀⠁⠀⠀⠀⢠⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠘⠛⢁⠀⠀⡇⠀⡇
⠀⠀⠶⢀⡵⠀⠀⠀⠀⢨⣼⠁⠀⠀⠀⠀⠀⠀⢀⣠⣖⣮⣽⣦⣤⣴⣦⣤⣇⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⢄⡀⠸⡄⡇
⠀⢘⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠛⠷⠟⠀⢻⡇
⠀⢨⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠻⣅⣀⠤⢊⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠁⠀⠀⠀⠰⠀⣠⡇⣸⡃
⠀⣈⡀⠠⠀⠀⠀⣸⡿⢳⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠒⠈⠙⠲⠤⠤⠤⡤⠤⠤⠔⠒⠚⡉⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⡷⠀⢀⡴⠏⢀⣼⣿⢠⠟⡇
⠀⡍⡄⠰⣴⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠐⠀⠀⠉⠀⠀⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡤⠖⠋⢀⣤⣿⣿⢿⣿⢘⡇
⠀⠳⡙⡄⠈⢿⣷⣤⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠤⣄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣠⣒⠋⠀⢀⣠⣴⣿⣿⣟⣵⣿⡇⢨⡇
⠀⠀⢹⣿⣷⣦⡙⢿⣿⣷⣤⣘⢮⣭⡹⠳⠂⠀⢀⢾⣷⠀⠀⠀⠀⠀⠀⠸⠶⠾⠿⠋⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠚⠀⢀⣴⣦⣴⣿⣿⣿⡅⣴⣿⣿⣿⠇⠰⡇
⠀⠀⠈⡄⠙⠿⣿⣜⠿⠿⠀⠙⢻⢞⡂⢀⡀⠀⠻⠞⣡⣶⡦⠀⠀⠀⠀⠀⠄⠀⣀⠠⠤⠤⠀⠀⠀⠀⠀⣀⣀⣠⣤⢰⣿⣿⣿⣿⡿⠟⠃⣼⣿⣿⣿⠏⠀⢸⡇
⠀⠀⠀⠱⡀⠀⠈⠿⢿⣶⣦⡀⠈⠛⢿⡿⣷⠿⠷⣾⣿⣯⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣯⣤⣴⣾⣿⣿⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠛⠿⣿⣦⣤⣤⣤⣤⣀⣀⡀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠟⠻⠛⢛⣿⣋⣽⣷⣶⣬⣟⣛⣉⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠈⠉⠉⠛⠟⠿⢿⣿⣿⣦⣄⡀⠉⠛⠛⢉⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣴⠀⠀⠀⠀⠉⠈⠛⠿⣿⣷⣦⣤⣶⣿⠿⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⣤⣤⣀⠀⠀⠑⠛⠛⢿⣯⣶⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠾⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠶⠧⣷⣤⢜⣫⣧
⠒⢬⠂⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢨⠃⠠⠀⠀⠀⠀⠀⠀⠹⣿⣶⣄⣠⠀⠉⠻⠶⠻⠆⠀⠀⠀⠀⡀⠀⡀⢠⡄
⠒⠉⠱⣣⠆⡀⠀⠀⠀⠀⠀⠀⢠⣤⠠⢘⠃⠠⠴⢣⠿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣿⡟⣴⣤⡀⠀⠀⠀⠀⠀⠶⢀⡉⠉⣱⡂
⢁⠀⠈⠡⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢌⠁⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⡇⠏⢻⡿⢿⡦
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣷⣤⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠒⢩⣤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠟⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣀⣀
⠀⢰⠁⠈⡙⢿⣷⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⢹⣏⣉⣀⣢
⠀⠈⠀⢇⠀⠹⣿⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢤⢀⡄⠀⠀⠈⠛⢯⡙
⠀⠀⠀⠀⠀⡀⠙⠛⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣴⠖⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠙⢥⠖⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⢺⣽⠀⠀⠀⠀⠀⠹⠁⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⢲⠟⠛
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⡶⠠⡄⠀⠀⠀⠀⠀⠀⠀⣋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⢸⠁⢀⠆⠀⡐⠂⠲⠀⠀⠀⣿⠁⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠚⢢⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢷⠀⠀⠉⠁⠀⠀⢣⣠⣀⠀⠀⠀⠋⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⡀⠶⣀⠺⢂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠀⠀⠀⠀⠳⠄⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠃⠀⠀⠀⠀⠘⠋⡤⠉⠴⢏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣠⡀⠛⠴⠎⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡅⠀⠀⠀⠀⢀⡌⣹⣦⡀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠇⠀⠀⠀⠀⠀⠤⠉⠁⠛⣷⣟
⠀⠀⠀⠀⠀⠀⣠⠾⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢩⠉⠀⠈⠹⡿⠉⠉⠉⠙⠉⠯⠉⠉⢈⠍⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠸⠦⠴⠶⣷⣦⣶⠶⢷⣦⡰⡂⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡴⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⡷⠶⠶⠶⠶⠶⠶⠦⠾⠄⠀⠀⢰
⠀⠀⠀⢀⠠⢼⣄⠸⡄⠀⠉⠹⠉⢻⡿⠛⠛⠛⠿⠿⠿⢯⢭⣤⣾⣶⣶⣤⣀⣀⠀⣀⡠⠾⣦⣄⣀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⢀⣿⣿⣿⣿⣷⡷⠀⠀⠀⠀⠛⣏⠁⠀⠀⠀⠒⠒⠒⡾⣾⡶⠖⣤⠤⡤⢼⣿⣯⣷⣿⢾⣿⣶⣿⠀⢀⣴⡖⠻⡇
⠀⠀⢸⣿⣶⣮⣿⣿⣧⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⢰⣦⣷⣿⣥⡟⠀⡀⠈⢸⣿⣿⡧⣿⠰⢿⣿⡟⣀⣼⡇⠭⠀⠇⣄⣀
⠀⠀⢸⣿⣿⢷⣿⡇⢸⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣘⡛⠻⠼⢿⣧⣈⣀⡀⢸⣿⣿⣟⣿⢸⣿⣿⣿⣿⡿⢿⣷⣾⣿⣿⣿⠗⢀
⠀⠀⠀⣿⣿⢿⡿⡧⠄⣿⢿⡅⢢⠀⣶⠆⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠛⣿⡇⢸⣿⣿⣷⣾⣸⣿⣿⣿⠿⢿⣷⣿⡿⣿⣿⣿⡷⠁
⠀⠀⠀⢹⣿⡞⢣⣿⣦⡛⠯⠿⣓⡦⠄⡄⣰⣿⣿⣿⣿⣿⣿⣿⣷⡤⣤⠟⣇⢸⣿⣿⣿⣿⢿⣿⣿⣇⢠⡄⢨⡟⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠈⢿⣿⣾⣿⠁⠈⠂⠠⡀⠉⠛⣿⣿⣿⣿⡿⢭⠀⠸⣿⣿⣷⣷⣾⣿⢿⣯⡿⠷⠿⠾⠛⠛⣿⣦⣤⣾⣷⣶⡄⠀⣻⣿⣇
⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⣿⠟⠈⠀⢆⠙⣷⢿⣿⣿⣿⣮⣷⣲⣦⣼⣿⣿⣿⣿⡿⠿⠿⢿⣿⡯⠀⡬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡗⠄⣘⡑⠧⢿⢸⣿⣿⣷⡿⢿⢿⣿⢻⣏⣉⣄⣤⢴⣶⣶⡿⠟⠁⢠⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡗⡘⢿⣿⣍⢹⣞⣿⢿⣿⠗⠚⠛⠛⠚⠋⠉⠁⠀⠈⢻⣿⣿⣦⣶⣯⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡆⡘⠙⠀⣸⡽⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠓⢄⠀⡿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣾⣡⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⡤⠤⠤⠤⠤⠤⣄⣀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠄⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⠖⠉⡉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⢡⠖⠛⢛⢲⣞⠒⢇⠁⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠋⠉⠉⠓⢦⠤⠤⣤⠶⠤⣤⠤⠖⢉⣨⠽⠛⣿⣆⠀⠀⠇⢴⠊⢱⠛⠉⠁⠀⢓⠩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⢀⣀⣈⣷⢖⣚⡷⢀⡔⠒⠛⠉⠀⢀⣴⡿⠋⠀⠀⠀⠘⠑⠎⡳⠖⢟⡠⠳⡄
⠀⠀⠀⠀⣠⣶⠿⠷⠈⠙⢷⣦⣀⡏⠉⠉⠁⠀⠀⠛⠉⠳⣄⣀⣤⡶⠛⢁⠀⣀⠤⠤⢤⡀⠀⠀⠁⢴⠈⢱⡆⣿⠐
⠀⠀⠀⠀⠛⣿⣄⠀⠀⠁⠀⡀⠉⠛⠷⠶⠶⢤⠶⠶⠶⠛⠋⢉⣀⣤⠖⠃⠰⢑⡄⠆⣤⢹⣄⡤⠦⠉⠙⠈⠡⠃
⠀⠀⠀⠀⠐⠈⠛⢷⣤⡀⣎⠉⠉⠀⠀⠀⠠⠀⠠⣄⢀⣠⣴⠿⠋⠁⠈⠀⠀⣈⠃⣄⠿⣸⠃⣠⠰⢤⡘⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣤⣤⣤⠴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⢿⢦⡀⠗⣤⡺⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣴⣻⡌⡇⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠚⢉⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⡒⢲⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣉⣉⣉⣉⣁⣉⣉⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠛⠻⠿⠿⠿⠿⠿⠷⠾⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠦⠀⠀⠐⠶⠶⠶⠶⠶⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣧⡀⠀⣀⣤⡄⠀⣤⣼⢀⣴⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⢹⣿⡇⠀⣿⣿⠸⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠇⢰⠀⢸⣿⡇⠀⣿⣿⡄⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⣿⠀⢸⠀⣿⣿⡇⠀⢸⣿⣿⢨⡉⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⣾⣿⠀⡄⢀⣿⣿⡇⠀⢸⣿⣿⡼⣇⠀⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣼⣿⣿⠀⠇⢸⣿⣿⡇⠀⢸⣿⣿⡇⢿⡄⠀⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⣿⣿⠇⡆⠀⠈⠛⢿⡇⠀⠸⣿⣿⣿⡘⣿⣄⠰⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⢀⠃⠀⠀⠀⢸⡇⠀⠀⠙⢿⣿⡇⠻⣿⡷⠆⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣭⣤⣤⣤⣿⣤⣀⣀⣀⣤⣶⣄⣀⣀⣀⣀⣿⣦⣤⠠⡶⠿⠿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣛⣛⣛⣋⣩⣿⣿⣿⡿⠯⣍⣹⡛⠉⠩⣭⣽⠟⢿⣿⣿⣿⡿⠟⠓⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⠟⠋⣉⣿⣿⣋⣴⣤⣴⡶⠊⠀⠙⣿⠗⠈⠳⣦⣬⣿⣿⢿⣿⣿⠿⠛⠺⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠟⠛⣹⡿⢁⣴⡟⢁⣈⠉⠁⢾⣿⣋⣀⡹⠷⠂⠙⣅⡀⢱⣦⣸⣿⣿⣿⠶⠾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⢛⣿⢿⠟⢉⡽⠃⣹⣿⣾⡿⠿⠛⠛⠛⠿⢿⣷⣦⡙⣷⡞⣭⡉⢉⡹⣿⣿⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⣩⡟⣥⣿⡷⠻⣷⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢿⡿⢺⣷⠾⣿⣶⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⢛⡟⠁⢻⣾⣀⣾⡟⠁⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣿⣧⡜⢿⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⣿⣸⣷⠚⠙⢿⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⣴⣿⣿⡷⠀⠀⠀⠀⢽⣿⣿⠿⠟⢿⣿⣧⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡟⠁⣼⡄⠹⠀⠀⠀⠀⠀⣀⡀⢀⣻⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠻⣿⡔⢷⣼⠙⣿⡉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⡟⣧⡰⠿⠻⡆⠀⠀⠀⠀⣼⣿⡇⠘⢿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠙⣷⢠⡟⢳⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⡇⠙⡇⠸⣇⠀⠀⠀⠐⣶⣿⣿⣿⡖⠈⢻⡿⣿⡿⢿⣿⠟⠋⠁⠀⠀⠀⠀⠸⢻⣿⢧⣦⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⢇⠀⣿⡆⠉⠀⠀⠀⠀⠙⣿⣿⡿⠇⠀⡸⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣀⢻⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢾⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⢿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣷⡈⢡⣄⢻⠀⠀⣤⣤⣤⣄⣤⣄⠀⣤⣤⡄⠀⢠⣤⣴⣤⠰⣿⠆⣠⣤⣤⡀⠀⢸⣿⡇⣿⣇⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣿⠃⢼⠋⢸⠀⠀⣿⡏⢸⡿⢸⣿⢠⣶⣿⣷⣤⢸⣯⣸⣿⠀⣿⠆⣿⡷⢻⡇⠀⢸⣿⣿⠏⣿⣿⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣇⣼⠉⠳⠀⠀⠈⠁⠈⠃⠈⠋⠀⠙⠋⠙⠉⠸⠿⠉⠀⠀⠿⠆⠙⠛⠛⠁⠀⣾⣿⠉⣼⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⢻⡿⠁⢸⡆⣿⠀⢰⣿⣛⠂⠺⣷⣀⡾⢰⣿⠾⣷⢴⣿⠀⣾⡆⢰⣿⠿⣷⡄⠀⣿⣿⡾⣛⣿⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠻⣿⣀⡇⠠⠀⡆⢠⣍⣿⡿⠀⢻⣿⠃⢸⣿⡇⠁⠘⣿⣦⣿⡇⢸⣿⣤⣿⠃⣸⣿⣅⣴⢿⣿⠛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⣯⣝⠿⣤⣤⠁⡄⠀⠀⠐⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⢰⣿⣿⢿⡏⣼⡿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣣⣼⣿⣄⣽⡉⣆⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⡁⣼⣷⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⢿⣏⢻⡷⢿⣦⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣩⡿⠿⠏⢡⣿⠋⣡⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣝⣛⣿⣿⡿⣶⡤⢶⣌⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⣴⡟⣠⣾⣷⣿⠷⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣯⡉⠻⣷⣈⣣⣈⣽⠂⣨⡐⠂⣀⠀⠀⠀⠀⠀⢠⡔⢒⣩⡶⠿⣿⠗⣭⣽⡿⣡⣴⠾⣫⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣍⣀⣈⣿⣿⡟⠻⣷⣬⣿⡿⠒⢶⣦⣬⡵⠶⣶⣾⣻⣥⣤⣾⡿⢾⣿⠏⠘⠛⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠻⠍⠉⠛⢿⣄⣹⣏⡈⠻⣷⡦⣉⣭⣿⣿⣿⡿⣫⣽⣷⣿⣶⣟⣥⣤⣶⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣶⣶⠀⠀⢈⣿⣿⣿⣷⣉⣲⣿⣿⣷⡛⢛⣾⣿⣿⠟⠋⣀⣩⣽⡿⠿⠛⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡜⠛⢻⣶⣾⣿⣟⣙⣻⣿⣿⣿⣿⣿⣿⠿⣛⣻⣿⣿⣏⣉⣠⣤⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣾⣯⣍⠉⠯⠛⠛⠛⠛⠛⠛⠛⠛⠛⢯⣭⣭⣟⣛⣿⣯⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠶⠲⠶⠖⠲⠶⠖⠲⠒⠚⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠁⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⣁⣀⣀⣀⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⢫⣀⠀⠀⠀⢨⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠉⣳⠀⢀⠼⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠁⡎⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⣀⣀⡀⠈⠊⣁⣸⡤⢄⠤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠉⠁⠀⠀⡭⡉⠁⠀⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠑⠁⠀⠀⢸⠀⠀⠀⠙⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠤⢄⡀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⣀⣤⠜⠐⠄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢑⠄⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠲⠤⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠤⡀⢠⠄⠠⢀⡄⡀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠘⠁⡜⠀⠀⠀⡇⢦⠈⠀⠘⠂⣸⠃⡃⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠀⠔⠀⠐⣇⡰⠄⠸⣆⠏⠀⢀⣉⣇⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠦⠎⠑⠃⠈⠉⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠄⠒⡄⢔⠢⠔⢱⠞⢁⠈⠁⡔⠂⡤⢤⠖⠒⠒⡄⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠣⢄⡈⠁⡄⢱⠀⡀⡀⠈⡀⠚⢥⠀⠀⢸⠀⡍⠠⠖⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢰⣊⣁⠴⠲⣅⡜⠀⠧⢠⠍⠦⠚⠂⠀⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠸⠄⠀⠀⠀⠀⠀⡰⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠠⡀⢀⡠⠂⠀⠀⠀⡔⠁⠀⠀⡷⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠲⠶⡶⠶⠀⠠⠤⠤⠼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠶⠶⠤⢤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠃
⠀⠀⠀⠀⠀⢀⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠒⠀⠀⠀⠀⠳⣀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠺⠀⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠙⠂⠠⠤⣤⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠀⠀⠀⠈⢿⠛⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠿⠉⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠂
⠀⠀⠀⠁⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠼⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡠
⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠢⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡼⠁
⠀⠀⠀⠀⠀⠀⣒⠀⠉⠀⠠⠤⠤⠤⠶⣤⣿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠰⠂⠀⠠⠀⠐⠋⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⠀⠀⠀⠠⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠈⠀⠆⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠈
⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⢄
⠀⠀⠀⠀⠀⠈⠉⢳⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣈⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⣀
⠀⠠⠤⠤⠤⠤⢀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⡀⠉⢓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⣤⣤⣤⣤⣾⣾⠏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠙⠋⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢐⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⢀⡠⢀⠙⠊
⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣝⢌⠎⣆⡆
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠙⡜⣾⣼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠠⠀⢠⣄⠉⠁⠀⠮⢼⠀⠤⠶⠆⢠⣤⢂⠀⢀⡀⢀⠀⠀⠠
⠀⠀⠀⠀⠁⠐⠂⠀⠀⠙⠉⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢈⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠐⢄⡀⠀⠀⠀⣀⣀⡉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠇⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠂⠁⠀⠊⠉⠲⡀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠡⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⣈⠉⡆⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢠⡤⠄⢇⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠆⠠⠂⡐⢦⣍⣠⢟⡀⠀⠀⠀⠐⠐⠠⠀⢀⡔⠁⠀⠐⠀⢑⢠⠔⠀⠀⠀⠀⠑⠄⠀⠀⢁
⠀⠀⠀⠀⠀⠘⠀⠀⢈⡈⠦⡄⠀⣄⠈⠂⡤⠀⠀⠀⠀⠀⣹⡌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠐⢢⢨⣰⠂⠠⠴⠁⢤⠀⠀⡀⢀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⡙⡘⢣⡗⠀⠰⠀⠀⠠⠁⠀⠀⠁⠀⢄⠤⠟⠃⠐⠐⠀⢀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠦⠀⡀⠈⠁⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⣠⠄⠃⠀⠀⠀⠀⠀⠉⠠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂⠐⠒⠒⠃⡀⠂⠀⠐⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠐⡀⢠⠖⠀⠀⠀⠈⠀⠈⠑⠖⠀⠠⠛⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠱⡈⠢⠀⠀⠁⠀⠀⠀⠀⠁⠀⡄⠀⠁⠈⠑⠒⠣⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠂⠄⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠠⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠔⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠁⠒⠒⠈⠀⠂⠀⠀⣀⡠⠤⠚⠋⠀⠁⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠁


⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠤⡤⣤⡤⠶⠴⡦⡤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣤⣤⣤⣤⣤⣾⣴⣿⣤⣤⣄⡐⡶⠂⣀⣀⠭⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⣿⠉⢻⣯⣥⣽⣿⠛⠋⠉⠉⠁⠀⠉⠙⠿⡇⢷⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⢻⡇⢀⣼⣟⣀⣘⠿⢿⣿⡿⠤⣤⣀⢀⡀⢀⡔⢀⡀⣇⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⠾⠐⢛⣛⠉⢛⡛⣵⠟⠉⠉⠙⠛⠛⠛⠛⠀⠀⢱⣋⠁⠈⢈⡉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠠⣾⠙⠀⢀⣻⠆⢀⣰⠋⠀⠀⠀⠀⣠⣤⣴⠦⣄⡀⠀⠀⠉⠀⠀⠐⠆⠈⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡰⡛⠓⣄⢀⠀⠀⠀⠀⢤⡄⣶⡇⠀⠀⠠⠚⠉⡙⣷⠀⠀⣼⣷⣤⣤⠖⢢⣤⣴⠇⠀⠐⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠐⣇⠻⠀⠞⠈⠀⠀⠀⠀⢸⢠⣿⠀⠀⠀⢈⠻⠆⡘⠋⠀⣸⣿⢿⣿⣇⢙⣡⣿⣿⠀⠀⢸⣯⣿⣇
⠀⠀⠀⠀⠀⢀⣲⣮⣭⢋⣰⣄⣡⣤⣤⣶⣾⣼⣿⣄⠚⣴⣦⡥⠼⣦⠀⢠⣿⡏⢸⣿⣿⣍⣽⣿⡏⠀⠀⢸⡟⢸⣿
⠀⠀⠀⠀⠀⠛⣿⣭⣽⣞⣛⡛⠛⠛⠛⠛⣟⢻⡟⠛⣾⣛⣈⣽⣿⣿⣿⠃⣿⡇⢸⣿⣿⡿⠿⢿⣧⣦⣀⣼⡅⢸⠏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠻⠿⠛⠃⠉⠉⢿⣿⣿⣿⡿⠿⠿⠿⣿⣀⣿⡀⣼⠿⠿⠿⠿⠿⠉⢿⣟⣻⣧⡸
⠀⠀⠀⠀⠐⠒⠚⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠓⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⣀⣀
⠀⢠⣤⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⢹⣿⡋⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⢛⣿⣿⣿⣿⣷⣶⠆
⠀⠈⠋⢰⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠭⠿⠿⠿⠿⠿⠁
⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⡿⣷⣄⠀⠀⢻⣿⣿⣿⣯⣙⣿⣿⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⢳⣶⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣾⣿⣧⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⠃⠠⣾⣿⣿⣿⣷⣦⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣢⣤⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠁⣼⠏⠉⠛⠙⢻⣿⣿⣿⣿⠏⠀⢻⣿⣿⣿⣿⣆⠈⠛⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠉⠀⠀⠀⠀⠸⡇⢠⣭⡍⠀⣠⣾⣿⣿⣿⣿⣿⡏⢈⡉⠉⠉
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣿⢁⣤⣴⣿⣆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣸⠇⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣛⣷⣿⣿⣿⡿⢠⡟⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢉⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣾⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣿⠟⣿⣿⣿⣿⠻⠿⠋⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⢈⣿⣿⡇⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠻⠇⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⡖⢾⣋⣉⣙⣙⣉⣙⣉⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠂⣠⡂⣤
⠀⢀⠤⣀⣀⢐⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⡀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢈⣻⠔⡻
⠀⣧⣿⠿⠿⢷⣅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⠖⠋⢀⠠
⠀⠻⣏⣀⢀⡸⣫⠁⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⢰⠠⠆⠀⠠⠄⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡚⠍⠀⠀⠀⣀⣀
⠀⠀⠈⠉⠗⠋⠁⠀⠀⠀⠀⠀⠀⣀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀
⠀⠀⠀⠀⠀⢐⠀⣠⡤⠴⠖⠒⠂⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡡⠌
⠀⠀⠀⠀⢀⡿⠊⠁⠀⣄⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠗⠒⠀⠀⠀⠀⣤⣳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠠⠀⠀⠀⠀⣀
⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⢄⠑⠠⡀⢤⣀⣸⣿⡎⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠿⣿⣦⡱⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣄⣌⣙⣿⡇⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣧⠀⠀⢸⣿⡇⡁⡀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠈⠄⠋⠀⠀⠀⠀⠀⠀⢀⠎⠙⢻⡀⢸⣿⡟⣰⠀⠠⠀⠀⠀⠀⢀⡴⠀⠀⠀⠘⢮⡳⢴⣶⠿⠊⠀⡈⠀⠀⠀⠀⠶⠄⠀⠄⠀⢰⣍⣳
⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⣈⠂⡂⢻⣷⣿⠏⡔⠉⠀⠀⠀⠀⠠⠂⠙⠁⠀⠀⠀⠀⠀⠉⠉⠉⠒⠉⠁⠀⠀⠀⠀⠄⣀⠀⠄⠀⢀⣮⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠚⠶⠿⠶⠿⠛⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠒⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⢬⡍
⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠈⠃⠀⡠⢀⣠
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⠀⠀⠂⠄⠈⠀⠀⣈⠅⠀⢠⣾⣦
⠀⣀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠖⠀⢀⠌⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠙⡬⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠶⠖⠒⠛⠛⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣈⣠⢤⡤⠤⠤⠤⠤⠴⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⢀⠣⢤⡀⠀⠀⠜⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠈⠀⠀⠀⢀⠇⣀⡴⠋⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡌⢸⠁⢰⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢄⣀⠤⠦⢄⣈⣁⠤⢼⡖⠶⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠢⠀⠀⠀⡄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⢁⠀⠀⠀⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠤⢀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢀⡤⠤⠇⠐⠤⢠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠱⡄
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢣⠀⠀⠀⢹⡀
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠲⠴⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢷
⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡠⠤⠀⡤⠖⠠⣀⡤⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢸⡆
⡼⠀⠀⠀⠂⠀⠀⠚⢠⡋⠀⠀⠈⠠⡦⠀⡀⠰⠀⣸⢀⡁⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣧
⡇⠀⠀⠀⠀⠀⠀⠀⡄⢠⠦⠈⢢⣀⡶⣀⣹⣀⡏⢠⣈⢉⡇⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠠⠤⠋⠼⠀⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸⡀⠀⠀⣿
⢷⠀⠀⠀⠀⠀⠀⣠⠤⢄⢠⡴⢤⠔⢦⠒⢉⠉⠂⡤⢀⡤⣤⠴⠒⠦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡿
⢸⡀⠀⠀⠀⠀⠸⣁⠀⠚⢄⠑⣄⢀⠃⠀⠘⠀⢞⡇⠈⠁⢸⠀⣘⣀⡼⠀⠀⠠⠀⠀⠀⡴⠀⠀⢰⠆
⠈⣇⠀⠀⠀⠀⠀⠴⠚⢀⡴⣐⢁⡎⠀⡀⢈⠳⣠⠟⠤⠤⠻⢤⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡾
⠀⠘⣆⠀⠀⠀⠀⠁⠉⠁⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠃
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠀⢰⡀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠇
⠀⠀⠀⠈⠳⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⠀⠀⢀⠀⠀⠀⠀⢠⠏⠀⠀⢺⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠓⢦⣤⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠑⠐⠉⠀⠀⢀⡴⠁⢀⡠⠔⠋⠉⠉⠉⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠶⠶⠶⠦⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣰⡯⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣀⣀⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣠⠤⠶⠒⠚⠋⠉⠉⠓⠒⠶⠦⠤⠴⠶⠖⠚

⠀⠀⠀⠀⢀⣠⣴⣶⣦⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧
⠀⠀⠈⢿⣿⣿⣿⡿⢋⠙⣦⠦
⠀⠀⠀⠀⠙⡿⠣⢡⠀⣫⣶⡀⡀
⠀⠀⠀⠀⠀⠀⠬⠲⣿⣿⣿⣇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢽⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣖⣠⣦⠶⣿⣥⣽⢶⡈⣷⠟⠧⠴⠠⠤⢲⠄⠂⠀⣙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⡟⣿⠿⢧⣤⣶⣋⢳⢸⠁⡝⠉⢧⡀⢀⣠⢤⡔⢧⣀⣒⡡⠎⠙⡖⢤⠰⡤⡀⠀⠀⠐⣶
⠀⠀⠀⠀⠀⠀⠀⣰⠦⠥⡾⢩⠇⢸⡄⠀⠹⣦⣍⣹⡿⢿⣷⣤⣤⣟⡻⠴⠒⡲⠶⣆⣛⣀⣀⣄⡹⣆⠢⣠⠀⠀⠀⠘⠿⡍
⠀⠀⠀⠀⠀⠀⠈⢡⠞⠛⠀⠛⠀⢸⠷⣄⣴⣻⡞⠋⠻⢬⣙⡿⣮⣙⠛⠛⠛⠲⢤⣤⠀⠬⢉⢬⠡⠁⠁⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⡛⢇⠀⠃⣔⢰⢀⡟⠗⠝⣷⣠⢿⣀⣂⣀⣸⣉⢿⠿⣻⣄⢥⣂⣀⠜⠷⠖⢀⣀⠁
⠀⠀⠀⠀⠀⠘⠀⠸⡚⡩⠃⣤⠎⠙⠀⠀⣼⢸⢻⡿⣅⣐⢋⡀⠀⢷⠀⠊⢢⢈⠁⠀⡸⢄⠀⠈⠳⣤⠤
⠀⠀⠀⠀⠀⠈⠀⠤⢁⢆⢟⡎⠀⠀⠢⣍⠀⡱⡇⢳⡄⠙⣝⠲⠶⠶⢤⣀⡀⠀⠀⢀⠁⠬⡄⡀⠀⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⠤⠀⠐⠒⠲⠋⠀⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠾⠤⣤⡄⠀⠀⠀⠀⠀⠇⠀⠁⢀⣀⣀⡀⣠⠤⠤⠤⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⠋⠀⠀⣀⣭⣷⣶⡀⠀⠀⢀⠒⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠽⣧⣄⣀⣀⣀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡖⠁⠙⣷⡄⠀⠀⠰⠁⠤⠈⡿⡁⢀⣀⡬⠶⠯⣤⣀⠀⠀⠀⠀⠀⡐⠁⠈⣶⣮⣏⠉⠛⠒⠒⠞⠿⣾⡄
⠀⠀⠀⠀⣀⣤⣶⣽⣿⣇⠠⢀⣷⣧⠀⠀⠸⡀⢀⣠⣿⣿⣷⣿⣿⣾⣿⣾⣿⣿⣦⣀⠀⢰⠃⢀⣺⣹⣿⡹⡆⠀⠀⠀⡴⠛⢺⣿⡄
⠀⠀⠀⣾⣿⡛⣿⣿⡍⣿⡿⣿⣿⣿⠀⠀⢀⣽⣿⣻⣿⠟⠉⣠⡿⠿⠛⠛⠿⣿⣿⣿⣷⣿⡀⢠⣾⣿⣿⠆⡇⠘⠀⢰⣇⠀⣾⣿⣷
⠀⠀⠈⠉⠠⠞⣿⣿⣧⣾⣿⣿⣿⣿⠀⢠⣌⣿⣿⣿⣿⣾⣿⠋⠀⠀⣰⢮⣕⠪⢻⣿⣿⣿⣧⢸⣿⡙⠃⢠⣧⣴⣦⣾⣿⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣤⣌⣻⣿⣿⣿⣿⣿⠁⠀⠀⣎⣤⡶⣻⣯⡄⣿⣿⣿⣿⣿⣿⢷⣆⣿⣿⡿⠿⠛⢛⣻⣿⣹
⠀⠀⠀⡄⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢜⣟⣽⣾⣵⢸⠀⠙⠋⠉⠉⠙⠛⠿⠿⡿⠤⠒⠊⠉⠀⢿⣿⠟
⠀⠀⠀⢁⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃⠀⠀⢨⣽⣿⣏⣼⢫⠈⡀
⠀⠀⠀⠈⢄⠀⠈⠻⢿⣻⣿⣿⡿⠛⠻⠿⠿⠿⠿⠿⢿⡄⠀⠀⠨⡖⡻⡟⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠑⠢⠅⠠⠤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣳⡧⠥⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠳⡌⢀⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠙⠀⢀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠠⡇⠀⠂⠀⠀⠀⠈⠃⠔⠀⠠⡔⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⢠⠀⠀⠀⠀⠀⠐⡀⠀⢠⠔⠁⠀⢈⠜⠏⠀⢀
⠀⠀⠀⠀⠀⠐⠢⡀⠁⠀⠈⠀⡀⠈⠠⠀⠀⠐⠄⠸⢀⠄⠈⠀⠘⠀⡀⠿⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡁⠈⠃⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⡡⢁⠊⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡁⠀⠀⠀⠈⢀⣂⠔⠁⠀⠓⠁⠀⠀⣀⠀⠀⡐⠢⢤⡤⠄
⠀⠀⠀⠀⠀⠰⠄⣀⣀⣀⠀⠈⠀⠱⡄⠀⠀⠂⠘⠁⠀⠀⠀⠀⢀⣀⠀⠁⢀⠠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠐⠢⠀⡱⣐⠀⠄⠿⢀⠔⢀⡤⠴⠉⠀⠂⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡠⠳⢌⢆⠐⠎⣠⠔⣋⣀⣤⡄⡀⠠⠦⣄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢱⣶⡾⠃⠉⠁⠀⠂⠀⡀⢀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢂⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡗⢺⢣⠏⣠⡀⣀⢠⣠⢏⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠈⠈⠀⢡⠏⠉⠈⠈⠉

⠰⢀⠉⠀⠀⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣉⠒⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠥⢄⣉⠈⠁⠀⢠⡘⠀⠇
⠀⠀⠂⠀⠐⡀⠁⠠⠁⠀⠀⠁⠀⠀⠀⠀⢉⠈⣉⠂⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠲⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡆
⣠⡐⠃⢀⢁⡄⠀⠀⠀⠀⠀⠁⠤⣀⠀⠠⠀⣠⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀
⠋⠁⡉⠡⠀⢠⣄⣐⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠦⠊⠀⢀⠀⠀⠀⢆
⠀⠤⠁⠐⠃⠀⠀⠁⡠⡙⠄⠀⠀⠀⠀⠀⠀⡀⠐⠀⠒⠀⡤⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠤⣄⠀⢠⡆⡆⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢳⠀⡿⣤⡈⠻⢅⡀⢸⡄⢸⠀⣿⢸⡇⣰⠛⢸⡇⢠⡄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢳⡒⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⣧⠀⠈⣿⡄⢸⡇⢸⣇⠾⠼⡟⣼⠁⣿⠀⢸⡧⢸⡇⠀⠐⣄⠙⠦⠤⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠸⡆⢄⡿⠃⠈⠁⠀⠁⠀⠀⠀⠻⠀⠉⠀⠘⠇⢸⡇⠀⠀⠀⠁⠢⠀⠊
⠀⠀⠀⠀⠀⠁⠂⠠⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠄
⣄⠀⠀⠠⠤⠀⠀⠀⢠⠀⠀⠲⢘⠉⠀⠀⠈⠉⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠄⠈⠀⠀⠀⠀⠀⢠⠈
⡍⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠀⠀⢀⣀
⠈⠂⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⠾⠄⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⡀⠁⠀⠀⢢⠋⣹⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠱⠀⢁⡂
⠔⣂⠸⢟⢠⠄⢀⢹⢰⠄⠉⠴⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠂⢀⠈⣣⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠙⢍⣘⠁⠀⠀⢘⣸⡀⠠⠀⢈⣫⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⢶⠈⣁⠌⠀⠀⠀⠌⠀⠀⠠⠀⠀⠀⠀⠀⠐
⠢⣀⠉⠀⠀⠜⢻⣤⡵⢶⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠁⢈⠈⠁⠀⠀⠀⠊⠘⠄⠀⠐⠀⠀⠀⠀⠀⠁⡀
⠀⠈⠪⡰⠄⡀⡊⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⠒⢦⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⢄⠐⠀⡀⠈⣆⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠖⠀⢀⡆⠀⠐⠒⠂⠰⠦⠂⠀⠀⢸⡶⣌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢺⣀⠐⣯⠂⠾⠋⠀⡀⠲⣀⣤⠀⠀⠀⠀⢀⣼⢡⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠈⢄⠉⢰⡞⠀⡾⢛⠛⠁⢀⣤⣤⣾⡿⠃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠰⠉⠀⠀⣷⣾⣶⣶⣿⣿⡿⠟⠁⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠁⠀⠀⠀⠀⡦⠒⡂⢶⢤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠊⠢⡀⠀⠀⠀⠀⠀⠡⠀⢓⢈⠈⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠀⣀⠔⠉⠁⠂⠀⠀⠑⣀⡀⠰⠋⠢⡘⠀⠣⠁⠀⠀⡌⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠖⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠅⠀⠢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠚⡁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⡶⠧⠶⠓⠒⠒⠛⠒⠒⠷⠴⠂⠀⠀⣀
⠀⠀⠀⠀⠀⡠⠃⠀⡇⠀⠀⠀⠜⡄⠀⠀⡲⣤⡀⠀⠺⡄⠀⠈⠀⠀⠀⠀⠁⠒⠻⢧⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⠞⠃
⠀⠀⠀⠀⡰⠁⠀⠀⢡⠀⠀⡄⠀⢱⠀⠀⠙⣈⠻⣶⣄⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⡒⢲⠶⠒⠒⡒⠂⠢
⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⡐⠀⢰⠸⡄⠀⢄⠈⠓⠀⠙⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱⠈⠢⠌⠀⣰⡇⠀⠀⢁
⠀⠀⠀⡇⠀⠀⠀⠀⠘⠴⠁⠀⠞⠀⠈⡲⢈⠏⢀⠀⠀⠀⠈⢃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢢⡀⠈⡁⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⢶⡖⠒⣋⡵⢲⠀⠀⠙⢶⣄⠀⠈⠓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠐
⠀⢀⡜⣀⣀⣀⠤⣤⠒⠉⠀⠞⢁⡔⠁⠀⡎⠀⠸⡄⠀⠈⠻⢦⣀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠠⡄⠀⠀⠈⢠⡐⠂⠀⠀⠀⡀
⠀⠀⠀⠈⡁⠀⠀⠉⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠊⠈⠑⠒
⠀⠀⠀⠀⠑⠲⠄⠤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⡀⠀⠀⠀⠀⠀⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⠀⡘⠀⠰⠀⠀⠀⠀⠆⠀⢀⠀⢀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠁⠀⠀⢣⠀⠀⡜⠀⠀⠈⠉⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⡀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⢀⠄⠀⠃
⠀⠀⠀⢀⡤⢀⡀⠀⠀⠀⠀⢀⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡀⠀⠀⠀⠀⢀⡀⢤⡀
⠀⠀⠀⠀⠆⠀⢉⠙⠲⠀⢤⠞⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠳⡄⠄⠂⠋⡉⠀⡰
⠀⠀⠀⠀⠘⡄⠈⠑⠦⡀⠀⠀⠀⠰⡀⠀⠑⠢⢄⡀⡠⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣄⠀⡠⠔⠋⠀⠀⠂⠀⠀⠀⢀⠴⠊⠁⢠⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠳⢄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡠⠞⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠱⢄⠘⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠁⣠⠞⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠤⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠠⡄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢠⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⡤
⠀⠀⠀⠀⠀⡀⠀⠂⠀⣀⡀⠀⠀⠀⠀⠈⢆⠀⠀⢰⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡆⠀⠀⡰⠃⠀⠀⠀⠀⠀⣀⠠⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠁⠒⠂⠤⠄⣀⣱⡀⠈⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠁⢀⣎⣀⠠⠤⠐⠒⠉⠁⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠇⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠁⢀⡀⠊⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠐⠒⠠⠤⠤⠄⢀⣀⣀⣱⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣏⣀⣀⡀⠠⠤⠤⠄⠒⠂⠀⣴⡁
⠀⠀⠀⠀⣀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⣀
⣤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠉⠑⠒
⠀⠑⠢⣀⠀⠀⠀⠀⢀⣀⣀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⢉⣩⠟⠃⠀⠒⠒⠒⠒⠒⠂⠀⠀⠠⠄⢠⡷⢾⣤⠤⠄⠀⠀⠀⠐⠒⠒⠒⠒⠂⠘⠻⢍⡉⠁⠀⠀⠀⠀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⣰⣆⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠎⠁⡌⠳⣄⠀⠑⢄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡠⠊⢀⠴⠊⠀⠀⠀⡇⠀⠀⠑⠦⡀⠑⢄⡀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠔⣋⠤⠚⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠓⠤⣙⠢⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢡
