Indicate Emojis & Symbols

⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠈⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⡇⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣠⣤⣴⡶⡶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠟⠛⠛⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣄⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⠀⠀⠀⠀⡞⠑⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠞⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⣴⡷⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠙⠛⠻⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁


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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠒
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠘⠀⠀⠀⢂⣴⣟
⣿⣿⣿⣿⡟⠛⠃⠙⢿⣿⣿⣿⣿⣿⠛⠿⣿⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄⠀⠀⢀⣠⣤⠄⠀⢃⣤⣤⠀⠀⠀⠀⢂⣴⣿⣿⣿⠀⠀⠐
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⡀⠀⢻⡇⠀⡀⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⢂⣼⣿⣿⣿⠀⠀⢉⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣴⣿⣦⡀⢳⣾⣿⣦⡈⣿⣿⣿⣿⣿⠀⠀⡀⠈⢸⡿⢿⣿⣿⣿⡿⣿⣿⣿⠀⣱⣿⣿⣿⣿⣿⠟⠉⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠈⠙⠛⠛⠋⠻⣿⣿⣿⣟⠀⠀⠀⢀⣼⣶⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⡀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⠔⡀⢦⣄⠀⠀⠀⢸⣿⣿⣿⠀⠀⣠⣿⣿⣿⣿⠿⢛⡛⠿⢿⣿⣿⣿⣿⣿⠟⠁⠀⡀⠀⠀⢀
⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣦⡀⠀⠀⠸⣿⡘⠿⢂⡿⠀⠀⣠⠸⣿⣿⣿⠀⠀⢻⣿⣿⣿⣧⣇⠻⠟⢀⣸⣿⣿⣿⠟⠁⠀⠀⠀⣀⠀⢀
⣿⣿⣿⣿⡷⠚⠓⠾⠛⠛⠿⢿⣿⣦⡀⠀⠙⠻⠟⠁⠀⠀⠀⣿⣼⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣤⣤⣶⣤⣤⣯⣤⣿⠁
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠉⠙⢿⣿⣦⡀⠀⠀⠀⠀⢀⣴⡿⣹⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣷⣦⡐⠿⠿⠟⠛⢠⣿⣿⣿⣿⠒⠀⠀⠹⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠉⠉⠀⠈⠉⠀⠀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣿⣿⣿⣿⣆⠀⠀⣠⣿⣿⣿⣷⢿⡍⢡⣄⠀⠘⢿⣿⣿⠋⠀⠀⢠⠀⠀⠐⠀⠀⠀⠂⠀⣴⠛⢦⠀⠀⠐
⣿⣿⣿⡟⢿⣿⣿⡿⢻⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣉⣿⠲⠖⠐⠂⠀⠀⠻⠇⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⣦⠙⠒⠋⣵⡀
⣿⣿⣿⣇⠀⢹⣛⡇⠘⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⢻⣿⣿⣿⣿⣶⠿⣾⣭⣬⢋⣀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⣿⣷⣶⣿⣿⠃
⣿⣿⣿⣿⣷⣼⣫⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣻⣿⣻⠆⠰⠶⠀⠀⠀⣸⣿⣄⠊⠁⠀⠈⠀⠀⠀⠘⢀⣾⣿⣿⡟⠋⠉
⣿⣿⣿⣿⣿⣼⣭⣇⠀⠀⠨⠛⠿⣿⣿⣿⠿⠛⠁⠀⠀⣸⣿⣿⣿⠿⣤⢿⡙⢋⣀⠀⠀⠀⣿⣿⣿⣷⣶⣀⣀⢀⣈⣴⣶⣿⣿⣿⠟⠈⠀⢀
⣿⣿⣿⣿⣿⣵⣯⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⢿⣿⣿⣿⣛⣿⠴⠶⠁⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠀⢀
⣿⣿⣿⣿⣿⢾⣶⣿⡿⠙⢿⣶⣦⣄⠀⣠⣴⣶⣿⣿⡏⠀⠘⣿⣿⣿⠿⣽⣃⣘⠓⠀⢠⣿⣿⡇⠀⢸⣿⠛⢿⣿⠟⣻⡏⠀⣀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣾⣾⣿⡃⣀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣻⠌⠡⠄⠀⣾⣿⣿⡇⠀⠰⣿⠀⠈⠁⢀⣾⡇⠀⠀⠀⠀⠀⠀⠁⠀⢠
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣭⢿⣚⠓⠀⢰⣿⣿⣿⡇⠀⠀⠻⣧⣤⣤⣴⡟⠀⠀⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣟⣾⢥⡄⠀⣿⣿⣿⣿⡇⠀⠀⠀⠈⠉⠉⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⣿⣿⣽⢒⠀⢸⣿⣿⣿⣿⣧⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠰⠄
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠸⣷⣻⡭⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀
⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣾⢃⡀⠀⠀⢿⣿⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⣠⣌⣉⣉⣉⣉⣉⣩⣥⣾⡇⠀⠀⠸⣿⢹⣿⣿⣿⡇⠙⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠋⠓⠀⠀⠀⠀⠀⠐⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣾⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠀⣠⣾⡿⠋⠙⠀⠀⠀⠁⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⢺⣏⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⣾⡿⠋
⣿⣿⣿⣿⣿⡿⠟⠛⠻⠛⠛⠛⠛⠛⠛⠿⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣆⠁⢀⠀⠉⠀⠀⢠⣿⣿⣯⣤⣤⣤⣦⣤⣤⣤⣤⣥⣄⡀⠈⠁⢀
⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⡆⠈⠀⢀⠀⠈⠛⡛⠛⠛⢛⠛⠛⠛⠉⠉⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣟⣇⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠭⠯⠽⠭⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⠀⠀⢠⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⢠⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣤⢽⡞⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠐⠐⢾⣽⣿⣿⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⡇⠻⠘⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢙⣹⣿⣿⠇⠰⠀⠀⠀⠄⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⣠⣾⣶⣿⣾⡂⠀⠰
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠘⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠏⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠂⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣤⡄⠐⣶⣶⠿⠛⠉⠙⢿⣶⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⣿⣥⣶⠟⢋⣀⣠⣶⣶⡄⠀⠙⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⢿⣿⣿⣿⠿⠿⢻⡟⢻⢿⣶⣄⣀
⠀⠀⠀⠀⠀⢿⣷⠀⢻⣿⣻⣿⣿⣿⣿⡿⠛⠉⢹⣿⣿⣿⡇⠘⣿⣧⣤⣀⣈⣿⠁⣿⣤⣟⣠⣀⣾⣿⣿⣻⣿⣦⣄
⠀⠀⠀⠀⢠⣤⣽⣷⣎⠉⠙⠛⠟⠛⠋⠀⠀⢀⣿⣿⣿⣿⣧⡀⠻⣿⣿⣍⣿⣿⣤⣿⣯⣿⣿⣯⣷⣼⣿⡻⣿⣯⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠙⢻⣿⢿⣶⣦⣴⣶⣶⣤⣼⣿⡿⠛⢫⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⡏⠉⠉⠁⠉⠉⠉⠉⠀⠀⢾⣽⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣼⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡶⡿⢷⣄
⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⣤⣿⣦
⠀⠀⠀⠀⣀⣀⣿⣀⣀⣿⣿⣿⣿⠁
⠀⠀⠀⣾⡟⢉⠉⡉⠉⠀⠉⠉⠉⠀⠀⠻⡇
⠀⠀⠀⢿⣿⣿⡍⠉⢸⣟⣻⣋⣿⢿⡿⣿⠃
⠀⠀⠀⠘⣿⣷⣄⠀⠀⣟⣹⣉⣿⣼⣿⠃
⠀⠀⠀⠀⢹⣿⣏⡀⢤⣯⣭⣥⢤⣤⣿
⠀⠀⠀⠀⠀⢻⣿⣇⠀⢚⠓⢾⡖⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣻⣆⣹⡟⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣉⣈⣹⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣏⣈⣹⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⡠⢸⡄⠀⠀⠀⠀⠀⢀⣴⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⢄
⠀⠀⡜⣉⠉⠱⣄⠘⠛⠋⠀⠀⣤⣤⢧⣤⣆⣷⠀⠀⠀⠀⠀⢸⡇⢉⢛⡇⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣆⡀⠀⠀⠀⠀⠀⡇⠰⠶⠶⣀⡃
⠀⢔⠁⠿⠇⣰⠃⠀⠀⠀⠀⠀⠈⢿⠀⠉⠙⠙⠀⠀⠀⠀⠀⠀⣻⠀⢿⠀⠀⠀⠀⠀⠀⠘⢿⠷⠷⠷⠿⠇⠀⠀⠀⠀⠀⠘⠀⠀⢾⣿⣿
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠠⠊⢉⡉⠑⠀⠀⠀⠀⠀⠀⠀⡇⢀⡖⢋⡙⢦⠀⠀⠀⠀⣰⠋⠀⠀⠈⢢⠀⠀⠀⠀⠀⣿⢛⣛⢛⣛⡳⠀⠀⠀⠀⠀⠀⡞⣭⣭⣷⡀
⠀⣶⣶⣬⣥⣶⣶⠀⠀⠀⠀⠀⠀⡇⠠⠧⠼⠷⠋⠀⠀⠀⠀⣿⣷⣦⣤⣾⢟⢦⡀⠀⠀⠀⣿⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⢸⠻⣏⣀⣹⣷
⠀⠻⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⢬⠴⠃⠀⠀⠀⠑⠀⠉⠉⠉⠑⠀⠀⠀⠀⠀⠘⠂⠀⠉⠛⠻


⠀⠀⠀⢀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤
⠀⠀⢀⣿⣥⠿⠃⠀⠀⠀⠀⠀⢠⣶⢶⡶⠶⠶⣦⠀⠀⠀⠀⠺⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⢶⡆⠀⠀⠀⠀⠀⢰⡃⢰⠂
⠀⠀⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣗⣸⣟⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⢋⣤⡀⠀⠀⠀⠀⠀⠈⢉⡞⠛⣆⠀⠀⠀⠀⠀⠀⠀⢳⡈⢆
⠀⠀⠀⠏⠇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠿⠧⠤⠿⠇⠀⠀⠀⠀⠀⠀⠀⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⣁⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢁⣤⣶⣿⣿⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣶⣿⣿⣿⣿⡟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠘⠛⠿⢿⣿⣿⣿⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡿⠋⠁⠀⠙⠿⠃
⠀⠀⠀⠀⠀⠚⠛⠛⠒⠒⠶⠶⣤⣄⠈⠙⡿⠀⣾⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⢰⡇⠀⠀⠀⠀⠀⣠⣾⣁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡟⠀⠀⠀⠀⢠⠞⠙⡿⠋
⠀⠀⢠⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⣴⣿⣷⡞⠁
⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠏
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⢆⡇⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣯⡇⡾⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠀⣀⣀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠠⢿⣶⣶⣶⣶⣾⣧⡀⠀⠀⠙⢄⣀⡁⠄⣀⣴⣤⣠⣀⣀⣠⣤⣦⣀⣀⣀⣀⣰⠇
⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠉⠀⠉⠉⠛⠶⢤⣄⣀⣀⡴⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣧⣴⣤⣦⣤⣤⣄⣄⣀⣀⠈⢁⡀
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠏⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠋
⠀⠀⠀⢳⠀⣀⠀⠀⠀⠀⠀⢀⣴⣿⣶⡿⠷⠶⠦⢦⣀⣀⡀⠀⠀⣠⠇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⢄⠀⠈⢣⡘⠀⠀⠀⠀⠀⠈⠉⠻⣿⠾⠛
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣦⣤⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⢧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡿⠿⡿⢧⡴⠃
⠀⠀⠀⠀⠈⠻⢷⣤⣤⣀⣀⠀⣀⣴⣄⣀⣤⣀⣀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠛⠛⠛⠉⠉
⣿⣿⢟⠿⢛⠿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⣿⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆⠀⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⢀⡤⠞⠉
⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠛⠁
⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣄⣀⣀⣠⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣿
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠿⣿⢿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⠋⠉⠉⠁⠀⠸⠉⠉⠓⠢⢍⡛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠳⡈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠻⣑⢤⡀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⢣⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡳⣄⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠨⡄⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⠝⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡠⠋⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⡀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡵⢊⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⣷⣖⣠⠤⠤⠤⠼⠤⠄⣒⣩⣥⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⠀⠠⠂⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠻⢿⡿⠛⠀⠀⠂⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠡⠉⠀⠰⢦⡰⠃⡀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⡀⠀⢁⠀⠴⡂⢰⠃⡀⠀⠔⠀⠄⠀⠀⠰⠀⠀⠠⡄⡂⠀⠠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠈⠁⠀⠙⠢⡑⠊⣿⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠈⢳⣿⠀⠀⠀⢠⠄⠀⣀⠄⡀⠀⠷⠟⠀⠀⢂⡀⠀⠘⠑⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠂⠀⣴⡆⠀⠀⠈⠀⠀⠀⠘⠈⢿⡄⠀⠄⢹⣤⢚⣁⡀⢁⠀⠢⢰⡀⠀⢢⣰⣀⠀⠈⠉⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠋⠉⠰⢉⠳⠀⠱⠦⢄⣀⣈⣧⠠⢁⡾⠁⠀⠙⠁⠜⢁⠀⠀⢀⣤⡔⠊⠁⣀⡀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⣄⠀⠀⠀⠄⠀⠀⠸⣆⣾⠀⠀⠀⠰⠀⢠⠞⢻⣴⡏⠀⠀⠀⢸⠭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⢢⠰⣄⣀⠇⠀⠀⠀⠀⡀⠀⠹⣇⠀⡀⠀⠙⠁⢨⠨⠸⢾⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠑⠷⠤⠴⠶⢶⡶⠶⠿⣶⣤⣻⣦⠁⠐⠀⠀⠩⢛⠓⢾⡇⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠔⠀⢀⠞⠁⢄⠀⠀⠀⡬⠙⢿⣷⡀⠀⠀⣐⡙⠁⢈⡇⢠⠀⠵⢀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⡀⠰⣠⠇⠈⢉⡠⠃⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠕⠀⠄⢀⠂⠀⢁⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣧⢋⢁⠁⢠⠿⢶⠶⠛⠁⠑⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠐⠃⢀⡀⠒⡀⠤⢢⣀⣀⣠⣄⣀⡛⢸⣿⣆⣡⡾⠃⢀⠄⠠⣎⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠤⢐⠚⠉⠉⠉⠀⠈⠙⠻⢿⣿⡿⠋⠀⠐⠊⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠏⠑⢀⠰⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢀⣄⡰⠾⠂⠈⢠⠰⡴⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠁⠀⣤⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠶⠶⠿⠿⠿⠿⠶⠶⠒⠒⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣸⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢃⡤⣄⢀⣣⣃⠀⠠⠟⠁⠀⠀⠀⠀⠀⠀⠻⡆⠀