
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 that $y+1$ is directly proportional to $x-3$, and when $x=2$, $y=-2$.  
$(1)$ Find the analytical expression of $y$ in terms of $x$;  
$(2)$ In the Cartesian coordinate plane, if the graph corresponding to this function has intersections with the $x$-axis and $y$-axis at points $A$ and $B$ respectively, does there exist a point $P$ on the line $AB$ such that the area of $\triangle APO$ is equal to $2$? If it exists, find the coordinates of point $P$; if not, explain the reason.

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