World No. 1 Tiger Woods was just two shots back in his bid to win for a fifth time on the PGA Tour this year.
With most of the world's top players gathered at the Firestone Country Club, Simpson birdied four of the last six holes to move to the front. He is seeking his first victory since capturing last year's U.S. Open.
Henrik Stenson, who finished second to Phil Mickelson at the British Open two weeks ago, was alone in second with a 65 and Woods was joined at 66 by Ryan Moore, Chris Wood and defending champion Keegan Bradley. Woods has won this event seven times.
Mickelson, who captured the third leg of a career grand slam with his win at Muirfield, settled for a 2-over 72. Rory McIlroy, who missed the cut at the British Open and is trying to find some form leading up to his defense of the PGA Championship next week, had an even-par 70.
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