
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 $g(x) = x^2 - 6x + 8$, consider the sequence defined by $x_{n} = g(x_{n-1})$ for all $n \ge 1$. Determine the number of real numbers $x_0$ for which the sequence $x_{0}, x_{1}, x_{2}, \ldots$ takes on only a finite number of different values.

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