Plague doctor Emojis & Symbols

⣿⣿⣿⣾⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣴
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⣘⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢠⣤⠄⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⢀⣠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⢰⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⣴⡾⠛⠋⠉⠀⠀⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⠞⠉⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⠗⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⡿⣏⢙⣻⣿⣿⣿⣿⣿⣿⡴⢨⡔⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣿⣿⣿⣿⣿⣿⣧⢿⣾⣿⣿⣿⣿⣿⣿⣿⣟⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠐⠘⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢘⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡢⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣯⠿⣳⢹⡯⣿⣿⣿⣟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⡅⠀⠈⠈⢹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⣿⣦⠀⠀⣼⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡧⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣤⣶⣶⣶⣿⣿⣿⣿⣷⣸⣿⣿⣿⣣⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⡽⠾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠛⢻⣵⠰⣿⣿⣗⣶⢮⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣶⣾⣿⣴⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⣺⣿⣯⣙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣿⣿⣿⣾⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⣿⣿⣿⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣵⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠟⠋⠉⠀⠀⠉⠉⠛⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣃⣘⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⣏⣡⣤⣿⣿⣷⣶⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣶⣿⣛⣻⡿⢿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠟⢛⣉⠟⡙⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡟⢻⣿⠀⡿⠋⠉⠳⣏⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⡰⢟⣿⢻⣾⣯⣽⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠠⠤⠶⣿⣿⠿⢿⠟⠁⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣨⣴⣶⢎⣽⢀⣾⣿⣿⣿⣿⣿⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠛⠋⠉⠉⠉⠙⠛⢿⣭⣍⡉⢠⡽⢛⣵⣿⣿⣿⣏⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⠀⣀⣤⣼⡿⠿⡛⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣇⢉⣩⣽⣷⠶⢆⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣸⡏⢀⣽⣧⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣻⣿⣧⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠸⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠾⠛⠛⠛⠛⠘⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠶⠚⠛⠛⡛⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⣿⣿⢛⣡⣿⣿⣿⣿⣿⣿⣇
⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣠⣠⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠰⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠰⡆⠀⠘⠿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢱⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢹⡿⢻⣿⣿⡿⣿⡟⢹⢹⣿⡇⣿⣿⢸⣿⣿⡇⢸⢹⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠘⠙⣿⠀⠁⠃⠈⠀⠙⠃⠘⠿⠀⡿⢻⡇⢸⢸⡿⡿⢿⠟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠁⠀⠈⠀⠀⠀⠙⠿⢻⡿⣿⣿⡿⣿
⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠛⠁⣿⠆
⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⢀⣴⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⠿⠿⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠿⣿⠛⠉⠩⠉⢛⣍⢻⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣡⣾⣷⡄⣀⣬⡜⣓⣅⣯⣾⣻⢿⣿⣿⡶⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡻⣿⣿⡷⢋⣴⠟⢹⢹⣿⣿⣿⣿⣿⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⠛⠛⠻⣇⣾⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣀⠂⠀⠀⣠⣴⣶⣯⣨⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣀⣴⡾⢟⣫⣭⢭⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡂⠈⢀⣼⢟⣿⠾⠻⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⢋⢾⣿⣿⣻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⢀⣴⡿⠗⠉⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⢟⣵⡟⣾⣯⣿⣿⣿⡟⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⠟⢁⣿⣿⣡⡾⣻⢿⣿⣿⠟⠀⠘⣛⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠗⣷⣶⣼⣿⣿⠏⠀⣠⢾⠋⠁⠀⠰⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⡈⢩⣴⢿⣿⣷⠆⣴⡫⢃⣔⣴⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢰⣼⠟⣡⣿⣿⣿⣿⣿⣇⣾⣿⡿⣿⣿⣿⢿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣛⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⡏⣿⠟⠘⣿⣿⣏⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠿⠿⠉⠀⠉⠉⢿⣿⡿⠃⠀⢰⠁⡟⠀⠀⢹⡉⠋⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⠸⠇⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠖⠒⣒⣲⣏⣹⣿⣿⣿⣿⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡒⢓⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠠⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⣠⣴⡖⠛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠿⢷⣶⣦⣧⠰⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣾⣿⣷⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢻⣦⣤⣬⣭⣭⣽⣥⣶⣿⣿⡿⠿⣛⣼⣾⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣀⠠⠿⣿⡝⠛⣭⣛⡭⠿⢛⣋⣤⣶⣛⣛⠉⣹⣿⡟⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣤⣀⣀⣀⣀⣀⣈⣭⣤⣶⡾⠛⡟⣫⣷⣿⣿⢯⣯⡝⠛⢿⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣇⣶⢸⣇⣿⠉⠟⠛⢈⡿⣿⣿⣿⣶⣤⣀⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠟⢛⣛⣩⡘⣿⢿⣝⢧⣄⣤⡾⠛⠁⠈⠛⠉⠉⠉⠛⠀⠀⠈⠙⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠷⠿⣿⣿⣿⣷⡜⢆⠈⠓⠂⠀⠀⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣾⣶⡟⠦⣑⢄⠀⠀⠀⠐⠒⡒⣶⡖⢶⣤⣭⣯⣙⡳⠶⢤⣀⡀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠻⣿⣿⣿⣿⣿⣷⣤⣨⡗⠮⣄⣀⠀⢀⣉⣉⣀⣤⣤⠤⠤⠿⠿⠿⣿⣾⣭⡓⠦⣀⠈⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠥⠶⠶⢯⣄⣉⣿⢻⠿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣿⡓⢄⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠠⣒⠤⠶⢶⣤⣬⣭⣿⣭⣬⣍⣹⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣥⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡴⡟⠁⠀⠀⠀⠈⠓⠶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣼⠁⣴⡔⣼⣷⡿⡀⡴⣠⣸⣿⡿⣿⡿⣽⣿⣿⠿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡇⣸⡏⢠⣿⢿⣷⢻⡇⣿⣷⡏⢻⣿⣷⠠⠾⣿⣦⣿⣿⣦⡽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣸⢹⡟⠂⢼⡿⣿⡉⣿⣿⣿⣿⣧⠈⢿⣿⡆⣷⣿⣿⣿⣯⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⣼⠱⠀⣿⠃⣽⣿⣿⢹⣿⣿⣿⠀⡞⣿⣷⠹⣼⣿⣿⢿⡇⢛⣽⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠰⠂⠀⠀⠰⣻⠋⣼⣿⣿⣸⣿⣸⣿⢘⣹⣿⣿⣧⡛⠿⠿⣊⣡⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠲⡖⢶⣶⣦⣭⣥⡻⠻⣿⣿⣿⣿⣟⣥⣬⣿⣯⣽⣿⣷⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠰⠇⢸⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠀⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣼⣿⠎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣟⣻⣆⣠⣤⣴⣶⣾⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡍⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣭⣭⣭⣭⣭⣭⣭⣉⣉⣛⣛⣛⣛⡻⠿⠿⠷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢟⣛⣟⡻⣷⠔⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⡇⣿⣿⣿⣿⢹⠅⣀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣴⠼⠋⠀⢽⣧⡻⠿⠿⣣⠆⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠷⠋⠀⠀⠀⠀⠀⠍⠛⠻⠻⠋⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢏⣁⣠⣤⣤⣆⣄⣠⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⣿⣯⣉⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠘⠻⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢻⣿⣿⣿⣿⣿⣿⣶⣯⡝⢹⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣇⠀⠀⠀⢀⣴⣿⣿⣿⣿⠛⠛⠉⠉⠀⠀⠀⠀⠀⣈⣿⣿⡟⠿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣾⡟⢿⣽⣿⡄⠀⢀⣾⡿⣯⣙⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⡌⢛⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣧⣤⣼⡟⠀⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣾⣿⡟⠻⣿⣿⣿⡏⣰⣿⣿⣿⣠⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿⣻⣿⣿⠁⢰⣿⣿⣿⣇⣄⣠⣴⣾⢿⣿⣿⣿⣿⡷⢋⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣷⡈⣿⡧⢤⣾⣿⣿⣿⣿⣿⣿⣿⣧⡠⢿⣿⣿⡟⣰⣿⣿⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣦⣉⣻⣿⣯⣿⣿⣿⠟⢋⣴⣾⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣡⣴⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣓⣲⡶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣿⣿⣿⡿⣿⣷⣶⣿⣭⣿⣻⡿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⡿⣭⣯⣿⣟⣛⣿⠷⢾⣿⣽⣿⣻⣯⢷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢹⣿⣿⠿⠿⣿⣿⣯⣿⣶⣶⣯⣽⣟⡻⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣿⣱⣿⣿⢌⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣞⣿⣿⣿⣿⣿⣷⣭⣥⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⠃⠹⠋⠙⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣏⣾⣿⣿⣫⣾⣿⡏
⠀⠀⠀⠀⠀⣠⣯⣿⣻⣿⣿⣿⣿⣿⠿⠶⣶⣶⣿⣿⣿⡿⣟⣼⣿⣿⢓⡫⣷⣿⠇⠀⢠
⠀⠀⠀⠀⣼⣿⠿⠛⠋⠉⠀⠀⠁⠁⠀⠀⠀⠉⠿⣿⣿⣛⣯⣿⢟⣡⣮⣻⢷⣾⣦⡄⠈
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣞⢿⣿⣿⢿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡟⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⡟⠀⠁⡀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈
⠀⠀⠀⠀⠀⢀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠐⢿⣿⣿⡿⠷⣤⠔⢀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣓⡄⠀⠀⢻⣿⡟⠉⢸⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢅⢤⣤⠾⣿⣧⣤⣼⣿⣿⣿⣿⣿⢿⣿⣿⣿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣷⣵⣼⣿⣿⣶⣿⣿⣿⣿⣿⣟⣻⣿⣿⣟⣻⠸⣟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣆⡀⢳⣽⣷⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⡏⣷⡀
⠀⠀⠀⠀⠀⣠⡟⠀⢀⣵⣞⠿⣟⠙⠛⠿⣟⣿⣿⣿⣿⣿⣿⣿⠏⢫⠈⠻⣷⡄⢘⡇⠸⣇
⠀⠀⠀⠀⠀⣿⣃⡶⢫⡛⣿⣷⣼⣿⣿⣦⣙⣿⣿⣿⣿⣿⣿⣿⡤⠾⡄⠀⠀⠙⣾⡇⢰⣿⡆
⠀⠀⠀⠀⠀⣽⡟⣓⣴⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠿⠶⣦⣤⡜⣷⡌⣿⡁
⠀⠀⠀⠀⢸⢿⣟⠃⠀⠀⠀⠀⠑⢦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣬⣿⣎⢿⣿
⠀⠀⠀⠀⠀⠘⣿⡷⢄⣀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⡈⢛⣿⡟
⠀⠀⠀⠀⠀⠀⢹⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⢿⣿⣯⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⢿⠿⠛⠛⠉⠋⠀⠀⠀⢸⣿⣇⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⠶⠶⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠟⠀⠄⠀⠀⠀⣾⠿⠟⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣟⣛⣛⣻⣿⡿⠿⢶⣦⣄⣈⣽⡛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠈⢻⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠶⢈⣿⠻⣿⣿⣇⣽⠀⠀⡞⣿⣿⢡⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢀⣄⣰⠦⠞⠂⠉⠈⠀⠀⠋⠉⡉⠞⠀⡼⠁⣿⢇⡾⣿⡿
