AMC 12 2011b Problem 22

Let $T_1$ be a triangle with side lengths $2011, 2012$, and $2013$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ 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 $\left(T_n\right)$?

\textbf{(A)}\ \frac{1509}{8} \textbf{(B)}\ \frac{1509}{32} \textbf{(C)}\ \frac{1509}{64} \textbf{(D)}\ \frac{1509}{128} \textbf{(E)}\ \frac{1509}{256}