Celestial object Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀⣀⣀⣈⣁⣉⡉⣉⣉⢉⣈⣉⢉⣉⣀⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠈⠓⠀⠀⠀⠈⢻⡍⠀⠐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥
⠂⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠄⠀⢀⠀⠐⠄⣀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠠⡬⣤⠄⠲⣖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠈⠉⠀⠐⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⣀⣀⠀⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠤⠀⡆⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⡀⠀⠘⠄⠀⠀⠉⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠡⡀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⢀
⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀
⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣀⣀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠈
⠀⠀⠀⢀⣀
⠈⠁⠀⠀⠀⠀⠀⠐⡠
⣀⣀⠀⠠⣀⣀⣀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄
⣀⡠⠤⣄⡀⠀⠀⠀⣁⠉⠙⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠟⣭
⣁⢠⣀⣈⣈⣠⣰⣚⣉⠳⠆⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠀⠀⣿⡓⣀⣞
⣟⣿⣿⣻⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣻⣿⣟⣻⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠌⣀⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣇⢀⣀⢠⣤⣄⣤⣶⣶⣤⣄⠀⠠⡿⢯⡽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⡀⠠⣀⣄⣎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣥⣤⣿⣿⣷⣿⣿⣿⠓⠆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣯⣿⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⡔⠳⣶⢂⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡯⣧⠻⢿⣿⣿⡿⣶⣽⣿⣿⣿⣿⣿⣿⣿⡗⣿⣆⠈⢹⣿⣧
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡿⠿⣷⣇⠌⠰⣢⡤⠐⢿⣿⣿⣿⣿⣿⣶⠟⣻⣿⣿⣿⣿⣿⣿⣤⡙⢿⡧
⠀⠀⠀⠀⠀⠐⠹⣿⣿⣿⣿⣿⣧⣶⣍⣿⢟⠝⡇⢰⠅⢿⠿⣆⠛⢋⣶⠙⢛⣽⣿⣿⣿⣿⣿⣿⣿⣷⣦⡗⠗
⠀⠀⠀⠀⠀⠐⣰⢿⣿⣿⣿⣿⣿⡟⣧⠋⢩⣶⣶⡄⠰⢿⡄⣙⡓⠉⠀⠀⠀⠉⠙⠛⣿⣿⡏⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠘⢿⡏⢿⣿⣿⣯⣶⣁⣾⣷⣄⡀⣽⡦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⡏⠛⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⠘⠋⢛⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠘⣻⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⡿⣿⣏⣛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⢿⣿⣼⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢽⠉⠆⠙⠻⢿⡿⢁⡙⠟⢿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⢫⡀⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠈⠉⠢⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠁⠁⠁⢨⡀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠨⠆⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢠⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠈⠐⡄⠄⠀⠀⠀⠀⢀⠠⠘⠁⢀⠀⡄⠀⠃⠁⠀⠀⠀⠀⢠⡁⠔⢀⠀⠃
⠀⠀⠀⠀⠠⠄⠠⠄⠀⠤⠠⠸⡀⠀⠀⠀⠀⠸⡀⢀⠀⠀⠀⡀⢃⠆⠀⠀⠀⠀⠀⠂⠁⠈⠉⡚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠡⣬⠴
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠨⠀⡀⠀⢀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡤⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊⠔⡡⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠊⠀⢠⠀⠀⠀⠀⠀⠈⠉⠙⢅⠀⠠⡢⠀⠀⠀⠀⠀⠀⠀⡐⠁⡀⠀⢄⠀⠈⢀
⠀⠀⠀⠀⠄⣄⡀⠉⠰⡀⠊⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡀⡝⠀⠀⠀⠀⠀⡶⠀⠨⣢⠄⡁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⣀⣀⠈
⠀⠀⠀⠀⡀⡀⠀⠀⢀⠀⠌⠀⠀⠀⠀⠀⠀⠠⠂⠐⠆⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠀⠈⠹⠐⠀⠀⠀⠀⢄⠀⠊⠑⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠆⠀⠐⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠖⠇⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠊


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠀⡅⠀⠀⠀⠀⠠⠂⠐⠄⠀⠂⢄⠑⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⡁⠀⠀⠀⡅⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠔⠀⠂⠉⢀⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⣣⠀⡆⢠⠀⠀⠀⠀⠀⠈⠀⠠⠂⠁⡠⠄⡴⠁⠀⠀⠀⠀⠀⢐⠂⡀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠠⢀⣂⠡⠊⠀⠀⠀⠀⠀⠀⠠⠀⠄⡈⠊⢁⠀⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⠁⠃⡘⠂⡐
⠀⠀⠀⠀⠠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⡌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠉⡁
⠀⠀⠀⠀⡠⠀⢀⠊⠀⣉⠑⡠⠁⠀⠀⠀⠀⡠⠀⠀⠂⠁⠀⠐⡀⢄⠀⠀⠀⠀⠀⠀⠀⠑⢊⠗⠁⡱⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠅⠀⠀⣰
⠀⠀⠀⠀⠉⠁⢂⡀⠀⡀⠔⠀⠀⠀⠀⠀⢸⡀⠀⠁⠂⠀⠐⠈⠀⢀⡄⠀⠀⠀⠀⠋⠉⠐⠈⠀⢈⠓⣍⠢⠀⠀⠀⠀⠘⠀⠃⠀⠀⡠⠊⠰
⠀⠀⠀⠀⠀⣁⠠⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢐⡀⠀⢀⡀⠀⠋⠀⠀⠀⠀⠀⢁⣠⠀⠀⠁⠀⠁⠠⠂⠀⠀⠀⠀⠂⠁⢀⡀⠊⣀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⡠⠷⠖⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⣠⣴⡾⠿⠉⠩⠻⢷⣦⣄⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⠟⠀⠀⠀⡖⢢⠀⠀⠀⠻⣷⣄⣀⣀
⠀⠀⠀⠀⠀⣀⣤⡶⣽⠿⠛⣯⣽⣟⠁⠀⢐⡂⠤⡈⢁⠤⢔⠂⠀⠀⢻⡯⢛⣟⣹⡷⣦⢄⡀⣀
⠀⠀⠀⡤⣾⡕⠁⡞⢁⢠⠁⠀⡟⠇⠀⠈⠩⠏⠓⢀⠂⠊⠼⠝⠁⠀⠰⢳⠀⠙⠁⠉⣂⠑⡾⢭⠇
⠀⠀⠈⠳⠾⡣⣀⠹⡋⣹⠕⠠⠷⣆⡀⠀⢞⠆⠀⠼⡧⠀⠰⡱⠀⢀⣴⠿⠐⠀⣀⠔⢋⡠⣃⠞
⠀⠀⠀⢀⠀⠈⠑⠫⢿⣮⣭⣑⣒⣂⡤⠬⠬⠍⠁⠀⠁⠈⠭⠭⠥⢄⣐⣲⡾⢧⣖⠪⠕⠊⠁
⠀⠀⠀⣉⢠⠂⠀⠀⠀⠀⠈⠉⢿⣉⡿⣿⡿⠟⠿⠿⠿⠿⠻⢿⣿⠗⠛⠩⢮⡴⠍
⠀⠀⣄⠈⠈⠳⠖⠦⠃⢠⠄⠀⠀⠁⠀⠈⠙⠺⠶⠦⠤⠶⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠞⠻⠂⢧⣌⠀⢀⣴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠗⠚⢧⣭⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⣿⡯⣿⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⢦⡀⠉⠙⡤⠲⢄
⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣦⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠋⠻⡄⠀⠀⡇⠀⠀⠇⠀⠸⠋⠳⡄
⠀⠀⢀⣀⡴⠁⠀⠀⠀⢀⠀⠀⠹⣦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⠶⠛⠉⣸⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠙⣆⠀⢠⠀⠀⠀⡠⠁⠀⡐
⠀⣔⠉⢀⡇⠀⠈⢤⡠⠈⠀⠀⢀⣯⠤⠋⠀⠀⠀⠀⠀⣠⡤⠶⠒⠃⡄⠀⠐⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠈⢦⣼⠀⠀⠀⠀⡠⠂
⠀⠉⠙⠛⣟⠛⠛⠛⠛⠛⠋⠉⡹⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⠀⠶⠞⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠈⢧⡌⠴⠎
⠀⠀⠀⠀⠈⠳⢤⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⠛⠶⣄⣸⡆⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠘⢯⡉⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜⠁⠀⠀⠀⠉⠁⠀⠀⡆⠙⠶⠒⢫⠃⠀⠀⠀⠀⡀⠀⠛⠦⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⠁⠦⡀⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠤⠤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠁⠈⠉⠉⠙⢶⡄⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⡇⡠⠞⠉⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣄⣀⠐⠁⠀⢿⠤⠔⢒⠞⠃⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢦
⠀⠀⠀⠀⠈⠛⢦⡀⢀⠒⠐⠂⢀⡔⠃⠒⠒⠒⠒⠦⠤⢤⣀⣀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡙⠈⡉⣭⡀⠸⡧⠤⢀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⣐⠠⠈⠀⠈⠛⢦⣷⠀⠀⠀⠉⠒⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠐⠿⢦⣸⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢠⣤⠚⢻⠻⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠀⠀⢀⣀⡴⠋⠠⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⠀⢾⠄
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣄⡇⠘⠦⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⡠⠊⠁⠀⢀⡠⠤⣶⣊⠁⣂⡁⠸⡄⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣀⣤⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠈⠀⠀⡠⠂⠁⡠⠊⠠⠋⡇⠐⠛⠲⠧⠀⠀⠀⠀⢸⡀⠙⠛⠛⠛⠋⠁⠀⠈⠙⠛⠒⠂⠀⠀⠀⠀⢀⠜
⠀⡀⠀⠀⡀⠈⠀⠀⠈⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⡐⠁⢀⠊⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠁⠀⠀⣀⣠⠤⠶⠶⣤⣄⡀
⠉⠋⢧⣀⣀⠔⢧⣀⡠⠂⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠒⠛⠋⠉⠁⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈⣿⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢑⣿⣖⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⢻⠁⠀⠀⠐⠂⠀⠀⠋⠀⠀⠀⠀⢸⡆⠀⣀⣠⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠁⠀⣿⡉⠱⠇⠀⢶⡀⠶⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⡄⠀⠀⠀⠐⣓⣂⣀⣤⠤⠴⠖⢺⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣤⠴⠿⢾⡛⠁⠀⠀⠀⠀⠈⠛⠛⠒⠒⠒⠚⢿⡛⠛⠛⠉⠉⠉⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠉⠁

⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣦⣤⣴⣴⣆⣄⣀⣀⣚⣛⣛⣓⣶⣄⣠⣤⣶⡶⠀⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣾⣿⣿⣿⣿⣏⠁⠙⠿⡿⠏⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣧⣶⣾⣿⡿⢋⣛⠉⠉⠀⠀⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⠁⣨⣀⣤⣴⡶
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⣷⣿⠟⠛⠁⠀⠀⠋⢀⡿⠟⠁
⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣻⣿⣿⣿⣿⣿⣟⣨⠿⡿⠋⠀⠀⠀⠀⠀⠀⠈
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⠿⠛⠉⠀⢼⣿⣿⠿⠏⠾⠡⢾⡅
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⡠⠀⠀⢈⡄
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣷⡟
⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠛⠛
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿⠟⠁⠀⢠⣾⢠⠐⡆⠀⢀⡄
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⠿⣯⢰⣶⠶⠈⠁⠀⠀⠀⠀⠚⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣷⣬⣄⣀⣀⣀⠲⠄⠀⠀⡀
⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⠟⢻⠟⢻⣿⡿⢧⡀⠀⡀⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆⣼⠟⠀⠀⠉⣣⠀⠉⠠⡀⢮
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣸⠋⠀⠀⠐⠀⠀⠂⠲⠃⠤⠃
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⡼⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢠⡟⠀⠀⠀⠀⠀⠀⢀⡠⠂
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣴⠛⠁⠀⠀⠂⠀⠀⡰⠋
⢸⣿⣿⣿⣿⣿⣿⡿⠋⠈⡴⠛⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠶
⢘⣿⣿⣿⣿⣿⠋⠀⠀⠊⠀⠀⠀⠀⠀⢘⣀⣀⡀⠀⠀⣀⣠⣦⣤⣄⠀⠰⠦⡀
⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠈⠁
⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢹⣿⣿⣿⡇
⠈⣿⣿⣿⣇
⢰⣿⣿⣿⣿⡀
⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠐
⠼⠿⠟⠋⠁⠀⠖⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠎⠉⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠋⠈⠀⠀⠀⠓⠒⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⡤⠤⠏⠀⠛⠋⢉⡇⠀⠀⠀⢰⠇⠀⠀⠀⢀⠀⠀⠐⡜⡆⠀⠀⠀⠀⠀⠀⣠⣏⠘⡆
⠀⠀⠀⠘⢤⡀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢸⠀⠀⣀⡴⠋⢣⡀⠰⠇⡇⠀⠀⠀⠐⣏⠉⠁⠁⠀⠉⠋⡹
⠀⠀⠀⠀⢰⠃⠀⣠⣠⣄⢳⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⠙⠢⢤⠇⠀⠀⠀⠀⢈⡗⠀⢀⠀⠲⣏
⠀⠀⠀⠀⠘⠶⠚⠁⠈⠙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡴⠋⠳⣄⣻





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

⠀⠀⠀⣏⠋⡛⠻⡋⡍⢻⠀⢎⠀⠀⠀⡀⢸⠀⠁⡄⠇⠸⠸⢇⢀⠀⢸⢸⢌⢈⠆⠀⡗⡇⠀⠘⡄
⠀⠀⠀⣷⡌⢛⣈⣶⠒⢿⠀⠀⠀⢀⢀⢀⠀⢀
⠀⠀⠀⡛⡉⢛⠀⣹⡒⣿⠀⠀⠃⠘⠘⠀⠈⠘⠂
⠀⠀⠀⠓⠒⠛⠛⠛⠓⠚
⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠄⡀⠀⠀⠀⠀⡀⠒⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠠⠤⠤⡶⡂⠀⠀⠀⠀⣠⣔⠶⠶⠶⣄
⠀⠀⠀⠀⠀⢠⡞⡩⡶⢒⡄⠀⠀⠀⠀⡌⠁⠀⠀⠐⠀⠂⣃⠀⠀⠀⠀⠐⠀⠀⠀⡈⡌⠀⠀⠀⡼⣁⣁⣀⣀⣀⣀⣥
⠀⠀⠀⠀⠀⠻⢃⣯⡾⣺⠇⠀⠀⠀⠀⢇⠀⠐⠂⠐⠐⠀⠚⠀⠀⠀⣤⠀⠀⡔⠃⠀⠀⠀⠀⠀⠯⢠⣶⡄⠀⠀⠀⠬
