Let $T_1$ be a triangle with sides $100$, $101$, and $102$. For $n\ge 1$, if $T_n=△ABC$ and $D,E,F$ are the points of tangency of the incircle of $△ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?