Class Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠉⢻⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠛⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣟⢹⡇⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡈⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢃⠀⠀⠀⠙⠛⠛⠛⣿⣿⡿⠛⠛⠛⠛⠛⡃⠃⠀⠈⣼⠃⡀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢁⠀⠀⠀⠀⠔⣈⠗⠒⠒⠲⠀⠉⠉⠙⠋⠉⣩⣿⣿⣏⡈⣠⠌⠉⣀⣯⣠⠐⡀⠂⠀⠀⠀⠃⠀⠀⡸⡇⠁
⠄⠀⠀⢀⣭⣤⣤⣄⠀⠀⠀⠀⠀⠂⢤⠀⠀⣳⢞⠁⡁⢈⠀⢀⠀⣀⣀⣀⡀⠘⠛⠛⣫⣏⣸⠁⠀⠀⡉⣉⣼⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡶⣤⠁⣺⣿⣿⣿⣿⡀⠂⢀⠀⠀⡉⠀⠀⠀⠛⠚⢉⡁⠀⢁⡤⠥⠌⠉⠀⠀⠠⡅⠀⠠⠿⠓⡀⠀⠘⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠸⠇⠀⣾⣿⣿⣿⣿⠇⡆⠀⠀⠀⠀⠀⠀⠀⢠⠔⣄⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⢄⠀⠄⠁⠀⠀⠂⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠤⠀⣘⢉⢿⣛⠻⢟⠈⠁⠄⠀⠀⠀⠀⠠⠀⠀⠀⠈⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣴⣴⣿⣿⡿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⡿⣿⠁⠈⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⢿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠀⠴⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⢀⠠⢀⣀⣀⣀⣀⣤⣀⣀⣠⣤⡤⢤⣤⣤⡤⠀⠀⢀⣀⡀⡽⠿⠛⡇⢀⣉⣀⣀⡀⡇⠀⠀⢀⡾⠃
⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡡⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⡵⢀⠞⠉⠀⠀⣷⣤⡀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠖⠃⢀
⠂⠀⠀⠀⣴⣷⣶⣶⡄⠀⠂⠀⠀⠀⠀⠈⠀⠀⢨⠂⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠉⠉⠉⠎⣠⠜⠀⠀⢀⡀⠆⡿⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⢚⣤⠀⢸⣿⡏⣿⣿⣷⠀⠁⠤⠀⠀⠄⠀⠀⠀⠈⠁⢀⠤⠤⠄⠤⠤⠄⠀⡀⠀⠀⠐⠤⠀⠀⠐⡕⠙⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠠⠗⠀⣼⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⠀⢀⠀⠀⢀⡔⠀⣄⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠲⠐⠀⠂⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠤⠀⢰⢁⡽⠟⠻⠿⡉⠉⠱⠄⠀⠀⠠⠀⠀⠄⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣈⡀
⣿⣿⡿⠿⣿⠿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣶⣶⣷⣶⣷⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠐⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠏⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠢⣀⡤⠓⠀⢠⠂⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢠⡤⠤
⠀⠀⢠⠂⠈⠉⠑⠿⠒⢤⠴⠃⠀⠀⡰⠋⠈⠑⠣⣀⠀⡀⢀⠀⠀⠋⠂⢀⠀⢀⣁⠀⠿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢⡄⠘⠁⠀⠀⠀⠀⠜⠀⠀⠀⠓⠀⠀⠀⠌⠀⡔⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡀⠀⢀⣀⣀⣀⡀⠀⠀⢠⢣⡶⢤⡶⢤⣄⠰⢊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠁⢨⡇⠀⢸⠀⠀⠈⢿⡿⠋⠛⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠆⢀⠤⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣄⠀⠀⠀⢷⠄⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⢀⣀⣿⣳⣦⣤⣀⣾⣽⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⢘
⠤⠤⠤⠤⠤⠤⠤⠼⠃⠈⠛⢿⣙⣟⣍⣝⣾⣿⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⢸⠀⠀⠀⠀⢀
⠀⠀⠀⢠⠀⠀⡀⠀⢀⣀⠤⠚⠛⠛⠯⣽⣿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⠉⡏⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⠠⡀⠀⠀⠀⠱⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠎⠀⠣⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⢤⠆⠀⠀⠀⠀⢀⡄⣠⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⢠⠀⠰⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠦⡄⠀⣀⠀⠈⣀
⠀⠀⠀⠀⠀⠈⠉⠁⢀⠀⠀⠀⠀⠀⠖⠙⠀⠀⠀⠐⢚⠂⠄⠀⠀⠠⠉⠂⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠐⡎⠈⠀⠀⠁
⠀⠀⠘⣾⠀⠁⢀⣤⡋⠀⠁⠀⠀⠀⠈⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⡂⠐⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠙⢁⡀⢀⠁⠀⣤⠶
⠘⠃⠚⠋⠀⠠⣄⣤⠄⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠁⠀⠀⠀⠀⢠⣥⣄⣦⡖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠈⠁⠀⣤
⠀⠐⠀⠀⠀⠀⠀⠐⡌⠀⠃⠀⠀⠀⠊⠁⠈⠙⢃⠀⠀⠂⠀⠀⠐⠃⢠⡏⠁⢠⠀⠠⠀⠀⠀⠐⠀⠉⠸⠉⠁⠀⠀⠠⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠐⠢⡄
⠀⠀⠀⣤⠤⠄⠄⢲⠀⠀⠀⠀⠀⠀⡔⠆⣠⠀⠀⢀⠖⠀⠀⠠⠀⠐⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⠀⠉⡀⠀⠀⠀⠀⠁
⠀⢀⠀⣧⡀⠀⠐⠲⠤⠄⡀⠀⠀⢄⢠⠄⠃⠈⢲⠀⠀⠀⠀⠀⠠⠴⠄⠓⠊⠸⡁⠃⠀⠀⠀⠀⠁⠛⠔⠓⠂
⠄⠀⠰⠋⠉⡗⠠⡄⠀⠒⠀⠀⠀⠈⠁⠀⠈⠤⡎⠀⢫⡶⠀⠘⠀⠙⠃⠀⠀⠘⠃⡤⠁⠀⠀⣷⠀⡷⠀⠀⠀⢠⡤⢴⡆
⠈⠃⠀⠀⠠⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢈⣀⣈⠇⠀⠀⠰⠄⠀⣀⠀⢄⣉⣱⡄⠀⠀⠀⠀⠀⠀⢰⣳⠀⠀⠀⣽⢷⠈⠙⣄

⠈⠀⠚⠀⡀⠉⠚⠐⠀⠇⠀⠀⠀⠸⠉⠀⠃⠒⠀⠀⠗⠀⠂⠀⠋⠒⠂⠀⠐⠂⠀⠀⠀⠀⠀⡴⢾⢹⠌⠭⠉⠏⠘⡀⠀⡋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣴⣦⣤⣤⣤⣤
⣿⣭⣏⣹⣍⣯⣥⣭⣹⣯⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣯⣵⣿⣽⣿⣯⣭⣽⣭⣭⣭⣭⣭⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⠉⠐⠒⠢⠄
⠀⠀⠀⠀⠀⠀⣀⢀⡀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠀⠀⠠⢀⠀⠀⡀⠈⠀⡠⠤⠤⠒⠒⠢⢄⠀⠈⠤⡀
⠀⠀⠀⡠⠂⢁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈
⠀⠀⠈⢀⠔⠈⠉⢢⠀⠀⠀⠀⠀⠀⠀⢣⠀⡐⢁⠀⣀⣀⣠⣀⣀⡀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰
⠀⠠⠔⠁⠀⠀⠀⠀⠣⠠⠤⢀⡀⠀⠀⢸⢐⠀⡈⠉⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡠
⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠄⠀⠈⠑⢤⡜⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢰⢓⠅
⠀⠀⠈⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠠⠀⠘⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢪⠦⣀⠀⠀⠄⢐⠈⠀⣀⠔⠁
⠘⠣⢄⣀⡑⠄⣈⡐⠒⠒⣁⡠⠔⠁⡠⠶⠆⠢⠠⠼⠤⡀⠀⠁⠘⢢⣤⣀⣀⣀⡠⠊⠀⠈⠙⢒⠶⠒⠶⣋⠁
⠀⠀⠀⠠⢠⠜⠁⠉⠉⠘⠉⢦⡈⡉⠀⢀⡦⠤⠤⡄⠀⠋⠉⠉⠋⠟⠐⢤⠤⠄⠒⡣⠤⠎⠀⠁⠐⠄⠀⠈⢆⡑⠒⠢⢤
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡒⠁⠀⠘⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⢀⡠⠬⢁⡀⠂⠠⣤⠤⠤⠄⠤⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢇⡀⠀⠀⠄⠠⠤⠤⠤⢼⣀⡀
⠀⢀⣳⡀⠀⠀⠉⠈⠈⢂⡔⠐⠒⠂⠀⠀⠀⠚⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⢀⡢⠀⠀⠀⢰⣀⣀⣀⣀⣦⠤⠔⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡖⠒⠊⠉⠈⠳⣄⡴⠀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⠀⠀⠀⠈⠁⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠈⢉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠫⡉⢢⡉⠢⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡄⢄⠀⡄⠲⡈⠣⠑⢲⠤⡄⠀⣼⠖⠀⠀⠀⡠⢺⠋⢍⠒⠀⠀⠁⠀⠁⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣸⢸⡀⠣⠘⠀⠁⠀⠄⠈⣏⣥⠾⠃⠀⠀⠀⠀⡔⡏⢀⡀⠀⠀⠀⠀⡀⠀⠀⢸⡱
⠀⠀⠀⠀⠀⠀⠀⢠⠿⡀⠀⠂⠀⠀⠀⢀⡄⠀⡏⢀⣀⠀⠀⠀⠀⠀⠈⢇⠀⠁⡀⠀⠀⢀⡁⠀⠀⡬
⠀⠀⠀⠀⠘⣷⣀⣈⡽⠳⡀⠀⠒⠒⠒⠉⢀⡴⠡⡋⠈⠃⢀⠐⠠⡆⠀⠈⠢⡀⠙⠂⠐⠊⠀⢀⠔⠁⢠⡤⢤
⠀⠀⠀⠀⠀⣈⠉⠁⠀⠀⠈⢓⢶⡴⠦⣚⣉⣠⠔⠉⠁⠀⠈⠓⠚⠳⢄⣠⢤⡌⠓⡴⢤⣤⠖⠥⣤⠴⠚⢦⡼
⠀⠀⠀⠀⠸⣉⡹⠒⠒⠾⡉⠉⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢤⠀⠀⠀⠀⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣀⣀⣀⠠⢧
⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⠀⠀⠀⠀⠀⢀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⡀⠀⢀⡱⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠒⠒⠒⠲⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠒⠃⠑⠘⡀⣠⣻⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠦⠠⠃
⠀⠀⠀⠀⠀⠀⠰⢏⣈⡹⠃⠀⠰⣛⣃⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠁





⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠻⠛⠲⠧⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠄⢠⠀⠠⠄⡠
⠀⠉⠀⠔⢀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢫⣿⠓⡖⠀⢀⠔⠂⠉⡀⠀⢠⠀⠀⢀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠘⠲⠖⠀⠀⠀⢀⠇⠀⢸⠤⡰⢃⣅⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡏⠀⠀⠀⠀⠁⠃⠌⠏⠙⠞⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢙⡀⠀⠀⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢾⡔⢷⣄⣸⠇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢠⠀⠀⠀⠀⠀⢄⠀⣰⠂
⠀⠈⠀⠀⠀⠀⠀⠀⢾⡆⢿⣤⡼⣧⠀⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡀⠀⠀⡆⢀⠀⢀⠀⢲⠃
⠀⠀⠀⠐⢶⡀⢠⣤⣸⡇⠸⣏⠀⠹⡆⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⣷⡼⢦⡇⠸⠄⢿⠀⢸⡄
⠀⠁⠀⠀⠀⠳⡼⠉⠻⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠶⠀⢸⠃⠈⠃⠀⠀⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠐⡀
⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⠀⠄⠂⠀⠘⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠀⢈⡀
⠀⠀⠀⢤⢠⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠠⠀⠀⠰⣤⠀⣀⠀⣾⡄⣶⡟⠙⡆
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡆⠰⠂⠴⠄⠁⠀⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠙⣷⡿⣶⠃⠏⠻⠹⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣠⡄⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⣀⢄⡄
⠀⠀⠀⠀⢈⡟⠓⠙⠛⢷⣷⠁⠀⡰⠋⡹⠟⠟⠛⠳⢽⠯⡁⢱⠀⢀⡤⠞⣫⢶⣤⡘⠙⣶⢶⣶⢿⢿⣿⢿⣧⣤⣀
⠀⠀⠀⢀⡎⠀⠀⠈⠁⠀⢸⡀⠀⡇⣼⠁⡆⠀⠀⠀⡀⣇⢇⠇⠘⡏⢠⠐⠡⠀⠉⢻⢺⣿⡗⢁⠀⠀⠋⡀⢿⣿⣿
⠀⠀⠀⠙⢇⠘⢄⣀⡤⠂⡼⠋⢠⡷⠉⢆⠈⠒⠒⠊⣠⢿⠘⡄⠀⢣⡘⢤⠤⠂⢀⡼⠂⠛⣇⠘⠤⣀⠴⠁⡎⠹⡍
⠀⠀⠀⠀⢀⣱⠦⠤⢴⠊⠀⠉⠉⠀⠀⣀⣵⠒⠒⢎⠁⠀⠣⠝⢀⣠⡼⠒⢲⣒⡋⠀⢰⠷⠈⣲⠤⠤⢔⠊
⠀⠀⣀⢴⠉⡜⢦⡔⢺⢱⡄⠀⠰⣶⠊⢡⠟⢼⡀⢿⠣⡠⡦⠐⣇⡴⠁⠀⢸⣀⡟⠊⣁⣀⠼⡧⢖⠦⡜⢄⡀
⠘⠛⠀⢘⡟⠀⠀⠀⠈⡏⠈⢢⡄⠀⢠⠋⠠⠀⠀⠸⡀⠹⠖⠠⡎⠀⠀⠀⠸⠀⠁⠀⠟⣠⠊⠀⠀⠀⢰⡀⠸⠓
⠀⠀⠀⠀⣱⠲⠄⠤⢤⠃⠀⠈⠀⠀⠀⡽⠒⡶⡄⢴⠓⠀⠀⠀⡜⠉⠀⠀⢸⠁⠀⠀⠀⠙⠿⢖⠶⣲⡿⠵
⠀⠀⠀⣠⠃⠀⠔⡄⠘⡄⠀⠀⠀⠀⢠⠃⡸⠀⢧⠘⡄⠀⠀⠘⢦⠴⠓⠤⡜⠂⠀⠀⠀⠀⡤⣇⠀⠀⣣⢄
⠀⣀⠼⠤⣀⡞⠀⣟⣀⣸⡀⠀⠠⣊⣉⡻⠁⠀⠸⡟⠓⣄⣠⣒⣳⠀⠀⠀⣱⠒⣢⠀⢤⣚⣙⠇⠀⠀⠙⣉⣲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠶⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠛⠳⡟⠀⠀⠀⠀⠘⣆⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡄⠀⠤⠬⠠⡀⠀⠀⢠⠿⠚⠛⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠁
⠀⠀⠀⢀⣴⣄⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁
⠀⠀⣼⢾⡁⡉⠉⠁⠀⠀⠔⠉⡁⠀⠀⠀⢀⠉⠀⠀⠀⠀⠉⠒⢄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣿⠀⠀⠀⠈⠙⠁⠀⠀⠀⠁⢠⠀⠀⠈⠁⠀⠀⠸⡄⠀⠀⢀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠸⣆⠀⠀⠀⠈⢃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⢡⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠐⠕
⠀⠀⠀⠈⠳⠶⢶⣀⠈⠑⢄⡀⠀⠒⠒⠒⠊⠁⣠⢴⣥⠴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠁⠄⡀⢀⠠⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡇⠐⠀⣿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⠇⠀⠀⠹⠶⣤⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠈⠀⢀⠀⠀⠀⠀⢱⠀⠙⠦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⢛⡔
⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡆⠀⠀⠼⠧⣤⣀⣀⣤⡤⠶⠖⠛⠉⡀⠈⠀⠁⠱⡄
⠀⠀⣠⡾⠋⠀⠢⠀⡀⠀⠀⠄⠀⠀⠈⠠⠁⠀⠀⠹⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⣀⡤⠖⠃
⢀⡴⠋⠀⠀⢀⡴⠋⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢦⣄⣀⣀⣠⣤⠔⠀⠀⠀⠉⠋⠉⠁
⣿⠁⠀⠀⡴⠋⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠈⠉⠉⠩
⠘⣧⠀⠀⠹⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠀⠂⠀⠀⢠⡔⠋⠉⠉⠉⠉⠳⡄
⠀⠈⢷⡀⠀⠘⢦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠹⣦⠀⠈⢧⡤⠶⠧⢀⠀⠀⠀⠀⠀⠐⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⣀⣀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠹⣤⠤⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣾⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⢀⣀⣤⣶⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⡿⠛⠛⠛⠛⠉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⢹⣿⣶⣤⡀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡏⠙⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠹⠿⣿⣿⣿⡿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠈⠻⠟⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣼⣿⣤⣿⣤⣿⣿⣿⠇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠛⠋⠉⠉⠉⠙⠷⢤⣤⣤⣤⣤⣐⣞⠁⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⠉⠙⠛⢿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⡿⠛⠋⠉⣿⣿⣿⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠛⠛⢻⡏⠉⠙⣿⣿⡀⠀⢸⡇⠀⢀⣿⣿⠏⠉⢹⡟⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣿⣷⡀⢸⡇⢀⣾⣿⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣿⣧⢸⡇⣼⣿⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠈⠁⠈⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣵⣶⣯⣶⣶⣤⣤⣀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢠⣄⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢿⢿⣶⠄⠐⠒⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠈⢀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠸⢇⠈⠁⠀⠐⠒⢲⠿⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⠨⠅⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣆
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢆
⠀⠀⠀⢠⡏⣀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢳⠄
