Math Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⡇⢹⡃⣸⢿⡇⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣹⡇⠉⠁⣿⣼⣧⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠿⠗⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠶⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⢸⣿⣛⠃⢦⡆⣾⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⡛⢹⡇⠞⠇⣿⢻⣧⢨⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠿⠾⠃⠀⠀⠻⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡗⠀⠀⣴⢶⡄⢀⡀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣬⡅⣸⡇⣿⣧⣭⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠛⠛⠛⢻⣿⣿⣿⠀⠀⠀⣿⣼⡇⠀⠀⢿⣧⣿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣺⣿⣿⡏⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡀⠘⠓⠒⠚⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠺⠿⠿⠷⠶⠾⠿⠿⠷⠾⠃

⠀⢀⠔⠀⠐⡄⠀⠀⠀⠀⠀⢀⠔⢉⢉⠑⡄
⢠⡡⠊⠉⡆⡀⠀⠀⡎⠃⠀⠈⠒⠁⢠⢃⠇⢀⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡖⠛⡄
⠀⠀⠀⡜⡰⠁⣖⣂⠃⢘⣒⡆⠀⡠⢃⠎⠀⢈⣉⣁⣈⡩⠖⢊⡩⠟⠋⠉⠉⠉⠉⠳⣄⠀⠀⠀⠸⡀⢱
⠀⣠⠊⡴⠁⠀⠀⠀⡁⡏⠉⠁⡰⢁⠎⠀⠀⠘⠒⠒⢺⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣠⢷⡈⡟⠓⠲⡄
⢴⣁⠚⠀⠤⢤⠀⠀⠉⠁⠀⠴⣀⣉⣐⠒⠀⡄⠀⠀⢸⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠱⠇⠀⠀⠉⠳⡆
⠀⠀⠈⠉⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠤⠀⣠⡼⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠶⠚⠛⠋⠉⠙⠛⠛⠿⢤⣀⠀⢰⠇⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⡴⠚⠁⣰⠉⠉⠋⠉⠉⠉⠉⠋⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠠⡀⢀⣀⠠⠤⢤⣀⡴⠀⠀⠀⠀⠸⡏⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⠀⢠⡇
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⣇⠀⠀⠲⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠽⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⠀⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⠀⡏⢀⡠⠄⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⣷⠃⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢿⠀⠀⠀⠀⠀⠀⡠⠊⠙⠒⠒⠒⠉⠁⠈⠂⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠛⠶⣦⣀⡀⠀⠀⣀⣀⣤⠶⠋⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⢹⠋⠉⠉⠉⡇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠤⣴⣄⣀⣴⣤⠼⠇⠀⠀⠸⠧⢤⣦⣤⣤⠼⠦⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⣧⢰⡴⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⣧⠸⠳⢰⡟⣿⡇⢘⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣷⣦⠀⠀⠀⠀⠀⠀⠘⢿⣾⠏⠀⠀⠘⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢛⠉⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣭⣅⠘⣿⠸⣿⣹⡿⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣭⢸⣿⠘⠙⢰⣿⢹⣷⠨⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠈⠻⠿⠏⠀⠀⠈⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⠀⠀⢠⣤⣤⡀⡀⡀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣾⣧⣍⡁⣹⡇⣿⣇⣍⠁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠠⠿⠿⠿⣿⣿⣿⣿⠃⠀⠀⢿⣏⣿⡇⠁⠁⣿⣏⣿⡇⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠀⠀⠀⠈⠛⠛⠁⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣤⣤⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣄⡀⢀⣴⣦⣤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⢀⣀⣀⣠⣀⠀⠀⣀⡀⢀⡀⠀⠠⢄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠈⠉⠉⠉⠃⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠓⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠇⠀⠒⢀⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⢠⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣄
⠀⠠⡀⠀⠀⠘⠛⠻⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠒⠀⠀⠀⠁⠙⠘⢁⠐⠀⠐⠀⠀⠓⠃⠺⠿⠛⠋
⠀⠀⠹⢒⠤⠄⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⡀⣈⠁⠃⡀⢀⢀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠁⠀⠈⠀⡢⠀⠆⠐⠐⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠈⠀⠈⢡⠀⠀⢠⠀⠀⠀⠀⠀⠂
⠀⠀⢀⣠⡀⠀⠙⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠄⠀⠐⠂⠰⠀⠀⠀⠈⣀⠀⠀⠀⠈⡊⠉⠉
⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⢠⣷⣄⠀⠀⠀⣠⣄⠀⠀⠀⠐⠘⠂⠀⣾⣻⡷⠀⠀⠀⠐⠆⠀⠀⠂⠀⠀⣤
⢀⠐⠈⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠐⠀⠊⠀⠀⠀⠈⠉⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠈⠐⢢⠀⢀⠀⠀⠤⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠈⠘⠀⠀⠀⠀⠀⠀⠅⠀⠠
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠈⠛⠛⢁⠀⠀⠀⠀⠀⠀⢠⠲⡄⠀⠐⠀⠀⠀⠀⠀⢀⠒⢲
⠀⠀⠐⠀⠐⠈⡈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⣉⣈⠉⠀⣀⣴⣆⣀⠀⠀⠉⠀⠀⠂⠐⠂⣀⣤⡀⠀⢀
⠀⠀⢀⢀⢀⡀⠀⠀⢀⠠⠸⠐⠐⠀⠀⠀⠀⠀⢸⠯⡟⡧⠀⣿⣿⣿⣿⣿⣶⣶⠀⠁⠀⠀⢸⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠚⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠈⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⣿
⠀⠀⣤⡤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠴⠧⠤⠶⠀⠀⠀⠀⠻⠤⠤⠤⠴⠶⠶⣦
⣠⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠞⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡆⠀⠀⠀⣏⠀⠸⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⢸⡆⠀⠧⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠚⠛⠛⣿⠃⠀⠀⠀⢹⡀⠀⠉⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡏⠉⠉⠉⠉⣩⠟⠁
⠀⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠈⠧⠤⠤⠤⠤⠴⢦⠀⠀⢸⠀⠀⠀⠀⢷⠖⠒⠒⠛⠁
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠓⠚⠛⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠳⣤⡀⠀⠀⠀⠀⢀⡞⠁⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠈⢙⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠙⠲⣄⠀⣠⠏⠀⠀⠀⣠⢾⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣙⣶⣾⣯⣤⡤⠤⠴⠶⠶⠶⠶⠶⣆⠀⠀⠀⠀⠀⠀⡼⠉⠓⠦⣀⠀⠀⠀⠈⠙⠃⠀⠀⢀⡜⠁⣼
⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠈⠓⢤⠀⠀⠀⠀⠀⢰⣋⡠⠞⠁
⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡴⠃⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀
⢸⡁⠀⠹⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠚⠋⠉⠉⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢠⠞⠲⢤⣀⠐⠀⠀⠀⠉⢹
⠈⣧⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠴⠶⠶⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⡰⠃⣀⡀⠀⠈⠑⠦⣀⠀⡠⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣤⠞⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠢⡀⢀⡜⠁⡼⠋⠙⠲⣤⡀⠀⠈⣹⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠢⢤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⠋⢀⡞⠁⠀⠀⠀⠀⠉⠓⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢀⡞

⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⡠⠖⠀⠀⠀⢠
⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠃⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⡰⠃⢀⣈⠀⢀
⠀⠁⢀⢀⡀⢠⡶⣤⢹⡆⠰⠒⢨⢀⢁⢡⡂⠦⣤⣤⢀⡂⠢⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡼⠀⣾
⠀⠀⠎⠀⠱⡀⠙⠋⢸⡇⣀⡁⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠀⠀⠄⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡣⢴⠇
⠀⡆⠀⠀⠀⢇⡠⠂⢸⡇⢿⣿⠀⡀⠈⠀⠀⠀⢀⠀⢰⡁⠀⢠⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠸⣁⡜
⠐⡃⠀⠀⠀⠞⠄⠀⠈⠃⠀⠠⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⡈⠤⠀⠀⠠
⠀⠆⠀⠀⠘⠀⠑⠘⠀⠰⠂⠰⠀⠰⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠠⠀⠀⠀⠀⠈⠀⠐⢠⠐⠃⠰⢍⣁⠀⠒⠈
⠀⠄⠀⠂⡀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠐⠀⠀⠀⠀⠙⠛⠟⠛⠛⢁⠀⠀⠠⡄⠆⠀⠀⣀⠒⠀⢌⢀⠒⠠⠀⠁
⠀⠀⠀⠀⠣⠄⠀⠀⠄⠉⠀⠀⣀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⣠⠀⠠⠄⡁⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠄⠀⠀⠀⠠⠀⠀⢀⠀⢰⠋⡆⡿⠀⣼⠉⠧⡏⠙⢲⣶⡄⢸⢻⠀⣸⢻⡆⣀⠄⠠⠂⠀⠐⠃
⢀⡈⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⣼⡇⡇⠀⡇⢲⡆⡧⠄⢨⡿⣅⢸⣏⠀⡿⠶⡇⠠⠆⠀⠴⠀⠐⡁⠄
⠘⠁⠀⠐⠇⠐⠀⠀⢀⠁⢸⡇⢸⠇⠧⠤⢻⡼⢹⣇⣀⢸⡇⣸⢸⠈⣧⡇⠀⠷⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠
⠀⣐⣀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⢉⣀⣉⣅⠈⠀⠀⠀⠀⠀⠀⠂⠀⠃⠂⠀⠀⠁⠀⣤⠤
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡏⠘⠁⠀⠀⠀⡀⠒⠀⠀⠀⢠⠀⠇⠈⠀⠀⣈⡉
⠀⠀⠀⠀⠀⠞⠆⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠒⠂⣄⣰⣇⡠⠀⡀⠐⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣈⡉
⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⡀⣈⡁
⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠃⠀⣠⠖⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠖⡳⢶⣤⡤⠁⠀⠀⠀⠀⣄⣀
⠀⠑⠄⠀⠀⠀⡄⣀⠄⡈⠁⠀⠈⠀⠀⢠⠀⠁⠈⠀⠀⢀⠀⠀⢀⡀⠀⠀⢀⡼⠁⡎⡀⠀⢡⠀⠀⠸⠀⠤⣅
⠀⡄⠀⠀⡀⠀⠘⠄⠚⠀⢀⠀⠀⠀⠀⠁⠈⠀⢀⠤⠤⣬⠀⠀⠎⠀⠔⠈⠉⠀⠘⠛⠁⠀⠀⢆⢀⠂⠀⠢⢄
⠀⠀⠀⠰⠀⠀⠆⠈⢄⠀⡈⠀⠈⠀⠀⠀⠀⠄⠀⣠⠞⠁⠀⠀⠀⠀⠐⠃⠐⢀⠔⠂⠀⠀⠀⠀⠁⠀⠀⠠
⣀⣀⣈⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣂⣀⣀⣈⣀⣊⣙⣲⣛⣀⣀⣃⣀⣐⣃⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠁⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⡠⠒⣶⠀⠘⡄⠀⠀⢀⠄⠊⠢⡀⣠⡶⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠉⢉⢤⠾⠋⠄⡰⠁⠀⠀⢿⣦⡀⠀⢈⠁⠀⠀⡠⠂
⠀⠀⠀⠀⠀⢠⡤⠤⠛⠃⠀⠀⠀⠈⠈⠩⠍⡇⠀⠀⠀⢺⡇⠀⠀⢂⣀⠀⠤⠄⠠⠇⠀⠀⠀⠀⣰⣕⠁⢀⡴⠊⠀⠀⠀⠀⠀⣨⡟⢀⠀⠀⠀⢪⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢙⣣⠤⠌⠀⠀⠀⠀⠀⠠⢾⡁⠀⢀⠤⣦⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠹⣧⡠⠤⣶⡆⠀⠁⢸⠉⠉⠁⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⡀⠈⠀⠜⠀⠀⠀⠀⠻⣁⡂⠆⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠈⠛⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣴⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠈⢣
⠀⠀⠀⠀⠀⢰⡟⠹⢑⡄⣴⣻⠋⠐⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣸⡇⠀⣶⠶⠒⡄⠀⠀⠀⠀⣿⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠑⠒⢋⣴⡷⠁⠰⢀⠜⠀⠀⠀⠀⠀⠀⢀⣄⡤⢤⠴⠶⠒⠚⠛⠛⠛⠓⡄⠀⠸⢇⣀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⢎⠞⠀⠀⢠⠊⢀⡀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣼⠟⠀⣠⡐⠑⡄⠀⠀⠀⠀⠀⠈⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡴⡳⠋⠀⠁⡔⠁⠠⡟⠭⠳⡄⠀⠀⠀⠀⠙⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠲⠊⠈⠻⠒⠁⠀⠀⠀⠀⠀⠀⠙⣒⣒⠁
⠀⠀⠀⠀⠀⠀⠻⡁⠀⢤⠎⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⢀⣠⣤⠤⢀⠀⠀⠀⠀⠀⠀⡴⡒⠀⠀⢠⣴⡂⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⢆⠀⠀⠀⣾⡏⠁⠂⠤⢀
⠀⠀⢠⢛⠋⠀⣠⡀⠙⡆⠀⠀⠀⢳⡇⠀⡁⠀⠀⠉⢹⡆⢸⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠄⣠⠖⠁⠀⠀⠀⠈⠁⠒⠤⢔⡀⠀⠑⢐⠄
⠀⠀⣇⡎⠀⠌⣸⣿⠀⢸⠀⠀⢀⢾⠋⢐⠁⠀⠀⠀⢅⡄⠸⢄⠀⠀⠀⠀⢠⣾⠟⠁⠀⢠⣋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡮⠟⢁⡴⠋
⠀⠀⣿⡇⠸⣴⡿⠃⢀⡎⠀⠀⠀⢀⣣⠘⡆⠀⠀⠀⠜⠟⠀⡎⠀⠀⠀⠀⠀⠉⠓⠦⢅⣀⠈⢉⠓⠶⢤⠀⠠⣴⡿⠛⠙⢀⡴⠋
⠀⠀⠈⠳⣄⣀⣀⡤⠊⠀⠀⠀⠀⢸⡟⠰⠇⠀⠀⣤⣤⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣤⠞⠀⠀⠘⢆⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⠤⠃⠀⠙⠃⠥⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⣀⢀⠀⣄⡐⠀⢀⢀⢀⠢⡐⠂⠀⠙⠙⠉⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠚⠀⠁⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠠⡀⠀⠐⠁⠢⠀⠀⠀⠁⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠀⠠⠀⠈⢠⡄
⠀⠀⠀⠀⠀⠀⢈⠂⢀⠀⠀⠂⠐⠈⠀⠀⠀⠀⡀⠀⠀⡆⢀⠢⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⢸⡇⡀⠀⠈⣤
⠀⠀⠀⠀⡀⠸⠮⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠠⠊⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠄⢸⡇⢰⣶⠀⣿
⠀⠁⠀⠀⠁⠃⠀⠀⣀⣤⠄⠀⠑⢄⡴⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠈⠉⠉⠈⠉⡉⠉⢩⠉⢩⡍⠉⡏⠉⠃⠐
⠀⢠⣂⡀⠀⠀⠀⠀⠛⠁⠀⠀⡰⠋⡠⠒⡀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⠠⣉⠕⠊⠝⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠊⠀⢈⠁⠥⠐⠂⠃⠀⠀⢀⡡⠤⠔⠒⠓⠋⠀⣀⠀⠀⢀⠀⠰⠢⠀⠒⠀⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢁⢊⡀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠰⠉⠀⠀⡀⠀⠐⠀⠆⠀⠀⠠⠀⡀
⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠘⠀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⠀⠀⠀⠦⣀⠘⠀⠀⠀⠀⠀⠀⠀⢠⢹⣿⣷⡄
⠀⠀⠃⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠀⠁⠈⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠈⠻⣿⠄
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠻⠶⣲⠴⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⠑⠒⣀⣤⣧⡄
⠀⠀⠀⡀⠀⠀⠂⣀⠀⠃⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠐⠫⡨⠆⠀⠀⠀⠀⠀⠸⠄⠨⠜⠉⠛⠛⠁
⠀⠀⠰⠀⠐⠚⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠈⠃⠀⠀⠀⠀⠀⠰⠱⠀⠀⠀⠀⢀⡀⠀⢠⠀⣀⠀⠀⡀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠄⡒⠇⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠘⠀⠙⠃⠐⠂⠇⠘⠘⠃
⠀⠀⢠⠂⠀⠀⠀⠀⢸⠀⠀⠃⡀⠀⡜⢀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡰⢢⠕⠂⠀⠀⠀⠀⠀⠀⢀⡈⠀⡈⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠈⠒⠀⠀⠀⠀⠈⢸⠀⣦⠆⠀⡏⠀⠑⠀⣴⣁⣀⣩⠆⠀⡠⢊⠔⠁⠀⢀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⡂⠀⠀⠀⠐⠀⢀⣠⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠛⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⣀⠀⡰⠁⠀⠀⢠⢠⠀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠘⡀⡄⠀⠀⠀⠀⠀⣮⣤⣼⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⢀⠀⡠⡀⢠⡄⡄⠀⡄⢠⡄⠀⠀⠀⠀⠈⠱⠃⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢀⡡⠶⠂⡀⢀⠀⠀⠀⠀⠀⠀⡀⡀⠤⢀⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⡠⠔⠁⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⠠⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠸
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠊⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠀⠐⠀⠀⠀⠈⢀⠈⡄⠀⠀⠀⠀⠀⠠⠈⠀⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⡇
⠀⠀⠀⠀⣠⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⢀⡴⠖⠒⠒⠛⠒⠃⠀⠀⠀⠳⠒⠒⠒⠒⢲⡄
⠀⠀⠀⡞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢳⠀⢇⣀⣀⣀⡀⠤⠤⠤⠤⠤⠤⢤⡇⠀⠀⠘⡆⠀⠓⠂⠉⠉⠉⠙⠀⠀⠀⢠⠖⠒⠒⠒⣲⠏
⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠎⠀⠀⠀⠀⠳⠤⠤⠤⠤⢤⠀⠀⡇⠀⠀⠈⡶⠶⠒⠚⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢃⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣈⣉⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢇⡀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠙⠲⣄⠀⠀⠀⢠⠞⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⠽⠋⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡧⣀⠀⠀⠈⠙⠦⣴⠋⠀⠀⡠⢺⠁
⠀⠀⠀⠀⢠⡴⠖⠒⠒⠒⠛⠛⠛⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⠀⢼⡀⠀⠑⠢⣄⠀⠀⠀⠀⢀⠞⣡⠞
⠀⠀⠀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠉⠲⣄⣠⠎⠀⠀⠀⠀⠈⠛⠧⣄⡀
⠀⠀⠀⢾⠀⠘⠦⠤⠤⠤⠒⠒⠒⠒⠒⠒⠚⠉⢉⡽⠋⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⡰⠒⠢⢄⡀⠀⠀⠉⣹
⠀⠀⠀⠘⣦⠤⠤⠤⠤⠤⠤⠶⠖⢒⣶⡚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⢀⠞⣠⢤⣀⠀⠉⠲⢤⣔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢏⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠙⠦⠊⡰⠃⠀⠈⠓⠦⣄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠤⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⣀⠀⣀⣂⣄⠀⡈⠀⣀⠀⠀⠀⣄⠀⠀⠀⠀⠁⡘⠃⠀⢀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠉⠏⠥⠈⠉⠀⠀⠀⡃⠀⠉⠠⠄⠁⠀⠀⠀⠀⠀⠠⡏⠠⠔⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⢂⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⠀⡀⠂⠀⠄⠠⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠓⢾⡇⠀⠄⠀⠰⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠡⠀⠂⠀⠀⠀⠐⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠠⠐⠀⢀⠀⠀⢠⡥⠀⢀
⠃⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠈⠀⡡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠐⠂⠄
⠀⠀⠂⠀⠀⡠⠊⠀⠀⠀⠀⠘⢘⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⣀⠀⠀⠀⣀⠀⠀⠠⠒⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠊⠀⠐⠀⠀⠀⠀⠁
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠸⠀⠅⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠴⠶⠶⠄⡂⠀⠂⠶⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠃⠴⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠠⠄⠙⠢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡉⠀⢀⣀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠠⠀⡐⠚⡃⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠻
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃
⠀⣀⡀⠀⠀⠀⠀⡀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⢀⠂⡀
⠀⠙⠟⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠇⠘⠡⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠀⠀⠜⠁⠀⠠⠀⠐⠀⠀⠀⠂⠀⡠⠠⠀⠁⠻⠀⠑⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰
⠀⠀⡀⠀⣀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣋⣁⣆⡀⣀
⠀⢳⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⣏⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⣿⣿⣼⣯⣟⣿⣽⣿⣿⣿⣿⢿⢫⣿⣷⣾⣷⣶⣶⠄
⠢⡈⠒⠀⡀⠀⠀⠀⠀⠠⣀⡎⠀⠀⠈⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣙⣙⢳⣇⢹⣿⣿⣿⣿⠟⠃
⢀⡀⠄⠋⠃⠐⠃⠠⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡉⠀⠛⠛⠫⠙⠉⠻⠛⠿⠛⠋⠛⠟⢉⡏⠉
⠈⠟⠀⠀⠰⡀⠁⠠⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠒⢀⣀⠄⠓⠀⠀⡀⠀⠀⢠
⠀⣷⠂⠀⠀⡇⠀⠀⢀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠃⠀⠀⠉⠓⠂⠀⠠⠄⠀⠀⠐⠂
⠀⠉⠁⠀⠀⠀⠀⠀⠂⠐⠀⠂⠉⢀⠐⠒⠀⠀⠀⣀⡀⢀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡜⡇
⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⣶⠚⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⢳⣄
⠀⠀⠀⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢸⡇⠀⠀⠀⠀⠀⢸⣤⠤⠶⠶⡆⠀⠀⠀⡖⢒⣒⣲⣧⡿
⠀⠀⠀⠸⣗⡚⢛⣛⣋⣉⣉⣉⣩⣭⣥⣼⠃⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⣧⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠶⢶⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠦⣄⡼⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠀⠀⠀⢀⠞⣻
⠀⠀⠀⠀⠀⠀⢀⣀⣈⣛⣛⣋⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠻⣾⡁
⠀⠀⠀⠀⣾⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⣿⣀⣀⣀⣤⠤⠤⠤⠤⠤⠤⠾⡿⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡴⣣⣝⡲⢄⡀⣠⣾⠇
⠀⠀⠀⠀⠈⠿⠗⠒⠒⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢻⣦⣄⣠⢞⡴⠋⠈⠙⢦⣍⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣭⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⢄
⠀⠙⠋⠂⠀⠀⠀⠀⠈⠰⢄⠀⠐⠄⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠲⣤⣐⣀⣀⣀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠄
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠢⡊⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠒⠈⡁⠀⠀⠈
⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⢄⠀⠀⠀⣀⡀⡀⠀⠠⠤⠠⠤
⠀⠈⠀⠈⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠂⠤⠄⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⡀
⠀⠐⠀⠒⠃⠈⠁⠀⡀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠉⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠂⠈⠓⠒⠉⠉⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠲⣤⠀⢠⡄⠈⠢
⠂⠀⠀⡀⢀⡀⠀⠀⠀⠀⠈⠁⠁⠈⠑⠋⠂⠊⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠳⠀⠀⠜⡀⢀⠳⢀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠐⠲⠦⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⠀⠂⠀⠀⠀⠀⡂⠘⠀⠀⠀⠀⠈
⠕⠀⠀⠄⠁⠀⠀⠀⡀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠠
⠀⠐⢆⠀⠀⠀⠀⠀⠐⠂⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢊
⠀⠀⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠄⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⡠⠃⠀⠀⠀⡐⠀⠀⢀⡀⠀⠉⠉⠙⠀⠀⠘⠀⠈⠆
⠀⠄⠀⠀⢀⠀⠀⢠⢀⡀⠀⠀⠈⠰⢤⡀⠈⠀⠀⠐⠀⠀⠀⢀⠉⠈⠓⠒⠛⠢⠤⢌⠀⠀⠀⠁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠂⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠱⠁⠀⠈⠙⠦⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠔⠒⠒⠊⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠐⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⣿⢢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣿⢸⣼⣿⡇
⠀⠀⠀⠀⠀⢸⠃⡟⠀⠀⠰⠄⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠃⠀⠀⠀⣿⠸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠐⠀⠀⠇⠰⠂⠀⠁⠀⠀⡀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⣿⢀⣼⣿⡇
⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠛⠀⠀⠀⢀⠀⡀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⣿⡇
⠀⠀⠀⠀⠀⡟⠸⠇⠀⠠⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⡠⢀⠀⠀⠀⠀⠀⣿⢀⠀⣿⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠀⠠⠂⡐⢀⠂⠀⣄⢀⠀⢠⡟⢸⢠⣿⡇
⠀⠀⠀⠀⠀⡇⢀⠀⠀⣠⡤⠀⠀⠀⢰⡿⠀⡖⢰⠀⠀⠀⢠⠄⠀⠈⠁⠀⠘⠁⠀⠀⢸⡇⣼⢘⣿⡇
⠀⠀⠀⠀⠀⡇⢸⠀⠈⠁⠂⠀⠀⢠⡿⠀⠀⢷⠈⣹⠀⠀⠨⠀⠀⠈⠐⠠⠀⠀⠀⠀⢸⡇⡋⣼⣿
⠀⠀⠀⠀⠀⡇⢸⠀⡀⠀⣄⠀⠀⣼⠁⠀⠀⠸⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠆⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⠀⡇⣾⠀⠁⣀⠂⠐⠻⠃⠀⣀⠀⠀⢰⠋⠉⡆⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⠀⡇⠛⠀⠀⠀⠀⣸⠏⠀⠿⠷⠀⠀⠈⠂⠀⠁⠇⠀⠀⢀⠔⠩⡇⡠⣁⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⡎⠀⠀⢀⢀⣤⡆⣴⢠⢤⡰⠤⣀⠰⠁⠀⠀⠔⠁⠾⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⢸⠇⢠⠀⠀⢘⡟⠀⣴⣷⢻⣾⣿⡇⡏⣿⢸⢃⡠⠊⠀⠀⠀⣀⣀⡠⠂⠀⢸⡇⡇⢸⣿
⠀⠀⠀⠀⢸⠀⣾⠀⢤⡾⠁⡸⠟⣛⣸⣿⣿⠇⢿⣿⢸⣨⣶⠀⠀⠐⢻⠀⢸⠀⠀⠀⣸⡇⡇⢸⡇
⠀⠀⠀⠀⢸⠀⡏⠀⢀⠤⡀⠃⠀⠀⣠⢄⣹⣿⢘⡩⠮⣬⠒⡆⠀⠀⡾⠀⢬⠀⠀⠀⣿⡇⣇⣼⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡄⠀⠐⢸⣿⣾⢡⣿⠜⡬⠿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠰⠐⠁⠀⠀⠈⠉⠁⠉⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⠶⢿⠁⣿⣿⡇
⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠻⡿⠿⡟⣞⠿⠯⠽⣦⣾⣶⡘⢦⣿⣿⡇
⠀⠀⠀⠀⠘⣴⣖⡺⢿⣿⣿⣻⣻⣝⣯⡧⠽⠦⠤⣄⣻⣿⠶⠤⠤⠐⠒⣊⣉⣩⡿⠥⠴⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠪⡻⣾⡉⠡⠤⠄⠒⢒⣂⣉⣉⣁⣭⣄⣀⣀⣤⣤⣼⣷⣾⠷⠶⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣿⡿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠐⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢴⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⡠⠒⣶⠀⠘⠀⠀⠀⡀⠐⠁⢄⣀⡴⠚⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠂⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⢰⠞⠉⢀⡜⠀⠀⠀⠿⣦⡀⠀⠈⠀⠀⢀⠜
⠀⠀⠀⠀⠀⣠⠐⠒⠛⠁⠀⠀⠀⠉⠉⠥⢹⠀⠀⠀⠘⣷⣀⣀⠀⠠⠤⠤⠄⠀⠊⠀⠀⠀⠀⣰⡴⠁⢀⠖⠁⠀⠀⠀⠀⢀⡼⠋⡀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⠀⠀⠀⢀⣀⣀⣸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⣈⡃⢒⠂⠀⠀⠀⠀⠀⠐⢿⣀⠀⡠⠺⣦⡀⠀⡳
⠀⠀⠀⠀⠀⠻⠧⠔⠒⣾⡀⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠠⠀⠀⠚⠀⠀⠀⠀⠛⠤⠥⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⡀⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣤⠤⡀⠀⢀⣴⡟⡁⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠒⡀⢀⡀⠀⠀⠀⠀⢀⣾⠁⠀⡇
⠀⠀⠀⠀⠀⠸⣇⢈⡼⢠⣯⠏⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢠⡶⠲⠾⠇⠀⠛⠋⣉⠄⠀⠀⠀⠈⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⡴⡻⠃⠀⢁⠔⠁⠀⠀⠀⠀⠀⢠⣶⠞⠛⢛⠉⢉⠁⠀⠀⠐⠀⡇⠀⠈⠃⢢⣤⠀⠀⠀⢜⠁⠀⠀⠀⠀⠀⢹⡆⠀⢰
⠀⠀⠀⠀⠀⠀⢀⢞⡞⠡⠄⢠⠊⢀⡴⢤⡀⠀⠀⠀⠀⢾⣀⣀⣀⣀⠀⠀⠤⠤⠤⠤⠖⠀⠀⠀⠻⡁⡠⢶⣦⣈⠆⠀⠀⠀⠀⠀⠸⣇⣀⡸
⠀⠀⠀⠀⠀⠠⣧⠊⠀⠀⡔⠁⠀⠸⣄⣁⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠖⠦
⠀⠀⠀⠀⠀⠀⠉⠢⠨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠤⡀⠀⠀⠀⣠⡤⠄⡀
⠀⠀⢀⣴⡿⠋⠉⠐⢤⠀⠀⠀⠀⣤⠁⠠⠀⠀⠻⠷⣄⠈⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⣡⠟⠀⠀⠀⠛⠧⣀⠀⠉⠐⠀⠄⢀
⠀⠀⡌⡜⠀⡔⢹⣧⠀⡇⠀⠀⢀⣿⠇⡸⠀⠀⠀⠰⢛⠀⡇⠀⠀⠀⠀⠀⣠⣾⢟⠁⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⣉⣵⡦⠀⣠⠞
⠀⠀⣶⠃⢰⣠⣿⠇⠀⡇⠀⠀⠛⠣⡀⢥⠀⠀⠀⢈⢿⠆⢠⠂⠀⠀⠀⠈⠛⠧⣄⡠⠈⠓⠢⠤⣄⡀⠀⠀⣠⢴⡯⠻⠉⣠⠞⠁
⠀⠀⠻⢇⠈⠉⠁⣀⠜⠀⠀⠀⠀⣿⡇⣸⠀⠀⠀⣈⣠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢬⣀⠀⡼⠀⠈⢻⡁⠀⣠⠞⠁
⠀⠀⠀⠀⠉⠒⠉⠁⠀⠀⠀⠀⠀⠻⠇⣀⠱⠀⠘⠿⣁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠖⣂⣀⣀⣀⣀⣀⣐⠢⠀⠀⠐⠒⠖⠂⠀⠀⠀⠀⠄⢀⡂⠠⠄⠀⠀⢰⢈⠉⠭⠭⢉⠒⠊⡉⠭⠭⠍⢰⠀⠀⠀⠀⠙⢮⢄⠀⠀⣀⠀⠀⣀⠀⡇
⠀⠀⡀⣇⣀⣀⣂⣀⣀⡸⠘⠀⠀⠘⡉⠉⠃⠀⠀⠀⠁⠜⣁⣀⠑⡌⠀⠀⢐⠐⢒⣀⡒⠒⢨⠀⠒⠒⠀⠒⢸⠀⠀⠀⠀⡠⣄⡙⢦⡄⠈⠲⡀⢸⠀⡇
⠀⠀⠇⠖⣤⠒⣤⠢⡔⢢⠘⠀⠀⠃⡇⠘⠀⠀⠀⠈⠀⢆⠃⠀⠠⠃⠁⠀⠠⢈⠥⠤⠤⢌⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠑⢢⠇⠀⠙⢖⡀⠈⠋⠀⡇
⠀⠀⡆⠟⣿⠛⣽⠛⡝⢫⢘⠀⠀⢂⡇⢸⡀⠀⠀⠀⠁⠀⡶⣖⡇⠈⠀⠀⠈⠐⠂⠐⠒⠒⠘⠀⠒⠀⠐⠒⢸⠀⠀⣴⠏⣴⠏⠀⠀⠀⠀⠙⢵⡀⠀⡇
⠀⠀⠇⡛⣿⠛⣿⢛⡇⢸⠠⠀⠀⠘⢭⡭⠃⠀⠀⠀⠀⠀⠛⠒⠁⠀⠀⠤⠘⠮⠍⠭⠭⠉⠾⠏⠭⠩⠭⠭⠜⠀⠸⣟⣾⠏⢀⢠⠤⠤⠠⣄⠀⠙⢇⡇
⠀⠀⠣⠭⠤⠭⠤⠥⠬⠥⠞⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠜⣌⡣⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠉⠀⡔⠃⢀⠤⠤⢤⡀⠙⢆⠀⠁
⠀⢀⣴⠖⡖⠸⠰⢒⢲⣦⠀⠀⠀⡠⠒⠉⣻⡯⢋⢓⣄⠀⠀⠀⠀⠀⡧⣭⢴⠀⠀⠀⠀⠀⠀⡜⢉⢣⠀⠀⠀⠀⠀⠀⣨⠀⢰⠁⠀⠀⠀⢙⡄⠘⡆
⠀⢸⠉⠉⠉⠁⠉⠉⠉⠩⠀⠀⡜⠀⠀⠀⡿⣗⣕⣕⣝⡆⠀⠀⠀⡸⣠⠍⣆⢧⠀⠀⠀⠀⡸⠀⢨⠀⢣⠀⠀⠀⠀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⠀⠙⠉⠉⠉⢙⠀⠀⠀⢠⢡⠃⠀⠘⡌⠆⠀⠀⡰⠁⠀⠘⠀⠀⢣⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣄⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠖⠋⠀⠀⠀⠈⠺⠀⠐⠻⢿⣯⣏⣿⡽⠾⠂⠀⠀⠀⠀⣤⠄⢠⠀⢶⠌⢠⢠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠑⠒⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠘⠧⠠⠄⡀⠀⣤⠤⠜⠀⠀⠀⠀⠀⠀⠉⠉⡉⣱⠀⠀⠘⠃⠈⠃⠸⠋⠀⠁⠃⠀⠀⠀⠀⠀⠁⠖⠒⠒⡒⠒⠒⢢⢹⠀⠀⠀⣀⠤⠦⣼⢒⢶⣄
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⠁⠀⠇⠾⠀⠀⠀⠀⢀⣤⠤⡤⢄⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠂⠚⢸⠀⢀⠞⠀⠀⠀⣿⢕⢕⠝⣧
⠀⠈⡎⢁⠤⣁⠤⣉⠤⡌⠁⡇⠀⠀⠀⢰⠀⢠⠉⡇⠀⠀⢀⠚⠁⣀⠤⠤⣀⠈⠳⡄⠀⠀⠀⠀⠐⣁⢿⣨⢯⣹⢏⡹⢸⠀⢸⠀⠀⠀⠀⠿⠷⠿⠿⡞
⠀⠀⡏⠻⡠⠻⠄⠟⠄⠏⠙⡇⠀⠀⠀⢈⠀⢸⠠⡇⠀⠀⣌⠀⡜⠀⠀⠀⠀⢱⠀⢙⠀⠀⠀⠀⠸⣥⠿⣬⢯⣼⡇⢹⢸⠀⠘⡄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⣇⣰⠓⣦⠚⣦⠚⣆⣨⡇⠀⠀⠀⠋⠀⠘⢰⠃⠀⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠈⠉⠀⠀⠀⡘⠤⠿⠬⠧⠼⠧⠜⣸⠀⠀⠈⠳⠤⠀⠀⠤⠚⠁
⠀⢀⠣⡘⢒⡉⢒⣉⠒⣁⠜⡇⠀⠀⠀⡆⠀⡆⣹⠀⠀⠉⠉⠁⠉⠉⠉⠉⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢁⣀⣀⣀⡀
⠀⠀⢩⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠾⣥⣁⣇⡏⡯⠽⡆⠀⠀⣋⣙⣋⡄
⠀⠀⠀⠘⡦⡀⠀⠀⠀⠀⠀⢀⠀⡠⠬⠄⡀⡀⠀⠀⠒⣉⠭⠤⠍⡁⡒⢈⠉⠤⠭⢉⡂⠀⠀⢀⠞⣘⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢡⠀⡌⡃
⠀⠀⠀⠀⠀⠹⢢⡀⠀⠀⠀⡀⢸⢠⠒⢦⢹⢀⠀⠀⠀⠐⠂⠀⠒⠂⡁⠐⠒⠀⠀⠒⠆⠀⠀⠈⡮⣊⢴⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡇⡁
⠀⠀⠀⠀⡷⢄⠀⠩⢢⡀⠀⠠⠌⢆⠀⢠⠋⠄⠀⠀⠄⡉⠤⠤⠭⠁⠃⠐⠀⠐⠀⠂⠀⠀⠀⣸⣠⣹⣄⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣀⣇⠃
⠀⠀⠀⠀⣇⣀⣑⡄⠀⠊⢦⡀⠀⢸⡻⢹⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⡃⠐⠒⠒⠒⠂⠆⠀⢰⢡⠃⠀⠰⡘⡄⠀⠘⢇⣀⣉⡀⢈⣁⡠⠇⠀⠀⠑⠶⠊
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠬⠼⠦⠀⠉⠁⠀⠀⠀⠀⠻⠭⠭⠉⠩⠅⠿⠨⠍⠉⠭⠡⠟⠀⠗⠃⠀⠀⠀⠑⠧⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⢀⠤⠐⢶⣟⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡔⠒⠒⢌⠄⠀⠀⠀⡜⢸⠈⡄
⠀⠀⢠⠃⠀⠀⣿⣷⣿⣛⣧⠀⠀⠀⠛⠀⠉⠀⠛⠁⠐⠘⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⣀⢀⡀⣀⡀⠀⠤⢸⠠⠋⠱⢨⠠⠀⠀⡜⠀⢸⠀⠘⡄
⠀⠀⢸⠀⠀⠀⠉⠉⠉⢙⠇⠀⠀⠀⣠⢲⠆⠤⠤⠤⠄⠒⠐⡄⠀⠀⠀⡇⠥⠤⠤⠥⠤⠤⠡⠄⠀⠀⠐⡇⠀⠐⠂⢱⣀⣠⠁⠂⠀⣼⣀⣀⣸⢀⣀⣘⡄
⠀⠀⠀⠳⣄⠀⠀⢀⡠⠊⠀⠀⠀⠀⠙⢼⢂⠂⢒⡒⡂⣠⡀⠇⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠐⠃⠀⠀⠀⠸⢝⡻⠀⠀⠀⠙⠛⠓⠷⠖⠒⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠚⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠈⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⣿
⠀⠀⣤⡤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠴⠧⠤⠶⠀⠀⠀⠀⠻⠤⠤⠤⠴⠶⠶⣦
⣠⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠞⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡆⠀⠀⠀⣏⠀⠸⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⢸⡆⠀⠧⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠚⠛⠛⣿⠃⠀⠀⠀⢹⡀⠀⠉⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡏⠉⠉⠉⠉⣩⠟⠁
⠀⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠈⠧⠤⠤⠤⠤⠴⢦⠀⠀⢸⠀⠀⠀⠀⢷⠖⠒⠒⠛⠁
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠓⠚⠛⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠳⣤⡀⠀⠀⠀⠀⢀⡞⠁⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠈⢙⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠙⠲⣄⠀⣠⠏⠀⠀⠀⣠⢾⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣙⣶⣾⣯⣤⡤⠤⠴⠶⠶⠶⠶⠶⣆⠀⠀⠀⠀⠀⠀⡼⠉⠓⠦⣀⠀⠀⠀⠈⠙⠃⠀⠀⢀⡜⠁⣼
⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠈⠓⢤⠀⠀⠀⠀⠀⢰⣋⡠⠞⠁
⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡴⠃⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀
⢸⡁⠀⠹⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠚⠋⠉⠉⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢠⠞⠲⢤⣀⠐⠀⠀⠀⠉⢹
⠈⣧⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠴⠶⠶⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⡰⠃⣀⡀⠀⠈⠑⠦⣀⠀⡠⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣤⠞⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠢⡀⢀⡜⠁⡼⠋⠙⠲⣤⡀⠀⠈⣹⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠢⢤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⠋⢀⡞⠁⠀⠀⠀⠀⠉⠓⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⠀⠷⠿⠖⠒⠛⢻⣦
⠀⠀⠀⠀⣶⠖⠒⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⣷
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢹⠆⠀⠀⠀⠀⠀⠀⠀⢹⣦⡶⠶⠶⠶⢶⠀⠀⠀⠀⠀⡟⠛⣋⣛⣻⣭⣤⡿
⠀⠀⠀⠀⠸⣷⡶⠶⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⢋⣉⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠛⢻⡆⠀⠀⠀⠀⡇⢸⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠙⠷⣶⡶⠶⠶⠶⠿⠿⠟⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣉⣉⣉⣥⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⡴⠟⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠙⠳⣦⣀⣴⠏⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⡴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣛⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣬⣭⣭⣥⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡆⠀⠀⠀⠀⠀⠀⠙⠻⣯⣀
⠀⠀⠀⠀⠀⢰⡟⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⡞⠙⠶⣄⡀⠀⠀⠀⣰⢿⡆
⠀⠀⠀⠀⠀⠘⣷⡴⠶⣖⣒⣛⣛⣋⣉⣉⣉⣉⣩⣭⣭⣵⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠀⠀⠀⠀⣰⢋⣴⠟⠷⣦⡙⠲⣤⡼⣱⡟
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⣉⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡶⣄⣠⠞⢡⡾⠃⠀⠀⠈⠻⢶⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣥⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡶⠶⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⠒⠚⠛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⠤⠿⠤⠼⠀⠀⠀⠀⠘⢧⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⣠⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⣆⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢻⠀⠸⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠒⠒⠒⠒⣺⠇⠀⠀⠀⢸⡄⠀⠉⠉⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⣹⠟⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⡄⠀⠐⡇⠀⠀⠀⠘⡶⠶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⡇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠸⠶⠖⠒⢲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣤⣤⣤⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⣄⡀⠀⠀⠀⠀⠀⢀⡞⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠙⠦⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠙⢦⣀⠀⠀⣠⠏⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣤⠞⢋⣀⣀⣀⣀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⡟⠦⣄⡀⠀⠀⠀⠈⠳⠶⠃⠀⠀⠀⡴⠋⣿
⠀⠀⠀⠀⠀⢀⣴⠶⠒⠒⠒⠚⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⣠⠞⢀⡴⠃
⠀⠀⠀⠀⠀⡼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⢀⡠⠃⠀⠀⠀⠀⠀⠙⠺⣯⣀
⠀⠀⠀⠀⣴⠃⠈⣇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠶⠖⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠉⠛⠲⢦⡀
⠀⠀⠀⠀⢻⡀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣠⠋⠀⠈⠙⠲⣄⡀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠈⠷⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⢋⣩⠽⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢄⠀⠀⠀⡼⠁⣴⠷⣤⣀⠀⠀⠉⠲⣤⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡁⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠑⢤⠞⢀⡼⠁⠀⠀⠉⠳⢦⣄⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠖⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⢸⣿⢱⢤⣤⣄
⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡖⢃⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⣀⣤⠀⠀⠀⢸⣿⢸⣼⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠤⠀⠀⠀⠀⡅⠀⠰⠀⠈⠀⠀⠀⡇⠀⠀⠀⠆⠈⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⠀⡏⠀⠀⠀⠀⠀⠠⠀⠀⡇⠀⠦⠐⠂⢃⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣿⢀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⢰⡇⠀⠀⠀⠀⠁⠘⠀⠀⠀⠨⠄⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠉⣿⡿
⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⠀⠀⠀⠀⠰⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⠀⣿⡇
⠀⠀⠀⠀⠀⢠⡇⠈⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠃⢠⠊⠡⠀⠀⠀⠀⠀⢸⡏⢰⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⣁⡔⠀⢀⡵⠐⠀⠀⢸⡇⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⣠⡴⠂⠀⠀⠠⣐⡿⠀⢸⠁⣇⠀⠀⠀⠀⡖⠤⢀⠀⠀⠀⠈⠣⠀⠀⠀⣾⡇⡟⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠁⠐⠀⠀⠀⣠⡿⠀⠀⠸⡇⢀⡽⠀⠀⠀⠁⠀⠀⠈⠐⠠⢀⠀⠀⠀⠀⣿⡇⡇⣴⣿⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢀⢀⡀⠀⢀⣜⠁⠀⠀⠀⠇⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⡶⠀⠀⣿⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⡟⠀⠉⠀⡤⠁⠠⢿⠇⠀⠀⠀⠀⠀⡰⠚⠛⡄⠘⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⣿⠀⡇⣿⣿⠃
⠀⠀⠀⠀⠀⢸⠃⣯⠀⠀⠈⠀⠀⣴⡇⠐⣷⣿⡄⠀⠀⠣⡀⠀⠇⡆⠀⠀⠀⠀⡠⢚⢦⠀⡔⠀⠀⣿⠀⡇⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠉⢀⠀⢀⠀⣀⢀⠀⠀⡁⠀⠀⡠⠊⠀⠈⢋⠞⣙⡆⠀⣿⠀⡇⣿⣿
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⣰⡿⠀⢠⣄⣱⣴⣟⣻⢰⣧⣾⢹⡦⠉⢱⠖⠁⠀⠀⠀⠉⠀⠈⠀⠀⣿⠀⡇⢻⣿
⠀⠀⠀⠀⠀⣿⠀⣾⠀⠀⠀⣼⠁⣰⣷⣿⠸⣿⣿⣿⢸⢻⣿⢸⢠⠔⡁⠀⠀⢀⣠⠤⢤⡖⠁⠀⠀⣿⠀⡇⢸⣿
⠀⠀⠀⠀⠀⣿⠀⡿⠀⠲⠾⠃⠀⠝⢋⣉⣰⠿⢾⡟⡸⢸⣿⣸⣤⡾⠇⠀⠀⠁⡿⠀⢰⠁⠀⠀⠀⣿⢰⡇⢸⣿
⠀⠀⠀⠀⠀⡿⠀⡇⠀⠀⡠⢄⠈⠁⠀⠀⣠⢖⡥⣿⡇⡏⡔⢳⣬⡞⣳⠀⠀⠀⡇⠀⠼⠄⠀⠀⢀⣿⠸⣇⣼⣿
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠈⠀⢀⠀⠀⠂⢸⣿⣾⡇⣿⣧⢇⡝⠯⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢰⣿⣿⣿
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢠⡤⠔⠀⠀⠀⠉⠛⠓⠘⠋⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣾⣿⢸⣻⣿⣿
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡶⢶⡶⣿⣿⣯⢿⣿⡛⢻⣏⣉⣧⡀⢿⣸⣽⣿⣿
⠀⠀⠀⠀⠀⢇⢀⣥⣤⣤⣴⣶⣶⡾⣿⠿⡟⢟⠻⣏⢉⣉⣉⣤⣤⣭⡉⢈⣈⣃⣭⠭⠥⠟⢿⡟⢋⣀⠙⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠺⡳⣬⣙⣿⡿⠷⠿⠚⢓⣛⣛⣩⣭⡭⠭⠶⠞⠛⠁⠉⠉⠁⣤⣄⣶⣖⣚⣛⣭⣭⣭⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠪⡻⣦⣀⣚⣋⣉⣭⣭⣤⣴⣶⣶⣶⣿⣧⣴⣶⣶⠿⠿⠿⠟⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⣀⠀⢠⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣠⠖⠂⠀⠄⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠈⠀⠄⠀⠃⠈⠈⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⢱⡄⠀⠈⠁⠀⠀⡆⠀⠀⠉⠀⢶⠦⠄⠀⠑⠒⠊⠉⠀⠀⢀⡆⠀⡀⠸⢠⠈
⠀⠄⠈⠂⠠⠇⠐⠃⠀⠀⠠⢤⡀⠀⠀⠀⠀⡄⢠⠀⡄⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠐⢀⠄⠣⠀⡴⢶⡀⠀⣴
⠀⠀⠀⢀⠤⣄⠀⢰⡛⢳⡄⣾⡇⠀⢱⠹⢰⠅⣸⢠⠁⣮⢀⠷⢆⣶⢲⢠⠇⠘⢂⠀⠀⠀⢀⡆⠐⠀⠀⠀⠀⠆⠀⠁⠘⠀⠀⠀⠀⠀⠁⡞⠀⢸⡏
⠀⠀⢠⠋⠀⠈⣆⠈⠻⠿⠃⣿⡇⠀⠈⠃⠀⠀⠀⠀⢀⠁⠈⠀⠋⠁⠘⠘⠈⠀⠀⠀⡀⠐⠈⠁⠆⠁⠀⠀⢀⢤⠂⠀⠀⠀⠀⠀⠀⠄⠘⢧⣤⣾⠃
⠀⢠⠀⠀⠀⠀⠸⡆⠀⣀⠄⢸⡇⣼⣿⣦⡀⠀⠀⠠⠀⠑⠄⠀⠀⠀⠀⣀⡄⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡠⠀⠀⠒⠀⠀⠀⣰⠖⢰⡏
⠀⣿⠀⠀⠀⠀⠀⢧⡜⠁⠀⢸⡇⠈⠻⠛⢁⡔⠆⠀⠀⠀⠀⠀⠀⠄⠀⠜⠦⠀⡀⡸⢻⠉⠀⠀⠀⠀⠀⠀⠀⢸⡔⠀⠀⠀⡀⠀⢠⣄⠘⠶⠚
⠐⠃⠀⠀⠀⠀⢠⠋⡆⠀⢀⠄⠁⠀⡀⠰⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠂⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠋⠀⡁⠀⣀⠁⠀⡈⠲⠤⠤⠀⠀⢰
⠀⢦⡀⠐⠒⠒⠟⠀⠘⠐⠒⠀⢠⡧⠉⢠⡀⠄⢦⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠆⠀⠂⢀⠀⠜⠀⠰⡉⠙⠒⠂⠰⠤⡌
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⣈⠣⠈⠀⣀⠀⠀⠀⢀⠀⠰⢾⣿⣿⣷⣶⣶⠾⠀⠀⡆⠄⠀⠐⡀⠀⠀⠀⠘⢂⠄⠀⠢⠀⢉⣉⠀⠂⠠⠁
⠀⠓⠂⠈⠀⢦⠀⢀⠀⠐⠀⠀⣀⡔⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠋⠈⠉⠀⠰⣤⠂⠡⡀⠓⠈⠁⠀⠀⠠⡴⠁⠀⠀⠒⠂⠄⠈⠉⣁⠂
⠀⠀⠀⡀⠀⠈⠢⠀⠀⠀⠀⡆⠘⠀⠀⠀⢈⡄⠈⣅⠀⠈⢀⣀⠂⠀⠀⠇⠈⠠⠄⠀⠀⠈⠒⠀⠀⢀⣄⠀⠰⠤⠊⡤⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⡄⠈⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⢠⡟⣷⢀⡇⠀⢀⡿⠉⣷⢸⠛⠲⠆⣶⢦⡀⢸⡟⣧⠀⢀⡿⢻⠀⠀⢀⠀⢀⡔⠀⠀⡩⠖⠀⠀⠀⠈
⠀⠈⠦⠁⠀⠀⠀⡀⠀⠑⠀⠀⣴⠀⠀⡿⠀⣿⢸⡇⠀⢸⡇⣄⡀⢸⠀⠀⠀⣿⣼⠃⢸⣧⡟⠀⣸⣇⣸⡇⠙⣉⡄⠀⢀⢀⠀⢀⠞⠀⠀⠀⡆
⠐⡦⠀⢀⠀⣀⡀⠉⠀⠀⠀⠸⠧⠀⢸⡷⠚⣿⢸⡇⠀⢸⡇⢨⡇⣾⠙⠃⠀⣿⠋⢷⣸⡟⣦⠀⣿⠉⠹⡇⠐⠋⢀⠀⢚⠁⠄⠀⠐⠊
⠈⠁⠂⠈⠀⠛⠀⠐⠂⠀⠀⣡⠄⠀⢸⡇⠀⡿⠸⠧⠴⠘⢧⠾⠁⣿⣀⣀⡀⣿⠀⣸⢛⡇⢸⣇⣿⠀⠀⠟⠀⠡⠨⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠇⣧
⠀⠄⠰⠀⠠⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠛⡁⠈⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠠⠄⠀⠀⠈⠀⠀⣀⣀⡀
⠀⠃⠘⠘⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡠⠒⠈⠉⠁⠂⠀⠀⡾⠋⢳⡟⣹⡆⠀⠀⠀⠀⠉⢠⣀⡀⠀⠀⠀⠀⠀⢠⡀⢰⠀⠀⠀⠙⠲⠤
⠀⠀⠰⠀⠀⠀⠑⣄⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢠⠀⠂⠀⠸⠀⠀⠀⠀⠀⠀⠀⡆⢘⠁⠀⠀⠀⠀⠬⣤⡄
⠀⠀⠀⡀⠀⡀⠰⡞⢇⠀⠀⠀⡀⠀⠀⠀⡰⠃⠀⠀⠀⠈⠀⠀⠘⢋⢠⣆⣠⢿⡀⣠⠂⢀⡀⠉⣀⣀⣠⠄⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⡐
⠀⢀⠰⠀⠀⠀⠀⠀⠀⠓⠄⡀⠀⣀⠀⠊⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⠀⠉⠀⠀⠀⠛⠉⠁⠄⠀⠡⠇⠐⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠨⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⡀⠀⢔⡤⠠⠄⠀⢀⡤⠀⠀⠀⠁⠰⠀⢰⠀⠆⠂⠐⠀⢀⣠⡄⣀⡀⠀⠀⣴⠀⠀⠀⠀⠀⠃⠉⠛⠂
⠀⢤⠤⠤⠤⠄⠀⠀⠀⠀⠀⢠⠄⣑⠀⠈⢀⠎⠀⠀⠁⠀⢀⢶⢉⡠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠁⡴⠋⣱⠓⠞⢱⠀⠀⠀⠀⢠⠀⠲⢤⡥
⠀⠀⠳⡀⠀⠀⠀⠐⡄⢀⣠⠄⣈⠠⡀⠀⡌⠀⢀⠀⢠⡄⠀⠀⠀⠀⠀⠄⡀⡤⠀⠀⠀⣀⠀⠀⠀⠀⢀⡼⠁⣰⠁⡀⠀⠀⣤⠀⠀⠀⡜⠀⠦⢤⡄
⠀⢀⠎⠀⠀⠀⠀⠀⠈⠎⢰⠫⠑⠀⠑⠀⠀⠀⠀⠂⠃⠁⠘⠀⠀⣠⣄⣀⣉⡁⠀⢀⡞⠁⠀⡠⠀⠓⠉⠀⠐⠾⠟⠁⠀⠀⢸⡀⠀⢠⠁⠀⢶⣒
⠀⠉⠀⠀⠀⠚⠀⠀⡀⠈⢆⢂⠃⠂⠳⠈⡠⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⣤⠟⠁⠀⠀⠁⠂⠘⢄⡴⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠐⠃⠀⠀⠀⠈⠉
⠀⠀⡄⢐⠈⠁⠀⠀⠇⠀⠀⠣⣀⡨⠐⠉⠀⡌⢀⠂⠜⡀⢰⠀⢀⣴⣟⠁⢀⡀⠀⠄⡄⠄⠀⡀⡀⠈⠁⠰⢋⢁⠄⠀⠀⠠⠐⠀⠀⡀⠀⠀⠀⠓⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣉⣀⣀⣀⣀⣐⣁⣈⣀⣁⣀⣀⣋⣉⣙⣓⣚⣃⣀⣀⣁⣀⣀⣑⣃⣀⣀⣈⣉⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠘⠛⠛⠟⠛⠿⠃⠀⠀⠈⠛⠒⠚⠓⠀⠀⠈⠉⠁⠳⣆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢁⠀⠀⢁⠀⢠⡄⠀⠀⠀⢠⡀⠀⣀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⡄⡀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠂⠀⠀⣠⠈⠁⠀⠀⠀⠆⠀⠀⠀⠀⠁
⠀⠀⠠⠄⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠀⠀⠀⠀⣰⢃⠰⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠀⠀⠀⣤⣤⣤⠄⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠁⠀⠁⠉⠉⠀⠰⠀⣠⡀
⠀⠀⡀⠀⠀⠀⠀⣿⣿⣿⡔⠀⠀⠚⠚⠀⠀⠀⠀⠁⠁⠀⠀⠂⠠⠠⠀⠀⠤⠄⠀⠀⠀⠘⠓⠷⠸⠛⠀⠴⠀⠂⠄⠀⠀⠦⠀⠀⣾⣶⣮⠿⠃
⠀⠀⠸⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠠⠆⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣠⠟⠉⠀⠄⠀⠰⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀⠀⡀⢖⠀⠀⠀⠐⠈⠁⠀⠃⠀⠀⡄⠄⠄⠀⡀⠤⠠⠠⠰⠀⠀⠂⠀⠀⠘⠘
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⡩⡀⠐⠂⠐⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠘⠃⠀⠀⠀⠀⠘⠀⠀⠉⢁⠀⠀⠀⡀⠀⠀⠀⠠⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠄⠀⠀⠀⠀⠄⠀⠀⠀⠐⠓⠀⠀⠐⡀⢤⡀⠀⡄⠀⠀⠀⠀⠆⠀⠊⠁⠀⠀⠩⠴⠖⠒⠃⠀⠀⠀⠀⠈
⠀⢠⣤⣴⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡄⢀⠀⢠⣴⣦⡄⠀⠀⠀⢀⢆⣀⢀⠀⠀⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⣿⣷⣤⢀⠀⠀⢠⣴⣦⡀⠀⠀⠈⠈⠀⠈⠁⠀⢻⣯⣼⠟⠈⠀⠀⠀⠁⠋⠈⠀⠀⠀⠀⠀⠀⢶⠂⡀
⢀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡾⠇⠀⠂⠐⠊⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡀
⠘⠂⡄⡄⡀⢀⠀⠀⠀⢀⠕⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⡄⠀⠀⠶⠀⠄⡄⠀⠀⠀⠈⠐⠁⠠⢀⡄⠀⠀⠃⠀⠈⠉
⠀⠀⠀⠉⠀⢠⡃⠀⠀⠉⠀⠀⠎⢀⣀⠀⠲⠒⠀⠀⠀⠀⠀⠘⢿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⢀⡀⠀⠀⠂⠀⡀⢀⠀⠄⠀⠀⠐⠐⠀⠀⠂⠀⠀⠀⢀⡄⡀⣀⣈⢀⡀⡄⠀⠀⠀⠀⠄⠀⠀⠰⣏⣽⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠰⠈⠀⠿
⠀⠀⠀⠀⠈⠊⡀⢀⠌⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⣀⠀⠀⢀⣀⣾⣿⣤⣄⠈⠂⠄⠀⠀⠀⠒⠀⠊⠀⣠⣤⣤⡀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣀⠸⡇⠆⠐⠌⠀⠈⠀⠀⠀⢰⡿⣭⡝⣷⠀⢸⣿⣷⣿⣿⣿⣾⣦⣶⡆⠈⠃⠀⠀⠀⢸⣷⣟⣿⣿⠀⠀⠁
⠀⠀⠀⠘⠘⠄⠙⠀⠀⠀⠀⠁⠀⠀⠀⠆⠴⠀⠀⠀⠀⠀⠀⠈⠳⢼⡷⠋⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆⢀⡀⠀⠈⠻⠿⠷⠋

