Equality or Inequality, That is the question!

How many $\textbf {unordered}$ triplets $(x,y,z)$ , subject to constraints, $(x^4-2x^3)_{\text{cyclic}}\leq0$ , satisfy the system of equations:

$\left\{\begin{array}{l}(4x^2-8x+3)\sqrt{(2x-x^2)}+1=y\\ (8y-4y^2-3)\sqrt{(2y-y^2)}+1=z\\ (4z^2-8z+3)\sqrt{(2z-z^2)}+1=x\end{array}\right.$

Details and Assumptions :

$\bullet$ $(\cdots)_{\text{cyclic}}$ means that the relation holds true for all individual $(x,y,z)$

$\bullet$ An $\textbf{unordered}$ triplet means that $(1,2,3)$ is the same as $(3,2,1)$ or $(1,3,2)$ .

It is inspired by one of my favourite problems by Daniel Liu .