
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 $f(x)= \frac{e^{2}x^{2}+1}{x}$ and $g(x)= \frac{e^{2}x}{e^{x}}$, determine the range of the positive number $k$ such that the inequality $\frac{g(x_{1})}{k} \leqslant \frac{f(x_{2})}{k+1}$ holds for any $x_{1}, x_{2} \in (0,+\infty)$.

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