Not symmetric

Consider the following equations: $\left\{\begin{array}{l}\dfrac{1}{x^2}=\dfrac{1}{y}+\dfrac{1}{z}\\\dfrac{1}{y^2}=\dfrac{1}{x}+\dfrac{1}{z}\\\dfrac{1}{z^2}=\dfrac{1}{x^2}+\dfrac{1}{y^2}\end{array}\right.$ Find the sum of $(x+1)^4(y+1)^4$ over all possible ordered triples $(x,y,z)$ that satisfy the above three equation simultaneously.

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