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣟⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⣿⢧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⠀⣇⢀⡾⡆⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢄⣀⣠⡗⠈⠀⢣⡾⢀⡽⠛⢁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢆⠈⠀⠉⢄⠀⠈⠋⠞⠁⠐⣈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠣⡀⠀⠀⠑⠘⡞⠀⣠⠎⡠⡎⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⢦⡀⣀⠀⠀⣰⢦⡀⠈⢻⣦⡉⠐⠠⡀⠀⠗⠊⠁⠀⢀⢋⡿⠀⠀⣠⡾⡇⠀⠀⡀⠀⣀⢠⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣆⠈⠈⡙⠶⠋⠀⠻⣄⠀⣿⠁⡀⠀⠈⠒⠀⠀⣀⣴⠋⢸⡇⢠⡞⡹⠠⠉⠑⠉⡈⢉⡵⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⢹⣆⢀⡇⠀⠀⠀⡀⠙⠛⢃⠀⠈⠠⡀⠀⠀⣷⠚⠁⠀⠈⠙⠋⢀⡇⠀⠀⠀⠀⣦⡽⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠓⠌⢿⣄⠀⠀⠀⢠⠀⠀⣄⠀⠀⠀⠀⠙⠺⡇⠢⡰⠀⡄⠌⠀⠘⠀⠀⠀⠰⣒⠼⠰⣯⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢯⡐⠦⠤⢬⠶⣄⡀⢸⠀⠀⢹⠀⠀⠑⢄⠀⢸⣿⡘⡐⠂⠐⠀⠀⢃⠀⠷⠛⠶⠀⠐⠋⠷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⠇⠠⢤⣀⢤⠬⠟⠾⣀⠀⢸⠀⠀⡆⠀⠙⠺⣿⢰⠇⠀⠀⡦⠀⣨⠃⠄⠀⠠⠤⠀⠐⢋⢉⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣐⠬⣅⣀⢀⡀⣀⣀⣴⠾⠻⣾⡀⠀⣧⠀⠀⢀⣿⠞⠠⠀⠀⣆⡜⠁⠠⠀⠤⠤⠶⠴⢶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⡭⢁⠀⢀⠐⣀⣨⣷⣤⡏⠀⠀⢸⣿⡆⢀⠆⣰⠏⡀⠀⠀⠀⠀⣠⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠄⢬⠹⠀⠈⢰⣼⢿⡤⡀⢸⣷⡆⢊⡶⣻⣾⣄⣤⡴⠽⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠛⠓⠲⢤⡀⠀⢆⠘⠙⠿⣯⣿⣏⣴⠟⣱⡧⠚⠁⠀⠀⠀⠐⢛⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢢⣤⡤⠴⢲⠟⠛⠉⣩⠞⠛⣿⣿⣿⣿⠿⠿⣤⣄⣠⢚⣁⢀⣔⣉⣈⣭⣛⣳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣿⣖⢚⡟⠋⠀⢀⠐⠁⠀⠀⠔⢁⣠⠞⣻⠿⣴⠁⠉⠓⢦⡄⠀⠉⠙⠯⣉⠉⢉⡭⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⠶⠖⠛⠛⡶⠾⠞⠋⠁⢠⡿⠀⠈⠙⢦⡠⢤⣙⢦⣀⣴⠒⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠉⠛
⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠁⠛⢿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⢻⣿⣿⣿⡿⠛⠛⠉⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⡑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠱⡀⠙⠦⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠼⠀⠀⠀⢱⡀⠀⠀⠹⠶⡦⠀⠀⠀⢀⠀⠀⢀⣴⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠐⢤⡀⠀⠀⢣⠀⡇⠀⠀⣧⠀⠀⢠⠏⠓⠖⠉⡼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠈⠳⣄⠈⢷⡇⠀⠀⠘⢦⡴⠋⠀⡄⠀⣸⠁⠸⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠈⠓⢾⣇⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⡤⣄⡀⢸⡄⠀⠀⠀⠀⠹⡄⠀⠀⢠⠀⠀⠀⣧⠏⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠸⡿⢍⡉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢳⠀⢠⠏⠀⠀⣰⢋⡤⠖⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠓⠤⣀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⢢⡀⠈⣧⠏⠀⠀⣰⠟⠉⠀⠀⢠⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠈⠉⠒⠦⢤⣀⣈⢣⡀⠀⠀⠀⠀⠙⢄⣸⡄⠀⣰⠃⠀⠀⠀⠀⠘⠒⠒⠋⠉⠩⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⢉⡹⠿⠦⢄⣀⡀⠀⠀⠙⣇⣰⠃⠀⠀⢀⣠⣤⣒⣁⣀⠤⠤⠤⠜⣻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢦⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠈⠉⠙⠲⢤⣼⣧⡤⠔⠚⠛⠉⠉⠉⠀⢀⣀⡀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡴⢺⠟⣿⣍⠒⢤⣀⠀⠀⡴⠉⠁⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠙⣆⠀⠀⠀⠀⡠⠋⠀⡞⠀⠟⣇⠱⡄⢈⡑⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣠⠞⠁⠀⢸⠃⢸⠀⢿⡀⠘⠏⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⠀⠀⣠⠞⠁⡴⢦⣄⠘⣠⠞⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡠⠊⠀⢀⠞⠀⠀⠸⣤⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠤⠖⠒⠒⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣷⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢻⣿⣿⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣴⣿⣿⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣟⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⡿⢡⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣷⣿⣿⣿⢉⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣯⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⡀⣐⣶⡾⠿⠿⠿⠿⠿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣀⣤⣤⣽⣷⣍⢿⣯⣥⣤⣤⣤⣤⣾⣭⢭⣥⣶⣶⠶⢶⣶⡿⢉⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣈⣤⣞⢿⣿⣟⣙⣿⣿⣆⢻⣿⣿⣯⣯⣽⣫⣵⣯⣥⣾⣧⣴⣾⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⣿⡿⣿⡟⠛⠛⠛⠛⠛⠛⠻⣦⢹⣟⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⢇⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⡿⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣼⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢀⡿⢡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⡄⠀⠀⠀⠀⠈⣿⣾⣿⣿⣷⠘⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⣼⢃⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢷⡀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣧⠹⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡘⣧⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢹⡆⠀⠀⠀⣹⣿⣿⣿⡇⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣿⡀⠀⠀⣿⣿⣿⡏⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢹⡇⠀⠰⣿⣿⣿⡃⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⠀⠀⢻⣿⣿⣧⢹⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⡄⠀⣼⣿⣿⣿⡜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣏⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠸⣇⢸⡇⠀⢻⣿⣿⣿⣿⣿⣿⡟⣿
⠀⣀⣀⣤⣴⣶⠶⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠶⠾⣿⡈⣿⣤⣼⣿⣿⣿⣿⣿⣿⣦⣿
⣾⣟⠉⠁⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⣽⣧⢸⣷⡄⠉⠉⠛⢻⣿⣏⢿⣿
⠉⢻⣿⠿⣷⣶⣶⠶⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⠉⠆⢿⣷⣿⡿⣿⣿⣿⣿⣾⣿
⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣈⣙⢻⣿⣶⣴⣿⣭⠄⣏⣿⣿⣿⣿⡿⣿
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⣭⣽⣻⣿⣿⣿⣿⣥⣴⢄⣻⣿⣿⣿⣿⣗⣿
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣤⣽⣯⣭⣿⣿⣿⣿⣯⠅⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⣾⣯⣿⣿⣿⡿⠶⠶⠂⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⣶⣿⣿⣿⣋⣉⡁⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢉⣿⣿⣿⣿⠿⠏⠉⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣽⣿⣿⢿⡶⠦⠀⣸⣿⣿⢿⣿⣿⣿⣿⠉⣿
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⣶⢾⣿⣿⣿⣿⣶⠶⢀⣿⣿⣿⣾⣿⣿⣿⡏⢸⣿
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⡿⣿⣿⣷⣏⣿
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣧⣿⡿⢡⣿⠿⣿⣿⣿⣿⣿⣿⡿⣿⣾
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠻⣶⣿⣿⣿⣟⣛⡀⣼⡿⠀⣿⣿⣏⠹⣯⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣛⣻⣻⣿⣿⣿⣿⠯⢠⣿⠃⠀⢻⣿⠙⣷⣿⣿⣷⢻⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⢴⣶⣿⣿⣿⣾⣛⠁⣾⡏⠀⢀⣼⣿⣷⡘⣿⣿⣿⡞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣶⣿⣿⣿⠛⠁⣰⣿⠁⠀⢸⣿⣿⢏⣿⣿⣿⣯⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣻⣿⣿⣶⣶⣟⢀⣿⡏⠀⠀⠀⢻⣿⢾⣿⣿⣿⣻⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣭⣭⣽⣿⣯⣭⣤⣶⣼⣿⠁⠀⠀⠀⣿⣿⣾⣿⣿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣛⣉⣉⣉⣉⣉⣭⣭⣭⣭⣍⣉⣩⣭⣭⣭⣥⣥⣼⣿⠇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿

⠀⠀⣠⣤⣼⣿⣿⣿⠶⠀⠀⠀⠐⠁⠐⡄⠀⠀⠀⡠⠤⠶⢶⣦⣀⡀⠀⠀⠀⠶⡄⠀⠀⠀⠀⠠⠞⠀⠠⠞⠁⠀⠴⠋⠀⠄
⠀⠀⠀⠉⠉⠋⡁⠰⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠄⠀⢾⣼⣿⡇⠀⠀⠀⠀⣿⠀⠀⢰⣦⣤⣴⣦⣀⣰⣤⣀⣠⣤⣄⣒⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⠃⠀⠀⠰⠀⡿⠀⠀⠘⠛⠈⠡⠜⠆⠛⠰⠼⠧⠴⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⠟⠛⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠇⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⠀⠀⢀⡀⠀⣠⡆⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⠀⠀⠀⠈⠁⠀⠀⢀⣴⡿⠛⠛⠻⠛⠢⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣽⣶⣿⣿⠻⠿⠇⠀⠉⠉⠀⠁
⠀⠀⢀⠐⠄⠀⠀⠀⠀⠂⢈⠄⠀⠀⠀⢀⣽⠏⠁⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⡄⠀⡀⠀⠘
⠀⠀⣄⠀⠁⠒⠠⠁⠀⠖⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠷⠀⠀⠀⠀⠇⠀⠀⢰⠘⡆
⠀⠀⠓⠺⢿⣈⡢⠞⠁⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⠀⠀⠔⠀⠀⠀⠀⠂⠀⠀⣠⠁⢀⠘⠞⢣⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⡷⠈⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⡅⠀⢠⡇⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⣻⣿⣿⣟⣩⠁⠀⠀⠈⢂⠀⠀⠸⣿⣏⠰⡇⠀⠀⠀⠀⠀⢁⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⡀⠀⠓⠂⠀⠀⠀⢀⡆⠀⠀⠀⠀⠿⠉⠉⠉⢙⠉⠹⠀⠀⠠⠀⠀⢻⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠇⡁⣹
⠀⠀⠀⠀⠈⢡⡆⢈⠜⢧⠠⠤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣄⣠⣄⠀⢀⠀⠀⣦⠀⠀⢆⠃⣿
⠀⠀⠀⠀⠀⠉⡤⠋⠀⠈⡆⠒⡀⠀⠀⠀⠀⢀⣠⠄⡐⢻⠷⠢⠀⠀⠀⠀⠀⠀⣸⠀⠀⡟⠛⠋⠉⠻⠀⠈⠙⠂⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢰⣮⣿⣷⣽⣭⣦⣴⣿⡤⠀⠆⠀⣠⠋⠀⠀⠷⠂⠀⠀⠀⠀⢀⡀⠀⠀⣀⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣐⠶⢿⣯⣝⡳⣆⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠰⠃⠀⠚⠀⠀⠘⠆⠀⠀⠀⠄⠀⣹⠀⢀⡳⢶⣶⡼⢷
⠀⠀⠀⠀⣀⠀⠀⠛⢷⣦⠋⠟⠁⠛⠁⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⠀⣴⠏⠀⠀⠉⠀⠋⠑⠚
⠀⠀⡆⠀⢹⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⢀⣤⢴⡶⠀⠀⠀⠀⠜⠀⣼⠇⠀⠀⠀⢀⡀
⠀⠀⡇⠀⠈⡱⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠜⠋⠀⠀⠀⠀⠀⠤⣴⠏⠀⠀⠀⠠⢸⢃⠐
⠀⢀⡇⠀⠀⡷⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠄⠀⠀⠀⢀⡴⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⡀⡀⠀⠀⠀⡘⠘⠀⣇⡁⡐
⠀⠸⡇⠀⢀⡘⠂⡁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠁⠁⠈⢹⡷⠁
⠀⠀⡇⠀⢠⠟⠀⠀⠀⠀⣄⡀⣔⡾⢤⠗⣤⢠⣆⠤⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣶⡄⠄⠆⢂⢐⡐⡀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⣾⡾⠂⠀⠀⢃⣀⠐⣛⣂⢛⡚⣘⣋⢳⠞⠋⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸⡟⠒⠐⠂⠂⠃⠐⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠐⠒⢶⣦⣤⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣤⣬⣹⣿⣿⣿⠖⣦⣾⣿⡿⣿⣿⡟⢺⢾⢶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠠⣤⣭⣽⣿⡿⢋⣵⣾⣿⣿⣿⢳⣿⡿⡇⢹⣙⡢⠋⠁⡼⠛⠁
⠀⠀⠀⠀⠠⠤⠶⣾⡿⢋⣴⣿⣿⣿⣿⣿⠃⠙⡛⠀⠀⠠⡀⠀⠀⠈⠉⠙⠇
⠀⠀⠀⠀⠤⠲⣿⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⡀⠀⠀⠨⡀⠀⢁⠤⠃
⠀⠀⠀⠠⠴⣾⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⢡⡀⠀⠒⠈⠳⡤⠝⠁
⠀⠀⠀⢀⣴⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠋
⠀⠀⠀⢠⣼⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⢀⡠⠄
⠀⠀⠀⢀⣼⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣴⣠⣶⣿⠋
