Postal Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠒⠂⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣶⣶⣶⣶⣶⣶⣶⣶⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢛⠛⠻⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠃⠀⠀⠀⠨⠀⠀⠀⠈⢢⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠁⠀⠑⡲⠞⠁⠀⠀⠀⠀⠀⠑⡅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠐⢀⠀⠀⠂⠀⠀⠀⠐⠁⠀⢀⠐⠀⠀⢃
⠀⢰⠀⠀⠀⠀⢀⣀⠀⣀⣼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠄⣀⢀⣀⠄⠀⠀⠀⢰
⠀⠐⠄⠀⠀⠀⠀⢀⡜⠀⠇⠣⡀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡠⠋⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠐⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡠⠂⠉⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢘⡀⠀⠀⠦⠦⠀⠀⠀⣀⣀⣀⣀⡀⡈⠉⠀⣀⢀⣀⣀⣀⣠⣤⣤⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠉⠛⠛⠛⠛⠛⠿⠿⠿⠜⠸⠿⠿⠿⠛⠛⠛⠛⠋⠉⠙
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠐⠠⠀⠐⠂⠀⠀⢨⣁⣐⣂⡠⠤⠤⠤⠤⠤⢤⣀⣒⣀⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠟⠂⠘⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠣⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘

⣴⣿⣿⣦⣤⣾⣿⣷⣤⣾⣿⣿⣦⣴⣿⣿⣧⣴⣿⣿⣷⣤⣾⣿⣿⣤⣼⣿⣿⣦⣴⣿⣿⣷⣤⣾⣿⣷⣤⣼⣿⣿⣦⣴⣿⣿⣷⣤⣾⣿⣷⣤⣾⣿⣿⣤⣴⣿⣿
⠻⣿⠃⠀⠈⠛⠋⠀⠀⠛⠋⠀⠀⠙⠋⠀⠀⠙⠋⠀⠀⠙⠋⠀⠀⠙⠋⠀⠀⠙⠋⠀⠀⠙⠋⠀⠀⠙⠛⠀⠀⠙⠛⠀⠀⠙⠛⠁⠀⠘⠛⠁⠀⠙⠛⠁⠀⠘⣿
⣴⣿⡀⠀⢠⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠿⣇⡀⣹⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣠⣿⡄⠀⢸⣿⣿⠿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡛⠻⢤⣴⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢿⣿⠃⠀⢸⣿⣿⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠂⠀⠘⠻⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⣠⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣤⡤⢤⣤⣼⣿⡀⢈⡿⠁⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⡿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠃⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠘⢿
⢈⣿⡀⠀⢸⣿⣿⣿⣇⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠰⣾⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢀⣿
⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿
⢈⣷⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⠞⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⠀⢀⣾
⣿⡿⠃⠀⢸⣿⣿⣿⣿⡯⠁⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⡀⢀⡀⠈⠙⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢿
⠙⣿⡀⠀⠸⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⡇⠀⢸⣇⠀⠙⠶⣀⣨⣃⣰⠟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣾⣿⠃⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣶⣿⣿⣿⣷⠀⠀⢿⣭⣁⡶⠟⣩⡉⠛⣿⣯⣤⣤⠌⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠘⢿
⠙⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⠍⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠸⢹⣿⣷⠞⠃⡴⢲⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣾
⣾⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⢿⣿⡟⠁⣦⠀⠀⣼⣿⣿⣆⠀⠀⣾⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⠀⠘⢿
⠻⣿⡀⠀⢸⣗⡀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣰⡟⠀⠈⠁⣠⣾⠿⠿⠛⡿⠈⠙⢿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⠀⢀⣼
⣶⡿⠃⠀⠘⠻⢿⣿⣿⣿⣿⣿⡟⠋⢿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠟⠁⢰⡃⠀⠀⠀⠀⢚⣻⡍⠙⢿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠻⣧⡀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣷⣦⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⡾⠀⠀⠀⠀⡴⠋⡴⠁⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣽
⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡾⠡⠂⠀⢠⡞⢁⠾⡷⠀⣿⣾⣿⡄⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠘⢿
⠻⣯⡀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣧⠀⠀⠀⣰⢋⣴⠏⢠⡁⢠⠙⣿⣿⣿⣦⡀⠀⠀⠈⠉⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣼
⣠⣿⠋⠀⢸⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠇⢀⣼⣡⠞⠁⠀⠀⠁⢸⠇⡈⣿⣿⣿⡅⠀⠀⠀⠾⠁⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⠀⠘⢿
⢿⣯⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⡄⠀⠀⠇⠀⢸⠂⣿⣾⣿⣿⣇⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠀⢀⣼
⣠⡿⠃⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⣼⠏⠀⡰⠁⠀⠰⠀⠀⢸⡗⢡⣿⣿⣿⡏⢀⣶⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣜⠃⠀⣠⠇⠀⠀⡄⠀⢰⠁⣶⠀⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⢀⣼
⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡍⠀⠀⠀⠁⠀⢸⠀⣿⠀⢸⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠘⢿
⣿⣯⡀⠀⢰⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⡏⠀⡮⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⣿⣿⠋⠁⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣼
⢈⣿⠃⠀⢸⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠸⠀⠀⠀⠀⠀⠀⠀⠞⠀⣿⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⠀⠘⣿
⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⣿⠀⠀⢸⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢀⣼
⢈⣿⢷⠀⠠⣬⣭⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠀⡀⠀⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⣼⠀⠀⡼⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⣰⠟⠙⠀⠀⡁⠀⣿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⢀⣼
⠙⣿⠿⠀⢰⡇⠀⠀⠀⣶⡆⠀⠀⠀⣉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠚⣵⠆⠀⠀⢀⡇⠈⢁⣤⣶⣿⣿⡏⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠘⣿
⣾⣿⣶⣶⣾⣿⣶⣦⣄⡀⠀⢀⣴⡞⠉⢷⡈⢻⣷⡈⠙⠻⠿⣿⣿⡇⢠⡞⠀⠀⠀⠀⠈⠷⠾⣿⣿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣼
⠙⣿⣧⣀⣸⣿⣿⣿⣿⣿⣷⣿⣥⣙⡢⠞⠁⠀⠙⣿⣄⠀⠀⠀⠉⠀⠁⣀⣀⣀⣀⣀⣀⣀⣤⣿⣡⣿⣧⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⠀⠘⣿
⣾⣿⡙⠁⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⢛⡿⣿⣿⠛⠉⢉⣿⣏⣹⣿⣿⣿⣤⣄⣀⣀⣀⣀⣀⣀⣈⡉⠈⠉⠉⠀⢀⣽
⠻⣿⠇⠀⠑⠲⠾⠿⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣿⣿⣶⣶⠶⠶⠶⠾⢿⠿⢿⠿⣿⣿⠿⠟⠛⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠉⠉⠛⠛⠛⠓⠒⠺⣿
⣼⣿⣶⣄⠀⠤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣮⣿⣿⣥⣾⣿⣿⣿⣿⣷⣦⣤⣴⣶⣴⣿⣶⣿⣿⣿⣷⠾⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠶⠤⢤⣽
⠻⣿⡏⠁⣀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⡉⢹⠿⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠘⠁⢠⣿⡿⠙⡗⢰⠏⠀⠀⡆⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿
⣴⣿⣿⣿⣌⢻⣦⠀⠀⠀⠀⢴⡶⠀⣿⡇⢹⡀⣸⠁⣸⠛⣿⡟⠁⣀⣀⣀⣀⣀⣀⣻⣋⣀⣀⣉⣀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⣿⣿⣿⠀⠀⣼
⠿⣿⠛⢹⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠀⠁⠀⣿⡇⠀⠉⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠀⠘⣿
⣠⣿⣀⣼⣿⣿⣿⣿⣷⣦⣤⡀⠀⣠⣿⠃⠴⣤⣤⣀⣸⣂⣿⡇⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⢀⣽
⢿⣿⣿⡿⢿⣿⣿⣿⠿⣿⣿⣷⠾⢿⣿⣷⡶⠿⣿⣶⣿⠿⣿⣷⣶⠿⢿⣶⣶⡿⢿⣶⣶⡿⠿⣶⣶⣿⠿⣶⣶⣿⡿⢶⣶⣿⡿⠷⣶⣿⣿⠷⣶⣿⣿⡷⢶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣰⠳⢤⣀⣤⠞⠉⠙⢧⣀⣀⡴⠋⠉⠉⠳⣀⣀⣠⠞⠉⠉⠙⢄⣀⡴⠟⠋⠙⠳⠤⠤⠾⢻⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⢸⡆⠀⣶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⣧⠀⢸⡀
⠀⠀⠀⠀⣴⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣧
⠀⠀⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠏
⠀⠀⠀⠀⠈⢳⡄⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢯⡀
⠀⠀⠀⠀⠀⣼⠇⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻
⠀⠀⠀⠀⢸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⠞
⠀⠀⠀⠀⠸⢦⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠻⣄
⠀⠀⠀⠀⠀⠀⢹⡆⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠿⠃⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣴⠟⢁⣠⣤⣄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⣠⡶⠛⠃
⠀⠀⠀⠀⠀⣧⣠⡾⠁⠀⠈⢷⣄⣠⣴⠋⠀⠈⠳⣤⣤⣼⠋⠀⠀⠙⠦⣤⡼⠃⠀⠀⠻⠤⠤⠴⠏
⠀⠀⠀⠀⠀⠀⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣿⣿⡿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣉⠑⢒⡦⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠤⡨⠿⠟⠁⠀⢀⢄⠀⠠⢤⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⢋⠁⠀⢹⠀⠀⡏⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢓⡢⠂⠀⢰⠉⢃⠄⢉⠙⠍⢳⠀⠀⠀⠀⠀⢠⣿⣿⠊⢂⡀⣞⠉⠀⢀⠁⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣅⡞⠁⠀⠀⢠⢾⠚⠸⢇⠃⠘⠇⢸⣦⠀⠀⠀⠀⣾⣿⡏⢀⡤⠴⠋⠀⠀⡘⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠘⢿⠀⠀⢨⠥⣄⠀⠸⡇⠀⠀⠀⠀⠀⠈⠉⠛⢲⢪⣏⡀⡰⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⢸⠀⠀⣀⣠⡠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⣃⠜⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣶⣶⣶⣿⣿⡛⠋⣿⣏⠉⣹⡟⣰⣦⣤⣤⣀⣀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣿⣯⣼⣧⣿⠋⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⢹⣦⠐⢲⡜⠿⣿⣿⣿⣿⣿⣿⣟⣡⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡁⢈⡗⠒⢢⠘⠁⡰⠻⢿⣿⣿⣻⡿⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣖⠼⣄⣸⣦⣼⣩⡤⣾⢛⣩⢾⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠉⠓⡞⠒⠒⢒⣻⢟⠝⣡⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠜⠀⠀⠀⡥⡞⠁⣰⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠎⠀⠀⠀⠀⢠⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⣿⣿⣿⠛⢻⣿⣿⡿⠛⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⢸⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠈⠁⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠒⣛⠛⠁⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢾⣿⣿⣿⣶⣦⣤⣀⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⠈⠉⠫⣿⡟⠂⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⢀⣾⡟⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠹⡄⠀⠀⠀⠀⠀⢀⣴⡿⡋⣿⠋⢁⡺⠟⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⢆⠀⠀⠀⣰⣾⣿⣿⣷⡯⢊⣼⠿⣻⠋⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠈⢳⣤⡾⠿⡟⢸⣿⡏⣼⡟⠘⢠⠇⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢀⡿⣴⢛⣧⢺⡿⠃⣿⠁⢰⠏⠀⠀⡘⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣮⡾⠿⣿⠁⢠⡏⢀⡟⡀⠀⠲⣌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⣿⢀⣾⣿⣸⡷⠶⢦⠈⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣼⠿⠋⠁⠈⠖⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡤⠤⠤⢤⡴⠟⠋⠁⢀⣠⠴⠶⠿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠿⣞⣕⣤⣀⡴⣶⣿⣿⣟⣟⣒⣮⣭⣵⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣧⣠⣾⣿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠟⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣀⣸⣏⠠⠀⠆⠀⠘⠀⠐⠙⠟⠗⠘⢷⡀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣫⡴⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣏⢻⣿⡀⡇⣀⠀⣄⠀⢀⡀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣠⡾⣡⡶⠟⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢀⣠⣴⣿⣶⣯⣦⣤⣶⣶⣶⣶⣶⣶⣿⣶⣿⣷⣿⣶⣆⣀⡾⢋⡴⠋⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣴⣿⣿⣿⢹⣿⣿⡿⢣⣿⣿⡿⣋⣉⣉⣉⣉⣉⣉⣹⣿⣿⣏⣰⠏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⡟⣿⠈⣿⣿⢁⣿⣿⣿⣿⡛⠋⠉⠡⠦⠹⠿⢿⡍⢿⣿⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣷⣿⡇⠹⢃⣾⣿⣿⡏⣿⣿⣄⠲⣦⡀⣀⣐⣠⣷⠐⢹⣇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⣿⣿⣿⠏⠸⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⣿⣿⣿⡿⣿⡿⠀⣿⣿⣿⣿⣷⢹⣿⣿⣿⣶⣬⣭⣭⣴⣶⣾⣧⢿⡀⠀⠀⠀⠀⢨⣿⠛⢿⣿⣿⣿⠃⠀⠀⢿⣿⣿⣿⡿⢿⡟⠁
⣼⣿⣿⣿⠇⣿⢣⣆⢹⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢠⣿⠃⣠⠏⠀⠀⠀⠀⠀⠀⠈⠉⠈⢿⠀⢸⣿
⣿⣿⣿⣿⢰⣿⠟⢻⣦⡙⠻⠿⠿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠹⡆
⢿⣿⣿⡿⢀⣿⡄⠈⢙⡋⠉⠶⣶⣶⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⡼⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣿
⠘⠻⠿⠛⠛⣿⣀⣀⣶⣶⣆⢀⡼⠿⣿⣷⣿⣭⡻⢿⣿⣿⣿⣿⣿⣿⠃⢠⣾⠁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⡇
⠀⠀⠀⠀⠀⣹⡟⣿⠟⠻⠿⠻⣿⠿⠿⠿⠿⠿⢿⣾⡿⠿⢿⡿⣿⢃⣴⣟⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⡀
⠀⠀⠀⣶⣾⡟⣼⢿⡀⠀⠀⣠⣿⡷⣄⠀⠀⠀⣠⣿⣷⣦⣼⢷⡘⣿⠟⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⢛⣿⣿
⠀⠀⢸⣿⣿⣽⣿⣿⣷⣄⣾⠉⢻⣷⢸⡇⠀⣾⠋⢾⣿⡄⣿⠸⣷⣼⠇⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣗⣶⣿⣿
⠀⠀⠈⠻⣍⣛⣿⡼⠋⠈⢻⣽⣈⣥⡾⠃⠀⠙⢶⣶⣶⠾⠛⢷⣴⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠛
⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣿⣶⣶⣾⣶⣶
⣿⣿⣧⣥⣬⣧⣴⣤⣽⣬⣴⣯⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠻⣦⡀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⠤⣤⣤⠤⠴⠚⠉⠀⠀⠀⠈⣷⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣤⡀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠙⠳⣯⡀⢀⡤⣄⠀⠐⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣧⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠉⠻⢤⣠⡿⠖⠋⠉⠁⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⢀⣀⣀⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⡠⢒⡦⣀⠀⠀⠀⣏⣀⣀⣠⣴⡞⠋⣯⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢀⡴⠊⣵⠋⠀⢀⡜⠁⢸⡇⠈⢱⠀⠀⠀⡽⢛⡧⠤⠟⠚⠁
⠀⠀⣠⠖⠋⢀⡾⠁⠀⠀⠈⠦⣘⠛⠋⢠⠎⠀⢀⡞⠛⠉⠙⠲⣄
⣤⣞⣁⣀⣠⠟⠀⠀⠀⠀⠀⠀⠈⠙⠒⠃⠀⣠⣏⣀⣀⣀⣀⣀⣈⣳⣤
⡏⠉⠓⢶⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢀⣠⡴⠞⠋⢹
⡇⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⢀⡞⠁⢀⣠⡴⠞⠋⠁⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⠴⠒⠻⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢸
⡇⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸
⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾
⣻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⠟⠚⠟⠓⠿⠓⠺⠛⠺⠟⠲⠟⠓⠟⠓⠾⠛⠶⠛⠳⠟⠳⠞⠛⠞⠛⠶⠛⠷⠚⠻⠚⠻⠖⠻⠗⠛⠷⠚⠿⠚⠻⠓⠻
⣾⠆⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠰
⣼⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⡿⣿⣿⣿⠀⢰
⣹⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢟⡫⠱⠵⡮⢅⠞⣻⣿⠀⢠
⣻⡄⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣫⣾⣼⣛⣛⣿⣛⡛⠛⠛⢛⣿⣟⣿⣷⣯⣎⣹⣷⣊⣓⣷⡿⠼⠙⠀⢠
⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣛⠛⠛⠛⠿⠿⢿⣿⢿⣿⠿⠿⠿⠿⠟⢹⣧⣉⠹⡯⠀⢨
⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢣⣤⠀⠀⠀⠈⠉⠲⣶⣀⣀⢤⣙⠱⣄⡇⠘⠀⠇⠀⢈
⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢹⣷⠖⠐⣿⣶⣀⣿⢷⡈⡻⠿⢀⣮⠱⠶⡓⢀⠀⢈
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⣶⡆⠀⠀⡴⠒⢻⡀⢶⣤⢹⡏⠁⣇⠀⠈⢿⣷⡇⢡⠀⠀⣿⣄⠁⠙⠀⠀⠘
⣼⠃⠀⠀⠀⠀⠀⠀⠀⢠⣶⠊⠁⠀⣼⡏⠉⡠⠔⠚⠀⠀⠘⡇⠈⠉⢺⣧⠀⡇⠀⠀⠘⡟⣇⣇⠀⠀⠘⡿⣧⣘⣀⣀⠘
⣹⠇⠀⠀⣠⠄⠀⠀⣘⣿⠇⠀⠈⣶⡿⠀⠀⢹⠀⠀⠀⠀⠀⠿⡏⠀⠀⢻⢷⣇⠀⠀⢰⡇⣟⣿⣀⡤⠶⣻⢻⠿⠍⠉⣿
⣻⠇⠀⠀⠀⠀⠀⣀⣰⡏⠀⠀⠀⢧⡁⠀⠀⢸⠁⠀⠀⠀⠀⣈⡇⠀⠀⢸⢸⣿⣀⠀⠈⢷⣿⢿⣯⣰⠁⣿⠠⣏⠗⠀⠿
⢻⠆⠀⠀⠀⡴⠋⠁⠀⣷⠀⢀⣰⣈⣧⣀⣠⣾⣻⣀⣀⣀⣀⡀⢻⡀⠀⠘⣾⣿⣻⠷⣤⠟⡓⢭⣿⣙⣭⡽⠶⢛⡛⠛⠻
⢿⡟⠛⠛⠛⣛⣛⣛⣛⣋⣹⣿⣿⣍⣭⣽⣿⣭⣭⣭⣭⣭⣉⣉⣙⣹⡖⢄⠈⣀⣽⢳⣟⡄⣦⠴⣿⠟⣿⠘⠃⠈⠁⠀⣾
⣾⣏⣉⣉⣉⣉⣉⣩⣽⣯⣿⣯⣿⣴⡧⠷⠾⠷⠶⠤⠤⠤⠤⠤⠤⢽⠋⠫⠆⠓⠀⡟⡬⢀⡀⢰⠏⠤⠀⠀⠀⣠⢄⣾⣿
⣾⣍⣉⣍⣁⣤⣀⣤⣀⣠⣀⣠⣄⣀⣄⣀⣤⣀⣤⣀⣠⣄⣠⣄⣠⣄⣀⣤⣀⣤⣸⣧⣤⣤⣄⣿⣴⣆⣄⣠⣦⣵⣿⣉⣿

⠀⠀⠀⠀⠀⠀⠀⠺⣉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠿⢕⡢⢤⣀⡈⠙⢦⡀
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠹⢕⡲⢭⡟⠉
⠀⠀⠀⠀⠀⣏⠀⠀⠀⠘⠃⠀⠸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⣼⣄⠀⠀⢦⣀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠉⢫⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢠⣾⡞⣦⠉⢷⠶⢶⡾⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠈⣿
⠀⠀⠀⣿⣿⣀⣿⡀⠀⣠⢞⠝⠁⠈⠙⠓⠒⢲⡖⠋⠄⠀⠀⢈⠉⠈⣿⠀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⡞⠁⠀⠀⢈⣷⣯⢚⠟⠙⡁⣀⠀⠐⣆⠀⠀⠉⠉⠁⠀⠁⠀⢠⠋⠉⠙⠓⠒⣿⠒⢻⣿⠛⢻⠉⠉⠉⠉⠉⠉
⠀⡞⢧⣀⢀⣐⠞⠀⡟⢻⠠⠄⢀⡀⠀⢈⡼⠂⠀⠀⠀⠀⠀⠀⠀⡏⣀⠀⠀⢀⣼⠏⠀⢸⣿⠀⡼
⠀⠳⡘⠻⠋⠁⠀⣰⠃⡰⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠾⠛⠁⠀⠀⢸⣿⠀⡇
⠀⠀⠈⠙⣶⠚⠉⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣇⣠
⠀⠀⢀⡾⠓⢢⡄⣰⡷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠿⢻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠢⠤⢤⣤⡤⠴⠚⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠖⠋⠀⠀⠀⠀⢀⣴⠞⣫⣷⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⣀⡤⡤⣶⣶⡿⢿⣛⡿⠯⠤⠤⠤⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠠⠴⢭⣤⣴⣖⠿⠯⣗⣻⣭⣵⣒⣋⣉⣀⣠⡤⢴⢶⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢲⢒⣾⣿⣷⡇⠀⠀⠀⠀⠀⠀⢿⠀⣼⠂⠉⠉⠉⠙⠒⠲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⡏⠱⡟⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⣀⣀⣠⡤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠇⢧⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠸⡆
⠀⠀⠀⠀⠀⠀⡖⠒⢐⠀⢳⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⣾⡿⢿⠻⠿⣿⠉⢩⠃⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠴⠶⠶⠶⢶⣶
⠀⢎⠹⣆⠘⡄⢰⠯⢥⣾⡶⠛⠻⢗⢶⠦⠤⠤⣤⣄⣀⣀⣀⣠⣤⡤⣴⠋⠀⠀⠀⠀⡼⣛⣛⡶⠀⠀⠀⠀⠀⢀⡀⠓⠊
⠂⠰⡯⠭⢚⣷⣚⣒⣺⠟⠀⡔⠀⢈⢀⠆⠀⠀⠐⡌⠑⢤⣀⠈⢹⡢⡟⣷⢄⣠⣴⢻⡀⡏⠁⠀⠈⠭⣭⣠⣇⣐⡀
⠀⠀⡇⠀⠀⠀⠀⢠⡏⠀⢸⠁⢀⣜⡞⠀⠀⠀⠀⠘⠤⢦⠉⠲⢼⣿⡽⢹⡀⠀⠸⣣⣽⣇⣀⣠⣤⡤⠤⠤⠤⠴⠶⠶⠶
⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⢸⣴⠿⠋⡾⣿⡖⠢⢤⡀⠀⠀⠉⠉⠐⠻⡀⠀⡏⠉⠉⠉⠀⠒⠤⠀⠀⠉⠉
⠀⠀⠸⡆⠀⠀⠀⠈⢧⠀⢀⢝⡶⠾⠥⣬⠀⢀⣾⣷⠀⠀⠀⠀⠀⠰⠁⠀⢿
⠀⠀⠀⠘⠦⣄⡀⠀⠈⣿⢧⢞⡀⢨⣭⡷⢄⣾⡟⠁⠀⠀⠀⠀⠀⠀⡶⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⣏⠀⣰⠋⠓⠚⢃⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣸⠇
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠉⠙⠧⣤⣀⢀⣾⠯⠤⠤⠤⠤⠤⠶⠒⠒⣻⠏⠉
⠀⠀⠀⠀⠀⠀⠈⠀⣰⣄⠀⠀⠈⠓⢯⡍⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠒⠶⠒⠋⠀⣰⠛⣦⡀⠺⣉⠁⠀⠀⢀⡴⠋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⠒⠦⠤⠴⠚⠁⠀⢌⠛⢦⡈⠙⠒⠴⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠳⢦⣄⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⠒⠶⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢰⡉⣉⡇⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⣀⣭⠿⠛⣉⣙⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣠⡤⣖⣯⣵⣶⠻⠿⠍⢻⣔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠋⠉⠈⣁⣦⠀⠼⢓⡆⢻⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠒⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠫⡥⠀⠀⠀⣀⠄⠀⠐⢋⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⣄⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢮⣷⡄⠀⠀⠛⢦⣤⢴⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡎⢠⠶⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢻⠲⡄⠀⣷⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⣿⡀⠀⠀⠀⢀⣀⡤⣾⠀⢹⣯⡽
⠀⠀⠀⠀⠀⢀⣀⣰⡞⠛⠙⣷⠤⠞⠛⠹⡆⢸⡄⠀⠀⠀⣠⢒⣁⣤⡞⢻⡧⢻⡖⠒⠚⠉⠀⣀⡿⠚⠉
⠀⠀⠀⠀⢀⣸⢧⡀⣿⣀⡄⢿⠒⠛⢻⣆⣷⡼⣟⣒⠲⠾⢯⣉⡼⠻⣏⠹⡟⣆⢷⠴⠖⠛⠉⠉
⠀⠀⠀⠀⠘⢦⠀⢳⣿⠀⣷⣼⠶⣶⡿⠋⠁⠀⠀⠉⠙⠛⢳⡛⠳⣄⢹⣆⠹⣿⢸⡄
⠀⠀⠀⠀⠀⠈⢳⡤⠟⠋⠁⣠⠞⢁⠀⢀⣠⣴⣶⠶⠀⠀⠀⠈⠙⠻⡆⢿⣆⢻⠘⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡅⠀⠙⣿⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡸⣿⣆⣧⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢷⠖⠛⢳⣤⡄⢻⠀⠀⠀⠀⠀⠀⠀⠘⣟⣋⣥⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡖⢻⢠⢦⢹⡀⡇⢸⠀⠀⢀⣀⣠⠄⠀⠀⡏⠁⢀⣾⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧⡼⠾⠿⠷⢛⣣⣾⠀⠀⠉⠁⠀⠀⠀⣸⠇⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢀⣠⣴⠾⠛⠉⣿⠀⠀⠀⠀⠀⠀⣰⡏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠋⠉⠀⠀⠀⢰⡇⠀⠀⠀⣀⣤⠞⠉⠉⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣸⣧⢶⣾⣋⣉⣤⡤⠶⣶⠄⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡷⢦⣤⡼⠋⢉⡽⠃⠀⢠⡟⠀⣾⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢽⣅⣀⡀⣀⣤⠞⠁⠀⠀⠾⣿⣿⣭⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠀⠀⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⡀⠀⠀⠀⠀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣦⣀⣴⣾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠀⠀⠙⠛⠋⠁⠈⠛⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣶⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣤⣾⠟⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠉⣻⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠃⢠⡏⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠿⠿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠇⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣿⣶⣮⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠙⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠛⠋⠀⠀⠈⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⠀⢡⠀⠉⠙⠉⠭⣀⡚⠛⣀⠀⠀⠒⠒⠃⠀⠠⠀⠀⠀⠈⠉⠁⠐⠒⠀⠀⢀⣠⢄⢄
⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⠀⡐⠀⠈⠒⠤⠄⠀⣀⣀⣈⣁⣛⠲⠄⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡏⠉⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠀⠘⠒⠒⠀⠀⠴⠀⠤⠤⠀⡀⠓⠒⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠤⠤⠤⠼⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠠⠀⠀⠀⠀⡠⢲⡯⡉⢀⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡞⡉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣌⠀⠀⠀⠈⠱⠤⠀⠂⠀⠀⠀⠉⠀⠀⢠⣤⣶⠪⢶⠈⣆⠴⠋⠀⠸⢤⡀⠘
⠀⠀⠀⠀⠀⠀⡈⠁⢈⠀⣇⠈⠀⠒⠋⠍⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣰⣦⣝⡄⢹⠀⠀⠀⡮⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠫⠶⠃⠀⢸⠀⠀⢀⡏⠉⢀⠆
⠀⠀⠀⠀⠀⡀⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⣠⣄⡤⠤⠰⠖⠒⠈⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠂⠀⣠⡐⣀⡀⠤⠄⢀⣤⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢙⣒⠀⠩⠍⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⡀⠀⠂⠈⠨⠥⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠤⠀⠐⠐⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠒⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⢈⡡⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠈⣀⠤⠔⠊⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⣉⠠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢠⠞⠻⠦⠶⠤⠤⣤⣀⠀⠀⢀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢽⣾⣿⣿⣿⣿⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⢿⣿⢻⠉⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢠⡤⡉⠀⣋⡋⢾
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⣷⠀⠀⠀⠀⠀⠀⣀⡠⣤⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠘⠃⠁⠺⠬⠋⣼⡄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⣦⠴⠖⠋⠁⠀⠈⠳⣖⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠘⠗⠀⠀⢠⣿⡆
⠀⢀⣼⣿⣿⡿⣿⠿⠛⠩⡿⣻⢛⡾⣏⡈⠣⡀⠀⠀⠀⢀⡠⠜⠳⣄⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⣠⣴⣿⣿⡇
⠀⣼⣿⣿⣷⣿⣟⡀⠀⠀⠀⠳⠼⣇⣀⣈⠆⢈⠦⡖⠈⠀⠀⠀⠀⢈⣳⡄⠀⠀⠀⠀⢸⣞⢻⡰⣯⠉⠉⠛⡇
⣠⣽⡗⣯⡟⠘⠋⠁⠀⠀⠀⠀⠀⠬⠉⡟⠈⠀⠀⠈⢢⣀⡠⠶⠮⣉⣼⠀⠀⠀⠀⠀⢸⣷⣼⣵⣿⣷⠀⠀⡇
⠀⠀⠀⢿⡟⠒⠃⠀⠀⠀⠀⣰⠒⠒⠒⠓⠖⠒⠒⠒⠛⣦⣤⣶⣞⡉⢉⡟⠐⡹⣆⠀⢸⣿⣿⡿⢻⣿⡇⠀⡇
⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⠠⡿⠤⠤⠤⢠⣄⣀⣀⣀⣸⡏⠙⢹⡇⠈⢹⠃⡧⣸⣹⠂⢸⣿⣿⣧⠀⣿⣧⠀⡇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⣠⣀⣀⡀⠀⣽⠤⣤⢸⣏⡙⠛⠀⠳⠤⠇⠀⢸⣿⣿⣿⡆⢻⣿⠀⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣭⠛⠛⠉⠉⠁⠀⣀⠴⣿⢋⣉⠀⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⢸⣿⠀⡇
⠀⠀⣾⣿⣿⣿⣿⡿⢿⣿⣿⣶⣶⣶⣿⣭⣤⣴⡞⠵⠓⣚⡷⣠⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠠⢾⣴⠶⡇
⠀⠀⣿⣿⣿⣿⠋⠙⣾⡞⠛⠛⠛⠛⠿⠿⠿⠽⢧⣴⣯⡽⠿⣿⣆⡷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠉⠀⡇
⠀⢐⣿⣿⣿⣧⣀⠜⠀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⡇⠺⢳⣾⠁⠀⠀⠀⠀⠀⠀⠀⣯⠃⠀⠀⠀⠀⠀⡇
⠀⠈⣿⣿⣿⣿⣿⣷⣶⣿⡄⠀⠀⠀⠀⠀⢀⣤⠞⠉⠙⠓⠒⠾⠉⡄⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣾⣶⣶⣶⣦⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⠯⣿⣿⣿⣿⣿⣿⡥⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣤⡤⠦⠠⢶⠤⠶⠶⠲⠒⠒⠒⠶⠶⠦⠶⠶⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣀⢠⣄⢠⣤⣤⢠⣤⣤⣄⡀
⠀⠀⠀⠀⣠⠞⠃⠀⡀⠄⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⠄⠄⠤⠤⣀⠈⠛⢦⡀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⢻⡀
⠀⠀⠀⡾⠁⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢻⡄⠀⠀⠀⡇⠀⠈⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⠃⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⣴⠿⢦⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣿⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⡀⠀⢀⣀⣀⠀⠀⡅⠂⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣤⣀⡀⢸⠄
⠀⠀⢸⡆⠆⢂⣀⣉⣉⣉⣀⣈⣁⢀⡀⠀⠀⠀⠀⢀⣀⣠⣍⣤⣬⣥⣄⣠⡄⠀⢠⡏⠀⠀⠀⡇⠀⠀⠀⠛⠀⠀⠀⠘⠙⠛⠻⠙⢓⢸⡆
⠀⠀⠀⠻⣆⠀⠙⠿⣿⣿⣿⣿⣍⣻⣤⣤⣤⣄⢠⡄⣽⣷⣿⠿⣿⠿⠟⠋⠀⣀⠟⠀⠀⠀⠀⡇⣶⣦⣤⣤⣥⣤⣄⣬⣴⣰⣄⣠⣾⢸⠃
⠀⠀⠀⠀⠈⠳⢦⣀⠀⠈⠩⠍⠉⠩⠍⠁⠠⠈⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠉⠀⠀⠀⠀⠀⠧⡙⠟⠿⠿⠿⠾⠾⠿⠷⠿⠿⠿⢋⡼
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠋⠛⠋⠙⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠲⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠰⠆⠰⠶⠆⠰⠶⠶⢤⡀
