AMC 12 2023b Problem 13

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 \textbf{(B)}\frac{3}{8} \textbf{(C)}\frac{9}{8} \textbf{(D)}\frac{9}{4} \textbf{(E)}~\frac{3}{2}