
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 $y=ax^{2}+bx+c$, where the set $M=\{x|ax^{2}+bx+c\leqslant 0\}$.  
$(1)$ If $a=c=1$ and $M=\varnothing$, find the range of real number $b$;  
$(2)$ If $M=\{x|-1\leqslant x\leqslant 3\}$, find the solution set of the inequality $-cx^{2}-bx-b \gt cx+4a$ in terms of $x$;  
$(3)$ If $M=\{x|x=t$, $t$ is a constant$\}$ and $a \gt b \gt c$, find the minimum value of $\frac{{a+4c}}{b}$.

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