Unconscious Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⢤⣤⢤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣤⣤⣤⣤⣤⡤⠤⣶⣢⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⢸⣿⣿⣾⣿⣿⣼⣿⡇⠀⠀⠀⡀⠐⠀⢀⣿⣿⣿⣿⣿⣟⣉⣉⣭⡍⣿⣿⣿⡙⢀⡉⠛⠀⢀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣹⡗⢠⠍⣿⣿⡿⠟⠉⠈⠀⠀⠀⠦⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢘⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⡿⠿⢁⡀⠰⣉⡙⣏⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣤⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠴⠂⠀⠈⠁⠁⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⣿⣽⡿⠢⣼⠡
⠀⠀⠀⠀⠀⠀⠀⢰⠀⣧⢀⠰⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠁
⠀⠀⠀⠀⠀⡗⢀⣿⡀⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⣰⣶⣶⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣿⢿⣿⣿⣿⣾⣧⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣾⣿⣿⣿⣿⣿⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⢩⣿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣶⣬⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣀⠀⠰⡆⠀⠀⠀⠀⠀⢀⠂⡀⢀⣿⣿⡟⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠒⠱⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠈⠀⠀⠀⢀⣾⣯⣤⣋⣼⣿⠟⠉⠻⢿⣿⡏⠉
⠀⠀⠀⠀⢀⣤⣄⣷⣖⣤⣤⣤⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠠⡀⢴⣿⡗⢉⣿⠟⢹⠟⠀⣀⣷⣿⣿⠁
⠀⢀⡄⢐⠊⣄⠉⠉⠉⠉⣴⡀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⠐⢶⣷⡞⠁⠠⡀⢠⣼⣿⣿⠋⠀⡀⢀
⠀⠀⠛⠋⠀⠈⠃⠀⢀⡀⠈⣹⡇⠞⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⣾⡀⠀⠈⢻⡆⠀⠀⣷⣾⠿⠟⠁⠈
⠀⠀⠤⢨⣅⠀⣀⣀⢿⣷⡴⠞⠶⠛⠀⠀⠀⠀⠀⠀⡀⢀⠘⠀⠀⠀⢀⡧⠀⣠⢦⣿⣷⠀⠀⠙⣿⣶⣾⡟⠁
⠀⣤⡤⣀⣼⣟⣧⡤⢹⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠋⠃⣰⢿⢳⣿⣿⣤⡄⣄⣿⡿⠃
⢀⣿⡄⠙⠁⠀⠀⠀⠀⣽⡿⠃⠖⠀⠀⠀⠀⠖⢠⡀⠀⠀⠀⡀⠀⣨⣷⢻⣯⣀⣿⡿⢏⣳⢠⡼⣿⣯
⢸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠉⠓⠐⠂⠀⠀⠀⠠⠟⣠⡴⢀⠄⠅⠘⠟⣧⡜⠛⢛⣽⣴⣾⣿⣿⣿⣿⣿⡆
⠉⣿⣿⣿⣿⣥⣠⣤⠖⠀⠀⠀⠀⠂⠀⠀⣠⠴⠛⡉⠀⠈⠀⢀⢀⣱⡟⣠⣿⣿⣿⠁⠀⣨⣿⣿⣿⣿⣷⡀
⠀⠛⠿⠿⠿⢿⣭⣤⣶⣶⣆⣀⣤⣆⣠⣼⣧⣴⣧⣤⣤⣤⣼⣿⣿⡏⣰⣿⣿⣿⡋⢀⡤⢿⡃⣙⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠿⠿⣿⣿⣿⣷⣬⣿⣿⣋⠀⢬⣴⣿⣿⣿⣿⣿⣿⣿⣦⣦⣤⣤⣤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠶⠤⣾⡷⠶⢦⣬⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠄
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠤⠐⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠨
⠀⠀⠀⠀⠀⠞⠂⠀⠙⣧⠶⣳⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣤⣤⣤⠀⠱⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣔⢶⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⢋⠛⡉⠉⠙⠉⣹⣿⣿⣿⣆
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠠⠾⡿⠿⣿⣿⣿⣿⣿⣿⣿⡎⣤⣧⣤⣬⣅⣀⣿⣿⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⡦⠀⠀⢰⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⠇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣅⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢼⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣆
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⣠⣿⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠙⣿⣿⣿⣿⡿⠋⠀⣴⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠈⠻⣿⣿⣇⠠⣾⣿⣿⣿⣿⣧⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⣀⣸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣽⣛⣛⣛⣫⣾⣿⣿⣿⡿⠋⠉⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣀⣠⣿⣿⣿
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠊⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⣶⣶⣦⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⢶⡶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡮⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣠⡿⠁⠀⠀⠀⠀⠀⠀⡈⠀⠁⠀⠈
⠀⠀⠀⠀⣤⠴⠀⣀⣾⣿⣿⣦⠀⡀⠠⠌⠉⠑⠢⠀⠀⠀⠀⠀⢠⡷⠀⠀⢠⣾⣿⣿⠁⠀⠀⠸⠄⠂⠈
⠀⠀⠀⠀⡅⠀⣿⣿⣿⣿⣯⣿⠆⠀⠀⠠⠄⣀⣴⠄⠀⠀⠀⢀⣿⠗⠶⣤⣿⣿⣿⣿⠀⠀⠀⠳⠀⠀⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠁⠘⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠰⠟⠃⠀⠀⠀⣴⣿⣿⣤⠀⣿⣿⣿⡟⠻⣆⠀⠀⠀⠀⣘⠃⠀⠀⠈⡄⣀⠤⢤⣤⣀⠀⠠⠄
⠀⠀⢀⣄⡀⡀⠛⣙⠉⣻⣿⣿⡇⡀⠀⠀⠀⠀⠀⣾⠁⠸⣿⣿⠈⠛⣻⣿⣿⡿⠁⠀⢻⡄⠀⠀⣀⣉⡠⠶⠖⢛⠟⠉⠀⠀⠀⠉⠁
⠀⠀⢿⡿⣧⣿⣿⠛⡟⠿⣽⣾⣿⠀⠀⠀⠀⠀⣸⡏⠀⣰⣿⠇⠀⢠⣿⣿⡿⢷⡄⠀⢸⡇⢀⡾⠛⠉⠀⠀⠀⣿
⠀⢰⣿⣧⣈⡉⢉⠁⣶⠖⠿⢿⡷⠾⡤⠀⠀⠰⠆⠀⣀⣸⡿⠀⣰⣿⣿⣿⠃⠀⢻⣆⢸⡇⣸⡇⠀⠀⠀⠀⠀⣿
⠀⢸⣿⣿⣿⣿⡛⢿⣿⣽⣿⣿⣿⠒⠀⠀⡙⠁⢀⣴⢿⣿⠁⣴⣿⣿⣿⠏⠀⠀⠀⠻⣾⣧⢹⡇⠀⠀⠀⠀⣰⡟
⠀⢸⣿⣿⣿⣾⣿⣿⣻⣿⣿⣿⣿⢷⣀⣸⠟⢳⡄⠙⣿⣏⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⠀⢠⡞⠋
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟⣷⡄⠳⣟⠻⣾⣿⣿⣿⠋⠀⠠⣿⠀⠀⠀⠀⠀⠹⣿⣷⣴⣿⠀⠀⠀⠀⠉⠙⠟⠛⠢
⠀⠀⢿⠜⣿⣿⣿⣿⡬⠉⡩⡀⠀⠀⠙⣿⣦⠀⠀⣿⣶⣿⣿⡿⠃⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠠⣴⡟
⠀⠀⠀⢸⣿⠛⠀⢿⣦⠀⠀⠈⡀⠀⠀⣸⣿⠇⣴⣿⣿⣿⡟⠁⠀⠀⣠⡾⢿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⣀⠉⠁
⠀⠀⠛⢋⣿⣷⣶⣾⡿⣆⠀⠀⠀⠀⢸⣿⣿⣤⣟⣿⣿⣿⠃⠀⢀⡀⢿⡤⠀⠁⠀⢀⣴⠄⠀⠀⢸⣿⣿⣧⣶⣶⣿⣯⣤⡀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠉⠉⠙⡋⢹⡷⢨⣤⠀⠀⠀⠉⢻⣯⠉⢿⠋⠉⠻⣝⡀⢸⣿⣯⣿⢻⣰⣦⣾⣯⣀⠜⠀⠈⣿⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⢀⠀⢸
⠀⠀⠀⠠⣶⣿⣅⣸⣇⣼⣯⣆⠀⠀⠀⢸⣷⠀⠘⡇⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⠟⣿⣿⣿⣶⣧⣴⠿⠛⠋⢹⡉⡀⠀⠀⠀⠀⠀⠉⢳⠟
⠀⠀⠀⠀⣻⣿⣥⣤⢿⣿⣿⠃⠁⠄⠀⢰⣿⣧⠠⠄⠈⠀⠀⠙⢿⣾⣿⠟⣾⣿⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⣰⣿⣦⣤⠀⣀⠀⠰⠀⠸⢷⡄
⠀⠀⠀⠀⢻⣿⡿⢿⣿⣛⢿⣇⣀⠀⢀⣼⣿⣿⣗⠀⠀⣰⡗⠀⣴⢿⣧⣾⣿⣿⡿⠿⢿⡟⠁⠀⢠⡾⠀⢀⣿⣯⣿⣷⣴⣿⣴⡇⢲⠀⢸⡇⢠⣆⠀⢀⣹
⠀⠀⠀⠀⠀⠋⠀⣀⣻⣏⣷⣾⣏⣳⣾⠛⠛⠻⢟⠇⠀⠙⢣⣾⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⢻⣷⡿⠃⢠⣿⡟⠁⣈⣿⣿⣿⣿⣰⣿⠀⢈⣿⡿⢻⣴⣤⡗
⠀⠀⠀⠀⠀⠀⣠⠈⣿⣦⣏⠿⠛⣿⣯⣻⣤⠀⢈⣀⣀⢀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠠⠤⣿⠃⢠⣿⣯⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⢉⣠⣾⠟⠻⣿⣄
⠀⠀⠀⠀⠀⣠⠀⠘⠁⠈⠈⢱⣦⣩⣽⣟⠁⠛⠛⠻⣿⣭⣷⣶⣿⠇⠀⡴⠀⠞⠀⠀⠀⠀⠀⣻⣶⢿⣿⣷⣾⣿⠉⢿⣿⠿⣿⣿⢿⣠⣿⣿⣿⠇⠀⠀⢻⣿⣯
⠀⠀⣀⠀⠒⠁⠀⠀⠠⠼⠷⠀⣿⠋⠁⣌⠁⠀⣤⠀⠀⠙⢿⣿⣿⣿⣼⠇⣼⠇⠀⠀⠀⠀⠀⢹⡇⠀⠈⠙⢿⣿⣿⣿⡇⠀⠸⡇⢘⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿
⠀⢸⣿⣿⣼⣿⣷⣶⣶⣤⡀⠀⢈⠀⠂⠠⠄⢠⠀⠰⠂⠀⢸⣿⡿⢫⡿⠖⠿⣆⠀⢀⠀⠀⠀⣿⣵⠀⠀⢠⠀⢙⠛⣿⣿⣤⡾⡗⢸⣿⣿⣿⡟⠁⢠⣾⣿⠏
⠀⢸⣿⣿⣿⣉⢘⣿⣿⣿⣿⡆⠀⣀⣠⣄⣀⠈⠀⠘⣦⠀⣿⡿⠁⠈⠀⠀⠀⠛⣀⢨⡔⣄⡾⠃⣿⠀⢀⠑⢿⣯⣙⡿⠿⣋⢴⣿⣿⠏⣻⣿⡾⠃⢸⣿⠃⠀⣠
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⣧⣴⣿⣿⣿⣿⣿⣦⣄⡁⠀⠺⢡⠀⣀⠀⠀⠀⠐⠻⢦⡀⠻⣿⡍⢿⡄⠀⠀⠀⠉⠈⠑⠠⣉⠻⠍⣷⡾⣿⣿⣯⣤⣾⣯⣴⣾⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⠾⠙⠿⢿⣿⣿⣿⣿⣿⡄⠐⢃⠄⠉⠀⠀⠀⠀⠀⠀⢹⣶⣾⣄⢾⡇⠀⠀⠀⠀⠀⡀⠀⠛⠁⠀⠀⠓⢾⣿⣿⣏⣈⠉
⠀⠀⢻⣿⣿⣿⣶⣭⣼⣿⣿⡟⠣⣽⣻⣿⣿⣿⣿⣿⠿⣣⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠻⡿⣾⣧⠀⠀⠀⠀⠀⠁⠀⡀⠀⣀⣀⠀⠈⠱⠿⠿⠟⣷⢷⣦⣤
⠀⠀⠀⠻⢿⣿⣿⣿⣶⣿⣹⣧⣤⣮⣻⣿⣿⣿⣿⡿⠲⣽⠃⠠⠀⠈⠀⠀⠀⠀⠀⣼⡇⠠⠀⢻⣮⡻⣆⠀⠂⠀⠒⢺⠀⠁⠀⠀⠉⠀⠀⣀⡀⢀⣠⡌⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⠛⠛⢻⣿⣿⡟⠋⠁⠀⡀⠀⢰⠃⠀⡀⠀⠀⠀⢠⣿⣧⠀⠀⢸⣯⣿⣿⠀⠇⠀⠀⢸⠁⣄⢀⣤⣾⣷⣿⣿⣝⠿⣿⣿⠟⢉⣭
⠀⠀⠀⠀⠀⠹⣿⣶⣿⣿⣿⣿⡀⠃⣉⣿⣏⠀⠀⠀⣰⠀⠀⠆⠀⡄⠀⠀⠀⠀⠘⣽⠉⠀⠀⠘⣷⢿⡏⠀⠠⣔⢠⣷⣷⣿⣾⣿⣿⣿⣿⠟⠉⢠⣸⡇⢠⣾⠛
⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⠻⢿⣿⣿⣿⣿⣿⠄⠀⢠⡗⠀⠀⠀⣾⠇⠀⠄⠀⡠⠀⡏⠀⠀⠀⠀⣿⢸⡷⠀⠀⢹⣿⣿⣿⣿⣿⡏⠁⠀⠙⣿⣷⣘⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠘⢛⠿⣿⠋⠁⠀⠀⠸⠷⣶⣦⣾⡟⠀⠀⠀⡈⠃⢈⡃⠀⠀⠀⠀⠙⠃⠁⠰⣤⣿⣿⣿⣿⣿⣿⣧⡄⠀⣠⣿⣿⣿⣷⣴⣦⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠰⠎⠀⢠⡝⠓⢦⣀⣤⣿⣧⣤⣆⣈⡀⢀⠘⡁⠀⠀⣄⢀⠀⢠⣴⡖⢿⣿⣿⣟⠛⠙⣯⣹⢷⣴⡛⣿⣿⣿⣿⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠄⠁⠸⠚⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠇⠉⢉⢁⠐⠈⠐⠻⣼⠿⣿⡏⢿⣿⣿⣿⣿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣉⡛⠛⢋⣹⠚⣛⡿⠏⠙⠛⠋⠉⠨⠅⠀⠀⠀⠀⠋⠈⠀⠀⠀⠂⠁⠈⠀⢀⠛⣿⣿⣿⣿⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⣉⣥⠔⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠉⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣁⣭⣽⣧⣬⣭⣥
⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢀⣀⡀⣆⣀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢀⠀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠄⠋⠁⠀⠀⠀⠀⢀⣠⣴⣶⣾⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠉⠑⠠⢀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠔⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⡀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠠⠠⠤⠤⠤⠀⠐⠂⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣛⣛⣋⣆⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠟⠟⡟⠻⢻⠁
⠀⠀⠀⠢⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⢄⡘⢦⡽⣄⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣻⣿⡏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡴
⠀⠀⠀⠀⠉⠲⡟⠛⠻⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⢯⣋⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢀⡔⣋⡴⠋⣠⠔
⠀⠀⠀⠀⠀⠘⢼⣷⣛⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⣷⠛⣛⢓⡇⢀⡀⠀⠀⠀⠀⠀⠀⠹⣶⣻⡯⢻⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⢤⡀⠀⠀⠀⠦⢸⣿⣿⣮⡽⡿⡙⣿⣏⣾⣿⣿⢠⠆⠀⠀⠀⠀⣿⣿⣿⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂⠀⠀⢸⣝⣞⣻⡽⣿⣟⣺⣿⣿⣽⣛⣟⣶⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡶⢾⣟⣁⢀⣘⣿⠦⢿⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⡖⠉⢱⣿⠿⠿⢿⣷⠉⠑⡜⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡳⣶⡫⣔⣫⢭⣳⢽⣲⠴⣣⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣟⣋⣱⢞⡝⣮⣉⣛⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢹⠹⡟⢟⡿⠽⣻⠽⡯⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡰⠛⠦⣭⣭⡥⠞⢳⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠖⠒⠒⠒⠒⠒⠮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⡆⠀⠀⠀⠀⢿⡲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⣨⠷⠀⠀⢴⡶⠦⣷⣞⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠉⢻⠷⠬⣷⣄⡀⠈⣷⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠸⡄⠀⠀⠈⠻⣍⡿⣗⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠙⠲⢤⣙⠲⠤⠤⠶⠋⠓⢮⡧⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⣴⡋⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡞⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡴⠋⢀⡴⢛⣉⡁⠉⠙⠳⣄⠀⠀⠀⣰⠷⢤⣼⠁⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡾⠁⣏⠀⠹⣤⠤⣄⠸⡆⠀⢠⡏⠀⢠⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣧⠀⢸⡄⠀⠀⣀⡼⢀⡇⠀⡼⠀⢀⡞⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⠀⠀⠀⠀⠀⠘⢦⡀⠓⠛⠉⠁⢀⠞⠀⣸⠃⠀⣼⠁⠀⢰⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⣤⡀⠀⠀⠀⠉⠓⠒⠒⠋⠀⠀⢠⡿⣤⡴⠃⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡼⠳⣄⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠀⢀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡼⠁⠀⠈⢙⡦⣤⣀⣀⣠⠏⠀⠀⢸⠀⢸⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡞⠁⠀⠀⢠⠟⠀⠀⠀⠉⠉⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢀⡞⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠹⣄⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⡄⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠐⠀⠄⡆⠐⠂⠀⢆⠀⣀⠄⡂⠀⢐⠀⠀⢄⠀⠀⡆⢠⠀⠄⡔⠐⡄⣆⠄⢐
⣀⣀⣀⣀⣀⣀⣀⣡⣄⣠⣤⣂⣇⣀⣀⣌⣈⣨⣤⣇⣤⣄⣠⣤⣀⣤⣝⣀⣇⣰⣈⣆⣣⣤⣁⣇⣣⣨⣤⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀⣴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⢠⡦⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴⣿⣯⣿⣗⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣴⣿⣺⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⣿⣿⣿⣧⣔⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⣇⢨⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣦⣴⣿⣿⣿⣿⣿⠿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣧⣶⣶⣟⣢⣼⣴⣮⢌⣹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠄⢁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣍⠀⠀⠀⠀⠀⠉⠛⠛⡿⢿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣯⣿⣿⠻⠏⠀⠀⢨⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠙
⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⢈⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠇⠄
⠀⠀⠀⠀⠀⢰⣽⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⣸⡅⠀⠀⠀⢘⣠⣦⠀⠀⠀⠀⠉⢼⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣾⣿⣶⣤⣴⣿⣿⠁⠀⠀⠀⠀⢈⣿⡿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⣾⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠠⢼⣿⢿⣿⣿⠻⣤⠄
⠀⠀⠀⠀⠀⠈⢙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⠟⢿⡟⣿⣽⣿⣿⡿⣃⠿
⠀⠀⠀⠀⠀⠀⣠⣀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣯⣧⣽⣿⡾⢹⣷⣿⣿⣿⣽⣷⣌⠁
⠀⠀⠀⠀⠀⢩⠀⡋⣰⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⠇⢿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠘⡄⣷⢻⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣏⡁⡄⢀⣶⣾⣿⣿⣿⣿⣻⠿⣿⡅
⠀⠀⠀⠀⠀⠀⠐⡦⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠓⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣫⠄⡉⢻⣿⣿⣿⣿⣿⣿⣿⣤⣯⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠻⠉⠉⠛⠉⠉⠸⣿⢻⢿⣿⡟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⣀⡀⠀⣀⡀⠀⣀⡀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⠛⠃⠈⠛⠃⠀⠛⠃⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⠀⣠⣾⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⣀⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣟⣠⣴⡾⠟⠁⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣠⣴⠿⠋⠡⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠈⠻⢷⣦⡀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢠⣿⠋⢀⣴⡾⠿⠻⠿⠗⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⣶⣄⠀⠹⣷⡀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠿⠃⠀⣈⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣋⣀⠀⢻⡇⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣇⣤⣶⣿⠿⠿⠟⠻⠿⢿⣶⣄⠀⠀⠀⠀⠀⢠⣖⠟⠟⠛⠛⠛⠿⢿⣿⣶⣄⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠘⣿⣿⠏⠁⠀⠀⠀⠀⣀⡀⠈⢻⣷⣶⣶⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡏⠀⠀⠀⠀⠀⣼⣿⣿⡆⢸⣿⠏⠉⠉⢿⡿⠀⣾⣿⣿⡆⠀⠀⠀⠀⠘⣿⡇⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢸⣷⠀⠀⠀⠀⠀⢻⣿⣿⠃⢸⣿⢀⣀⠀⢸⣿⠀⢻⣿⣿⠃⠀⠀⠀⠀⢠⣿⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⣿⣧⣀⠀⠀⠀⠀⠈⠀⣠⣾⠏⠈⠻⠃⠈⢿⣦⡀⠈⠀⠀⠀⠀⢀⣠⣿⣇⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣿⣶⣶⣶⣶⡿⠟⠃⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣶⣾⣿⡿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠂⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣦⣤⣤⣤⣤⣶⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⣿⣿⣿⣿⣿⣿⣽⣿⡁⣈⣥⣄⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⣿⣿⣿⠿⠻⣿⣇⣿⢻⡿⠟⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣧⣬⣿⣥⣿⣿⣿⣿⣧⣿⡿⠿⠿⣿⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣵⣿⣿⢏⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡿⡟⣺⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣧⣤⣤⣭⣤⣿⣷⣿⡏⣬⣿⣿⣿⣿⣿⣿⣾⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⢿⣿⣿⣿⠿⣿⣿⣿⣿⡽⣿⠿⠿⢿⣿⡟⣛⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⣏⢭⣾⣇⠰⢿⣿⣿⠆⢓⣛⠀⢀⣠⣼⣿⢯⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣭⣭⣥⣶⣶⣀⣀⣿⣿⣿⣷⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡉⡏⠁⠀⢉⣹⢻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣻⣟⣛⣛⣻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠛⠛⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⠋⠀⠀⢀⣴⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠁⠀⠀⣼⣿⡿⠁⠀⢀⣴⣿⡿⠟⠁⠀⠀⢀⣠⣶⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⣼⣿⠟⠀⠀⣠⣿⡿⠋⠀⠀⣀⣴⣾⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⢰⣿⡿⠀⠀⣸⣿⠏⠀⢠⣾⡿⠋⠀⣀⣴⣾⠿⠛⠉⠀⠀⣀⣀⣤⣴⣶⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⣿⠇⠀⢰⣿⠏⠀⣰⣿⠏⠀⣠⣾⠿⠋⠁⣀⣤⣶⣾⠿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⡀⠀⠀⢸⣿⣇⠀⠀⣿⣿⠀⢀⣿⡏⠀⣼⡟⠁⣠⡾⠛⢁⣤⣶⠿⠛⠉⣁⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⣰⣿⠀⠀⠀⣿⣿⠀⠀⣿⡏⠀⣸⡟⠀⣼⠟⢠⣾⠋⣠⡾⠛⣉⣤⡶⠾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢿⣿⣆
⠀⠀⠀⢿⣿⣇⠀⠀⢻⣿⠀⠀⣿⡇⢀⣿⠃⣸⡏⣠⡟⣡⡾⢋⡴⢟⣩⡤⠶⠞⠛⠛⠛⠛⠛⠛⠿⠿⢷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⢿⣿⡆⠀⠸⣿⡇⠀⣿⡇⢸⡿⢠⡿⢠⡟⣰⢋⡴⣫⠶⣫⠵⢖⣛⣋⣭⣉⣉⣛⠛⠷⢶⣦⣄⣀⠉⠙⠛⠿⣿⣷⣶⣤⣀
⢀⣦⠀⠀⠈⢿⣿⡀⠀⢿⣷⠀⢻⡇⢸⡇⢸⡇⡾⢰⢃⡞⡴⣡⢊⠵⣪⠭⣶⣒⡲⠶⠬⣍⣛⠳⢦⣄⡉⠛⢿⣶⣤⡀⠀⠉⠛⠿⣿⣿⣶
⢸⣿⣷⡀⠀⠈⢿⣷⡀⠈⣿⡆⠸⣷⠸⡇⢸⠁⡇⡟⣼⣸⢱⢱⢣⢊⢔⡫⠭⣛⠭⢭⣓⠦⣍⠻⢦⡉⠻⣶⣄⠈⠙⢿⣷⣤⡀⠀⠀⠉⠻
⠀⠙⢿⣿⣆⠀⠈⢻⣷⡀⠘⣿⡀⢻⡄⢿⠸⡇⣇⢇⢧⢻⢠⢠⠉⠀⠪⠊⠥⠶⠭⣕⢮⡑⢮⡻⣦⡙⢷⣄⠙⢿⣦⡀⠈⠻⣿⣷⣄
⠀⠀⠀⠻⣿⣷⡄⠀⠹⣿⣄⠘⣷⡈⢷⡘⣇⢳⡸⡜⣌⢮⡣⠁⠀⠀⠀⠀⠑⡪⢝⠢⡱⡝⢦⢳⡌⢷⡌⠻⣦⡀⠻⣿⣆⠀⠈⠻⣿⣿⣦
⣷⣤⡀⠀⠈⠻⣿⣦⡀⠈⢿⣦⡈⢿⣌⠳⣌⢧⡱⣜⠮⣓⡨⢀⠀⠀⠀⠀⢀⢮⢢⢳⢹⡸⡌⣧⢹⡄⢻⡄⠹⣷⡀⠈⢿⣷⡄⠀⠈⠻⣿
⠻⣿⣿⣦⣄⠀⠈⠙⢿⣦⣄⠙⢿⣦⡙⢷⣝⠳⣝⠢⢝⣒⡒⣒⡨⢂⢂⢆⢈⢘⡆⡆⡏⡇⣷⢸⡄⣿⠈⣿⡄⠹⣿⡄⠀⠻⣿⣦⠀⠀⠈
⠀⠀⠉⠻⢿⣿⣦⣄⠀⠉⠻⢷⣤⡈⠻⢦⣍⡓⠦⣍⣒⣺⠯⣖⣚⠵⣫⢞⡜⡼⣸⢸⢣⡇⣿⢸⡇⢹⡇⢸⣷⠀⢹⣿⡄⠀⠹⣿⣷⡀
⠀⢀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣀⠉⠛⠷⣦⣬⣙⡓⠶⠦⠭⠭⠵⣒⣫⠵⣫⢞⡵⢣⡟⡚⠳⠟⠸⠇⣸⣧⠸⣿⡆⠀⣿⣷⡀⠀⠹⣿⣿⣆
⠀⠘⣿⣷⣦⣤⣀⡀⠀⠈⠙⠛⠿⢷⣶⣤⣌⣉⣉⣛⣛⣛⣛⣩⡭⠶⣋⡵⢞⣴⢋⣼⢃⣚⢃⣿⠃⢛⡛⠀⣻⣏⠀⢸⣿⣧⠀⠀⠹⣿⠏
⠀⠀⠈⠛⠛⠿⠿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣉⣉⣉⣉⣉⣉⣭⣴⠶⠛⣩⡴⠟⣡⡾⢁⣾⠃⣸⡟⠀⣿⡇⠀⣿⣿⠀⠀⢿⣿⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⢉⣉⣡⣴⡶⠟⢉⣤⡾⠋⣠⡿⠃⣰⣿⠁⢰⣿⠃⠀⣿⣿⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣶⣶⣶⡾⠿⠟⠛⠉⣁⣤⣶⠟⠋⣠⣾⠟⠁⣰⣿⠃⠀⣾⡿⠀⠀⣿⣿⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⢀⣀⣤⣶⣿⠟⠋⠁⣠⣾⡿⠋⠀⣴⣿⠃⠀⣸⣿⠇⠀⢀⣿⣿⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣿⡿⠟⠛⠉⠀⢀⣴⣾⡿⠋⠀⢠⣾⡿⠃⠀⢰⣿⡟⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠿⠟⠛⠉⠁⠀⠀⣀⣤⣾⣿⠟⠉⠀⢀⣴⣿⡟⠁⠀⢠⣿⣿⠁⠀⠀⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠋⠁⠀⠀⣰⣿⣿⠋⠀⠀⢠⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠸⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⡟⠃⠀⠠⣶⠶⠾⠂⢭⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢠⣝⣂⠙⠿⢿⠄⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠘⣿⣿⣿⣦⡐⠀⢿⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣄⡀⠉⠉⠉⠀⠙⠳⠷⠀⣠⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠐⢻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢋⣴⡿⢛⣩⣭⣭⣙⠻⣿⣿⣿⣿⠏⠿⣿⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣰⠋⣤⡍⢿⢿⣷⡜⣿⣿⡟⣸⣷⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⣿⡆⢿⣿⣶⣶⠌⡇⣹⣿⢡⣿⡟⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⠹⣿⣘⣋⣭⣴⡾⢣⣿⠇⣾⣿⠁⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣮⣙⣛⣛⣩⣴⣿⡟⠸⠿⠇⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⣈⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⠀⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⣴⣿⣷⡮⠙⠻⠿⢿⠟⠀⠀⠀⣿⠀⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⢋⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠺⠂⠀⢀⣤⣤⡀⠈⢀⣴⣤⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⠆⠀⠀⣿⣏⢈⣿⣶⢿⣿⣿⣇⠀⠀⠀⠀⠀⠾⠇⠀⡀⠀⢠⡄⠀⢀⡀
⢠⣶⡄⠀⣨⣿⣿⣿⣿⡿⢿⣿⣿⣿⣷⣀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣤⣤⣄⡀⠀⠀⢰⣿⣿⣷⣤
⣾⣿⣷⣶⣿⣿⣿⣿⡟⠀⠀⠹⢿⡿⠛⠻⠿⠿⢿⣿⣦⣄⣠⣤⣀⣀⣠⣾⠿⠛⠻⣿⣦⣴⣿⡿⠟⠛⠿
⣿⣿⣿⡟⠋⠻⠿⠏⠀⣠⣶⣄⠀⠀⢀⠀⠀⡄⠀⠉⠛⠛⠛⢿⣿⠟⠁⠀⠀⢀⣤⣄⠈⠉⠉
⡙⠛⠈⠀⢠⡄⠀⠿⠀⠻⠿⠏⠀⣤⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠿⠟

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⣿⣿⠀⠀⠀⠀⢰⢶⣶⣶⡀⢀⣀⠀⠲⣶⠶⣶⠶⠦⠤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⢸⣿⣿
⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⡛⢃⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠛⠛⢿⡿⠿⠛⠻⠛⠛⠻⠿⠟⠿⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⣄⠀⢸⣿⣿⣿⣿⣯⣤⣀⣠⣤⣤⣤⡄⠿⠧⠀⠀⠀⠀⣤⣴⣦⣤⣤⣾⣿⣿⡍⠀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠘⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡖⠰⠠⠉⠀⠀⠈⠉⠉⠛⢻⣿⣿⣿⣿⣧⡄⠀⠀⠀⢸⣿⣿
⣿⣿⡇⢤⣈⣻⣿⣿⣿⣿⠿⣿⣿⣿⠿⠏⠁⣉⣧⣴⣶⣿⣿⣶⣦⣤⣶⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠙⢻⣿⣿⣿⣿⣿⡿⠵⠆⠀⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⣀⣻⡟⠋⠉⠁⠀⠀⠀⠀⢀⣀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠈⠙⢻⣿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿
⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣽⣾⣿⠆⠀⠀⢠⣄⡀⠀⠀⠀⡇⢸⣿⣿
⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⢛⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⣇⢸⣿⣿
⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿
⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠃⠀⠛⠛⠃⠀⠀⠀⠀⠀⣿⢸⣿⣿
⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠈⠛⠛⠛⠛⠛⠛⠛⢻⣿⣟⣸⣿⣿
⣿⣿⠀⠀⠂⠀⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠤⠦⢀⣼⣷⢿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⡀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠆⠀⠀⠀⠀⠄⢠⠀⡀⠀⡄⢸⢀⢏⣾⢇⢻⣼⡎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷
⠀⠀⢠⣼⣧⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠠⢂⠀⠘⠸⢹⠇⡀⠚⠀⠈⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣄⠀⠸⣿⣯⢆⠙⠹⣄⠀⢀⠀⢀⠀⠀⠀⠀⢰⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⢀⠀⠏⠃⠁⠀⠀⠀⠁⠀⠀⢰⣿⠿⠶⠄⠴⠖⣶⠀⠀⢀⠃⠀⠀⣴
⣿⠐⠆⡹⣿⣾⣇⢣⠸⣂⣄⡿⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢿⣦⠀⠀⠀⠀⠀⢤⣶⣦⣄⣧⠀⣀⡀⣀⠞⠁⠀⠀⠈⢧⡠⣼⣿
⣿⡆⠣⡄⠙⣿⣿⡞⣧⡘⣌⠴⢖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠀⠀⠀⠀⠀⠀⣿⠄⠀⣹⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⢳⣸⣿
⣿⣿⣦⡈⢦⡈⢿⣿⣦⡄⣿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣏⠀⢀⣱⣿⡴⢀⡴⣊⠀⠀⠀⡌⠀⢀⡿⠀⠀⠀⠀⣄⠀⠀⢀⡆⠀⠀⠙⣿
⣿⣿⣿⣷⣎⢿⣷⣿⡷⢾⣿⣻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠆⠈⠁⠃⠀⠀⠀⢸⣥⠞⠋⠀⠀⠈⠁⠶⠟⠉⠀⠘⠛⠏⠁⢀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣤⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠐⠒⠆⠀⣤⡶⠤⢼⣆⠖⠀⠾⡍
⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⡅⣾⣣⠂⣀⢠⣾⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠒⠻⢆⠠⠙⠛
⠸⣯⡻⣿⣿⣿⣿⡻⢿⣿⣟⡁⢳⡝⠂⠽⢿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣦⣤⣤⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠛⠀⢀⠀⠀⠀⠀⢀⡀
⠜⠘⢿⣮⣻⣿⣽⡿⣴⠊⢿⣷⢿⡏⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠹⡿⠉⣿⠟⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀
⠀⠀⠀⢙⣿⣿⣿⣁⡈⠆⠘⡿⢘⣷⡀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⡰⢇⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠠⠿
⠂⠀⠐⠉⠀⠘⢿⣧⡙⣿⢿⢆⣼⣿⠇⢠⡟⠁⠀⠀⠀⠀⠠⠄⢸⣟⣿⡋⠒⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣷⠀⠀⠠
⠀⠀⠀⠴⣣⠀⠀⠙⢻⣦⢳⡚⣶⣿⣿⣁⡴⢞⠉⠀⠀⠀⠀⢠⢋⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣯⠻⣿⣿⣯⠞⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣀⠶⡀⡀⣄⣀⠀⠙⢷⡝⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⡴⠀⠀⣿⣦⡀⠀⠀⠠⠖⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⣄⡠⠄
⠀⠀⠀⠈⣿⣷⡄⠺⠋⠉⠀⠀⠀⠻⠉⠀⠈⠇⣴⣄⡆⣠⣶⠞⠀⠀⢸⠿⠟⢁⠠⠆⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡏⠀⣀⠀⢠
⠀⠀⠀⠀⢻⣷⡘⢿⣆⠀⠀⠈⢀⣀⠀⢠⠴⢗⣿⣾⡿⢡⣿⣤⡀⢀⣼⡤⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢨⣿⣡⡠⢟⣀⡤⠤⠤⠆
⠀⢠⣤⠀⠘⠿⠆⠀⠘⠳⠀⠰⣯⣿⣿⣿⣿⣯⣷⡜⠃⣀⣄⣼⡟⠚⠁⣡⠤⣄⡙⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠜⠘⢿⣿⠯⡽⡆⠀⡆
⢀⡘⠙⠀⠀⠀⠀⠀⣀⣴⣦⣼⣿⣿⣿⡏⣹⡏⢸⣧⣸⡿⣷⣤⡽⠶⢾⡱⠷⣄⡁⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⠀⢈⣶⣤⣷⡄
⢺⣿⠗⠀⠀⠀⡀⢀⡹⣿⣿⢿⣾⣿⣧⠠⣾⣷⣯⣽⣿⣳⣿⣿⣧⣾⣿⣷⡶⡟⠇⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠸⣆⡞⠛⠏⠰⣷
⠀⠈⠀⠀⠀⠀⠉⢰⡾⠛⠳⠿⣏⣿⣥⡔⠛⣿⣦⣾⡛⣷⠂⣿⣿⣿⣿⠯⣷⡀⢠⡃⠀⠸⣶⠴⠂⠀⠰⠷⠐⠃⠁⠐⠃⠆⠀⠀⠁⠀⠀⢣⡅⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣘⢿⣯⠳⣾⣍⣭⣤⣿⣿⣷⢬⣙⠟⢻⣽⣿⣁⠀⡿⠤⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⠀⣴⣿
⠀⢀⠀⠄⠀⠀⠰⠤⣤⣀⣀⠀⠉⠀⠻⣄⢉⣹⣿⡟⣻⣛⣋⣤⣙⣦⡀⠙⠋⠈⠀⠀⠀⠀⠀⠀⣠⣴⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣯⡀⠙⠋⠁
⠀⠀⠀⠀⠀⠄⠀⢀⡀⠉⠍⠻⡓⠈⣠⠼⣟⣿⡿⣿⣿⣿⣿⣿⣿⣻⣿⠏⠀⠀⠀⢀⣤⣤⣦⠘⠙⠻⡾⠋⠀⠀⠀⠀⠘⠶⣦⣶⣾⣶⣤⡀⣿⠀⠀⣂⡖
⡠⡇⠀⠠⢀⡄⠀⠀⠀⢀⣰⡿⠋⠀⣡⠞⠉⠀⡴⢿⠟⠉⠉⠉⠉⠉⠈⠂⠀⢀⡐⣿⣿⣿⣿⣇⠀⠀⠀⣦⣤⡤⠄⠀⠀⠀⠈⠻⣿⣿⣿⣷⣿⡀⠀⡇
⠀⠀⠀⠀⠚⠐⠀⠀⣠⣿⠟⠀⠀⠐⠁⠀⠀⠞⠁⠉⠀⣔⠀⠀⠀⠀⠀⠠⠀⠀⣁⣸⣿⡟⣿⡟⠀⠀⣼⣿⣭⣤⣄⣀⡀⠀⠀⠀⠈⠻⣿⣿⣿⣷⢦⠁
⠀⠨⠀⠂⠀⠀⢀⣰⠟⣱⡇⠀⠀⠀⠀⡀⠀⢄⣀⡀⡰⠏⠀⠀⣁⠀⠀⠀⠀⠀⠉⠻⠁⢸⡏⠁⠀⡿⡀⠀⠉⠉⠉⠉⠉⠉⠒⠀⠀⠀⠘⢏⠁⠱⣼
⠀⠀⠀⠀⠀⠀⠀⠟⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢍⡹⠁⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⡌⠀⠀⠀⢷⣌⡱⠀⠂⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⢂⠀⢹⣄
⠀⠀⠀⠀⠀⣤⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠟⠀⢠⣾⡔⢠⣾⢻⡌⢷⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⢸⣿⣿⣧⡀⠀⠈⠄⠀⠀⠀⠀⣹⣿⢀⣤⣶⣾⣿⣿⣷⣶⣤⣀⢸⣿
⠀⠀⠀⠀⠘⠀⠀⠀⡌⠀⠀⠀⠀⠀⣠⣾⣧⢠⣶⣶⣶⣷⣄⠀⠐⡀⠀⠀⠀⢿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⣴⣿⣿⣿⢸⣿⣿⣿⣿⣿⢰⣄⡀⠂⢠⣾⡛⠙⠻⣿⣿⣶⡄⠈⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⡔⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⠏⠠⠠⣿⠋⣡⡀⠀⠙⢿⣿⡄⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢼⣩⡅⣩⣭⣭⣭⣭⢈⣭⣭⣭⣭⣭⡈⠀⢀⠀⠀⠀⠄⠀⠀⢆⢠⠋⠀⠀⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠸⢀⣠⢠⠤⣬⣿⣇⡿⠿⠟⠛⠛⠚⢛⡟⠛⢿⡿⣁⣀⠄⠀⠐⠀⢀⠀⢀⠀⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⠀⠀⠀⠐⢀⠈⢰⡏⠁⠐⡄⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡡⠄⠀⠀⠀⠙⠄⠀⠀⢉⡀⠈⡷⢖⣶⣞⣻⣄⣁⠔⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⠉⠭⣽⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠀⠀⠀⠀⠀⢀⠓⢒⡄⠀⠒⠦⡤⠤⠤⠴⠒⠋⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠒⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⢃⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⠀⠤⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠴⠀⠀⠀⠀⠀⠀⣿⣿⢿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⠠⢲⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣯⣷⡄⢠⡩⠻⠃⠀⠀⠀⣠⠔⠚⠉⠀⠀⠙⠬⠱⠊
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠿⢥⣆⡀⠀⠀⢀⢴⣞⠉⢠⣐⡀⠀⣿⠉⢺⣤⠏⣀⠀⢀⡠⠋
⠀⠀⠀⠀⠈⡳⠉⠀⣢⡄⠀⠀⠸⡀⠀⡈⠀⡇⢷⠳⡄⣰⠿⠟⠅⢀⣄⡙⠙⣶⣿⡠⣾⠏⠘⠁⠀⠀⠁⠀⠤⠀⠲⠲⡽⠈
⠀⠀⠀⠀⠀⠙⠁⠀⡉⢉⢢⠀⠀⢣⠀⣧⠀⡇⢸⠀⢸⣿⡀⣀⠤⢴⡛⢿⣷⣿⣿⣿⣿⠴⢃⣠⠊⠀⠀⠀⠀⢰⠢⠦⡶⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⡄⠀⠈⠉⣹⡃⠀⡈⣧⣼⣀⠧⣾⠄⣯⡟⠓⠋⠹⠭⣭⣭⣨⣿⣿⣿⣇⡀⠈⠀⠀⠀⠀⠀⢈⠼⠷⣿⣻⣿⡿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⣨⣍⢶⣬⣾⣯⠟⠲⢿⡏⠋⠀⡄⠐⣀⠀⠲⣶⢾⠿⣿⣿⢻⣓⣋⠩⠂⠀⠀⠘⠋⢍⡿⠋⠉⠀⠀⠈⠂⣶⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠙⣽⣽⠋⠙⢳⡀⢸⣷⢸⡆⢣⠀⠘⠁⠐⢹⡯⢶⡟⠋⠙⢉⠉⣠⠀⠀⠀⠀⡆⡸⡇⠁⢀⣤⢊⠠⠎⡍⡽⢀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣘⠿⠶⡒⠻⣿⣿⣽⣿⡔⢻⣦⡀⣀⠀⠀⠱⣸⢷⣄⣶⡏⠐⣁⠀⠃⠌⢀⣼⣰⣿⣤⠞⠛⣍⡠⠞⣼⠃⠈⢾⡇
⠀⠀⠀⠀⠀⠀⡶⡦⢀⣠⣄⣾⡃⢀⠶⠶⣿⣿⣿⣿⣿⡿⠛⠀⠀⠌⣭⠎⣹⣼⢿⣷⣿⣎⢚⡡⠂⠀⠀⠀⠀⠙⢯⣠⣤⣼⣯⢶⡿⠃⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢘⣤⠴⠻⠛⠻⠉⠀⠀⡱⢒⣫⠿⣿⣿⢧⣀⡀⠀⠠⠦⣄⣠⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⢀⣠⣾⠿⣡⠭⠄⠚⠟⠀⠀⢀⣀⠀⠀⠀⣌⠃
⠀⠀⠀⠀⠀⢀⡟⠱⠂⠀⠀⠰⢤⣰⡛⠙⠋⠁⠚⠫⢨⠜⠁⠀⠀⠀⠀⠒⠭⣙⡿⣻⡿⠿⠃⠤⣤⣶⣶⢻⢿⡆⠀⠠⠖⠀⠀⠀⠂⠀⠸⠋⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠐⠀⢀⡅⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠁⠀⠁⣀⡘⠁⠁⠙⠀⠀⠀⠀⠙⢷⠀⢤⢳⠄⠀⠀⠀⠀⠀⠀⡔⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣁⠀⠦⣀⣀⠀⠦⠀⠀⡀⠀⠀⠀⠀⡔⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠈⠀⠐⠀⠀⠂⠀⠀⠦⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠉⠡⠀⣀⠨⢧⣜⣿⣄⣀⠀⠀⠀⠤⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠲⠄⠘⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢉⠓⢂⣈⣉⣼⣿⣿⣿⣿⢵⡤⠠⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⣑⠦⣀⠀⠀⠀⠘⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⢄⠿⣏⠈⠙⢛⡿⣿⣿⡿⣿⣆⡅⠈⠁⠀⠀⠀⠐⠀⠀⠂⢤⣄⣤⢴⣾⣧⣤⡴⠾⠥⠈⣆⣀⠑⠢⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡱⠀⠀⠀⠈⠰⣾⣷⡯⠏⣝⢀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠘⠰⠈⠉⠺⡿⣿⡿⠙⠋⠁⠀⠀⠙⣌⠑⠲⠤⣬⠭⢄⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⠌⠀⢈⡍⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠙⠚⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⡀⠈⣷⣀⡀⠀⡈⠐⢲⠖⠒⢲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡿⡀⡼⠛⢏⣿⣾⡇⠈⡿⠈⠀⡾⢡⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡮⢞⠵⠋⠀⠀⠈⢻⣿⡇⢠⠇⠀⡴⠛⢀⠠⡀⠀⠀⠀⠀⢀⠔⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠁⠀⠀⠀⠀⠀⠀⢹⠀⣾⣀⡼⣣⣤⡤⠰⠆⠀⠂⠠⠐⠉⠀⠦⢶⠑⠒
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡅⣫⠞⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⡛⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢀⣿⣿⣿⡟⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⠇⠴⠶⠉⠙⠟⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡷⠈⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⣰⣿⣿⣿⣿⡆⠓⠚⢸⠀⠓⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣴⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀⠀⠦⠴⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠛⣻⣿⣿⣿⣿⣶⠀⠀⣶⣿⣿⣿⣿⣿⣿⡟⠉⠙⣿⢻⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠈⠙⠻⢿⣿⣿⣭⣧⣄⣃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⠃
⠀⠀⠈⠻⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡎⠀⠰⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠇⢠⠁⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠒⠀⠀⠀⡄⠀⠀⠀⡠⡋⠀⢠⣁⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣠⠋⠀⠀⠀⡀⠀⠀⠀⢠⣮⡄⠛⣀⠀⢠⣷⠞⠁⠈⠪⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣦⣄⣈⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⢀⣀⠀⣀⠠⠚⠟⠁⠀⠋⠉⠛⠉⠛⠿⠦⠌⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡐⠁⢸⡇⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⡠⣥⣄⡀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠀⠀⣼⡇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢿⣻⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠇⠀⠀⢰⡏⠀⢠⡇⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⣿⡇⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡀⠀⢸⠃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠇⠀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠈⣦⠀⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⠃⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠙⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠠⣸⠃⠀⠀⢀⢀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈⡄⠀⣿⢦⠀⠀⠀⠀⠀⠀⠀⠈⢇⠜⠁⠀⠀⠀⢀⡌⢀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢹⡜⠳⣀⡀⢀⡠⠀⠀⠘⠁⠀⠀⠀⠀⠀⢿⡿⠫⢔⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀⡴⡸⣼⠃⠀⣇⣴⣉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⢆⠗⣎⢀⣄⣿⡉⡁⠈⠧⠜⣀⠀⠀⢀⣤⣠⡶⣺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢮⠔⢹⣟⣟⢁⠈⠈⢋⣑⠟⣁⠽⠭⠩⠤⠿⠾⠻⠙⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢇⡄⠞⢻⠀⠃⠀⠀⠈⠁⠈⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠁⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⠉⢹⠂
⠀⢠⠖⢦⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡛⠛⡇⠀⠀⡞⠉⠰⠿⡄
⣠⠞⠛⠉⠛⡏⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠾⣇⠀⠀⠛⠛⠛⠛⠁
⢷⣄⣠⡤⣴⠥⡴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠖⣦⠀⠀⠘⠒⠒⠒⠋
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⠴⢧
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛
⠀⣿⠀⠀⢸⠀⡇⠀⠀⣠⡄⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⣤⣤⣀⡀
⠀⣿⠀⠀⢸⠀⣇⣀⣰⠃⠸⠋⡰⠟⢛⣧⣠⡞⠉⠙⠋⡉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠉⠓⢦⡀
⠀⣿⠀⠀⢸⢠⠋⠘⠉⠁⠀⠀⠀⠐⠋⢱⡜⠀⠀⠀⡐⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠙⣄
⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣆
⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠈⢀⠇⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠶⠤⢤⣀
⠀⣿⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⣿⠀⠀⢸⠈⢢⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢠⣞⡀
⠀⢻⠀⠀⢸⢰⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠈⡅⢹
⠀⣿⠀⠀⢸⠸⠤⠤⠤⠤⠤⠤⠤⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢸⠀⠄⠀⡧⠼⢶
⠀⣿⠀⠀⢸⢤⣤⣤⣤⣤⣤⣤⣤⣬⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⣣⠀⣠
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⠀⢀⣸⣀⣀⣀⣀⣀⣀⡀⠀⠀⣇⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⣿⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⢸
⠀⠙⢶⡶⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⠞
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠂⠽⠶⣶⣦⡭⠉⢉⣉⠛⠉⣀⣈⣛⣈⣉⣠⣤⣾⣷⣽⣦⣬⣭⣉⣀⣠⣶⠶⣲⣾⣻⡓⠿⢿⣴⣶⣤⣤
⣭⣛⣧⣤⣤⣤⣤⠈⣉⡴⠛⠉⠀⢀⣒⣉⣹⣟⠉⠉⣽⣿⣿⢿⣿⢿⡷⠯⠛⠓⠈⠈⠛⣙⣣⡔⢶⠏⠹
⠮⣭⣭⣭⠍⣀⣠⣬⣽⣿⠽⢭⣭⢩⠁⠛⣻⣟⠉⠙⠉⠉⠁⠉⠃⡄⠀⣠⣁⣰⣆⣤⡤⠴⣶⣧⣉⣐⣀
⣉⣉⣀⣿⣿⣥⡴⠟⠿⠿⠶⠾⢿⠏⠀⠀⠙⢻⣷⣦⡀⠀⠀⠀⣀⣀⠀⠥⠌⣁⣙⡻⠶⢖⣛⣻⢏⡉⢴
⢙⣿⣯⠁⠀⠀⠀⠙⠛⠒⠶⠦⠤⠀⠐⠒⠞⠛⠉⠉⠉⠀⠐⣛⠻⢿⣤⣤⣶⠟⢃⣀⣤⠶⢞⣫⣤⢈⡵
⠀⠻⣷⣄⠀⠀⠰⣶⣶⣾⡿⠿⠛⣛⣃⣀⣤⣤⣤⣤⣤⡤⢤⣭⡄⢈⣻⣛⠛⣛⠋⢭⡴⠞⠋⠉⠤⠷⠞
⢄⣄⢀⣙⣷⡦⠀⠀⣤⣶⣛⠛⠛⠛⠛⢿⣿⣿⣿⠷⢖⣒⣺⣭⠴⠿⠶⠛⠛⣻⣥⣴⣿⣶⡶⠒⠚⠻⠿
⣤⣬⣷⡦⠉⣤⡶⠶⠦⢬⠭⢭⠭⣽⣛⣛⣛⡉⠁⣀⠭⠷⠶⠶⠶⠶⠚⠻⣿⣿⣿⣿⣿⣿⣷⣶⡶⠚⢲
⠀⠈⠉⢁⣤⣬⣍⣀⣂⣘⠧⠤⢄⣀⣒⣀⣩⣴⣿⣁⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⣉⡁⠀⠘
⠀⠀⠰⣿⣿⣿⡉⠉⠉⠉⠉⢙⣿⣿⣿⣿⣿⣯⣭⣭⡿⣿⣇⠀⠀⠀⢀⠈⠉⠛⠁⠀⠑⠀⠸⠁⠀⠘
⠶⠦⠄⠈⠉⠛⢛⣿⣶⠶⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⣠⣴⣿⣿⣏⣉⣛⣛⣻⣿⣿⣿⣷⣿⣿⣷⣶⣾⣷
⠒⠒⠒⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣤⣤⡴⠶⠟⠛⠉⠉⠉⠉⠉⠩⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢦⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡉
⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠶⠶⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⠸
⠀⠀⠀⠀⠀⠉⠻⣿⣿⡃⠀⠀⠀⠈⠉⠈⠀⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣻⣿⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠀⠀⠀⠒⠒⠒⠛⠛⣛⣿⣿⣷
⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠚⠿⠷⠶⠶⠃⠀⢀⣠⡤⣤⣴⣿⠟⠋⠈⠁
⠉⠛⠛⠻⠶⢦⣤⣤⣤⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠶⠶⠶⣶⣶⣶⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⠿⠟⠛⠁⠀⠀⠀⣀⣴⣿⣿⣿⡃
⣀⣀⣀⣀⣀⣀⣀⣤⣶⠶⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⣀⢠⣤⣴⣶⣶⣶⣦⣄⣀⣀⣀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡴⠊⣇⠑⢄
⠀⠀⢀⠤⠒⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢆⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢣⢂⣀⣁
⠀⠎⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⡄⣀⢈⣄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⠀⠀⠀⠠⠤⠤⠀⠀⢀
⢰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⠀⠩⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠖⡀⠀⠠
⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠁⠀⢀⠔⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢈⡴⢨
⠀⡄⠀⠀⢹⣿⣿⣿⣿⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣠⣶⠟⠃⠂
⠀⠘⡄⠀⠀⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢊⣴⣾⠟⠁
⠀⠀⠈⠢⡄⠀⠀⠹⣿⣿⣿⣆⠀⠀⠈⠀⠀⠀⠀⡠⠃⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣾⡿⠊
⠀⠀⠀⠀⠸⠒⠠⠀⠀⠉⢩⠍⠋⠖⡀⣶⡒⠒⡘⠁⠐⠀⠶⣓⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⠀⠀⢰⠀⢠⠄⡄⠀⠀⠀⡜⠀⠀⠩⠄⠀⢀⠀⠀⠀⠀⠉⠉⢀⠒⠄⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢰⠀⢸⣴⡾⡄⠀⠀⠹⣶⣦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠆⢁⡠⡆⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠻⡀⠀⠔⠛⢿⣿⣿⣿⣿⣿⣿⣿⡖⠒⢶⣻⣿⠿⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣈⠩⠀⠉⠉⠛⠛⠿⠿⡇⠀⡟⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⡀⠀⠀⠀⢀⠀⠀⠀⠗⠀⠁⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢁⢀⣙⣀⣂⣶⡟⠦⢄⢤⡀⣀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⢭⣋⣉⣻⡿⠘⠋⠁⢠⣶⣯⣨⣿⣶⣄
⠀⠀⠀⠀⢀⡠⠉⠂⠀⠈⠃⢀⡄⠀⠴⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡹⠉⠢⣿⣎⠛⣿⣾⡧⠾⢿⣟⠻⣟⠻⣝⣿⣯
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡒⣛⣈⣉⣰⠿⠭⠽⠦⠿⠷⠶⠶⠾⢿⣿⣆⣉⣉⡁⠲⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣿⣟⠑⠲⣦⣤⣤⣄⡀⠀⠀⠒⣢⠀⢤⣬⣭⣭⣍⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠙⠻⠿⣿⣿⡿⣷⢦⡍⢻⣿⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠙⢉⡀⠨⣟⠉⣀⣠⡴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⣠⠄⣠⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⠛⣿⠲⣿⣿⣿⣿⣶⣷⣤⣤⣦⡀⠀⠀⠀⠀⠀⠀⠬⠆⠇⠸⠀⠛⠁⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣶⣾⣷⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⣿⣿⣟⣉⡀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠟⠛⠋⠉

⠀⠀⣀⣤⡶⠶⢶⣤⣄⣴⠾⠛⠛⠿⢶⣦⣄⣠⡶⠾⠛⠷⣶⡶⠾⠟⠛⠻⢶⣤⣤⣤⠶⢶⣤⡀
⠀⣼⠟⠁⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠻⣷⣄⣀
⣸⡏⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⣦⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠘⢿⣄⠀⠀⠀⢀⣤⣄⠀⠀⣀⣴⡾⢷⣄⣀⣀⣀⣠⣴⠾⠛⢷⣄⣀⠀⢀⣀⣠⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⣿⠷⠟⠋⠁⠉⠛⠛⠛⠋⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⠤⠀⠀⠀⠀⠀⠀⢺⣏
⠀⠀⠀⢸⣿⠀⢀⡴⠚⢶⡀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⢸⡏⠀⣼⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⢸⡇⠀⠅⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡿⠀⣤⣤⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢨⡿⠃⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⣾⡇⠀⠀⣿⡷⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠿⠿⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⠃⢰⡆⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⠒⠉⠉⠉⠚⢿⣆
⠀⠀⠀⣿⠀⠸⣧⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⣿⠣⣀⣀⣀⡠⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⠀⠀⠹
⠀⠀⠀⣿⠀⠀⠣⣈⣉⡿⠁⡇⠀⠀⠀⠀⠀⢻⡀⠀⠁⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠊⠉
⠀⠀⢸⣿⠀⠀⠀⠞⠋⢀⡜⠀⠀⠀⠀⠀⠀⠘⢿⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸
⠀⠀⢸⣿⠀⢀⣄⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠀⠀⠀⠀⠀⣼
⠀⠀⢸⣿⠀⠀⠙⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈⠧⣄⣀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⠿⠟⠛⠉
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⡄⠠⠤⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⢰⠣⣤⠤⠤⠤⢀⣀⣀⣀⣀⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠳⠧⠤⠤⠤⢤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠳⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠈⠻⠷⣶⣤⣤⣤⣤⣴⣶⠶⠾⠟⠛⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⣀⡀⠀⠀⠀⣀⣠⣾⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠂⠀⠀⠀⠀⢠⣄⣠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⣤⣤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣴⣾⠋⠇⠳⢠⠏⢭⢿⣄⠄⠒⣀⠀⠦⠀⠀⢀⠀⠀⡀⠀⢠⠄⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣤⠜⠛⠻⢒⡖⠲⠻⢷⣀⠒
⠀⠀⠀⢠⡐⢄⣶⠉⠙⠷⣼⣜⡅⠀⠈⠙⡦⣌⡠⠖⢫⡡⠀⠁⠔⠀⣀⠀⢸⢴⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣯⠛⠛⠶⣼⣜⠄⠀⠀⢠⣉⠅
⠀⠀⠀⡀⢹⣷⣈⢆⠀⠀⠈⣙⣣⣀⠄⢘⡱⠉⠀⢰⣾⣁⠠⡀⠁⡀⣥⡀⢨⢰⣿⠁⠠⠀⠀⠁⠄⠀⡘⢷⣞⡙⣄⠀⠀⠀⣩⣶⣤⣒⢨⢎⡄
⠀⠀⠀⠠⠀⢿⣿⣿⣓⡁⣞⠭⠼⢛⣳⠎⠁⠙⠲⣜⣝⣦⣶⢷⠀⠰⣿⡇⠈⣼⣿⠀⠀⠀⠀⠀⣀⡤⢓⠈⣿⣷⣮⢓⣃⣾⠫⠔⢛⣽⠃
⠀⠀⠀⢢⡄⠘⡼⠔⠋⢻⣷⣄⡠⣪⠃⠀⠀⠀⠀⡘⠩⢒⣲⣼⣧⠀⢻⠀⠀⣿⠋⣰⣶⣶⠂⠘⡁⠐⢤⡂⠸⡉⠀⠝⢻⣷⣤⡐⠹⠁
⠀⠀⠀⢴⣿⣄⠸⢌⣠⣔⡽⣿⣿⣷⣄⡀⠀⠀⢀⣁⢉⣀⠉⣼⣿⡆⠀⠀⠀⠁⠐⢋⡩⠽⠗⠿⠿⣤⣬⣿⣄⡡⣔⣀⣄⡙⣿⣿⡷⣤⡄
⠀⠀⠀⢠⡟⢷⣀⡨⠍⢁⡈⠬⢿⢀⠈⠃⠀⡍⢈⢉⣯⣭⣛⣍⠻⣷⠀⠈⠁⠹⢆⡞⠛⢦⣄⠀⠀⠀⢈⡿⢷⣀⣈⠏⢀⠀⠬⢿⢅⢀⣂
⠀⠀⠀⠘⠀⠂⠁⠀⠈⠀⢀⠀⢰⠀⠁⡀⠀⠃⠘⠙⣻⣿⣿⣿⡷⠈⠁⠀⢢⠠⢀⣬⣀⣀⣬⠟⣠⢀⠼⠁⠘⠉⠀⠈⠀⢀⠀⠠⠀⠁⡀⠀⡇
⠀⠀⠀⢀⠀⡐⢀⠀⠀⠀⠁⠀⠀⠀⠀⢀⢠⠀⠟⠛⠛⠛⠻⠛⠁⠀⠀⠓⢈⡴⡯⠼⡏⠉⠀⢁⡈⣵⡄⠂⠀⠂⠀⠀⠀⠁⠀⠀⠃⠀⢠⠠
⠀⠀⠀⠘⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠤⠤⠄⠀⠠⠀⡀⠀⠔⠧⠄⠠⠀⢈⣦⣊⣥⠞⢋⠞⠑⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠘⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠰⢰⠀⠁⡠⠁⠀⡠⠡⠊⣀⠀⠂⠀⠐⢦⡰⠿⠿⢿⡧⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⢈⠀⢀⠠⠀⢀⠔⣤⢇⣔⠠⣄⣀⢠⢬⢏⠒⠦⣄⡊⠀⠈⢃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠂⠀⠀⢻
⠀⠀⠀⠐⣄⡤⢀⠀⠀⠐⢀⠀⠀⠈⢆⢻⠸⠈⠰⠄⢁⠄⡈⠸⣿⡆⠀⠙⡆⠀⠀⢀⡀⠀⠁⠀⠀⠀⠑⢄⡤⠀⠀⠀⠀⢀⠀⠀⠈⢄⢸⠸
⠀⠀⠀⢸⡟⠘⢦⡀⠀⠐⠁⠀⠐⠀⢘⢻⡇⠀⣶⢀⠂⢰⣶⣦⣝⡧⠤⠚⠋⠲⣴⡶⢶⣤⣀⠐⠀⣠⣶⡿⠰⢧⡀⠔⠀⠀⠀⠀⠄⠸⢻⣷
⠀⠀⠀⢘⠁⠀⠀⡈⠒⠄⣀⠀⠐⠀⢸⡌⣇⠀⣿⠇⢀⣶⣮⠍⠉⢁⠀⠀⠀⠀⠘⡏⠀⠙⣤⣷⠞⣿⢟⠃⠀⠀⣈⠒⠄⣀⠀⠐⠀⢰⡄⣿
⠀⠀⠀⠀⣠⣠⡞⠀⢻⣶⣏⠩⠙⠛⠠⢄⣿⠀⠏⠀⣼⣧⣈⠋⠁⠚⠀⠀⠀⠀⢀⣸⣴⣾⠟⠈⠁⢀⠃⣠⣠⡞⠁⢺⣶⣞⠋⠉⠛⠀⢤⣿
⠀⠀⠀⢰⣿⣿⣿⡟⢻⣒⣫⣧⡢⢎⢠⠖⠈⠀⠀⢰⣿⣏⣛⣭⡭⠉⠃⢃⠀⠈⠉⠉⡿⠑⠲⠶⡤⠃⢰⣿⣿⣿⡿⢻⣄⣫⣧⡂⢞⣀⠞
⠀⠀⠀⣾⡿⠛⠀⢷⢈⢂⠀⠐⠚⣠⡑⢄⣠⠀⠓⣾⠏⣴⣿⣟⣛⠛⠓⠀⠀⠀⠀⠁⠙⠒⠀⠠⠒⠦⣾⣿⠿⠁⢻⠰⣂⠀⠒⠚⣢⡁⢤⠠⠄
⠀⠀⠀⠟⠀⠅⠀⡈⢈⢽⡋⠓⡭⣉⣹⣦⡑⠴⠃⠀⠘⠻⠿⠿⠷⠶⢾⠈⡄⠐⠁⢀⠀⠀⠄⠀⠀⣼⠟⠁⠄⠀⠈⢃⠽⡏⠓⡭⢍⣍⡦⡙⠔
⠀⠀⠀⢀⠀⠐⠀⣜⠷⣔⠺⣄⠀⢀⢀⣀⣩⣄⠐⡀⠀⠄⠈⢀⣉⡀⢀⠀⠗⠃⠀⠀⢀⠀⠀⠀⡼⠋⠀⠀⠀⢀⣔⢿⣗⡻⢦⠀⠀⣀⣠⣬⠦
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠢⠠⡁⠀⠀⠀⡠⠀⠒⠲⠶⢦⡄⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⡈⠀⠀⠀⠀⠀⠈⠙⠱⡄⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠁⢈⠀⠄⣄⠠⠀⡐⠀⠀⡀⠈⡀⠃⠐⠠⡀⠀⢐⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠁⠴⢡⡄⣀⢠⡄⠲⠀⠀⠀⠀⠀⠄⡀⣀⣴⠇⢀⠀⠀⠉⠁⠐⠓⠄⡀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠃⠀⠘⢺⠟⠋⠙⠙⠿⡗⠃⠀⠀⠀⠀⠀⠀⠉⠁⢆⡭⠌⠀⠂⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠂
⠀⠀⠂⠀⠀⠀⣎⠀⠀⠴⠀⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢎⡀⣀⠀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠿⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⢒⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠒⠒⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⣀⡠⡀⠀⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠋⠈⣭⢀⠂⠁⢀⣗⠒⠄⣀⠀⠀⠀⠀⢀⡠⠒⠉⢁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣽⣿⣷⣀⣀⡙⠘⠀⠀⠀⠘⠋⠲⡄⠹⠶⠤⢐⠏⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡀⠀⠀⠀⡠⠊⠀⢰⣿⣿⣿⣿⣿⣿⢿⠟⠀⠀⠘⠃⠀⠦⡀⣀⠔⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⢻⣝⠣⠤⢶⡎⠀⠀⣀⠼⣿⣿⣿⣿⣿⣿⣶⣶⣿⡄⠀⠀⢀⢀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⣷⣻⡿⠿⢿⣏⣶⣯⡀⠀⢙⡿⠛⣻⠋⠀⠀⠀⠉⠳⡤⠊⠁⠀⠀⠀⠀⠀⠈⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠀⣀⠀⣾⣗⠋⠉⠿⡍⠁⠲⠶⣅⡀⠀⠀⡀⠀⠀⠈⢦⠀⠀⠐⠂⠤⢀⡀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⠞⣋⣭⣄⣯⢄⠀⠀⠀⠘⠲⢤⣙⢦⡙⢄⠀⠇⢀⠔⠀⠀⢣⠀⠀⠀⠀⠀⠈⠉⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠴⡻⠃⢠⠄⣈⠑⠛⠒⠶⠤⡀⠀⠀⠉⡿⢋⠑⢄⣡⡊⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡻⣄⣀⠛⠐⠉⠀⠀⡀⠀⣆⡈⣑⠤⣀⢁⠃⠀⠀⠀⠙⠀⠀⢦⠀⠃⠀⠀⠀⠀⢠⠔⠒⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣛⣆⡙⢯⣳⣤⣦⠮⠤⠤⠋⠙⣌⠠⡺⠦⠤⡀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⡠⠄⠈⠙⠓⠒⠚⠻⡒⠒⢦⡀⠀⠙⢄⢀⠜⠀⠑⠒⠤⣈⡀⢀⡠⠊⢀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⡔⠁⠀⠴⠋⠀⠀⠀⢰⠀⡴⣞⣂⡟⣽⣦⡙⣄⢀⡔⠁⠀⢀⡠⢤⠆⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠳⢭⣥⠤⣿⡏⣧⠜⠁⠠⠔⠊⢁⠔⠁⠀⠀⡠⠊
⠀⠀⢀⣌⡠⠔⠒⠃⡉⠉⠐⠒⠒⠤⢄⡀⠀⠀⠀⡿⣇⡏⠀⠀⠀⠀⠀⠊⠀⠀⠀⠊
⠀⠀⠛⠒⠨⠷⠤⣀⠉⠂⠀⠀⠀⠀⠀⠈⠑⠤⡀⠈⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⣀⠀⠀⠤⢀⣀⣀⠈⢦⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠐⠒⠠⠤⠼⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣴⣶⣶⣄⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⢇⠖⠦⣄⠀⠀⠀⣀⡤⢤⡸⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢀⣔⣬⣆⠁⠀⠈⣡⣤⣄⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠻⠿⠧⠂⠀⠐⠿⢷⠿⠁⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⣀⣀⣀⡀⠀⠀⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⠻⠿⠿⠃⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠈⠢⢀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⢒⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⢀⣀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠴⠒⠋⣹⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣼⣀⠤⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠉⣀⣠⠤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⡤⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⢢⡤⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣴⣋⣀
⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⡞
⠀⠀⠀⠙⢶⣖⠒⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠈⢻⡳⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠾⢶⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢳⡘⢦⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⠋⠀⠠⡏⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢧⠘⣷⡄⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⢄⡔⢆⡇⢱⡄⠀⠀⠱⣄⣀⣄⠀⢲⣠⣾⣟⣤⠖⠀⠀⠀⠀⣀⠔⢹
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠙⠋⢈⣡⠬⠽⠯⠥⠧⢞⠓⠂⣸⠋⢉⣿⣿⣿⣟⠁⠀⣀⣠⣴⣊⠅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡴⠊⠁⠀⠀⠀⣀⡠⠴⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢸⡍⣛⣛⣓⣲⡶⠤⣄⣀⡤⠖⢿⡇⠀⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⡯⣟⡻⣿⣽⡟⠛⠉⠁⠀⠀⠀⠁⢀⡤⠚⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡧⣔⣲⢏⢌⡝⣳⣢⣤⣀⣀⣠⣶⣯⣴⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡵⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⡿⠛⠓⢤⡀
⠀⢶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠿⣛⡛⠛⠒⠲⠶⣶⣤⣤⣤⣴⣶⠟⣛⣛⣛⣛⣛⣿⣿⣿⣛⣛⣛⣛⣛⣛⣭⣭⠤⠤⠤⠴⠶⠞⠳⣄
⠀⠠⠤⠤⠬⠭⠭⠿⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠉⠉⠉⠉⠹⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⡀⢳⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠁

⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡴⢶⣶⡄⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠾⠿⠷⡄⠀⠀⠀⠠⠀⠀⡀⠀⠀⢰⣷⣶
⠀⡄⢀⡄⠀⠀⠀⠀⠀⠠⣶⣎⡉⡇⡓⢒⠷⣄⠐⢀⢤⣆⠐⠱⠀⡀⠀⣠⢀⡴⠀⠀⠀⠀⠀⠀⠰⣿⣭⡄⣫⡍⠻⠻⢦⡀⡡⢴⣣⠀⢀⠂⣠⢸⣿⣿
⢈⢱⡏⠀⠀⠀⠀⠀⣄⡠⠇⠨⠛⠿⠃⠀⢁⢄⠙⠁⣼⠃⢀⠐⠈⣅⠀⢣⣿⠀⠀⠀⠀⠀⠰⣤⠼⡁⠀⠙⠛⢃⢀⠀⡢⠋⠀⣼⠃⢠⠀⠀⣬⢸⣿⣿
⠈⣾⡇⠀⠀⠀⠀⡰⠘⣷⣌⡤⠀⡴⢚⡷⡤⠃⠲⢄⣫⣢⣼⡄⢰⣿⠀⢸⣧⠀⠀⠀⠀⣠⠇⢸⣷⣴⠤⢠⠞⠫⠗⠜⠁⠒⠤⣽⣶⡾⣇⠀⣿⢸⣿⣿
⠀⡿⢁⣦⡆⠐⠉⢄⡀⠩⠛⠱⣷⣅⠀⡼⠃⠀⠀⠌⢉⠨⡱⣷⠀⠿⠀⣿⠃⣴⣶⠒⠉⠠⣄⠀⢝⠭⠺⣷⣅⡀⡞⠁⠀⠀⢠⡍⢒⣞⣿⡄⠹⠸⣿⣿
⠀⡠⠚⡻⠷⠶⢧⣤⣷⡀⢠⠀⡈⢿⣷⣦⡀⠀⠀⢘⡁⣋⢢⣿⡇⠀⠀⡀⠘⡉⠽⠲⠿⢦⣬⣧⠀⠄⣀⡜⢿⣿⣦⡀⠀⢀⢀⣉⣩⡀⣿⣧⠀⢰⣿⣿
⠀⠣⣘⠛⢦⡀⠆⠈⡿⢧⣀⠛⢁⠀⢿⠅⠉⠀⠄⢈⣙⣙⣭⡙⢷⠀⠀⠘⢦⠋⠳⣄⢈⠀⢨⠿⣄⡈⠋⠀⠈⢿⠀⠁⡀⠅⣈⣽⣯⣍⣍⠻⡀⢸⣿⣿
⢀⠀⣰⣀⣤⡞⢑⠤⠁⠈⠁⠀⠀⠀⠀⠀⢀⠠⠀⢉⣉⣿⣿⣿⠈⢁⠀⠔⡠⣇⣤⠴⠃⡢⠘⠀⠉⠀⠈⢀⠀⠀⠈⢀⠀⠁⠏⣉⣽⣿⣿⠂⢉⣸⣿⣿
⠀⠚⠓⢇⠀⠀⢈⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠀⠉⠉⠉⠉⠁⠀⠈⠁⠼⠳⠺⡀⢀⡤⣼⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠀⠋⠉⠉⠉⠁⠀⠀⢹⣿⣿
⠐⠂⠀⣠⣦⡴⠋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡠⠛⠒⠀⢀⣶⣤⠞⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢺⣿⣿
⠀⢨⣞⡛⠛⠗⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠁⠀⠄⠃⠀⠀⠀⠀⣶⡛⠛⠛⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠅⢸⣿⣿
⠐⠚⠀⠉⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⡀⠠⠀⠚⢾⡉⠲⡤⠒⢁⠉⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢠⠂⠞⢶⡅⠒⣼⣿⣿
⠤⡄⣀⣂⠀⠀⠀⢀⣴⢪⠀⠀⠀⢂⠀⠀⢦⣇⠀⡃⠐⢀⣀⡺⡇⢀⡴⢤⢀⡂⠀⠀⠀⠀⣀⡶⢁⠀⠀⠀⠀⠀⠀⢤⣇⠀⡋⠠⢀⢀⠘⢇⠀⣸⣿⣿
⠀⠘⡷⠉⠟⣤⣤⣿⠏⠀⠑⠄⠀⠀⠀⠀⡀⣿⢸⣷⠁⣸⠻⠛⠎⠁⠀⠈⢿⠌⢻⢂⣄⣴⣿⠁⠈⠂⢁⠀⠀⠀⠀⡈⣿⢸⣷⠁⣘⣛⠿⠶⠉⢸⣿⣿
⠀⠀⢨⣠⣾⠋⠒⠱⢀⢀⡞⢠⣤⡲⠶⠦⠉⢸⠸⠏⢰⣏⠿⠆⠤⠀⠀⠀⠈⣄⣴⡿⠣⠊⠇⠀⣠⠖⢤⣐⢦⠤⢄⠃⣹⠘⡏⢠⡟⠷⠤⠠⠀⢸⣿⣿
⠀⠘⠋⣿⠣⢤⡴⠀⣼⣿⣷⠾⣛⣳⡂⡄⢠⠋⠀⠀⣾⡟⢂⠰⠲⠀⠀⠠⠞⢻⡟⠠⣄⠈⢠⣿⣿⣶⢿⢛⣦⢀⠄⣨⠛⠀⠀⣼⡿⢶⡴⠶⠂⢸⣿⣿
⠀⠀⠋⠐⠀⠀⠐⢴⣿⠿⠹⡐⡉⠀⠥⣈⠡⡀⠀⢰⡟⣹⣿⣷⠶⠆⠀⠀⠐⠉⠂⠀⠈⢤⣼⣿⡏⢷⠘⠉⠨⠥⠈⢋⠀⠀⢤⡿⢫⣿⣶⡶⠦⢸⣿⣿
⡀⠂⠀⠂⠀⠀⢀⡾⠁⠀⠀⢁⠽⠘⠪⠉⡵⡈⠰⠂⠈⠻⠿⠿⠶⠆⠀⠀⠀⠀⠀⠄⠀⢰⡟⠁⠀⠈⢊⢗⠒⢙⣛⣦⠁⠴⠊⠀⠿⡿⠷⠶⡄⢸⣿⣿
⢉⠁⠀⠀⠀⢀⠞⠀⠀⠀⢰⡻⢤⠳⠀⠀⠠⠮⢄⠀⠀⠀⠀⢬⠄⠀⢈⠅⠀⠀⠀⠀⣠⠏⠀⠀⠀⢔⢷⡂⢦⠀⠉⣊⣅⡄⠂⠀⠀⠀⢀⡁⠐⢘⣿⣿
⠘⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠻⠌⠳⠾⠆⠔⠂⠀⠄⠀⠐⠀⠀⠀⠀⠈⠄⠀⠀⠀⠰⠃⠀⠀⠀⠀⠈⢧⠙⢦⣶⡄⠤⠄⠌⠄⠀⠰⠀⠀⢀⠀⠸⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠙⠋⠀⠀⠉⠛⠉⠀⠈⠉⠀⠀⠉⠋⠉⠉⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⡤⠤⣤⡴⠖⠦⠤⠤⠖⠛⠦⢤⠤⢴⡏⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⣳⠄⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⠋⢹⡄⠀⠀⠀⡞⠀⠀⠸⡆⠀⠀⠀⢿⣀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⡇⠀⠀⢰⠁⣶⠀⠀⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⢠⡇⠀⠀⢸⡆⠀⠀⣠⠃⠀⠀⠀⠀⠘⠖⠈⠉⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣀⠟⡀⠀⠀⠘⣇⣁⡨⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠤⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠑⠢⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣫⠤⠠⠠⠤⠬⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠁⠈⠙⠰⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠖⠒⠛⠋⠉⠛⠳⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⡿⢀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⡠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣤⣄⣀⣄⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣷⣷⣾⡟⠀⡟⠚⢿⣿⣿⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣿⠁⢹⣿⠇⠠⠄⣀⠀⠈⠛⣿⣿⣿⡟⢿⣿⠓⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⣿⣿⣿⣿⡇⠙⣿⣷⣶⣿⡟⢀⢀⣾⣿⣿⠁⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠘⡹⢿⣿⣿⣅⣀⣩⣭⣿⣭⣄⣀⢻⣿⣿⡟⠠⢠⠚⠁⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠀⠀⡿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⠀⠀⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢀⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠘⠛⠛⠙⠟⠛⠹⠏⠉⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡋⠉⠻⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠃⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠞⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠸⣘⠛⢿⣷⡄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣉⢂⢀⡎⣀⣁⣤⠂⠀⠀⠀⢀⣀⣀⢀
⢀⠠⠐⡠⠈⠉⠁⠀⢤⠴⠒⣉⡭⠄⠒⡒⠀⠀⡀⠃⠘⣓⠒⠶⠤⢭⣉⡕⠲⣄⡀⠀⠁⠀⠀⣁⠂⠠⢀
⠀⠀⢼⠀⠀⠀⠀⠀⣷⠀⠀⢯⡀⠀⠀⠂⠀⡀⠀⠀⠰⠪⠵⡒⠈⠆⣹⢳⠀⠈⠻⡆⠀⠀⠀⠈⢳⠀⠀⠈
⠀⠀⠈⠢⣄⡀⠀⠀⠙⢆⡠⠀⠉⠂⠢⠤⢤⣤⣭⣬⣷⣶⣧⣤⢔⠠⠅⠊⣠⠔⣈⠆⠀⠀⠀⣢⠞
⠀⠀⠠⡀⠀⠁⠀⠐⠢⠠⢈⣁⣉⠀⠐⠧⠶⠖⢾⣯⣴⢦⣤⣀⡤⠶⠪⠑⠐⢉⣀⠠⠴⠒⠉⠀⡀
⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠄⠀⠈⠉⠉⣉⠐⠒⠚⠛⠙⠃⠒⠓⠒⠒⠉⠁⠉⠡⠀⠀⠒⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠁⠐⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡤⡀⠈⠀⢸⠀⠀⡇⠀⢀⠚⠠⡀⢃⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣑⠀⠀⠢⢄⠀⡀⢟⢦⡁⠡⣉⠄⠬⠓⠂⠉⠉⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢋⡀⠀⠠⣤⠴⠁⠈⠀⠙⠁⠀⢀⢀⡀⠠⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⠉⠐⠀⠬⠄⠒⠠⠒⡡⠚⢀⠔⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢠⡦⡄⠀⠀⢩⡉⠉⠉⠀⠀⠀⠐⠒⠠⠀⢀
⠀⠀⠀⠀⠀⢀⡐⠇⡣⠈⠁⢀⡆⠤⠜⠋⠁⠀⡀⠀⠀⠀⠀⠞⠀⠄⠀⠀⠠⠶⠦⢤⣀⡀
⠀⠀⠠⠄⠀⠊⢡⠈⠁⢀⠄⠉⡀⡀⠀⠀⣄⡴⠗⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢀⣀⡄⠤⠉⠀⠐⠀⠄
⠊⠀⠀⠀⠀⠀⠃⢀⠴⠃⢠⠀⠀⢐⠀⠈⠁⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⢉⡉⢳⡀⡀⠀⠀⠁⠢⢌⡁⠀⠠
⠀⠀⠀⠀⠀⠀⣠⠆⢀⢠⠤⠥⢠⠃⠀⢀⡼⠠⠯⢴⡞⠀⠀⠀⠀⢴⡋⠀⠀⠐⠀⢁⡌⠀⢱⡄⠀⠀⠀⠀⠙⠦⡀
⠀⡀⠀⠀⢀⠔⠀⠀⠀⠈⠑⠦⢄⠀⢠⠆⠀⠀⠂⡀⠀⠀⠀⣀⠈⠀⠁⠀⠠⠀⡀⢀⡆⢀⡴⠀⠀⠀⠀⢀⣠⣤⠕
⠸⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠤⣀⡠⠐⠒⠓⠻⠖⡲⠎⠀⠑⢄⣀⣤⠲⢖⣿⠁⠀⠀⠀⠀⡠⢂⠞⢁⠃
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠁⠀⠀⡠⠛⠈⠉⠙⠓⠀⠀⢀⡠⠈⠐⠁⡠⠞⠠⢄
⠀⠀⠈⠉⠉⢀⡠⠖⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠂⠁⢺⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡁⠈⠁⠀⠠⡀⠀⢠⣇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠟⠀⢼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠐⠄⠀⠀⠀⠀⠐⠀⣤⡄⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠔⠋⠑⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⠀⠀⠈⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⢸⠀⠀⠀⡇⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⣽⡁⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⢶⣶⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⢺⣷⡈⢿⣧⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣃⣵⣸⣿⣿⣦⣉⡛⠿⠿⠟⣢⣄⣀⢀⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠋⠀⠀⠀⠄⠂⠰⠀⠀⠀⣠⣾⣿⣿⣠⠐⣿⣿⣿⣿⣿⠿⠻⣾⡛⡿⠿⢒⣛⣿⣟⣂⠐⠁
⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠤⢤⡀⢠⣾⣿⣿⣿⣿⣾⣴⣴⣶⣼⣿⡿⠒⠒⢿⣷⡼⣿⠋⠙⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠛⠂⣀⠪⠙⣿⣿⣿⠏⠈⠻⣿⡿⣿⣏⣀⣀⣽⣷⡶⠶⠻⠏⠀⠈⠀⠀⠔⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠐⠀⣨⠃⠄⢀⠀⣀⠁⠀⣰⣿⣿⣶⣼⣿⣿⣃⡀⠀⠹⣾⣿⢿⡿⣿⣿⣇⡶⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⠙⢋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠇⢄⠀⠀⠀⠀⠉⢤⡴⣾⣾⣿⣿⣿⣿⢿⣽⣿⣿⡆⠀⠸⣿⡟⣿⣯⢈⣿⣿⡟⠡⣃⢠⠀⠀⠀⠉⠙⣿⢻⡆⠈⢢⠹⣿⣷⡄
⠀⠀⢄⡀⠂⠀⠀⠀⠠⠠⠓⠀⠀⠀⢑⠻⢳⣛⢿⣿⣤⠝⣵⣿⣿⣿⣽⣷⣄⠀⢻⣿⣿⣿⢨⣿⡿⠃⠀⠸⠁⠀⠀⠀⠀⢰⡈⢾⡇⠀⣩⣧⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠸⣿⣿⣟⣶⡿⠻⠿⠻⠛⡿⠿⠀⣸⠟⠛⠿⢘⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⡜⠂⠉⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡄⠩⠐⠀⠸⡿⠿⢿⣿⣾⣿⣿⣿⣷⡲⠾⣿⡕⢀⡴⢸⡇⢰⣇⢀⠀⠀⠀⠀⠀⠀⢀⢹⣿⠻⡄⠀⠈⠘⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢉⠙⠒⠀⢀⣁⣉⣨⠷⠶⢶⣶⠄⠀⠀⠀⠈⠉⠉⠛⣻⣿⣿⢳⣭⠁⣈⣰⣟⡷⠈⠀⠀⠀⠀⠀⢀⠀⣼⣿⣷⢱⣄⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢵⣖⣃⠉⠐⠂⠚⠓⠠⣤⣤⣤⢀⡀⠐⠀⡄⠘⠴⢦⣀⠨⠡⣜⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣈⣍⣻⣿⣿⣷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠙⠲⢞⣳⣶⣤⣉⣛⣻⣿⣿⣿⣿⣟⢀⣠⣼⣷⣷⣬⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠬⠥⠬⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠯⣽⡿⠿⠿⣿⢿⣷⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣶⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣷⣶⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀






⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⠀⠀⡀⠀⠀⠀⠀⠒⠂
⠀⠀⠘⠛⠛⠋⠛⠋⠁⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠈⠀⠉⠙⠛⠉⠉⠉⠉⠀⠀⠀⢀⣠⣤⣴⣶⣿⣤⣤⣀⠀⠈⠉⠉⠉⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⠶⡶⠶⠶⠦⠀⠠⣂⣤⣤⣤⣶⣦⣤⣤⣄⠀⠀⠀⡀⠐⠀⠒⠛⠷⣀⣈⣉⠀⠀⠀⣀⠉⣄⣛⣛⣿⣭⣍⣉⣑⣆⢀
⠠⠤⠈⢉⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⣿⢿⣿⣿⣿⣿⣿⣿⠿⠛⣛⣥⣤⣦⡤⠂⠀⣈⣹⣆⣀⠀⠀⡈⠙⠛⠛⢿⣿⣿⣿⣿⣿⠿⠋
⣴⣆⣤⣀⣬⣿⣩⣿⣦⣤⡀⠀⣀⠀⠀⠀⠀⠀⢀⠠⣤⡭⠤⣤⣀⣀⡐⠿⠛⠛⢋⣩⣴⣿⣿⣿⣿⣿⣿⠿⣶⣿⣶⣮⣭⣶⣃⡄
⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣤⣶⣾⠋⠀⢀⣠⠀⠀⠉⠉⠙⠻⠿⠧⣤⣀⣈⣉⠉⠉⠉⠁⠀⠀⠉⠁⠀⠈⠉⠁
⢨⡿⣿⡿⣿⣿⣽⣿⣿⣿⣿⣿⡞⠳⣾⣴⡿⠃⣀⢰⣿⣿⣾⣇⣰⣶⣤⡀⠀⠀⠀⠉⠍⠻⠓⠆
⢉⣷⣿⣿⣿⣿⡍⠥⣾⠏⠋⠁⠐⢚⠟⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣠⣤⣀⣀⣀⣤⣤⣦⠀⠀⠀⠀⢀⡀
⣴⣄⣠⣀⣰⠊⠢⣅⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠿⢿⠿⠿⠟⠻⠟⠁⢀⣠⣾⣿⣷⣤⣀
⢻⡿⠟⠋⠀⠀⡄⠈⠻⢶⣤⣀⡀⠀⠈⣉⡟⢿⣿⣛⠻⠿⠿⡏⠛⠫⢤⣤⡀⡀⠀⠁⠘⠳⢶⠀⠀⢀⣀⣀⣤⣖⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⢠⡶⠃⠈⠛⠀⢀⠀⠈⠉⠙⠛⣂⡈⠙⠛⠿⠯⠠⠀⠂⠀⠀⢀⣀⣤⣤⣶⠶⠶⢚⣛⣂⡴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⢰⡄⢠⣦⡐⣶⣾⣴⣾⣾⣷⣀⠀⠀⠈⠉⠛⠲⢤⣤⣀⡰⢦⡄⠀⠛⠛⠛⢿⣿⣿⣯⣉⣉⣉⣉⡁⠀⠈⠉⠛⠛⠿⠿⠿⠿⠟⠿⠛⠛⠁
⣿⢱⣿⣿⣿⣮⣻⠿⠛⢿⣿⣿⡇⢸⣦⣠⣤⣀⡀⠈⠙⠛⠷⣦⣄⠀⠀⠙⠂⠘⠻⡿⣿⣿⣯⣄⣀⡀⠐⠒⠲⠒⠠⠄⠐⠒⠀⠄⠂
⣿⣾⣿⣿⣿⣿⣿⣶⣿⣯⣿⣟⣱⣿⣿⣿⣝⢿⣿⣶⣶⣤⣀⡀⠈⠈⠳⢦⣤⣀⣀⠀⠀⠀⠉⢉⣉⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾
⢹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣎⢉⣿⣿⣿⣿⣿⠶⠶⢧⠀⣩⣉⡛⢟⠻⠷⠾⠿⢿⣿⣷⣿⣿⠤⠀⠀⠞⠛⠉
⠈⢻⣮⣟⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣥⣥⣤⣙⠷⠋⣠⣤⣶⣶⣦⡀⣹⣛⠿⠿⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⢷⣧⣽⣿⣿⣷⣦⣉⠻⣿⣿⡿⣿⣿⣿⣿⡻⣿⣿⣿⣷⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀
⣨⣙⠹⣧⣍⢹⣿⣿⣇⡈⢉⠁⠜⠻⣿⡿⢧⣉⠛⠿⠟⠋⠀⠙⠿⠛⠿⠛⠃⠉⠉⠙⠻⠛⠉⠁⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠁⠙⠓⠶⢦⠀⠤⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⣠⣾⣿⣤⣿⣿⣿⣃⣰⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣼⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣿⢉⠟⠀⠀⠈⠉⠙⠿⢿⣿⣿⣿⣏⣹⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⡀⣄⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢷⣤⣤⣄⣠⣉⣈⢳⡿⠿⠿⠟⠀⢀⣴⢿⣿⣿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⡿⣿⣿⠁⢸⣿⣿⣷⣧⣌⡓⠒⣿⣿⣿⡟⣻⡿⠟⠋
⠀⠀⠀⣀⣴⢲⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣟⣻⣿⣿⣅⠋⢻⣿⣷⣀⡀⠈⠉⠉⠃
⠀⢦⡾⣿⣿⣿⣻⣿⣿⣿⣹⣿⣿⣶⣿⣿⣿⣟⣿⣿⢻⣶⣿⣿⣿⣿⣿⣆
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠈⠻⣿⣻⣿⡿⢿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣿⣿⣿⣟⣹⢿⣿⣿⡿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⣿⣿⣷⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠉⠁

⣿⣿⣿⣿⡿⠟⣁⣴⣿⣟⠁⠀⠀⠀⠉⠙⠻⠿⣯⣁⣠⣤⣶⡾⠿⠛⠛⠉⠁⠀⠈⠉⠉⣴⣾⡿⢡⣿⡏⣿⣷⡻⠯⠻⣷
⠟⠛⢉⡀⠔⢊⣡⡀⠉⢉⣉⠛⠲⠶⠤⣤⣀⠀⠀⠉⠉⠉⠀⢀⣀⣤⡤⠶⣶⣿⣿⣦⣤⣤⣈⢡⣿⡛⠃⢿⣿⣧⣤⠀⢈
⠀⠀⢁⠤⠚⠉⠀⠉⠀⠀⠀⠉⠒⠒⠀⠀⠶⣤⣴⣾⣿⠿⠟⠋⠁⢀⣀⣀⣉⣉⠤⠤⡈⠙⠋⠳⠹⣟⣦⡀⠙⢿⣿⡆
⠘⠓⠀⠀⢀⠔⠀⣶⣶⣶⣦⣀⣀⣤⣶⣾⠟⣿⣍⣉⣉⣉⠉⠉⠉⠅⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢠⠈⢻⣿⣶⡈⠻⣧
⢀⣶⠋⡴⠁⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡌⣿⡿⢿⣿⣿⣦⣤⠀⣀⣠⣴⣤⠀⠀⠘⠀⠀⠐⠀⠀⠘⣿⣿⣥⠀⠙⡆
⡈⠡⠞⠀⠀⠀⠀⠀⢰⡶⡿⠟⠿⠛⡏⢹⣿⣽⡄⠈⠲⠽⠛⠙⠃⠀⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣷⣄⠈
⠱⣀⣀⠀⠀⠀⠀⠀⠸⠇⠿⠈⡡⢊⣴⣾⠿⠿⠷⠀⠀⠀⠀⠀⠀⠘⠋⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⣷⣺⡽⠁
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⣿⠏⠀⠀⠀⠴⢶⡆⠀⠀⠀⢰⣦⡄⠻⡿⠀⠀⡀⠀⠀⠀⢀⣠⡟⠀⡝⢻⡇⢀⡇
⡀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠲⣀⣙⣻⠄⠀⠀⠣⡬⡇⠈⣩⣄⣠⣥⣔⡛⠛⠉⠀⡷⢞⣠⠞⠀⣸⡇
⠉⠀⠓⠀⠀⠀⠀⠀⠀⠀⣾⣹⠗⠀⢾⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢹⣙⣿⣿⠟⠀⠀⠘⠛⠿⣯⣴⣠⣿⡇
⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣺⡟⠿
⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⣿⣼⡇⣸
⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠿⡤
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢀⡴⠋⢀⠜⡕⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⡠⠂
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠜⢟⣫⡿⠋⠀⠀⡠⠋⢼⡀⠀⠀⠀⠀⠀⡞⠉⠁⠈⡈⠜⠁⠀⠀⠀⠀⣠⠔⠈
⣿⣿⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠔⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢀⠠⠐⠉
⣿⡏⣷⡀⢀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠦⠤⠤⠖⠚⠁⠀⠀⠠⠇
⣿⡇⣿⣷⣆⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⢨⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⣀
⢿⣦⡤⠐⠉⠐⠂⠀⠀⢀⢀⣀⡀⣤⣾⠿⠃⠀⠀⠀⠀⣖⡀⠀⠀⠀⠀⠀⡀⠔⠀⠿⠅⡓⢣⡞⠀⠀⠀⠀⠀⡀
⠂⠀⠀⠀⠠⠆⠈⠀⠰⠛⠘⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠠⠄⠀⣀⠀⠀⠁⠣⠀⠈⠉⠳⠛⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠢⡀⠀⡜⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠂⠌⠁⠉⠈⠑⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠋⠐⠂⠰⠀⠀⠒⠓⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠃⠘⠀⡄
⠀⠀⠀⠀⠀⠠⠀⠁⠃⠀⠀⠀⠀⠀⡀⠐⡒⠀⠐⠒⢲⠂⠠⠄⠄⠒⠒⣖⡲⢊⠍⠉⠀⠈⢱⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡥⠆⠒⠂⠒⠀⠐⠓⠊⠀⠘⠒⠚⠓⠒⠒⠀⠉⠘⢒⠞⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠠⡇⠀⠀⣰⢠⠄⠀⢀⠀⡄⠀⠰⡇⢰⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⢰⡀⠀⠀⠀⡇⠁⠀⠀⡍⠰⠁⠐⠐⠈⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢀⣠⠃⠀⠀⠀⡫⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⢀⠀⠃⠀⠀⠀⠚⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠀⠂⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠂⠇⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠁⠠
⠀⠠⡴⣄⠀⠁⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠤⠀⠉⠬⠄⠇⠀⠀⠂⣀⣀⣀⣈⡈⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢐⣀⣃⣤
⠀⠀⠀⠈⠀⠁⠉⠁⠈⠉⠈⠀⠀⠄⠠⠀⠈⠉⠀⠀⠃⠁⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠀⠀⠐⠈⠉⠀⠀⠈⠀⠉⠉⠉⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠟⠛⠿⡿⠟⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠛⠃⣿⣿⡿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠈
⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣤⣄⡀⠀⣄⡀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⣀⠀⠠⣤⣾⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⠁⠀⠀⠀⠀⠠⠤⠼⣿⡇⠀⣠⣾⡿⠃⠀⠈⠉⠉
⣷⣦⣤⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⣀⣀⡀⠀⠀⠀⠀⣀⣿⣿⣾⣿⣿⣧⣤⣤⣤⣤⡀
⣿⣿⣿⣿⡇⣀⠀⠀⢠⣿⣿⣥⣿⣿⣤⣿⣿⣽⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣀⣠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⡠⠞⢉⣤⣴⣶⣾⣿⣶⣬⡓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⢀⡀
⠀⣼⣯⣾⣯⣹⡿⡿⣿⣿⣿⣿⣿⣝⡆⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠟⠸⡣⠔⠀⠀⠀⠀⠤⠉⢷⡀
⠀⢸⣿⣿⣿⣾⣶⣿⣾⣾⣷⣿⣿⣿⣷⠀⠀⠀⠀⢀⣠⢴⡿⠷⠶⠀⠘⠀⠀⠀⠀⠀⣰⠀⡀⣴⣧
⠀⢸⣿⣿⣿⡿⢿⣿⣿⢫⢾⣿⣿⣿⡟⠀⠀⢀⡴⢩⡞⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⣸⣿⣦⡀
⠀⠀⠹⣯⡻⢿⣿⣿⣿⣾⣿⡿⠟⠿⠃⠀⣴⠃⠈⣸⠧⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⠶⠖⠀⠉⠀⠉⢇⡀⠔
⠀⠀⠀⠈⠛⠖⢒⠛⣻⣭⠥⠐⠁⠀⢀⠚⢁⣷⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣅⣀⣴⠃⠀⠀⢀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣾⠿⠏⠛⠛⠋⠀⡉⠉⢩⣀⡀⠀⢀⡀⠀⠀⠀⠚⠑⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠋⠁⠀⠰⠂⠀⠒⠀⠙⠓⠀⠉⠉⠀⠠⡔⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⡏
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⢶⣿⡇
⠠⣠⡄⢀⠀⠸⠟⠁⠀⠀⠀⠀⠀⢀⣠⣦⣾⠋⠀⠀⠀⠀⠀⠀⠙⢢⠉⠀⠀⠈⠳⣄⠀⠀⠀⠀⠸⡿⣥
⠀⡵⣕⠘⠄⠀⠀⠀⠀⠀⠀⢀⠸⣞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⣷⣤⣀⠀⠀⠃⣿⡆
⠀⠉⡄⠁⠀⠀⢀⠀⠀⠀⠐⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣿⡛⠿⣧⠀⠰⢻⢸
⠀⠢⣑⠀⠀⠀⠀⠀⠀⠀⡇⣠⠜⠂⠀⠀⣀⣀⠀⠀⠠⠐⠀⠀⠀⠀⠀⣧⡀⠀⠀⡠⠋⢷⡀⠈⠇⠀⠨⠀⠄
⠀⠐⡄⠀⢀⡀⠀⠀⠁⠀⣷⠇⠠⠁⠀⠀⡏⡯⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠉⠀⠀⠀⠀⠈⠻⡄⢀⠀⠀⠣⣸⡄
⠀⠀⠉⣖⣒⡀⠁⠀⠀⠴⠁⠀⢀⠈⣄⠀⠐⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣼
⠀⠀⠀⠀⠙⠋⢁⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠠⠀⠠⢄⡈⣷⣤⣀⣀⣀⣿⡏
⠀⠀⠀⠒⢒⡿⠁⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⣿⡟⠛⠛⣿⣛⣋⡀⠂⡀
⠀⠀⠀⠀⠀⠃⠤⠄⠀⠶⠒⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠿⠻⠛⠉⠃⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⡀⠀⠀⠘⢳⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠘⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠓⠒⠒⠚⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣓⡦⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡴⠖⠚⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠁⣠⡶⠉⠀⠙⠳⣝⡆⠀⠀⣠⡴⠋⠉⠙⠋
⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡟⠀⠀⣦⠀⠀⠀⠉⠲⣴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠈⠁⠀⠚⠁⠀⠀⠀⠀⠀⠘⣾⡄⠀⢸⠁⠀⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣟⠛⠃⢀⣴⠀⠲⠦⣀⠀⠀⠀⠀⣸⠀⠀⢹⣷⡀⠘⣇⠀⠳⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⠃⣀⡼⠀⠈⠓⠀⠀⣠⠏⠞⠀⢀⡇⠱⡄⠈⠳⢦⡙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡀⢀⣀⣀⡀⠔⠁⠀⠀⢀⡼⣧⠀⠙⣆⠀⠀⠙⠲⣝⣦⡙⠦⣄⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣝⣒⣾⠋⠉⠛⠲⢾⣉⡁⠈⠳⢄⡈⠳⢤⡄⠀⢀⣹⣿⣷⣄⣙⣦⣤⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣒⣒⣒⣒⣀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠷⠶⠶⢄⠁⠉⢙⡛⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⠀⠀⢀⣀⣀⣀⠹⣎⢻⡄⠈⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠉⠉⠉⠙⠇⠹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠎⠀⠀⠀⠉⠉⠉⠉⠙⠒⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⣠⣶⣿⣿⣷⣈⣿⣝⢿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣀⣀⠀⠀⣠⣾⣿⣿⣟⠷⠞⠉⠉⠉⠀⠟⠿⢿⣿⣷⡄⠒⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢸⣿⣯⣹⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠐⢶⣷⣦⣿⣿⠀⠀⠉⠉⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡜⠁⢀⢀⣿⣿⣿⣿⣿⣏⣻⠀⠀⠀⣀⠈⠑⡄⠀⠀⠀⠀⢙⣿⣻⣧⡤⠖⠚⠉⠁
⠀⠀⠀⣠⠴⠚⠛⢛⣷⣶⣾⣅⡀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⡿⢿⠛⠉⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⣠⠟⠋⠁
⠀⠀⠀⢀⣐⠔⣀⠄⠉⠛⣿⣟⡿⣶⣄⠀⠀⠀⢸⠙⢿⡿⠻⠷⢋⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀⠉⠣⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠉⢻⣷⣶⣤⣠⠄⠘⠿⣾⣿⣷⣄⢀⡼⡆⠠⢐⠀⠀⠸⡄⠀⠀⢸⣶⣄⠀⠆⠀⠙⢢⠖⡠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣁⠄⠈⠿⣭⡻⣿⣼⠃⠄⠨⠀⠀⠀⠙⣆⠀⠀⠙⠛⣀⠤⣢⠖⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠘⢶⣤⡤⢀⡀⠙⠿⣿⣦⡀⢀⠈⠹⣯⣿⣶⣂⠎⠀⠀⣀⣤⣾⡷⠒⠒⠉⠁⣘⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣔⢀⡀⠀⠘⠿⣷⡴⠂⠀⠈⠉⠛⠦⣀⠹⠿⠛⠁⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡤⠀⢀⠀⠀⠉⠑⠲⢶⣄⣀⠄⠈⠙⠲⢤⣤⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠥⣖⠁⠀⠀⠀⠀⠉⢻⣿⣦⣴⠄⣀⠈⢙⣦⣀⠈⠑⠶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠮⣁⠤⠀⠀⠹⢿⣿⣯⡁⠀⣀⠘⢿⣷⣄⠀⠀⠉⠓⢦⡀⠀⠀⢴⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⠙⠻⠿⣌⠄⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⠐⠀⢀⣴⠄⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠁⠀⠉⢉⡉⢬
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣄⣠⠠⢄⠠⣀⠀⢠⠄⣤⡠⡀⠄⠀⠀⠀⠦⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠀⠤⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⡐⠀⢘
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠂⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣤⠀⣠⠀⢀⠀⠀⠠⠠⠤⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⣠⣄⣤⣀⣀⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣴⣦⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣶⣠⠔⠋⣽
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠿⣿⣿⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠉⠀⠀⢸⣿⣦⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⠀⣀⣆⣛
⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣿⣤⣼⣿⡿
⠀⠰⣿⣏⡁⢳⣀⣠⣶⣶⣦⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋
⠀⢸⣿⣭⣴⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣴⣿⡗
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠁⠂
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡴⠒⠚⢫⣥⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣴⠶⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⡀
⠀⠀⣤⣴⠶⠚⠋⠁
⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⢀⠀⣀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠚⠀⠁⠀⠉⠁⡤⠀⠅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠀⠀⠂⢙⠃⢀⡴⡆⠒⠂⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠛⠁⣴⡞⠁⠀⠀⠐⠄⣠⣥⣀⢈⣻⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠠⠅⠀⠀⢀⠐⣐⡶⡄⠀⠀⢹⣯⣀⠀⠉⠁⢂
⠀⠀⠠⠀⠀⠠⡆⡜⠋⠻⡀⠚⢿⣟⠀⠁⣾⠦⣿⣿⢯⠁⠒⠀⠘⠂⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣴⣄⡀
⠀⠀⠀⠀⠠⠠⠄⠴⠀⠀⢬⠱⡴⢏⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠼⠃⠁⠂⠋⠛⠃
⠀⢀⣲⣶⡖⡠⣤⣤⣤⢲⣀⣀⣤⢈⣤⣴⣦⣮⣀⣠⠠⠄⠀⣀⡀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠘⠓⠂
⠀⠸⠶⠄⡻⢻⣿⢏⡟⠇⢛⠷⢶⣿⡛⠃⠘⠛⢟⣭⡔⠀⢩⠥⠀⠀⠀⠀⠀⠈⠀⠉⠋⠀⠀⠀⠀⠀⠀⢠⡠⢄⣀⡀
⠀⠐⠐⠒⠂⠘⠋⠉⣉⠭⠵⣚⠯⡇⠀⠀⠀⠀⡌⠙⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠤⠄
⠀⠈⠉⡩⠂⠠⢀⠀⠀⣀⠀⠀⠈
⠀⢐⠚⠀⠀⠀⠻⢅⣀⡄⣈⡛⢂⠀⠀⠀⠀⠀⠂⠀⡀⠄
⠀⠈⠀⠂⠀⠂⠀⠁⠐⠀⠐⠘⠉⠈⠁⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠰⠀⠀⠀⠄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣁⠀⠀⢉⡛⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢡⣌⡉⠀⣉⣠⡆⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡏⠔⠚⠓⠶⡄⠀⡶⠖⠛⠲⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠈⣀⣸⣈⠁⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⠇⠸⣶⣤⣤⠀⡀⣠⣤⣶⡿⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⠟⢿⣾⡿⢣⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡁⣆⠀⠀⠉⠀⠀⢹⠀⠉⠁⠀⢀⡆⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣿⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠳⣿⡀⢀⣴⡶⠶⠿⢶⣶⡄⠀⣼⡷⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠛⠿⠋⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣷⡼⠋⢾⣛⣛⣻⠎⠻⣴⣿⠇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡋⠛⣒⠛⢉⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⣽⣯⣡⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠟⠉⠛⢦⣛⡶⠋⠉⠛⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣤⣌⡉⠓⠀⠘⠋⣁⣤⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⡟⠆⢠⡐⠻⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢀⣾⠿⣛⣛⣛⡿⣷⡄⢠⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢤⣄⣀⡀⣀⣠⣾⣿⣿⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣜⠁⠻⣯⣿⣭⡟⠈⣣⣿⡿⠀⠀⠀⠀⠀⠀⠀⠳⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⣬⣿⣭⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣮⣭⣭⣝⣛⣩⣭⠍⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⢸