⠀⠀⠀⠀⢀⣔⣿⣿⠶⠶⠶⠶⠶⢶⣶⣶⣦⣤⣌⣀⡼⠁⠀⣡⠏⠀⣻⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣯⣷⣒⣚⡥⠀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣷⣾⣿⣿⠿⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⠿⠟⠛⢛⢻⣇⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠉⠉⡙⠛⠁⠀⣠⢖⣽⣿⠏⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣀⠀⠀⠉⠻⣿⣿⣿⣎⠂⠀⠀⠀⡿⣱⣿⠟⠁⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣧⠀⠀⠀⠑⠙⢿⣿⡄⠀⣀⣸⣷⣿⠏⠀⢀⣠⣤⣤⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⣁⣠⡀⠀⠀⠀⠀⠳⠻⢤⡽⠟⣹⣿⣶⣿⣿⣿⣿⣿⡿⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠈⠉⠙⠛⠷⠀⠀⠀⠀⠀⠀⠀⠁⠐⠙⣿⣿⣿⣿⡿⣋⣩⣿⠶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠷⣤⣿⣶⣶⡆⠀⠀⠀⠀⠀⠰⠿⢻⣿⡿⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢿⣿⣤⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠖⠒⠀⠄⠀⢀⡀⠀⠀⠀⠀⠉⠉⠛⢻⣿⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣧⣀⣠⣤⣄⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣈⣤⡄⠀⠀⣐⢿⣿⣿⣿⣿⣿⣄⠀⣸⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⣿⣧⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡁⠙⠛⢿⣿⣿⣿⡿⠛⠛⢫⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿⣷⠞⡁⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⡶⢶⡿⠿⠟⣀⠤⣤⣼⣧⣬⢟⣽⣏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⢀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢈⣀⠀⣸⠟⢻⣿⣿⣿⣷⠿⢻⣿⠄⠐⣀⠄⠀⠀⠀⡀⠀⢠⣎⣿⣿⣾⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠟⠁⠠⢙⣛⠟⠛⣡⣾⣿⢡⠀⣸⡇⢰⣤⠘⠛⠛⢳⣾⠻⢿⣿⣿⣷⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠺⠁⣠⡎⠀⢠⣀⡉⠃⠠⠧⠾⠇⠎⣰⠿⠷⣨⡏⠀⠀⣼⣿⣷⣶⣴⣧⣙⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠉⠀⠀⡼⠋⢹⣆⠘⠃⠈⢃⣀⣠⣴⣶⣼⣿⣿⣷⣿⢠⣤⣤⣽⣟⡏⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠘⠀⠀⣸⡇⠀⠻⣿⡿⠟⢻⡿⣿⡙⢿⣿⣿⡏⡿⠆⠀⠰⢾⡇⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡀⠀⠀⣰⡏⢰⢻⡞⠛⠃⢰⠈⠇⢿⣷⠈⠋⢿⡟⠀⠀⠀⠀⢸⡇⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣡⣿⡓⠾⢿⠀⠸⣷⠀⣤⣠⡟⠀⢰⠘⣿⣧⠐⣾⡅⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡄⠀⠀⠈⢸⡿⠋⠀⠀⢸⡧⠙⢻⡆⠿⢱⣄⡠⠤⠄⣼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⢸⣿⣷⣿⣿⣧⣢⣼⣟⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⣿⣿⣿⣶⣤⣀⣀⣀⠀⣁⣀⣠⣤⣎⣿⣿⠖⢶⣤⣭⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢸⠙⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠉⠛⠛⠛⣳⣴⡗⠀⠐⠒⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠘⣿⣷⡹⢿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠈⠉⠀⠀⡉⠉⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘⢿⣷⡈⠛⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣀⡤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣔⣲⣶⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⣽⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⢿⣿⠿⣟⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⠙⢷⡶⠜⠀⠀⠀⠀⣀⣠⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⣡⣼⣇⠀⣾⣃⣀⣠⡴⡾⠟⠛⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣆⠀⠈⠉⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢦⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⡖⠺⠷⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠃⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣟⡻⠿⣿⣿⣿⣯⣿⠧⠈⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡅⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⢛⠛⢒⣢⣤⣤⣤⣿⠀⠀⠀⠄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢳⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣁⠀⣈⣷⣾⣿⣿⣿⣿⣿⣰⣷⣠⠀⢀⢹⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣧⡈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣆⣿⠛⢷⣄⠀⠀⠀⠀⢠⠏⣄⣸⣷⣼⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠙⢧⣀⡀⢀⡏⢀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣽⣙⢋⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢢⣄⣤⡖⣡⣴⡆⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠽⡮⣉⠁⠙⠇⠀⣼⣉⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣌⣱⣦⢖⣀⠤⣹⠟⢿⣿⡇⠀⠈⠉⠛⠛⠻⠿⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣠⠐⣤⣦⣾⠋⠉⢉⠑⠰⠋⣼⣾⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⣠⣤⣾⣿⣿⣿⣿⡇
⠀⡀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠛⠹⡿⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠇⠙⠻⠀⠁⠻⠙⠘⠛⠈⠛⠋⠃⠙⠀⠀⠀⠀⠀⠀⠇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⡦⠁⠘⠛⠛⠛⠋⠻⢿⣿⣿⠻⣴⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡄⠀⠀⠐⡀⠀⠀⠀⠈⠁⠀⠸⣿⣿⡆⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣯⠄⠀⠀⠀⠠⣤⠄⠀⢀⣀⡀⠴⣿⣿⣷⡹⠿⣛⣛⣫⣽⣷⣶⣛⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡧⠠⠆⣀⣤⡶⠾⠂⣉⡡⠀⠀⣀⣹⣿⣿⣿⣷⣿⣿⣿⣿⡿⠿⣋⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⢿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣩⣤⣶⣶⡿⠿⣛⠛⠛⠻⣿⣿⣿⢟⣫⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠁⠀⢉⣡⢴⣦⡈⠣⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⠀⠁⡈⣷⣦⣬⣿⣀⣠⠮⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠾⣻⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡄⠀⠀⢇⠹⡿⣿⡟⠛⠒⠠⢤⣈⡐⠭⢛⡿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡟⣿⣿⡿⣿⡭⠀⠻⡄⠀⢈⠅⠈⠁⠀⢀⡀⠀⠈⠐⠉⠇⠢⢌⡚⠝⡻⣿⣿⣿
⠀⠀⠀⠀⠀⣽⣤⣼⣶⣶⣶⣿⣿⡿⣿⣿⣷⡈⠛⢿⣤⡹⡰⣦⣽⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡈⠠⠁⡠⡙⢟⢿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣿⣿⣶⣬⡀⢿⣷⣼⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⡨⢁⡀⠀⠁⠈⠆⢑⡁
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢴⣭⣟⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣭⣵⣶⣶⣶⣾⣭⣽⣛⣛⡿⣾⣴⣤⡀⠠⡠⠛⣄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⣸⣿⣟⢠⣧⠉⠻⢿⣿⣿⣿⣿⣿⡻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢷⣦⡐⠹⣆
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢟⠜⠁⠀⢨⠟⠓⠆⢀⡰⣟⠻⣿⣿⣿⡿⠎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠙⠳⣼⡄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⢟⠟⠉⠀⠀⠔⠁⠀⠀⠀⠳⡙⢮⣿⣿⣿⣿⡿⣮⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣅⠁⠁⠀⣤⠊⠀⠀⠀⠀⠀⡄⠙⣄⠻⣿⣾⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠹⡀⠘⣆⠙⢟⢏⠣⡈⠻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⠀⠘⣆⠀⠂⠀⠈⠢⠙⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠘⠀⠀⠀⠀⠀⠘⣆⠘⣦⠀⠀⠀⠀⠐⠀⢘⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠂⠀⢹⠀⢈⠧⠀⠀⠀⠀⠀⢾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠂⢄⠀⠀⠀⠀⢂⠀⠀⠀⡆⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧⣤⣐⣛⣀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢹⣿⣿⣿⣿⣿⡦⠄⠀⠁⠉⠙⠛⠶⣆⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣿⣿⣿⣿⣿⠷⠦⠠⠤⠤⠀⡀⣀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠀⢈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣷⣶⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣟⠟⠋⠀⢼⣿⡿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀⠀⠂⠉⠹⣿⣿⣿⣿⠀⠀⢀⣾⣿⣇⣿⣰⠃⠘⠛⠛⠛⠻⣟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢠⠀⠂⡠⠠⠀⠀⠐⢿⡉⠛⠋⠀⠀⣾⣯⣿⣿⡟⠁⠀⠆⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣤⠔⣋⣠⡾⠿⣆⠈⣄⢠⣀⠠⠉⣶⠀⠀⣼⣿⣿⣿⡿⡅⠾⣿⣷⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⢀⣶⡿⢟⣩⡷⠾⠖⠉⠑⠚⠲⢶⣶⣶⣦⣼⣿⣿⣿⣿⠟⡐⠛⠒⠘⠃⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢠⣿⣿⡾⠟⠻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣩⣟⣋⣿⣿⠛⠻⣷⠦⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣭⣉⡒⠶⠿⠥⢴⡶⢒⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣿⣿⢿⣿⡋⣻⣿⣿⠿⠿⠿⢿⣿⡿⠷⠒⠲⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣹⣿⡿⠟⠛⠻⠶⣄⠀⣸⣿⣦⡀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠌⢠⣿⣿⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⢠⡿⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣸⣆⠀⣼⡇⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣖⣒⣂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣯⡀⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣯⣤⣶⣶⣶⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⠿⠛⠛⠛⠛⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣟⠀⡴⡶⠲⣖⣛⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⣋⣤⠘⢤⣀⡀⠛⠁⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢰⣶⠶⠶⡶⠆
⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⡿⢿⣿⣿⡀⠀⠉⠡⣀⣤⠶⣶⣶⡶⠶⣿⣿⣆⠀⠀⠀⠀⠀⠀⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠛⠉⢀⡎⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠠⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣻⣿⣿⣿⠿⣻⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢞⣿⣔⣻⡖⡖
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⢿⡘⢿⣟⡛⠋⠴⠚⠛⠀⣿⡟⣷⣄⠀⠀⠀⠀⠀⠀⢀⡆⣀⣉⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣯⣤⡀⠈⢷⣄⣌⠉⠉⡀⠀⡋⣴⣿⣿⣯⣽⠧⡐⠀⠀⠀⠀⣼⣿⣏⣿⣧
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣻⣦⣥⣿⣴⣷⣧⣿⣿⣿⡿⠟⡛⠁⣾⣷⡀⠀⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢢⠀⣽⣿⣧⠀⢠⣟⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣤⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢀⣿⣿⣿⣿⣿⠯⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⣧⣿⣿⣿⣿⡿⠟⠉⣴⣷⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣴⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⠿⠟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⠁
⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠛⠁⠙⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⠻⠟⠋⠋⠘⠀⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉⠁⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣷⣭⣿⣻⣿⣿⡿⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⠋⠙⢿⣿⣼⡿⢡⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣯⣿⣿⣷⣶⣾⣿⠿⡰⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⢻⣿⡿⠆⠰⠷⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢻⣿⣿⣿⣿⣿⣿⣦⡈⢛⣀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣦⢻⣿⣿⣿⣿⢿⣿⣿⣦⡉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢷⡝⢻⣿⣿⣿⣶⣶⣌⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡻⠆⣿⣿⣿⣿⣿⣿⣿⣿⣆⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣜⢷⣿⣿⣿⣿⣿⣿⣿⣿⡄⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠹⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠊⠁⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣁⣴⡶⠃⠀⠀⣤⠀⠂⠖⢀⠀⠀⣳⣤⠟⣧⣌⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣥⣶⣟⣭⣴⣾⣏⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣏⣠⣎⣙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣋⣥⣾⣿⣿⣿⠿⣻⣽⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢠⣿⢿⣿⣿⣿⣿⣿⣿⠛⣡⣴⣾⣿⣿⣿⣿⣟⣵⣾⣿⣿⣿⢿⣿⣷⣿⣿⣿⣿⣿⡿⠟⢁⣤⣄⣰⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃⣀⣴⣿⣵⣿⣿⣿⣿⡿⣋⣴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠃⢀⣠⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣾⡋⣠⡾⢋⣼⣿⣿⣿⣿⣟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⢻⣏⠁⢘⣷⢾⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⠏⠘⠻⣉⣶⣿⣿⣿⣿⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⡃⢾⡀⠻⣷⣾⣿⣿⣽⣤⢆⣸⣿⣿⣿⣿⣿
⠀⠀⠀⣦⣤⣴⣶⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣃⣴⣻⣿⢈⣇⣾⡟⠻⢿⡿⣿⣿⣾⣿⣼⡛⠀⣿⣿⣿⣿
⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣉⠄⠀⢩⡿⣴⣿⢻⣿⠿⣏⣄⡀⠀⠀⠉⠛⣉⣿⣿⣿⣦⡙⣿⣿⣿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢫⠙⣤⣶⠏⢀⠀⣿⣇⣿⣇⡸⢻⣾⣻⣘⣧⣴⢇⣘⣿⢉⡀⠛⢿⣿⣿⣮⣿⣿
⠀⠀⠀⠟⢋⣥⣼⣽⣿⣤⣿⣿⣿⣿⡟⠉⣙⠟⠉⠉⠀⢀⣴⣿⣾⣿⡅⠀⠈⢰⣿⣿⣿⣿⣇⣿⣿⣿⣉⡉⠻⣿⣝⣿⣛⣃⡶⢤⡈⠉⢹⢿⣿
⠀⠀⣼⣼⣿⡿⢛⣿⣿⠟⠛⠛⡛⣿⢧⡜⢁⠀⠀⠀⢠⢿⣿⣿⣿⣾⠇⠀⠀⣸⣿⣿⣿⣿⣧⣿⠿⢿⣿⣿⣦⣼⣿⡉⠉⠙⠷⢦⠛⢸⡷⣄⢈
⠀⣸⣿⣾⢣⣴⠿⠃⣠⣴⣾⣿⣿⣿⡟⣰⣿⡆⢸⣤⡟⣿⣿⣿⣿⣿⠀⣿⠀⣿⣇⣿⣟⣿⣿⣥⡌⣠⡙⠛⢿⣿⣿⣿⣿⣭⣷⣌⢿⣙⠧⠰⠿
⢀⣿⡿⢙⣿⣅⠐⣀⣿⣿⣿⣿⣿⣿⣼⣿⣿⠃⢸⣯⣽⣿⣿⣿⣿⣿⢰⣿⡇⢹⣿⣿⣾⣛⣿⣿⣷⣼⣾⠛⣯⣭⣭⣿⣄⣉⣈⡻⢟⣛⠛⢛⣻
⢸⣿⣅⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⢿⣿⠀⣿⡇⢸⣿⣽⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⡟⢧⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣠⣸⣷⣀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⡿⣿⠹⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢸⣿⣿⣿⡟⣿⡆⠹⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⡇⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⡼⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢹⣿⣿⣿⣿⣿⣿⡉⢀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚
⢰⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠻⣿⣿⣿⣿⣿⣿⣷⣄⠛⢷⣤⣉⡛⠛⠿⠿⠃⣾⣿⣿⡿⣯⣿⣿⣿⣿⣿⡏
⣸⣿⣿⣿⣿⣿⡟⣷⠸⣿⣿⣿⣿⣿⣿⠛⠟⠛⠿⠿⠛⠃⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⠟⣃⣴⣿⣿⣡⣾⣿⣿⡟⠿⢿⣿⡟⢡
⠀⣿⣿⣿⣿⣿⡇⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣾⠛⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣧⣨
⢠⣿⣿⣿⣿⣿⢳⠉⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣾⣿⣽⣿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡿⠿⠟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⣿⣿⣿⠿⢿⣿⡿⣫⣥⠐⣮⡻⣿⣿⣿⣏⡿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣎⢻⡇⠀⠀⢹⣿⣿⣿⡿⢸⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠘⣿⣿⣿⣷⡆⣹⠏⡿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⣿⣭⠟⠃⣸⣣⠈⡷⢝⣿⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⣻⣿⣧⠀⢠⣿⣿⣿⠏⠀⣴⣿⢋⡞⣡⣿⣿⣿⣮⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣧⣿⣿⡆⣸⣿⣿⠏⢀⣾⣿⣧⣿⣴⡿⠾⢿⣿⣿⣿⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⡸⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⡿⠶⣶⣤⣌⣻⣿⣇⡀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣝⣿⣿⣿⣿⣿⣿⣣⣾⢟⣩⣽⣿⣻⣿⣿⠿⠿⠿⢿⣿⣿⣿⣄⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣀⣬⡙⢿⣿⣿⣿⣯⣧⣿⣿⡿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠻⡇⢠⡀⠠⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡞⣽⣿⣿⣿⣷⣶⡿⢿⣭⡏⢉⡀⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⢠⣤⣤⣀⡀⠀⠀⠙⠊⠑
