The Republican rout was pretty decisive in Tuesday's midterm elections, and opinion polls show that President Obama has an approval rating about as low as his immediate predecessor.

Yet the president, in remarks Wednesday at the White House about the shellacking by Republicans, vowed to reach across the aisle and get both parties in Congress to work together.

"The American people are watching to see if we are serious about being able to compromise, and anywhere we can find common ground, I am eager to pursue it," the president said.

Going forward, however, Obama can either retreat into a lame duck cocoon, policy experts say, or he can work to forward his administration's agenda. His quiver will contain relatively fewer arrows, though. Chiefly those include his power to veto, issue executive orders, and formulate compromises with Republicans. And political observers are expecting a good fight from the president going forward.

"The president is young and energetic, and he is not going to sit out his last two years," says Dean Baker, co-director of the Center for Economic and Policy Research, a policy think tank.

Another Government Shutdown

In fact, Obama is going to have to act pretty fast in his newly diminished role. One of the biggest issues likely to set off partisan triggers is what will happen with the federal budget. A temporary budget extension, passed by a House that wanted to avoid a showdown during the election cycle, expires next month, which raises the specter of yet another government closure.

"Congress has a responsibility to pass a budget and to pass spending bills, and a veto from the president does not trump that," says Dan Holler, a spokesman for Heritage Action, the lobbying arm of the Heritage Foundation. Holler notes the likelihood of a government shutdown as a distinct possibility if a Republican spending bill gets tossed out.

Now that Republicans have firm majorities in the House and Senate, that budget is likely to look a lot like the one Paul Ryan promulgated in 2012, observers said. If you'll recall, it held steep cuts to entitlements including Medicaid and a repeal of key elements of the Affordable Care Act, President Obama's chief policy achievement.

Nevertheless, shutdowns are deeply unpopular with U.S. voters and small business owners, and Republicans would risk a rebound effect, policy experts say. They may not be willing to risk that heading into the 2016 elections.

"Republicans will be blamed for a government shutdown, not the president, and there will be a lot of pressure to avoid one, which gives the president a little bit of leverage," says Robert Shapiro, a senior policy scholar at Georgetown's Center for Business and Public Policy, and former economic adviser to President Clinton.

Health Care, Still a Hot Button

Yet on key issues, like health care, the president may have to compromise, says Todd McCracken, president of the National Small Business Alliance. Among the things that may disappear from the health care overhaul are a deeply unpopular medical device sales tax, instituted at the end of 2012, and the employer mandate. The employer mandate specifies that business owners with 50 or more full-time employees are required to offer their employees health insurance or pay a fee.

"Repealing the employer mandate is something that would not fundamentally alter the health care law, and it would give Republicans a victory on the ACA and it would allow the president to look like he's being flexible," McCracken says.

More Executive Orders Likely

In his remarks today, the president said it's now up to Congress to craft another plan on immigration, given the way the House blocked a Senate bipartisan compromise on the issue last year.

Short of their action, the president said today he will definitely issue an executive order before the end of the year on immigration. Late this summer, the president suggested his executive order might include lifting caps on green cards, among other things.

"If in fact there is a great eagerness by Republicans to take a broken immigration system, they have every opportunity to do that," President Obama said. "My executive actions don't prevent them from passing a law that supersedes those actions." 

While political observers are divided on whether a far-reaching executive order on immigration would enrage Republicans and stymie work in Congress, Republicans would be more likely to compromise if a path to citizenship for undocumented workers is taken off the table, Holler says.

Such a path to citizenship has been a cornerstone of Democratic immigration efforts. "If Obama takes amnesty off the table, you could bring most folks together on immigration reform," Holler says.

Not a Lame Duck

For his part, Obama rejected the notion that he would become a do-nothing president in his last two years.

"The history of every president shows that in the last two years all kinds of stuff happens, including stuff you can't predict," the president said. "What is most important is keeping an eye on the ball and getting some good stuff done."