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The influence of the Tea Party is about to endure a critical test, as lawmakers enter the second chapter of their deficit-reduction plan after approving a record increase in the debt ceiling.

Conservative Republicans have been conflicted over the 11th-hour compromise and the course of the debate as a whole. While they claim credit for shifting the conversation in Washington from stimulus to deficit reduction, Tea Party lawmakers and activists were disappointed with the result of the debt-cap talks.

In the end, 66 House Republicans and 19 Senate Republicans, many of them Tea Party aligned, opposed the package out of concern that it didn't go far enough.

But conservatives will have a second chance to at least ensure the next -- and arguably more important -- phase of deficit reduction is conducted to their liking. After enacting $900 billion in spending cuts, the bill signed Tuesday will pave the way for a bipartisan committee to find another $1.5 trillion in savings.

GOP lawmakers say they're going to press for serious entitlement reform as part of that process. But already, Democrats are vowing to press just as hard for revenue on that committee -- and challenge the Tea Party ideology.

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Senate Majority Leader Harry Reid indicated Tuesday that more will be on the line than just deficit savings.

"I ... think the joint committee will set a tone for what happens in the near future," Reid said, when asked whether Tea Party lawmakers could continue to fight increases in the debt ceiling.

"Tea Party direction of this Congress the last few months has been very, very disconcerting," Reid said, and he predicted things are going to start to change around Capitol Hill.

"There must be a sharing of sacrifice," Reid said. "We've had too much talk the last few days of Republicans ... saying there will be no revenue. That's not going to happen."

While Republicans hold a wide majority in the House, Reid noted there's "no supermajority" on the committee, where each party will have six members. The committee's recommendations would still need to clear Congress, but the law signed Tuesday has built-in disincentives aimed at preventing Congress from stonewalling the committee's report.

Though the Tea Party felt snubbed at the end of the debt talks, Democrats claimed the movement has had far too much influence. The frustration with the Tea Party's power was evident during a private meeting with Vice President Biden on Monday, when House Democrats compared Tea Partiers to "terrorists" -- the vice president later denied using the term himself, but explained in an interview that some lawmakers felt like they were "being held hostage by terrorists."

Democrats are now determined to change the dynamic.

President Obama said the committee must tackle entitlements but also reform the tax code "so that the wealthiest Americans and biggest corporations pay their fair share."

"Everyone's going to have to chip in. ... That's the principle I'll be fighting for during the next phase of this process," he said.

GOP lawmakers claim the bill didn't contain tax hikes and the committee won't allow tax hikes.

But while Democrats dispute that, it's unclear what exactly GOP leaders define as a tax hike. Democrats are pushing hard for the elimination of preferences for oil and gas companies and corporate jet owners, and potentially, certain income tax benefits for high-income Americans. But Republicans, like Senate Minority Leader Mitch McConnell, say they're in favor of tax reform provided overall rates go down -- leaving open whether the GOP could cede ground on those Democratic pet items and still claim a victory on taxes.

While GOP leaders are voicing confidence about their no-tax pledge and keeping the emphasis on spending cuts, Tea Party leaders are on edge. After all, Democrats say they want to use the committee process to raise revenue, not just simplify the tax code.

"I fear that this could lead to tax hikes," Sen. Mike Lee, R-Utah, a Tea Party-aligned lawmaker who opposed the bill, told Fox News.

A Weekly Standard report said Tuesday that GOP senators who voted against the deal would not be able to serve on the joint committee. A McConnell spokesman told FoxNews.com the story was not "at all" true.

But Tea Party Nation founder Judson Phillips seized on the report, writing on his website that the committee could exclude "strong fiscal conservatives."

"We are going to see an implosion later this year," he wrote.

Separately, Tea Party-aligned FreedomWorks issued a statement saying it opposed the debt deal but vowing to fight for its principles in the months ahead. President Matt Kibbe said the movement would continue to flex its muscle on the campaign trail.

"The Tea Party movement has changed the debate. Next up is changing out more congressmen. That's where we will get our big policy changes," Kibbe said.