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Sinn Féin vice president Michelle O'Neill said she has told the new secretary of state that the absence of an executive in Northern Ireland is no longer tolerable.
Ms O'Neill said the government needed to stop "pandering" to the DUP.
After a meeting with Chris Heaton-Harris, she also accused the government of "being in hock" to the the right-wing European Research Group (ERG).
The new secretary of state will also meet the DUP later.
But he is not seeing the other main parties, Alliance, the Ulster Unionists or the SDLP.
Mr Heaton-Harris said he looked forward to meeting "all political parties" over the next couple of days.
Speaking to the media after the meeting in Dungannon, Ms O'Neill said: "There must be a deal. We must find a way forward.
"We have a new prime minister and a new secretary of state but it's still the Tory party; different people at the helm but it's still the Tory party.
"My focus and I suppose my message to the new secretary of state this morning was very clear: 'It's get down to business; work with others, find an agreed way forward.
"'Bring that certainty and stability that is required. Get an Executive up and running and stop pandering to the DUP'.
Mr Heaton-Harris and the new minister of state at the Northern Ireland office, Steve Baker, are past chairmen of the pro-Brexit group ERG within the parliamentary Conservative Party.
"The ERG have been dictating the pace and bringing a very toxic influence to our local politics," Ms O'Neill said.
"So, regardless of who's there, that's a feature. My concern is that I stand up for people here and the people want an executive up and running.
"They want us to put money into their pockets to help them get us through this nightmare of a cost of living crisis."
First official visit
Mr Heaton-Harris was in Dungannon for his first official visit to Northern Ireland.
He has been criticised for only holding meetings with Sinn Féin and the DUP on the first day of his visit.
Mr Heaton-Harris added that he wanted to "deliver enough pressure" to get an executive restored and resolve the dispute over the Northern Ireland Protocol.
He did not take any questions from journalists.
At his first NI Questions on Wednesday Mr Heaton-Harris said the best way forward was to have a functioning executive.
Ms Truss told the Commons the Northern Ireland Executive must return - and at the end of next month the new secretary of state will have to decide what to do if this does not happen.
If Stormont has not returned by 28 October, a 12-week period will be triggered during which he will have to call a fresh assembly election, unless the government instead chooses to push back the deadline.
On Wednesday, Mr Heaton-Harris said he will "work with everybody" to resolve problems over the Northern Ireland Protocol.
He also told the Commons that he wants to negotiate a solution with the EU but that the UK government would legislate at Westminster if this did not happen.
He called on the DUP to return to power sharing as soon as possible. But he is seen by some as an ally of the party because of his support for Brexit and his desire to change the protocol.
Sinn Féin's Conor Murphy, who met Mr Heaton-Harris alongside Ms O'Neill, earlier said he would encourage him to meet all Northern Ireland parties.
He said: "We will deal with any British secretary the way we've dealt with previous ones. British government policy in Ireland is the problem and particularly in relation to the protocol and other issues affecting Brexit.
"We need to see them back in proper negotiations with the EU so the benefits of the protocol can be felt here and we stop suffering the full effect of Brexit as is happening right across Britain at this moment."
'Stormont doesn't have the money'
DUP MP Sammy Wilson said the outcome of what Ms Truss's government does about the protocol was "the most important thing".
"Until that is dealt with, people will find that their cost of living goes up even more than the rest of the UK, due to additional costs which the protocol is imposing on goods coming into Northern Ireland," he said.
"The cost of living issues are not going to be dealt with by Stormont - Stormont doesn't have the money.
"It is going to come from Westminster so we'll be looking out for what's announced today in terms of energy help for people and then of course later on when the government brings its budget."
Meanwhile, SDLP MP, Claire Hanna, has described the appointments of Mr Heaton-Harris' as secretary of state and Steve Baker as Minister of State for Northern Ireland, as "concerning".
Mr Heaton-Harris will not be meeting Ms Hanna's party later.
"It is fair to say Liz Truss has appointed two men known for being very much associated with having a very hardline Brexit view and they're former chairs of the ERG (European Research Group)," she said.
"I would suggest it is that ERG ideology that has got Northern Ireland into the place we're in. Six years of not properly acknowledging, understanding and processing the realities of Brexit with the geography we have.
"We do think it is a concerning appointment - there are optimistic interpretations that she's sending a hardliner in to look tough and to do the deal but at the moment it doesn't send a good signal about her willingness to work with people who do not support a hardline Brexit and to work with the EU," she added.
Mr Heaton-Harris is the third Conservative MP to hold the role of Northern Ireland secretary in just three months, and replaces Mr Vara, who was sacked on Tuesday.
Mr Vara was in post for 62 days, making him the shortest-serving Northern Ireland secretary.