System of a zero

Algebra Level 4

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

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 \(x^4+y^4+z^4\).

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