
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 the function $g(n)$ as:
$$
g(n) =
\begin{cases}
n^2 + 1 & \text{if } n \text{ is odd} \\
\dfrac{n}{2} + 3 & \text{if } n \text{ is even}
\end{cases}.
$$
Determine the smallest positive integer $n$ such that $g(g(\dotsb g(n) \dotsb )) = 7$ for some number of applications of $g$.

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