Feat Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢶⣽⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⡿⠛⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⣹⣿⣿⣿⣆⠸⠇⣸⣿⣿⡆⠀⠀⢾⠻⡏⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⡟⠿⠿⠿⣿⡇⠀⠀⠘⣦⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⠀⠘⠻⢿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠻⣿⣿⠖
⠀⠀⠀⠀⢀⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠉⠿⢿⣿⣿⣦⡀⠀⠀⠀⢀⣀⣼⡁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢾⢟⣿⣿⣦⡉⠛⠋⠉⠁⠉⠟
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣿⣿⣷⣶⡖⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠛

⠀⢀⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇
⠀⢸⡇⠀⠀⠀⠀⣀⣠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠚⠝⡍⢐⠑⠉⠗⢀⣤⣤⣤⡔⠛⠿⠏⢀⣤⡶⠟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⢀⠐⠂⢹⡅⠀⠀⠈⠙⠛⠃⢠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠘⡀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠘⠂⠆⠈⠁⠠⠌⠀⠀⠄⠀⠁⠀⠶⡀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⠋⢉⣤⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠿⣿⣷⣾⡿⢛⣿⣿⢿⣿⡟⠃⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠨⠍⠰⠇⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⣰⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠐⢠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠦⠤⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠐⣆⠉⠉⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⠶⢶⢤⣟⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠺⠿⠛⠻⠟⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⣤⡀⠀⠤⠗⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠾⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⢄⠀⠀⠀⠀⠐⣣⡄⣤⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⠀⢸⠀⠀⠀⠴⠀⢁⣤⠶⡲⠀⠀⢐⣶⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠛⠹⢦⡀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠊⡰⠂⠀⠀⣿⠀⢾⣀⠀⠀⠀⠀⠻⡿⠗⠃⠀⠀⢸⡟⠀⠀⢠⡷⠆⠀⠀⠀⠺⣿⡿⢷⡌⡄⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠂⠀⠠⢾⣿⣷⠀⠀⡀⠀⠁⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⣿⠃⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⡀⠀⠈⣶⡀⢠⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣓⣀⢀⠒⣇⠀⠀⢠⣄⣀⢀⡀⢠⣤⣠⣾⣿⣿⣦⡀⠰⣯⡄⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠘⠛⠛⠛⠛⠛⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠤⠏⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠚⠓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡁⠀⠈⠀⠐⠀⠒⠒⠲⣦⣀⡀⠀⣀⣀⣀⣀⣠⡶⠟⠛⠳⠄⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡆⠀⣤⠀⠊⠀⠚⠋⠃⠀⠀⠀⠀⠀⠂⠈⠁⠈⠋⠉⠉⠛⠻⣿⡿⠋⠁⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⡇⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠅⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⡷⣴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣀⣿⣷⠃⠸⢦⠀⠀⠀⠐⠀⠀⠀⠀⢠⡿⣀⠡⡏⠻⣷⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⢖⣿⠒⡒⠶⠍⠉⠀⢰⡇⠀⠀⠀⠀⢀⢰⡆⠀⠑⠺⠛⠿⠃⣰⣄⡟⠗⢿
⠀⠀⠀⠀⠀⠀⠛⠱⣄⣀⠺⠋⠂⠠⣽⡦⠸⢦⣩⠄⠀⡉⠁⠀⠀⠀⠠⡟⠘⠀⠀⠰⣿⢛⠛⠇⣽⠟⠒⠀⢸
⠀⠀⠀⠀⠀⠀⢰⠏⣹⣄⠀⣃⣠⠰⠈⠂⠓⢠⢸⠃⠰⠀⠀⠀⡤⡄⠀⠀⠀⢀⣀⠀⠈⢐⡆⣰⣯⣿⠯⠀⢸
⡄⠀⡀⣀⠀⣠⠄⢸⡏⠀⠸⣿⣿⡍⠀⠀⠀⠈⠈⠀⠀⠀⢠⠈⢀⣇⣰⣤⣠⣤⣼⡤⠈⠋⣠⠃⣘⣻⣿⡷⢸
⠁⠀⠈⠻⡧⠁⢀⣼⣰⣶⣐⣿⠏⠉⠀⡇⠘⢆⠀⠀⠀⠀⣤⠀⣸⣧⣽⣿⣿⣏⠀⡿⢷⣠⠋⢀⡛⢿⣿⣿⢿
⢀⡀⠀⠀⠁⣴⠟⢸⠃⠹⢿⡏⠂⠰⢤⡇⠀⠸⡆⢠⡄⠀⠂⠀⢋⣿⣿⣿⣹⣿⣷⡀⢀⣷⣄⠻⠁⢙⣏⠙⢸
⠃⠑⣂⠀⠰⢿⡀⡋⠀⢀⣫⣅⣀⣴⣾⡇⠀⣶⡶⠛⢳⡇⢀⣡⣾⣿⣿⣿⠉⡿⡟⣧⣾⣿⠁⢘⠃⢨⣷⡇⢸
⠀⠘⡟⡲⢤⣼⡏⣼⡖⢉⣿⠙⠻⣿⣿⠅⠀⠀⣿⡖⢸⣧⡈⠙⡷⡩⠿⠙⣿⠃⢀⣾⣽⣿⣷⣿⣾⣿⣉⣿⣸
⠀⠀⠈⠀⠀⣿⡍⢿⡇⣸⡏⠀⠀⢡⢸⠁⠀⠀⢰⠎⡉⠹⠃⠀⠀⠀⠀⠀⠋⠀⢨⡿⢿⣿⣿⣿⣿⣿⣿⡯⢽
⠀⠀⠀⠈⠚⠉⠑⠈⠓⠙⠃⠐⠀⠘⠙⠂⠒⠠⠒⠜⠳⠦⠴⠰⠆⠰⠤⠄⠠⠴⠟⠤⠎⠥⠼⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡎⡷
⠀⠀⠀⠐⣄⠀⠙⢦⡀⠀⠀⠀⠀⣀⣀⣠⠔⠀⠀⢸⡅⠆⡼⠀⡇
⠀⠀⠀⠀⠈⠳⣶⣾⣷⡀⡄⠀⡞⠁⢄⠀⠀⣀⠀⠀⢹⣆⠇⠀⣿⠀⠀⠀⠀⢀⠄
⠀⢀⣠⣤⣀⣀⠈⠙⠻⠷⢐⠂⢳⡄⠀⢱⡞⠁⢀⣠⡾⣿⠀⢀⡟⠀⠀⢀⠔⠁⡜⠁⡄⢀⡇⢀
⠀⠀⠤⢄⣀⣬⠿⠗⠂⠀⠈⠑⠚⠛⠷⣾⣧⡶⠛⠁⠀⢿⣤⡞⠁⠀⣼⣧⢀⢰⡇⠀⡰⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠛⣿⠀⠀⠀⠀⠨⠋⠀⠀⠀⠘⡟⠉⣻⣥⡔⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠳⣴⠿⢄⣉⣻⠀⢀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠏⡸⠈⢀⡜⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠛⠶⠶⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡏⠰⠁⠀⢸⠁⠀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⠁⡟⠀⠀⠀⠱⢤⣄⣀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠂⠛⠈
⠀⠀⠀⠀⣀⠀⢐⡀⠀⡇⢀⡇⣷⠀⠒⠒⠒⢾⡟⢸⣇⠀⠀⠃⠀⠀⠀⠀⠀⠀⣠⣾⣧⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⣠⣤⣷⠬⡇⢸⡷⣄⠀⠀⢿⡁⣼⠟⡜⠀⢰⠲⣄⠀⠀⠀⠀⢈⠉⣛⣛⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠒⠛⠉⠳⠌⡉⣿⣿⡇⢀⡟⠀⠀⠹⠀⠈⡇⠀⣀⠴⠋⠛⠻⣿⠋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣹⣿⣧⣿⡆⠀⠀⠀⠰⡀⠗⠋⣀⣠⠗⠀⢢⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠩⢿⣿⣿⣿⣷⡀⠀⠀⢀⣧⠶⠛⠉⠁⢰⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢰⠱⣄⠙⢿⣿⣿⣿⣷⣄⠘⠿⠒⠉⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠳⢤⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠑⠶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠷⢾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠲⢦⣶⣬⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⡟⣿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢿⠈⠇⠹⡇⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡅⠈⠇⠀⠀⠙⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠄⠀⠀⠀⠀⠀⠐⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣮⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠃⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡐⠀⠄⠷⣡⢼⣿⡀
⠀⠀⠀⠀⢀⠀⠀⢀⡌⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠐⠂⠀⣄⢀⠀⠀⠀⠀⠈⠬⠁⠀⠀⢸⠛⠻⠧
⠀⠀⠀⠀⢩⠃⠀⣹⡍⢶⣂⠀⠀⡴⠢⢤⡀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⣞⢂⡨⣧⣀⣠⣄⣴⣿⣆⠀⠀⠀⢠⣙⠀⢀⣦⣀
⠀⠀⠀⠀⢸⡀⠐⠑⣿⠘⠇⠀⠐⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡄⠀⣽⣿⠟⠻⠛⢿⣿⣿⣿⣿⡀⢲⣾⣽⡧⢶⣿⣷⣿⣷⠄
⠀⠀⠀⠀⠀⡇⢀⠀⠈⠠⠝⠓⠒⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣽⡆
⠀⠀⠀⠀⠀⠃⠉⠲⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⡿⠀⠀⣼⢋⡿⣿⡿⠇
⠀⠀⠀⠀⠀⢸⡆⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⠇⠀⢀⣯⠈⢡⣿⢿
⠀⠀⠀⠀⠀⣛⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠘⠛⢰⠾⡭⠛
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠄⠸⠿⠿⠿⢿⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠀⠀⢀⡀⠤⠤⠂⠀⠉⠁⠀⠀⠐⠫⠻⠦⠵⠒⢯⠟⢿⡏⣽⠋⠂⠀⠀⠈⠀⠘⡇⠀⠀⠀⠐⠠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡠⠶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣰⠀⠀⠀⠀⠀⡰⠂⠀⡀⠀⠉⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⢠⠘⢿⠆⠰⣤⡆⠀⠀⠀⠀⠀⡖⠀⠀⠈⠋⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⣤⠂⠀⠄⠀⠘⣡⣴⠭⡭⠄⣧⠀⡀⠀⠀⠀⢙⣦⣦⡕⢶⡀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣌⢡⡇⢤⣻⡀⢐⠀⠀⢃⣜⣛⣿⡙⢡⣼⣰⣇⣤⣀⣀⣫⣽⣶⣦⣾⠁⠰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠿⠷⠾⠓⠛⠋⠃⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠾⠝⣀⣠⣤⡄⢀⣤⠀⠀⠀⠀⠈⠀⣉⣻⠉⠉⠉⠙⠋⢓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠓⠒⠚⢀⠖⠓⠀⠀⠀⠀⡜⠀⠀⠀⡇⣿⣿⣿⣦⡉⠘⡛⣶⠞⠿⡄⠀⠠⡶⢀⠂⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢰⢁⣠⡄⠀⡇⣿⣿⡟⣚⢻⣽⣤⣹⡄⣦⣄⣴⣄⣙⣽⣾⡇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠦⠤⠤⠀⠄⠫⠤⠤⠀⠀⠀⠹⠄⠈⠤⠀⠃⠙⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠶⠶⢿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠘⠃⠐⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠⣀⣤⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠓⠦⢼⣾⣿⡿⠟⠛⠿⣿⣷⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⠁⠊⠀⡀⠘⣿⡇⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡤⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⢸⣿⣿⣆⠘⠟⠁⣸⣿⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣾⣿⠿⣧⣀⣠⠀⠀⠀⠀⢀⠀⣿⣿⣿⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⣿⣿⣿⣿⣦⣶⣿⡯⣬⣹⡟⠙⣧⠤⣄⢈⣉⠀⠀⠀⠀⠀⠀⠀⣀⢛⡊⣙⣉⣛⡊⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣹⣿⣟⡇⠀⠀⠀⠀⠀⠀⣿⢀⡀⣿⣶⡗⠀⠻⠿⠿⠶⠿⠷⠶⠾⠿⠿⢿⡿⣿⣿⣿⣾⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢻⣜⣣⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠾⠭⠭⠉⣀⡀⠉⠉⠉⠍⠉⢉⠻⠶⠒⠶⢄⣉⣙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠒⠂⠀⣀⣭⠁⠀⠀⠀⡀⠈⠀⡄⠀⠀⢈⠀⢄⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⠠⠤⠀⠀⠀⡠⠐⠉⠀⠂⠀⠀⠒⠦⣄⠀⠀⠀⠀⠂⠀⠀⠢⠀⠀⠊⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠖⠉⠀⠀⠀⠀⢀⠈⠀⠀⣠⠴⠶⡾⠟⠛⠲⣌⠻⣄⠐⠀⠀⠀⢠⠀⠀⠰⡀⠈⠻⡝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠁⠀⡴⠋⠁⠀⠐⠠⢢⡂⣹⣾⣧⠙⣬⠀⣂⠀⠰⠀⠀⠀⠀⠐⠘⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡠⠂⠁⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢀⡇⠹⣿⡿⠏⢹⡂⠱⠆⢈⠁⠲⡀⡀⠀⠀⠠⠉⠺⢶⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣷⠊⠀⠀⠀⠀⠀⠆⠀⣼⠁⠀⠀⠀⠀⣾⣳⣀⠀⠀⠀⣨⣳⣄⠲⣤⡃⠶⠀⢀⠀⠀⠀⠀⠡⢼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⠄⠀⠀⢀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣯⡿⠉⠛⡐⠤⣍⡒⢂⢀⠐⠀⠀⠂⢀⠂⡽⣅
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡠⠀⠀⠀⠀⢸⠀⢸⠃⠀⠀⢠⠖⠛⠛⠒⠦⣌⡉⠀⠀⠀⠐⢈⡗⢆⡑⠢⢠⡁⠀⠂⠀⠂⠀⠉⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡏⡰⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠈⠢⣌⢓⣀⣀⠡⠄⠀⠂⠀⠀⠀⡑⠈⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⠀⠀⠀⠘⠀⢸⡆⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠜⠉⠉⠉⠙⠶⣄⠈⡀⠀⠀⠀⠄⠀⠀⢹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠂⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠈⠑⢤⡀⠒⠠⠐⠒⠒⠒⠒⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡄⡆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢳⡀⠀⠀⠀⠀⠀⠀⠙⢦⣷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⣄⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢢⠈⢳⡀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠹⣄⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣈⠙⠶⠤⣤⣤⡤⠴⣞⡿⠋⠁
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⢩⡉⢛⠓⠶⣶⠶⠶⠶⠞⠛⠛⠛⠓⠒⠚⠛⠋⠉
⠀⠀⠀⠀⠀⣰⠋⠀⠠⠂⠘⣆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢤⡀⠀⠀⠀⠀⠀⠀⠑⢄⣈⠁⠈⠛⣦
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢍⡲⣄⡀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣸⠡⣀⠀⠀⠠⢀⣤⠞⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠓⠿⣶⣄⡀⠀⠀⠀⠙⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢻⠀⠈⠠⢀⡴⠋⠁⢀⡼⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠙⠳⢦⣄⠀⠨⠀⢄⠀⠈⢻⡄
⠀⠀⠀⠀⠸⡇⠀⣴⠟⠀⠀⢰⠏⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣆⠀⠑⢄⠀⣷
⠀⠀⠀⠀⠀⢻⣼⠃⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⠣⣸⡆
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠛⠓⠲⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢸⡇
⠀⠀⠀⠀⢠⡏⠀⠀⡄⠀⣼⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⢸⡇⠀⠉⠀⡀⣿⠈⠀⠀⠀⠀⠀⠑⠀⠠⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠤⠤⣄⣀⡂⠀⠹⣄
⠀⠀⠀⠀⢸⡇⠀⢤⡀⠀⢻⡀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠛⠲⢿⣆
⠀⠀⠀⠀⠀⣷⠀⠄⠀⠢⠼⣧⡀⠀⠀⠀⠂⠐⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣦⣀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠙⠃⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣶⣦⣄⣒⡒⠒⠛⠛⠛⠓⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠈⠻⠟⣭⣭⠿⠟⡲⠛⠿⣿⡷⠀⠀⣈⡁⠤⣲⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⣀⣴⠾⢿⣦⡀⠀⠀⠀⢤⣀⠁⣹⣿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠰⠟⢩⣴⣿⢦⠉⠛⠷⠀⢄⠀⠙⠃⠘⠿⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣿⣶⣶⣦⡀⠀⠀⠀⢢⡀⠀⠀⠀⠸⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣉⣉⣾⣃⣉⡉⠉⠉⠀⠀⠸⠆⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⠙⠛⢋⣟⠙⠛⢋⡛⠀⠀⢠⡄⠀⣼⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠉⠠⠖⠋⠙⠻⢀⠈⠉⠀⠀⣼⠛⠀⢉⠀⠀⣄⠀⠀⢂⠀⠀⠀⠀⠀⠡⠀⠀⢀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡵⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠤⠽⣴⣆⠹⡇⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⠒⠀⢰⠀⠀⠀⠀⠀⠀⣠⠂⡰⢚⡍⣏⠀⠇⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣿⡌⢻⣾⠀⠀⠀⠀⠀⣰⣿⠛⣡⣿⢀⡈⠀⠀⠀⣿⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠳⠀⣼⡟⠀⠀⠀⠀⠀⠻⡇⠀⠈⢀⠋⢀⣤⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣤⣼⣆⠰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⣉⣼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄⠈⠁⣉⣄⢀⣠⣄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⡆⠀⣊⠄⢀⠇⣠⡄⠀⠀⠀⠻⠃⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠑⠀⠀⠶⠋⠀⠠⠐⣪⡴⠀⠈⠉⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣸⣶⣄⣀⠀⢻⡩⠨⣝⠆⢀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠁⠀⠐⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠓⠟⣧⣈⣧⡟⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠃⡰⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢦⣄⠀⠀⠀⠀⠙⢿⡅⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⣈⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⠀⠀⠇⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢠⣤⣴⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⣈⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡍⠉⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⣄⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⠲⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢶⣶⣞⢿⣿⣞⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢂⣰⢿⣮⣿⢷⣽⣿⣷⣿⣿⣿⣿⣯⢷⣄
⠀⠀⠀⠄⠀⢠⠀⠀⠦⠀⢰⠀⠀⠀⠀⠀⠀⣰⣿⡟⣱⣿⡿⣿⣮⡟⢷⣝⢿⣾⣿⣾⣿⣿⣿⣯⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢿⡷⣰⢋⡽⣡⡶⣻⣴⢶⣻⣤⣿⡷⠹⣷⣿⣯⣿⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢻⣷⡁⠁⠉⠈⠉⠚⠋⠾⢫⡿⠶⣿⣿⣿⣿⢮⠿⣿⣿⣿⣧⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣿⡻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣿⣛⣛⣻⡷⣮⣻⣿⣿⣸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢹⡿⣏⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣾⣯⣭⣟⣛⠷⣾⢧⢯⣟⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢱⣿⣾⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣭⣽⣛⡶⡌⡿⣽⣻⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢄⢠⠀⠀⣸⠀⠀⠀⠀⠈⠹⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣿⡷⢼⡳⠃⣠⡄
⠀⠀⠈⠀⠀⢀⢄⡑⣤⣠⣧⡔⢀⠄⣂⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⣯⣟⠒⢡⡏⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠏⣼⠑⢍⡁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣝⣁⣾⣗⡒⠀⠁⠀⠤
⠀⠀⠀⡠⠀⠛⠃⡩⡢⢧⡑⠤⣈⣑⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠔⠊⠔⠐⠋⣇⠀⠐⢌⠂
⠀⠀⠂⠀⠀⡠⢀⠞⡐⡧⣈⠢⠄⠈⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢈⠐⠁⠜⡠⣳⢌⡑⠢⢄⣀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠌⠜⡼⢀⢦⡉⠒⠤⣀⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠂⢤⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⣞⠀⣸⠇⠈⠀⢢⡉⠒⠤⢍⣉⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⠄⡄⣡⠾
⠀⠀⠀⠀⠀⠸⣀⢻⠀⠀⠀⠀⠙⣁⠂⠤⠤⠀⡈⠧⣄⠀⠀⠀⠀⠀⠀⢸⡇⢄⠙⢁⠼⡇
