
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
Let the functions $f(x)= \frac{{x}^{2}+1}{x}$ and $g(x)= \frac{x}{{e}^{x}}$. For any $x_{1}$, $x_{2} \in (0,+\infty)$, the inequality $\frac{g(x)}{k} \leqslant \frac{f({x}_{2})}{k+1}$ always holds. Then, the range of values for the positive number $k$ is ______.

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