
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 a positive integer \( n \geq 2 \), determine the maximum value of \( m \) such that \( m \) mutually distinct subsets \( A_{1}, A_{2}, \ldots, A_{m} \) can be selected from an \( n \)-element set, satisfying \( \left(A_{i} \cap A_{k}\right) \subseteq A_{j} \) for all \( 1 \leq i < j < k \leq m \).

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