The length of the longer diagonal of the parallelogram constructed on the vectors $5\vec{a} + 2\vec{b}$ and $\vec{a} - 3\vec{b}$, if it is given that $|\vec{a}| = 2\sqrt{2}$, $|\vec{b}| = 3$, and $\vec{a} \cdot \vec{b} = \frac{\pi}{4}$ is: