
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 $|\overrightarrow{a}|=4, |\overrightarrow{b}|=8$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{2\pi}{3}$.

(I) Find $|\overrightarrow{a}+\overrightarrow{b}|$;

(II) Find the value of $k$ such that $(\overrightarrow{a}+2\overrightarrow{b}) \perp (k\overrightarrow{a}-\overrightarrow{b})$.

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