Nostalgic Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠰⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡙⢿⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠀⠀⠀⠀⠈⠹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢘⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠤⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠀⠛⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠀⠀⠒⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠐⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠠⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠐⠉⠀⠀⠠⠀⠀⠀⢸⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⠆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠀⡇⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠛⣶⠤⠀⠛⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⠀⠀⣀⠘⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⡶⡇⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣤⣤⣀⣠⣄⣙⣾⣏⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⠋⣠⠘⠀⠆⠀⠀⠀⠀⠀⠠⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠉⠄⠀⠀⠀⢀⡀⠠⡬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⢀⡀⠀⠀⠀⣠⠞⠁⠀⠈⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣾⣾⡅⠐⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠲⢶⣦⠴⠃⠀⠀⠀⠀⠀⠘⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢀⠀⠀⠀⠀⠀⠂⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀
⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⣺⡦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣄⣀⣀⡀⠙⣶⣶⡶⢀⣤⡀⢨⣅⣀⡤⣦⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠛⠛⠒⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠍⠀⠶⠯⢤⣤⣙⣛⠟⠐⠛⠛⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠸⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣷⡌⢻⠚⠀⣠⣄⣤⣀⡀⠀⠀⢠⣰⠀⠀⠀⠺⢶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⠀⣼⣿⣿⣿⡿⠇⠀⠀⠀⢾⡇⠀⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠄⢩⣿⣿⠟⠁⠀⠀⠀⠀⠿⠀⣰⠋⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣤⣀⠀⠘⠿⠃⠀⠀⠀⣤⠀⠀⠆⣰⠃⠀⣀⣀⣁⡀⣠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⣉⡽⠓⠂⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠞⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣙⣉⠓⠀⠀⠀⠀⠀⠀⢀⡀⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠠⡀⠘⠋⠀⠀⠀⠀⠀⣷⡄⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢙⡇⠀⠼⠛⠛⠛⠦⠀⠀⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡀⠀⠘⢿⣤⣤⣶⣤⡀⠀⠀⠀⢀⣠⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⢭⣙⡿⣿⣤⠀⠀⠀⠀⠀⢀⣀⣄⣀⣿⣷⠾⣿⢄⣀⣀⡉⠲⠿⠙⠛⠆⠀⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢉⣃⠀⠀⠀⣼⣿⡿⠟⢉⣸⣏⡠⠈⠸⢭⠽⣯⡄⢀⡀⠀⠈⠘⠁⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡦⠤⠛⠉⣴⣾⢛⣿⣿⡁⠀⠀⠀⠀⠀⠛⠀⠀⠘⠂⢀⣤⠖⠺⠦⠜⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡄⠀⠀⣴⠯⠶⢽⣣⣿⣿⡍⠀⠀⠀⠀⢛⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢶⣟⡛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⣾⢿⣻⡟⠀⣀⣤⡴⠞⢻⡂⠀⠀⡁⠀⠀⣷⣀⡀
⠀⠀⠀⠀⠀⠈⠙⠷⣤⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠘⢣⡿⠊⠡⣤⣾⣿⢯⣤⡄⠈⠀⠀⠠⠃⠀⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠦⠄⠀⠀⠀⠠⠈⢀⣰⠁⠀⠠⠴⠉⢇⠘⣷⠏⠀⠀⠟⢹⣿⣠⡀⢈⠀⠀⣤⡦⢶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠀⠀⠀⣡⠤⠜⠁⠀⠀⠀⣀⣀⡈⠠⢿⠀⠀⠀⠀⠈⣿⣷⡇⠈⠻⣶⣿⣷⡄⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢨⣿⣿⢠⡏⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣀⡀⠀⢰⣶⣽⣯⡏⠀⣼⠇⠀⢠⣶⣤⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⣿⣿⣿⣿⠻⣿⣿⡙⣿⡇⢆⡿⣷⡀⠠⠉⠸⣿⣿⡶⠀⢰⣶⢞⡿⠒⡀⣠⣤⣄⠲⢄⣀
⠀⠀⠀⠀⢀⣴⡶⠿⣛⣛⡻⠃⣤⣤⣦⣬⣼⣿⣿⣄⠘⠻⣿⠀⠳⢸⠁⢸⣷⠀⠀⠀⢻⠟⠁⠀⠸⠡⠤⠀⠚⠿⣿⣯⣼⣿⡋⠉⠉
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠉⠉⣉⣽⣶⠖⠃⠐⠦⠀⠾⡀⠀⢿⣦⠃⢀⠴⠶⣷⣶⣾⣿⣷⣦⡀⠀⠀⠉⠛⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠖⠁⠀⠀⠀⠀⣇⠀⠀⢋⡄⠀⣀⣀⣤⠷⠧⢿⠟⠛⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠠⢠⡀⢀⠏⠀⠙⠟⢿⡿⢿⣿⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡎⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇



⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⢿⡿⢿⡿⠿⣿⠿⠿⡿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣷⣾⣿⣿⣿⣶⣶⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣹⡿⠋⢠⣿⡿⠗⣿⣿⡟⠩⣿⣿⡏⠹⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⣴⡟⠀⢠⣾⠏⠀⢰⣿⣿⠛⠀⣿⣿⡗⠀⢸⣿⣷⡊⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⢰⣿⠏⠀⣠⣿⡏⠀⠀⣾⣿⡏⠀⠀⣿⣿⣷⠀⠀⣿⣿⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠓⢐⣿⡟⠀⢰⣿⣿⠁⠀⢸⣿⣿⠃⠀⢸⣿⣿⣿⠀⠀⢹⣿⣿⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⢀⣿⣿⠀⢀⣯⢠⡏⠀⠀⣾⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢨⣿⣿⡆⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⠁⢸⣿⡗⠀⢸⣟⣿⠇⠀⠀⣿⣿⡯⠀⠀⢸⣿⣿⣿⠄⠀⠐⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⡿⠀⣿⣿⠁⠀⣿⣿⣿⠁⠀⢠⣿⣿⡇⠀⠀⢸⣿⣿⣿⡁⠀⠀⣿⣿⣇⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠂⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⠆⠀⠀⣿⣿⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢺⣿⡇⠀⢿⣿⣿⡀⠀⠘⣿⣿⣇⠀⠀⠸⣿⣿⣿⠁⠀⠀⣿⣿⡇⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⣿⡇⠀⢸⣿⣿⡆⠀⠀⢿⣿⣷⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⠃⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣼⣿⣿⣗⠘⣿⣿⣧⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⠂⣀⣼⣿⣿⣡⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣆⠀⠀⣿⣿⣇⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣶⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠘⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠁⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⣿⠇⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣿⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⢄⢀⡄⣠⣠⣤⣀⡀⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠄⠀⢻⣿⣷⣿⣿⣿⡿⢳⡀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⠀⠁

⠀⠀⠀⢀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠬⢾⣷⠓⠂⠀⠀⣀⣠⣀⠀⠀⠀⠤⠤⠘⠀⠀⢹⣿⠃⠀⠠⠿⠏⠀⠀⠀⠀⠀⢺⡁
⠀⠀⠀⠀⠀⡇⡀⠀⠀⠘⣿⣿⣿⡀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⢈⡗⠒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠋⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣦⠖⠀⠀⢄⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣾⠅⠀⡀⢄⣄⠠⠀⠀⠂⠀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠰⠂⠀⠀⠀⠆⢤⠀⠀⠿⠃⠀⣼⡄⠋⠀⠀⠀⠀⠀⠀⣤⡄⠀⢈⣄
⠀⠀⠀⠀⠐⢨⠴⠀⠀⠶⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡞⠚⠦⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣹⡅
⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠄⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⡉⣵⣍⣦⣾⣧⡄⠀⠀⠆⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠀⠀⠀⠠⠀⠀⠀⠀⣀⡽⠾⣛⠿⣿⣳⣌⡀⠐⡿⣴⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⣠⡿⢿⣿⠉⡿⣟⣻⣄⣠⠤⡄⣙⢡⣶⡮⠉⣠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠁⠀⡐⠀⠘⢿⣿⣶⣏⣤⣲⣻⣽⣿⢻⣿⣯⣶⣎⣸⢷⣤⡈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⡡⠀⠀⠀⠀⢀⣔⣷⠿⣿⣿⣫⠫⢒⡿⠽⣿⣿⣯⢤⣾⡿⡿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⣇⠉⡉⠟
⠀⠀⠀⠀⠀⠀⣄⠄⠀⠠⠀⣤⠀⠀⠀⡘⢿⣿⣴⣿⢷⣎⢹⣿⣿⡀⢿⣮⣿⣾⣿⣿⣾⣿⣿⣇⠀⠀⠀⢀⣴⡶⢟⡀⣀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡠⠄⠂
⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⢸⠷⣏⣠⣎⡂⠀⢻⣿⣿⣿⣿⣿⡙⢿⡇⣀⣿⡫⢃⣿⠟⣿⣯⣼⣿⣆⠀⣠⡟⣡⣴⡟⠁⠁⠀⠀⡀⠀⣀⣤⠀⡎⢩⡇
⠀⠀⠀⠁⠷⠀⠀⠈⠀⢜⠁⡓⠹⣿⣿⣿⣷⣾⣿⣿⣋⣿⣿⣷⡶⠟⢛⣻⣭⣿⣿⣻⡯⠟⡋⠁⣙⣶⣧⠜⢡⢁⠃⠔⢁⣠⡤⣔⠏⠀⠁⢨⠃⢸
⠀⡀⠀⠀⠀⠁⠀⠀⠀⠨⠀⠡⠀⢿⣿⣿⣿⣿⡿⣷⣽⣿⣽⣯⣶⣿⡛⣻⡢⢊⡩⣥⣶⣿⣶⣿⣿⣿⠀⠠⢘⣬⢼⠒⠛⢗⣱⠀⡇⢰⡄⢸⠀⠸
⡀⢀⣠⡀⠀⠀⠀⠀⠀⠀⡎⠀⠠⡆⣿⣿⣯⣿⣿⣿⣛⣫⣭⣴⣯⣶⣏⣽⣾⣾⣿⣿⣿⠿⠛⠛⣿⣿⣴⡾⣏⠙⢺⡀⣀⡀⠻⣀⡇⠸⡅⡎⠢⡂
⢹⠀⠀⠀⣀⠀⣀⠀⠀⠀⣴⡶⣦⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⡿⣄⠀⠀⣛⣿⣿⠁⠟⠠⡌⠇⣸⠃⡀⢳⡣⣞⢙⣏⠁⡁
⠀⠂⠀⠀⠀⠀⢊⡶⠄⢀⢿⣃⣸⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠟⢉⣉⣼⣶⣿⣿⣿⣧⠠⠘⢼⣿⣿⡔⣇⠐⢧⣃⡏⢀⡘⣯⡉⠉⠉⠹⣧⡀⠠
⠀⠳⢀⠟⡄⠀⠉⠀⡰⡍⠘⣏⠀⠙⠺⠛⣾⣿⣿⠛⠻⢻⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢄⠈⠟⠿⣿⣡⣠⣇⣿⡻⠿⠄⢹⣷⡀⠀⠀⠘⣷⡀
⠀⠆⠀⠘⡀⠆⣀⠀⠠⠀⠈⣵⣄⠶⣼⣸⠟⢿⣿⣇⠀⠀⣿⣿⣧⣻⣿⣿⣿⣿⡿⡛⠋⢻⣿⣦⣔⣲⣋⣿⢹⣿⡿⣿⣷⡄⠀⠀⠙⢿⡄⠀⠀⠈⢿⡄
⠀⠀⠐⠀⠁⢠⣗⣤⠖⣪⡵⢾⣮⡄⠙⠛⠀⣿⣿⣿⣿⡎⣿⣿⣿⣽⣻⣿⣉⠇⠹⣽⣄⢴⣹⣧⢿⣿⣿⣿⣧⠈⢣⣻⠟⢿⡄⠀⠀⠈⢿⡄⠀⠀⠀⠻
⠀⠀⠀⠤⢖⣾⡵⢿⡋⢹⣇⣸⣿⡇⣤⡬⢷⣿⣼⣿⣿⣹⣸⣿⣿⣿⢿⣿⣿⣖⣻⣼⣿⣿⣿⣹⣯⠛⣿⣿⣟⣷⠑⠿⠀⠉⢿⡄⠀⠀⠀⠻
⠀⠀⠰⣞⡏⢽⡇⢸⠇⢸⡅⢿⡟⢏⠀⢤⡈⠛⣽⣿⣿⣷⣵⣻⣼⣿⣿⣿⣿⣿⣿⣷⣿⡿⢿⣟⢿⡆⠈⣿⣿⣿⣷⠀⠀⠀⠀⠛
⠀⠀⠈⢸⡇⢸⡇⣽⠄⣸⡇⢾⡧⢠⠀⢺⣾⠇⢸⣧⣼⣾⣿⣿⣿⣿⣿⠻⣿⣿⣿⢻⣯⢶⣰⣿⣿⣿⡄⠈⠛⠿⠋
⠀⠀⢀⣿⠀⣿⢁⣿⡦⣿⠀⡏⣙⣺⣄⣿⡏⠀⣸⣿⣿⣿⢿⣿⣿⣿⣿⡖⠺⣿⣿⣿⣿⣞⡄⠋⠻⠋⠁
⠀⠀⠈⣼⡖⣿⢞⣿⣷⣿⡖⣿⡝⠋⢸⣿⣿⣾⡿⢟⣫⣿⡀⠀⣾⣿⣿⣿⠄⠉⢿⣿⡿⢿⠂
⠀⠀⠀⣽⣿⣿⣹⢿⣿⣿⡇⣿⠁⣿⣿⠉⢹⣤⣾⣿⣿⡿⣏⠄⠿⣿⣿⠘⠃⠁
⠀⠀⠀⠛⢿⣿⣿⠘⢿⡿⠇⢸⢢⠨⠙⣷⣼⡆⣅⣏⠋⠁⣿
⠀⠀⠀⠀⠀⠉⣻⠀⠣⣆⢯⢸⠟⠀⠀⠿⠃⡃⠹⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠘⠁⠛⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣷⣤⣶⡄⠀⠀⠀⠤⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠉⣽⢻⣿⣄⡀⠀⠀⠀⠆⠀⠄⠶
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠐⢠⣿⣿⡀⠀⡀⢀⢀⣝⣿⣿⠫⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣟⠁⠀⣀⣶⣽⣿⣿⠀⢰⣿⣷⡄⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠓⠢⠖⠐⢿⣿⣿⠀⣿⣿⣿⡗⢀⠀⠀⠀⣀
⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⡀⠀⠋⠛⠒⠌⡃⠀⠀⠈⠁⡌⢻⠏⣤⠻⡿⢿⠛⢻⡅⠘⠁⢻
⠀⠀⠀⣀⠀⠠⠤⣤⣠⣤⣀⣀⣠⣤⣤⣤⠀⠉⠘⣤⠠⠤⢤⣤⣤⣹⣦⣀⣤⡆⠴⠇⠀⢸⣇⠀⠀⢉⣿⣿⣢⣰⣟
⠀⠀⠀⢸⠀⠀⠀⠄⢰⡄⢘⠉⣉⠁⡶⠗⠀⠀⣀⣽⣧⠲⣆⣸⣬⣿⣤⡿⠋⠀⠀⠀⠀⢸⡟⢷⠀⠀⢻⡝⢿⣿⣿⡿⠛⠛⠚⠓⢈⡛⠛
⠀⠀⠀⢸⠇⠂⢒⠀⠀⠓⠠⠀⡨⠿⠃⠀⠀⠈⣻⢛⣷⣴⣯⣽⣿⣿⢯⡈⠃⠀⠀⠀⠠⠞⠁⠰⠇⠀⠀⠄⣾⣿⣿⣯⡈⠤⠒⠀⠀⠀⠄
⠀⠀⠀⣿⣄⣤⣼⣤⣤⣄⣴⠎⠀⠀⠀⠀⠀⢠⣿⣾⣿⣶⣾⣿⡿⠁⠀⠉⠀⠀⢠⣄⠀⠀⠀⠰⣿⡇⠀⠀⢿⣿⣿⣇⡤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢠⠀⣿⠉⠙⠛⣭⠟⠋⠁⠀⠀⠀⠀⠀⣈⣿⣄⣀⠀⠀⠈⠀⠀⠀⣈⣟⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠠⡆⢱⡐⠦⢸⣦⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⠿⢄⡀⠀⠀⢀⠀⠀⠈⢹⢾⡇⣌⣁⣀⡀
⠀⠀⠀⢀⠄⣀⠀⣤⡠⠀⠄⠀⠀⢀⠀⣿⣲⣎⣻⣿⣸⣿⣦⣤⣲⣶⣾⣿⣿⣿⣿⡟⠳⡦⠉⠐⠠⡀⣀⠀⣀⣾⣿⣿⣿⣅
⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠿⠿⣿⣿⣿⣿⣧⣦⡉⠀⠳⠶⣄⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⠿⠿⠷⠶⠟⣿⠛⢿⣿⣿⣿⣿⡟⢛⣶⣄⡈⡇
⠀⠀⠀⢿⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⣾⡗⠹⢂⡴⠞⠀⠀⠀⠚⠀⢠⡅⢻⣿⣿⣿⣿⣿⣯⣥⡆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⢁⣀⡴⠛⠛⠁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡄⢹⡟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇⠠⠐⠻⠋⠁⠘⠁⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠸⠀⢸⣿⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡟⠏⠩⣿⣿⣿⣿⡿⠿⢣⡼⠛⠗⢤⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡟⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣶⣶⣿⣿⣿⡿⠋⠀⢀⡉⠀⠀⢠⣼⣯⠃⠘⠀⠁⡀⠀⠀⠃⠐⠂⣀⡀⠀⣀⠐⠠⠄⠐⠀⠁⣀⣴⠇⡹⣏⢻⣿⣿⣿⣷
⠀⠀⠀⣩⣥⣽⣿⣿⣿⣷⣾⠛⢤⡖⠊⠁⠀⠀⢸⣏⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠉⣶⡶⠟⠛⢀⠐⠀⢀⣶⣿⣯⢀⣼⣇⣿⣿⣿⣷⡷
⠀⠀⠀⣿⣷⣶⣿⣭⣝⠿⢿⣷⡜⠳⢤⣠⣴⣒⢻⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡗⠺⠀⢠⢠⣶⣿⣿⣼⣿⣿⣿⠛⣿⣿⡟⠍⠉
⠀⠀⠀⣿⣿⣿⠒⠈⢻⣤⠀⠙⢿⣷⣿⣿⣶⣭⣿⣛⣶⣿⠤⣤⣘⡶⠶⠖⠖⠢⠾⣅⣀⣠⣝⠀⢲⣾⣿⣿⣶⣿⣿⣾⣻⣶⣴⣿⡷⠠⠅
⠀⠀⠀⣿⠉⣿⠀⠀⠀⠈⡃⠀⠈⠻⣿⡿⢛⠷⣻⣿⣿⣿⣶⣁⣶⣿⡿⢿⣷⣾⣿⣿⣿⣿⣻⣿⣿⣿⣽⣷⣾⣿⣿⣿⣿⣿⡿⢻⣇
⠀⠀⠀⣿⠟⠁⢀⠀⣀⣾⣥⣤⠼⠶⣯⣶⠟⠋⠑⠛⠉⠉⠉⠉⠉⠉⠉⠐⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠉⠈⠙⠛⠛⠻⠛⠻⡷⣸⣿⣿⡇
⠀⠀⠀⠏⠀⡀⢉⣵⣿⣿⡋⢀⠀⡀⢻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠇
⠀⠀⠀⣤⣵⣿⣿⠿⠟⠛⠋⠙⠉⠛⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠆⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣠⣾⣗⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣿⣿⣷⡞⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⡏⠀⢈⣉⠻⣿⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡃⢈⡛⠃⠈⠁⠀⣿⣿⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠻⡦⡈⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠛⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠛⠖⠀⢀⣶⣬⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⡄⠋⢀⠀⠀⣹⡆⠀⠸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡅⠘⣆⠀⠀⠛⠀⢸⣷⠀⠾⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⣷⣀⠀⠀⢈⣿⡆⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢁⡴⠒⢴⣬⣿⣷⣶⣾⣿⣿⣿⢂⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠉⠀⠀⡈⠉⡹⠟⠛⠁⠛⠃⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⢀⠟⠁⡼⠠⢄⠀⠉⠀⢀⣀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡶⠀⣠⡶⠀⡄⠀⢀⣀⢀⣼⣷⣀⡄⢸⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⠏⠁⢚⠁⢀⣤⡀⠀⣾⠉⠻⣿⡟⢻⡇⠀⠛⠰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⣿⡇⠀⣴⠋⠀⢀⣽⠁⢰⡟⠀⠄⢹⣷⠸⣷⡄⣦⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⢋⣤⡾⠁⠀⠐⠚⠛⠀⣿⣷⠀⢢⢼⣿⠈⢻⣧⠈⡀⠀⢹⣷
⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣿⡿⠷⣶⣾⡿⠋⢠⣀⠔⢻⠆⠁⣰⣿⣿⢠⠀⣬⣿⡄⢠⣿⡌⠋⢡⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠒⠀⠀⢃⣾⣿⡿⠗⠚⢿⣿⠏⠁⠐⠚⣧⠓⠁⢰⡿⣿⡟⠘⠃⠉⣿⡇⠸⠿⣿⡄⠀⣴⢹⣿⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡆⣿⣷⣿⣛⣛⣻⡿⠃⠀⠀⠈⣀⠘⠀⢀⡾⠟⡻⠃⠀⠀⡀⢹⣇⠀⢀⢻⣿⡈⠻⡇⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⣿⣿⣿⡿⠋⠀⠀⠀⢆⣻⡼⠃⢀⣌⣉⢀⠀⠀⠐⠀⠀⢸⣿⠀⠉⠦⢻⣧⡀⠙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣼⣿⣷⣿⡿⢋⣤⡾⢋⣴⡶⠾⠛⠁⠀⣼⣿⠏⢸⠀⠀⠀⡀⠀⢸⣿⠀⠀⡄⠀⢹⣷⡀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢫⣴⠟⠉⠀⠺⣄⣼⣇⣀⣴⣾⣿⣿⡄⠘⠀⠀⠀⢉⡆⢸⣿⡀⣧⠀⠀⠸⣿⣿⣄⢻⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⡁⠠⠀⠀⣠⡟⠿⢀⣩⣿⡏⣾⠋⣧⡄⠀⠀⠀⠘⡋⢸⣿⡇⢸⣷⣄⠀⢹⣿⣿⣮⢿⣿
⠀⠀⠀⠀⢸⣾⣟⣻⢛⡴⣿⠧⠀⠸⠟⠉⢡⢶⣾⡿⣻⠃⢶⠀⢹⡇⠀⠀⠀⠘⠁⢰⣿⡇⣸⠚⢿⡶⠘⣛⣇⠹⣷⣿⣧
⠀⠀⠀⠀⢠⣿⣿⣳⣿⡗⣸⠃⠲⠀⠀⢀⣴⠿⢋⢿⡄⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⢸⣿⡇⢨⡀⠀⠀⢀⣭⣿⡇⠸⣿⣿⡀
⠀⠀⠀⢀⣸⣿⡝⢿⣷⣏⡿⢄⡃⣠⣾⠟⣁⠸⣿⠘⣷⣀⣷⣦⣬⣤⠀⠈⠀⠀⠀⢸⡟⣷⢸⣥⠀⠀⢈⠿⣿⠇⣷⠈⣿⠇⡀
⠀⠀⠀⣼⣧⠹⢿⠗⢀⣿⡷⢟⣵⣿⣿⠆⠹⣦⠙⠂⣼⣿⡄⠑⠀⠌⢃⣄⣰⡂⠀⢸⡇⣿⠈⠻⣧⡀⠺⠃⢰⣿⣿⠀⢨⣾⠁
⠀⠀⢠⣿⣿⣷⣤⣴⣦⣶⡄⠺⣿⣿⣿⡄⢷⣝⣷⣤⣿⣿⡿⣦⡀⠘⠛⢉⣽⠉⡂⢸⡇⣿⠀⣻⢻⡗⠂⣠⠌⠀⣶⠆⠀⣿⠆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡌⣿⡄⢸⣿⣯⠀⠈⠻⣿⣿⣿⣿⠇⣿⣿⣦⣀⠀⠀⠰⠇⢸⡇⢸⡄⠉⠉⣠⡜⠁⠀⠀⣿⡇⠀⣿⣦⡀
⠀⠀⣾⣿⣿⣿⣿⣿⠟⣧⠈⢁⣾⣿⣿⣦⠀⠀⠸⣿⣿⣿⢀⣿⠇⠈⠽⠿⠶⢦⣄⣸⡇⢸⣧⡴⠟⢻⠀⢠⠔⠋⣿⣿⠃⢸⣿⡇
⠀⠀⢿⣿⣿⣿⣋⡁⠀⠈⠁⢀⣿⣿⣿⣿⡀⠂⠀⢹⣿⣿⠘⠋⠀⠀⢀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠠⣿⠁⠀⠖⠀⣿⡗⠀⢸⣿⣁
⠀⠀⠀⠀⠀⠉⡿⣿⣿⢶⣾⣿⣿⣿⣿⠿⣿⠆⠀⣨⣿⣿⣆⣤⣀⡀⠀⠐⠃⠀⠀⠀⠀⠀⠐⠀⣨⣿⡇⠈⠀⢀⣽⣇⣀⡼⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣍⣛⣿⡿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⠿⠿⣴⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠀⠀⠠⠄⠀⠉⠉⠙⠉⠙⠿⠛⠛⠿⠻⠿⠶⠶⠆

⠀⠀⠀⠀⠀⠄⠀⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⣄⡀⠀⠀⣠⡤⣤⣤⣤⣀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠩⢶⣶⣤⣽⣿⣿⣿⡿⠏⣳⣾⣿⣿⡞⣿⣿⣿⡟⢈⡿⢛⡛⣿⣿⣿⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠤⢤⣭⣭⣿⣿⣿⠿⣉⣴⣿⣿⣿⣿⣿⡇⣯⣿⣿⡷⠘⣇⠻⢆⡟⠁⠀⣌⠿⠋
⠀⠀⠀⠀⠀⠀⣀⣉⣭⣭⣽⣿⠟⢡⣼⣿⣿⣿⣿⣿⣿⢟⣲⠽⠟⠀⠑⢄⠈⠉⠁⠀⠀⠘⠑⠶⢴⠄
⠀⠀⠀⠀⠀⠀⠊⣉⣩⣾⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠲⠄⠀⠀⠀⠉⠦⠀⠀⢠⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠚⠍⢿⣿⡟⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠄⠀⠀⢀⠰⢄⣀⢀⡠⠞⠁
⠀⠀⠀⠀⢀⢴⢾⣿⠏⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢙⣄⠀⠀⠐⠍⠉⠲⣦⠼⠋⠁
⠀⠀⠀⠀⠀⢀⣼⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠈⠖⠀⠠
⠀⠀⠀⠀⠐⠹⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣦⠄
⠀⠀⠀⠀⠠⢾⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣠⣄⣤⠖⠒
⠀⠀⠀⠀⢀⣴⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡿⡆⢠⣖⣰⣶⣿⡿⠃
⠀⠀⠀⠀⠉⢙⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⡽⡼⣾⣟⠯⢮⣰⣼⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠋⢣⢸⣿⣿⣿⣿⣿⣿⣿⣯⠘⠆⠘⠷⠻⠶⠽⠋
⠀⠀⠀⠀⠐⠀⢿⡀⣿⣿⣿⣿⣿⣿⣿⡷⢁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⢤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠘⠃⠀⠷⠹⣿⣿⣿⣿⣿⣿⣜⢠⠌⠀⠀⠀⠀⠀⠀⣰⠾⠛⠋⠉⠉⠛⠻⢮⡳⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠘⣏⡐⢇⣹⣿⣿⣿⣿⣿⣿⢫⣖⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⠫⠻⣽⣆⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠸⣧⡘⢁⢻⣿⣿⣿⣿⣿⣿⣎⠄⠀⠀⠀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠏⣧⣇⠀⠀⠀⠀⠀⢠⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠉⣷⣸⣯⠙⢿⣿⣿⣿⣿⣿⣳⡂⠰⣾⣟⣟⣡⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⣿⠀⠀⢀⢀⣴⡏⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⡯⠸⣧⠹⢿⣿⣿⣿⣿⣟⣄⠀⠉⠀⠀⠙⠛⠚⠀⠀⠀⠀⠀⠀⠀⠂⠀⢋⣌⣿⣶⣶⣿⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⡻⢳⢪⣭⠻⢿⣿⣿⣿⣦⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣼⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠙⠪⣟⠃⡀⠉⠻⠿⢿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣤⣶⣥⣤⣠⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠷⠶⠦⣭⣩⣭⣿⣻⣿⣟⣿⡿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣾⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣰⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⠁⢸⠀⠀⠀⠀⠀⠀⠈⣉⣙⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣀⡾⣿⣿⠆⡀⠀⣿⣿⣶⣠⣾⣧⡀⣀⣬⠀⠀⠀⠀⠀⠠⠟⠟⡿⣟⠻⢻⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⢙⣳⣿⣷⣆⡈⠉⠻⣿⣿⣷⢾⣇⣨⣿⠶⠄⠀⠀⠀⠀⠀⠠⣧⣿⣾⣩⣢⣆⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡮⢍⣿⣿⡿⢿⣿⡦⢩⣵⡑⠄⠙⠛⡳⠄⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⣿⣿⣿⡇⢀⠙
⠀⠀⠀⠀⢄⡀⠀⣾⣿⣿⣛⢻⡏⣅⣩⣿⣟⠸⣤⡌⡂⠀⠀⠑⠖⣀⣶⠆⠀⣀⠉⠉⠛⠛⠛⠛⠛⠛⠃⢀⣀⣀⣀⢀⣀
⠀⠀⠀⣼⣿⣿⣿⣿⣭⣝⣛⣛⡿⠿⠿⢿⣿⠀⣿⣿⣷⢀⣆⢰⣿⢏⢤⣤⠶⠷⠶⠖⠒⠒⠚⠛⠛⠉⠉⠉⠉⠀⡀⠉⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣶⣿⣭⣬⣭⣌⣭⣬⣭⠥⠴⠶⠖⠒⠚⠛⠳⠒⠀⠀⠀⣀⠸⠷⠃
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠘⢿⣿⣿⣿⣿⣯⠅⣀⣿⠀⠀⠀⠀⠀⣠⣤⠀⠀⠐⠀⠈⠉⠁
⠀⠀⠀⠀⢀⠀⠀⠍⠉⠙⠻⠿⠿⣿⣿⠛⣿⠀⠠⠄⠉⠉⠙⠛⠶⢤⣿⠀⠀⠀⠀⠀⠙⠑⠢⢾⡈⠀⠁
⠀⠀⠀⣀⣤⢀⣄⡦⢀⡅⢠⠀⠀⠀⠉⠓⢾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣶⣿⣿⣿⣶⡖⠶⠄
⠀⠀⠀⠘⠿⠻⠯⠠⠟⠂⠆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⠿⠷⠖⠖⠀⠀⠀⠚⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣤⣾⣦⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⠟⡛⣿⡿⠚⠿⠿⡿⠛⠿⢿⠿⠿⣿⠿⢿⣿⣿⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣾⣿⣿⣿⣭⣽⣿⣿⣏⣼⣗⣿⣀⣠⣤⣴⣻⣀⣀⣈⣂⣀⣉⣣⡀⢀⣛⣦⡀⠀⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠈⠢⡉⠈⠉⠫⣍⠉⠙⠛⣏⠉⠉⠙⣿⠋⠻⠹⠉⠉⡟⢹⢿⡛⠉⠙⢻⠋⠉⢛⡟⠟⣩⠟⡻⠛⠛⢛⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠈⠊⠒⢴⣬⣦⠀⠀⠈⢆⠀⠠⠀⠀⡇⠘⠸⠀⠈⠀⡁⢀⢴⡪⢔⡫⠔⠋⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠄⠈⠙⢷⣄⡀⠪⡀⠀⡄⠀⡇⠀⡃⠀⢀⡾⣠⣴⠯⠒⠉⠀⠀⢀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠙⠦⣇⠀⣆⠀⣿⢠⣦⣾⡿⠗⠉⠁⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠻⢵⣿⠿⠋⠁⠀⠀⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠄⠀⠀⢀⣰⣋⣟⣷⠀⠀⡰⠁⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠘⣌⣏⢸⣙⠻⣧⡐⣁⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⣾⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠔⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡯⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠊
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⢀⡼⡿⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠁⠄
⠀⠀⠔⣁⣤⣤⠑⠰⣿⣇⠀⢳⡄⠀⠀⣰⣶⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢆⡀
⠀⠀⠀⠈⠀⣤⣀⣀⣈⠙⠒⠺⢷⠒⠿⠯⢍⡙⠒⠒⠂⠀⠀⢰⡶⠀⠈⠙⠛⠓⢒⡰⠞⠛⠿⠆⠀⠀⠀⠄
⠀⠀⠀⠀⢀⣂⠀⠛⢍⠻⢭⠒⠾⠧⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⢀⣀⣀⣒⡋⣂⠀⠀⠀⠀⠤⠀⠀⠈
⠀⠀⠀⠀⠘⢿⡿⠋⠈⠑⣦⢬⣭⣖⣶⡄⠀⠀⠀⠀⠀⣠⠶⠻⠿⢿⣭⠤⠦⠿⠋⠍⠶⡂
⠀⠀⠀⠀⠀⠀⠁⢼⣿⠂⠐⠚⡋⠀⣐⠉⠢⡀⠀⡠⠾⠃⠀⠀⠢⣀⠀⠄⠀⡄⠀⠀⣤⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⠀⠐⠘⠂⠀⠀⠓⠋⠁⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠒⠀⠀⠀⣠⣴⠶⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠈⠀⠀⠀⠀⠈⠉⢁⣶⡆⠀⠀⠉⠐⠆⠈⠉⢛⣿⠶⣥⣈⡠⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣠⣄⣰⡤⠀⣠⠀⠀⠀⠀⠀⠀⠉⠻⠄⠀⠀⠀⠀⠈⣿⣿⣿⣶⠄⢀⣀⡀⠠⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⢀⣴⠳⠔⠛⣠⣤⣌⣼⣿⠂⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⣻⣿⠟⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣼⣏⢹⣵⣾⣿⣿⣿⠿⠣⣄⣀⣠⡴⣶⣶⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⠟⣿⠁⠀⣀⢻⣿⠟⠘⠋⠁⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠉⠀⠂⠀⠹⣤⢂⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⣶⠀⣠⣄⡤⠀⡀⢠⣇⠀⢺⣿⣿⣿⣿⡿⢀⡤⠶⠒⠀⠀⠲⣿⣦⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠈⠛⠛⠛⠁⠀⠀⠀⠈⠞⢵⡉⠀⠀⠀⠀⠀⠀⢾⣿⡿⠟⠓⠢⠤
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠂⣧⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⢀⡀⣸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⣶⣶⣾⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣦⣡⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠹⠟⡿⣷⣶⡄⠀⠀⠀⠀⠈⢷⣶⣿⣶⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⡾⣿⢈⣰⣿⠟⡋⠀⠘⠒⠢⣝⣷⣿⡆⠀⣠⣄⣘⣉⡀⠀⠀⠀⠀⠀⠀⣼⣯⣿⠟⢄⣀
⠀⠀⠀⠀⠀⠘⣯⡍⠓⠲⢍⡩⣌⣀⣬⣳⡙⣦⣀⡀⠀⠘⠻⠿⣷⢸⣿⣄⠉⠀⢝⡳⣶⣄⣀⣴⠊⠉⠉⠉⠀⠠⣍⠙⠓
⠀⠀⠀⠀⠀⠠⠋⠀⠀⠉⠢⣙⢾⡿⠛⠉⠁⠘⡏⠙⠣⠀⠀⠀⠿⢸⠋⠻⣧⡀⠈⠻⣾⣿⡿⠿⠷⡄⢀⣀⡀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡁⠀⢀⣀⣀⡅⠠⢶⡿⣛⣠⣴⣿⣤⣤⣬⣷⣦⡀⣰⠋⠀⠀⠀⠀⠁⠺⣽⣢
⠀⠀⠀⠀⣀⣀⠀⠀⣀⣤⣄⣰⣶⣧⣤⡿⢱⡿⠁⢀⣤⡶⠟⠛⠫⠯⡭⣤⢄⡈⠉⣹⣯⣇⠐⠂⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠐⠋⠉⠀⠀⠀⠀⠢⢤⢬⣙⣿⣿⣇⡿⠁⢀⡘⢹⢇⡀⠐⢛⣍⠉⠛⠛⠙⢿⣿⣿⣛⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠻⣄⠀⢸⣧⣴⠂⠀⠈⢲⡦⣄⣺⡁⢀⠟⠈⠉⠁⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣼⣷⣼⣿⣿⠇⣠⢴⠟⣿⠈⣿⣿⣿⡶⠄⠀⢀⠀⠀⠀⠀⢇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⠜⠛⠁⠀⠀⠀⣩⣿⣶⣯⣿⠀⢹⣄⣸⣿⡿⠛⠀⠀⠚⢷⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠁⠀⠀⠀⢀⡤⠒⠉⠙⠛⠛⢫⣿⣯⣿⣿⣿⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡿⠁⠈⢿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⢀⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠉⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠝⠒⠙⠨⣿⣔⣦⠀⠀⡀⡀
⠀⠀⠀⠀⢀⣤⠔⠶⠒⠂⢀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠈⠁⠪⢿⣿⡀
⠀⠀⠀⠀⡇⡔⠀⠀⠀⠀⠈⣷⠋⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣀⣤⣄⠀⠈⣷⡀
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠙⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡦⠀⠉⠉⢙⠂⠄⢉⠃
⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢹⡢⠁⠀⣀⠀⣌⡞⠢⡀
⠀⠀⠀⠄⢧⢀⠀⠀⠀⡇⠀⢠⠀⠀⣀⡄⠠⠄⠀⠀⠠⠀⠀⠀⠉⠂⠈⠒⣯⡀⠀⠨⡁⠀⠀⠔⠙⣆
⠀⠀⠀⢰⢸⡀⡀⢀⣤⣿⠀⢹⢩⣤⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢵⢑⠀⠀⠹⠀⠃⠈⠀⠘⠂
⠀⠀⠀⡼⠂⣇⠣⠉⠠⠌⠄⠘⠿⢿⡆⠉⢿⣇⠀⠀⠀⠀⠀⢠⡆⠀⠈⠄⠅⢺⣰⠓⣍⡂⠀⠀⠀⠄⠈
⠀⠀⢸⠐⠁⠹⣷⢦⡄⡀⢀⠰⠒⠶⣾⡶⠖⠛⣂⠀⢀⣠⣴⠁⠀⠀⠀⢀⠀⢸⡃⠸⡀⠀⠀⠀⠀⠀⢨⣀
⠀⠀⠀⠐⠀⢈⡿⣭⡽⠋⢸⠀⠇⠀⠀⠀⠈⠂⠈⠉⠹⠘⠂⠀⠀⠀⠀⠀⠀⠈⣃⠀⠀⠀⣠⠀⠀⠀⠀⠀⠉⠂⠄⡀
⠀⠀⠀⠈⠀⢸⡇⠁⠀⠀⢸⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⣏⣠⠏⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⡀⣀⠇⢀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠃⠈⡴⠁⢸
⠀⠀⠀⠀⠀⠀⠁⠋⠀⠈⠀⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠇⠐⠀⠀⡜
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠐⠄⠀⠀⠀⠀⠀⣀⠤⢀⡚⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠲⣯⢀⠀⠂⠤⡼⢷⡧⠀⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠦⣤⡀⠀⣀⣤⣶⣶⡶⠿⣝⣷⠽⠗⠒⢶⡆⠀⠀⠀⢰⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠙⢳⡟⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣠⣧⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠟⠛⠿⠶⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠠⠤⠀⠀⢨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⠀⠀⠀⢁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣀⠀⠈⠀⢺⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⢻⡿⠿⠀⠀⠀⠀⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠞⢳⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠐⠂⠂⢀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣢⠀⢲⠾⠛⠀⢰⣇⠀⠀⠀⠀⠀⠀⡀⢰⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⢤⡄⣶⣶⠾⠋⠀⠙⢶⣤⣄⡤⠉⠀⢸⡇
⠀⠀⠀⠀⠘⠀⠀⠊⠉⣖⣠⣦⠌⠁⠉⠀⠀⠀⠀⠀⠘⣷⠉⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡤⠈⢟⣋⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⢸⡿⣶
⠀⠀⠀⠀⠀⣀⠐⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣤⡾⠃⠀⠀⠀⠀⠀⢼⡿⠞⡀⠀⠀⣨⠖⣄⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⡟⠀⠀⠀⢀⠀⣠⣾⣿⠄⠀⠀⠀⠀⠀⢀⣿⡃⢸⡇⢀⡸⢦⢸⡄⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢂⣼⠿⣿⠟⠠⠀⠀⠀⠀⠀⢠⠿⣷⣄⣿⠋⢀⣰⡞⠁⠀⠀⠀⠈⠻⣿⡁
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡇⠀⠀⢰⣿⢏⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠡⣟⠿⠿⠛⠛⠁⠀⢰⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠁⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⢙⣷⡏⠀⠀⠀⠘⣿⢋⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⡅⢠⠀⠀⠀⠀⠀⠁⠀⠀⡄⠈⠄⠀⠀⠀⠸⡆
