
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 \( A = \{1,2,3,4,5\} \). Determine the number of functions \( f: \mathscr{P}(A) \backslash \{\varnothing\} \rightarrow A \) that satisfy \( f(B) \in B \) and \( f(B \cup C) \in \{f(B), f(C)\} \) for all non-empty subsets \( B \) and \( C \) of \( A \).

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