Fibonacci Sequence as Linear Recurrence

(2016 A-Star Math Tournament Guts Round #18) Let $F$ be the sequence of Fibonacci numbers, where $F_1 = 1, F_2 = 1,$ and $F_n = F_{n-1} + F_{n-2}$ for all $n\ge 3.$ The sum $\sum_{i=1}^{2016} F_{3i}$ can be written in the form $\frac{1}{a}\left(F_b - c\right),$ where $a, b, c$ are positive integers, $a\gt 1,$ and $b$ is minimal. Compute $a+b+c.$