⠐⠂⠀⠀⠀⠀⣿⣿⠟⣄⢀⣸⣏⠁⣾⣯⠀⢠⠀⠀⠐⡆⠀⣰⠀⢲⣿⠻⣀⣀⠀⠀⣠⠀⢱⠀⠸⡀⠀⠀⡸⣄⠀⢻⣄
⠀⠀⠀⠀⠀⢰⣿⡿⠀⢀⣾⠏⠁⢰⡿⠀⠀⡊⠀⠀⠈⠀⠀⠁⠀⠃⡀⠀⠁⠀⠀⠀⠀⡅⠀⠁⠀⢳⡀⠀⠁⠈⠂⠈⢻⡀
⠀⠀⠀⠀⠀⣾⣿⡟⠀⣼⠏⠀⢀⡟⢠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⢠⣾⣿⠁⢰⣿⢳⠇⣾⠁⡎⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⢸⣿⡇⢀⣿⠏⡟⢸⣏⠘⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢸⣿⣷⣾⡟⣰⢀⡿⠙⢿⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢤⡀⠀⠄
⠀⠀⠀⠀⢨⣿⣿⣿⣷⡏⢸⠇⠀⠚⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠈⢛⣩⣿⡍⠀⣿⠀⠀⠀⢸⡏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⢐⡆⠀⠀⠀⢸⠀⠀⢠⡷⣤⠀⠀⠨⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢱⣿⣿⣿⣇⣸⣧⡄⠀⠀⡿⠁⠀⣀⡵⢦⡀⠀⠀⠀⠀⢀⣿⡆⢸⠻⣦⣄⣀⡀⢰⠈⢇⠀⠉⠀⢤⣆⡠⠐⠮⠀⠉⠀⡄
⠀⠀⠀⣿⠸⣿⣎⣹⣿⣿⣽⣿⡄⠀⢺⣴⠞⣿⠁⢸⠋⠲⣦⡤⣀⣼⣿⡇⠈⡇⢸⡌⢻⣯⡸⣧⠘⡆⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⣿⣷⡘⢿⣿⣿⣿⡿⠟⢡⣾⢿⡇⢠⡿⠀⢸⠃⠀⣿⠀⠉⠉⣿⣧⠀⢿⠀⠃⠘⡄⢣⣿⡆⢹⠀⠀⠀⢠
⠀⠀⠀⣿⣿⣿⣷⣤⣷⣶⣶⡇⣿⡟⣼⡇⢸⣿⡀⢸⠀⠀⣿⠀⠀⠀⣿⢹⡀⢸⡆⠀⡀⢡⠀⢸⣿⡄⣇⠀⡀⢠⡀⠠
⠀⠀⠀⣿⣿⣿⣿⢹⡟⢀⣿⠀⣿⡇⣿⠃⢸⣇⠀⠘⡁⠀⢸⡆⠀⠄⠛⠸⡇⠀⣿⡀⢧⠀⠀⠈⣿⡇⠸⠀⢧⠈⠇⠀⠆
⠀⠀⢰⣿⣿⠛⠃⠛⠀⣸⠃⢠⡟⢠⡿⠀⣸⣃⠀⢰⣧⠀⠸⠧⠀⢠⠀⢰⠛⠀⡿⠃⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⣠
⠀⠀⠈⣿⣿⣷⣤⣄⣀⣏⣀⠋⢀⠚⠁⣠⡟⠛⠁⡈⠋⠀⢰⢀⣀⡁⠀⣸⠛⠀⣉⣀⣰⣤⣠⣶⣾⣿⣶⣾⣷⣾⣿⣷⣾⣿⣿⠟⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣶⣶⣿⣶⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠈⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡋⠐⠂
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⡿⢿⠿⠿⣿⡿⣿⣿⣿⣿⠿⠿⠿⡿⠟⠻⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⣩⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠗⣀⣰⢱⣶⣦⣤⠛⢧⡀⡴⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡃⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠒⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⢸⣷⡀⣾⣿⣿⣿⣿⣿⡿⢭⡿⠿⠿⠿⠇⢀⠀⠚⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡀⣿⣿⣿⣿⠦⣿⣿⣿⣿⣿⣿⠟⣟⢆⡀⡶⢦⡰⠿⠀⠀⠀⠀⠘⣆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⣐⢿⣷⠄⣿⡿⢿⣿⡄⣿⣿⣿⡿⣿⠵⠶⢿⡶⠅⠋⠨⠁⠀⢀⠀⡄⠀⢀⡷⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠐⠂⣸⠇⠐⣿⣿⣿⣿⣷⣿⣿⣿⣷⣯⣬⣴⣈⡋⠱⢰⣄⡀⡔⠡⢉⡀⢀⠸⣧⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⠤⣿⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠤⠁⠀⠐⠁⠀⠀⡷⠒⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣾⣿⣿⠀⠁⠀⠀⢀⡇⠀⣿⣿⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇⠀⠠⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣾⣿⣿⣿⣖⠀⠀⠡⢾⡗⢘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⡇⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⣿⣿⣿⣿⢺⣿⣿⣿⣿⣹⣿⣷⣦⠀⣈⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠃⠀⣘⣿⣷⣶⣷⣿⠻⣿⣿⣿⣿⡿⣿⢹⣤⡗⢫⣾⣿⣿⣿⣿⣿⣿⣯⣭⣭⣉⣉⣡⣾⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⡏⣯⣿⢸⣽⣷⣼⣾⣏⣿⣿⣿⣿⣿⣟⣟⣻⠛⣿⢻⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⣿⡿⠤⣀⣿⣿⣿⡟⣿⠾⠬⠿⣯⣥⣟⣶⣾⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⣿⣿⣿⡇⣼⣧⡀⠀⣤⣀⣀⣀⣁⠀⠛⠛⠉⠉⠉⠀⠀⠈⢻⣿⣝⣿⣿⣿⣿⡹⡇
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⣹⣿⣿⣿⣿⣾⣿⣟⣷⡄⠈⢷⣶⣼⠉⣧⡀⢆⠀⠀⠀⠀⠀⠀⠀⠐⣘⣻⣿⣿⣿⣿⡍
⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⠇⢀⣱⣿⣿⢸⣿⣿⠿⣿⣽⣿⣿⣄⠀⢻⣿⣧⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣄
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⡍⠐⢶⣾⣿⣿⡟⣿⣦⠀⠙⣼⣠⣿⣿⣆⠐⢄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣄⡀
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⣿⣿⣯⣧⡟⣿⣷⡀⠈⠛⠛⠛⠛⠳⠀⠣⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠈⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⢿⣿⣟⠧⣤⡿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⣽⣤⣴⣶⣶⣶⣶⣶⣾⡿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠤⣤⣉⣀⠉⠙⠛⠛⠛⠛⠻⠿⡿⠦⢶⣶⡀⢉⣦⣷⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠦⠴⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⡙⠛⣛⣻⣉⡉⠉⠉⣥⣤⣀⠉⠉⠉⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠶⠶⢦⣤⣤⣤⣄⠞⠾⣿⣿⣿⣿⣿⣖⠴⠿⣿⠿⠄⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠿⡿⠭⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⡦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⠃⠀⠐⡿⠖⠒⣉⣀⠠⠴⠶⡄⠀⢀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠿⡠⠄⢨⡷⢧⠘⠁⠀⢸⠟⢀⡀⠀⠔⠃⢠⣿⣿⣦⣠⣾⣯⡐⠒⢀⣀⢠⠟⠛⠓⡀
⠀⠀⠀⠀⠀⣀⠀⠄⢀⣀⡼⡃⢀⠀⠀⠀⠰⠾⠀⡸⠃⠊⠁⠘⠄⠰⠏⠉⠐⡦⠎⢻⡏⠈⠙⠿⠿⢇⠄⠀⠛⠉⠃⠲⣼
⠀⠀⠀⠀⠀⠉⠀⠀⢸⡿⠤⠉⢠⡇⠀⢲⣆⡀⠀⡀⡀⠀⢐⠀⠀⢸⢂⠓⠀⠸⡀⢹⡧⠤⢤⣖⠐⣠⣀⠀⠀⣔⠀⢀⣩
⠀⠀⠀⠀⡄⠁⠄⠊⢿⠀⡐⠀⠀⠁⢠⡏⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣘⠁⠈⠁⠈⡉⠙⠻⡻⠏⠙⣇⣴⡗
⠀⠀⠀⠀⢀⠀⠀⠀⢸⢤⢰⠀⠀⡠⠘⣤⠤⠀⢨⣤⠀⡄⢦⡄⢀⠰⡀⣤⠀⡄⡀⠖⠸⠙⠠⠀⠀⠈⠀⠀⢸⡇⢈⠐⠂⣿⡏⠇
⠀⠀⠀⠀⠀⠁⠀⠀⠚⠈⠋⠀⠀⠉⠀⠈⠒⠃⠀⠀⠀⠁⠈⠀⠘⠸⣀⠿⠀⣿⠀⠀⠀⠀⠈⠋⠀⠛⠈⠁⠈⠀⠀⠁⠍⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣷⣿⣿⣶⣶⣶⣤⣲⣞⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡾⡻⣿⡁⢿⣿⣿⣿⠿⡌⠙⠉⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣥⣤⣠⣏⣳⣾⣇⠈⠁⠱⢷⣶⠃⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣿⡇⠻⢿⣿⣿⣷⣵⣤⣀⣀⠀⣴⣿⣿⣿⣿⣿⡿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡅⠀⠻⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠢⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⣄⣠⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⠛⡿⣿⣿⣷⡄⠀⠀⠈⠛⠻⢿⠿⠿⣿⠘⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠘⠀⠀⢸⣿⣿⣿⣶⣷⣶⣿⡆⣠⣶⣾⣿⠰⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠉⠁⠀⠛⠈⠉⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣳⣷⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣷⠟⠷⣿⣿⣯⣽⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⢸⡟⡿⠿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢀⠀⢘⡀⢠⣴⠟⠋⠀⠑⠆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣿⣷⣄⣈⣛⣿⣏⠆⠀⠀⠀⠀⠀⠀⠀⠒⠿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠐⠂⡀⠰⠀⠀⠀⠀⢰⠁⣴⣿⣿⣯⡙⠿⠿⣻⣤⣙⣲⣦⣄⡀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠂⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⢿⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⡿⢛⣦⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⢷⢽⣿⢇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠷⠋⠀⠀⠹⠠⠄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢘⢉⡉⣻⣟⠉⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣼⢿⡷⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⣟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣐⠛⠿⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠂⠀⢀⣈⠁⠀⠀⠀⠂⠀⠀⠀⠄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⠶⠿⠟⠁⠳⠶⢶⣀⡤⠀⠀⠛⢷⣦⣭⣼⣷⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠴⢧⠠⣤⠀⢸⣷⣽⣿⣿⣿⣷⡀⢃⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠿⠷⠢⠀⠀⠠⠿⠶⠼⠿⠿⢿⣿⣿⣿⣷⡀⢷⡄⠀⠀⠀⠀⠀⠈⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣉⡀⠀⠀⠀⢠⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⡀
⠀⣲⣄⢤⡄⠀⠀⠀⠀⠀⠀⠂⠹⣄⠀⠁⢻⡄⠀⠀⠀⠀⢀⠩⢄⡀⠀⠀⠀⢠⠀⠀⠀⢈⣄⠀⠀⣦⠈⣿⡀⢒⠀⣤⡶⠶⢦⣤⡀
⡆⣿⣿⢸⡇⣥⠀⢀⡄⣰⡀⣸⣆⣿⠀⠀⢘⡇⠀⠀⠀⢀⣸⡇⣹⣞⣿⣦⠶⠾⠷⠒⠒⠘⠿⠶⢦⣿⡇⣼⣇⡄⣷⢼⣧⣄⡀⣿⡅
⠀⢿⣟⠼⠛⠛⠛⠻⠿⢿⣿⣿⣿⡟⢀⣼⣿⣿⣤⢁⣴⡼⠿⢟⣾⡿⡭⠐⠀⠈⠉⠀⠀⠀⠀⠁⠀⠙⠻⣿⠿⣿⣿⣸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⢿⣿⣿⣿⡟⣉⠴⠚⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣍⠉⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⠀⠇⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⢟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠘⢸⠀⢠⠾⠟⠩⠄⠤⢤⣌⠹⢦⡀⣀⡀⠀⠀⣠⣴⠚⠉⠁⠀⠉⠑⠢⠀⠘⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⠢⢄⡆⢘⢀⣏⠀⢀⡪⢀⡦⣆⠔⠳⠀⢷⣤⣤⣤⣾⡟⠁⢀⣴⣩⣧⡽⠇⡄⠈⣆⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣏⠀⠥⡘⣻⣿⠀⠘⢁⠻⣾⣿⢏⢹⡇⢸⠟⠛⠛⣿⡃⠀⢘⡀⠼⣭⣿⣎⡁⠀⣟⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠛⠦⠤⢾⢿⣧⠀⠲⣄⣸⣀⣴⡾⢃⠈⠀⠀⠀⣘⣧⡀⠘⢄⡀⠛⠉⡤⢁⣤⠉⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⡷⡦⣤⣈⣀⣡⠴⠿⢿⣿⣿⣿⡿⠿⠷⢦⣀⣈⡉⣀⣠⣴⣷⣶⣶⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠧⠐⢧⣠⠴⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠰⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠳⠄⡨⢤⣹⣤⠍⠦⢧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⠟
⠀⠀⠀⠀⠀⡤⣦⠘⠇⡞⠀⢀⠃⠘⠆⠀⠀⠀⠠⠀⣀⣘⡗⢠⡙⠾⣻⠟⡶⣖⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠛⣀⡴⣤
⠀⠀⠀⢀⡶⠋⢡⡔⠎⣉⢀⡠⠀⠀⠁⠀⠀⠈⣿⠧⣀⣨⡯⢰⣿⣾⣿⠙⠉⠉⠋⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠥⠀⠻⠿⠟
⠀⠀⠀⡿⠁⢀⡏⢀⣼⠤⢤⢄⠀⠀⢴⡀⠀⠈⣷⠶⢧⡄⠒⠾⠲⣿⢿⢿⣿⠿⠿⠄⢧⠀⠀⠀⠀⠀⠀⢠⣴⣇⣠⣤⣄⣀
⠀⠀⣰⠇⢰⡶⢙⣾⠓⢒⠤⠀⠇⢀⠔⠀⠀⠀⢹⣿⣿⣉⡉⠉⡛⠛⢻⢏⠾⡀⢀⠄⠀⠀⢀⣀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⠎⣶⢠⡀
⠀⠀⣷⣼⠟⣦⡾⣯⡾⠻⠀⠀⢀⣥⣀⠀⠀⠀⠀⠑⡿⣻⣿⣿⡷⠚⡏⢀⣆⠀⢠⡀⢰⠦⠈⠁⠀⣀⣿⣿⣿⣿⡿⣿⠟⡿⠀⠁⢸⢓⡄
⠀⠀⣿⡿⢟⡿⣷⣞⠑⠠⠔⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠉⠻⢻⡷⢿⣿⡏⠈⣗⠻⢏⠁⢀⠀⣦⣼⣿⣿⣿⣿⣿⣷⣿⣞⠀⠀⠀⠠⠾⠃
⠀⠀⢹⣿⣿⣿⣷⣿⢷⡅⠐⠛⠀⠀⠀⠀⠈⠀⠀⠰⠈⠀⠀⠄⠈⠈⠋⠃⠠⣿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⠟⠜⢳⣻⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠻⠿⠛⠟⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣜⣠⣾⡟⠀⢀⣦⠀⠀⢻⠀⢠⠇⠀⠿⢿⣿⣿⢿⣻⣥⣬⣬⣽⣿⠚⠁⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣦⣤⡀⢀⣷⣄⠀⠛⠿⣿⠿⣿⢠⠀⣸⣿⡇⠀⠸⢠⡏⠰⡄⠸⢿⣿⣷⣿⣿⣿⣿⣿⡿⠉⠀⠱⣬⡉⠛⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣶⣾⣿⠒⠤⡘⡄⠸⡆⣿⣿⣿⣿⣿⢰⡏⠀⠀⢹⡤⠛⣿⣫⢝⡿⠁⠻⣸⢰⠀⠄⡀⣍⣿⡷⠆
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⣆⣶⡈⠻⡄⢷⣿⣿⣿⣿⡿⢸⠀⠀⣴⠋⠀⢀⠟⠻⣯⣴⡇⢠⡇⣼⡀⣆⠷⡽⠞⠁
⠀⠀⠀⠀⢿⣿⡂⠀⠐⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡾⣯⠹⣿⣿⣿⡇⣿⠀⡼⠃⠀⢠⡜⣀⢤⡄⠉⠈⠋⠛⠙⠃
⠀⠀⠀⠀⠉⠛⠳⣦⡖⢀⠈⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠹⡄⢻⣿⣿⣀⢟⡿⠃⢈⠀⢸⡇⢻⡶⠇⠀⢀⠐
⠀⠀⠀⠀⢼⣿⡿⠋⡿⠫⢀⠈⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣵⠊⠀⣠⣾⠀⢠⢠⣮⠀⣠⣤⣯⠒
⠀⠀⠀⠀⠀⠀⠀⠺⠓⣄⣀⣾⣀⠤⢙⠛⠛⠛⣛⠙⠛⢿⣿⣿⣿⣿⣿⡿⠃⠀⠄⢀⣼⣶⣿⣿⣷⣴⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⢻⠏⣤⢤⣤⣴⣷⣶⣦⣴⣶⣿⣿⣿⣿⣿⣿⡗⢈⣤⣶⣿⣿⠋⡟⢻⣿⣿⡻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠚⠉⠋⠀⠙⣠⣼⣷⣿⣿⣿⣿⣿⣿⣿⠛⢿⣯⡿⠛⠉⠁⢸⡟⣿⣿⡏⣾⠡⢋⣁⠈⣙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠿⣿⣿⣿⣿⠟⠋⣹⣷⡾⠋⠀⣰⣿⣷⣮⠓⣿⡟⣠⣣⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⢠⣾⣿⡿⠗⠊⠀⢠⣌⣭⣻⣇⠹⢡⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⠀⠿⢿⣿⣿⢀⣿⣿⣟⣛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣴⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣄⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⢀⣠⡾⣟⠁⠰⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⣀⣤⣾⣿⣗⣨⣿⣦⣌⣿⣿⣿⣿⣿⣿⢸⣿⡟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⣿⣷⣤