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣽⣿⣟⣼⣿⣿⡏⣿⣿⣿⣿⣦⡘⣧⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣎⠙⠻⣿⣿⣷
⠀⠀⠀⠀⠀⢹⣿⣿⠟⠁⣼⢻⡿⣹⣿⣿⣿⣧⠹⣿⣿⣿⣿⣷⡜⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⢿⣽⣿⣿⣿⣿⣦⣰⣿⡏⣿⣇
⠀⠀⠀⠀⠀⠘⣿⠀⢀⣾⢃⣾⢣⣿⣿⣿⣿⣿⣆⢹⣿⣿⣿⣿⣿⣿⡜⣿⢿⣿⣿⣿⣿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣤⣿⠏⣼⢃⣾⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣇⣿⣾⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⣿⠏⣼⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠙⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⢿⣿⡻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢿⡾⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⠿⠃⠀⠘⣿⣿⣿⣿⡼⣿⣿⣿⡿⣿⣿⣿⡄⢸⡿⠀⣿⣿⣮⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢼⣿⣿⣿⠏⢀⣿⣿⣿⣿⠇⠀⠀⣠⣤⣾⣿⣿⣿⡇⣿⣿⣿⡇⠈⣿⣿⣧⣿⣡⡾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡀⠘⣿⡿⠟⠁⣀⣀⣴⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣷⣰⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠛⠻⠻⠿⠿⠿⠛⠟⠛⠋⠁⠙⠃⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣶⣶⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣻⣿⣿⣿⣿⡄⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣄⣀⣤⣤⣏⣰⣿⣿⣿⣿⡿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠾⠦⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣿⣿⣿⣽⢷⣿⣿⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠡⢄⠻⣄⠀⢀⢻⣿⣿⢄⣼⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠃⢠⣥⠘⢓⠒⢴⣜⢾⣾⡿⣿⡿⢿⣿⣧⣄⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⣀⣴⠶⣊⠁⠈⠘⢿⡗⠂⠉⠀⠈⠉⠛⠿⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠟⠀⢀⡴⠚⢁⣠⠶⣛⡭⢖⣿⢏⣶⣴⣷⡈⣧⠀⢦⠀⠀⠀⠀⠀⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣤⣦⡔⠋⣠⡶⠋⠁⠋⢁⣴⢟⣵⣿⣿⣿⡿⣷⠘⡆⠠⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠷⡀⠊⠁⠀⢀⣤⣴⣿⣿⢟⣽⡏⣿⣿⣧⢹⣧⢸⡆⠙⢦⡳⢄⡀⠀⠀⠀⠀⠀⠀⠄⡛⢀⠀⠀⠀⢀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠠⠾⣿⣆⠑⣿⡿⠿⠛⣹⣥⣾⡿⢠⣿⣿⣿⡄⢿⡆⢣⠰⣄⢙⡦⠃⠰⠄⠀⠀⠀⣄⣠⡴⣺⣤⣀⡀⢀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠐⠁⡿⠀⣲⠞⢻⣿⡷⠦⠤⣴⣿⣿⣿⡇⢸⣿⣿⣿⣧⠸⣷⠀⢃⠘⡏⠀⢀⡀⠄⠖⠚⣣⡀⠀⠀⠀⠉⠥⠒⢛⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣇⠙⢻⣾⣟⣦⣀⣴⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣴⣿⣶⣿⡏⠁⠈⢀⣠⣤⣴⠶⠛⠉⠉⠀⠀⠛⠉⠈⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⢞⣸⣿⣿⡟⣿⢿⣀⣹⣿⣿⣿⣿⣿⣿⣿⠿⡻⢿⢻⣿⢿⣶⣶⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⣗⣾⠇⣿⣿⣿⣿⡾⡍⣿⣿⣿⣛⣿⣶⣾⡗⢺⠁⠚⢢⡿⠘⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⢼⡟⣿⣿⣿⣿⣿⠉⠀⡿⢸⣿⣿⣏⢻⡏⢿⣿⡗⠈⠁⡀
⠀⠀⠀⠠⠀⠀⠅⢀⠜⠀⣼⡟⣿⣿⣿⣿⡿⢀⢠⡇⣿⣿⣿⠈⣧⡡⠀⢻⣿⣦⡀⠀⠂
⠀⠀⠀⢠⠂⠤⠘⣁⣤⣾⠟⠀⢸⣿⣿⣿⡇⢸⡘⠃⣿⣿⣿⡆⢿⣿⣄⢧⠹⣿⣆⠠⣀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢆⠂
⠀⠀⠀⠀⣈⠶⠶⢻⡿⠋⠀⠀⢻⣿⣿⣿⡇⣾⠇⠀⣿⣿⡟⣷⡜⢿⡿⣎⢧⠙⣿⣷⣄⡑⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠆
⠀⠀⠈⠛⠷⠶⠟⠃⠛⠀⠀⢀⣘⠻⣿⡟⣇⣿⣧⠀⢹⣿⣧⢻⣿⣦⡈⡙⢧⡀⠐⡘⢻⡿⣷⣤⣈⡒⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠛⢷⣿⣿⣇⣿⣿⣿⣷⡌⢿⣿⣦⣿⣿⣭⣭⡛⠿⣦⡀⡀⡇⣿⢧⣄⠉⢁⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣞⢿⣷⡈⢹⣿⣿⣿⣿⡜⣿⡇⣿⣿⠟⣩⡻⣷⣌⢳⡘⣦⣜⠀⢻⣦⠀⠉⠲⠦⢀⠀⠀⠀⠀⢤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠙⡛⠃⡿⢃⣼⣿⠿⣫⣿⣿⣽⣿⢻⣇⣴⣧⡱⡹⣿⡄⢱⠙⠻⣮⣿⣧⡷⡀⠁⠢⣄⠀⢄⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡈⠓⠀⠱⣌⣋⣤⣴⣿⣿⣿⣿⣿⣯⡓⢿⣿⣧⠰⣽⣿⣜⣦⡱⣬⣿⣿⡆⢸⡄⢄⠈⠱⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠡⠀⠙⠛⠿⠿⣿⣿⣿⣾⣯⣾⠃⠀⠈⠙⠳⢬⣃⣛⣛⣳⡟⠛⡧⢻⣬⡿⢬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠻⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⠿⠂⠀⠛⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡒⠒⠒⠂⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠃⠛⠓⠀⠀⠀⠀⠀⠐⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣦⣴⣶⣶⣖⡢⢤⣤⣄⣠⣞⠉⠉⠛⠓⢤⣄⡀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣦⣬⣙⣹⠻⢷⣦⣄⣀⠈⠉⢰⡶⢦⣄⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡟⠿⠛⠿⠶⣧⣝⡛⠿⢿⣿⣷⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⢁⣶⣶⡞⢻⡍⢹⣿⣷⣦⣭⣟⢹⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠈⠟⣿⡵⠏⠂⣾⣿⣧⣼⣿⣿⣿⣿⣶⣬⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⣸⣿⣿⢿⣿⣿⣿⠛⠛⠻⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⠈⠁⠐⠃⠐⠀⠰⠊⢉⠈⡀⢀⣿⣿⣿⠛⣾⣿⠃⠀⠀⠀⠀⠀⠈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣈⣠⡶⠤⠶⠴⠶⠶⣶⣳⣴⣶⣤⣴⣿⣿⡛⣡⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠯⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠛⠉⠀⠀⠀⠲⠟⠛⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠴⠶⢾⢿⣿⣿⣝⠿⢖⠂⠀⠀⠀⠀⢠⣴⣟⣭⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠀⠀⠀⠀⣀⣴⢿⣿⣿⣿⠷⣦⣐⣀⣤⣴⣿⣛⣻⣿⣯⣭⡭⣽⡷⢶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⡀⠀⠀⠚⠉⡤⠴⢛⣩⣶⣿⣿⣿⣿⣍⠻⠿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠈⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢻⢀⣤⠞⠋⠀⠀⠀⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠉⠿⣿⣿⡉⠹⢯⣏⡐⠢⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣵⠟⠁⠀⠀⣠⣴⣾⣿⣿⣿⣿⠿⣿⠿⣿⣿⣿⢙⣿⣿⣿⣿⣦⣄⠈⠫⣄⠂⠘⢿⡆⢡⡈⢧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣾⠏⠀⠀⣠⡾⢋⣾⡿⢿⣿⡿⠃⢳⠏⣰⣿⣿⡷⣾⣟⠻⢿⣻⣿⣿⣿⣦⡀⠹⢶⡀⢳⣼⣧⡈⢧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣾⠇⠀⢀⡼⠋⣠⡿⣿⢃⣿⡟⢡⡄⠀⣴⢫⣽⣅⣿⡚⠛⠀⠘⣿⡿⢿⣿⣿⣷⡀⠈⠻⣆⢻⣿⣧⠘⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠸⢸⠁⠀⡰⠁⢀⣼⠋⣸⢯⣾⠟⠄⠜⠁⠐⠁⢠⣿⠿⣯⠛⠁⠀⠈⠿⠁⢸⡇⢈⢿⣷⡄⠀⠙⢆⠻⣿⡄⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠁⢠⠞⠁⡐⢣⣿⣿⠀⠀⢀⣤⠀⠀⢸⡇⠐⠟⠀⠀⠀⠀⠠⡀⢾⡇⠈⠾⣿⣷⡀⠀⠈⢷⣿⣧⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣰⡿⢿⠃⠀⠰⡋⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣠⠀⠋⢹⣧⠀⠀⠀⢻⣿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠓⠀⠀⠀⣿⠀⠀⠀⠀⠰⠇⠀⠂⠀⠒⠃⠀⠀⠀⠘⠀⣼⡀⠀⠀⢹⡆⠀⠀⠀⠿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢹⠉⠉⣻⠛⠛⠛⡟⠛⠛⢿⠛⠛⢻⠛⠛⢻⠛⠛⢹⡏⠉⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣼⣄⣀⣽⣀⣀⣠⣇⣀⣀⣼⣀⣀⣸⣀⣀⣸⣄⣀⣼⣄⣤⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⢤⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⡠⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⠷⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⣼⠋⠉⠉⠉⡏⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⢿⣤⣤⣤⣤⣿⣤⣤⣤⠾⠁
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⣤⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡴⠶⢶⣤⡀
⠀⠀⠀⠀⣟⠀⠀⠀⡼⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠈⢹⣧⠀⢀⡼⠛⢶⣄
⠀⠀⠀⠀⠙⢦⣄⣀⣳⡀⠀⠀⢀⣰⠟⢀⣀⠀⠀⣿⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠙⢦⣠⡾⠋⠳⣮⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢛⣻⣷⠞⠋⠉⢻⡄⠹⣦⡀⢠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢿⡀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠻⢦⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⢧⠀⠀⣠⠟⠀⠈⠻⣧⡀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣼⠷⠛⠁⠀⠀⠀⠀⠈⠻⢦⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⠿⢰⣇⠸⠿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣇⣛⢻⡿⢛⣃⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣘⣃⣸⣿⣿⣿⡇⠀⠀⣿⡴⠖⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣴⣾⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⡴⠶⠶⠶⠶⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠳⠶⠶⠦⠴⢿⣿⣿⣿⡿⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⠁⢀⠤⠚⣛⡛⢟⣟⣻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡄⠀⠀⠀⠁⡰⠟⢛⣿⣮⣭⣧⣤⣤⣝⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢱⣤⣆⡴⢊⠔⢺⣿⣏⡙⣿⣿⣿⣿⣿⣿⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⢿⡴⠁⠀⠀⢻⠀⠀⢹⣿⣿⣿⣿⣿⡇⡟⢶
⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠺⠤⠈⠷⡦⠾⠆⠀⣠⣿⣿⣿⣿⣟⡏⢱⣾
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⠄⠀⠄⠀⠘⠻⣿⣿⣿⣿⡟⠀⣾⡟
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣉⢿⣿⣀⠀⢀⠀⠀⡀⠀⠈⠙⣿⣿⣷⡦⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠆⠀⠀⠀⠀⠀⠀⠻⢆⣿⡿⠋⠉⠁⠉⠁⢀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠈⠳⣦⣌⡉⠙⠛⠛⠛⠛⠛⢉⡉⠤⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⡀⠀⠀⠀⠐⡆⠀⠀⠀⢹⡤⡤⠐⠁⠀⠀⠀⣆⡀⠀⠀⠀⢸⠤
⠀⠀⠈⠙⠒⠷⠠⠀⠠⠈⠄⠄⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⢀⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠐⢶⠶⠆⠀⠠⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠻⢦⣤⣀⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣬⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠶⣶⣤⣬⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⡇⢸⡏⢹⡏⢹⡏⢹⡏⢹⡏⠉⡏⠉⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣄⣸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⡇⢀⣇⣠⣧⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠙⠛⠛⠻⠿⠷⠾⠷⠾⠷⠾⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣦⣤⣤⣀⣀⣀⣀⣤⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣉⣉⣉⣛⣛⡋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⢻⡇⠀⠀⠀⠀⢀⣴⣟⡱⠋⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠻⡿⠷⠶⣦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢰⡿⠃⠈⠹⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⣠⡇⠉⠛⢸⡇⠀⠀⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡇⠀⠸⣧⣀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣷⣾⠟⠁⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣛⡛⣟⠛⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⣀⣿⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣭⣽⣗⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣀⣤⣤⣤⣤⣄⣄⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣩⡿⠟⠛⠛⠛⠛⢿⣯⡛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⢰⢁⡏⠀⠂⠀⠀⠀⠀⠀⣠⠔⠀⠀⠙⡆⢡
⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⣦⡎⠁⣴⣲⠀⠀⣽⠘⣯⡇
⠀⠀⠀⠀⠀⠀⣼⣿⡦⢟⡿⣿⣿⣾⣷⣶⣿⣯⢟⡻⣼⡿⡇⠀⠀⠀⠀⠀⢠⢸⡄⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠀⠀⡾⢸⠛⠃
⠀⠀⠀⠀⠀⢠⡗⠋⣪⠗⠋⠁⠀⠀⠀⠀⠀⠉⠑⠪⣗⠝⢿⡀⠀⠀⠀⠀⠈⢎⢷⡄⠀⠀⠈⠁⠈⠉⠉⠀⠀⢀⣾⢇⣿⠇
⠀⠀⠀⠀⣠⠋⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠳⣖⣦⠀⠀⠀⠈⢯⣿⣦⣄⡀⠀⠀⠀⠀⣠⣴⣿⣷⠋⠈
⠀⠀⠀⢠⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄⢹⡿⠂⠀⠀⠀⠘⣿⢻⠿⢿⣿⣿⣿⡿⠿⠟⢻⡏
⠀⠀⠀⡞⢀⡏⠀⠀⠘⠲⠤⣀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣷⣤⡄⠀⠀⠀⠻⠼⠀⠀⠀⠀⠀⠀⠀⢠⠼⠁
⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⣀⣀⣉⠻⣷⡄⠀⠀⣏⣹⠀⠀⠀⢸⠀⣿⣷⡇
⠀⠀⠀⢧⢸⡇⠀⠀⠀⣾⣿⣿⣿⣧⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⠛⠃⠀⠀⠀⣴⢲⠀⠀⠀⠀⠀⠀⠀⠐⢲⡀
⠀⠀⠀⠘⡌⣿⡄⠀⠀⠻⣿⣿⣿⡟⠀⠀⠁⠀⠀⠀⠐⠀⣰⡿⣰⢷⠄⠀⠀⠀⢠⣿⡼⣶⣾⣷⣿⢿⣿⣶⡦⣾⣇
⠀⠀⠀⠀⠙⣜⢿⣄⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡱⠿⠏⠀⠀⠀⢀⡞⢡⠞⠉⠀⠀⠀⠀⠀⠉⠛⢆⠹⡄
⠀⠀⠀⠀⠀⠘⡦⣟⢿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣠⣴⡿⣫⣾⠁⠀⠀⠀⠀⢀⠎⡴⠁⠀⠄⠂⢁⠀⠀⠈⠂⠀⠀⠳⡘⡆
⠀⠀⠀⠀⠀⠀⢻⣿⠗⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣷⡇⠀⠀⠀⠀⠀⠸⢰⠁⠀⠈⠀⠀⠈⡄⠀⠀⠀⠈⠀⠀⢣⢸⣤⡄
⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣯⣭⣭⣭⣽⣿⣿⣿⠰⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠻⠃⠐⠐⠀⠂⠀⢸⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠊⠁⠀⠀⠀⠀⠀⠀⠸⡸⣆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡟⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡻⣦⡀⠉⠂⠀⠀⠀⠐⠀⠀⣠⡾⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣤⣄⣀⣀⣤⣴⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣽⣟⣛⣻⡯⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢨⠉⠙⠛⠛⠛⠛⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠞⣧⠀⠀⢀⣀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⣠⠋⠀⠘⠚⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠲⢤⠞⠉⠉⠙⠣⡀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⣧⠀⠀⠀⠀⡰⣋⣞⠉⣱⣄⡀⢸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠙⣄⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⢠⡾⣽⠣⣼⠃⢌⣇⡜⡸⠀⢀⡤⠲⡈⠳⣄⠀⠀⠀⠀⠘⠧⠤⠤⢤⣤⣀⣀⣀
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⠲⢧⡦⢴⠛⣬⣶⠁⡰⠛⢢⠰⢇⣠⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢾⣠⣿⠏⠻⣅⣀⠼⣀⡠⠃⢱⠛⡆⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⣨⠆⠀⠀⢠⡇⠀⠀⠀⠀⠠⠁⠀⣼⠟⠁⠀⠀⠘⠷⣤⢼⠒⠒⡇⢀⠇⠀⠀⠀⠀⠀⠐⠛⠲⠤⣄⣀
⠀⢀⡠⠖⠋⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠽⠗
⠶⣉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉
⠀⠀⠉⠙⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡛⢏⣙⡦⢵⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢫⠈⣇⣸⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡏⣧⣞⡁⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣈⣶⡄
⠀⠀⠀⣴⣁⣤⠤⠤⠶⠒⣶⠂⠀⠀⠀⡠⢿⡀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠉⠙⠲⢤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠔⠋⠀⠀⠙⠓⠒⠚⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠢⢌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡠⠞⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡽⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⢀⡴⠋⠸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠉⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠒⠉⠀⠀⠀⢦⠾⠃⠀⠉⠁⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣴⡀⠀⣀⣀⢀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠻⠋⡏⡏⡏⠍⠋⢀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣤⣤⣤⣤⣤⠤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣂⣀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣟⣚⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠻⠛⠛⠛⠿⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⠀⢀⠰⠂⠀⠐⠄⠀⠰⠖⠄⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⣏⣸⡃⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠇⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⡀⠀⠂⣶⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⡄⠠⠀⡠⠀⠀⠀⢡
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠛⠑⠊⠀⠀⠀⠄⢀⡤⢸⢁⠠⡀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⣥
⠀⠀⠀⠀⢸⣿⠀⡠⢄⠀⠄⠺⢸⠀⠀⠀⠀⠠⠖⢡⠆⣤⣞⣻⣆⠀⣱⣤⡀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⣿⠀⠁⢀⠀⠀⠀⠀⠐⠒⠚⠛⠙⠀⠂⢀⠀⠈⠉⠀⠸⠷⡽⡧⡀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⣿⡇⢰⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁⠛⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠃⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠹⣿⡄⠆⠀⠠⠀⠀⣼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠌⣯⠀⡀⠠⠘⠂⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢻⣷⡀⢂⠀⠤⠀⠈⠀⠀⠀⠤⠄⠀⠀⠀⠰⡆⠀⠀⠀⠀⠤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠐⡀⠀⠀⠀⠀⠀⢀⡎⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡈⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠙⠢⣄⡀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⡀⠤⠚⠁⣠⡴⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠈⠉⠒⠒⠂⠘⠧⠒⠐⠒⠈⠁⣀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠴⠴⠤⠤⠴⠶⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⣀⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣐⡀⢀⣿⣯⣽⡆⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⣿⣿⣿⣿⣿⠇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⡇⡃⠘⣿⣗⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣥⣥⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⠤⠭⠿⠧⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⡀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢀⠔⢈⢠⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡆⠔⠉⠀⠀⠙⣧⠀⠀⠘⠛⠀⠀⠀⠎⠀⠀⠀⠈⠿⣷⣄⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠋⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⡤⠀⠻⣿⣿⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠈⠁⠀⠀⠀⢻⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⢤⡔⠋⠀⠀⠀⠀⢀⡀⣀⡀⢸⣿⣿⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠻⡾⠃⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⣿⢼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣧⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣼⣿⣟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣟⣆⠀⠾⡫⠋⠀⠀⠀⠀⠂⢰⡇⠀⠠⠀⠀⠀⠙⠻⠿⠆⣰⡿⡿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣟⢟⣈⠁⠀⠀⣰⡗⠀⢠⢸⡦⠀⠐⠂⢄⠀⠀⠀⢀⣴⣟⡕⠈⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⢿⣷⣿⣷⣄⠘⠃⠀⠀⠀⠀⠀⠀⠠⠂⠈⠁⣠⣼⣿⣫⠞⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⣯⣳⣖⠂⠀⠀⢀⣉⣀⣠⣤⣶⠿⣛⣥⠞⢁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⣦⣬⣭⣭⣽⣷⣶⣿⣫⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠛⠻⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⣿⣦⣄⣠⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠻⣿⣯⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣽⣿⣷⣄⡀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣟⡛⠛⠋⠉⠉⠉⠙⠻⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠈⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣠⣾⠟⣿⣿
⠀⠀⠀⠀⠘⣿⣧⡈⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⠟⠋⢠⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣬⣭⣥⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠿⠟⠋⠉⠉⠉⠉⠉⠛⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣼⣧⣴⣿⣤⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⢸⣷⣷⣷⣧⣿⣿⣿⡀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣏⣉⣻⣿⣿⣿⣟⣥⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⡿⠿⠛⢛⣛⣛⣛⣛⣛⡛⠉⢿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⣉⣥⠶⠞⠛⠋⠉⠉⣉⡉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⣡⡴⠟⢉⡀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠁⠀⠀⠀⢠⠐⠁⣀⠀⠙⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣾⣿⡟⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠀⠀⠀⠀⢸⠀⣠⠊⠁⡀⠀⠙⢿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣿⡟⢀⡾⠁⠰⡄⢁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠞⠀⠀⠀⠀⢂⡠⠈⢻⣿⣿⣿⣷
⠀⠀⠀⣾⣿⡿⢁⡾⠁⠀⡐⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣰⡿⠃⠀⠀⠈⠠⠇⠀⢁⠀⠀⢿⣿⣿⣽⡇
⠀⠀⢠⣿⣿⠇⣸⠃⠀⢰⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠒⠂⠀⠀⠂⠀⠘⣿⣿⣿⣿
⠀⠀⢸⣿⣿⠀⣿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡺⠋⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢸⠀⠀⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠀⣿⠀⠰⠆⠘⠇⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡤⠀⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠀⣿⠀⠀⣇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠈⣿⣿⡄⢹⡄⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⠔⣶⠾⠿⡋⠛⠟⢦⡀⠀⠀⢀⡀⠀⢀⠄⠀⢀⣿⣿⣿⣽⠁
⠀⠀⠀⢻⣿⣷⣌⣷⠀⠀⢁⡀⡘⢦⠀⠀⠀⠀⠀⣼⡁⠀⠀⢉⠀⠀⢈⣦⠀⠀⠈⠑⠄⠈⡇⠀⠄⠀⠀⣼⣿⣿⣯⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣧⡀⠋⠡⡁⠈⠀⠀⠀⡀⢸⠈⠀⠀⠀⣨⠀⠀⠀⠈⣇⣀⠀⠀⠀⠀⢁⠸⠓⢀⣼⣿⣿⢿⡿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⠀⠈⠀⠀⠀⠀⡩⠻⣀⡀⠀⠀⠉⠀⠀⢀⣀⠏⢠⠀⠀⢀⠐⠃⠀⢠⣾⣿⣿⢫⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠃⠀⠙⠄⡀⠀⣀⠀⢀⡨⠉⠁⠈⣀⠐⠀⠀⣀⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠊⡏⠂⠀⠀⠀⠀⠀⠈⣁⣤⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣈⣁⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠰⣶⣶⣶⣤⣤⡤⠆⠄⠀⠀⠛⠛⠛⠛⠛⠛⠛⢿⣿⣟⡛⣛⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣤⠤⠤⠀⢠⠀⠀⠀⠀⠈⠭⢉⣠⣦⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣾⣾⣾⣶⣶⣶⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠂⡆⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣀⣁⣃⣘⣘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠉⢉⣉⣉⣉⣈⡀⠀⠀⠀⡴⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢨⡇⠀⢀⠎⠈⠢⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠤⡄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⡇⢠⠊⠀⠀⢱⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠣⢀⣀⣀⣀⠠⠔⠊





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠠⠦⣄⣉⣉⣛⣉⣉⣉⣉⠀⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣌⣷⣼⡿⠿⠟⠙⠋⠉⠉⠉⠉⠿⢿⣿⣦⣄⠉⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣑⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠙⡆⠀⠀⠀⠀⢠⣶⠾⢿⣷⣶⣶⣶⣤⠀⠀⠀⠀⡀⢃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⡈⠳⡄⠀⢀⣼⡇⠀⢸⣸⣿⣿⣿⣛⢠⡀⣀⠚⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣼⣯⣀⣸⣷⣤⣼⣿⢿⡿⢿⣛⣀⣤⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠶⠶⠶⠶⠶⠾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠒⣶⡶⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢈⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣬⣿⣿⣶⠖⠻⠓⠛⠷⠄⣨⡿⠿⠿⠟⠛⠳⣶⣤⣤⣤⣀⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣀⠤⠘⠛⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠁⠐⠒⠉⠽⢛⡻⢶⣤⣌⡓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⡋⠁⠊⠁⡀⢠⡀⠢⠈⠉⠒⠀⠄⠀⠠⠤⠠⠤⠔⠒⠋⣉⠤⠖⣀⡤⠘⡻⠶⣹⢿⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡉⠀⠀⢀⡠⣔⠲⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⡆⠀⠀⠀⠀⠀⠉⠁⠚⣣⢄⡈⠸⣋⢜⡽⣿⣶⣅⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑⠋⠀⣠⠶⠓⠉⠀⣀⢶⣄⠀⠀⠀⠀⠀⢀⠀⢠⠀⢠⡄⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡉⠙⠫⣦⡉⠺⣿⣽⡽⢿⣷⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢀⡨⠂⠀⠁⠀⠀⠉⠆⣈⠀⠠⡄⠘⠂⠈⠀⢀⣀⣈⣁⡀⠀⠀⠁⠘⠃⠰⠆⢀⡘⠃⠀⠀⠈⠻⣷⡌⠻⣿⣷⡽⣿⣧
⠀⠀⠀⠀⣰⢋⠄⠀⠀⡰⠋⠠⠀⠀⠀⠀⠀⢠⡀⠙⠃⣀⡀⠀⠀⠀⠀⠀⢻⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⠙⠃⢀⠀⠀⠀⠀⠈⠻⣷⡘⢻⢿⣞⣿⣧
⠀⠀⢀⡴⢡⠆⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠻⣿⢧⠀⠀⠀⠀⠀⣞⣦⣿⠀⠀⠀⢠⠀⠀⢠⠟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠻⣦⢫⢻⡾⣻⣧⡄
⠀⠀⣼⠃⠀⠀⣠⡞⠀⠠⣆⡄⠀⢠⠈⠃⠀⠀⠀⠀⠀⠙⣧⡦⠄⠀⠀⠀⠉⠁⠀⠀⠀⢀⡞⠀⢴⠏⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣀⠙⢷⡋⠾⣣⢿⣎⡄
⠀⣾⡇⠁⠀⢰⡿⠀⠐⠶⠷⠀⣤⡄⢀⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠆⠘⠋⠀⠈⣿⡜⢩⡞⣿⣤
⢠⣿⠀⠀⢀⠻⡄⠀⠀⠀⠀⢤⡀⠀⠾⠷⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⡽⠆⠀⠀⠂⠀⠀⠀⢸⣷⠋⣿⢹⣿⡆
⢸⡇⠀⠀⡘⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢙⡿⢂⠀⠀⠀⠀⠀⠀⠀⠻⠞⠉⠀⠀⠀⠀⠒⠂⠀⠀⠀⣿⡞⠛⣿⣿⣧
⣾⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢸⣷⢸⣿⣿⣿
⣿⠀⠀⠀⢿⠀⣄⣸⠏⠉⠉⠙⣶⣶⢶⣶⣶⣶⣦⣤⣼⣯⣁⣀⣀⣀⣀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⣿⡄⣿⣿⣿
⣿⠀⠀⠀⢳⠀⠐⠲⠆⠙⠃⠀⣿⠭⠭⠷⠆⠀⠁⠀⠸⣏⡍⠙⠻⠿⣿⣿⣯⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣓⣾⡇⠀⠒⠀⣭⠿⠀⣿⡇⢿⣿⣿
⣟⣹⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿
⢻⡏⠀⢀⡿⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣿⢷⣼⣿⣿
⢹⡇⡆⠀⢡⡶⠀⠀⠀⠀⣠⡄⠀⣿⣻⡞⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢆⣤⢀⡀⠀⠐⠂⠀⠀⠀⢸⣇⣸⣿⢿⣿
⠈⣿⠀⠀⠈⣰⠆⠀⠀⢠⡀⣠⠄⠈⣧⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⣈⣔⡿⠁⠀⠀⠀⠀⠀⢀⣿⣉⣿⡛⣾⡿
⠀⢹⣇⠀⠀⢡⠞⡀⠉⣩⠆⠀⢠⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠸⠷⢀⣛⡦⠀⣼⣏⢻⣞⣿⣿⠃
⠀⠀⢻⣆⠀⡀⠚⡱⠂⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣼⠍⣩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠈⠐⠀⣼⠇⣼⣿⣽⣿⠏
⠀⠀⠀⢻⣦⣉⠄⠰⣫⢆⠀⠀⠀⠀⠰⠆⠀⠀⠻⢮⣰⠋⠀⠀⠀⠀⠀⠀⡦⢠⡀⠀⠀⠀⠀⠀⠘⡷⣀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⣾⢃⣰⣾⣿⣿⠛
⠀⠀⠀⠀⢻⣷⣻⢆⡀⢀⢁⠀⠀⠀⠀⠀⠋⢀⣤⠀⠋⠀⠀⠀⠀⠀⠀⠀⣷⡸⠃⠀⠀⠀⠀⠀⠀⠰⠍⠃⠀⠠⠄⠈⠀⠀⠀⢀⣴⡿⢡⣿⣟⣿⣿⠃
⠀⠀⠀⠀⠀⠹⣷⣾⢦⡄⠲⡝⢦⡀⠀⢸⡅⡠⠀⠀⠆⢀⡀⠀⠀⠀⠀⠀⠷⠥⠶⠀⠀⠀⠀⠀⠀⢀⡀⠸⠆⢀⠀⠀⠀⢀⣤⡾⢋⢔⣵⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢻⣷⣽⣦⡈⠀⣙⠔⣈⡐⠁⠀⠀⠀⠈⠁⠀⠿⠀⢀⡄⠀⣤⠀⢀⠀⠀⡀⠐⠆⠀⠁⠀⠶⣬⠀⢀⣴⠿⠋⠔⣵⣿⣿⢟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣽⣦⡠⠂⢈⠟⢢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⢏⠉⣢⣴⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⣕⡣⣎⡳⢮⡽⠻⣶⠶⣀⣤⣀⠀⠀⠈⠉⠓⠀⠀⣀⣠⣤⣴⡶⢿⠛⢏⡡⣜⣶⣿⣯⣾⠛⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠩⣿⣿⣿⣣⡶⣍⡐⠚⠋⠥⠿⠟⠛⠛⠛⠛⠛⠛⠛⠶⢄⣉⣓⣦⣽⣾⣻⣿⣿⡿⠯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠟⢿⣿⣷⣿⣿⣶⣶⡾⣿⣿⣿⣭⠷⠶⠿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣩⣵⡶⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⢤⣶⣤⣤⣼⣦⣤⣤⣤⣤
⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡆⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣹⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠿⠶⠶⠶⠤⠶⠶⠶⠶⠶⠶⠶⠶⠿
⠈⢅⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⡄
⠀⠀⠑⠢⠤⠤⠤⠆⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠉⢉⣀⠀⠀⠀⠀⠀⠀⢀⡴⢿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠤⠀⢀⡀⠀⠀⠀⠀⣀⣈⣀⣼⠁⠀⠀⠀⠀⢀⡤⠒⠉⢸⡀⠈⠙⢆
⠀⠀⠀⠀⠀⣇⢠⢄⡠⠀⠘⠀⠀⠈⠦⡄⠀⠀⠈⡇⠀⠀⠉⠀⠀⠀⠀⠀⠜⠁⠀⠀⠘⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠉⢠⠋⢰⣄⠀⢀⡀⠀⠀⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠘⠁⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⢺⠐⠀⠈⠓⠋⠀⠐⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⠀⠀⢀⣀⣔⠇
⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠈⠀⢀⠀⠀⠀⠀⠀⠈⠚⠶⢾⢿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠑⠀⠀⣀⡀⠀⠀⠀⠀⠀⣆⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄
⢠⣠⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⠃⠈⠉⠳⡄⠀⠀⢠⣿⠋⠀⠀⠀⣀⣤⣬⣤⣾⣜⣂⣐⣒⣢⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠘⣆⢀⣾⡏⠀⠀⠀⠀⠈⠥⢤⣤⣤⣤⣤⣤⣤⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢹⣾⠋⠀⠀⠀⠀⠉⡉⠡⠉⠉⠁⠀⠉⠉⠙⠉
⠀⠀⠀⠀⣀⠄⢂⠄⠀⠀⠀⠀⠀⠀⢴⡈⣿⣷⣄⣠⠾⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂
⠀⠀⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⡠⠉⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠁⠈⠁
⠀⢈⠀⠀⠀⠈⠀⡂⠀⠀⠀⠀⡜⠀⠀⢴⠀⠐⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣀⣂⣀⠂
⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⠀⢦⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⠀⢃
⢠⠆⣶⣔⡐⠚⠒⠐⡆⠀⠀⠀⠁⠌⣀⣤⠶⠗⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠻⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢴⠊⢿⣿⣿⣿⣧⡈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⢁⠍⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⣅⡐⣿⣿⣿⣿⣯⡐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡨⢉⣤⣿⠏⠙⣿⣿⣿⣿⣿⣇⠀⠀⠰⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⡾⢃⣾⣿⠤⠤⣾⣇⠉⠻⢿⣿⣿⣷⡀⠀⠐⠆⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠸⠇⠸⣿⣠⠀⠠⠋⠙⠃⠀⠀⠉⠻⣿⣷⣶⣾⣶⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⣬⣿⣄⣀⣤⡾⡿⢿⣾⠃⣀⣉⠐⢆⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢻⣦⢉⣿⣿⠀⠀⠰⢾⡏⠀⠛⠉⠀⠀⠀⠀⣀⣄⠘⣿⣿⣿⣿⣷⣾⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣿⡷⣾⠋⠉⠶⠄⠀⡸⠰⡟⠀⠀⠀⢀⣦⣄⣤⠀⣄⠀⠟⠻⣿⣿⣿⣆⠀⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠘⡟⢀⣴⣤⣷⡴⣿⠈⢻⣶⣶⣄⣠⣿⣿⡶⠀⢸⡇⠀⠀⠉⢹⣿⣿⣿⡟⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠘⢷⣏⣙⣿⠋⠛⠷⣤⣤⣄⢿⣿⠛⠛⡿⢿⢷⡎⠀⠈⠰⣤⣷⣶⢿⣿⣿⣄⢀⠰⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠛⠻⠿⣤⣶⣤⠀⠈⠟⣠⣿⠇⣠⣿⠒⡿⠃⣀⣛⣿⠿⠿⠏⠀⢹⣿⡿⠿⢷⣴⠖⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆⠀⠀⢀⡀⢀⣄⠀⣀⣤⠀⢺⡿⢿⣿⣿⣿⣧⡀⢰⣿⡟⡟⠃⠀⠐⠛⠿⣿⡇⢀⣾⡅⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣤⠀⣴⠞⣿⢧⣤⣶⣿⣏⡉⠻⣧⠽⣿⣿⣿⣿⣿⣿⣿⣽⢻⣤⢛⣼⣿⣷⠋⣠⣾⣿⣷⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣿⡄⠉⠐⠉⢿⣷⡿⠋⣀⣹⣶⣾⣿⣿⣽⠟⣭⣿⠏⠸⣿⣿⡝⠋⣹⣤⣽⡛⠛⣯⡀⠀⢈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣿⣿⣷⢆⣤⣥⣿⣿⣿⡀⢉⡀⢺⣿⣟⣩⣿⣿⣿⣿⣶⣿⣿⣶⠷⢎⡁⠀⠚⠉⠙⠛⢿⡿⠄⢁⣤⠺⣿⠆
⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠈⣿⣿⣿⣤⣤⣈⣀⣨⡉⢿⡿⠉⢻⡿⢿⣿⠟⠛⠿⢟⢻⡇⠻⠛⢠⣀⣿⠀⠀⠀⢺⡆⢼⣧⣄⡈⢉⣥⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣯⣥⠘⣷⣀⣼⠓⢂⣠⣶⠶⡶⠋⠈⠿⠀⠀⣟⠻⢻⣤⠄⠀⠀⠛⣮⣉⣾⣧⣤⣽⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠘⠙⢿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⣿⡷⢏⣹⣿⣿⣧⣼⣇⠀⠀⠾⣧⣴⣿⡁⠀⠀⠀⠀⠉⠁⢩⣙⣿⠟⠉⠀⣡
⠀⠀⠀⠀⡀⢀⡇⣀⣼⣧⣷⣤⣿⣿⡿⠛⣿⣿⠉⣿⣿⡿⠛⢿⣤⣾⣿⣿⣿⣿⣿⡻⣿⣶⣾⡏⢁⣄⠙⢓⣤⣤⠄⣤⣴⠛⠛⠻⣶⣾⣷⡿
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⡙⣻⣿⢁⣀⠁⠙⡟⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⠟⢻⣧⣤⣶⣤⣉⣋⣛⣯⣤⣾⡇⢠⣿⣿⣿⣇
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣉⣿⣿⠖⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣷⠂⣹⣿⢉⠽⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣷⣄
⠈⠿⠿⢻⣿⣿⣿⠛⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡿⠋⠙⣿⣶⣿⡁⢙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠉⠙⠃⠰⠿⠿⢿⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣾⠿⠿⣿⡟⣛⣀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠈⠛⠿⠟⠿⠿⠛⠛⠛⠻⣿⣿⠷⠿⣿⣟⣿⠖⠋⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠏⠙⢻⡏⠉⣙⠛⢛⣏⠛⢹⣽⠟⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡏⢉⣹⣾⢋⡻⣦⡶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢾⣧⣴⡿⠻⠶⠟⠛⠷⠾⢿⣔⣶⡿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣷⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢦⣷⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡩⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣔⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢠⠹⣏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠏⠆⣿
⠀⠀⠀⠀⠀⠀⢰⡏⠄⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⠏⠀⠰⢹⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠄⠀⠀⠈⢷⣍⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⡾⠁⠀⠀⠀⠜⣧
⠀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠀⠀⠀⠙⠷⣤⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣤⠾⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣄⣂⡀⠄⣶⣿⣿⣷⠤⢀⣂⣨⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣧⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣧⣤⣥⣤⣄⣐⣀⣀⣀⣒⣀⣐⣀⣀⣁⣀⣈⣤⣬⣥⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣟⣻⡿⣿⠿⠿⠿⣿⡿⢿⣟⣻⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠶⢮⣭⣭⣭⣭⣭⣭⣭⣭⡵⠶⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⠛⠛⢶⡞⠉⠀⠈⠉⢉⣭⣭⣿⣿⡛⠉⠉⠉⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣼⣷⡀⠀⠀⠹⢦⣤⣤⣤⣴⣿⣿⣿⣿⣇⣀⣀⣤⣿⡿⣿⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠻⢶⣄⣠⣶⣽⣶⡉⠁⢻⣿⣿⣿⡟⠉⠉⠀⣨⣽⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣀⠀⠀⠀⠀⣩⣛⣿⣿⣇⣠⣴⠿⠿⢿⣻⣿⣶⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣙⠻⠶⣦⣤⣿⣿⣿⣿⡟⠉⠀⠀⠀⢻⣿⣿⣿⠇⠀⠉⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣌⣉⡙⠛⠋⣄⣀⣠⣤⠶⠟⢛⣉⣀⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠍⠲⡲⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡭⣿⣝⢟⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠤⣄⡀⢸⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢟⣛⢿⢿⡧⡇⣿⣿⢸⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⣿⣿⣿⣿⣿⣿⡏⢹⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣷⣿⡇⡏⠀⠀⠀⠀⠀⠀⢰⡞⣿⣿⣿⣿
⠀⠀⠀⢰⣶⣾⣿⣶⣾⣿⣿⢹⣿⣿⣿⣿⣷⠘⡀⠀⠀⠀⢸⣿⣿⣿⡏⠉⠁⠁⠉⠁⠁⠀⡆⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⢹⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⠀⢣⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣾⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡆⢸⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠿⠿⠿⠿⠧⠞⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣀⣛⣯⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⢈⣵⣶⣶⡶⢒⡶⠦⣴⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⣞⡻⠯⢩⠩⠍⠭⠉⠛⠛⠒⠶⠤⣄⡀⠀⠀⠀⢻⣶⣬⡿⢼⣷⣇⣼⡇⢺⠉⣽⣶⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⢉⠋⠓⠒⠦⢄⣀⠉⠛⠢⣄⣀⠿⡛⢦⡀⡇⠈⠻⣿⣿⣾⣡⡿⢹⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢪⣿⡷⠛⠁⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠉⠒⢄⡈⠙⢆⢨⣾⣷⣿⣾⣿⢋⣛⣻⣟⡃⣀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠫⢱⡿⠋⠀⠀⠀⠀⠳⣦⢄⠙⢶⣄⡀⠀⠀⠈⠁⠀⠀⠀⠄⠴⠖⠀⠀⠀⠙⢆⢸⣿⣿⣿⠛⠻⠿⠿⠯⣿⣿⣯⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢇⣿⠁⠀⠀⠦⢤⠤⠤⠀⠀⠀⠀⠹⢯⣿⣄⣀⡀⣀⣴⢶⡾⠋⠀⠀⣀⡀⠀⠀⢹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡀⣇⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠨⣿⣿⣿⡿⣯⠁⠈⠁⠀⠉⠁⠀⠀⠀⠀⠈⣿⡇⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣇⠹⡄⠀⢀⡖⠒⠀⠀⠐⠂⠀⡲⠌⠁⠐⠶⠖⠛⠁⠀⠀⠀⠀⠂⠐⠲⡆⠀⠀⢀⡿⢰⣿⠹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢤⢱⠀⢀⠙⣦⡀⠀⠄⣀⠀⠀⠀⢠⡤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⢦⣀⠀⠀⠀⠀⣠⠞⣱⡿⢧⠘⡌⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢣⡘⢀⢈⠛⢶⣤⣀⠑⠢⢀⣰⠀⠀⠀⢀⡜⠀⠀⠐⣦⠒⡆⠀⠀⠐⢋⡁⠤⠒⣋⣥⡾⠋⡅⢈⡜⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡙⢆⠱⣄⠈⠀⠀⠈⠛⠿⣶⣤⣈⡁⠒⠒⠠⠧⠤⠤⠤⠮⠦⠛⠒⠂⣉⣡⣤⡶⠟⠋⠀⠠⠞⣠⠎⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⢬⡓⢤⣀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠶⠶⢶⣶⣶⣶⠶⠾⠿⠛⠋⠉⠰⢀⡴⠓⢈⡴⢚⣡⠞⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⡙⠲⢬⣕⡲⠤⣀⡀⠀⠀⠀⠀⠄⠰⢠⣖⠀⣤⡔⡀⠺⠢⠘⠶⢃⣠⣴⣚⣽⡾⠋⣡⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⠀⠉⠙⠷⢿⣭⣍⣙⡓⠒⢶⡶⠶⢶⡶⠶⢶⡶⣶⣶⣾⣿⣿⠿⠟⣋⠁⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⠀⠀⠈⠉⠙⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣡⡴⢖⣫⣵⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⣶⣭⣭⣭⣄⣀⣀⣒⣒⣂⣛⣹⣭⣭⣶⣾⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠈⠁





⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡟⣉⣿⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠤⠤⠤⢤⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠉⠉⠉⠈⠹⢿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠤⠤⠤⠶⠾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⡌⠐⢲⡶⠲⠂⠒⠒⢒⣾⢻⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡔⠺⣾⡟⢀⣦⣀⡸⠟⡀⣸⠀⠀⠀⠀⠀⠀⠀⣥⣿⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⢈⣿⡋⢁⣼⡁⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠓⠲⣶⠶⣾⠟⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⢤⣧⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⡆⠀⠀⣞⠉⣷⠀⠀⣚⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠈⠋⠁⣀⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢸⡇⣼⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢀⠀⡤⠠⡄⠀⠀⣠⠒⢠⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⢸⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠀⠢⠴⠃⠀⠀⠙⠶⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣋⣸⡇⠀⠀⢨⣀⣵⠀⠀⠺⣤⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠐⠀⠀⢠⣤⣤⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡄⠀⠀⣔⠀⣆⠀⠂⢠⡁⣰⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢹⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠃⠀⠀⠈⠛⠋⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣈⣀⣀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⣷⠛⠛⣶⢶⡶⣶⠶⠶⣾⠶⢶⠶⠶⣾⠷⠶⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⢶⡶⡶⣾⢿⣿⣿⠾⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠛⠿⠿⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣶⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣾⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⡶⢒⠀⠈⢰⢠⣷⣄⠴⠖⣲⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⠫⣅⠢⣷⡓⠀⠀⠃⠠⣼⠟⢿⣿⣿⣿⠿⣍⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢝⣥⣏⡙⠛⣿⣧⣽⡧⢤⡤⢀⣽⣧⠀⣉⠁⡈⢰⣿⡿⡟⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣥⡾⣿⣦⣩⡿⢶⠏⣻⣷⣽⣿⣓⣚⠋⠻⣷⡟⢻⡇⠀⣗⠀⣼⣉⣳⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⠜⡋⣹⣾⡟⣃⢌⣹⣽⠟⣿⠂⠿⣿⡟⢿⣿⢿⣿⣹⣬⡻⣄⣷⣤⣉⡄⠙⣿⣏⢅⠙⢿⣶⡄
⠀⠀⠀⠰⠟⠩⣍⣩⡿⢧⡡⣿⡿⢿⣼⠃⠀⣀⣨⠀⣿⡜⣿⣷⡷⣾⣿⣿⠿⣿⣿⣿⣮⣿⡟⠻⠿⣶⣎⢻⣎⡙⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠛⠀⠀⠀⠀⠙⠛⠋⢗⢈⣇⠸⣿⣷⡹⢻⣿⣶⣮⣿⡿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣩⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⢻⡓⠉⠹⢿⠿⣿⣿⣞⣷⣶⣌⠻⠿⠿⣿⡿⠯⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠨⠀⠀⢀⠘⠀⠛⠩⠻⢿⣏⡿⢷⣿⠓⠪⠱⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠉


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣦⣷⣴⣤⣶⣤⣾⣵⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡡⠤⠐⠒⠀⠠⠤⣍⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣠⣋⣀⡀⣀⠀⠀⠀⠀⠀⠙⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡠⠊⡼⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⢠⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⡆⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠋⠀⠀⡌⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⢆⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⡠⠊⢁⡠⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠲⢄⣹⡛⣻⣒⣒⣊⡡⠤⠒⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠤⠒⠒⠒⠂⠤⠭⢉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⡀⠀⠀⠀⠀⠀⢸⠀⠀⡰⢂⡲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⢸⠀⢰⢱⠁⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢸⠀⣌⡌⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⡟⠁⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠱⣄⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⢸⠀⡇⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣑⠢⠤⣉⣒⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⢀⡆⢸⠀⢸⠀⡇⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⠬⠭⢑⣒⡠⠤⠬⢉⣑⡒⠒⠒⠉⣠⠏⠀⠘⡀⢱⡀⠀⡠⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠒⠒⠩⠭⢍⣄⣀⡀⠀⢣⠀⠈⢉⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣷⡼⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⣒⣛⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠿⣦⣄⠂⠄⠴⡤⢄⠀⠀⠀⠀⠀⢀⣾⠋⡀⠠⠀⠄⠉⠑⡍