⠀⠀⠀⠀⣠⠔⠋⠀⠀⢀⠀⠀⠀⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠁⠠⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⢰⠋⠀⠀⠀⣀⣤⡿⣷⣤⡄⠀⠀⠀⠨⠆⠀⠀⠀⢀⡴⠛⠁⠀⢀⠀⡀⠀⠈⠙⠶⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⠀⣩⣵⣶⢿⣾⣤⡄⠀⠀⠀⠄⠀⠀⢠⡟⠀⠀⠐⠁⠀⠀⠀⠀⠐⢄⠂⠙⣆⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⣈⣽⠶⠷⠾⠷⠿⠦⠄⠀⠀⡂⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⡀⠀⢦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⣀⣀⣀⡦⠀⠆⠠⠶⠤⠤⠄⠀⣷⠀⠀⡟⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠱⠦⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠀⠀⠀⠀⣟⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡤⠒⠛⠛⠛⠛⠛⠉⠋⠉⠉⠳⡄⠀⠀⠀⡟⠀⠀⠀⠀⢸⠇⢀⣸⡅⠀⠀⠀⠀⢸⡂⠀⢸⠀⠀⠀⠀⣿⣿⠀⠒⠂⠀⠀⠀⣾
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠘⢶⠁⠷⠄⠀⠀⠀⠀⢸⠇⠀⢸⠀⠀⠀⠀⣿⣿⠀⠀⠄⠀⠀⠀⣹
⠀⠀⠀⠀⡇⠀⠀⢀⠈⠄⠘⠀⠀⠀⠀⢀⠃⠀⠀⠀⡇⠀⠀⠀⠀⢀⣨⢀⣇⡅⠀⠀⠀⠀⢸⠇⠀⢸⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⡧⠀⣠⣾⣷⣶⡖⠖⠰⠶⠆⠀⣤⠀⠀⠀⣿⠀⠀⠀⠀⢸⠓⢈⣷⡆⠀⠀⠄⠀⣼⠀⠀⢸⠀⠀⠀⠀⣿⣿⡀⣀⡀⠀⠀⠀⢼
⠀⠀⠀⠀⣧⠀⠘⠉⠉⠛⠷⠿⢿⡿⠇⠀⣿⠀⠀⠀⠸⣧⠀⠐⠄⣼⣿⠉⣿⣇⠀⠀⠀⣰⠏⠀⠀⢸⠀⠀⠀⠀⠉⠉⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡿⠀⠀⠀⠀⠈⠳⣄⡐⠠⠁⠈⠉⠀⢀⣠⡾⠃⠀⠀⠀⢰⡀⠀⠀⠠⠇⠀⠐⠠⠀⠱⠀⠀⡼
⠀⠀⠀⠀⠈⠓⠒⠒⠚⠛⠓⠚⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠦⣄⡴⠚⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣸⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧⠀⠀⢾⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣈⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠲⡖⢦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠶⠤⢤⣀⡀
⠀⢀⣄⠐⠃⣠⠄⠁⢀⠁⡀⠀⠀⠀⠀⠀⠄⢥⠓⠃⠀⠃⠀⠞⣩⠦⠀⠀⠀⠀⠀⣠⠖⠋⠁⠐⠖⠀⢠⣂⠀⠀⣉⠢⣄
⠀⣢⠍⠃⢸⣇⣀⣠⠀⠁⡈⣄⡀⠀⠀⡀⠘⠀⢀⣀⣠⡅⠁⠀⠁⠀⠀⠀⠀⢀⡾⠁⠰⡈⠦⠀⠠⠤⠤⠈⠀⠯⡀⠀⠈⠳⡀
⠀⠀⠰⠀⠘⢿⡟⡉⠀⠀⠂⠀⠁⠀⠀⠀⠐⠛⠉⠛⠻⠻⢿⡗⠾⠧⢀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠖⠈⠀⢡
⠈⠄⠀⠲⠲⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⡇⠐⢀⠀⠀⠀⠀⡀⣀⡀⣠⣔⠀⠀⠀⠀⠄
⠀⠈⠢⢀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⡀⢁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠿⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠉⠂⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡴⠃
⠀⢀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⣠⣆⠯⡆⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠈⠀⡬⡆⠈⠇⠀⢀⡠⠊
⠀⠋⠄⠀⠀⠀⠠⠛⢇⠈⠄⠰⠀⠀⠀⠀⠙⠻⠁⠑⠀⣃⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠤⠴⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠇⠀⠈⡀⠀⠀⠇⠀⠀⠀⠀⡁⠂⠀⠛⠛⠀⠀⠀⠀⠀⠡⠀⠀⣀⡀⠀⠀⠀⠠⠄⣀⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠬⠉⠀⠒⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⢠⠀⠤⠀⠄⠀⠀⡤
⠀⠀⢀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⢨⣭⠙⠀⠂⢄⠀⠀⠀⠀⠋⠃⠘⠤⠀⠀⠀⠻⠼⠘⠧⠀⠈⠁⠘⠦⠆
⠀⠀⠃⠘⠀⠁⠀⠀⠁⠂⠀⡐⣄⠀⠀⠀⡔⠁⠤⠀⠀⣰⡀⠀⠀⠐⠳⡀⠀⠀⡤⠄⠠⠀⠀⠀⣢⡂⠀⠀⠀⠲⠀⠐⢀⡂⡆
⠀⢃⠀⠀⠀⠶⣾⡧⠤⠤⠴⡇⠘⠀⠀⠀⠀⡀⠀⠀⠰⣿⣣⠀⠀⢀⠄⠁⠀⠀⣀⠀⠀⡀⠀⢀⠁⠉⠀⠀⠀⠀⢀⠀⠀⠀⡁
⠀⡄⠀⠀⠐⢲⣿⡷⠶⠗⠂⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠻⡇⠀⠀⠀⠄⠀⠀⠀⢁⠀⠰⠘⠀⠀⡇⢰⠐⠀⠀⠀⠀⠀⠘⢀
⠀⠘⢄⠀⠀⡀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠠⡀⠀⡀⠐⠀⠁⠀⠀⠀⡀⠀⠀⠀⣄⢀⠀⠄⢠⠀⠀⢀⡀⠀⠀⠀⠄⡀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠏⠀⠘⠆⠞⠀⠈⠃⠀⠈⠀⠚⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠆⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡤⣄⡠⢤⣠⠤⣀⡤⢤⣀⡤⣄⣠⠤⣄⣠⠤⣄⣠⠤⣄⡤⢤⣀⡤⣄⡤⢄⣠⢤⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠉⠀⠈⠀⠀⠁⠀⠉⠀⠀⠉⠀⠉⠁⠀⠈⠁⠀⠈⠉⠀⠉⠀⠀⠉⠀⠈⠀⠀⠁⠀⠉⠀⣙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⠀⢰⠋⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⠀⢸⡅⠊⠀⠀⠀⠀⠀⠀⣀⡴⢒⠫⠉⠁⠀⠀⠈⠉⠝⡲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠀⢸⡄⠀⠂⠀⠀⠀⣠⢞⠅⠺⢄⣀⡤⣧⣴⣤⣤⢴⣀⠈⠑⢭⡳⣄⠀⠀⠀⠀⠀⠀⡇⠀⣖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡀⢸⡇⠀⠀⠀⢀⡜⡵⠁⠀⠀⣇⣿⡏⣽⣿⣿⠑⣸⣿⣷⣤⠠⠀⢌⢦⠀⠀⠀⠀⠀⡇⠀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⢸⡇⠀⠀⢀⡞⠜⠀⠀⠀⠀⣧⣤⣄⣈⣉⠉⠐⠿⣿⡿⢐⣴⣀⡀⠀⢳⡀⠀⠀⢀⡇⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⢸⠇⠀⠀⡼⠈⠀⠀⠀⠀⣰⡋⠒⣂⣙⠉⠙⠳⢤⣄⡁⠻⡏⠂⠀⠀⠈⣇⠀⠀⠀⡇⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢸⡅⠀⢰⠃⠀⠀⠀⠀⠀⣿⠴⠶⠼⢿⣿⣷⣶⣬⣙⠻⡶⠁⠀⠀⠀⠀⢸⠀⠀⠨⡇⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⢸⡇⠀⢸⠘⠀⠀⠀⠀⠔⠁⠈⠋⠀⠀⠙⠫⠚⢿⣿⣷⣧⣀⠀⠀⠀⠀⣸⡆⠀⠠⡇⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⢸⡇⠀⣺⢀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⢫⡇⠄⢰⡇⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⠀⢸⡇⠀⢹⢸⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⠃⠀⢨⡇⠀⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢸⡇⠀⠸⡌⡀⠀⠀⠀⠐⢄⣀⣀⣠⠀⠀⠈⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⣸⠀⠀⠈⡇⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠀⢸⡇⠀⠀⢧⢀⠀⠀⠀⠀⠀⢀⣠⡴⡶⠀⠟⠛⠉⠉⠙⠳⢦⠀⠀⠀⢤⡇⠀⠀⠀⡇⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⢸⡇⠀⠀⠈⢧⢂⠀⠀⠠⣞⡄⠁⠀⠀⠀⠀⠀⠀⠠⠘⡠⠎⠗⢀⣌⡞⠀⠀⠀⠠⡇⠀⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⢸⡇⠀⠀⡴⠬⠶⠷⠒⠂⠒⢌⠙⠓⠄⡀⠀⢀⡠⠚⡡⠒⠐⠒⠾⢾⠴⡆⠀⠀⠀⡇⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⢸⡅⠀⢶⡉⠀⠀⠀⠀⠀⣄⣿⠀⠀⠀⠘⠖⠁⠀⠀⢃⣠⠀⠀⠀⠀⠀⣉⠶⠀⠈⡇⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⡅⠀⠀⠉⠙⠒⠛⠉⣻⠀⠉⠉⣴⡒⢒⣲⣒⠒⢋⠉⠘⡏⠉⠙⠓⢋⠁⠀⠀⠐⡇⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣹⠇⣾⠃⠈⣿⡄⠀⠀⢣⠀⠀⠐⠀⠀⠀⠀⢀⡇⠀⢞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣐⠛⠆⠙⠳⠴⠟⣀⣠⡤⠞⠁⠀⠀⠀⠰⠄⠁⠐⡇⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣁⣁⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⡇⠀⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⢀⡀⠈⡈⠁⢀⡁⢀⣁⠀⢁⡁⠀⣀⡀⠀⣀⡀⠁⢀⣀⠀⣀⠀⠁⣈⠀⢉⠈⠈⣙⠁⢀⡀⢑⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠚⠙⠒⠋⠓⠊⠈⠓⠋⠙⠚⠃⠙⠚⠁⠙⠚⠃⠙⠚⠉⠓⠚⠉⠓⠋⠑⠚⠉⠒⠃⠑⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⣄⡤⠶⣤⣀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⠛⠶⣤⡉⠻⢶⣶⠛⠁⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣈⠛⠶⣤⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠉⠻⢦⣌⠙⠳⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠙⠻⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢖⣚⣛⣛⣭⣭⣭⣭⣭⣛⣛⣛⡲⠶⢤⣄⡀⠀⠈⠙⠷⣤⣀⢠⡞⠁
⠀⠀⠀⠀⣠⠞⣋⡵⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⡉⠛⠳⢮⣝⠳⣄⠀⠀⠀⠙⠛
⠀⠀⠀⠰⣏⢰⡏⠀⠀⠀⠀⠐⢦⣄⡀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠙⢦⣝⠶⢤⣄⣀⠀⠀⠈⠙⠳⠛⠉⠀⠀⠀⠀⣀⣀⣤⠶⣋⡴⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢭⣭⣽⣛⣛⣛⣛⣛⣛⣛⣿⣭⡭⠷⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⣿⣿⣷⣿⣾⣟⣿⢿⣿⣿⡷⣿⣾⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠰⠀⠑⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⡛⠛⠛⠛⠛⠛⠻⠿⠟⠛⠟⠛⠻⠻⠻⠻⠻⠻⠿⡏⡉⢉⢉⡉⠙⠉⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄⠀⠀⠁⠉⡀⠀⠂⠀⠀⠀⠁⠀⠀⢂
⠀⠀⠀⠀⢀⣀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⠸⡀⢸⠀⠀⡃⠀⠀⠁⠀⠀⠀⢐
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⡄⠇⠞⡴⠃⡀⠀⠀⠂⠅⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠉⠀⠄⠄⠸⠉⠁⠀⠀⣀⠀⡀⠀⠀⡐⡒⡂⡁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠁⠀⡀⠀⠄⠀⠀⠄⠄⠄
⠀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢀⠀⠄
⠀⠀⠀⠀⠄⠂⠈⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠒⠙⠒⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠐⠐⠀⠀⠐⠀⠀⠀⠂⠂⠀⠀⠀⠀⠂⠐⠐⠀⠐⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠂
⠀⠀⠅⠀⠀⠀⡀⡀⠀⠀⠀⠠⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠁⠁⠁⠁⠁⠁⠀⠁⠁⠁⠁⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠠⠤⠄⠨⠤⠀⠀⠤⠤⠄⠀⠀⠀⢀⢀⢀⠀⢀⠀⠀⡀⡀⡀⡀⡀⡀⠀⡀⡀⡀⡀⡀⠀⠀⠀⠀⠀⠒⠒⠒⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠉⠁⠉⠀⠉⢃⡀
⠀⠀⠀⠀⠠⠂⠠⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠄⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠇⠀⠀⠀⠀⠐⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠆
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⣠⠤⠤⢤⣤⡄
⠀⠀⠀⠀⠀⡏⠉⠉⠉⠹⣇⠀⢀⡟⠉⠹⣇⡀⢀⣸⠏⠉⠙⢧⣄⣠⡾⠁⠈⢷⣄⣀⡴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢧⣀⠀⠀⣤⣤⡤⣤⠤⠤⠤⠶⠶⠶⠖⣶⣶⣶⣶⣾⣷⣶⡶⠞⠛⠛⠛⢻⡀⠀⢀⡴⠞⠃
⠀⠀⠀⠀⠀⠀⢹⡆⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠁⠀⠀⢀⣠⣴⣾⡇⠀⢺⡀
⠀⠀⠀⠀⠀⡴⠞⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣶⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠛⢾⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⢿⡇⠀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⡷⣤⡀⠀⣿⣶⣾⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⢸⡇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣸⠇⠀⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⢾⠀⠀⠻⢦⣤
⠀⠀⠀⠀⠘⡟⠋⠀⠀⣿⣿⣿⠿⠋⠀⠀⠀⢀⣠⡴⡶⢶⢶⣤⡤⣤⣴⢾⠏⡇⡇⣿⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⣧⣄⡀⠀⣿⠀⠀⠀⠀⠀⣠⠞⣿⢸⠁⡇⢸⢸⣿⣀⣿⢸⢸⢰⣇⡇⣿⢸⠀⠀⣠⠞⠛
⠀⠀⠀⠀⠀⠁⠈⣷⠀⡧⠀⠀⠀⣠⡞⢻⠀⡇⢸⠀⡇⣸⣸⠋⢉⣳⣾⣸⠾⠛⠳⣿⣸⡀⠀⣯
⠀⠀⠀⠀⠀⣴⠾⠃⠀⡿⢲⠶⡟⢹⣇⣼⠞⠛⠿⣄⣧⣿⣁⢀⣾⣿⣿⣧⣴⣶⣷⣿⣿⠀⠀⠙⠓⢷⠄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣏⢸⠀⣧⡼⠋⢀⣤⣄⢠⣬⣽⣿⣿⣿⣿⣿⣿⣿⠟⣉⣿⣿⣿⠀⠀⠀⣠⣼⠆
⠀⠀⠀⠀⠀⠛⠛⣦⠀⣧⣼⣿⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢿⣿⣿⣿⡀⠀⣾⠁
⠀⠀⠀⠀⠀⢀⣀⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠇⠀⠻⣤⣤
⠀⠀⠀⠀⠀⣿⠉⠀⠀⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠶⢦⡀⠀⠀⣠⠶⠶⣦⡀⠀⣠⡴⠶⣄⠀⠀⢀⡶⠒⢶⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⣄⣀⣠⣼⠃⠀⠀⣿⣤⣴⠏⠀⠀⢸⣧⣴⠏⠀⠀⠿⠶⠖⠾⠁⠀⠀⠿⠶⠖⣲⢿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢒⡲⢶⣆⣾⣶⡀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣦⣤⠟⠉⠙⢧⣤⡾⠉⠉⢷⣤⡼⠋⠉⠻⣦⣴⠟⠉⠙⢧⣤⡾⠉⠉⢷⣤⡼⠋⠉⠻⣤⣴⠟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣧⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡟⠉
⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣏
⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣧⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡟⠉
⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⠟⠻⣦⣀⣠⡞⠛⢷⣀⣀⡶⠛⢳⣄⣀⣴⠟⠻⣦⣀⣠⡞⠛⢶⣀⣀⡾⠛⢳⣄⣀⣴⠛⠻⣦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣿⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⢯⣄⣀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡧⠶⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠛⠻⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⠉⠉⠉⠉⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣿⣿⡿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣴⣶⣾⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⢔⣦⣬⣿⣯⣥⣖⢄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⣿⡛⣷⠶⣄⡀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣜⣷⣿⣿⣿⣿⣿⣿⣿⣿⣟⣄⠀⠀⠀⠀⣴⣿⣷⠟⠋⠉⠉⠙⠶⣿⣿⣄
⢠⡿⡽⣯⢿⢿⡭⣭⢯⡽⡭⢯⢿⡄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀⣸⡿⣿⣷⡶⣶⡶⣶⢶⣶⠾⠿⣿
⢸⣔⣂⣃⣘⣀⣀⣀⣈⣀⣀⣒⣸⡇⠀⠀⠟⣇⣣⣠⣴⣦⣦⣤⣤⣡⣤⣧⡿⡇⠀⠀⢿⡇⠿⢿⡇⣿⡇⣤⠻⣿⠀⠭⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠀⠀⠀⠸⣿⣿⣏⠉⠉⠉⠉⠉⢉⣽⣿⡿
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠑⢝⡿⣿⣿⣿⣿⣿⣿⢟⠕⠁⠀⠀⠀⠀⠙⢿⣏⣿⡶⣶⠶⣾⡙⣿⠟⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠉⠉⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠛⠉⠁
⠀⠀⢀⣤⠶⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠐⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠛⠛⠒⠦⣄
⠀⣴⠋⠐⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⣸⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⡇⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⣸⢷⡶⠶⠶⠶⢶⠶⡶⠶⠶⣶⠾
⣿⠘⠿⠿⠓⠭⠾⠏⠇⠷⠺⠏⠃⡟⠀⠀⠿⠿⠿⠾⠿⠴⠾⠶⠼⠟⠹⠏⠿⠇⠀⠀⣇⢹⡧⡾⢦⡿⠾⢼⠟⢻⣤⣿⡁
⢸⡇⠐⠒⠀⠀⠐⠂⠀⠀⠐⠀⣺⠇⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠸⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺
⠀⠹⣦⡐⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠁⡄⡀⠀⠀⠀⠀⡀⠤⠃⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠙⠳⠶⠤⠤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠖⠚⠁
⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⡇⣿⣾⣿⡇⡇⢸⣿⣿⣸⣸⣿⡇⢸⠀⠀⣿⣧⡓⣿⢘⣸⠏⠜⣿⢀⠛⡀⣴⡆⠀⢸⣿⢀⡼⠈⣧⢺⡆⣿⠠⠉⠀⡁⣾
⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⠀⠿⠷⠶⠿⠾⠿⠾⠿⠾⠾⠾⠷⠿⠃⠀⠈⠛⠿⠷⠿⠷⠞⠷⠛⠖⠛⠖⠓⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠀⠀⡤⣶⣦⣶⣦⣴⣶⣴⣦⣦⣴⣦⣦
⢰⣶⣶⠐⣶⡆⡆⢶⣐⣶⣶⡆⣶⡆⠀⢸⡗⣾⣲⣶⣶⣶⡶⣆⡆⣷⣾⣾⡆⡇⠀⠀⣿⡏⣹⡏⢽⢙⡭⣏⢹⢉⣿⣭⣿
⠨⠬⠤⠄⠤⠥⠥⠤⠤⠤⠤⠤⠤⠅⠀⢸⣷⣬⣤⣥⣤⣤⣤⣥⣥⣤⣬⣬⣤⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⢻⠛⣿⠛⢙⡟⡟⢻⠛⡟⣿⣿⠀⣼⣿⠛⡟⣻⠻⠛⠟⠿⣟⡻⢛⡛⢻⣿⡀⢰⠋⣦⣦⣦⣦⣴⣦⣦⣴⣴⣤⣴⡎
⣿⣧⣼⣤⣿⣶⣤⣧⣷⣼⣤⣥⣽⣿⠀⢹⣿⣶⣷⣾⣶⢶⣾⣷⣷⣶⣾⣷⣾⣿⠀⠘⣄⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⢤⡀⡀⠢⣀⠄⠑⠤⠊⠙⠒⠁⠀⠉⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄⡠⠈⠓⠔⠁⠈⠁⠁⠀⣈⣀⡀⠤⠤⠤⠀⣒⠒⢀⠉⢉⠁⡀⢧
