
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  $S$  is the set of permutation of {1,2,3,4,5,6,7,8}
(1)For all  $\sigma=\sigma_1\sigma_2...\sigma_8\in S$ Evaluate the sum of S= $\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$ . Then for all elements in  $S$ ,what is the arithmetic mean of S?
(Notice  $S$  and S are different.) 
(2)In  $S$ , how many permutations are there which satisfies "For all  $k=1,2,...,7$ ,the digit after k is **not** (k+1)"?

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