A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter of triangle $\bigtriangleup AEC' ?$ $\textbf{(A)} ~2 \qquad$ $\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad$ $\textbf{(C)} ~\frac{13}{6} \qquad$ $\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad$ $\textbf{(E)} ~\frac{7}{3}$