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

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