WASHINGTON -- In a closed strategy session of House Republicans, Speaker John A. Boehner on Friday shared letters of encouragement he received from schoolchildren who urged him to withstand shutdown stress by taking a nap, listening to music – Lady Gaga was suggested – and other tips, according to those in the room.

The light moment quickly faded to Boehner’s denial of media reports that he was willing to “roll over” in coming negotiations to raise the the nation’s debt limit. Boehner told lawmakers he has no intention of leaving his conservative Republican majority behind to rely overwhelmingly on Democrats to raise the nation’s debt limit in coming weeks and avoid what could be a catastrophic default.

“He said that’s not going to happen,” said Rep. John Fleming (R-La.), a conservative. “He was basically blowing it out of the water.”

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Boehner’s hardline approach was met by applause in the room.

The House is planning to remain in session at least through Saturday to continue approving legislation to reopen specific parts of government – an approach President Obama and Democrats in the Senate have rejected, saying they have no interest in picking winners and losers among federal operations.

Republicans had almost no choice but to stay in Washington over the weekend once Obama canceled his trip to Asia because of the shutdown. The president had been set to depart on Saturday for the Asia-Pacific Economic Cooperation summit.

Unable to stop the president’s healthcare law as part of a tea party strategy that led to the shutdown, Republicans now want to pivot to the broader debate over the need to raise the $16.7-trillion debt limit or face a risky federal default.

Some more moderate-leaning Republicans have tried unsuccessfully to reopen the government while talks shift, but Boehner has encouraged those lawmakers to “have patience” as he pursues a broader strategy that widens the battlefield beyond the shutdown into the debt ceiling.

A pair of members from more competitive general election districts did speak up to remind their colleagues that the fallout from a prolonged shutdown would be different for them than those who are in more reliably Republican seats. Among them was Rep. David Reichert (R-Wash.), who represents a district that President Obama narrowly won in 2012.

Rep. Richard Hudson said the message from leadership was “stick together.”

“He said: ‘Look, the Democrats are going to do everything that they can to try and portray me in the worst light, to try and split us. But we’re sticking together. And there are no backroom deals going on, there are no meetings happening. The president and Harry Reid are not negotiating,’” Hudson said.

FULL COVERAGE: The U.S. government shutdown

The Treasury Department warned this week that a default could trigger a worse financial crisis than in 2008.

“A default would be unprecedented and has the potential to be catastrophic: Credit markets could freeze, the value of the dollar could plummet, U.S. interest rates could skyrocket, the negative spillovers could reverberate around the world, and there might be a financial crisis and recession that could echo the events of 2008 or worse,” Treasury said.

Democrats are skeptical that Boehner can make a big budget deal in the short time remaining, especially after past attempts at a “grand bargain” failed. They have publicly said they would not negotiate any substantive changes until after Republicans agree to reopen federal agencies and raise the government’s debt limit. That all but ensures there will be no quick resolution to the federal government shutdown, which has furloughed 800,000 workers and closed all nonessential government operations, including popular national parks and museums.

In the last major standoff over the debt ceiling, in 2011, the two sides agreed to paper over their inability to agree by creating a new committee to debate budget changes. Whether they can find a similar mechanism that would save face for both sides may determine whether the government can avoid an economically damaging default on its debt.

Top House Republicans are working on the demands the GOP will make in exchange for raising the debt limit and reopening government, according to those familiar with the internal strategy.

Still smarting from Boehner’s past failed efforts to negotiate a budget deal with the White House, Republicans have shied away from calling any new deal a “grand bargain.”

Rather, they see it as a “big down payment” toward trimming the nation’s deficits, preferring the language of Rep. Paul D. Ryan (R-Wis.), the GOP’s budget guru and the former vice presidential nominee.

Following Ryan’s lead, Republicans are expected to revisit the components of past budget battles: cuts to Medicare, Social Security and other entitlement programs, as well as reforming the tax code, a long-standing interest. They may also seek to gain the Obama administration’s approval of the Keystone XL pipeline between Canada and the United States, as well as pursue smaller changes to the healthcare law, including repeal of the tax on medical-device makers and an end to the individual patient advisory board.

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Lisa.Mascaro@latimes.com

Twitter: @LisaMascaroinDC

Michael.Memoli@latimes.com

Twitter: @MikeMemoli