Let $x, y, z \in \mathbb{R}$ and the equation $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$ holds. Find the minimum value of $x^2 + y^2 + z^2$, which can be expressed as $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $\gcd(a, b) = 1$. Determine the value of $a + b$.

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