GWEN IFILL: And we return to last night's debate by looking more closely at some of the debatable arguments made by both candidates and sorting through the rhetoric.
Ray Suarez picks up that part of the story.
RAY SUAREZ: The confrontations in last night's debate came early and often, with charges and denials flying. So we spent part of this day examining the candidates' statements for context and contradictions on three key issues.
First, Libya, and the attack that killed the U.S. ambassador and three other Americans on September 11. President Obama and Mitt Romney sparred heatedly over whether the administration was honest about what happened. The president strongly criticized Romney's initial response to the attack.
PRESIDENT BARACK OBAMA: While we were still dealing with our diplomats being threatened, Gov. Romney put out a press release, trying to make political points.
And that's not how a commander in chief operates. You don't turn national security into a political issue, certainly not right when it's happening.
RAY SUAREZ: Romney fired back that the president's team blamed the attack on an anti-Muslim video, even after they knew it wasn't so.
MITT ROMNEY (R): It was a terrorist attack and it took a long time for that to be told to the American people.
Whether there was some misleading, or instead whether we just didn't know what happened, I think you have to ask yourself, why didn't we know five days later when the ambassador to the United Nations went on TV to say that this was a demonstration? How could we have not known?
RAY SUAREZ: That was a reference to Ambassador Susan Rice's statement on September 16.
SUSAN RICE, U.S. ambassador to the United Nations: Our current assessment is what happened in Benghazi was in fact initially a spontaneous reaction to what had just transpired hours before in Cairo, and almost a copycat of the demonstrations against our facility in Cairo, which were prompted, of course, by the video.
RAY SUAREZ: State Department officials said last week they never believed the video had caused the attack. The president said last night the attack is still under investigation, but insisted that there was no attempt to deceive.
BARACK OBAMA: The day after the attack, Governor, I stood in the Rose Garden and I told the American people in the world that we are going to find out exactly what happened, that this was an act of terror.
And the suggestion that anybody in my team, whether the secretary of state, our U.N. ambassador, anybody on my team, would play politics or mislead when we've lost four of our own, Governor, is offensive.
RAY SUAREZ: And that led to this exchange, ultimately involving the moderator, Candy Crowley.
MITT ROMNEY: I -- I think interesting the president just said something which -- which is that on the day after the attack he went into the Rose Garden and said that this was an act of terror.
BARACK OBAMA: That's what I said.
MITT ROMNEY: You said in the Rose Garden the day after the attack it was an act of terror?
It was not a spontaneous demonstration? I want to make sure we get that for the record because it took the president 14 days before he called the attack in Benghazi an act of terror.
BARACK OBAMA: Get the transcript.
CANDY CROWLEY, moderator: It -- it -- it -- he did in fact, sir. So let me -- let me -- called it an act of terror...
BARACK OBAMA: Can you say that a little louder, Candy?
CANDY CROWLEY: He -- he did call it an act of terror. It did as well take -- it did as well take two weeks or so for the whole idea there being a riot out there about this tape to come out. You are correct about that.
RAY SUAREZ: According to the White House transcript, Mr. Obama said the following on September 12:
"No acts of terror will ever shake the resolve of this great nation, alter that character or eclipse the light of the values that we stand for."
The "two weeks later" reference was to a statement by White House press secretary Jay Carney. On September 26, he said for the first time, "It is certainly the president's view that it was a terrorist attack."
As contentious as the Libya discussion was last night, the exchanges over energy policy proved equally sharp.
The Republican nominee pointed to the price of gasoline as evidence of presidential failure.
MITT ROMNEY: The proof of whether a strategy is working or not is what the price is that you're paying at the pump. If you're paying less than you paid a year or two ago, why, then, the strategy is working. But you're paying more.
When the president took office, the price of gasoline here in NassauCounty was about $1.86 a gallon. Now it's $4.00 a gallon.
