Stop going in circles

Geometry Level 5

Suppose a regular octagon is inscribed in a unit circle. If $$3$$ of the vertices of this octagon are chosen uniformly at random, then the expected area of the triangle formed by these $$3$$ vertices is

$$\dfrac{a*(b + \sqrt{c})}{d}$$,

where $$a,b,c,d$$ are all positive integers with $$a,d$$ being coprime and $$c$$ square-free.

Find $$a*b*c*d$$.

Note: $$a,b,c$$ may not necessarily be distinct.

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