
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 $|\overrightarrow{a}+\overrightarrow{b}|=4\sqrt{3}$.

1. Calculate: $\overrightarrow{a}\cdot\overrightarrow{b}$ and $|4\overrightarrow{a}-2\overrightarrow{b}|$.
2. If $(\overrightarrow{a}+2\overrightarrow{b})\perp(k\overrightarrow{a}-\overrightarrow{b})$, find the value of the real number $k$.

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