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President Barack Obama’s Mideast envoy turned his attention to the Western-backed Palestinian government in the West Bank on Thursday as rockets thudded into southern Israel and Israeli warplanes attacked new targets in Gaza.

George Mitchell was seen entering Palestinian President Mahmoud Abbas’ headquarters in the West Bank town of Ramallah and was due to meet later in the day with Prime Minister Salam Fayyad.

He is not meeting with Hamas, the Islamic group that is Abbas’ rival and the ruler of the Gaza Strip. The U.S., Israel and European Union have blacklisted Hamas as a terrorist group.

Mitchell held his first round of talks with regional leaders in Cairo and Jerusalem Wednesday to determine the next steps toward reviving peace negotiations following Israel’s blistering military offensive Hamas.

But a flare-up of violence in Gaza underscored the more immediate priority — shoring up a 10-day-old cease-fire. Palestinians fired a rocket into Israel early Thursday, and residents of the south Gaza town of Khan Younis said an Israeli airstrike there wounded an unidentified man on a motorcycle and five passers-by, among them children walking home from school.

The Israeli military said it targeted the motorcyclist because he was involved in a bomb attack Tuesday on the Gaza-Israel border which killed an Israeli soldier and wounded three others.

On Wednesday Israeli warplanes struck Gaza smuggling tunnels and a weapons factory. There were no reports of casualties.

Israel and Hamas separately declared a cease-fire on Jan. 18, ending a three-week Israeli offensive that killed nearly 1,300 Palestinians and caused widespread destruction in the Hamas-controlled Gaza Strip. Thirteen Israelis also died in the fighting.

Israel launched the operation to end years of rocket fire on its southern towns.

International diplomats have been trying to work out arrangements for a longer-range truce. Israel wants an end to Hamas rocket attacks and weapons smuggling. Hamas wants Israel to end a crippling economic blockade of Gaza’s borders, imposed after the Islamic militant group seized power in June 2007.

After talks in Jerusalem Wednesday with Israeli Prime Minister Ehud Olmert, Mitchell said consolidating the cease-fire was “of critical importance.” He said a longer-term truce should be based on “an end to smuggling and reopening of the crossings” into Gaza.

Mitchell’s Mideast foray comes just a week after Obama took office, signaling the new U.S. administration’s willingness to make the region a priority. Mitchell, a former Senate majority leader and a broker of the 1998 Northern Ireland peace deal, said that after finishing his consultations in the region and with European leaders, he would report his recommendations to Obama and Secretary of State Hillary Clinton.

Mitchell was silent on the details of his meetings, and he has no news conferences planned during his seven-day tour.

In a meeting Wednesday night, Foreign Minister Tzipi Livni told Mitchell that only a peace agreement which guaranteed Israeli security would win the approval of the Israeli public.

“In order for the peace negotiations to succeed, Israel must continue its war against terror wherever it exists and is directed against us,” her office quoted her as saying.

Livni has been Israel’s chief negotiator in the past 15 months of peace talks with Abbas’ government. Those talks have been put on hold due to the fighting in Gaza, and Israel’s political campaign ahead of a Feb. 10 parliamentary election.

The recent Israeli offensive has fueled new calls for Abbas’ Fatah movement and Hamas to reconcile. The sides met in Cairo this week for their first talks since Hamas took control of Gaza, and Egypt hopes to hold more substantive reconciliation talks by mid-February.

The Palestinians want to establish an independent state in Gaza and the West Bank — areas located on opposite sides of Israel. Israel says a peace agreement is impossible as long as Hamas remains in charge in Gaza.

In the meantime, it would be hard for the cease-fire to hold unless arrangements are made to stop the flow of arms to Hamas and end the blockade of the tiny coastal territory, which has deepened the deprivation there and trapped 1.4 million people inside.

Mitchell said the crossings should be opened on the basis of a 2005 agreement brokered by the U.S. that put the main crossing — the passage between Egypt and Gaza — under the management of Abbas’ Palestinian Authority, with European monitors deployed to prevent smuggling.

However, Hamas wants a role at the crossings in recognition of its power in the territory. Israel and Abbas do not want Hamas there.

Olmert told Mitchell Wednesday that Hamas’ power in Gaza “must diminish” and Abbas must “gain a foothold” there, an Olmert aide said, speaking on condition of anonymity because their meeting was closed.

Olmert said crossings between Israel and Gaza “will only open permanently” after the freeing of Sgt. Gilad Schalit, an Israeli soldier Gaza militants captured in June 2006, the aide said.

In Qatar on Wednesday, Hamas’ supreme leader, Khaled Mashaal, said the group would not link the opening of crossings to the release of the Israeli soldier.

Hamas wants Israel to free hundreds of Palestinian prisoners in exchange for Schalit.

Egypt has been exploring the possibility of including some Hamas personnel in a Palestinian Authority presence at the border, but that would require some form of reconciliation between the factions, which remain bitter rivals.