253. A tournament

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The diagram shows an elimination tournament amongst six teams, A-F.  The winner of each game follows a blue line.

After the tournament, each team made a statement as follows:
A: We lost our first game, but to win the tournament, we would have had to win three games.
B: We did not play A.
C: We lost to D.
D: We played B in our second game.
E: We did not play C.
F: We won our first game.

Determine the team bracket, α to ζ.

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