Northern Ireland's power-sharing arrangement means nationalists and unionists, who want to remain part of the United Kingdom, have to work together, with the First Minister and Deputy First Minister leading the executive.
The British government was due to decide on Monday (March 27) whether to take back direct rule of Northern Ireland for the first time in a decade or call a new election after the collapse of power-sharing between Irish nationalists and British unionists.
Britain's system of "direct rule" ran from 1972, the deadliest year of Northern Ireland's conflict, until the Good Friday peace accord of 1998 paved the way for the first of several Catholic-Protestant coalitions in Belfast.
'We are just disappointed that Sinn Fein did not come to the talks in the same spirit as we came to the talks.
"Everyone owes it to the people of Northern Ireland to grasp that and provide the political leadership and the stability that they want", Brokenshire added.
Problems have been brewing since January when former Deputy First Minister Martin McGuinness - who died last week after a short illness - stepped down, triggering an election.
The controversial energy scheme which prompted Sinn Fein to pull the plug on the executive was introduced by Foster when she was economy minister.
'We are standing firm - previous agreements need to be implemented, ' she said.
Mrs O'Neill gave a very different assessment of the breakdown.
Mrs Foster claimed while her party entered talks in "good faith", Sinn Fein were not in "agreement-finding mode".
'We came at the negotiations with the right attitude, wanting to make the institutions work, wanting to deliver for all citizens.
Republicans have also been seeking movement on issues such as legislation to protect the Irish language, a hugely symbolic measure but deeply problematic for some unionists.
"The talks process has run its course and Sinn Fein will not be nominating for the position of speaker or for the executive office tomorrow (Monday)", said Michelle O'Neill, the party's leader in Northern Ireland.
New mechanisms for dealing with the legacy of the Troubles also remain a source of vexed dispute.
The Irish foreign affairs minister Charlie Flanagan said: "Despite constructive engagement by all of the parties and important progress being made during these discussions, it has not yet been possible to make the necessary breakthroughs on a small number of core issues". Adams said unionism was at a crossroads.
At the last round of elections Sinn Féin secured its best ever performance to cut the DUPs 10-seat advantage to one.