⠀⢀⠄⠂⠒⢄⠀⠀⠀⠀⠀⠀⡠⠊⡉⣉⢢
⢠⢂⠔⠉⡇⠰⠀⠀⢰⠐⠀⠀⠑⠊⠀⡜⢸⠀⡠⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠚⢆
⠀⠁⠀⡰⢡⠇⣐⠒⠚⠀⠒⢦⠀⠀⡔⣠⠃⠀⢉⣀⣀⣀⡩⠖⠂⣩⠗⠚⠉⠉⠉⠙⠲⣄⠀⠀⠀⠘⡆⠸⡄
⠀⢀⠜⡠⠃⠀⠀⠀⡇⢸⠉⠉⢀⠎⡰⠁⠀⠀⠓⠀⠀⡞⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡼⡄⢳⠒⠲⢤
⣰⠋⠘⠡⠤⢀⡀⠀⠉⠉⠀⢠⣊⡘⠃⠀⠄⢠⠀⠀⠀⣧⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆⠙⠾⠀⠀⠘⠓⢦⡀
⠀⠉⠉⠓⠒⠲⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠊⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣰⠏⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡤⠴⠶⠒⠒⠲⠶⠶⢿⣅⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡞⠁⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣤⠞⠁⡤⠔⠲⠖⠲⠖⠒⠲⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⢠⠞⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⣗⠛⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⠁⠀⠠⡀⢠⡟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠙⢯⡀⠀⠀⠀⠀⢻⡤⠤⠤⠤⠤⠷⠶⠶⠴⠦⠤⠤⠤⠤⠾⠥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡾⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢧⠀⢰⠇⠀⠀⡀⠀⠀⠀⠤⢼⠀⠀⠀⠀⠀⠀⠀⢠⡷⠒⠢⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠛⠀⢸⣠⠖⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⢸⡇⠀⠀⠀⠀⠀⠀⢀⡼⢆⣀⣀⣀⠤⠖⠻⠄⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠛⠷⢦⣤⣤⣤⣤⠶⡞⠋⠁⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⡀⠸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡛⠛⠛⠛⠛⠋⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣠⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⡤⠶⠾⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⡆⠀⠀⠀⠘⠁⡼⠀⠀⠀⠶⠒⠒⠀⠀⠀⠾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠰⢻⠁⠀⠀⠐⢻⡏⠉⠀⢀⣠⣞⡁⠀⠀⠘⠛⠉⠉⠀⣀⣀⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢋⣽⠀⠀⠀⡤⠴⠦⠀⠀⠀⢀⡼⠁⠀⠀⢀⣉⣠⡀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠋⢩⠇⠀⠀⠀⠀⠀⠀⣠⡶⠿⢤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠾⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⠀⢀⣇⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠒⠒⠒⢒⣚⣛⣛⣛⣛⣋⣉⡉⠉⠙⠓⠒⠒⠒⠶⠶⠇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⣴⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⠶⠶⠒⠚⣻⡆
⠀⠀⠀⣠⠶⠛⠉⢉⡭⠭⠭⠭⠭⠭⠭⠤⠤⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠸⣧⣀⣀⣀⣈⣓⣒⣒⣒⣒⣒⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⢀⡀⡀
⠀⠀⠀⠀⡄⠄⠰⡄⠂⠀⠠⠀⢒⡌⠀⠈⠋⠁⠁⠉⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠐⠀⠉⠀⠋⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄⠀⠈⠀⠂⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠂⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢐⣀⠀⣶⠄⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠄⠐⠀⡀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠨⠀⢈⠆⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⣿⡆⢀⡀⢰⡆
⠀⠄⠀⠰⢠⠉⠀⠀⠀⠈⠐⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠀⠀⠀⠀⠤⠿⠀⠬⠿⠥⠿⠧⠼⠧⢼⠇⣀⢀
⠀⢀⠄⠀⠀⠀⠀⢴⠞⠃⠀⢀⠔⠁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠤⣀⠤⡄⠀⠀⠀⠂⠀⠐⠀⠘⠀⠘
⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⡠⠊⠰⠊⠐⢀⠰⠄⠀⠀⠀⣀⣀⣅⠤⠴⠀⠈⠊⠀⠁⠀⣀⠀⢄⠈⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠡⠀⠀⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠶⡄⠀⡕⠀⠀⠀⠀⠠⢀⡀⠀⠀⡁
⠀⠀⡠⢄⠀⠀⠁⠉⠀⠀⠀⠀⠀⡀⠀⠪⢩⠔⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠀⠙⠀⠀⠀⠀⠀⠀⠀⡘⣶⣶⡄
⠀⠀⠃⠘⠒⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠁⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⠿⡧
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠻⢶⡦⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠓⠂⣠⣼⣤
⠀⠀⠀⠀⠀⠐⢠⠀⠀⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢚⠠⠇⠀⠀⠀⠀⠀⠡⠤⠥⠋⠙⠛⠋
⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠃⠀⢀⠀⠀⠀⣂⠁⠀⠀⠀⢀⡀⠀⢠⠀⣀⠀⢠⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠀⠙⠁⠉⠘⠈⠉⠁
⠀⠀⣔⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⣎⠦⡀⠀⢀⠤⠤⡀⠀⠀⠀⢈⠵⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⠄⡀⠀⠀⠀⠀⠒
⠀⠀⠀⠁⠀⠀⠀⠀⣄⢇⣼⠃⠀⡇⠀⠈⠀⠧⢦⠴⠇⢀⠄⡠⠂⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠐⠂⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠊⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⢠⠀⢰⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠰⣰⠀⠀⠀⠀⠀⠛⠿⠟
⠀⠀⠀⠀⠀⠖⠀⠀⠂⠐⠀⠳⠈⠀⠉⠂⠈⠈⠀⠙⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂⠈⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠂⠉⠀⠀⠀⠀⠀⠀⠀⢄⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠚⠛⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠶⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡠⠤⠤⢤⡀⠀⣖⣛⣛⡒⣆⠀⠀⠀⠈⣇
⠀⠀⠀⢠⠖⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⡇⠀⢰⢻⠋⠉⢳⡀⡇⠀⠈⠀⠀⢹⠈⡇⠀⠀⠀⢹⡄
⠀⠀⢀⡏⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠾⡇⡇⠀⠘⠛⠁⠀⢠⢧⡇⠀⠰⣶⣖⠋⣴⠃⠀⠀⠀⠀⢷
⠀⠀⣼⢀⠞⡄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⣧⠀⠀⢀⣠⡴⢫⡾⢤⠀⢠⣄⡤⠇⣸⠀⠀⠀⠀⠀⠘⡆
⠀⠰⡇⡎⠀⠘⠌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠚⠃⠀⠘⠷⠶⠒⠒⠚⠀⠘⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢷⠧⠔⠉⡤⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠘⡆⠀⡜⠘⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⢷⠘⠃⠀⠴⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠘⡆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢹⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀⣸⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢣⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠺⡿⡇⡏⡟⢰⣲⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠘⡆⠀⠀⠸⡅⢧⣠⠴⡄⠀⡀⠀⠀⠀⠀⠉⠉⢰⢣⠇⠹⠽⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢳⠀⢰⡖⢁⡀⢴⠒⠃⠀⡏⠙⠒⠒⢶⡀⠀⠙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⢇⠀⠛⠉⢣⣘⡦⠀⠀⠉⠛⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠒⠚⠋⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠘⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠔⠒⠒⠋⠉⠁⠀⠀⠀⠀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⢱⣀⡠⠤⠶⠒⠒⠒⠚⠉⠉⠉⠉⠁⠀⠀⣀⣀⠤⠤⠴⠄⠒⠒⠒⠀⠉⢉⣁⣀⠤⠤⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⡞⠁⣀⣀⡀⠤⠤⠤⠄⠒⠒⠒⠚⣉⣉⣁⡠⠤⠤⠐⠒⣒⣂⣚⣩⡭⠭⠥⠖⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠁⠀⠀⠀⠒⠒⠂⢉⣉⣉⣉⣭⠤⠤⠶⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠶⠶⠒⠒⠒⠚⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⠀⡠⠀⠀⠀⠄⡠⠄⠒⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠅⠀⠀⢰⢠⣀
⠠⢊⠷⠲⠄⠀⠀⠀⠀⠀⠀⢦⡀⠀⠐⢄⠠⠊⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡦⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠢⡀⠀⠀⠀⠀⠀⠐⢒
⠉⠊⠛⠛⠅⠔⠅⠲⠄⠀⢀⠐⣠⠞⠁⢀⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⠄⠄⣀⣤⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠊⠍
⠀⡠⢤⣄⠀⠀⠀⠘⠀⠀⠈⠘⣅⠀⠔⠁⠙⢦⣀⠔⠀⠀⠀⠀⠄⡀⠈⠀⠊⠁⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⣜⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠶⠦⢄⠀⠀⠀⠄⣀⡀⡀⠂⠐⠔⠤⠤⠦⠄
⠀⠈⠁⠐⠈⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⢲⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⡠⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⡀
⠀⠠⠀⠐⠤⠤⠐⠒⠈⠉⠈⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠉⠐⠒⠠⠠⢄
⠀⠀⠀⠀⠀⡠⡄⠀⢤⢐⡪⠭⠤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠀⠃
⠀⠀⠒⠅⠐⠃⠀⠈⠀⠀⠀⠀⠀⠂⠀⠈⠝⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⣰⣄⠀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠂⠀⠙⣴⠀⠀⣧⠁⠀⠑
⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠒⠒⠒⠚⠂⠀⠀⠀⠁⡀⠀⠉⠀⠀⠀⠘⠆⢀⠠⠇⡆⠀⠈⠢⡄⠔⠀⠀⡀
⠀⠀⠢⠠⠂⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⡲⠶⢄⠀⠀⠁⠀⠀⠀⠀⠄⠂⠀⠀⠰⠐⠤⠄⠀⠀⠀⠅⡇⠀⠀⠀⠀⠀⠀⠀⠲
⣳⠄⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡇⠀⠀⠀⠈⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⠀⠀⡀
⠀⢀⣄⠀⠁⡀⠀⠀⠀⠐⣆⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢩⠔
⠂⠁⠈⢧⣀⠠⠀⠀⠀⠢⡀⠀⠁⠑⠂⠤⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡰⠀⠀⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠐⠢⠀⠠⠀⣄⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢧⠆
⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠂⡐⠁⠀⠀⠀⢀⠆⢀⡀⠀⠀⠀⠉⠉⠁⠓⠀⠀⠀⢃⠀⠈⡄
⠀⡀⠀⠀⠅⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠖⣄⡀⠀⠙⠢⠈⠀⡠⠁⠀⠀⠀⠒⠒⠰⠭⠡⠰⢠⣐⣒⠐⠀⠀⠀⠘⡄⠀⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠉⢂⠀⠀⠀⠀⠈⠉⡶⣄⠀⠀⠪⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⡀⠀⠉⠀⠀⠀⠀⠐⠀⠀⢣
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠠⢊⡔⠁⠀⣄⣄⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠁⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⡀⠀⠊⠀⠉⠓⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀
⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠒⠒⠒⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⢿
⠀⠀⠀⠀⠀⠀⢸⠙⢂⠤⠴⠶⠶⠶⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸
⠀⠀⠀⠀⠀⠀⣸⢠⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢠⠖⡄⠀⠀⣀⣀⠀⠀⢋⡝⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠴⡏⠀⠀⠴⡷⠆⠀⢀⡔⠁⠀⠠⠖⠲⠀⠀⢈⡽⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢀⣴⣧⠀⠀⠀⠁⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⢀⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇
⠀⠀⠀⠀⠀⢸⠁⡿⠀⠀⠀⠀⠀⠠⢖⡆⠀⢀⣀⣀⠀⠀⠛⡹⠀⠀⠐⠒⠒⠀⠀⢶⠏⠀⠀⠀⠀⠀⠀⢠⡇⢠⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢋⡗⠀⠀⠀⠀⠀⡤⠾⠥⠀⠀⠈⠉⠁⠀⠤⠾⠆⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⡾⠀⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸
⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿
⠀⠀⠀⠀⢠⡇⢸⣇⣀⣀⣀⣀⣀⣀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠀⠒⢒⣒⣒⣒⣒⣒⡒⠒⠒⠲⠦⠤⠤⠏⢀⡇
⠀⠀⠀⠀⣸⣶⣋⣉⣀⣀⣀⡤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠐⠒⢲⣿⣿⣿⣿⣿⣿⡟⠒⠒⠒⠒⠒⠒⠒⢛⡷
⠀⠀⡴⠉⠁⠀⢖⣒⣒⣒⣨⣯⣭⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣤⣤⣤⡴⠟⠁
⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠶⠒⠒⠒⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣤⠶⠶⠦⣄⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⣠⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⣦⠀⠀⣰⣯⠗⠲⢄⠸⡆⠀⠙⠋⠉⠳⡌⢧⠀⠀⠀⠈⣧
⠀⠀⠀⣰⠃⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⡆⢹⠀⠀⢧⠿⠀⠀⢸⢠⡇⠀⢠⣄⣀⡴⠃⡼⠀⠀⠀⠀⠸⡆
⠀⠀⢰⡇⢠⡄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡇⢸⠀⠀⠀⠀⠀⢀⡜⣼⠁⠀⠈⠙⠓⣦⠸⡅⠀⠀⠀⠀⠀⢻
⠀⠀⡾⢠⠂⠱⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣏⣘⣆⠀⠀⣶⠚⢉⣚⣋⣻⠀⢸⣗⣚⣁⡴⠃⠀⠀⠀⠀⠀⠈⣇
⠀⠀⣇⡇⠀⣠⠔⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⢹⡗⠊⠀⡜⡆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢧⠀⡸⠀⢱⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢸⡄⠉⠐⠒⠋⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢵⣲⣇⡾⢸⢀⡶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢹⡀⠀⠀⠰⡏⢻⠀⣀⣄⠀⠀⠀⠀⠀⠀⠘⠒⠒⢁⡇⡇⠸⣜⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⣇⠀⢀⣤⠟⠈⢋⣡⡽⠀⢰⠶⠦⠤⣄⣀⠀⠀⠸⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠸⡄⠈⢧⠴⢶⠈⣇⠀⠀⠸⠦⢤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⢤⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢳⠀⠀⠀⠘⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⢇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠤⠤⠴⠶⠒⠒⠛⠋⠉⠉⠀⠀⠀⣀⣀⣀⡀⠤⠤⠤⠶⠖⠒⠒⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡤⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠴⠖⠒⠒⠉⠉⢉⣀⣀⡠⠤⠤⠤⠶⢒⣒⣺⣭⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢰⠃⠀⡤⠴⠖⠒⠒⠒⣋⣉⣉⣉⣀⣠⠤⠤⠴⠒⠒⣒⣉⣩⣭⠤⠴⠶⠖⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣾⠀⠀⠀⠀⠀⠀⢉⣉⣁⣀⣤⡤⠤⠴⠶⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠒⠛⠛⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⣀⠀⢀⡀⠀⠀⠀⣤⡤⠤⠤⠤⠤⠄⢤
⠀⠈⡔⠒⠒⠒⠒⠊⠀⠀⡮⠭⠅⠀⠀⠄⡠⠀⢄⠠⠀⠈⢐⠀⠀⡀⠁⢐⠀⠀⣘⠀⠀⠀⠈⠳⣄⠀⠲⡐⢢
⠀⠀⡨⣬⢥⠤⡠⡅⠀⠀⢆⠀⠄⠀⠀⠀⠄⠉⢠⠒⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠪⢹⠀⠓⢄⠈⠚
⠀⠠⣚⣿⣻⣛⡟⡇⠀⠀⣧⡀⡄⠀⠀⠀⢰⡶⡆⠀⠀⢀⠐⣀⣀⠂⡀⠐⣀⡀⢐⠀⢀⠎⣴⠃⠀⠀⠈⠳⣄
⠀⢠⠢⠛⠙⠉⠣⢃⠀⠀⠈⠛⠁⠀⠀⠀⠀⠉⠀⠀⡔⢌⠃⠀⠉⠀⠛⢀⠉⠀⠊⠀⠸⠟⢃⠄⠂⠐⠐⠢⡈⠱
⠀⢀⡤⡤⠄⠤⢤⡀⠀⠀⡠⠀⢲⡗⡲⣄⠀⠀⠀⠀⣜⣢⠀⠀⠀⠀⢀⠚⠆⠀⠀⠀⠀⢠⠃⡐⠉⠀⠁⠆⠐⡀
⠀⡏⠉⠉⠉⠉⠉⠁⠀⡌⠀⠀⢸⣧⣞⣽⠆⠀⠀⠬⠌⢦⢇⠀⠀⢀⠆⠨⠈⢆⠀⠀⢰⡁⠀⠀⠀⠀⠀⠁⠀⢉
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡀⠀⠀⢐⠐⠀⠀⢂⡀⠀⣎⣤⡤⣤⣌⡄⠀⠀⠀⠀⠀⠀⢀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠁
⠀⠑⠠⠤⠀⠤⠄⠃⠀⠀⠀⠀⡊⢉⢰⠀⠀⠚⠀⠁⠚⠀⠐⠀⠀⠀⠀⠊⠒⠒⠀⠒⠒⠜⠀⠀⣀⠄⢤⡖⢦⣄
⠀⢀⠤⠦⠤⠤⠴⠤⢄⠀⠀⠀⠃⠸⢘⠀⠀⢀⡄⠖⠒⠐⠤⡀⠀⠀⠀⢀⠁⡀⣄⢀⡀⡀⠀⡐⠁⠀⢸⣗⣱⣼⡆
⠀⢸⠴⠱⠎⠱⠉⠦⡇⠀⠀⢀⠀⡀⡺⠀⢀⠂⠠⠂⠉⠐⠄⠰⠀⠀⠀⠠⣚⣗⣿⣺⠗⠀⠀⢃⠀⠀⠈⠁⠁⠨
⠀⢸⢠⢲⡔⢢⠒⡄⡇⠀⠀⠘⠀⠃⡆⠀⠉⠀⠈⠁⠁⠀⠁⠀⢉⠀⠀⢐⠭⠯⠿⠽⠦⢠⠀⠀⠑⠤⠀⠠⠐⠁
⠀⠨⠤⠤⠬⠥⠥⠤⠕⠀⠀⠠⠠⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⢄⠀⠀⣀⣀⡀
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠀⣀⡀⠠⠄⠀⣀⠀⢄⠀⠀⠠⠰⡀⠀⠀⠈⠉⠁⠙⠈⠉⡅⠀⠒⠂⠂
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠠⠢⢱⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠨⠀⠀⠠⣩⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈
⠀⠀⠀⡗⢄⠈⠳⣄⠀⠀⣆⣠⠃⠀⠀⠀⠀⣒⡒⠀⠀⠀⣀⡀⠐⠀⠠⠵⠱⠴⡀⠀⢀⠀⠀⠀⠀⠀⡇⠀⢴⢴⡆
⠀⠀⡀⠓⠒⠓⠀⠈⢳⣄⠑⠙⠀⠀⠀⢠⠀⠄⢄⢠⡄⠄⠀⠀⡭⢀⡧⠃⠀⠐⢴⠀⠈⠐⠒⠀⠒⠊⠀⢀⡄⠉
⠀⠀⢀⠠⠠⣶⢦⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠠⡀⠀⠀⢀⠊⠁⡀
⠀⠀⠂⠀⢸⣿⣿⣿⠀⠀⠈⠁⠈⠀⠁⠀⢁⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠄⠄⢄⠀⠀⠠⠈⠑⠔⠀⢀⠂⠀⠀⠐⡀
⠀⠀⢂⠀⠀⠀⢀⠌⠀⠀⠰⣎⡁⠉⠉⠁⠁⢸⠀⠀⢀⠈⠈⠈⠁⠁⠀⠀⠀⡸⠀⠀⠀⣷⢾⠀⠀⠺⣶⣶⣰⣶⠖
⠀⠀⠀⠁⠒⠂⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⠀⠀⣠⡤⠖⠛⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠚⠛⣷⡀⢰⠏⠀⠀⠀⠀⢹⣀⡼⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣠⡶⠦⣤⣀⡀
⠀⠀⢀⣀⡤⣄⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⡼⢳⣼⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⠀⠀⠐⣟⡆⠀⠀⠀⠀⣿⢁⡇⠀⠀⠈⠉⢹⡆⢀⣤⣀⡀
⣶⠚⠉⠀⠀⢋⠓⢦⣄⠀⠀⢻⠀⠀⠀⠀⠏⠀⠀⠀⢻⠀⠀⢰⢦⠀⠀⠈⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢸⠃⠀⠀⠀⠀⠘⣇⡏⠀⠀⠉
⠘⡆⠀⠀⠀⠘⡄⠀⠈⠳⣄⡸⡆⠀⠀⠀⠘⠲⡄⠀⢸⠀⠀⠈⠉⠀⠀⠀⢸⡄⡏⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⣸
⠀⢹⡄⠀⠀⠀⠰⡀⠀⠀⠀⠙⣷⠀⠀⠀⠀⢰⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⢷⠀⠀⠀⠀⠳⠴⠀⠂⠉⢹⡆⠀⠀⠀⠀⠉⠉⢿⠀⠀⠀⢀⣦⠀⢀⣀⡀⢿⠀⠀⠀⡟⢆⠀⠀⠘⡆⢸⠇⠀⠀⢠⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡸⠈⡶⠁⠠⠌⢿⠤⠤⠤⠇⠀⠲⠤⠤⢽⣾⠀⠀⠀⡇⠱⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⣀⠸⠷⠤⠤⠒⠒⠒⢉⡉⠁⠀⢀⣤⣧⡘⠠⢊⣼⠦⣤⣤⣀⡶⣄⣀⣀⣤⠀⠈⠉⠚⢀⠀⢦⣀⠀⢀⡏
⠀⠀⠀⠀⢳⡀⢀⡠⠤⠒⠋⠉⠀⠀⣐⣶⠤⠶⠶⠶⠚⠙⠛⠋⠉⠁⠈⣯⠀⠸⡇⠀⠀⠀⠈⠀⠀⠈⠉⠉⠛⠛⠓⠶⠟⠳⣄⣈⣉⡿⠃
⠀⠀⠀⠀⠈⠻⢥⣀⣠⡤⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠈⠷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠰⠂⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡖⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⢷⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡞⠈⣇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠇⠀⠸⡄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡸⠀⠀⠀⢱⠀⠀⠀⡇⠀⠀⠀⠀⢀⡠⠒⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⣌⣙⠒⠦⠤⢀⡼⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⠀⠸⡤⠴⠖⢋⣁⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⡄⠀⠀⢘⡛⠛⠛⠛⢋⢙⡀⠀⠀⢢⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⢣⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⣻⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣰⢃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣰⢃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⣿⡿⠿⠿⣿⣿⠿⠿⢿⣿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡐⠒⢂⣽⣀⠂⠂⣸⣇⠒⠒⣂⣯⡀⠑⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⣉⠙⢿⠋⢩⠉⢻⡟⠉⡉⠙⡿⠋⠉⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣤⣿⣦⣬⣤⣾⣷⣤⣥⣴⣿⣤⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⠴⠀⣹⡀⠰⠄⢸⡇⠐⠆⢀⣏⠀⠲⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⣛⠛⢿⠟⢛⠛⢻⡟⢛⡛⠻⣿⠛⠛⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣈⣠⣾⣤⣈⣡⣼⣧⣄⣁⣤⣷⣄⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠠⠄⠌⣻⠁⠠⠄⢹⡏⠀⠀⠈⣟⠁⠠⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣾⣿⣷⣶⣶⣿⣿⣶⣶⣾⣿⣷⣶⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠈⠁⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡤⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠻⡅⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⢀⣀⡤⠶⠚⠳⢤⠼⠃⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⡄⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢻⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠾⠀⠀⠘⠧⠤⠤⠤⠴⠇⠀⠀⠿⠤⠤⠤⠤⠟⠀⠀⠸⠦⠤⠤⠤⠾



⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄⠀⢻⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠃⢀⡼⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡴⠶⠶⢦⠀⡴⠶⠶⢦⠀⡴⠶⠶⢦⠀⡴⠶⠶⢦⠀⡴⠶⠶⢦⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⢸⠀⡇⠀⠀⢸⠀⡇⠀⠀⢸⠀⡇⠀⠀⢸⠃⡇⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢛⣛⣓⡋⠀⢛⣚⣓⡋⠀⢛⣚⣓⡛⠀⢛⣛⣓⡛⠀⢙⣛⣛⡛⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⡏⠉⠉⢹⢀⡏⠉⠉⢹⠀⡏⠉⠉⢹⠀⡏⠉⠉⢹⡄⡏⠉⠉⢹⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣧⣀⣀⡼⠀⣧⣀⣀⣼⠀⣧⣀⣀⣼⠀⣧⣀⣀⣼⠃⢧⣀⣀⣼⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡴⠶⠶⢤⠀⡤⠶⠶⢦⠀⡴⠶⠶⢦⠀⡤⠶⠶⢦⠀⡤⠶⠶⢦⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⢸⠈⡇⠀⠀⢸⠀⡇⠀⠀⢸⠀⡇⠀⠀⢸⡁⡇⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢛⣒⣒⠛⠀⠛⣒⣒⠛⠀⠛⣒⣒⠛⠀⠛⣒⣒⡛⠀⠛⣒⣒⠛⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡏⠉⠉⢹⠠⡏⠉⠉⢻⠀⡏⠉⠉⢹⠀⡟⠉⠉⢹⠄⡟⠉⠉⢹⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣇⣀⣀⣼⠀⣧⣀⣀⣼⠀⣧⣀⣀⣼⠀⣧⣀⣀⣼⠁⣧⣀⣀⣼⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐
⠀⠀⠀⠀⡶⠒⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠐⠒⠦⠖⢀⣤⠇⠀⠀⠀⠀⣀⣠⡖⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⢀⠄⠀⠀⠀⠀⠀⡇⠀⠐⠀⠀⠠⠋⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⠛⡁
⠀⠀⠀⡀⠙⠀⠀⠸⠀⠀⠠⠃⠀⠀⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡾
⠀⠀⠀⠈⠉⠉⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠄⠀⠀⢀⣀⠈⠳⡄⠀⠑⠂⠤⠚⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⣀⣠⡁⠀⠀⢀⠀⠀⠐⠦⠾⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠐⠚⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠞⠂⢈⠛
⠀⠀⠀⠀⠤⠤⠤⠀⠀⡤⠇⠀⠀⠀⠀⠀⣀⡤⠾⠀⠀⡀⠀⠁⠀⢀⡆⠀⠀⠀⠀⠠⠤⠴⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⢀⣉⡀⠀⡀⠀⠈⠉⠉⠉⡉⠁⠀⡀⠀⠀⢠⡂⠐⠂⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⢁⠀⠀⡀⠘⠅⠉⠀⡽⠀⠀⠀⠀⠲⠁⠂⢀⡇⠀⠀⠀⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⢸⠀⠀⠙⠢⠤⠴⠐⠁⠀⠀⠀⠀⢀⢀⡀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠢⠄⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣈⡢⠀⠀⠀⠀⠠⠄⠀⠆⣶⠀⠍⣻⠀⢾
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠒⠐⣆⣤⣰⣹⡳⡢⠹⠞⡞⠉⠁⠀⠀⠀⣠⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣄⡌⢄⣂⠈⣀⠠⠀⢀⡀⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠁⠀⠈⠒⠂⠄⠐⣄
⠀⠀⢌⠉⠃⠀⢀⠸⠤⠞⢷⠈⠹⣏⠀⢸⡶⣿⡀⠀⠄⠀⠄⠂⠔⠀⠐⢀⡂⠒⢄⣠⠀⠰⠤⣤⠐⠂⠋⠉
⠀⠀⠀⠈⠂⢀⠀⢀⠀⠉⡩⠉⡈⠻⡀⠸⠿⡟⠉⠤⠆⢈⠂⠀⡌⠊⠀⢀⡈⣰⠁⡌⠩⢬⠭⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠥⡲⠞⠀⠐⠑⠀⣷⣟⡿⠢⣀⠀⣩⣁⣀⣀⣠⢀⣤⢤⠆⠔⠀⠲⡆⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⢀⣀⡸⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠴⠒⠃⠈⠁⠁

