
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
Each of the numbers \(x_{1}, x_{2}, \ldots, x_{2004}\) can be equal to \(\sqrt{2}-1\) or \(\sqrt{2}+1\). How many distinct integer values can the sum

\[ 
\sum_{k=1}^{1002} x_{2k-1} x_{2k} = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2003} x_{2004} 
\]

take?

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