Don't be more, be less!

Algebra Level 4

Let \(x, y, z, t\) be positive real numbers such that \(\dfrac { 1 }{ 1+x } +\dfrac { 1 }{ 1+y } +\dfrac { 1 }{ 1+z } +\dfrac { 1 }{ 1+t } \ge 3\).

If the maximum value of \(xyzt\) can be expressed as \(\frac pq \), where \(p\) and \(q\) are coprime positive integers, then find \(p + q\).