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

\[\dfrac{m}{n}(p-q\sqrt{r})\]

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

\[m+n+p+q+r.\]

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