
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 \( g_{1}(x) = \frac{1}{3} \left(1 + x + x^{2} + \cdots \right) \) for all values of \( x \) for which the right-hand side converges. Let \( g_{n}(x) = g_{1}(g_{n-1}(x)) \) for all integers \( n \geq 2 \). What is the largest integer \( r \) such that \( g_{r}(x) \) is defined for some real number \( x \)?

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