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“All of the shows this afternoon will be talking about this,” he said. “The Democrats create messes for themselves. When the president gave himself an ‘incomplete’ grade, we were off to the races with that.”

For Mr. Chaffetz and other Republican surrogates, three rules apply: Stay on message. Attack opponent’s gaffes. Do not, under any circumstances, make gaffes yourself.

According to Mr. Chaffetz, the third rule is the toughest to follow — in part because jet-lagged surrogates give dozens of interviews a day and in part because even the smallest verbal misstep can dominate a news cycle.

“Somebody is always saying something stupid, and it’s not just Democrats,” he said. “In some ways, [being a surrogate] is sort of like the holder. You catch the ball, put it down. Mitt Romney has to make the kick. But if you don’t get the hold down? Look out.”