
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 $A=\{x|2 \lt x\leqslant 6\}$, $B=\{x|x^{2}-4x \lt 0\}$, $C=\{x|m+1 \lt x \lt 2m-1\}$. Find:<br/>
$(1)$ $A\cap B$, $\complement _{R}(A\cup B)$.<br/>
$(2)$ If ____, find the range of real number $m$.<br/>
Please choose one from the three conditions $C\subseteq \left(C\cap B\right)$, $C\subseteq B$, $B\cap C=C$ to fill in the blank in (2) and complete the solution for (2).

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