Let $\left\{a_n\right\}_{n=1}^{\infty}$ be a sequence such that $a_1=1, a_2=1$ and $a_{n+2}=2 a_{n+1}+a_n$ for all $n \geq 1$. Then the value of $47 \sum_{n=1}^{\infty} \frac{a_n}{2^{3 n}}$ is equal to $\_\_\_\_$ .
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