Let $S = \{ 1, 2, \ldots, 15 \}$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:

1. $|A_i| = 7$, for $i = 1, 2, \ldots, n$;

2. $|A_i \cap A_j| \leq 3$, for $1 \leq i < j \leq n$;

3. For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$.

Find the smallest possible value of $n$.

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