⣿⡇⠘⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⣸⠶⠛⠉⠉⠭⣭⡳⣄⠀⢠⡖⠲⡦⣄⣠⠴⠒⠒⠛⢿⣴⣛⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣠⠤⢶⡾⠃⠀⠉⣻⡷⣾⡁⠀⠀⠀⠀⠀⠈⠛⠧⡀⢠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⣾⠀⢀⣠⠖⢻⣽⡶⠞⠓⠒⠛⠛⠻⠿⢯⣉⠉⠳⣄⠀⠀⠀⠀⠀⢢⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠯⣶⠞⠋⢀⣀⣤⡤⠤⠤⢤⣤⣀⠀⠈⠙⠦⣿⣿⣦⣄⡀⠀⢸⠂⡇⠐⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⣠⠞⠁⣠⡾⠟⠉⠀⠀⠀⠀⠚⠀⠈⠙⠲⣄⠀⠈⠳⡄⠀⠙⠳⣦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⠃⣠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⣆⠀⢆⠘⣆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⢣⠏⠀⠀⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠘⣆⠘⡆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡇⡾⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡀⢻⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⠶⠀⠀⠀⠀⠀⠈⡶⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⣇⠀⠀⠀⠀⠀⠀⠀⣱⠶⠃⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⡇⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣇⢸⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣆⢳⡀⠀⠀⠀⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡞⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢦⡙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⡽⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢⣍⡒⠤⣀⡀⠶⠀⠀⠀⢀⣠⠖⠋⠀⢀⣾⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⣀⠉⠛⠒⠲⠶⠒⠒⠛⠉⠀⣀⣤⡖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠈⠙⠓⠶⠶⠶⠶⠶⠖⠚⠋⠙⣦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠒⠂⠉⠉⠉⠉⠉⠑⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⣠⣤⡶⣲⠶⠻⢟⠟⠛⠒⢲⠶⠦⣤⣀⡀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡤⡾⢛⢹⣴⣿⠶⠿⠻⠿⠛⠿⠻⠳⠻⡿⢾⣵⣬⣝⡛⠲⢤⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡞⡙⣤⣷⠿⠋⠁⡀⣤⢰⡆⢰⣶⡄⢠⣦⠀⣠⡄⢀⡀⠊⠙⠫⢖⣬⣋⡳⢦⣤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⡏⣨⣶⡟⢉⣥⡄⢶⡌⠟⠉⠀⠀⢸⣿⡇⠀⠁⠀⠉⠁⠘⠏⢴⡗⢀⣤⡈⠙⠿⣯⣭⢛⣶⣦⣄⣑⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⣴⠻⣠⣾⠏⠁⣴⡹⣧⢻⣆⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣤⡄⠀⠽⢗⣴⣚⡻⣮⡝⡻⣦⡠
⠀⠀⠀⠀⡠⠃⢀⡞⢀⣾⠟⢁⣀⠻⠎⠀⠸⣧⠹⡧⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠉⠰⠿⢀⡀⠉⠿⢶⣛⡻⣦⡟⢷⣆⠒
⠀⠀⠀⡐⠁⢠⠏⢤⡿⠃⠀⣮⡙⠀⠀⠀⠀⠀⠡⠘⣆⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢡⡶⠀⠙⢿⣯⣃⠙⣎⣿⣆⣐⡀
⠀⠀⡐⠀⣰⠏⣄⡞⠁⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠁⡘⣆⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆⠀⠘⢿⠁⠈⢏⠹⣷⡤
⠀⠠⠀⢰⠿⠐⣾⠅⣀⠉⠻⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠟⠋⢡⣤⠀⠀⢂⠀⠘⣧⢸⣧⠄
⠀⡄⠀⡞⢂⣻⠇⢀⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠈⠄⠀⠸⣧⢻⡇⠀⠅
⢰⠀⢸⣏⣾⡛⢀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠸⠀⠀⢳⢿⡿⠂
⡈⠀⣬⢋⣿⠃⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⢸⣱⣿⣭⡄
⡇⠀⣯⢪⣿⠂⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⠀⠁⠀⢸⢕⣿⡂
⡇⢰⡇⠐⣿⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⡛⠋⠀⠀⠀⢸⢺⣿⣒⡀
⠀⠐⢧⠪⣷⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠆⠀⣞⢽⣿⠒
⢰⠀⠷⢌⣻⣆⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⡸⠀⢀⠗⣹⡿
⠈⡆⠘⣗⠨⢧⡀⠀⡶⠆⢀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣈⠋⠀⢠⠁⠀⣘⡩⣾⣯⡭⠩
⠀⢡⠀⢹⡬⢹⣧⡀⠰⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣶⣄⡈⠛⠀⠀⡌⠀⢠⠇⣺⡟⠉⠁⠄
⠀⠘⡄⠀⢻⣆⠻⣧⡀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⠀⠀⡜⠀⢠⢛⣭⣟⣃⠈
⠀⠀⠈⢆⠸⢳⡌⠹⣯⡄⠀⠺⠓⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⢀⠈⠀⡰⢋⣿⡟⠓⠒
⠀⠀⠀⠈⠣⣞⡻⣶⡞⢷⡢⡀⠈⠋⣴⡆⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠘⠷⠀⠀⠀⡠⠋⢀⡜⢭⣵⡟⠒⠒⠂
⠀⠀⠀⠀⠀⠙⣧⡟⢷⣔⣻⣷⡀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢴⣆⠛⠃⠀⠀⣠⠞⠀⣰⢓⣰⡾⠯⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣎⣹⢶⡀⠛⠦⣀⠀⠈⠁⠾⢂⣶⠀⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⠀⣤⡙⠿⠟⠈⠁⠀⣀⠔⠊⠀⣠⠾⢉⣰⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣅⠋⢦⣄⠈⠑⠢⢤⣀⠀⠁⠘⠛⠸⡷⠘⣿⣷⢰⣆⢸⡆⠙⠃⠉⠁⠀⣀⡤⠒⠊⠁⢀⣠⠾⢍⣽⠟⠓⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⢻⣧⡨⢱⠢⣄⡀⠀⠉⠐⠢⠤⠄⢀⣀⣈⣁⣀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠀⣀⣤⡶⣯⣥⡶⠿⠡⠬⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠽⠷⢧⣡⢉⢉⠱⡰⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢤⡔⠶⢙⣩⣤⡜⠛⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣙⣛⡳⣶⡤⣬⣌⣏⣫⡹⡙⢝⠏⡫⡙⠬⢑⣬⣦⠶⠽⠛⠛⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠖⠚⠛⠛⠛⣛⡛⠛⢛⣛⣛⣛⣛⡡⠬⠌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⣤⠀⣥⣴⠀⢷⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠙⣀⠀⠪⠠⡁⠄⢽⠗⣶⠊⠑⣟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠉⠄⢬⠀⠀⠀⢤⡋⢘⡤⣿⡀⣀⣿⡜⠃⠠⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠞⠀⠰⠂⢠⡄⠆⣈⠀⠝⣿⡤⠘⣌⠓⢼⡀⠀⣶⣾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂⠉⠀⢖⠀⠀⠀⢬⡀⢼⣷⣰⠧⢿⢸⡆⢻⣀⠉⣸⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠜⡳⢀⡀⠁⣀⠉⠁⠉⠻⣾⣿⠫⣄⠛⠔⢣⠸⢾⢤⠀⢸⡇⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢰⠻⠆⢋⡦⢡⠈⠉⠠⠜⠀⠀⢹⣿⡅⠘⣦⡄⣽⠀⠚⠀⠓⠈⠁⠋⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣀⢴⢛⠆⠀⠈⠁⠼⠪⠄⠒⢠⠸⣶⣷⠎⠹⠇⣾⠀⢁⡼⠉⢧⠹⡄⡁⢹⣞
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⡃⢠⣠⡈⠙⢰⡌⠘⠂⢃⡀⠙⢀⢸⣿⡿⠀⣠⢆⠀⣲⡞⣀⣀⠸⡆⡿⡗⠃⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⡈⠛⡁⠰⠈⣤⡁⢀⡒⠀⠀⣀⢸⣿⣷⣿⣷⢻⡀⣹⠀⢸⣇⣽⣓⠲⠧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⢦⠚⡰⠚⠁⠒⡈⡀⠩⠠⠀⠙⢿⣿⣿⣯⠀⢷⣛⣿⠠⣸⠇⠳⢤⣖⣲⡇⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠩⠀⠙⠃⠄⠗⠀⠀⠸⣿⣿⡄⠀⠐⠜⠻⠴⠛⣾⣉⡉⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠖⠃⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⡶⣿⣿⣿⣧⣿⣿⣿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠉⠀⣀⢀⡀⣀⠀⡉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⠁⢠⡀⠆⠘⠀⠁⠈⡘⣃⠁⠈⠀⠃⠰⢀⡄⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢄⠙⠈⠓⡄⡄⠀⠀⠀⣇⣜⠁⠀⠀⠀⢠⠚⠀⠋⡠⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠟⠀⣄⠑⠀⠀⠀⠀⠇⠇⠀⠀⣤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⣠⠀⠻⣿⣦
⠀⠀⠀⣸⣿⠏⠀⠲⢦⢀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠖⠀⠹⣿⣇
⠀⠀⢠⣿⡟⠀⣈⠁⠀⢸⠻⡸⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠈⣁⠀⢻⣿⡄
⠀⠀⣾⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣧
⠀⠀⣿⣿⠀⢘⣂⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣐⡃⠀⣿⣿
⠀⠀⣿⣿⠀⢨⠍⠀⠛⢀⣀⣠⣤⣤⡶⠶⠟⠛⠛⠻⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠩⡅⠀⣿⣿
⠀⠀⢿⣿⡄⠀⠂⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⡿
⠀⠀⠸⣿⣧⠀⢉⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣰⡆⠀⢀⡉⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣆⠀⠴⠚⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠓⠦⠀⣰⣿⡏
⠀⠀⠀⠀⠻⣿⣦⠀⠋⡠⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢄⠙⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠊⣠⠀⡤⠁⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠐⢥⠀⣄⠑⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠘⠁⠆⢠⠀⡀⢀⢨⡅⡀⢀⠀⡄⠰⠈⠃⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠉⠈⠁⠉⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣀⠤⠄⠒⣂⣀⣀⣐⡒⠠⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠔⡫⠔⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⢳⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠇⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⡆⢇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢦⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡡⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⠓⢬⣓⠦⢤⣀⣀⣀⣀⣀⣀⡤⠔⣊⡵⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠈⠉⠒⠲⠮⠭⠭⠴⠖⠚⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⢿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠈⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠒⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡠⠗⠁⠀⠀⠀⠀⠀⠀⠄⠀⠸⠘⡜⡆
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠦⢸⣇
⠀⠀⠀⠀⠀⢼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠂⢀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢢⢈⣽⡿
⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠇⠀⡤⡆⠁⠄⠠⠀⠀⠠⠐⠀⠀⠀⠐⠀⠀⠀⠐⡂⢸⠈⡈⣯⡇
⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⣚⣉⣍⣭⡅⠀⠀⠀⠡⠀⠠⠀⠀⠠⠀⡀⠀⠀⠀⠀⠂⢸⡏⡿⣾⡂
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠨⠀⠀⠀⠀⢥⡤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⣆⢷⢘⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡃⣨⢸⡏
⠀⠀⠀⠀⢸⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠺⡅⣧⡇⠁
⠀⠀⠀⠀⢸⠘⢣⣖⣒⣒⣢⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⢄⣄⣔⡈⡇⣷⠃
⠀⠀⠀⠀⠀⢇⣉⠉⠉⠉⠉⠙⠉⠛⠛⠻⠿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⠿⠿⠿⠛⣛⡃⡛⠉⠘⠋⠘⢑⣱⣿
⠀⠀⠀⠀⠀⠀⢀⣈⠙⠩⠮⠁⠀⠀⠀⠀⠀⠀⠒⣊⡹⠿⡿⣶⣶⣾⢟⡿⢝⡱⠶⠀⠒⠪⠍⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡟⡀
⠀⠀⠀⠀⠀⠐⠛⠒⠉⠲⠴⠶⠦⠤⠤⠤⠶⠖⠒⠲⠾⠷⢴⣿⣹⣽⡾⠿⠛⠒⠛⠓⠒⠂⠂⠄⠰⠰⠦⠶⠾⠛⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣯⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣀⣀⣀⣀⣀⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠉⠉⣿⡏⠉⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣠⣤⣤⣶⣿⣷⣤⣤⣤⣀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⣴⣾⣿⡿⣿⡛⠿⠿⠿⠿⠿⠟⣻⠿⣿⣿⣶⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋⢁⣴⣿⡿⡿⠟⠂⠉⠀⠐⠀⠁⠸⠀⠁⠐⠀⠈⠁⠋⠟⡿⣿⣷⣄⠉⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢀⣴⣿⣟⠟⡈⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠙⢟⣿⣷⣄⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢆⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠐⠦⢄⢼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠠⣪⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣗⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢔⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣦⢜⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢘⢤⣪⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣦⣤⣂⣠⣡⣬⢤⣥⣀⣲⣬⣶⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⡴⠊⠉⣉⠉⠑⢄⠀⠀⠀⠀⠀⠀⢠⡴⠊⢉⣉⡉⠛⢧⠀⠀⠀⠀⠀⠀⠀⠐⠀⣉⡉⠑⠆
⠀⠀⠀⠀⡼⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⠀⠀⢰⡿⠀⠀⠈⢀⡇⠀⠈⣧⠄⠀⠀⠀⠠⠁⠀⠀⢁⡷
⠀⠀⠀⠀⢳⠀⠀⠀⣿⠀⠀⠀⡞⠀⠀⠀⠀⠈⢧⠀⠀⢸⠋⠀⠀⠀⡿⠀⠀⠀⠀⠐⠀⠀⢀⡈⡷⠀⠀⠘
⠀⠀⠀⠀⠀⠳⣄⣀⠉⣀⡠⠞⠀⠀⠀⠀⠀⠀⠘⣷⣄⣘⠛⣃⣠⡞⠁⠀⠀⠀⠀⠀⠔⢄⣀⠉⢁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⠴⠚⠉⠓⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠛⠳⢤⡀
⠀⠀⠀⠀⣰⠃⠀⢸⠀⡇⠀⠈⢳⠀⠀⠀⠀⢠⡟⠀⠀⣾⠉⠀⠀⠹⡆⠀⠀⠀⠀⣰⠃⠀⢰⡏⠃⠀⠀⢳
⠀⠀⠀⠀⢿⠀⠀⠸⠶⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠙⢹⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⢸⡟⡆⠀⠀⢸⠃
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠇⢀⡠⠋⠀⠀⠀⠀⠀⠳⣄⠀⠙⠞⠀⣀⡜⠁⠀⠀⠀⠀⠈⢧⣀⠘⠷⠃⢀⣠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠋
⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠲⢦⣦
⠀⠀⠀⠀⣰⠃⠀⠐⢲⠂⠀⠘⡆⠀⠀⠀⠀⠀⡜⠀⠀⡖⣦⠀⠀⢣⠀⠀⠀⠀⠀⣼⠏⠀⠀⡞⣦⠀⠈⢦
⠀⠀⠀⠀⣯⠀⠀⠀⡾⠀⠀⠀⣹⠀⠀⠀⠀⠸⡅⠀⠀⡷⣿⠀⠀⢀⡆⠀⠀⠀⠀⢻⠀⠀⠀⠧⣿⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠀⠰⠇⠀⠀⡠⠃⠀⠀⠀⠀⠀⢣⡀⠀⠧⠟⠀⢀⡜⠀⠀⠀⠀⠀⠘⣧⡀⠀⠦⠟⠀⣠⠏
⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⢀⣤⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣈⠉⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⣤⡔⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣩⡦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣦⠀⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⡶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠰⣦⣄⡀⠀⠀⠀⠀⠈⣡⣀⣀
⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⡀⠀⠀⠈⠛⠛⠿⠿⣶⣄
⠀⠀⠀⢰⠇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣷⣶⣤⣤⣀⡀⠀⠈⠙⠿⣷⣶⣶⠶⠶⣦⣄⠀⢹⡆
⠀⠀⠀⣿⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⠿⠛⠛⠛⠿⢶⣦⣄⠈⠙⢿⡆⠀⠀⢿⡆⠘⣿
⠀⠀⠀⢿⡇⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡧⣤⣤⣤⣤⣶⣶⣶⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⠀⠀⢸⣧⠀⢿
⠀⠀⠀⠸⣿⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⢡⡆⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⡇⠀⢀⣼⠇⠀⣼
⠀⠀⠀⠀⣿⡇⠀⢿⣇⢲⡀⠀⠀⠀⠀⣰⣿⢃⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣿⠃⠀⠺⣷⣄⡼⠟
⠀⠀⠀⠀⣿⣧⠀⢸⣿⣾⠇⠀⠀⢀⣼⠟⢁⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡉⠀⣀⣼⠏⠀⠀⠀⠈⠉
⠀⠀⠀⢠⣿⡏⠀⢸⣿⡏⠀⠀⢠⣾⠋⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠘⠋⠁
⠀⠀⣰⢿⣿⠃⠀⣸⣿⠀⠀⢀⣿⠃⠀⠀⣿⡇
⠀⠈⣡⣿⠏⠀⢠⡿⢻⣆⠀⠘⣷⡰⣄⠀⠘⣿⡄
⣴⠾⠛⠁⢀⣴⠟⠁⠈⢿⣆⠀⠹⣷⡜⢷⣄⠈⢿⣆
⠀⠀⠲⠚⠋⠁⠀⠀⠀⠀⠹⣧⠀⠘⢿⣎⡻⠂⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⠇⠺⠟⠛
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠟⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠛⠛⠛⠻⠿⠶⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠾⠿⠛⠛⠛⠋⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣶⠀⢰⠏⠁⠉⣧⠀⢰⡖⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢹⡇⢸⡄⠀⠀⣼⠀⣾⠇⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠈⣿⠛⠙⠒⠚⠉⢻⡿⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢹⣇⠀⠀⠀⠀⣾⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣈⣿⣀⣀⣀⣰⣟⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⣛⣛⣛⣛⣛⣛⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⢶⡶⢶⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⣿⡟⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠛⠛⠻⠿⠿⠶⠶⢶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⠶⠿⠿⠛⠛⢻⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠿⠶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⣿⡇⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠶⢶⣦⣤⣤⣄⣀⡈⠻⣷⣿⣷⡿⠋⣀⣀⣤⣤⣤⡶⠶⠾⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠟⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡉⡤⢤⠀⠀⠀⠀⠀⡠⢔⣬⢖⡐⡀⠀⢨⢙⡟⠂⡇
⠀⠀⠀⠀⢀⣠⣄⣤⣖⡔⡔⠀⠀⡠⠨⡠⡞⠁⠀⠈⠙⠾⢶⠀⠀⠀⠀⢠⠇⠀⡥⣕⣋⠉⣷⠃⢁⠈⢹
⠀⠀⢀⡴⠋⢀⡤⠤⡑⠗⢣⢀⡬⠣⢽⡁⠘⠚⠉⢹⠀⠀⢸⠀⠀⠀⠀⣼⠀⠘⠧⣀⠀⡹⠁⢠⢺⠀⢸⡄
⠀⠀⡞⠀⣰⠃⠀⠀⠘⣆⠼⠋⠀⢀⠼⡇⠀⣀⠴⠋⠀⣠⢏⠀⢀⡀⠀⠉⠉⢢⠀⠈⢻⠁⠐⠓⠚⠀⠘⠓⣦
⠀⢸⠃⠀⣇⠀⠀⠀⢰⠛⢦⡖⢻⠀⠀⡗⠉⠁⠀⣤⡚⢁⣮⣴⣥⣵⠖⠲⠤⠞⠀⠀⢸⠦⠤⠤⢤⠀⢠⡖⠟
⠀⠘⣇⠀⠘⠦⠤⠖⠃⠀⡼⠁⢸⠀⠀⣧⣄⣀⠀⠀⡸⠉⢈⠥⢍⠉⢳⣄⣀⣀⣠⠴⠃⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠈⠲⣄⣀⣀⣀⣠⠞⠁⠀⢸⠀⠀⢿⠀⠈⠉⠓⢧⡀⠘⠲⠚⢀⡼⠀⠀⠀⠀⡀⢦⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢳⡖⠚⠀⠀⠀⢠⠖⠉⢀⣀⣀⠉⠲⣄⡠⠶⠶⣤⡸⣤⡀⢠⠚⢄
⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠲⠴⢞⠥⢹⣀⠀⠀⠀⣏⠀⠐⣇⣀⣸⠆⡰⠋⠀⣠⠴⠦⠤⣧⢆⡺⠫⠎⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠾⣤⣀⣀⡀⠈⠉⠁⣸⠤⢤⢡⡛⣐⣇⠈⠉⠁⣰⠁⠀⠀⠉⠉⠓⢤⡇⠀⡔⠉⠙⡆⡸⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⠔⠿⠤⣀⡉⠑⠚⡩⠋⠉⠉⣿⠀⢠⠋⠉⢱⠀⠘⣇⠀⠙⠶⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡴⠋⠀⠀⢠⠔⠊⠀⣠⠊⠀⠀⠀⠀⢹⡀⠈⠳⠴⠊⠀⣸⠉⠲⠤⡤⠴⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡞⠁⠀⠀⠀⠘⠶⢤⡀⠈⠱⡄⠀⠀⠀⠀⠙⠲⠤⠤⠴⠚⠁⠀⠀⣀⡤⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠻⠤⢤⡞⠀⠀⠀⠀⡴⠤⣤⠼⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⡿⠋⠻⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠿⣭⡿⠟⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⢀
⠀⠀⠀⠀⠀⠀⣿⣏⠀⠈⢉⣩⠭⠖⠒⠉⣉⠒⠒⠀⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠀⡁⠀⢠⣀⢀
⠀⠀⠀⠀⠀⠀⣿⣿⡄⢠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⠈⠉⢀⣌⣽⣷⣿⣶
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⢀⣀⡤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠛⠾⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢠⣶⣤⣡⠄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠉⠉⠉⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠉⠐⠚⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠁⢸⣇⠀⠀⠀⠘⡟⠛⠛⠛⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠢⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⡆⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣇⢸⡿⠇⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠊⠉⠈⠁⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣀⡤⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⣤⣀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠀⢈⠆⠀⢄⢀⠀⠠⡊⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⢊⠡⠀⠀⠀⠃⠀⠀⠄⠑⠠⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠒⡁⠔⢉⣠⡤⠖⠒⠒⡒⠒⠲⢤⣄⠈⠂⠈⠂⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡴⠋⠁⡆⠀⠀⡄⠱⠄⠀⠀⠌⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠏⡀⠀⠀⠇⠸⠀⠁⠈⠀⠀⠇⠀⢀⠀⠹⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠀⢹⢰⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠁⠹⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⣠⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⡀⢠⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠁⠘⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⡈⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⠀⢠⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣆⠁⠀⠀⢨⠁⠀⢀⠀⠀⠰⡂⠀⠀⠈⣠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠳⣄⡐⠂⠀⠀⢛⡃⠀⠀⠰⢂⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢠⡈⠙⠲⠦⠤⠨⠄⠤⠴⠖⠋⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠤⠀⠀⠂⠈⠀⠑⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣮⣉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⢣⠈⠙⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠈⢧⠀⠀⠘⢿⣿⡄⠀⠀⢀⣀⠤⠒⠒⠉⠉⢱⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⣠⡔⠒⠒⠒⠒⠒⠒⢺⣿⣷⢶⠊⠉⠀⠀⠀⠀⠀⠀⢸⡽⣤⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⢷⡄⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠓⢄⠀⠀⠙⢶⣿⣇⣀⣀⣀⣤⣾⣿⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢄⠀⠀⣹⡿⠿⠿⠿⠟⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣸
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠜⠓⠊⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡷⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⡎⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⣸⡿⣿⡇
⠀⠀⠈⠢⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣤⠶⠯⠓⠒⠒⠚⠒⣿⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⡖⠒⢲⠶⣶⣾⣿⣷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠾⠀⠉⠉⠉⠁

⠀⣠⣤⡀⠀⠀⣠⣶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠂⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠙⠋⡇⠀⣾⡿⠃⠀⠀⠀⠘⠆⣀⡀⠀⠀⣼⣿⠟⠁⠀⠀⠀⢀⠀⣴⠏⠀⠀⠀⠀⠀⢀⣾⢃⣒⡀⣚
⠀⠀⢰⠁⡸⠋⠀⠀⠀⠀⠀⢠⣾⣿⣿⡦⢰⠿⠃⠀⠀⠀⡀⣶⠈⡄⣀⡀⠀⠀⠀⠀⠀⠘⠁⠸⣛⡧⡟
⠀⠀⠈⠦⠄⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠃⠀⠀⠀⠀⠀⠙⠲⠬⠥⠴⠒⠁⠀⠀⠠⠟⠀⠀⠸⠿⠇⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣄
⡖⠒⠒⠚⠉⠈⠓⠦⠤⡄⠀⠀⠀⡹⠋⠉⢯⠀⠀⠀⠀⠀⠀⢷⣶⠆⠀⡀⠀⠀⠀⠀⠀⣸⣿⣽⣿⣀⣠
⠀⠘⠀⠈⠀⡆⣁⠐⠀⠂⠀⠀⢰⡇⢀⡀⢈⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⢻⣿⠛⠉⠙
⠈⠢⠐⢛⠜⠠⠈⠤⠂⠀⠀⠀⠀⢙⣿⣿⡏⠁⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠀⠀⠀⠀⠀⢠⠞⠙⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠰⢤⣀⡤⠂
⠀⠐⠀⠀⣶⣆⠀⠐⠀⠀⠀⠀⠙⠳⣶⣏⣲⠞⠀⠀⠀⠈⠙⠦⠍⠉⠉⠁⠀⠀⠀⠀⠀⠋⠁⣤⠉⠉
⠀⠀⢰⣾⣿⣿⣷⠄⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠐⠓⢺⣿⡗⠚
⢀⣤⣼⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⢨⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⢀⡤⠀⠤⣀⡀⣤⠠⣄⠀⠀⠀⠀⡰⠊⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠖⣒⠲⡄⠀⠀⠀⠀⠹⣿⣿⣿⠃
⢸⡇⠀⠀⢸⠀⠿⠀⣿⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠀⠀⢀⡴⢉⡊⠷⡬⠼⠀⠀⠀⠀⠀⠀⢹⣿⡇
⢸⣇⣀⡀⢸⠀⣀⣀⣿⠀⠀⠀⠀⠀⢳⣷⠀⠀⠀⠀⠀⠠⢇⠘⠟⡈⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠈⠉⠉⠉⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠋⠑⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⢚⠷⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⡀⠀⣠⠴⠦⡶⣥⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠉⠓⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⣧⣛⣇⡞⠁⠀⣀⣀⠀⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⡻⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢹⡉⠲⠋⠁⢸⡆⠀⢠⡇⠀⠀⠀⣀⠊⣰⢊⡙⢆⠀⠀⠀⢀⠔⠁⠀⠐⠣⠤⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡖⠁⢀⡄⠘⠉⣇⠀⠀⢀⠞⠀⢀⡾⠀⣾⠉⠉⠉⠑⠉⠮⠴⠋⢹⠀⠀⠈⠛⠲⠦⢤⣀⠀⠀⠀⠉⡶
⠀⠀⠀⠀⠀⠀⠀⠹⠶⠋⢹⡀⠀⢸⣄⠴⠃⠀⣠⣾⣤⣀⣿⠤⠤⠤⣤⠄⠀⠐⢁⡼⠀⠀⢀⠀⡠⠀⠀⡾⠀⢠⡟⠒⠃
⠀⠀⠀⠀⠀⢀⠄⣀⠴⡆⠈⣇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠃⠀⢀⡖⠉⣠⣅⣶⢇⠷⣸⡀⠘⠧⣤⡼
⠀⠀⠀⠀⠀⣢⣤⢥⣄⡰⣤⢹⡄⠀⢸⡗⠒⠒⠒⠒⠒⠒⠃⢀⡞⠀⠀⣰⠋⢠⠞⠁⢀⡤⠤⣍⠀⠙⢦
⠀⠀⠀⣰⠋⠀⠀⣀⣡⣀⡏⠀⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠇⢀⡏⠀⢰⠃⠀⠀⠀⢳⠀⠘⡇
⠀⠀⣼⠁⠀⠠⠯⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⢸⠃⡀⣿⡐⠆⠀⠀⢸⠀⠀⣿
⠀⠀⡇⠀⠀⢀⣠⣄⠀⠈⢳⠀⢠⠀⢀⣰⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⡼⣇⢤⡋⣌⡦⢤⣠⠞⠀⠀⡟⠀⠀⠀⢀
⠀⠘⡇⠀⠠⡏⠀⠀⣷⢀⡬⠧⡄⢮⡡⢮⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢁⣭⣬⣀⣸⠁⠀⢀⡼⣃⡤⠶⢶⠡⣰⠐⢶
⠀⠀⢷⡀⠀⠓⠦⠖⠃⢸⣇⣀⣈⡁⠈⠉⡴⠁⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⡇⠀⢸⣳⣦⣤⣤⠤⠶⠋⡼⠁⠀⣀⢬⣉⠫⠥⣎
⠀⠀⠀⠳⢤⣀⣀⣀⣤⠖⠃⡠⠊⠁⢀⡞⠀⠀⠀⣰⣤⣭⣤⣵⠀⠀⠀⠀⠧⠤⣄⡀⠈⠑⣆⠀⠀⢸⡃⠀⢸⡁⠀⢈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢺⣀⣀⡀⠀⠉⠳⣤⠊⠁⣉⠭⡉⠉⢳⡀⠀⠀⠀⠀⠀⡷⠀⠀⢸⠂⠀⠈⢧⡀⠀⠙⠒⠋⠠⠔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣿⠀⠀⠧⡤⠟⠀⣸⠃⠀⡸⠉⠙⠉⠀⠀⢀⡼⢠⠶⠤⣤⣩⠷⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠁⠀⠀⣠⣯⠷⠀⠀⠀⠀⠾⢅⠀⠀⠙⠶⠤⠤⠤⠖⠋⠀⢿⣀⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣤⠤⠴⠚⣽⠁⠀⣰⠚⠛⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⠤⠤⠞⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡤⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⢠⡂⢲⡀⣴⣶⣦⡀⣶⣒⠂⢰⠶⢶⡖⠀⠀⣰⠐⢲⡴⠶⣶⡖⢠⠰⣆⠆⡐⣄⠀⣴⠖⠒⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠈⢁⣼⠹⡿⠀⣽⡇⢬⣉⡗⢰⠀⣾⠁⠀⠀⣿⢠⣬⠀⠾⢿⣄⢐⢸⣿⠇⠒⢸⣧⣹⡗⠒⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⣿⣿⢀⡀⢹⡄⢀⣾⠟⠀⣠⣷⣦⡀⢸⣿⠀⠀⠰⣿⠀⢶⠐⢶⡾⠇⢨⢘⠿⡀⠍⣸⣿⠘⢓⣒⣚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⢸⣿⠈⠃⠚⠰⠿⠿⠿⠇⣿⡇⡘⠃⢨⠭⡄⠀⠀⢸⠀⣿⢸⡇⢸⡇⠑⢾⣶⣷⣾⣿⠏⠸⠿⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⠟⢰⣋⣵⠐⠒⠂⠈⠀⣿⡟⢻⡇⢸⡀⡿⠀⠠⢖⡦⡾⡌⢛⢻⡗⠷⣶⢪⣍⢹⠓⠀⠐⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⡀⣰⠞⢻⠀⡆⠀⠀⠰⠂⣿⣇⣼⡇⢸⠛⣧⠀⢀⡘⣧⠙⠁⡀⡨⡁⢰⡟⠈⠛⣈⠙⡆⢸⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡽⠙⣒⡋⢀⣥⣝⡀⣾⢸⡎⣉⣋⣀⢸⡀⣿⠀⠀⢩⠱⣢⢀⡇⢀⡇⣿⠃⣾⠃⠤⠴⠃⣐⣦⢠⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⢸⣉⡁⣿⠋⣹⣷⠈⠼⠃⢉⣿⠃⢈⣛⡁⠀⠀⠸⠴⠥⠨⡌⠉⠧⣖⠂⣿⠻⣦⡆⣷⠀⣿⣸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣸⠇⣿⠿⢻⣿⢸⢿⡂⠈⢹⡇⠸⢤⡄⠀⠀⣺⣼⠃⡜⢱⡾⠀⣿⠃⣿⠀⣿⠤⠟⠉⠁⠘⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠅⠀⠻⠿⠿⠏⠸⠼⠑⠦⠾⠃⠯⠼⠃⠀⠐⠧⠼⠁⠐⠸⠯⠘⠤⠏⠻⠾⠟⠰⠂⠀⠀⠄⠅⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⢀⣀⠀⣀⠀⣀⣀⢀⣀⠀⣀⣀⢀⡀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡤⠘⢧⣜⠸⣿⡌⣑⡆⡏⢹⢺⣿⢨⡕⣺⣭⠃⠀⢰⡋⠀⢹⠤⣿⡇⠈⠀⠐⠛⠸⠟⠛⣿⡆⣭⣍⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠣⢰⡟⢹⣧⠈⣴⠒⠀⣥⠾⢀⠋⠰⠲⠶⣽⠧⠀⠀⡑⣲⠋⠀⢺⢶⡶⠶⢸⣻⣴⡟⠻⠿⠳⠆⠹⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠄⠠⣼⡟⢰⣿⢹⡆⣧⣴⠸⠠⠇⢁⠎⣿⣷⠀⢀⡴⣧⠀⣦⡉⠸⠟⣷⠰⡿⠻⠿⡿⠿⠇⢈⡟⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣦⣄⢸⣿⠀⠻⠞⢡⣶⠀⡄⢦⢀⣿⢀⣭⡍⠀⠘⠧⣿⠸⠷⡷⠰⣦⣿⠸⣿⢰⣿⠅⢀⠘⠋⢸⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⠻⠿⠋⢸⠀⡇⢻⣿⣦⣅⡜⠀⠋⠞⣶⡄⠀⠀⡉⠵⢚⣶⢴⠶⠶⠶⣶⣴⣆⠀⠀⠔⠀⠤⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣶⠑⢹⡴⣿⢈⡐⡇⣴⠙⠻⢿⣿⣆⠺⠴⠟⠀⠀⢀⡅⢘⣿⢅⡆⠈⣱⢞⣁⣿⣟⣩