Following a Friday meeting with congressional leaders, an impatient and annoyed President Barack Obama said it was "mind boggling" that Congress has been unable to fix a "fiscal cliff" mess that everyone has known about for more than a year.

He then dispatched Senate Majority Leader Harry Reid, a Democrat, and Minority Leader Mitch McConnell, a Republican, on a mind-boggling mission: coming up with a bipartisan bill to break the "fiscal cliff" stalemate in the most partisan and gridlocked US Congress of modern times - in about 48 hours.

Reid and McConnell, veteran tacticians known for their own long-running feud, have been down this road before.

Their last joint venture didn't turn out so well. It was the deal in August 2011 to avoid a U.S. default that set the stage for the current mess. That effort, like this one, stemmed from a grand deficit-reduction scheme that turned into a bust.

But they have never had the odds so stacked against them as they try to avert the "fiscal cliff" - sweeping tax increases set to begin on Tuesday and deep, automatic government spending cuts set to start on Wednesday, combined worth $600 billion.

The substantive differences are only part of the challenge. Other obstacles include concerns about who gets blamed for what and the legacy of distrust among members of Congress.

Any successful deal will require face-saving measures for Republicans and Democrats alike.

"Ordinary folks, they do their jobs, they meet deadlines, they sit down and they discuss things, and then things happen," Obama told reporters. "If there are disagreements, they sort though the disagreements. The notion that our elected leadership can't do the same thing is mind-boggling to them."

The core disagreement between Republicans and Democrats is tough enough. It revolves around the low tax rates first put in place under Republican former President George W. Bush that expire at year's end. Republicans would extend them for everyone. Democrats would extend them for everyone except the wealthiest taxpayers.

The first step for Reid and McConnell may be to find a formula acceptable to their own parties in the Senate.

While members of the Senate, more than members of the House of Representatives, have expressed flexibility on taxes, it's far from a sure thing in a body that ordinarily requires not just a majority of the 100-member Senate to pass a bill, but a super-majority of 60 members.

With 51 Democrats, two independents who vote with the Democrats and 47 Republicans, McConnell and Reid may have to agree to suspend the 60-vote rule.

Getting a bill through the Republican-controlled House may be much tougher. The conservative wing of the House, composed of many lawmakers aligned with the Tea Party movement who fear being targeted by anti-tax activists in primary elections in 2014, has shown it will not vote for a bill that raises taxes on anyone, even if it means defying Republican House Speaker John Boehner.

Many Democrats are wedded to the opposite view - and have vowed not to support continuing the Bush-era tax rates for people earning more than $250,000 a year.

Some senators are wary of the procedural conditions House Republicans are demanding. Boehner is insisting the Senate start its work with a bill already passed by the House months ago that would continue all Bush-era tax cuts for another year. The Democratic-controlled Senate may amend the Republican bill, he says, but it must be the House bill.

For Boehner, it's the regular order when considering revenue measures, which the U.S. Constitution says must originate in the House.

As some Democrats see it, it's a way to shift blame if the enterprise goes down in flames. House Republicans would be able to claim that since they had already done their part by passing a bill, the Senate should take the blame for plunging the nation off the "cliff."

And that could bring public wrath, currently centered mostly on Republicans, onto the heads of Democrats.

Voters may indeed be looking for someone to blame if they see their paychecks shrink as taxes rise or their retirement savings dwindle as a result of a plunge in global markets.

If Reid and McConnell succeed, there could be political ramifications for each side. For example, a deal containing any income tax hikes could complicate McConnell's own 2014 re-election effort in which small-government, anti-tax Tea Party activists are threatening to mount a challenge.

If Obama and his fellow Democrats are perceived as giving in too much, it could embolden Republicans to mount challenge after challenge, possibly handcuffing the president before his second term even gets off the ground.

It could be a sprint to the finish. One Democratic aide expected "negotiation for a day." If the aide is correct, the world would know by late on Saturday or early on Sunday if Washington's political dysfunction is about to reach a new, possibly devastating, low.

If Reid and McConnell reach a deal, it would then be up to the full Senate and House to vote, possibly as early as Sunday.

Reid and McConnell have been through bitter fights before. The deficit reduction and debt limit deal that finally was secured last year was a brawl that ended only when the two leaders agreed to a complicated plan that secured about $1 trillion in savings, but really postponed until later a more meaningful plan to restore the country's fiscal health.

That effort led to the automatic spending cuts that form part of the "fiscal cliff."

Just months later, in December 2011, Reid and McConnell were going through a tough fight over extending a payroll tax cut.

In both instances, it was resistance from conservative House Republicans that complicated efforts, just as is the case now with the "fiscal cliff."