WASHINGTON » With his troops anxious for a way forward, Speaker John A. Boehner opened a closed-door meeting of House Republicans on Friday morning with a recitation of letters from local schoolchildren on how they deal with stress.

A shower helps, one child counseled. So does a nap.

But on the matter causing all that congressional stress, the speaker offered no clue as to how he expected Congress to get out of the dead end it has found itself in, with the government shut for a fourth day and no clear path to raise the federal debt limit to avoid a first-ever default. "We are locked in an epic battle," the speaker told his rank and file, those who attended the meeting said, urging them to "hang tough." The overarching problem for the man at the center of the budget fight, say allies and foes, is that he and his leadership team have no real idea how to resolve the fiscal showdown.

They are only trying to survive another day, Republican strategists say, hoping to maintain unity as long as possible so that when the Republican position collapses, they can capitulate on two issues at once — financing the government and raising the debt ceiling — and quickly head off any internal party backlash. Republican lawmakers say that Boehner has assured them privately that he will not permit a federal default.

Backers of the speaker say he does not have to fear a coup. His obvious successor, the majority leader, Rep. Eric Cantor of Virginia, is now so caught up in the current legislative battle that he would probably be washed away with Boehner in a Tea Party putsch, and with no other obvious candidates in waiting, no such uprising is likely.

"I do not believe he’s primarily concerned with saving his speakership. He’s concerned with saving the House Republican Conference," said Vin Weber, a former Republican House member who advises the current House leadership.

If the speaker were to move on a stopgap spending bill now, without conservative policy priorities attached, it would likely pass with Republican and Democratic votes. But the ensuing Republican uproar — on and off Capitol Hill — would ensure that there would be no Republican votes to raise the debt ceiling. "It’s common-sense strategy," one Republican strategist said. "If you’re going to take a bullet, you want to take just one."

It has been a long time since a House speaker has faced the many challenges confronting Boehner. On paper, he is the perfect man for the moment, a veteran congressional institutionalist with a history of working with Democrats on big problems and someone bruised by the memory of the last government shutdown and the harm it did to his party.

That John Boehner has faded, carried away by the Tea Party current that swept him to power and is now pulling him from the moorings of his past. His troops are badly fractured, with conservatives advocating one strategy and a growing band of pragmatists demanding the opposite. Aides to Boehner, however, say the speaker has a clear vision of where he wants to end up in the current crisis, but is unwilling to lay out his goals until he has a Democratic partner with whom to negotiate. Democrats are openly disrespecting his leadership and disregarding his demands. Sen. Harry Reid of Nevada, the majority leader, suggested again Friday that Boehner was putting his speakership over his country.

Even Republican allies say whatever strategy exists seems to be dictated not by the speaker, but by Sen. Ted Cruz of Texas, the hard-line Republican who helped launch the "defund Obamacare" movement.

Asked what the House was doing, Rep. Devin Nunes, R-Calif., and a Boehner loyalist, said: "You really have to call Cruz, I’m not even joking about that. That’s really what you have to do, because he’s the one that set up the strategy, he’s the one that got us into this mess, and so we’ve got to know what the next move is."

Democrats say they simply cannot trust the speaker to deliver. Reid said in an interview in his office Friday that Boehner came to him at the end of July with a proposition: If Senate Democratic leaders could accept a stopgap spending measure in the fall at levels that reflected across-the-board spending cuts, the speaker would refrain from adding extraneous measures that could precipitate a clash. Reid was leery, since that level – $988 billion in discretionary spending for fiscal year 2014 – would be $70 billion less than the Senate-passed budget. "I didn’t like it. I’ve got a couple of tough women to deal with," he said, referring to Sens. Patty Murray of Washington, the chairwoman of the Senate Budget Committee, and Barbara Mikulski of Maryland, chairwoman of the Senate Appropriations Committee.

On Sept. 12, in a meeting of the top four congressional leaders, Boehner said he was running into problems with a conservative groundswell demanding that a gutting of the health care law accompany any spending measure. Reid and Sen. Mitch McConnell of Kentucky, the Republican leader, both suggested a procedural maneuver that would allow the House to vote on a stopgap spending bill with a side provision removing funds from the health care law that the Senate could strip out before sending the spending measure to the president. Again, the speaker agreed. And again, he could not carry through, Reid said.

"If I told him I would do something, I would do that," said the Senate leader.

At a White House meeting with the president this week, Boehner twice brought up quiet talks between Murray and Rep. Paul D. Ryan of Wisconsin, the House Budget Committee chairman, as a way to end the impasse with a broad budget deal. On the third time, Reid laughed out loud.

"I couldn’t take it any longer. I said, ‘Stop it. They’ve met a few times and talked about nothing. The meetings were only for you, for show,’" Reid said.

If Boehner is feeling pressure, he is not showing it, sticking to his long-standing routines. He began most mornings this week as he always does, with a long walk and breakfast at the counter of Pete’s Diner a few blocks from the Capitol.

He has spent his days largely hunkered down inside his speaker’s suite, shuttling between his office and his conference room, meeting with small groups of lawmakers. He talks, he listens, and he tries to buck them up: Hang in there and stick with the team has become something of mantra.

Rep. Tom Cole, an Oklahoma Republican close to Boehner, said he believed that the speaker would like to see a deal that included a new way of calculating inflation that would slow the growth of federal benefits; a means testing for Medicare, as well as some other Medicare savings; and at least some slight changes to the Affordable Care Act, such as repeal of a medical device tax unpopular with some Democrats.

Cole said that Boehner wanted a broader negotiation that would involve current federal spending and the debt limit, one that "would move us towards some sort of — if not grand bargain — then big down payment on our fiscal problems," Cole said.

As for Boehner, on Friday he played down any personal animosity between him and his Democratic adversaries, but did show a flash of temper when he referred to suggestions from the White House that Democrats had the advantage. "This isn’t some damn game," he said, his voice rising. "The American people don’t want their government shut down and neither do I."