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The elders nodded. They promised to escort U.S. troops through Senjeray, where attackers hidden on rooftops tossed grenades at U.S. patrols nearly every time they passed.

But in the weeks that followed, nobody turned up.

Mr. Qari, the local mullah, said Senjeray’s residents were caught in the middle and could not control the insurgency.

“We told the Taliban we don’t want your support, and we don’t want the support of the U.S. Army,” he said. “We told them: ‘We can ensure our own security. Just leave us alone.’ “