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Mr. Liljenquist’s best shot at the nomination would be to capture at least 60 percent of the delegate vote. If the race goes to a primary, Mr. Hatch would benefit from his widespread name recognition and funding advantage.

In Mr. Liljenquist’s corner is his compelling personal story. In 2007, after winning the Republican primary, he was flying to Guatemala on a humanitarian mission when his plane crashed in a farm field, killing 11 of the 14 people onboard.

The experience made him determined to use his life to accomplish as much as possible. “He was vividly reminded that life is short and he did not have time to waste,” according to his campaign website.