⠀⠀⠀⠁⣽⢸⣿⣿⣿⣿⣿⣿⣟⡻⣿⣗⣣⣴⠟⠛⠉
⠀⠀⠀⠀⢠⡎⣿⣿⣿⣿⣿⣿⠂⠁⠈⠉⠉⠁⠀⠀⠀⢀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠐⠀⠱⠹⣿⣿⣿⣿⣯⢪⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠻⠯⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠸⣄⢣⢿⣿⣿⣿⣿⣯⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢯⠛⢯⣧⡀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠹⣆⣬⡿⣿⣿⣿⣿⣷⣄⢀⣠⢶⠿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⣽⡇⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠁⣿⡑⣮⢿⣿⣿⣿⣿⣀⠉⠋⠉⠓⠦⠆⠀⠀⠀⠀⠀⠀⠐⣙⣧⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠜⡓⢮⡟⢿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠀⣤⣌⣉⡻⠿⠿⣿⣶⣶⣶⣶⣾⣷⣶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠙⠙⠛⠛⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⣦⣄⡀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣤⣶⣶⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⢿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⡘⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣿⣿⡿⢻⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⣾⣿⡿⢿⣿⠿⢿⣿⣿⣿⣿⣿⡏⠀⠉⢹⣿⠇
⠀⠀⠀⠀⢠⣿⠀⠀⢀⣿⣿⣿⠇⠀⡄⠀⠀⠀⠀⢸⣿⠋⠁⠀⠈⠀⠸⣿⣿⣿⡇⢻⡏⠀⠀⠀⠻
⠀⠀⠀⠀⠘⠃⠀⠀⢸⣿⣏⣤⣤⣴⣧⠀⠀⠀⠀⣾⣿⠀⠀⢀⣠⣼⣀⡘⢿⣿⣿⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠻⣮⠀⣿⡟⣿⣿⡷⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢘⡛⠿⡿⠀⠀⠀⠀⠈⠀⠈⠀⡙⠿⠿⢟⠃⢀⡞⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣤⣤⣭⣽⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢊⡞⠀⢠⡇⢀⣼⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣟⡀⣠⡟⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣼⣿⠗⣿⠃⠛⠉⢻⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⠐⠻⠛⠋⠈⠉⣥⣾⣿⣿⣷⣤⣴⣿⡿⠻⠎⠁⠀⠀⠀⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠿⣿⣿⣿⣅⠀⠀⠀⠀⠀⢠⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠿⠀⠀⠀⠀⠰⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠃⢀⣀⡀⡀⠀⠀⠀⠀⠈⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠉⠑⠦⡀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀⣀⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢰⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⠁⠲⣄⠀⠀⠈⣄⠀⠀⠈⠁⠸⠄⠀⠀⠀⠀⠀⡈⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⣤⠀⠀⣠⣤⣤⣤⡉⠢⢀⠘⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⣼⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠍⢠⡄⠀⢈⣤⠀⠀⠉⠂⠀⢷⡄⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠁⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃⠀⠀⠉⠂⠀⠀⠀⠀⢸⣧⠀⠀⠀⠸⠐⠢⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠁⠈⠄⠁⠤⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢄⠀⠀⠀⢀⣠⣤⣴⣆⣀⡀⠀⠀⠀⠀⢸⣤⠀⣈⠆⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠂⠠⠄⠀⡀⠂⠀⠈⠙⠻⠯⣭⠍⠿⠛⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢀⠴⠋⠀⢰⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠦⠤⠶⠒⠁⠀⠀⡰⠋⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⢲⣿⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠓⠦⠤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠒⠁⢠⠋⠀⠈⠸⣖⣤⣤⣤⠠⠄⡐⡅⠀⢀⣠⠂⣠⡊⠀⠀⠈⡉⣲⠶⢄⠀⢰⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠞⡵⠃⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⢉⠀⠐⡎⠃⢰⡟⠀⣦⠀⡾⢰⠃⠀⠀⠑⢬⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠔⡀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠀⢰⠀⣤⣼⠁⠀⠁⢸⠃⢀⡔⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠁⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⣿⡏⡜⠀⡠⡿⢸⣿⡧⠊⠀⣀⣈⢢⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⡟⢹⢹⡇⣼⢡⠇⡾⡏⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠶⠀⢸⠚⠀⠀⢸⣿⠃⠸⠈⢱⡇⠀⠁⡀⠀⡂⠀⠰⡀⠀⠀⡰⠀⠦
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠄⢰⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡏⠀⠆⠀⠀⢧⡀⠋⣀⣾⡗⠄⠀⢈⠉⠉⠀⢀⢰⠂
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠙⡀⣿⢣⠐⣄⠀⠻⣆⠀⠀⠈⠈
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡏⠀⡀⠀⠀⠀⠀⠀⠇⠹⡈⡄⠈⢗⠀⠙⠂⠀⠀⡐
⠀⠠⠇⠀⠀⠀⠀⠀⠀⢀⢮⠀⢈⡇⠘⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡇⠀⠀⠈⠀⢰⠁⠀⠀⠀⠀⡞⠀⠀⠀⠛⠒⠠⠀⠀⠀⠐⠈
⠀⡧⠘⠀⠀⠀⠀⢀⡔⠁⠰⠰⠊⡅⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏⠀⠀⠀⠀⣽
⠀⠸⡀⠃⠀⠀⡠⠋⠀⠠⠃⡦⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠃
⠀⠀⠑⣦⠤⠐⠁⢀⡴⠕⠊⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠈⠒⠶⠒⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⣀⠀⠀⣀⡀⠀⠀⢀⣀⢀⣀⡀
⠀⢠⣿⣿⣿⠆⢸⣿⡆⠈⠋⣀⡀⢀⣸⡇⠀⠛⠛⠛⢛⣻⣿⣧⡀⢀⠀⢶⣴⣿⣿⡿⠿⠃⠀⠛⠁⠸⠿⠟⠀⣼⣝⣿⣿⡟⣷⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠐⠄
⠀⠘⠿⠿⠋⠀⠈⠉⠁⠀⠈⠁⠽⠏⠋⡡⠀⠀⠀⠀⢈⢿⣿⠍⠀⡸⠁⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡀⠀⣤⠘⠿⣿⣼⡽⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠴⠒⠖⣋⣻⣾⣗⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠉⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣧⣀⣿⣄⠀⠀⠀⠀⠀⠀⣤⣶⣤⣶⣶⣶⣾⣶⣄⡀
⠀⢀⡆⠀⢠⣟⢙⠿⣤⣿⡆⠻⢀⠀⠰⢶⣦⣿⣄⢀⣾⣿⣿⣿⣿⣶⡄⢀⠀⢠⡔⠊⠒⠄⠀⠀⡀⠰⢽⡼⣿⣹⣦⠀⠀⠀⠀⣦⣋⣻⣿⠏⠛⠚⣉⣉⣩⣧
⠀⢸⣿⡀⠀⣋⠀⠀⠀⠈⠃⢉⣹⠀⠀⠛⢶⣯⠭⣭⣭⣭⣭⣽⡷⠋⠀⠸⡄⠸⣇⣀⣀⡞⠀⢀⠃⠀⠞⠛⠛⠋⢡⣤⡟⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⢧⣙⣦⣌⣀⣀⣀⣠⣴⣿⡟⠀⠻⠇⠀⠉⠉⠙⠛⠛⠉⣩⢤⡤⠄⢠⠁⡀⠈⠉⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇⢠⣼⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠙⠻⠭⠉⢀⣩⠥⠴⠋⠠⣶⠤⠄⠀⠀⠀⠀⠒⠒⠚⠷⣶⡶⠗⠘⠦⠤⣄⣤⡴⠴⠶⠖⠀⠀⠀⠦⠤⠤⠼⠿⠀⠀⠐⠿⠻⠿⣿⠟⠛⠛⠋⠉⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣤⢠⡄⢀⡄⢠⡀⢀⡄⢀⡄⢠⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠁⠈⠉⠈⠉⠈⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠨⠬⠬⠼⠼⠭⠧⠥⠥⠥⠽⠇

⠀⠀⣠⣤⣶⣶⣶⣶⣤⣤⡀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⣶⣦⡀⠀⢀⣀⣀⠀⠀⡠⠢⣀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⣤⢀⣤⣄
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⡇⣠⠒⠒⠤⣀⠀⣀⠀⠰⠀⠙⠻⠟⠠⣿⣿⡿⠀⢀⠈⠠⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣍⣐⣶⣶⣷⣶⣶⡇⠀⢤⣙⣺⣻⢿⡀
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⣠⠤⢄⠀⠀⢡⠄⠀⠻⣃⣤⣤⣤⡤⣄⡀⠛⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣹⣿⣿⣿⠀⠀⠈⠈⠉⠒⢻
⠀⠀⠸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢧⣀⡀⠁⠀⢀⠀⣠⣾⣁⣿⣿⣿⣟⠉⠹⣔⡀⠸⢿⡿⣿⣿⣿⣿⣿⣿⡏⠀⠐⠐⠚⠛⠟⠟⠛⠀⢠⣤⠀⠀⠀⠁
⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠛⡠⠄⣸⢿⣃⠀⠑⢝⠿⢿⣽⣿⡿⠋⠀⠀⠀⠉⠗⠀⣠⡶⠶⠂⠀⢀⣀⠀⠸⠟
⠀⠀⠀⢑⠀⠀⠀⠀⠀⡀⢀⣄⡀⠀⠀⢀⡀⠀⠀⠄⠀⠈⣿⣿⣿⣷⣶⣶⡴⠟⣱⣾⡟⠀⠀⢀⣀⠀⢀⣠⠴⠒⠀⠀⠀⢀⣔⣛⡋⠉⣳⣦⡑⣬⣷⣄⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣟⣷⠀⢿⣿⠗⠀⠰⣶⡀⠘⣯⣿⡯⢭⣤⡶⣾⡽⠟⠁⣴⣾⠿⠿⡟⢉⠀⠀⢴⡶⠀⠐⢉⢳⣄⣿⣦⡸⣿⣿⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠈⠛⠒⠀⠀⠈⠀⠀⠙⠛⠁⠐⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉⠁⠓⠘⠛⠉⠁⠀⠀⠛⠓⠒⠉⠀⠀⠀⠀⠀⠈⠀⠈⠘⠛⠛⠻⠟⠛⠛⠛⠛⠋⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣤⣝⣥⣮⣭⣿⣥⣿⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠃⠀⠀⠀⠀⠄⡀⠀⠀⢇⢎⣾⠎⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣾⣷⠘⠷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣾⡄⠀⠀⠀⠀⠀⠀⢠⠈⠀⠈⠘⣼⠠⠈⠀⠀⠀⠀⠀⣦⣀⠀⠀⣀⡀⠀⠸⡀
⣷⢤⠹⣿⣵⡠⠹⡄⣐⡶⠂⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣦⣁⠀⠀⠀⠀⣀⣄⡘⡏⠉⠁⠈⡴⠃⠀⠘⣄⢀⣾
⣿⡐⢄⠹⣿⣧⢶⡹⣉⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⡏⠉⣹⠾⠛⠉⠀⠀⠀⠀⠈⣎⣿
⣿⣿⣌⢢⣘⢿⣷⢄⣿⣏⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⣬⠾⢔⢠⢮⠀⠀⢰⠀⣀⠟⠀⠀⠀⣄⡀⠀⣄⣀⠀⢻
⣿⣿⣿⣷⣻⣿⣟⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⠁⠀⢀⣀⠘⠁⡀⠀⡉⡃⢀⡼
⣿⣿⣿⣿⣿⣿⣿⣬⢿⢉⡿⠁⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠄⠀⠀⠉⠀⠀⠀⠀⠀⢀⣤⠠⢿⡉⡹⠿⠀⠈⠁
⢸⣟⣿⣿⣿⣟⣿⣿⣟⡘⣝⠆⠳⣿⠁⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠂⠀⡀⠀⠀⠀⡀
⠀⠙⢿⣿⣯⣿⢷⠊⢿⡟⣟⠀⠀⠀⢀⡜⠀⠀⠀⠀⢹⠏⡹⠟⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠄⢀⡀
⠀⠀⡠⠟⢿⣷⢠⣥⣈⡃⣿⡄⢀⡴⠋⠀⠀⠀⠀⢠⣞⣶⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢀⣤⠀⠙⢷⣍⢟⢾⣿⣧⢎⣠⠤⠀⠀⠀⠀⡼⠟⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠈⠀⠀⠀⠹⣿⡟⢿⣿⣿⠟⠁⠀⠀⠀⠀⠃⠁⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⣤⡾⠀⣤⣄⠀⠙⣾⣿⠁⢀⠀⠀⠀⠀⠀⡔⠀⢸⣷⡄⠀⠀⠒⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢸⣦⠄
⠀⠀⠀⢿⡗⢄⡍⠀⠀⠄⠈⠀⠀⢨⣰⣶⣣⢺⠟⠀⠀⡜⠋⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⡏⢀⣄⠀⠄
⠀⢀⡀⠸⣿⠈⠓⢦⠀⢠⣶⣦⣾⣿⣿⡹⠏⢋⢹⡷⠴⠛⣀⡀⢤⡒⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠟⢿⣾⢖⣏⠈⡍
⢀⠛⠃⠀⠀⠀⢀⣠⣤⣼⣿⣿⡏⣿⢉⣧⢸⣿⣭⡹⢤⣞⠷⣌⠃⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠐⣦⠈⠉⠀⣑⣠⣧
⠺⡟⠂⠀⠀⡀⠼⣿⡿⣿⣿⡷⠰⣿⡾⢽⣿⠿⢿⣷⣾⣿⡶⠟⠁⠀⢀⠀⠃⠀⠀⡀⢀⡀⠀⡀⠀⠀⢨⠦⠸⣴⠛⠻⠸⠇
⠀⠀⢀⡄⠀⠀⠛⠉⠙⠛⣾⣷⣎⣽⠷⠾⣿⣯⡸⢿⣿⡯⣽⡆⠰⡄⠀⠻⠛⠀⠀⠛⠈⠀⠀⠁⠁⠀⠀⠀⠀⢸⠀⠀⢀⡄
⠀⠀⢀⠀⠀⣀⣀⠀⠀⠐⠚⢿⡈⠻⣾⣿⢿⡿⢋⠳⣅⠙⠿⠒⠈⠉⠁⠀⠀⢀⡄⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠻⢺⠁⠿⠇
⠀⠀⠀⠀⠄⠀⡉⠙⠛⢖⠀⢀⣵⣻⣿⢿⣿⣿⣿⣿⣿⡷⠂⠀⠀⣀⣤⣤⠐⠛⢷⠟⠁⠀⠀⠐⢦⣤⣴⣦⣀⠐⡏⠁⣀⠄
⠴⠃⠀⢠⠄⠀⠀⢀⣼⠟⠀⡡⠚⠁⡠⢻⠏⠉⠉⠉⠁⠀⠀⠀⡼⣿⣿⣿⣇⠀⠀⣶⡦⠄⠀⠀⠀⠈⠻⣿⣿⣿⣇⠀⡇
⠀⡀⢀⠈⠀⠀⣰⣟⡅⠀⠀⠀⠀⠈⠀⠀⢠⠇⠀⠄⠀⠀⠀⠀⠴⣿⠟⡽⠇⢠⡔⠿⠶⠶⠶⢦⢄⡀⠀⠈⢻⡿⠿⢻
⠀⠀⠀⠀⠀⢰⠏⣾⠇⠀⠀⠀⠀⠈⠒⢲⠊⠀⡤⠂⠀⠀⠀⠀⠀⠀⣠⠃⠀⢨⣑⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⠀⢻⡀
⠀⠀⠀⠀⣤⠄⠀⠉⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠀⠐⠀⣰⣧⢀⣶⢦⢹⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠐⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠄⠀⠀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠁⠀⠡⡀⢀⠌⠀⠈⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢄⠀⠀⠀⠘⠃⠀⠀⠀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠒⠒⠠⢀⡀⠀⡀⠑⢄⠀⠀⠀⠀⢸⠀⠀⢢⠀⠀⠀⠀⠀⠀⡜⠀⠀⠃⠀⠀⠀⠀⡠⠊⢀⠀⢀⡀⠄⠒⡂
⠀⠀⠀⠀⠀⠀⠈⠄⠐⠠⡀⠀⠀⠀⡀⠑⠢⢄⡀⡸⠀⠀⠀⠡⡀⠀⠀⢀⠌⠀⠀⠀⢆⢀⡠⠔⠈⢀⠀⠀⠉⢀⠄⠂⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠐⢄⠀⢡⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠊⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠘⠃⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡥⠊⠀⠀⠀⠠⡁
⠀⠀⠀⠀⠀⠀⠣⠀⠠⠀⠀⠀⠀⠀⠈⠢⡀⠀⢃⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡘⠀⢀⠔⠀⠀⠀⠀⠀⠀⠄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠉⠐⠂⠤⠀⣐⣄⠸⠀⠀⠀⠈⠆⠀⠀⠀⠀⠰⠁⠀⠀⠀⠃⣠⣂⠀⠤⠐⠂⠉⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠈⢲⠀⠀⠀⠀⠈⢆⠀⠀⡰⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠱⠎⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠀⠉⠀⠀⠀⠉⠉⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠉⠁⠀⠀⠀⠙⠀⠀⠄⣀
