Athena Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⠆⠀⠀⠰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠛⠁⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡃⣰⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣟⠓⡆⠞⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⠀⡴⣇⣹⣄⡻⢤⠀⠀⠀⣥⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⡀⠀⠀⠀⠀⠈⠛⡈⢳⠈⣷⣶⡆⣸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⠞⠛⠛⢷⣦⠀⠀⠀⠀⠙⢯⠈⡿⣏⠛⡏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠃⡀⠀⠀⠀⢻⣧⠀⠀⠀⡀⣫⢾⡇⢘⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡜⠉⡥⠀⠀⢸⣿⠀⠀⠀⠖⢠⢏⡇⠀⡄⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠈⠊⠃⠀⠀⣸⡟⠀⠀⢰⡇⢁⡾⡇⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⣀⣀⣾⠟⠃⠂⢐⡿⣴⠟⠁⡇⡂⠃⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡭⠝⠋⠙⠄⠀⣄⣫⡴⠃⣀⣴⡇⡅⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢛⣩⣴⡾⣿⣿⣷⠳⠾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠘⠁⢠⣿⡏⣿⣠⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⡴⠀⢠⢻⠇⢸⡗⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢀⠆⣾⠀⢸⣿⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⡏⡆⢸⣿⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⣾⢸⠀⡟⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⣸⡇⡞⢠⡇⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⣴⡿⣸⠃⣼⠀⡇⠀⢠⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠛⢋⡽⣡⡟⢰⠏⢀⡇⡇⠀⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣶⣾⣿⣾⡿⣠⡟⠀⢸⢷⣿⠀⢸⣎⠣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣈⣽⣿⠿⠛⣴⣿⠆⣠⣟⣾⣿⣤⣾⣿⣿⣦⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠉⠙⠷⠶⠛⠛⠻⠾⠿⠟⠛⠋⠉⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠟⠛⠋⠙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡛⠁⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠉⠁⣼⢯⣼⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⢀⣀⣰⣿⣿⣋⠀⠀⣠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⢀⠘⣿⡟⠃⠀⠁⣶⡀⣾⣿⣷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⣿⣷⣶⣤⣾⣿⣤⣤⡀⠀⠈⠉⠀⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣉⠿⣿⡿⠿⣦⣉⣿⠿⠟⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡖⠙⣧⢀⣽⠛⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣶⣶⣾⠛⣻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡟⠻⡟⣿⠏⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠦⠿⣷⣦⣀⣷⡄⠈⣦⡄⢀⡀⢉⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠈⣿⣿⡟⠻⣦⠈⢾⣿⡀⠛⠋⣀⣰⣦⡄⠀⠀⣼⣦⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⣀⣽⣿⣴⡀⠘⠁⢐⣯⠀⠀⠀⢿⠇⠈⣿⣄⣠⡈⠙⠃⢀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠄⢸⣿⠃⠻⠟⠻⠆⠀⠻⡿⠃⠀⠀⢠⡄⢰⡟⠁⠛⠡⠀⠞⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⡟⠀⣀⠀⢀⣤⣄⣤⣤⣤⠜⠀⣴⠁⠘⠃⢰⣿⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⣷⣿⣷⣾⣿⣿⣿⣅⣨⣿⣿⣿⣷⣾⣯⣠⣼⡿⠚⣿⣧⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣼⣿⣿⠌⣭⠿⣿⣿⣷⠀⣻⡛⢿⣿⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⣿⣿⣋⠛⠿⠿⠋⠀⠈⠉⣥⡀⣴⡿⢶⣿⡿⣿⣦⡿⡇⠐⠈⠃⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣟⣿⣧⣤⠚⠿⠿⠿⣦⣄⡿⠁⣸⠗⠛⢯⡀⠘⣿⡇⢼⣀⣸⠇⢨⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠹⣿⣘⡿⣿⣿⣿⣄⡸⢆⠀⠘⢿⣷⣾⡇⣶⡄⠠⢿⡟⠻⣏⠀⠻⣿⡇⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠸⠏⠀⣤⡀⠀⠀⠙⢿⡟⠙⣿⣏⡷⠞⢃⢠⣿⣿⣿⡀⣿⣷⠀⠈⠋⣀⠿⣷⣶⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠁⠀⠀⠀⠙⠁⠀⠀⠀⠻⠃⠀⢹⣿⢧⠀⠸⠀⣿⣿⣿⣵⣟⣿⣷⡇⠀⠟⠀⠈⠁⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠙⠃⠀⠀⠀⣠⣿⡀⠙⣿⡇⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⢠⡄⠀⠀⠀⢰⣟⣿⠿⣿⣿⡁⢸⣿⡆⠀⠀⠀⠙⣾⡇
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢁⠀⠀⠀⠈⠉⢩⣴⣿⣿⡅⠈⣿⠀⣦⠀⢠⠀⠹⡇⠀⠇
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢿⣿⠷⣄⣈⣳⣹⣿⣿⣿⡄⠀⣿⠀⣿⣶⠀⠰⢤⠁⢠⡄
⠀⠀⠀⠀⠀⣿⠂⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣵⡄⠈⠻⢿⣿⣿⣿⣿⡇⠀⢹⣷⢸⣿⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⢸⡇⠀⠀⢰⣦⡘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰⣿⠇⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠏⠘⣿
⠀⠀⠀⢀⡜⠀⠀⠀⢺⡍⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢿⡞⠀⠀⠀⣿⣿⣿⣿⣿⣄⠀⠃⠘⣿⠀⣶⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⣶⣿⡆⠈⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢹⣿⢃
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⣠⣾⣿⣿⡟⢻⣿⡇⠀⠸⣿⣿⣷
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⡿⠁⣿⣿⡇⢸⣿⡇⠀⠀⠟⠉
⠀⠀⢨⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣿⡿⠀⠀⠀⣿⣿⣿⣭⣿⣷⠀⠀⢸⣇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⠀⠠⣄⠈⣿⡟⠙⠻⣿⠆⠀⠘⣿⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⡿⠁⠀⢰⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠈⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⣿⢡⡔⢀⣼⣿⠿⠟⠀⠀⠀⠀⣗⠀⢀⣤⠄
⠀⠀⢸⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⢀⣾⡟⠁⣿⡄⠀⠀⣷⠀⣿⠀⢸⠏
⠀⠀⠘⠇⠀⢻⡆⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⣼⣟⣤⣾⠟⠀⣿⠻⡇⢰⣆⡼⠐⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⠿⣿⣿⣦⣾⡿⠀⡧⢻⣿⠁⢠⣿⠃
⠀⠀⠀⢸⡄⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣾⠁⣸⡟⢸⡟⣿⣇⠀⠀⠘⣿⠀⢈⢠⣶
⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢰⣿⡷⠀⣷⡈⣿⠀⢰⡆⢹⠀⣿⠀⠉
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠄⣾⣿⣇⠀⠻⠁⣿⡀⠀⠀⠈⠀⡿
⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⣿⣿⣿⠃⠀⠀⣿⠇⢀⡀
⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠰⣿⣼⣿⡄⠀⠀⠀⢠⡼⠃⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠀⠀⠈⠀⠀⠀⠀⣠⣾⠟⡘⠃⠀⠀⢻⣿⡿⢧⡀⢠⠀⠘⢁⡀⠉⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⢀⡠⠄⣀⡼⠟⠉⠀⠟⠉⠀⠙⡷⠙⢧⣜⠃⠀⢀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠘⠉⠀⠀⠀⠀⠛⣡⠶⠖⠦⣤⠀⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠛⡛⠛⠛⣛⠛⢛⣛⣋⠀⠙⠛⢛⡛⠛⠋⠛⠛⣻⣿⣿⣿⣿⣿⣿⣥⣭⣭⣄⣈⣉⣉⣭⣅⣈⣿⣿⣯⣤⣉⣭⣉⣉⣉⣭⣭⣉⣉⣉⣉⣉⣉⣉⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢸⣿⡿⣿⣿⡟⠉⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢸⡟⣀⣼⣏⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⢤⣤⣤⣾⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣿⣶⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡿⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⠻⢷⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠋⠀⠀⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡟⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿⠿⠿⠟⢻⣿⠿⣶⣤⡄⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠉⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣽⠷⠶⣀⠶⣟⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠈⡃
⠀⡄⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠶⠤⠈⠦⢄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠡⣤⠄⡀⢠⡄⣿⡇
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣠⢀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠜⠣
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⣜⠳⣦⡀⣤⣤⣶⠏⠀⠀⠀⠀⠀⣦⡤⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠉⠛⠾⣿⠣⡘⢧⣿⣀⠀⠀⠀⣀⣠⡟⠷⠏⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠻⣿⣄⢀⡉⠻⣿⣙⢦⡿⢿⡧⠖⠀⠀⢠⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠛⡡⠀⠉⣝⣛⣿⡄⠙⠟⠰⢇⠀⡀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠸⣇⠀⢠⡞⢁⣴⣥⣾⣄⡀⠀⢀⣋⠓⠀⠀⠀⠀⠀⢠⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡿⠋⠈⠉⠉⠃⢹⣥⠂⠌⠙⠁⠀⠀⠀⠀⠀⢸⡇⠀⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠇⠀⠀⠀⠀⣼⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠇⠀⠀⢰⠀⠀⠀⠀⠀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠈⠀⠁⢈⡐⠀⠐⢲⠆⠀⠀⣿⠀⠀⠀⠀⠤⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠀⣦⣀⠈⠳⠛⠒⠞⢂⣀⠀⠅⠠⠀⣀⠠⢀⠀⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡃⠀⠉⢙⣲⣶⣾⣿⣶⡎⣭⢷⡀⠀⢠⣤⠶⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣴⡶⣿⣿⣿⣿⣛⣾⡟⠃⠉⠀⠀⠔⠃⠀⠐⠋⡀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢿⣠⠉⠩⠍⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢣⣤⡏⠷⠒⠳⠦⠤⠐⠒⠀⢀⠀⠀⠀⢀⠜⠀⠂⠀⠈⠉⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠦⠀⠀⠀⠀⠀⠄⠂⡠⠴⢀⡀⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⣤⣤⠤⠀⠀⠀⠀⣀⠤⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡂⡘⠣⠒⠀⠤⠤⠄⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠙⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠙⠀⠈⠐⠂⠀⣀⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠽⣲⠆⠀⠀⣀⠜⠋⠀⠀⠂⢀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣤⠝⠣⠤⠖⠁⠀⢀⠞⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠐⡀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠛⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠀⠀⠀⡄⠀⠀⣾⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠀⠀⠀⣶⠀⠀⠀⢿⠲⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠐⠎⠀⠀⠀⢰⠃⢀⡀⠀⢀⠈⠷⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠱⣤⡰⢠⡏⠀⡈⠁⠀⠀⠀⣤⠀⠀⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠠⡾⠂⠀⠀⠑⣿⢡⣾⠁⠀⠀⠀⠀⣿⠀⠀⠀⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠋⢹⡏⠀⠀⢰⠀⠀⣼⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠆⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⢀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠠⠀⠀⠀⠰⣦⣿⢎⠀⠀⠀⢨⣿⣂⣠⣴⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⡤⠤⠼⠁⠀⠀⠀⠀⠠⡟⠋⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠎⠀⠀⠠⠆⠏⠁⠀⢸⠀⠂⠀⠀⠀⠀⣧⠀⣾⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠋⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢣⡀⢾⡐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠀⠀⠀⠀⠀⠙⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠃⠈⣄⡀⠀⠀⠀⣄⠀⠀⣼⠀⠀⠀⠀⢠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⣈⠁⠀⠺⢿⡇⣠⣄⣤⣀⣀⣿⡆⣼⣠⣇⣠⣤⣤⣿⠋⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⡗⠘⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡻⡇⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⢁⣉⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣼⠁⠄⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣯⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠈⠉⠻⠏⡷⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⣀⡀⢰⣷⠀⠈⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠻⣿⢤⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠄⢀⠐⠈⠄⠄⠀⠈⢎⠃⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣀⠀⣀⠀⠀⠘⡄⠀⠘⡆⠀⠀⢰⠀⣸⠁⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠈⠃⠀⢢⡗⠀⠀⠘⢠⡿⢁⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠒⠀⡐⠀⠀⣼⠁⢀⠜⠀⢿⡇⠀⠙⠂⢄⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠈⠀⢀⡴⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⡀⣀⠴⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠛⠚⠛⠊⢉⠠⠔⠁

⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠉⣿⠋⠁⠀⢸⣤⣿⠀⠀⢺⡏⠀⠀⣿⣦⣾⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠛⠉⠛⠀⠀⠀⠻⠀⠀⠀⠸⠀⠛⠀⠀⠙⠓⠀⠀⠛⠈⠻⠀⠀⠚⠉⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠤⠴⢦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠘⠀⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠑⠀⢸⢂⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠱⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⣄⡠⢤⡿⡄⠀⠀⠀⠀⠀⣩⣿⡛⡟⠁⠀⠀⠀⠀⣜
⠀⠀⠀⠻⠀⢱⠂⠀⠀⠀⠀⠀⠀⠀⠬⡁⠃⠀⠃⠘⣀⠀⠀⠀⢀⣹⣮⠁⣷⡤⠤⠤⠀⠀⡏
⠀⠀⠀⠀⠀⠘⢂⣂⠀⠀⠀⠀⠀⠠⡘⢃⡀⡀⠀⡘⡁⡒⠀⣀⣹⣿⡿⠐⡧⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⢸⡄⠉⣀⠀⠀⠀⠐⠃⣀⣘⣿⣥⣿⣿⣿⣟⣻⣟⣿⣿⣰⣧⠜⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡅⣠⠈⠀⠀⠀⠀⢀⣢⡟⢾⢾⢻⢿⣿⠻⠽⣹⢿⢿⡵⠃⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠈⠌⠀⢸⢘⢸⠀⠈⠳⠀⠈⠻⢿⡃⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠐⠂⠂⠋⠁⠀⠀⠀⢰⡀⠀⢸⠈⠀⡀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠀⠀⠀⠄⣀⠈⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⠘⡀⠀⠀⡠⠄⠈⠗⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣄⠀⠀⠸⠀⠱⠀⠀⠈⢁⡚⠃⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⡇⠀⠀⠀⠀⠱⡄⠀⠀⠠⣝⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠘⠃⠀⠀⠀⠀⠀⠈⠐⢤⡀⢘⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣾⢶⣶⣦⣤⣀⣤⠄⠙⢿⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣥⣴⣿⢻⣯⠀⠀⠈⠁⢸⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⣿⣿⣿⢸⡟⠀⠀⠀⠀⢸⠇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠅⣿⣿⢹⢸⠀⡄⠀⠀⠀⠸⠆⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡇⣼⠀⠘⠀⠇⢀⢀⡠⡼⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⡇⢬⣐⣦⡴⢶⡄⠠⠜⠃⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⡋⠃⠀⣋⢸⠑⠂⠀⠀⢄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢼⣂⣆⢘⣇⣈⣀⣠⣤⡤⠄⠐⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠐⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢿⡇⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠃⠙⡶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⠊⡁⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠍⠓⠚⢷⡄⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠋⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠐⠀⠾⣏⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠹⡄⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢡⠀⠁⠀⠀⠂⠀⡤⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠀⠬⠀⠀⠀⣇⣀⠀⢸⠀⣴⡀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡅⠀⠀⠀⢀⠀⠀⢰⠇⠀⢀⠀⢠⡟⠑⢄⡀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠠⠀⠀⢀⡞⠀⡠⠋⠀⠸⠇⠀⠀⠀⠑⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⠴⠚⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠁⠉⠉⠉⠉⠉⠀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣟⣧⠀⠀⠈⠉⣯⠉⠀⠀⣿⠤⣿⠀⠀⣺⡏⠀⠀⢸⣷⣼⡇⠀⠀⣼⣻⡄
⠀⠀⠀⠀⠀⠀⠛⠉⠙⠃⠀⠀⠀⠛⠀⠀⠀⠛⠀⠛⠀⠀⠉⠓⠀⠀⠘⠀⠛⠃⠀⠘⠋⠉⠛

⣿⣿⠻⠿⢻⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⣤⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⡁⢊⣐⣿⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠙⢛⣟⠛⢛⣿⣿⡦⢄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠈⠻⢿⣿⣿⣿⣦⣤⣠⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣼⣿⣷⣬⡛⠿⣿⣿⣿⣿⠿⢛⣡⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣇⡘⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡟⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢾⣿⣿⠟⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⣻⡶⠣⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣟⣽⣿⠻⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣗⣹⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣯⣿⣿⣿⣿⣇⠿⢿⡿⣵⣿⣟⠁⠙⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣝⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢽⣿⠽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢻⣿⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢾⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣤⣤⡶⠿⠿⠿⠟⠉⠹⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀⣤⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠉⠁⠈⠉⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠂⢀⠀⠀⠚⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣦⣾⠇⠠⠶⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⣿⣶⡤⠤⠀⢠⣾⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⣯⣱⣆⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠚⠲⣾⣦⣴⣿⣍⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⠀⠙⣿⡟⣿⠿⠇⠀⠀⠀⢠⣾⠛⠛⢻⣿⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣧⡀⢠⣄⠀⠸⣿⡗⠂⠀⠛⠁⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⣿⣧⣀⣼⣷⣶⣷⣤⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠶⠀⠀⢷⡿⠋⠉⢹⣿⣿⡟⠛⣿⡄⠻⠧
⠀⠀⠀⠀⠀⠀⠰⠟⠀⠛⢻⡟⠀⣶⠀⠀⠀⠘⢇⡀⠀⡀⠀⠈⠱⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⣿⡀⠀⠀⠀⠘⠁⣀⠉⠠⡄⠀⠀⠀⣤⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠇⠀⢻⣿⠀⠀⢀⡀⢺⣅⠀⢀⡅⠀⠀⣠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠺⣍⣷⣾⣿⡿⣿⡿⣿⣿⣿⣿⣤⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣀⣴⠆⢹⣿⣿⡃⢿⠀⢸⠦⢸⣿⠿⠿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠃⠀⠚⠉⢻⡇⠀⠀⠈⡀⠈⠃⠀⠀⠈⠀⠁⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⣤⠟⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠹⣄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠀⢹⣀⠀⠀⠀⢸⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠻⣇⠀⠀⢘⠀⠼⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣀⡀⠀⠸⡏⠀⠀⢈⣀⣤⣀⣤⣴⡧⢠⣤⣤⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⢸⣿⣧⠀⠀⡇⠀⣾⠟⠉⠉⠁⢘⡟⠀⠈⠀⠈⠉⠈⠀⠰⡆⠀⠋⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢉⡀⠀⠀⠀⠸⠇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢠⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⢀⡀⠀⣀⣄⡀⢠⠀⠀⣤⣀⡀⠀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣶⣦⣀
⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⠈⠀⠀⠈⠉⠁⠀⠘⠋⠀⠐⠶⠾⠷⠾⠛⠒⠀⠠⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⢀

