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Darren Stevens scored his first century of the season before Kent declared on 533 for eight to set Hampshire a target of 461 to win the LV Championship match at Canterbury.
The visitors were 18 without loss at stumps from four overs and will have a minimum of 96 overs on the last day to score a further 443 runs.
Stevens built on the solid foundation laid during the second wicket stand between Rob Key and Martin van Jaarsveld the previous day.
That partnership was not to prosper further after they resumed on the third morning on 223 for one.
Van Jaarsveld, 109 not out overnight, failed to add to his score before he was caught behind from his airy drive at a ball well outside off stump from Chris Tremlett.
It was only the third ball of the day and meant the pair failed to beat Kent's second wicket record against Hampshire of 231 which has stood since 1913 when Wally Hardinge and James Seymour set the mark at Portsmouth.
Key, 96 overnight, wasted little time in reaching his second century of the season, square driving Sean Ervine to cover point for his 11th boundary.
An hour later, Hampshire struck again, with Dimitri Mascarenhas, having replaced Stuart Clark, trapping Key (120) lbw.
Matthew Walker had some luck in scoring 46. He was nearly run out on 20 when he slipped in mid pitch after being sent back by Stevens.
The return from Michael Brown, at short mid-wicket, was too high for the wicket-keeper to take and Walker made his ground.
On 38, he was dropped, off Clark, by Michael Lumb at first slip, in the second over with the new ball after lunch, which had been taken on 329 for three.
Five overs into the second session, Walker was dismissed when Clark had his top-edged hook shot held by Ervine at long leg. Along with Stevens he had put on 76 and Kent were 344 for four.
Six overs later, Geraint Jones, having played a glorious extra cover drive for four, off Ervine, was bowled by Clark. His poor batting form continues with just 66 championship runs in seven innings.
Stevens, scoring steadily on the excellent batting wicket, got up his 50 from 99 balls and along with Andrew Hall, who hit 48 from 71 balls, added 83 for the sixth wicket. Shane Warne had toiled for 26 overs, costing 94 runs before he claimed a scalp, that of Hall when he was held at cover.
Stevens, who was dropped at short cover from a firmly struck drive on 67, later reached 76 with a lovely square drive to cover point off Warne and eventually brought up his first century of the season from 179 balls with 11 fours.
But three balls later and having added another boundary he was bowled by Jimmy Adams as he missed a slog shot.
Ryan McLaren and James Tredwell were 25 not out and 27 not out respectively at the declaration with Clark claiming 2-80 and Adams with figures of 2-37.
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