
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
During a discussion about polynomial roots, Sarah considers a quadratic polynomial \[x^2 - tx + q,\] with roots $\alpha$ and $\beta$. She notes that \[\alpha + \beta = \alpha^2 + \beta^2 = \alpha^3 + \beta^3 = \cdots = \alpha^{2010} + \beta^{2010}.\] She aims to find the maximum possible value of \[\dfrac{1}{\alpha^{2011}} + \dfrac{1}{\beta^{2011}}.\]

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