⠀⠀⠀⠀⢹⠟⠛⠛⠶⣦⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡶⢶⠊⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⡸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠎⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠈⠀⣐⡀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⢼⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠨⠆⠀⠀⠄⠀⠀⠀⠀⠀⠏⠀⠸⠄
⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠈⠒⠂⠤⣤⠄⠘⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣀⣀⣀⠀⠀⠀⠀⠈⠐⠂⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⡀⠀⠀⠀⠘⠓⠒⣂⣉⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠀⣠⡩⠏⢌⠙⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢦⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠳⢶⠋⠀⠀⠀⠀⠹⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣀⣉⣉⣑⣤⠀⠀⠀⢠⡷⠾⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠑
⠀⠀⠀⠀⣠⡆⠀⢠⡟⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠢⢤⡊⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄
⠀⠀⢰⣴⡏⡛⠛⠋⠀⠀⢀⠜⠓⠀⠀⠀⠀⠀⠓⠀⠀⠁⠉⠒⠦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈
⠀⠀⣿⠈⠁⠀⠐⡖⡆⠀⠀⠀⠘⠀⡀⠀⠀⠚⠀⠀⠀⢳⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⡆⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠁⢂⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠖⠀⠀⠀⡀⠅⠀⠐
⠀⠀⠀⠹⢦⣤⣠⡤⠐⠳⢄⠀⠀⠠⢄⣠⡤⠔⠀⢀⣠⣂⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠰⠱⢀⡀⠀⡠⢈⠔
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠾⠗⠂⣤⠤⠤⠤⣤⠒⠉⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠰⠴⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣧⣀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⡟⠋⠀⠀⠀⢨⠉⡏⠻⢦⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⡄⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠃⠀⢀⠀⠀⡀⠀⢹⡀⠀⠉⣻⣆⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠀⠐⠳⠺⣄
⠀⠀⠀⢀⡴⠟⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠈⣇⠀⡀⠉⠉⠛⠲⠖⠚⠛⠉⠉⠀⡠⠐⠀⠀⠀⠀⣸
⠀⢀⡴⠋⠀⠀⠀⣠⡤⣄⠀⡈⠀⠀⠀⠀⠃⠀⠀⠀⢸⡶⣤⡀⠀⠀⠀⠀⠀⠀⡀⠐⠑⠳⣄⣤⠴⠒⠋⠁
⣠⠞⠁⠀⢀⣠⠞⠁⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢹⠶⡦⠴⢶⠞⠉⠂⠀⡀
⢻⡀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢸⠀⠀⠀⠈⠁⠀⣀⣀⣤⣄⣀⣀
⠀⢻⡄⠀⠈⢷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⣄⣉⣉⣉⣡⠏⠁⠀⠀⠀⠀⠀⠹⣌⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀
⠀⠀⠹⣦⠀⠀⠹⣄⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠳⣄⠀⠙⠶⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣀⣀⣙⣦⣀⣠⣤⣀⣤⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠧⢤⡄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⣀⣀⣀⣀⣀⣀⣸⣣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⣿⠿⡏⡎⡇⡿⢙⠼⠏⠉⢯⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠤⠠⠤⠄⠠⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⠶⠤⢤⠀⠀⣀⣀⣀⣀⣀⡼⠈⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠟⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠿⢿⣿⣿⠿⠟⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠀⠄⢤⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⠉⠉⠉⢹⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⡀⠀⣸⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣝⣦⣞⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣤⣶⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⢠⠠
⠀⠀⠀⠀⠀⢸⣿⣆⡀⢿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣼⣟⣻⠿⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠙⠁⢀⣀⠀⠀⠀⠰⠂⠀⠀⠀⠀⡀⠁⠀⠀⠀⢸⠆⠀⠶⠀⠰⠆⠀⣤⠀⠀⠀⠀⠂⠀⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⣀⡀⡀⠀⢀⠐⠀⠀⠀⠀⢀⣴⣴⣿⠶⠠⠎⠀⠀⠈⢓⢂⡀⠀⠔⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠐⠺⠉⣇⠀⢀⠈⡄⠀⠰⠒⠀⡠⠙⣿⡷⢄⠀⢠⣀⣀⣱⠌⠙⢱⣀⠀⠀⠀⠉
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⡷⢤⣀⡀⠀⡿⠠⠀⠀⡿⠓⠀⡤⠀⠀⠿⠻⡁⠐⢈⢿⣿⣿⣿⡓⣀⢛⣧⣤⣷⡦⢀⡀
⠀⠀⠐⠀⢹⠀⠀⠀⢂⣸⡇⠀⠈⠉⠉⠄⠬⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⡑⠠⡄⠀⢠⡟⣿⣿⣿⣿⣿⣿⡭⠏⢈⠃
⠀⠀⣄⠀⢾⡤⠀⠘⢽⣿⣿⡔⢶⠶⠶⠲⠲⠶⠒⠒⠶⠖⣶⣾⣿⠻⢾⣶⣶⠿⠿⢿⣷⣶⣾⠟⠙⠛⡟⠡⢠⠀⡀
⠀⠀⡄⠀⣸⠟⠁⠀⢚⢹⢿⣿⠎⠉⠀⠂⠀⠀⠀⠀⣠⣾⡿⠛⢿⡶⠒⢲⡖⠲⣶⡿⠿⣿⠶⠖⠀⣈⡽⢶⡰⠋
⠀⠀⢀⠀⣹⡣⣐⠨⠐⣺⣿⣿⡇⠀⠐⠀⠀⢀⣴⣿⣿⣿⡇⠀⠈⠋⡀⡈⢀⠀⠄⠁⠀⠀⠀⠀⠀⠉⠸⠎
⠠⣀⣂⣽⣿⣾⢳⣾⣽⣾⣿⣿⣏⣀⣀⣠⣴⣿⣿⣿⣿⣿⠆⠐⠐⣄⢩⢀⠀⢀⢀⠀⣀⠀⠄⠀⠰⠶⠆⠄
⠀⢻⣿⣿⣿⡟⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣐⡀⠜⡄⢰⡌⠀⠦⠡⠼⠐⣤⣤⡴
⠀⢸⣿⢿⣿⣵⣿⣟⣯⠜⣿⣿⢿⣿⣿⡿⠛⠉⠐⢿⡿⣿⠓⠒⠒⠒⠒⠀⠀⠐⠂⠀⠀⠀⠰⠿⠋
⠀⢸⡇⠀⣿⢹⡏⡟⢻⠿⣿⡇⠀⢸⡏⠈⠀⠀⠀⠈⡟⡛
⠀⠈⠁⠀⠋⠈⠀⠂⠈⢲⣿⣏⠀⠀⠁⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠈⡁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣾⣿⣧⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠠⣿⣿⣿⡆⠀⠀⢀⣤⣌⠛⢦⣬⣿⣿⣥⡤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⢀⣠⠦
⠀⠀⠲⣄⣙⣿⣿⠏⠀⠀⠺⣿⡿⠃⠀⢨⣿⣿⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠑⠲⢦⣭⣿⣿⡶⠋
⠀⠀⠀⠀⠉⢻⣿⣿⠆⢀⣾⣿⣿⣄⠀⣸⣿⣿⡇⠀⠀⠀⠀⣹⣿⣯⣤⠦⠀⠀⠀⢨⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣿⣿⡏⠻⠆⣹⡿⣿⡇⠀⠀⠀⣰⡿⣿⣷⡀⠀⠀⠀⢀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⠉⣿⠰⣿⠉⣿⡇⠀⠀⣿⣇⣾⡇⠀⠀⠀⠩⣾⠿⠿⢿⡆⠀⠀⠀⠈⡿⢸⡇
⠀⠀⠀⠐⣿⣿⢀⣿⠀⠀⠀⣽⡁⠀⠀⠿⠇⣿⡷⠀⠀⠀⠀⠙⠀⠀⠘⠁⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⣼⡇⠀⣿⠀⠀⠀⢸⠁⠀⠀⢠⡄⢸⡦⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡉⠘⡇
⠀⠀⠀⠀⠈⣿⣠⣿⠀⣯⣤⣿⡇⢀⠀⢿⡇⣹⡇⠀⠀⠀⢠⡡⡶⣶⠞⠁⠀⠀⠀⣀⣷⣸⣄⡀
⠀⠀⠀⠀⢀⢹⣿⣷⠄⠘⠿⣿⡷⠛⠀⢨⣻⣿⠇⠀⠀⠀⠀⢛⣿⡆⠠⠄⠀⠀⠀⠘⣿⣿⠋
⠀⠀⠀⠀⣡⣾⣿⡆⠀⠀⠰⠿⡷⠀⣀⣈⣿⣿⣀⣀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⢭⣿⣿⠓⠢⢀
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠐⠊⠀⠰⢾⠿⡂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠹⡆⠘⢦⡴⠃⢰⠏⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠘⠶⠶⠶⠶⠃⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⡀⢈⡷⡶⠖⠛⢷⣄⠀⠀⡀⣀⡀⠀⣠⡾⠙⠲⢶⠾⡁⢀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡨⠟⠁⠀⠀⠀⢀⣀⣨⣟⣉⠉⠉⣉⣻⣅⣀⡀⠀⠀⠀⠈⠻⢄⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⣠⠞⢳⡶⠋⣀⣀⣰⢯⡛⢞⡽⣆⣀⡀⠙⢶⡞⠳⣄⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡚⠙⣧⠞⠁⠀⣾⠀⣰⣏⠙⠛⣷⢄⡠⣾⠛⠋⢹⣆⠀⣷⠀⠈⠳⣼⠋⢓⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⢹⡇⠀⠀⠸⡌⠁⠉⢛⡻⠅⠋⠙⠈⢟⡛⠉⠈⢡⠇⠀⠀⢸⡏⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⢤⡤⠤⣽⠘⢚⡿⠃⢠⡆⢰⡄⠘⢯⡓⠃⣯⠤⢤⡤⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡿⠰⠟⠠⣄⡈⠛⠛⢁⣠⠄⠻⠆⢿⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢀⠀⠀⠙⢛⡛⠋⠀⢀⡀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⡌⢛⣒⣚⣏⣹⣒⣒⡋⢡⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣌⠀⠘⠶⠶⠃⠀⣡⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⠉⠛⠛⠛⠛⠉⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣆⣰⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⡿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠳⣄⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢙⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⡾⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢠⠋⠀⠞⠁
⠀⢀⡤⠄⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⢤⣤⡀⢀⣤⣤⣤⡤⠤⠤⠤⠤⠤⠠⠶⠶⠶⠶⠶⠶⠶⠦⠤⠠⣄⡀⠀⠀⣰⠋⠀⠀⡰⠁
⢰⡇⢀⡀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⣀⡀⠀⢀⣷⣞⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⣤⢤⡤⡤⣤⣀⠀⢳⢀⡼⠁⠀⣠⠎
⢸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⣸⠟⠁⢀⣼⠋⡀
⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢙⣑⡠⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⡶⢿⢛⡷⠃⠀⢠⡞⣡⠾⠁
⢸⡇⠀⣰⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣉⣉⣙⡁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠍⠀⣸⠟⠀⠀⣴⢋⡞⠕⠁
⢸⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠤⠤⠤⠤⠤⣴⠋⠀⢠⣾⠁⠘⠋
⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠒⢒⡾⠁⠀⢠⠟⢽
⢸⡇⢸⢿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣰⠛⠀⢰
⢸⡇⢸⣺⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⡞⠁⠀⢠⠞⢸⡋⠀⢸
⢸⡇⠀⠹⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⢠⠎⠈⠀⡀⠃⠀⠘⠁⠀⣸
⠀⡇⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⡞⠁⠀⠀⣿⣉⠀⣿
⠈⡇⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠔⠛⠁⠀⠀⠀⠀⣿⠇⠀⠛
⠀⡇⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⣿⠃⠀⣶
⠀⡇⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⡇⠀⡧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⠀⢀⣀⣀⡇⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⡿
⠀⢧⡀⠈⠛⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠀⠀⢀⣽⠻⣍⠉⠀⠈⠉⠁⠀⠈⠉⠁⠉⠉⠙⠛⠛⠉⠉⠉⠉⠁⢀⡆
⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠋⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠚⠛⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠐⠀⠐

⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠈⠁⠈⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡠⠀⠀⡀⠀⠀⢱⢤⠒⠀⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠉⠀⠈⠑⠒⠠⠀⣸⠃⠄⠰⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⢷⠈⠂⠃⠊⢈⠡⠃⢀⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡤⢀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠆⢀⠴⡟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱⣤⠕⠐⠠⡀⣀⡀⣀⠀⠠⠛⣦⣨⢆⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠈⠀⠀⠀⠀⠣⠘⢸⠔⠃⠀⠀⠀⠑⠄⠄⠀⢠⠄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠰⡀⠀⢁⡀⠀⠀⠀⡖⠠⠀⠀⠀⠀⣧⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠃⠀⠀⠀⣣
⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⠀⠉⠂⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⡤⠄⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢥⡴⠡⠐⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢢⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠐⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⡠⣠⡃⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢠⡏⣾⣷⣷⡟⠋⠀⢣
⠀⠀⠀⠀⢰⣼⠱⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢈⠉⠉⠉⠉⠁⠀⠀⠈⢇
⠀⠀⠈⠀⠞⠁⠂⣀⣤⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠓⡖⢲⢶⢶⣄⡀⠀⠀⠀⠈⠣⡤⢄
⠀⠀⠀⢠⢦⠐⣎⠹⡉⣿⣿⠦⠤⠖⠃⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠚⠤⠃⡑⡖⠀⠀⡀⢀⠎⢠
⠀⠀⠀⣰⠁⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠑⠒⢯⡜⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠋⠙⠋⠉⣁⣀⡒⣦⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⣂⢈⡐⣈⣀⣨⣄⣤⢰⣂⣀⣠⠀⠘⣉⣀⣀⠀⠀⠲⠐⠒⠠⠀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣀⣤⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⢬⣩⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠷⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠩⠉⠉⠁⠁⠉⠈⠈⠙⠻⢿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⠀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠙⢿⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣄⠀⠹⣿⣿⣷⣔⡀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⠃⠻⠿⢏⣾⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢛⣇⠈⠙⢷⣆⡈⠻⣿⣿⣿⣥⣦⣴⣾⣶⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⣶⣶⣶⣾⣿⣧⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⠛⠶⣬⣙⠛⠻⠻⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⠀⠐⢿⣷⠛⠛⠿⣿⣾⣁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣶⣿⣿⣿⣛⠿⢷⣤⣀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣈⣉⠙⠓⠒⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠻⢟⡭⠈⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⢦⠉⠒⠆⠀⠤⠔⢊⣀⣀⣀⣀⠀⠀⠀⠀⢠⠞⠉⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠉⠢⡀⠀⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠄⠀⢀⠇⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⡄⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠀⠀⠉⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡟⢦⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠈⠑⠋⠉⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠫⠁⠀⣼⣇⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⢸⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡏⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⡄⠸⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠽⡷⠒⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⡉⢆⢺⡒⠶⢌⣪⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢫⠢⣝⣋⣾⣥⡿⠟⠋⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣽⡿⢋⣩⡁⠀⠀⣤⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣾⣿⡆⠈⠉⠀⠀⠀⠉⠁⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠋⠉⣧⠞⢳⡶⠋⡿
