Distinct sums of Fibonacci numbers

The Fibonacci sequence is defined by \(F_1 = 1, F_2 = 1\) and \( F_{n+2} = F_{n+1} + F_{n}\) for \( n \geq 1 \).

Consider the set \(S_{15} = \{F_2, F_3, \ldots, F_{16}\}\) of 15 terms of the sequence. If three distinct elements are chosen from the set \(S_{15}\), how many different possible sums could these numbers have?

Details and assumptions

You may use the fact that \( {15 \choose 3 } = 455 \).

Note that \( F_1\) is not included in the set, so you can only use the number 1 once.

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