
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
 $2021$  points are given on a circle. Each point is colored by one of the  $1,2, \cdots ,k$  colors. For all points and colors  $1\leq r \leq k$ , there exist an arc such that at least half of the points on it are colored with  $r$ . Find the maximum possible value of  $k$ .

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