⣭⠙⢬⠋⣈⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡶⡾⢿⡃⣿⢈⣌⡈⢿⡐⠇⠀⣿⠇⠀⡖⠒⣇⠀⠈⠲⠛⠋⢈⠋⢺⡇⢸⢲⣬⠁⢋⣿⠀⢢⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇⠿⣿⣸⡇⠀⢁⣀⠈⣥⡄⠀⣇⣀⡄⠀⠐⠟⣿⠈⢸⠰⣿⣿⡏⣭⣐⢠⣆⢻⡗⡃⠸⢷⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⢠⡞⠙⡆⣿⢻⡇⠀⠅⠄⠄⠰⡇⠀⡇⠀⡇⠀⢠⡾⢋⠀⢸⢼⣏⢹⡧⡾⢠⡏⡛⡋⢸⢾⠰⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠛⠁⠈⠛⠁⠀⠂⠈⠀⠛⠃⠘⠓⠚⠃⠀⠈⠛⠛⠀⠛⠈⠛⠛⠁⠃⠘⠁⠁⠈⠈⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⣌⠻⣄⣀⣀⣀⣀⣀⣀⣁⣀⣈⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠋⠀⠈⢦⡘⢦⡤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⠞⠁⠀⠀⠀⠀⠙⢦⠙⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠻⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣴⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣬⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡖⢒⠒⠒⠒⠒⠒⡆⠀⠀⢸⡇⠀⢰⠒⠒⢲⠒⠒⣶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⡏⠀⠀⠀⠀⡇⡇⠀⠀⢸⡇⠀⢸⣀⣀⣸⣀⣀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⢸⡇⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⡇⠀⠀⠐⠆⡇⡇⠀⠀⢸⡇⠀⠘⠒⠒⠚⠒⠒⠛⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣀⣀⣇⣇⣀⣀⣀⣀⣇⣇⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠟⠚⠟⠀⠀⠀⠀⢀⡠⢤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣈⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⠀⠀⠀⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠟⠀⠀⠒⠂⠀⠀⠀⠿⠕⠈⠀⠸⣧⡄⠀⠀⠀⠀⠀⠀⢆⡀⠂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⠸⠄⠀⡄⠀⢴⡌⠁⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣦⠾⠃⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⡤⠀⠀⠀⠉⠉⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⣠⣿⣭⠴⠟⣿⣤⣤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⠿⠿⠿⠿⠯⠥⠭⠷⠶⠄
⠀⠀⠀⢰⣴⣆⢠⣿⠿⠿⠀⠀⠀⢀⡶⠦⡀⠀⠔⠶⠞⠒⠀⠀⠸⠾⠷⠟⠀⠀⠀⡤⠤⠤⠤⢄
⠀⠀⢠⡟⠛⢻⡎⠟⠠⡊⠀⠰⠷⠿⠿⠗⠹⣣⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢰⣤⣶
⠀⠀⠀⠃⠀⠘⠁⠐⠀⠉⠀⠀⠀⠀⣀⣀⣤⡻⢻⣀⡀⠀⢀⣀⣀⣀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣾⡀⠀⠀⠀⠀⠀⠀⠀⣈⡉⠁
⠀⠀⠀⢀⣈⡁⠀⠁⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠆⠀⣤⣤⠄⠀⠀⠀⠈⠙⠿⢿⣿⡿⠟⠃⠀⠂⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣯⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⢠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⣿⣷⠆⠀⡀⢀⠀⢨⡛⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣧⣆⠚⠉⠁⠐⠆⢀⠉⣀⣀⠘⠀⢤⣼⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠂⢰⣎⢀⣺⠋⡤⠄⢹⣿⢺⣿⣀⣆⢈⢋⠡⣾⢉⡁
⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⣠⡄⢻⡿⣷⣤⠀⠀⠀⢹⣿⣷⡧⠴⠿⣷⢶⠿⣋⠁⡀
⠀⠀⠀⠀⠀⠀⣸⣿⢄⣦⣘⣃⣤⠼⢿⣇⠠⠰⣿⡐⣿⣾⠁⠀⠀⠀⠙⣴⣟⢁⡤⢚⡂
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⢀⣩⣷⡦⣴⡟⠀⢀⠀⠀⢈⣄⣴⣿⣦⣀⡀⢻⡋⡉⡁⢵⣶⠂
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢘⠛⠅⠀⠒⠂⠀⠋⢁⡄⠋⠛⠛⠙⢛⣅⠁⠈⣀⠈⠃⠀⠹⠓
⠀⠀⠀⠀⠀⠀⢸⣿⡠⠞⡋⠙⠻⣷⣾⠦⣤⣤⣍⡀⠀⠀⠵⠀⠉⠋⠁⣺⣿⠓⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠐⣀⡠⠀⠀⢉⠚⠛⣶⣿⡿⠃⠒⢂⡁⠀⣬⡏⠹⣯⣄⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣮⡗⢻⡀⠀⠄⢛⡿⠓⣠⡃⢶⡷⢒⣀⣤⡔⠀⢀⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡷⣶⣟⠆⠀⠛⢩⢃⠒⢿⣯⡀⣀⣠⠏⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣦⣅⠻⡟⠀⣰⣠⡇⠙⠟⠀⠈⠉⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣬⣥⣤⣤⣤⣤⣴⣾⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⣺⣿⣿⣿⣿⣿⣖⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣬⣭⣍⣉⣩⣭⣥⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠯⠿⠟⠛⠛⠉

⠀⠰⠠⣿⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠄⠒⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠐⠒⠢⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠂⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠶⠿⠿⠿⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠱⠦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠦⣤⣴⣶⣶⡶⠿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠶⣶⣖⠛⠂
⠀⠀⠀⠀⠀⠀⡠⣟⡂⢤⣀⠀⠀⠈⠙⠓⠦⢤⣀⠀⢹⣯⣄⣀⣀⣀⣠⡤⠤⠶⣒⣚⣩⣭⣽⡿⠷⠶⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢐⠀⠀⠉⠂⠈⠑⠢⢄⡀⠀⠀⠀⠈⠙⠚⠿⢿⣿⣿⠷⠶⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⣘⣇⡀
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠠⣌⠑⠢⢄⡀⠀⠀⠀⢀⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢶⣒⣛⣯⣽⠅
⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣈⠑⠢⢀⣸⣿⣖⣀⣀⣀⣤⣤⣶⣖⣻⣽⣿⡶⠾⠛⠛⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡄⢈⣙⣻⣭⡽⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠂⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡊⣟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⣿⣧⠄⠀⠀⠀⢀⣀⣀⡤⠴⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡷⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠑⣿⠿⣿⠿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠄⢴⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⢺⣟⢿⡆
⠀⠀⣠⠸⠂⠛⢘⡗⠄⠀⠀⠀⢸⣿⣯⣿⡟⢂⠒⠈⠀⠀⠀⠀⠉⠀⣤⡶⣄⢹⣿⠃⠀⠀⠀⢀⣴⣾⡿⠎⠛⠊
⠀⠁⣬⠀⠀⠀⠘⠃⡀⠀⠀⠀⠀⠛⠶⠟⢅⠒⠂⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⢘⠃⠀⠀⠀⡔⠛⠋⠁
⠀⠐⠄⠛⠠⠂⠘⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⡶
⠀⠀⢀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠐⠻⡿⣿⠗⠀⠄⠀⠀⠀⢠⡶⠶⠿⠶⠶⢶⢶⡄⠀⠀⠀⠀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁
⠀⠀⢠⠓⡾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡞⡀⠀⠀⠀⠀⠀⠀⢠⢀⣠⣿⣿⡁⠂⢀⡀
⠀⠀⢘⠁⠘⠀⠀⠂⠀⠀⠀⢸⣇⣀⣀⣀⣀⣸⣿⡇⠀⠀⠀⠀⢀⣸⠇⠙⠲⣌⠀⣄⠀⠀⠈⠂⢕⢠⠀⡀⠔⠈
⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠛⠿⠆⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⣀⡀
⠀⠀⣀⢠⡠⠄⠂⠓⠀⠀⠀⠀⣉⠇⠀⢏⡅⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⡇⡤⣀⠀⠀⠀⠀⠀⠀⣹⢺⠗⣯⠤⢠
⠀⢰⣿⣴⣦⣦⣴⣴⠀⠀⠀⠀⡀⠀⢀⣀⣤⣀⣀⣀⠀⠀⠀⢶⡁⢾⡠⢇⣹⢎⡱⠆⠀⠀⠀⢠⡇⢸⠀⢸⡆
⠀⢸⣿⡿⣿⡿⣿⣿⠀⠀⠀⠀⠠⡀⠀⠙⢻⡟⠋⠁⠀⠀⠀⢸⡇⠸⠃⠘⠛⠈⣿⠀⠀⠀⠀⠘⠧⣼⣤⠼⠃
⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠂⠀⠃⠐⠓⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢫⠏⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠂⠀⠈⠉⠁⠀⠀⠀⡞⡸⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠠⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⢢⠀⡀⠀⠀⠀⠀⠀⣴⡇⠀⠀⡆⣥⣀⡀
⠀⠀⠀⠀⠁⠒⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠐⠁⠀⠀⠀⠤⠄⢠⡤⡀⠀⠀⠀⠀⢱⢙⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⢀⣴⣡⠊⠀⢀⡀⠀⠀⠀⠀⠀⠢⠷⠄⠀⠀⠀⡖⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣮⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠉⠉⠈⠈⠉⠁⠀⠁⠀⠉⠉⠀⠀⠀⠀⠉⠐⠐⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠊⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠖⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢒⠆⠀⠀⠀⠀⠀⠀⠒⢀⡤⡀
⠀⠀⠀⠀⠀⢀⡤⣠⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡌⠀⠀⠀⠀⡠⡴⠀⠀⠀⢱⢹
⠀⢀⠄⠀⠀⠀⠀⠀⠉⠙⢢⠢⡀⠀⠀⠀⠀⠀⠀⢀⣎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠜⠀⠀⠀⠀⠀⠢⠧⠄⠀⠀⡨⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠱⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⡖⡶⠀⠀⠀⠀⠱⡈⡄⠀⠀⠀⠀⠀⠀⡜⠐⠁⠀⠀⠀⠀⠀⢀⠔⡶⠀⠀⠀⠈⢢⠱
⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠎⡐⠀⠀⠀⠀⠀⡇⢇⠀⢀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡸⠠⠁⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⢨⠆⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⢜⡴⠁⠀⠀⠀⠀⠀⠉⠐⠓⠁⠀⠀⣰⢁⠆⠀⠀⠀⠀⣰⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠚⠂⠀⠀⠀⢀⠔⡸
⠀⠀⠀⠀⠀⠀⠴⠶⠾⠷⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣠⡤⠾⠵⠁⠀⠠⠤⠤⠤⠯⠾⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⠤⠤⠤⠴⠶⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠑⢬⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⠀⠀⡠⡺⠋⠀⠀⠀⠀⠀⠀⠀⠱⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣱⣄⠀⠀⠀⢀⢇⡇⠀⠀⠀⠀⡘⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠊⡘⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠎⠀⠀⠀⠀⠀⠀⢡⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⠤⠤⡄⠀⠀⠀⠀⢀⡌⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⢀⣠⣣⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⣿⡏⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⣸⣿⣿⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣴⡾⣿⣟⡀⣠⣿⠿⣿⣇⠀⣹⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣛⣫⢮⣛⣯⢮⣛⡿⢮⣛⡿⢮⣛⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣿⠓⠒⠛⠓⠚⢿⠛⠛⠛⠓⠒⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠙⠛⠓⣦⣀⠀⠀⢸⠀⠀⢀⠔⢶⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣿⠑⠏⢡⢚⢷⣾⡤⣀⠈⢉⣣⢤⠤⣹⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡄⢤⠢⣻⠓⠀⠁⢸⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⡄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⣠⣦⣈⣠⣦⣉⣠⣦⣉⣴⣦⣉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣀⡤⠖⠛⠉⠉⠀⠀⣄⢀⡀⠀⠉⠉⠛⠲⢤⣀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠚⠁⢀⢀⠀⠀⠀⠀⠀⠟⠰⠧⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠛⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢦⡀⠙⢷⡄
⠀⠀⠀⠀⣰⡿⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢿⡄
⠀⠀⠀⢠⡿⠁⢠⠏⢠⢠⣶⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢻⡄⠈⣿⡄
⠀⠀⠀⣾⠃⢀⡟⠀⠈⠀⠉⠁⠀⠀⠀⠀⠘⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢷⠀⠸⣷
⠀⠀⢰⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡄
⠀⠀⢸⡇⠀⣾⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡇⠀⢹⡇
⠀⠀⢸⡇⠀⢻⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠾⠃⡇⠀⢸⡇
⠀⠀⠸⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠀⢰⡇⠀⣿⠃
⠀⠀⠀⢿⡆⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠘⣿⡀⠘⣧⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡇⠀⣼⠃⢀⣿⠃
⠀⠀⠀⠀⠘⣷⡀⠘⢧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣄⠈⠳⣄⠀⠀⠀⣲⡖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢀⣰⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⢦⣀⠙⠁⠀⠀⠀⠀⢼⣧⡀⠀⠀⠀⠀⠈⠟⠀⣀⡴⠛⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠉⠛⠶⢤⣄⣀⡀⠈⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠣⠀⠀⠻⠋⢀⠀⢀⠀⠀⠀⠀⠀⣤⣾⣷⣤⡀⠀⠀⠀⣌⠀⠀⠀⠸⠟⠃⢰⠀⠀⠀⡀
⠀⢀⠠⣑⠦⠥⠯⢔⡨⠀⠐⠀⠀⠀⢠⣴⣶⣴⣶⡄⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠁⠀⠀⢠⣿⣿⣿⣶⣧⣆
⠀⠀⠑⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠟⠍⠩⠯⠩⠻⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⢾⡭⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣗⣛⡯
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠂⠀⠀⠀⠀⠘⠋⠁⠈⠁⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣾⣿⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣵⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣷⠦
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣯⣟⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠿⠿⠿⢀⣃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⠧⢼⠤⢤⢤⠧⢼⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣷⣿⣛⣿
⠀⠀⠀⢺⣿⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⣚⢞⡒⣖⢞⡲⠃⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⡿⠿
⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢄⡠⠄⢀⠀⠄⡀⠀⠀⠀⣀⣠⣄⣀⣀⠉⠉⠉⠀⣀⣀⠀⠀⠀⠀⠄⢀⡤⣀⠀⠀⠄⠠⠁⠈⠀⠀⠀⠀⠀⠸⢂⠀⢀
⠀⠀⠀⠛⠀⠄⢠⠀⠀⡞⠁⢘⠛⢻⠗⠷⠀⠀⢰⠉⠉⠙⡄⠀⠀⠀⡸⠉⠈⢣⠀⠀⠀⠀⠀⡀⢈⣁⣀⣀⣠⡠⠀⠐⢣
⠀⠀⠀⠁⠀⠀⠈⠀⢀⠀⠀⠀⢀⠀⠀⡘⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠁⠀⠀⠈⠀⣀⡀⠠⡇⢀⡀⢸⠒⠒⠒⡃⠀⡀⠈
⠀⠀⠀⠀⣄⣠⠀⠀⢨⡀⡀⡅⠀⢺⠀⠀⡀
⠀⢠⠒⠀⠈⠁⠀⠠⡀⢵⡃⠀⠀⠀⠒⠂⠀⠀⠀⠈⠉⠉⠀⣠⡄⢀⠀⠀⠀⠀⢠⠶⠦⡀⠰⠂⡴⢒⣤⡤⠀⡀
