Matrices and Determinants: JEE Main 2021 Mathematics question

Let $A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$, where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If ${A^2} = {I_3}$, then the value of ${x^3} + {y^3} + {z^3}$ is ____________.