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Since then, Taliban fighters have waged a pitched battle punctuated by attacks like the suicide bombings in Kabul on Wednesday, just after Mr. Obama’s visit. The attack left seven people dead and wounded some schoolchildren.

The militants planned the attack “hurriedly after finding out about Obama’s ‘surprise visit’ to Afghanistan,” Taliban spokesman Zabiullah Mujahed said in a statement.

The Taliban broke off peace talks with the United States in March after the Obama administration failed to release five high-value Taliban operatives from the U.S. military detention facility at Guantanamo Bay, Cuba. Their release was a key demand of the Taliban when it signaled its willingness to start official peace talks with the United States in January.

The delay in releasing the prisoners has strengthened the hands of Taliban members opposed to peace.

However, a U.S. official who briefed reporters Tuesday said the prisoners’ release had been put on hold for “reasons that appear to have to do with internal political turbulence among the Taliban.”

“It’s quite clear to us that there’s a range of interest among Taliban in reconciliation, and there’s quite a bit of internal political turbulence within the Taliban on that score,” the senior Obama administration official said.

The Taliban said Wednesday’s bombing was the start of the spring offensive.

After more than a decade of war, the U.S. has tallied nearly $570 billion in military, reconstruction and aid spending on Afghanistan, according to www.costofwar.com, run by the National Priorities Project. For 2012, $111.1 billion has been allocated, and Mr. Obama has requested $86 billion for 2013.

The group estimates the cost of war in Iraq since 2003 at $807 billion, pushing the total for both wars to $1.38 trillion. Of those funds, some $30 billion to $60 billion was squandered in waste and abuse, the nonpartisan Commission on Wartime Contracting reported last year.

Jo Comerford, the project’s executive director, said the continued U.S. presence in Afghanistan is expected to be larger than in Iraq, where the U.S. is spending $10 billion this year, so the spending in Afghanistan could balloon far past the $4 billion annual estimate.

“With all eyes on the federal budget right now, that price tag is going to have to mean budget priorities are going to have to shift to accommodate it,” she said. “Right now, the Congress is not in a position where Congress and the American people can just borrow a bit more money, we are at a moment where spending is finite.”