
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 three points $A$, $B$, and $C$ in the Cartesian coordinate system that lie on the same straight line, where $\overrightarrow{OA}=(-2, m)$, $\overrightarrow{OB}=(n, 1)$, $\overrightarrow{OC}=(5, -1)$, and $OA \perp OB$, with $O$ being the origin of the coordinate system.

(I) Find the values of the real numbers $m$ and $n$.
(II) Let $G$ be the centroid of $\triangle OAC$, and assume $\overrightarrow{OB}=\frac{3}{2}\overrightarrow{OG}$. Find the area of $\triangle AOC$.

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