If $a^4+\frac{1}{a^4}=50$, then find the value of $a^3+\frac{1}{a^3}$.

$\sqrt{2(1+\sqrt{3})}+(-1+2 \sqrt{13})$ $\sqrt{2(1+\sqrt{3})}(3-2 \sqrt{13})$ $\sqrt{2(\sqrt{13}+1)}(3+2 \sqrt{13})$ $\sqrt{2(1-\sqrt{3})}(-1+2 \sqrt{13})$

$\sqrt{2(\sqrt{13}+1)}(3+2 \sqrt{13})$