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠛⠛⠿⠿⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⢀⣀⡀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠶⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡿⠃⠀⢣⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢹⣿⡇⢀⣤⠉⠉⠙⠛⠂⢠⡀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣿⡇⢿⣟⣿⣿⡷⢶⡤⠈⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢷⡁⠙⣿⣧⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣶⡶⢆⣼⡇⠀⠰⣤⠄⠀⠀⠀⠀⠀⠀⠐⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠧⢽⣦⣄⣷⣌⢻⡗⠠⣶⠟⣗⠀⢀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⢻⡈⢻⣦⡄⢀⣿⠆⠿⠀⠾⣿⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⢻⣾⣍⠀⠸⡿⠀⡀⢸⠆⣠⠆⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠘⠃⣴⣌⢻⣮⠳⠾⠇⢀⡿⢸⣾⠃⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡖⢿⣯⣅⣀⣽⣿⣶⣦⣈⣠⣿⣿⣤⣈⢳⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣿⣿⣿⣧⣦⣤⠾⠿⠉⣹⡿⣾⠟⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⣼⣿⣿⣿⣧⠌⠩⣶⣦⣄⣤⡽⣿⣿⡿⣾⣿⣿⣿⣿⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⠻⣿⣿⣿⣿⣄⠀⠀⠛⠉⢿⡇⢻⣿⢿⡈⢿⣟⠻⠿⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡅⢰⡞⠃⡈⣿⣿⣿⢻⡆⠀⣀⠀⢸⣧⣼⣿⡆⠷⠀⢻⢷⣦⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠘⢧⣼⣧⣴⡀⠛⣿⣿⣧⠙⢦⣿⠇⠘⣿⣿⡿⢇⠀⠀⠀⣆⠁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠈⠉⠙⢧⡀⢹⣿⣿⣷⣤⣿⡇⢺⣿⡏⡇⢸⡀⠀⠀⢸⠇⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣿⣿⠙⣿⣿⡇⣾⣿⠀⢻⠀⣿⡄⠀⢷⡀⣿⠁
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢛⣿⠀⠘⣿⣧⢿⣿⡇⢸⡇⢸⣿⡀⠘⠁⠟
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣿⠀⠀⢸⣿⢸⣿⣿⠀⢱⠀⢿⡅⠀⠁
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸⣿⣾⠆⠈⢿⣾⣿⣿⡆⠈⠀⠀⠻⡦
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠀⠀⢸⣿⣿⣟⡀⠀⣷⠀⢀⡄
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⢀⣼⣿⠻⢿⡇⠀⢸⣇⣿⠇
⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⣀⣼⠃⢻⡀⢾⣇⠀⢸⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⢸⣿⡿⠁⠀⠠⣷⣾⢿⡀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡟⢀⣤⣾⡀⣿⣿⠀⣧⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠐⣿⣠⡾⢋⡙⣧⠹⡟⠀⣿⡆⢹⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⣀⣿⠋⠀⡌⢳⡟⠀⣤⡆⣿⠇⠘⠁
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣿⣧⣤⣾⢀⣿⠃⠀⣿⠀⣿⠀⡀
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⣿⣡⣿⡿⢸⣿⠀⠀⠟⢸⡇⣰⠃
⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢠⡟⢳⣿⣿⡇⢸⣿⠀⠀⠀⠘⣷⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢁⣿⠛⣿⣿⣿⡇⢸⣿⠀⢸⠀⠂⢹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⢀⣴⡿⢁⣾⡇⢠⣼⣿⢿⡇⢸⡿⠀⠸⣦⣇⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⢀⣼⠟⣡⣿⡿⢡⢄⣈⣿⡈⢡⡘⢇⣠⣤⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡶⢾⣥⠞⠉⠀⢦⣤⠀⣽⠙⠿⠈⠁⠀⣦⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠈⠉⠉⠁⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣾⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣿⣦⣤⣄⣀⣐⣀⣼⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⠿⠿⠿⠿⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠙⢿⣦⣀⠀⠀⠀⠀⠀⢾⣿⠿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠒⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⣿⣷
⣿⣿⣷⣄⣶⣶⣤⣤⡄⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⣿⣿⣿⣿⣿⣿⡟⣹⣷⣿⣧⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣴⣿
⠿⣿⣿⣿⣿⣿⣴⡟⠻⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿
⠀⠀⠉⣿⡝⠻⠟⠀⠀⢸⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⣿⡿⣿⣿
⠀⠀⠀⢻⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣾⣷⣴⣿
⠀⠀⠀⠀⠙⣦⠄⠀⠀⣀⣻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⣿⣿⡏⠁
⠀⠀⢀⣤⣾⣿⣧⣄⣼⠿⠛⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⡀⢸⣿⣿⣿⣿⡟
⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⢹⣿⣿⣿⣆⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣷⣸⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⢀⣼⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠻⢿⣿⣿⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡇
⣶⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⣤⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠿⠿⢿⣿⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠄⠀⠀⠀⠀⠆⢀⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⣴⣶⣿⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣁⣼⣿⣿⣿⣿⣿⣿⠿⠻⠿⠟⠻⣿⣿⡟⠛⠉⠁⠈⠉⢠⣄⢸⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⣱⣿⣿⢿⣿⣿⠟⠀⠘⠃⠀⠀⠈⠙⠿⣶⣦⡀⠀⣠⣾⣿⣿⣿⣿⡟⠛⠉⠁
⠀⠀⢀⢀⡀⢀⣠⣤⣠⣦⣿⣿⣿⣿⣿⣌⣻⣧⣄⠀⢀⣀⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⣀
⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠉⠹⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠈⠙⠿⣿⡿⣿⣿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠙⠿⠾⣿⣿⣿⣶⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠶⢋⡼⣁⠀⠀⠀⢀⡹⣌⠳⢦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⣭⠉⠉⢉⡤⢾⡁⣠⠎⠑⠮⡑⠲⠒⡩⠒⠉⢦⡀⡹⠦⢌⠉⠉⢩⡉⠉⠱⡄
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢻⠀⢰⠃⠀⢀⡽⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠻⣄⠀⠀⢷⠀⢰⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡸⢤⡏⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠘⣧⠼⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠠⣮⢁⡀⠀⡼⡄⢀⣉⣉⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣉⣁⠀⡼⡄⠀⣀⢹⡆
⠀⠀⠀⢀⡾⠁⠒⡞⡟⢸⠄⠤⠤⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⠤⠄⢴⠘⡟⠲⠂⠹⣆
⠀⠀⠀⢸⠃⢠⠎⠰⡇⠈⣇⠐⠂⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢒⢀⡏⠀⡗⠈⢦⠀⢳
⠀⠀⠀⢿⠀⠀⠀⠀⢻⣿⣿⡿⢿⣍⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⣽⡿⣿⣷⣾⠃⠀⠀⠀⢸⠂
⠀⠀⠀⠈⢧⣀⠀⠀⠘⣿⡟⠀⠀⠙⠷⣌⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠋⠀⠘⡿⡟⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠳⣜⢦⠀⠀⠀⠀⠀⠀⠀⢀⢜⣵⠟⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢵⣄⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⣀⣴⣾⠿⠀⣲⣤⣀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢘⣻⣷⣿⡟⠁⠀⢰⣝⣿⣿⣾⣟⡂⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣶⣿⠿⠛⠉⠁⠀⠀⣾⣿⠈⠙⠛⢿⣿⣾⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢹⣿⣿⠿⠩⡀⠀⡓⢰⠑⠲⣿⣿⣇⠀⡠⠁⠘⠻⣿⣿⡏⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡿⠉⠀⠀⠀⠀⣈⣽⣷⡀⣿⣿⣿⣆⣥⡀⢠⠒⠽⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠅⠀⠠⣄⡘⢦⡀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⡏⠀⠉⣻⣦⣉⠲⣍⣷⣀⣙⣹⠿⢛⣿⡿⠿⣿⡏⠁⡠⢻⣿⡿⣃
⠀⠀⠀⠀⠀⠀⠀⢉⣾⣿⡇⠀⠂⢀⣫⣿⣷⣬⡳⣿⠩⡶⡶⢚⣿⣿⣿⣿⣿⠀⠁⢸⣿⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⢀⣤⣽⣿⠿⣿⢮⣿⣿⠇⣮⣾⣿⣿⣿⢿⣿⣿⣧⡀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠃⠊⣼⣿⠋⠀⠸⢷⣿⡿⢖⣫⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠩⠈⠇⣸⣿⡃⠀⠀⣸⣿⣿⣿⢛⣿⣿⣿⣿⣇⢀⠛⣻⡿⠟⣇⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠞⢿⣿⣦⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡆⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⢹⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠁⠀⠉⠹⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠟⠀⠈⣸⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠲⢌⠢⡠⠤
⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠠⣴⠋⡁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢟⡲⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⡀⠐⣃⠜⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠡⠴⠉⠲⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠀⡀⠀⠐⢄⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢢⡀⠀⠢⠀⠀⢠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢳⡄⠀⠁⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠠⡠⠋
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠠⠳⣀⠀⠀⠀⢀⡶⢱⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠠⠊⢃⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠈⠑⠢⠤⠬⢧⡳⢾⡛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡸⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⡛⡽⠀⣗⠄⠀⠀⠀⠀⠀⠀⢰⡟⠶⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢇⠋⠈⠀⠀⠀⠈⠓⡦⢸⡖⡇⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠺⠯⠒⠒⠛⢤⡤⠤⠤⠐⠊⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠸⢷⡘⡟⢠⠿⠂⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⣉⣙⠲⣅⠒⡵⢚⣉⣙⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡟⢩⣤⡀⢈⡉⢳⣿⡋⢉⡀⣠⣬⡙⡷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣠⠞⠁⠀⣧⣘⣁⣿⣃⡴⢫⢂⠱⣄⣻⣇⡙⣡⡏⠀⠙⢦⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣲⣦⣤⠤⠼⠷⠚⠛⠛⠛⠉⠉⢉⠉⣉⢉⣉⣀⣀⣀⠀⠈⠉⠉⣉⡉⠉⠉⠛⠛⠛⠛⠲⠿⠤⢤⣤⣴⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠛⢋⣭⣥⣦⠀⢠⡶⠿⣶⡄⠈⢻⣿⠃⠘⣿⠋⠈⣿⣿⠉⣿⣷⠀⠀⠀⣹⣿⣿⡃⠀⠀⣴⡾⠳⣶⠀⠠⣤⣤⣭⣉⡛⠂
⠀⠀⠀⢆⠀⠀⠀⠀⢠⣿⠁⠀⠛⢀⣿⡇⢰⣿⣷⡀⣸⣿⡄⢠⣿⡄⠠⣿⣿⣴⣿⡃⠀⠀⢠⣿⠻⣿⣇⠀⣸⣿⡇⠀⡈⠁⠀⣧⣿⠀⠹⠇⠀⠀⠀⠀⡰
⠀⠀⠠⠁⢠⡄⠀⠀⣾⣿⠂⠀⠀⢸⣿⣷⠃⣿⡿⠀⢸⣿⠁⠀⣿⠀⠀⣿⣿⠀⣿⣿⠀⠀⣸⡿⠾⣿⣿⠀⢸⣿⡇⠈⣿⡟⠀⣿⣿⠶⠆⠀⠀⠀⢠⡄⠈
⠀⠀⠠⠀⠀⠔⠀⠀⢻⣿⡀⢀⣿⠀⢿⣯⣠⣿⠃⠀⠸⢿⣦⣴⠟⠀⠠⠿⠿⠄⠻⠿⠦⢠⠿⣧⠀⣽⣿⣆⠈⢿⣧⣠⣿⡇⠀⣿⣿⠀⢀⡄⠀⠀⠢⠐⠀⠄
⠀⠀⠀⠡⠈⠃⠀⡆⠈⠻⠿⠟⠁⠀⠰⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⣤⠒⣚⠁⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠠⠘⠛⠛⠿⠟⠁⢲⠀⠚⠁⠄
⠀⠀⠀⠀⠈⠦⣄⠃⠀⠀⠀⠀⠀⢠⠄⠀⣀⣀⣀⡀⠀⠀⠀⠋⠀⠀⠐⠁⠤⠘⠃⡙⢙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠃⠸⣠⠴⠁
⠀⠀⠀⠀⠀⠀⠈⡥⠀⠀⠀⠰⠈⠀⣴⠟⢉⣿⠏⢹⣿⠀⠀⠀⠀⠀⠋⠀⠀⠈⠉⠃⠈⠁⠀⠀⠀⠚⠂⠀⠀⠀⣾⡇⠀⠀⠀⠐⠑⠆⡀⠀⢬⠁
⠀⠀⠀⠀⡠⠒⠲⢤⣦⡟⣇⠀⠀⠀⠛⠀⣸⣿⠀⣾⡿⢻⣷⠀⣿⡗⠻⢦⣾⠛⣷⣰⣿⠃⣼⡟⢠⣾⠛⣻⡟⢸⣿⠛⣿⡆⠀⠀⠀⣰⢛⣴⡤⠖⠒⢤
⠀⠀⠀⢸⡀⠀⢐⠀⡿⡏⢹⡁⠀⠀⠀⠀⣿⡇⢠⣿⠁⣼⡟⢸⣿⠀⠀⣿⡇⢀⡟⣸⡿⢠⣿⠃⣿⡇⢀⣿⠇⣿⡇⢸⣿⠀⠀⠀⢈⡏⢹⣿⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠉⠀⠁⢀⠃⠃⠙⣁⠀⠀⠀⠘⠛⠀⠘⠛⠀⠙⠛⠛⠋⠀⠀⠙⠛⠛⠁⠙⠛⠋⠛⠛⠙⣻⣿⣿⠚⠛⠁⠘⠛⠃⠀⠀⣈⡉⠈⠘⡀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⢸⢀⡜⠉⠀⠙⡆⠀⠀⠀⠠⠔⢀⠐⠀⠒⠈⡀⣀⣂⠩⠤⠔⠒⠈⠁⠀⠀⠀⠐⠷⠿⠃⠀⠀⠀⠀⠀⠀⢰⠋⠀⠉⢳⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢨⣼⠀⠀⠤⠔⠃⠈⠀⠒⠙⠂⠢⠍⠁⠀⠠⠁⠀⠀⠀⠒⠄⠀⠀⣀⣀⣀⣀⣤⣤⣄⢠⣤⣤⡄⢀⣀⣀⠘⠦⠤⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⡠⠀⠘⣶⢶⡶⣦⣤⡄⠀⠀⣤⣤⣶⡆⠀⠀⢻⡟⡟⠁⡿⠉⢫⣹⡇⠀⠛⠀⣷⣿⠀⠈⣿⣿⠃⠀⢄⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⢷⠇⢀⣿⣴⡀⠀⠘⠃⠀⢀⣿⢿⣿⣇⠀⠀⣸⣿⣧⠀⠀⠀⣾⣿⡗⠀⠀⠐⣿⣿⡶⣶⣿⣿⡀⠸⡾⠆⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⣿⣿⠟⠿⠀⠀⠀⣾⣿⣘⣿⣿⠁⠀⢹⣿⡏⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣀⣿⣿⡀⠀⠀⠀⢠⣿⠋⠉⢻⣿⣇⠀⣸⣿⣧⡄⠀⠠⠾⠿⠷⠀⠀⠘⠛⠛⢃⠰⠿⢿⡄⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃⠀⠐⠛⠛⠃⠀⠛⠋⠉⠁⠉⣀⢀⣀⠠⢂⡀⠀⢂⣁⠀⠀⠒⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⠀⠐⠠⠀⠂⠀⠀⠨⠉⠉⠉⠂

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠁⠀⠛⣀⣠⡤
⠀⠀⠀⠀⠀⠉⠍⠀⠄⡀⠀⢀⡀⠀⠀⠀⠠⠌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣉⠆⠀⠀⢉⣤⠖⠉⠛⠛
⠀⠀⠀⠀⠀⠒⣂⠛⣦⣮⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠢⠐⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠻⢛⣾⣿⣯⣠⣫⡕⠂⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⢣⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠞⢡⡄⠀⢄⣀⡀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⡾⠛⢻⣏⠠⡀⠀⠔⠀⠀⠀⠀⠀⠐⢀⢀⡀⠀⠀⠀⠀⠁⡴⣦⠀⠀⠀⠀⠀⠀⠀⣀⡶⠉⡟⠧⠎⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠁⣿⣠⣘⣟⡳⠏⠰⠄⠀⠔⣣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠷⡄⣀⠀⢤
⠀⠀⠀⠀⠀⠠⠀⠀⢈⣹⣦⣤⣿⡤⠀⠀⠄⠸⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⠀⠀⠀⠀⠀⠀⢂⣠⣴⡆⠀⠴⠷⠖⠒⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⢟⣳⣘⠻⠎⠀⠲⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠈⠛⣛⣙⣃⠀⠰⢿⡎⠦⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠐⣉⠛⡉⠹⠂⠀⣸⣈⣀⡄⢠⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣯⣽⠀⠀⡿⣷⡀⢂⠀⠀⠀⡀⠀⠀⠀⠿⠐⠧⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡼⠃⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⢹⡆⠀⠐⠋⠁⠈⠀⠀⠘⡅⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⡀⢀⣀⠠⠤⠄⠒⣀⠀⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙⢦⣤⣤⠤⠤⠤⠒⠀⠀⠀⠠⠀⠓⣈⡁⠴⡿⠀⠈⠉⠫⡉⠁⠂⠠⣤⡴⠶⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⢀⣀⠀⠊⢀⡜⠋⢠⠙⠋⠀⠦⡄⢀⡀⠉⠀⢀⠀⡀⡀⠤⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢈⠀⠢⢉⡙⠏⡯⢉⣐⢀⠘⠄⠙⠒⢤⡀⠈⠁⢀⡀⠄⠂⠀⠀⢀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠄⠤⠊⠀⠀⠀⠀⠀⢀⠀⠀⢰⡏⣳⣆⣸⣿⠄⠀⠐⠄⠀⠀⠉⠀⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣉⣩⣠⣿⢀⠀⢀⠄⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⡤⠀⠀⠀⠀⠐⢻⡿⠥⢄⣈⠛⠭⠉⠥⠀⢰⠎⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠾⣷⢶⠶⠤⠛⡦⠰⠖⠀⠘⠂⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠦⠀⢀⡀⠐⢁⣠⠶⡄⠀⠐⠒⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠠⠞⠉⠉⠉⠁⠀⡔⣴⡄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣿⠉⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⡄⠀⠀⠀⠀⠀⠛⢠⣼⠋⢲⣄⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⢘⡃⠀⠀⢿⣧⢈⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⠁⠀⠀⢸⣏⠀⠀⢀⠀⠈⢣⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⢨⡻⠄⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠈⠀⠀⡿⠀⠀⠀⠘⣵⠆⠀⠀⣿⠀⠘⠠⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠤⠆⠀⠦⠆⠴⠒⠠⠿⠀⡜⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⠆⠀⠀⢹⠄⠀⠴⠆⠛⠷⠀⠤⠀⠐⠠⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠂⠀⢀⣬⠀⠀⠀⡀⣿⠇⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠋⠀⠀⠉⢁⠙⠋⠹⠀⠀⠀⢁⠀⢠⡀⠉⠁⠀⠀⠀⠀⢠⠀⠂⠀⠀⢆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠾⠀⡀⢀⡆⠘⠆⢨⠧⠀⢀⠀⠀⠰⢜⡧
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠠⠂⠁⠠⢀⠁⠜⠁⠀⢀⡁⠀⠈⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣄⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠄⠀⠀⠆⠀⠋⠀⠈⠡⠀⠘⠉⠀⠀⠀⠀⠠⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠒⠛⠊⢠⡀⠲⠶⢄⠀⠁⠲⠊⠑⢀⡀⠀⣀⠴⠾⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆⠀⠀⠸⡅
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃⠀⣼⡟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠔⠂⠘⠈⡄⠉⠡⢅⠀⠄⠈⢁⣀⡀⠀⠀⢸⣧⡀⠀⠛⠄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⣤⠘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃⢤⣄⡀⠘
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢸⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠃

