
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
Let \( O \) be the origin in the complex plane, and let \( Z_{1} \) and \( Z_{2} \) be two points in the complex plane that satisfy the following conditions:
1. The arguments of the complex numbers corresponding to \( Z_{1} \) and \( Z_{2} \) are constants \(\theta\) and \(-\theta\), respectively, where \( 0 < \theta < \frac{\pi}{2} \).
2. The area of the triangle \( O Z_{1} Z_{2} \) is a constant \( S \).

Find the equation of the locus of the centroid \( Z \) of the triangle \( O Z_{1} Z_{2} \) and determine the minimum modulus of the complex number corresponding to the point \( Z \).



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