Fibonacci Sums

\(a\), \(b\), and \(c\) are distinct positive integers, and the sum of any two of these integers is a Fibonacci number.

What is the smallest possible value of \(a+b+c\)?

If you think there are no possible values for \(a\), \(b\), and \(c\), please put the answer as 0.


The first few Fibonacci numbers are \(1,1,2,3,5,8,13,21, \ldots\).

In general, they satisfy the following recursive relation: \[F(0) = F(1) = 1,\quad F(n) = F(n-2) + F(n-1) \ \text{ for } n > 1.\]


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