VN-TST 2008 Problem 6

Consider the set $M = \{1,2, \ldots ,2008\}$ . Paint every number in the set $M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:

$S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$ ; $S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$ .

Prove that $2|S_1| > |S_2|$ (where $|X|$ denotes the number of elements in a set $X$ ).