Let
$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix}$
and
$B = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}.$

Let $S$ be the set of all matrices that can be generated from taking products of $A$ and $B$, in any order. For example, the matrix $A^2B A^4B$ is in the set $S$. Find the number of distinct elements in $S$.

Question

Let
$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix}$
and
$B = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}.$

Let $S$ be the set of all matrices that can be generated from taking products of $A$ and $B$, in any order. For example, the matrix $A^2B A^4B$  is in the set $S$. Find the number of distinct elements in $S$.