
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
Let \( S = \{1,2, \cdots, 15\} \). From \( S \), extract \( n \) subsets \( A_{1}, A_{2}, \cdots, A_{n} \), 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 3-element subset \( M \) of \( S \), there exists some \( A_{K} \) such that \( M \subset A_{K} \).

Find the minimum value of \( n \).

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