
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 $A=\{x|x \lt -3$ or $x \gt 7\}$, $B=\{x|m+1\leqslant x\leqslant 2m-1\}$.  
$(1)$ If $(\complement _{R}A)\cup B=\complement _{R}A$, find the range of $m$;  
$(2)$ If $(\complement _{R}A)\cap B=\{x|a\leqslant x\leqslant b\}$, and $b-a\geqslant 1$, find the range of $m$.

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