
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 $f\left(x\right)=a\ln \left(x+1\right)+\ln x\left(a\in R\right)$.  
$(1)$ If $a=2$, find the monotonic interval of the function $g\left(x\right)=f\left(x\right)-ax$;  
$(2)$ If the function $f\left(x\right)$ is monotonically increasing on the interval $\left[1,4\right]$, find the range of real number $a$.

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