
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

Given the infinite sequence $\left\{a_{n}\right\}$ that satisfies $a_{0}=x$, $a_{1}=y$, and $a_{n+1}=\frac{a_{n} a_{n-1}+1}{a_{n}+a_{n-1}}$ for $n=1, 2, \cdots$, 

(1) For which real numbers $x$ and $y$ does there always exist a positive integer $n_{0}$ such that $a_{n}$ is a constant for $n \geq n_{0}$?  
(2) Find the general term $a_{n}$.

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