⠀⠀⠻⠆⢀⣤⣭⢙⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣿⣭⣍⡀⠀⠀⠰⠿⣿⣿⣿⡟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠘⠿⠓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣤⣄⣀⣤⣿⣿⣿⣿⣴⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⡾⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⡀⠀⠀⠹⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⠋⣸⡿⠁⠀⠀⠀⠀⢀⣀⣀⣠⡤⠀⠀⣽⣿⡿⠀⠀⢄⣨⣶⣿⣿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣟⣿⣿⣿⠷⠶⠿⠿⠛⠛⠛⠛⠙⣫⡿⣅⣮⣾⣫⣤⠦⠂⠈⠹⢿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣁⣭⣿⣿⣾⣿⣿⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣉⢻⣿⣿⣿⡿⠟⠉⣙⡛⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡆⠻⠞⢛⣋⢠⣴⣾⣿⣿⣿⣷⣜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢀⠛⢻⡿⠟⠋⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠑⠽⠿⠿⠖⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠙⠲⠀⠀⠀⠀⠤⠖⠢⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⣤⣤⡲⠀⠀⡴⣠⣤⣀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠛⠁⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣁⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣏⣹⣷⡆⠀⠀⠰⣞⠛⠛⢳⣦⠀⠀⠀⣷⣿⣿⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡄⠀⠀⠉⠛⠛⠋⠁⠀⠀⣰⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣟⠻⣿⣶⣬⣝⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠙⣿⣧⠀⠀⠀⠉⠉⠉⠙⠋⠻⣿⣿⣷⣞⠙⢿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣽⣿⣿⣏⣸⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡻⢿⣇⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠉⠺⢿⣿⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣴⣦⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠿⠿⢿⠓⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣆⣠⣦⡜⠛⢂⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡌⣻⣿⣠⣠⣤⠎⠙⠉⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣶⠛⠋⠀⠉⠙⠉⠉⠿⣿⣿⣶⣶⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⡀⣷⠃⢀⣠⣦⣤⣤⡦⠤⣤⣿⣿⣿⣿⣿⣿⣥⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⡤⠼⢿⡆⢉⠹⣿⠿⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣀⠀⣿⡄⠀⠀⠀⢀⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡿⠋⠁⣠⣦⡿⣹⡔⠻⡄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⠀⢸⣿⣤⠀⣨⠀⢿⠷⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠋⠁⠀⠸⠿⣿⠃⣿⣿⣦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇⠈⠙⣇⣠⠏⠘⢺⡟⠋⠂⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠐⢾⣿⣶⢏⣻⣿⣿⣿⣤⣠⣶⠃⠀⢀⣀⠀⠀⢸⡟⠂⠘⠻⣷⣾⡇⠾⠿⠟⠛⠋⠉⣁⠀⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⡄⠀⠀⠹⢿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⣿⠟⠃⠀⢀⡏⠀⣼⣧⠖⠴⠔⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠠⣖⣿⡖⠀⠀⠘⢿⣿⣄⢛⠛⣿⣿⣿⣿⣿⣿⣿⣏⢽⠿⠛⠁⠀⣤⠞⠀⠀⠙⢿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠚⠛⠛⠻⠦⣿⣄⢳⣾⣆⠀⠘⠛⢛⣿⣿⣧⣤⣉⣹⠏⠉⠁⢀⡘⠻⣷⣤⣴⣾⠋⠀⠀⠀⢐⡈⠻⣍⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⡶⠛⣡⠤⠶⠿⢌⠹⣿⡷⠀⠀⠈⠁⠊⠸⠏⠙⠻⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣷⠀⠀⣀⠈⠉⠙⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠥⠶⠋⠁⠀⠀⢀⠀⠡⡟⠁⠀⢿⡏⠀⠀⠀⠀⢻⣿⣿⣿⠟⠄⠀⠀⣼⣿⣷⡵⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠛⠋⠄⠰⡄⠀⣿⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠐⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⡰⠚⠁⠀⠀⠈⠆⣿⠀⣀⡀⠤⠤⣀⠠⠤⢤⢤⣤⣴⣾⣾⣿⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⣿⣞⠁⠀⠀⠀⠈⠳⣦⣦⡀⠀⠉⠛⠷⣶⣷⣿⠛⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠚⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⢀⡄⠈⠻⡛⢦⠠⣄⠀⢄⡈⠉⣉⣉⡭⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠈⠀⠀⣄⠙⠾⣧⡀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠛⠟⠳⠤⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠤⠄⠠⠤⠤⠤⠄⠰⠢⠖⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⣲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢟⠿⠟⠋⢡⣴⣞⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠠⡑⡄⢈⠿⠋⠙⣷⣄⠀⠀⠀⠀⠀⠌⠀⠀⠖⠀⠐⢛⣻⢯⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠈⠀⠀⠀⢀⣾⠿⠆⠀⠀⠀⠀⣀⡀⠀⠀⠐⣤⣀⢈⠉⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠸⢫⠉⠀⠀⠀⠛⠋⠐⡄⢸⣿⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⢘⠁⠨⡽⡆⠀⣠⣤⣄⣸⣿⣿⡟⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣹⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣸⡋⠑⠚⢿⣿⣿⡆⠀⠀⠀⢈⣀⠔⢀⡀⠀⠀⠀⠀⢠⡝⠛⠉⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠀⠘⠀⠠⣯⡇⠀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠈⠠⢤⠄⠀⠀⠀⠀⣼⠇⠀⠀⣿⣿⣿⠿⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣄⠋⠄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣠⣿⡏⠀⠀⠀⣿⠿⠋⣰⡿⠿⠦
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠐⠿⡿⢱⣦⣀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⠾⠿⢿⠁⠀⠀⠘⠃⠀⣰⡿⠁
⠀⠀⠀⠀⣀⣤⠀⡎⠏⠀⠔⠺⠛⠁⢸⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠑⢤⠀⠀⠀⣰⡿⠁
⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡸⣼⠆⠀⣰⡿⠁
⠀⠀⠀⣯⣿⣇⠀⠀⠀⠂⠠⠀⡠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣷⣷⠀⣰⡿⠁
⠀⠀⠀⠉⢻⣿⠀⠀⠀⠀⢀⣾⣣⣿⠷⣿⢯⠯⢭⣵⣿⢿⣫⣿⣿⣿⠟⠋⠀⣼⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣾⢝⣾⡟⣸⣇⠃⠠⠿⠟⠖⠋⢱⣟⠟⠁⢀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢆⠑⠀⢀⣶⠛⣯⣿⣿⠃⢹⡽⡣⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣸⢻⣻⡿
⠀⠀⠀⠀⠀⠢⡀⠸⣏⣾⣿⣇⢟⣴⡷⠉⠈⠁⠀⠀⠀⠀⡄⢀⠀⡐⠀⠀⡏⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠈⠉⠿⢿⢶⠫⠿⠁⠀⠀⠀⠤⠊⡀⠂⠈⡆⠀⠀⠀⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠉⢳⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠈⠋⠀⠒⠉⠉⠉⠘⠛⠛⠿⠿⠿⠿⠿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡀
⠀⠀⠀⠀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖⠉⠉⣿⣿⣿⣶⣖
⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣿⣿⣿⣧⠀⢀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⣻⣿⣿⣿⡄⠀⠀⣿⣿⠃⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠛⠿⠇⠀⢀⢀⣁⣄⣿⣿⡟⠀⠀⠀⢀⡀⠞⠻⣿⣿⣿⠏⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⠺⠋⣺⣿⣿⠉⠁⠀⠀⢀⣤⡻⡿⠶⠌⠉⠉⠿⠿⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⣻⣿⡄⠀⠀⠀⢈⣙⠛⠳⠀⠀⣀⠀⠀⢀⢙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢴⣿⣷⠀⠀⠀⠈⠛⠠⠶⠀⠀⠉⢳⠀⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣦⣄⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣻⣿⣧⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⠉⢘⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⠀⠀⠀⠀⢈⣿⠆⠀⠀⠘⠿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢿⣿⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣩⣿⣿⡀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣿⣿⣇⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠿⣿⣿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠄⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢛⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⢯⡘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢷⡟⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢟⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣈⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⢁⠄⠉⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡷⠋⠀⠘⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠰⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠐⠁⣠⠊⠀⠀⠀⢢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣧⠀⠀⠀⠙⠀⣃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣮⡂⠀⠀⣷⡤⠶⠀⠶⠶⠶⣿⣿⣿⣷⣄⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠞⠁⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡡⠤⠲⢤⣀⢿⠋⠋⠀⠀⠀⠀⠀⣿⡟⢻⣉⡛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠻⣏⢈⠀⠀⠀⣀⡠⠾⣍⣸⡿⣋⣡⣤⣤⣄⡀⢀⡀⠰⣄⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣀⣤⣄⠀⠀⣰⣿⠐⢃⠀⣆⡈⠀⠀⠙⢁⣼⣿⠷⠚⠙⠺⢿⣦⠁⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠿⠻⠙⠁⠙⣿⠾⠿⠟⠓⠋⠀⠁⠙⢦⡀⠀⢸⣿⣟⠀⠒⠀⡄⣸⣿⣦⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡤⠋⢀⡐⠀⠀⡁⠀⠀⠀⠀⠂⡈⠛⢷⣦⣉⠀⢠⣿⣿⠀⠀⢀⣴⣿⠋⣈⣆⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠈⠀⠀⠙⢸⠁⠠⠀⢈⠸⠶⢸⣦⣽⣦⢸⣿⣿⣗⣿⣿⠟⠀⢀⠘⠻⢧⣀⠾⠿⣦⢻⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⣸⣿⣿⣦⣤⣤⣶⣤⣾⣿⣷⠀⠙⢿⡉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠡⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢃⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡎⢧⢹⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠃⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⣷⠈⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣋⢻⡟⠣⠿⢿⣿⡷⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⢠⣾⣿⣿⠋⣙⣿⣿⣿⡍⣿⡷⠂⡄⠰⣦⣶⣄⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡞⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⠀⢸⣿⣿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⡜⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠀⠐⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡼⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⡀⠂⣴⣤⠐⠃⠀⠀⠀⢠⣤⡶⠂⠀⠀⢠⣜⠏⣀⠀⣾⢋⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⣿⣯⡼⣿⣏⠙⠟⣠⣦⡀
⠀⠀⠀⢼⢁⣦⠈⠁⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⡰⠋⢁⡴⠃⠀⠇⢾⣥⣾⡏⢠⣿⣿⡇⠙⣻⢿⣏⡑⠀⡀⠙⢊⣉⣰⣾⡿⠋
⠀⠀⠀⢈⠉⠁⢀⠀⠀⠀⠀⡀⠉⠀⠀⠀⢀⠀⠁⠀⠞⠀⢀⡄⠀⠀⡄⠉⢤⢸⡿⢿⣁⣀⣛⣿⣯⡷⠺⢷⡀⠹⠛⠋⠹⠄
⠀⠀⠘⡵⠚⠋⠀⠀⠀⠀⢀⠤⠒⢀⡰⠟⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣤⣴⣦⣿⡉⠀⢻⡶⠀⠀⡤⠈
⠀⠀⠀⠰⢦⠤⢶⣤⣄⣀⣤⣄⡀⠀⣀⣠⣤⣤⣶⣄⣀⡀⢁⣄⣽⣶⣿⣦⣀⣀⣁⣀⣀⣧⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⢀⣬⣿⣿⣷⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⢶⣤⣤⣤⣤⣔⣺⠋⠏⠁
⠀⠀⠀⠀⠀⢾⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⠿⡏⣛⣙⠋
⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠛⠛⠛⠀⣾⠛⢫⣤⣿⣦
⠀⠀⠀⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡻⠿⢿⣿⣿⠘⠛⠢⠀⠲⠘⠣⠶⢿⣿⣿⣿⣤⠖⠐⠂
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢩⣽⣿⡆⠀⠀⠀⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢀⣴⣶⣎⡟⢀⣼⣿⣧⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⣴⣿⣿⣿⣿⣦⣬⣭⣿⡏⠀⢀⣀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⣾⣿⣏⣽⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢋⣙⣦⣄⡀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠋⠁⠀⠀⠸⣿⡟⠉⢉⠶⣄⡉⢀⣴⠈⠛⢻⣶⣼⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⡀⠀⠀⠀⠈⠙⣦⣎⡐⢾⣿⣿⣿⣷⡀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡜⠁⣻⣿⣿⣿⣟⡿⢟⡅⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⣺⣿⣿⣿⡿⠿⠟⢿⠃⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣰⡟⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢸⣿⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏
⠀⠀⠀⠀⣰⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠁⢼⣿⣿⡿⠛⠛⠂⡀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⢰⡟⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⡞⣿⣿⣧⠀⠠⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢿⡆⠀⠀⣠⡼⠛⠁
⠀⠀⢠⣿⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣬⣫⣽⣆⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅⠀⠀⠀⠃
