
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 each \( k \in \mathbf{N}^{*}, k > 1 \), find the smallest positive integer \( m > 1 \) such that there exists a polynomial \( f(x) \in \mathbf{Z}[x] \) where \( f(x) - 1 \) has at least one integer root, and \( f(x) - m \) has exactly \( k \) different integer roots.

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