The Scantron bubbles were gone. So were the page-long passages and the pressure to speed-read them. No. 2 pencils? Optional, and only for taking notes.

On Saturday, students in America took the newest version of the SAT, which was shorter, faster — and most notably, all online.

Some exams were briefly mired by technical glitches, but even so, test takers generally had positive views about the new format. They were especially relieved with the brevity of the exam — which dropped from three hours to a little over two hours — as well as the ability to set their own pace as they worked through the questions.

“It’s here to stay,” said Harvey Joiner, 17, a junior at Maynard H. Jackson High School in Atlanta, referring to the digital format. “Computers are what we’re more comfortable with.”

Given on paper for 98 years, the SAT was updated to reflect the experience of a generation raised in an era of higher anxiety, challenged attention spans and remote learning. The change comes as the College Board, which administers the test, and proponents of standardizing testing say that the exams still have a place in determining college acceptance and aptitude.

Disrupted by the pandemic and rocked by concerns that the tests favor high-income students, the SAT has had a shaky few years, with many colleges removing standardized tests as a requirement for admission. Some selective universities, including Brown, Yale, Dartmouth and M.I.T., have since reinstated the test, but at most schools, it has remained optional.

The current iteration of the test aims to drain some of the intimidation out of the process and evaluate modern students with tools to which they are more accustomed. The test has been trimmed, and students have been given more time for each question. The reading passages are much shorter, and an online graphing calculator is built into the application for the math section, which some see as a way to level the playing field for low-income students.

The tests also are harder to cheat on, with “adaptive” questions that become harder or easier, depending on a student’s performance. Students can bring their own laptops or tablets or use school-issued equipment, but cannot have any other application running in the background, and must take the test at a public test center with a proctor roaming the room.

Many students seemed to welcome this new format on Saturday. Naysa Srivastava, a 17-year-old who took the test in Chicago, found that the brevity of the reading passages and the built-in calculator better reflected her experience as an online learner. “Almost all my classes are digital,” she said.

Elijah McGlory, 18, a senior at Druid Hills High School in Atlanta, said taking the test digitally was “way better” compared with the paper version. “I got more questions done online,” he said.

Sharen Pitts, a retired schoolteacher who has worked for four years as a proctor in and around Chicago, noticed several of her students echoing the sentiment after the test she oversaw on Saturday. But she added that some “preferred paper because digital was harder on the eyes.”

Ms. Pitts said that the main difference she noticed with the new format on Saturday was the shortened test time, which some teachers see as a negative change for students. Critics of the new SAT have said that the shorter exam and reading passages do not help students develop the greater reading stamina they need amid constant distractions from technology.

But the test’s speed was offset by a range of technical issues.

The start of the exam was set back at some test centers, as students had problems connecting to the Wi-Fi. Specifically, test takers at Oak Park River Forest and Georgia State University experienced 30- to 45-minute delays because of connectivity issues.

“It took a little while for everyone to get on the internet,” said Matthew Schmitt, a 16-year-old junior from the Lincoln Park neighborhood of Chicago. “But this is the first time they’re doing the digital SAT.”

On social media, students and parents reported other glitches, including math answers that seemed incorrect and frozen on-screen calculations. In New York, Lida Safa, 15, noticed technical issues such as one student needing a charger at her test center. And she brought her own calculator, just as a backstop in case the online one felt too unfamiliar.

This is not the first time test takers have encountered glitches on digital versions of standardized exams. In recent years, several high school students taking Advanced Placement tests online have had problems with functions like submitting their answers and logging in.

Priscilla Rodriguez, senior vice president of college readiness assessments at the College Board, said that “a vast majority of students” were able to complete the new SAT on Saturday.

“As with paper-and-pencil testing, individual student or test center issues are possible with digital testing,” Ms. Rodriguez said. She added that those who had problems with testing would be able to retake the exam if needed.

And students seemed not too bothered by the snags on Saturday. Naysa, in Chicago, regarded the bugs as an inevitable feature of any new system. And Danny Morrison, 16, who tested in Atlanta, said, “I think as they keep going, they’re going to get more efficient.”

Several also liked a function of the test that puts each student on an automatic timer, rather than leaving stop and start times up to the proctor.

“Before, it was your teacher that had to have all the timing right, and you had to wait for everyone to finish to go on breaks,” said Lora Paliakov, 16, of Atlanta.

Matthew, the 16-year-old in Chicago, noted that “you could work more at your own pace.” This, some found, made the whole testing experience less stressful.

Nerves, however, were another matter. Lida, the 15-year-old in New York who goes to the Razi School, a private Islamic institution, had taken the test on paper in December, and she had a good sense of what to expect. “But I didn’t know about this one,” she said, referring to the new format.

So she leaned on a few home remedies before going into the exam. A light breakfast. A trick she has used to calm her mind — counting her fingers by touching each one to her thumb in order. And a little prayer before opening her MacBook for the test that her math teacher had taught her.

“To be honest? It wasn’t as bad as I thought it would be,” she said. “I feel like I probably did better this time.”

Dana Goldstein contributed reporting.

