Lost in space

Geometry Level 3

A point \(P\) in \(\mathbb{R}^{3}\) has distances of \(3,7,8,9\) and \(11\) from five of the vertices of a regular octahedron. If the distance from the sixth vertex of the octahedron to \(P\) is \(\sqrt{a}\), where \(a\) is a positive, square-free integer, then find \(a \).

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