Let $S$ be the solid in three-dimensional space consisting of all points ( $x, y, z$ ) satisfying the following system of six simultaneous conditions:
$$
\begin{aligned}
& x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0 \\
& x+y+z<11, \\
& 2 x+4 y+3 z \leqslant 36 \\
& 2 x+3 z \leqslant 24.
\end{aligned}
$$
Determine the number $e$ of edges of $S$.

Let's think step by step and output the final answer within \boxed{}.
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