
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  $A = {1, 2, 3, 4, 5}$ . Find the number of functions  $f$  from the nonempty subsets of  $A$  to  $A$ , such that  $f(B) \in B$  for any  $B \subset A$ , and  $f(B \cup C)$  is either  $f(B)$  or  $f(C)$  for any  $B$ ,  $C \subset A$ 

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