
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 \( n \geq 2 \) be an integer and \( X \) be a set with \( n \) elements. For a sequence of non-empty subsets \( A_{1}, A_{2}, \ldots, A_{k} \) of \( X \), if the union \( A_{1} \cup A_{2} \cup \cdots \cup A_{k} \) is a proper subset of \( X \) and no element of \( X \) appears in exactly one of the subsets \( A_{i} \), then the sequence \( A_{1}, A_{2}, \cdots, A_{k} \) is called "tight". Suppose \( A_{1}, A_{2}, \cdots, A_{k} \) is a sequence of non-empty proper subsets of \( X \) such that the entire sequence and any of its subsequences are not "tight". Find the maximum value of \( k \).

Note: For a subset \( A \) of \( X \), if \( A \neq X \) and all sets in the sequence are distinct, \( A \) is called a proper subset of \( X \).

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