Let \(n\) be a positive integer, and let \(z \in \mathbb{C}\) satisfy the equation

\[ (z-1)^{n}+(z+1)^{n}=0 \]

(a) Show that \(z=\frac{1+w}{1-w}\) for some \(w \in \mathbb{C}\) such that \(w^{n}=-1\).

(b) Show that \(w \bar{w}=1\).

(c) Deduce that \(z\) lies on the imaginary axis.