
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
Let $y=ax^{2}+bx+c$.  
$(1)$ If $a \gt 0$, $b=-2a-2$, $c=3$, find the solution set of the inequality $y\leqslant -1$;  
$(2)$ If $c=2a=2$, and the inequality $y≥\frac{3}{2}bx$ always holds when $1\leqslant x\leqslant 5$, find the range of real number $b$.

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