
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
Let the polynomial $f(x) = x^{2005} + 13x^{2004} + 2$ have distinct zeroes $s_1, s_2, ..., s_{2005}$. A polynomial $Q$ of degree $2005$ is defined such that for each root of $f(x)$,
\[ Q\left(s_j+\dfrac{2}{s_j}\right) = 0 \]
for $j = 1, ..., 2005$. Determine the value of $\frac{Q(1)}{Q(-1)}$.

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