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Three times in six matches this season they have been pegged back from that position only to score added-time winners, but when Ryan Gravenberch and Hugo Ekitike scored inside 29 minutes at home to Everton it appeared it would be plain sailing.
But, as this campaign has proved, nothing is straightforward for Arne Slot's side, which is still adjusting to the influx of summer signings, and a strangely lacklustre second half allowed the visitors to significantly change the dynamic with Idrissa Gueye's 58th-minute strike.
Conor Bradley, making his first start of the season, had a relatively easy job in the first half but faced a more difficult job after the break.
Ryan Gravenberch steered home a fine opener
Another fast start saw Liverpool take the lead in the 10th minute
Beto, whose only sight of goal was a difficult one when Iliman Ndiaye crossed behind him, was replaced by Thierno Barry at half-time, but it was Liverpool's slow start to the second half which gave Everton hope when there appeared little.
Everton's season, meanwhile, has stalled after the international break but Moyes will take encouragement from their second half even if it did not amount to anything.