
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
It is given that the roots of the polynomial  $P(z) = z^{2019} - 1$  can be written in the form  $z_k = x_k + iy_k$  for  $1\leq k\leq 2019$ .  Let  $Q$  denote the monic polynomial with roots equal to  $2x_k + iy_k$  for  $1\leq k\leq 2019$ .  Compute  $Q(-2)$ .

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