
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 the function $f(x) = axe^{x}$ where $a \in \mathbb{R}$.  
$(1)$ If $|a| \leq e$, and the maximum value of the function $g(x) = f(x) - \frac{1}{2}x^2 - x$ is $\frac{3}{2}$, find the value of $a$.  
$(2)$ If $f(x) + 1 \geq 0$ holds for all $x \in [-2, 2]$, find the range of values for $a$.

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