Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$ and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$? $\textbf{(A)}\ \frac{1509}{8} \qquad$ $\textbf{(B)}\ \frac{1509}{32} \qquad$ $\textbf{(C)}\ \frac{1509}{64} \qquad$ $\textbf{(D)}\ \frac{1509}{128} \qquad$ $\textbf{(E)}\ \frac{1509}{256}$