I lie on U

A cube \(U\) in \(\mathbb{R^3}\) has its vertices at \((0,0,0)\), \((8,0,0)\), \((0,8,0)\), \((0,0,8)\), \((8,8,0)\), \((8,0,8)\), \((0,8,8)\), and \((8,8,8).\) How many lattice points lie on the surface of \(U\)?

Details and assumptions

A lattice point has integer coordinates.

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