
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 \( S = \left(x_{1}, x_{2}, \cdots, x_{n}\right) \) be a permutation of the first \( n \) natural numbers \( 1, 2, \cdots, n \) in any order. \( f(S) \) is defined as the minimum absolute value of the difference between any two adjacent elements in \( S \). Find the maximum value of \( f(S) \).

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