
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 $A=\{x|\frac{x+3}{x-1}\leqslant 0\}$ and $B=\{x|x^{2}-mx-2m^{2}\leqslant 0, m \gt 0\}$.  
$(1)$ When $m=2$, find $A\cap B$ and $\complement _{R}B$;  
$(2)$ If $x\in A$ is a sufficient and necessary condition for $x\in B$, does such a real number $m$ exist? If yes, find the range of $m$; if not, explain the reason.

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