
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 which positive integer $n$ does there exist a permutation $x_{1}, x_{2}, \cdots, x_{n}$ of the set $\{1, 2, \cdots, n\}$ such that the absolute differences $\left|x_{k}-k\right|$ for $k=1, 2, \cdots, n$ are all distinct?

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