CUMBERLAND, Maine — Sara Gideon, her voice hoarse on a cold Friday night, stood in the center of a fairgrounds stage as the headline act at a drive-thru rally, making a closing pitch to a chorus of appreciative car honks and headlight flashes for a government dominated by Democrats that would move aggressively to address climate change, economic and racial inequality and out-of-control health care costs.

A day earlier, Sen. Susan Collins, R-Maine, had barreled across the state in her signature campaign bus with a very different message, highlighting the billions of dollars she had directed toward Maine businesses during the pandemic and her lifetime of connections built across the state, barely mentioning President Donald Trump or her party leaders as she played up her brand as a moderate pragmatist.

The appearances reflected the contrast between the two women waging the most expensive Senate race in Maine history. It has barely shifted since Gideon entered the fray more than 16 months ago, hoping to capitalize on liberal anger against Trump and outrage over Collins’ vote to confirm Justice Brett Kavanaugh to the Supreme Court to deny the Republican senator a fifth term.

But through a presidential impeachment trial, a deadly pandemic and yet another historically partisan Supreme Court confirmation battle, neither candidate has been able to maintain a steady advantage in the race. Instead, because of a relatively new voting system in Maine, the outcome of the contest — and potentially the power balance in the Senate — may come down not to whom Maine voters name first, but whom they name second.

The contest on Tuesday is likely to be the first time that Maine will count second choices in a Senate race using a ranked-choice voting system that has been in place since 2018. It allows voters to list a second candidate and counts those preferences as votes if no one reaches 50% when the first-choice votes are tabulated. The system could prove particularly dangerous for Collins — who like Gideon has consistently drawn below 50% in public polls conducted in recent months — because Lisa Savage, a progressive running as an independent in the race, has urged her supporters to list Gideon second.

“It’s obviously a very close race, but I feel that momentum is breaking my way,” Collins said on Thursday, after munching on an ice cream cone as she concluded a whirlwind string of rainy visits to local businesses across two counties. “My goal is to get 50% on Election Day, and ranked-choice voting would not come into play. So that’s what I hope.”

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But there is little evidence that Collins has been able to take a commanding lead in recent weeks. Even after she became the only Republican to break with her party and Trump last week to vote against confirming Justice Amy Coney Barrett to the Supreme Court, citing proximity to the election, voters appeared unmoved. In interviews across the state, her supporters and opponents both deemed it a necessary political maneuver to woo moderate voters, with Democrats noting that it had done nothing to affect the outcome.

“It’s hard to buck your party, and I do give her credit for that,” said Lara Rosen, 39, who was bundled in her car with a cup of haddock chowder and her 5-year-old son, Isaac Rosen-Murray, to support Gideon. “It’s not enough. It’s not the only thing that I care about.”

Maine first rolled out its ranked-choice voting system statewide two years ago, allowing voters to rank their preferences instead of choosing a single candidate. If the election concludes without any candidate attaining at least 50%, the one with the least number of votes is eliminated, and those ballots are redistributed to the remaining candidates based on voters’ second choice. The process of elimination continues until one candidate has broken through the majority threshold.

The system, which is also used in Australia, Ireland and the race for best picture at the Academy Awards, proved consequential in Maine’s 2nd Congressional District in 2018. After picking up more votes as a second or third choice, Jared Golden, a Democrat, unseated Rep. Bruce Poliquin, a Republican who had been the first choice of more voters. (Sen. Angus King, the independent who caucuses with Democrats, easily cleared the bar with more than 54% of the vote that year.)

“It’s not as simple as you might think — there isn’t a clear policy flow from the minor-party candidates to the majority candidates,” said Daniel M. Shea, a government professor at Colby College and the lead researcher on the college’s polling of the Senate race. In the college’s final poll, which called the race a “statistical dead heat,” Max Linn, a brash businessman, secured 1.7% of the vote while Savage, a teacher who has ties to the Maine Green Independent Party, secured 4.7%, behind Gideon at 46.6% and Collins at 43.4% The poll had a 3.3% margin of error.

Linn, an often belligerent debate presence who cut surgical masks in the middle of one exchange to illustrate opposition to a mask mandate, said in an interview that he was not working to influence whom his supporters ranked second on their ballots. But Savage, who supports several progressive causes like “Medicare for All” and a Green New Deal, has built her campaign partly around explaining ranked-choice voting — and urging her supporters to “vote blue No. 2” and direct their secondary votes to Gideon.

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“Our platform and issues are the ones that most young voters resonate with, but they say, ‘I don’t believe in electoral politics; I don’t think it changes anything; I’m not that inclined to vote,’” Savage said on Saturday. She was seated at a table at the Portland farmers’ market that offered condoms branded with “Medicare for all,” rainbow “Lisa for Maine” pins and multiple explainers about the voting system. “So now our pitch to them is: ‘But we have ranked-choice voting. It amplifies the power of your vote,’” she said.

Savage emphasized that she was not looking to undercut Gideon in her bid to unseat Collins, but instead to help attract otherwise reluctant, young and first-time voters who were discomfited by the bitter campaign and wary that Gideon was not liberal enough. Multiple experts say that Savage’s supporters could tip the scales and give a victory to Gideon.

“We want to send a signal to Democrats that we are on the ‘retire Susan Collins’ team with them,” Savage said. Her campaign, she added, approached Gideon’s team with a suggestion that the women campaign on ranking the other second, but did not receive a response. (In an appearance at Bates College on Friday, Gideon told reporters she would encourage her voters to consider ranking Savage second.)

But in search of a clear path to victory, both Collins and Gideon have delved into a flurry of last-ditch campaigning, doling out elbow bumps and platitudes in an effort to galvanize their supporters and persuade the state’s remaining undecided voters. The Colby College poll found that 3.6% of the 879 likely voters surveyed had not made a decision.

“There are a lot of people who have made up their mind, some of whom maybe made up their minds 10 months ago, and some of whom came to that place in the last two months,” Gideon said during a stop at a logging site in Oxford County, as machines felled trees behind her. “I do think that there are some people out there who still are not sure of what to do. They think about the balance of the presidential election and the Senate, and they struggle to understand exactly who is going to do what or who has done what.”

On a four-day tour of the state, Gideon frequently summoned the specter of Trump and Sen. Mitch McConnell of Kentucky, the majority leader, to cast the race in national terms and argue that it was vital for Democrats to control the White House and Congress to set the agenda in Washington.

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For her part, Collins spent the final days of the campaign highlighting the financial support she had given small businesses across the state by championing the Paycheck Protection Program, a popular federal loan program that her campaign says funneled more than $2.3 billion to nearly 30,000 businesses.

Ultimately, her final pitch for a fifth term depends on voters who still value the might of a Maine voice in the top position on the powerful Appropriations Committee, which allocates federal spending; the few remaining split-ticket voters in the state like Bill Green, a retired journalist and longtime fixture on Maine television.

Green, a registered Democrat who voted for former Vice President Joe Biden, the Democratic presidential nominee, endorsed Collins in a series of campaign ads.

“She’s gone to work every day, and whoever the people elected to be president, Susan Collins worked with him,” he said. “It’s her job to go down there, and do the best job that she can for Maine, and hold your nose and work with the guy.”