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The Ukrainian offensive amounts to the first time Kiev's troops have used lethal force to recapture territory from the fighters, who have seized swathes of eastern Ukraine since April 6 and proclaimed an independent "People's Republic of Donetsk".
Ukraine's acting president accused Moscow of supporting "terrorism at the state level" against his country for backing the rebels, who the government blames for kidnapping and torturing a politician found dead on Saturday.
The Ukrainian Interior Ministry said its forces backed by the army had removed three checkpoints manned by armed groups in the separatist-controlled city of Slaviansk.
"During the armed clash up to five terrorists were eliminated," it said in a statement, adding that one person had been wounded on the side of the government forces.
A rebel spokeswoman in Slaviansk said two fighters had died in a clash in the same area, northeast of the city centre. Slaviansk's separatist self-proclaimed mayor, Vyacheslav Ponomaryov, quoted on a local news site, said one man was shot dead and another badly wounded on the northeastern outskirts of the city. He said the dead had been unarmed.
The Kremlin, which says it has the right to invade its neighbour to protect Russian speakers, has built up forces on Ukraine's border - estimated by NATO at up to 40,000 troops.
Defence Minister Sergei Shoigu announced Moscow had launched military drills near the border in response to "Ukraine's military machine" and NATO exercises in eastern Europe. Kiev demanded an explanation within 48 hours of action on the border.
Russia already seized and annexed the Crimean peninsula from Ukraine last month after President Vladimir Putin overturned decades of post-Cold War diplomacy by announcing the right to use military force in neighbouring countries.
An invasion of mainland Ukraine's industrial heartland would be a far more serious action. It had seemed beyond contemplation only weeks ago but now looks like a real threat, although the full extent of Putin's territorial ambitions remains a mystery.
In St Petersburg, Putin said that if the authorities in Kiev had used the army in eastern Ukraine, it would be a very serious crime against their own people.
"It is just a punitive operation and it will of course incur consequences for the people making these decisions, including (an effect) on our interstate relations," Putin said in a televised meeting with regional media.
Washington criticised the Russian drills on the frontier. It was "exactly the opposite of what we have been calling on the Russians to do, which is to de-escalate the situation," Pentagon spokesman Colonel Steve Warren said.
Russia's Foreign Ministry responded that it was up to Washington to halt the Ukrainian military action, and "force the Kiev authorities, who are wards of the United States, to restrain themselves and not use force."
Reuters journalists saw a Ukrainian detachment with five armoured personnel carriers take over a checkpoint on a road north of Slaviansk in the late morning after it was abandoned by separatists who set tyres alight to cover their retreat.
However, two hours later the troops pulled back and it was unclear if Kiev would risk storming Slaviansk, a city of 130,000 that has become the military stronghold of a movement seeking annexation by Moscow of Ukraine's industrialised east.
At another checkpoint set up by the Ukrainian military, a soldier said they were there to instil law and order.
"Those separatists, they violated the constitution, they are torturing the country, they violated laws, they do not recognise the authority of police, so the army had to move in and we will finish what we have started so help me God," he said.
Under an accord signed by Russia, Ukraine, the United States and European Union in Geneva last week, illegal armed groups are supposed to disarm and go home, including the rebels occupying about a dozen buildings in the largely Russian-speaking east.
However, the rebels have shown no sign of retreating. U.S. President Barack Obama blamed Russia for failing to carry out the Geneva deal and said he was ready to impose new sanctions.
Putin said sanctions were "dishonourable" and destroyed the global economy but that so far the damage had not been critical.
Moscow also flexed its economic muscles, with the government suggesting foreign firms which pull out of the country may not be able to get back in. A source at Gazprom said the Russian exporter had slapped an additional $11.4 billion bill on Kiev. Ukraine is negotiating to reverse east-west pipelines so that it can receive gas from Europe if Moscow cuts it off.
Washington accuses Moscow of sending agents to coordinate the unrest in the east, as it did before seizing Crimea last month. Russia denies it is behind the uprising and says the separatists are responding spontaneously to hostility from Kiev. Russia made similar denials over Crimea until Putin acknowledged last week his troops had indeed acted alongside local militia.
