Key lawmakers and aides on Capitol Hill say they don't know how the battles over funding the government and increasing the nation's debt limit might be resolved. In interviews, they lay out several possibilities, all of which face huge political impediments:

BOEHNER YIELDS

House Speaker John Boehner could pass bills to reopen the government and raise the debt ceiling — with few or any concessions by Democrats — if he decided to anger many conservatives in his 232-person caucus and rely heavily on Democrats' votes.

That's what Boehner did to mitigate massive tax increases at the beginning of the year and to give aid to victims of Superstorm Sandy. Most House Republicans opposed both measures.

But if Boehner, R-Ohio, were to enact something as contentious as a debt-ceiling hike with a "minority of the majority," he would face a GOP insurrection that could cost him the speakership. Many Democrats say he should do that.

Rep. Chris Van Hollen, D-Md., said Boehner "is going to have to decide to stand up to a reckless faction of his party for the good of the country. That's just the way this ends."

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BOTH SIDES YIELD A BIT

Democrats conceivably could offer a few concessions that might help Boehner attract a slim majority of his House Republicans. For instance, they could agree to lift a tax on medical devices that helps fund the new health law or approve the Keystone pipeline to carry oil from Canada.

Any such decisions, however, would violate President Barack Obama's repeated vow not to negotiate on the debt ceiling and the government shutdown.

Besides, it's far from clear that such limited compromises would win most House Republicans' support.

Rep. Tom Cole, R-Okla., said Republicans would demand much deeper spending cuts and other concessions before raising the debt ceiling.

"You can't ask those Republicans to just put their political life on the line for nothing," Cole said.

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GRAND BARGAIN

The political gridlock has revived talk of a possible bipartisan "grand bargain" on major budget issues. Republicans would have to agree to higher taxes, which they fiercely oppose. And Democrats would have to swallow cuts in the growth of Social Security and Medicare benefits, which most Democrats are strongly against.

Obama and Boehner failed to reach such an accord in 2011 and again last December. Leaders of both parties say problems that killed those negotiations remain, and it's nearly impossible to resolve them before Oct. 17.

Obama told The Associated Press in an interview Friday he would be willing to negotiate with Republicans on health care, deficit reduction and spending — but only if Boehner allows votes to reopen the government and increase the nation's borrowing limit.

Rep. Mick Mulvaney, R-S.C., laughed out loud when told that Democrats will negotiate on budget matters after the debt ceiling is raised unconditionally.

"If they are judging whether or not they're going to move based upon politics, and we're judging whether or not we're going to move based upon principle, it's going to be very, very difficult to find a way out of this," Mulvaney said.

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OBAMA YEILDS

Many House Republicans predict the president will give ground to avoid a government default. They point to his past concessions, such as agreeing to raise income taxes only on households earning $450,000 or more, rather than the $250,000 cutoff he had campaigned for.

Democrats believe Obama will hold fast. The president "is not going to negotiate the full faith and credit of the United States," said Rep. David Price, D-N.C.

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THE GOVERNMENT DEFAULTS

If both parties stick to their promises, default appears inevitable. The economic impact and public backlash might prompt lawmakers to hastily agree to raise the debt ceiling and start paying all bills again.

Democrats note that lawmakers quickly reversed course five years ago when the Dow dropped 778 points after the House rejected a bank bailout bill.