
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 two linear functions $y_{1}=-x+1$ and $y_{2}=-3x+2$.  
$(1)$ If the solution to the equation $y_{1}=a+y_{2}$ is a positive number, find the range of values for $a$.  
$(2)$ If the point $\left(x,y\right)$ with coordinates $x$ and $y$ lies on the graphs of the two given linear functions, find the value of $12x^{2}+12xy+3y^{2}$.  
$(3)$ If $\frac{4-2x}{(3x-2)(x-1)}=\frac{A}{{y}_{1}}+\frac{B}{{y}_{2}}$, find the value of $\frac{A}{B}+\frac{B}{A}$.

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