
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 \(a_{1}, a_{2}, \cdots, a_{10}\) of the integers \(1, 2, 3, \cdots, 10\), find the arithmetic mean of the sums \(\left| a_{1} - a_{2} \right| + \left| a_{3} - a_{4} \right| + \left| a_{5} - a_{6} \right| + \left| a_{7} - a_{8} \right| + \left| a_{9} - a_{10} \right|\).

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