Quadratic System of Equations

Algebra Level 4

How many unordered triplets $(x,y,z)$ are in the range $[0,2]$ and satisfy the system of equations: $\left\{\begin{array}{l}2x^2-4x+2=y\\ 2y^2-4y+2=z\\ 2z^2-4z+2=x\end{array}\right.$

Details and Assumptions:

An unordered triplet means that $(1,2,3)$ is the same as $(3,2,1)$ or $(1,3,2)$ .