
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
Define the sequences \( x_i \) and \( y_i \) as follows. Let \( (x_1, y_1) = (0.8, 0.6) \) and let \( (x_{n+1}, y_{n+1}) = (x_n \cos y_n - y_n \sin y_n, x_n \sin y_n + y_n \cos y_n) \) for \( n \geq 1 \). Find \(\lim_{n \to \infty} x_n \) and \(\lim_{n \to \infty} y_n \).

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