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Talks are now expected to take place over the weekend which will determine the shape of the next administration. An SNP-Green coalition was looking the most likely, but it would not have an overall majority, so might seek a less formal deal with another party.
The SNP won 19 of the council's 63 seats - the same as at the last elections five years ago. Labour, who have been in coalition with the Nationalists since 2017, increased their tally by one to 13 and the Greens went up from eight to ten. The Tories' contingent of councillors is just nine - half what it was in 2017.
The Tory losses included the defeat of sitting councillors Cameron Rose in Southside/Newington, Mark Brown in Drum Brae/Gyle and Jim Campbell in Forth.
The SNP finance convener Rob Munn was a rare SNP defeat, losing his seat in Leith Walk in a surprise win for the Lib Dem candidate Jack Caldwell. The Lib Dems took three out of the four seats in Almond ward and two out of three in both Drum Brae/Gyle and Corstorphine/Murrayfield. They also won seats in Forth and Southside/Newington.
The Greens gained seats in Forth, Inverleith and Sighthill/Gorgie but lost the Fountainbridge/Craiglockhart seat which Gavin Corbett had held until he stepped down on becoming a Scottish Government special adviser.
SNP group leader Adam McVey said: “The right-wing agenda of the Tories has been systematically rejected across communities. I think the voters have given all parties the result they deserve.
"We will be working with other parties, working out where there is synergy in our programmes. We need bold action on climate change, bold action on poverty reduction and we need to think outside the box on improving some of our local services in what can be a challenging financial situation - so parties that are willing to work with us on that agenda, we will absolutely work with them."