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“Right now, House Republicans continue to tie funding of the government to ideological demands like limiting a woman’s access to contraception, or delaying the Affordable Care Act, all to save face after making some impossible promises to the extreme right wing of their party,” Mr. Obama said.

Mr. McHenry said that although Mr. Obama’s job-approval ratings are poor, and he isn’t leading on the budget crisis, tea party conservatives are inadvertently strengthening his position.

“He hasn’t done anything to distinguish himself by trying to lead on this issue,” Mr. McHenry said. “The president seems to have exhausted his ideas for leading the country. But by pushing this shutdown possibility as far as it’s gone, the Republicans have pushed the president to the one talking point he has left — railing against Republicans.”