
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  $n$  be an integer greater than  $1$  and let  $X$  be an  $n$ -element set. A non-empty collection of subsets  $A_1, ..., A_k$  of  $X$  is tight if the union  $A_1 \cup \cdots  \cup A_k$  is a proper subset of  $X$  and no element of  $X$  lies in exactly one of the  $A_i$ s. Find the largest cardinality of a collection of proper non-empty subsets of  $X$ , no non-empty subcollection of which is tight.

*Note*. A subset  $A$  of  $X$  is proper if  $A\neq X$ . The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

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