
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 $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear vectors. If $\overrightarrow{OA}=m\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{OB}=({m+1})\overrightarrow{a}+\overrightarrow{b}$, and $\overrightarrow{OC}=\overrightarrow{a}-3\overrightarrow{b}$.  
$(1)$ If $m=-\frac{1}{2}$, $|\overrightarrow{a}|=2\sqrt{2}|\overrightarrow{b}|$, and $\overrightarrow{OB} \perp \overrightarrow{OC}$, find the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$.  
$(2)$ If points $A$, $B$, and $C$ are collinear, find the value of $m$.

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