
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 $\angle A O B=\theta$ (where $\theta$ is a constant and $0<\theta<\frac{\pi}{2}$), points $P$ and $Q$ are moving along the rays $O A$ and $O B$ respectively such that the area of triangle $\triangle P O Q$ is always 36. Let $G$ be the centroid of $\triangle P O Q$, and point $M$ lies on the ray $O G$ such that $|O M|=\frac{3}{2}|O G|$.

(1) Find the minimum value of $|O G|$;
(2) Derive the equation for the locus of the moving point $M$.

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