Luggage Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠰⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠴⠶⠋⠉⠙⠶⠦⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⣹⠟⢉⡟⠉⠉⠉⠉⠉⠉⠻⡉⠛⣏⠙⠻⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣧⣤⣼⡄⠀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣧⡀⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠿⣿⣿⢿⠗⠒⠒⠒⠒⠒⠒⠺⢿⣿⣿⣿⠇⠚⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃

⡶⣶⢶⢶⡶⠶⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⡶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠿⠿⠿⠷⢶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⢀⣀⠀⢀⣀⣀⡀⣾⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣿⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⣧⠈⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣤⣤⣀⣀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢶⣾⣿⣤⣄⣸⣿⡀
⠀⢀⣤⣶⣾⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣿⣿⣷⡀
⣰⡿⠟⠛⠛⣿⡟⠛⠻⣶⠶⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⠶⠛⠉⠀⣿⣿⣇
⣿⡇⠀⠀⠀⡿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠉⠉⢻⡟⠷⠶⢦⣤⣀⠀⠀⣀⣀⣤⣤⡶⠾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿⢻⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⠹⣿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡇⠀⠀⠈⣿⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆
⣿⡇⠀⠀⠀⣿⣀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢠⡿⣻⣿⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢸⡇⢿⣹⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⣿⡇⠀⠀⢸⣇⠘⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠘⡯⢈⣉⡉⠉⢹⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⣿⡇⠀⠀⠀⢻⣶⢶⣶⡾⠃⠀⠀⠀⠀⠀⠀⢸⡇⣿⣉⣿⡄⠘⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢶⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟
⣿⡇⠀⠀⠀⠠⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣍⣉⣀⣀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⠀⠀⢰⣤⣦⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢙⣿⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠛⠻⣷
⠀⠀⠀⠻⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡟⠀⣀⠀⢹
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⣠⣤⣶⣾⣿⡿⠿⠛⠋⠉⣿⣿⡇⠀⠿⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣤⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣶⣶⣶⣤⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠛⠒⠒⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠛⣻⡇⠀⢠⣧⣤⣄⣀⣉⣉⣙⠛⡦⠀⠀⡾
⠀⠀⠀⠀⠀⢀⡤⠚⠫⠤⢤⡤⠖⠉⣀⣇⣀⣸⠁⠀⣠⠖⠛⠛⣿⠙⣇⠀⢸⡧⣤⣀⣀⣀⡀
⠀⠀⠀⣠⠖⢫⠀⠀⣀⠔⠋⢀⡴⠚⠧⠬⣭⣁⣀⠀⠛⠉⠉⠙⠃⠀⣇⠀⣼⠀⣠⠞⠉⢉⡽⠛⠓⠲⢶⣤⣀
⠀⠀⢻⠛⢲⠶⠷⣾⢤⣤⣞⣁⣀⣀⡀⠀⠀⠀⠈⠉⣩⠿⣖⣒⠒⠶⠬⢭⡵⠊⢀⣠⠞⠉⠀⠀⠀⢀⣸⡶⢿
⠀⠀⢸⡴⠋⠀⠀⡇⠀⢸⠁⠀⠉⠉⠙⠛⠛⠓⠶⢾⠧⣤⣤⣏⣀⣀⡴⠋⢀⡴⠋⠈⢉⣑⣒⣤⡴⠋⠁⠑⣾
⠀⠀⢸⡇⠀⠀⠀⠁⠀⢸⠀⣴⠒⠋⢻⠀⠀⠀⠀⠘⣆⠀⢹⡀⠈⢹⠉⠛⡿⠶⡿⠿⣭⡽⢻⠃⠁⠀⠀⠀⣾
⠀⠀⠘⣧⠤⡄⠀⠀⠀⢸⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠸⠤⠶⠃⠀⢸⠀⠀⡏⠀⠱⡀⠀⣇⡞⠀⢸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⣧⠀⢸⠀⢸⡇⢸⡇⢀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠙⢦⣿⠃⠀⠸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠶⠒⠦⣀⠀⣸⠀⠀⡇⣤⠖⢲⠀⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣿⠤⠼⠀⢸⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠀⢦⠈⣇⣿⠀⠀⢷⢹⠀⢸⡄⣿⠀⠀⢠⠀⠀⠀⢸⡀
⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⣿⠉⠉⠉⠉⠉⠉⡇⠘⣄⠢⠤⠜⢀⡏⣿⠀⠀⢸⠈⠷⠴⠇⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⢹⠒⠒⠒⠒⠒⠚⠁⠀⠈⠓⠒⠚⠋⠀⣿⠀⠀⢸⠑⠲⡄⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣧⡀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⡄⢸⠀⢻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⣧⠈⢢⡀⢻⠀⢸⡄⠀⠀⡏⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⣀⡤⠋⠀⢸⠀⠀⠀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠙⠓⠲⠷⢼⣤⣤⣇⣀⠀⠻⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⡇⠀⢀⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠶⠤⣤⣀⡀⠀⠀⠀⣾⠀⠀⢸⠀⠀⠀⢀⣼⣇⠀⠀⡇⢀⣾⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣿⣀⡀⢸⡆⢀⡴⠋⢸⡇⢳⣤⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠶⠦⠼⠗⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠤⠤⠤⢀⡈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠤⣤⢤⣤⣠⣀⣀⣀⣀⢀⠂⣐⣁⣀⠀⠀⠀⠈⡄⠰
⠀⠀⡠⠒⠚⠛⠙⠛⠒⠚⠛⠻⠯⠿⠲⠶⠤⠤⠤⠤⣛⣲⣿⣃⠉⡉⣉⢉⣉⣷⣼⡷⠓⠒⠒⠒⠶⠴⠀⠒⠖⡦⠤⠤⠤⠤⢤⣤⣀⠀⠀⡀
⠀⠘⠐⠈⠉⠉⠁⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠤⠤⠤⠤⠀⠀⣀⢀⣉⣉⣁⠉⠈⠁⠉⠉⠃⠓⠒⠚⠛⠿⠿⠻⠘⢛⠀⢤⣀⡠⠤⠔⠈⠀⢮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠂⠐⠒⠒⠀⢄⠀⠈⢢⠈⠆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⢸⠀⠀⠀⠀⠀⡇
⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢺⠀⠀⠀⠀⠀⠠
⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⠀⠴⠀⠀⠀⠀⠀⣷
⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡄⠀⠀⠀⠀⢀⢸
⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⢇⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⢸⠸
⠀⠀⠁⠐⠨⠭⠍⠅⣐⣒⠂⠤⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⣸⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠒⠒⠬⠭⠄⣁⢐⣒⠀⠤⠤⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⡇⢀⣠⠴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠒⠂⠨⠭⠤⢀⣀⣒⠀⠠⠤⠀⠀⠴⠀⢀⣄⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⣤⡤⠤⠤⠶⢶⣦⣤⣶⣤⣤⣤⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣶⣯⣽⣽⣃⣀⠀⢀⣀⣉⠶⠖⠒⠈⠷⠿⠾⢭⣤⣞⣻⠿⢷⣶⣦⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⣸⡍⠉⢻⣿⣿⠷⠶⢦⢤⡤⢤⣀⣀⠀⠀⠈⠉⠙⠒⠢⢍⣟⣻⣿⠦⢀⡀⠐⠀⠲⠄⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠎⠯⠀⢸⢯⠀⢠⣴⣖⣶⣿⣿⠋⠙⠉⠺⠶⢦⣀⢀⣀⠄⢘⠩⠐⠒⠛⠀⠠⢄⡀⠀⠀⡀⢭⣽⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢹⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡾⣁⣀⢴⣮⣭⣤⣭⣶⣾⢿⣵⡚⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠄⡐⡬⢖⣦⡀⣼⣬⡀⠈⠁⠀⠀⠀⠀⠀⠀⣴⣿⢯⠡⠀⠸⠿⢧⣿⣿⠈⢿⣿⣿⣿⣿⠃⢸⡿⠃⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣄⢛⣩⣥⠄⠿⢿⠇⠀⠀⠀⠀⠀⢠⡄⠀⠰⣼⣧⠜⠀⢸⠀⠄⠈⠃⠀⠀⢹⢻⣿⡿⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢨⠈⠘⠛⠙⠀⢠⣤⠂⠀⠀⠀⠀⠀⢳⣬⡀⠀⠀⠀⠀⠀⣻⣿⡇⠀⠀⠀⠀⢼⢸⣿⠁⠀⠸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠸⡟⡳⡄⠀⠀⠀⠀⢈⠙⡃⠀⡶⠟⠀⠘⢸⣷⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⢀⣴⡇⠀⣬⠀⠀⠀⠀⠀⠀⠀⢹⡯⠖⠀⠀⠀⠀⢸⣾⠃⠠⢿⢟⠠⠀⢸⡏⠁⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢐⠞⡿⣿⡇⠀⣟⡇⢠⣶⣄⠀⠀⠀⣸⣔⠑⠾⠆⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⢸⡧⠄⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠿⠙⠗⠀⣏⡷⣿⡬⢉⢿⣦⡀⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⣿⠄⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⡀⠀⠀⠀⣧⡇⠈⠑⠿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣵⣀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢲⠦⡄⠸⠀⠀⢾⢿⣦⣄⠰⢸⣏⡀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣄⣀⠀⠀⠀⠀⠀⣆⣻⡄⣼⠀⢠⠀⠀⡴⣿⣄⣠⠞⠀⠀⣿⣗⠂⠀⣿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⣤⢀⣀⠀⠈⠉⠉⠉⠛⠀⡄⡇⠀⠈⠻⠀⠀⠀⣯⣅⠀⠀⡏⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠒⠆⠤⣄⡀⢀⣄⡇⠀⠀⠀⠀⡠⣶⣿⣬⡤⢠⡏⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠢⢤⣠⣎⣀⣿⣿⣿⣷⣼⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⠤⠤⠤⢄⣈⢑
⠀⠀⠀⠀⠀⣀⢀⣤⣤⢶⡤⢔⣠⣶⠤⢤⣼⣧⠀⢀⣀⣀⣨⣧⣀⣤⣄⣀⣤⣤⡤⣤⣤⣤⠄⡀
⠀⠀⠀⠀⣎⠀⢀⡠⠊⠁⠀⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣴⠋⠁⣄⠀⠀⠀⠆⡇⢱
⠀⠀⠀⠀⡁⡔⠁⠀⠀⣂⣀⣓⠀⠀⠈⢄⠁⠀⠀⠀⠀⠀⠀⠰⣂⡀⣹⠀⠀⠈⠑⢔⠁⢸⡇⢛
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠨⡂⠊⡇⢸
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠓⠀⠘⠀⢸
⠀⠀⠀⠀⡇⠀⠈⠚⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠩⢄⠄⠐⠞⠀⠀⢸
⠀⠀⠀⠀⠸⢄⠀⠀⢂⠀⠀⢸⠀⠀⠀⠀⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠑⠄⠀⠈⠚⠀⠀⠀⣆
⠀⠀⠀⠀⡆⢀⠠⡀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠄⠀⢠⡀⢠⡄⠀⢲⠀⢠⠀⠀⠀⠀⠀⠂⢱⡄⣾
⠀⠀⠀⠀⠑⠒⠒⠞⠒⠒⠀⠘⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣀⠀⡀⠀⠀⠄⣨⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠉⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡖⠒⠒⠒⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠶⠶⡶⠂
⠀⠀⠀⠀⠔⠁⢸⠀⠸⠉⠁⠀⠀⠈⠉⢙⠀⢁⠀⠀⠂⠀⠀⠀⠀⡰⡆⠀⠒⠒⠛⠓⠒⠛⠛⠛⠛⠃⠘⠓
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡆⠀⠀⠀⠉⢢⣠⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠁⠓⠒⠀⠀⠐⠂⠐⠀⠀⠀⠒⠚⠉⠈⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣾⡿⡿⣷⣆⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣶⣶⣶⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠊⠉⠁⠉⠐⠉⠈⠁⠈⠁⠈⠀⠈⠫⡉⢻⡇⠀⠀⠀⢠⡟⠉⠉⠉⣿⠛⠉⠉⠉⠙⠋⣿⠋⠋⠉⠂
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⢀⣸⡗⠈⠀⠈⣉⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠙
⠀⠀⠀⠀⡇⠀⠀⡟⡷⠀⠀⠀⠀⠀⢟⢻⠀⠀⠀⢠⠄⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⣤⣤⠀⢠⣤⣤⣤⣤⣤⡄⢀⣤⣤⣤⢠⠀⠀⠀⠘⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢨⡆⠀⠀⠀⠘⡧⠁⠀⠀⣉⡀⠀⠀⠀⠀⠀⣉⡁⠀⠠⢤
⠀⠀⠀⠀⠄⠙⠿⠇⠸⠿⠿⠿⠿⠿⠇⢸⠿⢟⣁⠝⠁⠀⠀⠀⠀⠑⠀⠀⠀⠿⠒⠂⠀⠀⠀⠀⠿⠗⠂

