
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
Find the smallest \( k \in \mathbf{N}^{*} \) such that for any polynomial \( f(x) \) with integer coefficients (leading coefficient 1) of degree \( n > k \), if there are at least \( \left\lfloor \frac{n}{2} \right\rfloor + 1 \) distinct integers \( x \) such that \( |f(x)| = 1 \), then \( f(x) \) cannot be factored into the product of two non-constant polynomials with integer coefficients.

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