
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 $A=\{x|2 \lt x\leqslant 6\}$ and $B=\{x|x^{2}-4x \lt 0\}$.  
$(1)$ Find $A\cap B$ and $\complement _{R}(A\cup B)$;  
$(2)$ Given set $C=\{x|m+1 \lt x \lt 2m-1\}$, if satisfying ______, find the range of real number $m$.  
Please choose one from $C\subseteq \left(C\cap B\right)$, $C_{R}C\supseteq C_{R}B$, $B\cap C=C$ to fill in the blank in (2) for the solution.

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