
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 $|\vec{OA}|=4, |\vec{OB}|=2, \angle AOB= \frac{2\pi}{3}, \vec{OC}=x\vec{OA}+y\vec{OB}$, and $x+2y=1$, find the minimum value of $|\vec{OC}|$.

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