IR-TST 2019 Problem 18

$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$ . Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ Proposed by Navid Safaei