Boring Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⢀⠠⠐⠀⢹⠟⠭⠹⠋⠈⠁⠂⠄⠀⠀⠀⠀⢫⢉⠄⣿⡇
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂⢴⡥⠀⠀⠀⠀⠰⠀⠀⠀⠀⠛⠚⠋⠕⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠄⠀⡀⠀⠀⠀⠀⠐⢱⠤⠁⠢⡀⠀⣠⣾⣿⣿⣦
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠂⢀⠠⠀⠀⠈⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⢘⠏⠒⢸⠿
⠀⠀⠀⠀⠀⠈⢀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⢘⠀⠀⠠⠈⠀⡠⢞⡇⠐⠒⠢⡀
⠀⠀⠀⠀⠀⢁⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⡡⠀⠈⠀⠀⢇⠀⢀⠀⠘⠀⠀⠂⠀⠸
⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠄⠀⢄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣎⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣀⠄⠐⠀⠀⠀⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠐⠉⠙⠩⠀⠀⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠃⠁
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⣿⣅⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣼⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡦⠀⠈⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⣄⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣏⣩⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣈⣿⡏⠀⠘⣛⢻⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠉⠀⣿⠆
⠀⠀⠀⠀⠀⠀⣼⡟⢡⡤⣤⡄⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠏
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠘⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠈⠛⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠁⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠟⠋⠁
⠀⠈⠛⠻⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠛⠷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠿⠛⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣄⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡷⢾⣷⠶⢶⣤⡾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠈⠙⠻⣶⣄⣴⠃⠀⠀⠀⣸⣟⠉⠀⠹⣷⠀⢿⣆⠀⣿⡄⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠸⠿⣧⣤⣤⣤⡶⠿⢿⣇⣀⣠⣿⣀⣼⣿⣶⠿⠁⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣦⣄⡉⠛⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣉⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠈⠉⠙⠛⠛⠻⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣶⣶⣶⠆⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣿⠂⣿⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⣿⠀⣿⡟⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢠⣿⠀⢻⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢤⡼⢿⣤⣤⣤⣤⣼⣿⠀⢸⣧⣤⣤⣤⣤⡾⢷⣤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⢻⣷⣄⣷⣤⠏⠉⠉⠉⠙⣿⠀⣸⡏⠉⠉⠉⠹⢧⣴⣅⣾⡽⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⡀⣿⣥⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠃⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠁
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠂⠀⠀⠀⠀⠀⢀⡤⢂⣤⣤⠀⠀⠀⠀⠘⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡀⢀⢀⠀⢰⣄⡦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠠⠄⠉⠁⠀⠀⣀⡀⠀⢠⣄⠀⠀⠀⠀⢁⠱⠍⠚⠒⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠘⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠢⠶
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠤⡛⢀⠀⢀⡀⠀⠀⠈⠣⣄⠀⠑⠀⠀⠂⠀⠀⢤⣳⠂⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠁⢆⡀⠀⠀⠀⠉⠐⠒⠲⠒⠒⠈⠀⠀⠐⠀⠴⠇⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡄⢠⣤⡄⠀⠀⣾⡷⠀⠀⢠⣤⡄⠀⠀⠀⠘⣀⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠛⠊⠀⢸⡻⡇⠀⡀⢸⡆⢆⠀⠸⣟⡟⠀⣀⡀⣀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⣾⡇⠘⡄⠀⠈⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡀⠀⠀⢸⣿⡿⠀⠘⠀⢀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣉⣉⣉⣉⣈⣀⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠊⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠠⠇
⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⢿⠿⡿⢟⠻⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡿⠛⣟⣿⣿⡿⠿⠿⣿⢻⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⢋⡽⢋⣀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣛⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡿⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢸⣧⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠲⠤⢤⣄⣀
⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠖⠒⠊⠉⠉
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠂⠙⠃⠤⣀
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⢉⡑⠢⠤⣄⡀⠀⠀⢠⡵⠆
⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡽⠟⠃⠀⠀⠈⠙⠲⠤⣄⣉⠙⠶⣤⣙⢦⡀
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠈⣶⠆⠤⣀⡉⠙⠀⠀⠉⠛⠿⣄⡀
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⢸⣷⠀⢠⣀⠀⠀⠀⢰⠉⠋⠀⠈⠉⠛⠲⣤⠄⠀⠀⠈⠙⢦⡀
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⢸⡏⠀⡟⢹⣿⠓⢄⡄⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⢦⡀
⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⡿⠁⢸⡇⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⣀⣀⠀⠻⡄
⣰⠞⢉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⠷⠆⠀⠀⠀⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢸⠇⢀⡾⠋⠉⠙⢷⡀⠹⡄
⣧⣴⢋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠐⠛⢧⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⢀⣠⠏⠀⡾⠀⠀⠀⠀⣸⠃⠀⢻
⠘⠁⢸⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢠⡟⠀⠉⠉⠉⠀⢀⠈⠁⠀⣦⡤⠞⠁⠀⠀⢸⡄
⠀⠀⠀⠉⠉⠉⠀⠀⠈⠓⡄⠀⠀⠀⠀⠀⡇⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠒⠴⢖⡫⠃⠀⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⠦⠀⠠⠤⡖⠉⡇⠀⠘⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⣀⣀⣀⣡⠀⠀⠀⠀⠀⠀⣀⣱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠐⠿⣭⣤⠏⢹⡄⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⣈⣙⣿⡒⠲⠦⠤⣤⠀⠀⢈⡷⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⡽⠓⠶⡴⠛⠓⣶⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠚⠉⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣴⠃⠀⣰⠉⠉⠉⠙⠛⠛⠃⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⡰⠁⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⡴⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣀⡼⠁⠀⠀⣼⠁⠀⠀⢀⣤⠖⠛⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡼⠁⠀⠀⣰⠃⢀⡤⠞⢉⡀⠀⠀⠀⠀⠈⠑⠦⣀⠀⣀⣠⡤⠶⠒⠛⠉⠁⠈⠓⢮⡙⠓⢿⡀
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⣿⠃⠀⠀⣴⠟⠒⠻⣤⠖⠁⢀⡴⠋⠀⡀⠀⠀⠀⠈⠉⣻⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣽⢷⡬⠷⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠁⠀⠀⠀⠈⠛⠲⣾⣅⣀⣤⡼⠳⠦⠶⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠶⡇⠼⠆⠇⠷⠶⠰⣿⠐⠆⠸⡶⢿⡇⢼⡀⠻⠰⠶⠶⢎⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠄⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡠⡜⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⣀⠆⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⣤⣤⣄⡀⡀
⠀⠀⠀⢨⠳⣩⠀⠀⠉⠈⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡴⢒⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⢱⡁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⡞⢡⠈⡆⠀⠀⠀⠘⢡⡄⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠙⢀⠀⠠⠖⠈⠀⠀⠀⠀⠀⠀⢀⠎⠀⠐⠑⣈⠟⠱⣤⡄⢦⣀⠞⣿⣖⠚⣆⠀⠀⠀⠀⠀⠀⠣⡀⠀⢼⠿⠿⠁
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠰⣦⡀⠀⠀⠀⠀⢠⢋⠀⠔⠦⢔⣻⣰⢂⣿⡄⢸⠙⢿⡙⡉⠀⠈⣆⠀⠀⠀⠀⣠⡆⠀⠔⠘⡀
⠀⠀⠀⠀⠀⠠⠌⠒⠠⠀⠙⢮⣀⠲⡤⠀⢄⠈⣃⠀⠀⠀⡼⠉⠘⠛⠣⠤⠦⠤⢯⡁⠀⠀⠈⡄⠀⢤⣾⠿⣄⠀⠂⢸⢹⠙⢦
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⣿⠂⠀⠀⠀⠈⠉⠀⠂⠓⠂⠄⠀⠁⠀⠀⠀⠀⠑⠒⠒⠊⠀⠀⠀⠻⡔⠉⠀⠀⠀⣼⠀⡼⠃
⠀⠀⠀⠀⠀⢀⡴⠶⢖⣷⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠦⠀⠠⢏⡐⠀⠀⠆
⠀⠀⠀⠀⢰⠟⠁⠆⣾⣿⠁⢉⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠀⠀⣤⠀⣌⠠⣄⣀⠀⠀⠑⠦
⠀⠀⠀⠀⡞⠀⠈⡜⡏⠙⠙⢅⣀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠁⠓⠀⠀⠈⡀
⠀⠀⣰⠄⡇⠀⠀⠐⣷⣦⣴⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⠶⣤⡔⠞⢻⡥⠏⠀⠀⠀⠀⠀⢃⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠒⠀⠴⠶⡄⠙⠲⠦⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠠⠤⢤⡇⠀⠀⠀⠀⠀⠀⢐⠃⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣐⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣊⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠀⣹⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠸⢇⣀⣀⣀⡤⢤⣔⣀⣀⡁⠤⣤⡀⠀⠀⠀⠀⠀⠀⠠⢤⡽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⠄⠀⠀⠀⢀⡤⠒⠒⢢⡀⠉⠉⣹⡋⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⠤⢔⣢⣤⡀⠀⢀⠔⠙⢲⢄⡀⢰⠃⠀⠀⠙⡇⢀⣴⣎⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠐⢏⣉⣀⣉⡴⠁⠀⡎⠀⠀⠀⣀⡵⠁⣠⡤⣄⠀⠛⡥⣄⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡶⣄⡀⣔⠂⠤⠀⠀⠀⠑⠒⠈⠉⠉⣀⡏⠃⠁⠸⠖⠲⡶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣉⡷⢬⣁⡀⠀⠀⠀⠀⠈⣉⡀⢼⠀⢀⡤⠤⠴⠃⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢩⡜⠉⠝⡛⢿⠩⡀⠀⢸⠀⠘⠗⣄⡴⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠤⠤⡞⠉⠉⠉⠑⠦⡦⠈⠀⠀⡷⠒⠻⢤⡤⠖⠳⠤⢤⠖⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠁⠒⠒⠒⠠⢷⡀⠀⠀⠀⢠⣈⣛⡗⡾⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⡿⠋⠉⠉⠉⠀⣒⡲⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⠀⠀⠀⣀⡴⠚⠻⢶⠦⠤⣾⣿⠇⠹⣇⠀⠀⠀⠀⠀⠀⣹⣿⠻⠔⠂⠰⡖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⠀⠀⠀⠀⠳⣄⣀⣀⣳⣾⣿⠟⠀⣰⠉⠉⠲⣶⠒⢚⣿⣿⡏⡀⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⢠
⢶⠶⠶⠤⠭⢉⣉⣀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠀⠨⠭⠭⣉⣀⣀⡀⠸⣦⡀⢀⣸⣶⣿⡿⠋⠀⠀⡌⠀⢸⡀⠀⠀⠀⠀⠀⠀⢨
⠈⢧⠀⠠⠀⠤⠀⠈⠉⠉⠒⠂⠤⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠙⠛⠛⠁⠀⠤⠤⠤⠄⢀⣸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠘⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠖⠢⠤⠤⢤⣄⣀⣀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠣⠬⣀⣀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠠⠤⢀⣀⡀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠂⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠠⠤⢀⢰⣇⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⡤⠖⢋⣩⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠸⠋⠀⠀⠈⢡⣀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⢹⡁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣘⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⢸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⢀⡞⢇⠀⢀⠤⢄⡀⠀⠀⠀⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠁⣳⠀⠳⣈⡄⠀⠙⣄⡔⠋⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⢠⠀⠄⠀⢴⣴⢰⡄⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠐⠃⠀⠀⠀⠀⠀⠛⡆⠀⠈⠱⡀⢀⡜⣆⣍⡑⢋⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⠈⠈⠀⠸⠚⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠖⠀⠀⠀⠀⠀⢠⢧⠀⠀⠀⠱⣏⡁⡼⠀⢀⡀⠀⠀⠀⠀⠉⠉⠉⠉⡳⠀⠀⠀⠀⠀⡶⢧⡔⣤⢄⣤⡂⠀⡀⣴⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠐⠃⠀⡼⠉⠑⠒⠀⠄⠆⠀⠒⠉⠀⠀⣠⠖⠀⠀⡉⡁⠈⠈⠁⠀⠀⠀⢉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠅⣀⣤⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣏⠕⠮⠚⣥⠒⠷⣴⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠍⣠⡄⠀⢰⠲⣄⠀⡌⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠖⠻⢠⢨⠏⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠞⣳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣆⠀⢼⠁⢀⡠⠐⠉⠀⠀⠀⠀⠀⡠⣤⣶⠒⠒⠠⢀⡐⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠈⢆⠘⢿⢁⡁⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠉⠁⠒⠢⠼⡶⠦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⠚⠛⡡⢊⡔⠀⠈⣆⠰⡜⡎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⡜
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⢆⠀⡜⡰⠁⠀⠀⠀⠘⡄⣇⠠⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⡠⢁⡌
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⡟⢰⣁⡀⠀⠀⠀⠀⢠⡟⡀⢦⣀⣀⡜⠀⠀⠀⠀⠀⠀⢠⠻⠋⠗⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠴⣃⠼⢲⠣⢄⡀⠀⠀⠀⡇⠧⠴⠤⠤⠤⡏⠈⠉⡏⡒⠒⠟⢸⣰⠀⠃
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢣⠀⠀⠀⡸⠀⠀⠀⠈⣿⢻⠀⠀⠀⠀⠀⠀⢰⠀⠀⠶⠁⠀⠀⠉
⠀⢀⣀⠀⠀⠸⠀⠀⠀⠘⠒⠒⠉⠀⡄⠀⠀⣼⡟⡎⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊
⢀⡟⠸⡀⡔⠃⠀⠀⠀⠐⠆⠀⠀⣀⠃⢀⠼⢟⠈⠇
⢸⠁⢠⣝⡻⠂⠠⠀⠀⠀⠁⠀⠉⠀⢀⠊⠀⠌⢸
⡏⠉⠙⠄⠀⠀⠀⠀⠈⢠⣶⠆⠤⠴⢃⣀⡾⠤⠊
⡇⠀⢠⠀⠀⠀⢀⡤⠖⣡⠇⠀⠠⠭⢌⡁
⠘⠢⠂⠀⠀⠈⠙⠛⠛⠣⠤⠄⠀⠀⠀⠼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⢀⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠢⢤⡰⠋⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣀⠤⠓⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠳⠴⢒⠛⢧
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠠⢤⠖⠒⠊⡹⡴⠒⠢⣄⠀⠉⡠⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠀⠀⠀⠱⡼⠁⠀⠀⢸⠄
⣠⡤⠀⠐⠒⠒⢺⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢣⠀⠀⠀⢳⠀⠀⠀⣠⣿⣦⠤⣄⣸⢧⡀
⣷⠐⠥⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⢣⣀⠜⠁⢨⡧⠤⠔⠃⠀⠱⡄
⢱⠀⢠⠄⠀⠐⢇⣰⠂⡄⠀⠀⡰⠊⣉⣗⡲⠞⠀⠀⠀⠀⠀⠉⠒⠶⡺⠃⠀⠀⠀⠒⢦⢹⡀
⠸⣇⠈⠀⣀⣀⣀⣬⡾⢧⡀⣀⣳⣄⣀⣁⡭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢋⡸⠃
⠀⠉⠉⠉⡁⠀⠀⠃⣇⡠⠞⠉⠀⠀⠀⠀⢀⠈⠑⠲⠤⣄⣀⣀⣀⣀⣀⣀⣀⣠⠤⠖⢋⣀⣀⡀
⠀⠀⠀⡆⡇⠀⢀⡼⠋⠀⠀⠀⡆⠀⠀⠀⣼⠀⠀⢀⠔⢡⣃⡠⠜⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢐⠒
⠀⠀⠀⢀⣠⣀⡞⠑⠤⣀⠀⠘⢦⣤⣄⣀⣻⣀⠴⠗⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡀⠂
⠀⠀⠀⢫⡀⠀⠓⢄⡀⠀⠙⠒⢾⠉⣹⣘⣍⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢊⠀⠂⡃⠁
⠀⠀⠀⠀⡌⠉⠒⠦⢬⢑⣺⠀⠎⠀⡇⠉⠢⡏⡗⠒⠠⠄⣁⠀⠂⠠⠀⣀⠀⠀⠀⠀⣀⠄⢊⠄⠊⠀⠄⠀⢠
⠀⠀⠬⡔⠀⠀⠀⠰⡎⠀⠉⣦⣠⠔⠁⠀⠀⡁⡷⠤⢀⡀⠀⠈⠉⠒⠀⠤⢀⠉⢰⢊⠄⠊⢀⠄⠊⠀⠀⠀⠰⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢾⢠⣀⣀⣀⣾⡁⣁⣠⠜⠁⠉⠒⠠⢄⣀⠀⠀⣽⠹⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⢼⡙⢚⠏⢻⡁⡅⣱⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠈⠉⠈⠙⠧⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠛⠋⠉⠉⠙⠛⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⡤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠠⠾⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠻⢥⡤⠤⠤⠤⠴⢶⣂⣀⠤⠥⠐⣶⣀⣀⡀⠀⠀⠀⠀⠀⠲⢦⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⢀⠤⠀⠀⠀⠀⣠⠔⠒⠒⠲⡀⠈⠉⢩⣋⠁⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⠤⢔⣲⣤⡀⠀⠀⣠⠊⠓⣦⣄⠀⢰⠃⠀⠀⠘⣿⠀⣀⣶⣊⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢸⡛⠉⠉⢛⡠⠃⠀⣰⠁⠀⠀⠀⢈⡷⠃⣀⣠⣄⠀⠸⢋⣥⡀⠀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⣤⠀⢉⡭⠭⢍⠀⠀⠀⠙⠤⠒⠚⠉⠉⢀⡖⠇⠱⠘⡦⠤⡤⠤⠽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠸⠛⣆⡑⠂⠀⠀⠀⠀⠀⠠⠒⠢⠄⣸⠉⠁⣀⣀⣀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡏⠉⢿⣲⠒⢒⠶⠶⢖⠒⠊⠉⡽⠀⠸⡯⡉⢉⡔⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠷⠤⠾⢧⡀⣇⠘⠘⠃⢑⣄⣠⣿⠀⠀⣁⣽⡋⠀⣀⣠⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠶⢶⣶⣒⣒⣊⠉⢹⡇⠀⠀⠀⠀⠈⠳⣥⣂⣂⡿⠉⠀⠈⠙⠋⠁⠀⠉⠙⠋⣠⣮⡤⠤⠤⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡀⢠⠀⠀⠀⠀⠉⢙⣷⣀⠀⠀⠀⢘⣶⡦⠿⣼⠁⠀⠀⠀⠀⠀⠀⢰⣧⣴⠾⠟⠋⠀⠀⢀⣀⠤⠴⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠠⢴⠋⠉⠙⠿⡛⢛⣿⣿⡇⠀⢻⣄⣀⠀⠀⠀⢀⣀⣼⣿⠏⠛⠉⠉⢹⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠧⢤⣄⣀⣀⣀⠀⠈⠷⣦⣤⣤⣿⣿⡿⠟⠀⢰⡇⠀⠈⠻⡟⠉⢩⣿⣿⡟⢀⠀⡀⠀⢸
⢺⡛⠻⠿⠭⠥⢤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠦⠤⠤⢄⣀⣀⡀⢳⣆⣀⣤⣽⣾⣿⡿⠋⠀⠀⠤⠀⠀⢸
⠀⢳⠀⠀⢄⠰⠄⠀⠀⠉⠉⠑⠒⠒⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠓⠒⠶⠤⠤⠤⠄⣀⣸⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠲⠶⠦⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠧⣀⣃⡀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠉⠉⠓⠒⠠⠤⣄⣁⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠀⠀⠀⠀⠀⠀⠠⡉⠐⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠤⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠠⠤⠤⣀⣀⡀⠀⠀⠈⡇⠀⢀⣀⠤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠁⠙⠒⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢂⣨⠴⠞⠀⠈⠁⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⠤⠐⠺⢿⠀⠀⠀⠀⡄⠀⠁⠠⡀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠰⠁⠀⠀⠀⣠⠖⠛⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡜⠰⠛⠓⣄⢠⠀⠀⠀⡰⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠀⠐⠄⡀⠈⠀⠑⠒⠒⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠔⠐⠚⠉⠉
