
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 $a_1, a_2, a_3, \ldots, a_8$ of the integers $1, 2, 3, \ldots, 8$, form the sum
\[|a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8|.\]
Find the average value of all such sums, and express it as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Determine $p+q$.

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