
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 $f(x)=\frac{ax^2+1}{e^x}$ (where $e$ is the base of the natural logarithm), the function $g(x)$ satisfies $g'(x)=f'(x)+2f(x)$, where $f'(x)$ and $g'(x)$ are the derivative functions of $f(x)$ and $g(x)$ respectively. If the function $g(x)$ is a monotonic function on $[-1,1]$, determine the range of values for the real number $a$.

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