
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 \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), and let \( A_{1}, A_{2}, \cdots, A_{k} \) be subsets of \( S \), satisfying:
(1) \( |A_{i}| = 5 \) for \( 1 \leq i \leq k \);
(2) \( |A_{i} \cap A_{j}| \leq 2 \) for \( 1 \leq i < j \leq k \).

Find the maximum number \( k \) of such subsets.



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