
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 $M$ be a set with $n$ elements. If $M$ has $k$ distinct subsets $A_{1}, A_{2}, \cdots, A_{k}$ such that for every $i, j \in \{1, 2, \cdots, k\}$, $A_{i} \cap A_{j} \neq \varnothing$, determine the maximum value of the positive integer $k$.

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