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WASHINGTON, Nov 4 (Reuters) - Bipartisan efforts in the U.S. Senate to reopen the federal government as early as this week gave way to fresh signs of frustration on Tuesday, even as Republicans and Democrats aired details of a possible path out of the five-week-old impasse. The shutdown, which is set to become the longest in U.S. history on Wednesday, gave way to the first glimmers of a breakthrough in closed-door bipartisan discussions that lawmakers said were aimed at agreeing on a new short-term funding bill to reopen the government and give Congress additional time to agree on full-year appropriations bills. Sign up here. "We're all being very careful to say that nothing is a done deal at this point, because it's not," said Republican Senator Mike Rounds. The Senate voted for a 14th time to reject existing legislation to fund government operations through November 21, with Democrats demanding an extension of expiring federal tax credits to help Americans pay for private health insurance under the Affordable Care Act. Republicans say the government must reopen before healthcare talks begin. "I think there are people who realize this has gone on long enough, there's been enough pain inflicted on the American people, and it's time to end it," said Senate Majority Leader John Thune, who told reporters he remained optimistic about finding an "off ramp." CLOSED-DOOR TALKS Senate Democrats and two independents who caucus with them spent hours behind closed doors on Tuesday for a meeting that Senate Democratic leader Chuck Schumer said "discussed all of the options." But Schumer showed no sign of compromise. "We're going to keep fighting day after day, vote after vote, until Republicans put working families ahead of the wealthy few," Schumer told reporters. The shutdown, which began October 1, has cut off food assistance for low-income Americans, led to the furlough of about 750,000 federal employees while others work without pay, and has disrupted U.S. air travel in some cities. President Donald Trump has stepped up pressure on Republicans to end the shutdown by jettisoning the Senate filibuster that requires most legislation to pass with support from 60 of the 100 senators. On Tuesday, the president also threatened to withhold food assistance until the government reopens, while his administration warned that it could close some national airspace to air traffic if the impasse continues for another week. Senate Republicans, who hold a 53-47 seat majority, were invited to have breakfast with Trump at the White House on Wednesday, according to a source familiar with the matter. Aside from Republicans, only Democrats Catherine Cortez Masto and John Fetterman and independent Angus King have been willing to support stopgap funding legislation. With Republican Rand Paul also opposed to the measure, Republican leaders need five more Democrats to reopen the government. "We're still in the throes of a give-and-take," Democratic Senator Mazie Hirono said of Tuesday's closed-door party discussion. Hope of a breakthrough rests on groups of Senate Republicans and Democrats who have held private meetings to look for ways to resolve the gridlock. One topic up for discussion is when a new stopgap funding measure would end, with views ranging from December to January and March. Reporting by Richard Cowan, Katharine Jackson and David Morgan, writing by David Morgan; ediiting by Scott Malone and Stephen Coates