
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 $A=\{x|2\leqslant x\leqslant 7\}$, $B=\{x|-3m+4\leqslant x\leqslant 2m-1\}$, and $B\neq \varnothing$.  
$(1)$ If the proposition $p:\forall x\in A$, $x\in B$ is a true statement, find the range of real number $m$;  
$(2)$ If the proposition $q:\exists x\in B$, $x\in A$ is a true statement, find the range of real number $m$.

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