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In response, Mr. Romney acknowledged that he hired a lawn company that had illegal immigrants working on the crew and “when that was pointed out to us, we let them go.”

Then he tried to turn the table on Mr. Perry, highlighting his support of in-state tuition for some children of illegal immigrants and a controversial report from the Center for Immigration Studies that suggested 40 percent of all the new jobs created in Texas went to illegal immigrants — a charge Mr. Perry called “a falsehood on its face.”

At one point, Mr. Romney put his hand on Mr. Perry’s shoulder, saying: “It’s been a tough couple of debates for Rick. And I understand that so you’re going to get testy.”

Topics in the fifth debate in six weeks ranged from questions about the candidates’ personal faith and the Wall Street bailout to the Occupy Wall Street protests and the call to build a border fence.

Mr. Paul questioned how candidates who support a border fence, including Mr. Cain, Mrs. Bachmann and Mr. Romney, would pay for it. Mr. Paul later suggested that the nation could free up money to bolster the southern border by ending assistance to foreign countries and bring the troops home from overseas.

“I want to hear somebody up here willing to cut something — something real,” he said.

Mr. Cain found himself backtracking on two other issues during the two-hour debate.

He backed away from earlier comments about building a lethal electric fence on the Mexico border and also said he wouldn’t be willing to negotiate with terrorists, even though he suggested earlier in the day that he would be open to such a decision.

With all the fire trained on one another, the Republicans’ big target, President Obama, was left relatively unscathed.

Mr. Gingrich tried to steer the conversation back to the incumbent and voiced concerns that the intraparty squabbling will hurt the GOP’s nominee in the long run.

“Let me just point out a second that maximizing bickering is probably not the road to the White House,” Mr. Gingrich said.

Former Utah Gov. Jon Huntsman skipped the debate, part of the CNN/Western Republican Leadership Conference, in protest of the Nevada GOP’s decision to move its caucus date up to Jan. 14. That move has threatened to upend the rest of the primary schedule, with New Hampshire’s secretary of state warning he might push the first-in-the-nation primary to early December.

Last week, several of the candidates, including Mr. Gingrich and Mr. Cain, said they will join Mr. Huntsman and boycott Nevada’s caucus if the state party refuses to push it back three days, as requested by New Hampshire.

Some see the boycott effort as an attempt by cash-strapped candidates to devalue Nevada’s contest in the face of strong support here for Mr. Romney, though an uproar from some elected leaders in the Granite State and editorial in the influential New Hampshire Union Leader could turn up the political heat on the Romney camp.