As shown in the figure, there is a rhombus-shaped piece of paper ABCD with side length 4 and $\angle \mathrm { BAD } = \frac { \pi } { 3 }$. Let M and N be the midpoints of sides BC and CD respectively. The paper is folded along the three line segments $\mathrm { AM } , \mathrm { AN } , \mathrm { MN }$ to form a tetrahedron PAMN. The area of the orthogonal projection of triangle AMN onto the plane PAM is $\frac { q } { p } \sqrt { 3 }$. Find the value of $p + q$. (Here, the thickness of the paper is neglected, P is the point where the three points $\mathrm { B } , \mathrm { C } , \mathrm { D }$ coincide when the paper is folded, and $p$ and $q$ are coprime natural numbers.) [4 points]
As shown in the figure, there is a rhombus-shaped piece of paper ABCD with side length 4 and $\angle \mathrm { BAD } = \frac { \pi } { 3 }$. Let M and N be the midpoints of sides BC and CD respectively. The paper is folded along the three line segments $\mathrm { AM } , \mathrm { AN } , \mathrm { MN }$ to form a tetrahedron PAMN. The area of the orthogonal projection of triangle AMN onto the plane PAM is $\frac { q } { p } \sqrt { 3 }$. Find the value of $p + q$. (Here, the thickness of the paper is neglected, P is the point where the three points $\mathrm { B } , \mathrm { C } , \mathrm { D }$ coincide when the paper is folded, and $p$ and $q$ are coprime natural numbers.) [4 points]