
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that triangles $\triangle Z_{1} Z_{2} Z_{3}$ and $\triangle Z_{1}^{\prime} Z_{2}^{\prime} Z_{3}^{\prime}$ are similar in the same orientation, find the necessary and sufficient conditions for the complex numbers corresponding to their vertices (e.g., $Z_{1}$ corresponds to $z_{1}$, etc.). From these, derive the equation of the perpendicular bisector.

Present the answer in LaTex format: \boxed{Your answer}
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