Let $\overline{\mathrm{u}}, \overline{\mathrm{v}}$ and $\overline{\mathrm{w}}$ be vectors in three-dimensional space, where $\overline{\mathrm{u}}$ and $\overline{\mathrm{v}}$ are unit vectors which are not perpendicular to each other and $\overline{\mathrm{u}} \cdot \overline{\mathrm{w}}=1, \overline{\mathrm{v}} \cdot \overline{\mathrm{w}}=1, \overline{\mathrm{w}} \cdot \overline{\mathrm{w}}=4$. If the volume of the parallelelopiped, whose adjacent sides are represented by the vectors $\overline{\mathrm{u}}, \overline{\mathrm{v}}$ and $\overline{\mathrm{w}}$, is $\sqrt{2}$, then the value of $|3 \overline{\mathrm{u}}+5 \overline{\mathrm{v}}|$ is $\_\_\_\_$ .