
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 \( f(x) = \frac{x + 8}{x} \). Define the sequence of functions \( (f_n) \) by setting \( f_1 = f \) and
\[ f_n = f \circ f_{n-1} \]
for all \( n \geq 2 \). For instance,
\[ f_2(x) = f(f(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
and
\[ f_3(x) = f(f_2(x)) = \frac{\frac{9x + 8}{x + 8} + 8}{\frac{9x + 8}{x + 8}} = \frac{17x + 72}{9x + 8}. \]
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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