Skepticism Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢂⢰⣦⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠆⠀⠀⠸⢃⠂⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠸⠇⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠨⣆⠀⠐⢀⠜⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣥⣿⣽⠀⡀⢀⡇⠘⡄⠀⠀⢀⠀⠀⠀⠀⢹⠒⠀⢊⠠⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣿⢧⣿⣿⢰⠊⣺⣣⣾⣿⣷⡶⠀⠀⣤⠀⠀⣸⣧⣶⣶⡄⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣔⣿⣿⣿⣿⣿⠀⠛⢁⣽⠀⣾⣿⣿⣿⣿⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢻⣻⣿⢭⢻⣿⣿⣿⣿⣿⣏⣆⠀⠀⢳⣾⣿⣿⣿⣿⣿⡇⢘⢀⣜
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣏⣶⣾⣿⣿⣿⣿⣿⣊⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⠅⠈
⠀⠀⠀⢀⠀⠀⠀⢺⠞⣴⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣾⡿⠟⠙⢤
⠀⠀⠀⠀⠀⠀⠀⣿⡗⣽⣿⢏⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢈⣹⣿⢹⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡆
⠀⠀⠀⠀⠀⠰⡟⣿⣸⣯⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠠⠀⠤⡴⡘⣡⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⢄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢸⣿⣿⣿⣿⣿⣇⠀⣨⣻⣿⣿⣿⣿⣟⡁⠀⣽⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⡛⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠂⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠰⡉⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠢⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡿⠿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠟⠛⠁⠀⠀⠀⠉⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⠀⠀⠀⠀⠀⠀⠀⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠶⣶⡦⢀⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠸⡟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⡁⠐⠶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣎⢢⣀⠁⠀⢽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣦⠙⣷⣶⠖⡁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡖
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⢍⣡⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢠⣾⣷⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⢿⡿⠿⠿⢿⣿⢿⣿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠉⠀⠤⠤⠀⣀⠈⠛⠿⠛⣁⠈⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⣶⣶⣴⣶⣤⣤⠀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠃⠀⢎⠻⠿⣋⠤⠀⠈⡏⢹⣿⡿⢉⠿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠈⠁⠀⠀⠀⢰⠀⠀⠛⠛⠋⠀⢸⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣅⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣤⣴⣾⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣤⣤⠀⠀⠀⠀⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠶⠟⡛⠻⣿⡇⣀⣀⣀⣀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡶⠊⠻⠃⢹⡀⠀⠀⡇⠀⠈⣄⣀⣤⡀⠈⠙⠉⠉⠉⠙⣷⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠈⠀⠀⠈⠀⠀⠇⠀⠀⡇⠀⠀⠘⣇⠀⠹⢤⣠⣆⣼⡄⣀⣿⣀⣄⣀⣙⡓⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⣴⣦⣤⣤⣶⣤⣾⣦⣴⣦⣦⣄⣀⣀⣸⡇⠀⠀⣀⣹⠿⠛⠛⣿⠉⡏⠉⠉⣿⣆⠀⠋⠉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⠉⠈⠙⠉⠉⠙⠛⠛⠻⠛⠻⠟⠿⠛⠛⠛⠛⠀⠀⠀⠀⠈⣡⠀⠀⣿⠀⠀⠀⠀⡇⠙⣷⠀⠠⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢧⡀⡇⡏⢠
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠃⠀⠰⠋⠀⠀⠈⠓⠒⠚⢛⣤⣶⣿⣤⣤⣤⣠⣧⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣀⣠⣶⠶⣶⠷⠾⡿⠶⠶⠶⡷⢾⠋⠀⠀⠈⠉⠁⠀⠐⠛⠁⠀⣿⠉⢹⠿⣿⣿⣷⣤⠀⠀⠻⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣠⣶⣶⠟⠋⠁⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⡀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠁⠀⠀⠀⠀⡆⠀⢸⠀⠀⡇⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡿⣿⣿⣧⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠈⠑⠂⢠⣤⣤⣿⠀⠀⢸⡇⠀⠀⠀⠁⢹⣿⣿⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⡧⣴⠄⣶⠠⠤⠴⢶⠖⠒⠿⠒⠺⡗⠒⠻⠷⣾⠋⠀⠉⠓⠤⠀⢀⠠⠄⢿⠀⠛⢸⡇⠀⠀⠀⢠⠀⢿⣿⣿⡇⠀⡿
⠀⠀⠀⠀⠀⠀⣿⡁⡀⢀⡀⣀⣀⡄⣴⠦⠀⠶⠄⢶⠧⠤⠶⠶⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⢺⣿⠶⠶⣦⣼⡆⠘⢿⣿⣧⢸⡗
⠀⠀⠀⠀⠀⠀⢿⡏⠉⠘⠁⠀⠀⠀⠈⠀⠀⠀⠀⣾⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⡁⠘⣿⠀⠈⠉⢹⡇⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⡔⠈⠉⠠⢀⣀⠤⠐⠒⠤⣧⡀⢽⡀⠀⠀⢸⣇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣼⡇⢈⠁⠀⠀⠁⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⢧⠄⠀⠠⢄⣀⣀⡠⠤⠤⣈⣧⣸⠃⠀⠀⠸⣿⠛⠄⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢻⠋⠙⠑⠒⠶⠂⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠷⢠⣌⠙⣿⣟
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣴⣿⡆⠀⠉⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡌⠀⢀⠀⠀⠀⠀⣿⠀⢀⠀⠀⣀⠀⣀⣀⠀⠀⠤⣀⠀⠀⠀⠀⢀⣀⣈⡅⢀⣽⠉⣿⡟⢀⡀⠀⢸⣿⠑
⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣤⣤⣤⣤⣌⡉⣙⣋⣉⣟⣉⣹⣋⣉⣏⣉⣻⣧⠀⠀⠀⠀⠉⠒⠂⠉⠁⠀⠈⡇⠈⢻⠀⣿⠙⢢⣿⣄⡎⢻⡀⠠
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠉⠉⡉⠉⠉⠉⠁⠉⠁⠈⡏⠈⠙⠉⠙⠟⠁⠀⡁⠀⠀⠈⠐⠤⠒⠉⠁⠉⠀⠒⡇⠀⢸⠀⣿⠀⠀⠙⢿⣷⣈⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣟⣀⣀⣀⣠⣅⣀⣠⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⢽⣷⡀⠀⣴⠋⠻⣇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣿⠤⠤⠾⡷⠠⣤⠤⣾⡇⢠⣶⠀⢠⣤⡀⢀⣶⠄⠀⠄⠀⠀⠀⠀⠀⢀⡄⠠⣸⡀⢀⠀⢸⠻⠄⢹⣿⡀⠀⠙⢦⡀⠈⡧
⠀⠀⠀⠀⠀⠀⣯⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⡇⠀⢀⡍⠀⠀⡇⠀⠀⠀⠈⠁⠒⠂⠈⠁⠀⠀⠈⠃⢺⠀⠸⠂⠀⢸⣿⠉⠰⣧⠈⢍⠢⣤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠈⠋⠛⠃⠘⠟⠃⠘⠟⠀⠸⡷⠄⠰⠿⠃⠀⠙⠓⠦⢀⡀⠄⠀⠲⠤⣄⣿⠀⠀⠀⠀⠀⠈⠋⠣⡀⠈⠑⠬⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠛⢛⣿⣿⣿⣿⣦⡀⢠⣾⣿⣿⠿⠟⠛⢛⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⣰⣿⣿⣿⣿⢿⣿⣿⡄⠹⠏⠁⠀⠀⣼⣿⣿⣿⡟⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠹⣿⣿⣿⠟⠀⢿⣿⣿⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣀⠀⠀⠀⢀⣠⣴⣿⣆⠙⣿⣿⣷⣤⣀⠀⠀⠀⢀⣠⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣀⠀⢸⣧⠖⠀⢠⣒⣲⠀⠀⡖⠒⡄⠀⢺⠒⠀⢰⠀⠀⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜⠀⠰⠏⠢⠀⠘⠤⠤⠀⠀⡧⠤⠇⠀⠸⠤⠀⠸⠀⠀⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⣠⣴⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⣀⡤⠚⠉⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⣀⠴⠛⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢀⡠⠔⠋⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⡴⠖⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢁⣀⠄⠂⠁⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠠⠐⠉⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣋⣤⣊⣁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⢻⡿⣿⡻⣛⣿⠉⣿⡿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⣿⣿⣾⣟⣻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⠾⡿⠛⠓⣀⠙⠻⣿⣿⡿⠟⠛⠛⠉⠙⠿⣿⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢩⣿⣿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⣜⠛⣿⠿⠟⠻⣩⠟⠻⠀⢘⠋⠀⢀⣤⡶⢶⡇⠀⠀⠙⢿⣿⣿⣿⣎⣉⣻⣯⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠒⡿⠦⠂⠀⠀⠒⠒⠁⠀⢔⡢⠍⡙⢄⠀⠁⣰⢶⣶⣾⣿⣏⣩⣿⣯⡹⣏⣹⣿⡿⣿⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣩⣧
⠀⠀⢀⡐⠀⠀⠐⢄⠁⢀⣤⡴⠂⡀⠁⠀⠙⢦⣄⠀⠀⠒⠉⠸⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣧⣴⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣭⠁
⠀⢀⠊⠀⡀⠀⠀⠀⠉⠀⣿⡀⡰⠁⠀⠀⠀⠀⠙⡇⢤⣿⣴⣿⣿⣿⡿⠛⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⣿⠷⠇⠀⠀⠀⠀⠀⠀⠈⠂⠙⣿⣿⠛⠟⠀⠈⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢰⠃⢠⠞⣦⣄⠀⠴⠋⠉⢠⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢿⣄⣨⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⣬⣼⣿⣿⣿⣍⣿⣿⣟⢛⣿⣿⡇
⠀⠈⣤⠁⢰⡏⣀⡠⠤⡀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡾⣿⣏⡅
⠀⢰⠂⠀⠈⠉⠀⠀⠘⢳⣶⢾⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠃
⠀⠘⢧⣄⣀⠀⠀⠀⠀⠘⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢿⡉⠛⠛⠒⠒⠈⠉⠉⡇⠀⠀⠀⠀⠀⢀⡠⠶⠃⠙⢀⠀⠀⡟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢻⡆⠀⠀⣼⠇⠀⠀⡇⢠⠀⠀⠀⠀⢠⡈⠃⠰⠂⠇⠀⠀⣿⠲⠶⢤⣄⣈⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⡄⠀⢰⢠⡇⠀⠀⡝⠛⡧⠴⣤⠀⠤⡗⠀⠀⠂⠀⢰⠰⠋⠑⠲⠦⢄⣉⡙⠓⠒⠦⠄⣀⡀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⢸⡇⠀⢸⠀⡇⠀⠀⡇⠀⢰⠄⠘⢆⠀⠓⠒⠈⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠤⢄⣉⠑⠒⠂⠤⢀⣀⠈⠉⠉⠛⠻⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠘⡇⠀⠀⡄⢿⠀⠘⡇⠀⢸⡄⠀⢻⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⢤⣄⡉⠑⠲⠂⢤⠀⠀⠀⠉⠛⠓⠒
⠀⠀⠀⠀⠈⠃⠀⠀⡇⢸⠀⠀⢧⠀⠘⡇⠀⢸⠀⠀⠀⡇⠀⠀⠀⡄⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⢀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠀⠀⠸⡆⠀⠁⠀⠨⠀⠀⠀⢁⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⡀⡄⡀⠀⠀⠀⢠⠤⣀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠄⠐⠌⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠀⠴⠀⠀⠀⠀⠐⠃⠰⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠐⠐

⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠒⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡔⠉⠁⢀⠀⣀⠽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⡀⢇⢨⣭⣴⠄⣷⠟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⡇⠙⢼⠄⠀⠀⢀⠼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⢢⠀⢀⣄⡠⠤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣼⠀⠘⣦⡉⠻⠗⢤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡳⠀⠀⠀⠀⢑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠢⣼⣅⡀⠀⠀⡠⠊⢈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠉⠉⠢⣄⠀⢩⠛⣍⢽⢁⡴⠁⠀⠳⡒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⣀⡌⠋⠙⡄⠀⠀⠀⠙⢄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢣⠀⠀⠘⢆⠀⠀⠀⠈⢣⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠣⠀⠀⠈⣆⠀⠀⠀⠀⠱⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠇⠀⠀⠘⡆⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⠀⠀⠀⢸⣄⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠘⠀⠀⠀⠀⢸⠈⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣷⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠘⠀⣾⣇⣿⣿⣿⣿⣿⣿⣿⣿⡄⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢤
⣿⣿⣿⣿⣿⣿⡇⢿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣺
⣿⣿⣿⣿⣿⣿⣖⢺⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⡅⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡀⢰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠟⢿⣿⣿⣿⣿⣿⡟⢻
⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣧⡔⠂⢠⡅⢸⡷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠘⠛⠿⣿⣿⣿⣷⣼
⠉⠙⠛⢁⣿⡷⠟⠋⠙⠛⠋⠉⠉⢹⣿⣀⡷⠸⡗⠘⡧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡏⠀⠤⠤⠄⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢻⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⢱⡆⣿⠀⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠐⠒⠘⠓⠘⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣳⠿⣿⠀⠘⢇⣀⣿⣿⣿⠛⠛⠛⠛⠛⠘⠿⠟⠛⠉⠛⠛⠂⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠿⠛⠉⠉⠉⠱⠀⡞⠉⠉⠃⠿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⡦⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠿⠇⢸⣦⣤⠀⣀⣀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠉⠉⠀⠛⠽⠛⠾⣷⡖⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣻⢀⣶⣤⣄⡀⠀⠀⠀⠀⠀⠈⠿⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠘⣇⣯⠙⡇⢠⡇⣶⠠⣦⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣀⠈⠁⠀⠧⠾⢧⣙⣶⡟⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠦⢀⣀⠀⠁⠘⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡀⠂⠠⠀⠀⠀⠐⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠁⠀⠙⠂⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠤⣀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣶⠄⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⠛⠛⠛

⣿⠟⠻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⢿⣿⢿⣿⣿⢿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⠀⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⢼⣿⣿⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡤⢤⣄⠀⠀⠀⠀⣿
⣿⠀⠿⠿⠿⠿⠿⠿⠿⠛⠛⠿⢿⠿⠿⠿⠿⠟⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⢟⡼⢿⣿⣿⣿⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⢀⡠⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠠⣤⠄⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠉⠉⠁⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⣴⠿⠿⠿⢷⣆⠀⠀⠀⠀⣿
⣿⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠉⠂⠀⠀⠀⠈⠀⢸⣿⡇⠀⢀⣤⣤⣄⠀⠀⢻⣄⣀⠀⠴⠞⢻⣿⣿⣿⣿⣿⠀⠀⢰⠁⠀⠀⠀⠈⣿⡆⠀⠀⠀⣿
⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣟⠃⠈⢌⢻⣿⣾⢳⡀⠀⣇⠀⠀⠄⠀⢸⣿⣿⣿⣿⣿⠀⠀⠹⠓⠀⠃⠀⠀⢛⠃⠀⠀⠀⣿
⣿⠂⠐⠒⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⢈⡀⢀⢸⣯⡳⢦⠀⠀⠙⠟⢰⣃⡴⢟⣦⠒⠖⢀⣼⣟⡿⢻⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⡉⠀⠀⠀⠀⣿
⣿⣇⠀⠠⡤⠀⠀⠀⠀⠀⠖⠒⠂⠀⠈⠉⢉⠁⠀⢸⣿⣿⣿⣄⠀⠀⠀⠈⠉⠀⣸⢏⠸⠟⢛⡽⠋⠉⣡⣻⣿⠀⠀⢰⣋⠀⠀⠀⢀⣏⠀⠀⠀⠀⣿
⣿⡃⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠄⠠⢸⣿⣿⣿⣿⠄⠀⠀⠀⠣⢞⣿⣷⣶⡴⠋⢀⡖⠘⠈⣿⣿⠀⠀⢸⠀⠀⠀⢀⡞⢩⣷⣄⣀⠀⣿
⣿⡏⠁⠒⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⢸⣿⣿⡟⠁⠀⠀⠁⠀⠀⠈⠁⢈⠏⢀⣴⠏⠁⠀⣠⣽⣿⣀⣠⣤⣤⡀⣰⡀⢠⣿⣿⣿⣿⣿⣿
⣿⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢀⠀⢀⣸⣿⡟⣰⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠛⢹⣿⣿⣿⣿⣿⡇⣸⣡⣿⣿⣿⣿⣿⣿⣿
⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣠⣤⣦⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⢠⣿⡉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣾⡿⠛⠉⠉⠙⠻⣿⣷⡀⠀⠀⣿
⣿⠀⠀⠀⠀⢸⡮⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠠⠀⠀⢄⡁⠀⠀⠀⠀⠀⣄⠀⣠⣴⣾⡇⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⣿
⣿⠀⠀⠀⠀⠸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⢀⠙⠋⠰⠒⠀⠀⠠⠟⠁⠾⣿⣿⡇⠀⠀⣿⠀⠴⣤⠀⣴⢿⣿⣿⣿⣿⡇⠀⣿
⣿⠀⠀⠀⠀⠰⣿⠀⠀⣼⣦⣤⡄⠰⠾⠀⠀⠀⠀⣆⡀⠀⠀⠀⣀⡈⡀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣿⠀⠀⠀⠀⣻⡀⢈⣽⣿⣿⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠊⡷⠀⠀⠀⠀⠀⠀⣿⡿⠂⠀⠘⠹⣷⣜⣷⣂⣴⣿⠣⡀⠀⠀⣠⣾⣿⡇⠀⢸⡷⠂⠀⠀⠉⣉⡓⢸⣿⣿⣿⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠈⠹⣆⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢚⠛⠿⠋⠙⠀⢀⠠⠀⠀⠉⢻⣿⡇⠀⢸⡗⡀⠀⡸⠉⠉⠻⠛⣿⣿⣿⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠏⠡⢼⣉⠃⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⣠⠄⠀⠄⠀⢠⠅⠀⠀⣶⣿⣿⡇⠀⢸⣿⣧⢿⠀⠀⠀⠀⣔⣼⣿⣿⠇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠠⠚⠀⠀⠀⢠⡄⠀⠀⡀⠀⣿⣿⣿⡇⠀⢸⣿⣷⣿⠆⣦⢁⠀⢉⣿⣿⣿⡀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠗⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⡄⢀⡾⠉⠀⠀⠈⢆⣿⣿⣿⡇⠀⠈⣿⣿⣯⢀⣿⢱⡗⣸⣿⣿⡿⠃⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡠⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⢸⣷⣾⣿⣤⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠈⠛⠻⠿⠿⢿⡿⠿⠟⠉⠀⠀⠀⣿
⣿⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣷⣤⣤⣤⣤⣤⣤⣤⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⡏⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⢀
⠀⠀⡆⢸⣀⢸⣋⠆⡞⢡⣿⢸⡇⢀⡆⠘⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢀⣀⣀⡀⢼⠿⠿⠛⡿⠛⠈⢁
⠀⠀⡇⠈⠁⠀⣀⠀⣉⠈⠀⠈⠉⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢛⣏⠉⢁⡼⠉⠄⠀⣴⠂⠤⠚⠀⠀⠉
⠀⠀⠇⠀⠀⢜⡷⢲⢿⢡⣛⠖⣦⣔⢺⣄⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠲⠶⠀⢐⡪⠗⠋⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣦⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢨⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⢀⠀⠀⣀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⡀⢶⣾⠇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠒⢒⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢟⢀⡐⢻⡄⠦⠽⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠉⠠⠄⠁⠠⢬⠉⠁⠛⠄⠒⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠔⣱⣿⠗⣴⡀⠀⠆⠀⠻⡀⠀⢀⠆⠀⠀⡄⠀⠀⠀⠀⢀⡐⠁⠐⡀⠀⠘⠀⠀⠰⠻⠀⠀⠀⠀⠶⠄⠀⠀⠀⠁⠆⠀⠀⠀⠂
⠀⠀⠀⣠⠦⢾⡿⡧⠸⠛⡛⢿⣯⣹⣿⠧⣜⢆⠀⠐⣿⠢⠀⣰⣣⣄⠀⠀⠀⢀⠈⠀⠐⠇⠀⢀⢀⠀⢁⠀⠃⠀⠒⠒⠒⠀⠀⠈⠉⠁⠐⠂
⠀⠀⠀⠀⠁⠠⠌⠀⠑⠃⣀⣀⣄⢨⣿⠄⢾⠀⠦⡀⠈⠀⠂⡟⠛⠛⠀⠀⠀⠈⠉⡋⢹⠀⠀⠀⠀⠀⠘⠀⢰⠀⡚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓
⠀⠀⠀⠀⠲⢦⣀⡀⠁⠘⢿⠿⠿⠙⠛⠒⢈⠀⠀⠈⠀⠁⠀⣥⣤⠶⠀⠀⠀⠀⠀⡁⢸⢠⡀⠐⠀⢀⡀⡀⠘⣦⣭⠐⣓⣂⡥
⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣀⣀⣠⣤⠴⡖⢸⠀⢀⡀⠀⠀⠀⢁⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣅⡀⠋⣀⣠⡀⠖⢫⣤⠟⣨⡍⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢹⠠⢂⣽⣾⣦⣼⣷⣄⣠⣤⡌⠀⠀⠀⠀⠀⠀⠀⠃⣆⠸⠉⠋⠈⠉⠉⠐⠀⠀⣀⣠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠘⠂⠠⠄⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⡄⢀⠿⠂⢲⢠⡯⡅⠀⠀⡀⢈⡹⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⢆⠚⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣤⠀⢹⡗⠒⠀⠀⡏⠒⠁⠃⠀⠀⣰⠀⢀⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣘⠃⢀⣛⡀⣿⣿⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠆⠀⠂⠀⠀⠀⢀⢠⡖⣩⢠⣾⠀⢰⣿⡀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠉⠁⢻⣿⣿⣿⣿⣿⢀⡀⠄⠀⠀⠀⠀⠈⠆⠄⠀⠀⢀⠠⠀⠀⢠⠀⠁⣙⠏⠸⣇⠴⠀⡋⡀⠆⢠⢤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠠⠄⠀⠀⡀⠀⠀⠈⠂⢸⠀⠚⠀⡇⠀⠘⠀⠀⡌⣳⠊⡹⡄⠘⢰⠃⣐⠷⠲⡭⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⠇⠀⠉⠀⢘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⡇⠀⠰⡀⠘⣉⠛⡩⠆⡠⢊⣱⠆⡱⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⢛⣻⣿⣿⣼⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣆⠀⠈⠀⠀⠀⠀⡄⢢⣽⢋⡀⠈⠶⣎⡹⣤⡛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢏⣉⣡⡽⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣄⡈⠶⠖⢋⣠⢤⢚⡆⠀⡼⠀⣾⠁⠮⠁⠀⠃⢒⣸⣍⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠜⢚⣉⠁⢠⡇⡿⡏⠁⡀⠀⠀⠃⢌⣸⡏⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠊⠐⠂⣤⡾⠓⠀⢸⢰⣾⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠻⠛⠘⠛⠻⠙⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠈⠛⠂⠀⠘⠛⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⡀⠀⠀⣀⠀⣀⡀⢀⢀⠀⠀⡀⠀⡀⢀⢀⠀⢀⡀⠀⢀⠀⢀⡀⠀⣀⠀⠀⣀⠀⠀⠀⢀⡀⣀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⠀⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⠿⠿⢛⣛⣋⣭⣭⣭⣭⣭⣥⣶⣶⣶⣬⣭⣭⣭⣭⣍⣛⡛⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣽⢟⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣝⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡎⣻⡟⢹⣯⣿⡷⡏⢷⡽⠙⢿⡏⣾⠻⣭⣾⡽⣽⣹⣧⣿⡛⢹⠙⠏⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢶⣝⠶⣷⣿⣾⣿⣿⣿⣿⣷⣿⣿⣾⣿⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣾⡶⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣶⣭⣟⣛⣻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣟⣛⣻⣭⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠛⢛⣿⣇⣛⣻⣽⡶⢶⣒⣒⣻⣟⣛⣛⣛⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠏⠉⠁⠀⠀⠈⠁⠛⠻⣿⡏⣴⣿⣿⣿⣿⣿⢛⠿⣻⢹⣷⣍⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⠀⢠⣤⣤⣤⣤⡀⠀⢶⣾⣧⠻⣿⣨⣍⣿⣰⣼⣾⣷⣿⣿⡟⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣾⡟⠙⠟⣻⣧⣾⣿⣿⣿⣿⣶⣭⣽⣛⣛⣛⣛⠿⡟⣵⡎⠁⠀⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣷⣽⣲⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣧⣲⣶⣦⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣳⣿⣿⣿⣶⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢿⡟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣻⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣷⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣸⡇⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠷⠀⠀⠈⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣿⣿⣿⣿⣶⣶⣶⣶⣾⣷⣤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⡿⠿⠿⠟⢻⣿⠿⠛⠛⠛⠛⠛⠻⣿⣍⠉⠙⢷⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢠⣿⣇⡠⠤⢤⣤⣤⣄⣀⣘⣿⡆⠀⠀⠉⠛⠷⢿⣆⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠸⣿⣷⡞⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠏
⠀⠀⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠁⠀⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣆⠀⠀⠀⢸⡟⠁
⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣄⣀⣹⣦⡀
⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠉⠉⠉⠁
⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣶⣄⡀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠋⠙⢿⣿⣿⡀
⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣿⡇
⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣾⣿⠃
⠀⠈⠻⢿⣿⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⢿⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠈⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣀⣠⣴⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⠾⣿⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⡿⠉⠉⠉⠛⠛⠻⣿⡟⠛⠛⠿⢿⣿⣿⡿⠿⠿⠛⣿⣿⠛⠛⠛⠋⠉⠉⠉⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠛⣿⣿⠃⠀⠀⠀⠀⠀⣼⠟⣷⡀⠀⣠⠞⢹⠏⢻⡄⠀⣴⠟⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠛⠋⠉⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢀⣿⠀⠘⢷⣴⠋⠀⡿⠀⠀⣹⡾⠋⠀⢸⡇⠀⠀⠀⠀⠀⣴⡄⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⣄⣀⣃⣤⠾⠋⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢹⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣆⣰⣿⡇⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠉⣿⠋⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠸⣿⡀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣿⣶⡿⢻⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣿⠀⠿⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠿⠂⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢹⣷⣘⣿⡇⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣦⣤⣤⣤⣄⣀⣀⣼⣇⣀⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⣠⣤⣤⣤⣿⣧⣤⣶⣶⣶⣶⣿⣿⣿⡿⠿⠃⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢻⣿⣿⠋⠉⠉⠉⠉⠁⠀⠀⠹⣿⣶⣶⣾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠉⠙⠉⠉⠙⠛⠻⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⠛⠋⠉⠉⣙⠛⠛⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣄⣤⣀⣀⣤⣤⣶⡆⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣴⣤⣄⣀⡀⣿⠀⢀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠉⠘⠿⠶⠾⠛⠀⠀⠀⠀⠀⠀⠐⠿⣶⣴⣿⠀⠉⠙⠛⠛⠛⠛⠋⠁
⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⣶⢶⣶⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⢻⡏⢸⠂⡇⠐⠐⠀⡏⣿⣿⢹⢙⣿⢶⣤⣀
⠀⠀⠀⠀⣠⣶⠿⡏⠋⢀⡀⠀⢸⡇⢸⠀⡇⠀⠀⠀⡇⣿⠛⠀⢸⣿⠂⣿⡟⢷⣦⣀
⠀⠀⣠⣾⣿⡅⢰⡃⠀⢸⠁⠀⠸⡇⣸⠀⡅⠀⢃⠘⡇⣿⠀⡄⢸⣿⠐⣿⠃⢸⣿⠻⣷⣄
⠀⢰⣿⣹⣿⠀⠀⢀⡄⢸⠀⠀⡀⡇⠀⢠⠃⡀⢰⠀⡇⣿⢂⡇⢸⣿⣀⡙⢷⢸⣿⠀⠀⡟⣷⡀
⠀⢻⡿⣿⣿⠀⡇⠈⠀⡞⠀⠀⣧⠀⣀⣼⣦⣷⣾⣤⣤⣽⣾⡇⣿⣿⣿⡄⢸⣿⡿⠀⠀⡇⢸⣿⡄
⠀⠘⢿⣿⡇⠀⠇⡄⢸⠀⣾⣠⣿⠿⠛⠁⠀⠀⠀⠀⠈⠙⠻⣷⣿⡾⣇⣀⠀⣿⡇⠀⠀⣇⠈⠆⣿⡆
⠀⠀⠈⠛⠿⣿⣦⣧⣼⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠹⢶⠀⡇⡃⠀⠈⠀⠀⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⠁⡇⠀⢠⡀⠀⠀⢣⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⡇⠀⢸⡇⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⢸⡇⠀⢠⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⡤⢠⠀⠁⠀⢸⢈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⡇⠀⣿⢸⡇⠀⠀⢸⠈⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⢘⠀⡁⠀⣿⢸⠀⠀⠀⠀⢄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⢹⠀⡇⠈⢀⠀⠀⡽⠘⠀⠁⠀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠀⠃⢸⠀⡇⠀⢰⠀⠀⣾⡌⢠⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⢻⠆⢹⠀⡆⢸⠸⠀⡇⠀⢰⠀⣧⣘⣧⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⡻⣿⠀⡏⢰⣿⠀⡇⢸⠀⡀⠃⠀⢸⢀⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢸⠁⣿⠸⠃⣸⢸⠀⡇⢸⠀⠁⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⠀⡌⢠⢻⠆⢰⡏⠘⢠⡇⣸⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢸⡄⠃⡈⢸⡇⡿⠀⢇⣼⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡶⢀⡇⢸⢤⣅⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣤⣼⣾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁⠀⠀⠀⠀⠉⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠈⠲⢄⠀⠀⠀⠀⣀⣀⡀⡀⠈⣿⡆
⠀⠀⠀⠀⣸⡇⠀⠀⠰⣿⣗⣢⠅⠀⠀⠰⣿⣉⣐⠆⠀⢸⡇
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠟⠋⠁

⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⠒⠒⣲
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠁⠀⢺
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠁⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⣰⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢘⣰⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⣿⠃⠀⠂⠀⠀⢀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⠃⠀⠈⠁⠀⠁⡀⠁⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢠⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠄⠀⢠⠈⢿⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠛⠃⠀⢾⠟⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡦⠀⠀⠀⡆⠚⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡟⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠚⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⡟⢁⡄⠀⠁⠀⠀⠀⡘⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠓⠢⠤⠤⠐⠊⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸
⠀⠰⠉⠛⠿⠏⢠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠠⠀⠀⢀⠀⢄⠀⠀⠁⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠀⠋⠉⠑⠒⠒⠒⠹⠁⠉⢦⡀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⢠⠁⡀⠀⠠⠑⠄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⡸
⠀⢸⠀⠀⡈⠀⡇⠀⠀⡄⠀⠀⠸⠄⠀⠀⠀⠀⠠⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢈⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠚
⠀⢸⠀⠀⡇⠀⡇⠀⠀⠁⠀⠀⠠⠀⡀⠀⠀⠀⢀⣻⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠔⠀⠀⠈⠑⠒⠀⠀⠀⠘⣄⠀⢰⠀⠀⠀⠀⢣⠀⠀⠀⠀⢁⡀
⠀⢸⠀⠀⠇⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠊⠁⠀⠄⠀⠒⡦⠄⢐⢀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠄
⠀⢸⠀⠀⢁⠀⠈⠂⠤⠤⠀⠀⠰⠂⠀⠠⠀⠀⠰⠁⠰⠀⠀⠀⢀⣀⢨⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣾⡇⠀⠀⠀⢓⠆
⠀⢸⠀⠀⡸⢠⠀⡄⢠⠔⠀⠀⠀⡇⢀⠀⠈⠀⡠⢀⢀⠀⠀⠀⠈⠉⠀⠇⠀⠀⠀⠀⠀⠀⠆⠀⠚⠆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣰⢸⡇⠀⠀⠀⢹
⠀⢸⠀⢀⠁⠀⠀⠇⢸⠀⠀⠀⠁⡇⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠄⣀⠈⠀⠀⠀⠀⢀⠸⢁⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠁⠀⠀⠀⢸
⠀⢸⠀⠈⠀⠀⠀⠜⢀⡇⠀⠀⠀⠳⠘⡄⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠈⠠⡄⠀⠀⠈⠁⢼⠁⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⢀⡠⠀⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠛⠠⠤⢀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢨⣀⣀⣳⣶⣶⣦⣤⣴⣶⣾⣻⣇⣀⣃⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣴⣁⣀⣀⣀⣀⣀⣀⣀⣑⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠰⣀⢇⣠⢀⠄⢀⢀⣄⣀⡠⢀⠀⡰⣠⡀⡀⠀⣄⡀⣄⡄⠤⡄⠀⠀⡆⡄⡄⠀⠀⡄⣠⢰⡀⡄⠀⡀⣄⣄⠀⢀⠀⣄⠀⡀⡤⠀⣀⢀⡄
⠀⠀⠀⠀⠀⢀⢀⠀⡀⠄⠀⡀⢀⠀⠀⢀⠀⢀⢀⡀⠀⠀⠀⢠⠀⡀⢀⢠⣀⡀⠀⠀⢀⡀⢀⠀⠀⠀⡀⡠⠀⡀⡄⠀⢀⢀⡀⠀⠀⠀⣀⠀⠠⠈⠄
