
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 $f(x)=x(x^{2}-3x+a)$, where $a\in \mathbb{R}$.  
$(1)$ When $a=-9$, find the interval where the function $f(x)$ is monotonically increasing.  
$(2)$ If the function $f(x)$ is decreasing on the interval $(1,2)$, find the range of values for $a$.  
$(3)$ If there exist two extreme points $x_{1}$ and $x_{2}$ in the interval $(0,2)$, and $|f(x_{1})-f(x_{2})| \gt |f(x_{1})+f(x_{2})|$, find the range of values for $a$.

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