
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
A non-empty family of subsets $\mathcal{A}$ of an $n$-element set $X$ is called a filter family if for every pair $A, B \in \mathcal{A}$, there exists a $C \in \mathcal{A}$ such that $C \subset A \cap B$. Find the number of such families.

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