⠀⠀⠀⠀⠉⠉⠉⠓⠉⠈⠁⠉⠈⠈⠁⠈⠁⠈⠉⠁⠀⠉⠀⠈⠉⠈⠉⠉⠁⠁⠉⠀⠉⠉⠀⠈⠁⠀⠉⠀⠀⠉⠉⠁⠉⠈⠉⠁⠈⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣾⣿⣂⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠉⠉⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢶⣶⣶⣶⣶⣦⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠤⣤⡄⠀⢸⠀⠀⠀⠀⡏⠈⣉⣉⡉⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠁⠀⠀⠉⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡚⡀⠀⠀⡘⠇⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⢿⣷⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠉⠁⠀⠀⠀⠀⠀⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡀⠂⣸⣿⣧⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣷⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡟⠉⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠿⠏⠙⠿⢿⣿⣶⣤⣀⣀⠀⠀⠀⠀⣀⣠⣤⣾⣧⠀⠠⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠂⠀⠰⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠁⠛⢻⣿⣤⣤⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣀⣤⣀⣿⣿⣿⠟⢿⠟⢻⣿⣿⣯⠀⠀⠂⠀⠙⣿⠿⠛⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠀⢀⡀⠀⢾⣿⣿⣶⣦⣤⣤⣿⣧⡄⠂⢈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣖⢀⡀⢲⣶⣿⡿⠋⠙⢻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⣠⠴⢲⡞⠉⣉⡉⢱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⢠⠃⢰⢏⠀⢞⣋⠁⠐⠏⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⡟⠀⡀⠀⣚⣀⡙⠒⠆⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⣧⠀⣅⣀⣀⣨⡇⠀⣠⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠉⠉⢉⣠⣥⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⣀⡀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠗⠀⠀⠀⠀⠀⠘⠋⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⡔⠀⠰⢦⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠰⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⢆⠀⡏⠉⠇⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠄⣀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡈⢳⡇⠈⣰⠋⡥⢤⡉⢧
⠀⠀⠀⠀⠀⠀⣠⠖⢄⠀⠀⣀⠱⠀⠻⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡀⠘⢄⢀⠇⢻⠸⡀⢠⡇⠸
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠁⠀⠈⠑⠒⠚⠀⠀⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠾⠀⠘⢦⡈⠉⡠⠁
⠀⠀⠀⠀⢠⡏⠀⠀⠀⡠⠚⣁⡄⠀⠀⣰⢆⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣼⠀⠀⠀⠀⡱⠛⣉⠀⠀⠀⡏⠘⣄⠀⠈⢧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⢋⣭⠄⠀⡀⠃⠀⠘⣦⠀⠀⢳⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢵⡆⠀⡁⡏⠀⠀⠿⣄⡀⠀⠳⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠘⡀⠀⠀⠁⠞⣱⠃⢠⠇⠀⠀⠙⡀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠱⡄⠀⠰⣶⣯⡴⠋⠀⠀⣰⡟⠁⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡘⢏⠁⠀⣠⠖⢁⣠⠞⠵⠆⠀⣽⠀⠀⣤⠤⠿
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠻⡄⠀⠹⠋⠀⠀⠀⢠⣿⣥⠴⠷⠚⠁⠀⠀⠀⠀⠀⠀⡖⠀⣠⠞⠁⣿⣰⠃⣤⠀⠀⡈⣀⠀⢰⡋⣷⠀⣠⣤⠀⠘⡆⠀⣤⠼⠂
⠀⠀⠐⠒⣆⢱⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣄⣴⡆⢹⡇⢀⡤⠤⢄⠘⢧⡀⣸⠀⠸⣀⣀⣀⣠⠼⢧⣤⣀⣸⡇
⠀⠰⡐⡫⣥⠈⣇⠀⠀⠀⠀⠀⠈⢦⣀⣠⡾⠿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣿⠁⢧⣼⣧⡾⠁⠀⠀⠉⠉⠉⠉
⠀⠀⠱⡔⡾⠃⠸⡆⠀⠀⠀⠐⠒⠒⠛⠓⠒⠓⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢹⡙⢣⢀⠹⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠈⠉⠂⠙⠄

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣦⡑⢄
⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠿⣿⡆⢸⣿⠿⠿⠇⠀⣸⣿⣇⠀⢸⣿⠀⠀⠀⠀⠈⠻⣿⣿⣄⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣶⣿⠃⢸⣿⣶⣶⠆⢀⣿⠛⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⢘⣿⣿⣷⠌⡣
⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠘⣿⣆⢸⣿⣤⣤⡄⣸⡿⠿⢿⣇⢸⣿⣤⣤⡄⠀⠀⣠⣿⣿⠟⣡⠞
⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠀⠈⠉⠈⠉⠉⠉⠁⠉⠁⠀⠈⠉⠈⠉⠉⠉⠁⣠⣾⣿⡿⢃⠜⠁
⠀⠀⠀⠀⠀⠀⠉⣒⣒⣒⣶⣶⣶⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣶⣶⣶⣒⣒⣚⡁
⠀⠀⠀⠀⠀⣠⠊⣴⣶⣶⠖⠀⢀⡀⣀⠀⠀⢀⡀⠀⠀⢀⠀⠀⢀⡀⠀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⢠⠞⣡⣾⣿⡟⠁⠀⠀⣿⡿⠿⠇⢀⣿⣿⡀⠀⣿⡇⣴⣿⠋⢸⣿⠿⠿⠇⠀⢸
⠀⠀⢰⠁⣾⣿⣿⡋⠀⠀⠀⠀⣿⣷⣶⠆⣸⡟⢻⣧⠀⣿⣿⣿⣧⠀⢸⣿⡿⠿⠆⠀⢸
⠀⠀⠀⠑⢌⠻⣿⣿⣦⡀⠀⠀⣿⡇⠀⢠⣿⠿⠿⣿⡄⣿⡏⠘⣿⣆⢸⣿⣦⣤⡄⠀⢸
⠀⠀⠀⠀⠀⠑⢌⠻⣿⣿⣦⠀⠉⠁⠀⠈⠉⠀⠀⠉⠁⠉⠁⠀⠉⠉⠈⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠞⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⢻⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣭⡍⠉⠉⠉⠉⠁⠨⢧⣉⣩⣿⣿⣿⡿⣿⣿⡯⠛⣻⠛⠿⣻⠿⣿⣿⣿⠿⣿⣿⠿⠿⣟⣿⣿⡿⣿⣿⡿⠿⠿⠿⣿⣿⠿⡿⠿⢾⣿
⣿⣿⣿⣇⡀⣀⣀⡀⢀⣀⠤⠒⠛⠛⢉⡉⠿⠛⠛⠛⠿⠟⠛⠛⠛⠛⠓⠂⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢤⣿
⣿⣿⣹⣟⠛⠉⠉⠉⠉⠙⠛⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣧⣼⣿
⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿
⣿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢿⢿⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢠⣿⣿
⣿⣿⣯⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿
⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣷⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿
⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⡿⡶⠴⡶⢤⣤⣤⣒⣊⣉⣁⣀⣀⣩⣽⡛⡒⣦⣶⡖⣲⣤⡤⣤⣖⣓⣀⣭⣽⣿⣚⠛⢭⣍⣉⣉⣃⣒⣠⣤⣤⢒⡚⢁⣭⡬⢿⣿⣿
⣿⣿⣟⣓⣤⣖⣊⣿⣿⣿⣿⣿⣽⣭⣽⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⣷⣿⣾⣿⣿⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿⣾⣤⣾⣥⣤⣴⣶⣶⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣯⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⢿⣿⡺⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⡿⠿⠟⣫⣾⣿⡿⠟⠙⠀⣬⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣷⣾⣿⠛⣿⠋⠀⠀⠀⣸⣿⣿⢟⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢹⣿⡿⠉⠁⡀⠀⠀⠀⢀⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⣧⣄⠀⢩⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢾⣶⣾⣾⣿⣿⣶⣶⣶⣶⣶⡶⣾⣿⣷⣶⣾⣿⣿⣿⣿⣾⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣭⣭⡯
⠀⠀⠀⠀⠈⠁⣿⣿⣿⠃⠈⠛⠀⡀⠃⣿⣿⣿⠀⢿⣿⣿⡟⠙⣿⣿⣿⠁⢀⠀⠃⠹⢸⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠸⡇⠀⣿⣿⣿⠀⢸⡇⢹⡏⠀⣿⣿⡏⠀⢼⠆⠀⠀⢸⣿⣿⡇
⠀⠀⠰⠿⠿⠟⣿⡏⠉⠩⠟⠛⠛⠛⠛⠛⡋⣉⣉⣿⣿⣿⣿⣓⡛⠛⠓⠒⠻⠷⠶⣆⣚⢻⣿⡷⠶⢶⡶
⠀⠀⠞⠛⠛⠟⠛⠛⠛⠳⠶⠲⠶⠶⠒⠿⠿⣟⣿⢿⣿⣿⡿⣿⣿⠿⠷⠶⠶⠶⠶⠒⣚⠳⠿⠿⠶⠶⠶⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⢤⡤⠖⣺⡎⠁⣀⣈⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠜⢀⡟⠄⠘⢥⡤⣈⠃⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢸⠁⠀⠀⠭⠴⡄⠉⠂⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠦⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠰⠖⠒⠲⠇⠀⠒⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡐⠋⠁⠀⠀⠐⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡵⠀⠀⠀⠀⠀⠹⠛⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢀⠆⠀⠦⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⢒⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠳⡀⡿⠉⠇⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠠⢄⡀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢄⠰⠇⠈⣼⢉⠔⢦⡙⡄
⠀⠀⠀⠀⠀⢠⠎⠂⢄⠀⢤⢚⠈⣻⠋⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠀⠡⣀⠆⢹⠐⣄⡼⢁⠃
⠀⠀⠀⠀⢠⠏⠀⠀⠀⣀⣀⠀⠉⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠤⠄⠂
⠀⠀⠀⠀⡾⠀⠀⠀⠜⣡⠦⠂⠀⢀⠏⡆⠀⠘⡆
⠀⠀⠀⢀⡇⠀⠀⠀⠈⡴⢖⠀⠀⠸⠀⠘⣆⠀⠘⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⣫⡀⠈⢢⡂⠀⣹⡆⠀⠈⢆
⠀⠀⠀⠈⡇⠀⠀⠐⡀⠀⠈⢆⡜⣼⠀⣸⠀⠈⠳⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣧⠀⠀⠀⠱⡀⠀⢠⣼⣇⠴⠁⠀⢀⡾⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⢆⠂⠀⢀⠔⢁⡴⠫⡥⠀⣹⠀⠀⣀⢨⠇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠱⡄⠀⠝⠁⠀⠀⢀⣾⣧⠴⠷⠊⠀⠀⠀⠀⠀⠀⡶⠀⡠⠚⠘⣇⡞⢠⡈⠀⡀⡄⠀⣴⠋⡇⢀⣤⠀⠘⣆⠀⣤⠷
⠀⠀⠒⣺⡌⣆⠀⠀⠀⠀⠘⢄⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣤⡴⠀⣿⠀⣤⠤⠄⣈⣧⣀⣿⠀⠃⣀⣀⡠⠴⠧⣤⣤⣼⠂
⠀⠐⠈⠷⡆⠹⡀⠀⠀⠀⠀⠈⢦⣠⡴⠟⠛⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡿⠀⠶⠿⠾⠇⠀⠀⠈⠉⠁
⠀⠀⠱⡸⣏⠀⢳⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠑
⠀⠀⠀⢳⠘⢆⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠈⠙⡧⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⣠⣤⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣴⠛⠉⠉⠻⡄⢰⠏⠁⠀⠈⢻⡄⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⣟⠻⣿⣿⣿⡇⢸⡛⢿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡥⠀⠀⠀⠀⠀⣙⣦⣬⣴⣏⠀⠠⢷⣤⣉⣭⡟⠁⠀⠀⠀⠀⠈⢉⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⡃⢀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀⠀⠀⣀⠘⣃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠉⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣿⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣶⣿⠟⠁⠀⠀⠀⣿⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠿⣿⣶⣦⡄⠀⣦⠀⢰⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡿⢀⡾⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⢷⡀⣀⣤⡀⢀⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡼⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣻⠃⡰⠋⣿⠀⢸⡏⢻⣤⠴⠖⠒⠓⢒⣦⠤⠤⠤⠦⡼⠃⠀⢠⣿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⡏⢀⡇⢠⡇⠀⢸⡇⠘⡇⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⣁⣀⣤⣿⡀⢳⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣇⣸⡅⢸⡇⠀⢸⡇⣰⠿⢯⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠉⢸⣿⢷⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠙⠛⠻⡶⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⡟⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⣿⠋⢻⡍⠁
⣠⣄⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⡟⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣼⣧⣤⣤⣀⡀
⢧⡈⠙⠶⣦⠴⠾⠛⠋⠉⠉⠀⠀⠀⠀⠈⠉⠘⣷⡀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠉⠉⠻⣷⡄
⠈⠳⣦⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⡀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠛⠷⢶⣦⣤⣤⣤⣤⣶⡶⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⠧⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣿⣷⣶⣤⣤⣼⣦⣀⣀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠿⠟⠛⠛⣻⠟⠋⠉⠉⠙⢿⡍⠙⠦⣄⣠⠀⠀⠀⠀⢧⣀
⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠰⣿⠉⠉⢿⣿⠛⢺⡇⠀⠀⠀⠉⢧⣀⠀⠀⠀⠉⠙⣻⠃
⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣀⣤⠾⠁⠀⠀⠀⠀⠀⠉⠙⢦⣀⠀⢻⡁
⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠒⠛
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣷⣦⡄
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢻⣿
⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣾⠏
⠀⠀⠙⠻⠿⠶⣶⣶⣶⡶⠶⠶⠿⢿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠔⠒⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡿⠁⠉⠉⢉⣿⡉⠙⣻⢿⢿⠛⢋⣿⡏⠉⠉⠉⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⠛⠉⠁⣿⠃⠀⠀⠀⣸⠁⢳⡴⠃⡎⠈⣷⠞⠀⣇⠀⠀⠀⢠⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀⢸⡟⠀⠀⠀⠀⡿⠀⠀⠹⢦⡵⠞⠁⠀⠀⣿⠀⠀⠀⢸⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣿⣿⠃⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠿⢻⣿⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⠘⠃⠀⠀⠀⢿⠀⠀⠀⠀⢹⡆⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠰⠆⠀⠀⠀⢸⠀⠀⠀⠀⠘⣧⣹⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⢶⣶⣶⣦⣾⣦⣤⣤⣤⣾⣤⣤⣶⣶⣶⣿⠷⠶⠿⠿⠛⢿⣟⣁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠒⠿⢧⡀⠀⠀⠀⠀⠀⢠⡿⠓⠚⠛⡛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠦⠶⠶⢷⣀⣿⠀⠀⠀⠀⢰⣏⣰⡶⠦⢤⣧⣤⠾⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢴⣶⠄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⡶⠲⠆⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⣁⠀⢀⣾⣿⣿⣷⢶⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣹⣿⠆⠈⢻⣿⠏⠻⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠦⢿⣿⣏⠀⠀⠀⠚⠋⢀⣼⠚⢿⡶⢦⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣞⣧⡄⠀⠀⣰⡿⠗⠈⠻⢿⣷⣶⣶⣶⣾⣿⡟⠀⠈⠳⢿⣧⡀⠀⢠⣼⣣⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢒⣾⠀⠀⣾⣝⠇⠀⠀⠀⠀⢹⣿⣿⡍⣿⣿⠁⠀⠀⠀⠘⣟⣽⠀⠀⣿⡒⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢽⣟⠀⢀⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢘⣿⣄⠀⣻⡧⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡨⣷⠀⢸⣭⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣭⡿⠀⣺⢍⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣚⣿⡀⠀⣿⣧⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣿⡀⢀⣿⣓⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡦⢚⣷⠀⠳⣶⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣷⡞⠁⣼⡓⢼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡥⢺⣧⡀⠈⣿⢛⣦⠀⠰⣿⣿⣿⣿⣿⣿⠿⠷⢠⣟⢿⠇⢀⣼⡟⢬⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣭⠞⡷⠀⠈⠛⠻⣿⢟⣦⣍⣭⢭⣍⣵⡞⣿⡟⠛⠁⠀⢾⠳⣭⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠈⠙⠉⠙⠻⠾⠛⠉⠉⠉⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⡀⣠⠀⢠⣤⣤⣤⣄⣤⡀⣤⣤⣄⣤⠀⣤⡀⣤⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣯⣀⣿⢸⡏⣸⣯⣷⣿⣀⣸⡇⢸⡇⣼⣿⣷⣿⠶⣟⣿⣼⣯⣷⣿⠻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠈⠁⠉⠀⠈⠉⠉⠉⠁⠈⢁⠉⠀⠉⠉⠀⠉⠉⠉⠀⠈⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠋⠛⠸⠋⠘⠘⠁⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠉⠀⠁⠈⠀⠁⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠖⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⢲⡄⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⢀⡖
⠀⠀⠀⠘⣿⣦⡈⠻⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣄⣠⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠟⢁⣴⣿⠃
⠀⠀⠀⣤⡈⠻⣿⣦⢰⣌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋⢋⣀⠀⡈⣈⡉⠙⠛⢾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣫⠆⣴⣿⠟⢁⣠
⠀⠀⠀⠙⢿⣶⣄⡁⢤⡙⠿⣝⢦⠀⠀⠀⠀⠀⠀⢠⡿⠋⠁⠔⢈⠄⣀⣤⠀⠀⠀⠁⠂⠄⠛⢧⡤⠀⠀⠀⠀⠀⠀⣰⣫⠾⢋⡴⢈⣠⣶⡿⠋
⠀⠀⠀⠰⣤⣈⡙⠋⢌⡛⠶⣬⣛⢷⣄⡀⠀⠀⢤⡾⠁⣴⣷⣮⣿⡀⠉⠀⠀⣠⣿⣷⣶⣶⡣⡈⢿⣦⠀⠀⢀⣠⡾⣛⣥⠶⢛⡁⠙⢋⣁⣤⠆
⠀⠀⠀⠀⠈⠙⠻⠷⠈⣙⠓⠶⠬⢽⡏⠀⠀⠀⣾⠇⡐⢸⠿⠟⠁⠀⠀⠀⣉⣿⣿⣿⣿⡿⣷⢡⠀⢿⠃⠀⠀⢹⡯⠥⠖⢚⣋⠁⠾⠟⠋⠁
⠀⠀⠀⠀⠈⠙⠳⠶⠶⠈⣭⣭⣭⠽⡆⠀⠀⢰⡏⠀⡇⠈⠤⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⡍⠨⠧⠀⢹⡃⠀⠀⢸⠿⢭⣭⣉⠅⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⡲⢒⣫⠝⠒⠀⠘⣿⠀⠇⠀⠀⢿⣿⣦⡀⠀⠀⠀⠸⣿⣿⠁⠀⢠⠀⣽⡉⠀⠖⠫⣝⡲⢖⡀⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠐⢙⣇⠈⠄⠀⠈⣿⡿⠃⠀⠂⠀⠰⣿⡏⠀⢀⠂⢠⣿⠁⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠈⠢⡀⠹⠀⠀⠀⠀⠀⠀⠉⢀⠴⠃⣰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠻⣦⣈⠐⠀⠢⠤⠤⠴⠂⠂⢁⣰⡞⠛⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡄⠀⠀⠁⠈⠛⢶⣤⡴⠴⡶⠾⠛⠛⠋⠀⠀⠀⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣷⠾⠷⣾⣀⠀⠀⠀⠀⠀⠀⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣷⡀⠀⠀⠀⠀⠈⣹⡇⠀⠀⢸⣏⡀⠀⠀⠀⠀⢀⣾⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⣫⣴⣿⣷⣄⠀⠀⠉⡷⢦⡴⢿⠉⠀⠀⣠⣾⣿⣦⣍⡻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣿⡟⣽⢿⣷⣦⣄⣀⣈⣁⣀⣠⣴⣾⣿⠙⣻⣿⠟⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣿⣰⣿⢋⡝⣿⡏⢍⣿⣎⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢷⣷⣿⡷⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠉⠀⠀⠀⠈⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣠⡶⠲⣦⡀⠀⠀⠀⠀⣠⡶⠲⣦⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⢧⣤⠏⣿⠀⠀⠀⢸⡏⢧⣤⠏⣿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣇⠀⠀⢀⡿⠀⠀⠀⠸⣇⠀⠀⢀⡿⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠓⠖⠛⠀⠀⠀⠀⠀⠉⠓⠖⠛⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣠⣞⣋⣉⣉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠉⠉⠉⠉⠉⠙⠳⢦⣈⣧⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⣀⣀⣀⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⠀⠀⣿⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡀⠀⠀⢠⣿⣿⣿⠀⠀⣿⣿⣿⣇⠀⠀⠀⣠⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⢟⣛⣭⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⢟⣫⣭⣶⣿⣿⣿⣿⠇⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡞⣿⡿⢿⣛⣭⣵⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣾⣿⠿⣛⣫⣭⣶
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠛⠋⠁⠰⣿⡇⠶⢾⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠹⠿⠛⠃⣅⠀⠀⠀⢀⣠⠤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣴⣶⣶⡎⣿⣿⣯⢰⣾⣿⣿⣿⣿⣿⣶⣶⡆⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠻⠿⢛⣁⡸⣿⣿⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷⠹⢟⣼⣶⣭⣛⣛⣿⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢫⣥⣶⣶⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣛⣭⣜⢻⣿⣿⣿⡿⠟⠋⠀⠀⠀⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉⠉⠉⠛⠻⣿⣿⣿⣿⡿⠟⠋⣉⣉⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣿⣿⣶⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠙⠛⠁⣿⣿⣿⣿⣿⣿⠉⠛⠃⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⢀⣿⣿⣿⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠉⣛⣻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢎⠁⠒⠀⠀⠂⠏⠰⠰⣁⠀⢰⠀⠒⠂⠐⠙⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢺⣷⣄⠀⣀⣀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⢰⣿⠀⠐⢿⣿⢃⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣤⣍⣏⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣸⣯⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣶⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣶⣷⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣷⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠚⢀⣤⣻⣿⣣⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⠀⢀⣿⣿⡇⠀⣸⡇⠠⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣶⣶⣾⠇
⠀⠀⠀⠀⠀⢶⣼⣿⣿⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣶⣿⣿⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⣭⣯⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⠋⠛⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠁⠰⠿⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⡆⠀⣀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡅⢿⣿⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣯⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣻
⠀⠀⠀⠀⠀⠀⠈⢻⣶⣷⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠿⣿⢿⣷⣾⣷⣶⠶⠶⢶⢶⡶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠿⠿⠿⣿⠿⠿⠿⠿⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿⣿⣿⣿⢿⣿⢷⣦⣿⣿⡟⢋⣽⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⠇⣸⣿⣿⣿⢃⣀⠹⢿⣧⣿⡿⠿⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡾⣥⣴⡿⠋⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠁⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢿⣿⣿⣿⣿⡿⣿⣿⣿⣟⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣄⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠙⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡿⣧⠀⠀⠀⠠⡅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⣠⠴⠶⠶⣶⣶⡄⠀⠀⣠⡞⣛⣉⣍⣉⣻⠘⡦⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⠿⢻⣿⣿⣇⠀⠊⢀⣠⣤⣦⣤⣝⠋⠀⠀⢿⣾⣿⣿⠛⢻⣿⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡆⠀⣿⢰⣏⢻⣿⣿⠀⠙⢏⣿⣿⣿⠛⣛⡷⠀⠀⠈⢿⠿⠿⠚⠋⠘⣧⠀⠀⠀⢀⠆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣧⠀⠈⢿⡇⠄⣿⣧⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢸⡇⠀⠀⠘⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⠘⣿⢦⡘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⡸⠃⠀⠀⢸⡇⠠⠶⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢠⡄⠀⠀⠘⣦⡓⠿⣆⠀⠀⢘⡄⠀⠀⠀⠀⠀⠈⠀⣁⣬⣥⣤⣀⠀⣾⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣆⢻⡀⠀⠀⠹⣷⣦⣼⢧⡀⠈⠻⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⢱⠇⠀⠀⣰⣦⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣧⣿⡀⠀⠀⠘⢿⣿⣧⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢠⡏⠀⠀⢰⣿⣿⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⡿⡇⠀⠀⠳⠷⠞⣿⡄⠈⢝⡳⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⡑⣼⣿⠟⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢬⣧⢣⡀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢙⣻⣟⠻⠿⢷⣶⣶⣦⣤⡾⠃⢰⣿⡟⣿⠇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣎⣷⠄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠛⠋⠀⠀⣼⡿⣴⣶⣶⣤⣾⣿⣿⣿⣿⣷⡆⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⡿⠁⠀⠀⠀⠀⣾⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣷⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⠀⠀⠀⢀⣠⣶⣿⣿⣟⣀⣀⣀⣀⣠⣤⣀⣤⣼⣿⣿⣿⣿⣿⡿⠏⢻⣿⣨⣿⣷⣶⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠀⣀⣴⣿⠿⢯⣙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⡀⣼⣿⠉⠹⣿⠿⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⣿⣿⡿⠁⠀⠀⠈⢙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣼⣿⣷⣦⣼⣦⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁⠀⠀⣀⠠⠀⠰⠤⠈⣉⣀⣀⣨⣭⣿⡾⠿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠴⠶⠶⠞⠛⠋⠉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠷⠶⠾⠶⠿⠿⠷⠶⠶⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣘⣛⡃⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠋⠀⣠⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠀⠀⠀⢀⠀⣾⡧⠘⢻⣿⠻⠟⣄⡀⠀⠀⢠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣽⠏⢀⡔⢃⢿⣿⡃⠀⠀⠚⠃⣠⠗⠶⢿⣀⠀⢻⣅⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣰⡏⢀⣨⠛⠁⠀⠛⢿⣷⣶⣿⣿⣿⠀⠀⠀⣏⣳⠀⢻⡜⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡯⣾⠁⣼⣹⠁⠀⠀⠀⠀⣿⣿⣦⣿⡇⠀⠀⠀⠈⣻⡀⢸⡧⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣸⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣯⡗⢠⡟⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢺⡆⢹⣼⠄⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⢠⣾⠀⣸⣗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠗⠀⢸⡶⣄⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⢐⣧⠞⠀⠿⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠑⠾⢧⣤⣜⡻⡿⢿⣛⠟⣵⣚⡿⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣭⣟⡀⠈⠓⠾⠓⠺⠯⠽⠋⠉⠁⣀⣻⣽⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⣷⡶⣶⣶⣶⣶⣶⡞⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢾⣿⣷⣽⣧⣼⣿⡷⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⢰⠛⢳⡜⣻⠃⡼⣆⢰⠀⠀⡆⢻⡟⣰⢷⠀⣟⣳⢴⡆⡼⣆⢰⣦⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠓⠚⠀⠛⠘⠋⠙⠚⠒⠂⠃⠘⠃⠋⠉⠃⠃⠙⠚⠓⠋⠙⠚⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠀⠀⠀⠐⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠈⠀⣀⣴⣿⣶⣿⣦⣶⣄⣀⠀⠀⠀⠂⡠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣶⢸⣿⣿⣾⣾⣯⡝⣟⣻⣯⣿⢰⣦⣄⡀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⣿⠟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡸⣇⡀⠉⠀⣠⣿⡿⠁⠀⠀⠀⠈⢻⣿⣿⣟⢿⣿⣿⠃⠀⠀⠈⢻⣷⡄⠀⠉⢀⡸⢆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⡭⣽⠅⠀⠀⣽⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣾⣿⣿⠀⠀⠀⠀⠀⢿⣷⠄⠀⠨⣯⠭⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⢒⣿⠀⠀⣘⣿⡏⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡂⠀⠀⣿⡲⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠦⠵⣗⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⣻⠬⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣊⣿⠀⠀⠘⣻⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣸⣿⠅⠀⠀⣿⣑⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠴⢻⡅⠀⠀⠘⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⣸⣿⠋⠁⠀⢼⡗⠤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡡⢿⡄⠀⠈⠙⢿⣧⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⡇⠀⣠⣾⣿⠁⠀⠀⢠⡿⣉⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣖⣋⢿⣄⠀⠀⠈⠉⢿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣧⣶⣾⠿⠟⠀⠀⠀⣤⣿⣑⣲⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣘⡹⠃⠀⠀⠀⠀⠘⠛⠻⠿⠿⢿⠿⡿⠿⠿⠉⠁⠀⠀⠀⠀⠸⢏⢳⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡶⠆⡤⠶⣄⠰⢶⠶⢀⣤⡀⢠⡄⠀⢠⡤⢶⡶⠄⣴⡄⠀⡶⠶⡄⢤⢠⡶⢦⠀⣦⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣧⣀⡼⠀⢸⠀⣼⠿⢧⢸⣇⣀⢸⡁⢸⡇⣸⠷⢿⡀⡟⢻⡅⣿⢠⣙⣳⢤⡿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠓⠸⠃⠃⠋⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⠀⢀⡠⣤⣤⡄⠈⠢⣶⣶⠾⠿⠿⠿⢿⠿⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣦⢰⣿⣿⣷⣶⣶⣤⣶⣾⣾⡿⢿⣏⣽⣛⣡⣾⣧⡀⣴⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⡉⠙⠛⠛⠋⠈⠉⠉⠻⣇⠀⠉⠀⠉⢩⡄⠀⠈⠉⠉⠉⠉⠉⠻⢿⠋⠈⠹⢷⣶⣦⣤⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠛⠻⢿⣿⣦⣤⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠉⣠⠀⢤⠀⠀⠰⠀⠐⠃⠀⠂⠀⢀⣀⠑⠀⣼⢛⣷⣄⣀⣀⣉⣷⡄⠀⣀⡀⠈⢛⠀⡛⠀⡟⢁⣤⣤⣤⣤⠄⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣤⣿⣦⣾⣶⣶⣶⣶⣶⣶⣶⣾⠿⣿⣿⠷⠿⠀⣉⣤⣝⢋⡙⠛⠉⠉⠿⠿⠛⠙⠻⠿⠿⠿⠻⠟⠛⠛⠛⠻⠲⠿⠃⠈⠑⠿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⠿⠿⠿⠿⠟⢿⡿⢿⣿⠟⠋⠁⠀⠀⡟⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠓⠒⠛⠛⠓⠒⠒⠒⠒⠚⠓⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣤⣤⣤⣤⣤⣧⡄⠻⠁⠀⠀⠀⠀⢠⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣿⣠⠶⠶⠾⠷⠾⣿⡆⠁⠀⠀⠀⠀⠀⠀⢸⡇⠐⠂
⠀⠀⠀⠀⠀⠀⠐⢿⣯⣤⣤⠀⣀⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣛⣩⡍⣠⣀⣿⣿⡇⠀⠀⣠⣶⣾⠿⢷⣦⣁⣀⣀⣀⣠⣶⣶⣦⡴⠲⢶⣶⣆⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣡⣭⡉⢩⣭⣠⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣟⠉⠉⠉⠁⠀⢿⡇⠀⣿⡏⠉⠀⠤⣤⠀⠀⠉⠀⠀⠠⡤⠙⢿⡷⠀⠀⠈⠛⣤⣄⣹⡆⠀⠀⠀⠀⠀⠀⠀⠹⣀⡿⢋⣽⢿⡿⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣿⣿⠶⠶⣿⡇⢰⣿⠀⠸⣄⠀⠿⠀⠀⠀⠀⠀⢀⣴⡴⠆⠀⠀⠘⠳⣶⡄⠘⠋⢠⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠙⠿⠂⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠉⣹⠇⠀⣿⡇⢸⣿⢷⣄⣀⣀⣀⣀⣤⣤⣴⠾⠟⠉⠁⠀⠀⠀⠐⠆⠀⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⣷⣶⣶⡶⢾⠿
⠀⠀⠀⠀⠀⠀⠀⠘⡿⠘⣿⣿⣧⣴⠿⠿⠀⠙⠋⠉⠋⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣏⡹⠁⣶⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢿⣿⡇⠺⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣄⠀⠀⣽⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠁⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣶⣤⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣶⠊⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣹⣿⡇⠀⠀⠻⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠎⠁⢠⠀⠀⠀⠀⠠⠄⠾⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠘⣿⡗⣀⣀⣀⣀⣀⠀⠀⠒⠂⠈⠀⠀⠀⠈⠲⠲⣶⣶⣾⣶⣶⣤⣤⣤⣤⣿⠇⠀⠀⠀⠀⠀⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⠀⠀⠀⠀⠀⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⠏⠈⣿⠀⠁⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⣿⡀⢄⠀⠀⠀⠀⠀⠀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢀⣿⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣀⣀⣀⣀⣠⣾⠃⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡗⠀⣸⡿⠃⣼⣿⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣅⡀⠀⢀⣀⣍⠁⠠⠄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⢁⡀⠀⠘⢿⣌⡑⢄⡀⠀⠀⠀⡀⠀⢠⣽⡾⠛⠉⠛⠛⠛⠛⠛⠛⠲⠴⠶⠦⣼⣧⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠈⣧⠀⠀⠸⠋⠳⣦⡹⠶⣄⣀⠻⠿⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢻⡆⠀⠀⠀⠀⠈⠙⠶⠉⣉⣤⣤⣞⠁⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠈⠙⠻⢦⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⠀⠀⢸⡇⠀⠀⠀⠀⡁⢸⡁⠀⠀⠀⠀⠈⠉⠀⠶⠶⠶⠶⠶⠶⠶⠶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢼⡀⠀⠸⠇⠀⠀⠀⠀⠰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⢸⣧⠀⠀⠀⣿⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠻⣇⡀⢸⣿⠀⠀⢠⠀⠘⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣄⠀⢸⠁⠀⠀⠀⠀⠀⠘⠀⠀⣠⣤⡄
⠀⠀⠀⢀⣠⣤⣤⡴⢿⣏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣇⡀⠀⢸⠀⣦⣠⣶⣾⠟⠃⠀⠀⠘⢷
⠀⢀⣶⠟⠁⠠⠀⠀⠈⢻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣰⠀⢸⣀⣿⡟⠁⠀⠀⠀⠀⠀⠀⣸⣇
⠀⣿⠋⠀⠀⢀⡀⠀⠀⠈⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣻⣶⣿⠂⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⠀⠒⠂⣤⡀
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠐⣠⡿⠋⠀⡈⠙⠛⠛⠉⠿⠁⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠈⢿⣄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⣀⣾⠟⠁⠀⠀⠁⠀⠀⢀⣾⣇⠀⠄⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠟⠁⠀⠀⠀⠀⠃⠀⢀⡘⠀⢿⣧⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⡶⠀⠀⠀⢸⡇⠀⠀⠙⢦⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠂⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠁⠀⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠛⠁
⠀⠀⠠⠦⠠

