
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 2022 lines in the plane, such that no two are parallel and no three are concurrent. We denote \( E \) as the set of their intersection points. We want to assign a color to each point in \( E \) such that any two points on the same line, whose connecting segment does not contain any other point from \( E \), have different colors. What is the minimum number of colors required to achieve this coloring?

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