Triangle-Ception IV

Geometry Level 5

In the diagram above, \(OAD\) is a quarter circle of radius 2.\(OB\) and \(OC\) trisect arc \(AD\), and \(DE\) bisects segment \(OA\). If the area of quadrilateral \(WXYZ\) can be expressed in the form


for positive integers \(m,n,p,q\) and \(r\) with \(\gcd(m,n)=1\), \(\gcd(p,q)=1\), and \(r\) is not divisible by the square of any prime. Find the value of



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