⠀⠀⣼⠇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠡⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡿⢏⣴⣿⡿⠿⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⡛⢻⣿⣳⣦⡀⠀⣽⣿⠋⠁⠀⡠⠂⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⢺⣭⣿⣿⣷⣜⣿⠋⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠲⡦⢀⠀⠀⠀⠀⠀⠀⢳⠈⢿⢯⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⢤⣀⡀⠀⠀⢀⡀⠀⡇⢠⣿⡿⣛⣿⣿⡿⣿⣿⣿⣿⣷⣦⣀⡸⠃⠀⢀⣠⣴⣤⣤⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢠⠀⠀⠘⣌⣻⡅⠻⣿⣯⣾⣭⣫⣿⣿⣿⣿⣿⣿⣒⣺⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠿⣷⣜⣻⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣯⣽⣿⣿⣿⡛⠛⠋⠉⠡⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢙⡿⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⡔⣊⣉⣭⠶⠃⠉⠛⢿⣿⣿⣿⣷⣦⡀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣴⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣀⣼⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠸⣦⣵⣦⣄⡚⠀⠀⣀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⢵⡄⣴⣿⣿⣿⣿⣿⣿⣷⣌⡡⠿⠛⠛⠟⠤⣌⠉⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠉⠙⠉⠻⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⢠⣿⣿⣿⣥⠀⠀⢀⡀⡏⣾⠧⢀⡴⠄⠁⠒⠀⠀⠀⣶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⠀⢀⣄⣀⠂⠀⠀⡰⣿⣿⡏⠀⠀⠀⠀⠀⣷⡿⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠈⠞⢀⠀⠀⢧⡄⠈⠁⠀⠀⠀⠀⠀⠀⢀⡴⠤⠤⣤⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠃⣀⡼⠟⠀⠂⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠚⠀⢆⠉⠉⢿⡆⠀⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠃⠰⢨⣟⠀⠀⢹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠆⠡⠄⣠⠸⠀⣀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠙⢦⢀⠀⠀⠂⠙⠟⠁⠠⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣴⣾⢋⡟⠯⣩⡕⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣤⠶⠺⣤⣤⣦⣶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠀⠣⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣶⡾⢿⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠨⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣙⣊⣹⣭⣉⣋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣟⣛⣛⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠦⢤⣤⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣦⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⢠⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠀⠐⠆⣀⣤⠀⠀⠠⠅⠉⠀⡴⣦⣤⠚⠗⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⢰⠀⠀⠘⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⣶⣷⣶⠄⠀⠀⣀⠀⡤⠀⣄⠀⠀⠀⡄⢀⢠⣄⣠⣿⠀⠠⡌⠠⣴⣾⣷⣤⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢰⡄⡀⣿⠀⡿⠄⠰⣷⣷⠀⢨⢯⡟⢻⡦⠲⡇⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⣀⣤⣾⣕⣾⣿⣧⣟⠀⠀⣿⣇⡀⢸⣾⣷⣴⠇⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⢿⣿⣿⣧⣬⡿⢏⡁⠈⣿⡇⠁⠨⠋⠙⣿⣿⣿⣶⣾⣿⠿⠆⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠂⠊⠙⠂⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⡿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⠀⢹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣠⣴⣤⡈⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣇⢨⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢩⡛⢷⣄⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⢩⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣯⠈⠁⠀⣻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⠀⠀⠀⠀⠲⠶⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢴⣾⣆⠀⠀⠀⡀⠀⠀⣤⡀⠰⣤⠀⢀⣤⣤⣶⣿⣶⣶⣿⣿⢖⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⡻⢿⣧⡀⠀⠀⠀⠀⢼⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⡅⠈⠿⠿⢷⣶⣤⣴⣾⣧⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⠙⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣷⣤⠀⠀⣼⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⣧⣄⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⡀⠀⠀⠀⠀⠀⢀⡠⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢿⣧⡄⠐⠁⣰⣴⠋⠀⣠⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣆⢀⣽⣷⣿⣴⣿⡁⠀⣰⠋⢀⡟⠉⣿⣛⣿⠩⣵⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠏⣿⣿⣿⠹⣿⣾⠃⢠⢞⣀⣼⣿⣿⡇⠀⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢻⠀⠀⠀⠀⣿⣿⣟⡀⠉⢁⡴⠻⢾⣿⣿⣿⣿⠁⠀⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠘⠀⠀⠀⠀⢼⣿⣧⠀⠉⢹⣛⣀⣶⣿⣿⣿⣿⠀⣀⡄⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⣸⡿⣿⠀⢀⠀⢉⣿⣿⣿⣿⣿⣿⠀⡿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⢰⠀⠀⠀⣿⡇⣿⡇⠈⣇⠀⠙⢿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣟⣿⣿⠇⠀⠀⢸⠀⠀⠀⠈⠉⣿⡇⠀⠇⠀⠧⢼⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⣼⢹⣿⡏⠀⠀⠀⢸⠀⠀⠀⠀⠂⢸⣿⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣾⣶⡏⠐⢶⣶⣶⣤⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⢹⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣡⠆⠀⡀⡀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠉⠙⠛⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠿⠷⠶⠦⣤⠄⠀⠀⠀⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡏⠙⠛⢻⣷⠶⠶⠂⠀⠀⠀⠀⠙⣷⠒⠖⢠⣤⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣄⠁⠀⠀⠀⠀⠙⠛⠶⠤⢀⣀⣀⡀⠈⢳⡀⠀⠀⠈⠛⢧⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠚⠛⠀⠀⠀⠀⢀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠘⢇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡆⠀⠀⡠⠀⠐⠒⠀⠀⢤⡤⠄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⡏⠀⠈⠃⠀⠰⣶⣶⠆⠈⠀⠀⠀⠀⠀⠀⠀⠐⣶⡶⠀⠈⠀⠀⠀⠀⣿⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠾⠀⠀⠸⡇⠀⠀⠀⠀⠀⠴⠿⠉⠀⠀⠀⠃⠀⠀⠈⠠⠄⠻⠟⠁⠀⠀⠀⠀⠀⠏⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡤⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠃⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢶⡿⣧⠉⠙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⣿⠁⠉⠉⠀⠉⣲⡞⠋⠀⠁⠈⠁⣠⣾⢳⠏⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠇⣶⠀⠀⠀⠊⠁⠈⠀⠀⠀⠀⠊⡳⣽⠇⠀⠀⠀⠀⠀⡰⠏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠉⡿⠀⠀⠸⠋⣦⠀⠀⠀⡆⠀⠀⠀⠀⣠⢞⡷⠋⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⢸⡇⠀⠀⠀⠻⡈⢧⣀⡀⠀⠀⣀⣠⡞⣡⠎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠘⠃⠀⠀⠀⠀⠙⢦⡉⠛⠒⠚⣉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣾⠉⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠠⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠟⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⠑⠒⠒⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠠⠛⠷⠖⠀⠸⠁
⠀⠀⠀⠀⢠⣿⠦⠛⠛⢻⣦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⢀⣞⣁⡆⢰⠀⡿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠉⠉⠙⢧⣠⣧⢸⠃⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠷⠶⢀⣤⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠿⠾⠓⣿⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡿⠶⠶⣶⣄⠈⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⡀⢀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠰⢤⣤⣄⣘⣻⣷⣦⣤⣿⠀⠀⠀⠀⢙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠙⠃⠀⠀⠈⠀⠀⠀⠉⠛⠛⠲⢶⢂⣉⣉⠉⠉⠉⠙⣿⣿⣦⣶⣴⠟⠁⢾⣇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⡶⣶⢠⣶⢤⣤⣤⣄⣀⣀⣀⣀⠐⠚⠛⠛⠛⠛⠛⠛⣛⣛⣉⣉⣿⣿⣿⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⡇⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⢿⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⠁⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠤⠤⠤⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡇⠀⠀⠐⠒⠒⠂⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣄⣤⣤⣤⣄⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣧⡴⠆⣈⣉⣀⠴⢄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣉⣉⡉⣉⣙⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠂⠉⠉⠁⠐⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠤⣶⣶⣶⠤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠉⢹⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠃⠀⣢⡄⠠
⠀⠀⠀⠀⠀⢁⠻⣿⡿⠁⠀⡀⠀⠁
⠀⠀⠈⢳⡀⠋⠒⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣤⠄⠻⢿⡀⠀⠀⠀⠀⠀⢸⣏⠘⡛⠻⣿⣿
⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⢀⠀⠀⢘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠉⠀⠁⣨⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠟⠳⡀
⠀⠀⠀⠀⠀⠀⠃⣠⠄⣠⠘⣦⡴⠖⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⠁⠈⠟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠀⠐⣈⣢⡟⢠⠆⠀⠁⠀⠀⠀⡀⢰⡦⠀⠀⢈⡀⢰⣧⠀⠹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠒⢶⣽⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠀⠀⠀⠀⠀⢴⡀⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡞⢸⡇
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣧
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠈⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠛⠂⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠘⠃⢀⣤⣶⣶⣶⣤⡁⠀⠀⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⢚⡉⠡⢻⡟⣽⢿⡟⠀⠀⠀⠀⠈⠙⠀⠄⠠⠀⠀⡀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠺⠋⠀⣼⠁⢰⡇⠀⣿⢸⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡁⠘⠃⠀⣿⡇⠀⡿⡜⠀⠀⠀⠀⠃⠀⠸⠀⠀⠰⠀⠳⣶⠀⠀⠄⠉⠞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⢀⣴⣿⠇⣴⣿⡁⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣴⣿⠇⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣻⣿⣿⣾⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣹⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠙⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⣿⣽⣿⠏⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠟⣀⠀⠀⠐⣎⡆
⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐⣿⣽⣿⣿⣿⡿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣼⠓⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⠁⠀⠀⢾⣷⡆⠀⢨⡿⠀⠀⠀⠀⠀⡘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣽⣿⣽⠇⠀⠀⣠⠐⠋⢤⡅⠌⢁⣴⡀⡀⠳⡄⠙⢷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡶⢢⣿⠀⠀⠚⠁⠀⠀⠀⠀⠄⠀⠛⠃⠀⣾⣶⣄⡄⠉⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠺⡿⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠉⠉⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠁⠀⢀⠈⢀⠈⠀⠀⡀⠁⡀⠀⠀⢐⠈⢐⠀⠀⠀⡀⠁⡀⠀⠈⢐⠈⢀⠀⠀⠁⡀⠁⡀⠀⠨⢀⠈⢀⠀⣀⣁⣠⣤⣦⣤⣬⣀⣨⣀⣥⣀⠁⡀⠁⠈⠀⠨⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢈
⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠠⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠀⢀⢠⣼⣿⠿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣯⢻⣆⠀⢐
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⢀⣾⡿⣣⣾⣿⣿⠟⠋⠉⠉⠙⠛⠛⠛⢿⣿⣝⢿⣰⣿⠀⠐
⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡠⠤⡀⠀⣼⢻⣳⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢾⣿⣿⡇⠠
⠄⠀⢐⠠⢐⠠⠀⠄⠀⠄⠀⠀⠀⢐⠠⢐⠀⠀⠀⠂⠄⠀⠀⠠⢐⣀⣐⠀⠀⠸⣄⣀⡸⢀⣿⣼⡿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⠃⠠⠀⠠
⠀⠀⠐⠠⠐⠀⠀⠀⠀⠀⣀⡴⠶⠒⠛⠛⠛⠒⠂⠂⠄⡠⠖⠉⠉⠉⠉⠉⠒⢄⡂⠁⠀⠰⣿⣿⢧⣿⣿⠠⠦⣄⡀⠀⠀⣠⠴⠛⠃⠀⠀⢿⣿⢿⣿
⠀⠀⠠⠀⠀⠐⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢨⣿⣿⣼⣿⡷⠖⣾⡟⠣⠀⠈⠑⠺⣿⠟⠗⠀⠈⡅⠉⡟
⠀⠀⠈⠀⠈⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡟⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣷⠀⠐⠀⠐
⠀⠀⠀⢀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣵⣿⡆⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢿⣿⣆
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⣾⣷⣿⣿⣿⠨
⢱⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⢿⢿⠟⠁⠀⠀⣠⣾⣿⣿⣿⣇⠛⠈⠀⠈
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣠⢾⣿⣿⣿⣿⣿⣿⣦⢈
⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⢰⣝⣯⣻⣿⠟⠻⡎⢳⠒⠒⠋⠀⢸⣿⣯⢿⣿⣹⣿⣿⣷⣨⣀
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠁⠀⠀⠀⠀⠉⠉⠛⠻⠟⠃⠀⡴⠁⠀⠁⠀⠀⠀⠀⠙⠻⣧⣿⣷⣿⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠒⠒⠒⠒⠒⠒⠈⠁⠁⠀⠘⠟⠀⠀⠀⠀⠀⠀⣈⣿⢹⣿⣧⠟⣿⣷
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡀⢀⣀⠀⠤⠤⠔⠒⠒⠈⡅⠀⠀⠀⠀⠀⠀⣠⡴⢁⡼⢿⡏⠀⠛⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⣀⣀⠄⠴⠈⠀⡄⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⢾⣿
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⣀⠠⠠⣶⣷⠀⠀⠀⠘⠄⢔⡈⠒⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠉⠉⠀⠀⠀⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷
⠀⠀⢀⠈⢙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⣿⠛⠢⣀⠀⠀⠀⠀⡄⢀⠞⣿⣿⠋⠙⣿
⠄⠀⢐⠠⢀⠠⠀⠍⡉⠍⡃⠘⠊⢐⠠⢘⠢⣄⡀⠀⠀⠀⠀⢀⣠⠠⢀⠤⠤⠤⡤⠶⣷⠀⠀⢸⠀⠀⠀⠀⠀⢳⣿⣿⣷⣶⣿⣷⣄⢣⣸⢠⣷⣾⣿⣿⣷⣾⣿
⠀⠀⠐⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠉⠛⠛⠛⠙⠉⠐⠀⠐⠀⠀⠀⠀⠀⠘⡀⠀⢸⠀⠀⠀⠀⠀⠈⡋⠉⣿⣿⠋⠉⣿⣦⣿⠉⣿⣿⠋⠙⣿⣿⠋

