
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 $f(x)= \frac {\ln (ax)}{x+1}-\ln (ax)+\ln (x+1)$, $(a\neq 0,a\in R)$  
(I) Determine the domain of function $f(x)$.  
(II) Find the intervals of monotonicity for function $f(x)$.  
(III) When $a > 0$, if there exists an $x$ such that $f(x)\geq \ln (2a)$ holds, determine the range of values for $a$.

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