
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 $y=ax^{2}+bx+c$, where $a$, $b$, $c\in R$.<br/>$(Ⅰ)$ If $a \gt b \gt c$ and $a+b+c=0$, and the distance between the graph of this function and the $x$-axis is $l$, find the range of values for $l$;<br/>$(Ⅱ)$ If $a \lt b$ and the solution set of the inequality $y \lt 0$ is $\varnothing $, find the minimum value of $\frac{{2a+2b+8c}}{{b-a}}$.

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