Mike Pompeo is believed to have met with Kim during this trip. In August, Trump canceled Pompeo's trip to North Korea because they were "not making sufficient progress" in the denuclearization of the Korean Peninsula, he said.
Pompeo said he and Kim discussed denuclearization steps to be taken by the North and the issue of USA government monitoring of those actions, which Washington sees as vital, as well as the measures the United States would conduct in return, Moon's office said.
Pompeo tweeted he had "a good trip to Pyongyang" and "continue to make progress on agreements made" at Singapore summit held between U.S. President Donald Trump and Kim. Progress made on Singapore Summit Agreements!
Pompeo told South Korean President Moon Jae-in, "I will surely tell you in private about our conversation".
South Korea has moved forward with various joint projects, but the US has remained skeptical of North Korea's intentions, and officials have criticized what they view as a slow pace to Kim Jong Un's denuclearization.
In their meeting, Abe and Pompeo agreed to urge North Korea to take concrete steps toward denuclearization and to coordinate their policies aimed at compelling that outcome.
Pompeo and Kim had lunch together at a guesthouse, where the North Korean leader said through an interpreter, "It's a very nice day that promises a good future, for both countries".
In addition to government officials, the South Korean delegation includes lawmakers, civic and religious leaders and the son of late South Korean President Roh Moo-hyun, who participated in the 2007 summit with North Korea's then-leader Kim Jong Il, the father of current ruler Kim Jong Un. Details of the discussions weren't immediately available. On Wednesday, he said there was unanimous support for this at last week's UN General Assembly, even if Russian Federation and China "had some ideas about how we might begin to think about a time when it would be appropriate to reduce them".
Pompeo said in Tokyo that it was important to hear from the Japanese leader "so we have a fully coordinated and unified view".
"Then we are also going to set up the next summit", said Pompeo.
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He said Amazon needs to communicate to workers better if overall compensation is higher "or you're going to have morale problems". Workers in different parts of the world went on strike during the company's Prime Day promotion over the summer.
However, he played down expectations for a major breakthrough. "I look forward to seeing Chairman Kim again, in the near future".
"Past experience shows that the list and the verification about the list takes a lot of back-and-forth, and I think the last time things broke down precisely as we were working out a detailed protocol on verification after we had gotten the list", she said. Prior to this year, leads of North and South Korea had only met twice, but since April, they have met three times-most recently last month, when they drew up a new agreement.
Moon and Kim held their third summit in Pyongyang last month, where Kim reaffirmed his willingness to achieve denuclearization on the Korean Peninsula.
Trump scrapped a previously planned trip by his top diplomat to Pyongyang after what he said was insufficient progress towards implementing the terms of the Singapore declaration.
But the unorthodox USA president has also since declared himself "in love" with Kim.
"None of these have addressed the core task on denuclearization, which is declaration, verification and timeline", said Victor Cha, the Korea chair at CSIS who was at one point Trump's leading candidate for USA ambassador to South Korea.
But clearly, the "corresponding steps" from the United States are critical: Foreign Minister Ri told the UN General Assembly last month that there was "no way" it would denuclearize without U.S. concessions.
Analysts say Washington may now consider new options as China, Russia and South Korea seek to relax sanctions.
The North has said this can't be used as a bargaining chip for denuclearization.