Conciliatory messages on social media. Open invitations for coffee. Zoom calls inviting attendees to unload.

Even before she announced her campaign for Senate in Arizona, Kari Lake, a Republican and a favorite of former President Donald J. Trump, has been on a mission to make peace. Her failed bid for governor two years ago was defined by her fervent embrace of Mr. Trump’s false claims of a stolen election, and by her relentless attacks on the party’s establishment figures who criticized her for that dishonesty. But now, looking to wrest a seat from Democrats in a key presidential battleground, Ms. Lake is courting former foes and trying to mend fences.

In addition to her public overtures, Ms. Lake has reached out privately to establishment Republicans in the state in recent months — including some she personally offended — seeking their support. The list includes Doug Ducey and Jan Brewer, two of the state’s former governors; Karrin Taylor Robson and former Representative Matt Salmon, two of her 2022 primary rivals; and Meghan McCain, the daughter of the longtime Arizona senator John McCain, according to six people with knowledge of the outreach, some of whom insisted on anonymity to discuss private interactions. In some cases, Ms. Lake has expressed regrets about her past behavior, one of the people said.

“There was some damage done from that primary that clearly bled into the general election,” said Daniel Scarpinato, a Republican consultant in Arizona who worked as a top aide to Mr. Ducey years ago. “I think you clearly see a genuine effort to bring more Republicans into the fold.”

Ms. Lake, a former television anchor and political newcomer in 2022, conducted a scorched-earth campaign to win the G.O.P. primary for governor. She appealed to Mr. Trump’s supporters by championing his baseless theories of election fraud, while lacerating her opponents. She accused Ms. Robson of “trying to buy the election with her 95-yr-old husband’s millions,” and blasted Mr. Ducey as “do-nothing Ducey.”

Perhaps most critically, she angered the family of Mr. McCain, who died in 2018, by declaring that her political rise “drove a stake through the heart of the McCain machine” and by inviting the voters in the state who admired him to “get the hell out.” The divisiveness caused some Republicans to balk at backing Ms. Lake, even if it meant a Democrat would win.

She now says her insults to Mr. McCain were meant “in jest.”

“Things have gotten so much worse under Joe Biden that we’re at a point where we don’t have time to have past arguments getting in the way of us moving forward as a country,” Ms. Lake said in an interview in Phoenix last month. She described herself as someone “who enjoys talking to people and bringing people together.”

If Ms. Lake wins her primary, she can expect a tight race against Representative Ruben Gallego, who is running essentially unopposed for the Democratic nomination. Ms. Lake has a primary opponent, Sheriff Mark Lamb of Pinal County, but she leads him by a wide margin in polls ahead of the contest on July 30. Senator Kyrsten Sinema, a former Democrat who left the party in 2022, is not running for re-election.

Some early signs suggest her effort is bearing fruit, at least on the national level. The National Republican Senatorial Committee, which works to elect Republicans to the Senate, endorsed her, and Senator John Barrasso of Wyoming, the third-ranking member of the Senate Republican leadership, campaigned alongside her in Phoenix last month. She met with Senator Mitch McConnell of Kentucky, the longtime Republican leader, on Capitol Hill on Wednesday, before attending a fund-raiser with about 20 senators in Washington.

Newt Gingrich, the former Republican speaker of the House, said in an interview that he was also endorsing Ms. Lake and that she had “matured” in her approach.

Still, her success appears mixed at home, where some of her overtures have been rebuffed. Ms. Lake sent Ms. McCain, whom she once compared to a “rabid dog,” a public message on X last month inviting her to lunch. Ms. McCain responded: “NO PEACE,” punctuated by a vulgarity.

“These are wounds that are unable to heal for me and my family because people like Kari Lake and Trump continue to debase my family and dad,” Ms. McCain said in an interview. “What she has asked of me is to give her cover for her hideous commentary and her hideous statements about my family, and I would rather die than do that.”

Mr. Ducey is not expected to endorse in the Republican primary, according to a person familiar with his plans. A recent conversation between Ms. Lake and Ms. Robson was productive, according to advisers for the two women, but nothing was decided.

By her own account, Ms. Lake’s efforts to make peace have included meeting with skeptical Republicans at their offices, taking them to coffee or drinks and making calls. In an interview at the Conservative Political Action Conference near Washington last month, Ms. Lake said some of the conversations had been difficult, describing occasionally intense phone and Zoom calls that began with outrage from the other side. She has also held Zoom meetings courting Republican donors and consultants, and attended a lunch with members of the Arizona Chamber of Commerce last month.

“I’m willing to continue to extend olive branches. If somebody does rebuff it and say, ‘No, I’m not interested,’ that’s fine,” she said in the interview in Phoenix. “The olive branch is still out. My door is still open.”

But fresh reservations about Ms. Lake have also cropped up among some grass-roots Republicans, many of whom are vocal supporters of Mr. Trump. Many were dismayed by the publication in January of an audio recording that Ms. Lake secretly captured during a conversation she had last year with Jeff DeWit, the chairman of the Arizona Republican Party, during which Mr. DeWit appeared to offer her a bribe to persuade her not to run for Senate. In the recording, which Mr. DeWit said was “selectively edited,” Ms. Lake can be heard rejecting his request.

