Example 7 Let $S=\{1,2, \cdots, 15\}$, and take $n$ subsets $A_{1}, A_{2}, \cdots, A_{n}$ from $S$, satisfying the following conditions:
( i ) $\left|A_{i}\right|=7, i=1,2, \cdots, n$;
(ii) $\left|A_{i} \cap A_{j}\right| \leqslant 3,1 \leqslant i<j \leqslant n$;
(iii) For any three-element subset $M$ of $S$, there exists some $A_{K}$ such that $M \subset A_{K}$.

Find the minimum value of the number of such subsets $n$.