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣄⠀⠠⣤⡤⠀⣰⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠃⠀⣰⠏⠀⡾⠁⣸⠃⣤⣶⡄⣠⠶⣦⠴⡆
⠀⠀⠀⠀⠐⢿⣯⣱⣿⣿⣿⣿⣿⣿⣷⡄⠄⣤⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⣼⠁⢰⡇⠸⣁⡾⢳⠏⠰⠃⢸⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣀⣀⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⢦⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠹⣿⣿⣿⣿⣿⢿⣿⣿⠏⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢈⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⠻⢿⣿⣿⣿⡟⠁⠀⣤⠀⠀⠸⡀⠀⠀⠀⠀⣸⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣰⢷⡀⠈⠙⠿⠋⠀⢀⡼⠙⣆⠀⠀⢻⡀⠀⣠⡾⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⢻⠀⠀⠴⠀⠀⢨⡇⠀⠸⡄⠀⢀⣷⡞⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⠤⣾⡀⠀⠰⠂⠀⠈⢧⠀⠀⣹⣶⠟⠉⢳⡄⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠈⠙⠓⢦⡀⣀⢠⣞⣦⡾⠋⢱⡀⠀⠀⠹⣄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢄⣀⡀⠀⡸⠁⠀⢸⡿⠛⣆⠀⠀⢳⡄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠛⣻⣦⣤⠿⠏⠀⠈⢦⡀⠀⠱⡀⠀⠀⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⡾⢿⣿⣿⠀⠀⠀⠀⠀⢹⣤⠀⠙⠶⠞⠁⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣀⣸⣿⣿⣦⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⢀⣀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⠛⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠻⠿⠛⠛⡿⠛⡿⠛⠻⡟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⢰⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⢸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣷⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣾⠇⢸⣿⠟⣿⣌⣳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠈⠛⠀⠙⠻⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣤⣤⣤⣤⣄⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⢞⣷⣹⣮⣽⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠐⠀⢢⠊⠑⡄⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠈⠃⣄
⠀⠀⠀⠀⡄⠒⠁⠀⠀⠀⠀⠀⠧⢤⢤⡙⠢⢄⠀⠀⠀⡠⠊⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡰⠃⢀⠏⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠇
⠀⠀⡸⠉⢹⡀⠒⠂⠀⠀⠂⠐⠂⠇⠀⠈⠁⡀⠀⠀⠙⠤⡤⠄⠀⠄⠒⠂⠊⠉⡦⡀⠀⠀⡆
⠀⢘⠂⠀⠘⠄⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⢀⠋⠑⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠈⣦⡜⢄
⠀⠈⠀⠽⠛⡀⠀⠀⠠⠀⠀⠀⠀⠒⢒⠶⠂⠀⠀⠈⡂⠐⢄⠀⠤⠄⠀⠀⠀⠀⠑⣉⠘⡄⠀⠱⡀
⠀⠀⠀⠀⠀⠈⢒⣤⣀⣀⣀⣤⡄⠚⠡⢠⠶⠀⡠⡘⠀⠀⢀⠡⣶⣤⢤⡤⢠⡶⢥⡒⢄⠘⠦⠴⠁
⠀⠀⠀⠀⠀⢠⠃⠀⡿⠀⡠⠒⢌⠫⡁⠐⢶⠋⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠸⡀⠀⢱⢸⠦⢠⡄
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡆⠀⠀⠄⠈⢂⠀⢂⠀⠀⠀⠀⠠⣰⡇⠀⠀⠀⠀⠙⠎⠉⡈⢆⣤⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⠢⢀⣀⠠⠞⠀⠀⠂⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠠⠃⢀⠁⠀⠈
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⣀⠐⠁⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⣄⣀⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢁⡖⠒⠒⡔⢲⠛⡫⢴⡁⠂⠀⠀⠀⠀⠀⠀⠐⡴⠒⢄⢀⠝⠣⠤⡀
⠀⠀⠀⠀⠀⢀⣤⠪⣀⠀⣰⣁⠎⡰⠄⠀⠈⢆⠀⠔⠀⣀⣀⣀⢎⠀⠀⡘⢣⡔⠁⠀⠈⢱
⠀⠀⠀⢠⠊⠩⣀⡩⠂⠈⠁⠈⠧⠤⠊⠀⠀⡸⠀⠀⠐⠁⠀⠀⠁⠁⢸⠁⠈⠁⠐⠀⠀⡸⠀⢄⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⡰⠀⠀⠈⠦⠄⠤⠐⠁⠀⠀⠰⢄⠀⠀⠀⠀⡌⠀⠀⠢⢀⡠⠞⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠈⠈⠁⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠋⠡⡀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠⣾⣿⡇⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣻⣼⣿⣿⣿⠛⠋⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣬⣿⢿⡿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠐⠀⠀⠉⠁⠀⠐⠠⡀
⠀⠀⠀⠀⡀⠄⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⢀⣀⢀⠀⢀
⠀⠀⠀⢀⠣⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠅
⠀⠀⠀⢀⠀⡅⠈⠁⠀⠈⠀⢣⠀⠠⠤⠤⠤⠤⠤⠤⠄⢀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠤⠄⡀⣎⠐⠾
⠀⠀⠀⠀⢄⠂⠀⠠⡤⡔⠀⢌⡀⠠⠠⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠂⠀⠀⠂⠀⠂⠀⠃⢀⣀⠀⠈⢱⡠
⠀⠀⠀⠀⠀⡑⠠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠐⢉
⠀⠀⠀⣠⠎⢌⠐⠒⠉⢢⡀⠀⠀⠀⠀⠀⠀⣴⠁⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠈⠀⡑⠄
⠀⠀⡐⠁⠀⠀⠄⠀⠀⠀⠏⢂⠀⠀⠀⠠⢞⢫⠛⠁⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠡⠔⠁⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⣳⡆
⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⠐⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠠
⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣎⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠬⣿⣮⣧⡙⣦⠦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣿⣄⠀⠀⡨⠙⠻⢾⣷⣤⣴⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⣠⣴⣿⣿⣿⣗⣋⠀⠀⠀⠀⠈⠙⠛⠷⣤⡀⠈⠳⣄
⠀⠀⠀⢀⣀⣠⣤⣤⡤⠤⠴⠶⠒⠛⠉⠀⠀⠀⠘⣻⠉⠁⠀⠈⠉⠧⠠⣦⣀⡀⠀⠀⠀⠀⠀⠉⠳⣦⡀⡡
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣼⣿⣄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣆⡶⢀⣆⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⢿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⢥⣌⣀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣶⡔⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣙⢿⣦⣀⠀⢀⠆⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⠀⢀⠀⠂⠰⢿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣌⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠠⠈⠉⠛⠿⣿⡏⢳⡄⡀⡜⠀⠘⠛⡿⣦⣟⡔⠀⠀⠀⠀⠀⡿⠛⢿⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀⢽⡆⢰⢰⣷⣾⠀⠀⠀⠈⡻⣷⣿⣴⡀⢬⡁⠀⣋⣿⣿⣷⣶⣶⣴⣥⣄⣈⣿⣧⣄
⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣍⡻⢿⣷⣶⡄⣀⠀⠋⠿⣿⣻⣿⣷⣯⣼⣿⣿⣿⡿⠿⠿⠟⠋⠉⠀⠈⠻⣷⣌
⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣮⣼⣤⣤⣋⣏⣹⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠉⠩⣭⣭⣭⠭⠽⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠛⠛⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠂⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⢢⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡠⠂⠁⢀⠔⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⠀⠤⠀⠈⠀⠀⡰⡄⠀⠀⠀⢀⠊⠀⣀⠖⣠⡄⠐⠈⠤⢂⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡄⠁⠀⠁⡹⠏⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠉⠀⠀⠘⢿⠁⠀⠢⠔⠈⠀⠀⠀⠀⠠⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⢠⣄⠀⠀⠀⢰⣦⠀⡀⠀⠀⠘⠀⡠⠋⠀⢀⠀⠀⢀⣅⠀⠀⠀⢾⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠛⠀⠀⠐⡌⠁⠀⠁⠀⠀⠀⠀⡇⠀⡆⠄⠀⠀⠀⠛⠁⠀⠂⡌⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢀⣠⠄⠁⢀⠀⠀⠀⠀⠀⢡⢣⠀⣸⡄⠀⠀⠀⠀⠀⠄⠄⠀⢀⢜⠎⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⡀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠡⢳⣇⡈⠀⢄⡀⠀⢀⣀⡠⠔⠃⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⢂⢸⣟⠦⠤⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠆⠈⠀⢀⠉⠀⣠⡄⡄⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⠢⡑⠜⠘⠚⠒⠂⠀⠀⠶⠂⠠⠀⠲⠄⠀⠀⠀⠒⠒⢑⠁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠢⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡒⠒⠲⠒⠲⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠀⠀⢺⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠺⠂⣠⡉⠰⠦⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡁⠐⠗⢀⡉⠀⡦⠀⠘⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠜⢁⠤⠄⠸⠟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡐⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⡆⠀⠀⠀⠀⠸⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠁⠀⠀⠀⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⡈⠉⠡⠃⠀⠀⠀⠀⠀⠏⠉⠁⢈⡤⠆⠀⠀⠀⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⢈⣁⣀⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆

⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣽⡆⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⣰⠶⡄⠀⠀⠀⠀⣤⡙⢤⠈⠉⠐⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠠⣙⣿⡥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⡀⠀⠀⠈⠉⠁⠐⠂⠀⡠
⠀⠀⡤⠀⢤⡞⣶⣤⣀⠀⠀⠈⠻⣆⣰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠺⠫⠟⠠⣿⡇⠉⠻⢷⢤⡀⠀⠘⣷⡀⠀⠀⠀⢀⠴⢮⡀⠀⠐⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣀⣀⣘⡂⠀⠀⠀⠀⠐⢾⠉
⠀⠀⠀⣠⠀⣸⡇⠀⠀⠀⠘⢻⣷⣤⣿⢴⣀⣢⡠⠋⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠀⠜⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣿⣿⣷⠃⢀⢠⠀⠈⠓⣄⢀⣀⠈⠃⠀⠁⠀⠓⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣬⣾⣷⢄⠤⠄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢈⠀⠀⠀⢸⡷⠀⣹⣾⡍⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⣿⣾⣿⣿⣮⠳⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣟⠨⠇⠠⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⢈⣿⡟⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠀⣺⣿⠠⣨⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣧⣼⣁⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⣤⠀⢠⡀⢀⠎⡀⠀⠀⠥⣼⡙⢷⣀⡘⠃⠘
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⡋⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠞⢃⡌⠀⠀⠀⢸⠃⡈⢇⠈⠹⠿⣀
⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓⠉⠉⠙⠚⠋⢮⢴⡜⠁⠈⠀⣠⣤⢀⡀
⠀⠀⠀⠴⠀⠿⠇⠐⠀⠀⠀⠀⠀⠻⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏⠁⠀⠀⠀⠀⠀⠀⣨⠀⡃⠐⡀⠙⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⣄⠀⠀⠀⠀⠀⢀⠏⣀⢀⣽⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠈⠳⠻⣏⣄⣣⡐⠂⠠




⣀⣀⣄⣀⣀⣀⣀⣄⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⣙⣿⣉⡛⣻⡟⣻⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣤⠠⠀⠠⢂⢀⡔⠂⣀⢀⠞⠁⠀⠀⠃⣠⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠠⠐⠂⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠋⠀⠔⢠⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⣀⠞⡐⠀⠀⠀⠀⠀⠀⠂⢀⠀⡀⢠⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠉⠀⠀⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⡀⣀⠀⣀⠀⣀⠀⣀⢀⣀⢀⡀⢀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠋⠙⠛⠛⠙⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠶⠶⠿⠶⠶⠿⠶⠶⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⣔⣴⣶⣢⣢⣶⡴⠴⠶⠦⠆⠲⠴⠴⠦⢢⣶⣴⣔⣴⣶⠂⠀⠀⢠⠄⡔⢠⠢⡔⡤⠢⠤⠄⠀⠢⠄⠤⠤⢀⡔⢤⢢⡄⡤
⠀⠀⠀⣻⣞⡿⠓⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢞⢾⣿⡁⠀⠀⢡⡘⣜⠃⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢫⡜⣤⠃
⠀⠀⠀⣽⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⢿⡅⠀⠀⢡⡜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠃
⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⠀⠀⠀⠉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃
⠀⠀⠀⢫⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡄⠀⠀⠉⠜⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠡⠃
⠀⠀⠀⣷⣻⣿⣟⣾⣷⣠⣤⣤⣄⣀⣤⣤⣤⣤⣖⣾⣷⣳⣻⣟⠆⠀⠀⠈⠤⠡⠉⠌⠱⠠⠄⠤⠠⢄⠀⠤⠤⠄⠴⠡⠉⠌⠤⠡⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣀⣀⣄⣀⣤⣀⣠⣄⣠⣄⣀⣤⣀⣠⣀⣀⣄⣀⡄
⠀⠀⠀⠀⠒⠀⠒⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠒⠀⠂⠀⠀⠀⢻⣿⠿⣿⠿⠟⠋⠉⠉⠉⠁⠈⠉⠉⠉⠙⠿⢿⣿⢻⣿⠇
⠀⠀⠀⠙⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠃⠀⠀⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡇
⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⢿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁⠀⠀⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠤⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡇
