JEE Advanced 2021 · Paper 1, Mathematics question 19

Let $\vec u$, $\vec v$ and $\vec w$ be vectors in three-dimensional space, where $\vec u$ and $\vec v$ are unit vectors which are not perpendicular to each other and
$$\vec u\cdot\vec w = 1,\qquad \vec v\cdot\vec w = 1,\qquad \vec w\cdot\vec w = 4$$
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors $\vec u$, $\vec v$ and $\vec w$, is $\sqrt2$, then the value of $|3\vec u + 5\vec v|$ is ____.