
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
Consider the set of all points $(x,y)$ in the coordinate plane for which one of the coordinates is three times the other, plus a constant for one of the relationships. Specifically, if $y$ is three times $x$ or $x$ is three times $y$ plus two. Plot all such points and determine into how many regions the resulting graph would split the plane.

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