Among $100$ points in the plane, no three are 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 $50$.

Let's think step by step and output the final answer within \boxed{}.
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