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π° The maximum height is $ \boxed{10} $.
π° Question: A museum curator catalogs a 3D artifact with three vertices of a regular octahedron at $ (1, 0, 0) $, $ (-1, 0, 0) $, and $ (0, 1, 0) $. Find the coordinates of the fourth vertex, given all coordinates are integers.
π° Solution: A regular octahedron has six vertices, with pairs at $ (\pm1, 0, 0) $, $ (0, \pm1, 0) $, and $ (0, 0, \pm1) $. Given three vertices, the missing vertex must be $ (0, 0, 1) $ or $ (0, 0, -1) $. Since the problem specifies integer coordinates, both are valid, but the fourth vertex (completing the octahedron) is $ (0, 0, 1) $.
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