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 $p + q$ .