
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
For every point on the plane, one of  $ n$  colors are colored to it such that:

 $ (1)$  Every color is used infinitely many times.

 $ (2)$  There exists one line such that all points on this lines are colored exactly by one of two colors.

Find the least value of  $ n$  such that there exist four concyclic points with pairwise distinct colors.

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