Fancy dress Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡠⠒⠉⠁⠉⠐
⠀⠀⠰⠁⠀⢀⠰⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠔⠃⠀⠀⠀⠂⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠆⠈⠀⠀⠁⡉⠀⠀⠀⠀⠀⠀⠄⡠⠔⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠂
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⡀⠀⣤
⠀⠀⠀⠆⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣤⣀
⠀⠳⢄⣸⣶⣷⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢻⣿⡿⠋⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣧⣽⠿⠟⢹⣿⡿⠋
⠀⠀⠀⣤⡘⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠆⡀⢸⣿
⠀⠀⢀⣿⣿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣷⡀⢻⠎⠁
⠀⠀⣾⢻⣿⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣠⡄⢀⣀⡖⠀⢹⣿⣿⡿⠟⠋⠁⠤
⠀⠀⡿⠘⣿⡄⠁⠀⢀⡀⠀⣀⣶⠿⢛⡍⢁⣾⣿⡇⢸⣿⣇⣴⣿⣿⠀⠀⠀⠐⠿⠶⣤⡄
⠀⢠⠁⠀⢿⣷⡘⣐⠛⣩⣤⣤⣤⣾⣿⢁⣾⠛⣿⡇⢸⣿⣿⣿⣿⡏⠀⠀⢀⣴⡿⠖⣴⡶⠀⣠⡄
⠀⠀⠀⣄⠈⣿⡿⠿⠿⠟⠋⢉⣉⣁⡟⢸⠇⣰⣿⡇⠈⣿⣿⣿⡿⠁⢀⣶⡿⠋⢀⣴⣿⠇⢠⣿⡇
⠀⠀⠀⣿⣄⣨⣥⣶⣶⣶⡿⠿⠛⠛⠁⡜⢀⣿⢿⣷⠀⣿⣿⣿⠃⣰⡿⠋⢀⡶⠋⣿⡟⢀⣿⣿⣧
⠀⠀⠈⠛⠿⠿⠭⠅⠀⠀⠀⠀⠉⠙⠀⡟⢸⡏⢸⣿⠀⣿⣿⣏⡾⠋⢀⣴⡿⠁⣸⡿⢀⣼⡟⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣼⠇⣼⣿⣆⣿⡟⠉⠀⠀⣻⣿⠃⣰⡿⢁⡾⢻⡆⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢰⡿⠀⣿⠛⠿⢁⣤⡆⢀⣾⡿⠁⣼⠏⣷⣽⣷⣾⣿⠀⢿⣿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡿⠀⢠⣿⡟⠀⣼⡿⠁⣾⠁⣾⡇⣿⡏⢷⣿⣧⠘⣿⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢰⡟⢀⣿⠃⠀⣼⣿⠀⢸⣿⠃⣰⠏⣰⡟⣴⡿⠃⠀⠈⢿⡆⣹⠛⢾⣏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣿⠁⣼⡟⠀⢀⣿⡏⠀⣿⡟⠀⣿⠀⣿⣇⣿⡇⠀⠀⠙⡜⠻⣿⣆⠈⠻⢦⣄⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢸⡏⢀⣿⠃⠀⣸⣿⠃⢸⣿⠇⢰⡏⢰⣿⣨⡟⠁⠀⠀⠀⠸⣧⡘⢿⣷⣦⣄⢩⣖⣷⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢀⣿⠁⣼⡟⠀⠀⣿⡏⠀⣾⣿⠀⣸⡇⢸⣇⠹⠁⠀⠀⠀⠀⠀⠈⠙⠢⠄⠉⠻⡦⠉⣻⣭⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣸⡇⣸⣿⠁⠀⠀⣿⡇⢠⣿⡇⠀⣿⡇⢸⣿⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⢀⠈⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣰⡿⢀⣿⠏⢀⡆⠀⣿⡇⢸⣿⠀⢸⣿⠃⣼⡟⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣰⣿⠁⣼⡿⠀⣾⡇⠀⣿⡇⠈⣿⡆⢸⣿⡀⢿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡥⠰⣿⡃⢰⣿⠇⢠⣿⡇⢰⣿⡇⠀⣿⡇⠘⣿⡇⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⣿⠃⢸⣿⡇⢸⣿⡇⠸⣿⡇⣠⣿⣇⢀⣿⡇⠘⣿⡇⢹⡆
⠀⠀⠀⠀⠀⠀⢠⡟⢠⣿⠏⠀⣿⡿⠀⣼⣿⠃⢸⣿⡄⢸⣿⡏⠘⣿⡇⠀⣿⡇⠸⡇
⠀⠀⠀⠀⠀⢠⡟⢀⣾⡟⢀⣼⡿⠁⣰⣿⠏⢀⣾⣿⡆⠈⣿⡇⠀⣿⣷⠀⢹⣿⠀⢿
⠀⠀⠀⠀⢀⡟⢀⣿⡟⠀⣼⡿⠁⣰⣿⠏⢀⣾⠏⠹⣷⠀⢻⣿⠀⢸⣿⡄⠘⣿⡇⠈⣧
⠀⠀⠀⠀⢸⠀⣾⡟⠀⣼⣿⠁⢰⣿⠏⢠⡿⣿⠀⠀⠹⡇⠸⣿⡇⠈⣿⣧⠀⣿⣿⠀⢿⡄
⠀⠀⠀⢠⠀⣸⡿⠀⣸⣿⠃⢠⣿⡟⣠⠟⣳⠏⠀⠀⠀⠿⡀⢿⣷⠀⢿⣿⠀⢸⣿⡆⢸⡇
⠀⠀⠀⠀⢠⣿⠃⢠⣿⡏⢀⣾⠏⣰⠏⣸⠏⠀⠀⠀⠀⠸⢿⣼⣿⡀⢸⣿⡇⠀⣿⡇⠸⡇
⠀⠀⠀⢀⣾⠏⢀⣿⡟⢀⣾⠏⣼⡏⣼⠏⠀⠀⠀⠀⠀⠀⠞⢳⡙⣷⠀⣿⡇⠀⢿⣿⠀⡇
⠀⠀⠀⣼⡟⠀⣼⣿⠀⣾⠏⣰⣿⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣆⢹⣿⠀⢸⣿⡀⡆
⠀⠀⢰⣿⠇⢸⣿⡇⢸⣿⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⡎⢿⡇⠘⣿⣇⠁
⠀⠀⢸⣿⠀⢸⣿⠀⣿⡇⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣧⢸⣿⠀⢻⣿⢠⡗
⠀⠀⠸⣿⠀⢸⣿⠀⣿⡇⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠸⣿⢠⣿⡀⢸⣿⣤
⠀⠀⠀⣿⡇⢸⣇⠀⣿⡇⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣿⠈⣿⡇⢸⣿⠀⣄
⠀⠀⠀⢹⡇⢸⣿⢠⣿⡇⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢿⣆⣿⡇⠀⣿⠀⡏
⠀⠀⠀⢸⡇⢸⡟⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⠿⣿⣷⠀⣿⠀⡇
⠀⠀⠀⢸⣇⠈⣿⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢿⣿⠀⠟⢸⡇
⠀⠀⠀⠈⣿⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢸⣿⡇⠀⢸
⠀⠀⠀⠀⢿⠀⢻⣿⢹⡇⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⣤
⠀⠀⠀⠀⢸⠀⢸⣿⡀⢿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⢹⣿⠀⢸⡄
⠀⠀⠀⠀⢻⡷⣾⣿⡇⢘⣷⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣼⣿⡗⢸⣿
⠀⠀⠀⠘⠃⢸⣿⡏⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡘⢿⣶⣾⣇
⠀⠀⠀⢰⣿⣾⣿⣤⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣿⣿⣿⡄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⣠⣶⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷
⠈⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠚⠉⠉⠉⠙⣲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠃⢸⣟⢧⡀⠀⢀⡴⣻⣇⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣆⠹⣤⠞⢠⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡴⠛⢦⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣸⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡖⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⡿⡣⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢼⢿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠜⡡⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄⠧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠿⣏⡹⠛⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠙⢏⡹⠟⠻⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠽⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠻⠻⠟⠛⠻⠿⠛⠙

⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡛⠛⠛⠛⢛⣛⣿⣿⠛⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠟⢤⣿⣿⣿⣿⣧⡀⠀⠀⠀⠻⣿⣿⣧⣾
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠘⠻⣿⣿⣿⣿⠿⣶⣿⣿⠁⠀⠈⠉⢹⠋⢿⣿⠷⣦⣶⣄⠀⠃⢬⠉⣿
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⡀⠀⠀⣤⣭⠁⠀⠀⠀⠀⠾⠀⣼⣿⣿⣿⣿⣿⣷⣶⣄⠀⣿
⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠉⠉⠙⢿⣿⣿⠆⡀⠀⣤⣤⣴⣦⣼⠛⠁⠀⠀⠀⠠⣄⣠⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⢴⡆⢰⣿⣿⣿⡀⠄⠀⠉⠙⣿⡏⠉⠀⠀⢀⣤⣴⣶⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢈⣿⣿⣿⠄⠀⠀⠀⠸⣿⡀⢿⣄⣼⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠻⣿⣿⣿⡀⠀⠀⢀⡤⢤⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡝⠛⠃⠰⠶⣾⡿⢿⣿⣷⣄⣀⣀⣰⠾⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠧⠃⠀⠀⣀⡎⠀⢸⡟⢹⡅⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⢠⣧⠀⠀⠋⠀⠀⣾⠁⠘⠁⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣿⠆⠀⠀⠀⢰⠏⠀⡗⠀⢠⡄⡈⢻⡟⠸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡀⠀⠃⠀⠀⠀⢠⡞⠀⢼⣷⢀⣾⣿⣿⠈⣀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣀⡀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⣾⠁⢀⣾⠿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⣹⡅⠀⠀⠀⡿⣿⣿⡇⠀⠀⠀⣾⠀⠀⣸⣿⠀⢈⣿⣿⣿⣷⣤⣀⡄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠁⠀⠀⠀⠸⣿⠏⠀⠀⠀⣤⠋⠀⣼⣿⡿⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠃⠀⠀⠠⠶⠁⠠⢠⣿⣿⠇⣿⣿⣿⣟⠋⢹⣇⠀⠀⠀⠀⡼⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠈⠀⢀⠀⠀⣸⣿⠏⢈⣿⣿⣿⣿⣇⠀⢸⣄⠀⣤⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠴⣤⡾⠀⠀⠦⠈⠁⢸⣿⠃⠀⣿⠿⣿⡿⣿⣿⠀⠘⣿⣦⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⠃⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⡀⠙⠃⢀⣼⠿⢷⡈⡷⢺⣿⡆⠀⢹⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⢀⣤⡀⠀⠀⠀⠘⢰⣟⣿⡇⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⠇⠀⠀⣸⠿⡀⠀⠿⠇⠘⢻⣿⣤⣾⣿⣿⣿⣿⡏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠹⡿⠹⠷⣄⡀⢾⣿⣿⣿⠁⠀⠀⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣯⢑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠸⣧⠀⠰⣿⣿⣿⣿⣿⣿⠀⡀⠃⣠⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢗⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⡄⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⣿⣿⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣷⡀⠀⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⢠⣶⣄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣷⠀⠤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣴⢾⣿⠎⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢸⠃⢀⡄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠁⡌⠀⠀⠀⠀⠀⠀⠂⢿⠀⠀⠀⠀⠀⠀⢸⡇⠘⠇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣾⡇⠀⠀⢼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢩⡄⠀⠀⠈⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⣿⣿⠀⠀⠀⣾⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆⠀⠀⠀⠀⠈⣿⠇⠀⠀⣸⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆⠀⠀⠀⠀⣿⣿⡆⠀⢰⡄⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⠀⠀⠀⠀⠀⠀⣼⠏⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷⠀⠈⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠂⠀⠀⢀⠀⢀⣿⣷⡄⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣷⣶⣄⠀⠀⠀⠀⢀⣠⣬⣿⣿⣿⣦⡀⠀⢀⡀⢀⣯⣭⣤⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠠⠀⠀⠀⣬⡿⠷⡤⠄⢀⣄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⠀⢀⠀⢀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠀⢐⡀⠀⠈⠀⠀⠁⠀⠀⢠⡀⠀⠀⠿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢺⠀⠀⠀⢀⣀⢠⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠘⠇⠘⠀⣾⡇
⣶⣶⣿⣿⣶⡄⠀⠹⠘⠀⣨⣣⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡆⠀⠀⢹⠃
⠿⣿⣿⣿⡛⠀⠀⠀⠀⠈⢿⣟⣿⣯⣷⡆⠀⠀⠀⢄⠶⣶⡤⠀⠀⠀⠀⢶⣾⡦⠂⠀⣤⣤⡀⠄⠀⠀⠀⢠⣅⠀⠻⠇⠀⠀⣜
⠀⠀⣿⠁⠀⢦⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠛⢧⠄⠀⠀⠀⠘⣛⠀⣀⡸⡄⠀⠗
⠀⠀⣿⠁⣤⢨⣄⣄⣀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠀⠀⠀⠀⠐⢟⣀⡼⣴⣷⣤⣄⣀⣀⡀⠀⠀⠀⣀⠀⠀⢀⡀⢀
⣶⡷⣿⠘⣻⣾⡏⠛⢿⣇⣤⣬⣿⣿⣴⣦⣶⣤⣀⠀⣼⡇⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⢀⠀⢚⡷⠃⠀⢺⢋⡻⣿⣿⣿⢿⣿⣷⣷⣶⣿⣷⣿⣷⣶⣤⣧⣼
⡿⣃⣿⠘⠏⠙⢷⡄⡀⢻⡇⣻⣿⣿⣰⣟⣻⣿⡟⢸⣿⣧⣄⡀⠀⠛⢤⣼⣶⣶⣾⣿⣶⣿⣿⣷⣿⣿⣿⡿⣿⡟⠓⠈⠛⠀⠛⡟⠋⠛⠛⠉⠙⠛⣿⣿⣿⡿⢿
⣦⠟⣿⣤⣨⣄⡉⡁⢀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣽⣟⣷⣿⣿⢉⠿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣄⣿⣡⣐⣫⣎⡀⢘⣄⣀⡠⡮⠀⢀⣀⣠⣀⢙⣁⡇⣴
⠛⠠⣿⣿⣿⠟⠙⣻⣤⣛⠇⣛⣿⣿⣿⣿⣿⣛⣟⣿⣿⣯⣭⣭⣿⣿⣾⢤⠀⣧⣿⡿⢿⡿⣿⡏⢻⣿⣿⡿⡟⡿⠿⣿⢿⣼⡿⠿⢿⣿⣷⠈⣩⣽⠿⢿⡿⡀⢴
⠀⠰⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⠳⡆⣼⣿⡌⣿⠧⣿⣿⣿⣻⣿⣿⡇⢳⠇⠠⠄⠈⠻⣄⠀⠛⠡⢈⡃⠙⠋⣠⣿⣿⡷⢈
⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⡿⠿⢛⣿⣿⣤⣿⣿⢿⣿⣶⣾⣿⣿⣿⣿⣼⣷⣿⣿⣄⣿⣿⣷⡎⠀⠀⣠⣀⢻⡀⠀⠀⠀⢠⣷⣼⣷⣿⣿⣿⠿⢻
⣻⣿⣿⠻⣿⣿⣧⣿⣿⢿⣿⡅⠈⢿⡿⠿⢳⡖⣻⣿⠧⣤⣟⣳⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡃⠰⣾⣿⣏⣠⡀⠀⠀⠀⣾⣿⣿⠿⠟⣽⡇⣀⠈
⢹⣿⣿⡦⠹⢻⣿⢁⠵⢀⣈⣧⣖⠀⠀⢰⡽⠃⢸⢿⠏⡉⣿⣿⣷⣾⣿⣇⣿⣵⣫⣿⣿⣿⣿⣿⣿⣿⣟⡿⣀⠡⢾⣯⣟⣣⡏⠀⠀⣠⢹⡟⠿⠞⢠⣿⢿⠾⠤
⠀⠳⣛⠀⠀⠁⢨⣿⠆⠺⢙⡋⠁⠀⢄⠊⢷⣶⣿⣄⢷⣿⣯⢹⣿⡍⠃⠝⢛⣿⣷⣿⣿⣿⣿⣿⣿⠋⣽⡧⡌⠑⣾⠿⣿⣿⣿⣷⣿⣿⠆⠳⡖⢸⡿⠛⠀⠒
⠀⣸⣿⣿⠖⠳⠛⢻⡟⠁⠀⠹⣦⡀⡏⠀⠀⠈⠛⣦⠁⢹⠿⣷⡉⠻⣤⣾⣿⢹⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣇⠀⢸⣧⡾⠟⢠⣿⣿⣿⣿⣆⣧⣾⣿⣇⠂⣀
⣾⣿⣿⣿⣷⠍⠀⠈⠀⠀⠀⠀⠉⠹⠃⠀⠀⠀⠁⠙⠷⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠛⠉⠙⢢⣿⣏⡀⠀⠀⠸⣿⣿⣿⣿⣿⣧⣿⣿⣤⣴⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢳⡦⠚⠂⠀⢆⣃⣭⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⡤⣿⠒⠂⠰⣿⣿⣆⠀⠀⠀⠘⢋⣹⣾⣿⣶⣶⣶⣶⣯⣬⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠠⢿⠟⠘⠆⠀⠈⠉⢁⢀⣠⣴⣾⣿⣿⣭⣍⣀⠤⢈⣿⢿⣻⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⠐⠺⠧⢤⣈⣿⣿⣌⡛⠛⠛⠛⢻⡏⠁⣀⠘⣇⢻⣿⡇⠸⣿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠻⣷⣄⣀⠀⠀⢿⡿⠟⠉⠁⠀⠀⠰⢿⡏⠀⠉⠀⣼⢮⣼⡧⣹⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣧⣄⣀⡀⠀⠀⠀⠀⠀⣀⠋⣡⠌⠀⠠⠶⠘⠛⠃⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠦⠀⠀⢀⡀⠀⡄⠀⠛⠉⡟⠁⡀⠀⠐⠀⠀⢠⢶⡶⣤⠶⠺⢛⠀⠒⢄⣱⣆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢁⠀⠐⠙⠡⡄⠀⡀⠀⠀⠀⠀⠘⠀⠁⣨⠑⠀⠀⠙⢻⣿⠃⡀⠀⠀⠠⢈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠠⣿⠚⠀⠀⠀⢀⠂⢤⡀⠀⠀⡀⠈⢠⡁⠖⠀⠀⠁⠀⠀⢸⣄⠃⠒⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠞⡀⠀⢀⠀⠀⢙⣦⣀⣡⣄⣠⣵⡟⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣻⡍⠐⣦⡀⢠⣤⡀⠤⣤⣀⡀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⣹⣄⡀⣠⣶⣾⣏⣉⠁⢤⠘⠳⢤⣤⡦⠄⠀⣼⣷⣶⣦⡤⠶⠛⠋⠉⠀⠉⠉⠻⣤⢀⣀⣉⡉
⠀⠀⠀⠀⠀⡴⣿⠾⠗⠀⣠⣾⣿⣟⡝⠩⠙⠋⢹⣷⣋⠀⠀⠀⠀⠀⠀⣼⣿⡟⢿⠈⠁⠀⠀⢠⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⠰⠗⠀⣄
⠀⠀⠀⠀⠸⡿⠿⠤⠶⣛⣿⡟⣿⡞⠸⠀⠀⣤⡆⠸⣬⣷⡶⣆⠀⠀⢰⣿⡟⣧⠘⠠⠘⠀⠀⠈⠇⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠉⠰⣤⡀
⠀⠀⠀⠀⠀⣇⠀⠀⢰⠎⣫⣿⡿⠀⡇⠀⢨⣅⠁⠀⠀⠈⠀⠈⠓⣤⡌⣿⡇⠉⠀⡆⢧⠀⢸⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠈⠵
⠀⠀⠀⢀⡞⢹⡄⠚⠡⢾⣿⣿⣷⠀⡇⢀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠈⡄⠀⢳⢹⡀⢳⠀⠳⡈⠷⠀⠈⠀⠀⠀⠀⠀⠄⠀⠈⠦
⠀⠀⠀⠁⠁⢸⡃⢠⣴⣿⠋⣿⣿⠀⣡⢸⡄⠀⠀⠸⢠⠀⠙⡈⠀⠰⠘⣦⡹⠿⠇⠘⡇⠀⢳⠀⠁⠄⠀⠀⡆⠀⠀⠀⠢⣀⠀⠀⠈⠳⠟
⠀⠀⠀⠀⣰⡟⠀⠀⣴⠏⠰⣿⡿⠀⡏⢸⠃⠀⠀⠀⠀⠀⠀⢸⠷⠀⢳⡀⠀⠀⠰⣄⠤⠁⠈⢣⣄⢻⠀⠀⠀⠀⠀⠀⠀⠲⣲⡄
⠀⠀⠀⠠⢼⣧⠀⢀⣿⠀⠰⡟⠅⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠸⠰⡦⠀⠀⠀⢰⡄⠈⢰⡄⠀⠀⣻⢰⡀⠀⠀⠄⠀⠀⠀⠀⠈⢦⠉⠐
⠀⠀⠀⠀⠀⡟⠀⢰⡁⠀⢰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢻⣶⣾⢻⠀⠀⢹⠀⡃⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠁⠤⣼⣁⠀⡷⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠐⣦⢾⡇⠈⠛⠄⠸⡆⠁⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣻⠰⢤⠀⠀⠀⠐⠀⠀⠁⠜⠀⠀⠈⠐⠀⠀⠀⠀⠀⣤⡶⠺⣄⠀⠀⠀⠀⠈⠁⠀⡀⠄⠂⠒⠈
⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣾⣿⣧⡦⢼⣶⣶⡶⢲⡶⠿⠇
⠀⠀⠚⠛⠛⠿⠿⠿⢿⣿⣿⠋⠉⣹⢶⣶⣶⠶⢾⣿⣻⣏⡙⠁⠈⠈⠉⠉⠛⠛⠦⠶⠿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠟⠃⠀⠀⠉⠛⠁
⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⣆⣴⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠲⠐⠲⢶⣶⣦⣤⣀⠀⠀⠀⠂⠲⣤⣤⣀⣶⣴⣾⣿⣿⣿⣿⣿⣷⣦⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢉⠉⠉⠙⢛⣛⡋⠉⠿⠿⠿⡿⠿⠿⠏⠽⠝⠋⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠋⠉⠁⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣦⣀⠀⠀⣠⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⣦⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣄
⠀⠀⠀⠀⠀⠱⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡗
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⠿⠿⠿⠇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⡤⣀⠀⠀⠀⠀⣐⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⡞⠁⠀⠀⣶⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡅⠀⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡆⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡺⠤⠀⠀⠀⡀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣿⢿⣷⣄⣀⢀⣼⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⡍⠀⢼⣿⣿⣿⣿⣷⣭⣌⣛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠿⢬⠽⠗⣺⣿⣿⡿⢳⣿⣿⣾⣭⣙⡛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣤⠶⣻⣿⢿⡿⣱⣿⣿⣿⣿⣾⣭⢟⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣿⣭⣴⡾⢻⠏⣼⢡⣯⡿⠙⠻⣿⠘⢝⠈⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⣿⠏⠀⠀⣌⣴⡏⠸⢧⠁⠀⡀⠛⣧⠈⢢⠙⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣷⠿⢤⠖⠺⢧⣾⠃⠀⡀⠈⠀⠠⠠⠜⡆⠀⠳⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⠟⠁⢸⣿⠀⢘⠁⠙⠀⠈⣀⣠⢹⡄⠀⠹⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣾⣿⠀⠘⠁⠠⠀⠤⡂⢀⠀⢣⠀⠀⠹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠰⡧⣿⠀⠘⠃⠀⠀⠀⠀⠈⠀⠘⣦⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣇⢿⡇⠀⠸⠠⠀⠀⠄⠀⠂⠀⢹⣀⣰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣷⣾⣿⡄⢀⣦⡀⠀⡀⠀⠰⠀⠸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠀⠀⠰⠀⡄⠀⠀⠀⠀⠙⠛⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⢰⠻⠋
⠀⠀⠀⠀⠀⠀⣤⣖⢻⣷⠃⠀⣰⠀⠲⠀⠀⠀⠀⠀⠀⢀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠠⠘⣆
⠀⠀⠀⠀⣴⠿⡡⠔⣧⠃⢀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡄⠀⠀⠀⠀⠀⢢⣀⣀
⠀⠀⠀⡜⠀⠀⠅⠈⠀⠴⠷⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⣦⣤⣤⣄⣤⣠⣤⣶⣼⣶⣿⣿⣿⣷⣀⠀⠀⣀⣤⡿⠛⠀⠀⠄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⣧⣿⣬⣿⣯⡻⢿⠛⢿⡙⠛⠋⠉⢀⠠
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⡿⣿⠽⡿⠿⠚⠟⠉⠋⠈⠁⠀⠀⠀⠳⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠷⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢺⡏⠀⠀⠀⠀⠀⢈⣿⠒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣸⣿⡄⠀⠀⠀⠀⣼⣿⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡿⠿⠟⠟⠙⠻⢿⢿⢩⣿⣿⣿⣿⣿⣿⣿⢿⠋⢁⣿⠟⠋⠉⠛⠿⠛⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠉⠉⣹⣿⣿⣿⣿⣿⡄⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠈⠁

⢀⣀⡀⠀⡀⢀⡀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⣿⣏⢹⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢻⢿⣿⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠓⠀⢹⡿⣼⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡚⢀⠜⡷⣯⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⢿⡇⠀⠑⢿⠓⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⢀⠁⠀⣀⠘⠀⢰⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠠⠈⡦⠧⡀⠘⠙⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⢣⠄⢄⠀⣘⡓⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠖⠀⠀⠂⠀⣀⣅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠠⠆⠘⠃⠀⣙⠈⠀⢆⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠃⠀⠠⠵⠛⠇⢆⣙⠀⠒⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠆⠀⣀⡘⠀⠀⢆⡚⠀⠀⣰⠌⢧⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡤⠀⡄⠘⠋⠁⠔⢊⡭⠀⣾⠿⠠⢶⢰⣤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠂⢘⣠⡀⠠⢖⠄⢀⡀⠖⡾⠴⠆⣀⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢧⠀⣠⠇⠀⠒⠾⠓⠀⠐⠴⡆⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢱⡄⠤⠉⠀⠀⠀⣺⠁⡴⡗⣀⠶⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠸⣆⠸⠞⠒⡖⢠⢠⠀⠀⠋⠠⡤⡋⠁⠆⠀⡀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⡞⠡⢀⠀⠀⠁⠀⠬⡤⠀⠁⠴⢶⡇⠀⣀⠃⠸⠇⢀⠋⣿⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⠀⠀⠂⡶⠄⠀⠁⠀⠀⠴⣸⢂⢈⠹⢿⠀⠲⠁⡀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⢀⠀⠀⠸⠃⣠⠐⠀⠏⠐⠈⠹⠀⠈⠀⢘⠁⢀⠃⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠢⠀⠔⠂⠀⠀⠤⠘⠀⠀⠀⢠⠀⠖⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⢃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⢛⢻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⠋⠀⠀⠀⠀⠀⠘⢏⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠈⠁⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢯⡏⠩⠉⢻⣿⣿⡿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣿⣿⣿⣿⢿⣿⣥⠆⠀⢰⣦⣤⣀⣬⣽⣧⣤⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣿⣿⣏⣙⣿⣿⣿⣿⣇⠀⢼⣿⡟⠈⠉⠛⠋⠉⠇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡟⠛⠋⣿⣿⣯⠉⣋⠻⡻⠇⠐⠋⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠈⢙⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠤⠚⠛⠛⠆⠀⠀⢀⡀⣠⣀⣼⣿⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆
⠀⣀⣿⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢃⠙⠿⣿⡿⠇⠈⠀⠀⠀⠀⣀⡠⣀⣤⣬⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⠁⠁⠘⠛⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠈⡀⠀⠀⠀⠀⠀⢠⣿⡎⢉⠉⡄⠀⢩⡅⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣾⣷⠏⠀⠀⣀⠀⠀⢀⠰⠆⠀⠂⢠⡟⢳⡀⠀⠀⣤⡞⠀⢀⣰⡀⠿⣷⠄⠛⠛⠶⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢹⣾⠀⠀⠀⠘⣿⣿⣦⡵⠶⡴⠖⠁⠀⠀⠀⠀⠀⠻⣶⣾⣿⣿⣿⣿⡆⠀⣿⣿⣿⡳⣦⣄⣤⠀⠀⠀⣤⣤⣤⣾⣿⣦⣤⣴⣾⣵⡿
