# 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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