
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
Determine the largest positive integer  $n$  for which there exist pairwise different sets  $\mathbb{S}_1 , ..., \mathbb{S}_n$  with the following properties:   $1$ )  $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$  for any two indices  $1 \leq i, j\leq n$ , and  $2$ )  $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$  for any  $1 \leq i < j < k \leq n$ *Proposed by Ivan Matic*

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