A Basic Product

Algebra Level 3

If $x, y,$ and $z$ are positive real numbers such that $x + y + z = 1,$ find the minimum value of

$\left( 1+\frac { 1 }{ x } \right) \left( 1+\frac { 1 }{ y } \right) \left( 1+\frac { 1 }{ z } \right) .$