
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
Given a function $f(x) = ax^3 + x^2$ (where $a \in \mathbb{R}$), it has an extremum at $x = -\frac{4}{3}$.
(1) Determine the value of $a$ and the extremum of $f(x)$.
(2) Let $g(x) = f(x)e^x$, discuss the monotonicity of $g(x)$.

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