
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 the line segment \(x + y = 1\) (with \(x \geq 0\) and \(y \geq 0\)), which contains 2020 points, find the smallest positive integer \(k\) such that for any selection of these 2020 points, there always exists a way to divide these points into two groups. In one group, the sum of the y-coordinates of the points does not exceed \(k\), and in the other group, the sum of the x-coordinates of the points does not exceed \(k\) (the 2020 points may coincide).

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