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:

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

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

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

• $V$, a reflection across the $y$-axis.

Each of these transformations maps the square 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 22 transformations chosen from $L,R,H,V$ will send all of the labeled vertices back to their original positions?