⠀⠀⠀⠀⠀⠀⠀⣔⠃⠀⡠⠤⠤⠖⠒⠒⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⢆⠒⣐⢠⡄⠄⠀⡅⢔⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⠀⠀⠀⠔⠁⠀⠉⠂⠀⡠⠀⠀⠀⠀⠀⠀⢨⡘⢀⠐⠗⠠⠃⢠⢀⠕
⠀⠀⠀⠀⠀⠀⠀⠈⣑⠀⠀⠀⠄⠢⠑⠋⠷⠴⠋⠀⠈⠡⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢓⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢁⠀⠀⠀⠌⠴⠐⠂⠂⠀⠤⠍⢠⠀⢀⠤⠒⠒⠒⠠⢄⠀⠀⠀⠆⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢖⠁⠘⠀⠀⠀⢠⠔⡒⠒⠒⠢⢄⡀⠀⡔⡡⢊⠄⠚⠀⠀⠀⠑⡀⠀⢰⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⢰⢁⠊⡔⠁⠀⠀⠀⠑⣴⠀⠓⠁⠀⠀⢀⠀⠔⡀⡆⠀⠀⠀⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡀⠃⠀⠸⢈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠡⢈⠈⠀⠀⠀⠀⡄⠑⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡀⠐⠀⠀⠄⠒⠁⠀⠀⠀⡀⠀⡠⢀⠔⡡⢊⠔⠑⡡⠊⠌⠀⠀⠀⠇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠈⢆⠈⠲⢑⡨⢐⡩⢒⠅⡪⢔⠅⡠⠊⠀⡐⠀⠀⠀⠀⠠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠀⠆⠀⠀⠀⠑⠢⡁⠂⠥⡂⠕⡪⢔⠡⠊⢈⢀⠌⠀⠀⠀⠀⠀⠐⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⢠⠀⠀⠀⡤⠄⠈⠐⠄⡀⠉⢂⠥⡪⠄⣡⠮⠶⠀⠴⢦⠀⠀⠰⠆⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠘⡋⠂⠂⠁⠘⢰⠀⠀⠈⠑⠢⢈⣊⠔⢡⠔⠴⠢⡖⠲⠄⠆⠀⠆⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠃⠒⠂⠀⠉⠀⠀⠂⠄⠀⠀⠀⠁⠀⠈⠛⠋⢁⠀⠠⠤⠒⠒⠋⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⢰⠀⠀⠀⠀⢀⣀⠀⠤⠄⠀⠐⠒⠈⠉⣀⣀⠀⠠⠤⣄⡡⠂⠢⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣀⠀⠉⢁⣀⠀⡠⢄⠀⡔⢢⣠⠊⠡⠜⠀⠈⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠃⠈⠒⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⠀⡀⠀⡀⢀⠀⣀⠄⡀⠀⡀⠀⡠⢀⡠⢀⢠⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐⠩⠂⠅⠂⠕⠈⠒⠉⠒⠁⠒⠁⠒⠁⠂⠁⠊⠁⠈⠀⠉⠀⠁⠀⠁⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢿⡿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢦⣄⠀⠀⠀⠀⠀⠀⣀⣴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠛⢦⣄⣠⡴⠛⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣇⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⠶⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠶⠶⠶⠾⠿⠷⠶⠶⠶⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣴⠟⠉⠛⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⢷⣦⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡄⠀⢀⣿⡀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠈⢿⣄⡀⠉⠛⠋⠉⣉⣭⠁⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⠀⠀⠀⠀⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠈⠛⢷⣤⣀⣠⣴⠟⠋⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠇





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣀⠀⠈⠛⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣚⠯⠽⠗⠒⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⣙⣛⠛⠿⠿⣷⣦⣄⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢏⡀⠀⠀⠀⠀⠀⠠⠤⣄⣀⣀⣀⣤⣤⠶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⢀⡹⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠮⣭⣓⣒⡶⠤⠤⠤⠤⣌⣉⣁⣀⣀⣀⣀⣀⡤⠤⠤⠤⣶⣖⣶⣯⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣿⠀⠀⠀⠀⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣤⡀
⠀⠀⠀⠀⠀⢰⣿⣯⣀⢸⡇⠀⠾⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠒⠒⠖⠲⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠀⠀⣿⢀⣨⣿⣷
⠀⠀⠀⠀⠀⢸⡇⠈⠙⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠉⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣬⣁⠈⠉⠉⠉⢉⣉⣉⠉⠉⠉⠉⢀⣩⣼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⡾⠛⠉⠙⠻⣦⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣇⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⡀⠂⠁⠀⠀⠉⠐⢄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠉⠙⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢀⠆⠀⠂⠀⠀⠀⠀⠘⡆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠐⠒⠀⠀⠁⠀⠀⠀⠀⠈⠑⠢⠤⠤⠆⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠋⠁
⠀⠀⠀⠀⣀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠉⠉⠉⠙⠤⣄
⠀⠀⠔⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⡔⠉⠀⠈⠀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠘⠀⠀⠀⠀⠀⠈⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠀⠀⠀⠀⠀⠠⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⡤⠞
⠀⠀⠱⢀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠳⢄⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠈
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⠔⠋⠉⠁⠀⢤⡀⠀⠀⢀⠜⠁⣠⢤⢄⢀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢳⠀⠀⢸⠁⠀⠁⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠘⠇⠀⠀⠄⠦⠖⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⣀⠌⠁⠀⠀⠀⠀⠓⠤⢤⡤⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⢀
⠀⠂⣀⠖⠉⠁⠀⠈⠑⣄⠀⠀⠀⢀⠴⠛⠉⠉⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢶⠉⠁⠀⠘⠉⢪⡢
⠀⢗⠆⠀⠀⠀⠀⠀⠀⠈⡂⠀⢰⠃⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠴⠚⠏⠲⠄⠀⠀⠀⠠⡯⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠀⠄⠐
⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠈⠦⡀⠀⠀⢀⡠⠈⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠙⠳⠌⠛⠁⠀⠀⠀⠀⢁⣤⠀⠀⠀⠀⣀⠰⠃⠁⠀⠀⠀⠂⠠⠀⠂
⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠞⠒⠊⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣤⠤⣍⠉⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠸⣇⣀⡼⢁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⣤⠿⠛⢋⣉⣭⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⢀⣠⡴⢖⣋⣭⣶⠾⡿⠿⠉⢿⣵⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠷⠟⠋⠙⠋⣭⠀⢀⣶⠤⡄⢿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠀⠀⠀⠐⠚⠉⠃⠀⣠⠴⠃⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣟⠀⠀⠀⢀⣴⠋⠀⠀⢀⡔⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⢓⡗⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠈⠳⠾⣧⠀⠀⠈⠁⠳⠶⠶⢏⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣾⣛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠁⠈⣿⠻⡄⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣨⣷⡀⠀⠀⠀⠀⢀⣠⡤⢾⠁⠈⣹⣥⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠚⠛⠛⣷⣦⡤⠿⠚⠛⣇⠀⣷⠀⠀⠀⠀⢀⡤⠚⣁⣤⡶⠛⣿⡼⢷⠶⠶⠖⠛⠉⠀⠀⣿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣏⡉⢹⡇⠀⠀⢹⣧⠴⠶⢾⡄⢿⣀⣿⠶⠦⣤⣤⣾⠗⠋⣽⢿⣿⠲⣿⢦⡌⣧⣠⣤⠴⠶⠚⠛⠉
⠀⠀⠀⠀⠀⢶⡛⠉⢳⡘⡟⠛⣷⢸⣇⣀⣤⣼⡷⠟⠉⠉⠛⠲⠦⣤⣭⣽⣾⣇⠀⢹⣆⠘⣆⡇⢸⡅
⠀⠀⠀⠀⠀⠀⠹⣆⠀⢳⣿⣠⡿⠞⠛⣩⠿⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠳⢤⣘⣷⡈⣿⣆⠹⣷⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠛⠉⠀⢀⡴⠞⢁⣠⣀⣠⣤⠶⢿⣟⠋⠀⠀⠀⠀⠀⠈⠙⢧⠸⣿⣆⢻⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⣇⠀⠀⠹⣟⠉⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⣿⣤⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠳⣦⠖⠛⢳⣄⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣋⣩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⢴⡇⡀⡄⢺⠁⢹⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠛⠉⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⡧⠀⣧⣳⣹⣌⣧⣾⢀⣿⠀⠀⠶⠖⠛⠋⠀⠀⠀⣿⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠻⠚⠋⠉⠉⣉⣥⣶⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣼⣏⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠉⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣀⣠⡶⠛⠉⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣿⣤⣴⣶⣛⣉⣁⣀⣤⡴⣶⡆⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣌⡉⢁⣼⠟⠛⢻⡿⠉⠀⠀⣼⠃⢀⣾⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⡟⠉⠙⠛⠁⠀⣠⡟⠁⠀⠀⣼⣯⣤⡾⠇⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠙⠻⢶⣤⣤⣴⠾⠋⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢺⢩⢣⢣⢃⠇⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⢇⢆⠃⡞⠆⠎⡌⡘⡜⡘⡳⣄
⠀⠀⠀⠀⠀⡤⠄⣀⣀⣀⣀⣀⣴⡞⡝⡜⡌⡎⡎⡜⡜⡸⡘⢰⡸⡰⠱⢣⢫⢦⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠓⠓⠳⠧⠷⠧⣧⣧⣧⣧⣇⢛⢦⡀
⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠤⠤⢤⡀
⠀⢀⡠⢶⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢰⠀⡄⠀⡇
⠀⢸⠳⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠇⠀⡇
⠀⡄⠀⠈⠙⢷⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡄
⠀⡇⠀⠀⠀⠀⠉⠻⣥⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣝⣙⣆
⠀⡇⠀⠀⠀⠀⠀⠀⠈⠳⢖⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣿⡼⠟⢻
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⡥⣒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣲⠯⠒⠉⠀⠀⡆⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣚⠭⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠿⠚⠉⠀⠀⠀⠀⠀⠀⡁⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣒⣤⣀⣤⣀⠄⣰⣶⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡇
⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠉⠑⠿⣅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⡿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⢆⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⣒⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡥⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣷⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣂⡀⠀⠀⠀⠀⠀⠀⡆
⠙⠫⠤⣔⣲⣂⠤⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠬⠬⣐⣐⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡠⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀⣀⠀⡀⠀⠉⠉⠐⠒⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠄⠀⢸
⠀⠈⠑⠉⠚⠕⠫⠚⠝⠫⠒⠩⠖⠭⠪⠕⢪⠕⡢⢂⢔⡤⣋⢝⡱⢒⢦⡤⣤⣀⣀⡀⠀⠀⠀⠀⠻⣆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠉⠘⠁⠊⠐⠉⠋⠝⠹⠳⠖⠾⠷⡫⠪⠿⠯⠗⡵⡪⢆⢴⠖⠄⡤

⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢘⢢⣥⡤⠄⠠⠔⠒⠒⠒⠒⠞⠥⣐⠀⠀⠀⠀⠀⠀⢠⠈⣆⠀⠀⠀⠀⠀⠀⠀⣶⠛⠉⠁⠠⠀⠀⠀⠀⠀⠉⢻⣶
⠀⠀⢸⣿⠀⠀⠠⠶⡀⠘⡄⠀⠀⠀⠒⢾⠀⠀⠀⠀⠀⢠⠾⠛⠿⢆⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠘⣹⠀⣀⣀⣿⡁⠀⢰⣷⣤⣙⣶⣾⠀⠀⠀⠀⠀⣹⣼⣀⣦⣼⠁⠀⠀⠀⠀⠈⣵⣀⡤⢀⢸⡦⠴⣀⠐⢒⡠⣀⣿
⠀⠀⠸⠩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠩⠀⠀⠀⠤⠋⠙⠛⠉⠁⠫⠜⠣⠀⠀⠀⠀⠛⠻⠿⠷⠩⠤⠤⠍⠸⠾⠿⠿⠟

⠀⠀⠀⠀⠰⠶⢠⣤⣤⣤⣤⣤⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⣤⣤⣤⣤⣤⣤⠄⠄
⠀⠀⠀⠀⠀⠒⠊⢉⠉⠉⠉⠉⠘⠂⠀⠀⠀⠀⣾⢒⡀⠤⠄⠀⠀⠀⠰⢲⡆⠀⠀⠀⠀⠒⠈⠉⠉⠈⠉⠉⠉⡙⠒⠆
⠀⠀⠀⠀⠈⠀⠀⠘⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠉⠉⠉⢉⣉⣀⠈⠛⠀⠀⠀⠀⠈⠒⠒⠒⠆⠀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⢳⡀⠠⠤⠀⠄⠈⠂⠀⠠⡆⠀⠀⠀⠀⢰⠀⠲⣤⣀⡐⠒⠀⠂⢑⡃
⠀⠀⠀⠀⢀⡀⠿⡇⠃⠀⠀⠁⠃⡄⠀⠀⠀⠀⢻⡧⠀⠀⡀⠀⠀⠀⠀⠒⡇⠀⠀⠀⠀⢠⣼⣴⣎⡑⠛⠀⠀⠂⡯⠇
⠀⠀⠀⠀⠘⠒⣗⣛⣋⣛⣛⣛⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠿⠿⣉⣀⣁⣁⡀⠀⠣⡄
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⢿⡛⠯⠙⠛⠚⠛⠓⠚⠛⠩⠟⣻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣟⠝⠊⠁⡤⣶⠂⢻⡇⢸⡧⠁⣼⠇⣠⡄⡉⠚⢝⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⠞⠁⢶⣝⣧⡀⠻⠆⠘⠃⣘⡃⠐⠿⢰⡟⡒⢠⡶⢆⠙⢮⡻⣷⣄⣠⣤⣴⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠝⠡⣮⡡⢀⡻⠊⣡⠴⠚⠉⠉⠉⠉⠉⠙⠲⢤⡁⠛⠵⠟⠀⠀⠐⠙⣻⠭⠭⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠎⣐⠂⠈⠻⠊⡴⠋⠀⠀⠀⢼⣷⡆⠀⠀⢴⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣤⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡿⡌⠀⠘⠿⠶⢀⠞⠠⣾⡖⠀⠀⠘⠉⠁⠀⠀⠀⠀⢀⣀⡀⣤⣤⠀⢲⣿⡎⢙⠂⢹⣿⡆⢿⣿⡄
⠀⢀⣀⣤⣤⣴⣶⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡄⢤⣶⡂⠸⣿⡇⢉⠃⢸⣿⡇⠈⣿⣧⠻⢀⠈⣿⣧⢸⡿⠁
⠀⠈⡯⠭⠕⠒⠚⠉⠀⠀⢀⣀⢀⣤⣄⢶⣶⡄⣿⢹⣿⡆⠓⢸⣿⡇⠀⣿⣿⠙⠂⠈⣿⣷⡀⣻⡿⠠⠾⠂⠛⠉⠈
⠀⠀⠀⢀⣤⠠⢴⡐⣾⣿⠈⡙⠀⢻⣿⣀⣿⣃⠀⠀⣿⣷⠀⠀⣿⣿⡀⢸⠿⠇⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠤⠤⣖⣒⣢⣥
⠀⠀⠀⣾⣿⠀⠈⠁⢸⣿⡇⠳⣀⠸⣿⣇⢹⣿⡦⠀⠿⠛⠃⠀⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠹⣿⡇⠀⡰⢀⣿⠷⠰⠛⠀⠋⠉⠀⠀⠀⢀⣀⠀⠀⠀⣄⣤⠀⠀⠐⠻⠃⣠⠏⡐⠶⣦⡀⢠⢳⣿⠃
⠀⠀⠀⠀⠈⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠿⠁⠀⠀⠛⠟⠁⠀⠀⣠⠔⢁⢶⣄⠀⠔⢁⢪⣿⠏
⠀⠀⠀⠀⡠⠤⢴⣒⣺⣭⣥⡠⡀⠀⢠⡾⣣⡦⣉⠒⠦⢤⣀⣀⣀⣠⠤⠖⠋⣁⢶⣌⠐⣙⠋⡠⣣⣿⠏
⠀⠀⠀⠀⠸⠟⠛⠛⠉⠉⠹⣿⣬⡢⡈⠺⠋⡠⣴⡟⢰⡆⠀⣤⠄⣶⠀⢿⡄⠉⣯⡻⠂⡠⣪⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣮⣖⠤⣁⠛⠀⠾⠁⡌⣿⠀⢿⡄⠜⠓⢃⡠⢔⣮⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣾⣯⣷⣒⣠⣤⣤⣤⣤⣐⣲⣯⣿⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠛⠛⠛⠉⠁



⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠁⠐⠂⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⢀⡐⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠹⣓⢲⣤⣄⡀⠀⠀⡞⠤⡠⠊⠀⠀⠀⠈⠢⡐⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢰⠀⠀⡇⠀⠀⠀⠀⣤⠀⢳⠉⢰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠆⠀⠈⡆⠀⢃⠀⠀⠀⠀⠀⢀⠏⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁
