
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 study session, Lucy considers a quadratic polynomial \[x^2 - tx + q,\] with roots $a$ and $b$. She notes that \[a+b = a^2+b^2 = \dots = a^4+b^4.\] She is now curious about the minimal possible value of \[\dfrac{1}{a^5} + \dfrac{1}{b^5}.\] Help Lucy by determining this minimum value.

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