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President Barack Obama and Russian Prime Minister Vladimir Putin shake hands during a meeting at the near Moscow Tuesday.

Soul mates? No. But President Obama and Russian Prime Minister Vladimir Putin could be called frenemies after their first high-stakes meeting outside Moscow Tuesday.

The two leaders shared breakfast at Putin's dacha, where they praised each other - but also acknowledged deep divisions.

A key flash point is their differing view of neighboring Georgia. The Kremlin backs separatists there and crushed the country's tiny army during battles a year ago, while the U.S. supports its admission to the NATO alliance.

"I don't anticipate a meeting of the minds anytime soon" on Georgia, Obama told Fox News.

Putin noted the two nations have been at loggerheads over "events of different, shall we say, color" - a reference many experts saw aimed at the U.S.-backed pro-democracy "color revolutions" in the former Soviet states of Georgia and Ukraine.

Putin, extremely popular and widely regarded as the true ruler of Russia, consolidated his power dramatically while opposing those revolutions, branding them Western power grabs and attempts at regime change.

Obama still delivered a pro-democracy message after his meeting, though, leveling careful criticism at Russia's autocratic tendencies and widespread corruption.

"The arc of history shows us that governments which serve their own people survive and thrive," Obama told the progressive New Economic School. "Governments which serve only their own power do not."

Analysts saw the careful language and the acknowledgment of lingering differences as a good sign the two leaders may be able to build on eight agreements Obama announced Monday, including a deal to cut nuclear weapons.

"Obama came away with a better impression [of Putin] than he had going in," said Sam Charap of the White House-friendly Center for American Progress think tank.