
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 
$$
g(n) =
\begin{cases}
n^2 + 1 & \text{if } n \text{ is odd}, \\
\frac{n}{3} & \text{if } n \text{ is even and divisible by } 3, \\
\frac{n}{2} & \text{if } n \text{ is even and not divisible by } 3 
\end{cases}.
$$
For how many integers \( n \) from 1 to 100, inclusive, does \( g ( g (\dotsb g (n) \dotsb )) = 1 \) for some number of applications of \( g \)?

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