
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 $f(x)={x}^{3}+3ax-1$ with its derivative $f'(x)$, and another function $g(x)=f'(x)-ax-3$.

(I) Find the monotonic intervals of $f(x)$ when $a=-2$;

(II) If $g(x) < 0$ holds true for all real numbers $x$ when $-1\leqslant a\leqslant 1$, find the range of $x$;

(III) If $x\cdot g'(x)+\ln x > 0$ is always true for all $x\geqslant 2$, find the range of real number $a$

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