
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 the set \( S = \left\{y \mid y = \sum_{k=1}^{1004} x_{2k-1} x_{2k} \right\} \), where \( x_1, x_2, \cdots, x_{2008} \in \{\sqrt{2} - 1, \sqrt{2} + 1\} \). How many different integers are there in the set \( S \)?

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