$ABCDEFGH$ is a cube with edge length equal to 1.\\
The space is equipped with the orthonormal coordinate system ($D; \overrightarrow{DC}, \overrightarrow{DA}, \overrightarrow{DH}$).\\
In this coordinate system, we have:\\
$D(0;0;0)$, $C(1;0;0)$, $A(0;1;0)$,\\
$H(0;0;1)$ and $E(0;1;1)$.\\
Let $I$ be the midpoint of $[AB]$.\\
Let $\mathscr{P}$ be the plane parallel to the plane $(BGE)$ and passing through the point $I$.\\
It is admitted that the section of the cube by the plane $\mathscr{P}$ is a hexagon whose vertices $I, J, K, L, M$, and $N$ belong respectively to the edges $[AB], [BC], [CG], [GH], [HE]$ and $[AE]$.

\begin{enumerate}
  \item a. Show that the vector $\overrightarrow{DF}$ is normal to the plane $(BGE)$.\\
b. Deduce a Cartesian equation of the plane $\mathscr{P}$.
  \item Show that the point $N$ is the midpoint of the segment $[AE]$.
  \item a. Determine a parametric representation of the line $(HB)$.\\
b. Deduce that the line $(HB)$ and the plane $\mathscr{P}$ intersect at a point $T$ whose coordinates you will specify.
  \item Calculate, in units of volume, the volume of the tetrahedron $FBGE$.
\end{enumerate}