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:

\(\quad\bullet\qquad\) \(L,\) a rotation of \(90^{\circ}\) counterclockwise around the origin;

\(\quad\bullet\qquad\) \(R,\) a rotation of \(90^{\circ}\) clockwise around the origin;

\(\quad\bullet\qquad\) \(H,\) a reflection across the \(x\)-axis; and

\(\quad\bullet\qquad\) \(V,\) 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. Find how many sequences of \(20\) transformations chosen from \(\{L, R, H, V\}\) will send all of the labeled vertices back to their original positions. The original answer is in the form \(x * 2^y\), please give the value of \(x + y\).

Let's think step by step and output the final answer within \boxed{}.
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