
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  $ n\ge2$  be a positive integer and denote by  $ S_n$  the set of all permutations of the set  $ \{1,2,\ldots,n\}$ . For  $ \sigma\in S_n$  define  $ l(\sigma)$  to be  $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$ . Determine  $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$ .

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