
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

Given the set \( S=\{1,2,\cdots,13\} \) and subsets \( A_{1}, A_{2}, \cdots, A_{k} \) of \( S \) such that:
1. \( |A_{i}|=6 \) for \( i=1,2,\cdots,k \);
2. \( |A_{i} \cap A_{j}| \leqslant 2 \) for \( 1 \leq i < j \leq k \).

Find the maximum value of \( k \).

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