⠀⠀⠀⠀⠸⡀⠉⢁⡾⠋⠀⠀⠀⠀⠀⠈⠢⠀⠉⠉⢡⠖⠀⠀⠀⠰⠉⠓⠹⠂⠀⠀⠀⠀⠀⠀⠈⠲⡤⠤⠤⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢰⣺⡆⢀⡊⠳⡄⠀⠀⠀⠀⠂⠂⠀⢂⡀⡄⠀⠀⠀⠀⠠⢠⠂⠀⠹⠑⠄⠀⠀⠀⠀⠀⠀⠠⡴⠶⠀⠞
⠀⠀⠀⠀⢰⠀⠙⠁⠠⠀⠂⠀⠀⠀⠀⣠⠀⡤⠈⢉⡁⠔⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠃⠀⡖⠂
⠀⠀⠀⠀⠘⡀⠂⠀⡠⣄⠀⣰⠀⠀⠀⠈⠉⠓⠘⠁⠚⢉⠁⠀⠀⠀⠀⠀⠠⡄⠀⠀⠠⠀⠀⠀⠐⠄⠁⠤⠀⠆⠈
⠀⠀⠀⠀⠀⠘⠶⠤⠅⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠆⠙⠷⠀⠁⠀⠀⠀⠘⠶⠡
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠘⣧⠀⠀⡀⠀⠀⠀⠀⠀⢀⣾⡽⠛⠊⠉⠛⣄⠀⠀⠀⠀⣀⢀⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⠉⠈⠉⢀⡴
⠀⠀⠀⠀⠄⠀⠀⠶⠠⢀⠀⡀⠀⠀⠀⠈⠒⣦⠀⠆⠄⣂⢨⠀⠀⠀⣀⠃⡀⡇⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂
⠀⠀⠀⠀⠀⡄⠰⠧⠀⠀⡘⠋⠀⠀⠀⠘⠖⠙⣶⠀⠂⢈⠏⠀⠀⠀⠡⠘⠀⣠⡀⠀⠀⠀⠀⠀⢠⠛⢀⣀⡂⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠀⠀⠀⠀⠈⠙⠉⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⡠⠀⡈⠓⢲⠀⠀⠀⠀⠀⢀⣀⣤⣤⣢⡄⡤⠀⠀⠀⠀⠀⠠⠶⠬⠥⠂⢄⠀⠀⠀⢀⠀⢄⠀⡀⠠⠰⡀
⠀⠀⠀⠀⠀⠁⠀⠠⠈⠂⠀⠵⠀⠀⠀⢀⣰⠋⠀⠀⠙⣶⠀⠀⠀⠀⡦⣀⣐⠒⣒⡒⠬⠇⠀⠀⠀⢄⠀⠀⡀⠀⠰⠁
⠀⠀⠀⠀⠰⠀⠐⠈⠈⠠⠀⣀⠀⠀⠀⢠⠻⡀⠀⠀⢀⡏⠏⠀⠀⠀⠀⠒⠢⠤⠤⠔⢒⡀⠀⠀⠀⠀⠂⠀⡀⠀⠫⠇
⠀⠀⠀⠀⠀⠒⢀⡀⠀⠂⠐⠀⠀⠀⠀⠠⠒⠩⠶⢒⠍⠬⡀⠀⠀⠀⠀⠢⣤⣬⠉⠉⠙⠀⠀⠀⢰⣉⡂⠘⠚⠀⠐⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠐⠂⠉⠁⠀⠀⠀⠀⢿⡐⢿⣿⣽⡿⠀⠀⢸⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⣀⣰⠀⠀⠀⣢⡡⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠘⠛⠚⠊⠁⠀⠀⣉⠀⠀⢠⠀⢁⡀
⠀⠀⠀⠀⠀⣰⢋⡟⣸⠀⠀⠀⢯⡼⠀⠀⠴⠀⠀⢀⠀⢾⠋⠄⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠂⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠳⡀⠠⠋⠀⠀⠀⢨⠆⠀⠀⠀⠀⠀⠺⠀⠶⠀⠀⠀⠀⢰⣢⠂⠀⠀⠀⢚⠠⠀⠀⠀⠐⠀⠀⠄⠀⡁⡄⢀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠀⠀⡟⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠀⠄⡘⠒⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣴⢀⠈⠀⠀⠀⠀⢠⢀⠂⠇⠀⠀⡀⠀⠀⠀⠀⡴⠒⢌⠀⠀⠀⠈⡀⠁⣠⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢤⡀⠀⠀⠀⢠⠈⠈⠹⠋⠀⠀⠀⠀⢀⡘⢈⢢⠀⠀⠀⢷⠀⠀⠀⠨⠀⠛⠀⠀⠀⠀⠉⠀⠄⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢙⠦⠀⠀⠀⠢⠠⠔⠀⠀⠀⠀⢀⠎⠐⢘⠠⡀⠀⠀⠘⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⠚⠀⠠⠄
⠀⠀⠀⠀⠀⠠⡜⠂⠀⠀⠠⠹⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠀⡇⠀⠀⠐⣷⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⡁⠀⠀⠀⡇⢇⠀⠀⠀⢠⡆⠀⠀⠀⠀⢄⡀⠉⢅⡀
⠀⠀⠀⠀⠀⠀⠠⢈⠀⠀⠀⠈⣠⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠒⠀⠃⠀⠀⠀⠳⠞⠀⠀⡀⠈⠀⠀⠀⠂⣀⠄⠐⠊
⠀⠀⠀⠀⠀⠀⢛⠼⠆⠀⠀⠀⢐⡦⠖⡅⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⠟⡘⠳⠀⠀⠀⠀⠀⠀⣻⠃⠀⠀⠄⠀⠀⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢈⡄⠾⠀⡀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡄⠠⠃⠀⠀⠀⢀⠀⠓⠁⠀⠀⠀⢀⡂⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠢⣌⡥⠀⠀⠀⠀⠀⠘⡄⠀⠀⡀⠀⠀⠀⠀⡰⠓⣅⠢⠃⠀⠀⠀⣀⠌⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⢋⠠⠢⠀⠀⠱⠀⠀⠩⣄⠀⠀⠀⠱⠄⠛⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡑⡄⠀⠈⢲⢦⣀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠂⣀⣛⠆⠠⡤⠄⠀⠀⠐⠒⠂⠀⠈⠀⠀⠀⠀⠀⢇⢀⠀⠀⠈⣂⢁⠂⠄⠈⠉⠉⠈⢹⣤⣱⠁⢂⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠀⠀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠒⠒⠒⠒⠒⠀⠈⠀⠈⠁⠈⠉

⠀⠀⠀⠀⠀⠀⡀⠀⢰⡟⠛⣿⡝⢹⡇⢸⣯⠓⣿⣿⠚⣿⠻⢺⡃⣼⣷⢸⡇⠀⠀⣿⣽⣰⡏⣷⢸⠛⡆⣿⢸⡽⠲⣯⣗⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠚⠘⠓⠛⠓⠛⠓⠛⠀⠛⠀⠘⠓⠋⠙⠚⠓⠃⠀⠛⠚⠋⠓⠋⠘⠛⠃⠛⠚⠚⠛⠓⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⢭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠠⣀⣠⣄⠀⢀⠂⠁⠀⠀⢀⠀⢰⡴⡄
⠀⠀⠀⠀⠀⠠⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⡆⢺⣶⡀⠀⠠⠀⡌⠋⠀⠂⣀⠀⠀⠴⠙⠚⠺⢿⡇
⠀⠀⠀⢀⠀⠀⡈⢃⠀⡀⠀⢀⠀⠐⠂⡴⢋⣧⠔⠂⢠⢘⣻⣶⠓⢦⣥⠸⡋⢦⠀⣘⠀⠀⠈⠱⠚⠀⣸⡇
⠀⠀⠀⢸⢠⣿⢀⡠⠞⢰⣆⣤⣀⠀⣸⡕⢡⢻⡀⢀⣃⣸⡻⠉⢧⡀⠘⢿⣧⡀⣗⡼⠀⠀⢀⠀⠐⠘⠋⡅
⠀⠀⠀⠀⠠⠍⣿⣶⠄⣀⠉⠁⣠⣶⣿⣿⡛⠛⢿⣹⡟⣿⣶⠀⠐⣿⣻⣿⣿⣷⣾⡴⠦⠼⠢⠖⣻⣶⡆⠄
⠀⠀⠀⠀⢠⣴⠋⠆⠀⠀⠀⢀⣿⣿⡿⣿⣿⡏⠉⢿⣿⣿⡟⠉⡽⢻⡏⢻⣿⣿⣿⡍⠀⣔⠲⠊⢋⣽⢧⠆
⠀⠀⠀⠀⢺⣏⣄⡀⠀⠠⠀⠈⢛⣻⣿⣿⣿⣧⣄⣹⣿⣿⣧⣎⣤⣼⣿⣿⣿⣟⣉⣀⠀⠀⠆⢩⢉⢡⣸⡇
⠀⠀⠀⠀⠘⢛⡋⠛⠃⠐⢠⣾⣿⢷⣼⡥⣾⡿⠛⠿⢿⣿⣿⡿⠿⡿⢿⣿⣿⣿⢿⣿⣷⣤⠔⠈⡰⣶⡞⠁
⠀⠀⠀⡀⢻⡿⠿⠄⣠⢡⣿⣾⠑⣼⣿⣵⣶⠃⣀⠀⢸⣿⡄⠀⠀⠘⣆⡇⣿⣿⣤⠹⣿⣿⠀⠘⢙⣿⠇
⠀⠠⡄⣿⣿⡀⣀⣲⡅⣿⣿⣷⣿⣿⣯⣿⢣⣲⣦⣭⣸⣿⣇⣐⣑⣒⣘⣧⣸⣿⣿⣴⣿⣿⡇⠀⣾⣿⡄
⠀⠀⢸⣼⣏⢼⡿⢊⢹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣻⣡⢺⣿⡋⣯⡟⣇⣄⣽⣿⣿⣽⣿⣿⣿⡧⢀⠈⠈⠁
⠀⠀⢸⣿⣿⣿⠝⣠⣼⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⣿⣿⣿⣿⢿⣿⣿⣵⣾⣿⣿⢿⣾⣶⣀
⠀⠀⣸⣿⣷⢺⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣾⣿⡿⣿⡿⠛⣻⣿⣿⣿⣿⠂
⠀⠀⢹⣿⣾⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣽⣿⣿⣿⣯⣴⣿⣻⣿⣙⢯⣉
⠀⠀⠀⠈⠙⠋⠛⠛⠛⠛⠉⠛⠿⠿⠻⠟⠿⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠏⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⡿⠿⢿⠛⠿⣿⣿⣷⡶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⢿⣃⡾⠀⡐⠌⠐⠋⠈⠀⠙⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⢛⡛⣉⣉⣁⣄⣀⣤⣀⣀⣀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⢻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣬⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⠿⠻⠛⠛⠛⠃⠒⠟⠛⠹⠟⠟⠻⠿⠟⠻⠟⠻⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠹⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⣤⣤⣤⣄⠀⠀⠀⠀⢀⢛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠝⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⣿⣿⣿⣿⣷⣾⣿⣿⣧⡄⠀⠀⡤⢈⢩⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣉⠛⠂⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⢎⣉⣉⡿⠿⠷⠾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣘⣛⣿⣷⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡠⡀⣀⢀⣤⢀⣄⡄⣠⠾⠆⠤⠐⠐⠒⠊⠁⠆⠀⠀⠀⠉⠛⡿⠉⠋⢭⣿⣿⣧⣍⢋⡉⠹⣿⣿⣷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⠟⠙⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠈⠙⠧⡄⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣋⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣨⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣾⣿⣿⣭⣤⣴⣤⣶⣶⣶⣶⣶⣾⣷⣶⣪⣴⣶⣶⣆⠴⣦⡶⣦⣤⣄⣀⣐⣠⣤⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠋⣨⣴⣿⣿⣿⣿⣿⣿⠿⡛⠛⠙⠉⠋⠉⠉⠉⠉⠛⠉⠀⠁⠀⠘⠻⠻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣡⢀⣂⣤⣠⣤⣴⣶⣶⣶⣤⣤⣴⣤⣀⣀⠀⠀⠀⠀⣹⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣹⢻⣿⣿⣷⣾⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠏⢛⣿⣛⡛⠻⠙⢟⢿⣿⣿⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣽⣀⡆⠠⠦⡌⠠⡝⢛⡚⢢⣮⣾⣿⣿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣥⣶⣷⣿⣿⣷⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠿⣷⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣶⣤⣀⡀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣄⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⢿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠈⠙⠛⠦⣄⠀⠀⠀⠉⠂⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢷⣶⣤⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⡀⠀⠐⢶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⠛⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠶⠶⠶⢲⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠯⠛⠉⠀⠀⠀⠀⣀⢀⣠⣭⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⠷⠶⠾⠿⠶⠾⠿⣿⣟⣿⣽⡿⢟⣉⣡⣿⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠈⠉⢈⣁⣈⣋⣩⣉⣉⣭⣽⣟⣾⣧⢿⣿⡽⠿⠋⠀⠀⠒⠒⠒⠐⠢⢤⡀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠛⠓⠛⠿⠿⠭⠽⠿⠥⠼⠥⠿⣶⣺⠟⠉⠀⠀⠀⣠⣼⣇⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⣶⠿⠋⣙⣿⠀⢀⡠⠖⠁⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠟⣿⣅⣀⣀⡀⠶⠶⠶⠶⠿⠧⠶⠶⠿⠟⠛⠚⠛⢉⣡⣤⣶⣾⡿⠟⠋⠁⠀⠀⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⣠⣶⠿⢿⡿⣻⠟⠁⠀⢿⣧⣤⣄⠉⠉⠉⢛⣒⡒⠒⠒⠒⠒⠲⣚⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⣠⣤⠖⠿⠛⠉
⠀⠀⠀⣠⣾⠋⠁⠀⢿⣟⣅⡀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣷⣶⢶⠶⠶⠚⠋⠉⠀⠀⠀⠀⣀⣀⣤⣶⡶⠛⠉
⠀⠀⢸⣿⡁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⡿⠟⣻⠏
⠀⠀⠈⠻⢿⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣄⠖⢒⠿⠿⠿⠿⠛⠛⠛⠛⠯⠽⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠉⠙⠋⠈⠙⠿⣧⣶⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⠶⠶⠶⠖⠛⠋⠁

⠀⠀⠀⢀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⡴⣔⡁⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⢃⠀⠀⠄⠀⢀⠠⠀⠀⠀⠀⠀⠀⠔⠖⠚⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⡀⠂⠃⠀⠀⠁⢢⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢐
⢀⠑⠚⠰⠦⠖⠁⠲⡤⢮⡆⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⢀⠰⣠⠃
⠘⠈⠰⠐⠠⢀⠠⢶⠄⠄⠋⠀⠀⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡉⠰⠂⠂⠈⠒⠀⡈
⠸⡷⠄⠄⠀⠀⢲⣦⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠀⡄⠊⠠⠔⠁
⠀⠘⢤⡠⠔⠂⠀⢤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡄⠒⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠖⠀⠀⠀⠂⡀⢀⠱⡄⠒⠄⠤⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠁⠀⡠⠏⠀⠒⠀⠈⡠⣜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠔⠀⠡
⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠁⣐⢠⠀⠠⣅⣉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⢀⠀⠀⠄⠀⠀⠀⠉⠉⠉⠀⠃⠀⠀⠀⠀⠉⠔⠆⠀⠘⠄⠀⠈⣁⠋⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠔⠃
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⢠⣤⢄⡀⠒⠒⠒⠒⠒⠖⠶⠒⠒⠲⠶⠖⠒⠒⠒⠴⠐⢲⡶⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⣒⡒⠀⡒⢂⢦
⠘⢿⣻⣿⡋⠐⠒⠖⠓⠂⣤⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⣛⠃⠀⠤⠄⠀⠀⠀⠀⣀⠀⠀⠴⡀⢁⣘⡃⠉⣲⣌⣪⠘
⢠⢀⡄⣹⣿⠓⣚⢶⢡⣺⠋⠃⠈⢳⣾⣄⣐⣋⡂⠀⢀⢣⠀⠀⠀⠑⠶⠶⠿⠖⠂⠀⠀⠀⠐⡿⣉⠀⢨⠻⠘⢻
⠸⣾⠀⡗⡎⣠⠀⠸⡯⠁⠠⠂⠁⠀⠀⠈⠉⠉⠁⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢋⡀⣰⣀⡇⠘⢟
⠀⠈⢁⠳⢭⡡⠁⢈⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⢹⣽⠃⢠⠉⠘
⢠⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠁⠀⠸
⢸⡀⢰⣄⠐⣀⣤⣤⠶⠖⠀⠁⠈⣀⡠⠤⠔⠒⠒⠀⠀⠙⠀⠀⠀⡦⢀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⣠⡀⡀⣄⠀⢸
⠀⠁⠈⠻⠈⠉⠀⠀⡀⠖⣾⣶⠄⢀⡆⠠⠄⢠⠤⠐⠀⠘⠀⠀⠐⠀⠉⠠⠤⠀⠀⠈⠁⠒⠄⡒⠛⠛⠛⠃⠀⠀⢸
⢸⡀⢀⠃⠀⠀⠀⠉⠀⡠⠘⠀⠉⠀⣀⣀⠤⠈⠀⠛⠁⠖⠀⠀⠀⠠⠤⢀⣀⡀⠀⠀⣦⣄⠀⠀⠁⠂⠀⠀⠀⠀⢸
⢘⢠⣠⠔⠀⢀⠀⠘⠀⠀⡠⠔⠒⠉⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠁⠀⠉⠃⠀⠈⠀⠠⠶⠃⢠⡴⢼
⢘⢘⡁⡀⣠⡄⢀⡠⠆⣵⠄⠀⣀⣀⣀⣀⣀⣀⣸⣿⡿⠟⠿⠿⣿⠀⡀⣀⢀⣀⡀⡀⠀⠀⣠⣴⢄⠀⠀⠂⣬⡏⢸
⠘⣾⠛⠁⢀⡴⢥⡀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⣿⣿⡿⠁⠀⠁⠀⠀⠀⠀⠈⠁⠻⠋⡀⢹⠂⡀⢸⠿⢾
⠀⣧⣠⢶⣻⣒⠀⠉⡖⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠁⠀⠙⠀⠈⢾⣠⢺
⠀⡮⠉⠘⠿⢿⡧⠁⠦⠄⣀⣬⠶⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠔⠂⠁⠀⠀⠔⠊⠀⢀⠀⠊⠀⠈⢸
⠀⡇⠘⣧⡀⠀⠁⠀⢄⡀⠀⠉⠒⠀⠠⣀⡀⠀⠀⠀⠀⠈⠁⠀⠿⠿⠂⣀⠀⠄⠂⣨⣏⣀⡀⢒⣀⣀⣀⡀⠀⠀⢸
⠀⣀⠀⠙⠁⠻⠶⢤⣤⣉⣁⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠀⠉⠀⠈⢀⡀⠴⠀⠀⠀⠀⠀⠀⠈⠀⠉⠋⠁⠉⠀⢸
⠀⣇⢠⡤⡀⠀⠐⠄⠀⠉⠉⠙⠛⠃⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈
⠀⣧⣼⣠⣶⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⢀⡘⠀⢹
⠀⣭⡏⣟⠙⡔⠈⣰⡀⠀⠑⠠⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⠀⠊⢁⠀⣨
⠀⡟⣣⢹⣆⡀⠚⢽⠍⠑⣆⢼⡜⠡⢶⡾⠿⢛⠋⠀⠤⢝⠀⠀⠀⢀⡤⢤⣤⠶⠄⣔⠀⢀⠀⡉⠉⠀⠀⢩⢀⠴
⠀⡇⢻⣿⡯⣉⣫⣁⠀⠐⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡅⠀⠐⠀⠀⠀⠀⠀⢀⡈⠘⠡⣤⢄⣥⠀⣠⠔⠫⡄
⠀⠚⠓⠉⠁⠉⠈⠁⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠐
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣶⣴⠂⠀⠀⠒⡒⠒⠦⣄⡀⠀⠀⠀⠀⠀⢀⣸⣦⠆⠀⠒⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣸⣦⡆⡀⠚⠈⠀⠀⠀⠀⠠
⠀⠹⠉⠃⠀⢀⠠⣌⠙⠢⣄⠙⠳⠀⠀⠀⠀⠀⠹⠋⠃⠀⠓⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠘⠋⠁⠀⠺⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⢱⡀⠑⣄⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠡⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣠⠤⠤⢤⣄⣀