⠀⠀⠀⠀⠀⠀⢰⡇⣰⣯⢁⢛⣤⣤⣌⢘⣷⠀⠀⠀⠀⣾⡇⢀⠉⢲⡄⠀⣀⡤⠈
⠀⠀⠀⠀⠀⠀⣸⡃⣧⣽⡀⠨⠤⢽⠈⢧⠄⢌⡀⠀⠀⣿⡇⠰⠀⠀⠛⠉⠁⠐⠀⠄
⠀⠀⠀⠀⠀⠀⢹⡇⢻⣧⣀⡠⠀⡠⠤⠞⡀⣼⡇⠀⠀⠘⣷⣄⠘⠠⠄⠠⠈⠀⠈⠀⠀⠀⠀⣀⢀⠀⠀⠀⣼⣙⠢⡀
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠻⣿⣿⣾⡥⣬⣤⣾⡷⠃⠀⠀⠀⠈⠙⠿⢶⣶⣶⠶⠏⠀⠀⣠⠖⠉⢡⠀⡤⠁⠀⣿⠿⣶⡃
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠈⠙⠛⠿⠿⠟⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣌⠃⠀⡄⠀⠘⠃⢉⢧⠙⣧
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⠛⠋⠉⠉⠙⠛⣦⠀⠀⠀⢀⣄⠀⠀⣯⢈⡁⠀⠀⠷⠄⣀⣈⣈⢸⠀⢸⡆
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣠⠴⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⠟⠀⠀⠀⠸⣿⣿⣄⣿⣎⢡⡀⠀⠀⠀⢠⣄⡹⡾⢀⡞⠁⢀⡴⢦
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢉⣰⠖⠛⠛⠛⠛⠛⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⢌⠻⣯⣻⣶⣄⣀⣈⣛⣀⣼⠟⠁⢼⣁⣀⡜⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⢐⡋⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣻⣿⣍⠉⠛⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠶⠚⠒⠲⠦⠤⠦⠤⠤⠴⠶⠖⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⣀⣼⣍⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣝⣫⣍⣉⣏⣹⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠛⠛⠛⠓⠂
⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠈
⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠊⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄
⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢧⠴⠋⠀⠀⠐⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⢦⢤⣄⣀⠀⣸⠁
⢠⠴⠒⠒⠒⠒⠚⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⢄
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂
⡇⠀⠀⠀⡟⠆⠀⠤⠤⠀⠀⠐⠁⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⡇⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⠤⠤⢤⣀⠀⢀⠀⠀⢀
⡇⠀⠀⢠⣧⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢀⠀⠀⢸⡟⠦⠤⣒⣒⣨⡥⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠔⠀⠀⠀⠀⠀⠉⠒⠲⠤⠔⠁⠸⣀⠀⠄⠀⠀⠀⡆⢧
⠀⠀⠀⠀⠙⠿⠛⠉⠉⠉⠀⣠⡤⠶⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠈⠉⠒⠒⠲⠾⣅⡼
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠿⠴⠟⢶⠗⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⢉⣉⣉⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢤⢄⠀⡀⢢⣀⣠⣿⠤⠤⠤⠒⢳⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢳⣡⠸⡀⠀⠻⠀⠀⠀⠐⠂⠈⠀⠈⡇⠀⠀⠀⢰⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⣇⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠒⠀⠁⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠰⣏⣝⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣍⣹
⠀⠈⠻⠻⣿⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⡿⠟⠟
⠀⠀⠀⠀⠀⠻⢯⡷⣄⡀⢀⢠⠠⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡠⠤⡄⡀⢀⣴⢮⡽⠋
⠀⠀⠀⠀⠀⠀⠈⠹⢿⣏⠟⣴⠟⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⠻⣙⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣣⣼⢟⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣻⣧⡜⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠋⠈⢻⣯⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠙⣌⠂
⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠑⢄⠀⠀⡠⠊⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢎⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣧⣀⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⢻⣧⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠙⢿⣦⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⢠⣾⢞⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡳⣷⡄
⠀⠀⠀⣰⠟⣱⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣎⠻⢆
⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂

⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢰⡇⠀⠀⠀⠀⠉⢳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡐⡇⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠻⣤⡀⠈⠳⣶⣤⣈⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡥⠀⠀⢀
⠀⠀⠀⠈⠙⠷⣾⣿⣿⣿⣷⣦⣉⣻⣟⠂⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣼⣷⣔⡁⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⣿⠡⣓⣤⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢜⠁⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⢀⣤⠏⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢷⠀⢀⠀⠀⢀⠀⣀⣴⣿
⣿⣶⠦⠄⠀⠂⠀⠈⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣻⣿⣿⡿⠃
⠙⠻⠶⣤⣤⠤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠄
⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠒⠐⠚⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⠞⠓⠂⠐
⠀⠀⠀⠉⢁⢤⠁⠀⠀⠀⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣟⡽⠁
⠀⠀⣾⡤⠴⠐⢦⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⡀⢸⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⢳⡴⠀⠚⠇⠰⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠸⡇⢹⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⢹⣿⢀⣄⣄⣰⡶⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠻⣏⡭⠵⢿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⣭⣿⡇⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢀⣀⠀⠀⠀⠀⡴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠺⢍⠀⠀⡄⠀⣀⡈⠃⠀⢀⢿⣦⡆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢇⣀⠙⠓⠀⠉⠀⠛⠀⣤⡀⡩⡏⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣾⠃⠼⠟⠀⠀⠀⣼⠀⠀⠀⠉⠁⢑⢽⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢄⠐⣿⡰⣶⠂⠀⠀⠀⠟⠦⣀⡠⠈⠉⢹⢸⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢿⣧⡱⣿⣧⠁⢔⠀⠀⠀⠀⠀⠈⠉⠺⠗⡌⣸⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢿⣷⣜⢿⣷⣍⠀⢴⠀⣤⣤⠐⠿⣢⠞⣹⠃⠀⡶⠋⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠻⣿⣦⣻⠷⣵⣢⡤⣭⣍⣐⣬⡴⠛⠁⢀⡿⠁⠀⠀⣴⠏⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡿⣿⣦⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠈⣷⣦⣤⡾⠁⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⢠⣤⠿⠝⠛⠛⠺⢶⣦⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⢠⣿⠁⠀⠀⣖⠛⠓⢦⡀⠉⠉⠙⠓⠿⠿⠷⠧⠶⠶⠶⢶⡶⣶⠶⠽⠟⠛⠋⠁
⠀⠀⠀⠀⠻⠷⣾⣿⠿⣷⣀⣼⠇

⠀⠀⠀⢀⣠⠖⠛⣿⡆
⢀⡴⣾⣿⣷⣶⣾⣿⣦⡀
⣿⡟⠁⠀⠀⠘⣧⠤⣾⣿⡄
⢹⣦⠀⠀⠀⠀⠘⣷⠟⠋⣹⡄
⠀⢻⡀⠀⠀⠀⠀⠘⢷⣽⣿⣿⡄
⠀⠀⢻⣆⠀⠀⠀⠀⠈⢻⣦⡿⠟⣛⣙⣛⡓⢦⡀
⠀⠀⠀⠹⣕⡤⣤⣤⡴⢞⣩⣶⣿⣿⣧⡀⠉⣳⠇
⠀⠀⠀⠀⠈⠙⠛⠿⠚⠋⠁⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠻⣟⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⢦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠦⡀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠢⡀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠢⡀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⠒⢄⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡁⠢⣄⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣙⠢⣄⣙⠲⠦⣍⡒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣛⣶⣬⣟⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣶⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⡴⠛⢉⡻⣟⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠞⠛⠛⣯⠉⠉⠳⣏⠀⣴⠋⠀⠸⡆⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢀⡴⠞⣩⠞⠉⠀⠀⠀⠀⢸⣆⣤⣤⡽⠶⠷⠶⠖⠒⠓⠒⠲⠾⢦⣄⣀
⠀⠀⠀⠀⢀⡴⠋⢀⡼⠃⠀⠀⢀⣤⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⢠⠞⠀⣠⠟⠀⠀⠀⣴⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⣰⠏⠀⣸⠏⠀⠀⠀⣼⡟⠉⠁⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⢠⡏⠀⣰⡏⠀⠀⠀⣸⡏⠀⠄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⡾⠀⠀⣿⠀⠀⠀⢀⡿⠀⠈⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢰⡇⠀⣸⡇⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠀⠀⣿⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡏⠀⠀⣿⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣇
⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⡀⠀⠀⠀⢠⠞⠋⠙⣧⠀⠀⠀⠀⣼⠃⢻⡄
⣷⠀⠀⢹⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠹⣄⠀⠀⠁⠀⠀⢸⡄⠀⠀⢹⠚⠛⠛⠋⠁⠀⠈⣧
⣿⠀⠀⢸⡆⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⠦⠤⠴⠞⠛⣷⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹⡄
⢸⡀⠀⠸⣇⠀⢀⡟⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠛⠛⠀⢀⣀⣤⡤⠶⢤⡈⣧
⠘⡇⠀⠀⣿⠀⢸⠇⠈⠇⠈⠉⠉⠉⠁⠀⠀⠀⢸⠃⣿⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⢳⢹
⠀⢻⠀⠀⢻⡆⣾⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣿⠀⠀⢹⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇
⠀⠘⣇⠀⢘⣷⡇⠸⣶⠶⠶⠶⠦⢤⣤⣄⣀⣀⢀⡷⣿⠀⠀⢸⣇⠀⠀⠀⠀⠄⠀⢀⣀⣈⣤⣤⣤⡤⠶⠺⡇⠈⡇
⠀⠀⢻⡄⠀⢸⠃⠀⡟⠀⠀⠀⠀⠀⠀⠈⠉⢹⢻⠃⣿⠀⠀⣾⢧⣀⣤⡤⠶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢻⡀⡇
⠀⠀⠀⣧⠀⡿⠀⢠⡇⢀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⣿⠀⠀⣿⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢧⡇
⠀⠀⠀⠘⣦⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⣿⠀⣸⠃⠀⠀⢀⣀⣠⣤⡤⠶⠶⠶⠚⠛⠋⣡⡞
⠀⠀⠀⠀⢸⡇⠀⠘⢷⣄⣀⣀⡀⠀⠀⠀⢀⡿⢸⡇⡟⠀⠀⠉⠓⠿⠶⠶⠛⠋⠉⠀⠀⠀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠓⠋⣸⠇⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⣤⣤⣄⣀⣀⣠⣴⣯⣤⡤⠶⠖⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠐⠲⠤⠤⠤⠤⢤⠤⠤⠴⣶⢶⢾⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡽⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣟⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠛⣛⣿⢿⣿⠑⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣙⡎⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠹⠀⡆⡆⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣱⢳⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠦⠴⠶⠞⠳⠻⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡼⠇⠸⣤⣄⣄⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣤⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣮⠉⢧⠀⠀⡎⠘⢿⣿⠉⠉⢏⠉⠉⠉⠉⢩⠉⣴⣦⡶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣡⡀⣷⠀⣸⣿⡖⠀⢈⢄⠀⠀⠀⣸⢴⡿⠈⢅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣿⡀⠉⠻⣿⡻⣿⡇⣠⢞⣺⣧⣤⡰⠣⠋⠁⡐⠕⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠈⠻⣾⣿⡿⠋⠁⠀⣨⡧⠁⠀⡰⠋⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣟⣷⠤⣠⣿⣿⣀⡠⣤⣾⢟⣷⣀⣠⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠶⠾⠿⠛⠛⠛⠉


⡿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣤⣤⣠⣴⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣍⣿⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠻⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⢦⣤⣤⣤⣤⣀⣀⠀⠀⣼⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠟⢷⣄⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢰⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠈⠉⠛⠛⠳⠶⠶⢶⣤⣤⣤⣤⡀⠀⠀⣿⠀⢠⡿⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡟⠀⢸⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⡿⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢸⡇⠀⠀⢠⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣼⠃⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⣦⣤⣄⣸⣇⣀⣀⣿⣤⣤⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡡⠊⠁⠀⠉⢆⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠈⠉⠉⠳⣍⠉⠒⣄⠀⠀⢸⠀⡇⠀⠀⠀⠀⢨⠀⠇⠀⠀⢀⠔⠚⣩⢝⡉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣠⠖⠉⠀⠈⠆⠀⠘⡆⠀⢀⣃⣧⣀⣀⣀⣀⡎⡜⠀⢀⡔⠁⠀⡔⠁⠀⠈⢢⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠜⠁⠀⠀⠀⠀⢸⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⢎⡀⠀⡜⠀⠀⠀⠀⠀⢱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⠊⠀⠀⠀⠀⣠⠚⠁⠀⠀⢀⡠⠤⠐⠒⠒⠂⠤⣀⡀⠀⠀⠀⠉⢲⡁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⠎⠀⠀⠀⠀⡔⠁⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⡜⠀⠀⠀⠀⡜⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃
⠀⠀⠀⠀⢠⠁⠀⠀⢠⠁⠀⠀⠀⢰⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠸⡆⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠸⠀⠀⠀⡌⠀⠀⠀⠀⡞⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢻⠀⠀⠀⠀⠀⢡⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⢠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡌⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢲⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⢠⠁⠀⠀⣀⠠⠤⠤⠤⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠄⢀⣀⡀⠀⠀⡆⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⢀⠴⠾⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠗⠒⡄⠀⡇⠀⠀⠸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⡜⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠁⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠃⠀⠀⠃⠀⡇⠀⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠄⢰⠀⠀⢰⠁
⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠁⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡇⠀⢸⠀⠇⡘⠀⠀⡸
⠀⠀⠀⠀⠈⡄⠀⠘⡄⡇⠀⠀⠀⢸⠀⠉⠉⠈⠉⠉⠁⠀⠀⠐⡖⠒⠒⢐⡆⠈⠉⠉⠉⠉⠉⠉⠀⠀⠃⠀⢸⠀⣷⠁⠀⢠⠃
⠀⠀⠀⠀⠀⢃⠀⠀⠱⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣠⠃⠀⢀⠏
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⣄⡇⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⢃⠀⠀⠀⠉⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠄⠒⠊⠁⠀⠀⡦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⢤⢤⣀⣀⣀⣀⣀⣀⣀⣠⡽⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⢧⡀⠻⣤⣀⣀⠀⠀⠀⠁⢈⣀⣀⡴⠃
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠙⣆⡧⢀⣈⡉⠉⠉⠉⠉⠁⠀⣶⣛⣉⢉⣙⣦
⠀⠀⠀⢠⣶⠋⠙⢫⡳⡀⠀⠀⠘⡇⠀⠀⢩⢭⠉⢹⠀⠀⢸⡷⠤⠤⠤⠤⢾⣀
⠀⠀⠀⠇⡇⠀⣠⠤⣷⢹⠀⠀⠀⣷⠀⠀⠘⡫⠆⢸⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇⠉⠓⠢⢄⡀
⠀⠀⠀⡇⢧⡠⠗⣶⣹⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⣾⠑⠒⠲⠶⠒⠊⡇⠀⠀⢀⠔⠉⡆
⠀⠀⠀⠘⢾⣤⣞⣡⡧⠏⠀⠀⠀⢸⠉⠒⠦⠤⠴⠚⠀⠀⢻⠀⠀⠀⠀⠀⠀⡇⡌⡀⡜⢰⠁⢩⣦
⠀⠀⠀⠀⠀⠈⢉⣈⣱⡦⠤⠤⠴⠚⠓⠤⣄⣀⣀⣀⡤⠔⢻⡄⣀⣀⢀⣀⣠⠏⠙⠓⠷⠞⠛⠛⠉⠑⠢⢄⡀
⠀⠀⡠⠤⢤⠴⢅⣀⡤⠟⠀⠀⠀⠀⠀⡴⠶⠶⣆⠀⠀⠀⠀⠉⠛⠛⠚⠛⠁⠀⢠⠤⠖⠢⠤⢤⡀⠀⠀⠀⠀⠁
⠀⠀⠑⠲⠦⠤⠤⠃⠀⠀⠀⡆⢉⠉⣹⡯⠭⢭⡿⠶⡉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⣤⡤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠭⠭⠟⠁⠀⣎⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠙⠉⠉⠠⠦⠄⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠉⠙⠉⠛⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠰⠧⠤⠄⣀
⠀⠀⢀⣠⡄⠒⠛⠓⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠑⢢
⣀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣌⠳⠆⠈⠉⠉⠉⠉⠠⠾⢋⡴⢆⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠖⠉⠁
⠀⠀⠀⠀⣿⠛⡟⠀⠠⠤⠤⠤⠤⢀⡴⣛⡾⠁⣀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠠⠴⠶⢌⡷⡄⠚⠛⠋⠁⠈⠻⢄⣘⡷
⠀⠀⠀⡌⢿⠏⠀⠀⠀⠀⠀⠀⢀⢮⡙⠁⠀⠀⠀⠀⠀⢰⠄⠀⢴⠄⠀⠀⠀⠀⠀⠈⠳⣡⡄⠀⠀⠀⠀⠀⠀⠹⠇⣴
⠀⠀⣴⠃⠘⠀⠀⠀⠀⠀⠀⢠⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⡸⠂⠀⠀⠀⠀⠀⠀⢸⠉⡆
⠀⠀⣅⢠⡆⠀⠀⠀⠀⠀⠀⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣴⠀⠀⠀⠀⠀⠀⢸⠀⡘
⠀⠀⠈⠲⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠻⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢴⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠃⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠓⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣀⠀⠀⠀⠀⠙⠿⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⣴⠷⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀⠀⠀⠀⠈⠛⠛⢷⡾⠷⠾⠟⠛⠉⠀⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⢤⣤⡄⠀⢸⡂⠀⠀⣿⠂⢠⣤⡤⠰⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢹⠃⠀⠀⣟⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣼⡁⠀⠀⠉⠉⠉⢹⡇⠀⠀⣿⣉⡉⠉⠉⠀⠙⣧⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢸⠉⠳⣄⠀⠀⠀⠘⣇⠀⠀⠟⠈⢡⠀⢀⡴⠛⠋⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡌⠀⠀⠈⢳⠄⢀⡜⠛⠳⠤⠀⠀⠈⠾⠯⠤⢀⣸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⣧⣤⡤⠰⠛⠲⠞⠙⠲⠀⠀⠀⠀⠸⠀⠀⠀⠈⠉⠑⠒⢠⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠷⠶⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⠤⠤⠀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠈⢤⠤⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⠀⡀⠀⠀⣇⣀⣀⣀⡈⠉⠿⢂⠒⠒⠀⢤⣤⣄⣀⡀⡀⢀⡀⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠉⣟⠉⠉⢻⠋⠹⠟⠛⢩⣽⠛⠏⣭⡅⠴⡉⢹⠖⠒⠐⡟⠈⢿⡁⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠋⢉⠉⠹⢇⠿⠲⠶⢶⡄⠀⠀⣰⠴⠶⢧⣸⠇⠀⠈⠙⠆⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣄⣠⣾⡦⢾⣿⣦⡀⠀⠀⠀⣠⡟⢹⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠠⣿⡟⠛⠿⠀⠙⠁⣠⡛⠀⠱⠀⠀⠀⡛⠁⡾⢷⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣥⠄⠒⠒⠈⢀⣍⠉⠀⢿⠀⠀⣠⠃⢸⠀⠀⠳⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠂⠈⠉⣹⠏⠉⢹⣿⠀⢠⡋⠀⠿⢢⡄⠀⢣⣀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠛⠉⠠⠷⠤⠰⠾⠛⠳⡄⠙⠾⠆⠾⠇⠀⠈⠁⠖⠁⠶⠶⠄⠀⠀⠀⠀⠠⠶⣷
⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠀⠀⢀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄
⠀⠀⠀⠀⠀⡌⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⡀
⠀⠀⠀⠀⠀⠏⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⢃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠧⠤⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠐⠀⣠⣄⣠⣀⣀⣀⣀⣀⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⢞⣛⡿⢿⠿⣛⣛⣫⣿⣬⡁⣿⡏⣨⣿⣯⣛⣛⣛⠿⠿⢿⣛⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠶⢂⣀⣀⣀⣀⡰⠶⠬⠓⠉⠘⠫⡴⠖⣀⣀⣀⣀⣀⠶⢦⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣓⠲⢄⠈⠑⢤⣀⣀⣠⢤⣤⡴⢶⡶⠖⠲⣶⡦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡴⠊⠹⠀⠉⣳⠵⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢙⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⢀⡷⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⢢
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⠒⠂⠤⣘⡴⢋⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇
⠀⠀⠀⠀⠀⣠⡫⠒⡖⠀⠀⠐⠒⠢⡬⠔⠒⠂⠛⠛⠒⠢⠭⣀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡄
⠀⠀⠀⠀⠠⡝⠀⡸⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧
⠀⠀⠀⢀⠃⠁⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⣦⠀⣀⡟
⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡡⠀⠀⠀⠀⢴⣛⠽⢻⣧⠖⢒⠁⡨⢿
⠀⠀⠀⠰⠘⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡴⠒⠒⠛⠛⠲⡄⠀⠀⠀⠀⠀⡙⠦⣴⣒⣀⣠⣤⣄⣼⡽⣷⠛⠉⠀⠈⡇
⠀⠀⠀⠀⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⢠⠀⢰⠀⠀⠀⠀⠀⡇⠀⠀⢇⠀⠀⢶⠄⠀⠀⡿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⡤⠔⠒⠒⢶⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⡇⠘⡆⠀⠀⠀⠀⡇⠀⠀⠈⠓⠒⠂⠀⠠⢺⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⡁⠀⠀⠀⠀⣼⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢣⢹⠀⠱⡀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠒⠒⠒⠉⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡈⡆⠀⠱⡀⠀⠀⢻⡀⠀⠀⠈⠙⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⡼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢣⢳⡀⠀⠘⢄⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠣⡑⢄⡀⠀⠉⠒⠒⣳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠉⠉⠙⠛⠒⠉⠙⠲⠤⣀⣀⣀⠀⡀⢀⣀⡼⠥⠤⠄⠐⠒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣂⠭⠉⠉⠩⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢸⡐⣲⣙⣹⣗⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠿⣉⡙⣉⡭⠷⢽⠿⣆⣀⠀⠀⠀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⣥⠀⠀⠈⠛⣿⣉⣷⣯⡿⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠻⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢓⣬⣿⡐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠄⣂⠈⠩⢭⠁⣒⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢔⡠⠂⠁⠀⠀⠀⣎⠁⠀⡌⠰⢅⠢⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⢢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠗⢄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⠌⠃⠀⠀⠁⠀⠀⡀⠴⠞⠋⠈⠁⠠⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⣄⡀⠠⠤⠤⠤⠴⣾⠛⠐⠀⠀⠀⠀⢘⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⢃⠘⠀⠀⠀⠀⣀⠀⠙⣇⠀⠀⠀⠀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⡌⡇⢸⡇⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⣯⢀⠁⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡕⢌⠪⣁⠀⢲⠆⠀⣠⡄⠀⠙⢇⢀⡴⡡⡁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢮⡒⢆⠤⠀⡙⠃⠠⠈⣁⣈⠊⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⣁⡉⠀⣀⠀⠨⢝⠒⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠾⠿⠿⢿⡿⣿⣿⣿⣉⡉⡉⣉⠒⡒⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠍⡉⢉⢁⢀⡐⠒⠒⠂⣀⣀⡤⠐⠀⡈⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠂⠐⠀⠐⠊⠁




⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⣠⣴⠶⠞⠛⠛⠛⠛⠛⠛⠶⢶⣤⣀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣤⡾⠋⢉⣠⣴⠶⠟⠛⢛⠛⠛⠷⢶⣤⣀⠙⠻⣦⣄⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣴⠟⢁⣤⠾⠋⣅⠀⠀⠀⠀⣿⡅⠀⠀⠀⢀⡍⠻⢦⣄⠙⢷⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠃⣠⡟⠁⠀⠀⠹⠇⠀⠀⠀⢿⠇⠀⠀⠠⠿⠁⣀⡀⠙⣦⡀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣼⠏⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡌⢿⡄⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣸⠏⠀⠈⠙⠃⠲⣦⣀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠛⠉⠀⠈⢿⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⣿⠰⠶⠶⠶⠆⠀⠀⠀⠀⠈⢹⣏⣿⠁⠀⠀⠀⠀⠰⠶⠶⠶⠶⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢹⡆⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢀⣾⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⢻⣄⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢃⣾⠃⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣦⡀⠀⠀⣰⡆⠀⠀⠀⣾⡆⠀⠀⠐⣶⡀⠀⠀⣠⡾⠁⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣦⡈⠻⢶⣄⡋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣃⣴⠾⠋⣠⡾⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠈⠛⢶⣄⡉⠛⠳⠶⣤⣤⣬⣤⣤⡶⠶⠛⠋⣁⣴⠞⠋⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠉⠛⠻⠶⣦⣤⣤⣤⣤⣤⣤⡶⠾⠛⠉⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣸⣿⣀⣀⣸⣇⣀⣀⣿⣇⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣴⠟⠋⠉⠉⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠶⠶⣶⠀⠀⣶⠶⠶⠿⠀⠀⠀⠀⠀⣿⣷⠀⠀⢀⣀⣀⣤⣤⣴⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⠛⠛⠋⠉⠉⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⣿⡀⢶⡄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣧⠈⠻⣦⣤⣤⡀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠁⠀⠀⠙⢷⣤⣀⣀⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⡀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⢰⢻⡇⣾⣩⠆⣶⠙⡇⢰⢠⡎⠁⢐⣩⡆⢹⠋⢷⢠⠎⢱⡄⡖⠙⡆⢾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘⠃⠙⠒⠂⠙⠒⠃⠘⠀⠓⠒⠙⠒⠃⠘⠂⠙⠈⠓⠚⠁⠃⠀⠃⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠂⠀⢀⠀⠀⠀⠀⠀⠘⢣⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠉⠉⠀⠂⠀⠀⠠⢄⣀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢤⣀⠡⠄⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠒⠛⠂⠐⢢⣀⢈⢁⡒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠭⠀⠀⠀⠀⠄⠐⠀⠒⠐⠶⢶⢎⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠉⠙⣫⣍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡦⠄⠠⠠⢠⣄⡀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣔⣤⣀⣩⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠦⠉⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠁⠐⠶⣀⠂⢀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠥⠌⠀⠀⠀⠸⠀⠀⠈⠠⡥⡍⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣃⣥⡠⠤⠤⠀⠀⢀⡤⠔⠋⢠⠤⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣔⣓⡶⠒⠒⠀⠀⡄⠄⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠰⣯⡽⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠓⠒⠐⢊⠔⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⢣⡀⠀⠀⠀⠀⢀⠈⠒⠄⠠⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢆⠀⠀⠀⠀⠀⠀⠁⠀⠲⢜⡀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣚⡂⣀⣀⢀⣀⡀⠀⠀⠀⢙⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠕⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄

⠀⠀⠀⠀⠀⢀⣴⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄
⠀⠀⠀⠀⠀⡀⠤⠥⠤⠬⣙⡆⠀⠀⠀⠀⠀⠀⣾⡇⢀⠩⠐⠶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠞⠉⠰⡄
⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⣰⡿⠀⠃⢀⡴⠒⣈⣉⢲⠀⠀⠀⠀⢠⡇⢀⠀⠤⠀⣠⣽⣷
⠀⠀⠀⣠⣄⣀⣀⣀⠀⠀⠀⢦⣇⠀⠀⠀⢰⠉⠀⢠⠀⣸⠀⠀⢈⣃⣸⠀⠀⠀⠀⢸⠀⢰⠖⠒⠐⠂⡟⢿⡄
⠀⠀⠀⢩⡖⠀⡠⠠⠄⠀⠀⠈⡟⡀⠀⠀⣿⣀⠈⠛⠂⡇⢠⠀⢠⠂⢀⢴⠀⠀⠀⠛⢠⢀⡄⠀⢠⠀⡅⠀⣿
⠀⠀⣏⣸⡇⠀⠑⠄⡀⠄⠀⠀⣇⠁⠀⠀⠉⠛⠀⠠⡆⠀⣄⡶⠂⠀⠙⢺⠀⠀⠀⠀⢀⢂⡀⠀⠠⠄⣃⣨⡽
⠀⠀⠈⠈⠃⠠⠀⠠⠀⠀⠒⠂⠉⠀⠀⠀⠀⠈⠓⠤⠇⣄⣏⠀⢈⡤⠴⠊⠀⠀⠀⠙⠉⠈⠀⠤⠤⠬⠥⠞⠁

⠀⠀⠀⠀⠀⣀⡤⠶⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣒⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣀⡀
⠀⠀⠀⢠⠏⢩⠖⠒⠲⣿⠿⣦⠀⠀⠀⠀⠀⢠⠎⠕⣒⣒⣒⠫⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⢸⠀⡇⠀⣀⠠⠉⡀⠬⣇⠀⠀⠀⠀⡘⢘⡏⠀⠀⠀⡿⡕⡇⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠨⡇⡇⡿⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⣧⡞⠁⠀⠀⠀⠀⢹⢿⠀⠀⠀⠀⢀⡿⠃⠀⠀⢸⢸⢀
