What body/group did the Kennedy administration work with to come up with solutions to the crisis?

The Executive Committee of the National Security Council

Explanation:

The Executive Committee of the National Security Council (commonly referred to as simply the Executive Committee or ExComm) was a body made up of United States government officials that worked together to advise President John F. Kennedy during the Cuban Missile Crisis in 1962. It was made up of 12 members as well as the president, with the meeting being recorded by the President without any of the committee members knowing.