
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 $f(x)=(x-2)e^{x}$, where $x\in(0,+\infty)$.  
$(1)$ Find the interval of monotonic increase for the function $f(x)$.  
$(2)$ If $g(x)=f(x)+2e^{x}-ax^{2}$, $h(x)=x$, and for all $x_{1}$, $x_{2}$, $[g(x_{1})-h(x_{1})][g(x_{2})-h(x_{2})] > 0$, find the range of the real number $a$.

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