
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
Let integer $n \geq 2$. For any permutation $\sigma$ of $\{1, 2, \cdots, n\}$, where $\sigma = (\sigma(1), \sigma(2), \cdots, \sigma(n))$, define $S=\{\sigma \mid \sigma(i) \neq i, \; i=1, 2, \cdots, n\}$. Find the minimum value of $\sum_{i=1}^{n} |\sigma(i) - i|$ over all $\sigma \in S$, and determine the number of permutations $\sigma$ that achieve this minimum value.

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