Let $x$, $y$, and $z$ be real numbers such that
$$12x - 9y^2 = 7$$
$$6y - 9z^2 = -2$$
$$12z - 9x^2 = 4$$
Find $6x^2 + 9y^2 + 12z^2$.

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