
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 $f(x)=\ln x+ax$ where $a\in \mathbb{R}$.  
$(1)$ Discuss the monotonicity of the function $f(x)$.  
$(2)$ If $a\geqslant 0$ and there exist $0 \lt m \lt n$ such that the domains of both $f(x)$ and $f(f(x))$ are $\left[m,n\right]$, find the range of real numbers for $a$.

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