⠀⠀⠀⠀⢠⡟⠿⢻
⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⢸⡇⣶⠿⠯⠭⠭⠽⠿⣆
⠀⠀⠀⠀⢸⡇⢻⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⣠⣿⣿⠟⠻⠛⠛⠻⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡴⠒⢺⢛⣡⢄⠀⠈⠀⠀⠀⠈⠁⠀⣠⠈⡛⣗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠉⠙⠻⣿⣿⠟⠿⢿⣿⣿⡄
⢧⡀⢻⠀⠉⠉⠀⠀⠢⠄⠠⠔⠁⠀⠈⠉⠁⣽⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⠀⠰⠀⠀⠀⠰⠆⠀⠘⠷⢀⣀⣿⠿⣿⠃
⠀⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⡁⠀⠐⠤⠔⠀⠀⠘⠑⠀⠀⠁⠀⠀⢸⠆
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣾⣄
⠀⠀⠀⠀⠐⡿⢻⠛⡖⠒⠒⣲⠛⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣛⡖⠒⠒⠒⢛⡏⠛⣿⣿⡿⠟⠛
⠀⠀⠀⠀⢰⠃⡞⠀⠘⠒⠒⠁⠀⢻⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⢳⡙⣟⠿⢿⠋⠙⠦⡀⢸⡿⠿⠄
⠀⠀⠀⠀⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⢘⡇⢸⠀⠀⠀⠀⠀⠸⡆⢧
⠀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⡼⢁⠞⠀⠀⠀⠀⠀⠀⣇⠘⡄
⠀⠀⣠⠃⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢶⠿⡖⠉⠀⠀⠀⠀⠀⠀⠀⢹⠀⠳⡀
⠀⠀⠙⠒⢺⡗⡶⢶⡦⠤⠴⢶⡶⢶⡞⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⣷⡶⣶⠤⠤⠴⢶⡶⢶⠛⠒⠚⠁
⠀⠀⠀⠀⢸⡇⢧⠀⠈⡗⢺⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⢹⡒⢺⡅⠀⢸
⠀⠀⠀⠀⢸⡇⢸⡄⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢻⠀⢸⡇⠀⡇⠀⢸
⠀⠀⠀⠀⢸⡇⢈⣇⠀⣇⣸⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡆⠀⡇⠀⣧⠀⣸
⠀⠀⠀⣴⠚⠉⠉⣽⣀⡿⢹⣀⣺⠉⠉⠛⣢⠀⠀⠀⠀⠐⠒⠀⠒⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⡗⠛⣇⣀⡏⠉⣿⣀⡸⠓⠒⠢⣄
⠀⠀⠀⡏⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡗⠒⠒⠲⠶⠶⠶⠒⠒⠒⠚⠉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡈⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⡎⢹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠏⢣⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣘⣒⣒⣒⣒⣒⣒⣒⣒⣚⣀⣀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣱⠀⠤⢀⣀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠐⠒⠒⠒⡖⡴⣊⠅⠒⠉⠉⠁⠒⠭⡢⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠟⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢎⢆⠇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⡼⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡊⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⡄⠀⣤⣤⣤⣤⣤⣤⣤⣦⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠃⠀⠛⠛⠛⠛⠛⠛⠛⠛⡜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡏⠉⣾⣿⣿⣿⣿⣿⣿⣷⠘⠪⡓⢄⡀⠀⠀⠀⣀⠤⡪⣞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⡉⢉⣉⣉⣉⣉⣉⣉⣉⡁⠀⠀⠈⠑⠂⠥⠥⠌⠒⠫⣀⠜⢆⡃
⠀⠀⠀⠀⠀⠀⠀⢸⢨⠀⣇⣀⢿⣿⣿⣿⣿⣿⣿⡯⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⡽⢄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡖⠒⣴⣶⣶⣶⣶⣶⣶⣦⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠆⢘⢣⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠓⠐⠛⠛⠛⠛⠛⠛⠛⠓⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⡳⡄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣀⠇⠈⢆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⣶⠶⢶⣿⡏⣶⠶⢶⣶⢹⣿⡾⢶⢶⣾⣿⣵⣞⣾⣶⡞⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣼⣯⣿⣿⣿⣯⣿⣽⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⣿⡿⣿⣽⡟⣽⡿⣿⣯⢻⣯⣿⢿⢿⣯⣿⣫⣿⣿⣿⡝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣽⣿⣿⣇⣿⣗⣺⣿⢸⡿⣿⣞⣼⡿⣿⢿⣿⣿⣿⣇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢛⣾⣿⣷⣻⡿⣻⣾⣿⣟⢿⣟⣿⣿⣿⣟⣿⢟⣿⣿⣿⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣁⣿⣿⡇⣿⡧⢝⣿⢸⣿⣿⡨⣻⣿⣿⣿⣿⣽⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡮⣭⣭⡭⢿⣷⢯⣭⣭⡽⣾⡷⢭⣭⣭⠵⣿⡮⣭⣭⣭⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⡭⣿⣿⡇⣿⣯⣿⣿⢸⣿⣿⣿⢿⣿⣿⣾⣿⣛⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⣛⣻⣛⣿⣧⣛⣷⣿⣛⣼⣿⣛⣻⣛⣫⣿⣝⣛⣛⣛⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⡿⣟⣿⣾⡏⣾⣿⣿⣷⢹⣷⣾⣿⣿⣾⣿⣵⣿⠿⣿⡎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡻⠷⢭⠿⣿⣇⠿⠿⠿⠿⣸⣿⠿⠿⠿⢟⣿⡻⠿⠶⠿⢧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠤⠶⠶⠶⠶⠶⠦⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣚⣻⡍⠉⠁⠀⠲⠖⠒⠛⠓⠀⠀⠀⠀⠀⠲⠶⠦⣭⡉⠙⠓⠦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⢯⣍⠉⠉⠉⠛⠛⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣤⣤⣤⡤⠴⠶⠶⠖⠛⠛⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣧⡆⠀⠰⠖⠛⠛⢻⡆⠀⠀⠀⠀⠀⢀⡏⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣻⣏⣀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⣸⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⢀⣄⣀⣤⣤⣼⡇⠀⠀⠀⠀⢀⡟⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣌⣽⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠐⠛⣻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⠁⠀⠀⠀⠀⣼⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⠀⠘⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⣿⠄⢠⣤⣤⣤⡤⣿⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⣿⠒⠀⠀⠀⠀⠀⠾⠧⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣁⡀⠀⠀⠀⠀⠀⣼⡇⠀⠴⠶⠶⠚⠋⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣤⣴⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠖⠛⠛⠉⠉⠁

⡀⠈⠁⣀⠊⠀⠀⠀⡠⠈⡱⠈⠠⡀⠈⠀⡀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⢉⠦⡁⠀⠀⢀⠀⡑⢂⠐⠁
⠁⢠⠊⠀⠀⢀⠠⠚⡄⠊⠀⠀⠀⠈⠢⠄⠁⠀⠀⠀⠡⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⢆⠀⠀⡀⠈⠀⠄⠀⠔⠁⠀⠈⢢⡔⡤⢊⠔⠁⠀⢠
⠊⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠈⠁⠈⠠⡀⠀⠂⠀⠀⠀⠄⠀⠔⠁⠀⠀⠀⠀⠄⠁⠀⠀⠊⠀⠀⠀⠀⠁⠀⠀⢀⠄⡡⠊⠒⠁⠀⡀
⠀⠀⢠⠂⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⠔⠀⠀⠀⠄⠁⢀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡠⠊⠀⠀⠀⠀⢀⡐⣢⠂
⡀⠀⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠄⠀⠀⠀⡀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⡠⠈⠢⡀
⡑⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠉⢠⠔⠁⠀⠀⠀⠒⠈⠂⡀⠀⠆⠀⠀⠰⢆⠀⠀⢀⠐⠀⠀⠀⢀⠀⢤⠊⠁⢀⠀⠀⠀⠀⢡⠊⠀⠀⠀⡀
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠠⡔⠁⠀⠔⠁⠘⠀⡀⠀⠀⠀⠀⠠⠌⠁⡀⠀⠂⠀⠀⠉⠄⠁⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠂⡠
⠀⠀⠈⠆⠁⠀⠀⠀⠐⠁⠀⠒⠁⠀⠀⠀⠀⠁⠠⠀⢀⠀⠀⠀⠀⠈⠂⡄⠀⠀⠁⠀⠐⠀⠀⡠⠊⠁⠀⠀⠀⢀⠐⠀⠀⠁⠀⠀⠀⠊⡱⠂
⠀⠈⠁⠀⠈⠠⠐⠁⠀⠔⠀⠀⠀⠠⠌⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠔⠣⡀⠀⠀⠔⡠⠊⠀⠀⠀⠀⠀⠤⠀⠐⠀⠀⠀⠀⠐⠄⠊⠀⠐⠁
⠁⠀⠀⠀⠠⠀⢀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢀⠐⠁⠀⠀⣈⠲⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠠
⠀⠀⠡⠆⠀⠀⠁⠀⠀⠠⡐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠜⠂⡀⠀⠀⠐⡠⢂⠌⠀⠀⠀⠀⠀⡠⠒⢀⠀⠀⡠⠑⡄⠀⠀⠀⠀⠀⠁⠆
⡀⠀⠀⠀⠀⠌⢀⠄⠂⠀⠀⠢⡀⠆⠀⠀⠀⠀⢀⡰⣀⠄⠀⠀⠀⢀⠖⠡⢊⠔⠁⠀⠠⢄⠲⡀⠂⠀⠀⠀⡰⢎⡠⠊⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠁⢀⠀⠀⠠⠄⠁⠂⠀⠀⠀⠀⢀⠐⠈⠂⠀⢀⡄⠀⠀⠔⠁⠀⣀⠀⠀⠨⠒⠈⠢⠀⠀⠂⡠⠊⠁⠄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠠⠆⠀⠀⠠⠀⠀⠤⡉⠄⠀⠀⠀⠀⢜⠁⢀⠄⠁⠀⠰⠄⠀⡡⠀⠀⠀⠀⠠⢌⡠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠱⠆⠀⠀⠀⠀⠉⠀⡀⢀⠀⡡⢊⠔⠁⠀⠐⠎⠀⠀⠓⠀⠀⠀⣀⠈⡠⠊⠀⠈⠢⡀⠀⠆⡀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠄⠁⠀⡠⠀⠀⠀⠃⠔⠁⠀⠀⢀⠘⠣⡊⠔⠁⠀⠀⠘⠄⡠⠙⠀⠀⠀⠀⢌⠠⠋⠀⠀⠀⠀⠠⠌⠃⡀⠀⠀⡀⠈⠁⠄
⠀⠀⠡⠄⠑⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠐⡠⠊⠀⠀⠀⠀⢆⡀⡠⠂⠁⠈⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠁⠔⠁
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠘⠢⠘⠀⠀⠀⡠⠈⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⠀⠀⠀⠐⠁⠀⠐
⠁⢄⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⡀⢂⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⣀⠠⡀⠈⠀⡀
⠠⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠀⠁⠄⠀⠀⠀⠘⠀⠀⠀⠠⠂⠀⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠘⠠⠀⠀⠀⠀⠀⠀⠈⠠⠄⠁
⠀⠀⢀⠀⠁⠀⠀⠀⠀⡐⢎⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠱⢎⡠⠊⠀⡀⠈⠢⡀⠆⠀⠀⠀⢀⠜⠃⡀⠐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠁⠈⠠
⠢⡄⠁⠀⠀⢀⠐⡠⠊⠀⠀⠑⠄⠀⠐⠀⠀⠀⠀⠁⢄⠊⡠⠊⠁⠄⠀⠀⠠⠄⠉⠠⠀⢀⠔⠁⠀⠀⠁⠂⡀⠀⠈⠀⠀⠀⠀⠐
⠁⠈⠢⣀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠈⠂⠀⠈⠁⢀⠄⠠⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⡌⢪⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠐⢄⡀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢊⠐
⠄⠠⢀⠄⠁⠀⠀⢀⠀⡠⠊⠀⠀⠀⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⢀⠰⠁⠀⠠⠈⠱⣨⠢⡀⠀⠀⡠⠊⡱⠂⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢑⠄⠁⠀⠀⠀⠀⠆⡁⠀⠀⠀⠀⠐⢪⡲⡁⠀⠀⢀⠐
⠃⠄⠀⠂⠔⡀⠊⠀⠀⠈⠠⠈⠠⠊⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠁⠄⠀⠌⠁⠀⠈⠠⠀⠤⠀⠄⠁⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢿⣟⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠙⠳⠆⠀⠈⠀⠀⢉⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠷⠄⠀⠀⠀⣀⣴⠿⠛⣋⣉⣉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⢀⣼⠟⣡⡾⠟⢛⣙⠛⠿⣶⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠙⠀⠀⢀⣾⠃⣾⠏⣴⡾⠛⠛⢿⣦⠘⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠲⠀⠀⢸⡏⢸⡏⢸⡏⠀⠀⠀⠀⢻⡇⣹⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⡶⠶⢶⣶⠶⠶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠙⢷⣦⡀⢸⡇⢸⣇⠸⣷⣄⠀⠀⢀⣾⢃⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⢿⡄⠀⠉⠳⠀⠀⠙⠃⠀⠈⠙⠂⠈⠻⠟⠛⢿⠶⣶⣤⣤⣭⣿⣿⣿⡄⠻⣷⣬⣉⣠⣴⠟⣡⡾⢃⡾⠃
⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⠉⠙⠷⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⡙⠊⠉⠻⠍⠙⠿⣶⣬⣉⣉⣩⣥⡾⢋⣴⠟⠁
⠀⠀⠀⠀⠺⣿⡁⠘⠿⠆⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢷⣶⣤⣤⣄⣀⣉⣉⣉⣩⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⢀⣠⣶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣛⣿⣷⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⢻⣿⣿⣿⣿⡿⠛⢉⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠋⣠⠾⠋⣁⣠⣄⣀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣿⣿⣿⡟⢀⡾⠃⣰⠞⢋⣉⡉⠙⢻⣄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠁⣼⠃⣼⠃⣰⣿⣿⣿⣷⠀⢻⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⣿⠀⣿⠀⢻⣿⣿⣿⣿⠃⣼⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠐⢶⣶⣦⡐⣶⣶⣄⠠⣤⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⡄⠘⢷⣄⣸⣿⡿⠋⣰⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠲⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣷⣾⣿⣦⣶⣦⣀⢤⣀⠀⣀⣀⠈⠻⣦⣄⣉⣉⣁⣤⠞⢁⡴⠋
⠀⠀⠀⠀⢀⣠⡀⠘⢿⣿⣿⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣾⣿⣷⣦⣀⣉⣉⣉⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠈⢻⣿⣶⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠾⠿⠿⠿⠿⠿⠿⠷⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⢶⣶⣶⣶⡾⠿⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⣶⠶⠶⠶⣶⣆⠀⣴⡶⠶⠶⠶⣶⡆⠀⣴⡶⠶⠶⠶⣶⡆⠀⣶⡶⠶⠶⠶⣶⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣷⣤⣤⣤⣾⡿⠀⢿⣧⣤⣤⣤⣿⡇⠀⣿⣧⣤⣤⣤⣿⡇⠀⣿⣧⣤⣤⣤⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣬⣥⣭⣬⣥⣄⠀⣠⣭⣭⣬⣤⣥⡄⠀⣠⣭⣥⣥⣬⣤⡀⠀⣤⣭⣤⣭⣬⣥⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡏⠉⠈⠁⢹⣿⠀⣿⡏⠉⠉⠁⣿⡇⠀⣿⡏⠉⠉⠉⣿⡇⠀⣿⡇⠉⠉⠉⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣇⣀⣀⣀⣸⣿⠀⣿⣇⣀⣀⣀⣿⡇⠀⣿⣇⣀⣀⣀⣿⡇⠀⣿⣇⣀⣀⣀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢈⣛⣛⣛⣛⣛⡁⠀⢉⣛⣛⣛⣛⣛⠁⠀⢙⣛⣛⣛⣛⣛⠁⠀⢙⣛⣛⣛⣛⣛⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⡟⠛⠛⠛⢻⣇⠀⣿⡟⠛⠛⠛⣿⡇⠀⣿⡟⠛⠛⠛⣿⡇⠀⣿⡟⠛⠛⠛⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣽⡇⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠏⠀⠻⠿⠿⠿⠿⠿⠃⠀⠻⠿⠿⠿⠿⠿⠃⠀⠻⠿⠿⠿⠿⠿⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⡿⠿⠿⠿⢿⣦⠀⣾⡿⠿⠿⠿⣿⡆⠀⣾⡿⠿⠿⠿⣿⡆⠀⣾⡿⠿⠿⠿⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣷⣶⣶⣶⣾⡟⠀⢿⣷⣶⣶⣶⣿⡇⠀⢿⣷⣶⣶⣶⣿⠇⠀⢿⣷⣶⣶⣶⣿⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢶⣶⣶⣶⣶⣶⣶⡶⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢠⣤⣤⣤⡄⠀⣾⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁

⣿⠋⠙⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⣯⣿⢉⢉
⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠦⠀⢰⠀⠀⠠⠄⠀⠀⠠⠄⠂⠀⠀⠀⠒

⠀⠀⠀⠀⢠⡶⣦⠀⠀⣴⡄⠀⢠⡶⣦⠀⠀⣠⡄
⠀⣴⠀⠀⠀⠙⠉⠀⠀⠈⠀⠀⠀⠙⠉⠀⠀⠈
⠀⠘⠓⠀⢀⣤⣄⠀⠀⣀⡀⠀⢀⣠⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠘⠳⠟⠀⠀⠉⠁⠀⠈⠳⠟⠀⠀⠘⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠸⣏⡷⠀⠀⠾⠆⠀⠰⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠸⣯⡿⠀⠀⠺⠆⠀⠀⠀⠀⠀⠀⠼⠇⠀⠀⢯⡽⠂
⠀⣻⡅
⠀⠉⠈⠁⠀⠀⠀⠀⠀⣤⡄⠀⢠⡖⣦⠀⠀⢠⡆⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠠⠀⠠⠀⠠
⢠⡤⣄⢀⡴⣄⢠⡴⣄⢀⡴⣄⢀⡴⣤⢀⡴⣤⢀⡴⣤⢀⡴⣤⢀⡴⣤⢀⡴⣤⠀⡴⣤⠀⣴⣤⠀⠀⠀⠐⠀⢐⠀⠐
⠈⣛⠋⠈⣛⡋⠈⢓⡋⠈⣛⡋⠈⢛⡋⠈⢛⡋⠈⢛⡋⠈⢛⡋⠀⢛⡛⠈⢛⡋⠀⢛⡛⠀⢛⡛⠀⠀⠀⠠⠀⠠⠀⠠
⠰⣯⡷⠸⣯⡷⠰⣏⡷⠸⣯⡿⠰⣯⡽⠐⣯⡿⠰⣯⡽⠰⣯⡿⠐⣯⡽⠰⣯⡽⠀⢏⡽⠀⣯⡽⠀⠀⠀⢐⠀⢐⠀⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⠀⣀⣀⠀⠀⢠⡄⠀⠀⠀⠀⡞⣩⣭⣭⣍⡍⠆⠀⠀⠀⠀⢠⠶⠶⠶⠶⡄⣚⡇⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⢰⢫⣩⢩⢩⢳
⠀⠀⣿⢉⣛⣛⣉⣉⢹⡇⠀⠀⠀⠀⡇⣉⣉⢉⢉⡁⠀⠀⠀⠀⠀⢸⠀⣒⣒⠀⡇⢹⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠘⢸⢸⢸
⠀⠀⣿⣨⣭⣉⣩⣭⣸⡇⠀⠀⠀⠀⡇⠒⢲⢰⢸⡇⠀⠀⠀⠀⠀⢸⢀⣂⣀⡀⡇⡼⡇⠀⠀⠀⠀⣸⢻⠛⠛⡟⡏⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠙⠀⠁⠀⠀⠉⠘⠃⠀⠀⠀⠀⠳⠭⠬⠬⠬⠥⠆⠀⠀⠀⠀⠸⠦⠤⠤⠤⠇⠘⠀⠀⠀⠀⠀⠿⠿⠧⠤⠼⠿⠿⠀⠀⠀⠀⠀⠘⠦⠤⠥⠤⠞