⠀⠀⠀⠀⠟⠛⠀⠀⠀⣰⣿⣘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⢤⣄⣠⣾⣿⣿⣿⠶⣾⣷⣶⣄⣀⣬⠻⣿⣿⣿⣿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠀⠙⠿⣤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⠀⠀⣸⣿⣿⣿⡇⠀⠰⢤⠉⠻⢿⣡⣞⣨⠙⣿⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣬⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠈⠙⠛⠃⠰⠿⠿⠻⢿⣇⠀⣼⡃⠀⠀⠈⢻⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠙⣆⠀⠀⠈⢳⠀⠀⠈⠂⠀⢦⢄⣈⣳⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢘⣿⣴⣰⣿⣶⣿⣇⠀⢿⣿
⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣹⣧⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⢻⡄⠀⠀⠀⠈⠇⠀⠀⡆⠀⠈⠂⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠻⣿⣿⣿⣿⣿⡏⠀⢺⣿⣿⠀⡟
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠙⢿⣿⣿⣿⡇⠀⣾⣿⠋⠻⣧⣼
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⠃⠀⠀⣤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡧⠀⠀⠈⢻⣿⣿⡇⢠⣿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⠀⢀⣾⣧⠀⡀⠀⣀⣷⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣨⣋⡟⢡⣶⣆⠀⠀⠀⠹⣿⡇⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢿⣿⣶⣶⣯⠀⠉⠉⠁⠀⠀⠀⠀⣠⡤⠴⠒⢤⣄⠀⠀⠀⢀⡉⣉⣀⣾⣿⣿⣄⡀⠀⠀⢹⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣿⠟⡹⠆⠀⠀⠀⠀⠀⠀⣴⡿⢃⠔⠒⡿⠀⠈⢳⠀⠰⠞⢠⣿⣿⣿⣿⣿⡟⠀⠀⠀⣀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⣴⢳⠀⢀⣴⠀⠀⢀⣰⣿⣇⡇⠀⠘⠡⣶⡟⣸⣿⣶⣾⣿⣿⣿⣿⠋⠀⠀⠀⠀⡾⠋⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⡏⠀⢸⡏⠀⢸⠁⠀⠀⠈⣿⣿⣿⣿⣶⡶⠿⢋⣴⣿⣿⣿⣿⡿⠋⠉⠁⠀⠀⣠⣶⠀⣧⡀⣠⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣸⡇⠄⢠⡀⠀⠀⠻⣿⣄⣽⢻⣿⣿⠋⠉⠿⠟⢻⣿⠋⠁⢀⣀⣠⣴⣿⣿⡏⠀⠀⠀⢩⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⢱⣾⣿⣷⣤⣀⡀⠉⠉⠀⣀⡀⠀⠀⠀⠀⣈⣿⣤⣶⣿⣿⣿⣿⣿⣿⣧⠀⣶⣶⣾⡟⢁⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢾⣿⣯⣿⡿⣿⣿⣷⣶⣶⣶⣶⣶⣶⠟⠛⠉⣿⠛⣿⣿⣿⣿⣿⣿⡇⢩⣀⠻⣿⠏⠀⠾⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⣻⣿⣿⣿⣿⣧⠠⠁⠀⡌⣀⠀⢹⣿⠀⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠏⠑⢀⠀⠀⢠⠟⠀⢸⣿⡴⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⡀⠀⠀⠞⢷⣄⣸⣿⣿⣼⣿⣧⣀⡄
⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠙⣷⣦⡀⠑⠀⠀⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠉⣻⣿⡿⠿⠋⠀⠀⠈⠹⢿⣿⣿⣿⣽⣿⡏
⠀⠀⠀⠀⡄⠉⢠⣀⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠠⠘⠋⠀⠀⠀⠀⢤⣶⠀⠀⠀⠀⣿⡿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠙⠀⢿⣿⣟⣰⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡣⡀⠀⠀⢙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⡄⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠧⠺⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⠻⠿⠿⠿⠿⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣗⠉⠁⠀⠀⠀⠉⢹⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢰⣿⡄⠀⠀⠀⠀⠀⣾⣇⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣮⡿⢻⣿⢿⣿⣻⠯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠈⣿⣹⣧⢸⣿⣾⠏⢁⣴⣻⣏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢳⣿⣿⡟⠿⡋⣡⡶⢿⣿⣿⡿⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢏⣿⣿⡏⠁⣰⣿⣯⡷⠀⣉⣿⣿⣟⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⣿⣿⣯⡆⠀⠿⣿⣿⡿⠀⢹⣿⣿⣿⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡴⢿⣤⣿⣿⠇⠀⣿⣿⣿⣷⠀⠘⣿⣿⣯⣼⡦⠽⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣿⣇⡿⡟⠀⠀⣿⣿⣿⡟⠀⠀⠙⢿⣭⣿⣷⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⢠⣿⣿⣿⡿⠀⠁⠀⢠⣭⣿⣿⣿⠀⠀⠀⠅⢹⣿⡿⢿⡀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡤⣿⠁⠈⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⡧⣞⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠈⢧⠀⠀⠀⠀⠀⠀⢸⣿⣿⡍⠀⠀⠀⠈⠀⠰⡤⠈⠹⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⠇⠀⠀⠀⠀⠀⠀⠑⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠙⠲⣄⣀⣀⣤⠞⠉⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⠀⠀⠀⢠⡞⣉⠩⠙⣯⣝⠩⣉⠙⢦⠀⠀⠀⠈⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣦⠀⠀⠈⠧⠤⠴⢻⡇⢸⠳⣤⣉⣨⠃⠀⢠⡔⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣧⢸⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⢹⢸⡀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡟⠘⡞⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠾⣧⡤⠤⠤⠴⠒⢻⡄⠀⡇⠈⠓⠲⠤⠤⠤⣤⡿⢦⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡟⠀⠀⠀⠀⠀⠀⠷⠲⠇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⢀⡼⠧⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠷⣄⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⡴⠋⠀⠀⢀⠌⠉⠉⠋⠉⠛⠛⠛⠛⠋⠙⠉⠉⠀⠀⠀⠈⠳⣄⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⠖⠋⠀⠀⠀⢠⠞⠁⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⠓⢦⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠘⣦⣶⣈⠁⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠘⠈⣁⠀⡀⠈⠉⠀⠀⠘⡽⣭⢃⡞⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⣀⠁⠉⠐⠉⡠⠆⠀⠀⠀⠠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⠀⠀⠀⢸⠐⡃⠀⠀⢀⠄⠀⢙⠀⡄⠛⠤⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣈⠁⠉⠂⠀⠉⠐⣶⠁⣀⠀⠀⡁⠎⠁⢀⡀⡐⠀⣄⠀⠀⠐⠢⡔⠂⠀⡀⢈⠃⠀⠊⠁⢶⠀⣘⠁⢠⠡⠆⡇⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢮⡀⢱⠀⢀⠂⠑⠖⠁⢑⠁⢈⠆⠀⠀⢀⡄⠈⡧⠠⡄⢠⠤⢀⠀⠀⠀⢀⣄⢀⠅⡀⠙⠶⠈⡁⢀⠐⢋⣷⠤⠖⠚⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠒⠶⠶⠖⠚⢧⡌⠉⢡⠀⠀⠸⠀⢈⠃⠠⠀⠈⢆⢠⡈⠀⠀⠐⢄⣀⡃⢀⡘⠒⡇⢈⣀⣠⠴⠊⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠤⣤⣉⣨⣄⡤⠴⠢⠦⢄⣨⣀⣀⣀⡮⠥⠒⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⡞⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠚⠉⠙⠲⠦⠤⢤⡤⠤⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⢸⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢻⡇⠀⢀⣠⣴⣶⣿⣿⠿⠿⠛⠻⢿⣿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣤⣼⣿⣄⠀⠀⢸⣷⣾⡿⠟⠛⢧⣄⠈⠙⠲⢄⡀⠀⠈⠳⣄⠙⢿⣿⣄
⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⠋⠉⢉⣩⠭⠿⠿⢿⣿⣿⣿⣷⣶⣿⢿⡇⠀⠀⠀⠀⠈⢧⠀⠀⠀⠙⢦⡀⠀⠈⢦⠀⠹⣿⣧
⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⢀⡴⠞⠉⢀⡤⠖⢋⡽⠋⠉⢀⡟⠛⠁⠀⢷⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠙⣄⠀⠈⢧⠀⠹⣿
⠀⠀⢠⣾⡿⠋⠀⠀⠀⣴⠋⠀⢀⡴⠋⣀⣰⡋⠀⠀⠀⣼⠀⠀⠀⠀⠈⣇⠀⢠⣴⠾⠛⠿⣦⡀⠀⠀⠀⠘⣆⠀⠘⡆⠀⢿
⠀⢠⣿⡟⠁⠀⠀⢀⡾⠁⠀⣰⢋⣴⠟⠋⠉⠙⠳⣦⠀⡇⠀⠀⠀⠀⠀⠸⡄⣿⠁⠀⠀⠀⠸⣷⠀⠀⠀⠀⠘⣆⠀⢹⡀⢸
⢀⣿⡿⠁⠀⠀⢀⡾⠀⠀⢰⠃⠸⣿⡀⠀⠀⠀⣀⣼⢳⡇⢀⣾⣷⡀⠀⠀⢷⠿⠿⠿⢿⣷⣿⡿⠃⠀⠀⠀⠀⢸⡄⠀⣇⠸
⣸⣿⠇⠀⠀⠀⣸⠃⠀⢀⡟⠀⠀⠙⢻⡟⠛⠛⠉⠁⢸⠀⣼⠏⠈⢻⡄⠀⢸⡄⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧⠀⢹⣠
⣿⣿⠀⠀⠀⠀⡿⠀⠀⢸⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⢠⣿⣤⣤⣶⠷⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⣀⣤⣴⣆⢸⠀⢸⣿
⣿⣿⠀⠀⠀⢰⡇⠀⠀⣼⠀⣶⣦⣤⣼⣤⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣤⣷⣿⠿⠋⢸⣿⢸⡄⠀⣿
⣿⣿⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⡅⠉⠉⠉⢹⡇⠀⢸⡿⠿⠶⣶⣶⣶⣶⣶⡶⠿⣿⡄⠀⠀⢹⡟⠉⠀⠀⠀⣼⡏⢸⡇⠀⣿
⢸⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⢻⣇⠀⠀⠀⠸⠿⠶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⠶⠿⠇⠀⠀⠀⣴⡿⠁⢸⠃⢰⣿
⠀⢿⣿⡀⠀⠈⣇⠀⠀⢸⡀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⣸⢀⣾⡿
⠀⠈⢿⣷⡄⠀⣿⠀⠀⠸⡇⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠕⠀⠀⢀⣿⣾⣿⠃
⠀⠀⠀⠻⣿⣦⣸⡆⠀⠀⢿⡀⠀⠀⠙⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠟⠋⠁⢸⡇⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⠈⣧⠀⠀⠀⢻⡀⠉⠙⢛⣿⣿⡶⠶⠶⠿⠿⠟⠛⠛⡏⠁⠀⠀⠀⠀⣼⠁⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣈⣷⡀⠀⠈⢧⡀⠀⠀⠀⠈⣏⠉⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣰⠏⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⣄⣈⢳⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⣴⣯⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣷⣶⣶⣤⣽⣦⣤⣤⣤⣤⣿⣶⣶⣾⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣆
⠀⠀⠀⣠⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠁⡀
⠀⠀⠀⠀⠀⣴⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠘
⠀⠀⠀⠀⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠙⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠁⢀⣀⣤⡤⠰⢂⡄
⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⢀⠀⠁⠀⢀⣶⣿⡆⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⢠⣈⡁⠱⠒⠂⢰⣿⣿⣦
⠀⠀⣀⣀⣄⠀⢠⠀⢀⠐⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠒⡀⠐⠀⠀⠀⠀⠑⠀⠈⠀⠐⠀⠄⠀⠀⠁⠃⠀⠀⠀⣾⣿⣿⡄⠀⠀⠸⡿⠿⠛
⠀⠀⠿⣽⠛⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠈⠁⠉⠁⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠨⢍
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⢀⠰⠀⠄⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡆⣥⡄⠀⠀⠀⠀⠘⠂⠁⠀⠀⠚
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡈⠐⠀⠤⠒⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⢀⠒⣀⠀⢀⡄⠀⠠⢄⠆⠀⡀
⠀⠀⠀⠀⠀⠈⠃⠀⠈⠀⣁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⣤⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⡼⠀⠀⠛⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡣⡝⡒⡂⠄⠀⠀⢰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⣧⣷⡀⠑⠶⠀⠀⠀⢀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⡀⣀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠙⠮⠁⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⣦⡀
⠀⠀⠀⠀⡀⠀⠀⠘⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠰⡤⠀⠀⠄⠄⠠
⠀⠀⠀⠀⠑⠒⠈⠊⠁⠀⠀⡀⠖⢠⠖⢰⡴⡄⢠⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢂⠀⣾⡟⢁⠀⢌⠀⠀⠀⠀⠀⢀⣠⣌⣿⣿⣿⣿⣯⣾⣿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢆⣹⡇⠀⠁⠀⣿⢀⠠⠀⠀⠀⠀⠀⠾⠿⠿⢿⡿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣡⠆⠀⠀⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⠎⠐⠀⠧⠈⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠈⢇⠀⢺⠀⡀⠀⠀⢀⡄⢠⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⢸⡀⠀⠀⠀⡀⠀⢸⠀⢸⠀⣇⠘⡇⠈⡇⠀⠀⠇⢀⠀⠀⠀⠀⣿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠐⠈⠁⠀⠴⠊⡁⠀⢻⠀⢸⠀⢻⡀⠀⢠⣇⢠⠀⢀⡏⠐⠂⠶⡀⢿⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡠⢦⠠⢀⡀⣡⣄⠈⢷⠈⡏⠈⠋⡇⠁⠟⠈⢳⢸⡁⠛⠳⠀⠤⡞⢸⡄
⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⣇⠀⠁⠈⣇⠀⠙⡇⠹⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠘⠉⠙⣦⠸⣦⣠⣇⡼⣗⠀⠈
⠀⠀⢠⣿⣾⣷⣄⠀⠀⠀⠀⠸⡐⠃⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠶⠀⠀⠀⠀⠀⠈⠀⠀⠹⡇⠀⢻⡄
⠀⠀⠘⢿⣿⣿⣿⣷⣦⡀⠀⠀⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣿⣿⣷⡶⣤⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣭⣈⠀⠀⢀⣁⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣠⡴⢒⣄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⣘⣛⠃⠀⡘⠛⢋⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣌⢸⡟⠛⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣽⣿⢿⣻⣿⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣋⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⡿⣿⣿⢿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠨⠀⠀⠀⠀⠀⢀⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⢀⢀⠀⠀⠈⠉⠺⠷⠴⣁⣻⡆⣇⣼⡅⠩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣶⠖⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠷⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠴⠿⢴⢤⠤⠤⢀⣸⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⢀⣾⣿⣿⣿⣿⡟⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⢿⣿⣿⣿⣯⣽⣿⣿⡄
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⠦⠿⢟⠛⠻⢿⣿⣿⣿
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠓⠒⠒⠊⠉⢉⠉⡀⠙⣿
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠰⢀⣿⣆⠀⠀⡀⠀⢠⣿⣦⡈⢆⠘⣧
⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⠿⣛⠛⣿⣿⣿⠃⢰⡇⠉⠉⠉⢀⣪⣿⣆⠀⠀⠀⠀⢸⠀⣿⠀⢀⣤⡀⠀⠀⠀⣴⣶⣄
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠿⠿⢿⡀⣻⣿⣿⡀⢸⡇⢠⣄⠀⢈⣉⠉⠉⣀⡀⣰⡆⢸⠀⡿⠀⣹⠿⣿⣂⣷⢠⣭⣿⠟
⠀⠀⠀⣰⣿⣿⣿⣟⣻⣿⡏⠀⠀⠀⢸⠹⣿⣿⣧⠀⠣⠀⢿⣷⡿⢿⣷⣾⡿⠿⠿⠃⠊⣰⠃⡜⠁⠴⢀⣷⣠⠞
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠘⠆⣽⣿⣿⣷⣄⡀⣠⡀⠠⡀⠉⠁⠀⠀⠀⣠⣾⠇⠴⡛⠂⢀⣤⣿⣥
⠀⠀⢻⣿⣿⠋⠙⣿⣿⣇⠀⠂⠀⢐⠃⣶⣿⠿⣿⣿⡟⠛⢻⣿⣿⣅⡀⠜⣶⢶⣿⣿⡟⠠⣦⠘⠧⠺⠿⠿⣛⡄
⠀⠀⠈⢿⣿⣷⢹⣿⣤⣿⣽⣶⣿⣯⣽⡿⣿⡶⣿⣾⣿⠀⣶⣿⣿⣿⡿⣝⠎⣻⣿⣿⣧⣂⡌⠁⠲⣶⣾⠚⣿⡧
⠀⠀⠀⠀⠙⡿⢠⣿⣤⣯⣠⣧⣸⣧⣼⣦⣼⡶⣿⣿⡟⢸⣿⣿⣿⣿⣿⣾⣿⣦⡤⠛⢛⣿⣕⠓⢾⣻⡜⡷⠋
⠀⠀⠀⠀⢠⡇⡎⠁⡞⢿⣈⣯⣨⣏⣸⣧⣽⣦⣝⢻⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢳⡾⠶⢶⢶⣿⣭⠏
⠀⠀⠀⠀⠸⡗⠀⡼⢡⠞⢽⣯⣩⣏⣹⣧⠚⠁⣾⠀⠁⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⣛⣥⣄⠺⢟⣽⣧⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠙⠞⠁⣿⠠⣄⡻⣭⡯⣽⡵⠞⠀⣿⣷⣴⣿⣿⣿⣿⣧⣠⣿⣿⣿⡿⠉⠴⠉⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢺⡤⣏⡐⣒⠂⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂⣀⠜⢻⣿⣿⣿⣿⡟⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⣠⣾⡙⡇⠈⠻⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⢺⠋⣤⠊⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣴⣞⣙⣯⣉⡿⠁⡇⠀⠀⠉⣫⢙⣯⣨⣿⠿⣿⣿⣿⣿⣿⣿⡏⣄⡻⡊⠀⡱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣏⣿⣉⣿⣉⣿⣽⡃⠀⡇⣠⡴⠛⣿⢉⣿⣄⣿⣶⣿⣿⣿⣿⣿⣿⠋⣻⠀⡱⢎⢀⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣉⣿⣭⣿⣉⠛⣦⣿⣥⠴⣶⣼⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⢹⡧⣀⠴⢿⣿⣿⡅⣀⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣇⠘⠑⡶⠛⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡹⡗⠉⢢⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡇⠀⢀⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⢻⣁⠶⡀⢻⠴⠁⠈⢿⡟⠛⠛⠋⠉
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡗⠀⡇⠄⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣸⣇⠀⠜⡿⣄⡰⢦⡀⣿⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠃⠀⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡇⡙⢃⣤⠃⠘⣦⢀⠛⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⡆⠀⡇⠀⠀⣿⣿⣿⣿⣿⠿⠋⠀⣤⡫⠛⢤⣮⡏⠀⠀⢹⠳⣤⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢇⠀⢸⠉⠉⠉⠁⠀⠀⣠⣼⣣⢽⢢⣯⣽⠀⠀⠀⢠⣗⣀⣱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸⠀⡀⠀⠀⠀⠀⠀⠀⠳⠤⢝⣻⢿⣭⠇⠀⠀⠀⢈⣿⡿⣿⡂⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣷⣧⡘⢀⡳⢰⠆⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⢛⡋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⠧⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠻⣝⠛⠒⠤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠚⠉⠉⠀⣠⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣄⣼⠠⡀⢠⠏⠀⠀⠀⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⡤⠶⣻⠫⡀⠀⠀⠈⠓⠢⠤⠤⠤⠤⠤⠤⢖⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢀⡿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠉⠀⠀⡰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢫⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣼⡄⡏⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠉⠀⠀⣀⡤⣾⠀⠀⠀⠀⠀⠂⢤⣀⡀⠀⠀⣠⠖⠁⠀⠸
⠀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇⠀⠀⠘⢷⣦⠶⠋⠁⠀⠀⠠⠔⠒⠋⠉⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡽⡿⠁
⠀⠈⠉⠛⠲⡦⣤⣀⠀⠀⣠⠀⠀⠹⣇⣸⣧⠀⠀⣰⡟⠉⠳⣤⠤⠦⠤⠶⠒⠂⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⡤⠒⠁⢸⠃
⠀⠀⠀⠀⠀⠈⢢⡈⠙⠛⠻⣶⢶⣶⣿⣿⣿⣿⣾⠟⠀⠀⣀⣌⣷⣤⠤⢤⠤⠴⠃⠀⢠⡀⠀⠀⣉⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣴⣾⣿⣿⣿⣿⣿⠿⢷⣞⣛⡯⠋⠁⣠⠈⠶⠤⢤⠀⢳⣄⡤⠽⠖⠉⠀⢀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⡴⠛⠉⣾⣿⣿⣿⣿⣿⣿⣿⡁⡀⠀⠉⣍⠙⢻⢓⣶⣾⣾⠾⠯⠥⠦⠤⣼⣄⣀⣀⣘⣆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡠⠞⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠉⢷⣸⠀⠀⡼⠀⢈⠈⢦⠏⠉⡄⢀⡁⠠⣴⡁⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠲⠿⠦⠤⣤⡀⣀⡙⠢⣄
⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⢸⠇⠀⢀⣿⣆⠁⠀⠀⠀⣷⠌⡀⠀⠙⡉⠀⠀⠀⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⡠⠒⠁⠀⠈⠃⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣀⠠⠚⣿⠶⢄⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠰⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾
⠀⠀⠀⠀⠀⣀⣾⣣⠤⠶⠏⠑⠠⠄⡏⠀⠀⠁⠀⠀⠀⠀⠀⢻⡾⠋⠀⠀⠀⠙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠟⠳⢤⣀⠀⠀⠀⠀⢠
⠀⠀⠐⠚⠛⠉⠉⠛⠛⠛⠓⠤⡀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠸⡄⠀⠀⠀⠤⠒⠒⠒⠲⢦⣀⣤⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣙⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢾⡏⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⣠⠏⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢳⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⡀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⣀⣠⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢟⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⣿⣿⣿⣿⣿⣷⡈⠻⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⠻⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⡀
⠐⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠁⠀⠀⠀⢙⣛⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠲⣿⣶⣶⣯⣭⣾⣡⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠉⠻⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣤⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠧⢤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⢻⡇⠀⠀⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠛⠀⠀⠀⠀⢀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠶⢺⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⢆⢰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢻⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠴⢒⡾⢧⠀⣀⣘⣷⣿⡿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠈⢿⡥⠼⣿⣿⣿⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⠞⠁⠀⣼⣷⠶⠦⠤⠤⠤⠤⠶⠶⣺⠿⣆⠀⠀⠈⠳⡄⠉⠉⡇⢹⢄
⠀⠀⠀⠓⠲⢤⣀⡤⠊⠁⠀⠀⣀⡴⠋⠁⠀⣠⠜⠁⢈⣷⡤⣤⣤⣄⣀⣤⣴⡋⠀⠘⢷⡀⠀⠀⠘⠧⣄⠁⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⡏⠉⠛⢶⡶⣏⣁⠀⣀⡴⠞⠁⠀⣠⠞⠙⣧⠀⠀⠀⢀⡾⠉⠑⢤⡀⠀⠙⠦⣄⡀⢀⣈⣳⣤⠤⠤⠶⢻⡖⠒⠚⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇⠀⠉⢻⡟⠦⣤⣴⣞⠁⠀⠀⠘⣧⠀⢠⡟⠁⠀⠀⠀⣙⣦⡤⠶⣾⠏⠉⠁⣼⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠇⠀⢸⡏⠉⠛⠶⢦⣄⣼⠷⠛⣦⡤⠖⠛⠉⣹⠋⠀⣰⠏⠀⠀⡼⠃⠀⠀⠀⡜
⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠇⠀⠀⣼⠀⢀⣾⣁⣤⣤⠴⠶⢺⣧⣄⣴⠿⣄⠀⠀⢰⡏⠀⣰⠏⠀⠀⡼⠁⠀⠀⢀⡜
⠀⠀⠀⠀⣠⡏⣀⣀⣠⣿⣤⠴⠖⠻⣟⠉⠙⢷⡀⠀⢀⡴⠟⠁⢸⣿⠀⠈⠛⣦⡟⠀⣸⡏⠀⠀⣼⠃⠀⠀⢀⡾
