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Hamas has threatened to execute Israeli hostages if air strikes on homes in the Gaza Strip are carried out without warning.
Nearly 700 people have been killed in retaliatory Israeli strikes following an incursion by Palestinian militants from Gaza on Saturday.
Abu Obeida, a spokesman for Hamas's armed wing, the Ezzedine Al Qassam Brigades, said the group would kill one hostage for every bombing and broadcast footage of the executions.
The group claims to be holding more than 100 hostages who were seized when hundreds of its fighters crossed the border into southern Israel early on Saturday.
Hamas attacked towns and settlements as thousands of rockets were launched into Israeli territory.
More than 900 people were reported to have been killed in the attack, including 260 at a music festival near the Gaza border.
Mr Abu Obeida said Israel had conducted strikes on civilian areas, destroying homes.
“We have decided to put an end to this and, as of now, we declare that any targeting of our people in their homes without prior warning will be regrettably faced with the execution of one the hostages we are holding,” he said.
Israel has established a command centre on the hostage issue and said it had information on "every person" believed to be held in Gaza.
Hamas said earlier that four of the hostages died in Israeli air strikes on Gaza. The fighters guarding them were also among those killed, the group said.
Regional attempts to mediate an end to the conflict and the release of the hostages have so far been unsuccessful.
Israeli Foreign Ministry spokesman Lior Haiat told The National that the country was at war and rejected any mediation until it destroyed Hamas's capabilities and infrastructure in the Gaza Strip.
Defence Minister Yoav Gallant announced a total siege on the Gaza Strip on Monday, which has been under an Israeli blockade for 16 years.
"We are putting a complete siege on Gaza. No electricity, no food, no water, no gas – it's all closed," Mr Gallant said in a video message on Monday.
"We are fighting animals and are acting accordingly."
UN Secretary General Antonio Guterres warned Gaza faced a humanitarian disaster due to the latest escalation.
“The humanitarian situation in Gaza was extremely dire before these hostilities,” he said in New York on Monday.
"Now it will only deteriorate exponentially."
Conditions in Gaza have been deteriorating under a 16-year Israeli blockade, imposed after Hamas took control of the Palestinian enclave.
The blockade has been a major factor in forcing most of the two million people in Gaza to depend on UN food aid and other assistance.
Mr Guterres called for the protection of civilians and for Israeli military operations to be conducted in strict accordance with international humanitarian law.
He condemned the “abhorrent attacks by Hamas” in Israel, which have left more than 2,500 injured.
“Nothing can justify these acts of terror and the killing, maiming and abduction of civilians,” Mr Guterres said, repeating his call for an immediate cessation of violence and the release of all hostages.
He also expressed alarm at reports of deaths and injuries in Gaza.
The Palestinian health ministry said on Monday night that 687 people had been killed in the Israeli strikes including 140 children and 105 women, and that that 3,726 were wounded.
“Some 137,000 people are currently sheltering in UNRWA facilities, with the number increasing as heavy shelling and air strikes continue,” Mr Guterres said, referring to the UN Relief and Works Agency for Palestinian refugees.
He urged all sides and the relevant parties to allow the UN access to deliver urgent humanitarian assistance to Palestinian civilians trapped in the Gaza Strip.
Four Palestinian Red Crescent workers were among those killed in an Israeli attack on a school in an area of Gaza where hundreds of people were taking shelter.
Ahmad Jibril, head of the Ambulance Department at the Palestinian Red Crescent, told The National that four other workers were wounded.
The UNRWA said a school where displaced families in Gaza were sheltering “was directly hit” in the Israeli attacks, but there were no casualties.
Forty-five of the UNRWA schools in Gaza are equipped with emergency shelters.
Hamas's political leaders living abroad had no advance knowledge of the surprise attack by the militant group's armed wing, Egyptian security officials said.
