
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 an infinite sequence \(\left\{a_{n}\right\}\) satisfying \(a_{0}=x, a_{1}=y,\) and \(a_{n+1}=\frac{a_{n} a_{n-1}+1}{a_{n}+a_{n-1}}, n=1,2,\cdots\):

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

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