Counting Strategy

China 1999, by Zhusheng Zhang

A $4×4×4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be colored red. A coloring is called interesting if there is exactly $1$ red unit cube in every $1 × 1 × 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colorings. (Two colorings are different even if one can be transformed into another by a series of rotations.)