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A rescuer wipes their face while working in a residential building heavily damaged by a Russian missile strike, amid Russia's attack on Ukraine, in Kryvyi Rih, Dnipropetrovsk region, on June 13.ALINA SMUTKO/Reuters

Day 475 of the war for Ukraine began with yet more tragedy for this country. But it ended with mounting hope for a counteroffensive aimed at driving the invading Russian army out of territory it has held for most of the past 15 months.

An early Tuesday morning cruise-missile strike slammed into a five-storey apartment building and nearby warehouse, killing at least 11 people in the southern city of Kryvyi Rih, the hometown of Ukrainian President Volodymyr Zelensky. The attack joins a long list of atrocities that have killed large numbers of civilians over the course of this war, including blasts at a crowded train station in the eastern city of Kramatorsk and a busy shopping mall in central Kremenchuk, as well as previous strikes on apartment blocks in the cities of Kyiv, Dnipro, Odesa and Uman, where – as in Kryvyi Rih – residents were killed as they slept.

“More terrorist missiles, Russian killers continue their war against residential buildings, ordinary cities and people,” Mr. Zelensky wrote Tuesday on his Telegram channel. “Terrorists will never be forgiven, and they will be held accountable for every missile they launch.”

Accountability feels a long way off in southern Ukraine. Kryvyi Rih sits a short drive from the shores of the Kakhovka Reservoir, which continues to drain one week after an explosion destroyed the massive Nova Kakhovka dam, unleashing a torrent of water that drove thousands of people from their homes. Ukraine has accused Russia of intentionally destroying the dam, and Mr. Zelensky has called it one Russian war crime among many.

The growing optimism in this country comes from a large-scale counteroffensive that began earlier this month. The push, which involves tens of thousands of freshly trained troops backed by donated Western tanks and long-range missiles, has resulted in only incremental gains toward Ukraine’s apparent goal of reaching the shores of the Azov Sea.

Ukrainian Deputy Defence Minister Hanna Maliar said in a statement that the country’s forces pushed south another kilometre in the direction of the Azov Sea port of Berdyansk on Tuesday, a day after geolocated videos showed Ukrainian troops moving through a series of small villages that had previously been on the Russian side of the front line in the Donetsk and Zaporizhzhia regions. Ukrainian forces have gained about 90 square kilometres of territory since the offensive began, Ms. Maliar said.

The progress is slow, and there’s no indication the main Russian defensive lines have been breached, but the push toward the Azov Sea coast – about 100 kilometres away from the pre-counteroffensive front line – raises the possibility of cutting the so-called “land bridge” connecting Russia to the Crimean Peninsula.

Russia has occupied Crimea since 2014. Reaching Berdyansk or the larger port of Mariupol would allow Ukraine to sever the land bridge, a vital supply line, and undo the main strategic gain Russia has achieved in more than 15 months of war.

Ms. Maliar described the fighting as fierce. “The enemy is doing everything to keep the positions they have captured. They actively use assault and army aviation, and conduct intense artillery fire. During the offensive, our troops encounter continuous minefields, which are combined with anti-tank ditches. All this is combined with constant counterattacks by enemy units on armoured vehicles and the massive use of ATGMs and kamikaze drones,” Ms. Maliar wrote, using the acronym for anti-tank guided missiles.

The two main axes of attack appear to be south from the Donetsk region town of Velyka Novosilka and south from the Zaporizhzhia region town of Orikhiv.

NATO Secretary-General Jens Stoltenberg also predicted that the battles ahead would be difficult. “What we have seen is that Ukrainians have the courage, the determination to liberate land … and push back the Russians,” Mr. Stoltenberg said during a visit to Washington, where he met with U.S. Secretary of State Antony Blinken.

Mr. Blinken announced a fresh US$325-million package of U.S. military support for Ukraine, which he said in a statement would include “critical air defence capabilities, additional munitions for High Mobility Artillery Rocket Systems, artillery rounds, anti-tank weapons, armoured vehicles, and other equipment essential to strengthening Ukraine’s forces on the battlefield.”

Russian President Vladimir Putin denied on Tuesday that Ukraine had made any gains since the counterattack began, and claimed that Ukraine’s losses were 10 times higher than Russia’s. Both sides regularly make unverifiable claims about the level of casualties suffered by the enemy.

Reporters from Reuters and the BBC reached the village of Neskuchne, in the Donetsk region, on Tuesday, where they confirmed that the tiny settlement – which was under Russian occupation for more than a year – was now under the control of Ukrainian forces. But it was devoid of residents, and fighting was still audible nearby.

The Ukrainian military posted a video on Tuesday showing a column of armoured vehicles from its 35th Marine Brigade moving through the nearby town of Makarivka, which Ukraine captured on Monday along with Neskuchne. Both villages are on the Velyka Novosilka front.

The Ukrainian counterattack follows a months-long Russian offensive that finally resulted in the capture of the eastern city of Bakhmut, which had been besieged since last summer. But Russia otherwise failed to make significant progress, despite tens of thousands of casualties on both sides.

In a televised exchange with Russian military bloggers and war correspondents, Mr. Putin said he was open to negotiations over Ukraine, but said the West must stop supplying Ukraine with weapons for peace to be achieved.

Russia’s defence ministry released videos on Tuesday of what appeared to be destroyed and captured Western military equipment, including Leopard 2 tanks – over 100 of which were given to Ukraine by a coalition of countries, including Canada, which gave eight – and U.S.-donated Bradley Fighting Vehicles. It wasn’t clear where or when the footage was shot.

In his remarks, Mr. Putin made several threats aimed at Ukraine and the West. He suggested that Russia might need to create a “sanitary zone” on Ukrainian territory to prevent cross-border attacks, after a series of raids into Russia’s Belgorod region last month, or that it might even try again to capture the Ukrainian capital of Kyiv, which Russian forces attempted and failed to do at the start of the invasion. Mr. Putin said he “has plans,” but that his next move would come after the end of the Ukrainian counteroffensive.

Mr. Putin also said Russia might abandon a Black Sea grain deal that allows Ukraine to export some of its harvest through a Russian blockade.

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