2023 AIME II Problem 4

Let x, y, and z be real numbers satisfying the system of equations \[ \begin{aligned} xy + 4z &= 60, \\ yz + 4x &= 60, \\ zx + 4y &= 60. \end{aligned} \]

Let S be the set of possible values of x. Find the sum of the squares of the elements of S.

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.