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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