Let $T_1$ be a triangle with side lengths $10$, $11$, and $12$. For $n\ge 1$, if $T_n=△ABC$ and $D$, $E$, and $F$ 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 $AD$, $BE$, and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?