
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 $g(x) = \frac{x + 8}{x - 1}$. Define a sequence $(g_n)$ of functions where $g_1 = g$ and
\[ g_n = g \circ g_{n - 1} \]
for all $n \geq 2$. Find the number of distinct real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$.

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