
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

Construct a sequence of polynomials that results from applying the Lobachevsky method (see problem 61333) for approximating the roots of the polynomial \( x^{2}-x-1 \). Which sequences will converge to the roots \( x_{1} \) and \( x_{2} \), if \( \left|x_{1}\right|>\left|x_{2}\right| \)?

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