HMMT 2020 U16

Determine all triplets of real numbers $(x, y, z)$ satisfying the system of equations

$$\begin{aligned} x^{2} y+y^{2} z & =1040 \\ x^{2} z+z^{2} y & =260 \\ (x-y)(y-z)(z-x) & =-540 \end{aligned}$$