⢀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠹⢌⠀⠁⠂⠤⢄⠂⠁⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⠔⣧⠀⠠
⠺⠄⢁⣀⠀⠀⠀⡀⠀⠄⠀⠠⠀⢀⡇⠀⠀⠀⠀⡠⠜⠒⣷⠥⠭⠝⠋⢉⡑⠂⠄⠀⡐⠉⠉⠉⠉⠁⠈⠉⠲⢎⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠈⢁⣒⣀⣬⣀⡂⠀⠀⡜⠁⠀⠀⠠⠊⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠣⠀⠀⠀⠀⠀⠀⡐⠁⢀⡈⢦
⠀⠀⣠⠒⠉⠉⠀⠀⢀⣬⠉⠉⠛⣷⣬⣀⡀⢘⡀⠀⢀⣬⣷⠀⠈⠀⠀⠀⠈⠀⠀⠈⢲⠀⠀⠀⠀⠀⠐⣤⠀⠀⠁⠀⢧
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠀⠀⠈⡙⠢⣈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠣⠀⠀⠀⠀⠀⠈⠓⠒⠂⠀⢀⠇
⠀⠀⢆⠀⠀⠄⡀⢠⡄⠐⠠⠀⢀⠀⠀⠈⢻⠀⠈⠳⡄⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠒⠮⣨⡙⣉⣀⣀⣀⣈⠏
⠀⠀⠈⠢⡀⠘⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠸⡄⠀⠈
⠀⠀⠀⠀⠈⠢⣌⠁⠞⠶⠀⠸⠠⠀⢺⠄⣼⠠⡆⠀⠀⣷⠒⠲⠤⠤⣤⣀⣁⠀⠀⠀⠀⢀⠀⢠⠆
⠀⠀⠀⠠⠤⠤⡀⠭⢶⠠⢤⣀⡀⠀⢼⢧⠺⣄⣱⣤⢟⡉⢷⠀⣠⠄⡀⡠⠀⠹⠉⣓⡖⠤⢄⠈⠂⠀⠀⠀⠀⠤⠒⠒⠒⠒⠤
⠀⣤⠊⠀⠄⠀⠀⠀⡀⠑⢄⠀⠈⡙⠉⠏⠉⠉⠁⠐⠆⠀⠀⢆⠀⠄⠀⠀⠀⠀⠐⠢⠀⠀⡜
⢀⡇⠀⠂⠀⠀⠐⠀⠈⠄⠈⡄⠄⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⡀⠀⠀⠀⡐⠂⠀⡀⣪⠞⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠈⡄⠀⠄⠀⠀⠀⠀⢀⠀⢱⠁⠚⠀⠀⠀⠀⠀⠀⢀⠰⡒⠒⠀⠲⣦⠠⡄⠀⠈⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡠⡤
⠀⠘⢄⠀⠄⠀⠀⠐⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢫⡀⠠⠁⢀⠔⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠁⠒⠠⠤⠖⠨⢀⣀⣀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⢀⢀⠀⡊⢧⢀⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡀⢄⡤
⠀⠀⠔⠈⠀⠀⠀⠰⠆⠀⠀⠀⠈⠑⠒⠄⡀⠀⠀⢠⠁⠀⠀⠀⠐⠓⠚⢛⠁⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⡀⢩⡆⠄⠀⡫
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⢤⡀⠀⠀⠀⠀⠈⠀⠈⠀⠞
⢀⠀⠶⠀⠀⠀⠀⠈⠀⣈⠀⠀⠠⠀⠀⠀⠁⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡂⠀⠀⠉⠢⣀⠀⠣⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠢⢀⠀⠀⠀⠀⠉⠒⠂⣤⡌⣄⣄⣠⣠⠤⢤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠆
⣠⣄⣀⣀⣀⣀⣀⣀⣈⣀⣀⣁⣀⣀⣀⣈⣀⣈⣁⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⢀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⡷⡾⠷⠾⠷⠿⠷⠿⢶⠿⡶⠿⠶⠿⠾⢷⠾⣷⣾⡀⠀⠀⠀⢨⣿⠿⠿⠿⠿⠿⠿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⢴⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠴⣿⠀⠀⠀⠀⠀⠀⠠⢸⣷⠄
⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠰⣿⠀⠀⠀⠠⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠂⠀⠀⠀⠸⣿⣤⣤⣤⣦⣤⣤⣤⣼⡷⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠈⠋⠙⠁⠙⠁⠙⠁⠙⠁⠉⠉⠋⠈⠋⠈⠋⠈⠋⠉⠂⠀⠀⠀⠚⠁⠘⠁⠉⠁⠙⠁⠉⠀⠙⠂
⠀⠀⠀⠀⠀⠀⢠⣄⠀⣠⠀⣠⡀⢀⣄⠀⣄⠀⣠⡀⢀⣄⠀⣄⠀⣠⡀⢀⡄⠀⣄⠀⣠⡀⢠⡄⢀⣄⠀⣠⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠐⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠠⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠄⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠉⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢺⣿⣤
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⣼⣿⣠
⠀⠀⠀⠀⠀⠀⠈⣹⣿⣴⣶⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣴⣤⣤⣤⣴⣦⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠛⠥⠙⠁⠈⠋⠀⠛⠀⠙⠁⠘⠋⠈⠛⠀⠙⢋⠉⠋⠈⠋⠁⠙⢁⠘⠃⠈⠋⠀⠙⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⢀⠀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠀⡐⠢⡀⠀⠀⠀⠀⠉⠁⠈⠁⠁⠁⠈
⠀⠀⠀⢀⠀⠀⠀⠄⠀⠐⠀⠰⠁⠁⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⣀⣀⡀⠠⠤⠔⠀⠀⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢰⡒⠒⠒⠒⠒⠒⣲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⠐⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⣄⠀⠀⠀⠀⢰⠆
⢺⠃⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⡀⠀⠀⠀⠀⠀⠀⠁
⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠠⡀⠀⢠⡄⠀⠀⠀⠀⠀⢠⠀⣠⡒⠒⠦⠄⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠀⠐⠒⠒⠒⢲⠀⠀⠐⣄⠐⠦⠔⠲⠊⠀⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠈⠐⠀⠀⠀⠒⠀⠀⠀⠀⣟⣟⣿⣿⣿⣿⣿⣟⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠈⠢⠀⠀⠀⢀⠠⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⠶⠶⠶⠾⠿⠶⠶⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠶⠂⠀⠀⠀⣿
⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠻⣶⣄⠀⠘⣛⣛⣃⢀⠛⠛⠛⠛⠒⠒⠶⠶⠆⠀⠀⠀⣠⣴⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠻⣮⣉⠉⣉⣉⣉⣙⣁⣀⣀⣀⣀⠀⢀⣠⣶⠟⠋⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⠳⣮⣉⠉⠉⠉⠉⠉⢉⣩⡶⠟⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⢠⣴⣦⣄⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡤⢞⣛⣯⠭⠭⠽⠯⠍⠩⣽⣛⡿⢤⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠾⣫⠵⠚⠉⠀⠀⢀⠀⣠⠀⣀⠀⣀⠀⠉⠓⠮⣝⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢟⡵⠂⠀⢀⢠⠀⢉⠆⠐⠂⠓⠀⠙⠀⠁⢦⠆⢀⣀⠈⠙⢮⡻⣷
⠀⠀⠀⠀⢀⣴⡆⠡⠋⠀⠠⡺⡀⠁⠉⠀⢀⣀⣤⠤⠤⣤⣀⡀⠀⠀⠑⢍⡰⡄⠀⠹⣜⢷⣦⡀
⠀⠀⠀⠀⢘⡟⣴⠃⢀⢄⠐⠈⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠉⠃⢤⡀⠀⠉⠀⠀⠀⠈⢎⢿⡃
⠀⠀⠀⣴⣾⢱⠃⠀⣌⡙⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⣤⡀⠘⠎⢷⣦
⠀⠀⠀⢻⡏⡏⠀⠰⣨⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⣦⠰⣟⡉⠀⢷⠀⣹⠀⢰⢸⡟
⠀⠀⠀⣾⣷⡇⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⠀⠴⡖⠂⢻⠈⣧⠄⠀⠸⡄⢯⡤⠤⠘⠛⠁⠀⢸⣿⣷⡀
⠀⠀⠀⢿⣿⡇⠀⣀⣄⡀⢴⢒⠃⠸⣯⢼⡃⠀⢻⡀⠘⠇⠹⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⢸⣿⡿⠃
⠀⠀⠀⣸⡇⣇⠀⣏⠀⠀⠸⣏⣠⡄⠷⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠂⠆⠀⠘⢸⣇
⠀⠀⠀⠻⣿⣹⡄⠙⠓⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣌⠹⠀⢀⣏⣯⠟
⠀⠀⠀⠀⢀⣧⣳⡀⠀⠀⠀⣀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⢄⡙⠁⢀⡞⡼⠅
⠀⠀⠀⠀⠈⠿⣷⡹⣄⠀⠘⠜⢑⣄⠀⠀⠉⠑⠀⠒⠒⠒⠋⠉⠀⠀⢀⠐⡏⠊⠀⣠⢏⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣮⡳⣄⠀⠉⠊⢤⢦⢀⠀⣄⠀⠀⠠⡄⢰⣲⠁⠇⠁⠀⣠⢘⣵⣯
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢮⣝⠦⣄⠀⠀⠉⠀⠛⠀⠛⠐⠃⠀⠀⢀⣠⠴⣪⡵⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⢠⣼⣓⣲⠀⠠⠴⣶⣖⣒⣫⣽⠶⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠙⠾⠿⠋⠉⠛⠛⠁

⠀⢠⣾⣿⡟⠁⠀⣴⣿⣿⠃⠀⢀⣼⣿⡿⠁⠀⢠⣾⣿⠟⠀⢀⣼⣿⡿⠃⠀⢀⣾⣿⡟⠁⠀⣠⣿⣿⠟⠀⢀⣼⣿⡿⠁⠀⢠⣾⣿
⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠤⡠⠒⠖⡙⢲
⠀⠀⠀⠀⠀⠀⢀⡴⠤⠞⣒⣋⠬⠥⠐⢒⡉⢭⢱⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢒⣉⣒⣉⣒⣋⣒⣛⣞⣛⢹⠆
⠀⠀⠀⠀⠀⠀⢘⡆⢚⢩⠄⠂⠀⠀⠀⣠⡄⠀⠧⡆⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣮⢸⡧⠆⡴⠒⣦⠴⣲⡢⡸⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠱⡌⣴⠀⠀⣾⣿⣶⣿⡇⠀⠸⣱⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠓⠉⢯⣼⠂⢊⠓⠙⢹⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣄⢧⡇⠀⠈⠻⢿⠟⠀⠀⢀⡃⡇⣞⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡰⢞⣙⡎⡆⢸⠀⡀⢸⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⡘⣀⣀⠠⠄⢐⡒⠬⣉⣂⢰⠈⡵⠀⠀⠀⠀⠀⠀⠐⡧⢸⠣⡢⠭⠤⣣⢾⣶⡸⠸⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢣⢑⠂⠭⣌⠒⠈⣍⡑⠬⠌⡆⢣⡀⠀⠀⠀⠀⠀⠘⡖⠘⠒⠈⠿⠯⠥⠤⠭⠤⠞⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⡀⢌⣓⣂⠬⢅⡒⣂⣍⡥⠦⠎⠁⠀⠀⠀⠀⠀⠀⠉⠚⠙⠙⠚⠉⠚⠓⠚⠚⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢥⣉⠭⡤⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⡠⢆⣀⠴⣀⣠⢦⣀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⣶⠒⠒⠒⠒⠊⠑⠲⣑⡶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢒⣞⣥⠽⠿⣟⣷⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢾⠉⠛⢒⢂⣀⢀⠀⠀⠘⡗⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢛⣽⢟⡥⣀⣰⣒⣒⠉⠻⣾⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⣼⠞⡩⣌⡙⢶⡀⠀⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⠃⣩⢪⠛⣉⠉⠙⢕⢎⢪⣷⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣻⠐⡏⠌⠀⠀⢹⠨⡗⠀⠟⡧⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⡏⡆⡇⡇⠘⣌⢟⡹⠀⣿⠾⢺⣣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣻⠀⢹⣨⣒⣒⣩⡼⠉⠀⢎⡧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⢠⡱⣱⣀⠘⠁⣀⢜⡜⠀⡾⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣽⠀⠰⣒⣋⣙⣒⣒⡂⠀⢧⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣘⢷⣇⡈⠿⣿⡿⠓⠉⣠⣾⣡⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡼⢄⠀⠉⠋⠋⠙⠉⠁⡠⠱⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠽⢿⣗⣦⡴⠖⣛⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠳⠎⠉⠶⠋⠱⠞⠉⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠋⠁

⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀
⠀⣴⣿⣿⠏⠀⢀⣾⣿⡟⠁⠀⣠⣿⣿⠟⠀⠀⣴⣿⡿⠃⠀⢠⣿⣿⡟⠀⠀⣰⣿⣿⠏⠀⠀⣼⣿⡿⠃⠀⢠⣿⣿⠟⠀⠀⣴⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠈⠉⠉⠉⠉⠀⣠⣤⣀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣤⡀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⡏⡿⣿⣿⢿⣿⠛⠛⠛⢿⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢰⣿⣇⣿⣿⣿⣿⣶⡶⠶⠿⠿⠿⠿⠿⠇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣨⣯⣧⣿⣿⣿⣿⣿⢀⡐⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⢂⡀
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣶⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣡⣾⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣦⣍⠻⣿⣿⣿⣿⣿⠟⣩⣴⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⠓⣠⡙⠻⢋⣤⡘⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠟⣉⣴⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠿⢋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠻⠇
⠀⠀⠀⠿⠿⠿⠇⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗

⠀⢸⣿⣿⣿⠛⣿⡇⠀⢸⣿⣿⣿⡿⠇⠀⠲⠖⠂⣀⠒⡒⠒⠒⣒⠂⠀⣶⣶⣶⣶⣶⣶⡄⠀⡰⠂⣄⠀⡀⠀⢸⡿⠿⠿⠿⣿⣿⣿
⠀⢸⢟⣳⠿⣼⠿⡇⠀⢸⣇⢹⣿⠺⡆⠀⢠⠀⢀⡟⣆⢀⢃⡅⠑⡄⠀⣿⣿⣿⣿⣿⣿⡃⠀⣷⣿⣿⡀⡇⠀⢸⣥⣂⣙⣋⣿⣿⣿
⠀⠘⠉⢣⣶⢿⠀⠀⠀⠸⡏⢠⡟⠻⡇⠀⢨⠀⠀⠁⠙⠂⠈⠑⢀⠁⠀⠻⠿⠿⠿⠿⠿⠃⠀⣿⣿⣿⣿⡇⠀⢸⣍⠻⠿⣿⡿⠂⣠
⠀⠀⠠⠈⣷⠻⡆⠀⠀⠀⠙⠛⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠋⠛⠉⠉⠉⠉
⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⣀⠀⠘⣋⡇⣁⡉⠉⠘⢣⣿⡁⠀⢸⣏⢠⣤⡎⠃⠀⠂⣤⣶⣄⠀⠀⠀⣶⣶⣷⣒⣖⡖⢲
⠀⠀⠀⠀⠀⣀⣤⡶⣿⠁⠀⠄⠀⠤⡄⡁⠀⢰⣯⣵⢁⢈⣶⢀⡹⠾⠃⠀⢸⣿⣾⣿⣧⠄⠀⢸⣷⢁⣿⠆⠀⠀⣿⡿⠖⠈⠉⠹⠹
⠀⠀⠀⠐⠛⠛⠙⠋⠛⠀⠀⠀⠈⠀⠃⠋⠀⠀⠋⠋⡀⠉⠁⠀⠀⠐⠀⠀⠘⠉⠛⠛⠛⠀⠀⠀⠉⠘⠋⠀⠀⠀⠙⣡⡄⠀⠒⠘⠙
⠀⣤⠀⢀⡤⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣜⠇⠀⠀⠀⠀⢀⡖⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⡳⠁⠀⢠⠆
⠀⣿⠠⢊⢡⣤⠄⡾⠁⡤⣀⢀⡤⡀⢠⡤⠀⠀⣼⢋⠀⣼⠋⣠⢀⡀⡴⠄⣼⣣⡄⠀⠀⢸⡍⠀⠀⢀⡀⠀⢀⢰⢟⡤⣘⢡⡏⢀⣤
⠀⡿⠁⠳⠚⠙⢺⠡⠺⠞⡧⠾⣸⠃⢿⠄⠀⠀⠟⠼⢻⡇⡰⠷⢻⠴⣇⠘⠟⠧⠀⠀⠠⣏⠜⠀⠀⠸⠰⣇⠞⠛⠸⡷⠓⣿⡀⢿⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢠⠀⢈⡴⠀⣀⠀⢰⢃⡀⣰⠃⠀⣀⣀⣀⡀⢰⢃⡀⢀⡀⡀⠀⡀⢰⠃⣀⢀⣀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠓⠃⠉⡟⠉⠇⠈⠛⠇⠀⡜⠊⠉⠀⠟⠇⠙⠙⠻⠉⠃⠀⠞⠧⠈⠙⠁⢙⡟⠋⠜
⠀⠀⣤⣤⣤⣤⣤⣤⡄⠀⢀⠀⣠⣄⡡⣄⠀⠀⠀⣤⠀⣤⣄⢠⠄⠀⠠⠀⠀⠁⠀⠠⣄⣤⡄⠀⣤⣤⣄⣤⣤⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⣿⣷⣿⣿⣿⣭⡗⠀⠀⠀⢘⣿⡧⠀⠀⠀⠀⣿⣤⣿⠷⣿⠃⠘⠁⠀⠀⠀⠒⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠉⠉⠉⠉⠉⠉⠃⠀⠀⠀⡵⠛⢳⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⢸⣿⡟⠛⢻⣾⡆⢰⣶⣶⣶⣶⡆⠀⠀⠰⣶⣶⡄⠀⣿⣿⣿⣿
⠀⠀⣤⣤⣤⣄⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠃⠛⠀⢸⣿⡉⠀⣿⣿⡇⢸⣿⣷⣼⣿⡇⠀⠀⠀⠹⠟⠁⠈⠿⠿⠿⠿
⠀⠀⣿⡟⢹⣯⣼⢿⣧⠀⠀⣶⠶⣶⣶⣶⡄⠀⠀⢠⠠⣄⣴⣄⠀⢸⣿⣷⣀⣹⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠐⠊⠀⠀⠀⣀⣀⣀⣀
⠀⠀⣿⣇⣾⡟⡁⢸⣿⠀⢰⣿⢉⣿⡍⣿⡇⠀⠀⢸⣾⣿⣿⣿⠀⢸⣿⣟⢫⡈⣿⡇⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣿⣿⡟⣿⣿⡼⣄⠀⠀⣯⠀⠛⡃⣽⡇⠀⠀⢼⣿⣿⣿⣿⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠄⠀⠀⠀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣐⠀⣀⣤⣽⣴⣾⡄⢠⢀⣤⣼⠃⡐⠂⡀
⠀⠀⠀⠀⠀⠀⢠⡤⢤⣄⡀⠰⢆⣽⣷⢿⣿⡿⠿⠿⠿⢿⣿⣯⣸⣗⡘⢫⡷⡑⠢⠐⣒
⠀⠀⠀⠀⠀⠀⠀⡜⣸⣷⣿⣿⡮⠟⠋⣁⣤⣤⣾⣶⣶⣥⣤⣌⡙⠻⢿⣿⣿⣿⣶⢶⠿⠘
⠀⠀⠒⠀⠤⢸⣶⣤⣽⣿⡿⠋⣢⣶⢟⢻⢙⡸⣝⣗⣷⢛⡞⡩⣿⢷⣄⡙⢿⣾⣿⡏⠀⢀⡀
⠀⠀⠀⠤⣤⠀⠻⢿⣿⠏⣠⣾⡫⢽⣦⣿⣿⣿⡿⣿⣿⣿⣷⣿⣵⣟⣿⣷⣄⠙⣿⣶⣶⣿⣦⣤⡀
⠀⠀⠀⢀⠀⠉⢻⡟⠏⣰⣿⡍⣩⣾⣿⢿⣛⣿⣤⣿⣧⣿⣿⣿⠿⠿⠟⠛⠛⣂⣈⡉⠉⢿⠟⠉
⠀⠀⠀⠀⢡⣶⣶⡞⢀⣿⣿⣿⠿⠿⠿⠛⢛⣛⡉⠉⣭⣤⡀⣆⠀⡆⢿⣋⡁⢻⠉⢻⠀⢸⣷⣤
⠀⠀⠀⡜⠿⠙⠉⢁⣄⠀⢴⠶⣆⢺⠛⣷⢹⣉⡹⠘⡇⠀⣷⠸⣦⡇⢸⣏⣁⢸⣆⣼⠃⠘⣷⣤⣀⡀
⠀⠀⠀⠐⣹⡇⠀⢸⡟⣆⢸⡶⠟⢸⡟⠋⠸⡏⠻⣄⠳⠤⠟⠀⠙⢃⣈⣋⣩⣤⡄⢠⣶⣶⣿⡿⠿⠆
⠀⠀⠀⠘⣟⣧⠀⢸⠛⠻⠎⠇⠀⠀⣃⣀⣠⣥⣤⣴⣶⣶⣾⣿⣿⣿⡟⣛⢿⡿⠁⣾⣿⣧⡀⢀⠀⠆⠁
⠀⠀⠀⠀⡈⢻⡤⢄⣤⠀⢶⣿⣿⣿⣿⣿⣯⣿⣽⣿⣉⣿⣶⣿⣿⠫⣷⢮⡿⠃⣼⣿⣿⣟⠛⠂
⠀⠀⠀⠀⠉⠂⢩⣜⣶⣷⡄⠻⣿⣭⢢⡟⡟⡿⡿⡿⡿⡿⣛⢫⣓⣻⣶⠟⠁⢘⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠈⠃⠰⣿⣿⣿⣦⣌⠙⠻⢾⣿⣧⣷⣇⣧⣷⣭⣾⢿⠋⣁⣴⣿⣿⣿⣿⣛⡏⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣶⣬⣔⣉⣉⣉⣉⣉⣰⣬⣶⣿⣻⢿⢿⡿⠿⠏⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⡿⢛⣿⠏⠁⠹⠎⠡⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟⠁⠘⢫⡹⠙⣿⡿⠀⡛⠂⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⠀⠈⠱⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⢠⠠⠰⢠⢀⠀⠀⠀⠀⠠⡂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠀⢀⠀⠀⠂⠈⣂⠀⠀⡈⠐⠀⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⣀⣀⡠⠬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⢀⠀⠈⠐⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⡁⠀⠁⠄⠀⠀⠂⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⢁⢀⠀⠉⠁
