# Triangles with Proportional Sides

**Geometry**Level 5

Let \(S\) be the set of all triangles which have side lengths \( a, b, c\) which satisfy the equation \( \frac {a}{b} = \frac {b}{c}\). For each triangle \(ABC\) in the set \(S\), let \(M_{ABC}\) denote the value of \(\frac {a}{c}\). Let \(R\) be the smallest real number such that \( M_{ABC} \leq R\) for all triangles \(ABC\) in the set \(S\). The value of \( 2R\) can be expressed as \( a +\sqrt{b}\), where \(a\) and \(b\) are positive integers. What is the value of \(a+b\)?

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