Octagon in an Octagon

Geometry Level 3

A regular octagon ABCDEFGHABCDEFGH has squares ACEGACEG and BDFHBDFH inscribed in it. These squares form a smaller octagon as shown.

Let the the area of octagon ABCDEFGHABCDEFGH be ALA_L and the area of the smaller octagon be ASA_S. Then for some integers aa and bb, where bb is square-free, ASAL=ab.\large \dfrac{A_S}{A_L}=a-\sqrt{b}. Find a+ba+b.

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