⠀⠀⠀⠀⠀⠑⠄⠄⠐⠈⠀⠀⠀⠀⠂⠀⠄⠀⠐⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⡠⠒⢉⢉⠉⠑⠠⡀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠠⠐⠀⠆⠀⠄⠀⠀⠀⠀⠀⡀⠐⠐⠰⠤⢀⠀⠀⠀⠀⡠⣀⢀⣁⣀⡀
⠀⠀⠀⠈⠀⢀⢀⠀⡀⡀⠀⠘⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⢀⠀⡀⡀⠀⠁⡀⠀⢀⠠⠍⠀⢀⠁⠀⠡⠆⠀⠀⠀⡀⢈⠈⠈⢉⠁⡈
⠀⠀⠀⢄⠈⡈⠀⠃⠁⠀⡀⢠⠁⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠄⠨⠈⠁⠉⠉⠀⠁⠀⠐⠀⡀⠈⢈⣀⡈⢀⠀⠀⠀⠀⠁⠈⡀⡀⠈⡀⠁
⠀⠀⠀⠈⠂⠁⣀⣀⡀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠐⠀⠀⠁⢁

⠀⠀⠀⠀⠀⡀⠠⠀⣄⠀⠀⠀⠀⠀⢀⢠⠄⠄⠤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⣀⣀⣀⢀⣀⣀⠀⠀⠀⠈⠀⢠⡄⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠉⠐⠄
⠀⠀⠀⠀⡀⠀⠀⠠⠖⠶⠀⠦⠀⠶⠀⠀⠀⠀⠀⠐⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠋⠈⠘⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⢠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⣈⠉⠭⠭⣍⡒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠐⠈⠁⠀⠀⠀⠀⠀⠙⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠔⠁⠀⠀⠀⠀⠀⣀⡤⠴⠒⠲⢦⡄⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠔⠁⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠊⠀⠀⢀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠟⢻⡄⠀⠀⠀⡀⢤⣔⣲⣤⣉⠉⠛⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠶⣶⣺⣷⡖⠀⠀⠀⠰⢋⡽⠾⢿⡧⣤⢤⣤⣾⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠀⠀⠀⠸⣿⡇⢀⣀⢹⣧⣤⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡍⠻⣝⠳⠟⠀⠀⠀⠠⠿⢿⣯⣿⠰⠛⠓⠚⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⡀⠀⠀⠀⠀⠀⠐⡇⢠⣀⡿⠃⠀⠀⣠⡄⠀⠀⠀⢁⡝⠁⣀⡀⠀⠀⠀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⢴⢦⠻⣄⠀⠐⢷⡀⠀⠙⠦⠭⣄⠀⠀⠀⠋⠻⣦⡀⠀⠘⠳⠄⠈⠙⠓⠀⠂⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢑⡵⠛⠋⢀⡀⡖⡀⠻⣖⠀⠉⡇⣿⠀⠀⠀⠀⠀⠹⣿⢷⡤⣄⣀⣀⣀⠠⠐⠒⠒⠀⢨⡇⡖⣒⣒⣒⡒⠒⠒⠒⠂⠀⠀⢠
⠀⠀⠀⠀⢀⠔⢀⡠⠀⢈⠠⠐⣂⣀⠺⡵⡜⢜⢄⠙⣦⣀⡻⠿⠒⠀⠉⣿⠀⠀⠘⢦⡙⢦⣄⡼⢻⡀⠀⠀⠀⣠⠊⢸⡗⣁⡀⢴⠃⠀⠀⣴⣷⡖⢀⡎
⠀⠀⠀⣔⠁⠀⠈⢷⡘⢄⢀⣎⠘⠫⡆⠱⡘⣌⢫⢆⠈⢷⡀⠀⠠⡟⣆⢻⠀⠀⠀⠈⠳⣤⠦⠶⠺⣧⠀⢀⠜⠁⠀⣼⠃⠀⠿⠤⠃⠀⠐⢾⣏⠁⣸⢇⣀
⠀⠀⠀⠈⠛⣦⠔⠉⠹⣎⠋⠛⠒⠀⠀⠀⠙⠌⠦⠃⠁⠀⠻⣄⠀⡁⡟⢸⣄⡀⠀⠀⠀⠈⣙⠂⢈⠁⢠⣇⣀⠔⡁⣾⢀⡤⣤⣀⠀⠀⠻⢼⠉⢠⡟⠲⣭⠙⡻
⠀⠀⢀⠠⠊⠴⠋⠒⡒⠘⢧⡀⢀⠀⠐⠒⠒⣦⠤⣄⠀⠀⠀⠙⢶⡴⡇⣀⡆⣏⢷⠒⠲⢤⡈⠉⠁⢠⠧⠈⣁⠀⢻⡏⠘⢧⡒⠉⠙⠳⣄⠀⠀⡾⠢⢞⡁⠊
⣠⣂⣁⣐⣂⣤⡴⠎⠀⢀⠜⠋⠀⠀⠀⠤⠤⣤⣤⠊⠀⠐⡈⢀⡈⠻⣆⡇⣷⠹⣾⡆⢠⣎⣹⣗⣶⡇⡆⠀⣼⡉⡉⠓⠒⠲⢽⣶⣄⠀⠈⢷⣼⠃⡴⠊⢉⠱⠖
⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⢀⠔⠀⠀⠀⠀⠺⠌⢹⠆⠀⠀⠒⠀⠄⠈⢷⣼⡆⢻⡷⠊⣽⢷⡌⠔⢿⡱⠀⣼⡙⠳⠖⠤⣄⠀⠉⠻⣆⠀⠀⢻⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⣤⢤⣭⣁⣈⡀⠀⠀⠀⠀⠁⠀⣤⠞⠉⢳⠈⢷⣤⡋⢠⣦⡄⠈⣧⠰⣮⡛⢧⣄⡀⠈⠳⢦⠀⠀⢀⡖⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡄⠀⠀⠀⠀⠀⠀⠸⢄⠝⠀⠀⣀⡴⠚⠁⠘⢧⡀⠘⣇⠈⠿⢿⣄⠹⣿⣦⠸⡆⢻⣿⣦⣀⠙⢳⣄⠀⠀⢠⠎⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣠⠶⠭⠤⡤⢤⣀⣴⡞⠉⠀⠀⠀⠀⠈⢻⡌⠁⠀⢀⠜⢉⢳⡌⢿⠀⢹⡀⢿⣷⠉⢷⡀⠀⠠⠂⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣈⣁⡰⢫⠏⣿⠀⠀⠀⠀⠀⠀⡠⠛⠀⠀⠘⠦⣜⢷⢝⢦⡀⠀⢷⠀⡡⠂⢹⡇⠀⠀⠀⠀⡰⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⣻⠁⠈⣧⠸⠘⠁⠀⠀⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠉⠻⣄⠘⠋⠀⠀⣴⡷⣄⣀⠀⣴⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣧⡀⠧⠐⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠈⠳⣄⠘⢧⣀⠜⡴⠈⠛⠦⠀⣉⣄⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⢦⣍⡒⠀⠁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠃⠀⠀⠈⠳⣌⠋⣸⠁⠀⠀⠀⣼⡻⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⡴⡆⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠳⠋⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⡠⠋⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧⠀⠀⠀⢰⡶⢤⣄⡀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠊⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠉⠀⠀⠀⠀⠻⣄⠈⠙⢦⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠙⣆⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣻⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠀⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⢠⡄⡀⠀⠂⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⠋⠉⠉⠉⠒⠐⠒⠒⠁⠉⠩⠟⡲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⠯⠂⠁⠀⠀⢠⡀⠀⠀⠀⠀⠀⢰⣦⣀⣀⠀⠈⠐⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠊⠁⠀⠀⠀⠀⣤⣾⣿⣿⠏⠀⠀⠶⣿⣷⣿⡏⠀⠀⡀⠀⠈⠐⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡫⠈⠀⢀⣧⣄⣠⠀⠀⠈⣿⠿⠿⠆⠀⠀⠀⠸⠟⠙⠓⠀⣠⣿⣦⣴⠂⠀⠑⠙⣆
⠀⠀⠀⠀⠀⢀⡼⡕⠁⠀⠲⢿⣿⣿⣏⠀⠀⠀⢀⡀⠤⠄⠀⠀⠀⠤⢀⡀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠈⠊⢣⣀⡀⣀
⠀⠀⠀⠀⠀⡾⠌⠀⠀⡀⠀⠸⠟⠉⠉⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠋⣀⣀⣤⠤⠶⠚⠋⠉⠁⠀⠈⡆
⠀⠀⠀⠀⣸⠁⠀⢀⣰⣿⣦⡶⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠒⠛⠉⠁⣠⡄⠀⠀⢸⡆⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢠⡇⠁⠀⠉⢻⣿⣿⣧⠀⠰⠁⠀⠀⢀⣀⣤⠤⠶⠒⠋⠉⢁⣀⠀⠀⢶⡄⠀⠀⣿⠻⣦⠀⠀⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⢸⢧⠀⠀⠀⠘⠁⣀⣀⡤⠷⠒⠚⠉⠉⢀⣤⣤⣤⠀⢻⡆⠘⣿⠷⣤⡘⣧⠀⢰⣿⠶⠾⢷⣄⠸⠷⠿⠛⠃⣀⡽
⠀⠀⢀⣸⣼⠤⠶⠒⠛⢉⣁⣤⣤⣤⠀⢸⡆⠀⣿⠁⢀⣠⣄⠈⣷⠀⢹⡆⠀⠻⣿⠄⠸⠃⠀⢀⣀⣀⡤⠴⠖⠞⡏⠉
⠀⡞⠉⠁⢠⣴⠶⣦⡀⠘⣿⢉⣀⣼⠇⠀⣿⡀⢻⣦⣈⣩⡟⠀⠺⠆⠈⠃⣀⣀⣤⠤⢶⠒⠛⠉⢁⡀⠀⠀⠀⠃⡇
⠀⢹⡀⠀⣿⠁⠀⠈⣿⡄⢻⡟⠉⠻⣦⡀⠸⠇⠀⠈⠉⣉⣀⡤⠴⠖⠚⠋⠉⠀⠀⢀⠆⠀⢻⣶⣿⣇⣀⠀⠀⢸⠃
⠀⠈⡇⠀⠻⣦⣄⣴⡿⠀⠘⠃⠀⢀⣀⣠⠠⠄⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠾⠿⣿⠟⠉⠀⢄⡟
⠀⠀⢹⡀⠀⠈⢉⣁⡤⠴⠖⠚⠋⠉⢀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣀⣀⣠⡄⠀⠈⠀⠀⡐⡼⠁
⠀⠀⠀⠑⠛⠉⢯⡣⠀⠀⠀⠀⢿⣶⣿⣆⡀⠀⠈⠁⠒⠠⠤⠀⠀⠒⠊⠁⠀⠀⠀⢹⣿⣿⣷⡤⠀⠀⢔⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠱⣔⢄⠀⠠⠿⠿⣿⠋⠁⢀⣀⣴⡆⠀⠀⠀⢰⣤⣴⣧⠀⠀⠐⠛⠛⢿⠁⠀⡠⣢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠄⠀⠀⠉⠀⠀⢸⣿⣿⣷⠶⠀⠀⢠⣿⣿⣿⠟⠂⠀⠀⠀⠀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣀⠄⡀⠀⠉⠉⠻⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠠⣐⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣠⠀⠀⠀⠀⠀⠤⣀⣀⣀⠬⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢉⠉⠉⡡⠀⡄⠈⠈⠉⠹⠡⣈⡙⣇⠀⠀⠀⠀⠃⠒⠠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠒⣀⣤⡤
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠙⠲⠤⠀⠀⠀⠠⢍⣉⣒⣀⣀⣀⠀⠀⠀⠀⡈⡀⠀⢐⠀⠀⠀⠀⢸⠇⠈⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠐⠒⠀⠤⠤⠀⠈⡉⠙⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠤⠠⠄⠒⠒⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠐⠐⢄⠀⠀⣀⠔⢱⠧⠅
⠀⠀⠀⠀⠆⠀⢀⠀⡀⢰⠁⡀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣤⠄⠀⠀⠠⠈⠒⠀⠀⠀⠈⠁⠀⠀⠀⢰⠶⣟⠁⣃⠈⣦⠊⠁⠀⡞⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡄⣀⡀⠀⠇⠀⠨⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣾⢷⡜⠀⢹⠀⠀⢠⣇⡀⠀⠃
⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠛⠁⠀⢸⠀⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⡄⢈⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠒⠲⠂⠂⠉⠉⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡀⠠⠦⠄⢖⠐⠂⠄⠒⠂⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢁⣉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⢩⡃⠀⠆⠄⠀⠀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠄⠤⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠈⠓⢦⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⡀⠀⠀⠀⠀⠀⠈⢓⡴⠋⠙⢯⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠔⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠙⢦⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⢙⡆
⠀⠀⠀⠀⣠⠞⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠾⡀
⠀⠀⠀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡜⠃⠀⠙⣆
⠀⠀⠀⡇⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡴⠋⢀⣠⠔⠋⢹
⠀⠀⢠⡇⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣠⢎⡠⠖⠉⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠈⠓⢦⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡼⠗⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡥⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡤⠖⠒⠤⣠⡴⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠠⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢹⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⣸
⠀⠀⠀⠸⣤⣚⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣙⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⢀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⣼⣯⣭⣷⠀⡞⢥⠍⢣⠀⡀⠀⠀⠀⢰⣿⣭⣿⡄⡘⣁⣄⠙⡀⣌⣤⣤⡐⠀⠊⡀⠈⢱⢠⢡⣶⡆⠂⢠⣤⣶⣤⡄⢠⣁⣀⣁⠀⣘⣛⣛⡇
⠀⠘⠷⠶⠋⠀⠑⠦⠴⠃⠀⠀⠀⠀⠀⠈⠻⠶⠟⠀⠘⠦⠥⠞⠀⠀⠄⠤⠈⠀⠳⠤⠤⠊⠀⠂⠉⠁⠁⠀⠈⠉⠀⠀⠘⠢⠤⠒⠀⠘⠿⠿⠃
⠀⠀⣀⢀⠀⠀⣴⣶⣶⣄⢀⡔⠶⠖⠄⢀⠀⡀⠠⡀⣠⠚⡟⢢⡀⠠⣴⢤⡄⠀⠐⢀⡀⠄⢀⣔⣒⣒⡄⠠⣀⣀⣐⠀⢠⠒⠁⢢⠀⡴⣂⣅⢢
⠀⠀⠙⠂⠀⠀⢿⣶⣿⠞⠘⢄⣈⣠⠆⠀⠘⠓⠀⠀⠹⣌⣉⡸⠁⠠⠾⠽⠇⠀⠈⠑⢂⠄⠘⢟⠛⣛⠇⠠⡘⠂⢁⠀⠹⣈⢘⡲⠁⠣⡸⢟⡸
⠀⢀⣠⣤⡀⠀⢀⠀⠀⠀⠀⣠⣬⣄⠀⢀⠀⠀⡀⠀⢀⣤⣅⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⠀⠀⠀⣁⠀⠀⠀⠀⠄
⠀⣾⣭⣭⣿⠀⠁⢨⠄⠀⢸⣿⣽⣿⡇⠈⠒⠒⢈⠀⣿⣮⣿⡷⠀⢠⣷⠌⠀⢸⡵⠦⣽⠀⠀⠀⠀⠀⠀⢈⡇⠷⣁⡇⠀⠀⠑⠀⠀⠀⠀⠐
⠀⠈⠛⠛⠋⠀⠈⠛⠋⠀⠈⠛⠛⠋⠀⠈⠂⠀⠂⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠂⠀⠈⠙⠐⠋
⠀⢀⣔⣭⣷⣄⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣶⣛⣷⡄⢀⣔⣬⣖⣄⠀⣤⣲⢔⣄⠀⠠⠞⠛⡦⠀⢀⣤⣤⣤⣤⡀⢠⣖⣉⠲⡄⢀⣴⣤⣔⣄
⠀⠉⠫⠭⠭⠋⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠳⠭⠿⠃⠈⠩⠭⠭⠉⠀⠉⠽⠯⠉⠀⠘⠷⡤⠟⠀⠈⠉⠉⠉⠉⠁⠈⠦⡥⠜⠁⠈⠫⠭⠍⠋
⠀⢠⣤⣄⠀⠀⣀⣄⡀⠀⢀⠤⠤⡄⠀⢀⣤⣄⠀⠀⢀⣤⣄⠀⠀⣠⣤⣄⠀⠠⢶⠆⠀⣤⠀⣀⣀⣀⣤⡄⣤⠤⠤⢴⠄⠀⠀⠀⠀⠀⠄⡄
⠀⢯⠀⣸⠇⠀⢿⣿⠗⠀⢿⡒⠒⡽⠀⠸⣿⣿⠇⠀⢿⣽⡿⠀⠸⣯⣯⡿⠀⠈⣭⡁⠀⢠⡴⠿⠿⠿⠇⠰⠶⠶⠶⠆⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠸⠿⠇⠀⠀⠷⠰⠇⠀⠈⠙⠋⠁⠀⠰⠗⠿⠀⠀⠾⠎⠷⠀⠀⠈⠉⠁⠀⠐⠶⠗⠀⠠⠖⠶⠲⠆⠰⠾⠶⠿⠿⡿⠷⠿⠿⠷⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣾⣋⣁⣀⣀⣈⣙⣷⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⠀⠀⠀⠀⠀⠀⣴⠛⢉⣭⣍⣉⠙⣦⠀⠀⠀⠀⠀⠀⣿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣀⣿⠀⠀⠀⠀⠀⢰⡇⠀⡏⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⣀⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠿⣤⡀⠀⠀⠀⠀⢷⣄⠙⠒⠚⠛⢒⡛⠀⠀⠀⠀⢀⣤⠿⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢶⣄⠀⠀⣠⡶⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⡆
⠀⠀⠀⡇⠘⠀⠁⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⣀⣠⣿⣿⣿⣷⡄
⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠿⢟⣅⠀⠻⠟⢻⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢇⠈⠉⠀⠄⢠⠜⣤⡀
⠀⠀⠀⡄⠀⠀⠇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡾⠖⠂⠀⡀⡀⡉⠈⠓⡦⡶⠤⣀
⠀⠀⠀⡇⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡤⣞⠋⢀⠇⠀⢸⠠⠁⠀⠈⠳⢄
⠀⡠⢶⢃⠼⡒⢓⣼⣶⡶⡶⡤⢶⡖⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡗⣿⡓⠹⠀⢴⡎⠀⡄⠀⠀⠀⠼⣇
⢸⠁⠀⠳⢾⡓⠒⠒⠐⠒⠒⡛⡛⠛⠊⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⠸⠛⣿⣷⠋⠀⠀⡀⡀⠁⠀⠀⠀⠀⠸⣦⡀
⢸⠀⠀⠀⠀⠹⡄⠀⠘⠲⠀⠑⠟⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠀⠀⡄⠘⠃⡀⠀⡼⢃⠁⠀⠀⡀⢀⠀⠠⠈⠛⣦
⢸⠀⠀⠀⠀⠀⡟⠒⠐⠒⠒⠛⠛⠛⠛⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣆⠀⠀⣷⠰⠀⠀⡴⠀⠀⣆⠆⢀⣿⣿⡓⠂⣀⠀⣿⣧
⢸⠀⠀⠀⠀⠀⠁⠀⠐⣶⠶⠲⠶⢶⡖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⡄⠉⠀⠀⠌⠁⢀⡄⠿⢬⣈⢹⣿⣖⠊⡭⠶⣾⣿⠁
⢸⡀⠀⠀⠀⠀⠐⠀⠴⡇⠘⠚⠘⠨⢠⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠹⣇⢻⣴⢲⠃⢀⠠⢠⣈⣀⣀⠀⠀⠟⠃⠀⠈⣙⣼⣿⠀⠢
⢸⡃⠀⠀⠀⠀⣼⠀⠀⠮⠖⠘⠶⠞⠞⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣾⣿⠈⢸⢠⣀⠸⡇⠀⠉⠉⠙⡳⠋⠓⠢⣄⣹⣿⠿⠷⢾⠄⠓⡦⣤
⢈⡇⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢈⣿⠳⢨⣀⠀⢠⣷⠰⠆⠀⡤⢄⣀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⣠⣯⡀
⢀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣇⠀⢀⣀⣀⣤⠚⠁⠈⢣⣀⣈⣉⣽⡿⠀⣸⠉⡉⠘⠉⠓⠄⣠⠀⠀⢹⣦⡾⠋⠀⠀⠀⠀⣠⣴⡞⠉⠀⠙⣄
⠀⠧⠤⣤⣄⣀⣼⣀⣠⣤⣤⠤⠤⢴⣚⠁⡠⠀⡉⢻⢤⣀⡈⢗⠀⠀⣠⣯⡿⠛⠋⠁⢃⣷⣾⠀⡀⠑⠀⠤⡀⠒⢀⡿⠲⠶⣦⣤⣤⣴⠞⠋⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠿⠟⠛⡟⠛⠻⣿⣿⠀⠀⠺⠷⠶⢄⡈⠉⠀⢀⡤⠷⠘⠧⡿⠟⠋⠀⢀⠎⠀⢸⣿⠛⠻⢿⣶⣆⡠⣀⠶⣿⣇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⡆⠀⠀⡇⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠙⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⡇⠀⠀⠈⠿⣿⣿⡆⠀⠹⣿⣷⣄⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⣠⡿⠃⠀⠀⠀⢠⡏⣻⣷⡄⠀⠘⣏⠛⢻⠇⠀⠈⢆⠀⠀⠀⠀⠀⣤⠞
⠀⠀⠀⣧⠀⠀⡇⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡻⠁⠀⠀⠀⠀⠀⡇⠁⠙⢿⣄⣤⣼⣿⣦⣀⣀⡀⠀⠱⢀⣀⡤⠞⠁
⠀⠀⠀⠟⠀⠀⡇⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠃⠀⠀⠀⠀⠀⣰⠛⡄⠀⠘⡋⠛⠿⣿⣿⣿⢿⣁⡤⠞⠉⠁
⠀⠀⠀⢠⠀⠀⠂⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⣠⣶⠟⠀⠸⡄⠀⢱⠀⠀⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠁⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡌⠀⠀⠀⣰⢿⡏⠀⠀⠀⢱⠀⠀⢧⠀⠀⡿⣿⡆