⠀⠀⠀⠀⠀⠀⠉⠘⠀⠰⠀⢠⠀⠀⠰⡶⠒⠒⠒⢂⣈⠵⠂⢤⣀⣀⠀⢸⡇⢀⡍⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠆⠀⡄⠐⡀⠱⡈⠻⠋⢥⡤⠐⣊⣡⢚⣡⢋⡝⠱⣈⡩⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⡀⢀⠀⠐⡀⠑⡄⠑⣄⠱⣄⠈⠑⠀⣭⠞⢇⣠⡯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢷⡀⠂⠀⠘⠄⠈⢢⡈⠢⡀⢑⣤⢞⣁⠠⠐⠉⠀⠀⠀⢀⢰⠠⠀⠄⠀⠄⡀⢠⠀⠀⠀⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠼⠦⠄⠓⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠂⠀⠁⠈⠈⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣧⠀⠈⢉⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠿⠟⠻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡆⠀⠀⢻⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠟⠨⡁⢀⣠⣄⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠋⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⡉⠀⠀⢄⣠⣼⣾⣿⣿⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠋⠀⠠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⠟⢉⣀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠘⣿⣿⣿⣷⣦⢀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣹⣾⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⣦⣈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⡀⠀⢬⣿⣿⣿⣿⣿⣿⡃⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⣁⣬⣿⣿⣿⣿⣿⠟⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠍⡑⢦⣜⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣶⡾⠟⠛⣿⣿⣿⣿⣿⡗⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⡿⠻⣀⣸⣦⣜⢾⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣋⠝⢻⣿⣿⣿⣿⣯⣴⣶⣿⣿⣿⣿⡇⠉⠀⠀⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⡿⠟⠓⠈⣷⣾⣿⣟⢯⣓⠮⣓⢭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣛⣋⡉⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⠷⢤⣤⣼⣿⣿⣿⣿⡻⠗⢪⡁⠨⣷⣹⣿⣿⣿⣿⣿⣿⣫⢶⣛⡭⢭⣭⣭⣿⣶⡬⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣿⣿⣧⣭⣤⣾⣿⣿⣿⣿⣿⣿⣟⠳⣄⣩⣦⣿⣿⣿⣿⣿⣿⣿⣟⡵⠋⠡⡶⢻⣛⣿⣿⣿⣿⣦⣌⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢞⡉⣀⣿⣿⣿⣿⣿⣿⣿⣷⣄⠐⠊⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⡬⢀⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠄⢐⣿⣿⠿⡎⠛⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣩⠵⠒⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⣻⠟⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⠛⠒⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⡟⠁⠀⠀⠀⠁⠠⠖⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠒⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣷⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠋⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⣙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢠⠀⠀⠈⠿⠀⠀⠀⢠⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⣠⣴⣾⣿⣿⣷⣤⣀⣀⣠⣤⠤⠤⠒⠂⠈⠙⠋⠉⠙⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣧⣶⣒⡀⠀⠀⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⢀⠇⢀⣴⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣤⣌⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣄⠀⢘⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣠⣴⡿⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⠿⣿⣿⣿⣷⣦⡤⠤⠶⣛⣍⣭⣮⣦⣤⣤⢀⣠⣄⣠⣄⣠⣤⣦⣤⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣷⣦⣭⣭⣥⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⡽⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⡞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣤⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣤⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⣿⠹⣷⠿⠛⢋⡉⠉⠉⠉⠉⢉⡙⠛⢿⣾⠏⣿⠃⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⢿⠀⠉⠛⢷⡺⣤⣀⠠⠂⣀⣤⢖⡾⠛⠉⠀⡿⢀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠻⣦⡳⢕⣦⡀⠱⠹⣾⣦⣴⣿⠏⠎⢀⣴⡮⢎⣰⠟⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣴⠀⢀⣾⠛⣦⣝⠻⣷⣄⣻⣿⣿⣟⣠⣾⠟⣫⣴⠛⣷⡀⠀⣦⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⡇⡇⠀⢸⡇⢸⡟⠈⠛⢮⡙⢿⣿⣿⡿⢋⡴⠛⠁⢻⡇⢸⡇⠀⢸⢸⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢷⣷⠀⢸⣧⠘⣿⣄⣀⣸⡿⡌⢻⡟⢡⢿⣧⣀⣠⣿⠃⣼⡇⠀⣾⡿⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡌⣿⣆⠀⢿⣷⣬⣉⣛⣉⢀⣷⢈⠁⣾⡀⣉⣛⣉⣥⣾⡿⠀⣰⡿⢡⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡄⠘⢌⢿⣦⡀⠻⢿⣿⣿⣿⣿⣿⢰⡆⣿⣿⣿⣿⣿⡿⠟⢀⣴⡿⡡⠃⢠⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠻⡀⠈⡈⠿⣿⣶⣤⣈⣉⣉⣁⣼⣸⣇⣧⣈⣉⣉⣁⣤⣶⣿⡿⢁⠁⢀⠟⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠠⠘⢆⠻⡘⠌⣿⡟⢉⠋⣩⠞⠋⢻⡏⠙⠳⣌⠙⡉⢻⣿⠡⢃⠞⡰⢃⠄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠱⡀⠑⢝⣦⣼⠃⣦⣼⠃⠀⣰⢟⠻⣆⠀⠘⣧⣴⠘⣧⣴⠏⠊⢀⠎⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⡑⢄⡀⠙⣿⠀⠁⢸⣴⠞⢁⡼⢧⡈⠳⣦⡇⠈⠀⣿⠃⢀⡠⢊⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠙⠲⣾⡆⠆⢸⠁⢰⠋⢠⡄⠙⡆⠈⠇⢸⠘⡿⠾⠋⠱⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⢷⠘⡄⢧⠈⢦⠞⠳⡴⠁⡼⢠⢇⡾⠐⠀⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⢰⣧⠸⡌⢢⡀⣿⣿⢀⡔⢡⠇⣼⡇⢺⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⡇⠈⢦⡹⠃⠐⢏⡴⠁⢸⣿⣧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧⠀⢿⣿⣆⣰⣿⡿⠀⣼⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⡟⠻⠟⠻⣷⠾⠋

⠀⠀⠰⠆⢀⠒⢲⢦⡍⣿⢃⡞⢀⣐⠠⠞⡠⡢⠂⡴⡲⠢⢂⣰⢤⠒⡂⠒⠀⠐⠀⣦⡴⢿⣤⣷⣾⡇
⠀⠀⠐⠂⣅⠀⠀⠁⠁⢣⠋⡠⣿⡿⡡⣠⠞⣡⢂⠊⢦⡎⣥⣶⡼⠈⢁⣀⠙⠀⡦⣅⣽⢾⡛⠨⢞⡃
⠀⠀⠈⠉⠠⢿⣿⣂⠂⣰⢞⡵⣏⣤⠞⣡⠞⠑⠧⢰⣷⠆⠀⠞⣣⠀⠈⠉⣴⠿⠛⠻⣿⠓⡉⠐⠛⠃
⠀⠀⠀⡁⢰⣮⡽⣫⠞⣡⢎⣵⡿⢋⡶⣥⠨⠾⠀⠀⢀⣤⢶⡿⣿⠘⠘⢃⣄⣢⡴⠤⠾⣞⣛⣛⣋⡂
⠀⠀⠀⠄⠠⣫⠞⢁⡜⠁⡶⠋⡴⢫⣚⡁⠀⢘⣡⢾⠻⣝⣦⠷⠟⠀⢘⣲⡬⠶⠶⢛⡏⠽⠉⠾⠇
⠀⠀⠀⡀⡔⢽⡢⢉⡠⠛⢠⠖⡴⢢⡆⢠⣄⠙⠉⡦⠓⡉⢰⣠⣶⠺⣏⣩⠱⠚⠉⣀⠠⢐⣀⣀⣶⠁
⠀⠀⠐⠘⡉⠚⣤⠚⢁⡔⣡⣊⠀⢸⣿⢸⣿⡷⡠⢁⣭⣾⣻⣿⠾⡋⠡⢰⠶⡲⠀⠒⠛⣿⠭⠯⣿⠅
⠀⠀⠠⠑⡀⣚⢁⡤⢢⢞⠕⠃⠘⠘⣧⡎⢫⢊⡴⠋⡊⢋⣵⠷⠒⢣⢻⣇⡋⢴⢂⠎⠀⢐⠻⣷⣿
⠀⠀⢰⠈⠐⠰⡛⡫⠆⢀⠤⠐⣴⡄⠈⠙⡵⡏⠈⠀⠹⢌⣧⡴⣾⣯⣥⢊⠀⠈⠁⠿⠃⠘⣇⢿⣽
⠀⠀⠰⠈⠒⢾⡿⢁⢊⣤⣭⣄⠤⠄⠔⠎⡥⡂⢸⣴⢃⡎⠃⠿⠫⣕⣠⠧⡲⣄⠠⠄⠀⢴⢻⢸⣿
⠀⠀⠠⢌⠂⠀⠃⠅⣿⡐⠸⣿⣷⣶⢶⡊⢂⡴⠻⠀⢈⠧⡘⣤⡞⢻⣯⠦⠗⢌⠻⣶⠆⠀⠉⠜⠡
⠀⠀⠰⠈⡘⠀⠀⠀⠊⠙⠛⠻⡭⢖⢫⣶⢋⣾⡇⢀⠜⢦⠑⢯⣷⢊⠔⢀⣼⠛⣑⢍⣺⢀⠀⠶⠆
⠀⠀⠀⡑⡄⢀⡀⣀⢰⠆⡔⡠⣿⡀⠸⣧⣉⡝⢀⢮⡴⣯⢳⣸⠶⠇⣴⡿⠱⠀⢀⣁⣿⡟⢀⢂⡛
⠀⠀⠐⡰⡸⡸⠗⣇⡟⡼⠁⠙⣮⡻⣷⣬⠋⢠⠃⠘⣹⣭⠀⠻⣶⡾⡿⢡⠁⠀⠀⠰⠋⠓⡘⠊⣵⡆
⠀⠀⠐⡡⡱⢱⠀⢿⣟⡰⡀⠀⠨⠙⠊⠁⣠⢿⡦⣄⣑⠚⠂⢰⣭⣥⡇⡆⠀⠘⣶⣦⠀⣤⣀⣷⡏⡇
⠀⠀⢄⡇⠃⢀⠀⠚⠺⠓⠊⠉⢀⡀⢲⣿⡇⣴⢮⣝⣚⡿⠽⡟⣀⣀⠚⢀⡠⠰⠟⠋⠐⣣⡿⡇⠁⡇
⠀⠀⠈⠀⢀⣒⣀⠠⣢⢀⢼⣷⢋⡅⠀⠑⠀⢧⠻⣿⠋⢁⢌⡼⡩⣃⠀⠸⠛⠀⠈⠟⢩⠟⢠⡇⠀⡇
⠀⠀⠀⠁⠒⠒⠒⠂⠰⣷⠞⣿⣏⢉⠞⢀⠀⢀⠀⢀⣐⣵⢟⣪⣶⠟⠀⠀⢀⠀⠀⡴⠃⣬⢹⡟⢠
⠀⠀⠉⠈⠁⠉⠠⠭⢭⣟⣒⠲⣤⣎⡠⢅⠀⠀⠴⢶⢖⣚⣫⡅⠀⡇⣿⢃⣾⢄⠖⡐⠘⢯⢼⡧⠤
⠀⠀⠚⢸⣋⡭⢿⣻⠾⣋⡵⢊⠕⡒⠩⠓⢮⡂⠀⠀⠙⠺⠆⠀⣠⠀⠁⣼⠓⠭⠀⢰⠛⠳⢼⠆⡟⡂
⠀⠀⠘⠉⢀⣨⠽⢒⡏⣁⣤⣥⣭⢀⠁⠰⠄⢨⠷⣄⠀⠀⠀⠈⠙⠂⠼⠉⠇⠀⠦⠸⠰⠄⠘⠀⠃⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠙⠲⣤⠀⠀⠀⠀⣠⠖⠋⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣧⣔⣿⢣⠤⣼⣧⢀⡀⣼⣧⠤⡜⣿⣢⣼⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠋⠉⠉⣉⣿⣍⠀⢯⠘⢙⡿⢸⡇⢿⡋⠃⣽⠀⣩⣿⣉⠉⠉⠙⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢇⢐⡀⣾⡉⢉⠉⠻⣮⣻⣯⠀⢨⣇⠀⣹⣟⣵⠟⠉⡉⢙⣷⢀⠀⡸⠈⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⡾⠛⠓⠶⢻⡗⠚⢀⣄⣤⣽⡿⣷⡙⠋⣾⠿⣯⣤⣤⡈⠓⢻⠟⠶⠚⠛⢷⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⢴⢓⣆⣼⠃⣼⣫⣖⣋⡻⣞⣕⣕⣺⣣⣷⢟⣛⣺⣽⢧⠸⣆⣰⡒⡆⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣌⠉⢩⣄⠸⣿⣿⣽⢲⢽⣟⠝⣫⣝⠫⣻⣯⡖⣯⣿⣿⡇⣨⡍⢉⣡⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣬⡿⢿⣿⣏⡈⢳⣿⡻⠱⣒⣫⣿⡟⠉⠉⢻⣿⣝⣲⢎⢟⣿⡟⢁⣹⣿⡿⢿⣥⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣄⠈⣿⡄⢿⣯⣞⣷⢞⡻⣷⡄⢠⣼⢛⡳⡞⣱⣍⡿⢠⣟⠁⣠⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢤⠉⢀⣿⢃⠎⠻⣿⣾⣿⣫⣿⠀⠘⣿⣟⣿⣷⣿⠟⠰⠘⣿⡀⠉⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣴⠟⣡⣾⡀⢀⠎⠻⣿⣿⣿⣤⣴⣿⣿⣿⠟⠱⡈⢀⣧⢘⠿⣦⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣉⣛⠻⠶⣿⠁⢌⠙⢿⣿⣿⡿⠋⣡⠈⣷⠶⠛⣛⣁⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣤⣭⣷⣬⣧⡀⠀⠀⢿⡯⠀⠀⢀⣼⣥⣾⣭⣤⡤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣛⣿⣿⡄⠀⣺⣗⠀⢠⣿⣟⣛⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣩⠝⢷⠀⢻⡇⠀⡿⠫⣭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠸⠀⣸⣇⠀⢇⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠉⠉⠁

⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⣠⣤⡀⠀⠀⢸⠀⡇⠀⢀⡀⠀⠀⠀⣀⡀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢶⣾⣿⣿⣷⣶⠂⠀⡇⢸⠀⢀⣴⣿⣿⣿⣿⣤⡀⢸⠀⡇⠀⢸⢹⠀⠀⢰⠋⡇⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠂⠐⠀⠀⢸⠀⡇⠀⣿⣿⡿⠿⣿⣿⡆⠀⡇⢸⠀⠻⠿⠿⠿⡿⠿⠟⠁⢸⠀⡇⠀⠀⣿⣤⣤⣼⡞⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⡇⢰⣿⣿⣿⣿⣿⣿⣧⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⡇⠀⢰⣿⣿⣿⣿⣿⡆⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠸⣿⣿⣿⣿⣿⠇⠀⡇
⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⣇⣾⣿⣿⣿⣿⣿⣿⣿⣄⡇⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⣇⣀⣀⣼⣿⣿⣿⣇⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣴⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣄⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⢀⣠⡾⣟⠁⠰⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⣀⣤⣾⣿⣗⣨⣿⣦⣌⣿⣿⣿⣿⣿⣿⢸⣿⡟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⣿⣷⣤
⠀⠀⠻⠆⢀⣤⣭⢙⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣿⣭⣍⡀⠀⠀⠰⠿⣿⣿⣿⡟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠘⠿⠓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣤⣄⣀⣤⣿⣿⣿⣿⣴⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⡾⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⡀⠀⠀⠹⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⠋⣸⡿⠁⠀⠀⠀⠀⢀⣀⣀⣠⡤⠀⠀⣽⣿⡿⠀⠀⢄⣨⣶⣿⣿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣟⣿⣿⣿⠷⠶⠿⠿⠛⠛⠛⠛⠙⣫⡿⣅⣮⣾⣫⣤⠦⠂⠈⠹⢿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣁⣭⣿⣿⣾⣿⣿⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣉⢻⣿⣿⣿⡿⠟⠉⣙⡛⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡆⠻⠞⢛⣋⢠⣴⣾⣿⣿⣿⣷⣜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢀⠛⢻⡿⠟⠋⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠑⠽⠿⠿⠖⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠙⠲⠀⠀⠀⠀⠤⠖⠢⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⣤⣤⡲⠀⠀⡴⣠⣤⣀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠛⠁⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣁⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣏⣹⣷⡆⠀⠀⠰⣞⠛⠛⢳⣦⠀⠀⠀⣷⣿⣿⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡄⠀⠀⠉⠛⠛⠋⠁⠀⠀⣰⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣟⠻⣿⣶⣬⣝⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠙⣿⣧⠀⠀⠀⠉⠉⠉⠙⠋⠻⣿⣿⣷⣞⠙⢿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣽⣿⣿⣏⣸⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡻⢿⣇⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠉⠺⢿⣿⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣴⣦⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠿⠿⢿⠓⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣆⣠⣦⡜⠛⢂⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡌⣻⣿⣠⣠⣤⠎⠙⠉⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣶⠛⠋⠀⠉⠙⠉⠉⠿⣿⣿⣶⣶⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⡀⣷⠃⢀⣠⣦⣤⣤⡦⠤⣤⣿⣿⣿⣿⣿⣿⣥⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⡤⠼⢿⡆⢉⠹⣿⠿⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣀⠀⣿⡄⠀⠀⠀⢀⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡿⠋⠁⣠⣦⡿⣹⡔⠻⡄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⠀⢸⣿⣤⠀⣨⠀⢿⠷⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠋⠁⠀⠸⠿⣿⠃⣿⣿⣦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇⠈⠙⣇⣠⠏⠘⢺⡟⠋⠂⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠐⢾⣿⣶⢏⣻⣿⣿⣿⣤⣠⣶⠃⠀⢀⣀⠀⠀⢸⡟⠂⠘⠻⣷⣾⡇⠾⠿⠟⠛⠋⠉⣁⠀⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⡄⠀⠀⠹⢿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⣿⠟⠃⠀⢀⡏⠀⣼⣧⠖⠴⠔⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠠⣖⣿⡖⠀⠀⠘⢿⣿⣄⢛⠛⣿⣿⣿⣿⣿⣿⣿⣏⢽⠿⠛⠁⠀⣤⠞⠀⠀⠙⢿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠚⠛⠛⠻⠦⣿⣄⢳⣾⣆⠀⠘⠛⢛⣿⣿⣧⣤⣉⣹⠏⠉⠁⢀⡘⠻⣷⣤⣴⣾⠋⠀⠀⠀⢐⡈⠻⣍⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⡶⠛⣡⠤⠶⠿⢌⠹⣿⡷⠀⠀⠈⠁⠊⠸⠏⠙⠻⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣷⠀⠀⣀⠈⠉⠙⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠥⠶⠋⠁⠀⠀⢀⠀⠡⡟⠁⠀⢿⡏⠀⠀⠀⠀⢻⣿⣿⣿⠟⠄⠀⠀⣼⣿⣷⡵⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠛⠋⠄⠰⡄⠀⣿⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠐⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⡰⠚⠁⠀⠀⠈⠆⣿⠀⣀⡀⠤⠤⣀⠠⠤⢤⢤⣤⣴⣾⣾⣿⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⣿⣞⠁⠀⠀⠀⠈⠳⣦⣦⡀⠀⠉⠛⠷⣶⣷⣿⠛⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠚⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⢀⡄⠈⠻⡛⢦⠠⣄⠀⢄⡈⠉⣉⣉⡭⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠈⠀⠀⣄⠙⠾⣧⡀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠛⠟⠳⠤⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠤⠄⠠⠤⠤⠤⠄⠰⠢⠖⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⣲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢟⠿⠟⠋⢡⣴⣞⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠠⡑⡄⢈⠿⠋⠙⣷⣄⠀⠀⠀⠀⠀⠌⠀⠀⠖⠀⠐⢛⣻⢯⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠈⠀⠀⠀⢀⣾⠿⠆⠀⠀⠀⠀⣀⡀⠀⠀⠐⣤⣀⢈⠉⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠸⢫⠉⠀⠀⠀⠛⠋⠐⡄⢸⣿⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⢘⠁⠨⡽⡆⠀⣠⣤⣄⣸⣿⣿⡟⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣹⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣸⡋⠑⠚⢿⣿⣿⡆⠀⠀⠀⢈⣀⠔⢀⡀⠀⠀⠀⠀⢠⡝⠛⠉⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠀⠘⠀⠠⣯⡇⠀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠈⠠⢤⠄⠀⠀⠀⠀⣼⠇⠀⠀⣿⣿⣿⠿⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣄⠋⠄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣠⣿⡏⠀⠀⠀⣿⠿⠋⣰⡿⠿⠦
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠐⠿⡿⢱⣦⣀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⠾⠿⢿⠁⠀⠀⠘⠃⠀⣰⡿⠁
⠀⠀⠀⠀⣀⣤⠀⡎⠏⠀⠔⠺⠛⠁⢸⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠑⢤⠀⠀⠀⣰⡿⠁
⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡸⣼⠆⠀⣰⡿⠁
⠀⠀⠀⣯⣿⣇⠀⠀⠀⠂⠠⠀⡠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣷⣷⠀⣰⡿⠁
⠀⠀⠀⠉⢻⣿⠀⠀⠀⠀⢀⣾⣣⣿⠷⣿⢯⠯⢭⣵⣿⢿⣫⣿⣿⣿⠟⠋⠀⣼⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣾⢝⣾⡟⣸⣇⠃⠠⠿⠟⠖⠋⢱⣟⠟⠁⢀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢆⠑⠀⢀⣶⠛⣯⣿⣿⠃⢹⡽⡣⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣸⢻⣻⡿
⠀⠀⠀⠀⠀⠢⡀⠸⣏⣾⣿⣇⢟⣴⡷⠉⠈⠁⠀⠀⠀⠀⡄⢀⠀⡐⠀⠀⡏⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠈⠉⠿⢿⢶⠫⠿⠁⠀⠀⠀⠤⠊⡀⠂⠈⡆⠀⠀⠀⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠉⢳⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⡤⠀⠀⠀⠂⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⢤⡀⡀⠀⠀⣀⡀
⣀⣾⡞⣧⣀⠀⠀⢀⠀⣠⢹⡆⠀⢸⡄⠀⠀⠀⣶⠹⡦⣤⢀⣠⣧⡄⠄⣾⣆⣀⣷⢸⡇⠀⡄⣿⠉⠙⣿⠄
