
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 \( f(x) = x^{2021} + 19x^{2020} + 1 \) with distinct zeroes \( r_1, r_2, \dots, r_{2021} \). A polynomial \( Q \) of degree 2021 has the property that 
\[ Q\left(r_j + \frac{1}{r_j}\right) = 0 \] for \( j = 1, \dots, 2021 \). Determine the value of \( \frac{Q(1)}{Q(-1)} \).

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