
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 function $f(x)=\ln x+\frac{2}{x}$,  
$(1)$ Find the minimum value of the function $\sigma(x)=f(x)+\frac{{e}^{x}}{x}-x-\frac{2}{x}$.  
$(2)$ Let $g(x)=-xf(x)+x^{2}+4$. If there exists an interval $[a,b]⊂[\frac{1}{2},+\infty)$ such that the range of $g(x)$ on $\left[a,b\right]$ is $\left[k(a+2),k(b+2)\right]$, find the range of values for $k$.

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