
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 \( x_{1} = a \), \( x_{2} = b \), and \( x_{n+1} = \frac{1}{2}\left(x_{n} + x_{n-1}\right) \) for \( n \geq 1 \). Calculate \(\lim_{n \rightarrow \infty} x_{n}\).

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