
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 $f(x)=(ax^{2}+x+a)e^{x}$, where $a\neq 0$.  
$(1)$ When $a=-1$, find the extreme values of the function $f(x)$.  
$(2)$ When $a \gt 0$, find the maximum value of $f(x)$ on $x\in [-2,-1]$.

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