U.S. Secretary of State Mike Pompeo on Tuesday urged Armenia and Azerbaijan to pursue a diplomatic solution to the Nagorno-Karabakh conflict as fighting in and around the region continues as the ceasefire brokered in Washington failed again.

Armenia acknowledged overnight that Nagorno-Karabakh forces had withdrawn from a strategic town between the mountain enclave and the Iranian border.

Both sides accused each other on Tuesday of striking targets outside Nagorno-Karabakh itself in defiance of a truce brokered by Pompeo at the weekend.

Pompeo, in India on Tuesday, spoke separately with Armenian Prime Minister Nikol Pashinyan and Azeri President Ilham Aliyev over phone and "pressed the leaders to abide by their commitments to cease hostilities and pursue a diplomatic solution," the State Department said.

Azerbaijan rejects any solution that would leave Armenians in control of the enclave that is part of Azerbaijan but populated and controlled by ethnic Armenians.

Armenia says it will not withdraw from territory it views as part of its historic homeland and where the population needs protection.

The ethnic Armenian-controlled Nagorno-Karabakh "defense ministry" said its military had recorded 1,009 deaths since the fighting erupted on September 27. Azerbaijan has not disclosed its military casualties. Russia has estimated as many as 5,000 people have been killed in the conflict.

What the major powers say?

World powers want to prevent a wider war that might suck in Turkey, an ally of Azerbaijan, and Russia, which has a defense pact with Armenia. The conflict is also close to pipelines that carry oil and gas from Azerbaijan to international markets.

U.S. President Donald Trump said on Tuesday that it was disappointing to see the U.S.-brokered ceasefire collapsed. "It's disappointing to see that, but that's what happens when you have countries that have been going at it for a long time."

Iran's foreign ministry said on Twitter that Deputy Foreign Minister Abbas Araqchi would travel to several countries including Turkey and Russia to discuss the crisis.

And Iran's Foreign Minister Mohammad Javad Zarif said on Tuesday that Iran has prepared a peace proposal for the regional conflict. "Iran's proposal for permanent resolution of the conflict will be tabled either today or tomorrow."

Turkish Foreign Minister Mevlut Cavusoglu and his Russian counterpart, Sergei Lavrov, discussed Nagorno-Karabakh in a phone call. Moscow said they discussed an immediate ceasefire.

The OSCE Minsk Group, formed to mediate the conflict and led by France, Russia and the United States, is scheduled to meet the Armenian and Azeri foreign ministers in Geneva on October 29. Turkey has demanded a bigger role in the mediating body.

(With input from agencies)