Inscribed Sphere, Octahedron and vectors

Sphere $O$ is the inscribed sphere of a regular octahedron whose side length is $2$ . Point $M,N$ are on the sphere so that $MN$ is one of the diameters of the sphere and point $P$ is an arbitrary point on the surface of the octahedron. Find the range of $\overrightarrow{PM} \cdot \overrightarrow{PN}$ .

The range can be expressed as $[M,N]$ . Submit $\lfloor 10000(M+N) \rfloor$ .