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡄⡦⢠⠐⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠦⠀⢠⠤⠔⠓⠂⠁⢀⠏⠫⠵⠂⢀⡇
⠀⠀⠀⠀⠀⠀⢀⠎⠈⠒⠀⠤⠀⢀⣀⠀⡇⢸⠩⠉⠙⠋⠓⠀⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠎⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⣄⠀⠀⠀⡄
⣀⣀⣀⣀⣀⡆⠀⠾⢀⠀⠀⢀⠀⠐⠒⡀⢇⠐⠀⡜⠀⠀⡇
⠀⠀⠀⠀⠀⠉⠉⠒⠒⠬⣁⡠⢤⡂⣀⡿⢾⡠⠮⠤⠤⠤⠧⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⡔⠈⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢘⣽⣿⣿⣿⡿⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⠐⠲⢮⣼⣿⣿⡟⣱⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣤⣄⣀⣀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡂⠉⠄⠀⠻⣿⣯⠻⣿⣿⣿⣿⣿⣿⣵⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢿⣧⡀⠘⠀⠀⠈⠀⠀⢛⠁⠈⢹⣿⣿⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⢷⠀⠀⡀⠀⠀⠠⣼⣇⣠⣿⠛⣁⣾⡿⢿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⡀⣠⡀⠀⠀⠀⠒⢺⣿⠿⠋⠸⢗⡳⢝⣸⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣷⠀⠀⣀⢤⠈⠁⣠⣶⣫⡷⠿⣓⣷⣚⡭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⡿⠋⠀⢰⢧⣿⡄⠀⠙⢋⠒⡁⢬⡭⠭⠅⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣠⢎⣿⣉⣳⡀⠀⠀⠙⢿⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣳⣿⣿⣿⣿⣷⣄⠀⠀⠀⠙⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⠟⠉⠁⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣠⠞⢋⣽⣿⣿⣿⣿⣿⣖⡺⠭⠒⠃⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢌⣁⣴⣿⣿⣿⣿⣿⣿⣿⣷⣖⣮⣭⣥⠀⠀⠀⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠫⣕⠪⠿⠛⠉⠀⠉⠙⠻⠿⣿⣿⣿⣿⣧⣧⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣵⠶⠚⢉⣀⣤⠴⠀⠈⠉⣽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⠙⠋⠐⢊⣉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⣠⣤⣾⣵⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣶⣦⡷⢷⡏⠉⠈⠐⡀⠂⠔⠁⠀⠀⠀⠀⠀⠁⠠⢾⣿⣿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠋⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡆⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡐⠐⠂⣠⣀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣮⡀⠘⠆⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠛⠿⠿⣿⣿⣾⣺⣇⠛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠦⠀⠀⣠⣀⣀⣙⢿⣿⣿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⠟⠀⠒⠶⠤⢤⣄⣸⢿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⢀⠀⣦⠀⠀⠀⠀⠀⠸⡁⣉⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣢⠾⠦⠄⢀⣤⣤⠀⢾⣿⡻⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⠋⠀⠀⠀⠀⢸⣿⠀⣰⠋⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⠀⠀⢀⡰⡄⠀⠧⠞⠑⠁⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀⣀⣘⠦⠐⠁⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠠⠉⠀⠀⠀⢀⡀⢀⡎⠀⠀⠀⠀⢠⠆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠑⣾⢦⣀⠀⠀⢠⠇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠀⠈⣹⡿⠛⠊⠙⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⢷⠀⡀⠀⠀⠀⡞⠉⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠒⣄⡀⠀⣈⣿⠁⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣶⣿⣾⣿⠏⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⠋⠀⣰⣿⣷⣤⣤⣤⣤⣤⣶⣶⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡡⠀⠀⢀⡴⠟⠛⠛⠛⠉⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡑⢄⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠳⡌⢳⡜⠀⣤⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠘⣄⠳⠇⡄⠀⠀⠀⠀⠀⢀⡜⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠓⣄⢀⡠⠴⠚⢲⡴⠁⣠⠎⠈⠆⡜⠀⠀⠀⠀⠀⢀⠞⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀⠀⠣⣍⣳⣢⡼⠉⠀⠀⢀⡼⠵⠊⢀⣄⢒⠋⢉⣀⠄⠀⠀⢠⠎⠀⣠⠋
⢀⣀⣀⣀⣀⣀⡄⠒⢤⠀⠀⣀⣀⣀⡠⠤⡴⠋⠀⠀⠀⠀⠀⠉⠉⠉⣡⠈⡿⠧⠤⠖⠋⠀⠀⠴⠛⢫⠀⠑⠆⠤⠀⠀⢠⠏⠀⡴⠁
⣬⣿⣿⣿⢿⣟⠏⣷⠈⠀⠀⠀⠀⠀⡆⠀⢇⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⠀⠈⠀⠀⠀⠀⠀⣠⠴⠋⠉⠁⠉⠑⠢⡀⢠⠋⢀⡜
⠈⠀⣌⡳⠒⠀⠉⠉⠀⠤⢀⠀⠀⢠⠃⠀⠀⢙⠷⠋⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠹⡎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠠⢁⠶⠎⣀⣠⠔⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣀⠠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠶⠒⠉⠁⠐⠀⠀⠀⠀⠀⠈⠦⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡄⠀⠀⠀⠀⠳⣄⢠⠤⠤⠔⠒⡃
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢠⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠙⢄⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣀⠤⠒⠉⠁⠈⠑⠂⠀⠀⠄⠀⠀⠀⢠⠎⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⢀⠧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣗⠂⠤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡴⠃⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡰⠁⢀⡠⠀⠐⠠⢶
⠀⠀⠀⠀⠀⠀⠈⢌⠉⠙⡦⢀⠀⠀⠀⠀⠀⣤⡤⣴⣶⡶⠖⠻⠁⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⠐⠉⠢⡱⠊
⠀⠀⠀⠀⠀⠀⡔⠈⢣⡈⠷⡀⠉⠉⠉⠉⣉⠥⡖⠉⠻⢄⡀⠃⠀⠀⢠⠋⠀⠀⠀⠀⣀⡠⡤⠲⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⢨⠟⠙⠫⣁⠄⠀⠀⠀⠀⠀⠉⠳⡦⣀⠥⠄⠒⠒⠉⠉⠀⠀⠘⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠁⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⢀⣙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⢤⡴⠋⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡠⠤⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠽⠳⠔⢒⡉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠠⢶⠖⠒⠉⡽⡴⠒⠲⢄⡀⠉⡡⠓⠒⢦
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠙⡞⠀⠀⠀⢸⠄
⡠⠔⠂⠀⠀⠐⢺⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⠀⢹⡀⠀⠀⣰⣿⣶⠤⠤⣼⢧⡀
⠃⠀⠏⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠳⣠⠞⠁⢠⡧⠤⠒⠁⠀⠹⡄
⢸⠀⢰⡂⠀⠀⠣⣴⠃⡄⠀⠀⡰⢉⣩⣭⣓⠎⠀⠀⠀⠀⠀⠈⠓⠒⠻⠁⠀⠀⠀⠀⢢⢹⡀
⠘⣃⣀⣀⣀⣀⠠⣤⡿⠣⣀⣀⡰⠤⣀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢁⡴⠁
⠀⠀⠈⠉⠁⠀⠀⠈⣃⡤⠚⠁⠀⠀⠀⠀⢀⠀⠉⠓⠢⣤⣤⣀⣀⣀⣀⣀⣠⠤⠴⠾⠭⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠇⠇⠀⢀⡜⠉⠀⠀⢀⡇⠀⠀⠀⣼⠀⠀⡠⠊⢠⡧⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐
⠀⠀⠀⢠⠰⢤⡎⠉⠢⢄⡀⠘⢲⣦⣤⣤⣽⠤⠞⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄
⠀⠀⠀⠩⢄⡀⠙⠦⣀⠀⠉⠑⢺⠀⣹⣬⢇⡀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⡅
⠀⠀⠀⠀⠌⠈⠉⠒⠢⠍⣻⠌⠋⠀⡇⠈⠲⡏⡏⠐⠒⠠⠄⣉⠐⠂⠠⠄⣀⡀⠀⠀⡠⠐⡁⠐⠀⠀⠀⠀⠀⢀
⠀⠀⡘⡄⠀⠀⠀⢐⡇⠀⢀⣯⢤⠚⠁⠀⢀⡅⡗⠂⢤⣀⡀⠀⠀⠁⠒⠂⠄⣀⣹⡡⠐⠉⡀⠔⠀⠀⠀⠀⠸⠸
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣾⣺⢸⣤⣤⣤⢾⡄⣥⣤⠎⠀⠈⠁⠒⠠⠀⡀⠀⠸⢈⣀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢼⣍⣨⠃⣹⣅⣇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⢻⠿⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⣙⡒⠀⢀⣉⠛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⢹⣷⠀⠀⠉⠁⢰⢣⠄⡴⠛⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠈⣿⣆⠀⠰⠄⠀⣠⠞⠛⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣄⠀⠀⠘⢿⣷⣶⣶⠾⢧⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⢀⣴⣾⣶⠆⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠤⠤⠴⣿⠀⠀⠀⠀⢀⡔⣿⣿⣿⣿⠀⠀⠘⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣿⣄⡤⠤⠤⠞⠀⠀⠀⠀⣿⣆⠀⠀⠘⡆⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⣿⣿⡄⠀⠀⢸⠉⠁⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣷⣀⡭⠟⠒⠒⠋⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣤⣤⣴⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠿⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠒⠛⠉⣩⣟⣁⣤⣤⣤⣤⣤⠤⢤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⡭⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⡹⢁⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣖⣛⡉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠀⠠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣩⣽⡿⠶⠆
⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣴⣒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠖⠿⢭⣅⣀⠀⠀⠈⢙⣦⡀
⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⡷⠆⠀⠀⠘⠶⣤⣉⡛⠶⢦⣌⡻⣅
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢑⣶⢤⣉⡙⠛⠂⠈⠙⠻⣧⡀
⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣶⡆⢠⣀⠀⠀⠀⣼⠛⠃⠈⠙⠓⠶⣤⠄⠀⠀⠙⠳⣄
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁⠀⠀⠀⠀⠀⣿⠀⣼⠹⣿⠓⢦⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⠳⡄
⠀⠘⠷⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⢤⡀⠀⠀⠀⢸⠇⢀⡏⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣀⣄⡀⠘⣆
⠀⢠⠟⢁⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⠛⠂⠀⠀⠈⠀⣸⡅⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⣸⠀⣰⠋⠁⠈⠙⡆⠘⣇
⠀⠸⡶⠃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠁⠹⣆⠀⠀⠀⠀⠀⢸⡧⢤⣤⡤⠖⠃⢠⠃⢀⠀⠀⡸⠃⠀⢸⡄
⠀⠀⠀⠘⠷⠶⠶⠶⠒⢤⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⣾⠇⠀⠀⠀⡀⢴⠀⠀⠘⠒⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣀⣀⣀⣠⣷⡞⠒⣾⠦⣄⠀⠀⠀⠀⠀⠈⠉⠑⠋⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⣿⠀⡇⠀⡇⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⣖⣛⡻⢻⠀⠁⠀⠁⠀⠈⠺⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠈⠉⠁⠘⣳⣶⠦⠤⣤⣀⣀⠀⠙⣶⠀⠀⠀⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠛⠉⠁⢈⡿⠲⢤⡞⠓⢶⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⡞⠀⢠⠏⠉⠉⠙⠛⠒⠒⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⢀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡞⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢀⠎⠀⠀⣸⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢴⠎⠀⠀⣰⠃⠀⢀⣠⠖⠋⠉⠙⠳⢄⡀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠿⣶⠤⣄⣘⡆
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣧⡏⠀⠀⢠⣏⣠⡖⢋⡠⠀⢀⡄⠀⠀⠀⠉⠲⢤⣤⠖⠚⠉⠉⠀⠀⠀⠈⠙⢦⡈⢹⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣀⣀⡴⠋⠉⠉⠹⢯⣀⣠⠏⠀⠐⣦⣀⢀⣀⡤⠞⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠂⠂⠀⠀⠛⠛⠛⠛⠁⠈⠙⠁⠀⠂⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠀⠀⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⣤⡤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠐⠛⠉⠙⢷⣄⠲⠶⢟⣥⡶⠶⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠤⣤⡆⠀⠀⠠⣤⡤⢹⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⢀⣀⡀⠀⠀⣻⣿⢾⡇⣿⠀⠀⠀⠀⠀⣿⢿⠉⠉⢳⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠶⣤⣤⣬⣥⣤⡴⢾⠟⠉⢸⠃⣿⠀⠀⠀⠀⠀⠘⣎⢧⠀⠀⢳⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠒⠒⠒⠒⢲⡞⠀⠀⢴⡼⠐⣯⡿⣼⡇⠈⣇⠀⠘⢦⣇⠀⠃⠐⠂⠒⠀⠹⣎⢧⠀⠀⢳⣸⠃
⠀⠀⠀⠀⠀⠀⢰⠇⢲⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠘⣿⣄⣿⠀⡟⢿⡄⢿⡀⢀⣀⣭⡉⠉⠉⣉⣉⣉⣉⣹⡌⣇⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠸⣇⢸⡇⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⣿⣿⣀⣇⢈⣻⣿⣴⣯⣭⣿⣿⣤⣤⣯⣭⣭⣭⣭⣥⣬⣦⣤⣤⣼⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠧⣤⣤⣤⢤⣿⠀⠀⠀⠀⠀⢰⠇⢸⣹⡏⠉⢻⣥⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⣩⣿⠀⠀⠀⠀⠀⣏⠀⢈⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⡶⢿⣆⠀⠀⠀⠀⠈⠶⢊⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⣛⠉⠉⠙⠛⢦⣄⣤⣤⣀⣴⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣍⣉⣙⠛⠓⠀⠀⠈⠉⢻⣿⡿⠹⡷⣶⢦⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⠛⢻⠛⠛⠉⢻⣧⣞⡔⠁⢸⠇⢀⣴⡿⠟⠛⣿⠀⠀⠀⠀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣼⣀⣸⣀⣀⡀⠘⣿⡟⠀⣠⢿⣶⡿⠋⢀⣠⠾⠁⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣠⣾⡛⠛⠛⠛⢛⣦⡙⣦⡿⣧⠞⠁⠀⠿⠖⠚⠋⠁⠀⠀⠀⢀⣀⣀⣿⣼⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣁⣼⠃⠀⠀⠀⠈⢷⣉⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣉⣀⣀⣀⣀⣠⣤⣬⣉⣳⣄
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⠿⠿⠿⠿⢿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠋⠉⠙⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⢀⣤⣶⣾⠿⠿⠟⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣹⣿⣇⠙⢿⣦⣀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠹⣿⡄⠀⠙⠻⣷⣦⡀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠀⠀⠙⣿⣦⠀⠀⠈⠻⣿⣆⠀⢠⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⣾⣿⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠸⣿⡄⢸⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣼⣿⡿⠋⠁⠀⢀⣀⣨⣭⣭⣽⣏⣉⣉⠉⠁⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠈⠙⠿⣷⣄⣿⣧⣾⡟
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠃⣠⣴⣾⠿⠛⠋⠉⠉⠉⠉⠛⠻⢿⣶⡄⠀⠀⠐⠿⠟⠋⠉⠉⠉⠉⠛⠻⢿⣶⣤⡈⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⣿⣿⠃
⠀⠀⠀⠀⠀⣀⣤⣿⣿⠀⠀⣀⣤⣴⣶⣾⣿⡿⣶⣦⣄⠀⠀⠀⢹⣿⡀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣦⣤⣀⡀⠀⠀⣿⣏⣤⣴⣦⣄
⠀⠀⠀⠀⣾⡟⠛⢻⣿⠀⠸⣿⣏⠉⠙⠿⠟⠀⢀⣽⡿⠃⠀⠀⠀⣿⡇⠀⠀⢺⣿⣏⠁⠘⠿⠿⠋⠙⢻⣿⠆⠀⣿⡿⠛⠉⠉⣿⡇
⠀⠀⠀⠀⣿⣆⠀⢸⣿⠀⠀⠘⢿⣷⣄⣀⣤⣶⡿⠛⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠙⢿⣷⣤⣀⣀⣀⣴⣿⠏⠀⠀⣿⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠙⢿⣦⣼⣿⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠋⠁⠀⠀⢠⣿⡇⢀⣤⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣶⣤⣄⣀⣀⡀⠀⣀⣀⣀⣤⣤⣶⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠟⠿⠿⠿⠿⠟⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⣿⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⡿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⣴⡿⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⢿⣷⣄⠀⠀⠀⠀⠙⣿⣦⠀⢀⣾⡿⠁⠀⠀⠀⠘⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡿⠿⠛⠉⠀⠀⠈⢻⣷⡄⠀⠀⠀⠘⣿⣦⣿⠟⠀⠀⠀⠀⣠⣾⠟⠛⠻⠿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣶⣤⡀
⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠘⣿⡆⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣇⢀⠾⠽⣝⡀⠀⣀⡀⠀⡇⠀⠀⠀⢀⡠⢒⡪⣉⠭⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⡏⠀⠈⠉⠹⠋⠋⡇⠀⢨⠂⠀⢠⠎⡫⣡⣤⣐⠶⢢⣾⣵⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡁⠠⠁⡄⠈⠀⠁⣃⡠⠎⠉⠐⢼⢈⢸⠩⠜⡝⢃⠀⢠⠘⢢⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠒⠺⠙⡀⠀⠓⠀⠀⢠⠋⠀⠀⠀⠀⠈⡾⡘⡄⠀⠑⡄⠀⢀⠤⠄⠾⢠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠇⠱⣀⠙⠁⡠⠃⠀⠀⠀⠀⠀⠀⠁⢹⢈⠆⠀⠸⠀⠀⠀⣠⠃⡇⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠃⠀⣰⠇⠀⠀⠀⡏⠀⢡⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁⠀⠰⢣⠑⢄⢀⡰⠁⠀⠀⠱⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠉⠐⡖⠒⠒⠂⠒⠃⠀⠀⠀⢸⠀⢠⢇⣸⠤⠤⣀⡀⠀⠀⠀⠀⠑⣄⠀⠀⠙⡄
⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⢆⠀⠀⠀⣀⡱⠖⠰⢄⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣿⡒⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢱⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⠴⠓⠈⠁⠈⠉⠋⠒⠈⠉⠁⠀⠒⠒⠒⠒⠒⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠢
⠀⠀⠀⠠⡄⠀⢠⠴⢄⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⠀⢄
⠀⢦⡀⠀⠈⠁⡈⠀⠤⡀⡱⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠶⠖⠋⠉⠐⠀⢄⠀⢀⢤⣤⠀⠀⠀⡄⠀⠀⠀⡀⠁⠀⠀⠁
⠀⠀⢳⠀⠀⠀⢀⡀⠠⠸⡅⠀⠀⠨⠂⠄⠀⠀⠀⠘⠀⠠⠁⠀⠀⠀⠀⡤⠀⠀⠘⠤⠏⠀⠀⠀⠀⠀⠓⠂⠈⠲
⠀⠀⠀⠆⢀⣀⣉⡒⠪⢷⢱⠀⠀⠀⠀⡀⠀⠀⣀⣐⣒⣁⠤⠂⠀⠀⢰⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠠⢠⡀⠀⡄⠀⠀⠀⡸⢸⠀⠴⠶⣾⠶⠀⠀⠉⠁⠀⠀⠀⠀⡀⢀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠈⡇
⠀⢥⡷⠄⢽⣛⠙⠀⢠⠃⣼⠀⠀⠀⠀⠁⠒⠀⠄⠀⠀⣤⣄⣈⡈⡼⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢊⠤⡄⢈⡉⠐⠀⢸⣰⠃⠄⢔⣚⣂⡤⠄⠀⡆⠁⠀⢠⣤⣏⡀⡇⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⢲⠒⠒⠃
⠀⢨⠉⠀⠀⠀⢸⠅⢀⣳⣄⠀⠰⠤⣀⣤⡆⠀⡇⠂⠘⠠⠼⡅⠉⢻⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⠌⠒⠈⠈⢧⡈⣡⠀⡇⡇⠀⠀⠀⠘⠵⠚⡆⣀⣀⣠⠤⠤⠖⠃⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠸⠇⠘⠒⡇⡇⢀⣄⣀⣴⣥⡈⠙⠙⡃⠢⣄⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠃⠀⣠⠞⠉⠉⠁⠘⢦⡘⠋⠀⠀⠀⠀⠀⢁⡌⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⡳⡀⠀⠀⠀⠈⠉⠁⠀⠀⠾⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠐⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣤⠀⠀⢀⡀
⠀⠀⢨⣿⣇⢀⣿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⡆⠀⣠⣽⣯⣀⠀⢰⣿⡆⠀⠀⠀⠀⠐⣿⡷⠀⠈⢿⡿⠁⠀⢿⣿⠆
⠀⠀⢠⡏⠁⡄⣴⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢈⣭⡁⢸⣿⡏⣿⣿⡆⢈⣭⡁⠀⠀⠀⠀⠀⣶⡆⢰⣿⣷⣼⣿⡆⢰⣶
⠀⠀⠘⢃⣀⠁⠻⢻⡟⠛⠛⣻⡛⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⣿⣧⣤⠀⠀⠀⠀⣤⣼⣿
⠀⠀⠀⠸⠿⠃⠀⠻⠟⠀⠘⢿⠇⠀⠀⠀⠀⠀⠘⠿⣿⣿⢸⡏⣿⠈⣿⣿⠿⠃⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠠⣿⣿⡿⢸⣿⣿⡇⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⡇⣿⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠛⠃


⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢀⣸⣇⡀⢴⡄⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣠⣰⣧⠚⢛⡛⠓⣼⣆⣤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⢸⣿⠇
⠀⠀⠀⠀⠙⡿⢿⣿⣿⡿⢿⠋⠀⠀⠀⠀⠀⠀⠀⣼⡼⠁⢀⣻⣟⡀⠈⢧⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⣶⡄
⠀⠀⠀⣤⡄⠁⠀⠈⠁⠀⠈⣠⣄⠀⠀⠀⠀⠀⠀⣟⠁⠀⣾⣿⣿⣷⠀⠈⣻⠀⠀⠀⠀⠸⢧⣤⣤⡤⣤⣤⣤⣤⡤⢼⣿⣿⡇
⠀⠀⠀⡻⢇⠀⠀⣿⣷⠀⠀⡸⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⢸⡷⠀⠰⣿⠀⢸⣿⡏⠁
⠀⠀⠘⠛⠛⠃⣠⣜⣣⣄⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⣾⣷⡆⣶⣿⣆⢸⠿⡇

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡤⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠠⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣄⣾⣿⣆⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⣴⣽⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⣶⡆⠀⠀⣿⣿⠀⠀⢰⣦
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⢀⡀⠀⠀⣀⠀⢸⠀⠀⠀⠀⢠⣾⣧⠀⣴⣾⣷⣦⠀⣼⣷⡄
⠀⠀⠀⠀⠀⠀⢿⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣼⣧⡀⣼⣷⡄⣤⣿⣄⣸⠀⠀⠀⠀⠸⣿⣿⠃⣿⣿⣿⣿⠸⣿⣿⡇
⠀⠀⠀⠀⣶⡆⠀⣴⣦⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠋⢿⣿⠏⢿⣿⠟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠉⣿⣿⠉⠀⢸⣿
⠀⠀⠀⣠⣾⣦⢠⣾⣷⡄⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠸⠿⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿
⠀⠀⠀⠈⠉⠉⠈⠉⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣄⣠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠉⠁⠉⠳⡌⠛⡵⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⣀⠀⠀⢀⣀⢹⢀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠉⠀⠀⠈⠉⢸⢹⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀⠀⠀⡤⠄⠀⣸⣿⢿⢰⡇⠀⠀⠀⢻⣯⡉⠙⡄⠀⣸
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡴⠁⢉⢻⣲⢶⣺⡏⢯⠀⢸⣸⠁⠄⢀⢀⡀⢷⢧⠀⠘⣄⡏
⠀⠀⠀⠀⢠⠏⣅⠀⠀⠀⠀⡼⠁⠀⠘⣧⣠⡏⡽⣧⢸⡆⢀⣉⡓⠒⠒⠒⠒⠚⢯⢧⠀⠈⢧
⠀⠀⠀⠀⠘⣆⢹⠀⠀⠀⢰⠇⠀⠀⠀⠈⡿⣧⣇⣈⣿⠷⠯⠽⠿⠦⠼⠯⠭⠭⠭⠤⠧⠤⠬⠧⠤⠤⠤
⠀⠀⠀⠀⠀⠈⠙⠒⣶⠲⣿⠀⠀⠀⠀⡸⠁⢧⡏⠀⠓⠒⠶⠶⠶⠶⠶⠶⡶⠶⣶⠶⠶⠶⠶⠶⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣚⣿⡆⠀⠀⠀⠳⣤⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⠇
⠀⠀⠀⠀⠀⠀⣠⢖⣛⡛⠛⠛⢦⣠⣤⣀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠙⠦⢬⣭⣭⣀⣀⣈⣽⣿⡏⡿⡒⢢⠀⠀⣠⣤⣤⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢻⠀⠀⢸⣿⠏⢀⣾⣠⡿⠋⢁⡼⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡞⣩⣟⠛⠛⠛⢯⡝⣤⣿⡠⠊⠘⠯⠴⠚⠉⠀⠀⢀⣀⣸⣸⣀⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣯⣥⣞⣀⣀⣀⣸⣮⣽⣁⣀⣀⣀⣀⣀⣀⣀⣀⣾⣥⣤⣤⣤⣤⣤⣭⣷⣄⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⢤⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣠⡤⠤⠤⠴⠶⠶⠒⠚⠛⠉⠉⠉⠁⠀⠀⠀⠘⡇⠀⠀⠀⠠⡷⠤⣸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⡇⠀⠀⠀⢀⣶⣴⡧⠄⡀
⠀⠀⠀⠀⣿⠀⠆⠀⠀⠀⠀⠀⠀⢀⢆⠠⠁⠀⠈⠓⢦⡤⣷⡀⠀⢠⡊⢊⡏⡁⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄
⠀⠀⠀⠀⢹⡄⢸⠀⠀⠀⠀⢀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠒⢧⡀⠁⠉⡇⠈⣧⠧⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⡎⠳⣄
⠀⠀⠀⠀⢸⡇⠈⡇⢀⣀⠢⠉⠑⠃⠀⠀⣀⠀⠀⠴⠆⠀⢸⠀⠉⠒⠇⠐⠈⠖⢲⠔⠀⠀⠀⠀⠀⠀⢀⠊⠑⠐⢻⠙⢦⠀⠀⢰⠒⣶⡀
⠀⠀⠀⠀⠘⣧⠀⢷⠄⠈⠀⠀⠐⠂⢉⣀⣀⣤⠤⠴⠖⠒⢻⠀⠀⡀⠴⠴⠖⠛⠚⠓⠲⣴⡆⠀⢶⠂⡉⣶⠶⠇⢰⠆⣾⡆⠀⢎⣀⣀⢘⡀
⠀⠀⠀⠀⠀⢿⣤⡤⠶⠶⠖⠛⠛⠋⠉⠁⢠⠂⠠⡀⡀⠀⠚⠋⠁⠀⠀⠀⢠⣤⡄⢠⣄⡛⠛⠀⠀⠂⠰⡕⠖⣀⡡⠀⠀⠑⢤⡞⠉⠉⠅⡅
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣤⡋⡹⢄⠆⣤⢺⡤⠀⢄⠀⠘⠛⢻⣞⠓⠂⠀⠀⢰⠉⠉⠑⠄⠫⢿⡂⠦⠀⠚⠟⠀⢰⠂⠇
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠉⠐⢖⠀⠀⠙⢷⠺⠷⢏⣼⠁⠀⠀⠄⠀⠀⣎⣉⠆⠀⠀⠀⢈⠦⠰⠒⠓⠄⡀⡀⢉⠐⠰⠆⠤⠆⢀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠳⢤⣑⠂⠀⣼⣀⠀⠀⠳⣄⡗⠀⠁⣻⣽⡀⡐⠁⡠⠀⠀⠂⠈⠁⠀⠀⣋⡠⠞⠊⢀⠂
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢂⠀⠶⣿⠿⠎⠀⠀⠈⠉⠉⠉⠳⡀⠀⢀⣁⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⢰⠒⠁
⠀⠀⠀⠀⠀⣠⡿⢧⡀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠁⠒⠓⠂⢶⠀⠀⠀⠀⠀⠀⠀⠇⠰⣶⡶⣸⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣦⡀
⠀⠀⠀⠀⢠⡿⠁⠀⠙⠆⠀⠀⠀⠀⠀⠀⠁⠐⢲⡆⡶⠂⠘⠇⠀⠀⠀⠀⠀⠀⢨⢰⣽⣧⡘⢷⣶⠀⠀⠀⠀⠀⠀⢀⠜⠪⠁⠛⠰⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⣻⠋⣳⣤⠈⡄⠀⠀⠀⠀⠀⣸⠎⠀⠀⠰⢲⣌⣀⣀⣀⡀⡠⠐⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠁⠙⠛⠀⠙⠓⠶⣶⠶⠟⠁⠀⠀⠰⣀⠓⠏⠉⠉⢉⣧⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⠏⣷⡆⠀⠀⠀⠀⠉⠁⠀⣤⡏⣸⡅⠸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠃

⣿⢿⡟⡛⠛⠉⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠘⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣤⣀⣀⣀⢠⠄⠈⣽⣇⡤⠀⠀⠀⠀⠀⠀⣍⠂⠀⢀⠀⢀⠀⠀⠀⠀⠀⢁⣳⡒⠚⣿⡏⠙⠻⢿⣿⣿⣿⣿⣿
⣿⣿⡿⠁⢽⣿⠠⠀⠐⣻⣯⠀⠀⠀⠀⠀⠀⠀⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⣽⣿⡗⠒⠒⠢⣾⣿⣿⣿⣿
⣿⣟⠀⠀⠀⠀⠀⠀⠸⣽⣇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢀⣠⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⢀⢰⣷⣿⣿⣿⣿⣿
⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⡎⠛⠯⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢼⣿⡇⠀⡀⢸⣿⣧⠸⣿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⢰⣿⣷⣄⡀⢰⣿⡇⠀⣸
⠀⢿⣇⠀⠀⣠⣿⡆⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣷⣿⣶⠒⠒⠒⢻⠒⠒⠒⢺⣿⣿⣿⣗⣺⣿⣷⢶⣿
⠀⣬⣿⣸⣿⣿⣿⣧⣤⣿⣿⣰⣷⣶⣶⣾⣷⠂⣾⡗⣸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣷⣿⣿⣿⣡⣥⣴⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢟⣿⣿⣯⣷⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢘⡿⠿⠏⣋⠀⢒⣛⣳⣶⣶⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠉⠉⠉⠉⠒⢦⡀⢀⣀⣀⡀⠀⡀⠤⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠀⠀⢀⣀⣀⠀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⡷⢿⡄⣇⠀⠀⢠⡿⣧⠀⣼⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁⠀⠀⠈⠁⠁⠀⠈⠀⠀⠀⠀⠀⡀⠀⢦⠀⣧⠀⠀⠈⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢦⠀⣆⠀⠀⣿⢦⠘⠏⠙⠆⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢨⣟⡷⠀⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡳⠘⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠺⠬⠃⠰⣧⣀⢀⡼⣿⡀⢰⣃⣰⠆⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠀⠿⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠬⣟⢸⡀⠀⢠⠿⣄⢻⠖⡇⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠀⠀⠀⠀⠁⠘⣆⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠁⠐⠉⠀⠝⠛⠲⢦⡒⠂⠀⠀⠀⠈⠙⠒⠒⠒⠊⠁⠘⠦⣄⣀⠠⠚⠦⣄⣀⣀⣀⣠⠔⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⠊⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠠⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⢳⡀⠀⡀⠰⡌⠳⢤⡤⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠤⠤⠤⣤⠤⠤⠀⠀⠒⠀⠀⢀⠎
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⢀⡠⠙⢆⠀⠀⠀⠀⠉⠓⠒⠦⢤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠬⠁⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⢼⠦⡄⠀⠀⠀⠀⠀⠀⠀⠔⠙⠄
⡆⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠀⠐⠒⠒⠒⠒⠒⠂⠀⠒⠒⠂⠒
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠙⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⣀⡀⢀⡀
⣀⣀⣨⣍⣭⣁⣉⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣽⣭⣭⣭⣀⣠⣭⣭⣭⣭⣭⣍⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣲
⠀⠀⠀⠀⠼⠷⣅⣠⡆⣤⣄⠀⣆⠀⠀⠀⠀⠀⠀⠈⠙⠱⡄⢠⡖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣦⡻⣈⠁⡄⠀⠿⠼⠛⠆⠀⠀⠀⠀⠁⠙⠎
⠀⠀⠀⠀⣠⠄⡄⠐⣄⡘⠀⠀⠸⠗⠈⠁⠀⠀⠀⠀⠀⣠⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠈⠻⣭⠇⠀⠀⢀⠀⠀⠒⠉⠫⡉⢻⡓⢦⣄⠉⠊⠳⠀⢴⢇⡀⡀⠀⠀⢀⠄⠈⠀⠙⠢⣄⠀⠀⠀⠈⠓⢤
⠀⠀⠀⠀⠀⣀⠠⣠⢆⡐⠁⠀⠀⠀⠀⠀⢹⠀⢣⠀⠈⢳⡄⠀⣤⡀⠈⠋⠀⠀⢀⠊⣠⡀⠙⠓⠀⠈⢢⠀⠀⠀⠀⠈⢳
⣀⢄⡀⣄⢀⠹⠃⢁⡼⠀⠀⠂⠀⠀⠐⣒⠀⠀⠘⠀⠀⠀⢹⡄⣟⠳⠤⠾⠦⣄⡦⠀⠛⠃⠀⠀⠀⠀⠀⢧
⠘⣎⠑⠈⠁⠀⠀⢦⡇⠀⠀⡐⠟⠀⢀⠛⠁⠀⠀⠀⠀⠀⢰⡀⢸⠂⠀⠀⠤⠄⠙⠂⠠⠤⣀⠀⠀⠀⠀⠈⡄
⠀⠀⢀⢶⣹⠀⠀⢸⡇⠀⠀⣉⣀⣀⡀⣉⡀⢀⠤⡀⠀⠀⠀⡏⠹⡀⣦⣦⡀⠀⣦⠀⣤⡆⠠⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠹⣏⠁⠀⡌⢱⡀⠀⣇⣹⡠⠧⠼⠒⠚⠂⠁⠀⠀⡘⠀⠀⢇⠓⠊⠓⠊⠉⠁⠁⡑⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠈⠀⢹⣇⡄⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠾⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳⠦⣤⣀⣀⡀⣀⣅⠀⠀⠀⠆
⠀⠀⠀⠀⠀⣄⡞⣄⣿⠂⠀⠈⢹⣶⠶⠶⠶⠖⠚⣩⡨⠃⣴⠄⠀⠀⡇⠀⠀⠀⠀⠰⠎⠃⠀⠀⠈⡉⠉⠀⢰⠀⠀⢸
⠀⣖⣄⢦⡀⢻⠁⠈⠀⠀⠀⠀⣐⡇⠀⣅⠀⠀⠙⠋⠀⠈⠉⠐⠒⠇⠻⢥⠄⠲⣾⠊⠀⠀⠀⠀⢠⣿⠀⠀⢸⡀⠀⢸
⠀⠘⠊⠀⠀⡆⠀⠀⡀⣀⡀⢤⢙⡃⠉⢹⡙⠒⠤⣄⣀⠀⠀⠀⠀⠀⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⠀⠀⠀⠳⣄⠸⡄
⠀⠀⠀⠀⠸⠆⠚⠈⠉⠀⠴⣦⡍⠀⠀⠀⣇⠀⠀⠀⠈⠉⠛⠲⠦⢤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⠀⠀⠀⠀⠈⠙⠓
⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⡄⣄⠀⡶⠜⠎⠃⠀⠀⠀⠀⠀⠀⠰⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠿⢯⡀⠀⣠⠀⠀⠀⠀⡿⠀⠀⠀⠀⠳⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠰⢧⠂⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣧⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡌⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠂⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣲⠞⠀⠀⠀⠀⢰⡂⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣦⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡆⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣥⣬⣝⢻⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⡟⣸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢰⣿⣿⣿⣿⣆⢿⣿⣥⣴⣶⣾⣿⣷⣾⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⣠⣿⣿⣿⣿⣿⠘⠿⣿⣿⣿⣿⡿⠟⣫⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢋⣼⣿⣿⣿⣿⡿⢣⣾⣷⣶⣶⣴⣶⣶⣿⣿⣿⣏⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢡⣾⣿⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⣠⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢿⣿⣿⡿⢣⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⡿⢟⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣬⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⡿⢛⣩⣴⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣉⣭⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠛⠛⠿⠿⣿⣿⣿⣿⠟⠉⠿⣿⣿⣿⡿⠿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⡀⢰⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠋⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠙⠿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣶⣶⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣾⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⢿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⢿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣄⠀⠀⠀⠀⢠⠰⠿⢿⣿⡷⣠⠀⠀⠀⠀⢀⣤⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢫⣍⣉⠁⠀⠀⠀⢸⡐⣿⣶⣭⡄⣸⠀⠀⠀⠀⢉⣩⣭⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢹⣿⣿⠀⠀⠀⠀⠘⢿⣌⢛⢋⡼⠏⠀⠀⠀⠀⢸⣿⣿⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⡉⠁⠀⠘⣟⣟⣟⡇⢴⣶⣶⡔⣟⣟⣟⡏⠀⠀⠉⣩⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⣰⣧⡹⣿⣄⡀⠀⠀⠉⠉⡍⣡⣾⣿⣿⣧⠉⡍⠉⠁⠀⠀⣀⣼⡿⣡⣷⡀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣮⠙⠛⠛⠛⣂⣛⣂⣙⣛⣛⣛⣛⣁⣚⣃⠚⠻⠛⠛⣩⣾⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠙⣛⣛⣛⣀⣀⣀⡀⣿⣿⣋⣙⣙⣋⣛⣛⣉⣻⣿⠇⣀⣀⣀⣈⣙⣛⡛⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣙⣛⣛⣛⡛⢿⣿⡜⣿⣿⣿⣭⣭⣭⣽⣿⣿⡏⣼⣿⠟⣛⣛⣛⣛⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⡿⠿⠿⠿⠿⢇⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣇⠿⠿⠿⠿⠿⡿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠿⠂⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠻⠿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠷⠖⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠀⠀⠀⠐⠾⠳⠂






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

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⣀⠀⠀⢀⣄⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡟⠀⣸⣿⠀⢀⣿⣿⠀⢸⡿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣿⢀⣿⡇⠀⣼⣻⣿⠀⣿⣧⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣴⠏⢸⣿⣠⢰⡏⢹⣿⢰⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⠀⢠⣷⠀⠀⢠⣿⡇⠀⣰⡆⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢧⢀⣾⡟⠀⢠⣿⣿⡇⢠⣿⣇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣰⡟⣸⣿⠃⢀⣿⢾⣿⡇⢸⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠀⢿⣯⠤⠸⠃⠸⠿⠃⣿⡿⠸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠀⣼⣿⠀⠀⢠⣿⡇⠀⣾⡇⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢧⢀⣿⣿⠀⢀⡾⣿⡇⢸⣿⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣴⠟⢸⣿⡇⡀⣼⠛⣿⡇⣼⣿⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⠛⠛⠁⠋⠀⠻⠁⠛⠃⠘⠟



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⢠⣤⣤⣤⣤⠀⣤⣤⠀⠀⠀⢀⣤⣤⣤⠀⠀⣤⣤⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⣼⣿⣿⠟⠀⠀⠀⠀⠀⢸⣿⣧⣿⣿⠀⣿⣿⠀⠀⠀⣸⣿⢻⣿⡇⠀⣿⣿⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣏⣽⣿⡇⣿⣿⣤⣤⢀⣿⣿⣿⣿⣿⠀⣿⣿⠉⣿⣿⠀⣴⣦⢠⣶⡄⣴⣦⠀⠀⠀⠀⢹⣿⡄
⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠉⠉⠉⠋⠈⠉⠁⠀⠈⠙⠁⠉⠋⠀⠙⠉⠀⠙⠋⠈⠋⠁⠙⠉⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⢰⣶⣶⣶⣤⠀⣴⣦⠀⠀⠀⢠⣶⣶⣦⠀⠀⣶⣦⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⣟⠀⣿⣿⠀⠀⠀⣼⣿⢻⣿⡇⠀⣿⣿⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣯⣽⣿⡇⣿⣿⣤⣤⢠⣿⣿⠿⣿⣿⠀⣿⣿⠉⣿⣿⠀⣴⣦⢠⣶⡄⣴⣦⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠈⠉⠁⠀⠈⠉⠁⠉⠉⠀⠉⠉⠀⠉⠉⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⠀⣶⣶⠀⠀⠀⢀⣶⣶⣶⠀⠀⣶⣦⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣧⣾⣟⠀⣿⣿⠀⠀⠀⣼⣿⢻⣿⡇⠀⣿⣿⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⢸⣿⣯⣽⣿⡇⣿⣿⣤⣤⢠⣿⡿⠿⣿⣿⡀⣿⣿⠉⣿⣿⠀⣶⣦⢰⣶⡄⣶⣦⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀⠀⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠈⠉⠁⠀⠈⠉⠁⠉⠉⠀⠉⠉⠀⠉⠉⠈⠉⠁⠉⠉⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣶⣄⡀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠿⠛⠋⠉⠉⠛⠿⣿⣷⣶⣦⣤⣤⣤⣽⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣔⢀⣰⣃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠑⢄⠀⠀⠀⠄⠀⠑⠀⢆⠀⡤⠀⠀⠀⠀⠉⠀⠴⠀⠀⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠦⠐⠁⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠰⠎⠈⠘⠁⠋⠀⠀⠀⠀⢀⡄⠀⢠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠁⣦⠀⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠠⠃⠀⢀⠔⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⣶⢤⢌⡀⠄⠀⠀⠀⠒⠦⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠠⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠒⢷⣺⡀⢁⡪⠃⠻⡾⠆⠐⠀⠔⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠱⡒⠠⠐⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠸⠁⠄⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⠀⠀⢱⣄⠀⠀⠀⠉⠂⠀⢀⠀⠀⠀⠀⠀⠀⠊⠈⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠣⠤⠊⠉⠚⢦⣻⡀⠀⠀⠀⢀⠜⠁⠀⠀⠐⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠁⠉⠀⠈⠈⠁⠀⠀⡠⠃⠀⠀⠀⢀⠀⢹⠉⢃⠀⠀⠀⠀⠀⠀⠄⠘⠳⠞⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡖⢀⡀⠀⠐⠒⢒⣒⣰⠊⠀⠀⠀⠀⠀⠂⠀⠀⢦⠤⠀⠀⠀⠀⢀⠀⠀⢀⣀⡃⠀⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠂⢀⡀⠀⠀⠤⠄⠒⠉⠀⠈⠂⠀⠀⡠⢣⠀⠀⠀⠸⠀⢣⠀⠀⠀⠀⠆⠠⠤⠤⠄⠀⠀⡠⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠀⠀⠂⠀⠀⢰⡅⠀⠂⢠⠁⢸⢁⣀⡀⢀⠄⢀⠄
⠀⠀⠀⠀⡠⠀⡌⠀⠄⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⢀⣀⣈⣄⣈⡁⠀⠀⠀⠀⠀⠂⠀⠤⠀⠀⠀⠀⢀⡀⣈⠈⠀⠐⠂
⢀⡀⠁⠐⠁⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠃⠀⠀⠘⠀⠰⠁⠤⠤⠤⠤⠤⡀
⠀⠁⠀⠀⠀⠀⠈⠀⢀⠺⣤⠤⣤⣀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢀⡀⢀⣀⣀
⠂⠆⠀⠑⠤⠀⢄⠠⠎⡜⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈
⠀⠔⠂⡀⠀⡀⢀⠤⠔⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠁⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠤⣴⠀⠑⣠⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⣄⣠⣠⡄⣮⣀⣀⠀⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⡾⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⣼⠛⠻⣄⠀⢀⣤⣀⠀⢀⡴⣶⠀⢀⡄⠀⣤⡀⢀⡦⢀⣴⠞⠷⠀⠀⢠⡏
⠀⠀⠀⠀⢠⣸⠃⠀⠀⣸⠀⣟⢀⣠⠗⣾⠛⢧⠀⡾⠁⣸⠏⢧⡾⠀⣾⠁⢶⡶⠀⠀⡈
⠀⠀⠀⠀⠀⠙⠓⠛⠋⠁⠀⠈⠉⠁⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠛




⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠐⠓⢈⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⣀⠀⠀⠀⡀⠀⠈⣁⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠂⠀⠁⠀⠶⡐⢠⠀⠸⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⢬⣙⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁
⢰⡀⠀⠀⠀⠀⠀⠐⠉⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⡄⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡠⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠇⠀⠀⠀⠀⠀⣀⠴⠃⠀⠀⠁⠀⠀⠀⠉⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⡀⠄⠀⢂⠀⠀⠀⠈⠀⠀⣀⡠⠒⠈⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠆⠠⠔⠈⠀⠀⠀⠀⠀⠈⠈⠀⠂⠀⡄⠀⡖
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⢣⠀⠂⡄⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠈⠂⠀⠂⠂⠀⠀⠈⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣶⣦⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣴⣾⣶⣦⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢹⠃⠈⠙⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢸⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠈⠛⠛⠁
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⣠⣴⣶⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤
⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣿⣿⠏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⠇
⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⠛⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢚⣇⣀⣀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠘⠂⠀⠀⢀⠴⣄⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄
⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣄⠀⠀⠀⠀⠀⠸⠃⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⠹⣿⣿⣿⡇⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠸⡄⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠙⢿⣿⣿⠟⠁
⣿⣷⣌⢿⣿⡇⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⢠⣾⣿⡏
⣿⣿⣿⣮⢻⡇⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⣹⣿⣿⡇
⣿⣿⣿⣿⣧⣻⣦⡄⠀⡀⠀⠀⠀⠁⠀⠀⠀⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣶⣶⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠿⣧⣤⣄⣠⠀⣤⡄
⣿⣿⣿⣿⣿⣷⡻⣿⠀⢀⠀⠀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⣿⣿⣿⣿⣿⣿⣷⣌⠀⠀⢠⠀⠀⠀⠀⠀⣸⣿⣿⡟⠿⠿⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠶⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠄⠀⠀⣰⣿⣿⣿⣷⣶⣶⣖⠀⢰⣶⣷⡖⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⢀⡖⠀⠀⠀⣶⢶⣶⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⠀⢀⣴⣿⣿⣿⣿⣿⡿⢻⣿⣆⣿⣿⣿⡇⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣴⣿⡌⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠋⢻⠿⠛⠉⢡⣿⠃⣼⠈⣿⠛⠛⠻⡇⠀⢹⡿⠿⠟⠧⠀⠀⠀⠀⠀⢠⣄⣀⣀⣰⡇⠀⣸⣅⣀⣀⡽⠈⣇⠘⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⠃⠀⡟⠀⠀⠤⠀⠀⣿⠀⠈⣷⠀⠀⠀⠀⠀⠄⠀⠂⠀⠈⠉⠉⣿⠀⠀⡿⠈⠀⠀⠁⠀⢿⠀⠸⡇
⣿⣿⣿⣿⣿⣿⠿⠟⠋⠙⣿⣿⠀⠀⠀⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠀⠀⠘⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠃⠀⠀⠃⠀⠀⠀⠀⠀⠘⠂⠀⠛
⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠋⠉⣹⡇⠀⣿⠁⠀⠀⠀⠀⢠⡿⢻⡇⠀⢀⣿⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣀⣴⠟⠀⢸⡏⠀⠀⠀⠀⢀⣾⠃⢸⡇⠀⢸⣇⣠⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣠⣾⣿⠷⢶⣄⣿⠁⠀⠀⠀⠀⣼⣷⠶⢿⡗⠀⣿⠋⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⢀⣴⠟⢹⡏⠀⠀⠀⠀⢰⡟⠀⠀⣾⠇⢰⡟⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⣠⡾⠛⠁⠀⣿⠁⣀⣀⣤⡀⡿⠁⠀⠀⣿⠀⣾⠃⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠉⠀⠀⠀⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⠀⢠⡄⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⢀⣄⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠀⠀⣸⡟⠀⣾⠃⠀⠀⠀⠀⠀⣾⠃⣿⠀⠀⢸⡏⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⡾⠋⠀⢰⡟⠀⠀⠀⠀⠀⣼⠏⠀⣿⡀⢀⣿⣥⣴⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⡿⠿⠿⢿⣦⣾⠃⠀⠀⠀⠀⢰⡿⠿⠛⣿⠁⢸⡟⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⣠⣴⠟⢱⡟⠀⠀⠀⠀⢀⣿⠁⠀⢠⣿⠀⣼⠇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣀⣴⡾⠋⠁⠀⣾⣃⣀⣠⣤⡄⢼⠇⠀⠀⢸⡏⠠⣿⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠁⠀⠀⠀⠀⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣸⡇⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠟⠛⠛⢛⣷⠀⣸⡇⠀⠀⠀⠀⠀⣰⡟⣿⠀⠀⢰⡿⠀⠀⢰⡟⠀⠀⢀⣿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣠⡾⠃⢀⣿⠀⠀⠀⠀⠀⢠⡿⠀⣿⠀⠀⣼⡇⣀⣀⣼⡇⠀⠀⢸⡇⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⣴⣿⣯⣤⣄⣸⡇⠀⠀⠀⠀⢀⣿⣥⣴⣿⠆⢰⣿⠛⠛⠉⣿⠁⠀⠀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠋⠉⣠⣼⢟⣿⠀⠀⠀⠀⠀⣼⠏⠁⠀⣿⠀⢸⡇⠀⠀⢰⡟⠀⠀⢸⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣤⡾⠋⠁⣸⡇⠀⠀⠀⠀⢰⡟⠀⠀⢸⡟⠀⣿⠃⠀⠀⣼⠇⠀⠀⠾⠃⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠟⠉⠀⠀⠀⠿⠶⠾⠟⠛⠋⠙⠁⠀⠀⠸⠇⠀⠋⠀⠀⠀⠉⠀⠀⣤⣦⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠶⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⣒⣒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠿⠟⠿⠯⠽⠿⠶⠶⠦⠴⠶⠶⠿⠽⠯⠿⠿⠿⠿⠭⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣤⡄⠀⠀⠒⠂⠈⠉⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⠾⣛⠿⣿⣿⣿⣿⣿⣄⠀⠙⠂⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯⣟⡅⢬⡝⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⠤⠤⠤⠄⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⣁⠀⠙⢿⣿⣿⣷⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⡃
⠀⠀⠀⣠⠀⠀⠹⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢠⣶⣗⡀⠀⠀⠈⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣄
⢀⣿⣿⣿⣿⣶⣤⠀⠀⠹⣿⣿⣿⣷⠀⠀⠀⣤⠀⠀⠀⠀⠈⡀⠀⠀⠠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣽⣿⣷⡄
⠘⣿⣿⣿⣿⣿⣿⣿⣶⣄⡈⠻⣿⣿⣧⠀⠀⠙⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢹⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣇⡀⠀⢀⣠⠄
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣶⣤⠀⡀⠀⠀⠸⡆⠀⠀⠀⠀⢸⣿⣯⡁⠀⠀⠙⢻⣿⣿⡿⣾⣽⣾⣿⣿⣍⣻⣿⣿⡄
⠀⠀⢿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣎⣿⡇⠉⠁⠒⠀⡻⠀⠀⢠⠀⢸⣿⣿⣿⣄⡀⠀⠀⢹⠏⣠⡄⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠘⣿⣇⠀⠀⠈⠉⠻⣿⣿⣿⣯⡻⠿⣫⣿⣷⣶⣶⣶⡆⠀⠀⠀⢈⠆⠘⣿⣿⣿⣿⣿⣦⣤⣴⣾⣿⠷⢛⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣧⣀⠀⠀⠀⠙⠋⣁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⠘⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⡏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄
⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⣶⣾⣿⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣀⡀⢀⠸⣿⣔⡉⣿⣿⣿⣟⣴⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⡿⣿⣿⣿⣿⣿⣿⡟⠈⠁⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡷⡄⡌⢻⢆⣿⣿⣿⠿⠛⣿⡃⠀⠀⠀⠀⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠹⣷⠀⠹⣿⣿⣿⡿⠀⣠⣾⡿⠛⠛⢿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣶⣠⢣⡿⠉⠀⠀⠀⣿⡆⠀⠀⠀⠀⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠀⢀⣿⢿⣿⡤⠴⠟⢿⠀⠀⠀⠀⢹⣿⣿⣿⠇⠀⡀⢸⣿⣿⣿⣿⣿⡇⣿⡓⡜⠀⠀⠀⠘⣷⡀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢸⡆⠀⠀⠀⢸⣿⣿⣿⠀⠀⡇⢸⣿⣿⣿⣿⣿⣧⣿⣿⣷⣶⠀⠀⢂⠸⣿⡠⠀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠘⣿⣿⣿⠀⣴⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⠉⠉⠉⠉⠈⠉⠘⠛⠉⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠐⣿⣿⡿⣠⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢹⣿⣇⠧⢻⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢠⠖⣿⡀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠸⣯⢋⡀⣴⡇⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣽⡀⠂⡹⢂
⠀⠀⠀⠀⠀⡀⣧⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠠⠈⣿⡀⠀⣿⡇⢸⠛⠛⠛⣿⣿⣟⣿⣟⢻⠛⠛⠻⠟⠛⢻⢻⡻⠻⣾⣻⠟⠀⠁⢋
⢰⠀⠀⠉⠉⠙⠱⠘⣿⠃⠀⠀⠀⠀⠀⠀⠈⢹⣿⢀⢠⣿⣷⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠀⠴⠤⠤⠄⠀⠀⠿⠀⠀⠀⠀⠀⠀⠤⠼⠾⠿⠯⠾⠼⠿⠿⠯⠇⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚

⠀⠀⠀⠀⢄⠀⡀⠀⣀⡀⢀⠀⡀⠀⣄⠀⠀⠀⠀⢀⡄⠀⣀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡦⡆⢠⡄⡄⡄⡄
⠀⠀⠀⣦⢾⡀⣷⢠⣏⡇⢸⣀⡇⠀⣿⠀⠀⠀⠀⢸⠖⣠⡇⣠⠖⡄⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡁⡇⡼⣧⢷⢠⠇
⠀⠀⠀⡇⣀⡟⣇⠸⠁⠇⠘⠀⠇⠀⠅⠀⠀⠀⠀⠙⠜⠙⠅⠻⠘⠃⠛⠸⠇⠠⠠⠀⠀⠀⠀⠀⠀⠀⣧⠟⠧⠃⠟⠸⢘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡸⠀⡐⠀⠀⠀⠀⠀⠀⡆⣰⠀⣶⠀⣰⡄⢰⡆⣠⠀⣸⠀⠀⠀⠀⠀⠀⣰⠇⢠⡞⠀⠀⢠⡆
⠀⠀⠠⣀⠀⠀⠀⠀⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠸⠇⠠⠗⠳⠐⠏⠃⠸⠁⠇⠀⣉⠀⠀⠀⠀⠀⠀⠟⠀⠞⠰⠫⢃⡿⡰
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡞⠹⢰⠄⢠⣦⢀⡄⢠⠀⢀⠄⠀⠀⠀⠀⣤⠖⣠⠀⣴⠀⠀⠀⢀⡤⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠠⠀⢠⡄⠘⠁⠀⠀⠀⠀⠀⠀⢰⣏⡵⣼⡀⡼⢽⢸⡇⡏⠀⡜⠀⠀⠀⠀⢰⡿⠚⢹⣴⡏⡴⢳⠄⣾⠀⡄⠀⢰⡟
⠀⠀⠀⠀⣦⢘⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠈⠈⠀⠁⠀⠁⠀⠀⠀⠀⠘⠃⠰⠞⠈⠃⠑⠙⠚⠟⠸⠅⠀⡛
⠀⠀⠀⠀⠛⠈

⠀⠀⠀⠀⣰⠀⢀⡟⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢠⠤⡄⢀⡄⠀⢀⣦⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡰⣴⠀⠀⢀⡄
⠀⠀⠀⠀⡿⣶⣾⢱⡏⣇⢸⣧⠄⠀⠀⠀⠀⠀⣾⡤⡇⢸⠁⠀⣼⡿⢸⣵⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣾⢳⣿⡆⣾⣄
⠀⠀⠀⠀⠁⠁⠈⠀⠀⠉⠀⠻⠀⠀⠀⠀⠀⠀⢃⣠⠇⠾⠚⠘⠁⠃⠘⢸⠀⠤⠀⡤⠀⣀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠛⠙⢅⡐⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢙⢉⣭⣺⠷⠖⢲⠿⣾⠦⠶⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢨⠟⣿⢉⣑⣢⢟⢈⣟⣅⣠⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⣧⢤⣛⡿⡤⣿⠟⠻⡻⠿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢹⡿⢾⣰⡮⢽⣄⣠⣓⣺⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢑⣴⡽⣖⣾⠋⠉⠻⡿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢠⣉⣻⠟⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣄⣀⠀⠀⠀⠀⢀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠻⡁⠣⠐⠈⠑⢄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢘⡄⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣴⢫⣵⠀⠀⠀⠀⠐⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡈⠈⠈⢀⡆⠀⠀⠀⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⢉⡠⠀⠀⠀⢹⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⢀⠈⢦⠀⠀⠀⢀⣉⣀⠀⠈⡄⠀⠀⠀⠈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠂⠀⠀⡱⣄⡀⠀⠀⠀⢉⡠⣷⠀⠀⠀⠀⠘⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⠌⠑⠒⠒⢉⡜⠈⡆⠀⠀⠀⠀⢀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⠚⠒⠐⣒⣉⡀⠀⢡⠀⠀⠀⠀⠈⠑⠺⠷⢶⣄⣀⣀⣀
⠀⠀⠐⠒⠀⠀⠀⢂⠀⠀⠀⠀⡐⢀⣆⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠒⠒⠚⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣂⣄⣀⣀⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠂⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠁⢀⠄⠐⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡄⠀⠑⠀⢀⠀⠀⠐⠃⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢠⣤⠀⢶⠀⢠⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⣿⠀⠈⠁⠘
⠀⠀⠀⠀⠀⠠⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠁⠈⠉⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⠈⠆⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠒⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠐⠊⠀⠂


⠀⠀⠀⠀⠀⣐⢄⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂⠀⠀⠀⠀⠁⠉⠁⡢⠀⠀⠀⠀⠈⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠉⠠⢤⣀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠒⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠐⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠹⠀⣀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡇⠀⠀⠀⠀⠀⠛⠂⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣱⠣⠘⠣⠀⣤⡀⠀⠀⠀⠘⠤⠀⡀⠀⠀⠘⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠤⢀⠀⠀⠻⠟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠰
⠀⠀⠀⢠⡤⠀⠀⢀⣶⣇⢸⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢀⠀⠀⡀⠀⠠⣄⠀⠀⣀⠀⠀⠀⠘⠂⡀⠤⠄
⠀⠀⠀⠿⣿⠀⠀⣾⣿⣿⣾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢰⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⢿⡇⠀⣿⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠁⠀⢸⠃⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡾⠁
⠀⠀⠀⠀⠘⣷⠀⠻⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡙⠁⠈⠈⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠑⠄⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⠀⣀⢀⢀⠀⢀⠀⣀⣀⡀⠀⢀⡀⠀⠀⠀⡀⠀⣀⣀⢀⣀⢀⡀⢀⠀⠀⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣿⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⡀⠀⠀⠀⠄⠀⠈⠙⠛⠿⣿⣿⠷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠵⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⣤⣤⣄⠀⠀⠀⠀⠐⣶⡟⠋⠀⠀⢰⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠻⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣄⡀⠀⠀⢤⠀⠀⠀⣀⣾⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⢉⠈⡙⢲⠀⠀⠀⠈⣉⣻⠟⠛⠋⠛⠲⣦⡤⢒⣉⣤⢾⣟⣿⣄⣙⡂
⠀⠀⠀⠀⠀⠀⢸⠋⢀⣴⠀⠀⡀⠀⠋⠉⠳⡿⢭⡭⠉⠁⠂⢀⠀⠀⠀⠀⠉⠪⠁⠺⣮⡿⠿⠾⠯⠝⢟⡲⣶⣦⡄
⠀⠀⠀⠀⠀⠀⢫⡀⠘⢿⠀⠀⠇⠀⠀⠀⠀⢳⣊⣀⠀⣀⣰⣏⡰⠁⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⣰⠞⠙⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣧⣄⠈⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣷⣄⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠢⠤⢤⣄⡀⠀⠀⠀⠀⠀⠘⣯⡾⣦⡀⠀⠀⠀⠀⠀⢸⣀⣻⡀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣀⣀⣀⣠⢤⡼⣷⠈⠻⣳⣆⣀⡠⠴⠞⣿⣿⡅⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣀⣀⣀⣀⣘⣟⣿⣇⣀⣿⣉⣀⣀⣀⣀⣻⣻⣷⣀⣀⣀⣀⣀⣺⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⡄⢀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠷⡷⢟⡇⡇⡇⠀⠀⠀⠀⠀⠀⢐⡷⣀⡾⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣶⣇⣤⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠈⠙⠋⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣇⣶⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣦⣄⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⠿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⡃⡇⣒⡄⡗⣦⠀⠀⠀⠀⠀⠀⠀⠰⢄⡀⠐⢢⠤⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠉⠓⠃⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠉⠀⠁
⠀⠀⠀⠀⠀⢠⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠠⠄⠀⠀⣀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⢀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡧⠀⠀⢻⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘
⠀⠀⢀⣠⣤⠀⣀⣶⣶⡄⠀⣀⡀⠀⢀⣀⠀⠀⣠⣤⡀⠀⣠⣤⠀⠀⣀⠀⠀⢰⣶⣄⡀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣠⡀⠀⣠⣤⡀⣤⣄
⠀⠀⣿⣿⣿⣻⣿⣿⣿⣷⣾⣿⣷⣤⣸⣿⣷⣦⣿⣿⣿⣦⣿⣿⣤⣼⣿⣷⣶⣼⣿⣿⣿⣷⣴⣿⣿⠀⣴⣿⡆⢸⣿⣿⣀⣽⣿⣿⣿⣿⣶⡄
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠼⠠⠴⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠑⠄⠀⠀⠊⠀⢈⠠⢣⣐⡆⠀⠀⠀⠀⠀⠙⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠊⠙⡲⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠓⠈⠀⠀⠀⠀⠀⡀⠄⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⠀⠻⠀⠀⠈⠂⠄⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠠⠊⠀⠀⠀⠈⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢀⢤⡤⡎⡟⣹⡟⣧⣡⡀⠨⠀⣨⡡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠘⠌⠿⠑⠘⠊⠀⠈⢏⠄⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠠⠂⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⢧⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⠞⠒⠶⡜⣧⠀⠀⠀⠀⠰⠊⠁⠈⠠⠀⠀⡄⠀⠀⠈⠀⠀⠀⠠⠀⠜⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠀⠀⠀⠉⠉⠁⠉⢀⠜⠀⠀⠀⠀⠀⢳⠒⠇⠀⠀⠀⠤⠠⠤⠼⠮⢴⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⠄⠀⠂⠐⣒⢒⡒⠒⠒⠒⡒⣒⡐⠃⠀⠀⠀⢀⠈⠀⠈⡤⢤⠀⠀⠀⠀⡀⠀⢀⣀⡘⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡀⠀⠤⠤⠒⠂⠁⠀⠁⠀⠀⡠⠋⡀⠀⠀⠰⠀⠄⠀⠀⠀⡲⠀⠀⠀⡄⠀⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠒⢀⣂⣠⠀⣩⡠⠌⠈⠄⠀⠥⠀⠀⢀⠃⢀⠠
⠀⢀⡀⢀⠐⠰⠁⠐⠀⠀⠀⠀⠒⠀⠠⠀⠀⠀⠀⠠⠤⠤⠧⠤⠀⠀⠀⠀⠀⠈⢠⠐⠂⠀⠀⠀⣠⠄⠄⠀⠁⠈⠁
⢰⠅⢀⠂⠄⠀⠀⠀⢠⡞⠂⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠁⠀⠂⠈⠀⠀⠀⠃⠐⠐⠒⠒⠒⠒⠒
⡀⡀⠸⡀⠀⠀⠀⠀⢎⠓⠒⠒⢲⠐⠀⢰⠒⠀⠀⠀⠰⠐⠒⠒⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠔
⠀⢁⡀⠈⠁⠀⠒⠈⡈⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠡⠔⢣⠐⠠⠊⠉⠀⠀⠀⠀⢸⠀⠀⠘⢠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣀⣀⡠⠀⡄⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⢠⣉⣉⣁⡀⡂⠉⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠉⠉⠀⠉⠉⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⣀⠀⣀⠀⢀⡀⠀⣀⠀⣀⠀⠀⢠⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇⣸⡟⣿⠛⣿⠛⢻⡟⠛⣿⠛⣿⢻⣇⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⢻⣇⣿⣄⣿⣀⣸⣇⣀⣿⣠⣿⣸⡟⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠈⠉⠙⠉⠛⠉⢹⡏⠉⠛⠉⠋⠉⠁⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢾⡇⢸⠟⠛⠛⠛⠛⠛⠃⢸⡇⠘⠛⠛⠛⠛⠛⠻⡇⢸⡷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢈⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣬⣉⠀⠛⠛⠛⠿⠇⠹⠿⠿⠿⠏⠹⠿⠿⠿⠏⠸⠿⠟⠛⠛⠀⣉⣥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣉⣙⠀⠻⠿⠿⠿⠆⢰⣶⣶⣶⡆⢰⣶⣶⣶⡦⠰⠿⠿⠿⠟⠀⣋⣉⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡙⠛⠀⠿⠿⣶⣶⡆⢴⣶⣦⣤⡄⢠⣤⣶⣶⡦⢰⣶⣶⠿⠿⠀⠛⢋⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠛⠿⠂⢶⣶⣶⣦⡄⣠⣤⣤⣤⡄⢠⣤⣤⣤⣄⢠⣴⣶⣶⡶⠀⠿⠛⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀⣴⣤⣤⣬⡄⢨⣉⣉⣉⡁⢈⣉⣉⣉⡅⢠⣥⣤⣤⣦⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠷⠾⠿⠿⠿⣷⣾⡿⠿⠿⠷⠾⠿⠿⠛⠛⠛⠉⠉

⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠
⠀⠀⠈⣿⠉⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⢰⣠⣄⠀⠀⠀⠀⠀⠠⣦⡇
⠀⠀⠈⠛⡤⠟⠻⣿⣯⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⣼⣧⣄⠀⠀⠀⣠⣿⣿⣧⣤⣤⣤⣤⡤⠴
⠀⠀⢰⡆⠠⣶⠀⢱⡧⠀⣷⠀⠀⠀⠀⠿⠶⠒⠚⠋⠉⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠰⣾⣷⣾⣿⡶⣾⣿⣶⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⢀⡻⣀⠀⢠⡀⠀⠀⠀⠈⠉⠉⠉⠉⡉⠉⠉
⠀⠀⣠⣼⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⣿⣦⣾⣿⣿⣶⣾⣶⣇⣀⠀⠀⠀⠀⠀⠸⣿⠂
⠀⣠⣼⣻⣞⣋⣀⣀⣀⣀⣀⣠⠀⢀⣸⣿⡿⣷⣿⣿⣿⣿⣿⣿⡟⣿⣯⣄⠀⠀⠀⣴⣧⣯⣿⡆
⠀⣻⣿⣿⣐⣇⣸⣂⣸⣂⣾⡀⠀⠈⠙⠛⠹⠿⠿⠿⠿⠿⠿⠿⠿⠏⠉⠁⠀⠀⢰⣿⣿⣿⣿⣷⡆
⠀⠈⠉⠉⠉⢉⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⢰⣦⠀⢀⣼⣯⡀⠀⣾⠆⠀⠀⠀⠀⠠⣿⠆⢠⣶⣯⣶⠀⢾⡷⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣷⠄
⠀⢔⣿⣿⣾⣿⣿⣷⣿⣿⣖⠀⠀⠀⢠⣶⣿⣦⣾⣿⣿⣿⣷⣾⣷⣶⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠘⠿⠿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠈⠉⠋⠽⠿⠿⠿⠿⠿⠯⠉⠉⠀⠀⠀⠈⠿⢿⡿⠿⠿⠿⠟⠉

⠀⠀⠀⣴⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣆
⠀⠀⠀⣿⠁⠀⠀⣤⣀⠀⣀⣤⢤⡄⠀⠀⠀⠀⣤⡀⢀⣠⣤⢤⡄⠀⠀⠀⢀⣀⡀⢀⣠⡤⢤⡀⠀⠀⠉⢹⡿
⠀⠀⠀⣿⠀⠀⢺⡇⡘⢈⣥⣶⠀⡽⣀⠀⠀⠸⡏⡝⣘⣷⣮⠀⡽⣀⣀⠀⠺⣥⡇⣸⣤⡌⢠⢇⣴⡶⡄⢸⣷
⠀⠀⠀⣿⠀⣠⡌⢳⠰⣍⣻⡟⡱⠛⡅⢫⢳⠈⢳⡙⢍⣻⣿⢋⡷⡱⢢⢱⠈⢧⠳⣝⣻⣟⣾⣯⣹⢙⣷⢸⡧
⠀⠀⠀⣿⠈⢷⢼⣚⠛⢋⠉⣿⢃⣆⢇⠜⡼⣀⣈⣹⠞⣙⣿⡇⣗⠇⡈⣸⣠⣈⡿⠶⣻⣿⣏⣼⢟⣜⡼⢸⡇
⠀⠀⠀⣿⠀⠀⠀⣉⣙⡻⡄⢸⠈⡵⠋⣾⠼⢧⡾⠲⣛⣇⠀⡇⡡⢪⡴⠳⠷⡼⢳⣟⢯⡉⡏⢩⢊⣼⠗⢸⡇
⠀⠀⠀⣿⢠⡟⢦⠻⡀⣨⠟⣸⡄⣅⢸⡷⢀⣀⣀⢸⠉⡹⢃⣅⢣⡘⡷⢀⣤⡜⢏⡉⢹⠇⢰⠘⣞⢳⣤⢸⡇
⠀⠀⠀⣿⡆⠛⣮⣍⣵⡇⢠⢿⡣⣸⢞⠄⢸⣋⣹⣀⠆⡇⢸⢿⣆⣱⢱⡹⣦⣽⣒⣃⣾⠀⢻⣷⣘⣾⠋⣾⡇
⠀⠀⠀⢿⡇⠀⠀⠈⠁⢱⢈⡿⠙⠓⠋⢀⣀⡸⠧⣍⣼⠳⣘⢦⡝⠓⠋⠀⢀⡬⢭⠉⠋⢧⠸⣟⣒⣮⠀⣿⡇
⠀⠀⠀⢸⣧⠀⢠⡦⢶⣥⣏⡽⢶⠀⣾⢍⠃⢙⠄⠈⢉⣠⠎⣼⢰⠞⡃⢸⣋⣍⣈⡗⣠⢬⡙⠛⣖⣯⠀⣿⠁
⠀⠀⠀⠘⣿⠀⠘⠛⠛⠁⠀⣽⠸⡘⢻⡿⣣⣯⡤⣄⠘⣧⡼⠏⠨⣇⢧⣚⢿⣿⣷⡟⢸⢳⣱⢾⣼⠇⢸⡿
⠀⠀⠀⠀⢻⣇⠀⠀⣴⠶⣤⣬⡷⠏⢿⣟⡝⣏⣼⡎⡇⠀⢰⣟⣑⠬⠷⠾⢿⣻⡟⢸⣿⡸⢾⡀⠁⠀⣾⠇
⠀⠀⠀⠀⠈⣿⡄⠀⠉⠛⠙⠞⠳⢾⠀⢨⠙⠟⡫⡞⠁⢠⠤⡿⣭⠓⣛⠒⣣⡉⢇⢈⠟⣡⠞⠁⠀⣸⡿
⠀⠀⠀⠀⠀⠹⣷⢰⡞⢳⠀⢏⠉⢐⠇⣸⠠⣼⠰⡋⡇⣸⣶⣧⡸⢻⣍⠚⢉⡿⡌⡞⢦⡸⣷⠏⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣧⠉⣧⣍⣉⡾⡏⠀⣿⣦⢝⡗⣧⢣⡬⣿⣥⢃⡤⢼⡋⠳⣿⠈⣿⣦⣙⢾⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠟⠋⠀⢇⠸⣄⡉⣭⣄⡹⢋⣼⠾⢇⣦⢟⠓⣧⡀⢧⠸⠤⢽⠓⢃⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢀⣈⣹⢠⡇⠛⠶⠛⠋⢳⡀⡌⠱⢁⡶⢛⣁⣀⡹⢰⠟⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⢻⣥⣼⠙⠃⠀⠀⣟⠉⣹⠃⣹⠸⡈⣶⠸⣧⣶⠛⠛⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣠⡿⢦⡀⢘⠄⣧⢀⡷⡧⣘⠎⣶⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⠷⢺⣭⣭⣬⡿⢌⠓⢺⡖⠚⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⣩⡥⢞⣠⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡸⠶⠞⣉⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡾⠛⠁

⣶⣖⢲⡶⢶⣶⠶⢶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠋⠉⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⠋⠀⠀⠀⡴⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢀⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣿⠞⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠒⢩⣬⣤⡄⠀⠀⠀⢀⣤⡠⠈⣆⠈⠁⠀⡀⠀⠀⢀⡸⣦⣁⢀⣀⣬⣵⣿⣷⣖
⠀⠀⢠⣤⠦⠌⠀⠀⠀⠀⠈⡀⠄⠸⠁⠀⢤⠀⠀⠀⠁⠘⠉⠈⠂⢠⡁⠀⠠⡌⠀⠀⠙⢄⡀⢀⠇⠀⠆⢠⣤⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⢰⠀⠀⠀⠀⠀⣆⠀⢰⣀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⢈⣀⣜⠀⢀⣄⠉⠁⠀⠀⠀⡈⢀⣿⣿⣿⣿⣶⣽⣿⣿⣿
⠀⢸⠀⠀⠀⠀⠀⠹⠶⡿⠫⢁⠤⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠨⠀⠀⠀⠀⠠⣄⣀⣴⣴⣶⣷⣿⣾⣶⣿⣶⣶⣾⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡶⢻⣷⣶⣾⣦⣤⡴⣦⣼⡇⢸⣿⣿⣿⣿⣿⡏
⠀⠀⡇⠀⠄⣷⢤⣷⣼⠉⣯⣁⣸⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠉⣿⣿⣿⣿⣿⣏⠨⠄⠰⣿⣿⣿⣿⣿⠇⡟⣹⠇⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠘⠂⠀⢶⢾⣶⣿⣷⠏⢡⠏⠃⠀⠙⢻⡛⢿⡅⠀⢀⠀⠀⣿⠛⠹⠉⠉⢿⡄⡄⢠⣿⡿⣿⢿⡿⣷⠃⣿⠀⠀⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⡾⢼⣟⣿⣭⣤⠟⠀⠀⠀⠀⠀⠀⢸⠇⠀⠌⠆⠀⡏⠀⠁⠀⢀⣀⣹⡇⢸⠋⠐⢛⣱⠗⢿⠀⣿⠀⠀⠈⢻⣿⣄⣿⣿
⠀⠀⠈⡤⢀⡠⠖⠶⠤⢹⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠇⠀⠀⠀⠘⠶⣴⣀⠜⠀⠿⠟⢳⡵⠂⠀⠆⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⣋⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠖⠀⣠⢆⠐⡢⠀⢀⠂⠀⣢⡀⠐⢦⡀⢤⡤⣤⣴⡟⣷
⠠⠲⠈⠛⠫⣭⣍⣉⡙⢻⡞⢁⠚⣁⠟⢌⡤⣡⢈⢸⠄⠀⠉⣁⣠⠿⢟⡧⣂⢿⣤⡟⢧⣨⣿⡉
⢠⠚⠀⠉⠛⡳⡶⣿⢡⠏⣠⣤⠸⠣⣪⠎⠰⡷⣃⡡⠴⣓⡉⡄⠀⢠⡂⠀⠽⢯⣙⣷⠻⢿⣿⠟
⣀⠛⢨⣀⡀⠁⠀⠡⠃⡐⣿⠏⣔⡴⢡⢞⡌⢀⠮⠰⢿⣴⠀⠀⣴⠄⣀⣽⣦⣾⡹⣆⢘⣭⣬⠳
⠃⠉⠁⢸⣿⣆⠀⣰⢞⡴⣣⣠⠏⡴⠋⠺⠆⢿⡖⠀⠘⣹⠐⡀⠀⠾⠟⠉⠙⠿⠉⣄⢄⣀⠙⠄
⠟⢘⠀⣬⣹⢏⡼⢣⠞⣵⡿⢉⡾⡥⠡⠢⠀⠀⣠⡴⡟⢿⠀⠀⠉⣤⣠⣜⠶⠟⣛⣛⣩⣭⣥⡄
⠠⢀⠀⡝⣱⠏⡰⠋⣾⠋⡴⣋⠚⠀⠀⢚⣴⡟⢏⢳⡹⢼⠐⠀⣒⣬⠵⢾⢛⣍⠩⠁⠶⠇⠰⠁
⠸⠀⢠⣜⠧⡘⢁⡼⠁⡤⣡⢍⡇⣑⠄⠛⠳⡹⠎⣁⢔⣡⠶⡿⢙⡢⠴⠛⢀⣀⠰⢒⡂⣤⠻⡀
⠀⡈⡿⠨⠋⣴⠋⠠⢞⡼⠁⢸⡇⣿⣷⡄⡘⠐⣫⣾⣯⣽⡶⢚⢉⣴⢶⠀⢡⠶⢾⣛⣿⣿⣲⣦
⠀⡀⡐⣠⠞⠁⡀⣣⣮⠖⢠⢹⣿⢹⢟⠕⣡⢞⢿⠟⣻⣁⣥⢸⣞⢱⢊⡀⡤⠄⠩⢽⣦⢹⢉⡉
⠀⡡⣐⠰⣢⡾⢌⠑⠁⢁⠀⠈⠷⣦⢪⡞⠇⢰⡄⢾⠋⣡⣤⠆⠻⢞⠘⠝⣤⡦⠰⡆⣿⢸
⠀⡂⡌⣀⣥⡖⣁⠀⢽⠀⠿⠒⠒⣰⠛⠀⣀⠀⡟⢾⢿⣻⠿⠚⡅⢀⠀⠀⠈⠀⢀⣿⢸⣾
⠀⠃⡌⠘⠾⠠⣡⠶⢺⣶⣒⠚⠛⣲⠻⢂⡿⠘⣅⠈⡘⠁⣺⢷⡚⢎⢳⣜⠒⠀⠚⣼⣸⣿⠀⠘
⠀⠆⡄⠀⠀⠀⢻⣌⢘⣿⣿⠿⣫⢈⡡⢊⠇⠀⡟⢆⢱⣿⢙⡧⠗⢊⣣⡙⢿⣏⠀⠀⠃⠰
⢀⢇⠘⠀⠀⠀⠀⠉⠉⠹⣔⡋⢵⡟⣴⡟⠀⡨⠻⡄⠻⡼⢋⠔⢉⣴⠛⢈⣪⣹⣤⠀⠺⢆
⢰⢂⢣⢨⡄⢠⢰⠂⣜⡠⣿⡀⢸⣇⣩⠂⡰⢥⢺⡹⣆⡷⠧⣰⣿⢃⠀⢀⣉⣸⡿⢀⣃⡙⢱⡀
⢸⢆⢇⠁⠻⣿⣾⡴⠃⠻⣌⠻⣦⣹⠃⠰⠁⢚⣋⡅⠹⡇⡖⣿⠃⠆⠀⠀⠺⠭⠻⡘⠊⣣⡾⡇
⢠⠌⢌⠜⠄⢿⣿⢱⡐⢄⣈⠷⠊⠀⣰⣧⣄⡈⠱⠧⠀⣗⣃⡉⠘⠀⠰⢦⣄⡀⣀⠅⣦⡟⡶
⢨⠘⠸⠈⠀⠸⢡⡷⠞⠃⠀⠀⣦⢿⠃⣭⣚⠯⢷⣶⣶⠉⠉⢹⠆⠀⢰⣾⡟⠁⣟⣿⣟⠁⡆⡀
⠊⠋⠀⠨⠅⠀⡀⡀⣠⣤⡞⠄⠘⡏⠰⡷⣿⡿⠒⢀⢁⡞⡽⠀⠐⣾⠄⠁⣤⠆⣾⠋⡿⠀⡿⠁
⡔⠀⢀⣯⣭⣄⠈⣼⣽⣿⡾⠇⡄⠀⠀⠁⠈⠃⡠⣢⣾⢞⣵⡄⠈⢀⠐⠀⠃⡼⠣⢴⡇⠀⡇
⢰⠄⠤⠰⠆⠤⠤⢭⣁⡛⠓⡼⡀⡄⠀⠀⠰⠶⣞⣫⡴⠟⡏⢀⣀⢠⡆⢀⠞⣁⣷⢺⡇⠰⠓⡀
⢑⡀⢨⠤⣬⣭⢙⣶⠶⣩⢟⡳⢍⡢⢐⠈⠉⢯⡯⠴⠲⠀⢠⣿⢃⣾⢧⠏⠌⢈⣹⢾⡇⢠⠀⣧
⠸⡀⠷⢞⡫⠵⣫⡵⢚⡁⡪⠌⡳⠙⠲⣅⠀⠀⠙⠛⠀⣰⢸⠀⣼⠋⠂⠀⢸⡚⠻⢾⠇⠟⣄⠉
⢜⠁⢠⠶⠺⠩⠦⠠⠶⠶⠶⢠⠀⠬⡀⣼⠳⣄⠀⠀⢀⠈⢸⡰⠋⠏⠀⢶⢨⡰⡆⡘⣀⠃⢁⣴
⠰⠇⢉⣉⣈⣛⣿⣻⣿⣿⣿⣻⣓⣶⣶⣿⣦⣘⠦⠀⢄⠡⢘⢳⠀⠀⡄⠀⢬⠀⢀⢼⠎⢠⣾⠋
⠀⠀⠈⠀⠀⠀⠁⠈⠀⠉⠀⠙⠛⠋⠉⠛⠉⠑⠘⠂⠈⠀⠙⠀⠈⠀⠀⠀⠀⠀⠘⠁⠰⠿⠡⠏

