# Ball of chi in a pyramid

Geometry Level 5

$$ABCD$$ is a convex quadrilateral such that $$[ABD] = [CDB]$$, $$|AB| = 1$$ and $$|BC|=|CD|$$. $$S$$ is a point in space such that $$|AS| + |DS| = \sqrt{2}$$ and the volume of the pyramid $$SABCD$$ is equal to $$\frac{1}{6}$$.

The surface area of the largest ball that can fit inside such a pyramid can be expressed as $$\frac{a-\sqrt{b}}{c} \pi ,$$ where $$a,b,c$$ are positive integers, with $$c$$ the smallest possible. What is $$a+b+c?$$

Details and assumptions

The ball need not touch all 5 faces of the pyramid.

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