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The frenzied pace of Monday gave way to legislative lethargy Tuesday.

The House spent most of the day in recess but returned late in the evening to try its piecemeal strategy with three bills. One of them would have funded the Department of Veterans Affairs; another would have restored money for the National Park Service, the Smithsonian Institution, the United States Holocaust Memorial Museum and the National Gallery of Art; and a third would have given the District of Columbia the right to spend its own tax money during the shutdown.

Democrats said they supported all of those ideas but couldn’t accept choosing priorities.

“As outrageous as all of this is, the worst tragedy is what it does to our capital city,” said Rep. James P. Moran, Virginia Democrat. “In the District of Columbia, D.C. residents have paid for their government and yet this shutdown refuses to even let them use their own revenue to run their government.”

Delegate Eleanor Holmes Norton, a Democrat who represents the District, said Mayor Vincent C. Gray has “cobbled together” the city’s contingency funds to keep the government open, but those funds will run out in a couple of weeks if Congress fails to act.

“If federal workers should not be punished because Congress did not do its job, surely a shutdown of the District of Columbia would have punitive effects no one can condone,” Ms. Norton said in her prepared remarks. “No Republican or Democrat desires the unintended effect of shutting down the local government only because D.C.’s local balanced budget has not cleared the Congress.”

Mr. Gray has deemed all city employees essential, meaning he will try to keep everyone including janitors and librarians on the job.

But the bill in Congress to grant him more authority failed to garner the two-thirds vote needed to pass under special expedited rules. Most Democrats opposed the bill.

Democrats also opposed the veterans and parks bills.

“This isn’t about the parks. They’re using parks as a pawn. This is about defunding the Affordable Care Act,” said House Minority Leader Nancy Pelosi, California Democrat. “It’s not going to happen.”

Republicans said they had to resort to the piecemeal approach because the Senate rejected all of their other offers — including a request to enter into formal negotiations in a conference committee.

Senate Democrats rejected that proposal early Tuesday in another party-line vote, saying they will not begin to negotiate until the House relents and agrees to a funding bill with no strings attached.

“All they have to do is accept what we already passed,” Mr. Reid said.

House Republicans have long argued that the annual spending bills — and the debt limit fight — are leverage points that should be used to overcome a Democratic Senate that has blocked most of the Republican agenda the past two years.