JBMO 2023 Problem 2

Prove that for all non-negative real numbers $x, y, z$, not all equal to 0 , the following inequality holds

$$\frac{2 x^{2}-x+y+z}{x+y^{2}+z^{2}}+\frac{2 y^{2}+x-y+z}{x^{2}+y+z^{2}}+\frac{2 z^{2}+x+y-z}{x^{2}+y^{2}+z} \geqslant 3$$

Determine all the triples $(x, y, z)$ for which the equality holds.