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Here’s how we analyzed in real time the Republican Nevada caucuses.

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Senator Ted Cruz’s presidential campaign erupted in turmoil on Monday as Mr. Cruz fired his chief spokesman for spreading a misleading video of Senator Marco Rubio — an ill-timed shake-up for a campaign already under duress on the eve of the Republican caucuses in Nevada.

For Mr. Cruz, who has vigorously defended himself in recent weeks as rivals accused him of running an underhanded race, the episode cuts at the premise of his bid. He has cast himself as the candidate of honesty and integrity, with a faith-based pitch and a backdrop that reads “TrusTed” during speeches.

On Monday, amid persistent sniping from Mr. Rubio and Donald J. Trump about the Cruz campaign’s penchant to mislead, the Texas senator was at last forced to acknowledge his team had gone too far.

At issue was a video of a hotel lobby encounter between Mr. Rubio and a Cruz staff member who had a Bible with him. The video’s subtitles suggested Mr. Rubio had said there were “not many answers” in the Bible. Mr. Cruz’s communications director, Rick Tyler, posted the video on Facebook and Twitter on Sunday, needling Mr. Rubio for an “awkward remark.”

In fact, Mr. Rubio had said the Bible had “all the answers.”

Mr. Cruz said Monday that he had spent the morning investigating the episode.

“I have made clear in this campaign that we will conduct this campaign with the very highest standards of integrity,” he told reporters during a news conference in Las Vegas, describing his decision to request Mr. Tyler’s resignation.

He called Mr. Tyler a “good man,” but said that his sharing of the video was “a grave error in judgment.”

“Even if it was true, we are not a campaign that is going to question the faith of another candidate,” Mr. Cruz said. “Even if it was true, our campaign should not have sent it.”

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Mr. Cruz took no questions about Mr. Tyler before making a hasty exit.

The campaign’s self-inflicted drama came as Mr. Cruz and Mr. Rubio, who edged past him in South Carolina, strained to emerge as the chief challenger to Mr. Trump. The two are also tussling over evangelical support, which is particularly essential to Mr. Cruz, making the dust-up over a remark about the Bible especially tense.

But Mr. Tyler’s dismissal, and Mr. Cruz’s acknowledgment of a campaign misstep, were a clear boon to Mr. Rubio on another front, creating the kind of cudgel that could prove powerful at a crucial stage of the race.

It represents a public admission of what Mr. Rubio and Mr. Trump have angrily asserted for weeks: that Mr. Cruz has employed deceitful tactics to undermine his rivals, including a photograph doctored by his aides that suggested Mr. Rubio was shaking hands with President Obama in front of the Capitol in Washington.

After Mr. Tyler’s dismissal, Mr. Rubio’s team moved quickly to focus attention on his boss.

“Rick is a really good spokesman who had the unenviable task of working for a candidate willing to do or say anything to get elected,” said Mr. Rubio’s spokesman, Alex Conant. “There is a culture in the Cruz campaign, from top to bottom, that no lie is too big and no trick too dirty. Rick did the right thing by apologizing to Marco. It’s high time for Ted Cruz to do the right thing and stop the lies.”

Mr. Trump has, less credibly, attacked Mr. Cruz on similar grounds, threatening a defamation suit over an ad reminding voters of his past support for abortion rights. He was likewise eager to seize on Monday’s news.

“Wow, Ted Cruz falsely suggested Marco Rubio mocked the Bible and was just forced to fire his Communications Director,” Mr. Trump wrote on Twitter. “More dirty tricks!”

Privately, Mr. Cruz’s advisers have acknowledged that perceptions about his integrity had taken a toll since his victory in Iowa. On caucus night, his team spread word to voters — with a liberal interpretation of a CNN report — that Ben Carson planned to suspend his campaign.

The mild-mannered Mr. Carson, widely admired even by non-supporters, seethed with anger, and much of the Republican field came to his defense.

Days earlier, Mr. Cruz had earned a rebuke from Iowa’s secretary of state for mailers that accused residents of nonexistent “voter violations.”