⠀⠀⠀⢠⠐⠂⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠉⠐⠂⡄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠠⠤⠌⢦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠔⠠⠤⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠠⠔⠈⠀⠀⠀⠀⠀⠉⠉⠉⢉⠝⢛⡛⠫⡉⠉⠉⠉⠀⠀⠀⠀⠀⠁⠢⠄⡀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⠋⡗⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠔⡁⠔⠉⠀⠀⡇⠀⠑⠢⣈⠢⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠊⠔⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⠢⠵⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠳⢤⡼⠀⡀⠹⣤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢟⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠅⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⠓⢦⣄⣀⢀⣴⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⢸⡇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠠⠈⠉⠉⠀⢆⠛⠏⠉⠀⠀⠀⠈⠄⠀⠈⠛⠋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠈⠢⠀⠀⠈⢧⡀⠈⠢⡀⠀⠀⠈⡄⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠭⢥⣀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡷⠀⠀⠀⠐⠀⠀⢸⠀⠀⢠⢆⠀⢸⠷⢤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠊⠀⠉⠁⢀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⣧⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡦⠀⠀⠀⠀⠀⠀⠀⡠⠂⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠁⠐⠒⠚⠋⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠃⠀⠀⠀⣀⠤⠖⠉⠀⠀⠓⣶⡆
⠀⠀⠀⠀⠀⢦⡴⠶⠖⠚⠁⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⣀⠂⢄⢠⢀⠄⠂⠉⠀⠠⠄⠀⠀⡠⠞⠉
⣀⣀⣤⣤⠴⠞⠁⠀⠀⢀⠠⠤⠤⠶⠂⠀⢀⣠⣤⣤⡀⠀⠀⠀⠈⠁⠸⠛⢳⡶⣶⣦⡤⣤⣤⣬⣵⡦⠄
⠈⠉⠉⠉⠉⠁⠲⠀⠀⠀⠀⠀⠘⠁⢠⠞⠉⠀⠀⢈⣿⠀⠀⠀⠴⠀⠀⠀⠀⢳⠈⠙⢯⡲⣄⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠤⠒⠛⢷⢀⠖⠁⠀⠀⢀⣠⠾⠋⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠱⡄⠻⣆
⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠘⠋⠀⠀⠀⠐⠿⣧⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠹⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠀⠀⢠⠀⣠⠔⠁⠀⠀⠀⠀⠀⢧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣀⣠⡀⠈⠀⢀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⣧⠀⣰⠋⠳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⣤⣶⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣻⣾⠿⠏⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣛⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠋⠁⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡛⠿⣦⣌⣉⣉⣉⡉⣻⠉⠉⠉⠉⠉⠁⠀⠀⠀⣉⣈⣀⣀⣠⡤⠤⠖⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣀⡍⠉⠉⠉⠉⠉⠀⠀⠉⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦⣀⡀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣿⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢷⣶⠿⠶⢷⢶⣶⠶⠾⣿⠿⠋⠉⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣅⣀⠀⠈⠉⠀⠀⠁⠠⠴⠀⠈⠀⠰⠒⠀⠀⠳⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣾⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⢻⡛⠭⠽⠋⠿⠿⣷⣦⣀⢀⡀⠀⣀⠀⢀⣀⣠⣤⡶⠖⠛⠛⠛⠉⠉⠁⡀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡶⣦⠄⣀⡀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣤⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣴⣶⣾⡿⠛⠛⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⡿⢿⡟⠅⠁⠀⠀⠀⠀⠀⠀⣠⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣭⣟⠉⠉⠙⠙⠀⠀⠀⠀⠀⠛⠋⠉⠉⠀⠀⠀⣁⣤⣄⣤⣤⠶⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⡳⠦⢙⠻⢿⣿⣿⣿⣿⣾⣾⣶⣶⣶⣶⣦⣶⣶⣶⣾⣷⣾⣿⣿⠿⠛⠋⠡⠀⠐⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣈⡙⠛⠲⠶⠍⢉⡛⠛⠛⠛⠉⠉⠋⠙⠛⠛⠛⠛⠛⠋⠀⠐⠊⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⠛⠛⠉⠉⠉

⠀⠈⣄⠀⠀⠀⢤⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠯⢬⣟⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠨⠀⠀⠀⠙⠣⡀⠀⣀⣤⣀⠈⣿⡄⠀⠀⠀⠀⢠⣠⡀⠀⠀⠀⠀⠀⣰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢣⡀⣼⠀⠀⠀⠀⠈⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣤⠹
⡹⠥⠀⠀⠀⠀⠀⣆⣺⠀⠉⡟⠻⣿⣦⡀⣁⠀⠀⠀⠀⠀⠀⢠⡴⢇⠠⠀⠀⠀⠀⠁⠂⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⣦⣾
⢣⠟⠂⢺⣶⡂⠀⠀⠀⠀⠀⠉⣐⠜⣿⣿⡏⠠⠀⠀⠀⠀⠀⠈⠑⠿⠂⠀⠀⢀⣤⣤⡀⠀⠀⠀⢠⡀⠀⠾⠛⠁⠀⣇⠀⠀⠐⠀⢸⠀⣠⡚⣏
⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣇⠜⣠⠃⠀⣷⠀⠀⠀⠀⠀⠤⠀⠀⠿⠿⠇⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠀⠉⠒⡀
⠀⣀⣤⠀⠀⠀⠈⠀⠀⢴⡄⠀⠀⠀⢨⠷⠋⠂⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⡐⠁⠀⠀⠂⠀⠀⠀⠀⠀⠰⡀⠀⠢⣿
⣿⡿⠋⠀⡶⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⠀⠀⠄⠀⣤⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⠀⠀⠤⡁⠀⡄⠀⠀⡀⠀⢥⣦⠀⠀⠸⢦⣀⠸
⠛⠷⠄⠀⠀⠀⠀⣱⠂⠀⢠⡀⠀⠀⠀⠀⠈⠇⠠⠂⠀⠀⠼⡟⢻⡅⠀⠀⠀⠀⠁⡀⢰⣦⣜⢿⠀⠀⠈⠆⠀⠔⠉⠀⠀⠀⠉⠀⠀⠀⠀⢹⢦
⠃⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠻⠆⢰⠀⠀⡴⠁⠀⠀⠀⠀⠀⢗⠿⠀⠀⣤⠔⠀⠀⢹⣾⣯⡿⣿⣷⡀⠀⠀⠌⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠟
⠐⠂⠁⠈⠉⠐⠀⠀⠀⠀⠶⠀⠈⠁⢠⡞⠀⠀⠀⡀⢤⣄⡀⠀⠀⠀⠀⢃⠀⠀⠙⠛⢯⣟⣁⣮⣿⠃⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠸
⢄⣀⣀⣀⠀⣉⣶⡆⠀⠀⠐⠆⠆⠀⠊⠀⠀⠀⠈⠀⣃⢌⠽⣟⣦⣀⠀⠈⢀⡀⠃⣠⠀⠈⠛⠋⠁⠀⠀⠀⠀⢸⡏
⠒⠬⢻⣟⡿⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⢀⠋⡟⠰⢾⡿⠳⣦⠀⠀⠀⠀⠀⠯⠀⠀⠀⢤⠀⠀⠀⠀⠀⠘⢷⠀⠀⠈⢀⠀⠊
⠀⠀⣸⠁⠁⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡆⠀⠎⣟⣉⡸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠸⠀⠀⡷⠊⠀⢀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⢀⠀⠐⠹⠉⢰⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⡄⠀⣤⠄⠀⠀⠀⠀⠀⠀⠁
⠈⠀⠀⠀⠈⠀⣀⣄⠀⠀⢧⠀⡠⣾⡿⠉⣈⠧⠀⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠟⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠡
⠀⠂⠀⠀⢠⡄⢻⡧⢀⠀⠈⠀⢿⣿⣿⡷⠟⠀⠀⢀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⡰⠇⠀⣯⠀⠀⠀⠀⣐⢒⠗⣠⣿⡷⠦⠀⠈⠁
⠀⠀⠀⠀⠀⠉⣀⠂⠀⠂⠀⠀⠈⠋⠀⠀⣠⠀⠀⠀⠉⠀⠰⠆⠀⠈⠋⠁⠀⠀⠈⠳⠀⠀⠐⠛⠉⢀⠀⠀⠀⠀⣴⡌⣠⢖⢮⠟⢣⠆⡄
⢀⠀⠀⢀⣀⠘⠁⠀⠐⠀⠀⠀⡤⠀⠀⠐⠂⠐⡀⠀⠀⠀⠀⠀⠀⢐⣾⠃⣧⡄⠐⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⢡⣹⠦⡾⢣⡞⢿⠃
⠀⡤⠀⠀⠀⠀⣀⣄⣴⠀⠀⠡⠁⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠈⠛⢊⠉⠀⠀⠈⠀⣀⡀⠀⣀⠀⠀⠀⢰⡆⠀⠹⠮⠳⣛⡻⠛⠋
⢀⣏⣀⠈⣦⡀⣿⣿⣏⠀⠀⠀⠀⠀⠰⠏⠠⠀⠀⠄⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠰⠌⢰⣛⣛⡄⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⡠⢀⡀
⢺⣇⠀⠀⢸⠏⠉⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠐⠏⠽⠀⢀⠀⠸⣿⣇⠀⣠⡄⣤⢤⣀⣀⠀⠀⠉
⠀⡍⠡⡀⢸⠴⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠘⠋⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢷⡄⢻⣛⣶⣿⣿⣿⡟⠛⠋
⠀⡃⠀⠈⠢⡄⠀⠀⠀⡀⠀⠀⠘⢳⡆⠙⢠⡄⢵⡄⠀⠀⠀⠀⠢⡀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠇⠀⠀⠀⢹⣶⡇⠀⠀⠀⠀⠀⠈⣠⣤⣟⣼⠾⢄⠀⠀⠀⠀⠀⢸⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡤⠀⠀⠀⢀⡾⠉
⢠⡾⣋⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⣾⡻⠧⠟⠱⡦⠠⠀⠀⠀⠀⠀⠈⠁⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠟⠈⠀⠀⠀⠀⠰⡄⠀⠀⠓⠂⡤⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣠⡄⠀⠀⠀⠀⠒⠀⠀⠀⠀⣀⣠⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠠⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⡀⢀⠈⣀⠄⠀⠀⠡⡈⢤⠀⠀⠘⠋⠀⢀⣠⡄⠀⠀⣦⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠀⠒⠈⠱⠊⣄⣁⠈⣁⠀⠋⠁⢠⡵⢢⡅⣀⣲⢻⡿⠃⠀⠀⢯⣹⡟⠉
⠀⠀⠀⠀⢀⠄⠀⠀⠂⢀⡀⠀⠀⡀⠤⠀⠂⢂⣉⠛⠠⠚⠃⣵⡄⠁⠀⠻⢿⣿⢆⣷⠀⠙⣦⠈⠁⠀⠄⠀⠀⠀⠂⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠲⠀⠁⠀⠀⠀⢀⠈⠙⠨⣉⠖⠀⠂⡀⢈⠙⠆⠀⠀⠀⢙⣀⠙⠀⠂⠁⠀⠀⠄⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠠⠤⣶⡈⡙⢁⠀⠀⠀⢰⠂⡥⠡⣔⡶⣄⣀⡴⠎⠁⠀⠀⢰⣆⡀⠀⠀⠄⠀⠉⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⣙⢄⣿⣳⣧⡶⠀⠑⠘⠀⠰⡿⠁⠀⠈⢹⠀⠀⠀⡀⢙⠣⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⠘⠛⠛⠀⠀⠀⠀⠀⡠⠁⠠⡘⠀⠈⢀⠀⠠⠈⢻⡄⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠛⠂⠈⠀⠀⠀⠀⠀⢰⡀⠀⠀⠂⠆⠠⡌⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠻⠹⣦⣠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⢀⢀⡀⠀⠀⠀⠀⢩⣏⣴⣤⡾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢖⣻⡁⠀⢀⣯⡥⠀⠀⠀⠀⠚⢿⣿⡶⣿⣧⠀⣣⡴⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⣵⠾⣿⣲⣶⣿⣷⣶⡶⣶⠂⠘⣷⣶⣿⣿⡿⢶⡾⠷⠖⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢩⣤⣬⣿⣿⣿⣿⣟⣿⣟⢰⣿⣿⣿⣿⣿⣶⣶⣿⣶⣠⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣶⣿⣷⣿⡿⣿⣍⣿⣿⣿⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣼⡿⢻⢏⣽⣿⠣⣏⠘⢃⡀⣤⣴⠂⠸⣯⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡟⢻⣿⣿⣿⠑⣯⣿⣾⣻⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⡟⢃⡞⢾⡛⠿⣿⡌⠶⡾⣿⠧⢙⣤⢼⣿⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⡿⣷⣺⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣼⣯⣠⢮⣿⣿⣿⣾⣯⢀⣠⣋⣽⣶⠟⣿⡿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⠸⢸⣿⣧⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣵⣿⣷⣭⣫⣾⣿⡍⡪⣻⣿⣿⣹⣾⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣤⡿⣿⣿⣿⣿⣿⣷⣿⣯⡿⣿⣿⣿⡋⢃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣽⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⢧⣾⣿⣿⣿⠿⣿⣿⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣿⣯⠭⠷⢿⣿⣿⣿⣿⣿⠃⢠⣾⣷⡀⢀⡖⣰
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠰⢾⣿⣿⣇⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⡿⣿⣿⡶⢷⣾⣽⡶⢯⣁⣧⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣟⣿⠿⣬⡗⠊⠁
⠀⢀⠀⠀⠀⠸⣣⡆⢀⣴⣆⠀⢉⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣯⣿⣿⣷⣼⡟⢀⣽⡿⠛⠀⡜⢿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢛⣿⠟⢦⣤⡀
⠀⡿⣦⡆⠀⠀⢼⣷⣿⣷⣿⣷⣤⣧⡽⣻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣏⡘⣿⣇⡼⠛⠷⣿⣾⣷⣷⣿⣿⢿⣿⣿⣿⡷⣻⣿⣾⣿⣷⣼⣿⠋
⠀⢸⡓⢾⣷⣾⣿⣿⣿⣿⣿⢟⣻⣿⢢⣿⣟⣩⣿⣿⣯⣝⣻⣟⣿⣟⡃⢹⣿⣧⣀⡸⡿⣿⣽⣿⠟⠻⠸⣇⣿⣉⣷⣟⢿⣿⣿⢿⡿⠉⠰⠄
⠀⢀⣫⣿⢈⣩⣿⡿⢿⡿⣿⣾⣿⣟⠑⢉⢹⣿⢿⣿⣻⡿⣿⣾⣿⣝⣧⣼⣿⡍⢩⣷⡾⣿⡿⡿⠋⣀⣎⣽⠿⢭⣿⣿⣿⣿⣿⣾⣷⣿⡔⣤
⠀⠈⠁⠀⢙⣿⣿⡷⢛⣿⣿⣾⣿⣎⣷⢺⡧⡉⠱⠝⠋⢴⣷⣿⣿⡿⣟⢿⣿⠅⣼⡟⢱⣿⣿⡿⣯⣿⣿⡔⣻⠷⣿⣿⣿⣿⡿⢺⣿⣿⣿⣿⠗⢢⣀⣠⣂
⠀⠀⣌⣾⣿⣿⣿⢷⣠⣿⣿⣿⢿⡙⢻⣿⡷⣿⣦⡀⣀⣺⣿⡿⣻⡟⠿⢾⣿⠀⠸⡇⠘⣿⣿⠟⢻⣿⣿⣿⣿⣽⣿⡶⠛⢽⢷⡰⢿⣛⣘⣇⣠⣿⡜⠓⠌
