
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
On a circle, 1000 points are marked, each colored in one of \( k \) colors. It turns out that among any five pairwise intersecting segments, whose endpoints are 10 different marked points, there will be at least three segments, each of which has endpoints of different colors. What is the smallest \( k \) for which this is possible?

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