
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 there're  $1000$  marked points, each colored in one of  $k$  colors. It's known that among any  $5$  pairwise intersecting segments, endpoints of which are  $10$  distinct marked points, there're at least  $3$  segments, each of which has its endpoints colored in different colors. Determine the smallest possible value of  $k$  for which it's possible.

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