Culture Secretary Maria Miller has said editors meeting at Number 10 pledged to meet the challenge of setting up an independent press regulator "swiftly".
"They want to come forward with those plans as quickly as possible," Ms Miller said, adding that the industry would announce a deadline for its proposals over the next two days.
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163,000 French and 143,000 German soldiers were killed in the 10-month battle in 1916.
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The fire broke out on Sunday morning at the home for 35 residents in Litochki, a village 25 miles north of Kiev.