
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. $Z_{1}$ and $Z_{2}$ are two moving points in the complex plane that satisfy the following conditions:
(1) The arguments of the complex numbers corresponding to points $Z_{1}$ and $Z_{2}$ are fixed values $\theta$ and $-\theta$ respectively, where $0 < \theta < \frac{\pi}{2}$.
(2) The area of the triangle $\triangle O Z_{1} Z_{2}$ is a constant $S$.
Find the equation of the locus of the centroid $Z$ of $\triangle O Z_{1} Z_{2}$ and the minimum modulus of the complex number corresponding to point $Z$.

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