Don't be more, be less!

Algebra Level 4

Let x,y,z,tx, y, z, t be positive real numbers such that 11+x+11+y+11+z+11+t3\dfrac { 1 }{ 1+x } +\dfrac { 1 }{ 1+y } +\dfrac { 1 }{ 1+z } +\dfrac { 1 }{ 1+t } \ge 3.

If the maximum value of xyztxyzt can be expressed as pq\frac pq , where pp and qq are coprime positive integers, then find p+qp + q.

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