
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 a permutation  $\sigma = (a_1,a_2,a_3,...a_n)$  of  $(1,2,3,...n)$  , an ordered pair  $(a_j,a_k)$  is called an inversion of  $\sigma$  if  $a \leq j < k \leq n$  and  $a_j > a_k$ . Let  $m(\sigma)$  denote the no. of inversions of the permutation  $\sigma$ . Find the average of  $m(\sigma)$  as  $\sigma$  varies over all permutations.

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