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WASHINGTON — For all they fail to agree on, Republicans in Congress and President Obama have come to see eye to eye on at least one thing: Four years of relatively little contact is no way to run the country.

So for about 90 minutes Wednesday night, a dozen senators and the president gathered on neutral territory — a private dining room at one of this city’s most elegant hotels — and tried to work out their frustrations over beef and wine.

Aside from the issue of how to handle the check, what was described by all as a convivial dinner raised difficult questions about how effective the new White House campaign to woo Republicans will be and whether Obama, even at his most contrite and conciliatory, can bridge the gap between two parties that remain deeply divided over fundamental questions of policy and the role of government.

Lawmakers in both parties say the president’s efforts may make him a few new friends, but he is not going to change ideologies.

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Others privately complained that convening such a high-profile meeting seemed like an effort to distract from his failure to help forge a solution to avert the automatic budget cuts that went into effect last week.

Next week Obama will take the extraordinary step of traveling to Capitol Hill to hold four meetings with members of Congress — one with Democrats and one with Republicans in each chamber.

The last time he visited the Capitol to meet with the House Republican conference was January 2009; with Senate Republicans it was May 2010, although the president has met with them on occasion since.

And Thursday, Obama hosted a lunch at the White House that included Rep. Paul Ryan of Wisconsin, the House Budget Committee chairman.

One theory that has gained currency in recent years is Washington would be a much more civil and productive place if there were more bipartisan social gatherings like the one Wednesday.

But political scholars say bipartisan camaraderie is no substitute for the most crucial factor in advancing a president’s agenda: large majorities in Congress.

“Stories about dramatic interaction between big personalities make for excellent reading, but they do tend to downplay the structural constraints against which that drama is playing out,” said Andrew Rudalevige, a professor of government at Bowdoin College in Maine. “What does the Congress look like? Who holds seats? And what does public opinion look like?”

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