
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 real numbers $\alpha$ and $\beta$ such that $\alpha \beta > 0$, find the set of real numbers $r$ for which there does not exist an infinite sequence $\left\{x_{n}\right\}$ satisfying
$$
x_{0} = r, \quad x_{n+1} = \frac{x_{n}+\alpha}{\beta x_{n}+1}, \quad n = 0, 1, 2, \ldots
$$

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