
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
We apply Newton's method (see problem 61328) for approximate finding of the roots of the polynomial \(f(x)=x^{2} - x - 1\). What sequences of numbers will be obtained if
a) \(x_{0}=1\);
b) \(x_{0}=0\)?

To which numbers will these sequences converge?

Describe the continued fraction expansions of the numbers \(x_n\).

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