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⣠⢶⡿⠇⣠⡔⠶⣖⠀⠀⠀⠀⠀⢀⣤⣶⡶⠆⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠈⣁⠉⠚⢠⣦⠈⠀⠉⠀⣶⣿⣿⡻⠆⠀⠙⠃⠀⣶⠟⠋⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⢠⠈⠉⠠⡿⠀⠀⠀⠠⠀⠉⠉⠁⠰⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠈⠄⠀⠈⡧⡀⠀⠀⢀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠈⠃⠰⠃⠈⡀⢀⣬⠀⠀⠠⡄⠀⠁⠀⠰⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡟⠁⢁⣤⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣴⣾⣿⢟⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠛⠛⠁⠚⠟⠋⠁⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠⢠⡄⠤⡄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠗⠈⠛⠘⠇⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⢨⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠐⢠⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠠⣤⠉⠙⠋⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦⣤⣤⣼⡇⠀⢠⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⠀⠀⢁⣼⣿⡶⢾⣯⠟⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠞⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠦⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⠴⠦⠶⠶⠾⠿⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⣣⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠋⠀⢁⡄⠀⣷⠀⠀⠀⢠⡤⠘⠉⣀⣭⢴⠀⠀⠀⢰⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⢷⣄⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠄⠀⠀⢺⡇⠀⣿⠀⠀⠀⢀⠀⢠⣾⣿⣦⡇⠀⠀⠀⢸⡟⠀⠀⠀⣠⣄⠈⠛⠛⠃⣤⣶⣤⣤⣄⠘⡃⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠠⣄⠸⠣⠀⣿⣶⣄⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠛⠉⠁⠀⢀⠀⠀⠛⠛⠉⠻⡧⡇⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡄⠀⢠⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠁
⠀⢸⣿⠀⠀⠀⠀⣤⠀⠀⠸⠧⠀⠰⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⣬⣍⣄⣀⣉⣧⡀⠀⠀⢸⣠⣤⣀⣤⣀⣶⣦⣼⣿⣿⣿⣿⣦⡀⠙⢷⣆⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠟⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠏⠀⠠⠐⢂⡀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠲⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡆⠀⠀⠀⠈⠀⠈⡀⢠⣈⠻⣶⣤⣤⣤⣤⣦⣴⣤⣴⣾⠏⠁⠀⠈⠃⠀⠀⣿⠄
⠀⢸⣿⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠀⢰⣷⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠇
⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⢦⣀⣀⣠⠔⠋⠁⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⣠⠾⢉⢙⡙⢶⡄⠀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠐⠀⠨⢍⠲⠦⣄⡀⠀⠀⠀⠀⢹⡆⢿⣷⣄⣶⣴⣿⠇⢠⡿⠁⠀⠀⠀⣀⣠⠔⠊⠀⠄⠀⠈⠈⠑⡆
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠁⠈⠁⠀⠀⠐⠀⠉⠓⠤⢄⡀⠀⢹⣿⣩⡿⣿⣻⠻⣿⠛⠁⣀⡤⠖⠋⢁⠀⡄⠀⠐⠀⠂⠁⠈⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡐⠐⠀⢀⢀⡀⠀⠀⠀⠀⠉⢙⡚⣷⣿⣇⠍⠁⠈⣛⢐⣋⠁⠀⠀⠀⣁⡀⠁⡀⠠⠂⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢒⠊⠔⠐⠒⠒⠐⠂⢂⣠⢼⣿⣿⣿⡾⣞⣾⣧⢄⣀⡀⠁⠒⠒⠢⠤⠀⠁⣤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⢚⣿⠋⢀⣙⣿⣿⣿⣟⣿⣿⣗⡈⠹⣯⠓⠂⠤⠤⠤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠁⣸⡟⣿⣿⡿⣿⣿⣽⣿⣿⡟⢷⡉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⢰⣻⠀⣿⣿⣷⣟⣛⡛⣹⣹⡇⢸⣱⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣻⠀⢻⣿⣟⣷⣿⣓⣿⣿⠇⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠃⠀⠘⣿⣿⡷⣿⣷⣿⡟⠀⠀⠀⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⠿⣯⣿⡿⠟⠀⠀⠀⠀⠀⠀⠙⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⠶⢏⡠⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣼⣷⣶⣶⣾⣷⣿⠀⠀⡄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⠿⢿⣿⣿⣷⣤⣥⡀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣧⣘
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣌⣿⣀⡹⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⣯⣋⣽⣯⡁⣉⠛⣋⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣯⡔⢊⡽⠟⠁⠙⠫⢤⢋⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣵⠃⠀⡀⠀⣤⠄⢋⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣟⣿⡄⠛⠁⠀⠀⠀⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⢻⣯⣷⣄⠀⠐⠀⢀⣮⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠉⢮⣽⡏⠀⠀⠀⠀⠿⡃⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠏⠁⢀⠰⠷⠊⠀⠀⠀⠀⠐⣷⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢘⠀⠦⠞⠀⠀⠀⣀⠀⠀⠘⣅⢀⡀⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠂⠀⠙⠢⡘⠳⠄⠀⢹⣇⠀⠀⢿⡠⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣦⡀⠐⣤⠀⠀⢻⡷⣞⢻⣏⣿⣿⣿
⠀⠀⠀⠀⠀⠂⠀⢀⠀⠘⣿⣿⣿⣿⣿⣿⣶⡞⣧⠀⣘⣅⣿⣿⢿⣿⡿⣿⣧⡀
⠀⠀⠀⠀⠰⠀⠀⡊⠀⢃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⢷⣾⣿⠷⢾⣷
⠀⠀⠀⠀⠀⠀⣀⣤⠾⢷⣟⣨⣻⣿⠟⣿⣯⣿⣿⣿⠟⠁⠀⠀⠀⠹⣟⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠛⢰⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⠻⣿⣶⣄⡀⣀⣀⣀⡬⣥⣤⠄
⠀⠀⠀⡄⠀⠀⣰⣿⣿⢧⣾⣿⣿⢿⢿⣧⡀⠀⣤⠈⠙⠻⣛⣉⣡⣠⣴⣲⢿⡆⠀⠐
⠀⠀⠀⠀⠀⣠⣽⣿⣿⣾⣿⣻⣿⡀⣾⣿⣿⣷⣬⣞⠃⠀⠈⠙⢤⣈⠍⠁⡼⡁⢀⠀⡄
⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠛⠓⠄⠤⠾
⠀⠀⢀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⣤⣤⣀⣿⡤⣤
⠀⣼⠛⡻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣿⣿⣿⢿⣿⣿⣿⣷⡄⢀⠀⣿⡿⢛⣿⠀⠈⣤⠤⠤
⠀⣷⣾⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⡟⠸⣿⠀⢠⣤⣀⣤⣀
⠀⠉⣻⣿⡟⠉⣿⣦⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⣿⠀⠀⣿⣿⣿⣦⣀
⠀⢀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡈⢻⠀⣿⠆⠀⣿⣿⣿⣿⡯
⠀⢹⣿⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣗⠃⠈⣄⢸⡄⠰⣿⣿⣿⣿⣷
⠀⠀⢹⣿⡿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠀⣿⡈⠁⠀⣿⣿⣿⣿⣟
⠀⠐⠛⢿⡇⠘⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢗⣠⡄⣾⣿⣿⢟⣣⡆
⠀⠀⣶⣿⡇⠈⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⣸⣛⠁⠘⣿⣿⣤⣽⡅
⠀⠘⣿⢿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣼⣿⣿⣿⠆
⠀⠀⠀⠀⠃⠀⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠘⠁⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣮⣻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠚⢀⣹⠈⣻⢿⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣝⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠻⠿⠿⠿⠿⢿⣿⣿⣇⡼⠿⠿⠿⠿⠛⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⢀⢐⠊⠙⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⢀⠀⠂⠀⡀⢚⠋⡀⠰⣦⡇⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣯⡷⣰⡾⠁⢀⡈⠁⢲⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠻⢗⠏⡀⠰⣿⣆⠄⠉⣉⣿⣷⡶⠃⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⡇⠀⠀⠀⣹⠀⠈⠘⢁⡒⠿⠛⠉⠀⣲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡢⡝⢼⡲⡠⢞⡵⠖⢲⡟⣷⣀⠀⡴⠁⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣗⡠⢳⡸⠸⣵⡇⢿⠃⠈⠻⣙⠇⢹⠉⣠⢀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡓⢉⣠⠷⢹⣿⡜⢦⣤⣄⣈⡼⣴⠈⠛⠤⡤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠞⠘⠁⠨⣼⣧⡀⠂⢈⣉⠙⢈⡉⢀⠈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⢠⠀⠈⠠⠖⢍⠒⠿⠙⠉⠀⠀⠀⠀⠠⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠄⡀⢡⡴⠬⣾⠦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣞⢔⣫⣽⡓⣮⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣍⠨⣿⣟⡵⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠏⠸⣥⣴⣄⣀⣀⣀⠀⡄⡀⣺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢀⣼⣿⣥⣾⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠿⢿⣿⠿⠛⠁


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

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣴⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣄
⠀⠀⠀⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡟
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⢀⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⢠⡀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣧⣤⣄⡀
⠀⣸⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⡿⠁⠀⠀⠁⣰⠀⠀⢀⣤⡤⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢿⣇
⠀⣿⠁⢻⣿⣿⣿⣿⠀⠀⠀⠀⣰⡟⠁⠀⠀⡀⠐⠻⣇⣰⠟⠀⠀⠀⠀⣿⡟⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢰⡿⠉⣿⣿⣿⡏⠈⣿
⠀⢿⡄⠈⠟⣛⣿⣿⡆⠀⠀⠀⠘⣷⠀⠀⠁⠙⢰⡖⠻⠁⠀⠀⠀⠀⠀⣿⡁⢹⣿⣿⣧⣀⠀⠀⠀⠀⠀⢸⣷⠀⣹⣿⡟⢃⡀⣿⠇
⠀⠈⣿⠀⢼⣿⣿⣿⡧⠀⠀⠀⠀⠈⢧⣀⠀⠀⠸⡇⢠⠀⠀⠀⠀⠀⠀⢹⣷⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⣯⠀⠋⠉⠠⡐⡟
⠀⠀⣿⡆⠈⣽⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣁⡀⠁⠀⠀⠁
⠀⠀⠈⠋⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠨⣀⡀⠘⠒⢠⠀⢢
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠉
⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢰⡟⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⢀⡾⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⣿⠃⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠰⣿⠃⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣦⠀⠀⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣉⡉⠻⢧⣶⡇⠀⠀⡔⠀⠀⢾⣤⡒⠁⠀⡰⣿⡿⡏⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠐⠲⣦⣄⡀⢺⣿⣿⣿⣷⡈⣿⢿⣿⣆⣷⡀⠀⠈⠛⠳⢾⣻⡗⠛⡁⠠⢈⣉⡁
⠀⠀⠀⣤⣶⣶⡤⠈⠙⣃⣈⡑⣼⣿⠿⠿⢿⣷⡏⣽⡏⢷⣤⡀⢀⣀⣴⡏⢁⣉⡈⢱⣌⢻⣯⡀
⠀⠀⠐⠛⠛⠉⠐⠁⠾⣿⣿⣿⣿⣆⣤⣀⠀⠙⣿⣿⣷⡤⠈⠋⠙⠋⠉⣠⣾⣿⣷⢸⡟⡧⠉⢓⠂
⠀⠀⠀⠀⣠⣤⣴⣶⣤⠙⢻⣿⣿⣿⣿⣿⣆⣦⣿⣿⠉⣼⠇⢠⣶⣶⣤⡈⠻⠿⠿⠀⠳⠇
⠀⠀⠀⠀⠀⠘⠛⠛⠁⢠⣿⣿⣿⠛⠿⣿⡿⠿⠟⠁⣸⡟⠀⡌⠻⣿⣿⣿⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠆⠀⢹⣷⠖⠁⠀⠀⠀⠉

⠀⠀⠀⠀⣀⣠⣤⣴⢶⣤⡶⢦⡤⢦⣀⣤⡀⣀
⠀⠀⣤⣾⣯⣿⣶⣿⣶⡿⣿⣿⣿⣾⣿⣴⣿⣹⣶⢦⣀⣄
⠀⠀⠹⣿⣿⣯⣹⣿⣺⡿⣿⣼⣿⣿⣯⢻⣟⣿⣿⣾⣯⣽⣿⣇⣀
⠀⠀⠐⡇⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠛⡿⠛⠛⠙⡛⠙⠃⠒⠠⢄⡠⠀⢀⠀⢀⡀⢀⢀⠤⠄⠒⠐⠉⢐⠉⠒⠖⠠⣤
⠀⠀⠀⢠⢐⡀⠄⢩⡈⠉⠈⠉⡈⠋⡀⠁⠀⢀⣂⣀⡤⠌⢧⠝⡄⠸⠁⠶⠈⡡⠀⢠⡄⣴⡆⣀⠀⠶⠀⠿⠀⠛⠤⢯⠴⣷⠰⡆⡄
⠀⠀⠀⠀⠎⣿⠲⠀⠤⡄⠁⡀⠧⢤⠀⠐⠈⠋⠙⢡⠆⠐⢺⠀⡀⠀⠀⠀⢸⡔⣇⣤⠀⣼⡇⣬⣗⠲⣼⠁⠀⠀⠀⠀⠀⢹⠤⠆⣧
⠀⠀⠀⠀⠸⠸⠃⡰⡆⢱⠈⡁⠀⠻⡄⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠄⠀⠀⠸⠁⠀⡽⠀⡏⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠞⣠⢹⣆⢶⣡⡀⠀⢳⠀⠀⠀⠀⠀⡴⠀⠀⡇⠀⠀⠀⠀⡆⢀⢠⡇⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇⠀⢸
⠀⠀⠀⠀⠀⠈⠀⢡⡎⢻⡘⣧⣧⠀⠀⢧⠀⠀⠀⠀⠃⠀⠀⢹⠀⠀⠀⠠⠃⠀⠘⡇⠀⡧⢀⠀⠄⢸⠃⠀⠀⠀⠀⠀⡾⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⢣⠘⣏⢧⠀⠈⢣⠀⠀⠀⢰⡆⠀⠸⡀⠀⠀⢠⠀⠀⢸⠁⢸⣤⣼⠆⠀⢸⠃⠀⠀⠀⠀⡼⢃⢠⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⢃⠘⣌⡄⠐⠘⣦⠀⠀⠀⢁⠀⠀⡇⠀⠀⢸⠀⠀⣿⠀⢸⠀⢰⡇⠀⢸⠀⠀⠀⠀⡼⣁⣆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠸⡄⢹⡾⡥⠈⠈⢆⠀⠀⠛⠀⠀⢹⡄⠀⢸⠀⠀⢻⠀⢸⠀⠘⠃⠀⣾⠀⠀⢀⡾⣻⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⡄⢣⡹⣂⠐⠌⣆⠀⠨⡇⠀⠈⢠⠀⢨⠀⠀⢸⠀⢠⠀⠀⠀⠀⣿⢠⣀⣾⢷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠁⠌⠷⢻⡄⠘⡘⣧⠀⢠⠀⠃⠙⡀⢈⠀⠀⢸⠀⢸⠀⠀⠀⠀⡿⢬⣿⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠐⢃⢿⡀⠐⠘⣧⠘⠀⢠⠀⡇⢸⠀⠀⣸⠀⢿⠀⢀⠀⠀⡇⣸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⢣⠈⡌⢳⠀⢢⠙⡄⡇⠀⠀⢴⢸⠀⠀⣿⠀⢸⠒⠺⠀⠀⣇⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠸⠀⢷⠈⡁⠸⢠⠁⠂⠈⡾⠀⠀⡏⠀⢸⠀⢸⡄⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡀⢃⠼⡗⠀⠀⠻⡠⠀⠀⢷⠀⠀⠀⡄⠈⠀⠰⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⠘⡆⢳⡀⠀⠀⡇⠀⠀⠈⡇⠀⢠⡇⠀⡇⠀⣤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠃⡌⣷⠰⠀⢰⠀⠀⠄⠇⠀⠀⢸⠀⢋⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠃⠘⠀⢸⡆⠀⠈⣴⠀⠃⢺⠀⠁⣸⡀⢸⠂⠀⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠇⠀⠀⡄⠈⣿⠐⠇⠻⡄⢠⠸⠒⠀⠈⡧⠀⣟⠀⠠⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣾⠀⠀⢱⠀⠉⡟⠠⠸⠇⠃⡆⡧⠐⠀⢷⠀⢻⡄⠸⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣘⣧⣿⠀⠀⠈⠀⠐⢁⠆⡆⢼⠀⢡⢡⡀⠆⠸⡄⠘⢇⠈⡇⡼⣱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣧⣱⢊⢡⡇⠀⠀⢀⠀⠀⠸⢆⠁⠀⡆⠈⠸⡇⠁⠀⡇⠀⠸⠀⢸⢀⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢿⣥⣷⡅⠀⢰⡇⠀⠀⣾⠀⠀⠀⣏⠀⠀⢃⠀⠀⢿⠀⠀⢸⡀⠀⡇⠈⣾⡸⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢫⡾⠉⠩⡳⠁⠀⠀⢈⠇⠀⠀⣿⠀⠀⠀⢿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⠀⢹⠠⢿⡄⠾⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⡰⢋⣴⡿⠋⠎⡴⠃⠀⠀⠀⣾⠀⠀⢡⠅⠀⠀⠀⢸⡄⠀⠈⡆⠀⠀⣧⠀⠀⢹⠀⠸⡀⠘⣧⠀⠋⡿⣿⣿⣿⣷⡀
⠀⠀⠀⢠⠞⡡⠛⠋⠀⢀⠞⠁⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡅⠀⠀⡇⠀⠀⢺⠀⠀⠸⡇⠀⡇⠀⢿⠀⠀⢱⠈⠉⣿⣽⣿⡄
⠀⠀⢠⣏⢰⠏⠀⠀⡀⠎⠀⠀⠀⠀⠀⡘⠇⠀⠀⡌⠀⠀⠀⠀⠈⡇⠀⠀⢷⠀⠀⢸⡆⠀⠀⣧⠀⢳⠀⢸⡇⠀⢸⠀⠀⡟⠉⣹⣷
⠀⠀⣿⣵⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠈⡇⠀⠀⢼⠀⢸⠀⠘⣇⠀⠘⡇⠀⠀⡇⢸⣿
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠁⢰⠁⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⡄⠀⣿⠀⠀⡇⠀⠀⡇⡟⢹⠃