⠠⣿⣿⣡⠰⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⡂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢾⣿⡇⠀⠀⠹⣟⣒⠲⢦⣌⠁
⠀⠈⠱⠆⠀⡅⠈⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠁⠢⡑⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢳⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢄⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿
⠠⣿⣾⡳⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣋⡼
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢷⣴⡾⠛⣿⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠀⠀⠀⠀⢸⣇⠀⠈⠻⣦⡀⢀⣀⡀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣯⣅⣀⠀⠀⠀⠀⠀⠙⢳⣦⡀⠙⣷⡄⢻⡟⠓⠶⠞⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣴⠟⠛⠛⠋⠙⠁⠀⠈⢿⣄⣿⠄⠀⠀⠀⠈⠛⠳⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠋⢀⣀⠀⠀⢠⡶⠞⠋⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣄⠀⣀⠀⠀⠀⠀⠀⠸⣿⡟⠛⠋⠉⠳⣄⠸⣧⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢷⣄⠀⠀⠀⠀⣈⣿⣦⡀⠀⠀⠘⢿⣿⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⠶⠞⠛⢩⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠃⠀⠀⠀⠀⠀⣠⡟⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣯⣉⡀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⡄⠀⠀⢀⣀⣀⡀⠈⠛⣷⡄⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣰⠿⠋⠉⠛⠛⠛⠛⠛⠂⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠈⢿⣆⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⢻⣆⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⢿⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢿⡆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⢿⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠴⢋⠁⣤⣤⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⡖⢋⡧⢸⠁⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⢈⡠⢋⠝⢦⡀
⠀⠀⠀⠀⠀⢰⠏⢂⡈⠀⢨⠁⡀⠘⠃⡈⣧⠀⠀⠀⠀⢰⠇⠀⠂⠠⣄⠉⡉⢉⣉⡹⡆⠀⠀⠀⠀⡼⢃⠄⠂⡀⠤⠤⠊⡠⠊⢹⡄
⠀⠀⠀⠀⠀⢾⠀⢅⡨⠀⢀⠤⡄⠻⠃⠀⢸⠀⠀⠀⠀⣿⠤⠤⠄⠒⠃⠘⠋⠘⠒⠂⣻⠀⠀⠀⠀⣟⣁⠄⢊⡀⠄⢂⡡⠔⢊⢵⡇
⠀⠀⠀⠀⠀⠸⡆⡀⡄⠀⠘⠶⠋⠀⠤⢀⡞⠀⠀⠀⠀⠸⡦⠴⠀⠚⠀⠫⠭⠥⠤⢠⡏⠀⠀⠀⠀⢳⡦⢊⠅⠂⠉⠐⠒⡈⠐⣹⠇
⠀⠀⠀⠀⠀⠀⠙⢮⡁⠀⠁⠀⠤⢀⣠⠞⠀⠀⠀⠀⠀⠀⠙⢮⣭⠭⠥⠤⠶⢐⡶⠋⠀⠀⠀⠀⠀⠀⠻⢵⣨⠔⡂⠉⠍⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⢞⡉⠒⠀⠤⠤⠬⠓⢦⠀⠰⠁⠄⡐⢉⣻⣟⣉⠁⠀⠂⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⡴⠋⣡⠀⠄⣒⡢⠉⠳⣄
⠀⠀⠀⠀⠀⢰⡇⠀⠉⠒⠤⢀⣀⣐⠍⣈⣧⠈⢄⠡⠑⢄⠑⢬⣉⣉⣉⣉⣉⣉⠉⠼⣇⠀⠀⠀⠀⡼⢱⠄⠀⢀⠂⠁⠑⠈⠛⠹⡆
⠀⠀⠀⠀⠀⢸⠉⣑⡒⠀⠤⠤⠤⠤⠤⠀⢾⠀⠀⠑⢄⠙⠟⠢⣄⠀⠲⠄⡀⠰⠐⠒⣿⣷⣄⠀⠀⡇⠀⠠⡠⠌⠀⠀⠂⠘⢃⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠓⡁⠄⠀⠀⠀⢀⡀⢰⡏⠀⠀⠀⠀⠹⡶⢌⡁⠸⠃⡀⠠⠉⡉⠀⠛⢀⠙⠱⡄⢱⡀⡑⠉⠂⠀⠰⣂⢠⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠐⠀⠀⠐⣢⡵⠋⠀⠀⠀⠀⠀⠀⠈⠶⣌⡳⠶⣤⣁⣀⠀⠛⠁⠀⠈⢀⠄⠀⠙⢦⣀⠐⠀⠈⢀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⢤⣤⣀
⠀⠀⠀⠀⠀⠀⣰⠊⠀⠈⠉⡝⠀⢀⡟⢦⡀⠀⠀⠀⠀⠀⡰⠛⠡⠡⠈⢈⡤⠸⢛⢦⠀⠀⠀⠀⠀⢀⡴⠯⠥⠬⠥⠤⠔⢋⠷⣄
⠀⠀⠀⠀⠀⢰⢣⠀⢤⠐⠁⠀⢀⡬⠛⠼⣷⠀⠀⠀⠀⢸⠃⡀⠰⡅⠴⢀⣀⢣⠱⠬⣇⠀⠀⠀⠀⡾⠬⣉⣑⠒⠀⠒⢊⣁⣠⢼⡆
⠀⠀⠀⠀⠀⢸⠀⠁⢰⠉⠉⠲⡌⠒⢦⠀⢸⠀⠀⠀⠀⢟⠢⡍⠀⡊⠉⠁⢎⠘⠲⢒⡿⠀⠀⠀⠀⡯⢑⣒⠂⢀⣩⡽⠴⢒⡪⠭⡟
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢡⠀⡰⠁⠀⠘⣠⠏⠀⠀⠀⠀⠸⣆⠃⠈⢀⠆⢀⠀⠉⠉⣴⠃⠀⠀⠀⠀⠹⣆⢀⠀⠉⠁⣒⡒⠉⠀⣺⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣘⡘⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠣⣤⣀⡠⣈⣳⡽⠞⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⣠⡬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠸⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣠⠄⣀⡒⠂⠂⡿⡄⢀⠂⠀⠄⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠩⠓⠂⠀⠔⠀⠀⠀⡆⡄⠀⠀⢦⠒⠀⠉⠓⠤
⡇⠀⠀⠀⠀⠀⠀⠠⢔⠋⠀⠀⠀⡔⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀
⡇⠀⠀⠀⠀⢀⢆⡐⠀⠀⠀⣀⠚⠁⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀
⡇⠀⠀⠒⢠⠾⠶⠶⣈⣀⣭⠭⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⣤⣿⡿⠟⢉⠤⠘
⡇⠀⠀⢠⡳⠁⠀⠀⠀⠀⡌⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠀⠀⢀⠀⠈⢰⡤⠀⠡⠔⠀⠀⢄⣄⠠⠧
⡇⠀⢀⢯⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠐⠀⠀⠄⠀⣀⠔⢦⡀⠄⠈⣀⠀⠁
⡇⠀⡸⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣤⣶⣿⣾⣶⣦⡲⡿⠄⠋⡀⠀⠈⠀⠠⠂⠁⠀⠀⠀⠀⠀⡀
⣇⠀⣇⡂⠂⠀⠒⠂⠦⠨⠀⠉⠉⢥⣭⣩⣾⣿⣿⣭⡽⠛⠁⣠⠐⠈⠀⠤⠖⠁⠰⠀⠒⠐⠀⠀⠀⠠⡄
⡏⠀⣿⠡⠤⠤⠒⠒⡂⣴⠀⠂⡠⠬⠆⢽⣿⡟⠉⢀⠤⠆⠀⣘⠴⠂⠀⠀⠀⠀⢐⠀⡀⠀⠀⠀⠀⢹⠁
⡇⠀⡗⡀⠀⠀⣀⠔⠃⣶⠀⠈⠟⠛⣉⠀⠘⠉⢁⣀⣴⣾⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠃⣄⢰⣧⣴⣿⡷⠐⠛⡉⢀⣠⠀⠀⢉⣄⠀⠀⠉⣹⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⡇⠀⠈⣧⠤⠘⠛⠈⠋⣀⣠⠟⠒⠛⠛⠂⠀⠀⠀⢰⠲⠀⠀⠐⠒⠒⠒⠒⠀⢈⡤⠤⠀⠀⣤⠀⢰⡜⠲
⡇⠋⡀⠘⣤⡖⠒⠒⠛⠤⢶⣠⡀⠀⠀⠀⠀⢀⣀⣸⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠹⡻⣀⡀
⡇⠀⡠⣤⣌⠊⣀⠀⠀⠀⠀⠢⠁⠀⠉⠉⠉⠉⠉⠉⠘⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣿⣷⣷
⡇⠐⠀⠿⣾⡆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⣾⣿⡟⣟⣿
⡇⣤⡀⠛⣉⠐⠂⠈⠙⠖⠶⠶⠒⣀⠰⣀⠀⠀⣠⡄⢠⣄⠀⠀⠐⡰⠀⠀⠀⣩⡄⣴⠂⠠⢹⣿⡹⠍⢸
⡇⠈⠀⣠⡭⡀⠛⠿⠤⢀⠷⠢⠀⠠⢀⠀⠀⠉⠉⡁⣈⠉⠉⠔⠁⠀⡀⠤⡠⠔⠋⡈⠁⢀⣙⡛⠻⣿⣿
⡇⠀⢸⣿⡖⠛⣆⠙⠃⢠⠐⠀⠀⢴⠅⠀⠁⠀⠐⠿⠗⠁⠀⠀⠀⣰⠀⠀⠘⠃⠀⣉⣴⣿⣯⠁⣴⠈⠉
⠁⠛⢈⣿⣿⠀⣟⢳⣀⠈⠁⢀⡄⢀⣤⡀⠀⢀⠛⣠⣶⣶⣤⠀⢀⡈⠀⠘⠃⠀⣄⢸⡟⣿⣿⡀⠀⢠⡄
⡇⠀⢸⡏⢹⢶⠈⠯⣿⡄⠀⠈⢡⣴⣿⣧⡄⠋⠀⢿⢷⠾⣿⠀⠘⠃⠀⠴⠂⠀⣀⢨⡿⠀⣿⣿⠦⠀⣄
⡇⠀⢸⡇⣼⣆⠀⠀⢻⣿⣴⣀⣿⢿⣿⢿⣷⣶⢤⣼⠛⠘⢿⣰⣆⢀⣶⣀⣰⣆⣩⡇⢠⣤⠅⠀⠀⢤⡄
⡇⠀⣾⠃⣿⣿⣦⠀⠾⡟⢷⠀⠘⠾⠞⠟⡒⠀⣴⠛⣭⣽⠫⠹⡆⢀⠀⣀⣴⡦⢾⣇⠀⡔⢶⠠⣿⣆⠇
⡇⠀⣿⣷⡈⠋⠈⠁⠐⣐⠇⠀⠀⠀⠀⠀⡀⠀⠸⣶⣿⣿⣿⡆⠹⠀⠀⣿⡍⠓⠛⣭⣷⣷⡦⣷⡙⠡⡹
⡇⠀⡿⣿⣿⣦⠲⣯⠈⠤⡗⠉⠀⠀⠈⠀⠁⠠⣶⠛⢿⡿⢿⠟⢠⠃⠄⠉⠙⠂⣽⠿⣿⡿⠛⣿⣿⡺⣅
⡇⠀⣷⠿⣿⡏⠻⡏⠀⢰⣿⠈⠉⠉⠉⠉⣠⡖⠃⢀⣸⣛⣇⣴⢯⡈⠉⠁⢠⣾⡁⠀⢾⣠⣿⠿⣿⡿⢾
⠁⠀⠟⠀⣼⡇⠠⠇⠀⠟⠇⠀⠀⠀⢀⠊⠈⠳⡈⢡⠿⠯⠹⠇⠘⣷⠠⠀⣾⡧⠰⠆⣻⡿⠩⠊⢼⡏⠈
⠀⠀⠈⢸⠉⠀⠃⠃⣰⠀⠀⠀⠀⠀⢀⢸⠀⡖⠈⠄⢉⠹⠀⠀⠀⡙⠃⠐⠛⠑⢃⣼⢏⡠⠀⠀⣸⡧⢠
⠀⢾⠀⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⢷⠰⠴⠚⠀⣼⠀⠀⠀⠇⡃⠐⠿⠺⣏⠅⠊⠀⠞⠸⠿⣷⣀
⠀⠀⠀⠀⠈⡄⠀⠂⠀⠀⠀⠀⠀⢀⡈⠀⠀⠂⠀⣀⡀⠏⠀⠀⠀⠐⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣬⣤⣤⣤⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠔⢂⠒⡰⠻⠉⠏⠉⡙⠛⠻⢿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣲⠟⢋⠀⠂⡠⠊⢁⠞⢀⠃⢸⠀⢢⠈⢆⠁⢦⡙⠻⣿⡿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⠋⢀⠔⠁⢠⠞⠁⣰⠏⠀⣜⠀⢈⠀⠘⡄⠀⢧⡀⠱⡄⠈⢿⡍⠻⣾⠉⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣼⠏⠀⣠⠋⠀⣰⠏⠀⣰⠏⠀⢰⡇⠀⣽⠀⠀⡇⠀⠘⣧⠀⢹⡄⠀⢻⣦⡈⢿⣿⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⠟⢠⡿⠁⠀⣼⠃⠀⣰⡏⠀⠠⠿⠀⠀⢛⠀⠀⠛⠀⠐⠛⠀⠀⢿⡀⠈⢿⠀⢈⣿⡆⠁⢿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣟⣴⣿⡁⠀⢚⣛⣀⣠⣤⣤⣴⣶⣶⠾⠿⠛⠛⠛⢛⡛⢛⡛⣛⠛⠿⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⠥⠐⠂⡀⠉⠀⠈⠉⠻⢏⡉⠭⡀⠐⠂⠄⢈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣿⠟⡡⠐⠉⠀⢡⣄⠈⠀⠋⠀⠀⠀⠀⠀⠘⢦⡈⠏⠐⢷⣦⣈⠒⢌⡻⣿⣿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠀⠀⠀⠀⢀⣡⣶⣄⢰⣤⠀⣀⠀⠀⠀⠀⠳⡀⠀⠀⠹⣿⣿⣦⣄⢪⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠢⡄⠀⠀⠀⠈⠀⡿⠯⠦⣡⣄⢀⡴⠀⠘⠀⠀⠘⡄⠀⠀⢹⣿⣿⣿⣷⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠈⠀⣀⠀⠂⠀⢀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣉⣿⣿⣿⣿⣿⣦⠑
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⢀⡀⠀⠠⠀⠀⠀⠠⡀⠀⠀⠑⢮⢣⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠻⣯⢿⣩⣿⣿⣿⣧⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠂⢰⡷⣮⣳⠜⠀⣾⠀⠀⠀⠀⢦⡀⠀⠀⠻⢧⣄⠀⠈⠀⠀⠀⠀⠀⠈⣧⠰⣿⣾⣿⣿⣿⣿⣿⣧⠐
⠀⠀⠀⠀⠀⠀⠘⠀⣿⣟⠀⣽⠃⢰⡇⠀⠀⠸⠀⠀⠙⢶⣞⠀⠀⢩⠑⠀⠀⠀⣠⠀⠀⠀⢸⡀⢿⡿⣿⣿⣿⣿⣿⣿⡆⠃
⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣤⡶⠀⢸⣧⢦⡀⠀⠆⠀⠀⠀⠿⣦⡀⠈⠢⠀⣰⡟⣿⠂⠀⠀⠈⡇⠈⢥⡿⣿⣿⣿⣿⣿⣷⣘
⠀⠀⠀⠀⠀⠀⣄⣺⣿⣿⣿⣀⣴⣿⣣⣖⡁⠀⢦⠀⠁⠀⠀⡎⠙⢦⡐⢳⠃⠞⠛⠓⠒⠒⠻⡇⠀⣾⣷⣿⣿⣿⣿⣿⣿⣟⣀
⢠⡄⠀⠀⠀⠀⠉⠉⠛⠙⠉⠛⠛⠁⠿⠋⠓⠆⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠻⠓⠆⠠⠤⠀⠀⠧⠀⠻⣟⣻⠋⠛⠛⠋⠟⠉⠉⠀⠀⠀⣠
⠀⠙⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠁
⠀⠀⠀⠈⠉⠙⢡⠀⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣲⣐⣓⣒⣶⣶⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠸⣿⣿⡏⠹⣿⣿⣙⠪⡛⣿⡿⠟⠛⢰⣾⣷⠀⠀⠀⠀⠀⣷⠀⠻⣦⡾⢻⣿⡏⢨⣿⣿⣿⣿⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠹⣿⣿⡄⠹⣿⣿⣷⡈⠳⣦⣶⣶⠟⣱⣧⡀⠀⠂⡀⠀⢸⣷⡾⣋⣰⣿⣿⠃⣼⣿⣿⣿⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠙⣿⣿⣄⠹⣿⣿⣿⣷⣌⠲⣄⢀⢀⢠⣿⣶⣦⣶⣾⡿⣫⣾⣿⣿⣿⠃⣰⣿⣿⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣽⣿⣆⠙⣿⣿⣿⣿⣷⣈⠻⣏⣳⣯⣽⣿⠶⣩⣾⣿⣿⣿⣿⠃⣴⣿⣿⠟⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢌⠻⢷⡌⠻⣿⣿⣿⣿⣷⣌⠻⠿⢟⣫⣾⣿⣿⣿⣿⠟⣡⣾⡿⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠍⠂⠘⢻⣾⣿⣟⣯⡇⣀⣬⣿⣿⣿⣿⠟⣡⠾⠋⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠌⠻⠟⣉⠚⠛⠛⢈⠻⢋⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⣈⣹⣷⣾⣶⣦⣁⡀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣤⣿⣿⠿⠺⠿⣿⠛⠷⠶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣿⣩⠄⣰⣿⣿⣾⣿⣦⠀⠐⣶⣶⣤⣀⣀⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡟⠛⠋⣠⠞⠛⠛⠋⠀⠻⠛⠛⠉⠁⠈⠀⢀⠉⠉⠛⠛⠛⠿⣦⡛⢿⣷⣶⣄
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣾⣿⢗⣁⣾⢃⣤⣶⣶⣿⣿⣤⣴⣄⠀⠶⠶⡆⠠⠰⣦⣤⣤⣄⠀⢀⠉⢨⡟⣻⣿⣿⢷⣶⣦⠀⣀
⣠⠴⠚⠉⠉⠉⠉⠉⠀⠀⢰⡏⢸⣿⣧⣄⣉⡉⠉⠛⠛⠛⠃⠒⠒⠂⠀⠃⠸⣶⣶⣶⣆⢉⣉⣩⣤⡾⣿⣿⣿⠀⢸⡿⢿⣿⠛⣿⣷⣤⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠉⢻⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣰⡇⢠⣿⣿⣿⣿⣿⣿⠿⠿⡏⠉⢻⣿⣷⣄⠉⠀⣻⡶⠘⠿⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢉⣀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⠋⠙⣿⣿⡧⠄⠀⠀⠀⠀⣺⣿⣿⡿⠶⠆⠀⠈⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠚⣠⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣛⡀⠀⠘
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⡀⠈⠈⠙⠂⠀⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⠀⠁⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠘⠳⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠰⠀⠉⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⣠⠀⠈⠏⠀⠀⠀⠀⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀⠀⠄⠀⠄⠒⢁⠀⠀⠀⠐⠊⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄
⠀⠀⠀⠀⠀⠐⠔⠀⠤⠋⠀⠀⠊⠙⡇⠀⠐⠰⣦⠸⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡠⡤⣄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢎⠀⠀⠀⠠⡄⣀⠀⠈⠃⠀⠜⠒⠃⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠱⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣈⣁⠙⠃⠀⠀⠀⡠⠀⠀⠀⢤⡆⠀⠀⠀⡀⠒⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠘
⠀⠀⠀⠀⠀⠘⣦⠀⠐⠁⠈⢧⠀⠀⠀⢰⠇⠀⠀⠈⠁⢿⡄⠀⠀⠀⣀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⢸⡇⠀⢻⣀⡀⠀⠀⢀⡴⢚⠁⠀⠔⠋⠀⠈⠉⠻⣻⣄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠈⡑⠂⡊⠀⠢⠟⠫⠉⠁⠊⠐⠻⠁⠀⠀⠀⠀⠀⠁⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⢀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡀⠆⠀⠀⠉⠛⠃⠠⣤⡐⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⣁⠀⠈⣀⠀⠠⣀⠈⠂⠀⠀⠀⢀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⢸⡉⠙⢿⢥⡀⠀⢀⣀⣀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠀⠀⢙⠟⠀⠐⠠⠉⠝⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠛⠛⠛⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⣙⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⠶⠶⠶⠶⠖⠚⠛⠛⠋⠉⢉⣀⣠⡴⠞⢋⣠⡼⢻⣆⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠒⠒⠲⠶⠶⠶⠶⠖⠒⠒⣛⣛⣉⣭⣤⠶⠞⠋⢁⣤⠞⠉⢿⡴⠶⠶⠶⠦⢤⣤⣀⠀⠀⠈⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⢿⣿⣟⣛⣛⣛⡛⠛⠛⠋⠉⠉⠉⣁⣀⣀⣤⠴⠞⠋⠁⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⡴⠞⠙⣇⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣚⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠶⠚⠋⠁⠀⢀⣠⣿⣠⡴⠞⠋⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⣀⡤⠞⢋⣥⠶⠛⠉⣿⢬⣭⣍⣉⣉⡙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠉⠁⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⣠⡴⠛⠉⢠⡾⠋⠀⠀⠀⠀⢿⣤⣤⣀⣉⠉⠉⠛⠛⠛⠓⠒⠲⠶⠶⠶⠶⣒⣺⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⣠⠞⠁⠀⠀⠀⣯⡀⠀⠀⠀⠀⠀⠸⣷⠶⢬⣍⣉⠛⠛⣒⣲⣶⣶⠶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠋⠁
⣼⠁⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⠶⠲⠛⠒⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⣟⣯⣿⠏
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣟⣛⣿⣭⠿⠖⠛⢉⣴⠋
⠀⠈⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⣶⡖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠒⠉⠉⠛⠛⠻⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠑⢴⣷⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠘⢻⡏⣿⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢦⡄⠀⠀⠀⠂⠀⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢸⡧⠀⠀⠀⠀⠂⠀⠀⢀⣿⣿⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⠴⠟⠁⠀⠀⠀⠀⠀⠂⠘⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣻⣷⡆⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣺⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢷⣿⣷⣾⣯⣀⠀⠈⠲⣄⠀⠀⢀⡀⢀⣩⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣶⣦⣤⣽⣾⣷⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠯⠿⣿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠔⠉⠉⠙⣻⣶⡄⠀⠀⠀⠀⠀⠀⠘⣏⣿⣦⡀⠀⠀⠀⣠⢾⣭⠋⠙⢷⣄⠀⠀⠀⠀⠀⠀⠠⢼⣿⣷⣄
⠀⠀⡖⠀⠀⠠⠀⠐⣽⣿⡀⠀⠀⠀⠀⠐⡀⡦⠀⣿⣷⠀⠀⢠⡁⠀⠀⠈⢲⣾⣿⡇⠀⠀⠀⠀⡀⢠⣀⣈⣽⣿⡆
⠀⠀⠁⠀⠃⠀⠉⠦⣹⣿⠀⠀⠀⠀⠀⠀⢁⣱⣮⣿⡿⠀⠀⠀⠻⣿⡆⢠⣿⣿⣿⡇⠀⠀⠀⠈⡩⢭⣹⣯⣽⣿⠃
⠀⠀⠀⠣⣀⣠⣤⣶⠟⠁⠀⠀⠀⠀⠀⠐⠈⣿⣿⠟⠁⠀⠀⠀⠀⠹⣧⣾⣿⡿⠋⠀⠀⠀⠀⠀⠂⣀⣩⣵⠿⠃
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠘⠛⣲⣦⡀⠀⠀⠀⠀⠰⠛⠹⣿⢶⣶⠻⣿⠀⠀⠀⠀⠘⠀⠘⢢⣅⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠠⣤⣤⡤⣤⡼⣷⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡄
⠀⠀⠀⠠⢶⣶⡴⢶⣶⣿⠁⠀⠀⠀⠀⠀⠈⡵⢟⣥⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⣤⣬⡽⠋⠀⢀⠀⣀⠀⣀⣤⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠈⢉⣻⡿⠏
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠋⠉⠉⠛⠓⠶⣤⣠⡤⠶⠖⠛⠛⠋⠉⠉⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡴⠶⠶⠶⣤⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⢠⡿⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠋⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢉⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣉⣤⠞⣿⠁
⠀⠀⠀⠀⢀⡼⠋⠀⢠⡾⠁⠈⢷⡀⠀⠀⠀⣀⣠⡴⠶⠛⣉⣤⠶⠛⠉⠀⣰⠏
⠀⠀⠀⣰⠏⠀⠀⠀⠸⢧⣤⣤⣬⠿⠶⠚⠛⣉⣥⡴⠞⠛⠉⠀⠀⠀⢀⡼⠋
⠀⠀⠀⢿⣀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠺⢿⣭⣀⣀⠀⢀⣀⣠⣤⠾⠋
⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⢤⡀⠀⠀⠀⠄⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣽⣿⠂⠀⠀⠀⣰⡄⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⡤⠀⢀⢠⡜
⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠉⠁⠀⠀⠘⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠃⠓⠋⠁
⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⣀⢘⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠒⠲⠖⠊
⠀⠀⠼⣿⠷⠀⠀⠀⠀⠀⢨⡀⢰⠹⡄⠀⠀⠀⠃⢐⡄⠀⠀⢤⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢻⣖⠚⠖⠓⠉⠀⠀⠀⠸⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠈⠁⠄⠀⠀⢀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢽⡿⠄
⠀⠨⠄⠀⠀⠀⠀⠆⠀⠈⠛⠃⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⢤⠀⠁⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠐⠀⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄⡐⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⢹⠀⠀⠀⠀⠘⣄⠀⠀⠀⡀⠀⢀⣄⠀⠀⣿⡟⠡⣾⣿⣷
⠀⠀⠀⣼⣿⣅⠀⢀⠄⠀⠀⠀⠀⣀⡖⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠈⠙⠁⠀⠱⣦⣄⠀⡉⠁
⠀⠀⡜⠘⠁⠀⢰⠁⠐⣄⣀⡤⠚⡡⠀⠀⣤⡤⠀⠀⠀⠀⠀⡄⠀⢸⠉⠐⢢⣤⡀⠀⠀⠈⠀⠀⡀
⠀⠀⡂⠀⢂⠀⠈⠓⠒⠋⠉⠈⠁⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⢸⣠⠖⡋⢀⠙⢄⠀⠀⠀⠿⠃
⠀⠀⠠⠀⡂⠀⠀⠀⠀⠀⡔⠉⠈⡁⣐⡀⠀⠀⠀⠈⠀⠀⠀⣠⣀⠈⠣⡀⠀⠊⠀⡈⠢⣀⣀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠐⠀⡀⡀⠀⠀⠀⠰⠆⠀⠀⠛⠇⠀⠀⠙⢆⠈⠢⢃⣔⠁⠈⢢⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⡨⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠚⠹⡀⠙⣄⡠⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢶⣾⣄⠀⠀⢰⠉⠢⡀⢈⠦⠜⡆
⠀⠀⠀⠐⠦⣠⡠⠔⠁⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠑⠢⢬⣣

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣛⡿⠿⠹⡩⣽⣿⣿⣿⣿⣷⣦⣀⠀⠀⢀⣀⣠⣤⢐⠀⠡⠘⠪⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⠀⠒⠋⠀⠐⠈⠹⣾⣿⣿⡿⡟⡓⢮⠄⠰⠋⠀⠋⢁⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⡀⢀⢡⡄⢀⣠⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠗⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠃⠐⣁⠀⢐⣚⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠗⡂⠀⠀⠀⠀⢠⠐⠂⠑⣀⣁⠂⣈⣗⣂⣴⣿⣭⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣴⣦⢠⠐⠑⠈⢂⠄⡈⣣⣴⣶⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢋⠉⣉⠣⠆⣸⣡⣴⣻⠽⢇⣉⠘⣋⣽⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣾⣶⢩⢥⣶⣿⣿⣿⠟⠁⣸⣋⣵⡾⠟⠑⡊⠅⣈⡉⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⢀⠐⢻⣿⣿⣷⣮⣾⣿⣿⣶⣾⣿⣿⡛⠋⡊⠘⠩⠀⢀⣇⡠⣽⣿⡿
⠀⠀⠀⢀⠠⠘⠛⢀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡟⢫⠔⠱⠂⣨⠷⢀⣤⠂⣀⣻⣾⣿⡟⠁
⠀⢀⡠⣛⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣟⣩⣷⡷⢪⣦⣴⢿⣿⣭⠾⠿⢿⣿⡿⠋
⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣪⣾⣿⣾⣋⣡⣆⡽⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⢠⠘⢇⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⠋⠉⠝⠘⠸⣾⡧⠤⣀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣸⣤⠖⠃⠀⠠⠈⠋⢿⣖⠈⠳⢦⣀⠀⢠⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢠⠋⡤⠊⣽⡇⠀⠀⠀⠂⠀⠰⣯⠙⣷⡀⠀⠈⢷⡤⡃⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣮⣠⣼⣤⣿⣤⣄⣄⠡⠈⠀⠉⠀⠈⠁⠀⠘⢀⣳⣹⡃⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣿⠿⣏⣙⠉⠉⠉⣧⢈⢹⡁⣸⣹⣙⠀⠀⠙⣿⡉⣿⣿⢛⢛⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣉⣿⣟⡿⣿⡏⠉⠙⢿⡟⠻⣿⡟⠋⠁⠉⡸⢸⡟⠐⢿⣿⡆⠙⢷⣀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣯⣿⣧⣜⣿⠿⣧⠀⠀⠸⣿⣧⣿⣿⠀⡠⠊⠀⢸⣇⣀⡛⣻⣿⣾⡿⠃⠀⠀⠀⠀⠀⠠⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠐⠀⡀⠀⢀⡠⠽⢿⣿⡿⢿⣿⡶⢀⡀⡀⣀⡀⡷⢛⣀⣀⣤⣺⣿⣿⣏⣩⣥⣤⢄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠃⠀⠐⠀⣠⠞⣛⣾⣆⣠⣁⣤⣿⣿⡟⠻⠶⣿⣿⣿⣿⣻⣿⣤⠿⡿⢿⣿⣿⣿⡗⣾⢿⣿⠱⣖⣤⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢤⣦⣾⣶⠄⠀⠀⠀⠂⣴⠟⣼⢋⠄⣼⣿⡟⠀⢁⠘⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⢡⡀⠀⠈⢿⣿⡇⠀⠹⣿⣿⡞⠆⠀⠀⠖⠄⢹⡇
⠀⠀⠠⣄⣀⡿⠛⠉⠀⠀⠐⠂⣸⣿⣿⠃⠂⢰⣿⣿⢡⢄⡼⠀⠀⠀⠀⠀⣿⣿⡳⠀⠀⠀⠀⠀⢣⡄⠀⢸⣷⣿⣾⠀⠹⣿⣷⡐⠀⠀⠀⠄⢿⡇
⢠⡄⠐⣽⣿⣏⡀⠀⠀⢪⣤⢀⣿⣿⣿⢶⣗⣿⣿⣗⣈⡾⠁⠀⢄⢀⣄⠀⣿⣿⣦⢀⠀⠀⠀⠀⠀⠹⣄⠀⣿⣿⣿⣤⣶⣿⣿⣷⠀⠀⠀⠀⠮⡆
⠀⠀⠀⠈⠉⠀⢠⢠⡤⠘⠁⣸⣿⣴⡟⢤⢾⣿⣿⣿⣿⣥⣾⣃⡿⢞⠁⠀⣿⣵⡆⠀⠉⠠⠂⠀⠀⠀⠊⠗⠛⠀⠙⣿⣿⣿⡉⢻⠀⠀⠀⠀⠈⠃
⠀⠀⣴⢀⣤⣤⡀⢛⠀⠀⠀⣿⣿⣏⣷⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣠⡌⠃⣸⡝⣢⣠⣶⣣⣀⣤⠄⢄⣰⣶⣶⣶⠈⠛⣿⣿⣿⣿⡁⠀⠄
⠀⢰⣿⣽⣿⣶⠉⠄⠀⡰⣶⣿⡯⢙⣙⡻⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣶⣾⣿⣧⠁⣹⣿⣿⣿⣯⣿⠿⣿⣿⠿⣪⣾⣷⡟⢻⡛⠛⢷⣮⣶⣆⣀⡀
⠀⠘⣿⣿⣯⠀⢰⣆⠘⣿⣿⣿⣿⣿⢿⣿⣶⣺⡿⣷⣬⢹⣿⣿⣋⣿⣿⣿⣿⡛⠄⢘⣿⣿⢫⣕⣸⣄⡘⣳⣮⠩⣤⣼⡷⠛⠛⣿⣷⣹⣯⣿⣿⡦
⠀⢹⢿⣷⠉⢂⡠⠒⢁⣸⣦⣿⣿⣿⣿⣿⣿⡷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡷⢈⣿⡿⢛⡟⠙⢿⢛⣿⣿⣴⡟⠋⢀⢠⣼⣿⣿⠿⠟⠿⠿⠄
⠀⠀⡽⣿⢿⣯⣴⣽⣿⣿⣿⣿⣹⢹⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⡮⢿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣲⣤⣘⣶⣬⣿⣿⣦⡊⠲⢦⡀
⣀⣀⣀⣉⣁⣙⣀⣙⣿⣛⣛⣛⣻⣇⣃⣻⣿⣿⣿⣿⣿⣛⣿⣟⣿⣟⣻⣋⣻⣻⣻⣿⣛⣛⣻⣿⣿⣿⣟⣏⣀⣿⣛⣟⣛⣻⣟⣛⣻⣛⣛⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡴⠀⠊⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡅⠀⠀⣀⣠⣀⣤⣤⣤⣄⣀⠀⢁⣶⣞⣻⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠁⠀⠁⠀⡀⡀⢠⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⣀⣼⣿⣶⣽⣿⣿⡷⣤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣤⣄⡤⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣟⣬⣸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣆⠈⠓⣾⣷⡀
⠀⠀⠀⠀⠀⠀⡞⣿⣿⣿⣿⣿⣿⣿⣿⣥⠉⠿⣁⣇⡈⢘⣿⣿⣯⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣾⣿⣇⡀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⢋⣽⢿⣯⡥⣷⡞⣽⠄⣿⣿⡛⣿⡿⢛⠻⢵⢶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣻⣧⡄
⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢖⣀⡇⠛⢡⣿⠙⢷⣞⡴⠟⠀⠀⠾⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠻⣿⣿⣿⣿⣿⣿⣞⣻⠄⣾⡟⠛⠣⣬⣙⠇⠒⠻⠀⠀⠀⠀⠀⠀⢉⣼⣿⣿⣦⣽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠙⠻⣿⢿⣿⣿⣾⣿⣿⣷⣦⣿⠏⠀⠙⠂⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⡄⠘⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣦⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣤⣤⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣥⡀⢸⣿⣿⣿⣿⣟⣿⣧⣭⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡟⢻⡿⢿⣿⠏⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠙⠂⠉⠙⠻⠋⢴⢬⡍⠻⣛⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢈⣶⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠃⠀⠀⠀⠈⠑⠘⢁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣰⠮⣡⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡄⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⢀⣤⣶⠿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡿⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋
⠀⠀⠀⠀⠀⣄⣋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣁⠀⠀⠀⠀⠀⠀⠀⢀⣸⡋
⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠺⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣖⣭⠵⠶⠆⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⡋
⠀⠀⠀⠀⠀⠠⣿⣗⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠇
⠀⠀⠀⠀⠀⠠⣴⣦⠁⠀⠀⠀⠀⠀⠀⠀⣴⣷⠍⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⣤⣦⠍
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠂⠀⠀⠀⠀⢀⣰⣋⡀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣶⣯⠀⠀⠀⠀⠀⠀⠀⢀⣠⣜⠁⠀⠀⠀⠀⠀⠀⢠⣤⡄⠈⠈⠀⠀⠀⠀⠀⣴⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠟⠓⠀⠀⠀⠀⠀⠀⠀⠈⠟⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡆⢀⣠⣤⣤⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣩⣥⣶⣶⣶⣶⣶⣶⡶⣆⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⡟⣿⣷⡤⣷⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⡠⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠳⠄⠄⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠆⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⠀⠐⠀⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠈⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡴⢢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣼⡏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣄⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⡌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣤⡀⠀⠀⠀⠀⠠⡠⠖⠉⡁⢠⠖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣄⣄⠠⣈⠕⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠉⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣃⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣤⣁⣚⠿⠻⢿⣿⡿⠻⠟⢛⣛⣭⣽⣔⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣯⣹⡿⠿⠟⠛⠉⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢨⠅⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⡆⣨⣿⡶⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠿⢿⣿⠿⠿⠿⠋⠁⣀⣤⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠛⢹⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠿⠟⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠙⠋⠍⠙⠛⠿⠛⠛⠟⠉⡁⠈⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢶⣶⣦⣦⣤⣤⣤⡤⠤⠶⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