As the race reached South Carolina, Mr. Rubio’s team assailed Mr. Cruz and his allies for myriad alleged mistruths. After a legal review, some television stations in South Carolina pulled an ad from a pro-Cruz “super PAC” that had criticized Mr. Rubio’s record on immigration. Mr. Cruz also faced criticism from National Right to Life for what it called “inaccurate and misleading” remarks about Mr. Rubio’s record on abortion.

There have been lesser unforced errors, as when Mr. Cruz pulled one of his campaign ads after learning that an actress in it had a prior career in soft-core pornography.

Mr. Cruz, once one of the most broadly popular candidates in the field, has seen his favorability ratings slip as the race has taken a sharply negative turn. A Monmouth University poll of South Carolina Republicans showed Mr. Cruz with a 45 percent favorability rating and a 39 percent unfavorable rating, a steep fall from November figures of 52 and 21.

The latest episode is unlikely to help, even as Mr. Cruz’s team made clear it was moving on. Shortly after the announcement, the campaign circulated a memo arguing Mr. Cruz is “the only candidate who can defeat Donald Trump.”

Bob Vander Plaats, an Iowa evangelical activist and national co-chairman for Mr. Cruz, said the senator was determined to show his campaign operated at “the highest level of integrity,” after weeks of accusations from rivals.

Mr. Vander Plaats said he believed Mr. Tyler’s misstep would not have been a major controversy, if not for recent history.

“At another time, it’s probably not a big deal, but this time it’s a big deal,” Mr. Vander Plaats said, adding, “I love Rick, but it’s something that probably had to be done.”

After posting the video about Mr. Rubio and the Bible on Sunday morning, Mr. Tyler apologized late that night. “I would not knowingly post a false story,” he wrote on Facebook, adding that the initial article, from the University of Pennsylvania’s student newspaper, The Daily Pennsylvanian, had misquoted Mr. Rubio in its video. “But the fact remains that I did post it when I should have checked its accuracy first. I regret the mistake.”

Earlier on Monday, Mr. Rubio said he had accepted Mr. Tyler’s apology, but raised questions about the “accountability” of the Cruz campaign.

Mr. Tyler had been scheduled to be interviewed on MSNBC moments before Mr. Cruz began his news conference about the move, the cable network said. Mr. Tyler left abruptly just before his planned appearance, Kate Snow, the news anchor, wrote in an email to colleagues. Calls to Mr. Tyler were not returned.

Though Mr. Tyler, a former top aide to Newt Gingrich, was not a longtime member of Mr. Cruz’s orbit like some other advisers, he was one of the earliest hires for the campaign and was known for his colorful quotes and aggressive attacks on rivals. He attained a level of cult status in 2011 for a creative news release in praise of Mr. Gingrich, which was later the subject of a dramatic reading by John Lithgow on “The Colbert Report.”

“The literati sent out their minions to do their bidding,” it began, before a flourish on the “billowing smoke and dust of tweets and trivia” from which Mr. Gingrich had emerged unscathed.

But in this campaign, Mr. Tyler had a tendency at times to get ahead of Mr. Cruz. When Sarah Palin was expected to endorse Mr. Trump last month, Mr. Tyler said the move would damage her and her reputation as a conservative champion. Mr. Cruz later said he would not have been a senator without her help.

Until now, though, Mr. Cruz had vigorously defended his team, instead accusing journalists or rival campaigns of seeking to generate controversy at his expense.

On Monday, Mr. Cruz did not entirely abandon the instinct to hit back, suggesting that Mr. Rubio was seeking to deflect attention from his record.

“They have a long record they’ve earned in South Carolina for engaging in this kind of trickery, of impugning the integrity of whoever their opponent is to distract attention,” Mr. Cruz said. “We are going to remain focused on issues.”



Reporting was contributed by Maggie Haberman, Jeremy W. Peters, Jonathan Martin, Alexander Burns and Michael Barbaro.