
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  $F$  be the set of all polynomials  $\Gamma$  such that all the coefficients of  $\Gamma (x)$  are integers and  $\Gamma (x) = 1$  has integer roots. Given a positive intger  $k$ , find the smallest integer  $m(k) > 1$  such that there exist  $\Gamma \in F$  for which  $\Gamma (x) = m(k)$  has exactly  $k$  distinct integer roots.

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