
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  $a_1 ,a_2 ,\ldots, a_n$  be a permutation of the integers  $1,2,\ldots, n.$  Call  $a_i$  a *big* integer if  $a_i >a_j$  for all  $i<j.$  Find the mean number of big integers over all permutations on the first  $n$  postive integers.

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