In a regular tetrahedron ABCD with edge length 4, let O be the centroid of triangle ABC and P be the midpoint of segment AD. For a point Q on face BCD of the regular tetrahedron ABCD, when the two vectors $\overrightarrow { \mathrm { OQ } }$ and $\overrightarrow { \mathrm { OP } }$ are perpendicular to each other, the maximum value of $| \overrightarrow { \mathrm { PQ } } |$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]