
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 \(g(x) = \frac{2x + 9}{x}\). Define a sequence \((g_n)\) of functions where \(g_1 = g\) and
\[g_n = g \circ g_{n - 1}\] for all \(n \geq 2\). Let \(T\) be the set of all real numbers \(x\) such that 
\[g_n(x) = x\] for some positive integer \(n\). Find the number of elements in \(T\).

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