What did JFK and his brother leak to the media to try to de-escalate the situation in Cuba?

The story of this leak is quite fascinating, and actually led to an enduring misconception in most Americans' understanding of history to this day.

On October 26, 1962, Nikita Khrushchev of the USSR sent a letter to John F. Kennedy agreeing to withdraw missiles from Cuba if he agreed to cease all plans for military action against Cuba.

On October 27, Khrushchev expanded his demands, in addition demanding that JFK also withdraw US nuclear missiles from Turkey.

The rumor that John and Robert Kennedy leaked was called the "Trollope Ploy"; it said that JFK simply agreed to the first deal and ignored the second letter, pretending he had never received it. That way, the US wouldn't have to make any real concessions and the crisis could be defused. The media latched onto this as a brilliant diplomatic move and a great success for JFK in de-escalating the Cuban Missile Crisis.

But in fact, this leak was actually itself a ploy, used by JFK to conceal the fact that he had in fact agreed to the second letter and removed US nuclear missiles from Turkey. The removal of the missiles was kept top secret for many years, perhaps because JFK feared that it would be unpopular or seem "weak" to the American public. In fact it had very little strategic significance, as by that point submarines were the core of the US nuclear arsenal, and would soon be complemented by ICBMs that could hit Russia even when launched from the safety of Colorado and Montana.