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There are 10 flood warnings in the north-west of England, five in Wales, three in the Midlands, and one each in the North East and Scotland.
There has also been a Met Office warning for localised flooding in western counties of Northern Ireland.
A search has resumed for a 21-year-old woman who was swept along the River Usk in Brecon, Powys, on Saturday night.
A man who was with her raised the alarm. He has been helping police with their inquiries as they try to establish exactly what happened.
Work continues on a safety review of all 1,800 bridges in Cumbria, where six crossings have already collapsed and Calva Bridge in Workington, which is also in danger of crumbling, has been condemned.
Its closure means people in Northside face a 40-mile road detour just to get to the town centre.
Twenty-five soldiers from 39 Regiment the Royal Artillery have been sent to flood-hit areas to assist the emergency services.
Cumbria Chief Constable Craig Mackey said it could take years for Cumbria to recover from the damage.
Cumbria County Council chief executive Jill Stannard, who took up her post only last Thursday, said the damage from the floods had run into "tens of millions of pounds" after more than 1,300 homes were flooded by record daily rainfall last Thursday and Friday.
Liam Dutton with the BBC weather forecast
Ms Stannard said the authorities were trying to establish the whereabouts of people who were unaccounted for, although she stressed many of them might have gone to stay with relatives and friends outside the area.
She told BBC Radio 4's Today programme: "We are confident we can reach everyone. People get very frightened - totally understandably because this is very traumatic.
"It is important that people listen to advice through the media and don't listen to rumour and gossip."
An Environment Agency spokeswoman said the additional rain on Tuesday would fall on already saturated ground.
"We are keeping a very close eye on the developing situation and we ask the public to stay tuned to local radio stations and call the Floodline on 0845 988 1188 for the most up-to-date information," she said.
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