
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 vector $\overset{→}{a}$ with magnitude $| \overset{→}{a} |=2$, vector $\overset{→}{b} =(-\frac{1}{2}, \frac{\sqrt{3}}{2} )$, and an angle of $\frac{2π}{3}$ between $\overset{→}{a}$ and $\overset{→}{b}$:

(1) Find the magnitude of $(\overset{→}{a} +2 \overset{→}{b} )$;

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

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