House Republicans insisted Thursday they will settle on a health care overhaul of their own, saying Democrats who argue the GOP cannot repeal and replace Obamacare with something better are resorting to “scare tactics.”

Rep. Kevin Brady, chairman of the tax-writing Ways and Means Committee, made his pledge during a break in closed-door meetings with GOP members who are brainstorming health and tax policy ideas amid Congress’ holiday break, after the November elections handed their party the keys to reform.

“On Election Day, Americans made it clear that Obamacare is not working for them,” the Texas Republican told reporters at the Library of Congress, citing double-digit rate increases on the law’s insurance exchanges. “Those premiums hikes are unacceptable, and they’re not slowing down, they’re accelerating.”

Mr. Brady reminded Americans that Obamacare will remain law beyond Jan. 20, when President-elect Donald Trump is sworn into office, and that it will take time for Republicans to gut the Affordable Care Act through a fast-track budget process and then hammer out a replacement with Mr. Trump’s team.

Republicans haven’t described how long of a transition period they will need to reach consensus on a replacement and win over enough Senate Democrats to bypass a filibuster of the plan.

Analysts and segments of the health sector say repealing parts of the law without an adequate safety net in place could cause insurers to flee the marketplace and disrupt coverage for 20-plus million Americans who gained coverage under Mr. Obama’s program.

Mr. Brady said the idea that 20 million Americans will lose their health care under GOP leadership is the “new big lie,” after President Obama’s famous promise that “if you like your health care plan, you can keep it” under the 2010 overhaul.

“There’s a lot of scare tactics out there on this,” “My point is this — we can assure the American public that the plan they’re in right now, the Obamacare plans, will not end on Jan. 20, that we’re going to be prepared and ready with new options tailored for them.”

Yet right now, he said, the timing and sequence of repealing and replacing the law “haven’t been determined.”

In the meantime, the administration is urging millions of Americans to sign up or reenroll in exchange coverage before critical deadlines, hoping to place Obamacare on firmer footing and make it more difficult for Mr. Trump and his GOP allies to unravel it.

Customers had until midnight Thursday to enroll in coverage that’s effective Jan. 1. The administration said signups through Dec. 10 exceeded last year’s by 250,000, though updated figures after the Dec. 15 cutoff will shed light on whether the administration and its partners have been able to root out the remaining uninsured in spite of political threats to the law.

Customers have until Jan. 31 to enroll in 2017 coverage, though the administration warned customers not to delay, noting there is a baseline tax penalty of $695 for remaining uninsured during 2017.

Republicans have said they will strike the “individual mandate” requiring Americans to hold coverage or pay a penalty.

Instead, they want to entice customers into coverage with “market-oriented” reforms, such as an age-based tax credit that allows people to purchase any plan they want or shop for coverage across state lines.

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