
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
Given 100 different points on a circle, determine the largest positive integer \( k \) such that if any \( k \) of these points are arbitrarily colored either red or blue, the remaining points can be suitably colored red or blue in a way that 50 line segments can be drawn between these 100 points as endpoints, with no two line segments sharing a common point, and each line segment having endpoints of the same color.

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