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(CNN) -- Scores of pro-Russian protesters storm a Ukrainian police headquarters and confront the officers inside. Hundreds outside back up the intruders, chanting. A Russian flag flies from the structure as the crowd listens to activists with a megaphone at the entrance.

Monday was the day the Ukrainian government demanded pro-Russian activists give up. Instead, they appear to have overrun another key building in yet another eastern Ukrainian city.

Protesters forced police officers out of their four-story headquarters in the eastern Ukrainian city of Horlivka, witnesses said, adding a new complication for a Kiev government worried about demonstrators' escalating activity in the east and a Russian neighbor that it accuses of fomenting the discontent.

Acting President Oleksandr Turchynov had given pro-Russian protesters in other eastern Ukrainian cities until 2 a.m. ET to disarm or face a "full-scale anti-terrorist operation" by Ukraine's armed forces. But the deadline passed with no sign that it was heeded, including in the eastern city of Donetsk, where protesters have held the regional government building for more than a week.

Similar deadlines in the past have come and gone with no consequences.

Speaking to CNN's Christiane Amanpour on Monday, Ukraine's U.N. Ambassador Yuriy Sergeyev said it was his understanding the "anti-terrorist operation is in the process."

He talked about the difficulties involved in finding a solution to the crisis, given that women and children are inside the buildings, which -- in turn -- are surrounded by peaceful protesters.

"It's really very difficult to find a solution to avoid the blood. A combination of different approaches should be used," said Sergeyev, calling for dialogue.

A video on online streaming service Ustream purports to show scores of people entering the police headquarters in Horlivka, a city of 300,000. CNN could not independently verify the video.

Outside, a crowd chanted "Referendum!" and "Russia!" A severely beaten man in a police uniform was taken to an ambulance as onlookers shouted at him.

Demonstrators -- some masked, and a few wearing camouflage, but many unmasked and wearing street clothes -- crammed the halls in the video. One room with desks was littered with broken glass.

At one point, a group of men confronted what appeared to be a police officer trying to block their progress in a doorway. The men eventually backed him down, and one of them swung a baton-like object at someone or something, his intended target obscured by onlookers as people shouted and sounds of a struggle ensued. It wasn't clear what happened to the officer.

Officers warned activists to avoid looting. But no officer appeared to stop the masses from walking through the halls and rooms for long.

Horlivka would be at least the 10th city or town in eastern Ukraine where activists have taken over security or government buildings in recent days.

Also Monday, in Slaviansk, Ukrainian authorities appeared to take no action to retake a police station that was occupied earlier. Pro-Russian protesters milled around with makeshift shields outside the building.

When asked why the Ukrainian government had made no apparent move to evict protesters after the deadline passed, Ukrainian Deputy Foreign Minister Danylo Lubkivsky told reporters that the government still believes in a political solution. Although it wants to avoid bloodshed, the government still will protect the country's territorial integrity, he said Monday in Kiev.

Also Monday, U.S. President Barack Obama and Russian President Vladimir Putin spoke about the worsening crisis. The two leaders spoke last March 28.

According to a Russian statement, Putin stressed the protests "are the result of the Kiev authorities' unwillingness and inability to take into account the interests of the Russian and Russian-speaking population," and called on Obama to help prevent the use of force.

"In response to the U.S. President's expressed concern about Russia's alleged interference in southeastern Ukraine, the President of Russia noted that such speculations are based on inaccurate information," it read.

According to the White House, Obama urged Putin to withdraw Russian troops from Ukraine's border and to use his influence to persuade armed pro-Russian groups to leave the buildings they have seized.

"The President noted Russia's growing political and economic isolation as a result of its actions in Ukraine and made clear that the costs Russia already has incurred will increase if those actions persist," the White House said.

Kiev blames Moscow

The unrest is the latest in a series of events ratcheting up tensions between Ukraine and Russia, which Kiev accuses of fomenting trouble in Russian-speaking eastern Ukraine.

After then-Ukrainian President Viktor Yanukovych backed out of a deal with the European Union in November in favor of closer ties with Russia, months of protests in Kiev led to his ouster in February.

Distrust among the population in eastern Ukraine, the base of Yanukovych's power, grew as the new national government shifted rapidly in a pro-Western direction. A short time later, pro-Russian elements occupied the Black Sea peninsula of Crimea, which Russia annexed in March. Since then, pro-Russian protesters have taken to the streets in eastern Ukraine and in some cases stormed and occupied buildings.

Kiev's fragile new government and the West accuse Russia of destabilizing the region as a pretext to potentially send in troops to protect the local Russian-speaking population.

NATO says Russian armed forces are massing on Ukraine's eastern border, while Moscow says they are merely carrying out military exercises.

