Shocked Face

Geometry Level 5

Let \(R\) be the region in the first quadrant outside of the circles \(x^{2} + y^{2} = 4\) and \(x^{2} + (y - 3)^{2} = 1\) but inside the circle \(x^{2} + (y - 1)^{2} = 9.\)

The largest circle that can be inscribed in \(R\) has radius \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

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