How far to a pyramid

Geometry Level 5

A point \(P\) in \(\mathbb{R}^{3}\) has distances of \(3,7,9,11\) (not necessarily in that order) from the base vertices of a square-based pyramid with all edges equal to \(s\). The distance of point \(P\) to the apex of the pyramid is \(e\). The maximum value of \(e\) can be expressed as \(\sqrt{a+ b \sqrt{c}}\) and its minimum as \(\sqrt{a- b \sqrt{c}},\) where \(a,b,c\) are positive integers and \(c\) is square-free. Find the value of \(a+b+c\).

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