
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 $f(x)= \frac {x^{3}}{3}-x^{2}-ax+\ln (ax+1)$ $(a\in\mathbb{R})$.
- (I) If $x=2$ is an extremum point of $f(x)$, find the value of $a$;
- (II) If $f(x)$ is monotonically increasing on $[3,+\infty)$, find the range of $a$;
- (III) When $a=-1$, the equation $f(x)= \frac {x^{3}}{3}+ \frac {b}{1-x}$ has a real root, find the maximum value of $b$.

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