Mr. DeWit resigned shortly after the recording surfaced, and Ms. Lake framed the episode as an example of her independence. Some members of the state party reacted with anger, expressing worry about other private conversations Ms. Lake might have recorded. Ms. Lake was greeted with boos at a meeting to elect a new chair.

“Is it really the way we should all be behaving, even as Republicans? To tape-record somebody that trusts you?” asked Jeanne Kentch, the chair of the Mohave County Republican Party. “I love Kari, don’t get me wrong. But I think that’s what people are concerned about.”

Ms. Lake denies that she regularly records private conversations. Still, Mr. Lamb, who trails far behind her in fund-raising, has sensed an opening. Mr. Lamb is the only candidate who “can appeal to all Republicans, conservative independents and disaffected Democrats,” said Ed Morabito, a senior adviser to his campaign.

Despite a newfound desire for party unity, Ms. Lake has not shied away from extreme views, continuing to push baseless theories of voter fraud in news media appearances and telling reporters in Washington last week that “we had a really rigged election in Arizona.” (After losing to Ms. Hobbs, Ms. Lake falsely claimed fraud, filing fruitless lawsuits in attempts to overturn the outcome.) She has also sympathized with and appeared alongside people convicted of crimes related to their participation in the Jan. 6, 2021, attack on the Capitol.

However, she has not made these stances a centerpiece of her Senate campaign — one of a handful of departures from her campaign for governor. On abortion, which she once called the “ultimate sin,” she now opposes a federal ban.

“Kari Lake will say or do anything to gain power,” Hannah Goss, a spokeswoman for Mr. Gallego, said in a statement.

And for some Republicans, the scars that Ms. Lake left may be too deep. Sharon Harper, the chief executive of the real estate firm Plaza Companies who was close friends with Senator McCain, supported Ms. Hobbs in 2022 and has no plans to back Ms. Lake this campaign, either.

“I think people understand who Kari Lake is,” Ms. Harper said. “We’ve seen what she has demonstrated, and I don’t think an opinion changes if someone says, ‘I didn’t really mean what I said.’”

Michael C. Bender and Kayla Guo contributed reporting.

The United States has a history of using its military to get food, water and other humanitarian relief to civilians during wars or natural disasters. The walls of the Pentagon are decorated with photographs of such operations in Haiti, Liberia, Indonesia and countless other countries.

But it is rare for the United States to try to provide such services for people who are being bombed with tacit U.S. support.

President Biden’s decision to order the U.S. military to build a floating pier off the Gaza Strip that would allow aid to be delivered by sea puts American service members in a new phase of their humanitarian aid history. The same military that is sending the weapons and bombs that Israel is using in Gaza is now also sending food and water into the besieged territory.

The floating pier idea came a week after Mr. Biden authorized humanitarian airdrops for Gaza, which relief experts criticized as inadequate. Even the floating pier, aid experts say, will not do enough to alleviate the suffering in the territory, where residents are on the brink of starvation.

Nonetheless, senior Biden officials said, the United States will continue to provide Israel with the munitions it is using in Gaza, while trying to deliver humanitarian aid to Palestinians under bombardment there.

So the Pentagon is doing both.

For decades the Army Corps of Engineers, using combat engineers, has built floating docks for troops to cross rivers, unload supplies and conduct other military operations. Maj. Gen. Patrick S. Ryder, the Pentagon spokesman, said on Friday that the Army’s Seventh Transportation Brigade (Expeditionary), out of Joint Base Langley-Eustis, near Norfolk, Va., would be one of the main military units involved in the construction of the floating pier for Gaza.

The dock will be built and assembled alongside an Army ship off the Gaza coast, General Ryder said. The ship will need armed escorts, particularly as it gets within range of the coast, Defense Department officials said, adding that they are working through how to ensure its protection.

A U.S. Army official said that typically in these operations, a large vessel sits off the shore of the desired location, and a “roll-on-roll-off discharge facility” — a big floating dock — is constructed next to the ship to serve as the holding area. Cargo driven or placed on the dock is loaded onto smaller Navy boats and moved toward a temporary pier or causeway anchored ashore.

The 1,800-foot, two-lane temporary causeway is built by Army engineers, flanked by tugboats and driven, or “stabbed,” into the shore. Cargo aboard the smaller Navy boats can then be driven onto the causeway and onshore.

General Ryder insisted on Friday that the military could build the causeway and stab it into the shore without putting any American boots — or fins — on the ground in Gaza. He said it would take up to 60 days and about 1,000 U.S. troops to move the ship into place from the East Coast and to build the dock and causeway.

After the ship arrives offshore, it will take about seven to 10 days to assemble the floating dock and the causeway, a Defense Department official said.

“This is part of a full-court press by the United States to not only focus on working on opening up and expanding roads via land, which of course are the optimal way to get aid into Gaza, but also by conducting airdrops,” General Ryder said.

The floating pier will allow for the delivery of “upward of two million meals a day,” he said. The Gaza Strip has a population of about 2.3 million people.