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

⠀⠀⠀⢀⣤⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦⣤⡀
⠀⠀⠴⠟⠛⠉⠉⠹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣯⣄
⠀⠀⠀⠀⢰⣿⣿⡋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣏⠉⢦⣀
⠀⠀⠀⠰⣿⢿⣿⡅⠀⠘⣿⣷⣦⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⠉⠢⣾⣿⣷⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⢹⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠠⠤⠀⠀⠀⠀⠠⠤⠀⠸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⠀⠀⠀⠀⣷⣤⠀⢻⣿⣿⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣛⣟⣜⢳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡞⠋⣩⠉⠙⢿⣧⣽⣻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠻⠻⣦⡀⠠⠀⠀⠈⠡⢘⡁⠄⠙⠻⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣦⡀⢀⡔⠒⠰⠂⠀⡠⢾⣿⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠟⠿⣿⣿⡋⠀⡀⢴⣼⠿⠛⠋⠉⠹⣿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡆⢈⠟⠈⠀⢀⠻⣤⠃⠀⢀⠀⣄⢘⠁⠆⡜⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠁⡺⠀⢸⠛⣄⠸⣏⢸⣎⢺⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠠⡄⢿⠑⢿⡕⠬⣿⣽⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠈⢀⡘⠳⡄⣻⣄⣻⣮⣽⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠴⡶⠹⣦⡀⠻⣿⣏⣺⣿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⣄⣠⣴⣷⣄⣿⢻⣶⣿⠙⣿⣻⢟⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡍⣿⡿⠿⣥⡾⠹⣷⣴⣿⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣲⣷⠟⠛⢾⣼⠷⣎⠻⢽⣿⣾⣼⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣼⢟⣄⣀⣾⢻⠛⠿⢿⣵⣶⣭⣻⣿⣟⠻⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢹⡟⠛⠠⢄⠀⠉⠛⢿⣿⣷⣬⣵⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⣿⡻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡿⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢻⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣶⣂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡼⠛⣿⣛⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⢢⣧⢸⣿⣇⣿⣏⣿⢻⣿⡗⠯⡂⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠡⢹⡿⠋⠉⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠐⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢠⡀
⠻⠿⠛⠷⠋⠀⠈⠛⠻⣿⣿⣿⡿⠛⢻⢿⣟⠛⡛⠋⣁⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡇⠀⠀⠈⠉⣿⣿⡶⣿⣷⣶⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠆⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⢣⣦⣾⣿⣿⣤⣴
⣶⡀⠀⠀⠀⠀⠀⢀⣀⣼⣿⡧⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⢠⢏⣿⣿⣿⣿⣿⣿
⣿⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⢠⡆⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⢀⣤⡞⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟⢡⣿⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣄⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⡀⠿⠿⠦⠾⠿⠯⡄⠀⠀⠀⠀⠐⣿⣿⡁⠀⡀⠀⠈⠙⠛⠉⣀⣾⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⣶⣶⣶⠤⠄⠒⠂⠐⠒⠀⠀⠀⠀⠀⠬⠉⠉⠀⠂⠀⠀⢀⠀⣤⣉⠉⣉⣭⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠝⠛⠛⠛⠛⠋⡉⠭⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣻⣾⣿⣿⣿⣿⣷⣦⡄⠃⠀⠀⠸⠁⠀⠀⠀⣠⣤⣤⣶⣶⣶⣿⣾⣦⣦⣶⣿⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣴⡀⣠⣶⡀⠠⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣻⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣻⣻⣻⣟⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠦⣤⣀⡴⠞⠋⠙⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣀⠤⠤⠒⠒⠒⠒⠒⠒⠲⠦⠤⣤⣀⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⠛⠭⠥⣤⣄⣈⣹⣦⣀⣠⡤⠤⠤⢤⣤⠤⠴⠟⠛⠉⢹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣀⣠⡤⠶⠖⠒⠛⠛⠛⠒⢶⣤⡤⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠗⠋⠉⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⣠⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⡴⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡄⠀⠀⠀⠀⠀⠀⢀⣎⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢰⠀⠀⠀⠀⠀⠀⢠⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡏⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡇⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⡜⠀⡠⠄⢰⠇⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠈⠳⢦⣠⠞⣀⡼⢄⣡⣞⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠠⠄⠙⡄⢠⠜⠁⣀⡟⠛⣛⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠐⠋⠀⢠⣼⣾⠀⣴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣰⠏⣐⠁⠀⠃⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣠⡴⠊⠀⠀⡟⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠏⠀⠀⠠⠚⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⡀⠀⢀⣠⠶⠾⡇⠀⠀⠀⠀⠀⢀⠇⠀⠀⠤⠀⠠⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠠⡏⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⣄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠲⢦⡀
⠀⠀⣀⣤⠤⠄⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢆
⠀⣀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚
⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠉⠑⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠁
⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠙⠓⠒⠒⠚⠛⠓⠒⠒⠒⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠀⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢰⣂⣀⣀⣀⠀⢐⡆⠀⠀⠀⠀⠐⠀⠉⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠉⠉⠉⠉⠓⠃⠀⠀⠀⠀⠀⢀⠎⢠⠎⠀⠀⠀⠀⠀⠔⠒⠒⠒⠒⠠⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡜⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠔⠀⠀⠀⠀⢀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⣿⣿⠟⠀⠀⢮⣀⣈⣉⣉⣉⣩⡇⢀⣄⠀⠀⠐⠀⠢⠤⠤⠤⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠳⢄⠁⠀⠀⠀⠀⠈⠀⠀⠀⠛⣻⡟⠀⠀⢻
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠀⠀⠀⠀⠀⠀⠀⣴⣏⣀⠀⠀⣸
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣠⣿⣿⣿⣧⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣾⣧⣄⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠹⣇⠀⠀⣴⣏⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣀⣀⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⠀⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⣴⣋⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉⣉⠡⠊





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠒⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠦⢤⣀⡀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢧⡀
⠀⠀⠉⠉⠉⠉⠉⠛⠒⠲⢤⣉⠙⠲⣄⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠂⠀⠀⢄⡁⠢⣀⠀⠀⢇⢀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠙⢦⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⢪⢂⠈⠳⡀⠘⡈⡄⠀⢧
⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠒⠲⢤⣌⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠇⢆⠀⠹⡄⡇⣷⠀⠸
⠀⠀⠀⠠⠁⠂⠈⠉⠉⠉⠓⠢⢄⡀⠈⠙⢦⡀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠆⡀⠀⠀⠀⠀⢸⢸⠀⠀⣷⡇⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢦⡀⠘⣎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⢀⡇⣸⠀⠀⣿⠃⣹⡇
⠀⠀⠀⢀⣠⣤⣤⡀⡀⠀⠀⠀⡀⠀⠀⠹⣄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡜⢠⠇⠀⢀⡟⢠⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡈⢦⡀⠀⠙⡄⠀⠀⠈⢧⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⠎⠀⠀⡜⠀⡎⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠑⢆⠀⢹⡀⡀⠀⠀⠙⣦⡀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⢀⣀⡤⠚⠁⡴⠋⠀⢀⠞⠀⡜⢠⡇
⠀⠀⠀⠀⠄⡀⠀⠀⠀⠙⠒⠋⠀⠈⣇⠐⢄⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⢀⡴⠊⠀⢀⡴⠋⢀⡜⠁⡾⠀⠀⣸
⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠸⡄⠀⠑⠤⣀⡀⠀⠉⠓⠶⢤⣤⣀⣀⣤⡤⠴⠚⠉⠀⢀⡴⠋⠀⣠⠎⠀⡼⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠙⢦⡀⠀⠀⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⣠⠞⠁⢠⠞⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣄⣀⠀⠉⠙⠚⠉⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⠀⢠⡎⠀⣠⠴⠋⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠶⠶⠶⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠶⢲⡆⠀⠘⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣤⣀⣤⠴⠦⣄⡀⠀⠀⠀⠀⠀⢁⣴⠋⡀⠀⣴⣏⣤⡄⠀⣴⣋⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠙⠏⠁⠚⠉⠀⠀⠀⢸⡇⣀⡀⠀⠀⠀⠛⠛⠋⠁⠀⠉⠉⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⣀⡀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠙⢳⡄⠀⠀⣠⡤⠴⠶⠶⢦
⠀⠀⠀⠀⠀⣽⠆⠀⠀⠈⢮⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡶⢿⣤⣤⣤⡄⠀⠀⠀⠘⢿⣦⣼⠃⠀⠀⠀⠀⢘⡷⠶⠤⣤⣀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⢸⡷⠶⠶⠞⠋⠁⢀⠀⠤⣄⡀⠈⠙⢦⡤⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠈⢷⣶⠒⠒⠦⢤⣀
⠀⢀⣠⣾⣅⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⣾⠁⢠⠶⢦⣄⠀⢀⣀⣠⡾⠁⠀⠀⠀⢻⡄⢀⣴⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠸⡏⠳⡄⠀⠀⠈⠉⠓⠦⣄
⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡇⠀⣿⠀⠀⣹⡦⠈⠉⠁⠀⠀⢢⡶⠃⠘⣷⡾⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⡄⠀⠀⠀⠀⠀⠈
⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠛⠶⠚⢿⠃⠀⠀⣰⠀⠀⢸⡇⠀⢰⡿⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷
⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠘⠦⠤⠟⠁⠀⠀⣸⠇⢀⡾⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠸⡆
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣷⠀⠀⠀⢀⣀⡠⠋⠀⣾⠃⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡉⠂⠛⠋⠉⠀⠀⣸⠏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋⡳⣶⣶⣤⣤⣴⡿⠀⠀⠀⠀⠀⢀⡍
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣴⠾⠛⣃⠀⠉⠙⠛⣿⠃⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣠⡶⠟⣉⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠛⢷⣿⣋⣴⠿⣫⣾⡿⠛⢾⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠦⣤⣄⣀⣀⣀⣈⣿⣿⡿⠿⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⡴⠖⠤⠶⠠⠀⠀⢀⠤⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠁⠔⠋⠙⡆⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠠⠃⠀⠀⠈⠉⠚⠉⠁⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠀⠀⠋⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠁⣤⣴⣤⡄⠀⢈⡈⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢈⡷⠞⠛⠛⢶⣄
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣨⠷⠖⠲⡀⠀⠀⠀⠰⣾⣿⢿⣿⣿⣴⣿⣿⣦⣤⣤⣶⣘⡚⠳⠀⠀⠀⠋⠀⠀⠀⠀⠈⠉⠛⠆⣠⠆
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠠⢤⡇⠀⠀⠀⠀⣴⣶⣶⣾⣿⣙⣛⣛⣿⣛⣛⣛⣿⣛⣛⣻⣿⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠠⣀
⠀⠀⠀⢠⠖⢫⣻⣆⠀⠀⠀⠀⡀⠀⠀⠀⢁⣠⣿⣿⣿⣾⣤⣤⣤⠥⠤⣤⣬⣤⣴⣿⣿⣿⣷⡄⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⠃⠀⠀⠀⢸⡀
⠀⠊⠉⠛⢢⠀⢧⢻⣷⣄⣠⣤⣤⣄⣀⣀⣸⣿⡿⣿⣿⡵⢋⣥⣴⠖⢲⣶⣶⣬⡙⢿⣿⣿⣿⠃⠀⣂⣀⣠⣤⣤⣀⣰⣿⢏⡏⠀⠀⠐⠒⠋⠙
⠢⣤⠀⠀⠀⠀⠘⡆⢋⠻⢭⣍⠉⢻⣿⣿⣿⣿⣿⣿⡏⣰⠏⢰⡟⠛⢣⣤⣸⣿⣿⡆⢻⣿⣿⣷⣞⣿⣿⠋⢁⣭⠽⢋⠂⡼⠀⠀⠀⠀⠠⡤
⢠⠞⠁⠀⢸⡀⣴⣌⢟⣷⣶⣿⣧⣺⣿⢿⣿⣿⣿⡿⢰⡏⣠⣿⡟⢛⣃⡀⢸⣿⣿⣿⠀⣿⣯⣿⣿⢿⣿⣰⣾⣧⣶⣞⢛⣄⡆⢸⠀⠀⠛⢦⡀
⣏⠀⠀⠀⢸⣇⣿⣿⣦⠘⢿⣿⣿⣿⣿⣅⠀⠘⣻⡇⣼⣿⣿⣿⡀⠸⠁⠇⢀⣿⣿⣿⡀⢻⡟⠁⢀⣹⣿⣿⣿⣿⡟⢈⣾⣿⣧⣸⠀⠀⠀⠀⡇
⢨⡀⠀⠋⣾⣿⣿⣿⣿⣧⠈⢿⣿⣿⡈⠻⢇⣶⣿⠀⣿⣿⣿⡾⠷⣶⣶⠲⠾⣿⣿⣿⣇⠸⣷⣶⠿⠋⣸⣿⣿⡟⢠⣾⣿⣿⣿⣿⡇⠃⠀⢈⠁
⠀⣽⣠⢰⢿⣿⣿⣿⣿⣿⣧⠺⣿⣿⣿⣶⣾⣿⣿⠀⣿⣿⡿⣷⣴⠋⣈⠑⣦⡾⣿⣿⣿⣄⣿⣿⣷⣾⣿⣿⡿⢦⣿⣿⣿⣿⣿⣿⣳⢀⣹⡋
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢿⢿⣿⣿⠟⢫⣤⣯⢭⣿⣟⣧⠀⠭⠐⣼⣿⣿⣭⣯⣭⠙⡿⢿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⡛⠽⠶⠿⠛⠉⠙⠲⢿⡿⠿⠧⠤⠾⠿⠿⣿⡟⠙⠋⠙⠛⢻⢯⣟⡻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⢀⠙⢿⣿⣿⣿⣿⡿⢡⣶⣶⣿⠋⢀⣀⠓⣤⠈⠻⠆⠀⠀⢺⠀⠀⠘⡿⣿⣷⣦⣾⠋⣁⡤⠉⣀⣈⣉⣅⣉⣉⢿⣿⣿⣿⣿⡿⠉⠁
⠀⡰⢣⣴⣆⠸⢽⣿⣿⠡⣿⣿⣿⣿⣿⣷⠉⠀⣀⠙⠀⠀⢀⡟⣿⠀⠀⠀⣿⣿⣿⣿⣤⠞⢉⣤⠞⢩⣿⣿⣿⣿⡿⠈⣿⣿⡟⠁⣼⣤
⠀⠁⢻⣿⠃⠀⠀⠈⠁⠀⠈⢿⣽⡿⠟⠁⠀⠀⣈⣡⣤⣤⣬⣥⣥⣤⣤⣤⣭⣭⣭⣥⣬⣭⣭⣤⣤⣿⣛⣿⠿⣿⡃⠀⠙⠀⠀⠀⠹⣿⡇
⠀⠀⠈⠃⢦⠀⠀⠀⠀⠀⠀⠩⣴⣴⣶⣶⣿⡿⠛⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⣟⡿⠿⣿⣿⣷⣤⠀⠀⠠⡆⠋
⠀⠀⠀⠀⢧⣀⡀⢠⠀⠀⠀⠀⠉⡟⠉⠙⠁⠀⠀⠿⢿⣿⣿⡟⠑⣿⣿⣿⠟⠁⠀⠿⣿⣿⠿⠿⣿⠟⠛⠛⠿⠋⠁⠹⠿⡇⢀⣀⠇
⠀⠀⠀⠀⠀⠉⠙⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢙⡏⠀⠀⠀⣿⠃⠀⠀⠀⠀⠈⠀⠀⠀⢺⠀⠀⠈⠀⠀⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠁⠀⠀⠀⠈⠳⠖⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠉⠉⠑⡄⠀⢠⠖⠉⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⡄⢁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣠⠜⡇⢸⠀⠀⠣⡀⢰⣄⡀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⣀⡤⠋⢁⡀⢀⠘⠀⠀⡀⢣⢸⠀⠉⠉⠣⡀⠀⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣀⣀⠀⠀⣸⡠⠦⠤⠥⣶⢸⠀⠀⠀⠀⠈⠁⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠈⠉⠺⣷⣤⠖⠒⠒⠺⢍⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⢀⣹⣿⣇⠓⠀⠀⠀⢀⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣿⡿⠿⠋⠝⠋⠉⠛⠇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣇⠀⢫⠙⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡄
⠀⠀⠀⢀⣀⣀⣀⡀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢻⠀⠀⠀⠀⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡄⣀⠀⠀⡠⠋⡠⠁
⡠⠂⢉⣀⡀⠀⠀⠉⠑⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢀⢸⡀⠀⠀⠀⣇⣰⠙⣢⠓⠂⢄⣠⠖⠋⠀⠀⠀⡇⠈⠙⣏⡠⠞
⡇⠸⡁⠀⢉⡆⡠⢄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⢠⣿⡄⠀⠀⠀⣇⢀⠉⢩⡜⠭⢊⠈⢦⠀⠀⠀⣠⣇⢄⠀⢹
⠘⢄⠈⠉⠉⠀⠑⠊⠀⠀⣸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣁⡠⣿⣿⣿⡇⠀⠀⠀⡇⢻⣌⣹⢃⢤⢸⠀⠀⠑⠒⠚⠁⠳⠴⠁⢸⠃
⠀⠀⠙⠒⠤⢄⣀⣀⠠⠾⢅⣀⣐⠀⠋⠀⠀⠀⣀⣀⣀⠤⠔⠊⠁⠀⠀⣵⢸⡟⢙⡁⠀⠀⢠⠀⠀⠓⠑⠮⢥⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⠀⢸⡇⠀⠀⡸⠀⠀⠀⠀⠀⢸⣷⠦⠀⢠⠤⢠⣤⡴⠚⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⠁⠀⠀⠇⠀⠀⠀⠀⠀⡾⢿⠸⡆⢸⠀⠀⣿⣧⠀⢱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣼⣀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠈⠀⡇⠨⠀⠀⠸⡜⢆⠀⢣⠘⠎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠋⠀⠀⢀⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⢠⡀⢣⠀⣄⡀⢀⡴⠈⠉⣦⠗⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢰⡥⠞⠀⢠⠋⠀⠀⠀⣠⠴⠦⠰⠃⠀⠙⣼⠄⠀⠈⢾⠀⠀⢰⠃⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⢀⠃⠀⠀⠀⠀⠀⢘⡀⣠⠴⠃⠀⠀⠀⠀⡇⡠⠀⡠⠄⣠⠞⠁⠀⠀⠀⠈⠀⠀⢇⠀⡄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⢙⠀⣸⠁⢰⡃⢀⠀⠀⠀⠀⢸⡆⡇⠈⣲⠳⣄⣈⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠈⠓⠞⠂⠀⠓⢾⠀⠀⠢⢤⠞⠉⠉⠉⡁⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡛⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡴⠁⠀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⠑⠂⠐⠚⠁

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣦⣠⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠀⢀⣀⣤⣾⣿⣷⡚⠓⠂⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⡤⠊⠁⢈⠂⠀⠀⠙⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣴⡧
⠀⠀⠀⢀⠀⠉⠘⠒⠤⠀⡤⠀⡀⠴⠦⠀⠀⠀⠀⠀⠀⠸⣄⠀⡇⠀⠀⠈⠳⡄⠀⣀⡴⠋⠀⠀⠀⢀⠀⠀⠘⢛⣟⠃⠟⠙⠛⠛⢻⣿⣦⣄⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠷⠶⠶⠶⠓⠚⠉⠀⠀⠀⠀⠚⠰⣦⠀⠀⢹⡶⣤⣀⣀⢢⣄⣮⣵⣿⣷⠶⠶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣀⣀⣠⣀⣾⣿⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢿⣿⣯⡉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⣇⢁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⢻⢠⣏⠂⡴⢐⢸⡇⣴⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠋⠀⠙⠊⠋⠉⠉⣇⠘⢺⡗⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡾⠋⠡⠊⡀⠀⢸⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠘⢷⣶⣆⡘⠁⠀⠸⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠻⢿⣤⣴⣿⣾⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢤⣰⡄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣂⣩⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠶⠚⠙
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢿⢻⣽⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⣀
⠑⠃⠀⡰⠁⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢸⡞⠋⠁⡢
⠀⠀⣸⡇⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⢰⡞⠂
⠀⠀⢿⡇⠀⡞⠀⢀⡔⠀⠀⢠⡄⠀⢡⠀⣧⠀⢠⠰⡟⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠁⠀⣻⡇⢠⡇⢀⡿⠁⠀⠀⠀⠀⠘⡉⣇⣻⠀⡆⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠄⠀⣿⣇⡼⠁⣼⠇⠀⠀⠀⠀⠀⠀⢹⣿⡟⣸⣏⣴⣿⢃⡀⠀⠀⣀⣀⡤⠘⢿⣼⣸⠟⠶⣶⣶⣦⣤⣀⠐
⠀⢠⣿⡟⢁⣼⠇⠀⠄⠀⠀⠒⠀⠀⠀⠙⠷⠟⠿⠿⠛⠚⣠⣴⣾⣷⣶⣶⣆⣀⣿⡗⠇⢤⣼⡾⡿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠⣤⣀
⠀⣸⠟⡴⠟⢁⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡏⢀⣹⣧⡷⣀⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⣰⠟⣭⣷⣦⣽⣧
⢰⣿⣫⠔⠚⠋⠉⠀⠈⠉⠙⢶⡀⠀⡀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠛⠒⣶⣿⣶⣿⡿⠙⢿⣮⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣘⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⡖⠔⡄⠀⠀⠐⠄⠀⠀⠀⠈⠀⠀⠐⠀⠀⠠⠀⣾⣿⣿⠟⠫⠁⠈⠲⡟⢿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠿⠿⠋⠈⠉⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢀⣀⣀⡀⠀⢤⢀⣀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠐⠀⠀⢀⣀⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣛⣋⣭⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠙⠻⣭⠭⠿⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠉⠀⠀⠀⠠⢤⣤⣄⡀⠀⠀⠙⢿⣿⣦⣴⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⡏⠀⡀
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠄⠀⠀⠀⣴⣀⣀⣀⠀⣀⠀⠀⠴⠤⠀⠀⠀⠀⢀⡒⠛⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⣠⢠⣾⢀⡄
⠀⠀⠀⠀⠀⢠⡦⠀⡀⠀⡜⠛⠀⠀⠀⢰⡇⢿⣇⡞⢀
⠀⠀⠀⠀⠀⠋⢤⠀⠛⣾⡟⠆⠀⠀⠀⢸⣃⣇⢸⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠠⠀⠀⠀⠀⢠⣿⡧⠀⠀⠀⠀⠘⡋⠉⠻⣣⣾⣀⡀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⡇⠀⠀⠀⠀⠀⠁⠀⢠⠈⠹⠿⠃⠀⠀⠀⠀⡠⠊⠁⣀⠴⠂
⠀⠀⠀⡤⢀⡀⣿⠀⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⢀⣼⠋
⠀⠀⠀⠀⣀⠙⡷⣴⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣄⣤⡐⠀⠀⢠⣼⠉
⠀⠀⠀⢱⠋⣼⣿⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠝⠛⠙⠀⠀⠈⠊⢅⣀⣀⣠⠤⠐⠀⠀⠂
⠀⠀⠀⠘⣰⡟⢻⡏⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⡵⠂⠀⠀⠀⢤⣤⣤⣬⣿⣿⣧⣀⠀⠀⠀⠈
⠀⠀⠀⢰⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣏⣹⠛⢿⠷⠤⠄⠀⠀⢀⠠
⠀⠀⠀⢸⢷⣾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠖⡠⠐⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡇⠀⡜⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⡃⠀⡃⡾⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠈⠳⢦⣙⣽⠾⠧⠇⠀⠀⠀⠀⠀⠀⣶⣥⡒⠂⡐⢾⣶⢄⣀
⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠛⠛⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⢸⠀⠀⠛⡀
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠁⠉⣴⣯⡆⠀⠀⠀⠀⠀⠰⣀⠿⣿⣦⡟⡂⢴⠈⠖
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠂⠋⠈⠁⠀⠄⠱⠈⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠤⡀⠈⠁⠀⠀⠀⣠⠱⢦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠁⠀⠀⠀⠀⠀⠸⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠒⢄⡜⠁⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠤⡀⠀⢀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠈⡄⠀⢣⡀⠀⠀⠁⠀⠀⠀⠀⡤⠊⠉⠀⠈⠉⠓⠦⣀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⠼⢒⡉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡤⠊⢡⣾⡿⠃⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢧⡀⠀⠠⠀⢀⡴⠋⠀⠀⢙⠋⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠉⠓⠒⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠘⡔⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠹⣄⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠀⠀⠀⠀⠀⠀⢈⣷⠤⠤⡴⠋
⠀⠀⠀⠀⠀⠀⢣⠀⢹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠡⢐⣀⣂⣀⣀⣠⡤⠤⢒⠭⠊⠀⢀⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠳⣄⡉⠀⠄⠁⠀⠀⢀⡠⠖⢋⠀⠀⠈⠀⠒⠒⠒⠒⠒⠀⠉⠀⠀⠀⢠⠞⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡳⠒⢒⠒⠉⠁⠀⠀⣠⠲⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⢸⠀⠀⢀⠖⠒⢇⠀⠀⠈⠉⡙⠛⠙⠛⣋⡉⠉⠁⠀⠀⠀⣀⣀⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸⠀⠀⠸⡄⠀⠀⠁⢀⣀⣀⣁⣁⡰⠋⠀⠙⡆⠀⢀⡴⠋⠉⠉⠉⠳⣄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡼⠀⢀⠖⠛⠂⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠐⠅⢀⡞⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠒⠒⠚⠛⠛⠹⡀⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢄⠆⠀⠑⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠆⠀⠀⠀⠀⠀⠀⠢⢤⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⠔⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⣄⣀⡤⢦⡤⢄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠴⠚⠁⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠁⣿⠀⠀⠀⠀⡎⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⢤⣦⡤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⣀⡏⠀⠀⠀⠀⢇⡄⡀⡜⠑⠢⠤⣤⣄⣀⣀⣀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠳⠦⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣾⡿⣽⣿⣿⣶⣷⣤⡄⠀⠀⢠⣿⣿⣿⣶⡆⢸⣿⣿⠀⠘⢿⣿⣷⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢿⢽⣿⠀⠀⠸⣿⣿⣿⣿⣳⣸⣿⣿⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣿⣿⣿⣿⡟⢼⣿⣿⣿⠀⠀⢠⢻⣿⣿⣿⣿⠻⣿⣿⡆⠰⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣾⣿⣿⠁⢀⣲⣶⣷⠀⠀⠀⣾⣿⣿⣿⡏⠀⢻⣿⣷⡦⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⢸⣼⣿⣿⠀⠀⢷⣿⣿⣿⣿⣇⣸⡎⢻⣿⣷⡠⡄⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡿⣿⠣⠀⠀⢸⢺⣾⣿⡅⠀⢸⣿⣿⣿⣿⡟⡿⣧⣾⣿⣿⢗⡂⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠠⠤⠀⠸⠾⢿⣿⡇⠀⠸⠿⠟⠛⣿⢧⡟⣸⣿⣿⡿⠀⠀⠀⢯⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣿⠀⢰⣶⣾⣿⠃⠀⠀⡀⠀⢀⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠈⠻⣿
⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠨⠍⠀⠸⢿⠿⠅⠀⠀⠀⠇⠘⠀⣿⣿⣿⣿⣿⣿⠇⠀⠆⠆⠀⠀⣿
⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⢈⡁⠀⠀⢨⡉⡁⠀⠀⠠⣀⣰⡀⣿⣿⣿⣿⣿⠏⠀⠀⡀⡃⠀⠀⣿
⣈⣶⣦⠀⠀⠀⠀⠀⠀⢾⣏⢸⡟⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠙⠈⣿⡆⠀⠀⠀⠀⠀⠀⠐⢨⣍⠒⠉⠻⠁⣺⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣿
⠀⠈⢧⣇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡍⠈⠀⢀⣄⣀⣠⣤⣤⣀⡄⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿
⠠⠀⠀⣿⣦⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⠀⠀⢸⡟⣿⣿⣿⣷⣶⣶⠸⣿⠟⠃⠀⠀⠀⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡇⠀⠀⠀⠀⠀⠨⠽⠟
⠀⠀⠀⢙⠻⣷⡀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠉⠻⠿⢿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠠⣶⣄⠀⠀⠀⠀⠀⢀⣼⡇⠈⠁
⠰⣦⣤⣤⣿⣿⣿⣿⡆⠀⣀⡀⢀⡀⢀⡀⠀⠀⠀⣠⡀⣀⠀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠐⠒⠀⠀⢀⣾⠇
⠀⠀⠀⠈⠉⠁⠀⠉⠁⠀⠀⣰⡄⢰⡌⠉⠁⠀⠁⠉⠁⠉⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠉⠳⠀⠀⠀⠲⣶⣂⡆

