Circles in a circle

Geometry Level 4

Given that there are three identical circles with radius \(r\), each touches each other on their respective circumferences and they are all inscribed in a circle with radius \(R\). If the ratio of \(r\) to \(R\) can be expressed as

\[ \large \dfrac1{\frac a{\sqrt b} + c} \]

for positive integers \(a,b\) and \(c\) with \(b\) square-free, find the value of \(a+b+c\).

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