
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 $\left\{a_{n}\right\}$, which satisfies $a_{0}=x, a_{1}=y, a_{n+1}=\frac{a_{n} a_{n-1}+1}{a_{n}+a_{n-1}}, n=1,2, \cdots$:
1. For what real numbers $x$ and $y$ does there always exist a positive integer $n_{0}$ such that for $n \geq n_{0}$, $a_{n}$ remains constant?
2. Find the general term $a_{n}$.

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