Positive rational numbers $x < y < z$ sum to $1$ and satisfy the equation:
\[ (x^2 + y^2 + z^2 - 1)^3 + 8xyz = 0. \]
Given that $\sqrt{z}$ is also rational, it can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. If $m + n < 1000$, find the maximum value of $m + n$.

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