
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]

How many different positive integer values can \( S \) attain?

Present the answer in LaTex format: \boxed{Your answer}
user