
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
Given the polynomial $f(x) = x^{2009} + 19x^{2008} + 1$ with distinct zeroes $r_1, \ldots, r_{2009}$, a polynomial $Q$ of degree $2009$ satisfies 
\[Q\left(r_j + \frac{1}{r_j}\right) = 0\] for $j=1, \ldots, 2009$. Determine the value of $\frac{Q(1)}{Q(-1)}$.

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