
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  $\mathcal{P}$  be a set of monic polynomials with integer coefficients of the least degree, with root  $k \cdot \cos\left(\frac{4\pi}{7}\right)$ , as  $k$  spans over the positive integers. Let  $P(x) \in \mathcal{P}$  be the polynomial so that  $|P(1)|$  is minimized. Find the remainder when  $P(2017)$  is divided by  $1000$ . 

*Proposed by **eisirrational***

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