
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 998 red points on a plane, with no three points being collinear. Blue points are to be placed such that every triangle formed by red points contains a blue point inside it. Determine the positive integer \( k \) such that it is always possible to place these blue points using \( k \) blue points, but there exists a configuration of red points where \( k-1 \) blue points are not sufficient.

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