
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
On the hyperbola \( xy = 1 \), let \( A_n \) be the point with the abscissa \( \frac{n}{n+1} \) and \( B_n \) the point with the abscissa \( \frac{n+1}{n} \), where \( n \in \mathbf{N}_{+} \). Furthermore, let \( M \) be the point with coordinates \((1,1)\). Let \( P_n (x_n, y_n) \) be the circumcenter of \( \triangle A_n B_n M \). When \( n \rightarrow +\infty \), find the coordinates \((a, b)\) of the limit point of \( P_n \), 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}
user