Let $z$ be a complex number with non-zero imaginary part. If $$\frac { 2 + 3 z + 4 z ^ { 2 } } { 2 - 3 z + 4 z ^ { 2 } }$$ is a real number, then the value of $| z | ^ { 2 }$ is $\_\_\_\_$.
Let $z$ be a complex number with non-zero imaginary part. If
$$\frac { 2 + 3 z + 4 z ^ { 2 } } { 2 - 3 z + 4 z ^ { 2 } }$$
is a real number, then the value of $| z | ^ { 2 }$ is $\_\_\_\_$.