The length of the longer diagonal of the parallelogram constructed on 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 the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\), is