For a triangle with sides 13, 15 and altitude 12, find the radius of the circle that passes through the following points:

- the midpoint of each side,
- the foot of each altitude, and
- the midpoint of the line segment from each vertex to the orthocenter.

If the radius can be written as \(a + \frac{m}{n}\), where \(a,m,\) and \(n\) are positive integers, \(m<n,\) and \(m\) and \(n\) are coprime, find \(a+m+n\).

**Note**: Assume the given altitude to be through the vertex common to both the given sides.

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