smc

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.)