⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡧⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠂⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡼⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠘⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠠⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣾⣅⢀⣰⣾⠂⠀⠀⠀⠀⠀⢘⣿⢀⠀⢀⠀⠀⠈⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⢀⠂⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⡾⣷⣺⢿⠿⠀⠀⠀⠀⠀⠀⣼⡟⠃⠠⠀⠀⠀⠀⠁⠀⠄⠂⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣷⣇⡄⠀⠈⠀⠀⢀⠆⠘⣷⠀⠀⠀⡀⠀⠠⠐⠀⢀⠠⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢻⡟⢻⠛⠀⠀⠀⠀⠀⢸⠆⢠⣿⠠⢈⠐⡀⠡⠀⠀⠄⠀⡀⠐⠈⠀⢀⠐⠀⠀⠀⢀⠀⠄⠀⠠⠐⠀⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣄⡛⠁⠂⠀⠀⠀⠀⠸⠠⠂⣿⠀⠀⠐⡀⠀⠐⡠⠀⢂⠀⢂⠡⠈⡀⠠⠐⠈⠠⠀⠠⠀⠂⠀⠄⠂⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣌⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⡟⠈⢠⡀⠀⠆⠀⠘⢆⠠⠈⠠⢀⠡⠐⡀⢂⠁⢂⠁⠂⠄⡐⠀⡀⢂⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠱⠄⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠑⡆⠀⠀⠀⢸⠐⠈⠂⠀⠂⢡⠐⠠⠌⠠⠌⡐⠠⢀⠡⠐⠠⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠻⠿⣿⣶⠀⠀⠀⠸⡀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠈⠠⡈⡁⢂⠤⢁⠂⠤⢁⠂⠡⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⣇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠻⣿⣄⠀⠀⠰⠠⠀⢠⣤⣄⠙⢦⡀⠀⠙⣇⠀⢧⡘⠀⠀⠀⠀⠀⠀⠀⠀⣹⡐⡈⠐⡄⢊⠐⡠⢊⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⠀⢘⣻⢧⡠⠙⠻⣵⡄⠱⣦⡀⠀⠀⢺⠃⠀⠀⠀⠀⠀⠀⠀⠠⠋⣷⠠⢁⠒⡈⠔⡐⢂⠩⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢠⡗⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠤⢙⡕⠀⠀⠀⠉⠶⠾⠀⠀⠀⢀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⡈⠆⢡⠊⠤⣁⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣼⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠆⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠊⣸⡇⡱⠘⡄⢊⠔⢠⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠚⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⣹⡇⡐⠡⡐⠌⡐⢂⡉⠐
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡄⠀⠀⠀⠀⠀⠠⣼⣇⠤⢁⠐⡈⠐⠠⢀⠃
⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢱⣿⠀⠂⠐⢠⠁⠂⡄⠈
⠀⠒⠲⠶⣶⡆⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠘⣧⣽⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠢⢻⡅⠀⠀⠀⠀⠀⠀⠀
⣀⠀⣀⣀⣀⣀⣀⠾⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀
⠿⠿⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀
⣀⣀⣀⣀⣀⣀⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠂⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