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Andrei Berezhkov is home in Moscow, where he still denies that he had written to the New York Times and President Reagan asking to be allowed to stay in the US. ''It was all a big misunderstanding,'' he said in excellent English to the crowd of Western journalists at Moscow's Sheremetyevo Airport. State Department officials had demanded to interview Andrei about the letters, but the Soviet Embassy refused. Eventually a solution was found whereby State Department officials were present but silent while three journalists interviewed him at Washington's Dulles Airport before he left for Moscow.