⠀⠉⠉⠉⠉⠛⢍⡀⠀⠈⠳⣄⠀⠀⠈⢷⡀⠀⣷⡶⠯⢤⣤⣀⢸⣿⣠⠴⠚⠉⠙⢦⣿⠀⠀⢠⡇⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢧⡀⠀⢈⣷⣾⣭⣄⣀⠀⠀⠈⠻⣿⠁⢀⣠⠶⠛⠉⠙⢷⣄⡸⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢿⣠⣞⣁⣀⡀⠀⠉⠙⠳⠦⣄⣿⡶⠋⠀⠀⢀⣀⡠⠴⠛⢷⣄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⡾⠋⠀⠉⠉⠉⠙⠓⠲⢤⣀⠈⣿⠀⣠⡴⠚⠉⠀⠀⠀⠀⣀⣨⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠒⠦⠤⣤⣀⡀⠀⠀⠀⠀⠈⠙⢻⠟⠁⠀⠀⢀⣠⠤⠒⠉⠁⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢤⡀⠀⢸⡄⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢼⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡠⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼⣀⡄⠀⠀⣉⡒⠠⠤⢀⣀⢀⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠃⠐⢡⣵⠆⣭⢹⠂⠀⠀⠀⠹⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⠀⠘⠺⠋⠢⠭⠟⠀⠀⠓⠀⠀⠋⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⡐⠁⠀⠀⢙⠆⠀⡀⠁⠀⠀⢶⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠃⠱⣄⣤⠴⠋⢀⠔⠀⠀⠀⠀⣿⡤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣾⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⡏⣇⢰⡆⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠞⠁⠰⢰⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢻⠀⠃⠀⠀⠀⠀⠀⠹⡍⠒⠤⣀
⠀⠀⠀⠀⣠⠔⢋⡡⢴⡏⠀⠀⡄⣼⠀⠘⢧⣄⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⢻⠑⠦⡀⠑⣄
⠀⠀⣠⠞⣡⠖⠁⠀⡸⠀⠀⠀⢇⡇⠀⠀⠀⠪⠉⠓⠒⠒⠋⡉⠀⠀⠀⠀⢸⢸⠇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠘⣆⠈⡆
⠀⢰⠁⢸⠁⠀⠀⢀⠇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡏⠀⣼
⠀⢸⠀⠈⢦⡀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣀⡼⠔⠋⠀⣰⠇
⠀⠈⢧⡀⢀⡟⠑⡾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⠠⠐⠒⢫⡉⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠙⠟⠀⠀⣷⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡔⠁⠀⠀⠀⠀⢹⠖⠛⡇
⠀⠀⠀⠒⠿⠄⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⣇⠀⠀⠀⣀⠴⣡⡜⠀⡇
⠀⠀⠈⠣⠴⠖⠒⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⠓⠦⢬⣍⣉⡴⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠶⢦⡶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⠶⠶⠶⠖⣶⠖⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠤⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠰⡤⠤⠤⠶⠶⠚⢻⠇⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣷⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠻⡍⠉⠉⠉⠙⠳⠚⠓⣦⠴⢾⠉⠙⠙⠛⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠐⢯⣀⣀⠀⢐⠀⣀⣀⣤⠴⠦⣼⠁⠀⠀⠁⠀⠀⣀⣴⣆⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠒⠒⠚⠚⠋⠉

⠐⡆⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠰⣿⠃⡀⢙⠁⠀⠘⠀⠀⠈⠀⠺⡿⠛⡁⠙⣿⡷⠀⠈⠉⠀⢀⣿⣿⡇⠀⢀⠀⠀⠀⢀⠀⢀
⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⢢⠐⠂⠄⠀⠀⢠⢂⠀⠀⠀⠀⠤⠀⠀⠀⠁⠀⠀⠀⠴⠿⠿⠋⠀⠸⠿⠂⠀⠀⠀⢐⡴⡄⠀⠀⠀⣾
⣿⣿⣿⡿⠁⠀⠀⢺⠀⠀⠂⠂⠀⠀⠐⠁⠀⣀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠈⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠂⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⢀⠀⠀⠐⠐⠈⠀⠁⠈⠈⠈⠀⠈⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠆⠀⠘⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠠⣀⠠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠀⠀⠀⠀⡀⠠⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⡆⢀⣤⣄⣸⠀⠀⠀⠀⢀⠀⠉⠀⠈⠁⠀⠀⠀⠀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⢠⣄⠀⠁⠀⠀⠀⢀⠀⢨⠀⢀⠀⠀⠀⠿⠂⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⢶⣶⣄⡀
⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⡀⣀⣀⣀⣀⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⢠⣀⠐⣀⠰⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⢹⣿⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⡀⣤⡀
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⢿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠈⠀⠈⠛⠀⠀⠀⠀⠈⠋
⣿⣧⡀⠀⠾⡶⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⣶⡶⠆⠀⠂⠈⠀⠀⢀⡐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠋⠻⣿⡄⠀⠀⢀⣿⠀⠠⢿⠄⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⣰⠀⠀⠀⠠⣾
⢉⠀⠄⠀⢀⣤⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠛⠃⠀⣠⣿⣷⣄⡀
⢈⡀⠀⠀⠙⠏⠁⠘⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠆⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣧⣶⣤⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⢿⣿⡆
⠠⡄⠁⠀⠀⠀⠬⡴⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠋⠀⠐⣀⡈
⠃⠃⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠁⠂⠀⠀⠂⠧⠆⠀⠀⣴⡄⠀⠀⠀⠀⢠⡶⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠑⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠂⠐⠁⠀⠀⠀⠁⠀⢁⠐
⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⣤⣄⢠⣿⣿⡆⠀⠀⢀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⣠⠀⠀⠀⠘⠃⠀⠀⠀⡀
⢩⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡉⠁⡀⠂⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⢈
⢒⠀⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡿⡟⠁⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⡈⠀⠀⢰⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠨⠀⣨⠀⠀⠀⠀⠀⠙⠛⡂⠠⠀⠀⠀⠀⠓⠃⠀⠀⠈⠙⠃⡀⠀⠀⠀⠆⠀⡀⠀⠀⠘⠀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠁⠀⠀⠀⠀⠀⠀⠃⢈
⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣤⣶⣶⣶⣦⣦⣄⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠙⠻⣶⣦⣀⠀⠀⠁⠀⣀⠆⡀
⠀⢌⠀⠀⠐⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠈⠿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠐⠋⠈⠉⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⡆⠀⠠⣶⡄
⠀⠈⠀⠈⠀⠀⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣠⣤⣯⣀⣀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠆⠀⣠⠀⠀⢰⠀⠀⠀⠀⡀
⠀⡌⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠀⠀⠀⠛⠛
⠀⠁⢀⡀⠀⠘⠻⠿⠿⠿⠟⠁⠀⠀⠀⢀⣿⣿⡇⢀⣴⣷⣿⣶⣶⡄⠀⠀⡀⠘⠃⠀⣠⣶⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠄⠀⠀⠄
⡇⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⠿⠋⠀⠘⢿⣿⣿⣿⣿⠃⠀⢸⡇⠀⠀⠘⠋⠉⠃⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⡇⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⣠⣿⣿⣦⡀⠀⠀⠀⠀⡗⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⣿⣿⣤⣾⣿
⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣿⡀⠀⠀⠸⡿⠋⣈⢻⠿⠀⠀⠀⣠⣇⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠺⠿⠗⠀⠀⢸⣿⠟⠛⠻
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⡀⠀⠀⠀⠀⠀⣰⣿⣿⣦⣄⠀⠀⢠⠀⢰⠀⠀⣠⣾⣿⣿⣦⠀⠀⢀⣄⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣈⣍⠀⠀⠀⢘⣁⡀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣥⠂⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣌
⠀⠀⠀⠰⠋⠈⢁⣀⣀⠈⠉⠳⠀⠀⠀⠀⠀⢀⠀⠉⠙⠃⣀⠀⠛⠉⣀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠠⠾⠏⢀⠀⠙⠿⠆
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⣀⠀⠀⠴⠄⠀⠉⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣈⣁⠀⣠⠞⠁
⠀⠀⠀⢸⣿⠇⠙⠛⠛⠁⢹⣿⡄⠀⠀⠀⠀⠀⠻⣷⣾⣿⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠘⢶⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣨⡅⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⣈⣉⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⢢⡀⣐⠀⠀⢀⢀⠖
⠀⠀⠀⢀⠀⠀⣿⠃⠘⠟⠀⠀⡀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⡀⠀⠼⠿⠃⠈⠻⠏⠀⠠⡀⠀⠀⠀⢠⡀⠙⠉⠀⠀⠈⢉⡄
⠀⠀⠀⠸⣧⡀⡀⢈⠀⠀⣰⣶⠇⠀⠀⠀⠀⠀⠈⠛⠛⠁⠤⠈⠛⠁⠀⠀⠀⠀⠀⠰⣀⠀⠀⠁⠈⠀⠀⣀⡴⠁⠀⠀⠀⠀⣿⣶⣄⣰⣶⣶⣿⠃
⠀⠀⠀⠀⠙⠿⠻⢿⡼⠾⠛⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣀⣀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠙⠋⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⢂⣀⠀⠀⠀⣁⣌⠑⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⣿⡋⠁⠀⠀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⠶⠂⠀⠲⠾⠋⠀⠀⠀⠀⠀⠀⡓⠀⠀⢀⣤⠀⠀⠀⡒⠀⠀⠀⠀⠀⠉⠈⠃⠀⠘⠁
⠀⠀⠀⠀⠈⡉⣀⢀⣀⣉⠉⠀⠀⠀⠀⠀⠀⠈⢳⣤⣄⣀⣁⣠⣤⡶⠁⠀⠀⠀⠀⠀⠛⣷⣤⣤⣀⡤⣤⡿⠃⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣶⡄
⠀⠀⠀⠐⠶⠿⠛⠛⠛⠻⠻⠧⠄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠿⠷⠀⠛⠨⠿⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠲
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢚⡓⠀⠀⠀⣠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⣠⡆⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣷⡄
⠀⠀⠀⠈⠛⠛⠒⢠⡄⠴⠛⠛⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⡀⢨⡿⠻⠄⠀⠀⠀⠀⠈⠲⢦⣤⠄⠐⢶⠶⠋⡀⠀⠀⠀⠀⣠⣄⡉⠉⡤⠀⠛⠁⠀⣠
⠀⠀⠀⠠⣶⣶⣶⣶⣦⣀⣴⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⣦⡀⣀⢘⣃⢀⣄⢾⠇⠀⠀⠀⠀⠁⠘⣿⣤⣴⣦⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠘⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢸⣿⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠉⠓⠂⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠟⠒⠦⣄⣀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠲⡄⠙⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⣠⠊⣨⡆⠀⢣⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠘⠒⠛⠢⠿⠃⠀⠈⠙⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⣀⣠⠖⠀⠀⠀⠀⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠚⢻⠉⠀⡠⠄⠀⢲⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣺⣿⣿⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⡐⣚⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⢤⠤⠭⣤⠤⠺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠸⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠐⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠛⡉⠀⡧⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣃⣠⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⠁⠃⠀⠈⠲⡇⠠⠤⠄⣀⡀⠉⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⠤⢘⣡⠞⠀⠀⠀⠀⠀⠀⢀⠌⡔⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⢄⠈⠀⠀⠄⣀⠃⠘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡘⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣘⠂⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣄⡸⡀⠀⠘⢦⡀⠉⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠁⠒⠒⠒⠒⠁⠀⠉⠒⠦⣤⣀⣀⣀⣀⣀⣀⣳⠀⠀⠀⠀⠀⠱⠤⠤⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⠀⠒⠒⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠀⠉⠉⠉⠉⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠤⢤⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢦⣀⠀⣀⠠⠄⠒⠀⢦⠼⡄⠀⠀⢡⣀
⣀⠠⠀⠂⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣀⠀⠀⠀⠀⠀⠀⠈⡦⠲⣄⠀⠀⢣
⠉⠙⠓⠲⠶⠦⠤⠤⠶⠶⠚⠛⠒⠢⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠛⠁⠒⠶⠷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠋⠁⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡆⢰⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⢸⡇⢸⡇⢸⡇⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⢸⡇⢸⡇⠀⠀⢿⣿⣿⡿⠋⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠰⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣠⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⡀⠠⠀⠂⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⢰⣷⠀⠀⠀⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠄⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⣸⡿⠀⠀⢀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⣼⣿⣧⡀⠀⠁⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠻⣿⣿⡿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠠⣦⠉⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⢿⣏⠙⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣾⣷⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣀⠉⠁⠀⠀⠀⠄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⣿⣿⣿⣷⡀⠘⢣
⠀⠀⠀⠀⠀⠠⣶⡄⠻⣿⣿⣿⡿⠇⠀⠀⠄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠃⠀⠈⠛⠛⠉⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⢹⣷⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣶⣾⠗⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⣿⣿⣿⣇⡀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⠇⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠰⣶⣦⡀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠄⢸⣿⡆
⠀⠀⠀⠀⠈⠳⣯⠀⢸⡿⣦⡀⠀⠀⠀⠀⣰⡿⠟⠻⣿⡿⠀⠀⠀⢀⡀⠀⠀⣴⡞⠀⠀⠀⠀⠀⠂⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢧⡘⠃⠘⠁⠀⠀⠀⠘⠋⠀⠀⢀⠛⠁⠀⠀⠀⣾⠿⣦⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠀⣠⢾⡿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⣠⡐⣦⣀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠴⠞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠠⠀⡄⡀⠀⠀⡄
⠀⠀⣤⣼⣀⡀⢀⣤⠀⣸⠋⢠⡾⠉⠀⢠⠧⣄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠁⠰⠀⡄⢡⠀⣴⠁
⠀⠈⠁⠀⠀⠀⢀⣍⠀⣿⣄⠘⠀⡆⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠐⠠⢀⣠⡾⠃
⠀⠐⠀⠁⢀⢴⣿⠇⢀⣀⣤⣤⣄⠘⣦⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⡔
⠀⠀⢀⠐⢀⠎⡎⠀⠀⠛⠛⠛⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠄⢣⡀⠀⠈⢁⣀⠚⠁⠀⠀⠂
⠀⠐⠀⣠⠂⠸⠀⠀⠀⠀⠨⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⡽⠠⢆⠟⢦⣀
⠀⠀⠀⠀⢀⡤⠚⠁⠴⢀⡄⠀⠄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠨⡄⢨⣛
⠀⠀⢀⡔⠋⠀⠀⠀⠰⠃⠁⢠⠀⠀⣴⠀⠀⠀⡠⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⢀⣀⣀
⠀⣴⡇⡀⠀⠀⠀⠃⠀⠹⢀⠀⠀⠀⠀⠀⢀⣌⣀⣀⠁⠀⠂⠀⠰⠆⠀⠇⠀⠀⡀⠛⠈⠋⠐⠄⡧⢴
⠀⠀⢾⠎⡌⡐⠀⠀⠀⠀⠉⠀⢷⡄⢀⠐⠋⠉⠉⠉⠀⠀⡀⠀⠀⠲⠶⢞⠀⡀⣠⣁⣈⠀⡀⠀⢸⣽
⠀⡀⠀⠀⠀⠁⠁⠀⠀⠀⣠⠞⠁⢀⣡⣄⠀⡄⠆⠀⠀⠀⠈⢀⣈⣁⠀⠀⠀⠀⢐⠊⠋⠐⠄⠀⠈⢯⠂
⠐⣷⠀⣸⠠⠤⣂⡥⠄⠈⢀⠆⠀⠈⢹⣿⠀⠐⠶⠞⣀⡤⠄⢓⡌⠚⠀⠘⠿⠂⠀⠠
⠀⠀⠵⠶⠀⠀⢠⠀⠂⠀⠊⠉⠀⠀⢸⡿⠀⠀⠠⡎⢒⡡⠖⠁⠀⠆⠿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠐⢦⣴⣞⣷⣾⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡤⠄
⠀⢀⣴⣦⣤⣤⣼⣿⣿⣿⡿⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃
⠀⣾⣿⡿⠟⠻⠁⠀⠙⠉⠀⠀⠀⢀⡀⠀⠀⠛⠳⠶⠀⠀⠀⣸⣧⠤⣿⣧⡀⠴⢢⡀⢰⡆⢶⠄⣴⠰⣦⠀⢀⡀⢠⡤⠀⠀⢀⡄⢠⣿⣿⣿⣿⡀
⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⡏⠃⣤⢹⡇⢸⡇⢺⡆⣿⠀⣿⠀⠈⣧⡼⢡⣤⣤⣤⢶⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡟⠀⠾⠆⠀⠙⠘⠁⢿⠁⠛⠁⣿⠐⠋⠀⠀⣸⠁⠈⢿⣿⣿⣧⣝⠻⠿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠀⣹⡇⠀⠀⠀⠘⠀⢀⠠⠀⠀⠀⠀⠀⠝⠁⠀⠀⠛⠀⠒⠥⠴⠃⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣶⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣿⡇⠀⠀⠀⠀⠘⣿⢠⣷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡏⠉⣹⡿⣦⠰⢒⡆⠀⣿⠈⡏⢠⣶⢿⣿⣶⣶⡀⢺⠃⣸⠇⢸⡇⣴⠛⣦⠠⡾⢻⡆⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⢴⡀⣷⠘⣿⢸⣿⢸⣿⣶⣿⣾⣿⠇⣿⠂⣹⡀⢸⡇⣿⠀⠈⢰⡇⠀⠁⢸⡇⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠠⠟⠳⠄⠈⠉⠁⢀⡏⢸⣟⠀⠙⠛⠛⠛⠁⠀⠀⠈⠉⠉⠉⠁⠈⠓⠁⠀⠙⠊⠀⠚⠇⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆⣿⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠛⠧⠀⠀⠉⠛⠶⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣄⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⠿⠿⠛⠛⠛⣻⡿⠟⠉⠉⠉⠉⠙⠻⣷⡄⠀⠙⢷⣄⡀⢀⡄⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⢰⣿⡷⠶⠒⢲⣶⠶⡶⢤⣼⣿⠀⠀⠀⠈⠙⠻⣷⡀⠀⠀⠀⠀⠀⠙⠳⠶⠶⢶⡆
⠀⠀⠀⠀⢀⣤⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠘⠿⠿⠃⠀⣸⣿⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠀⣤⣠⠾⠁
⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⣀⣀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⣀⠀⠀⠹⣅
⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣤⣼⡷⠆
⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣀⡀
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⠀⠀⠀⠹⣿⣿⡀
⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣤⣾⣿⠏
⠀⠀⠈⠛⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⠛⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣶⠶⢶⣶⣶⣤⣤⣤⣤⣬⣷⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠃⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠋⠛⠛⠛⠛⢿⣿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⢿⣿⠿⠛⠛⠛⠛⠛⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠟⢿⣿⠇⠀⠀⠀⠀⠀⣾⠻⣦⠀⠀⣴⠛⡍⠙⣦⠀⣠⡟⢻⡇⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠉⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⢰⡿⠀⠹⣧⡞⠁⢠⠇⠀⢙⣶⠟⠀⢸⣷⠀⠀⠀⠀⠀⣴⡄⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⡀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⣸⡇⠀⠀⠘⢷⣄⣿⣠⡴⠛⠁⠀⠀⠈⣿⠀⠀⠀⠀⠀⢿⡇⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀⣿⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡟⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⣶⡿⠋⣿⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠸⠗⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡆⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠛⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣹⣧⣼⣿⠀⢠⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣶⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣶⣶⣿⣷⣶⣶⣿⣿⣿⠿⠿⣿⡟⠛⠀⣳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⣿⣿⠉⠉⠁⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⣶⠶⣶⣤⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠠⡆⠀⠀⠈⢉⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⢸⠀⠈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⣷⣶⣶⠾⢿⣇⣀⣸⡇⠀⠀⠀⠀⠀⠀⣼⣏⠀⣸⡿⠶⣶⣤⣾⣤⣤⣾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⡀⠶⣦⣤⢀⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣀⠀⠀⠀⠤⠛⢿⠓⢆⡈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢸⡁⠑⠄⠀⠀⠀⠀⢀⡀⠈⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⢸⠀⠀⠤⠀⠙⠒⢂⠤⡀⢶⡊⠉⣱⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⢀⡠⠤⠼⡄⡄⣓⡘⡆⣀⡳⠤⣂⢀⣈⠉⡁⠀⠀⠀⠀⡸
⠀⠀⠀⢠⠖⠀⠈⡉⠛⠚⠁⣀⡀⠀⠙⣮⢸⡑⣣⠤⠐⠚⠤⣌⡠⠀⠁⠀⠀⠀⡰⠃⠂⠄⡀
⠀⠀⢠⠃⡀⠀⠯⠬⢂⡤⡘⠓⠚⡤⠒⣶⣿⡏⠀⠀⠀⠀⠀⠘⣄⣀⣄⡨⠔⠊⠀⢀⡄⠀⠀⠢⡀
⠀⠀⢸⠀⠁⢰⠖⣆⡤⠭⠤⣄⡜⠀⢠⠄⣬⠁⢠⠀⠄⠤⠀⠀⡏⠉⡀⢀⡀⠀⡄⠠⠃⠀⠀⠀⢣
⠀⠀⠈⢧⡂⢀⠧⢀⠀⠀⠀⣀⡇⠀⢈⠭⢤⠤⠭⠀⠀⠀⢄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠙⠾⡆⢈⠩⠛⠉⠉⠉⠉⠙⠲⡊⠑⠐⠚⠑⠒⠈⣼⠀⠀⠈⠠⠈⠁⠀⡀⠀⣼⢀⡼⠓⠒⠲⢤⡀
⠀⣠⠔⠊⠉⠁⠉⠬⠆⠀⠀⡀⠀⠀⠀⠀⠘⣄⣀⣀⣀⠤⠞⠉⠉⠑⢦⡁⠐⠂⠁⣀⡼⠛⠋⠀⠀⠀⠀⠀⠙
⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢃⠀⠀⠀⠀⣀⠠⠒⠂⠠⢹⠒⠚⠉⠁⠀⠀⠔⠊⠉⡂
⡇⠀⠀⠀⠀⢀⢀⠀⢠⠀⠤⠂⠀⠀⠀⠀⠀⡮⢉⠩⠉⢁⡑⠒⠋⣤⠀⢠⠂⠄⠀⠀⠐⠀⠈⠉⠉
