
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are not collinear, and $\overrightarrow{OA}=2\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{OB}=\overrightarrow{a}+2\overrightarrow{b}$, $\overrightarrow{OC}=m\overrightarrow{a}+n\overrightarrow{b}$.  
$(1)$ If $2\overrightarrow{OA}-\overrightarrow{OB}=\overrightarrow{OC}$, find the values of $m$ and $n$;  
$(2)$ If points $A$, $B$, and $C$ are collinear, find the maximum value of $mn$.

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