A cube of edge length 1, and an octahedron of edge length 1.5 share the same center, and are oriented such that all vertices of the octahedron lie on the normals to the faces of the cube drawn from the cube's 6 face centers, as shown in the figure below.
The region in space that is common to both the cube and the octahedron (their intersection) is shown below.
Find the volume of their intersection.
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