General Ryder acknowledged that neither the airdrops nor the floating pier would be as effective as sending aid by land, which Israel has blocked. “We want to see the amount of aid going via land increase significantly,” General Ryder said. “We understand that is the most viable way to get aid in.”

But, he added, “we’re not going to wait around.”

The United States will work with regional partners and European allies to build, fund and maintain the corridor, officials said, noting that the idea for the project originated in Cyprus.

On Thursday, Sigrid Kaag, the U.N. humanitarian and reconstruction coordinator for Gaza, welcomed the Biden announcement. But, speaking with reporters after briefing the Security Council, she added, “At the same time I cannot but repeat: Air and sea is not a substitute for land, and nobody says otherwise.”

The Biden humanitarian efforts in Gaza so far “may make a few people in the United States feel good,” Robert Ford, a former U.S. ambassador to Syria, said in an interview. But, he added, “this is applying a very small Band-Aid to a very big wound.”

The humanitarian aid will probably be gathered in Larnaca, Cyprus, some 210 nautical miles from Gaza, officials said. That would allow Israeli officials to screen the shipments first.

While the temporary port will initially be military-run, Washington envisions it eventually being commercially operated, the official said.

Officials did not go into detail about how aid delivered by sea would be transferred from the coast farther into Gaza. But the assistance will be distributed in part by the Spanish chef José Andrés, founder of the nonprofit World Central Kitchen, which has served more than 32 million meals in Gaza.

Two diplomats briefed on the plans said the port would be erected on Gaza’s shoreline slightly north of the Wadi Gaza crossing, where Israeli forces have erected a major checkpoint.

The central problems, however, remain unsolved. Aid officials say that delivering supplies by truck is far more efficient and less expensive than bringing them to Gazans by boat. But trucks are still unable to deliver goods amid Israeli shelling and ground fighting, which is fierce in southern Gaza.

And delivering assistance by sea may not prevent the chaos that has accompanied deliveries.

More than 100 people in Gaza were killed last month, health officials there said, when hungry civilians rushed at a convoy of aid trucks, leading to a stampede and prompting Israeli soldiers to fire at the crowd.

The U.S. military has airdropped aid in the Middle East and South Asia during previous conflicts, even during wars in which the United States was directly involved.

In 2014, President Barack Obama ordered military aircraft to drop food and water to tens of thousands of Yazidis trapped on a barren mountain range in northwestern Iraq. The Yazidis, members of an ethnic and religious minority, were fleeing militants who were threatening genocide.

In 2001, President George W. Bush ordered British and American troops striking the Taliban in Afghanistan to airdrop daily rations to civilians trapped in remote areas of the country.

President Biden was standing in an Upper East Side townhouse owned by the businessman James Murdoch, the rebellious scion of the media empire, surrounded by liberal New York Democrats who had paid handsomely to come hear optimistic talk about the Biden agenda for the next few years.

It was Oct. 6, 2022, but what they heard instead that evening was a disturbing message that — though Mr. Biden didn’t say so — came straight from highly classified intercepted communications he had recently been briefed about, suggesting that President Vladimir V. Putin’s threats to use a nuclear weapon in Ukraine might be turning into an operational plan.

For the “first time since the Cuban Missile Crisis,” he told the group, as they gathered amid Mr. Murdoch’s art collection, “we have a direct threat of the use of a nuclear weapon if in fact things continue down the path they’ve been going.” The gravity of his tone began to sink in: The president was talking about the prospect of the first wartime use of a nuclear weapon since Hiroshima and Nagasaki.

And not at some vague moment in the future. He meant in the next few weeks.

The intercepts revealed that for the first time since the war in Ukraine had broken out, there were frequent conversations within the Russian military about reaching into the nuclear arsenal. Some were just “various forms of chatter,” one official said. But others involved the units that would be responsible for moving or deploying the weapons. The most alarming of the intercepts revealed that one of the most senior Russian military commanders was explicitly discussing the logistics of detonating a weapon on the battlefield.

Fortunately, Mr. Biden was told in his briefings, there was no evidence of weapons being moved. But soon the C.I.A. was warning that, under a singular scenario in which Ukrainian forces decimated Russian defensive lines and looked as if they might try to retake Crimea — a possibility that seemed imaginable that fall — the likelihood of nuclear use might rise to 50 percent or even higher. That “got everyone’s attention fast,” said an official involved in the discussions.

No one knew how to assess the accuracy of that estimate: the factors that play into decisions to use nuclear weapons, or even to threaten their use, were too abstract, too dependent on human emotion and accident, to measure with precision. But it wasn’t the kind of warning any American president could dismiss.

“It’s the nuclear paradox,” Gen. Mark A. Milley, the chairman of the Joint Chiefs of Staff until he retired in September, told me over dinner last summer at his official quarters above the Potomac River, recalling the warnings he had issued in the Situation Room.

He added: “The more successful the Ukrainians are at ousting the Russian invasion, the more likely Putin is to threaten to use a bomb — or reach for it.”

