
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 the set $A=\{ x | m-1 \leqslant x \leqslant 2m+3\}$, the domain of $f(x)=\lg (-x^2+2x+8)$ is $B$.

$(1)$ When $m=2$, find $A \cup B$, $(\complement_R A) \cap B$;  
$(2)$ If $A \cap B = A$, find the range of the real number $m$.

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