
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 sets $A=\{x|a-1\leqslant x\leqslant 2a+1\}$ and $B=\{x|-2\leqslant x\leqslant 4\}$. In conditions ① $A\cup B=B$; ② "$x\in A$" is a sufficient but not necessary condition for "$x\in B$"; ③ $A\cap B=\varnothing$, choose any one of these three conditions and complete the blank in question ②, then solve the following problems.<br/>$(1)$ When $a=3$, find $\complement _{R}(A\cap B)$;<br/>$(2)$ If ____, find the range of real numbers $a$.

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