Not symmetric

Algebra Level 4

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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