
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 $g : \mathbb{R}^+ \to \mathbb{R}^+$ be a function satisfying $g(1) = 1$ and 
\[g\left(x^2 g(x)\right) = x g(x^2) + g(x),\]
for all positive real numbers $x$. Determine the number of possible values of $g(4)$, denoted as $n$, and the sum of all such possible values, denoted as $s$. Calculate $n \times s$.

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