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To protect the president, that requires the Secret Service to screen hundreds of would-be visitors against preapproved lists.

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But to protect his business, it has also required the Secret Service to defer to Mar-a-Lago staffers and allow in some visitors who are not on the list.

Last weekend, that complex system of lists and exceptions broke down.

When a visitor approached the club, officers found she was not on the approved list — but let her in anyway after a Mar-a-Lago staffer suggested she might be the relative of a club member.

The woman, identified as ­Yujing Zhang, a Chinese national, was later arrested inside the club’s main building. Authorities said she was carrying four ­cellphones, a laptop and a thumb drive with malicious software.

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“I’m surprised that she got in. But then again, I’m not surprised,” said Shannon Donnelly, the longtime society columnist for the Palm Beach Daily News who has covered Mar-a-Lago for years.

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She described a situation in which the Secret Service is dealing with two missions, to keep the president safe and to keep his customers happy.

“It’s bound to happen” that people will slip through, Donnelly said. “There’s hundreds of people coming and going when there’s an event, and half of them are members — they’re not used to being stopped.”

On Wednesday, Trump said he had a brief meeting about the incident but said he was not concerned about potential ­espionage efforts aimed at Mar-a-Lago. He praised the Secret Service as well as the receptionist who first noticed something was amiss with Zhang.

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“We have very good control,” he told reporters at the White House. “The person at the front desk did a very good job, to be honest with you.”

Zhang is in jail, charged with making a false statement to a federal officer and entering a restricted area. On Monday, a federal judge will decide whether she should remain in custody.

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Counterintelligence agents at the FBI are also looking at Zhang to see whether they can find any information that would explain her behavior, according to people familiar with the matter.

On Wednesday, three top Senate Democrats asked FBI Director Christopher A. Wray to investigate whether foreign spies could exploit weaknesses at Mar-a-Lago to steal classified information. Zhang’s arrest “raises very serious questions regarding security vulnerabilities at Mar-a-Lago, which foreign intelligence services have reportedly targeted,” wrote Senate Minority Leader Charles E. Schumer (D-N.Y.); Sen. Dianne Feinstein (Calif.), the ranking Democrat on the Judiciary Committee; and Sen. Mark R. Warner (D-Va.), vice chairman of the Intelligence Committee.

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Bernd Lembcke, Mar-a-Lago’s longtime managing director, did not respond to questions about the club’s security procedures, including whether members are checked to see whether they might be foreign agents. Neither did Trump Organization executives in New York.

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Mar-a-Lago stretches the full width of narrow Palm Beach island, off the coast of South Florida. It features a beach club, a main building with dining and living rooms, two ballrooms, six hotel suites and an attached house where Trump lives.

There is a cap of 500 members. As of last year, joining required sponsorship by an existing member and a payment of $200,000 — an initiation fee that doubled the year Trump took office. The annual dues are about $14,000, according to members.

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Trump has been to the club more than 20 times since he became president, according to a Washington Post tally.

On busy Saturdays in the winter and spring — like this past Saturday, when Zhang got in — there are hundreds of people arriving. Some are members, coming to swim, eat or play tennis.

Others are attending luncheons and galas, holding tickets that cost them hundreds or thousands of dollars. At first, these galas largely drew guests from Palm Beach’s pastel-colored social scene. These days, after a decline in that traditional banquet business, the galas are more commonly aimed at Trump’s fervid political fan base, which extends beyond the clubby island.

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That busy schedule is what Trump wanted for Mar-a-Lago, according to one former senior Trump administration official. Even after he became president, Trump did not want Mar-a-Lago to become a place where visitors became uncomfortable.

So he kept it as it was — and made his aides uncomfortable instead.

“The president has no idea who most of the people around him at the club are,” said another White House official, speaking on the condition of anonymity to describe private conversations. “You pay and you get in.”

When Trump is present, guests say, the first stop for visitors is a security screening in a parking lot across the street from the club.

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There, past visitors said, guests give their names and identification to Secret Service agents or police officers, who check them against a list supplied by the club. The checks are strict: One member said her 11-year-old grandson brings his passport with him when he comes to use the pool.

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But visitors also described instances in which — if a name was not on the list — Mar-a-Lago security personnel would make exceptions if they knew the guest or found another staffer to vouch for them.

“Usually it’s the Mar-a-Lago people that are giving the go-ahead,” said one person familiar with the property who spoke on the condition of anonymity to avoid angering management. “If [the guest is] a familiar face, they would let them in.”

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The Secret Service confirmed as much in its statement about Zhang’s arrest. “The Mar-a-Lago Club’s management determines which members and guests are granted access to the property,” the agency said.

The Secret Service has additional layers of protection around Trump. Agents stand outside the door to his residence, cordon off his table at dinner and surround him if he drops in to weddings or galas in the ballrooms. Guests cannot approach unless Trump waves them over.

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“There’s no more access than they’d have than if he was in a restaurant,” said Ronald Kessler, an author who has known Trump for two decades. Kessler said his wife was yanked back by a Secret Service agent when they approached Trump’s table at Mar-a-Lago two years ago.

But, intelligence officials have said, a foreign spy might find Mar-a-Lago a gold mine — even if the spy never laid eyes on Trump. The club is full of Trump’s friends, aides and hangers-on; it could be bugged, or its computers hacked, if someone could get in the door.

In the case of Zhang, the Chinese woman arrested Saturday, she arrived at the first security checkpoint, in the parking lot across the street, and said she was headed to the club’s pool. She was not on the list. According to charging documents, a Mar-a-Lago staffer still allowed her in because the club “believed her to be the relative” of a club member whose name was also Zhang, prosecutors said.

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Zhang was picked up by a club employee and driven in a golf cart to the main building. There, prosecutors said, a club receptionist stopped Zhang and asked her why she had come to the club.

Zhang said she had come from Shanghai to attend a “United Nations Friendship Event” at the club, at the invitation of a friend named “Charles.” But there was no such event scheduled, according to charging documents. The receptionist called over a Secret Service agent, the documents said, and Zhang then became “verbally aggressive” and was arrested.

Trump was in Palm Beach this past weekend, but at the time of Zhang’s entrance he was out of the club playing golf.

On Wednesday, authorities were still trying to understand her motivations.

One possibility: She really thought she had a ticket to an event at the club.

There is a Chinese entrepreneur named Charles Li, with a group called the United Nations Chinese Friendship Association, who has sold package tours in China that included tickets to galas at Mar-a-Lago, according to reporting by the Miami Herald.

The Washington Post sought to reach Li at the Beijing address listed for the association, but the building’s management said it had no such tenant.

The Post also sent messages through the Chinese social media network WeChat to a number listed for Li. When The Post asked whether this number belonged to Charles Li, the user sent back a photo of Trump doing a thumbs-up.

But then, when The Post asked about Zhang, the account did not respond, and then it blocked The Post from further contact.

Could Zhang have been at Mar-a-Lago as part of a foreign intelligence operation? Former U.S. counterintelligence officials said that was possible — but noted that it appeared to be an unsophisticated effort, lame enough to be foiled by a receptionist.

“I don’t know if it’s a sanctioned activity by the Chinese government, but there’s no doubt it’s some type of potential intelligence operation,” said Robert Anderson, a former senior FBI counterintelligence official. He is now chief executive of Cyber Defense Labs in Dallas.

He also called it “very disturbing” for someone “with that shoddy of a story to get by two or three levels of security” at a facility where the president could be in attendance. “How in the heck does that happen?”