
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
Find the smallest integer  $n$  such that each subset of  $\{1,2,\ldots, 2004\}$  with  $n$  elements has two distinct elements  $a$  and  $b$  for which  $a^2-b^2$  is a multiple of  $2004$ .

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