
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 $\{a_n\}$ such that $a_0 = x$, $a_1 = y$, and $a_{n+1} = \frac{a_n a_{n-1} + 1}{a_n + a_{n-1}}$ for $n = 1, 2, \ldots$:
1. For what real numbers $x$ and $y$ does there always exist a positive integer $n_0$ such that $a_n$ is constant for all $n \geq n_0$?
2. Find the general term $a_n$.

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