Let $n \geqslant 2$ be a positive even integer. Consider the subspace $R$ of $\mathbb{C}^{2}$ given by the equation $w^{2}=z^{n}-1$, where $(z, w)$ are coordinates in $\mathbb{C}^{2}$, and let $\pi: R \rightarrow \mathbb{C}$ be the restriction of the projection map to the first factor. Show that $R$ has the structure of a Riemann surface in such a way that $\pi$ becomes an analytic map. If $\tau$ denotes projection onto the second factor, show that $\tau$ is also analytic. [You may assume that $R$ is connected.]

Find the ramification points and the branch points of both $\pi$ and $\tau$. Compute the ramification indices at the ramification points.

Assume that, by adding finitely many points, it is possible to compactify $R$ to a Riemann surface $\bar{R}$ such that $\pi$ extends to an analytic map $\bar{\pi}: \bar{R} \rightarrow \mathbb{C}_{\infty}$. Find the genus of $\bar{R}$ (as a function of $n$ ).