Medial Ellipse?

Geometry Level 5

Given a 3-4-5 triangle \(ABC\), find the area of the inscribed ellipse that is tangent to all three sides at their midpoints. If this area can be expressed as \(\frac{a\pi\sqrt{b}}{c}\), where \(a\), \(b\) and \(c\) are positive, \(\gcd(a,c)=1\) and \(b\) is square-free, find \(a+b+c\).

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