Geometry Level 5

Let $$f$$ be a function from the quadrilaterals to the quadrilaterals such that for a quadrilateral $$ABCD$$ such that $$f (ABCD)=EFGH, E, F, G, H$$ are the circumcenters of triangles $$ABC, BCD, CDA, DAB$$ respectively.

Consider the sequence of quadrilaterals as follows:

$$X_{1}$$ is a convex quadrilateral.

$$X_{n+1}=f (X_{n})$$ for all positive integers $$n$$.

If $$X_{100}$$ is made up of vertices $$A_{i}, i=1, 2, 3, 4$$, find the minimum integer $$R$$ such that

$\sum_{i=1}^4 i \sin A_{i-1}A_{i}A_{i+1}$ is always strictly less than $$R$$ over all $$X_{1}$$ such that $$X_{100}$$ has nonzero area. $$A_{i}=A_{i-4}$$ for all integers $$i$$.

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