⠙⣿⡧⠿⠟⠶⠿⣾⣷⣿⣿⠃⣴⣾⣇⠀⣠⣼⢿⣾⡿⡋⠅⠀⠀⠀⠀⠀⠀⠉⠛⢾⣿⣷⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⢿⣿⡿⣋⠔⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣍⢹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇⡇⠀⣀⣠⠴⠶⠤⠤⡀⠀⠀⠀⠀⠀⠀⣠⡤⠄⠀⢉⠛⠋⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡢⣄⢀⡀⠇⡼⠉⢀⠃⣃⡝⡲⡙⢦⠒⣀⣀⣼⠏⠀⡠⣤⣤⡄⠁⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣖⠍⡂⢶⣿⠀⠛⡰⣯⣷⡒⣦⢸⠿⠿⢿⡇⠀⢘⠉⣟⣷⡘⠀⢸⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠺⠤⠼⢿⣆⠐⢆⣸⣈⡴⠏⡈⠀⣀⣈⣧⡀⠘⢄⢘⡉⠌⢠⠈⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠲⠤⠄⠐⠛⠛⠿⠿⠟⠛⠛⠢⠤⠤⡤⢴⠾⠾⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⢁⠄⠉⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡷⠋⠀⠘⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠰⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠐⠁⣠⠊⠀⠀⠀⢢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣧⠀⠀⠀⠙⠀⣃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣮⡂⠀⠀⣷⡤⠶⠀⠶⠶⠶⣿⣿⣿⣷⣄⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠞⠁⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡡⠤⠲⢤⣀⢿⠋⠋⠀⠀⠀⠀⠀⣿⡟⢻⣉⡛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠻⣏⢈⠀⠀⠀⣀⡠⠾⣍⣸⡿⣋⣡⣤⣤⣄⡀⢀⡀⠰⣄⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣀⣤⣄⠀⠀⣰⣿⠐⢃⠀⣆⡈⠀⠀⠙⢁⣼⣿⠷⠚⠙⠺⢿⣦⠁⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠿⠻⠙⠁⠙⣿⠾⠿⠟⠓⠋⠀⠁⠙⢦⡀⠀⢸⣿⣟⠀⠒⠀⡄⣸⣿⣦⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡤⠋⢀⡐⠀⠀⡁⠀⠀⠀⠀⠂⡈⠛⢷⣦⣉⠀⢠⣿⣿⠀⠀⢀⣴⣿⠋⣈⣆⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠈⠀⠀⠙⢸⠁⠠⠀⢈⠸⠶⢸⣦⣽⣦⢸⣿⣿⣗⣿⣿⠟⠀⢀⠘⠻⢧⣀⠾⠿⣦⢻⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⣸⣿⣿⣦⣤⣤⣶⣤⣾⣿⣷⠀⠙⢿⡉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠡⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢃⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡎⢧⢹⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠃⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⣷⠈⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣋⢻⡟⠣⠿⢿⣿⡷⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⢠⣾⣿⣿⠋⣙⣿⣿⣿⡍⣿⡷⠂⡄⠰⣦⣶⣄⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡞⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⠀⢸⣿⣿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⡜⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠀⠐⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡼⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⡀⠂⣴⣤⠐⠃⠀⠀⠀⢠⣤⡶⠂⠀⠀⢠⣜⠏⣀⠀⣾⢋⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⣿⣯⡼⣿⣏⠙⠟⣠⣦⡀
⠀⠀⠀⢼⢁⣦⠈⠁⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⡰⠋⢁⡴⠃⠀⠇⢾⣥⣾⡏⢠⣿⣿⡇⠙⣻⢿⣏⡑⠀⡀⠙⢊⣉⣰⣾⡿⠋
⠀⠀⠀⢈⠉⠁⢀⠀⠀⠀⠀⡀⠉⠀⠀⠀⢀⠀⠁⠀⠞⠀⢀⡄⠀⠀⡄⠉⢤⢸⡿⢿⣁⣀⣛⣿⣯⡷⠺⢷⡀⠹⠛⠋⠹⠄
⠀⠀⠘⡵⠚⠋⠀⠀⠀⠀⢀⠤⠒⢀⡰⠟⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣤⣴⣦⣿⡉⠀⢻⡶⠀⠀⡤⠈
⠀⠀⠀⠰⢦⠤⢶⣤⣄⣀⣤⣄⡀⠀⣀⣠⣤⣤⣶⣄⣀⡀⢁⣄⣽⣶⣿⣦⣀⣀⣁⣀⣀⣧⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⢀⣬⣿⣿⣷⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⢶⣤⣤⣤⣤⣔⣺⠋⠏⠁
⠀⠀⠀⠀⠀⢾⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⠿⡏⣛⣙⠋
⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠛⠛⠛⠀⣾⠛⢫⣤⣿⣦
⠀⠀⠀⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡻⠿⢿⣿⣿⠘⠛⠢⠀⠲⠘⠣⠶⢿⣿⣿⣿⣤⠖⠐⠂
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠸⣦⣵⣦⣄⡚⠀⠀⣀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⢵⡄⣴⣿⣿⣿⣿⣿⣿⣷⣌⡡⠿⠛⠛⠟⠤⣌⠉⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠉⠙⠉⠻⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⢠⣿⣿⣿⣥⠀⠀⢀⡀⡏⣾⠧⢀⡴⠄⠁⠒⠀⠀⠀⣶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⠀⢀⣄⣀⠂⠀⠀⡰⣿⣿⡏⠀⠀⠀⠀⠀⣷⡿⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠈⠞⢀⠀⠀⢧⡄⠈⠁⠀⠀⠀⠀⠀⠀⢀⡴⠤⠤⣤⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠃⣀⡼⠟⠀⠂⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠚⠀⢆⠉⠉⢿⡆⠀⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠃⠰⢨⣟⠀⠀⢹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠆⠡⠄⣠⠸⠀⣀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠙⢦⢀⠀⠀⠂⠙⠟⠁⠠⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣴⣾⢋⡟⠯⣩⡕⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣤⠶⠺⣤⣤⣦⣶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠀⠣⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣶⡾⢿⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠨⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣙⣊⣹⣭⣉⣋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣟⣛⣛⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠦⢤⣤⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣦⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⢠⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠀⠐⠆⣀⣤⠀⠀⠠⠅⠉⠀⡴⣦⣤⠚⠗⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⢰⠀⠀⠘⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⣶⣷⣶⠄⠀⠀⣀⠀⡤⠀⣄⠀⠀⠀⡄⢀⢠⣄⣠⣿⠀⠠⡌⠠⣴⣾⣷⣤⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢰⡄⡀⣿⠀⡿⠄⠰⣷⣷⠀⢨⢯⡟⢻⡦⠲⡇⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⣀⣤⣾⣕⣾⣿⣧⣟⠀⠀⣿⣇⡀⢸⣾⣷⣴⠇⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⢿⣿⣿⣧⣬⡿⢏⡁⠈⣿⡇⠁⠨⠋⠙⣿⣿⣿⣶⣾⣿⠿⠆⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠂⠊⠙⠂⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢩⣽⣿⡆⠀⠀⠀⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢀⣴⣶⣎⡟⢀⣼⣿⣧⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⣴⣿⣿⣿⣿⣦⣬⣭⣿⡏⠀⢀⣀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⣾⣿⣏⣽⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢋⣙⣦⣄⡀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠋⠁⠀⠀⠸⣿⡟⠉⢉⠶⣄⡉⢀⣴⠈⠛⢻⣶⣼⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⡀⠀⠀⠀⠈⠙⣦⣎⡐⢾⣿⣿⣿⣷⡀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡜⠁⣻⣿⣿⣿⣟⡿⢟⡅⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⣺⣿⣿⣿⡿⠿⠟⢿⠃⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣰⡟⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢸⣿⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏
⠀⠀⠀⠀⣰⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠁⢼⣿⣿⡿⠛⠛⠂⡀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⢰⡟⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⡞⣿⣿⣧⠀⠠⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢿⡆⠀⠀⣠⡼⠛⠁
⠀⠀⢠⣿⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣬⣫⣽⣆⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅⠀⠀⠀⠃
⠀⠀⣼⠇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠡⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡿⢏⣴⣿⡿⠿⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⡛⢻⣿⣳⣦⡀⠀⣽⣿⠋⠁⠀⡠⠂⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⢺⣭⣿⣿⣷⣜⣿⠋⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠲⡦⢀⠀⠀⠀⠀⠀⠀⢳⠈⢿⢯⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⢤⣀⡀⠀⠀⢀⡀⠀⡇⢠⣿⡿⣛⣿⣿⡿⣿⣿⣿⣿⣷⣦⣀⡸⠃⠀⢀⣠⣴⣤⣤⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢠⠀⠀⠘⣌⣻⡅⠻⣿⣯⣾⣭⣫⣿⣿⣿⣿⣿⣿⣒⣺⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠿⣷⣜⣻⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣯⣽⣿⣿⣿⡛⠛⠋⠉⠡⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢙⡿⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⡔⣊⣉⣭⠶⠃⠉⠛⢿⣿⣿⣿⣷⣦⡀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣴⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣀⣼⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⡿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⠀⢹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣠⣴⣤⡈⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣇⢨⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢩⡛⢷⣄⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⢩⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣯⠈⠁⠀⣻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⠀⠀⠀⠀⠲⠶⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢴⣾⣆⠀⠀⠀⡀⠀⠀⣤⡀⠰⣤⠀⢀⣤⣤⣶⣿⣶⣶⣿⣿⢖⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⡻⢿⣧⡀⠀⠀⠀⠀⢼⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⡅⠈⠿⠿⢷⣶⣤⣴⣾⣧⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⠙⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣷⣤⠀⠀⣼⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⣧⣄⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⡀⠀⠀⠀⠀⠀⢀⡠⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢿⣧⡄⠐⠁⣰⣴⠋⠀⣠⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣆⢀⣽⣷⣿⣴⣿⡁⠀⣰⠋⢀⡟⠉⣿⣛⣿⠩⣵⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠏⣿⣿⣿⠹⣿⣾⠃⢠⢞⣀⣼⣿⣿⡇⠀⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢻⠀⠀⠀⠀⣿⣿⣟⡀⠉⢁⡴⠻⢾⣿⣿⣿⣿⠁⠀⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠘⠀⠀⠀⠀⢼⣿⣧⠀⠉⢹⣛⣀⣶⣿⣿⣿⣿⠀⣀⡄⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⣸⡿⣿⠀⢀⠀⢉⣿⣿⣿⣿⣿⣿⠀⡿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⢰⠀⠀⠀⣿⡇⣿⡇⠈⣇⠀⠙⢿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣟⣿⣿⠇⠀⠀⢸⠀⠀⠀⠈⠉⣿⡇⠀⠇⠀⠧⢼⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⣼⢹⣿⡏⠀⠀⠀⢸⠀⠀⠀⠀⠂⢸⣿⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣾⣶⡏⠐⢶⣶⣶⣤⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⢹⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣡⠆⠀⡀⡀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠉⠙⠛⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠿⠷⠶⠦⣤⠄⠀⠀⠀⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡏⠙⠛⢻⣷⠶⠶⠂⠀⠀⠀⠀⠙⣷⠒⠖⢠⣤⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣄⠁⠀⠀⠀⠀⠙⠛⠶⠤⢀⣀⣀⡀⠈⢳⡀⠀⠀⠈⠛⢧⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠚⠛⠀⠀⠀⠀⢀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠘⢇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡆⠀⠀⡠⠀⠐⠒⠀⠀⢤⡤⠄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⡏⠀⠈⠃⠀⠰⣶⣶⠆⠈⠀⠀⠀⠀⠀⠀⠀⠐⣶⡶⠀⠈⠀⠀⠀⠀⣿⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠾⠀⠀⠸⡇⠀⠀⠀⠀⠀⠴⠿⠉⠀⠀⠀⠃⠀⠀⠈⠠⠄⠻⠟⠁⠀⠀⠀⠀⠀⠏⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡤⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠃⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢶⡿⣧⠉⠙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⣿⠁⠉⠉⠀⠉⣲⡞⠋⠀⠁⠈⠁⣠⣾⢳⠏⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠇⣶⠀⠀⠀⠊⠁⠈⠀⠀⠀⠀⠊⡳⣽⠇⠀⠀⠀⠀⠀⡰⠏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠉⡿⠀⠀⠸⠋⣦⠀⠀⠀⡆⠀⠀⠀⠀⣠⢞⡷⠋⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⢸⡇⠀⠀⠀⠻⡈⢧⣀⡀⠀⠀⣀⣠⡞⣡⠎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠘⠃⠀⠀⠀⠀⠙⢦⡉⠛⠒⠚⣉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣾⠉⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠠⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠟⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⠑⠒⠒⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠠⠛⠷⠖⠀⠸⠁
⠀⠀⠀⠀⢠⣿⠦⠛⠛⢻⣦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⢀⣞⣁⡆⢰⠀⡿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠉⠉⠙⢧⣠⣧⢸⠃⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠷⠶⢀⣤⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠿⠾⠓⣿⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡿⠶⠶⣶⣄⠈⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⡀⢀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠰⢤⣤⣄⣘⣻⣷⣦⣤⣿⠀⠀⠀⠀⢙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠙⠃⠀⠀⠈⠀⠀⠀⠉⠛⠛⠲⢶⢂⣉⣉⠉⠉⠉⠙⣿⣿⣦⣶⣴⠟⠁⢾⣇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⡶⣶⢠⣶⢤⣤⣤⣄⣀⣀⣀⣀⠐⠚⠛⠛⠛⠛⠛⠛⣛⣛⣉⣉⣿⣿⣿⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⡇⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⢿⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⠁⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠤⠤⠤⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡇⠀⠀⠐⠒⠒⠂⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣄⣤⣤⣤⣄⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣧⡴⠆⣈⣉⣀⠴⢄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣉⣉⡉⣉⣙⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠂⠉⠉⠁⠐⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠤⣶⣶⣶⠤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡌⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢶⣄⣰⡀⠀⠀⠀⢠⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠭⠍⢻⣧⣤⣀⣀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣿⣿⡿⣿⠛⢶⣄⣴⣶⣤⣤⣤⠀⠙⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠉⠀⢺⣿⣿⣿⣿⣿⣿⣠⣄⠀⠿⠲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⣷⣦⣀⠀⠀⠀⠈⣿⣿⡿⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣤⣀⡟⠛⢃⣠⣤⣀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⡉⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠓

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⡦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⠃⠀⠐⡿⠖⠒⣉⣀⠠⠴⠶⡄⠀⢀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠿⡠⠄⢨⡷⢧⠘⠁⠀⢸⠟⢀⡀⠀⠔⠃⢠⣿⣿⣦⣠⣾⣯⡐⠒⢀⣀⢠⠟⠛⠓⡀
⠀⠀⠀⠀⠀⣀⠀⠄⢀⣀⡼⡃⢀⠀⠀⠀⠰⠾⠀⡸⠃⠊⠁⠘⠄⠰⠏⠉⠐⡦⠎⢻⡏⠈⠙⠿⠿⢇⠄⠀⠛⠉⠃⠲⣼
⠀⠀⠀⠀⠀⠉⠀⠀⢸⡿⠤⠉⢠⡇⠀⢲⣆⡀⠀⡀⡀⠀⢐⠀⠀⢸⢂⠓⠀⠸⡀⢹⡧⠤⢤⣖⠐⣠⣀⠀⠀⣔⠀⢀⣩
⠀⠀⠀⠀⡄⠁⠄⠊⢿⠀⡐⠀⠀⠁⢠⡏⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣘⠁⠈⠁⠈⡉⠙⠻⡻⠏⠙⣇⣴⡗
⠀⠀⠀⠀⢀⠀⠀⠀⢸⢤⢰⠀⠀⡠⠘⣤⠤⠀⢨⣤⠀⡄⢦⡄⢀⠰⡀⣤⠀⡄⡀⠖⠸⠙⠠⠀⠀⠈⠀⠀⢸⡇⢈⠐⠂⣿⡏⠇
⠀⠀⠀⠀⠀⠁⠀⠀⠚⠈⠋⠀⠀⠉⠀⠈⠒⠃⠀⠀⠀⠁⠈⠀⠘⠸⣀⠿⠀⣿⠀⠀⠀⠀⠈⠋⠀⠛⠈⠁⠈⠀⠀⠁⠍⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⢀⣠⣀⠀⢑⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡄⠀⣰⢰⢠⡄⣉⠶⣏⠁⢈⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣶⣿⣯⣿⣿⣿⣿⣦⣼⣷⣧⡀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢂⢠⣿⡈⢨⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣶⠀⠀⠀⡴⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣯⣰⣿⣗⢸⣿⣿⣿⣿⣿⣿⣯⣽⡋⠁⣉⣉⠉⣠⣆⠀⠀⠀⠀⢸⡃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣼⣗⠆⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣟⡁⣷⣋⠖⣹⠭⡗⠉⠁⠀⠀⠀⠀⠀⣗⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠎⢻⡟⠃⣿⣿⢹⣿⣷⣸⣿⣿⣿⢛⡷⢩⡉⠹⣿⠓⠲⠀⠃⠀⡀⠂⡜⠀⠀⢠⡏⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠀⣾⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣤⡆⠈⠁⠷⠄⠸⠀⠒⢶⠀⠴⠙⡷⠀⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡗⠒⠚⣿⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠂⠀⢀⣐⣀⡀⢀⠌⠀⠀⠐⡟⠉⣿⣿⣷⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣸⣿⣿⡇⠂⠀⠀⡀⠠⡇⠀⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡇⠀⠰⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠏⠀⣾⣿⣿⣿⣿⡂⠀⠀⠴⢻⡏⢉⣿⣿⣿⣟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡄⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⣿⣿⡻⣿⠚⣿⣿⣿⣿⣯⣿⣿⣿⡆⠀⣸⣿⣿⣿⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠚⠀⠀⣹⣿⣷⣶⣷⣿⠻⣿⣿⣿⣿⣿⢿⡏⣧⣼⠇⣣⣾⣿⣿⣧⣿⣿⣿⣿⣍⣉⣉⣉⣉⣠⣿⣿⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⢀⢠⣼⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⢻⣸⡇⢿⣿⣧⣿⣾⡏⣿⣳⣿⣿⣿⣿⢻⣛⠟⢿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⢰⣴⣿⣿⣿⣿⣿⣿⣿⡿⣀⠀⣼⣿⣿⣿⢾⣯⢧⣹⣿⣽⣬⡟⣭⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣽⢿⣿⣿⣿⢈⣷⡀⠀⢀⡉⠉⠁⠉⠋⠈⠻⠿⠛⠛⠛⠋⠉⠉⠻⢿⣿⡻⣿⣿⣿⣿⡟⣿
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⢲⣿⣾⣿⣿⣯⣾⣿⢿⣄⠈⠻⣏⣍⢹⠳⣆⠐⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢿⣿⣿⣿⣿⣝⠂
