
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 in his math club, Alex is pondering about the properties of a quadratic equation, \[ x^2 - tx + q, \] where the roots \( a_1 \) and \( a_2 \) satisfy the equalities \[ a_1 + a_2 = a_1^2 + a_2^2 = a_1^3 + a_2^3 = \cdots = a_1^{2010} + a_2^{2010}. \] Intrigued, Alex decides to find the maximum possible value of \[ \frac{1}{a_1^{2011}} + \frac{1}{a_2^{2011}}. \]

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