
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 largest natural number  $ n$  for which there exist different sets  $ S_1,S_2,\ldots,S_n$  such that:

 $ 1^\circ$   $ |S_i\cup S_j|\leq 2004$  for each two  $ 1\leq i,j\le n$  and
 $ 2^\circ$   $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$  for each three integers  $ 1\le i<j<k\le n$ .

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