Secretary of State Mike Pompeo has wrapped up his fourth visit to North Korea after meeting Kim Jong Un to seek elusive progress in efforts to persuade him to give up his nuclear weapons.

Pompeo tweeted on his arrival in Seoul he had met with Kim and they "continue to make progress on agreements made at Singapore Summit."

Had a good trip to #Pyongyang to meet with Chairman Kim. We continue to make progress on agreements made at Singapore Summit. Thanks for hosting me and my team @StateDept pic.twitter.com/mufyOKkDLw — Secretary Pompeo (@SecPompeo) October 7, 2018

He said, "Thanks for hosting me and my team."

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The two met for roughly two hours during their private meeting away from cameras. Their meeting comes as President Trump and the North Korean dictator intend on sitting down for another summit to continue work on verifiable denuclearization. The secretary told CBS News late last month that while a location and date have not been set yet, he would personally be traveling to Pyongyang in order to get the conditions for the next summit "just right."

In a readout of the meeting, State Department Spokesperson Heather Nauert said the meeting one of "productive conversions" and says the two leaders "refined options for the location and date" of the next Kim Jong Un-Trump summit.

"Secretary Pompeo and Chairman Kim also agreed to instruct their respective working-level teams to meet soon to intensify discussions on the key remaining issues to deliver on the Singapore Summit Joint Statement. In addition, Chairman Kim invited inspectors to visit the Punggye Ri nuclear test site to confirm it has been irreversibly dismantled," said Nauert in a statement Sunday.

Following private talks, the two exchanged pleasantries as Kim told Pompeo he was "really pleased" for the opportunity to share a meal together after having such a "nice meeting."

"This will be great," Pompeo told Kim in agreement. Kim remarked it was the top U.S. diplomat's fourth time visiting the region, noting, "so I am sure that you wouldn't have those first strange feelings that you would have when you were first coming." Pompeo agreed. Pompeo and Kim did not meet during Pompeo's last trip to North Korea in July.

As Pompeo and Kim seated for their lunch meeting, Kim, without specifically mentioning any topics up for discussion, said it was a "very nice day that promises a good future for both countries."

North Korean leader Kim Jong Un (right) seated with Secretary of State Mike Pompeo, flanked by translators, for lunch in Pyongyang on Sun., Oct. 7, 2018 CBS News/Kylie Atwood

"We had a great, great visit this morning. President Trump sends his, his regards. And we had a very successful morning so thank you and I am looking forward to our time here at lunch as well," Pompeo replied.

Pompeo had flown to Pyongyang from Tokyo after talks there with Japan's prime minister during which he pledged the Trump administration would coordinate and unify its strategy for denuclearization with allies. Japan has been wary of the initiative but South Korea has embraced it.

While in Seoul, Pompeo will meet with President Moon Jae-in and Foreign Minister Kang Kyung-wha.

CBS News' Kylie Atwood contributed to this report.