
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
Given  $n$ , let  $k = k(n)$  be the minimal degree of any monic integral polynomial  $$ f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0 $$  such that the value of  $f(x)$  is exactly divisible by  $n$  for every integer  $x.$  Find the relationship between  $n$  and  $k(n)$ . In particular, find the value of  $k(n)$  corresponding to  $n = 10^6.$ 

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