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UPDATE 1-Oil prices fall on relentless rise in U.S. crude output


* EIA says U.S. shale oil output to rise from last record * U.S. April shale output to be 6.95 mln bpd -EIA * March shale output already expected at record 6.82 mln bpd * Shale pushes overall U.S. output to more than 10 mln bpd * U.S. crude output to break through 11 mln bpd this year -IEA (Adds shale comment, updates prices) SINGAPORE, March 13 (Reuters) - Oil prices fell on Tuesday, extending losses from the previous session, as the inexorable rise in U.S. crude output weighed on markets. U.S. West Texas Intermediate (WTI) crude futures were at $61.25 a barrel at 0414 GMT, down 11 cents, or 0.2 percent, from their previous close. Brent crude futures were at $64.85 per barrel, down 10 cents,