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The value of the infinite sum
∑k=1∞Fk2k=12+14+28+316+532+864+…,\sum_{k=1}^\infty \frac{F_k}{2^k} = \frac{1}{2}+\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{5}{32}+\frac{8}{64}+\ldots,k=1∑∞2kFk=21+41+82+163+325+648+…,
where FkF_kFk represents the kthk^{th}kth Fibonacci number, can be written as ab\frac{a}{b}ba, where aaa and bbb are positive coprime integers. Find a+ba+ba+b.
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