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Striking Boeing workers have rejected a new offer from the plane-making giant, which included a 35% pay rise over four years.

The International Association of Machinists and Aerospace Workers (IAM) union said 64% of its members voted against the proposed deal.

More than 30,000 of Boeing's employees have joined the walkout, which started on 13 September, after an initial offer was rejected.

Hours earlier Boeing's boss Kelly Ortberg warned that the company is at a "crossroads" as losses at the firm surged to roughly $6bn (£4.6bn).

“After 10 years of sacrifices, we still have ground to make up, and we're hopeful to do so by resuming negotiations promptly," union representatives said in a statement.

"This is workplace democracy – and also clear evidence that there are consequences when a company mistreats its workers year after year," it added.

Boeing has declined to comment on its latest offer being rejected.

It is the second time that the striking workers have rejected a proposed deal in a formal vote. The previous offer was turned down last month by 95% of workers.

Earlier, Mr Ortberg, who took over as chief executive in August, said he had been working "feverishly" to stabilise the firm, as it worked to repair its reputation, which has been hit by manufacturing and safety concerns.

"This is a big ship that will take some time to turn, but when it does, it has the capacity to be great again," he said.

The latest crisis at Boeing erupted in January with a dramatic mid-air blowout of a piece of one of its passenger planes.

Its space business also suffered a reputational hit after its Starliner vessel was forced to return to Earth without carrying astronauts.

The strike has compounded the problems, leading to a dramatic slowdown in production.

Mr Ortberg said the firm was "saddled with too much debt" and had disappointed customers with lapses in performance across the business.

Boeing's commercial aircraft business reported operating losses of $4bn in the last three months, while its defence unit lost nearly $2.4bn.

The strike "is costing them $100m a day so the cash burn is really significant... This is getting to a pretty severe situation for Boeing,” said Anna McDonald from Aubrey Capital Management.

Mr Ortberg argued the firm was in a strong position, with a backlog of roughly 5,400 orders for its planes.

But he warned investors that restar7ting the firm's factories, whenever the strike does end, will be tricky.

"It’s much harder to turn this on than it is to turn it off. So it’s critical, absolutely critical, that we do this right," he said.

"We have a detailed return-to-work plan in place and I’m really looking forward to getting everybody back and getting to work on that plan."

The company announced plans earlier this month to cut roughly 10% of its workforce. Thousands of other staff are already on a rolling furlough due to the strike, which has also hit suppliers.

Mr Ortberg told investors that his first priority was a "fundamental culture change".

"We need to prevent the festering of issues and work better together to identify, fix and understand root cause," he said.

Boeing's suppliers are also feeling the impact of the strike.

Spirit AeroSystems, which makes plane bodies, has already announced a 21-day furlough for 700 of its workers.

It has also warned it could have to lay off staff if the Boeing strike continues beyond next month.