⠀⠀⠉⠿⠛⠳⠃⢴⣿⣿⡿⠛⣍⣛⣮⣹⠗⢴⣯⣿⣿⣿⣿⣷⡿⣄⠀⣿⣿⠀⢠⣧⡾⠛⠛⣿⣏⣿⣿⢟⡿⡿⢿⣶⣏⣾⣿⡶⣟⣋⣿⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢹⣯⣵⠘⣊⢋⣉⣁⡽⠿⠻⢧⣄⠈⠻⣿⣿⣶⠿⠋⠀⠀⠌⣙⣻⣿⣿⣿⣶⣿⣿⣉⣻⣿⢾⣿⣿⣿⣿⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡲⢤⡌⣿⠛⠑⣏⢛⡛⣁⣄⠀⠀⠀⣈⠙⠦⣿⣿⡇⠀⣀⡤⠾⠛⢋⢹⣅⠈⣋⡉⡸⢷⣿⣿⣿⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠚⠛⠃⠠⢾⣿⣿⣟⣿⣝⡟⡴⠆⣃⡭⠀⣻⣿⡷⠞⣩⡴⢆⣀⣊⠀⠺⣿⠏⢠⣿⣦⣼⡿⡿⣷⣾⡿⣿⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠑⢿⣽⠟⠿⣄⣍⠛⠾⠵⣾⣽⣿⡇⠀⠀⠀⠉⠈⡉⠀⠛⢿⠘⣷⠟⠀⢿⣧⣖⣦⠀⠀⠃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⢏⡙⠷⠆⠠⢌⣹⣿⣷⠀⢤⠼⢳⡏⣭⠗⠂⠀⠀⠁⠀⠀⠉⠉⠣⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡻⠛⠃⠀⠀⠀⢸⣿⣿⠖⠛⣀⠒⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠗
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣁⣍⣻⣀⣍⣉⣽⣉⣨⣟⣩⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⡙⢦⠯⠟⡠⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡼⠧⠤⠄⠂⣆⡀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢰⣄⠀⠘⠒⠣⠴⣧⡴⠷⣦⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡀⢈⣀⣀⣀⠀⠀⠀⠛⠁
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠘⡉⠩⠗⡆⠀⠀⡸⠛⠚⠣⡴⠀⠀⠀⠀⠰⢯⣬⡷⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠸⣤⣈⡁⠀⠀⠋⡿⢃⡤⠁⠀⠀⠀⠀⠆⢹⣻⡆⠦⠀⠀⠀⠀⠀⠀⢱⣤⡄⠀⠀⠀⢀⠀⠰
⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠉⠓⠒⠄⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⣤⣛⠀⠀⠀⠀⠀⠀⠛⠂
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠋⠙⠛⠛⠉⠀⠀⠀⠀⠀⣠⣔⣲⡇⠀⠀⢀⡂⠀⠀⠀⠀⠖⠁⠁
⡷⠉⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⢶⣰⣤⡀⠀⠀⠸⢿⡽⠻⠄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⢦⡙⠾⢷⡷⡽⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠭⣄⣾⠯⠐⠀⡀⢷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠠
⡄⡨⠤⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⠶⣄⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠰⠗⠄⢀⣤⡺⡄⠀⠀⠈⣧⠀⠀⠀⢠⠀⡀
⢦⣤⣠⠆⠀⠀⠀⠀⠀⠀⠀⣟⠳⣶⢶⣆⠀⠀⠀⢰⣶⣾⢿⠿⠂⠀⠀⠀⠀⠀⠰⠂⠀⠱⡤⠀⠀⣴⡇⠀⠀⠀⠀⣆⠙⢏
⠀⣀⣤⠤⠒⠋⠀⠀⠀⠀⠀⠙⠳⠿⠚⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⡀⣤⡴⠟⠁⠀⠀⠀⠀⠘⠀⠴⠖
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⡡⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠐⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⡡⠎⡱⢎⡱⢆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠌⠡⢞⣹⡃⣱⠎⣡⡶⡀⠀⠀⢠⠀⠀⠀⠀⣠⢖⣶⡶⢶⣄
⠤⢤⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠱⢎⡹⠟⣹⣾⣷⣿⡇⠀⢠⣾⠀⠀⠀⠠⠿⣾⣻⣿⣿⣽⠇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣽⣦⣌⣩⣾⡿⣯⣿⡤⠄⠀⠈⢸⠀⠀⠀⢠⣲⣈⠉⠙⠻⣬⠀⠰
⠀⠀⠾⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢫⠎⣹⠿⠋⠀⠀⠀⢀⣦⣀⠀⠀⠀⠈⠙⠿⠛⠙⠇⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠴⠼⣣⠿⠗⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣿⣭⣯⡥⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣤⠐⢌
⠁⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⢫⠐⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣿⣄⠀⠀⠀⡄⠐⠒⢣⡌⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⢁⣀⠀⠀⠐⢄⠀⠀⠀⠀⠐⢿⣿⣿⡻⠃⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠾⠿⣯⠋⣉⡙⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⠈⢢⠱⠀⠄⠀⠀⠰⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⠤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠂⠀⠀⠂⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⣀⣀⣐⣒⣒⣂⣀⣀⣠⣠⣤⣤⣤⣀⡄⡀⢠⣤⠤⠤⠤⠤⠐⠒⠉
⠀⠀⠀⠀⠠⠔⢶⠀⠀⠀⠈⠙⢲⣤⢀⣀⣀⣤⣤⠶⠛⠉⠉⠁⠀⣀⣴⡀⠀⠀⠀⠀⠐⠉⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⢍⡉⠁⠀⠉⠉⠀⠀⢀⣀⣤⣤⣖⣋⣁⡀⡁⠂⢐⠀⣋⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡎⢭⠀⠀⠒⠬⠍⠉⠓⠒⠒⠊⢉⡀⠤⠛⠉⠉⠩⠉⠉⠻⠄⠀⡠⠂⠀⠹⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠀⢀⣠⣴⠶⡋⠉⠉⠉⢀⠀⢀⠀⠀⠀⢀⣀⣀⠰⠀⢠⠀⠀⢀⡀⠄⠁⢀⣀⣀
⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⠴⢋⠉⠙⠂⠈⠛⠛⠒⠀⠀⠘⠁⠤⠄⠈⠀⠀⠂⠀⠐⠶⠚⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠔⠛⠇⠀⠀⢠⠶⠶⣿⡆⠀⠀⠀⢀⠴⠖⠶⠄⠀⢀⠞⠳⠄⠀⠟⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢾⠀⠀⠀⠐⠤⠐⠁⠀⠀⠈⠃⠐⠂⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⢀⣀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠐⠒⠦⣀⡼⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⣥⣤⣧⣿⣷⣾⣉⣳⣤⠋⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣿⡄⢘⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣮⡀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣇⣿⣗⢸⣿⣿⣿⣿⣿⣿⣿⡋⠉⣋⠙⢠⡄⠀⠀⠀⢠⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣼⣷⠒⣿⣿⣿⡏⢹⣿⣿⣿⣿⢿⣁⣷⣱⠪⠏⢷⠈⠁⠀⠀⠀⠀⣟⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡘⢸⡏⢠⣿⣇⣿⣿⣼⣿⣿⣏⣿⡁⢅⠙⠏⠉⢀⠉⢀⠄⠠⠂⠀⢰⡏⠀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣀⣿⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠈⠉⡘⠈⠉⠇⠈⠀⣯⡤⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣶⣶⡆⠐⠀⠀⠀⣇⠀⣿⣿⣿⣔
⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⣿⠇⢠⣾⣿⣿⣧⠄⠀⢈⠼⡗⠒⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢃⣿⣧⠀⠏⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢺⣾⣿⣿⣟⣟⣷⣶⠀⢀⣷⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠀⢀⣿⣿⣶⣿⣿⢻⣿⣿⣿⣿⢿⡯⣧⡿⢸⣼⣿⣿⣾⣿⣿⣿⣭⣭⣭⣭⣡⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠃⠠⣿⣿⣿⢹⣽⡇⣯⣿⡾⣷⣇⣿⣿⣿⣿⣿⣹⣹⣛⣻⣻⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠀⢴⣿⣿⣿⠙⣟⠓⠛⠷⠶⠯⢿⡿⠞⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⣿⡿⣱⣿⣆⠀⠲⡦⠤⢤⣤⡀⠍⠁⠀⠀⠀⠀⠀⠙⠛⢮⣿⣿⣿⣷⡼⡄
⠀⡀⢻⣿⣿⣿⣿⣿⣿⡿⠁⣴⣾⣿⣿⣿⣿⣿⡿⣿⣧⡀⠙⣿⣾⡀⢷⡌⠢⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⡄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠣⣤⣾⣿⣿⣼⠿⢛⣙⣿⣿⣿⣷⡄⠘⢿⢷⢈⣿⣆⠀⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣟⣸⣿⣆⠀⠻⣾⣿⣿⣷⡈⢢⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣤
⠀⠉⠉⠛⢻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⢟⣋⣸⣿⣦⣀⣀⣀⣀⣀⣀⣀⣱⡄⣀⣀⣀⣤⣤⣤⣽⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢀⣈⡙⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣉⣥⡉⠱⢂⣻⣋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠈⠉⠉⠐⠂⠤⠄⡠⠀⠀⠈⠁⠀⠉⠛⠒⠛⠛⠿⠿⢿⡛⠛⠛⢋⣍⣉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠦⣤⣤⣤⣀⣀⠴⠾⣾⣿⣿⣿⣷⡄⢼⣿⣿⣇⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠛⢿⣿⡭⠁

⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣹⣿⠟⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⡇
⠀⠀⣸⣿⠁⡇⠀⠀⠀⢸⣤⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⢠⣶⣴⠄⠂⠆⠀⠸⣆⣠⠃
⠀⠀⢻⣿⡗⠁⠀⠀⠀⠸⢂⠖⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⣿⣾⡀⠀⣃⣀⠀⢸⠁
⠀⠀⣿⣿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⡀⡏⠀⠀⢸
⠀⠀⠙⠉⠉⠈⠁⠀⠀⠀⠳⡶⢄⢀⠀⠘⠆⠠⡆⢀⢏⠀⠀⠀⢀⠄⠀⠀⠀⠀⢸⣿⡇⠀⠃⠁⠀⠀⢘
⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠁⠠⡄⠀⠁⠉⠀⠀⠀⡘⠀⡄⠠⠆⠠⠀⠀⠀⠀⠀⢸⣿⡷⠖⠀⣇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⢄⠀⠀⠀⠠⠀⠀⢀⠉⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠈⠈⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⠈⠁⠀⠐⠂⣀⠀⠀⡐⠁⠀⠀⠈⠁⠀⢀⠔⠒⠊⠉⠓⢦⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡎⠀⠀⠀⠀⢀⡀⢀⠂⠀⠀⠀⠀⠀⠈⢳⣄⠀⠐⢶⣿⠷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠀⢠⠀⠚⠁⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠶⢿⠦
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡄
⠀⠀⢠⡟⠀⠀⢸⣇⠀⠀⠝⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣿⡇
⠀⠀⠘⠋⠀⠀⢸⡁⠀⠠⠀⢷⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⠀⠈⠐⠆⠀⠒⠛⢻⡟⠿⠿⠿⠿⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠁⠊⠀⠀⠄⠀⠆⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⣤⣄⣀⡐⠀⠒⠀⠀⠀⢁⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣟⣒⡮⠭⡒⠦⠖⠈⠩⢉⡁
⠀⠀⠀⠠⡀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠇⠀⠀⠂⠘⢿⡫⠭⠑⣒⡒⠒⢒⣋⡩⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠁⠂⠍⢉⡁⠚⠛⠉⠉⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠓⢄⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣀⠄⠀⠀⠍⠄⠠⣝⠅⠀⠀⣦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠷⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠠⠀⢀⡐⡁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⡏⣼⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢂⣄⠀⠀⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠆⠀⠀⠋⡋⠉⣠⣤⣴⣿⠃⠸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⢿⣷⣾⣿⠿⠟⠃⠀⢸⢇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⡔⠁⠀⠢⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠋⠀⠀⢠⠄⠀⠐⡂⡘⠀⠘⠧⠤⠤⣀⣰⡀⢀⠄⣠⠀⠀⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠴⠃⠀⠀⣠⢆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡀⠀⠜⠁⠀⠐⠀⠀⠀⠀⠰⠀⠁⣰⠇⠘⢰⠁⠀⠀⠀⠀⠙⠂⠠⢶⣎⡁⠀⣤⡶⠿⠿⠉
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⣠⡏⡄⠀⠀⠀⢰⣿⣦⣡⣾⢿⡇⠀⣸⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣿⠀⠀⠀⠤⠀⠀⠀⠀⢠⠀⠰⠃⠉⣷⠁⠙⠁⠀⠀⠀⣿⢏⣿⣿⡿⠀⠙⠔⢁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡇⠀⠀⠸⢇⠀⠈⣿⣿⣿⣿⣧⢀⡄
⠀⠀⠀⠀⠀⢀⠔⠟⠁⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠠⠁⠀⠹⣿⡷⠿⠿⠞
⠀⠑⠄⣀⣤⣶⣊⡤⠖⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠄⠀⠀⠀⠀⡀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠶⠒⠶⠶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⢁⣀⣀⣠⣒⣀⣠⡴⠶⠶⠶⠷⠦⢤⣄⡀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠜⣁⠤⠶⡶⠛⠍⠁⠀⢠⣭⣽⣳⣶⣶⣤⣈⣩⣷⡶⣦⣍⠙⠲⣤⡀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⣬⢪⠅⠁⠀⣠⣤⣶⣶⣾⠟⠛⠿⠽⠻⠯⠿⠿⢝⠿⣄⠠⢿⣷⣷⣄⠹⣦⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢠⠞⣱⣁⠀⣠⢊⡄⠈⠁⠐⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠷⣜⣷⡌⠉⠉⢻⣷⠈⢳⣄⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠃⠈⣭⣿⣿⣱⠇⠀⠀⠀⠀⢀⠐⠄⠲⠠⠿⠌⠀⠀⣄⠀⢿⣿⣿⣶⣶⡿⠛⠀⠀⠹⡆⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⡹⠀⠀⠃⢀⠀⠀⠉⣀⠀⢀⣘⠨⠀⠋⢠⠁⢰⣦⣹⡿⢿⣷⡀⠀⠀⠀⣹⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠃⠀⠀⠀⣾⡇⠀⡀⠀⠀⠁⠀⢀⠀⠀⠁⠊⠀⠤⢠⢀⢦⣼⡿⠟⠁⢸⣿⣗⠁⢁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠙⢿⣿⣦⠶⠶⢦⣄⠀⠐⠆⠸⠀⠋⠲⠆⣰⡿⠋⠁⢈⠀⣼⠿⢃⣠⠞⣼⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⡿⣆⡂⠀⠀⠀⠈⠉⠉⠉⠛⢿⣛⣗⣚⣿⣿⣿⣟⢭⣿⡇⠀⣠⢈⣾⣯⠴⠛⣱⣿⣿⣷⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⢿⣿⡂⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⢷⡹⣷⣨⠕⣿⣿⣥⣶⣿⣻⣶⡿⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⠻⣝⡶⣤⣑⣶⠦⣤⣤⠴⠤⣤⣤⣤⣄⣀⣤⣿⣷⣻⣿⣾⣿⢿⣿⡿⣻⠵⠋⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣉⣉⣛⠛⠛⠛⠓⠒⠒⠒⢛⣋⣥⣿⣽⣿⣿⣿⢾⣟⣶⠁⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⢿⡻⣟⡻⠿⠿⢿⡿⢿⣿⠿⠿⡛⠲⣶⣾⣿⠿⣊⡵⠃⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⡀⠀⠀⠉⠛⠻⠿⣶⣶⣶⣶⣶⣶⣶⣾⣯⡽⠷⠚⠋⠁⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢥⣄⣀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣢⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠲⠶⠶⠒⠒⠒⠋⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠂⠀⠐

