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RB Leipzig finished third after a 2-1 defeat at home to the group-winners Besiktas.
If Napoli were to win at winless Feyenoord while Shakhtar lost at home to Manchester City, Maurizio Sarri’s side would be through.
But against a much-changed Man City, Shaktar was superb.
Liverpool had missed its last three spot kicks, but Philippe Coutinho converted, the first penalty scored by the home side at Anfield in 2017.
Coutinho added a second on the quarter-hour mark after some slick interplay on the break.
Despite the recent criticism he's been fielding, he was the top scorer of the group stage with eight goals.