A Fibonacci Sum

A sequence \(\{a_i\}_{i=1}^n\) has the property that \(S_n=F_n\) where \(S_n=\displaystyle\sum^n_{i=1}a_i\), \(F_1=1\), \(F_2=1\), and \(F_n=F_{n-1}+F_{n-2}\). The closed form of \(\displaystyle\sum^n_{i=1}a_{2i-1}\) can be represented as \(S_{f(n)}+c\) where \(f(n)\) is a function of \(n\) and \(c\) is a constant. Find the last three digits of \(f(2013)+c\).

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