
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 $A$, $B$, $C$ in a plane rectangular coordinate system lying on a straight line, satisfying $\overrightarrow{OA}=(-3,m+1)$, $\overrightarrow{OB}=(n,3)$, $\overrightarrow{OC}=(7,4)$, and $\overrightarrow{OA}⊥\overrightarrow{OB}$, where $O$ is the origin.<br/>$(1)$ Find the values of real numbers $m$ and $n$;<br/>$(2)$ Let $G$ be the centroid of triangle $AOC$, and $\overrightarrow{OG}=\frac{2}{3}\overrightarrow{OB}$, find the value of $\cos \angle AOC$.

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