Kaleidoscope Octagon

Geometry Level 5

A regular octagon $$ABCDEFGH$$, which shall be named $$O_1$$, has squares $$ACEG$$ and $$BDFH$$ inscribed in it. These squares form a smaller octagon, $$O_2$$.

$$O_2$$ has squares inscribed in it in the same manner, and the resulting hexagon formed is named $$O_3$$. This pattern is repeated until infinity.

Let the area of octagon $$O_i$$ be denoted by $$A_i$$. Given that $$A_1=1$$, for some integers $$a$$ and $$b$$, where $$a$$ is square-free, $\large \sum _{ i=1 }^{ \infty }{ { A }_{ i } } =\sqrt { a } +b$ Find $$a+b$$.

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