⣷⣿⣿⣷⣿⣿⣿⣿⣷⢿⣭⣿⣼⣤⣾⣵⣷⣿⣿⠿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣾⣼⢞⢾⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣧⢽⣿⣿⣿⠿⣿⣿⣿⣟⡿⣾⣿⣿⣿⣯⣿⣻⣿⣟⡺⢫⣻⣿⣿⢹⣿⣿⣿⣿⣧⣿⣿
⡿⠿⣫⡟⣾⡽⣿⣛⣿⡿⢿⡻⠏⣑⣯⣿⣿⣿⣿⣽⣿⣿⣿⣿⣟⣿⣟⣁⡛⢻⣿⠛⠉⠙⠛⠻⣯⣽⣿⣻⣿
⠛⠀⣠⣇⣿⣿⣾⣾⣥⣄⣴⣾⣿⣿⢹⣿⣿⣧⣝⣾⣿⣿⣿⣿⣿⣿⣿⣍⣇⣛⡧⣀⠀⠀⠀⠀⠈⢿⡿⢿⢿
⡶⠟⠛⣛⢳⣿⣿⣿⣷⣿⣿⣿⣿⣯⣸⢿⣿⣿⣟⣻⣿⣿⢿⣭⢻⣿⠟⠉⠻⢿⣿⣮⣕⢢⡀⠀⠀⠘⠛⠻⡿
⠀⠀⠀⢠⣬⣿⣿⡿⣿⣿⣿⣋⠤⠋⠀⢸⡇⠈⠻⣿⣿⣯⣬⣬⠿⠃⠶⠀⠀⠀⠙⢿⣿⣿⣿⡱⡀⡀⠀⠀⠁
⣿⡾⠃⠀⠖⣻⣿⣷⣟⣽⡟⠁⠀⠀⠀⠈⠁⠀⠀⠙⢿⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⡀⠹⣿⣿⣿⣿⣿⢾⣄⠤
⣦⣤⣷⠀⠁⣿⡿⢿⣿⠏⠐⠊⠳⠖⠂⠀⠀⠀⠐⣗⢾⡄⠀⠀⠀⠀⠀⠀⠲⠿⠀⠀⠁⢹⣿⣿⠟⠚⠋⠉⡄
⠛⢫⣅⠘⡁⠈⢿⣯⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡗⢻⡉⠀⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣀⣡⡀
⣿⠾⠛⠆⠉⢠⢀⣾⣿⣷⡀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣾⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⣿⣰⠝⡆
⣄⢰⡄⢠⠀⢸⣿⡿⣿⣿⣷⡄⠀⠠⣤⣿⣯⣩⣿⣾⣿⣎⣿⣄⠀⠀⢤⣤⡆⠀⠀⠀⢿⣧⢻⢼⣿⣿⣖⠢⠃
⡛⠋⠁⠀⠀⢿⣿⣷⣬⣿⣿⠀⠀⠀⡟⠋⢁⣴⢏⢛⢿⣾⣿⣿⣷⣦⣤⣄⣠⣀⠀⡄⠨⠷⢿⣿⠟⢉⣿⣱⢢
⠣⠂⠀⠈⠒⠚⣿⣿⣿⠟⠻⠀⠀⠀⠙⠀⡞⠈⠻⡿⡯⣌⡿⣯⣛⣿⠻⡾⣾⡾⢁⢀⣀⣀⣠⠧⣤⣿⣶⣽⠎
⣶⣾⣿⡶⠖⠊⠉⠘⡖⠐⠀⠀⠀⠀⠀⠀⠟⡀⠸⢙⠰⠋⠀⡘⣧⠱⣦⠈⠊⠀⠀⠀⠈⠈⠘⠛⠿⠿⣿⣿⣿
⡿⠯⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠠⣆⣀⣠⠌⢷⣻⡆⠀⠀⠀⠀⣀⠰⠜⢡⡝⣃⠺⣀⢩

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡃⢽⣿⣦⣀⠀⠀⠀⠀⢀⡃⠈⠉⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣷⣆⣳⣦⣈⣽⠓⠲⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⠛⢻⣶⢾⣿⡖⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣾⣷⣿⢊⡿⠏⠀⠙⣷⡀⣸⣿⣿⣿⣇⣿⠁⠀⢰⣆⠀⢲⣦⢲⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢳⡌⠯⢸⡟⠩⢙⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠿⠷⠦⢾⣿⣀⣀⡭⠿⠒⠂
⠀⠀⠀⠀⠀⠀⢀⢠⣤⠀⠒⢹⡄⠉⠃⠰⣿⡅⠀⠀⠀⠀⠀⠼⢉⣹⣿⡿⠋⠁⠀⠀⠀⠐⠙⠻⣷⢤⡀
⠀⠀⠀⠀⠀⠄⢁⣴⠟⠉⠒⠛⠃⠀⠀⢀⠀⢓⣨⣵⠀⠀⠀⠀⠀⠀⢈⣻⡶⠦⠐⠋⢙⣇⠐⢻⡾⢢⢷⡈⠂
⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣀⡀⠀⣀⣀⡾⣡⣾⠋⠰⠀⠀⠀⠀⠀⠀⠀⠩⢿⣷⣦⣄⢄⡟⠐⢞⡷⠈⣿⣻
⠀⠀⣠⠂⠀⣀⣤⡶⠟⠀⢈⣯⠿⠿⣿⣿⣿⣿⣂⡤⠴⠤⣴⣄⢀⠀⡄⠀⠈⠙⢿⣷⣵⣅⢤⣄⣅⣸⣿⢯⣷⢄
⠀⠀⠁⣴⠚⠋⠈⡇⠀⣠⡾⠃⠀⢑⡿⣿⣿⣿⡏⠀⠀⠀⠀⠐⢄⣉⡉⠒⠀⠤⠈⣿⣿⣿⣗⢽⣟⣛⣿⣶⣿⣶⣋
⠀⢠⢶⣿⠀⠀⠀⣤⣾⣿⠁⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣯⠉⠀⠀⠈⢳⣿⣿⣿⠷⣝⠙⢿⢻⣿⣿⠿⠿⠞⠛⣻
⠀⠀⠀⠀⣧⡀⠀⢹⣿⡏⠀⢀⣾⡿⢋⣵⣿⣟⣿⣿⣿⣦⣤⢄⡀⠘⣿⣿⣤⣀⠀⠈⠻⣿⣿⣷⣮⠑⠪⣁⡛⠿⢿⣷⡄⠘⠋
⠀⠀⠀⠀⠈⠳⢦⡵⡿⣥⣀⢿⣋⣴⣿⣿⣿⣡⣿⣿⣿⣿⣿⣷⠳⠤⣿⣿⡙⣿⣿⣿⠷⢆⠀⠉⠛⣷⡀⡀⠉⠛⠳⢶⣿⠅
⠀⠀⠀⠀⠀⠀⠠⣷⡼⣍⢻⣿⡿⠛⣽⣿⣿⣿⣿⡏⢛⠷⢿⣿⣷⣖⡻⠿⠿⠿⠿⠿⠷⣄⠀⠀⠀⣁⠻⣤⣍⣁⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠒⠛⠛⠋⠀⠈⠳⣤⠤⣶⣿⣮⣽⣿⣿⠶⠶⠶⠶⠾⢷⣿⣗⣒⢲⣀⣀⣀⠲⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠿⢿⡏⣳⠒⠒⠒⠐⢿⡿⠻⢿⡿⠦⠹⢿⡗⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠙⢆⠠⣸⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣈⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀
⣿⣿⢋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢀⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡴⠒⢢⡰⠊⢀⣠⠔⠊
⠀⠀⠀⠈⠳⣶⢤⣀⠀⢀⡠⠔⠊⠉⠈⠱⡀⠀⠀⢠⠂⡄⡀⠤⢀⠀⢠⠃⠀⠀⢘⣦⠎⢀⡴⠊⣠⣶⠟⠓⣄
⠀⠀⠀⠀⠀⡨⠓⠪⠙⠫⣀⠀⠀⠀⠀⠀⢇⣰⠉⠈⠁⢱⡇⠀⢰⠀⡌⠀⠀⣠⡾⠅⢊⣁⣤⡾⠟⠁⠀⠀⣸
⠀⠀⠀⠀⠘⢅⠀⠀⠀⠀⠀⠉⣉⠩⠭⢔⠺⣿⣀⡖⠊⠉⠈⠁⠺⠖⠁⡠⢊⣥⡶⠿⠿⠟⠉⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠉⠉⣉⡵⠶⢷⣌⡏⠂⠀⠁⠀⠈⠙⢺⣾⣰⣿⣿⣒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡓⠂⠀⢄⠘⢣⠁⡀⠀⠀⢀⡀⢺⢻⣿⣿⣿⣏⡙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⠐⠊⠻⢶⣾⡄⠀⠻⡄⠙⠛⠈⠋⠁⣼⣾⣿⣿⡟⣡⡤⠼⢗⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡜⠠⣌⠓⢄⣹⣧⠈⠳⢧⡀⠀⠒⠂⠀⣼⣿⣿⣿⢿⣄⠄⠒⠈⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣎⡀⠤⣈⠑⣦⢇⣿⣄⠀⠀⠈⠀⠤⠴⠛⢋⣿⣿⣿⠘⣤⠐⠊⣁⡀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠢⠤⠟⠛⠋⠀⠉⠛⠒⠶⠤⠞⠓⠖⠛⠛⠉⠉⠓⠋⢳⣁⣤⡴⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣻⣀⡅⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⣿⡷⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⣿⠁⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⠉⠀⣰⣿⠓⠂⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠃⠀⢺⣆⢈⣿⣏⠀⠀⠙⠟⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡟⢀⡀⣸⣿⣏⠀⣠⣀⣀⠈⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠀⠈⠁⢻⡿⠿⢛⣻⠛⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⡀⢀⣿⣷⠶⠋⠀⠛⠁⢺⣏⠱⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣧⠀⡄⣀⡆⠀⣿⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠁⢠⢀⠻⣶⡀⠈⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣀⢈⣸⡀⢿⣷⠀⣧⣀⣠⣤⣣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⢿⣿⡿⣬⡟⠈⠇⣼⠁⠀⢠⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿⣷⡌⠀⠀⠀⢻⡄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠘⣿⣿⡇⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠇⢀⣿⡞⠀⠀⠀⠀⠀⣷⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠈⣿⡇⠀⠀⢸⠀⠘⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣼⣆⠀⣿⣿⡀⠀⠘⠀⠀⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⣿⣼⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⡄⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿⣿⡄⣻⣿⣿⣿⠿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⠛⠛⠛⠛⠉⠛⠛⠋⠛⠀⠛⠃⠛⠻⠛⠛⠂⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠐⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠀⠀⠀⠀⣤⠀⠀⠀⠸⠀⠀⡀⠠⠀⡀⠀⠀⠀⢂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁⠀⠀⠁⠈
⠀⠀⠀⣷⣄⠀⠀⠀⠀⣠⣿⣿⠀⠀⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣄⣠⣤
⠀⠀⠀⣿⣿⣦⠀⠀⠀⢹⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣴⣤⣦⣴⣶⣶⣶⣦⣶⣤⣤⣶⣾⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣷⠀⠀⢸⣿⣿⠀⠉⠉⠁⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⡶⠄
⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⠗⠐⠒⠐⠂⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣉⣙⣿⣿⣿⣿⣿⣧⣴⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠚⠛⠉⠉⠉⠁⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛
⣿⣿⣿⡿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠛⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢀⠀⠀⢸⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠷⠀⢀⡀⠰⠏⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢲⡶⠾⠋⠻⠶⣶⠂⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣷⠀⠀⠀⢰⡇⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⡆⠀⠀⣾⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣧⠀⢠⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⠀⢸⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣛⣉⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⡀⠈⠲⠊⠀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠰⠶⠶⠶⠒⠚⠛⠋⢉⣉⣠⣤⠤⠤⠤⣤⣈⡙⢶⡀⣰⠞⣉⣤⡤⠤⠤⢤⣤⣌⣉⠉⠛⠛⠒⠲⠶⠶⠶
⠀⠀⢀⣀⣀⣤⣤⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⡷⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⢤⣤⣄⣀⣀
⠀⣠⠟⠁⠀⠀⠀⠀⠠⠶⠶⠖⠒⠛⠛⠛⠛⠛⠓⠲⠦⣤⡉⣡⡤⠶⠒⠛⠛⠛⠛⠛⠛⠒⠶⠶⠦⠀⠀⠀⠀⠀⠙⢧⡀
⢜⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣾⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⢻
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣇⠻⠶⠾⢃⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁



⠀⢸⣏⣹⠆⣰⠏⠉⢷⠀⡾⠋⠙⣆⢸⣇⡴⠃⠀⠘⣇⢰⢿⠀⡿⢸⡇⢰⣏⠉⠀⡟⠙⢷⠀⣾⠋⠙⣦⢰⣿⡄⢰⢿
⠀⢸⣏⣹⠇⠸⠆⣀⠾⠀⢷⣄⣠⠏⢸⡏⠻⡄⠀⠀⢻⡾⠈⣷⠇⢸⡇⢀⣈⡷⠀⣇⣠⡾⠀⢿⣄⡠⠟⠘⠇⢳⡏⢸
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⠟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣌⢿⡂⣿⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠈⠁⢈⣛⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡼⠊⡁⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡇⠈⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⠿⠿⣭⣱⣤⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⡙⠃⣾⠓⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠠⠎⡀⠀⢀⣿⠆⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠈⠁⢻⣿⣇⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠄⠀⠀⠂⢀⠀⠀⠉⢇⠱⡀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠀⠰⠂⠀⠀⠀⠡⠀⠀⠐⠡⠀⠁⠀⠀⠘⠀⢀⡤⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢀⣤⠀⣠⡀⠐⠀⣀⡇⠀⠠⢹⡇⠈⠀⠀⡇⠀⣸⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠁⠀⢀⠶⠀⠀⠀⠁⠈⠇⠀⠀⣰⡟⠈⠁⠀⠃⢠⣿⢃⡀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢃⠁⠘⠀⠁⠀⠀⢾⡿⠀⠨⠄⠀⠰⠀⠀⢀⡼⠀⠀⣠⠂⠀⣾⡇⠀⠉⠲⢄⡀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠡⠀⠈⠠⠀⡐⢀⠔⠁⠀⢀⣼⠃⢀⠜⠁⠀⠀⠉⠁⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠦⠄⠀⠀⠀⠀⠀⣠⠟⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⢀⣠⡴⠚⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠉⠉⠉⠉⠉⣀⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠠⠶⣶⡶⠖⠀⠀⣴⡆⢰⡆⠀⠀⣤⡶⠆⠀⠀⣴⣆⢠⡆⠀⠀⠀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠽⣧⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡏⢻⡇⠀⠀⢼⣟⠀⠀⠀⣿⠛⣿⣏⠀⠀⢠⡿⠽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠉⠀⠈⠁⠀⠀⠈⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠺⠃⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⢀⠀⡀⠀⠐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⣶⣎⠁⡄⠂⠀⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣶⣭⡟⠀⠒⣠⡔⠋⣣
⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⢸⡿⣤⠋⠘⠛⢉⠠⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠠⠀⠀⠘⣷⣿⡜⠃⠺⠟⠀⠞⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⠘⣿⡇⠀⠀⠀⣦⡀⠸
⠀⠀⠀⠀⠀⠀⠀⣤⣼⠃⠀⠀⠀⢺⡟⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⡼⢷⣀⢀⣀⣀⠳
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⡴⠚⠛⣤⣾⠁⠀⠀⠀⠙⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⣀⣿⡶⣿⠉⠛⠰⠶⢀⣀⠀⠀⠛⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣯⣰⣿⠁⠀⠀⠀⢈⡹⠿⠻⠦⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢿⣿⣿⠋⠳⣤⣧⠀⠚⠑⢀⣠⠀⡀⠐⠋⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⡇⠰⠀⠉⠙⠖⠀⡾⣦⠟⣶⡷⠾⠂⡀
⠀⠀⠀⠲⠶⠒⠲⠀⢨⣍⣴⣿⡇⠰⠀⠀⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⠙⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢽⣏⣿⠇⡠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢸⡆⢰⠆⣼⡇⠈⢁⠀⠀⣸⣇⣤⣀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⣿⠀⠀⠘⠶⣴⣿⡿⠿⠟⢻⡶⠿⣿⣶⠋⣴⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠠⠄⢿⡄⢠⡾⢻⣿⣿⠇⠀⠀⠫⡟⠓⢼⡟⠁⠉⣐⡀⠀⠀⠆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣿⢡⣼⣣⣿⣿⣏⠀⠀⠀⠀⣤⠀⢸⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡏⢻⡅⠀⠀⠀⠠⡇⠀⠈⠀⠀⠀⢠⠀⠀⠄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠟⣹⣿⡞⢧⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⢿⣧⣈⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣶⠀⣄⣠⡇⣨⡍⢻⡅⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣽⣿⠿⠃⠀⠀⣇⣠⠶⡦⠴⠇⡾⠛⠻⠋⠰⣦⣀⣤⠀⣷⡄⢸⣉⣻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣸⠨⡅⢹⣇⠰⠁⠀⠉⣿⠀⠆⠀⠀⣿⠀⡆⠀⠀⠀⠈⣿⢤⣿⣤⠟⠉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⠃⡿⢠⡀⠀⠀⡿⠀⠀⠀⠀⢻⠀⠁⠀⠀⠀⠀⠈⠆⡀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣻⠀⠢⣾⣧⠈⢷⡄⣼⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⣠⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣿⠀⡇⢸⠆⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⡇⠘⠀⡃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣧⠀⡇⢰⡶⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠀⣧⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣾⠿⣿⠷⡄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠈⣿⠀⠈⢲⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠻⡀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠇⠸⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⣄⠀⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣸⠄⠠⠄⣀⢇⡀⣹⣄⠀⠀⣠⡀⢀⣀⡙⠃⣤⡀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠉⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠒⠈⠀⠠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠞⠛⠷⠿⠿⠾⠗⠀⠺⠀⠿⠟⠻⠿⠿⠟⠛⠻⠇⠹⠆

