
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
Let $f(x) = 6x - x^2$. Given $x_0$, consider the sequence defined by $x_n = f(x_{n-1})$ for all $n \ge 1$. Determine the number of initial values $x_0$ 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}
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