
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 \( n \) be an odd number not less than 3, and \( x_{1}, x_{2}, \cdots, x_{n} \) be real numbers. Find all bijective functions
$$
f:\left\{x_{1}, x_{2}, \cdots, x_{n}\right\} \rightarrow\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}
$$
such that \(\left|f\left(x_{1}\right)-x_{1}\right|=\left|f\left(x_{2}\right)-x_{2}\right|=\cdots=\left|f\left(x_{n}\right)-x_{n}\right|\).

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