Rewinding a Volume-Doubling Transform

A 3-D modelling tool applies the same linear transform to every vertex of a scene:

$M = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & p \\ 3 & 7 & 11 \end{pmatrix}$

The gain $p$ was never written into the project file, but the tool's spec sheet recorded two facts about $M$:

After the transform, one vertex of the scene sits at

$M\mathbf{x} = \begin{pmatrix} 13 \\ 27 \\ 46 \end{pmatrix}$

Recover the original vertex $\mathbf{x}$ and report its distance from the origin, $\lVert \mathbf{x} \rVert_2$.

Round your answer to 2 decimal places.