⠀⠀⠀⠘⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⠁⠀⠀⠘⠃⣾⠁⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀⠀⢠⣤⣤⣤⣄⠀⠀⣤⣤⣤⣤⡀⠀⣠⣤⣤⣤⡄⠀⢀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠙⢷⣶⠿⠉⠉⠉⠿⣶⡾⠏⠉⠉⠹⢷⣶⠿⠉⠉⠉⠿⣶⡾⠋⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠇⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠀⠀⠸⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡆⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⢰⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣠⡾⠿⣶⣀⣀⣀⣶⠿⢷⣆⣀⣀⣰⡾⠿⣶⣀⣀⣀⣶⠿⢷⣄⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠀⠀⠙⠛⠛⠛⠃⠀⠈⠛⠛⠛⠛

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠥⢀⠠⠄⠒⠂⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡔⣡⠄⠀⠀⠈⢢⡃⢫⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠁⠀⠀⠀⠀⠀⢐⠂⢢⢣⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣄⠀⠀⢈⢂⢢⢢⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⢄⣬⣧⠀⠠⡢⢣⡶⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠉⠀⢀⢜⢁⠍⠔⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣐⢅⠝⡡⣊⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢒⡥⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠡⢊⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠕⡪⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡨⡪⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣂⣀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠮⣰⠽⣇
⠀⠀⢔⡀⠀⠀⠀⣀⠨⠅⡢⣖⠒⠶⠠⠤⠤⣀⣀⣡⣤⣅⣪⣀⣭⣿⡀
⠀⠠⢊⣴⡂⠉⠁⠀⠈⠁⠈⠀⠁⠈⠀⠀⠀⠀⠨⠀⠉⠉⠛⢋⠉⢋⢻⡤⣀⣀
⠀⠀⠊⠒⠨⠉⠑⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠸⣾⣭⣲⣒⣂⣈⣢⣿⢏⣭⣽⣟⣛⣛⠒⡲⠶⠤⠤⠀⣀⣀⣀⠀⠀⠀⠤⢍⡑⠦⡀
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠖⠠⠤⢀⣉⡉⠉⠉⠩⠧⠒⠒⠂⠀⠐⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠉⠁⢉⣠⢶⡈⠂⠈
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠄⣀⣀⠠⠄⠀⣠⣤⢐⠠⠒⣀⡤⠲⢊⠫⢋⠈⠻⣆
⠑⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠠⣔⡺⡩⡓⠐⡡⡑⡡⠢⠈⠀⠈⣧⡀
⠀⠀⠉⠛⠛⠻⠿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠔⠜⠐⡄⠐⡔⠐⡑⡐⠁⡑⣐⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢎⠌⠌⢀⠀⢌⠌⡐⣠⣴⣷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢷⣖⣦⡤⣤⣀⣀⠀⠆⠀⠀⡄⡄⣐⡆⠀⠸⣃⢎⠜⢀⣁⡶⣞⡿⠗⠋
⠀⠀⠀⠀⠀⠀⣀⠠⠕⢒⠈⠍⠐⣂⠩⠀⢒⡀⠀⡀⠀⠀⠀⠈⠉⠛⠓⠛⠻⠿⠿⠶⣶⣆⣤⣤⣄⣀⣧⠶⣞⡯⠑⠉⠁
⠀⠀⠀⠀⠀⠀⠐⣈⠡⠀⢂⠈⠥⢐⡂⠭⠐⡈⠍⠐⢊⠠⢄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠔⠂⡩⠄⢒⠈⠀⠀⠈⠁⢀⠀⠅⠒⠈⠠⠐⡀⠡⠐⠂⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠤⠐⡀⠤⠐⡀⠤⠐⡀⠠⠐⢂⡀⠤⢐⠨⠀⡂⠭⠐⣂⠠⠄⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠄⠒⡈⠤⠄⡒⠬⢑⡂⠥⠐⣂⠡⠔⠂⡉⠤⠐⠮⢑⠪⠑⣊⠡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠊⠥⠐⣈⠡⢐⡀⠁⠐⠒⠈⠁⠂⠁⠀⠨⠄⢂⡉⠤⢒⠈⠄⣒⠡⢔⡂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⢀⠤⢂⡡⠔⢊⡉⠄⢂⡩⠄⢐⡩⠄⣂⠥⠑⡈⠅⢘⡁⠄⢒⡩⢐⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠊⠡⠔⣈⠠⠐⡈⠅⢂⡡⠄⣊⠥⠒⣉⠥⢒⡈⠄⢊⠡⢔⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠔⡪⠔⣊⠅⢔⣂⠥⠖⡊⠥⠒⠈⠄⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠊⠡⠒⡈⠅⢂⡡⠔⣊⠕⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡩⠀⡠⠤⠀⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡰⢫⡤⠀⠀⠀⠱⣘⡎⣏⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠟⠀⠀⠀⠀⠀⠈⠜⠌⠌⢧⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⢎⠎⢌⢎⢎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣎⢉⢳⡀⠀⢨⢪⢎⢮⢪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠈⠛⠛⠁⠀⡵⡡⡻⢡⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⡪⣬⠎⡴⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡪⢞⡡⣪⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠨⠔⡩⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣋⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠔⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣪⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⢋⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡩⠥⠤⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⡺⢓⠥⣻⣄
⠀⠀⠔⡡⠀⠀⠀⣀⡈⠥⢐⡲⣒⢒⠒⠢⠤⠤⢄⣀⣸⢤⣤⣎⣵⣩⣮⣿⡀
⠀⠠⠊⣤⡒⠈⠉⠀⠀⠉⠁⠈⠀⠁⠈⠀⠀⠀⠀⢠⠁⠈⠉⠙⠛⡛⡉⣍⢷⡤⣀⣀
⠀⠀⠌⡐⠄⠍⠁⠒⠒⠠⠤⣀⣀⠀⠀⠀⠀⠀⠀⠘⢦⣂⣕⡕⠔⡕⡘⣬⣿⣏⣙⣛⣛⣛⠿⢖⣲⣶⡤⠤⠄⣀⣀⣀⠀⠀⠀⠀⠤⢌⡑⠢⣄
⠀⡌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⠤⢄⣀⣀⠉⠉⠙⠛⠛⠛⠓⠲⠒⠂⠤⠤⠤⠤⠤⠤⠤⠀⠈⠀⠀⠀⠉⠉⠁⠒⠒⣤⡌⠓⠀⠁
⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠂⠤⠤⢀⣀⣀⠀⠀⠀⣀⠀⠀⣀⡀⠄⠠⠤⠀⣀⢤⠖⠞⠝⠑⠟⣦
⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠢⠬⠍⣉⡠⡴⠺⠏⠣⢂⠮⠋⠌⢀⠋⡜⢳⡀
⠀⠈⠻⠿⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⡑⠢⠡⢊⠁⢂⢢⠈⡌⡐⡀⡹⣆
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⣰⣰⣁⣥⣡⣀⢂⣌⣔⣸⣴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢶⣦⣤⣤⣄⣀⡀⡀⠀⠀⠀⠀⠀⣀⠀⠀⢀⢀⣀⢀⢀⢹⡝⡽⣳⣿⣿⣿⣫⣿⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⣠⠠⢀⢀⡤⠀⠀⠀⠀⠀⠉⠉⠛⠺⠿⢯⣷⣖⣶⣤⣬⣬⣉⣙⣙⠈⠛⠙⠙⢉⡉⣙⣉⡙⢛⡋⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠒⡂⠉⠅⠐⣀⠉⠀⢒⣀⡉⠀⣠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⠿⣿⣿⣖⣞⣛⣯⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠥⠂⠀⡀⠀⠀⣀⠨⠄⢒⡈⠄⠂⠈⠁⠀⡌⠴⠐⡀⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⠒⢁⠥⠐⠀⠀⠐⢀⠀⠂⠀⠨⠀⠂⢈⠅⠖⣈⡬⠐⢂⡁⠄⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠖⣈⠁⠐⠂⠩⠐⣀⠡⠄⠂⣈⠡⠄⠂⠨⠄⡀⠁⠐⢂⣉⠤⠖⣀⠠⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⢈⡡⠄⢒⠈⠁⢂⠈⠄⠒⣈⠠⠐⠂⣉⠤⢐⠆⠑⣂⠭⠒⣈⠡⠔⢀⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠒⢉⡠⠔⢈⡡⠄⢒⠉⠅⢀⠉⠀⣀⠡⠐⠂⣁⠤⠐⣊⠡⠔⣊⠥⢒⣉⠤⣂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢈⠤⢒⡡⠄⠂⣉⠤⠐⣊⠡⠔⢊⠡⠄⣂⠡⠐⣈⠥⠂⣉⠤⠐⣊⠥⢒⡡⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠒⠁⠤⢒⡉⠤⢒⡩⠄⢂⡡⠄⣂⠥⠐⢊⡡⠄⢂⡩⠔⣊⠠⢐⠨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢡⠔⡪⠥⢚⡁⠤⢒⡉⠥⢒⡩⠄⢒⠩⠔⣈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠡⠔⢊⠥⠐⣈⠤⠐⣈⠤⢒⠭⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠀⠂⠈⠙⠲⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠶⠿⢆⡀⠀⡐⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⣦⡤⠒⠊⢉⣙⣉⣙⣹⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢼⣿⣿⣿⢿⠿⠿⡿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⡃⠰⢶⡦⠀⢨⣤⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠁⠀⠈⠉⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠀⠀⠀⡀⠀⢈⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠁⠀⠌⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠠⢐⠂⠀⡴⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠩⠀⠀⠀⠑⠀⠀⠑⣶⡄⠆⠙⢶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠁⠀⠀⠀⠱⡀⠀⠀⡈⠀⢀⠙⣿⡓⠀⠲⠃⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠑⢄⠀⠀⠀⠙⠊⢉⡁⠀⠈⠰⣿⣿⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠈⠐⠢⢤⣈⠻⢀⠆⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⣀⣀⣀⣀⣀⣀⣨⡙⡎⠀⠰⠶⠒⣦⡬⢓⣒⡈⣉⣅⣤⠴⠙⡄
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⢀⠇⠈⠉⠉⠉⠀⠀⠀⠀⢀⡀⠇⠀⠘⠄⢼⠁⡈⠉⠀⠀⠀⠀⠀⠂⠀⠘⡄
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘⠁⠀⠀⠀⠤⠀⠈⠀⠀⢠⠀⠀⠀⠀⢤⠀⠁⡀⢲⠄⠀⠀⠀⢀⠀⠼
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢲⢏⠀⠈⠀⠠⣀⣀⣀⣧⠊
⠀⠀⠀⠀⣿⠠⠀⠀⠀⠀⠀⢀⡾⠀⠀⠠⡀⠀⣠⠖⠙⠍⠉⠉⠉⠉⠱⢦⠝⠀⢐⠀⠆⠀⠀⠀⠀⡩
⠀⠀⠀⢠⡿⠛⠉⠉⠛⢻⠶⠶⡷⠊⠀⢤⣈⠘⡅⠀⠂⠂⡀⠐⠀⣠⡄⠁⠀⠀⢿⢆⡠⢤⣤⡖⠊
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠓⣀⠶⠂⠀⠀⠂⠀⠊⠀⡆⠀⠀⠀⠀⣀⠀⠈⢠⠇
⠀⠀⠀⢻⠀⠀⠀⠀⡀⢄⠀⠘⢿⣥⠈⠉⠁⠩⠇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⣦⣤⣭⠴⠋
⠀⠀⠀⠈⠓⠦⣄⣳⣌⣦⣦⣔⣀⠏⠙⠂⠒⠺⠶⠆⠀⠀⠀⠒⠀⢘⠁⠈⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠆⠘⠒⠒⠊⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⡀⢀⣀⢀⣀⢀⣀⠀⣀⠀⣀⠀⣀⠀⣀⡀⣀⡀⢀⡀⢀⡀⢀⡀⢀⣀⢀⣀⢀⣀⠀⣀⠀⣀
⠀⠀⠀⠀⠐⢇⠈⠁⠈⠉⠈⠉⠀⠉⠀⠉⠀⠉⠀⠉⠁⠉⠁⠈⠁⠈⠁⠈⠁⠈⠁⠈⠉⠈⠉⠀⠉⠀⠉⢀⠕
⠀⠀⠀⠀⠐⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵
⠀⠀⠀⠀⠰⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵
⠀⠀⠀⠀⠰⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲
⠀⠀⠀⠀⠠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢
⠀⠀⠀⠀⠠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢
⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥
⠀⠀⠀⠀⢀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅
⠀⠀⠀⠀⠀⠑⠊⠑⠚⠙⠚⠙⠒⠉⠒⠉⠒⠉⠒⠉⠓⠋⠓⠊⠑⠊⠑⠊⠑⠊⠑⠚⠙⠚⠙⠒⠉⠒⠉⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣾⣷⣾⣷⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⣿⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⢸⣿⠈⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠷⣦⣹⠇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠸⣏⣴⠞⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⢶⣄⣤⡾⢷⣤⣠⡶⠛⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠾⠋⠈⠻⠶⠶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠠⠐⠴⠒⠢⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⢠⠄⠖⢠⠀⠀⢸⣇⠀⠀⠀⠀⠀⢠⠂⢀⣀⣀⣤⣀⣀⡈⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣶⣶⣶⣶⣶⣶⣦⠀⢙⣷⠄⠀⠀⠀⡄⠀⢠⣄⣬⣭⣥⣤⡄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢠⡄⡀⢠⠀⠀⢠⡟⠁⠀⠀⠀⠀⠃⠀⢈⣉⣉⣭⣉⣉⡁⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⠀⠁⠀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠁⠄⠀⠉⠉⠉⠉⠉⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢷⣤⡾⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠠⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⣴⠞⢉⠀⣀⠀⣤⡄⢀⣀⠀⡉⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⣟⣿⠀⠀⠀⠀⠀⠀⢸⡏⠀⢈⢀⠉⡁⠉⠁⣈⢉⣈⡁⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣧⣿⣿⣿⣿⡿⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠤⠤⠤⠤⠤⠤⠄⠀⣰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠒⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠑⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⣠⣤⣤⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠠⢂⠹⠄⠌⠐⢄⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣤⣭⣭⣭⣤⡄⠙⣦⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⠤⠤⠖⣶⣚⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠉⠛⠛⠛⠛⠃⡼⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⡷⠶⠖⠛⢛⠏⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣄⠈⣀⡤⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⠈⠢⢀⣀⡆⠤⠊⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⠤⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⡀⠀⠀⡀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠄⢁⣠⣤⣤⣄⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢠⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠿⠿⠿⠿⢿⠷⠀⠀⠀⠓⠀⠀⠀⠀⠀⠃⠀⠀⢠⢦⣤⣤⢤⣤⣤⡄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠘⠛⠓⣐⣛⣊⣂⣈⣘⠒⠓⠁⣘⠀⠀⠀⠀⠀⢆⠀⠀⠈⠋⠉⠋⠋⠉⠋⠁⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠙⠿⣿⣿⠟⠋⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠑⠦⢄⠀⠀⠀⠀⠀⠀⣠⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠐⠀⠤⠤⠀⠈⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⣠⠤⠚⠃⢉⠁⠚⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠆⠒⠛⠂⠴⢦⡀
⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠈⠁⠠⠘⢆⡀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢰⠞⡁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆
⠀⠀⠀⠀⢀⠃⠐⠀⢀⣀⣀⠀⡀⢀⠀⠀⠀⠀⢃⠀⠀⠀⣾⡿⢛⣉⣙⠻⡟⠉⡿⢰⣿⠃⢸⣿⡄⠀⠀⣴⠃⠀⠀⢀⣀⡀⠀⠀⡀⠀⠀⠀⠱⣦
⠀⠀⠀⠀⡛⢠⠀⠀⢯⣴⣬⣟⡇⢯⣴⡏⠀⢠⠈⡃⠀⢸⣿⡀⠿⠩⠟⡃⢰⢸⣷⣌⣡⠆⣾⣿⡇⠀⠀⡽⢠⠀⠀⢯⣦⢽⢾⠱⣧⣾⠁⢀⡄⣇
⠀⠀⠀⠀⢻⠀⠀⠀⠴⢥⠼⠹⠷⠦⠞⠀⠀⠀⢰⠃⠀⠘⣿⣿⡆⣰⣿⠇⣾⢸⡟⠹⢋⣼⣿⣿⡇⠀⠘⢷⠀⡀⠀⠾⠥⠞⠜⠰⠴⠃⠀⠀⠀⠟
⠀⠀⠀⠀⠈⢆⠁⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⡎⠀⠀⠀⢻⣏⣠⣭⣥⣾⣿⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠂
⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠆⢀⡴⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢳⣤⣀⠀⠀⠀⠠⠐⣀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠲⠂⠲⠂⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠒⠻⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣟⣿⣷⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠠⢠
⠀⠀⠀⠀⠀⣠⣾⣻⠿⠋⠉⠉⠉⠉⠛⢾⣽⣦⡀⠀⠀⠀⠀⠀⢀⡴⢒⠲⣄⣴⢠⡆⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⢿⣷⣶⣌⠐⡀
⠀⠀⠀⠀⢠⣿⣿⠋⣠⡤⣄⠀⣀⢀⠀⢀⢻⣿⣷⠀⠀⠀⠀⠆⠈⢳⡿⠚⡏⡿⠘⠞⣿⠁⠰⠂⠀⠀⠀⠀⠀⢡⣿⢟⣛⢻⡿⢿⢿⣿⢿⡌⠁
