
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 $f\left(x\right)=\frac{e^{2}x^{2}+1}{x},g\left(x\right)=\frac{e^{2}x}{e^{x}}$, for any $x\_1$, $x\_2\in(0,+\infty)$, find the range of positive values for $k$ such that the inequality $\frac{g\left({x}\_{1}\right)}{k}\leqslant \frac{f\left({x}\_{2}\right)}{k+1}$ always holds true.

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