
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 distinct points in the plane with rectangular coordinates $A(3, a-1)$ and $B(b+1, -2)$.
$(1)$ If point $B$ lies on the $y$-axis, find the value of $b$;
$(2)$ If point $A$ lies on the angle bisector in the first and third quadrants, find the value of $a$;
$(3)$ If the line $AB$ is parallel to the $y$-axis and $AB=5$, find the values of $a$ and $b$.

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