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To them, Mr Meshaal said this was not a battle against Hamas alone, but against the entire umma, or nation - analysts say an apparent reference to a populist Islamist idea that the Palestinians are defending the Muslim world against a modern form of Crusades.
The UN said the Israeli military escalated its offensive against the Hamas leadership in Gaza on Friday, targeting the homes of more than 20 Hamas officials in its latest air strikes.
In response, Palestinian militants fired on Israel, launching more than 60 missiles in 24 hours, injuring four people in the southern city of Ashkelon.
Four Israelis have been killed so far by militant rocket fire.
Earlier on Friday, five Palestinian civilians - including three children - were killed in an Israeli strike on Gaza.
The UN says up to 421 Palestinians may have been killed by Israeli action so far and more than 2,000 injured - though it says it cannot confirm those figures.
The humanitarian crisis in Gaza is worsening, UN officials say
Israel is refusing to let international journalists into Gaza, despite a Supreme Court ruling to allow a limited number of reporters to enter the territory.
The UN says the week-long assault has worsened the crisis in Gaza, despite an increase in humanitarian shipments.
Israel tightened its control of what gets in and out of the crowded coastal Strip after Hamas, the elected power, seized control of the area from rival Fatah forces 18 months ago.
Since then, the UN says there has been a significant deterioration in infrastructure and basic services, with 80% of the 1.4m population unable to support themselves.
In a statement, the Israeli foreign ministry said it was working with international organisations in Gaza as well as various governments "in order to assess the humanitarian needs... and make the necessary response".
All reports indicate that there is sufficient medicine and food in Gaza, the statement read.
In his weekly radio address President Bush said Hamas was responsible for the latest violence and rejected a unilateral ceasefire that he said would allow Hamas to continue to fire on Israel.
And he called for tougher action to prevent Hamas and other groups from receiving weapons.
"There must be monitoring mechanisms in place to help ensure the smuggling of weapons to terrorist groups in Gaza comes to an end," he said.
"I urge all parties to pressure Hamas to turn away from terror and to support legitimate Palestinian leaders working for peace," Mr Bush added.
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