Another Problem using Heron's

Geometry Level 3

Let \(\triangle ABC\) have a circle inscribed inside of it such that the circle is tangent at points \(X\), \(Y\), and \(Z\) along sides \(AB\), \(AC\), and \(BC\), respectively. Let \(AX=3\), \(BZ=4\), and \(CY=5\). Find the area of \(\triangle ABC\).

If your answer can be expressed as \(a\sqrt{b}\) where \(a\) and \(b\) are positive integers with \(b\) squarefree, find \(a+b\).