
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 \( f(x) = x^{2023} + 19x^{2022} +1 \) have distinct zeroes \( r_1, r_2, \ldots, r_{2023} \). A polynomial \( Q \) of degree 2023 satisfies the property that
\[ Q\left( r_j^2 + \frac{1}{r_j^2} \right) = 0 \]
for \( j = 1, 2, \ldots, 2023 \). Determine the value of \(\frac{Q(2)}{Q(-2)}\).

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