KABUL, Afghanistan -- The U.S. military has identified an American soldier killed fighting likely ISIS militants on New Year's Day in eastern Afghanistan.

In a statement released Wednesday, the Pentagon said Sgt. 1st Class Mihail Golin, 34, of Fort Lee, New Jersey, was killed, "after being engaged by enemy small arms fire while on a dismounted patrol." Golin was assigned to the 2nd Battalion, 10th Special Forces Group (Airborne), based at Fort Carson, Colorado, the statement said.

A military statement issued Tuesday said four other soldiers were wounded in the Jan. 1 battle in the Achin district of Nangarhar province. It said two of the wounded were in stable condition and the other two have returned to duty.

The United States formally concluded its combat mission in Afghanistan at the end of 2014 but still carries out operations against the Taliban and the Islamic State of Iraq and Syria (ISIS) affiliate, known as ISIS Khorasan. While both militants groups are active in Nangarhar, it is largely ISIS territory.

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ISIS Khorasan claimed to have killed a number of Afghan troops in an attack on a military post in Achin district on Jan 1. A statement by the group said three ISIS commandos armed with light machine guns engaged troops at a military post, killing and injuring 10 "apostates." ISIS frequently exaggerates its own victories on the battlefield, but there was no mention in its statement of any U.S. troops being present during the attack.

A U.S. military official tells CBS New Radio correspondent Cami McCormick that American forces have had recent success in Achin -- "aggressively" clearing a significant amount of territory over the past weeks. He said the U.S. and Afghan Special Forces would continue to apply pressure on the ISIS faction in the country and force them to move back from their front line positions.

He said the fighting was taking place in very rugged territory with poor weather conditions, but added that the "ground movements" would continue, and there would be absolutely no respite for the militants over the winter months.

"It will be an extremely cold winter for ISIS-K because they will be on the run," he said.

During an unannounced visit to Afghanistan last month, Vice President Mike Pence told CBS News White House and senior foreign affairs correspondent Margaret Brennan that American forces would stand "shoulder to shoulder" with their Afghan counterparts as they fight the Taliban and other terror groups.

"It's important for the American people to know that President Trump's view of this is that the Afghan people are leading this fight," Pence told Brennan. "Our armed forces here, with new freedom, new flexibility and new resources, are supporting their efforts to take the fight to the enemies of both of our countries."