AMC 12 2020b Problem 19

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:

$\bullet$ $L,$ a rotation of $90^{\circ}$ counterclockwise around the origin;

$\bullet$ $R,$ a rotation of $90^{\circ}$ clockwise around the origin;

$\bullet$ $H,$ a reflection across the $x$-axis; and

$\bullet$ $V,$ a reflection across the $y$-axis.

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)

\textbf{(A)}\ 2^{37} \textbf{(B)}\ 3\cdot 2^{36} \textbf{(C)}\ 2^{38} \textbf{(D)}\ 3\cdot 2^{37} \textbf{(E)}\ 2^{39}