Suppose $x$, $y$, and $z$ are integers that satisfy the system of equations:  
\[ x^2y + y^2z + z^2x = 2186 \]  
\[ xy^2 + yz^2 + zx^2 = 2188. \]  
Evaluate $x^2 + y^2 + z^2$.

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