
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 a permutation \(\sigma\) of \(1,2, \ldots, 7\), a transposition is a swapping of two elements. (For instance, we could apply a transposition to the permutation \(3,7,1,4,5,6,2\) and get \(3,7,6,4,5,1,2\) by swapping the 1 and the 6.) Let \(f(\sigma)\) be the minimum number of transpositions necessary to turn \(\sigma\) into the permutation \(1,2,3,4,5,6,7\). Find the sum of \(f(\sigma)\) over all permutations \(\sigma\) of \(1,2, \ldots, 7\).

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