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UPDATE 4-Oil prices drop as fears wane about fallout from Syria strikes


* U.S., France, Britain launched missiles on Syria * Oil prices climbed in run-up to the Syria strikes * Rising U.S. drilling activity drags on oil prices (Updates prices) LONDON, April 16 (Reuters) - Oil eased on Monday after U.S. drilling activity rose and fears waned about escalating tensions in the Middle East following air strikes on Syria over the weekend. The United States, France and Britain launched 105 missiles on Saturday, targeting what they said were three chemical weapons facilities in Syria in retaliation for a suspected poison gas attack on April 7. The oil price had risen nearly 10 percent in the run-up to the strikes, as investors bulked up on assets, such as gold or U.S. Treasuries,