
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]


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On the hyperbola \( x y = 1 \), the point with abscissa \( \frac{n}{n+1} \) is \( A_{n} \), and the point with abscissa \( \frac{n+1}{n} \) is \( B_{n}(n \in \mathbf{N}) \). Let the point with coordinates \( (1,1) \) be \( M \). Also, let \( P_{n}\left(x_{n}, y_{n}\right) \) be the circumcenter of \( \triangle A_{n} B_{n} M \). As \( n \rightarrow \infty \), find the coordinates of the limit point \( P_{n} \), denoted as \( (a, b) \), where \( a = \lim _{n \rightarrow \infty} x_{n} \) and \( b = \lim _{n \rightarrow \infty} y_{n} \).

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