For decades, the West - and Washington particularly - has asked itself the question: how do you solve a problem like North Korea?

Now it may be more urgent than ever as Kim Jong Un and Vladimir Putin's heavily sanctioned regimes enter what appears to be a new era of friendship. But the US, it appears, is out of ideas.

Details of any deal Moscow and Pyongyang may have struck are scant, but the biggest concerns are over the possibility of Russia sharing technology, particularly involving ballistic missiles or nuclear submarine systems.

So far Washington has responded with derisive swipes - Mr Putin is so desperate he is "begging" a pariah nation for help, and had to travel "hat in hand" across his country - and warnings of serious, and as yet unspecified, consequences.

But there isn't much left in the American economic toolkit to lob at North Korea. As one State Department reporter joked, there may be a corner shop or two in Pyongyang that haven't been targeted, but that's about it.

So how can President Joe Biden, who has been busy building alliances to counter China, bring the mercurial North Korean leader to the table?

During a visit to Seoul in May last year, President Biden was asked by reporters whether he had any message for Mr Kim. He responded: "Hello. Period."

"If Mr Biden was indeed open to talks, as the State Department seems to suggest, this was a funny way of showing it," said Frank Aum, a northeast Asia expert from the United States Institute of Peace.

He says this was yet another missed opportunity: "A series of missteps and missed opportunities by both sides over the last seven decades cumulatively led to the intractable situation we have today."

Mr Kim may seem uninterested. He has not responded to Washington's current offer to engage in talks anywhere and at any time. But he has also done his best to prove that he is a worthy threat as he continues to build his nuclear arsenal. He has carried out more than a 100 missile tests since 2022 and has twice tried, and failed, to launch a spy satellite - all this while he was under the strictest sanctions ever imposed on his regime.

"I think the US underestimated how resilient and determined North Korea was to counter international pressure," Mr Aum said. "Many people across different administrations probably characterised the North Korea issue as a third-tier security concern and didn't accord it the full attention it required, either because it was a small country or constantly appeared on the brink of collapse."

Experts believe this put Mr Kim further down the priority list than he would like.

"I think the US has just decided that a strategy of deterrence and containment is the best that they can do. And I can understand why. We are in a situation where North Korea can rely on Russia and China more than has been the case in decades. What exactly does the US have that North Korea would respond to?"

He says "creative diplomacy would be welcome", but adds, "I don't think there's a way forward right now".

In 2017, Pyongyang claimed it had successfully created a miniaturised hydrogen bomb designed to fit inside its missiles - a major step in its nuclear ambitions.

Then US President Donald Trump threatened North Korea with "fire and fury like the world has never seen", ramping up tensions.

Mr Kim then declared he had a complete nuclear arsenal and a button on his desk - a deadly threat that he hoped to leverage down the line to get relief from sanctions.

Mr Trump eventually offered him unprecedented talks and the two first shook hands in Singapore in June 2018.

While this may count as "creative diplomacy", both leaders appeared to have come to the three meetings they had - in Singapore, Hanoi and at the Korean border - ill-prepared to negotiate.

But those summits transformed diplomacy between North Korea and the US. They raised expectations in Pyongyang that all it took for a deal to happen was face-to-face conversation. And it offered Mr Kim a diplomatic spectacle just a year after he had shown off his military might.

The pivotal moment came at the Hanoi summit in February 2019.

Mr Trump reportedly offered Mr Kim partial sanctions relief - in exchange North Korea would have to give up its major nuclear facility in Yongbyon. Mr Kim ignored that offer, according to John Bolton, Mr Trump's then national security advisor. He then had a very long train journey back home with nothing to show his people.

There was a third meeting later that year in the Demilitarized Zone (DMZ), an area that divides the two Koreas. It was no less of a spectacle but it yielded little. The damage had been done.

"After the bruising failure of the Hanoi talks, I believe Kim was even more determined to emerge in a stronger position — with a bigger, more threatening arsenal," said Jean Lee, a North Korea expert who opened the Associated Press' first bureau in Pyongyang.

"He was surprised that his arsenal wasn't enough in 2018 and 2019 to compel the US to make a deal. He has used the time during the three years of Covid isolation to recalibrate and build up an arsenal that he believes will put him in a better position and give him better leverage. And he may be waiting to see what happens with the next US election in the chance President Trump — with whom he has a strong rapport — may be back in the White House."

Since the Trump-Kim summits, China has undoubtedly emerged as the thornier concern for Washington.

Mr Biden and his administration have made a huge diplomatic push in Asia - from restarting high-level dialogue with China, to strengthening alliances and increasing their influence in the region. But it appears to have come at the cost of ignoring North Korea for too long - and to the point where there seem to be no diplomatic channels.

Meanwhile Beijing, which still has a relationship with both Moscow and Pyongyang, is also a crucial player in US calculations regarding North Korea.

For one, it likes stability in the region. Greater cooperation between Russia and North Korea could bring more US military assets to East Asia, which Beijing will not want. Mr Xi has also tried to position himself as a global peacemaker, and has outlined a Chinese solution to the war in Ukraine. He would not want to be seen supporting Pyongyang or Moscow in any efforts which might prolong that war.

On the other hand, China might also see a Russia-North Korea alliance as a way to counter US influence in Asia.

"I can imagine a lot of the reporting around a possible North Korea, China and Russia relationship could revolve around a "new Cold War" narrative," Mr Green said, but dismissed it as an "oversimplification".

"I suspect that Kim Jong Un would like to diversify his relationships and play China and Russia off against one another to get what he needs," he added. "What the US needs to do now is look at what opportunities that may offer them."