
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
Among  $100$  points in the plane, no three collinear, exactly  $4026$  pairs are connected by line segments. Each point is then randomly assigned an integer from  $1$  to  $100$  inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least  $50$ .

*Proposed by Evan Chen*

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