Complicated Tangency of a system of Circles!

Geometry Level 5

Nine circles of radius $$\dfrac12$$ are externally tangent to a circle of radius $$1$$ and also are tangent to one another as shown. If the distance between the centers of the first and the last of the circles can be expressed as:

$\large{\dfrac{A\sqrt{B}}{C}}$

where $$A,B,C$$ are positive integers satisfying $$\gcd(A,C)=1$$ and $$B$$ is squarefree, find the value of $$A+B+C$$.

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