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

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⣾⣿⣿⣿⠟⠛⠛⠛⠛⠒⠶⠦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠘⢿⣿⣿⣄⣀⣀⣀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣧⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠙⣻⠿⣾⣽⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣯⣟⡻⢷⣾⣭⣛⣛⠿⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣶⣯⣽⣛⡿⢷⣶⣬⣭⣅⣀⣀⣀⠀⠀⢀⣀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣄⠈⠙⢦⠒⢦
⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣦⡀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⠀⠀⢸⠀⠀⠀⠀⢐⠋⠙⡆
⠀⠀⣀⣤⡜⠀⢀⠀⠀⡀⠀⠀⢻⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⠖⠋⠁⡼⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠳⡀⠀⠀⠀⠀⠠⠂⠀⠀⠁
⠀⣞⣁⣰⡁⠀⠀⣦⡄⣀⣀⣀⣸⡥⠊⠀⠀⠀⠀⠀⢠⣴⠒⠚⠁⠄⠀⢈⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠱⣤⡄⠀⠀⠀⡠⠂
⠀⠈⠉⠙⣏⠉⠉⠉⠉⠉⠉⣉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡄⠀⠛⠚⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡂⠹⠮⠔⠒
⠀⠀⠀⠀⠈⠛⠦⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡴⠛⠳⠦⣼⡆⠀⠀⠶⡄⠀⠀⢀⠀⠀⠀⠙⢦⠠⢤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠀⠀⠀⠀⠀⢀⣀⣰⠀⠙⠂⠉⠜⠀⠀⠀⠀⠠⠀⠈⠑⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⠲⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠺⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠊⢀⣀⣀⣀⡤⠤⠤⠤⠤⣤⣤⣤⣔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠰⣖⠒⠖⠀⠀⢈⠓⠉⣉⠝⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⢦
⠀⠀⠀⠀⠀⠈⠳⡄⠦⠥⠤⠀⣴⠁⠀⠀⠉⠉⠉⠉⠛⠒⠢⠤⣄⣀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠳⣄⢸⡂⠀⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠲⡍⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⣤⡀⢸⡇⠀⣤
⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⣙⣿⣷⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠄⠀⠐⠋⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠟⠁⢸⡃⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣆⣀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠚⠁⠀⠈⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠠⠒⠉⠉⠁⣉⣩⠾⠁⠨⠁⡎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠔⠁⠀⢀⠤⠂⢁⠜⠙⢶⠈⡻⢤⣹⡀⠀⠀⠀⢸⡉⠲⠦⠤⠤⠶⠚⠉⠛⠶⠤⠤⠄⠀⠀⠀⠀⢀⠜
⠀⢀⠀⠀⠠⠂⠁⢀⠔⠁⠀⠀⠸⠆⠀⠀⠈⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁
⠀⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⣀⣀⣀⣀⡀
⠓⠒⣎⠀⢀⠰⣁⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠤⠶⠖⠛⠉⠀⠀⠀⣠⠖⠋⠁⠀⠀⠈⠙⠳⣦
⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⣀⡀⢻⡂⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠖⠛⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢼⣧⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⣾⠀⠀⠀⠰⠀⠀⠐⠀⠀⠀⠀⠸⡇⠀⠀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠊⠁⢹⡍⠲⠶⠀⢤⡀⣶⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢿⠀⠀⠀⠀⠲⠖⣀⣀⣠⣤⠤⠶⡟⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⢠⣤⡽⢶⡚⠃⠀⠀⠀⠀⠘⠓⠶⠶⠶⠶⠾⣶⠒⠒⠛⠋⠉⠉⠁⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⣀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠒⠒⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⣠⠾⠋⠈⠳⣄
⠀⠀⠀⢤⡒⢊⣉⡡⠬⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠋⠁⣠⠞⠁⠀⠀⠀⠀⠈⣣⡀
⠀⠀⠀⠀⠙⠣⣤⠤⠒⢶⡉⢑⣦⡀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠟⠋⠁⢀⡴⠊⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠈⠳⢖⠉⣙⣧⣤⠼⠢⣄⠀⠀⢀⣎⣀⣠⣤⣴⣶⣿⣿⣿⠿⠟⠋⠁⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢋⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡤⠜⠳⣎⡉⣑⣄⣼⣿⣿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢋⣴⡿⠋⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣲⣊⠉⠙⣦⣠⡽⠂⠀⠀⢀⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⢁⣴⠟⠋⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠈⠻⡯⣥⠖⠻⡗⣒⣈⣭⣭⡳⡀⠘⡏⢳⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠟⠉⣠⠴⠋⠁⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡜⣧⠀⢱⢸⣟⢻⡍⠁⢡⠀⡗⠊⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⣀⠴⠊⠁⠀⠀⠀⠀⠀⣠⡾⠟⢁⡇
⠀⠀⠀⠀⠀⠀⠸⡁⣀⡠⠤⠖⡄⠀⠀⢱⢹⠀⢸⠊⣉⣀⣹⣶⣧⣴⠿⣿⣿⡿⠿⠛⠋⠁⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣴⣿⠃
⠀⠀⡠⢤⣤⣖⠮⡭⠒⠒⠒⠉⠁⠀⠀⢸⡿⠒⠺⣿⣿⠿⢍⣉⠬⠷⣖⠋⣣⣄⡤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⢀⣴⡿⠟⡏
⠀⠀⠉⠊⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠱⣤⣤⣤⣷⠶⡒⠊⣙⣧⣠⠵⢤⡀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⣠⣶⠟⠋⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠔⠉⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠈⠫⣄⡠⠽⠶⣖⠙⣢⣀⡤⠔⠊⠁⠀⢀⣠⣴⠿⠋⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⣀⣠⣵⣶⡒⠉⣻⣄⠬⠷⢄⣠⣴⡾⠟⠋⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⡉⠉⠀⠀⣙⣣⣀⣀⣤⠴⠒⠋⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣟⡛⠉⠁⠀⠀⠀⢀⣀⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⢶⣞⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⡟⣿⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠾⠿⢛⣛⣛⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣟⣟⣟⣿⢯⡇⣿⡧⠽⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⣟⣽⣭⣭⢿⣾⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⣽⡗⢒⣹⢸⡏⠉⠉⢹⠁⣿⣿⣶⣶⠇⠀⠀⠀⠀⠀⠀⣠⣮⡭⡥⠶⠶⠶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣿⣚⡺⣺⠠⢬⣸⣸⠇⠀⠀⢸⠀⣿⣿⢸⣿⠀⠀⠀⠀⠀⢀⡶⠷⠒⢒⣚⣛⣻⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⢍⢭⡭⡭⡏⠀⠀⡉⠁⠀⠀⠀⣾⠘⠉⢛⢻⣿⡌⢠⡄⠀⣰⠛⠛⠛⣛⢋⣩⡏⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢖⢓⢒⢖⠓⠂⠁⣀⣴⣽⣿⣿⢿⣿⣿⣷⣶⣶⡟⠁⠀⠀⢠⠾⢟⠛⢛⣟⣟⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣯⣩⣭⣉⣭⠉⠁⠀⢛⣿⣿⣿⣿⠆⢰⣿⢻⡹⣿⠟⠀⠀⠀⣴⣛⣛⣋⣹⣯⣭⣍⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢤⠭⠭⠦⠤⣦⠖⠀⠀⢀⠔⢩⣏⣟⣹⣯⣴⣾⣥⣮⣿⠏⠀⠀⢠⣾⣭⣭⣩⣭⣬⣬⣭⠋
⠀⠀⠀⠀⠀⠴⠷⠴⠶⠶⠴⠶⠊⠀⢠⠔⠁⢠⣶⣿⡿⣿⢸⡟⢻⠟⣏⠋⠀⢀⣴⣯⣤⣴⡤⢤⠶⢶⠔
⠀⠀⠀⡴⠶⠒⠒⣒⢓⢗⠒⠀⡠⠂⠀⠀⠀⠀⣿⡿⢹⣷⣚⡧⢾⣦⣏⠀⣠⠾⠶⠶⠶⠶⣶⠚⡷⠋
⠀⣠⠶⢛⠛⣋⣋⣛⠙⣁⠔⠉⠀⠀⠀⠀⢀⣬⣶⡏⣿⡟⣽⠛⢻⠟⣿⠒⣿⠻⣿⣿⢿⠛⠛⠋⠁
⠚⠊⠙⠉⠙⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣳⣽⣧⣿⠅⢸⠀⢾⢰⣿⠀⢹⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡹⠋⠟⢳⡿⠤⣾⣾⢾⠦⠺⡟⠉⠇⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡹⢁⣀⣀⣸⣧⣤⣴⣿⣤⣤⣀⣀⣀⡀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠛⠋⢹⠉⢸⣿⡏⠀⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠀⠀⠀⠀⢆⣘⣂⣚⣿⣷⣶⣿⣿⣿⣹⣾⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢃⣠⣴⣾⣿⠭⢿⣿⣿⣻⣻⣮⣫⣝⣯⣀⣩⣭⣽⣯⣻⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣯⡿⡿⠷⠿⢿⡿⠩⣿⡄⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡀⣿⣿⣿⠿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣄⣾⣿⣦⣣⣾⡏⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠽⠿⠮⢿⣿⣝⣿⣿⣿⠿⠿⠿⢿⡯⢽⣿⣏⣀⠽⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠋⣩⡶⠔⢋⡽⠋⠀⠀⠀⠀⠀⠀⠀⢠⡞⠿⠿⠛⠛⠻⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣠⡴⠞⢉⡤⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⣤⣤⣤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⣀⠼⠉⣁⠴⣚⠽⢊⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⣾⡥⠖⠛⠟⣋⣥⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣶⣖⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⣀⣾⢻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠁⠀⠀⠀⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠆⠙⢳⣿⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠍⢲⣀⢀⠏⣛⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠑⠀⠐⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠿⢿⣷⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠤⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⣀⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⡶⡀⢀⠆⠀⠀⠀⠀⠈⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠐⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⢀⠀⠀⠀⡢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⡂⠀⠀⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢁⡇⠀⠀⠀⠂⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣁⣈⣐⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢠⣊⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣴⣿⣿⠿⢿⡶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠁⠀⠀⠀⠀⠠⠴⣾⣿⣿⣿⣷⣶⡿⣪⣿⠃⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻⢿⠿⠟⠀⠀⠀⠀⠀⠀⣐⣺⣿⣿⣿⣿⣿⣯⠾⠋⠁⠀⠛⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠀⡹⢿⣔⣄⣀⣀⣀⣀⣀⣤⣀⣼⣿⣿⠿⠟⣛⣋⣥⣶⠀⠀⠀⠀⠀⠉⠉⠻⣿⡻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣉⣉⣭⣥⣶⣶⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⣶⢸⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⢀⡴⢫⣾⣷⣿⡿
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣇⢣⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢀⡪⠶⢋⣵⣿⠟
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣯⣳⣵⡀⢀⠀⠀⠀⠑⠻⠿⠿⠿⠿⢿⣿⠿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⢀⣈⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣷⣮⣤⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣷⣶⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⠀⠀⠀⠀⠀⠀⠀⢻⡍⠉⠉⠙⡧⠀⠈⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠃⣰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⠀⡞⠹⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠶⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠀⣐⠰⣟⠀⠀⠀⢰⠉⢻⠇⢠⠀⢀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣰⠋⠙⣿⠀⠀⠀⠀⠐⠪⠃⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠠⡀⠳⡴⠛⠀⢀⡈⢁⣀⡪⢊⣠⠾⠛⠟⣻⣿⡷⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣖⣫⣿⣶⣮⣝⢿⢜⣋⣠⢦⣿⠁⠀⠀⠀⢱⣿⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⡆⢿⡘⣿⣧⠻⢆⡀⣀⣠⣬⠁⠃⠂⠀⠀⠀⢀⣴⢿⣿⣦⠀⠀⠀⠀⣀⠁⠈⡇
