Trigonometry and Algebra Crossover

Geometry Level 5

Consider the system of equations

\[ \begin{align*} \frac{x}{y} &= xy + 2 \\ \frac{y}{z} &= yz + 2 \\ \frac{z}{x} &= zx + 2 . \end{align*} \]

Let \((x, y, z)\) be the solution to this system with the minimum value of \(x\). Then, \(x + y + z = - \sqrt{a} \) for some positive integer \(a\). Find the value of \(a\).

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