"It was another step forward", Pompeo said in a meeting with Moon at the Blue House.
"It's a very nice day that promises a good future for both countries", Kim told Pompeo through a translator as they sat down for the meal. President Trump sends his regards. Both men spoke through translators.
Pompeo and Kim also discussed "denuclearisation steps that will be taken by North Korea and the issue of attendance by the USA government", as well as "corresponding measures" to be taken by the U.S., the statement said.
Last month, Abe hinted that he was prepared to meet with North Korean leader Kim Jong Un "to break the shell of mutual distrust".
She also said she believed the two countries "wouldn't be here without Trump".
He was referring to the meeting between Kim and President Donald Trump in June in Singapore.
In a more upbeat note, Pyongyang's state media said on Monday Kim lauded his talks with Pompeo, where Kim "explained in detail the proposals for solving the denuclearization issue".
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In recent weeks, USA officials have pointed to a recent meeting in Pyongyang between Kim and Moon, when North Korea committed to dismantling a missile-engine-testing site and raised the possibility of doing the same at its Yongbyon nuclear-enrichment facility. He said he would brief Moon in private but added he had "a good, productive conversation" with Kim.
"As President Trump said, there are many steps along the way, and we took one of them today". "It was another step forward".
But even on that, he said, "I doubt we'll get it nailed" down on this trip.
Moon expressed hopes that Pompeo's trip and the proposed second meeting between Kim and Trump would make "irreversible, decisive progress in terms of denuclearisation as well as the peace process".
Pompeo said it was important to hear from the Japanese leader "so we have a fully coordinated and unified view".
"They also discussed the upcoming second summit between President Trump and Chairman Kim and refined options for the location and date of that next summit", the statement read.
North Korea so far has suspended nuclear and missile tests, freed three American prisoners and dismantled parts of a missile engine facility and tunnel entrances at a nuclear test site.
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Pompeo was joined by US special representative on DPRK issues Stephen Biegun, and Kim was joined by Kim Yo Jong, first vice department director of the Central Committee of the ruling Worker's Party of Korea (WPK), the statement said.
Pompeo also discussed additional denuclearization steps the reclusive regime can take, as well as Washington's monitoring of the verification process and "corresponding measures", according to Yoon.
"South Korea seems overly eager to improve relations and offer benefits to North Korea without requiring commensurate progress in reducing the North Korean security threat", said Klinger. Instead, they want both sides to take a "phased" approach, where Pyongyang is rewarded as it takes gradual steps to roll back its nuclear program. "We continue to make progress on agreements made at Singapore summit". USA officials have maintained, however, that verified denuclearization remains a precondition for sanctions relief.
While North Korea blew up the entrances of the tunnels in May and removed the diagnostic equipment, the shafts could be reopened or new tunnels could be excavated if Kim chose to restart underground test blasts, he said. The war concluded with an armistice but no peace treaty.
With talks largely deadlocked, at the heart of this round of negotiations is a debate over what the USA should demand from North Korea and what it is willing to give in exchange.
At his last meeting with Moon, he expressed willingness to allow outside observations of key missile facilities and, for the first time, to "permanently" scrap North Korea's main nuclear complex in Yongbyon.
Also on Monday, South Korean President Moon Jae-in said he expects Kim to travel to Russian Federation and for Chinese President Xi Jinping to visit North Korea soon amid a global diplomatic push to resolve the nuclear crisis.
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