# 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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