# 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.

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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