Question 27
Consider the following equations: $$\begin{aligned}\frac1{x^2}&=\frac1y+\frac1z,\\\frac1{y^2}&=\frac1x+\frac1z,\\\frac1{z^2}&=\frac1{x^2}+\frac1{y^2}.\end{aligned}$$ Find the sum of $(x+1)^4(y+1)^4$ over all possible ordered triples $(x,y,z)$ that satisfy the above three equations simultaneously.