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. How many sequences of $22$ transformations chosen from $L,R,H,V$ will send all of the labeled vertices back to their original positions?