⠀⠀⠀⠠⠠⠲⠆⠀⠀⠐⠶⠤⠤⠀⠀⠀⠀⠠⠤⠤⠦⠀⠀⠠⠶⢤⢠⢀
⠀⠀⠈⠀⠐⠐⠀⡀⠀⠀⠂⠠⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⣀⢁⣭⠉⠉⢉⣥⢉⡈⡂⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⢒⢒⠠⠀⢂⠀⣒⡒⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠉⠉⠉⠉⠁⠈⠀⠅⠀⠀⠀⠁⠉⠀⠀⠈⠉⠁
⠀⠀⢀⠀⠈⠀⠀⠠⠀⠠⠈⠀⠀⢀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠄⠦⡆⠀⠀⠀⢠⠄⢠⡄⠠⠀⠠⠀⠦⠀⠀⠀⢢⠠⠄
⠀⠀⠀⠰⠤⠰⠀⠀⠀⠈⠇⠤⠤⠀⠀⠀⠘⠀⠤⠀⠀⠀⠀⠀⠠⠰⠠⠁⠀⠀⠀⠄⠀⠀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠜⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠐⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢀⡇⠀⠀⠀⠀⠀⠀⢀⢀⡇⠀⢀⣇⠀⡀
⠀⠀⠀⠤⠐⠦⠂⠀⠀⠀⠦⠆⠠⠄⠘⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠡⠔⠀⠀⠐⠦⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠤⢶⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠠⠀⠈⠆
⠀⠀⠀⠀⢠⠤⠤⡾⠟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠰⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠘⠈⠂⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠛⣝⢛⣉⡛⣻⣛⣹⡟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣶⣶⣶⣶⣶⣶⣶⡦⠀⣀⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣤⠾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠙⠛⠋⠁⠀⠀⠙⠛⠋⠁⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⣠⣤⣈⣉⠛⠲⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣄⡉⠙⠳⠦⣤⣀⡉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠈⠙⠒⠦⣤⣈⠉⠓⠦⠤⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠉⡿⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⣸⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⡶⠖⠒⠲⠶⠶⣟⠙⢧⣼⡀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠈⠙⠳⢤⣀⠀⠀⠈⠳⢦⣈⠙⠳⢦⣀⠀⠀⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠈⠙⠢⢄⣉⠛⠶⢿⣼⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠉⠛⠶⣤⣀⣽⠂
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⠳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⠉⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⡶⠚⠛⠋⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⠶⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⣆⣀⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⣤⣿⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣴⠒⠲⠶⣶⠀⠀⠀⣿⠀⠀⠀⠀⣿⢁⣀⣽⡆⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⣻⡆⣀⣀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠸⢧⣤⣹⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⡀⠰⢯⣀⠉⠻⣷⣜⡳⠶⠶⠛⠀⠀⠀⣿⠀⠀⠀⣼⢛⢳⡄⢸⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠉⠉⠉⣇⠀⠀⠈⠓⠶⣬⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⢽⢸⡇⢸⣿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣌⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣿⢟⣦⣾⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⢤⡀⠀⠀⠀⠀⠀⢰⠛⢉⠙⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡿⣏⣁⣾⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⠷⢦⣀⠀⠀⠘⣧⣍⣠⠟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣧⣬⡽⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢘⣇⣠⢶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣄⠀⠀⣿⠀⠀⢀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⠿⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠄⣐⣂⣴⣦⠀⠀⢀⣀⣠⠤⢴⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠟⢛⣋⠩⠥⠶⣿⣈⠙⣻⣥⣜⣷⢞⣿⠛⠛⡶⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⣴⠖⣿⡉⣉⣠⣤⠴⠖⠛⠋⠛⠦⣄⠉⠫⣤⠀⠂⠚⠉⢻⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠶⢒⢞⡽⣷⣤⡔⠛⠻⡛⠉⢁⣀⣠⠤⠖⠒⠛⠀⠉⢻⠀⠙⠀⠀⠀⠀⠀⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣧⡶⢺⢿⡿⠏⠘⢙⢦⡀⠈⠋⠉⢁⣠⣤⣄⠀⠀⠀⠀⠸⡀⠀⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣿⡿⡔⢀⠢⠉⠀⠀⠈⣧⠀⠠⠀⣏⢘⣧⢼⡆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢹⡯⣶⠶⠂⠀⠀⠀⠀⢸⠀⠀⠂⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⠸⡇⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⣠⠦⡀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢻⡄⣷⠀⠀⠀⠀⠀⠀⢹⠀⠘⡄⠀⠀⠀⡴⡋⢙⣿⣿⡆⢠⠀⢸⠀⠀⢰⣐⣶⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠜⡇⢹⡆⠀⠀⠀⠀⠀⠈⡆⠀⠇⠀⠀⠀⠙⢾⣿⠟⠋⠀⠘⠀⠘⡀⠀⢾⣻⣿⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣿⠘⣇⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⢀⣀⣀⣀⣁⣀⣀⣀⠀⡀⠀⢇⠀⠈⠉⠉⡠⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡀⢻⠀⠀⠀⠀⠀⠀⠘⡀⠀⠇⢹⣐⣒⣢⣄⣹⣜⣿⠀⢸⠀⢸⠀⢀⣀⣸⣦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣧⠈⣇⠀⠀⠀⠀⠀⠀⠇⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠧⠜⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣆⣻⣄⡀⠀⠀⠀⠀⢸⠀⠘⣄⣠⡤⠴⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⡟⠉⢠⣤⡤⠶⠾⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⢀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠈⠃⢉⠡⠒⠄⢀⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠠⠙⠒⠐⠂⠀⣀⠀⠤⠂⠀⣠⡀⠉⠉⠉⠁⠰⠀⠉⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠀⠀⠀⡙⠄⠀⢀⣠⠆⠁⠀⠀⠀⠈⠀⠛⠉⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⣀⣤⣀⠤⣀⣀⠈⠚⠛⠋⣉⡀⠀⣠⡦⣤⣧⡤⣠⣔⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠤⡀⡄⣶⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣸⡏⠉⠉⠀⠀⠐⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠈⠙⠛⠛⠓⠀⢀⠀⣾⠒⠖⠶⢒⡲⠒⠒⠐⠶⠶⠶⢶⠖⠒⢀⡃⡀⠓⢶⡶⢶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡘⣀⣀⣻⠀⣟⠀⢠⠋⣧⠤⣶⢤⣤⠤⠶⠿⠴⠛⠀⠸⣇⠂⠐⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀⡀⠀⡀⣯⠀⣟⡀⠉⣼⣷⡄⠀⠀⠀⠀⠀⡀⠀⢀⣧⡄⠉⠉⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⢼⣦⣤⣤⣾⣷⣿⣁⣿⣾⣭⠀⢀⠀⣿⣷⣦⣤⣦⣤⣤⣷⣬⣸⣿⣿⣿⣷⣾⣾⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠌⠀⠘⠀⣯⣌⠁⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠉⢉⠉⠀⢀⢻⡃⠐⠺⠃⣔⠚⠟⠒⠒⠀⢀⡄⠀⠰⠒⠒⠂⠶⠶⠶⠶⠶⠟⠓⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠈⠀⠃⠉⢿⠁⠀⠈⠺⡙⠓⠂⠐⠓⠚⢓⣿⣦⣁⡼⠃⠀⣈⣀⣴⣸⣿⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢧⡄⠀⠰⢀⠀⡀⠀⠀⠀⠀⠀⢠⣲⠨⠙⢿⣿⠟⠛⠋⠀⠀⠀⠀⠀⠀⡉⢙⣿⣩⡜⠀⡛⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⣿⠟⡟⠛⠟⠷⠒⠶⠶⠶⠿⠷⠶⠶⠶⣴⣷⣾⡶⠿⢿⣵⣤⣶⡧⠿⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⡀⠀⢀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠄⣀⣀⣀⡰⠄⠀⢐⣀⣀⢘⢀⣟⣲⡀⡀⠀⣀⣀⠐⠀⡈⢷⣤⣤⣤⣄⡦⣷⣬⣧⠤⠤⡰⠚⡓⢺⣶⡾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠉⠉⠩⠀⣠⢻⡟⠋⢹⠀⣿⣿⠁⠀⠀⠤⢠⡀⠀⣥⢫⢨⣶⠀⠀⡆⢸⠗⠀⠀⠀⠄⠘⢋⠸⡿⣑⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠐⠀⠒⠓⠻⠋⠀⠘⠀⣿⡷⠀⠀⠀⠸⡿⡄⠈⠿⠾⣼⠟⠀⠀⣆⠈⢀⠀⠀⠀⠀⠀⠀⣴⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣾⡀⠀⠀⠀⠀⠀⠄⠂⠀⣰⡀⣿⣷⡄⠀⠀⠀⠈⠀⠒⠒⠫⡉⣀⣤⣿⣿⣿⣿⣿⣿⣴⣴⣾⣶⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣄⣀⣀⣄⣤⣤⣶⣴⣿⣆⣿⣿⣿⣷⣦⣦⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠟⠛⠛⠛⠛⠋⠽⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣋⣉⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣀⠀⠀⢄⠾⡄⠀⠀⠼⠖⠀⠄⠀⠐⠀⣠⠀⠀⠀⣿⣿⣿⠿⠿⣿⣿⣿⣛⣻⣿⣿⣇⣩⣿⣟⣿⣿⣿⠟⣿⣿
⠀⠀⠀⠉⢰⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⣿⣿⣿⣾⡏⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿
⠀⠀⢸⠀⠀⣧⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⣿⡧⣿⣿⠸⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠸⠀⠀⡜⣄⡮⢆⠈⢧⣀⠀⠠⠄⢀⠀⠘⠦⠛⠈⠀⠀⣿⣿⣇⣿⣿⣷⡻⣛⡹⣷⣜⢿⣿⣟⣻⣿⣿⣧⣛⣵⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⠄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⠼⠄⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠋⠁⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢠⣿⣄⡀⢄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⠀⠠⠀⠐⠀⠉⠉⠁⠘⠛⠛⢃⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣀⣔⠉⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣠⣶⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠰⠛⠛⠋⠀⠀⠀⠐⣀⣤⣥⣴⣾⣿⣿⣿⣿⣿⢿⠟⠟⠙⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⣾⠀⠀⣀⣀⣤⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⠀⢸⣿⣦⡙⢿⣿⣿⣿⣿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⠘⠀⢸⣿⠏⢙⡦⠈⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⠀⠀⠀⢸⣿⣇⣢⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⡀⠀⠀⠆⢻⣿⣷⣧⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠉⠉⠀⢀⣀⣽⡄⠉⠛⠲⠶⠓⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⠛⣿⡇⠀⠀⠀⠀⠀⠰⣿⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢠⠀⢰⢰⡄⠀⢹⣇⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣸⢀⠘⠀⠀⠀⢾⣟⠀⠀⠀⠀⠀⢸⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⢄⡎⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⣯⣿⠀⠀⡆⠀⠀⢸⡟⠀⠀⠀⠀⠀⢰⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣬⣾⡇⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣝⢿⡪⡇⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡸⣷⠃⠛⠀⠀⠞⡇⢠⠀⠀⠀⠀⢹⣿⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠚⠷⣦⡀⠘⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠑⣤⣠⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠀⠀⣼⡇⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠒⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⣤⠤⠶⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣫⡿⢳⣴⡶⠶⢶⠶⠶⢶⣶⣸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣡⢾⠏⠀⡞⠉⠉⠉⠉⠉⠉⠙⢻⡻⣄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⣞⣁⡾⠀⠀⣧⠤⠤⠤⠤⠤⠤⢤⣤⣧⣘⣧⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⣉⣩⣤⠶⠴⠼⡇⠀⠀⡷⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠞⡆⠈⣿⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢟⠉⠽⢛⣛⠿⣤⣀⠀⣀⣸⡇⠀⠀⣿⣖⠒⠖⠒⠖⠐⠖⠒⠖⠒⢒⣧⠀⠘⣷⢖⠲⠾⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣾⡎⠀⠀⠀⠀⠉⠒⠉⢿⠛⣆⣓⣒⠊⠯⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢌⣀⠩⠥⠞⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠼⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠎⠃⠂⠁⢠⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠙⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⢸⠁
⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⢸⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡖⡆⠀⠀⠀⠀⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠇⡇⠀⠀⠀⠀⡷⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢸⠀⠰⠀⣿
⠀⠀⠀⠀⠀⠀⠈⡇⠷⠀⠀⠀⠀⠀⡇⠀⢃⢸⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢧⢐⣦⡀⠀⠀⠀⠃⡇⠈⠈⡆⠸⡀⠘⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡉⠁⠀⠀⡄⠠⡀⠂⢁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠤⠤⠤⠤⣐⣒⣲⣿⣤⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠬⠥⠤⠤⠬⣽⣤⣥⡤⠶⠶⠿⠿⠗⠒⠒⠒⠒⠒⠒⠉⠉⠉⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⡶⠟⠉⠉⠙⣿⢶⠒⠋⠀⠀⢀⡀⠀⢀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠀⠀⠀⠀⠀⠙⠞⢑⡲⡤⠄⠲⣾⠀⠀⣴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠂⠀⠀⢀⠐⠀⢀⣀⣤⡴⠖⠻⠿⠁⠀⠀⠀⡠⢾⠀⠀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡠⢀⣀⣥⣦⡀⣿⠻⠍⠉⠀⠀⠀⠀⠀⠀⢰⠄⠀⠑⢾⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡟⡆⠀⠀⡀⣱⣾⣿⣿⣯⣧⠀⢿⠀⠀⣠⣾⠃⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⡇⠅⠀⠀⠆⢿⣿⣿⡟⢻⣧⠤⠞⠀⠸⠟⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⢸⡿⡟⠃⠀⢻⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣧⠶⠞⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣷⡧⡪⠀⢺⠀⠃⠀⠈⠹⠮⣄⡀⡄⠀⠐⠀⠀⠀⠄⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⢸⣿⠣⡗⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⢠⠀⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⡯⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣈⣼⡃
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣿⠠⡄⠀⢀⠀⢰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣶⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⡄⣿⣼⣧⡀⣾⡇⢸⡇⠈⠀⠀⠀⢀⣀⣤⣴⣾⣿⢿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣀⠃⣿⣿⣿⣠⡟⡇⠸⣧⣀⣤⣶⣾⣿⡛⠿⠍⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣙⠛⢿⣾⣿⣯⣯⣷⣷⣾⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⠈⠙⠻⠿⠿⠿⡛⠙⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠀⠀⠉⠉⠉⠉⣭⠺⣶⣶⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⣴⠦⠀⠠⡄⣵⠞⠛⠭⣵⣶⣦⣿⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⣀⠀⠀⠀⠀⠐⣿⣶⣴⠰⠂⢂⣬⣣⡐⣄⠀⠀⠉⠛⠿⣿⣷⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠯⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⣴⣿⣿⣿⣶⣿⣿⣶⣭⣉⣀⣰⣇⣉⠉⢻⣿⠪⡓⣄⠀⠀⠀⠀⠈⠑⠢⢌⣢⡀⠀⠀⠀⠀⢀⠔⠉⠈⠉⠁⢆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠅⠰⢿⣿⠗⠈⢪⡑⣄⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡠⠊⠁⠀⠀⠀⠀⠀⠸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡿⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢼⣷⡆⣁⠀⠀⠈⠻⠦⡱⢄⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣾⣿⣿⣿⣆⠀⠀⠀⠀⠈⠪⡓⣄⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠹⠄⠀⠀⠀⠀⠀⠈⠪⣁⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⢣
⠀⠀⠀⠀⠀⠀⢸⡠⠔⠂⠐⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⢆
⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠘⡄⠘⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠈⢆
⡔⠚⠓⠒⠒⠒⠒⠒⠒⠛⠛⠓⠚⠒⠚⠻⣿⣿⣿⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣠⣾⣧⡳⣄⠀⠀⠀⠀⠀⠈⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡇⠻⠿⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⡰⠋⠈⠉⢐⠎⢀⣤⠀⠀⢀⣴⣿⣿⣿⣿⣿⡮⡳⣄⠀⠀⠀⠀⠘⡄
⡇⠀⠀⠀⢀⡴⣶⢻⡟⢶⠦⣄⠀⠀⢸⡇⡇⠀⠀⠀⠈⠙⢧⡀⣀⣄⣠⣤⣋⠀⠀⠀⠠⢃⣴⠟⢁⡠⠊⠁⠈⠻⠿⠛⠛⢿⠃⠈⠪⡳⣄⠀⠀⠀⠘⡄
⡇⠀⠀⣴⠛⢶⠗⢺⡷⠚⣷⠚⢧⡀⢸⡇⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣙⠓⣛⠥⠊⠁⢁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠪⡳⣄⠀⠀⠘⡀
⡇⠀⢰⣇⣀⣿⣀⣸⣇⣀⣼⣄⣨⡇⢸⡇⡟⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄⠀⠰
⡇⠀⠈⣇⠀⢻⠀⢸⡇⠀⣼⠀⢰⠇⢸⡇⡇⠘⡄⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⡀⡠⠊⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣁⠔
⡇⠀⠀⠘⢿⣙⣏⢹⡏⣹⢏⡽⠋⠀⢸⡇⡟⣀⡼⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠁⠉⠀⠀⠀⠀⠀⠀⣠⠎
⡇⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⢸⡇⡏⡥⠢⡈⡄⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⡇⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸⡇⣷⡘⡄⠱⡈⠄⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⡘⣄⡱⢘⡀⠀⠀⠀⢿⡿⠀⠀⠀⣀⣀⠤⠒⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⢛⡡⠔⠊⠁⠈⠉⠁⠀⠈⠉⠈⠉⠁
⢇⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣠⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣩⣽⠅⠀⠀⢺⣷⡀⣀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡇⠤⠤⠄⣜⢸⣿⠙⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⡟⠀⠀⢀⣤⡄⠀⢹⣇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢸⢀⣿⢀⣀⣠⠤⠰⡖⣄⠀⢹⣄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡄⢈⢦⢚⣻⠏⠑⣷⠓⠀⠀⡉⠗⠀⠈⢷⠀⢸⠀⢀⣀⣀⣀⣠⢐⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢆⡻⢸⢃⠤⠐⢻⠒⠂⠉⠙⠛⠛⠒⢺⣷⠋⠡⠀⠀⠀⡀⠳⠃⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣙⡟⣿⣄⠤⡞⢠⠀⠀⠀⢸⠀⠀⠀⢫⡐⠌⠘⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣑⣾⣁⡿⠅⠒⡇⢸⠀⠀⠀⢸⡇⠠⠀⢾⡁⠀⣸⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢘⣽⠏⠈⣇⠀⠀⣁⡸⠤⡖⠒⠊⠉⠉⠀⣈⣿⡉⣉⡁⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⣟⡠⠤⠛⢃⣁⣀⣀⠀⠀⡦⠍⠊⠉⢠⠀⢹⡅⠠⠤⣱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢓⣯⣥⠴⡴⠋⣿⡵⠾⢻⡀⠀⢹⠁⠀⠀⢀⣴⢺⡷⠙⠂⠌⣇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢾⠟⠟⠚⡇⡄⠆⠁⠀⢸⡃⠀⠈⣇⣒⣚⣫⣅⣸⣿⣀⣠⣀⣞⡄⢃
⠀⠀⠀⠀⠀⠀⠰⢁⣆⣄⣸⠤⠦⠔⠃⠑⠋⠉⠉⣩⣁⣀⣀⣀⣀⡀⠀⢀⣸⣿⢀⣀⣀⠀⠀⠈⣆⡀
⠀⠀⠀⠀⠀⠤⠀⢈⣉⣠⡤⠀⠀⠀⠀⠀⠒⢀⣿⠂⠼⢖⡂⠔⠂⠀⠐⠒⢘⣿⣛⠂⠀⠀⠀⠀⢻⠤⡀
⠀⠀⢀⣰⡷⠚⠛⠛⠃⢤⡀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢣⠀⢔⠀⠀⠞⠶⡶⠀⠃⠀⠒⠒⠀⢺⡀⢰
⠀⠀⢾⠋⠁⣀⠀⡀⠀⠀⠱⡌⠀⠀⠀⠐⡞⠀⡠⡄⠀⠀⠀⣧⢀⣀⡀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠨⡇⢸
⠀⠂⢾⠀⣰⢇⢴⠘⡄⠀⠀⢳⠀⠀⠀⢺⡇⢰⢃⠿⠀⠀⡀⠛⠀⠘⡇⡟⠉⠀⠀⠀⠈⠘⡆⠗⠒⠣⡜
⠀⠀⠚⠀⢻⠸⣸⠀⠋⠐⡶⠊⠀⠀⠈⠉⠉⢹⠘⣤⠃⠒⠛⢲⠖⠒⠻⠷⠖⠒⡖⠒⠂⠲⠓⠓⡖
⠀⠀⠀⠀⣀⡄⠁⣰⣁⣼⣇⣀⣀⣀⣀⣀⣀⣀⣇⡈⣠⠃⣀⣼⣀⣀⣀⣀⣀⣀⣠⡀⣠⠃⢀⣼⠁
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠚⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡶⠶⣶⠴⢦⣆⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⣴⣆⣀⣀⢶⣶⣤⠤⢤⣤⣤⡠⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠅⠀⠈⠉⠉⠉⠉⠉⠹⠉⠁⢠⡾⠋⠡⡄⠀⠈⠀⠐⠢⡈⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣂⠀⢀⣧⠀⠀⠀⠠⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⢸⡇⠀⠀⠘⢦⡀⣈⡆⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠎⠀⠀⠀⠀⢥⠤⠤⣤⠀⠀⠀⠀⠓⠄⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢠⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢘⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠀⠈⠐⠉⠀⠀⠂⠀⠀⠀⠀⠀⡁⠀⠀⡼⠇⠀⠀⠀⠀⢀⢤⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢡⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠰⢨⢦⠀⠀⡇⠁⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⠀⢫⠀⠀⢀⠀⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠀⠁⠂⠀⠤⠤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⡳⠀⠀⠀⠠⢤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⡆⡀⠈⠑⠪⡅⢠⠌⠀⠀⠁⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢂⠀⠀⠀⣡⡀⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠈⠀⠀⠀⠁⠁⠀⠀⠀⠈⢂⠀⠀⠀⠉⠀⠀⠀⠀⢼⡄
⠀⠀⠀⠀⠀⢸⡀⢠⠑⢄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡠⡀⠑⡆⠀⠀⡆⠀⠐⠀⠀⠀⠀⠂⠈⢲⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣶⠤⢬⣀⡀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠀⠀⠀⡇⠀⠀⡇⠈⠀⠀⢀⠀⠀⡀⠀⢛⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠒⠒⠀⠀⠀⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⢴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠲⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠈⢣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠣⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠈⢺⠤⠤⠤⠤⠤⢤⡟⣻⣟⠛⠛⠛⠛⠛⠛⠛⢛⣿⣓⣆⣀⣀⣀⣀⣀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⣄⡞⠀⠀⠀⠀⠀⠀⠙⢣⣀⣓⣒⣒⡒⠒⠒⠒⣋⣀⡸⠃⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⡀⠈⠻⡲⠶⠦⠤⠤⠤⠤⣤⣤⣬⣭⣭⣭⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣠⣴⡧⠽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⣠⠴⠞⠻⡄⠈⢢⡀⠀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⣀⣀⣀⣠⣤⠤⢤⣶⣶⣄⣠⠤⠤⠒⢺
⠀⠀⠀⠀⡾⣽⡄⠀⠀⠀⠙⣆⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⢚⡿⠯⣍⡁⣀⣀⣠⠤⠼⠃⣈⡇⠀⠀⠀⢀⡼
⠀⠀⠀⠀⡗⠊⠉⠓⠶⢤⣀⠘⢦⠀⠈⠒⠒⠒⠒⠒⠂⠉⠉⠙⠺⠳⠤⣤⣭⠥⠴⠶⠖⠛⣋⣁⡤⡤⢴⡖⠋
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠙⠺⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠶⠒⠚⠋⠉⠁⣀⡤⠶⠏⣿
⠀⠀⠀⠀⡏⠑⠢⢄⡀⠀⢀⡀⠀⠀⠙⠶⠤⠤⢴⣖⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⠀⠀⢹
⠀⠀⠀⠀⢻⣆⠀⠀⠈⠑⡯⣭⠛⠶⢦⣄⡀⠀⠀⠈⠉⠓⢶⣮⣉⣑⣦⠴⠒⠉⠁⠀⠀⠀⠀⢀⣠⠤⠒⣻
⠀⠀⠀⠀⠀⠙⠲⠤⣄⡀⢣⡙⠤⣄⡀⠈⣙⣷⣦⠀⠀⠀⠀⠓⠤⠚⠁⠀⠀⠀⠀⣀⡠⠴⠚⠉⠀⠠⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣕⣒⠬⣍⣁⡼⠋⠑⠲⠤⣀⣀⣀⣀⡠⠤⠖⠚⠉⠁⠀⠀⠀⠀⣀⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣯⣅⣈⣹⣤⠤⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣒⠒⠲⠤⠄⠀⠁⢴⠀⠠⠴⠖⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⣀⣀⡤⠄⢛⣠⣶⠲⠒⠀⠀⠉⢶⢶⠖⠶⠂⠠⠜⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡀⡀⠐⡡⠀⠌⢂⠀⠀⠀⠀⠀⠀⠀⡐⠡⠀⠄⠢⠤⠀⢀⡀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⢆⠃⠀⢀⠞⠈⠉⠉⠉⠉⠉⠉⠑⡘⠀⠀⡀⠃⠀⠂⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢢⠑⠄⠠⠄⠠⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠠⠰⠆⠀⠀⠀⠈⣄⢀⠨⠋⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⠀⠀⢀⠔⠈⠀⠉⡠⠈⠀⠀⠀⠀⠀⢀⠠⠒⠈⣡⠔⠋⡙⠋⠙⠷⢦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠊⠀⠀⡔⠀⠀⠀⠀⠀⠱⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⢀⡁⣀⠀⡰⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢌⣀⣠⣤⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠁⠠⣄⡈⠉⠁⠀⠑⠒⠠⠤⠤⠤⢀⡀⠀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣿⡘⣷⠚⠹⣟⣿⣻⣭⡆⠀⠀⠀⠀⠐⡀⠀⠀⠀⠰⠀⠀⠉⠛⠃⠀⠄⢧⡄⢀⡀⠀⠀⠈⣉⠐⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⣀⡿⢹⣷⡈⣯⣀⢸⣿⠋⠀⠑⠀⠠⢀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠀⢈⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⢤⡄⠉⠀⢸⣿⣷⣿⡀⠈⡁⠀⡀⠀⠀⠀⠀⡟⡄⠀⠀⣸⡀⠀⠀⠀⠀⠀⡄⠃⢠⠯⢧⡀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠃⠀⠛⠒⠒⠚⠛⠛⠛⠛⠓⠀⠐⠀⠁⠂⠀⢰⠁⠈⠂⢚⢡⠀⠀⠀⢀⠀⠀⢱⢄⡴⢠⠏⠀⢪⡁⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠄⡀⠀⠅⠀⠀⠀⠀⠀⠀⠈⢣⠜⣥⢋⣠⠏⢀⡤⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠺⢦⣂⡰⣃⣪⠵⠚⢊⣴⠋⠀⠌
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡃⠀⠀⠀⠀⠀⠀⠴⡮⡟⠲⢤⡒⠮⣟⣼⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠂⣙⠀⠂⠐⣶⠉⠉⢙⣶⠋⠉⢹⣦⣀⣤⣤⣤⣤⣤⠧⠊⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⢤⠌⠑⠂⠈⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠓⠒⠉⠉⠓⠒⠟⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⠀⠀⠁


⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠰⠶⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠀⣴⣶⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⡄⠩⠩⠛⠿⢿⡿⠿⢿⣟⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢠⣤⣶⠶⠶⠿⠟⠛⠛⠛⠋⠉⠉⠉⠁⠀⢀⣀⡤⠤⠾⠷⠤⣄⣀⠉⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠶⢶⣶⣶⡶⢶⣦⠅⠀⠀⠀⠀⠉⠙⠲⢦⣄⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣖⣒⠚⠛⠛⠓⠶⢶⣶⣶⣄⣀⡀⠀⠀⠀⠀⠈⠛⣶⡟⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢭⣭⢭⣶⣶⣾⡛⠋⠙⠒⠦⠄⠸⢷⣿⣿⣽⡏⢳⡄⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢠⡞⢡⠎⢉⣯⡉⢸⢀⣄⢀⣀⣀⠁⠀⠈⠙⠿⡋⠉⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡰⢋⣤⠃⠀⣾⣿⢁⡇⠆⢸⡘⠀⠀⢃⠏⠙⠆⢠⢌⠓⣄⠀⣰⠂⢀⡿⢠⢖⢲
⠀⠀⠀⠀⠀⠀⠀⣰⠅⢠⢜⠁⡼⠁⢀⣼⣿⠁⡸⡸⢑⠸⡇⠀⠀⢼⢀⠀⢸⢸⠄⠁⠸⠿⠟⠛⠛⠛⠻⣄⣀
⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠛⠓⠒⠚⠿⢥⣰⠃⣃⣤⣪⠇⣀⣀⣸⢰⣢⣨⠰⠤⠋⢑⢀⡀⠤⠐⠂⢻⣷
⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⣤⣴⠖⠺⣿⣦⠀⢀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣤⣾⣿⣧⡀⠄⠉⠠⠤⠤⠄⢴⡒⠒⠨⠉⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣶⣶⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢈⡟⠉⠉⣽⠋⠉⠉⡟⠙⢏⡀⠀⠀⠁⠕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡼⠁⠀⢶⣾⡧⠀⠀⢪⢆⠀⠀⠈⠮⠄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠿⠶⠶⠾⠷⠶⠶⣶⣶⡿⠶⠶⠶⠶⠶⠶⠶⠾⠾⠞⠁



⣿⣿⠛⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣦⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⠈⠀⠈⢸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠛⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠛⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀
⠀⠠⠾⠿⡤⢮⡌⠷⠺⠦⠶⣰⢾⡝⠿⠛⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠉⠉⠃⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠂⠀⠀⠀⠐⠚⠡⠋⠀⠉⠁⠁⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠤⠀⢄⣀⢀⠀⠤⠤⠤⠀⠄⠀⣠⣄⣤⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⡤⠄⢀⠀⠀⢀⠙⠉⠉⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢽⣯⡳⠀⠀⠀⠀⠂⠒⠀⠉⠀⠛⠃⠀⠈
⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢛⡷⡆⠁⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣆⠀⠀⠀⠀⠀⠈⣃⠀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣢⣍⡰⠀⠀⠀⠀⠠⠀⠀⠀⢤⣤⠀⠀⠘
⠀⠲⠄⠒⠠⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠉⠁⠈⠙⠀⠁⠀⠀⢀⡤⠤⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠀⠀⠀⠀⠤⠤⠤⠂⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦⠀⣀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠲⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⢠⡔⠒⣆⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠊⠅⠂⠀⠀⠀⠀⠀⢠⣌⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣤⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢀⠀⠀⠄⠠⠠⢤⢠⠀⠀⠀⡀⡒⡤⠀⠀⠀⠐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠶⠀⠈⠈
⠀⠀⠀⢀⠠⠄⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣀⠀⠴⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒
⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠄⠀⠒⠦⠤⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠐⢈⠔⠀⠀⠈⠐⢄⠀⠀⡐⠁⢀⠀⠀⠀⠀⠢⡀⢀⠄⢈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⡠⠔⠁⠀⠀⠀⠀⠀⠀⢈⠊⡀⠐⠂⠀⠀⠀⠀⠀⠸⠁⢀⠇⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⠂⠁⠀⠀⠀⠀⠀⠀⡀⠀⠰⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⡐⠁⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠪⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡠⠑⠄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠀⢀⠠⠎⠠⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠈⠴⠋⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡂⠀⠀⠄⠻⠠⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠰⡗⠀⠀⠀⠐⡛
⠀⠐⡂⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⠀⠀⠀⠀⣀⠀⣀⣠⣀⠀⢀⠀⠀⠀⠐⠀⠀⢠⣦⠀⠀⠀⠊⠉⠀⠀⠀⢿⠅
⠀⠼⠷⠀⠀⢨⣿⠀⠀⢰⡆⠀⠺⠃⠀⠀⠀⢠⣞⠱⠀⠈⢰⠁⡌⠈⠀⢰⠀⠀⠀⡆⠀⡄⢁⠀⠀⣠⡀⠀⢸⡯⠀⠀⠀⢀⡄⠀⠀⠀⣹⡁
⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠋⠈⠉⠉⠉⠉⠆⠉⠉⠉⠉⠄⠈⠉⠉⠉⠁⠸⠀⠀⢾⠆⠀⠙⠏⠀⠀⠀⠼⠿⠀⠀⠀⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⣤⣀⠒⠀⠀⠀⠀⠆⠃⠀⠀⠀⠀⣀⡄⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠚⣇⢈⡿⠂⠀⠀⠀⠠⠄⢻⠉⢹⠓⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠐⠒⠒⠛⠚⠓⠀⠀⠂⠀⠐⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄






⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡶⠛⠛⠛⠛⠛⠛⠛⢳⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⡇
⠀⣠⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⢦⡀
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢹
⣿⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⢰⡏⠉⠉⠉⠉⠉⠉⣷⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢾⣧⣤⣤⣤⣤⣤⣤⣿⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⡴⠶⠶⠶⠶⠬⠭⣭⡉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢸⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⠛⠿⠿⠿⣭⣿⠿⢷⣶⣶⣶⣶⣤⠤⣤⣤⣧⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣤⠤⠤⢤⣤⣤⣄⣀⣈⣉⠁⠒⠒⠚⠯⠭⢍⣓⣒⣶⣬⣍⡙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠴⣃⡟⢀⡤⠤⠤⠤⣤⣈⠉⠉⠉⠛⠛⠒⠶⠦⣤⣄⣀⣀⡴⠟⢻⢩⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠉⠀⠈⠉⠛⠛⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢏⣿⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⣠⡴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠦⠤⠤⢽⡀⣸⣏⡉⠉⠒⠒⠒⠲⠦⢤⡾⠻⣦⣀⣀⡀⠀⣿⠀⡀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠙⠉⠀⠉⠉⠉⠉⠐⠒⠒⠺⢧⣤⠿⠤⢀⣉⣉⣿⠀⡇⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡧⠠⠤⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡇⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡗⠒⠂⠰⡇⢀⣿⠀⠀⣀⠁⠀⠐⠒⠀⣴⠶⢶⡄⠀⣀⠀⢸⡇⠀⡇⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⢻⣀⣼⠃⠤⠀⠀⢼⡇⢰⢷⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡇⢸⢸⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠱⠋⡗⠀⠀⠀⢸⡇⢸⢸⠃⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢫⣱⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠞⠀⠀⠀⢀⣸⠀⠘⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢲⠒⣖⠒⡶⠒⠒⠶⠤⢤⣄⣀⡀⢀⡴⢚⡭⣿⠀⡇⢸⠀⣆⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⢧⣸⣍⣹⠟⠓⠦⠿⣤⣅⡼⠶⠯⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠴⡶⠚⠻⠷⢶⡶⡿⣯⣿⡧⠓⠆⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⢴⣿⣶⠖⠲⣶⡟⠷⡮⡦⢶⠟⠙⡷⠦⠀⢿⣃⡿⠛⠁⠈⡇⠀⢀⣤⢶⣶⢶⡄⠤⠔⠐⡦⡔⠂⠈⠀⡼⠉⠉⠨⠏⣩⡟⢓⣖⣲⡶
⠈⠛⠉⠉⠙⠯⣁⡀⠄⡐⠙⣦⣸⠛⣢⣤⠀⠉⡇⠀⠀⠀⠰⣦⣈⡹⠜⠋⠀⡠⠀⠠⠾⢋⠐⡡⢂⠈⠀⡡⠊⢱⢾⡉⠉⣵⠏⠁
⠀⠀⠀⠀⠀⠀⢌⠀⢈⡐⠷⢀⢀⡼⠁⠉⠀⢀⣠⠀⠠⡀⠘⠚⠉⠓⠲⣤⣄⡀⠀⠀⠁⠀⠀⡤⠁⢀⠜⢑⣡⠡⠏
⠀⠀⠀⠀⠀⠀⠈⢺⡌⢉⠤⠥⡎⠀⠀⠀⠀⠩⠁⠉⠀⡙⡇⠀⣀⠀⠤⠀⠀⠀⠑⣀⡀⠀⠊⢀⠞⠁⠀⢨⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢤⡄⠀⠀⠀⠀⠀⠀⠄⠈⠀⢰⡇⠀⠠⠴⠂⠀⠀⠀⠀⠀⠈⠱⣴⠉⠠⣀⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣖⢖⣄⡀⠀⠀⠀⠀⠘⣿⠀⠂⠀⣠⠄⠒⠒⠉⠁⠀⠀⠂⠀⠀⠈⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡱⠋⡤⢚⠀⠀⠀⠀⢀⢿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠞⠀⠈⠅⠀⠀⠀⠀⢼⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⣢⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠂⠁⠀⠀⠀⠀⢺⡇⢀⠀⣇⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢵⣋⣀⠐⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⠀⠀⠀⠀⠸⣥⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠁⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠂⣻⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣾⠀⠅⢸⡀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡜⢧⣀⢀⣉⣄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠒⠒⠒⠓⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⠂⠐⠒⠒⠛⠻⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠤⠤⠤⣶⣿⣵⡶⠤⠤⠴⠶⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠳⢦⣄⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣔⣀⣈⣁⣀⣀
⠀⠀⠀⠈⡿⣿⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⣭⣉⣉⣉⣉⣩⣭⣭⡭⠥⠶⠶⠖⠂⡀
⠀⠀⠀⣸⠁⠙⣧⣀⠹⠦⠀⠤⠤⠤⠤⠤⠤⠤⠤⠿⠟⠛⠛⠋⢉⡉⠁⠀⣿⣿⠛⣿⣿⡄⠀⠀⣿⣷⠤⢄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠒⠲⠶⠶⠤⢤⣤⣤⣤⣤⣤⣼⣿⣷⣤⣤⣄⣀⣹⣛⣂⣙⣹⣇⣀⣻⣤⣤⣴⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢛⡿⠿⠿⠿⠛⢻⡟⡭⠛⠀⠙⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⢿⣿⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠤
⠀⣀⡀⠀⠀⣸⠃⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠛⠛⣛⣻⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠈⠙⠳⢶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⣀⡀⢀⡀
⠀⠀⠀⢀⡟⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣦
⠀⠀⠀⠚⠁⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⢿⡟⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠯⠭⠍⠉⠉⠙⠋⠘⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⣀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⢀⡔⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠉⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⡀⠐⠁⠀⠀⠔⠃⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠒⠒⠚⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠠⠄⠐⠈⠀⢀⠠⠐⠉⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠈⠀⢀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠶⠆⣢⣀⡄⣤⡲⠄⠀⠀⠀⡆⠀⡠⠊⡠⠐⢀⠈⢀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡤⠤⠞⠀⠐⠀⠈⠁⠰⠀⣀⡁⢨⠀⠀⢠⠌⠠⠊⠀⠀⠀⠐⢄⠹⠀⢀⡌⠀⠐⡀⡀⠐⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⣨⠶⠛⠓⠶⣕⠉⠀⠀⠀⠀⠀⠘⢀⠁⠀⠀⠀⠀⠀⡐⠙⠰⠒⠆⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠆⠀⠈⣼⠃⠀⠀⠀⠀⠹⡷⠀⠀⠀⢀⠀⠐⣎⠂⠀⠀⠀⠀⠀⠑⣣⠒⠐⠀⠀⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣸⢗⠀⣀⠀⠤⡄⠀⠉⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢀⣀⣝⣲⠦⢴⠞⠋⠈⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⣀⢀⡁⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢏⠁⠀⠁⠈⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡑⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠛⠛⠚⠓⢤⣀⠀⠀⠁⣀⡤⠛⠒⠚⠛⠓⠒⢶⣄⠠⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠈⠈⠀⠉
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣠⣜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⡄⠀⠀⠀⠀⠈⠂⠘⠁
⠀⠀⠀⠀⠀⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⠤⠴⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⣁⣀⣀⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣦⣀⠀⠀⠀⢀⣀⣴⢛⣀⢂⡀⡛⣦⣀⡀⠀⠀⠀⣀⣴⣃⣀⣀⠀⡐⡀⢀⡀⣀⢐⣀⣀⣐⣀⣂⡐⣀⡂⣐⣀⣀⡀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠨⠀⠜⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠄⠄



