Octagon in an Octagon

Geometry Level 4

A regular octagon \(ABCDEFGH\) has squares \(ACEG\) and \(BDFH\) inscribed in it. These squares form a smaller octagon as shown.

Let the the area of octagon \(ABCDEFGH\) be \(A_L\) and the area of the smaller octagon be \(A_S\). Then for some integers \(a\) and \(b\), where \(b\) is square-free, \[\large \dfrac{A_S}{A_L}=a-\sqrt{b}. \] Find \(a+b\).

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