
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 $A=\{x|y=\sqrt{(2+x)(4-x)}\}$ and $B=\{x| -1 \lt x \lt m+1\}$.  
$(1)$ If $m=4$, find $A\cup B$ and $(\complement _{R}A)\cap B$;  
$(2)$ If $B\subseteq A$, find the range of real number $m$.

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