
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 $|\vec{a}|=2$, $\vec{b}=(-\frac{1}{2}, \frac{\sqrt{3}}{2})$, and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{2\pi}{3}$,  
(1) Find $|\vec{a}+2\vec{b}|$;  
(2) If $(\vec{a}+k\vec{b})\perp(2\vec{b}-\vec{a})$, find the value of the real number $k$.

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