Square A B C D 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:
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L, a rotation of 90^{\circ} counterclockwise around the origin;
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R, a rotation of 90^{\circ} clockwise around the origin;
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H, a reflection across the x-axis; and
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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 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.)
Answer Choices
A. 2^{37}
B. 3 \cdot 2^{36}
C. 2^{38}
D. 3 \cdot 2^{37}
E. 2^{39}