
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
Sarah is examining a problem related to her polynomial quiz which features a quadratic polynomial \[x^2 - tx + q.\] She notices an odd pattern:
\[r_1 + r_2 = r_1^2 + r_2^2 = r_1^3 + r_2^3 = \cdots = r_1^{13} + r_2^{13}.\]
Curious about consistency in this sequence, she calculates the maximum possible value of \[\frac{1}{r_1^{14}} + \frac{1}{r_2^{14}}.\]

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