
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 $A=\{x|(x-3)(x-3a-5) < 0\}$, the domain of the function $y=\lg (-x^{2}+5x+14)$ is set $B$.  
$(1)$ If $a=4$, find the set $A\cap B$;  
$(2)$ If "$x\in A$" is a sufficient condition for "$x\in B$", find the range of the real number $a$.

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