⠀⠀⠀⣠⠖⠛⠓⢦⡄⠀⠀⠀⠀⠀⢀⡴⠚⠛⠳⣤⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠶⣄⠀⠀⠀⠀⠀⠀⡴⠞⢛⠓⢦⡀⠀⠀⠀⠀⠀⢠⡴⠚⠛⠲⣄
⠀⠀⢸⡇⢤⣤⣤⡄⢻⠀⠀⠀⠀⠀⣾⠀⣤⣿⣤⠈⡇⠀⠀⠀⠀⢠⡏⠀⣷⣞⠀⢹⡄⠀⠀⠀⠀⢸⠁⣤⣭⣤⠀⣷⠀⠀⠀⠀⠀⡟⠐⠒⠒⠒⢸⡆
⠀⠀⠈⢧⣀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠹⣆⡀⠛⢀⡼⠃⠀⠀⠀⠀⠀⢳⣌⠁⠈⣠⡞⠀⠀⠀⠀⠀⠘⢦⡀⠚⢀⣰⠏⠀⠀⠀⠀⠀⠻⣌⠉⠉⣉⡼⠁
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡂⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆
⠀⠀⢸⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣙⣿⣧⠀⠀⠀⠀⠀⠀⠀⢴⠟⣧⠀⠀⠀⠀⠀⠀⢰⡇⢀⡶⠳⣆⠀⡇
⠀⠀⢸⣿⣤⣽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠠⡌⠈⠹⣧⠀⠀⠀⠀⠀⠸⣇⣀⣻⣞⣃⣀⡇
⠀⠀⠀⠁⠁⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⡤⢤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢰⢋⠓⢸⠐⢉⢳⠀⠀⠀⠀⠀⠰⣦⡖⠐⣆⢠⠆⠀⠀⠀⠀⠀⢀⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣡⠟⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠁
⠀⠀⢸⠀⣱⣸⣰⠃⢘⠀⠀⠀⠀⠀⠰⠛⠧⠀⢹⡞⠀⠀⠀⠀⠀⠀⠀⣿⠀⢄⣠⠄⠀⠀⠀⠀⠀⠀⢹⣶⠋⠀⠉⠁⠀⠀⠀⠀⠀⠉⢳⣼⠁
⠀⠀⠈⠣⣃⢸⠐⡲⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⢳⡀⠀⠀⠀⠀⠀⢠⡾⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣠⠔⠀⠉⡁⢶⠆⠉⠉⠀⢤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡎⠀⣠⠂⡈⢠⠄⠉⠠⢀⣰⠈⠉⢤⠈⡀⢠⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠀⣀⡰⣀⢀⡟⠉⡱⠖⠯⣀⡨⠖⠶⡁⠉⣇⢀⡰⡀⣀⢀⠀⠑⠄
⠀⠀⠀⠀⢀⠄⠾⠇⢡⡁⠀⣟⢀⡬⢣⣵⣭⠥⠿⠿⠖⠶⠿⠧⢭⣵⣥⠫⣀⢘⡇⠀⣁⠀⠿⠆⢄
⠀⠀⠀⢀⠁⢀⠄⠰⡊⠉⡙⣬⣦⠖⠋⠙⡄⠀⠀⡇⠀⠀⡇⠀⠀⡜⠉⠓⢦⣦⠜⡉⠙⡒⠂⢄⠀⢁
⠀⠀⡠⠾⠀⢠⣄⢤⠓⣌⡴⠋⠉⢦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠎⠉⠣⣌⡔⢣⢄⣴⠀⠸⠦⡀
⠀⡀⠀⢠⠐⢪⡀⡰⢨⠟⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⢯⠰⡀⣨⠒⢠
⠀⣠⡀⠠⣀⠼⢍⣶⣏⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⡀⠀⠐⠀⠀⠄⠀⠀⠀⠀⢈⣷⣎⠽⠄⡠⠀⣠⡀
⠐⠑⠀⠶⣇⠠⠄⡆⠀⠉⠀⠀⠀⠀⠐⠀⠀⢄⢢⣸⢸⢸⣠⢂⠄⠀⠐⠀⠀⠀⠀⠊⠁⠀⡄⠠⢀⡷⠆⠀⠑
⠇⠠⢃⡀⡸⡅⣿⡇⠤⠤⠀⠠⠀⠀⠀⠠⣈⡲⣵⣷⣿⣷⣷⡵⣊⡡⠀⠀⠀⠀⠀⠀⠤⠄⣿⡇⡽⡀⣀⠣
⠸⠂⢀⣱⡀⡸⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⣒⣚⣿⣿⣿⣿⣛⣟⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⣽⠸⡀⣰⣁⠁⠾
⡇⠀⢆⠀⢹⡅⢿⡇⠉⠉⠀⠀⠀⠀⠀⠈⠤⢚⢝⣿⣟⡿⣟⢝⠢⠈⠀⠀⠀⠀⠀⠈⠉⠁⣿⠇⣽⠁⢀⡆
⠀⣄⡀⠓⡇⠈⢇⣷⣀⡠⠄⠀⠀⠀⠀⠀⠀⠁⠎⡸⢸⠸⡈⠆⠁⠀⠀⠀⠀⠀⠀⠤⣀⣰⡇⠏⠀⡗⠃⣀⡔
⠀⠊⠠⣈⠉⢹⡥⡛⢧⠀⠀⢀⠀⠀⠀⠄⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀⠀⢠⠟⠣⣽⠉⢉⡠⠈
⠀⠀⡀⠈⠰⢺⡀⢙⡨⣷⡎⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣠⡯⣘⠁⣨⠲⠈⠀⡀
⠀⠀⠈⢻⠀⢘⠉⠉⡖⠩⡙⢦⣠⠋⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡄⠀⠈⢣⣠⠞⡩⠑⡎⠉⢙⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠁⠒⢣⠤⢴⣉⢋⠓⢦⣼⣁⠀⠀⠇⠀⠀⠇⠀⠀⣱⣤⠖⢋⢋⡱⠤⢠⠓⠂⠁⢀
⠀⠀⠀⠀⠀⠁⠺⠇⠈⢄⢀⣵⡀⣉⣌⡙⠭⡙⡾⡷⡒⡲⡿⠞⡩⠝⣉⣌⡁⣀⣅⢀⡌⠀⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⣀⡀⠘⣀⠀⡣⢄⠼⠓⢧⠉⢡⠖⠻⢄⠤⡃⢀⡘⠀⢀⡀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠂⣠⠈⠁⠤⠘⠉⠙⠠⠄⠉⢠⡀⠈⠀⡠⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣉⣃⣄⣀⣀⣚⣂⣀⣀⣄⣋⣉⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣦⡀⠀⣠⠀⠙⢿⣿⣷⣄⠀⣠⡄⠘⢿⣿⣿⣦⡀⢀⣤⠈⠻⣿⣿⣦⡀⠀⣠⠀⠙⢿⣿⣷⣄⠀⣠⡄⠙⢿⣿⣿⣦⡀⢀⣤⠈⠻⣿
⠟⣡⣾⣿⠀⠀⠀⠙⠿⣫⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⣿⠀⠀⠈⠻⠟⣡⣾⣿⠀⠀⠀⠙⠿⣫⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⣿⠀⠀⠈
⠈⠻⣿⣿⣧⡀⠀⣠⡄⠙⢿⣿⣷⣄⠀⢀⡄⠙⢿⣿⣿⣦⡀⠀⣠⠈⠻⣿⣿⣧⡀⠀⣠⡄⠻⢿⣿⣷⣄⠀⢀⡄⠙⢿⣿⣿⣦⡀
⠀⠀⠈⠻⠿⣫⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⡇⠀⠀⠙⢿⠟⣡⣾⣿⠀⠀⠈⠻⡿⣫⣾⣿⡇⠀⠈⠹⡿⢋⣴⣿⡇⠀⠀⠙⢿⠟⣡⣾
⣦⡀⠀⣠⠈⠻⣿⣿⣧⡀⠀⢀⡄⠙⢿⣿⣷⣄⠀⠀⣠⠘⢿⣿⣿⣦⡀⠀⣀⠈⠻⣿⣿⣧⡀⠀⢀⡄⠻⣿⣿⣷⣄⠀⠀⣀⠘⢿⣿
⡿⣣⣾⣿⠀⠀⠈⠻⡿⢋⣴⣿⡇⠀⠀⠙⢿⠟⣡⣾⣿⠀⠀⠙⢿⡿⣃⣾⣿⠀⠀⠈⠻⡿⢋⣴⣿⡇⠀⠈⠻⣿⠟⣡⣾⣿⠀⠀⠙
⠘⢿⣿⣿⣦⡀⠀⢀⡀⠻⣿⣿⣧⡀⠀⠀⡀⠸⣿⣿⣿⣄⠀⠀⣀⠘⢿⣿⣿⣦⡀⠀⢀⡀⠻⣿⣿⣧⡀⠀⠀⡀⠸⣿⣿⣿⣄
⠀⠀⠙⢿⡿⢋⣴⣿⡇⠀⠈⠻⣿⠟⣠⣾⡇⠀⠀⠹⣿⡿⢃⣴⣿⠀⠀⠙⢿⡿⢋⣴⣿⡇⠀⠈⠻⣿⠟⣠⣾⡇⠀⠈⠙⢿⡿⣣⣾
⣄⠀⠀⢀⠐⢿⣿⣿⣧⡀⠀⠀⡀⠺⣿⣿⣷⡀⠀⠀⣀⠸⢿⣿⣿⣄⠀⠀⢀⠀⢿⣿⣿⣧⡀⠀⠀⡀⠺⣿⣿⣷⡀⠀⠀⣀⠘⢿⣿
⡿⢃⣴⣿⠀⠀⠙⢿⣿⠟⣠⣾⡇⠀⠈⠻⣿⡿⢂⣴⣿⠀⠀⠙⣿⡿⢃⣴⣿⠀⠀⠙⢿⣿⠟⣠⣾⡇⠀⠈⠻⣿⡿⢂⣴⣿⠀⠀⠙
⠐⢿⣿⣿⣄⠀⠀⠀⠁⢾⣿⣿⣧⡀⠀⠀⡈⠰⣿⣿⣿⡀⠀⠀⢀⠰⣿⣿⣿⣄⠀⠀⠀⡁⢾⣿⣿⣧⡀⠀⠀⠈⠰⣿⣿⣿⡀
⠀⠀⠙⢿⣿⠗⣠⣾⡇⠀⠙⢿⣿⡿⢠⣾⡇⠀⠈⠻⣿⡿⢂⣴⣿⠀⠀⠛⣿⣿⠗⣠⣾⡇⠀⠙⢿⣿⡿⢀⣴⡇⠀⠈⠻⣿⡿⢆⣴
⡀⠀⠀⠀⠁⢾⣿⣿⣇⠀⠀⠀⠉⢶⣿⣿⣧⡀⠀⠀⠈⠰⣿⣿⣿⡀⠀⠀⠈⠁⢾⣿⣿⣇⠀⠀⠀⠉⢴⣿⣿⣧⡀⠀⠀⠈⠱⣿⣿
⣿⠆⣠⣾⠀⠀⠙⢿⣿⡷⢀⣼⡇⠀⠙⢿⣿⡿⢂⣴⣿⠀⠈⠻⣿⣿⠆⣠⣾⠀⠈⠙⢿⣿⡷⢀⣴⡇⠀⠙⢿⣿⡿⢂⣴⣿⠀⠈⠻
⠡⣾⣿⣿⡄⠀⠀⠀⠉⢶⣿⣿⣇⠀⠀⠀⠉⢰⣿⣿⣿⡀⠀⠀⠈⠡⣾⣿⣿⡄⠀⠀⠀⠉⣴⣿⣿⣇⠀⠀⠀⠉⢰⣿⣿⣿⡀
⠀⠈⠻⣿⣿⡦⣀⣼⡇⠀⠙⢿⣿⡷⢀⣴⡇⠀⠙⢿⣿⣿⠆⣠⣾⠀⠈⠻⣿⣿⡦⢀⣼⡇⠈⠻⣿⣿⣷⢀⣴⡇⠀⠙⢿⣿⣿⠆⣠
⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠀⠉⢴⣿⣿⣇⠀⠀⠀⠙⢡⣾⣿⣿⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠈⠛⣵⣿⣿⣇⠀⠀⠀⠙⢡⣾⣿
⣿⡆⢀⣴⠀⠈⠻⣿⣿⡦⢀⣴⡇⠀⠹⢿⣿⣷⠄⣠⣾⠀⠙⢿⣿⣿⡆⢀⣴⠀⠈⠻⣿⣿⡦⢀⣴⡇⠀⠹⣿⣿⣷⠄⣠⣶⠀⠙⢿
⢫⣴⣿⣿⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠀⠙⢡⣾⣿⣿⠀⠀⠀⠙⢋⣴⣿⣿⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠈⠛⢡⣾⣿⣿
⠀⠙⢿⣿⣿⡦⢀⣴⡆⠈⠻⣿⣿⣦⠀⣠⡆⠈⠛⣿⣿⣷⡄⢀⣴⠀⠙⢿⣿⣿⣦⢀⣴⡆⠈⠻⣿⣿⣦⠀⣠⡖⠈⠻⣿⣿⣷⡄⢀
⠀⠀⠀⠙⠋⣴⣿⣿⡇⠀⠀⠈⠛⣡⣾⣿⣇⠀⠀⠈⠛⢋⣴⣿⣿⠀⠀⠀⠙⠛⣴⣿⣿⡇⠀⠀⠈⠛⣡⣾⣿⣇⠀⠀⠈⠛⢋⣴⣿
⣷⡄⢀⣴⡂⠙⢿⣿⣿⣦⠀⣠⡆⠈⠻⣿⣿⣦⡀⢀⣴⠈⠻⣿⣿⣷⡄⢀⣴⠀⠙⢿⣿⣿⣦⠀⣠⡆⠈⠻⣿⣿⣦⡀⢀⣴⠈⠙⢿
⢋⣴⣿⣿⠀⠀⠀⠙⠟⣡⣾⣿⡇⠀⠀⠈⠻⢋⣴⣿⣿⠀⠀⠈⠻⢋⣵⣿⣿⠀⠀⠀⠙⠟⣡⣾⣿⡇⠀⠀⠈⠻⢋⣴⣿⣿
⠈⠻⢿⣿⣷⣄⠀⣠⡆⠙⢿⣿⣿⣦⠀⢀⡄⠈⠻⣿⣿⣦⡀⢀⣴⠈⠻⣿⣿⣷⣄⠀⣠⡆⠙⢿⣿⣿⣦⠀⢀⡤⠈⠻⣿⣿⣦⡀⢀
⠀⠀⠈⠛⠟⣡⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⡇⠀⠀⠈⠻⢟⣵⣿⣿⠀⠀⠈⠻⠟⣡⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⡇⠀⠀⠈⠻⢟⣵⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠒⠊⠉⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡦⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡴⢲⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠘⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠴⠊⠢⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⠔⠉⠀⠀⢠⣤⣴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣠⠖⠁⠀⠀⠀⠙⢆⠀⠳⡄⠀⠀⠀⠀⠀⢀⠔⠁⢀⠔⠁⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⢦⡀⠀⣠⠔⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠈⢦⡀⠑⠞⠁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠑⢄⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡤⢤⠤⠤⠤⠤⣿⣿⡿⣿⢤⠤⠤⢿⣿⣿⣿⠤⠤⠤⢼⣿⣿⡿⠤⠤⡤⢼⣿⣿⣿⠤⠤⠤⢼⣿⣿⣿⠥⠤⡤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⣿⠛⠉⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣽⡏⡉
⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠁⠁


⠀⠀⠀⠀⣴⢶⡄⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢠⡖⣦
⠀⢴⡄⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠁
⠀⠘⠑⠀⢠⣤⡀⠀⠀⣀⠀⠀⢠⣤⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠙⠾⠃⠀⠈⠉⠁⠀⠘⠷⠃⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⣯⡿⠀⠀⠺⠗⠀⠀⢿⡿⠂⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⣶⣶⠄⠀⢠⣤⠀⠀⣴⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⠊
⠀⣀⡀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠸⣯⠇⠀⠀⠿⠇⠀⠘⢧⠝⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣻⣇
⠀⠀⠀⠀⢾⣛⡆⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⣶⡂⠀⢰⣟⣷
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢠⡖⣦⢠⣶⣦⢠⡶⣦⢠⡶⣦⢀⣶⣶⠀⣶⢶⠀⡶⢶⡀⣶⣶⡄⣴⢲⡄⣴⢶⡄⣴⢶⡄⣴⣶⡄⠀⠀⢈⠀⢈⠁⠀⡁
⠈⣙⡁⠀⣙⡁⠀⣙⡉⠀⣙⡉⠀⣙⣋⠀⣙⣋⠀⣉⣋⠀⣉⣋⠀⢉⣋⠀⢉⣋⠀⢉⣋⠀⢈⣋⠁⠀⠀⠐⠀⠐⠂⠀⠂
⠘⠧⠟⠘⠧⠟⠘⠧⠟⠘⠧⠽⠘⠷⠽⠀⠿⠼⠀⠿⠽⠃⠿⠾⠃⠿⠼⠃⠿⠼⠃⠫⠼⠃⠻⠼⠃⠀⠀⠨⠀⠨⠁⠀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠤⠤⠼⠃⠙⠤⠤⠴⠃⠘⠤⠤⠴⠃⠘⠤⠤⠤⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⠤⢄⡤⠤⠤⣠⠤⠤⣄⡤⠤⢤⡠⠤⠤⣄⠤⠤⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⠭⠭⢍⡩⠭⠭⣩⠭⠭⣍⡭⠭⢭⣩⠭⠭⣍⠭⠭⢭⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢀⡇⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⡭⠭⢭⡩⠭⠭⣍⠭⠭⢍⡭⠭⠭⣩⠭⠭⣍⡭⠭⢭⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣧⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡭⠭⢭⣩⠭⠭⣍⠭⠭⢍⡩⠭⠭⣩⠭⠭⣍⡭⠭⢭⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⣀⣠⠚⣿⣷⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⡠⠊⠁⢸⣧⠀⠙⠛⡿⠀⠀⠀⠀⠀⣿⣿⢀⣀⠠⡤⠀⠀⠀⠀⠀⠀⠀⢠⣭⠀⠀⢀⣀⢸⣿⡇
⠀⠀⠅⠀⢀⣸⣿⣿⣶⠋⠀⠀⠀⠀⠀⠀⣿⣿⠈⣿⠰⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⢸⣿⢸⣿⡇
⠀⠀⢳⣾⣿⣯⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣿⢸⣿⢰⣶⠀⠀⠀⠀⠀⢸⣿⢸⡏⢸⣿⢸⣿⡇
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠈⠉⠈⠉⠀⠀⠀⠀⠀⠈⠉⠈⠉⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⡄⢀⣤⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣆⢠⣄
⠀⠀⢐⡀⠀⠂⠂⡎⠒⠀⠂⠀⠀⠀⠀⠀⡇⢀⣄⠙⢡⠾⠁⠀⠀⠀⠀⠀⠀⢸⠀⡟⢇⢋⡞⠹⡦
⠀⠀⢸⡇⣤⡀⠀⡇⡂⠉⠀⠀⠀⠀⠀⠀⣇⡞⡋⠳⠃⢐⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠘⠿
⠀⠀⠸⠃⣿⡧⣶⠙⠁⠀⠀⠀⠀⠀⠀⠀⠿⠴⠶⠤⠤⠶⠦⠆⠀⠀⠀⠀⠀⠸⠧⠦⠤⠤⠤⠤⠴
⠀⠀⠀⠀⠈⠠⠧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠅⣼⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⣷⡡⠃
⠀⢠⡾⣆⠀⠂⣼⠏⢴⠆⠀⠀⠀⠀⠀⠀⠀⣠⡀⣸⣷⣴⣿⠀⠀⠀⠀⠀⠀⠸⠀⠙⢂⠜⣫⣾⣷
⠀⠀⠀⠹⡆⢠⡏⠈⣠⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⠀⠀⠀⠀⠀⠀⠘⢲⣾⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠹⡟⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠤⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⠉⠉⠉⠉⠉⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠸⣿⡿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⢉⡉⠀⠙⠻⠃⠀⠀⠀⠀⢸⣿⡇
⠀⢰⣿⣷⠀⠀⠀⠀⠀⣿⣿⠀⡿⠋⢀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⢀⣼⣿⡿⠋⠉⠀⣤⣤⠀⠉⠙⢿⣿⣦⡀⠀⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⡏⢠⣿⣿⠋⠀⠀⠀⠀⣿⣿⠀⣿⣷⡄⠙⣿⣿⡄⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⠀⣾⣿⠇⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⢸⣿⣧⠀⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⢨⣿⣿⠀⣿⡇
⠀⢸⣿⣿⢀⣀⣀⣀⣀⣿⡀⢻⣿⣇⠀⠀⢀⣀⣀⣿⣿⠀⣿⣿⡇⠀⣸⣿⡏⢀⣿⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⢿⣿⣦⡈⠻⢿⣿⣿⣿⠀⠿⠟⢁⣼⣿⣿⣀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣀⣀⣀⣀⣤⣶⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠟⠛⠉⠀⠈⠛⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠀⠉⠐
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠃⢀⠀⠀⡂⠁⠀⠈⡀⠅⠀⠀⠀⠀⠀⠛
⠀⠀⠉⢀⠈⠉⡁⠀⢉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠆⠀⠀⠐⠀⠈⠤⠀⠀⠊⠀⠀⠀⠀⠀⡀⡅
⠐⡘⠄⠀⠀⠀⠙⡀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠶⢂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠻⡁⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠂⢁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢠⠦⢸⡀⠀⠀⠀⠋
⠀⠘⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡢⣀⠈⠁
⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠓⣾⠎⠀⠀⠀⠠⠈⠉⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⡄⢠⡆
⡌⠀⡀⠀⠀⠴⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠠
⠀⠀⠈⠀⠀⠈⠀⣼⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⡆⠀⠀⢀⡄⠠⠀⠃⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⣂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠉⠀⠀⢀⠒
⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⢀⠀⠈⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡀⠀⢄⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⣀⣀⣀⣀⠀⢀⣀⠈⠁⠀⠀⠀⠀⠀⠸⠀⠀⡀
⠀⡀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⢤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠎⠐⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠄⢈⢤⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⠀⠂⠂⠂
⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠠⡆⠀⠀⠀⠈⠈⠉⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡆⠂⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠀⠀⠀⠀⡀
⠀⠀⠈⡁⠀⢀⠀⠀⠠⠄⠐⠀⣠⠀⡀⠀⠞⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⢄
⠐⠀⠁⠢⣤⠋⠀⠀⠀⢠⠸⠈⠉⠀⠩⠄⠀⠐⠐⠚⠀⠀⠒⠀⡀⠀⢀⠘⢩⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⠀⢀⠀⠀⠑⡄
⠀⠀⢀⢀⡄⡀⠀⠀⠀⢰⡀⠀⠄⢃⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂⠄⠠⠀⠀⠈
⠰⠀⠃⠀⠀⠀⠄⠠⠂⠀⠁⠀⠀⠀⢃⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠂⠀⠃⠀⣠⠄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⢡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⡀
⠀⠶⢤⣦⠂⠀⠀⠀⡈⠐⠲⠒⠂⢸⠀⠀⠀⠀⠁⠀⡠⠀⢀⡄⡀⢀⠉⠀⠠⠄⠀⠀⠀⠀⠒⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠐⠀⠀⠀⠀⢀⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠂⠀⠀⠄⠀⠉⠐⠀⠀⠓⠀⣧⡄⠀⠀⠀⠀⠀⠀⠀⣀
⡠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠐⠀⣠⠀⣀⠀⠀⠀⠀⠩⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠂⠉⠱⠢⣤⣤⡐⣤⣤⡄⠂⠉⠀⠖⣈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⣀⠀⠀⠟⠓⠛⡆⠀⢠⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⢀⣐⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠊

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

⠀⠀⠀⠀⠀⢀⠴⠒⠒⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠐⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⣶⣶⡶⠶⢶⣶⣖⣂⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⢠⠃⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⣿⣿⣿⣿⣓⣒⣚⣿⣭⣭⠹⡇⢠⠬⠴⢿⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢾⡶
⠀⠀⠀⠀⢸⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠭⠴⠿⣉⣉⣈⣿⣉⣀⣀⣯⣿⣭⣭⣽⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠆⢸⣻⡯
⠀⠀⠀⠀⢸⠀⡃⣾⡟⠛⠛⠛⠹⠭⠿⠿⠾⠶⠶⢶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⢸⣭⡯
⠀⠀⠀⠀⠸⢀⠀⣿⠃⠀⢀⠄⡲⠐⠀⠀⠀⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⡙⡛⠛⠚⠛⠛⠛⠻⠿⠿⠿⠿⡆⢸⢸⠻⣏
⠀⠀⠀⠀⡆⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⢰⡟⠉⣿⠀⠘⠛⡶⢠⡆⠀⢰⠀⠈⣥⣬⡄⠀⣠⣤⣄⡀⠀⣀⣀⡀⠀⡇⢸⢸⡳⣗
⠀⠀⠀⠀⡇⢸⠀⣿⠀⠀⠀⠌⠂⠀⠀⠀⠈⠓⠒⡇⠈⠷⠶⣟⠀⢤⢴⡇⠘⣧⣀⡿⠀⢸⣇⡀⠀⠀⣿⠀⢸⡇⠀⠉⠙⣿⠀⡇⠈⢸⣿⡓
⠀⠀⠀⢀⠁⡌⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡇⠀⣀⣠⡟⠀⣀⣸⡇⠀⠀⠀⡟⠀⢸⡏⠉⣧⠀⣿⠉⢹⡀⢸⡟⠛⠋⠀⡇⡀⢸⣼⡧
⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠐⠒⠒⠒⠒⠒⠀⠀⠨⠥⠤⠤⠭⠉⢀⣀⣀⣀⣃⠀⠘⠓⠾⠃⠀⠛⠒⠚⠀⠘⠳⠶⠆⣸⠀⡆⢸⣼⡏
⠀⠀⠀⠸⠀⡇⣐⣉⣉⡉⠉⠑⣶⡀⢠⠤⠤⠠⠤⠤⢤⡀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠚⠁⠀⡇⢸⢾⡒
⠀⠀⠀⡇⢰⠀⡇⠋⠛⠃⠪⠄⡏⠇⡌⠀⠶⠇⠤⠄⢸⡳⢠⠃⣤⣤⠀⡀⠀⡝⡆⢰⠊⠉⠉⠉⠑⢒⡄⠀⡤⠤⠤⠤⠀⠀⢤⠀⠇⣾⢺⡅
⠀⠀⠀⡇⢸⠀⠓⠛⠛⠏⠭⠭⠴⠁⠳⣶⣒⣒⣂⣤⣜⡇⢸⣀⣁⣈⣈⣋⣀⣿⠁⡇⠀⠒⠛⠛⠀⢸⡇⢰⠁⢶⡀⣶⡆⠀⣽⢸⠀⣧⣹⠭
⠀⠀⠀⠃⠸⠑⠒⠒⠒⠒⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⣀⣉⡉⠛⠚⠛⠛⠚⠀⠙⠛⠭⠽⠭⠶⠿⠃⠘⠶⠶⣒⣦⣶⣾⡟⢸⢀⠳⣿⣒
⠀⠀⢸⠀⡇⠸⠋⠑⠒⠒⠒⢲⡄⢠⠤⠤⠤⠤⢀⣀⡀⠀⠀⣀⣀⣈⠉⠉⠉⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠤⠠⠤⠤⠤⠤⠤⠤⡎⢸⡑⣿⠠
⠀⠀⠸⠀⠃⡇⠀⠀⢸⠃⠀⣜⡇⢸⠀⠀⣖⣦⠀⢰⢹⠀⡞⠁⢠⠤⡍⠉⠻⡆⢰⠉⢐⡒⠒⠒⠒⣦⠀⡴⠒⠀⠀⠀⠤⢤⡀⡇⢸⡙⣯⣉
⠀⠀⠀⢸⠀⠧⣒⣤⣥⣤⠤⣯⠃⢸⡀⡘⣦⣜⣀⣸⡸⠀⡇⠀⠠⠩⠇⠀⣇⡇⢸⠀⠈⣡⣯⡀⠠⣿⠀⡇⠠⢤⡎⠉⠀⢸⣧⡇⣦⣈⣗⣒
⠀⠀⡇⠘⢠⠤⠤⠤⠤⠤⠤⣀⠀⣈⣉⣉⣉⠒⠓⠒⠃⠀⠛⠛⠛⠛⠯⠯⠿⠃⠸⢶⣖⣠⣾⣥⣼⡿⠀⣇⣤⣌⣀⣀⣀⣸⡏⡇⡗⢼⡗⠂
⠀⠀⠇⡀⢸⠀⠀⣴⣆⠀⠀⣿⢀⡇⠀⣤⠤⠈⠉⡵⠄⢰⠒⠒⠦⠤⠀⢤⡄⠀⡤⠤⠤⠤⠄⣀⣀⠀⢀⣀⣉⣉⠉⠉⠙⠉⠁⣇⡟⢼⠯⠥
⠀⢸⠀⡇⢸⣤⣀⢀⣀⣀⣸⡇⢸⠀⠀⠠⠜⠀⠀⣿⠀⡝⠀⠀⡿⢥⠀⢸⡇⢰⡃⠀⢀⣀⠀⠀⣿⠂⡏⠀⠀⠄⠀⠉⣵⠀⢠⢹⡍⢺⠁
⠀⢸⢠⠁⠀⠈⠉⠉⠉⠉⠚⠁⠘⠛⠛⠛⠛⠿⠿⠛⠀⠷⣶⣲⣚⣛⣦⣾⡇⢸⣀⡤⣀⣀⣀⣠⢻⠠⡇⠈⠉⠍⠉⠀⣿⡆⠘⢸⢮⣿⣁⡀
⠀⠘⢸⠀⡞⠉⠉⡁⠒⠒⣤⠀⢰⠒⠒⠒⠤⠤⢄⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠉⠁⠉⠉⠑⠒⠋⠀⠛⠯⠥⠭⠭⠿⠿⠀⡇⠸⢦⡗
⠀⡆⠀⠀⡇⠀⠀⠇⠀⠀⣿⠀⡇⠀⢨⠷⠂⠀⡎⡇⢰⠃⠀⠤⢤⠀⠈⣷⠀⢰⠋⠉⠈⠉⠁⡒⡄⠀⡴⠦⠤⠤⠤⠤⡄⠀⡇⡟⠲⡯⠭⠁
⠀⡇⡄⠀⠻⠭⠤⠴⠶⠮⠽⠀⢧⣤⠬⠭⢥⡠⢧⡇⢸⠀⣀⠚⠻⠀⢀⢿⠀⡇⠀⠀⠀⠀⠀⡏⡇⢸⠁⠀⣢⡔⠂⢸⡟⢠⠁⣏⢺⡏⠉
⠀⡇⡇⢀⠤⠤⠤⠤⠤⢤⡀⠀⢀⣀⣉⠉⠉⠉⠉⠀⠈⠓⠓⠒⠓⠚⠛⠚⠀⠇⠀⠀⠀⠀⠀⣧⠃⢸⣀⣈⣉⡑⠀⣸⡇⢸⢠⣦⣹⣋⣁
⢀⠀⡇⢸⠀⢠⠖⢄⠀⠸⢣⠀⡏⠀⠀⠈⠉⠉⣶⠀⢠⠒⠒⠒⠂⠤⠤⡀⢸⠀⠐⢺⠗⠂⢰⢸⠀⠀⠉⠛⠛⠓⠛⠛⠀⢸⢸⡒⢽⡆⠂
⢸⠀⡇⢸⠀⠈⠒⠋⠀⡐⡎⠠⠁⠀⠀⠂⠀⢠⢻⠀⡌⠀⠛⡰⢃⠀⢸⡇⢸⠀⠀⠀⠀⠀⢸⢸⠀⡴⠒⠒⠒⠒⠒⢤⡀⢸⢸⡕⣾⠡⠤
⢸⠀⡇⠈⠛⠛⠛⠛⠛⠛⠁⠀⠶⢶⣒⣒⣰⣚⡞⠀⣇⡠⢀⣁⣈⡠⢞⡇⢸⠀⠀⠀⠀⠀⣸⡆⠀⡇⠀⢒⣒⡂⠀⣷⠁⡎⡚⢮⣿⣒
⠘⠀⠈⠑⠒⠤⠤⠤⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠉⠀⠈⠻⠽⠵⠶⠊⠽⠃⠰⣅⣤⣀⣀⣀⣰⣻⢀⠃⣏⢢⡧⠤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⣀⣀⣀⣀⣀⠈⠉⠉⠛⠙⠛⠁⡸⠀⠯⣰⣇⣀⠂
⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⢸⡳⣹⠓⠒
⠈⠛⠿⢯⣹⣙⡗⢲⠲⡒⢶⢠⠤⡤⢤⠤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⢼⠭⠍
⠀⠈⢑⣒⣒⣒⣉⠩⠭⠭⠭⡍⣋⣛⣚⡒⠓⠚⠒⠷⠼⠬⠯⣿⣙⣏⣳⣲⡖⢶⢤⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣸⣝⣿⠯⠭
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠒⠒⠚⠛⠋⠩⠭⠭⠤⠖⣒⣒⣒⡒⡃⠭⠭⠭⠭⠙⣛⣛⣒⣓⣛⣦⢧⣽⢭⣯⣽⠭⠯⠿⠏⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠐⠒⠒⠒⠒⠀⠺⠭⠭⠭⠍⡉⢉⡓⠒

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠦⠾⣽⣷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠠⠄⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣤⠁⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣺⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠋⢋⠉⠄⠰⠿⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣉⢠⡤⠀⠆⣒⣂⣩⡤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⣶⢀⣃⡴⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣾⣟⣷⣶⣶⣶⣶⡶⡶⡖⣖⣒⣚⣛⣟⣋⣭⡩⡴⠦⠶⠒⡛⢋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠟⠛⢛⡛⢉⡉⢉⡉⢩⡅⢡⡅⠠⣄⠀⡆⠰⠇⠘⣃⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⡄⠰⠆⠘⣁⣘⣃⣀⣥⣬⣥⣤⠥⠬⠥⠶⠶⠖⠒⠚⠋⠉⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣶⣾⣿⣿⣿⣯⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠖⡒⠋⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠿⠛⠛⢛⡋⠉⡭⠉⢩⠉⠁⠄⠀⢰⠀⠀⠇⠀⣓⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢩⠄⠀⡆⠀⠩⢀⢀⡃⣀⣁⣀⣨⣡⣤⣤⠤⠴⠴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣂⣠⠤⠶⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣷⣤⣤⣤⣤⣤⣤⡤⡤⢤⠤⠤⠤⠴⢶⢶⣶⣶⡶⢶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣷⡿⣶⣺⢹⣦⣞⣼⣣⣷⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠛⠛⠛⠚⠓⠓⠚⠛⠛⠛⣛⣛⣉⣩⡭⠿⠿⣛⠻⢴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⢖⠒⢋⢉⣩⠉⠤⡀⡶⠀⠿⢀⣚⡤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢐⠹⢂⣚⣡⣭⠴⠴⠴⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⡛⡟⣻⠟⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠛⠉⠛⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⣿⣹⡏⡉
⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠀⠘⠊
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⢿⡼⠃⠀⠀⠿⠀⠀⠸⢧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈
⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢴⡛⣦⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⢰⣞⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠐⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠦⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠻⠾⠃⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠻⠾⠃
⠀⠀⠀⠀⠠⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⠄⠀⠰⠶⠀⠀⢸⣛⡳⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠐⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠸⣯⡿⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠨⠿⠀⠀⠸⣯⡽
⠀⣻⡇
⠀⠉⠀⠁⠀⠀⠀⠀⠀⣤⡄⠀⠀⣴⢶⡄⠀⠀⣤⠀⠀⢠⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⠉⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢁⡀⠀⡀
⠀⣠⣀⠀⣠⣄⠀⣀⣄⠀⢀⣄⡀⢀⣠⡀⠀⣠⣀⠀⣠⣀⠀⣀⣄⠀⢀⣄⡀⢀⣤⡀⢀⣠⡀⠀⣠⣀⠀⠀⠀⠠⠀⠠⠀⠀⠄
⠘⠧⠟⠈⠿⠿⠀⠿⠼⠃⠻⠼⠇⠸⠷⠟⠘⠧⠟⠘⠷⠿⠀⠿⠾⠃⠻⠼⠇⠻⠶⠇⠘⠧⠟⠘⠿⠿⠀⠀⠀⢈⠀⢈⠀⠀⡁
⢠⡖⣶⢠⣖⣶⠀⣶⢲⡄⣴⡖⡆⢰⣒⣦⢠⡖⣦⢠⡶⢲⠀⡖⣲⡄⣴⢲⡆⣴⡒⡦⢠⠖⣢⢠⡖⣶⠀⠀⢀⠠⠀⠠⠀⠀⠄
⠀⠙⠋⠀⠙⣋⣀⣉⣋⣀⣈⣛⣁⣈⣛⣁⣀⣙⣉⣀⣙⣋⣀⣙⣋⣀⣈⣋⣁⣈⣛⣁⣀⣛⣁⣀⣙⣋⣀⣀⣀⣈⣀⠘⠀⠀⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⣀⣁⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠿⠟⠛⠛⠛⠻⠿⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣤⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⡿⣦⠿⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⣿⠉⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⣴⡟⢻⡆⠀⠀⠀⣠⣾⠛⠛⠁⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⠛⠛⠛⢷⡟⢻⡟⢀⣤⣤⣤⠀⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣠⣤⣤⡀⠈⠛⠛⠁⢸⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⣿⣿⣿⡇⢀⣀⣀⡀⢸⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣄⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⢿⣿⣭⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠙⢿⣶⣿⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠄⠀⢀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⢄⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠖⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠋⠀⠀⠈⢱⠀⡖⠋⠉⠙⢢⢀⠎⠉⠈⠈⠠⠀⠋⠁⠀⠀⠱⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢰⠂⢨⠄⠀⠀⡬⡄⠀⢸⠀⠀⢉⡄⠀⡆⠀⠀⠸⠤⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠃⠀⢸⠀⡆⠀⠀⠀⢀⡈⡇⠀⠈⠀⠀⡅⡄⠀⠘⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣑⣶⠖⣖⠋⠀⢉⣒⣶⣒⠋⠀⠈⠒⠒⠒⠊⠀⠉⠒⠒⠒⠚⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⢱⠀⠃⠀⡀⠀⢹⠄⡏⠀⠀⠈⠑⠀⡞⠉⠉⠉⠑⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠸⠼⠀⠀⠀⡀⠀⠒⡄⠀⠀⡄⠀⡾⠀⠀⠁⠄⠐⣦⠂⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣄⠀⠀⢀⡰⠀⠀⠀⠉⠀⢠⠀⣇⠀⠀⠁⠀⡄⢃⠀⠀⠁⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠬⠭⠍⡀⠀⢀⡬⠭⠭⢅⠀⠈⣁⠶⠶⢉⠀⠈⣉⣁⣉⣉⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠁⠀⡀⠀⠀⠀⠎⠀⡠⠄⠀⠇⢸⠀⠀⠄⠀⢱⠸⠁⠀⠀⠈⢧⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⡇⠀⢰⠀⡇⠀⣠⠂⠀⠀⢸⠀⢀⣉⠀⢸⠐⠀⠄⠀⠀⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢧⣀⣀⣀⡼⠀⠱⢄⣀⣀⣠⠆⠸⣄⠀⠀⢀⡌⠀⡀⠀⠀⠀⡸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠒⠒⠂⢄⠀⣠⠔⠒⠒⠠⡀⢀⡤⠤⠭⢅⡀⠀⣠⠭⠭⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠤⠀⠀⠀⡅⠀⠀⠀⠀⢡⠘⠀⠀⠀⠀⠁⠘⠁⠀⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡄⠀⠀⠀⠀⡇⡇⠀⠐⠀⠀⡀⣠⠀⠂⠀⠀⢀⠀⠀⠒⡗⠂⢸⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠣⣀⣀⣀⡠⠃⠳⣀⣀⣀⣠⠇⠹⢄⣀⣀⣠⠎⠈⢄⣀⣀⣀⠜⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⠤⠤⠔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠋⠉⠀⠉⠙⠛⠻⢿⣿⣿⣿⣷⡄⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⢉⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠛⠿⠏⠀⠀⠀⠀⠀⡀⠿⠇⡀⠀⠀⠀⠀⢀⣴⠟⠁⠸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⠀⠀⢀⡀⠾⠋⠀⠀⠻⣦⠀⠀⣴⠟⠁⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠶⠶⣤⣄⡀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣦⣌⠻⣦⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠛⠛⠛⢛⣛⣛⣛⣛⡛⢿⠀⠀⠀⠀⠙⣿⣿⣿⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠀⠀⠀⠀⠀⠈⠻⠟⣡⣾⣿⣶⣤⣤⣀⣀⣀⡄⢠⣶⣿⣿⣷⣌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣍⣉⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⣌⡙⠿⢋⣴⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣦⠹⡿⢃⣤⠙⡿⢃⣾⣿⣷⣦⣄⠀⠀⠀⠀⢹⣿⣿⣷⣄⡀
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠙⢿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⡆⢸⣯⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿
⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣈⣉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⢸⣿⣟⣿⡟⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⣤⠶⢆⡴⠶⣄⢴⣄⢰⢀⡴⠆⢰⠀⢰⠀⣶⡀⣴⡄⣶⠶⢠⡶⣦⠀⠀⣴⣆⠀⣦⡀⡆⢀⣶⡄⢰⡄⠰⣄⣴⢠⡶⠆⣴⢠⡶⠆
⠀⢯⣀⡸⣇⣠⡟⣼⠙⣿⢀⣙⡷⢸⣅⣼⢀⡟⣷⢻⡇⣿⣋⢸⡟⣿⠀⢰⡷⢿⡀⡏⢻⡇⣸⠷⣧⢸⣇⡀⢹⠃⢈⣹⡆⣿⢈⣙⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠀⠀⠓⠈⠅⠀⠀⢰⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠍⣾⠈⠙⠒⠒⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⡀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡄⠀⡥⠀⠀⠀⣰⠊⠁⠈⠻⣆
⠀⠀⠀⠀⠤⠤⠤⣤⡤⠴⣦⣤⡤⠤⢆⠄⡀⠀⠰⣄⣀⣀⡈⡟⠉⣍⠂⡺⢀⡀⠀⠀⢠⣴⠇⠀⠀⠀⠀⠈⠉⠳⢦
⠀⠀⠀⣀⣀⢀⣌⠁⠀⠘⢻⣿⡿⠆⠈⠃⣄⣀⡤⠻⣿⣿⡐⡟⠀⡈⣠⠇⠀⢉⡆⠀⠈⢿⣆⠀⠀⡀
⠀⠀⠀⠀⠀⢺⣤⡀⣶⣿⣿⠆⠠⢀⠀⠀⠸⠀⠀⠇⢸⢸⣽⣷⣰⡷⢺⠀⠀⢸⢁⠀⠀⠀⠉⠳⠮⠤⠖⢂⣢⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠟⠸⠉⠀⠀⡀⠀⠀⠘⠢⣂⡀⠀⢘⢻⣇⣻⡷⠞⠅⠀⠈⠈⠑⠠⡀⢀⣀⣣⣼⡿⠿⣯⣙⠓
⠀⠀⠀⠀⠀⠀⠘⢯⠀⠀⢀⡀⠀⡠⠀⠀⠀⠀⠀⠉⠓⠺⠞⠋⠘⣏⠔⠁⠀⢂⡀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠈⠛⠕
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠚⣮⡉⠉⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⣧⠀⠀⠶⠀⢺⣦⠤⢤⠤⠶⠶⠛⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢧⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⡅⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣆⢡⠀⠀⠀⠀⠀⠠⣠⡴⠞
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡙⠳⣥⣀⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢷⡙⢏⠛⠛⠯⣉⠁
⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⠀⠁⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣽⣌⣳⣄⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢤⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠟⣯⠁⠠
⠀⠀⠀⠀⠀⠀⢀⡯⡁⠀⠈⠈⠉⠉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡻⡇⣿⡀⠁
⠀⠀⠀⠀⠀⠀⣟⠁⠚⡇⠀⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⢄⠨⠫⢸⠄
⠀⠀⠀⠀⠀⠰⣧⠀⠨⠇⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⡄⠈
⣶⣶⠶⠶⠶⣶⣿⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢠⠶⡄⠀⠀⠀⣀⠤⠖⠒⣶⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⠾⡅⠀⣠⠊⠁⠀⠀⠀⡯⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⠶⡇⡰⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⣶⣣⡇⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠸⠤⠻⡏⠉⠉⠉⠉⡩⢽⠏⠉⠉⠉⠉⠉⠉⢹⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢣⠀⣀⠔⠋⠀⡞⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠒⠒⠻⣦⡀⠀⠀⠀⣀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠀⡼⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢸⠀⠀⢀⡇⠙⣦⠔⠋⡥⠤⠤⡌⠢⣄⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠥⣀⣀⣀⠤⠔⠊⠁⠀⠀⠀⢸⠉⠉⠉⠗⠋⠉⠙⡆⣇⣀⣀⡇⠀⠈⠛⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠚⠚⡇⠓⠒⠚⡇⣇⣀⣀⡇⣎⣉⣉⣹⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠴⡏⡋⠉⠉⡇⣇⣀⣀⡇⡯⠤⠤⢽⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡎⣉⣉⣉⡇⣇⣀⣀⡇⣟⣐⣂⣻⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠀⠉⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠋⠊⠁⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧
⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⣿⣿⣷
⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢩⣶⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠮⠍⠐⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⣿⣿⡿
⠀⠀⠸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠹⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡄⡆⠆⠀⠠⡄⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⣀⣮⠄⠀⠀⠀⠀⠀⣀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄
⢸⡇⠇⢰⣷⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣭⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⠐⣾⡇⣢⡄⠀⠀⠀⠀⠀⠀⡝⠛⠛⠓⠒⡄
⢸⡇⣄⢸⣿⢀⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢈⣿⢀⣿⡇⣻⡇⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⡦



