
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 $g(x)=f^{′}(x)-ax-3$.
(1) Find the monotonic intervals of function $f(x)$ when $a=-2$;
(2) If $g(x) < 0$ for all values of $a$ satisfying $-1 \leqslant a \leqslant 1$, find the range of real number $x$;
(3) If $x \cdot g^{′}(x)+ \ln x > 0$ holds 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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