
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
In the Cartesian coordinate system, points where the horizontal and vertical coordinates are opposite are called "Li numbers," such as $\left(2,-2\right)$ and $\left(-2003,2003\right)$.  
$(1)$ Find the Li number points on the curve $y=-\frac{16}{x}$.  
$(2)$ If $y=ax^{2}+bx+c$ has exactly one "Li number" and shares the same Li point in the second quadrant with $y=-\frac{16}{x}$, find the value of $\frac{c}{a}$.

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