TOKYO/GYEONGJU, South Korea (Reuters) -U.S. President Donald Trump heads to South Korea on Wednesday for the final leg of his Asia trip, with high-stakes meetings expected with Chinese President Xi Jinping and South Korea's Lee Jae Myung.

After arriving on a flight from Tokyo, where he signed a rare earths deal with Japan's new prime minister, Trump is due to address a summit of CEOs and meet with Lee in Gyeongju, a sleepy South Korean town filled with historic tombs and palaces.

At the top of the agenda will be the unresolved trade agreement between the U.S. and South Korea.

The two allies announced a deal in August under which South Korea would avoid the worst of the tariffs by agreeing to pump $350 billion of new investments into the United States.

But talks over the structure of those investments have been deadlocked, and officials from both sides have said Trump and Lee are unlikely to finalise an agreement.

Trump has also pressed allies like South Korea to pay more for defence, and South Korea has sought reforms to U.S. immigration laws to allow for more workers to build factories after a raid on a Hyundai Motor battery plant in Georgia.

NORTH KOREA TEST-FIRES MISSILES

Trump and Lee are likely to discuss efforts to engage North Korea, which announced early on Wednesday that it had test-fired a nuclear-capable cruise missile the day before.

"It is our responsible mission and duty to ceaselessly toughen the nuclear combat posture," the North Korean official who oversaw the test said, according to state news agency KCNA.

Last week, the North fired its first ballistic missile since May.

Trump has made repeated calls for a meeting with leader Kim Jong Un, including during this trip, but there has been no public comment from Pyongyang. Kim has previously said he could be open to talking if Washington stops pressing him to give up nuclear weapons.

TRADE AGENDA

Filled with thousands of police and soldiers for security, Gyeongju will host the Asia-Pacific Economic Cooperation forum this week, but Trump will skip the leaders' summit scheduled for Friday and Saturday.

"Trump dislikes large international gatherings and prefers to have one-on-one meetings with key leaders," said Christopher Padilla, senior adviser at advisory firm Brunswick Group in Washington. "But while the U.S. steps back, most of the world has continued to work through such institutions, finding them a useful source of cooperation on international problems."

Instead, Trump will address the APEC CEO summit, hold bilateral meetings with several countries' leaders, including China's Xi, and have dinner with Lee.

The Xi-Trump meeting, expected on Thursday, is overshadowing the rest of the week's busy diplomatic schedule.

Negotiators from the world's top two economies hashed out a framework on Sunday for a deal to pause steeper American tariffs and Chinese rare earths export controls, U.S. officials said. The news sent Asian stocks soaring to record peaks.

Taiwan Foreign Minister Lin Chia-lung said on Tuesday that he was not worried that Trump would "abandon" the island in his meeting this week with Xi.

Since taking office in January, Trump has vacillated on his position towards China-claimed Taiwan as he seeks to strike a trade deal with Beijing. Trump says Xi has told him he will not invade Taiwan while the Republican president is in office, but Trump has yet to approve any new U.S. arms sales to Taipei.

FINAL STOP IN ASIA TRIP

Trump is arriving in Gyeongju after a whirlwind swing through the region, among the hardest hit by his tariff policies and increased U.S.-China competition.

In Malaysia, he announced a slew of trade agreements on the sidelines of the 11-member Association of Southeast Asian Nations summit and oversaw the signing of an expanded truce between Thailand and Cambodia after a border conflict.

In Tokyo on Tuesday, Trump lavished praise on Japan's first female Prime Minister Sanae Takaichi, welcoming her pledge to accelerate a military buildup and signing deals on trade and rare earths.

Takaichi applauded Trump's push to resolve global conflicts, vowing to nominate him for the Nobel Peace Prize, according to Trump's spokeswoman, Karoline Leavitt.

The U.S. and Japan also released a list of projects in the areas of energy, artificial intelligence and critical minerals in which Japanese companies are eyeing investments of up to $400 billion.

Tokyo pledged to provide $550 billion of strategic U.S. investments, loans and guarantees earlier this year as part of a deal to win reprieve from Trump's punishing import tariffs.

Washington has pressed South Korea to make a similar arrangement, but Seoul says it cannot afford to pay the $350 billion it pledged upfront. Instead, South Korea has offered a mix of phased investments, loans and other measures.

On Tuesday, South Korean Foreign Minister Cho Hyun said a last-minute concession by the United States could lead to a deal.