SEAMO 2027 - Paper F - SET 03 - Self-study Question 8

Let $x_i\in\{\sqrt2-1,\sqrt2+1\}$ for $i=1,2,\ldots,2012$. Define
$$S=x_1x_2+x_3x_4+\cdots+x_{2009}x_{2010}+x_{2011}x_{2012}.$$
How many different positive integer values can $S$ attain?