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Problem

Let $S = 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ... + F_k / 2^k + ...$, where F_$k$ represents the $k$th term of the Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, ...$ .

Find the value of $S$.