AMC 10 2014a Problem 16

In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?

$\textbf{(A)}\ \dfrac1{12} \textbf{(B)}\ \dfrac{\sqrt3}{18} \textbf{(C)}\ \dfrac{\sqrt2}{12} \textbf{(D)}\ \dfrac{\sqrt3}{12} \textbf{(E)}\ \dfrac16$