
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 $f(x)=e^{x}(ae^{x}+1-x)+a$ where $a\in \mathbb{R}$.<br/>$(1)$ When $a=\frac{1}{2}$, find the monotonic interval of $g(x)={f'}(x)e^{-x}$;<br/>$(2)$ If $f(x)$ has two extreme points $x_{1}$, $x_{2}(x_{1} \lt x_{2})$, find the range of $a$.

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