
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 $g(x) = \frac{x + 8}{x}.$  The sequence $(g_n)$ of functions is defined by $g_1 = g$ and
\[g_n = g \circ g_{n - 1}\]for all $n \ge 2.$  For example,
\[g_2(x) = g(g(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8}.\]
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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