2026 IMC keystage3-IWYMIC team Question 4

Let real $x,y,z$, not all equal, satisfy $x+y+z=0$ and $x^3+y^3+z=y^3+z^3+x=z^3+x^3+y$. Prove $(x^2+y^2+z^2)^2/(x^4+y^4+z^4)=2$.