Strange cyclic quadrilateral

Geometry Level 5

Let \(ABCD\) be a cyclic quadrilateral, where \(AB=a\), \(BC=b\), \(CD=c\), \(DA=d\) and \(a<b<c<d\). Let \(\angle BCA=\angle BDA=\alpha\), \(\angle BAC=\angle BDC=\beta\), \(\angle ABD=\angle ACD=\gamma\) and \(\angle CAD=\angle CBD=\theta\).

Let the equation \(x^4-24x^3+201x^2-698x+844=0\) has roots \(a\), \(b\), \(c\) and \(d\).

If the value of \(\sin \alpha + \sin \theta + \sin \gamma + \sin \beta\) can be written as \(\dfrac{m}{\sqrt{n}}\), where \(m\) and \(n\) are positive integers with \(n\) -square-free, find \(m+n\).

Note: Such quadrilateral indeed exists, the image is up to scale.


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