Parallelogram on the Coordinate Plane

Geometry Level 4

A parallelogram on the coordinate plane is defined by the following lines: $\left\{\begin{array}{l}y=4x+5 \\ y=4x+3 \\ y=nx+5\\ y=nx+3\end{array}\right.$

If the area of the parallelogram is $16$ , then $n$ can be expressed as $\dfrac{p}{q}$ , where $p,q$ are coprime positive integers. Find the smallest possible value of $p+q$ .