
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
Let $a \gt 0$, and the function is given by $f\left(x\right)=\left(x-2\right)^{3}-ax$.  
$(1)$ Find the monotonic intervals of the function $y=f\left(x\right)$;  
$(2)$ For the extreme point $x_{0}$ of the function $y=f\left(x\right)$, if there exists $x_{1}(x_{1}\neq x_{0})$ such that $f(x_{1})=f(x_{0})$, is $x_{1}+2x_{0}$ a constant for any positive number $a$? If so, find this constant value; if not, explain the reason;  
$(3)$ If the maximum value of the function $g\left(x\right)=|f\left(x\right)|$ on the interval $\left[0,6\right]$ is $40$, find the set of values for $a$.

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