The dot product (also known as the scalar product) is an operation on vectors that can tell you the angle between the vectors.

If a line makes angles \(\alpha\) , \(\beta\) , \(\gamma\) and \(\delta\) with the four body diagonals of a cube and the value of

\[\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) + \cos^2(\delta)\] can be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, find the value of \(p + q\).

**Clarification:** Body diagonals of a cube are the diagonals which do not lie along any face of the cube.

Let \(a_1, \ldots, a_{16}\) be the list of \(2^4 = 16\) distinct vectors which have 4 coordinates, whose values are either 0 or 1. What is the maximum possible value of \(a_1 \cdot a_2 + a_3 \cdot a_4 + \cdots + a_{15} \cdot a_{16}\)?

**Details and assumptions**

\( u \cdot v \) represents the dot product of vectors.

Examples of **vectors which have 4 coordinates and whose entries are either 0 or 1** are: \( (0, 0, 0, 0), (1, 1, 1, 1), (0, 1, 0, 1), (1, 0, 0, 0) \).

The list is a set of all the 16 distinct vectors which satisfy the condition.

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