
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
The domain of the function $f(x)=\frac{\lg (x-1)}{\sqrt{2-x}}$ is $A$, and the solution set of the inequality $x^{2}-(2a+3)x+a^{2}+3a\leqslant 0$ with respect to $x$ is $B$.

$(1)$ Find the set $A$;  
$(2)$ If $A\cap B=A$, find the range of the real number $a$.

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