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 \(p+q\).