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Continuing to deal with the shutdown fallout, the House passed a bill to restore more Border Patrol operations, adding to its list of piecemeal bills.

The Senate, meanwhile, tolled its 10th day without holding a floor vote on any of those proposals. The chamber did, however, clear a bill to ensure families of troops killed in action will be able to immediately receive the $100,000 death benefit they are due.

The White House said that was unnecessary since it had found a work-around that had a private foundation pay the benefits, with a promise that it will be reimbursed after the shutdown ends.

Still, Mr. Obama will have to decide whether to sign the new bill.

The Obama administration also signaled it is willing to relent on some of its closures of national parks, telling the Associated Press it would seek to allow states to use their own money to fund operations at some sites.

It remained unclear late Thursday whether the GOP was still focused solely on a debt deal or whether it would accept adding a shutdown deal into the mix.

House Republicans’ debt proposal was vague when Mr. Boehner laid it out for reporters Thursday morning, after conferring with the House Republican Conference.

“I don’t want to put anything on the table, I don’t want to take anything off the table,” he said.

The White House reacted cooly but didn’t rule out that Mr. Obama might go along with the GOP proposal.

“The president is happy that cooler heads at least seem to be prevailing in the House,” said White House press secretary Jay Carney, who added that the GOP proposal “seems to be a recognition that default is not an option.”

“It is certainly at least an encouraging sign … that they’re not listening to the debt-limit and default deniers,” he said.

But Mr. Carney also said the president “strongly prefers” a Senate bill that would raise the borrowing limit beyond the 2014 elections.

“We’ll see what they’re able to pass, and consider it then,” he said of House Republicans’ proposal.