⠀⡤⢀⡆⣶⢰⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⣦⠀⠀⠀⠀⠴⡀⢀⡇⢠⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡐⣄
⠀⠓⠊⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠾⠛⠈⠈⠀⠀⠀⠀⠐⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠿⠁⠁⠈
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡆⠀⣸
⠀⠀⠀⠀⡷⢁⠾⠟⠃⠃⠃⠈⠀⠀⠀⠀⠀⢸⣦⡀⣰⢦⡄⠀⡀⠀⠀⠘⡗⢱⠷⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⣼⣦⠃⣴⣠
⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⣀⢀⠀⠀⡀⠀⠀⠈⠁⠈⠉⠁
⠄⠀⠀⡤⢀⠀⠀⠀⠀⠀⠠⣯⢾⣷⣦⣯⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⡄⡀⠀⠀⠀⠀⠀⢸⣯⣿⣶⣴⣇⠀⠀⠀⠀⠀⠀⢀
⠃⡧⠰⠃⠾⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠰⣓⢣⡥⠜⠃⠗⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰
⢳⢂⡄⠀⠀⠀⠀⠀⡇⠆⣇⡆⡆⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⡴⢳⣠⠀⠀⠀⠀⠀⠀⡄⢘⡦⡄⠀⠀⠀⠷⠀⠉
⠈⠉⠺⢃⣖⡀⣀⠀⠹⠓⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠃⠾⣡⣆⠀⡀⠀⠗⠂⠁⠁⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠸⡡⠇⡖⢦⠷⠆⠀⠀⠀⠀⠁⠜⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢸⠰⠡⠶⠸⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠜⠤⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠎⠤⡼⠄
⠀⡤⢀⡆⣏⢰⣰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⣦⠀⠀⠀⠀⠴⣀⢀⡇⢀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⡘⣄
⠀⠑⠊⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠋⠈⠀⠀⠀⠀⠀⠐⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠁⠁⠈
⠀⠀⠀⠀⠀⠀⢠⢀⠀⡆⡀⢰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡆⡀⣼⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡷⢉⠾⠛⠓⠃⠃⠀⠀⠀⠀⠀⠀⢸⣢⡅⣰⢦⣀⠀⡀⠀⠀⠘⡇⢱⠗⠛⠛⠋⠂⠀⠀⠀⠀⠀⠀⣼⣦⡃⣼⣠⠠
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠁
⠄⡀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣯⠾⣶⣦⣧⠀⠀⠀⠀⠀⠀⠀⡤⠄⡄⢀⡄⡀⠀⠀⠀⠀⠀⠸⡧⠿⣶⣼⣇⠀⠀⠀⠀⠀⠀⢀
⠃⡧⠐⠃⠾⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⢃⠣⠅⠘⠃⠟⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰
⢳⣂⠄⠀⠀⠀⠀⠀⣇⠆⣧⡆⡇⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠴⢳⣠⠀⠀⠀⠀⠀⠀⡄⠘⡧⠄⠀⠀⠀⠿⠀⠉
⠈⠉⠚⢣⢆⠄⡀⠀⠸⠓⠘⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠈⠁⠚⣡⣦⠀⡀⠀⠟⠃⠁⠁⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠘⡡⠇⠞⠦⠷⠆⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠸⠰⠡⠶⠸⢳
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣇⣜⣴⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡣⣋⣆⣼⡀
⣿⣿⣛⣛⣛⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆
⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⠶⠶⠶⣿⣿⣿⣿⠶⠶⠶⠶⠶⠶⣶⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠛⢻⡟⠛⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⣼⡟⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠰⢾⣟⡁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠐⢻⣯⡅⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢉⣻⣧⠤⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣈⣽⡷⠆⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⣼⡟⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣄⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⣠⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠈⠉⠀⠀⣀⡀⠀⠀⠀⠀⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⢾⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⠢⠀⠀⠀⢠⣶⣦⣭⣵⣶⡆⢠⣶⣦⣭⣥⣶⣆⠀⠀⠀⠀⢺⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠻⣦⡀⠀⠀⠸⣇⠻⠿⠿⢱⡏⠘⣧⠹⠿⠿⢡⡟⠀⠀⠀⣠⡾⢣⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡗⠈⠉⠛⢂⣀⣉⣁⣒⣚⣋⣀⣀⣉⣁⣒⣚⣋⣀⣀⠛⠋⠉⠰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠧⠤⢤⡄⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⣤⠤⠴⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡛⠛⠛⠛⢛⢂⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣇⢻⠛⠛⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢸⡎⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⡇⡾⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠧⠹⠇⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠟⠰⠇⠀⠀⠀⠀⠘⠇







⣿⠿⠿⠿⡿⠟⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⢿⣿⡿⠋⠻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⠈⠁⠠⢤⡀⠈⠙⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠛⢿⣇⠀⠈⠿⠖⠀⡴⣒⠂⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠸⣿⡀⠀⢀⠀⠘⠁⢉⣼⠥⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢻⣷⣄⣈⣉⣠⣴⠋⠀⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡇⠀⠀⠀⠙⠿⠿⠿⢋⣘⣄⡀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡏⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⢻⣿⠀⠀⠀⠀⢠⠎⠿⠿⠿⢻⡆⠀⠀⠘⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠿⠖⠒⠒⠚⠉⠀⠀⠀⠀⣼⣿⡄⠀⠀⠘⣄⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠻⢿⣷⡀⠀⠀⡏⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣦⡤⠞⠉⠓⠋⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢶⣤⣤⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⢁⣁⣀⡀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣫⣾⣿⡟⠉⠁⠀⠈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠗⠂⢻⣿⣿⣄⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣭⡟⠃⠀⠙⠛⠛⠛⠁⠀⠀⠀⠘⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣟⣟⡏⠉⠉⠀⠉⠉⠈⠉⠉⠉⠉⠙⠛⢻⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣯⣋⢉⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠆⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣭⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⣛⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⠄⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣦⣁⠀⠀⠐⠻⠛⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢇⡐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠡⠀⠄
⠀⠀⠀⠀⠀⢀⣀⡘⣊⡁⢈⣑⢁⣀⠀⠀⠀⠀⠀⠀⢠⠒⠊⠉⠉⠉⠉⠑⠒⡄⠀⠀⠀⠀⠀⠀⠒⠒⠂⠊⠘⠂⠒⠒⠂
⠀⠀⠀⠀⠀⢸⣪⢌⡵⡇⢰⣡⢊⣔⡃⠀⠀⠀⠀⠀⢸⣲⠖⡲⠀⠀⢒⠖⡲⠇⠀⠀⠀⠀⠀⠀⠃⠀⢀⠃⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢘⡡⢊⠔⠀⠀⡡⢊⡴⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠘⠉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠉⠉⠁⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⡢
⠀⠀⠀⠀⠀⠨⢤⡠⠠⠠⠄⠄⠄⠄⠅⠀⠀⠀⠀⠀⠀⠀⠈⡏⠩⠍⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡼
⠀⠀⠀⠀⠀⠀⡄⠏⢰⠠⢄⢀⣹⡈⡆⠀⠀⠀⠀⠀⢰⠀⢀⢁⢉⡉⠈⡀⠈⡆⠀⠀⠀⠀⠀⠀⠒⠊⠉⠀⡇⠈⠉⠒⠆
⠀⠀⠀⠀⠀⢘⡒⢚⡓⣒⡚⢆⠼⣀⡄⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠐⢈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢃⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠪⠔⠀⠁⠁⠐⠄⠃⠁⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⠤⠤⠤⠤⠃⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠆


⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⢨⠤⡜⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡃⢘⠀⠀⠄⡆⠀⠀⠀⠀⠀⠀⠀⢀⡔⢡⢌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢁⠇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠁⡎⠀⠀⢱⠈⠠⡇⠀⠀⠀⠀⠀⠀⡀⠈⢆⠡⠜⣠⠅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣖⢀⢀⠀⢀⡣⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⢆⠀⠀⡰⠢⠤⡇⠀⠀⠀⠀⠀⠀⣀⠀⡀⡸⢩⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠂⠒⢀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣸⣃⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠀⠀⠰⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⣤⣤⣤⡴⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠚⠉⠁⠉⠳⣄⠘⠿⢛⡥⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠠⣤⣴⠆⠀⠀⢠⣤⣤⢸⡇⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⢸⡇⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣧⣿⢰⡇⠀⠀⠀⠀⠀⣶⢶⠒⠒⠲⡆⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⣤⣄⣀⠀⠚⠋⣁⣠⣴⣿⠟⠁⡏⢸⡇⠀⠀⠀⠀⠀⠸⣎⢷⡀⠀⠹⡄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣴⠋⠀⢀⡘⣏⣽⡟⣏⣹⡇⠸⡇⠀⠀⣇⢸⠅⠠⡄⢠⡄⣠⠀⠹⣎⢧⠀⠀⠹⣄⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢋⡁⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠸⣿⡅⢈⡟⠙⣿⣧⠀⢿⡀⠀⠈⠛⠶⠶⠶⠶⠶⠶⠤⠤⢿⡌⣧⠀⠀⠘⢾
⠀⠀⠀⠀⠀⠀⢸⡇⠈⣧⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠘⢿⣾⡇⢠⡇⠹⣧⣸⡇⣠⠶⢿⣷⡄⠀⢠⡶⠶⠶⠶⠶⠿⠘⢧⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢷⣀⡿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⠟⣷⢾⣧⣤⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠓⠒⠛⠓⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢲⣖⣚⣻⡇⠀⠀⠀⠀⠀⢀⡟⠀⢻⢸⠃⠀⠸⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⠶⠶⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠤⠼⣷⠀⠀⠀⠀⠀⢼⡀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣖⣚⣻⣆⠀⠀⠀⠀⠀⠙⠞⣱⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣛⡉⠉⠉⠉⠛⢷⣄⣤⣦⣄⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⣉⡉⠛⠛⠓⠀⠀⠈⠉⠛⣿⣿⠏⢻⡷⣶⢤⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⣿⠛⢻⠛⠛⠛⠙⣿⣠⢟⡜⠁⢸⡇⢀⣠⣞⡿⠟⢻⡇⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⠀⢸⠀⠀⠀⠀⢿⣿⡎⠀⢠⣿⣦⣾⠟⠉⠀⣠⠞⠁⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣩⡽⠿⠿⠿⠿⠯⣭⡛⢦⣸⣿⢀⡴⠋⠈⣿⣥⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⣸⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠾⠋⣻⠄⠀⠀⠀⠀⣯⠳⠞⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠉⠉⠉⠉⠉⠙⠛⠶⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⣟⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠐⠂⠊⠉⠁⠀⠀⠁⠡⣿⣻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢀⡶⢶⠶⡾⣻⣦⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣳⣻⣿⣿⣿⣿⡿
⠀⠀⢠⠋⠠⠁⠈⠀⠀⠐⣽⣻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣣⣿⢿⡿⠟⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠰⡷⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡵⣾
⠀⠀⢠⠀⠀⠀⠀⠀⠀⢁⢻⣻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣦⢘⣧
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠢⡱⣽⠿⣿⣿⣿⣷⡀⠀⠀⠀⣰⡼⠋⠀⠀⠈⠻⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⢽⣿⣿⣿⣷⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠙⣷⡛⣆⠀⠀⠀⠀⠠⣄⠶⣖⢦⡀
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠐⠨⣾⣿⣿⣿⣿⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡾⣦⠀⠀⡀⠀⠀⠀⠈⢋⡇
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠠⣀⠐⣤⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣇⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⡤⠼⣧⣾⡗⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠈⠈⢹⡗⣿⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡄⠀⠀⠀⠀⢸⣧⣿⠛⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢮⣤⣀⢀⢀⠀⢺⣫⡵⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠛⡉⠐⣿⣵⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⡀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⢰⣿⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⣠⣀⠙⢷⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⡻⣾⠷⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⣸⠏⠙⠦⣀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡥⢀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣿⠂⠀⡀⠈⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠧⡼⠵⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣾⣿⡟⠀⠒⠁⢄⣿⣧⣶⣶⣶⣾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣤⣤⣤⣀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣿⡿⠏⠁⠀⣄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢠⣿⠟⢿⣿⣿⣿⣟⣿⣶⣴⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣠⣾⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣽⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠰⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣽⠿⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣽⣿⣿⣿⡿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠴⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⠀⠀⠤⠤⠤⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣤⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠀⠀⢸⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡤⠄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⢸⣿⣿⣿⠀⠀⠀⠸⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⠠⠁⠀⠀⠀⣿⣿⣿
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⡄⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⠻⣷
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠂⠹
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢰⣶⡾⠿⠿⠿⠿⠶⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁⠒⠀⠀⠁⠒⠠⠀⠀⠀⠐⠠⣀
⠀⠀⠀⠀⠀⠄⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⡏⠁⠠⠀⠜⠇⠗⠖⠀⠀⠀⠠⠆⠀⠀⢠⡀⣀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠄⠐⠀⠀⠀⠐⠠⠄⠀⠀⠒⠐⠀⠤⢀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠰⠆⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠆⢠⡀⠰⠆⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠣⠀⠀⢀⢫⡄⠤⠤⠀⢠⢀⠀⠀⠀⡀⠀⠀⠀⢡
⠀⠈⠆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⡆⡀⠀⣴⠀⣆⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠤⠄⠰⠧⠐⠒
⠀⢐⠀⠀⠀⠀⠶⢆⠀⠀⡼⡄⠢⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⢠⠀⡀
⠀⠈⠦⠀⣀⠀⠁⠀⠈⠁⠀⠀⠀⢲⣀⠀⠀⣠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠘⠂⠀⠘⠊⠉
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⢀⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⢀⠀⠀⡄⠀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠠⠄⠀⠀⠠⠀⠉⠢⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠸⠑⠚⠀⠀⠛⠀⠁⠀⠀⡛⠀⠀⠠⠘⠂⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠉⠀⠐⠦⣀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢠⠄⠲⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠀⠀⠀⠀⠉⠈⠁⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⣀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠠⠀⡘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠰⢃⠀⠀⠙⠁⠈⠃⠀⠀⣀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠤⠄⠉⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠤⠤⠄⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠤⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠠⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡦⣀⣀⣀⣀⣠⣤⢰⣶⢰⠀⠀⢠⡀⡀
⠀⠀⠀⠀⠰⠖⠶⠒⠒⠖⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⡙⠛⠛⠛⢀⣾⡿⢀⣬⡍⣉⣛⣛⣉⣼⣿⢼⣤⣶⠤⠭⠭⠭⠭⠤⠤⣠⠂⠀⡀
⠀⠀⠀⡆⢀⠀⢀⠛⠛⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛⣳⢖⣁⢸⣿⡟⢸⣿⠃⢀⣀⣀⣬⣥⣾⣏⣸⣿⣤⣿⣧⣤⣤⠀⡴⠆⢠⡄⠁⡔⠴⠃
⠀⠀⢰⠁⡆⢰⠀⠓⣈⠉⢉⣉⣉⣉⣋⣉⠉⢩⣌⣉⢻⣟⢸⣿⠇⣿⡟⢠⣭⣭⣥⣤⣤⣤⣤⣼⣿⠘⠀⠄⠀⣠⡄⢈⠁⠀⠀⡇⡄⠰⡀
⠀⢀⠀⠃⢠⠋⡄⢀⠉⢰⠈⢉⣉⣉⣉⠉⠡⢃⠸⣿⢸⣿⢸⣿⢰⣿⠇⠀⠀⠦⢠⣶⣶⡄⢸⡇⣿⠀⠀⠀⢰⣿⣷⡈⠞⠀⠸⢡⠃⠞⠁
⠀⠞⢠⠃⠆⠰⠀⣼⠄⠀⠀⠀⠠⠦⠉⠀⠀⡸⠀⣿⢸⣿⢸⡿⣸⣿⠀⠐⢦⣶⣤⠉⣿⡇⢸⡇⣿⠀⡇⠠⣿⣿⣿⡷⠀⠃⠈⡄⡄⢹⠄
⠀⠀⠀⡆⠀⠀⢀⣿⠀⠀⠰⡚⣶⠂⠀⠀⣤⢇⡇⢹⡾⣿⢸⡇⣿⡟⠓⢷⡆⢹⡇⠀⠉⠁⢘⡃⣿⠀⠁⠀⠛⠛⠛⠃⡀⠤⠟⢰⠁⡏
⠀⠁⢰⠁⡀⠆⢸⡇⢀⠀⠑⠐⢟⠦⡀⠀⠈⠘⠸⠆⢧⣿⣸⡇⣿⡇⠀⣌⣁⣀⣈⣉⣉⣁⣈⣁⣿⣇⡀⠒⠛⠀⠀⠀⢇⠣⡘⢀⠆⣸⠆
⠀⡀⢠⠀⠀⠀⣿⠁⡬⠀⠰⠀⠀⢶⠏⢠⡀⠀⠀⠀⠘⢻⡅⣇⣿⡇⠰⢄⠉⠉⠉⠉⠉⠉⠉⠙⣿⡟⠛⠛⢿⣿⣿⠃⡎⡆⢹⠈⠐⡁
⠀⠀⠀⡀⠀⣸⣿⡆⠀⢰⢀⢇⠀⠀⠸⢇⠐⢆⠀⠀⠀⠈⣽⡇⣿⠁⡆⣾⠠⠶⠶⠶⠶⠶⠀⡇⣿⡇⠀⢀⠀⣿⡿⠀⡰⠀⢀⡆⡆⣨
⠀⠀⠐⠀⢀⢹⣇⠉⡁⠈⡘⡌⢃⢠⠀⠀⠳⣄⠺⡁⡰⣀⠘⢀⣿⡀⡇⣿⢐⣛⠛⡏⠉⠀⠀⡅⠈⢀⢄⣥⡄⢻⠇⠀⢣⠃⠘⡈⠐⢥
⠀⣄⠀⠀⠈⠀⠀⣄⠀⠚⠧⠀⠈⢸⣇⠀⢤⠹⣆⠐⣱⠋⠃⢸⣿⡇⡇⠸⠈⠉⡁⣤⠞⠀⢀⢁⢴⣵⠟⠙⡇⠘⠀⠀⠇⡜⢰⢡⠃⡼⠃
⠀⠀⢀⡠⠇⠀⠀⣷⠀⢘⠢⠉⠛⡛⠋⢀⡈⣧⢹⣆⢡⢏⠉⠸⢿⣿⠇⡄⠉⡠⢊⣤⢎⡴⠷⠉⠞⢡⣾⡇⢻⠘⠃⠀⢀⠇⠐⡀⡈⢲⡀
⠀⡩⣤⣤⠰⢒⡄⣿⠀⠈⠱⢀⠀⠱⢆⠀⣷⢸⡆⣿⡆⠼⠃⣠⣾⣿⢠⠓⠈⠰⡿⢡⡌⢊⠸⠃⠀⠻⠿⠿⠈⣘⣁⠲⣏⠀⡼⠱⢃⡴⠁
⠀⠄⠙⡱⣅⣨⠀⡇⣰⣶⣶⠀⣆⣦⠈⠀⡿⠀⡇⢸⣿⡀⢿⣛⣿⣿⡄⠐⠀⠀⡠⢊⡀⢋⣴⣿⣦⠀⢀⣶⠉⠘⣿⣷⡌⠘⢁⠁⠘⢄
⠀⠠⢟⣜⣺⣛⡀⡧⠘⢿⢃⡶⣿⡭⣤⡄⠀⣆⡇⢸⣿⣇⠸⣿⡿⢿⡇⠘⡣⢈⡤⠈⡄⢸⣿⣿⠏⣰⣿⣿⡄⠀⢻⣿⡇⠀⢸⠰⢢⡞
⣄⣠⣤⣤⢀⣌⣀⣷⣀⣼⣉⣉⣉⣛⣁⣄⣀⠀⢀⣿⠟⢋⠤⢫⢀⠜⣷⠊⣀⠈⠀⠼⢃⣼⠟⢃⡴⠉⣿⣻⣧⠀⢸⣿⠁⠀⠀⠀⠘⢦
⣤⣤⣀⣀⣀⣀⣼⣿⣿⣯⣀⡀⠀⣭⣿⣇⣀⣀⠀⣤⡀⢁⠴⠂⣾⠶⣽⣄⢻⣄⠲⢤⣭⣤⢴⣿⠇⠀⢹⣿⣿⠆⠂⣡⢆⠀⠀⠰⢰⠟
⣀⣼⣿⣛⣉⣉⣉⣿⡜⢻⠉⠭⠀⠍⠯⠉⠉⠉⠀⢉⢕⠜⢣⠜⢏⡀⠆⢯⠼⣋⠳⢄⣉⣁⠈⠛⠒⠒⠈⠉⠀⠶⠮⣍⡩⠀⠀⠀⠈⣣
⣉⣭⣭⣩⣉⣿⣟⣿⣉⣹⣉⣉⠉⠉⠁⠀⠀⠀⢰⠟⣁⠶⡁⣤⠸⠆⢀⡸⣃⠽⡗⠦⢠⣄⠉⣉⣉⣿⣿⣿⡟⠐⠆⣨⠓⠀⠀⠀⠈⠋
⠙⠛⡛⠻⠛⢁⡏⣹⣿⣿⠉⠿⠐⠆⢠⣴⡄⠐⣡⠞⢁⡼⠃⠸⡆⠐⠀⣁⢻⠀⢉⢷⠶⠴⠶⠛⢋⣵⣿⡿⣿⠀⢻⡐⡝⣦⠀⡴⡄⣤
⣀⣀⢉⣉⡉⢉⣉⣻⣉⣻⡀⣀⡀⢀⠀⠉⣠⡞⢡⣾⢋⡀⣴⣤⡇⢸⠀⠈⠈⣇⢁⡄⢢⣤⣴⣾⠟⣿⡟⢀⡿⠀⠀⢹⠁⢿⣷⡘⡗
⠀⠀⠀⠉⠁⡉⣉⣻⣉⣹⣭⠏⠙⠁⣠⠎⠁⣷⢻⡿⠛⠣⣠⣜⡀⠠⠀⠀⠻⠻⡆⠰⠀⠙⣮⠕⡪⠎⠀⣸⠇⠀⠀⢸⡇⣸⣿⢧
⠀⠀⠠⣤⣤⡄⢸⣿⣤⢀⣁⣤⣠⢎⣵⠞⠀⠀⠈⢀⡀⠀⢤⠘⠃⠀⡆⠀⠀⣦⠹⡄⠋⠰⡈⣿⡀⠀⢠⡟⠀⠀⠀⢼⣵⣿⠃⠀⢸⡆
⠀⠀⠀⠹⣿⣦⢸⣏⣧⡼⣫⢚⣵⣻⡶⡃⠀⣿⢀⠀⡟⠃⢸⡄⠇⠀⡇⣀⡀⢿⣧⢻⡌⢒⡽⢋⡼⣢⣾⣤⣤⣶⣶⡿⠟⠁⠀⠀⡈⡗⣠
⠰⠀⢨⠂⢻⡿⠀⣿⣩⠞⠁⠋⠀⠉⠸⠅⢀⡿⢿⡂⣇⣤⢸⡇⢰⠀⠁⡏⠁⠸⣿⡧⠻⣏⣠⢞⡶⠋⠠⡍⣉⠭⠤⠤⠄⠂⠤⠔⠕⣀⣾⠃
⠰⢁⡀⠀⣀⠛⣠⣿⠁⡘⢢⠀⠀⠀⠀⠀⢸⡇⢸⡇⡇⠀⢸⡇⢸⠀⠀⣧⡤⠀⢉⠀⣦⢹⣴⠋⠁⢠⢱⠀⣿⣶⡄⠀⠸⡀⠀⠠⠀⠉
⠐⠀⢃⡁⠼⠚⠄⢸⢀⡷⠋⡀⠀⠰⠀⠀⠺⠇⠸⠆⡓⠒⢸⡇⢸⠀⡆⣿⣄⠀⠋⠐⣚⣭⣿⣆⠘⣦⡘⣆⠹⣿⡇⠰⡶⣦⠀⣒⡀
⠀⣀⠠⢁⠐⠀⢀⡿⠆⡰⠦⡈⢢⠈⠰⣷⠀⠀⠀⠀⣁⡄⢸⡇⠸⠀⠀⣿⢸⡆⡀⠀⠙⠻⣿⣟⢧⡈⠳⡌⢦⡙⢿⣄⠹⣿⣆⠩⣉⠀⢶⠄
⠀⠀⠀⣠⠚⠃⢈⢹⠀⣀⡴⠃⠈⠰⠀⣀⠐⢦⡀⢄⠻⠆⢸⠇⡇⠀⢰⣬⠸⡇⢹⠀⠈⢱⣤⠙⢷⡙⢦⡙⢮⣵⣌⠻⢧⣌⠻⣧⡈⠻⢦⡄
⢀⡀⠊⣴⠜⠁⠀⢸⠃⣉⡀⠢⡈⢀⠀⠈⠳⡄⠹⣎⢣⡄⣾⢠⠃⠀⢸⣿⢰⣿⠈⣧⠀⠘⠀⠀⠀⠉⠀⢙⢦⣌⠛⢷⣄⡙⠓⠈⠙⠳⠄
⢸⣷⠶⠙⠘⠀⠀⢸⠀⠈⣩⠆⠰⠄⠀⠀⠘⢮⠧⡘⢧⣸⠇⡸⢠⢠⡆⢩⢈⣿⠀⣿⠀⠀⠀⢠⠀⠀⠀⣿⠀⣌⠻⢦⣌⣙⠳⠆⠰
⠸⢹⣇⣀⣐⠀⠀⢸⣦⡞⢋⠀⠀⠘⣄⠀⠀⠃⢓⡘⡌⠋⢠⠃⡆⣼⣿⡄⣸⣿⠠⠿⠀⢈⠀⣴⠀⠄⢺⣷⣡⠤⠀⠀⠈⠉⠛⠓⠂⡰
⢉⣽⣿⠍⠈⠀⠀⢸⠏⠶⢦⠁⠀⠰⠏⠀⠀⠀⠀⢿⡘⣤⠋⡜⡐⠻⣿⡇⠿⠿⠀⠀⠀⢀⠔⠉⠀⠀⠈⣟⠁⠀⠀⠰⣇⠀⢠⠀⣏
⢰⢨⡟⠀⣒⠀⠈⢸⡀⣄⡴⡶⠀⠃⡄⠀⠠⠜⠀⢸⣷⠀⠜⠀⠁⠀⢿⡇⠀⠀⠀⠐⠚⠁⢤⣷⣈⠢⣀⣉⠀⠀⠀⠈⣕⣀⠈⠀⣌⠀⡇
⢸⢻⣿⠀⠈⠀⠀⣿⡷⢋⠐⠀⠘⠰⠇⣴⠀⠀⠀⢸⣿⣷⣿⣿⡿⠶⠦⣀⠀⠀⠀⢠⡀⠀⠀⣽⠋⢐⠉⠀⠀⠀⣀⡈⠋⠉⠀⠀⣿⠠⡇
⠈⣾⡇⠀⠤⠀⠀⣽⠀⠈⢱⠀⠀⢀⡆⣿⢰⡇⡆⣸⣿⣿⣷⣶⠿⠿⠛⠳⠆⠀⠰⣿⡷⠂⠀⣠⠀⠳⠄⠀⠀⠘⠛⠂⠂⠃⢀⠀⣿⠀⠇
⠀⣿⡇⠀⠛⠀⠀⢻⣤⣄⣉⣤⣤⣿⣧⣿⣸⣟⣛⣉⣈⣛⡉⠰⠟⢛⣿⠓⠂⠀⠀⠘⠀⠀⠀⠛⠀⠀⠀⠤⠄⢤⣤⣬⣬⣥⣭⣤⣿⠀⡇
⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢀⣾⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣧⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠉⠉⠉⠉
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠹⣿⣿⣿⣿⣿⣷⡄
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⣿⣿⣄
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠰⡄⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢦⠀⠀⠀⣿⣹⡆⠛⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⣿⣩⣿⡁⠘⣧⠶⠃⠉
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣫⠴⠂⠘⡧⠞⠁⢀⣀⡤⠴⠒⠉⠁
⠀⠀⠀⠀⠀⣟⠁⢀⡠⠖⢦⠷⢀⣠⠤⠖⠊⣉⡀⠀⣀⠀⠀⢀⣦⠀⠀⣇⢀⡇⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠸⡆⢯⠀⠀⠀⠀⡉⠀⠀⠀⣏⣭⣿⡀⢿⣀⣀⢸⠿⢧⠀⠛⠉⠃⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⡟⢲⡄
⠀⠀⠀⠀⢀⡖⢻⡜⡆⠀⠀⢧⢻⡀⠀⠀⠻⠶⠚⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠻⣆⡇
⠀⠀⠀⠀⡟⠀⠀⢷⣹⡀⠀⠼⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⢷⠀⠀⠈⣧⢳⡀⠀⢘⠸⡄⠀⣴⣲⡆⢠⡄⠀⠀⣠⡀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡾⣷
⠀⠀⠀⠀⠈⠳⣤⣀⣼⣇⠣⠐⠦⢴⣅⣀⠻⠶⠟⠸⠧⠄⣴⠿⣇⢸⣅⡗⠀⣿⠀⠀⠀⠀⣸⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⣆⡴⠋⠀⠀⠉⠙⠛⠲⠶⢤⣄⡀⠀⠈⠀⠀⠁⠀⢿⡀⠀⠀⠶⣬⡛⠋⠉⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⡀⠀⠘⠿⠷⠦⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⠀⠀⠀⠁⠀⠀⠀⠀⢀⣠⡶⠛⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠉⠛⠶⢦⣤⣤⣤⡶⠞⠋⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⡁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠚⠛⠉⠉⣽⠋⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠴⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉

⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣀⣀⣀⣀⡀
⠀⢸⣿⠛⠛⠛⠛⠛⠉⣿⠛⠛⠛⠛⠛⢻⣟⠉⠉⠉⠉⠉⣿⠛⠛⠋⠉⠉⢻⡟⠉⠙⠛⠛⠛⣿⠛⠛⠛⠛⠛⢿⠟⠛⠛⠛⠛⠛⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠗⠒⠛⠛⠛⠛⣿⠟⠿⠛⠛⠛⢻⡿⠻⠛⠛⠛⠛⣿⠟⠛⠛⠟⠛⢿⡿⠟⠛⠛⠛⠻⣿⠷⠶⠶⠿⠷⢿⠿⠿⠿⠿⠿⠾⣿⡇
⠀⢸⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⣶⣶⡶⢶⡶⢶⣿⡶⢶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣿⠶⣶⣶⣶⣶⣾⣷⣴⣶⣶⣴⣴⣿⣤⣤⣴⣤⣤⣿⣦⣤⣤⣤⣤⣤⣿⡇
⠀⢸⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⣦⣀⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣄⣠⣄⣀⣀⣼⣇⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣠⣿⡇
⠀⠈⣿⡏⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⡇
⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⢹⣿⡶⠶⠶⠶⣶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⣶⡶⠶⠶⢶⣿⡟
⠀⠀⠀⠀⢿⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠆⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠘⣿⣇⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠸⣿⣦⣤⣿⡤⣤⣤⣤⣤⣼⣷⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣴⣿⣤⣤⣤⣤⣤⣿⣦⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣄⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣤⣀⣀⣀⣼⣧⣀⣀⣀⣀⣀⣿⣄⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣍⠀⠉⢹⡏⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠙⣿⠉⠉⣩⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢰⡿⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣇⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣉⠉⠉⠛⣿⠛⠛⠛⠛⠛⢻⡟⠛⠛⢛⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⣿⠀⠀⠀⠀⠀⣸⡇⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄⠀⠀⠀⢀⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣾⠿⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠄⢠⡤⠀⠴⢀⠆⠀⠀⠆⠀⠀⠦⠀⢀⠲⣆⠀⠀⣤⡠⣀⢄⣤⡾⠻⣶⣷⣦⣤
⠀⢀⣉⠠⢤⠤⠤⣤⣤⢤⡤⡄⣦⢖⣰⡶⠀⣴⠆⢀⡤⢀⠄⠰⠏⠰⠆⠰⠁⠀⠀⠰⢆⠀⠐⣿⣷⣦⠑⡌⢷⣄⠈⣻⣴⠛⠙⣮⣶⢿⣿⣿⣿
⢐⠈⠉⠀⢈⣉⡐⠲⠶⠶⣤⣤⢀⣿⠋⠀⡌⠁⣠⠍⡠⢁⡼⢃⡶⢠⢀⡴⠁⠀⢀⣠⠤⠼⡭⠉⠉⠥⠊⣤⣐⣹⣾⡯⢈⣷⣶⣿⣷⣴⣿⣷⣿
⢀⡔⠋⠀⡈⠉⠙⠛⢶⡆⣿⢃⡾⠃⢠⣆⡄⠺⢃⡜⣱⠏⠠⢟⡕⠁⢈⡠⠴⡟⢉⣁⣤⠀⠀⣀⡀⠀⠀⣼⠿⣅⢈⣷⣟⠫⠝⣿⡿⠻⣿⣿⣿
⢈⠰⠆⢐⠁⠀⠀⠋⠈⠰⢃⡾⠁⢰⣟⣾⠏⡠⢀⡾⠁⣴⠂⠚⠨⢍⡁⣤⣶⣏⡯⠀⠀⢀⣀⠈⠁⠐⢫⢔⣠⡟⢿⣅⠀⠴⢟⣉⡳⣤⣿⣿⣯
⡰⢠⣄⠈⢁⣤⣤⡀⠀⠀⠈⣀⡶⢀⡿⠃⠐⣵⠏⢠⡞⣡⠋⢀⢄⣀⠀⠈⢩⣿⠁⠀⠀⠛⠟⢀⢴⣻⠾⣿⣿⣿⣦⠙⠆⠰⠟⢿⡇⠈⣿⣿⣿
⢃⠀⠀⠀⢚⣛⣛⣋⡀⠀⣼⠏⣱⠟⣥⣦⡞⠁⣴⠋⡀⠁⠛⠀⠿⠿⠀⠀⠈⣠⢰⠁⣄⠀⡀⠗⠋⠁⢀⡀⠈⠤⢄⣒⣒⣂⣀⡀⠐⠸⣿⣏⣿
⡸⠀⠀⠀⣤⡍⣹⡟⣡⠾⢁⡼⠋⣜⣷⠏⢠⡾⣁⠘⡡⢻⠀⠀⠀⠀⣠⣴⡿⢻⡄⠀⠁⠘⠁⠀⢀⣀⠶⣴⠾⠳⠛⠉⢍⣉⣉⣁⣀⣀⣿⣿⣿
⠐⡀⠑⠀⠟⣵⢋⡴⠋⣠⡟⣡⢞⡾⢁⣴⢛⡴⠋⠂⠁⣤⠀⣀⣴⡿⢻⡅⢷⠘⡇⠀⠀⢒⣒⠩⠍⣈⣢⣤⣶⠶⠛⠟⢋⣙⣉⢉⣙⢻⣿⣻⣿
⠰⠀⢤⠀⠞⣡⠟⠁⣠⡋⠀⣹⠏⢠⡞⢡⡆⣠⡈⠢⡀⣡⣾⠿⣅⠹⣆⡿⠞⡛⠀⠀⢈⣥⡴⠞⢛⠉⣭⠀⡶⠆⠛⠃⠈⠉⠁⢈⣉⠠⣿⡿⣛
⠀⠆⠀⢁⡄⣹⠆⡘⠃⣠⡾⠁⢀⡆⣠⠆⣤⠇⣈⡂⡈⠛⠙⢧⡸⠖⠁⣰⠂⣡⣶⢞⣫⠅⢚⡩⡴⠛⠉⠀⠀⣀⡐⠂⠀⠄⣶⡈⣓⢐⣿⣿⣿
⠀⢄⠰⡙⡄⣱⠆⢀⡴⠋⠀⣰⢋⡴⠋⠀⣇⡇⣿⣿⣌⠢⡐⢎⢡⠖⣫⡴⣿⠷⣧⣤⠶⠚⣉⣀⣠⣄⠠⢋⣡⣤⣶⡶⢾⡖⢿⠶⣤⣩⣿⣿⣿
⠀⠔⠀⢃⠀⠁⣴⠏⠁⢀⡄⣡⠞⡁⠀⢰⣿⠇⢹⣿⣿⢇⡤⠀⣢⣞⡯⣺⣵⡺⠛⠅⢰⠒⣸⠟⣉⠞⠀⠈⠁⢠⡿⠷⠟⠛⢻⡷⠾⣿⣿⣿⣿
⠀⠂⠐⡈⡀⠾⠃⠀⡰⢀⡴⣣⡞⡁⢠⡞⣽⢼⠘⠿⣣⠊⣠⠞⠣⠻⠞⢁⣾⣥⠶⠾⠈⣿⡇⠻⠀⣤⠄⣴⠃⠀⠰⠚⣿⡇⢸⡇⠤⢨⣷⣿⣿
⠀⡎⢠⠁⡀⠶⣠⢞⣁⠊⠠⠉⠀⡀⠀⠀⠘⡿⢀⡰⢡⡾⠧⠀⢰⡄⢰⡟⠋⢀⣀⣤⣇⠙⠿⠞⠀⢇⡟⣠⣴⡄⢰⣦⠸⣇⢸⡇⠀⠀⣿⣿⣿
⠀⢨⠀⠀⠇⠸⠋⢨⡴⠃⠀⡶⢆⠈⣶⡠⣤⡄⠋⣰⢯⠀⠀⠀⠀⠿⣠⣷⣶⢟⣿⣣⣬⡞⢁⠈⠀⠀⠀⠹⠏⠀⠀⢿⡀⣿⢸⡇⠀⢀⣿⣿⣿
⠀⠡⠀⢢⠐⡜⣳⠏⠠⡙⠀⠙⠁⠀⠉⢁⢀⡠⠘⢃⣠⠀⢸⡇⡼⢠⠘⢋⢩⡼⠟⣩⠤⠘⣄⢰⡀⠀⣀⠀⠀⠀⣰⢸⡇⢻⣼⡇⠀⢠⡿⣿⣿
⠀⠐⠀⡆⠀⠛⡟⢀⢎⣶⠟⠛⣿⣯⣍⠉⠚⠀⢉⠟⠱⠂⣼⠟⠃⢸⠀⠘⡌⡄⢊⣽⢶⣟⠙⢦⠙⣦⣌⡉⠀⠀⠛⣦⢷⢸⣿⠅⠀⢹⣧⣿⣿
⠀⣠⠆⡑⠀⠀⠀⠸⠸⣧⡘⠀⣿⣟⣿⣿⡶⢞⡁⠛⣠⠞⡡⠀⠀⡘⢧⡀⢣⣴⡏⠻⣟⣼⠶⠞⠳⡌⢻⣿⣿⢁⠀⠀⠸⠸⢠⡄⠀⢰⣿⣿⣿
⠀⢣⡀⢣⡀⠀⠀⠀⠠⠙⠿⠶⠾⣿⢛⣩⡶⢛⣥⡾⢁⡷⠇⠀⢀⢷⡀⠑⣜⢋⣧⡴⢋⣠⠒⢈⣿⠿⢦⡙⠷⡌⠀⠀⠀⡄⡀⠀⠀⢸⡗⣿⣿
⠀⠊⢠⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠹⣎⡡⠒⣽⡟⢰⣿⠑⠀⠀⡎⢀⢿⡄⠈⡏⠻⡥⠋⢀⣴⡿⢃⢀⠲⣝⢤⣷⢶⠀⠀⠙⣁⣀⡄⢠⣿⣿⣟
⢀⠖⠀⢃⠃⣣⠀⢠⡀⢯⠀⠰⡑⢠⣿⡄⠀⢹⣧⣈⣡⠃⢀⠞⣤⣎⡆⢻⣄⢱⣾⡇⢀⣾⡿⠁⠆⠀⠀⣀⢀⣿⡴⠀⠘⠃⢸⠁⢖⢸⣿⣿⣿
⢈⠂⠙⠄⡆⠘⡇⣼⢸⡌⣠⡏⠘⣧⡙⢿⣦⡀⠙⣿⠃⢀⠎⠀⡷⡸⢛⡀⢻⣧⣠⡠⣾⡿⠁⠀⠀⠀⠈⠍⣛⠻⣷⠐⢏⡸⠃⢠⡿⣹⣿⣿⣿
⠰⠋⠐⢠⠃⡙⠀⣿⣼⣳⠏⠀⠀⠈⢻⣦⡙⣿⠞⠁⠀⡎⠀⠀⢰⡷⢻⠁⠀⠻⣿⠁⢿⠃⡘⠁⠀⠀⠀⠘⠋⠀⠈⠡⢀⡀⣲⡟⠑⠻⢿⣿⣯
⢠⡗⢨⠂⡇⠙⠀⠛⣿⢇⡸⡇⠡⢄⣀⠹⠛⠁⠀⢠⣼⣳⣦⣄⡈⠘⠻⠆⠀⣠⣉⣁⣨⢀⠋⠀⠐⠛⣶⣤⡄⠀⣤⡀⢈⣷⡟⠻⠇⠀⣿⣿⣿
⠨⡀⠁⠸⡀⠉⠀⠀⠏⣸⢧⡽⠓⠆⠀⠀⠀⣤⡞⣧⠁⣙⠓⢾⣏⣳⣶⢤⣤⡏⠉⠉⠙⣜⠀⠀⠀⣆⣿⡧⠟⢀⡿⣱⣿⡟⠃⢀⡀⠀⣿⣿⣿
⡰⠃⡞⠀⠃⠲⠀⠈⠀⠙⠀⠀⠀⢀⣠⡆⠀⠉⣿⠀⢰⠏⣻⣶⣦⣭⡉⠛⠻⢠⣤⣤⠄⠁⡀⣰⠂⠹⠏⠁⠀⠈⣜⣿⢻⡇⠀⢸⣇⠿⣿⣿⣿
⠀⡄⠁⠀⣘⣛⣀⠀⠴⣁⠎⣰⣽⣔⠪⣅⠀⠀⠽⠀⢸⡀⢿⣿⡗⠀⠀⡠⢡⡿⣡⠟⠀⡇⣹⠮⠂⠀⠈⣷⠏⢈⡾⠃⣸⢃⠀⡆⢸⠀⣿⣿⣿
⡴⠃⠘⢀⣭⣭⣭⣄⠐⢡⣞⡠⣭⣭⡛⠯⢄⡆⠀⠈⠀⠑⠀⠙⠃⢀⠌⣠⣋⠜⢃⣶⡆⠇⠁⠀⠤⠀⠐⠃⢠⡟⠡⡆⣿⢸⠀⡧⠘⢀⣿⣿⣿
⠑⣆⠀⣈⢁⣀⡀⣈⣀⣐⠻⠭⠝⣻⣃⠄⡼⠀⣀⠀⠀⠄⠀⣐⣠⣴⠾⢛⣡⣶⡿⠛⠁⠀⠀⢭⡐⠀⠀⣴⠏⡰⣅⠆⣾⠘⢀⠟⠀⢀⣿⣿⣿
⠘⠃⠀⠉⠀⠉⠠⠩⣭⣍⣛⠛⠶⠦⣥⣞⠰⣅⠁⠀⡀⠠⢤⣬⠴⠶⠾⠛⣋⠁⠀⣄⢰⣿⠇⣼⣇⢀⡦⢁⡔⢴⡟⡀⣿⠀⠈⠉⠁⣿⣿⣛⣿
⢈⡥⠀⡰⠶⠶⣶⣶⡄⣸⡿⠟⣁⡴⠋⣉⡓⢦⣉⠢⣀⠀⠀⠀⠲⣛⠱⠞⠽⠀⡀⠨⣸⠏⢠⣿⠃⡾⠁⠊⡀⣤⣌⠋⣿⠁⢀⡀⠀⣿⣿⣿⣿
⠸⣅⢠⣷⠶⠟⣋⡥⠞⣋⣴⠾⠋⡔⢋⣤⡉⠂⠈⠳⢮⡀⠀⠀⠀⠉⠻⠗⠀⢀⡁⣼⠈⢀⣿⠏⠂⠠⠂⠰⡗⠛⠳⠶⣿⠠⡿⢛⠈⠛⣿⣿⡿
⢀⡶⠁⠀⢀⣈⡥⠶⣛⠉⠀⣀⣈⣈⣭⠀⠈⠁⣀⣀⠀⣙⢦⡀⠀⠀⠀⠀⠀⠘⠃⣿⠀⡼⠳⡦⠀⠀⣀⠐⠓⢀⡀⠀⣟⢠⡧⠘⠃⣠⣿⣿⡟
⠹⢆⠀⣼⣍⡅⢙⣀⣛⣁⣙⣛⣙⣙⣛⣁⣇⠐⠶⠶⠆⣿⠀⠙⣦⠀⠀⠀⠐⡠⠀⢸⡘⠁⠄⠀⢰⠀⡟⠠⡧⢸⠁⡄⡀⢠⠀⠀⣰⣿⣿⣷⣷
⠰⠞⠁⢀⣈⣁⣈⣉⣹⣟⣭⣯⣛⣿⣽⣏⣿⣛⣿⡟⢓⣿⣷⣆⡈⠷⠀⠰⠄⠐⠁⠈⠙⠃⠂⠀⣤⠀⡇⢴⡆⢸⢀⠚⣙⠎⢀⣾⡿⢉⣿⣿⣿
⠀⠀⠀⠈⠀⠈⠁⠈⠉⠉⠩⠉⠉⠋⠉⠍⠙⠩⠉⠋⠉⢹⠏⠛⢡⠈⣧⠀⠘⠀⠈⢠⠀⠀⠛⠀⠉⠀⠁⠀⠂⠈⠸⠇⠉⣠⣿⠟⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢛⡟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⣾⠁⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⣿⣿⣿⣿⣿⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧
⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠁⠛⠒⠢⠤⣀⠀⠀⣤⣾⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣀⣀⣀⠙⠛⠿⣽⡯⣿⣿⣷
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢾⡟⢋⣹⣛⣷⣤⣼⠿⢛⠋⣿⡆
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠘⣷⡈⣿⡿⠉⠈⢁⠀⠀⠀⠀⢷
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⠈⠳⣼⢇⠀⠀⠀⠀⡀⠀⠀⠈⡆
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣭⠻⠋⣁⢦⡘⢷⣄⡀⠀⠀⠀⠄⠀⠀⢳
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⢞⠋⠠⠀⢀⠖⣺⣫⡟⣿⠢⠈⠛⠷⣦⡀⠀⠀⠀⠸⡆
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡄⠀⣠⣾⡿⠟⠀⠀⢀⣼⠷⠛⡀⠀⣸⣷⡀⠀⠢⢄⡙⢳⣄⡀⢀⣱
⣿⡿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢾⣿⣿⡀⠀⠐⢂⠙⠃⠄⠀⠀⢀⣼⢯⣽⣷⣤⣠⣙⣤⡈⠙⠻⢿⡇
⣿⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠈⢿⡜⠁⠀⠈⠁⠀⠂⠨⠇⢈⣱⠞⠋⣹⠟⠁⣈⣉⣁⠉⠑⢄⣄⢿
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠷⠀⠀⠙⢧⣄⠀⢀⣤⣾⣷⣾⠿⢁⣠⠞⢃⣴⣿⣿⣿⣿⣿⣦⡀⠈⠻⣇⠀⣤⣶⣿⡄
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡻⡿⢏⡀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠟⠛⠉⠁⢠⣾⣿⣿⣿⣶⣯⠿⣿⣿⠠⡦⢻⣾⣿⣿⣿⣿
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣵⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣷⣿⣻⢿⣿⣷⣼⡟⠐⢷⣼⣿⣿⣿⣿⣿
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⡟⠀⢀⣀⡤⣤⣤⣦⣤⢤⣤⢀⡀⠀⠀⠀⠀⠁⠙⢿⣿⣷⣿⡿⠛⠁⢀⣼⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡞⡏⣉⠰⠶⣏⠠⠍⠾⡟⢮⠻⣷⣤⡀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣍⡀⢤⠀⢐⡄⢰⢶⣄⠈⢻⣄⡀⠮⢻⣦⣀⣤⣾⣿⣧⣾⣻⣾⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⠚⡿⢀⠈⡉⠀⠙⠿⠿⣤⠈⠈⢣⡘⠻⠚⣿⣿⣿⣿⣿⣛⣿⣁⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⣷⣦⠈⠁⠀⠀⣀⠰⢿⠀⡀⠈⠛⢦⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⠀⠀⣀⡞⢋⠰⡞⣧⠀⢰⣾⣻⡝⠛⠉⠛⢀⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣅⣠⣴⣒⡒⠶⡟⠜⠛⠻⣿⠏⠀⠶⡾⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣛⣽⣿⣿⣿⣿⣗⣤⣾⣛⣿⢋⣠⡘⠳⡴⣿⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠭⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣄⣾⣿⣿⣿⣿⣿⣿
⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⢸⣿⣋⣿⣷⠀⠀⣿⣿⡆⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⣿⣷⠀⢸⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡿⠿⢷⣦⠀⣿⣿⠀⠀⠀⠀⣸⣟⢸⣿⡆⠀⠀⣿⣿⣤⣾⣿⡿⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⣿⣇⣴⣾⡿⠡⣿⣿⣀⣀⡀⢀⣿⠛⠛⣿⣧⠀⠀⣿⣿⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠘⠿⠛⠋⠁⠀⠀⠙⠿⠛⠋⠀⠘⠋⠀⠀⠙⠟⠀⠀⠻⠟⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣻⣶⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢸⣷⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⢸⣿⣇⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣠⣾⡿⠀⢺⣿⣇⣤⣤⡄⠀⣿⠋⢻⣿⡀⠀⢸⣿⡟⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠈⠛⠿⠛⠉⠀⠀⠋⠀⠈⠻⠃⠀⠘⠿⠃⠀⠛⠃

