
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 functions $f(x)=x^{2}\ln x$ and $g(x)=\frac{x}{e^{x}}$, if there exist $x_{1}\in[e,e^{2}]$ and $x_{2}\in[1,2]$ such that $e^{3}(k^{2}-2)g(x_{2})\geq kf(x_{1})$ holds true, determine the range of the positive real number $k$.

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