# Comfy cozy

**Geometry**Level 5

Let \(R\) be the region in the first quadrant that is

(i) inside the circle \(x^{2} + (y - 3)^{2} = 9\), and

(ii) outside the circles \(x^{2} + (y - 2)^{2} = 4\) and \(x^{2} + (y - 5)^{2} = 1\).

(The boundary lines of \(R\) are included as part of the region.)

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