
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 a quadratic function $y=ax^{2}+bx+c$.  
$(1)$ If the solution set of $y \gt 0$ is $\{x\left|\right.-3 \lt x \lt 4\}$, find the solution set of the inequality $bx^{2}+2ax-\left(c+3b\right) \lt 0$ in terms of $x$.  
$(2)$ If for any $x\in R$, $b=2$, $a \gt c$, and the inequality $y\geqslant 0$ always holds, and there exists $x_{0}\in R$ such that $a{x_0}^2+2x_{0}+c=0$, find the minimum value of $\frac{{{a^2}+{c^2}}}{{a-c}}$.  
$(3)$ If for any $x\in R$, $a \lt b$, and the inequality $y\geqslant 0$ always holds, find the minimum value of $\frac{{a+2b+4c}}{{b-a}}$.

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