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢾⠊⠙⡂⠂⠤⠀⢰⠚⠂⠐⢳⠂⠀⠄⠚⠃⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⡄⠀⠀⡌⠀⠀⠀⠀⢃⠀⠀⠃⠀⠰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⣦⣄⣤⣤⣼⣄⣴⣥⡄⠀⠀⠀⣼⣤⣴⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣇⡇⠀⠀⠀⠀⢀⣘⠀⠀⠀⠀⠀⠀⢡⡏⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢀⠀⣼⡉⣿⡈⠃⠀⠀⠀⢀⡈⠋⢀⣀⣀⣀⣀⣀⣘⣓⡀⢀⣀⣰⣛⣱⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠐⡡⠀⠀⠒⠀⠀⠀⠠⠀⣤⢤⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠈⣰⡁⠞⠃⢐⡠⠤⠤⣶⡄⢾⢈⠀⠠⡠⠀⠀⠀⡠⠄⠀⠀⠀⠀⠁⡀⢀⡄⠀⡤⢤⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠌⣰⠂⢠⣯⡄⠈⢧⠀⣄⠈⣏⣿⠑⠀⣦⣀⠀⢀⠂⢲⣀⡠⠆⣀⠀⠀⣵⠂⠁⠷⢄⠰⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠌⠰⠃⠀⠀⣋⣀⣀⡴⠀⠇⣴⠁⣼⡎⢻⠀⣏⡀⠈⡶⠦⢼⡕⠓⣀⡆⢀⣿⡦⠀⣈⡼⠃⠰⠌⡁⠀⠀⠀⠀⠈⢄
⠀⠘⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠈⠛⠛⠛⠉⠛⠛⠛⠓⠓⠚⠛⠂⠚⠃⠀⠛⠂⠀⠈⠀⠀⠀⠀⠀⠐⠀⠐⠃⠀⠀⠈⠐⠀⠄⣀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⠀⠀⠀⢈⣰
⠀⣿⣿⣿⠀⠀⠀⣰⣾⣿⣿⠿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣦⡀⣤⣾⣿⡄
⠀⠛⠻⠿⠀⠀⠐⢹⣿⣿⠀⠀⠀⣿⣿⡗⠀⠀⠤⠠⠤⠤⠤⠤⠀⠀⠀⠀⠠⠦⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⢹⣿⣿⠁⠀⠈⢻⣿⡧⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣤⣿⣿⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠁

⣠⣾⣿⣿⣷⣶⣶⣦⣤⣤⣤⣀⣀⣀
⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠳⢶⣦⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⢀⣨⣭⣭⣛⣛⣶⠶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣥⣶⣿⢹⣿⣿⣿⣿⡉⢢⣐⢄⠉⠙⠛⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠟⢻⣿⡈⠹⡆⠹⣿⣿⠿⠿⢛⣛⣓⡱⣤⡀⠀⠀⠀⠉⠙⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠙⣻⡚⣿⣟⠛⠟⣂⡥⣴⣾⡿⠋⠀⣠⣴⡿⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠛⠛⠛⣡⡴⠚⡧⣾⣿⣋⣠⣴⣻⣿⣿⠟⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⡿⠛⣠⣴⣿⣷⣶⣴⣾⣮⣿⣿⣿⣿⣿⣿⠯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣀⢴⣫⠴⣻⢹⣁⢴⣿⣿⣿⣿⣿⣿⣟⣿⣿⣉⣹⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⢛⣷⡾⣟⣭⠶⣿⣯⣧⡱⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠻⢉⠀⠈⠳⡄⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠛⠉⠀⠀⠀⠀⡤⠛⠛⠻⠿⢿⠟⣿⠟⠉⠉⠲⡀⠀⠀⠀⠀⣈⡦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⣈⡦⠄⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣇⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠠⠴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠇
⠀⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⣀⣍⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢐⣟⡀⠀⠀⠀⠀⢀
⡀⣁⠑⠀⠀⠀⠀⢠⣽⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠸⣅⠀⠀⢠⠀⣼
⠟⠋⣢⠈⠁⠀⠀⣘⠾⠑⠀⢀⣿⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠐⠟⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢨⣿⠄
⣤⠄⢹⠃⠀⠀⠠⣍⣲⠛⡀⠘⠛⠛⠄⡀⠀⠀⠀⠀⠀⠀⡆⠉⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢲⡗⠀⠀⣾
⣧⣠⣾⠗⠘⠀⣘⣧⣿⢉⢄⠐⠀⠀⠀⠉⠀⠀⠀⠄⠀⠈⠛⠃⠀⠚⠚⠚⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⢶⡈⣠⣿
⣬⣤⠀⠀⠀⠐⣻⢃⣿⣸⠉⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣞⡿⢦⠀⠈⠉
⠁⢘⡀⠐⠒⠈⠛⠺⠒⠙⣿⣄⣀⣁⡀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁
⠋⢻⠉⡀⣀⡄⠁⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣥⡄
⠒⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈

⣀⠀⠁⠠⠀⢨
⠀⠀⠀⠀⠀⠀⠰⢦⠄⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢀⣹⣇⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠀⣰⣾⣿⣿⡶⠲⠖⠒⠀⠿⠿⠿⠟⠉⢭⣿⣦⠈⣉⣉⣛⡻⠶⢶⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⠟⢉⣻⠿⠿⠿⠛⢃⣀⣀⣉⣻⣿⣷⣬⣴⣶⠶⠶⠿⠿⠛⠛⠛⠉⢹⣿⡏⠉⠉⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣸⣿⠡⣿⠃⣼⣿⣿⣷⡞⠛⠛⠛⠛⠋⠉⠉⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠀⢿⡿⠿⢿⠇⠀⠀⠀⠀⡀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢀⠔⠀⠀⠀⠂⡀⢼⣿⠀⠂⠉⠀⠆⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠈⠈⠀⢰⣿⡇⠀⠀⠀⠀⢰⣾⠛⠐⠆⢀⠇⣿⣿⠀⠀⠂⡀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡯⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠁⠀⢀⣯⡇⠀⠀⠀⠀⠀⠑⠠⠤⠔⠉⠀⡟⣷⠀⠘⠃⡴⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢤⠴⠊⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡠⠖⠋⠉⠉⢠⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣷⡇⠀⡞⠁⠀⠀⠄⠀⠊⠀⠀⠀⣿⡇⢠⠂⡀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢂⠂⢸⡏⡇⠀⠃⠀⠈⢠⡠⠀⠀⠀⠀⠀⣿⡇⡆⢀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⣀⠠⠋⠀⢸⡇⡇⠀⠀⠓⠊⠁⠀⣀⣀⠀⠀⠀⣿⡇⠳⡊⡠⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⡇⠀⠀⠀⠀⠀⠞⠁⡠⠠⠀⠀⠾⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠀⠀⠀⢀⠔⣩⠀⠈⠁⠀⠈⡈⠇⠀⠀⠀⠀⠀⠤⢼⡄⠤⢀⣤⣴⠀⠀⢀⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⡇⠀⠀⠀⠀⠀⠎⢸⠁⠀⢠⠀⠂⠀⢧⠀⠀⠀⠀⠀⠀⠛⠁⠀⢀⣸⣿⣿⡾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⠿⠀⠀⠀⠀⠀⠀⢄⣀⣈⠀⠀⠀⢠⣶⡄⠀⠀⢀⣀⣤⣴⣶⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣤⣀⣀⣀⡀⠀⠀⠀⣀⣀⣠⣼⣿⣷⡶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⠶⠾⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠄⡀⠠⠡⠑⠀⠀⠀⠀⠀⠀⡤⠒⠈⠉⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠂⠀⠀⠂⢄
⠀⢠⠀⡠⠐⠉⠀⠈⣁⠀⣀⠀⢁⠄⠀⠀⠀⢀⠊⡠⠐⢏⠉⣉⢰⣴⠌⠆⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠈⠀⢂⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠃⠀⠀⠀⠀⠈⣩⣿⣷⣦⠀⠐⠀⠀⠀⠀⠀⡀⢀⠜⠀⠸⡓⠉⡑⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠨⠰⠀⠀⠀⠈⠀⠋⣷⠀⠀⢃⢁⠀⠀⠐⡠⢰⠦⢀⣀⢀⡨⠆⢃⠆⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⠄⠀⣆⠁⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠇⢁⠀⢀⠠⠀⠰⠋⠃⠁⢈⡄⠀⠀⠀⠇⠆⠄⣀⠀⠀⢀⠰⡘⠀⠀⠀⠀⡁⠠⠀⠉⠀⠀⠂⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠸⠰⠆⠁⣀⠠⠄⠀⠒⢈⣁⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠁
⠀⠀⠀⠀⠃⠉⣁⠠⠄⠀⠒⠉⠀⠀⠀⠀⣖⣦⠀⠤⠄⠀⠠⢀⠀⠀⠀⠀⢀⣠⡖⠂⠉⠉⠙⢂⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⢊
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⡰⠋⢰⡁⠀⠀⠀⢀⡟⠉⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢺⡉⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⢠⣔⣖⡐⢀⢍⠁⠀⠀⠀⣰⠁⠀⠀⢙⡂⠀⠀⢑⡄⡀⢀⠀⠸⡄⠀⠀⠀⢀⣀⣀⣀⣀⡉
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⡀⠀⠀⠠⠖⠈⠉⠀⠀⠀⠀⠘⡀⠀⠀⠀⡋⢦⢠⠤⠤⠇⠀⠀⢈⠤⠆⠐⠁⢠⡇⠀⢰⠑⠂⠀⠀⠀⠉⠂⡆
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⠆⠀⠓⠤⠀⠠⣤⠴⠖⠒⠂⠁⠀⠀⠀⢡⡜⠛⠒⣲⠀⠀⠀⠨⣂⠀⠀⢠⢊⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡀⠄⠀⠊⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠀⢃⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢸⠀⠀⠀⠀⠈⠳⢠⡊⠊⠀⡠⠘⠆⠀⠀⠀⠀⠀⠰⠇⢄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠈⣆⣈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠴⢄⣀⣔⣢⡤⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀
⠀⠀⠈⠐⠄⠀⢀⣀⠤⠐⢈⡀⠤⠊⠑⡀⠀⠀⠀⠀⣀⡀⢀⡜⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠐⠒⡄
⠀⠀⠀⠀⢸⣾⡏⣀⣴⣿⣾⣕⠄⠀⠀⢰⡀⠀⡠⠎⡡⠀⢀⣀⠉⠀⠚⠀⠀⠤⢀⣀⢀⡩⠂⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠰⠔⠊⠈⢟⢿⣿⣿⠇⠀⢀⠄⠁⠘⢄⡀⠀⠀⢀⠭⠬⢛⠀⠶⠀⣤⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠡⣀⢀⠤⠛⢋⡡⠄⠊⠀⠀⠀⠀⠀⠈⠁⠒⠾⠀⠀⠀⣇⣀⣀⠀⠤⠤⠐⠋⠢⠀⠀⠀⠀⠀⠀⠣⢀⣉⣉⣁⣀⣀⣀⣈⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣁⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠰⣉⠌⢀⠄⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⣀⡀⠀⠀⠀⠙⢄⠀⠀⠀⡀⠜⠐⠁
⠀⠀⢀⣀⠀⡠⠊⠓⢄⠸⠃⠀⠀⠀⠃⠀⠀⢀⣀⣀⡾⠉⠉⠙⠓⠐⠛⠑⠆⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠱⠠⠊
⠀⠀⠈⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠤⠤⠩⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠔⠁⠊⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠀⡊⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠉⠂⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠤⠄⠐⠒⠐⠀⠉⠀⠀⠀⠀⠛⠻⠶⠶⣦⣤⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⠿⠿⠿⠿⠿⢿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣀⣀⣠⡤⠔⠒
⠀⠀⠀⠀⠀⠀⣠⣾⡟⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠞⠛⢿⣧⡀
⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⡶⠾⠿⠿⠛⠛⠛⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠙⣿⣆
⠀⠀⢀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠹⣿⣿⣦⣤⡤⠴⠶⠖⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠘⣿⣆
⠀⢀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣶⣿⣷⣀⡀⠀⠀⣀⣠⣾⣿⡀⠀⠀⠀⠀⠘⣿⡆
⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠻⣷⡄⠀⠀⠀⠀⢹⣿⡄
⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⢳⡀⠀⢸⣟⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠿⠛⠛⠛⠛⢿⣿⣿⣿⣿⠟⠁⠀⠀⠘⠧⠀⠀⠀⠀⠀⣿⣧
⢸⣿⣿⠈⢿⣿⣿⡇⠀⠀⠸⣧⠀⢸⣿⡘⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡿⠻⠀⠀⠻⣿⡇⠀⠀⠀⣿⣧⢸⣿⣿⣿⣿⣿⡿⣿⣷⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⡇⠀⠀⠀⠀⠹⡇⠀⠀⠀⢹⣿⣾⣿⣇⢻⢻⣿⣷⠘⠏⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢇⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡈⡇⢿⣿⡀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠸⠸⣿⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢸⣿⣷⡀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⣿⡇⠀⠀⣿⣷⠸⣧⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⣿⣿⣷⡀⠀⠀⠀⠀⣧⠀⠀⠸⣿⣿⣿⡇⢠⠀⢸⣿⡄⣿⣧⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⢻⣿⣿⣷⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣷⢸⠀⢸⣿⣇⢿⣿⣧⡀⠀⠀⠈⠻⣿⡟⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠈⢿⣿⣿⣧⠀⠀⠀⣿⡇⠀⠀⢻⣿⣿⣿⠈⠀⢸⣿⣿⢸⣿⣿⣷⡀⠀⠀⠀⠙⣷⡀⠀⠈⠙⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠈⢿⣿⣿⣆⠀⠀⣿⣷⠀⠀⠸⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠛⠻⣿⣶⠲⠶⠶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠈⢿⣿⣿⡆⠀⢻⣿⡀⡀⠀⠈⢻⣿⣀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⣈⠻⣿⣦⡀⠈⠙⢿⣿⠳⣤⡀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⢻⣿⣿⣆⠀⢸⡏⠉⠓⠤⠤⠬⢿⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡌⠻⣿⣷⣄⠀⠈⠳⠀⠙⢲⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣤⣧⣍⣻⣷⣼⣇⣀⣀⣀⣀⡴⠋⠁⠀⠉⠛⣿⣿⣿⣤⣤⣄⣠⣀⣀⣀⣀⣴⣍⣻⣷⣴⣼⣿⠿⠿⠦⠤⠴⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣷⣶⣶⣶⣿⣿⠿⢿⣿⡿⠟⠉⢉⣁⣀⣠⣤⣤⣤⣬⣿⣿⣟⡛⠻⠿⣶⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣀⣉⠉⠉⠛⠛⠛⠛⠶⠶⠶⠶⠶⠶⠾⠿⠟⠛⠛⠛⠋⣉⣉⠭⠭⠿⠿⠿⠿⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣛⠛⠿⠿⠷⠶⠖⠚⠛⠛⢉⣉⣁⣠⣤⣤⡭⠶⠶⠂⠀⣀⣀⣀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⢉⣉⣩⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣶⣶⣶⣶⣶⣾⡿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⠂⠀⢋⢉⢀⣀⡨⠇⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⡤⠤⠠⢬⠁⠀⢠⠊⣙⢀⠙⡏⠉⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⡔⠐⠒⠰⠆⠘⠂⢩⠹⠤⠭⠭⡤⠷⠠⠴⠦⠀⠀⠀⠀⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣠⡀⡤⠦⠐⠒⢲⠀⠀⠀⠀⠇⠀⠀⠀⠛⠭⠝⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣅⠠⠤⠄⠴⠤⠶⠶⠚⠒⠒⠒⠒⠛⠒⠈⠉⠉⠉⠉⠉⠁⠒⠚⠣⠤⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑
⠐⠂⡀⠠⠔⠄⢀⡤⢀⡀⡔⡩⠭⠤⡤⡴⣒⠒⠒⠒⢢⠀⢀⡠⠐⠀⠉⠉⢉⠝⡉⠉⡉⡑⡄⢚⠉⠉⣉⠙⠛⠒⠦⡙⠢⡀⠀⠂⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠋⣀⡀⡟⠰⠡⠂⢉⡄⣀⡁⠑⢢⢠⠃⠠⢃⠔⠉⠀⠀⠙⣥⠉⠁⣀⢸⠈⢡⠈⣞⠉⡩⢓⡈⢁⠒⢜⢆⠘⣆⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢀⠀⠀⠀⠀⢘⠃⣟⠅⡔⠋⠀⣀⢈⡵⠩⡎⠀⣤⢀⠔⠚⠁⠈⠓⢿⠰⠷⠾⢼⠤⠺⡀⢉⡸⡐⠁⢀⠉⠑⡈⡌⡆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠻⣦⣛⡍⢋⡎⠀⣀⠇⠀⣅⠃⢄⠰⠆⠲⢣⣜⠀⠀⡀⣨⠦⠚⡇⠈⠀⠡⡋⠒⠞⡤⣊⠤⢾⡸⢇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣔⣞⣋⠈⠁⠨⢋⠔⡀⠀⢧⡳⡤⠔⠀⠐⠇⢠⡀⠀⢀⠉⢱⠉⡇⠐⠀⠐⢃⠴⣋⢭⣖⠬⠜⣃⠄⣈⠧⠤⠄⠐⠒⠢⢄
⠀⠀⠀⢀⡠⠤⠐⠂⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠃⠀⠀⣀⡇⠀⠠⠐⠀⢉⣁⠁⠬⠀⠼⠧⠳⠖⠒⠚⠒⠒⠒⠚⠒⠒⡛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⣰⣅⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠐⠉⢩⠋⠀⣌⠁⠀⠀⠀⣀⠒⠂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠒⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⢇⢀⠤⣄⠀⠀⠉⠉⠀⠀⠀⠀⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠤⢚⣓⣀⡒⠢⣄⠀⠀⠀⠀⢸
⠀⠀⠘⣄⡎⣹⡆⡆⠂⠂⠐⠀⢖⡭⡑⣼⠀⠀⠀⠀⠀⠀⢀⡠⢶⡶⠿⠿⠿⠶⣶⠤⡀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠜⡡⠚⠉⠁⠀⠉⠳⡌⢣⣀⣀⠀⠬⢀⡀
⠀⠀⠀⢹⠈⠛⠰⠅⡰⠀⠀⠀⢜⠠⡳⢹⡆⠀⠀⠀⠀⢠⢋⡜⠁⠀⠀⠀⠀⠀⠀⠙⣜⣆⠀⠀⠀⢸⠀⠀⠀⠀⡈⣼⠁⠀⡤⠒⠲⡄⠀⠸⡘⡇⠀⠀⠀⠀⡁
⠀⠀⠀⢀⠦⠤⠤⣀⡀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡞⠀⠀⡠⠒⠒⠒⡄⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠸⡁⠀⢀⡇⠀⢠⣇⡇⠀⣀⡠⠞
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⢒⠖⠂⠙⠒⠒⠒⠚⡷⡇⠀⠀⢇⠀⠀⡠⠁⠀⠀⢸⢸⠤⠤⠤⠴⠖⠒⠒⠂⠹⡹⡀⠀⠈⠛⠉⠀⠀⣸⠉⠉⠉
⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣁⣧⠀⠀⠀⠉⠉⠀⠀⠀⢠⣯⡋⣀⣀⣀⣀⣤⠤⠤⠤⠄⠓⠙⠢⢤⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠚⠦⠤⠤⠤⠔⠒⠊⠁⠈⠳⢤⣀⡀⠀⢀⣀⡴⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣠⢁⠾⠤⣀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⡀⣠⡄⢀⣸⡟⣤⠽⠷⠶⣄⡑⠢⡀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡪⠤⣄⠠⢸⣋⡻⢾⢳⣛⢅⠈⢢⠀⠀⠈⠿⠢⣌⠢⡀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡯⠊⠀⠀⠀⠀⠀⠉⢉⡡⣟⢴⢈⡦⢈⠀⠀⣠⠷⠀⠈⢢⡈⢆⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠐⠙⡄⠀⠀⠀⠀⠀⠧⡀⡨⠙⠭⡀⠈⠲⠋⠀⠀⠀⠀⠀⢱⡈⠆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠘⠁⠰⡆⢶⠀⠀⠀⠀⠀⠀⠰⣦⣱⣧⠸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡠⢼⠀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⢠⠋⡠⠤⠤⠤⣄⠉⡆⠀⠀⠱⣦⠀⢠⠊⠑⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠉⠙⠂⡀⠀⠂
⠀⠀⠀⠀⣠⡖⠚⠒⠒⠒⠒⠒⠚⠒⠓⠲⣄⠀⠀⠑⠒⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢰⠀⡇⠀⠆
⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠈⠻⠉⠉⠁⠂⢄⡀⠀⠀⠀⠀⠀⠈⠁⠈⡏⢰⠀⠰⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⡤⣽⣦⣤⠤⠤⠤⠤⢼⣤⠶⠶⠿⠒⡾⣛⣛⡛⢏⠉⠉⠉⠉⠉⠙⢹
⠀⠀⠀⡀⠀⠐⢄⣿⣀⣀⣀⣀⣀⣿⡠⠇⠀⠸⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣅⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⡊⣼⠋⡹⢻⣇⢱⠀⠀⠀⠀⠀⠏⡄
⠀⠀⠀⠇⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡇⠀⡤⠊⠸⢠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⠩⣾⣀⣼⠇⢸⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠒⠠⢀⣀⠀⠀⢃⡘⠀⣀⣨⡈⡁⠒⠠⠄⢀⡀⠀⢼⠄⠒⠊⠉⢆⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⢨
⡠⠒⢺⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠛⠛⢲⠢⠤⠄⠉⠉⠉⠡⠤⠄⠆⡗⠒⠀⠈⠁⠀⠐⠺⠀⠀⠀⠀⠈⢆⠀⠀⢠⠁⠀⠀⠀⢀⡠⠴⡞
⣗⡂⡈⠀⠀⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⡆⠀⢸⠐⡒⠂⠤⠤⡄⠀⠀⠤⡤⠐⢲⢁⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⡠⠧⡀⢾⠀⠀⠉⠁⠀⠀⠀⢁⠇
⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢨⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠈⡼⠀⠀⠀⠀⠀⠀⠀⢠⠠⡄⢀⠌⠰⠶⣿⣤⣄⠀⠀⠀⠀⠀⠀⢠⢠
⢇⢠⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡄⡠⢀⠄⠀⠇⠀⠀⠀⠇⠠⡀⠆⡆⠀⠀⠀⠀⠀⠀⢸⣀⡴⠋⠉⠁⠰⠏⠀⡁⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠉⢄⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠉⠀⠘⠄⠀⠀⠀⠀⠄⠄⠠⠃⢁⢢⣄⣀⣀⣀⣄⣒⣈⣅⣀⣀⣀⣀⣤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠧
⣶⣶⣾⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⣶⣾⣿⣿⣿⣷⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣾⡿⠿⠿⢿⣿⣶⣶⣶⣶⣤⣤⣤⣤⠤⠤⠴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⣰⣿⣤⣤⣶⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠹⠋⠛⠛⠋⠙⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⢀⣀⡀⠀⣿⠀⠀⠀⠀⢸⣿⡃⠀⣴⡆⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾
⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⡅⠀⠀⠀⣽⣿⠀⢸⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⡇⠀⠈⣉⣀⡄⠀⢸⣿⡇⠀⣿⡄⢀⡀⣤⣼⣿⡇⠀⣿⡃⠀⠀⠀⣿⣿⠀⢸⣿⢰⡇⠀⠀⣯⠀⠰⡀⠀⠀⣿⣿
⠀⠀⡯⣿⣷⣿⣿⣿⣷⠀⣸⣿⡇⠀⣽⣿⣿⣷⣿⣿⣿⡿⠀⣿⡃⠀⠀⠀⣿⣿⣀⣼⣿⣸⠿⠀⠀⣿⠀⢐⡌⠀⢀⣿⠛⡄
⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣬⣤⣤⣀⣀⣿⣿⣿⣿⣿⣿⣦⣤⣸⣿⣿⣿⣇⣸⣿⣿⣀⣷⣄
⠛⢛⣛⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⠿⠛⠛⠛⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⠿⢿⡟⠻⢿⣿⠛⡿⠛
⠀⠀⠁⡏⠉⢉⣛⠛⠏⠀⢸⣿⡇⠀⣿⠟⠉⠉⠛⢿⣿⡏⠀⣿⡃⠀⠀⠀⣿⣿⠀⢸⣿⠺⠟⠀⠀⣿⠀⠠⠈⠀⠀⣿⣿
⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠘⠛⠃⠀⣿⠀⠀⠀⠀⢸⣿⡅⠀⠿⠇⠀⠀⠀⠙⠉⠀⢸⣿⠄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠿⠿
⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃
⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⢀⡀
⣶⣾⣥⣤⣤⣤⣷⣶⣶⣾⣦⣤⣤⣼⣷⣶⣶⣾⣦⣤⣤⣴⣷⣶⣶⣾⣦⣤⣤⣼⣷⣶⣶⣾⣤⣤⣤⣴⣷⣶⣶⣶⣦⣤⣤⣴⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⠂⠀⢋⢉⣀⣀⡀⠇⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⢈⠀⠀⠀⠀⡤⠄⠤⢬⠀⠀⢠⠂⣉⢀⠉⡏⠉⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡔⠐⠂⠐⠄⠘⠂⠉⠹⠄⠭⠭⡤⠶⠀⠴⠆⠀⠀⠀⠀⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣠⡀⡤⠦⠐⠒⢲⠀⠀⠀⠀⡇⠀⠀⠀⠛⠭⠙⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡅⠠⠤⠤⠴⠀⠴⠶⠚⠒⠒⠒⠒⠛⠂⠈⠉⠉⠉⠉⠉⠁⠒⠚⠣⠤⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠐⠂⠀⠀⠔⠄⢀⡤⢀⡀⡔⡩⠭⠤⡤⢔⣒⠒⠒⠒⢠⠀⢀⠠⠐⠂⠉⠉⢉⠝⡉⠉⡉⠑⡀⢘⠉⠉⣉⠙⠛⠒⠦⡙⠢⡀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠉⣀⡀⡟⠰⠡⠂⢉⣄⣀⡁⠑⢢⢠⠃⠠⢃⠔⠉⠀⠀⠙⣥⠉⠀⡀⠸⠈⢡⠈⣞⠉⡩⠓⡉⢁⠒⢌⢆⠘⣆
⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⣟⠅⠔⠋⠀⢀⢈⡽⠩⡎⠀⡤⢀⠔⠊⠁⠈⠓⢷⠰⠷⠾⢼⠤⠚⡀⢉⡸⡐⠁⢀⠉⠁⡈⠌⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠻⢦⣛⡁⢋⡎⠀⣀⡇⠀⣧⠁⢄⠰⠄⠲⢣⠜⠀⠀⡀⣨⠦⠚⡇⢈⠀⠠⠃⠒⠖⡤⣊⠤⢲⡨⠇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣐⣞⣋⠈⠁⠨⢋⠔⡆⠀⣧⡳⡤⠜⠀⠐⠗⢤⣀⠀⢀⠉⢰⠉⡇⢸⠂⠀⢂⠴⣋⢭⣔⠬⠜⣃⠄⣈⠧⠤⠄⠐⠒⠢⢄
⠀⠀⠀⢀⡠⠤⠐⠒⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠃⠀⠘⣄⡇⠀⠠⠐⠀⢁⣁⠁⡼⠄⠼⠧⠳⠞⠒⠚⠒⠒⠒⠚⠒⠒⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⣰⣅⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠐⠉⢡⠋⠀⣋⠉⠀⠀⠀⢀⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠒⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠇⠀⠤⢄⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⠤⢘⣃⣀⡒⠢⣄⠀⠀⠀⠀⢸
⠀⠀⠘⡄⡎⢹⡆⠆⠂⠂⠐⠀⢖⡭⠑⣼⠀⠀⠀⠀⠀⠀⢀⡠⢶⡶⠿⠿⠿⠶⣖⠤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠜⡡⠚⠉⠀⠀⠈⠳⡌⢣⣀⣀⠠⠤⢀⡀
⠀⠀⠀⢹⠘⠛⠀⠄⠀⠀⠀⠀⢜⠠⡢⢹⡆⠀⠀⠀⠀⢠⢋⡞⠁⠀⠀⠀⠀⠀⠀⠑⣜⣆⠀⠀⠀⢸⠀⠀⠀⠀⡈⣼⠁⠀⡠⠒⠲⡄⠀⠸⡘⡇⠀⠀⠀⠀⡁
⠀⠀⠀⢀⠦⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡞⠀⠀⡠⠐⠒⠒⡄⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠸⡁⠀⢀⡇⠀⢠⣇⡇⠀⢀⡠⠞
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⢒⠖⠂⠙⠒⠒⠒⠚⠷⡇⠀⠀⢇⠀⠀⡠⠁⠀⠀⢸⢸⠤⠤⠤⠴⠖⠒⠒⠂⠻⡻⡀⠀⠈⠛⠉⠀⠀⣸⠉⠉⠉
⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠈⠉⠉⠀⠀⠀⢠⣯⠏⣀⣀⣀⣀⠠⠤⠤⠤⠄⠓⠙⠢⢤⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠚⠦⠤⠤⠤⠔⠒⠊⠁⠈⠳⢤⣀⣀⠀⢀⣀⡴⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡤⠴⠖⠒⠦⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣁⣠⠤⠦⠤⣤⣀⣙⡆⠪⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⢔⢄⣠⣤⣴⡶⡟⣷⠾⣛⣽⡿⠶⢖⣿⣷⠲⠶⢶⣶⣾⣿⣬⡶⠮⣁⣾⣶⠶⠶⠴⠶⡶⢦⡀⢠
⠀⠀⠀⠀⠀⡁⣿⣀⠀⡠⠜⠉⠀⢸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢰⠏⠁⢂⠀⠈⠉⢐⣐⠀⣷⢀
⠀⠀⠀⠀⠀⠀⣗⢠⠋⠀⠀⢐⣀⣘⡃⠀⠀⠡⣈⠁⠀⠀⠀⠀⠀⠀⠀⣂⣀⣘⡇⠀⠀⠑⢄⠊⠈⣿⠐⡏⢸
⠀⠀⠀⠀⠀⠃⣼⠁⠀⠀⠀⠈⠀⢸⡇⠀⠀⠂⠀⠀⢊⠀⠀⠀⠀⠀⠠⠁⠀⡸⠄⠀⠀⠀⠀⣑⠰⠹⠀⣿⢸
⠀⠀⠀⠀⠀⠄⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢱⠈⠀⠀⠀⠀⠀⠼⠄⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡆⡇⠀⠈⠶⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⢑⡀⢀⠀⢤⠂⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡂⢻⠄⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠂⠀⠈⠑⠲⠆⠀⠀⢀⡇⢸
⠀⠀⠀⠀⠀⡃⣧⠀⠢⡀⠐⠃⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⢀⠀⠄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠈⣷⢸
⠀⠀⠀⠀⠀⠃⢿⣐⣡⣈⣦⣤⣄⣀⣧⣀⡀⠀⠀⠀⠀⠀⠚⠃⠈⠁⠀⢸⠀⠀⠆⠀⠀⠀⠀⢀⠀⣹⣤⡟⢸
⠀⠀⠀⠀⠀⠠⠲⠠⠄⠘⠐⠀⠉⠘⠅⠐⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠺⠦⠴⠗⠒⠶⣤⣤⣥⡼⠋⠁⠖⢸
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠖⠲⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠳⣤⠶⠶⠶⢤⡞⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⢹⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢼⣇⡀⠀⠀⠀⢸⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⠖⠛⠓⠚⠓⢷⠒⠚⠛⠛⠛⠛⢛⠳⠶⢤⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡌⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠐⠈⡡⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠁⠀⠠⠀⣀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡷⠖⢦⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⠴⠒⠀⠈⣧
⠀⠀⠀⠀⠀⠀⡇⠀⣠⠏⠐⠛⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⡇⢠⠏⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡤⠀⠘⡇
⠀⠀⠀⠀⠀⠀⡇⠈⠛⠒⠦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠤⠤⠤⠤⠤⣶⣶⣒⣒⣦⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣼⠵⢒⣛⣉⣉⣉⣉⣉⣉⣀⣈⠉⢻⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠏⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡈⡇⠀⠀⠀⢀⣴⡚⠒⠒⣒⡒⣶⠶⣄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠘⡇⠉⣿⠉⠉⠙⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡧⠷⡶⡶⢶⠶⠷⠶⠿⠶⠶⠶⠿⠶⠿⢶⠶⣶⠶⠶⠶⡶⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠦⣄⣁⣠⣬⣭⡭⠭⠭⠭⡾⠃⠀⡜⢰⠃⠆⠉⠀⠀⠘⠀⠀⠘⠁⠀⠁⠈⡆⢸⠀⠀⠀⠹⡆⠹⡆
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠀⢿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠀⣿⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡛⢳⡆⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠞⠑⣠⠖⡁⠀⠀⠀⠀⠀⠀⠈⡄⠀⡇⠀⠙⡷⣟⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠘⡟⢻⠀⠀⠀⢸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠠⠊⡠⠊⡡⠂⡠⠀⠀⠀⠀⠀⡇⠀⣧⠀⠀⡇⢿⠀⠀⢰⠞⠋⢁⣹⠄⠀⠀⠀⠀⡇⢀⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣈⣀⣈⣀⣀⣀⣀⣀⣀⣷⠀⢸⡄⠀⣇⢸⡀⠀⠀⠳⠚⠉⠀⠀⠀⠀⠀⠀⡇⢸⠇⠀⠀⣾⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣭⡏⠉⠉⠉⠉⠉⠀⠘⢆⠀⢳⡀⢹⠈⡇⠠⠄⠀⠤⠀⠀⠤⠀⠀⠤⠀⡇⡼⠀⠀⠀⡟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠑⠦⠷⠼⢤⣽⣶⣶⣶⠒⠒⠛⠛⠛⠛⠛⣾⣿⣿⣶⠚⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣀⠀⠀⠠⠀⠀⠀⠂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⣀⡄⠂⠀⠀⠐⠀⠀⠀⡊⠀⠀⢆⡄⠀⠀⡇⠀⠢⠤
⠀⠀⠀⠀⠀⠄⠈⠀⠁⡠⡀⠀⠋⠡⠠⠂⠀⡀⠂⠈⠈⠁⠐⢀⣀⡝⠥⠄⣠⣇⠀⠀⠠⠆
⠀⠀⠀⠀⠀⢀⠁⠶⠩⠀⣰⠈⠀⠐⢈⡁⠀⠄⠀⠈⠀⠀⠀⡉⣛⠚⢂⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠠⠀⠈⢁⡈⠁⠨⠐⠂⠈⠍⠉⠭⠅⠩⠤⠀⡽⠁⠄⡠⡎⡅⠺⠏⢛⠓⠀⠔⠀⡀⠠⣠⡀
⠀⠀⠀⢠⠀⠀⠄⠈⢁⣠⠤⠐⢀⠑⠀⠁⠀⣠⡀⠀⠀⠀⠀⡄⢰⠅⢙⣋⠊⠉⠉⠫⡀⠈⠀⠉⠀⠉⠋
⠀⠀⢁⠀⢦⡀⠂⠡⠠⠧⠤⠄⠀⠁⠀⢁⡄⠉⠁⠀⠀⠀⠀⠁⢈⠀⠅⠿⠀⣀⠀⠱⡀⠀⢁
⠀⠀⠀⠐⠀⠈⢀⢰⠛⣻⣧⣃⢀⣚⢁⢈⠁⠐⠲⡉⠀⠈⠙⢓⣂⡖⢀⠀⠐⣩⠀⡲⠚⠊⠀⣀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠠⣀⠀⢀⠀⠀⢨⠀⠜⠨⠄⢣⢠⠁⠈⠈⠁⢆⠀⠠⢰⠅⠀⠘⣤⣧⠃⢡
⠀⠀⠀⡀⠆⠀⠀⠆⠈⠂⠐⠠⡴⠢⠀⢀⡄⠂⠐⠘⠆⢄⠐⠨⠀⠀⠡⣤⠦⠴⠆⠋⠉⣡⣼⣻⡀
⠀⠀⡜⠀⠑⡀⠀⠑⢀⢀⠀⢸⠂⠀⠀⠈⠉⠱⡠⢀⢀⣼⡆⣁⣤⣶⣦⠹⡄⠐⠠⠄⢪⣄⠀⠀⠅
⠀⠀⠇⠀⠀⡏⢰⠌⠆⡚⣎⠠⠀⠀⠀⠀⠀⢀⡀⠐⠈⠊⠑⠘⣷⣙⡼⠀⠹⣄⠒⠀⢀⠉⠵⠖
⠀⠀⠀⠀⠀⠀⠘⡆⠆⢱⡶⢴⠀⣠⡄⣀⠈⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⢆⢠⣀⣬⣱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⢁⡄⢠⣯⠀⡇⠀⠂⠈⠰⠤⢀⠀⠀⠁⢀⣀⣠⡤⠴⠀⠉⠂⠁⠃⠃
⠀⠀⠀⠀⠄⠠⡞⠉⠁⠀⠻⡀⣸⠇⠂⠀⠀⡰⢀⠀⠀⠈⠀⠐⠀⠭⠁⠀⠀⢀⡀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠈⠀⠎⢘⡈⠉⠠⡤⠂⠀⠀⠄⠀⠀⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠠⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠚

⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⣀⡀⠀⡀⠀⠀⡀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⡀⠀⠀⢀⡄
⠀⠀⠀⠀⠨⠀⢰⠀⠇⠀⡞⠀⠙⠀⡇⡰⠁⡇⢸⡀⢸⢠⠁⠈⠂⠀⠀⠀⠀⠠⡒⠞⠁⠀⠻⠱⠏⠁⠀⠁⠀⠀⠈⠀⠙⠾⠹⠄⠀⠈⢡⠀⡀
⠀⠀⠀⠀⢰⠀⠌⣼⢸⠀⡇⠀⠀⠀⡷⡄⠀⡇⢸⢁⠈⢸⠀⣀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⡀
⠀⠀⠀⠀⢨⠀⠀⡏⠈⡇⣷⠀⠀⡀⡇⠸⡀⡇⢸⠘⣇⢸⠀⠉⢣⠀⠀⠀⠲⠇⠀⠀⠀⠄⠎⢚⠀⡐⠀⠀⠀⠀⠀⠠⠔⠁⠀⠄⠆⠀⠀⠀⠦⠶
⠀⠀⠀⠀⠸⠀⠠⠇⠀⢱⠹⣄⡼⠃⠇⠀⠇⠇⢸⠀⠿⠘⢦⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⣀⠀⣀⠀⣀⠀⣠⠀⣈⡀⢈⣴⠀⢀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⡄
⠀⠀⠀⠀⠀⠏⢀⠳⢂⡞⡀⠀⡜⣠⢀⠜⠃⣠⠂⣰⢁⠜⡾⢁⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠉⠁⠀⠁⠉⠑⠉⠉⠉⠁⠈⠈⠈⠁⠉⠉⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡤⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⠀⠘⠡⠈⠀⠉⠩⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠘⠁⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠁⠀⠀⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⢲⠀⠰⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠐⣀⠀⠀⠠⠞⣃⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣤⢀⣢⡄⣀⣋⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⢾⣿⡃⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⡉⠀⠃⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠰⡀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⠄⢀⡤⣀⣀⣀⣀⠈⡀⠀⠀⠄⡨⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⡌⠉⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⣹⣦
⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣀⢁⣀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣀⠀⢎⡠⠚⡡⠀⠈⠉⠁⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠠⠇⡆⠀⢠⡄
⠀⠀⠀⠀⠘⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢣⡄⠀⠀⠀⠀⢘⡿⠿⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⢾⣇⠀⠠⠈⠀⠁⠀⠀⡠⢻⡄⠠⠤⣆⠁⠀⠁⢰⡄⠀⢸⡇⠀⠀⠀⠀⠠⠁⣶⣦⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡄
⠀⠀⠀⠾⡃⠘⠛⡀⠀⠁⠀⠀⠀⠀⢡⠿⣳⡶⠂⠠⡀⠄⢠⠁⠀⡖⠚⢽⡀⠀⠀⠴⢰⠀⠀⢹⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡀
⠀⠀⠀⢀⠰⢀⣢⡀⠠⣄⠀⠀⠠⠀⠀⡴⠿⠷⣀⣠⡇⡦⠸⠀⠀⡤⠀⠜⠘⡃⠀⠀⠘⢀⠀⢸⣿⡇⠀⠂⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠙⠒⠛⠓⠐⠊⢉⡉⢉⢉⠁⠈⠉⠉⠀⡤⠤⢥⡁⠉⠀⠀⠐⠂⠒⠀⠀⠠⠀⠀⠀⢸⣿⣇⠀⢀⠀⠀⠸⣚⢳⠦⢄⣀⠀⠀⠀⠀⠃⠄⠀⠂⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠏⠀⢀⠀⠀⠀⠉⡟⠀⠀⠈⠙⠛⠉⢇⠘⡂⠐⣏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠂⠐⠃⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡏⠉⠉⠉⠉⢹⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⢩⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡍⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⡾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⡀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⣿⣿⠏⠁⠀⠀⠘⠻⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡷⠀⢠⣿⣄⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢿⡇⠀⠀⠈⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢺⠁⠀⠀⠀⠀⢀⣀⣀⣀⢀⣤⣶⣴⣶⡇⢸⢟⣴⡾⢟⣥⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⠛⠾⠿⠿⠇⠰⠿⠿⢿⡾⠿⢯⣶⠆
⠙⠫⢿⣿⣻⣿⣿⡟⢋⡜⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢐⡺⠿⠏⠁⠀⠋⠀⠠⠊⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠼⠛⠀⠀⣤⠀⠀⠀⣀⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠉⠂⠀⢠⡿⠀⠀⠚⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠓
⠀⠀⠀⠀⠛⠃⠁⠤⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠈⠀⣀⣀⣀⠀⠀⠀⠀⠓⠀⠀⠐⠁
⠀⠀⢀⡉
⣀⣀⣁⣀⣀⣀⣀⣈⣉⣙⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠛⠛⠓⠒⠒⠶⠦⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠙⠛⢓⡿⠀⠀⣯⣤⣄⣈⣉⣉⣙⠛⢳⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢀⣠⠞⠯⠤⢤⣠⠔⠋⢁⣇⠀⢰⡇⠀⢀⠼⠛⠛⢻⠟⢻⠀⠀⡿⢤⣀⣀⣀⡀
⠀⠀⠀⣠⠴⢯⠁⠀⢀⡴⠊⢀⡠⠞⠧⢬⣍⣉⡀⠀⠛⠒⠛⠛⠛⠀⢿⠀⢠⡇⢀⡴⠋⠉⣹⠿⠓⠶⠶⣤⣄
⠀⠀⢻⠓⢲⡶⠷⣶⣥⣤⣔⣋⣀⣀⠀⠀⠀⠀⠉⠉⢙⡷⣒⣒⠶⠦⠬⠭⣩⠖⠉⣀⡴⠋⠁⠀⠀⠀⣀⣧⠾⡇
⠀⠀⢸⡤⠊⠀⠀⠇⠀⢸⡍⠉⠉⠉⠛⠛⠓⠶⠶⢶⣧⣤⣤⣞⣁⣀⣠⠞⠁⣠⠞⠉⠉⠑⠒⢤⣤⠞⠋⠙⢶⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢠⠖⠚⢳⡀⠀⠀⠀⠈⢧⠀⠘⡌⠉⢹⡟⠛⢾⠷⣶⣯⣭⣭⠿⡟⢸⠀⠀⠀⢨⡇
⠀⠀⠈⣇⠤⡄⠀⢰⠀⠀⡇⢸⡀⠀⠘⡇⠀⠀⠀⠀⠘⣦⠤⠿⠀⢸⡇⠀⢸⠀⠘⣄⠀⣿⢰⠃⠈⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⣷⠀⢸⠀⠀⡇⠀⡇⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠀⠀⠈⠦⣸⡟⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⢸⠀⢸⡄⠀⡇⠀⠛⠋⠉⠁⠀⠀⠀⢀⡤⠶⠤⣀⠀⢸⠀⠀⢸⢀⡤⠖⡄⢸⡇⠀⢸⠀⠀⠀⠘⡇
⠀⠀⠀⣿⣀⣼⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡎⡀⠀⢤⠈⢧⢸⠀⠀⢸⠸⡄⠀⡇⢸⡇⠀⢸⠀⠀⠀⠀⣇
⠀⠀⠀⢸⠀⠀⠀⠸⡇⠀⢸⠒⠒⠒⠒⠒⠉⡇⠀⣇⠣⣀⡠⠃⣸⢸⠀⠀⢸⡆⢳⠤⠿⢸⡇⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⢸⠶⠶⠶⠶⠶⠶⠇⠀⠈⠓⠤⠴⠚⠁⢸⠀⠀⢸⡗⠢⣄⠀⢸⡇⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠈⣧⡀⠀⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣷⣧⢸⡄⢸⡇⠀⠀⢸⠀⠀⠀⢻
⠀⠀⠀⠀⣏⠙⢦⠀⢸⠀⠸⡇⠀⠀⡞⠛⠉⠁⢸⡆⠀⠀⠀⠀⠀⢸⠀⠀⠸⣇⣁⡼⠀⠈⡇⠀⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠘⠶⠤⣷⣼⣄⣀⡇⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠀⠀⢰⡏⠁⠀⠀⠀⡇⠀⠀⢸⠀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠚⠶⠦⣤⣄⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢠⣧⠀⠀⢸⠀⢰⣧⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣼⡀⠀⠀⡇⠀⢀⡴⠃⡿⠳⣄⣸⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠷⠶⢧⣤⣀⣿⠴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⠤⠴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠀⢀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠠⠂⠀⠀⠸⠀⣰⠀⠀⠃⠀⠰⠀⠆⠠⠄⠀⠀⠀⠀⠤⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣚⣛⣿⣭⣍⣿⣿⣭⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⠞⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠡⠾⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⢤⣀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⠙⡷⢴⣿⣿⣿⣿⣷⠀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⡌⣀⡀⢀⣾⣿⣿⣿⣿⣶⠔⢂⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⣿⠁⠀⣿⣿⠲⠿⠿⠶⠶⠿⠿⠿⠿⠗⠒⠒⠒⠒⠺⣿⣏⠀⢈⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⣡⠀⠀⣱⠀⠀⠀⢀⠀⠀⠀⠀⣼⠏⠴⠶⠶⠶⠶⣾⠷⠶⠖⠖⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣿⠶⠛⣱⡟⢩⡾⢋⡿⣵⢟⣿⠃⢀⠆⠀⠀⠀⣼⢯⡷⢟⡟⣴⡯⠊⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠓⢤⣝⣀⣘⣣⡄⠁⠙⠁⡏⡠⠋⠀⠀⠀⢰⠃⠉⠀⠘⣱⡟⠑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⣦⠀⠀⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⢿⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡑⢦⡀⠈⠁⠀⠀⠀⠰⣿⢿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣠⡞⠁
⠀⠀⠀⠀⢠⡞⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠙⠲⣝⢶⣄⠀⠀⠀⢰⠶⠾⢾⢻⣶⣶⠶⡶⣤⡀⠀⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⣻⣦⡀⠸⠟⡟⠻⢺⢻⣿⠶⡧⣧⡇⠀⢀⣠⠾⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣶⣿⣿⣿⣿⣿⣿⣷⣷⣷⣾⣾⣿⣴⣶⣦⣤⣶⣿⣿⣦⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⢶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⢷⣤⡀⠈⢿⡏⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⡳⣤⡀⠀⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣼⣇⣴⠞⢟⣀⣸⡾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣛⢦⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣼⣶⣻⣿⣄⢻⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⣿⡽⣟⠉⠃⠉⠛
⠀⠀⠀⠀⢀⠀⢀⣼⣷⣤⣤⣀⣀⣀⣤⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢙⣦⠼⠛
⠀⠀⣄⡀⣬⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡶⢶⣦⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠉⢻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣭⢐⣦⠽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣶⣶⠿⢿⣿⣿⡿⠋⠁
⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⣠⣿⣍⣽⡶⠟⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⣍⠙⠛⠛⠙⠛⠁⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡃⠀⢀⣹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠻⢦⡈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢛⣏⣽⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⣏⡙⠷⣄⠙⢶⣼⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠟⢻⡯⠟⣪⣿⣿⡿⣿⣿⣷⣶⣦⣤⣤⣴⣶⣶⣿⣿⣿⣿⠛⢧⣌⠛⢦⣈⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⣱⣿⣿⡽⠁⣾⣿⡟⢻⣿⠟⣿⡟⢿⣿⣿⠿⠋⠳⣄⡙⠳⣤⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⠞⠀⢰⣿⡿⠀⣾⡟⢠⡿⠁⠈⣩⠴⠒⣉⡄⠈⠙⠾⡿⠋⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⢸⣿⣅⢀⣿⡇⣸⠿⠀⠀⠀⠒⣋⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣿⣿⡿⠋⠀⠀⢀⣾⡿⠇⠘⠃⠀⠈⠀⠀⠀⠀⠈⡡⠔⢊⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣩⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⡧⢤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠐⠐⠐⠋⠀⠀⠤⠀⣴⠒⠂⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠠⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⣋⠹⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠉⠉⠙⠻⠻⢟⣛⣃⣤⡔⠰⠶⠶⠂
⠀⠀⠀⠀⣹⠶⠶⠶⠶⠦⢤⣀⣉⣩⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠤⠴⠶⣾⡿⠛⠻⡛⡏⠓
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠈⠁⠁


⠀⠀⠀⠀⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⢂⣀⠀⠀⠀⠙⣠⣀⠀⠀⠀⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣄⣰⣆⣀⣤⣤⣶⡶⠿⠛⠁⠀⠀⠀⠀⢐⠀⣀⣀⣀⣀⣉⣡⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠠⢤⣄⣘⠃⣀⣩⣿⡿⠿⣿⣿⡍⣥⠀⠀⠀⠀⠀⠀⠀⠀⠦⠰⠃⠀⠉⢲⣼⣿⣶⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠏⠁⠀⠀⠀⠀⠉⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠠⠩⠅⠙⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⡠⠐⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠈⠀⠀⠂⠈⠉⠉⠐⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢱
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⡆⠄⠈⠀⠔⠒⠀⠀⢱⠒⠒⠒⠂⠈⡈⡆⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠣⣀⠀⠀⢀⣠⠏⠀⠈⠉⠁⠀⠠⣇⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠁⠓⠲⠬⠭⢭⣭⠥⣤⡤⠤⠤⠤⠀⠐⢰⠀⠐⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠁⠂⠠⣀⣠⣀⣤⣤⣈⣁⣀⣀⣃⣀⣀⣀⣀⣀⡸⠀⠁⠇⠀⠌⠀⠰⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⡄⠀⠀⠉⠉⣁⣀⣀⣀⣾⡇⢰⣷⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣶⡆⢸⠇⢀⡀⢠⡟⠛⠉⢉⢹⣿⠛⣿⡿⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⣥⣿⡄⣿⢃⣴⡟⢁⢰⡟⢹⡟⢀⣿⠃⠓⣺⡆⣾⠂⣾⣦⢾⣦
⠀⠀⠀⠀⠀⢠⣄⣀⢀⣀⣀⠹⣏⢸⣧⡻⠴⠛⠧⠿⠜⠧⠼⠧⢺⣿⢀⠀⠿⠄⠻⠤⣿⡗⠾⠟⠒⠶⠤⢴⠶⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇



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

⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠈⠁⠀⠁⠀⠀⠀⠈⢁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠀⠋⠀⠄⠨⠁⠀⠑⠃⠀⠀⠀⠀⡠⠐⠃⠉⠁⠄⠉⠉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠉⠉⠁⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⢀⡈⠀⠠⠠⠤⠠⠧⠤⠤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢐⠀⢨⠀⠤⠀⢀⢄⠀⠀⠠⡀⠀⢀⠀⣱⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⠀⠐⠤⢀⣐⣀⣀⣁⣀⣈⡀⠀⠂
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢢
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠎⠉⠉⢀⣴⣶⣶⣤⠀⠉⠉⠈⠙⢬⠉⠉⠙⡆⠀⢰⠉⣩⣿⣿⣍⠉⠉⠉⠉⠉⢢⠀⠀⠈⢆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠫⠄⣸⡄⠀⠆⠀⠠⠀⣥⠄⠀⡇⠀⢸⠸⣿⡿⠟⡙⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⢦
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠄⢠⣾⡅⠀⠀⠀⠀⠀⠊⠁⠀⡇⠀⢸⠀⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⢣
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⢹⡧⡄⠀⡀⠀⠀⠀⠄⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣋⠀⠀⣀⠀⢸⣁⣇⡀⣀⠀⣀⣤⣀⣠⠇⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⡁⠌⠃⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲
⠀⠀⢰⠀⠀⠀⠀⠀⢀⠠⢐⣂⠠⠤⢀⣐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⢶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠹
⠀⠀⢸⠀⠀⠀⡠⢊⠔⠊⠁⣀⣀⣤⣀⡀⠉⠒⢌⠢⡀⠀⠀⠀⠀⠀⢀⠔⠊⠁⣀⣀⣀⣀⡀⠉⠛⢕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⡏⢹⢸⠀⠀⡐⡰⠁⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠳⡐⡀⠀⠀⡐⡔⠁⢀⣴⣿⣿⣿⣿⣿⣿⣶⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣭
⣇⣈⣹⠀⠀⣱⠁⠀⣾⣿⣿⡟⠉⠉⠙⢿⣿⣿⡆⠀⠃⠀⠀⢠⢸⠀⢀⣿⣿⣿⠟⠉⠉⠻⣿⣿⣿⡄⠀⠃
⠀⠀⠀⠉⠉⡋⠀⠀⣿⣿⣿⡂⠂⡀⠀⣸⣿⣿⡇⠀⢰⠒⠒⠚⡇⠀⢸⣿⣿⣿⠀⠄⡀⠀⣿⣿⣿⡇⠀⢸⠒⠂⠀⠀⠀⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠘⠒⠒⠻⣿⣿⣿⣦⣴⣶⣿⣿⡿⠓⠈⠉⠀⠀⠀⠙⠒⠒⢻⣿⣿⣷⣦⣶⣾⣿⣿⡟⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢁⣀⣀⣀⣀⠉⠳⣆⠠
⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⣀⣤⣤⣤⡤⠴⣤⣤⣤⣌⣿⣏⣁⣀⣉⣉⣹⣾⣃⣀⣤⠦⢠⣤⣀⣀⣐⣀⣠⣠⡀
⠀⠀⠀⠀⠀⠀⣰⠚⠉⠉⢉⡵⠞⠉⣽⠟⠛⠁⠈⠐⠛⠛⠀⠀⠙⠛⠛⠛⡿⠛⠉⡽⠟⢋⠉⠉⠛⠉⠉⢣⢹⡆
⠀⠀⠀⠀⠀⠀⣻⠓⡣⠚⠉⠀⡄⠀⢿⠀⠀⠀⢄⠰⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠳⣀⠄⠆⢸⡎⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡴⠁⠀⠀⠀⠖⠒⢶⠀⠀⢀⡉⠒⢀⠲⠀⠀⠀⠀⠀⠀⠗⠒⢲⡇⠀⠀⠀⠈⠳⣀⣸⢇⢸⡇
⠀⠀⠀⠀⠀⢐⢸⠂⡔⠀⠐⠀⡀⠀⠾⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠠⠀⠈⢄⡠⠘⠁⠀⠀⠀⠰⡐⠂⠉⢸⠸⡇
⠀⠀⠀⠀⠀⠈⣸⠀⠀⢀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⠄⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠋⠀⠇⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⢹⠀⠀⠠⠠⡀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡘⠇⠀⠠⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠉⠣⣀⢄⣉⠏⠀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠇⡀⠀⠈⢂⠀⠀⢸⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠀⠀⠀⠘⠀⠀⠉⠠⠀⠀⠉⠛⠀⠄⡇⢸⡅
⠀⠀⠀⠀⠀⠠⢸⡀⢈⢀⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠠⠁⠀⢠⡀⢆⡄⡀⢰⡀⠀⡆⠀⠃⠀⠀⠀⠂⢳⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠸⢦⠥⢤⣵⠦⢶⣀⣈⡦⢄⣀⣀⣀⡀⠀⠀⠉⠁⠈⠀⠀⢸⡇⠀⡟⠀⠀⠀⠀⢀⡄⢂⣷⡼⠇
⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠀⠀⠁⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠛⠛⠛⠷⠶⠦⠴⠏⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⢦⣄⡀⠀⢀⣀⠠⠄⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡩⣶⠪⠋⣩⠤⠾⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠀⠑⠄⠀⠀⠢⠠⠠⠌⣂⡤⢊⣿⣖⠯⠿⠑⠒⢥⡀⡆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠔⠠⠂⠀⡈⠁⠀⢀⣤⠞⠁⢀⠠⠂⢉⣀⣀⣀⣀⠀⠙⢷⣶⣷
⠀⠀⠀⠀⡀⠀⠐⠠⠀⠀⠀⠀⠀⠈⠈⠈⠀⠂⠀⠀⠠⠊⠀⢀⠆⠱⡾⠏⠉⠙⠋⠀⠈⠂⠈⠀⠻⣽
⠀⠀⠀⠀⠀⠀⠂⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠉⠀⠀⠈⠀⠀⠀⠀⠂⠀⠄⠂⠀⠀⠀⣘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠤⢄⠒⡄⠘⠀⠀⢧⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠩⠄⢀⠘⠩⡆
⠀⠀⠀⠀⡀⣠⠴⠶⠶⠶⠶⠒⠲⠿⠗⠒⠒⣧⣿⣂⡑⣰⣶⣷⣶⣥⡀⡐⢆⠀⠀⣤⠀⠀⢌⠀⠀⠀⠁⡼⣠
⠀⠀⠀⠸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡌⡂⠁⠐⢮⠉⠀⠀⠀⠀⠀⠰⢀⠀⢀⡌⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢠⢐⡄⠀⠀⠀⢀⣤⣠⢻⠆⡼⡈⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⢀⠀⠠⢈⡟⠀⠀⡂⢑⠁⠢⠈⣀⡘⣼⢈⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠡⠘⣅⡱⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⢘⠂⠠⠄⠤⢠⠐⠠⠔⠂⣇⡴
⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀⢸⢟⠒⡒⠒⠖⠐⠂⢫⡴⠋
⠀⠀⠀⠸⠷⠦⠤⠤⣤⡤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠁⢞⣤⣥⡲⣯⡥⠬⠵⠖⠛⠉⠈⢀⣀⣤⡴⠶⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠑⠛⠒⠒⠺⠿⠖⠛⠋⠋⠁⠀⢀⠀⠀⠀⠒⠈⠀⠈⠁⠀⠀⠀⠂⠀⠀⠉⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡋⠉⠉⠉⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⠀⢸⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⢟⣛⣃⣙⣛⣋⣀⣼⣄⣀⣀⣠⣇⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡇⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⡿⠿⠿⠿⠿⠷⠙⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢈⠉⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠒⠀⡄⠀⢈⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣄⣤⣶⣶⣶⣤⠀⢀⣠⣤⣤⣤⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣵⣿⢱⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠛⠃⠚⠛⠛⠛⠛⠛⢸⠿⠿⠿⠿⠿⠿⠿⠆⠑⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠛⠒⠒⢄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠶⣌⣀⡀⠀⢠⣾⣿⠿⣿⣷⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⣸⣿⠏⠀⠒⠒⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⣿⣏⠀⢨⣿⡗⠀⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣁⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⣾⣟⣀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢸⣿⠀⣸⠇⠀⢁⣀⠈⢁⡀⠀⠁⣀⣰⣿⠗⣾⡿⠋⠉⠉⠀⢴⠌⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⣿⣶⣿⣴⢀⣿⢋⣴⡿⢉⡀⣾⠋⣿⠇⢠⣿⠃⠓⣺⡗⣰⡗⢰⡿⡰⣻⡆
⠀⠀⠀⠀⠸⢶⣤⣤⡤⠴⠖⠒⠛⢿⠀⣿⡎⢯⡼⠻⣧⠾⠰⢯⠴⢿⠔⣿⡏⠀⠰⣿⠡⠿⡅⣾⡿⢴⡿⠳⠦⢤⣤⣴⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⣀⣀⠠⠤⠤⠄⠀⠐⣒⣒⣒⣒⠀⢠⠤⠤⠤⢤⣄⣀⣀⣀⣴⣭⣭⣵⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⡀
⠶⠭⠤⠐⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠴⠿⠿⢻⣽⣧⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠒⠒⠒⠒⠂⠠⠯⠭⠥⠶⠒⣂⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⢣⡤⣴⠶⠶⠚⠛⠉⠉⠁⠀⢠⠖⡨⠟⢁⣀⣀⣈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠙⡁⢃⠀⠀⠀⠀⠀⠀⠀⣀⡴⠗⢋⡠⠤⠭⡓⠈⠉⠉⠉⢉⡉⠿⠿⠯⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡀⣰⣨⣦⣤⠴⠶⠶⠞⠛⠓⠂⠈⡽⠉⠉⠑⢾⣰⠉⠉⠳⣏⠀⠀⠀⠀⠈⠪⡑⢌⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢏⡶⠞⠛⢛⠧⠴⣤⣄⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡏⡆⠀⠀⠇⢣⠀⠀⠀⠀⠀⠘⢆⠑⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠹⡀⠀⠀⠘⠃⠶⠀⠠⢼⡀⢣⠀⠀⠀⠀⠀⠈⢆⠀⠳⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠉⠁⠀⠀⠈⠢⠤⠤⠛⠂⠷⠤⣄⣀⣀⣀⣠⠂⢀⣀⡳⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠛⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠤⠭⢌⠀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⠉⠐⠋⣭⢽⣶⣤⣤⡤⠤⢤⣤⣤⣵⣦⣠⣤⡤⢤⣚⣊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⠛⠳⠦⣤⣄⣈⣿⡉⠉⠛⠛⠛⢻⢍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⢤⣤⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠶⢲⣂⡠⠤⣀⣀⣻⠉⠉⠛⢞⡂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣑⣒⣩⣭⣴⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⡣⠤⠀⠀⢦⢠⠀⠀⠀⠀⢠⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣧⡄⠁⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⢀⠛⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⢟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠇⠶⠄⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⡛⡵⡄⠀⢀⠠⠦⠦⠋⠡⠟⠉⢽⣛⡿⠃⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠋⡋⣝⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠂⠘⡽⠿⠻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⢀⠀⠀⠀⠀⠀⡴⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢳⣣⣱⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣧⠁⡀⣄⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠾⠾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠓⠒⠒⠂⠠⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣄⠀⠀⠀⠀⠀⢀⣾⣿⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⢿⣦⠀⠀⠀⢠⣾⣿⣏⠁⠈⢿⣆⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠆⠈⢿⣷⡀⣠⣿⣿⣿⡉⠀⠀⠈⢿⣦⠀⠀⣼⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⡀⠀⠀⠈⢿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠈⢻⣧⣼⣿⠋⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣒⣿⣿⣿⣿⢿⣿⣒⠮⣍⠻⣿⡟⠋⠁⠀⠀⠀⢼⡿⠿⠛⠃⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⠁⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠈⠱⣌⢳⡘⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠉⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡷⠖⠒⠒⠒⠒⠒⠒⣒⡒⠒⠒⠶⠶⠤⠤⠄⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠿⠿⠿⠶⠶⠿⠿⠷⠶⠶⠶⠾⠀⡇⠺⠽⠃⢀⣀⣤⣬⣯⠭⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢸⣀⠤⢤⠀⣠⡖⣾⡗⣲⣄⢀⡤⠤⡀⡇⠀⣠⠚⠉
⠀⠀⠀⠀⠀⣠⠖⢻⠻⢄⠴⠃⠈⠛⠧⠜⠒⠁⠘⢦⡠⠏⡇⠀⡇
⠀⠀⠀⣠⠞⠁⠀⢾⣻⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣛⡷⠀⢳⡀
⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⠀⠀⡄⠀⢹⣿⡇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣉⣉⣀⣀⣀⣀⣉⣉⣉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢠⠄⣮⣨⡖⢢⡀⢀⣀⡀⢀⡔⣾⣧⣷⢠⡀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡠⣖⠾⠿⠁⠘⠓⠛⠒⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣙⡿⢟⠓⠒⠓⠚⠂⠹⠿⠷⡢⡄
⠀⠀⠀⠀⡘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃⠠⡢⠑⢄⠀⠀⠀⠀⠀⠀⠈⢎⡆
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⢀⠈⠀⠀⠑⢄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢸⡇
⠀⠀⢸⠢⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡠⠊⠀⠀⠀⢸
⠀⠀⡘⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠊⠀⠀⠀⠀⠀⢸
⠀⢠⠃⡼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡸⢰⠁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠃⣾⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡄⢇⡜⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠉⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠈⠺⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠹⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠟⠿⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⣤⣶⣤⡵⠤⠤⠤⠤⠤⠤⠤⠄⣀⡀⠀⠀⠀⠀⢀
⠈⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⠤⠤⠖⠒⠚⢉⠁⠴⠛⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠘⠉⠏⠁⠀⠀⠀⢀⣀⠴⠂⣡⠀⢇⠈⠉⠈⡙⠿⠶⠦⣶⢄
⠀⠀⠀⠀⠀⠠⣀⣦⣴⣚⡒⠚⠉⠉⠉⠀⠀⠀⠀⢸⡏⠀⠘⠆⠀⠀⠀⠐⠕⢄⠢⡀
⠀⠀⠀⠀⠀⠈⠁⠀⢀⠉⠈⠙⠢⣄⠀⠀⠀⡀⠀⠘⠦⡄⠀⣄⠀⠀⠀⠀⠈⠀⠑⢌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⣧⠀⠀⠀⠉⠀⠀⠁⠐⠈⠁⠒⠂⠤⠀⠤⠀⠠⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠉⠒⠲⢀⣠⣄⣀⣀⣀⣀⣐⣄⢀⣀⣀⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⠞⠉⠉⠛⠲⠤⣌⣯⣉⡉⠛⢻⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢘⣅⣀⡀⠀⠀⢲⠷⠤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢰⣖⣲⣅⣐⠠⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠥⠀⠴⣞⢻⣶⠻⡈⠢⠀⠈⠓⠀⠀⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⡀⠀⠀⠀⠀⠄⢻⢦⠔⠀⣀⠄⠃⠀⠐⠌⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢐⠀⠀⠀⠀⠑⠎⢠⢱⡀⠀⠀⠀⠀⣤⣺⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⢀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⢜⡅⡆⠀⠈
⠀⠀⠀⢀⢀⢀⠂⠀⠀⡆⣁⢀⠀⠓⢀⣂⡀⠃⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢀
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠖⠒⠀⠀⡀⠀⠀⠀⠈⠂⢐⠇⠀⠀⠀⠀⢀
⠀⠀⠨⠀⠀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠐⠟⠒⠒⠒⠠⠾⠚⠒⠓⠙⣛⣝⡋⠉⠉⠉⠉⠡
⠀⠀⠀⠀⠁⠿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⢈⡷⠀⠀⠀⠀⠀⠃
⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠀⠄⢀⡀⢀⡈⣀⠠⣙⠂⠠⠀⠀⠀⠤⠀⠂⠣⠈⠉⠀⠀⠀⠀⠀⠀⢂
⡀⠈⠀⠰⢁⣀⣀⣀⣀⣀⣀⢁⠀⠈⠉⣃⠀⠂⠀⠀⠂⠀⡣⠈⠀⠀⠀⠈⢁⠀⠀⠀⠡⡀⢠⠄⠀⠤⠀⠀⢁
⡗⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠀⢸⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠈⢀⡀⠠⢪⣤⣌⡀
⠃⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠐⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⡀⠔⠉⠀⠸⠁⠁
⣀⣐⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣶⣶⣶⣤⣥⣦⣤⣖⣒⣒⣊⣉⣉⣉⣉⣉⣉
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠉⠁⠀⠐⡦⡀
⠀⠀⠀⠀⠀⣠⡄⣤⣤⣤⣤⣀⣀⡅⣼⣖⠛⠛⠒⠄⢊⠁
⠀⢀⢀⣤⠴⢯⡵⠙⣿⣶⠟⢩⣽⣿⣮⣿⣟⣛⣿⣅⣾⣷⠶⠲⣶⡤⢤⣤⣤⣤⣄⣀⣀⢀
⢀⠄⣄⠉⠀⠐⠉⠉⠭⠉⠉⢁⠈⠉⠉⠉⠛⠛⠛⠛⠻⡾⠟⠘⣷⣶⣞⡃⠀⠀⠨⢙⣤⠠⣬
⢸⣠⠏⠉⡉⠓⠒⠐⠦⠤⠄⣄⣀⣀⠁⠀⠀⠀⠂⠠⠀⠉⠭⠘⢉⡱⠦⠹⣋⢵⣦⣤⢾⣤⢾
⠀⢃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠺⠤⠤⠀⣀⡀⠀⠀⠀⠈⡾⣿⠀⠀⢸⢸
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⢀⡴⢿⣗⠀⠀⣽⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈⣩⡇⣰⣯⠀⠀⣏⡌
⠀⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⢕⣯⠀⠐⡇⡇
⠀⠰⠆⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠖⠲⠿⠗⢾⣿⣷⣾⣷⣷⣤⣄⡀
⠀⡐⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠰⡑⡒⣘⣒⣛⣋⣉⣛⣛⣛⣙⣿⣿⣷⣶⣤
⠸⣖⢸⣒⡀⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣽⠅⣸⠛⡉⡋⠉⠉⠁⠈⠉⠁
⠀⠀⠉⠉⠉⠛⠓⠓⠶⠦⢤⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣼⣿⣞⣴⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠑⠒⠢⣤⣤⣠⣄⠀⣴⡖⠀⠙⠚⣹⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⠬⠽⠞⠃⢀⣀⣀