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⠂⠖⣿⠀⠀⣷⠀⢰⠇⢃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠷⠦⢔⠢⠀⠀⡧⢸⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⠀⠀⣿⠀⠀⣷⠈⠀⢋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠌⢃⠀⡁⡾⠀⡀⢀⡇⠀⠀⣸⠀⢸⠀⠀⣟⠠⠀⣏⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡃⠀⠉⣸⠧⢀⢀⡟⠀⣼⠀⢰⡏⡀⢌⠅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⢀⣀⠨⠅⠉⡇⢀⡪⠅⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣷⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣶⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢿⣿⣿⣄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣿⣿⡿⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠟⢠⣿⣿⣿⣧⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣼⣿⣿⣿⠄⠻⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣷⢠⣿⣿⣿⣿⡄⠹⣿⣲⢤⣠⣤⣤⣄⣤⣖⣿⠏⢠⣿⣿⣿⣿⡄⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣿⣿⣿⣿⡄⢀⣀⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣄⡀⢀⣿⣿⣿⣿⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⣿⣿⣿⠟⣻⣿⣿⣿⠿⣟⣿⣿⣻⠿⣿⣿⣿⣟⠻⢿⣿⣿⣿⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣽⣿⡟⣯⣶⡿⠛⠉⠀⠀⣤⣻⣿⣷⣏⣤⠀⠀⠉⠛⣿⣶⣽⢛⣿⣯⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⡯⢼⣿⠿⠃⠀⠉⡳⠄⢿⣿⣿⣿⣿⡿⠠⠞⠋⠀⠘⢿⣿⣧⢽⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢉⢠⣾⠟⣀⡀⠀⣾⣇⠀⢸⡟⣿⣿⢻⡇⠀⣸⣷⡄⢀⣈⠻⣿⡄⡉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⣩⣤⣴⣿⣡⣤⣶⢴⣿⡐⢾⢡⣿⣿⡌⣷⢆⣿⡦⣶⣤⣭⣿⣦⣤⣍⢻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠵⢻⣿⣿⣿⣏⡙⢿⣮⣾⣷⣫⠞⣿⣿⠳⣝⣾⣷⣴⡿⢋⣹⣿⣿⣿⡟⠮⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏⢀⢽⣿⡇⠀⠉⣿⣦⣀⣸⠟⠁⠀⠈⠁⠀⠈⠻⣇⣀⣴⣿⠉⠀⢸⣿⡿⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣴⢿⣿⣿⡇⠀⠈⠛⢻⣿⡔⠀⠀⠤⡤⠀⠀⢲⣿⡟⠛⠁⠀⢰⣿⣿⡿⣦⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡏⠰⢻⣿⣿⣦⡀⠀⣾⣿⡇⠀⠀⠨⠅⠀⠀⢰⣿⣿⠀⢀⣴⣿⣿⡟⠇⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣇⢀⢀⡟⣿⣿⣿⣾⣿⠟⢀⣴⣶⠿⣿⢶⣦⡀⠻⣿⣶⣿⣿⣿⢻⡄⡄⣹⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡀⡜⢿⣿⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⠃⠀⣽⣿⣿⡿⢫⢀⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡆⢠⣿⣿⣿⣁⠀⠀⢙⣻⣟⡉⠁⠀⣈⣿⣿⣿⣆⢠⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣹⣿⣿⣿⣯⣝⣛⣛⣻⣽⣿⣿⣿⣏⣿⣿⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⢿⣿⣟⢿⣿⣿⣿⣿⣿⣿⡿⢻⣯⡿⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣽⠋⠙⣫⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢿⡿⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⢽⠙⠫⡍⢙⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠀⢺⢐⡆⡈⣇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⠶⠛⠀⢀⣂⣩⣜⠈⣰⡇⡈⠡⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢧⣀⢀⣀⣀⣠⣤⣿⠟⣿⣰⣿⣇⣸⠖⡴⢡⣾⠳⡄
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⣀⣈⣂⡀⠉⠁⠈⠉⠉⠈⠛⠀⠉⣿⣿⣿⣾⡿⣡⣾⣯⠂⡁
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠙⠙⠙⠻⠿⡿⣷⣶⣶⠤⣤⢄⣠⣤⢰⣿⣿⣿⡟⣼⢿⣿⠄⠀⢸
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣠⣤⣶⣿⣿⣿⣽⣿⣿⣿⠟⠁⠈⣿⣧⡀⠈⠓⢄⣀
⠀⠀⠀⢠⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⠏⠁⠀⠀⠀⠘⠿⣧⣇⡄⣀⣈⠑⡦⠄
⠀⠀⣠⣀⣀⡁⠀⠀⠀⠙⠻⠿⠛⠛⣻⣿⣿⣿⣿⡏⢿⠫⣿⣿⣿⣿⣧⣀⠀⢂⡦⣄⡀⠀⠀⠉⠛⠿⠇⠎⣑⣷⠶⣄⡀
⠀⢰⣿⡟⡁⠀⠀⠤⢤⡀⠀⠀⣠⠞⡿⠉⠁⠔⠋⣠⣶⠾⡛⠿⣿⣿⣽⣇⣀⡀⠀⡈⠻⠶⡄⠀⠀⠀⠀⢸⣏⡥⢲⠯⠴⢻⡏⠀⠆⣠
⢠⣿⣿⠿⠷⠂⢀⡀⡀⠀⢠⣾⣡⠀⠀⢶⣶⣶⡿⢋⠀⣀⣶⡟⠉⠉⠉⠛⠛⢓⣦⠀⠀⠀⡀⠀⠀⠀⠀⢉⠁⣀⣀⡀⢠⣿⣿⣀⣄⣯⠇
⣾⣽⠏⠀⠀⠀⠘⢷⣿⣦⣌⣙⡙⣗⣿⡿⠛⠁⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠉⠈⢥⠻⣿⠿⠟⠋
⢻⠟⠀⠀⠀⠰⣄⡀⠈⠙⠻⢿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⠀⠀⠀⠀⠀⢀⡼⠷⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡟⠻⣿⣦⡀⠀⡀⢤⣤⣤⣠⣤⣀⣴⠦⡀⠀⠀⠀⠀⢀⣼⡇⠏⠀⠀⠀⠀⠀⠀⠀⠐⠶⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣤⣀⣀⠉⠀⣀⣺⡿⣿⠾⣿⠢⡝⣯⠰⣿⣷⡧⢘⣃⠀⠀⠀⢀⣾⣽⠃⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡁
⠀⠀⠈⠉⠀⠘⠃⠉⠛⠙⠒⠂⢹⣷⣿⠾⣥⣟⠛⠜⠏⠋⠀⠀⠀⠾⣏⣐⣷⠞⠁⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢧⣍⡛⠛⠒⠦⠄⠀⠀⢸⣿⣿⣿⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠖⠛⠁⠁⠀⠄⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣶⣦

⠀⠀⠀⠀⠀⠀⡠⣶⣶⣿⣍⠉⠐⠒⠂⠤⣄
⠀⠀⠀⠀⠀⡀⠀⠈⠙⢿⣿⣿⣶⣦⣐⢀⡤⣠⠀⡄⠀⠀⠀⠄
⠀⠀⠀⠀⡒⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠒⠒⠒⠒⠆⠀⡀
⠀⠀⠀⣔⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠈⠩⠐⣂⡲⠆
⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣠⣤⢾⣿⡿⠃
⠀⠀⠀⠸⡀⠀⠀⠀⠐⠐⠒⠂⠀⢽⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⠟
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⣄⠂⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠈⢦⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣼⣻⡅⡮⣀⢘⢏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣿⣛⡾⣿⡽⡒⡋⠀⡃⠀⠀⠀⠀⠀⡠⠞⠉⠈
⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⣀⣠⣄⣀⠉⠓⢾⢿⣿⣜⠧⢶⡠⠓⢠⠟⠀⠀⠄⣀⣀⣠⠄⠀⠂
⠀⠀⠀⠀⠀⠉⠉⠉⠻⠿⢿⣿⣿⣿⣿⠿⠛⠒⣾⣿⣿⣿⣿⣛⣁⣤⠟
⠀⠀⣰⡇⠀⠁⠂⢀⣤⣴⣾⣿⣿⡟⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⢸⣆
⠀⢰⣿⣃⣠⣴⣾⣿⣿⣿⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⢸⣿⣿⣿⣿⡿⠟⠋⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠱⡀⠈⠙⠲⠀⠀⠀⢀⣀⡀⠤⠤⠀⠚
⠀⠘⠟⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠂⣬⡶⠿⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡄⠀⠀⠀⠐⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣉⠛⠒⠶⣶⣶⡶⠾⠿⠛⠒⠶⠶⠤⢤⣤⣴⣤⣄⣀⣀⡤⠖⢛⣛⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣼⣿⣿⣿⣆⣀⣀⣬⠿⠖⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠙⠳⣄⡀⣿⣿⣿⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⣿⣿⣿⣿⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡇⠀⠀⠀⠈⠙⢿⣿⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡌⢙⡿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠲⢤⣤⣀⣠⣈⢳⡀⠀⣇⠀⢠⢞⡁⣀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠔⠊⠉⢿⣿⣿⣿⡌⠓⠚⠙⠇⠻⣾⣿⣿⠆⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⣾⠇⡆⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡿⠁⠀⠀⠀⠀⠀⠀⠙⠿⣄⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⢸⣿⠀⣇⣼⡇⠀⣰⠀⠀⠀⠀⠀⠀⠈⣡⠔⠀⠀⠀⠀⢀⠀⠀⢀⣀⣌⣻⣾⡃
⠀⠀⠀⠀⠀⠀⠀⣿⡶⠀⠀⢸⣿⠀⢿⣿⡁⠀⡇⢰⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⢱⣴⣿⣿⣷⣶⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣦⢸⣿⠀⡈⢿⡇⠀⢿⣿⠀⠀⠀⢸⣿⠃⠀⢀⣤⣄⠀⠀⠀⠰⡸⣿⣿⣿⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡌⣿⡇⠘⣄⠙⢆⠈⢿⡄⠀⠀⠸⣿⠀⠀⣴⣿⣿⣷⣄⡀⠀⣉⣛⣿⣿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠘⣿⣼⣿⡀⠈⢳⣤⡀⠀⠙⠀⠀⠀⠹⣆⠀⢻⣿⣿⣿⣿⣟⢩⣯⣭⣭⣍⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣦⠹⣿⣿⣷⡀⠀⠙⠻⣶⣦⡀⠀⠀⡀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠸⣷⣜⣿⣿⣷⡄⠀⠢⣌⢻⣿⣿⣶⣿⣶⣄⠀⡘⣿⣿⣘⣿⣿⣋⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⢿⣿⣿⣿⣿⣦⡘⢾⣷⣿⣿⣿⣿⣿⣿⣦⣷⡘⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⢿⣿⣯⠻⢿⣦⣙⣿⣿⢻⡏⠹⣏⠙⠿⠻⣦⣍⣉⣩⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠻⣷⣄⠉⠻⢿⣿⣆⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠀⠉⠓



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠴⡶⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠾⠁⠹⠂⢈⡁⣠⡤⡄⠀⠀⠀⠀⠀⡀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢂⣾⡆⠀⠀⠀⡂⣤⠀⠀⢠⠀⠶⠉⠌⠠⠊⠠⠀⠀⣶⢷⠆⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣤⣶⣚⣿⡏⣧⣰⡄⣰⣯⣆⢠⣦⠸⠀⠐⠂⠀⠀⠂⠀⣶⣚⣿⣿⠶⠲⢶⣾⠇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢆⢹⣿⣟⣛⣛⣿⣿⣿⣿⢸⣿⣿⡟⡦⣁⠵⠀⠀⢀⣦⣤⣴⣶⣼⡟⠉⠀⠀⠀⠈⢻⢇⠁
⠀⠀⠀⠀⠀⠀⠈⠲⣤⣾⡿⢿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣿⣶⣶⣶⣾⣷⣿⣿⠟⠉⠀⢀⠤⠴⠄
⠀⠀⠀⠀⠀⠀⢈⣿⣿⠟⠀⠀⠰⠄⠀⠈⠉⣷⣍⣛⣛⡛⣿⣿⡛⢻⣿⣿⡿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⢀⠐⡦
⠀⠀⠀⠀⠀⢀⡞⠁⠁⠀⡤⠊⢲⠀⢀⠀⠄⠈⣿⣿⣿⢿⡿⣿⣿⣗⡿⠛⣿⣏⣀⠀⠀⠀⠀⡌⠀⠀⠀⠻⠱⠀⠂⠈
⠀⠀⠀⠀⠀⣸⡇⠀⠀⢨⡟⠟⢿⠃⠘⠀⠳⠿⣿⣿⣿⡿⣻⣙⢯⣼⣿⣦⠙⠛⢿⣷⡀⠀⠀⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠋⠀⠰⠀⢸⠙⡶⠈⠀⠈⠀⣠⠢⠘⣟⣛⣛⣄⣛⡿⣿⡗⠀⠀⠀⣀⢹⣷⠀⠀⠀⠈⠃⠀⠀⠄
⠀⠀⠀⠀⠀⢴⡆⠀⠀⠈⠛⠁⠀⠀⢀⠄⠀⢀⣴⣯⡴⣀⠛⣤⣄⣯⣷⣄⠀⠀⠘⢿⣿⠃⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⢳⣆⠀⠈⠖⠀⠤⠀⠀⠀⢰⣿⣿⢿⣿⣿⣧⡆⠀⠉⢻⡿⣿⣿⣿⣾⣿⣆⠀⠀⠀⢀⣀⡀⢵⣀
⠀⠀⠀⠀⠀⠀⠀⠹⢦⠀⠐⠂⢀⣤⣶⡶⠎⣂⣧⣤⣤⣾⣷⣶⣶⣦⣄⣸⠿⣿⡟⢻⣿⣿⣿⣿⠛⠉⢻⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⢿⣿⣿⣶⣼⣯⣅⣉⣩⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⢳⣿⣿⣿⣿⣿⡿⠿⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠁⠙⠻⠿⣿⣿⣿⡿⠿⠿⠿⠿⠏⣭⡹⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡧⣕⡐⠶⢦⡄⠀⠀⢤⡶⠀⠀⠀⠀⣿⣧⡬⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠘⠫⠟⠋⠋⠀⠀⠰⠹⠷⠶⠀⠀⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠛⠗⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡎⠀⠀⠀⠀⠀⡀⠀⠀⠁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⣿⣿⣿⣿⣿⣿⡷⠌⠉⠓⠮⣝⠳⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠿⠿⠿⠶⣦⣤⣀⠀⠀⠀⠀⠁⠈⠙⠎⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⣿⣿⣿⣿⣿⡿⠷⠶⠶⢤⡄⠈⠙⠳⣦⣄⠀⠀⠀⢄⠀⠐⢦⡙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣺⣿⣿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣌⣙⢷⣦⣀⠀⣷⡀⠈⣷⣌⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣀⣤⣤⡀⠙⢿⣿⣿⣿⣿⣿⣟⣛⣛⠛⠛⠻⠿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠊⠁⠼⠻⣿⡆⠈⢿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠓⠢⠄⡉⠻⢿⣿⣾⣿⠟⠻⠿⠿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⣾⡿⠋⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣄⠀⢸⣿⣧⣰⡿⢟⣿⣿⣿⡥⠀⢠⣾⣿⣿⣿⣶⣄⡀⠀⠀⠀⢠⠀⠀⣀⣴⣶⡆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⢻⣷⡈⠻⠇⢡⣾⣿⣿⣿⣿⡇⠀⠘⢿⢿⢿⡿⣻⣿⣿⣿⣶⣾⣿⣷⣟⡿⣿⡟⢀⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⠉⠃⠀⢰⣿⣫⣵⡿⢿⣿⣇⠀⠀⠈⠀⠀⢉⣁⣽⣿⣿⠯⠀⣻⣏⠙⠉⠁⠀⢸⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⡿⢉⣠⡿⢿⣿⣦⡀⠀⢀⠀⢀⣲⡿⠋⠀⠀⠈⠁⠀⠙⣖⡀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣱⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣟⣴⡿⠁⠀⠀⢻⣿⣿⣆⠈⠙⣿⣿⡄⠀⠀⠶⣦⡀⣠⠆⢸⣧⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⢸⢳⣿⣿⢋⣿⣿⣿⠇⣰⡏⠀⠈⢁⣿⣿⡆⣠⠟⠉⠙⠳⠤⠀⠀⠁⠁⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣽⣿⣿⣿⡇⠛⢸⣿⠇⣼⣿⣿⣿⣴⣿⡇⠀⠀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣧⣦⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⣿⣿⣷⡀⠀⢿⠀⢻⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣇⠚⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠈⢿⡎⢿⣿⣿⣿⣿⣷⣿⣿⣿⡄⠀⠀⠩⣭⣭⣭⣍⣉⣩⣍⣭⣭⡭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣦⡀⠀⠀⠳⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠉⠉⠉⠉⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⡀⠀⠀⠀⠉⠋⠻⢿⣿⣿⣿⣿⣿⣿⣷⣰⡄⢰⡄⢀⣤⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⠀⠀⠙⢧⠙⠏⠉⠛⠿⣿⣿⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣹⠟⠫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⡠⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀⠀⠀⠀⠀⡏⠁⠁⠀⠀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠘⠁⠀⡠⠽⣶⡀⠀⠀⠐⡄⠀⠀⠀⠈⠀⠀⠈⡷⣄⣀⣀⡀⠀⣀⢀
⠀⠀⠀⠀⢀⣀⣼⡇⠀⢻⡆⠀⢀⠀⠂⢸⣧⣤⣴⣛⣓⣦⣄⡀⠀⢀⡈⠀⣼⣷⣆⠈⠉⠉⣉⢿⣆
⠀⠀⣠⣾⢿⣿⡿⣷⣴⣾⡇⠀⠀⣢⡶⣿⡿⠛⣻⣿⠿⢿⣿⣿⣦⣼⠀⢸⣿⣿⢸⠀⠀⢠⡁⢸⣿⡆
⠀⠀⠉⠩⠺⣿⣧⣿⣿⣿⣿⠀⣄⣿⣿⣿⣶⡿⠉⠀⡴⢶⡊⢻⣿⣿⣇⢸⣟⠃⣸⣤⣤⣾⣷⣾⣿⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⡟⠀⠀⣶⣾⣫⢷⠈⣿⠿⣿⣿⣿⣧⡿⠟⣛⠉⢽⣿⡿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⢟⡗⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⠁
⠀⠀⠀⢀⠀⠻⢿⣿⣿⣿⠿⠿⠿⠿⣿⣿⡇⠀⠀⡿⣿⡿
⠀⠀⠀⠀⠑⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠥⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠌⠢⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣶⠛⠻⢿⣷⣤⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠠⣠⣾⡟⠁⠀⠀⢘⣷⣿⢿⡄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⡃⠀⢀⣴⣿⣿⠅⣸⣿⣄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠻⣿⠛⠒⢃⠘⣿⣿⠀⣿⣿⣿⣀⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠁⡠⠀⠀⠉⠉⠨⡀⠀⠀⠐⣨⠃⢶⡙⠛⠫⣽⣆⠀⢠
⠀⠀⠘⠆⠈⢉⠛⣋⠲⠖⠛⢋⣽⠀⠀⢘⣯⡡⡉⣛⢻⣟⣲⠿⠉⠁⠀⢸⣷⡄⠀⢾⣿⣾⣿⣦⠤⠤⠤⠤⠤⠄
⠀⠀⠀⡇⠀⠄⠀⠱⠌⠈⡹⠎⠀⠀⠘⡿⢿⣥⣿⣿⣿⣿⡑⠂⠀⠀⠀⠜⠀⠹⠀⠈⠅⣿⣿⣿⣐⠤⠀⠀⠁⠄
⠀⠀⢸⢤⠤⠴⠤⣤⣴⠏⠀⠀⠀⠀⣠⣷⣿⣷⣿⣿⠏⠀⠉⠀⠀⣤⡀⠀⠀⢺⡧⠄⠘⢿⣿⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⡀⠟⠀⢉⡳⠋⠁⠀⠀⠀⢀⣠⣼⣧⠤⠀⠀⠀⠀⠀⢼⣿⣿⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢤⣧⡸⣬⡀⣿⣀⠀⣀⣠⣶⣿⣿⣿⣭⡑⠂⠀⠀⠀⠀⢀⣿⣿⣷⡒⠐
⠀⠀⢰⣷⣶⣶⣿⣶⣷⣶⣶⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣤⣵⢶⣶⣾⣿⣿⣩⠋⠒⠤⣄⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠷⠖⢿⠛⢿⣿⣿⣿⡟⣶⣤⡈⠅
⠀⠀⢘⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣿⡿⠛⡿⠉⠃⠔⠋⠀⠀⠘⠀⠰⢸⣿⣿⣿⣿⣿⣷⡆
