System of a zero

$\large \begin{cases} \dfrac{x^4}{2x+yz} + \dfrac{y^4}{2y+zx} + \dfrac{z^4}{2z+xy} = 1 \\ & \\ x+y+z = 0 \end{cases}$

Let $x,y$ and $z$ be real numbers satisfying the system of equations above, where $2x+yz \ne 0$ , $2y+zx \ne 0$ and $2z+xy \ne 0$ .

Find the value of $x^4+y^4+z^4$ .