⠿⠿⠿⠿⠶⠶⠶⠶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⣷⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠙⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣷⡿⠋⠛⣿⠟⠛⠛⠛⢻⠟⠛⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⠿⠏⠀⠃⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠰⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠋⠙
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⡀⠀⠉⠁⠉⠙⢿⣿⠟⠛⠉⠈⠹⠷⠀⠀⠀⠀⠀⠀⠀⠆⢀⣠⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣾⣿⣯⣛⡻⠷⣦⣀⣀⠀⠠⡶⠂⣠⣤⣤⣬⣠⣶⣿⣀⣰⣦⣤⣄⣀⣤⣴⣦⣦⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠉⠉⠛⠃⠈⠉⠛⠁⠀⠀⢀⣠⣾⠟⠋⠉⢉⣟⡀⠀⠈⠛⠛⠋⠀⠘⢿⣷⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡦⠤⠶⠲⠟⠉⠀⠘⠛⠛⢛⠟⠀⠀⠀⠀⠀⠀⠐⠻⠋⠛⠻⠶⠊⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠟⠋
⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉⢁⣀⣀⠀⠀⠀⠀⠀⠀⠠⢀⣠⠄⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣤⣀⣀⣀⣠⣤
⠀⠀⠰⠶⠶⠦⠤⠄⠀⠴⠿⠿⠿⠶⠴⠆⠀⠀⡔⣾⢻⣿⠀⠀⠀⣀⣤⣶⡏⢸⠉⠉⠉⠛⢛⡛⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣴⠇⠛⠈⣀⣴⣶⣿⣿⣿⣿⣤⣼⣁⣛⣉⣸⣉⣛⣛⣛⣛⢻⣿⢻⣿⡿⢿⣿⣿⣿⣿
⠁⠀⠠⠀⠀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣠⣇⣤⣶⣿⣿⣿⠿⠛⠋⣁⡤⠿⠏⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠉⠙⠋⠉⠉
⣀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠉⠉⠉⠈⠁⠉⠉⠟⢛⡛⠛⠉⠀⠀⢶⠊⣁⡤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⣤⣭⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⣀⣁⠀⠀⠀⠀⠔⠊⠉⠉⠉⠉⠋⢤⠼⠛⠉⠀⠤⠤⠀⠄⠤⠤⠀⠀⠀⠀⠠⠀⠀⠉⠁⠀⡀⠀⠘⠒⠂
⣬⠥⠀⠁⠴⣷⣾⣿⣿⣿⣿⠿⠿⠟⠃⠐⠀⠀⠀⠀⠀⠀⠀⢢⠶⠂⡀⣀⣀⠂⠀⠀⠐⠦⠄⠀⠠⠤⣀⡤⠠⠆⠀⠀⠐⢲⣴⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⢀⣠⡤⣀⡀⠀⠀⠄⢡⡇⠀⠀⠀⠀⠀⢀⡀⠄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠘⢷⣄⠀⠀⠀⣸⣿⣿⣷⣄⢀⠀⠛⢳⣷⠚⢷⡄⠐⠰⠸⠀⠀⠀⠀⠀⠘⠿⠦⠺⠘⣾⣿⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣵⣟⠞⠿⣷⣦⣴⣿⣏⠀⠾⣻⣾⣷⡁⣏⢸⡇⡾⠀⠈⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠓⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃⠀⠀⣨⢛⡷⠿⣛⣢⣭⣭⣴⣶⣶⣶⣬⣭⣍⡛⠤⢤⣴⣤⡀⠀⠠⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣟⣷⣤⠶⢋⡠⠴⠊⠉⠙⠋⠉⠚⠛⣫⡯⠛⢿⣿⢿⣷⣶⣬⣙⠧⣀⠖⢋⣠⡼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣓⠃⣈⡷⢭⠟⣥⠞⠉⠀⠀⠀⠀⠀⠐⢿⣿⠟⠁⠘⠒⣿⣿⣷⣼⣿⣿⣿⣷⣌⠢⣨⣏⡀⠚⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠯⠟⢲⣦⡝⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢻⣁⣠⠈⠁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⡀⠠⣬⠟⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣤⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣍⢋⣄⠂⠀⠀⠀⠀⠲⠆⠄
⠀⢀⣀⣀⣀⡀⠀⠀⠉⢀⡀⣁⡎⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠹⣿⣿⡀⠀⠐⠒⠐⠀⠊⠁
⠀⠘⢿⣁⡏⣿⣷⣶⣠⣬⣉⡿⣸⣿⠀⢀⣾⣿⣶⣷⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢳⡄⠙⢦⡤⠔⠚
⠀⠀⠀⠉⠁⠛⠿⡏⣿⣿⠿⠇⣿⣿⣄⡙⠛⢙⣿⣭⣝⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢸⣯⡄⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⢲⣿⣿⣿⣿⣤⡘⣻⣿⣿⣼⣦⣤⣝⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⣿⡗⠳⣤⠥⡄
⠀⠀⠀⠀⠀⠀⠈⡄⠀⢸⣽⡸⣿⣿⣿⣿⣿⣧⣻⡿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⢾⠷⠶⢽⠦
⠀⠀⠀⠀⠀⠀⠀⢀⠄⢐⣻⡆⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢼⣴⣀
⠀⠀⠀⠀⠀⠀⡐⠡⠈⠀⡀⣷⢹⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠚⢃⢻⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⡿⡰⠉⣹⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣦⣤⣤⣤⣤⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⢰⣦⣼⣏⣹⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣡⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣴⣿⣿⣿⣿⣿⡿⢋⠜⢀⣽⣿⣿⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠉⠪⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣵⠣⢠⣾⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⡙⠢⠀⠈⠒⠭⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⢩⣶⡛⡶⠄⠀⠛⠛⠿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠙⠈⡁⣤⡄⠈⡑⣶⣭⡝⢛⡛⠻⠿⠿⢿⡛⠛⠋⠩⠐⣾⡀⠈⣸⡿⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠛⠀⠀⠋⠁⣾⢀⡟⣞⠋⠉⠁⣄⡀⠀⢀⣿⡗⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡸⠿⣡⣬⠀⠀⢀⡟⠁⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠤⠴⠶⠒⠚⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⡀⣠⣤⣴⣄⡀⠀⠀⢠⡆⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⣀⣀⣀⣀⣀⠠⠤⠴⠚⠛⠛⠛⠉⠉⠉⠉⠀⣨⢉⡩⠿⠣⡖⠾⠿⠃
⠘⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣼⠏⠁⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠸⣿⣿⣿⡇⣤⣄⡀⠀⠀⠀⣀⠀⣴⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⣿⣿⡇⣶⣿⣿⣿⣷⡘⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⢠⡀⠀⡼⠆⠀⠀⠈⠸⣿⣿⡇⢿⣿⣿⣿⣿⣷⡈⢿⣿⣿⠆
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠂⠙⠿⡇⢸⣿⣿⣿⣿⣿⣿⡌⠛⠋
⠀⠀⢀⣴⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠁
⠀⠀⠛⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⢀⡿⠓⠂⠀⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⠃

⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⠀⣤⣤⡀
⠀⠀⠀⠈⠿⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⣁⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⣁⣈⣿⣿⣃
⠀⠀⠀⠀⠀⢸⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⣿⣿⣿⣷⡀⣠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢀⣠⣼⣿⣿⡻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆
⠀⠀⠀⠈⠉⣰⣿⡿⠀⠹⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠃⣿⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣷⠋⠉⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣷
⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣶⣷⡆⣿⣿⣿⣿⢿⣿⣿⡿⢿⣽⣄
⠀⠀⠀⠀⣿⡟⢻⣿⡀⠀⠹⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣿⣿⡟⢹⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⢸⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠰⣿⠇⠀⠻⣿⡆⠀⢟⡿⠛⠋⠁⠀⠀⠀⠀⢿⡿⡿⣿⣿⡿⢿⣿⢼⠇⢸⣿⡿⢿⡇⢿⣿⠿⣿⠃⠀⠀⠀⠀⠛⠛⠃⢿⡿⢿⡿⠸⣿⢿⡇⠈⠛⠉

⠀⢀⣠⣤⣾⣦⣐⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣦⡄⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄
⠀⣿⠉⠿⡿⠏⢾⣿⡿⠠⣤⡀⠀⠀⠀⣼⠗⠾⠿⠇⢸⡟⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⣼⠿⡿⠇
⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣾⠃⠀⠀⣸⣿⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣄⡀
⠀⠀⣿⣿⣿⣏⣿⣿⣿⣷⠈⠀⠀⠀⠉⠁⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⢈⣭⣿⣿⣯⣄⠀⠉⠛⣻⠇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣭⣿⣿⣟⢓⠀⣀⣀⠀⣠⡿⠃⠀⠀⠀⣠⣤⣤⣤⡞⠉⣿⣿⢻⣿⣿⣻⣥⣤⣤⣀
⠀⢠⣿⡇⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⢸⣿⡆⠀⠀⠀⠀⠈⠛⣿⣿⠏⣾⣿⣿⣿⣿⣿⣷⡟⠁⠀⠀⠀⡾⠉⠀⣠⠏⠹⡆⢹⣿⢾⡿⣿⠋⢹⡄⠀⠙
⠀⢸⣿⠁⣿⡏⢹⣿⣿⡏⠁⠀⠀⠀⠀⢀⣿⡇⠈⣿⡇⠀⠀⠀⠀⠀⠘⠿⠁⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⣧⠀⠀⠉⠉⢙⡟⠻⠿⠉⠀⣿⠀⠀⠉
⠀⣾⣿⠀⣿⡇⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡴⠟⠁⠀⠀⠀⠀⠈⠳⢦⣤⡴⠞
⠀⠙⠁⠀⠙⠁⠈⠋⠙⠁⠀⠀⠀⠀⠀⠈⠛⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⣼⣷⣴⡲⢶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢫⣾⣿⣿⣿⣿⣶⣝⢿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣽⣿⣿⣯⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣏⣾⣿⡿⢿⡟⠛⠛⠛⠃⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⠿⠿⠘⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣛⣿⣿⣯⣿⣽⣿⣿⣛⣃⣀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣹⣿⣿⣦⠀⠀⣿⢸⣿⣿⣿⣿⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⣀⢀⣼⣿⣟⠁⠀⠀⣿⣀⣹⣯⣿⣁⣀⣀⣀⡙⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⣷⡀⣾⣷⣆
⢠⣴⣶⣶⣾⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀⠀⠀⠀⠸⡇⠘⢿⡜⢿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⢸⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡗⠀⠀⠀⣿⠀⠈⢻⣆⢉⣛⣛⣭⣭⣭⣍⡛⠿⠃
⣤⣿⣿⣿⣿⢋⣴⣾⣷⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣶⣿⣷⣮⠻⣿⣿⣿⣥⡀⠀⠀⢸⡇⠀⢀⡿⠛⠋⠉⠉⠈⢷⡉⠛⠷⢶
⠉⠉⠉⠉⠁⢸⣿⣀⣨⣿⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣿⣿⣀⣸⣿⠇⠉⠉⠉⠉⠀⠀⠀⠀⠗⠀⠞⠁⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠒⠶⠶⠶⠶⠶⠿⠖⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠙⠛⠛⠉⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠶⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣤⣤⣤⣤⣤⣤⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠀⣿⣿⣿⣿⣿⣿⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣍⣉⣉⣉⣉⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣟⣹⣯⣹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀⡀⠀⠀⣠⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⣄
⠀⢀⣴⡆⣀⡒⣞⣇⣀⡀⣠⢤⣶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠠⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⢆
⠄⠀⠉⠉⠤⢋⠖⠘⠚⠙⠉⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠑⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠤⠀⣀⣀⣀⣄⠈⠠⠠⠀⢀⣀⣠⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣈⠉⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⢸⣯⠑⠀⠀⠀⠁⠀⠀⠀⠈⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢹⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⠪⢟⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠤⠄⠤⠀⠂⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠷⠁⠀⠀⢀⢈⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠉⠁⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠦⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢠⡎⠑⣆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠲⠔⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣤⠀⠀⠀⡄⠠⢤⣤⣀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠗⠒⠒⠒⠚⠛⠛⠋⠉⠁⠀⠀⠧⠤⣄⣀⡀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣈⠀⠉⠉⠐⠒⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

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

⠀⠀⠀⢀⣠⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⣀
⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢸⢀⣀⣀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠣⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⢂
⠀⠀⠀⠋⠉⠙⠉⠉⠁⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣴⣿⣶⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠛⡋⠀⢘⡄⠀⡤⢤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⡡⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⣷⠀⠀⢀⣀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠠⣿⠀⠀⡇⠀⠀⡀⠀⠀⠀⣀⠀⠀⠐
⠀⠁⠊⠀⠀⠀⠀⠀⢸⡟⠀⠀⣷⠦⠖⢂⣄⣁⢈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠂⠀⢰⢀
⠀⠀⠀⠒⠀⠀⠀⠐⠃⠀⠀⠀⠈⠉⠛⠛⠛⠓⠚⠁⠀⠐⠛⠙⠋⠀⠀⠙⠙⠋⠂⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠌⠐⠀⠀⠀⠈⠀⠀⠁⠀⠀⠤⢤⡤⢴⠶⡶⠄⠼⢰⠆⠐⠚⠛⠛⠛⠧⠇
⠀⠀⠁⠁⠀⠀⣀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⢀
⠀⠈⠉⠀⠀⠀⠀⠉⠙⠊⠀⠀⠀⠀⠀⠐⠒⠶⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⢀⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠂⠀⠀⠀⠄⠈⠀⠀⠀⠀⠠⠀⣀⠀⠀⠸⠁⡇
⠀⠀⠀⠀⠀⠈⠉⡏⠉⣄⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⣀⣀⣁⣁⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡼⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣥⠤⠤⠤⢤⣌⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡅⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠠⠀⠀⠀⠠⠶⠶⠦⠀⠀⠀⠈⠿⣧⡀⠀⠀⠀⠀⠭⠧⠠⠀⠠⠠⠄⠴⠛⠻⠂⠀⠀⡂⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣐⡨⠃⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠈⢿⣀⠤⡀⠀⠀⠀⠀⠁⠀⠁⠠⢀⠁⠀⡄⠄⠀⠈⠪⠬⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢀⠄⠀⠁⠀⠑⠀⠄⠀⠀⠀⠀⠀⡀⡞⠷⠄⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣄⠀⠠⠂⠢⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⣚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠑⠁⠀⡑⠘⢄⠀⠀⠀⠀⣰⠀⢀⡇⠀⠀⢄⠈⢈⡥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡣⡈⠂⣈⠤⠎⠢⢂⠊⢈⢸⠀⢘⡃⠀⠠⡀⠁⢠⣐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⡀⠂⠠⠢⠈⢀⡠⡹⠢⢡⠞⠔⡡⡡⡠⠠⢈⠀⢸⢇⢐⠄⠀⠐⡠⢭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠊⠀⢁⢔⠽⠀⢸⠀⠊⠨⠂⢀⢔⠡⠂⢄⠄⢴⢃⢊⢄⠄⠄⠐⠀⠵⠀⠸⡣⠠⠊⢊⢌⠔⢮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠊⢀⢔⠅⡡⢚⠀⢨⠀⠀⢔⠔⢐⢅⠔⡁⠑⠔⡑⡡⡡⠠⢢⠂⠊⠈⢺⠀⠘⠖⠐⠑⠡⠁⣪⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⢔⠐⠁⡨⡠⢘⠀⢺⢄⢔⢕⠕⠁⡡⡪⠢⠊⢊⢀⠔⢔⠔⡑⠀⠀⠀⠠⠀⠠⠂⡀⢀⢄⢔⢷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠓⢤⡨⡊⢈⢀⢰⠀⢸⡅⠑⡁⢠⡊⢊⡄⢐⡔⢡⡡⢨⠂⢪⡈⢤⠆⢄⢴⠀⠀⢖⡔⢡⢌⡱⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣬⠄⠖⠒⠚⠚⠛⠀⠘⠓⠒⠓⠓⠓⠓⠓⠒⠒⠒⠒⠒⠛⠙⠘⠚⠙⠚⠛⠒⠒⠛⠚⠚⠚⠤⣤⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠤