Equality or Inequality, That is the question!

Algebra Level 5

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.
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