
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]


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A configuration of  $4027$  points in the plane is called Colombian if it consists of  $2013$  red points and  $2014$  blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied:

i) No line passes through any point of the configuration.

ii) No region contains points of both colors.

Find the least value of  $k$  such that for any Colombian configuration of  $4027$  points, there is a good arrangement of  $k$  lines. 

Proposed by *Ivan Guo* from *Australia.*

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