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣇⣛⣽⣿⣿⣿⣿⣿⣿⣏⡚⠋⡁⡘⠉⠁⠁⢀⡀⠀⠀⣼⠀⠀⠘⡿⢃⠀⡄⠀⠘⢆⠀⡄⠰⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣹⣿⣿⣿⣷⡔⠜⠆⣤⠝⠳⠟⠀⠀⠈⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣀⢄⣀⣠⣤⣴⣄⣤⡤⡄⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠙⢿⣿⣿⣿⣿⣿⣿⣅⠹⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠰⠆⠀⢠⠉⢸⠟⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⡄⠀⠛⠁⠀⣀⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣒⡔⣄⣶⣶⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣿⣿⣿⣿⣿⣿⣿⣆⠹⠛⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣛⣉⣩⣭⣽⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠞⠋⣩⡿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⣁⣀⣠⣤⠶⠛⢉⣠⠶⠋⣷⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣉⣉⣛⣛⡛⣛⣛⣛⣛⣛⣛⣉⣩⣭⡤⠶⠖⠛⠉⢀⣤⠞⠁⠙⣶⠒⠒⠒⠲⠶⠶⢤⣄⠀⠀⠀⠀⠉⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠿⠿⣭⣭⣭⣉⣉⣉⣉⣉⣉⣀⣀⣀⣠⣤⡤⠶⠚⠉⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⢻⡀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢞⣫⣿⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠶⠖⠛⠉⠁⠀⢀⣤⢾⣧⡴⠞⠋⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⢀⣠⠶⠛⣉⡴⠞⠋⠁⣿⠶⠦⢤⣭⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣀⣤⣴⡾⠿⠛⠉⠁⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⢀⣤⠞⠋⠁⣠⠞⠉⠀⠀⠀⠀⢹⡶⠤⣤⣄⣀⣉⠉⠉⠉⠛⠛⠛⠛⠛⠛⣛⣛⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉
⢀⡴⠋⠁⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠈⣿⠓⠶⣤⣍⣉⣉⣛⣿⣿⠿⠿⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠛⠋⠁
⣾⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠾⣛⡯⢿⡟
⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣛⣿⣿⣭⠽⠷⠚⠋⠉⣰⠏
⠀⠈⠙⠛⠲⠶⠶⠤⠤⠤⣤⡤⠤⠤⠴⠶⠶⠶⠒⢾⣟⠛⠉⠉⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠒⠒⠶⢦⣄⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⣮⠉⢉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣩⡗⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡘⣇⠀⠀⠀⠀⠀⢀⣠⡤⠖⢋⣡⠴⢚⡉
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠺⠤⠤⠤⠴⠖⠚⣋⣩⠤⠖⠛⠉⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠤⠤⠴⠶⠒⣛⠉⠁⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠖⠚⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⡤⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣤⣤⣦⠀⢀⠋⠙⠉⠐⠲⠽⡟⠋⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠹⣿⣿⣰⣃⡀⢀⣀⣠⡞⠸⣍⡉⠿⡀⢰⡶⠆⢀⣤⣄⠀⠀⢀⡀⠀⠀⢠⠄⠀⣀⠀⠀⣀⡠⠖⠂⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣄⣲⣦⡽⠶⠛⠋⠉⠀⠈⠈⠙⠻⡀⣿⣤⠀⢀⡀⡄⠀⠀⠙⠒⠋⠰⢋⣽⠟⠀⠀⠀⢀⣾⠤⠀⣤⠘⢶⣤⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣀⣼⣿⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠞⠈⠁⠀⠰⣿⠿⡟⠘⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣟⠲⣶⣤⣄⡀⡀⠀⠉⠐⠠⠎⠀⠀⠀⢀⣶⠋⣠⡇⠀⢀⡀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠤⠒⠊⠙⠻⡏⠃⠠⢄⣐⠀⠀⠀⢠⣾⠁⣾⣏⣷⣠⠉⣁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠓⡤⣤⠖⠋⠈⢣⣄⣠⠴⠒⠉⢻⣷⣶⣦⣿⣄⣀⣀⠀⣿⢐⡏⡿⠗⠸⢁⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⢀⠀⠼⠂⢞⡉⠀⣈⡱⠶⠒⠉⠁⠀⠀⠀⠀⠀⠘⠉⠛⢿⡿⣶⣬⣽⣟⠈⠁⠘⠒⢳⣿⢁⠆
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠘⠲⢎⠀⠀⣀⣤⠴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⢿⡀⠟⠏⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠁⠀⢀⣀⡤⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠴⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠛⠛⠛⠛⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⢀⣴⠾⠛⢋⣩⣭⣽⡷⠶⠾⠿⠿⠿⠿⠿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⣿⠁⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠹⣧⡀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠈⠻⣶⣌⠻⣦⣀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠙⠻⣮⣝⣷⣦⣄⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡟⠿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣦⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠉⠛⠿⣯⣛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠈⠙⢷⣮⡛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠙⠛⠿⢿⣿⣿⣷⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠃⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣛⣛⣋⣩⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⡀⠀⢀⣀⣀⣠⣤⣶⡿⠿⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⠢⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠂⠀⠈⢄⠀⠀⡠⠁⠀⠀⠉⠃⠀⠨⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠈⡖⠀⠀⠀⠑⠤⠁⠀⠀⠂⢄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢤⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⢀⡔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⠄⠀⢀⠄⠀⠀⠀⠀⠀⠈⢡⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠁⠀⠀⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠⠀⠀⠠⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠅
⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡠⢅⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠁
⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠧⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠩⠨⠄⠀⢀⢀⠔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠅⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⠦⠀⢀⠀⠀⠀⠀⠀⣤⠾⣿⠋⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⢀⣛⣩⡥⠄⠐⠒⠐⠆⡹⠀⠀⠀⠀⢸⣿⣤⡏⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣛⠛⠛⠛⠃⠠
⠀⠀⠀⠀⠀⣾⣸⣿⣿⡇⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠘⣿⣬⣇⠠⠄⠀⠁⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠑⢀⠀⠀⠀⠀⠀⠀⠘⣋⣮⣅⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠿⠿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣖⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠟⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠐⠄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢠⣿⣯⣥⡴⠶⠒⠛⢻⡆⠀⠀⠀⠀⢰⣿⣿⡇⢀⣀⡤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⠀⣀⡾⠀⠀⠀⠀⠀⠀⣿⣟⠛⢃⠀⠀⠀⠈⡇⠀⠀⠀⠀⣠⡾⣿⣿⠇⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠘⢿⣿⣧⣉⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣴⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠙⠳⠿⠿⣿⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣻⠟⢉⣠⠀⠀⠀⠀⠀⢠⠛⠺⡏⠀⣾⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣁⢴⣏⢹⡏⠀⠀⠀⠀⢠⣾⣿⣟⡐⢄
⠀⠀⠀⠀⠀⠀⢠⣿⣭⣿⡏⠒⠉⣀⠀⠀⠀⠀⠀⠀⢴⡞⠀⠀⡀⠠⡦⣄⠀⠀⠀⠀⠀⠀⠀⠠⠋⢀⣼⠋⣡⣾⠏⠀⠀⠀⠀⠀⢸⣿⣿⠉⣄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣭⡟⠛⠋⠀⠀⠀⠀⠀⠀⢀⣼⣧⠀⢠⠏⡀⠉⠹⣷⡄⠀⠀⠀⠀⠀⠀⢀⣏⡻⣟⡯⡴⠃⠀⠀⠀⠀⠀⠘⢿⣿⣷⣾⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠉⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠤⠀⠄⠀⠠⠀⠤⠠⠀⠀⠠⠀⠠⠄⠠⠠⠠⠀⠄⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⡄⢐⡒⣂⣀⠀⠀⠀⠀⠀⠀⣤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡂⠝⢂⠉⠤⠔⠂⠀⠀⠀⠀⠀⠐⠢⠤⠈⠉⢘⢊⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠵⠖⠈⢁⣀⡀⠠⠄⠀⠀⠈⠀⡀⠤⠦⣠⠀⠀⠀⠀⠀⠀⠄⡐⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠤⠄⠂⠈⠀⠀⢀⡈⠁⠀⠀⠂⠌⠁⠂⠀⠀⠀⠀⠀⠁⢘
⠀⠀⠀⠀⠀⡐⠁⡀⠀⠀⠀⠀⢀⠀⠀⠠⠀⢐⡀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠄⢀⠐⡃⡄⠃⠀⠀⠀⠀⠀⠀⠀⠁⠅⠠⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢸⢄⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠠⢀⡰⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠠⡀⠀⠆⠁
⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠠⠀⠀⠀⠈⠀⠐⢁⠁⢄⡌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠁⠀⠈⡄
⠀⠀⠀⠆⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣄⡤⣀⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⢾⡾⠷⠽⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠛⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠜⣿⠛⠛⠓⠒⠃⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⣶⢧⣘⡋⣳⣾⣿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣻⣿⠭⠛⠅⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠐⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠆⠀⠃⢼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡀
⠀⠀⠀⠀⠘⢄⠀⠐⠆⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢄⠀⠀⡀⠠⠆
⠀⠀⠀⠀⠀⠈⢦⠆⠀⢀⠀⠀⠀⡅⠀⡀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠐⠀⡀⢒⠀⠆⢠⠀⠠⠀⡀⠀⠰⠀⠀⢀⡀⢀⠐⠀⠈⠃⠀⠀⠀⠀⠀⠀⠱⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢗⢦⣀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠘⠀⠃⠂⠈⠁⠀⠀⠀⠸⠀⠀⠀⡀⠐⠈⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣓⠂⢄⡂⠄⠀⠀⠀⠀⠀⠉⠁⢄⡀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠻⠋⠓⡖⠂⠀⠠⠤⠤⠤⠙⠑⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢳⡔⠁⡹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⢉⠶⡁⡨⠻⣀⠔⠙⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⡠⠚⢢⡔⠉⡲⡊⢠⠟⢦⡀⠀⢀⣴⣿⣷⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠁⡸⠪⣠⠞⢳⡴⠻⣄⣴⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠖⢉⡶⠋⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠛⢦⣴⣷⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⢿⠟⠳⡤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣦⣤⣀⠸⣿⣿⣿⣿⣿⣿⠗⠀⡠⠾⣀⠔⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠛⠿⣿⠟⠙⣶⠚⢣⡔⠉⡲⡎⢀⠝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣧⠀⠀⠀⠈⠳⣔⠁⡨⠮⣀⠜⠣⣤⠊⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠈⠻⣄⠔⠉⡦⡊⢈⠲⣅⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⣀⠜⢣⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡿⠛⠉⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀
⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣼⣿⣿⣶
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⠛⠛⠋⠉⠉
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣾⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⣿⣿⣿⡿⠁⡰⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿
⠸⣿⣷⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⡰⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆⠀⠀⠀⠀⠀⣿⡇
⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⣹⣿⣿⣧⡀⠀⠀⢸⣿⠇
⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠛⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣼⣿⣿⢿⣿⣷⡄⠀⢸⣿
⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⢤⣀⡀⠐⠛⠛⠋⠉⣉⠠⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⣴⣿⣿⣿⠋⠀⠹⣿⣿⣆⣿⡏
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣤⣈⠑⠪⠍⢈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠒⢉⣠⣶⣿⣿⣿⠟⠁⠀⠀⠀⠈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⣦⣤⣄⣉⣀⡐⠒⠒⠒⢂⣈⣉⣠⣤⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⣓⠀⠤⠤⠤⠤⠄⠀⣀⡀⠀⢀⡿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⠁⢸⠁⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠉⣷⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣋⡷⠀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⣖⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⠛⠐⠆⣄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣿⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⢳⡀⠀⠂⠀⠀⠀⠀⠀⠁⠄⠀⠈⢧⣹⣾⣿⣹⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⠂⡀⠀⠀⠀⢳⡀⡁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠘⢷⡿⣿⣿⠯⣾⣿⣿⣿⣷⣄⢄⡯⠛
