
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, and \( f(x)=1 \) has an integer root. The set of all such polynomials is denoted as \( F \). For any given integer \( k>1 \), find the smallest integer \( m(k)>1 \) such that there exists \( f \in F \) such that \( f(x)=m(k) \) has exactly \( k \) distinct integer roots.

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