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)