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While both men are still generally viewed by Virginia voters favorably, Mr. Bolling appears interested in testing whether the high-profile issues Mr. Cuccinelli is best known for might work against him on the campaign trail.

He cautioned that for Republicans to succeed in November and beyond, they cannot fall prey to becoming a party of rigid ideology.

“Governing has to be about more than breaking the dishes,” he said. “It has to be about building consensus and finding solutions to problems. … We have to be viewed as a conservative party whose focus is on results.”