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⠿⣿⠿⡿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡪⠕⠲⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠉⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠶⠶⣦⣤⣄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣀⠀⣸⣿⣿⣿⡀⠀⣸⢀⠀⠀⢀⣤⣦⣤⡶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠿⠛⠻⠿⣿⣿⣿⣷⣿⣾⣦⣴⣿⣯⣬⣤⣤⣤⣤⣭⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⢚⣹⣿⡷⣾⣿⣿⡟⠛⠛⢻⣿⣿⣟⠵⣲⣶⡶⣍⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠛⠏⠈⠛⠛⠁⠻⣻⣿⡇⠀⠀⢸⣿⣿⠿⠀⠿⠿⠃⢸⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠈⠱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢺⣧⠀⠀⢀⣾⣟⣦⡀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⣿⣿⠿⠛⠹⡀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⡏⠀⠀⠀⠀⠙⢦⡀⠀⢴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⣀⣀⡀⠀⠀⢳⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠿⠿⢿⠟⠛⢿⢶⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠳⣤⣀⣀⣀⣈⣠⣤⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠔⠁⠀⠀⠠⠿⠟⠛⠛⠿⢿⡇⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠻⣿⣉⠀⠀⠈⠉⠙⠛⠛⠻⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡜⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠀⣀⣠⡴⠀⣀⣀⠀⡀⣀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠏⠠⠀⠀⠙⠛⠿⠟⠁⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡶⠄⠀⠀⠐⣿⣿⣿⠿⠀⠀⠀⠀⠀⣿⣿⣿⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣤⠤⠥⠤⢄⠀⠀⠀⢸⣿⣿⣿⣷⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⢀⣰⣦⡀⠉⡀⠀⢀⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠁⠀⠀⠀⢀⣴⠂⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠃⠀⡀⠀⢀⡿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣯⣄⡹⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⢀⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠉⠻⣧⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⣃⣼⣿⡿⠋⠀⠻⣿⢿⡦⠈⠹⣯⠹⡇
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⢻⣄⡀⠙⢿⣿⣿⣿⣿⣷⣾⣿⠟⠛⠇⠀⠀⠀⠙⠳⣌⠀⠀⠈⠙
⠀⠀⠀⠀⠀⢇⡉⠀⣇⠀⠀⠀⠀⠀⠘⡿⠀⠀⠈⠙⠛⠛⢿⠟⠁
⠀⠀⠀⠀⠀⠚⠂⠠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠁⡀⠀⠉⢻⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠸⣿⣀⡀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⡱⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢀⠐⠉⠟⣿⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⠁⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠐⢀⣃⠀⠀⢦⠾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⣄⠾⣿⠇⠉⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣼⡋⢷⡀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢀⣼⠟⢣⣿⣲⠀⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣿⡷⣾⣿⡄⠀⠀⠀⠀⣼⣾⣴⣖⠟⣿⠀⡾⠟⣹⣰⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢉⠀⠀⠀⠀⠀⢠⣿⣿⣷⣉⠠⡟⢘⢱⣦⣵⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣟⣿⣏⣿⠀⠀⠀⠀⠀⢸⣿⠃⠈⠁⠊⣽⣧⣠⣿⣾⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡟⠋⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣰⡟⣿⡏⢙⣿⣿⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣠⣿⠏⣿⣧⣼⣿⠛⠃⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡅⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡟⠀⢸⣇⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⠿⠿⠷⠾⠟⠀⠀⢰⡿⠛⠛⠿⠿⢿⣿⣿⣿⣿⠿⠋⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣙⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣾⣷⣄
⣿⠀⢸⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦⡀⠀⡇
⡇⠀⢰⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡇⠀⠀⠀⠀⠀⣠⡀
⣿⠶⢼⡮⣿⣿⣷⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡇⠀⠀⠀⠀⠀⢹⡇
⡿⠶⢼⡇⣿⣿⣿⣷⣴⣿⠙⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⢈⡁⠀⠀⢠
⣉⠑⠸⡇⣿⣿⣿⣿⣿⣿⣴⣿⣴⣿⣿⢿⣿⡿⣿⣿⣿⣋⣉⡉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢘⡇⠀⠀⢹⡇
⡉⠁⢰⡆⣿⣿⣿⢿⢿⣿⣿⠀⣿⣿⣿⣿⣿⡉⣿⣿⣿⣿⣛⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠀⠀⠀⠀⠘⡇⠀⠀⢰⢆
⣿⣿⣾⣷⣀⣀⠀⢐⠀⣿⣿⣿⡿⣿⣿⣿⣿⣷⡋⢿⣟⣧⣤⠾⠷⠀⠿⢹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⠀⠀⠀⡇⠀⠀⠸⢾
⣿⣿⣿⡏⠉⢻⣿⣿⣷⣀⣀⣀⠀⠀⠿⣿⠿⠿⠇⡀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⡇⠀⠀⢠⠈
⡟⠁⠈⢷⠀⠘⢻⣿⡇⠀⠉⠙⣿⡿⢤⣠⣀⣀⢰⡟⣋⣈⠉⣁⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡀⢸
⡛⢠⣴⣿⠀⠀⢸⣿⣇⣀⠀⠀⣿⣿⠀⠈⣿⣿⠺⠛⠿⠛⠀⠙⠻⠛⠀⠂⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣏⣽⣿⣿⣿⣷⣄⡇⢸
⢿⣀⠸⣿⡇⠐⢺⣿⣿⣇⠀⠀⣿⣿⠀⠀⢸⣿⠀⡄⠀⢠⣦⠄⠀⠀⠀⡀⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣧⣼
⢸⡷⣾⢸⣿⠀⢸⣿⡿⠀⠀⠀⣿⣿⠀⠀⢸⣿⠀⣧⡀⢶⣷⡖⠀⠀⢠⡄⢹⣿⣶⡆⣠⢛⣻⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿
⡈⠗⠲⠼⢿⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⢸⣿⠀⣻⣿⣤⣤⣤⣠⣴⣿⡃⢸⣿⣿⡇⠁⢸⣿⣿⣿⡷⠀⣿⣿⡿⣿⣿⣿⣿⡿
⣴⣿⣷⠂⠨⠀⢸⣿⣿⣀⠀⠀⣿⠿⠀⡀⢸⡿⠀⣿⣿⣿⣿⣿⡋⠡⣿⡇⢸⣿⣽⠇⣴⢸⣿⣿⣿⠃⢺⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⡀⡀⠀⠸⢿⣧⣿⣠⠄⣻⣦⣸⣗⣛⠾⠥⠁⠀⠀⠀⠉⠀⠀⢡⠀⠈⠟⢛⠀⢻⣸⠇⣿⣿⠀⡙⢿⣿⡏⣿⣿⣿⣿⣄
⢈⣹⡍⣁⣾⣗⢷⡜⢙⣉⢁⣀⢻⢻⣿⣿⣿⡦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣶⣦⣍⢀⢰⡏⣧⡀⣸⣿⣾⣷⣠⣤⣑⣿⠨⣿⣿⣿
⢿⠛⠷⠉⣛⣿⠟⠛⣮⣉⣨⠍⣠⣴⠈⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠳⠺⣿⣟⣴⡿⣿⣿⣿⣿⣿⠻⡿⢺⣿⡿⠙
⡄⢁⢰⣿⣿⣿⣿⣶⣮⣽⣿⣦⢿⣿⣤⣽⣿⣿⣶⣶⣭⢹⣿⣟⢿⣿⣿⣿⣿⣿⡇⣶⣺⣿⢷⣄⢰⣿⠇⠈⠛⠰⣷⠸⠲⣤⣤
⠀⠀⢹⣿⣿⣿⣿⣿⠻⣿⠟⣿⣿⡁⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⡇⢻⣿⡉⢸⠌⠁⠀⠀⠀⣶⡤⠒⠚⠁
⠀⢃⣾⣿⣿⣿⣿⣿⣷⡀⢰⠸⣿⣣⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⡟⠛⠄⠳⣸⣿⢻⣾⣄⣀⣤⣄⡀⣿⠇
⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿⡎⣿⡏⣽⣿⣷⣶⣦⡈⠯⠇⣿⣉⣇⣹⣿⣿⣿⢸⣿⣿⡶⢿⡿⠈⢸⣦⣤⡶⠟⠀⣿⠆
⣈⢻⣿⣿⣿⣿⣿⣿⣠⣿⣿⣧⣿⣇⣿⣿⣿⣿⣿⡇⣄⠀⣿⣇⠀⢸⣿⣿⠇⣼⣧⣿⣅⣿⡃⠀⢸⡏⠁⠀⠀⠀⣾
⠀⠘⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣷⣿⣿⣶⣾⣾⣿⡗⣿⣿⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⣿⣿⣿⣿⣟⢹⣾⣹⡿⠞⠟⢻⣿⣿⣿⣿⣿⣤⡛⠛⠛⢛⣛⣛⣻⣿⡿⠁⠁⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⣿⣤⣴⣤⠤
⡄⠀⢿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⡛⢛⣏⠛⠛⠀⠀⠀⠀⠀⠀⢸⣧⣀⢤⠀⠀⣿
⡇⠀⠀⠀⠈⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠛⠀⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣞⣇⡀⠿⠀⠀⠀⢠
⡇⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡛⠉⠁⠀⠀⠀⢠⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠁⠠⠤⠀⠀⠙⠛⠛⠁⠙⢿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠰⢿⣶⣶⠶⠄⢠⣴⣶⣶⢶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠁⠈⠢⠙⠑⠃⠀⠘⠸⡿⠧⠊⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠈⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣴⡶⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠋⠉⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠿⣿⡟⠀⠀⠀⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡤⠖⡶⠹⡏⠀⢈⠀⠈⣅⢈⣉⠉⠻⠉⠛⠛⣶⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠉⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠘⢅⠈⠢⢤⣆⣷⣀⣽⣠⣄⣌⣟⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠿⢿⡿⢶⣿⣶⣶⣷⣾⣷⣾⣶⣶⣦⣦⣾⠁⠐⠠⠼⠛⠉⣿⠁⠁⠀⠘⢷⡀⠁⠸⢦
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠉⠁⠀⠀⠁⠀⠀⠀⠀⣷⠀⣾⠀⠀⠀⠀⠀⠻⡄⠀⠘⠃
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠀⠚⠀⠐⠂⠤⠴⢛⣠⣿⣄⣠⣀⣠⠀⠀⠙⢶
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣴⠶⣶⠶⢿⠶⠶⠶⠶⠞⠀⠀⠈⠀⠀⠐⠛⠉⢹⡋⠻⠿⣿⣷⣄⠀⠘⢷⣤
⠀⠀⠀⠀⠀⢀⣠⣦⡶⠾⠋⠁⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⡇⠀⠀⣿⣿⣿⡀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⣼⡟⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠠⢀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠘⢻⣿⣧⠀⠀⣿⠁
⠀⠀⠀⠀⠀⣽⣀⡀⣠⠀⣀⣤⡤⠤⣦⠤⣶⠤⢴⣤⡷⠂⠒⢄⡀⠈⢉⡩⡇⠰⠄⡇⠀⠀⠀⡈⣿⣿⣷⠀⣿
⠀⠀⠀⠀⢀⣿⠀⠀⣀⣀⢀⢤⡦⠠⠤⠠⡷⠤⠴⠼⡇⠀⠀⠀⠀⠀⠀⠀⠱⠒⠲⣿⠤⢤⣤⡆⠹⣿⣿⡀⣿
⠀⠀⠀⠀⠀⢿⠉⠀⠁⠀⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⠁⠀⢀⣀⠀⠀⠀⠀⠀⢃⠀⣿⠀⠉⠈⡇⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⣸⠤⠤⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠖⠀⠀⠐⠤⠐⠈⠁⠺⠆⢸⡄⠀⠀⣿⣄⠀⣿⣿⣿
⠀⠀⠀⠀⠰⣿⠤⠦⠤⣀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⠀⢸⠁⠀⠈⠒⠒⠈⠀⠁⠚⣾⠀⠀⠀⣿⣬⢉⠿⣿⣿
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⠂⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣿⡄⠈⠓⢿⣿⠆
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⡀⣀⡀⢸⣄⣀⡀⢀⣀⣀⣀⠀⠠⢄⡀⠀⠀⢀⣀⢨⡍⣩⡏⢹⡗⢀⡀⢸⣿⠁
⠀⠀⠀⠀⠀⣿⠿⠿⠶⣶⢶⡶⠥⠼⠅⢼⡧⢽⣥⣭⡯⠽⡷⠀⠀⠀⠀⢀⡨⠀⢀⢀⡇⠈⠁⢸⠉⠪⣷⣤⢹⠆⠠
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠐⠀⠈⠁⠀⡆⠀⠀⠀⠉⠁⠀⠀⠀⠈⢁⠀⠀⢰⣄⠀⠉⡿⢷⡄⠀⢡⡀
⠀⠀⠀⠀⠀⣿⣟⣛⣿⣋⣨⣿⣄⣀⢀⣿⠀⣤⡀⢀⣀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠸⢿⠐⢾⡇⠀⢻⣄⠀⡇
⠀⠀⠀⠀⠀⡏⠉⠀⠉⠀⠀⠀⠀⡀⠀⡋⠀⡉⠀⢈⠁⠈⡏⠀⠀⠀⠀⠂⠀⠈⠀⠀⠙⡐⠇⠀⠃⠀⣿⡏⠰⣆⠙⡓⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠘⠋⠈⠛⠀⠻⠃⠐⠟⠁⠉⠉⠒⠀⠐⠈⠁⠲⠠⡇⠀⠀⠀⠀⠈⠑⢄⠀⠑⢶⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠁⣀⠄⠉⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣀⣤⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠑⠶⣤⣀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢠⣭⡳⢦⠀⠀⠀⢀⣀⣠⣤⣤⣀⠀⠈⠓⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠋⠀⠀⠀⠀⠉⠝⣿⣦⡄⠈⠀⠀⢠⠀⠐⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣾⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠀⠀⠀⠴⠶⠶⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠐⠒⠻⠿⠷⣦⣄⠀⡄⠀⠀⣠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠉⠛⠀⠀⠀⠀⠁⢀⡴⠋⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⡦⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠑⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢀⡤⠘⠁⠀⠀⠀⠆⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⡀⠀⢀⢰⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⡀⠔⠉⠀⠀⠀⠜⠃⠀⠀⠀⠀⡀⠠⡀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⠠⠒⠁⠠⠖⠋⠄⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣠⡴⣚⡁⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⢤⣀⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠱⡟⡁⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡸⠁⠁
