
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  $n$ ,  $(n \geq 3)$  be a positive integer and the set  $A$ ={ $1,2,...,n$ }. All the elements of  $A$  are randomly arranged in a sequence  $(a_1,a_2,...,a_n)$ . The pair  $(a_i,a_j)$  forms an  $inversion$  if  $1 \leq i \leq j \leq n$  and  $a_i > a_j$ . In how many different ways all the elements of the set  $A$  can be arranged in a sequence that contains exactly  $3$  inversions?

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