
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 $f(x) = \frac{2x + 6}{x}.$ The sequence $(f_n)$ of functions is defined by $f_1 = f$ and
\[ f_n = f \circ f_{n - 1} \]
for all $n \ge 2$. For example,
\[ f_2(x) = f(f(x)) = \frac{\frac{2x + 6}{x} + 6}{\frac{2x + 6}{x}} = \frac{8x + 6}{2x + 6},\]
\[ f_3(x) = f(f_2(x)) = \frac{\frac{8x + 6}{2x + 6} + 6}{\frac{8x + 6}{2x + 6}} = \frac{14x + 42}{8x + 6}.\]
Let $S$ be the set of all real numbers $x$ such that
\[ f_n(x) = x \]
for some positive integer $n$. Find the number of elements in $S$.

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