This account of what happened in those October days — as it happened, just before the 60th anniversary of the Cuban Missile Crisis, the closest the United States and the Soviet Union ever came to a nuclear exchange in the Cold War — was reconstructed in interviews I conducted over the past 18 months with administration officials, diplomats, leaders of NATO nations and military officials who recounted the depth of their fear in those weeks.

Though the crisis passed, and Russia now appears to have gained an upper hand on the battlefield as Ukraine runs low on ammunition, almost all of the officials described those weeks as a glimpse of a terrifying new era in which nuclear weapons were back at the center of superpower competition.

While news that Russia was considering using a nuclear weapon became public at the time, the interviews underscored that the worries at the White House and the Pentagon ran far deeper than were acknowledged then, and that extensive efforts were made to prepare for the possibility. When Mr. Biden mused aloud that evening that “I don’t think there’s any such thing as the ability to easily” make use of “a tactical nuclear weapon and not end up with Armageddon,” he was reflecting urgent preparations being made for a U.S. reaction. Other details of extensive White House planning were published Saturday by Jim Sciutto of CNN.

Mr. Biden said he thought Mr. Putin was capable of pulling the trigger. “We’ve got a guy I know fairly well,” he said of the Russian leader. “He is not joking when he talks about potential use of tactical nuclear weapons or biological or chemical weapons because his military is, you might say, significantly underperforming.”

Since then, the battlefield advantage has changed dramatically, and October 2022 now looks like the high-water mark of Ukraine’s military performance over the past two years. Yet Mr. Putin has now made a new set of nuclear threats, during his equivalent of the State of the Union address in Moscow in late February. He said that any NATO countries that were helping Ukraine strike Russian territory with cruise missiles, or that might consider sending their own troops into battle, “must, in the end, understand” that “all this truly threatens a conflict with the use of nuclear weapons, and therefore the destruction of civilization.”

“We also have weapons that can strike targets on their territory,” Mr. Putin said. “Do they not understand this?”

Mr. Putin was speaking about Russian medium-range weapons that could strike anywhere in Europe, or his intercontinental ballistic missiles that can reach the United States. But the scare in 2022 involved so-called battlefield nukes: tactical weapons small enough to be loaded into an artillery shell and designed to eviscerate a military unit or a few city blocks.

At least initially, their use would look nothing like an all-out nuclear exchange, the great fear of the Cold War. The effects would be horrific but limited to a relatively small geographic area — perhaps detonated over the Black Sea, or blasted into a Ukrainian military base.

Yet the White House concern ran so deep that task forces met to map out a response. Administration officials said that the United States’ countermove would have to be nonnuclear. But they quickly added that there would have to be some kind of dramatic reaction — perhaps even a conventional attack on the units that had launched the nuclear weapons — or they would risk emboldening not only Mr. Putin but every other authoritarian with a nuclear arsenal, large or small.

Yet as was made clear in Mr. Biden’s “Armageddon speech” — as White House officials came to call it — no one knew what kind of nuclear demonstration Mr. Putin had in mind. Some believed that the public warnings Russia was making that Ukraine was preparing to use a giant “dirty bomb,” a weapon that spews radiological waste, was a pretext for a pre-emptive nuclear strike.

The wargaming at the Pentagon and at think tanks around Washington imagined that Mr. Putin’s use of a tactical weapon — perhaps followed by a threat to detonate more — could come in a variety of circumstances. One simulation envisioned a successful Ukrainian counteroffensive that imperiled Mr. Putin’s hold on Crimea. Another involved a demand from Moscow that the West halt all military support for the Ukrainians: no more tanks, no more missiles, no more ammunition. The aim would be to split NATO; in the tabletop simulation I was permitted to observe, the detonation served that purpose.

To forestall nuclear use, in the days around Mr. Biden’s fund-raiser appearance Secretary of State Antony J. Blinken called his Russian counterpart, as did Defense Secretary Lloyd J. Austin III and the national security adviser, Jake Sullivan. Germany’s chancellor, Olaf Scholz, was going on a planned visit to Beijing; he was prepped to brief Xi Jinping, China’s president, about the intelligence and urge him to make both public and private statements to Russia warning that there was no place in the Ukraine conflict for the use of nuclear weapons. Mr. Xi made the public statement; it is unclear what, if anything, he signaled in private.

Mr. Biden, meanwhile, sent a message to Mr. Putin that they had to set up an urgent meeting of emissaries. Mr. Putin sent Sergei Naryshkin, head of the S.V.R., the Russian foreign intelligence service that had pulled off the Solar Winds attack, an ingenious cyberattack that had struck a wide swath of U.S. government departments and corporate America. Mr. Biden chose William J. Burns, the C.I.A. director and former U.S. ambassador to Russia, who is now his go-to troubleshooter for a variety of the toughest national security problems, most recently getting a temporary cease-fire and the release of hostages held by Hamas.

Mr. Burns told me that the two men saw each other on a mid-November day in 2022. But while Mr. Burns arrived to warn what would befall Russia if it used a nuclear weapon, Mr. Naryshkin apparently thought the C.I.A. director had been sent to negotiate an armistice agreement that would end the war. He told Mr. Burns that any such negotiation had to begin with an understanding that Russia would get to keep any land that was currently under its control.