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠉⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣴⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣰⣾⣷⡆⠀⣿⣿⣿⣿⠀⢠⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠙⠿⠿⠃⠀⣿⣿⣿⣿⠀⠘⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡄⠀⣿⣿⣿⣿⠀⢠⣶⣶⣄⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠃⠀⣿⣿⣿⣿⠀⠘⢿⣿⠟⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠐⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠾⠂⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⡈⢺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⣈⣹⣥⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⡉⠀⢰⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡜⢿⡇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡸⠇
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⣽⣿⣿⢧⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⡤⢀⠀⢀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⣠⡿⠂⠀⣼⠛⣿⠈⠷⠿⠂⡜⠛⠉⢳⠶⠆⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇
⠀⠀⠀⠀⠸⣆⡘⠃⠣⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠁⡐⢨⠛⠠⠎⢁⠈⠁⠀⠀⢠⡀⢸⣷⡇⢨⡀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⣦⠈⡆⢸⠏⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⡇⠀⠀⠀⠀⢰⡄⣡⠶⢀⡀⠀⠀⠀⣴⠋⡀⣷⣼⡆⣷⡌⣿⠂⠑⠒⠒⠒⠛⠛⠃
⠀⠀⠀⠀⠠⠏⠈⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⡈⢆⣴⡄⠀⢀⢠⣿⣏⠀⠀⠘⡟⠀⣠⡾⣿⢋⣿⣿⣏⣷⢿⢿⡏
⠀⠀⠀⠀⠀⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣿⡾⣧⠘⠃⣾⣷⢿⡿⠓⢶⡐⣟⣷⡆⡶⢻⣿⣿⣿⣿⣿⣠⣸⠇
⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⣾⠇⠸⠀⡀⡉⠿⠶⣤⣾⣿⢯⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠁⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠀⡇⢈⠻⡟⢠⠀⠛⠛⠁⠀⠈⢙⠿⣉⢋⢿⣿⣧⠿⠁
⠀⠀⠀⠀⠠⠀⠀⠈⢹⡷⣶⣦⣄⣶⡤⠄⢰⣿⣷⠛⢰⡁⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠱⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢀⣀⣀⣬⠴⠋⠉⠉⠁⠀⠀⢸⣿⣿⠀⠸⠀⢸⡀⠀⠈⡄⠀⠀⠀⠀⠀⣤⠂⣽⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⡆⠀⠈⠀⠀⠀⡦⠀⠀⠀⢻⣄⠼⣷⣌⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⡇⢠⠸⡇⠀⠀⠇⠀⠀⢠⣰⣿⣶⣿⢻⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⡇⠀⠀⠀⠀⠀⠀⢹⣬⡒⠛⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠸⠀⠀⠀⠀⢀⠀⠀⠈⠺⠛⢷⣦⣞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⢾⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⡤⠼⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠛⠀⠀⠈⠀⠀⠀⠀⠀⠧⠀⠛⠠⠘⠢⣄⣠⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡏⠁⢺⠁⠘⠀⠀⠀⠀⠈⠀⢁⠀⠀⠈⠐⣤⣤⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⣴⣾⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠈⠀⠀⡇⠀⠀⡄⠀⠲⣄⠀⠀⠡⠈⢩⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣄⠀⠀⢸⠀⠀⠀⢀⠀⠈⠻⢦⣀⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡟⢉⣿⠒⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠻⠗⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠛⠙⠓⢹⡇⣿⡿⢻⣷⣶⣶⣶⣿⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏⢰⣄⣠⣾⡇⠏⠁⢨⣿⣿⣿⣿⡾⠓⠂⢀⣿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⢹⠀⡀⢠⣽⣿⠃⠀⠀⠀⣿⣿⣿⣼⠟⣷⣾⣿⡏⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠀⡇⢸⡇⢸⠀⠀⠀⠈⣿⣿⡟⢿⣧⣤⣹⣟⣷⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡘⢰⡇⢸⡇⠘⠀⠀⡂⠐⣿⣿⣿⣎⣻⢿⣽⣿⡟⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢾⡇⢸⡇⡇⠀⢈⡇⠀⣿⣿⡿⢿⣿⣦⣭⣥⡄⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠃⣸⡇⡇⠀⢸⡇⠀⣿⣿⣿⣮⣻⣿⣿⡿⠁⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⣿⠇⣗⠀⠸⣇⣀⢻⣿⣿⡿⣿⢿⣿⡇⠀⠀⠀⠀⠀⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠟⢀⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣷⣾⣾⣿⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡏⣿⣯⡏⠙⠋⠈⡉⠉⠉⣿⣿⣇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡿⡏⣤⡽⠇⢀⡆⠀⣁⠀⠀⢿⣿⠛⢳⣄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡿⣷⣂⣠⠝⡆⢾⡇⠀⢫⠀⣀⠈⡟⢠⠉⠉⠿⣦⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢸⣷⠾⠷⠔⠛⠆⠛⠀⠛⠛⠀⠀⠒⠿⠿⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠁⠤⠤⠤⣶⣦⣉⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠚⠉⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡀⠀⠘⠛⠢⠤⠤⠤⠤⠀⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠟⠛⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢾⣷⡀⠀⠀⠀⠹⢿⡁⠈⠐⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠂⠈⠀⠰⠁⠹⢆⡀⠀⠀⠈⠙⢦⡀⠀⠄⠉⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⠀⠀⠀⠀⠀⢠⡆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⢤⠘⢣⡐⢣⠜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠤⠀⣠⡄⠀⠀⠀⠀⠀⠉⠀⠉⠠⢶⡉⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠓⠀⠀⡟⢠⡶⢀⡀⠀⠀⠀⠀⠈⠀⢀⠙⢦⡀⢾⣱⣄⡀
⡄⠀⠀⠀⢀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⣶⠞⣿⠀⠈⣻⡆⠀⠀⠀⠀⠀⠈⠐⠈⠑⠀⠙⠊⣷⣆⡀
⠇⠀⠐⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠉⣴⠛⢀⡼⠇⣰⠗⠂⣀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠄⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⢰⣾⠃⠀⣿⠀⣠⠛⠣⡄⠀⠀⠀⠀⠀⠈⠳⢶⣀⠀⢈⣿⣷⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠉⣴⠋⢀⣤⣁⡴⢦⠀⠀⠀⠀⠀⠀⠉⠿⡏⠉⠁⡀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⣀⡀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣤⠾⣿⠏⠀⣼⠁⣴⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣆⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡾⠃⣾⡏⢀⣴⠀⢀⡘⠁⣄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⡈⣿⣦⡿⠁⠠⡞⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡿⠀⢀⠞⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⣶⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢰⡄⠈⠳⣄⠀⠀⠀⠀⠀⣠⡟⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠘⠁⠀⠀⠀⠛⢤⡀⠀⣰⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠤⠤⠤⣤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣷⣤⣴⣶⣾⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣳⣤⣶⣤⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡯⠀⠀⠉⠁⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠈⠻⠶⣦⡶⠾⠋⠀⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⣠⡶⠟⠛⢿⣶⠶⠶⣾⠟⠛⠶⣤⡀⠀⠀⠀⠈⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠻⣦⣾⠃⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⡾⠛⠋⠛⠻⣦⡀⠀⠀⠀⢠⡿⠀⢀⣾⠀⠀⠀⠙⠁⠀⠀⢸⡆⠀⠸⣧⠀⠀⠀⠀⣠⡾⠛⠋⠛⠻⣦⡀
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠈⠻⠶⣶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢶⡶⠾⠋
⠀⠀⣠⡶⠟⠛⢿⡶⠶⠶⣾⠟⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⢿⣶⠶⠶⣾⠟⠛⠷⣦⡀
⠀⣼⠏⠀⠀⠀⠀⢻⣦⣾⠋⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣦⡀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢻⣆⣼⠋⠀⠀⠀⠈⢿⡄
⢸⡏⠀⢀⣶⠀⠀⠀⠹⠃⠀⠀⢰⣆⠀⠈⣷⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢰⡏⠀⢀⣶⠀⠀⠀⠹⠃⠀⠀⢰⣆⠀⠈⣷
⣾⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻
⠿⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⠇⠀⠀⠈⠻⣦⣀⣀⣀⣠⡾⠋⠀⠀⠀⠿⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣌⣉⣉⣉⣉⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠈⠻⣏⠉⢉⣿⠋⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠹⣧⡾⠁⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣼⠇⠀⠀⠀⠘⠁⠀⠀⠀⢿⡄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠯⡀⠀⠀⠀⠴⠚⠛⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣟⣯⣥⣤
⠀⠀⠀⢀⡞⠳⣤⢦⡀⠸⣶⡋⠀⠀⣠⠞⠁⢹⣆⠀⠀⠀⠀⢀⣦⡀⣀⠀⠀⠙⠵⣿⣿⣿⢟⣾⣷⣄
⠀⢀⡴⠋⠀⢢⡈⠻⡀⠀⠀⢈⣦⡞⠁⠀⢀⡏⠸⡆⠀⢀⣴⣿⣯⣻⣯⡀⠀⠀⠀⠘⢟⣵⣿⣿⢧⣿⡄
⠀⠘⢧⡀⠀⠀⣹⠖⠃⢀⡴⣫⠎⠉⢢⣠⠞⠀⢀⡇⠀⠘⢿⣿⣿⡷⠉⠁⠀⢀⡴⠋⠀⠈⢿⣫⣿⣿⠇
⠀⠀⠀⠙⠒⢾⢳⢀⡴⣫⠞⠁⠀⠀⠈⠛⠢⡀⡈⠀⠀⠀⠀⠙⠉⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠉⢻⡟
⠀⠀⠀⠀⢀⡞⣯⡻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣼⣦⡛⠁
⠀⠀⠀⠀⠈⢧⣛⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠕⠁
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⢸⣟⣯⣼⣿⣿⣿⣿⣷⠛⡻⢶⡶⠋⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡿⣺⢿⣿⣿⣿⣯⣜⣧⢸
⠀⠀⠀⢸⣿⣥⣿⣿⣿⣿⣯⠟⠧⣼⠏⠀⠀⠀⠀⢾⣽⣷⡟⣛⠁⠿⢊⣷⣶⣤⡀⠀⠀⠘⣿⡥⢄⣹⣿⣿⣿⣬⣽⢸
⠀⠀⠀⢸⠘⣷⠿⣿⣿⣿⢣⠪⢵⡏⠀⠀⠀⠀⠀⠈⠿⢩⣿⣿⣷⣆⣀⠉⣹⣿⡆⠀⠀⠀⠸⡇⠼⢋⣿⣿⠓⣿⣷⢸
⠀⠀⠀⢸⡀⣯⣠⣼⣿⣿⢽⡦⣾⠇⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⠙⢿⣿⣿⠂⡙⣟⢸
⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⢿⣂⢻⡆⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠛⠁⣬⣿⠃⠀⠀⠀⠀⠀⣿⣐⡿⢿⣿⣶⣿⣾⣼
⠀⠀⠀⢸⡟⠿⠟⡛⢺⣿⣼⣗⠺⣧⠀⠀⠀⠀⠀⠀⠀⠑⠒⠄⠀⠀⠹⠟⠛⠁⠀⠀⠀⠀⣸⣧⠼⣯⣼⣿⡟⣿⣿⢸
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠷⢀⡀⢀⣠⣤⣄⣀⡀⠀⠀⠿⣿⣛⣿⣿⣿⣛⣿⣿⢸
⠀⠀⠀⢸⡿⣛⣿⣿⣷⠛⢿⣿⡿⠋⠁⠀⠀⠀⠀⣠⠾⢻⣧⡘⠡⠀⣬⢏⠾⡋⠉⣡⢜⣦⠀⠈⠻⡿⠋⣿⣶⣿⡿⢸
⠀⠀⠀⢸⠑⣾⣿⡏⢹⠇⢸⠈⠀⠀⠀⠀⠀⢀⡼⢱⠆⠄⠀⣀⠀⠚⠉⠿⢀⡤⠚⠵⠶⠿⡇⠀⠀⠀⠀⣿⣾⣷⡧⢸
⠀⠀⠀⢸⢼⣄⣿⡗⢀⡀⠾⡀⠀⠀⠀⠀⢠⣞⣿⣠⣿⣤⣼⠷⠆⠀⣨⡷⣫⢔⣠⡴⠖⣛⣻⠀⠀⠀⠀⣿⣯⡈⣷⢸
⠀⠀⠀⢸⣼⣻⣿⡇⠈⠣⡀⠈⠳⣤⡀⠀⣾⣿⣿⣿⢬⣿⣍⠁⠀⠰⢿⠃⢐⣭⢷⣚⣩⡉⢼⠀⠀⠀⠀⢿⣬⣿⣽⢸
⠀⠀⠀⢸⢿⠷⣿⡇⠀⠀⠈⢦⡀⠀⠉⠑⢿⡿⢡⠿⠀⠉⢂⣙⣡⡾⠓⠀⠰⢚⣩⣵⠶⣤⣏⣧⡀⠀⠀⣸⣿⢻⣿⢸
⠀⠀⠀⢸⢸⣿⣿⡇⠀⠀⠀⠀⠙⢦⡀⠀⠀⠱⡞⢶⣠⣶⣾⣯⣷⠾⣿⠳⣿⣏⠉⢷⣟⠩⡙⠏⡇⠀⠀⢹⡟⣺⣿⢸
⠀⠀⠀⢸⣼⠻⢿⡇⠀⠀⠀⠀⠀⠀⠉⠳⠤⠞⣱⣽⡷⢂⣍⣡⠀⣛⠀⢾⣵⣿⣀⠀⢙⢧⡙⢦⡃⡀⠀⣾⣧⣿⣿⢸
⠀⠀⠀⢸⢻⢶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⣋⠹⢭⡤⠭⡽⣖⠫⣽⣧⠀⠀⠀⡆⢿⢮⡉⠃⠀⣿⣼⣇⣿⢸
⠀⠀⠀⢸⠙⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⠀⡰⠞⠃⠀⠈⠡⠀⠙⠛⣻⡄⠀⢹⠈⣧⣿⠀⠀⣹⡿⢯⣿⢸
⠀⠀⠀⢸⡴⢿⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⡄⠴⠟⢋⠀⠤⣀⡲⠆⠰⣺⣷⠀⢸⡇⠈⡋⠀⠀⣿⣿⣿⣿⢸
⠀⠀⠀⢸⡇⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢊⣁⠙⣛⣭⣶⡶⠛⠗⠨⡥⠼⣿⣧⠀⠉⢲⠏⠀⠀⣿⣼⠛⣯⢸
⠀⠀⠀⢸⣿⡋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⡞⢹⢻⡇⢨⡷⢔⠠⠐⠚⠀⠁⣴⣿⣿⠋⠀⡟⠀⠀⠀⢻⡿⢿⣿⢸
⠀⠀⠀⢸⡅⠛⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⠇⣾⢿⢠⣜⡧⠀⠑⣎⠀⢠⡀⠈⣽⢷⣠⡞⠀⠀⠀⠀⣿⡇⡘⣹⢸
⠀⠀⠀⢸⣿⠟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢴⡿⠸⠀⠙⡇⠰⣄⠈⢷⡔⠹⣆⣿⣴⣿⠀⠀⠀⠀⠀⣿⣷⣷⣿⢸
⠀⠀⠀⢸⢸⣧⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹⠀⡇⠀⢱⡆⠘⣧⠀⠈⠀⠉⡿⣠⡇⠀⠀⠀⠀⠀⣧⡟⠰⣿⢸
⠀⠀⠀⢸⣤⢾⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⣴⠃⠀⠠⣇⠀⡄⠜⡆⠠⡉⠻⣆⠀⠞⣰⢿⠀⠀⠀⠀⠀⠀⣿⣟⣷⣿⢸
⠀⠀⠀⢸⣿⣟⣻⡇⠀⠀⠀⠀⠀⠀⢀⡇⣿⠀⠀⠀⢻⡌⡇⢸⡇⠀⠡⡆⠈⠸⠀⢋⢈⠀⠀⠀⠀⠀⠀⣿⣤⣴⣿⢸
⠀⠀⠀⢸⣿⣉⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠘⣧⠁⠈⣷⠀⠑⢝⣶⣀⣴⢯⣾⠀⠀⠀⠀⠀⠀⣿⣧⣹⣿⢸
⠀⠀⠀⢸⢻⡿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⡦⡀⠀⠠⠇⠹⣦⠀⢸⡆⢻⣄⡉⣭⡵⣫⠜⠀⠀⠀⠀⠀⠀⣿⣾⢹⣿⢸
⠀⠀⠀⢸⣈⣷⣻⡇⠀⠀⠀⠀⠀⠀⠀⢿⣆⠓⡀⠀⠀⢻⡿⡆⠈⣿⡤⣿⠙⠛⠟⠋⢰⠀⠀⠀⠀⠀⠀⣿⣿⢎⢻⢸
⠀⠀⠀⢸⣯⡭⢿⡇⠀⢀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠻⡤⠐⠸⣇⢹⡷⡌⠛⣴⠞⠀⠀⠀⢠⠀⠀⣏⢿⣥⣿⢸
⠀⠀⠀⢸⢻⣷⣾⡇⢸⣸⡇⠀⠀⠀⠀⠀⢻⡀⠀⢄⠀⠀⠀⣷⠀⠁⠘⢿⣷⣄⠀⣤⠀⠀⠀⠀⣾⣰⡇⣿⣾⣿⢻⢸
⠀⠀⠀⢸⢸⡏⢻⣧⣿⢟⣧⠀⠀⠀⠀⠀⠰⡇⠀⠘⣆⠀⠀⢸⡀⡀⠀⠀⠳⣝⡷⡟⠀⠀⠀⢠⣟⡻⣧⣿⡟⣻⣿⢸
⠀⠀⠀⢸⣬⣙⣹⣿⣨⠃⢿⡆⠀⠀⠀⠀⠀⢠⠀⠀⠻⣀⠀⠸⡇⢠⢸⣆⠠⣈⣱⠃⠀⠀⠀⣾⠃⠃⡈⣿⣸⣁⢿⢸
⠀⠀⠀⢸⣛⣻⣽⣿⣿⣦⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢳⠀⠨⠙⣷⠟⠁⠀⠀⠀⠀⣽⣶⣶⣤⣿⣿⣿⣿⢸
⠀⠀⠀⢸⡤⢾⣿⣿⣿⣷⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢹⡄⢸⢇⠀⠢⡿⠀⠀⠀⠀⠀⣼⣷⣶⣶⣿⣿⣿⢿⣿⢸
⠀⠀⠀⢸⢄⠺⡟⢫⣿⣿⣴⣯⢷⣄⠀⠀⠀⠀⡆⠀⡠⣷⢸⡀⠈⣿⣦⣴⠃⠀⠀⠀⢠⣾⣿⣿⣷⣾⣿⢚⣦⠖⣼⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⡆⢠⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣄⣿⡷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⢸⡗⠀⠀⠺⠷⠞⠃⠸⠷⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠻⢶⣿⣇⣀⣀⣠⣤⣤⡴⠶⠶⠾⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⣹⡆⣴⣿⠉⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⡀⠉⣼⢻⣿⠘⢺⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢶⣄⠙⢿⣷⣾⠃⢸⡏⠀⡞⢸⣿⢀⣾⣿⣤⣴⣟⣴⠛⣷⠀⣠⣾⣿⣿⠀⢃⡾⠟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡦⠤⠀⣿⣿⣆⢸⡇⣸⢃⣾⠏⣾⡟⢹⠁⣿⣿⠃⢰⣿⢸⣿⡏⢹⣿⡆⢈⣐⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⣠⠄⣼⠃⢹⣿⠘⣷⡏⣼⣟⡴⢻⣧⡞⠀⣿⣿⡀⢸⡇⣼⣿⡇⣸⢿⣷⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠸⣧⠴⠿⠟⠀⠉⠀⠻⠛⢀⣀⣉⣀⣀⣈⣋⡁⠈⠛⠉⠙⠛⢁⣼⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⣠⣶⣿⠟⢻⣿⠿⣿⠉⠟⠛⠿⣶⣄⠀⣠⣿⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢻⣍⣿⡄⠀⠀⠀⠐⣿⣿⠇⠤⠘⢇⢀⡈⠀⠀⠿⢆⣼⣿⢰⣿⠁⢀⣿⠁⠀⢰⣿⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠙⠉⠀⠀⠀⠀⠀⠙⠿⣶⣿⣶⣿⣾⣿⣿⣶⣾⣿⠿⠋⠘⠷⠶⠟⢁⣄⢰⡈⢿⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣴⣿⣷⡀⢀⣤⢀⠀⢠⣦⡈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⠀⢠⣶⢦⣸⣿⠘⣧⣾⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣰⣿⠋⣿⠛⣿⡟⣹⡇⢀⡟⢹⣷⢀⣴⡄⠀⣰⣶⡄⣼⣻⠇⣼⡟⢺⡟⢿⣤⠿⠃⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⢿⡴⠏⢸⣿⣠⣿⢃⡾⢁⣾⣿⣿⢻⡇⣸⡿⢹⣧⣿⣏⣼⠛⠿⠾⠀⡄⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣄⡀⠀⠙⠉⠘⠟⠁⣾⡿⠻⣿⡼⢧⢿⣧⡞⣿⡍⠉⠁⢀⡀⣴⢘⡧⠚⠉⣤⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠙⠳⣄⡀⠀⣄⠀⠈⠀⢀⠈⠀⠀⠀⠉⡼⢹⣧⠀⢔⣋⣀⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣌⡃⠙⠦⠈⠓⢆⡠⠆⣼⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠷⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠄⠀⠀⠈⠁⠐⠢⢄⠀⠐⠀⠐⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠚⠃⢱⡄⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠄⠀⠀⠀⠀⠀⡀⠀⢁⠀⠀⠀⡀⠇⠀⢀⡀⠀⠀⠀⠀⣨⠆⠀⠀⠀⠀⠳⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢠⡿⠀⠁⢀⠉⢀⣀⠴⠞⠛⢢⡈⠓⠈⡄⢱⡎⠀⢀⡀⠀⠀⠀⢦⡈⢻⡄⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠘⠃⡌⠀⣡⠞⠋⠀⠀⠀⠀⠁⠀⠋⠀⠁⠃⠈⠁⠊⠉⠳⠄⠀⠈⢱⡈⢷⣄⡀⠀⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠒⣀⡁⠰⢎⡌⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠛⠦⠯⡝⣦⡀⣠⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠔⠈⠀⠀⠐⡉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⣠⣤⣶⢿⡏⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⣷⣯⠄⠀⢐⡉⢤⠄⠒⡛⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠂⢀⠀⠀⠠⠀⠈⠻⣆⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡋⠠⡔⠁⠀⠀⠀⠈⡀⠀⠘⠋⠗⣉⣉⡉⢉⡳⡄⠀⢠⡾⠛⠉⣀⣈⣙⠒⠀⢠⠀⠀⠄⡀⠀⠀⠀⠀⠀⢹⠀⠉⠑⠂
⠀⠀⠀⠀⠀⡠⠄⠚⣿⡇⠀⠀⠀⠀⣀⠀⣴⠇⠀⠀⠐⠪⠿⠿⠇⠐⠀⠀⠀⠸⠇⠐⠹⠧⠿⠉⠛⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⣼⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣎⠦⠀⠀⠫⣄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠓⠀⠛⠀⠀⠀⠘⠀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠄⢣⡀⠈⣷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠔⠀⢈⠀⡠⠃⢀⣼⠃⠀⠠⠀⠀⠸
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢸⣐⣴⠇⠀⢻⢾⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢰⢃⠎⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠆⠀⠀⢠⡔⠶⠛⠻⠋⠚⠻⠋⢀⠇⠀⠀⠀⠀⢀⡤⠀⠈⠁⢤⡔⠁⠠⠀⠀⠀⠀⠀⠃⠀⠐⠀⠐⠸⡦⢤⣜⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠲⣶⣤⣴⠾⠳⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⡄⠐⠒⠿⠋⠙⡆⠀⠈⠀⠀⢸⢠⡖⠺⢋⣉⣉⠝⢳⠀⠀⠀⠀⠀⠀⢸⡆⠀⠤⠤⢤⠤⠀⢈⡙⢦⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢠⠲⡃⠀⠀⠀⣆⡄⠀⠈⠛⠀⠁⡄⠀⠈⠀⠁⠀⠸⡀⠀⠀⠀⠄⠀⠘⠤⡀⠀⠈⠉⠉⠉⠀⠀⢠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠘⢆⠀⢀⡣⠀⠀⠀⠈⢥⡀⠀⠀⡀⢠⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢀⡀⠀⢲⡄⠀⠙⢠⣴⡥⠈⠄⠀⠙⠀⠀⠐⢃⠜⠀⢀⡴⠾⠛⠁⠀⠀⠀⡀⠄⣐⠭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⣀⠈⠱⠀⠁⢱⡀⠀⠀⠈⠁⠀⢸⡧⡾⠂⠸⣀⣤⣿⠶⠊⠁⠀⠀⠀⠀⠁⠘⠀⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⢸⡟⠁⠀⠀⣀⠄⢠⠎⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠚⣗⠣⠠⠤⠴⠊⠠⠄⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠀⢀⡴⣍⣩⣿⣾⠿⠋⠉⠒⠀⠤⢀⡁⠀⠀⠀⠀⠐⠤⠤⠉⢀⡀⠨⠅⠀⠀⠀⠈⠳⣤⣴⡾⣷⠀⠀⢀⡼
⠀⠀⠀⠀⠀⢰⣶⠁⠠⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠴⠀⠢⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⣉⢠⢤⡄⡀⠄⠂⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢀⠔⢁⠆⡇⢸⡿⠃⠀⠀⢀⣴⡤⠀⠀⢀⡥⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡠⠃⠰⠋⠀⠣⠸⡇⠠⡆⢀⡿⡋⠠⠀⠀⣿⡁⠀⠀⠀⠉⠙⠓⠄⢀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⡰⠁⠀⣜⣭⣀⣰⣯⡛⠗⡒⠀⡐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⣄⠀⠱
⠀⠀⢠⠃⠀⠀⡿⡟⠉⢸⠀⡈⠹⢡⠔⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠞⠋⠉⠀⠀⢁
⠀⠀⡜⠀⠀⠀⠇⠀⠀⠸⡀⠣⠘⣨⡇⢠⠇⠀⠀⠀⠀⡀⠤⠀⠴⠒⠉⠡⢤⡌⠁⠀⠀⠀⠀⠀⠈⡀
⠀⠀⡇⠀⠀⠀⠀⢄⠀⠂⠑⢠⣀⣹⣿⣿⠀⠀⠀⠸⠓⠒⠤⠀⠀⠀⠘⠡⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠁⠢⣱⡀⢻⠛⠛⠛⠧⣈⣡⠞⣳⣽⡂⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⠀⠀⠀⢀⠃⠀⢠⠟⡇⠀⠀⠀⠀⠀⠀⣈⡘⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⠀⡴⠁⢀⠞⠀⠀⣀⢀⠠⠒⠁⠈⠁⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠱⡀⠀⠀⠀⠀⡇⢰⡏⠀⠀⣸⢻⡁⠀⠀⠀⠀⠀⠀⠈⠳⣶⠦⠤⠔⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠱⡄⠀⠀⢀⠃⣸⠈⠛⣡⣈⡄⠉⣀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠘⢦⠀⠈⠛⠁⠠⠂⡚⣣⡀⣾⣭⣉⠒⠤⣄⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠉⠀⠁⠁⠀⠉⠉⠛⠲⢄⡁⠆⢩⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢨⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⠤⠤⠤⠤⠤⠴⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣷⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⣿⣿⣿⠿⢿⣿⣯⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣶⡄⢰⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⣇⠘⡿⠛⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄⠘⠿⡿⠃⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢙⣿⣿⣿⣿⣿⣴⣾⣿⣷⠂⣠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣟⠉⣀⣾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡿⠁⠾⣿⣿⣿⢿⣿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣦⠀⠀⠁⠀⠉⠀⠀⠀⠨⣿⣿⣷⣦⠀⠈⠻⣤⣽⡿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣽⣿⣝⢳⣋⣠⡀⠀⠀⣄⠀⣤⠘⠻⢿⣿⣷⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣋⣶⣰⣁⡶⠀⣇⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣧⣾⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣯⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣶⣿⣷⣶⣮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣟⣁⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣼⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠛⠛⠓⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠆⠀⠀⠀⠀⢈⢹⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⣠⡄⣀⣀⣸⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣞⢑⠄⣀⠀⡎⢱⠀⣀⠠⡊⣳⠀⠀⠀⠀⣿⣿⣿⣿⣻⡯⣿⣿⣿⢹⡏⣿⣿⣿⢽⣟⣿⣿⣿
⠀⠀⠀⡤⠚⠓⠸⣄⠗⡺⢗⠺⣠⠇⠚⠓⢤⠀⠀⠀⣿⣿⣿⢿⣭⣬⣯⡿⣺⢇⡸⣗⢿⣼⣥⣯⡿⣿⣿
⠀⠀⠀⡇⡇⢰⡉⠉⢁⣳⣞⠈⣉⡉⣂⡸⢸⠀⠀⠀⣿⣿⣿⢿⣿⣯⣷⣷⡾⢜⡫⣷⠶⣾⡽⢿⡇⣿⣿
⠀⠀⠀⠈⡇⡸⡄⠀⢡⠀⢠⠊⢠⠔⢤⠈⢣⠀⠀⠀⣿⣿⣿⣷⣿⣟⢿⣿⣾⣿⡟⣵⡿⣫⡻⣷⣝⣿⣿
⠀⠀⠀⠀⠇⠇⡅⢠⢲⠀⢎⠀⢈⠓⢢⠀⢀⠇⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⣟⣿⣿⣿⣿⣯⣟⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠈⠦⣈⠳⠊⣠⠞⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢷⣯⣷⢿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣤⡿⢿⣧⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣌⣽⣫⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠱⠄⠰⠎⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣤⡀⠀⠀⠀⠀⣀⡈⢛⡧⠀⠀⣞⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣙⠟⠃⠀⠠⣦⣍⣡⣤⣾⣷⣶⣶⣾⣷⣿⣩⢭⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⡤⠀⠀⠀⢫⣽⣽⣯⣙⢧⣿⣷⣿⣿⣿⡻⣼⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣿⣾⠇⠀⠀⠀⢸⣿⠿⣿⣿⠿⠆⠿⣿⣿⣿⠗⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠰⠋⠀⣿⣿⠆⠀⠀⠀⠈⣉⠛⣛⣿⣷⣿⣿⣿⣿⣿⢻⡏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⣿⠀⠀⠀⠐⢺⣿⡻⢯⠿⡽⠷⠿⠿⠶⣿⣿⣷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣘⣿⠀⠀⠀⠘⢿⡟⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠟⠉⣧⡀⠀⠀⠀⠀⡟⠀⠀⠀⡇⠀⠀⠀⠀⢸⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠾⠛⠛⠋⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣶⣶⠀⠀⠀⢠⣶⢶⣿⣽⡆⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣾⣿⣿⣷⠀⢀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣿⣾⠏⠉⣿⣄⣨⣿⡏⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⣿⠀⠈⣾⡁⢸⡟⠀⠉⠛⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⢰⡇⠀⠀⠘⠿⠿⠇⠀⠀⣾⣿⣿⠀⠀⠀⣀⠀⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣟⣿⠁⠀⠀⠀⢠⡤⠒⠀⠀⣿⢿⣿⠄⢰⣷⠀⣽⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠚⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣀⡗⠀⠀⠀⠀⣀⠀⠀⠀⢸⣷⣿⣷⣿⣿⡏⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣾⠃⠀⠀⠀⠀⠉⠀⠀⠀⢠⠛⣿⡻⣿⣿⣧⣿⣿⣿⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡏⠐⣿⣿⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⡟⠀⠀⠀⠀⠀⠙⠀⠁⠀⠀⢼⣧⠼⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡇⠀⠀⠀⠀⠀⠑⠀⣄⠀⠀⣸⣯⠐⣼⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⡟⣿⣷⣿⡿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠁⠀⠀⠀⠀⠀⠰⠀⢸⠀⠀⠁⣼⢻⡿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠸⣆⣷⣿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⡤⠀⠈⠁⠀⠐⢀⣿⣿⣿⣿⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢰⠀⠀⠀⠻⡄⠀⠘⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⡄⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⠈⠁⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣿⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠶⠆⠒⠶⠓⠒⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣦⡾⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⢿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⠟⠛⠁

