The European Union needs an ambitious grand bargain at its next summit to rescue itself from an accumulation of crises that threaten to blow apart its model of integration.
Like children at a birthday party, each leader has to get a going-home present. And as with many children's parties, there may be a tantrum along the way.
German Chancellor Angela Merkel is at risk at home from a backlash against a mass influx of Syrian refugees. British Prime Minister David Cameron is trying to win a referendum on staying in the EU and cannot help her due to public hostility to immigration. Nor can French President Francois Hollande, who is struggling for re-election in a country transfixed by the threat from Islamist militants.
Berlin, the EU's pivotal power, sees controlling migration as the central priority as it tries to cope with one million asylum seekers who have arrived in the last year.
Ideally, EU leaders need to forge a deal on Feb. 18-19 that encompasses effective action to strengthen Europe's external borders and share the refugee burden, and a mutually acceptable solution to Britain's demands to change its membership terms.
A comprehensive package would also need to address Italy's political and economic frustrations with the EU, Greece's quest for substantial debt relief, and Poland's wish to see NATO raise its military presence in eastern Europe to deter Russia.
"These deals are only possible when countries are in a state of symmetrical despair," said Laszlo Andor, a professor at Brussels' ULB University and former Hungarian EU commissioner.
A package deal might yield a more integrated "core Europe", adding an EU border and coast guard and a common asylum policy, and a looser union for countries like Britain that opt to stay outside the 19-nation euro single currency area.
A deal seems within reach with London that would formally exempt the UK from the EU goal of "ever closer union", shield its financial sector from being regulated abusively by the euro zone and let it withhold some benefits from new migrant workers if its social welfare system were under great strain.
The EU would cease to be a "two-speed Europe", with all 28 members converging at different paces toward the same goal, and become a permanent two-tier or multi-tier construction, possibly with an outer circle of associates such as Turkey and Ukraine.
While European leaders' despair may not be symmetrical, the refugee crisis is concentrating minds on a threat that could break the union, fan populism and alienate British voters.
European Council President Donald Tusk has warned that the EU is close to a tipping point, saying it has only six to eight weeks left to save the Schengen zone of passport-free travel or see national barriers slam shut for the duration.
That raises the urgency of a deal with Greece and Italy, the main arrival points for migrants from Turkey and North Africa.
Horse-trading to accommodate multiple national interests is a classic technique for advancing European integration and seems designed to break logjams in what Juncker calls a "polycrisis".
Rome is blocking an EU aid package for Turkey to help curb the influx of migrants into Europe; Athens stands accused of failing to guard its borders or to register and retain asylum-seekers on its soil; and Warsaw is under scrutiny in Brussels over laws shackling the judiciary and the media.
Each of those countries has hinted it is willing to do more to help others if its own interests are taken into account.
Merkel is keen to help Cameron win his planned referendum on whether Britain should stay in the EU, but she is increasingly fighting for her own political skin in the refugee crisis.
Berlin feels it has received little solidarity from its EU partners, while elsewhere there is a sense that reluctance to share the refugee burden is partly payback for perceived German bullying during the euro zone crisis.
Germany and its Dutch, Austrian and Finnish allies are more inclined to use sticks than carrots with Greece.
The European Commission this week gave Athens three months to fix "serious deficiencies" in its management of the bloc's external frontiers or face suspension from the Schengen area.
That deadline expires just as Greek debt talks are due to start.
Domestic headwinds may make it harder for Merkel to use her dwindling political capital to offer concessions to Greece on rescheduling its debt to euro zone countries.
Many in Berlin and Brusssels doubt Greek Prime Minister Alexis Tsipras can deliver tighter border controls or keep tens of thousands of migrants penned up in Greece until they can be relocated to reluctant EU countries.
Yet the chancellor has voiced understanding for Greece's plight as a frontline state and says she wants a "European solution" to the refugee crisis. No one wants another "Grexit" crisis this year on top of the EU's other woes.
If Greek compliance is uncertain, Turkey's cooperation in preventing migrants leaving its shores for Europe in return for 3 billion euros in aid is subject to deep scepticism.
The Italians want more EU cash to cope with migrants landing on their own shores and more fiscal leeway from Europe's budget supervisors to stimulate their sluggish economy.
Poland is the biggest supplier of migrant labor to Britain. Its foreign minister has hinted it could acquiesce in London's needs to curb in-work benefits for new arrivals if Cameron sends troops to provide a long-term NATO presence on its soil.
The makings of trade-offs are easy to see, but in the fraught state of relations among EU leaders, it is a tall order.