
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 $A=(x+y)(y-3x)$, $B=(x-y)^{4}\div (x-y)^{2}$.  
$(1)$ Simplify $A$ and $B$;  
$(2)$ If the variable $y$ satisfies $2y+A=B-6$, find the relationship between $y$ and $x$;  
$(3)$ Under the condition of $(2)$, find the value of $(y+3)^{2}-2x(xy-3)-6x(x+1)$.

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