Let $x, y$, and $z$ be positive integers satisfying the following system of equations:
$$x^2 +\frac{2023}{x}= 2y^2$$
$$y +\frac{2028}{y^2} = z^2$$
$$2z +\frac{2025}{z^2} = xy$$
Find $x + y + z$.

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