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A number of people have been credited with the political rise of Donald Trump, including Roger Stone and Steve Bannon. But perhaps nobody has been more influential than Mark Burnett, the English reality-TV producer. After the massive success of his show “Survivor,” Burnett could have made virtually anything, and he chose “The Apprentice.” His task was to transform a New York real-estate developer who was a fixture in the tabloids into a national celebrity, a tycoon, and a decisive leader with unerring judgment. The New Yorker staff writer Patrick Radden Keefe interviewed a number of people who worked on shaping Trump’s image on “The Apprentice,” including the show’s supervising producer, Jonathon Braun. Braun told Keefe that Trump’s quick, instinctual decisions complicated the work of the show’s editors, who would often have to recut the episodes to find material that seemed to justify those decisions. And Braun argues that the White House and the news media now often play the same role that the “Apprentice” crew did: isolating Trump’s most coherent statement within a long string of improvised iterations.