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The former Florida governor wrote that Trump, who beat out more than a dozen established Republicans in the party's primary, "successfully tapped into the anger and deep distrust that voters feel toward Washington."
"These voters believe the American dream is increasingly out of reach," Bush wrote. "They believe our system is skewed in favor of the powerful and the connected. They believe that the politicians elected to right these wrongs have only made them worse. And they overwhelmingly voted for Mr. Trump, giving him a hard-won victory."
Bush noted that this election cycle, people seemed to be voting against something - including the Washington establishment - more than they were voting for something.
"Republicans need to do more than oppose things," he wrote. "We have to be for a few big ideas and show that we can put them into action."
Bush advocated for sending power "back to the people and back to the states" rather than concentrating it in "broken" Washington. He wrote that the federal government has become "too unwieldy, too powerful and too distant."
And he spelled out a way forward for Republicans to create a sustainable party that has a broader appeal.
"This is our chance to contrast our approach, which trusts people to make good choices, with the failed Democratic and progressive top-down approach, which prefers government to decide almost everything," he wrote.
He also warned against "stooping to the identity
"Let's not focus on angst, grievance and division over race, class or gender. Our party must be big-hearted and creative and opportunistic," he wrote. "We must make it clear that there is no room in our tent for despicable bigotries like racism, misogyny or anti-Semitism."
Bush and Trump sparred often during the Republican primaries. Early on, Bush was the favorite to win the Republican presidential nomination, but Trump quickly surged in the polls and defeated Bush handily in state contests.
Trump also mocked Bush often, calling him "low energy." Bush, in turn, endorsed Sen. Ted Cruz for the Republican nomination.
But Bush closed out his Friday editorial with good wishes for Trump.
"I want him to know that I hope for his success," Bush wrote. "I hope that he broadens the GOP, works across the aisle and governs with pragmatism and compassion. There are many who did not vote for him who agree that what matters most now is that this nation unites and moves forward together. I will work to support those goals."