
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 a polynomial $f(x) = x^{2013} + 19x^{2012} + 1$ with distinct roots $r_1, \dots, r_{2013}$. Another polynomial $Q$ of degree $2013$ satisfies:
\[ Q\left(r_j+\dfrac{1}{r_j}\right)=0 \] for each $j=1,\ldots,2013$. Determine the value of $\frac{Q(1)}{Q(-1)}$.

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