Let $x$, $y$, and $z$ be positive integers satisfying the following system of equations:

\[
\begin{align*}
&x^2 + \frac{2023}{x} = 2y^2, \\
&y + \frac{2028}{y^2} = z^2, \\
&2z + \frac{2025}{z^2} = xy.
\end{align*}
\]

Find $x + y + z$.

Let's think step by step and output the final answer within \boxed{}.
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