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Turkish President Tayyip Erdogan vowed to press on with rescue and recovery efforts more than a week after a powerful quake ripped through his country and neighbouring Syria, with an elderly woman the latest to be pulled from the rubble.

The combined death toll in Turkey and Syria has climbed over 41,000, and many survivors are enduring near-freezing winter temperatures, having been left homeless by the devastation in cities in both countries.

"We will continue our work until we remove the last citizen left under the collapsed buildings," Erdogan said late on Tuesday after a cabinet meeting held at the headquarters of the Disaster and Emergency Management Authority (AFAD).

Damage assessment of buildings, of which tens of thousands were destroyed, will be completed in a week and reconstruction will begin within months, he said.

"We will rebuild all the houses and workplaces, destroyed or made uninhabitable by the earthquake, and hand them over to the rightful owners," he added. More than 105,000 people were injured in the quake, he said, with more than 13,000 still being treated in hospital.

Overnight, a 77-year-old woman named Fatma Gungor was pulled alive from the rubble of a seven-storey apartment block in the city of Adiyaman, some 212 hours after the first earthquake, media reports said.

Wearing an oxygen mask, covered in a gold foil blanket and strapped onto a stretcher, Gungor was carried by rescue workers down from the ruins of the building to a waiting ambulance, footage from state broadcaster TRT showed.

Afterwards, Gungor's relatives hugged the rescue team, made up of military personnel and members of the disaster management authority AFAD.

Nine other survivors were rescued in Turkey on Tuesday as the focus of the aid effort shifted to helping people now struggling without shelter or enough food in the cold.

Erdogan has acknowledged problems in the initial response to the 7.8 magnitude quake that struck early on Feb. 6 but he has said the situation is now under control.

"We are facing one of the greatest natural disasters not only in our country but also in the history of humanity," Erdogan said.

More than 2.2 million people have left the worst-hit areas already, Erdogan said, and hundreds of thousands of buildings have become uninhabitable.

Those rescued on Tuesday included two brothers, aged 17 and 21, pulled from an apartment block in Kahramanmaras province, and a Syrian man and young woman in a leopard-print headscarf in Antakya.

UN authorities have said the rescue phase is coming to a close, with the focus turning to shelter, food and schooling.

"People are suffering a lot. We applied to receive a tent, aid, or something, but up to now we didn't receive anything," said Hassan Saimoua, a refugee staying with his family in a playground in Turkey's southeastern city of Gaziantep.

Saimoua and other Syrians had found refuge in Gaziantep from the war at home. Now, rendered homeless by the quake, they have assembled makeshift tents in a playground using plastic sheets, blankets and cardboard.

"The needs are huge, increasing by the hour," said Hans Henri P. Kluge, the World Health Organisation's director for Europe. "Some 26 million people across both countries need humanitarian assistance."

"There are also growing concerns over emerging health issues linked to the cold weather, hygiene and sanitation, and the spread of infectious diseases - with vulnerable people, especially at risk."

Families in both Turkey and Syria said they and their children were dealing with the psychological aftermath of the quake.

"Whenever he forgets, he hears a loud sound and then remembers again," Hassan Moaz said of his 9-year-old in Aleppo, Syria. "When he's sleeping at night and hears a sound, he wakes up and tells me: 'Dad, aftershock!'"

A first convoy of UN aid entered rebel-held northwest Syria from Turkey via the newly-opened Bab al-Salam crossing.

The search for survivors was about to end in the northwest of Syria, said the head of the White Helmets main rescue group, Raed al Saleh.

Russia also said it was wrapping up its search and rescue work in Turkey and Syria and preparing to withdraw.

The Turkish toll was 35,418 killed, Erdogan said. More than 5,814 have died in Syria, according to a Reuters tally of reports from Syrian state media and a UN agency.

Survivors joined a mass exodus from the disaster zone, leaving their homes and unsure if they can ever come back.

Hamza Bekry, a 22-year-old Syrian has lived in Antakya, in southern Turkey, for 12 years, having fled the conflict in his homeland, but he was now preparing to follow his family to Isparta in southern Turkey.

"It's very hard ... We will start from zero, without belongings, without a job," Bekry said.