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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