It took some time for Mr. Burns to disabuse Mr. Naryshkin of the idea that the United States was ready to trade away Ukrainian territory for peace. Finally, they turned to the topic Mr. Burns had traveled around the world to discuss: what the United States and its allies were prepared to do to Russia if Mr. Putin made good on his nuclear threats.

“I made it clear,” Mr. Burns later recalled from his seventh-floor office at the C.I.A., that “there would be clear consequences for Russia.” Just how specific Mr. Burns was about the nature of the American response was left murky by American officials. He wanted to be detailed enough to deter a Russian attack, but avoid telegraphing Mr. Biden’s exact reaction.

“Naryshkin swore that he understood and that Putin did not intend to use a nuclear weapon,” Mr. Burns said.

President Biden expressed regret on Saturday for using the word “illegal” to describe an undocumented immigrant who has been charged in the killing of a 22-year-old nursing student in Georgia, agreeing with his progressive critics that it was an inappropriate term.

Mr. Biden used the word during an unscripted colloquy with Republicans during his State of the Union address on Thursday night, and then came under fire from immigration supporters who consider the term dehumanizing. Among those who said he should not have used it were several congressional Democrats.

“I shouldn’t have used ‘illegal’; it’s ‘undocumented,’” Mr. Biden said on Saturday in an interview with Jonathan Capehart on MSNBC, during which he addressed his disagreements with former President Donald J. Trump.

“And look, when I spoke about the difference between Trump and me, one of the things I talked about in the border was his, the way he talks about ‘vermin,’ the way he talks about these people ‘polluting the blood,’ ” he said, adding, “I talked about what I’m not going to do. What I won’t do. I’m not going to treat any, any, any of these people with disrespect.”

He continued: “Look, they built the country. The reason our economy is growing. We have to control the border and more orderly flow, but I don’t share his view at all.”

Mr. Capehart asked if that meant he regretted using the word “illegal.”

“Yes,” Mr. Biden answered.

The president’s reply went further than when he was first asked about the matter by reporters on Friday. He did not explicitly take back the term at that point, noting that the immigrant charged in the murder in Georgia was “technically not supposed to be here.”

The president’s use of the word came on Thursday night when he was pressing Republican leaders to stop blocking a bipartisan agreement to toughen border security. Representative Marjorie Taylor Greene, a Republican from Georgia who enjoys the role of provocateur, shouted at him about the case of Laken Riley, the student who was killed last month by, according to the authorities, a Venezuelan migrant who had entered the country illegally. The case has become a cause célèbre among hard-liners critical of illegal immigration.

“What about Laken Riley? Say her name!” screamed Ms. Greene, who was wearing a T-shirt that read “Say Her Name,” and had been handing out buttons in the chamber with the same slogan.

Mr. Biden interrupted his speech to comply, holding up one of the buttons and saying Ms. Riley’s name, although he mispronounced her given name.

“Lincoln Riley, an innocent young woman who was killed,” Mr. Biden said.

“By an illegal!” Ms. Greene shouted.

“By an illegal, that’s right,” Mr. Biden agreed. “But how many of thousands of people are being killed by legals?” he added in mangled syntax, making the point that crime rates among undocumented immigrants have historically been lower than among others living in the United States.

“To her parents, I say my heart goes out to you,” he went on. “Having lost children myself, I understand.”

He then argued that Republicans could do something about illegal migration by passing the compromise legislation. “Get this bill done,” he told them. “We need to act now.”

Ms. Riley’s mother, Allyson Phillips, was not comforted by the president’s words and expressed indignation that he had mispronounced her daughter’s name.

“Biden does not even KNOW my child’s name,” she wrote on Facebook, adding that it was “pathetic.” She continued: “If you are going to say her name (even when forced to do so) at least say the right name!”

Mr. Trump met with Ms. Riley’s parents before a campaign rally in Rome, Ga., on Saturday, according to a senior campaign adviser, Chris LaCivita. And he seized on Mr. Biden’s comments once he took the stage, where the crowd held up signs with Ms. Riley’s photograph and the words “Say Her Name.”

“They just told me prior to what I’m doing right now that Joe Biden went on television and apologized for calling Laken’s murderer an illegal,” Mr. Trump told the crowd, adding that the immigrant in Georgia “shouldn’t have been in our country, and he never would have been under the Trump policy.”

Michael Gold contributed reporting.

Dr. Ronald V. LaRoche has not been able to cross into dangerous territory to inspect the hospital he runs in Haiti’s Delmas 18 neighborhood since it was ransacked by gangs last week, but a TikTok video he saw offered clues to its current condition: It was on fire.

He learned from neighbors and others who dared venture into gang territory that Jude-Anne Hospital had been looted and cleared of anything of value. It was the second hospital he has had to close.

“They took everything — the operating rooms, the X-rays, everything from the labs and the pharmacies,” Dr. LaRoche said. “Imagine! They are taking windows from hospitals! Doors!”