⢀⣴⡿⠻⢿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣺⡭⣭⣶⣄⠀⠀⠀⠀⠀⠀⣠⠖⣶⢐⡦⣄
⣿⡏⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢸⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⠀⢸⢹⠋⠀⠀⠁⠸⠀⠀⠀⠀⠀⢰⠧⣉⠿⢠⣇⢸
⠻⣷⣄⣀⣠⣬⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⣷⠗⡄⠀⠀⠰⠀⠀⠀⠀⠀⠈⢋⢵⡫⠝⣾⡝
⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉


⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣾⣾⡆⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠄⢸⡇⡜⡆⠀⠀⠀⠀⢲⢂⠒⡰⡾⢛⢟⣌⠀⠀⠀⠀⢠⢄⣀⣤⣎⢀⣱
⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠀⡀⠘⠃⠣⠂⠀⠀⠀⠀⠐⠘⠻⠓⡐⠠⠂⠙⠃⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿
⠉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢰⣶⣶⠶⢶⣶⣶⠀⠀⠀⠀⠀⢀⣀⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣦
⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡿⢠⡜⠛⣿⠀⠀⠀⠀⠀⠠⠤⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⡇⣿⠘⣿
⢻⡿⡫⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⠏⠹⣿⣷⣶⣿⠀⠀⠀⠀⠀⠐⠒⠐⠒⠐⠒⠒⠒⠂⠀⠀⠀⠀⣿⢸⡇⣿⢸⣿
⠀⠎⣀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠
⠶⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖⠀⠀⠠⠂⠰⠀⠄



⠀⠀⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⣤⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢤⡀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠖⢀⢀⠂⠀⠀⠀⠀⠈⠣⠄⠒⠄⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢤⡄⠁⠀⣋⡐⠁⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⡃⢀⠀⠀⠀⠀⡸⣼⠂⠈⠀⠀⠀⠀⠀⠀⠀⡄⠸⣷⠐⣿⠆⠠⠀⠀⠀⠀⢠⡀⠂⠁⠀⠀⠀⠤⢶⡁⢄⠀⠈⠁⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠕⠂⠟⠛⠓⠋⠑⠀⡤⡆⠀⠀⠀⠀⠀⠀⠄⠄⠀⠠⠦⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡵⠀⠀⠐
⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣄⠐⠀⠀⠀⠑⠂⠀⣤⠀⠀⢀⠄⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠒⠤⠤⠐⢋⠀⠀⠀⠀⠀⢑
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⠀⠀⠀⠀⢀⠈⠀⠀⠀⠈⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠚⠀⠀⠀⠀⠀⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠁⠀⠐⠀⠠⠀⠈⠳⠀⣀⢀⣀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠋⠀⠫⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠐⠲⠂⠀⠨⣁⠀⢀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠄⠀⠀⠃⢀⡀⠀⠀⢀⠀⠀⢀⠘⠂⠤⠤⢤⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠂⠀⠀⠀⠀⠁⠘⠛⣀⠈⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠂⠀⡍⣅⠀⢀⡀⠀⠈⠀⠀⠀⠀⣤⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⠃⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⢀⠠⠈⠐⠠⡀
⠀⠀⠀⠀⠐⠀⢀⠂⠀⠐⠖⠉⠉⠙⠀⠀⠀⠁⢰⣿⡦⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡃⢀⠀⠀⠀⠀⠀⠈⠉⢐⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠀⡴⠀⠀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠂⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣀⡠⠊


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⢀⣴⣲⣦⠀⣠⡶⢶⡄⠀⣴⣖⣦⡀⢀⣶⢲⣄⠀⢠⡶⢶⡄⠀⣴⢒⣦⡀⢀⡴⢶⣤⠀⣠⣖⢢⡄⠀⣴⠶⣦⡀⢠⣶⠲⣄⠀⣠⡶⣶⡄⠀⣴⣶⣦
⠀⠘⢯⣻⠿⠀⠸⢿⡹⠇⠀⠿⣡⡿⠃⠘⢿⣡⠟⠀⠹⣷⣼⠏⠀⠿⣳⡾⠃⠘⢧⣲⠿⠀⠹⣧⣿⠇⠀⠿⣖⡽⠁⠘⢿⣰⠟⠀⠹⣦⡾⠇⠈⠿⣶⡿
⠀⠀⣠⣤⣄⠀⢀⣤⣤⡀⠀⣠⣤⣤⠀⠀⣤⣤⣄⠀⢀⣤⣤⡀⠀⣠⣤⣄⠀⢀⣤⣤⣄⠀⢀⣤⣤⡀⠀⣠⣤⣄⠀⢀⣤⣤⡀⠀⢀⣤⣤⠀⠀⣠⣤⣄
⠀⠸⣷⣐⡿⠀⢸⣰⣎⠿⠀⢧⣺⡸⠇⠸⣟⣷⣿⠀⢸⣶⣬⡟⠀⢿⣰⣺⠇⠸⣆⣱⡿⠀⢻⣸⣳⡇⠐⢷⣸⣾⠃⠸⣷⣆⡿⠀⢺⣀⣺⠇⠐⣿⣤⣿
⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠒⠒⠚⠛⠛⠛⠛⠛⠛⠓⠒⠒⠲⠶⠶⠶⠤⢤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠖⠲⠶⠶⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⣦⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠲⠶⠶⠦⠤⠤⠤⠤⣤⣠⠏⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡈⠉⣉⠀⢲⣤⠐⠒⠀⠠⢤⣀⠄⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⠀⠰⠁⠀⣼⡏⠀⢿⠇⠀⣿⠇⠀⣶⡆⠀⣿⠇⠀⣀⡀⠀⣿⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠨⠟⠛⢻⣟⠋⠳⠶⠶⣦⣾⠟⣤⣤⣤⣄⣸⣿⣀⣀⣉⠀⢰⡟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡸⠀⣴⡆⠀⣼⡏⠀⣤⡅⠀⣷⡆⠀⢉⠀⠙⣭⡠⠉⠉⠩⠿⣛⠁⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢧⣤⣬⣀⣰⣿⣁⣀⠚⠁⢠⡟⠀⠐⠹⠀⢠⣿⠃⠀⡴⠀⠀⣿⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠐⠀⢛⠉⠉⣭⡀⠋⠙⠻⢟⡛⢙⣳⣶⠶⠶⠾⠻⣤⣤⣤⣄⣼⡇⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠃⠀⠽⠀⢰⣿⠀⢀⡳⠂⢰⣿⠁⠀⣤⠀⢈⣷⠂⠀⠐⠒⠛⣯⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢈⠳⣶⡶⠶⠿⠳⣤⣬⣍⣀⣾⣏⡀⠐⠛⠀⣼⡟⠀⠐⠒⠀⢠⡟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⠁⢠⢤⠀⢸⣦⠂⡀⠒⠛⢩⣅⠬⠝⠛⠛⢛⠛⢙⡶⠶⠦⣤⣾⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣈⠀⠙⠋⠀⣾⡏⠀⠉⣤⠀⣼⡏⠀⠀⠀⠀⢸⣷⠁⢀⣀⠉⠙⣦⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠛⠛⠛⠋⠳⠶⢦⣤⣴⡿⣇⣀⣀⣀⢀⣾⡇⠀⠚⠛⠀⢸⡏⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠈⠉⠉⠙⠛⠋⠙⠛⠶⠶⠶⡾⠁⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠟⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠒⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠆⠀⠀⠆⠰
⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠈⣯⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠶
⠀⠀⠉⠀⠀⠀⠀⠰⠖⠀⠀⠰⡄⠀⠀⠀⠀⡈⠀⡀⠀⠀⣈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣘⠀⠈⡁⠀⠀⠲⠀⠰⠂⠒⠴⠆⠀⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⡋⠀⠀⢹⡃⠀⠛⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠙⢀⠐⠃⠘⢣⡆⠀⠠⠆⡄⣠⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠼⡆⠀⠀⣰⠿⠀⡀⠉⣿⠀⠀⠸⠇⠃⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡗⠀⠀⡇⠘⢀⡄⠀⠄⠑⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⠁⠀⢈⠀⠀⢉⠘⠁⠀⠀⠃⣀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠿⠂⠐⠇⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠂⡀⠛⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠰⢶
⠀⠠⣯⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⣀
⠀⠀⢻⡄
⠀⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⢠⣶⣶⡄⣰⣶⣦⠀⣶⣶⡄⢠⣖⢦⠀⣴⢶⣆⢠⣶⣶⡄⣴⡶⣦⠀⣶⢶⡄⢠⡶⣦⠀⣴⡶⣄⢠⡶⣶⡀⣴⣶⣦⠀⠀⠆⠀⠀⠦⠀⠀⠤
⠈⠛⡛⠁⠘⠛⠋⠀⠛⠚⠁⠈⠓⠋⠀⠙⠛⠃⠈⠛⠚⠁⠙⠓⠋⠀⠛⡛⠁⠈⠳⠛⠀⠙⠚⠃⠈⢛⠛⠀⠙⡛⠋⠀⠀⡀⠀⠀⣀⠀⠀⣀
⢰⣿⣿⡆⣼⣿⡷⠀⣾⣿⡆⢰⣿⣷⠀⣾⣟⣧⠰⣿⣿⠆⣾⣛⣷⠀⣿⣿⡆⢰⣟⣷⠀⣾⣟⡆⠰⣟⣷⠀⣾⣟⣷⠀⠀⠁
⣀⣉⣉⣀⣈⡉⣁⣀⣉⣉⣀⣀⡉⣁⣀⣈⣉⣁⣀⣉⣉⣀⣈⣉⣁⣀⣉⣉⣀⣀⣉⣁⣀⣈⣉⣁⠀⢉⣁⣀⣈⣉⣁⣀⡀⢃⣀⣀⠛⣀⣀⣛⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⡙⢻⡋⣙⠛⢻⡟⠋⣿⠟⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⡇⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠸⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢻⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⣾⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣭⣷⣶⣦⡝⠛⠘⠛⠛⠛⠃⠛⠛⢉⡤⠖⠒⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢀⣠⣄⣀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⢿⣿⠟⣁⠀⠀⠘⠲⣄⡀⠀⠀⠀⣀⡤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡗





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠋⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠶⠶⣶⠀⢴⠶⠶⢶⠀⢰⠶⠶⢶⠄⢰⡶⠶⢶⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣀⣀⣿⠀⣿⣀⣀⣺⠀⢸⣀⣀⣸⡇⢸⣇⣀⣸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠀⠈⠉⠉⠉⠀⠈⠉⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠉⠉⣿⠀⣾⠋⠉⢻⠀⢸⠋⠉⢻⡆⢸⡏⠉⢹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣤⣤⡿⠀⢻⣦⣤⡾⠀⠸⣦⣤⣼⠃⠸⣧⣤⣼⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣤⣤⠀⣠⣤⣤⣤⠀⢠⣤⣤⣤⡀⢠⣤⣤⣤⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⢼⠀⢸⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠛⠛⠛⠀⠙⠛⠛⠛⠀⠘⠛⠛⠛⠁⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⠒⠒⣶⠀⣴⠖⠒⢶⠀⢰⠖⠒⢶⡄⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣀⣀⣿⠀⢿⣄⣀⣼⠀⢸⣄⣀⣼⠇⢸⣇⣀⣸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠁

⠀⠀⠀⡠⠤⠤⠄⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠒⠒⠒⠒⠂⠠⢤⡀
⠀⠀⢀⠁⠀⠔⠒⠒⠒⠒⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠐⠒⠲⡄⠈⣵⡄
⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡧⡇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢰⢭
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⢸
⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⡠⠤⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠸
⠀⠀⢸⠀⠀⡇⠀⠀⡔⠁⣠⠤⢆⡀⠐⡃⠀⠀⠀⠀⠀⢀⠊⢀⠴⠒⠢⡀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠲⡀
⠀⠀⢸⠀⠀⡇⠀⡜⠀⡜⠁⠀⣰⠀⢸⠃⠀⠀⠀⠀⠀⠱⢄⠊⠀⠀⢀⠇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⡇
⠀⠀⢸⠀⠀⡇⠀⠓⠚⠁⠀⡰⠁⡰⠃⡠⢢⡀⡠⠢⡀⠀⠀⠀⠀⡠⠃⢠⠋⠀⠀⣋⠙⡁⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⠞⠁⡴⠁⠀⠑⡄⠈⠀⡔⠋⠀⠀⡠⠊⢀⠔⠁⠀⠀⠀⡭⢉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠸⠀⠀⠇⠀⠀⡰⠃⢀⡜⠀⠀⠀⢼⢁⠤⢄⣹⠆⣠⠊⠀⡴⠁⠀⠀⠀⠀⠀⢦⣀⣤⡨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡄⠀⡰⠀⢀⠜⠀⠀⠀⠀⠀⠁⠀⠀⠁⠰⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⡇
⠀⠀⠀⡇⠀⡇⠀⡇⠀⠈⠀⠊⠀⠙⠁⠐⠆⠀⠀⠀⡇⠀⠈⠉⠉⠙⠂⠀⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇
⠀⠀⠀⡇⠀⡇⠀⠙⠀⠈⠁⠀⠁⠀⠈⠁⠉⠀⠀⠀⠘⠃⠀⠂⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇
⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⡇
⠀⠀⠀⠶⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠇⢀⠣⡇
⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠑
⠀⠀⠀⠀⠈⠓⠛⠩⠯⠥⠴⠰⠆⠀⠖⣲⣀⣂⣀⣂⣐⣂⣐⣂⣐⣂⣐⣂⣀⣂⣐⣂⣀⣒⣒⢒⠀⠲⠀⠴⠨⠯⠭⠭⠭⠥⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡏⠉⠉⠀⠉⠉⢱⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⡇⠘⠶⠴⠶⠶⠶⠶⠶⠶⠶⠴⠴⠃⡗⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠁⠢⡄⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⡀⠤⠇⠓⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢏⠉⠉⠉⡏⠉⠉⠉⣇⡤⠒⢞⡄⠀⠀⠀⠀⠀⠀⠀⡼⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⣀⡾⠁⠀⠀⠈⢞⠄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣀⣄⣀⣀⣀⣀⣍⢉⠉⡔⠀⠀⠀⠀⠀⠈⢾⣄⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢸⠸⠤⠿⠿⠿⠿⠿⠿⠿⠸⢴⠁⠀⠀⠀⠀⠀⠀⠈⢋⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠉⣿⣿⣿⣿⣿⣿⣿⣸⡽⠞⠒⠳⢶⣄⠀⠀⠀⠀⢻⡻⡄⣀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⢸⢠⠄⣤⣤⣤⣤⣤⣤⣼⠋⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠉⠑⠋⠥⢖⡤⠄⣀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⢘⣒⣛⣛⣛⣛⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡪
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣀⠿⣿⠿⣿⢿⡿⡟⣷⣄⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⢨⡘⠒⠒⠒⠒⣒⠖⢊⡤⠚⠓⠛⠛⠛⠛⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⢀⣀⣀⡀⠠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠔⠊

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣲⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡦⡤⢄
⠀⠀⠀⠀⢠⣶⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣸⠆⠀⠀⠀⠀⠀⠀⣿⢠⡝⠛⠉⠈⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠃⠀⠱⡀
⠀⠀⠀⠀⠙⠿⠟⠀⠅⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠦⠾⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠙⠗⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⣭⣲⡤⢄⡇
⠀⠀⢀⣼⣿⣿⣟⣛⣿⣿⣿⣅⠀⠀⠀⠀⠈⠛⠛⢛⣿⣿⡛⠙⠛⠦⠀⠀⠀⠀⠀⠀⣿⢀⣤⣠⣴⠆⡇⠀⠀⠀⠀⠀⠀⠈⢿⣵⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁


⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡴⢶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⢤
⠀⠀⢸⡏⠉⠩⠉⠉⠍⡉⠉⡇⠀⠀⠀⠀⠀⣸⣇⣠⡿⣤⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢻⡇
⠀⠀⢸⡇⢐⢸⣦⣥⣿⣶⠀⡇⠀⠀⠀⠀⠀⠸⣏⡀⣹⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⣤⣾⡇
⠀⠀⢸⣇⣘⣛⣛⣛⣛⣛⣀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠉⠋⠛⠙⠃
⠀⠀⠀⠈⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠚⠛⠛


⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⢰⣤⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢀⡞⠉⢠⣭⣷⣴⣦⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠈⠟⢿⡄⠀⠀⠀⠀⠀⠀⠀⡆⡀⢈⠁⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⠶⢤⡀⠄
⠀⠀⠐⠻⠟⢿⡻⠃⢀⡾⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⣀⡀⣤⣾⢈⣿⠀⠀⠀⠀⠀⠀⠀⠇⡅⢨⡁⢨⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣴⣿⢹⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠉⠙⢦⡀⠀⠀⠀⠀⠀⠈⢿⣜⠙⠙⠙⢛⣾⠏⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠞⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠛⠃


⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣤⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⣤⡀
⠀⠀⢸⠿⠿⠿⠿⠿⠿⠻⠿⣇⠀⠀⠀⠀⠀⣼⠋⠀⣠⣾⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠎⠾⠀⡧⠀⠦⠢⡀
⠀⠀⢸⢀⣄⢀⣄⡔⢉⣉⢦⣿⠀⠀⠀⠀⠀⢻⣜⠿⠿⠟⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⡦⢰⡆⠀⡧⠀⢰⠀⠆
⠀⠀⢸⠈⠉⠉⠉⠙⠯⠽⢇⡟⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠰⡔⠦⠀⣗⠉⢁⡴
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠷⠟⠙⠂

⠀⠀⠀⡇⡠⠒⠁⠀⠉⠐⠂⠤⣀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣇⠤⠄⣀⡀⠀⢸⠀⢀⣠⡼⠂
⠀⠀⠀⠙⠢⠤⣀⠀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠀⠀⠈⠣⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠉⢹⠋⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠧⠤⠤⠤⠤⠤⠤⠤⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⢀⡧⠊
⢀⡠⠤⠠⣤⢀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠁⠀⠀⠀⠈⡆⠙⢆⠀⠀⠀⡞⠥⣀⠀⠀⠀⠀⠀⠈⣑⢦⠀⠀⠀⠀⡠⠒⠉⠈⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⡴⠣⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⠀⠘⡆⠀⠀⡇⠀⠀⠑⠢⠴⠒⠊⠁⠀⢸⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠜⠀⠀⠱⡀⠀⠀⠀⠀⠀⡜
⣀⠀⠀⠀⠀⢸⣀⢴⠇⠀⠀⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⠊⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢳
⡀⠉⠉⠉⢉⠏⣠⠎⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⢀⡀⠼⠀⠀⠀⢇⠉⠐⠒⠒⠒⠒⠈⣡⠃⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈
⠈⠐⠒⠒⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⠤⠒⠁⠀⠀⠀⠣⢄⣀⠀⠀⠀⠀⣀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠤⢀⡀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡏⢆⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢩⠉⠑
⠀⠀⠈⢳⠀⠀⠀⠀⢀⠎⠀⠀⠧⠀⠣⡀⠀⠀⠀⠀⢸⠑⢄⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⡴⠊⡇⠑⠢⣀⠀⠀⠀⠀⠋⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⢠⠃⠀⠀⠀⢸⠀⠀⠑⡄⠀⠀⠀⢸⠀⠀⡏⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠋⠀⠀⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡨⢅⡀
⠒⠢⠄⣸⠀⠀⠰⣃⡀⠀⠀⠀⠀⡆⠀⢀⠜⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⠔⠊
⢀⡠⠖⠁⠀⠀⠀⠀⠈⠉⠐⠂⠤⠥⠚⠁⠀⠀⠀⠀⠈⠣⡀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⣀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⣂⠔⠁⠀⠈⠒⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢠⠦⡐⠒⠀⠂⠐⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⠤⣀⠔⠋⢸⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢢⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠱⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡶⡈⠉⠀⠀⠀⠀⠀⠀⣈⢶⠄⠀⠀⠀⠀⡔⠁
⡆⠀⠘⣄⠀⠀⠈⠲⣀⠀⠀⢸⠀⢀⠞⠀⠀⠀⠈⡇⠂⠤⠤⠤⠤⠤⠐⢊⠇⠀⠀⠀⡇⠈⠑⠤⡀⠤⠐⠊⠁⠀⠀⠀⠀⠀⢀⡞
⠃⢀⡠⠊⠀⠀⠀⠀⠀⠉⠢⢼⠔⠁⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠤⢀⡀
⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⡇⠀⠀⢀⡠⠤⠚⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡤⠚⡇⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠒⠗⠒⠉⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⡖⠁⠀⠀⡇⠀⠀⠀⠈⠉⢲⠀⠀⠀⠀⠀⢸⠙⠢⡀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠸⡈⠢⡀
⠀⠀⠀⡇⠀⠀⢀⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠈⡇⠁⠀⠀⠁⠀⠀⠈⠀⢹⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⣇⠀⠈⠢⡀