
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  $S$  be a set. We say  $S$  is  $D^\ast$ *-finite* if there exists a function  $f : S \to S$  such that for every nonempty proper subset  $Y \subsetneq S$ , there exists a  $y \in Y$  such that  $f(y) \notin Y$ .  The function  $f$  is called a *witness* of  $S$ .  How many witnesses does  $\{0,1,\cdots,5\}$  have?

*Proposed by Evan Chen*

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