Turchynov on Sunday issued a promise of amnesty for the activists -- including protesters as well as armed militants -- in eastern Ukraine but warned that anyone who continued to support the takeover of government buildings would be held responsible for their actions.

"We'll not allow any repetition of the Crimean scenario in the east of Ukraine," Turchynov said. "I have signed a decree that would allow those who did not shoot at our officers to lay down their arms and leave the occupied buildings by Monday morning without fear of being prosecuted."

Referendum proposed

After the deadline passed Monday, Turchynov said the majority of Ukrainians would support keeping Ukraine in one piece if a referendum is held along with presidential elections set for May 25.

"We do not mind having a referendum," Turchynov told members of parliament.

"Moreover, if there is (a) parliamentary decision to hold one together with the presidential elections, I am sure most people will express support of independent and unitary Ukraine."

Turchynov has said Russia was responsible for the bloodshed related to the most recent protests. At least one Ukrainian soldier was killed in Sunday clashes between pro-Ukrainian crowds and pro-Russian protesters, a high-level source in Ukraine's Security Services told CNN.

In Moscow, Russian Foreign Minister Sergey Lavrov said any use of force against pro-Russian demonstrators in Ukraine will make it more difficult to create a lasting solution to the crisis.

A stable Ukraine is in Moscow's best interest, Lavrov said, also denying reports that Russian forces are active in the east. He refused to speculate about what events or actions would spur a military intervention by Russia into Ukraine.

Lavrov said he would hold off judgment on Turchynov's proposed referendum until Moscow sees the outline of the questions to be asked. He added that pro-Russian activists in the east must be given an active role in shaping a new constitution.

'Evidence of Russian involvement'

Ukrainian officials have placed blame for unrest in the eastern section of their country squarely on Russia. The new Ukrainian government said it was launching security operations against terrorists who are attempting to "destroy our country."

Giving no further details, it also said it had "concrete evidence of Russian special service involvement" in the pro-Russian protests and storming of buildings and would present it at an international meeting on the Ukraine crisis on Thursday.

Samantha Power, the U.S. ambassador to the United Nations, said Sunday the attacks in Slaviansk were "professional" and "coordinated," similar to Russia's incursion into the Crimean Peninsula last month.

The United States is prepared to step up sanctions against Russia if the recent actions in Ukraine continue, she said. Speaking on ABC's "This Week," she said the latest events in Ukraine bore "the telltale signs of Moscow's involvement."

"I think we've seen that the sanctions can bite. And if actions like the kind that we've seen over the last few days continue, you're going to see a ramping up of those sanctions," she said.

Separately, the United States said a Russian plane made 12 passes near a U.S. Navy warship in the Black Sea on Saturday in the most direct confrontation between the United States and Russia in years.

The plane appeared to be unarmed but passed near the USS Donald Cook several times at both sea level and at several thousand feet.

Pentagon spokesman Col. Steve Warren called the Russian actions "provocative and unprofessional."

On Monday, a Russian Krivak-class frigate was shadowing the same Navy warship in the Black Sea, a U.S. military official said. The frigate remained within sight of the Cook but was not threatening it, the official said.

EU moves to expand sanctions, cutting tariffs for Ukraine

The European Union on Monday moved to widen sanctions and took another step to help Ukraine economically as Kiev's new government seeks closer ties to the West.

"In the European Union we are very strongly committed, as we are in the UK, to the sovereignty and the independence and the territorial integrity of Ukraine," British Foreign Secretary William Hague said in a statement.

"It is very hard to believe after recent events that Russia shares those objectives. I said this morning it is important to have a response and we have agreed among foreign ministers to expand the sanctions -- the second tier of sanctions - and to add further names to those sanctions. So we have taken that decision in the council today," Hague said about the EU's Foreign Affairs Council.

"We have also stressed the urgency of completing the work on a possible third tier of more far-reaching sanctions should those become necessary and the European Commission have assured us that that work is almost complete," he added.

The EU's Foreign Affairs Council also approved tariff cuts for most industrial and all agricultural goods, which will make it cheaper for Ukraine to export them to the European Union, saving Ukraine about €500 million a year, the council said.

The cuts will start at the end of April and last until November 1, by which time a fuller free-trade deal is expected to take effect.

European Union foreign policy chief Catherine Ashton is to meet this week with foreign ministers from the United States, Russia and Ukraine in Switzerland to discuss efforts to de-escalate the situation.

On Sunday night, the U.N. Security Council held an urgent, previously unscheduled meeting to discuss the worsening crisis, where strong condemnations and accusations were traded.

CNN's Arwa Damon, Khushbu Shah, Steve Almasy and Nick Paton Walsh; journalist Victoria Butenko in Kiev; and journalist Lena Kashkarova in Donetsk contributed to this report

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