
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
Define a function \( g(n) \) as follows:
\[
g(n) = 
\begin{cases}
3n^2 + 3 & \text{if } n \text{ is odd} \\
\frac{n}{3} & \text{if } n \text{ is divisible by } 3 \\
n-1 & \text{otherwise}
\end{cases}
\]
For how many integers \( n \) from 1 to 100, inclusive, does \( g(g(\ldots g(n) \ldots)) = 1 \) for some number of applications of \( g \)?

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