
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, it is known that the point $P(3m-6, m+1)$. Find the coordinates of point $P$ according to the following conditions:<br/>$(1)$ If point $P$ lies on the $y$-axis, find the coordinates of point $P$.<br/>$(2)$ The coordinates of point $A$ are $(1, -2)$. If $AP$ is parallel to the $x$-axis, find the coordinates of point $P$.

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