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Mediators hope any truce in the Gaza Strip can coincide with a Muslim festival that starts next week, and are looking to overcome seemingly irreconcilable demands from Israel and Hamas-led Islamist fighters, locked in conflict since July 8.
As the diplomacy continued, so did the fighting.
Gaza officials said Israeli strikes killed 33 people on Friday (local time), including the head of media operations for Hamas ally Islamic Jihad and his son. They put the number of Palestinian deaths in 18 days of conflict at 822, most of them civilians.
Militants fired a barrage of rockets out of Gaza, triggering sirens across much of southern and central Israel, including at the country's main airport. No injuries were reported, with the Iron Dome interceptor system knocking out many of the missiles.
The Gaza turmoil stoked tensions in the nearby occupied West Bank, where U.S.-backed Palestinian President Mahmoud Abbas governs in uneasy coordination with Israel.
Medics said five Palestinians were killed in separate incidents near the cities of Nablus and Hebron, including one shooting that witnesses blamed on an apparent Jewish settler.
On Thursday night, 10,000 demonstrators marched in solidarity with Gaza near the Palestinian administrative capital Ramallah - a scale recalling mass revolts of the past. Protesters surged against an Israeli army checkpoint, throwing rocks and Molotov cocktails, and Palestinian medics said one was shot dead and 200 wounded when troops opened fire.
Israel said an army reservist was killed in Gaza on Friday, bringing to 34 the number of soldiers lost in a ground advance it says aims to destroy dozens of cross-border tunnels used by Hamas to threaten its southern farming villages and army bases.
It also announced that a soldier unaccounted for after an ambush in Gaza six days ago was definitely dead, although his body had not been recovered. Hamas said on Sunday it had captured the man, but did not release a photograph of him.
Three civilians have also been killed in Israel by rockets from Gaza - the kind of attack that surged last month amid Hamas anger at a crackdown on its activists in the West Bank, prompting the July 8 launch of the Israeli offensive.
Israeli Prime Minister Benjamin Netanyahu convened his security cabinet on Friday to discuss a limited humanitarian truce under which Palestinian movement would be freed up to allow in aid and for the dead and wounded to be recovered.
A Palestinian official close to the negotiations said Turkey and Qatar had proposed a 7-day halt to the fighting, which had been relayed to Israel by Kerry while Hamas considered it.
An Israeli official acknowledged that the proposal had been received, but said any decision by the Netanyahu government would likely come after Hamas had delivered its own response.
Israel insists that, even if such a ceasefire is agreed, its army will continue digging up tunnels along Gaza's eastern frontier, a mission that could take between one and two weeks.
Netanyahu has said a truce should also lead to the eventual stripping of Gaza's rocket arsenals - something Hamas rules out.
"We must stop the rocket launches. How this is done - whether through occupying (Gaza), or broadening (the operation), or (international) guarantees, or anything else, I have to see it with my own eyes," said police minister Yitzhak Aharonovitch.
The rockets have sent Israelis regularly rushing to shelters and dented the economy despite Iron Dome's high rate of success.
A Hamas rocket intercepted near Ben Gurion Airport on Tuesday prompted the U.S. Federal Aviation Administration (FAA) to halt American commercial flights to Israel's main international gateway. Some European carriers followed suit.
Jolted by the blow at the height of an already stagnant summer tourism season, Israel persuaded U.S. authorities to lift the flight ban on Thursday, after which the European aviation regulator removed its own advisory against flying to Ben Gurion.
In the second such salvo in as many days, Hamas said it fired three rockets at the airport on Friday, an apparent bid to cripple operations there again. There was no word of impacts at Ben Gurion, whose passenger hall emptied at the sound of sirens.
HAMAS WANTS GAZA OPENED UP
Hamas leader Khaled Meshaal had on Wednesday voiced support for a humanitarian truce, but only if Israel eased restrictions on Gaza's 1.8 million people. Hamas wants Egypt to open up its border with Gaza, too, and demands that Israel release hundreds of prisoners rounded up in the West Bank last month following the kidnap and killing of three Jewish seminary students.
Such concessions appear unlikely, however, as both Israel and Egypt consider Hamas a security threat.
One Cairo official said next week's Eid al-Fitr festival, which concludes Ramadan, was a possible date for a truce. But U.S. officials were circumspect on progress made by Kerry, whose mediation has involved Egypt, Turkey, Qatar and Abbas, as Washington, like Israel and the European Union, won't deal directly with Hamas, which it considers a terrorist group.
"Secretary Kerry has been on the phone all morning, and he will remain in close touch with leaders in the region over the course of the morning as he continues work on achieving a ceasefire," said a senior U.S. State Department official in Cairo, which has been Kerry's base over the last four days as he has tried to bring about a temporary end to the conflict.
On Thursday, a US official said Kerry was seeking a way of bridging gaps between Israel and Hamas but that the diplomat would not stay in the region "for an indefinite amount of time".
More than 140,000 Palestinians have been displaced in Gaza by the fighting, many of them seeking shelter in buildings run by the United Nations Relief and Works Agency (UNRWA).
An UNRWA spokesman said the agency had tried in vain to arrange with Israel to evacuate civilians from the school in northern Beit Hanoun before it was shelled on Thursday.
Scores of crying families who had been living in the school ran with their children to a hospital a few hundred metres away where the victims were being treated. Laila Al-Shinbari, who was at the school when it was hit, told Reuters that families had gathered in the courtyard expecting to be evacuated shortly in a Red Cross convoy.
"All of us sat in one place when suddenly four shells landed on our heads ... Bodies were on the ground, (there was) blood and screams. My son is dead, and all my relatives are wounded, including my other kids," she said, weeping.