
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
Given a positive integer \( n \), find all real-coefficient polynomials \( f(x) \) such that \(\operatorname{deg} f = 2n + 1\) and \((x - 1)^{n} \mid (f(x) + 1)\), \((x + 1)^{n} \mid (f(x) - 1)\).

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