Factorization is an art (6)

Algebra Level 4

\[\large \begin{cases} (xy+yz+zx)^2 & = 3(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}) \\ x+y+z & = 2016 \\ xyz & = w^{3} \end{cases}\]

Let \(x, y, z\) and \(w\) be positive real numbers satisfying the system of equations above, enter your answer as the value of \(w\) .

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