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

1. $|S_i \cup S_j| \leq 2004$ for each pair $1 \leq i, j \leq n$.
2. $S_i \cup S_j \cup S_k = \{1, 2, \ldots, 2008\}$ for each three integers $1 \leq i < j < k \leq n$.

Let's think step by step and output the final answer within \boxed{}.
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