Limit distribution of a dyadic sum

Let $\{X_n\}_{n \ge 1}$ be a sequence of independent random variables with $\operatorname{Pr}(X_n = -\frac{1}{2^{n}}) = \operatorname{Pr}(X_n = \frac{1}{2^{n}}) = \frac{1}{2} \; \forall n \in \mathbb{N}$. Suppose that $\sum_{i=1}^{n} X_i \xrightarrow{d} U$ as $n \to \infty$. Then $6 \operatorname{Pr}(U \le \frac{2}{3})$ equals __________ (answer in integer).