Монгол Бодлогын Сан

Find all \( n \in \mathbb{N} \) such that the set \( S = \{1, 2, \ldots, n\} \) can be colored with two colors in a way that \( S^3 = S \times S \times S \) contains exactly $2007$ triplets \( (x, y, z) \) satisfying the following conditions:

1. \( x, y, z \) are all the same color.
2. \( n \mid x + y + z \).