"At the state level, Russia is supporting terrorism in our country," Ukraine's acting President Oleksander Turchinov said in an address to the nation on Thursday. "Armed criminals have taken over buildings, are taking citizens, Ukrainian and foreign journalists, hostage and murdering Ukrainian patriots."
Turchinov called for the eastern offensive this week after the apparent torture and murder of a pro-Kiev town councilor whose body was found on Saturday near Slaviansk.
Volodymyr Rybak had disappeared after being filmed trying to take down a separatist flag while trying to enter the rebel-held town hall where he worked in Horlivka, a town near Slaviansk.
"He was bruised and punctured from head to toe...it's clear they tortured him," said Aleksander Yaroshenko, a family friend who accompanied Rybak's widow when she identified his body at the morgue. "The police have lots of details, they have CCTV footage, they should know who did this," he told Reuters.
Rebels in Slaviansk released U.S. citizen Simon Ostrovsky, one of three journalists they are believed to be holding.
Moscow called for Kiev to release "political prisoners", including a pro-Russian activist named Pavel Gubarev.
So far, the United States and EU have taken only mild action against Moscow, imposing visa bans and asset freezes on a few Russians, measures Moscow has scoffed at as meaningless. Washington and Brussels both say they are drafting more serious sanctions and will impose them if the Geneva deal collapses.
Even without serious sanctions, Russia's confrontation with the West has hurt its economy as fearful investors send their money abroad. Mutual funds specialising in Russia and Eastern Europe were the 30 worst performers out of 3,489 equity funds for sale in Britain in the three months ending in March.
In NATO member Poland, the first group of a contingent of around 600 U.S. soldiers arrived on Wednesday, part of an effort by Washington to reassure eastern European allies who are worried by the Russian build-up near Ukraine's borders.
However, NATO and the United States have made clear they will not use military force to protect Ukraine itself.
Shoigu, the Russian defence minister, said aircraft would fly increased patrols near the Ukrainian frontier as part of the new exercises. Two local residents in the area told Reuters they had seen attack helicopters flying in formation.
Kiev said the city hall in another eastern town, Mariupol, which had been seized, was back under central control. A separatist crowd later surrounded the building, patrolled by police but otherwise apparently empty.
Kiev also reported a shootout overnight in another part of the east where a Ukrainian soldier was wounded.
Ukraine, a former Soviet republic, slid into unrest late last year when Moscow-backed President Viktor Yanukovich rejected a pact to build closer ties with Europe. Protesters took over central Kiev and he fled in February. Days later, Russian troops seized control of Crimea.
The Ukrainian defence ministry confirmed its involvement in the operation around Slaviansk on Thursday, saying the troops involved were airborne units with experience of such tasks from international peacekeeping missions.
"The morale of our forces will allow them to completely fulfil their task of defending Ukraine," it said.
Unarmed mediators from the Organization for Security and Cooperation in Europe are in eastern Ukraine trying to persuade pro-Russian gunmen to go home, in line with the Geneva accord.
Reuters reporters have not been able to establish that any Russian troops or special forces members are on the ground, though Kiev and Western powers say they have growing evidence that Moscow has a presence. Masked gunmen in the east, widely referred to as "green men", wear uniforms without insignia.
Kiev has been cautious in taking action for fear of triggering a Russian invasion. Last week it ordered an "anti-terrorist operation", but this quickly fizzled out when an armoured column of paratroopers surrendered weapons to a separatist crowd. (Additional reporting by Alexander Reshetnikov and Gleb Garanich near Slaviansk; Alissa de Carbonnel in Donetsk; Pavel Polityuk, Natalia Zinets, Richard Balmforth and Alastair Macdonald in Kiev, Denis Dyomkin in Birobidzhan, Russia, Mark Felsenthal in Tokyo, Alessandra Prentice and Vladimir Soldatkin in Moscow; Writing by Christian Lowe, David Stamp, Philippa Fletcher and Peter Graff)