
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  $n>1$  be an integer and let  $f_1$ ,  $f_2$ , ...,  $f_{n!}$  be the  $n!$  permutations of  $1$ ,  $2$ , ...,  $n$ . (Each  $f_i$  is a bijective function from  $\{1,2,...,n\}$  to itself.) For each permutation  $f_i$ , let us define  $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$ . Find  $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$ .

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