$\cos 2C$ from a Circumcenter Relation

Let $O$ be the circumcenter of $\triangle ABC$, and suppose $x\overrightarrow{OA}+y\overrightarrow{OB}+z\overrightarrow{OC}=\vec 0$ with $xyz\neq 0$, where $C$ is an interior angle of $\triangle ABC$. Express $\cos 2C$ in terms of $x,y,z$.