⠀⠀⠀⠀⣾⢻⡇⠀⢧⣶⠼⡟⡇⢿⢴⡟⠀⣿⢻⡆⠀⠀⠀⠀⠠⠿⠶⠞⠁⠇⠻⠞⠁⠀⠀⠀⠀⠀⠀⠠⠄⣿⣧⡛⣘⡁⡄⣿⣘⡁⣼⡇⠤
⠀⠀⠀⠀⢹⣿⣧⠀⡾⣥⠼⠹⠧⠦⠞⠁⢠⣿⣿⠁⠀⠀⠀⠀⠀⠐⠐⠐⠐⠂⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⢀⢹⣟⣘⣛⣴⣧⣇⣋⣼⣿⠇
⠀⠀⠀⠀⠈⢷⡟⣧⡀⠀⠀⠀⠀⠀⠀⣠⣿⣾⠏⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠤⠙⢿⣿⣿⣹⣿⣿⡿⢋
⠀⡀⠀⠀⠀⠀⠙⢿⣿⣷⣶⡶⣶⣶⣻⣿⠞⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⢉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⢾⠟⠺⠛⠺⠛⠺⠛⠺⠛⠺⠛⠲⠛⠲⠛⠳⠛⠳⠛⠳⠛⠳⠛⠳⠛⠳⠚⠳⠚⠻⠚⠻⠚⠻⠚⠻⠚⠻⠚⢿
⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣾⠂⠀⠀⢀⡠⠤⣤⡤⠤⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣾⠂⠀⣾⣻⢷⣄⡛⠁⠼⣿⢠⡿⠁⢸⣿⣶⣤⣄⡀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣾⠂⠀⠙⢷⣯⡿⣿⣷⣄⠁⠈⠁⠂⠞⠻⣿⣿⣿⠟⣰⠏⣄⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣼⠂⠀⠀⠀⠙⢿⣯⠿⣿⣷⣄⠀⠀⠀⠀⠿⠟⣿⠃⠎⣠⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣽⠂⠀⠀⠀⠀⠀⠙⢿⣹⠿⣿⣷⣀⡀⠀⣠⣾⣿⣶⣴⡏⢿⣿⠟⢻⡟⠀⣴⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣽⠂⠀⠀⠀⠀⠀⠀⠀⢨⡓⢾⢿⠁⠁⠁⠋⣿⣿⣿⣿⣧⠈⠁⠀⠀⠈⠒⠟⣠⣿⡑⢦⡀⠀⠀⠀⠀⠀⠀⢺
⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣙⠛⡿⢧⣤⡄⠀⠃⠀⠏⠐⠙⢀⡀⠀⣀⣀⡀⣀⣈⣘⠀⠿⠛⢄⠀⠀⠀⠀⠀⢺
⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⣿⡶⠋⠃⠀⣀⣀⣀⠀⢀⣼⣿⢿⣿⣿⣿⡿⣿⢿⢷⣶⣾⡟⢛⠿⠉⠹⣿
⣹⠂⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠙⠦⣧⠀⠀⠀⢀⣯⣼⡛⣿⣿⣛⡶⠿⠼⠶⠷⣿⣾⣿⠿⣿⢟⠀⢭⠇⠠⢿
⣻⠂⠀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣠⣤⣤⣤⣤⣴⣾⣭⣏⣻⣥⡜⠉⠁⠀⠀⠀⣴⠋⠘⢜⢀⣠⠶⡛⢫⠉⠙⢻
⣻⣏⣉⣭⣭⣭⣭⣭⡭⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠼⡾⠤⠀⠀⠀⠀⠀⣼⠁⠐⠂⢠⠟⠱⠀⠁⠀⠀⣰⣿
⣻⠧⠤⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠗⠒⠂⠀⠀⠀⢀⡇⠓⠒⢠⡏⠘⠀⠀⡀⢜⣶⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠊⠉⠉⠉⠙⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠠⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠠⡐⠾⠝⠓⠒⠒⠒⠀⠀⢷⠀⠀⡇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡀⠀⣠⠆⠔⠉⠀⠁⠀⠀⢀⣀⠀⠀⠀⠀⢀⣸⣄⡼⠁
⠀⠀⢀⡴⠊⠉⢳⡀⠀⠀⠀⠳⢄⣸⡀⢀⠀⠀⠐⠈⠁⠀⠀⠀⠀⢀⡤⠊⠀⡏
⣠⠖⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢹⠎⠙⠳⠶⠤⢤⣀⣀⣀⣤⢖⠁⠐⠀⢂⡗⠦⡀
⠹⡀⠐⠀⠄⡀⠀⠀⠳⡀⠀⢠⡞⢉⡄⠀⠀⠀⣶⣄⡄⠀⠀⠬⣸⡷⠈⠀⠈⡇⠀⡇
⠀⠹⣤⠒⠤⡀⠀⠀⠀⠙⢆⠘⣇⠀⢃⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⠃
⠀⢠⠇⡀⠀⠉⡆⠐⠂⠀⠀⠱⣌⠲⢬⣄⠀⠀⢀⣠⣼⣦⣴⣷⣦⣄⣀⢀⡟⠉⠁
⠀⢸⠀⠀⠀⠀⡇⠀⡆⠀⢀⠴⠋⠀⠀⡨⢧⣘⠻⠿⠿⠟⠛⠛⠛⢛⡩⢺⣦⡀
⠀⠈⣆⠀⠀⠀⠈⢦⣥⡞⠁⠀⣠⠔⠋⠄⠀⢁⠑⢄⡀⠀⠀⠀⢀⠜⠁⡜⠀⠉⠲⢄⡀
⠀⠀⠘⣆⣀⠤⠔⠚⠉⢻⡤⠚⠁⠀⠀⠈⠄⠀⠢⠀⠈⡹⠛⢟⡁⠀⣼⠁⠀⠀⢀⠀⠙⢦⡀
⠀⠀⠀⢳⡀⠀⠀⠀⢀⡰⠆⠀⠀⡆⠀⠀⠈⢄⠀⠹⡖⠉⠀⠈⠉⠛⠋⠀⠀⠀⠘⡄⠀⠀⠹⣄
⠀⠀⠀⠀⢳⡀⠒⠈⠁⠀⠸⠀⢀⠁⠀⠀⠀⠀⠢⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠙⡱⣁⠐⢠⡀⠀⠀⠀⠀⠀⠀⢠⠛⠉⢀⠉⢺⣧
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠤⠖⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠈⠂⢤⡀⠀⣀⣠⢿⠀⠀⠀⡀⠸⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡑⢢⡔⠉⠉⠀⠀⠀⠁⠈⠉⠁⡁⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣀⠀⠈⠉⠀⠀⠀⠉⠀⣤⣾⠋⢃⠐⡀⠀⠀⢶⣾⠾⠷⠟⠛⠂⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠒⠾⠃⠙⡄⠐⡀⠀⠀⠀⡀⠀⠀⠠⢤⠐⠂⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠉⠉⠈⠁⢒⠂⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠉⠉⠁⠀⠒⣀⣀⣈⣉⣉⠀⠉⠆⠀⠀⠀⠀⠀⠀⠁⢚⣀⣀⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠘⡄⠀⠈⠢⠤⠤⠒⡶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠈⠉⠉⢻⠀⠀⠀⠀⠀⠀⢨⡏⠉⠉⠁⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⣀⡀⠀⠀⠀⣈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠖⠒⠛⠋⠃⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠦⠴⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠒⠛⠁⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠘⠀⠀⠐⠀⠀⢀⡄⠉⠉⠐⠦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢠⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡄⠉⠓⢦⡀
⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⢰⡏⠛⠀⢰⡀⣦⢰⠂⠀⠀⠀⠀⠰⡐⠂⠀⡞⠁⡆⠀⡟⠲⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⡇⢈⡋⠉⠀⠈⣷⢹⡟⠀⠀⠀⠀⠀⠀⡅⠀⠀⣇⢀⡇⠀⡷⣆⠀⢸⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠉⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠈⠀⠈⠀⠈⣀⣀⣀⡀⢠⣄⣀⣴⣃
⠀⠀⠀⠀⠀⠀⠑⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠠⡴⠊⠁⠈⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢆⣀⡀⠀⠀⠀⠀⠀⠄⠀⠠⠤⣤⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠀⢀⢀⡤⢰⡶⠾⠛⠛⠀⠒⠒⠀⠠⠄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠶⠀⢠⣄⣀⡀⣀⡀⢀⠛⠒⠒⠒⠊⠒⠒⠒⣉⣉⣀⣠⣤⡴⠾⠛⠋⠉⢁⣀⡀⢀⣄⣀⣀⣀⡀⠀⠀⠉⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⢭⣍⣁⣉⣉⣉⠉⣁⡭⠤⠖⠉⠀⠀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠀⢀⣠⠄⠐⠒⠒⠛⠒⠛⠒⠂⠲⠤⡄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠒⠒⠂⠀⠈⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⢀⣀⣀⣀⣀⣀⣀⡤⠄⠖⠉⠁

⠀⠀⠀⠀⣀⣴⡿⠿⠿⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠈⠒⢤⡀
⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠎⢀⡴⠋⠉⠉⠳⡄⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣾⠀⠀⠀⠀⠀⢹⣤⣆
⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⢿⣿⣧⣀⡀⣀⣤⣿⣿⠇⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠤
⠀⠀⠀⠈⠙⢿⣶⣴⣦⣾⠟⠋⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠈⠱⢤⣀⣀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣀⡀⠙⢦⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⠛⠛⠛⠛⠀⠀⢧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠃⠀⠀⠐⠛⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣀⣀⣀⣀⠀⠀⡼⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⣀⣀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠛⠛⠋⢀⡴⠁⠀⠀⠀⠀⠠⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠶⠚⠉

⠀⠀⠀⠀⠀⢀⣠⣤⣤⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣥⣀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠁⠒⠤⡀
⠀⠀⠀⠀⣴⡿⠟⠋⠛⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠉⠉⠉⠻⠿⠆⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⣿⣿⡙⠋⠙⣛⣿⡟⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢀⣀⡄⡀⠀⠀⠀⠘⡀⠀⠀⠀⢀⠀⠀⠀⠀⠊
⠀⠀⠀⠈⠈⠛⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⢠⡄
⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠰⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠤⢄
⠌⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢳
⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣄⣁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠞⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠈⠀⣀⠜⠁⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠤⠚⠀⠙⠓⠒⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠠⡅⢀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠠⠀⠓⠘⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡄⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠂⠀⠠⠀⠀⠀⠀⢀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⡀⠐⠀⠀⠄⠀⠁⢀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠁⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠘⠠⠀⠀⢀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠔⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡰⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⡀⠄⣀⠀⠺⣦⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠠⠐⠀⠀⠠⢐⠠⢴⣄⣉⠿⢚⣅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣤⡀⢙⡶⢔⡋⠑⢷⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⣦⡤⠟⠹⣦⣈⡽⠛⣯⣀⣠⡤⠄⠀⠀⠀⠄⠒⠀⢠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⠀⢶⣄⣩⠟⢫⣅⣈⣳⠶⣿⠉⠻⣦⠴⠋⠉⠀⠀⠀⣠⠄⡀⠰⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣙⡷⢚⡁⠘⣷⣤⣾⠙⢷⣄⡬⠟⠋⢤⡀⠀⢀⣠⡴⠎⠋⠀⡁⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠐⠆⠀⠀⠙⠦⢴⠋⠰⣦⣠⡽⠫⣯⣀⣹⠗⠊⠁⠀⠀⠀⠀⠉⠓⠍⠁⠀⠀⠐⠀⠁⢠⠐
⠀⠀⠀⠀⠈⠀⠀⠈⠐⠀⠀⠈⠙⠒⣏⠈⢻⡦⠖⠉⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠂⠀⠀⠀⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⡀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡀⠀⠄⠈⢦⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣴⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡤⣤⠤⣤⡤⣤⣤⣤⣤⣤⢤⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣧⣿⣴⣿⣧⣿⣴⡿⡮⢿⢼⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡟⣾⠛⣾⡞⣿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⣇⣿⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡠⢄⠀⡠⢤⡀⠀⠀⠀⠀⢀⡤⣀⠀⢀⠤⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡀⢺⣿⡅⢀⣟⠀⠀⠀⠀⣼⡀⢻⠀⣻⠀⣽⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⠀⠁⠛⠁⠀⠙⠀⠀⠀⠀⠛⠀⠉⠃⠋⠁⠘⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡖⣢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣎⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣜⣛⣤⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢟⣽⠿⣽⢿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⡀⠀⠀⠀⠀⢀⢺⣏⣿⣿⣾⠹⡟⠾⣸⣿⡇⣻⢂⠀⠀⠀⠀⠀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣷
⠀⣿⣶⣝⡻⣿⣿⣿⣿⣿⡿⢟⣭⣾⣿⠀⠀⠀⠀⢸⣷⣦⣝⢿⣮⣟⣻⣿⣿⢟⣣⣶⣿⠀⠀⠀⠀⣿⣮⣝⢿⣿⣿⣿⣿⣿⡿⣛⣽⣿⡿⠿⠃
⠀⣿⣿⣿⣿⣮⠝⠿⠟⢫⣾⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⢎⣝⣛⣍⠾⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣮⠛⠿⠟⣽⣾⣿⣿⣿⡇
⠀⣿⣿⡿⣫⣵⣿⣿⣿⣷⣯⡻⢿⣿⣿⠀⠀⠀⠀⢸⣿⡿⣫⣶⣿⣿⣿⣿⣿⣮⡻⢿⣿⠀⠀⠀⠀⣿⣿⢿⣫⣶⣿⣿⣿⣷⣭⡻⣿⣿⡇
⠀⠛⠵⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠯⠛⠀⠀⠀⠀⠘⠩⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠝⠀⠀⠀⠀⠫⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠝⠃

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠶⠚⣛⣋⡉⠉⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣆⠀⠀⣏⣀⣧⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣿⣿⣿⣿⠟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠀⠠⠆⠀⠐⣙⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⠀⠀⣀⣈⠥⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠠⣀⣀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣧⡀⠀⣀⠀⡿⢶⡶⢶⡆⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠻⡄⢈⠞⣡⠟⠀⢸⡇⠀⠀⠀⠀⠀⢸⣽⠀⠀⠉⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡀⠀⠀⣹⠋⣴⠃⠀⠀⢸⣇⠀⠀⠀⢠⠤⠜⢻⡄⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣾⠁⠀⣀⣠⣼⣥⣤⢾⠛⠉⢽⡆⠀⠀⠀⣿⠋⢻⢻⡾⠀⣼⠉⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡟⠀⠀⠁⠀⠀⠀⢻⠸⡄⠀⣹⡇⠀⠀⠀⣿⠀⢸⢸⣥⣠⡏⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣀⣸⣦⡟⠉⠉⠀⠀⠀⠀⣿⠛⠛⠛⠁⠀⠙⠒⠲⠤⡾
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⠚⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⡿⢿⡿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡜⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⠞⠀⠀⠀⠀⠀⢠⣶⠿⣇⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣠⠋⠀⠀⠀⠀⢀⣴⠟⠁⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⢀⡾⠁⡴⠃⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢠⡟⢁⡜⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⣰⠏⢠⠎⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠸⣿⣴⠋⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢻⣤⣴⠆
⠀⠀⠀⠀⠈⠛⠳⣤⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⣀⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠚⠛⠛⠛⠛⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣀⡤⠒⠉⠉⠉⠈⠉⠉⠉⠓⠠⣄⡀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⠞⠁⠀⢠⡀⠐⡷⠂⠰⠖⢀⡄⠀⠀⠙⢦⡀⢀⣀⣀⣨⣿⡤⠴⠶⢦
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣴⠃⠀⠀⠄⠈⠁⠀⢀⣀⣀⣤⡤⠤⠶⠶⠒⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢀⡿⠀⠀⣀⣀⣸⣧⡤⠴⠶⠖⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢠⡴⠶⠞⠛⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡼⠁
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠴⣶⠖⠒⠛⠉⠉⡟
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠶⠶⠚⠛⠋⠉⠉⠁⡄⠀⠀⣠⠏⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠹⣄⢠⡤⣴⡶⠖⠚⠛⠋⠻⣍⠀⠀⠀⠆⢀⣤⠀⣠⣄⠀⠟⠂⠀⢀⡴⠋⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠁⠀⠈⠁⠀⣀⣠⠖⠋⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⣀⣀⣀⣀⣠⣤⡴⠆⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