After weeks of long discussions and little progress, cross-party talks between the government and Labour are over.

What’s the next step?

Theresa May has committed to giving MPs a vote on her key piece of Brexit legislation, the withdrawal agreement bill (Wab), in the first week of June.

It is unclear whether there will be any votes before that; but one proposal presented to Labour – and rejected – was for a series of indicative votes, next week.

That could have included a free vote on the prospect of submitting any deal passed by parliament to a referendum.

Could that lead to a breakthrough?

It’s hard to see how. The first two attempts at such a plan, instigated by backbenchers, ended in deadlock. And we now know that Theresa May seems set to leave Downing Street next month. Would her successor – who might have a very different idea of Brexit – want to walk through the door of No 10 with their hands already tied?

But the idea would be that whatever emerged from the process of votes would then be incorporated into the Wab, helping to build a majority for it.

How else could May get her deal through?

Downing Street sources said the government would incorporate some of the “progress” made in talks with Labour, most likely on issues including workers’ rights and environmental standards, into the Wab.

They hope that could win over a bloc of Labour MPs from leave seats who couldn’t vote for an unchanged deal.

At the same time, the government is reopening discussions with the DUP about how to reassure them the deal would not lead to a border down the Irish Sea; and also speaking to Tory leavers with concerns about the backstop.

What if that’s not enough to secure a majority in June?

Focus would then shift to a likely brutal Tory leadership election, possibly taking up much of the summer. The odds are narrowing on a hard Brexit-backing MP, such as Boris Johnson, winning, especially if, as expected, Nigel Farage’s Brexit party win seats in next week’s European elections, and the Tories are pushed to fourth or even fifth. A new leader would certainly want to spend some time drawing up a new departure plan, talking it over with Brussels, and then potentially putting it to MPs. Someone like Johnson could soon declare a structured departure impossible and move to no deal.

Will that finally settle things?

Not really. Both the EU’s position and the arithmetic of the Commons will be the same, whether it is May or someone else inside No 10. Any Brexiter demands for Brussels to ditch the Irish backstop plan will be given short shrift, while MPs have already rejected the idea of no deal.

How could the deadlock be settled?

Perhaps a general election, something many people think is possible once a new prime minister reaches their own impasse. This would then raise other imponderables, for example, whether Labour would fight the election promising a confirmatory Brexit referendum, something backed by the bulk of its members and many MPs, but resisted by Jeremy Corbyn.

It also remains to be seen whether the Conservatives might seek some sort of pact with Farage’s new party. Brexiters would be keen, but some Tory MPs have already declared this to be impossible for them.

Another possibility is that a Tory Brexiter who wanted to leave without a deal could even go over the heads of the public, and embrace the idea of a referendum to endorse their plan.

So where will it all end?

As ever with Brexit, no one really knows, but the current firm deadline is 31 October – the extended departure date offered by the EU and agreed by the prime minister in April. At that point, the default is, again, that the UK will leave without a deal. But much will happen before then – not least the selection of a new prime minster.