A regular octagon in the coordinate plane has vertices $A(1,0)$, $B(\cos (45^{\circ }),\sin (45^{\circ }))$, and so on, labeled counterclockwise. Consider the following four transformations:

• $L$, a rotation of $45^{\circ }$ counterclockwise around the origin;

• $R$, a rotation of $45^{\circ }$ 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 octagon onto itself, but the positions of the labeled vertices will change. How many sequences of 20 transformations chosen from $L,R,H,V$ will send all of the labeled vertices back to their original positions?