⠀⠀⠀⠀⠀⢸⣿⢻⣶⡄⠀⠀⣶⣦⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⣿⣿⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⢸⣿⡆⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡏⢙⣷⡄⠀⣿⣿⠀⠀⠀⠀⣼⣟⣻⣿⡀⠀⢸⣿⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠘⠿⣷⣿⡿⠁⠀⣿⣿⣤⣶⠀⢠⡿⠛⢻⣿⡇⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⠃⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⢀⣀⡀⠀⣀⢀⢀⡀⠀⣀⢀⢀⠀⢀⣀⢀⢀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣉⣀⡀⠀⠀⠀⠀⢠⠋⠀⠫⠽⠺⠔⠚⠘⠛⠙⠣⠀⠯⠽⠺⠔⠟⠚⠛⠙⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠲⣤⠎⠀⠀⣙⠀⠀⠀⠀⢸⠀⠀⡺⢸⣲⣂⣰⢳⣴⢴⡆⠀⡾⢼⣰⣀⡴⢳⣴⢴
⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⣀⣨⣶⣶⠉⣸⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⣠⠖⠋⠉⢧⠒⠉⠁⠙⡷⠖⢊⣁⡉⠉⠑⠲⢤⡀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⢸⠟⠀⠀⠀⠈⠓⢤⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠉⢒⡦⠀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠔⠒⠉
⠀⠀⠟⡿⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⡢⢘⡦
⢀⣀⣀⡇⠀⠀⠀⠀⡾⠀⢀⣀⠤⠤⣔⠲⠢⠤⢄⣀⠀⠀⠀⠀⣧⢠⠾⠓⠁
⠘⡦⠹⢇⡀⠀⠀⠀⢧⠀⠫⣸⣀⠄⠈⠙⢲⡄⠀⠈⠙⠲⢄⡀⢹
⠀⠉⢲⢄⡈⠑⠢⢀⠈⠢⣀⣀⣀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⡏⠀⠈⠲⢄⡀⠈⠑⠢⡀⢨⠦⠧⠤⣀⣀⠀⠰⣏⠲⡀
⠀⢠⢡⠀⠀⠀⠀⠙⠲⠊⠀⠨⣿⡖⠐⡀⠀⢠⣝⣀⣨⠇⠉⠀⢀⣐⠠⡀
⠀⢸⢸⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠁⠀⠘⡄⢸⡿⠤⠤⠒⠢⡀⠒⣄⢙⡇
⠀⢸⠸⡄⠀⣀⣠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⠈⠒⠊⠁
⠀⢸⡀⠉⠹⡟⢳⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⡄
⠀⠈⡏⠉⠉⢳⠈⣏⠉⠉⠉⠉⢉⠃⡇⠀⠀⢸
⠀⠀⢷⠀⠀⠈⠉⠉⠁⠀⠀⠀⢸⢸⠀⠀⠀⡾
⠀⠀⠸⡄⠀⠑⠢⠀⠀⠀⠀⠀⡎⡞⠀⠀⢠⠃
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠠⠜⢛⡟⠛⠀⠀⡎
⢆⣙⣻⣓⣲⠤⠤⠤⠤⠤⠄⠚⠂⠐⠲⠯⠯⠭⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠙⠛⢿⣿⣿⣿⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣦⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⣀⣤⣴⣿⠳⠶⢦⡤⣠⣀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠏⠁⠒⠶⠀⠉⠙⠛⠛⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠀⢸⣿⣇⠀⠀⠀⢾⣿⣿⡆⠀⠈⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣄⠀⠑⠈⢿⣿⣄⠀⠀⠀⠈⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠓⢄⠀⠠⠹⣿⣷⡤⠀⠁⢿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠑⢤⠃⠹⣿⣿⡆⢀⣼⣿⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠓⣄⣿⠿⠓⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣀⣠⣴⣶⣶⣤⣄⠀⢀⣴⣦⠀⢀⣤⣄⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢠⣿⡟⠋⠉⢹⡇⢀⠀⢹⡇⣼⠋⣿⣀⣾⠋⣿⠈⣰⡟⠉⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⣿⣷⠶⠶⢾⣧⣈⣤⡟⢡⣿⠀⣿⣿⠃⢸⡏⢠⡿⠀⢠⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡄⣿⣏⠀⠖⢸⣿⠿⣿⡇⢸⡟⠀⣿⣏⣀⣼⠃⣿⢃⠀⣾⠃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⢿⣿⡟⠿⠶⢿⣿⠀⢿⡇⢸⡇⢰⣿⣟⣻⡿⠛⠿⢶⣼⡟⢠⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⡇⠀⠀⠸⣿⠀⢸⡇⠸⢳⡆⠉⠙⢿⡟⠿⢿⠶⣿⠃⣼⠇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣤⣤⠈⠁⢰⠉⠀⢠⣤⠀⠻⣦⡾⠃⢠⣤⣦⣦⣄⡘⠘⠟⠀⠿⠿⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⣿⣷⠶⠶⠈⠘⠺⠿⣿⠿⠿⠷⢶⣤⣴⠶⣾⣿⠉⡛⠷⣤⢰⡆⣤⡀⡆⢰⡇⣠⡶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣉⣤⣤⣶⠶⢶⣦⠄⣿⡄⢀⣀⣠⡿⠃⣤⠄⠹⣧⣀⣤⡿⠘⠃⣿⣇⠀⣿⢣⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⡏⢉⣠⣴⣶⡿⠃⡇⢸⣿⠿⠿⣧⣄⠀⠿⠀⢠⣿⣿⡇⢰⡟⢸⡟⣿⣸⡟⣾⣃⣀⣘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣈⠋⡏⠁⠀⠀⠃⢸⣿⠀⣀⣀⣹⣷⣤⣴⣿⠁⠸⣧⠘⠃⢸⡇⢸⣿⡇⣿⡉⢻⡏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠋⠀⠁⡆⠐⠿⠟⠛⠛⠛⠋⠉⢩⠀⡅⠈⣿⠀⢰⠙⠷⠆⠈⠀⠀⠛⠁⠘⢷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠂⠀⠀⠀⠀⠈⢸⠈⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡂⢚⡛⠛⣦⡙⢂⠀⢀⣘⣢⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⢀⠀⠀⠀⠀⠀⠀⣤⡹⠖⢒⡅⠈⠉⠉⠉⠀⠉⠉⠍⠃⠀⠀⠀⣰⠂⠰
⠀⠀⠀⠀⠀⠀⢀⡔⣣⣾⡾⠶⠶⡈⠢⡀⠀⠀⠀⢸⣽⣮⣴⣳⣶⣶⣖⢲⣾⣶⣶⣶⠞⢀⣴⡶⣯⣾
⠀⠀⠀⠀⠀⠀⣾⡁⣾⡗⠛⢛⠉⢻⠀⢡⠀⡞⢩⣭⣭⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣀⢈⡍⣧⠹⠔⠇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠹⣇⠻⣷⣓⣊⡠⠞⣠⣿⡧⡇⣿⡿⠿⢁⠀⡉⣶⡆⠈⡏⠉⢉⡩⠉⢉⠈⡇⠛⠀⠀⠀⣠⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠑⢶⡶⣴⡶⠮⠋⠁⢉⡇⢹⣿⠀⠅⣣⡀⢦⠁⠀⡇⠀⡏⠀⠀⠀⠇⡇⣦⠗⠺⠚⠿⠉⠉⠁⠤
⠀⠀⠀⠀⠀⠀⠐⠂⠔⠡⠐⠀⠀⠤⢤⡢⢾⣄⢸⠟⣄⠀⠁⡶⠇⠈⠁⠓⠖⠚⠲⣒⣾⡆⢃⡟⠀⠀⠀⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠈⢿⠐⣦⣸⡶⠺⠓⠂⠀⠰⠒⠓⠤⠼⠛⢻⣷⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣄⣛⣉⣁⣘⣀⣐⣒⣶⣿⣯⣭⣭⡴⠽⠷⠾⠦⡄⣄⣀⣀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣒⣒⣂⠰⠳⣷⠴⣿⠾⣿⠄⠐⠚⠚⠘⠺⠂⢁⣁⣁⣀⣈⣠⠠⣍⣉⡁⠰⠶⠾⠟⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣈⣛⠋⠀⢈⣴⡿⠶⠿⠿⠿⠛⠟⠛⠛⠛⠛⠿⠛⠛⠛⠛⡛⠉⠉⠩⠭⠭⢥⠤⠤⠥⠤⠤⠶⠦⠬⠙⣳
⠀⠀⠀⠀⠀⠀⠀⠀⢫⣤⣀⠠⣵⠆⠀⠰⠴⠆⠀⠠⠖⠀⠀⣲⣗⣀⣀⣒⣃⣀⣙⣃⣀⡈⠘⠀⢀⣛⣃⣀⣐⣃⣀⣸⡏⢸⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣶⣶⣦⠶⠶⠊⠋⠉⠉⠉⠙⠛⠛⠃⠤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠲⠾⠟⠛⠉⠉⠉

⡀⣠⣤⢄⣠⡴⣠⣴⠀⠀⡴⠀⢀⣴⠆⡠⢦⣶⡶⢒⣠⣄⣴⠆⣰⠖⢠⡾⠃⣴⠟⢠⡾⣣⡟⢿⡄
⢠⣿⡟⣴⢆⣼⣽⠃⢀⠞⠀⣰⠟⣡⠞⣡⢔⣡⡾⠿⣥⡞⢁⡼⠃⣰⡟⣁⠼⣯⡴⣿⣥⣶⣦⣾⠇
⣼⣿⠃⣳⣿⣽⠇⢀⠎⠀⡼⢣⠞⣡⢞⣵⣿⠿⢴⣾⠋⣠⠟⣡⣼⡛⣶⣝⠠⠿⠟⠛⣉⣉⡼⢣⡄
⣿⡟⣰⣿⣯⡏⠀⡞⠀⢘⣵⣃⠔⣵⡿⠋⣀⣠⠞⣁⡼⠿⣿⣿⡭⠟⢊⣁⢠⣶⣾⣿⣿⣿⣵⣛⡃
⣿⠇⣿⡟⣹⠁⠠⢀⣴⣿⣿⢏⣾⠋⣰⢟⡿⣫⣾⣿⣿⠟⢋⣡⠶⠚⡫⡟⣼⣯⣾⣾⡿⣛⡟⣛⡃
⣿⢰⡿⣵⡏⠀⣠⡞⢻⣯⢯⡿⠏⢸⡷⣫⡾⣫⠷⢋⣤⣾⣿⣾⣿⣦⣡⡇⢋⢙⣹⣿⣡⠼⢱⢻⠃
⣿⢘⣬⡟⣡⣾⣿⣿⣯⠋⠞⣴⣷⢟⣴⣫⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⢀⣡⣿⣿⡿⠿⢇⣧⢎⡄
⣿⢠⣤⣬⣽⣿⡿⣿⣿⣠⡾⠋⢡⣿⡿⣣⣾⣿⠛⣯⣭⡛⢿⣿⣿⣿⡇⣾⡿⠟⠉⣠⡤⡼⣣⣾⡇
⣿⠐⣿⡿⣧⢿⣷⣿⡇⢉⣠⣾⣿⡿⣱⣿⣿⠿⠧⠼⣶⣶⣤⣿⣿⡿⠀⠉⣀⡴⠟⢹⢛⣵⣿⡿⡁
⣿⢸⢿⣿⣿⣾⢻⣼⠱⠛⠁⣼⣿⢱⣿⡏⡏⣠⣤⡄⠈⣿⣿⣿⡿⢀⣤⠞⠉⣀⡰⣣⣾⣿⠟⠰⠆
⣿⢸⣎⠉⣿⣾⣿⢿⣥⡶⠿⢻⡇⣿⣿⡿⡆⣿⣿⡟⢀⣿⣿⡟⠱⠋⠀⡠⠚⢉⣴⠿⠛⠁
⣿⢸⣇⠀⠈⢿⣯⣿⡁⣠⣴⠻⠅⡿⣿⣷⣿⣮⣭⣴⣿⡿⠟⠀⠀⡰⠋⠀⠀⠊⠀⠀⠀⠀⣀⣠⡆
⣿⡿⣟⣀⠀⠈⢽⣻⢷⢉⣀⣿⡆⣽⣋⠿⠿⢿⠿⠟⠋⡞⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⣩⡄
⣭⣥⠉⠙⣦⠄⣈⣓⣚⣋⣫⣭⣧⢻⣽⣧⣴⣾⣿⠀⣸⡇⠀⡾⢀⣴⡅⠀⠀⠀⠀⣼⡿⢋⣾⣿⠇
⢶⣴⡄⠀⠱⡄⣦⠤⠤⠤⣤⣝⣾⡾⢟⣿⣯⣻⡿⠀⣻⣶⢊⣴⣿⡿⠃⠀⠀⠀⠀⣿⡇⣸⠟⠁
⢿⣝⡣⠀⢠⣿⣮⡳⣔⢶⠒⢰⣍⠐⢟⣳⣟⣞⣷⡌⣋⣴⣿⡿⠋⠤⢛⡀⠀⠀⠀⣿⠟⠁⢀⣴⡇
⣷⣯⣛⠦⢼⣿⣿⣿⣮⡳⣍⣉⣤⣴⢿⣷⣿⣟⣡⣞⡿⣛⠥⠖⠚⠋⠹⣿⡄⠀⠀⠀⢀⣴⣿⣿⡂
⣿⣟⣿⣷⣆⠙⠻⢿⣿⣿⣾⣝⣉⣤⠾⣺⢛⣥⣞⡯⠚⠁⠀⠀⠀⣀⣤⣿⡷⡀⠀⠀⠙⢿⡿⢋⡀
⣿⣿⣻⣛⠻⣦⠀⠀⠀⠉⠙⢛⣫⣵⣿⣿⣿⣎⠁⢀⣠⣤⡶⢖⣻⡿⣿⣽⣯⣿⣦⡀⠀⠈⢺⣿⠇
⡛⣷⢿⣿⣷⡿⠒⠀⣤⣴⣾⣿⠿⢯⣭⣽⣶⣿⡿⢻⣯⣵⢾⣻⣽⣿⣿⣏⣵⣟⣻⣷⣄
⠙⠛⠋⠉⠀⢀⣤⣶⡎⠻⣮⣍⣻⣿⣿⠿⠤⣾⣼⣿⠿⠾⠿⣿⣟⣬⣿⠟⡻⠿⠋⠚⠉⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠠⡄⠀⠀⠀⢀⠠⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⢀⠃⡔⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡠⠀⠃⠀⠀⠀⠠⡀⠐⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⠀⠁⠠⡄⠈⠀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⡠⠀⠀⠄⠉⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠄⠀⠀⠅⠂⠠⠐⢰⠇⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠁⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⢰⠄⢀⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠑⢄⡀⠀⠀⠁⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⢀⠀⠀⠠⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠃⠀⠀⠰⠂⠀⠀⠀⢀⠄⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠈⠁⠀⠀⠘⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