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Geometry Level 3

A lattice cube is a cube in 3-dimensional space, whose vertices are lattice points. Describe the set of possible side lengths of a lattice cube.

Note: A lattice point is a point whose coordinates are all integers. \( (0,0,1) \) is a lattice point, but \( (\frac{1}{2}, \sqrt{2}, -0.2 ) \) is not a lattice point.

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