
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
If the function $y=f(x)$ has a maximum or minimum value at $x=x_0$, then $x_0$ is called an extremum point of the function $y=f(x)$. Given the function $f(x)=ax^{3}+3x\ln x-a$ ($a\in\mathbb{R}$).

$(1)$ When $a=0$, find the extremum of $f(x)$;

$(2)$ If $f(x)$ has only one extremum point in the interval $\left( \frac{1}{e},e \right)$, find the range of the real number $a$.

(Note: $e$ is the base of the natural logarithm)

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