Length of Median?

Geometry Level 5

\(\triangle ABC\) has an area of \(15\sqrt{3}\) with \(\angle BAC =120^{\circ}\) and \(\angle ACB < \angle ABC\). The distance from \(A\) to the centre of the circle inscribed in \(\triangle ABC\) is \(2\).

If the length of the median drawn from \(B\) is \(\sqrt{ab}\), where \(a\) and \(b\) are positive coprime integers and square-free, find \(a + b\).

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