For each real number $x$, let
$$f(x)=\sum_{n\in S_x}\frac{1}{2^n},$$
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x) ≥ L$ for all $x ∈ [0,1)?$ The original answer is in the format \frac{m}{n}, please provide the value of m + n.

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