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Maidstone will face AFC Fylde in the semi-finals of the FA Trophy if they get past Stockport.
Leyton Orient meet AFC Telford in the other two-legged semi-final.
United will be away in the first leg, on March 16, if they beat County in tomorrow night's replay at the Gallagher, with the return a week later on March 23.
The semi-final winners pocket £20,000 in prize money and meet in the final at Wembley on Sunday, May 19.
While Maidstone and Stockport meet again, Fylde and Barnet agreed to settle their quarter-final on the day.
It went the distance after a goalless draw, with the Coasters winning 4-1 on spot-kicks.
Fylde, fourth in the National League, started their Trophy campaign with a 5-1 victory over Stratford and have also beaten Biggleswade (1-0) and Ramsbottom (4-1 in a replay).
Orient will be at home in the first leg of their semi-final against Telford.
The O's, second in the National League, ended Brackley's reign as Trophy holders at the weekend.
Justin Edinburgh's side trailed at half-time but late goals from Matt Harrold and Dan Happe won it for the O's.
Orient have also knocked out Beaconsfield (4-0), Wrexham (1-0) and Blyth Spartans (1-0).
Telford upset National League title-chasers Solihull 2-1 in the quarter-finals.
Daniel Udoh scored an 89th-minute winner for the National North club after Amari Morgan-Smith's opener was cancelled out by Jamey Osborne.
Telford, who are sixth in their league, have also seen off Darlington (2-0), Farsley Celtic (4-3), Hartlepool (2-1) and Spennymoor (2-1).