⢇⠀⠀⠀⠀⠀⠈⡉⠀⠀⠄⠀⠀⠀⠀⠀⡰⠁⢀⢀⠀⠃⢞⢀⣀⡏⠀⡾⠀⠀⠀⢀⠀⠈⠠⠀⠀⠀⠀⠀⢀
⠘⢄⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⣀⠠⠊⠱⢄⣀⠀⠄⢀⠡⠴⠋⣠⠞⢧⡀⠀⠁⠀⠠⠂⢀⡀⠁⣀⣀⠴⠋
⣀⣀⣑⣢⣤⣄⣠⣤⣒⣉⣉⣉⣁⣀⣀⣀⣀⣀⣉⣑⣒⣒⣒⣒⣉⣀⣀⣀⣉⣓⣶⣦⣴⣒⣉⣉⣉⣉⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣠⣴⡶⠄⠀⠀⣠⣀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢰⠟⠁⠀⠀⠀⠀⠈⠙⢷⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⡴⢚⣷⢆⣠⢖⣶⣦⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢿⡄⠀⠀⠸⡇⠘⢿⣿⣧⠈⠻⢻⠇⠀⠀⠀⢸⣧⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⡂⠀⠀⠀⠀⠛⠶⠖⠋⠈⠛⠚⣉⣠⣤⡶⠾⠟⠛⠉⠉⠁⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣭⡄⠀⠀⢀⣀⡀⠀⠀⠀⠐⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣼⠋⠙⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡿⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢠⠀⠈⠙⠻⠶⣶⣤⣤⣤⣤⣶⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢷⡀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢦⣄⠀⠀⠈⠛⠶⠤⠶⠋⠀⠀⠀⠀⠀⢀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣸⡇⠀⠉⠛⠶⣤⣷⣦⣶⣖⣀⣀⣠⣤⣶⡾⠟⠋⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣠⠞⢉⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⡙⢷⣄
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡴⠟⢁⣤⠞⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄⠙⢷⣄
⠀⢀⣈⢣⣉⠓⠤⠖⠛⠛⠋⢀⡴⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢷⣄⠙⠳⣄
⢰⡭⠭⠭⠭⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠙⠷⣄⠈⠳⣤⣀⣀⣀⠔⣹⡤⢤
⠳⣤⡤⠤⠤⠀⣀⣠⠤⢦⡀⣸⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⢳⡄⠉⠁⠈⠉⢘⣡⣴⣫
⠀⠳⠤⠔⠒⠉⠁⠀⠀⠀⣇⠘⠦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢤⠖⠒⠉⣰⢦⣀⠀⠀⣤⣤⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠋⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠓⠖⠚⠁⠀⠈⠳⢤⣀⡉⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠈⠙⠳⢶⣤⣤⣄⣀⣀⣀⠀⡀⢀⣀⣀⣀⣠⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢸⡀⢿⠉⠛⠛⠛⠛⠛⣿⠛⡟⠉⠉⠀⠀⠀⠀⠀⣠⡶⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸⡷⢶⣤⣤⣤⣤⣤⣼⡇⢸⡆⠀⠀⠀⠀⠀⣿⠀⣇⢀⣀⣀⣀⣀⣠⡞⠁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠘⢷⡀⠀⠀⠀⣴⠋⠘⠋⠉⠉⠉⠉⠉⠋⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⣶⣤⣤⣤⣤⣤⣤⣀⣤⣼⡇⠀⠀⠀⢷⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡤⠤⠤⠤⢷⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣍⠈⣧⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠙⠲⠶⠦⠤⠼⠦⠤⠤⠤⠤⠤⠤⠶⠖⠒⠊⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⢼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣿⣿⠟⠻⣿⣧⡀⣠⣿⡇⠀⣹⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⢼⣿⣿⣧⣀⣨⣿⣿⣄⣤⣿⣿⣟⠁⠈⣿⣿⣟⠁⢙⣿⣿⣿⣿⠟⠙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠛⢻⣿⣿⡿⠛⢻⣿⡟⠓⢿⣿⡟⠛⢳⣿⡿⠛⠻⣿⣷⣴⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣴⣿⣿⣷⣤⣴⣿⣧⣤⣾⣿⣷⣤⣾⣿⣷⣤⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣟⣛⣿⣻⣿⣿⣶⣶⡆⢀⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⡯⠡⠏⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⢛⣵⣿⣿⣿⡟⠿⡧⠴⠶⠿⠛⠻⠟⠟⠋⠁⠀⠀⠘⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠿⠟⠿⣿⠀⢿⣾⣿⣿⠉⠀⠠⠤⠀⠠⠄⠀⠀⢀⣶⣿⣷⣤⣶⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⣟⣿⣶⡘⣦⣌⠛⢃⣀⢀⣠⣀⣄⣀⣀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢹⣿⡿⠟⠁⣨⣿⠁⠰⠿⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡏⠀⠀⠀⣠⣶⣿⢠⡇⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣰⣾⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣿⡗⣾⣿⢟⣀⣈⣠⣾⣿⣛⣟⡃⠿⣿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠆⠈⢻⣿⡟⠋⠉⠀⠍⠻⠉⠙⠻⢿⣷⣦⣄⡀⠀⠀⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡁⢰⢃⣼⡟⠀⠐⠲⠦⠄⠀⠀⠀⢊⠄⣩⡙⠟⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡷⢸⠿⡃⠀⠘⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⣡⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣷⡖⠛⢀⠀⠀⠀⠄⡀⠀⠀⠀⠄⠀⠀⠀⠒⠀⠀⠀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢛⣿⠀⢸⣿⡦⡀⠀⠀⡐⢦⣄⡀⠀⠀⢀⣠⣄⠀⠁⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠀⢸⣿⡅⣄⠀⠀⠀⠀⠉⠙⠛⠉⣈⣻⡏⠧⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⢸⣿⣗⡙⣺⣶⠶⠦⣀⡀⢀⣈⡿⢻⣿⣀⠀⠋⡄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢿⣿⢿⣿⣿⣿⣿⣿⢇⡠⠟⠋⣿⣿⣿⣾⣿⢏⠠⡈⡇⡔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡟⠾⢿⡧⠿⠟⢽⡿⣿⡿⠛⠙⠙⣷⣾⣿⣿⡷⣾⠀⠫⠐⣢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣦⠤⣤⠀⣌⠉⢮⡉⠁⠀⠈⠙⠛⠛⠻⠇⢿⡄⢆⣀⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⠛⠉⡰⠛⠂⠛⠀⠀⠘⠛⢶⠀⠀⠀⠀⠀⢺⡏⠉⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣇⣀⣄⣀⠀⠀⠀⠀⢠⡀⣷⣶⣤⣶⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠛⠻⠏⠙⡟⠣⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠆⠀⠀⠀⢀⡀⠀⠀⠸
⠀⠀⠀⠀⣀⠀⡏⠀⠀⠀⠀⠀⠘⢡⠟⠉⠳⡄⢀⠖⠉⢻⣦⠀⠀⢇⠀⠀⠀⡠⠀⡀
⠀⠀⠀⠀⠑⡄⠁⡰⡊⠀⠀⢀⠇⠸⣄⣘⡠⠃⣈⢆⣛⡸⠃⠀⠀⡞⠀⠐⢀⣀⠂⣐
⠀⠀⠀⠈⢐⠂⣀⡀⠓⡄⠀⠨⣄⠤⠤⠴⠒⢏⠈⠑⠢⠤⠤⠥⣤⠜⠀⠐⠤⡍⠑⢠⠼
⠀⠀⠀⠀⠀⠀⠀⠳⡏⢄⠀⠀⢲⠀⠀⠑⠤⢤⣤⠤⠂⠀⠀⢀⡟⠀⠀⠀⢺⡀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠢⡀⠀⠳⣤⡀⠈⠃⠀⠀⠀⠀⣠⠞⠀⠀⢀⡠⢊⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡲⢤⠜⣍⠓⠲⠤⢤⢶⣿⠉⠒⡤⢒⡩⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏⠀⠀⢹⠁⠙⡌⠇⠈⡗⠒⠊⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠏⠀⠀⡀⣠⠦⠠⣬⠀⠀⢸⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣸⠀⠀⠀⡇⡁⡈⢳⠘⡇⠀⠘⡀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⡇⠀⠀⠀⠻⣬⣁⣩⡾⠁⠀⠀⡇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠈⢆
⡰⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡇⠀⢰⠶⠶⠶⢶⠂⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠣⣑⠒⠊⠀⠀⠀⠀⠀⠀⠀⢉⠏⠉⠀⠀⠀⠈⠉⡏⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠾⡝
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⠁⠒⡇⡦⠤⠤⠤⠤⠤⠇⠗⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⡇⠀⠀⠀⠀⠀⠒⡆⠀⠀⠀⠀⠈⠉⠉⠒⠲⠤⣄⣀⠀⠈⠓⠚⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⣇⠀⠀⠀⠀⢀⠇⠈⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣐⣮⣤⣤⣤⣔⣚⣀⣀⣀⣀⣈⣒⣒⣠⣤⣤⣬⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢉⣿⣿⠋⠻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣾⣻⣿⡀⣰⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣈⣉⠉⣻⡟⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠒⠚⣼⣯⣭⣭⣤⡛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⣼⣿⣿⣿⣿⣿⣿⠀⠀⢈⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣯⣶⢀⣼⡟⠉⠙⢿⠟⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢾⣟⣠⣴⣶⣶⠾⡷⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⠿⠛⢶⣿⣷⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡓⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⠀⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡾⣾⡅⠀⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠈⠀⠀⠀⢻⣿⠖

⣿⣿⣿⣿⣧⢻⣧⡙⢿⣆⣻⣷⡘⢷⡌⠙⠿⣶⣶⣤⣬⣿⣿⣿⣿⣷⣿⣿⡏⣠⣾⣿⣽⡾⠟⣩⣾⣯⣴⣾⣿⠟⢀⣼⢟⣴⠟⠀⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡆⢻⣷⡄⠈⠻⣿⣿⣶⣿⣷⣦⣀⠉⠙⠛⢿⣿⡙⢻⣿⣿⣿⣰⣿⣿⢟⣭⡾⠟⣛⣿⣿⣿⣿⣷⣾⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⢻⣿⣆⠀⢹⣿⡟⠛⠻⠿⣿⣿⣦⣄⠈⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⡿⠿⠛⠛⢻⣿⠏⣴⣿⣿⠟⢫⣿⡏⣿⡿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⠀⢻⣿⡆⢸⣿⢠⡶⠶⣤⡀⠉⠻⣿⣧⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⣀⣤⠶⢶⡘⣿⣿⣿⠟⠁⣠⣿⠏⢸⣿⢣⣿⣿
⣿⣿⣿⣿⣯⢻⣿⣧⡀⢻⣿⣿⡏⣼⢲⣶⣂⣙⢷⡀⠀⠹⠿⠛⠋⠉⠉⠉⠉⠉⠙⠻⠿⠋⠀⣠⢞⡉⣦⡕⢺⡇⢿⣿⠃⣠⣾⡿⠃⢀⣿⢃⣾⣿⣿
⣿⣿⡹⣿⡻⣷⡙⣿⣷⡌⣿⣿⡇⣿⢸⠸⢮⣽⣂⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡡⣿⣺⡜⡀⡇⢸⣿⣾⡿⠋⠀⣠⣿⣷⣿⣿⣿⣿
⣿⣿⣧⠹⣷⡙⢿⣮⣿⣿⣾⣿⡇⣿⢸⣷⣯⣭⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣝⣿⣿⠆⡇⣼⣿⠏⣀⣴⣾⣿⣿⠟⣩⣾⣿⣿
⣿⣟⢿⣧⢻⣷⡀⠙⢿⣿⣿⣿⣿⡽⣎⣾⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣾⣗⣼⣿⣿⣿⣾⣿⣿⠿⣋⣵⣾⣿⢟⣿⣿
⣿⡿⣷⣿⣿⣿⣿⣶⣤⣙⣿⣝⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⢛⣻⣿⣿⣷⣿⣿⡿⣋⣴⣿⣿⣿
⣿⠷⠦⠈⠙⠻⢿⣿⣍⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣽⣿⣿⣿⣿⠿⠿⠛⢿
⣿⠷⠛⣛⣛⣻⣿⡿⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⡾⠋⠉⣹⡟⢦⡀⠀⠀⠀⠀⠀⣠⡾⣻⡍⠉⠙⡆⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠷⠖⠖⠉⠃⠀⠀⠀⠀⠀⢷⠀⠀⣿⡇⠀⢷⠀⠀⠀⠀⢠⡏⠀⣿⡇⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠖⠶⠿⣿⣿⣿⠿⣿⣿⣷⣿
⣿⡿⢋⣵⣿⣯⣷⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢿⠇⢀⡞⠀⠀⠀⠀⠈⢧⡀⢻⡇⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣶⣿⣿⣿⡶⣬⣙⣿
⣿⣴⠿⢋⣽⣿⡟⠁⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠤⡤⢄⠀⠀⠉⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠙⢿⣿⣿⣿⣶⣍⣻
⣿⣧⣾⣿⣿⡟⣠⡶⠛⣩⣷⠿⠛⠋⣻⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡟⠉⠛⠿⣯⡛⠳⢦⡈⢿⣿⡿⣿⣿⣿
⣿⣭⣾⣿⣿⣾⠋⢀⣾⠟⠁⠀⠀⣴⡿⡿⠀⠈⠙⠳⢦⣄⣀⠀⠀⢀⣠⣷⣀⠀⠀⢀⣀⣤⡶⠟⠋⠁⠘⡟⢿⣄⠀⠀⠀⠹⣦⠀⢻⣾⣿⣧⣾⣿⢿
⣿⣿⣿⠟⣿⣿⠀⣾⠃⠀⠀⠠⠚⠁⣠⣷⠀⠀⠀⠀⠀⠈⠉⠉⠙⢧⡼⡟⣦⠟⠉⠉⠉⠀⠀⠀⠀⠀⢰⣷⡀⠉⠳⠀⠀⠀⠈⠃⠀⣿⣿⡟⣿⢃⣾
⣿⣧⣿⠀⢹⣿⠀⠃⠀⠀⠀⠀⠀⠀⢁⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣇⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⣦⣧⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣿⣾⣿
⣿⣿⣿⣇⠀⢻⣇⠀⣆⠀⠀⢀⣀⣴⠟⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⣿⣄⠹⣆⠀⠀⠀⠀⠀⠀⠘⣿⡟⠷⣤⣀⠀⠀⠀⡄⠀⣾⠃⡟⢀⣿⣿⣿
⣿⣿⣿⣿⣦⡀⠻⡄⢿⣶⣄⣉⣁⣠⣾⡿⣿⣰⠀⠀⠀⠀⠀⣼⠃⡼⠁⡇⠹⣆⠹⣆⠀⠀⠀⠀⢸⣦⡿⢿⣦⣄⣀⣀⣤⣾⡇⡸⠋⠀⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⣀⣈⣳⣬⣭⣽⠟⠋⠀⢹⣿⢠⠆⠀⠀⣸⢃⡾⠁⠀⡇⠀⠘⢦⠹⡄⠀⠀⢢⢸⣿⠃⠈⠙⢿⣯⣭⣴⣋⣀⣀⣠⣴⣿⣿⣿⣿⣿
⣿⡻⢿⣿⣏⠛⠻⠿⢿⣿⣿⣿⣛⠛⠻⣦⡀⢻⣿⠀⠀⢀⣧⡏⠀⠐⠀⣇⠀⠄⠈⣷⣿⠀⠀⠈⣿⠏⣠⡶⠟⢛⣻⣿⣿⣿⡿⠿⠛⠋⣽⣿⡿⣫⣿
⣿⣿⣆⠻⣿⡆⢀⣾⣿⣿⣿⡿⣿⣿⣦⡘⣷⠀⣿⣄⠀⠈⣿⣷⠀⢀⠈⠛⢀⡀⢀⣿⡿⠀⠀⣰⣿⢠⡿⢠⣾⣿⣿⢿⣿⣿⣿⣦⠀⣼⡿⢋⣴⣿⣿
⣿⣿⣿⡆⠘⣿⣾⣿⣿⢨⣿⣇⣿⣿⢿⣷⣽⣇⣿⠉⢳⡀⢹⣿⡄⠒⠀⣆⠐⠈⢸⣿⠇⣠⠞⠹⣿⣼⣷⣿⢿⣿⡇⣿⡏⢰⣿⣿⣷⡿⠁⣼⣿⣿⣿
⣿⣿⣿⣷⠀⢸⣿⡟⣻⣷⣿⣿⣿⣟⠀⣿⣿⣿⡏⠀⠀⠹⣼⣿⡇⢠⡊⠈⣠⠀⣾⣿⣰⠃⠀⠀⣿⣿⣿⣿⠄⣿⣿⣿⣷⣾⢟⣻⣿⠁⢀⣿⣿⣿⢿
⣿⢿⣿⣿⡄⠈⣿⣧⣃⣿⣿⠁⡻⠿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣷⠈⠀⠀⠀⠀⣿⣿⠏⠀⠀⠠⠙⣿⣿⣿⣿⢿⡇⢹⣿⣗⣠⣿⡟⠀⣸⣿⣿⡇⣸
⣿⡈⢿⣿⣷⡀⠘⢿⣿⣿⣥⣄⣸⣶⣾⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣌⣽⣝⣿⣿⠿⠃⣠⣿⣿⠟⣠⣾
⣿⣿⣷⣿⣿⣿⣷⣤⣭⣝⣛⡻⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣹⡆⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⠿⠿⣛⣋⣭⣤⣴⣾⣿⣿⣿⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⢖⠉⢰⠀⠾⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣍⠉⠐⠐⠂⠤⠄⣀⢀⣠⣇⣨⠤⠬⠤⢥⢼⣳⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠀⠠⠤⠄⠒⣮⢹⠀⠀⢀⠭⢼⡴⠛⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣺⡐⢀⠢⠄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⢄⠀⠀⠞⠀⠈⢾⠀⠀⠀⢸⠀⠀⢠⠂⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⢗⢆⢸⠀⣀⡓⠀⣴⠊⠂⠐⠒⠁⣀⡀⠸⠀⠔⡕⠉⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣒⣉⡈⠛⢨⣏⡠⣌⠖⠁⠀⠀⠀⢸⠋⠴⣁⡠⠃⠊⢔⠩⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⡠⠄⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⢸⠨⡕⢢⠁⠀⠀⡖⢆⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⡖⠘⡄⠀⠀⠀⠀⠀⢀⠌⢠⠌⢲⢦⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁⠀⠈⠒⠤⢤⠤⠔⠉⠀⡘⠀⡸⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⠀⠀⡤⠒⠒⢳⣄⠔⠢⡴⢡⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⡀⢸⠀⢀⠄⢁⠀⠀⠀⡔⠁⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⢢⡍⠁⠀⢸⠀⢠⠊⠀⡀⠀⣠⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⠲⢦⠎⢠⠧⠤⠚⣋⡭⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⠤⡀⠀⠀⠉⠉⠀⠀⠀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣔⠀⠢⣄⠀⠀⢠⠒⢄⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠒⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠈⡓⢤⠼⣶⣠⠇⠀⠀⠈⢺⠤⣀⠀⠀⠀⠀⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢝⠳⣄⠀⠀⠀⠀⠀⠀⠉⡒⡖⠂⠐⠒⠊⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣿⢦⡀⠈⡗⢄⣀⢰⢀⠀⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⣕⢝⣦⠍⠀⠈⡏⠉⠂⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀⠈⠒⠋⠀⠀⠀⣱⣄⣀⣀⣀⣀⣀⣳⣤⡶⢶⡖⠲⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠉⠛⠉⠉⠀⠈⠒⠉⠉⠭⠝⠛⠂⠐⠒⠂⠛⠛⠬⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠂⠈⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⢁⠛⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠉⠀⠕⣪⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀⠐⠆⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠰⠄⠀⠠⠠⠀⠀⠀⠀⠀⠊⠥⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠐⠃⣀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣤⣄⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⢠⣴⣠⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣀⣤⠤⠤⠄
⠀⠀⠀⠀⢠⣶⣿⣏⣿⡜⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⡞⢁⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠈⠿⢋⣫⣬⣷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣾⠟⢋⣉⣉⡛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠛⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠸⣿⣿⣏⡜⣁⠤⠄⡉⠳⣌⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣀⣀⣀⣀⣈⣙⣿⣷⣦⣹⣿⣿⣕⡡⠒⠒⢌⢦⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣶⠿⠿⠛⠛⠛⠛⠙⢙⣿⠟⠛⠛⠛⢿⣿⠿⠿⣶⣾⣦⡱⡀⠑⢄⠙⢿⣦⡀
⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⡶⠆⢻⡆⠀⠀⠈⠙⢿⣿⣢⣌⠳⡌⢻⣷
⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠈⠙⢁⣿⠃⠀⠀⠀⠀⠀⠙⢿⣮⢆⢹⢸⣿
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠶⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡞⢸⡿⣇⣀⡄
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡸⣷⡈⠉
⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣶⣶⣤⡀⠀⠀⠀⢀⣿⣿⣿⣾⣿⠖⡀
⠀⠀⠹⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢿⡄⠁⠀⠀⢰⠟⠛⠻⣿⣿⠟⠋
⠀⠀⠀⠀⠉⠛⠻⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠗⠀⠀⠀⠀⢀⣼⠋⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣀⣀⣀⣀⣤⠶⠛⠁⠀⠀⣴⣿⠀⠀⠀⠸⣷⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⢿⡇⠀⠀⠀⢀⣤⣾⣿⡏⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣶⣶⣾⠟⠋⢁⣾⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⣤⣀⣠⣿⣡⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠛⢿⣥⣤⣴⠟⠁⣠⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⢿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣴⡾⠛⠿⠛⠁⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠶⠤⢤⣤⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠃⠀⠀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠓⠲⠶⠶⠤⠤⣤⣭⣷⣤⣟⣉⣉⣉⣉⣉⣉⣁⣸⣿⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⣏
⠀⢀⣴⠟⢳⡄⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⣸⠇⠀⢸⡇⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⡼⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⣴⠿⠇⣰⣃⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠶⠞⠛⠛⠻⠿⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠷⠶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣍⣉⠉⠉⠉⣿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣈⠉⢿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣴⠇
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠢⢶⡀
⠀⠀⠐⠒⠈⠛⢱⠦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠒⠒⠚⠈⠠⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠏⠀⠀⡀⠘⢇
⠀⠀⡂⡀⠀⠖⠂⠂⠙⢂⠂⠀⠀⡀⠀⢀⢸⡀⠀⠀⠀⠀⠈⢀⠀⠀⠸⠛⠶⠿⠿⠗⢦⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠘⠈⠀⡘⠙⠙⠚⠃⠍⡄⢈⠃⢹⠀⠀⣀⠀⠀⠸⠀⠀⠀⢀⡀⢐⢀⠈⠀⠀⠀⡰⠁⠀⠠⢀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠛⠉⡄⡄⡁⣠⠀⠗⠉⠁⠀⢨⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⢀⠈⠠⠊⠀⠀⠱⡁⠐⠂⡀⡒⠬⠍⠴⠁
⠀⠀⠀⠀⠀⠀⢀⠂⠁⠁⠉⠈⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠇⠀⠀⠀⠀⣄⢀⠀⢀⣔⠀⠀⠀⠰⣤⠤⠁⠦⠠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠤⡀⠄⠙⡀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⣀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⢻⠀⠀⡐⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡵⠉⠁⠀⠈⢿⡇
⠀⠀⠀⠀⠀⢀⠴⠋⢄⠙⠋⠱⣷⡀⠀⠀⠀⡇⠀⠀⡀⠀⢸⠀⠀⠁⢰⠀⣀⣀⢀⣀⡀⡄⡠⠀⠀⠁⠴⠄⠐⠶⠀⡇
⠀⠀⠀⠀⢰⢉⠲⣀⠀⠰⠶⠀⣿⡇⠀⠀⠀⡇⠀⠐⠃⠀⢸⠀⠀⠀⠈⠒⠒⠒⠒⠒⠀⠞⠀⠀⠀⡆⡀⢶⠄⣤⠀⠇
⠀⠀⠀⠀⠈⠁⠊⠉⡄⠀⠀⠀⣿⠃⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠇⠁⠒⠀⠛⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⠀⠀⠀⠀⠑⠒⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⣾⢳⣄⠴⠲⢶⢶⠀⠀⠀⠀⣀⢄⡶⠃⡅⢰⡆⠀⠀⢨⡀⠿⠸⠇⠟⢘⣄⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⢸⠊⠁⠀⡀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠿⣯⠀⢀⣤⠄⠐⢹⠿⠋⠁⠻⡀
⠀⠀⠀⠀⠀⠀⡘⠈⢀⠀⠁⠀⠀⢹⠀⠀⠀⡀⠀⠀⠆⣀⡀⠐⠄⠀⠈⠁⢀⠄⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⢀⣧⠀⠹⠀⠠⡦⠀⠘⡇⠀⠈⠗⠀⠀⢻⣿⡇⠨⠀⠀⠀⢀⠀⠀⠳⠀⠀⠀⠀⠀⠄⠀⠔⠀⡄⠀⠀⢀⠂
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⣀⣀⣀⡀⢉⣀⡀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣦⠀⠀⠈⠀⡀⠀⠈⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠖⠒⠚⠁⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⠁⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣽⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣯⣽⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠛⠐⠂⢀⡔⠉⠉⠐⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠠⠊⠁⠀⢀⡀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⣀⣀⡀⠀⠀⠀⠈⠓⠢⠄⠀⠐⠬⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⣀⣑⡄⠀⡀⠄⠒⠀⠀⠒⠂⠄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠤⡽⠊⠀⠔⠂⠉⠉⠁⠐⠠⢌⠑⢄
⠀⠀⠀⣀⠔⡫⠕⠒⢒⣢⣼⠇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠬⠵⢶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁⢆⢱
⠀⢀⠞⠁⠈⠤⠂⠉⠁⠀⡏⠀⠄⠤⠤⠤⠤⠄⣠⠤⠔⠒⠉⠀⠀⠀⢠⠃⠀⠀⡠⠂⠀⠀⠀⠒⠠⢀⠀⠀⠑⡌⣠
⣰⠃⠀⠀⠀⠀⠀⠀⠄⠀⣣⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⡰⠁⠀⣀⣠⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⣇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠘⠦⣄⣀⣀⣀⣀⣀⣀⡠⠴⡾⠷⠚⣿⠁⣠⠞⠛⠛⠛⠻⣿⣦⠀⠀⠀⢸⢀⠔⠋
⠈⠳⣄⡀⠀⠀⠀⠀⠈⠒⡤⢄⡀⣀⣈⣉⣉⣉⠀⢀⣀⣤⣴⣾⡿⠋⢁⣴⡶⠂⠄⠀⠈⠻⢀⣆⠤⠛⠁