⠀⠈⠄⡠⠀⠀⠀⡀⣈⣈⣹⠀⠀⠀⣄⠁⠁⠀⠀⠀⠀⠀⡐⠀⠀⠙⡀⠀⠀⠀⡇⠈⠀⢧⠀⠀⠀⢨⠁⠈⠉⡇
⠀⠄⠀⠁⠀⠀⠠⠡⠠⠄⠈⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠁⠠⠦⠀⠃⠀⠀⠀⣇⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠨⠀⠁⠀⠀⠀⠀⠀⡆⠀⣆⠀⠀⠀⠀⣾⠁⠈⣧⠀⠀⠀⠀⠀⠀⠠
⠶⠶⠾⡖⠷⠀⢢⠀⠀⠀⠀⠰⠆⠀⠀⠀⠁⠀⠀⡤⡐⢄⠀⠀⠀⠀⣴⡀⠀⢢⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠀⠠⠆
⠀⠁⠀⠖⠂⠐⠸⠀⠀⠀⡀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡂⠀⠀⠸⠑⠅⠊⠀⠀⠀⠠⢠⠀⠀⠠⠅⠀⠀⢠
⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠰⠀⠀⠀⠃⠐⠀⠀⠀⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠭⢗⡆⠀⠀⠀⡀⣆⠈⠉⠀⠀⠀⠠⠘⠁⠀⠁⠀⠀⠀⠀⠀⠁⣲⠄⠀⠀⠀⢠⡤⠲⣤⠂⠀⠀⠀⡰⠀⠒⡄⠀⠔⠀⣴
⠀⠀⡆⠀⠀⠀⡇⠀⠀⢲⠀⠀⠀⠀⡃⠀⠀⢠⠀⠀⠀⢸⡿⠄⠼⡦⠀⠀⠀⢺⣖⣀⣻⡇⠐⠒⠂⡔⠀⠀⠘⡂⠀⠀⢋
⠐⠀⠀⠀⠀⠀⠡⡀⢂⡄⠁⠀⠀⠀⠓⠈⠠⠎⠀⠀⠀⢨⠠⠀⢠⠂⠀⠀⠀⢧⠀⠄⠀⡄⠀⠀⠀⠘⠐⠓⠈
⠀⠠⠀⠀⠀⠀⠀⠳⠾⠗⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠁⠀⢀⠀⠪⠄⠀⠀⠂⠀⠀⠀⠀⠄⠀⠄⠀⠀⢀⣀
⡄⠀⠀⠀⠀⠁⠢⠀⠀⠀⢠⠒⠴⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠘⡄⣤⠀⠀⠠⠂⢠⡄⠀⠀⣤⠜⠍⣭⣀
⠀⠀⠀⢱⠀⠀⠀⡆⠀⠘⠧⠀⠀⠀⠠⠀⠀⠀⠀⡄⢀⠀⡞⠀⠀⠀⠁⡀⠀⢀⠃⠀⠀⠀⣿⡆⠨⣦⠀⠀⠙⣷
⠀⠀⠀⠈⢦⢂⣸⡃⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⢀⠉⢨⠀⠀⠀⠀⠀⠐⠈⠈⠀⠀⠀⢠⠁⠀⢀⡌⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠂⠂⠀⣀⠀⢀⠀⠁⠠⡀⠀⢀⣀⠈⠘⠀⠈⠏⠀⠀⠀⠒
⠀⢰⠁⠀⢈⣤⢌⠀⠀⠀⠤⠀⠀⠀⡄⠀⠰⠶⣄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠘⢡⠀⠑⠀⠀⢰⡀⠀⠀⠀⠄
⠀⠀⢀⠘⠀⠀⠁⢇⢠⡀⠖⠀⠀⠀⢄⠀⠀⠀⡟⠀⠀⠀⢉⠀⡀⠀⡇⠀⠀⠈⠅⠀⠀⠘⠁⠀⠀⢰⠀⠀⠀⡞⠁⠀⠰
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⠀⠀⢀⠀⠀⡂⠠⠀⠀⠀⢆⠀⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏⣤⣤⣤⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⠟⠛⠏⠀⠈⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣀⣤⣶⣶⣶⣤⡛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⢿⣿⠟⠋⠉⠉⠀⢀⣤⣾⣿⣿⣿⣿⡿⢛⠛⣦⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣟⢿⣌⠛⠛⠁⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠿⠋⠀⠈⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠻⠳⣦⣤⣤⣴⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠉⠙⢿⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣀⣠⣤⣤⣤⣤⡖⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣧⡤⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⢄⡁⠠⠴⢋⣁⣠⣴⣾⣿⠀⠒⠒⣒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠠⠶⠿⠿⠿⠻⠟⣷⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠁⡄⠂⡀⢘⠃⠌⠁⠀⠀⠀⠉⢻⣿⣶⣶⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡯⠎
⠀⠀⠀⠀⠀⠀⢀⠾⠀⣧⣄⣄⣤⠀⠀⢅⢀⣀⣠⣼⣿⣿⡿⢡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣟⡧⠀⠀⣰
⠀⠀⠀⠀⠀⠀⣘⣀⣀⣛⣛⣛⣁⣀⣀⣀⣀⣉⣉⣉⣉⠁⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠓⠒⠈⠉⠉⠉⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣃⡀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠁⠀⠀⠀⠀⣀⣴⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⢀⢀⢀⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠈⢠⡆⢵⢫⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⣀⠀⠄⠠⠄
⠀⠀⠀⡥⢈⢷⡘⣇⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡇⠁⠀⠀⠀⠠⠈⠀⠀⢀
⠀⠀⠀⠘⢧⣰⣦⣲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠷⠀⠀⠀⠀⠀⠚⢀⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⢡⣿⠈⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣌⣨⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠂⠀⡸⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠓⠒⠓⠺⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣽⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡂⠀⠀⠀⡀⢃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠤⠧⠤⠤⠤⠄⢷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⢠⠿⣻⠷⠶⠶⠶⠿⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠠⣀⠸⠆⠃⠀⠀⠀⢠⠟⣲⠋⠀⠀⠀⠀⠀⠳⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠠⠒⠂⠁⠀⠀⠀⠀⠀⣠⣏⣠⡏⠀⠀⠀⠀⠀⠀⢀⣇⣜⣄⠀⠀⠀⠀⠀⠖⢦⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⢂
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣩⣤⣾⣟⠃⠂⠀⠀⠀⠀⠀⠀⠀⢠⢠⠄⠤⣶⣿⣿⠇⠀⠀⡀⣷⣿⠀⠀⠀⠀⠀⠀⠠⢰⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠟⡉⠀⠀⠀⠀⡀⢀⣀⣷⡙⠉⡸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠉⠀⠀⠀⢠⢁⡈⣁⣷⠀⠀⠀⠀⠀⠀⠓⢂⡄
⠀⠀⠀⠀⢀⢀⣄⣄⣀⡔⠐⠁⠀⠀⠀⠀⠀⡠⠋⢙⣥⣤⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢈⣂⠀⠀⠀⠀⡏⠀⠀⢋⣼⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⢠⡟⠃⠁⢠⠟⠹⣧⢱⠠⡤⠀⠀⠊⠀⣰⠟⣧⠀⠁⠈⢳⡄⠀⠀⠀⠈⠀⠀⢉⡍⢉⣈⡛⠀⠀⠀⠀⠃⠀⠆⠛⠛⡃⠀⠀⠀⠈⢀
⠀⠀⠀⡎⠀⠀⣴⣷⣄⣀⣿⠀⣯⣵⠂⠀⠀⢘⡇⠀⠘⢧⡇⠀⠀⡗⠀⠀⠀⠀⠂⠀⢘⣀⣨⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢶⣶⡟
⠀⠀⠀⢧⠀⠀⠨⠀⠀⢠⠅⠀⣙⠉⠀⠀⠀⠀⢧⡀⠀⠈⠑⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠑⠦⠤⠤⠖⠃⠀⠀⠛⠃⠀⠀⠀⠀⠀⠙⠶⠦⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠭⠭⠽⠶⠃⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡂⠀⠀⠀⠀⣜⡁⠔⠻⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡦⠀⠀⠀⠀⠀⠀⠀⠱⢶⡏⢣
⠀⠀⠀⠀⣠⠦⠿⠯⣛⡸⢄⠀⠀⠀⠀⠀⢠⣐⢇⣚⡇⠀⣠⠖⠋⠛⠛⠾⠟⣆⠀⠀⠀⣤⠤⠴⣰⣎⠶⡼⢤⠀⠀⠀⠀⢠⠞⠒⠓⠲⡎
⠀⠀⢀⡞⠁⢠⠴⠒⠦⡀⠀⢳⡀⠀⣀⡴⠋⠀⠀⢸⡇⠰⢇⡀⣠⠶⢤⠀⠀⢸⡆⠀⠀⢧⣀⣀⣈⠀⠉⣰⠏⠀⠀⠀⢠⠏⠀⣰⠂⠀⡇
⠀⠀⡾⠀⢰⠇⠀⠀⠀⢸⡄⠀⣧⠀⣇⣀⣤⡄⠐⢹⡇⠀⠀⠉⠁⠀⣸⠀⠀⣸⠁⠀⠀⠀⣠⠔⠁⠀⣴⠃⠀⠀⠀⣰⠃⢀⡜⢸⠀⠀⡇
⠀⠀⡇⠀⢸⡀⠀⠀⠀⢸⠇⠀⣹⠀⠉⠁⢸⡇⠀⢸⡇⠀⠀⠀⢀⡴⠁⠀⣰⠃⠀⠀⠀⠈⣧⣠⣀⠀⠀⠙⢦⠀⣰⠃⠀⠈⠉⠉⠀⠀⠙⢻⡆
⠀⠀⢷⠀⠀⠳⣤⣠⡴⠋⠀⢀⡟⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⡔⠋⠀⠀⠚⠓⠒⢲⡆⠀⠀⠀⠀⢈⡇⠀⠀⢸⡖⠳⠲⠶⠶⠶⢶⠀⠀⡖⠚⠃
⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⡇⠀⢸⡇⠀⠐⣇⣀⣀⣀⣀⣀⣀⣸⡇⢀⡏⠙⠛⠉⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠒⡤⠒⠄⠀⠀⠀⠀⠀⠀⠠⠁⠐⠈⢣⡀⠀⠀⣠⠤⠤⠴⣐⣾⠱⠄⡧⣤⠀⠀⠀⢠⠀⡠⢦⠀⠀⠀⠀⠀⠀⡐⢁⣶⡉⢃
⠀⠀⠀⠀⡏⠑⠙⠣⠭⠏⠉⡇⠀⠀⠀⣠⠞⠛⠛⠓⠊⣹⠀⠀⣿⠀⠀⠀⠀⠀⠙⠋⠅⢸⠀⠀⣠⣧⣼⣶⣴⣄⠀⠀⠀⡴⠋⠙⢒⣙⠒⠋⠳⡀
⠀⠀⠀⠀⡇⠀⢰⠋⠉⠉⠉⠁⠀⠀⡼⠁⠀⣠⣒⡛⠙⠃⠀⠀⠛⠒⠚⢛⡷⠃⠀⢀⡴⠛⠀⡞⠀⢠⠴⢬⡀⠘⣇⠀⢸⠃⠀⣸⠉⠈⢱⠀⢰⢷
⠀⠀⠀⠀⡇⠀⠈⠑⠒⢦⡀⠀⠀⢸⠁⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⣠⠏⠀⢀⡴⠋⠀⠀⠘⣇⠀⠙⠦⠞⠁⢠⠏⠀⠸⡄⠀⠘⠲⠶⠋⠀⠉⢸
⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⠹⡄⠀⣿⠀⠀⡴⠉⠉⢣⠀⠘⡇⠀⠀⣰⠏⠀⢀⡾⠁⠀⠀⠀⣠⠞⠃⠀⠀⠀⠘⠳⢄⠀⠀⠙⠦⢤⣤⣄⠀⠀⠀⡟
⠀⠀⠀⢠⠤⠤⠤⠞⠀⠀⢠⠇⠀⢸⡀⠀⠳⣄⣠⠞⠀⢠⡇⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⢠⡎⠉⠙⡆⠀⠈⡇⠀⣠⣤⠤⠖⠁⠀⢀⡼⠁
⠀⠀⠀⢾⣀⡀⠀⠀⣀⡤⠋⠀⠀⠀⠳⣄⣀⠀⠀⣀⡤⠏⠀⠀⠸⣥⣤⣤⠇⠀⠀⠀⠀⠘⣇⠀⠀⠓⠒⠚⠁⠀⣠⠇⠀⣷⠀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠴⠚⠁⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⠋⠳⡀
⠀⠀⢀⢦⡌⠙⢏⠱⠦⡀
⠀⢀⣦⠠⠝⣄⠈⠳⡀⠈⠢⣄
⠀⠘⠈⡃⡄⠈⢧⠀⠙⢦⡀⠈⠳⣄⠀⠀⠀⠀⣠⢤
⠀⢠⣤⠳⠻⡄⠀⠳⡀⠀⡙⣆⠐⠞⢧⣀⡤⠶⠁⠣
⠀⡏⠀⣇⠀⢳⡀⠀⠙⣄⠉⠈⢦⠀⠀⠹⣄⡆⠀⣀⡄
⢰⠳⣄⠻⣄⠀⠳⡌⠓⠛⢄⠀⠈⣷⡀⠀⢹⡇⠀⢹⡄
⠀⠀⣿⢀⠹⣆⠐⠙⡆⠀⠘⣆⠀⠈⠛⠀⠀⢿⡄⠀⣧
⡆⢀⣿⡈⠄⢹⡀⠀⢹⡄⠀⠈⠁⠀⠀⠀⠀⠈⠻⣦⠘⣇
⠱⡈⠁⠱⡸⡌⣇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢱⡀⠀⠡⠽⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⢿⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⣀⣀⡀
⠀⠀⠀⠸⣇⠀⠀⠀⠀⢋⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⠋⠉⢉⣽⣿⣿⣦⡀⣀⣤⣄
⠀⠀⠀⢀⣽⣧⡀⠀⠀⠘⠹⣿⣿⣿⣿⡿⠋⠀⠀⠀⣰⣾⠟⠀⠀⠀⠀⠀⣹⣿⣿⣿⡿⠋⠀⠙⣷⣤⡀
⠀⠀⢠⠞⠀⢘⡿⠷⠀⠀⠀⠉⠉⠉⢈⣿⣆⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⣽⣿⣿⠿⠋⠀⠀⠀⠀⠀⠙⢿
⠀⣰⠋⠀⠀⣾⡄⠀⠀⠀⠀⢀⣠⠞⣿⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⢾⣿⣿⠋⠀⠀⠀⠀⠀⣀⡤⠖⠈
⢠⡟⠀⠀⠀⢻⣿⣶⣶⣶⣿⠿⠁⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣺⣿⠃⠀⠀⠀⢀⣴⡞⠁
⢸⡇⠀⠀⠀⠀⠉⠛⢛⣋⣥⠶⠀⠛⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⢀⣴⡿⠋
⣾⢁⣄⣰⣷⣶⣾⣿⣿⡿⠋⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⣰⣿⠟
⣿⣎⠻⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⠀⢀⣴⣿⡿⢁⣠⣾⡿⠃
⠘⠿⣷⣦⣀⠀⠀⠀⣀⣀⣀⣄⢀⣾⠿⢻⣿⠇⠀⣀⣤⣶⣿⣿⣿⣷⠿⠟⠉
⠀⠀⠈⠻⣯⣥⣤⣼⣿⣿⠿⠛⠋⠁⠀⠀⠉⠳⠾⠽⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⢿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠿⠛⠉⠀⠀⠀⣿⢸⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠞⠁⠀⠲⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠖⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⠤⠀⠀⠀⣠⣤⣤⣤⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⣎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠎⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣀⡀⠀⠘⠃⠀⣀⣠⣴⣾⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⣀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠁⠠⠁⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠴⠀⠺⢠⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡡⠀⠄⠀⣔⣭⣄⡁⡀⠀⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣳⡀⢀⠀⠠⠄⠀⠀⠀⠀⢀⠀⢠⠉⠛⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠈⠠⠼⠀⠀⠓⠲⡖⣦⠀⠊⣴⠆⢀⠀⡔⡸⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⠂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⣀⢉⠗⢦⠀⠁⠀⡠⠋⠀⠆⠀⢀⡀⠀⡀⠀⠄⠀⠀⠚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠐⠁⠀⠀⠀⠀⠧⠆⠀⡀⠀⠀⢈⠙⠒⠊⣐⣢⡞⣒⢂⣀⢈⡤⢦⡍⠀⠈⠁⠀⢰⠖⢺⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡠⠀⠀⢀⠀⠀⣤⠀⢀⠰⣾⡷⠄⢐⠉⠙⣇⣭⣒⣩⡀⠉⠁⠁⠀⠀⡔⠂⠘⠢⠜⠳⣆⢱⡦⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠟⠑⢄⠀⠀⠛⠀⠃⢀⡀⠀⠠⢻⡍⠀⠀⣿⠂⠀⠀⠀⠀⠐⡇⠀⠢⢰⣦⣄⡀⢠⠀⠈⠀⠀⡂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢈⡉⠐⠀⣠⢁⠄⠤⠀⠀⠙⢆⢀⣏⡴⠶⢗⠀⠄⠀⠀⠀⠆⠈⠛⠋⠁⢀⡖⡚⠂⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⣀⠈⢉⡀⣲⢤⣾⡀⢘⣿⡋⠀⢀⣨⠟⠂⢀⡀⠀⠈⠊⡰⠀⡄⠜⠀⠀⠠⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠰⠥⠤⣀⠴⠛⠃⠣⠀⠀⢹⡟⠁⠉⠛⣠⣥⣄⠀⢀⣉⣀⡀⠀⠀⠀⠏⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢦⠀⢀⣠⠖⠚⣧⡀⠀⠀⠀⠖⢮⡇⢠⠤⠤⡻⠿⠟⢀⡀⣸⠋⠙⣀⣠⡞⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⡀⠈⢰⠊⠉⠓⢤⣾⡇⠀⢀⡇⢈⠂⠀⡧⠀⠀⠣⢸⢹⡞⠉⠁⠀⠈⡆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠿⣿⡇⠀⠀⠀⠀⠘⠛⠛⠛⣶⣾⣇⠹⠶⣶⣷⠶⠶⣤⠾⠋⠙⠦⠆⠁⠐⠁⠈⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠋⠀⠀⠀⠀⠀⣠⠆⠄⠠⠈⢿⣿⣦⣼⠇⠀⠀⠀⠘⠓⠀⠀⠀⠀⣒⣦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⢼⣷⣦⣌⡙⠂⠄⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠈⠭⠍⠁⠀⠀⠀⠐⠀⠀⠀⠠⢄⠀⣀⣀⢀⡠⠼⠛⠛⠿⣧⣤⠀⠄⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠐⠂⠀⠈⢉⣀⡠⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠀⠉⢉⣉⣁⣀⣀⣀⣀⣐⣒⣀⣒⣀⣤⣤⣤⣿⣗⣋⣐⣿⡿⠦⠁⠀⠀⠀⠉⠩⠭⠭⠭⠭⠥⠄⠙⠛⠛⠓⣒⠨⠇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠉⣀⡴⠐⡖⠮⣄⡀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠶⡿⠛⢀⣀⣉⣀⡘⠫⡓⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠊⡁⠀⠀⠀⣩⠶⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠙⢦⣤⠞⠉⠁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡀⠀⠀⠈⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢄⠀⠀⠀⠀⢂⠄⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣦⣌⡁⣀⡉⢄⣶⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠀⠀⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡾⣜⡚⡬⣞⣖⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⠶⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣲⢿⠟⠋⠁⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⠀⠀⠀⠉⠻⣻⣢⢄
⠀⠀⠀⠀⢀⠎⠇⠋⠁⠀⢀⣤⣶⣿⣿⣿⣿⡿⠿⠞⠿⠿⣟⡿⡿⣟⣿⡦⣄⡀⠀⠀⠩⠯⢗⡀
⠀⠀⠀⡔⡘⠌⠀⠀⠀⣴⣾⣿⣿⠿⠛⣿⠀⠀⠀⠀⠀⠸⠁⠀⠉⠙⠸⠽⡘⡘⠤⠀⠀⠀⠛⣾⡤
⠀⠀⣠⢵⠂⠀⠀⢐⢨⣽⡿⠋⠁⠀⠀⢹⡆⢀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠀⠑⢂⠡⢑⣂⠀⠀⠘⣿⣧
⠀⣔⡯⠅⠀⠀⢤⣖⣯⠿⣇⠀⢀⣤⣶⣿⣿⣿⣿⡝⡒⡀⠀⡀⠀⠄⠀⠀⠀⠀⠩⢅⣂⣧⠀⠀⠩⣿⣇
⢠⣿⡏⠀⠀⠘⠽⢞⡟⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡡⠰⣐⠨⠠⡀⠀⠀⣀⡤⣼⡚⠿⠇⠀⠀⢻⣿
⢸⣿⠃⠀⢀⣶⣿⣷⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠓⣧⢧⣳⣲⢿⣷⣻⣽⣳⣶⣷⠀⠀⠘⣿
⣿⣿⠀⠀⢸⣿⣿⣯⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿
⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⢻⣿⡆⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢰⣿
⠘⣿⣧⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣼⣿
⠀⠹⣿⣆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⢿⢿⣿⣿⠟⠀⠀⣰⣿⠏
⠀⠀⢻⣿⣆⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⡿⣟⣟⡿⣿⣾⡿⠏⠀⠀⣰⣿⡏
⠀⠀⠀⠹⠿⣧⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⣀⡰⠽⠚⠹⠙⡘⠨⠙⠚⠀⠀⠀⡼⠝⠊
⠀⠀⠀⠀⠀⠑⢻⢤⡀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣗⠃⠁⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠩⢿⣶⣤⣀⠀⠀⠈⠉⠛⠛⠛⠛⠙⠒⠈⠈⠀⠀⠀⠀⠀⠀⢀⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⣀⠀⣀⣠⡤⣐⡸⠌
