
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 \( S = \{1, 2, \cdots, 10\} \). The sets \( A_{1}, A_{2}, \cdots, A_{k} \) are all subsets of \( S \) and satisfy the following conditions:
1. \( |A_{i}| = 5 \) for \( i = 1, 2, \cdots, k \);
2. \( |A_{i} \cap A_{j}| \leqslant 2 \) for \( 1 \leqslant i < j \leqslant k \).

Find the maximum value of \( k \).

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