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠄⠒⠂⠠⠤⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⢀⣀⣀⣀⣀⣀⣀⡠⠤⢄⠀⠠⢀⣤⣤⣠⠤⣀⡀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⢀⣠⠂⠀⠠⡐⢿⣿⣿⢣⠎⡽⠟⠒⣄⠀⠑⣄
⠀⠀⠀⠀⢀⠜⠀⠀⡀⢄⡠⠉⠘⠤⠃⠀⠀⠀⠀⠛⠀⠀⠀⢓⠀⣿⣿⣿⣸⣷⣯⣭⣭⣷⣄⠈⢣⡀
⠀⠀⠀⢀⠎⠀⡔⢈⡼⢳⢌⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢰⠿⠷⣌⢿⡿⢒⡒⠦⣭⣿⠧⠀⢳⡀
⠀⠀⠀⠎⠀⠀⠀⠑⠨⠑⢎⣩⠶⣬⡐⢄⡀⠀⠀⠀⠀⠀⠈⢠⢡⣤⡄⠺⢷⣷⣄⠨⠳⡌⢿⣧⠀⠀⢳
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠑⠋⣴⣋⠿⣿⡙⢛⣡⣀⠀⠐⠐⠘⠜⠧⠾⠀⣾⡿⢃⣷⠀⠹⣾⡿⠀⠀⠈⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣐⡀⠈⠙⠫⣄⣀⣰⠽⢛⡳⣄⢂⠀⠀⠄⠀⣛⣋⣵⣿⣿⢀⣇⣿⡇⠀⠈⠀⢹
⠀⠀⡇⠀⠀⠀⠀⠀⡐⠀⠿⠘⠀⠀⠀⠈⡉⣶⣶⣿⡖⣾⣷⣷⣀⣀⣾⣿⣿⣿⡿⢃⢎⣾⠟⠀⠀⣄⠀⢸
⠀⠀⡅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠱⠿⢷⣧⣿⣿⣿⣿⣿⣿⡿⢋⢔⣥⡿⠃⠀⠀⠀⠀⠁⢸
⠀⠀⢁⠀⠀⠀⠀⠘⢄⣤⡤⠀⠀⠀⠀⠀⠀⠀⠁⢤⣾⣿⣿⣿⣿⣿⣿⡟⢠⢃⡎⡞⠃⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠄⠀⠀⠀⠀⠀⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⣯⡥⠋⢘⡟⣿⣿⣿⣧⠸⡸⡇⠩⡄⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠐⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⢠⡞⠐⢠⡎⠡⣇⢿⣿⣿⣿⢯⡊⢢⡘⡆⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⠢⢤⣤⣤⣤⣤⡶⠚⠁⠈⠂⠈⠐⡆⢹⣦⢻⣿⣿⡎⣽⠀⡇⡄⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⠛⠟⠉⠀⠀⠀⠀⡀⡣⢌⡄⠁⢧⡼⣿⣿⠧⠀⢺⣷⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⣨⣤⣾⣷⣾⣾⣿⣿⣿⡄⠹⠟⢃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠐⢠⠴⠾⠿⠟⠛⠉⠉⠁⠙⠋⠁⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢤⣀⣀⣀⣀⣀⣀⣤⠤⠴⠒⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⢰⡦⠀⠀⣤⣴⣶⣦⠀⢠⣦⡀⠀⣴⣶⣦⡄⠀⣦⣾⣿⡄⣴⣶⣦⣄⠀⣴⣶⣄⠀⣴⡄⣴⡆⢠⣦⢠⣦⣦⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣯⣭⡅⠀⣾⣿⣇⠀⣿⣿⣿⣿⠀⣿⣭⣭⣿⣿⣏⣹⡿⣿⣿⣬⣉⠀⣿⣿⣿⡇⢸⣿⢸⣿⣉⣿⡇
⠀⠀⠀⠀⠀⢸⣧⣀⡀⣿⣏⣉⣁⢀⣿⣾⣿⠀⣿⣿⣼⡿⠀⣿⣉⣉⣿⣽⡟⢻⣷⣹⣿⣙⣿⠀⣿⣿⣿⡇⢸⣿⢸⣿⠛⠛
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠙⠛⠛⠋⠘⠋⠀⠛⠃⠛⠛⠛⠁⠀⠛⠛⠛⠃⠛⠃⠘⠛⠋⠛⠛⠋⠀⠛⠃⠛⠃⠘⠛⠈⠛







⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣵⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⠅⠀⠀⠀⠀⢀⡀⠂⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠆⠸⣿⡆⠀⠈⠀⠀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⡘⢄⠀⢻⣿⡀⠆⠀⠀⠉⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢎⢂⠀⢻⣧⠀⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠉⠐⠢⢕⡀⠱⣕⢄⠻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠶⣤⣌⡐⠌⠓⠤⡙⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠈⠙⠓⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡈⡀⡌⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠁⣥⠁⠀⠀⠀⢀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢷⢠⠇⠀⣠⣴⢿⣿⣿⣿⣿⣿⢧⣠⣤⡀⠀⠀⠀⠀⠀⢴⡿⢿⣶⣾⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⢸⢠⣾⣿⣿⣿⣿⣿⣻⣷⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠀⢸⣿⣿⣷⣿⣿⣿⣿⡿⠟⢋⣉⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠯⣙⣿⠿⢟⡋⠵⠒⠉⠁⠀⠀⠀⠀⠈⠳⣎⠉⠉⠛⠛⠋⠉⠉⠀⢀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢷⡐⢄⠀⠀⠀⠀⢀⣮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡢⠀⠘⣸⡸⡀⠀⠀⠀⠀⢀⡠⣈⠔⠀⠀⡼⠀⢰⠀⠀⠀⢣⠈⢷⡈⠳⠄⢀⣀⡾⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⢂⠁⠳⣍⣢⣤⢔⡢⠕⠊⠀⠀⠀⡰⠁⠀⢨⠀⠀⠀⠈⡆⠘⣧⠀⠀⠛⠉⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡀⠄⠈⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⢸⣴⠀⠀⠀⠀⠰⡀⠸⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠓⠋⠉⠉⠉⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡄⡁⠀⠀⠀⠀⠀⠡⠀⠹⡀⠀⢀⣪⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠡⣀⠠⡈⠳⣄⠀⠀⣀⠠⠔⠊⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠁⠀⠱⣶⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡒⢄⡉⠐⠂⠐⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣤⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣀⠀⠀⠈⠒⢬⡓⠤⣈⠁⠒⠒⠒⠁⢸⠀⠀⢈⠉⠐⠒⠂⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣀⡈⠑⠢⣑⠢⣄⠀⠀⠀⠣⠔⠉⠑⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣺⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⠙⠢⢍⡒⠄⠀⠀⠀⠘⠀⠀⢀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃

⠀⣀⢀
⡰⡄⠂
⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣤⡤⠤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠁⠰⠖⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠻⠀⢿⣆⣤⣤⡜⠁⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉⠺⠟⣠⣀⠀⠈⠱⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⣀⠀⠀⠾⢘⣷⠾⠷⣿⣿⣷⡲⣿⣿⣿⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠸⢏⡇⠀⠘⠿⠟⠁⠉⣽⣿⢰⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⢀⢨⢁⡀⠀⠀⠀⠀⠀⠀⠁⢈⠘⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣎⠸⣧⡶⠀⡄⠀⠀⠀⣾⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡷⠸⡇⠀⠘⢳⡆⠀⢶⠀⡄⢷⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠁⡄⠿⣦⣀⡀⠱⠀⠘⣂⣇⣼⠸⣇⡤⠼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣽⣿⣿⣾⣿⠟⢿⣟⣭⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⡷⠀⠀⢀⠲⠛⠙⠀⠀⠀⠀⣀⡶⢯⠔⢫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣷⣴⣧⣚⢟⠤⠔⠂⠀⢠⠴⠮⠍⠇⠈⡀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⣻⣿⡛⠀⠀⠤⠴⠤⠀⣛⠤⠂⠈⢴⠀⡿⠀⠀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢦⠙⠳⢖⣂⣀⡈⠁⠀⠄⠁⠉⡀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠷⡀⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⢆⡹⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠐⠂⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢠⠇⢀⠀⡐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢞⡋⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⡎⢹⠀⢘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⢁⢠⡐⠖⠉⠀⠀⠁⢸⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢋⣹⠍⠀⠀⡇⠠⠀⠀⠀⠈⠀⠰⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠈⠉⢥⠀⠀⠀⣿⢰⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⡷⢂⣼⠆⢾⢠⡄⠀⢩⡴⣸⣶⣿⡛⢓⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠾⣿⣯⡖⠌⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠀⠙⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣤⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢈⣙⢷⡄⠀⣴⢟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠻⠄⠀⠙⡾⠁⠸⠧⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠿⣶⠞⠛⠛⠛⠒⠒⠛⠉⠀⢸⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣸⠇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣰⣏⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠥⠴⠒⠒
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠔⠉⠀⠐⠤⣀⡀⠀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⣴⣡⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⢹⡏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣌⣦
⠀⠀⠀⠀⠀⠒⠒⠶⠶⠮⠭⣭⣽⣟⣓⠶⢦⣄⡀⢸⡇⢀⣠⡴⠶⣚⣻⣯⣭⠭⠵⠶⠶⠒⠒
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠓⠶⠾⢭⣝⣓⠶⣍⠻⠟⣩⠴⣚⣫⡭⠷⠶⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣁⣈⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠉⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣾⣿⣥⣶⣾⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡫⢹⣿⡿⣿⣿⡿⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣰⣿⣿⣿⣿⣟⠀⠀⠹⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⠀⠀⢀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⣿⣿⣿⣷⣼⣿⣷⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠗⠻⣿⣿⣿⣿⣿⡏⠋⢸⣿⡟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠁⠀⠀⠛⠿⠟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿

⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠄⡀
⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣍⠋⣀⣀⡀
⠀⠀⠀⠀⠀⢐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠉⣙⠻⣍⡁⠰
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠀⠙⠁⠘⣏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠰⠂⠀⠀⠿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠒⠂⠀⠐⠇
⠀⠀⠀⠀⢂⠀⠀⠀⢠⠀⠀⠀⠀⣀⣠⢤⡄⠀⠤⠴⢆⠘⠁⠀⠠⠀⠀⠀⠀⠀⠚⠖⢠⡀
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠉⠑⡆⡼⠂⠁⢺⡀⠉⠂⠤⣀⠄⠐⠲⠆⠀⠀⠂⠰⠆⣀⠔
⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⣠⠏⢛⠀⠄⢸⠃⠀⡁⠐⢏⢡⠼⠄⠒⠀⠀⢆⠀⡐⠫⠄⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⠛⠀⢿⣦⡄⠨⠀⠀⠰⠀⣠⢦⣧⣴⢦⣤⠈⠀⠀⣤⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠸⠇⣉⠰⡆⠀⠀⢀⠠⣔⠓⢤⡄⠄⢐⣀⠂⣝⠣⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠈⡃⣀⠀⡏⡛⠦⠠⠄⠈⠀⠉⠀⠀⠘⢠⡜⠋⠙⠛⢻⠛⣀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠣⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠳⣄⠀⠞⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡠⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⡀⢛⠛⣰⡀⠰⣦⠤⣤⠀⠀⠀⠐⢀⠀⠀⠐⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠈⠣⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⡄⠑⢤⠀⠘⢀⠀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡐⣄⠀⣠⠙⣆⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡆⠀⠀⠘⠃⠀⢀⠀⠀⠀⠀⠄⠈⡀⠀⠡⠈⠓⡀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠰⠀⠐⠀⠀⠀⢄⠀⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢦⡜⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⢄⠑⠆⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⣆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢦⠀⠈⠢⡀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⡀⠘⢦⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⢄⠀⠙⢦⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠁⠐⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⢀⡀⠀⢀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣷⡀⠀⠙⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡛⠀⠳⠀⠀⠀⢀⡄⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⣆⠀⠤⠸⡇⠀⠈⠀⠳⠆⢤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣾⡇⠀⠀⠇⠀⠀⠀⠀⢀⠀⠘⠃⠈⠛⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⣰⡏⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢶⠖⠲⠬⠥⠄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠴⠖⠒⠛⠛⠂⢴⠄⠆⠀⠠⠦⠲⠄⠠⠠⠶⠃⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠀⠠⠄⠠⠤⠤⠀⠀⠄⠠⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢤⣤⣠⣬⣿⣧⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠈⣿⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⡀⣀⠀⢀⣠⣄⣀⣠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⢿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣛⠿⣿⣿⣿⣿⣿⡜⣿⢻⣿⣿⡿⣫⣭⣭⡛⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⢿⣿⣮⢻⣿⣿⣿⠿⣀⢿⣿⡟⣾⣿⣿⣿⣿⣎⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣭⣝⣇⢩⣭⣶⣿⣿⣷⣝⠸⣿⣿⣿⣿⣿⢏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡟⣾⣶⣭⣭⣭⣵⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣿⣾⣿⣿⣿⣧⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠤⠤⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠒⠀⠀⠀⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠀⠀⠤⠄⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠉⠁⠀⠉⠀⢀⢀⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⠀⠐⡒⠖⠈