RAY SUAREZ: But President Obama warned that promises of lower gas prices are not risk-free.
BARACK OBAMA: Think about what the governor just said. He said when I took office, the price of gasoline was $1.80, $1.86. Why is that? Because the economy was on the verge of collapse, because we were about to go through the worst recession since the Great Depression, as a consequence of some of the same policies that Gov. Romney's now promoting.
So, it's conceivable that Gov. Romney could bring down gas prices because, with his policies, we might be back in that same mess.
RAY SUAREZ: One of those watching that back and forth, as well as the argument of drilling on federal land, was Juliet Eilperin, a national environmental reporter with The Washington Post.
JULIET EILPERIN, The Washington Post: Obama did made a legitimate point, that we were in the depths of recession worldwide, and so there was a reduction in demand, which led to a lower gas price.
RAY SUAREZ: Eilperin said Romney's charge that production fell during the president's tenure was partly correct, but that was primarily for a time after the Gulf oil spill. Overall:
JULIET EILPERIN: The first three years of the Obama has seen an increase in both oil and gas production on public lands compared to the last three years of the Bush administration.
You have seen an uptick of 13 percent in terms of oil production, 6 percent in terms of natural gas production. So there actually has not been a falloff over the course of his administration.
That said, it is true that the Romney administration, if elected, would pursue policies that are more favorable to oil and gas companies, and could open up areas for production which the Obama administration has closed off. So, you could see an expansion in oil and gas exploration in the United States. That's pretty clear.
Whether that would shift the price at the pump remains unclear.
RAY SUAREZ: On another pocketbook issue, tax cuts, the two men also presented sharply competing claims. Gov. Romney insisted he can cut marginal tax rates across the board by 20 percent, while limiting some deductions and loopholes that would be both revenue-neutral and benefit the middle class.
MITT ROMNEY: Why am I lowering taxes on the middle-class? Because under the last four years, they've been buried. And I want to help people in the middle class.
And I will not -- I will not under any circumstances reduce the share that's being paid by the highest-income taxpayers. And I will not, under any circumstances, increase taxes on the middle class. The president's spending, the president's borrowing will cause this nation to have to raise taxes on the American people, not just at the high end.
RAY SUAREZ: But the president pointed to analyses that found Romney's initial tax proposals could cost nearly $5 trillion over 10 years. That, he argued, didn't take into account at least $2 trillion more in spending and other tax proposals from Romney.
BARACK OBAMA: Gov. Romney was a very successful investor.
If somebody came to you, Governor, with a plan that said, here, I want to spend $7 or $8 trillion, and then we're going to pay for it, but we can't tell you until maybe after the election how we're going to do it, you wouldn't take such a sketchy deal.
And neither should you, the American people, because the math doesn't add up.
RAY SUAREZ: Today, we caught up with Donald Marron of the bipartisan TaxPolicyCenter. He's an author of one of the analyses quoted by the president and others.
DONALD MARRON, TaxPolicyCenter: Gov. Romney has laid out four to five different principles that he would like to accomplish in his tax reform. And what our study concludes is that it is not possible to accomplish all of them.
If he is willing to compromise in one dimension, allow the deficit to be bigger or raise taxes on capital gains and dividends, it may be possible. Someone may be able to put forward a plan that would do it, but, thus far, why we haven't seen one.
RAY SUAREZ: For his part, Gov. Romney has said there are six studies that show he can fulfill his tax cut pledges.
But while some of those analyses suggest Romney's plans might be plausible, under certain conditions not acknowledged by the governor, four of them are either op-ed columns or online posts written by conservative economists.
GWEN IFILL: You can watch the entire debate on our home page, or, if you choose, eight minutes of highlights. We posted the most memorable exchanges to give you a chance to vote on how the candidates performed.
Plus, you can watch our after-hours live-stream conversation, where political editor Christina Bellantoni and Hari Sreenivasan answered the questions you asked on Twitter and Facebook.