
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 \(0 < m < 10\), let \(S(m, n)\) be the set of all positive integers with \(n\) 1s, \(n\) 2s, \(n\) 3s, \ldots, \(n\) \(m\)s. For a positive integer \(N\), let \(d(N)\) be the sum of the absolute differences between all pairs of adjacent digits. For example, \(d(122313) = 1 + 0 + 1 + 2 + 2 = 6\). Find the mean value of \(d(N)\) for \(N\) in \(S(m, n)\).

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