
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 a set of 4027 points on a plane, called a "Colombian point set," where no three points are collinear. Among these points, 2013 are red, and 2014 are blue. A set of lines drawn on the plane can divide the plane into several regions. A set of lines is considered a "good line set" for a Colombian point set if it satisfies the following two conditions:

1. The lines do not pass through any of the points in the Colombian point set.
2. Each region formed by the lines does not contain both red and blue points simultaneously.

Find the minimum value of $\mathrm{k}$ such that for any Colombian point set, there exists a "good line set" formed by $\mathrm{k}$ lines.

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