⠀⢠⢈⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢮⣿⣿⣻⣿⣿⣿⣿⡟⣺⣿⣆⠀⠙⣿⣿⡆⢹⣧⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⡄
⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⠟⢤⣶⣿⣿⣿⣯⡿⠛⣏⣹⣿⣿⣿⣿⣷⡀⠘⢿⢿⡄⣽⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣀⠼⣿⣿⣿⢿⢡⣿⣷⡄⠀⠻⣾⣿⣿⣿⣦⡈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠛⠛⠛⠿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣻⣿⣿⣿⠟⠃⣿⣿⣆⠀⠀⠀⠀⠈⠉⠁⠀⠱⣄⠀⠀⠀⠀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠙⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⢻⡅⢻⡿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠒⠲⠶⠤⢄⣀⠈⠈⠉⠉⠉⠙⠛⠿⠛⠻⠿⣧⡀⣴⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣿⣻⣿⠿⠟⠿⠿⠿⠿⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⢂⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣉⣉⣩⣭⣤⣄⡀⠀⢰⣶⣦⡄⠀⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⠶⠶⢶⣦⣤⡐⠙⠿⣿⣿⣿⣿⣿⣷⠰⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠶⠒⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠭⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢣⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠙⢄⠀⠀⣰⠉⠉⠀⠘⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⢙⡴⠁⠀⠀⣀⠀⣀⣠⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢓⣞⣋⠤⠤⠤⠤⠼⠿⣏⡀⣻⠁⠀⣀⣀⣀⣀⢀⣀⣀⣀⡀⠤⠤⢀⣀⣀⣀⣠⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⣇⣀⡠⠤⠒⠢⢄⣀⣴⣿⠛⢶⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡶⠒⠒⠂⠈⠉⠁⠒⠀⠉⠁⠀⠀⠀⣀⠤⠖⠁⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠇⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠸⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠑⣶⠒⠂⠉⠉⠉⠉⠉⠉⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⠴⠮⠅⠈⠉⠲⢄⣸⠀⣀⠤⠔⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣨⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠁⠀⠀⣔⣠⠖⣧⠠⡀⢤⢠⢠⢠⢠⢀⠀⢠⢠⠐⠀⡠⡄⠀⣠⡄⠔⠂⢠⠀⠠⢀⡄⠄⡆⠀⡉⠢⠉⠤⢡⠘⢱
⠀⠀⠀⢀⡠⠖⠊⠉⠀⢀⡠⠔⠒⠈⠉⠀⠀⠁⠈⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠈⠀⠀⠋⠀⠈⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠉⠀⠁⠀⠁⠀⠁
⠀⡠⠚⠁⠀⠀⠀⢠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠉⢹⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠃⠀⣢⡄⠠
⠀⠀⠀⠀⠀⢁⠻⣿⡿⠁⠀⡀⠀⠁
⠀⠀⠈⢳⡀⠋⠒⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣤⠄⠻⢿⡀⠀⠀⠀⠀⠀⢸⣏⠘⡛⠻⣿⣿
⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⢀⠀⠀⢘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠉⠀⠁⣨⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠟⠳⡀
⠀⠀⠀⠀⠀⠀⠃⣠⠄⣠⠘⣦⡴⠖⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⠁⠈⠟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠀⠐⣈⣢⡟⢠⠆⠀⠁⠀⠀⠀⡀⢰⡦⠀⠀⢈⡀⢰⣧⠀⠹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠒⢶⣽⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠀⠀⠀⠀⠀⢴⡀⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡞⢸⡇
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣧
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠈⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠛⠂⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠘⠃⢀⣤⣶⣶⣶⣤⡁⠀⠀⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⢚⡉⠡⢻⡟⣽⢿⡟⠀⠀⠀⠀⠈⠙⠀⠄⠠⠀⠀⡀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠺⠋⠀⣼⠁⢰⡇⠀⣿⢸⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡁⠘⠃⠀⣿⡇⠀⡿⡜⠀⠀⠀⠀⠃⠀⠸⠀⠀⠰⠀⠳⣶⠀⠀⠄⠉⠞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⢀⣴⣿⠇⣴⣿⡁⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣴⣿⠇⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣻⣿⣿⣾⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣹⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠙⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⣿⣽⣿⠏⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠟⣀⠀⠀⠐⣎⡆
⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐⣿⣽⣿⣿⣿⡿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣼⠓⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⠁⠀⠀⢾⣷⡆⠀⢨⡿⠀⠀⠀⠀⠀⡘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣽⣿⣽⠇⠀⠀⣠⠐⠋⢤⡅⠌⢁⣴⡀⡀⠳⡄⠙⢷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡶⢢⣿⠀⠀⠚⠁⠀⠀⠀⠀⠄⠀⠛⠃⠀⣾⣶⣄⡄⠉⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠺⡿⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠉⠉⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢷
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⡆
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⢠⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢋⡗⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⣀⡤⡁
⠀⠀⠀⠀⠀⠀⠀⢀⡜⡁⠎⡀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⠛⢙⠊⣓⠄
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⢀⣀⠔⠁⠀⠀⠀⠸⠉
⠀⠀⠠⣀⣄⡰⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⣰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⢦⠶⢆⣤⡀⢄⣠⣤⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠒⠚⠋⠁⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡀⠈⠲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠈⠀⡀⠀⠀⠀⠀⠀⠀⠈⠡⠉⠈⠁⠺⢧⣶⡆⠀⡀⢄
⠀⠀⠀⠀⡠⠄⠚⠛⠚⠀⡌⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠈⠀⠾⢿⡄
⠀⠀⠀⢀⣣⠇⠀⠀⠀⠀⢹⣏⡀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣴⣦⠆⠀⣿⡄
⠀⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡼⢂⠀⠀⠨⠅⢁⠰
⠀⠀⠀⠘⡇⡇⠀⠀⢀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠖⢯⡂⠀⠠⠌⠲⠛⠀⣠
⠀⠀⠀⡀⢧⠁⠀⠀⢸⡀⢨⡦⠄⣀⠆⠘⠁⠀⠀⠉⠀⠀⠀⠈⠀⡌⠹⠢⠀⠀⢧⠀⡄⠪⠀⢣
⠀⠀⠀⢙⠸⡐⡀⠔⠚⡇⠈⣡⣾⡄⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠸⢣⠂⣀⡬⡀
⠀⠀⢰⢡⠘⣧⣁⠀⠉⠈⢀⣈⣙⣃⣄⣙⣷⡀⠀⠀⠀⣠⠚⠀⠀⠈⠈⠀⡗⠣⠝⠂
⠀⠀⠈⡀⠀⢸⡯⡟⣴⠀⡆⢣⠈⠙⠛⠡⠄⠲⠤⢶⠺⡉⠀⠀⠀⠀⠂⠀⠷⠀⠃⠀⠀⠀⠀⠀⠈⠒⠄
⠀⠀⠀⠁⠀⣼⠹⠋⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⢰⠂⣸⠀⢨⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢙⠀⢠⠀⠀⡅⠀⣣⠀⠀⠀⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⠀⠘⠚⡃⡆⢸
⠀⠀⠀⠀⠀⠘⠸⠇⠀⠂⠣⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠁
⠀⠠⠄⠀⠀⠀⠀⠀⣀⠄⠠⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢴⡆⠀⠀⢀⣄⣮⣶⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣰⣯⣅⡀⠀⢀⣠⣤⣤⡴⢿⡿⣮⡿⠧⠤⣤⠀⠀⠁⠀⣷⠏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠛⢶⠟⠁⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣾⣿⣿⣿⠃⠀⠀⢈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠉⠉⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠈⠉⠉⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣤⣄⣠⣄⠀⠀⠈⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣧⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣿⡄⠀⠀⠀⠀⠙⠁
⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢩⣿⠋⢠⣽⡿⠝⣻⣿⣿⠀⣻⣿⡇⠙⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⣴⡿⠁⢠⣾⡟⠀⢠⣿⣿⠉⠀⣿⣿⡇⠀⢹⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣿⡟⠀⢠⣾⡿⠀⠀⣾⣿⡇⠀⢀⣿⣿⣿⠀⠀⣿⣿⡆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣽⡿⠁⢀⡿⣿⡇⠀⢠⣿⣿⠇⠀⢐⣿⣿⣿⠀⠀⢻⣿⣷⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠁⣼⣿⡇⠀⣼⣆⣿⠀⠀⢸⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⢸⣯⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠠⣾⡟⠀⣿⣿⠁⢀⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠨⣟⣿⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⡇⢀⣿⣗⠀⢸⣿⣿⡇⠀⠀⣿⣿⣯⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⡇⠐⣿⣿⠀⢸⣿⣿⡇⠀⠀⣿⣿⣯⠀⠀⢸⣿⣿⣿⠁⠀⢐⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⠘⣿⣿⡇⠀⠀⢻⣿⣿⠀⠀⠈⣿⣿⣿⠀⠀⢸⣿⣿⠃⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣄⣹⣿⣤⡀⣿⣿⣿⠀⠀⢸⣿⣿⡀⠀⠀⣿⣿⣿⠀⠀⣼⣿⡿⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⢼⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⣿⣴⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣆⣠⣼⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⣿⠇⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣿⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠛⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠄⠀⢠⣀⡄⣤⣤⣦⣠⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠆⠀⠸⣿⣷⣿⣿⣿⠟⠣⠄⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠁⠀⢀⠈⢀⠈⠀⠀⡀⠁⡀⠀⠀⢐⠈⢐⠀⠀⠀⡀⠁⡀⠀⠈⢐⠈⢀⠀⠀⠁⡀⠁⡀⠀⠨⢀⠈⢀⠀⣀⣁⣠⣤⣦⣤⣬⣀⣨⣀⣥⣀⠁⡀⠁⠈⠀⠨⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢈
⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠠⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠀⢀⢠⣼⣿⠿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣯⢻⣆⠀⢐
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⢀⣾⡿⣣⣾⣿⣿⠟⠋⠉⠉⠙⠛⠛⠛⢿⣿⣝⢿⣰⣿⠀⠐
⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡠⠤⡀⠀⣼⢻⣳⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢾⣿⣿⡇⠠
⠄⠀⢐⠠⢐⠠⠀⠄⠀⠄⠀⠀⠀⢐⠠⢐⠀⠀⠀⠂⠄⠀⠀⠠⢐⣀⣐⠀⠀⠸⣄⣀⡸⢀⣿⣼⡿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⠃⠠⠀⠠
⠀⠀⠐⠠⠐⠀⠀⠀⠀⠀⣀⡴⠶⠒⠛⠛⠛⠒⠂⠂⠄⡠⠖⠉⠉⠉⠉⠉⠒⢄⡂⠁⠀⠰⣿⣿⢧⣿⣿⠠⠦⣄⡀⠀⠀⣠⠴⠛⠃⠀⠀⢿⣿⢿⣿
⠀⠀⠠⠀⠀⠐⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢨⣿⣿⣼⣿⡷⠖⣾⡟⠣⠀⠈⠑⠺⣿⠟⠗⠀⠈⡅⠉⡟
⠀⠀⠈⠀⠈⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡟⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣷⠀⠐⠀⠐
⠀⠀⠀⢀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣵⣿⡆⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢿⣿⣆
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⣾⣷⣿⣿⣿⠨
⢱⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⢿⢿⠟⠁⠀⠀⣠⣾⣿⣿⣿⣇⠛⠈⠀⠈
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣠⢾⣿⣿⣿⣿⣿⣿⣦⢈
⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⢰⣝⣯⣻⣿⠟⠻⡎⢳⠒⠒⠋⠀⢸⣿⣯⢿⣿⣹⣿⣿⣷⣨⣀
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠁⠀⠀⠀⠀⠉⠉⠛⠻⠟⠃⠀⡴⠁⠀⠁⠀⠀⠀⠀⠙⠻⣧⣿⣷⣿⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠒⠒⠒⠒⠒⠒⠈⠁⠁⠀⠘⠟⠀⠀⠀⠀⠀⠀⣈⣿⢹⣿⣧⠟⣿⣷
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡀⢀⣀⠀⠤⠤⠔⠒⠒⠈⡅⠀⠀⠀⠀⠀⠀⣠⡴⢁⡼⢿⡏⠀⠛⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⣀⣀⠄⠴⠈⠀⡄⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⢾⣿
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⣀⠠⠠⣶⣷⠀⠀⠀⠘⠄⢔⡈⠒⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠉⠉⠀⠀⠀⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷
⠀⠀⢀⠈⢙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⣿⠛⠢⣀⠀⠀⠀⠀⡄⢀⠞⣿⣿⠋⠙⣿
⠄⠀⢐⠠⢀⠠⠀⠍⡉⠍⡃⠘⠊⢐⠠⢘⠢⣄⡀⠀⠀⠀⠀⢀⣠⠠⢀⠤⠤⠤⡤⠶⣷⠀⠀⢸⠀⠀⠀⠀⠀⢳⣿⣿⣷⣶⣿⣷⣄⢣⣸⢠⣷⣾⣿⣿⣷⣾⣿
⠀⠀⠐⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠉⠛⠛⠛⠙⠉⠐⠀⠐⠀⠀⠀⠀⠀⠘⡀⠀⢸⠀⠀⠀⠀⠀⠈⡋⠉⣿⣿⠋⠉⣿⣦⣿⠉⣿⣿⠋⠙⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⡠⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠲⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⠻⠿⠿⠿⠛⠋⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⣀⠙⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⣰⡀⠀⢸⣦⡀⠀⠈⠈⠀⠀⠀⠿⠋⠀⠁⠀⠹⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⡿⠻⣆⠈⢿⣿⣷⣤⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣤⣦⣀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣷⣤⣼⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⣼⣿⣿⣿⣷⣦⣄⠀⣠⣿⣆⠀⠙⢿⣿⣿⣿⣿⡿⠟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⣀⣀⣀⣠⣤⡴⠾⠿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣄⠀⠀⠀⣀⡤⠚⠋⠉⠉⠉⠉⠉⠁⠀⢰⡿⠛⠻⠟⠓⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠻⠆⠀⠜⠻⣷⠄
⠈⠓⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠙⠁



⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⠶⢏⡠⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣼⣷⣶⣶⣾⣷⣿⠀⠀⡄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⠿⢿⣿⣿⣷⣤⣥⡀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣧⣘
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣌⣿⣀⡹⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⣯⣋⣽⣯⡁⣉⠛⣋⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣯⡔⢊⡽⠟⠁⠙⠫⢤⢋⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣵⠃⠀⡀⠀⣤⠄⢋⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣟⣿⡄⠛⠁⠀⠀⠀⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⢻⣯⣷⣄⠀⠐⠀⢀⣮⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠉⢮⣽⡏⠀⠀⠀⠀⠿⡃⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠏⠁⢀⠰⠷⠊⠀⠀⠀⠀⠐⣷⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢘⠀⠦⠞⠀⠀⠀⣀⠀⠀⠘⣅⢀⡀⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠂⠀⠙⠢⡘⠳⠄⠀⢹⣇⠀⠀⢿⡠⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣦⡀⠐⣤⠀⠀⢻⡷⣞⢻⣏⣿⣿⣿
⠀⠀⠀⠀⠀⠂⠀⢀⠀⠘⣿⣿⣿⣿⣿⣿⣶⡞⣧⠀⣘⣅⣿⣿⢿⣿⡿⣿⣧⡀
⠀⠀⠀⠀⠰⠀⠀⡊⠀⢃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⢷⣾⣿⠷⢾⣷
⠀⠀⠀⠀⠀⠀⣀⣤⠾⢷⣟⣨⣻⣿⠟⣿⣯⣿⣿⣿⠟⠁⠀⠀⠀⠹⣟⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠛⢰⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⠻⣿⣶⣄⡀⣀⣀⣀⡬⣥⣤⠄
⠀⠀⠀⡄⠀⠀⣰⣿⣿⢧⣾⣿⣿⢿⢿⣧⡀⠀⣤⠈⠙⠻⣛⣉⣡⣠⣴⣲⢿⡆⠀⠐
⠀⠀⠀⠀⠀⣠⣽⣿⣿⣾⣿⣻⣿⡀⣾⣿⣿⣷⣬⣞⠃⠀⠈⠙⢤⣈⠍⠁⡼⡁⢀⠀⡄
⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠛⠓⠄⠤⠾
⠀⠀⢀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⣤⣤⣀⣿⡤⣤
⠀⣼⠛⡻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣿⣿⣿⢿⣿⣿⣿⣷⡄⢀⠀⣿⡿⢛⣿⠀⠈⣤⠤⠤
⠀⣷⣾⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⡟⠸⣿⠀⢠⣤⣀⣤⣀
⠀⠉⣻⣿⡟⠉⣿⣦⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⣿⠀⠀⣿⣿⣿⣦⣀
⠀⢀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡈⢻⠀⣿⠆⠀⣿⣿⣿⣿⡯
⠀⢹⣿⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣗⠃⠈⣄⢸⡄⠰⣿⣿⣿⣿⣷
⠀⠀⢹⣿⡿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠀⣿⡈⠁⠀⣿⣿⣿⣿⣟
⠀⠐⠛⢿⡇⠘⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢗⣠⡄⣾⣿⣿⢟⣣⡆
⠀⠀⣶⣿⡇⠈⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⣸⣛⠁⠘⣿⣿⣤⣽⡅
⠀⠘⣿⢿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣼⣿⣿⣿⠆
⠀⠀⠀⠀⠃⠀⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠘⠁⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣮⣻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠚⢀⣹⠈⣻⢿⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣝⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠻⠿⠿⠿⠿⢿⣿⣿⣇⡼⠿⠿⠿⠿⠛⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡀
⠀⠀⠀⠀⠀⠀⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮
⠀⠀⠀⠀⢠⠀⣇⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣰⢀
⠀⠀⠀⢀⡠⠟⠘⠋⣩⠻⢲⢠⣄⠀⢠⡄⠀⠀⠀⠀⠀⡀⣀⢀⣠⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⡉⠀⠁
⠀⠀⠠⠋⣠⣤⣶⣶⠬⢁⠩⣿⣯⠀⠀⠻⡆⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠙⢃⡀⣀
⠀⠀⠀⠀⠈⠁⢀⣀⠀⣀⡀⠉⠘⠒⠒⠺⣿⡶⠾⢿⠧⣍⡑⠒⠒⠶⠐⠂⠀⠀⠀⢴⣶⠂⠀⠘⠚⠛⠛⠒⠲⠢⡶⠾⠛⢛⠿⠄⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⡉⠉⣛⣿⠻⢶⡖⠦⣄⡘⣧⡀⠀⠀⠀⠉⠀⠀⠒⠀⠀⠀⠐⠒⠚⠋⠀⠐⢀⣠⠤⠤⡀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠀⣴⣶⣀⣀⡈⠧⣀⠙⠃⢤⣉⣉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⡶⠶⠶⠶⠠⢤⣤⣤⣄⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠘⢿⣿⠋⣀⠀⠈⠻⡶⢶⣴⣮⣹⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠛⠛⠻⠷⠶⠖⠒⠾⠯⠁⠢⠤⢛⣁
