
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
For each permutation \(a_1, a_2, a_3, \ldots, a_{12}\) of the integers \(1, 2, 3, \ldots, 12\), form the sum
\[
|a_1-a_2| + |a_3-a_4| + |a_5-a_6| + |a_7-a_8| + |a_9-a_{10}| + |a_{11}-a_{12}|.
\]
The average value of all such sums can be written in the form \(\dfrac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

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