State the Lefschetz fixed point theorem.

Let $n \geqslant 2$ be an integer, and $x_{0} \in S^{2}$ a choice of base point. Define a space

$$X:=\left(S^{2} \times \mathbb{Z} / n \mathbb{Z}\right) / \sim$$

where $\mathbb{Z} / n \mathbb{Z}$ is discrete and $\sim$ is the smallest equivalence relation such that $\left(x_{0}, i\right) \sim$ $\left(-x_{0}, i+1\right)$ for all $i \in \mathbb{Z} / n \mathbb{Z}$. Let $\phi: X \rightarrow X$ be a homeomorphism without fixed points. Use the Lefschetz fixed point theorem to prove the following facts.

(i) If $\phi^{3}=\mathrm{Id}_{X}$ then $n$ is divisible by 3 .

(ii) If $\phi^{2}=\operatorname{Id}_{X}$ then $n$ is even.