(CNN) California Sen. Kamala Harris has had trouble with the question of whether she would retain a role for private insurance in a "Medicare for All" system.

At a CNN town hall in January, Harris said she supported Sen. Bernie Sanders' Medicare for All bill, and that she would be in favor of eliminating private insurance. She then spent months waffling on the issue, explaining that she would actually like to keep private insurance.

She faced a similar situation in Thursday night's Democratic presidential debate, when NBC's Lester Holt noted that many Americans get their insurance through their employers. He then asked for the 10 candidates onstage to raise their hands if their health care plans would "abolish their private health insurance in favor of a government-run plan?" Harris and Sanders were the only ones who raised their hands.

Harris and her campaign have been quick to clarify that. Harris said she thought the question was whether she'd be in favor of getting rid of her own private insurance plan and has insisted that supplemental private insurance would still be available under Medicare for All.

"I am supportive of a Medicare for All policy, and under a Medicare for All policy, private insurance would certainly exist for supplemental coverage," she said Friday morning on CBS.

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