Kyiv, Ukraine(CNN) As Ukraine ramps up its offensive to retake Russian-occupied territory in the country's south, its forces have regained four villages and broken through Russian defenses at multiple points of the frontline in the Kherson region, according to Ukrainian officials.
Ukraine's President Volodymyr Zelensky also issued a warning to Russian forces during his evening address Monday, saying it was time for them to "run away, go home," and that "the occupiers must know we will chase them to the border."
Yet one of his advisers cautioned that the operation will be a slow one, that would "grind the enemy."
"This process will not be very fast," Oleksiy Arestovych, adviser to the head of the office of the President of Ukraine, said in a statement posted on Telegram late Monday, "but will end with the installation of the Ukrainian flag over all the settlements of Ukraine."
"[The Ukrainian Armed Forces] have broken through the frontline in several sectors," Arestovych said in a video interview posted on YouTube, adding that Ukrainian forces were shelling ferry crossing points that Moscow is using to resupply Russian-occupied territory over the Dnipro River in the Kherson region.
"We suppress their attempts to supply their groups on the west bank [of the Dnipro]," he said. "We strike at reserves, including reserves that are on the east bank and are trying to cross."
Earlier Monday, a Ukrainian military source told CNN that Ukrainian troops had taken four villages -- Novodmytrivka, Arkhanhel's'ke, Tomyna Balka and Pravdyne -- back from Russian occupation, adding that the main target was the city of Kherson.
"The operation began at night with massive shelling of Russian positions and the rear," the source, who is not being named for security reasons, told CNN.
Russia's defense ministry acknowledged Kyiv's operation but claimed that Ukrainian troops had "suffered heavy losses" and "failed miserably" in their "attempted" offensive. CNN has not been able to verify both Russian and Ukrainian claims.
The Kherson region, crucial to controlling Ukraine's southern coast and access to the Black Sea, was one of the first areas to be occupied by Russian forces when they invaded more than six months ago.
In recent weeks, Ukrainian forces have been making advances around Kherson, including the targeting of critical bridges to disrupt Russian supply routes.
At a Tuesday briefing, Natalia Humeniuk, spokeswoman for the Ukrainian military's Operational Command South, said damage to bridges across the Dnipro river means that Russian forces are unable to assist units on the frontlines further north.
"Now they [the Russians] do not have the transport ability to pull up reserves from the Left bank. Therefore, they create the illusion of pulling up reserves by moving military units between the settlements ... All bridges have been worked by us as impassable," she said.
"They may continue to try to set up a ferry or pontoon crossing, but the whole area where it can be deployed is also under our fire control and will be hit."
Ukrainian forces also launched an attack at the Russian-held town of Nova Kakhovka on Monday, knocking out its electricity and water supply, according to Russian state media RIA Novosti, who cited a Russian-appointed local authority.
RIA quoted the head of civil-military administration in the town, Vladimir Leontyev, as saying the consequences of the strike "will probably be terrible."
"You can't leave now. There was a lot of shelling, something exploded. There were strikes in the city and near the hydroelectric plant. We can see the fire in both places," Leontyev added.
The counteroffensive comes as a delegation of 14 international experts from the International Atomic Energy Agency (IAEA) arrived in Kyiv ahead of a planned visit to the Russian-held Zaporizhzhia nuclear power plant in southern Ukraine later this week.
The nuclear plant, which is Europe's largest, has been under Russian control since March. Clashes around the complex have sparked widespread concern and fears of a disaster.
Kyiv has repeatedly accused Russian forces of storing heavy weaponry inside the complex and using it as cover to launch attacks, knowing that Ukraine can't return fire without risking hitting one of the plant's six reactors. Russia has claimed that they do not have any "heavy weaponry" at the nuclear power plant.
Satellite imagery from Maxar Technologies showed four holes in the roof of the plant's building, near at least three Russian armored personnel carriers.
In the satellite images, at least three Russian armored personnel carriers are seen sitting underneath a large structure with pipes, which feed from the building into all six of the nuclear reactors.
The Russian-appointed leader for the Zaporizhzhia region, Vladimir Rogov, claimed that the holes were the result of a Ukrainian military strike on the complex.
CNN has geolocated and confirmed the authenticity of the photos, but not the claims that the holes were the result of a Ukrainian military strike.
Ukraine and Russia continued to blame each other Tuesday for shelling the area around the plant. CNN is unable to verify who is responsible.
Russian authorities -- who now control the plant and the nearby city of Enerhodar -- reported artillery strikes early Tuesday morning close to a spent fuel storage building.
Kyiv blamed Moscow for the attack. Russia "is deliberately shelling corridors for [the] IAEA mission to reach ZNPP [the Zaporizhzhia nuclear power plant]," Mykhailo Podolyak, an adviser to Zelensky's Head of the Office, alleged in a tweet that reiterated Ukrainian demands for Russian troops to withdraw from the plant.
Concerns over a possible nuclear accident saw Zaporizhzhia city authorities handing out iodine pills to residents. On Tuesday, the European Union said it was donating 5.5 million potassium iodide tablets to Ukraine to protect people from potential radiation exposure.
"No nuclear power plant should ever be used as a war theater. It is unacceptable that civilian lives are put in danger. All military action around the Zaporizhzhia nuclear power plant must stop immediately," EU Commissioner for Crisis Management, Janez Lenarčič, said.
Meanwhile, US officials believe Ukraine's long anticipated counteroffensive will include a combination of air and ground operations.
John Kirby, the communications coordinator for the National Security Council, said Monday that regardless of the size, scale and scope of the latest counteroffensive, the Ukrainians "have already had an impact on Russian military capabilities."
"Because the Russians have had to pull resources from the east simply because of reports that the Ukrainians might be going more on the offense in the south," Kirby said. "And so they've had to deplete certain units ... in certain areas in the east in the Donbas (region), to respond to what they clearly believed was a looming threat of a counteroffensive."
Kirby also said that Russia "continues to have manpower problems" in Ukraine and is trying to expand its recruitment of fighters inside Russia as well as "entice" some of their conscripts and contract soldiers to serve beyond their timeframes.