Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \geq 2$. For each $i$, where $1 \leq i \leq 200$, calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. Determine the sum of the distinct values of $g_i$.

Let's think step by step and output the final answer within \boxed{}.
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