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UPDATE 1- In shift, South Korea and U.S. to push forward on trade talks

(Adds background on trade pact, timeline for next talks)

WASHINGTON, Oct 4 (Reuters) - South Korea indicated on Wednesday it was open to talks on revising a 2012 trade pact with the United States after initial differences that followed President Donald Trump's threat to terminate the accord unless it was renegotiated.

After a day of talks in Washington, South Korea's trade ministry said in a statement, "The two sides recognized the need to amend the FTA to enhance mutual benefits of the KORUS FTA," as the pact is called.

The U.S. trade representative, Robert Lighthizer, said the United States looked forward to stepped-up talks "to resolve outstanding implementation issues as well as to engage soon on amendments that will lead to fair, reciprocal trade."

The statements mark a shift from an initial meeting in August, when the two sides failed to agree on next steps after Lighthizer had made demands to amend the agreement to reduce the U.S. trade deficit with South Korea.

Since the trade agreement went into effect in 2012, the U.S. goods trade deficit with South Korea more than doubled to $27.6 billion last year. But through July 2017, the bilateral trade deficit fell to $13.1 billion from $18.8 billion during the same period of 2016, according to U.S. Census Bureau data.

No date was given for a third round of talks between the two countries, which comes as Lighthizer is also focused on revamping the North American Free Trade Agreement with Mexico and Canada. (Reporting by Lesley Wroughton; Additional reporting by Jane Chung in Seoul and David Lawder in Washington; Editing by Jonathan Oatis and Leslie Adler)