⠀⠀⠀⠀⠀⠀⠀⠈⠱⢰⣿⣷⠟⠀⠔⠛⠋⠤⠄⢐⠋⠱⣄⠀⠀⠀⣠⣾⠓⠀⠀⠀⢶⡀⠀⠀⡤⠀⠀⡀⠀⠀⠀⣰⠻⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠀⠒⠂⠉⠀⠈⠉⠀⠀⠈⠓⠀⠾⠤⠀⠀⠀⠀⠀⢀⣁⣧⠀⠀⠂⠸⣥⣓⡒⠐⠃⠀⠄




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

⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣤⡀
⠀⠀⠀⢰⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⠤
⠀⠀⠀⢰⣿⠀⣿⢀⣴⣦⠀⠀⠀⢠⣤⣴⣶⣶⣾⠿⣿⣿⣿⣿⡿⠿⣿⣟⠋⠙⠻⠿⣿⣿⣿⡄⠀⠀⠀⠻⣿⣿⣏⠁⠀⠀⠉⠁
⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠨⠽⣷⣶⠾⠿⠿⠛⠉⠀⠘⣿⣿⣿⣦⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠙⣿⣿⣷⣄
⠀⠀⠀⢸⣿⡄⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡆
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⢸⣿⡇⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⣿⣿⣇⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⢿⣿⣿⣆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣄⣽⣿⣿⣇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡇⠀⠀⠀⠀⠀⢙⣿⡿⢿⣿⣿⣿⡏⣿⣿⡀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⣭⣭⠿⠇⠀⠀⠀⠀⠀⣸⡿⠁⢸⣿⠟⠋⠀⠻⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣭⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⡇⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠶⠾⠟⣻⠛⠛⠛⠻⣿⣿⣿⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣿⠁⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠀⠀⠘⠻⢿⠟⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠿⢤⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⢀⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠸⠿⠷⣄⠀⢀⣾⣶⣄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠈⠙⠛⠻⠿⠿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣰⣿⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡾⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣿⣿⣿⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣽⣿⣿⣿⡿⠉⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⣴⣿⣿⣿⠿⠿⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠋⠀⠀⠀⣹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣿⣿⣿⣏⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⠻⢿⡿⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠒⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣇⣼⡄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠞⠉⠙⠛⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠒⠂⢀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡿⠀⠀⡄⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣆⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣰⡆⠁⠛⠉⠁⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⡀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠀⢤⣴⠾⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣋⣀⡈⣠⣤⣴⠎⠑⢦⣄⣀⢀⠐⠂⢘⡇
⠀⠀⠀⠀⠂⠀⠐⠒⢠⢀⣤⡈⠃⠙⠉⠀⠀⠀⠀⢻⡍⠋⠀⠀⢰⣷
⠀⠀⠀⠀⠀⢀⡀⢲⣛⣻⡯⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣿⣶⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠲⠷⠿⠟⠃⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⠀⠀⠀⢰⡿⠞⡀⠀⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⠀⠀⠀⢠⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⣾⡇⠰⡇⢠⡸⢆⣣⠀⠀⠀⠴⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⢠⣼⢿⣿⠃⠀⠀⠀⠀⠀⠀⡟⣷⣤⣿⣁⣀⣴⠏⠀⠀⠀⠀⠙⢿⣇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⠷⠀⠀⢠⣿⣣⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠁⠿⠛⠟⠋⠉⠀⠀⠆⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⢘⣿⢿⡏⠀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⡀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⠀⣤⡿⣱⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⣢⣰⠀⠀⠀⠀⠀⠀⡀⠰⡀⠐⠀⠠⡀⠈⣧
⠀⠁⠀⠀⠀⢸⣿⡏⠰⢀⣶⠷⢄⣿⠒⠀⡆⠀⠀⠭⠀⠰⠃⣨⠛⠈⠶⠒⠀⠀⠲⠀⠡⠀⢱⠀⠀⠆⠳⡀⠸⣧⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣿⣿⡁⠀⣾⠇⠀⣸⠟⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢐⣿⡿⠀⣼⣇⡄⢠⡏⡸⠀⢸⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⢻⣿⠃⢰⣿⢹⠁⡿⠠⠃⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣀
⠀⠀⠀⠀⣸⣿⣤⣿⠇⡆⣸⡟⣴⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⢹⣿⣿⣿⣼⢹⡏⠀⠸⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠘⢟⣻⣛⠋⢸⠃⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠿⠀⠀⣰⣦⡀⠀⠘
⠀⠀⠀⠈⣴⣿⣻⣟⣀⣮⡄⠀⠀⡾⠁⠀⣀⡧⣄⠀⠀⠀⠀⠀⣾⡆⢹⠿⣤⣀⣐⠀⣆⠹⠀⠈⠃⢠⣆⡠⠐⡎⠀⠁⢀
⠀⠀⠀⣇⢻⣿⣉⣿⣿⣯⣿⣤⠀⢺⣶⠚⣿⠀⢿⠑⢶⡤⣄⣠⣿⡿⠈⡇⢸⠈⣿⣅⢻⡄⢳⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠂
⠀⠀⢠⣿⣦⡙⢿⣿⣿⡿⠟⢡⣾⢿⡇⢸⡇⠀⠸⠀⢸⡇⠈⠉⣽⢷⠀⢹⠀⠂⠸⡈⠶⣿⠈⡇⠀⠀⠠
⠀⠀⠀⣿⣿⣿⣶⣶⡶⣶⡏⢾⡟⣼⡇⢸⣷⠀⢰⠀⠸⡇⠀⠀⢸⢸⡇⢸⡄⢀⡀⢡⠀⣿⣧⢸⠀⡆⠰⡄
⠀⠀⢰⣿⣿⣿⢇⡿⠁⣾⠁⣿⠇⣿⠁⢸⠇⠀⢸⡀⠀⡗⠀⠂⠈⠈⣇⠀⣿⠀⡇⠀⠀⠸⣿⠀⠃⠰⠀⠃⠀⠃
⠀⠀⢼⣿⣏⡉⠀⠁⢰⠇⢠⠏⢰⡟⠀⣿⣥⠀⣾⡇⠀⠛⠂⢰⠀⢸⣉⠀⡟⠀⠀⠀⠀⠀⣻⡀⢐⡀⠀⣀⣀⣀⣀⣤⣤⠄
⠀⠀⢰⣿⣿⣷⣶⣦⣾⣤⣎⣤⣌⣀⣴⣃⣀⣀⣄⣀⣀⣧⣴⣆⣀⣼⣅⣠⣶⣤⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠉
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣤⣾⣦⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⠟⡛⣿⡿⠚⠿⠿⡿⠛⠿⢿⠿⠿⣿⠿⢿⣿⣿⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣾⣿⣿⣿⣭⣽⣿⣿⣏⣼⣗⣿⣀⣠⣤⣴⣻⣀⣀⣈⣂⣀⣉⣣⡀⢀⣛⣦⡀⠀⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠈⠢⡉⠈⠉⠫⣍⠉⠙⠛⣏⠉⠉⠙⣿⠋⠻⠹⠉⠉⡟⢹⢿⡛⠉⠙⢻⠋⠉⢛⡟⠟⣩⠟⡻⠛⠛⢛⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠈⠊⠒⢴⣬⣦⠀⠀⠈⢆⠀⠠⠀⠀⡇⠘⠸⠀⠈⠀⡁⢀⢴⡪⢔⡫⠔⠋⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠄⠈⠙⢷⣄⡀⠪⡀⠀⡄⠀⡇⠀⡃⠀⢀⡾⣠⣴⠯⠒⠉⠀⠀⢀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠙⠦⣇⠀⣆⠀⣿⢠⣦⣾⡿⠗⠉⠁⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠻⢵⣿⠿⠋⠁⠀⠀⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠄⠀⠀⢀⣰⣋⣟⣷⠀⠀⡰⠁⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠘⣌⣏⢸⣙⠻⣧⡐⣁⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⣾⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠔⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡯⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⣻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⠉⠙⠛⠻⠷⣦⣤⣀⣠⣴⠾⠛⠁⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⣀⣴⣤⣀⠀⠀⠀⢠⡿⣿⡏⠁⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⣀⣤⡾⠟⠁⠈⠙⠛⠷⣶⣿⠃⠹⣷⣄⣀⣴⠿⠋⠁
⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⣹⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⡿⢰⡿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⠛⢷⣿⠿⠷⣦⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⣿⡆⠀⠘⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣿⣇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣯⣴⠿⠁

⢀⣀⡀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠳⢦⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⣦⣤⣤⠾⠋⠀⠀⠙⠷⣤⣤⡴⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⣤⠶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⣤⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠸⣧⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⡿⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣿
⢀⣾⠃⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠏⠀⠀⠈⠻⠶⠶⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠘⣷
⣾⠃⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠘
⢿⡀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢀
⠈⢷⡄⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢠⡾
⠀⠈⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣠⡶⠋⠉⠻⢦⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⢠⡏⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠈⠻⠶⢤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠶⠛⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠾⠛⠻⢦⣄⠀⠀⣠⡶⠟⠛⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠷⣄⣀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀⣤⡾⠃⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⣿⠀⠀⠀⠙⠷⣦⣤⣠⡴⠞⠋⠁⠘⣧⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠐⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠐⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠘⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⣀⣠⣴⠞⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠹⣦
⠀⠈⡤⠾⠿⠶⠶⠾⠛⠛⠛⠉⠁⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠈⠉⠉⠙⠛⠛⠻⠶⠶⠾⣷
⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡏
⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢰⡟
⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠻⠂⠀⣿⠀⣤⡶⠿⠿⠿⣷⣆⢠⣿⠏⠀⠀⠈⣿⣧⢸⣿⣆⠀⣾⡗⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣄⣀⠀⠀⣿⠀⣿⣁⣤⣤⣴⡿⠏⣼⡏⠀⠀⠀⠀⣿⡟⢸⣿⣿⡆⣿⡇⢸⣿⢠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠉⠙⢿⡄⣿⠀⣿⡿⢿⣿⣁⠀⠀⢻⣧⠀⠀⠀⣼⣿⠃⢸⣿⢹⣿⣿⣧⢸⣿⡀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣴⠿⠁⣿⠀⣿⡇⠀⠙⢿⡷⠀⠈⢿⣷⣶⣾⡿⠃⠀⣸⡟⠀⢻⣿⡇⠘⣿⣷⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢠⡿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⡄⠀⠘⢿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣤⣿⣿⣷⠀⣠⡄⠈⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣡⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠈⣿⣿⡇⢿⣧⣄⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠁⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠛⠃⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⣀⣀⣀⣀⠀⠀⣿⣷⡄⢰⣿⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⣿⡄⢰⣿⡇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⣴⣦⡀⢀⣿⡏⢠⣾⠟⠋⠙⢿⣷⠀⣿⣿⣧⣿⣿⡇⠀⠀⠀⣼⣿⣿⣆⠀⠀⣿⣿⣷⣸⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢰⣿⣿⣷⣼⣿⠃⣿⠃⠀⠀⠀⢸⣿⠇⣿⢻⣿⣿⣿⣷⠀⠀⣼⠏⢹⣿⣿⡆⠀⣿⡏⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠹⣿⣿⣿⠀⣿⣄⠀⠀⢠⣾⡟⠀⣿⠀⠉⠁⢸⣿⠀⣰⠿⠟⠛⢻⣿⣿⠀⣿⡇⠹⣿⣿⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃⠀⠻⠿⠇⠀⠙⠿⠷⠾⠿⠋⠀⠀⠿⠀⠀⠀⠸⠿⠠⠏⠀⠀⠀⠀⠻⠿⠆⠿⠃⠀⠻⠿⠇⠀⠤




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

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣴⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣄
⠀⠀⠀⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡟
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⢀⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⢠⡀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣧⣤⣄⡀
⠀⣸⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⡿⠁⠀⠀⠁⣰⠀⠀⢀⣤⡤⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢿⣇
⠀⣿⠁⢻⣿⣿⣿⣿⠀⠀⠀⠀⣰⡟⠁⠀⠀⡀⠐⠻⣇⣰⠟⠀⠀⠀⠀⣿⡟⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢰⡿⠉⣿⣿⣿⡏⠈⣿
⠀⢿⡄⠈⠟⣛⣿⣿⡆⠀⠀⠀⠘⣷⠀⠀⠁⠙⢰⡖⠻⠁⠀⠀⠀⠀⠀⣿⡁⢹⣿⣿⣧⣀⠀⠀⠀⠀⠀⢸⣷⠀⣹⣿⡟⢃⡀⣿⠇
⠀⠈⣿⠀⢼⣿⣿⣿⡧⠀⠀⠀⠀⠈⢧⣀⠀⠀⠸⡇⢠⠀⠀⠀⠀⠀⠀⢹⣷⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⣯⠀⠋⠉⠠⡐⡟
⠀⠀⣿⡆⠈⣽⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣁⡀⠁⠀⠀⠁
⠀⠀⠈⠋⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠨⣀⡀⠘⠒⢠⠀⢢
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠉
⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢰⡟⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⢀⡾⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⣿⠃⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠰⣿⠃⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⣀⣠⣤⣴⢶⣤⡶⢦⡤⢦⣀⣤⡀⣀
⠀⠀⣤⣾⣯⣿⣶⣿⣶⡿⣿⣿⣿⣾⣿⣴⣿⣹⣶⢦⣀⣄
⠀⠀⠹⣿⣿⣯⣹⣿⣺⡿⣿⣼⣿⣿⣯⢻⣟⣿⣿⣾⣯⣽⣿⣇⣀
⠀⠀⠐⡇⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠛⡿⠛⠛⠙⡛⠙⠃⠒⠠⢄⡠⠀⢀⠀⢀⡀⢀⢀⠤⠄⠒⠐⠉⢐⠉⠒⠖⠠⣤
⠀⠀⠀⢠⢐⡀⠄⢩⡈⠉⠈⠉⡈⠋⡀⠁⠀⢀⣂⣀⡤⠌⢧⠝⡄⠸⠁⠶⠈⡡⠀⢠⡄⣴⡆⣀⠀⠶⠀⠿⠀⠛⠤⢯⠴⣷⠰⡆⡄
⠀⠀⠀⠀⠎⣿⠲⠀⠤⡄⠁⡀⠧⢤⠀⠐⠈⠋⠙⢡⠆⠐⢺⠀⡀⠀⠀⠀⢸⡔⣇⣤⠀⣼⡇⣬⣗⠲⣼⠁⠀⠀⠀⠀⠀⢹⠤⠆⣧
⠀⠀⠀⠀⠸⠸⠃⡰⡆⢱⠈⡁⠀⠻⡄⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠄⠀⠀⠸⠁⠀⡽⠀⡏⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠞⣠⢹⣆⢶⣡⡀⠀⢳⠀⠀⠀⠀⠀⡴⠀⠀⡇⠀⠀⠀⠀⡆⢀⢠⡇⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇⠀⢸
⠀⠀⠀⠀⠀⠈⠀⢡⡎⢻⡘⣧⣧⠀⠀⢧⠀⠀⠀⠀⠃⠀⠀⢹⠀⠀⠀⠠⠃⠀⠘⡇⠀⡧⢀⠀⠄⢸⠃⠀⠀⠀⠀⠀⡾⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⢣⠘⣏⢧⠀⠈⢣⠀⠀⠀⢰⡆⠀⠸⡀⠀⠀⢠⠀⠀⢸⠁⢸⣤⣼⠆⠀⢸⠃⠀⠀⠀⠀⡼⢃⢠⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⢃⠘⣌⡄⠐⠘⣦⠀⠀⠀⢁⠀⠀⡇⠀⠀⢸⠀⠀⣿⠀⢸⠀⢰⡇⠀⢸⠀⠀⠀⠀⡼⣁⣆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠸⡄⢹⡾⡥⠈⠈⢆⠀⠀⠛⠀⠀⢹⡄⠀⢸⠀⠀⢻⠀⢸⠀⠘⠃⠀⣾⠀⠀⢀⡾⣻⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⡄⢣⡹⣂⠐⠌⣆⠀⠨⡇⠀⠈⢠⠀⢨⠀⠀⢸⠀⢠⠀⠀⠀⠀⣿⢠⣀⣾⢷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠁⠌⠷⢻⡄⠘⡘⣧⠀⢠⠀⠃⠙⡀⢈⠀⠀⢸⠀⢸⠀⠀⠀⠀⡿⢬⣿⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠐⢃⢿⡀⠐⠘⣧⠘⠀⢠⠀⡇⢸⠀⠀⣸⠀⢿⠀⢀⠀⠀⡇⣸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⢣⠈⡌⢳⠀⢢⠙⡄⡇⠀⠀⢴⢸⠀⠀⣿⠀⢸⠒⠺⠀⠀⣇⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠸⠀⢷⠈⡁⠸⢠⠁⠂⠈⡾⠀⠀⡏⠀⢸⠀⢸⡄⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡀⢃⠼⡗⠀⠀⠻⡠⠀⠀⢷⠀⠀⠀⡄⠈⠀⠰⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⠘⡆⢳⡀⠀⠀⡇⠀⠀⠈⡇⠀⢠⡇⠀⡇⠀⣤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠃⡌⣷⠰⠀⢰⠀⠀⠄⠇⠀⠀⢸⠀⢋⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠃⠘⠀⢸⡆⠀⠈⣴⠀⠃⢺⠀⠁⣸⡀⢸⠂⠀⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠇⠀⠀⡄⠈⣿⠐⠇⠻⡄⢠⠸⠒⠀⠈⡧⠀⣟⠀⠠⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣾⠀⠀⢱⠀⠉⡟⠠⠸⠇⠃⡆⡧⠐⠀⢷⠀⢻⡄⠸⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣘⣧⣿⠀⠀⠈⠀⠐⢁⠆⡆⢼⠀⢡⢡⡀⠆⠸⡄⠘⢇⠈⡇⡼⣱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣧⣱⢊⢡⡇⠀⠀⢀⠀⠀⠸⢆⠁⠀⡆⠈⠸⡇⠁⠀⡇⠀⠸⠀⢸⢀⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢿⣥⣷⡅⠀⢰⡇⠀⠀⣾⠀⠀⠀⣏⠀⠀⢃⠀⠀⢿⠀⠀⢸⡀⠀⡇⠈⣾⡸⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢫⡾⠉⠩⡳⠁⠀⠀⢈⠇⠀⠀⣿⠀⠀⠀⢿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⠀⢹⠠⢿⡄⠾⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⡰⢋⣴⡿⠋⠎⡴⠃⠀⠀⠀⣾⠀⠀⢡⠅⠀⠀⠀⢸⡄⠀⠈⡆⠀⠀⣧⠀⠀⢹⠀⠸⡀⠘⣧⠀⠋⡿⣿⣿⣿⣷⡀
⠀⠀⠀⢠⠞⡡⠛⠋⠀⢀⠞⠁⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡅⠀⠀⡇⠀⠀⢺⠀⠀⠸⡇⠀⡇⠀⢿⠀⠀⢱⠈⠉⣿⣽⣿⡄
⠀⠀⢠⣏⢰⠏⠀⠀⡀⠎⠀⠀⠀⠀⠀⡘⠇⠀⠀⡌⠀⠀⠀⠀⠈⡇⠀⠀⢷⠀⠀⢸⡆⠀⠀⣧⠀⢳⠀⢸⡇⠀⢸⠀⠀⡟⠉⣹⣷
⠀⠀⣿⣵⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠈⡇⠀⠀⢼⠀⢸⠀⠘⣇⠀⠘⡇⠀⠀⡇⢸⣿
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠁⢰⠁⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⡄⠀⣿⠀⠀⡇⠀⠀⡇⡟⢹⠃
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⠂⠖⣿⠀⠀⣷⠀⢰⠇⢃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠷⠦⢔⠢⠀⠀⡧⢸⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⠀⠀⣿⠀⠀⣷⠈⠀⢋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠌⢃⠀⡁⡾⠀⡀⢀⡇⠀⠀⣸⠀⢸⠀⠀⣟⠠⠀⣏⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡃⠀⠉⣸⠧⢀⢀⡟⠀⣼⠀⢰⡏⡀⢌⠅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⢀⣀⠨⠅⠉⡇⢀⡪⠅⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣦⠀⠀⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣉⡉⠻⢧⣶⡇⠀⠀⡔⠀⠀⢾⣤⡒⠁⠀⡰⣿⡿⡏⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠐⠲⣦⣄⡀⢺⣿⣿⣿⣷⡈⣿⢿⣿⣆⣷⡀⠀⠈⠛⠳⢾⣻⡗⠛⡁⠠⢈⣉⡁
⠀⠀⠀⣤⣶⣶⡤⠈⠙⣃⣈⡑⣼⣿⠿⠿⢿⣷⡏⣽⡏⢷⣤⡀⢀⣀⣴⡏⢁⣉⡈⢱⣌⢻⣯⡀
⠀⠀⠐⠛⠛⠉⠐⠁⠾⣿⣿⣿⣿⣆⣤⣀⠀⠙⣿⣿⣷⡤⠈⠋⠙⠋⠉⣠⣾⣿⣷⢸⡟⡧⠉⢓⠂