Haiti is in the throes of an uprising not seen in decades. As politicians around the region scramble to hash out a diplomatic solution to a political crisis that has the prime minister, Ariel Henry, stranded in Puerto Rico and gangs attacking police stations, a humanitarian disaster is quickly escalating. The food supply is threatened, and access to water and health care have been severely curtailed.

André Michel, an adviser to the prime minister, said Mr. Henry has refused to resign, and has demanded that the international community take all necessary measures to ensure his return to Haiti.

The United States and Caribbean leaders have been trying to convince Mr. Henry that to continue in power is “untenable.” An international security mission led by Kenya has been stalled. The United States has offered to finance the mission, but showed little interest in sending troops of its own.

While gangs expand their territory and band together in concerted attacks against the state, millions of people throughout the country are caught in the middle. Many are afraid to leave their homes for fear of getting caught in the crossfire. They are hungry. They are running out of clean water and gas. They are desperate.

“Around me everyone is running,” said Dr. LaRoche, who packed up and closed three more medical facilities to avoid more looting. “Women, children and elderly have bags on their heads, and by foot they are fleeing. It is a war zone.”

Gangs that in the past year have spread throughout the country joined forces last week to attack state institutions, releasing thousands of prisoners. They are demanding the resignation of Mr. Henry, who was prevented from returning to Haiti as violence surrounded the airport and grounded all flights.

The chaos has left people to protect themselves as best they can.

“The biggest fear is stray bullets,” said Nixon Boumba, 42, a Haiti-based consultant to American Jewish World Service, an international aid and human rights organization.

Last weekend he called the motorcycle taxi driver he uses on a regular basis to go shopping. “He told me, ‘I can’t come now. My brother was hit by a stray bullet,’” Mr. Boumba said.

The driver’s brother was struck in the stomach and is recovering at a hospital. The daughter of another friend was hit in the jaw by a bullet on the campus of the city’s main public university, he said.

Blondine Tanis, 36, a radio broadcaster who was kidnapped for ransom in July by people on her street who then sold her to another gang that held her for nine days, said the violence in Haiti was nothing like she had seen before. She compared it to the 1991 coup that led to three years of military rule, but she was a baby then.

“There are young kids in the streets with heavy automatic weapons,” she said. “They shoot people and burn their bodies with no remorse. I don’t know how to qualify that. I ask myself what happened to this generation. Are they even human?”

Ms. Tanis said she has applied to enter the United States through the Biden administration’s humanitarian parole program.

As the security situation worsens, so does the food insecurity. Nearly one million of Haiti’s 11 million people are on the brink of famine, according to the U.N. About 350,000 of them are on the run, living on the streets, in tent cities or in overcrowded schools, as gangs invade their neighborhoods.

Most people now only leave their homes to do essential things, like go to the bank or shop for food and water. They take advantage of a lull in the violence to buy groceries. But experts fear that stocks will soon begin to dwindle as more and more goods pile up on the docks, because transportation by road is too dangerous and gangs have seized ports.

One person described the scene at a supermarket Saturday as a “carnival,” because so many people spent hours in line to stock up on supplies. Zanmi Lasante, a health organization affiliated with Partners in Health, which has worked in Haiti for decades, said it has enough fuel to run its generators for about a week.

Doctors Without Borders had to increase its hospital bed capacity from 50 to 75, as more and more people unable to access the closed public hospital showed up with gunshot wounds. One patient arrived at 3 p.m. for treatment of a gunshot wound from that morning. He died minutes later, said Dr. James Gana, who treats patients and helps run the clinics.

Doctors Without Borders recently reopened an emergency medical clinic in the city center after it had been closed for several months because gang members had removed patients from an ambulance and then killed them in front of the organization’s staff. Blood and oxygen supplies are running low.

“We are going very soon to have shortages of everything,” said Jean-Marc Biquet, head of mission for Doctors Without Borders in Haiti. “There is no more petrol in the petrol stations. People are selling fuel in small buckets, and nobody knows where that fuel is coming from.”

With no supply of clean drinking water, there is an increased risk of cholera, he said.

Mario Delatour, 68, a filmmaker, said he has not found bottled water in three days. A generous neighbor with a water-treatment system filled a 5-gallon bottle for him on Saturday, but he still needs gas for the generator that powers his home. His neighborhood, a relative safe haven, has not had electricity in three months.

“I have enough fuel for tonight, but I don’t know about tomorrow,” Mr. Delatour said. “I’m a little bit on edge. It’s a hell of a thing, man.”

Julio Loiseau, a community activist in Port-au-Prince, said that with the power out, groceries spoil quickly, when you can find them.

“To have bread, one needs to get in line very early in the morning,” he said. “The only bread factory cannot cover its demands because of supply scarcity. My supplies ran out.”

Jean-Martin Bauer, country director in Haiti for the U.N. World Food Program, noted that the financial situation for many people is especially precarious because it has been too dangerous for people to go outside to work, and many people make their money on a day-to-day basis.

“What’s going on in Haiti is a protracted episode of mass hunger,” Mr. Bauer said. “This is probably one of the causes of what’s going on. We know hunger is related to instability and is a breeding ground for conflict, a breeding ground for strife and mass migration.”

