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updated 1:19 PM EST, Sun December 28, 2014

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Washington (CNN) -- If Republicans don't wield their congressional majority next year to pass immigration reform legislation, a GOP takeover of the White House in 2016 will be "difficult, if not impossible," Sen. Lindsey Graham of South Carolina said during a CNN interview released on Sunday.

Graham, a Republican who has long-favored comprehensive immigration reform, said he believes the GOP has hurt itself with Hispanic voters due to its resistance to reforming the current system. And without a major change, Democrats will get another four years in the Oval Office, Graham told CNN's Dana Bash on "State of the Union."

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"If we don't at least make a down payment on solving the problem and rationally dealing with the 11 million [illegal immigrants believed to be in the U.S.], if we become the party of self-deportation in 2015 and 2016, then the chance of winning the White House I think is almost non-existent," he said.

But Graham, who is mulling a 2016 presidential run, slammed President Barack Obama's executive action last month to safeguard as many as 5 million undocumented immigrants. He accused the President of "acting in a rogue fashion" for "political reasons."

Republicans, though, need to "do more than just fight the executive order," he added.

Graham supports giving a pathway to citizenship to the so-called DREAMers -- undocumented immigrants who crossed into the U.S. illegally as children and have lived in the U.S. since. And in 2014, Graham showed the political viability of his position by successfully beating back a tough primary challenge in his conservative state by reaffirming -- rather than running away from -- his stance on immigration.

"If the Republican Party cannot muster the political courage to deal with the DREAM Act children in a fair and balanced way after we secure our border, that says a a lot about the Republican Party's future regarding the Hispanic community," Graham said. "I don't believe most Americans would fault the Republican Party if we allowed children who have been here since they're babies to assimilate into society with a pathway to citizenship after we secure our borders."

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Graham: 2016 hinges on immigration