
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
Let 

$$
x_{n}=1+\rho \cos \alpha+\rho^{2} \cos 2 \alpha+\ldots+\rho^{n} \cos n \alpha
$$

where \(0 < \rho < 1\) and \(n=1,2,\ldots\). Find \(\lim_{n \rightarrow \infty} x_{n}\).

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