Four circles $$\omega,$$ $$\omega_{A},$$ $$\omega_{B},$$ and $$\omega_{C}$$ with the same radius are drawn in the interior of triangle $$ABC$$ such that $$\omega_{A}$$ is tangent to sides $$AB$$ and $$AC$$, $$\omega_{B}$$ to $$BC$$ and $$BA$$, $$\omega_{C}$$ to $$CA$$ and $$CB$$, and $$\omega$$ is externally tangent to $$\omega_{A},$$ $$\omega_{B},$$ and $$\omega_{C}$$. If the sides of triangle $$ABC$$ are $$13,$$ $$14,$$ and $$15,$$ the radius of $$\omega$$ can be represented in the form $$\frac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers. Find $$m+n.$$

(第二十五届AIME2 2007 第15题)