
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 where one of the coordinates is either three times the other. Specifically, for the situations where $y$ is three times $x$, and $x$ is three times $y$. Plot the lines representative of these conditions in the coordinate plane. Determine how many distinct regions the graph splits the plane into.

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