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“What is absolutely the case is the private sector does some things very well and better than the government can, and running effective websites may be one of them, is one of them,” he said, after explaining why many Americans seeking health insurance through Obamacare’s flawed website continue to run into trouble.

Officials also changed the overall look of HealthCare.gov. Now, it features a blue motif with graphics that let people calculate costs, compare plans before they apply or head straight into the application.

But if patients go to the doctor and find out they don’t really have coverage, Mr. Laszewski said, then “nobody’s going to care what tone of blue it is.”

Ben Wolfgang contributed to this report.