There’s a buoyant sense at the White House this week – a feeling that a much-embattled President Barack Obama has achieved the goal he set in January 2009 of engaging Iran on the basis of “mutual interest and mutual respect.”

But like the dog who catches the car he’s been chasing, Obama must now worry about what to do next.

The first priority is pinning down the deal that Secretary of State John Kerry reached last week so that it’s not a fuzzy framework, but an actual, enforceable agreement. There are many details left to clarify, and U.S. officials aren’t yet sure they actually have clinched the deal that they appeared to have won.

Problem areas include limits on Iranian research and development of advanced centrifuges buried underground at Fordow; the mechanism for removing sanctions and then reimposing them if Iranis thought to be cheating; and the procedures for inspecting supposedly “nonnuclear” sites where covert research might be taking place.

These are big holes in the framework. Its unfinished nature is a sign that the administration wants the final pact so much that it will offer compromises that allow the Iranians to save face, even at modest cost to U.S. interests. The administration’s goal, over the next three months, appears to be gaining the best final accord possible – that the Iranians can also sell back home.

Obama’s comfort level has been boosted by the presence at the negotiating table of Energy Secretary Ernie Moniz, one of the world’s top nuclear physicists. Moniz can signal compromises that, while appearing generous, have little practical consequence, for technical reasons.

Obama’s outreach to Iran has been shaped from the beginning by his effort to understand how Iranians see the world – and to distinguish between truly dangerous, aggressive actions and more comprehensible defensive moves. This empathetic view is part of what irks Israeli Prime Minister Benjamin Netanyahu. But centuries of diplomatic history suggest that such an ability to see the world through the adversary’s eyes is essential for effective negotiation.

If there has been a surprise in the Iranian negotiating style, it’s that they have adhered so closely to the terms of the initial framework reached in November 2013, rather than cheating at the edges. President Hassan Rouhani sought to underline this theme of trustworthiness (contrary to what the Israelis and many Arabs see in Iran) when he said last Friday: “If the other side honors its promises, we will honor our promises.”

Obama rejects the case made by Netanyahu and congressional critics that if the United States just keeps squeezing, the Iranians will capitulate. The White House thinks too much pressure could backfire. U.S. officials agree that Supreme Leader Ali Khamenei’s ultimate goal is regime survival, but officials saw the crowds in the streets of Tehran last week cheering the deal as a check on Khamenei and other hard-liners.

The most delicate test ahead may involve, not the Iranians, but Saudi Arabia and the other Arab Gulf States. Obama knows that the metastatic danger for the Middle East is a post-agreement scramble by Iran’s Sunni rivals, such as the Saudis, Egyptians, Emiratis and Turks – to achieve their own versions of the Iranian “threshold nuclear capability” envisioned in the agreement.

The White House is still mulling the details, but officials are contemplating a kind of “dual engagement” approach. Even as it negotiates with Iran, the administration might extend security guarantees to the Gulf States, pledging to come to their defense if attacked by external powers. (Tricky question: Would that include a strike from Israel?) In exchange, the Gulf states hopefully would agree to forgo or limit their nuclear programs, keeping some lid on proliferation in the region.

Obama’s challenge is that the Sunni nations have been suffering a kind of vertigo since the Arab revolutions of 2011, doubting themselves and the United States even as they reel from Iran’s proxy wars. Somehow these Sunni nations need to find the will to push back, so that there could eventually be a security balance between Iran and its neighbors. Because Obama understands the need for this pushback, he has supported the Saudi assault on Yemen, and might even endorse a Turkish military move into northern Syria.

Dealing with Congress will be its own special nightmare, as always for this administration. Obama needs a formula that allows members to reassure Netanyahu of their toughness, while keeping what many see as a pretty good Iran deal – and accepting that it’s the president, not Congress, who conducts foreign policy.