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣿⣯⣗⣃⣯⣗⣓⣏⣓⣃⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⢸⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠄⠠⢄⠀⢀⠤⠀⠠⢄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠊⠉⢨⠀⠀⠀⠀⢱⠃⠀⠀⠀⢀⠍⠉⠲⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⢼⢆⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⡴⠋⠈⠢⢄⡀⠀⠀⠀⡀⢸⠀⡄⠀⠀⠀⢠⠠⠔⠈⠑⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⢰⠀⠀⡄⠀⠀⠁⠀⠀⠀⠃⢸⠀⠁⠀⠀⠀⠊⠀⠀⡀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠸⡀⠀⣈⢀⡀⠀⠴⠂⠀⠀⢸⠀⠀⠀⠐⠤⠀⢰⢀⡁⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠑⡌⠁⠀⠁⠀⠀⠀⠎⠠⢼⡦⠄⠢⠀⠀⠀⠉⠀⠈⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣄⠠⡀⠀⢀⡀⠀⣠⠋⠳⡀⠀⢀⠀⠀⣠⠄⡰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣤⠕⠊⠁⠀⠀⠈⠒⠤⠤⠤⠔⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢸⠇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣼⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠤⣒⣉⠡⠤⠄⠒⠒⠢⢤⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⡴⠖⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡖⠒⠒⠒⠒⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⠲⡀⢄⠂⠀⢠⢲⣤⡟⢇
⠀⠀⠀⠀⠀⡀⠀⣀⢀⢄⠀⠀⠀⠠⠀⣠⠴⠒⠺⢶⣜⡍⡇⠀⠀⠀⠀⢠⠏⠑⠛⢔⡦⠧⣄⡜⠲⠓⠤⡎
⠀⠀⣠⠴⠚⠺⠛⣵⣆⡇⢠⢠⡎⢬⡎⠣⣀⡤⢤⡀⠉⠙⡆⠀⠀⠀⠀⡼⠀⢰⠛⠒⠲⢤⡟⠀⣰⡆⠀⡇
⢀⡜⠁⢠⠔⠚⠓⢎⠀⢹⡤⠋⠉⠹⡇⠀⠀⠀⣠⠃⠀⢠⠇⠀⠀⠀⠀⠧⣄⡈⠑⠦⣀⠞⠀⡼⢡⡇⠀⡇
⣼⠀⢠⠇⠀⠀⠀⢸⢶⠉⢀⣀⠠⠂⣇⣀⠤⠚⠁⢀⡴⢋⠤⣀⠶⡀⣀⠀⠀⣱⠀⠀⢹⠀⠀⠉⠉⠀⠀⠉⢹
⡇⠀⠸⡄⠀⠀⢀⡼⠘⢻⠏⢹⠀⠀⣏⠀⠀⠀⠲⠯⣄⠼⡶⢷⣶⢿⡉⠉⠛⠁⠀⠀⡼⠛⠛⠛⢻⠀⠀⡏⠉
⠹⣄⠀⠈⠓⠒⠉⠀⢀⡞⠀⢸⠀⠀⣿⠲⠤⣄⣀⢰⠁⠀⡖⠒⡄⠀⢳⠦⠤⠤⠴⠊⠀⠀⠀⠀⢸⣄⣠⡇
⠀⠈⠳⠤⣤⣤⣤⠴⠋⠀⠀⢸⠀⠀⢸⠀⠀⠀⠈⠙⢆⠀⠙⠋⢁⣠⠋⠀⠀⡔⠠⠄⢷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⣏⠉⠀⠀⠀⢀⠖⠁⢀⡤⠤⡀⠈⠳⣤⠔⠛⠓⠶⠜⣦⡤⢠⣊⢣⣀
⠀⠀⠀⠀⠀⠀⣼⠉⠓⠚⠼⢏⠶⢨⣄⡀⠀⢀⢸⠀⢀⢻⣀⣀⡿⢀⠞⠁⢀⣤⡒⠒⠶⣣⠦⠽⢛⡊⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠧⢤⣤⣀⡀⠀⠉⠑⣸⠦⠤⡠⣜⢩⣻⡀⠀⠀⣀⡞⠀⠀⢀⣀⠉⠑⢤⡇⠀⡜⠉⠉⡇⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⣠⠖⠻⠤⢄⣉⠉⠒⣩⠟⠉⠉⢩⡇⠀⢰⠉⠈⢳⠀⢸⣇⠀⠙⠲⠖⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠋⠀⠀⢀⠴⠊⠁⢀⡔⠁⠀⠀⠀⠀⢧⠀⠘⠢⠔⠃⠀⣸⠉⠲⢤⣤⠤⡦
⠀⠀⠀⠀⠀⣰⠃⠀⢀⡞⠁⠀⠀⠀⠸⠤⢤⡀⠀⠙⣆⠀⠀⠀⠀⠈⠓⠦⣤⣤⠤⠞⠁⠀⠀⢀⡠⠔⠁⠀⣠
⠀⠀⠀⠀⠰⠧⣤⣤⡞⠀⠀⠀⠀⡤⣤⣀⣠⠇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠊⠉⠉⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠉⠉⠉⠉⠉⠉⠫⣅⠀⠀⠀⠀⣀⡤⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⣠⠖⠲⣄⣈⣳⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⢀⡠⡾⠁⠀⠀⠀⠀⠀⠀⣠⢊⣡⠦⠒⠀⠉⠉⡆
⠀⠀⠀⠀⢠⠔⠒⠚⠉⠉⢠⣾⠿⠛⠉⢰⠁⠀⠀⠀⠀⠀⠀⠁⡼⣏⡀⣀⣀⠀⠀⠸⡅
⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠎⢢⡴⠋⠁⠀⠀⠀⢠⡇
⠀⠀⠀⢀⡇⠀⠀⠀⠀⢿⡀⠠⠤⢐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠳⠴⠒⠊⠁⠀⢀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣧⠀⠀⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣤⡤⠔⠋
⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠸⣆⠀⢀⣤⣴⠾⠦⠤⢤⣄⣀⣀⣀⣀⠤⠗⠋⠁
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⠻⠿⠟⡟
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣄⣀⡐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠟⠋⠉⠉⠉⠉⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠃⠀⠀⡀⠀⠐⢶⣶⡆⠤⣄⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⢄⠀⢀⢸⣹⡇⣶⣬⣇⠀⢨⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⢂⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣿⠛⣿⠛⢻⣿⣿⡏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⠀⣇⠀⢸⣿⣿⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⢀⡋⣀⣐⣛⣟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠻⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠤⠐⠶⠖⠾⠷⠷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄
⠀⠀⠀⠀⠀⢀⢤⡶⠂⠀⠀⠀⢀⠤⠒⡉⠍⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣀⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⢀⠀⠀⠀⠀⠀⠸⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣇⢄⡀⠀⠀⠙⢿⣄
⠀⠀⠀⢠⠟⠀⠀⠀⣠⠀⡠⢊⠐⠀⠈⢦⡳⡀⠀⠀⠀⠀⣆⣸⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⢷⣕⣦⠀⠀⠈⢻⣦
⠀⠀⢀⠋⠀⠀⠀⣴⢃⠞⠁⠀⠀⠀⠀⠀⠑⡷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣄⡀⠈⢻⣧
⠀⠰⠏⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⣤⠀⠘⣿⣿⣿⣷⡈⢿⣆
⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⡴⠟⠋⠀⠀⠀⠘⣿⣿⣿⡷⠸⣿
⠈⠃⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠀⣻
⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⠁⢸
⠀⠀⠀⠄⣿⠀⠄⢠⣒⡒⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⢷⠀⠀⣿⡇⢹⣇⢨
⠀⠀⠀⢻⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣟⡘⠇⢸
⢠⠀⠀⠈⣮⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣸⣿⠷⡏⠀⢸
⠀⡆⠀⠀⢿⠟⡄⠀⠀⣤⡲⡫⢵⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠒⠄⠀⠀⠀⠀⠀⠰⢿⣷⣦⢦⡄⠀⢠⣿⡿⠾⠋⢀⣾
⠀⠈⠀⠀⢹⣧⢹⣆⣀⠙⣫⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠒⠙⠁⣠⡟⡟⢡⠋⢀⣾⠃
⠀⠀⠀⢀⠀⣻⣦⡛⡗⢀⠉⠀⠀⠀⠀⢀⣾⡃⠀⠀⣀⠀⣷⣤⠀⠠⠄⢀⢠⡀⠀⠀⠀⠀⠀⠀⣴⣏⠟⡴⠃⠀⣼⠋
⠀⠀⠀⠀⣦⠘⢿⣷⣌⢻⣷⣄⠂⠀⠠⡾⢉⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⡀⠀⠙⢆⠀⠀⢚⣤⣿⡷⣣⡾⠀⢀⡜⠁
⠀⠀⠀⠀⠈⠃⢈⠻⣿⣷⣍⡻⣷⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⠀⠀⠀⠀⣠⣼⣿⠿⣫⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣧⣝⡻⠳⠲⠤⠀⣀⡀⠀⠀⠅⣀⣀⣀⣤⣴⣖⣭⠮⣋⣡⣾⠛⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢻⣿⣿⣿⣿⣾⣯⣤⣄⣀⡀⠀⠽⠛⠛⠛⣛⣛⣋⣩⣶⣾⠿⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡵⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣶⣪⠃⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡉⢁⣀⣀⣼⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠭⣅⢀⣴⠾⣧⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡜⢡⣴⡄⠛⢸⡟⢀⡾⠀⠀⢩⢿⣿⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡷⢾⡟⠀⠀⣠⠝⠏⠓⠲⣾⡗⠚⠀⢹⣦⡀⣠⠖⢉⣠⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⠛⣿⣟⣷⣌⠁⣠⡴⡆⠀⠀⠙⣦⣶⣻⠟⣷⡏⠀⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⢾⣿⢱⣿⣏⣹⠋⠶⠽⠞⠁⠀⠀⠀⠉⢧⠀⣀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡈⣋⣛⣣⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⢿⡏⠃⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⣟⢋⡟⢠⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡿⡀⠈⠳⢦⣤⣀⣀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠃⢸⣸⠇⠀⠁⠀⠀⠀⠀⠀⠀⢠⡶⢀⣻⢯⣤⠿⢇⠁⠀⠀⢀⠀⡠⢻⣦⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⠦⡀⠀⠀⠀⠀⠀⠀⠈⢙⠟⣫⣻⣶⡖⣠⣶⡿⠿⠀⠐⠣⢾⡛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣦⡈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠛⢛⣁⡉⠟⣇⠤⠤⠶⠖⠚⠛⢉⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠋⠑⠶⠉⠙⠒⠿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠙⠻⢯⣹⠿⠃⠛⠛⠀⠀⠀⠀⠀⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢀⡄⠀⠇⠀⠀⠀⠀⠘⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠿⠞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠘⠞⠀⠀⣶⣆⡴⠾⠗⠉⠈⠀⡅⠀⠀⠀⠈⠣⡀⠀⠀⠀⣠⣤⠀⢱⣤⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⢴⠟⢿⠀⠀⠀⢠⡄⢠⠃⢀⠃⠀⢴⠶⢷⠛⠀⠈⠉⠉⢿⡟⣃⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣡⠃⠀⠀⠀⠀⠀⠂⢈⢣⠀⠀⠸⢻⣧⣿⣙⣛⡤⠀⠀⠀⠀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣧⡀⠻⠀⢠⠀⠀⡀⠈⠀⣣⠞⠁⢀⣼⡀⠠⡀⠀⡷⢸⠠⢀⣠⠞⠋⢀⡟⢉⣽⣋⠳⡆⠀⠘⠿⢇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠳⣄⣀⠀⠀⢀⣣⣴⡾⡁⢀⣴⣿⣿⣿⣶⡄⣷⣾⡏⠸⣇⣶⠄⢤⣶⣿⣟⣻⣿⣿⣥⠤⡴⠚⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣀⠈⢩⣛⡋⠯⢿⣽⣡⣾⣿⣿⣿⣿⣿⣬⠧⣿⡿⡱⠁⠘⠿⣟⠉⡺⢿⣤⣀⣄⣄⡀⣾⢺⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⣰⠿⡷⣹⠿⣧⣼⠟⠋⠉⠉⠀⠉⣈⠁⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠓⠊⠁⠀⢸⣏⠀⠀⠀⠀⣴⢋⣀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣷⣌⣋⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣈⠓⣠⠏⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣝⣧⡶⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡀
⠀⠀⢀⠱⣀⣠⣄⣤⠔⣂⠥⠤⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⡈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣤⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣓⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠒⠓⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣟⣛⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⣀⣤⣶⣄⡀⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⢽⣽⢽⠤⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠞⠻⠻⠛⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣛⣿⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣩⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣶⣶⣦⣶⡾⠿⡇
⢀⡀⠀⣀⣾⣿⣀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣡⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣼⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⡀⡂⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠠⣴⠂⠀⠀⠀⠀⠀⠠
⠀⠀⡀⠔⠁⡃⠀⠀⠀⡴⠉⣉⠙⠱⠀⠀⠀⠀⠸⠀⡈⠉⡝⠀⠀⠀⠀⡐⢁⠁⡇⠀⠀⠀⠀⠈⢈⠽⠅⠅
⠀⠀⠦⢲⠀⠁⠀⠀⠀⠀⠉⢀⠃⢰⠁⠀⠀⠀⠀⣊⡀⠘⢄⠀⠀⠀⡜⠠⠨⠀⠇⡀⠀⠀⠀⢤⣈⠐⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠔⠁⠐⠃⢤⠀⠀⠀⢠⠤⠜⠀⢸⠀⠀⠘⠀⠀⢤⠀⡔⠃⠀⠀⠀⡤⠤⠃⠀⡇
⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠚⠀⠀⠀⠓⠂⠐⠒⠁⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠈⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡠⠤⡌⢄⠀⠀⠀⡤⠤⠠⠮⡡⡄⠀⠀⠀⣀⣠⣩⡀⠀⠀⠀⠀⡠⢠⠌⢄⠀⠀⠀⠀⢀⡤⢤⣬⡀
⠀⢀⠎⠀⠔⣀⠂⠀⠀⠀⠦⠤⡤⠀⣠⠇⠀⠀⢘⠀⣆⠄⢹⠀⠀⠀⡎⢠⠂⢲⢀⡇⠀⠀⡰⢁⠔⠒⢄⠑⡀
⠀⢸⠀⠠⠀⠄⠱⠀⠀⠀⢀⠞⢀⠞⠁⠀⠀⠀⠜⢃⠀⡐⠣⡀⠀⠀⠱⢄⣉⠁⠀⡆⠀⠀⡇⢸⠀⠀⢸⠀⡇
⠀⠘⣄⠐⠒⠃⡸⠀⠀⠀⡞⠀⡞⠀⠀⠀⠀⠘⡄⠘⠤⠂⢀⠇⠀⠀⢰⠒⠊⢀⠔⠁⠀⠀⠳⡀⠑⠒⠁⡠⠃
⠀⠀⠀⠉⠂⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠊⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣀⣀⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠁⠀⠀⢈⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⠤⣖⣲⡭⢿⣛⣋⣁⣀⣠⡶⠦⣤⠀⠈⢙⣁⣀⣀⣀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⣶⡭⠷⢚⣻⠿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠓⠿⣄⡀⠸⣏⢆⠀⠉⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡴⠞⠉⢀⡠⠤⣒⣺⣭⡭⢭⣭⣙⣒⠤⣄⠀⠙⠳⣜⢯⣣⡀⠀⠈⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⠀⡠⣖⡽⠞⡋⠁⠀⠀⡆⠀⢀⡆⢭⡛⢦⣑⢄⠀⠈⢿⣷⡵⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⣪⠞⡉⠀⠀⠛⠀⠀⠀⠀⠀⣨⡕⠾⠃⠀⠙⢦⡳⡀⠀⢻⡽⣞⢆⠀⣼⠰⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⡜⡴⢃⣺⣿⠂⠀⠀⠀⠀⠀⠀⣼⢫⠇⠀⠀⠀⢠⠌⢳⡙⡄⠀⢷⠘⢮⡾⠃
⠀⠀⠀⠀⠀⢀⡟⠀⡜⡼⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⣰⣷⠏⠀⠀⠀⠀⠀⠀⠀⢷⢱⠀⠸⡇
⠀⠀⠀⠀⠀⣸⠃⢰⣿⡇⠀⠀⠀⠀⠀⠀⣠⠦⢤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⡇
⠀⠀⠀⠀⠀⢹⠀⠘⣿⡇⠚⠁⠀⠀⠀⠘⢧⡴⠞⠛⠻⡀⠀⠀⠀⠀⠀⠀⠘⠃⣸⢣⠃⢀⡇⢰
⠀⠀⠀⠀⠀⠘⣇⠀⢳⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⡏⡜⠀⣼⠁⠏
⠀⠀⠀⠀⠘⠄⢹⡄⠀⢿⣧⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣀⡾⢣⣞⡜⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⠻⣷⣄⠀⠀⣴⣶⠆⠀⠀⠀⠀⢀⠀⠀⠀⢉⡵⣫⠎⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣠⠾⣝⣦⡀⠈⠻⣷⣤⣛⡛⠀⠀⠋⠀⠀⢈⣁⡤⢖⡫⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⠴⠞⠉⠙⢦⣄⠈⠙⠻⠭⠭⠽⣛⣛⣛⠭⠕⠚⠉⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣀⣀⣀⣀⣀⣤⣤⣶⡟⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠹⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠛⠛⠛⠻⠿⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣧⠀⠐⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣇⡈⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⠦⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡻⣿⣿⡇⡀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⣿⣿⣾⣄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⡙⢿⣿⣿⠻⣿⣿
⠀⠀⠀⠀⠀⣀⣾⣷⣿⡟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣷⠹⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⣀⣀⣀⣸⣿⣿⣿⣿⣿⠁⣸⣿⣿⠏⠀⢻⣷
