
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
Lucky and Jinx were given a paper with  $2023$  points arranged as the vertices of a regular polygon.
They were then tasked to color all the segments connecting these points such that no triangle formed
with these points has all edges in the same color, nor in three different colors and no quadrilateral
(not necessarily convex) has all edges in the same color. After the coloring it was determined that
Jinx used at least two more colors than Lucky. How many colors did each of them use?

*Proposed by Ilija Jovcheski*

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