A rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is $13$, the sum of the areas of all $6$ faces of $P$ is $\frac{11}{2}$, and the volume of $P$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $P$? $\textbf{(A)}~2\qquad$ $\textbf{(B)}~\frac{3}{8}\qquad$ $\textbf{(C)}~\frac{9}{8}\qquad$ $\textbf{(D)}~\frac{9}{4}\qquad$ $\textbf{(E)}~\frac{3}{2}$