
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
The function $f(x)=e^{2x}+2me^{x}+2x$, where $m\in \mathbb{R}$.  
$(1)$ If $m=0$, find the area of the triangle formed by the tangent line of the function $f(x)$ at $x=0$ and the coordinate axes.  
$(2)$ If the function $f(x)$ has two critical points $x_{1}$ and $x_{2}$, find the range of $\frac{f(x_{1})+f(x_{2})}{e^{x_{1}}+e^{x_{2}}}$.

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