Fibonacci Sums

Probability Level 2

$a$ , $b$ , 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: $F(0) = F(1) = 1,\quad F(n) = F(n-2) + F(n-1) \ \text{ for } n > 1.$