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

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢰⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠃⠀⠀⠀⠀⠀⢤⢰⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⢀⡎⠠⠀⢸⡁⠀⠀⠀⠀⠀⣄⠀⠺⢢⠚⠃⢀⡄⠀⠀⠀⠲⠀⣀⣀⠐⠃⠀⠀⠀⠠⠴⠀⣀⠄⠀⠀⠤⠀⠒⠂⠀⠀⠀⠈⡟⢻⠀⢐
⠀⠀⢶⣶⠟⡇⡏⠀⠀⠀⠀⠀⠘⡗⣬⣿⡹⡄⡯⠁⠀⢀⣐⠀⠃⢴⣄⠘⡁⢂⡀⠀⣰⣦⠀⣤⠟⠀⠚⡴⠂⠈⠀⠀⠀⠑⢘⣓⣘⡣⠊
⠀⠀⠨⣐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠗⠉⠀⠀⠀⠀⠀⠠⠧⣙⣁⠴⠑⠂⠀⠈⠘⠁⠀⠀⠉⣀⡘⠁⠀⠀⠀⢠⠀⠀⠀⠏⠹⠄⠀⠀⢀⡀
⠀⠀⠀⢿⡵⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠘⠖⠀⠀⠀⡀⠀⠀⠀⠀⢭⡥⠀⠀⠀⠀⣡⢣⠀⠀⢀⣀⠀⢠⠌⠙⠆
⠀⠀⠀⣘⢠⠀⠀⠀⢀⡷⢯⡀⠀⠀⠀⢀⢀⠀⠀⣀⠤⠄⡄⡀⠀⠸⠗⠀⠀⡀⠠⠀⡁⠠⠀⠀⠀⢰⡆⠀⢀⠀⡘⠔⢥⡏⠀⠨⠍⠀⢰⢚⠀⠈⠀⠀⡀
⠀⠀⠀⢸⡖⠀⠠⣁⠁⣷⣯⡇⠀⠤⠀⠘⢉⠀⠮⢰⡶⠳⣶⠹⡄⠀⡂⠀⠀⠀⠛⠀⡅⠘⠃⠀⠀⠈⣷⠒⠋⠀⠀⣄⣢⠀⠀⢀⣆⠀⠀⠙⠶⣀⡈⠟⠁
⠀⠀⠀⢰⢺⠀⢄⡅⢰⢾⠿⡇⠤⠄⠀⠸⠚⠀⣇⢹⣦⣤⣟⢁⠃⠈⠋⠀⠠⠀⣀⠈⠂⢀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠰⠖⠀⠀⠘⡞⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢬⡷⣾⠔⠡⠀⠀⢰⣠⠀⠀⠃⠠⡅⠄⠋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠸⠁⠂⠀⠀⡟⠀⠀⠀⠀⠈⡏⠀⠀⢀⢦⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣴⠄⠀⢐⣸⠀⠐⠀⠀⢠⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠰⠇⠀⠀⠀⡏⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢐⠀⠀⠀⠘⠿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠘⠦⠆⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡄⡄⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⡆⠀⠀⢠⠉⠄⠀⠀⠀⠘
⠀⠀⠀⢸⢉⠀⠀⠀⠖⠐⢠⠁⠀⠀⠀⠀⠇⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠷⡆⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⡆⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢐⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠰⠀⢠⢠⠄⢄⠄⠀⠀⠀⠀⠀⠀⢠⢠⠄⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠰⠒⠚⠀⠀⠀⠀⠀⠞⠀⠐⠀⠰⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣰⣆⠀⠀⣀⣠⣆⠀⠀⠀⠀⠀⠀⠙⣤⣈⣀⡴⠃⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⣧⣤⣠⡴⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠐⡦
⠀⠀⠀⠀⣹⣿⠀⠐⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣾
⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⢹⣧⠺⠛⠒⣾⣿⣿⡷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢼⣿⠀⡟⢸⡇⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠾⠿⠀⠀⠀⠼⠇⠀⠀⠀⠀⠀⠀⠀⠸⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠙⠠⠃⠘⠇⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣤⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠐⠟
⠀⠀⠀⠀⢼⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟
⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⡿⢿⡿⠃⠀⠀⠀⠀⠀⠈⣿⡏⠀⣿⢻⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠘⢻⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⢸⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⣧⢸⣿⣹⡇⣸⠻⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⠃⠿⠁⠉⠀⠀⠀⠀⠀⠀⠸⠿⠟⠸⠿⠾⠇⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣬⠀⣀⠉⠀⣘⢁⣤⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⡄⠐⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣁⣰⠆
⠀⠀⠀⠞⢹⣿⣶⣿⣧⣸⣿⣯⡄⠀⠀⠀⠀⠀⠈⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢾⣻⣿⡿⡴⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣷⡟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣏⢻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠰⡆⠦⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⢠⡞⠁⣸⠏⣹⡿⠋⠉⠙⣆⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⣿
⠀⠀⠀⠋⠀⢾⠻⠟⠉⠧⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⢼⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠠⠄⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⡌⠙⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣶⡀⠀⢰⣶⠶⠖⠂⠀⠀⠀⠀⠈⢳⣶⡞⠁⠀⠀⠀⠀⠀⠠⠄⠠⣿⣿⣿⡇⠀⠀⠙
⠀⠀⠀⠀⠀⢻⣶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⣶⣄⡀⠹⣿⣿⠓⠀⢀⣾⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣿⡇⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⠀⠀⠀⠀⢺⣿⡟⠿⠶⡻⠟⠈⠾⢻⣿⣧⠁
⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡗⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢀⡿⠺⣇⠀⠀⠀⠀⠀⠘⠛⠛⣿⡿⡇⢹⢀⣿⠛⠛⠛
⠀⠀⠀⠀⠀⣸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⣿⡇⠀⠰⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣼⣷⡄⠀⠀⠀⠀⠀⠀⣿⡇⠃⠘⢻⡿⠄
⠀⠀⠀⠀⠀⠁⠀⠈
⣿⢿⠿⠿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣿⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⡀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⡃⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣟⣫⣍⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⢳⣿⠏⠀⠀⢰⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⣿⣿⢾⣿⠇⢰⣶⣈⢀⡀⠙⠉⢀⡤⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣮⡿⡆⢿⣷⡷⢾⡛⠓⢶⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢷⡉⢀⣿⣡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢠⠧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣦⡿⢒⣨⡇⠀⢠⡆⡄⠤⠀⠀⠀⠀⠀⠀⠘⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠦⢽⣦⣈⣷⡘⢿⣿⠤⠼⠷⢻⡂⠈⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⡟⢿⡈⢻⣦⡄⠀⣾⡇⢸⠃⣰⢿⣶⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠈⢿⣜⣛⠁⠀⣿⠁⢸⠀⣿⠄⣡⠘⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠹⠏⣠⡌⢿⣟⢷⣤⡯⠀⣸⠆⣹⣾⠇⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⣿⣷⠦⣴⣛⠛⠈⢛⣷⣤⣤⡀⠁⣰⣿⣇⡀⢲⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡃⣚⣛⣿⣿⡟⢛⣋⣿⡿⢿⣿⣿⣿⣿⣿⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣤⢰⣿⣿⣿⣏⡻⢛⣉⣅⠀⠀⠚⣭⣽⣷⣌⣿⣿⣿⣿⡿⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⡁⠀⠹⣿⠟⣿⡇⢻⣿⣿⡌⢻⣿⣿⣿⡇⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠄⢐⡿⠀⣆⢶⡘⠻⣿⣿⣿⣿⡀⠓⠈⡞⢻⣷⢸⣿⡏⢷⠀⢻⣶⣈⡁⢸⡄
⠀⠀⠀⠀⠀⠀⣡⡞⠋⣿⠀⣿⡄⡀⠐⣿⣿⣿⡘⢧⡘⣷⡇⢸⣿⣿⡟⣿⠈⠀⠀⣎⠻⠇⣿
⠀⠀⠀⠀⣄⣾⠋⠀⠀⠈⠻⠿⡿⣷⡀⠘⣿⣿⣷⣄⠙⣿⡕⢀⣽⣿⢷⢻⡄⠀⠀⠘⣦⠀⣿
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢹⣿⡟⢿⣿⣯⡁⢨⣿⡇⢸⡀⣧⡀⠀⠀⡙⢀⣿
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣷⠈⢿⣿⣇⢻⣿⣇⠘⡇⠘⣷⡄⠘⣿⢸⡟
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⠀⠈⢿⣿⢸⣿⣿⠀⢿⠀⢿⡟⠀⠈⠘⠃
⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢸⣿⠀⠀⢸⣿⡈⣿⣿⡆⠸⡆⠈⣿⡀
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠀⠀⢹⡇⣿⡿⡷⠀⢁⠀⠀⢿⠆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⢸⢿⣿⠀⠀⢸⣷⢿⣷⡇⠀⠸⣆⢰⣠⠀⡆
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⢻⣿⠀⠀⣼⣿⠘⢿⡇⠀⠀⣿⡸⣿⠀⡇
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢸⢸⣿⣀⣾⠃⢹⡄⢸⣿⠀⠀⢻⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢸⢾⣿⡿⠃⠀⠀⣷⣼⡿⣇⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⡟⢀⣠⣾⡇⣿⣿⠃⢿⡀⢸⡇⠂⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⢠⣾⠏⡝⣡⠇⣿⠀⢸⣧⠸⣧⠆⢘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣤⣿⠟⠁⡴⢱⣿⠀⣿⡀⢸⡟⠀⠁⠀⠈
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣻⡇⢠⣼⠃⣿⡇⠀⣿⡇⣼⡇
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣸⢿⢋⣿⣿⢠⣿⠀⠀⣿⠀⣿⠀⡾⠀⠀⡀
⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⡿⢎⣿⣿⡇⠈⣿⠀⠀⣿⠀⣿⡜⠁⠀⠀⠇
⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢠⣿⣧⣼⣿⣿⠇⢸⣿⠀⠀⡇⢸⠈⡇⠀⠀⢀
⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣰⡿⢁⣾⠇⣀⢿⣿⣿⠇⢸⣿⠀⠀⣇⢨⡀⠇
⠀⠀⠈⢿⡄⠀⠀⠀⠀⢰⣿⡟⣡⣿⣿⠀⠈⠻⣿⠙⡇⢸⡿⠀⠀⠻⠮⠁
⠀⠀⠀⠈⠻⣦⣀⣀⣠⡾⢋⣴⠟⠛⡁⠛⠐⢟⢿⣆⠱⢆⠘⣛⠛⠓⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠙⠋⠀⠘⠋⠀⠀⠈⠳⢛⣤⡴⡤⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⡿⣍⡛⠮⡓⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢹⣷⣟⣛⠑⠀⠁⠁⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⡹⠖⣠⣉⣓⠀⠀⠐⢄⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣠⣾⣿⣿⠿⣿⣿⣿⣷⣤⡈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⣿⢸⣿⣯⣅⠀⠀⠈⠉⣿⣿⣿⣿⢸⣇⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠄⣽⣠⣦⡏⣿⣿⣿⠛⠲⢿⡟⠻⠷⠶⢿⣿⣿⣼⣿⣴⣆⣿⠠⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠲⣧⣼⣆⣤⣾⣿⡇⠋⣿⣿⣿⣆⠀⠚⠓⠀⣸⣿⣿⣿⡛⢹⣿⣷⣤⣰⣧⣼⠖
⠀⠀⠀⠀⠀⠀⣄⣸⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣏⠀⠀⢀⣿⣿⣿⣿⣧⠍⣿⣿⣿⣿⣿⣿⣇⡠
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣌⢻⢻⣿⣿⣿⣀⣀⣴⣿⣿⣿⣿⣿⣦⢸⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⣿⣿⣿⡙⠿⣿⣷⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⣾⣿⣿⣿⣿⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿⣷⣮⣤⣋⠻⣾⣿⣿⣿⣿⣿⣿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⣿⣟⣿⣿⣿⣿⣿⣾⣧⡻⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢋⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠙⠻⢿⠛⠿⡛⣿⣿⣿⣿⠟⠿⠛⡿⠟⠛⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⢻⡿⠉⠋⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠋⠀⠀⣠⣤⡶⢒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⠉⠀⠀⠀⠔⠒⠲⢮⡙⠛⢋⣠⣤⣤⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⣿⣦⢻⣿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠺⣿⣿⡇⠀⣿⣿⣷⣝⣿⣿⣝⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠙⠋⠃⠀⣿⣿⣿⣿⣿⣾⣿⣷⡀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠈⠻⣞⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠈⠀⠉⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠁⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⢠⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⢀⡏⠀⠀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⣰⣦⣼⣿⡟⠀⢀⣴⠇⠀⠀⠀⠀⠀⢀⣾⡿⠁⢀⡾⠁⢀⣦⣿⣿⡆⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠁⣤⣴⣾⣿⠏⠀⠀⠀⠀⠀⠠⠞⠁⠀⢠⣾⠃⠀⣼⣿⡿⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⡄⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⣼⢋⣿⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡇⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢀⡞⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⢙⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣽⣿⣷⣄⠀⠀⠀⠘⠛⠛⠛⠛⢿⠟⠛⢻⡟⠉⠉⠉⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⣀⣀⡀⣼⠀⠀⢸⡇⢀⣴⣿⡿⠟⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣷⣾⣿⣿⠀⢀⣴⣿⠟⣿⣿⣿⡀⠀⢸⣧⣾⣿⣿⣿⣦⣿⣿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡍⠙⡏⠉⠉⠀⣴⣿⣿⣷⣦⣾⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⣄⠀⠈⡇⠀⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣷⣼⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣿⣦⠀⣷⠀⢻⠀⣼⣿⣿⣿⣿⣯⡸⣿⣿⣿⣿⣿⡇⡿⠛⠛⣻⠛⠛⠛⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣙⣿⣟⣡⣿⣿⣷⡰⡇⠘⠀⠘⠛⠛⠛⠻⡿⠿⠿⣿⠿⠿⠟⢡⡇⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠉⠉⠀⣃⣄⣀⣾⠂⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠠⢸⠇⠀⢸⠱⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠁⢣⠀⠙⡆⠀⠀⠀⢰⡇⠀⢠⡇⠀⢀⡀⣾⠀⢀⡏⠁⢀⣴⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣴⣶⣦⡘⡆⠀⢸⡀⠀⠀⢸⠀⠀⠘⣡⡾⣿⣿⡆⠀⡼⠀⢠⣾⣟⣿⣅⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣷⠀⠈⡇⠀⠀⡞⠀⢰⣾⣿⣿⡿⣿⣿⡌⠁⠀⠘⣻⠛⠿⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⣿⠿⡟⠋⡇⠀⢸⣶⣶⡇⠀⢸⡟⣻⠛⢻⠿⢿⠇⠀⠀⢠⠇⡌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⢹⡄⢹⢀⣿⣧⣿⣿⠆⢸⠁⠇⠀⣾⡴⠃⠀⠀⢠⠏⣸⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣄⢧⠘⢷⡘⡆⠀⡏⢸⠁⠀⣿⡜⠀⣴⠋⠀⠀⡠⣠⢳⣌⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣟⠈⢆⠀⢳⣇⠀⡇⢸⠀⢠⡿⠁⢰⡏⠀⣠⢞⡴⠇⡁⠈⢳⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠈⢦⠘⣆⠀⠙⣄⡷⢸⠀⡼⠁⢀⡞⠀⡪⣐⢩⠴⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⠀⠀⠈⠣⡈⠧⠈⢧⠀⠈⠃⠘⠋⠀⠀⠚⠋⠉⠈⠀⠀⣠⠔⠋⠀⠀⢠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠆⠀⢸⣿⣷⡄⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠰⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⡀⠀⣼⣿⣿⣿⣿⣷⠀⢀⠀⠸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡦⢤⣤⣀⡀⣀⡴⠛⠛⠋⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⣠⣾⣿⣈⣿⡷⣾⣿⣽⣿⣿⡧⠈⠀⠀⠀⠀⠀⠃⠀⠰⠶⠄⠠⠶⠖⠶⠺⠅⠶⢿⡿
⠀⠀⠀⠀⢰⣿⢿⣭⢋⡀⠐⠀⠧⢢⠙⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠚⢹
⠀⠀⢀⣠⣿⣿⠉⡗⢛⠄⠀⡀⠀⢈⡁⢸⣿⣶⣿⡇⠀⠀⠐⣾⠀⠐⣦⠛⠀⠀⠘⠃⠀⠀⠀⢀⣿
⠀⠠⣿⣿⣿⣟⣶⣧⡀⠄⠚⠡⢀⠀⣡⣾⣿⣿⣟⡇⠀⠀⠀⣿⠀⠀⣿⡇⢠⡀⠀⠀⣠⡄⠀⠸⢻
⠀⠀⡙⠛⢿⣿⢿⣧⡔⠤⣶⢆⣄⡧⠈⢿⣿⠹⠟⡇⠀⠀⡸⡇⢀⠀⣿⡇⠈⠉⠀⠀⠈⠁⠀⠀⠘
⠀⠀⠈⠀⠘⣿⣞⠙⣷⡿⠀⢀⠀⠁⠂⣸⣿⣆⣃⡇⠀⠀⠐⠷⡦⠶⠿⠷⠀⠀⠠⠦⠀⠀⠀⠈⠂
⠀⠀⠀⠀⡀⠹⣮⣗⣻⣧⣀⠐⢂⣇⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⣷⡀⠘⢿⣿⣿⣿⣿⣿⣯⣹⠏⢙⢛⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢿⣟⠠⣀⠀⠉⠙⠛⠋⠉⢀⠀⢸⣋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⣺⣿⠿⣿⡿⣦⢶⣶⡶⢾⣏⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴
⠀⠀⠀⠀⢩⣥⣆⢿⣷⣴⣽⣟⢲⡄⢉⠀⢰⣾⣷⡆⠀⠀⡖⢂⣶⢰⠀⡆⡖⡶⣦⣖⠠⡆⠀⢀
⠀⠀⠀⠀⢰⣮⣉⣽⣿⣉⣹⣿⣈⣑⣾⠀⢈⣴⡛⡇⠀⠀⠳⡌⣽⢾⡠⡧⡧⢿⠏⣯⠸⠇⠀⠘⣧
⠀⠀⠀⠀⢸⡿⠿⣿⣷⠿⢻⣿⢿⠋⢠⣤⡀⣿⣿⡇⠀⠀⣀⣀⡀⣀⣀⢀⢀⣀⣀⡀⡀⢀⡀⢠⠹
⠀⠀⠀⠀⢸⡇⣿⢿⣿⣷⣿⣷⢾⡦⠀⣸⠳⢻⣏⡁⠀⠀⣟⣿⣿⡏⣿⢸⢸⣉⣈⡀⠉⠘⠃⠀⢰
⠀⠀⠀⠀⢸⣧⣽⣿⣿⣭⣻⠟⢁⣠⡞⠋⢰⡬⡿⠁⠀⠀⠟⠟⠛⠳⠟⠘⠚⠟⠛⠇⠛⠚⠂⢰⣿⡆
⠀⠀⠀⠀⢹⡟⡿⣿⣷⠟⠁⣴⢿⡿⢆⣀⣰⠟⠁⠀⠀⢠⢤⠄⣤⠤⡤⠠⣤⣤⣤⡤⠀⡤⠀⢸⠻⡇
⠀⠀⠀⠀⢱⢶⣿⣻⡟⠀⣼⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠚⠘⠙⠘⠛⠊⠉⠉⠉⠁⠉⠉⠀⢸⣀⡇
⠀⠀⠀⠀⢸⣷⣶⡿⠇⢀⣅⣀⠁⠀⠀⠀⣲⠆⢐⢒⣖⡒⣲⠒⡒⢲⣶⣶⡖⢲⣂⠀⠀⠰⠆⠾⠟⠃
⠀⠀⠀⠀⢸⡟⣃⠀⠰⠾⠿⠋⣀⡀⠀⣀⣀⡈⡀⠀⠈⠁⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠸⠟⠻⠷⠖⠤⠶⠾⠿⠿⠦⠬⠿⠦⠥⠾⠶⠤⠬⠷⠿⠿⠷⠦⠼⠿⠿⠿⠿⠦⠼⠿⠿⠇

⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣦⣿⣷⣀⠀⠢⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⢿⣿⣿⣿⣿⣧⡹⣦⡹⠿⢶⡶⣶⣶⡤⠀⠀⠀⠀⠀⢸⡀⢠
⣝⣿⣿⣿⣿⣿⣷⣜⡋⠐⣛⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⣇⣼⡀
⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣷⣾⣿⣧⠘⣷⠄
⣿⣿⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠉⠀⢹⡆
⣿⣿⣿⣷⡙⠻⣿⣏⠈⠉⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡀⣿⡀
⢻⣿⣿⣏⠳⢦⣤⣉⡢⣄⡀⢻⠁⠀⠀⠈⠉⠻⢿⣿⣿⣿⠀⣿⣿⣧⡀
⠈⠻⣿⣿⣧⠀⠈⠛⢿⣮⣛⢌⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠘⣿⣿⣷
⣧⡀⢌⢻⣿⠄⠀⠀⠈⠙⠿⣿⣯⣑⡦⡀⠀⠀⠀⠉⠀⠀⠀⠀⠸⣿⣿⣧⡀
⣻⣿⣦⣦⠉⠀⠀⠸⡤⠀⠀⠀⠉⠉⠙⠛⠻⠶⣦⣷⣦⣀⣠⣾⡆⢹⣿⣿⣇
⠈⠉⠛⣡⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠙⠋⠉⢀⣼⣿⣿⣿
⠀⠀⠀⠻⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠠⠤⠴⠈⢿⡿⠛⠛⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠘⠛⢛⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⣿⣿⣷⣤⣀⡀⠀⠐⠒⠺⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠶⠶⠶⣶⣤⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠉⠛⢿⣦⠈⠉⠉⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⠿⠿⠿⠿⠷⢶⣶⣦⣤⣀⡀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⢀⣀⡀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⢀⣠⣤⠀⠈⣉⡛⠿⢿⣯⡀
⠀⠀⠀⠀⢀⣼⣿⠾⠛⠛⠉⠉⠁⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣶⣿⣿⣿⡿⠀⠉⠛⣷⣴⣶⣄
⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢀⣤⡾⠟⠁⠘⢿⣷⣄
⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⣴⠟⠋⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⣿⠁⣀⣀⡀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣫⡶⠟⠁⣠⣶⣶⣶⣶⠤⠀⠀⠈⠻⣷⣄
⠀⢠⣿⠿⠋⠁⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣠⡾⠋⠀⢀⣼⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⢿⣦
⢠⣿⠏⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣠⡾⠋⠀⠀⠐⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡈⢻⣧
⣾⡟⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⣠⡾⠋⠀⠀⠀⠀⢀⣠⡤⠶⣶⣿⣿⠿⠿⢿⡿⠛⠛⠛⠛⠛⠛
⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⡾⠋⠀⠀⢀⣤⣶⣿⣿⣗⣒⡭⠟⢋⣀⣀⠀⠘⣿
⢻⡇⢠⣶⠂⠀⠀⠀⣿⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣠⣾⠋⠀⢀⣴⣾⣫⠞⣩⠟⠛⠉⠁⠖⠛⣉⣉⠉⠁⢠⡇
⠘⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣶⠟⢋⡉⠛⢿⣏⣹⢿⡶⠟⠋⠀⠀⠀⠀⣤⣊⣻⡇⠀⠀⠘⣧
⠀⣿⡏⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⡏⠀⠛⠙⢦⠀⠹⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⢧⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⣡⣴⠞⠉⣷⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⣷
⠀⢹⣷⠀⠀⣼⠀⠀⠀⠀⠸⣿⣿⣿⣯⡷⠟⠋⢀⣤⣾⣟⢷⡀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠒⣢⡴⠋
⠀⠀⠻⣧⢰⣇⠀⠀⢀⠀⠀⢻⣿⣿⣁⣤⡴⠾⠛⠉⠈⣿⡎⢷⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⢿⣿⡀⠀⢸⡀⠀⠘⣿⢻⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡏
⠀⠀⠀⠀⠘⢿⣷⣤⣿⡇⠀⢰⡟⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⠟
⠀⠀⠀⠀⠀⠀⠉⣿⣿⠁⢀⡾⠁⣸⣿⣤⣤⣤⣤⣶⣾⠿⠃⢀⣴⢻⡇⢸⡄⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⣿⠏⣠⠞⠁⣠⣿⣇⣀⣉⣉⣉⣉⣀⣤⣴⣿⡏⢸⡇⢸⡇⠈⢻⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣴⣿⡿⠞⢁⡴⠟⢋⣡⡼⢿⠭⠿⣛⣿⡿⠋⢀⣿⡇⢸⡇⢸⣷⠀⠀⠹⣿⣿⣿⡿⠷⠶⢦⣤⣤⠴⠞⠋
⠀⠀⠀⠀⠊⠉⠁⠀⣴⠏⣠⡾⠋⣡⠴⢒⣋⣩⣭⣤⣴⠶⠟⠛⢁⣾⠃⢸⣿⡆⠀⠀⢻⣿⣿⣇
⠀⠀⠀⠀⢠⣀⣀⣼⠇⡼⢻⡇⣸⣧⡾⠟⠛⢉⣁⣤⡤⠶⠶⠚⠛⢁⣠⣿⢻⡇⠀⠀⠀⠻⣿⣿⡄
⠀⠀⢲⣦⣾⣯⣽⢋⣼⢁⣿⢀⣿⠏⢀⣴⣾⣿⣽⠶⠶⠶⠶⠿⠿⠛⠛⢁⣼⠇⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⣈⣻⣦⣭⣥⣾⠏⣼⡇⣸⡟⢠⣿⠋⠁⣀⣠⣤⣤⣤⣴⣶⣶⣶⠟⣻⡟⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠙⣿⡻⢯⣭⣁⣴⠟⢠⣿⠃⣾⡇⢀⣾⡿⠿⣿⣦⣭⣙⣋⣉⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠈⠛⣶⣬⣥⣤⣴⡿⢁⣰⡿⠀⣸⡟⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠷⠶⠶⢶⣤⣤⣄⣀⡈⢻⣆
⠀⠀⠀⠀⠘⠛⠿⠭⠤⠴⣶⣿⢟⣡⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢡⣶⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣧⣄⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⠀⣨⣿⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⡝⣭⣟⣇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠘⣿⣿⣟⠋⠉⠻⣿⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣿⠀⢻⣿⠛⠛⡜⠛⣿⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠰⣿⢡⡿⠃⠀⢸⣿⡀⠸⠟⢠⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣴⠟⠉⣴⠛⣹⠀⠀⠿⠛⠃⠀⣠⣾⣿⡿⠓⠂⢹⠅⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⣼⠏⠠⣜⢶⡟⠁⠀⠀⣤⠴⠾⠿⣿⣿⡿⢀⠀⣀⠸⣶⡀⠦⠤⣤⡀⠀⠀⡇
⠀⠀⣠⡾⠟⠂⣰⠈⣨⠇⢀⠀⢢⡟⢧⣀⠀⢿⡏⠉⠀⠦⡄⠀⠈⠻⠀⠀⢿⡿⠖⠀⠁
⠀⠀⠉⠀⠀⠘⢁⣾⢁⠴⠋⠀⠘⠁⢀⣛⣿⡀⠀⠀⣀⠂⡑⠀⠀⠀⣀⣠⣏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣿⠙⡁⠀⠀⢀⣴⠤⣸⣿⣿⡧⣤⡄⣻⡦⣬⣷⠇⣴⣿⣿⡝⢇⠀⠀⠘⡂
⠀⠀⠀⠀⢰⣿⣆⢀⡄⠀⠀⠀⠁⠀⣸⣿⣿⡗⢫⠁⢣⠟⠉⢠⠀⡌⠙⢿⣇⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠐⠷⢾⣧⣇⡀⠀⠀⠀⣿⣿⡟⣉⣋⣷⣴⣦⣀⣾⣄⣀⡀⠈⣿⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⢺⣿⣿⣿⣿⡇⢹⠀⠀⠀⢸⣍⠏⢱⣶⡄⢿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⠈⢀⡀⠀⢷⠸⡆⠹⢣⡄⠘⠆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢸⡇⠀⢼⡚⣁⠀⠘⣃⠀⠈⢄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣇⡀⠀⢸⡗⠀⢸⡗⠉⠀⠀⠇⡀⠀⠀⢄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠁⠀⢾⡇⠀⣸⣿⣀⠦⠀⢸⡇⠀⠀⣠⣁⢸⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢿⣿⣿⢿⣿⢿⣿⣿⠓⠂⠀⢸⣿⣷⣿⣿⣿⣷⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⢢⠸⣿⣿⣿⣿⣿⣧⣿⢸⣷⣿⡯⣿⡇⠀⠀⠸⣷⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⢀⣿⣿⣿⣿⣿⣿⢻⢸⡇⢸⡇⣿⠗⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡟⢸⣿⣯⣿⣿⣿⠏⢸⠈⡀⢸⡇⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣿⣿⣿⣿⣿⡁⢸⠀⠀⢸⠃⣿⢹⠀⠀⠀⣿⣿⣿⣿⣿⣿⢻⣾⣿⣿⣿⣾⡟⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣿⣿⡀⢸⠀⡆⣼⠀⣿⡆⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⡟⢿⣿⣿⠟⠻⣯⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⠀⠀⣿⣿⣿⣿⣿⠧⢸⠀⣧⢼⠀⣿⣷⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣶⣮⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡿⠀⠀⣿⣿⢿⣿⣟⢠⢸⠀⣏⢸⡄⢸⣿⠆⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣟⣿⣿⣿⣶⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣀⣸⠆⢸⡇⠀⠀⣿⣿⠸⣿⡟⣻⢸⠀⠇⢸⡇⢸⣿⢲⠀⠀⢸⣿⣿⡻⣹⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣤⣾⣿⣯⠤⢼⣿⣦⡀⣿⣿⣧⡿⣇⣿⢸⠀⠀⢸⡇⢸⣿⣿⠀⣤⣸⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⢹⡇⣿⣿⢸⠀⠀⣾⣆⠘⣿⣿⠀⢿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢠⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣷⣯⣿⢸⠀⠀⣿⢻⢀⣿⣿⠄⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⢸⠀⠀⣿⢸⢸⣷⣿⡆⣾⣿⣿⣿⣿⣛⣿⣛⣿⣿⣇⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⢸⣄⣶⣿⣹⢨⣿⣿⡇⣻⣟⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣦⣽⣇⣹⣿⠸⣿⣿⠃⠿⢻⣿⡟⠿⢿⣿⣯⣿⡿⢿⣟⠋

