
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 \( g(x) = \frac{x + 5}{x} \). Define the sequence \( (g_n) \) of functions where \( g_1 = g \) and
\[ g_n = g \circ g_{n - 1} \]
for all \( n \geq 2 \). For example,
\[ g_2(x) = g(g(x)) = \frac{\frac{x + 5}{x} + 5}{\frac{x + 5}{x}} = \frac{6x + 5}{x + 5} \]
and
\[ g_3(x) = g(g_2(x)) = \frac{\frac{6x + 5}{x + 5} + 5}{\frac{6x + 5}{x + 5}} = \frac{11x + 30}{6x + 5} \].
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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