
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
Set $A=\{x|-2\leqslant x\leqslant 5\}$, $B=\{x|m+1\leqslant x\leqslant 2m-1\}$.  
$(1)$ If $B\subseteq A$, find the range of the real number $m$;  
$(2)$ When the elements of $A$ are $x\in\mathbb{Z}$, find the number of non-empty proper subsets of $A$;  
$(3)$ When $x\in\mathbb{R}$, if $A\cap B=\varnothing$, find the range of the real number $m$.

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