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢷⣼⡷⣤⣾⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡴⣘⣷⠟⣿⣷⣿⣿⣿⣷⡀
⠀⠀⣿⡷⢰⡃⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠜⢿⠿⣦⣿⣿⠿⠿⣿⣿⣷⡀
⠀⠈⠋⢠⣯⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢢⡅⡇⠁⠀⠀⠀⠀⠄⠈⠀⠀⠂⠳⣎⠴⣯⣀⣀⣀⣤⣼⣷
⠀⠀⠀⡾⢹⡀⠀⠀⠀⠀⠒⠀⠼⢩⠏⠀⠹⣄⠀⢀⠀⣀⠀⠄⠠⠄⠠⠀⠆⠀⠈⠦⠳⡄⠄⠀⠄⠈⠁⢀⠀⠀⠀⠀⠁⠈⠻⣿⣿⣟⣿⣿⣿⣿⡆
⠀⠀⢰⡇⠈⢧⠀⠀⠀⢀⡀⠠⠐⢄⠀⠈⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠘⢦⠀⠀⠀⢀⠠⠀⠀⠈⠀⠀⠐⠆⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⢀⠈⢷⠈⠁⠀⠀⠀⠀⠀⠁⢄⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠳⣄⠀⠟⠁⠃⠀⠀⠀⠀⡾⣴⠐⠀⠈⠛⢿⣿⣿⣿⠀⡀
⢠⣦⢸⣆⠀⣀⠈⣧⢀⡀⢀⡀⢀⡀⢀⠀⢰⡄⠀⠈⢧⡀⠠⠀⠄⠠⠄⠠⠴⣀⠤⠠⠤⠠⠤⠠⠌⢣⡄⠀⠴⠀⠀⠀⠀⠁⠃⠀⠂⠐⠂⠐⠈⠛⣻⣠⠻
⠓⠛⢘⠷⣤⡀⠀⠈⢣⡀⢀⣀⡀⠀⠀⠀⠀⠀⠂⡀⠀⠱⣄⠀⠀⠠⠀⡀⠸⡠⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⢸⠂
⠀⠀⠸⡇⠈⠙⠳⠦⢤⣳⣘⡸⠃⠀⠀⠀⠀⠀⠀⠈⢄⠀⠘⢦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⢦⣀⣀⣠⣤⠴⠶⠛⠉⠁⠀⠀⠀⣼
⠀⠀⠀⣽⠄⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠶⠤⠤⠤⣤⣤⣩⣀⣀⣱⣄⣀⣀⣀⣤⣤⣤⠤⠤⠤⠴⠶⠒⠒⠛⡛⠉⣹⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢹⡄⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⢸⢍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢳⠈⢄⠀⠀⠀⠀⠀⢀⠀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⣰⢃⣤⠖
⠀⠀⠀⠀⠀⢣⠈⢢⠀⠀⠀⢀⠠⠓⡉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠋⠻
⠀⠀⠀⠀⠀⠀⠣⡀⢁⡠⠒⠁⠀⠀⠈⢌⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⢠⣆⣤⠘⢄⠀⢂⡀⠀⠀⠀⠀⠡⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠺⠁⠀⠀⠈⠳⣄⠙⢄⠀⠀⠀⠀⠱⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡀⠀⠀⠀⠀⠀⢥⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠓⣄⠀⠀⠀⠀⡀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⡔⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢑⠤⠀⠀⠘⡀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕⠛⢸⣉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠶⢄⣁⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⠤⠒⠋⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠠⠤⠤⠤⠶⠤⣤⣄⡐⡂⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀⠀⢀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡚⠁⠉⠈⢱⠤⡀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠌⠀⠀⠀⠠⢫⠐⢌⠐⢄⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠂⠀⠀⠀⡠⠁⣀⣄⠀⠈⠢⣈⠢⡀⠀⠀⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡡⠁⠀⠀⠀⡔⠀⠀⠙⠛⡘⠁⢩⠋⠁⠀⠑⠄⣀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠑⢤⡴⠊⠁⠐⠂⠠⠌⡀⠀⠀⠀⡐⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢅⡀⢀⠄⠔⠓⠒⠛⠭⢥⣒⠠⠈⠂⢄⡜⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⡊⠀⠀⠀⠈⢃⠀⠈⠆⠀⠀⠀⠀⠈⠙⠦⣧⡀⠁⢠⠃
⠀⠀⠀⠀⠀⡔⠁⠀⠈⠐⠤⡀⠀⠀⠢⡀⠈⠆⠄⠀⠀⢀⡠⠒⠈⢳⡜⠃
⠀⠀⠀⠀⠀⠀⠑⠠⣀⠀⠀⠈⠐⠄⡀⠈⠂⢄⡖⣤⠒⠁⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⢈⡰⠔⠈⠐⠉⠛⠁⠒⠤⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠏⢹⡀
⠀⠀⠀⠀⠸⡟⠁⠓⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠁⠀⠈⠓⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠰⠗⠢⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠦⠎⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠶⠚⠶⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡀⢠⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡇⠀⠀⠀⠀⠛⠉⠉⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠴⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠶⠤⠤⠤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⣨⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠛⠳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠀⠀⢀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⢀⣀⡤⠴⠒⠉⠉⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⢀⡠⠔⠚⠉⠀⠀⠀⣀⣀⡠⠤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣠⠔⠊⠁⠀⢀⣠⠤⠖⠚⠉⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡠⠖⠉⠀⣀⠤⠖⠋⢁⣀⡠⠤⠖⠒⠛⠉⢹⠏⢀⣀⣠⡤⠶⠞⠛⠳⣄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠔⠋⢀⡤⠖⠋⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡠⠞⢁⡤⠚⣁⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣼⡇
⠀⠀⠀⠀⠀⠀⢀⠜⡡⢊⡠⢚⡡⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣠⣫⢊⢔⡫⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⢣⡀
⠀⠀⠀⠀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠜⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣀⢠⠟⠁
⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⡬⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙
⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠲⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠋⠛⢿⣯⠁⢸⣾⣷⡶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁⠀⠀⠀⠀⢄⣀⠹⢻⣶⡿⣿⣿⣃⣿⣿⣿⣯⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠀⣤⣤⣀⡘⠛⠛⢛⣽⣤⢾⡿⢿⣿⣿⣿⣿⣿⡻⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣦⡤⢄⠀⠀⣉⡿⠓⠡⣄⣄⠂⣮⠉⣻⣿⠗⣺⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡠⣾⣿⡷⠚⡀⠸⢯⢗⣧⠀⠿⠿⠋⠀⠀⠈⠪⠀⠛⡻⠀⠾⢿⣿⡿⠯⣷⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠳⣹⡟⣛⢈⠦⢄⠄⣠⣞⢿⠀⠀⠀⡀⠀⠀⡀⠠⠄⠄⠀⠀⠈⠉⠉⠑⢊⣿⣧⣄⣀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⣿⣯⣽⢼⣿⣾⣿⢼⣷⣻⣾⡓⣬⠖⠶⢀⡠⣤⠆⡴⠠⠀⠁⠀⠐⠏⠁⠈⢾⡊⣿⣿⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠟⣸⠟⠋⢱⣿⣿⣿⡿⢿⣿⣿⢿⠋⠉⠂⠀⢺⡿⠃⠍⠁⠀⠀⠀⣠⠐⠉⠀⠀⠙⠿⢿⣿⣿⣻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠽⠢⣰⣿⣿⣿⣧⣝⣿⣿⣷⠇⢀⢄⠀⠀⠀⠰⠀⠀⠠⠀⠀⠆⠀⠀⢀⠀⠰⣿⠁⠐⠩⠹⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⡀⠈⣉⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣶⣄⢀⣤⡾⠂⠀⢠⡄⠒⠂⠀⠀⠻⠀⠁⢰⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡧⣼⢲⣽⣶⡿⠸⣿⣿⣟⣿⣯⣼⣇⣠⢤⣆⠉⣻⣷⣿⡟⢡⠋⢴⠇⠀⠀⠀⠀⠐⠂⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⠈⣿⡐⠦⣍⢶⣭⣿⣿⣿⣯⣿⣶⣿⣿⣿⣿⣿⣆⠂⣈⣶⡄⠀⠀⠀⢠⣲⠄⠀⠠⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡘⠀⠇⡿⡸⣯⣶⣿⣿⣿⣿⣿⣟⣸⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⡶⡂⠀⡀⢽⣿⣧⣀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠀⠉⣹⡿⣟⣻⣿⣿⣿⡿⠛⠛⠛⠿⣿⣿⣿⡝⣬⡛⠉⢡⠀⠀⠀⠈⢹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣎⠉⠀⠀⠬⢃⡺⠿⡿⣿⡏⠰⠂⢀⠀⠀⠈⢿⣿⠏⠁⡴⠀⡤⠐⣀⠀⠀⠘⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠤⠄⠸⠽⢇⠶⡿⣧⠀⠀⠀⠀⢠⣀⣿⣟⡀⣀⠀⠀⠀⡀⠀⣰⣼⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣀⠀⠀⠁⠙⡳⠶⠤⡄⢴⣿⣿⣿⣿⣦⡿⣦⣴⣴⣿⣿⣧⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢭⣿⡄⠦⢿⣷⣾⣂⣰⡾⣿⣿⡍⣿⣟⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠪⢯⣍⡹⡛⣶⣿⣛⣵⣶⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠉⠑⠢⢄⠀⠀⠀⠀⠯⠕⠀⠀⠀⢸⠀⠀⡇⠀⠀⢰⠋⠀⠀⠈⠉⠒⠤⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢰⣿⡧
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠉⠑⠒⢄⡀⠀⠀⠉⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢣⠀⠀⡠⡎⠀⠀⡠⠄⣀⡀⠀⠀⠀⠀⠙⢆⠀⠀⣀⠀⠀⠀⠀⢇⠀⠀⢀⣴⣿⣿⢟⡿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠣⣰⢅⠘⢄⡀⠘⠢⣄⠈⠑⢄⡀⠀⠀⠀⢃⠈⠒⠁⠀⠀⠀⠈⢦⢀⣿⣾⡿⣿⣮⣟⢿⣷⣝⠿⣿⣿⣿⣧⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡿⡄⠳⢄⠙⡄⠀⠈⢢⡀⠀⠀⠉⠙⠢⣸⠀⠀⠀⠀⠀⠀⠀⠀⢫⡳⣝⡿⣾⣿⣿⣯⡻⣿⣿⣿⣷⣯⣻⠆
⠀⠀⠀⠀⠀⠀⠀⣇⠈⢢⡀⠉⠁⠀⠀⠀⠉⠢⡀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣟⢿⣶⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡌⠱⡀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣕⢝⢿⣿⣿⣿⣿⣿⣝⢿⣿⡟⠈⢆
⠀⠀⠀⠀⠀⠸⡱⠀⠙⢄⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⣱⠊⠀⠀⠱⡀⠀⠀⠀⠰⠆⠀⠈⠻⣿⣷⣕⣝⣿⣿⣿⣿⣿⣷⠉⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠢⠄⣀⣈⣲⣄⡠⠔⠊⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠻⠿⠯⠪⢟⠛⠉⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢠⠒⡄⠀⠀⠧⠼⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⢀⠇⠀⠀⠀⣤⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⢀⡸
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⢀⣀⣀⣀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠬⣙⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠘⣺⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣉⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣛⣩⣥⠶⡞⠋⢉⣠⣴⠇
⠀⠀⠀⢀⣾⡟⢤⣄⡀⠀⠈⠀⠀⠻⠛⠛⣛⣛⣉⣭⡥⠴⢖⣚⣫⠽⠟⣋⣭⣴⣾⣿⣿⡟
⠀⠀⠀⢿⣿⣧⣀⠈⠉⠉⠉⠙⠋⠉⢉⣉⣉⣭⣤⠴⠶⢛⣋⣭⣴⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠩⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⡿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠉⣠⣴⠞⠋⠁⠀⠀⢀⡤⠊⠀⠀⠄⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⡴⠞⠋⠉⣴⣷⡶⠟⠉⠀⢀⡠⠒⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠺⣿⣿⠋⠁⠀⠀⠀⠀⠘⠉⠃⣠⡴⠞⢉⡤⠞⣁⠄⠀⠀⠠⣷⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⣀⣤⣶⠟⡉⣠⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠾⣿⣿⡋⠀⠐⡿⠏⣠⠞⠁⠀⠀⠂
⠀⠀⠀⠀⠀⢲⣶⣶⣿⣧⣀⠀⠋⠀⠁⠀⢀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⠀⢰⣄⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠘⠋⠉⠻⣿⠀⠀⠰⢿⣿⣿⡄⠀⠀⠀⢀⠠⠀⠀⠀⢾⢿⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⣶⠶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠒⢂⠋⠈⠉⠉⠉⠉⠉⠋⠛⢰⣶⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡐⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠉⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢦⣙⡻⠟⠓⡄⣀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣠⣴⡤⠙⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁⣿⣿⣿⣷⡄⠀⠀⠉⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠟⠛⠻⣄⡀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣡⠀⠀⢈⣉⠉⠁⠹⠄⠀⠀⠀⠀⠁⠂⡀⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠈⠑⠀⠀⠄⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠘⣆⠀⢸⣿⣿⣶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⣀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠸⣆⠈⣿⣿⣿⡆⠀⠈⢳⡄⠀⠀⠰⠆⠀⠈⠂⡀⠀⠀⡠⠀⠀⠀⠀⢦⣠⣄⠀⠀⠈⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠈⢇⢸⣿⣿⣿⡀⠀⠀⠙⠀⠋⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⠀⠈⠋⡀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠉⠒⠄⠀⠉⠹⣿⣿⣷⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠂⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠘⡧⠸⣧⡄⠀⠀⠈⠁⠀⡀⠈⠻⣏⣄⡀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠄⠀⣴⠇⣶⢀⡀⠀⠀⠤⡀⠉⣷⡄⠀⠀⢼
⠀⠀⠀⠀⠀⢾⡇⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⢤⡀⠙⠷⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠛⣡⣌⡏⠀⠀⠀⠀⠀⠈⢻⡀⠀⠲⠆
⠀⠀⠐⠀⡀⠀⠀⠀⠂⠀⠸⠷⠄⠀⠀⠀⡤⠀⡙⣷⡄⡈⢀⡀⣀⠀⡒⣁⢀⡀⣀⢀⠀⠴⠦⠷⠶⠀⠀⠀⠀⠒⠂⠈⠁⠀⠀⢀⣠⠄