⠀⠀⠀⠙⠲⠤⣄⣂⠀⠔⠀⠩⠭⠁⠀⠐⠶⡒⢚⣛⣭⣷⠟⠋⣀⣵⣿⣿⣿⣦⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣟⣶⣶⣷⣿⣷⣿⣿⡿⠋⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⠿⠟⠁⠀⣴⡆⣶⡄⠀⠙⠿⢿⣿⣿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢱⡇⠀⠀⠀⠀⡆⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣿⣇⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠟⢷⣤⡀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⣰⡶⠻⠀⡜⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡿⠦⣴⣤⣴⣧⣼⡧⢼⠗⠚⢏⢁⠅⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣰⣥⣄⠀⢀⢀⠉⠀⢨⠀⠀⡠⠈⢀⣴⣿⡿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣯⣿⣿⣿⣿⣧⣭⣽⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠂⢠⣀⡄⣀⠒⡄⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⡀⠠⡀
⠀⠀⠀⠀⠀⠦⠈⠋⠁⠆⠺⠀⠀⢰⣇⣘⣋⣃⣘⣀⣷⠀⠀⢹⠎⠈⠁⠀⠈⢿⡄⠀⠀⢸⣹⠆⠊⠉⠹⡇⠀⠀⠀⢀⠖⡩⠤⠌⠩⡙⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿⠉⠉⠉⠉⠙⠿⠀⠀⠀⠠⠤⠀⠰⠤⠘⡇⠀⠀⢸⠁⠀⠀⠄⠀⠱⡀⠀⠀⡄⠁⠠⠄⡀⢸⠁⡚⠂
⠀⠀⠀⠀⢠⡀⠀⠈⣥⣤⡀⢉⠆⠀⠀⠀⠂⢀⠐⠀⢐⠀⠀⣦⠀⢠⣤⠀⡀⠀⡇⠀⠀⣜⠀⢴⠀⢀⡀⠀⣇⠀⠸⠀⠀⠀⠀⠁⡸
⠀⠀⠀⠀⠀⡅⠀⠀⠻⠿⠀⠁⠀⠀⠀⡀⠀⠘⠀⠀⢀⠀⠀⣿⠘⠠⡇⠘⠟⠀⢁⠀⠀⡏⠁⠈⠀⠈⠁⠀⣈⠀⠀⠀⠀⠁⠀⡘
⠀⠀⠀⠀⠀⢦⣶⣦⣤⣴⣶⠀⠀⠀⢠⣧⣤⣤⣤⣤⡼⠆⠀⠢⠁⣀⣀⡀⠠⠤⠜⠀⠀⠓⠦⠤⠤⠤⠤⠖⠁⠀⠐⠉⠉⠒⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠓⡀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⢀⠀⣴⢆⠀⠀⠀⠀⢾⣿⣿⡿⠿⢿⠆⠀⠀⠀⡀⠀⡠⢴⣶⡖⠊⢷
⠀⠀⠀⠀⡠⠆⠢⠆⠀⢠⠀⠇⡀⠀⠟⠛⠿⠿⠟⠛⡷⠄⠀⢀⣿⣷⣿⣸⣷⣤⡀⠀⡸⠰⠤⠀⠒⢸⠀⠀⠀⠨⠟⠁⠀⠈⠁⠀⠀⠈⠆
⠀⠀⠀⠀⡀⠰⠀⡀⠀⡀⢠⠀⡄⠀⠘⡈⠀⠀⠀⣠⠵⠃⠀⢺⣾⣿⣿⣿⣿⣿⠃⠀⢄⡀⣳⣤⡍⡼⠀⠀⠀⠀⠀⢀⡠⠀⡀⠀⠊⠀⠄
⠀⠀⠀⠀⠀⠐⠤⡄⠉⠀⠘⠉⠀⠐⢏⣙⣢⡦⡒⣉⡱⠀⠀⠀⠻⣿⡟⠿⣿⠟⠀⠀⠀⡇⠀⠉⠀⡆⠀⠀⠀⠀⠘⢄⠀⠀⠑⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠸⡥⠛⠒⠋⠀⠀⠀⠀⠙⠂⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢸⠉⠀⠀⠉⠙⡇⠀⠀⣠⣤⣠⣴⣶⣦⠀⠀⢀⣄⠀⣤⣤⣴⠀⠀⠀⠀⢹⡉⣈⣉⣉⣹⡄⠀⢸⣿⣿⣶⣤⡿⢺⡿⠾⡷
⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠈⡇⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠈⠩⡙⠛⠛⠋⡉⢇⠀⡴⢍⣇⣒⠠⠤⠼⢺⡆⠸⣿⣿⣿⣿⣿⣷⣸⢾⡇
⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⢰⠁⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡄⢀⡁⣄⠀⡘⠀⡇⠀⢩⠐⠀⠀⠉⢸⡇⠀⢿⣿⣿⣿⠿⣤⣥⣨⠇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠈⠒⠀⠤⠀⠠⠔⠁⠀⠲⢄⣸⠀⠐⢠⣊⣸⠇⠀⠀⠻⣿⠏⠀⠀⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠴⠀⠀⠁⠈⠀⣀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⡀⢠⠄⠀⠀⠀⢸⡟⣿⣷⣾⠿⠛⠛⢶⣶⠶⣛⣓⣶⡆
⠀⠀⠀⠀⠀⡄⠀⠀⠂⠀⠀⠀⢀⠀⠴⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⡁⠠⠿⠛⡟⢡⡏⠀⠀⠀⠀⢸⡟⢉⣀⣙⡿⢰⡄⠀⠘⣷⣽⣡⢿⡇
⠀⠀⠀⠂⣠⡄⠀⠀⡢⠀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠔⠃⠀⢀⡞⢁⣾⢀⠀⠀⠀⠀⠸⣤⣤⣙⢯⢁⡎⢻⡀⠙⣽⣶⣶⣟⡇
⠀⠀⠀⠀⠀⠃⠀⠀⢁⣀⣄⣀⣀⠈⣀⠐⠈⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⢠⠤⠤⢤⢎⡀⠼⠧⠼⠇⠀⠀⠀⢰⣿⠘⣿⡇⣼⣁⡾⣧⠈⢻⣟⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⡼⠁⠀⠀⠰⠄⠀⠀⡀⠀⢸⡷⠀⠀⠀⠀⠀⠀⢸⠟⢦⡟⢀⣛⣛⡚⢟⣀⠀⢻⡼⣥⡇
⠀⠀⠀⡀⡄⠀⠀⠂⠀⠀⠀⠈⠀⠀⡉⠚⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠞⠁⠀⣿⠃⠀⠀⠀⠀⠀⠀⢸⡇⣸⢁⡿⣟⣟⣰⠻⣻⡄⠈⣿⡀⡇
⠀⠀⠀⠠⢤⡆⢀⠀⢀⡁⣈⣡⣄⡩⠀⠋⢁⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠔⠉⠀⠀⠀⣿⣤⠞⠀⠀⠀⠀⠀⢸⣷⡃⢾⣧⣻⣸⣞⣋⣡⣽⠀⠘⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠰⢦⡄⡀⠀⠛⠃⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⢰⣼⣶⠾⠛⠳⢛⣋⣛⣛⠾⠿⣷⠾⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⢲⡀⠈⣋⣥⣾⣿⣿⣿⣷⣄⠀⠆⡄⠀⠀⠀⠀⠈⢋⡷⠂⢠⣧⡄⣿⡛⠛⣷⡄⢿⣼⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠑⠀⠙⠿⠿⠟⢋⠅⢄⠉⠻⣿⡆⠈⠁⠀⠀⠀⠀⢀⡾⠀⠀⡾⣻⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢆⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣄⠾⡁⠀⠘⣿⠄⠀⠀⠀⠀⠀⢸⡇⠀⠰⣿⠟⣩⠄⢆⣘⣿⢻⡟⢿⡇⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⢈⠀⠀⢀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠙⡳⡻⣿⠰⡟⠀⠀⣿⠀⠀⠀⠀⠀⠠⢟⣷⠀⠀⣯⠼⡁⢀⠄⠈⣰⡼⢻⣼⡇⠀⠀⠀⠸⢀⠄⠀⠀⢀⠠⠄⠐⠓⠒
⠀⠀⠀⠀⠀⢀⠤⣬⣅⣱⢇⠚⠁⠀⢠⠏⠐⠀⠀⠀⠀⠈⢻⣿⠀⣄⠹⣦⣁⡘⢂⣿⡿⠁⣟⠉⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠁⢦⡈⠛⠿⣿⣶⣦⠶⠋⡴⠀⠀⠀⠀⠀⠀⢨⡤⠑⠈⢳⡮⣭⣛⣉⣥⣴⣾⡿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡀⢀⠤⠀⠀⠀⡀⠀⠀⢤⡴⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠈⠉⢛⠐⠆⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⣌⠙⠒⠛⠓⢒⣶⡒⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡞⠃⠀⠀⠃⠀⠀⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠐⠄⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠⡔⠀⠀⠰⢩⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠠⡀⡀⢀⠀⠜⠁
⠀⠀⠀⠀⠰⠀⠀⠠⠤⠄⠀⠀⠠⠀⠁⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⠛⠀⡂⠀⢈⡈⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⠁⠀⣰⠀⠋⢐⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠰⠆⠀⢠⡈⢀⣀⢠⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠈⠉⠂⡀⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⡈⢁⠈⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠊⠀⠢⠜⠃⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠄⠀⠒⠁⡖⠁⠁⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠆⠁⠀⠀⢐⠀⠘⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠀⠀⠀⠼⡀⠈⢀
⠀⠀⠀⠀⠀⠀⠠⠀⢀⠐⠂⠀⣤⡀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠀⠖⠀⠀⡀⠀⢀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠒⢾⡟⠂⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⣠⠞⠙⢶
⠀⠀⠀⠀⠀⠀⠀⠼⠻⣿⡷⣿⡇⠀⠀⠀⢹⣏⣽⣶⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣀⣤⠄⣀⣤⣴⣤⣿⣧⡎⠀⠀⠀⠀⠉⢉⣿⣠⣄⡀⢠⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠰⣁⠀⠀⣾⠁⠀⠀⠀⡖⠒⠒⠒⠂⢐⠄⠀⠀⠐⠀⢸
⢠⣾⡿⣿⠦⢿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⣿⡿⢟⣿⢻⣧⠀⠀⢘⠀⣦⡀⠀⠀⣠⣿⡇⠀⠘⢷⡀⢀⠼⠁⠀⠀⠠⠖⠉⠀⠀⠀⢠⢰⣏⡆
⠸⠿⣧⠽⣷⠂⠉⠛⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠈⡀⠘⢿⠇⠀⠙⡛⠁⠀⠀⠘⠉⠁⠀⠀⠀⢀⣀⣹⠀⠀⠀⠀⠈⢁⢈⣰
⠀⠀⠀⠀⠈⠀⠀⠀⣀⡤⠀⣤⢣⣴⠾⢂⠀⢀⡀⠀⠀⠀⠀⢀⠀⣀⣀⣴⠇⠀⠀⠀⢤⣴⠇⠀⠀⠀⠀⠀⢠⣞⣁⣸⠉⠉⠀⠀⠀⠀⠀⠀⠸⠹⣿
⠀⠀⠀⠀⢠⠆⠁⡠⠘⠛⠻⠛⠂⠉⠉⠉⠂⡀⠈⢦⡀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠉⠉⠀⠴⢦⡄⠀⠀⠀⠀⠉⠀⠻
⠀⠀⠀⢠⠇⠀⡼⢁⣄⠀⠁⠀⠀⠀⠀⠀⣤⠈⡄⠀⢣⠀⠙⠃⠀⢀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠒⠊⠀⠐⠁⠀⠀⠀⠀⠀⠀⢭⡎⠁
⠀⠀⠀⢸⠀⢰⡇⠈⠹⣷⡄⢀⠀⠀⣤⣾⡇⠀⢹⠀⢸⡆⠀⠀⠓⠾⠣⣄⣼⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠰⠀⢧⡀
⠀⠀⠀⢸⡄⢸⠀⠀⣄⠋⠀⠾⠷⠀⠙⠋⠀⠀⢘⠀⣸⠃⠀⠀⠀⠀⠀⠈⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠳⠒⠃⠀⢠⣄⡀⠀⠀⠙⠃⠶⠒⠋⠁
⠀⠀⠀⠀⠹⢿⣧⡀⢈⠶⢦⣠⣦⡴⢶⡎⠀⢀⣼⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⣠⣄⠀⠀⣀⡔⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⢄⣁⣀⡉⠴⠦⠤⠚⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠍⠈⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⢀⣶⣢⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢠⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣼⣿⠆⠈⠛⠻⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠁⠀⢤⡀⠀⠀⠀⠀⠀⢠⣦⣾⣿⡿⣿⣿⣏⠀⠀⠀⢰⣿⡿⠋⢽⣿⡏
⠀⢸⠁⠀⠀⠀⢈⣿⡆⢸⣿⠆⢠⡆⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠉⠙⣿⣟⡯⣿⣿⡔⠠⠀⣼⣿⠀⠀⢸⣿⣡⡀
⠀⢿⠀⠀⢀⠂⠸⣿⢅⣌⠻⠆⠈⢁⠀⠀⠀⢀⡼⡇⠀⠀⠀⠀⠀⠀⠐⢸⡇⠀⠀⠀⠀⣠⣤⣷⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⠁
⠀⠘⢇⣀⣋⣤⢀⣀⢈⣉⡀⠤⠂⠈⠀⠀⠀⢻⡆⠁⠀⠀⠀⠀⠀⠀⠀⢺⣽⣦⠀⠀⠀⠈⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣼⣿⣷⣄⣻⡆⠄
⠀⠀⠀⠉⠉⠉⠐⠖⣖⡶⣗⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡞⠀⠀⢠⣄⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⡿⢿⠟⠻⡀⠒⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⡀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠀⠀⠀⠘⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⡿⠁⢹⡤⠀⣤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠦⢤⡀⠀⠀⠀⠛⠂⠐⠒⠲⠄⣀⢀⣀⣼⣾⡧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⣹⣿⣟⣋⣉⢻⣿⡇⣿⣿⣧⣄⢻⣄⣀⡟⢰⡎
⠀⠀⠀⠀⣠⠶⣄⢀⡴⠁⠀⠀⠀⠀⠑⠀⡀⠀⠈⠀⠂⠀⠀⠀⠂⢉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣼⡿⠓⠰⠒⢒⢸⣿⣿⣿⡟⢻⣾⡏⠀⣴⠯⠤
⠀⠀⠀⢰⣣⣄⠈⣉⣤⣤⣀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⣿⣿⡇⠀⠈⠀⠀⠘⣿⣿⣿⣧⡌⠋⠙⢏⡀⠀⠒
⢀⡼⢶⡞⠁⠸⠷⠋⠀⣼⣿⣶⢤⡖⢀⡄⣤⢁⠑⢀⣀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣶⣿⣿⣿⣶⣿⣿⣯⣝⠛⠿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⣧⡀⢀⣵⣤⢲⠬⠘⠛⠊⠭⣥⣴⣒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢃⣴⣾⣿⣿⣇⣀
⠀⠘⠶⢾⣿⣏⡁⠈⠀⢀⣥⠉⠉⠉⠀⣠⣄⣀⠤⠀⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⣀⡀⠉⡉⠙⠛⣿⣿⣷⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⠀⠀⠀⠀⢀⣤⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⣽⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⠉⠉⠛⠛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠿⠀⠈⣿⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡌⠁
⢀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⣤⣴⡾⣿⣷⣂⡤
⠀⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠃⠀⠉⠀⠀⣀⣿⣿⣿⣷⣄
⠀⠀⠙⣿⣿⣧⠀⠀⠀⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢻⣿⣿⣷⣄⢀⣽⣤⣠⣾⣿⣿⣿⣿⣿⣿⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⠀⠸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠟⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡙⢿⣿⣾⣿⣿⣿⣷⣿⡿⢿⣿⣿⣿⡏⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡉⢻⣿⡏⠉⠀⢀⣀⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣧⡀
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣥⣄⣻⣷⣄
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠰⠇⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢚⠀⠀⢀⣼⡿⣹⣁⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⡟⢋⣴⡿⢋⣿⠁
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠤⠚⢷⢠⣾⣧⡌⢹⣦⣀⡽⢿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠦⡀⠀⠺⠛⣿⣿⣷⣶⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣟⠻⣿⣿⣾⡄
⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠠⢀⣀⡀⠠⠚⠻⣩⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠈⠉⠙⡶⢪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣷⡟⠻⣿⣿⠿⠋⠁⠿⣻⡇⠘⣋⣀⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠓⣦⡙⠿⢀⡀⠀⢠⣿⣧⣴⡭⠋⢠⠟⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⢿⣿⣿⡇⠀⢹⣿⣶⣄⣁⣴⠄⣹⣅⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣾⡹⣿⣧⠀⣾⠋⢸⠉⠻⣿⡟⠻⠛⢿⣿⠉⠀⠙⢿⣆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⣷⡏⠀⠀⠀⠈⣿⡇⢠⡄⠈⣿⡄⢣⣤⠛⣿⣆⣉⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⣿⣿⣿⣿⣿⣿⠻⣷⣤⣀⣀⣿⠃⠾⠃⠀⣸⣇⣨⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⡏⠀⢯⡻⣿⣿⣿⣷⡶⠿⢷⣿⣿⠛⠋⠉⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣏⣿⣿⣿⣿⡀⠀⢊⣠⣿⣿⣿⡿⠀⠀⠀⢿⣿⠀⠀⢁⡈⣿⡟⡇⠐⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣘⡁⠙⣿⠏⠁⣄⢰⡆⠸⣏⢸⡆⠈⣳⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡿⠀⠙⢿⣿⣶⣾⣧⣤⣭⣿⣧⣴⣿⣿⣿⢿⣿⣿⠛⡿⢿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡃⠀⠘⠁⠘⠏⠈⣿⡟⡿⠛⠛⢻⣿⡇⠠⠀⢻⣿⡇⠀⣠⠻⣿⣈⣛⣵⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣴⠀⢠⣿⠃⣿⠀⠀⠈⣿⣇⢠⢴⠏⢻⣿⣄⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠘⢻⣿⣦⣄⣈⣿⠀⣿⡇⠀⣀⣹⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠃⠛⠒
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠋⠛⠛⣿⣿⢿⣿⠿⣿⣿⡿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⢿⣿⣶⣿⣿⣿⣿⣆⠘⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠳⢤⡀⠀⣼⣿⢰⣿⣆⢈⢹⣧⠸⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣺⠃⠸⣿⠁⠺⣿⡏⢸⡤⠃⠀⢁⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⢿⣿⣿⠺⠅
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣶⣿⣧⣴⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣥⠼⣿⡞⢿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢈⡈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠈⠶⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⡿⢋⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠠⠤⠴⠮⠉⠀⠀⣈⠟⢛⣛⣿⣿⣿⣿⣿⠀⠀⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣥⡤⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠉⣈⣿⣿⣿⣾⣿⣿⣿⣿⠟⠛⠛⠷⠈⠉⠉⠛⢋⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠿⠋⠙⠉⢉⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡤⠤⢀⣤⠤⠾⠷⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⣀⠄⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣠⢖⠩⠤⠀⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⠞⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⡏⠀⣶⠋⠉⠙⢦⡀⢀⡴⠂⠀⠲⣆⡈⢷⡄⠀⠀⠀⠀⠠⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁⠀⣿⠀⠐⠗⢸⠇⢸⡀⠀⠺⠀⡿⠁⠈⣷⠀⠀⠀⠀⡀⠂⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡀⠀⢻⠀⠀⠈⠓⠒⠒⡩⠀⠀⠓⠦⠤⠞⠁⠀⠀⢸⡇⠀⠀⠀⢠⠘⠀⠸⡇
⠀⠀⡟⢦⣀⠔⠆⠀⠀⠀⠈⣇⠀⠀⠘⣇⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠘⠀⡀⠀⣷
⠀⡠⠇⠀⠀⢸⠀⠀⠀⠀⣠⠃⠀⠀⠀⠹⣦⠀⠀⠀⢦⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⠀⡇⠀⣿
⠺⠤⢤⠀⣠⡤⣧⠀⠀⢰⠁⡐⠁⠀⠀⠀⠈⠳⣦⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⠘⠀⠀⠀⢸⡆
⠀⠀⠈⡶⢳⠀⠀⠀⠀⣸⠀⠁⠰⠀⠀⠀⠀⠀⠀⣙⠷⠶⣤⣤⣤⣤⣤⣶⡶⠛⠋⡿⠀⠀⠀⠀⠀⠀⡆⠀⠈⣷⡀
⠀⠀⠀⠀⠀⡎⡀⠀⢠⠟⠀⢸⠀⠂⠀⠀⣀⣤⠞⢁⠴⠚⠉⣤⡼⠙⡷⣄⠻⣤⡈⠀⠀⠀⠀⠀⠘⡀⠐⢄⠀⣈⣿
⠀⠀⠀⠀⠀⠸⢠⠀⠙⠲⠤⣦⣤⡦⠶⠛⢉⠤⢂⡏⡀⣠⠔⠉⠳⣄⣧⠈⢣⡈⠙⠳⠶⢤⣤⣤⣤⣴⡶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⢃⡤⢄⠀⠀⠀⠀⠀⢀⡠⢚⣄⣼⠀⠉⠀⠀⠀⠀⠈⠉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣈⡠⠆⢁⠀⠐⢀⡉⠤⠒⠉⢉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⢀⠔⡢
⠀⠀⠀⠀⠀⠸⡀⠐⠂⠘⠁⠉⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠣⣄⠀⠀⠀⢣⠈⠑⡦⡌⡀
⠀⠀⠀⠀⠀⠀⠙⠦⣤⠎⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠓⢬⣉⣛⣋⣥⠖⠳⣄⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠙⠛⠳⠶⠶⠶⠶⠶⠶⠚⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⠇⠀⠀⠀⡍⢨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢺⡇⠀⠀⠀⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠏⣈⣇⣀⣀⠴⠃⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⠧⣀⣰⠋⠀⠀⠱⠤⠤⠄⠒⢋⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠉⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠀⣰⠆⢠⡀⠀⠀⣠⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⡾⠁⠀⠀⠀⠁⠀⠉⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠐⠂⠀⠀⢃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢀⣴⠀⣠⠀⠀⠈⣦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⠚⢃⣠⠴⠛⡛⠀⠙⠦⣀⠀⠸⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣈⡷⠈⢥⠠⣾⡇⠀⣷⣤⠄⠙⠛⠋⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣻⠁⠀⠉⠀⠴⠆⠠⡄⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠⠏⠀⠀⠀⠀⣶⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⠏⢀⡴⠁⠀⢰⠏⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠐⠿⠖⠋⠀⣠⠔⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡏⢲⡖⠠⢤⣤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠿⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠡⠚⡅⠀⠰⡇⠀⠈⠉⠉⠀⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠉⠀⠀⠀⡇⠀⠀⡿⠣⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⠩⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠰⢇⡀⠀⠇⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⡟⣧
⠀⠀⠀⠀⠀⢀⡞⡀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⢦⡜⠀⠀⠀⠀⠀⠀⢸⡇⠈⣷
⠀⠀⠀⠀⢠⠟⢀⠇⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢈⠷⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧
⠀⠀⠀⢀⠎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡋⢢⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⣴⣁⠀⠀⠀⠀⠀⠀⢸⡿⡄⠀⠘⣆
⠀⠀⢀⡟⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠇⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡿⠀⢀⡏⠀⡜⠀⠀⠀⠀⠀⠀⢸⠇⢳⡀⠀⢻
⠀⠀⡞⠀⢀⠾⣽⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣠⠋⢀⡟⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⠳⣤⡌⢡⠀⡇⠀⠀⠀⠀⠀⠀⣾⠀⢸⡇⠀⠸
⠀⣼⠁⠀⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠘⠓⠉⠁⠀⠼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠀⠀⠀⠉⠉⠀⢱⠦⠤⣤⣤⣤⣤⣿⡴⠟
⢰⡇⠀⠀⠹⢦⣷⣤⣤⣤⣤⣤⣤⡤⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⣼⣁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡾
⢿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣨⣷⣄⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⣠⡞⠋⠉⠙⠛⠛⠛⠛⠛⣿⠛⠛⠉⠁
⠀⠉⠛⠛⠛⢻⡟⠛⠛⠉⠉⠉⠉⠉⠀⠀⠙⠓⠶⠶⠴⣶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⡏⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣠⣤⣤⣤⡴⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣩⠟⠋⠉⢉⡦⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⣄⠀⠙⢫⣉⠉
⠀⠀⢠⣤⡤⠤⠤⠤⠔⠒⠒⠈⠉⠀⢀⣀⡴⠋⠛⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠴⠟⠳⣤⣀⠈⠓⠢⠤⣀⣀⣀⣀⣀⣀⡀⣀
⠀⠀⠀⠉⠛⠒⠒⠲⠶⠶⠶⠖⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⣄⣀⣉⣉⣉⣩⠤⠟⠋
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡜⢳⡆
⠀⠀⠀⢠⠆⠠⣤⣤⡄⢀⡴⠖⠓⠲⣶⣤⠐⠀⠀⠀⠀⠀⠀⠀⢰⣒⣦⠤⣥⣬⣡⣤⣥⠀⢀⣒⣶⠀⠀⠀⠀⢀⠘⢀⣈⣳⣿⡞⣉⢀⣀⠀⣀⡉⠇⢰
⠀⠀⠀⠈⠁⠀⠈⣿⣦⡟⠁⠀⡀⠀⢸⣿⡍⠀⠘⠀⠀⠀⠀⠀⢸⠋⣀⣀⢈⣿⣿⣻⠉⠀⠀⠈⣿⠀⠀⠀⠀⠚⠢⠞⣿⣿⡏⣿⠋⠉⠉⠉⢙⡣⢐⡩⠄
⠀⠀⠀⠀⠇⠀⠀⣻⡿⠀⠆⠀⣀⣀⣸⡿⠁⠀⢀⡀⠀⠀⠀⠀⠈⠀⠨⠿⢁⣽⣿⣿⠀⢶⡄⠀⠈⠀⠀⠀⠀⠂⠀⠀⣿⣟⣣⡘⢦⣤⣶⣿⠿⠟⣁
⠀⠀⠀⠀⠀⣰⠆⣾⡇⡀⢰⡉⠀⣩⠟⢿⣷⣥⡀⠈⠀⠀⠀⠀⠀⣀⣠⠀⠈⢿⣽⣷⠀⠈⠁⡶⠃⠀⠀⠀⠀⠘⢳⣅⣿⣿⣤⣽⡄⢹⠿⡿⣧⠾⣯⣤⡄
⠀⠀⠀⠀⠈⠁⠀⣿⡇⠀⠀⠉⠁⢀⣁⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠓⠺⡀⠃⡀⣿⣶⡠⢀⣾⠟⠁⠀⠀⠀⠀⣀⠀⢀⣿⡏⠀⣴⣴⡶⠲⣄⠈⢀⣷⣻⡆
⠀⠀⠀⠀⠄⠀⢰⣿⣿⡀⠁⠀⠀⠀⣻⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⡦⠶⣿⡏⢨⣿⣟⠰⡀⠈⠠⣨⣋⣿⠇
⠀⠀⢀⠄⠀⠀⣸⣿⠿⣷⡀⠀⢠⡴⠿⠋⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⣄⣿⣿⣦⣬⣿⣧⣴⣶⣬⣾⠹⠁
⠀⠀⠘⠦⠄⠒⠻⠿⠇⠀⠙⠶⠤⠤⠤⠖⠋⡡⠠⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢦⣿⣽⣦⣄⠀⠀⠀⠀⠀⠀⠀⠓⠛⡄⠰⠴⠄⣠⡮⠶⠖⠶⠖⠋⣡⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⢀⠀⢤⣄⡀⠀⡀⠀⠀⢄⠀⣀⡤⠶⠤⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣀⠀⣠⠄⣀
⠀⠀⠀⠀⠐⣚⣿⣿⢁⣾⣇⣀⠀⠈⣾⠇⢸⡟⢁⠀⠀⠀⠀⠼⣠⣤⣤⠐⢃⡟⣿⣧⢦⣉⢋⢀⡠⠀⠄⠀⠀⠀⠀⢈⣬⡄⢀⣠⣥⣧⣄⣼⣀⣀⡖
⠀⠀⠀⠀⡌⡁⠼⢿⡎⠻⢿⡇⠠⡼⣏⢀⡞⠐⠋⡄⠀⠀⠀⣼⣿⡉⢉⡁⣼⢁⠸⣟⣛⢁⠠⠌⡅⠃⠀⠀⠀⠀⢰⠃⣦⡽⣽⣿⠉⠀⠹⣄⡉⢻⡇⢤⡀
⠀⠀⠀⠀⠡⠁⠂⠹⣿⡌⢾⡇⠀⣳⢭⡞⠐⠒⢦⡀⠀⠀⠀⠁⣁⢛⠣⢱⠇⢼⠇⢹⣿⢂⠠⠐⠀⠈⠁⠀⠀⢀⠈⢃⡀⠈⠿⠿⣷⣄⠀⠸⠰⡼⡧⣉
⠀⠀⠀⠀⠀⣼⣶⡦⠹⣿⡌⢻⣤⢦⡞⠀⡀⠘⠐⡇⠀⠀⠀⠆⢿⡆⢄⡟⠀⠋⣠⠄⢻⣶⡀⠲⠂⠌⠀⠀⠀⠀⣷⣦⠈⠉⠉⡜⠹⠿⣿⣻⢦⣄⠉⢁⡀
⠀⠀⠀⠀⢀⡴⠟⢿⣦⡹⣿⡌⢣⡞⠀⣧⠘⠒⠲⠁⠀⠀⠀⣤⣌⠃⣼⠷⠒⠀⢠⡐⠺⣿⣷⠈⢁⠋⠀⠀⠀⠀⠉⣹⠔⠒⠲⡾⠒⢡⡾⠉⠻⣿⣷⡄
⠀⠀⠀⠀⢾⠃⢰⡖⡟⢻⣿⣿⡞⢀⠑⣾⠀⠠⣥⡀⠀⠀⠀⣸⡟⣸⠇⡐⢄⢰⡏⣄⡀⠸⣿⣧⠀⣿⠀⠀⠀⠀⡜⠀⠀⢹⠀⡏⠀⡜⠀⡀⢑⣼⣿⠃
⠀⠀⠀⠀⢨⠃⠤⣀⡠⠴⣷⢟⣀⠁⣾⣿⣶⠀⠈⠇⠀⠀⠀⠘⢡⡟⠀⢧⠈⡌⢃⣨⣴⠧⢹⣿⣆⠘⠀⠀⠀⠀⢇⠀⠀⢸⡿⠟⢛⡷⠶⣾⡿⠟⠃⠘⠁
⠀⠀⠀⠀⠈⠑⠒⠋⠀⠘⠁⠉⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠢⠘⠁⠀⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠃⠀⠀⠀⢸⠶⠒⠲⠤⠤⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠛⠁⠀⠀⢀⡀⠀⠀⠀⠈⣀⢀⠀⠀⠄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⢀⡠⠀⠂⠀⠀⠀⠀⠈⠑⠒⠓⠚⠖⠉⠀⠀⠀⠀⠀⠀⢄⠀⠈⠓⠲⠦⣤⡀
⠀⠀⢀⡴⠚⠉⠉⠀⠀⡔⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢦
