Let $x$, $y$, and $z$ be real numbers such that:
\[
12x - 9y^2 = 7 \\
6y - 9z^2 = -2 \\
12z - 9x^2 = 4
\]
Find the value of $6x^2 + 9y^2 + 12z^2$.

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