During the darkest days of the Brexit crisis, Theresa May allies kept their spirits up by describing her “Bee Gees strategy” – a reference to the group’s hit “Stayin’ Alive”.

This goal is what lies behind Tuesday night’s announcement that MPs will hold a fourth vote on May’s Brexit deal in early June, when she tables the withdrawal agreement bill.

It will be her last stand. If she loses the second reading vote, she will have nowhere to go but to walk out of No 10’s heavy black door for good.

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Three immediate pressures led to May deciding on one last desperate roll of the dice.

On Thursday she meets leaders of the 1922 Committee of Tory MPs to answer their call for a firm departure timetable.

Tabling the bill will buy her a few more weeks. Secondly, May came under pressure at Tuesday’s longer than expected cabinet meeting to “do something” to answer the accurate charge that we have a zombie government and parliament.

Thirdly, May hopes that at least trying to leave the EU will stem Tory losses at next week’s European parliament elections. I doubt it will. Nigel Farage is on the rampage. His new Brexit Party will surely inflict grievous bodily harm on a governing “Brexit party” that has woefully failed to deliver it. No need for nuance there, which is just how Farage likes it.

With her own MPs and Tory grassroots leaders (who meet on 15 June) plotting to remove her, May did indeed need to do something. But, on the face of it, her gamble is reckless because she will lose the critical Commons vote.

Her deal lost by 58 votes last time. Some Tories who grudgingly backed it then say they will oppose it next time. Eurosceptics know May will have to resign if they defeat her and so have every incentive to vote against the bill.

Next week’s inevitable Tory meltdown will not (as she hopes) make them more likely to back her deal; they are already blaming her for it. The Democratic Unionist Party remains opposed to her deal.

So, to have any chance in the Commons vote, May will need Labour backing. Although her talks with the opposition limp on, they are on their last legs. Downing Street talks up their prospects but the Labour mood is downbeat. Jeremy Corbyn was never going to prop up an ailing Tory prime minister. He wanted to be seen to negotiate in good faith, but knew he could always find an escape hatch.

He now has several. May’s offer of a customs arrangement will not cover services, so she can keep a fig leaf of an “independent trade policy” and deals with non-EU countries.

Labour will argue that this would open up NHS contracts to the US health giants.

More fundamentally, May cannot “entrench” any agreement with Corbyn to prevent her successor ripping it up.

Labour’s fears of this are justified. Tory leadership candidates pander to the party’s MPs and members by stating their opposition to a customs union.

Even the sensible ones like Jeremy Hunt flirt with a no-deal exit. This will intensify when the contest begins officially.

Corbyn knows that doing a deal with May would widen the already big enough divide in his party. He is under mounting pressure from his MPs to attach a Final Say referendum to any agreement, and doesn’t want one. Why advertise Labour splits when he does not need to, and can allow the spotlight to fall on Tory divisions? It’s called the luxury of opposition.

So Corbyn will not back the bill or abstain, as some loyalist ministers hope. The real question is whether May could be rescued by Labour backbenchers representing areas that voted Leave in the 2016 referendum.

A group of 20 to 30 of them see circumstances in which they might support May’s deal. Next month’s vote will be their last chance. However, those most likely to rebel talk a good game but have not mustered more than a handful of votes for the deal so far.

May hopes that would change if the government accepted Labour backbench amendments to the bill. But she would probably need around 40 Labour votes to wipe out the Tory rebellion. The numbers do not add up for her.

May’s move will keep her stayin’ alive for a few weeks. But not much longer.

When the bill is sunk, I suspect MPs will face a stark choice in October when the UK’s Article 50 extension runs out: no deal or no Brexit. No deal is winning growing support on the Tory benches, and the incoming leader will almost certainly threaten the EU with that outcome.

Meanwhile, the nuclear option of revoking Article 50 and a more palatable Final Say vote is gaining ground in Labour and even Tory land. As one minister told me: “It will boil down to either no-deal or a referendum.”

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