AMC 12 2022b Problem 10

Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$?

$\textbf{(A) }\ 4\sqrt3 \textbf{(B) }\ 8 \textbf{(C) }\ 4\sqrt5 \textbf{(D) }\ 4\sqrt7 \textbf{(E) }\ 12$