
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 vectors $\left|\overrightarrow{a}\right|=2$ and $\left|\overrightarrow{b}\right|=1$, with an angle of $\frac{2\pi }{3}$ between them,

(1) Find the magnitude of $\left|\overrightarrow{a}+2\overrightarrow{b}\right|$;

(2) If $(\overrightarrow{a}+k\overrightarrow{b})\perp(2\overrightarrow{b}-\overrightarrow{a})$, find the value of the real number $k$.

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