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UPDATE 3-Oil slips further below $57 after third-quarter rally

* Brent posted biggest third-quarter increase since 2004

* U.S. drillers add oil rigs for first week since mid-August

* OPEC oil output rises in September - Reuters survey

* Libya's Sharara oil field closed since late Sunday-sources (Previous TOKYO, updates prices, adds quotes)

LONDON, Oct 2 (Reuters) - Oil dipped further below $57 a barrel on Monday as an increase in U.S. drilling and higher OPEC output put the brakes on a rally in which the market scored its biggest third-quarter gain in 13 years.

U.S. energy companies added oil rigs for the first week in seven and Iraq announced its exports increased slightly in September when OPEC overall boosted output according to a Reuters survey.

Brent crude, the global benchmark, was down 12 cents at $56.67 a barrel at 0846 GMT. It notched up a third-quarter gain of around 20 percent, the biggest third-quarter increase since 2004 and traded as high as $59.49 last week.

"I think it's going to be a struggle to move above $60 Brent," said Olivier Jakob, oil analyst at Petromatrix.

U.S. crude was down 17 cents at $51.50. The U.S. benchmark posted its strongest quarterly gain since the second quarter of 2016.

Oil's rally has been driven by mounting signs that a three-year supply glut is easing, helped by a production cut deal by global producers led by the Organization of the Petroleum Exporting Countries.

"Brent crude oil prices have gone from strength to strength as surplus oil stocks are being depleted," Bank of America Merrill Lynch said in a report. "Importantly, this rally is supported by a tighter physical market, providing a fundamental backbone that was not present before."

But a Reuters survey on Friday found OPEC oil output rose last month, gaining mostly because of higher supplies from Iraq and also from Libya, an OPEC member exempt from cutting output.

The Libyan gain appears short-lived, though. The country's largest oilfield, Sharara, has been closed since Sunday, an engineer at the field and a Libyan oil source said.

And in a sign U.S. oil output could rebound, energy services firm Baker Hughes said on Friday energy companies added oil rigs for the first week in seven after a 14-month drilling recovery stalled in August.

Middle Eastern oil producers are concerned the price rise will only stir U.S. shale producers into more drilling and push prices lower again.

(Additional reporting by Aaron Sheldrick; Editing by Louise Heavens)