
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 \( f(x)=\ln (a x+1)+x^{3}-x^{2}-a x \):
( I ) If \( x=\frac{2}{3} \) is a critical point of \( y=f(x) \), find the value of the real number \( a \);
( II ) If \( y=f(x) \) is an increasing function on \([1,+\infty)\), find the range of values of the real number \( a \);
( III ) If \( a=-1 \), and the equation \( f(1-x)-(1-x)^{3}=\frac{b}{x} \) has a real root, find the range of values of the real number \( b \).

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