
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
There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number  $k$  such that no matter how I select and color  $k$  points, you can always color the remaining  $100-k$  points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.

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