
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
$(1)$ Discuss the monotonicity of $f\left(x\right)=(1+x+ax^{2})e^{-x}(a\geqslant 0)$;<br/>$(2)$ Let $g\left(x\right)=e^{x}-mx-\cos x\left(m\geqslant 0\right)$, investigate whether there exists a value of $m$ such that $g\left(x\right)$ has a local minimum at $x=x_{0}$ and $g\left(x\right)\geqslant 0$ always holds. If it exists, find the value of $m$; if not, explain the reason.

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