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⡷⠈⢿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣿⡿⠿⠛⠁⠙⠛⠻⠿⠿⠟⠋⠙⢿⣿⠆⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣤⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⣉⠙⠻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⢿⣿⣶⣤⣈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣁⣤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣷⣶⣶⣦⣤⣀⣉⠙⠻⢷⣦⡈⠳⡄⠀⠀⠀⠀⠀⢀⡞⢋⣤⡾⠟⠋⣁⣈⠻⢿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠛⠛⠉⠉⠛⠛⠛⠛⠛⠿⠿⢶⣦⣌⡙⠦⠈⠀⠀⠀⠀⠀⠉⡴⢛⣡⣴⡶⠿⠿⠛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⣠⣤⣤⣶⣶⣄⠻⣿⣿⣿⣿⣿⣇⣀⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣦⣤⣴⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⠟⠘⢿⣿⣿⣿⣿⣿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡛⠻⠟⢛⣡⣾⡇⠀⠙⠛⠛⠋⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠃⠈⢿⣿⡿⠃⠀⠛⠉⠉⠉⠁

⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠁⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠁⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠂⠉⠈⠉⠁⠐⠢⢄
⠀⢀⠔⠁⠀⢀⠠⠤⠀⠠⠤⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢀⠔⢅⠠⣀⠠⠤⠀⠠⠤⣀⠀⢀⠕⢄⠀⠀⠀⠀⠀⠀⢀⣔⢧⠀⢀⡠⠤⠀⠠⠤⢀⠀⢀⢕⣄
⢠⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⢆⠀⠀⠀⠀⢠⠃⠀⢀⠝⠙⣦⠀⠀⠀⠀⠀⠙⢇⠀⠀⢣⠀⠀⠀⠀⢀⠞⢕⣷⠟⠁⠀⠀⠀⠀⠀⠀⣙⣷⣟⠼⣧
⡌⠀⠒⡶⠤⣄⣀⢀⣀⠀⠀⠀⠀⠈⡆⠀⠈⡄⠀⠀⠀⡀⠀⠀⡎⠀⠀⠈⢳⣄⠀⠀⠀⠀⠈⡆⠀⠈⡄⠀⠀⠀⣾⠄⢻⡏⠀⠀⠀⠀⢀⡠⡴⠞⠁⠈⣷⠒⠼
⠓⠒⠒⠓⠒⠒⠛⠛⠓⠓⠒⠒⠒⠒⠓⠒⠒⠁⠀⠀⠀⠑⠒⠒⠓⠒⠒⠒⠚⠓⠛⠒⠒⠒⠒⠓⠒⠒⠁⠀⠀⠀⠙⠚⠛⠓⠒⠒⠒⠒⠓⠛⠒⠒⠒⠒⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠈⠁⠀⠀⠀⠀⠉⠑⠢⡀
⠀⣠⠮⡢⣐⣤⠐⠂⠀⠐⠒⠤⣀⣠⡫⣦⠀⠀⠀⠀⠀⠀⢠⡮⡢⣀⡠⠐⠂⠀⠐⠒⠠⣀⡠⣪⣦⠀⠀⠀⠀⠀⠀⢠⡮⡣⣀⡠⠔⠒⠀⠀⠒⠤⣀⡠⡪⢦⡀
⢰⠧⡾⣿⠋⠈⢷⡀⠀⠀⠀⠀⠈⢻⣽⠦⢣⠀⠀⠀⠀⢰⠣⡾⣽⠋⠀⠀⠀⠀⠀⠀⢀⡼⢿⡯⡢⣷⠀⠀⠀⠀⢠⢫⣺⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⡦⢷
⣾⣛⣻⡇⠀⠀⠀⢫⡦⡀⠀⠀⠀⠀⣿⢪⣺⡆⠀⠀⠀⣾⣚⣸⡇⠀⠀⠀⠀⡠⣔⠞⠋⠀⠀⣿⣚⣻⡆⠀⠀⠀⣾⣊⣹⡟⠒⠶⣤⣄⡤⣄⠀⠀⠀⠀⣿⣚⣞
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⢀⠐⠋⠀⠀⠀⠀⠀⠀⠀⡉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⣴⣿⣷⣄⠔⠊⠉⠀⠈⢁⡾⢄⣴⣿⣷⡀⠀⠀⠀⠀⠀⡰⠁⠙⢿⠟⠋⠉⠀⠈⠉⠛⢿⢟⡁⠱⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠋⠉⠁⠈⠉⠛⢿⣿⣿⣷⡄
⣸⣿⣿⡿⠁⠀⠀⠀⠀⣠⡟⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠸⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⣠⠖⠻⡀⠀⠰⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣠⣔⣟⣀⣀⣀⣀⣿⣿⣿⡇⠀⠀⠀⣇⣀⣀⣇⣀⣀⣀⣀⣄⣾⣋⣁⣀⣀⣇⣀⣀⡇⠀⠀⠀⣿⣿⣿⣇⣀⣀⣀⣀⣔⣴⣶⣛⣋⣁⣻⣿⣿
⣿⣿⡿⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣷⠀⠀⠀⢀⣤⠶⣤⡃⡸⠆⠀⠁⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠋⠀⠀⠁⠀⠻⣦⠀⠹⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠄⡀⠰⢿⣿⣷⣿⣾⣆⠀⠀⠀⠀⠀⡀⠄⠰⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣦⣜⠻⠂⠸⠶⠄⠺⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠠
⠀⠁⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠛⠛⠀⢠⣿⣿⣿⣧⣘⠻⢆⣰⣶⣾⠃⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠇⠀⠄
⠀⠀⢤⣤⠀⠑⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡴⢦⠙⡿⡱⢲⡀⠙⠻⢿⣿⣿⣿⣿⣿⣿⠟⠀⠀⡀⠀⠉⠃⠈⠀⠀⠀⠀⠈⠋⠀⠀⠠⡀⠀⠀⠐
⠀⠀⠈⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⣿⠀⠀⠀⠙⠻⠿⠟⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⠈
⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣤⠀⣠⡤⠀⠀⢀⣀⠀⠐⠋⠁
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠅⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢀⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠰⣿⣿⣿⡿⠋⢠⣦⡘⠿⠆⢀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⣆⡆⠀⠀⠀⠐⠀⠃⡄⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠻⠿⠋⠀⠀⣼⣿⣿⣶⣾⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⡀⠀⠀⠀⠐⠀⠀⢺⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠄⠀⠀⠀⠈⠀⠰⠂⠠⣄⠀⡉⠀⠀⠀⠀⠀⠀⢻⣿⡷⠄⠀⣠⠀⠀⠀⣸⠆⠀⠀⣀⣀⡈
⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠀⢸⡂⠀⠀⠀⠀⠀⠀⠉⠀⢀⠐⠁⠀⠀⠀⠉⠀⢰⣇⠀⠠⢉⠄
⣮⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢸⣿⣿⣶⣶⣶
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⠀⠀⠀⠀⠀⠀⠀⠦⢠⡤⠙⠻⢿⣿⣿
⠉⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣿⣿⣷⣿⣿⣿⣿⣧⡀⠀⠀⠀⠇⢄⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠐⠀⠀⠂⠀⠀⠇⠀⢀⡀⠄
⡠⡀⠉⠈⠀⠀⠀⠀⠀⠀⡠⠶⠋⠁⠀⠀⠀⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠨⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⢈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠁⠀⠀⠀⠀⠈⠉⠉⣉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠂⠀⣐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡇⠀⠀⠀⠆⠀⠦⠈⢅⣀
⠀⠀⠀⠀⠀⠀⠠⠀⢰⡶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠏⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⡉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠡⠈⣿⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠠⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⠱⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⠋⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⡞⠀⠀⠀⢀⡴⠋⠙⢄⠀⠀⠀⠈⡳⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⣀⠀⠀⠀⠀⠘⡄⠀⡠⠋⠀⠀⠀⠀⠑⢄⠀⡰⠁⠀⠀⠀⢀⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠑⠒⠤⢄⡸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⣅⡤⠤⠒⠊⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠸⣿⠗⠀⢆⠀⠀⡰⠀⠺⣿⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠴⠒⠒⠲⢄⠀⠀⠀⠉⠉⠀⠀⠀⡠⠖⠒⠒⠦⡀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡎⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢱⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡘⢤⣀⣀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣀⡤⠃⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠈⠓⠦⢄⣀⡀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠉⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠉⠉⠉⠉⠑⠒⠒⠤⣀⠉⡏⠉⡏⡡⠖⠂⠉⠉⠉⠉⣉⡡⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠉⠙⠓⠒⠒⠤⣀⣑⣧⠀⣟⡠⠔⠒⠛⠉⠉⠉⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣄⣀⣀⣀⣀⣀⡬⢿⠀⠹⠤⣤⣤⣤⠤⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠲⠶⠞⠶⠒⠶⠶⠶⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣇⣀⣱⡄⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠀⠈⠀⠧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠄⠀⡞⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢈⢿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⠻⠟⠇⢰⡏⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⠀⢨⢁⢹⡘⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠮⡀
⠀⠀⠀⠀⠀⠀⠔⡐⡈⠂⠡⢘⡀⠘⡃⡠⠐⠈⠘⢯⠂⠀⠀⢠⡄⠀⠀⠀⢰⣀⠀⣀⣀⡩⢦⢀⡔
⠀⠀⠀⠀⠀⠀⠸⡷⠋⢀⡀⠀⠆⠈⢽⡄⠂⢁⠀⠎⡀⡸⣗⠠⣌⣴⡁⠑⡦⠈⠈⠍⠽⠓⡄⣩
⠀⠀⠀⠀⢀⣀⣄⠑⠐⣌⠉⡆⠌⢠⠈⠣⠄⠐⠊⠠⢘⠂⠀⠀⠒⠈⣘⣼⢦⠀⠀⠀⢀⠸⡯⠃
⠀⠀⠀⠀⠀⠈⣉⣳⠀⢧⠀⠃⠜⢇⡀⡀⢀⠀⠢⠔⡀⠸⠃⠈⠀⠀⢈⡵⢮⡅⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠈⠱⢟⠉⠀⠙⠈⠀⡠⣦⠅⠀⣍⡀⢴⡁⠡⢋⣄⠀⠚⡁⠀⡗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢠⠱⢣⠂⠀⢻⠀⢜⡠⢸⡖⠜⡟⣀⣖⡽⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠌⠀⣰⡴⠠⢰⠆⠠⠀⢛⢴⠠⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠇⠌⢴⢯⠀⠒⠐⠠⠀⡁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡊⠁⠁⣳⣥⠆⠐⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢨⣜⠀⢾⣿⢤⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣐⢉⡶⠜⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡻⠩⠁⡛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣮⣷⣶⣼⣧⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠛⠛⠃⠘⠛⠛⠛⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡇⠀⠀⣺⡅⠀⠰⣿⠄⠀⢘⣧⠀⢰⡟⣧⠀⠈⣹⠃⠀⢸⣿⡅⠀⢻⣽⠀⢈⣿⣏⡇
⢀⣶⠶⢤⠀⡠⠶⣄⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠌⠀⠀⡈⠁⠀⠐⠉⢁⠀⠀⠁⢐⠐⠂⢉⡒⢐⠨⠁⡀⢀⠡⢙⡀
⠈⠙⠶⠎⠉⠱⠶⠋⠸⠇⠺⠇⠻⠇⠿⠃⠿⠘⠿⠘⠿⠸⠗⠸⠟⠸⠇⠺⠇⠿⠇⠿⠘⠿⠸⠿⠸⠟⠸⠏⠹⠇⠸⠇⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠀⠀⢠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣁⡄⠉⠙⠢⣀⡤⠄⠁⠀⠈⠀⠀⠐⣘⢃⣴⣴⣄⠠⠄⠀⠘⠉⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠜⠛⠲⠁⠈⠳⢌⠀⠀⠀⠂⠂⠐⠁⠘⠛⣿⣿⡇⠆⠀⠠⠄⠘⠁⠀⠀⠠⠁⠀⡀
⠀⠀⠀⠀⠀⠴⠈⠁⠰⠛⠠⢍⠀⠓⠀⢤⠀⠚⠁⠨⠛⠀⠖⠉⠁⣠⣦⣬⣿⡷⠀⠀⠀⠀⠀⣴⣄⠀⠀⠁⠐⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠈⠁⠚⠀⠀⠁⢈⠁⠀⠖⢀⡀⠆⠀⠋⠛⠛⠀⢉⠂⡀⡈⠻⢻⣿⣷⡦⡆⠀⠈
⠀⠀⠀⠀⠀⢣⠰⢄⠠⠒⢄⡠⠤⡄⢂⡀⠤⣶⠀⠀⠄⠚⠁⡌⣠⢄⠀⠀⠀⠇⠈⠀⠓⢾⢤⣼⣿⣿⣷⣯⣀⡈
⠀⠀⠀⠀⠀⠠⡠⠖⢲⡤⠒⢢⠶⠨⣭⠉⠆⢀⠠⠆⣊⡁⠀⣄⠀⠀⣀⠢⠀⢀⠐⠂⠀⠀⢀⡌⠙⠛⠋⢛⡔
⠀⠀⠀⠀⠠⠁⡄⠀⠠⡆⢐⣄⢠⢁⠁⡤⢄⠠⣖⠀⠉⠀⠐⠐⠀⠀⠀⠀⠀⠂⡠⢬⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠂
⠀⠀⠀⠀⠀⠀⢹⠀⠀⡉⣗⠏⢘⠿⠀⣤⡊⠄⠃⠁⢠⠄⠀⠠⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣡⠄
⠀⠀⠀⠀⠀⢰⠸⣶⣤⢿⣿⣃⣤⠀⠀⣿⣀⠀⠀⠀⠤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠰⡀⠉⢸⣸⣾⣳⡭⡘⠁⢮⠙⠛⠒⡀⠀⠀⠀⢀⠀⠀⢠⡄⣅⠀⠀⠀⠀⠀⠈⠀⠀⠘⠁⢀⡀⠀⢦⢠⡄
⠀⠀⠀⠀⠀⠠⠄⠀⠃⠛⠛⠋⠰⠀⠀⣀⡀⢲⡄⠘⠀⠀⠊⠁⠀⠀⠀⢹⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣦⠹⣆⣋⣇⠁
⠀⠀⠀⠀⠀⠀⠀⡂⢶⠶⣤⠀⠒⡀⠀⢈⠋⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⣀⠀⠂⠀⠀⡠⡀⠘⠋⠉⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠈⣓⣈⡃⣀⡋⣴⢤⠤⢤⣄⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠉⢷⣢⡟⠈⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⢟⡛⢍⡹⠿⠿⠷⣿⡛⠻⠟⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠄⠀⠀⠀⠀⠀⠀⠙⠄⡬⠞⠁

⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠙⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⠛⠛⠛⡟⠋⠉⠻⠛⠉⠙⣿⡇
⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⡇⢹⠀⢸⠈⣹⠂⡧⡎⢹⠤⢰⠉⡧⣽⡇
⠀⣸⠃⠀⠀⠀⠀⠀⢀⣬⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⡇⠈⠀⣸⠘⠓⢀⣇⠙⠊⢀⡈⠓⠁⣾⡇
⠀⢹⡄⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣧⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣼⠟⠡⠀⠐⠀⠠⠉⠻⣆⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣾⠁⠂⠀⠀⠀⡔⠀⠀⠂⠘⣇
⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⣠⣤⡴⠞⠋⠀⠀⠀⠀⣿⠐⠀⠀⠀⠚⠁⠀⠀⠐⠂⣿⠇⠀⠀⢀⡴⠚⠋⠉⠉⠉⠉⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢠⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀
⠀⠀⠀⠀⠀⠀⡴⠚⠛⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣌⡀⠀⠀⣈⣠⣾⠋⠀⠀⣨⣿⣿⣿⣷⣦⠀⢤⡤⠀⣠⣶⣶⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠳⢴⣿⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⢟⣩⣤⣼⣧⣤⣝⡻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠟⠋⣡⣿⠟⡋⠉⠉⠉⠉⢙⠿⣿⣍⠻⢿⠇
⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇⠀⠀⠀⠀⢀⣸⡛⣅⠀⠉⠀⠀⢀⡴⠋⠀⣘⣿⣧
⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⠁⠘⣿⡇
⠀⠀⣿⠇⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⠁⠐⠒⠒⠚⠁⠀⠀⠀⠈⠁⣿⡇
⠀⠀⢿⡇⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣸⡏⠀⣴⣶⣶⢰⡶⣶⡄⣶⢶⣦⣴⡶⣶⠀⠀⠘⣿⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⡿
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣿⣾⣿⢸⣿⣼⡇⡗⣿⣿⣟⣿⢺⠀⠀⠀⠘⢿⣦⣀⠂⠀⢠⠀⠀⢣⣠⣾⠟⠁
⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣤⣴⡾⠋⠀⠀⠀⠿⠿⠿⠸⠶⠾⠃⠿⠾⠟⠻⠷⠿⠀⠀⠀⢀⡼⠋⠻⠿⢶⣶⣶⡾⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠾⠿⠟⠛⠛⠛⠻⠿⠷⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⢀⣀⣀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠛⠷⣦⡀
⠀⠀⠀⣠⣶⠟⠉⠀⠀⣠⡾⠟⠋⠙⠻⣷⡿⠛⠉⠉⠙⢿⡄⢀⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⣠⣴⣶⣶⣦⣀
⠀⢀⣼⠟⠁⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡿⠛⠛⠻⣦⡀⠀⠀⠀⠘⣷⡀⠀⣾⠏⠀⠀⠀⠈⢻⣾⠏⠁⠀⠀⠈⢻
⢀⣾⠃⠀⠀⢀⣶⠟⠛⠛⢷⣦⠀⠀⠀⢸⡇⠀⠀⠀⣠⡿⠁⠀⠀⠀⢸⣷⠀⠀⠀⠀⢹⣇⠸⣿⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀
⣸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣰⡾⣿⡀⠀⠀⠀⢀⣸⣟⠀⠀⠀⠀⢸⣿⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⠁⠀⠀⣠⡾⠿⢷⣦⠀⠀⠘⣿⡀⠀⠀⠀⠀⠟⠀⠈⠻⠂⠀⠀⠛⠋⢻⣇⠀⠀⠀⢸⣿⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⢀⣤⣶⠟⠁
⣿⠀⠀⢸⣟⠀⠀⠀⠙⠁⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠻⣦⣠⣶⠟⠉