⠀⠀⠀⠀⠤⠀⠴⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠤⠀⠄⠀⠀⠀⣸⣿⣭⣿⣦⣤⣀⠀⠀⣀⠀⠀⡀⠀⠀⣀⣤⣶⣿⣼⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠋⠛⠛⠙⠛⠋⠛⠃⠛⠛⠙⠛⠉⠛⠋⠙⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠀⠈⠠⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⣀⡀⣠⠀⠀⠢⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⡁⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠁⢨⠉⠋⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⣈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⡆⠐⠐⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⢀⠗⡠⠌⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⠀⠊⠀⠈⠂⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠒⠒⠒⠀⠣⠓⠂⠀⠀⠀⠉
⠀⡠⠛⠣⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠈⣤⡤⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁
⠀⠃⠀⠘⣹⡀⠀⡆⠀⠀⠀⠀⣌⠀⠀⠀⠘⣂⠂⠀⠐⢤⣀⠰⡀
⠀⢸⢠⠀⡇⡇⠀⠁⠸⣄⠀⠀⠈⡴⠀⠀⢀⠉⠓⢄⡀⠀⠀⠉⠱⡆⠄⠀⠤⠀⢀⠀⠀⠀⡠
⠀⠸⠈⠀⢸⢡⠀⠘⠀⠙⢆⠀⠀⠘⢄⡰⠁⠊⠀⠀⠈⠁⢀⠖⠒⠛⢆⠀⠀⠀⠀⠚⠂⢂⠒⠤⠦⠤⠤⢀
⠀⠀⡇⠀⢘⢸⠀⢰⠤⢀⣀⠡⡀⠀⡛⠃⠒⠐⠒⠒⠀⡋⠁⣀⠀⠌⠀⠀⠀⠀⠀⠀⠚⠃⣀⠤⠒⠁⠀⠉
⠀⠀⢸⠀⠀⡌⡆⡎⠀⠀⠂⠀⠘⢦⡀⣀⡤⠤⠰⢾⠐⠁⠀⠀⠀⠂⢀⢠⡢⢐⣂⠤⠒⠉⠀⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠃⣿⠠⠊⠀⠀⠀⠀⠀⠇⢀⣀⣀⠀⠀⠀⠉⠀⣄⠂⠠⠠⠭⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⢸⢦⣠⢻⠀⠀⠊⠉⠈⠍⠂⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⠤⢀⡀⠀⠀⣀⣀⡤⠤⠤⠖⠒⠚⠉
⠀⠀⠀⢸⢨⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⢸⢸⢈⡌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⢻⠈⠇⡷⢤⣭⣉⣛⣓⣒⣒⣀⡠⢄⡀⢠⣀⠀⢀⠤⠊⡔⢀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠇⠊⠑⢀⣇⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠨⣆⠤⠊⠁⡠⠊⠀⢸⠀⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠈⠀⠀⠈⠉⠁⠒⠒⠒⠒⠙⢏⠀⡆⠀⠀⠊⠀⠀⢠⢸⠀⠇⠀⣡⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣀⣀⣀⣀⣀⡮⢸⠰⠀⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠓⠋⠉⠀⠈⠙⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡜⢦⣀⣐⣠⣤⡼⣃⡤⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡸⠁⠀⠀⠀⠀⠀⠀⠉⢳⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⢿⠇⠀⠀⠰⡿⠀⠈⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠵⠛⣶⢤⢄⢀⡤⡔⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡷⠋⢠⢀⠋⠑⢶⡴⠒⢣⢸⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⠗⢄⡼⣘⠀⠀⠈⠀⠆⠻⣼⡦⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢀⢾⣧⠇⠀⠀⠀⠀⠂⠚⢿⣇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣾⢄⡿⠀⠀⠀⠀⠀⠀⠀⢸⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡨⢠⣧⣀⠀⠀⠀⠀⠀⣀⣤⡞⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⠈⠉⠉⣉⡋⠉⠉⠊⠀⠩⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⠿⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⠀⡿⣚⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠉⢑⣤⣷⠋⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠿⠂⠀⠀⠀⠀⠀⢀⠀⣀⡉⠙⠀⠀⢀⣀⠠⢤⣦⠀⠀⠠⣤⣄⡙⠿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣟⠁⠀⠀⠀⠀⠀⠈⠑⢾⡿⠏⢠⣤⡀⠋⠀⠁⠀⢹⡀⠀⠀⠙⠿⠡⠒⣀⠀⢠⠒
⠀⠀⠀⢠⣤⣴⣿⢿⣿⣷⣷⠀⠀⠀⠀⣴⠄⠖⠃⠂⠘⣿⡇⠐⠒⢠⣾⣿⢙⣠⡀⠀⠈⢦⡹⣿⠀⠀⠠
⢠⣤⣤⡄⣇⣿⢷⣼⣿⣿⣿⣧⠀⠀⢠⡿⡆⠀⠀⢀⠘⣿⡇⠀⠀⢸⣿⣟⣺⡿⡅⠀⡆⢸⣦⣿⡆
⠸⣿⠿⠧⠿⢿⣿⣿⠿⠛⢻⣿⡃⣤⠸⡇⢡⠀⠀⠼⠤⡇⠆⠀⠀⢸⣿⡿⢿⡇⡇⠠⣿⢸⣿⣿⣿⠀⢀
⠀⠐⠒⣀⣿⣿⣭⡙⢿⣿⣿⣿⣿⣿⡃⡛⣯⢷⣤⢀⣴⣶⢄⠀⠀⠨⣿⣷⢰⡧⡇⣰⠟⣿⡿⡟⣽⣴⣿
⠀⠀⠀⡇⣿⣿⣿⣿⣿⢛⡇⣼⣿⠛⠺⠂⡝⡿⣷⢿⡿⢃⠈⢹⠉⣹⣹⣿⣿⡇⠨⠿⢿⣯⣰⡿⣿⠿⠿
⠀⠂⣀⡇⣿⣄⣿⣿⣿⣸⣿⡏⣿⠀⠀⠄⢀⡇⣿⢠⣇⣾⢰⢸⣴⣸⣹⣿⣹⣿⡶⠀⢺⣷⡁⣷⠲⠶⠖
⠀⠘⠛⡃⡿⠿⢿⣿⣿⡿⠿⠿⣿⠄⠰⠖⠋⠇⡟⠀⡏⠿⢪⡾⠋⣼⡿⣿⣿⣿⣇⠀⠀⣿⣵⣿⡀
⠀⠀⠁⠀⠁⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⠀⠀⠀⠛⠉⠁⠰⡿⠀⢀⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠾⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠒⠒⠒⠂⠸⠿⣍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠄⠀⣠⠦⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣔⠤⢊⣉⠤⠚⠁⠀⠑⢦⣘⣳⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡍⠒⠉⠉⠂⠀⠀⠀⠐⣋⡉⠙⠀⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⣰⠀⠈⣴⣶⣷⠀⠀⠀⢸⣷⣦⡉⠀⢰⡶⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⡇⠀⠀⠻⠿⠏⢀⠀⠀⠘⠿⠿⠃⠀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⡤⠈⣁⠀⠀⠀⠀⣤⡠⠤⠤⢠⡆⠀⠀⠀⢀⡍⠠⠴⠥⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⢾⡓⣛⠮⠁⠀⠀⠀⡌⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡤⠄⠤⢤⠐⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡖⠒⠒⠒⠒⠂⠈⠉⠁⠢⠔⠈⠉⠁⠐⠒⠒⠒⠒⢲⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠀⠀⠈⠉⠀⠀⠀⠀⠚⠛⠚⠛⢻⠒⠂⠀⠀⠈⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⢀⠀⠀⠀⠀⠀⠀⠀⠠⣛⡓⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⡯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠃
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠀⠀⠠⠤⠤⠴⣈⢃⣊⡸⠤⠤⠄⠀⠀⠐⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⡖⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓
⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠤⢤⡤⣤⠤⢤⡤⣤⢠⣤⣀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠤⠤⠤⠤⠤⢤⣄⣀⣀⡤⢠⡤⣤⠤⢠⡤⣤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢰⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠠⠤⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢢⡤⠤⠄⢲⠊⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣦⠾⣇⠀⠀⠀⠀⠀⢸⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⣶⡌⣌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇⢱⣶⡈⠳⣄⠀⠀⠀⢸⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠓⠀⠀⢀⠞⠉⠐⢤⠟⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠹⡄⠂⠉⠳⠀⠀⠘⠉⢱⠀⠸⠀⠂
⠀⠀⠀⠀⢀⠦⢦⠀⠰⠀⠄⠀⠀⠸⠃⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢀⠇⠀⠠⠄⠼⠀⠰⠦⢆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⡆⠀⢘
⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⡀⠀⢸⢇⣠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠤⢄⣀⠀⠀⠀⠀⡟⣸⠀⠀⠀⣠⠝⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⣄⠀⢈⡷⢨⡦⢠⠞⠁⠀⡠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⢷⣠⣧⡌⢿⡟⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⢃⠈⣻⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⢳⣤⣴⠃⠀⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⡿⠁⠀⠀⡀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣟⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠀⠀⠆⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠐⡄⠀⠀⠀⠹⣟⡀⠀⠀⠀⠀⠀⣸⡇⣼⠀⠀⠀⡄⣰⠃
⠀⠀⠀⠙⣆⡀⠀⡇⠀⠀⠀⠀⠘⠷⠶⠶⠦⠤⠴⢿⣄⠎⠁⠀⠘⣷⠃⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠈⠛⠀⠛⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠰⠆⠀⠀⠃⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠦⢤⣄⣀⣤⣼⣯⢠⠀⠀⠀⢸⣥⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣾⣠⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢯⡋⠁⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣯⣿⡧⠉⣀⠀⡖⢜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣴⢀⣠⣿⣿⣿⣿⣷⣿⣭⣷⡽⠂⠑⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡲⢼⡗⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠉⠉⠉⠉⠙⠓⠦⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡘⠡⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡐⠀⠀⠣⡀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠃⠀⠀⠀⠀⠢⢀⠀⠀⠈⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢘⠦⠀⠀⠀⡑⡄⠀⠰⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⡕⠀⢱⣄⡂⠀⢠⣶⣶⣦⠘⠀⠀⠀⠀⠀⠈⣴⣶⣶⡄⢱⠈⣆⡎⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠀⢆⠀⠐⢜⣿⣏⡿⠀⠀⠀⢀⠀⠀⠐⢿⣹⣿⠃⠀⠀⡘⠀⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠁⠈⢆⡀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⡰⠉⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠜⠒⠤⣀⢀⠉⠂⠠⠀⠄⠀⠀⠀⢀⠠⠔⠫⢄⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡞⠁⠀⠀⠀⠀⠈⠃⠄⣀⣀⡀⠀⠈⢀⠎⠀⠀⠀⠀⠙⣄⠀⢻
⠀⠀⠀⠀⠀⠀⢀⠜⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠸⣄⠀⠣⡀
⠀⠀⠀⠀⠠⠒⠓⠒⠊⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠒⠚⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⡆
⠤⠴⠤⠤⠤⠤⠤⠤⡎⠀⠀⠀⠀⠀⠀⠲⠒⠒⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢷⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⢰⠀⠀⢀
⠀⠀⠀⠀⢠⣎⡉⠉⠁⠈⠣⡀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠸⠁
⠀⠀⠀⠀⠠⡷⠄⠀⠀⠀⠀⠐⢄⡐⢀⠖⢌⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠⢀
⠀⠀⠀⠀⠐⡟⠂⠀⠀⠀⠀⠀⠀⠙⡂⠄⠀⠢⠞⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⡄⣄⢂⠑
⠀⠀⠀⠀⠘⣟⡁⠀⠀⠀⠀⠀⠀⠀⠐⠄⣈⠡⠾⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣧⢸⠂
⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊
⠀⠀⠀⠀⠀⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⠀⢠⣀
⠀⠀⠀⠀⠈⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⡀⠀⠤⢄⣀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠹⠀⠀⠀⠀⠈⢹⠉
⠀⠀⠀⠀⠀⢗⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠁⠉⠉⠁⠀⠀⠉⠉⠁⠀⠁⠀⠃⠉⠉⠃⠀⠛⠐⠒⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⠦⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢱⣄⣀⣀⣀⣀⡀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⣒⣒⠒⡒⢒⣒⠒⣒⣂⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠈⠉⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⢂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠈⡎⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠾⢶⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡠⠭⡀⢣⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢀⠀⠀⠀⣠⠐⢀⠁⠀⢤⣽⣒⠶⣶⡄⠀⡦⠀⠀⢘⢸⠁⠀⢃⢸⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠄⠀⠀⠀⠈⠇
⠀⠀⢠⡁⠀⢾⡟⠐⠊⠁⠀⠀⠀⢀⠀⠀⠘⢄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣄⠀⠀⠀⡄⢂⠀⠀⠀⡀⠀⠀⠀⢀⡀⡐⠠
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢇⠀⠀⠑⢌⠑⠒⠉⠀⠀⠀⠀⠈⡡⠃
⠀⠀⠀⠀⠀⠛⢆⠀⠀⠀⡀⠀⠀⠰⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣀⠀⠀⠀⠁⣢⠄⣀⣀⡤⠔⠊⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠈⠐⠊⠀⡋⢄⠁⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠙⣦⣤⠴⢊⠛⠜⠁⢠⡃⠑⢆⠀⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠘⠧⠛⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⢸⡁⠀⠀⣀⣁⡠⠞⠐
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡧⡸⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡉⠁⢸⠁⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠘⠢⠀⡷⠃⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢣⡰⠀⡀⡄⠀⠀⠀⠘⠤⠛⢤⡼⠀⠀⠀⠀⠀⠠⢀⡏⢀⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢬⡀⡀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠪⡨⢛⢾⠇⠸⠃⡇⠀⠰⠒⠼⠀⣠⢤⡧⠀⢀⠀⠀⠀⠀⠀⢑⢀⡀⠎⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠁⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠹⠤⢅⠼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠵⠄⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠜

⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠏⠀⡐⡶⠀⠀⠀⠀⠀⠀⡀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢟⡿⠄⠓
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡉⠙⠁⠀⠀⠑⠲⣶⣟⣤⣤⣄⣠⣶⣶
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣤⣤⣤⡀⠀⠈⣉⣿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣠⣶⡌⠁⠀⠈⠡⣶⠀⠀⠙⣿⣿⡇
⠀⠀⠀⢀⡀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⡇
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⣶⣤⣦⠀⠀⠀⠀⣼⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⣀⣤⣤⣄⡀⠀⠀⢸⣿⠇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠉⠻⠿⠋⠁⢀⣴⣿⡿⢀⡟⡆
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣴⣿⣿⣿⠁⣸⠀⢁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣆⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢀
