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“You can’t allow someone to stand there and basically manhandle the truth and not deal with that,” Mr. Axelrod said. “I know that [the president] is very eager for the next debate.”

Mr. Obama insisted that Mr. Romney, who devoted much of the debate to promising to help the middle class, “cannot pay for his $5 trillion tax plan without blowing up the deficit or sticking it to the middle class.”

“Gov. Romney’s math just doesn’t add up,” he said, “and I had to spend a lot of time last night trying to pin it down.”

Mr. Axelrod also suggested that his boss was partly to blame for not thinking more quickly on his feet at the debate when Mr. Romney attacked his economic policies.

The president “made a choice last night to answer the questions that were asked and talk to the American people,” Mr. Axelrod said. “The president viewed it as a great opportunity to talk to the American people. He didn’t view it, as perhaps Gov. Romney did, as a performance.”

Mr. Axelrod claimed that staging a political performance in an election debate is not Mr. Obama’s “strong suit.”

‘Damage-control mode’

Romney campaign aides said the former Massachusetts governor was consistent in his positions on the issues in the debate, and they said the Obama camp was looking for a scapegoat for the president’s poor performance.

“In full damage-control mode, President Obama today offered no defense of his record and no vision for the future,” said Romney spokesman Ryan Williams. “Rather than a plan to fix our economy, President Obama simply offered more false attacks and renewed his call for job-killing tax hikes. Mitt Romney demonstrated why he should be president, laying out the clear choice in this election. We can’t afford four more years of the last four years. We need a real recovery — and Mitt Romney has a real plan to deliver it.”

More than 12 hours after the debate, the encounter still seemed to be eating at the president, who told supporters that Mr. Romney changed his positions for the TV cameras.

“The real Mitt Romney said we don’t need any more teachers in our classrooms,” Mr. Obama said. “But the fellow on stage last night, he loves teachers — can’t get enough of them. The Mitt Romney we all know invested in companies that were called ‘pioneers’ of outsourcing jobs to other countries. But the guy on stage last night, he said that he doesn’t even know that there are such laws that encourage outsourcing — he’s never heard of them.”

He added with annoyance in his voice, “The man on stage last night, he does not want to be held accountable for the real Mitt Romney’s decisions and what he’s been saying for the last year. So Gov. Romney may dance around his positions, but if you want to be president, you owe the American people the truth.”

Some attributed Mr. Obama’s debate performance to the absence of a teleprompter. At the rally Thursday, the president again was aided by the speech-reading device. But his confidence seemed shaken, as he told the Colorado crowd at one point, “I don’t know how many of you will be with me this time around. I still believe in you. I’m asking you to keep on believing in me.”

Looking ahead

Even some of Mr. Obama’s most ardent supporters acknowledged that he performed badly in the debate. Rapper tried to cheer up the crowd at the rally in Denver, which was held in frigid temperatures.

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