Suppose $A,B,C$ are points in the plane with $AB=24$ and $AC=32$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $△ABC$. Then the domain of $f$ is an open interval $(p,q)$ and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$?