
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  $n$  be a positive integer. For any  $k$ , denote by  $a_k$  the number of permutations of  $\{1,2,\dots,n\}$  with exactly  $k$  disjoint cycles. (For example, if  $n=3$  then  $a_2=3$  since  $(1)(23)$ ,  $(2)(31)$ ,  $(3)(12)$  are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \]*Proposed by Sammy Luo*

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