
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 points $A$, $B$, and $C$ are collinear on line $l$, for any point $O$ outside line $l$, it holds that $\overrightarrow{OC}=4m\overrightarrow{OA}+n\overrightarrow{OB}\left(m,n \gt 0\right)$. Find the minimum value of $\frac{1}{m}+\frac{2}{n}$.

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