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Dover crashed out of the FA Trophy with defeat at Boreham Wood on Saturday.
Nicky Forster's men returned to the scene of a 4-2 league defeat two weeks ago and were unable to break down the hosts, who never relinquished their 15th minute lead.
Justin Cochrane's unmarked far-post header from a Boreham Wood corner was enough to secure a place in the first round.
It could have been 2-0 moments later when Dover keeper Mikhael Jaimez-Ruiz missed his punch, but his blushes were spared by Olly Schulz who cleared off the line.
James Walker (pictured) hit the bar with a cross before the break, but Barry Cogan spurned a glorious chance to level in the second half. After Southam had played him in, he blazed over the bar when in space.
Tom Wynter also had to step in and clear off the line in the second half as Dover failed to press for an equaliser in a lacklustre display.
Folkestone Invicta bowed out of the FA Trophy to Blue Square South strugglers Staines at Cheriton Road.
Neil Cugley's side were the lowest-ranked Kent team left in the draw but were on the back foot when they conceded two first half goals in the space of three minutes. Gareth Risbridger put Staines 1-0 ahead on 38 minutes before Millwall loannee Jake Goodman doubled their lead before the interval.
Marc Charles-Smith put the tie beyond doubt after 64 minutes with an impressive run and shot for 3-0. Frankie Chappell pulled back a consolation for Invicta late on to make it 3-1.