It noted that most undocumented immigrants who arrived in the United States in the past decade came not over the border but on visas that then expired.

So what should be done to address visa overstays?

“I don’t know,” O’Rourke said, pausing in a lengthy interview.

O’Rourke, who represented a border district in the House for six years, talked through the issue and came up with a possible solution: The United States could harmonize its visa system with Mexico’s to keep better track of who is coming into the country and leaving it.

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“That’s an answer,” he said, “but that’s something that we should be debating.”

When it comes to many of the biggest policy issues facing the country today, O’Rourke’s default stance is to call for a debate — even on issues related to the border and immigration, which he has heavily emphasized in videos posted to Facebook and Instagram over the past month.

O’Rourke’s approach reflects how he is likely to handle issues should he launch a presidential campaign. Beyond a few mainstream Democratic stances — including closing private immigration prisons, allowing undocumented immigrants to become citizens and modernizing the work visa system — O’Rourke insists the thorny immigration answers will come from everyday Americans. It’s an approach that puts off specifics that might define him or narrow his appeal in a presidential race — but O’Rourke says he is being open-minded, as he wishes more politicians would be.

“That’s a problem when you’re like, ‘It will be a wall,’ or ‘It will be this,’ or ‘We can only do it with this,’ ” O’Rourke said when asked why he doesn’t have firm stances. “The genius is we can nonviolently resolve our differences, though I won’t get to my version of perfect or I, working with you, will get to something better than what we have today. . . . It’s rare that someone’s ever been able to impose their will unilaterally in this country. We don’t want that.”

He insists that once Americans are informed about “the facts and the story and the information and the opportunity,” they will come to the right conclusions about what to do about an issue that has divided the country for decades.

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“I trust the wisdom of people. And I’m confident — especially after having traveled Texas for two years — people are good, fundamentally, and if given the choice to do the right thing, they will. To do the good thing, they will,” he said, referring to his unsuccessful U.S. Senate campaign while giving a walking tour of El Paso and its Mexican sister city, Ciudad Juarez.

On other issues, his approach was similar.

When asked whether he agrees with Trump’s plan to quickly withdraw troops from Syria, O’Rourke said he would like to see “a debate, a discussion, a national conversation about why we’re there, why we fight, why we sacrifice the lives of American service members, why we’re willing to take the lives of others” in all the countries where the United States is involved.

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“There may be a very good reason to do it. I don’t necessarily understand — and I’ve been a member of Congress for six years,” O’Rourke said. “We haven’t had a meaningful discussion about these wars since 2003.”

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Asked about the “Green New Deal” being crafted by Democrats to dramatically curb emissions and heavily invest in clean-energy jobs and infrastructure, he praised it as a “bold” start that avoided “wishy-washy change.”

The details are apt to change, he said, adding, “But, thank God, the work has been done to articulate the goal, the vision, the means to achieve it, and that’s a perfect point from which to start a conversation.”

As O’Rourke’s decision on a presidential campaign nears, immigration is the issue in which he has chosen to invest his time — putting him directly at odds with Trump, against whom the next Democratic nominee will compete.

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For all his current focus on the border, O’Rourke played a negligible role in shaping immigration policy during his six years in Congress, which ended this month. Even now, he rarely uses his expanding national platform to call for specific legislation or transformative changes in the immigration system.

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He said he believes that the border is already fully secured and that further investment would take it further “past the point of diminishing returns,” pushing migrants seeking to cross the border illegally into more dangerous and desolate territory.

“You will ensure death,” he said of Trump’s proposed wall. “You and I, as Americans, have caused the deaths of others through these walls.”

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Just as Trump has used the heart-wrenching stories of Americans killed by undocumented immigrants to build support for his wall, O’Rourke leans on a narrative of migrants and those living along the border. In his unsuccessful Senate race last year, O’Rourke frequently compared Central American migrants fleeing violence and poverty to Jewish refugees fleeing the Nazis and being refused entry into many countries, including the United States.

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Luis V. Gutiérrez, the former Democratic congressman from Illinois who spearheaded immigration measures in the House for many years, said he was “very pleasantly surprised” to see O’Rourke suddenly interested in immigration last year. Even though O’Rourke represented a majority Hispanic district along the border, he was not deeply involved with immigration reform, Gutierrez said. But he praised O’Rourke for his recent efforts to demystify the border and bring attention to immigration issues.

“A lot of people want to talk about where people start,” he said, “and I like to talk about where people are at.”

The last major attempt at a sweeping immigration package came just after O’Rourke took office. In June 2013, the Senate passed a bipartisan immigration bill that would have allowed millions of undocumented immigrants to legally stay in the United States and eventually become citizens. It also would have doubled the number of U.S. Border Patrol agents along the southern border and authorized 700 miles of fencing.

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O’Rourke said at the time that he supported “a pathway to citizenship for immigrants that pay their taxes, obey our laws and learn English,” but he opposed efforts to “militarize our border against a threat that does not exist.”

Then-House Speaker John A. Boehner (R-Ohio) refused to take up the Senate’s immigration bill, at which point, O’Rourke said, the issue died.

“We could have these discussions in caucus meetings, but it’s like spitting in the wind if [Republican lawmakers] are not going to actually engage in the conversation,” O’Rourke said. “There was a huge missed opportunity, which created the opportunity — in some ways — for Trump.”

O’Rourke is trying to undo Trump’s image of the border by showing Americans what he sees.

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In the past month, he has introduced his social media followers to migrant families just released from detention centers, broadcast a rally held outside a tent camp that once housed thousands of detained migrant children, and showed the numbers written on the wrists of Guatemalan migrants waiting their turn to claim asylum in the United States. He has taken his viewers along on a late-night walk through his historic El Paso neighborhood and a Saturday night trip to Juarez for dinner with his family.

He has interviewed his neighbors — and, rather famously, his dental hygienist during a cleaning — about life on the border, reinforcing their feeling of safety in a zone the president has condemned as crime-ridden.

After Trump spoke to the nation about his demand for border wall funding in exchange for reopening the government, O’Rourke aired, to thousands, a conversation with two close friends discussing the president’s messaging.

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“He has seized this emotional language very effectively — completely irresponsibly, not tethered to the truth,” O’Rourke said. “But if I don’t live in El Paso, if I haven’t had the experience that we have, if I live in Michigan, Iowa, Oregon, the northern border, I may not know any better. . . . The president of the United States just said that there are rapists and criminals and murderers who will chop your head off coming to get us. . . . And so I can see responding that way.”

Throughout the two-hour interview — which was often interrupted by bystanders urging him to run for president — O’Rourke boomeranged between a bright-eyed hope that the United States will soon dramatically change its approach to a whole host of issues and a dismal suspicion that the country is incapable of implementing sweeping change.

When asked which it is, O’Rourke paused.

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“I’m hesitant to answer it because I really feel like it deserves its due, and I don’t want to give you a — actually, just selfishly, I don’t want a sound bite of it reported, but, yeah, I think that’s the question of the moment: Does this still work?” O’Rourke said. “Can an empire like ours with military presence in over 170 countries around the globe, with trading relationships . . . and security agreements in every continent, can it still be managed by the same principles that were set down 230-plus years ago?”