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Geometry Level 5

Let \(R\) be the fraction of the area of a regular octagon of side length \(2\) that lies within a distance of \(1\) of at least one of its vertices.

If \(R\) can be expressed as \(\dfrac{a\pi}{b}(\sqrt{c} - d)\), where \(a,b,c,d\) are positive integers with \(a\) and \(b\) being coprime and \(c\) square-free, then find \(a + b + c + d.\)

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