
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
Consider the polynomial $f(x) = x^{2007} + 19x^{2006} + 1$, with distinct zeroes $r_1, \ldots, r_{2007}$. A polynomial $P$ of degree $2007$ satisfies
\[ P\left(2r_j + \dfrac{1}{r_j}\right) = 0 \] for $j = 1, \ldots, 2007$. Determine the value of $\frac{P(1)}{P(-1)}$.

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