
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 $R$, set $A=\{x|x^{2}-x-6\leqslant 0\}$, set $B=\{x|\frac{x-4}{x+1}＜0\}$, and set $C=\{x|2-a \lt x \lt 2+a\}$.  
$(1)$ Find the set $(\complement _{R}A)\cap B$;  
$(2)$ Choose one of the following conditions and complete the answer below.  
①$A\cap C=C$; ②$B\cup C=B$; ③$C\subseteq \left(A\cap B\right)$.  
If _____, find the range of real number $a$.

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