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Demonstrators supporting an immigration overhaul stage a sit-down protest at the offices of Rep. Mario Diaz-Balart, R-Fla., on Oct. 11.

Lynne Sladky/AP

Demonstrators supporting an immigration overhaul stage a sit-down protest at the offices of Rep. Mario Diaz-Balart, R-Fla., on Oct. 11.

Lynne Sladky/AP

After successfully staring down congressional Republicans in the shutdown-debt ceiling fight, President Obama pivoted to immigration in a move with almost no downside.

That makes it perfect as the next vehicle for him to use to cause the GOP major indigestion.

Before being re-elected last year, President Obama said he hoped the Republican “fever” of opposition to him would break during his second term. But if the just-completed standoff is any indication, that temperature is still spiking.

The immediate shift to immigration could be Dr. Obama’s way of trying to effect a cure.

What the White House and immigration advocates working with it hope is that the political loss Republicans suffered in the recent fiscal fight will make GOP leaders desperate to show that the party can govern.

“One can hear the debate within the GOP, which is, ‘Do we continue confrontational tactics that make us look bad? Or do we find a way to pragmatically govern and work more cooperatively with Democrats to do so?’ ” said Frank Sharry, executive director of America’s Voice, an immigration advocacy group.

Even during the shutdown, groups backing the overhaul of the nation’s immigration laws never hit the pause button. They continued to stage demonstrations in Washington and elsewhere.

They were supported by House Democrats who introduced an immigration overhaul bill during a news conference that drew less attention than it might have otherwise because of the approaching fiscal Armageddon.

Sharry explained the strategy: “We just want…, if [House Republicans] decide that it’s in their political self-interest to govern responsibly and, when necessary, work with Democrats to pass legislation, that immigration reform will be first up. So we think we’ve got a shot. The central question is whether we’re dealing with a rational political party or not.”

Obama was certainly acting as though the House Republicans would view it in their self-interest when he said on Thursday:

“The majority of Americans think this is the right thing to do. And it’s sitting there waiting for the House to pass it,” he said, referring to a Senate-passed immigration bill. “Now, if the House has ideas on how to improve the Senate bill, let’s hear them. Let’s start the negotiations. But let’s not leave this problem to keep festering for another year, or two years, or three years. This can and should get done by the end of this year.”

That proposed timetable could be overly optimistic, to put it mildly, given the present mood among Republicans.

True, House Speaker John Boehner has said in the past that he favors fixing the nation’s immigration system but he supports a piecemeal approach, not the Senate’s comprehensive legislation.

In an emailed statement, Michael Steel, a Boehner spokesman, said: “The Speaker remains committed to a step-by-step process to fix our broken immigration system.”

But a senior House GOP aide who asked not to be identified suggested that by turning the screws on Republicans by refusing to negotiate until the shutdown ended and the debt ceiling was raised, Obama made immigration a heavier lift.

“The president’s attitude and actions over the past few weeks have certainly made getting anything done on immigration considerably harder,” the aide said.

Tea Party Republican Rep. Raul Labrador of Idaho echoed that sentiment Wednesday at a monthly Heritage Foundation forum.

Dripping with sarcasm, he told the journalists: “If the president is going to show the same kind of good-faith efforts that he has shown in the last couple of weeks, I think it would be crazy for the House Republican leadership to enter into negotiations with him on immigration.

“He has tried to destroy the Republican Party, and I think that anything that we do right now with this president on immigration will be with that same goal in mind, which is to destroy the Republican Party and not to get good policies.”

Coming from Labrador, that was ominous. He had been in a bipartisan House group working on immigration legislation but dropped out in June because of a disagreement over health insurance for immigrants.

To that Republican response, Obama and Democrats could easily respond fine, in the long run, we win either way.

It would obviously be a major domestic policy accomplishment for Obama’s legacy if he could sign into law a comprehensive immigration overhaul of the sort that has eluded Washington policymakers for the better part of a decade.

He also would be redeeming a campaign promise to Latino voters, an important part of the Democratic Party’s base, by making a major revision to the nation’s immigration system a top second-term priority.

Congressional Democrats could also benefit in the 2014 midterm elections by pushing immigration.

Those benefits to Democrats would be magnified if Republicans wound up in another internecine battle, this time over immigration. And such a fight seems nearly assured, with the establishment wing of the party favoring an immigration overhaul containing a path to citizenship, which congressional Tea Partiers generally oppose.

An ugly, divisive debate within the GOP could make the Republican Party even less attractive than it has been to Latino voters in recent years.

Heading into the midterms, those GOP divisions would put it at an obvious disadvantage versus a mostly unified Democratic Party.

Immigration advocates hope that the desire of GOP leaders to broaden the party’s appeal to minorities and remake its brand can overwhelm the opposition to an immigration overhaul.

Sharry said immigration groups plan to pressure a number of House Republicans who sit in districts where the fastest growing voter group is Latinos. They include Jeff Denham and David Valadao in California, Mike Coffman in Colorado, Joe Heck in Nevada, Mike Grimm on Staten Island, Peter King on Long Island and Daniel Webster in Florida. And they see a path to victory that runs through the same group of lawmakers who voted to end the fiscal stalemate.

The groups also plan actions and civil disobedience to raise the heat.

“If they want to take advantage of the get-out-of-jail card Democrats have offered them, this would be the perfect opportunity to do it,” Sharry said. “We’re going to throw down until they either say ‘yes’ or they make it clear they’re not going to get to yes and then we’ll pivot to try to un-elect them.”