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣖⣲⣶⣶⣶⣶⡄⢀⣶⣶⣾⣯⣴⣶⣶⣶⣶⣶⣶⣶⣶⠆⡼⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⡿⢟⣵⣿⣿⣿⡿⢿⣏⣭⣭⣿⠟⢻⡿⠿⠛⠛⠀⠈⠁⠚⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⢻⣿⢿⣿⡇⠰⣿⣿⣿⣿⠋⠁⢀⣠⠄⢀⣀⠀⠀⠀⣠⣶⣶⣄⣠⣄⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢣⣴⢶⢶⣄⢷⣄⠙⠿⠃⠉⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠸⣿⣿⣿⠟⠀⣻⡷⠀⣸⣷⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⢀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢹⠏⠁⠀⠀⣛⣥⡶⢰⠂⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⢠⣶⣤⣾⣿⡿⠙⠘⢀⣼⣿⣿⣿⡟⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⠈⠻⣯⣿⣿⠿⠛⠛⠻⣶⠳⠶⢾⣿⣧⣀⠀⠀⠀⠀⠘⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡼⠋⣠⠀⣹⡟⠁⢂⣀⡀⠈⠉⠀⠀⠠⠉⠛⠿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡀⢻⣾⠟⠁⢆⠀⠈⠁⠁⠀⠀⠀⠀⠁⠸⠗⡄⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣙⣡⡮⠨⠀⠈⢀⠀⠀⠀⠁⠀⠀⠀⠐⠀⣀⡐⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⣦⣤⠀⠀⠑⠂⢀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠠⢸⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣼⡇⠀⢿⣿⠓⢄⠀⠀⠊⠻⠷⣤⣤⡤⠠⣾⣷⣆⠀⠀⠙⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡇⠀⢸⣿⡔⣆⢀⢀⣀⠀⠀⠀⠀⠀⣤⣼⣿⡘⠂⢠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⣠⣼⣿⣥⣬⣿⣿⡛⠱⣤⣤⣠⣶⣿⢩⣿⣧⠄⠁⣬⠸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡟⣿⣽⢻⣿⣿⣿⣿⣿⣥⣌⣉⣁⡿⠿⣿⣿⣿⢻⡄⣢⠽⠺⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠙⠛⠿⠒⠛⠁⣿⡟⣿⣟⡉⠀⢈⣿⣿⣿⣿⣿⢻⡇⠀⠂⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣖⣲⡆⢀⣇⢀⡳⠄⠀⢀⣀⠈⠉⠉⠉⠋⠸⠇⡰⣤⣨⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠉⢡⠊⠉⠁⠉⠀⠀⠉⠉⣻⠀⠀⠈⠀⠀⢈⡟⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣥⣤⣤⣄⡀⠀⠀⠀⢠⡀⢽⣷⣶⣶⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠉⠙⠛⠏⠉⠏⠱⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⡟⠉⠉⠉⠉⠛⠛⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⠆⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⡤⠖⠋⠁⠾⠥⠤⠤⠔⠚⠉⢡⣞⣁⣠⠤⠖⢫⡇⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠋⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢀⣤⡀⣠⡀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠃⠀⠈⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣤⡀⠀⢷⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡀⣠⣄
⠀⣤⣼⡏⣿⠉⣇⣤⡄⠀⠀⠀⠈⡷⢦⣼⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠿⡇⠀⠀⠀⠀⣸⠇⣤⣤⡏⢻⡏⣿⣴
⣴⡽⣮⠓⠘⠀⡟⢸⡇⠀⠀⠀⢸⡇⢢⣟⠀⠀⠀⠀⢸⣷⣿⠀⠀⣠⠃⠀⠀⠀⠀⣾⣽⡆⠀⠀⠀⠀⠹⣀⡴⠢⣠⡟⠀⣿⠘⡇⠈⠐⢫⡾
⠈⠛⢶⡀⠀⠜⠁⣸⠇⠀⠀⠀⠀⣧⠀⡟⠀⠀⠀⠀⠀⠉⠁⠀⢞⠉⠁⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠛⡴⢫⡏⠀⠀⢾⡀⠹⠆⠀⣠⠞
⠀⠀⠈⠳⣦⠀⠀⢻⡄⠀⠀⠀⠀⠘⢦⣭⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣼⠃⠀⢠⡴⠋
⠀⠀⠀⠀⠹⡆⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⡟⠒⠂⠤⠤⠤⠤⢤⡶⠀⠀⠀⠀⠀⠀⠀⢀⣧⡼⠃⠀⠀⡼⠃⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣠⡞⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠙⢦⣀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠓⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⣠⠞⠃⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⣽⣷⣶⣤⣄⣈⣳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣡⣴⣶⣿⣇⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠲⠤⣀⣀⣀⣀⡀⠤⣴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⢧⡀⠀⠀⠀⠀⢀⡰⠋⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠓⠴⠶⠒⠋⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠫⠭⣉⡉⠛⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠛⢉⣉⠭⠝⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠉⠓⢦⣙⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⣋⡴⠚⠉⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠞⠁⠀⠀⠀⣀⣤⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣶⣤⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣦⣤⣶⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠠⡄⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣆⢐⠀⠀⠐⠀⠀⢀⣦⠀⢈⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠣⠁⠀⡀⠈⠀⠀⠀⠀⢀⡲⠋⠀⠀⢀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡰⠀⠀⠠⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠞⠒⠂⠀⠀⠘⠀⠀⢰⠠⣧⠀⠀⡰⠦
⠀⠀⠀⢿⡾⢥⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠊⠀⠀⠀⠀⠀⠀⢬⣿⡆⠀⢸
⠀⠀⠀⠀⠌⣢⣽⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠰⠀⠀⠠⠐
⠀⠀⠀⠀⠈⢻⢻⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠖⠀⡀⠀⠀⠀⠀⠀⠀⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠸⠈⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⢛⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠋⠳⠄⠀⠀⠀⠀⠀⠀⣀⣠⡁⠀⠐⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠓⡛⠃⠀⠜⢢⠛⢄⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⡈⠃⠈⢷⠀⠀⠀⠀⠀⢀⢙⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⣄⠠⠰⠀⢀⡀⠀⡀⢃⠀⠀⠀⠀⡘⠋⠀⡀⠄⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠋⠐⠈⠀⠀⠙⠦⣙⣌⡆⠀⠀⠀⠙⠂⠈⠳⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⠀⠸⡀⠀⡀⠐⣷⡄⠈⠙⠻⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠈⠊⠀⠙⠉⠀⠈⠀⢧⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⡀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢀
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠓⠀⣄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠂⠀⠘⠐⢳⣤⣬⣟⣿⣀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⣀⣀⣙⣡⣘⣿⣿⣷⡄⢀⠀⠀⠀⠀⠌⠢⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣿⣿⣧⣴⠿⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠰⠈⠉⠉⠉⠉⠠⠀⣀⠀⠠⣦⣀⣵⣿⣿⣿⣿⣿⣿⣷⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠀⠸⡅⠀⠀⢀⠀⣠⡜⢛⣿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠚⢳⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⡱⣴⠛⢉⡤⢿⡸⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⣭⡿⠋⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣤⡔⠍⣠⢾⣿⠚⠁⠀⠀⠠⠄⠂⠀⣠⣤⣴⣶⣾⣿⣾⣿⣬⣶⡿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⢟⣡⣾⣯⣿⣿⣦⣄⣽⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡻⢀⣼⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢿⣿⣏⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠄⢀⣼⣿⣿⡿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⢿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠁⠠⡄⠠⣤⠁⢀⣿⣿⣿⣯⡴⡦⢿⣿⣿⣿⣛⢿⣿⣿⣿⣷⣶⣾⣿⣿⢿⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⠛⣡⣿⣿⣤⣿⢸⣿⢹⣿⣿⣿⣿⡊⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢷⣽⣿⢹⣏⠛⣿⣿⠃⢻⣿⣿⣿⣿⠙⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣏⣘⣛⣀⣛⣛⣧⣚⣿⣿⣿⣿⣿⣞⣿⣿⣿⣛⣻⣿⣿⣛⣛⣉⣩⣼⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣿⣿⣿⣇⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡁⠀⡀⢰⡄⠀⠀⠀⠀⠀⠀⡃⢀⣀⠐⣤⣄
⠀⠀⠀⠀⢀⠀⡠⠴⣦⠀⠀⠀⠀⠀⠀⠴⠠⢄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠛⠀⠨⡏
⠀⠀⠀⢾⣿⡇⣠⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⣄⠀⠀⠀⠀⠀⠑⢁⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣰⠀⠈⠁⠢⢀⢀⣴⣞⣂⣀⠀⠀⡉⠀⠤⠄
⠀⠀⠀⠀⠀⠀⢀⣈⡁⠀⠀⠀⠀⠀⠀⢽⠄⠠⣥⠀⠀⠀⠀⠀⠁⢸⡟⠀⠀⠀⠀⣠⣴⣟⣿⢹⡿⠿⢡⠆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⡀⣠⡽⣥⡄⠁⠋⠐⣂⡀⠀⠂⠀⠈⠁⢂⠀⠄⠀⠀⠈⢠⠀⠀⠀⠀⣿⣿⣿⣿⢿⠁⠀⣄⠀⠁⠀⠀⢠⣀⣼
⠀⠀⠀⠀⠀⠘⠉⠈⣧⠀⠀⠀⢾⣿⣿⣿⣄⠀⢀⡄⣿⣿⡇⡆⠀⠀⢸⡀⠸⢀⡼⢿⣿⣻⡏⠀⠀⣰⣿⣦⣀⣈⠈⠻⠏⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢻⣿⣿⣿⣄⠺⡿⣹⡟⢹⠷⢀⣴⣿⣇⠠⠘⢷⣿⣿⣿⡇⠀⠀⣿⣿⠿⢋⡸⡄⠄⠴⠃
⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⢀⠀⠀⠘⣿⡻⠿⢿⣦⡀⣿⡇⣸⣾⣿⡿⣯⠁⠀⠀⣨⣿⣿⣿⠀⠀⢸⣿⣿⠀⠚⠓⠀⠀⡔⠀⠀⠀⣆
⠀⠀⠀⠀⢹⡀⠀⠀⠤⠷⠘⠃⠀⠀⢸⠿⢿⣿⣿⣿⣿⣷⠙⠻⠃⠠⡿⠀⠀⠀⣼⣿⣿⣿⣇⠀⢸⣿⣿⠀⡄⠀⠀⣼⠃⠀⠀⠸⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⡀⠀⢞⣼⣿⣿⣿⣾⣿⣿⣄⣀⣀⣀⡄⣠⣤⡄⣿⣿⣿⣿⢿⡴⢸⠘⣿⠀⢀⠀⢰⡇⠀⠀⠀⠀⠈⠿⠶
⠀⠀⠀⠠⠤⠄⠒⠚⠉⣩⠴⠂⡄⢠⣾⡿⢿⣿⣿⣿⣿⣿⡏⠙⠛⠛⠁⠛⠛⣹⣿⡏⢿⣿⣶⠶⠾⣰⠟⢀⢸⣍⠀⡁⠀⠀⠘⢉⡄
⠀⠀⠀⢰⠠⠤⣤⣤⣴⣶⣶⣿⣇⣿⣿⠏⣼⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⡿⠿⠛⢧⡀⣼⣿⣿⠀⠸⠄⠐⠚⠀⠀⠀⢂
⠀⠀⠀⣸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣿⣿⡇⣿⣿⣿⡇⠀⣀⠐⠀⠀⠀⠁⠀⣟⢿⣿⠦⣰⣴⣿⣿⣿⣿⡜⠀⠀⠂⠀⠀⠀⠀⣀⡜
⠀⠀⠀⡿⣻⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⣿⡛⠋⢠⣿⣿⣟⣀⡄⠈⣷⠀⠀⠀⠠⢸⡿⠸⣿⣿⡫⠙⢿⣿⣿⣿⠃⠁⠀⠁⠤⣤⣶⣿⡿⠃
⠀⠀⠀⡇⢸⣿⣿⣿⢿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⠀⠀⠀⠀⠀⢸⡄⣿⣿⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠇⣿⣿⣿⣿⣾⣿⣿⣿⡇⠈⣿⣿⣿⣿⠉⣿⣿⣿⣿⡇⠀⠈⡇⠀⠀⠀⠀⠘⠁⣿⣿⣿⣿⡆⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⠀⢘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣟⡇⠀⠀⢻⣿⠋⠀⠀⡿⢿⣿⠇⠀⠸⠀⠀⠀⠀⠜⠋⠘⢹⣿⣿⣿⡇⠀⢿⡇
⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⢿⡇⠀⠀⣾⣿⠇⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⢻⣿⣿⡇⠐⢸⣷⣤⣀⣀⣀⣤⠤⠶⠒⠚⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠄⣿⣿⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠛⢉⠀⠉⠋
⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⢹⣿⣴⠀⠀⠀⢹⣿⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⢰⢰⣿⡆⠀⠀⢀⠀⠀⠈
⠀⠀⠀⣷⣿⣿⣿⢸⣿⣿⣿⣿⠀⠀⠀⡎⢿⣿⡇⠀⠀⠀⠻⣿⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⡇⠸⢸⣿⣇⠀⠀⡆
⠀⠀⠠⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⠀⠀⠀⣴⣿⡇⠀⢀⣁⠻⢾⣿⣿⠇⠀⣴⣶⣿⣿⡀⠀⣁⣈⠙⢻⠇⢀⠘⢿⣿⠀⠀⠁
⠀⠀⠀⣧⣛⣿⣿⠀⢹⣿⣿⣾⣄⡀⠀⢸⣯⣭⣇⠀⠘⠻⠟⠻⠷⠶⣄⠀⠈⠙⢿⣿⣴⣿⣿⣿⣿⡿⠀⢰⣶⣴⡍
⠀⠀⠀⣧⣼⣟⢿⠀⠈⠿⠿⠿⢿⠿⠷⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⣼⡿⢛⣠⣴⣿⡇⠀⠀⠠⣴⡇
⠀⠀⠀⠈⠙⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⠲⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠹⡆⠀⠀⠈⠒⠸⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣇⠀⠀⠀⠹⢄⡀⠀⡠⠖⣡⠖⣄
⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠤⠤⠤⠀⠒⠒⠒⢂⡈⠉⠁⠉⠁⠀⠀⠀⠑⢶⠁⠀⠀⣀⣀⣹⠤⡄⠀⠐⣎⢴⣋⡴⠋⠀⠀⠘⡆⢀⡠⠤⠄⣀
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⢠⠟⣗⢒⣢⣉⠁⠀⠀⡇⠀⠀⣏⠇⠀⡀⠀⠀⠀⠀⣷⠊⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠉⠹⣄⣽⠧⠤⠤⠌⡤⣄⣗⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⣑
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡤⠜⠋⠤⢤⣤⡎⠀⠀⠀⠉⠉⠒⠲⠤⡟⠉⠉⢹⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠉⠙⠳⢤⣄⡀⠀⠀⠀⠈⠀⠀⡞⢘⡇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠠⠀⠒⠒⠂⠈⠉⠉⠓⠲⢤⣤⠞⠁⢸⢁⠤⢄⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣾⠿⠿⠛⠛⠁⠀⢸⡇⠀⠀⠈⠑⡄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⠶⠶⠒⠒⠒⠛⠻⣷⣤⣤⣤⣴⣶⣶⡿⠿⠿⠟⠛⠛⠛⠉⣉⣉⣉⣀⢠⡄⠈⠙⠲⣄⢀⣠⣤⣼⠀⠀⢄⡀⢀⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢹⡥⣤⣤⡤⠴⠖⠒⠒⠋⠉⡭⠷⠟⠉⠀⢨⡇⠀⠀⠀⠈⠉⣠⠞⠁⠀⠀⠀⠉⢉⡜⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⢰⡎⠳⣄⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢸⠻⠀⢀⣠⣴⣾⣷⣶⣖⣒⣒⣒⣒⠋⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⠀⢸⠁⠀⠈⠳⣄⠰⠁⠀⠀⠀⠀⠀⢀⣼⣾⠋⠁⠀⠉⢹⣿⠉⠉⠉⠉⠉⠉⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⢸⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠙⣿⣿⡗⠀⠀⢠⡻⢯⡄⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣾⠀⠀⠀⠀⠀⠀⠈⢣⡀⢀⣀⡤⢤⣿⡇⠙⣄⢠⣳⠃⠘⠃⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣇⣀⣉⣀⣠⣤⣀⣦⠤⠶⠾⠻⣟⠛⠉⢉⣿⠀⠀⢸⢣⠃⡤⠄⣀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠧⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⢦⡀⢸⣿⠀⣠⢣⡇⠀⣹⠀⡼⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⡷⢚⢫⣶⠿⡊⢡⡮⠈⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⢆⡾⠋⠀⡩⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⡜⠓⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠘⠌⠁⠀⠀⠀⠀⠈⠃⠐⠂⠁⠀⠀⠤⠄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠦⡀⠀⠀⠀⠀⣆⡙⢄⠐⠀⡀
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠡⠚⣁⠀⠀⠀⠀⠄⣢⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣶⠢⢹⡆⡴⢋⡩⠉⠻⣆⠴⢋⣽⡄⠠
⠀⠀⠀⠀⡅⠀⠈⡀⠀⠠⠀⠀⢠⣿⡄⢀⡀⠀⢻⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢧⡸⢟⣀⡼⢻⡇⣾⡿⠀⠀⣿⣤⣉⠈⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠑⠚⣢⠅⠀⠀⠨⠟⠁⠀⠉⡷⠀⠀⢀⠀⠀⠀⢐⡀⠀⠀⠀⠐⠀⠀⠀⠙⠙⠃⠀⠀⠳⢬⣀⣀⠴⠟⠀⢹⠆⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢅⠈⠥⡀⠀⠀⠀⠀⠦⢠⡦⠀⠀⠀⠌⠀⠀⠀⠘⣭⡀⠀⠀⠐⠠⠒⠃⠤⠀⠀⠘⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡤⢢⠴⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠤⡀⣀⣀⡀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢯⣤⣬⠦⠄⠠⠠⠀⠰⠀⠀⠀⠘⠭⠅⠀⠀⠀⢠⣀⡀⠀⢀⣤⢠⣅⠀⠡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⢀⠋⡍⢸⠈⣿⡄⠀⠠⠆⣼⢃⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⣷⡋⠀⠤⠀⠀⠛⢧⢊⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠀⠴⠀⠀⠋⢀⢨⡇⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⠤⠤⠤⠤⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡭⠥⠤⠤⠤⡯⢤⠌⠰⠇⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠈⢀⢄⡰⢀⣀⣻⣤⣤⡀⠀⠀⠶⢒⣲⣤⠴⢶⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⢥⡐⠅⠀⠁⠀⠉⠛⠛⠛⠁⠀⠀⠀⣀⣀⣀⠐⠨⠋⠉⠉⠙⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠩⠴⡮⢄⢀⡀⣂
