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Mr. Obama never mentioned the “47 percent” video — an omission that shocked strategists who said the president was missing major opportunities to dent the Republican.

Instead, Mr. Obama repeatedly attacked Mr. Romney for refusing to specify what tax breaks he’ll eliminate in order to make the math on his tax plan add up.

And Mr. Romney did not offer any new details, instead flatly vowing that he won’t allow the rich to get tax cuts at the expense of the middle class.

Mr. Obama said it was becoming a pattern for the Republican candidate to hide the details of his proposals.

“At some point, I think the American people have to ask themselves, is the reason that Governor Romney is keeping all these plans to replace secret because they’re too good? Is it because that somehow middle-class families are going to benefit too much from them?” he quipped.

But Mr. Romney was brutally clear on some specific budget cuts he’d make — including axing funding for PBS, telling debate moderator and long-time PBS newsman Jim Lehrer.

“I like PBS. I love Big Bird. I actually like you, too. But I’m not going to keep on spending money on things to borrow money from China to pay for,” he said.

As the debate began, the two men shook hands and exchanged what appeared to be friendly words. Mr. Obama then began his opening remarks by speaking to his wife, Michelle, who was in the audience and with whom he was celebrating his 20th anniversary.

“A year from now we will not be celebrating it in front of 40 million people,” he promised her.

“I’m sure this was the most romantic place you could imagine — here with me,” Mr. Romney quipped.