
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 \( f(x) \) be a polynomial with integer coefficients. For any prime number \( p \) and integers \( u \) and \( v \), if \( p! (uv + u + v) \), then \( p \mid (f(u) f(v) - 1) \). Determine all such polynomials \( f(x) \) that satisfy this condition.

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