
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 $A$, $B$, $C$ on a plane coordinate system are on the same line, satisfying $\overrightarrow{OA}=(-2,m)$, $\overrightarrow{OB}=(n,1)$, $\overrightarrow{OC}=(5,-1)$, and $\overrightarrow{OA} \perp \overrightarrow{OB}$, where $O$ is the origin of the coordinate system.  
$(1)$ Find the values of the real numbers $m$, $n$;  
$(2)$ Let $G$ be the centroid of $\triangle OAC$, and $\overrightarrow{OB}= \frac {3}{2} \overrightarrow{OG}$, try to find the size of $\angle AOC$.

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