ABC News’ Michael Falcone reports:

In their third debate in as many weeks, the two leading contenders for the Republican presidential nomination waged a war of authenticity, challenging each other on their records in government — even on the words printed in their books.

Former Massachusetts Gov. Mitt Romney, who has seen his lead in national and some early primary state polls evaporate, swiped at front-runner Rick Perry on the issue of Social Security. Romney has been trying to make the case that the Texas governor’s stance on the entitlement program is too extreme.

“There’s a Rick Perry out there that’s saying that — almost quoted it says that the federal government shouldn’t be in the pension business, that it’s unconstitutional, unconstitutional and to be returned to the states,” Romney said, referencing Perry’s book, “Fed Up!” “So you better find that Rick Perry and get him to stop saying that.”

Later in the debate, showing that he too has been carefully studying Romney’s writings, Perry accused his opponent of changing his tune between printings of his book, “No Apologies,” on whether the health care plan he signed into law in Massachusetts should be a model for the rest of the country.

“You said that it was exactly what the American people needed to have,” Perry said.

Romney shot back: “I said no such thing.”

“It’s fine for you to retreat from your own words in your own book, but please don’t try and make me retreat from the words that I wrote in my book,” he said.

The exchange at the debate in Orlando sponsored by Fox News and Google reflected the larger battle between the two campaigns, each trying to jockey for the top spot in the Republican field along with seven other candidates who shared the stage on Thursday night, who were just hoping to get some attention and airtime.

Despite some lines he might wish he could take back (“There are a lot of reasons not to elect me”), Romney delivered a mostly solid performance displaying his command of issues and even some sharp-edged responses while under attack.

At one point, Perry accused Romney of flip-flopping on President Obama’s “Race to the Top” program, an education initiative that seeks to incentivize improvements in public schools through competitive grants based on high-stakes testing.

“Being in favor of the Obama ‘Race to the Top,’” Perry said. “That is not conservative.”

“Nice try,” Romney snapped. “I don’t support any particular program that he’s described.”

At a town hall meeting in Miami earlier this week, Romney said programs like Race to the Top “make sense.”

Perry landed some punches on Thursday night, but his performance was uneven at times. He fumbled through an answer to a foreign policy question about what he would do if, as president, he received a middle-of-the-night phone call informing him that a Pakistani nuclear weapon had fallen into the hands of terrorists.

His answer — “Obviously before you ever get to that point you have to build a relationship in that region and that’s one of the things this administration has not done” — seemed to skirt the original question.

The Texas governor also appeared to lose his way while trying to portray Romney as a flip-flopper: “I think Americans don’t know sometimes which Mitt Romney they’re dealing with.”

“Is it the Mitt Romney that was on the side of against the Second Amendment before he was for the Second Amendment?” Perry said. “Was it before he was before the social programs from the standpoint of he was for standing up for Roe v. Wade before he was against Roe v. Wade — he was for Race to the Top — he’s for Obamacare and now he’s against it. I mean we’ll wait until tomorrow and see which Mitt Romney we’re really talking to tonight.”

But other candidates had their blunders too. When Michele Bachmann was asked to give a specific answer about how much out of every dollar earned, Americans deserve to keep, she said, “I think you should keep every dollar that you earn.”

A moment later, she added: “Obviously we have to give money back to the government so that we can run the government.”

Ever since Perry entered the race in August, Bachmann has had a difficult time combating the perception that the Republican primary has turned into a two-person race. She enjoyed something of a comeback in last week’s debate, calling Perry out on his decision to issue an executive order mandating young women in Texas receive a vaccine for a sexually transmitted disease.

“There’s the even more important point, because Governor Perry made a decision where he gave parental rights to a big drug company,” Bachmann said. “That big drug company gave him campaign contributions, and hired his former chief of staff to lobby him to benefit the big drug company.”

Perry was ready to defend himself.

“I got lobbied on this issue. I got lobbied by a 31-year-old young lady who had stage four cervical cancer,” the Texas governor said. “I spent a lot of time with her.”

Perry, however, did not meet the 31-year-old cancer patient, Heather Burcham, until after he had already issued his executive order mandating the HPV vaccine.

Among the lower-tier candidates on the stage Thursday night, including Texas Congressman Ron Paul, former House Speaker Newt Gingrich, businessman Herman Cain, former Utah Gov. Jon Huntsman and former New Mexico Gov. Gary Johnson, it was former Pennsylvania Sen. Rick Santorum who seemed to stand out. At times he criticized his better-known rivals for ducking questions. Both he and Romney also doggedly pursued Perry on immigration.

“I would say that he is soft on illegal immigration,” Santorum said.

The debate had its lighter moments too. Near the end, Fox News moderators asked each to say who they would choose as a running mate if they captured the nomination.

Perry suggested that he would like to take Cain, the former Godfather’s Pizza CEO who has never held elected office, and “mate him up” with Gingrich.

“There are a couple of images I’m going to have a hard time getting out of my mind,” Romney joked.