
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
Given an ordered  $n$ -tuple  $A=(a_1,a_2,\cdots ,a_n)$  of real numbers, where  $n\ge 2$ , we define  $b_k=\max{a_1,\ldots a_k}$  for each k. We define  $B=(b_1,b_2,\cdots ,b_n)$  to be the “*innovated tuple*” of  $A$ . The number of distinct elements in  $B$  is called the “*innovated degree*” of  $A$ .
Consider all permutations of  $1,2,\ldots ,n$  as an ordered  $n$ -tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to  $2$ 

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