⠀⠀⠀⠀⠀⠀⠰⠋⠀⠿⣶⣤⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠰⣄⠀⣈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠠⣤⣼⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣠⠄⠀⠉⠊⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠦⠗⣀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠠⣤⡐⠀⠀⠀⠀⠐⠑⢊⠀⠑⣄⠀⠘⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇⠀⠀⠄⠀⢠⡀⠀⣤⣤⠆⠀⠀⠀⢀⡴⠷⡦⠤⠙⠀⡰⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠠⠀⠀⠠⠊⢠⡾⣿⠃⠀⠀⠀⡰⠋⡈⢻⠀⠀⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣥⠀⣀⠞⠀⠀⠉⠈⠈⠀⠀⠐⠞⠤⠿⣷⡈⣇⠀⣰⠇⠀⣿⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⡸⠀⠈⠞⠁⠀⢀⠀⠀⠀⠀⠀⣠⠄⡤⠀⣴⠎⠁⠞⠒⠛⠐⠺⣿⠚⢡⣮⣿⣒⡦
⠀⠀⠀⠀⠀⠐⢧⡀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣶⠀⣤⡀⠘⠀⠀⢶⠶⠶⠾⠛⠻⢿⡿⣯⣷⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢠⣁⣠⣃⣀⣀⡈⠨⠀⠤⠃⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢠⣿⠟⠁⠀⠈⢷⣄⡄⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⠘⠙⠀⠀⠀⠀⠀⠙⢿⡄⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠐⠂⠁⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠈⠟
⠀⠀⠀⠀⠀⠐⠐⠈⢧⡀⣴⠶⠛⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊
⠀⠀⢀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐
⢶⣶⢯⣿⣶⣻⣄⠈⡠⢤⣤⣤⣄⠀⢀⠀⠀⡀⠈⠋⠰⠞
⠀⠈⠙⠛⣿⡿⣿⣟⣛⣻⣭⡶⢾⡻⢿⣟⣶⣽⢇⣀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠻⠟⣫⣭⣶⣮⣷⣿⡿⣝⡳⢯⣻⡯⠮⠟⠿⠧⠒⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠒⠒⠁⠀⠒⠒⠻⠶⠶⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠐⠀⠀⠀⠀⠀⠄⠀⠠⠠⢀⣠⢂⢄⡉⠛⠿⣶⢄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠆⠈⠑⠂⢌⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠐⠃⠊⢈⡀⣀⠔⠕⢄⠑⣕⡀
⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢐⠀⠈⡀⠕⢀⠜⣆⠈⣷
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⡠⠊⠂⠀⠠⠀⠀⢸⠀⠙⢮⡀⠀⠡⢚⠈⡄⢸⡇
⠀⠀⠐⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⢄⠄⠀⢀⠀⠀⠂⢒⡿⠀⠀⠀⠉⠢⡔⡡⡀⠰⠀⡿
⠀⠀⠀⠀⠃⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠐⠰⢅⡡⠒⠐⢁⡤⠴⠶⠶⠏⠀⠀⠀⠀⠀⠈⢪⡀⠈⠆⣊⡅
⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⢐⡇⡥⠆
⠀⠐⢠⠀⠂⠀⠈⠢⡀⠀⠀⠀⢈⠛⣲⢤⠀⠀⠀⠈⠀⠀⠀⣠⡴⡶⠂⠀⠀⠀⢀⡔⠀⠁⠠⣪⠠⣿
⠀⠀⢁⠂⠐⠄⠀⠀⠈⠢⡀⠀⢡⢞⢈⠀⠱⡀⠀⠀⠀⠀⣼⢣⠞⠕⡁⠀⢀⡴⠋⢃⠄⠀⢀⠇⣘⡇
⠀⠀⠈⢘⠄⠀⠀⠀⠀⡐⣉⠦⢈⣐⢑⡄⠪⡇⠀⠀⠀⢠⡧⠣⠠⠂⢦⠔⢋⠄⠐⠑⠀⣰⠋⡸⡁
⠀⠀⠀⠀⠪⣢⡀⢨⡊⢴⠞⠀⠀⡠⢄⠀⠀⡇⠀⠀⠀⢸⢁⢄⢔⠔⠐⡁⠡⠂⠢⠂⡚⠁⡴⠋
⠀⠀⠀⠀⠀⠈⠻⣄⡙⠣⣂⡈⠔⠁⡡⠀⠤⠀⠀⠀⠀⢼⠵⡥⠠⡪⢮⣴⣴⣦⢶⠞⣠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣆⣀⠉⠲⠤⣤⣒⡅⠂⠻⣻⠫⡧⢠⠤⢄⡴⣴⠶⠀⠁⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠭⣗⡲⠤⣌⣉⠉⠚⠛⠚⠓⠒⢒⣂⠥⠖⡖⡶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⢮⢭⣭⣩⢩⢧⡆⠉⠃⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠋⠹⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⢃⣀⣀⣀⡈⠙⢻⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠉⠑⣶⣿⣿⣿⣿⣿⣆⠀⠻⠿⠻⣿⣶⣤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣉⠛⣉⣿⣿⡆⠀⠀⠈⠀⠘⠧⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣯⠻
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠋⠁⠉⠀⣿⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢄⣾⣦⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⢿⣟⡋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⣧⠀⠀⠀⢤⡷⠾⠆⠀⠀⣶⣤⣶⣿⣿⠿⠛⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣸⡿⠻⠏⠀⠀⠀⠈⠁⣠⣀⣠⢾⡿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠀⠀⡉⠤⡈⢲⣶⣿⠿⠏⠛⠾⠿⠿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣍⡁⠒⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣯⣭⣍⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣻⡿⠉⣷⠟⢛⣿⣿⣿⣿⣆⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⢯⡽⠀⢀⣠⡿⣿⣿⣿⣿⣿⡿⠿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⢾⣤⣶⣿⣿⡃⠙⠛⣷⡦⠀⢠⣶⡾⢺⣷⡶⠛⠛⠛⠛⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢭⣷⣯⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⣿⢿⣿⣑⣾⣷⣾⡟⣡⣶⣶⡾⠋⣠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⡟⣵⠋⠘⢿⣿⣿⣿⣿⣿⣷⣏⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⢏⣾⣿⡏⣸⣯⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠙⢧⠤⡀⠀⢻⣿⣿⣿⣯⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⠇⠀⢡⣾⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⡂⢈⠓⣄⠐⣷⡄⠙⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠿⠀⠀⢸⣿⣧⣼⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣶⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣤⣿⣿⣷⣿⣦⢻⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣿⣿⣿⣿⣿⣿⡿⢿⡇⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣿⣿⣿⣿⣿⣿⡃⣾⠀⢻⣿⣿⠁⠀⠹⠿⠿⢿⠟⠛⠠⠉⠁⢈⢉⠛⢿⣿⣷⢻⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢿⣿⣿⠋⡟⡟⠁⢻⠀⠘⠯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠡⢿⡟⠻⢿⣿⣿⣿⣿⣿⢿⡟⡇⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⠻⠛⠂⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠾⠿⠛⠛⠿⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⣼⠋⠁⠈⠙⢿⡀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢠⡶⠛⠛⠳⡄⢀⡴⠋⠉⠉⠻⣆⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⡿⢴⣶⣦⣴⡇⢸⡷⢶⣶⣶⣦⢿⡄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⡛⠁⠀⠀⠀⠀⠻⣌⣙⣋⣟⠃⠈⣳⣄⡛⠛⣻⠟⠀⠀⠀⠀⠀⠚⡛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⠏⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠀⠈⠉⠉⠙⠛⠿⣦⣄⠀⠀⠀⠀⠀⢹⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠶⠆⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠲⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⠀⠈⠛⢿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠏⠀⠀⠀⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠷⠂⢸⣷⠀⣸⠇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠿⠋⣰⠇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢿⣄⡴⢶⣄⣤⠖⢦⣤⣄⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⡟⢀⡞⠀⡏⠀⠘⡇⠘⡷⠚⠋⠉⢉⣨⡽⠷⠚⠛⠛⢲⠟⠀⠀⣰⠿⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣼⡇⢸⠀⢠⡇⠀⢀⡇⢀⡇⣀⡴⠞⠋⠁⠀⠀⠀⢀⣀⣀⣤⣤⢾⣿⡀⠹⣎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠸⣧⣼⣆⢸⡇⠀⣸⣇⣸⠟⠻⠶⠶⠶⠾⠿⠛⠛⠛⠋⠉⠁⠀⢸⡏⠳⣤⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠙⠋⠻⠾⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡿⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⡟⠻⣿⠛⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⡏⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣿⣀⣀
⠀⠀⠀⠀⣿⠙⢷⣤⣀⣀⣤⣴⠶⠾⠛⠛⠉⠉⠉⠉⠛⠃⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠉⠉⠉⠙⠛⠻⣶⣄
⠀⠀⠀⠀⠙⢦⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠿⠿⠟⠛⠻⠿⠶⣶⣦⣶⣶⡿⠃⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠾⠟⠛⠛⠛⠛⠻⠷⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠋⠙⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⢳⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⢠⡿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⠛⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠟⠋
⣿
⣿
⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⣶⣶⣦⣤⣤⣴⣶⣶⠶⠟⠛⠋⠁
⣿⣿⡿⠛⠟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⠟⠛⢿⣧⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡏⠠⡀⠀⢰⢭⡸⠇⣼⣳⣒⠒⣶⣻⣟⣟⣿⡀⢇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⠑⠥⠧⠀⠸⠊⠁⣡⠇⠀⠑⠀⢳⡸⢗⣒⣋⠹⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠲⡆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⠺⡅⠀⠀⢀⠗⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣷⡀⠀⠀⠀⠀⠘⣄⠀⢑⠪⠄⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡒⢦⣠⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠻⣿⣿⣦⠀⠀⠀⠀⠈⠒⢤⠤⠚⢣⡔⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠙⢿⣿⣷⡀⠀⠀⢀⠊⢹⠒⡶⣊⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢿⡇⠁⠀⢠⠀⠀⠒⠀⠈⠀⠀⠠⣀⡀⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢱⣌⣿⣽⣶⠈⡀⠀⠀⣀⠤⠀⠀⠀⠾⡅⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⢫⡍⠧⢠⡊⢤⠊⡀⠀⣀⡤⠦⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠈⠤⠤⢀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⠀⠀⢸⡴⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠨⠥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠈⠀⠀⠒⠀⠀⠀⠈⠒⠂⠒⠂⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⢀⠤⡀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠾⠊⠿⠃⠰⣥⠜⢡⣃⡞⢀⢠⡝⠁⢈⡹⠀⣠⣆⡄⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠸⠥⠊⣀⣘⠖⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