
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  $2N = 2004$  real numbers  $x_1, x_2, \ldots , x_{2004}$  equals either  $\sqrt 2 -1 $  or  $\sqrt 2 +1$ . Can the sum  $\sum_{k=1}^N x_{2k-1}x_2k$  take the value  $2004$ ? Which integral values can this sum take?

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