\noindent\textbf{5} \hfill \textit{(See solution page)}

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In coordinate space, take three points $\mathrm{A}(1,\ 0,\ 0)$, $\mathrm{B}(0,\ 1,\ 0)$, $\mathrm{C}(0,\ 0,\ 1)$, and let $\mathrm{D}$ be the midpoint of segment $\mathrm{AC}$. Find the volume of the solid obtained by rotating the perimeter and interior of triangle $\mathrm{ABD}$ once around the $x$-axis.



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