⠀⠀⠀⠀⠀⠀⢀⠴⠚⠛⠛⠶⠚⠋⠙⠓⠤⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡀
⠀⣠⠔⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡀
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠀⣀⣀⢀⣀⣀
⠀⠙⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠙⠁⠀⠉⠋⠁⠈⠉⠀⠈⠉⠉⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠘⠦⣄⣀⡀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣀⡴⠃⣠⣤⣄⠀⠀⢀⣀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠘⢧⣄⣾⠀⠀⠳⠼⠀⠀⢳⡀⠀⣀⡀⢀⣤⡀⠀⣠⢤⡀⠀⣀⡴⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠉⠉⠀⠀⢀⣤⡀⠀⠉⠀⢧⡀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠉⢇⠀⠀⢰⠁⠀⠸⠀⠀⠀⠻⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠶⢸⠀⠀⢰⠶⠀⠀⠁⠀⠀⠀⢇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢠⡎⠀⠀⢸⠠⠀⠚⠀⠀⠀⠀⠈⢁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠉⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠏⠀⠄⣀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠖⠒⠚⠛⠲⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⢀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣹⣶⣶⣶⣯⣤⣤⣤⣤
⣿⣿⣏⣉⣩⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠖⠀⠀⠀⠀⠀⠀⢀⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠴⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢰⣿⡧⢀⣤⠀⠀⢠⠾⣅⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣷⣿⠏⠀⠀⡏⠀⠈⠳⣄⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣿⣿⣿⣿⣿⡶⠆⠀⢇⠀⠀⠀⠈⠳⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢹⠙⠛⠀⠀⠀⠈⠢⣀⠀⠀⠀⢈⡳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢄⡀⠀⠀⠀⠀⠀⢀⡀⠤⢄⣀⠀⠀⠀⠀⠀⢀⣉⣽⣭⣉
⠀⠀⠀⠀⠀⠀⢠⠾⣅⠀⠀⠀⠀⠑⣄⠀⢠⠾⣅⠀⠀⠀⠀⠙⣆⠀⢠⢾⣿⣿⣷⠀⠈⣹⢦
⠀⠀⠀⠀⠀⠀⡏⠀⠈⠳⣄⠀⠀⠀⠘⡆⡏⠀⠈⢳⣄⠀⠀⠀⠈⡆⡏⠈⠙⠻⣟⠙⠉⠁⠈⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠳⣄⠀⢠⠇⣇⠀⢀⠙⠟⠳⣄⠀⢀⠇⣇⢀⣤⣤⣌⠳⣄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢙⡷⠏⠀⠘⢶⡋⠀⠀⠀⢈⣳⠎⠀⠘⢾⣿⣿⣿⠤⣼⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣷⣞⡉⠀⠀⠀⠀⠀⢈⣙⣗⣞⣉⠀⠀⠀⠀⠀⠈⠙⠒⠛⠉
⠀⠀⠀⠀⠀⠀⢠⠾⣍⠀⠀⠀⠈⠓⣄⠀⢀⣶⣍⢠⣿⣦⠈⣱⣄
⠀⠀⠀⠀⠀⠀⡏⠀⠙⠳⣄⠀⠀⠀⠘⡆⡏⠸⣿⣿⣿⣧⣾⡟⠘⡆⠀⢠⢴⣶⡆⣶⣶⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢳⣄⠀⢀⠇⣷⣶⣿⣿⣿⣿⣿⣶⢶⡇⠀⢸⣬⡁⠻⢊⣥⡞
⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⢙⣷⠞⠀⠘⢦⡼⠛⢻⠛⠻⣷⠞⠀⠀⠀⢹⠛⠿⠟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉⠀⠀⠀⠀⠀⠉⠓⠚⠒⠉⠁

⠀⠀⠀⠀⠀⢠⠐⠀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⡂⣄⠀⠀⣸⠀⡈⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠁⠀⠀⣀⠀⠀⠀⢀⡶⠀⠀⡄
⣀⡄⠈⣿⠀⠀⠀⠀⠀⠀⠤⠀⡰⣄⠀⠸⠇⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠈⠐⢶⠗⠊⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⠰
⠶⠦⠛⠉⠹⣋⢛⣴⣦⣤⡆⠄⣐⠀⢣⠰⠛⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠔⠊⢁⠀⠀⡀⠀⠀⠈⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠙⠂⠘⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠮⠡⡀⠀⠀⠈⠉⠉⠻⠂⠀⢠⣄
⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⠄⣀⣀⡄⠔⠂⠈⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⣀⢀⣀⡀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠁⠉⠑⠖⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠠⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠰⠿⡓⠡⠆⠂⠤⣥⣖⡂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠑⠦⡖⠀⠀⠀⠀⠀⠀⠀⠃⠐⢠⡼⠉⢝⠲⠀⢀⣶⠠⠔⠀⠁⠀⠀⠆⠈⠁
⠀⠀⢠⢤⡤⣶⢏⡌⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠙⢤⠤⣶⢺⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢄⠚⣶⡄⠀⠀⠀⣆⡁⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠙⢤⡆⡀
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⠌⠠⠋⠀⠉⠀⠀⣰⢃⡃⠀⠀⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠠⠀⠀⠀⠘
⣤⣤⣦⣜⣀⡛⠀⠈⠈⠀⠐⢀⠊⠀⠀⠀⢰⡾⠞⠁⠎⠀⠀⠐⠀⠀⠀⡆⠐⠀⠈⠐⠁⠒⠀⠉⠀⢀⣠⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠤
⠈⠁⡉⠀⠐⠃⡤⠀⠤⠠⠞⠁⠀⠀⠀⠀⠛⠁⠈⠂⠀⠀⠀⠀⠀⠀⠍⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⣀
⠠⠐⢀⠄⠀⠀⡡⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⣠⡀⢄⠀⠠⠁⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⢰⡆⡀⠀⠘⠿⠃⠂⠀⢈
⠟⠀⢁⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠈⡴⠀⢀⡀⠀⠀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⢀⡀⠢⠠⠈⠓⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡄⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡄⠀⡀⢄⠀⠀⠀⠀⠀⠀⡂⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢡⠆⠀⠀⠀⠀⠀⠘⡂⠀⠀⠠⢾⡿⠈
⠀⠀⠀⠀⡨⠣⠄⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢣⠹⠌⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠈⠀⠀⣠
⠀⠀⠀⠘⠀⠀⠂⠀⠄⠀⠀⠀⢀⣰⠆⠀⠀⠀⣀⠀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⣀⡔⠁
⠈⠀⠀⠀⠀⠄⠤⠊⠈⠀⢣⡀⠠⠁⡆⣀⠖⠉⠀⠁⠀⠈⢋⡋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠔⠈⠀⢀
⠀⠈⠐⣄⠀⠀⠀⠀⠀⢈⠄⠈⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢰⡵⠂⠉
⠺⠃⠀⠁⠐⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠠⠀⠉⠊⠉⢳⠂⠀⢄⡰⢾⠀⠀⠚
⢂⠒⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈⠺⠻⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠃⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠐⠶⠠⠖⠀⠀⠀⠀⠀⠀⠀⢰⣤⣦⠄⠠⠦⢶⣦⠀⠀⠷⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠀⠀⠐⠁⠃
⠀⠀⠀⠀⠀⠀⠙⠳⣯⣥⡀⡾⢻⣿⣿⣸⡇⡇⠀⠀⠀⢸⡁⠀⠀⣤⠀⠓⠀⣴⠟⣠⡄⠥⣄⢦⢰⠀⠀⠂⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⢋⠺⡍⠹⢿⠇⡟⠈⠀⠀⠘⠂⠀⠀⠀⠀⠀⢰⣷⠿⣹⢠⡴⠞⡏⣠⡕⡞⣀⠀⠸⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣵⠀⢹⣇⣷⣠⣄⡔⣸⣆⠀⢠⣏⣤⠾⠟⠉⢲⡷⣿⠀⠀⣿⣷⣿⡾⠋⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⡙⢧⣟⠁⢉⠁⠨⠏⠋⣩⣾⡟⠋⠁⠀⠀⠀⢠⣧⣷⠊⠀⣻⣏⣴⠗⠛⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡼⢠⣀⣕⣦⡈⠳⣼⣥⠼⠛⢹⣿⢿⣄⠀⢀⠀⡶⠀⢸⣵⣦⠠⠀⠀⡸⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢴⣿⡗⠁⠀⠀⠈⠉⠛⢷⣾⣅⣀⣀⡤⡿⠛⠀⠈⠓⠦⣬⣁⣤⠟⡟⠏⠘⠃⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⣿⢿⡇⠀⠀⢀⠀⠀⠀⡦⡄⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢁⢀⣸⡈⠀⣦⡥⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠐⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⠀⠀⠀⠀⡤⢀⣀⡀
⠀⠈⣈⣤⣤⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣨⡝⢆⢀⡀⠀⠀⢠⡆⠀⠀⢈⣄⡀
⠀⠀⠉⠀⠰⠷⠄⠀⠀⠀⠈⠙⡏⢀⡅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⣀⡈⠩⢓⡏⡈⠀⠀⠀⢠⡀⠀⠕
⠀⠀⠀⢈⣵⡆⠀⠀⠀⠀⠀⠀⠘⣾⡇⠀⠀⠀⠀⠀⠡⠀⡀⠀⠈⠐⢩⢠⣶⠀⠉⠁⠀⠀⠀⠘⠃⠂⠒
⠀⠀⢈⡧⠄⣠⡔⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⡎⡔⠀⠀⣁⣤⠄⢅⠉⠁⠀⠀⠀⠀⠰⢢⠀⡀⠙⠷
⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⡀⠀⡿⠁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠈⣽⣶⠃⠀⠀⠀⠀⣾⡗⠀⠀⠀⠀⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨
⠀⠀⢀⠱⢟⣛⠀⠀⠀⠀⠈⣱⡀⠀⠀⠀⠀⠄⠀⢀⡄⠀⠀⠀⠀⠀⠀⢈⣻⡩⢊⠁⠄⠀⠀⠀⠠⡀⠠⠒
⠀⠀⠈⣇⠀⠉⡤⠀⢀⣶⣿⡿⠅⠀⠀⠉⠉⢀⠀⠰⠀⠆⡂⠀⠀⡰⠀⠾⢿⡿⠀⠀⠀⠀⠀⠉⠂⠀⢀⠈
⠀⠀⠀⢹⣶⡟⠁⠀⠘⢛⣿⣿⡆⠀⠀⠀⠀⠃⠀⠀⠀⠀⢳⠀⠀⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣌⡤⢀
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⢱⡁⠀⠀⠀⠀⠀⠀⠀⠉⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠙⠛⠿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠛⠀⣀⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⣿⡆⠀⠀⠀⠀⠐⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⡿⠋⠀⣠⣴⣆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠉⠛⠛⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⣿⣷⣄⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀
⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣾⣿⣯⡀⠀⠙⢿⣿⣿⡆⠀⠈⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦
⣾⡿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠻⣿⣿⣦⡀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢀⣴⣿⣿⣿⣧
⠙⣿⣦⠀⠐⢶⣾⠂⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣦⣤⣴⡾⠟⠀⠀⠘⠛⠉⣀⣼⣿⡆
⠀⠈⠻⣷⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣧⣤⣤⣀⡀
⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⣿⣶⣤⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠘⡷⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⣿⣿⠿⣿⣦
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣶⣤⣤⣶⠟⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠋⠀⠀⠸⠟⠁⢀⣾⣿⣷⡀
⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡁⠀⠀⠀⠀⠀⠀⠠⡿⠿⠛⣿⣷
⠀⠀⠀⠀⠀⠹⣿⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠋⠁⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡀⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣏⠀⠈⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⢻⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⠿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⠁⠀⠀⠀⠀⠉⠉⠁⡀⠀⣿⣿⠃⠀⣠⡀⠀⠀⣠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣾⣿⣿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⣷⣤⣶⣿⣤⣾⣷⣴⣿⣿⣷⣾⣿⣿⣶⣾⣿⣷⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⠿⢿⣿⣿⡟⠋⠀⢈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠙⠛⠛⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟

⠀⠀⠀⢀⠐⢉⠁⠀⠄⠐⠠⢤⡀
⠀⠀⠠⣈⣀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠠⠀⠐⠂⠄⣀⣀⠀⠀⠠⣤⣤⣀⣰⣀⠆⣤⠀⡦⡄⠄⡄⡠⠄⢰⠀⠄
⠀⠀⠒⠀⠙⠋⠛⣻⠗⣿⠧⠄⠀⡀⠀⠀⠀⠀⠈⠢⡀⠣⠀⠂⠀⠨⠂⢸⠀⠀⠙⠃⠉⠈⠉⠈⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣰⣶⣿⣷⣠⢢⣶⣿⣻⣷⣶⣄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⡆⠀⠀⠀⠁⠀⠀⢈⣠⣄⣬⢿⣛⣓⣤⣾⣦⣈⠂⠀⠸⢿⣽⣿⢿⣿⣿⣽⢿⣯⣽⣭⣿⣟⣳⣶⣶⣮⣧⣄⣠⡏⡍
⠀⠀⠈⢠⠻⠀⠀⠀⢄⣾⣿⣿⣿⣷⣈⣵⣍⡿⣿⡿⣓⠑⣄⣤⣉⣛⡻⠿⠿⠿⠿⠿⣿⢿⣟⠿⠿⠿⢿⣿⠿⠿⠷⠾⠶
⣀⡀⢠⡈⠠⠀⠀⠠⣻⣿⣿⠿⣹⣿⣿⣿⣿⣿⣿⣧⣰⣶⡆⠈⣭⣭⠛⠛⠉⠁⠒⠒⠃⠂⠯⣿⣛⡓⡎⠌⠆
⠛⢥⡼⣛⡃⣶⣻⣆⢐⣿⣛⣿⣛⣱⡶⠶⠾⠶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣛⣛⢛⡛⣛⣛⣛⣛⣳⡝⢒⣀⣀⣀⣀⡀
⠀⠀⠨⠍⠩⠉⠉⠉⣉⣉⣉⣙⠻⣿⡿⠯⠭⠭⠍⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⠛⢛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠂⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠴⠃⠀⠂⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⣓⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⢀⢴⠁⠀⠀⠀⢠
⠀⠴⠾⠄⠀⠀⠀⠀⠎⡤⠄⠀⣜⠀⠀⠀⠰⠄⠀⠀⠀⣌⢠⠤⠀⠠⡜⠀⠐⠚⠧⠠⡄⠀⠠⠒⢢⠄⠀⢠⡃⠀⠀⠀⠐⠆⡀⠀⠀⡂⡤⠄⠀⢤⠃
⣄⡠⣒⠴⠷⡄⠀⠀⢳⡏⠀⣧⠐⢦⣀⡤⡢⠼⢦⡀⠀⠀⢺⠃⢈⡆⠐⢤⣀⢔⡢⠾⢦⠃⠀⠐⣾⠁⢰⡄⠲⣄⡀⢖⠤⠷⣄⠀⠀⠐⡞⠁⣱⠀⠲
⠡⢍⠁⠀⣰⣡⡤⠰⠆⠀⢠⣈⠀⠀⠹⣉⠀⢀⣸⣥⠄⠰⠄⠀⣠⣁⠀⠈⢌⡉⠀⢀⣎⡤⠄⠶⠀⠀⣄⡁⠀⠈⢎⡁⠀⣀⣋⡤⠀⠦⠀⢀⣈
⠀⠁⡄⢀⣽⠀⢠⡀⢘⡅⠀⢀⡀⠀⠈⢁⡔⢀⣇⠀⢄⠀⢸⡀⠀⢀⠀⠀⠈⢡⠆⣨⡅⠀⣀⠀⣻⠀⠀⡀⠀⠀⠉⣠⠀⣸⡄⠠⡀⢀⣇⠀⠀⡀
⠀⠀⢛⠉⠉⢡⡀⣀⣋⣀⠀⣀⢯⠁⠀⠐⡏⠋⠉⣤⠀⣠⣃⡀⠀⡘⠏⠁⠀⠘⡏⠋⠁⣄⢀⣸⣁⡀⠀⡘⡍⠀⠀⢚⡹⠉⢡⡄⢀⣜⣀⠀⢀⠻⠉
⠀⡀⠲⠀⠀⠀⠐⠈⡱⠇⣀⡄⢙⡃⢀⡐⠢⠀⠀⠀⠂⢈⠹⠁⣠⠈⣛⠀⢀⠐⠂⠀⠀⠀⠂⢉⠾⢀⣠⠈⣛⠀⣀⠒⠀⠀⠀⠐⠊⡉⠟⢀⡀⢙⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⡆
⠀⠀⠀⠀⠀⠀⣿⢿⡿⠿⠏⣽⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⣄⡀⠀⠀⢀⠙⣦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢟⣼⣷⣇⢰⣶⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢸⣿⣿⣿⣿⣦⢻⣧⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠘⠿⠛⠿⢿⡿⠿⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣠⣤⣼⣿⣿⣿⣷⣬⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣤⡴⠾⠛⠛⠛⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠛⢷⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⡟⠁⠀⠀⠙⠀⠀⠀⠁⠈⠹⢿⣶⣤⣼⣿⣿⣿⡿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣼⠋⢩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡋⠻⣿⣿⠿⠀⠀⠈⠛⣉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣶⣆⠁⣦⡀⠠⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠚⠃⠀⠈⠉⠀⠙⠻⣿⣦⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣠⣾⣿⡧⠉⠀⠉⠀⠀⠆⠀⢀⠀⠟⣀⠀⠀⠀⠀⠀⢤⡤⣤⠀⠀⠀⠀⠐⠀⢁⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣡⡾⠍⢋⠉⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣏⡼⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠉⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⠒⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣤⡴⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢼⡿⠛⠀⠀⠻⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠚⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡀⠚⠁
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠰⠒⠀⠀⠀⠺⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⡠⠀⡤⠐⠠⠄⠀⠀⠀⠀⠁⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⣠⡤⠒⠀⠀⠀⠀⠘⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢿⡎⠰⣶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡀⠀⠀⠀⠀⠡⢰⣾⣿⡟⠀⠀⠀⠀⠀⠐⠷⠘⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⢀⠀⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁
⠀⠀⠀⠀⠀⢠⡄⠳⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢷⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣹⣿⣿⣷⡀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠐⠈⢷⣵⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠠⠦⢀⡼⠄⠉⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠰⣦⣄⣄⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠐⠠⠈⢿⣿⣿⣿⣿⣿⣿⠁⢀⣴⠟⠛⠋⠉⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢠⣄⣀⡀⠀⠀⠀⠀⠰⠾⢻⣿⣿⣿⣟⣴⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⡿⣿⣿⣿⣶⣀⠀⠀⠀⠀⣾⣿⠿⠃⣿⡟⠀⠉⠀⠀⠀⠀⠀⣀⣤⣶⠿⢛⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⣠⣿⣅⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⢀⣼⣿⠟⢛⣛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⣿⣿⣿⣿⠀⠀⣠⣾⣿⣟⣥⠀⠀⠀⣼⠟⠉⢣⣼⢋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⠀⣴⣿⡿⣭⣹⣿⣯⣽⣿⣯⣤⣴⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠆⢰⣿⠟⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠐⠿⠗⠒⠻⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠛⠟

⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠰⢦⣄⣀⣛⣙⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣭⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⣿⣿⣟⡽⠚⠋⠉⠀⠈⠁⠀⠀⠀⠠⣄⠀⠰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣇⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡲⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⡿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠴⠥⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣧
⠀⢸⡿⣿⣷⣶⣄⡀⠀⣾⣆⠀⠀⡀⢺⣿⣤⠀⠀⢀⣈⣿⠀⠀⠀⠀⠀⢠⠀⢠⣿⣿⣿⡇
⠀⢰⡿⠿⢿⡟⠙⣻⣶⣽⣿⣧⣀⣸⢿⣟⣟⠛⢠⡹⢿⣿⠟⠀⠐⢲⠿⠀⠀⢸⣿⣿⣿⡦⠤⢷⣴⡤⠄
⠀⢰⡖⢓⡦⢤⢾⠇⢾⣽⡀⠙⢩⡧⠚⣳⠁⠀⠡⠀⠀⠀⠔⠂⠠⠞⠲⢀⣠⣾⣿⣿⣿⠯⠏⠳⡙⢣⡄
⠀⢸⣿⣿⣷⡿⠶⢠⣼⡛⣇⠀⠙⠀⠠⢤⣶⣦⣤⠄⠀⠁⠄⠃⠀⠀⢠⠋⠁⠈⠉⠉⠛⡟⡂⠒⣓⠎
⠀⠘⠚⠛⠛⠛⠒⠂⠀⠓⠃⠀⠐⠃⠀⠛⠛⠉⠋⠓⠀⠘⠓⠓⠀⠀⠀⠑⠀⠀⠐⠈⠀⠀⠉⠐⠈⠁

⠀⠀⢠⠄⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⢠
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⠒⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠰⢁⠄⡈⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠻⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⢎⢸⠉⢡⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢡⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢧⠔⠲⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠔⠒⣄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡲⡀⠃⠀⠸⠀⠀⠃⠀⠱⠤⡀⠀⠀⠀⢸⡗⢄⡇⠀⠸⠀⠀⠀⠀⢱⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⡆⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠈⠀⠐⡇⠀⠀⠀⠀⠀⠀⠈⠀⢡⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣀⢀⡀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠤⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡤⠊⠑⢰⠀⡆⠨⢴⢰⠀⠮⠀⠀⠸⠁⣐⣃⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠈⠈⠀⠁⠀⠈⠈⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⡆⠀⡇⢱⢸⢼⢀⣟⡄⡋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠁⠀⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢨⠀⠀⠉⡅⠀⠀⠁⢤⠤⠤⠤⠤⠤⣤⠠⠀⠀⠀⠀⠀⢀⡀⠅⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠐⠀⠀⠀⠰⠸
⠀⠀⠀⠸⠀⠀⢈⠀⠀⠤⡀⠆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⡂
⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⢰⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣇⣀⡀⠀⠀⠀⡀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠆⠸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠠⠖⢠⠜⠉⠣⠀⠀⠉⠓⢤⠤⠥⢆⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡠⠊⢉⣰⠎⡀⠔⠉⠀⠀⠀⡀⠀⠈⠉⠐⢱⡆⡈⠉⠷⠒⠒⠛⠘⠤⠤⠄⠀⠄
⠀⠀⠀⠀⠀⣧⠔⠁⠀⡌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠑⡼⠁⢉⠁⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠺⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⡌⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠃⠀⠀⣰⠠⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⢄⡀⢀⣀⣀⣀⠀⠀⠠⠊⠀⠀⠀⠀⠸⣇⠘⣖⣲⠆⠁⠈⢂
⠀⢀⠀⠀⠀⠀⣿⠄⠀⠤⠂⢁⡁⡀⢀⢀⡀⠀⢉⠐⠖⡀⠒⠐⣃⣤⡈⠉⠉⠀⠊⠉⠉⣨⣀⣀⣀⣀⣀⣀⡀
⠀⠈⠀⠀⠀⠀⡏⠤⡎⠀⣀⠿⢣⠄⠀⠀⠄⣉⠞⠵⡀⢘⠀⠀⠉⠘⠁⠉⠉⠉⠋⠉⢻⣿⣉⣉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠰⠄⠲⠚⠚⢁⠄⠊⠐⠄⠂⢀⠀⠁⠀⠉⣛⡛⢛⠂⠻⣃⠀⠂⠂⠀⣶⣼⢇⡀⡈⣧⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠃⠀⠄⠀⢀⠥⢠⠀⠈⠁⠉⠂⠁⢤⠤⢄⠁⠈⠡⠃⠀⠩⡉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁
⠀⠀⠀⠀⡞⠄⠡⠔⠂⠇⢀⠌⠈⠓⠒⠐⠒⠂⠈⢄⢀⠃⠐⠀⠀⠄⠈⠡⠀⠐⠠⢤⠔⠈⠀⠀⠀⢠
⠀⠀⢠⠘⠀⠀⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠃⠀⠀⠈⠉⠀⠉⠉⠉⠁⠀⠀⠓⠀⠃⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠤⠤⠬⠥⠍⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠼⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡤⠃⠀⠈⢻⣿⣷⡄
⠀⠀⠀⠒⠿⠿⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠉⠉⢀⠀⢸⠀⠀⠀⠀⠀⢛⠛⣿⡆⠀⠀⠀⠀⣠
⠀⠀⠀⠒⠠⠄⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠀⠸⠀⠀⠀⠀⢀⣈⡤⠿⠷⠆⠀⠐⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⡩⠁⢰⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⠋⠀⠒⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡯⠀⡀⣀⠂⠀⠠⢀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢌⣳⣦⣴⣆⣼⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠋⣤⣎⡡⢧⡄⠛⣆⣀⣥
⣶⣤⣆⣠⣦⣴⣿⣶⣾⣿⣿⣷⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⣳⣾⢃⣸⠧⣷⡴⢧⡅⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣾⣿⣶⣾⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢀⡄⡁⡀⠀⠐⠮⠉⠃⠴⢤⣀⡀
⠀⠀⢠⣘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢩⠗⡤⠒⠐⠦⣄⢀⡀⠀⠀⠀⠀⣄⣀⣀⣀⣰⣆⣰⡄⣠⠀⣠⣤⢠⡤⠄⠀⣤⠀⣤⠀⡀
⠀⠀⠬⠙⠻⣿⡿⣷⣶⣶⣶⣶⣐⠂⠄⢀⠀⠀⠀⠀⠠⠖⢦⣉⠙⣆⡰⢦⠀⣉⡙⣏⢉⡏⠉⠡⠿⠿⠿⠋⠷⠳⠱⠿⠏⠐⠿⠞⠧⢲⠀⢂⣀⣀⡀
⠀⠀⠀⠀⠈⠀⠐⠔⢿⣍⠙⣿⠉⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠀⣄⣠⣴⣬⡱⣩⢃⣤⣶⣶⣄⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡀⠀⠀⠤⣤⣄⣀⣉⣇⡜⠇⠀⢠⣤⣀⣀⣀⠀⠶⡀⢀⣀⠀⢀⢸⣿⣿⣿⣿⣾⣧⣿⣿⣿⣼⣿⣿⣿⣶⣲⣶⣶⡶⠾⠶⠶⡤⠤⢤⣤⣬⣤⣤
⠀⠀⡇⠀⢀⡀⠀⠀⠀⠈⠀⣠⣿⣦⣶⣻⢭⣉⣝⣶⣾⣿⣦⣄⠓⠀⠘⠿⢿⣿⣿⢻⣿⣿⣯⣿⢿⣿⣶⣿⣾⣿⣯⣭⣽⣿⣿⣾⣷⣦⣤⣼⣡⡇
⠀⠀⠈⢩⠹⣇⠀⠀⢀⡄⣼⣿⣿⣿⣿⣿⣶⣾⣾⣏⣿⣿⣿⢿⣛⠙⣌⣻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⡆
⠀⠀⢀⠘⠦⣴⡀⠀⢨⡖⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡆⣭⣤⣤⠈⢉⣙⣛⠿⠿⠗⡒⠒⠶⠤⣿⣤⣼⣟⣛⡿⠷⢦⡏⡉⣧⠠⠤⠤⠤⠤⠤
⠾⠀⣿⣷⠾⣤⣫⣤⣀⢹⣿⣿⣿⣶⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣿⣿⣿⣟⣧⣁⠌⠀⢀⠀⠀⠀⠁
⠈⠻⠭⠶⠦⣽⣷⣾⣧⣤⣬⣽⣿⣯⣶⣶⣿⣭⡽⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠭⠉⠉⠯⠿⠿⠿⠷⠶⠶⠶⠶⠿⠾⠿⠿⠧⠵⠾⠿⠿⠷⠶⠆⠙
⠀⠀⠒⠒⠐⠒⠒⠛⠛⠻⠿⠿⠟⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠚⠛⠙⠛⠛⠛⠛⠛⠛⠃⠒⠀⠘⠋⠙⠀⠐






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⡀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠈⠀⠀⠈⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢌⠃⠀⠀⠀⠀⠀⠘⠀⠠⠀⢠⡴⠈⠁⠀⠀⠁⠐⠆⠀⢄⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢪⠀⠀⠀⠀⠀⢀⠀⢀⠷⠳⠀⠀⠀⢀⢆⡀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⡄⠀⠀⠀⢀⠉⠁⠀⣀⠀⠀⠀⠀⢈⠜⠐⠀⠁⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢥⡀⢀⡔⢓⠀⠢⠄⠴⢃⣀⡀⣠⡤⢦⠤⣤⣂⣽⣐⣂⠀⣐⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠫⣕⡋⠓⠀⢈⠥⣀⣈⡹⠂⠈⢉⣁⡠⠀⢀⡂⢀⣔⠀⡸⡼⢰⡿⠪⠄
⠀⠀⠀⡉⠀⡄⢀⠀⠀⠈⠲⡠⠾⠐⢊⡑⠀⠠⠐⣉⠀⡉⢚⡁⠈⠁⠼⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠂⠐⣺⣥⣀⡔⠁⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⣠⠂⠀⠀⠀⡄⣐⣐⡇⡀⢀⠀⠐⠄⠀⢀
⠀⠀⠀⠀⠛⢞⠀⠈⠉⠉⠊⠀⠀⠀⡊⠔⠉⠁⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠡⠐⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⣠⢾⠅⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠠⢎⠀⠄⠀⠀⠀⠀⠀⠈⠄

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡏⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡏⠁⠈⠙⠻⡆⠀⠀⠀⠀⠱⡀⠀⠀⠈⢢
⠂⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡼
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠠⠄⠀⠤⣤⠶⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠐⠂⠠⢀
⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠄⠀⠠⠀⠄⡀⠀⠀⠀⢀⡀⠠⠤⠤⠀⢀
⠀⠀⠀⠀⡠⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⣀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠌⠁⠐⢄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠠⠁⠃⠀⢐⠀⠀⠀⠀⠉⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⡌⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠁⠀⠀⠀⠀⠠⣀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠁⣀⠠⢄⡀⢢⡀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠤⠤⠤⠴⠦⠄⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⢐⠀⢰⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣧
⠀⠀⠀⠀⠾⣇⡸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠉⠉

⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠃
⠀⠀⠀⢀⡴⠛⠁⠀⠀⢀⣀⣀⡀
⠀⠀⣴⠋⠀⠀⢀⡴⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠲⢦⣄
⠀⡼⠁⠀⢀⡴⢋⡾⠁⣀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⣸⠃⠀⠀⡾⠁⠺⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⣿⠀⠀⢸⠇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸
⣰⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠉⠘⣇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠴⠶⠖⠛⠋⠁⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⡇⠀⠀⠀⠀⢸
⣿⠀⠀⠀⣶⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣀⣤⡤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⣤⣄⡀⠀⠀⠹⣆⠀⢸⠇
⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⢾⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢳⡿⢦⡀
⣿⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⡾⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇⠀⢷
⣿⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⢰⠄⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⢸⠇⠀⢸⠇
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡟⠀⠀⠀⢰
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠴⠋⠀⠀⠀⠀⣿
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠉⠛⠛⠛⠛⢿⠛⠛⠳⣤⣀⢰⡟⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠘⢷⣄⣀⣠⡴⠃⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣤⡤⠶⠛⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠠⠱⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢠⡶⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠋⠃⠀⠠
⠀⠂⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠄⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢧
⠀⠀⠀⠀⠀⢣⣥⡆⠀⠀⠀⠀⠀⠀⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠀⠈⠀⠲⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠘⡃⡔⠀⠀⢸⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠄⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠢⢞⠄⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠘⠁⢐⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡀⠁⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢁⡀⢀⠀⠀⠀⡀⠀⠐⠋⠘⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠠⠀⠀⠀⠀⠀⠀⠊⠉⠑⠀⠀⠈⠃⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠖⢰⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠰⠀⢀⡂⠀⠀⠀⣀⣀⠔⢠⠀⠀⠀⠀⠀⠀⠀⠀⣅⠉⠀⢀⠛⡔⠀⠁⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣶⣌⣠⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠘⠃⢋⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠂⠀⠀⢠⢷⡾⠋⠁⠄⠸⡷⠆⠀⠐⠃⠀⠰⠆⠈⠁⠀⣄⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⠿⠟⠿⡿⢷⠀⠀⠟⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢾⠿⠛⠀⣰⣶⣿⣷⣄⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠀⠈⠲⢿⣿⣿⣿⣿⡿⠛⠻⠟⠉⠛⠹⣿⡿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⣦⣀⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠶⠆⣠⣿⣴⡾⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡋⢰⣿⡿⠁⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠊⠉⠉⠉⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⡄⠀⢨⣿⣿⣿⣿⣿⠿⠃⠀⠀⢠⣴⣄⠈⠀⠀⠘⠠⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⣿⣿⡄⠘⠿⠿⢿⣿⡄⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣷⡄⠀⠀⠘⠋⠀⠀⣦⣶⣿⡿⢹⣷⣦⣀⡀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⢰⣿⡿⠋⠀⣼⣿⣿⡟⠁⠀⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠀⠀⠘⠿⠃⠀⠀⠀⠀⠈⠁⠐⢁⡖⣛⡟⠋⠀⠀⠀⠺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣦⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⣿⣿⣧⠀⠀⠻⢿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖

⣿⣿⣿⣿⣿⣿⣦⣴⣶⣦⣴⣴⣦⣄⣀⣀⣤⣴⡦⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⠛⢿⡟⢛⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠄⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠟⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⢀⡀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢫⣿⡏⠉⠉⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢈⠏⠁⠀⠀⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠛⠃⠚⠛⠋⠀⢒⢶⣦⣦⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⣵⣿⣿
⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠿⢿⣿⣿⣷⣶⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣿⡿⠷⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣤⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠲⠞⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠰⠀⠲⣀⣤⠤⠔⠀⠀⠀⠀⢀⠘⠛⠛⠛⠛⠉⣽⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠿⠷⠖⠂⢀⣄⣀⣀⣉⢁⣖⣾⣥⣶⣶⣿⠾⠋⠉⠀⢤⡌⢙⣿⡿⠛⠙⠛⠉⠀⣽⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣩⣔⣲⣴⣒⣲⣼⣿⣿⣒⣲⣶⣮⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⣶⣾⣿⣦⣦⣴⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠓⠛⠛⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠓⠛⠛⡛⣄
⠀⢠⡞⠀⠀⠀⠀⠀⡀⠀⠉⠒⠠⠄⠀⠀⠉⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⡀⠀⡙⠒⠠⠄⠀⠀⠉⠀⠀⠀⣡
⠀⣿⡃⠀⠀⠀⠀⢀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⣀⠀⢰
⢀⡿⢡⣤⡄⠀⠀⢀⡀⠉⠙⠱⣦⣀⠀⠀⠀⠀⠀⠀⠈⠉⢰⣇⠘⡇⠀⡿⢡⣤⡄⠀⠀⢀⡀⠉⠛⠰⣦⣀⠀⠀⠀⠀⠀⠀⠉⠉⢰⣇⠘
⠀⠀⠉⡉⠁⠚⠣⢤⣷⡄⠀⡀⠈⢻⣷⢦⡀⠀⠀⢀⠀⣄⢰⣿⡀⠀⠀⠀⢉⡉⠁⠚⠣⢤⣿⡀⠀⢀⠈⢻⣿⣦⡀⠀⠀⢀⠀⣄⢰⣿⡀
⠀⠀⠀⠈⠒⠀⠀⠀⡟⠣⠤⠀⠀⠀⠛⠀⠀⠀⠀⠠⣤⣤⣴⡌⠧⠀⠀⠁⠀⠈⠂⠀⠀⠀⡟⠣⠤⠀⠀⠀⠛⠀⠀⠀⠀⠤⣬⣤⣴⡌⠧
⠀⠀⠚⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠿⠋⠀⠀⠀⠀⠚⠒⠚⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠿⠋
⠀⠀⠀⢀⡄⣀⠔⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⠔⣨
⠀⠀⡰⠟⠛⠧⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠟⠛⠇⠼⠃
⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠐⠠⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⢀⠀⠀⠀⠀⠀⢀⡴⠁⡀⠀⠀⠀⠀⠀⠢⣆⢀⡄⠀⢀⣀⠘⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⡴⠁⡀⠀⠀⠀⠀⠀⠰⡆⢀⡄⠀⢀⣀⠘
⠀⠀⠂⠀⢠⡶⠒⠛⠃⠀⢀⠀⠀⠀⠀⠀⠄⣿⢸⡗⢀⡌⠀⠉⠀⠀⠀⠀⠀⠀⢠⣶⠖⠛⠃⠀⢀⠀⠀⠀⠀⠀⡄⣷⢸⡇⢀⡮⠈⠁
⠀⢀⠴⣾⠟⠀⠀⠀⣠⣴⣅⣸⠶⡍⠉⠁⠀⠿⠘⠁⣼⡧⠀⣀⢀⠀⠀⢀⡴⣾⠟⠀⠀⠀⣠⣴⣅⣸⠶⠈⠉⠁⠀⠿⠘⠁⣼⡦⠀⡀⢀
⠀⠀⠀⠉⠀⠈⠁⣠⣿⡿⢻⠁⠀⠐⠀⠀⠂⠀⠀⢠⡿⣣⣤⡤⠤⠀⠀⠀⠀⠋⠀⠈⠁⣀⣿⡿⢻⠁⠀⠐⠀⠀⠀⠀⠀⢠⡟⣣⣤⡤⠤
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠁⠃⠐⠀⠉⠀⠀⠀⠈⠘⠿⠷⠶⠤⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠁⠲⠐⠂⠈⠀⠀⠀⠈⠘⠿⠷⠶⠤
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡐⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠠⡑⢄⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⡀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⢂⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡀⠠⣀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⡄
⠀⠀⠀⠀⣠⠆⡀⠀⢰⣣⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢐⠏⠦⠃⠀⣏⣿⣾⠙⠁⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠀⢠⢿⡵⠀⠐⠈⠉⠩⠕⠒⠐⠒⠀⠂⠀⠀⠀⠀⠀⠠⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⢀⠄⠄⠒⡉⢍⠁⠈⠉⠛⠢⢄⠃⠀⠙⠁⠉⠁⠀⣀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠰⠀⠀⠀⠀⡇⠀⠄⠀⢀⡠⢀⠔⠁⠀⠀⠔⢀⣴⠶⢉⣭⣥⣭⣹⣿⣆⡴⢠⣄⠀⡩⠤⠶⠊⠙⠃⠀⠤⡔⣖⠲⠦⣉⡱⠦⣤⠛⠉⠁⠈⠉⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡰⠋⠀⠀⠀⣠⠂⡟⠃⡀⠈⢢⡴⠀⠲⢷⣍⣝⣵⠋⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⣷⠀⠈⠸⠇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⡜⠀⡀⢙⣦⠅⠀⠀⠄⠀⡀⠀⠉⠀⠁⠈⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠲⢼⡆⠀⠀⢀⢠⡀
⢀⠀⢸⠀⠀⠀⢸⡄⠀⠀⠀⢸⡆⠀⠀⣴⠁⠀⣃⠈⢤⡀⠀⠘⡀⠠⠃⠀⠀⠀⠀⠀⣧⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⢿⠀⡀⢸⢧⣿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡌⣇⠁⠂⠻⢰⣬⣿⣷⡟⠁⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⢸⠒⢨⠤⠙⣦⢚⡛⠁⠀⠀⠀⠀⢲⡟⣨⣎⣟⠀⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠙⣄⠀⢳⠌⣻⠿⡧⠁⠀⡄⠈⡀⠀⠚⠀⡿⠘⠀⠀⣾⠘⠀⠀⡸⠟⠉⠉⠂⠀⠀⠀⠀⢸⡇⡟⣿⣿⣿⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠃⣬⡟⠻⣶⠿⠦⣌⡀⠶⠀⠀⠀⠀⢀⡄⠠⠾⠿⠥⠞⠉⠀⠀⠀⠀⠀⠀⢲⡄⢀⣼⡿⠣⣿⢸⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⠀⠐⣀⠴⠀⠀⠀⠀⠉⠓⠒⠲⠴⢶⣥⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⢃⡌⠀⠋⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠉⠀⠒⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣀⣠⣤⡤⠤⠤⣲⣾⢻⠏⠀⠁⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠈⠉⠛⣼⠟⢡⣾⠇⠎⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢨⠎⠉⠀⠙⠒⠦⢤⣤⡤⠴⠛⠁⠀⠀⠀⡼⠋⠀⠸⠏⠈⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠄⡲⠀⠀⢄⣟⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⡀⠀⠀⠀⣼⡃⠀⢀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠉⠁⠀⠀⠀⠃⠀⠀⡾⠊

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣀⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