The Largest Sphere Within A Pyramid

Geometry Level 5

ABCDABCD is a convex quadrilateral such that [ABD]=[CDB] [ABD] = [CDB] , AB=1 |AB| = 1 and BC=CD |BC|=|CD| . SS is a point in space such that AS+DS=2 |AS| + |DS| = \sqrt{2} and the volume of the pyramid SABCDSABCD is equal to 16 \frac{1}{6} .

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

Image credit: Wikipedia Avishai Teicher
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