
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 points $A(a-1,2)$ and $B(-3,b+1)$, determine the values of $a$ and $b$ based on the following requirements:  
$(1)$ Line $AB$ is parallel to the $x$-axis;  
$(2)$ Line $AB$ is parallel to the $y$-axis;  
$(3)$ Points $A$ and $B$ are on the angle bisector of the first and third quadrants.

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