Let $\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$.  There are two circles located inside $\angle BAC$ which are tangent to rays $\overline{AB}$, $\overline{AC}$, and segment $\overline{BC}$.  Compute the distance between the centers of these two circles.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

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