
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  $N = 12!$  and denote by  $X$  the set of positive divisors of  $N$  other than  $1$ . A *pseudo-ultrafilter*  $U$  is a nonempty subset of  $X$  such that for any  $a,b \in X$ :
\begin{itemize}
\item If  $a$  divides  $b$  and  $a \in U$  then  $b \in U$ .
\item If  $a,b \in U$  then  $\gcd(a,b) \in U$ .
\item If  $a,b \notin U$  then  $\operatorname{lcm} (a,b) \notin U$ .
\end{itemize}
How many such pseudo-ultrafilters are there?

*Proposed by Evan Chen*

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