⠀⠀⠀⢀⡄⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠈⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠒⠀⠂⠀⡁⠈⠑⠈⡋⣂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠠
⠀⢀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠁⠚⠲⣦⡀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠑⠶⣦⢤⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡨⠽⣿⡛⠃⠀⠀⠀⠀⠀⡀⢸⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢠⡎⠀⠂⠙⠉⠀⠀⠀⠀⠀⠀⠙⣃⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠗⠗⢹⡇⠂⠀⠀⠀⠀⠀⢀⣀⠓⠦⠀⣨⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⡞⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠉⢁⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⢿⡉⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠋⠉⠀⠀⠀⠈⣷⠀⠀⠀⠹⡄⠀⠀⠀⣆⠀⠀⠀⠀⢠⢦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣤⡀⠀⠀⢹⡄⠀⠀⠀⢹⡄⢀⡤⠘⢦⣄⡀⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⢸⡇⠀⠀⠀⣀⣷⡏⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠏⠙⢱⣈⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢠⣏⡀⣀⠞⢿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠃⠈⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠼⣟⠀⠀⠴⠦⠀⠀⣹⠀⠀⠄⠀⠀⠀⣦⢁⠀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠆⠀⠰⣟⠀⠀⣀⢌⡴⠉⠀⠀⠀⢀⣀⣀⣿⣀⠀⠀⢀⠠⠤⠼⠷⢔⣦⡀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠴⠟⠒⠛⠛⠛⠛⠛⠓⠚⠛⠛⠉⠉⠁⠈⢿⡉⠀⠀⠀⠀⠉⠙⠛⠛⠶⣄⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠃⢸⡇⠀⠀⠀⠀⠀⠀⣈⡛⠦⣍⣳⡄⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠁⣼⣧⣄⣀⣀⣀⡀⠶⣤⡽⠟⠉⠉⢛⠦⠠⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠶⠛⠛⠛⠛⠋⠁⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⣉⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠘

⠀⠀⠀⠄⠐⠀⠠⠀⠀⠀⠁⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠄⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠤⠄⠀⡐⠤⢠⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡀⠀⠀⠚⠀⠏⠀⠃⢁⠀⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢃⣀⡀⠀⠀⠀⠀⢠⡸⠀⠆⡨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢠⠄⠇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠁⠀⠀⠀⠈⠀⠈⠉⠉⠀⠈⠉⠉⠉
⠈

⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠠
⠀⠀⠀⢰⣃⠒⠂⡀
⣤⣤⣤⣼⣤⣤⣤⣥⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣤⣀⣾⣿⣿⣷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⣤⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣠⣤⡀⠀⢸⠀⢀⣤⣄⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣤⣼⡇
⠀⠀⣿⣿⣷⠀⠀⠀⣾⣿⣿⠀⠀⠀⢰⣿⣿⡇⠀⢸⡀⢸⣿⣿⡆⠀⠀⢰⣿⣿⣗⠀⠀⠀⣾⣿⣿⡇
⢸⣿⣿⣿⣿⡍⣧⣿⣿⣿⣿⣏⢹⣾⣻⣿⣿⣿⡙⣏⣿⣿⣿⣿⣯⢹⣿⣻⣿⣿⣿⡝⣯⣟⣿⣿⣿⣯⢻
⣾⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿
⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿
⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⠤⠄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠒
⠀⠀⢐⠄⠂⠐⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠰⠻⠒⠚⠑⢀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠔⠁⠀⠒⠐⢀⠁
⠄⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣠⣤⣤⣀⡀⠋⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣠⣤⣤⣄⡀
⠠⠀⠀⠀⠀⠀⠰⠤⠶⠿⠾⠶⠦⠤⠤⠤⠝⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠶⠿⠾⠶⠦⠤⠤⠤⠘⠳
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠲⠦⠤⠆⠀⡆⠦⠤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠔⡄⣤⣤⣤
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠠⠤⠀⠁⠀⠀⠤⠀⠈

⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⢤⣄⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉⠋⠉⠛⠋⢻⠆⣠⠞⠉⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡼⠃⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣦⣼⠋⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣮⡙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⣫⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣸⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠉
⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠉⣿⡏⠙⠛⠛⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢿⡇
⢴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠃⠀⠀⠀⠈⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁⠀⠀⠀⠀⠀⠀⠀⠘⠆⢀⡔⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠂⠉⠉⢖⠄⠀⠸⠀⠀⠀⠐⠀⠐⠒⠂⢠⠔⠁⡼⠢⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢀⡀⢀⠀⢘⠀⢸⠀⣈⣀⣀⣀⡈⢁⠔⠁⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠄⠓⠋⡠⠂⠀⢈⠀⠀⠀⠀⡀⢀⠂⢤⠊⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢾⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠄⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠉⠀⠀⠀⠀⠀⠀⠀⠞⠀⠱⠂⠀⠀⠀⠀⠀⠀⠈⠂




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠂⠂⠄⡀
⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠁⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠔⠕⠀⠠⠊⠀⠀⠀⠀⠀⣴⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⣀⣀⣠⡤⠀⢀⠀⠀⠀⣀
⠈⠈⠀⠀⠀⠀⠀⠀⢀⠀⡴⠀⢓⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠘⠂⠁⠃⠄⠀⠈⠀⠡⡀
⡆⠀⠀⠀⠀⠀⡠⢀⠡⠚⠀⠀⠀⠁⡀⢦⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⢃⠀⣀⠀⠠⠁⠂⠀⠀⠀⠀⢠⠀⠀⠀⢹⢉⠀⣀⢀⠀⠀⢴⡃⠀⠀⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠂
⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⣀⠓⢄⠇⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘⠑
⠀⢽⢣⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⢀⢴⣡⠖⠁⢀⠂⠀⠀⠀⠀⠁⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠔
⠀⠀⠀⡗⠤⡀⠀⠀⠀⠀⠀⠄⡒⡟⢃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⡖⠂⣀⠲⣆
⠀⠀⠀⠃⠀⠁⠀⠀⢠⢧⣀⠘⠂⠇⠀⠓⠁⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠈⠂⠛⠊⠀⠁⠢⡀
⠀⠀⠀⠁⠀⡀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⢁⡀⡆⠀⠀⠀⠀⠀⠀⠧⡠⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠠⢆⣀⣀⠀⠄⠀⠀⠘⠠⢤⠀⠀⠐⠢⠀⢈⡁⡗⠐⠲⡀⠠⢰⣇⣣⡤⠒⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠉⠻⢦⠤⠠⠄⠀⡁⠀⠀⠤⠤⢖⣀⣓⣀⠀⠀⠀⢀⣁⡒⣒⡒⣒⣚⣋⣭⣭⣥⣤⣤⢤⣶⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠖⠶⠶⠶⠒⠒⠀⠉⠙⢌⠉⠉⠉⠀⠨⡁⠉⠉⠹⣔⠈⠉⠉⠀⢤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢖⡀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣤⠀⠀⢠⣀⣐⡄⠀⠀⡤⠄⠸⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠆⠶⡄⠀⢀⠠⠄⠀⡖⠀⡙⠀⠀⠉⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢄⠄⡆⠀⠀⡁⠨⠁⠀⠀⠀⠀⠀⠀⠂⣀⠠⠀⠈⠒⠐⠇⠐⠀⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡂⠄⠀⠀⠈⠢⢴⠄⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠉⠉⢬⣻⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣮⣀⣤⣺⣿⣿⣹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⠻⠿⠿⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠗⠒⢄⠀⢀⣤⣤⡀⠘⠿⠋⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠈⠛⠋⠀⠠⣤⡤⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠤⠠⠀⠀⠀⠀⠀⣼⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⠲⠶⠖⠂⠀⠀⣠⢞⡉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡟⣿⣿⣧⣀⣀⣤⡶⠛⠁⠘⠁⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⢡⠁⠁⠈⡿⠍⠛⢿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⠇⠀⠀⠃⢀⣴⠋⠀⠀⠀⢸⡆⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡾⠁⠀⠐⠂⠀⠀⢹⠀⠰⠃⢀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣸⡇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣿⡠⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡄⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣛⠳⠣⣤⣾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡆⠣⠀⠀⡶⢦⣦⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠦⣽⡏⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣧⠐⠂⠒⠂⠬⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠂⠐⠲⠄⡀⡀⢀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡠⠃⠀⣤⡈⢠⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣣⡀⠀⣾⠃⠀⠀⠀⢀⠆⠰⢠⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠈⠉⠲⠛⠇⠐⠀⠀⣬⡀⠀⠂⣫⡌⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢳⠀⠀⢘⣤⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠸⡏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⣃⣄⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣁⡴⠺⣭⣁⣀⣤⣤⣬⣭⣽⣿⣯⣽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣾⠻⠟⠛⠛⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣴⣮⣤⣤⡴⠶⠶⠿⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⢀⣰⡏⠀⠠⢀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠈⠻⢍⡒⠤⠄⠀⠀⡀⠀⡀⣀⣀⠤⢾⡟⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⠏⠀⠈⠐⠀⠌⢐⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⢀⣤⣤⣾⣀⡀⠤⠖⠋⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡠⣮⠕⠀⠀⠀⠄⠐⣲⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣝⡇⠀⠀⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⢄⡀⠘⢆⠀⠀⠀⠀⠉⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢂⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⠀⠀⠈⠂⢄⠐⠀⡀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⣀⣀⣠⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⢰⡿⠀⠀⠀⠀⠀⠀⠔⠋⠉⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢧⠜⠀⠘⢿⠄⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⡀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠈⠚⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢶⣶⠶⠠⠐⠂⢁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠆
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⢀⣀⠀⠈⠁⠀⠄⠈⢀⡀⠀⠄⠀⠀⠀⠘⠲⠦⠭⣀⣀⡠⠊⣡⣄⣀⣤⠐⠺⠟⠀⠀⠀⠠⣄⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡸
⠀⢀⣀⣀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡈⠛⣇⣀⣀⡏⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⢄⣀⣀⣀⣀⡤⢴⣪⠋⠀⢈⣆
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠛⠃⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠊⠉⠧⣤⣤⣚⢄⡀⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠈
⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣢⠀⠀⢀⠄⠒⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠈⠙⠛⣃⣠⣤⢦⠀⢀⣤⠀⠀⣠⣤⣀⡘⡟⠛⠃⢀⠃⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠂⠀⠉⠁⠀⠀⠨
⠀⠀⠀⠀⠉⠀⠀⠒⠿⠇⠀⠰⢟⡁⠉⠛⠇⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⣀⡀⠄⢢⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠀⣠⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣴⣾⠯⠙⠓⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣀⣽⣧⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣩⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢔⡀⠀⠀⠀⠀⠀⠄⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⢙⠿⣁⢀⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢸⣷⡀⢄⣇⣿⣛⠶⠷⠼⠟⣛⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢣⡘⠄⠁⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⣾⣿⣷⢀⡀⣹⡅⣐⣦⡞⠋⠁⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤⣭⡤⠀⠀⠀⠀⠀⠀⠀⠱⠶⣤⡐
⠀⠀⠀⠀⣿⣿⣿⢨⣿⣵⡆⢸⣯⣉⣵⠾⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡵⠛
⠀⠀⠀⠐⣿⡛⢹⣧⣿⣿⢿⣂⢛⣿⡙⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢄⠀⠀⠀⠀⠀⠄⠀⣟⠀⠀⠄⠈⠵⠁
⠀⠀⠀⠀⠘⣧⠀⠉⢩⣯⣦⠛⢽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠈⢂⠀⠀⠀⠀⠇⡙⢳⡌⠀⢖⢀
⠀⠀⠀⠀⣴⣿⣳⣦⣀⣙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣄⠀⠀⢀⣴⣾⣇⡨⡿⢀⣎⣺
⠀⠀⢠⣾⣿⣯⠙⠦⢝⣹⣏⢰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠒⠻⣿⣿⣿⡄⣡⠾⡣⠴⠊
⠀⠀⢸⣿⣿⣿⣷⡄⢸⣿⡿⢸⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢦⠶⠆⠂⠀⠈⠁⠀⣀⣩⠔⠀⠀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣄⣿⣿⣌⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣴⣟⡛⠛⠿⣿⡿⣀⡂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣠⣐⢀⡙⢿⣿⣿⡶⠦⠬⠄⠀⡀⠀⢳
