Question 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 $\tfrac{11}{2},$ and the volume of $P$ is $\tfrac{1}{2}.$ What is the length of the longest interior diagonal connecting two vertices of $P?$