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

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