⠀⠀⠀⠀⣠⣤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠘⡄⢠⢃⡨⠛⠧⣽⡤⠃⠠⠿⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣧⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠯⠤⠤⠔⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⣄⠫⠑⠀⡀⠀⠀⠀⠠⠀⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠉
⠀⠀⠄⠀⠀⠀⠀⠾⠆⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠄⢀⣴⣿⠟⠁
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠁⠤⢘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠘⠀⡒⠈⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⣾⡇⠀⠀⢠⡶⣆⠀⠀⠀⠛⠛⠛
⠀⠐⠂⠀⠀⠀⠀⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠈⠀⠀⠀⢸⣽⣿⡇⠀⢀⢀
⠀⠀⠀⠀⠀⣀⡀⠳⠆⠀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⣠⣶⡧⠀⠀⠀⠀⠀⠻⠓⠀⠙⠛
⠠⠀⠀⠀⠀⢙⡃⠀⠀⠀⠐⠂⠀⢠⣤⡄⠀⠻⡟⠀⠀⠘⠁⠀⠀⡀⠀⢀⡴⡶⢢⠀⢀⣦⡀⠀⠀⠀⠀⢴⡷
⠀⠐⠢⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣤⣤⡄⠀⣾⠿⠀⠘⢷⣏⡿⠁⠈⠰
⠀⢈⠡⠀⠹⣾⡿⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠑⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠾⠮⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠛⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⢰⠀⠀⠆⠀⠀⠈⠀⢀⡤⠒⡻⣏⠑⠢⣄⠀⠀⠀⠀⢒⠀⠗⠀⣀
⣐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣏⣤⣤⡧⠼⣦⠤⢬⣇⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⢀⠠
⠀⠅⠀⠂⠀⢲⡶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠉⠲⣾⣀⠀⣹⡖⠉⢸⠀⠀⠙⠀⠀⠀⠁⣰⣰⡀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢈⡤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡏⣤⣀⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠒⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣀⣸⡉⠻⠿⠿⠄⠀⠀⠀⢀⣠⣶⣿⡆⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠈⢠⣶⣶⣿⣿⣿⣿⣿⣏⠀⣠⣶⡶⢶⣿⣁⣄⡈⣶⡇⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣧⣾⢿⣿⣿⣾⣵⠂⣀⣤⣄⣀⣀
⣀⣀⣠⣿⣿⣽⣿⣿⣿⢏⡉⠉⣯⣉⣄⣤⣼⣿⣿⠝⠛⠻⠗⣶⠈⢳⣄⣤⣦⣼⣯⣿⣯⣿⣿⣯⣼⣿⣿⣷⣾⣿⣿⣿⣷⡽⣿
⣀⣤⣤⣼⣯⣯⣯⣥⣿⣬⣶⢱⣾⣷⣾⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣵⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠛⠛⠋⠉⠉⠉⠁⠈⠁⠈⠉⠀⠉⠉⠋⠉⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠘⠿⢿⣿⣧⣿⣾⣿⣿⣦⡀⠈⣌⣿⣿⣶
⢀⣴⣶⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠿⢿⣥⠶⠟⠓⢈⣵⣿⣿⣿⣩⠽⠿⢷⣄⣀⡹⢿
⣉⣍⣏⣿⣿⣿⣿⣿⣿⣿⣿⣶⣟⣷⣶⣶⣦⣤⣤⣤⣄⣤⣤⣄⣠⢤⣤⣤⣄⡀⣠⣶⣾⣿⣟⣯⡻⣿⣿⣿⣿⣖⣶⣥⣴⣮⣻
⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣉⣁⣀⣁⣈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠰⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠉⣁⣀⣀⣀⠁⠀⠀⠀⠀⡰⢿⠋⠁
⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⣠⡶⠛⠉⢍⠂⠀⠑⠂⣄⠀⢠⡏
⠀⠀⠀⠀⠀⣼⣿⡿⠁⢀⣾⡏⠀⠀⠁⠈⠀⠀⠣⠀⢨⣄⢾⡆
⠀⠀⠀⠀⢰⣿⣿⠁⠀⣾⣿⠀⠀⠈⠀⠁⢔⠐⠁⠀⡘⠿⠏⣀⠆
⠀⠀⠀⠀⢸⣿⣿⠀⢰⣿⡿⠣⠀⠂⡀⠀⠁⠀⠀⠀⠈⠈⠀⣿⣿⠶⠖⠦⣀⡀
⠀⠀⠀⠀⠘⣿⣿⡄⠘⡟⡁⠀⠀⢀⠄⢀⠤⠠⢀⡀⠀⠀⠈⠅⠀⠈⠀⠀⠀⡍⠳⠂⠄
⠀⠀⠀⠀⠀⢻⣿⣿⡄⠸⡀⠂⢡⣶⣄⢀⣾⣿⣶⣤⣉⠂⠆⠠⠔⢀⣀⠥⣴⣶
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣘⢧⡿⠿⢿⢸⣿⣿⡿⣻⣿⣿⣿⣿⡿⡿⢁⢠⡌⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⢃⣀⠃⠀⡙⣸⣿⣿⣿⣟⣩⣤⣤⣴⠊⠀⣠⢾⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠠⠐⣡⣷⣿⣿⣋⣭⣭⣶⡿⢧⣀⣈⠁⠈⠁⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⣋⣴⡿⣿⣿⡟⣿⣿⡿⠮⠿⠟⢻⡆⠀⠀⢨⠁⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣮⡾⠟⢉⣴⣿⣿⣷⣌⡉⠛⢶⡄⠀⡜⠁⠀⠀⠩⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠏⢀⣴⣿⠟⠋⣸⣿⢿⡏⠛⢷⡗⠀⡁⠀⠀⠀⠀⢠⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣧⢀⣸⣴⡿⠋⠀⣀⣼⣿⠋⢸⡇⠀⠘⢧⣀⢿⢿⣄⡀⢀⡾⢠⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠰⡅⣿⡈⣷⠋⢀⣴⠾⣻⣿⠃⠀⣼⠃⠀⠀⠀⢠⣧⡀⠙⡄⣼⢡⣿⠃⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣌⡃⣼⡋⢾⣵⢿⣿⡏⢀⡼⢣⡀⠀⠀⠀⠀⠈⣿⡀⢰⣿⣞⣧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣷⠏⣴⢹⣧⡿⠁⢿⣾⣇⢦⠈⡂⠀⠀⠀⢰⣿⣧⣿⠛⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡌⢸⣇⡯⡹⣧⠨⡄⠻⢿⣿⠿⢿⡆⠀⠀⠘⣿⣿⣘⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠁⢹⣷⡀⢠⣽⠤⢬⣢⠈⠁⡦⠀⠑⠤⠐⠀⣭⣛⠻⠿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣜⠃⠀⠀⠳⠠⡨⢁⣦⡀⢀⣀⠀⠁⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠻⠷⠿⠥⠄⠈⠔⠈⠉⠉⠉⠉⠉⠑⠒⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⠀⠀⠀⢠⣶⣏⡀⢀⠀⡀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠈⠀⠀⠀⣼⠏⣧⣶⣶⣇⢳⣳⠀⠀⠀⠘⣻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⣿⠀⠀⠀⢀⣼⡿⣾⣿⣿⣿⣿⣎⢻⣧⡀⠀⢀⢹⣿⡏⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠈⣿⢿⣿⣿⠃⢀⣼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣯⠳⣧⠁⠤⠝⠁⠃⢸⡿⣸⣿⣇
⠀⠀⠀⠀⠀⢀⣿⠁⢰⡏⢸⡿⣥⣶⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⡟⢀⣤⡤⠀⠸⡁⣹⣿⢸⡄
⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⢀⣾⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠁⡁⠿⣸⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣷⣿⣿⣿⣿⣾⣭⣭⣬⣍⣁⣠⠔⢁⣴⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣽⣿⣷⣿⣿⣿⣷⣾⣛⣛⣋⣉⡼⠿⣿⢿⣿⣿⣿⣧⣠⣽⣷⣷⣿⢹⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣯⣽⠤⣶⠛⣱⣾⣿⣿⣿⣿⣿⣯⣿⣿⡇⣨⡟
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⣿⠟⣛⡻⣿⣿⣿⣿⣿⣿⣿⡷⢾⣿⣿⣿⣿⣿⣿⣿⣿⠐⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠏⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⡟⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⠋⠀⡋⠉⠁⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡿⠉⠟⣋⣼⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡤⠄⣲⣆⡚⢿⡾⠿⠶⠂
⠀⡴⢦⣄⠀⠀⣀⠠⣤⣰⣖⡪⢝⡚⣖⣾⠕⠒⠛⠀⠉
⣸⡇⠀⠈⠙⠲⠄⠉⠛⠳⠌⠙⠀⠉
⢻⡆⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠘⠻⣶⣀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠛⠋
⠀⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣤⣴⡶⠞⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⣸⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠠⠔⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⣿⡇⠀⢀⣀⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠏⠛
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠘⠃⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡅⠐⣿⣷⡀⠹⣿⣿⠏⠀⠀⠘⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡀⠘⣿⣷⣶⣶⣶⣶⣄⠀⢴⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡟⠃⠀⣿⣿⣿⣿⠛⠁⢰⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⢿⣿⣿⡿⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠻⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠦⣄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢬⣅⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣤⠴⢤⣄⠀⠀⠀⠀⠀⠈⢳⡀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢁⡴⠚⠋⠙⠃
⠀⠀⠀⠀⢸⡇⣤⣄⠈⢷⡀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⠉⠓⠲⠶⠦⠤⠴⠖⠒⠊⠉⠁⠀⠀⢠⠶⠒⢦
⠀⠀⠀⠀⠈⠛⠒⠋⠀⠈⣇⢀⡴⠚⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡤⣄⠀⠀⣸⠀⢀⣄⠀⠀⢀⡴⠖⠛⠛⠳⢶⣄⠀⠀⢀⡾⠁⠀⠀⣾⡃⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢾⠀⠞⠀⣠⠏⠀⣾⠟⠀⢰⠏⠀⠀⠀⠀⠀⠀⠈⢧⢠⡞⠀⠠⡄⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠓⠒⠊⠁⢀⣰⢿⠀⠀⢸⠀⠀⣠⠤⡄⠀⠀⠀⢸⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣤⣤⡤⠖⣫⣥⠘⣧⠀⠈⢧⡀⠛⢀⡼⠀⠀⠀⣸⠃⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡞⠁⠸⣄⡘⢧⡀⠀⠉⠛⠋⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸⠇⠀⠙⠶⣤⣄⣀⣤⡴⠞⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣠⠀⠀⠀⣸⢁⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢰⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⡄⠰⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠻⣆⠈⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠈⠻⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢻⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⢀⣤⠞⠁
⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠈⣳⣤⡀⣀⣀⠀⠀⡀⠀⢀⣀⡀⢀⣀⣀⠀⣀⡀⠀⣀⣠⣴⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀⣤⠾⠋⠀⣠⡴⠛⠁
⠀⠀⠺⢿⡶⠶⠶⠶⠶⠶⠶⣿⠁⢀⡿⠉⠉⣿⠋⢹⣿⠋⠉⣽⠋⠁⣹⠟⠉⣽⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡝⠁⢀⣤⠞⠋
⠀⠀⣠⣼⣧⣤⣤⣤⣤⣤⣤⣿⣄⣸⣧⡀⠀⣧⡀⢸⣿⣄⢀⣿⡀⠀⢻⡄⠀⢿⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠛⠉⠙⠋⠁⠉⠉⠈⠛⠶⠋⠙⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢀⠀⠄⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠲⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣀⠤⣲⢖⣦⠀⣠⠔⠒⠉⠉⠉⠉⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠁⡠⢺⡏⢰⢣⣿⠗⠋⠀⠀⠀⠀⠀⣀⢐⠀⠀⠀⠈⠉⣓⡢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⢀⡴⠁⢀⣼⠁⢸⡇⢸⣾⠇⠀⠀⠀⠀⢀⢶⣫⡿⠛⠀⠀⠀⢠⣾⡏⠉⠙⠛⠺⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⢠⠞⠀⠀⡿⠀⠃⠀⢣⡘⢿⠀⢀⠴⣇⠠⢸⡼⢻⠁⢰⠀⠀⠀⣻⣏⡄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⣀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⢠⠏⠀⠀⣰⠖⡳⠀⢐⢶⣽⣸⢷⣞⣘⣡⠌⠉⢠⠃⠀⠀⠀⠀⠆⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⡀⠀⠀⠀⠳⡀
⠀⢀⡏⠀⠀⠠⣧⣾⢟⣤⣶⢾⡿⠿⢬⣭⣤⡆⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣵⡦⠀⠀⢱
⠀⡞⠀⠀⠀⠰⣿⢷⠏⡐⡔⣻⠀⢀⡀⢀⠎⢠⡖⠁⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⢧
⢰⠇⠀⠀⠀⠀⡇⢨⠀⡃⡁⣿⡀⢸⡏⢸⢧⢸⢧⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠛⢉⣉⡉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⠀⠀⠀⠀⠀⢡⠀⠇⢃⠁⠘⣧⠀⠉⠁⠀⣿⣧⣀⣼⠇⠈⠀⠀⠀⣠⡤⠤⠶⠚⢉⠀⠛⠉⠀⠀⢂⣩⣍⡭⠟⢸
⡏⠀⠀⠀⠀⠀⠀⠣⡀⠀⠙⠢⡈⠣⣠⣤⣀⣹⣿⣿⣿⠀⠀⠀⠀⢀⠹⠟⠉⠉⣱⣧⠀⠀⠀⠀⠈⠛⠲⡞⠀⠀⠀⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠱⡄⠈⢻⡿⣿⣿⣿⣿⣷⡀⠖⠲⠧⣰⠶⡦⢺⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠀⠀⢷⠀⠀⠀⠀⠈⣳⣴⣶⢾⠃⢰⢭⡯⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣄⣨⠃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⡀⣸⠀⠀⠀⠀⠀⢘⠁⢀⠂⢀⡾⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠧
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠯⠃⢠⢠⠏⠀⠀⠀⠀⡰⡡⠀⣁⡴⠋⠀⠲⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠁⡐⣱⠏⠀⠀⠀⣠⣬⣂⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢹⡀⢀⣾⠃⣀⠀⣀⢔⡟⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣯⣷⣤⡤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠘⡇⠈⡟⠸⢵⠶⠐⣈⠧⡔⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣴⣤⣼⠃⠀⠠⠀⣿⣥⢴⡧⠂⢠⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠙⠋⠁⠀⢰⠚⠐⡏⢣⡇⠠⢶⣿⣭⣉⠉⠛⠶⣤⣀⡀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠰⠟⠃⣺⠍⠧⠚⠙⠛⠻⠿⢷⣄⡀⠘⠳⡂⣄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣌⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣘⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣤⠶⠞⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣷⣶⣶⣶⣶⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀⠀⠀⠚⠏⠀⠀⠀⣄⣀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠛⠿⣶⣄⠀⠀⠀⠚⠻⠁⠀⠀⠠⠴⢿⡟
⠀⠀⣠⡾⠛⠉⠀⠀⢀⣠⡤⠶⠚⠛⣋⣉⣩⣭⣭⣭⣛⣦⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠁
⢠⣾⠋⠀⠀⢀⡴⠞⠉⢀⡤⢖⣪⠭⠒⠒⢋⣩⡭⠿⠛⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠏
⣿⠃⡎⣖⡆⡞⠀⢠⠞⡡⠚⠁⢀⣠⠶⠛⠉⠀⠀⠀⠀⢀⣀⣠⣤⣶⠟⠁⠀⠀⢀⡾⣻⠏
⣿⡄⢣⠳⠃⠳⢤⣏⣞⣀⡤⠞⠋⠀⠀⠀⠀⣀⣤⣶⠟⢛⣿⠉⠁⠀⠀⠀⠀⢀⡞⣰⠃
⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠁⣠⢎⣿⡄⠀⠀⠀⠀⢠⡞⣴⠃⢀⣸⣦⠄
⠀⠈⠻⢷⣤⣀⣀⠀⠀⢀⣀⣠⣴⡾⠛⠉⢀⣠⠴⢊⡵⢋⠘⣧⠀⠀⠀⢠⠟⡼⠃⠀⠀⠀⠙
⠀⠀⠀⠀⠈⠉⠛⣿⣛⠿⠿⠯⠗⢒⣒⣉⠥⠔⠊⡥⢖⡡⠂⠹⣆⠀⢠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐⡩⠔⣡⠀⢻⣴⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⣢⣴⢃⡞
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠚⣵⢣⡞
⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢡⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣠⣤⡴⠶⠛⡿⢡⠏⠀⠀⣰⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢀⣾⣿⠏⠀⠀⠈⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣹⣦⣴⣿⣶⣶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠛
⠀⠀⠀⠀⠀⢠⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⣿⡿⠛⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⢋⠉⠉⠈⠉⠀⠘⢿⣿⠋⠁⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠂⢸⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