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠁⣀⡴⠖⠫⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣷⢻⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⡧⠟⣿⣿⣿⣿⣿⣿⣹⢿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣵⣶⣿⣿⡿⠋⠀⡈⠁⠀⢠⣿⠃⠐⠀⢀⠀⠀⠢⠠⢀⡀⠀⡀⠀⡀⠀⠀⠀⣀⡏⢸⡟⣿⣿⣿⣧
⠀⠀⢨⣭⡾⣿⣿⣿⣾⠃⢴⠂⠁⠀⠀⣸⡙⠀⠀⠀⠈⠀⠀⠀⠀⠈⢱⡶⠟⠁⠄⠀⣠⣾⣿⣄⣴⣧⣿⣿⣷⡗
⠀⠀⢸⣿⣿⡷⢴⡟⠛⢿⣎⣓⣦⡾⣮⣽⣟⣤⠀⠀⣀⢀⣀⠀⠀⡴⠟⠏⣅⠀⢢⣿⣿⢷⣿⡿⣿⠉⢿⣿⠑
⠀⠀⢸⡟⣿⠀⠀⠛⠆⠀⠻⣿⣿⣿⣿⣿⣾⣿⣭⢟⣺⣭⣥⣥⣥⣿⣶⣶⣷⣴⣿⣿⣿⣻⣿⣿⣿⣿⡿⣿
⠀⠀⢸⡷⠋⠀⠀⣠⣇⣀⣤⣽⣣⡶⠖⠙⠛⠛⠛⠚⠛⠛⠉⠉⠉⠿⠟⠻⠟⠛⠛⠛⠛⠛⠻⠿⠟⢿⡇⣿⣷⡆
⠀⠀⠸⠁⣀⣩⣾⣿⣛⣁⢀⡘⣧⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠇
⠀⠀⢰⣾⡿⠿⠛⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣄⣀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⣅⠀⠀⠀⠀⠀⢀⣶⣿⣿⣶⡀⠀⠀⠀⠀⠀⣨⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⢏⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⣹⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣤⣼⣿⣿⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣿⣿⣿⣤⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⠸⣿⣿⠋⣡⠔⠀⠀⠀⠉⠙⠻⢯⣉⣉⡽⠟⠋⠉⠀⠀⠀⠠⣌⠙⣿⣿⡇⣰⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⣷⣿⡇⢸⣿⣴⣶⣶⣷⣦⣤⣄⣦⠙⠋⣰⣀⣤⣤⣶⣶⣶⣦⣾⣧⢸⣿⣷⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢿⣿⣿⣇⠸⣿⣿⣿⣏⣿⣿⣿⣿⡟⣠⣄⢻⣿⣿⣻⣿⣻⣿⣿⣿⡏⣸⣿⣿⣿⠣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢽⣿⣿⣇⢻⣿⡛⠂⠉⠉⣿⡟⡵⢻⡟⢧⡻⣿⡉⠉⠴⢛⣿⡿⣰⣿⢿⣯⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣟⣿⣿⣦⣙⠯⠴⠶⠞⣫⢊⡀⠈⠁⢀⡁⣌⡻⠶⠶⠿⢛⣵⣿⣿⡿⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⣾⡟⣿⣿⣿⣿⣿⣿⠏⠻⠈⠛⢶⡴⠛⠃⠞⢙⣷⣾⣿⣿⣿⣿⣿⣷⡕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢋⣼⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠜⢿⣿⣿⣿⣿⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢻⢿⣿⣿⣷⣄⠤⠀⠀⠤⠤⠤⠤⠤⠤⠤⠠⠶⣿⣿⣿⡿⣿⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⠘⡿⣿⣿⡟⠙⠋⠉⠉⠛⠒⠚⠛⠛⠛⢩⡿⢻⣿⣿⢧⠇⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣿⣆⢠⣠⡆⠀⣄⢆⢀⣀⣸⣿⣴⣿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⣿⣿⣴⣤⣿⣿⣼⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⣿⣿⡿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢿⡿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠲⡖⣒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⠎⣀⡀⠀⣾⣩⣽⠿⡦⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣋⣷⣿⣿⣿⣿⣯⣿⣦⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⣶⣶⣿⣿⣷⣯⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⡀⠀⠀⠀⠠⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⡟⣡⡆⠙⣿⣿⣿⣿⣿⣿⣷⣹⡆⠀⠀⠀⠀⠀⡙⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⠈⠉⠀⠈⢹⣿⣿⣿⡿⠋⣹⠇⠘⠄⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⠟⠛⠛⠛⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⣆⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢄⣀⣀⣬⣭⣴⡶⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡎⢠⡀⢂⢶⣖⡚⠉⠉⠉⠀⣠⢠⠁⠴⠐⠀⠀⠀⠃⢣
⠀⠀⠀⣀⣀⣤⣤⢤⡤⠴⠶⢒⣿⣿⣿⠧⢉⣄⡔⣣⡟⣽⣿⣻⣶⣶⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠈⠋⣋⣿⠏⠀⠀⠀⢠⣿⡿⣟⢫⣶⡛⠋⠀⠈⢠⢿⢧⢿⣿⠏⠀⣤⡂⠀⢰⠀⠀⠀⠀⢠⡇
⠀⠀⠀⢠⣿⡿⠀⠀⠀⢠⣿⣿⢏⠁⠀⢡⠂⠀⠀⠀⡾⢰⡿⣿⡞⢀⣾⣽⠇⠀⢸⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⡏⠀⠀⢠⠃⠀⠀⠀⢸⠁⢟⠇⡿⣠⠞⣿⣟⠀⠀⠘⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣀⣤⣾⣠⡂⣠⢦⡇⣼⠊⣿⡗⠃⡼⠿⠃⠀⠀⢾⠻⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿⣿⡩⢿⣿⣷⣿⣿⣼⠟⠘⢿⣷⡞⠀⠸⡄⠀⠀⠸⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠿⠿⣫⣻⣿⣿⣿⣿⡟⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠠⣷⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠉⠿⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⡧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⡴⠚⢉⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠶⠋⠘⠓⠛⢷⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣿⠠⣴⣶⠶⠒⢺⣿⣿⣿⣿⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢀⣽⣶⡸⠇⠀⠀⠀⠻⣿⣿⣿⠃⢼⣇⠀⠂⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⣛⣟⡸⣷⣾⣄⠀⣀⠀⠀⠀⠘⣧⠈⣥⣞⡈⣻⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠸⣿⣗⢿⣿⣿⣿⣿⣶⣶⣆⣰⡷⠀⢀⠋⢡⣭⠤⠤⠴
⠀⠀⠀⠀⠀⢀⣀⣤⣠⣀⡐⢾⣃⠀⢸⣿⣮⣍⠻⢿⡿⠿⠻⣽⣋⢁⣰⠞⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠉⢀⡤⢒⡨⠽⠆⢻⡧⠀⠐⠉⠻⠟⠾⣶⣶⠖⠈⢻⣿⣿⠀⠀⠀⠂⠈⠂
⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⢀⡄⠘⡁⢈⡏⠀⠀⠈⠻⠟⠋⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡠⠂⠀⠐⢼⡇⣀⠀⢀⡀⠠⠠⣤⣴⣿⣟⢃
⠀⠀⠀⠀⠀⠀⢄⣋⡠⠛⠀⠀⠀⠀⠘⣯⠀⠀⠀⠙⢶⣦⡀⠉⠛⠿⠿⠵⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠁⢀⡑⢵⡉⠒⠲⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠠⠀⠀⠠⠀⠐⠄⠀⠰⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⢀⠆⠰⠂⠄⠀⡴⠀⠲⣄⠀⠀⢀⣰⣶⡆⠀⠀⡿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⡀⠘⠲⢶⡐⢠⡄⢀⢚⠁⠹⣵⣾⡍⢯⠀⢤⡀⠳⠛⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠊⠀⠀⢠⠀⠂⠀⠀⠀⠀⠐⢄⡾⢧⠀⠀⠈⠛⢧⠀⠀⠀⠹⠇⠻⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠒⠆⠀⠈⠁⠀⡄⣃⢉⣤⣄⡀⢀⡤⠓⢁⠀⢠⡄⠀⠀⢀⠀⠘⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⣻⠰⣷⣴⣤⠀⠨⠰⠀⢸⡿⠉⠉⠙⡆⠀⠀⡀⠛⠎⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠛⠛⠃⠀⠀⠀⠀⡨⠁⠄⠂⠀⠁⡀⠀⠈⢻⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠑⠀⠁⠀⠀⠀⠀⢠⡀⠀⠐⠲⠀⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠻⠦⡄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣷⣶⣦⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣏⡉⠈⠛⢧⠙⡽⢉⠛⠁⠋⢹⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣿⣿⣿⣿⣻⣷⣤⣾⣧⣤⣿⣤⣾⣟⣿⣿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠐⢿⣶⣿⣿⣟⣁⣀⣠⣈⣙⠛⢿⡿⠛⣋⣡⣤⣀⣈⣻⣿⣿⣷⡿⠂⠐⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣧⣼⣿⡟⣩⣤⣶⣶⣲⠮⢭⡉⠙⠛⢉⡭⠥⣶⣲⢶⣤⣍⢻⣿⣧⣼⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡿⣿⡧⠙⢰⣿⣿⣿⣿⣿⣿⣿⡟⣠⣄⢻⣿⣿⣿⣿⣿⣿⣿⡆⠛⢼⣿⠿⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢿⣿⣿⡘⣿⣿⣿⣾⣽⣿⡿⣱⡗⠺⣎⢿⣿⣯⣷⣿⣿⣿⢇⣿⣿⡿⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⡏⣷⣜⠻⢿⣿⠿⢋⡴⠫⠿⠿⠝⢦⡙⠿⢿⡿⠟⣫⣾⢙⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣧⠸⢿⣿⣷⣾⢿⡏⠠⣄⡀⢀⣠⡅⢹⡿⣾⣾⣿⡿⠧⣼⣏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣾⣿⣿⠇⠈⠳⠄⠉⠻⠟⠉⠀⠞⠁⠘⣿⣿⣷⣤⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⣿⣿⣿⣦⣀⣀⣠⡤⠤⠤⢤⣄⣀⣀⣴⣿⣿⣿⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⣿⣟⠲⠖⠚⠛⠛⠛⠶⠖⣻⣿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣷⣷⣤⣀⣀⣤⣾⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠏⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⢾⠯⣠⣛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣪⡤⠥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⢫⣹⣳⣤⣤⢤⣤⣤⡶⠤⠤⠖⠚⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣀⣦⡶⠟⠛⠉⠁⣷⠣⣫⡿⡟⣠⠂⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢴⣿⠟⠋⠀⢀⡰⣶⣤⡘⣿⣿⣿⣿⠯⣠⣾⠋⠀⢀⠀⠀⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠋⠁⠀⠀⠀⠀⠙⠷⠎⣸⣿⣿⣿⠋⣠⣛⢛⠚⡓⠉⠀⢀⣠⣽⡁
⠀⢀⣤⣤⣤⣤⣤⡤⠤⢤⡜⡁⢀⡀⣀⣀⣀⣠⣤⣴⣶⡿⠟⠛⠉⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠹⠿⠃
⠀⠀⠉⠛⠳⣶⣿⣇⡈⠟⢛⠙⠛⠿⣋⣭⠉⠠⠀⠓⠁⠀⠀⠀⡀⠀⡠⠆
⠀⠀⠀⠀⣼⣷⡿⠟⠁⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠈⢀⠀⠀⠀⠀⠁⠴⠒
⠀⠀⠀⠸⠟⠋⠀⠀⠈⠀⠀⠉⠉⠀⠉⠉⠶⠤⢄⣀⣀⠀⠐⠀⣀⣀⠠⠐⡒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠣⣼⣿⡏⠀⣉⣽⣇⠀⠈⢧⠈⠙⢧⠙⡟⢿⣍⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠈⠻⡙⢛⠛⠿⡛⠛⠻⢿⡟⠛⠛⢿⡿⠻⠻⡟⠛⡿⢻⢿⠿⠿⠿⡿⠛⠿⣷⡾⢻⡿⣷⠶⠶⢿⠷⠆
⠀⠀⠀⠀⠀⠀⠈⠢⠀⢄⠙⠖⠠⣠⣱⡄⠀⠈⢣⠀⠀⡇⠀⡇⠚⠸⠀⠀⢔⠁⣀⣜⡡⣒⠥⠖⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡀⠉⠻⣦⡀⠰⡁⠀⣇⠀⡇⠀⡇⠀⣰⢟⣪⣾⠕⠊⠁⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠂⠀⠈⠉⠳⢮⡀⣧⡀⣿⣰⣷⣾⡿⠋⠏⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⢿⣿⠿⠋⠀⢀⠈⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣁⡀⢀⠠⣸⣧⣯⣷⠀⢀⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢣⣿⣼⣮⡻⣶⣦⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣻⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢟⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠁⣠⣤⣀⣉⡓⢤⡀⠀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⡀⣀⡤⠚⣉⣁⣤⣄⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢼⣿⣿⣿⣿⣿⣦⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣴⣿⣿⣿⣿⣿⡧⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⢾⣿⣿⣿⠟⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠻⣿⣿⣿⡷⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠸⣿⡿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⣿⣿⠆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⡿⠋⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢙⢿⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⡠⠞⠂⣉⣉⣉⣉⣙⠛⠶⣄⣀⣄⣠⠶⠛⣉⣉⣉⣉⣁⠐⠓⢄⠘⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣁⡎⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠘⢋⡙⠃⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⢳⣈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢘⡤⣧⢸⣿⣿⣿⡛⠛⢿⣿⣿⡿⢠⡟⢻⡄⢿⣿⣿⡿⠛⢛⣿⣿⣿⡇⣼⢤⡊⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡾⠀⠸⣎⢿⣿⣿⣿⣶⣾⣿⠟⣡⠋⠀⠀⠙⣌⠻⣿⣷⣶⣿⣿⣿⡿⣱⡇⠀⣷⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⠀⠈⣹⣦⣍⣛⣛⣛⣋⡥⠞⠁⠀⠀⠀⠀⠈⠳⣬⣙⣛⣛⣛⣩⣼⣏⠁⠀⠹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⣰⡿⣻⣿⣿⣿⣿⣿⠀⢀⣶⣿⣿⣿⣿⣶⡀⠀⣿⣿⣿⣿⣿⡿⢿⣄⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡾⢹⣷⠉⢰⠏⣟⣻⡗⠀⠘⣯⣿⣦⣴⣿⣽⠃⠀⢹⣟⣿⢹⡆⠉⣾⡏⢷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⣄⣿⣧⣿⣿⡇⠀⠀⠈⠛⣿⣿⠛⠁⠀⠀⢸⣿⣿⣼⣿⢠⣿⡷⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣼⣿⣿⣿⣿⣿⣿⣷⣀⢀⣠⣴⣬⣷⣤⣄⡀⣠⣾⣿⣿⣿⣿⣿⣿⣧⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠹⣿⣿⣿⣿⣿⣿⣿⡉⡀⠀⠁⢀⢉⣽⣿⣿⣿⣿⣿⣿⠟⣨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⣄⡙⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⢋⣠⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣌⠻⢿⣿⣿⣿⣿⡿⠛⣡⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⢷⣦⠹⣿⣿⠏⣴⡾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣷⣿⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢴⣶⣖⣾⠛⢲⣿⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⡏⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠀⢅⣽⣷⣿⣿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠯⣻⣻⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣛⡵⢸⣿⣿⠛⠁⠀⠀⠺⣿⣷⣶⣦⣀⠀⠀⠀⠈⠙⠃⢭⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢨⢫⣿⣿⠃⠀⠀⠀⠀⠘⠳⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡎⠘⢸⣿⣆⠀⠀⠀⢀⣀⣀⣘⣿⣿⣿⣿⣿⣆⠈⠐⠀⠁⠀⠀⠳⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣫⡔⢘⢿⣿⣷⣤⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⢀⣩⣤⣴⣿⣿⣯⣌⡛⠻⣿⣛⡛⠿⠿⢿⣿⣿⣿⣦⣴⣾⣿⣿⡿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣵⣾⣿⣿⣿⣿⠿⡿⣿⢿⢿⣿⡞⠙⣿⣷⣶⣤⡙⢿⣮⠛⢿⣿⣿⣟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡰⣣⡟⣵⡿⠋⠀⠈⠉⠸⠿⣿⡿⣽⣦⡍⢆⠙⣿⣿⣿⡿⣷⣿⣌⠛⠃⠛⠉
⠀⠀⠀⠀⢀⣠⣴⡖⡿⢛⣿⣷⣿⣿⣻⡾⣿⣙⣿⣆⣩⣭⢙⠛⠀⣴⣾⣿⣿⡿⠇⠀⠈⠉⠛⠻⠖⠀⣿⣿⢧⠁⠉⠿⣿⣷⣄
⠀⠀⢀⣴⡿⡹⣿⣿⣷⣿⣿⣿⣿⡿⢿⣿⣿⣽⣣⡲⣌⣏⠀⢹⣾⣟⢿⣛⣉⠀⡀⠀⠠⠀⠀⠀⠀⠀⣾⡟⠿⣿⣤⡔⠈⢿⣿⣧⡀
⠀⣰⣿⣿⣾⡿⢛⣿⣿⡿⣾⠟⠁⠀⠀⠀⠉⠻⢿⣾⣿⣿⣿⣯⣿⠿⠛⠉⠉⠉⠁⠈⠘⠀⠀⠀⣄⣸⡿⠀⠀⠈⢻⠟⠀⠐⢻⣿⣷⡀
⢰⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⠛⣿⠧⣾⢥⠀⠀⠀⠀⣀⣠⣶⣽⣿⠟⠁⠀⠀⠀⠀⢰⣾⡑⢸⣿⣿⣇
⢸⣿⣿⣻⣇⠀⢸⣿⣿⡽⣧⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣿⣷⠾⢠⣤⣶⣶⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢸⣟⢈⣾⣿⣿⣿
⢸⣿⠿⣧⣿⡀⠀⠻⣿⣷⣯⣽⣿⣶⣶⣤⣤⣄⣄⣈⣾⣿⣿⣿⣯⣾⣿⠜⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡋⣼⣿⣿⣿⡏
⠈⢿⡟⣿⣿⢿⣦⣄⡀⠉⠛⠛⠿⠿⠿⢿⣿⣿⣾⣿⣏⡼⠻⣯⣮⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢹⣿⣿⣿⡿
⠀⠈⠻⣯⣻⣾⣻⡛⣿⣿⣶⣶⣶⣶⣶⣶⣿⢿⣿⣯⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣶⣿⣿⣿⠟
⠀⠀⠀⠈⠻⢿⣿⣿⣯⡿⣿⣿⣝⣾⣾⣷⣿⣞⣿⣟⣫⠒⢚⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠳⠛⠛⠋⠉⠉⠅⠀⣼⣿⡞⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠷⠿⠿⠿⠿⢿⣶⣶⣦⣤⣤⣤⣶⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