
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
In the Cartesian coordinate system, there are three points $A(-2, a+1)$, $B(a-1, 4)$, $C(b-2, b)$.  
$(1)$ When $AB$ is parallel to the $x$-axis, find the distance between points $A$ and $B$.  
$(2)$ When $CD$ is perpendicular to the $x$-axis at point $D$, and $CD=1$, find the coordinates of point $C$.

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