
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 that the universal set is $R$, ${∁}_{R}A=\{x|2{x}^{2}-3x-2＞0\}$.  
$(1)$ Find the set $A$;  
$(2)$ Let the solution set of the inequality $x^{2}-2ax+a\leqslant 0$ be $C$. If $C\neq \varnothing$ and "$x\in C$" is a sufficient but not necessary condition for "$x\in A$", find the range of real numbers for $a$.

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