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:


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

Here is a junior version of this problem - Twisting the Problem - "Complicated Tangency of a system of circles

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