Let A = [x, y, z; y, z, x; z, x, y], where x, y and z are real numbers such that x+y+z>0 and x y z=2. If A^2= I_3, then the value of x^3+y^3+z^3 is \…

Let $\displaystyle \mathrm{A}=\left[\begin{array}{lll}x & y & z \\ y & z & x \\ z & x & y\end{array}\right]$, where $\displaystyle x, y$ and $\displaystyle z$ are real numbers such that $\displaystyle x+y+z>0$ and $\displaystyle x y z=2$. If $\displaystyle \mathrm{A}^2=\mathrm{I}_3$, then the value of $\displaystyle x^3+y^3+z^3$ is \_\_\_\_.