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠛⠛⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⣀⡀⣄⣀⣠⣤⡀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣀⠳⠌⠀⠉⠙⠓⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠉⠃⠀⠀⠀⠠⡄⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⠆⠀⠀⠀⢀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢹⠙⠲⠋⠉⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣁⠀⠀⠀⢀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠙⠆⠉⠉⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠱⣄⠙⠢⡞⢉⠉⠲⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠒⠊⠐⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢳⡀⣟⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⣀⠀⠀⠀⠀⠙⢛⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢹⡏⠁⠀⠀⠀⠀⠀⠀⣀⠀⣤⠖⠀⠀⠀⠀⢉⣡⠎⠀⠀⠀⠠⡤⢺⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣰⠊⢙⢮⣛⡆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡼⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡏⢰⢆⢳⡼⠀⠀⠀⠀⠀⠀⠓⠚⠓⢢⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⣸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⣹⠀⠀⣴⣏⣳⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⣖⡯⢿⡋⢿⣿⢙⡛⣽⣿⣷⣦
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⣰⠃⠀⢈⡗⠀⠀⠀⠀⠀⠀⢀⡼⣏⠍⢣⢛⢐⣾⣿⣷⣈⠐⡁⣲⠛⢤⡀
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢰⠏⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⡼⠁⠘⢞⣠⡶⣻⣿⢸⢿⠱⣎⣾⠃⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⡟⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣸⠁⠀⡇⠀⠉⠀⡛⠇⠘⠛⠀⠀⠀⠀⢀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠿⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡇⠀⢀⣾⠃⢀⣀⣤⡤⠀⠀⠀⣸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠃⡰⠋⠛⠩⢗⣻⣭⢾⡏⠀⢀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡞⠁⢀⣀⣤⠼⠿⠿⠟⣀⢀⣾⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢠⠏⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⡴⠋⡠⢾⠉⢠⣤⠀⠀⣠⣴⠃⡼⠉⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠶⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠞⢁⡞⠁⠀⢧⢈⡄⠀⠀⠀⡏⣼⠁⠀⣇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡖⠁⢠⠏⠀⠀⠀⢸⣇⠋⠀⠀⢻⠀⣿⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠣⡀⠟⠀⠀⠀⢀⡞⠸⠿⠀⠀⢻⠀⢹⡀⠀⡟⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡀⠉⠀⠀⠀⢀⡾⢁⠷⡄⠀⢠⣼⡂⠀⢷⡀⢹⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠠⡄⠀⠀⠀⣄⢇⠀⠈⢧⡘⠦⠤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠘⡆⠀⠈⠁⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣾⡿⠁⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣣⣶⣿⠟⠁⠀⠀⡀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⠛⠛⠛⠛⠉⠁⣀⡀⠀⠀⢹⡄⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠛⠻⢿⡷⠀⠀⢾⡿⠛⠛⠉⠙⡿⢸⣿⡄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣿⣧⡀⠀⠀⠻⠛⠛⢿⣷⡀⠀⠀⠀⠀⣼⣿⣷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠻⠿⠶⠀⠀⡀⢀⠈⠛⠿⠶⠄⠀⠀⣿⣿⣿⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⣸⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠰⣿⣭⣭⣿⠄⠀⠀⠀⢢⣿⣿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠉⠉⠀⠀⠀⢀⣞⣾⣿⡿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⣀⠀⠀⠀⠀⣀⣴⢋⣼⣿⣿⠁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣄⠀⠘⣿⣷⣶⣿⡿⠁⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢸⣿⣿⣿⠁⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⡇⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⠁⠀⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡇⡀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⣷⠀⠸⣇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⣌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⡶⡙⢾⡿⠋⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠉⢀⡴⢫⣾⣿⣾⢠⠄⡀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⣘⡴⢋⣴⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀
⠀⠀⠀⠀⠀⡰⠃⠀⡰⢋⡴⠟⢋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣄⡀⣀⡀
⠀⠀⠀⠀⡸⠀⢶⠊⡔⠋⠀⠀⠫⠇⢀⣀⣀⣤⣞⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⢠⠃⣰⡏⠀⠨⢀⣀⣤⣶⣿⣿⣿⡟⣿⣿⣦⡙⠛⢧⣄⣀⠀⠀⠀⠀⢀⠀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢠⣾⣵⠛⣿⠻⠻⢿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣷⣶⣤⣭⣯⣴⣠⡶⣿⣿⠀⢸⣿⣿⣏⠹⣿⣻⣿⠻⠘
⠀⠀⢠⣿⠟⠁⠀⠛⢶⡐⠀⣳⠛⢿⣿⠓⠨⢍⠓⢦⣁⠹⢿⣿⣿⣿⠛⠻⠹⣿⡇⠀⣸⣿⣿⣿⣦⣇⣿⣿
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠓⠻⣄⠀⠙⡦⠤⠴⠃⠀⠉⠙⠛⠛⠛⠛⠉⠛⠛⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⣼⡧⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠟⠁⠀⠀⠀⠀⠀⠱⡌⠛⣷⣤⣀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿⢰⡀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠈⠏⠟⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠨⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇⠀⠃⡀⠀⠀⠀⠀⠀⠀⡘⠀⠔⠋⠉⠙⠻⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼
⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠉⠒⠤⠀⠀⢄⠀⠠⠀⣰⠑⠃⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡏⡇
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣠⡶⠖⠛⠶⠷⢶⠶⠗⠚⠒⠲⢦⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡇
⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡇⢸⠀⠒⠛⠷⠆⠀⠀⠀⠀⠀⠶⠟⠓⠀⢧⢸⠀⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣰⡷⠚⣫⣿⣿⡲⣄⠀⠀⢠⠶⣺⣿⣿⣛⢿⣦⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢹⡃⠈⢳⣾⣿⡿⢸⡶⠲⣟⠀⢷⣾⣿⡟⠘⣿⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣸⢧⡀⠠⠍⠉⢀⡼⠁⠀⢻⣄⠀⠉⠭⢀⣸⣇⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⡉⠛⠛⠛⠉⠰⣦⣤⡶⠉⠛⠓⠛⢉⣴⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⠟⢿⣶⣤⣄⣀⠀⠈⠋⠀⢀⣀⣤⣴⡟⠟⢧⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣇⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣦⣤⣮⣽⣿⣿⠛⠋⠁⠈⠙⠛⣿⣿⣧⣵⣤⣤⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⠀⠀⢣⡤⣤⠴⠒⠲⣄⣀⣀⠀⠀⠀⢀⣀⣀⡈⠙⣋⡽⢁⣤⠀⠺⣶⡂⢀⠘⣏⠛⢋⣉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠤⡤⣬⣄⣀⣀
⠀⡴⠋⠉⠱⣏⠀⠀⠜⠁⠀⣼⠋⠉⠉⣟⣁⣀⢹⣰⠏⢠⡞⡟⠀⠈⠛⠁⢸⣧⠘⢦⡸⣹⠋⠀⠈⡆⠀⡀⡀⢠⠞⠉⠉⢳⠀⠀⣸⡏⠀⠀⠙
⠀⠼⢀⠀⣰⡇⠑⢤⡜⢌⠀⠼⡄⠀⢠⠏⠉⣼⠛⠋⣠⡟⣴⠇⠀⠀⠀⠀⠠⣿⢣⣀⠛⠼⣦⠀⢠⡧⠐⠄⠀⠘⡆⠀⠀⡴⡠⠺⠅⢧⣠⢀⠼
⠑⠒⠂⠀⢠⠀⠀⠸⠍⡀⠃⢈⡞⢱⢀⠓⠂⠙⠳⢶⢧⣀⣯⡀⠀⢀⢂⠀⠀⣸⠀⠙⣦⣴⠟⠈⡉⠋⠙⠀⠀⠀⠺⡙⠘⡅⠈⠀⠀⠠⠙⠁⡖
⠉⠀⢠⣤⡌⠀⠀⠀⠀⠙⠈⠈⠃⠐⠀⢲⣀⠀⠀⠀⠈⠉⢻⣄⡁⠁⡆⠁⣁⡿⠈⠉⠉⠉⠈⢑⠠⠀⠀⠀⠤⠄⠠⠁⠀⠁⠀⡄⠀⢠⣀⣬⡇
⠀⠀⡾⠉⠃⠀⠀⠁⠀⠨⢶⡀⣴⢲⠈⢷⠃⠸⠓⡇⠀⠀⠀⣿⣿⡟⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⢰⢲⠤⠃⠀⠁⠀⠘⠋⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⣿⡇⠀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⣿⣷⣶⣿⡿⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠁
⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠉⠁⠀⠀⠈⠁⠀⠒⠤⡀
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣢
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⠒⠙⠂⠈⠈⠀⢻⡀
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢾⣯⠁⠀⠀⠀⠀⠾⠁⣿⡇
⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠
⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⡀
⠀⠀⠀⠀⠈⠐⠀⠀⠠⢄⠀⠀⠈⣄⣀⣀⣀⣀⠤⠶⠃⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⡈⠁⠀⠀⢜⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠑⢜⣆⠀⠀⠈⢢⣳⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⡿⣧
⠀⠀⠀⠀⠀⠀⠀⢷⢄⠀⠀⠀⠀⠀⠙⢦⠴⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠑⢄⠀⠀⠀⠀⠀⠑⠄⠾⠤⠀⣀⡀⠀⠀⢀⡀⠀⢀
⡆⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠉⠐⠀⠀⣤⣤⠤⠤⠄⠒⠀⠰⠤⠤⠴⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠈⠁⠚⠂⠒⠚⠚⣉⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠉⠁⠀⠀⠈⠉⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠒⠒⠒⠒⠒⠂⠐⠒⠒⠂⠒⠆
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡆⠈⠈⠁⠉⠈⠉⠉⠉⠁⠁⠁⠈⠈⠉⠉⠉⠉⠉⠉⠉⠈⠉⠀⠁
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣁⣀⣀⣀⣀⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀
⣿⣿⡋⡛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⣁⣶⣭⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠱⣤⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠄⡐⢀⠀⠀⠀⠀⡀⢂⠜⢲⣿⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠤⠄⠀⠒⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⠊⡠⠀⣴⠋⣰⠋
⠀⠀⠀⠀⠀⠀⠰⡆⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠖⢀⠊⢠⠞⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠐⣔⡀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠄⠈⠋⠉⠁⠀⠀⠀⠀⡠⠀⠔⠁⡴⠛⡠⠋
⠀⠀⠀⠀⠀⠀⢠⣷⠐⡀⠀⠛⠃⠉⠉⠁⠀⢀⢀⣀⡠⡀⠀⠒⠂⠀⠀⠀⠠⠊⡠⠊⡠⠞⢰⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⡀⠀⠐⠒⠒⠂⠉⠈⠀⠀⠀⠀⠀⡀⣀⠤⠤⠔⢁⠔⢡⠟⠀⡴⠃⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢰⡀⠀⠀⡤⡀⠤⠒⠀⠋⠉⠉⠈⠀⠀⡠⠂⡠⣅⡴⡇⣨⠎⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⣀⡠⠠⠴⠀⠀⡔⢀⠌⢠⠎⢀⡗⠁⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⡀⠀⠉⠉⠉⠀⠀⠀⠀⣠⠊⡠⢡⡰⠻⡠⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠰⡀⠀⠓⠃⠁⠁⠀⠀⠌⠙⠬⣞⣣⣎⠠⠤⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠐⠒⠃⠀⠸⣄⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢦⠀⠀⠰⠴⠀⠾⠋⠄⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⡀⡒⠒⠛⠛⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣧⡀⠀⠀⠀⠀⣀⣀⡤⠄⠞⠋⠙⣉⣉⣬⣥⣤⣤⣴⣶⢿⣯⣽⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣜⠲⣐⣊⣩⣤⡤⠶⠶⠞⢉⠯⠍⠡⣈⣉⣀⣤⡬⣤⣤⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡿⠛⢋⠥⠒⣉⠴⠊⡤⠖⢺⠟⠋⠩⠭⠉⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣋⣥⡶⠚⠉⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⡀⣀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠋⠀⠁⠀⠈⠀⠀⠉⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢨⣧⣴⣿⣷⣄⣴⣦⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⡿⢿⣟⣿⣿⡿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣷⣦⣍⡻⢿⣿⣾⣿⣿⣿⡿⣷⣦⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣤⣬⣽⣿⢿⡿⣿⣿⣿⣿⣿⣿⣷⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⡮⣙⠻⣿⣿⣿⣸⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⣉⠺⣷⡹⣿⣿⣿⣿⣿⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣀⢙⣷⡹⣿⣿⢿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣿⣦⣽⣷⣻⣿⣾⣿⡟⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣉⣷⣻⣿⢁⣠⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣧⢻⣿⠁⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣭⣭⣅⣈⡙⣧⢻⣶⠀⣿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣛⣛⠛⠻⣿⣯⢿⣿⠃⠀⣹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢿⣿⣿⣿⢿⠿⣿⣎⢿⡿⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣛⡉⢺⣿⣞⣿⣿⠇⢘⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠋⢐⣦⣄⠀⠀⠀⠀⠘⠛⠛⠿⢷⣶⣤⣝⣿⣞⣿⣖⣿⣿⣿⡃
⠀⠀⠀⣠⣦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣌⠻⣾⣿⣹⣿⠃
⠀⠀⣰⣿⣿⣿⣷⣷⣶⡇⣀⣀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⢿⣷⢿⡿⣿⠆
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⡾⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠻⣿⠅
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⢸⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⢇⢰⡇⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀⠀⢄⡀
