\(a\), \(b\), 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.

**Definition:**

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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