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Neil Dexter scored only the second championship century of his career as Kent, who had started their second innings 230 runs behind, wiped out the arrears with eight wickets in hand before Rana Naved-ul-Hasan revived Yorkshire's victory hopes with an inspired spell of three wickets for no runs in 14 balls.
Yorkshire must win to climb out of the LV County Championship first division relegation zone following Hampshire's remarkable victory over Durham at Basingstoke and their shoulders were starting to slump when Dexter was sharing partnerships of 132 with Joe Denly and 107 with Martin van Jaarsveld.
Then Naved, who had been singularly unimpressive in his first spell, returned to produce a snorting delivery that hit the top of Dexter's off stump after he had made a chanceless 105 off 224 balls with a six and nine fours.
Naved went on to york Martin Saggers, the nightwatchman, and have Darren Stevens caught at short leg and, though Van Jaarsveld is still there, having added an unbeaten 61 to his first-innings 107, Yorkshire will begin the final day looking up rather than down.
Kent had not helped themselves when Yorkshire, resuming 182 runs ahead with three wickets in hand, were adding another 48, mainly through a last-wicket stand of 40 between Darren Gough and Deon Kruis.
Kruis was dropped at first slip off Amjad Khan before he had scored and Gough was put down at midwicket, completing a sloppy performance in the field from Kent that included at least six catches dropped and 38 runs conceded in no-balls and wides.
It did not get much better when they went in again, Rob Key going for 12 when he edged Matthew Hoggard to first slip and Denly continuing to play as though it was Twenty20 rather than championship cricket.
Denly should have been out for 17 when he cut Naved to third man but Andrew Gale let the catch slip through his hands and he had made 66 when his luck ran out and he edged Hoggard to second slip.
Dexter, who had made only 241 runs in 11 innings this season for Kent and Essex, with whom he had a spell on loan, and only got back in the side because of injury to Justin Kemp, was far more impressive and seemed to be turning the match Kent's way - until Naved struck.
Kent: First Innings 227 (M van Jaarsveld 107, J L Denly 50)
Second Innings
J L Denly c McGrath b Hoggard 66
*R W T Key c Rudolph b Hoggard 12
N J Dexter b Naved 105
M van Jaarsveld not out 61
M J Saggers b Naved 0
D I Stevens c Gale b Naved 0
G O Jones not out 4
Extras (b 5, lb 6, w 1, nb 13) 25
Total (5 wkts, 80 overs) 273
Fall of wickets: 1-23, 2-155, 3-262, 4-268, 5-268.
Bowling: Hoggard 15-0-46-2; Kruis 15-6-41-0; Rashid 22-2-70-0; McGrath 5-2-6-0; Gough 12-0-46-0; Naved 11-2-53-3.
Yorkshire: First Innings (overnight 409-7)
A U Rashid lbw b Khan 43
*D Gough c Key b Tredwell 33
M J Hoggard c Stevens b Khan 0
G J Kruis not out 17
Extras (b 8, lb 8, w 8, nb 30) 54
Total (143 overs) 457
Fall of wickets: 1-19, 2-37, 3-213, 4-273, 5-303, 6-351, 7-373, 8-410, 9-417.
Bowling: Joseph 33-2-112-3; Khan 26-8-79-3; Saggers 29-7-97-1; McLaren 21-4-70-1; Tredwell 19-4-46-1; Stevens 10-2-16-1; Van Jaarsveld 5-0-21-0.
Umpires: J H Evans and R T Robinson.
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