⠀⠀⠀⠀⣠⣤⣴⣶⣤⠙⢻⣿⣿⣿⣿⣿⣆⣦⣿⣿⠉⣼⠇⢠⣶⣶⣤⡈⠻⠿⠿⠀⠳⠇
⠀⠀⠀⠀⠀⠘⠛⠛⠁⢠⣿⣿⣿⠛⠿⣿⡿⠿⠟⠁⣸⡟⠀⡌⠻⣿⣿⣿⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠆⠀⢹⣷⠖⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⢶⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣩⣙⠻⠿⣿⣿⡇⠀⠀⠀⢳⣦⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣼⣿⣿⣶⣤⣉⠁⠀⠀⠀⠀⢿⣿⣷⣤⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣶⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⣿⣿⣧⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡝⠛⣶⣭⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣝⡛⠿⠿⠿⢿⣛⣈⡱⣾⣿⣦⣝⢿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣎⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⢙⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠘⠿⢿⠿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢰⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠙⠛⠛⠋⠙⠻⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣠⣤⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠆⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠴⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢿⣿⣿⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⢠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢿⣿⣄⣄⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⠩⠍⠉⠉⢻⣿⣦⣤⣀⣀⡀⠀⢀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠴⡶⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠾⠁⠹⠂⢈⡁⣠⡤⡄⠀⠀⠀⠀⠀⡀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢂⣾⡆⠀⠀⠀⡂⣤⠀⠀⢠⠀⠶⠉⠌⠠⠊⠠⠀⠀⣶⢷⠆⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣤⣶⣚⣿⡏⣧⣰⡄⣰⣯⣆⢠⣦⠸⠀⠐⠂⠀⠀⠂⠀⣶⣚⣿⣿⠶⠲⢶⣾⠇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢆⢹⣿⣟⣛⣛⣿⣿⣿⣿⢸⣿⣿⡟⡦⣁⠵⠀⠀⢀⣦⣤⣴⣶⣼⡟⠉⠀⠀⠀⠈⢻⢇⠁
⠀⠀⠀⠀⠀⠀⠈⠲⣤⣾⡿⢿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣿⣶⣶⣶⣾⣷⣿⣿⠟⠉⠀⢀⠤⠴⠄
⠀⠀⠀⠀⠀⠀⢈⣿⣿⠟⠀⠀⠰⠄⠀⠈⠉⣷⣍⣛⣛⡛⣿⣿⡛⢻⣿⣿⡿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⢀⠐⡦
⠀⠀⠀⠀⠀⢀⡞⠁⠁⠀⡤⠊⢲⠀⢀⠀⠄⠈⣿⣿⣿⢿⡿⣿⣿⣗⡿⠛⣿⣏⣀⠀⠀⠀⠀⡌⠀⠀⠀⠻⠱⠀⠂⠈
⠀⠀⠀⠀⠀⣸⡇⠀⠀⢨⡟⠟⢿⠃⠘⠀⠳⠿⣿⣿⣿⡿⣻⣙⢯⣼⣿⣦⠙⠛⢿⣷⡀⠀⠀⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠋⠀⠰⠀⢸⠙⡶⠈⠀⠈⠀⣠⠢⠘⣟⣛⣛⣄⣛⡿⣿⡗⠀⠀⠀⣀⢹⣷⠀⠀⠀⠈⠃⠀⠀⠄
⠀⠀⠀⠀⠀⢴⡆⠀⠀⠈⠛⠁⠀⠀⢀⠄⠀⢀⣴⣯⡴⣀⠛⣤⣄⣯⣷⣄⠀⠀⠘⢿⣿⠃⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⢳⣆⠀⠈⠖⠀⠤⠀⠀⠀⢰⣿⣿⢿⣿⣿⣧⡆⠀⠉⢻⡿⣿⣿⣿⣾⣿⣆⠀⠀⠀⢀⣀⡀⢵⣀
⠀⠀⠀⠀⠀⠀⠀⠹⢦⠀⠐⠂⢀⣤⣶⡶⠎⣂⣧⣤⣤⣾⣷⣶⣶⣦⣄⣸⠿⣿⡟⢻⣿⣿⣿⣿⠛⠉⢻⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⢿⣿⣿⣶⣼⣯⣅⣉⣩⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⢳⣿⣿⣿⣿⣿⡿⠿⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠁⠙⠻⠿⣿⣿⣿⡿⠿⠿⠿⠿⠏⣭⡹⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡧⣕⡐⠶⢦⡄⠀⠀⢤⡶⠀⠀⠀⠀⣿⣧⡬⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠘⠫⠟⠋⠋⠀⠀⠰⠹⠷⠶⠀⠀⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠛⠗⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡎⠀⠀⠀⠀⠀⡀⠀⠀⠁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡄⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣛⡻⣿⣦⣤⣾⡿⣱⡿⣶⡿⣿⣦⣤⣾⡿⣟⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⢀⣾⣟⣾⡟⠻⣷⣿⣿⣯⡾⠏⠀⠈⢻⣾⣿⣿⣵⡿⠛⢿⡞⣿⣇⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣛⣿⠿⣟⣼⡟⢀⣠⣤⣽⣿⣿⣷⣾⡷⢶⣶⣿⣿⣿⣤⣄⣀⠘⣿⣜⣿⣿⣿⣏⣉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠋⠛⠛⢻⣿⣿⣿⠿⣛⣻⣽⣽⡿⠶⠾⠿⠿⠶⠾⣿⣯⣽⣛⡻⢿⣿⣿⣿⠛⠛⠋⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⢷⣿⢀⣤⣾⡿⣻⣽⡾⠟⠛⠉⣁⣤⣤⣤⣤⣶⣶⣤⣤⣤⣀⡉⠙⠛⠿⣶⣝⡿⣿⣦⣄⢸⣿⢿⣷⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣼⣿⣫⣭⣭⣭⣵⣿⣿⣿⢟⣵⡾⠛⢁⣠⣴⣾⠿⣟⣛⣯⣭⣯⣭⣭⣭⣭⣟⣛⡿⢿⣶⣦⣀⠉⠻⣶⣝⢿⣿⣿⣾⣭⣭⣭⣽⣻⣿⡀
⠀⠀⠀⠀⠀⢻⣿⢿⣏⠉⠈⣴⣿⢟⣵⡟⠋⣠⣶⡿⢟⣹⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠉⠉⠛⠻⠿⣮⣟⠿⣷⣦⡈⠙⢷⣝⢿⣷⡄⠉⢉⣿⣻⣿⠃
⠀⠀⠀⠀⠀⢈⣿⣿⣿⢀⣾⡿⣱⡿⠃⣠⣾⡿⣯⡾⠟⠉⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⢿⣶⠀⠀⠀⠙⢿⣮⡿⣿⣦⡀⠻⣷⡽⣿⣆⢸⡟⣿⣟
⠀⠀⣠⣶⣾⠿⣟⣽⣿⣿⢟⣿⠟⢀⣼⡿⣣⣾⠋⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠻⣮⡻⣿⣄⠘⢿⣜⣿⣿⣿⣝⡿⢿⣶⣤⡀
⠀⢼⣿⢻⣾⠿⠛⢁⣿⡟⣾⠏⢀⣾⡟⣽⡟⠁⠀⠀⠀⠀⢰⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⡽⣿⣆⠈⢿⣏⣿⣏⠙⠻⢿⣾⢻⣿
⠀⠘⣿⣏⢿⣦⠀⣾⣿⣼⡟⠀⣾⡟⣿⡟⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡼⣿⡆⠘⣿⡼⣿⡄⢠⣾⢏⣾⡟
⠀⠀⠘⣿⣷⣿⣶⣿⣧⣿⠀⣼⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⢿⣆⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⢹⣧⣿⣷⣾⢳⣿⡟
⠀⣠⣾⡿⣻⣿⢿⣿⣹⡏⠀⣿⡏⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣿⣇⠘⣿⢸⣿⢿⣯⡻⣿⣦⡀
⣾⣿⣻⡾⠛⠁⢼⣿⣿⡇⢰⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⠀⣿⣾⣿⡆⠙⠻⣾⡻⣿
⠻⣷⣘⠳⣦⡀⢸⣿⢿⡇⠸⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⠀⣿⢻⣿⠇⣠⡶⢋⣴⡿
⠀⠙⠻⣿⣾⣿⣾⣿⣾⣧⠀⣿⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡇⢠⣿⣸⣿⣿⢿⣾⡿⠛⠁
⠀⠀⣠⣿⢟⣿⠟⣿⡟⣿⡀⢹⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣹⣿⠁⣼⡟⣿⡿⢿⣞⢿⣧⡀
⠀⢰⣿⢯⣾⠋⠀⢻⣿⣹⣷⠀⢿⣷⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣳⣿⠇⢰⣿⢹⣿⠃⠈⢻⣯⢻⣷
⠀⠹⣿⣼⣻⠷⣦⣄⣿⣧⢻⣧⠈⢿⣷⡹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣰⡿⣱⣿⠏⢠⣿⣧⣿⣏⣤⣶⢿⣻⣽⡿
⠀⠀⠉⠛⠿⣿⣿⣽⣿⣿⣯⡻⣧⡀⠻⣿⣟⢿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⣠⣾⣟⣽⡿⠃⣠⡿⣳⣿⣿⡟⣿⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⢠⣿⢿⣿⠀⢻⣷⣝⣿⣄⠘⢿⣷⣟⡿⣶⣄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣴⡾⣟⣵⣿⠟⢀⣼⣟⣵⣿⠋⢸⣿⢿⣷
⠀⠀⠀⠀⠀⣼⡟⣿⣏⣀⣀⣙⣿⣾⡻⣷⣄⡈⠻⣿⣾⣽⣻⠷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⡶⢿⣛⣽⣾⠿⠋⢀⣴⡿⣫⣾⣟⣁⣀⣀⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠹⣿⣯⣿⣿⣟⣛⢿⣿⢿⣧⣝⢿⣦⣄⠉⠛⠿⢿⣿⣿⣿⣿⣛⣛⣛⣻⣯⣭⣿⣿⠿⠟⠋⢁⣤⣾⢟⣵⣾⢿⣿⢟⣿⣻⣿⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠙⣿⡏⣿⡀⠙⠿⣿⣾⣝⣻⣶⣦⣤⣀⣉⠉⠙⠛⠛⠛⠛⠋⢉⣁⣠⣤⣶⡿⣟⣽⣾⡿⠟⠁⣸⡿⣿⡿⠙⢿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣤⣤⣶⠾⣿⣿⠿⣿⣾⣭⣽⣛⣛⣿⣿⠿⢿⣿⣟⣛⣛⣯⣽⣶⡿⣿⣿⡿⢶⣦⣤⣿⡇⣿⡇⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣿⣿⣿⣿⣷⣹⣷⠀⠉⢉⣻⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣋⠉⠀⢰⡿⣱⣿⣿⣿⣯⣯⣾⠿⠁⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠈⠉⠉⠁⠀⠈⢿⣷⠻⣧⣾⢟⣯⣿⣽⣿⣦⡀⣠⣾⣟⣽⣯⣟⡿⣦⣿⣻⣿⠇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣷⣽⣿⡿⠛⠉⢿⣷⣝⠿⢟⣵⣿⡏⠙⠻⣿⣿⣽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⢻⣷⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠸⣿⣶⠶⠿⠿⠟⠛⠛⠋⠉⠉⠉⢿⣦⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⢠⣾⠋⠉⠉⠉⠛⠛⠛⠿⠿⠷⢶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⣼⡟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠛⠚⠛⠛⠲⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⣠⣴⠤⣶⢦⣴⣄⣤⣄⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡾⠃⠈⠀⠈⠀⠁⠀⠉⠀⠛⠛⢶⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡇⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⣿⠀⠀⢉⣉⣛⠛⠀⠀⠈⣋⣉⡉⠁⠀⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡇⠀⠀⠀⠉⠁⠀⡆⠀⠀⠈⠉⠀⠀⠀⡿⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢳⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⡇⠀⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠀⣸⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠤⠤⠤⠤⠄⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⠒⠦⠔⠂⠀⣠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⢦⣄⣀⣀⣀⣠⠞⠁⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡇⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠁⠙⠳⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠉⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠒⠦⢤⣤⣤⣄⣀⣀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠹⡆⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠀⣠⠏⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣎⠁⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⣀⠀⠀⠀⠀⠀⠙⠛⡁⠀⠀⠁⠀⠀⡟⢻⡄⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣼⢹⠀⠀⠒⠒⠚⠛⠛⠉⠁⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⢸⡇⠘⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢸⡆⢀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣾⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡇⠀⣧⠀⠳⣄⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⣀⡀⠀⣼⠃⠀⠀⢀⡏⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣸⠇⠀⣿⠀⠀⣿⠀⠈⠛⠛⠉⠀⣿⠀⠀⠘⠛⠉⠁⠀⢠⡟⠀⠀⠀⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠸⣧⠀⢀⡀⠀⠀⣿⠀⠀⠀⣀⣀⠀⠀⣾⠁⠀⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠀⢸⡇⠀⠀⠉⠁⢀⡿⠀⠀⠉⠉⠁⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⢀⡿⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡧⠾⣦⣤⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣼⣿⣻⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⡿⠾⠿⣿⣯⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠹⣤⠀⠶⠈⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣶⣦⠀⠀⠀⣿⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠘⠶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⠄⠀⣴⣬⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢀⠀⠁⣀⠀⢹⡇⠀⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠘⣧⠀⠈⠁⠘⣷⠀⠟⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢂⡀⠘⣷⣄⣀⣀⣿⣦⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠋⠀⠟⠚⠛⠿⢿⣿⡿⠯⢦⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⢀⠜⢀⠎⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣥⡄⢀⡤⠀⡁⠀⡈⠀⣸⣆⢀⡀⣿⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⡿⠉⢐⠋⢀⣤⠀⢰⠉⠻⣿⡏⣿⠁⠙⠃⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⣰⠏⠀⢀⣏⠀⣾⠀⠀⢹⣇⢸⣧⢰⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⣉⣤⡾⠁⠀⣀⠚⡉⢸⣿⠀⠠⢼⣿⠈⢿⡆⣥⡀⠹⣷
⠐⠀⠀⠀⠀⢀⠀⠠⡀⣰⣿⣟⣛⣿⣿⣏⠀⢂⣎⣙⡁⢠⣿⣿⡆⡆⠼⣿⡀⣿⣷⡈⠐⢀⣿⣧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣭⣤⣽⠏⠀⠀⠄⠹⠀⠀⣾⣿⣿⠀⠀⠀⢿⡇⠉⢹⣧⠠⣿⠼⣿⡄
⠀⠀⠀⠀⠀⠀⣸⣹⣿⣟⣷⣿⡾⠋⠀⠀⠠⣔⡆⠀⡸⠃⠀⠁⠀⠀⠂⢸⣇⠐⢂⣻⣇⠈⢶⣼⣿⡀
⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡟⣁⣴⠒⣤⣤⠿⠛⠀⣸⣿⠆⠄⠀⠀⠀⠀⢸⣇⠀⢀⠀⢻⣧⠘⢿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⡿⣫⡾⠋⠁⠺⣁⣾⣀⣠⣶⣿⣿⡀⠁⠀⠀⠈⡄⢸⣿⠘⡄⠀⠰⣿⣷⡈⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⠿⣉⠀⠀⠀⣠⠟⠃⣈⣽⡟⢸⠛⣷⠀⠀⠀⠈⡃⢸⣿⠀⡿⣦⠀⢹⣿⣿⡝⣿⡇
⠀⠀⠀⠀⢿⣿⡿⣩⡾⣟⠂⠐⠋⠁⠐⣾⣿⡻⠇⠺⠀⢹⡅⠀⠀⠈⠀⢸⣿⡀⡏⠹⠗⠈⣭⣯⠻⣿⣿⡀
⠀⠀⠀⠐⣿⣿⣽⣿⣥⡎⢈⠀⣀⣤⠞⢫⡜⣷⠀⢄⡀⠄⠁⠀⠄⠀⠀⢸⣿⡁⡆⠀⠀⠠⣾⣿⢀⠻⣿⡇
⠀⠀⠀⣶⠻⣧⣝⠛⣾⣩⡦⣴⣿⡃⢦⡘⣧⢘⣷⡚⢟⠛⠣⠀⠀⡄⠀⢸⢿⡇⢿⡇⠀⣰⡍⢻⣼⡇⠘⣣⠆
⠀⠀⢸⣿⣧⣈⣁⣘⣟⠩⣾⣿⣿⡁⣌⠷⣀⣼⣿⣷⣄⠀⠰⠾⢺⠶⠄⢸⠀⡇⠰⡿⣧⠌⢀⠚⢉⡅⠀⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣹⣆⠈⣿⣟⠁⠹⢿⣿⣿⣿⡏⣿⣦⣄⠀⠉⠀⠇⢸⠀⣇⠘⠓⢃⡴⠋⠀⢸⣷⠀⢻⣤
⠀⠀⣿⣿⣿⣿⣿⠿⣇⠉⣰⣿⣿⣆⠀⠀⢿⣿⣿⢀⣿⠇⠹⠿⠶⢤⣀⣸⡀⢿⣠⠶⢻⠀⢠⠔⢹⣿⠇⢸⣿⠃
⠀⠘⠿⠿⢿⣏⣁⡀⠈⠀⣼⣿⣿⣿⡀⠀⠈⣿⣿⡜⠃⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⣾⡅⢐⠂⢸⣿⠀⠘⣿⣅
⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣿⣿⣧⣤⣤⣀⣀⠉⠀⢀⠀⠀⠀⣀⣠⣿⣇⣀⣀⣰⣿⡤⠼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠟⠛⠽⠟⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣹⣏⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠗⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣿⣿⠖⠀⠀⠀⣰⠃⣠⣶⠃⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣟⣵⠆⠀⣴⣿⣿⣰⣿⢇⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠝⢻⣵⡟⣨⣾⣯⣾⣿⣵⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠄⣴⡿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣤⠝⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣶⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡏⠘
⠀⠀⠀⢰⣶⣿⡟⠋⠉⠉⠉⠉⠛⢻⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠛⢿⣿⣿⣤⡀
⠀⠀⠀⠈⢻⣿⣧⣌⡳⠦⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⠉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣏⡙⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣶⣤⡀⢸⣿⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⡇
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠹⣿⠏⠀⠀⠠⠈⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢃⣇⠀⠀⣸⣿⣧⡀⠀⢀⣖⠲⢭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣤⣀⣰⡿⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢿⣿⣇⡀⠈⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟⠆⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠛⠛⠛⣾⡏⠑⣡⠟⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⢸⣿⣿⡿⠁⠀⠻⠻⠀⡏⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⠋⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃⢀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⣦⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠀⢹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣴⣶⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⢀⠤⠖⠒⠛⠲⠲⢤⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣀⡤⠶⠒⠛⠋⠉⠛⠛⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟
⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡆⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁
⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⢛⡶⠯⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠸⣄⠀⢰⠋⠀⠀⠀⣠⡴⠟⣁⣀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠈⠳⢦⡤⠤⠶⠚⢉⣴⠟⢉⡴⠂⠀⠋⠁⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠳⣄⠀⠀⣾⣁⡴⠛⠀⣠⠞⢹⠉⠉⡗⢦⡀⠀⠀⣿⡿⠂
⠀⠀⠀⠀⠀⠈⠉⠻⡏⠁⠀⠀⠀⠁⣰⠃⠀⠀⡇⠀⣧⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠤⠴⡋⠁⠀⢀⡞⠁⣠⡏⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣾⣭⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣴⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡠⠤⠤⣄⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠉⠒⠒⠒⠂⢤⡀⠀⠀⢸⡑⠒⠐⠊⠀⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠰⣕⡦⠀⢀⡀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠉⠒⠒⠒⠒⠉⠁
⠀⠀⠀⣠⠤⡀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠰⣒⡌⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⡤⣀⠀⠀⠀⠀⠀⡇
⠀⠠⣖⣢⡄⠀⢠⠊⠀⠀⠀⠀⠀⠀⠰⠖⠒⠶⢖⣒⠒⠪⠦⠑⢦⡀⢀⡼
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠴⠯⠭⣉⣉⠁⠀⠀⠀⠀⠑⡀
⠀⠀⢰⣂⡶⠀⠀⠑⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⢰⠀⠸⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⡠⠚⠢⢄⣀⡠⠋⠒⡇⠀⠀⡤⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠧⠇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⡠⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀⠀⠀⠀⠀⡏⠁⠁⠀⠀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠘⠁⠀⡠⠽⣶⡀⠀⠀⠐⡄⠀⠀⠀⠈⠀⠀⠈⡷⣄⣀⣀⡀⠀⣀⢀
⠀⠀⠀⠀⢀⣀⣼⡇⠀⢻⡆⠀⢀⠀⠂⢸⣧⣤⣴⣛⣓⣦⣄⡀⠀⢀⡈⠀⣼⣷⣆⠈⠉⠉⣉⢿⣆
⠀⠀⣠⣾⢿⣿⡿⣷⣴⣾⡇⠀⠀⣢⡶⣿⡿⠛⣻⣿⠿⢿⣿⣿⣦⣼⠀⢸⣿⣿⢸⠀⠀⢠⡁⢸⣿⡆
⠀⠀⠉⠩⠺⣿⣧⣿⣿⣿⣿⠀⣄⣿⣿⣿⣶⡿⠉⠀⡴⢶⡊⢻⣿⣿⣇⢸⣟⠃⣸⣤⣤⣾⣷⣾⣿⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⡟⠀⠀⣶⣾⣫⢷⠈⣿⠿⣿⣿⣿⣧⡿⠟⣛⠉⢽⣿⡿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⢟⡗⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⠁
⠀⠀⠀⢀⠀⠻⢿⣿⣿⣿⠿⠿⠿⠿⣿⣿⡇⠀⠀⡿⣿⡿
⠀⠀⠀⠀⠑⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠥⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠌⠢⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣶⠛⠻⢿⣷⣤⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠠⣠⣾⡟⠁⠀⠀⢘⣷⣿⢿⡄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⡃⠀⢀⣴⣿⣿⠅⣸⣿⣄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠻⣿⠛⠒⢃⠘⣿⣿⠀⣿⣿⣿⣀⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠁⡠⠀⠀⠉⠉⠨⡀⠀⠀⠐⣨⠃⢶⡙⠛⠫⣽⣆⠀⢠
