
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
Gina is studying a quadratic expression, \[ x^2 - ax + b, \] with roots $u$ and $v$. She finds that \[ u + v = u^2 + v^2 = u^4 + v^4 = \ldots = u^{18} + v^{18}. \] Curiosity piqued, she wants to calculate the greatest possible value of \[ \dfrac{1}{u^{20}} + \dfrac{1}{v^{20}}. \]

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