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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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