
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 $f(x) = x^{2} + x|x-2a|$, where $a$ is a real number.  
$(1)$ When $a=-1$, find the minimum value of the function $f(x)$.  
$(2)$ For $a=-9$, if there exist two distinct real numbers $x_{1}$ and $x_{2}$ ($x_{1} < x_{2}$) such that $f(x_{1}) = f(x_{2})$, find the range of values for $\frac{x_{1}}{x_{2}} + x_{1}$.

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