
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
Let $f\left(x\right)=\sqrt{2+x}+\ln \left(4-x\right)$, where the domain of the function is $A$, and the set $B=\{x\left|\right.m+1\leqslant x\leqslant 2m-1\}\left(m\in R\right)$.  
$(1)$ Find the set $A$;  
$(2)$ If $p:x\in A$, $q:x\in B$, and $p$ is a necessary but not sufficient condition for $q$, find the range of real number $m$.

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