
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 $f(x)=\ln x-ax$ $(a\in\mathbb{R})$.
- $(1)$ Discuss the monotonicity of the function $f(x)$;
- $(2)$ Let the function $f(x)$ have two zeros denoted as $x_{1}$ and $x_{2}$, with $x_{1} < x_{2}$. Given $\lambda > 0$, if the inequality $1+\lambda < \ln x_{1}+\lambda\ln x_{2}$ always holds, find the range of $\lambda$.

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