Let integers $\mathrm { a } , \mathrm { b } \in [ - 3,3 ]$ be such that $\mathrm { a } + \mathrm { b } \neq 0$. Then the number of all possible ordered pairs $( \mathrm { a } , \mathrm { b } )$, for which $\left| \frac { z - \mathrm { a } } { z + \mathrm { b } } \right| = 1$ and $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 1 , z \in \mathrm { C }$, where $\omega$ and $\omega ^ { 2 }$ are the roots of $x ^ { 2 } + x + 1 = 0$, is equal to $\_\_\_\_$ .
Let integers $\mathrm { a } , \mathrm { b } \in [ - 3,3 ]$ be such that $\mathrm { a } + \mathrm { b } \neq 0$. Then the number of all possible ordered pairs $( \mathrm { a } , \mathrm { b } )$, for which $\left| \frac { z - \mathrm { a } } { z + \mathrm { b } } \right| = 1$ and $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 1 , z \in \mathrm { C }$, where $\omega$ and $\omega ^ { 2 }$ are the roots of $x ^ { 2 } + x + 1 = 0$, is equal to $\_\_\_\_$ .