In the wake of a failed GOP-led effort to repeal and replace the Affordable Care Act, Republican Sen. Jeff Flake of Arizona and Democratic Sen. Dianne Feinstein of California say the path forward toward health care reform includes "regular order."

Speaking on "Face the Nation" Sunday, Flake said issues like health care, in addition to the nation's debt crisis, can't be tackled by one party alone.

"We've just seen the limits of what one party can do," said Flake. "I'm glad to see that now we're talking about sitting down with our colleagues, going back to committee, going back to what we call regular order, and letting the committees and the experts deal with it, and bringing the public in more than we have before," he added.

Flake, who details the divide among the Republican party in his new book, "Conscience of a Conservative: A Rejection of Destructive Politics and a Return to Principle," said that in the fallout over health care, the next best way to solve the issue requires

"both parties sitting together and sharing the risk."

"It's hard to imagine that can happen when we're ascribing the worst motives to our opponents," he said.

Flake's comments come after Senate Republicans' attempt to repeal parts of Obamacare failed in a drama-filled, middle-of-the-night vote.

President Trump issued a series of tweets following the bill's failure, saying, "As I said from the beginning, let ObamaCare implode, then deal. Watch!"

Meanwhile, Feinstein, also appearing on "Face the Nation" Sunday, appeared to echo Flake's call for bipartisanship in the health care debate -- saying, "You can't take a bill as big as this one, write it with a select group of people in a back room, not let one of the political parties even see it until the Friday before a vote comes up, and think that this bill is going to pass."

She also called for regular order, urging Congress to now hold a series of hearings on health care in order to achieve the best outcome possible.

The bipartisan effort, embraced by both Flake and Feinstein, is also a shared feeling among Americans. A new CBS News Nation Tracker Poll found that more Americans prefer that Republicans now work with Democrats to improve Obamacare (47 percent), rather than try to repeal it outright (21 percent) or replace it with something exclusively of their own (19 percent).

This sentiment is particularly strong among Democrats and independents, but a quarter of Republicans want to keep Obamacare, with bipartisan improvements.

While the two senators appear to agree on positive steps forward in the health care debate, Senate Minority Leader Mitch McConnell, R-Kentucky, made it clear that his chamber is moving on to other things.

"It is time to move on," McConnell said when the Health Care Freedom Act failed early Friday morning, noting the other unrelated topics the Senate would take up in the hours ahead.