⠀⡴⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡤⠊⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⢠⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠂⠐⠒⡄⢆⠀⠀⠀⠀⠀⠀⠀⣠⠁⡀⠠⠀⠑⠊⠁⠀⠀⡠⣿⠀⠀⠀⠰⡅
⠸⠀⠀⠀⠘⣷⣄⠀⠀⠀⠈⠉⠉⠁⠀⠑⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠄⠊⣠⠏⠀⠀⠀⠀⠙
⡸⠀⠀⠀⠀⠈⢆⠉⡒⠤⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡀⠀⠈⠀⡠⢾⠃
⡇⠀⠀⠀⠀⠀⠀⠑⢌⡢⣄⠑⣤⠊⠀⠈⠉⠉⠥⢀⠉⠉⠉⢠⠓⠂⠀⠀⠈⠓⠔⣨⠔⠁⠀⠀⡄
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠩⠔⢣⠤⢄⣀⣠⣤⣌⣀⣀⣠⣤⡶⠷⠶⠎⠑⠒⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⡞
⠈⢧⠀⠀⠀⠂⠀⠀⢠⠒⠒⠶⢠⡄⠄⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠒⠆⠠⠰⠚⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠑⠦⣄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠠⠀⠀⠀⠀⣀⠼⠁
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⢀⡀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠷⠲⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠛⢦⡀⠀⠀⢈⣙⠥⠤⠤⠭⠤⠤⠩⠥⠬⣋⡇⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠁⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠰⠀⣀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⣤⣤⣤⡤⣤⣤⣤⢤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠈⠉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠢⢼⣀⣀⣀⣀⡤⠤⠀⡼⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⡠⠼⣄⣠⣤⡤⠤⠤⠤⠗

⢶⣞⠓⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠚⣱
⢰⠈⠳⣄⠀⠀⠈⢳⠀⠀⠀⠀⠀⢀⡠⠀⠀⠈⠉⠉⠁⣰⡄⠉⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢠⠞⢁
⠀⢣⡀⠀⠐⠦⢄⣀⣇⠀⣠⣀⠀⠈⠀⠀⠀⠀⠀⠀⡠⢣⡘⢄⠀⠀⠀⠀⠀⠀⠑⠄⠀⠄⠀⣸⠀⡤⠤⠂⠀⢀⡜
⠀⠀⠑⢄⠀⠀⠺⡀⣽⡯⠂⠁⠀⠀⠀⠀⢁⠀⢀⠔⠁⢸⠁⠈⠢⡀⠀⠈⠀⠄⠀⠀⠀⠐⠽⣇⣠⠞⠁⠀⡠⠊
⠀⠀⠀⠀⠑⠒⠶⠋⠻⡦⣀⡀⠀⠀⠀⠀⠘⡠⠊⠀⠀⣘⡧⠂⠀⠘⣄⠃⠀⠀⠀⠀⠀⣠⢴⠎⠙⠲⠒⠋
⠀⠀⠀⠀⠀⠺⠀⠀⠀⢱⠀⠉⠁⠀⠀⠀⢠⣃⡠⠤⢒⡞⠳⡒⠠⢄⣸⡀⠠⠀⠂⠂⠉⢀⠎⠀⠀⠀⠷⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠇⠀⠰⣆⡤⠔⠉⠀⠀⢀⠎⠀⠀⠑⡄⠀⠀⠉⠲⢤⣰⡆⠀⡘⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠸⠀⡠⠊⢴⠲⢤⡀⠀⣨⠠⢤⡤⠄⣰⠀⢀⡤⠖⡆⠙⢄⠀⠇⠀⠀⢀⡠⠀⠀⠈
⠀⠀⠀⡄⠀⠀⠀⠑⡦⢄⣀⣴⡁⠀⠸⡄⠈⠻⠏⠀⢠⠇⠠⡀⠀⠙⠏⠀⢠⠇⠀⠈⣾⣀⡀⢴⡉⠀⠀⠀⠀⢁
⠀⢀⢩⠀⠀⠀⡠⠊⠀⠀⢠⠇⠈⠉⠐⣳⠀⡠⠴⠶⢅⠀⠀⡨⠶⠦⠄⠀⠾⠒⠈⠉⠙⡄⢀⠂⠙⢄⠀⠀⠀⡛⡀
⠀⠌⢸⣀⡤⠊⠀⠀⢀⣰⣾⣀⡀⠀⡰⠃⣀⡤⠒⢒⣄⡆⢰⡤⡒⠒⠤⣀⠘⢄⡀⢀⣀⣷⣧⡀⠐⠀⠑⠤⣀⡇⠡
⠀⢂⢸⠙⠲⢌⠁⠀⠉⢻⢿⠀⠀⠈⠑⡉⠀⠑⠠⠬⠼⢱⡈⡇⠥⠄⠊⠀⢉⡎⠁⠀⠀⡿⡏⠁⠀⠈⡡⠔⠉⡇⡰
⠀⠈⡼⠀⠀⠀⠑⠄⠀⠂⠘⣆⡠⠒⠉⢱⡀⠀⢀⡠⠂⠻⠗⠘⠄⡀⠀⠀⡜⠉⠒⢄⣰⠃⠈⠀⢀⠊⠀⠀⠀⢧⠁
⠀⠀⢀⠂⠀⠀⠀⢀⠖⠀⠀⡺⠄⠀⢠⠁⠙⠫⢅⠀⡄⢾⠷⢀⠀⣠⠝⠋⠈⡄⠀⢠⢿⠈⠐⠲⣁⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⢠⠁⠈⢦⡠⠒⠉⠁⠀⢉⠲⣄⡠⠞⠉⠀⠈⠉⠐⢄⣴⡁⠀⡆⠀⠀⠀⠑⠀⢀⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡌⠀⠠⠌⠛⠢⢄⡀⠀⠈⢆⠀⠀⡠⠁⠀⢀⣠⠔⠛⠩⠧⠀⢰⠀⠀⠀⠀⢀⠂⠁
⠀⠀⠀⠀⠀⢰⠄⠀⠀⡰⠁⢀⣀⠤⠀⠀⠘⡌⠉⠒⠢⢷⡼⠔⠒⠋⢁⠋⠀⠂⠤⢀⡀⠀⢇⠀⠀⠀⡤
⠀⠀⠀⠀⣠⠴⠤⣄⣴⠗⠊⠁⠀⠀⠀⠀⢀⠙⢄⠀⢀⠊⡑⢄⠀⢀⠚⡀⠀⠀⠀⠀⠈⠑⠾⣦⣀⠤⠧⣀
⠀⠀⣠⠞⠁⢀⣴⠋⢸⡒⠀⠀⠀⠀⠀⠀⡘⠀⠈⠲⡀⢠⡇⠀⡔⠁⠀⠃⠀⠀⠀⠀⠀⡠⣲⡏⠉⢦⠀⠈⠑⣄
⠀⡰⠁⠀⠠⠆⠘⠒⡏⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠘⣄⣀⠎⠀⠀⠀⠈⠀⠀⡠⢐⠍⠊⠀⢹⠛⠓⠐⠄⠀⠈⢦
⢰⠁⣠⠞⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⢀⠘⠇⣀⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢧⠀⠀⠀⠱⣦
⠠⠮⠥⠤⠤⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠬⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⢀⢀⣠⡄⣿⣿⣧⠀⠀⣴⣾⣿⣿⣷⣦⣄⣡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢻⣾⣿⣿⣿⡏⠀⠀⠀⣼⣿⣿⣿⣿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠀⢀⠴⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⢸⣿⡄⣿⡏⠀⠀⢾⣥⣤⣶⣶⣶⣾⣯⣄⣀⣀⡈⠀⠄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠁⢀⣴⣸⣿⢿⣿⣇⣴⡀⠀⠉⠉⠙⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⣠⡄⣸⣿⣿⣿⣾⣿⣿⣿⣧⣰⢄⡀⣰⠃⠀⣾⣄⠉⠛⠛⠛⢉⣿⡍⠉⡇⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⣏⣐⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢤⡈⠉⠁⠴⠦⠀⠉⠉⣀⠄⡇⠀⠉⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⢘⡃⣸⠇⡀⣿⠰⠆⣹⠰⠆⣷⠘⢠⣿⣆⠈⠻⢦⠶⣶⣶⡶⢦⠾⠋⢠⠇⠫⣽⠎⠸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠚⣿⣿⣿⣿⣿⣸⣇⣹⣠⣅⣻⣐⣿⣿⣰⣿⣿⡆⣿⣿⣿⣷⣤⣤⣀⣨⣍⣀⣤⣤⣴⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣿⠉⣿⠉⣿⡏⠉⣿⡏⢹⣿⡏⢹⣿⡟⢉⡉⢿⠉⡿⢹⡏⢹⡏⢉⣩⡏⢉⣭⣽⠉⢻⡏⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠠⣶⠀⣿⠁⡆⢹⡇⢸⣿⡇⢸⣿⡇⢸⡇⣼⡇⠁⡄⠀⣾⡇⢠⣾⡇⢠⣤⣿⠀⡄⠁⢸⣟⣁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⣨⣿⣠⣿⣤⣯⣴⣷⣨⣧⣈⣉⣧⣈⣩⣧⣈⣡⣾⣧⣠⣿⣰⣿⣇⣈⣭⣧⣈⣉⣿⣀⣿⣄⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠶⢊⠙⢿⣿⠛⠿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣤⣷⠶⠶⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⡟⠋⠉⠉⠁⠐⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠀⠀⡀⠀⠙⠟⠟⡟⠛⠻⣿⠿⣿⡛⢛⠛⠛⣛⠿⢿⠏⠉⠉⠉⠉⢉⠔⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠈⠁⠀⠀⠀⠀⠣⠀⠀⠈⠳⢄⠙⠛⠋⠉⠑⠒⠲⢖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠆⠀⠀⠀⠀⠀⠀⠁⠂⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠶⣤⣀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⡴⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠙⢷⣄⣴⠾⠛⠋⠉⠉⠀⠀⠀⠉⠉⠛⠷⣤⣠⡶⠛⠉⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⠃⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣀⣀⣠⣴⣏⣭⣥⣤⣤⣤⣤⣤⣠⣦⣀⣀⣀⠀⠀⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⡴⢆⠀⠀⠀⠀⠙⠋⠉⠉⠛⠻⢿⡀
⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⣠⠟⠁⠀⢀⣀⣠⣤⣤⠴⠶⠖⠛⠋⠁⠀⠈⠛⠶⠤⣤⣤⣤⣀⣀⠀⠀⠀⠹⣦⡀
⠀⡴⢤⡀⢣⡀⡇⠀⠀⣰⠃⣠⠶⡟⠋⠉⠀⡀⢰⡶⠛⠛⠛⠳⣄⠀⢀⡶⠚⠛⠶⣦⡀⠉⠉⠛⢳⣆⡀⠘⢷⡀
⠀⠙⢄⡈⠞⠁⠇⠀⢰⡏⠀⣿⢠⡇⠀⠀⠀⠚⢿⡅⠀⠠⡶⠀⣿⠀⣿⠀⠀⢶⠀⢹⡏⠀⠀⠀⠀⣿⡈⡧⠘⣷
⢖⣃⡚⠁⣠⠞⠻⢂⣨⣽⠆⠈⢻⡇⠀⠀⠀⠀⠈⠳⣤⣀⣠⡾⠃⠀⠙⢦⣀⣀⡴⠟⠀⠀⠀⠀⠀⣿⡏⠀⠀⣿⡄
⢀⡐⣭⡀⠁⠀⠀⡞⠀⢡⠀⠀⠘⣇⠀⠀⠀⠀⢠⣤⡤⠀⠀⢰⣥⢤⣤⠀⠀⠀⣠⣤⡤⠀⠀⠀⠀⣿⠁⠀⠀⣿
⠈⠁⠀⠛⠦⢴⡆⢻⡀⠀⢇⠀⠀⢻⡀⠀⠀⠀⠀⡠⠔⠂⠀⠈⠓⠚⠁⠀⠀⠒⠤⢄⡀⠀⠀⠀⣸⠇⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠹⣌⢳⣶⠈⢷⠀⠈⢷⡄⠀⠀⠀⠀⠠⠋⠠⡀⠀⠀⠀⡆⠘⢧⠀⠀⠀⠀⠀⣴⠏⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠘⣧⣸⡇⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠙⠶⠶⠞⠁⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠠⢧⣀⠀⠈⠛⢷⣤⣀⣀⠈⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠟⠙⢛⣿⢋⡬⠛⠻⠷⢶⣦⣤⣴⣶⡶⢿⡟⢧⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢤⡖⠋⠀⠉⠲⢤⣤⣴⣾⢇⠺⣄⠀⠀⠀⡴⠋⠈⢷⠀⠀⠀⢹⡌⢧⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⣼⠇⠀⠀⠈⠳⣦⣈⡇⠀⠀⢸⣤⡶⢛⣉⠁⠺⡇⠤⠙⠷⠶⣷⠀⠀⠀⢠⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⡿⠁⠠⢄⡀⠀⠀⠙⠃⠈⠉⠉⠋⠀⠀⠀⢀⣀⣽⣶⠿⠶⠚⠁⠀⠀⠀⣾⢀⠀⣷⠶⠒⠚⠋⠛⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠉⠚⠒⠠⠤⠤⠤⠤⠄⠀⠒⠛⠉⠁⠸⡟⢦⡀⠀⠀⠀⣀⠼⣣⠟⣄⣸⣦⠶⢦⣄⡀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣧⡄⢙⠶⣶⠋⣠⢴⠇⡞⢤⣥⠙⣦⠀⠈⠻⣆⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠙⠒⠒⠲⠦⠴⠶⠖⠒⠛⠉⠁⠀⣿⣇⣸⠀⣿⠉⠀⢸⡇⡟⠲⠤⢀⣿⠀⠀⠀⢹⣇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣿⠀⠻⣄⣿⠀⠀⢸⡇⠘⡶⠀⣾⠃⠀⠀⠀⠀⢿⣄⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠉⠉⠀⠀⠀⣀⣿⠀⠀⠈⠛⠀⠀⠈⣧⠀⡿⠲⠛⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠒⠦⠤⢤⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠞⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠶⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢾⡇⠀⠀⠀⠀⢀⣴⠟⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢻⣶⠶⠶⠾⠛⢁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣁⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⡴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⡏⠀⢈⡏⠈⠉⠉⠉⠉⠙⢻⡟⠛⢷⡷⠶⠶⣦⣤⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠛⠁⠀⢠⠂⠀⣿⠀⠀⠀⠀⠀⠀⣸⠃⠠⡄⠀⠈⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠿⣄⣀⣀⣤⣀⣤⣤⣤⣮⣤⣴⠿⠀⠀⠀⠀⠀⠀⠸⣧⣀⣾⠀⠀⠀⠀⠈⠉⠛⢦⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣽⣿⣿⣶⣶⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⡔⣶⣴⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣶⠾⠿⣿⣿⢭⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠋⠁⠀⣼⠏⠀⠀⠀⠀⢻⣆⠀⠀⠉⠉⠛⠛⠶⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⢿⡀⠀⠀⣴⣦⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠒⠛⢷⣦⣤⣬⣵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⢶⡶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⢶⡄⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⣿⡷⢶⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢠⠞⢉⣿⣦⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣶⠾⠟⠋
⢹⣇⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⢿⣦⣄⣤⡦⠀⢸⠀⠊⠀⠉⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠉⠉⠉
⠀⢻⣆⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢹⣿⠉⠁⠀⠈⠳⠦⠴⠞⠋⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠹⣷⡀⠀⠀⠈⢷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠉⠳⣄
⠀⠀⠀⠈⢻⣆⠀⠀⠀⠹⣦⣤⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠙⢷⣄⣀⠀⠻⢿⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⣤⡿⣿⣴⡄⠈⢻⣄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣷⣾⠟⢻⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠶⢾⠿⣷⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣠⣾⡟⣁⣠⣼⣧⣄⠀⠙⣦⣀⣀⣈⠉⠉⠉⠹⡍⠉⢉⡽⠋⢹⠉⠳⡄⢸⠀⠸⣿⣦⡀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠛⠛⠉⠀⢹⣷⣤⣾⡟⢿⣯⣿⣦⡀⠀⠹⣄⡞⠀⠀⢸⠀⢀⠹⠎⠀⠀⢿⡏⠛⠷⣦⣽⣆⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⡿⠏⡲⢶⣽⠟⠉⢧⠀⠈⠀⠀⠀⠈⠀⠲⠃⠀⠀⠀⠸⣷⠀⠀⠀⠉⠉⢳⣦⣴⣤⣿⡾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢿⣦⣬⡶⠋⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢦⠄⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠈⠻⠞⢿⣾⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠻⠤⠤⣤⣄⣀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣼⠉⣷⣤⣴⣶⣶⣶⠶⠶⠶⠶⠶⠶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⢦⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡉⠉⠉⠉⠛⠳⣦⠀⣰⠟⣋⠉⠉⠉⡉⠙⢳⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣷⣴⣟⣛⣶⣶⣤⣶⣛⣐⣳⣶⣏⣛⣳⣶⣷⣶⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⡛⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⡀⠀⠀⠀⠀⡙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⣿⣷⣾⣿⣿⣿⣶⣆⠀⠀⠐⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠉⣽⣿⣿⠛⠛⠛⢻⣿⣿⣷⣄⠀⠸⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢐⣫⣜⣿⣇⣀⣤⣄⠘⠿⠛⠉⢿⣦⡀⢸⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣈⣴⡟⠉⢉⣀⡈⠉⠁⠀⠀⠚⢻⣿⣧⠘⢧⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⣟⣿⠁⠚⠛⠿⠏⠀⠀⠀⠀⠰⣿⣿⣿⣀⣼⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⣄⠈⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⣿⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⠿⠟⠛⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣾⣶⡄⠀⠀⠀⠀⡀⢀⡤⠂⣠⡤⠀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢻⣿⣷⣄⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡌⣿⣿⡟⠙⠶⣄⡀⠀⠀⢀⡄⠀⠀⢠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠈⣹⢿⣿⣿⣿⣦⡀⠀⠉⠛⡿⠋⠁⠀⠀⠘⣿⣦⣄⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⡁⢀⣠⣾⡏⠀⣿⣿⣿⣿⣿⣆⠀⠀⠁⠀⠀⠀⠀⢾⣿⣿⣯⣿⣿⣦⣄⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢀⡶⠋⠹⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⠛⣿⣟⣿⣧⠀⠀⠀⠹⣄
⠀⠀⠀⢀⡟⠀⠀⠀⠈⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⢦⡀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⢿⣿⡿⠋⠛⠻⢿⢿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣷⣿⡄⢹⣿⣏⢉⣿⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⣿⣿⡇⠐⠟⠛⣢⣌⠛⠶⠤⣤⣄⣠⡤⠤⠖⣻⡅⣤⠈⠉⠁⠾⢹⣿⣿⣿⣧⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⠀⠛⠛⠀⠐⠗⢠⣤⡄⠠⡦⠀⢀⠀⠀⠀⣠⣄⠀⣨⣿⣿⣟⣿⣆⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣤⣄⠀⠛⠀⠀⠈⣛⡁⢠⣤⠀⠈⢛⡀⠀⠛⠛⢀⡉⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠘⣿⣿⠛⢻⣥⣤⠶⠖⢻⣏⣡⣤⣴⠖⢛⣉⣥⠶⠖⣀⠿⢿⣿⣿⣿⣯⣿⣿⣷⠀⠀⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣶⣾⣿⣫⣭⡶⠿⢟⣛⣭⠥⢶⣞⣩⣴⣶⣿⢿⣿⣿⡿⢿⣿⣿⡆⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣨⣿⣿⠿⣷⣾⣿⣭⣿⣿⡿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣷⣴⣿⣿⣇⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⣶⣿⣿⢿⣿⣿⣿⠿⣿⣿⣿⣾⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠘⢿⣷⣤⣿⣿⣿⣿⠿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠘⣿⠻⠿⠿⠿⠷⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠋⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢧⠀⠏⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⡰⣀⠈⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠘⣄⡘⠒⠂⠀⡀⢀⡀
⠀⠀⠀⠀⢤⠸⡢⣙⠖⠈⠀⣠⢰⠃⠡⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠘⠞⠣⠁⠀⠀⠀⠆⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣀⠈⡀⠀⠠⠀⠀⠀⠄⠀⡀⠠⢄⠀⠎⣍⠱⠄⠀⠀⢁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⡄⠀⠀⠀⠘⠗⡁⠀⡀⠀⠈⠙⠂⠁⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠂⠀⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢰⠀⠀⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶⠄
⠀⠀⠀⠀⢀⢀⣰⣀⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢠⣤⣀⣈⣿⣿⣿⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⣶⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⠉
⢠⡿⠿⠛⠋⠀⠙⠉⠀⠀⣠⠀⠈⠛⠲⠆⠀⠠⣿⠴⣿⣄⠰⣦⠠⡆⣶⢰⡖⡦⠀⣀⢠⠄⠀⠠⠄⣼⣿⣿⣇
⠈⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠈⢶⣸⢰⡇⡽⢸⡇⡿⠀⢻⡎⢰⣾⣿⡻⣿⣿⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⣶⠀⠀⠀⠃⠀⠉⠀⠀⠀⠸⠇⠀⠸⠇⠤⣀⠜⠀⠀⠙⠿⣿⣶⣯⣥⣵⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣿⠀⠀⠀⠀⡿⢸⡆⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⡀⠀⢉⢙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⢀⣿⠳⢘⢵⠀⣷⢸⣤⣿⣻⣟⣷⡄⡏⢸⡇⣿⢰⠏⠧⣾⠹⠆⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⢠⣿⡀⠹⠺⠂⡟⢸⡇⠻⠿⠿⠟⠁⠃⠺⠃⠟⠘⠦⠀⠙⠤⠀⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠂⡠⠶⠭⠭⠭⣉⡵⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡇⠂⡆⢀⠔⢤⡇⠀⣇⠀⠀⠀⠀⠀⡞⠿⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡉⠀⠀⠁⠁⡇⠀⢸⠀⠀⠀⠀⠀⣧⡘⠒⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠬⣖⣊⡯⠇⡧⠄⢸⠀⠀⠀⠀⣰⡑⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⣶⢲⠶⠶⠷⠊⠁⠀⠀⢀⡼⣑⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠒⠻⠶⠓⠒⠒⠦⣤⣄⣠⠮⣱⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡅⠈⠉⠉⠉⠉⠒⠒⡎⢹⡗⣉⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠲⡇⠀⡩⢩⠉⠉⢱⡀⡇⠸⠭⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢲⣷⠀⡍⣉⠀⠀⢸⡃⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⢺⢸⠀⢟⣒⣁⣀⣸⠇⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠸⡄⠀⠀⠀⠀⠀⠀⣇⡠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣈⡷⡀⠧⠤⣤⠭⠭⠭⢭⣷⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠁⢂⣱⢠⣉⡇⠀⠀⠀⠀⡦⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠈⠉⢸⠒⡇⠀⠀⠀⠀⣏⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢬⠃⠀⠀⠀⠀⡗⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢼⣺⢦⠀⠀⠀⡐⠫⡼⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡡⢼⠀⠀⠀⠹⣒⠠⠤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠊⠁⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⢀⡾⢛⣥⠤⠖⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢰⠈⠃⠀⠀⠀⢹⢻⡄⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢤⠀⠠⣤⡤⠰⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢤⡤⢸⣏⠻⣆⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⣴⠅
⠀⠀⠀⠀⠀⣀⡦⠀⠀⠀⠀⠀⣾⠋⢀⣿⣿⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠀⠠⠄⠤⠀⠤⡇⠀⠹⣿⣿⣏⠐⠂⠀⠀⣴⣶⣄⣀⠀⠀⠀⠈⠙⠉⠉⠀⠉
⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⢴⣇⣀⣰⣿⣿⣿⠃⠈⠀⢸⡿⣿⣭⡉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⣤⡄⠁⠀⠃⠙⣿⣿⣶⣤
⠀⠀⠀⠀⠈⠃⠠⠣⠀⠠⠐⠀⢸⣿⣿⣿⢿⠤⣏⣁⠀⣀⡁⠈⠿⠟⠋⡉⠛⠲⢄⡀⢀
⠀⠀⠀⠀⠀⣀⡴⢒⠀⢀⣤⣄⣈⣀⣀⣉⣀⣼⡿⠿⠈⠉⠁⡀⠙⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇⠀⡀⠀⠀⢀
⠀⠀⠀⠴⢎⣁⣰⣯⠶⠞⠛⢛⡉⠉⠉⠁⠀⠀⠀⠀⠈⠀⠋⠁⠀⠀⠠⠞⠀⢳⡀⠀⠴⣾⠀⠀⣰⠶⠀⡁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠳⠞⠛⣿⡖⠒⠾⣿⠛⣿⡆⣴⡆⣀⣶⡆⠀⠀⠀⠈⠁⠹⣄⠀⢹⣄⡼⠁⡀⡀⠉⠀⢀
⠀⠀⠀⠀⢦⣴⡀⠰⠛⠀⠀⠚⠋⢀⡿⡆⠉⠘⠚⠋⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠸⣆⣿⣿⠷⠚⠋⠉
⠀⠀⠀⠀⠘⢿⣿⣶⣿⡄⣤⡰⠄⠺⠿⠇⠀⢀⣠⣴⣿⡿⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠠⣘⢻⣿⡇⢘⢧⠀⠀⠀⠀⠀⠸⠟⣿⣿⣟⣘⣥⣾⣿⣿⣿⣧⠖⠛⠠⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠰⠿⠀⠉⢋⢸⣄⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⡃⠀⠀⠠⡏⣿⣧
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡿⠀⣤⠣⠀⠀⠀⠀⠀⠛⢿⡟⢻⠿⢿⣿⣿⠿⢁⣿⡃⢸⢳⣿⣶⣿⣿⠀⠀⣀⡀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣤⣤⣤⡤⠠⠂⠤⢸⡀⠀⠀⠠⣘⢾⠠⡔⢀⢀⠀⡀⠈⠀⠀⣘⠺⠧⠎⣼⡉⣘⣿⣿⡄⢺⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⢧⣤⣽⡟⣇⢺⠆⣸⣿⣦⡀⠀⠘⠉⠋⠁⠀⠀⠀⠀⠀⢀⣠⣭⣤⣷⣶⣤⠀⣼⣿⣧⣷⣿⣿⣇⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⠉⣠⢆⣁⣠⠀⠐⠀⣶⠂⠀⠀⠂⠀⠐⠀⠤⢈⣽⣿⡿⠿⠿⠿⠿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠾⠟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣬⣿⣯⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⣶⣾⣿⣿⣿⣿⣶⣤⣶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠏⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠈⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⣀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣡⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣌⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⣏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠻⣏⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠁⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⠛⠶⢀
⠀⠀⠀⢀⣀⣤⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣌⣙⣶⣶⣤
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⣾⣿⠻⣿⣿⣿⣿⣿⣧⣼⡿⢹⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠉⠿
