Quadrilaterals in quadrilaterals

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 $i$ .