This question is about complex numbers. Statements (9) The complex number $\left( e ^ { 3 } \right) ^ { i }$ lies in the third quadrant. (10) If $\left| z _ { 1 } \right| - \left| z _ { 2 } \right| = \left| z _ { 1 } + z _ { 2 } \right|$ for some complex numbers $z _ { 1 }$ and $z _ { 2 }$, then $z _ { 2 }$ must be 0. (11) For distinct complex numbers $z _ { 1 }$ and $z _ { 2 }$, the equation $\left| \left( z - z _ { 1 } \right) ^ { 2 } \right| = \left| \left( z - z _ { 2 } \right) ^ { 2 } \right|$ has at most 4 solutions. (12) For each nonzero complex number $z$, there are more than 100 numbers $w$ such that $w ^ { 2023 } = z$.
This question is about complex numbers.
\textbf{Statements}
(9) The complex number $\left( e ^ { 3 } \right) ^ { i }$ lies in the third quadrant.\\
(10) If $\left| z _ { 1 } \right| - \left| z _ { 2 } \right| = \left| z _ { 1 } + z _ { 2 } \right|$ for some complex numbers $z _ { 1 }$ and $z _ { 2 }$, then $z _ { 2 }$ must be 0.\\
(11) For distinct complex numbers $z _ { 1 }$ and $z _ { 2 }$, the equation $\left| \left( z - z _ { 1 } \right) ^ { 2 } \right| = \left| \left( z - z _ { 2 } \right) ^ { 2 } \right|$ has at most 4 solutions.\\
(12) For each nonzero complex number $z$, there are more than 100 numbers $w$ such that $w ^ { 2023 } = z$.