
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 positive integer  $j$  such that for every polynomial  $p(x)$  with integer coefficients and for every integer  $k,$  the integer 
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the  $j$ -th derivative of  $p(x)$  at  $k$ ) is divisible by  $2016.$ 

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