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Washington • Attempting to restart a shuttered government, House Republicans on Tuesday pushed a piecemeal approach to reopen segments of the government, starting with fully funding the Veterans Affairs and National Parks, a move that Democrats and the White House swiftly rejected as frivolous.

The idea was pitched by Republicans Sens. Ted Cruz of Texas and Mike Lee of Utah, who saw it as a way to force the Senate, controlled by Democrats, to reopen key government services after a budget impasse prompted the first shutdown in nearly 18 years.

"We ought to fund vital priorities." Cruz said to a question by The Salt Lake Tribune, noting that Congress made a special exemption to ensure active-duty soldiers got paychecks during a shutdown and other agencies could be spared, too. "We should reopen the national parks today; we should fund the VA today. ... The only reason it might not happen is if Harry Reid and the Democrats would object."

It marked the latest overture from Republicans who were still pushing a delay of the Democrats' signature health care law as part of each stopgap budget that passed the House.

Obamacare rolls on • On Tuesday, as hundreds of thousands of nonessential federal workers were furloughed and the Capitol ran on a skeleton staff, Obamacare — the focus of the shutdown — rolled out as planned because it was paid for with funds already authorized.

The Senate once again tossed back a House-passed measure and defied the Republicans to try again.

Eyeing a new game plan, Cruz and Lee pushed the plan to pass agency-by-agency, temporary budgets, and force Democrats to vote on each one separately.

"This is an effort to compromise, an effort that is badly needed," Lee said on the floor, adding later, "Let's leave Obamacare for another day and not hold the vast majority of government functions hostage when the vast majority of government functions don't have anything to do with the implementation and enforcement of Obamacare."

The White House and Democrats signaled quickly that they wouldn't entertain the idea.

"If they want to open the government, they should open the government," White House Press Secretary Jay Carney said of Republicans. "Then we can negotiate about how we fund our budget priorities in the future. A piecemeal approach to funding the government is not a serious approach."

Senate Majority Leader Harry Reid, D-Nev., backed by his leadership team and other Democrats, said Republicans were just trying to pick the more popular things to fund while ignoring other needs.

'Wacky idea' • "Now they are focusing on trying to cherry-pick some of the few parts of government that they like," Reid said. "Just another wacky idea by the tea party-driven Republicans. ... We support veterans, and parks, we support the FBI, we support the federal government, that's our job that's what we do. We can't and we won't be forced to choose between parks and cancer research or disease control or highway safety or the FBI, or national security agencies."

As the government closure plodded on without an end in sight, the law that Republicans were attempting to kill was being implemented, with individuals signing up for health care coverage through state and federal exchanges.

In a planned speech about the new health care enrollment, President Barack Obama paired the two issues together and noted that this government shutdown was not about deficits or spending but about "rolling back our efforts to provide health insurance to folks who don't have it," and halting the health care law.

"This, more than anything else, seems to be what the Republican Party stands for these days," Obama said. "I know it's strange that one party would make keeping people uninsured the centerpiece of their agenda, but that apparently is what it is."

Not ready for primetime • Republicans, though, were quick to jump on glitches being reported nationwide with the websites individuals were using to sign up for health care, and said that was proof enough that the system wasn't ready to go online.

"From reports around the country it seems that Obamacare is off to a very rocky start," Sen. John Cornyn, R-Texas, said.

Congress would have to repeal or delay the law to actually stop the implementation of the health care law.

The government shutdown, the first since 1996, came as polarized parties refused to retreat from their positions: Congressional Democrats unwilling to change or postpone health care reform and Republicans intent on using their leverage to get concessions on Obamacare.

In an interview with Fox News' Greta Van Susteren, Lee on Monday night claimed credit for kick-starting the effort to use the federal budget as a way to cut funding for Obamacare, a move that led to the impasse and the government shutdown after Democrats refused to yield to House Republicans' demands.