Tricky altitudes

Geometry Level 4

\( \Delta ABC \) is an isosceles triangle with \( AC = BC \) and \( AB = 2 \). \( D \) is the midpoint of \( AB \). The ratio of the radii of it's inscribed circle (radius is \(r_1\)) and circumscribed circle (radius is \(r_2\)) is \( 3:8 \).

Two different triangles can be formed under these conditions. In one triangle the square of length \(CD\) is an integer \(a\), in the other the square of length \(CD\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers.

What is the value of \(a + m + n\)?

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