
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
Let  $K$  be the set of all sides and diagonals of a convex  $2010-gon$  in the plane. For a subset  $A$  of  $K,$  if every pair of line segments belonging to  $A$  intersect, then we call  $A$  as an *intersecting set.* Find the maximum possible number of elements of union of two *intersecting sets.*

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