
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 $g(x) = x^{2005} + 19x^{2004} + 1$ with distinct zeros $s_1,\ldots,s_{2005}$. A polynomial $Q$ of degree $2005$ satisfies
\[ Q\left(s_j+\dfrac{1}{s_j}\right)=0 \] for $j=1,\ldots,2005$. Determine the value of $\frac{Q(2)}{Q(-2)}$.

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