AIME 2023a Problem 10

There exists a unique positive integer $a$ for which the sum $$U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor$$ is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$. (Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)