How will you do this elegantly?

Geometry Level 5

Consider a triangle whose semi-perimeter is \(\dfrac{23}{24}\) , sum of products of two sides is \(\dfrac{29}{24}\) , product of all sides is \(\dfrac{1}{4}\).If the sum of cosines of the angles of this triangle can be expressed as \(\dfrac{a}{b}\) where \(a,b\) are coprime integers , find \(a+b\).

Advice: Do this without actually calculating the sides of the triangle.

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