⢿⡇⠀⠈⢿⣤⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⡀⠀⠘⠋⠁⣿⡇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁
⠸⣷⠀⠀⠘⣿⡀⠀⠀⣠⣿⣃⣀⣀⣀⣀⣀⣘⣿⣀⣀⣠⣼⢿⣦⣀⣠⣴⠟⠁⠀⠀⠀⢹⣧
⠀⢿⡆⠀⠀⠈⠛⠷⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢀⡀
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣤⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⣴⠐⠒⠶⡶⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⣀⡀⠃⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠸⠀⣀⣀⢿⡀⢀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢰⡟⡉⡀⠀⠀⠀⠀⠀⠀⡐⢃
⠀⠀⠀⠀⠀⢀⡐⠒⠖⠀⠋⠀⠯⠐⠚⠋⠉⠈⠃⢿⡿⣿⣶⠄⠀⠀⠀⠀⣴⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠘⠀⠀⠀⠀⠀⠈⡧⣬⠁
⠀⠀⠀⢀⡴⢛⣗⣾⣶⡆⡆⠀⠀⠀⠀⠀⠀⠀⢠⣫⣶⣿⠒⣦⠀⠀⠀⠀⠈⠂⣹⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣠⠍⡉⠀⠀⠀⠀⠀⢀⠐⠉⠈⠂⢠
⠀⠀⠀⠀⠀⠈⠸⣷⡾⠇⠉⠉⠉⠉⠉⠀⠈⠉⠉⠿⣦⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⢮⡂⠀⠀⠀⢀⣴⣷⢼⣦⠀⠀⠛⡠⠴⠇⠀⠀⠀⠀⢸⣟⣛⣛⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠱⡀⠀⠀⠉⠀⠘⠁⠀⠀⠀⢀⡀⣰⡄⠀⠀⠀⡆⣿⣿⠿⣟⣦⠂
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠙⡧⠀⠀⠀⣿⣟⣷⣜⣋⣟⡀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⠀⢐⢳⡞⠂⠀⠀⠀⠙⢄⠀⠀⠀⠐⠲⣤⣰⣶⣤⡾⢷⣶⠄⠀⢙⢈⣽⢭⡭⣍
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣧⠀⠀⠀⠀⠀⠀⣐⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⣿⣷⣿⣷⣶⠉⠀⠀⢸⣾⣿⢸⣿⣿
⠀⠀⠀⢠⣤⣯⣄⣠⣤⣤⣴⣤⡄⠀⠀⠀⡞⠉⠀⠙⡆⠀⠀⠀⠀⠀⠂⠒⠊⣿⠾⠉⠑⠒⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢨⡿⢣⠼⠭⢥
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠓⢄⣄⠰⠃⠀⠀⠀⠘⣆⠀⠀⢀⡚⣷⠀⢀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣣⣬⢩⣭⡭
⠀⠀⠀⠀⠀⢏⣹⠶⢶⣛⡷⠠⡄⠀⠀⠀⢆⡉⠙⠛⡆⠀⠀⠀⠀⢙⣿⢧⠴⣇⢰⠦⠾⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠿⡿⢓⣬⣷⣙⠿⡟⡀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⢠⡇⠑⢶⣀⣶⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣦⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠐⠂⠀⠀⠀⠀⠀⢰⢴⣷⢰⣾⠆
⠀⠀⠀⠀⠲⠾⠟⠟⠻⠿⠳⠞⠀⠀⠀⠀⢶⡾⡿⠻⠻⡛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣤⣷⠸⠧⠙⠉⣳⠾⠂⠀⠀⠀⠀⣀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢐⡲⢤⡲⢶⣖⣾⣿⣶⠷⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣴⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣷⣟⣳⣭⣿⣭⡷⣻⣿⣿⡇
⠀⠀⠀⢴⡶⢦⣀⠤⢲⠾⠛⠒⣦⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣩⣍⣹⠿⣿⡿⣏⡻⣭⡛⠇
⠀⠀⢰⣿⡷⡌⢳⢶⡶⠾⢟⣉⠻⠄⠀⠀⠠⠤⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⠤⣶⠀⠠⠀⠀⠀⠒⠉⣍⡤⠀⠀⠀⢀⠉⠴⣫⣿⣇⣿⡗⣿⣿⣮⣙⡇
⠀⠀⢘⡭⣛⢴⠒⠒⠈⣁⣀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⠓⠛⠀⠁⠀⠀⠀⠀⠀⠠⠤⠿⠀⠀⠈⠁⠈⠉⠀⠁⠉⠉⠉⠉⠉⠉
⠀⠀⠈⣇⡈⢸⠀⠀⠘⡁⣯⣠⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⣀⠀⡀⠀⠀⠈⠃⠀⠐⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀⠀⡀⠀⣀⢀⣀⡀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣇⡙⣿⠀⠈⠙⠇⢹⣀⡈⠀⠀⠀⠀⠀⢠⣴⠛⡉⠉⡅⢉⡿⢷⣄⡀⠀⠀⠀⠀⠀⢸⣋⣁⡻⣋⣀⣈⣟⢉⡈⢹⡉⡉⡹⡍⠉⢉⠟⠉⠉⠈⡏⠁
⠀⠀⠀⣏⡙⣿⠀⠈⠈⠈⠸⠀⡆⠀⠀⠀⠀⣀⡎⠀⠲⡜⢼⢃⣞⡤⠞⠙⡇⠀⠀⠀⠀⠀⠹⠀⠀⠀⠐⠖⠒⠀⠈⠐⠂⠉⠀⠃⠘⠛⠛⠓⠛⠋⠉⠘
⠀⠀⠀⡏⠙⣷⠉⠈⠉⠀⠹⠀⡇⠀⠀⠀⠀⣸⠁⣀⡅⢘⣿⣿⡥⢀⠀⠀⢻⠃⠀⠀⠀⠀⠂⠀⣦⡶⠦⣤⣤⣄⡀⣤⠄⣤⣤⠴⣀⣴⣦⣠⡤⣤⣤⣤
⠀⠀⠀⣏⡛⣿⠀⠐⠠⠋⢱⠀⡇⠀⠀⠀⠀⠘⣇⠠⠖⢡⣿⣿⡙⠦⡈⢉⡟⠂⠀⠀⠀⠀⠀⢀⣥⡴⢃⣤⠤⣬⡇⢀⠀⣏⢀⣀⠙⠁⠀⢸⡏⠁⠀⠃
⠀⠀⠀⡏⠙⣿⠀⠰⠀⠋⢹⢉⠇⠀⠀⠀⠀⠈⠙⢧⣄⢰⠇⢿⡇⣀⡴⠛⠂⠀⠀⠀⠀⠀⠶⠿⠿⢟⡛⠠⠤⠤⡷⠴⠤⡷⠶⠦⢶⣤⣦⣼⣇⡀
⠀⠀⠸⣦⣕⣻⠀⠘⠊⡉⣉⣀⡠⠦⠀⠀⠀⠀⠀⠈⠉⡙⠒⠾⡏⠉⠀⠀⠀⠀⠀⠀⠀⠰⢰⡶⣶⣦⡿⠤⠴⠶⡧⢤⡤⣧⠠⣶⠾⣯⡽⣿⣿⣇⡠⢀⡀
⠀⠀⠀⠀⠉⠋⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⣠⣾⣿⡿⠋⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠙⢿⣿⣷⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⣼⣿⡿⠋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠙⢿⣿⣧⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠁⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⣄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣿⣿⠏⠀⠀⢠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄⠀⠀⠘⢿⣿⣷⣤⣀⣸⣿⣿⣿⣿⣿⣿⡇⣀⣤⣾⣿⡿⠃⠀⠀⢠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠙⣿⣿⠋⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣿⣿⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠋⠀⣀⣤⣶⣾⣿⣿⣷⣶⣤⣀⠀⠙⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣴⣾⣿⠟⠋⠀⣿⣿⠈⠙⠻⣿⣷⣦⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣇⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠀⠀⢀⣿⣿⡀⠀⠀⠸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠉⠀⠈⠉⠉⠉⠉⠀⠈⠉⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢸⡇⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⠶⠶⠀⠀⣤⣼⡇⣿⢸⣿⡶⡆⠀⠰⠶⠶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢷⡘⠇⠿⠸⠛⠇⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⡀⢰⡟⠛⠛⠛⠛⣇⠀⣀⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉



⣏⣈⣯⣭⣟⣯⣿⣽⣽⣿⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢾⣿⢷⣄⠀⢀⣠⣄⡀⠀⣠⡾⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⣿⡿⣟⣿⣿⡞⣿⡄⣿⣿⣿⣿⢀⣿⢳⣿⣿⣻⢿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣀⣤⣶⢾⣻⣽⣾⣿⣯⣷⣿⢿⣿⣽⣷⣟⢷⣼⣻⣟⣧⡾⣫⢿⣯⣿⡿⣿⣾⣽⣿⣷⣯⣟⡷⣶⣤⣀⣀
⠀⠀⠈⠉⠻⣭⣿⣿⣿⣻⣿⣷⣿⣿⣿⣯⣿⣿⣟⣿⣿⣿⣶⣯⣿⣿⣽⣶⣽⣿⣿⣻⣿⣿⣿⣻⢿⣿⣾⣿⣟⣿⣿⣿⣭⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣽⣿⣿⣿⣟⣯⣾⡿⣿⣿⣟⣿⣟⣽⣯⣻⣿⣻⣿⣟⢿⣿⣿⣻⣿⣿⣿⣿⣽⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣷⣯⣿⣿⡿⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢉⣩⣭⣭⣉⠀⢸⡇⠀⣉⣭⣭⣍⡉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡉⠛⠻⠗⢸⡇⠺⠟⠛⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣦⣼⣧⣶⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢻⡍⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣦⣘⣧⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⡛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⢸⡇⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡼⣧⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣜⣧⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣫⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠚⠉⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠤⡤⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣧⣴⣯⣤⣄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠰⠋⡀⠘⢶⠀⠀⢤⠶⠎⡐⢆⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡠⠖⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠈⠀⠀⡆⠀⠀⠈⢻⡆⠀⠀⠀⠀⢀⠴⠲⢦⡀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⢡⠴⠒⠲⢦⡀⠀⣬⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡇⠀⣴⠶⢄⣆⢸⣠⣶⣄⠀⢀⡇⠀⠀⠀⠀⡏⠀⠀⠀⢹⠀⠀⠠⠊⠁⠉⢢⠀⠀⢐⣴⡏⠀⠀⠀⠀⢳⣤⣾⠀⢠⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠱⣀⣹⣦⠞⠉⠉⢧⣼⠟⢀⡞⠀⢀⣠⠴⠚⠁⣠⣾⣦⠸⠦⡴⠁⠀⠀⠀⢀⡇⠀⠀⠈⢳⡀⠀⣼⡿⢶⠀⠀⣣⡬⠿⣤⠁⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢉⠉⢻⢀⡴⠋⠀⠀⠀⢣⣹⣇⡽⠃⠀⠀⠀⣾⠿⡇⡸⠀⠀⠀⠀⢠⠟⠢⢘⠿⠚⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢀⠤⠊⠀⠀⠙⣄⡀⠀⠀⣠⡏⠁⣼⠏⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠄⠿⠤⢿⡁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘⠳⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣴⠒⠋⠁⠀⠀⠀⠀⠀⠘⡈⢉⢉⡜⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠀⠁⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⡄⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸⡀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠒⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠀⠀⣀⣀⠽⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠋
⢸⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠁⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀
⠸⣤⣤⡤⠶⠚⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠉⠀⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣀⡤⠤⠿⣄⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠠⡾⠴⠒⠋⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠦⣤⣤⡤⠊⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠛⠉⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⣄⠀⣀⣠⠤⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣀⣠⡤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⡀⠀⠀⢠⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⢀⠀⡰⢣⠀⣠⣖⠠⠋⠐⡄⠀⠀⠀⠀⢰⠀⠀⠑⠺⢷⡀⢣⠑⠄⢂⡰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠹⣦⣎⣰⣧⣴⣯⠵⢤⣄⣤⠃⠀⠀⠀⠀⠀⠓⠒⠚⠛⠛⠛⠚⠷⣾⡿⢁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣹⡟⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠆⠀⠀⠀⢸⣷⠟⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠻⣇⠀⠀⠀⠈⣇⠀⠀⢰⠀⠀⡀⠀⠀⠀⡇⠀⠀⣾⠀⠀⠀⣠⣾⣅⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠞⠻⠷⣦⣄⣀⣹⣆⠀⢸⠀⠀⣷⠀⠀⢠⣇⣠⣾⣷⡶⠾⠛⠋⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣾⣧⣼⠇⢀⣤⡾⠟⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠿⣿⡋⣠⢞⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣁⡀⠙⠱⣏⡸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⢀⠘⠛⠃⠀⣦⢻⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢴⣯⣄⣤⣾⣴⡟⡸⣧⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠹⣽⠃⣰⢱⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡸⠏⣾⡟⡟⡟⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⡘⣿⣿⢻⡘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣾⣿⡜⢿⣾⣧⠃⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣝⢿⣿⡌⢿⡟⣰⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣍⠁⠙⢿⡌⣰⣟⡜⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡝⣷⣅⣿⠟⣸⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⠃⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣖⡶⠂⠒⠀⠀⠀⠀⠀⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⣡⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢒⡀⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣴⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣠⢖⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⡺⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⢎⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⢌⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣠⠞⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⡡⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⢜⠌⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⢞⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣣⡯⡁⠀⠀⠀⠀⠀⠈⡗⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⢞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡴⠏⠊⠀⠀⠀⡠⠂⠠⢊⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⢟⠞⠁⠀⠀⡠⢊⠔⠔⡵⠃
⠀⠀⠀⠀⠀⠀⠀⣠⢊⡴⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⡾⡳⠫⠊⠀⢀⢔⠔⠡⣪⠎
⠀⠀⠀⠀⠀⠀⡴⢡⣈⣾⢆⣠⡤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⡵⢊⠈⢀⠀⠔⡡⠃⠔⡵⠁
⠀⠀⠀⠀⠀⢠⣵⠟⢟⣿⡿⢻⡿⣿⣵⣶⣮⣴⣶⣶⣒⣂⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣡⣪⢞⠔⢁⠔⠁⡨⠊⡠⢢⠎
⠀⠀⠀⠀⠀⣿⡟⠐⠢⠀⠀⠀⠤⠅⡊⠠⢋⡉⠙⣻⢛⠿⡿⢿⣶⣦⣬⣭⣐⣒⠀⠬⠙⡫⣫⡞⣛⠁⠀⡀⠀⠀⠊⢀⢌⡜⠁
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠍⢈⡙⠚⠽⠾⣽⣿⣽⡷⡾⡫⣺⠋⡀⣈⠉⡠⢃⠐⡹⠋
⠀⠀⠀⠀⠈⡘⣿⡄⠁⠐⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢀⠁⠐⢀⠀⠀⢨⠋⢌⠔⢀⣌⠉⣁⠈⡠⣋⠉
⠀⠀⠀⠀⠀⠘⢎⡛⠋⡃⠠⣄⣀⣠⢄⣠⡭⢀⣀⡐⠠⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁⣽⠐⠁⢔⠕⢁⡔⢁⢊⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠠⠌⠉⢒⣀⠭⡁⣂⡉⢉⣐⡀⠂⠂⠄⣒⠒⠤⢄⠀⢺⠀⠀⠃⢀⢏⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠐⠋⠡⠭⢿⣲⡆⣀⢈⠥⠀⡐⠈⣇⠀⢔⡑⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠘⠿⠶⣒⡈⢐⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠠⠐⡀⠅⡀⠄⠀⠨⠀⡀⠁⡀⠄⢐⠀⠒⠠⠀⡀⠄⡀⠄⣀⠄⠀⠄⡀⢄⠠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠁⠐⠉⠀⠂⠡⠂⠁⠐⠉⠀⠂⠡⠂⠠⠂⠠⠔⠉⠔⠈⠔⠡⠂⠀⠊⠄⠊⠁⠊⠕⠀⠂