
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 $A=\{x|x\leqslant -1$ or $x\geqslant 3\}$, $B=\{x|1\leqslant x\leqslant 6\}$, $C=\{x|m+1\leqslant x\leqslant 2m\}$. Find:

$(Ⅰ)$ $A\cap B$, $(\complement _{R}A)\cup B$;

$(Ⅱ)$ If $B\cup C=B$, find the range of real number $m$.

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