⠀⠀⠘⠆⠈⢉⠛⣋⠲⠖⠛⢋⣽⠀⠀⢘⣯⡡⡉⣛⢻⣟⣲⠿⠉⠁⠀⢸⣷⡄⠀⢾⣿⣾⣿⣦⠤⠤⠤⠤⠤⠄
⠀⠀⠀⡇⠀⠄⠀⠱⠌⠈⡹⠎⠀⠀⠘⡿⢿⣥⣿⣿⣿⣿⡑⠂⠀⠀⠀⠜⠀⠹⠀⠈⠅⣿⣿⣿⣐⠤⠀⠀⠁⠄
⠀⠀⢸⢤⠤⠴⠤⣤⣴⠏⠀⠀⠀⠀⣠⣷⣿⣷⣿⣿⠏⠀⠉⠀⠀⣤⡀⠀⠀⢺⡧⠄⠘⢿⣿⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⡀⠟⠀⢉⡳⠋⠁⠀⠀⠀⢀⣠⣼⣧⠤⠀⠀⠀⠀⠀⢼⣿⣿⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢤⣧⡸⣬⡀⣿⣀⠀⣀⣠⣶⣿⣿⣿⣭⡑⠂⠀⠀⠀⠀⢀⣿⣿⣷⡒⠐
⠀⠀⢰⣷⣶⣶⣿⣶⣷⣶⣶⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣤⣵⢶⣶⣾⣿⣿⣩⠋⠒⠤⣄⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠷⠖⢿⠛⢿⣿⣿⣿⡟⣶⣤⡈⠅
⠀⠀⢘⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣿⡿⠛⡿⠉⠃⠔⠋⠀⠀⠘⠀⠰⢸⣿⣿⣿⣿⣿⣷⡆
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠁⣀⡴⠖⠫⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣷⢻⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⡧⠟⣿⣿⣿⣿⣿⣿⣹⢿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣵⣶⣿⣿⡿⠋⠀⡈⠁⠀⢠⣿⠃⠐⠀⢀⠀⠀⠢⠠⢀⡀⠀⡀⠀⡀⠀⠀⠀⣀⡏⢸⡟⣿⣿⣿⣧
⠀⠀⢨⣭⡾⣿⣿⣿⣾⠃⢴⠂⠁⠀⠀⣸⡙⠀⠀⠀⠈⠀⠀⠀⠀⠈⢱⡶⠟⠁⠄⠀⣠⣾⣿⣄⣴⣧⣿⣿⣷⡗
⠀⠀⢸⣿⣿⡷⢴⡟⠛⢿⣎⣓⣦⡾⣮⣽⣟⣤⠀⠀⣀⢀⣀⠀⠀⡴⠟⠏⣅⠀⢢⣿⣿⢷⣿⡿⣿⠉⢿⣿⠑
⠀⠀⢸⡟⣿⠀⠀⠛⠆⠀⠻⣿⣿⣿⣿⣿⣾⣿⣭⢟⣺⣭⣥⣥⣥⣿⣶⣶⣷⣴⣿⣿⣿⣻⣿⣿⣿⣿⡿⣿
⠀⠀⢸⡷⠋⠀⠀⣠⣇⣀⣤⣽⣣⡶⠖⠙⠛⠛⠛⠚⠛⠛⠉⠉⠉⠿⠟⠻⠟⠛⠛⠛⠛⠛⠻⠿⠟⢿⡇⣿⣷⡆
⠀⠀⠸⠁⣀⣩⣾⣿⣛⣁⢀⡘⣧⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠇
⠀⠀⢰⣾⡿⠿⠛⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⣀⣀⣀⣀⠀⣴⠶⢦⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠛⠷⣤⡾⠛⠿⠶⠶⠶⢶
⣴⠶⠞⠛⠋⠉⠁⠉⠻⠃⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡾
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡾⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣸⣀⣀⡀⣰⡉⠀⠉⡿⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⣿⠀⠸⡿⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣿⠀⢠⡟⠛⠛⠛⠛⠓⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠍⠀⡀⠿⠀⠀⠄⠀⠠⡄⠀⠀⠉⠙⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠃⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⡀⠀⠺⠂⠀⢀⡠⠄⠐⠀⠠⠤⠤⠤⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢠⡦⠈⠁⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣀⠀⠀⠀⠃⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡀⡀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢰⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠰⠂⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢡⠄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠔⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠰⠅⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠚⠁⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠀⠀⠀⢀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡀⠀⢠⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠹⣄⠀⠀⠨⠄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠁⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠗⢝⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣄⠀⠀⠀⠳⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢸⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⢣⠀⠀⠀⠀⢠⡄⠀⠀⠀⠙⠳⠤⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⣠⠞⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢄⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⢦⡀⠀⠀⠀⠑⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⠀⠀⠀⠉⠀⠀⣀⡴⠞⠁⢀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠙⠳⢦⣀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⠶⠛⠁⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠈⠉⠓⠶⠆⢠⣤⡴⠒⠦⣤⣤⣤⣤⠤⠶⠖⠊⠉⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣤⣀⣀⣀⣰⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠞⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣭⣷⣶⣶⣶⣾⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⢐⠀⠀⠤⠀⠀⠨⡀
⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⢀⠀⠀⠀⠀⠁⠀⠀⠤
⠀⠀⠉⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⢀⠀⠀⡠⠂
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⢠⠀⠀⠀⠀⡠⠀⡀⡄⠀⠀⠀⠀⠀⠀⡤⠀⠀⡄⡀⣀⠀⢠⢀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⡇⠀⡆⠶⢸⠠⣏⠀⠈⠳⣸⡁⠷⢸⢴⣳⢠⣻⠆⡇⠀⠠⡖⠁⡆⣔⣹⠹⡌⠀⠀⠀⠀⠀⠄⢀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠘⠈⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠆⠈⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⡀⠐⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠂
⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⡄⠀⠈⢁
⠀⠀⠉⠀⠀⠀⠄⠀⠀⠀⠀⡀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⠄⠠⠀⡀⢀⠀⠀⠀⠀⠀⠄
⢄⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠂⠐⠀⠀⠀⠀⢀⡄⠊⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠
⠀⣉⠢⠤⠄⢀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠌⠁⠁⠀⠄⠀⠀⠀⠀⠀⠀⠐⠛⢯⠀⠆⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⣀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠊⠁
⠃⠀⠀⠆⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠑⠄⠈⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂
⠀⠈⠀⠀⠀⢐⠀⠠⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣹⠐⠒⠚⠉⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠒⠀⠈⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⣿⣿⠇⠀⢿⣿⣮⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⢋⠀⠘⡏⠀⡉⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄⠁⠚⠜⠈⣁⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠉⠁⠀⠉⠁⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⠀⠀⢠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣠⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠉⠉⠋⠉⠻⠿⠿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠤⠄⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣿⣿⣷⣦⣤⣀⠀⠀⠀⣀⣠⡤⣤⣄⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⣤⣤⣭⣻⣿⣿⣿⣾⢝⣧⣶⣿⣷⡜⢿⣿⣿⣿⠁⣹⡿⠟⢿⣿⣷⣶⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣛⣿⣿⣿⡟⢋⣤⣾⣿⣿⣿⣿⡇⣼⣻⣿⣿⠀⢿⠸⠯⢸⠟⠉⢁⢹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣛⣛⣿⣿⠟⢉⣼⣿⣿⣿⣿⣿⣿⡿⠥⣛⡿⠉⠸⢀⠈⠛⠚⠁⠀⠀⠞⠧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠭⠛⠛⣫⣿⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡈⠙⢠⡈⠀⠀⠀⠑⢢⠀⠀⠀⣠⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠤⠖⣺⣿⣿⠏⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡀⠀⠀⠀⠈⣄⠀⠀⠡⣠⠔⠁
⠀⠀⠀⠀⠀⣀⣤⣴⣿⢃⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡀⠀⠀⠠⠴⠐⠦⣌⣙⡲⠞⠁
⠀⠀⠀⠀⠈⠁⠘⣽⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠿⠳⢤⡀⡈⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠴⣿⡟⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣧⣀⣀⠀⠁
⠀⠀⠀⠀⠀⣠⣾⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠈⢙⣯⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⣀⣴⣿⣿⠋
⠀⠀⠀⠀⠠⢻⣿⠨⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⢾⣿⣻⠿⡫⠃⣰⣿⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢠⢿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠭⣛⡈⢯⣳⣉⣑⢶⡿⠏⠉
⠀⠀⠀⠀⢀⠁⢸⡆⢿⣿⣿⣿⣿⣿⣿⣿⡓⠄⡁⠀⠉⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡄⠙⢣⠸⣿⣿⣿⣿⣿⣿⣿⢃⠢⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣳⢾⡶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠡⣀⠙⣁⢻⣿⣿⣿⣿⣿⣿⣧⡟⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠁⠀⠀⠈⠉⠙⢳⣝⢮⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⢿⣤⠑⣤⢿⣿⣿⣿⣿⣿⣟⡰⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠃⠙⡞⡶⡄⠀⠀⠀⠀⠀⠀⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⣌⣿⣿⣿⣿⣿⣿⣽⢳⡂⠀⠀⢀⣴⣶⡶⠦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠃⣳⣧⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡡⣿⡆⣙⢿⣿⣿⣿⣿⣿⣽⣄⠘⠿⣯⠭⠴⢦⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⠀⠀⣠⣀⣴⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢄⢻⣆⠛⣿⣿⣿⣿⣷⣯⣦⠀⠀⠂⠀⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣮⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢘⠏⡛⢶⣼⡻⢿⣿⣿⣿⣯⣴⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠙⠺⣏⢀⡀⠈⠙⠻⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣴⣭⣤⣄⣴⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠦⢭⣭⣭⣽⣿⣿⣿⣽⡿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⡞⣻⣏⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⢠⠀⠀⠰⣴⣶⡴⠀⠀⢰⣶⡦⠀⠀⠀⠀⠀⠐⢛⡴
⠀⠀⠀⠀⠉⠙⣿⠂⠉⠀⠀⢀⣾⣶⣿⣤⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⠻⠟⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⢧⠘⡀
⠀⠀⠀⠀⠀⠀⡜⢰⠀⠀⠀⠀⣿⣿⣿⢽⠄⠀⠀⠀⡤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠁⠂⠀⠀⠀⠀⠀⠀⢿⠈⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠩⠛⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⡄⠀⣲⣬⠃⠀⠠⢠⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣴⡇⠀⢀⢸⠀⡄⢀⡀⠀⠀⠄⠀⠀⠀⠉⠕⠃⠀⠁⠀⠀⠀⠀⠈⠁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠏⠉⠁⠀⠀⠈⠀⣘⠿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⠤⠀⠀⢀⣠⡀⠀⠀⡄⠠⣤⠆⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠃⠀⠀⠀⠈⠃⠲⠆⠀⠘⠛⠁⠀⢄⣷⠠⠁⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠠⣤⣄
⠀⠀⠀⠀⠀⠘⢠⡧⠖⠀⠀⣾⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠃⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⣦⠠
⠀⠀⠀⠀⠀⠀⠘⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠈⠻⣦⣄⣠⣤⣤⣴⡿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣻⣿⣿⣿⣿⣿⣇⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠐⠟⠋⠁⠉⠙⠙⠛⣛⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠀⢾⡿⠆⠀⠀⣾⠿⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣐⣀⡀⠏⣀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⠿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⢿⢿⣟⣿⡿⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡿⠟
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠛
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈
⠀⠀⢾⣿⣿⣿⣿⣿⠟⠛⠛⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣰⣿⡟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⣿⣿⡇⠀⠀⢰⡀
⠀⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠁⠃
⢰⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⣀⣀⣴⣧⣀⡀⣀⣠⡴⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠲⡖⣒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⠎⣀⡀⠀⣾⣩⣽⠿⡦⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣋⣷⣿⣿⣿⣿⣯⣿⣦⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⣶⣶⣿⣿⣷⣯⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⡀⠀⠀⠀⠠⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⡟⣡⡆⠙⣿⣿⣿⣿⣿⣿⣷⣹⡆⠀⠀⠀⠀⠀⡙⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⠈⠉⠀⠈⢹⣿⣿⣿⡿⠋⣹⠇⠘⠄⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⠟⠛⠛⠛⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⣆⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢄⣀⣀⣬⣭⣴⡶⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡎⢠⡀⢂⢶⣖⡚⠉⠉⠉⠀⣠⢠⠁⠴⠐⠀⠀⠀⠃⢣
⠀⠀⠀⣀⣀⣤⣤⢤⡤⠴⠶⢒⣿⣿⣿⠧⢉⣄⡔⣣⡟⣽⣿⣻⣶⣶⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠈⠋⣋⣿⠏⠀⠀⠀⢠⣿⡿⣟⢫⣶⡛⠋⠀⠈⢠⢿⢧⢿⣿⠏⠀⣤⡂⠀⢰⠀⠀⠀⠀⢠⡇
⠀⠀⠀⢠⣿⡿⠀⠀⠀⢠⣿⣿⢏⠁⠀⢡⠂⠀⠀⠀⡾⢰⡿⣿⡞⢀⣾⣽⠇⠀⢸⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⡏⠀⠀⢠⠃⠀⠀⠀⢸⠁⢟⠇⡿⣠⠞⣿⣟⠀⠀⠘⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣀⣤⣾⣠⡂⣠⢦⡇⣼⠊⣿⡗⠃⡼⠿⠃⠀⠀⢾⠻⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿⣿⡩⢿⣿⣷⣿⣿⣼⠟⠘⢿⣷⡞⠀⠸⡄⠀⠀⠸⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠿⠿⣫⣻⣿⣿⣿⣿⡟⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠠⣷⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠉⠿⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣌⡻⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣾⠀⣴⣿⣿⣿⡦⠀⠀⠘⣷⡀⠀⢀⣙⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡈⢻⣿⣿⠏⠀⠀⠀⠀⡿⠀⡰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣤⡀⠀⠸⠿⠂⠀⠀⢠⠀⠀⢆⡼⠁⠀⣀⣊⠁⣨⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠿⣉⠽⠒⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⢤⠉⠀⠀⠀⠀⠀⢠⣄⢁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣧⠀⢰⣿⣶⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⠏⣀⣉⡀⠀⠈⠁⠀⠀⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡻⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣴⣦⠀⠈⠛⠿⣿⠿⢦⡄⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠂⠀⠀⠀⣶⣿⠿⠛⢹⡏⠈⠃⢽⡻⣷⡀⠀⠀⠀⠐⠚⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡤⠜⠛⣡⣶⢟⣿⣿⠀⠀⠀⠀⠀⠙⠀⠀⠖⠀⢠⡤⠶⠦⠜⠃
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠄⠀⠠⠿⠖⢾⣧⣿⣯⠁⠀⠀⠀⢘⠀⠀⠀⡀
⠀⠀⠀⠀⠸⣿⡝⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⢻⡽⠇⣀⣤⡶⠛⠹⠇⠀⢠⠀⠀⢻⡦⠄
⠀⠀⠀⠀⠀⠀⠙⠓⠿⢄⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⡠⣾⢁⢸⡾⠻⢟⣶⣟⠀⠀⡀⠈⠀⠀⠛⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⢰⠟⠀⠀⠈⠁⠘⠀⣿⠀⠀⠈⠀⣿⣞⡆⠰⣄⣘⣿⡙⠛⠛⠛⠿⡦
⠀⠀⠀⠀⠀⠀⠠⠦⠾⠋⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⣿⣶⠈⡅⠀⠀⠀⠀⠙⢿⡇⠀⠈⠙⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢤⣜⣗⡛⠉⣸⠇⠀⣠⣄⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⢻⣿⣿⣷⢾⣿⣟⢫⡿⢀⣿⣦⠀⠜⠉⢿⣿⡷⠀⢠⣴⢿⠶⠀⢀⣤⣀⠤⣀⡀
⠀⠀⠀⠀⣠⡶⠿⢛⣛⠟⠢⣤⣴⣬⣼⣿⣿⡄⠙⠻⠇⠙⢸⠁⢸⣧⠀⠀⠸⠟⠁⠀⠸⠡⠀⠐⠻⢿⣯⣾⣿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣩⣷⣶⠚⠀⠓⠄⠸⡀⠈⢿⡎⠀⠴⠚⠿⡷⣾⣿⣷⣦⠀⠀⠈⠉⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠁⠀⠀⠀⠀⠣⠀⠀⡵⢀⣤⣤⣶⣟⣓⣿⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠄⡘⠀⠀⠈⠙⠛⠛⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣯⣩⣭⣯⣥⣍⣽⣿⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣭⣭⣭⣿⣿⣿⣭⣭⣭⣭⣭⣯⣭⣭⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣀⡀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢦⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣻⣿⣿⣿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠙⠉⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡏⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⢠⣿⣟⢀⣀⣠⣤⣶⠶⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣤⣴⣾⣿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡶⠾⣿⠿⠋⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣼⣦⣾⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣶⣾⣅⣠⡏⠀⢀⣀⠄⣀⠧⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣶⣶⣶⣦⣄⠀⠀⠀⡤⣟⣋⣹⡏⠁⢀⣄⢉⣇⣏⣀⣷⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢂⡤⠀⣈⣹⣿⣿⡿⠿⠛⢹⠋⠹⢐⣾⣿⣿⣿⣶⣤⢴⣈⣝⠉⣻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣵⣿⣷⣿⣿⡿⠛⠇⢠⠀⠠⢠⡏⠑⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣋⡵⢮⠀⡈⠀⢀⠬⣙⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣷⡾⣀⢈⣧⣻⠤⠞⠊⠉⠉⠉⠉⠉⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢷⣾⣿⣫⠖⢫⣭⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⢿⣿⣿⠿⣻⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⡟⣽⣿⣾⣿⠟⠁⠠⠶⠞⠛⠿⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⣯⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠁⠀⠀⠀⠀⣀⣀⣀⣈⣟⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢋⣤⠈⢿⠋⠀⠀⠀⠀⠰⢾⣿⡟⢿⣄⠀⠀⠀⠀⠀⠾⠟⠉⡉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠟⠀⠀⠀⣤⢾⣿⠛⠻⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣀⠐⡋⠉⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠐⡿⠟⠻⢿⠇⠀⠄⣠⣿⣿⡆
