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Let k1,k2,…, k_1, k_2, \ldots, k1,k2,…, be a sequence that is recursively defined as kn+2=kn+1+2knk_{n+2} = k_{n+1} + 2 k_{n}kn+2=kn+1+2kn, for all n≥1n\geq 1 n≥1, with k1=k2=1k_1 = k_2 = 1k1=k2=1. The infinite sum, S=k171+k272+k373…S = \frac{k_1}{7 ^1} + \frac{k_2}{7 ^2} + \frac{k_3}{7 ^3} \ldotsS=71k1+72k2+73k3…, is a fraction of the form ab\frac{a}{b}ba, where aaa and bbb are coprime integers. What is the value of a+ba+ba+b?
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