As shown in the figure, there is an equilateral triangle ABC and a circle O with diameter AC on a plane. Point D on segment BC is determined such that $\angle \mathrm { DAB } = \frac { \pi } { 15 }$. When point X moves on circle O, let P be the point where the dot product $\overrightarrow { \mathrm { AD } } \cdot \overrightarrow { \mathrm { CX } }$ of the two vectors $\overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { CX } }$ is minimized. If $\angle \mathrm { ACP } = \frac { q } { p } \pi$, find the value of $p + q$. (Note: $p$ and $q$ are coprime natural numbers.) [4 points]
As shown in the figure, there is an equilateral triangle ABC and a circle O with diameter AC on a plane. Point D on segment BC is determined such that $\angle \mathrm { DAB } = \frac { \pi } { 15 }$. When point X moves on circle O, let P be the point where the dot product $\overrightarrow { \mathrm { AD } } \cdot \overrightarrow { \mathrm { CX } }$ of the two vectors $\overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { CX } }$ is minimized. If $\angle \mathrm { ACP } = \frac { q } { p } \pi$, find the value of $p + q$. (Note: $p$ and $q$ are coprime natural numbers.) [4 points]