The socialite Anne said to fellow socialite Claire, "I have three daughters. Can you figure out the ages of each of them knowing that the sum of their ages is 11?"

"That is not enough information," replied Claire.

"The product of their ages is either 16 years less or, 16 years more than your age," added Anne.

"Still not enough information." replied Claire after careful thought.

"The daughter whose age, in years, is the greatest is learning to play chess." said Anne.

Claire was then immediately able to determine the ages of Anne’s three daughters.

What are their ages?