
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. Define $f(S)$ to be the minimum of the absolute values of the differences between every two adjacent elements in $S$. Find the maximum value of $f(S)$.

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