
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_{10}$  of the integers  $ 1,2,3,\ldots,10,$  form the sum
\[ |a_1 \minus{} a_2| \plus{} |a_3 \minus{} a_4| \plus{} |a_5 \minus{} a_6| \plus{} |a_7 \minus{} a_8| \plus{} |a_9 \minus{} a_{10}|.\]
The average value of all such sums can be written in the form  $ p/q,$  where  $ p$  and  $ q$  are relatively prime positive integers.  Find  $ p \plus{} q.$ 

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