Frantz Louis, 35, a security guard who was waiting for his shift on Saturday, said that like many Haitians, he feels Haiti has “completely collapsed.”

“The best solution for a young person for now is to leave the country,” he said. “If you want to stay in your country and you can’t eat and you can’t go where you want, what other choice do you have?”

Mr. Louis said he wondered what the gangs’ end game is. “Do they have an ideology?” he asked.

Robert, a 41-year-old furniture maker in Port-au-Prince, who did not want his name published for fear of reprisals, said he had been forced to sell his furniture for less than what it cost him to build.

“Sometimes you buy rice and you no longer have money to buy vegetable oil and spices, and that’s what happened to me last week,” Robert said, from his outdoor workshop. “Now the rice is finished, and I have to find another piece of furniture to sell at a low price — and also I need a customer.”

Robert has a wife and two children, a 7-year-old boy and 15-year-old girl. He avoids even looking at the large wardrobe he built in December that he has not been able to sell.

“The day I no longer have furniture to sell,” he said, “it will be hunger.”

Several fans of the Kansas City Chiefs who attended a playoff game on a bitterly cold January day in Missouri suffered frostbite that required amputations, according to the hospital that treated them.

Twelve people — including some football fans who were at Arrowhead Stadium on Jan. 13 — had to undergo amputations involving mostly fingers and toes, the hospital, Research Medical Center in Kansas City, said in a statement on Saturday.

The center said it treated dozens of patients who experienced frostbite during an 11-day cold snap. Not all of the patients who had amputations attended the Chiefs game. Some were people who worked outdoors in the extreme cold, the hospital said.

The exact number of fans who attended the game who had amputations was unclear. The hospital said there was some overlap among the fans and those who had also worked outdoors.

The hospital also noted that symptoms of frostbite can develop slowly, and that many of the frostbite patients it treated could not identify when their injuries occurred — when their pain, numbness and other symptoms began.

The hospital said it was a record number of frostbite patients since the burn center opened 11 years ago.

The National Weather Service had warned of dangerous temperatures that week, starting Jan. 6, with Arctic air spilling onto the Plains.

“Our specialized physicians and expert care team continue to treat and monitor patients’ healing to address long-term needs, and we expect more surgical procedures over the next two to four weeks as their injuries evolve,” the hospital said.

At kickoff of the game between the Kansas City Chiefs and the Miami Dolphins, temperatures hovered around minus 4 degrees, with a windchill of minus 26 degrees.

The helmet of Chiefs quarterback Patrick Mahomes cracked open during a tackle, a malfunction that the helmet’s manufacturer said was caused by the extreme cold.

Dr. Megan Garcia, the medical director of the hospital’s Grossman Burn Center, said in an interview with WDAF-TV that the Chiefs fans who came in with frostbite injuries had to schedule amputation surgeries after weeks of hospital treatment.

The treatment included rewarming the injured areas, applying antibiotics and thrombolytic therapy to dissolve blood clots and restore circulation, and hyperbaric oxygen therapy to boost oxygen to injured areas to reduce swelling.

Patients with frostbite experience “lifelong sensitivity and pain,” Dr. Garcia said, “and will always be more susceptible to frostbite in the future.”

During the cold snap in January, the medical center’s parent company posted information about frostbite on its website, warning that it can happen within minutes of skin exposure in freezing air, and in less time with wind chill.

People who work outside during the winter were especially vulnerable, the hospital said in its statement, as were people “attending football games, the elderly, pregnant women, and kids waiting at the bus stops to return to school.”

Frostbite happens in “extremely cold temperatures,” according to the Centers for Disease Control and Prevention, with injury often happening during the thawing process as vessels become damaged by clots and inflammation, strangling blood flow.

Although frostbite can happen anywhere on the body, it typically affects exposed areas like the nose, ears, cheeks, chin, fingers and toes.

Julie Loving, a physician assistant in the emergency department at Adirondack Medical Center in Saranac Lake, N.Y., said the hospital treats three to five patients for frostbite every winter.

After administering medication to expand blood vessels and generate new tissue, patients undergo a bone scan, she said.

“Sometimes it can take days, sometimes weeks, to make a decision that someone needs an amputation,” she said. “When someone presents at the E.R. that first day, there’s no way to predict.”

Instead, she added, members of the medical staff monitor how the tissue evolves. If the tissue does not regenerate, it becomes infected, and that is when amputation is necessary, she said.

Prolonged exposure to cold weather also puts people at risk of hypothermia, a sudden drop of body temperature, and lung diseases, such as pneumonia.

A representative for the Kansas City Chiefs did not immediately respond to a request for comment on Saturday.

Cold weather is often a feature at N.F.L. games, where fans bundle up but sometimes strip down, going shirtless to stand out in a crowd.

The coldest game recorded in N.F.L. history was the Ice Bowl of 1967, when the Green Bay Packers defeated the Dallas Cowboys in a New Year’s Eve game. Temperatures in Wisconsin were at minus 13 degrees at kickoff.