Let \(T_1\) be a triangle with side lengths \(2011, 2012,\) 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. Find the perimeter of the last triangle in the sequence \(( T_n )\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Let's think step by step and output the final answer within \boxed{}.
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