⠂⠈⠉⠀⢿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣠⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠇⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⠀⣸⣿⣿⠉⠛⠛⡟⢻⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡿⠛⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡧⣻⡶⠃⠀⡀⢀⠀⠈⠉⠀⠟⠉⠙⠁⠀⠀⠀⠘⢿⣿⢹⣿⣿⣿⣿⠟⢳⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣶⡆⢠⣶⣾⣶⣦⠀⠀⠘⠻⠆⢸⣿⣿⣿⣿⣿⢟⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠛⠛⠀⠀⠈⠙⠋⠙⠀⠀⠀⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⡀⢠⣤⣤⣄⣤⡀⢀⡀⢠⣄⣴⡄⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠁⠈⠛⠻⠿⠛⠁⠘⠿⣿⠿⠟⠋⠀⠀⢹⣟⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢿⣿⣿⠠⣤⣤⣤⣦⣄⣀⣶⣆⢀⣠⣤⣤⣄⡀⠀⠈⣿⡻⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠃⢸⡿⠋⢀⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢏⡉⢀⡀⣿⡇⠘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡇⠀⠈⣧⢰⣿⣷⣶⠀⠈⠹⠿⠿⠿⠟⠛⠁⠀⣾⣿⣿⣧⣿⡇⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠉⢙⠿⡿⠋⠀⠀⣀⠀⠀⠐⣤⡆⠀⠀⢀⡈⠀⠀⣻⠃⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⣠⣾⣿⣷⣤⣠⣿⣧⣠⣴⣿⣿⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡾⠛⠻⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⢻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⠀⠀⠀⠀⠀⠀⠻⠋⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠸⣤⡄⠀⣀⡀⠀⠀⠀⣠⣆⠀⣤⡀⢀⣤⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣿⣷⣾⣿⣿⣆⠀⣰⣿⣿⣷⣾⣷⣾⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡗⠀⠋⠙⠿⣿⣿⣿⠟⠛⠻⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠻⠛⠛⠋⠉⡁⠐⣶⠆⠀⡀⠉⠛⠛⢀⠀⠀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣸⣿⣀⣄⠀⣾⣷⣴⣿⣶⣾⡇⠀⣴⣦⣨⣷⣄⣿⣾⡇⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣶⣿⣧⡿⢇⠀⠀⠀⢀⣤⠀⠀⣤⣴⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⢹⡿⢟⢿⡿⠀⠿⠋⠀⠛⠛⣿⠃⠀⠙⠟⠛⠻⠏⠉⢁⣼⠃⢀⣾⣿⣿⣿⣿⣷⣾⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⢠⣤⣾⣮⣱⣶⣀⣴⣦⣤⡄⢠⣤⣄⡀⣿⣆⣤⣶⠆⠈⢻⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣌⠙⠻⢿⣿⣿⣿⡿⠟⠃⠀⠀⠛⠿⣿⣾⣿⠟⠉⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⢈⣉⣡⣟⣿⡶⣶⣾⣥⣥⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣿⣿⣿⣿⣟⣻⣶⣾⢻⡿⠭⠙⠩⡟⠋⠉⠉⠉⠉⣉⣉⠉⠛⠛⠻⠿⣿⣯⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠈⠉⠛⠛⠛⠛⠻⠟⢲⣶⠐⠃⠐⠂⠀⠀⠀⠉⠉⠁⠀⠀⠀⠄⠀⠉⠀⠀⠐⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡆⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⣠⣾⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡠⠛⠁⠀⠉⣿⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣥⠞⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣫⠂⠀⠀⠀⠀⠀⢼⣿⣷⣀⠀⠀⠀⠀⠀⢀⣀⠤⣴⣾⣿⣿⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣃⠔⠀⠀⠀⠀⠀⢨⣿⣿⣿⣧⡤⠀⠀⣨⣵⣶⣿⣿⡿⠟⠜⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⢯⡥⠀⠤⠀⠐⠈⣩⠭⠿⠛⠉⠁⣠⣴⣾⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⠚⠛⠛⠛⠓⠒⠒⠒⠛⠓⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⠅⠊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣧⠁
⠀⠀⠀⠀⢰⡤⠠⢤⠶⢯⣁⣈⡛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠏⣿⣿⣿⠛⡾⠿⢻⣿⣿⢷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠛⢻⠟⣿⣿⣇⠀⠀⣨⢽⣿⢿⣿⠃⠀⢡⡄⣠⣾⢿⢿⡀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢰⣿⣿⣿⠃⣿⣿⢂⠉⠀⠁⠁⢸⣿⠁⣿⡋⡚⢦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡏⡇⠉⠌⣿⣿⡻⣾⡶⣶⡖⣹⢿⡇⠹⡌⠡⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣡⠀⡖⠻⠃⣰⠻⡉⢷⡴⠃⠻⠜⢋⠞⠜⢦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣐⢀⢦⡏⠀⡈⠓⢄⠀⢀⠖⠁⠀⠀⠘⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⡎⠣⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠐⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠄
⠀⠀⠀⠀⠀⡘⠁
⠀⠀⢠⠀⣴
⠀⠀⠈⠀⠸
⠀⠀⠀⠀⣠⠟⠒
⠀⠀⠀⢰⡇
⠀⠀⠀⠘⣧⡤⣄
⠀⠀⠀⠀⡏⠁⠉
⠀⠀⠀⠀⢻⡄⠀⢠⣤⢛⣀⠤⣀
⠀⠀⠀⠀⣠⢿⠷⠛⠿⠋⠀⠀⠈⢧
⠀⠀⠀⡾⠁⠀⣤⡀⠀⠰⠿⢆⡀⢸⡇
⠀⠀⠀⣇⠀⠀⣉⣥⣤⣶⣷⡟⠁⡸
⠀⠀⠀⣸⣦⣀⠈⠉⠛⢈⣁⣠⠞⣡⡤
⠀⢠⠞⠁⠀⠘⠛⢾⠟⠛⠻⠟⠛⢿⣍⠉
⠀⡎⠀⠀⢀⣴⡆⠀⠀⠀⣠⡄⠀⠀⠹⣆
⠀⡇⠀⢰⣬⡛⣃⣀⠀⠀⠻⠇⠀⠀⠀⣿⠈⠂
⠀⠹⣄⠈⠛⢿⣿⡿⣷⣾⣿⠟⠀⠀⣰⣇⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⡈⢷⣤⣄⣈⣁⠈⢁⡀⢀⣠⣾⡟⠛⠉⠙⠳⣆⠀⢠⡄⠀⠀⢠⡶⢄⣤⡄⢸⢖⣋⣀
⠀⠘⢣⣾⠟⠉⣹⠛⠙⠛⠛⢻⡏⠀⡁⠀⠠⡀⠀⠙⣧⠛⠂⣠⡤⠶⢧⣾⣿⣿⠿⠿⢿⠛⠛⢶⣄
⠀⣸⡿⠁⠀⠀⠃⣀⠀⠈⠀⠀⠁⢸⣿⣦⠀⠀⣀⢰⣿⣧⣴⠃⠀⠀⠀⡁⠙⠁⠀⠀⢀⡀⠀⠀⠙⣷
⠀⣿⠁⠀⠀⠈⢠⣿⣷⡄⠀⠀⠀⠘⠟⢛⣡⠞⠉⠋⠿⢿⣿⠷⣶⣦⣼⣿⡀⠀⠀⠠⡿⠧⠀⠀⠀⢸⡇
⠀⢿⡄⠃⠀⠠⢀⣉⣤⣤⠀⣴⣶⣶⣿⣼⡇⠀⠀⣠⡆⠈⠀⠀⣀⠀⠈⢻⣤⣤⣄⣠⣶⣶⠆⠀⠀⣼⠃
⠀⠸⣷⡈⠄⠀⠈⠛⢿⣿⣿⣟⢻⣿⠿⠻⡆⠀⣶⣍⣃⣀⠀⠰⠿⠀⠀⠀⣿⠿⠛⠿⠛⠁⠀⢀⣴⠋⠀⠀⢠⠊⡈⢠
⠀⠠⠈⠻⣄⡂⡀⠀⠀⢀⠀⠀⢠⡄⠀⢀⣷⣄⠈⠻⢿⠿⣶⣿⠟⠀⠀⣰⣃⣀⣄⣀⣴⡤⠶⠻⠤⢤⡘⣀⡠
⠀⠀⠀⠀⠈⠙⠻⠶⠦⠾⠷⠶⠾⠛⠛⠫⠟⠉⠲⠦⢤⣄⣀⣄⣀⡤⠞⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣠⠴⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡾⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠊⠉⠁⠈⠙⢷⣌⠐⠤⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣁⢀⢀⠀⠈⢣⠀⠀⠀⠉⠳⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⠋⢹⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠉⢠⣾⡄⠀⠀⠀⠀⠀⠀⠸⡏⠛⠶⠄⠀⠀⠀⠀⠀⠀⢀⠀⠊⢀⡂⠰⠃⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡇⠀⠀⠀⠀⠀⠀⠙⠁⢠⣷⡀⠀⠀⠀⠐⡓⢄⠀⠀⠀⠀⠀⠀⢀⠑⠀⣏⣀⠀⠉⠓⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣧⠀⠀⠀⠀⠀⠀⠀⢠⠎⢸⣇⠀⠀⠀⠀⠜⣸⠀⠀⠀⠀⠀⢠⠞⠹⣦⠈⠉⠙⠆⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠳⠶⠶⠞⠀⠀⡰⠋⠀⣿⢹⠀⠀⠀⢸⡟⠁⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⣤⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⣼⡏⢸⠀⠀⠀⣼⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⣶⡿⠏⠀⢸⠀⠀⢰⠏⢀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⣠⠏⠀⣠⠟⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠤⢞⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠤⢤⠤⣶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⢀⢠⠀⣤⢤⠤⠤⠴⠖⠒⠛⠉⠀⠈⠳⢤⣀⣀⣀⣀⣀⣤⠴⠚⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠄⢢⡀⠢⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠉⡏⠀⠀⠀⠀⠀⢹⡀⠀⠃⠉⠉⠀⠀⠀⠀⠀⠀⠘⣧
⡔⠛⠀⠀⢠⠀⠀⡽⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠇⠀⢹⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢎⠉⠙⠀⢸⠀⠈⡇⠀⢀⡇⠀⠀⠀⢀⣀⣤⠴⠚⢻⡀⠀⠢⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢍⣁⣀⠀⠀⠤⠔⠊⠿⠶⠞⠛⠉⠁⠀⠀⠀⠘⡇⠀⠀⠑⠄⠀⠸⣆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⠂⠹⣆⠀⠀⠀⠀⠀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠙⣿⠒⠒⠛⠛⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⣠⣤⠶⠛⠉⠓⠢⠤⠤⠤⢤⡂⠁⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣤⣤⣤⣤⣤⣬⣷⠀⠀⠀⠀⠈⠷⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠴⠖⠋⠁⢸⡆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢻⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡶⠶⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣠⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣁⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠳⢦⣄⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢀⣠⣤⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠻⣦⣴⠾⠛⠋⠉⠁⠀⠀⠀⠉⠉⠛⠳⣦⣤⠞⠋⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠋⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⠤⢤⡴⠯⠥⠤⠤⠤⠤⣤⣤⣼⣷⣤⣄⣀⣀⣀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢦⣀⠀⠀⠀⠉⠈⠉⠉⠉⠛⢧⡀
⠀⠀⠀⠀⠴⢦⠀⠀⠀⢀⡾⠁⠀⣀⣤⡤⠶⠶⠶⢚⣛⣋⡉⠀⠀⠀⠉⠛⠲⠶⠶⢦⣤⣄⣀⠀⠀⠻⣦
⠀⡖⠲⣄⢳⡈⡄⠀⢀⡞⢀⡶⢻⠋⠀⠀⠤⣨⡶⠋⠁⠈⠙⣦⠀⢠⠞⠉⠉⠓⣶⣂⠀⠀⠉⢻⡦⣄⠘⣧
⢀⣈⢢⡘⠋⢀⡃⣀⢼⣁⠸⣇⣿⠀⠀⠀⠈⠙⣧⠀⠀⠻⠂⣼⠀⢿⠀⠀⠾⠀⣼⠃⠀⠀⠀⠸⣷⡸⠃⢸⡇
⠻⠀⡌⠀⠞⠋⠙⡡⠶⠞⠀⠈⣿⠀⠀⠀⠀⠀⠘⠷⠶⠶⠞⡁⠀⠈⠳⠶⠶⠞⠃⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⢠⡀⢤⡀⠀⠀⠰⡅⠀⢧⠀⠀⢹⡆⠀⠀⠀⠀⠒⠒⢒⡂⠸⣗⢒⡶⠀⠀⠒⠒⠂⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇
⠀⠀⠀⠉⠓⠺⣇⢳⠀⠈⢦⠀⠈⢷⡀⠀⠀⠀⠔⠊⣡⠀⠀⠈⠉⠀⡀⢢⡉⠓⠢⠀⠀⠀⢀⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⣦⡿⣷⠈⣧⠀⠈⢷⣄⠀⠀⠀⠐⠃⠀⢦⣀⠀⣠⡗⠀⠱⠀⠀⠀⠀⣠⡟⠁⠀⣰⠟
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⢷⣿⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠨⠛⠶⡤⣤⢤⣿⠿⣶⣤⣀⣀⠀⠀⠀⠀⣀⣠⣴⡾⠋⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡝⠳⣤⡀⠸⠀⣈⣾⢣⡖⠁⠈⠉⢛⣿⠟⣟⠛⠋⠙⢦⠱⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣅⠀⠀⠀⠙⠛⠿⣿⠓⠠⠙⣦⡀⠀⡾⠁⠀⢸⡄⣀⣤⡼⠇⣳⢀⠘⣧⣀⣀⡄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⣰⡇⡀⠀⠀⠀⠛⠶⡧⠄⠤⠼⠿⠏⠀⠒⠈⠉⣧⣁⣄⣉⡽⠃⠀⠀⠀⡞⠉⢷⣀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⠀⠈⠑⠢⠤⣀⣀⡀⡀⠀⢀⣀⣀⣀⡤⠴⠒⢻⣿⡉⠉⠀⠀⠀⠀⡸⣡⡆⠸⣏⣀⣀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠲⢤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⡇⠹⢦⣀⣤⠔⢊⣱⠻⣌⠒⠻⣏⠉⠛⢷⡄⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠈⠛⠲⠦⣤⣤⣀⣀⣀⣀⣠⣤⡴⠶⠛⠉⣷⡀⠈⢸⡷⠖⠋⣽⠰⣬⣙⠃⢸⡄⠀⠀⢻⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⣀⣴⣿⠳⣦⢸⡇⠀⠀⢿⠘⣦⡉⢠⡟⠀⠀⠀⠀⢿⡄⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⠑⠒⠠⠤⠤⠤⣀⠤⠤⠤⠤⠒⠚⠉⠁⠀⣿⠀⠈⠻⠇⠀⠀⢸⠀⢰⣤⣼⠇⠀⠀⠀⠀⠈⠻⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⡷⠀⠀⠀⠀⠀⠀⢸⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠁⠀⠀⠀⠀⢀⣴⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⣷⣤⣤⣤⡴⠟⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠉⡹⠉⠉⣿⠉⠉⠙⠛⠛⠛⢻⡿⠶⢶⣶⣦⣤⣤⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠶⠞⠋⠀⠀⡄⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠰⡄⠀⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣠⣠⣄⣤⣤⣤⣼⣤⣤⡾⠃⠀⠀⠀⠀⠀⠘⢧⣀⣿⠀⠀⠀⠀⡈⠉⠛⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⡇⠀⠁⠀⠀⢠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⠄⠀⠀⠬⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣧⡄⠚⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⢃⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠃⡌⠎⠀⠠⠀⠀⡏⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡇⢀⠓⣅⡀⠐⠀⠀⠀⠀⠀⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡇⢼⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠖⠐⠀⡇⠀⠀⠸⠀⡷⠀⠀⢆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣀⠀⢰⣷⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠈⠀⢠⡄⠘⠀⢸⡇⠀⠀⡇⠀⣆⠀⠀⠘⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠰⢶⡶⠷⣶⠉⡄⠠⠘⣟⠃⡇⠀⡄⠰⣿⠅⠀⠘⢀⠀⠰⠀⠉
⠀⠀⠀⠀⠉⠀⠉⠆⠀⡀⣀⠸⠷⣤⡄⠀⢠⠇⠀⠀⠐⠀⠀⠃⣾⡿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠉⠀⠀⠀⣁⢀⡞⠀⠀⠀⠀⠀⠀⠀⣶⣷⣶⣶⡶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣆⣀
⠀⠀⠀⠀⢼⠃⠚⠀⠀⠀⠀⠀⠀⠀⡉⠈⠀⠈⠀⠀⠀⠀⠀⢰⣾⣶⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠠⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⢀⣼⣤⣤⣬⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢈⣭⣉⣉⣉⣉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠛⠛⢛⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⣀⠤⢤⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠿⠿⢿⠀⠀⠀⠀⣿⣿⣯⣿⣯⣜⣅⣀⣠⣄⣶⠀⢠⣾⡀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⡾⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣯⣭⡽⠏⣩⠅⠀⣉⠙⠶⢤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⣤⣤⡄⠀⠀⠀⠀⢿⣿⣿⡿⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢿⣉⣉⡁⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⣘⡁⢀⡀⠀⡀⠀⣤⣾⣷⠄⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡝⣛⠛⢻⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠙⠛⠋⠉⠀⣸⡇⡀⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢿⠿⠿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣀⠀⠛⠛⠀⠀⣀⡾⠁⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⣄⠐⢶⡇⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⠻⠿⠾⣶⣾⡟⠉⠁⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠈⣻⠈⠢⣀⣈⣂⢀⡀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢀⡀⠀⠀⢀⣤⡤⠐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠰⣬⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣄⣀⣫⣥⢾⡄⠙⠀⣹⢹⣿⢧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠑⠾⢧⣙⢷⡀⠀⣨⣄⣼⣷⣿⣆⣣⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣌⣙⣂⣤⣽⣶⣿⣿⣿⣿⣷⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠛⢀⢝⢸⣿⣿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡠⣒⣤⣔⡦⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠨⣤⣾⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⡩⢲⡿⣿⡽⢋⠰⢳⡾⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⣀⢦⣋⣿⣾⣥⠾⣿⡟⠢⢶⠅⠊⣲⡞⠦⠈⠁⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢀⣮⣼⣿⠿⡟⠻⢠⡀⢓⠶⢄⣾⣟⠭⠙⠁
⠀⠀⠀⠠⣴⣿⣿⡿⢽⡧⢿⣬⡝⢀⡩⠗⠉⠑⠀⠉
⠀⠀⠀⣡⣿⣿⣭⠟⡇⠃⠀⠉⠀⠈⠁
⠀⠀⡰⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡤⢀⢃⣠⡴⠁
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣾⣭⡿⠀⠊⡀⠁
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠿⠷⠭⠙⠉
⡇

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⠀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠸⠛⠛⠛⠉⠛⠛⠋⠉⠩⠤⢡⣭⣯⣿⠿⣿⣻⣯⣻⣿⣽⣻⣿⣿⣟⣿⣿⣯⣭⠭⠭⠭⠭⠭⠉⠉⠅
⠀⠀⠀⢚⡛⠉⠉⠉⠑⠚⠁⠍⣹⣿⣿⢟⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡛⢿⣯⠭⠭⠥⠭⠤⠅
⠀⠀⠀⠐⠂⠀⠀⠀⠒⠀⢀⣾⣿⢟⣴⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⠓⠌⠻⣿⣿⣿⣮⢻⣷⡁⠈⠭⠤⠅
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⣾⣿⠏⣾⣿⣿⣿⠟⢰⣁⣈⢿⣿⣿⣿⣿⣇⣤⣴⣧⣾⣿⣿⣿⡎⣿⣇⠀⠀⢀⡁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣿⣿⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⢒⣒⡆
⠀⠀⠀⠒⠀⠀⠒⠀⠀⢀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣶⣾⣿⣿⢿⣿⡟⠍⣻⣿⣿⣿⣿⠒⠐⠒⠂
⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⣏⢍⠻⠯⠉⠻⢿⠋⠋⠛⠋⠀⠁⢸⣤⣵⣿⣿⣿⣿⠧⠤⠄
⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣠⣶⣅⢀⣠⣶⣦⣠⣾⣷⣦⣿⣿⣿⣿⣿⡿⢁⢀⣀⠀⠀⡄
⠀⠀⠀⣾⣿⣇⠀⣰⣿⣆⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⢂⡀⢐⡒⠒⠀⡂
⠀⠀⠀⣿⣿⣧⣾⣿⣿⣿⣧⡀⠀⠐⠂⠂⠒⠉⣻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠹⣅⠀⠀⠀⠀⣀⣲⠴⢐⡒⡃
⠀⠀⠀⣿⣿⣿⣿⡯⣭⢽⣿⣿⡄⠀⠈⠉⠡⡶⠁⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⡍⠈⠡⢾⠋⠀⢎⡠⢳⡀
⠀⠀⠀⣿⣿⣿⣿⣧⣭⡿⠿⠿⣿⣿⣿⣿⣿⠁⠀⢰⠃⠀⠀⠀⢰⠀⠸⡇⠀⠀⠘⣿⠭⠬⢉⣷⣾⣿⣦⣼⡇
⠀⠀⠀⣿⣿⣿⣿⡟⣡⡆⢠⠖⠘⠋⣿⣿⡏⠀⠀⣾⠀⠀⠀⠀⢸⡆⠀⣿⠀⠀⠀⣿⣧⣴⠟⠿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⣿⣿⠁⠿⠆⠀⠀⠀⠀⠉⣽⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⠀⠸⡆⠀⠀⢸⣿⡏⣈⠀⠀⠀⠉⣻⡇
⠀⠀⠀⢸⣿⣿⣿⡀⠐⡀⠀⠀⠀⠆⠀⡏⠀⠀⢸⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⣿⣻⡄⠠⠀⠂⢰⣿⡁
⠀⠀⠀⢸⣿⣿⣷⣴⠀⠁⠀⠐⠀⡀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢿⣇⢻⡄⠀⣠⣞⣹⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣄⣀⠙⠒⠛⡁⣼⠁⠀⠀⠘⣿⡇⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠸⣿⡄⢿⣏⢿⣾⡿⠃
⠀⠀⠀⢸⣿⣿⣿⣿⠻⣍⠣⣠⣟⣠⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢻⣧⠀⢻⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⡿⣬⣓⣾⠭⠭⠯⠶⣿⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⢿⣇⠀⠹⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣗⢶⣾⢿⣯⠏⣯⡭⢿⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠈⢿⡆⠀⠈⠛⠿
⠀⠀⠀⣾⣿⣿⣿⣯⣵⣻⡇⠀⢐⣛⣻⡄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣭⣽⣇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⠀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠴⠮⠄⢂⣤⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⠄⢀⡀⠀⠀⠀⠀⠀⣦⠈⠆⠀⢃⠘⠌⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⢂⣐⠷⡞⠉⠉⠡⠍⡏⢓⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣃⣇⡀⠀⠊⠉⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡁⠀⢢⠀⠀⡎⠀⠀⠈⡁⠀⠀⢀⠄⠀⠀⢐⠁⠀⡓⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡈⢏⢢⢀⢀⠤⠌⠀⢠⠀⠀⠀⠈⡇⣠⠤⠀⠘⡄⠎⠜⢀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠭⠌⠆⠉⢘⣤⠄⠢⠐⠀⠀⠀⠀⠀⡏⢀⠳⢀⡤⠃⠉⠠⠤⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠂⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⡇⠨⡉⢢⠀⠀⠀⠎⠳⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⡆⠈⠇⠀⠀⠀⠀⠀⠀⢠⠃⢠⠌⢺⢢⢀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠃⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⡄⠀⡼⢸⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠐⣄⡀⠒⢀⠘⠠⠊⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠁⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠦⣀⣸⠀⣰⣁⡀⠴⠕⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠓⠒⠃⠀⠀⢸⡏⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠠⢈⣂⠀⠀⠀⢀⡛⠀⠀⠀⠀⠀⠀⢀⡬⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡚⢆⠀⠙⣀⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠫⠈⢵⡖⠀⠀⠀⠀⠈⠉⠀⠤⣄⠀⠀⠀⠀⣀⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠂⠹⢄⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠁⠀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢾⡼⡳⣄⠀⢏⠐⢤⣤⣧⡃⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠑⢌⠪⡷⠀⠀⠀⢉⠀⠈⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣄⣀⣀⣉⣁⣀⣀⠀⢸⣶⣤⣤⣤⣤⣤⠼⠽⡶⠞⠺⡍⠉⢁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠉⠈⠀⠀⠉⠀⠉⠁⠁⠀⠀⠁⠀⠀⠒⠊⠉⠉⠀⠉⠉⠋⠉⠉⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠄⠲⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠂⠂⠐

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣶⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣟⣨⣿⡉⣋⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣸⣥⣿⡥⣤⡀
⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠰⣶⣼⣿⣿⡿⠀⠩⣶⡀
⠀⠀⠀⠀⠀⠀⠘⠛⢻⡏⠀⠸⣿⡿⠀⢻⣿⣿⣯⣤⣤⣶⣶⣶⣾⡿⡿⢿⡿⠃⢀⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣋⣉⣙⣻⣾⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣲⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⣿⡿⠛⢁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⢫⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⢻⣏⠀⠀⠈⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡶⠀⢀⣸⣷⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠘⠂⡀⣾⣿⣿⣿⣿⣿⡻⠟⠋⠻⢷⡀⠀⠀⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡛⠻⠿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠈⠻⣀⠲⠖⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣆⢢⣤⠙⠃⠀⠀⠀⠀⠀⠀⠀⢀⠿⣷⣄⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡏⠠⠀⠔⠀⠀⠀⠀⠀⠀⠀⢰⡟⣛⣿⠓⠶⢊⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠠⠠⠞

⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⠁⠀⠀⢀⢀⠀⡀⡀⠀⠀⢸⠁⠀⠀⠀⡠⠀⠠⠄⠀⠀⢸⢁⡤⠤⠠⠤⠤⣀⣀⡀
⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠁⠄⡁⠀⠈⡀⠀⢸⡄⠁⢢⠊⠀⠀⠀⠀⠆⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠈⢙
⠐⠐⡀⠀⠘⠀⠀⠀⢀⡀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢸⠑⢄⠀⢈⠐⠠⠀⢠⡀⠀⢸⠈⢶⠀⣔⡆⠀⠤⠒⠊⠁
⠀⢃⠃⠀⠀⠀⠀⠀⠈⠀⢨⠀⠀⠀⡀⠀⠀⠈⠀⠃⠀⢸⠀⠀⠀⠐⠈⢀⠐⠒⠀⡠⢹⠀⠀⠘⠏⠓
⠀⠀⡈⡀⠀⠀⠀⠀⠀⠀⠸⠀⠌⠀⠁⠀⠀⠀⠀⠀⠄⢸⠀⢺⠙⢙⡏⠋⡀⠀⠀⠀⢸⠀⠀⢀⡴⠛⢛⡓⠦⠴⠾
⠀⠀⠰⢁⡀⠀⠀⠀⠀⠀⢨⠊⠀⠘⠐⠠⠄⠀⠀⠦⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠀⠀⣾⠰⣖⡋⠖⢀⠀⡄
⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⢘⡄⠐⠀⠀⠀⠀⠀⠀⠂⠂⢸⠀⠀⠀⠀⡇⠀⣰⠀⠀⠀⢸⢤⡴⠏⠀⠉⠄⠀⡀⠀⠁
⠀⠀⠰⠛⠙⠓⠄⠀⠀⠀⢨⠃⠐⠅⠀⠂⠀⠊⠀⠐⠀⢸⠀⢶⡄⠎⠳⠴⠉⢀⠀⠀⢸⠀⠈⠀⡚⠀⠂⠁
⠀⠀⠠⣀⣀⡤⠞⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⢸⠀⠻⣦⣀⣀⣁⣀⣄⣁⡤⢸⠀⠀⠀⠠⠖
⠃⠀⠀⠈⡁⠀⡀⠀⠀⠀⢘⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠈⠉⠁⠀⢀⡀⢸⠂⠀⡀
⠀⠀⡐⠀⢀⣀⠀⠀⠀⠀⢸⠀⠀⠑⠀⠀⠠⠀⡀⠀⠀⢸⠀⢀⠀⠠⠀⠀⠠⠘⣇⣹⢺⠀⠀⠠
⠀⠀⡄⢀⡎⠉⢳⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠐⠀⢀⠀⠀⠀⠐⠀⢸⡇⢸⠀⠈
⠲⠬⣅⣀⣓⣒⣋⣀⡭⠴⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⡿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⢰⠶⢾⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡗⠒⠗⠀⠩⠍⢸⠀⠀⠐⠀⠀⠀⠄⠀⢮⡽
⠀⠀⠀⠀⠂⠱⠔⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠂⠙⠘⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡈⠭⢵⠢⠀⢰⡮⠭⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⠉⠀⠀⠈⠙⠢⡄⢸⠀⠄⠠⠀⠀⠀⠀⠠⠀⠄⢸⠀⠂
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⢰⡀⠤⣐⠄⡀⠆⠀⠂⠀⣠⡰⠃⠄⠰⠰⠂⠀⠀⠀⠀⠠⠀⡀⡀⠀⠀⠀⠀⠀⡀⠀⡀⡀⠀⡀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⢠⡀⠀⠀⠀⡠⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠠⠀⠘⡠⠋⠆⢠⠈⡀⢇⠐⢠⡆⠘⡈⠄⢰
⣀⣀⣀⣀⣈⣀⣈⣁⣚⣀⣁⣀⣉⣁⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⢉⠁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠑⢄⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⢆⠀⠀⠀⠀⠀⢀⠆⢠⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠉⠉⠉⢱
⠀⠀⡌⠀⠀⢀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣤⡞⠀⠀⠀⠀⠚⣦⣴⡾⣶⣀⡀⠀⠀⡀⢀⣤⡴⠖⠒⠂
⠀⠐⠃⠀⢀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠄⠂⣀⣀⣉⠀⣸⠀⠉⠻⢷⣤⣦⣶⠾⠛⠁⠀⠨⠄⣠⡀⠀⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠋⠉⠉⠀⠀⢀⣤⣍⡁⠀⠸⠀⠀⠀⠠⠈⠟⢣⡄⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢀⠀⠀⠀⠙⠂⠀⠐⢊⣠⣤⠄⠀⠀⠊⠀⠀⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⢤⢄⡀⠀⠀⠀⠀⠀⡆⠀⠙⠖⠀⠀⠀⠠⠐⠀⠈⠛⠁⠀⠐⠂⠀⠀⣨⠥⢸⠇⠀⠀⠀⠀⠀⠀⠂⠤⠆⠴⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠴⣆⠀⠙⢦⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⢀⡖⠀⠀⠀⠘⠀⠀⠒⠀⠂⠤
⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⡤⠶⠶⢄⠀⠳⡀⠀⠀⠀⠈⠷⣌⠀⠑⠂⢀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⡠⠊
⠀⠀⠀⠀⠀⢧⡀⣀⠔⠁⠀⠀⠀⠀⢳⡀⠇⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠈⠀⠀⠀⠈⠁⢀⣠⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠂⠂⠀⠐⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠐⠒⠺⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠖⢀⠤⢀⠂⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⠤⠤⣀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣣⡀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢀⠘⠁⠳⣄⡰⠃⠈⠂⣄⠀⠀⠀⠀⢠⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠁⡀⠀⡄⠀⠀⠻⢆⡠⠖⠁⠘⠒⠂⠞⠃⠀⢋⢀⡀⢈⠾⡁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠙⠒⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠈⢦⡈⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠌⠀⠀⡠⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠰⠀⠀⠀⣿⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⣠⠀⠁⠀⠀⠀⠀⡄⠀⢰⡌⢣⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠀⠀⠉⠀⠁⠀⠉⠀⠠⠃⠀⠈⡄⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⢀⠀⠐⠀⠀⠀⠸⠀⠀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠀⢠⣄⠀⠠⠜⠀⠀⠀⠀⠀⠆⠀⠀⡇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠸⠀⠀⠡⡀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⣷⠊⠀⠀⣐⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢯⡄⠀⠀⠲⠄⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠵⠀⠀⠀⠀⣠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⢀⣀⣀⣀⠀⢀⣀⣀⣀⠠⠤⠒⠋⠁

⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠐⡆⡇⠰⠄⠀⠀⢀⠂⠀⢠⠀⢠⡀⠀⡄⢀⡄⠀⠄⢠⠀⠀⠀⠠⠀⣸⠀⢧⠰⠆⡰⠄⢠⡃
⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⢦⣀⡤⠶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣆⣀⠸⠦⢼⡅⢀⣴⡃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⢯⠤⣄⣀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠈⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠋⠙⠃⠀⠀⠀⠀⠀⠀⠤⢶⡛⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠛⠛⠛⠶⣤⡀⠀⠀⣤⠖⠒⠢⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠗⢺⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣧⣾⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠉⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⢀⣤⡾⢷⣄⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠙⢷⣦⡀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⡾⠟⠉⠁⠠⠄⠒⠈⠇⠀⠀⠀⢠⡛⠂⠤⠄⠀⠙⠻⢷⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⠛⠉⠀⠀⠀⠀⠋⠀⠂⠉⠁⠀⠀⠀⠀⠀⠉⠐⠒⠚⠛⠀⠀⠀⠈⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⢘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣠⣤⢶⡶⠟⠛⠛⠋⠉⢻⡏⠀⠀⢠⡟⠀⠀⢀⡉⠉⠙⣿⠛⠛⠻⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠋⠀⠀⠰⣅⠀⠀⠀⠿⠆⢠⡇⠀⠀⠸⣧⠀⠀⠙⠋⠀⢠⡏⠀⠀⠀⢿⣤⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠄⠀⠀⠀⠻⣦⣀⣀⣀⣴⠟⢀⣀⠀⠀⠙⠷⣦⣤⣤⠶⠏⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠉⠒⠦⣤⣤⣤⣤⣤⣴⠾⠋⢁⡞⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣤⡾⠁⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠉⠛⠛⠷⠶⢶⣶⣶⣶⠾⠿⠛⠋⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠁
⠀⠀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢤⣧⣸⣀⣿⣀⣀⡀
⠀⣰⡏⠈⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠉⠙⠛⠿⠷⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⠤⠤⠤⣀⡠⢉⣉⣉⠁⠈⠙
⢠⡏⠙⢦⣄⠀⠉⠙⠻⢶⣤⣤⣀⣀⣿⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠢⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡈⠙⣯⠉⠛⠛
⢸⠇⠀⡎⠉⠻⣶⣤⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣄⣈⠉⠉⠒⠒⠒⠤⢤⣤⣤⣤⣔⠉⢉⠛⠶⢿⣄
⡇⠀⠘⠇⠀⠰⠀⠈⠙⠛⠛⠷⠶⠶⡶⠲⡶⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⠉⠛⠶⠶⠶⣤⣤⣤⣤⣤⠤⢿⣍⣉⡁⠀⠀⣹
⣧⠀⡆⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠇⣸⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠖⠀⣠⡼⠃
⣿⠀⠋⢠⠞⢁⣀⣀⠙⢷⣀⠀⠀⢠⢀⡿⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣹⡇
⣿⠀⢸⡇⢰⠁⠀⠈⣃⢸⡿⠀⠀⠀⢸⡇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⢻⡆
⡏⠀⠸⡇⠸⣀⣀⣠⠎⣸⠇⠀⠈⠀⡿⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠴⠟
⢗⠀⠀⠙⢦⣈⣉⣡⡴⡿⠀⠀⠀⣾⠁⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⢸⡇⠀⠀⠀⠉⠉⡏⠀⠀⠀⠀⣼⠇⢀⣾⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣾⠇
⠀⢿⡄⠸⠀⠀⠀⡇⠀⠀⢀⣼⠋⢀⡾⢿⡟⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⣻⠁
⠀⠈⢿⣶⣄⠀⣸⣀⣠⡴⠟⢁⣴⡿⠁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠈⠻⣶⣭⣉⣉⣡⣤⡶⠟⠉⠀⠀⣿⠁⠀⠀⢤⣤⣀⣀⣀⣀⣤⣤⣴⣶⡞⠋⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠀⠀⢸⡇⠀⠀⢀⣀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⣤⣤⣶⣶⠿⠃⠀⠀⠀⠀⠀⠙⠛⡿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠉⠓⠛⠳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠿⠶⠚⠷⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠛⠷⣄⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣨⣧⣤⠶⠖⠋⠀⠀⠀⠀⢘⡃⠀⠀⠂⣀⣉⣉⣉⣋⡻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠐⣿⣿⠀⠀⣴⡄⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢶⣤⡀⠀⠀⢀⣀⣤⣤⡀⠀⣿⡿⢠⠀⠈⠅⠈⡩⢿⠿⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⠿⠿⠃⣠⡿⢁⠞⠀⠚⠉⠉⠛⣿⡶
⠀⣀⡀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠘⠁⠴⠋⠀⠀⠀⠀⠐⠢
⠀⢿⣿⡰⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠈⠁⠀⠀⠟⠃⣄
⣤⡌⢿⣯⢆⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢀⠀⠀⠈⣿⣷⡀
⠈⠻⢦⣝⡳⠕⠢⠤⣠⠀⠄⠐⠀⠀⠈⣿⠧⠠⠶⠶⠾⠏⠉⠁⠠⠀⢸⡀⠀⠀⠁⠀⠀⠁⠹⢳
⠀⣀⣤⠄⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⣀
⠞⠋⢀⡞⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⢠⣿⢀⡟⠁⠀⠀⣦⣠⡀
⠀⠸⠇⢸⣧⡄⠀⠀⠺⠿⠃
⠀⠀⠀⠸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢟⢛⠛⠶⠶⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡔⠡⢀⠠⠄⠉⢀⣐⡒⢬⣻⣄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢷⣥⣰⣥⡶⠟⠛⠛⠛⠿⣷⣧⣝⡷⣤⣤⠀⠀⠀⣠⣤⣀⡀
⢰⣿⣿⣻⡇⠀⠀⣠⣤⣤⣤⡿⠛⡳⣤⠀⠀⠀⠀⠀⠀⠙⠻⣯⣿⣧⡀⠀⣿⢿⢟⣟⣟
⠻⣟⣷⣿⡇⠀⠀⣩⡼⢿⡏⠀⠘⠗⢸⡇⠀⠀⠀⠀⠀⠀⠸⠏⠀⠈⣿⠀⠈⠛⢿⡟⠋
⠀⠀⠉⠉⠻⣦⡾⠋⠀⠈⠳⣤⣤⡴⠟⠋⠀⢠⢾⡿⣅⠀⠀⠀⠶⠾⢿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣇⠈⠻⣆⠀⠀⠀⠈⢷⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡟⣧⠀⠀⠀⠈⢿⡄⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠸
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⢀⡞⠁⠀⡰⠁⣽⠀⠀⠀⠀⣨⢿⡗⠒⠲⠶⣤⣤⣀⣠⣤⠶⠋⠀⢀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢿⣄⠐⠉⠙⠳⠾⠴⠶⠋⠀⠀⣠⠞⣡⠼⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢻⣄⡀⠀⠀⣀⣤⡶⠛⠁⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⡤⠶⣎⣉⣻⠇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢀⢤⠊⢳⠞⠉⢉⡴⠶⣄⠀⠀⠀⠀⠻⣦⣠⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⣎⡴⠺⠔⣡⠞⠛⠛⠀⠀⢸⠀⠀⠀⠀⢀⠜⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠘⣷⠀⠀⠀⢿⡀⠀⠀⠀⢀⠎⠀⠀⢀⠔⠉⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡆⠀⠀⠀⠉⠓⠒⢐⣫⠊⢹⠂⢁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣇⠀⠀⣰⠃⢠⠞⣿⡀⠀⠀⠀⠀⢀⠧⠤⢐⣡⡶⠛⢻⣄⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠒⠚⠁⠀⡾⠀⠸⣧⠀⡀⠄⠂⢁⣤⡶⠛⠁⠀⠀⠀⢏⠳⠊⠁⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠴⠚⠁⠀⠀⠹⣇⣠⡴⠞⠋⠹⣧⡀⠀⠀⠀⠀⢸⣀⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣯⠙⠤⡀⠀⠀⠀⠀⠀⠀⢦⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⣠⠞⠒⠁⠀⡀⠀⢦⠀⠀⠀⢣⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠁⠀⠀⢦⠀⠙⣄⠈⢳⡀⠀⠀⠱⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠘⢆⠀⠱⡀⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⢦⠀⠑
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣄⣀⣁⣀⣀⣀⣀⣀
⣿⣿⢋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡴⠊⠀⢀⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠀⠀⢈⠥⠴⣶⣿⡆⠈⣧⠀⠀⠀⠤⠴⣀⠀⠀⠀⠀⣀⠴⠂⣈⡙⠳
⠀⠀⡇⠠⣆⠀⠸⠀⠉⠃⡤⠀⡟⠀⢠⠖⣛⣉⣉⣉⠄⢀⣠⠴⣺⠭⢯⣩
⠀⠀⠰⡈⠀⠛⢆⣟⡁⠔⠁⣸⡡⠄⠒⠛⢛⡠⡀⣙⣴⣾⠿⠃⠀⠉⠉⠢⢄⠠⣄⢀
⠀⠀⠀⠁⠛⡠⠟⠋⠢⣀⣼⡯⠖⠚⠛⡩⢹⠿⠻⢥⣿⣍⣴⣴⡒⠋⠉⠒⠢⣕⠌⢳⡳
⠀⠀⠀⠀⠀⠀⡀⠂⠀⡿⠋⠀⢂⡔⠈⠀⠀⠀⢀⠎⠔⠠⠀⠀⠈⠑⡄⠀⠀⠈⠳⣄⠇
⠀⠀⠀⠀⠀⠀⠀⡞⢸⠁⠀⢠⠎⢀⣵⣶⠆⢀⠏⠀⠀⠀⠀⠀⣶⣶⣄⠀⠀⠀⠀⠹⡇⠀⢠⠖⠉⠉
⠀⠀⠀⠀⠀⠀⢻⠁⡏⠀⠀⡞⠀⢿⣿⣿⣦⣾⣄⠀⠀⣠⣴⣦⣾⣿⡿⠂⠀⠀⠀⠀⢹⡄⡟⣳
⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⡇⠘⣆⡀⠉⠉⡏⠉⠀⠀⠈⠙⠋⠉⢉⣴⠇⡆⠀⠐⠀⢸⠃⠈⠡
⠀⢠⡴⣠⠶⢤⡀⠀⢳⠀⠀⢇⠀⠹⣿⣷⣦⣧⣤⣀⣠⣤⣤⣶⣾⣿⡟⢸⠇⠀⠀⠀⡾⢎⡁
⠀⠀⠙⢧⣀⢀⣷⡤⠋⢳⡀⠸⡄⠀⠈⠋⠁⢿⣿⣿⣿⣿⣿⠉⠙⠋⢀⡎⠀⠀⢀⠞⠑⠦⣼⠋⢠
⠀⠀⣄⠀⠉⣩⠏⠀⠠⣄⠉⠓⠿⢄⣀⠀⠀⠈⢧⠈⠉⠀⠀⠀⠀⣠⡊⢀⣠⠴⠁⡁⠈⠁⠈⠙⡏⢐
⠀⠀⠀⠉⣉⣥⡂⠀⠀⢨⣳⠤⣤⡀⠈⠉⠁⠙⠊⠑⠒⠂⠀⠚⠉⡀⠉⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠉⠁⠀⠀⠀⠈⠒⠲⣶⣶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠄⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠁⠉⠀⠀⣀⠀⠂⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠒⠀⠀⠇⡀⠀⠆⠀⠆
⠀⠀⠀⠀⠈⠋⠀⠀⢸⠀⠉⠃⢒⡂⠀⠐⣈⠀⠉⢀⣁⡀⠨⠖⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⣀⡒⠈⠐⣁⠀⠉⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠛⠃⠄⠀⠛⠁⠀⢀⡀⠃⢀⡤⠐⠀⠀⢀⢀⠀⠄⣅⠀⠀⢙⠁⠂⠀⢛⠁⠀⢀⡍⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠢⣀⠀⠁⠒⠂⠀⢀⠠⡊⠲⡔⠁⠀⠀⠀⠀⣟⠀⡘⠂⠮⠀⠀⡈⠉⠀⠈⢉⠀⡀⠼⠀⠆⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠴⢤⠤⠀⠀⣄⠀⠀⢀⠀⠰⡕⠋⠶⠐⠂⠀⠀⠀⠀⠀⠀⠉⠀⢤⠀⠀⡍⠳⡶⢞⡏⢀⠠⣤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠈⠰⠈⠁⢎⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡁⠽⠀⠀⠀⠤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⢊⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠆⠀⠁
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢠⠁⣀⡀⠀⢀⣀⣠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠖⠐⠂⠀⠀⠒⠂⠀⠀⠃⠸⠂⠒⠞
⠀⠀⠀⡤⡊⠀⠀⠀⠀⢀⠂⠀⠩⠩⠁⠥⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠀⠄⠠
⠀⠀⠀⠈⠙⢦⠀⠂⠉⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⠀⣀⣈⠐⠠⠌⠲⠅⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⣦⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⢀⠔⠠
⠀⠀⠀⠀⠀⠀⠐⢢⠀⠐⠆⠀⠒⠄⡙⢁⣄⠐⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠈⠁⠀⠀⠈⣁⣠⡄⠀⠀⠀⢠⡴
⠀⠀⠀⠀⠀⠀⠁⢨⡀⠂⠄⣈⠀⠂⠀⣈⣉⣀⠀⠼⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠖⠛⠛⠋⠀⠠⠂⠈⣃
⠀⠀⠀⠀⠀⠀⠤⠀⣙⠀⣀⣈⣉⠀⠀⠀⠀⡰⡖⠳⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠐⠃
⠀⠀⠀⠀⡔⠀⢀⠤⣇⢉⢀⡉⠳⠒⠀⠀⡁⠸⡔⠚⠁⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⢤⡀⢐⣢⡠⡄⠀⠀⠐⠃
⠀⠀⠀⠀⠁⠂⢈⠀⡀⠁⢉⡀⠀⠠⢂⡉⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡖⣛⣅⢋⣙⠰⡄⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⠀⠀⠀⣂⠉⣀⠰⣉⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣽⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠰⠉⣅⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠋⢉⠏⢸⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠀⠤⠤⠒⠈⠉⠈⠀⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⡇⠀⠀⢀⠀⡄⠀⠀⢸⣧⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣷⠁⠀⣠⡟⠀⢷⣄⠀⠈⡿⣯⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠏⢁⣼⡟⠀⠀⠈⢻⣧⡘⠽⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⣄
⠀⠀⢠⠴⠰⢴⡏⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠘⣧⠴⠴⣄
⠀⠀⢨⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣴⠶⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠲⣦⡀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠸
⠀⠀⠀⠁⠀⠀⠈⠳⢄⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠹⣆⠀⢠⣶⡄⠀⡰⠋⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠁⢀⣉⡀⠈⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠄⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣙⣒⡤⠤⠤⠴⣿⣿⣿⡦⠤⠤⢤⣖⡛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢀⡔⠀⠠⣾⠆⠈⠲⡄⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠈⠲⣄
⠀⠀⡐⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⢂
⠀⠀⠳⣄⡠⢾⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡧⣀⣤⠎
⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⢿⣆⠀⠀⠀⣠⣿⢇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⠃⠈⠻⣧⠀⣰⡟⠁⠈⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠘⠂⠏⠀⠀⢸⡟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡆⠀⠀⢀⠀⠀⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⣠⡞⠈⢷⡀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡷⣾⠦⠀⣀⡀⢻⠀⠀⡀⠀⢠⠇⣀⡀⠀⢴⡶⣾⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠙⠃⠀⣛⠀⠘⠉⠉⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⠀⠀⠀⠀⠲⠂⢺⣿⠇⠰⠆⠀⠀⠀⢀⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠻⢦⣄⣠⢴⠆⠀⣤⠀⢴⠤⣀⣤⡶⢛⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣾⠟⠻⠋⠀⠀⠀⣾⠀⠀⠀⠀⢈⡇⠀⠀⠀⠻⠟⠿⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠸⣦⡀⠀⢀⣾⢃⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠈⢷⢠⠟⠁⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃⠀⠀⠀⠀⠀⠛⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⣀⣀⡀⣸⡆⠀⡊⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣠⣶⣻⣿⣿⣿⣿⣿⣧⡼⢥⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠺⠄⠀⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠛⠛⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⡙⠋⠁⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⢩⡀⣀⣀⣠⣤⣶⣾⣿⢣⡆⡀⢀⣇⠄⣽⣿
⠀⠀⠀⠀⠀⠀⣼⠀⠀⣴⠀⣀⣸⡄⠀⠂⣤⠀⣀⠢⣔⠺⣿⣿⣿⠿⡿⣿⡿⠟⠁⣼⣇⠀⢸⣷⢈⡉⠉⠁
⠀⠀⠀⠀⠀⠀⣿⡀⣶⠿⠊⢀⠝⠓⠴⠀⠍⠉⣠⣐⡩⠌⠋⠉⣀⠀⢠⠀⠒⠒⠤⠄⣠⠉⠛⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠟⢁⠄⠀⠀⠉⠀⡆⡀⠀⠤⠀⠉⠀⡀⠀⠂⠁⢀⡄⣿⡀⠀⠂⠀⠄⠋⠑⡄⠀⠈
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⡛⠉⠠⡄⠀⣀⢠⣧⢤⡦⠀⠼⠗⠛⠓⠈⠁⠀⠁⠀⠀⠇
⠀⠈⠀⠁⠠⢀⠀⠀⢀⡀⠙⠛⠛⠋⠀⠀⠀⠀⠈⠈⠛⠀⠐⠡⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠙⠻⠷⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠷⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⡆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠤⠾⠛⣧⡄
⠀⠀⠀⠀⠀⠀⢸⣈⡹⠁⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢈⣿⣂⠾⠋⠀⠀⠀⠀⠀⠀⣠⡴⠀⠴⣦
⠀⠀⠀⠀⡤⠀⢁⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣌⢻⡾⠁⡜⢻⠀⠀⠀⠀⠀⠀⢀⠞⠁⣆⣀⣴⡇
⠀⢠⡴⠁⣀⡀⠘⠇⠤⠈⠋⣔⠠⣄⠀⠀⠀⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⢠⣯⠀⣼⣿⣿⣿⣦⡀⣀
⠠⡿⢡⡌⠁⠀⠜⠋⠒⠆⠶⠙⠀⠁⠀⠀⡼⣿⠀⣠⢿⣷⣿⠀⠀⠀⠀⢠⣿⣿⡇⠀⠈⠋⠀⠀⣶⣿⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢨⣼⡏⣽⢹⡆⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⡄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡋⠸⡇⢹⢯⠀⠀⠀⠀⠈⠋⢸⣶⠀⠀⠀⠳⣤⣿⣿⡇
⠀⠀⠀⠀⢀⡄⢀⣤⣄⣄⢻⠀⠀⠀⠀⢸⣿⠶⢷⢨⡇⢸⣿⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠐⠾⡟⠻⠛⢋⣏⠈⠀⠀⠀⠀⣸⣿⢘⡟⢸⡇⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⠀⠀⠀⠀⠀⠘⣿⠘⡟⣸⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡀⠀⠀⠀⠠⣿⡸⠇⢰⣦⡬⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠘⠶⠦⠀⠀⠀⠙⠀⠀⠀⠀⢸⣷⣴⣿⣿⣿⣟⣡⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠹⠋⠁⠁⠚⠙⠛⠃⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠰⠀⢆⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⠁⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠆⢻⠄⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⣘⢙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⢶⠀⠀⢀⢼⡀
⠀⠀⠀⢻⠴⠋⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣀⣟⣷⣄⠀⠀⠀⠀⠀⠀⢾⡟⠀⠀⠀⠀⢐⡉⠉⢂⠁
⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⢀⣿⡟⣫⣾⣟⢻⠀⠀⠀⠀⠀⠀⣬⣿⢶⣤⣠⣶⢸⡤⢃
⠀⠀⣸⡏⢀⡿⢷⣧⠀⠀⠀⠀⠀⠀⠀⠀⣬⣍⣾⣮⣿⣿⣿⡄⠀⠀⢀⠀⢰⣿⣇⣿⣿⣿⣿⡧⣿⠟
⠀⠀⣿⢸⠏⢰⢵⣿⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⢸⣿⡇⠀⠀⠉⠙⠙⢿⣺⣿⠋⠉⠀⢰
⠀⢸⣏⢸⢆⢼⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⠏⠀⠀⠀⠀⠀⠀⢸⡿⢿⣇⠀⠀⠘⠃
⠀⢾⡷⣾⢦⣽⡇⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⢻⡟⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠉⢻⣿⠉
⠀⠀⣿⢻⣟⣿⣷⣽⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢿⣧⣿⣿⠿⠟⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠸⠶⢦
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