2023 AIME I Problem 10

There exists a unique positive integer a for which the sum \[U = \sum_{n=1}^{2023} \left\lfloor \frac{n^2 - na}{5} \right\rfloor\] 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.)

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.