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Q: When did the Federation make first contact with the Cardassian Union? It cant have been later than 2347 (Setlik 3 massacre and beginning of Cardassian-Federation War) but is there any other reference to when the two governments made official first contact?

A: Theres no main canon description of the Federations first formal contact with the Cardassian Union. They do encounter Cardassian individuals in Star Trek: Enterprise — a Cardassian Borg, and what appears to be a Cardassian body at the automated repair station in Dead Stop — but these obviously dont answer the question of first official contact.
There are some (conflicting) accounts in various Trek novels and comics.
Per Memory Beta:

By some accounts, first contact between the Federation and the
  Cardassian Union occurred circa 2321, on the human colony world of
  Juhraya. (VOY - The Brave and the Bold, Book Two novella: The Third
  Artifact) 
This conflicts with Day of the Vipers, which states that the
  Cardassians were already engaged in pointless, unresolved conflicts
  with the Federation by 2318. This implies a first contact date
  sometime before or during 2318. 
The first official talks between the Cardassians and the Federation
  took place in 2327, however the Cardassians were using the talks as a
  pretense to disrupt relations between the Federation and Legarans to
  make Legara IV susceptible to annexation by the Cardassians.
  Fortunately the plan was uncovered and the Federation placed extra
  Starfleet protection around Legara. (TOS comic: Enter the Wolves)


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The text includes fourteen "y"s in total.
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