
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 $f\left(x\right)=(mx^{2}+1)e^{-x}(m\in R)$.  
$(1)$ Find the monotonic intervals of the function $f\left(x\right)$;  
$(2)$ If the function $g\left(x\right)=f\left(x\right)+nxe^{-x}-1$ has a root in the interval $\left(0,1\right)$, and $m+n=e-1$, find the range of real numbers for $m$.

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