Let \( ABC\) be a triangle with sides \( AB=7\), \( BC=11\) and \( CA=12\), and let \( \omega_a\) be the circle tangent to \(AB\), \(AC\) at the points \(D\), \(E\) and internally tangent to the circumscribed circle \( \odot ABC\). What is the length of the segment \( AD\)?

**Details and Assumptions**

Assuming that the length of the segment \(AD\) is the simple fraction \(\frac{a}{b}\) , type the sum \(a+b\) as your answer.

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