Find area of triangle

Geometry Level 5

Triangle \(ABC\) has an inradius of \(5\) and a circumradius of \(16\).

If \(2\cos B = \cos A + \cos C\), then the area of triangle \(ABC\) can be expressed as \(\dfrac{x\sqrt{y}}{z}\), where \(x, y\) and \(z\) are positive integers such that \(x\) and \(z\) are relatively prime and \(y\) is not divisible by the square of any prime.

Find \(x+y+z\).

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