⠀⠀⠀⢀⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⡀⠀⠀⠀⢠⡈⠺⡄⠀⠀⠀⢸⡇⠀⠀⠀⣸⢼⠈
⠀⠀⠀⠺⢻⡧⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⡼⠂⠐⠇⠀⠀⠀⠈⡇⠀⢱⠀⠀⠀⠸⡇⠀⠀⠀⠉⢸⠀⠠⠔⣎
⠀⠀⠀⠀⠀⢸⠸⠟⢀⣤⣤⢷⣾⠀⠀⠀⠇⡀⠀⠀⠀⠀⠠⠜⠁⠀⠸⠂⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠣⠤⢾⠀⠀⠀⠩⠛⠋⠁⠀⠀⠀⠘⠓⠶⠶⠤⠴⠶⠶⠖⠛⠀⠀⠀⠀⠀⠚⠁⠀⠠⠷⠒⠊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣿⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠂⠀⠀⠀⣀⡀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠤⠤⠤⠀⠀⢀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠩⠉⢉⣁⣀⠀⠠⢼⡯⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠉⠐⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣒⡒⠒⠆⠀⠀⡜⠧⡀⠰⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠄⠀⠠⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠀⠤⠭⠭⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠁⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⠉⢁⣐⡒⠢⠠⠭⣁⣺⢀⣀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⡒⠒⠢⠬⠍⣉⣒⠒⠼⣿⠉⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠤⠤⠤⠭⠍⣁⣒⠂⠠⠬⢹⠿⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠙⠒⠂⠤⢤⡨⣄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠠⣄⣀⡀⠀⢀⠀⢀⣴⢶⡚⠳⠀⢀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⢶⣦⣽⠀⠈⠉⠙⠁⠂⠉⠁⢱⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠱⠀⠀⣸⢳⡖⠀⠀⠈⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⢀⠃⠀⠀⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡤⢺⠁⠉⠙⠛⠁⠀⠀⢀⡎⠀⢠⠴⠃⠀⢉⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢠⠀⠛⠛⠟⠀⠀⠠⣀⠄⠀⠀⣸⠃⠀⠀⠀⠰⠋⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠠⢟⡧⣀⣀⣀⠰⡄⠀⠀⠀⠀⠐⣾⠶⠴⠛⠈⣳⢄⡀⡄⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡿⠇⠀⠀⢱⣌⣹⠋⠈⠙⠄⢰⣶⣄⣀⣅⠀⢀⣠⣴⢧⠀⠀⠈⠁
⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⠀⠀⠐⠤⡄⢶⠾⠛⠛⠀⠀⠀⠈⣟⠉⠉⢽⣛⣛⣯⡅⠀⢣⠁⣠
⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⣋⠙⠻⠿⢿⡄⠀⠈⠒⠁⠀⠀⡀
⠀⠀⠀⠤⠴⠂⠀⠀⠀⡠⠐⢢⣄⠀⣀⠀⠀⠀⠹⣀⠀⢠⠾⣿⢽⣯⡷⠄⠀⠀⠉⡄⠀⢠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⠀⠈⠀⠒⠒⢂⠀⠉⡁⠈⠀⠀⠀⡳⠳⠾⠛⠋⠀⠉⠀⠈⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠀⡠⡗⠀⠀⢁⡀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡔⠊⠀⠀⡀
⠀⠀⠀⠠⠂⠀⠀⠀⠹⠿⠄⠀⠀⠀⠀⠀⠀⠜⠀⠀⡁⠁⠀⠀⠀⣠⡴⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠁⠀⠠⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠔⠉⠀⢠⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡷⠾⠛⢹⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢇⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣰⣭⣜⣦⠀⠀⠀⠀⢀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠻⡀⠰⠠⠀⠀⠀⠀⠐⠊⠄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠀⡉⢉⠉⠁⠘⠂⠶⡀⢀⠊⠀⠀⠈⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠕⠀⠀⠀⠂⠠⢀⠀⠀⠀⢀⠀⠈⠢⡀⠀⠀⠀⠠⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⡄⠀⠘⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠁⠀⠁⢻⠄⠀⠐⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊⠉⠀⠓⠐⠠⠄⠠⠀⢀⠈⠗⠀⠠⠀⠈⢁⠀⠀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⡀⠀⠀⠇⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡎⠰⡆⠁⠀⠀⠰⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢙⡇⠈⠀⠠⠉⠀⠁⡀⠀⢀⠊⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠁⢸⡀⠈⢀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡐⠂⠊⠀⠀⢠⣖⠀⠀⠀⢸⣷⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡔⠂⠀⣐⢫⡘⢣⠈⠹⡿⡘⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⣀⣀⣀⠀⠀⠔⠀⠀⠀⠀⠘⢫⡰⠂⣐⢓⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⡠⠒⠠⢍⡽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢮⡷⢒⡤⢮⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡠⠀⠀⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⡩⠕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⢀⠤⠂⠁⢀⡠⠔⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠠⠂⠁⣀⠤⠒⠁⢀⠠⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠳⡔⠈⢀⠠⠐⠉⠀⡠⣴⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢹⣎⣀⡀⢄⣒⣥⣮⠶⠦⠤⠽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⡁⠉⡺⠽⠃⣀⡤⠄⠀⠲⠤⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⡼⠃⣠⠊⠀⠁⠀⠀⠆⠀⠀⠀⣀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⢠⡅⠀⢠⠀⠀⢸⠀⠀⠀⠈⠉⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠑⠒⠿⠀⠀⠀⠀⣀⠄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⢌⠢⢄⣤⣀⣀⡀⠦⠔⠊⣠⠊⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣞⠢⠤⠤⣤⠤⠔⣒⠩⠕⠚⠉⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡔⠅⠂⠁⠀⢀⣀⠤⠒⠈⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠣⣤⠤⠒⠈⠁⢀⠀⠄⠒⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠦⡔⠂⠉⠀⡀⠤⠒⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⡖⠈⠀⠀⢀⡠⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣼⣁⣈⡩⠥⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡉⠭⠍⣁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠁⠀⠀⠀⠀⠉⢢⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⣂⠉⠉⠉⠉⠉⠉⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⢌⠢⢄⣀⣀⠤⢊⠜⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣁⠂⠀⠀⢂⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⠉⡉⠉⢹⡁⠀⣀⠀⠀⠀⠀⢸⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠘⠋⠀⠀⠈⠁⠢⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠂⢀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠱⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⣀⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡎⢐⠄⢀⠠⡀⠀⠀⠀⠘⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⢀⢀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠠⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠉⠰⠒⠉⠑⠂⠀⠁⢠⡍⠉⠒⠊⠉⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢎⠈⡳⠸⠏⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠋⡀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠋⠑⢌⣉⡥⠚⠀⠤⢄⡀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠘⠤⢂⢌⠔⠀⢸⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⠌⠀⠀⠀⠀⠀⠈⠑⢠⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠢⣑⠤⣀⠀⠀⢎⡰⠀⢀⡠⢐⠥⠊⠀⠀⢸⠀⠀⠀⠀⢠⣎⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⡀
⠀⠀⠀⠀⠀⠘⡤⠤⠤⠤⣬⠃⠀⠀⠀⠁⠒⠨⠭⠬⠭⠭⠐⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠪⣓⢦⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠇⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠈⢁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀
⠀⠀⠀⠀⠀⡰⠡⢤⠬⠥⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠂⡶⡰⣒⠂⠲⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⡜
⠀⠀⠀⠀⡰⠱⠁⠀⠠⡄⠀⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠃⡣⡘⣢⡀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠤⣀⠀⢀⠔⠁⢀⠜
⠀⠀⠀⠰⠡⠃⢀⡀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⢊⡁⠈⠓⠤⣀⠜⠁⠀⠀⠀⠀⠀⠀⠑⡁⠀⡠⠊
⠀⠀⢠⢁⠂⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠲⢄⡀⠸⠀⠀⠀⠔⠁⢑⠀⠀⠀⢰⠈
⠀⢀⢂⠂⠀⠀⠀⠀⠰⣉⡁⠀⠀⠑⡐⡤⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠸⠤⠤⠀⠤⠄⢤⠀⠀⠀⢙⠀⠀⠀⠈⠐⠈⠀⠀⠀⢸
⢀⢂⠂⠀⠘⠃⠀⠀⠀⠀⠀⠰⠉⠆⠱⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣆⣘⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣁⣵⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣤⣴⣤⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⡿⠞⠛⠉⠉⠉⠉⠙⠛⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⢀⢀⣀⣀⣀⣀⢀⡀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⢰⣿⣮⣽⣯⣿⣧⣿⣿⠀⠀⢔⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠛⠉⠉⠉⠛⠙⠫⢿⣷⣷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣴⣤⣤⣤⣷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⣻⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠠⠤⠠⠘⠒⠛⠒⠻⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠉⣁⠤⣤⣒⣶⣶⣼⣼⣤⡭⣤⣶⣄⣢⣤⣞⣣⠄⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣂⣴⣶⣧⡶⠿⠛⠋⠉⠉⠀⢠⣆⣠⢲⡄⠈⠉⠉⠛⠻⠿⣿⣦⡌⠑⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡡⢞⡭⠟⣭⠁⠀⠀⠀⠀⢀⠀⠠⠀⠉⠌⠈⠄⠀⡀⠀⠀⠀⠀⡀⡽⠿⢶⣀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢊⣴⣿⠟⠋⢻⣤⡴⠞⠀⠀⠈⠀⠀⠀⢲⡆⠒⡗⢰⡆⠀⠀⠈⠀⠐⠀⡿⠋⠀⠀⠙⠳⣄⢀⡙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡠⢊⡷⠁⠀⠀⠀⡅⠀⣡⠰⣆⠀⠀⠀⠀⠀⣿⠀⡇⢸⡇⠀⠀⠀⠀⢀⡶⠂⠐⠀⡀⠀⠀⠈⢢⡹⣯⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⣾⣷⠏⠀⠀⢀⠀⠂⠠⠾⢤⡆⠈⢧⡀⠀⠀⣠⠇⠷⠷⠸⠇⠀⢀⠁⢀⡞⠁⠀⠀⠀⠀⠄⠀⠀⠀⣹⡜⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠸⣽⠃⣔⣦⠀⠄⠀⠀⠀⠀⠀⢙⢳⠾⠛⠀⠀⠀⠀⠀⠀⠀⣀⣰⠇⠐⠻⠀⠀⠀⠀⠀⠀⢀⡈⠐⠚⣉⢄⡀⢿⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠢⣰⣽⠷⠤⣲⠀⠂⠼⠄⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡍⠐⠀⠀⠀⠀⠀⢀⣠⠴⢋⣵⡈⠘⠓⠉⠀⠘⣧⢿⡄
⠀⠀⠀⠀⠀⠀⡈⢢⣧⠏⠀⠀⠁⠀⠸⠛⠛⠷⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢲⣤⣄⠀⠀⠀⠀⠀⠈⢻⡵⠞⠉⠀⠀⠀⡄⠀⠀⠐⠸⣿⣷
⠀⠀⠀⠀⠀⠀⢃⣹⣽⠀⠀⠀⠈⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠔⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⡇
⠀⠀⠀⠀⠀⢘⠀⡈⡟⠀⢀⡀⠁⢠⡀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠰⢸⣟⡷
⠀⠀⠀⠀⠀⢸⠀⠃⣏⠙⣀⢃⠀⢰⠶⠛⠛⢙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⣯⣭⣭⣭⣭⠀⠈⠏⢛⡃⢸⣟⡷
⠀⠀⠀⠀⠀⠸⠀⣤⣿⠀⠉⠁⠀⢲⠖⠒⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⢏⡘⠳⣄⠀⠀⠀⠀⠐⠟⠛⠛⠓⠛⠀⠈⠛⠉⠁⢸⣉⡿
⠀⠀⠀⠀⠀⠀⡴⢸⢸⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠩⢤⣑⣄⢀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡯⢽⠇
⠀⠀⠀⠀⠀⠀⢰⡼⡏⣧⠀⠀⠈⡀⠠⣔⣫⠵⣺⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣑⠦⣄⠀⠀⢀⠁⠀⠀⢀⢠⠳⣿
⠀⠀⠀⠀⠀⠀⠈⡴⢻⡹⣆⢠⢤⡄⠄⠩⠞⠋⣡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⣽⡺⣽⣦⡝⠃⢀⠀⠀⠀⢀⡎⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣥⠱⣹⡟⢉⣶⡀⠄⠀⢶⠋⠀⠀⢀⡼⣳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠶⠟⠀⢀⡛⡠⠀⢀⡞⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡦⢳⡟⣎⠉⠀⠀⠂⠀⠀⠀⣠⠞⡴⢁⡾⠀⠀⠀⢲⠂⣦⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⠠⣠⠟⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠻⣮⣧⡀⠀⠀⠀⠀⠀⠉⠺⢥⢀⡇⠀⠀⠀⢸⠀⢹⡄⠀⠀⠀⠠⠙⣦⠄⠀⠤⠀⠀⠀⠀⡀⢀⡤⢉⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢳⣍⢢⡀⠀⠾⠓⣀⠀⠀⠈⡁⠀⠀⠀⠾⠠⠀⠷⠀⠀⠀⠈⠀⠆⣨⠀⠀⠀⠀⡀⢈⡴⢏⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢘⢷⣾⣿⠇⢠⣅⠁⠀⠀⠀⠈⠀⠀⣀⠂⢀⠀⠀⠀⠁⠀⢴⡊⠹⡄⠀⠀⣠⡴⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣝⡳⢂⢤⢂⠤⢄⣀⠀⠀⠀⠁⠚⣍⠀⠀⠀⠀⣀⠀⠛⣂⣥⠖⣟⣧⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣭⣛⣛⠶⠖⠺⠠⣭⣤⣉⣩⣭⡍⠒⠶⣂⣻⣭⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠶⠶⠶⠿⠶⠶⠞⠛⠛⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠚⠛⠛⠛⠛⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠲⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⢿
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠫⣉⠀⠸⣄⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⠟
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⢸⠙⠛⠒⠒⠶⠦⠴⠤⠶⠶⠖⠚⠛⡃⣠⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⣜⣀⣤⣀⡀⠀⠀⠀⠀⠹⣼⡇⠂⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⢰⣿⠧⠤⠤⠤⠤⠴⢿
⠀⠀⡴⡿⠋⠉⠙⢦⡹⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⡤⠤⡄⠀⣿⠀⠀⠀⢸⡿⠶⠤⠤⠤⠴⠶⣏⣀
⠀⡼⢰⡇⠀⠀⢀⣈⣧⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠷⢮⡤⠀⣿⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⡏⠉⠓⠦⢤⣀
⠀⣇⢸⡇⠀⢾⣁⣈⢿⡄⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠀⢸⠀⠀⠀⣿⠤⣀⣀⠀⠀⢀⣀⡤⡇⠀⠀⠀⠀⠈⣹⠦⡄
⠀⢹⡀⢧⠔⠋⢩⠿⣼⢡⡏⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠁⠀⡧⠀⡀⠀⠀⡞⢁⡀⣿⡀
⠀⠀⠳⣜⡦⢴⣋⣴⣣⠞⠀⠀⠀⠀⠀⢿⠈⠙⠓⠶⠶⠶⠖⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠁⡄⣸⠀⡏⠀⢀⣽⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠳⢤⣀⣀⣀⣀⡤⠾⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⡀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠲⢧⣿⣤⠽⠳⠾⠋⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠤⠒⠻⠉⢹⡄⠉⠉⠀⠀⠀⠀⠀⣉⣉⣉⡛⠛⠋⠉⠀⠀⠈⠳⣭⣓⣒⣒⣒⡪⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄
⡞⠋⠉⡛⠛⠓⢦⡴⠒⠋⠀⠀⠀⢀⣀⣀⡀⡾⣒⣂⣉⣹⡆⠀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⢹⠉⠓⢦⡀
⠉⠒⠲⠧⠤⠤⠼⠁⠀⠀⠀⢠⡎⠁⡸⠉⢉⡷⠦⠭⠭⣾⠖⠫⡁⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣼⡤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣙⣛⣛⡩⠇⠀⠀⡼⠁⠀⠀⣰⠞⠁