
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 $g(x) = x^{2009} + 19x^{2008} + 1$ having distinct roots $s_1, s_2, \ldots, s_{2009}$. A polynomial $Q$ of degree $2009$ is defined such that 
\[Q\left(s_j + \frac{2}{s_j}\right) = 0\]
for $j = 1, \ldots, 2009$. Determine the value of $\frac{Q(2)}{Q(-2)}$.

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