⠀⠀⠀⠀⠀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⠒⢶⡖⠀⣠⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣦⣀⠀⠀⠀⢸⡁⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣶⣶⠟⠣⠀⠘⣿⡿⠋⣼⠃
⠀⠀⠀⠀⠀⠙⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠠⠻⠿⢛⡏⠁⣠⡤⣈⣐⠯⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠰⠁⣾⠃⢀⣀⣈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠞⠁⡉⠉⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⣿⣦⣄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣈⢿⡛⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠁⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣀⣀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⡻⣏⢹⡿⣿⢿⠉⢉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⢲⣿⣿⣦⣄⠙⠋⢀⣼⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣾⣿⣿⣿⠛⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⢿⣿⣿⣿⡿⣿⣳⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠙⣿⣿⣦⣤⣴⡀⣩⣽⡇⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁⠡⣾⣻⣿⣿⣿⣿⣿⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢨⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢀⣘⡀⠴⠟⠁⠑⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡈⣸⣿⡶⢶⣿⣷⠠⣄⡀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠁⢻⣿⣿⣶⣶⣿⣿⣿⣿⢷⣄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⢻⠛⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡛⢻⡟⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⡈⡀⠀⠀⠀⠀⣀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣾⣿⣟⣉⠉⠀⠉⢙⡖⠁⠀⢩⣿⣾⣿⡄⠢⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠰⣾⣦⣀⠀⠀⠀⣀⠛⠛⠛⠿⠻⠷⠦⠀⠀⠼⠦⠿⠿⠿⢿⣿⣆⢦⡀⠀⠀⠀⢘⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⠀⢠⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⣀⠔⠂⠉⠉⠉⠒⠦⡀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠜⠀⠀⡠⠁⠀⠀⠀⣀⠴⡒⣒⢒⡒⡢⢄
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠙⣿⣿⣿⡄⠌⠀⠀⡐⠁⠀⠀⢀⠞⣵⠝⠋⠀⠀⠈⠛⣴⠑⡄
⠀⢸⠀⣴⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠀⠀⠀⠘⣿⣿⣿⡄⠀⡐⠀⠀⠀⠀⡘⣸⡃⠀⣀⣴⣧⣄⠀⠘⣧⢸
⠀⢸⠀⢿⡀⠀⠀⠀⠀⠀⢀⡯⢀⠃⠀⠀⠀⠀⠘⣿⣿⣿⡔⠀⠀⠀⠀⠀⢃⢽⡄⠀⢈⡿⢿⡇⠀⠀⣯⢸
⠀⠀⢣⠈⠳⣤⣀⠀⣀⡴⠞⢁⠞⠀⠀⠀⠀⠀⠘⣻⠄⢖⡋⠀⠀⠀⠀⠀⠘⣔⢿⣄⠀⠀⠀⠀⣠⡼⢂⠎
⠀⠀⠀⢑⢦⣄⣉⣀⣁⣠⡴⠁⠀⠀⠀⠀⠀⣴⡿⢓⠀⢐⠻⡷⡄⠀⠀⠀⠀⠈⠲⣍⣛⠃⠐⢛⣩⠔⠁
⠀⠀⠀⡜⣾⣿⣿⠛⣿⣿⣷⢣⠀⠀⠀⠀⢸⣝⠐⠀⠀⠀⠑⢸⣽⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⣿⣷
⠀⠀⢰⠽⠻⣿⠏⠀⢻⣿⠛⠏⡆⠀⠀⠀⠸⣻⡔⢀⠀⢀⠔⣸⡻⠀⠀⠀⠀⢠⣾⠿⣿⠏⠸⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠑⠿⣶⣀⣲⡾⠗⠁⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⢾⠯⣠⣛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣪⡤⠥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⢫⣹⣳⣤⣤⢤⣤⣤⡶⠤⠤⠖⠚⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣀⣦⡶⠟⠛⠉⠁⣷⠣⣫⡿⡟⣠⠂⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢴⣿⠟⠋⠀⢀⡰⣶⣤⡘⣿⣿⣿⣿⠯⣠⣾⠋⠀⢀⠀⠀⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠋⠁⠀⠀⠀⠀⠙⠷⠎⣸⣿⣿⣿⠋⣠⣛⢛⠚⡓⠉⠀⢀⣠⣽⡁
⠀⢀⣤⣤⣤⣤⣤⡤⠤⢤⡜⡁⢀⡀⣀⣀⣀⣠⣤⣴⣶⡿⠟⠛⠉⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠹⠿⠃
⠀⠀⠉⠛⠳⣶⣿⣇⡈⠟⢛⠙⠛⠿⣋⣭⠉⠠⠀⠓⠁⠀⠀⠀⡀⠀⡠⠆
⠀⠀⠀⠀⣼⣷⡿⠟⠁⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠈⢀⠀⠀⠀⠀⠁⠴⠒
⠀⠀⠀⠸⠟⠋⠀⠀⠈⠀⠀⠉⠉⠀⠉⠉⠶⠤⢄⣀⣀⠀⠐⠀⣀⣀⠠⠐⡒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠐⠒⠐⠒⠤⢠⢀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣴⣿⡷⠾⢗⠒⣀⠠⠀⣀⠀⠈⠘⢷⣄
⠀⠀⠀⠀⠄⠂⢀⠀⠀⠀⠀⢰⣿⠟⠑⠈⡀⠀⠀⡀⠀⠀⠀⠀⠢⣄⠀⠙⢗
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣉⢉⣀⠀⠶⡾⠿⠿⣶⣥⡢⡀⠀⠀⠪⠑⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠈⠠⡠⠊⢁⠔⠳⢄⠈⠑⢄⠀⠁⠂⢍⠻⠦⠀⠀⠀⠀⢎⠀⠀⢆
⠀⠀⠀⠈⠀⠀⠰⡝⢀⠔⠃⠤⣶⣄⡑⠀⠀⠀⠀⠀⠀⠑⢄⠈⠀⠀⠀⠀⠢⠀⠈⠂⠀⠀⢀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿⣷⣦⡙⣿⠀⠀⠠⠀⢀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢌⠻⠿⠟⣡⠞⠁⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠑⢤⠞⠁⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣾⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣏⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⢀⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⢿⣿⣿⣇⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣾⣿⣿⠆⠀⠀⣰⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢸⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⡿⠁
⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⢿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⠿⠟⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⢤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢒⣒⠶⣤⠴⣒⠲⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢦⣾⠿⠄⣄
⠀⠀⠀⠀⠀⠀⠂⢛⡐⡄⠀⢀⢥⣧⠀⠀⠀⠀⢀⣴⣿⡧⡏⣴⣷⠀⡊⣴⣿⢸⣿⣦⡀⠀⠀⠀⢀⣴⣿⠼⠃⢺⠟⠀⢸⣿⣧⡀
⠀⠀⠀⠀⠀⢐⣀⣨⡵⠃⠀⠘⠒⠈⠀⠀⠀⡔⠁⡧⢻⣷⣍⣛⣵⠀⣬⣛⣫⣾⡟⢸⡏⢆⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⡇⠂⢌⡋⡋⠅⠀⠉⠛⢋⠡⠐⣀⣇⣸⠀⠨⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⡇⡒⠈⣡⠀⠃⣀⢘⢠⡍⠂⠃⣸⠏⠏⠀⠀⢿⣿⠇⢀⣴⣦⣤⣀⣠⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣮⠰⠖⠀⢠⡄⢉⢠⡄⠀⠲⢀⣯⡀⠀⠀⠀⠈⣿⢸⡿⠙⠛⢻⣿⣾⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠷⠨⠀⣤⠈⠅⠷⠈⣾⠏⠀⠀⠀⠀⠀⠈⢻⣧⡾⢆⣼⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⡄⠀⠀⠀⣤⣤⡜⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⠾⣿⣿⣷⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⠀⠀⠀⣤
⠀⠀⠀⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⣀⠉⠈⠀⠀⢀⠁⢩⣀
⠀⠀⠀⠀⡟⠁⣴⡆⢰⡆⣰⣦⠈⣿⡇⠀⠀⠀⠀⠘⣿⣿⡟⢁⣁⣈⣤⠘⣿⣿⠁⠀⠀⠀⡞⠁⡴⠶⡄⠰⢁⡴⢦⡈⢳⡄
⠀⠀⠀⠀⣧⡀⠛⢃⡜⢣⣈⣋⣠⣿⡇⠀⠀⠀⠀⠀⠘⠿⣧⣘⠻⡟⢉⣴⠿⠁⠀⠀⠀⠀⣷⡈⠳⠴⠃⣴⠘⠦⠼⢃⣼⠃
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠒⠋⠉⠘⠒⠒⠛⠁
⠀⠀⠀⠂⠈⢰⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀⠘⠇
⠀⠀⠀⠀⠀⢀⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡖⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠒⣧⣤⡀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⠁⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢶⡶⣴⣦⡀⡀⠀⡀
⠀⠀⠀⠈⠙⠁⠋⠙⠈⠙⠉⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠒⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⣀⡀⣀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠉⠉⢻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣄⠀⠀⣠⣿⣿⡿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡿⠀⠀⢿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠃⠐⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⣧⡠⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣽⣧⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢀⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⣿⣧⢸⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡛⣏⣿⣿⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⣻⣯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣿⣬⣿⣦⢹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣿⣻⣿⣄⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣈⡿⣿⣿⣿⣿⢦⣤⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢿⣧⠙⢹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣾⡿⠛⠚⠛⠛⠛⠛⠓⠛⠓⠲⠴⠦⠴⠶⣶⣶⣶⣶⣶⣶⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⣀⢀⣀⠀⠀⡀⠀⠀⢀⡤⣼⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣯⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠤⢼⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⡿⠟⠋⠀⠀⠀⠀⠰⠀⣀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣤⠈⠄
⠀⠀⠀⢔⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡱⠀⠀⢠⣿⣿⠿⠛⠉⣇⠸⡀
⠀⠀⠀⣀⣽⣿⠦⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠀⠀⠘⣿⣇⠀⠀⠀⡇⠀⠈⡄
⠀⠀⢠⣿⣿⡁⠀⠈⠢⡬⠒⢄⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠈⢻⣷⡖⠈⢳⠐⢄⡁
⠀⠀⠀⠙⢿⣥⣀⣀⠠⠧⡄⠀⠵⢄⠀⠀⠀⠀⠡⠘⠁⠀⠀⠀⡜⠁⠈⠁⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠔⠁⠀⠀⠀⠈⠢⡀⠀⠀⠙⠢⠄⠐⠘⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠔⠁⠐⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⡀⢀⠤⠒⠐⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⠀⠀⠘⡳⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠃⠂⠈⠉⠢⡀⠀⠀⢀⡠⠄⠂⠁⠀⢡⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠎⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⢀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡌⠀⠀⡠⠊⠉⢄⠀⠀⠃⠀⢀⠁⠀⡠⠂⠁⠈⠢⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠢⠄⠀⠀⡘⠀⢀⠎⠀⠀⠀⠀⠡⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠈⠢⡀⠆
⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⡰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠐⠦


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠒⢒⣿⣿⣷⣭⣭⣭⣭⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣵⣾⣿⣿⣦⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⣾⣿⣿⣿⣿⣿⣿⢾⣿⣿⣻⣿⢿⣿⣿⠿⢿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⣼⣿⣿⣿⣿⣿⣿⡿⣨⣿⣿⣯⡻⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣏
⠀⣀⣤⣶⣶⣶⣶⣶⣦⣤⣀⣀⣈⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀
⢰⣿⣿⣿⣿⣛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⠈⠉⠉⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣧⡄

⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣴⠒⠒⠒⠒⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠳⠋⠙⠓⢲⠆
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠋⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠶⢤⡄⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢀⠀⠓⢤⠄
⠀⠀⠀⠀⠀⠀⠁⡄⠐⠂⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠌⠀⠠⠤⠄⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢨⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠱⣄⣀⠀⠉⠊⠑⠒⢳⠀⣗⢒⠒⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠦⡀⢠⠀⠀⠘⠁⠉⠀⠀⠠⢊⡤⠖⠒⠊⠉⠻⣀⠇⠀⠀⠉⠉⠒⠦⢄⣀⠛⢄⠀⣀⠀⠀⠐⢀⠀⠀⠀⢙⠲
⠀⠀⠀⠒⠀⠀⠀⢅⠀⠀⠀⣠⠔⠊⡵⠛⠉⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠨⡁⠀⠀⠘⠁⠀⠀⢀⣨⠄⠒⠠⡄
⠀⠀⠀⠀⠰⠎⠀⠈⠀⠀⠀⠀⢀⠞⠀⠐⠀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠱⡚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⠈⢉⡏
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢒⠼⡆⢈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡆⠚⠀⠁⠀⠀⠀⢠⢤
⠀⠀⠀⠐⠀⡏⠀⠀⠀⠐⠓⣤⠘⠢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⢀⠀⠠⠂⠀⠓⡲
⠲⠦⠤⠀⠀⠥⠦⠀⠀⠀⠀⡏⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠠⠎⠙⡰⠂⠀⠀⠀⠀⡠⠄⣴
⠀⠀⠀⠰⠃⠀⠀⠀⠀⠲⢀⡇⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢠⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠂⠈⠉⢉⠝⠀⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⣈⣧⠀⠁⢀⠀⠑⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡐⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢨⢦⡐⢄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡀⠀⠀⠀⢀⡠⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⠦⣑⠢⠄⡀⠀⠀⢀⠀⠀⠀⠀⣀⠠⠀⠀⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⣤⣉⣈⣤⡉⠉⠉⢁⣀⣠⠴⠚⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠁⠀⡠⠋⠀⠀⠋⠉⠁⠀⠀⠰⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠸⡄⠀⠀⠀⠀⠀⠐⠂⠁⠀⠄
⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣈⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣴⣿⣿⣿⣿⡿⠙⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣀⣀⠀⣼⣿⣿⡯⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⣿⣿⠟⠋⢩⣿⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣿⠈⠑⠀⠀⠀⠀⢸⢿⣿⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠂⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠃⠀⣻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣤⡖⠖⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠳⠀⠙⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡻⣿⣧⠀⠀⠀⠈⣿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⠟⠛⠚⠁⠀⠀⠈⠛⠛⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠸⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠬⠙⠛⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠨⠭⠁⠀⠈⣈⡁⠀⠀⣈⡉⢀⣾⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠁⠀⠉⠀⠈⠀⠀⠀⠰⣿⣿⠿⠛⣽⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣶⡄⠀⣀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠛⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠤⠀⠜⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡔⠁⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠨⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠠⠂⠀⠀⠀⢀⠀⠈⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠤⠘⠀⠀⠀⠀⠙⠤⠃⡀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⢀⠀⠀⠀⠀⢂⡰⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠐⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⡀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⢀⠀⠁⠀⡨⠦⣀⠀⠀⠀⠀⢀⡠⠃⠄⠀⠀⠀⠢⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠀⢀⠎⠀⠀⠀⠁⠀⡈⠀⠉⠈⠋⠁⠐⠰⠂⠐⣀⠞⠁⠀⠀⠀⠀⠀⠀⠈⢈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡐⢰⠁⠄⠀⡘⠀⠉⠂⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠃⡘⠃⢠⠀⠃⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠄⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠆⠀⠀⠀⠂⠰⠀⡄⠀⠀⠀⠀⢀⠀⢀⠒⢠⠀⡀⠂⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠠⠂⠀⠊⢀⠈⠀⠀⠀⡀⠀⠀⠀⠈⠠⢊⠀⠀⣠⢀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠂⠀⢀⡄⠂⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡀⠀⡷⠊⢀⠊⢡⠄⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠐⢉⣀⠔
⠀⠀⠀⠀⠀⠀⠈⠉⣄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⠀⠃⠀⠀⠀⠂⠀⡠⠀⠀⠀⠈⠂⠌⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⢀⠀⠑⠂⠂⡀⠐⠂⠄⠀⢀⠀⠀⠀⠠⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⢁⠀⠂⡀⡆⠰⢰⠀⠄⠀⠒⠈⡀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⢀⠔⢣⠄⣀⠀⠀⣀⠴⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠸⠲⡀⠁⡤⠚⠀⡰⠀⠀⠀⠂⠀⠀⠀⣀⠆⠐⠴⢃⡠⠇⢈⠆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⡄⠠⠃⠐⡇⠀⢠⠃⠐⠈⠁⠀⠃⡀⠆⢀⠀⠈⠀⢀⠎⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠇⢀⠀⠀⢀⠀⠁⠀⡾⠀⠀⠀⠀⠄⠈⠂⡀⠀⠀⠀⠠⠂⠘⠀⡀⠠⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⢠⠐⣯⠀⢠⠀⡇⠘⠀⠔⠁⠀⠀⢰⠁⡆⠀⠀⠄⡴⠂⠁⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠈⠀⠄⠀⠀⢳⠀⢇⠀⠈⡄⢻⡄⠈⡀⠀⠰⠀⠐⠁⠀⠀⠂⠄⠀⢉⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣇⠘⡅⠀⠁⠈⠓⠃⠃⠘⠸⠅⠀⠀⠠⠃⠾⠥⠐⠀⠇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠂