day2-2.prdc

l]²¨Ûfμ™Îf‹m}L_Æ .. 16 bytes
.. (takes about a minute or two to run)

{
l] .. get lines
²  .. square i.e. cartesian product with itself
¨  .. flatten once
Ûf .. filter for pairs that are unique, to eliminate lines paired with themselves
μ  .. map by the following function
     ..   (which receives a length-2 list of two lines)
  ™  ..   transpose, to get a list of pairs
  Îf ..   filter for pairs that are identical
  ‹m ..   extract first element of each pair
}  .. (end filter)
L_Æ .. take max length
};