GAIC Math 2024 Problem 74

The system of equations $\left\{\begin{array}{l}x^{2} y+y^{2} z+z^{2}=0, \\ z^{3}+z^{2} y+z y^{3}+x^{2} y=\frac{1}{4}\left(x^{4}+y^{4}\right)\end{array}\right.$ has how many sets of real solution $(x,y,z)$ ?.