⠀⠀⠀⠀⠀⢀⣤⣔⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡀⡀⠛⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠦
⠀⠀⠀⢠⡶⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⣠⠞⠂⠀⠐⠋⠀⠀⠀⠘⠋⠉⠀⡀⢀⠄⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠁⠀⢀⣤⢀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⡀⠀⠀⢀⠀⠀⠘⡻⠋⠀⠀⡔⠁⠀⣾⠖⠀⠀⠀⠀⣀⡀⠉⠁
⠀⠀⢸⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣀⠴⠂⠀⠀⠀⠀⡛⠁⠀⣤⣄⣀⠤⠚⠁⠀⠀⢰⡗⠃⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⡀⣠⣆⢒⣈⣀⣤⡤⠟⠉⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⠀⠀⠀⠀⠚⠛⠃⠀⠀⠰⡿
⠀⠀⠘⠻⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠂⠀⠉⠁⠀⠀⠀⠀⠞⠀⠘⠟⠀⠀⠀⣠⣤⣄⠀⠀⣠⣤⣤⣤⣀⡀⢀⣾⢃⣼⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠘⠲⣿⠆⠀⠀⠈⡁⠀⠐⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣠⣿⠃⠘⠟⠛⠙⠛⠛⠛⣿⡟⣛⡁⣘⠁⣤⢤⢿⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢸⣿⠇⢀⣶⠲⡄⣼⡏⢹⢠⣿⣏⣼⢣⡿⢸⡧⢊⣔⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⢂⣾⣯⣠⣧⣿⠣⡼⢻⡟⢿⣄⠘⢓⣉⣙⡉⢸⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣠⠞⠻⣧⠟⠁⠙⠚⢁⣀⣥⣼⠿⢿⣍⣁⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠈⠿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠞⠃⠀⠈⠉⠁⢀⡀
⠀⠀⠀⠀⣴⠆⠀⠀⣤⠀⠀⠀⠈⠙⠁⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠒⠓⠚⠃⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⡾⠖⠋⠉⠁⠙⠒⠀⠀⠀⠰⠒
⠀⠀⠀⢠⠀⠘⠷⠀⠀⠀⠀⠀⣴⡶⠂⠀⠀⠉⠉⠀⠄⠀⠀⠀⡠⠂⠛⠁⠀⠐⠟⢃⡀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠘⢿⠁⣠⣤⠀⣀⣘⣀⣀⠠⣤⠀⠀⢤
⠀⠀⠀⣿⣧⠀⠀⣠⣄⢠⣴⣦⠀⠀⢠⣶⣤⠀⠔⢠⠆⠀⠀⣼⠃⠀⢛⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⢦⡤⠒⠉⠉⣉⠉⠉⠉⠓⢄⢀⡀⠁
⠀⠀⢰⣿⣿⣿⣧⣤⣤⣄⡀⢤⣄⣀⣀⣀⣀⣀⣴⡏⠀⠀⢀⡴⠀⠀⠙⠁⠀⠀⠀⠤⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⠀⠀⠋⠀⠀⢀⡀⠀⠈⠡
⠀⠀⠘⢿⣽⣋⡉⣉⡉⠹⠿⠟⠛⠉⠉⢉⣽⡷⣀⣶⠀⠀⠈⠳⠀⠀⠦⢀⠐⠀⠈⠀⠁⠚⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⣏⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⢰⣿⣿⣿⣭⢤⣦⣦⣤⣤⣴⡷⠾⠿⠛⠙⠋⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠉⠀⣀⠀⠀⠤⠀⠀⣀⡀⠀⠀⢸⡇⣴⠄⠀⠀⠀⠀⢴⣆⠀⠀⠀⡀
⠀⠀⠀⢨⣥⣌⣉⠀⠀⣀⣀⣀⢠⠾⠆⠀⣀⣠⣤⣴⡛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⣿⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡄⠛
⠀⠀⠀⢉⠾⠋⠉⢭⡉⠉⠋⢹⠟⢻⡏⠉⠉⠉⢈⣉⣤⠀⠀⠀⠀⠀⣀⣀⠀⠰⠖⠒⠒⠀⠉⠈⠁⠀⢠⡀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠶⠂
⠀⠀⢠⣿⡟⢛⡶⠶⣤⣤⣤⣤⡶⠶⠶⠞⠛⢋⡉⠡⢄⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢴⣦⡄⠀⠀⠀⠙⠁⢠⡆⠀⠀⠀⠈⠳⣄⣀⡀⠀⠀⢶⠆⣀⠤⠔⠁
⠀⠀⠀⠨⣷⣿⣿⣄⣀⠺⠷⠀⠀⠀⣀⡀⠀⣛⣣⠴⠊⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠉⠁⠀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠁⢃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⣀⠇⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⠢⠔⢢⢀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠢⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠉⠈⢆⣸⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠤⣀⠖⠦⠔⢊⠝
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⣀⣆⣠⠀⠑⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⢠⠋⠻⠟⡆⠀⢰⠁⠛⠋⠆⠀⠀⠀⠀⠀⠀⢎
⠀⠀⠀⠀⠀⠠⡤⠤⠔⠢⠼⠀⠀⠀⠀⠓⠤⠐⠁⠀⠀⠑⠒⠊⠀⠀⠀⠀⠀⠀⢀⡤⠃
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢱⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣐⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⣠⣀⠔⠊⢻⠐⠢⠤⠤⣀⡠⢀⣘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠊⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⣤⠤⢤⣀⢀⠀⣀⡤⠶⣦⣠⣄⡀
⠀⠀⠸⣟⣱⡟⣦⡀⠀⢀⣴⢻⣎⣻⠃⠀⣄⠤⣀⡀⠀⠀⠀⠀⣀⡤⣤⡀⠈⠳⢦⡀⠀⠈⠛⢻⡟⠋⠁⠀⣀⠴⠚⠁⣠⣤⡤⢠⣄⣀⢀⢀⣀⣤⠠⣦⢤
⠀⠀⠀⠙⢳⢿⡟⣿⣿⣟⣻⢿⡞⠃⠀⠀⠘⣶⡿⣿⣷⣶⣶⣿⡿⣷⡟⠀⠀⠀⠀⠘⣷⣄⣠⡞⢧⣄⣰⡿⠁⠀⠀⠀⠈⠟⢠⢾⣿⠷⢿⡟⢿⡟⢦⠘⠃
⠀⠀⠀⠀⣿⣿⣿⡁⡿⣸⡻⣿⡿⠀⠀⠀⠀⠀⡄⠛⠿⢿⡿⠟⢃⡄⠀⠀⠀⠀⠀⠀⢈⠿⠟⠀⠈⠻⢿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⠈⠘⠻⣿⠟
⠀⠀⠀⠀⠙⠻⠏⠁⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠙⠻⠖⠀⠘⠾⠛⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠔⠄⡀⠀⠀⠀⠀⠀⡀⠔⠆⠀⠀⠀⠀⠀⢀⣤⣶⣆⡀⠀⠀⠀⣀⣤⣤⣤⢤⣤⣤⣀⣄⣤⣤⡄⣤⣤⣤⣄⠀⠈⢿⣿⣿⠋⠁⣢⡖⠊⠙⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠁⠁⣦⣤⡄⠁⠀⠀⠀⠀⠀⢀⣠⢴⣿⣿⣿⣿⣿⡦⣄⡀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠙⢯⡶⠀⢸⡇⠀⢴⡽⠋
⠀⠀⠀⠀⠀⠀⡀⠋⠟⠣⠀⠀⠀⠀⢀⣴⣿⣷⣾⣿⣿⣿⣿⣿⣷⣿⣿⣷⡀⠀⠀⠸⣟⣉⡏⢙⣙⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡆⢰⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠂⠀⠀⠀⠀⠙⠉⠻⠿⠾⠿⠷⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠉⠁⠈⠉⠙⠄

⠀⠀⢀⠀⢀⣀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣷⣿⣿⣿⡇
⠀⠀⠈⠳⠄⢄⠸⣿⠁⡄⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⢉⣿⣿⡋⣽⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠴⠤⠁⠦⠀⠀⠀⠀⣤⡄⠀⠀⣶⠂⠀⠀⠐⠷⠀⠾⠂⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠐⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⠀⠀⢡⡸⠠⠄⠁
⠀⠀⠀⠀⠀⡠⠀⢐⡂⠀⠀⠀⠀⠀⠀⠀⠂⡀⡂⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠓⠋⠀⠀⠸⠗⠆⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⣀⢀⠁⣠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⠀⢀⢠⡤⠒⠉⠀⠀⠀⠀⠐⠄⠤⠤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⡆⢀⠄
⠀⠀⠀⠀⠸⠀⠠⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠀⠀⡄⢀⠀⡆⠀⠀⠀⠀⠀⢨⠈⡟⣀⣤⠛⣀⠸
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠈⠂⠀⠁⠉
⠀⠀⣠⢄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢒⠧⠏⠒⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⡤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠄⣀⠀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣙⠛⣁⡴⠞⡁
⠀⠀⠀⠀⠀⣀⠤⠛⠁⠀⠀⣠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠓⢁⣠⠔⠂⠀⠀⢄⠀⠀⠢⠀⠀⣥⢈⣥⣼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠢⢠⢈⠀⠀⠀⠀⠀⡰⠉⠂⠀⠉⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠘⡟⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠔⠂⡟⠩⠷⣤⣴⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠉⠒⢖⠃⠇⣷⣰⠆⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⠀⠀⢸⠑⢠⡿⢷⠴⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⢀⡤⣄⡀⠀⠀⣸⠀⢀⠀⠠⣱⢀⡇⢀⣠⣆⠀⢧⣀⣀⡴⠋⢛⣶⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⣝⠉⢀⠀⠉⠑⡒⠁⠀⠈⢂⢇⠈⢩⡃⠋⠉⠁⠉⣚⠯⢁⡬⠖⠉⠁⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣊⣨⠷⡦⣁⣇⠰⠂⠀⡠⣲⣀⠀⠊⣪⠁⡀⠀⠰⠟⣫⡶⠿⠗⢓⠗⠈⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠦⠴⢷⣋⢍⡳⠷⣦⡖⠧⢷⡄⠌⣟⠀⠁⣀⡠⢞⢥⠀⢀⡔⠃⢀⣀⣈⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠲⢤⡀⠴⣂⣀⡈⠈⠙⠓⢦⡉⢹⣃⡤⡶⠋⠀⣀⡀⠕⣛⣽⡏⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠉⠙⠢⠀⠠⠀⡀⠉⢾⣧⣾⣿⣦⣿⣷⣶⣿⣿⣯⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣮⡤⠴⠒⠒⠊⠉⡉⠉⠉⢹⢿⣟⠻⠿⠀⢀⡐⠀⠀⢋⠉⠙⢺⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢤⣀⡰⠶⠮⢄⣀⣔⣁⣾⡿⡄⠀⠀⠀⢈⠀⣐⣸⣆⡤⠴⠒⠒⠚⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣩⠟⢸⡇⠹⡄⢠⠞⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⡘⡇⠀⠘⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁

⣿⣿⡿⢟⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣄⣿⣾⣶⣿⣿⣟⣠⣴⣶⣆
⣿⠿⣿⢻⣯⣥⣄⣀⡀⠍⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠠⣾⣿⠿⠻⠿⠟⠿⠟⠿⠿⣿⣷⣶
⠃⠀⡏⠈⠿⣿⣿⣿⠉⢳⣦⣹⣿⠃⠀⠀⠀⣀⣀⣐⠋⠁⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠆
⠀⠀⠆⠀⠀⠈⠛⠫⠄⠈⠙⠋⠀⠀⣴⣿⣿⡿⠟⢻⠇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⢣⠀⠀⠀⠐⠲⠖⠃⠀⠀⠀⠀⡝⣿⣯⢙⣷⠋⠀⠀⠀⠀⠀⠀⢸⣧⢴⢯⣵⣦⣄⣀⣺⣯⣭⣦⣬⡅
⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢭⡌⠁⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠁⢈⡟⠻⠀⠉⠀⠈⡽
⣧⠀⠀⠹⡄⠀⠀⠀⡄⠉⠁⠀⠀⢰⠀⠙⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⡫⡇⠀⠙⠒⠒⠊
⣿⠁⠠⢤⣘⣦⡀⠀⠻⣿⣿⣦⣤⣴⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀⢈⠃
⡇⠀⠀⠀⢈⣻⣿⣦⣤⣜⣿⣿⡿⣰⡿⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡿⠂⠋⠁⠉⠳⠀⢀⣀
⣿⣶⠖⠀⠀⢨⡙⠿⣿⣯⣿⣿⣷⣿⣥⠤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠺⣿⡇⠐⡣⠸⠃⠀⠀⠄⠀⣸⣿⣿⡶
⣿⣿⠓⠀⠀⠚⠻⢶⣤⣌⣭⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣰⠧⢀⣤⣀⣀⠀⠀⠀⠀⠀⢀⣶⣶⢄⡀⡀⢰⠆⡀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢸⠀⠀⠸⣬⢻⣽⣥⣦
⠀⠀⠀⠀⢀⡜⠻⠶⣾⣿⣉⣀⠀⠀⠀⠀⠀⠈⣩⡝⠛⠻⡟⠙⠚⠋⠀⠀⢠⡔⢿⣯⣏⠀⠀⣦⣤⣼⣤⣴⡄⠙⢯⣘⣿⢿⣴⡆
⠀⠀⠀⠀⢺⣷⡶⢤⡀⢈⠿⣹⡏⠂⠀⠀⠀⠀⣭⠷⣶⠶⠶⢶⡷⠂⠀⠀⠀⠙⣿⣿⣿⡀⠀⠱⢄⣀⣨⠟⠀⠀⠀⠻⡤⠞⠋⠁
⠀⠀⠀⢰⣟⣈⣶⣀⣘⠿⠗⠈⠘⡀⡀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠀⠀⠀⢠⣾⠟⠛⠻⣷⣆⠀⠀⠀⠀⢀⡀⣀⣠⣤⣄
⠀⠀⠀⣸⠋⢉⣀⣸⡿⣷⠊⢻⠿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⣠⣶⡄⠈⣿⠀⠀⢸⣹⣿⡿⢿⢿⣿⣟⣣
⠀⠀⠀⢸⠿⣿⣇⣀⣉⣿⣿⣳⢶⣄⠀⠀⠀⠀⣞⣾⣿⣯⣙⣲⣦⡀⠀⠀⣧⢦⠘⠛⢃⣄⣿⠀⠀⠀⠻⣾⠃⢽⣷⣦⣿⣏⣰⡄
⠀⠀⠀⠹⣦⣬⠀⣿⣧⣤⣭⡿⣷⣤⠁⠀⠀⣨⣿⣿⣿⠿⢽⣿⣫⠀⠀⠀⢿⣿⣗⣒⣾⣿⡿⠀⠀⠀⠀⠈⠛⠻⢤⣼⣛⣫⡟
⠀⠀⠀⠀⠀⠉⠉⠉⢻⠋⠈⠉⠉⠀⠀⠀⠀⢹⣿⣯⣿⣿⣿⡿⢋⠀⠀⠀⠘⠿⠿⠿⠿⠿⠃⠀⠀⢀⣠⣴⣤⣤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠛⠛⢾⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣦⣿⣿⣿⣿⣛⣿⣿⣛⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⡶⣤⣈⢙⣯⣿⡯⡀⠒
⠀⠀⠀⠀⠀⠲⡤⡄⠀⣠⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣿⣿⡿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⢀⡘⢞⠋⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠁⠀⡁⢘⣿⠿⠂⡴⠄⢻⣀⠀⠀⢀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⡴⣶⣳⣿⣥⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠠⠴⢴⢻⣿⣄⢆⠀⠂⣴⣥⠛⠀⠠⡟⠈⢚⡅⢠⠂⠀⠀⠀⠀⡜⢲⠡⠀⠰⢤⣿⣿⣏⡂⠀⠀⢀⡀⣀⣦⣿⡨⡔⢠⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠠⢧⣔⠀⠀⠀⠀⠶⢗⣦⣟⣄⣁⠵⠆⠀⠀⠀⢳⡈⠛⠲⠾⢿⣥⣾⣿⠆⠀⠀⣩⣷⣿⢿⣿⡟⢷⣝⢤⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠁⠙⠫⢙⡈⠁⠀⠀⠀⠀⠀⠉⠿⣿⣾⣿⡿⠟⠁⠀⠀⠀⠴⢿⣹⣟⣹⣿⣿⡕⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⣸⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⣀⣀⣀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠛⠛⠛⠛⠋⠉⠉⢻⠋⠉⠉⠛⠛⠛⠛⠻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