Prime Minister Ehud Olmert met Defense Minister Ehud Barak and Foreign Minister Tzipi Livni Thursday night to discuss responses to Hamas's infractions of the cease-fire, amid agreement among the triumvirate that Israel's responses must go well beyond striking at the terrorists who violated the cease-fire, diplomatic sources in Jerusalem said. "We will refuse to play by the rules of the game that Hamas is trying to dictate," one senior government official said, "whereby we only hit those who shoot the rockets. The goal is that there will be no rockets on the South, and that means that in our response to the attacks we have to send the message that we will not allow this to continue." Israel's top security brass - Chief of General Staff Lt.-Gen. Gabi Ashkenazi, Mossad head Meir Dagan, and Shin Bet (Israel Security Agency) chief Yuval Diskin - briefed visiting US Mideast envoy George Mitchell Thursday morning on the situation in the Gaza Strip, and on Israel's security requirements. Following those meetings, Mitchell went to Ramallah and met with Palestinian Authority President Mahmoud Abbas, after which he said that the Gaza border crossings should be opened with a PA presence. "To be successful in preventing the illicit traffic of arms into Gaza, there must be a mechanism to allow the flow of legal goods, and that should be with the participation of the Palestinian Authority," Mitchell said after the meeting. He also met later in the day with PA Prime Minister Salaam Fayad. Mitchell went to Ramallah after hearing a briefing on the status of the diplomatic process the night before from Olmert who, according to a report in Yediot Aharonot, laid out the understandings he and Foreign Minister Tzipi Livni reached in their negotiations with the Palestinians: a nearly full withdrawal to the 1967 lines, with adjustments to incorporate major settlement blocks; the division of Jerusalem, with the east Jerusalem neighborhoods coming under PA and the holy places under international jurisdiction; and no "right of return" for Palestinian refugees. According to the Yediot report, Mitchell said the Obama administration stood behind the commitments president George W. Bush gave to prime minister Ariel Sharon in 2004, whereby the US would recognize that in any future agreement the changing realities on the ground, including existing major Israeli population centers, needed to be taken into consideration, and that the US would not accept a Palestinian right of return. Although diplomatic sources said it was clear the new Obama administration would stand by that letter, this did not mean that the US's recognition of changing realities on the ground meant it accepted Israel's interpretation of this to meant Washington would back the annexation of the large settlement blocs - something the US has never explicitly said. This topic is likely to come up on Friday morning when Mitchell meets Likud head Binyamin Netanyahu, who is expected to present Mitchell with a view of the diplomatic process significantly different from the one that Olmert presented. Mitchell will also meet on Friday with Welfare and Social Services Minister Isaac Herzog, to discuss humanitarian aid to the Gaza Strip, and with Lt.-Gen. Keith Dayton, the US security coordinator for the Israel and the Palestinian Authority. From Jerusalem Mitchell will travel to Amman for meetings, followed by visits to Riyadh, Paris and London, before returning to Washington and reporting back to US President Barack Obama and Secretary of Sate Hillary Clinton. Meanwhile, in New York, Israel's UN ambassador, Gabriela Shalev, had her first one-on-one meeting with new US envoy Susan Rice, who took over the post earlier this week. Rice told Shalev the Obama administration intended to focus first on stopping weapons smuggling to Hamas, along with alleviating the humanitarian crisis in the Gaza Strip. Rice said she understood first-hand the "fears Israel has," according Israeli mission spokeswoman Mirit Cohen. Rice traveled with Obama to Sderot last July. Allison Hoffman contributed to this report from New York.