
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 $h(x)=\ln x+\frac{\lambda}{x}$, where $\lambda$ is a real number.  
$(1)$ If $y=h(x)$ is a monotonic function on its domain, find the range of the real number $\lambda$.  
$(2)$ If the function $y=h(x)$ has two distinct zeros, find the range of the real number $\lambda$.  
$(3)$ Let $g(x)=h(x)-\lambda x$. If $p$ and $q$ $(p<q)$ are two stationary points of $g(x)$, when $\lambda$ varies in the interval $[\frac{4}{17},\frac{2}{5}]$, find the range of $|g(p)-g(q)|$.

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