SMO 2011 Open Round 1 Question 8

Let $x_i\in\{\sqrt2-1,\sqrt2+1\}$ for $i=1,2,\ldots,2012$. Define $S=x_1x_2+x_3x_4+\cdots+x_{2011}x_{2012}$. How many distinct positive integer values can $S$ attain?