Triangle Inside Parabola

Geometry Level 3

An equilateral triangle is inscribed in the parabola \(x^2=8y\) such that one of its vertices is at the vertex of this parabola.

The side length of the triangle can be written as \(q\sqrt{p}\) ; where \(q\) and \(p\) are natural numbers and \(p\) is a prime. Find \(p+q\).

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