
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  $k$ -element set.

*(a)* Find the number of mappings  $f : S \to S$  such that
\[\text{(i)  } f(x) \neq x \text{ for } x \in S, \quad  \text{(ii)  } f(f(x)) = x \text{ for }x \in S.\]

*(b)* The same with the condition  $\text{(i)}$  left out.

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