
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
In the method of finding the zero points of a function, there is a method called Newton's method. The process of this method is as follows: For the function \( f(x) \), given \( x_{1} \), let \( Q_{n}\left(x_{n}, f\left(x_{n}\right)\right) \) be the point where the tangent line of \( f(x) \) intersects with the x-axis. The x-coordinate at the intersection point is denoted as \( x_{n+1} (n=1,2,3, \ldots) \). When \( n \) is sufficiently large, we can use \( x_{n} \) as an approximate value of the zero point. Suppose \( f(x) = x^{2} - 2 \) and \( x_{1} = 2 \), find the recurrence relation of the sequence \( \left\{ x_{n} \right\} \) and determine its limit.

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