Let \( \vec{u}, \vec{v} \) and \( \vec{\omega} \) be vectors in three-dimensional space, where \( \vec{u} \) and \( \vec{v} \) are unit vectors which are not perpendicular to each other and \( \vec{u}\!\cdot\!\vec{w} = 1 \), \( \vec{v}\!\cdot\!\vec{w} = 1 \), \( \vec{w}\!\cdot\!\vec{w} = 4 \). If the volume of the parallelepiped, whose adjacent sides are represented by the vectors \( \vec{u}, \vec{v} \) and \( \vec{w} \) is \( \sqrt{2} \), then the value of \( \lvert 3\vec{u} + 5\vec{v} \rvert \) is ________.
