
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 $f(x) = e^x\left(\frac{1}{3}x^3 - 2x^2 + (a+4)x - 2a - 4\right)$, where $a \in \mathbb{R}$, and $e$ is the base of the natural logarithm.
1. If the tangent line to the graph of $f(x)$ at $x=0$ is perpendicular to the line $x+y=0$, find the value of $a$.
2. If the inequality $f(x) < -\frac{4}{3}e^x$ holds for all $x$ in $(-\infty, 2)$, find the range of possible values for $a$.
3. Discuss the number of extreme points of the function $f(x)$.

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