⠘⠛⠿⠿⣄⣨⣴⡏⠢⠄⢻⣿⠟⢉⣪⠞⣇⠀⠀⠀⠩⠛⠂⠀⠀⠐⠶⠉⠁⠀⠀⠬⡑⠢⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠒⠭⢍⣛⠲⠲⠿⡿⢻⠋⠀⠀⠉⠑⠢⢄⠀⠀⠀⠀⠀⣴⠈⠀⠀⣤⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠺⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⢄⠀⠐⠀⠀⠀⠀⢀⠀⠙⢲⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠠⠀⠐⠛⠀⠐⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⢿⣿⠿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣿⣶⣷⣾⣶⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣤⣿⡀⠀⡴⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢀⣴⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡖⠃⡮⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣸⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢒⣧⠀⠀⠠⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⢿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠧⠦⠔⠂⠉⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⢸⣿⣿⡿⠿⠟⠛⠛⠉⠉⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⢀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⣀⣠⣤⣴⣶⣿⣿⣿⡿⢃⣷⢠⡠⠀⣿⠀⠃⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⣰⠀⠀⡀⢰⡇⠀⠘⠂⢠⡀⠀⣐⣒⣔⠒⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⡛⠀⣸⣿⡐⠠⢰⣿⡄⣈⡩⠉⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⣿⡤⠎⠉⢻⠥⡤⢀⠀⢞⠑⠚⢉⢡⠘⠛⣓⠿⠛⠫⠁⠡⠈⡀⠪⠩⣖⢄⢀⠈⠉⡃⠶⠸⠿⡧⠘⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠰⠋⢉⠀⣠⢶⡁⢀⠈⠁⠀⠈⠐⠀⠾⠖⠒⠉⠀⢀⡠⠄⠳⠀⢰⣇⠀⢄⠀⠉⠑⠙⢸⡧⣄⠀⠐⠈⡶⠚⠁
⠀⠀⠀⠀⠀⢀⠀⠘⠫⣀⠔⠂⠀⠀⠠⠀⡀⣿⢠⡆⠈⠒⠂⠀⠀⠀⢠⠀⠂⠁⢀⠀⢠⡇⣸⣿⠀⠀⠁⠀⠀⠒⠠⠒⠉⡆⠀⠀⡇
⠀⠀⢄⠀⡠⠀⠀⢀⠀⠀⠀⠀⠁⠈⣀⣀⣀⡻⠊⠁⢀⣄⠄⠀⠀⠀⣼⡆⠤⣎⣀⡠⠼⠷⠿⠛⠓⠉⢋⠄⠂⠁⠀⠀⠁⡇⠀⠀⠃
⠀⠀⠋⠠⠦⠁⡂⠠⠀⠈⠁⠂⠀⠻⠿⠿⠽⠿⠂⠀⠀⠀⠀⠈⠙⠀⠿⠷⠈⣀⡀⠪⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠶⠄⠘⠀⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠃⠀⠠⠦⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⠭⢜⡀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⠟⠉⢀⠉⣀⢸⠸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⣿⣿⣿⣿⢁⠀⢀⡏⢀⡇⢨⣶⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⢎⡽⣿⣿⣿⣧⡟⢀⡞⣠⠞⣠⢎⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⢃⣾⣿⣿⣿⣿⣫⣴⣫⣴⣯⣾⡋⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⣾⢠⡏⣼⣿⢧⣿⣿⣿⡟⠉⠁⠀⣿⡦⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⡏⣼⢡⡟⣿⢸⣿⣿⠿⠗⠀⠀⠀⠁⠀⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠃⣿⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⢸⠁⡇⢸⠀⣿⠸⡇⣿⣦⡄⠀⠀⠀⠀⠀⠀⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⡐⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢻⠀⡇⠸⡀⢡⠀⠸⡈⢿⣤⡀⠀⣾⣶⣶⡶⠆⢰⠇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢸⡆⠱⢆⢣⠈⣷⡀⢣⠀⠈⣿⡇⢹⣿⣿⠁⢠⡞⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢳⣄⡃⠀⢆⡘⣷⣤⣷⡾⠛⠀⠀⠈⢀⡴⠋⠀⠀⠀⠀⠀⠀⡜⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢻⡿⠛⠉⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠋⠊⣞⢁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⠈⠙⠾⠀⢠⠆⢀⠀⠀⠀⠀⠀⢠⡟⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠈⠳⣄⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⠽⢷⢲⣄⢄⠈⠀⠃⠷⢢⡦⠀⢠⡟⣴⣿⠋
⠀⠀⠀⠀⠀⣀⣤⡾⢫⣽⣷⣄⠀⠉⠒⠤⠤⠔⠁⢸⡛⢶⣦⣄⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠜⠾⡖⠀⠀⠈⠂⢠⠟⣴⣿⠃
⠀⠀⢀⣴⠟⢿⣷⣮⣳⣿⣾⣿⢿⡖⠄⠀⠀⢠⠤⠻⣇⢿⣾⣿⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⠏
⠀⢀⡿⠁⣤⠈⠠⠙⣿⣿⣿⣿⢆⢻⣆⠀⠀⠀⠀⠀⢿⣧⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡼⣻⠋
⠀⣸⠇⣮⠁⡟⣷⣽⣹⣿⣿⣿⣜⡆⢻⡄⠀⠀⠀⠀⠸⣏⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢞⣿⠃
⠀⣿⢰⡟⢼⡿⣿⠟⢹⣿⣿⣿⣿⣷⠈⣷⠀⠀⠀⠀⠀⣿⠀⣿⡶⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⢠⡿⠱⣱⠇
⠀⣿⣼⣇⡾⢿⣿⣧⡏⢹⣿⣿⣿⣏⢀⢻⡄⠀⠀⠀⠀⣿⣠⡼⠷⠚⣻⡝⠳⢦⣄⡀⠀⠀⣤⡞⢺⡄⢻⡆⢳⡿⠡⢠⠏
⠀⢸⣿⣿⣀⠾⡦⢤⡄⢹⣿⢛⣿⣿⣹⣾⣧⠀⠀⠀⢀⡞⠁⠀⠠⠾⠿⠷⠛⣉⣽⠟⢷⣶⡆⢣⠀⢇⠀⠃⠈⣷⠁⠏
⠀⢸⣿⣹⣿⣿⡛⠈⢁⣿⣿⣿⣿⣿⣻⠏⢿⡀⡠⣄⡜⠀⠀⠀⠀⠠⠤⠒⢊⡡⢾⢶⣬⣙⣿⠘⠃⠈⠀⠀⠀⢸⠇
⠀⣸⠇⣿⣿⢻⡧⠘⠛⣿⣿⣿⣯⣬⣵⡾⠿⠻⣄⠈⢆⠀⠀⠀⠀⠠⠔⠒⣉⣴⣻⣿⡿⠿⡟⠀⠀⠀⠀⠀⠀⡼
⠀⢹⡆⢿⠛⠛⢗⣴⣿⣿⣿⠟⠋⠁⠉⠀⠀⠀⠘⡇⠈⠆⠀⠀⠀⣐⣂⣩⣴⠟⠿⡿⡂⢰⡛⢄⣀⣀⢀⣠⠞⠁
⠀⠀⢻⣾⣶⡰⣾⢿⣿⡿⠋⠀⠀⠀⠀⠀⣀⣀⡀⢹⡀⢸⠞⡛⢋⢹⣿⣷⣿⡇⠖⡓⣬⢦⠳⣤⣬⡭⠋⠁
⠀⠀⢸⡯⣿⣷⣴⣿⣿⣁⣄⣠⣤⣶⢿⣿⣿⣿⣿⢸⣿⣿⡀⠀⠸⠌⣿⡿⣶⣥⣤⣤⣬⡶⠟⠉
⠀⠀⠀⢻⡘⢟⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣯⣭⣴⣿⠋⣿⠃⠂⠂⠀⣿⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠉⠀⠈⠙⠛⠟⠻⠶⠟⠛⠛⠛⠉⠉⠉⠗⠀⠿⠀⠀⠀⠀⢺⡇


⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠉⠉⠉⠉⠀⠒⢤⡀
⠀⠀⠀⠀⡠⠂⠊⠁⠒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠈⢀⠀⠀⢀⡠⠄⠠⠤⣀⠄⠒⠂⢄
⠀⠀⠀⠘⣄⡀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⡇⢠⠁⠀⠀⠀⠀⢈⠀⠀⠀⢈
⠀⠀⠀⡘⠀⠸⠑⠂⡤⠤⠤⢤⠊⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⢄⠀⠀⠀⠀⢘⠒⠤⣰⣎
⠀⠀⠀⠇⠠⢺⢴⠁⠀⠐⡂⡜⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠘⡄⡛⠉⠒⠤⣸⢤⠀⡏⠁
⠀⠀⠀⠘⡌⡟⠈⠢⣀⣀⣀⠅⠀⠀⠀⠠⠃⠀⠐⠒⠒⠒⠒⠒⠒⠊⠛⠒⠂⢠⠄⠒⠂⠉⠢⠤⠬⠂⠠⡃⠀⡀⠀⠀⢸⠎⡰
⠀⠀⠀⠀⠈⣧⣀⠀⠈⠣⡍⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⡉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡥⠃⣀⡠⢺⠉
⠀⠀⠀⢀⡸⠀⠉⠙⠓⠏⠀⠀⠀⠀⠀⠐⡄⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⢀⠔⠢⡀
⠀⢀⠔⣉⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⢀⠆⠀⢀⣀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠈⢆
⢠⢣⠊⠀⠑⢄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠉⠒⠂⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡠⠐⠂⡀⠀⡆
⠀⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣆⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡐⠁⠀⠀⠰⠀⢠
⠀⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠕⡡⠋⠀⠀⠣⡘⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠀⠀⠀⢀⠀⢸
⠐⢠⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠤⢊⡵⢗⡝⠀⠀⠀⠀⠀⠘⢝⠢⠁⠢⡀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⡈⠀⢸
⠘⠘⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⣀⠔⠁⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⡈⠂⠠⠄⡀⠀⠁⢠⠀⠀⠀⠀⢠⠁⠀⢸
⠀⠀⠱⠀⠀⠀⠀⠀⢰⣀⠠⠄⠓⠉⠁⣀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠁⠑⢂⠀⠀⠀⢀⠄⠀⠀⠈
⠀⠀⠀⢣⠀⠀⠀⢀⠎⠀⠀⠀⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠆⠀⢀⠌
⡀⠀⠀⠀⢣⠀⠀⡸⢀⠤⠔⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠐⠈
⡇⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⡀⠀⢀⡀⠤⢀⣀⣀
⢠⠄⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠉⠁⠀⠀⠀⠀⠸
⠀⡇⠀⠐⠒⠒⠒⠒⠒⠒⠲⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⡁
⠀⢸⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢡
⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠺⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠁⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣦⡀⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡟⠂⢤⣤⡛⠂⠀⠀⠀⠀⠀⠀⠀⢨⣀⡀⠏⠉⠩⠭⠩⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⢄⠈⢄⠀⠢⠸⠢⡀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠡⠀⠈⠂⢕⣄⢣⠀⠱⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠑⢄⠀⠀⠉⠻⠀⠀⢁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠑⢄⠀⠠⠄⠰⢾⣆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠒⢄⡀⠀⠀⢀⡀⢤⠘⢉⡌⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠒⠈⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠉⠉⠉⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⢀⠀⠀⠀⡇⠀⣠⣤⣄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⡇⢸⣿⣿⣿⠆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣍⡛⣉
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⡇⣿⣷⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿⡇⣿⡏⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣇⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣷⣿⣿⣶⣶⣶⣤⣲⣞⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡾⡻⣿⡁⢿⣿⣿⣿⠿⡌⠙⠉⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣥⣤⣠⣏⣳⣾⣇⠈⠁⠱⢷⣶⠃⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣿⡇⠻⢿⣿⣿⣷⣵⣤⣀⣀⠀⣴⣿⣿⣿⣿⣿⡿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡅⠀⠻⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠢⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⣄⣠⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⠛⡿⣿⣿⣷⡄⠀⠀⠈⠛⠻⢿⠿⠿⣿⠘⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠘⠀⠀⢸⣿⣿⣿⣶⣷⣶⣿⡆⣠⣶⣾⣿⠰⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠉⠁⠀⠛⠈⠉⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣳⣷⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣷⠟⠷⣿⣿⣯⣽⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⢸⡟⡿⠿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢀⠀⢘⡀⢠⣴⠟⠋⠀⠑⠆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣿⣷⣄⣈⣛⣿⣏⠆⠀⠀⠀⠀⠀⠀⠀⠒⠿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠐⠂⡀⠰⠀⠀⠀⠀⢰⠁⣴⣿⣿⣯⡙⠿⠿⣻⣤⣙⣲⣦⣄⡀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠂⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⢿⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⡿⢛⣦⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⢷⢽⣿⢇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠷⠋⠀⠀⠹⠠⠄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢘⢉⡉⣻⣟⠉⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣼⢿⡷⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⣟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣐⠛⠿⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠂⠀⢀⣈⠁⠀⠀⠀⠂⠀⠀⠀⠄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⠶⠿⠟⠁⠳⠶⢶⣀⡤⠀⠀⠛⢷⣦⣭⣼⣷⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠴⢧⠠⣤⠀⢸⣷⣽⣿⣿⣿⣷⡀⢃⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠿⠷⠢⠀⠀⠠⠿⠶⠼⠿⠿⢿⣿⣿⣿⣷⡀⢷⡄⠀⠀⠀⠀⠀⠈⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣉⡀⠀⠀⠀⢠⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡍⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡄⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀
⠇⠀⠀⠀⠃⠀⡠⣀⡂⢀⣤⣀⣄⣤⣤⣴⣀⣀⠀⠀⠀⣀⣠⣀⣀⣤⣤⢤⡤⠀⢠⣤⠤⡤⠀⣀⣼⣶⣴⣶⣶⣖⠀⢀⠈⡄
⠀⠀⠀⠀⠀⠘⠃⠀⡀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⢿⣷⢽⡿⣿⢿⣿⣿⣷⣶⣿⣽⣿⣿⣿⢏⠹⠙⠛⢹⠗⠀⠈⠃⢄⡀
⠀⠀⠀⠀⠋⠀⠀⠀⠁⢸⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⠿⠹⢿⡿⠿⠟⠻⢿⢿⣿⡼⠛⡛⣋⣀⡀⠀⠢⠁⠐⠢
⠀⠀⢀⣠⡏⠀⠀⠀⠀⢸⡿⢾⣿⠿⠿⣿⣿⠟⢿⠛⢻⡋⢻⣿⣻⢻⣿⣿⠿⠿⢛⣺⢷⢾⢚⢿⣻⠟⢋⣭⣭⠉⣁⡙⣦⡄
⠀⢀⣾⣿⣷⡆⠀⠀⢀⣨⡵⠥⠍⠁⠄⠙⠫⠤⠀⠿⡻⢡⠤⠏⠩⠮⡟⠛⠁⠀⣀⠍⠤⢠⠀⠘⡓⢂⠀⠭⠩⠀⠽⢆⣊⠑⣵⡀
⢶⣿⣿⣿⣿⠇⠀⠀⠡⠨⠂⠈⣶⣤⣄⠀⢠⣰⠊⡉⣰⠀⣀⣀⠀⢠⠍⣹⣀⠀⣇⡀⠘⣃⣀⢠⣉⡈⠢⣄⣀⡀⠀⢨⣤⠠⠘⠟⢤⡀
⡿⢿⣾⣿⣿⠔⢁⠒⠀⡀⠀⣀⠹⢛⠯⠿⠿⣻⣦⣅⠈⢺⣿⣿⣿⡖⢧⣈⣨⠀⢠⢄⣺⣿⣿⡓⢀⣠⢦⣒⢊⣴⣶⣾⠐⠒⠒⠂⠀⠙⠆
⠋⡴⠘⣽⣦⣤⡄⠈⢈⡔⠁⠀⠀⠀⠀⡄⢨⢀⠀⠀⠎⠉⠉⠉⠀⠋⠀⠀⠀⡀⠀⡦⠈⠋⠒⠒⠐⠀⠀⡍⣈⠨⠄⠤⠒⠘⠋⣀⣀⣀⡁⠑
⠷⣠⠎⣻⢻⠏⠳⠀⢾⣴⣶⣦⣤⣀⣶⣗⡀⡀⠀⢤⠄⣠⣀⣀⡀⡀⠁⠁⠁⠃⠀⣇⡀⠀⣀⣀⠀⣈⠉⠁⢊⠢⣀⠀⠀⣀⣀⡉⡀⢀⡀
⠒⠁⠈⠹⠉⢃⡄⠤⠤⠿⢿⣧⣤⢿⠈⠎⠀⠁⠀⡀⢮⢹⣿⣿⡇⠠⠀⠤⠄⠀⠀⠈⢠⣾⣿⣿⣗⠈⠋⠉⣾⣐⠺⡋⠘⠿⣿⣛⢀⣀⢀
⠄⠁⠂⠐⡠⠀⠀⠀⠀⠀⠀⠀⡌⠈⠈⠁⠁⠀⠈⠁⠀⠀⠁⠉⠀⠈⠀⣀⡀⠀⠀⢰⠐⠁⠒⠃⠀⠂⠛⠉⠱⠈⠙⠃⡉⠋⠙⠐⠂
⠒⠠⢀⠜⣦⠀⠀⠀⠀⠀⠠⣼⠇⡐⠂⠂⠠⣤⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⣠⣀⡤⠄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠃
⠘⠀⠀⠀⠈⣃⠂⡶⢀⣒⡞⡄⡤⠡⠄⠀⠀⠀⠤⢢⠤⠤⠤⠀⠤⣉⡈⠀⠉⠀⠀⠈⣉⣋⠀⠀⣀⠀⢀⠈⠀⠤⠀⠀⢘⠠⣹⣩⡁⠀⠀⢀⡌
⠘⠀⠀⠐⠀⢹⠓⡐⠲⠿⡿⠻⠇⠀⠀⠀⠀⠀⢰⢯⡤⣿⣿⣴⣶⡦⠷⠤⠀⠀⠀⠀⠀⣿⡴⣶⣶⢠⡶⠄⠀⠀⠀⠀⡜⠓⣼⣴⣶⣶⣦⣼⠁
⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠁⠀⠀⠀⠘⠀⠐⠀⠙⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠛⠇⠝⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡅⠀⠀⠘⠀⡏
⠠⠤⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⡄⡄⠀⣀⠠⠀⠄⠀⠠⠀⠀⡄⠀⠀⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⡄⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠈⠉⠉⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⡀⠀⠀⠀⠬⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡝⡇⠀⠈⠁⠉⠉⠲⣬⣖⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠇⠀⠀⠀⣀⢉⣳⡀⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⡿⠁⠘⠀⠀⠀⠀⠉⠛⢁⠀⡰⣤⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠒⠉⠛⠉⣙⠀⡇⠀⠀⠃⠀⠰⣄⠠⡤⢹⠟⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⢀⠜⠀⠀⢣⠀⠀⣰⠤⡸⠗⠁⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡀⠈⠀⠚⠉⠓⠇⢸⡿⠖⠀⠰⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢀⡴⠚⠓⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⡀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⡽⣆⢋⠀⠀⠀⢠⠱⠷⠛⠉⠉⠁⠀⠀⠣⣤⠴
⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⠶⠖⠛⠛⠻⠿⢤⣀⡀⠀⣀⡀⠔⠁⠀⢀⣄⣀⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣙⣵⢴⣶⣤⣆⣀⣀⣀⠀⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⠤⠤⠀⠴⢧⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠤⠀⠀⠉⠉⠉⠉⠀⠒⠂⠐⠀⠒⠒⢀⣬⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠪⠕⣂⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣤⣤⣤⣀⣀⠀⠀⠀⠀⣠⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⡷⠖⠋


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