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Geometry Level 5

\[\large{\sqrt [ 3 ]{ \left( \frac { a }{ b+c+2a } \right) \left( \frac { b }{ c+a+2b } \right) \left( \frac { c }{ a+b+2c } \right) }}\]

Let \(a,b,c\) be the sides of triangle \(ABC\) with circumradius 2 and inradius 1. If the minimum value of the expression above is \(\frac { \alpha }{ \beta } \) for positive coprime integers \(\alpha\) and \(\beta\), find \(\alpha +\beta\).

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