CZ-SK-MO 2018 Problem 2

Let $x,y,z$ be real numbers such that the numbers $$\frac{1}{|x^2+2yz|}, \frac{1}{|y^2+2zx|}, \frac{1}{|z^2+2xy|}$$ are lengths of sides of a (non-degenerate) triangle. Determine all possible values of $xy+yz+zx$ .