Let f be an invertible function defined on the complex numbers such that
z^2=f(z+f(i z+f(-z+f(-i z+f(z+\ldots)))))
for all complex numbers z. Suppose z_0 \neq 0 satisfies f\left(z_0\right)=z_0. Find 1 / z_0. (Note: an invertible function is one that has an inverse).