
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 $g(n) = 
\begin{cases}
n^2 - 3 & \text{if } n \text{ is odd} \\
\dfrac{n}{2} & \text{if } n \text{ is even}
\end{cases}$.

For how many integers $n$ from 1 to 100, inclusive, does $g(g(...g(n)...)) = 1$ for some number of applications of $g$?

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