# 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 $$a + b$$.

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