
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 $f(x)=\frac{x^{3}}{3}-x^{2}-ax+\ln (ax+1)$, where $a\in R$.

(1) If $x=2$ is an extreme point of $f(x)$, find the value of $a$.

(2) If $f(x)$ is monotonically increasing in the interval $[3,+\infty)$, find the range of $a$.

(3) When $a=-1$, the equation $f(x)=\frac{x^{3}}{3}+\frac{b}{1-x}$ has real roots, find the maximum value of $b$.

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