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UPDATE 5-Oil steady, tempered by caution over rising U.S. output

* U.S. Dec shale output to rise for 12th straight month - govt

* Fitch says prices above $60/barrel "may not be sustained" (Updates with comment, refreshes prices; changes dateline from SINGAPORE)

LONDON, Nov 14 (Reuters) - Oil prices held largely steady on Tuesday as the prospect of further rises in U.S. output offset some of the optimism that OPEC-led production cuts would tighten the balance between crude supply and demand.

Brent crude futures were at $63.11 per barrel at 0942 GMT, down 5 cents, while U.S. West Texas Intermediate (WTI) crude was down 13 cents at $56.63.

Both benchmarks early in the previous week hit highs last seen in 2015, but traders said the market had lost some momentum since then.

Traders said they were cautious about betting on further price rises.

"Prices ... are starting to look like a pause or pullback is needed," said Greg McKenna, chief market strategist at futures brokerage AxiTrader.

This sentiment comes in part on the back of rising U.S. oil output <C-OUT-T-EIA>, which has grown by more than 14 percent since mid-2016 to a record 9.62 million barrels per day (bpd).

The U.S. government said on Monday U.S. shale production in December would rise for a 12th consecutive month, increasing by 80,000 bpd.

A cooling Chinese economy also stoked some concerns about demand, although so far the country's refiners are processing crude oil near record levels of 11.89 million bpd.

Fitch Ratings said in its 2018 oil outlook that it assumed 2018 "average oil prices will be broadly unchanged year-on-year and that the recent price recovery with Brent exceeding $60 per barrel may not be sustained".

So far in 2017, Brent has averaged $54.5 per barrel.

Despite the cautious sentiment, traders said oil prices were unlikely to fall far, largely due to supply restrictions led by the Organization of the Petroleum Exporting Countries and Russia, which have helped reduce excess stockpiles.

The International Energy Agency on Tuesday delivered a more cautious outlook for oil demand.

In a monthly report, the Paris-based agency cut its oil demand forecast by 100,000 bpd for this year and next, to an estimated 1.5 million bpd in 2017 and 1.3 million bpd in 2018.

The IEA said warmer temperatures could cut consumption, while sharply rising production from outside OPEC might mean the global market tilts back into surplus in the first half of 2018.

"You cannot have the same forecast at $60 as you have at $40. You need to address that and the IEA is starting to make that adjustment," Petromatrix strategist Olivier Jakob said. (Additional reporting by Henning Gloystein in SINGAPORE; Editing by Dale Hudson)