⠀⠀⠀⠀⣽⣿⣧⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣤⣤⠤⠀⠀⠒⠂⠀⠚⠻⣫⡅
⠀⠀⠀⠀⠹⣿⣾⡄⠀⠀⠀⠀⠀⢀⡻⠯⣆⠀⠀⠀⠀⠱⢏⡱⢛⢷⣧⣀⠀⠀⠈⠉⠉⠀⠈⠟⢿⣿⣧⠄⠀⠀⠀⠀⠀⢘⣈⣿
⠀⠀⠀⠀⠀⢻⣱⢷⠀⠀⠀⠀⠀⣾⠀⠀⠈⠳⣄⠀⠀⠀⠈⠇⠐⠀⡝⢄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⠀⠀⠀⠀⠀⢠⠘⣏⠇
⠀⠀⠀⠀⠀⠈⣿⡎⣃⡀⣀⠀⢀⡇⠀⠀⠀⠀⠈⠛⢦⠀⠀⠀⡖⡀⠈⠀⢑⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠄⠀⢠⣆⡼⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠁⣀⠁⠀⠀⠔⠀⠀⠀⠀⠉⠳⠄⠃⠀⠀⠄⠁⠀⠀⡿⣆⠀⠀⠀⣿⣿⣿⡄⠀⠀⢲⠌⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠂⠖⢦⣀⠈⠀⠀⢠⡇⠈⢠⡀⠀⣿⣿⣿⠃⠀⠴⠇⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣿⣤⣤⠤⠆⠀⠒⠉⠁⠀⠀⠀⠀⠀⠉⠳⢦⠀⣸⠁⠀⠘⣷⠀⣿⣿⣿⣶⣁⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣽⣳⣦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠈⣨⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⠿⣶⣬⣷⡷⢷⣤⣄⣀⣀⢀⡀⣀⠀⣼⡀⠤⠄⠶⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⣱⢾⣿⣬⣤⠙⢋⣽⢻⣶⡶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣿⠂⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣶⣶⡶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⢯⡙⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⠣⡐⢄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢄⣠⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡸⡧⠀⠚⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠿⠛⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣤⣴⣾⣀⣬⣵⣤⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣵⠿⠋⠁⠀⠙⣍⣋⡭⠀⠈⠙⢧⣨⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢁⡞⠁⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠙⢧⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠓⠀⠀⠀⠀⠀⠀⠉⠑⠦⣤⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠉⠛⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡖⠛⠓⠲⠦⣄⡀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣇⠀⠀⠀⣠⠤⢍⠑⠦⣄⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⣆⠀⢸⡁⠀⢀⡧⣀⠀⠙⠢⢄⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢣⡀⠙⢲⡛⠀⠀⠙⠲⣤⠴⠛⠚⠉⠉⠓⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⢆⠀⠙⢄⠀⣠⠞⠁⠤⠒⠀⠀⠀⠀⠀⢤⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠳⣄⠀⢱⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠳⣄⠈⠳⣏⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⣄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠈⠳⣄⡏⠳⣄⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣹⠚⠉⠉⠑⢦⡙⠢⣀⠀⠀⠀⠀⠀⢀⣏⠈⠳⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣷⣀⠈⠓⢤⡀⠀⣠⠎⠈⠳⡄⠈⠳⡀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠙⠦⠤⠀⣠⣏⣈⣑⣤⣠⠼⠻⢥⡀⠀⠀⠘⢦⠀⠘⢦⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠚⠁⠀⠙⠦⣀⠈⠳⢤⡀⠉⠓⠦⣤⣸⡇⠀⠀⢳⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠉⠓⠦⣀⠀⠀⠀⢀⡤⠶⢷⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠙⠒⠦⣼⠀⠀⠀⡷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣈⡓⠲⠚⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣍⡛⠛⢋⣩⣤⣴⣶⣄⠙⣿⣿⣿⡇
⠀⠀⠀⢠⣶⣿⣿⣿⣷⣦⠄⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣶⣄⡙⢿⣿⣿⣿⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠹⣿⡿⠋⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⡟⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⡿⢁⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⣿⡿⢁⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⠟⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⠋⣴⣿⣿⡟⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣴⣿⣿⡟⢡⣾⣿⣿⠏⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠏⣠⣾⣿⡿⢋⣴⣿⣿⡿⢁⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⡿⠋⣠⣿⣿⣿⠋⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⣾⣿⣿⠏⣠⣾⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⡿⠋⣡⣾⣿⣿⠟⢡⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣴⣿⣿⡿⠋⣠⣾⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠄⢻⣿⣿⣿⣿⣿⣿⠿⢋⣠⣾⣿⣿⠟⢋⣴⣿⣿⣿⠟⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢰⣆⠹⣿⡿⠟⢋⣠⣶⣿⣿⡿⠛⣡⣶⣿⣿⡿⠋⣡⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⣿⣷⣤⣴⣾⣿⣿⠿⠛⣡⣴⣿⣿⣿⠟⢋⣴⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠖⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⢿⣿⣿⣿⠿⠟⢋⣡⣴⣿⣿⣿⠿⢋⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⣿⣿⡿⠟⠋⠡⠶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣔⡒⠒⠒⠀⠀⠒⠒⠒⠂⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⢋⣉⣭⣵⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⢿⣿⣷⣄⡀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠋⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⠿⠿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣶⣤⣄⣀⣀⣀⣤⣤⣤⣄⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠿⠛⠛⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠸⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣶⣦⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⡶⠶⠇
⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠋⠉⠙⠛⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠟⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡶⣦⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⠶⠶⠿⠿⠿⠿⢿⣿⣿⣶⣶⣶⣶⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⠛⠛⣩⣤⣤⡤⠶⠾⠛⠛⠛⠷⢶⣦⣍⡉⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣀⠙⠿⠿⢶⣦⣤⣤⣤⣴⣶⠿⠟⢋⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠛⠛⠋⠉⠉⠉⠙⠛⠿⢿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⠾⢛⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠻⠿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣉⡙⠛⢛⣛⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⣈⡩⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢡⡁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠁⠀⠤⠀⠂⠀⡀⠀⠀⠐⡄⠀⠀⠀⠀⠉⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⡀⠄⠂⠁⢀⠠⠀⠂⠉⠄⠀⠀⢠⠜⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⠄⠊⠀⡀⠔⠈⢀⠄⠀⢁⠠⠐⠁⠠⠆⠀⠂⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠔⠁⠀⡠⠊⠀⡠⠊⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⠁⠀⡠⠊⠀⡠⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠊⠀⢀⠈⠀⢀⠄⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠠⠁⠀⠠⠂⠀⢀⠂⠀⢀⠊
⠀⠀⠀⠀⠀⠰⠁⠀⠠⠁⠀⠠⠁⠀⢀⠂⠀⢀⠂
⠀⠀⠀⠀⢀⠃⠀⢀⠃⠀⠀⠂⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⢀⠁⠀⠰⠀⠀⠀⠁⠀⠀⠆⠀⠀⡄
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡾⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣏⠝⣟⣿⡟⠛⠙⠿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣬⣭⣍⣻⠿⣿⣿⣿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣝⠙⣿⡿⠁
⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡿⠏⠾⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠐⠒⠒⠒⠒⠒⠒⠂⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣠⠶⠣⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠃⣄⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠸⡄⠈⠳⣄⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⢀⣠⣴⣶⠿⠛⠋⠉⠀⠀⠀⠀⠈⠢⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢳⡀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠈⠑⣦⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠳⡀⠀⠀⠈⠢⡀⠀⠀⠀⢀⣤⡾⠟⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡇
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠈⢢⣠⣾⠟⠋⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⣴⡿⠛⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠡
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⡯⡤⠤⠤⠬⢦⡤⠤⠤⠤⠤⠤⠤⠤⠤⡵⣤⣠⣀⣀⣀⣀⣀⣼⡟⠁⠀⠄
⠀⠀⠀⠀⢀⣴⠴⠒⠒⠋⠉⠉⠉⠀⢀⣴⡿⠋⠈⠳⣄⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣠⣿⠋⠉⠉⠙⠓⠒⠦⣄
⠀⠀⠀⠀⠘⢶⣤⣀⣀⡀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣾⠟⠁⠀⠀⠀⢀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠉⢉⣿⠟⠓⠒⠒⠒⠶⠦⠤⠤⠼⠷⣤⠤⠤⠤⠤⠴⢦⠤⠤⠤⠤⠴⣶⣾⠟⠛⣖⠒⠋⠉⠉⠩⠁
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠑⢄⠀⣠⣾⠟⠁⠀⠀⠈⢦⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⣀⣴⡿⢏⠁⠀⠀⠀⠀⠀⠀⢣⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠱⡀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡾⣟⠉⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠉⠀⠀⠙⠢⣄⡀⠀⠑⢄⠀⠀⠀⢠⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡑⣀⡔⢁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣶⣶⣶⡶⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠰⠤⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡠⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢀⡀
⠀⠀⠀⠀⠀⢀⠚⠀⠀⡉⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠀⠉⣂
⠀⠀⠀⠀⠀⢘⠀⠀⠙⠇⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠁⢨
⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠈⣀⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⣀⣤⡤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢
⠀⠀⠀⠀⠀⠀⡔⠊⠁⠈⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠃⠈⠁⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠭⠀⠉⠲⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡤⠀⣀⣀⢸
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣈⡂⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠁⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠗⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡠⢀⡂⢀⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀⢠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠐⠽⠞⠀⠀⠀⠀⠱⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠸⢨⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⣇⡀⠀⡀⠄⠀⢨⡉⠆
⠀⠀⠀⠀⠀⠣⡳⡀⠀⠀⣰⣭⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠚⠉⠒⠈⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠄⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠢⢄⠘⠚
⠀⠀⠀⠀⠀⢀⠃⠀⢀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡋⠀⠀⠀⠀⡸⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠒⢄⣠⠒⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠘⠢⠀⠄⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠓⠎⣀⣀⡱⠚⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡰⠊⠁⠀⠈⠓⢄⡄
⠀⠀⠀⠀⠠⣲⣇⡀⠀⠀⠀⣀⣸⣖⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣰⣀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠱⡉⠈⠉⠉⢁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠑⠦⢤⢜⣢
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠁