
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 the modified function $g(x) = 3x - \frac{1}{3}x^2$, define the sequence $y_n = g(y_{n-1})$ for all $n \ge 1$. For how many real numbers $y_0$ will the sequence $\{y_0, y_1, y_2, \ldots\}$ take on only a finite number of different values?

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