
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  $S = \{1, \cdots, 6\}$  and  $\mathcal{P}$  be the set of all nonempty subsets of  $S$ . Let  $N$  equal the number of functions  $f:\mathcal P \to S$  such that if  $A,B\in \mathcal P$  are disjoint, then  $f(A)\neq f(B)$ .
Determine the number of positive integer divisors of  $N$ .

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