
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
Suppose the polynomial $g(x) = x^{2007} - 2x^{2006} + 2$. The polynomial has distinct zeroes $s_1, s_2, \ldots, s_{2007}$. A polynomial $Q$ of degree $2007$ satisfies
\[Q\left(s_j + \frac{1}{s_j}\right) = 0\] for $j = 1, \ldots, 2007$. Determine the value of $\frac{Q(1)}{Q(-1)}$.

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